⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⣶⡷⣟⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢿⣿⣻⣽⢷⣿⣻⣽⡿⣯⡿⣷⡀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⢴⣞⣿⢽⣳⡿⣽⣾⣻⣾⢿⣽⣯⣿⣻⣯⣿⡻⠟⠬⡂⡂⡀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣽⣟⣯⣿⣻⣯⡿⣿⡽⣟⣾⣿⣻⡾⣷⣟⠗⠋⠀⠈⣸⣾⢾⣟⣿⣽⠯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⡀⡀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣷⣟⣯⣿⣽⣾⣟⣯⣿⣟⣿⠾⠏⠋⠁⠀⠀⣠⡴⣟⣿⣽⣟⣯⠗⠋⠀⠀⠀⠀⠀⡀⢀⢀⠄⠢⠑⡰⠨⠂⠒⠈⠀⡀⠈⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⢾⣟⣷⠿⠾⠫⠛⠑⠉⢀⠪⢀⣠⣴⣲⣿⣯⡿⣟⣿⢾⣻⣽⢷⡀⠀⠀⠠⠁⠁⠈⠢⠂⠀⠀⠂⠀⠀⠂⠀⠐⠀⠀⠀⠄⠀⡀⠀⡂⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠠⡩⢅⢁⠀⠀⠀⡀⣀⣤⣴⢶⣟⣿⢿⡾⣯⣷⢿⣻⢿⡽⡿⣽⢾⡻⣧⢀⢆⠬⡠⠀⠀⡀⠀⠄⠀⠄⠂⠁⠀⠠⠀⠠⠐⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣴⡵⣶⣶⣻⣟⣿⣻⣽⣾⢿⣯⡿⣟⡿⣯⣟⣯⡟⡽⣯⣻⢽⢝⢾⢽⡂⠑⠀⣐⢭⢫⡢⡀⠀⢀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠁⠀⠈⠀⢀⠀⠂⠀⠀
⢠⣾⣻⣽⣾⢷⡿⣿⡾⣯⣿⣽⣟⣯⡿⣯⡷⣟⡿⣯⡷⡿⠵⡑⡌⣷⢽⢽⢽⢽⣽⡂⠠⢋⠔⡈⠪⡊⠀⢕⠠⡠⣄⡈⠀⠀⠀⠀⠀⠁⠀⠂⠁⠀⠠⠀⠀⠈⡂⠀
⠈⠉⠉⠉⠉⣟⣿⢽⣯⢿⣞⣷⣻⢾⢽⡳⡯⣗⣟⢷⣻⣯⢦⣆⡈⢽⠃⡯⣯⣟⢾⠀⢈⠢⠀⡂⡅⠕⡄⣀⠘⠕⡕⡮⢀⠀⠐⠈⠀⠀⠂⠀⠀⠠⠀⠀⠠⠀⠘⡀
⠀⠀⠀⠀⠀⢯⣞⢿⣺⣟⣵⣻⣺⢽⢯⢯⢯⣗⠏⠛⠄⠀⣊⡯⡟⠌⡐⢹⡳⢁⠂⠀⠄⠧⡁⡂⠎⡆⡜⡄⢅⢃⠌⡰⡀⢑⡲⡲⣠⠀⠠⠐⠈⠀⠀⠠⠀⠀⡀⠐
⠀⠀⠀⠀⠀⡽⡺⣽⡺⣺⣺⣺⣺⢽⡽⣽⣻⢾⠀⠀⠄⠀⠐⠈⠀⠀⠡⠈⠀⠀⠀⠀⠀⠡⢂⡃⣐⣬⣌⡪⠸⡐⡔⡐⠌⠌⡪⠪⣪⠣⠀⠀⢀⢀⠀⢀⠀⠄⠀⠠
⠀⠀⠀⠠⠼⡽⣝⣧⢟⡵⣗⣗⡯⣯⣟⣾⢽⢯⠀⠄⠀⠀⠂⠁⠀⠂⠀⠁⠀⠀⠀⠀⠠⠠⠐⠀⠀⠊⡊⠋⢷⣬⡊⡢⡑⡁⡂⢱⢡⠈⠘⢠⢢⢲⢲⢠⠠⡠⢂⠊
⠀⠀⠀⠀⢸⢽⣳⣯⢯⢯⣗⣟⣞⣗⣿⣺⢽⣻⡔⠀⠀⠁⠀⢒⣤⢐⡆⠀⠀⠀⠀⠀⠠⠀⠀⠠⠐⠀⠀⡀⠀⠀⠀⢕⢁⠀⢂⠃⡂⠕⠢⡊⠌⡱⢙⠈⢈⠄⡅⠕
⠀⠀⠀⠀⠸⡯⣾⣿⡽⣽⢮⣿⣺⣞⣗⣯⣻⢮⡷⣡⡀⡂⠁⠀⣸⣯⡇⠀⠀⠀⠀⠀⠈⢤⠔⠀⠀⡀⠄⠀⠀⠂⢀⠣⢐⠨⠠⢁⠄⡅⡣⠠⡁⠢⠨⠹⡸⡪⠀⠀
⠀⠀⠀⠀⠀⢫⣿⡸⠟⠽⣻⢮⡚⡺⢵⠳⡯⣯⡇⢿⠛⢾⢶⢮⣟⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠄⠀⠀⠀⠂⣀⠣⠨⢐⠈⢌⢐⢜⢬⠪⡑⢨⢘⠌⡂⣇⢯⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠙⠂⢰⠨⡙⡪⢊⠪⡱⠱⣕⢆⠄⡁⠐⣹⣟⡦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠰⠠⣐⠌⢔⠡⡢⣑⢌⢂⡬⡂⡎⡇⡇⡇⡪⡪⡢⣡⢺⢜⡜⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡊⡪⢒⢌⢆⠣⡊⡌⡢⡑⡱⣜⡤⢷⡿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⢣⡑⠌⡎⣆⢗⢧⡓⡆⡯⡪⡪⢊⠜⡮⣺⢸⡱⡕⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠌⠀⡀⢈⠀⠂⠪⠨⡒⢌⠢⣪⣝⢯⢎⢺⣻⡗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡎⡧⡣⣇⣅⣳⡱⣝⡸⡪⡳⣹⢺⡪⡢⣨⡨⡎⡮⡚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠂⠐⠀⠈⠐⠠⠈⠨⢊⠕⠡⠑⢯⡳⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⢜⢎⠝⡜⢮⡪⡞⡖⣗⢽⡸⡳⡳⣝⢝⡆⣗⠽⢭⡳⡀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠀⠀⡀⠈⢞⡄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⢘⠜⡌⢎⢢⠣⡱⡙⡜⡵⡝⡮⡳⡼⢥⠱⢕⢅⠳⡑⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠜⢌⢂⠀⠀⠁⠀⠀⠀⠐⠀⠠⠐⠀⠀⠀⠈⢞⠄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⢇⢇⢣⢑⢕⠸⡘⡜⢜⡜⣎⢗⣝⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡑⠅⡒⠄⠀⠠⠀⠐⠀⠠⠈⡐⠀⠀⠀⠀⠐⠈⡳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⢄⢣⢣⢑⢕⠸⡘⡌⡪⢪⢪⢮⢳⢕⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⢢⠀⠀⠀⠀⠀⠀⠀⡀⢔⠀⠀⠀⠀⠁⠀⠂⠱⣅⠀⠅⠀⠀⠀⠀⠀⠀⠀⡌⢧⢳⢱⢣⠱⡘⡜⢌⢪⢘⢌⠆⡇⢯⢺⢕⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡘⡀⢀⠀⠁⠀⠀⠡⠠⠡⠈⠀⠀⠂⠀⠀⠀⢸⣪⠀⠅⠀⠀⠀⠀⠀⢄⢣⠪⡱⢱⢑⢅⢣⢱⢘⢌⢎⠜⢔⠕⡜⡔⡽⣱⡣⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠨⡢⠀⠄⠀⠀⠂⠁⠀⡅⠂⠀⠀⡀⠀⠈⢌⡞⣞⠐⠀⠀⠀⠀⠀⢜⠸⡐⡕⡱⡱⡡⢣⢱⢘⢜⢜⢦⢹⡸⡸⡸⡸⣝⡜⣞⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣻⣟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣿⣻⢿⣽⣟⣾⢷⣿⣽⣯⣷⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⣤⣶⢿⣻⡯⣷⢿⣽⣾⢿⡷⣿⣽⢿⣳⣿⢾⣯⠿⠪⡢⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⢷⣻⣾⢿⣻⣽⣿⣻⣽⣾⢿⣻⡿⣾⢿⣻⡽⠋⠁⠀⢣⣵⢷⣻⣟⣯⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣟⣿⣽⢿⣻⣽⣾⢿⣽⣾⢿⣻⠟⠟⠋⠁⠀⢀⢤⣶⢿⣻⣟⣯⡿⠏⠁⠀⠀⠀⠀⠀⠀⢀⠠⡐⡐⡑⢌⠢⠢⠑⠈⠈⠀⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⣾⢿⣻⡿⡾⠟⠟⠚⠉⢅⠂⢀⣀⣤⣶⡿⣿⣽⢿⣻⣽⣯⢷⡀⠀⠀⡠⠠⠐⡐⢌⠂⠁⠀⠐⠈⠀⠀⠀⢀⠀⠂⠈⠀⢀⠀⠐⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⢛⠝⠉⠉⠀⠀⠀⠀⣀⣠⡴⣶⣻⣯⡿⣯⣷⣿⢿⡾⣟⣯⡷⣟⣯⣧⠀⠄⢆⠀⠀⠀⠁⢀⠠⠀⠂⠀⠂⠈⠀⠀⠀⠀⡀⠠⠀⠀⢀⠈⠈⢀⠀⠀⠀
⠀⠀⠀⣀⣴⣡⣪⡬⣤⣴⣶⣾⢿⣯⣿⣻⣯⣿⡾⣿⣻⣽⢾⣻⢽⢯⣗⢿⢽⡺⣺⡯⡊⠐⢢⢠⢔⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠈⠀⠀⠀⠀⢀⠀⠀⢀⠠⠀⠀⠀
⢀⡴⣾⢿⣽⣯⣷⡿⣿⣽⣾⣟⣯⣷⣟⣯⢿⣺⢿⡽⣗⣿⡻⡃⢍⣟⣞⢯⣳⢽⣳⠇⢰⡀⡎⢎⢎⡏⢄⢀⠂⠈⠀⠀⠀⡀⠀⠀⡀⠠⠐⠈⠀⠀⠀⠀⠀⠨⡀⠀
⠙⠛⠛⠟⠳⣟⣷⢿⣻⣽⢾⣽⣽⢾⢯⣻⢽⣝⡯⣟⣯⣷⣁⡉⠑⣺⠞⢸⢽⢽⣺⡃⢈⢐⠐⢄⠱⡀⠀⠢⢱⢸⣒⢅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠨⡀
⠀⠀⠀⠀⠀⣿⢽⣻⣽⢾⣻⣳⢯⡯⣟⡾⣽⣺⢽⠽⡓⠉⢹⡽⣧⡸⢁⢾⣻⡝⠹⠀⡀⡇⡀⠢⡑⡘⢢⢐⠡⢑⠘⡌⠈⢄⢄⣀⡀⠐⠀⠐⠈⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣗⡯⣗⡯⣟⣞⡾⡽⣺⣗⣟⣾⣺⠁⠀⠁⠀⢒⢙⠅⠀⢂⠸⠊⠀⠁⠐⢑⠅⡢⠃⡓⠜⡌⡆⡌⠆⢌⠘⢔⠸⣱⢱⢕⢄⠠⠀⠀⡀⠄⠀⠄⠁⠀⠈
⠀⠀⠀⠀⣠⣞⡮⡷⣽⡺⡮⡯⡯⣷⣳⣳⣟⣾⠀⠐⠀⢀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠁⠸⠀⠲⣞⣦⣅⠪⢪⠢⡁⡂⠅⡂⢜⢌⠐⠔⢀⠥⡤⢤⢀⢀⠄⢄⠊
⠀⠀⠀⠈⢸⣪⢯⣟⡮⡯⡯⣟⡽⣞⣷⣻⣺⣳⡂⠀⠂⠀⠀⡈⡀⠀⠠⠈⠀⠀⠀⠀⠔⠀⠁⠀⠂⠁⠐⠉⠻⠦⠱⡡⠊⡐⠄⡣⠑⠤⠤⡘⡑⢝⠼⠈⠂⡁⠥⡑
⠀⠀⠀⠀⢸⣞⡿⣗⣯⢯⢿⡽⣝⣗⣿⣺⣺⢵⣳⢀⠄⠂⠁⠀⠻⣪⡧⠀⠀⠀⠀⠀⢁⠀⠈⠀⠀⠂⠀⠐⠀⢀⠀⠏⠠⠠⠘⠀⠅⡊⢌⢐⠐⢄⠑⢖⢎⠎⠁⠀
⠀⠀⠀⠀⠘⣗⡿⣿⣽⡽⡯⣟⢽⣗⣗⢷⢯⡯⡟⣶⢦⣂⡄⡀⣞⠯⠋⠀⠀⠀⠀⠀⠈⡲⠄⠁⠀⠂⠁⠀⠐⠀⡌⠌⠌⠌⠌⠌⡰⡸⡰⢂⠅⡢⢑⢐⢭⢳⠀⠀
⠀⠀⠀⠀⠀⠘⠿⣎⠊⠙⡻⣳⢕⢜⢜⢏⢫⣛⠧⠙⢅⠹⣫⣿⡽⠈⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠂⠀⡀⠂⡠⠌⠔⠨⠨⢐⠁⣕⢵⢹⢈⢔⠜⠌⡢⣸⢪⡪⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⢐⢔⢑⢌⠢⡑⢕⠱⡣⡑⡄⡀⣸⡾⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⢰⢅⢆⢑⠔⡹⡨⡥⡕⡮⢸⢸⢸⢰⢑⢕⡭⡳⡹⡜⣜⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⡑⠅⢕⠡⡒⠥⡑⢅⠕⢌⢜⣾⡨⡫⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣜⢕⡅⡊⣎⢎⡮⡇⣏⣎⣝⢎⠮⠄⢅⠫⡎⣇⢗⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⢀⠠⠀⠄⢈⠈⠪⢨⠢⡑⠵⣝⢮⡗⡘⢯⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⣺⢸⢜⢧⠦⡧⣳⡪⡜⡖⣕⢮⡫⣏⢮⢲⡹⣜⢮⣃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠈⠀⠁⠠⠀⠐⢅⠍⠊⠈⠳⢽⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⢧⠫⡊⡕⡝⢞⡜⡮⡻⣜⢮⡳⡍⡮⡋⡮⡺⡈⣗⠕⢄⠀⠀⠀
⠀⠀⠀⠀⠀⡸⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠠⠀⠀⢫⢣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠕⠁⡇⡕⢜⢌⢆⢣⢱⢹⢸⡱⡳⣹⣊⠓⠈⠊⠂⠸⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡪⢨⢂⠀⠐⠀⠀⠀⠀⠁⢀⠐⠀⠀⠀⠀⠀⢓⢆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⡣⡣⡑⡆⢕⠱⡘⡌⢎⢮⢺⡪⡺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡪⠂⡑⠄⠀⠀⢀⠀⠂⠀⠂⡈⠀⠀⠈⠀⠈⠀⢳⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⢸⢅⡣⡣⡪⢸⠨⡪⢪⠸⡘⡎⡧⣫⢽⡸⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⡐⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀⠀⠀⠂⠀⠀⠁⠨⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇⣗⢕⢕⠜⡌⢎⠜⡔⡱⡑⢜⢌⠺⣜⢮⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡣⠂⢀⠀⠈⠀⠀⠨⠀⢅⠀⠁⠀⢀⠠⠐⠀⢨⡺⡄⠀⠀⠀⠀⠀⢀⠔⡅⢇⢣⢪⢊⢆⢣⠱⡡⡣⡊⡆⡣⡱⡘⡜⣜⢮⢮⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⢅⠀⠄⠀⠀⠂⠁⠀⡂⢂⠀⠀⠀⠀⠀⠄⣗⡽⡂⠁⠀⠀⠀⠀⡆⡣⡪⡊⢆⢇⢕⠜⡔⡱⢱⢱⢕⢜⡜⡬⡪⣪⡳⣕⢗⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣟⣿⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⣟⣿⣽⡾⣯⣷⢿⡾⣯⡿⣯⣧⡀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣤⡶⣿⢽⣳⣟⣽⣯⣷⣟⣿⢯⣿⣻⣟⣿⣻⣯⡿⢗⢪⠨⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠨⣿⣯⣿⣻⣟⣯⣿⣽⣾⢷⣟⣯⣿⣯⣿⣽⡯⡟⠊⠀⠐⣡⣷⣳⡿⣟⣯⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠨⣿⢾⣯⣿⣽⣯⣷⡿⣾⢿⣻⣽⡷⠟⠚⠁⠀⠀⣀⣴⢾⣻⣽⣯⣿⡟⠏⠁⠀⠀⠀⠀⠀⠀⢀⠠⡠⠢⡐⢌⠢⠡⠑⠀⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣟⣷⢿⡾⣷⡟⠿⠻⠛⠉⠅⡂⠀⣀⣠⣴⣷⢿⣽⢿⣻⣽⣾⣻⣦⠀⠀⡀⡀⢄⠄⡂⡊⠀⢁⠔⠑⠈⠀⠐⠀⠀⠂⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡛⠝⠉⠉⠈⠀⠀⠀⡀⣀⣤⣮⣾⣻⣟⣯⣿⡾⣿⣟⣿⣻⣽⢾⣗⣟⡧⠨⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠂⠀⠀⠁⠀⠂⠈⠈⠐⡀⠀⠀⠀
⠀⠀⠀⣀⣤⣕⣱⣡⣥⡴⣶⡶⣿⣟⣿⣽⣾⣯⣿⣽⣟⡷⣟⣿⣺⣳⢟⡾⣽⡺⣕⢎⠢⠡⡀⡀⠂⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢀⠀⠁⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀
⢀⢴⣞⣯⣿⣽⣯⡿⣷⢿⣻⣽⣿⡽⣿⣞⣿⣺⣗⣿⣺⣟⣯⠷⡑⢜⡯⡯⣗⣽⣺⢵⡀⢸⢸⢪⢫⠂⡀⠁⠀⠈⠀⠀⠄⠐⠀⠀⠀⡀⠄⠀⠠⠀⠀⠄⠂⠁⠄⠀
⠙⠛⠏⠟⠳⡿⣾⣻⣟⣿⣻⣽⢾⣯⢷⣻⢞⡷⣽⡳⣟⣾⣝⣈⢈⠒⣯⠇⡯⣞⡾⣽⠊⠕⠠⢑⡃⠀⠪⡐⡔⣔⢔⠀⠀⢀⠀⠀⠁⠀⠀⠀⢀⠀⢀⠀⠀⠀⠕⡀
⠀⠀⠀⠀⠀⣟⢿⣺⢷⡯⣷⣻⣻⣺⣽⣺⢽⢽⣺⣝⢯⠃⠉⡯⣷⢦⠙⢠⢟⡷⠏⠵⠨⠂⡁⢆⢘⠤⣀⠤⠑⠕⠭⠃⠨⢀⠀⠀⠂⠀⠠⠐⠀⠀⠀⠀⠀⠂⠀⢂
⠀⠀⠀⠀⠀⢜⡯⣞⡯⡯⣗⣗⢷⡳⡷⡽⣽⣻⠎⠀⠀⠀⠐⡊⡫⠁⠠⡁⠸⠉⢀⠕⢭⢐⢈⠎⡔⡌⡔⡠⢑⠡⠁⢇⡨⢔⢯⢹⢢⠀⢀⠀⠀⠀⠐⠈⠀⠀⡀⠠
⠀⠀⠀⠀⢠⣞⡽⣺⢝⣽⣺⣺⢽⣝⡯⣯⢷⣟⡇⠀⠐⠀⢀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠡⠱⡐⢐⣨⣨⢒⠜⡌⡢⡡⠡⡐⠨⠊⡜⠕⢑⠠⠀⣂⣁⡠⡀⠀⡀⡀⠄
⠀⠀⠀⠈⢘⢮⢯⡿⣝⣞⡮⣷⡻⡮⣿⢽⢯⣗⡧⠀⠂⠀⡀⠀⢄⠁⠀⡐⠀⠀⠀⢀⠀⠔⠐⠀⠍⠞⠙⠶⣮⣘⢰⠡⠂⠅⠅⡎⠣⣀⢀⠕⠕⢎⢮⠢⠃⠢⢊⠢
⠀⠀⠀⠀⢸⡯⣻⣯⣗⡷⣫⡷⣯⢯⢿⣽⣳⢽⣞⡅⠄⠀⡀⠀⠑⢽⡼⡆⠀⠀⠀⢐⠀⠀⢀⠀⠠⠐⠀⠐⠀⠀⠸⡨⢀⢁⠪⠠⢁⠢⡘⠠⢑⠐⡑⡄⣔⠍⠆⠁
⠀⠀⠀⠀⠐⡯⣿⣻⣾⡽⣗⣿⢫⢿⣽⢺⡾⡽⣞⢳⣧⣥⣠⡀⣂⣯⠻⠁⠀⠀⠀⠀⢄⣈⠀⠀⠀⡀⠀⠄⠀⢂⢑⠐⡐⠠⢂⠂⡂⡎⡆⠕⡁⡢⢂⠌⡎⡧⠀⠀
⠀⠀⠀⠀⠀⠘⢟⣮⠂⠙⡻⡺⡕⡕⡍⡗⡝⣝⡻⠌⠳⠁⢻⣽⡽⡮⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠁⠀⠀⠠⢀⢊⠄⠊⢄⢑⢐⢸⡸⡪⢊⢰⢨⠢⠡⣸⢪⢺⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠕⢌⢊⠢⡑⢌⢊⠎⢞⢌⠢⢄⠀⣾⣽⠇⠀⠀⠀⠀⠀⠀⠀⢅⡀⠄⣢⡠⡑⠌⢜⠔⡌⣊⢔⡔⡰⡕⡕⡕⡱⡱⡕⣕⢏⢮⢣⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠅⢃⠣⡑⠕⠥⡑⢅⠪⠢⣑⣵⣗⠝⣟⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⢣⡣⡑⡱⡱⡕⣗⢵⢱⡸⢵⡱⡱⢐⢑⠝⣎⢎⢧⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⢁⠠⠀⠄⡈⠈⠊⢜⢘⢌⠪⢮⣺⣪⠣⡱⡿⠕⠀⠀⠀⠀⠀⠀⠀⠀⢠⡪⡎⡮⡺⡬⡮⡺⣎⢎⢎⡇⣯⡳⣝⢮⡢⡆⣏⢮⡺⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠁⠀⠀⠁⠄⠀⠢⡑⠍⠂⠊⢞⣇⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡪⡺⡜⢜⢜⢪⠫⡎⡗⡗⡧⡳⣝⢼⢕⠗⡝⡼⢑⣝⡎⢄⠀⠀⠀
⠀⠀⠀⠀⠀⠰⡐⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⠄⠀⠀⠀⠘⢜⡄⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⠓⠁⢕⢱⢨⠢⡃⡇⢕⡹⡸⡹⡜⡮⡫⡇⠑⠹⠤⠸⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢕⢌⠢⠀⠀⠠⠐⠀⠀⠠⠐⢀⠀⠀⠀⡀⠀⠨⡲⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⡥⠀⡱⡱⡡⢣⠱⡸⢰⠱⡘⣎⢮⡫⡺⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢕⢐⠡⠃⠀⠀⠀⠀⠀⠀⠐⢀⠈⠀⠀⠀⠀⠁⠈⡧⡀⠠⠀⠀⠀⠀⠀⠀⠀⢨⢝⢆⢸⢢⢱⠡⡣⢪⠢⡃⢇⢣⢳⡹⡹⣜⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠄⠂⠀⠈⠀⢄⠅⡀⠀⠀⠐⠀⠈⠀⢱⢅⠀⠀⠀⠀⠀⠀⠀⢀⢜⢕⢽⢸⠱⡡⢣⠪⡢⡃⢇⢣⢑⢕⠸⡹⣜⢵⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠕⡄⠐⠀⠀⠀⠀⠠⠨⠐⠄⠐⠀⠀⠄⠀⠀⡀⡪⡧⠀⠄⠀⠀⠀⢀⢔⢕⢑⢕⢱⠱⡑⡅⢇⢕⢌⢎⢪⢘⢔⠅⡇⣕⢧⡫⡂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢘⢔⠀⠂⡀⠀⠄⠀⠀⢸⠠⠀⠀⠀⠀⡀⠐⡠⡯⣫⠌⠀⠀⠀⠀⡕⡌⢆⢣⢊⢎⢪⠪⣘⢌⢆⢇⢧⢣⢣⢣⢕⠵⣕⢗⣝⡅⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣟⣷⢿⣽⢯⣷⡿⣾⣻⣿⡽⣟⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣠⣴⣞⣯⢿⣽⣳⣯⣿⣽⣟⣿⢷⣿⣻⣽⣾⢿⣻⡿⢖⠕⢌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠨⣿⣯⣷⢿⣽⣟⣷⣟⣷⢿⣾⣻⣾⢿⣽⣯⣿⢾⠟⠋⠀⠢⣱⣵⣵⣯⣷⣯⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠨⣿⢾⣯⡿⣯⣿⣳⣿⣻⣟⣷⣿⣽⢿⠽⠚⠉⠀⠀⢀⣤⣾⣻⣾⣻⡾⣷⠟⠁⠀⠀⠀⠀⠀⢀⠠⡠⢠⢐⢄⠢⡐⠄⠄⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣟⣷⡿⣿⢾⣯⣷⢿⠯⠗⠫⡨⠂⠀⢀⣠⢴⣞⣿⣻⣾⣯⣿⡽⣿⣅⠀⠀⡀⡀⢄⠢⠁⠁⡁⡪⠐⠁⠀⠄⠀⠀⠀⢀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⡝⠓⠛⠉⠉⠁⠀⠀⠀⡀⣐⣤⣷⣺⡿⣾⣟⣯⣿⣽⣾⣻⣞⣯⡷⡏⠈⠀⠈⠌⠢⠀⠀⠂⠁⠀⠀⠀⢀⠀⠀⠄⠁⠀⠀⠀⠂⠈⠂⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢨⡨⢌⡂⣂⣠⣤⣤⡶⣶⡿⣟⣿⣻⣾⣯⣿⣯⢿⡯⣷⣟⣾⣳⢯⣗⠍⢗⡁⠀⡀⠄⠀⠠⠀⠀⡀⠀⡀⠄⠀⠀⡀⠀⢀⠀⠐⠀⠀⡀⠀⡀⢁⠀⠀⠀
⠀⣠⣴⣾⢿⣻⣟⣿⣻⣽⣾⢷⡿⣟⣿⣻⣯⣿⣳⢿⡾⣽⢯⣟⡷⢑⢗⡯⣷⡳⣝⠀⢨⢒⢖⠦⡄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠄⠀⠀
⠶⣿⣞⣯⣿⣿⣽⣯⣿⣽⡾⣿⣻⢿⣽⣻⣞⣗⣟⣯⢿⣽⣻⢓⠑⠢⢽⡽⡆⣟⢷⠱⠪⢘⠸⡱⠁⠢⡠⡀⡠⣁⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠈⠀⠈⡢⠀
⠀⠀⠀⠀⠀⡷⡷⣷⣻⣞⣯⢿⣽⣻⣺⢵⣳⣳⢗⡯⣯⡷⠟⠷⡾⣤⡐⡏⢠⣯⣟⠨⠈⡐⢄⠱⡀⠀⠂⢇⢗⢕⢕⢀⠀⠐⠈⠀⠀⠀⢀⠠⠈⠀⠀⠂⠀⡀⠈⠄
⠀⠀⠀⠀⠀⢹⡽⣳⣻⢾⢽⢽⣺⣺⡽⣽⣺⢞⣯⠏⠁⠃⠀⢨⢫⢯⠃⠠⠊⣺⠎⡌⡐⠠⣑⢌⠨⡑⠅⡡⢂⠁⢇⠀⢊⢴⢰⢄⡈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣪⢯⣫⢾⢝⣽⢽⢮⣗⡯⣗⣯⢿⣽⠀⠀⠠⠀⠈⠐⠀⠀⠈⢂⠁⠁⡍⢢⠩⢀⠣⠣⡣⡱⡐⠅⠌⠌⠪⠘⡜⠎⡮⡢⡀⠠⠐⢀⢈⠀⠀⠈⠀⠀⢁
⠀⠀⠀⠀⢖⡯⣗⣟⡾⣽⣺⢽⣳⡳⣟⣗⣿⢽⢾⠀⠠⠀⠠⠀⠂⠀⠈⠀⢀⠄⠀⠀⠨⠂⠒⠺⡶⢶⣅⡑⢍⠎⠌⡌⢂⢂⢱⢱⠀⠈⠂⣔⢖⢔⢆⠆⢄⠢⠱⡀
⠀⠀⠀⠀⢰⢽⣳⣟⣞⢷⢽⢽⣞⡽⣽⣗⡯⣟⣽⡅⡀⠀⠄⠀⠐⣢⢅⣰⠁⠀⠐⠈⠀⠂⠈⠈⠈⠀⠨⠛⠓⢑⢕⠌⠐⢀⠎⢄⢑⠂⡎⠄⢅⠋⡎⠈⢠⠡⡑⠌
⠀⠀⠀⠀⢘⣟⡾⣿⢾⡽⣽⢽⡾⣽⣞⢷⢯⣗⡷⣯⡂⡄⡀⠐⠀⢈⣽⡾⠅⠀⠘⠀⠀⠄⠀⠐⠈⠀⢀⠠⠀⢀⢃⠄⡂⠢⠁⡐⠠⡑⠄⡅⡂⠅⡊⢎⢇⠇⠁⠀
⠀⠀⠀⠀⠀⢳⣟⡟⣟⢽⡯⡿⡍⡗⡯⣗⢟⣾⢽⡇⢿⡷⢞⣶⢤⡾⡃⠍⠀⠀⠀⢑⠦⠄⠂⠀⠄⠂⠀⠀⢀⠎⠠⢂⠂⠅⡡⢢⢹⢸⠑⡐⢄⢃⠆⡸⡜⡕⠀⠀
⠀⠀⠀⠀⠀⠀⠫⠓⠆⠀⡜⡙⡻⡘⢜⢸⢸⢨⢧⢅⢈⠃⠀⠏⣷⣯⡁⠀⠀⠀⠀⠐⡀⠀⠀⡀⠀⡀⠠⡐⡌⡐⡁⠆⠌⡂⡜⣜⢕⠕⡌⡜⡜⡠⢥⡺⡱⡕⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢱⠨⡢⡑⢅⠆⠕⢌⢊⢣⠱⡨⣄⣸⣿⣺⠃⠀⠀⠀⠀⠀⠂⠀⠑⡦⡕⡌⡊⢌⢎⢆⡕⣜⢞⠄⣗⢕⢱⠑⢕⢵⢳⢍⢧⢳⡑⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠆⠊⠂⠑⠔⢅⠣⠪⡪⡨⢊⠔⡵⣿⡲⡘⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⢇⢧⢫⡌⢜⢔⡇⡯⡎⣞⢬⢮⢳⢕⢅⢑⠨⣚⢜⢮⠊⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⠀⠠⠐⠐⡀⢂⠀⡈⠑⢔⢑⢅⠝⡝⡮⣧⢑⠝⠋⠀⠀⠀⠀⠀⠀⠀⠰⢵⢹⢜⢎⢞⡎⣗⢵⡣⣺⡸⡪⣗⢽⡹⣜⢮⢹⢜⣕⢯⡂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡂⠀⠠⠀⠀⠀⠀⠀⠀⠄⠐⡡⠊⠈⠀⠙⢮⠆⢃⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⡕⠹⡨⡢⡹⢸⠱⡹⡪⣺⣚⢮⢳⢝⢮⢣⢳⡱⠱⣝⠜⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠨⡂⡀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠱⣹⠀⠂⠀⠀⠀⠀⠀⠀⠀⢗⡍⠀⢨⢊⢆⠕⡅⡣⡱⡘⡆⡇⡯⡪⣏⢗⡂⠁⠓⠂⠕⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢕⠡⡢⡀⠐⠀⠀⢀⠀⠀⡀⡀⠂⠀⠀⠈⠀⠀⠨⣣⠀⠁⡀⠀⠀⠀⠀⠈⢀⢗⠀⢐⢕⠕⡅⡣⡱⡘⢜⢰⢑⡇⡯⡮⡳⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢅⠕⠐⠔⠀⠀⠀⠀⠀⠀⠀⡀⠁⠁⠀⠠⠀⠐⠀⠸⡥⠀⢐⠀⠀⠀⠀⠀⢸⢭⢣⢸⢸⡘⡌⢆⢕⠜⡌⢆⢣⢓⢝⢎⢯⡪⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⢕⠀⠀⠀⠀⠈⠀⠀⠈⠠⡐⡁⠀⠀⠀⠀⠀⠈⠀⢯⡄⠀⠀⠀⠀⠀⢠⢸⢸⢪⢣⢣⠣⡪⡊⢆⢣⠱⡑⢕⢸⠨⡊⡗⣝⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢅⡃⠈⠀⠀⠂⠀⠐⡈⠐⡅⠀⠂⠀⠁⠀⠈⠀⢀⣗⡅⠠⠀⠀⠠⡸⡐⢕⠱⡑⡭⢢⠣⡪⡘⡌⡆⡇⡣⢣⡑⡅⢇⢵⡳⣝⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⢱⠀⢁⠀⠀⢀⠀⠀⢀⠇⡀⠀⠀⡀⠠⠀⠌⡴⣳⢕⠈⠀⠀⡇⢎⢜⠸⡘⢜⢌⢆⢇⢎⢜⢸⡸⡜⡜⡆⣇⢎⢧⡳⣕⢧⠁⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣻⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⢿⡷⣿⢷⣻⡷⣿⢾⣯⣿⣽⢿⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⢿⣻⡽⣾⣻⣽⣯⣿⣯⣿⣻⣟⣷⢿⡾⣟⣿⠶⢌⢢⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣻⣯⣿⢾⣟⣯⣿⣻⣽⢿⡾⣷⢿⡾⣯⣿⣽⣟⡿⠋⠉⢈⢪⢦⣵⢶⡶⣖⣶⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣽⣾⣻⣟⣯⣿⣽⣯⣿⣻⣟⣿⣻⣟⡿⠞⠓⠁⠀⠀⣠⣴⢿⣻⣽⣟⣿⣻⠓⠁⠀⠀⠀⡀⡀⠄⢄⠄⡄⢄⠄⡄⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢱⣿⢾⣯⣿⣽⡷⣿⢾⣯⢿⠽⡋⡫⠀⠀⠀⢀⣠⣴⢿⣽⣾⢿⣿⣽⣯⣏⠀⢀⠀⢄⠢⠑⠈⡢⠩⠂⠑⠈⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣟⠿⠷⠟⠚⠋⠋⠁⠀⠀⠀⣐⣐⣴⣴⣟⣯⣿⡾⣟⣿⡾⡿⣗⡯⠊⠈⠈⠢⡑⠁⠀⠀⠊⠀⠀⠀⠄⠀⠀⠁⠀⠁⠀⠂⠀⠀⠂⢂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢕⢐⠅⠠⠀⢀⣀⣠⢤⣦⡷⣟⣯⣿⣽⡾⣿⣽⡷⣿⣟⣯⣟⡿⡹⢧⠂⠀⠈⠀⠀⠄⠀⠄⠀⠠⠐⠀⠀⠀⠁⠀⠐⠀⠠⠀⠐⠀⠐⠀⠈⠄⠀⠀⠀
⠀⠀⣀⣴⢾⣷⣳⡷⣷⡿⣿⣽⣾⢿⡷⣿⣟⣯⣷⣿⣻⣟⡾⣯⡷⣟⢞⢵⢯⡲⠈⠨⠢⡤⡰⡀⠀⠀⡀⠀⡀⠀⠀⡀⠁⠀⠐⠀⢀⠀⢀⠀⠄⠀⢀⠀⠈⡀⠀⠀
⣠⡾⣯⣿⣻⡷⣟⣿⣯⣿⣻⡾⣟⣿⣻⣯⢿⡽⣗⡿⣞⣯⣟⣷⠿⢕⢔⠹⡽⡽⡄⡜⡑⢇⢗⠅⢂⡀⢀⠀⠀⡀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠐⢄⠀
⠈⠉⠉⠊⠉⢹⣿⣻⢾⣽⣽⣻⣽⢷⣻⣞⡯⡯⣗⡿⣝⡷⣻⣾⣥⣤⣀⠩⣿⢹⢐⠐⠨⢐⠱⠀⠐⠨⢢⢣⢳⡰⠀⠀⠀⠁⠀⢀⠈⠀⠀⠁⠀⠀⠀⠠⠀⠀⢑⠀
⠀⠀⠀⠀⠀⢸⣗⢿⡽⣗⣿⣺⢽⢽⣳⣗⡯⡯⣗⣯⡗⠟⠏⠀⠰⢽⡽⡦⠃⡨⠰⡀⠡⢪⠈⠕⡤⠔⠂⢍⠪⢪⠂⠡⣐⣀⡈⠀⠀⠀⠈⠀⠀⠁⠀⡀⠠⠐⠀⢁
⠀⠀⠀⠀⠀⠰⣝⣗⡯⣗⡷⡽⣝⡯⡾⡮⡯⡿⣽⢮⠀⠀⡀⠀⠘⠌⠍⠀⠰⠁⡗⡌⢌⠂⠏⡆⡎⡌⢌⢂⠢⢈⠦⠰⡸⡢⡫⡕⡄⠀⠁⠀⠐⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣣⣗⡯⣳⢽⣝⣗⡯⣯⢿⣝⡿⣽⢗⠀⢀⠀⠠⠀⠄⠀⠄⠀⠁⠱⠘⠔⢈⢴⣴⢅⡃⢕⣑⠬⢐⠨⢐⠨⠊⢜⠎⠈⡢⠠⢈⣐⣀⢄⠀⠠⠐⡀⠌
⠀⠀⠀⠀⠁⣗⣗⡷⡯⣯⡻⡮⣗⡯⣗⣿⣺⡽⡯⣟⠄⠀⢀⠀⠄⢀⡀⠄⠐⡠⠄⠔⠁⠀⡐⢙⠩⠛⢶⣤⠩⣘⠌⠔⡈⠌⡎⠱⣀⢀⢸⠱⢪⢲⠱⠨⠊⠌⡌⡢
⠀⠀⠀⠀⠠⣗⡷⣿⡯⣗⡯⡿⣽⢽⡳⣯⢷⢯⢯⢯⢧⢀⠀⠀⡀⠀⠪⣳⢼⣇⠀⠀⢀⠀⠀⠀⠀⠀⠁⠁⠀⢱⠁⡂⠠⡑⢐⢁⠂⡜⠠⠨⢐⠑⢥⢠⢌⢕⠂⠀
⠀⠀⠀⠀⠀⢻⣺⣟⣿⣳⢿⡽⡯⡿⡽⣯⣫⢿⢽⢯⡟⣦⣕⡠⡀⢀⢰⣺⠿⠁⣄⡐⠀⠀⠈⠀⠈⠀⠐⠀⢁⠆⠡⢂⢁⠂⢄⢐⠜⡄⡅⠕⠠⡡⢊⠪⣣⡃⠀⠀
⠀⠀⠀⠀⠀⠈⢿⢶⡙⠝⠫⢿⢽⡪⡍⢗⢗⢝⠿⡽⡃⠻⡍⠛⣟⣷⣫⢧⠈⠀⠡⠉⠀⠀⠁⠀⠁⠀⠂⢀⠎⠠⢑⢐⠠⢑⢰⢸⢜⠪⢐⢡⢱⢐⠡⡸⣢⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢰⠡⡃⡓⢌⠕⡑⢕⢕⢷⡑⡄⢌⠀⢡⣿⡽⡆⠀⠀⠨⡀⠀⡁⡀⡁⢄⠢⠨⡪⡐⡡⢁⢂⡌⢔⢕⢇⢇⢕⢅⠧⡅⡦⡪⣫⢲⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⢌⢆⢣⠱⡨⢊⠢⡢⠢⡱⡘⣴⡡⣺⣞⣯⠅⠀⠀⠀⠀⠁⠀⢏⢞⢔⠡⡑⡕⢕⢵⡱⡳⡕⢸⢕⢕⢕⠡⢣⡫⡞⡜⡞⡜⠎⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢈⠀⠂⠑⠌⠜⢌⢪⢐⠕⡰⣝⣟⡮⢢⣻⣟⠦⠀⠀⠀⠀⢀⢰⢕⡝⡼⡌⡂⡗⡽⡸⢪⡣⡫⡮⡳⣳⣑⡊⡐⡌⡮⡳⡹⠌⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠁⠠⠈⠄⠂⠡⠐⢀⠈⠐⡡⡑⢅⠫⡚⡮⡾⡐⠌⠃⠁⠀⠀⠀⠀⠈⣪⢣⡣⢗⢝⢕⢯⡺⣹⢲⣹⡸⣕⢯⡺⣜⢮⢺⡸⣪⢯⡺⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠌⠊⠀⠀⠈⢮⣃⠡⠀⠀⠀⠀⠀⠀⡎⡮⠊⠸⡰⡡⡃⡇⢝⠜⡕⡵⡕⣗⢝⢞⡮⡋⢪⡺⡂⢳⡹⠐⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢈⢆⢂⠀⠄⠂⠀⠀⠄⠂⠀⠀⠀⠁⠀⠀⠀⠀⠘⣜⠄⠠⠀⠀⠀⠀⠀⠳⡁⠀⠸⡔⡜⢔⠱⡡⢣⢑⢎⢎⢎⡏⡧⣫⡙⠈⠊⠊⠂⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⡨⠢⡡⠀⠀⠀⠀⠀⠀⠀⠀⠄⡁⠀⢀⠀⠁⠀⡐⢝⠄⠈⠀⠀⠀⠀⡸⣢⠀⢸⢸⢘⢌⢎⢪⢘⢌⢆⢣⢣⢯⢺⡱⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠨⢢⠁⠊⠀⠀⠀⠐⠀⠀⠂⠁⠐⠀⠀⠀⠀⠀⠀⢀⢘⢵⠀⠐⠀⠀⠀⡧⡳⡄⡧⡣⡣⡊⡆⢕⢅⢕⠜⡌⢎⢎⢗⡝⣞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢢⠡⠀⠀⠂⠀⠠⠀⠀⡀⢰⠑⡀⠀⠂⠀⠈⠀⠀⠀⣇⠇⠀⠀⢀⠔⡕⡵⡹⡸⡪⢪⢨⢊⢆⢣⠢⡣⠪⡊⢆⢣⢹⢪⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣑⠀⠂⠀⠀⠀⠀⡀⠂⠂⠅⠀⠂⠀⠀⠠⠀⠀⡀⣮⡳⢀⢆⢣⠱⡑⡅⢇⢇⢣⢱⢘⠔⢥⢑⢕⢌⢣⢱⢑⢕⢬⡳⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡢⠅⠀⡂⠈⠀⠀⠀⠀⠌⡊⠀⠀⠀⡀⠀⡀⠅⡲⣳⢝⢆⢕⢅⢣⠱⡸⢸⡘⡌⣆⢣⢩⢪⢪⡣⣪⢪⡪⡲⡱⣕⢗⢵⠅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣤⣴⡶⣾⢾⣻⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⣟⣿⣻⡷⣟⣾⣟⣿⣻⣟⣿⣻⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⢿⣻⣻⢯⣯⢿⣾⣻⣿⣽⣯⣿⣽⣯⢿⣳⣿⢷⢔⠅⢕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣯⣿⣽⣾⣟⣯⣿⣻⣽⣿⣳⣿⢾⡷⣿⢾⡷⣟⡿⠛⠉⠀⠆⣧⣵⣲⡶⣖⣶⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣯⡿⣷⢿⡾⣯⣿⢾⣟⣷⢿⣽⣾⢿⣻⣟⠟⠋⠉⠀⠀⢠⣤⣿⣻⡾⣯⣿⣟⠟⠀⠀⠀⡀⠠⡠⢠⢐⢄⠢⡐⠄⠄⠄⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢱⣿⣟⣿⣻⡿⣾⣟⣯⣿⢿⠽⠞⡋⠅⠀⠀⢀⣐⣤⣟⣿⢾⣯⣟⣿⣻⠾⡁⡀⡄⠆⠊⠐⢁⠪⠐⠂⠂⠁⠀⢀⠀⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡿⡽⠯⠿⠙⠋⠋⠁⠁⠀⠀⢨⣐⣥⣴⡾⣯⣿⣽⣯⣿⣻⡷⣿⠅⠀⢀⠐⠨⠂⠀⠄⠀⠁⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠂⡂⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡱⢨⠠⠠⠀⢀⣀⣀⣤⡴⣶⣻⣯⣷⡿⣾⣟⣯⣷⡿⣞⣯⡷⣏⠚⢇⢂⠀⠀⠀⠀⠄⠀⠀⠂⠈⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠠⠀⠀⠀⠀⡂⠀⠀⠀
⠀⠀⣀⣴⣞⣷⣗⣷⡷⣿⢿⣾⣻⡷⣿⣻⣽⣷⢿⣻⢿⡽⣯⡷⣟⣯⢗⢯⣗⢇⠀⠱⡢⣓⢖⠄⠀⠀⠂⠀⢀⠀⠄⠐⠈⠀⠀⠀⠀⢀⠠⠀⠀⡀⠁⠀⠐⡀⠀⠀
⣠⣾⣽⣾⣯⣷⢿⡷⣿⣟⣿⣽⣯⢿⣟⣿⡽⣾⣻⡽⣿⡽⣯⣟⡯⠯⣐⡘⣮⢷⠢⢋⠘⢜⠎⠁⡊⢄⠠⡀⣀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⢀⠠⠀⠐⢄⠀
⠈⠁⠉⠊⠁⢻⣻⢿⣽⢾⣻⢾⢾⣻⣗⡯⡯⣟⡮⣯⢗⡿⣽⡾⣵⣤⡠⠈⣻⡅⡌⠐⠨⡠⠣⡀⠈⠀⡣⡫⡪⣣⠀⠁⠀⠀⠀⠀⢀⠀⠠⠐⠈⠀⠀⠀⠀⠈⢐⠀
⠀⠀⠀⠀⠀⢸⣻⢽⢾⣻⣽⡻⡯⣗⡯⣯⢯⣗⡯⡯⡯⠏⢏⠈⠀⠷⣻⠷⡈⢆⡊⠄⡑⡨⡂⢍⢒⠩⢐⠨⢑⢑⠀⠑⡄⣅⣀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠄⠡
⠀⠀⠀⠀⠀⢐⡽⡽⣽⣳⡳⣯⡻⡽⡽⣝⣗⡷⣯⢿⠈⠀⠀⠀⠈⠍⠣⠃⠊⢱⠑⡅⡂⠍⠪⠪⡸⡨⡰⡁⠢⠘⠢⠒⡝⡆⡧⡳⡀⠐⠈⠀⠀⠁⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⢀⣼⣪⢟⣮⣺⣺⢵⢯⢯⢯⣟⡮⣿⢽⣻⠀⠠⠐⠀⠀⠂⠀⢀⠀⠀⠃⢸⠄⠴⢲⣵⡤⣁⠒⢌⢇⠅⠥⠡⠨⢈⢜⠜⠐⢄⠄⢂⣁⢂⡀⠂⠁⢀⢀⠄
⠀⠀⠀⠀⠁⣳⢽⡽⣞⣞⢾⣝⣯⢯⢯⢷⣻⡽⡯⡷⡅⠀⠀⡀⠂⢀⢈⠀⠀⠖⠊⠀⠀⡀⠁⠂⠉⠙⡳⠦⠡⢕⠅⠅⠅⡂⠇⠣⣀⢀⠸⠱⡪⡣⡇⠕⠡⠑⢌⠢
⠀⠀⠀⠀⢀⡿⣳⢿⣻⣺⢽⢮⣯⢯⡯⣿⢽⡽⣝⡯⣗⠄⠁⠀⠀⠀⠌⠯⣖⡇⠀⠀⠂⠀⠀⠄⠂⠀⢀⠀⠀⡇⠅⡀⡁⠎⠨⠨⢐⢘⠨⢐⢐⠑⡤⣠⢪⠡⠃⠁
⠀⠀⠀⠀⠀⢽⢽⡿⣿⣽⢽⣳⢿⢽⡽⣾⢽⢽⣳⢯⡟⣮⣆⣢⢈⠀⢀⣽⢯⠗⠴⡠⠀⠂⠀⡀⠀⠄⠀⢀⠰⠨⢐⠐⠄⠅⡂⢡⢪⢐⠌⠔⢐⠨⡈⢪⡪⡃⠀⠀
⠀⠀⠀⠀⠀⠈⢯⣷⡙⠎⠻⡽⣻⡜⡜⡝⡳⡹⡫⡟⡧⠹⡏⠙⡯⣟⣯⢯⠀⠀⡅⠀⠀⠠⠀⠀⡀⠄⠀⡰⠁⠅⠢⠨⠨⠐⡔⣕⢕⠕⡁⡱⡘⢔⠈⣆⢗⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢐⡑⢌⠕⢜⠨⡊⢎⠎⢷⡱⡠⠠⡀⠈⣯⡿⡥⠀⠀⠃⠄⠈⢠⡠⡐⢄⢂⠅⡇⢬⢘⠨⡨⡔⡱⡹⡸⡸⡰⡸⡸⡨⡔⣎⢞⢜⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⡊⢆⠣⠥⡑⡌⢆⠕⡡⠪⡘⣬⡲⡸⣿⢽⡋⠀⠀⠀⠀⠀⢀⡳⡹⣔⢐⠅⡇⡣⡧⣫⢣⢇⢺⢜⢜⢌⠢⡣⡫⣞⢜⢎⢮⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠈⠐⠁⠣⢑⠜⡰⡑⢌⢪⢺⣫⢷⢘⣺⢿⠧⠀⠀⠀⠀⣄⢮⡪⡺⣸⣠⣑⢭⡳⡹⢸⢪⡺⣪⡫⣞⢎⢂⢂⢍⢮⢪⡳⠅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⠠⠈⠐⠈⠐⠐⠠⠀⠁⠆⢕⠱⢌⠝⢮⣻⡰⡈⠃⠁⠀⠀⠀⠀⡠⣪⢺⡸⡊⡆⢗⡳⣝⢝⢮⢇⢧⡳⣝⢮⡳⡝⡎⡧⣳⣫⡺⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⡀⠀⠠⠀⠀⠈⠀⠈⠢⠃⠁⠀⠈⢮⢲⠈⠄⠀⠀⠀⠀⢰⢕⠵⠑⡕⡱⡘⡌⢎⢪⢹⢪⡫⣇⢯⡺⣚⢮⡋⢎⢞⠆⠺⡼⠐⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢐⢅⠢⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢏⢆⠀⠄⠀⠀⠀⠈⠣⡁⢀⢇⢇⠎⡜⢌⢪⠢⡱⡱⢕⡳⣹⢪⣫⡚⠈⠊⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠢⡑⢅⠀⠀⠂⠁⠀⠀⠀⠀⠄⡁⡀⠀⠠⠀⢀⠐⡹⡄⠀⠂⠀⠀⠀⡸⣅⢸⡸⡐⡕⢅⠇⡕⢅⢇⢪⡣⡳⣕⢯⡪⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⢌⠂⠑⠀⠠⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⢀⠠⡫⡂⠀⠂⠀⠀⡧⡫⡪⡪⢪⢘⠔⡕⢅⢇⢪⢸⢸⢕⢗⣕⢗⣝⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢢⢑⠀⠀⡀⠀⠀⠄⠈⠀⠨⠌⠄⠀⠀⠂⠀⠂⠀⠀⢸⡱⠀⠀⢀⠔⡕⡝⢌⢪⠢⡣⡱⢸⢐⠕⡌⢆⢣⠹⣱⢕⡗⡵⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢐⢑⠀⠠⠀⠀⠀⠀⠠⠨⠈⡅⠈⠀⡀⠀⢀⠀⠀⡀⣜⢞⢀⢔⢅⢣⠣⡪⡊⢆⢣⠪⣘⢔⢅⢇⢕⠱⡡⢣⢱⡳⣹⡪⣳⡁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡱⡡⠀⠂⠀⠈⠀⠀⠀⠐⡆⠂⠀⠀⠀⠀⠀⠐⡰⡵⣻⢢⢱⢘⢔⢕⡱⡸⡸⡰⡩⡎⣎⠦⡕⡪⡪⡸⣸⡪⡞⡮⡺⡪⡆⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣴⣶⣶⣟⣿⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣴⡿⣿⣽⡯⣿⢾⡷⣿⢾⣯⣿⣻⣧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣞⣿⣻⢯⡷⣟⣿⣺⣟⣿⣻⣟⣿⣻⡷⣿⣽⣾⢷⠤⡑⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣻⣽⣾⣾⣾⡾⣿⣻⣿⣽⣯⣿⣽⣯⣿⣽⣯⣿⣻⠞⠋⠁⢅⢇⣵⢴⡶⣖⣶⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣯⣿⢾⡷⣿⣽⢿⣽⣾⢷⡿⣾⢷⡿⣾⠷⠛⠊⠁⠀⢀⣠⣾⣟⣿⣻⣟⣿⡽⠋⠀⠀⡀⡠⠠⡠⡐⢄⠢⡐⠄⠄⠄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⡾⣯⡿⣟⣿⢾⣿⣻⣾⢿⣻⠟⢏⠋⠀⠈⠀⢀⣠⣴⣿⣽⡾⣯⡿⣽⢯⠋⡀⢄⠢⠑⠐⡈⡪⠐⠈⠂⠁⠀⢀⠀⢀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣟⣯⡿⣟⠿⠛⠓⠋⠉⠁⠀⡈⠆⣂⣤⢶⡾⣟⣿⣽⣾⢷⣿⢿⠈⠀⠀⠑⠨⠂⠀⠀⠐⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠌⢂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡪⡑⡁⡀⠀⠀⠀⢀⢠⣠⢴⣴⣾⢿⣻⣽⣟⣿⣻⣽⡷⣟⣟⡇⡻⡅⡀⠁⠀⠀⠂⠀⠄⠀⠄⠐⠈⠀⠀⢀⠀⠀⠂⠁⠀⠁⠀⠀⡀⠀⠈⡀⠀⠀⠀
⠀⠀⠀⣠⣴⣦⣵⣴⡴⣶⢾⣾⢿⣻⣽⣿⣻⣾⢿⣻⣯⡿⣯⣟⣷⣟⡯⣗⣗⠄⠐⡡⡲⡲⡡⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠀⠀⠠⠀⠀⠁⠀⠀⠠⠐⠀⠀⠀
⢀⣴⡾⣿⢾⡷⣿⢾⣻⣟⣿⣽⣿⣻⣽⣾⢿⡽⣿⢽⣷⣻⢷⣻⣞⡗⡑⢜⡽⡕⢄⢃⠣⡳⡹⠐⡠⢀⠀⠁⡀⠁⠀⠀⠠⠀⠀⠀⠠⠀⠀⠐⠀⠀⠂⠀⠀⠡⡀⠀
⠛⠽⠿⠿⠻⢿⣻⣟⡿⣯⣟⡷⣯⣟⣷⣻⡽⡯⣯⡻⣞⣯⢿⣽⢊⠈⠘⠒⢿⢐⠐⢐⠨⠈⡆⠀⠈⠢⡕⣝⢜⢤⠀⠂⠀⠀⠠⠀⢀⠠⠐⠀⠀⠂⠀⠀⠂⠁⢌⠀
⠀⠀⠀⠀⠀⢺⣻⣞⣟⣷⣻⣽⣻⢞⣗⣗⡯⣟⢾⣝⢷⢽⡻⠚⠟⣟⣮⣄⠵⣘⠠⠐⠌⡅⡊⠲⢄⠒⠌⡊⠣⡓⠀⡂⣀⢐⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡀⠀⡃
⠀⠀⠀⠀⠀⢸⣺⢮⡷⡯⣗⡷⣽⣫⣗⡯⡯⡯⣟⣾⠩⠁⠁⠀⠠⡝⢞⠓⠃⡕⢌⢌⠘⡸⠸⡸⡐⡅⡕⠄⠅⠪⢄⠤⡺⡸⡪⣆⠐⠀⢀⠈⠀⠀⠀⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡸⡮⣳⢯⡫⡷⣽⣺⡺⡮⣯⢟⣽⣟⣾⠀⠀⢀⠐⠀⠀⡀⠀⠀⠘⠀⡇⡀⢀⡠⠀⠑⠘⢔⢱⠡⡡⡁⡂⡙⠌⡇⠗⡐⡀⡀⠄⠄⠂⠁⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠐⣯⢯⢯⣗⡯⣟⡮⣗⡯⣟⣽⡽⣳⣟⡾⡀⠠⠀⠀⡀⠄⠀⠀⢑⠂⠁⠈⠀⠙⠫⠷⡶⣦⣄⡑⡱⡐⠔⡐⠠⢱⠑⡄⠀⠁⢰⢪⢭⢫⡱⡐⢔⠡⡑⡁
⠀⠀⠀⠀⠠⣯⢯⣿⣺⣝⣗⡯⣯⢯⣗⣯⢿⡽⣺⡽⡆⡀⠀⠄⠀⠀⡲⣌⡠⠀⠈⠀⠐⠀⢀⠀⠀⠀⠀⠀⢨⠪⠂⠈⠨⡊⡐⠨⡑⡒⠩⢘⠸⡸⡈⠀⡠⢪⠨⠈
⠀⠀⠀⠀⠘⣗⣟⣾⣷⣳⣳⢯⢿⣽⣺⣞⢿⣝⣗⡯⣯⡂⢄⠀⠠⠀⠀⣣⢿⡠⣄⠈⠀⢀⠀⢀⠈⠀⠐⠀⣘⠨⠠⠡⠑⡀⠠⢡⡘⠠⡁⡂⠅⡘⢜⢭⢳⠁⠁⠀
⠀⠀⠀⠀⠀⢹⣞⣟⣽⢾⡽⣯⣟⡺⡳⣻⡞⣗⣷⣻⡇⣿⡷⣮⣴⣠⣰⢯⠛⢂⠀⠀⠠⠀⠀⡀⠀⡀⠂⡨⠂⢌⢐⠡⢁⠂⡕⢜⢌⠖⠌⠄⡕⢠⠡⡪⡎⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠽⣤⠁⢉⢓⢟⢎⢎⠖⡝⡜⡪⣇⡓⠈⠣⠀⢫⣾⢽⡓⠀⢁⣀⠠⢀⠠⢀⠠⠠⡠⢊⡂⠅⡂⠢⠨⢐⢸⢜⢕⠇⢅⢅⢇⠎⠢⣘⡎⡮⡂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢆⠪⡨⠢⡑⢌⠪⡘⡚⡜⡌⢆⠄⢠⡾⣟⡗⠀⠀⠀⠀⠁⢩⢲⢅⠕⡨⢢⢃⢇⢥⢳⣹⠰⡱⡣⡣⡫⡸⡰⡕⡮⡺⡜⣎⠎⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡌⠕⢌⠍⢜⢘⠌⡆⢕⢌⠢⡑⢜⣼⡮⡹⣽⣟⡇⠀⠀⠀⠀⡀⣇⢗⢕⢇⠌⡆⡗⣝⠮⡳⡸⣌⢞⣎⡪⡂⡑⢕⢝⡎⡞⡼⡘⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠠⢀⠈⠀⠁⠕⢌⢒⢔⠅⡕⣯⡳⡽⡰⠱⡻⡓⠀⠀⠀⠑⢑⢇⢗⢝⢜⢕⢞⣕⢧⢕⣹⢪⢮⡳⡵⣹⢬⢐⣨⢲⢹⢜⡎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡨⠀⠠⠁⠈⠀⠈⠈⠐⡀⠀⠱⡐⢕⠱⠘⢺⡝⡎⢕⠰⡈⡢⠠⠀⡰⡹⣸⢱⢱⢑⢅⢳⢱⢝⣕⢗⡗⣕⢗⢽⡺⣪⢇⢧⡫⢺⢕⣗⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠐⠈⠀⢀⠠⠨⠸⠐⠠⡀⠀⢓⢧⠑⠌⢜⠨⡊⢄⢏⢞⡜⢎⢎⢢⠣⡱⡘⡜⢜⢵⢝⢎⣏⢧⡫⣞⡨⣪⢺⠈⢧⠣⠈⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠨⡊⡢⠀⠀⠂⠁⠀⢐⢁⠂⠡⠠⠐⠀⠄⠀⡂⠀⠳⣅⠈⢢⠑⢌⠢⠡⢑⢌⢎⢪⠢⡣⡊⢆⢣⠱⡱⡱⡣⡳⡵⡝⣎⢇⠀⠈⠁⠘⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢌⠢⡊⡢⠀⠀⠀⠀⠀⠐⠒⠐⢢⠠⠈⠀⠀⠈⠄⠀⠳⣂⠀⢉⠢⠡⡃⣎⢧⢣⠱⡑⡌⢎⠪⡢⢣⢱⢕⣝⢕⡗⣝⢎⡗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠕⡀⠊⠀⠀⠐⠀⠀⡀⠀⠈⠠⠈⢆⠀⠈⠀⠈⡂⠈⡞⡄⠀⠌⠌⡢⡣⢣⢑⢅⠇⡎⡪⢪⢘⢌⢎⢗⢎⣗⢝⢮⢳⡹⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢕⠄⠀⠀⠂⠀⠀⡀⠀⠀⢌⠊⠄⠈⠢⡂⡁⡠⢊⠄⢸⡱⠀⠀⡜⡌⡪⠢⡃⡎⡪⡂⢇⠕⡅⡕⢅⢫⡣⡳⣝⡕⣗⢝⢮⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠡⡱⠀⠁⠀⠀⠀⠀⠀⠅⠂⢅⠀⠂⠀⢘⢔⢜⠬⡩⡒⢽⢀⢜⢆⡣⡪⡱⡑⢬⢢⡣⢣⢑⢕⢸⠨⡢⡹⡹⣜⢮⢮⢫⡣⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡆⠈⠄⠀⠁⠀⠈⠀⢀⡃⠄⠀⢠⠸⡨⢢⠣⡪⢸⢐⢕⢕⢕⢕⢵⢱⢕⡕⡇⣇⢧⢱⢊⢆⡣⡕⡵⣫⡺⣜⢮⢳⢝⡎⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢶⣶⢾⣻⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⡾⣿⣻⣽⣟⣿⣽⢿⣯⡿⣯⣿⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣻⡿⣽⢟⣿⣽⣻⡾⣯⣷⣿⣻⡷⣿⣻⡷⣿⡦⡊⡌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⡾⣿⢾⡿⣷⢿⣽⢿⡷⣿⣽⣟⣯⣷⣿⣽⣟⣿⡽⠛⠁⡐⢕⣬⣢⣶⡶⣶⢶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣻⣟⣿⣻⣟⣿⣽⢿⣻⣟⣾⣯⡿⣷⢿⠞⠏⠁⠀⠀⣀⣴⣻⣽⢿⣾⣻⣟⠏⠁⠀⢀⢀⠄⢔⠠⠢⡐⢄⠢⡐⠄⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣽⣯⣿⣽⣯⣷⢿⣟⣿⣽⡷⡟⠏⠋⠀⠀⠀⣀⣴⢾⣯⣿⣽⣟⣯⢷⠏⡅⢄⢄⠢⠁⠂⢁⠅⠕⠑⠈⠀⢀⠀⠀⠀⡀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢽⡷⣿⢾⡷⡿⡾⠟⠋⠋⠈⠀⡐⠅⢄⣤⡶⣿⣻⣽⣿⣳⣿⢾⣻⡕⠀⠀⠀⠡⠢⠀⠀⠠⠀⠀⠀⢀⠀⠄⠀⠀⠀⠁⠀⠀⠐⠀⠁⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠝⢝⠉⠉⠁⠀⠀⠀⠀⣀⣄⣤⣶⢾⡿⣷⢿⣟⣯⣿⢾⡯⣿⡝⡌⢳⠠⠀⠀⡀⠀⠀⠠⠀⠀⡀⠁⠀⠀⠀⠀⠈⠀⠀⠄⠂⠀⠠⠐⠀⠁⠄⠀⠀⠀
⠀⠀⠀⢀⣠⡕⣅⣂⣡⣤⣦⡶⣷⢿⣻⣯⣿⢾⣿⣻⣿⣻⣯⢿⡽⣯⢟⣗⢿⣌⠀⢨⢪⡓⡖⡄⠐⠀⠀⢀⠀⠀⠠⠐⠀⠁⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠡⠀⠀⠀
⠀⣠⣴⣿⣽⣟⣿⣻⣯⡿⣾⣻⣟⣿⣻⣷⢿⢿⡽⣷⣻⣗⣿⢽⣻⢝⢐⢜⣯⠪⠒⡨⠨⠪⠮⠁⠢⡂⢄⢀⡀⠠⠀⠀⠀⢀⠠⠀⠀⠂⠁⠀⠀⠈⠀⠠⠀⠡⡀⠀
⠾⣻⣷⣿⡾⢯⣿⣽⡾⣿⣻⡯⣿⢽⣗⣿⡻⣯⣻⢽⣞⢷⣻⠽⠃⠤⠄⠖⡷⠡⠁⢐⢨⠨⠸⡀⠀⠈⡲⡱⡍⡧⡀⠀⠈⠀⠀⠀⠀⠄⠀⢀⠀⠂⠀⡀⠀⠈⢔⠀
⠀⠀⠀⠀⠀⢺⣗⣯⡿⣽⢾⣻⣽⢯⣟⡮⡯⡷⣝⡷⡽⣽⡾⡳⢶⡦⣦⡀⡟⡬⠨⡐⠈⡦⢡⠱⡑⡊⢔⢈⠪⢪⠀⠅⡄⣀⠄⠐⠀⠀⡀⠀⠀⢀⠀⠀⠀⠄⠀⠆
⠀⠀⠀⠀⠀⢱⢯⣳⣟⡯⣟⢷⢽⢽⣺⢽⡽⣽⣳⢯⠋⠀⠃⠀⠴⠹⠋⠋⠂⡝⢢⡑⠐⢈⠑⠕⡅⢆⢕⠠⠨⠐⠥⡠⡪⡎⡮⡳⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⢸⣝⢞⡮⡯⡯⡯⡯⡯⣞⣯⣞⣷⣻⡽⠀⠀⠠⠀⢀⠀⠠⠐⠀⢘⡀⢊⢀⣤⣤⡀⠐⠈⠢⡱⢡⠡⡁⠢⡈⡪⠊⡞⡐⡀⠁⠠⠁⢀⠀⠂⠀⠐⠀⠠
⠀⠀⠀⠀⠠⣟⣼⢽⣳⢽⢽⢽⢽⣝⣗⣷⡳⣯⢷⢯⢀⠐⠀⠀⡀⠀⡀⠀⠕⠀⠀⠀⠀⠈⠑⠛⠳⠶⠤⠌⡒⡌⠄⠅⡂⡍⠢⡄⠐⠈⢢⢪⢎⢧⡱⠠⡂⡢⡑⡁
⠀⠀⠀⠀⢀⣟⡮⣿⣺⢽⢽⣝⣯⣞⣞⣷⣻⡽⣽⣻⡄⡀⠠⠀⠀⠀⢒⡤⡅⠀⢀⠁⠀⠂⠀⠠⠀⠀⠄⠀⢸⠐⠁⢁⠢⠡⠡⠨⢒⢒⢘⠘⢎⢮⠊⠈⡠⢄⠪⠐
⠀⠀⠀⠀⠘⣮⢯⣿⣗⡯⣟⡮⣷⣳⣳⢯⢷⢯⣗⣗⣷⢐⠄⡀⠈⠀⠀⣩⠖⠎⠄⠀⠀⠂⠈⠀⠀⠄⠀⣀⠣⢐⢁⠂⠅⠄⡑⡨⡂⡂⢂⢊⢐⠱⣹⢸⠌⠂⠁⠀
⠀⠀⠀⠀⠀⢫⣟⡷⣿⢯⡯⡿⣝⣳⢯⡿⡺⣻⣺⣞⡗⣿⡽⣶⣵⣤⣥⢿⠡⠀⠀⠀⠂⠀⠄⠂⠀⠠⡐⠔⠨⢀⠢⠨⠨⢐⢜⢌⢆⠆⠅⡂⠆⡌⠔⣇⢇⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢫⢿⣘⠊⢛⠻⣽⠴⡱⡩⡻⣘⡓⣏⡢⠘⠇⠈⡳⣽⢾⡙⠀⠒⠨⠂⡆⡢⢂⠢⡑⡑⡜⡈⢌⠔⠨⢠⢡⢳⡱⡕⠕⡨⡰⡑⠕⠠⣝⢜⢬⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⢌⠪⡐⢍⠢⡑⢌⠢⠣⡳⢕⢅⠢⡀⢐⡿⣯⡇⠀⠀⠀⠀⠀⡘⡆⢕⢐⠅⡎⡪⡢⣎⢞⡕⡸⡜⡜⡜⢜⢌⡮⣢⢳⡹⡪⡣⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠕⡑⢍⠢⡃⡎⡌⡪⢑⢌⠢⣡⣽⣜⠞⣿⣽⠆⠀⠀⠀⢀⢔⢵⡹⡔⣆⢇⡧⣇⢯⢺⡸⡲⡱⣣⡣⡪⠨⢢⢳⢣⢇⢗⢝⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⠠⢀⠀⠁⠊⠔⣑⢜⠔⢆⢕⡽⣺⡺⢨⠠⡩⢓⠀⠀⠀⠀⡬⡣⢣⠪⡨⠳⡕⣗⢵⢕⣕⢭⣫⡺⣪⢭⢡⣂⢆⢏⣎⢗⠅⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⠀⠐⠀⠂⠐⠈⠎⠠⠑⡀⡒⢍⢢⢑⢍⢷⢝⡆⠕⢌⠢⢂⠀⠀⣜⢪⢪⠪⡘⡌⢎⢪⠪⡳⡝⡮⣪⢮⡺⢵⡫⣳⢪⡪⡧⡳⣽⡂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢆⠀⠄⠀⠀⠀⠀⠑⠥⠄⠤⡀⠈⠐⠐⠁⠄⢳⢕⢡⢑⢌⠢⢂⠀⢪⢎⢎⠕⡅⢇⢣⢑⢅⢇⢏⢮⢳⢕⢽⢥⢫⢆⢣⡫⡆⢙⢖⠈⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⢌⢂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢅⠀⠄⠀⠈⢊⢚⢆⠐⡢⠡⡡⡀⡗⣕⢕⠱⡘⡌⡆⡣⣑⠢⣕⢵⢱⡫⡳⣕⢇⠁⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠢⡑⢌⠢⠀⠐⠀⠀⠐⠀⠀⠐⡈⡈⢢⠠⡀⢁⢐⠄⢫⢄⠐⡑⢔⠱⢕⢕⠸⡘⡌⢆⢣⢑⢢⢹⢸⢜⢮⡺⡹⣜⢵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⠌⠄⠑⠁⠀⠀⢀⠠⠀⠀⠁⠠⠀⠀⠘⢔⡡⡢⡓⡪⡳⡀⡜⡌⢎⢪⢂⢇⢕⢜⢸⠨⡪⡘⡌⢎⢮⢳⡹⡹⣜⢵⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢕⠅⠀⠀⠀⠀⠀⠀⠀⠀⢌⢊⢀⠀⡨⡪⢸⢐⢅⢇⠪⡪⡪⣊⠎⡆⡣⡊⡆⣕⢜⠌⡆⡣⢪⢊⠆⣗⣝⢽⡸⡵⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠡⢱⠀⠂⠈⠀⠀⠄⠠⠑⠐⠄⠀⡔⢕⠜⡌⢆⡣⡊⢎⢆⡏⣎⢮⢪⢎⢖⢝⢜⢆⢧⢱⢊⢎⡢⣣⡳⡪⡧⡳⡳⣹⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢜⠄⠐⢀⠀⠀⠀⠀⠀⠄⠕⠀⠈⠢⡣⣪⢣⢣⠱⡑⣕⢕⢜⢜⣎⢎⡇⣗⢕⢇⢗⢕⢵⡱⣕⢧⡫⣳⡹⡪⣏⢮⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡾⣟⣿⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⢿⣽⡾⣷⣻⣽⡿⣯⣿⢾⣷⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⢾⢿⣻⣻⣽⣯⣷⡿⣟⣯⣿⣻⡿⣾⢿⣽⠿⡢⡑⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⢾⣿⣻⣽⣟⣯⣿⣽⡾⣷⣟⣿⣻⣯⣿⣳⣿⡟⠏⠁⠐⢨⣪⣮⣾⢿⣻⣟⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣻⣽⣯⣷⡿⣯⣷⢿⣻⡿⣾⣯⣿⢾⠯⠛⠈⠀⠀⣠⣴⣯⣿⣽⣾⡿⣿⠝⠁⠀⠀⠀⠀⡀⢄⢂⠢⡑⢔⠡⡊⠔⠠⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡿⣯⡿⣾⢷⣿⣻⣽⡿⣯⣿⠻⢚⠉⠀⠀⠀⣠⣴⡾⣿⢾⡷⣿⢾⡷⣟⠆⠄⠐⢄⠢⠊⠀⠀⡠⠂⠑⠈⠀⢀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣯⣿⣟⣿⠽⠏⠓⠉⠁⠀⠀⠜⣐⣴⣼⢿⣻⡾⣟⣿⣻⣟⡿⣯⢟⣗⡀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⡀⠄⠀⠀⡀⠀⠠⠐⠀⠀⠨⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡙⢍⠁⠀⠀⠀⠀⣀⣠⣤⣖⣾⡾⣿⣞⣿⣻⣟⣿⣻⣯⢷⢿⣽⣳⢱⠈⢂⢂⣁⣀⠂⠀⠂⠁⠀⠈⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠄⠀⠀⠁⢂⠀⠀⠀
⠀⠀⠀⢀⣠⣜⣰⣥⣴⣴⡾⣯⣿⢾⡷⣿⣽⣿⣽⣾⢿⣽⣻⣞⣷⡛⠜⡯⣞⡾⡝⢄⠠⠳⡸⡢⡓⡀⡀⠀⡀⠂⠀⠠⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠂⠀⠀
⠀⣠⡶⣿⣽⡾⣿⣞⣿⢾⣟⣿⣾⢿⣟⣿⣳⣟⣾⢽⣟⡾⣗⣟⠎⠈⠈⡺⣗⣻⠨⠠⠡⡁⡊⢚⠀⠐⠌⡢⡢⢤⠠⠀⠀⠀⡀⠈⠀⠈⠀⠀⠂⠁⠀⠀⠁⠈⠄⠀
⠾⢟⠿⠻⠞⣻⢿⣾⣻⢿⡽⣗⣿⡽⡷⡯⣗⡯⡾⣽⣺⢽⠓⡁⣒⣊⠈⠩⡏⡰⠅⡂⡐⢱⠠⡁⡣⣄⢠⠰⠹⡸⡹⡄⠈⠀⠀⠀⠀⠂⠀⠄⠀⠀⠄⠈⠀⠀⠱⡀
⠀⠀⠀⠀⠀⣺⣟⡾⣽⡯⣟⣯⢷⣻⢯⢿⢵⣻⢽⡳⡙⠵⡚⢛⡫⠿⠝⠃⠔⠂⢇⡢⠨⠐⠍⢖⢌⢢⢐⠅⢌⢐⠡⡁⠨⠢⣠⡈⠀⠀⡀⠀⠄⠂⠀⠀⠄⠂⠁⠂
⠀⠀⠀⠀⠀⢰⣳⢯⡷⣻⢽⣺⢽⡽⣽⣫⣟⣾⣻⡇⠀⠀⠀⠀⠀⠀⠂⠀⠉⡠⠐⠨⠅⣀⣤⡀⠁⠡⠱⡘⢔⢐⠨⠘⠔⡱⡕⡮⡂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⣜⢮⣳⢏⡯⡯⡾⣽⣺⢵⣗⣗⣿⣺⡇⠀⠀⠂⠀⠈⠀⡀⠁⠨⠀⢀⠀⠁⠈⠉⠛⠷⣤⠀⠈⢪⢰⠈⠌⡂⡪⡨⠪⠢⡀⡁⠠⢁⠀⠀⠀⠈⠀⠀⡐
⠀⠀⠀⠀⠊⡾⣽⣺⡽⡽⡽⡽⣞⡾⣽⣳⣻⣞⣗⣧⢁⠀⠂⠈⠀⡀⢠⣀⢨⠀⠀⢀⠀⠁⠀⠄⠀⠀⠉⠀⠀⢇⢈⠂⡂⠇⠌⢆⠀⠈⡰⡹⣸⢱⡈⡠⢐⠨⡐⠁
⠀⠀⠀⠀⢠⢿⢵⡿⡽⡽⡽⡽⣗⣟⡮⣟⣗⡷⣽⣺⡔⡐⡀⡀⠂⠀⠀⢨⡏⠓⠀⠀⡀⠄⠁⠀⠀⠁⠀⠄⡡⡁⠄⡐⠜⠈⠌⡐⢍⠡⠡⠩⢪⡪⠊⠀⡡⡑⢔⠀
⠀⠀⠀⠀⠈⣯⢷⣿⡿⡽⣯⣻⣽⢾⣽⣫⢷⣻⣺⢾⠵⣮⢶⣴⣰⣠⡠⣽⢁⠀⠀⠄⠀⠀⢀⠀⠁⡀⡂⠰⢐⠠⢁⢂⢂⠂⢅⢎⢐⠨⡈⠌⢄⠣⡣⡺⠐⠌⠀⠀
⠀⠀⠀⠀⠀⠺⣽⣗⢿⠻⣗⣿⣺⢹⢚⢷⢭⢺⠽⣯⠇⠹⡏⠊⢷⣻⣽⠵⠀⠁⠑⠐⢐⢄⠢⡐⠌⠔⡌⠕⢐⠨⢐⠠⠂⡎⡎⣆⢇⢑⢠⢑⠡⢂⢕⢽⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠓⠦⠀⢐⢌⢛⢜⢌⢊⢎⢪⢪⢞⢆⢄⢄⡀⣁⡿⣾⠆⠀⠀⠀⠀⠀⠀⡊⢔⠡⡃⡕⡌⢆⢡⢢⢜⢨⡺⡸⣘⢠⢢⢱⠡⣑⢬⢳⡱⡁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣘⢢⠱⢔⢢⠣⣊⠪⢌⢪⢘⢔⣱⢌⠢⢻⣯⠃⠀⠀⠀⠀⡠⡢⠹⣕⣝⢖⢵⣑⢳⡹⡜⣅⢮⢪⢪⠪⡸⢸⡪⡣⡇⣗⢵⠑⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠊⠀⠈⢩⠜⡐⡑⠢⠩⡂⢅⠕⣞⡽⣝⠌⡂⡻⠇⠀⠀⠀⢸⠨⡊⢎⢢⠣⡏⣗⢵⢵⡹⣪⣚⢮⢳⡱⢕⠨⠘⡜⡵⡹⣸⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠄⠌⠈⠄⠄⠄⡈⠢⡈⠈⠢⢑⠸⣺⣣⡑⢌⠌⢅⠀⢀⡄⡇⡇⢇⢣⢑⠕⢜⢌⡳⡵⣹⣪⢲⢝⡵⣝⣕⢮⢪⢎⢞⢮⡣⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⠀⠀⢕⢕⡇⣕⠨⡂⡆⣪⢺⢌⢪⠪⡘⡔⢍⢆⠣⡪⡺⡜⣎⢗⢽⡺⢘⢮⢓⢵⢹⠉⣗⡝⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡑⢄⠀⠠⠀⠂⠀⠀⠄⠀⠄⠀⠈⢢⡑⢅⠥⡣⡣⡢⡹⡘⡎⡎⡎⢎⢢⠣⡱⡘⡌⡆⡣⡣⣣⢫⢎⣏⢧⡃⠀⠑⠇⠑⠹⠄⠘⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⢅⠥⢡⠀⠀⠀⠀⢀⠀⠀⠀⠄⡁⡠⡪⡱⡑⢕⠌⡎⢆⢇⢣⠣⡱⡑⡅⡣⡱⡘⡔⢅⢣⢹⢸⠵⣝⣜⢮⡲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠅⢑⠀⠀⠀⠄⠀⠀⠀⠈⠠⡰⡑⡌⡆⡣⡱⡑⡜⢬⢣⢑⢕⢸⢐⢅⢇⢪⢂⢇⢕⢱⢘⢌⢏⢮⡺⡜⣞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠢⡡⠀⠀⠀⠀⠀⠀⠀⠂⢡⠊⡈⢎⡢⡣⡣⡣⠪⢌⢧⢳⢱⢱⢱⢸⢸⢸⢜⢔⡱⡨⡢⡃⡎⣎⢧⡳⡝⣎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⠄⠠⠀⠈⠀⠈⠀⠌⠠⡁⠀⠀⢣⢫⢪⠸⡘⡜⡜⣎⢧⢳⢹⡸⡪⡣⡣⡇⣇⢗⢜⣜⢮⢮⢳⢕⢯⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡑⡆⠀⠄⠀⠀⡀⠄⠀⠐⠔⡀⠀⠀⢝⡜⣕⢕⢵⢱⡣⡳⡹⡕⣕⢝⢎⢇⣗⢵⡫⣣⢗⣳⢹⣪⢳⡣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢑⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠠⠀⢅⠒⡌⠰⡈⠆⠡⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠂⠈⠒⡄⠃⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠝⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣿⣻⣜⠀⠁⣂⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣷⡿⠏⢠⢻⣞⡷⠏⠆⡠⠑⢎⠶⡙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⣟⣿⣻⡽⠃⠀⣀⡸⣿⢫⠐⠀⡐⢈⠠⢙⡀⠀⠊⠔⣢⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣯⣿⣷⡿⣯⣟⡿⣞⣷⣻⡞⠃⣀⣨⣥⣀⡀⠏⢰⡀⢂⠠⠹⡄⢀⠱⠤⠠⠌⠑⠏⡶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⣿⡿⣿⣻⣿⢿⣽⣾⣟⡷⣯⢿⣽⡞⠁⠒⠛⠩⠉⠛⠙⠉⠘⠁⠽⡀⠎⠀⠘⢣⢃⠦⣑⠊⡐⠠⠈⡄⠐⢠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⣷⣻⢿⡽⣯⢿⡽⣾⣽⣻⣽⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡨⠈⠅⣠⣤⣄⠀⠑⠠⢓⡘⠠⢁⠘⠂⢣⡝⣎⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⡗⣯⢿⣹⣏⡿⣽⣳⢾⡷⣯⣿⣟⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠀⠡⣓⠀⢂⠁⡞⢀⠋⠠⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠰⣿⡽⣯⣷⣻⢾⡽⣷⢯⣟⣿⡷⣿⣽⣳⠠⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠂⠄⢂⠃⠄⢦⡀⠀⣸⡹⢬⢳⡀⢄⠠⡐⢂⠀
⠀⠀⠀⠀⠀⣷⣻⣽⡷⣯⢿⡽⣯⣟⡾⣿⣟⡷⣯⣟⣇⠰⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⡀⠋⠀⠌⣀⠌⠠⢀⡘⢡⠟⠀⢈⠠⣁⠆⠀
⠀⠀⠀⠀⠀⣿⣽⣿⣿⡽⣯⣟⣿⣾⣽⣷⢺⣿⣳⣯⡟⣿⣿⣿⣶⣶⣦⣴⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⢂⠐⠠⢁⠐⡠⢆⡈⡐⠠⢀⠂⡙⢶⡹⠐⠈⠀⠀
⠀⠀⠀⠀⠀⠸⣿⣎⠿⠻⣿⣽⣻⠼⡻⢾⣇⠹⣛⠷⠃⠘⠇⠉⢿⣽⣿⠵⠀⠈⠑⠐⠄⡢⠐⡄⢢⠑⡸⢀⠐⠠⢈⠐⡀⣎⠵⣪⠜⠁⡰⢊⠰⢀⡮⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠻⠦⠀⢀⠛⠻⢆⠡⣃⡙⠞⡤⢿⢆⠄⣄⡀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣇⣌⠰⡈⡕⢢⢊⣡⠦⡖⢸⢬⠓⡥⡐⢎⡱⣁⡢⢞⡵⢫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢪⡑⢎⣑⠢⢩⠒⡌⢢⢊⠒⣤⣱⢈⠻⣿⠆⠀⠀⠀⢀⡐⢬⠘⣮⢻⢵⣫⡖⢫⡜⢧⢇⢺⡌⢳⠰⢉⠾⣱⢣⡝⡺⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠈⠀⠩⠤⠠⢉⡀⠣⢘⡀⢣⢾⡿⣻⠀⢆⢹⡓⠀⠀⠠⢬⡘⠤⢋⡔⡙⢞⡶⣹⢫⡜⣧⡚⣧⣛⢮⠇⡀⢊⣑⠧⡞⡵⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠘⡀⠀⠘⠀⠹⣞⣷⡈⢂⠆⡀⠀⠀⣌⠳⢌⡃⠧⡰⢉⡒⡌⠷⣣⢏⡷⣙⡶⣹⣎⠿⣔⡫⣜⣳⡝⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⠀⠠⠌⢶⢫⠔⣢⣡⢄⣋⢧⡓⢌⠲⣡⠑⡣⠜⣨⢱⡍⢾⣙⢧⡻⡕⠸⣏⠳⡱⢧⠉⡾⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣍⠖⣩⠒⣣⠚⡄⢦⢣⠞⢢⠍⡜⠢⡅⢎⡱⡘⢤⠳⣎⢷⡹⣎⢷⡁⠀⠉⠣⠉⠚⠠⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⠢⡘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢤⠚⣄⠳⢠⢋⢼⡂⢇⠎⡥⢚⡐⢣⠜⣂⠖⣉⠦⡙⢮⡳⣝⢮⡳⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡱⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⡍⢆⢣⢆⡍⠦⣩⢞⡘⣌⠲⣁⠎⢬⡱⡘⢤⠚⢤⢒⡉⢆⠿⣜⢧⣛⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠼⣌⢳⠊⡔⠣⡜⢮⡱⢆⡻⢤⣋⠶⡱⣍⢦⡩⠖⡌⡜⣬⢞⡝⣮⢳⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⣘⠀⠀⠀⠀⠀⠀⠀⠀⠂⡅⠀⠀⢫⠖⡭⢌⢓⡜⣣⠟⣬⣓⠧⣎⢳⡱⢎⠶⣱⢫⡜⡵⣎⡟⣼⢣⡟⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⡅⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡟⡼⣩⠞⡼⣡⢟⢦⢫⠳⣜⢣⡝⣎⣳⣏⢷⣹⠳⣝⢮⣓⢯⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣻⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⢆⠱⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣈⣾⣿⣿⣿⣿⣿⡯⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⡀⠀⠀⠂⡘⠤⡑⠌⡐⠂⠂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⡄⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠡⢒⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢺⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⢀⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠎⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡐⢀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⢳⣟⡿⣽⣺⠀⠐⠢⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⢿⠋⠄⣹⢿⣝⠇⠢⠔⠋⠲⡥⠏⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⡷⣿⢯⣟⡿⠞⢁⡀⡀⠼⣿⢊⡐⠁⠠⢌⠐⠘⡀⠀⠉⠘⣴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀
⠙⠛⠛⠛⠋⢹⣿⣿⢿⣻⣿⣽⣾⣟⣿⡽⣾⣽⣻⣽⣻⠞⠉⠉⢀⣀⣀⠀⠇⢰⣂⠐⡀⠪⣄⡡⢙⠢⠔⠊⡐⠫⢖⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⢿⣻⣿⣟⣯⣿⣻⢷⣟⡷⣯⣷⡏⠁⠐⠒⠛⠛⠋⠉⠁⠘⠁⠜⢠⠅⠀⠐⠙⢦⡑⠬⡁⢄⠡⠐⣄⠈⢰⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢨⣟⣯⢿⣽⣻⣞⡷⣯⢿⡾⣽⣻⣽⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠨⠘⠂⢤⣶⣤⡀⠈⠢⢑⡊⠰⢀⠈⢉⢰⠳⣌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣏⣾⣛⣮⢷⢯⣟⡾⣯⣟⣷⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠐⡭⠀⠌⡀⠞⡀⠋⠐⢠⠀⣠⢄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠒⡿⣞⣷⣟⡾⣯⣟⡾⣽⣳⣿⣟⣾⣻⣾⡀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠐⢠⠃⠐⠣⣄⣀⠸⡱⢎⡽⡠⢄⠰⣀⠃⠀
⠀⠀⠀⠀⠀⣿⡽⣾⣯⣟⡷⣯⣿⣳⢯⣿⣟⣾⣳⢯⣧⢁⠂⠄⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠠⢀⠃⠀⢁⢂⠄⠠⠐⡈⠱⣎⠁⢀⠔⣠⠉⠀
⠀⠀⠀⠀⠀⣿⣽⣿⣷⣯⣟⣷⣿⣯⢿⣳⠼⣷⣿⣻⡏⢿⣿⣿⣿⣷⣶⣾⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⢠⡐⢀⠂⡐⢀⠂⠌⡤⢃⡌⡠⠁⢄⠡⢈⠻⣜⡁⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣎⠻⠹⢿⣾⢷⡸⣛⠿⣇⣙⢓⡻⠇⠈⠣⠀⢫⢿⣾⠓⠀⠀⠈⠁⠒⢐⠢⢘⠠⡑⢢⠐⡀⢂⠐⠠⢈⡞⡴⢳⠈⠄⣡⢊⠔⡀⡟⡴⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠣⠄⢀⠌⡛⢥⢂⡱⡨⠞⢢⠿⣅⠢⠤⡀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣇⣌⣒⠠⢇⠲⣉⡤⢞⡃⢼⡸⣑⠣⡜⡸⣄⢣⠴⣚⡵⣩⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢎⡱⠎⣘⢂⡑⠢⢍⢂⠲⣈⢑⣶⣍⠢⢙⣿⡆⠀⠀⠀⢀⢄⡓⡘⢮⡝⢯⢧⣏⣱⢻⢬⢳⡨⡗⣌⠓⢨⠱⣏⢧⣋⠷⡸⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠀⠀⠉⠀⠢⢄⡉⠰⠈⡔⠡⣾⢻⡽⠐⡂⢹⠓⠀⠀⠀⠼⡐⢬⠑⡆⣙⠫⣞⡼⢦⣛⣎⢧⢻⡝⣶⠛⡀⢂⣍⠶⣩⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠘⡀⠀⠈⠐⠘⢯⢿⡄⡑⢂⢂⠀⠀⣠⢋⡕⡊⡜⡰⢡⠚⡄⡏⣷⣚⡞⣮⢳⡝⣮⢻⣍⠯⣜⣳⢏⣯⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⢀⡀⢣⢙⣦⢑⡨⢄⣂⢜⡱⢎⠰⡑⢬⠑⡬⠱⡘⢴⢣⢞⡹⣜⢧⡟⠘⣧⡉⣞⣱⠈⠾⡅⠒⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠦⡘⡍⣩⠲⣡⠚⡌⡞⢬⠓⡌⢣⠜⠢⡍⡔⢣⠙⣎⡳⢮⡝⣮⢳⢦⠀⠀⠃⠈⠐⠓⠈⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡜⠠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢊⠖⡱⡘⣄⠣⢆⢳⡜⢡⠎⡜⡨⢅⢎⡱⢢⢑⣊⠱⡌⢳⣏⢾⡱⣏⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠣⢚⡤⠳⡄⠳⣌⢳⡜⢂⢞⡠⠓⣌⢦⡒⠥⣊⠤⠓⢬⠑⢮⡳⣝⢮⡳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢎⡳⡜⡱⢌⠣⣜⢣⡞⣭⢲⡱⢫⡜⢦⡹⡔⢦⡙⢜⡢⣝⢮⡳⣝⢮⡝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢆⠄⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠳⡹⣔⣊⡱⣌⠳⡞⡴⢧⡙⢧⣚⣥⢳⡙⡦⣝⢮⡵⣫⢞⡵⣫⢞⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⡳⡜⢦⠳⡜⣭⢻⡜⣭⢹⡒⢧⢎⠧⡽⣻⡜⣧⢳⡝⣮⢳⡝⣮⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⣟⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠱⡄⢣⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣡⣾⣿⣿⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡟⠉⢀⡀⢀⠀⠄⠐⠐⢠⠃⠜⠠⠃⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢉⡀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣏⠀⠁⠀⠐⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⢠⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠳⣟⡿⣶⡽⠀⠘⢔⡲⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⡟⠁⠢⢹⡿⣽⢈⠃⠘⠁⠹⣎⠁⢐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⡷⣿⣽⣻⡽⣋⢀⠠⠤⠝⡿⢸⡀⠀⠡⡌⠠⠘⡀⠀⠈⢡⡛⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠙⠛⠛⠛⠋⣽⣿⡿⣿⣻⣯⣿⣾⣿⡾⣽⣻⢷⣻⢷⡯⠋⠁⢀⣀⣤⣄⠈⢃⡂⡇⠐⡐⠸⢤⣁⡙⠒⢢⠁⠌⠓⢎⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢾⣿⢿⣿⣟⣿⣽⣷⡿⣽⢯⣟⣯⣿⡏⠀⠀⠛⠛⠉⠉⠈⠁⠨⠀⢙⠤⡁⠀⠐⠈⠵⡑⢆⢡⠈⠄⡈⢆⢀⡸⣒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣯⣟⣷⣻⣟⣾⣳⢿⣽⣻⢾⣽⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠑⠀⠃⠴⣶⣦⣀⠀⠉⠢⠍⡐⠠⠈⠄⢸⠱⢮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢞⡽⣞⣳⢯⡾⣽⢯⣾⣯⣟⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠄⠀⠘⡱⢀⠡⠈⡞⠠⠁⠈⠔⢀⡤⢤⣀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠔⣯⢿⣽⣳⢯⡿⣽⢯⣟⡾⣿⣾⣻⢾⣽⠀⠀⠀⠀⠀⠀⠀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⢡⠃⢈⠓⠤⠀⠼⣜⢣⢞⡰⢀⠆⡡⢃⡀
⠀⠀⠀⠀⢠⣟⡿⣾⡽⣯⢿⡽⣟⡾⣽⡿⣷⢯⡿⣽⡎⣁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠄⠠⠃⠀⠂⢄⠃⠌⠠⠄⡙⣎⡀⢠⡐⢤⠒⠀
⠀⠀⠀⠀⠘⣯⢿⣿⣿⡽⣯⣟⣿⣽⡷⣟⣹⣯⣿⣻⡟⢿⡿⣿⣿⣿⡿⢿⠠⠀⣀⠀⠀⠀⠀⠀⡀⡐⠤⡐⠠⠈⠄⡁⠂⢌⡰⢃⡌⡠⢁⠐⡠⠈⣝⢳⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠹⣿⡎⠿⠹⣿⣽⣧⠻⡻⢿⡤⣙⠓⣿⠃⠈⠣⠀⠟⣾⣿⡉⠀⠀⠀⠈⠑⠌⠢⡑⠤⠑⡜⠀⠡⡈⠐⠠⢁⢮⠴⡳⢈⠐⣠⠋⠔⢁⣎⠳⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠻⠦⠀⢨⠙⡺⣅⠒⠤⢗⡉⢞⠹⢣⡐⠢⡄⢸⣿⣿⠇⠀⠀⠀⠀⠀⢀⢧⡴⣌⡂⡝⢢⢣⢤⡳⠇⣜⢣⠏⡥⢢⣙⡤⢥⡜⣎⡎⠷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡉⢝⣐⠊⠭⡑⠒⡌⢢⠱⡡⢌⣷⣱⠂⠹⣿⡄⠀⠀⠀⢠⡘⢢⠎⢳⣭⠻⡵⣏⡹⢮⡱⢧⡸⣃⠞⡰⢡⢚⡼⣣⢞⡱⡚⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⠒⠐⠤⠉⠐⢂⠆⠱⣟⢯⡗⢨⠡⠸⠍⠀⠀⠀⢶⠨⢅⠎⡥⢌⠻⡱⣏⣞⡳⣝⢎⡷⣹⢞⡓⢀⢂⡱⢇⡮⡵⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠁⠀⠀⠀⠀⠙⡀⠀⠈⠐⠘⢯⠿⣄⠢⠑⡂⠀⠀⡴⣉⠣⢎⡘⡔⢊⡑⢦⡙⣶⡹⣞⣎⢷⡹⣮⡝⣯⠹⣜⣣⢟⣥⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⡠⠀⠌⣄⠻⣥⢂⣱⣀⢧⢸⡱⠆⡍⢆⡱⢌⠣⢜⡰⡜⢦⡝⣶⢹⣎⠟⠐⣯⠁⡟⣼⠀⢯⡇⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⢪⠔⣩⠒⣥⠒⡌⡶⢃⡕⠪⢜⠰⠒⡬⢑⡊⠴⣙⢦⡝⣮⢳⣎⢷⠀⠈⠃⠘⠀⠓⠈⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣃⠎⠥⠚⣄⠓⡤⣛⠬⣐⠣⢌⡓⢌⢣⡙⠤⢣⡘⡱⢌⠳⣞⡵⣫⡜⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠣⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠡⢎⡵⣋⠤⢋⡴⢭⣒⠡⢞⡰⢘⡬⢲⡌⡱⢡⠒⢥⢊⡱⢎⡷⢣⡻⢵⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠻⡜⢦⠑⣌⠣⣜⠳⣌⡻⣌⢳⢣⢞⡱⢬⡱⢎⣩⢒⡬⣲⢏⣞⡳⣝⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢛⡬⢳⡌⡲⣍⢳⣍⠶⣍⣎⠳⣎⠵⣣⠝⣎⡶⣝⡺⡵⣫⠞⣵⢫⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⣏⠳⣜⡱⢎⡳⣎⢳⡹⣌⠗⣎⠳⣥⢛⡯⣜⢧⠯⣵⢫⡝⣮⢳⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⢆⠩⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣈⣾⣿⣿⣿⣿⣿⠯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⡀⠄⠐⠨⢌⠱⡈⠱⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠐⠂⡡⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠊⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⡐⢀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⢷⣻⣟⣾⡹⠀⠈⡤⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣴⣦⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣾⢿⠃⠤⢹⡾⣝⠇⠣⠔⠊⢱⢣⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⢿⣾⣻⣟⣯⠟⢁⡀⡀⠼⣿⢋⠐⠁⠠⡈⠄⠸⡀⠀⠁⠣⣜⡲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠙⠛⠛⠛⠋⢹⣿⣿⢿⣻⣿⣽⣾⣟⣿⡽⣞⣯⣟⣾⣳⠟⠈⠉⢀⣀⣀⠀⠏⠰⣈⠄⢠⠙⣄⢂⡙⠢⠄⡊⠐⠹⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣻⣿⣟⣯⡿⣟⡷⣿⡽⣞⣿⡎⠁⠐⠒⠛⠛⠋⠉⠁⠈⠁⠙⣄⠃⠀⠨⠉⠶⡑⢬⠁⢌⠐⡈⡄⠈⡰⢖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⣯⣟⣯⢿⡽⣾⡽⣯⣟⣷⣻⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠠⠘⠀⢤⣶⣤⡀⠈⠐⢍⡊⡐⠀⠌⠉⢱⠫⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣸⢧⣻⡞⣯⣽⣳⣟⡶⣯⣷⢯⣿⣟⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠰⠱⢈⠠⠁⡞⢀⠃⠈⢄⠀⣠⢄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⡿⣯⢷⣟⡷⣯⢷⣯⣟⡷⣿⣟⡿⣾⣽⡠⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⢡⠃⡀⢣⣀⣀⠼⡱⢎⣭⠂⠤⡐⠤⡁⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣯⢿⡽⣯⣷⢯⣟⣿⢯⣟⡷⣯⣇⠢⢄⡀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠠⠀⠄⠃⠀⠐⣀⠆⠀⠄⡘⠸⣌⠁⢀⠔⣄⠑⠀
⠀⠀⠀⠀⠀⣿⣽⣿⣿⣯⣟⣷⡿⣿⣞⣿⡸⣿⣽⣷⡛⣿⡿⣿⣿⣶⣶⡾⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡰⠁⠠⠁⠌⡐⢈⠴⣡⢀⡬⠐⢀⠡⡈⢯⡹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣎⠻⠱⢿⣯⣿⠸⡝⢯⣇⡙⡛⠾⠁⠈⠃⠀⢫⣿⣽⠚⠀⠀⠀⠁⠘⢀⠒⢌⠰⢈⡒⢁⠠⠡⢈⠐⣀⠞⣦⢣⠋⢀⡰⡉⠔⣀⠧⣝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠙⠦⠀⢀⠩⡙⠧⣘⠤⡩⠖⡍⠿⣣⠄⠤⡀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠈⣌⣢⣘⠄⡝⢢⢩⠴⣆⡇⢸⡙⢆⡣⢔⢣⢖⡡⢖⢮⡛⡼⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡵⠎⣑⢢⡑⠢⢍⠢⠱⡠⢙⣴⣭⠂⠹⣿⡂⠀⠀⠀⢀⢄⡓⡌⠷⣭⣛⢮⣕⢫⢞⡲⢣⡹⣜⢸⠰⢉⣚⢮⡕⣫⢖⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠉⠀⠂⠤⡉⠐⢌⠡⡑⣯⡟⣷⠈⢅⠺⠓⠀⠀⠀⣜⠢⢜⡐⢣⠜⡹⢺⡜⡮⣞⡵⣋⢷⡹⣖⡓⡀⢄⡱⢞⡱⡞⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠈⠀⠐⠀⠀⠘⡀⠀⠀⠁⠘⣟⣳⡌⢢⠘⡀⠀⠀⡰⢌⠳⡨⠜⢢⢅⠣⣃⠽⣓⡞⣾⣡⢯⡳⣝⠾⣍⠯⡼⣹⢳⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⢀⡐⠤⢓⣧⢂⣅⠒⣀⠼⡱⢎⡑⠦⡙⣂⠎⡱⢌⡲⣍⠾⣱⢎⡷⡓⠨⣟⠁⣏⡳⠈⢷⡉⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠦⡒⠭⣉⠞⣄⠎⡙⣜⠮⡑⢎⡰⢃⡱⢂⡍⡒⢬⠳⣌⢯⣓⢯⢞⡥⠀⠈⠃⠈⠑⠃⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡜⠠⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢊⡒⣉⠖⡡⢎⡰⢹⡔⣡⢒⡉⠖⡰⢃⢆⠣⢒⡉⠦⡙⢮⡳⣝⠮⣏⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⢅⠣⡜⡰⢎⡑⠦⢱⢫⡔⢢⢎⢌⡱⣑⢮⡰⢉⠦⣉⠖⣉⢆⢻⣜⡻⡼⢼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⣜⡱⠣⠜⣌⢣⢏⣜⢣⠞⣦⠱⣎⠶⣡⢏⡴⢊⠴⣡⢎⡾⢼⡱⣏⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢣⡝⣬⢃⢦⡙⣎⢮⠳⣝⢦⡛⣬⢳⡱⢎⡼⣩⢾⡱⣏⡞⣧⢻⣜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⡴⣋⠶⡹⢬⣏⡳⢭⢲⡙⢦⡣⡝⢮⡟⣵⢫⡵⣫⡜⣧⣛⢶⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡓⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⢐⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠠⠀⢅⠒⡌⠰⡈⠆⠡⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠑⠀⠈⠒⡄⠃⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠄⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠝⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣿⣻⣜⠀⠁⣂⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣠⣎⣤⣧⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣷⣿⠏⢠⢻⣞⡷⠏⠆⡠⠑⢎⠶⡙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⢿⡷⣿⣻⡽⠂⠀⣀⡸⣿⠛⠰⠀⠐⣈⠠⢙⡀⠀⠊⠔⣢⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠘⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣯⣿⣾⢯⣿⣻⣞⣯⣟⡷⠋⠀⠈⢁⡀⠀⡏⠼⡀⠄⢠⠱⣀⠄⡱⠤⠠⠌⠑⠏⡶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡷⣿⣻⡽⣟⣾⣳⣟⡞⠁⠀⠴⠶⠛⠛⠛⠒⠘⠁⠹⡄⡃⠀⠊⠝⣢⠡⡜⢂⢁⠂⠌⡄⠐⢠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣳⢿⡽⣞⣷⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡨⠠⠅⣠⣤⣄⠀⠁⠚⠔⣊⠄⠂⢉⠂⢣⡝⣎⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⣭⣟⣿⡽⣿⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠀⠡⢎⠐⠠⠈⡖⢀⠋⠠⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⣞⣿⣽⡿⣽⣻⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠃⠄⢦⡀⠀⣸⡹⢬⢳⡀⢄⠠⡐⢂⠀
⠀⠀⠀⠀⠀⣯⢿⣽⡷⣯⢿⡽⣷⣻⣞⣿⣯⣟⡷⣯⣇⠒⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⡀⠃⠀⠌⣀⠌⠠⢀⡘⢡⠟⠀⢈⠠⣁⠒⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣽⣯⣟⣿⣷⣻⣾⢸⣯⡿⣷⡟⣿⣿⣷⣾⣶⣦⣴⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⢂⠐⠠⢁⠂⡰⢂⡌⡐⠠⢀⠂⡙⢶⡹⠐⠈⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠿⡻⣾⢿⣽⡸⡻⢽⣇⡙⢟⠯⠇⠘⠧⠉⢿⣹⣽⠯⠀⠈⠑⠐⠄⡢⠐⡄⠢⢑⡸⠀⡐⠠⢈⠐⡀⣎⠵⣣⠚⠀⡡⢎⠰⢀⡮⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⢀⢋⠛⢦⢁⣣⡙⠼⠬⢟⣆⠂⣄⡀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣕⣠⢃⠢⡕⢢⢌⡡⢦⡖⢸⢬⠓⡥⢄⢫⡔⣃⣠⢞⡵⢫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢢⠛⠤⣉⡒⠬⡑⠬⢒⡈⢣⣤⣱⠈⠻⣿⡇⠀⠀⠀⢀⡐⢬⠘⢧⣏⠷⣮⡱⢮⡝⣖⢣⡸⣎⠱⡜⢨⢺⣜⢣⠞⡮⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠙⠡⠄⣈⠐⠌⢢⠡⠸⣼⠿⣽⠈⡡⢹⡓⠀⠀⠀⢬⡘⠤⢋⠆⡭⢛⡶⣹⢎⡼⣬⢓⡷⣹⠶⠅⠂⠌⣘⢧⣛⡴⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠀⠄⠀⠀⠊⡀⠀⠁⠃⠼⢻⣳⡌⡐⠡⡀⠀⠀⣠⠓⣌⠣⢍⠲⢌⡱⣈⠷⣫⢼⡳⣍⢾⡱⣯⢏⡷⣚⡭⢶⡹⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡀⠩⢇⣯⢄⡃⡔⣀⠜⣥⠓⡌⡜⠢⢍⡒⠤⣃⠼⣱⢫⡵⣫⢞⡵⠙⣮⠳⡥⢏⠉⡷⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢑⢣⠚⡤⣊⠴⡉⣞⡜⠦⣙⠰⢌⡓⢌⠲⡑⢬⠳⣥⢳⣭⢳⣫⢆⠀⠉⠣⠈⠋⠂⠸⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢣⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⡔⡊⢆⢓⡰⢉⢦⡱⠢⢜⢢⠡⡍⡒⡌⢎⡱⢘⢢⡙⢶⢫⡖⢯⡖⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⠂⠡⠃⠀⠀⠀⠀⠀⠀⠀⠠⣃⠞⣠⢙⡌⢦⠡⢋⢮⠥⡙⣂⠎⡱⢰⡱⡘⢢⢌⢃⠖⡌⢎⡳⣝⢧⣛⠶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢶⡡⢏⡜⢢⠙⣬⢋⡞⡱⡜⢮⢥⢣⠳⣍⠲⣌⠎⢲⡘⢦⡝⣮⢳⣭⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢛⡼⣄⠣⣍⠲⣍⠾⣱⣙⡎⣎⢧⡛⣬⠳⣌⢏⡧⣝⢮⡝⣮⢳⠶⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠷⣌⡳⢬⡓⣎⢿⡰⢫⠼⣌⢧⡙⢦⡻⣞⡽⣚⡭⣞⡹⣜⠯⣞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢑⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠠⠀⢅⠒⡌⠰⡈⠆⠡⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠂⠈⠒⡄⠃⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠝⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣿⣻⣜⠀⠁⣂⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣷⡿⠏⢠⢻⣞⡷⠏⠆⡠⠑⢎⠶⡙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⣟⣿⣻⡽⠃⠀⣀⡸⣿⠛⠰⠀⠐⣈⠠⢙⡀⠀⠊⠔⣢⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣯⣿⣷⡿⣯⣟⡿⣞⣷⣻⡞⠃⠀⠈⢁⡀⠀⡏⠼⡀⠄⢠⠱⣀⠄⡱⠤⠠⠌⠑⠏⡶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⣿⡿⣿⣻⣿⢿⣽⣾⣟⡷⣯⢿⣽⡞⠁⠠⠴⠶⠛⠛⠛⠒⠐⠁⠽⣀⠃⠀⠊⠝⣢⠡⡜⢂⢁⠂⠌⡄⠐⢠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⣷⣻⢿⡽⣯⢿⡽⣾⣽⣻⣽⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡨⠠⠃⣠⣤⣄⠀⠁⠚⠔⣊⠄⠂⢉⠂⢣⡝⣎⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⡗⣯⢿⣹⣏⡿⣽⣳⢾⡷⣯⣿⣟⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠀⠡⢎⠐⠠⠈⡖⢀⠋⠠⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠰⣿⡽⣯⣷⣻⢾⡽⣷⢯⣟⣿⡷⣿⣽⣳⠠⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠃⠄⢦⡀⠀⣸⡹⢬⢳⡀⢄⠠⡐⢂⠀
⠀⠀⠀⠀⠀⣷⣻⣽⡷⣯⢿⡽⣯⣟⡾⣿⣟⡷⣯⣟⣇⠰⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⡀⠃⠀⠌⣀⠌⠠⢀⡘⢡⠟⠀⢈⠠⣁⠆⠀
⠀⠀⠀⠀⠀⣿⣽⣿⣿⡽⣯⣟⣿⣾⣽⣷⢺⣿⣳⣯⡟⣿⣿⣷⣶⣶⣦⣴⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⢂⠐⠠⢁⠂⡰⢂⡌⡐⠠⢀⠂⡙⢶⡹⠐⠈⠀⠀
⠀⠀⠀⠀⠀⠸⣿⣎⠿⠻⣿⣽⣷⠹⡳⢿⡆⣝⠻⠷⠃⠘⠇⠉⢿⣽⣿⠵⠀⠈⠑⠐⠄⡢⠐⡄⢢⠑⡸⢀⠐⠠⢈⠐⡀⣎⠵⣣⠚⠀⡡⢎⠰⢀⡮⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠻⠦⠀⢰⠙⠺⢇⢡⣃⡹⠦⠽⣳⠆⡠⢄⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣇⣌⠰⢈⡕⢢⢊⣡⠦⡖⢸⢬⠓⡥⢄⢫⡔⣃⣠⢞⡵⢫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡜⠳⢌⡓⡠⢅⡒⠅⢦⢉⢲⣬⢣⠘⢻⣿⠆⠀⠀⠀⢀⡄⢣⠺⣜⢯⡳⣌⢧⢛⣬⡓⢧⡸⣎⠱⡜⢨⢺⣜⢣⠞⡮⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠉⠪⠄⠤⡁⠒⡈⠜⡀⢎⡾⣟⡧⠘⡄⢻⡓⠀⠀⢠⠣⣌⢃⠖⡌⠳⡽⣹⠞⣮⢴⡙⣧⢳⡝⣞⠴⠀⢊⣘⢧⣛⡴⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠄⠀⠀⠀⠣⠀⠈⠐⠈⠻⣵⢧⠡⡘⢠⠀⠀⠀⣌⠳⡐⢎⡘⠬⣑⠢⢍⡟⣦⢯⡝⢦⡟⣼⡹⣎⡗⢮⣱⢺⡼⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠀⡐⠌⣯⡓⣄⡡⢂⣀⠻⣬⢑⡉⠦⣉⠖⣡⠚⡌⡼⡜⢮⡝⣧⢻⠎⠱⣏⠻⢆⢷⠉⡾⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠆⡕⡚⡰⣑⢢⠱⣉⣎⠳⡌⢆⢍⡒⡡⠎⢤⠓⡼⣱⡙⣮⡝⣮⢏⡇⠀⠉⠣⠉⠊⠂⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢨⠐⡩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠞⡰⢡⠱⢌⡒⡇⠖⡌⡱⡘⡌⠦⣘⢡⠚⡤⠓⣌⠱⣝⢶⡹⡖⢯⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠣⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢎⡱⣡⢃⠇⢎⡴⣛⢌⠲⣡⠱⣘⢢⡱⢨⡑⠦⡙⠤⠓⡬⣳⠽⣹⢣⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠣⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠑⢮⡔⡣⢍⡊⢆⡳⣍⢎⢧⡱⢲⢥⢣⡝⣡⢜⡢⢱⡉⣕⢲⣭⣛⡵⣫⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠈⢞⡱⢆⡩⢆⡳⢬⡏⢶⣩⢓⣎⡳⣜⡱⢎⡵⢣⡽⣜⡳⣞⣬⢳⡝⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠘⡵⣋⠶⣩⢞⡱⣏⢮⣍⠳⣌⡳⣌⢳⡻⣞⡝⣮⢳⡝⡶⢭⣞⡹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠓⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⢘⣼⣶⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⡀⠄⢂⠒⡐⢂⠆⡡⠂⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⠁⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣟⠃⠂⠐⠠⢂⠅⠀⠀⠠⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⠿⠟⠛⠋⠁⠀⠀⠀⢀⣃⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠝⣯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢩⠁⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⢷⣿⣻⢖⠈⠑⣀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣠⣜⣤⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⠏⡈⢿⡾⣽⠏⣄⡠⠓⢮⡜⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣿⣟⡾⣟⡿⠊⠀⠀⣸⣿⢍⠰⠀⠠⢐⠀⡙⠀⠀⠩⠐⣤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠺⠿⠿⠿⠟⢻⣿⣿⡿⣿⣿⣻⣽⡿⣷⣻⢯⣟⣷⣻⢾⡽⠋⠒⠈⠁⠁⠀⡏⢠⠒⡀⠠⢣⡐⢀⠳⢄⡀⠜⠢⣓⢎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣽⡿⣟⣿⢯⣟⡿⣞⣷⡏⠋⠠⢤⠶⠾⠟⠛⠂⠰⠃⠽⡀⡅⠀⠙⢚⠴⡂⣌⢂⠐⠠⢈⡄⠐⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢘⣯⢷⣯⣟⣯⢿⡽⣯⢿⣽⣻⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢐⠸⠄⣀⣤⣀⠈⠑⠰⡌⠜⣀⠂⠘⠒⢸⢲⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢨⣟⡻⡾⣝⡾⣯⣟⣭⢿⣞⡷⣿⣯⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠛⠷⢦⡀⠈⠨⡔⠠⠁⠌⡸⢁⠛⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⢾⡽⣷⢯⣟⣷⣻⣞⡿⣿⣽⡿⣽⣳⢀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡩⠀⡁⢂⠇⡐⢆⠀⠁⢰⢫⡛⡵⣀⠀⡠⢀⠔⠁
⠀⠀⠀⠀⠀⣯⢿⣽⣯⣟⣾⣳⣟⡾⣽⢿⣯⢿⣽⣻⡆⠤⢀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠀⠌⠀⠄⠂⡍⢈⠡⠡⢹⡜⠂⠁⣀⠡⢊⠀
⠀⠀⠀⠀⠀⣿⣯⣿⣷⣻⣞⡷⣿⣻⣽⣯⢽⣟⣾⣳⡿⣶⣷⣾⣴⣦⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⡈⠄⡈⠐⠠⣈⠜⡀⢄⠂⠡⢀⠹⢆⢾⠠⠙⠀⠀
⠀⠀⠀⠀⠀⢹⣾⡝⣿⢿⣽⣿⣻⠽⣳⣿⣊⠻⡽⣟⡇⠙⡏⠙⢟⣻⣻⠽⠀⠁⠒⠠⢄⠠⡀⢄⠒⡈⢬⠁⡐⢀⠂⠄⡁⡖⣡⢎⠵⠈⢠⡑⢢⠀⡞⡼⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣄⠀⠉⠞⠿⣌⠡⢚⡙⢦⡱⣶⢂⠄⣀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⡅⢊⠤⠑⡎⢤⠐⣈⡤⠖⣰⡙⢦⠋⡤⡘⡥⠊⣄⢲⢏⢶⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢌⠖⡘⠦⠩⣑⠢⡑⢌⡉⠖⣨⢣⠙⢻⣿⡅⠀⠀⠀⠀⢠⡐⡹⢏⡶⣳⣜⡢⢟⣱⠭⣇⢰⡝⣌⠳⢰⢹⣒⡟⣬⢛⡜⠎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠚⢘⣂⢉⡑⠠⢃⠌⢢⠑⣾⢿⣳⠈⠆⢻⠦⠀⠀⠀⣘⠆⡜⢡⠎⢳⢳⣎⠷⣍⢧⡛⡜⣧⢳⣬⠣⢀⢃⡹⡜⢦⣫⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠈⢡⠀⠘⠀⠎⣱⣯⢟⡠⢉⠢⠁⠀⠀⣀⠭⣒⡉⠖⣌⢃⠖⣌⡻⣜⢧⣻⡱⣭⢳⣎⠷⣆⠖⣮⢹⢮⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠜⣫⠗⡠⢃⠡⠄⡰⢣⠞⡠⢎⡑⢢⠎⡘⢤⢣⡝⢮⡳⣝⢮⡷⠩⣟⠮⡝⢮⠙⣮⡛⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢄⡣⢜⡰⢋⠵⡉⢞⡱⣩⢇⢣⡑⢆⡩⢒⠬⣑⢊⠶⣙⢎⡷⣹⢮⡅⠀⠛⠆⠙⠭⠄⢐⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢎⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢣⡘⠆⡜⡡⢓⡜⢢⡙⡔⡌⡒⡌⠦⣑⢊⠦⡑⢪⡙⢮⡝⣞⡵⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡄⠃⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠔⡣⡘⡜⡰⢡⢃⢾⢡⠒⡬⢰⡑⠬⣑⢂⢣⢒⡉⢦⢡⢋⢾⡱⣞⡵⣫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠙⣌⡱⢣⠝⣐⠣⣚⣬⠳⣬⠱⢦⣘⡱⣍⠎⠦⣑⡜⡰⢢⣉⢶⡹⢶⡹⣎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠈⢷⡩⢎⠔⢣⡜⢦⣛⢴⣋⠷⣌⠳⣌⢏⡳⢥⢲⡱⣣⢞⣧⡛⣧⢻⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⡟⡜⣎⠳⣜⢣⣏⠶⣹⢚⡬⣛⡜⢮⠵⣯⢖⣯⢓⡯⢶⡹⣎⣗⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⡀⠀⠄⠂⡘⠤⡑⠌⠐⠂⠂⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣯⠑⠀⠈⠐⠢⠄⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⠃⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢯⣷⣻⣽⡹⠀⠠⡡⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠋⠄⣹⢷⣛⠇⠣⠔⠊⠵⡲⠝⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢾⣻⣟⣯⠗⢁⡀⡀⠬⣿⢋⠐⠀⢂⠰⠀⠙⡀⠀⠉⠘⣴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣾⡿⣽⢯⣟⡾⣯⢷⡛⠈⠉⢀⣀⣀⠀⠏⠰⣁⠂⢄⠹⣄⣁⠙⠢⢄⠊⡐⠫⢖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⣯⣟⣾⣻⡭⠁⠐⠒⠛⠛⠋⠉⠁⠘⠁⠚⡄⠎⠀⠠⠉⢮⡑⢢⢃⢀⠂⠌⡄⠈⣰⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣻⣞⣷⣻⢾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠤⠘⠁⢤⣶⣤⡀⠈⠁⠎⠬⡐⠠⢈⠉⢰⠛⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⡵⣯⣷⢯⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠡⢓⠀⠂⠌⡚⠠⠉⠐⢠⠀⣠⢄⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⢷⣿⣯⢿⣽⣻⡀⡀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠐⢠⠃⡐⠣⣀⣀⠼⡱⢎⣭⠂⠤⡐⢄⠃⠀
⠀⠀⠀⠀⠀⣯⢿⣽⡷⣯⢿⡽⣷⢯⣟⣾⢿⣻⣞⡷⣧⠐⡠⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠠⠀⠄⠃⠀⠐⢠⠆⠀⠄⡑⢩⢎⠁⢀⠔⠌⠌⠁
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣽⣯⣟⣿⡿⣞⣿⢸⣷⡿⣽⡟⣿⣿⣿⣿⣷⣶⣶⠄⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡰⠁⠠⠁⠌⡐⢈⡔⣃⠠⣈⠐⡀⢂⡙⢮⡹⠈⠈⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠿⡻⣾⢿⣽⡹⠻⢯⣇⡚⡝⠻⠃⠈⠣⠀⢫⣿⣾⡓⠀⠀⠈⠁⠲⢐⠂⡜⢀⠎⠴⠁⠠⠡⢈⠐⣀⠞⡴⣡⠏⢀⡰⢌⠰⢀⡮⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⢀⢋⠛⢦⠙⣌⡙⠴⡩⢟⣆⠂⣄⡀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣎⣐⡌⡐⣋⠖⣩⠴⣒⡇⢸⡙⡖⢥⡐⢦⣑⢊⣠⢞⡵⢫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢢⠛⠤⣙⠒⠬⡑⠬⢒⡈⢣⣤⣱⠈⠻⣷⡇⠀⠀⠀⢀⠰⣨⠑⢯⡞⣵⢣⣞⢩⡞⣥⢳⣨⡓⣜⠢⢉⠾⣌⡏⣖⢫⡜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠙⠡⠠⢌⠁⢊⠔⠡⠸⣼⠿⣽⢀⠣⠹⠓⠀⠀⠀⢬⡑⢢⠍⣆⠹⢎⡷⣹⢎⣼⠲⡭⣖⡻⢦⠓⢈⠨⣑⢞⣬⠳⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠀⠄⠀⠀⠊⡀⠈⠁⠑⠸⢿⣵⡂⢌⠡⢂⠀⠀⣠⠓⣌⠣⢜⡠⠓⣌⠲⡹⣎⢶⣻⡱⢯⡝⣯⣛⢦⢫⡵⣺⡜⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣀⠪⢖⢯⣄⡃⢢⢄⡸⢥⡓⠬⣘⠢⠜⣑⠢⡱⢡⡛⢮⡵⢫⠷⡝⠰⣏⠳⣣⢳⠁⡿⡅⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡰⢔⢪⠘⢦⡒⢌⠣⣞⡸⢃⠬⣑⠢⢍⡚⢄⢣⢱⢫⡜⣣⢟⣭⠻⣅⠀⠉⠣⠁⠋⠂⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢣⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢎⡑⠎⡤⢋⠴⡘⢮⡑⢢⡑⢎⢒⡡⠚⡔⡸⢌⢢⢃⠣⣝⢧⡻⣜⡻⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⠂⠡⠃⠀⠀⠀⠀⠀⠀⠀⠠⣃⠖⡌⣓⠴⡉⢦⢱⢫⡜⢡⡜⢂⠣⡜⣱⡘⢔⡊⠦⡡⠓⣌⢳⡝⣮⢵⣫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢶⡸⢥⠣⢍⠢⢎⡳⣜⢣⡜⣣⠳⣜⡱⢎⡒⣬⠱⣁⠏⣤⢟⡼⣓⢮⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢹⢦⣙⠢⣍⢮⡱⣎⢳⡼⣡⢛⡴⣙⢮⡱⢎⡳⣬⢟⡼⣋⢾⣙⢮⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡷⣌⢳⡜⢦⢳⡹⢎⡽⣡⢏⡼⡱⢎⣳⣏⢷⡹⣎⢷⡹⢮⡝⣮⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠋⢀⠀⡀⠀⠄⠐⠐⡀⢣⠘⠄⠃⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⢇⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⡿⣮⠄⠈⡔⣲⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣯⢿⠃⢃⠐⠉⠲⡹⠌⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⠄⢱⠀⡘⡄⠀⠁⠐⣝⡚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡰⣄⠂⠤⠑⠦⣄⢊⡑⠒⡌⢀⠋⢧⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⢚⢂⠇⠀⠑⠈⠓⣌⠲⡐⢠⠂⡀⢆⠀⣸⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠂⠁⠴⢶⣦⣄⠀⠑⠈⡣⢐⠠⠈⠡⢰⠍⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠆⠀⠐⠭⠀⠌⡀⡏⢀⡁⠈⠔⢀⡤⢤⣀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠐⠀⢰⠁⠠⠑⢤⠀⠼⣘⢧⣚⠄⢢⠐⡡⢃⠀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠄⡈⠁⢀⠡⣈⠆⠐⠠⠌⠲⣍⡀⣠⠒⣠⠊⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠠⠀⣀⠀⡀⠀⠀⠀⡀⢄⠢⠔⠠⢁⠂⠄⡁⢂⠴⣡⢀⡬⠐⡀⠆⡈⢳⢭⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠁⠊⠔⡡⠌⢂⡜⠠⢁⠂⠌⠐⢠⢏⠶⣡⠋⠀⡤⢃⠔⣀⡏⣞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⣰⡈⠆⡝⢢⢣⡜⡞⡍⢸⢎⠳⣡⠒⣍⢖⣡⠴⣊⡗⠮⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡰⢨⢌⠳⣝⣛⢮⣳⢩⢞⡭⢳⡜⣎⢱⠢⠩⡜⢮⡕⣫⢕⡺⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢈⡖⣡⢊⡱⢌⠛⣮⢳⠧⣬⢞⡥⣛⢮⣳⢓⡀⢌⡱⢮⡱⠮⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⠜⠲⣄⠣⢒⡌⢓⡌⢳⢻⣜⣫⡖⣯⡳⣝⢮⡝⣭⢺⣱⢏⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⣋⢞⡡⢆⠹⡰⣈⠣⢌⠣⡞⣬⢳⢞⡵⣋⠸⣣⠛⡴⢣⠈⣟⡌⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⢸⡱⠎⠦⡑⡌⢣⠔⡡⢓⡌⡳⡝⢦⣛⢮⡳⣭⠀⠈⠃⠈⠙⠃⠸⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⡴⢊⠴⡉⠖⡱⣈⠇⡜⣡⠒⣌⠱⣙⢮⡝⣮⢳⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⣝⢣⠍⣢⠙⣌⠱⣐⠎⡔⢢⢍⡰⢃⡌⢳⡝⣮⢳⡝⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡭⢎⡳⢆⡻⢤⢫⡜⡹⣌⡒⠦⣑⢊⡜⣲⡝⣮⢳⡝⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣭⠳⣍⡞⣥⠳⣜⡱⢎⡝⡳⣌⢷⡺⡵⣛⡼⣣⢟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⣜⢧⡹⣙⢦⢫⠵⣙⢞⡾⣹⡝⣮⢳⡽⣱⢫⡵⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠑⠐⠤⢀⢀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠢⠘⡠⠑⢢⡘⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣧⣐⡉⢄⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⠔⣫⡝⣧⢳⣞⡸⢮⡱⠽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢩⠚⠤⡙⢎⠷⣮⡕⢦⣏⡳⣹⢎⡷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⢆⡙⠆⡭⢘⢢⡱⢏⡿⣜⡳⡽⣎⢷⡹⣎⠵⣚⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠤⣃⠭⡰⢉⢆⡱⢎⡳⣭⢳⡽⣉⠐⢯⡙⢮⡹⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢊⡕⠤⠓⣌⠱⡌⡜⢧⠳⣭⢳⢞⣭⠀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢣⠘⣌⠓⣌⠲⢌⡙⢎⡿⣜⣫⢞⡼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠦⣉⠦⢩⢄⠳⣈⠖⣡⠺⡵⣭⠞⣵⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣌⠶⣩⢞⡡⢎⠖⣡⠚⣤⢳⡝⡶⢻⡜⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⡹⣌⣳⠱⣎⠵⣋⠞⣴⢫⡞⢧⣛⣭⢳⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡝⢦⣍⠻⣌⡳⢯⡻⡵⣫⢞⡧⢯⡜⣧⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠑⠐⠤⢀⢀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠢⠘⡠⠑⢢⡘⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣧⣐⡉⢄⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⠔⣫⡝⣧⢳⣞⡸⢮⡱⠽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢩⠚⠤⡙⢎⠷⣮⡕⢦⣏⡳⣹⢎⡷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⢆⡙⠆⡭⢘⢢⡱⢏⡿⣜⡳⡽⣎⢷⡹⣎⠵⣚⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠤⣃⠭⡰⢉⢆⡱⢎⡳⣭⢳⡽⣉⠐⢯⡙⢮⡹⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢊⡕⠤⠓⣌⠱⡌⡜⢧⠳⣭⢳⢞⣭⠀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢣⠘⣌⠓⣌⠲⢌⡙⢎⡿⣜⣫⢞⡼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠦⣉⠦⢩⢄⠳⣈⠖⣡⠺⡵⣭⠞⣵⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣌⠶⣩⢞⡡⢎⠖⣡⠚⣤⢳⡝⡶⢻⡜⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⡹⣌⣳⠱⣎⠵⣋⠞⣴⢫⡞⢧⣛⣭⢳⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡝⢦⣍⠻⣌⡳⢯⡻⡵⣫⢞⡧⢯⡜⣧⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠳⡠⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢡⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠿⠋⠠⢀⠠⡀⠂⠈⠀⠔⠁⠂⠑⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⢤⠙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠱⣯⢿⣽⡞⡀⠀⣭⢚⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣦⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⠿⠁⠁⢺⣟⡾⢁⠡⠉⢀⠉⠞⠁⠈⠔⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⣳⢯⣟⡷⢋⠠⠄⠐⠚⡿⠐⡂⢀⠈⢆⠐⡈⢆⡀⠀⡰⢫⡝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣯⣿⣾⢿⡽⣯⣟⡷⣯⠿⠊⢀⣀⣠⣤⣤⣀⠣⠜⣡⠂⡌⠈⠳⢴⡈⢌⠑⡄⠐⡈⠱⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣽⣿⢿⣽⢯⣟⣿⣳⢯⣿⡗⠀⠈⠉⠉⠉⠁⠀⠀⠐⢀⠉⠱⡄⠀⡈⠀⠙⠢⣍⠰⡐⠠⢁⠣⠄⢞⡭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣽⣻⡽⣯⣟⣾⣻⣞⡷⣯⣿⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠙⠛⢷⣦⣀⠀⠁⠇⡜⠀⡐⠈⡼⢘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡷⣽⣞⡷⣯⣿⣳⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠈⡜⠀⠄⠡⠎⠰⡀⠀⠃⣠⢖⡲⣄⠀⠀⢀⢀⠠⠁
⠀⠀⠀⠀⠠⢾⣻⡽⣶⢯⣟⡷⣯⣟⡷⣿⣯⣟⡿⣾⡅⡀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⢀⡙⠀⢂⠉⡒⢀⠓⠮⣱⠮⠑⠈⠂⢌⢂⠀
⠀⠀⠀⠀⠀⣼⣷⣻⣯⣟⡾⣽⣷⢯⡿⣽⣻⢾⣽⣳⣷⣨⣔⣤⣀⣀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡀⢂⠀⠄⢂⡜⢀⠂⡐⠂⡑⢧⡄⡴⢉⠆⠀⠀
⠀⠀⠀⠀⠀⢻⣞⣿⣿⣾⣽⣳⣿⢯⣿⣽⠸⢿⣷⢿⡇⢻⡿⠻⣿⣿⢿⡿⠈⠐⠂⠄⡠⢀⠀⡠⢠⠘⠤⠃⡀⢂⠐⠠⠈⡔⢣⢒⡤⠒⢀⡡⠄⠂⣝⢲⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠻⡻⢾⣟⣧⣸⠻⣟⣧⢃⠎⣝⠃⠀⠁⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⠁⡊⠔⡠⢃⢎⡐⡠⠂⢌⡠⢱⡍⢧⡋⠄⡐⢎⠔⡈⣰⢏⡞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⢛⢦⠑⣌⣉⠮⠞⠼⣖⠠⢠⡀⢰⣿⣿⡃⠀⠀⠀⠀⠀⢀⣇⡘⡄⠌⡦⣑⢶⡫⢏⡅⢺⡜⡱⢢⡙⣜⢣⡖⢮⡵⣚⠜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡆⠦⣉⡒⠤⢃⠎⡱⢌⠱⣡⡞⡄⡙⣿⣅⠀⠀⠀⠀⣘⠩⣌⢻⣹⣛⠶⣭⠲⢛⣬⢛⡶⣥⠓⡅⠘⡜⡳⢎⡳⣜⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠈⠥⠄⢌⡁⠊⠔⡰⢈⢶⢿⣻⠆⢡⠘⠃⠀⠀⠀⢨⢆⠣⡔⢢⡑⢮⢻⡼⣙⣦⢧⣋⢾⡱⢯⡍⣄⡰⣜⢫⠶⡍⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠐⠁⠌⢷⣻⣌⠂⡑⢂⠀⠀⢠⠜⣊⠖⣡⢃⠜⣂⠣⣝⢳⣎⠷⣭⢳⡽⢧⣻⡜⣱⢎⠿⣹⢇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⠀⢀⠢⠱⣝⢢⣁⠎⣀⡰⣋⠞⠤⢚⠄⡎⡜⠤⠓⣌⡳⢎⢿⡸⢧⡇⠘⢵⡈⢧⢻⠀⢻⠆⠁⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣎⠴⢊⡕⢪⢖⠨⡜⣡⠳⢍⢪⠱⡉⡜⡰⢌⢣⠙⡦⡝⣎⢷⣹⠳⣎⠀⠀⠁⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⠞⡰⠌⡥⢊⠵⣢⠣⢍⡊⡔⢣⠱⡘⢤⠃⣎⢡⠓⡽⣜⡳⣎⠿⣜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢎⡒⡩⢜⡡⠞⡰⣉⠞⡥⠚⣤⠱⢌⡱⢢⡍⢢⠍⡔⣊⠜⡰⢩⠷⣭⢻⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠣⣕⢫⠜⣡⠱⢌⡻⣜⠳⣔⢫⠖⣜⢣⢞⣡⢚⠴⡨⢜⡡⣏⡟⣼⢣⡟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⣧⢚⡄⠳⣌⠳⣼⠹⣌⢧⡛⣬⢓⡎⢶⡩⢖⣹⢎⡷⣹⢺⡱⣏⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢸⢣⢞⡳⣜⡹⣜⢧⣋⢧⡹⣌⠯⣜⣣⡟⣯⢞⡹⣜⢧⡏⡷⣭⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠋⠠⢀⠠⡀⠂⠈⢀⠰⠁⠂⠑⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣯⢣⠘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠣⣟⡿⣞⡿⡀⠀⣍⢞⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⠿⠁⠒⠘⣿⡽⢁⠡⢈⠠⠈⠳⠃⠐⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⡽⣯⣷⢋⠠⠤⠀⠚⣿⠁⢂⠄⠠⣘⠠⠁⢦⡀⠀⡠⢞⡬⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣽⣳⢯⣟⣷⡻⠓⠀⣀⣠⣤⣤⣄⢃⠜⢦⡐⠤⠀⠳⢬⣂⠌⡉⡄⠠⢈⠓⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⣯⣟⣯⣟⣯⣿⣾⠀⠈⠉⠉⠉⠈⠀⠀⠀⠂⠈⠱⡂⠀⣈⠀⠊⠦⢱⠈⡴⠀⢂⠣⠤⢸⡱⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⢿⡾⣽⣳⣟⣾⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠘⠛⠷⣦⣀⠀⠑⠘⣌⠠⠐⠠⢸⠁⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣭⣟⣾⣳⣿⣞⣿⢿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢒⠠⠈⠄⡏⠀⡄⠀⠊⢠⢖⡲⢆⡀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⡻⣷⣻⣞⡷⣿⣯⣟⣯⣟⡆⠄⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡰⠀⠡⠘⡒⠐⠊⢮⢱⢫⠔⠈⠑⢨⠁⠄
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⣟⡷⣿⢾⣽⣻⢻⣞⡷⣯⣿⣈⣖⣠⣄⣀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⡐⢀⠠⠁⡒⠁⠌⡐⠂⠘⢧⡄⣴⠉⡅⠊⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣟⣾⡽⣿⣟⣾⢿⡩⢿⣟⣯⡇⢻⡿⠿⣿⡿⣿⣟⠉⠐⠂⠄⡠⢀⠀⡠⢀⠆⡑⠎⢀⠁⢂⠐⡀⢢⡙⠴⣡⠖⠀⡌⠌⠠⣙⠶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢯⣿⢷⡘⡟⢿⣇⠎⡭⣉⠃⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⠁⡊⠔⢂⠌⡜⣀⠢⠘⣀⣂⠰⢧⡙⢧⠁⣠⠚⡔⠊⢡⡏⣞⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⢧⠘⢤⣉⠺⠴⢹⣳⠠⢠⣀⢠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣇⣘⡄⢊⡵⣈⢧⣏⠧⡇⢸⢣⡙⢆⡚⢤⡏⣴⢚⡧⣙⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡜⠢⢍⡒⠢⢍⠢⠕⢤⠙⣤⣎⡤⠙⣯⣇⠀⠀⠀⠀⣈⠣⣌⠳⣝⡳⢶⣹⡰⢋⡞⣹⢎⡧⡜⢆⠘⠼⣱⢣⢏⠶⡉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠈⠭⠄⠌⡁⠊⠔⡉⢄⢳⡟⡿⡆⢡⠘⠃⠀⠀⠀⢈⡆⡓⠤⠓⣌⠛⣧⢻⡴⣣⢟⡴⣫⢞⡽⣩⢄⣢⠴⣋⣮⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠈⢅⠀⠁⠊⠈⠿⣽⣆⠡⡘⠠⠀⠀⢠⠜⠲⣉⠜⡱⢨⠱⣈⢧⢛⡵⣫⠶⡽⣎⢷⡹⣎⢖⣫⠽⣎⡷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⢀⠒⡱⢞⣡⣐⢡⢂⣠⢋⡞⡱⢨⠜⡡⢃⡕⠢⢎⣎⢳⣭⣛⡵⡋⠐⢻⡌⢮⡱⡀⢫⡆⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠦⠓⡔⠫⣔⢌⠪⡱⣆⢏⠒⣅⠣⢊⡱⢌⢢⡙⡜⣎⢳⡲⣭⢞⡽⡀⠀⠉⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢩⡘⡱⢌⠓⡤⢋⣖⠱⢂⠎⡱⢨⠜⣡⠒⣌⠒⡬⡘⢼⣣⢟⡲⣏⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢎⡜⢢⠑⣦⠩⡜⠰⣍⢦⡙⣌⠎⣡⢃⡎⡴⠑⡬⢑⠢⢍⢢⠱⣏⡳⣭⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠱⡎⡝⡆⡓⢌⠳⣌⠷⣘⢦⡛⡴⡸⣜⡱⣩⠔⣍⠚⡌⢦⡝⣮⢳⣭⢳⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠸⡱⢎⣑⠪⡱⢎⡽⡱⢎⡵⢣⢳⣌⠳⣥⢛⡬⣳⡹⢮⡝⣮⢳⣎⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⢎⡜⣣⠝⡮⣜⢧⢫⡝⣣⢏⡼⣙⠖⣿⠺⣕⢯⡳⣝⢮⡳⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⡜⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢱⣼⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⠠⢀⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⡠⠌⠑⠂⠘⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠗⣯⣟⣯⡟⡄⠀⣒⡝⡲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⢿⠋⠐⡸⣟⣾⠃⠅⢂⠁⡘⠵⠋⠀⢂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣳⣯⢿⡽⣞⠁⡀⠤⠔⣿⠃⢎⠀⢀⢒⠠⠘⢂⠀⠈⢀⢯⣙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣷⡿⣯⣟⡷⣯⢷⡯⠟⠁⠀⣀⣠⣤⡀⢃⡜⢤⠂⠤⠈⠶⢤⡉⠍⢂⠅⠠⠉⠓⡃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠸⣷⣿⢿⣽⣿⣯⢿⣯⣟⡷⣯⢿⣽⣯⠀⠀⠚⠛⠉⠉⠁⠉⠀⡀⠘⠣⡒⠀⠈⠐⠉⠞⣨⠂⠥⠈⠄⢣⡀⢜⡫⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⡽⣞⡿⣾⢽⣻⣽⣻⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⠁⠒⠿⢷⣤⡀⠀⠋⠜⣨⠐⢀⠐⢨⠓⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣫⣞⢷⣻⡽⣯⡽⣯⣷⢯⣿⣟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠧⠐⠠⠈⡞⠀⡆⠈⠐⢠⡔⡲⣄⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠠⢾⣻⢷⣯⣟⣷⣻⢷⣻⢷⣿⣿⣻⣞⡿⡔⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⢢⠁⠂⠑⢢⠄⠚⠼⣱⢚⡌⠢⠑⢌⡐⡀
⠀⠀⠀⠀⠀⣼⢯⣿⣷⣻⢾⣽⣻⣽⣻⢾⣟⡷⣯⣟⣿⣈⣔⣠⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠐⠠⠁⡀⢈⡰⠃⠠⠈⠔⡈⢧⣀⣠⠎⡰⠀⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣽⣻⣞⣿⢷⣯⣿⠜⣿⡷⣿⡇⢻⡿⢿⣿⣿⣿⢾⠑⠤⠠⣀⠀⡀⢀⠀⢠⠐⠢⠜⠀⠄⡁⢂⠡⢐⠦⡱⢄⡅⠊⢠⠐⡈⢲⠭⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⣽⢿⣾⡘⡟⢾⣗⠬⣙⢫⠃⠀⠑⠀⠸⢷⣿⡏⠀⠀⠀⠀⠉⠐⡂⢩⠐⠌⡱⡀⢌⡐⢀⢂⠄⣏⠶⣙⠂⢠⢌⢣⠘⢀⣏⠞⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⢧⡈⢅⣊⠳⠬⢞⣳⠠⢠⣀⠠⣿⣿⡗⠀⠀⠀⠀⠀⠀⣜⣠⢊⠔⣣⠙⣦⢲⣋⡇⢸⢎⢣⠇⣎⠱⣎⡤⢖⣫⢎⡽⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡜⠢⢍⡒⠌⡕⡊⠔⣌⠑⢦⡜⡤⠙⣿⡇⠀⠀⠀⠀⡒⢪⠅⢯⣫⢟⡲⣽⠸⡳⡬⢽⡜⣮⡘⡜⠠⢛⠶⣙⢮⡱⡎⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠈⠭⠀⢌⡁⠢⠑⡌⠤⣹⢿⣻⡅⠒⡨⠍⠀⠀⠀⢠⡙⡔⢊⢆⡱⢋⠷⣭⠗⡦⢟⢦⣛⠶⣽⣘⣀⢂⡮⣙⢦⡓⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠐⢁⠀⠐⠂⠌⢷⣛⣦⠑⡠⠡⠀⠀⢠⠔⠳⢌⠣⡌⢆⠭⡘⣬⠻⣝⢯⡖⣯⣛⢶⡹⣎⠳⣼⢹⣎⡷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⡀⢀⠠⠹⣜⣡⣂⠥⢡⡀⣏⠞⣑⢊⡱⢌⠪⡔⣑⠢⡝⣎⢷⡹⡖⡏⠈⢷⡉⢟⡼⠀⢳⣃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠢⣃⠳⡘⢤⠢⢍⢣⠳⣌⠳⣈⠦⣑⢊⡱⠰⢌⡳⡹⡜⣮⢳⡽⣹⠀⠀⠁⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣈⠕⣂⠇⡩⢆⡙⢦⡉⠖⣌⢒⡡⢒⡌⠦⣁⢏⡰⢡⠳⣝⢮⡳⢞⡵⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⡒⠬⡘⣔⢪⢑⠢⣝⡣⠜⣡⠒⣌⠒⣥⡘⢆⠥⡒⠬⡑⡜⡸⢧⣻⣙⠾⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡍⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠩⣖⡹⠜⢢⡉⠖⡥⢏⡳⢆⡻⢤⣋⠶⣙⣌⠲⣅⢣⠑⣦⡹⢧⡳⣭⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⣁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⡵⣋⢆⠥⣛⠼⣩⠳⣍⡞⡱⢎⡳⡱⢎⡳⡜⢮⡹⢶⡹⢧⣛⠶⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⢬⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⡱⢎⡳⣍⢞⡱⣏⡼⣍⠳⣍⡳⣍⢳⢷⣫⢷⡹⢧⣛⢧⢏⡻⣖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⡜⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢱⣼⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠋⢀⠀⡀⠀⠄⠐⠈⡐⢌⠰⠁⠊⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⢇⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠗⣯⣟⡿⣮⠄⠈⣑⢲⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⢿⠋⠐⡸⣯⢿⠃⢃⠐⠉⠊⣗⠎⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣳⣯⢿⡽⣞⠁⡀⠤⠔⣿⠃⡍⠀⠈⢤⠁⡐⡄⠀⠈⢀⡟⡜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣷⡿⣯⣟⡷⣯⢷⡯⠟⠁⠀⣀⣠⣤⡀⢃⡰⣘⠠⢈⠐⠦⣄⡘⡑⠒⡄⠈⠝⠲⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠸⣷⣿⢿⣽⣿⣯⢿⣯⣟⡷⣯⢿⣽⣯⠀⠀⠚⠛⠉⠉⠁⠉⠈⡀⢘⠢⡅⠀⠘⠀⠳⢌⠲⢠⢁⠂⡁⢇⡀⣘⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⡽⣞⡿⣾⢽⣻⣽⣻⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠡⠰⢶⣦⣄⠀⠉⠂⢣⢐⠀⢂⠁⢌⡛⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣫⣞⢷⣻⡽⣯⡽⣯⣷⢯⣿⣟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠛⠆⡀⠀⢏⠐⠠⠈⡜⠠⡁⠁⠆⢀⡤⢤⣀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠠⢾⣻⢷⣯⣟⣷⣻⢷⣻⢷⣿⣿⣻⣞⡿⡔⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⢢⠁⢂⠱⢄⠀⠼⣜⢣⢞⡰⢀⠆⡡⢃⡀
⠀⠀⠀⠀⠀⣼⢯⣿⣷⣻⢾⣽⣻⣽⣻⢾⣟⡷⣯⣟⣿⣈⣔⣠⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠄⡀⠃⠀⢂⠰⢈⠐⠠⠠⠙⣎⡀⣀⠔⣠⠒⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣽⣻⣞⣿⢷⣯⣿⠜⣿⡷⣿⡇⢻⡿⢿⣿⣿⣿⢾⠠⠄⡀⢀⠀⠀⠀⠀⡀⢄⠂⡔⠁⡈⠐⠠⠁⢌⠴⣉⢄⡨⠄⢁⡂⠌⢳⡹⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⣽⢿⣾⡘⡟⢾⣗⠬⣙⢫⠃⠀⠑⠀⠸⢷⣿⡏⠀⠀⠀⠈⠀⠣⠘⡐⠰⡈⡜⢀⠂⠤⢁⠂⢡⠞⣦⢓⠎⢀⡰⢊⠔⡀⣧⢳⡁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⢧⡈⢅⣊⠳⠬⢞⣳⠠⢠⣀⠠⣿⣿⡗⠀⠀⠀⠀⠀⠀⣱⣈⡡⠐⡍⢦⣩⠴⣎⡏⢰⢏⠖⣩⠐⣎⡴⣡⢦⣚⡥⡳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡜⠢⢍⡒⠌⡕⡊⠔⣌⠑⢦⡜⡤⠙⣿⡇⠀⠀⠀⠀⣠⢊⠅⢯⣝⡻⢞⣦⢹⠞⣦⢹⡜⣎⢜⠢⢙⡸⢖⡧⣚⡜⡲⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠈⠭⠀⢌⡁⠢⠑⡌⠤⣹⢿⣻⡅⠒⡨⠍⠀⠀⠀⢠⡅⠎⡜⢢⠜⡹⢭⡞⢧⡬⢧⡣⢟⣼⣚⡓⠠⢐⡩⢖⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠐⢁⠀⠐⠂⠌⢷⣛⣦⠑⡠⠡⠀⠀⢠⠔⣪⠱⠌⡥⢚⡐⢣⢜⢧⡻⣭⢳⡻⢶⣭⢻⡽⣩⢞⣱⢏⣯⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⡀⢀⠠⠹⣜⣡⣂⠥⢡⡀⣏⢞⡠⠣⢍⡔⠣⣌⢃⠎⡼⣱⢫⡳⣝⡏⠘⣧⠍⣖⢫⠄⢻⡌⠑⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠢⣃⠳⡘⢤⠢⢍⢣⠳⡜⠢⢥⠙⡢⢌⠣⡔⠪⡜⢧⢣⣏⢗⡳⣞⡀⠈⠑⠈⠁⠓⠈⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣈⠕⣂⠇⡩⢆⡙⢦⡉⢖⢡⢋⠴⣉⠴⣉⠲⢌⡱⢉⠾⣱⢎⣯⢳⡭⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⡒⠬⡘⣔⢪⢑⠢⣝⡣⠜⣌⠲⢌⠲⣌⡒⣌⠱⢊⡔⢩⢢⠹⣞⡼⣣⠷⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡍⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠩⣖⡹⠜⢢⡉⠖⡥⢏⡳⣌⠳⣎⠵⣪⢕⡢⣍⠖⡌⣃⠦⣝⡞⢶⣭⢳⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⣁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⡵⣋⢆⠥⣛⠼⣩⢳⣜⠳⣬⢳⡱⢎⡵⣊⠞⣼⣡⠿⣼⣩⢗⣮⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⢬⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⡱⢎⡳⣍⢞⡱⣇⠾⣙⢦⢫⡕⣫⢖⣯⢻⡵⣹⠞⡵⣎⢯⡖⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⡜⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢱⣼⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠋⢀⠀⡀⠀⠄⠐⠈⡐⢌⠰⠁⠊⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⢇⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠗⣯⣟⡿⣮⠄⠈⣑⢲⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⢿⠋⠐⡸⣯⢿⠃⢃⠐⠉⠊⣗⠎⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣳⣯⢿⡽⣞⠁⡀⠤⠔⣿⠃⡍⠀⠈⢤⠁⡐⡄⠀⠈⢀⡟⡜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣷⡿⣯⣟⡷⣯⢷⡯⠟⠁⠀⣀⣠⣤⡀⢃⡰⣘⠠⢈⠐⠦⣄⡘⡑⠒⡄⠈⠝⠲⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠸⣷⣿⢿⣽⣿⣯⢿⣯⣟⡷⣯⢿⣽⣯⠀⠀⠚⠛⠉⠉⠁⠉⠈⡀⢘⠢⡅⠀⠘⠀⠳⢌⠲⢠⢁⠂⡁⢇⡀⣘⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⡽⣞⡿⣾⢽⣻⣽⣻⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠡⠰⢶⣦⣄⠀⠉⠂⢣⢐⠀⢂⠁⢌⡛⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣫⣞⢷⣻⡽⣯⡽⣯⣷⢯⣿⣟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠛⠆⡀⠀⢏⠐⠠⠈⡜⠠⡁⠁⠆⢀⡤⢤⣀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠠⢾⣻⢷⣯⣟⣷⣻⢷⣻⢷⣿⣿⣻⣞⡿⡔⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⢢⠁⢂⠱⢄⠀⠼⣜⢣⢞⡰⢀⠆⡡⢃⡀
⠀⠀⠀⠀⠀⣼⢯⣿⣷⣻⢾⣽⣻⣽⣻⢾⣟⡷⣯⣟⣿⣈⣔⣠⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠄⡀⠃⠀⢂⠰⢈⠐⠠⠠⠙⣎⡀⣀⠔⣠⠒⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣽⣻⣞⣿⢷⣯⣿⠜⣿⡷⣿⡇⢻⡿⢿⣿⣿⣿⢾⠠⠄⡀⢀⠀⠀⠀⠀⡀⢄⠂⡔⠁⡈⠐⠠⠁⢌⠴⣉⢄⡨⠄⢁⡂⠌⢳⡹⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⣽⢿⣾⡘⡟⢾⣗⠬⣙⢫⠃⠀⠑⠀⠸⢷⣿⡏⠀⠀⠀⠈⠀⠣⠘⡐⠰⡈⡜⢀⠂⠤⢁⠂⢡⠞⣦⢓⠎⢀⡰⢊⠔⡀⣧⢳⡁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⢧⡈⢅⣊⠳⠬⢞⣳⠠⢠⣀⠠⣿⣿⡗⠀⠀⠀⠀⠀⠀⣱⣈⡡⠐⡍⢦⣩⠴⣎⡏⢰⢏⠖⣩⠐⣎⡴⣡⢦⣚⡥⡳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡜⠢⢍⡒⠌⡕⡊⠔⣌⠑⢦⡜⡤⠙⣿⡇⠀⠀⠀⠀⣠⢊⠅⢯⣝⡻⢞⣦⢹⠞⣦⢹⡜⣎⢜⠢⢙⡸⢖⡧⣚⡜⡲⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠈⠭⠀⢌⡁⠢⠑⡌⠤⣹⢿⣻⡅⠒⡨⠍⠀⠀⠀⢠⡅⠎⡜⢢⠜⡹⢭⡞⢧⡬⢧⡣⢟⣼⣚⡓⠠⢐⡩⢖⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠐⢁⠀⠐⠂⠌⢷⣛⣦⠑⡠⠡⠀⠀⢠⠔⣪⠱⠌⡥⢚⡐⢣⢜⢧⡻⣭⢳⡻⢶⣭⢻⡽⣩⢞⣱⢏⣯⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⡀⢀⠠⠹⣜⣡⣂⠥⢡⡀⣏⢞⡠⠣⢍⡔⠣⣌⢃⠎⡼⣱⢫⡳⣝⡏⠘⣧⠍⣖⢫⠄⢻⡌⠑⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠢⣃⠳⡘⢤⠢⢍⢣⠳⡜⠢⢥⠙⡢⢌⠣⡔⠪⡜⢧⢣⣏⢗⡳⣞⡀⠈⠑⠈⠁⠓⠈⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣈⠕⣂⠇⡩⢆⡙⢦⡉⢖⢡⢋⠴⣉⠴⣉⠲⢌⡱⢉⠾⣱⢎⣯⢳⡭⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⡒⠬⡘⣔⢪⢑⠢⣝⡣⠜⣌⠲⢌⠲⣌⡒⣌⠱⢊⡔⢩⢢⠹⣞⡼⣣⠷⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡍⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠩⣖⡹⠜⢢⡉⠖⡥⢏⡳⣌⠳⣎⠵⣪⢕⡢⣍⠖⡌⣃⠦⣝⡞⢶⣭⢳⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⣁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⡵⣋⢆⠥⣛⠼⣩⢳⣜⠳⣬⢳⡱⢎⡵⣊⠞⣼⣡⠿⣼⣩⢗⣮⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⢬⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⡱⢎⡳⣍⢞⡱⣇⠾⣙⢦⢫⡕⣫⢖⣯⢻⡵⣹⠞⡵⣎⢯⡖⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢑⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠠⠀⢅⠒⡌⠰⡈⠆⠡⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠂⠈⠒⡄⠃⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠝⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣿⣻⣜⠀⠁⣂⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣷⡿⠏⢠⢻⣞⡷⠏⠆⡠⠑⢎⠶⡙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⣟⣿⣻⡽⠃⠀⣀⡸⣿⠛⠰⠀⠐⣈⠠⢙⡀⠀⠊⠔⣢⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣯⣿⣷⡿⣯⣟⡿⣞⣷⣻⡞⠃⠀⠈⢁⡀⠀⡏⠼⡀⠄⢠⠱⣀⠄⡱⠤⠠⠌⠑⠏⡶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⣿⡿⣿⣻⣿⢿⣽⣾⣟⡷⣯⢿⣽⡞⠁⠠⠴⠶⠛⠛⠛⠒⠐⠁⠽⣀⠃⠀⠊⠝⣢⠡⡜⢂⢁⠂⠌⡄⠐⢠⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⣷⣻⢿⡽⣯⢿⡽⣾⣽⣻⣽⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡨⠠⠃⣠⣤⣄⠀⠁⠚⠔⣊⠄⠂⢉⠂⢣⡝⣎⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⡗⣯⢿⣹⣏⡿⣽⣳⢾⡷⣯⣿⣟⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠀⠡⢎⠐⠠⠈⡖⢀⠋⠠⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠰⣿⡽⣯⣷⣻⢾⡽⣷⢯⣟⣿⡷⣿⣽⣳⠠⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠃⠄⢦⡀⠀⣸⡹⢬⢳⡀⢄⠠⡐⢂⠀
⠀⠀⠀⠀⠀⣷⣻⣽⡷⣯⢿⡽⣯⣟⡾⣿⣟⡷⣯⣟⣇⠰⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⡀⠃⠀⠌⣀⠌⠠⢀⡘⢡⠟⠀⢈⠠⣁⠆⠀
⠀⠀⠀⠀⠀⣿⣽⣿⣿⡽⣯⣟⣿⣾⣽⣷⢺⣿⣳⣯⡟⣿⣿⣷⣶⣶⣦⣴⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⢂⠐⠠⢁⠂⡰⢂⡌⡐⠠⢀⠂⡙⢶⡹⠐⠈⠀⠀
⠀⠀⠀⠀⠀⠸⣿⣎⠿⠻⣿⣽⣻⠼⡻⢾⣇⠹⣛⠷⠃⠘⠇⠉⢿⣽⣿⠵⠀⠈⠑⠐⠄⡢⠐⡄⢢⠑⡸⢀⠐⠠⢈⠐⡀⣎⠵⣣⠚⠀⡡⢎⠰⢀⡮⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠻⠦⠀⢀⠛⠻⢆⠡⣃⡙⠞⡤⢿⢆⠄⣄⡀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣇⣌⠰⡈⡕⢢⢊⣡⠦⡖⢸⢬⠓⡥⢄⢫⡔⣃⣠⢞⡵⢫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢪⡑⢎⣑⠢⢩⠒⡌⢢⢊⠒⣤⣱⢈⠻⣿⠆⠀⠀⠀⢀⡐⢬⠘⣮⢻⢵⣫⡖⢫⡜⢧⢇⢺⡌⢳⡘⢌⢺⣜⢣⠞⡮⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠈⠀⠩⠤⠠⢉⡀⠣⢘⡀⢣⢾⡿⣻⠀⢆⢹⡓⠀⠀⠀⡱⢌⠦⡙⡄⡛⠾⣴⢫⢏⡼⣣⠞⣧⣛⡦⠇⢈⠢⣘⢧⣛⡴⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠘⡀⠀⠘⠀⠹⣞⣷⡈⢂⠆⡀⠀⠀⣠⠙⣆⠲⣑⠸⣐⠣⢌⡻⢮⡵⣻⢜⣧⢻⣜⡻⡖⢮⡵⢺⡼⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⠀⠠⠌⢶⢫⡔⣂⢡⡀⡜⣥⢋⠤⠓⣌⢒⡡⢚⠤⢣⢏⠶⣭⢳⣎⡗⠩⣗⠻⢆⢯⠉⡾⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣍⠖⣩⠒⡣⡜⠤⠣⣝⡜⠦⢩⡘⣑⠢⢎⠰⢃⢮⡙⣮⢹⣎⠷⢮⣅⠀⠉⠣⠉⠚⠠⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⠢⡘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢤⠚⣄⠣⡱⢌⢣⡳⠰⢌⡊⢥⠒⣡⠚⡌⠥⢋⠤⡙⢶⣫⡜⣯⢳⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡱⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⡍⢆⢣⢆⢣⡑⢪⢳⡡⠍⡆⣍⢢⢙⡤⠓⣌⠣⢍⡒⣉⠆⣷⡹⢮⡝⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠼⣌⢳⠊⢦⡘⢥⡛⣜⠳⣜⠲⣌⠞⣜⢣⢆⡭⢒⠌⢦⡱⢶⡹⢧⡻⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⣘⠀⠀⠀⠀⠀⠀⠀⠀⠂⡅⠀⠀⢫⠖⣍⠦⣘⠦⡝⣎⡳⡼⡹⣌⢻⣌⠳⣎⡜⣣⣛⢶⢫⢷⡹⢧⣛⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⡅⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡟⣬⠳⣍⢞⡱⣏⠶⣹⠱⣎⠳⣌⡻⣴⡻⢵⣫⢞⡽⣪⡝⣧⣛⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⡀⠀⠄⠂⡘⠤⡑⠌⠐⠂⠂⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣯⠑⠀⠈⠐⠢⠄⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⠃⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢯⣷⣻⣽⡹⠀⠠⡡⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠋⠄⣹⢷⣛⠇⠣⠔⠊⠵⡲⠝⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢾⣻⣟⣯⠗⢁⡀⡀⠬⣿⢋⠐⠀⢂⠰⠀⠙⡀⠀⠉⠘⣴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣾⡿⣽⢯⣟⡾⣯⢷⡛⢈⣉⣤⣤⣤⣀⠏⠰⣁⠂⢄⠹⣄⣁⠙⠢⢄⠊⡐⠫⢖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⣯⣟⣾⣻⡭⠁⠘⠋⠉⠒⠛⠉⠀⠘⠁⠚⡄⠎⠀⠠⠉⢮⡑⢢⢃⢀⠂⠌⡄⠈⣰⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣻⣞⣷⣻⢾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠠⠘⠁⢤⣶⣤⡀⠈⠁⠎⠬⡐⠠⢈⠉⢰⠛⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⡵⣯⣷⢯⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠡⢓⠀⠂⠌⡚⠠⠉⠐⢠⠀⣠⢄⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⢷⣿⣯⢿⣽⣻⡀⡀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠐⢠⠃⡐⠣⣀⣀⠼⡱⢎⣭⠂⠤⡐⢄⠃⠀
⠀⠀⠀⠀⠀⣯⢿⣽⡷⣯⢿⡽⣷⢯⣟⣾⢿⣻⣞⡷⣧⠐⡠⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠠⠀⠄⠃⠀⠐⢠⠆⠀⠄⡑⢩⢎⠁⢀⠔⠌⠌⠁
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣽⣯⣟⣿⡿⣞⣿⢸⣷⡿⣽⡟⣿⣿⣿⣿⣷⣶⣶⠄⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡰⠁⠠⠁⠌⡐⢈⡔⣃⠠⣈⠐⡀⢂⡙⢮⡹⠈⠈⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠿⡻⣾⢿⣽⡹⠻⢯⣇⡚⡝⠻⠃⠈⠣⠀⢫⣿⣾⡓⠀⠀⠈⠁⠲⢐⠂⡜⢀⠎⠴⠁⠠⠡⢈⠐⣀⠞⡴⣡⠏⢀⡰⢌⠰⢀⡮⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⢀⢋⠛⢦⠙⣌⡙⠴⡩⢟⣆⠂⣄⡀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣎⣐⡌⡐⣋⠖⣩⠴⣒⡇⢸⡙⡖⢥⡐⢦⣑⢊⣠⢞⡵⢫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢢⠛⠤⣙⠒⠬⡑⠬⢒⡈⢣⣤⣱⠈⠻⣷⡇⠀⠀⠀⢀⠰⣨⠑⢯⡞⣵⢣⣞⢩⡞⣥⢳⣨⡓⣜⠢⢉⠾⣌⡏⣖⢫⡜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠙⠡⠠⢌⠁⢊⠔⠡⠸⣼⠿⣽⢀⠣⠹⠓⠀⠀⠀⢬⡑⢢⠍⣆⠹⢎⡷⣹⢎⣼⠲⡭⣖⡻⢦⠓⢈⠨⣑⢞⣬⠳⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠀⠄⠀⠀⠊⡀⠈⠁⠑⠸⢿⣵⡂⢌⠡⢂⠀⠀⣠⠓⣌⠣⢜⡠⠓⣌⠲⡹⣎⢶⣻⡱⢯⡝⣯⣛⢦⢫⡵⣺⡜⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣀⠪⢖⢯⣄⡃⢢⢄⡸⢥⡓⠬⣘⠢⠜⣑⠢⡱⢡⡛⢮⡵⢫⠷⡝⠰⣏⠳⣣⢳⠁⡿⡅⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡰⢔⢪⠘⢦⡒⢌⠣⣞⡸⢃⠬⣑⠢⢍⡚⢄⢣⢱⢫⡜⣣⢟⣭⠻⣅⠀⠉⠣⠁⠋⠂⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢣⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢎⡑⠎⡤⢋⠴⡘⢮⡑⢢⡑⢎⢒⡡⠚⡔⡸⢌⢢⢃⠣⣝⢧⡻⣜⡻⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⠂⠡⠃⠀⠀⠀⠀⠀⠀⠀⠠⣃⠖⡌⣓⠴⡉⢦⢱⢫⡜⢡⡜⢂⠣⡜⣱⡘⢔⡊⠦⡡⠓⣌⢳⡝⣮⢵⣫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢶⡸⢥⠣⢍⠢⢎⡳⣜⢣⡜⣣⠳⣜⡱⢎⡒⣬⠱⣁⠏⣤⢟⡼⣓⢮⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢹⢦⣙⠢⣍⢮⡱⣎⢳⡼⣡⢛⡴⣙⢮⡱⢎⡳⣬⢟⡼⣋⢾⣙⢮⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡷⣌⢳⡜⢦⢳⡹⢎⡽⣡⢏⡼⡱⢎⣳⣏⢷⡹⣎⢷⡹⢮⡝⣮⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠋⢀⠀⡀⠀⠄⠐⠐⡀⢣⠘⠄⠃⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⢇⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⡿⣮⠄⠈⡔⣲⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣯⢿⠃⢃⠐⠉⠲⡹⠌⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⠄⢱⠀⡘⡄⠀⠁⠐⣝⡚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⣁⣤⡴⣶⣦⡄⢇⡰⣄⠂⠤⠑⠦⣄⢊⡑⠒⡌⢀⠋⢧⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠈⠉⠀⠀⠉⠈⠀⠈⠀⢘⢢⠃⠀⠑⠈⠳⢌⠲⡐⢠⠂⡀⢆⠀⣸⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠂⠡⠰⢶⣦⣄⠀⠑⠈⡣⢐⠠⠈⠡⢰⠍⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠆⠀⠐⠭⠀⠌⡀⡏⢀⡁⠈⠔⢀⡤⢤⣀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠐⠀⢰⠁⠠⠑⢤⠀⠼⣘⢧⣚⠄⢢⠐⡡⢃⠀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠄⡈⠁⢀⠡⣈⠆⠐⠠⠌⠲⣍⡀⣠⠒⣠⠊⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠠⠄⡀⢀⠀⠀⠀⠀⡀⢄⠢⠔⠠⢁⠂⠄⡁⢂⠴⣡⢀⡬⠐⡀⠆⡈⢳⢭⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠈⠂⠑⠌⡰⢐⠨⡰⠁⢂⠤⠈⡐⢠⢏⠶⣡⠋⠀⡤⢃⠔⣀⡏⣞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⡰⣁⠂⡧⡙⣤⢖⡳⡍⢰⣋⠞⣡⠒⣍⢖⣡⠴⣊⡗⠮⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡰⢨⢌⠳⣭⢻⡵⣹⢘⢮⡕⣳⠼⣌⠺⡄⠙⡼⢎⡵⢫⡕⡺⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢈⡖⣡⠊⠵⡘⢣⢯⡝⡧⢦⣛⢬⢻⡜⣷⠚⡀⢐⣩⢞⡱⡞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⠜⠲⣄⠋⢖⡉⠖⡌⣼⠹⣧⣛⢮⡳⣝⣮⢻⡝⣭⠲⣭⡳⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⣋⢞⡡⢆⡙⢢⠜⣡⠚⣄⡻⢴⢫⡞⣵⡋⠰⣏⠓⣬⢛⠀⠿⡅⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⢸⡱⠎⠦⣑⠢⢍⡒⢜⡠⢃⡎⢷⡩⢷⡹⢶⡹⡀⠈⠃⠈⠉⠓⠸⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⡴⢊⠴⡉⠖⡌⡜⢢⠜⢢⢡⠣⠜⣣⡝⣧⣛⢧⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⣝⢣⠍⣢⠙⣌⠒⣥⢊⡜⢡⠆⡙⡜⣠⠹⣶⢹⣎⠷⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡭⢎⡳⢆⡻⢤⣋⠶⣩⢔⡣⡜⣡⢒⡡⣾⡱⢯⣜⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣭⠳⣍⡞⡱⢎⡳⡱⢎⡵⣚⡴⣫⢞⣥⣛⢧⡞⣭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⣜⢧⡹⣍⠳⣍⢳⡙⣞⢶⣫⢵⣫⢞⢶⡹⡞⣼⡓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡡⡃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢰⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⡰⠠⠑⠈⠐⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢄⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⢿⡽⣞⡀⠀⢖⡝⡲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡿⠃⠤⢹⣯⢿⢁⠁⢊⠠⠙⢵⠋⠀⠤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠜⣿⠉⡄⠀⠂⡔⠠⠈⣄⠀⠀⢑⣎⡛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀
⠛⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣾⡿⣽⢯⣷⣻⢯⡿⠝⣁⣤⡴⣶⣦⢄⢃⠎⢴⠀⡡⠘⢣⡔⡌⡑⢊⡁⠠⠙⢢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣽⣿⣯⢿⣯⣟⣯⣟⣾⣽⣏⠀⠉⠉⠀⠀⠉⠈⠀⠘⡀⠸⠢⠅⠀⠐⠈⠱⠜⢢⢐⠡⠂⠄⢣⡀⣜⢫⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⡾⣟⣾⣽⣳⢯⣟⡾⣽⡾⣽⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠈⠁⠒⠿⣶⣤⡀⠈⠂⠝⣐⠈⡀⠄⣨⠓⡮⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣹⣞⡳⣯⣟⡾⣽⣳⣿⣳⣿⢯⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠈⠖⠠⠐⡀⠧⢀⡁⠀⠒⢀⡴⢲⣄⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠄⣿⢾⣽⡾⣽⣳⢯⣟⡷⣯⣿⣷⢯⣟⡿⡄⡀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⠀⠂⡘⠁⠠⠑⠢⠄⠺⠜⣣⠞⡌⠢⠑⢌⡐⡀
⠀⠀⠀⠀⠀⣯⣟⣾⣟⡷⣯⣟⣯⣟⡷⣿⢯⣟⣾⣽⣧⣐⣤⣀⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠂⠄⡁⢀⠡⣈⠃⠄⠡⠘⠐⢯⣀⣠⠎⡰⠀⠀
⠀⠀⠀⠀⠀⣿⣞⣿⣿⣽⣳⣯⣿⣾⣽⢿⠸⣿⡾⣷⡏⢿⡿⢿⣿⣿⣿⢾⠰⠠⠀⣀⠀⡀⢀⠀⢠⠐⠢⠜⠀⠄⡁⢂⠐⡠⢖⡡⢔⠌⠠⢁⠌⠄⢳⡜⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣇⠿⡻⣯⣷⣷⢸⠻⢿⣇⢎⡙⢏⠃⠀⠓⠀⠹⣾⣾⡍⠀⠀⠀⠀⠈⠐⣂⠩⡐⠌⣓⠈⣐⠐⡀⣂⠤⣋⠶⡹⢈⠠⣑⢊⠌⢠⡗⡼⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⠀⡛⠚⣇⠉⢆⣙⠢⡝⢾⣣⠄⠤⣀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣆⣂⠥⠘⢦⡙⣴⢲⡝⡃⢼⡱⢋⡕⢢⠳⣌⢦⡜⣣⡛⡴⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢃⠗⢪⢉⡒⠌⡕⡨⠐⡤⢙⣐⡮⡄⡙⣿⡇⠀⠀⠀⠀⡒⡩⢌⢏⡿⣹⢦⣻⢰⢫⡜⡽⣔⢧⠩⡜⠠⢛⣬⠳⡜⣥⠫⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠈⠘⠥⠠⢌⡁⠢⢁⠣⠘⣴⢿⣻⠔⡐⡘⠍⠀⠀⠀⢨⡔⠱⣊⠤⠓⡽⢎⣳⡓⢦⢟⡴⢫⡞⣯⣑⡠⣁⡬⢳⡹⡜⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠐⠆⠈⠀⠃⠘⢯⣞⡆⢡⠘⡠⠀⠀⢠⢆⡙⠦⢡⠎⡱⠢⢍⡲⣝⢯⡻⣜⡻⣜⢧⣳⡳⣱⢚⢷⡹⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⡀⢀⠢⢩⢽⢂⣡⢂⢥⡐⡭⢆⡍⢲⠡⢎⢡⠓⣌⠱⣎⠳⣝⢮⡳⠏⠈⢷⡉⢖⣫⠀⢻⡅⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠦⢓⡱⢊⠖⡤⢋⢲⡱⡙⠦⣘⢡⠚⡌⠦⢩⡐⢯⡜⣣⢟⢮⣝⡻⡀⠀⠑⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢡⠚⡔⢢⠍⡜⡱⢌⠎⡴⢁⡓⢌⠆⣓⠸⢌⡱⢌⠲⢹⣜⣫⢞⡼⣱⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⢢⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⡜⢡⠚⣌⠦⡩⠔⣭⢃⠞⣄⠣⡜⠌⣎⡔⢃⠎⡔⡊⢥⠃⡞⣥⢻⡜⣧⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠲⣥⢛⡌⢲⠡⣙⠦⣏⠞⡴⢳⢬⡙⢦⡹⣌⢎⠴⣉⠦⣩⡜⣧⢻⡜⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡁⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠸⢣⢎⡥⠚⡴⢫⡼⡹⣜⢣⢎⣝⡲⡱⢎⣎⢳⣌⢷⣣⢟⡼⣣⢟⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢫⢞⡸⡹⣜⢣⡗⡳⡜⣏⠞⣬⢳⡙⣞⡞⣏⡞⣧⢞⡭⣞⡵⣫⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠋⠠⢀⠠⡀⠂⠈⠀⠔⠁⠂⠑⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢤⠙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠱⣟⡾⣽⡞⡀⠀⣮⢹⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠿⠁⠡⢸⣿⡽⢁⠡⠁⡀⠁⢯⠁⠐⠤⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⣾⣳⢯⣟⢊⠠⠄⠐⠚⣯⠑⡄⠀⠐⢬⠐⠈⢦⠀⠀⡨⢳⣌⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣾⢿⣽⣻⢾⣽⣳⠯⢋⣠⣤⠶⣾⣶⠤⢃⠌⣲⠈⡔⠈⠶⢥⡢⢉⠡⡔⠀⠌⡙⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⡷⣯⣟⣾⡧⠀⠈⠁⠀⠉⠁⠀⠀⠈⠄⠈⢣⠂⠀⣈⠀⠉⠦⢓⠰⡨⠐⢀⠣⠤⢜⡹⣓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣻⢾⣽⣳⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠁⠚⠛⠷⣦⣀⠈⠁⠑⣌⠀⢂⠐⣨⠃⡽⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⣭⣟⣷⣯⣿⢿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠈⠖⢈⠀⢂⡇⠐⡀⠀⠊⢠⢖⡲⢆⡀⠀⢀⠀⡠⠂
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⣞⣿⣷⢯⣟⡿⡔⢀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⡘⠀⠄⡉⠲⠀⠋⢮⢱⢫⠔⠈⠂⠱⡀⠄
⠀⠀⠀⠀⠀⣯⢿⣽⡷⣯⢿⡽⣷⣻⣞⣿⢽⣻⢾⣽⣷⣠⣒⣤⣀⣀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡈⠐⡀⢀⠂⡜⠁⡈⠐⠂⡘⢧⢄⡴⢉⡱⠈⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣽⣯⣟⣿⣷⣻⣾⢘⢿⣿⣳⡏⢻⡟⠿⣿⣿⢿⣟⠙⠀⠂⠄⡠⢀⠀⡠⢠⠘⣐⠊⢀⠂⠄⠡⠐⣠⢋⡔⣢⠔⠁⣢⠐⡈⢺⡜⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠿⡻⣾⣟⣧⢺⠻⣽⡇⢎⠥⣋⠃⠀⠑⠀⠱⣿⣿⡌⠀⠀⠀⠀⠀⠈⢢⠁⢆⠡⢎⡐⡠⠌⢠⡁⢲⢥⢫⡜⠁⣠⠚⡄⢊⢠⡗⣺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⠀⣋⠻⢆⠩⢄⣙⠪⠶⠽⣖⠠⢠⡀⢰⣿⣿⡇⠀⠀⠀⠀⠀⢀⡇⣌⠢⠌⡣⢜⡴⣚⢏⡇⢜⠮⣑⠦⡙⢤⡏⡴⢎⣳⡙⡖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢃⡖⠫⣘⠒⡌⠕⣊⠱⠌⡑⣆⡞⡤⠙⣿⡇⠀⠀⠀⠀⣘⠩⡜⢭⡻⣝⢧⢯⡰⢻⡜⡼⣎⢷⡘⢆⠙⠬⣳⡙⢮⣱⠙⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠁⠈⠐⠍⠄⢌⡐⠃⠰⡈⠜⣴⢿⣻⠄⡃⠜⠋⠀⠀⠀⢬⡐⢣⠜⠤⡙⠮⣏⢾⡱⣦⢻⡔⣯⢞⡭⣍⣄⡢⢖⣭⢳⠞⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡽⣆⠡⢊⠄⠀⠀⢠⠆⡝⣂⢎⠱⡡⠓⣌⢣⢟⡼⣳⡜⣧⣛⢾⡱⣎⠵⣫⠼⣫⡗⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⡀⢀⠢⢙⡞⢢⣡⠘⣄⡰⣋⠞⢤⠱⣈⠣⠜⡱⣈⠥⣎⠳⣧⢛⡶⡋⠐⢻⡌⢳⡱⡀⢳⡃⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠦⢓⡑⠮⣑⠤⢋⡴⡱⢍⡚⠤⠓⡌⠥⢋⡔⢢⡛⣬⢳⣭⠻⣼⡹⠀⠀⠁⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢡⠚⡤⣉⠖⡡⢛⠴⢊⡑⠦⢡⢋⡱⢌⢃⠧⡘⢢⠙⢦⡻⣴⣛⢶⡹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠊⡔⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢎⡜⢡⠚⣔⠢⢎⢡⢏⡇⢣⠜⠬⣑⢢⡱⣌⠪⢔⡉⢦⠙⣢⢹⠶⣹⠮⣝⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠲⡥⣛⢌⠣⠎⡜⣎⢞⡱⢎⡳⢤⢣⠳⣜⣘⢢⠍⣆⡙⡤⣞⡝⣧⢻⡼⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢌⡂⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠱⣍⢦⡙⢢⡱⢎⡾⡱⢏⣜⢣⢏⡳⡜⣬⢣⡛⣴⡹⡞⣥⢻⡜⣧⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢩⠖⣭⢣⡝⢮⢵⣋⡞⡭⢎⡭⢳⡙⣖⡷⣫⣓⢷⡹⣎⢷⡹⣎⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠪⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⡀⢄⠠⢄⡀⢆⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⢁⠀⠄⡀⢄⠐⠀⠁⡐⠌⠒⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⡃⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⡿⣿⡿⣽⠧⡌⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠹⣷⣻⡽⡟⣆⠀⢨⡕⣫⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣽⢾⠏⠀⠡⢹⣷⢫⠇⡀⠡⢀⠈⠱⠏⠀⢂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣯⢷⠯⠥⠀⠐⠒⠹⡏⢐⢣⠀⠐⡌⡂⢁⠲⡄⠀⢀⠽⣌⡳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⢿⡽⣾⣽⡻⠝⣡⡴⠶⢻⡿⠶⠊⡐⠎⣞⡠⠡⠀⢙⠲⢆⡌⣉⠡⠀⠄⡙⠃⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣳⢿⣯⣟⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⢌⡃⢀⣀⡀⠁⠒⢡⠎⡜⢀⠄⠳⠠⢜⡹⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢨⡷⣯⡿⣽⢯⣟⡷⣯⢿⡾⣽⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠀⠈⠈⠙⠻⢷⣤⠀⠁⠘⡬⢀⠂⠡⢸⠁⣏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢨⡟⣵⡻⣝⣻⣼⣻⡽⣯⣿⣳⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠱⠀⠌⠠⡍⠐⡄⠈⠒⢀⡴⢲⡄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠐⣯⣟⡷⣿⣽⣳⢯⣷⣻⡽⣿⣯⢷⣻⣾⡈⠄⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⡰⠁⠠⠉⠦⢀⠚⢜⢣⢞⡡⠘⠄⠣⢄⡀
⠀⠀⠀⠀⠀⣿⢾⣽⡷⣯⣟⣯⣷⢯⣟⡿⣽⢯⡷⣯⣇⣲⣰⣄⣂⣄⣀⣠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠌⠠⢈⠐⡀⢀⠡⡘⠁⠄⠂⠌⠘⢮⣀⣠⢊⠲⠀⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⢷⣻⣞⡿⣟⣾⢿⠸⣿⣟⣿⠏⢻⡟⠻⣿⢿⣟⡇⠀⠈⠐⠂⠄⡠⢀⠄⡰⠈⢆⠜⢀⠁⢂⠐⡀⢢⢎⡱⣌⠒⢁⡈⠆⡀⢳⡜⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣿⣏⠿⡻⣯⣿⣇⢹⡻⢿⡱⡌⡜⣩⠃⠀⠁⠀⢋⣿⣿⡄⠀⠀⠀⠀⠀⠀⡃⠌⡄⠣⡜⣀⠢⠘⢀⣂⢰⢣⢎⡵⠂⣐⢢⡑⠊⢠⡗⢮⡁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⠀⡙⠻⢆⡘⢤⣙⠲⢕⠾⣳⠀⠤⣀⢼⣿⣿⠁⠀⠀⠀⠀⠀⢀⣆⡱⣀⠣⡜⢤⢳⢫⢏⣆⢸⢣⢎⡔⢣⢜⢦⡲⣍⣳⢹⡚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡎⠦⢉⣒⠢⢍⠢⡑⣄⠛⣰⣌⡆⠙⣿⡄⠀⠀⠀⠀⡈⢭⢘⢳⣭⢳⢯⡺⡘⠟⡼⡸⣦⢧⡚⣌⠐⠎⣳⠳⣌⢧⡓⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠁⠫⠄⠄⡁⠢⢑⠰⠈⢶⣻⣟⠆⡑⡘⠂⠀⠀⠀⠬⡜⢂⢣⠒⡜⢫⡞⣵⢛⡶⣫⠵⣹⢮⡽⣌⣄⣨⠴⣛⡼⡚⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠁⢆⠀⠁⠉⠌⢷⣞⡇⠰⢁⠂⠀⠀⢠⢎⡱⡉⠦⣉⠲⢡⠜⡱⣏⢾⣱⣛⢧⡻⢜⡳⣎⢖⡻⠼⣳⣏⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⡀⢀⢂⠹⣜⢣⣈⢌⣡⡐⣫⠖⡰⢉⡒⣡⢃⢣⢊⡕⣎⢳⣣⡝⣮⠟⠈⢷⡉⢮⣱⡀⢳⡆⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢖⠤⣊⠕⡪⢖⡈⢎⠴⣱⠣⢍⡒⠥⠚⡄⢎⢢⡑⡞⣬⢣⢷⡹⣎⠷⠀⠀⠉⠀⠀⠉⠐⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣉⠲⣁⢎⡑⠦⠹⣌⠚⣄⠳⢨⠔⣉⠧⡘⣌⠒⡴⠙⢦⣛⢮⡳⣭⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⢃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢎⠦⡑⡜⣤⠒⡜⢌⡳⢣⠩⣄⢣⢃⡚⣤⠒⡥⢢⠙⡤⢋⢆⠻⣎⣗⡳⣎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠳⣌⠳⣌⠣⠜⣌⢳⣋⠷⣌⠧⢦⡱⢎⡝⣤⠣⡝⢠⢃⡎⣽⣚⣬⢳⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣡⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠸⣙⢦⡉⢎⡴⢣⣞⡱⣎⡝⢦⡝⢮⡜⢦⡛⣜⣣⢏⡾⣱⢞⡼⣣⢟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠴⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢩⢖⡹⢎⣜⢣⣞⡱⣎⡝⢦⣋⠷⣘⢧⡟⣮⣓⢯⢞⡵⣫⢞⡵⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢆⠣⡁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢈⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⠠⡀⠤⠠⢄⡐⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡋⠄⠀⠐⡠⢄⠁⠈⠀⠐⠈⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢠⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⠿⠿⠟⠛⠋⠁⠀⠀⠀⢀⣀⣥⣶⣿⣿⣿⣿⣿⣿⣿⣻⢿⣻⣞⣯⢣⠈⠰⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠃⠹⣯⢷⢿⡼⠋⠄⠰⠙⢦⢣⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⢀⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⢷⣿⠻⠁⠀⢁⣻⡿⢸⡇⠂⠈⢄⠡⠈⢃⠀⠀⠊⢔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⣳⣟⡷⣯⡟⠂⢂⣉⣀⡁⢸⠁⡼⢃⡐⠠⡈⢥⣂⢌⠓⠤⠄⡉⠺⢥⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀
⠈⠉⠉⠉⠁⢸⣿⡿⣯⣿⢿⣾⡿⣿⣳⣟⣾⣳⣯⡟⠁⠲⠛⠩⠹⠛⠋⠁⠊⢘⠏⡳⠰⠀⠀⠈⠓⢎⡰⢡⠐⡀⠂⡅⠀⢢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠸⣟⣿⣻⣿⢿⣯⢿⣽⣻⣞⣷⣻⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠉⠠⠶⣶⣄⡀⠀⠣⠱⢠⠁⠌⠒⢩⠖⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⢀⡿⡾⣽⣞⡿⣞⡿⢾⣽⣞⣷⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠀⠀⢧⠈⡐⠈⠼⠀⡝⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣰⣯⢷⣻⣼⣳⢯⣟⡿⣞⣿⣾⢯⣿⡖⡀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⢡⠋⠀⣆⠀⠉⢰⢞⡹⢆⡀⢀⠀⡄⠐⠀
⠀⠀⠀⠀⠀⢻⣞⣿⡷⣯⣟⣯⡿⣽⣻⣿⣽⣻⢾⣽⡄⠡⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡀⢀⠊⠀⠡⢀⠇⢈⠁⠪⢱⠻⠀⠃⣈⠰⠡⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣟⣷⣻⣞⣿⣳⢯⡷⢻⣽⣟⣾⢽⣿⣿⣿⣿⣿⣷⡛⠇⠠⠀⡄⢀⠀⡀⢀⠀⢄⠢⢄⠃⡐⢀⠂⠄⡁⡢⢍⠀⢂⠌⠁⠌⠓⣦⢚⠄⠃⠀⠀
⠀⠀⠀⠀⠀⢹⣷⡛⣿⣷⡿⣾⡏⢻⣿⣻⢡⠛⠯⠿⠆⠙⠄⠘⡽⣷⣟⠂⠀⠀⠀⠀⠁⠒⠨⡐⠌⡂⢦⠁⡐⡀⠂⠌⡐⡼⣑⢮⠙⠁⣠⠚⡠⢁⠮⣭⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⣈⠀⠙⢻⣷⡌⢒⡙⠧⢋⣎⣹⡤⢀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⡅⠢⡑⠰⣈⠳⣌⣤⠳⡞⢰⢣⡝⢢⢌⠲⣡⢊⡄⣞⠳⣜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⢤⡑⢆⡚⢔⡋⢄⠪⡙⠦⣉⡱⡉⠻⣟⠀⠀⠀⠀⠀⢀⠛⢳⡳⣖⢧⣌⠷⡹⣜⡣⢷⣨⡓⢬⢣⢨⢹⢖⡫⣜⠮⡝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠃⡕⢂⡘⠂⠜⡠⠓⡘⣰⢾⡿⡅⢡⠩⠁⠀⠀⠀⡰⢡⠋⣆⠱⢎⡷⢎⣯⠵⣥⣛⢦⢳⡽⢦⠇⠠⠉⣊⡵⢎⡳⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⠀⠑⢂⠀⠱⠐⡘⢯⡾⣅⠢⢡⠀⠀⠀⢀⡝⢦⠩⣄⠳⢨⢌⡙⢮⡳⢞⡼⣣⡟⣼⣫⢞⡧⢫⡵⣹⠯⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⡈⢻⡱⢆⡁⠎⠠⢠⠞⡔⣊⠱⡄⣋⠔⣊⠜⣡⢻⣙⢾⣱⢻⠖⠸⣭⠳⣣⢗⠨⡻⡕⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡢⢕⠢⡝⣡⠙⡜⢫⢥⡛⠴⡁⢇⡒⣡⠚⢤⣉⢮⠳⣜⡳⣎⢯⠆⠀⠈⠣⠈⠘⠂⠱⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢠⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡔⣊⠱⠌⣆⠫⣔⢍⡒⠬⡑⡜⢢⡑⢢⠍⢆⡜⢊⡷⢭⡳⣝⢮⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠁⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⡴⢡⡍⢚⠤⣙⡆⠎⡔⢣⠱⡨⢅⡜⡡⢚⡰⠌⡥⠚⣧⢻⡼⢭⡞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠣⣍⢖⡣⠜⡡⢎⡵⣚⠵⣊⠧⣥⠳⣍⢖⣡⢃⠦⡙⢤⠳⣜⢧⢏⡷⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣘⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠘⣎⠶⣉⠲⣌⠶⣹⢎⡵⣋⠶⡹⡜⣎⠶⣩⢖⣹⢎⡿⣜⣫⢞⡵⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢘⣣⢝⡲⣍⢞⡱⣎⡞⣭⢚⡭⢳⡌⣷⡳⣏⢾⣙⢶⡹⣬⢳⡽⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢆⠣⡁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢈⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⠠⡀⠤⠠⢄⡐⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡋⠄⠀⠐⡠⢄⠁⠈⠀⠐⠈⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢠⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⠿⠿⠟⠛⠋⠁⠀⠀⠀⢀⣀⣥⣶⣿⣿⣿⣿⣿⣿⣿⣻⢿⣻⣞⣯⢣⠈⠰⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠃⠹⣯⢷⢿⡼⠋⠄⠰⠙⢦⢣⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⢀⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⢷⣿⠻⠁⠀⢁⣻⡿⢸⡇⠂⠈⢄⠡⠈⢃⠀⠀⠊⢔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⣳⣟⡷⣯⡟⠂⢂⣉⣀⡁⢸⠁⡼⢃⡐⠠⡈⢥⣂⢌⠓⠤⠄⡉⠺⢥⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀
⠈⠉⠉⠉⠁⢸⣿⡿⣯⣿⢿⣾⡿⣿⣳⣟⣾⣳⣯⡟⠁⠲⠛⠩⠹⠛⠋⠁⠊⢘⠏⡳⠰⠀⠀⠈⠓⢎⡰⢡⠐⡀⠂⡅⠀⢢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠸⣟⣿⣻⣿⢿⣯⢿⣽⣻⣞⣷⣻⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠉⠠⠶⣶⣄⡀⠀⠣⠱⢠⠁⠌⠒⢩⠖⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⢀⡿⡾⣽⣞⡿⣞⡿⢾⣽⣞⣷⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠀⠀⢧⠈⡐⠈⠼⠀⡝⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣰⣯⢷⣻⣼⣳⢯⣟⡿⣞⣿⣾⢯⣿⡖⡀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⢡⠋⠀⣆⠀⠉⢰⢞⡹⢆⡀⢀⠀⡄⠐⠀
⠀⠀⠀⠀⠀⢻⣞⣿⡷⣯⣟⣯⡿⣽⣻⣿⣽⣻⢾⣽⡄⠡⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡀⢀⠊⠀⠡⢀⠇⢈⠁⠪⢱⠻⠀⠃⣈⠰⠡⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣟⣷⣻⣞⣿⣳⢯⡷⢻⣽⣟⣾⢽⣿⣿⣿⣿⣿⣷⡛⠇⠠⠀⡄⢀⠀⡀⢀⠀⢄⠢⢄⠃⡐⢀⠂⠄⡁⡢⢍⠀⢂⠌⠁⠌⠓⣦⢚⠄⠃⠀⠀
⠀⠀⠀⠀⠀⢹⣷⡛⣿⣷⡿⣾⡏⢻⣿⣻⢡⠛⠯⠿⠆⠙⠄⠘⡽⣷⣟⠂⠀⠀⠀⠀⠁⠒⠨⡐⠌⡂⢦⠁⡐⡀⠂⠌⡐⡼⣑⢮⠙⠁⣠⠚⡠⢁⠮⣭⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⣈⠀⠙⢻⣷⡌⢒⡙⠧⢋⣎⣹⡤⢀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⡅⠢⡑⠰⣈⠳⣌⣤⠳⡞⢰⢣⡝⢢⢌⠲⣡⢊⡄⣞⠳⣜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⢤⡑⢆⡚⢔⡋⢄⠪⡙⠦⣉⡱⡉⠻⣟⠀⠀⠀⠀⠀⢀⠛⢳⡳⣖⢧⣌⠷⡹⣜⡣⢷⣨⡓⢬⢣⢨⢹⢖⡫⣜⠮⡝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠃⡕⢂⡘⠂⠜⡠⠓⡘⣰⢾⡿⡅⢡⠩⠁⠀⠀⠀⡰⢡⠋⣆⠱⢎⡷⢎⣯⠵⣥⣛⢦⢳⡽⢦⠇⠠⠉⣊⡵⢎⡳⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⠀⠑⢂⠀⠱⠐⡘⢯⡾⣅⠢⢡⠀⠀⠀⢀⡝⢦⠩⣄⠳⢨⢌⡙⢮⡳⢞⡼⣣⡟⣼⣫⢞⡧⢫⡵⣹⠯⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⡈⢻⡱⢆⡁⠎⠠⢠⠞⡔⣊⠱⡄⣋⠔⣊⠜⣡⢻⣙⢾⣱⢻⠖⠸⣭⠳⣣⢗⠨⡻⡕⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡢⢕⠢⡝⣡⠙⡜⢫⢥⡛⠴⡁⢇⡒⣡⠚⢤⣉⢮⠳⣜⡳⣎⢯⠆⠀⠈⠣⠈⠘⠂⠱⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢠⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡔⣊⠱⠌⣆⠫⣔⢍⡒⠬⡑⡜⢢⡑⢢⠍⢆⡜⢊⡷⢭⡳⣝⢮⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠁⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⡴⢡⡍⢚⠤⣙⡆⠎⡔⢣⠱⡨⢅⡜⡡⢚⡰⠌⡥⠚⣧⢻⡼⢭⡞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠣⣍⢖⡣⠜⡡⢎⡵⣚⠵⣊⠧⣥⠳⣍⢖⣡⢃⠦⡙⢤⠳⣜⢧⢏⡷⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣘⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠘⣎⠶⣉⠲⣌⠶⣹⢎⡵⣋⠶⡹⡜⣎⠶⣩⢖⣹⢎⡿⣜⣫⢞⡵⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢘⣣⢝⡲⣍⢞⡱⣎⡞⣭⢚⡭⢳⡌⣷⡳⣏⢾⣙⢶⡹⣬⢳⡽⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠪⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⡀⠄⡄⠠⢄⠂⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⠁⠠⢀⠠⣀⠂⠀⠁⠰⠁⠌⠑⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⢿⡄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡜⡄⠑⣠⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⠱⣯⣟⡷⣏⠇⣀⠦⠞⣦⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⠟⠁⠂⢻⣞⠿⠠⠀⠄⣀⠂⠱⡁⠈⠘⢄⡢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⣾⣳⣟⡾⠉⠤⠄⠐⢻⡟⢨⠐⡀⠐⢢⡌⢀⡱⠤⣀⠔⠩⢧⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⢯⣟⡷⣯⠷⠋⢀⣀⣤⣤⣤⡐⠰⠂⢳⡀⡍⠀⠘⠓⢦⡡⡐⢂⠄⡁⠸⠁⠐⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣽⣻⢯⣟⣾⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠀⠃⡱⠂⣀⣤⣀⠀⠀⠹⢈⡒⠠⠁⠳⠄⢺⣌⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢰⢯⡷⣯⢿⣽⣻⢷⣯⣟⡿⣾⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⢡⢃⠌⠐⡀⡳⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⡼⣯⢷⣞⣾⣟⣷⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢀⠂⢰⠃⢸⠀⠁⠂⣠⠴⣢⠄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⣿⢾⣽⡷⣯⢿⣽⣻⢾⡽⣿⣾⢯⡿⣾⠀⠄⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠈⠆⢈⠀⠓⠤⠠⢃⠻⣔⢫⠕⠈⠆⢩⢀⠀
⠀⠀⠀⠀⠀⣿⣻⣾⡿⣽⣻⣞⣯⢿⡽⣿⢯⡿⣽⢯⡗⣌⣦⣠⣀⡀⠀⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠁⠄⠡⠈⠄⠠⢈⠞⠀⠡⠠⢁⠸⣃⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⣿⣳⣿⣿⡷⣟⣾⡿⣯⢿⣻⠼⣿⣯⣿⡛⢿⡟⢿⣿⢿⡿⡏⠌⠀⠣⠐⡢⠐⡄⢂⠌⣂⢩⠂⠈⠄⠡⠈⢄⡱⢍⠲⡌⢂⠁⡢⢂⠌⡞⣆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣯⠻⠽⢟⣯⣇⢞⡛⢿⣆⠭⡱⢛⠁⠀⠃⠀⢏⣿⣿⡁⠀⠀⠀⠀⠀⢃⠰⢁⠒⡠⢣⢄⠱⢈⡤⡅⣸⠱⣎⡛⠀⡤⢊⠕⡀⣲⠹⣜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠝⠲⠄⠀⠎⡛⢆⡉⢆⣉⠶⠱⢟⣆⠂⡄⡀⣸⣿⣟⠃⠀⠀⠀⠀⠀⣰⣈⢆⡑⠄⡇⡮⡳⣝⠲⡅⢧⡛⢤⢣⡙⣴⢋⡖⢮⡕⣫⡜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣓⠜⠦⣉⡒⠬⣐⠩⠢⡌⠓⣤⡳⢀⠻⣿⡄⠀⠀⠀⢀⡘⠥⡙⢮⡝⣞⣳⣱⠫⢱⡫⡝⡶⣍⠦⡃⠘⢬⡻⣜⢣⢞⡱⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠈⠈⠘⠡⠀⢌⡁⠢⢁⠃⠬⣽⢿⣻⠀⢆⡙⠈⠀⠀⠀⢬⡐⢣⡉⢦⠙⢮⣓⢧⣫⣓⢷⡩⢷⡭⢯⢥⣈⡤⣚⢬⡳⠞⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠠⠀⢀⠀⠈⡄⠈⠘⠐⠸⢷⡻⡌⢄⠒⣀⠀⠀⣠⢣⠙⡤⣑⠊⡍⢆⡩⡞⣵⢫⡞⡽⢧⡻⣝⢮⡕⢮⡕⣯⣝⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡀⠬⢳⢭⣂⡜⣀⢂⠜⡱⡆⢭⡐⠦⡙⢌⠦⡑⣼⡘⢧⣛⣭⢻⡕⠪⣛⠘⣧⢻⠐⣮⠇⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡴⡘⠲⢌⢣⡒⢤⡉⣞⡜⣓⠸⡐⡜⡰⣉⠒⠦⡱⢎⡝⣮⢳⢎⡷⣂⠀⠉⠃⠁⠑⠂⠑⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⡰⡉⡕⢊⠆⣍⢲⡸⢄⠲⢌⡱⢘⠤⣑⢢⢙⢢⠱⡩⢞⡱⣏⢾⣱⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢎⠱⣰⢩⠜⠬⣰⢫⡔⡊⡕⣊⠴⣉⠖⣌⠢⢃⠎⡑⠦⣉⢷⡹⣎⢧⢯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢮⢱⢣⢃⢎⡱⢢⢏⠶⣱⢚⡴⢢⡝⡺⣄⢫⡜⠬⣑⠱⣬⢞⡵⣫⢞⣣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡃⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢫⠞⣌⢆⠲⣍⢎⡟⡴⣫⢜⡣⢞⡱⢎⡳⣜⠳⣬⢷⡹⣎⢷⡹⣎⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢻⡜⢎⡳⡜⢮⣝⡲⢭⢣⡝⣭⠳⣍⢶⣏⠿⣜⣣⢟⡼⣣⢟⡼⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣤⣀⡀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢿⡟⠈⠑⠂⠒⡄⢢⠐⡠⠒⡈⢌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⠂⢅⠢⠑⡎⢤⠑⡈⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⣠⡜⣠⡁⠣⡙⢦⡛⡽⣌⠇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⠜⣡⠙⢧⣛⢷⣫⡳⡙⢱⣎⠻⣭⢇⡹⢂⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠄⣃⠙⠈⠀⠀⠀⠼⡑⢢⠍⣆⡙⢮⡵⣫⣝⣲⣭⠳⣝⠾⣭⣍⣀⣡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⡔⢠⠉⡄⠀⠀⣠⡙⠥⣃⠎⡔⡸⠰⢌⡳⣎⡗⢾⡹⣎⡟⣶⡹⡜⢦⡏⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢹⣹⢂⣘⢠⢁⡰⢣⠇⣓⠰⢊⠴⣁⠏⠴⡱⢎⡽⣣⢟⡼⠗⠸⣳⠉⣧⢻⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⡱⠪⡔⡌⢎⡱⢎⠧⡙⠤⡙⣌⠲⠌⡜⢢⡝⣎⢶⡹⢮⣝⣃⠀⠁⠃⠁⠑⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⣐⠣⢜⡸⢢⢍⠢⢓⠬⣑⠡⢆⠣⢍⡒⠥⢚⡥⢯⣝⡳⣎⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡑⢦⡉⢦⡙⡇⠬⣑⠣⢜⣠⢋⡴⢉⡒⠬⡑⢣⡘⢷⣪⠷⣭⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⡣⠝⢢⠜⢢⡝⣎⢳⣌⢳⡌⢶⡩⢖⡡⢎⡕⡉⢖⡬⣞⣥⣛⠶⣭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠱⡹⣌⠣⡜⣱⢚⣬⠳⣬⠳⣜⢣⡝⢮⡱⣋⠶⣙⠾⣜⢧⡞⣭⢞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢳⢍⡳⡜⣥⢋⡾⡱⣍⠻⣌⢧⡹⣒⡳⣏⢿⣩⢟⡼⣓⠾⣥⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢿⡞⠌⠒⠰⠂⡔⡀⢆⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠒⠄⡃⡖⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⢠⣆⣘⠤⠑⡬⡱⢯⣙⠦⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⡜⡡⢎⢽⡺⣝⢧⣝⡲⢩⡳⡹⣎⢧⡙⠆⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⣜⠰⡱⣈⠦⡙⢮⡳⢮⣙⣖⡳⡵⣫⢗⡯⣍⣄⡡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⣠⠜⢣⠔⣡⠒⡱⢂⡍⠷⣹⢎⡷⢫⣗⢯⡞⣵⢪⡕⡯⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⡔⣣⠏⢢⠍⡤⢣⠱⣁⠎⣱⢭⢺⣹⢳⣎⡗⠸⣣⠱⣚⢳⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣱⢜⡱⢊⠥⠚⣄⢣⠱⠌⡼⡱⢎⢧⣛⠶⣭⢆⠀⠉⠃⠉⠘⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⢆⡒⡩⢌⠓⣌⠢⢣⠍⡔⡩⢏⡾⣩⢟⣬⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡓⡔⢢⢅⠓⣌⢣⢆⠹⡐⠎⡴⢑⡊⢷⡹⣞⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⣍⠞⡼⡱⢎⡵⣌⠳⣎⣡⢣⠝⣠⢣⡜⣮⢳⣣⠷⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⣌⠲⣍⢾⡱⣝⢎⠶⣩⠳⡜⢦⣋⢞⡴⡳⣞⡱⣏⢶⣫⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡽⣌⠳⣎⠷⡱⣎⢏⡞⣱⢫⡜⣳⡞⣽⠺⣵⣋⢷⡹⢮⡵⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢢⡁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣢⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠠⢀⠠⡀⠂⠈⠀⠰⠁⠂⠑⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⢿⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣜⡄⠉⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡍⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠸⣷⢯⣟⡗⠆⣀⠴⢊⠶⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⠟⠀⠃⢻⣯⢛⠠⠂⠀⡄⢈⠱⡁⠀⠃⠒⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⡽⣯⠷⠋⠤⠄⠐⢛⡏⢈⠆⡀⠡⢘⡄⠠⠱⢄⣀⠔⠣⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣟⣷⠻⠉⢀⣀⣤⣤⣤⡘⡠⠉⢶⡀⡃⠀⠊⠝⣣⠂⡔⢂⠐⠠⢘⠁⠐⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⣯⣿⡆⠀⠈⠉⠉⠁⠁⠀⠀⠐⠀⢃⡱⠁⣀⣤⣀⠀⠉⠸⡈⡜⠠⢈⠲⠐⢮⡜⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⣟⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠰⡃⠄⠂⠌⣒⠉⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⣞⣿⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠠⠁⢲⠁⣘⠀⠁⠂⣠⠖⣦⡀⠀⠀⠀⢀⠠⠀
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⣿⣾⣻⢾⣽⠠⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⡈⠆⠐⠀⠓⠤⠄⠣⢛⡴⢣⠑⠌⠢⠡⢄⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣷⢿⣽⣻⣞⣧⣢⣔⣀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⢁⠂⡐⠠⢀⠡⠞⠀⠄⡁⢂⠹⣅⣀⡴⠑⠌⠀⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣷⣿⣾⠍⢿⡟⢿⣿⣿⢷⡏⠈⠐⠢⠂⡔⢠⠂⠔⣈⠂⡍⢂⠐⡀⠂⠄⠡⡜⣡⢣⡌⠒⢀⠢⢐⠈⣖⣃⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣇⡛⡝⠿⣇⠎⡵⢛⠂⠀⠁⠀⠓⣿⣯⡁⠀⠀⠀⠀⠀⢡⠘⡰⢀⠣⡜⡠⠒⣀⣡⡌⢲⡙⢦⡃⢐⡨⠎⡔⠡⣸⠲⡥⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠣⠄⠀⡌⠛⠧⡈⢥⡙⠮⠴⢻⣆⠂⠤⡀⣸⣿⣿⠃⠀⠀⠀⠀⠀⣠⢆⣁⠆⡱⢌⡱⡟⡼⣱⠂⣏⡝⢢⡑⢣⡼⡱⢆⢯⡕⣫⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢓⡍⠶⢉⡒⠬⣑⠪⠡⡌⡑⣦⡵⣀⠻⣿⡄⠀⠀⠀⠀⡜⢡⠛⣼⢫⡗⣯⣱⠹⢱⣣⢛⡶⣌⢣⠌⡑⠮⣝⠺⡜⡼⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠐⠍⠄⢌⡁⠂⡅⢃⠴⣹⡟⣷⠀⢆⡙⠈⠀⠀⠀⡜⣌⡑⢎⡰⢋⠾⣱⢧⣛⣖⢧⣋⠾⣭⢏⣥⣀⡲⢼⡹⣼⠙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠈⡄⠀⠊⠀⠱⢿⡽⡌⢄⠒⣀⠀⠀⣠⠙⡤⢚⡐⠦⣉⠒⡍⢾⡱⣎⢷⣫⢟⡼⣫⢶⡱⡹⣆⡟⣶⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⡀⢢⠹⣍⢢⡘⡠⢄⡸⢥⡋⡔⢣⡘⠆⡥⢋⠜⣢⠝⣮⢳⣎⢟⡆⠹⣞⠱⣣⠇⠘⡷⠌⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣡⠲⢡⠚⡥⢆⡱⢉⡞⡼⢑⠢⢍⡒⠬⡑⠦⣉⢮⡱⢫⣜⡳⣎⢯⠆⠀⠉⠂⠁⠙⠂⠙⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣡⢃⡃⠧⡘⣌⠲⡥⠚⣄⢋⡜⢢⠌⡓⠬⣑⠢⢆⡹⢳⣎⢷⡹⣎⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢑⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⢤⢢⢍⠲⡡⢆⡻⡔⡩⢔⡊⡔⢣⡜⡡⢓⠤⢋⡔⢢⠓⣮⢳⡝⡮⣏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢩⠖⣭⠊⡕⢢⡙⢶⡙⢶⡩⢖⡬⢳⡜⣥⢃⢮⡑⡌⢦⡹⣜⢧⣻⡱⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠻⢴⣉⠖⡡⢞⡥⣛⢦⡝⢮⡜⣣⠞⡴⣋⠶⣱⢎⡷⣹⢎⡷⣲⡝⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠴⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢳⢎⡽⣱⢋⢶⡹⣆⡻⡱⢎⡵⣋⢶⡹⣟⡱⣏⢾⡱⣏⢾⡱⣞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢷⡟⠈⠐⠢⠂⡔⢠⢂⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠡⡐⢌⠲⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⡠⣆⣡⡘⠠⣇⡱⢯⡹⢆⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⢜⢡⡙⢶⣹⢳⢶⡹⡒⢹⡣⠾⣭⢇⡹⢂⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⢼⡈⢖⠨⢆⡹⢭⡞⣵⣋⣷⣹⢣⡟⣮⢯⣉⣄⡡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⣠⠣⡙⢤⠋⡔⢒⠢⢜⡣⣝⢶⡹⣎⢷⡹⣎⢷⡪⡕⡯⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⡔⣣⠇⡍⠦⡙⢌⠎⡱⢊⢴⡙⢮⡳⣝⠾⡕⠪⢷⠑⣭⢳⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣱⢜⡱⢊⡜⡰⣉⠲⢌⡑⢎⢮⣙⢮⡳⣝⠾⣁⠀⠉⠃⠁⠙⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⢆⠲⠡⢆⡑⠆⣍⠒⡜⣨⠲⢭⡞⣵⢫⣝⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡓⡔⣊⠥⢋⠴⣉⡜⢄⠫⢔⠢⡑⠎⡽⣎⠷⣎⢷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⣍⠞⣴⢩⠖⡭⢎⡵⣊⣌⢣⢎⡡⣙⠼⣜⢧⡻⣜⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⣌⠲⣍⢾⡡⣏⢞⡱⣋⠶⡱⢎⡳⢎⡶⡝⡾⢭⡞⣵⢫⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡽⣌⠳⣎⠷⡱⣭⢋⢶⡩⢏⡵⣛⡾⣹⠞⣭⡝⣧⢻⡜⣧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢿⡞⠌⠒⠰⠂⡔⡀⢆⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠒⠄⡃⡖⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⢠⣆⣘⠤⠑⡬⡱⢯⣙⠦⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⡜⡡⢎⢽⡺⣝⢧⣝⡲⢩⡳⡹⣎⢧⡙⠆⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⣜⠰⡱⣈⠦⡙⢮⡳⢮⣙⣖⡳⡵⣫⢗⡯⣍⣄⡡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⣠⠜⢣⠔⣡⠒⡱⢂⡍⠷⣹⢎⡷⢫⣗⢯⡞⣵⢪⡕⡯⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⡔⣣⠏⢢⠍⡤⢣⠱⣁⠎⣱⢭⢺⣹⢳⣎⡗⠸⣣⠱⣚⢳⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣱⢜⡱⢊⠥⠚⣄⢣⠱⠌⡼⡱⢎⢧⣛⠶⣭⢆⠀⠉⠃⠉⠘⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⢆⡒⡩⢌⠓⣌⠢⢣⠍⡔⡩⢏⡾⣩⢟⣬⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡓⡔⢢⢅⠓⣌⢣⢆⠹⡐⠎⡴⢑⡊⢷⡹⣞⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⣍⠞⡼⡱⢎⡵⣌⠳⣎⣡⢣⠝⣠⢣⡜⣮⢳⣣⠷⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⣌⠲⣍⢾⡱⣝⢎⠶⣩⠳⡜⢦⣋⢞⡴⡳⣞⡱⣏⢶⣫⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡽⣌⠳⣎⠷⡱⣎⢏⡞⣱⢫⡜⣳⡞⣽⠺⣵⣋⢷⡹⢮⡵⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣤⣀⡀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢿⡟⠈⠑⠂⠒⡄⢢⠐⡠⠒⡈⢌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⠂⢅⠢⠑⡎⢤⠑⡈⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⣠⡜⣠⡁⠣⡙⢦⡛⡽⣌⠇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⠜⣡⠙⢧⣛⢷⣫⡳⡙⢱⣎⠻⣭⢇⡹⢂⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠄⣃⠙⠈⠀⠀⠀⠼⡑⢢⠍⣆⡙⢮⡵⣫⣝⣲⣭⠳⣝⠾⣭⣍⣀⣡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⡔⢠⠉⡄⠀⠀⣠⡙⠥⣃⠎⡔⡸⠰⢌⡳⣎⡗⢾⡹⣎⡟⣶⡹⡜⢦⡏⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢹⣹⢂⣘⢠⢁⡰⢣⠇⣓⠰⢊⠴⣁⠏⠴⡱⢎⡽⣣⢟⡼⠗⠸⣳⠉⣧⢻⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⡱⠪⡔⡌⢎⡱⢎⠧⡙⠤⡙⣌⠲⠌⡜⢢⡝⣎⢶⡹⢮⣝⣃⠀⠁⠃⠁⠑⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⣐⠣⢜⡸⢢⢍⠢⢓⠬⣑⠡⢆⠣⢍⡒⠥⢚⡥⢯⣝⡳⣎⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡑⢦⡉⢦⡙⡇⠬⣑⠣⢜⣠⢋⡴⢉⡒⠬⡑⢣⡘⢷⣪⠷⣭⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⡣⠝⢢⠜⢢⡝⣎⢳⣌⢳⡌⢶⡩⢖⡡⢎⡕⡉⢖⡬⣞⣥⣛⠶⣭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠱⡹⣌⠣⡜⣱⢚⣬⠳⣬⠳⣜⢣⡝⢮⡱⣋⠶⣙⠾⣜⢧⡞⣭⢞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢳⢍⡳⡜⣥⢋⡾⡱⣍⠻⣌⢧⡹⣒⡳⣏⢿⣩⢟⡼⣓⠾⣥⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣤⣀⡀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⢿⡿⣟⠈⠑⠊⠰⡀⢆⠰⣀⠒⡈⢌⠃⠠⠈⠄⠡⢈⡴⢋⠦⣡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠙⣿⣿⡁⠀⠀⠀⠀⠀⢨⠐⡄⢊⠔⡩⢄⠱⢈⣄⡅⢢⠳⣍⠞⢀⡰⢊⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣸⣿⣿⠃⠀⠀⠀⠀⠀⣠⣒⣈⡔⡈⡗⡬⢷⡹⢆⡇⢭⠳⣘⢆⠣⣼⢣⡖⣺⡙⢦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⡄⠹⣿⡆⠀⠀⠀⠀⠜⣡⠙⢮⡽⣱⢧⣝⡲⢙⡮⡹⣎⡗⡬⢊⠱⠬⡳⢼⣡⢛⡼⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠄⢃⠜⠁⠀⠀⠀⠼⡑⢢⡙⢢⠙⡵⣎⢷⣙⣖⡳⡕⣯⢞⣭⢥⣀⣢⢞⡱⢮⡝⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣌⠂⡌⠄⠀⠀⣠⡙⠥⣃⠬⡡⢋⠴⣈⠷⡭⣞⡵⣛⢮⡻⣜⢧⡳⡜⡮⣝⢧⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣄⠣⢄⡘⡵⣊⡑⠦⢡⢃⠣⢎⡐⣣⢝⡲⡽⣹⢞⡕⠪⢷⠱⣍⠷⠘⣧⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡆⠦⡙⡼⣸⠑⣆⠱⢊⠥⣊⠱⢊⠴⣩⢎⡵⣫⢗⣮⢃⠀⠉⠃⠉⠘⠂⠱⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⠤⣩⢘⡱⡸⢰⡁⠞⣠⠙⣌⢒⡡⢍⡊⠖⡡⢏⡾⣱⢏⡶⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡑⡎⠴⣈⢞⡥⢃⣌⠓⡤⢋⡔⣊⠔⣊⠴⣉⠲⣉⢾⡱⣏⢾⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⡣⠝⣐⠣⡜⣎⢞⡱⣌⡳⣌⠳⡜⣥⢚⡰⠆⢥⢃⡖⣮⢳⡝⣞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠱⡹⡜⡤⢓⡼⣘⢮⠵⣎⠵⣪⢝⡜⢦⢫⡜⡹⣎⢷⢺⠵⣫⢞⣥⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢳⡹⣜⢣⠞⣥⢻⡜⢮⡝⢦⣋⠞⡭⢶⡻⣝⢮⢏⣭⢻⡱⣏⢮⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣤⣀⡀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⢿⡿⣟⠈⠑⠊⠰⡀⢆⠰⣀⠒⡈⢌⠃⠠⠈⠄⠡⢈⡴⢋⠦⣡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠙⣿⣿⡁⠀⠀⠀⠀⠀⢨⠐⡄⢊⠔⡩⢄⠱⢈⣄⡅⢢⠳⣍⠞⢀⡰⢊⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣸⣿⣿⠃⠀⠀⠀⠀⠀⣠⣒⣈⡔⡈⡗⡬⢷⡹⢆⡇⢭⠳⣘⢆⠣⣼⢣⡖⣺⡙⢦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⡄⠹⣿⡆⠀⠀⠀⠀⠜⣡⠙⢮⡽⣱⢧⣝⡲⢙⡮⡹⣎⡗⡬⢊⠱⠬⡳⢼⣡⢛⡼⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠄⢃⠜⠁⠀⠀⠀⠼⡑⢢⡙⢢⠙⡵⣎⢷⣙⣖⡳⡕⣯⢞⣭⢥⣀⣢⢞⡱⢮⡝⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣌⠂⡌⠄⠀⠀⣠⡙⠥⣃⠬⡡⢋⠴⣈⠷⡭⣞⡵⣛⢮⡻⣜⢧⡳⡜⡮⣝⢧⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣄⠣⢄⡘⡵⣊⡑⠦⢡⢃⠣⢎⡐⣣⢝⡲⡽⣹⢞⡕⠪⢷⠱⣍⠷⠘⣧⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡆⠦⡙⡼⣸⠑⣆⠱⢊⠥⣊⠱⢊⠴⣩⢎⡵⣫⢗⣮⢃⠀⠉⠃⠉⠘⠂⠱⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⠤⣩⢘⡱⡸⢰⡁⠞⣠⠙⣌⢒⡡⢍⡊⠖⡡⢏⡾⣱⢏⡶⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡑⡎⠴⣈⢞⡥⢃⣌⠓⡤⢋⡔⣊⠔⣊⠴⣉⠲⣉⢾⡱⣏⢾⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⡣⠝⣐⠣⡜⣎⢞⡱⣌⡳⣌⠳⡜⣥⢚⡰⠆⢥⢃⡖⣮⢳⡝⣞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠱⡹⡜⡤⢓⡼⣘⢮⠵⣎⠵⣪⢝⡜⢦⢫⡜⡹⣎⢷⢺⠵⣫⢞⣥⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢳⡹⣜⢣⠞⣥⢻⡜⢮⡝⢦⣋⠞⡭⢶⡻⣝⢮⢏⣭⢻⡱⣏⢮⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡱⢂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⡀⢄⠠⢄⠠⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠁⡀⢄⠠⣀⠐⠀⠁⠰⠈⠒⠈⠀⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣯⡗⡄⠙⢠⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡾⡝⡥⢀⠦⢚⡥⣛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣞⡟⡡⠀⠂⢄⠂⠘⡁⠀⠰⠁⣆⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠱⠀⡈⠒⣌⠠⡑⠦⣀⠰⠈⢧⡹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⢯⣷⣻⢷⠯⠋⢀⣀⣤⣤⣤⡈⠡⠌⢺⡄⠒⠀⠈⠳⢱⡂⢤⢃⠈⠄⠱⡁⠐⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⢸⣟⣯⣿⡿⣿⡽⣿⢯⣟⣯⣟⣾⣽⡧⠀⠈⠉⠉⠁⠁⠀⠀⠑⠀⢂⡩⠅⣀⣤⣀⠀⠈⠒⠌⡜⢀⠂⠱⠄⢮⢣⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⢾⣽⣻⣽⣻⣽⣻⣞⣷⣻⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⡀⠀⠡⢎⠠⢁⠐⡬⠁⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣟⡽⣞⢧⣯⢷⣻⢶⣯⣷⢯⣷⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢊⠄⠂⠄⡇⢘⡀⠀⠡⢠⡔⢦⡄⠀⠀⠀⢀⠠⠀
⠀⠀⠀⠀⠐⢻⣽⣻⣽⣻⣞⣯⣟⡿⣞⣿⣿⣻⣞⣷⡐⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠚⠀⠄⡑⠦⢀⠚⢜⡣⣝⠢⠑⠈⠆⠤⠀
⠀⠀⠀⠀⠀⣾⣳⣿⣷⣻⣞⡷⣿⡽⣯⢿⢾⣳⣟⡾⣧⣰⣡⣂⣄⡀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠆⠠⢁⠂⠄⡀⢂⡜⠀⢂⠈⠄⠸⢥⣀⢤⡉⠎⠈⠀
⠀⠀⠀⠀⠀⢻⣷⣿⣿⣳⣯⣟⡿⢽⣯⣿⠸⣿⡾⣟⡏⢻⡟⠿⣿⣿⢿⡏⠈⠁⠒⠰⡀⢆⡐⠄⡒⢨⢐⠂⠠⢁⠂⡈⠄⡴⢃⢦⡑⠆⠈⡄⠃⠄⡹⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣿⣧⠻⠝⢷⣿⣳⡘⠟⡽⣗⡌⣛⢹⠃⠀⠁⠀⠑⣿⣿⡉⠀⠀⠀⠀⠀⢐⠠⢃⡘⠄⡎⢤⠑⢂⣐⡤⢸⡱⣋⠖⡁⢤⡙⠤⠋⣰⢏⡽⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⡉⣔⣉⠲⠥⢟⡶⢀⠤⣀⢰⣿⣿⠃⠀⠀⠀⠀⠀⢠⣃⣢⡐⠌⡜⣢⢟⣭⢚⣄⢣⠳⣉⠖⡩⢆⡳⢦⡛⣜⠮⡜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡎⠦⣑⢒⠨⣑⠪⡐⡌⡑⢦⡵⡈⠹⣿⡄⠀⠀⠀⢀⡈⢇⡙⢶⣹⣛⢾⡱⡚⢜⡣⡞⣮⢳⡘⡌⠐⠫⣝⠶⡹⣌⡳⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠘⠡⠄⢌⡁⠂⢅⠢⠑⣼⢿⣻⠄⡃⠜⠉⠀⠀⠀⢬⡘⠤⣃⢆⢣⠻⣜⡳⣝⣲⡻⡜⣧⢏⡯⣍⣄⣡⢎⣳⡱⡝⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠈⠄⠀⠁⠃⠘⢯⡾⣄⠱⢈⢂⠀⠀⢠⢎⡱⢑⠢⢎⢢⠱⢌⡳⣝⢶⡹⣝⢮⡻⣜⣳⡜⢦⣛⡴⣏⡧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⢀⠢⢹⢭⣂⣅⠢⢄⡘⢧⡒⢬⠡⡓⡌⠦⣉⠦⡱⢎⡳⣝⢮⡳⠟⠨⣳⠙⣦⢻⠈⢾⡅⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣋⠤⡓⢆⠫⡔⣠⠋⣞⡜⢣⡉⢆⠓⡬⣐⠣⢒⣌⠳⣍⠾⣭⢞⡹⣇⠀⠈⠃⠈⠑⠂⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡑⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢡⠚⡌⢬⠱⣨⠱⡜⢢⠜⢢⡑⢪⠑⣆⠱⢌⠣⢌⠳⢭⣛⠶⣋⠷⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣈⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢏⡘⠆⣭⡘⠦⣑⢢⣛⠬⡡⢎⠥⣘⢡⢣⡌⢚⢌⡱⢊⠜⣂⢯⣛⣭⣛⢶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠸⡱⢦⡙⠆⡥⠲⣍⡞⡱⢎⠶⣡⢎⡳⣌⢣⡌⠖⣉⢦⡱⣎⠷⣜⢮⡳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠹⣆⢳⡘⡔⡹⢦⡝⣣⢏⡞⡱⢎⡵⣊⠷⣸⠹⣜⢶⡹⣎⠿⣜⢧⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢪⢧⡹⣜⡱⢣⡞⡥⢯⡹⣱⢋⢶⡩⣞⢷⣫⡝⣮⢳⣭⢻⣜⡳⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢿⡞⠌⠒⠰⠂⡔⡀⢆⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠒⠄⡃⡖⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⢠⣆⣘⠤⠑⡬⡱⢯⣙⠦⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⡜⡡⢎⢽⡺⣝⢧⣝⡲⢩⡳⡹⣎⢧⡙⠆⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⣜⠰⡱⣈⠦⡙⢮⡳⢮⣙⣖⡳⡵⣫⢗⡯⣍⣄⡡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⣠⠜⢣⠔⣡⠒⡱⢂⡍⠷⣹⢎⡷⢫⣗⢯⡞⣵⢪⡕⡯⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⡔⣣⠏⢢⠍⡤⢣⠱⣁⠎⣱⢭⢺⣹⢳⣎⡗⠸⣣⠱⣚⢳⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣱⢜⡱⢊⠥⠚⣄⢣⠱⠌⡼⡱⢎⢧⣛⠶⣭⢆⠀⠉⠃⠉⠘⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⢆⡒⡩⢌⠓⣌⠢⢣⠍⡔⡩⢏⡾⣩⢟⣬⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡓⡔⢢⢅⠓⣌⢣⢆⠹⡐⠎⡴⢑⡊⢷⡹⣞⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⣍⠞⡼⡱⢎⡵⣌⠳⣎⣡⢣⠝⣠⢣⡜⣮⢳⣣⠷⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⣌⠲⣍⢾⡱⣝⢎⠶⣩⠳⡜⢦⣋⢞⡴⡳⣞⡱⣏⢶⣫⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡽⣌⠳⣎⠷⡱⣎⢏⡞⣱⢫⡜⣳⡞⣽⠺⣵⣋⢷⡹⢮⡵⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢷⡟⠈⠐⠢⠂⡔⢠⢂⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠡⡐⢌⠲⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⡠⣆⣡⡘⠠⣇⡱⢯⡹⢆⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⢜⢡⡙⢶⣹⢳⢶⡹⡒⢹⡣⠾⣭⢇⡹⢂⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⢼⡈⢖⠨⢆⡹⢭⡞⣵⣋⣷⣹⢣⡟⣮⢯⣉⣄⡡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⣠⠣⡙⢤⠋⡔⢒⠢⢜⡣⣝⢶⡹⣎⢷⡹⣎⢷⡪⡕⡯⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⡔⣣⠇⡍⠦⡙⢌⠎⡱⢊⢴⡙⢮⡳⣝⠾⡕⠪⢷⠑⣭⢳⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣱⢜⡱⢊⡜⡰⣉⠲⢌⡑⢎⢮⣙⢮⡳⣝⠾⣁⠀⠉⠃⠁⠙⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⢆⠲⠡⢆⡑⠆⣍⠒⡜⣨⠲⢭⡞⣵⢫⣝⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡓⡔⣊⠥⢋⠴⣉⡜⢄⠫⢔⠢⡑⠎⡽⣎⠷⣎⢷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⣍⠞⣴⢩⠖⡭⢎⡵⣊⣌⢣⢎⡡⣙⠼⣜⢧⡻⣜⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⣌⠲⣍⢾⡡⣏⢞⡱⣋⠶⡱⢎⡳⢎⡶⡝⡾⢭⡞⣵⢫⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡽⣌⠳⣎⠷⡱⣭⢋⢶⡩⢏⡵⣛⡾⣹⠞⣭⡝⣧⢻⡜⣧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢷⡟⠈⠐⠢⠂⡔⢠⢂⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠡⡐⢌⠲⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⡠⣆⣡⡘⠠⣇⡱⢯⡹⢆⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⠜⣡⠙⢶⣹⢳⢶⡹⡒⢹⡣⠾⣭⢇⡹⢂⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⢬⡑⢆⡙⠢⣍⢛⣮⢳⣙⣧⢻⡱⣏⠾⣭⣍⣀⣡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⣠⢣⡙⢤⠩⡱⢠⠣⢌⠷⣹⣜⡳⣝⢮⣛⡶⣹⡜⢦⡏⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⡔⣣⢇⡘⢢⠱⣁⠇⡚⣌⢲⡓⢮⡳⣝⠾⡕⠸⢧⠙⣖⢻⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣱⢜⡱⢂⡍⢆⠓⡬⢘⡡⢆⡳⣙⢮⡳⣝⠾⣁⠀⠉⠃⠉⠘⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⢆⠲⡡⢜⡨⡑⢆⢣⠒⣡⠓⣭⢞⡵⣫⣝⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡓⡔⣊⠱⡘⡔⣢⠍⡒⠬⣑⠢⢍⠢⣟⡼⢳⣜⡳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⣍⠞⣴⢩⢧⡱⡜⣥⢋⡬⡱⠆⡍⢆⡳⣜⣞⡳⣎⠷⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⣌⠲⣍⢾⡡⣏⠶⣱⣙⢦⣋⠶⣩⢓⣎⢷⡹⣞⡴⢻⡜⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡽⣌⠳⣎⠷⡱⣭⢛⡴⣩⠖⣭⣚⡷⣫⢞⣣⢟⡲⣝⢧⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⣿⢿⡞⠌⠒⠰⠂⡔⡀⢆⠰⡐⠌⣌⠡⠀⠌⠠⠁⠌⡴⢋⢦⡡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠱⣾⣿⡉⠀⠀⠀⠀⠀⢡⢂⠒⠄⡃⡖⡠⠅⢊⣄⡥⢸⡱⣋⠖⠁⡤⢋⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣰⣿⣿⠃⠀⠀⠀⠀⠀⢠⣆⣘⠤⠑⡬⡱⢯⣙⠦⡇⢣⠳⡑⢎⡱⣸⢣⡖⣺⡙⡦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⠤⠙⣿⡆⠀⠀⠀⠀⠜⡡⢎⢽⡺⣝⢧⣝⡲⢩⡳⡹⣎⢧⡙⠆⠡⠳⣫⠜⣥⢛⡴⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠂⠥⠙⠈⠀⠀⠀⢬⠱⡑⡌⢆⡙⢮⡳⢮⣙⣖⡳⡵⣫⢗⡯⣍⣄⡡⢞⡹⢦⡛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣅⠊⡡⢂⠀⠀⡠⣍⠳⠌⡜⢢⡉⢆⢍⡳⣝⢮⡳⢏⣷⢫⡞⣵⢪⡕⡯⣼⢳⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣐⠢⣀⠴⣙⠆⡥⢋⡔⢣⡘⢌⠦⡱⢎⡳⣝⢯⡜⡗⠸⣣⠱⣚⢳⠘⡧⠏⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡖⣨⠱⣩⠞⣡⠋⡔⢣⡘⡔⡸⡈⢶⡙⣮⢱⣏⠾⣙⣆⠀⠉⠃⠉⠘⠂⠙⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⡄⣓⠸⢤⠳⣈⠖⣡⠚⣌⠒⡴⢡⢒⡉⢆⡙⢦⡻⣜⡏⡷⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡱⡰⢌⡱⢎⡳⢐⢎⠤⠓⡤⣋⠴⠡⢆⠍⣆⢩⠒⣽⣚⡼⢳⡹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⢣⢣⠱⢌⡸⢣⡛⡼⣘⢮⡱⡜⣥⢋⡕⣪⠜⡰⢌⡱⣞⡬⣝⢧⡻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠹⢎⢦⠓⡬⣘⠧⣝⡲⣭⢲⡱⣙⢦⢫⡜⡱⢎⡵⣏⡳⡽⢼⡹⣎⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢫⢎⡝⡲⣍⠞⣵⢣⡝⣣⠳⣍⢎⡳⣼⡻⣝⣞⡱⢯⡝⣧⢻⣜⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢢⡁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣢⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠠⢀⠠⡀⠂⠈⠀⠰⠁⠂⠑⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⢿⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣜⡄⠉⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡍⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠸⣷⢯⣟⡗⠆⣀⠴⢊⠶⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⠟⠀⠃⢻⣯⢛⠠⠂⠀⡄⢈⠱⡁⠀⠃⠒⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⡽⣯⠷⠋⠤⠄⠐⢛⡏⢈⠆⡀⠡⢘⡄⠠⠱⢄⣀⠔⠣⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣟⣷⠻⠉⢀⣀⣤⣤⣤⡘⡠⠉⢶⡀⡃⠀⠊⠝⣣⠂⡔⢂⠐⠠⢘⠁⠐⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⣯⣿⡆⠀⠈⠉⠉⠁⠁⠀⠀⠐⠀⢃⡱⠁⣀⣤⣀⠀⠉⠸⡈⡜⠠⢈⠲⠐⢮⡜⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⣟⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠰⡃⠄⠂⠌⣒⠉⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⣞⣿⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠠⠁⢲⠁⣘⠀⠁⠂⣠⠖⣦⡀⠀⠀⠀⢀⠠⠀
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⣿⣾⣻⢾⣽⠠⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⡈⠆⠐⠀⠓⠤⠄⠣⢛⡴⢣⠑⠌⠢⠡⢄⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣷⢿⣽⣻⣞⣧⣂⣦⣠⣀⡀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⢁⠂⡐⠠⢀⠡⠞⠀⠄⡁⢂⠹⣅⣀⡴⠑⠌⠀⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣷⣿⣾⠍⢿⡟⢿⣿⣿⢿⡏⠈⠐⠡⠂⡆⠰⣀⠢⠌⢂⢩⠂⠐⡀⠂⠄⠡⡜⣡⢣⡌⠒⢀⠢⢐⠈⣖⣃⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣇⡛⡝⠿⣇⠎⡵⢛⠂⠀⠁⠀⠓⣿⣯⡁⠀⠀⠀⠀⠀⠱⢀⠆⠩⢄⡣⢄⠱⢀⣡⡌⢲⡙⢦⡃⢐⡨⠎⡔⠡⣸⠲⡥⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠣⠄⠀⡌⠛⠧⡈⢥⡙⠮⠴⢻⣆⠂⠤⡀⣸⣿⣿⠃⠀⠀⠀⠀⠀⣰⣈⡔⡡⢂⡕⣪⠗⣏⠶⡄⢯⡙⢦⡑⢣⡼⡱⢆⢯⡕⣫⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢓⡍⠶⢉⡒⠬⣑⠪⠡⡌⡑⣦⡵⣀⠻⣿⡄⠀⠀⠀⠀⠜⣡⠙⡾⣹⢳⡞⣥⠛⡹⢎⡽⣲⣍⠦⡁⠣⢚⡽⣚⠼⣸⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠐⠍⠄⢌⡁⠂⡅⢃⠴⣹⡟⣷⠀⢆⡙⠈⠀⠀⠀⣌⠳⢠⢋⠴⠩⢗⡾⣱⣛⣜⢷⣘⠷⣎⢯⢥⣀⡥⣚⢥⡻⠜⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠈⡄⠀⠊⠀⠱⢿⡽⡌⢄⠒⣀⠀⠀⣠⠎⢣⢃⠎⡒⡍⡒⣌⠷⣙⢮⡳⣞⣻⡜⣯⢞⡱⢎⡵⣞⡽⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⡀⢢⠹⣍⢢⡘⡠⢄⡸⡱⢎⡡⢎⡘⢅⠲⢡⢊⡼⣙⢮⡳⢽⡸⠇⠸⢯⠙⣎⢇⠘⣳⠉⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣡⠲⢡⠚⡥⢆⡱⢉⡞⡴⡙⣂⠖⡡⠎⣌⢃⠇⣎⠶⣩⢶⡹⢧⡻⡅⠀⠉⠂⠉⠘⠂⠙⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣡⢃⡃⠧⡘⣌⠲⡥⠚⢤⡑⠆⡎⢱⡘⡰⢊⡜⢠⠫⡵⢮⡝⣧⣛⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢑⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⢤⢢⢍⠲⡡⢆⡻⡔⣩⠒⢬⠱⣈⢧⠰⡡⢃⡌⢣⠱⡘⣧⣛⢶⣙⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢩⠖⣭⠊⡕⢢⡙⢶⡹⢴⡙⢦⠳⡜⣎⢣⡱⣡⠎⡱⢌⣱⢎⣧⢻⡜⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠻⢴⣉⠖⡡⢞⡥⣛⠦⣝⢎⡳⡹⣌⢧⢳⡱⢎⣵⢫⡞⡽⣬⢳⡝⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠴⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢳⢎⡽⣱⢋⢶⡹⢎⡽⢪⠵⡹⡜⣎⡳⣏⠿⣬⢳⣝⣣⡝⣧⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⠠⡀⠤⡀⠤⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⡀⠄⡠⢀⠐⠀⠈⠄⠑⠂⠑⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡖⡅⠉⠤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⡷⣏⠥⣀⠜⢎⢧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠟⠁⠂⣹⣾⠏⠡⠂⠀⢄⠂⡘⠆⠀⠉⠔⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⣳⢯⡟⠊⠤⠄⠐⠚⡿⢀⠡⠀⠌⠘⣄⠐⡸⠤⣀⠔⠃⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣳⢯⣿⣳⠯⠋⢀⣀⣤⣤⣤⡐⠡⠌⢺⡄⡚⠀⠊⠙⢦⡡⣐⢂⠐⡀⠙⡁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⢯⣟⣯⣟⡿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠐⢀⠓⡸⠁⣀⣤⣀⠀⠀⠣⢌⠚⡀⢁⠓⠄⢮⢣⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⡽⣯⣟⣾⣳⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⡀⠀⠡⢣⠀⠌⠠⣜⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⢯⣟⡷⣞⡷⣿⢯⣿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢡⠆⢘⡀⠁⠒⢠⡔⡦⡄⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡽⣯⡿⣽⣻⣞⡿⣽⣻⢿⣯⣷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠆⠐⠠⠑⠤⠠⠓⢮⡱⢣⠇⠘⠄⢣⢀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣟⣷⣻⣞⣿⣳⢯⣿⢿⣞⡷⣯⣷⣠⣆⣤⣀⡀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠄⠡⠈⠄⠠⢁⠞⠀⠡⠐⠂⠩⣇⣀⢴⠉⡆⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣾⣳⣯⡿⢯⣿⣻⠸⣿⣽⣷⡋⢿⡟⢿⣿⢿⡿⣟⠈⠑⠊⠰⡀⢆⠰⣀⠒⡈⢌⠃⠠⠈⠄⠡⢈⡴⢋⠦⣡⠒⠁⡌⢂⠄⣏⢆⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣧⠻⠝⢿⣳⣏⡼⠝⠿⣗⢌⡛⡜⠃⠀⠁⠀⠙⣿⣿⡁⠀⠀⠀⠀⠀⢨⠐⡄⢊⠔⡩⢄⠱⢈⣄⡅⢢⠳⣍⠞⢀⡰⢊⠴⠁⣰⢭⣚⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⣈⠛⢣⢉⣌⣉⠦⢳⠼⣇⠂⠤⡀⣸⣿⣿⠃⠀⠀⠀⠀⠀⣠⣒⣈⡔⡈⡗⡬⢷⡹⢆⡇⢭⠳⣘⢆⠣⣼⢣⡖⣺⡙⢦⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⡎⠦⣉⠒⡌⢔⠃⡒⡌⡙⣤⡗⡄⠹⣿⡆⠀⠀⠀⠀⠜⣡⠙⢮⡽⣱⢧⣝⡲⢙⡮⡹⣎⡗⡬⢊⠱⠬⡳⢼⣡⢛⡼⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠈⠐⠍⠄⢌⡐⠊⠰⡁⠜⣴⢿⣻⠄⢃⠜⠁⠀⠀⠀⢬⡑⢢⡙⢢⡙⠵⣎⢷⣙⣖⡳⡕⣯⢞⣭⢥⣀⣢⢞⡱⢮⡝⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠠⠀⠀⠐⠆⠁⠈⠂⠘⢯⡾⣌⠂⡌⠄⠀⠀⣠⢣⡙⠴⢨⠅⣌⠓⣌⡳⣭⠞⣵⣛⢮⡻⣜⢧⡳⡜⡮⣝⢧⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢀⠡⢙⣧⢊⣄⠣⢄⡘⡵⢢⠜⡌⢣⠜⣠⠋⡔⣢⡍⣟⢶⡹⣞⡕⠪⢷⠱⣍⠷⠘⣧⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠤⡓⣃⠎⡆⠦⡙⡼⣸⢑⠣⡜⡘⠤⡓⠤⢋⠴⡳⡜⡼⣎⢷⢺⣄⠀⠉⠃⠉⠘⠂⠱⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠡⢎⠱⠤⣩⢘⡱⡸⢰⡁⢎⡒⡡⢎⠱⢌⢣⡉⠖⡱⢭⡳⣝⢮⡳⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜⣑⢪⡑⡎⠴⣈⢞⡥⢃⡜⢢⢡⡑⣎⡑⠎⡤⢃⡹⢄⠣⣟⡼⣣⢟⡼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠫⣔⡣⠝⣐⠣⡜⣎⢞⡱⣜⢣⢦⡙⢦⡙⣜⡰⢃⠖⡬⣱⠞⣶⡹⣎⠷⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠱⡹⡜⡤⢓⡼⣘⢮⠵⣎⠳⣎⡝⢮⡱⢎⡵⣋⡞⣵⢫⡝⣶⡹⣎⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢳⡹⣜⢣⠞⣥⢻⡜⣭⢛⡴⣩⢧⣙⡞⣾⣱⢻⡜⣧⣛⢶⡹⢶⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣄⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠐⣬⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⡀⢄⠠⢄⠠⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⡀⠄⡀⢄⠐⠀⠁⠠⠊⠐⠈⠂⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠻⣇⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⡿⣽⢯⠥⠘⢠⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⡍⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏⡹⣷⣻⡽⣏⡳⢀⠰⠼⣡⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⠋⠐⠀⣻⣷⢹⡇⠀⢂⠐⠠⠀⢇⠀⠘⠰⢠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⣾⣻⣽⠳⠥⠀⠐⠚⢩⠇⣸⠆⡁⠀⠜⣡⠈⠌⠦⣀⠠⠓⢼⡡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣻⢾⡽⣞⠓⢁⣀⣤⣤⣶⠦⡈⡔⠌⣞⠤⡑⠀⠐⠋⢯⡐⢤⢃⠐⠠⠈⡅⠐⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⢸⣿⢿⣻⣿⣿⡽⣿⣳⢿⣳⣟⣿⣽⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⠁⠢⢍⢀⣠⣄⡀⠈⠐⠎⡸⢀⠁⠒⠤⠲⣍⡛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣯⣟⣿⣳⢯⣟⡷⣯⢿⣳⣟⣾⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢄⠀⠁⡎⠄⡁⠂⢼⠀⡻⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣟⣮⢷⣫⣟⡾⣽⢯⡿⣷⣻⡿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠄⡁⡎⠠⡁⠀⠂⢠⡔⢦⡄⠀⠀⠀⠀⡠⠀
⠀⠀⠀⠀⠠⢾⣻⣞⣷⣻⢾⡽⣯⢿⡽⣿⣷⣻⡽⣯⡐⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠀⡔⠀⠡⠑⠦⠠⠚⢜⢣⢞⡡⠊⠰⠡⢄⠀
⠀⠀⠀⠀⠀⣾⣳⣿⡷⣯⢿⡽⣯⢿⡽⣟⣷⢯⣟⣷⢧⣘⣤⣡⣄⣀⣀⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠌⡐⢀⠂⠄⡀⢂⡕⠂⠡⠐⡈⠘⢦⣀⣠⢉⠲⠈⠀
⠀⠀⠀⠀⠀⢻⣷⣿⣿⣽⣯⣟⣿⣯⣟⣯⠼⣿⣻⣾⠏⢻⠛⠿⡿⣿⣟⡇⠀⠁⠘⠄⢒⠠⢂⠔⡠⢑⠨⠌⠠⠐⠠⠈⠄⣰⠣⡜⡤⠑⠠⣀⠃⠄⣣⢝⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣖⠿⡻⣾⣟⣇⢸⡻⢟⡖⢬⢩⣙⠂⠀⠁⠀⢛⣿⣷⠀⠀⠀⠀⠀⠀⠁⣊⠰⢀⠣⠜⡤⠡⢊⣠⠥⢂⢧⢳⡙⠠⣁⠖⡡⠊⣰⢧⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠦⠀⠀⢙⠻⢆⠱⣈⣙⠢⢇⠾⣖⠠⢠⡀⣼⣿⣿⠀⠀⠀⠀⠀⠀⢀⣄⣊⠤⡁⢏⡔⢯⣣⠧⣇⢸⣋⢆⢇⠳⣌⡞⡴⢣⡳⣍⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡎⠦⣑⡒⠌⡕⢊⠔⡌⡑⢦⡵⡠⠙⣿⡀⠀⠀⠀⠀⡨⢩⠜⣭⠻⣵⢫⡞⡜⢥⡛⡜⣧⢮⡘⣌⠰⡘⡳⢭⢳⡱⠮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠈⠐⠩⠄⢄⡉⠰⢁⠊⡔⣹⢿⣻⠅⡘⡐⠂⠀⠀⠀⢬⡁⢇⢎⠰⠛⡼⣣⢟⣜⣣⡻⣜⢮⡳⢯⡥⣀⡰⣜⢣⢧⡽⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠁⠂⠘⢷⣻⡄⠱⠐⢂⠀⠀⢠⢣⡙⢤⢊⠥⢋⡔⣡⠻⣜⢧⣛⢮⢷⡹⣧⢻⣱⠹⣬⣛⢮⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⠀⢀⠢⠱⢯⣡⣉⠢⢄⡘⢧⠣⠜⣂⠎⡜⡡⢒⠤⣛⢬⢳⡝⣮⢏⠗⠘⣧⡉⣗⢳⠈⠷⡍⠂⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠢⢓⡱⢋⠴⣠⠋⣞⡜⢣⡙⢢⢅⡚⠤⢃⡍⢲⡍⣎⢧⣛⢶⢫⠇⠀⠀⠃⠈⠁⠓⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⡩⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⢎⡱⠰⣉⠦⢱⠪⡔⠬⡡⠜⣡⠒⠬⡑⢣⡘⢆⡹⡜⣧⣛⢮⢏⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⢃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢎⡒⣉⠖⣬⡑⠦⣘⢭⡓⠬⡱⢌⡱⢂⡝⣢⠙⡔⣘⠢⢆⡙⢶⡹⣎⡟⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠳⣌⠞⡴⢘⠢⢍⠶⣍⡳⡜⢦⢥⢫⡜⣥⢚⡤⢃⠞⣠⢣⣏⢷⡹⣜⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⢂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⡹⢴⡉⡜⣬⢓⣮⠱⣝⢎⣎⠳⡜⢦⣋⠶⣍⡞⣵⢫⡞⣎⢷⡹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢹⢦⡹⡜⢦⣋⢶⣋⡞⡭⢎⡽⣙⢦⣛⡾⣱⢻⡜⣧⢻⡜⣧⣛⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⢿⣟⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢆⡑⠂⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢈⣾⣿⣿⣿⣿⣿⡯⠃⠀⠀⠀⠀⠀⢀⠠⠠⠄⠤⣀⠂⡄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡋⠠⠀⠐⣀⠢⠁⠈⠀⠀⠃⠘⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⢄⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⠛⣧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠿⠿⠟⠛⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣻⢿⣻⣞⣯⠦⠈⢐⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⡂⢆⠠⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡯⢃⠹⣯⢷⢿⡼⠙⠄⠰⠊⡷⣌⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⢀⣤⣾⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣟⠏⠁⠀⣐⣿⡏⢸⡇⠀⠈⢤⠁⡀⢃⠀⠐⠈⢦⡒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣯⣟⡾⣽⣳⢯⠟⠁⠂⠉⠀⠀⠼⠁⢾⣩⠐⠈⠄⢧⣐⣈⠣⠤⠔⠁⡛⢬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀
⠉⠉⠉⠉⠁⢸⣿⣿⣯⣿⣷⡿⣿⣟⣿⢾⣽⣻⣷⠋⠁⠤⠶⠶⠛⠛⠓⠠⡈⢹⠇⠣⡜⠀⠀⠂⠕⢆⠲⢨⠐⠠⠈⡆⠈⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢸⡿⣷⡿⣿⡾⣟⡿⣽⢾⣻⣞⣷⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠱⠰⢶⣶⣄⡀⠉⠢⡙⢠⠁⡘⠂⢹⢲⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⢸⡟⡷⣟⣷⣻⣽⣻⢽⣻⡷⣯⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠀⠀⢧⠐⠠⠈⡼⠀⠝⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣼⣽⣳⢯⣶⣻⢾⡽⣯⢿⣿⣽⢷⣟⡇⡀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠁⢂⠃⠌⣄⠈⠁⢴⢫⡝⣆⡀⢀⡀⢄⠐⠀
⠀⠀⠀⠀⠀⣿⢾⣽⣻⢾⣽⣻⡽⣯⣟⣯⣟⡿⣞⣿⡐⠤⡀⠄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⡐⠊⠀⢂⢠⠃⠌⢈⡑⢺⠴⠃⠂⣈⠂⠥⠀
⠀⠀⠀⠀⠠⣟⣯⣿⣯⣟⡾⣽⣿⣳⢯⡇⣿⣽⣟⣾⢽⣿⣿⣿⣿⣿⣾⠞⠂⠠⢄⠀⡀⢀⠀⡀⠀⡄⢢⠰⠁⠌⡐⢀⠂⣁⢢⢃⠐⣈⠄⠠⢈⠓⣦⠚⠤⠉⠀⠀
⠀⠀⠀⠀⠀⢻⣽⡻⣿⣽⣟⣯⡏⠿⣿⣳⢌⠻⢫⠿⡄⠙⠄⠘⢷⣿⡞⠁⠀⠀⠀⠀⠁⠢⢘⡀⢣⠐⢢⠁⠌⡐⢀⠂⡐⣜⢢⡍⠞⢀⢠⢃⡰⢀⢧⣛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢻⣷⡉⠈⠙⢻⣷⠘⡘⠹⠎⢇⡳⣸⡤⢀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⡆⡘⠄⡊⢥⢋⢬⠴⣒⡏⢸⣌⠳⡜⡠⢎⡆⣃⣠⢞⢦⡙⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠦⢤⠙⣔⠣⢚⡡⢑⠬⡙⠦⡉⡵⢉⠻⣧⠀⠀⠀⠀⠀⢀⢛⡱⣖⡳⣜⡦⡽⣙⢾⡱⢺⡰⣎⠱⡌⢱⢚⡼⢣⠞⣎⠮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠘⠊⠕⢂⡑⠢⠘⡄⢒⠡⣢⢿⣷⢃⠢⠩⠁⠀⠀⠀⣐⢊⢆⡱⢌⠳⣝⢮⣳⠱⣆⢯⢣⢻⣜⡳⢌⠀⢋⣘⢧⣛⡼⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⡀⠀⠀⠑⡂⠈⠂⠅⡙⣯⢾⡁⠆⡱⢀⠀⠀⢀⡜⢎⡰⠒⣌⠣⡌⠳⣭⢻⣜⣫⢧⡻⣼⡹⣏⠾⣌⡳⣎⠷⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠌⢫⠷⢌⠐⢢⠀⣠⠳⡌⠦⢡⢋⠤⠓⣌⠓⡴⣋⢶⡹⣎⢷⠓⠸⣭⠳⢎⢷⠨⡟⡅⠦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⣘⠜⡢⢟⡨⢍⣃⢻⡰⢫⠜⣡⠃⡎⠜⣑⠢⣙⠼⡱⢎⡷⣹⠞⡆⠀⠈⠃⠉⠘⠂⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⢑⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢤⠋⡔⢢⠍⣲⢈⠖⣡⢃⠎⡔⡱⢌⠓⣌⠱⢌⠳⣭⢻⡜⣧⢻⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠃⠆⠑⠀⠀⠀⠀⠀⠀⠀⠀⡜⢤⠋⡆⢭⡘⢆⡱⢎⡇⡚⢤⢊⡜⢰⡡⠎⡜⡄⣋⠆⡣⠜⣧⢻⣜⡳⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠍⡂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠢⢧⣙⠦⣉⠖⣨⢓⡮⡱⢎⠶⣌⡱⣍⠳⣄⠳⣄⢣⡑⢮⣜⡳⣎⠷⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⣎⢖⡡⠚⡴⢫⣜⢣⣏⠞⡴⢳⢬⡓⣎⠳⣆⢧⡽⣺⡜⣳⢭⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢎⡵⣛⡜⣣⢞⡣⣞⡹⡜⡭⢖⡹⣼⢻⡜⣧⢳⡇⣿⣡⢟⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡠⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢲⣿⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡀⢄⠠⢄⠠⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣟⠁⡀⠄⢠⢀⠐⠀⠁⡀⠎⠐⠈⠂⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⡿⣿⡿⣽⠯⣄⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠙⣷⣻⣽⡻⣜⠀⢨⡱⣋⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣇⣬⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡷⠟⠀⠁⢼⣷⡋⡇⠄⡈⠄⠠⠙⠞⠀⠰⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⣾⣳⢯⡷⠭⠀⠐⠒⠉⡟⢀⢧⠀⠀⡜⡄⠡⠘⣄⠀⢀⠼⣌⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣻⢾⡽⣾⠝⠃⣀⣠⣤⣶⠦⡌⠐⠜⡶⡌⠰⠀⠩⡒⢧⡈⣉⠡⢀⠈⠓⡅⠐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⣽⡿⣽⣻⢯⣟⡿⣽⡇⠀⠀⠉⠁⠀⠀⠀⠀⠐⠀⠃⠈⡗⠀⣀⡀⠁⠑⢌⡒⢆⠈⠄⠱⢄⠸⣍⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⣷⣻⢾⣽⣻⢾⣽⣳⢯⣟⡾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠙⠛⢷⣤⡀⠈⠨⣘⠠⠁⢂⢸⠂⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡟⣭⣟⢾⣹⣟⡾⣽⢯⣿⡽⣿⢯⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠱⠂⢁⠂⡜⠀⡇⠈⠡⢀⡴⢲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⣽⣞⣯⢷⣯⡿⣽⢯⣿⣿⢯⡿⣽⡌⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠬⠁⢂⠘⠢⠄⠚⢜⢣⢞⡰⠈⠆⠱⣀⠀
⠀⠀⠀⠀⠀⣼⣷⣻⣟⣾⣻⣞⣿⡽⣯⢿⢾⣯⢿⣽⣳⣌⣥⣒⣄⣠⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡜⠀⠌⡐⠠⠀⠄⡸⢁⠐⡈⠄⡘⢮⣀⣠⢊⠲⠀⠀
⠀⠀⠀⠀⠀⢻⣞⣿⣿⣞⣷⢯⣿⣽⣯⣿⢘⢿⡿⣾⠏⢻⡟⠻⡿⣿⢟⡯⠀⠈⠐⠀⠤⡀⢄⠠⡐⡈⠆⡜⠀⠌⡐⠠⠁⢌⢎⡱⢢⠔⢀⠢⠔⡀⡱⢎⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠿⡻⣯⣿⣇⢸⡳⢿⡇⢎⠭⣙⠃⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠈⡆⡁⢆⠡⢚⢄⡨⠔⢀⣡⠌⣏⢮⡱⠃⡠⢌⡱⠊⢠⣏⠳⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠦⠀⠀⢘⠛⣇⠘⢤⣙⠪⠖⡭⢷⡀⠤⣀⠼⣿⣿⠁⠀⠀⠀⠀⠀⢀⡖⣈⡔⡈⢎⠲⣴⢺⡍⣆⢸⡜⢢⠕⢪⣑⠮⡴⣚⡵⣩⠓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⣒⠢⢍⣒⠢⢍⠢⡑⢆⡙⣰⣌⡆⠙⣿⣅⠀⠀⠀⠀⡠⢋⠜⣣⠟⡽⣎⡟⡜⠳⣜⡹⢦⢯⡘⡜⠐⡜⢫⢗⡱⢎⡕⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠁⠁⠋⠄⠤⡁⠊⠔⠡⠌⢴⣻⣟⡆⠱⢈⠂⠀⠀⠀⢨⡔⣉⠦⣁⠛⡵⢫⡞⣭⢳⣞⡱⣛⢮⡽⣩⣄⣈⡖⣫⡜⣏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠄⠀⠀⠐⢡⠈⠐⠉⠠⠷⣞⣧⠘⢠⠂⠀⠀⢠⡔⠳⢰⡘⠤⢋⡔⢣⡹⢎⡷⢮⡵⣫⢷⢹⡳⣎⠵⡺⠵⣛⡶⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀⡐⠭⣚⣥⣂⢩⡐⣠⠳⣌⠣⣅⠚⡌⢥⡘⠆⣥⢛⡼⣣⢯⡵⡋⠀⢟⡅⢫⢵⡂⢹⡆⠉⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠦⢒⡑⢎⢴⢠⠃⡽⣰⢋⠆⡓⠤⢋⠜⡰⡘⠜⣦⢋⢶⡹⢶⡹⣝⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠘⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢋⠬⣑⠸⢌⡂⢧⣙⠴⡁⢎⠜⣡⠙⡌⢎⠱⢌⠣⠜⡭⣞⣭⢳⡝⣮⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⡰⣉⠲⣌⡱⢢⢉⠶⣉⠦⢩⠜⣈⠦⣉⠖⣩⢘⢢⠙⣢⢑⢺⡜⣧⣛⢶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢊⠅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠱⣤⢛⡴⠑⡌⢎⡳⣍⠞⣥⢛⡴⡸⢥⡛⣤⢊⠦⡙⢤⣊⡶⣹⠶⣹⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⡗⢦⡙⡌⢦⠳⣼⡙⣦⢋⠶⣙⠦⡝⢦⣋⢞⣹⢦⡻⣜⢧⡻⢵⣚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⢣⡝⣜⢣⡛⣴⢫⡜⢯⡹⣍⠻⣜⠳⣞⢯⢎⡷⣙⢮⡳⣝⠧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠋⠠⢀⠠⡀⠂⠈⢀⠰⠁⠂⠑⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣯⢣⠘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠣⣟⡿⣞⡿⡀⠀⣍⢞⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⠿⠁⠒⠘⣿⡽⢁⠡⢈⠠⠈⠳⠃⠐⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⡽⣯⣷⢋⠠⠤⠀⠚⣿⠁⢂⠄⠠⣘⠠⠁⢦⡀⠀⡠⢞⡬⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣽⣳⢯⣟⣷⡻⠓⠀⣀⣠⣤⣤⣄⢃⠜⢦⡐⠤⠀⠳⢬⣂⠌⡉⡄⠠⢈⠓⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⣯⣟⣯⣟⣯⣿⣾⠀⠈⠉⠉⠉⠈⠀⠀⠀⠂⠈⠱⡂⠀⣈⠀⠊⠦⢱⠈⡴⠀⢂⠣⠤⢸⡱⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⢿⡾⣽⣳⣟⣾⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠘⠛⠷⣦⣀⠀⠑⠘⣌⠠⠐⠠⢸⠁⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣭⣟⣾⣳⣿⣞⣿⢿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢒⠠⠈⠄⡏⠀⡄⠀⠊⢠⢖⡲⢆⡀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⡻⣷⣻⣞⡷⣿⣯⣟⣯⣟⡆⠄⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡰⠀⠡⠘⡒⠐⠊⢮⢱⢫⠔⠈⠑⢨⠁⠄
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⣟⡷⣿⢾⣽⣻⢻⣞⡷⣯⣿⣈⣖⣠⣄⣀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⡐⢀⠠⠁⡒⠁⠌⡐⠂⠘⢧⡄⣴⠉⡅⠊⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣟⣾⡽⣿⣟⣾⢿⡩⢿⣟⣯⡇⢻⡿⠿⣿⡿⣿⣟⠉⠐⠂⠄⡠⢀⠀⡠⢀⠆⡑⠎⢀⠁⢂⠐⡀⢢⡙⠴⣡⠖⠀⡌⠌⠠⣙⠶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢯⣿⢷⡘⡟⢿⣇⠎⡭⣉⠃⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⠁⢊⠔⢂⠌⡜⣀⠢⠘⣀⣂⠰⢧⡙⢧⠁⣠⠚⡔⠊⢡⡏⣞⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⢧⠘⢤⣉⠺⠴⢹⣳⠠⢠⣀⢠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣇⣌⠢⠌⡳⢌⢧⣫⢇⡇⢸⢣⡙⢆⡚⢤⡏⣴⢚⡧⣙⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡜⠢⢍⡒⠢⢍⠢⠕⢤⠙⣤⣎⡤⠙⣯⣇⠀⠀⠀⠀⣈⠣⢜⢪⡟⣽⢣⣟⡨⢓⣎⢞⡮⢧⡘⣆⠘⠼⣱⢣⢏⠶⡉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠈⠭⠄⠌⡁⠊⠔⡉⢄⢳⡟⡿⡆⢡⠘⠃⠀⠀⠀⢨⡄⣋⠒⡆⡙⢮⢳⣎⣗⣣⡽⢪⡝⣧⢻⢥⣈⣠⢖⡫⣎⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠈⢅⠀⠁⠊⠈⠿⣽⣆⠡⡘⠠⠀⠀⢠⡔⠳⣈⠵⣈⠕⡢⢃⡼⢮⢵⢫⢷⡹⣎⢟⢮⡗⡜⡮⢵⣫⠶⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⢀⠒⡱⢞⣡⣐⢡⢂⡠⢳⡜⣡⠒⣌⠒⡬⢡⢃⢆⡏⢾⡹⣎⢷⡋⠈⠷⡌⢳⡹⡀⢳⠇⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠦⠓⡔⠫⣔⢌⠪⣱⢣⠳⡘⣄⠳⢨⠜⣐⠣⠜⣬⢛⢦⡻⣜⢧⣛⡀⠀⠉⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢩⡘⡱⢌⠓⡤⢋⣖⠡⢎⠱⡘⠤⢃⡃⢎⠴⡉⢎⡔⢫⡞⣵⢫⡞⣵⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢎⡜⢢⠑⣦⠩⡜⠰⣍⢦⡙⢢⢃⠭⡘⣥⠚⡌⢆⡹⢠⠚⣄⠻⣜⢧⣛⠶⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠱⡎⡝⡆⡓⢌⠳⣌⠷⣘⢧⣋⠶⡱⢎⡝⣰⠣⡔⢣⡑⣢⠽⣎⠷⣭⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠸⡱⢎⣑⠪⡱⢎⡽⡱⢎⣜⢣⡝⢮⡜⣥⢛⡜⣣⠽⣎⠿⣜⣛⢶⡹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⢎⡜⣣⠝⡮⣜⢧⢫⡝⢦⣋⠷⣘⡖⣿⢺⢭⡻⣜⡻⣼⣩⢞⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠒⠠⠄⣀⠀⡀⢀⠀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠰⢂⠩⡐⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣆⣡⢂⠅⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣏⣛⢷⣺⢨⢳⡱⣹⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⠎⣒⠩⢞⣣⡝⡗⢦⣻⠴⣹⢎⣷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⠦⡘⠤⢋⡔⢢⠹⣝⠯⢧⡻⣵⣫⢖⣧⣛⠴⣫⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠴⣉⠜⡡⠜⣂⠳⣌⡻⢧⣛⠶⡍⠘⢶⡉⢗⣣⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢚⡰⢌⡊⠥⠓⣌⢳⢣⡝⣣⢏⡯⣝⡀⠈⠑⠀⠁⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡌⣒⢩⡑⢢⠍⢶⣹⢣⡟⣼⢓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⢦⡱⠌⢆⡜⢡⠚⣄⢫⠷⣹⢎⡟⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⣣⢳⡘⢦⠜⣡⠚⡤⣞⡽⢣⡟⣼⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⡼⣡⢧⡙⣎⠞⣥⢻⣜⡳⢞⣧⢻⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣎⠵⣞⢯⡳⢏⡮⣝⣏⢶⣫⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠒⠠⠄⣀⠀⡀⢀⠀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠰⢂⠩⡐⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣆⣡⢂⠅⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣏⣛⢷⣺⢨⢳⡱⣹⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⠎⣒⠩⢞⣣⡝⡗⢦⣻⠴⣹⢎⣷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⠦⡘⠤⢋⡔⢢⠹⣝⠯⢧⡻⣵⣫⢖⣧⣛⠴⣫⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠴⣉⠜⡡⠜⣂⠳⣌⡻⢧⣛⠶⡍⠘⢶⡉⢗⣣⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢚⡰⢌⡊⠥⠓⣌⢳⢣⡝⣣⢏⡯⣝⡀⠈⠑⠀⠁⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡌⣒⢩⡑⢢⠍⢶⣹⢣⡟⣼⢓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⢦⡱⠌⢆⡜⢡⠚⣄⢫⠷⣹⢎⡟⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⣣⢳⡘⢦⠜⣡⠚⡤⣞⡽⢣⡟⣼⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⡼⣡⢧⡙⣎⠞⣥⢻⣜⡳⢞⣧⢻⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣎⠵⣞⢯⡳⢏⡮⣝⣏⢶⣫⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠠⠠⢀⡀⢀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠢⡐⠡⢊⠰⣘⢀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣅⢣⡌⢂⡇⢎⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣝⡳⣎⢯⡰⢫⠖⡽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⢊⠵⡘⠳⣝⠾⣕⢮⢻⡜⣱⢮⣳⢍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⢆⡩⢢⢑⠣⡌⠽⢎⣯⢳⡽⣹⣎⢷⢫⡗⡱⡞⢵⣣⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠤⢃⡱⢊⡱⢌⠣⣏⠶⣫⢞⡵⡋⠈⡷⡙⢵⡹⠀⢗⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢊⡱⢃⠲⢡⠒⣌⠳⣍⢞⡵⣫⢞⡽⡀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡩⢌⢣⢩⠰⣉⠞⣮⢳⡝⣮⢳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⣜⡰⢃⡌⢆⡓⠬⣘⠱⣏⢾⡱⣏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⢦⡙⢦⡘⢦⡘⢥⢢⡽⢎⡷⣹⢎⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⣜⡣⣝⢲⡙⣦⣙⢮⡳⣭⢻⡜⣧⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣬⢓⡿⣲⡝⣮⢳⣭⠳⣝⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣁⡄⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣾⣶⢶⠣⠀⣀⠀⡀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣿⡏⠀⠀⠀⠀⠁⠱⢀⠃⡌⢢⡘⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⣌⡐⠂⡗⣌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⢜⡸⣝⡻⣖⢾⡘⢮⡱⢽⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡄⣋⠔⡢⢍⠳⣎⠷⡝⢦⢟⢦⢻⣜⡳⣍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⠔⠳⣈⠜⡰⢃⡱⢌⡹⡝⣯⣛⢮⡳⣮⢝⡶⡳⢥⡳⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⣋⢞⠱⣈⠎⡱⢌⡒⠬⡑⣮⢱⣎⢷⣹⡃⠈⣷⡙⢧⡹⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⢸⡱⢎⢊⡕⠢⢍⡒⣌⠲⣁⠟⡴⢣⡞⣧⢳⡝⡀⠀⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⡴⢊⡔⢊⠦⣘⠱⢊⡔⢢⠱⢌⠚⣭⢳⡝⣮⢳⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⣝⢣⠅⣎⠱⢢⢑⡪⢥⡘⢢⢍⠢⡍⢤⢛⡼⣣⢟⡼⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡭⢎⡳⣌⠳⡥⢎⡵⢣⡜⣡⠎⣒⡉⣆⢾⡱⣏⢾⡱⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣭⠳⣜⡹⣜⢣⢞⡱⢎⡵⢪⡵⣹⢎⡷⣹⢎⡷⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⣜⢧⡹⡕⢮⡙⢮⡱⣋⡾⢧⡻⣥⢻⡜⣧⢻⡜⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⢠⣁⡄⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣷⣶⣶⢃⣀⠀⡀⠀⠀⠀⡀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠓⠀⠹⢾⣿⡎⠀⠀⠀⠈⠁⠪⠡⡐⠡⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⣌⡡⠊⣕⢢⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠑⣲⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⢜⡸⣝⡻⢦⢯⡰⢫⠖⡽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⢟⡆⠱⢈⠭⠀⠀⠀⢨⡄⣋⠔⡢⢍⠳⣏⠾⣕⢮⢻⡜⣱⢮⣳⢍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠿⡾⣅⠊⡔⠠⠀⠀⢠⠔⠳⣈⠜⡰⢃⡱⢨⢹⢎⣯⢳⡽⣹⣎⢷⢫⡗⡱⡞⢵⣣⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠹⣞⣡⣐⢡⢂⣠⢋⡞⡱⣈⠎⡱⢌⡒⠥⢃⡞⡜⣧⣛⠶⡍⠈⡷⡙⢵⡹⠀⢗⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⡑⢎⢴⡈⢎⡱⣆⢏⠒⡅⠦⣙⠰⢊⠴⣉⢎⠷⣙⠶⣭⣛⡽⠀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⢎⠱⡘⣌⢢⠙⣴⠊⠴⣈⠣⡜⢡⠆⡹⣈⠖⡌⣊⠷⣭⣛⠶⣭⢞⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢃⡌⡳⡌⡔⣊⡝⢦⡙⢢⢅⡓⠬⣡⢎⠱⡐⠎⠴⣁⠖⣱⢫⡝⣞⡺⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡣⢞⡱⡘⢔⠢⣝⢣⠞⣥⡚⡴⢣⡓⢮⡡⣍⠮⣑⢢⢚⡴⣫⢞⡵⣛⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠙⣎⠶⣑⠊⡵⢪⣝⠺⣴⠹⣜⢣⡝⣲⠱⣎⠶⣡⢯⡞⣵⢫⡞⣵⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⣍⠻⣜⢣⡞⣧⠹⡝⣬⢣⠽⣌⠷⣞⢯⣳⢫⡜⣧⢻⡜⣧⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣁⡄⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣾⣶⢶⠣⠀⣀⠀⡀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣿⡏⠀⠀⠀⠀⠁⠱⢀⠃⡌⢢⡘⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⣌⡐⠂⡗⣌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⢜⡸⣝⡻⣖⢾⡘⢮⡱⢽⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡄⣋⠔⡢⢍⠳⣎⠷⡝⢦⢟⢦⢻⣜⡳⣍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⠔⠳⣈⠜⡰⢃⡱⢌⡹⡝⣯⣛⢮⡳⣮⢝⡶⡳⢥⡳⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⣋⢞⠱⣈⠎⡱⢌⡒⠬⡑⣮⢱⣎⢷⣹⡃⠈⣷⡙⢧⡹⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⢸⡱⢎⢊⡕⠢⢍⡒⣌⠲⣁⠟⡴⢣⡞⣧⢳⡝⡀⠀⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⡴⢊⡔⢊⠦⣘⠱⢊⡔⢢⠱⢌⠚⣭⢳⡝⣮⢳⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⣝⢣⠅⣎⠱⢢⢑⡪⢥⡘⢢⢍⠢⡍⢤⢛⡼⣣⢟⡼⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡭⢎⡳⣌⠳⡥⢎⡵⢣⡜⣡⠎⣒⡉⣆⢾⡱⣏⢾⡱⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣭⠳⣜⡹⣜⢣⢞⡱⢎⡵⢪⡵⣹⢎⡷⣹⢎⡷⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⣜⢧⡹⡕⢮⡙⢮⡱⣋⡾⢧⡻⣥⢻⡜⣧⢻⡜⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠠⠠⢀⡀⢀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠢⡐⠡⢊⠰⣘⢀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣅⢣⡌⢂⡇⢎⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣝⡳⣎⢯⡰⢫⠖⡽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⢊⠵⡘⠳⣝⠾⣕⢮⢻⡜⣱⢮⣳⢍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⢆⡩⢢⢑⠣⡌⠽⢎⣯⢳⡽⣹⣎⢷⢫⡗⡱⡞⢵⣣⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠤⢃⡱⢊⡱⢌⠣⣏⠶⣫⢞⡵⡋⠈⡷⡙⢵⡹⠀⢗⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢊⡱⢃⠲⢡⠒⣌⠳⣍⢞⡵⣫⢞⡽⡀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡩⢌⢣⢩⠰⣉⠞⣮⢳⡝⣮⢳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⣜⡰⢃⡌⢆⡓⠬⣘⠱⣏⢾⡱⣏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⢦⡙⢦⡘⢦⡘⢥⢢⡽⢎⡷⣹⢎⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⣜⡣⣝⢲⡙⣦⣙⢮⡳⣭⢻⡜⣧⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣬⢓⡿⣲⡝⣮⢳⣭⠳⣝⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠒⠠⠄⣀⠀⡀⢀⠀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠰⢂⠩⡐⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣆⣡⢂⠅⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣏⣛⢷⣺⢨⢳⡱⣹⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⠎⣒⠩⢞⣣⡝⡗⢦⣻⠴⣹⢎⣷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⠦⡘⠤⢋⡔⢢⠹⣝⠯⢧⡻⣵⣫⢖⣧⣛⠴⣫⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠴⣉⠜⡡⠜⣂⠳⣌⡻⢧⣛⠶⡍⠘⢶⡉⢗⣣⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢚⡰⢌⡊⠥⠓⣌⢳⢣⡝⣣⢏⡯⣝⡀⠈⠑⠀⠁⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡌⣒⢩⡑⢢⠍⢶⣹⢣⡟⣼⢓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⢦⡱⠌⢆⡜⢡⠚⣄⢫⠷⣹⢎⡟⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⣣⢳⡘⢦⠜⣡⠚⡤⣞⡽⢣⡟⣼⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⡼⣡⢧⡙⣎⠞⣥⢻⣜⡳⢞⣧⢻⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣎⠵⣞⢯⡳⢏⡮⣝⣏⢶⣫⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠒⠠⠄⣀⠀⡀⢀⠀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠰⢂⠩⡐⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣆⣡⢂⠅⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣏⣛⢷⣺⢨⢳⡱⣹⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⠎⣒⠩⢞⣣⡝⡗⢦⣻⠴⣹⢎⣷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⠦⡘⠤⢋⡔⢢⠹⣝⠯⢧⡻⣵⣫⢖⣧⣛⠴⣫⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠴⣉⠜⡡⠜⣂⠳⣌⡻⢧⣛⠶⡍⠘⢶⡉⢗⣣⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢚⡰⢌⡊⠥⠓⣌⢳⢣⡝⣣⢏⡯⣝⡀⠈⠑⠀⠁⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡌⣒⢩⡑⢢⠍⢶⣹⢣⡟⣼⢓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⢦⡱⠌⢆⡜⢡⠚⣄⢫⠷⣹⢎⡟⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⣣⢳⡘⢦⠜⣡⠚⡤⣞⡽⢣⡟⣼⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⡼⣡⢧⡙⣎⠞⣥⢻⣜⡳⢞⣧⢻⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣎⠵⣞⢯⡳⢏⡮⣝⣏⢶⣫⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠠⠠⢀⡀⢀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠢⡐⠡⢊⠰⣘⢀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣅⢣⡌⢂⡇⢎⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣝⡳⣎⢯⡰⢫⠖⡽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⢊⠵⡘⠳⣝⠾⣕⢮⢻⡜⣱⢮⣳⢍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⢆⡩⢢⢑⠣⡌⠽⢎⣯⢳⡽⣹⣎⢷⢫⡗⡱⡞⢵⣣⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠤⢃⡱⢊⡱⢌⠣⣏⠶⣫⢞⡵⡋⠈⡷⡙⢵⡹⠀⢗⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢊⡱⢃⠲⢡⠒⣌⠳⣍⢞⡵⣫⢞⡽⡀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡩⢌⢣⢩⠰⣉⠞⣮⢳⡝⣮⢳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⣜⡰⢃⡌⢆⡓⠬⣘⠱⣏⢾⡱⣏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⢦⡙⢦⡘⢦⡘⢥⢢⡽⢎⡷⣹⢎⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⣜⡣⣝⢲⡙⣦⣙⢮⡳⣭⢻⡜⣧⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣬⢓⡿⣲⡝⣮⢳⣭⠳⣝⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠳⡠⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢡⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⡀⠄⣀⢀⠠⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡿⠋⡀⢀⠀⠄⠀⠂⠁⠌⠒⠄⠊⠐⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡄⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣷⡽⠀⠀⡞⡼⣱⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣦⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⡟⠃⠤⢹⡾⣽⠊⠑⠊⠠⠑⢧⠃⠐⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣳⣯⢿⡽⡞⢁⠀⠤⠖⣿⠁⠎⠀⠁⢆⠡⠐⣄⠀⠀⢡⢏⡛⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣯⣿⣾⡿⣽⢯⡷⣯⢷⡯⠟⠁⠀⣀⣤⣄⡀⢃⠞⣬⠐⡈⠘⠦⣔⡨⠑⢊⠄⡈⠙⢪⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣽⣿⢿⣽⣷⣻⢯⡿⣽⣯⡟⠀⠀⠛⠛⠉⠉⠉⠁⠈⡀⠒⠣⠆⠀⠈⠐⠱⢉⠖⡰⢠⠁⠂⢦⡀⢽⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣽⣻⡽⣯⣟⣾⣳⢯⡿⣽⣻⣽⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠈⠁⠲⠿⣶⣄⡀⠈⠢⠅⡌⠐⡀⠄⢪⠙⡮⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡷⣽⢾⣭⢿⣟⡷⣿⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠘⡜⢀⠐⡈⠇⡘⡀⠐⠂⢠⢴⢢⡄⠀⠀⠀⢀⠠⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣶⢯⡿⣽⣻⣞⣯⣿⣿⣻⣞⣿⡁⠄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡜⠁⠠⠑⡢⠄⠫⢎⢧⢳⠁⠎⠰⢈⠤⠀
⠀⠀⠀⠀⠀⣼⣷⣻⣯⢿⣽⣳⣿⣞⣷⣻⢯⣷⣻⣞⣷⣈⡤⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢀⠀⡁⠲⠁⡐⢀⠊⡘⢧⣀⢤⡉⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣞⣿⣿⣟⣾⣳⣿⢾⣳⣿⠸⣷⣿⢾⡏⢻⡿⢿⣿⣾⣶⢶⠂⠀⣀⠀⡀⠀⡀⢀⠠⡐⢂⠎⠠⠁⠌⡐⢀⢢⠜⡡⢆⠔⡀⢂⠔⠈⡞⢦⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣗⠻⡻⣽⣿⣞⡸⢻⢟⣧⢊⠭⣛⠃⠈⠑⠀⠹⢾⣿⡍⠀⠀⠀⠀⠁⠣⢐⠡⢂⡑⢎⡀⡂⠅⢂⡐⢠⢏⡞⡱⢁⢠⢒⡡⠊⢰⡝⣺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠡⢊⣉⠮⠖⡽⣖⠠⢠⡀⢠⣿⣿⠇⠀⠀⠀⠀⠀⢀⡇⣌⡂⠜⣢⢑⡶⣚⢭⡃⡹⢎⡜⡡⢎⢦⢣⡴⣜⣣⢛⡴⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢓⠇⠮⣁⡓⠌⢆⠣⠔⡌⡑⢦⡝⡄⡙⣿⡃⠀⠀⠀⠀⡒⣩⠘⢧⡻⣝⢦⣏⠶⢫⢖⣳⣙⢦⠜⣁⠊⢞⡳⡜⢦⢣⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠈⠥⠄⢌⡁⠊⠔⡡⠘⣼⣻⢿⡂⢡⠘⠍⠀⠀⠀⢨⠔⣡⠚⡤⢙⠮⣳⠞⡵⢎⡷⢢⡟⣮⢯⣁⡌⣠⢥⡛⣼⡋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠁⠁⠌⢷⣻⣄⢃⡘⠄⠀⠀⢠⡌⠳⡄⡓⠬⡱⢘⠤⣻⢹⠯⣝⢧⣻⡜⢧⣳⡞⡱⣎⡽⣖⡧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⠀⢀⠢⢱⢫⢢⡐⣌⢁⡰⢣⢎⡱⢘⡌⡱⠢⢍⡒⣡⢏⡽⢎⡷⢣⡏⠘⣵⡉⠷⣹⠀⢯⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣎⠴⢊⠥⣋⢦⠱⢨⠹⣔⢋⠦⣑⢊⠴⢡⡙⢢⠜⢦⡫⣜⣏⡞⣧⢳⠀⠀⠁⠀⠁⠁⠘⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣁⠎⡱⠢⢅⠦⣙⠦⢋⠤⢋⠴⠨⡜⢂⠣⠜⣂⢎⠱⣓⠾⣜⡞⣵⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢎⠲⠌⡜⣡⡙⢢⠜⣬⢣⠍⡜⣌⠪⡑⣌⠣⢍⠚⢤⢊⡱⢌⠻⣜⠾⣱⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠱⣍⠞⡅⢎⠥⢚⡴⣋⠞⡜⢦⢣⡝⣬⢛⡌⣍⠖⠬⣐⢎⡵⣫⢏⣗⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⡹⣜⡂⢎⡱⢎⡽⣚⡝⣎⠳⣜⢦⢫⡜⡜⣎⣳⢭⡞⣵⢫⡞⣼⡱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢱⢎⡝⢮⡱⢫⡼⣱⢚⡭⢳⡍⣎⠳⣜⡿⡜⣧⢏⡾⣱⢏⡾⣱⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⢠⣁⡄⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣷⣶⣶⢃⣀⠀⡀⠀⠀⠀⡀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠓⠀⠹⢾⣿⡎⠀⠀⠀⠈⠁⠪⠡⡐⠡⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⣌⡡⠊⣕⢢⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠑⣲⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⢜⡸⣝⡻⢦⢯⡰⢫⠖⡽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⢟⡆⠱⢈⠭⠀⠀⠀⢨⡄⣋⠔⡢⢍⠳⣏⠾⣕⢮⢻⡜⣱⢮⣳⢍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠿⡾⣅⠊⡔⠠⠀⠀⢠⠔⠳⣈⠜⡰⢃⡱⢨⢹⢎⣯⢳⡽⣹⣎⢷⢫⡗⡱⡞⢵⣣⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠹⣞⣡⣐⢡⢂⣠⢋⡞⡱⣈⠎⡱⢌⡒⠥⢃⡞⡜⣧⣛⠶⡍⠈⡷⡙⢵⡹⠀⢗⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⡑⢎⢴⡈⢎⡱⣆⢏⠒⡅⠦⣙⠰⢊⠴⣉⢎⠷⣙⠶⣭⣛⡽⠀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⢎⠱⡘⣌⢢⠙⣴⠊⠴⣈⠣⡜⢡⠆⡹⣈⠖⡌⣊⠷⣭⣛⠶⣭⢞⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢃⡌⡳⡌⡔⣊⡝⢦⡙⢢⢅⡓⠬⣡⢎⠱⡐⠎⠴⣁⠖⣱⢫⡝⣞⡺⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡣⢞⡱⡘⢔⠢⣝⢣⠞⣥⡚⡴⢣⡓⢮⡡⣍⠮⣑⢢⢚⡴⣫⢞⡵⣛⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠙⣎⠶⣑⠊⡵⢪⣝⠺⣴⠹⣜⢣⡝⣲⠱⣎⠶⣡⢯⡞⣵⢫⡞⣵⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⣍⠻⣜⢣⡞⣧⠹⡝⣬⢣⠽⣌⠷⣞⢯⣳⢫⡜⣧⢻⡜⣧⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣁⡄⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣾⣶⢶⠣⠀⣀⠀⡀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣿⡏⠀⠀⠀⠀⠁⠱⢀⠃⡌⢢⡘⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⠀⣇⣌⡐⠂⡗⣌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⣂⠫⢜⡸⣝⡻⣖⢾⡘⢮⡱⢽⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡄⣋⠔⡢⢍⠳⣎⠷⡝⢦⢟⢦⢻⣜⡳⣍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⠔⠳⣈⠜⡰⢃⡱⢌⡹⡝⣯⣛⢮⡳⣮⢝⡶⡳⢥⡳⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⣋⢞⠱⣈⠎⡱⢌⡒⠬⡑⣮⢱⣎⢷⣹⡃⠈⣷⡙⢧⡹⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⢸⡱⢎⢊⡕⠢⢍⡒⣌⠲⣁⠟⡴⢣⡞⣧⢳⡝⡀⠀⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⡴⢊⡔⢊⠦⣘⠱⢊⡔⢢⠱⢌⠚⣭⢳⡝⣮⢳⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⣝⢣⠅⣎⠱⢢⢑⡪⢥⡘⢢⢍⠢⡍⢤⢛⡼⣣⢟⡼⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡭⢎⡳⣌⠳⡥⢎⡵⢣⡜⣡⠎⣒⡉⣆⢾⡱⣏⢾⡱⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣭⠳⣜⡹⣜⢣⢞⡱⢎⡵⢪⡵⣹⢎⡷⣹⢎⡷⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⣜⢧⡹⡕⢮⡙⢮⡱⣋⡾⢧⡻⣥⢻⡜⣧⢻⡜⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠠⠠⢀⡀⢀⠀⡀⢀⠠⢐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠢⡐⠡⢊⠰⣘⢀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣅⢣⡌⢂⡇⢎⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣝⡳⣎⢯⡰⢫⠖⡽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⢊⠵⡘⠳⣝⠾⣕⢮⢻⡜⣱⢮⣳⢍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⢆⡩⢢⢑⠣⡌⠽⢎⣯⢳⡽⣹⣎⢷⢫⡗⡱⡞⢵⣣⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠤⢃⡱⢊⡱⢌⠣⣏⠶⣫⢞⡵⡋⠈⡷⡙⢵⡹⠀⢗⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢊⡱⢃⠲⢡⠒⣌⠳⣍⢞⡵⣫⢞⡽⡀⠈⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡩⢌⢣⢩⠰⣉⠞⣮⢳⡝⣮⢳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⣜⡰⢃⡌⢆⡓⠬⣘⠱⣏⢾⡱⣏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⢦⡙⢦⡘⢦⡘⢥⢢⡽⢎⡷⣹⢎⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⣜⡣⣝⢲⡙⣦⣙⢮⡳⣭⢻⡜⣧⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣬⢓⡿⣲⡝⣮⢳⣭⠳⣝⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⠁⠀⣀⣠⣤⡀⠇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠀⠚⠛⠉⠉⠁⠉⠈⠀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⡷⣯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣡⣀⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⣿⣿⣯⣟⣷⣻⣯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⣿⣿⣿⢾⠒⠠⠄⣀⠀⡀⢀⠀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⣗⠻⡻⢾⣟⣷⡘⡟⢯⣧⢊⠭⣛⠃⠀⠑⠀⠹⢾⣾⡏⠀⠀⠀⠀⠈⠰⢂⠩⡐⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠧⠀⠀⡘⠛⣇⠌⣡⣉⠮⠳⠼⣳⡀⠤⣀⠠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣆⣡⢂⠅⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡖⠢⢍⡒⠢⢅⠣⡑⢤⠙⣰⣌⡆⠙⣿⡇⠀⠀⠀⠀⡒⣊⢌⠳⣏⣛⢷⣺⢨⢳⡱⣹⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠈⠐⠭⠀⠤⡉⠂⠱⢈⠢⢵⣻⣟⡆⠱⢈⠭⠀⠀⠀⢨⡔⢡⠎⣒⠩⢞⣣⡝⡗⢦⣻⠴⣹⢎⣷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠐⢁⠈⠀⠃⠌⠷⣞⣦⠑⡨⢀⠀⠀⢠⡔⡙⠦⡘⠤⢋⡔⢢⠹⣝⠯⢧⡻⣵⣫⢖⣧⣛⠴⣫⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⢀⡐⠭⢶⣡⣂⢅⣂⢠⠳⣜⡘⠴⣉⠜⡡⠜⣂⠳⣌⡻⢧⣛⠶⡍⠘⢶⡉⢗⣣⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⢒⠥⣋⢴⢐⢪⣘⢣⠛⡤⢚⡰⢌⡊⠥⠓⣌⢳⢣⡝⣣⢏⡯⣝⡀⠈⠑⠀⠁⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠎⢥⠒⡌⠦⡙⣤⠎⢆⡹⢠⢃⠖⡌⣒⢩⡑⢢⠍⢶⣹⢣⡟⣼⢓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢜⠤⢓⡘⢦⡉⢖⡡⡝⣆⠚⣤⢡⢃⠎⢦⡱⠌⢆⡜⢡⠚⣄⢫⠷⣹⢎⡟⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠱⡎⡝⢦⠙⡤⢒⡽⣌⠳⣆⠳⣎⡜⣣⢳⡘⢦⠜⣡⠚⡤⣞⡽⢣⡟⣼⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠘⡝⢦⡩⠔⣣⠞⣜⡳⣜⠳⡜⡼⣡⢧⡙⣎⠞⣥⢻⣜⡳⢞⣧⢻⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢣⡝⡹⢆⡻⡼⣱⢚⡭⢳⣍⠳⣎⠵⣞⢯⡳⢏⡮⣝⣏⢶⣫⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠡⢎⠰⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⣡⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠟⡁⠠⡀⢄⡀⠂⠈⠀⠔⠁⠂⠑⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢡⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣂⣥⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣳⢆⠙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⠤⢀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠱⣿⣽⣻⡞⡀⠀⣭⢚⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣠⣴⣮⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⡟⠁⠂⢼⣳⠿⢁⠁⡉⢀⠉⠞⠁⠈⠔⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⢷⣻⢷⣯⡿⢉⠠⠄⠐⠚⡿⠁⠆⡀⠐⠬⢀⠘⢆⡀⠀⡰⢫⡝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣯⣿⣾⣟⣿⣳⣟⡾⣯⣟⠿⠊⢀⣀⣠⣤⣤⣀⠣⠜⣥⠐⡈⠈⠳⢤⡊⢌⠑⡄⠐⡈⠱⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⢿⣿⣻⣿⣽⡿⣽⣟⡾⣟⡾⣽⣷⡯⠀⠈⠉⠉⠉⠁⠀⠀⠐⢀⠨⢡⠃⠀⡈⠀⠙⠢⣍⠰⡐⠠⢁⠣⠄⢞⡭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⢾⣽⣞⣯⣟⡷⣯⢿⣽⣻⣽⡿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠁⠙⠻⠷⣦⣀⠀⠁⠇⡜⠀⡐⠈⡼⢘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡟⣽⢾⣹⢮⣽⣻⣞⣯⣷⣯⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠈⡜⠀⠄⠡⠎⠰⡀⠀⠃⣠⢖⡲⣄⠀⠀⢀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡿⣽⡷⣯⣟⡾⣷⣻⣞⣿⣷⢯⡿⣾⡅⡀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⢀⡙⠀⢂⠉⡒⢀⠓⠮⣱⠮⠑⠈⠂⢌⢂⠀
⠀⠀⠀⠀⠀⣾⣽⣻⣟⡷⣯⣟⣷⣻⢾⣽⢾⣯⣟⡷⣷⣨⣔⣤⣀⣀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡀⢂⠀⠄⢂⡜⢀⠂⡐⠂⡑⢧⡄⡴⢉⠆⠀⠀
⠀⠀⠀⠀⠀⢻⣞⣿⣿⣟⣷⢯⡿⢯⣿⣾⠚⣷⣿⣻⡏⢽⡿⠻⣿⣿⢿⡿⠑⠀⠤⠀⡄⢀⠀⢠⠠⡘⠤⠃⡀⢂⠐⠠⠈⡔⢣⢒⡤⠒⢀⡡⠄⠂⣝⢲⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣿⣧⠻⠝⢯⣿⣟⠨⡟⠾⣏⠴⣡⢋⠃⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⢁⠊⠤⡁⢆⢣⡐⠤⠂⢌⡠⢱⡍⢧⡋⠄⡐⢎⠔⡈⣰⢏⡞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⠳⠄⠀⢨⠙⢧⢈⡥⣉⠶⢡⠾⣳⠂⠤⣀⠰⣿⣿⡃⠀⠀⠀⠀⠀⢀⣏⣐⡰⢈⠧⣘⠶⣫⢏⡅⢺⡜⡱⢢⡙⣜⢣⡖⢮⡵⣚⠜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⠞⣄⢓⡠⢒⠩⢒⡐⢆⡙⢢⣬⢃⠜⣻⣅⠀⠀⠀⠀⢈⠧⢌⠳⣝⢯⡳⣭⠳⢋⡞⣼⢲⣥⠓⡅⠘⡜⡳⢎⡳⣜⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠉⠀⠐⠤⣈⠂⠅⡌⠢⢜⣿⢻⡇⠢⠌⠃⠀⠀⠀⠠⣍⠲⢌⡱⢌⠳⣝⢮⣗⣣⣝⢦⡻⣜⢯⡍⣄⡰⣜⢫⠶⡍⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠠⠀⠀⠀⢆⠀⠀⠑⠈⠾⣯⢇⠢⢉⠒⡀⠀⢀⡜⢢⠓⡬⡐⢎⡱⡈⠶⢎⡷⣚⢮⡳⡽⢮⣝⡮⡕⣮⠟⡽⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⡀⢒⡈⢯⡓⣌⡰⢠⣀⠞⡼⡁⢞⠠⣍⠒⡔⡩⠜⣮⠱⣏⢾⡱⡏⠐⢽⡀⠟⡼⠈⢳⠇⠁⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢕⠪⣉⠞⣡⠆⡱⢣⢎⡝⢢⢍⠢⢓⡄⢫⠰⢡⡛⡴⣋⡞⣧⣛⡽⡀⠀⠁⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣑⢊⡱⢂⠞⣠⠹⣔⠣⣊⠔⠣⣌⢃⠣⡌⠥⣃⠣⢜⡱⣝⡺⡵⢎⡷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢮⠘⡤⢃⠶⣉⠦⢡⣛⡤⠓⣤⢋⡱⣐⢪⡑⢬⢑⠢⢍⡒⠴⢩⡳⣝⣫⢼⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠹⣔⢫⠓⣌⠲⡑⢮⡜⡳⢆⡳⠴⣌⢧⡙⣆⢎⡕⡊⢴⣉⡶⣝⢮⣓⣏⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣒⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠘⢧⢫⣄⠓⣜⢣⣞⡱⣏⡜⡳⡜⢦⡛⡜⣎⠶⣙⡶⣹⢮⡝⣮⣓⢮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢸⢣⢎⡽⣌⢳⡼⡱⣎⠽⡱⢭⢣⡝⡼⢾⡹⣇⢯⡳⢮⡝⣶⡹⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠡⡍⡘⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣱⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠁⠠⡀⠄⡀⠂⠈⢀⠰⠁⠂⠑⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠄⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣈⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠵⡈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡐⡀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠣⣟⣯⢿⡳⡄⠀⣍⢞⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⡿⠁⠂⠜⣯⡿⢁⠌⠐⠠⢈⠳⠃⠐⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣯⢷⣯⣷⠛⠠⠤⠀⠚⣿⠁⠤⠀⠈⠴⡀⠂⠧⡀⠀⡠⠞⣬⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣿⣽⣾⣿⡾⣽⢯⣟⡿⣞⡿⠚⠀⣀⣠⣤⣤⣄⢃⠜⢦⡁⠜⠀⢓⠦⣡⢉⢁⠆⠠⠁⢋⡁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠸⣿⢿⣻⣿⣾⣟⣯⡷⣟⣯⣟⣾⣽⡿⠀⠈⠉⠉⠉⠈⠀⠀⠀⠂⠈⢡⠃⠀⣈⠀⠃⠎⡸⢠⡑⢈⠠⠱⠠⢜⡱⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⢯⡿⣾⡽⣞⣷⣻⡽⣷⣻⢾⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠘⠛⠷⣦⣄⠀⠁⠚⣄⠂⠄⠡⢸⠁⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⡯⣯⣽⢳⣽⣻⣞⣷⣻⣿⡽⣿⣟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢒⠠⠈⠄⡏⠀⡄⠀⠊⢠⢖⡲⢆⡀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠐⢻⣽⣳⣯⣟⣾⣳⣟⣾⣳⢿⣿⣻⣞⡿⡆⠄⠀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡰⠀⠡⠘⡒⠐⠊⢮⢱⢫⠔⠈⠑⢨⠁⠄
⠀⠀⠀⠀⠀⣾⣳⣿⣷⣻⣞⡷⣿⢾⣽⣻⢻⣷⣫⢿⣽⣈⣖⣤⣀⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⡐⢀⠠⠁⡒⠁⠌⡐⠂⠘⢧⡄⣴⠉⡅⠊⠀
⠀⠀⠀⠀⠀⢹⣷⣿⣿⣳⣯⢿⡽⢿⣞⣿⡩⢷⣿⣟⡯⢹⡿⠻⣿⣿⢿⡿⠑⠐⠤⠀⡄⢀⠀⢠⠀⢆⡑⠎⢀⠁⢂⠐⡀⢢⡙⠴⣡⠖⠀⡌⠌⠠⣙⠶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣧⠻⠟⢯⣿⣟⡨⠟⢯⣷⠨⢥⢩⠓⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⠁⢊⠄⣃⠢⡜⣀⠢⠘⣀⣂⠰⢧⡙⢧⠁⣠⠚⡔⠊⢡⡏⣞⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠄⠀⢰⠙⣣⠌⡥⠬⡓⠣⠮⢷⡂⠄⣄⢠⣿⣿⡏⠀⠀⠀⠀⠀⢀⢧⣈⣄⠂⡗⣌⢧⣫⢇⡇⢸⢣⡙⢆⡚⢤⡏⣴⢚⡧⣙⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣃⡒⠩⠔⡱⢈⠱⢉⢆⡉⢲⣰⢣⠈⢻⣇⠀⠀⠀⠀⢈⠫⢌⠳⣎⢿⡱⣞⡰⢋⡞⡼⣎⢧⡜⢆⠘⠼⣱⢣⢏⠶⡉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠈⠒⠂⠆⢄⠡⢊⠰⢈⢴⣻⢟⡧⢘⡈⠃⠁⠀⠀⢠⢣⡉⢆⠣⣌⠳⣝⢮⣳⢣⣝⡲⣹⢮⡽⣩⢄⣢⠴⣋⣮⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐⠀⠠⠀⠀⠈⢂⠀⠀⠁⠈⠷⣯⢷⠠⠌⡡⠀⠀⠀⣔⠣⡜⢌⡱⢄⠣⠜⣨⢧⡻⣼⣱⣛⡮⢷⡹⣎⢖⣫⠽⣎⡷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⡀⠄⣒⡈⣞⡓⣤⠑⣈⣀⠻⣜⠢⣑⠪⢔⠊⡥⢋⠴⣌⢳⢳⣜⣣⢭⠐⢻⡌⢮⡱⡀⢫⡆⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡵⢊⡅⠦⡙⢆⠦⢩⡑⣎⠳⢌⡑⠦⣉⠦⡙⠤⢋⡴⣍⢎⡷⢎⡳⣏⠆⠀⠉⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠘⡌⢂⠀⠀⠀⠀⠀⠀⠀⠀⡠⢍⢢⠱⣈⠖⣑⠪⢼⢡⠚⢤⡙⢢⡑⠎⡔⢢⡙⠬⣁⠞⡸⣎⢷⢫⠷⣹⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⢆⡳⣌⠖⣡⢚⣥⠣⢍⡒⢬⠡⡜⣸⡈⢥⡘⢆⡱⢌⡑⢮⣫⡝⣾⡱⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠙⣦⠳⢌⡒⣡⢚⡴⣋⠶⣩⢖⡱⡜⣥⢋⡴⣘⢆⠲⢌⡲⣝⢶⡹⢶⣹⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⣝⠲⣌⡔⣎⠶⣭⠳⣕⢮⡱⣙⢦⣋⠶⣩⢎⡝⣮⢳⡝⣮⡝⣳⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡉⡆⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⣙⢦⡹⣌⢳⣭⢳⡙⢮⡱⣙⢦⡙⣞⡷⢫⡞⣵⢫⡞⣵⢺⡵⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠡⢎⠰⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⣡⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠟⡁⠠⡀⢄⡀⠂⠈⠀⠔⠁⠂⠑⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢡⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣂⣥⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣳⢆⠙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⠤⢀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠱⣿⣽⣻⡞⡀⠀⣭⢚⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣠⣴⣮⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⡟⠁⠂⢼⣳⠿⢁⠁⡉⢀⠉⠞⠁⠈⠔⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⢷⣻⢷⣯⡿⢉⠠⠄⠐⠚⡿⠁⠆⡀⠐⠬⢀⠘⢆⡀⠀⡰⢫⡝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣯⣿⣾⣟⣿⣳⣟⡾⣯⣟⠿⠊⢀⣀⣠⣤⣤⣀⠣⠜⣥⠐⡈⠈⠳⢤⡊⢌⠑⡄⠐⡈⠱⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⢿⣿⣻⣿⣽⡿⣽⣟⡾⣟⡾⣽⣷⡯⠀⠈⠉⠉⠉⠁⠀⠀⠐⢀⠨⢡⠃⠀⡈⠀⠙⠢⣍⠰⡐⠠⢁⠣⠄⢞⡭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⢾⣽⣞⣯⣟⡷⣯⢿⣽⣻⣽⡿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠁⠙⠻⠷⣦⣀⠀⠁⠇⡜⠀⡐⠈⡼⢘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡟⣽⢾⣹⢮⣽⣻⣞⣯⣷⣯⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠈⡜⠀⠄⠡⠎⠰⡀⠀⠃⣠⢖⡲⣄⠀⠀⢀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡿⣽⡷⣯⣟⡾⣷⣻⣞⣿⣷⢯⡿⣾⡅⡀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⢀⡙⠀⢂⠉⡒⢀⠓⠮⣱⠮⠑⠈⠂⢌⢂⠀
⠀⠀⠀⠀⠀⣾⣽⣻⣟⡷⣯⣟⣷⣻⢾⣽⢾⣯⣟⡷⣷⣨⣔⣤⣀⣀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡀⢂⠀⠄⢂⡜⢀⠂⡐⠂⡑⢧⡄⡴⢉⠆⠀⠀
⠀⠀⠀⠀⠀⢻⣞⣿⣿⣟⣷⢯⡿⢯⣿⣾⠚⣷⣿⣻⡏⢽⡿⠻⣿⣿⢿⡿⠑⠀⠤⠀⡄⢀⠀⢠⠠⡘⠤⠃⡀⢂⠐⠠⠈⡔⢣⢒⡤⠒⢀⡡⠄⠂⣝⢲⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣿⣧⠻⠝⢯⣿⣟⡨⠟⠞⠯⡰⢡⢋⠃⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⢁⠊⠤⡁⢆⢣⡐⠤⠂⢌⡠⢱⡍⢧⡋⠄⡐⢎⠔⡈⣰⢏⡞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⠳⠄⢀⡈⠬⠴⢉⠰⡁⠜⡣⠽⣳⠂⠤⣀⠰⣿⣿⡃⠀⠀⠀⠀⠀⢀⣏⣐⡰⢈⠧⣘⠶⣫⢏⡅⢺⡜⡱⢢⡙⣜⢣⡖⢮⡵⣚⠜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣨⢑⠢⠌⢒⠈⡔⠑⡆⠰⣉⢲⣌⢇⡘⢿⣅⠀⠀⠀⠀⡘⡩⢄⢫⡝⣯⢖⣫⠳⣙⡎⣞⠶⣥⠓⡅⠘⡜⡳⢎⡳⣜⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠈⠀⠈⠑⠈⡆⢡⠘⡐⢬⡑⣤⠿⣟⡦⠐⡌⠃⠀⠀⠀⠰⢡⡑⢎⢢⠙⡼⣹⢲⣙⡦⡽⣌⡿⣜⢯⡍⣄⡰⣜⢫⠶⡍⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⢈⠢⡕⢩⠒⢬⠰⣉⢯⢧⠘⡠⠑⡀⢀⠴⣉⠖⡌⢆⢣⠩⢔⠢⣛⢼⠳⣝⢮⡵⣫⢞⡽⣎⠵⣎⠿⣹⢇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡚⢌⠥⡙⢢⡑⢆⡊⣯⡓⣄⡣⣜⠣⡓⢌⠲⡘⡌⠦⡙⣂⢳⡜⣎⢻⡜⣧⢏⡧⠘⢵⡈⠷⣹⠀⢻⠆⠁⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⡄⠀⠀⠀⠀⠀⠀⠈⡕⢊⠦⡙⡰⡘⢢⠅⣧⡙⢤⡑⢢⠱⣉⠜⢢⠱⣘⢢⠱⣌⠷⣸⢌⡳⣺⠵⣎⢷⡀⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⣐⠂⠀⠀⠀⠀⠀⠀⠈⠕⣢⠱⢡⡑⢣⡘⣧⠍⡆⡜⢡⠣⢌⠎⣡⢃⠖⡌⠲⣍⠷⡱⢎⡵⣣⢟⡼⣣⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⡤⢃⠣⣌⠱⣌⢷⢫⡔⣩⢆⡝⢮⡜⠰⡌⣒⠩⡑⢬⢳⡙⣎⢾⡱⣏⢾⡱⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣅⢣⢂⠧⣘⢮⣳⠹⣆⠯⣜⢣⣌⠳⡴⢡⠣⣍⠾⣡⢛⡼⣣⢟⡼⣣⢟⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣒⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢏⠶⣡⠞⣥⡛⡴⢫⡜⡳⢬⠳⣬⡳⣜⣣⠟⣬⠳⣥⢏⡾⣱⢏⡾⣱⢏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠰⠁⠘⣧⢣⡛⡴⡹⣜⢣⢞⡱⣋⢷⣣⢟⡼⣣⢟⣦⢻⡜⣯⡜⣧⢻⡜⣧⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠡⡍⡘⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣱⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠁⠠⡀⠄⡀⠂⠈⢀⠰⠁⠂⠑⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠄⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣈⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠵⡈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡐⡀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠣⣟⣯⢿⡳⡄⠀⣍⢞⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⡿⠁⠂⠜⣯⡿⢁⠌⠐⠠⢈⠳⠃⠐⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣯⢷⣯⣷⠛⠠⠤⠀⠚⣿⠁⠤⠀⠈⠴⡀⠂⠧⡀⠀⡠⠞⣬⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣿⣽⣾⣿⡾⣽⢯⣟⡿⣞⡿⠚⠀⣀⣠⣤⣤⣄⢃⠜⢦⡁⠜⠀⢓⠦⣡⢉⢁⠆⠠⠁⢋⡁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠸⣿⢿⣻⣿⣾⣟⣯⡷⣟⣯⣟⣾⣽⡿⠀⠈⠉⠉⠉⠈⠀⠀⠀⠂⠈⢡⠃⠀⣈⠀⠃⠎⡸⢠⡑⢈⠠⠱⠠⢜⡱⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⢯⡿⣾⡽⣞⣷⣻⡽⣷⣻⢾⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠘⠛⠷⣦⣄⠀⠁⠚⣄⠂⠄⠡⢸⠁⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⡯⣯⣽⢳⣽⣻⣞⣷⣻⣿⡽⣿⣟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢒⠠⠈⠄⡏⠀⡄⠀⠊⢠⢖⡲⢆⡀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠐⢻⣽⣳⣯⣟⣾⣳⣟⣾⣳⢿⣿⣻⣞⡿⡆⠄⠀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡰⠀⠡⠘⡒⠐⠊⢮⢱⢫⠔⠈⠑⢨⠁⠄
⠀⠀⠀⠀⠀⣾⣳⣿⣷⣻⣞⡷⣿⢾⣽⣻⢻⣷⣫⢿⣽⣈⣖⣤⣀⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⡐⢀⠠⠁⡒⠁⠌⡐⠂⠘⢧⡄⣴⠉⡅⠊⠀
⠀⠀⠀⠀⠀⢹⣷⣿⣿⣳⣯⢿⠽⣿⣞⣿⡩⢷⣿⣟⡯⢹⡿⠻⣿⣿⢿⡿⠑⠐⠤⠀⡄⢀⠀⢠⠀⢆⡑⠎⢀⠁⢂⠐⡀⢢⡙⠴⣡⠖⠀⡌⠌⠠⣙⠶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣧⠻⢋⠅⣂⠒⠠⠛⢯⣷⠨⢥⢩⠓⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⠁⢊⠄⣃⠢⡜⣀⠢⠘⣀⣂⠰⢧⡙⢧⠁⣠⠚⡔⠊⢡⡏⣞⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠫⢐⠡⠌⢄⢊⡑⠌⠢⢌⠛⠤⢫⢷⡂⠄⣄⢠⣿⣿⡏⠀⠀⠀⠀⠀⢀⢧⣈⡄⢂⡗⣌⢧⣫⢇⡇⢸⢣⡙⢆⡚⢤⡏⣴⢚⡧⣙⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠌⡌⢂⠆⠤⢉⠒⣨⠱⣈⠆⢢⢉⢲⣰⢣⠈⢻⣇⠀⠀⠀⠀⠌⡍⠦⡙⢞⡳⡞⡼⡰⢋⡞⡼⣎⢧⡜⢆⠘⠼⣱⢣⢏⠶⡉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠌⠴⣈⠖⣩⢂⠓⡌⢎⡱⣌⡾⣟⡧⠘⡄⠋⠀⠀⡠⢌⡓⠬⡑⡜⢌⠳⣝⢶⣙⡶⣹⢲⡹⢮⡽⣩⢄⣢⠴⣋⣮⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠐⡎⢜⡰⢌⢓⡘⢢⡑⡌⠻⣵⢧⡡⡜⡴⢢⢛⠰⣡⠊⡕⡸⢄⡋⡔⢫⢎⡳⣭⢳⡳⣝⡧⢟⡵⡞⢦⡛⠽⣜⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢋⡔⢢⠎⢢⠍⡒⡌⠆⠀⠈⡇⠲⡑⠬⣑⢊⡱⠤⣙⠰⡡⢆⡱⢎⢧⢫⡜⡱⢯⡵⣋⢮⠘⢞⡉⢶⡹⡀⢛⡆⠉⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⠄⠀⠀⠀⠀⠈⢆⠣⡜⢡⠎⡱⡘⠼⠀⠀⡜⡡⢍⡒⡡⢎⠰⣑⠢⢃⡕⢮⣙⠞⣬⠳⣜⡹⢮⡵⣫⢗⡆⠀⠉⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠘⡌⢂⠀⠀⠀⠀⠨⡱⢌⠣⣌⠱⢌⢓⡀⠀⢧⡑⢎⡴⢣⡌⣃⠦⡙⢢⢌⠳⣎⡝⢦⡛⡴⣩⢗⡾⣱⢏⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠀⠃⠀⠀⠀⠀⠀⢳⠌⡲⢨⠜⣈⠮⣅⣠⢧⡙⢮⡜⣣⢒⡥⠲⣉⠖⣨⢻⠴⣙⠦⣝⢲⡝⡾⡜⣧⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡁⠀⠀⠀⠀⠀⠀⠈⡷⣡⠆⡍⢦⡛⡜⢦⢣⡝⢦⣙⠦⣍⠶⡱⢆⡞⣥⡛⡼⣡⡛⣬⠳⣝⡞⡽⣜⢧⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⢉⠀⠀⠀⠀⠀⠀⠀⢷⡱⢎⡝⣲⠹⣜⢣⡳⣜⢣⢎⡽⣎⢷⣛⠾⣜⢧⡝⡲⢥⡻⣴⠻⣜⠾⣱⢏⡾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡉⡆⠀⠀⠀⠀⠀⠀⢸⡱⢫⡜⣥⢛⡬⢣⢳⡌⣧⢻⡼⡹⠶⣭⢻⡜⣧⢻⡱⢯⡵⣎⠿⣜⡏⣗⣫⢞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠡⡍⡘⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣱⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠁⠠⡀⠄⡀⠂⠈⢀⠰⠁⠂⠑⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠄⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣈⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠵⡈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡐⡀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠣⣟⣯⢿⡳⡄⠀⣍⢞⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⡿⠁⠂⠜⣯⡿⢁⠌⠐⠠⢈⠳⠃⠐⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣯⢷⣯⣷⠛⠠⠤⠀⠚⣿⠁⠤⠀⠈⠴⡀⠂⠧⡀⠀⡠⠞⣬⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣿⣽⣾⣿⡾⣽⢯⣟⡿⣞⡿⠚⠀⣀⣠⣤⣤⣄⢃⠜⢦⡁⠜⠀⢓⠦⣡⢉⢁⠆⠠⠁⢋⡁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠸⣿⢿⣻⣿⣾⣟⣯⡷⣟⣯⣟⣾⣽⡿⠀⠈⠉⠉⠉⠈⠀⠀⠀⠂⠈⢡⠃⠀⣈⠀⠃⠎⡸⢠⡑⢈⠠⠱⠠⢜⡱⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⢯⡿⣾⡽⣞⣷⣻⡽⣷⣻⢾⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠘⠛⠷⣦⣄⠀⠁⠚⣄⠂⠄⠡⢸⠁⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⡯⣯⣽⢳⣽⣻⣞⣷⣻⣿⡽⣿⣟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢒⠠⠈⠄⡏⠀⡄⠀⠊⢠⢖⡲⢆⡀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠐⢻⣽⣳⣯⣟⣾⣳⣟⣾⣳⢿⣿⣻⣞⡿⡆⠄⠀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡰⠀⠡⠘⡒⠐⠊⢮⢱⢫⠔⠈⠑⢨⠁⠄
⠀⠀⠀⠀⠀⣾⣳⣿⣷⣻⣞⡷⣿⢾⣽⣻⢻⣷⣫⢿⣽⣈⣖⣤⣀⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⡐⢀⠠⠁⡒⠁⠌⡐⠂⠘⢧⡄⣴⠉⡅⠊⠀
⠀⠀⠀⠀⠀⢹⣷⣿⠿⠓⠯⢿⡽⢿⣞⣿⡩⢷⣿⣟⡯⢹⡿⠻⣿⣿⢿⡿⠑⠐⠤⠀⡄⢀⠀⢠⠀⢆⡑⠎⢀⠁⢂⠐⡀⢢⡙⠴⣡⠖⠀⡌⠌⠠⣙⠶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠏⢄⠢⠑⠢⠌⡙⢨⠟⢯⣷⠨⢥⢩⠓⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⠁⢊⠄⣃⠢⡜⣀⠢⠘⣀⣂⠰⢧⡙⢧⠁⣠⠚⡔⠊⢡⡏⣞⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠅⣊⠰⠉⡔⠡⣘⡠⢏⠢⡌⢟⡠⠯⣗⠆⡠⢄⢠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣧⣈⠄⡂⡗⣌⢧⣫⢇⡇⢸⢣⡙⢆⡚⢤⡏⣴⢚⡧⣙⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠄⢢⢑⣠⡓⡡⠜⣂⢓⡸⢄⢳⠐⡌⣉⢲⣸⢆⠈⢿⣇⠀⠀⢀⠠⣘⠡⢆⠜⡛⣦⢳⢺⡰⢋⡞⡼⣎⢧⡜⢆⠘⠼⣱⢣⢏⠶⡉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⢰⢣⣅⠳⣈⠖⡰⢊⠦⣑⠰⣠⣟⡿⠮⡜⢤⠫⡔⢎⠣⡑⢆⡍⠲⡌⡱⢌⡳⢧⣙⡶⣹⢲⡹⢮⡽⣩⢄⣢⠴⣋⣮⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠘⡧⢎⡕⣂⠎⡱⢌⠲⡅⢲⢉⠺⡝⢢⡑⣊⠱⡘⡌⢥⡉⢦⢘⡱⠰⡑⢬⠳⣍⢾⡱⣏⢷⡹⣧⠟⡵⡞⢦⡛⠽⣜⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢳⠒⡌⡜⢡⠎⡱⡘⠀⠀⠀⣇⠣⡔⣡⠣⢱⢘⠢⠜⠢⣅⠲⣡⣙⢎⡳⡜⣎⢳⡝⣮⢳⡭⠘⠽⡌⢣⡽⡀⢛⡆⠉⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⠄⠀⠀⠀⢘⠣⡜⣐⠣⡌⢥⢙⠀⠀⠀⣇⠲⡡⢆⡱⢃⠎⣡⢋⡱⢄⡳⣥⢋⢮⡱⡹⣌⢧⢻⡜⣧⢻⡄⠀⠉⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠘⡌⢂⠀⠀⠀⣗⠰⣡⠚⡌⠲⣌⡀⢀⠴⣩⢳⡡⢏⣜⠢⡍⡔⢢⡑⢮⡱⢎⡝⢦⢳⡱⢎⡎⣷⡹⣎⢷⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠀⠃⠀⠀⠀⢸⡅⠦⠱⡌⢣⠔⡣⠜⡸⢥⡣⡝⢮⡔⣣⢒⣉⢦⡙⡶⣩⢓⢮⡙⡦⣝⢺⡼⣱⢳⡝⣮⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡁⠀⠀⠀⠀⠘⡜⣎⡱⣌⠧⣎⠵⣫⢕⣣⢳⡙⣦⡝⣦⣫⣜⡦⢟⡱⢣⢏⠶⣙⠶⣩⢎⠷⣭⢳⡝⣶⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⢉⠀⠀⠀⠀⠀⢛⡴⢳⣌⠳⣬⢳⡱⢎⡖⡣⣝⢶⡹⣖⢣⠞⣜⢣⡝⣣⢞⣱⠹⣮⡱⣎⠿⣜⢧⣛⢶⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡉⡆⠀⠀⠀⠀⢸⢎⡳⣌⠻⣔⢣⡝⠮⠈⠁⠘⣧⢻⣬⢣⡛⣬⢓⡾⣱⢏⡶⣛⢶⡹⣎⠿⣜⢧⣛⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠡⢎⠰⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⣡⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠟⡁⠠⡀⢄⡀⠂⠈⠀⠔⠁⠂⠑⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢡⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣂⣥⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣳⢆⠙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⠤⢀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠱⣿⣽⣻⡞⡀⠀⣭⢚⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣠⣴⣮⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⡟⠁⠂⢼⣳⠿⢁⠁⡉⢀⠉⠞⠁⠈⠔⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⢷⣻⢷⣯⡿⢉⠠⠄⠐⠚⡿⠁⠆⡀⠐⠬⢀⠘⢆⡀⠀⡰⢫⡝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣯⣿⣾⣟⣿⣳⣟⡾⣯⣟⠿⠊⢀⣀⣠⣤⣤⣀⠣⠜⣥⠐⡈⠈⠳⢤⡊⢌⠑⡄⠐⡈⠱⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⢿⣿⣻⣿⣽⡿⣽⣟⡾⣟⡾⣽⣷⡯⠀⠈⠉⠉⠉⠁⠀⠀⠐⢀⠨⢡⠃⠀⡈⠀⠙⠢⣍⠰⡐⠠⢁⠣⠄⢞⡭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⢾⣽⣞⣯⣟⡷⣯⢿⣽⣻⣽⡿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠁⠙⠻⠷⣦⣀⠀⠁⠇⡜⠀⡐⠈⡼⢘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡟⣽⢾⣹⢮⣽⣻⣞⣯⣷⣯⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠈⡜⠀⠄⠡⠎⠰⡀⠀⠃⣠⢖⡲⣄⠀⠀⢀⢀⠠⠁
⠀⠀⠀⠀⠐⢻⡿⣽⡷⣯⣟⡾⣷⣻⣞⣿⣷⢯⡿⣾⡅⡀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⢀⡙⠀⢂⠉⡒⢀⠓⠮⣱⠮⠑⠈⠂⢌⢂⠀
⠀⠀⠀⠀⠀⣾⣽⣻⣟⡷⣯⣟⣷⣻⢾⣽⢾⣯⣟⡷⣷⣨⣔⣤⣀⣀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡀⢂⠀⠄⢂⡜⢀⠂⡐⠂⡑⢧⡄⡴⢉⠆⠀⠀
⠀⠀⠀⠀⠀⢻⡞⠛⡙⠻⣷⣯⡿⢯⣿⣾⠚⣷⣿⣻⡏⢽⡿⠻⣿⣿⢿⡿⠑⠀⠤⠀⡄⢀⠀⢠⠠⡘⠤⠃⡀⢂⠐⠠⠈⡔⢣⢒⡤⠒⢀⡡⠄⠂⣝⢲⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡖⢯⡹⡹⣌⡳⣟⡸⣙⠾⣏⠴⣡⢋⠃⠀⠑⠀⠱⣻⣿⡍⠀⠀⠀⠀⠀⢁⠊⠤⡁⢆⢣⡐⠤⠂⢌⡠⢱⡍⢧⡋⠄⡐⢎⠔⡈⣰⢏⡞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡞⣜⢣⢳⡱⢣⠽⡙⢣⠌⡌⡉⠖⡡⠾⣳⠂⠤⢀⢰⣿⣿⡃⠀⠀⠀⠀⠀⢀⣏⡐⠰⡈⢧⡘⡶⣫⢏⡅⢺⡜⡱⢢⡙⣜⢣⡖⢮⡵⣚⠜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣌⢧⢣⢍⡃⢖⡱⠢⠜⣠⡑⢌⠰⡁⢆⡙⢢⠝⡠⢆⡩⢥⣀⠤⣄⠒⣌⠣⡌⢍⠳⣌⢧⡹⢲⢋⡞⣼⢲⣥⠓⡅⠘⡜⡳⢎⡳⣜⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⢎⡃⢖⠸⠤⣙⡀⠃⠔⡨⠘⢤⡑⢢⢸⠡⢎⡑⠦⣑⠢⠜⡒⠤⢋⠤⠓⡌⢎⡱⣈⠞⣵⣓⢮⡵⣊⡷⣎⢯⡍⣄⡰⣜⢫⠶⡍⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠹⡜⣌⠲⡑⢢⡑⢢⡀⠀⢉⠦⠜⣒⠣⣍⢒⡸⠰⢡⠎⡱⢌⠣⣍⠸⡑⡜⢂⠦⣱⢋⠶⣹⢎⡷⣭⢳⡭⢷⣹⡎⡵⣎⠿⣹⢇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⢳⡜⢦⡙⡤⢃⠵⢨⡙⡌⢲⢉⠤⠓⡴⢫⠴⣉⠖⠬⣑⢊⡱⢄⣓⡸⣰⢩⠖⣥⢛⡼⣡⢟⡼⢎⡷⣩⠐⢳⡈⢗⡹⠀⢻⠆⠁⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠔⡀⢛⢦⡙⡔⣉⢆⠣⡔⢩⠆⡩⠜⣩⠞⣥⢋⡔⢪⢑⢢⡑⣎⢳⣬⢳⢥⣋⠞⣬⠳⣜⡱⢮⣝⣫⣜⣳⡀⠀⠁⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠆⡱⢈⢣⡓⡌⢆⡊⡕⠬⡡⢎⣡⢛⡴⣋⢦⠣⠜⢢⡉⢦⡹⣜⢣⢞⡣⢞⡬⣛⠴⣋⠶⣙⢦⣳⢓⡮⣵⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡔⠀⠃⠀⠻⡜⢦⢳⣜⡣⠝⠞⡒⡙⢌⡉⠮⢹⣙⢦⣙⢦⠳⣍⠞⣬⠳⣍⠶⣩⠞⣥⢫⡝⡶⢭⣛⡼⣱⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢊⠅⠀⠀⠀⠈⠉⠀⠀⠈⢈⠆⠁⠈⠀⠈⠀⢹⢎⡖⣫⢎⡳⢬⡛⡴⢫⡜⣣⢇⡻⣔⢣⠞⣝⢧⣏⢾⡱⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⣘⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠘⣮⢳⡱⢎⡵⢣⡝⣜⢣⡞⡵⢪⠵⣎⢧⣛⢮⡳⣎⢷⡹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸⢧⡻⣜⡜⣣⠞⣬⢳⣝⡳⣭⢞⡭⣞⢭⡞⣵⢫⡞⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⣟⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠡⡍⡘⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣱⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠠⡀⠤⡀⢄⠂⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠁⠠⡀⠄⡀⠂⠈⢀⠰⠁⠂⠑⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠄⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣈⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠵⡈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡐⡀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠣⣟⣯⢿⡳⡄⠀⣍⢞⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⡿⠁⠂⠜⣯⡿⢁⠌⠐⠠⢈⠳⠃⠐⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣯⢷⣯⣷⠛⠠⠤⠀⠚⣿⠁⠤⠀⠈⠴⡀⠂⠧⡀⠀⡠⠞⣬⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀
⠙⠛⠛⠛⠋⢹⣿⢿⣟⣿⣿⣽⣾⣿⡾⣽⢯⣟⡿⣞⡿⠚⠀⣀⣠⣤⣤⣄⢃⠜⢦⡁⠜⠀⢓⠦⣡⢉⢁⠆⠠⠁⢋⡁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠸⣿⢿⣻⣿⣾⣟⣯⡷⣟⣯⣟⣾⣽⡿⠀⠈⠉⠉⠉⠈⠀⠀⠀⠂⠈⢡⠃⠀⣈⠀⠃⠎⡸⢠⡑⢈⠠⠱⠠⢜⡱⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⢯⡿⣾⡽⣞⣷⣻⡽⣷⣻⢾⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠁⠀⠘⠛⠷⣦⣄⠀⠁⠚⣄⠂⠄⠡⢸⠁⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⡯⣯⣽⢳⣽⣻⣞⣷⣻⣿⡽⣿⣟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⢒⠠⠈⠄⡏⠀⡄⠀⠊⢠⢖⡲⢆⡀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠐⢻⣽⣳⣯⣟⣾⣳⣟⣾⣳⢿⣿⣻⣞⡿⡆⠄⠀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡰⠀⠡⠘⡒⠐⠊⢮⢱⢫⠔⠈⠑⢨⠁⠄
⠀⠀⠀⠀⠀⣾⣳⣿⣷⣻⣞⡷⣿⢾⣽⣻⢻⣷⣫⢿⣽⣈⣖⣤⣀⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⡐⢀⠠⠁⡒⠁⠌⡐⠂⠘⢧⡄⣴⠉⡅⠊⠀
⠀⠀⠀⠀⠀⢹⣷⣿⣟⡳⣏⣿⡽⢿⣞⣿⡩⢷⣿⣟⡯⢹⡿⠻⣿⣿⢿⡿⠑⠐⠤⠀⡄⢀⠀⢠⠀⢆⡑⠎⢀⠁⢂⠐⡀⢢⡙⠴⣡⠖⠀⡌⠌⠠⣙⠶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢿⡳⣜⢧⡝⣮⣟⡨⠟⢯⣷⠨⢥⢩⠓⠀⠑⠀⠱⢻⣿⡍⠀⠀⠀⠀⠀⠁⢊⠄⣃⠢⡜⣀⠢⠘⣀⣂⠰⢧⡙⢧⠁⣠⠚⡔⠊⢡⡏⣞⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⢳⡝⣮⡝⣶⣹⠣⡌⠡⢌⠓⠣⠮⢷⣂⠄⡎⡱⢢⢌⡛⠀⠀⠀⠀⠀⢀⢧⡈⢄⠂⡗⣌⢧⣫⢇⡇⢸⢣⡙⢆⡚⢤⡏⣴⢚⡧⣙⠖⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢪⠷⣙⠦⡙⢤⠋⠷⡘⣓⠚⡚⢣⠜⢲⢡⢚⠰⡑⠆⢦⢉⢏⡱⢩⡘⢱⢊⠤⡙⢢⢅⣳⢺⡰⢋⡞⡼⣎⢧⡜⢆⠘⠼⣱⢣⢏⠶⡉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣙⢮⡱⢊⡱⢢⢙⢢⡑⢆⡙⠬⡡⠎⢥⢃⣎⠱⠌⣍⠒⡌⢦⢘⠢⢅⢣⠊⢖⡉⠖⡌⡖⢧⣹⢲⣹⢲⡹⢮⡽⣩⢄⣢⠴⣋⣮⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣆⠳⣅⠲⢡⢎⡢⠜⢢⡉⠖⣡⠙⣢⠏⡜⢣⢍⡰⢩⡘⠔⣊⠱⢊⢆⡙⢢⠜⣡⢞⡹⢎⡳⣏⢾⣱⣛⡧⢟⡵⡞⢦⡛⠽⣜⡷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠙⠊⠽⢣⣎⡵⣛⢦⡱⣍⢤⡛⡴⢫⡌⠥⡒⡌⠥⣌⠳⣌⠥⡋⠴⣈⢧⡚⡵⢪⡕⣫⠖⡽⣎⡳⢮⡭⠘⢞⡉⢶⡹⡀⢛⡆⠉⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠔⡂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠚⢧⡚⣕⢢⡍⢶⣘⢳⡬⡳⣭⠳⡜⢦⡹⣜⢣⠞⣥⢛⡼⣱⢏⡷⣹⠂⠀⠉⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠐⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⣳⡜⢧⡚⢧⢳⡱⢭⣛⠼⣣⠳⣌⢧⡛⣴⢋⠶⣹⢎⡷⢭⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⢃⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡜⣣⠝⣎⢧⡙⣎⢳⡙⢦⡛⡜⢦⣙⢮⣛⢾⡱⣏⡞⣧⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠸⣎⠵⣫⠜⡦⡝⣬⢣⢏⢧⡹⣜⡣⢞⢦⡹⣎⢷⡹⣜⡳⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⣏⢷⣡⠻⡴⣙⢦⣋⠞⣦⠳⣌⡳⣭⢲⡱⣏⢾⡱⣏⡽⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⣳⢎⡷⡱⣍⠶⣩⢞⡵⣛⢦⡳⣭⢳⡝⣮⢳⡝⣎⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⢰⣼⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⢀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⡈⠆⠘⠠⠑⠈⠀⠁⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣠⣇⣦⣧⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⢯⣷⣟⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠘⠿⠿⠛⠛⢛⣿⣿⡿⣿⣻⣿⣽⣷⡿⣯⢿⣽⣻⢾⡽⠛⣁⣤⡴⣶⣦⡄⢇⡔⢤⠈⡤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢨⡿⣷⣿⡿⣟⣯⣿⢿⡽⣟⡿⣾⣽⣿⠁⠉⠉⠀⠀⠉⠈⠀⠈⠀⠘⠣⠆⠀⠀⠑⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣳⢿⡽⣿⡽⣞⣯⣟⣯⣟⣷⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠓⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡯⣝⣯⣛⡷⣽⣻⢮⣟⣷⣻⣾⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠀⢺⡿⣽⡾⣽⣻⢷⣯⣟⡾⣿⣷⣟⡾⣿⡔⢀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⣻⢷⣿⣳⢯⣟⣾⣽⣻⣽⢿⡾⣽⣳⣿⣀⣆⣠⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⡽⣟⢯⡝⠾⣜⠻⣷⣻⣾⠎⣿⣷⢿⡇⢻⡿⢿⡿⢿⣿⢾⠒⠠⠄⣀⠀⡀⢀⠀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⢈⡟⡜⢦⣙⠳⣬⠳⡌⢥⡉⢷⢪⢙⠫⢓⣀⠱⢢⠜⢢⠩⡏⠀⠀⠀⠀⠈⠰⢂⠩⡐⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⢎⡝⢮⡜⡳⣌⠣⠜⢢⠜⠢⡅⢎⡱⠢⣌⠣⢃⡜⡡⢣⢙⠶⣐⠂⡄⠄⠤⣆⠡⢂⠅⡳⢌⣲⢼⡩⡇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠺⡱⢎⡵⣊⠷⣩⠖⣌⠓⣌⠲⣐⠣⢦⡙⣆⠒⠥⡃⠎⡆⠥⢋⠜⣌⠣⡔⡩⠎⡄⣱⢯⠸⢮⡱⠽⣔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠲⢭⣚⢥⠻⣌⠷⣌⡒⣥⠺⣅⠛⡌⣍⢒⢩⢒⠩⡜⣡⠎⢤⠓⢬⠰⣉⠼⡱⣎⠗⡦⣏⣳⢹⣎⢷⣉⡄⣡⢬⠳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠓⠛⠚⠴⠽⣦⢛⡤⢋⡔⢢⣉⢆⠎⡱⢌⡱⢎⢢⢙⠢⡱⣌⢞⡱⢎⡿⣙⢧⡳⢯⣚⢧⡳⡝⢦⣫⢽⡲⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠹⣎⡜⣣⠜⣦⡙⣎⢳⡱⢬⡔⣊⠦⣉⢞⡱⢎⢮⡱⢫⡜⡽⢮⡝⣧⠏⠈⣷⡙⢧⢺⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⣥⡻⡴⣛⢬⢫⡝⣳⢺⢥⡛⡜⢮⡱⣋⠶⣩⠳⡜⣭⢳⡝⣮⣛⡀⠀⠑⠀⠉⠀⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠔⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⡷⣩⢎⡳⡜⣥⢫⣞⡱⡝⢦⠳⣍⢾⣡⡛⣜⠲⣏⢾⡱⡞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣳⠱⣎⠵⣙⢦⠳⡜⣧⡙⣎⢳⡜⢦⠳⣝⢮⡝⣮⢳⡝⣳⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡌⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢭⡳⣌⠻⡜⣎⢳⡙⢦⡹⣌⠳⣜⢣⡛⣬⠳⣝⢮⡳⣝⣣⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡵⣎⠽⣸⢌⢧⡹⢎⡵⣪⠝⣬⢳⣙⢦⡻⣜⢧⡻⣼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠱⣏⢾⡱⢎⠶⣙⢮⡵⣫⡝⣦⢏⡞⣧⢻⡜⣧⢻⡴⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⡜⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢱⣼⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⠠⢀⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⡠⠌⠑⠂⠘⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠗⣯⣟⣯⡟⡄⠀⣒⡝⡲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⢿⠋⠐⡸⣟⣾⠃⠅⢂⠁⡘⠵⠋⠀⢂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣳⣯⢿⡽⣞⠁⡀⠤⠔⣿⠃⢎⠀⢀⢒⠠⠘⢂⠀⠈⢀⢯⣙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣷⡿⣯⣟⡷⣯⢷⡯⠟⣁⣤⡴⣶⣦⡄⢃⡜⢤⠂⠤⠈⠶⢤⡉⠍⢂⠅⠠⠉⠓⡃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠸⣷⣿⢿⣽⣿⣯⢿⣯⣟⡷⣯⢿⣽⣯⠀⠉⠉⠀⠀⠉⠈⠀⠈⡀⠘⠣⡒⠀⠈⠐⠉⠞⣨⠂⠥⠈⠄⢣⡀⢜⡫⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⡽⣞⡿⣾⢽⣻⣽⣻⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈⠁⠒⠿⢷⣤⡀⠀⠋⠜⣨⠐⢀⠐⢨⠓⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣫⣞⢷⣻⡽⣯⡽⣯⣷⢯⣿⣟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠧⠐⠠⠈⡞⠀⡆⠈⠐⢠⡔⡲⣄⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠠⢾⣻⢷⣯⣟⣷⣻⢷⣻⢷⣿⣿⣻⣞⡿⡔⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⢢⠁⠂⠑⢢⠄⠚⠼⣱⢚⡌⠢⠑⢌⡐⡀
⠀⠀⠀⠀⠀⣼⢯⣿⣷⡻⢾⡽⣿⣽⣻⢾⢻⡷⣯⣟⣿⣈⡔⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠐⠠⠁⡀⢈⡰⠃⠠⠈⠔⡈⢧⣀⣠⠎⡰⠀⠀
⠀⠀⠀⠀⠀⢻⣯⢳⠮⣝⢳⡚⡝⡾⡽⣿⢩⢿⣷⢿⡇⢻⡿⢿⠿⢿⣶⢶⠆⣀⡀⠀⠀⠀⢀⠀⢠⠐⠢⠜⠀⠄⡁⢂⠡⢐⠦⡱⢄⡅⠊⢠⠐⡈⢲⠭⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣆⢏⡞⣬⢣⡝⠬⡑⠴⡉⢇⡋⢤⠛⠧⣄⠫⢔⠪⢔⡩⠏⠀⠀⠀⠈⠁⠃⢢⢉⠰⡈⡱⡀⢌⡐⢀⢂⠄⣏⠶⣙⠂⢠⢌⢣⠘⢀⣏⠞⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡜⢮⡜⢦⠳⣌⢣⡉⠖⣉⠦⣉⠦⣉⠖⡬⡑⢪⠜⢢⠜⣩⢓⠲⡐⡂⢄⠄⣆⠢⢡⠐⡥⡙⣦⢲⣋⡇⢸⢎⢣⠇⣎⠱⣎⡤⢖⣫⢎⡽⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠳⠜⣣⢛⡬⢳⡜⡳⣌⠲⡐⠦⡑⢮⠴⣩⠖⣨⢃⡜⣐⢊⡱⢌⠱⡊⡜⢄⡋⠦⡁⢰⠻⡬⠳⣎⠽⣜⢮⡘⡜⠠⢛⠶⣙⢮⡱⡎⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠓⠚⠵⣪⠵⣎⠳⣍⠶⣩⢎⡱⢃⠭⡐⠦⡘⢤⠃⡖⣌⢣⠒⣉⠦⠱⣡⢙⢮⡳⡱⢧⢏⡞⣹⢮⡽⣘⣀⢂⡮⣙⢦⡓⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠉⠉⠉⠚⢵⡳⢎⡴⢉⠦⡙⡴⣉⠦⢩⠜⡘⢆⡍⡒⣌⢓⡤⢏⠶⡹⣝⢯⢻⡜⣧⢻⣜⢧⣛⡜⢶⢭⡳⣯⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢹⢎⠶⣩⢖⡹⢲⡍⡞⢦⡱⣉⢦⡘⡱⢎⡳⡜⣭⢚⡵⢪⣏⢧⡻⣜⡏⠈⢾⡁⢞⣭⠀⢛⡦⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣞⡱⢮⡓⢧⢫⡝⣣⠟⣼⢢⡝⣱⢋⠶⣙⢦⢫⡜⣣⢞⣣⢟⡼⣆⠀⠀⠁⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⢣⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⣏⢧⠳⡜⣥⢛⡼⣓⢮⡱⣋⢞⡱⣎⡳⣜⡱⢎⣗⣫⢞⣭⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣎⢳⡙⢦⣋⠶⡹⣆⠳⣍⢮⡱⢎⡳⢧⡻⣜⢮⡵⣚⢶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡿⣌⢧⡹⢲⡍⣞⡱⣍⠳⣎⠶⣙⢮⡱⢫⢵⣫⢞⡼⣹⢎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢑⡂⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢱⣏⠶⣩⠳⡜⢦⠳⣜⢳⡬⢳⡍⣶⣩⢳⣎⠷⣎⢷⡹⣞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢾⡹⢶⡹⣜⢣⣛⣬⢳⡝⣧⢞⡴⣫⢳⣎⠿⣜⢧⡻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣟⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⡌⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢱⣼⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡀⠄⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⠰⠠⠑⠈⠐⠁⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠥⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⣯⡟⡄⠀⣚⠼⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⢀⣤⣧⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⢿⠃⠂⠼⣻⢾⠃⠅⢂⠁⡙⢦⠋⠀⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠔⣿⠃⣈⠀⢀⠲⠀⠄⢆⠀⠈⢀⠯⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠚⠿⠿⠛⠛⢻⣿⡿⣿⣟⣿⣯⣿⣷⡿⣯⣟⣷⣻⡽⣯⠟⣁⣤⡴⣶⣦⡄⢇⡔⣤⠈⠤⠈⠧⡴⣈⠍⢂⠅⡈⠙⠢⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣟⣿⢿⣻⣯⣿⢷⡿⣽⣷⣻⢾⣷⡿⠁⠈⠉⠀⠀⠉⠈⠀⠀⡀⠘⢣⠃⠀⠈⠐⠉⠎⡜⠰⡠⢁⠂⠧⣀⠼⣙⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣟⣯⣟⣯⣟⡷⣯⣟⣯⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⠁⠒⠿⢷⣤⡀⠈⠃⠱⣀⠂⡐⢀⢸⠁⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡿⣎⣷⣛⡾⣽⢮⣯⣽⣷⢯⣿⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠧⠐⢀⠂⡭⠐⡀⠈⠂⠤⣔⠲⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠠⢾⣻⡽⣾⣽⣻⣽⣻⣞⡷⣿⣿⢯⣟⡿⡄⠄⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠂⠸⠀⠄⡑⠦⠠⠚⢬⢛⡴⡁⠎⠰⢁⠆⡀
⠀⠀⠀⠀⠀⣼⢷⣻⣷⢯⠷⢯⣷⢯⡿⣽⢻⣟⡾⣽⣻⣠⣁⡄⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠐⠠⢁⠀⢂⡰⠁⢂⠡⢈⠘⣦⢀⣠⠊⠦⠈⠀
⠀⠀⠀⠀⠀⢻⣯⢻⡜⣫⢝⣣⢛⠯⣟⣿⢩⢿⣟⣿⡏⢻⡿⢿⠿⢾⣶⣶⢃⣀⠀⡀⠀⠀⠀⡀⢠⠐⠢⠜⢀⠡⢈⠐⠠⢈⠦⡱⣈⠄⠂⢄⠆⢈⢣⡝⡈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣌⠳⣜⡱⢎⠶⣉⠞⡠⢌⠏⢎⠴⡉⠧⣄⠳⢌⡒⡡⠜⣣⠀⠀⠀⠈⠁⠪⠡⡐⠡⠌⡱⣀⢂⠔⠠⣈⠤⣋⢶⡱⢁⡈⡔⢣⠘⢀⣗⢺⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢎⡝⢦⣙⢎⡳⣌⢒⠱⢊⡜⠌⢦⠑⡣⢜⡘⠢⣅⠣⡙⠴⢣⠒⢆⡂⢄⡀⣇⠌⡑⡈⡗⣰⣢⢞⡭⠇⢸⡱⢊⠖⣢⢱⢎⡤⢖⢯⢎⠧⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠮⡜⢮⡱⢎⣎⢳⢣⡜⣘⠢⣙⢰⢣⢎⡵⢠⠣⢍⡜⠡⣍⠒⡜⢢⢑⠢⢍⢢⠁⣳⠼⡰⢫⠼⣹⡔⣏⠼⡑⠠⢛⡮⢳⡍⡞⡬⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠓⠛⠼⣌⡳⢇⡞⡴⢣⡜⣎⢣⠚⡌⢆⡓⠬⣐⠣⢦⡙⢤⠃⣎⠱⢊⠦⣙⠦⣏⠵⣎⢷⢣⡝⣞⢧⣍⡠⣁⡬⡳⣼⡙⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠈⠓⠳⡼⢎⡣⢜⡘⠴⣈⡇⢆⡓⡌⠳⣌⠚⠤⣙⢢⡱⣍⠞⣭⣛⡭⢯⢧⣻⡜⢷⣎⡷⣱⢚⡵⢧⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠹⣎⠵⣪⢜⣣⠳⣜⢣⠖⣍⡲⣄⠫⡜⢥⣣⠳⣌⠻⡴⣙⢾⡹⢮⡵⠏⠐⣮⡑⢧⢫⠀⢟⡆⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡵⠎⡶⣛⠬⣏⠽⢮⡵⣊⠷⣙⢦⢣⡛⣬⢳⡱⣍⢶⣹⢳⣚⢧⠀⠀⠑⠀⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠑⣧⡙⡞⣬⢛⢦⡹⡝⢮⡱⢎⢧⣙⢦⣇⠳⡜⢮⣓⠯⣞⣣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣧⠹⡜⢦⣋⠶⣙⠾⣥⢛⡜⢦⡙⢮⡜⣻⡜⣧⢏⣟⡲⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⣏⡳⣙⢦⡙⢮⡱⣋⠶⣩⠞⣥⢛⢦⡙⢶⡹⢮⣝⡲⣝⢮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢩⢷⡩⢖⡭⢣⠳⣍⠞⣥⡛⡴⣋⢮⣙⢦⣛⡗⣮⢳⡝⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⡔⡂⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⢞⡽⢮⣜⢣⡛⣬⣛⢶⡹⢧⡝⣮⡝⣮⢳⠞⣵⢫⡞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡱⢂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⢀⡀⢀⠀⠄⠐⠐⠠⢃⠜⠠⠃⠘⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠐⠌⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣯⢇⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠳⣯⣟⣾⢮⠄⠀⣃⠶⣒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⠿⠁⠢⢹⡾⡝⠇⢃⠐⠉⠚⡵⠊⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⢎⠁⠠⠤⠞⡿⠁⡍⠀⠈⢤⠁⠘⡄⠀⠁⢊⡜⡳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⢯⣷⣻⡽⣯⠛⣁⣤⡴⣶⣤⠔⠃⡸⢤⠁⠌⠐⢮⣀⡜⡑⠒⡠⢈⠓⠭⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢸⡿⣯⣿⡿⣿⡽⣿⢯⣟⣯⣟⡾⣷⡏⠀⠉⠉⠀⠀⠉⠀⠀⠩⠀⠜⢢⠍⠀⠐⠈⠖⡔⢣⢐⠠⠂⠘⣄⠀⣸⡒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⣽⣳⢿⣻⡽⣯⣟⡾⣷⢯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠂⠴⢶⣦⣄⠀⠁⠊⡱⢈⠐⡀⢁⠲⢍⡳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡟⣮⡟⣯⣳⣟⡷⣯⣟⣯⣟⣿⡾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠤⠀⠐⠭⢀⠐⡀⢏⠠⠁⠁⠆⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠠⢾⣻⢷⣻⢷⣻⢾⣽⣳⢯⣿⣯⣷⣻⢿⡠⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠂⡸⠀⠌⠓⢄⠠⠝⣜⢣⡝⡂⠔⠠⢓⡀⠀
⠀⠀⠀⠀⠀⣼⣯⣿⣯⡿⡽⢯⣷⢯⣟⡿⢷⣯⣟⣯⢷⣈⣔⡠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠂⠄⡁⢀⠂⣌⠂⠄⠂⠌⠱⣎⡀⣠⠒⡤⠈⠀
⠀⠀⠀⠀⠀⢻⣞⣟⡲⣝⣚⠳⣎⠿⢾⢿⠹⣾⣽⣾⡏⢿⡿⢿⠿⠿⣿⡞⠠⠄⢀⡀⢀⠀⠀⢀⠀⡰⢀⠖⠀⢂⠈⡐⢀⢂⠴⣂⢌⡠⠁⢌⡐⢈⠳⣥⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⢧⢎⡵⢪⡜⡳⡌⣍⠲⣈⠻⡰⣁⠎⡣⢌⡓⢌⠎⣒⠡⣏⠀⠀⠀⠀⠁⢈⠒⠌⡰⠁⣎⠠⢈⠄⢂⠐⢠⢏⠶⡱⠊⢀⡜⢂⠔⢠⡟⡴⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢩⠞⣜⢣⢞⡱⡜⣠⠓⡌⡱⠰⢌⡒⡑⢮⢘⠢⢍⣂⠳⢌⠳⣒⠰⡀⠤⡈⣌⠂⠥⡘⢤⠣⣥⢜⡖⡋⡜⣎⠳⡡⢜⢢⡜⣡⠴⣫⠜⡵⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠬⣓⢮⡱⣙⢦⠳⣬⡑⡍⡒⣌⠱⣆⠧⣍⠲⢠⠓⣌⠣⠔⢣⡉⠖⡡⢌⡹⢄⠈⣥⢛⢼⠺⣜⢳⡼⣌⢣⠱⢈⠯⣜⢧⢫⡕⡫⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠓⠭⢎⡳⢦⡝⣲⠱⣤⢛⡌⠳⢌⡱⢃⠭⣠⠙⣬⢃⡌⢣⡑⢎⡰⢊⡗⣬⣋⠦⣟⡬⢇⡷⣹⢞⡓⠀⢌⡱⣎⢳⡜⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠉⠁⠉⠒⠻⣴⢫⡜⣑⢊⠴⣡⢞⡠⡙⠤⢳⠨⢥⢘⡂⣖⡱⣚⠴⣫⢻⡜⣽⢣⣻⡱⣏⣞⢯⡙⡶⣹⢧⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠨⡗⡜⢦⡙⢮⡱⢎⡵⢪⡕⣪⡑⣌⠲⣙⢦⠳⣍⢞⡱⢏⡾⣱⢏⡗⠙⠘⣮⠑⣭⢳⠈⢷⡍⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠧⢽⢖⡫⢭⣙⢧⢻⡴⡹⣌⠳⣍⢎⡳⡜⢮⡱⣋⢾⡱⣏⢶⡀⠀⠈⠃⠈⠑⠒⠘⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡑⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡵⢪⡕⡳⡜⣎⠳⣜⡳⢮⡹⡜⢮⡱⣝⢦⠳⣍⢾⡱⣏⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣈⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣜⡱⡹⣌⠳⣍⢳⣣⢳⡙⢦⠳⣍⣞⡳⣎⢷⡹⣎⡟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⡳⣌⢳⡱⢎⡽⣌⢳⡌⢧⣙⢮⠳⡜⢦⣛⡼⣣⢟⡼⢭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢌⠱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⡽⣬⢣⡝⣎⠶⣩⢖⡹⣎⡜⣎⢳⡝⢦⡳⣭⢳⣫⢞⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠹⡜⣧⢞⡬⢳⡱⢎⡷⣹⠼⣬⢇⡟⣧⢻⡜⣧⢳⣫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠀⠄⢡⠂⡅⢢⠑⠢⠉⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠂⠁⡒⡐⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⡗⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⣟⡷⣝⡆⠁⢂⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⠿⠃⠤⢹⡾⡝⡧⠣⠀⠞⠸⢦⡹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⣟⡿⢞⠁⡀⠤⠔⣿⠁⢇⠀⠈⢤⠁⠌⡁⠀⠐⠡⡲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣯⣟⣷⣻⣽⡻⠞⣁⣤⡴⣶⣦⢄⠃⠺⣨⠐⢈⠢⣌⠠⡙⠢⠤⡐⠙⠶⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⡿⣯⣿⡿⣿⣽⢿⣯⣟⣷⣻⢾⣷⣏⠁⠉⠉⠀⠀⠉⠈⠀⠈⡁⢘⢢⢂⠀⠈⠙⢎⡑⠦⣐⠠⠂⠈⡆⠈⢰⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⡽⣾⣻⢯⣟⣟⡾⣽⡾⣽⣻⣯⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠢⠠⣶⣦⣀⠈⠂⠡⢓⢨⠐⠈⠒⢩⠞⡴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡿⣹⢷⣫⣟⡾⣭⣟⡷⣿⣳⣿⢿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠳⢄⠀⠈⡖⠠⠁⠌⣸⠀⡓⠠⠄⡀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠠⢾⣻⡽⣷⣻⢾⡽⣷⢯⣟⣿⣷⢯⣟⡿⡄⡀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⢂⠃⡀⢣⡀⠀⡸⣍⢎⠷⡀⢄⠠⡐⢄⠀
⠀⠀⠀⠀⠀⣼⣷⣻⣷⢯⠿⣽⣯⣟⡾⣽⢿⣻⢾⣽⣳⣐⣤⣀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠠⠐⠊⠀⠐⣀⠎⠉⠑⡘⢮⡹⠀⢈⠠⣁⠆⠀
⠀⠀⠀⠀⠀⢻⡾⣹⢎⢯⡙⢶⡙⡾⡽⣿⠸⣿⣻⣾⡏⢻⡿⢿⠿⢿⣿⢾⠠⠠⢀⠀⠀⠀⠀⠀⠀⢠⠀⠴⠁⡈⠄⡁⢂⠁⡢⠅⡐⠈⠄⡁⠌⢳⡱⠎⠰⠈⠀⠀
⠀⠀⠀⠀⠀⢸⢱⢣⠞⣦⢛⢦⡙⠴⣁⠎⡝⢢⠱⣘⠣⣄⠳⠌⣆⠣⡘⡇⠀⠀⠀⠈⠁⠖⡐⢌⡘⠄⡹⢀⠂⡐⠠⠐⡀⡞⣡⠳⠜⠁⢂⡔⠡⢂⡽⣙⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢣⢏⡞⡴⣋⠶⣉⠒⣌⢒⡉⢆⡓⠤⢓⡌⢎⡱⢄⢣⠱⣩⢓⠲⣐⠠⢠⢈⡔⢂⠰⡈⡕⢢⢍⡤⡵⡚⢰⡙⠶⣉⡐⣌⠣⠘⣀⣼⠲⡭⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠼⡱⣍⠞⣥⢛⡴⢪⡘⠆⡜⣑⢪⠴⣪⠔⣊⠆⣓⠰⢊⡱⢄⣋⠒⢆⡜⡸⢄⠁⣹⢲⢹⡼⣱⢳⡨⡝⡸⢄⠳⡼⣩⢏⡵⢎⡳⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠛⠴⢫⡜⢧⣙⠞⡴⣡⢎⠳⡡⢋⡔⠪⢔⡩⢆⡱⢢⢌⡱⢊⡔⡡⢎⡵⣡⢏⠶⣣⢧⢣⢟⡦⣕⣊⠐⠱⠝⡮⣜⡣⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠉⠉⠈⠛⠶⣧⢋⡖⣡⠣⢌⡳⣌⠔⣊⠱⢣⠒⡔⢣⠰⣑⢎⠶⣩⢞⡻⣖⢯⣳⢮⣝⡳⢦⡬⢖⣫⢵⢮⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢱⡫⡜⢦⡙⢮⡱⢎⡳⢬⣑⢣⣩⢘⢣⡛⡜⣎⢳⡱⢎⡷⣹⠮⣵⠺⡜⢫⠷⢎⠧⣧⢛⢮⡳⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⡲⡭⢗⡹⢭⣹⢣⠟⣦⢣⢏⠶⡹⡜⣬⢣⡝⢮⡱⣏⠷⣭⡃⠁⠈⠻⡌⠓⢼⡀⢛⠂⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡡⢌⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣙⢮⡱⢣⢎⢧⡛⣼⣓⠮⣳⢱⡙⣦⢳⡜⣣⢝⣮⢛⡶⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠶⣩⢓⠞⡦⡝⢦⠏⢰⢩⠶⡹⣌⡳⣝⢦⡻⣬⢻⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢼⡣⢇⡇⠐⡣⠛⠀⠀⠀⠀⡯⣵⢪⡕⣎⢷⡹⣎⢷⡹⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠘⡽⣎⠆⠀⠀⠀⠀⠀⠀⠞⣡⠟⣦⡹⣜⢮⡳⣝⢮⡳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢘⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⣞⠀⠀⠀⠀⠀⠀⣈⡶⢧⡻⡴⣛⣬⢳⡝⣮⢳⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⠠⠐⡰⢈⠔⢢⠁⠆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⢂⠰⠠⠁⠀⠀⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⠷⢿⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⢿⣽⡳⡄⠃⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⠿⠃⠤⢹⣟⡞⡭⢃⡀⠎⢶⡱⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⣟⡿⢞⠁⡀⠤⠔⣿⠁⢇⠀⠐⣈⠀⠱⡁⠀⠃⢆⡰⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣯⣟⣷⣻⣽⡻⠞⣁⣤⡴⣶⣦⢄⢃⠸⢪⠀⡐⠌⣆⠁⡱⢄⡀⠄⠹⢦⡙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⡿⣯⣿⡿⣿⣽⢿⣯⣟⣷⣻⢾⣷⣏⠁⠉⠉⠀⠀⠉⠈⠀⠘⠊⠸⢄⠒⠀⠈⠝⠲⡄⠴⣁⠂⡁⠈⡅⠀⢂⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⡽⣾⣻⢯⣟⣟⡾⣽⡾⣽⣻⣯⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠜⢃⢠⣤⣄⡀⠁⠚⢄⢣⢠⠁⡘⠒⢨⡳⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡿⣹⢷⣫⣟⡾⣭⣟⡷⣿⣳⣿⢿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠛⠶⡄⠀⠀⡖⠠⠐⡈⢴⠁⠏⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⢾⣻⡽⣷⣻⢾⡽⣷⢯⣟⣿⣷⢯⣟⡿⡄⡀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡘⠀⠡⢀⠇⡀⢆⠀⠁⢱⢫⡝⣣⠄⡀⢠⠀⡔⠁
⠀⠀⠀⠀⠀⣼⣷⣻⣷⢯⠿⣽⣯⣟⡾⣽⢿⣻⢾⣽⣳⣐⣤⣀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡀⠀⠎⠀⠐⡈⠝⠈⢃⠳⣜⡱⠂⠁⡀⡣⢐⠂
⠀⠀⠀⠀⠀⢻⡾⣹⢎⢯⡙⢶⡙⡾⡽⣿⠸⣿⣻⣾⡏⢻⡿⢿⠿⢿⣿⡶⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠁⠂⠄⡁⠂⠄⡡⠎⢀⠁⢂⠁⠌⡳⢆⡞⠠⠑⠀⠀
⠀⠀⠀⠀⠀⢸⢱⢣⠞⣦⢛⢦⡙⠴⣁⠎⡝⢢⠱⣘⠣⣄⠳⠌⣆⠣⡘⡇⠀⠀⠀⠁⠒⢠⠂⡔⠨⡁⢆⠃⠠⢁⠂⠄⡁⡞⣡⢓⠦⠊⢠⡘⢠⠐⣏⠶⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢣⢏⡞⡴⣋⠶⣉⠒⣌⢒⡉⢆⡓⠤⢓⡌⢎⡱⢄⢣⠱⣩⢓⠲⡐⢄⡠⠐⡌⢄⠃⡔⣊⢆⡓⣠⢬⡖⢰⣙⢦⠋⣀⢆⠣⠔⢁⡼⣌⡳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠼⡱⣍⠞⣥⢛⡴⢪⡘⠆⡜⣑⢪⠴⣪⠔⣊⠆⡓⠤⢃⠣⢍⡒⠬⢱⡘⡜⣄⠁⢣⣎⢺⡵⢣⢶⡘⡜⢢⡙⡔⢮⢵⡚⣭⠞⡴⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠛⠴⢫⡜⢧⣙⠞⡴⣡⢎⠳⡡⢋⡔⡊⢥⡙⠬⣱⠒⡥⢋⢤⠙⢠⢊⢧⢳⡜⢮⣝⡳⢎⠷⣭⡒⣅⠘⡙⢮⡕⢮⡹⠌⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠄⠉⠉⠈⠛⠶⣧⢋⡖⣡⠃⢦⡙⣤⢊⠥⡡⠀⠧⠋⠀⠀⠀⠈⡼⣇⢟⡮⡼⣝⢮⡝⣶⢻⡤⡤⢬⡲⣙⣮⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢱⡫⡜⢦⡹⢦⣙⢦⣋⠖⡇⠀⠀⠀⠀⠀⠀⠞⣁⢾⡸⢳⡝⣮⢻⢜⡧⣻⠼⣕⡣⣏⢷⣚⠦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⡦⡽⢖⡹⢎⣭⢻⠀⠀⠀⠀⠀⠀⣈⢶⡙⡦⡝⣧⢻⡜⣧⡃⠀⠈⠟⡄⠷⣙⡀⢯⠃⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡔⢡⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣛⢬⢳⠭⠖⠀⢂⡀⠀⠀⢀⡼⡹⢦⡝⣲⠹⣜⢧⣛⠶⡃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡌⢢⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢎⠇⠀⠀⡀⠀⠀⠉⠁⡙⢶⣩⢳⣹⢣⡟⣼⢣⣏⢟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢚⡾⠀⠀⠀⠀⠁⠀⠀⠀⢀⡳⣌⠳⣌⡳⣝⢮⡳⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⢃⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⢀⢰⡎⡵⢪⡟⡴⣣⠟⣎⢷⡹⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⡌⢂⢆⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢠⢂⠱⠀⠀⢀⢀⠱⡈⢦⢻⡜⣧⢇⡻⡵⢫⡝⣮⢳⡝⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡡⢆⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⢀⠠⠐⠤⣁⠒⡌⢡⠂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⠓⠀⠐⢠⠐⡄⠁⠀⠠⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣐⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡟⢿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠳⣯⣟⣾⡴⠈⠃⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣬⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡟⠁⠢⢹⣾⡝⡞⠅⡀⠦⢫⡜⡳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⡾⣽⢯⣟⠞⢀⠠⠤⠝⡿⠀⡐⠁⢀⠢⢀⠙⡁⠀⠌⠰⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣻⣽⣳⢿⡽⠛⠀⢀⣀⣤⣄⠐⠣⢸⢐⠠⢀⠩⣄⠂⠵⣀⡀⠔⠣⢳⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⣯⢷⡿⣽⡏⠀⠀⠛⠛⠉⠉⠈⠁⠰⠂⠹⢄⡃⠀⠨⠙⣖⠠⡜⣀⢂⠁⡈⡄⠐⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢨⢷⣻⡾⣽⢯⡿⣽⣻⢾⡽⣯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠒⠄⣠⣤⣄⡀⠁⠘⢄⢎⠐⡀⠑⠒⢸⢦⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⣏⣷⡻⣝⣯⣽⣳⢯⣟⣿⣳⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠛⠶⡀⠀⠈⡖⠠⠁⠌⡼⠀⠏⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⣿⣞⡷⣿⡽⣞⡷⣯⣟⡾⣿⣷⢯⡿⣽⠠⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡱⠀⠁⢂⠇⠐⢆⠀⠈⢰⡹⢭⡓⠄⡀⢠⠀⡔⠁
⠀⠀⠀⠀⠀⣿⢾⣽⣷⡻⣽⢻⣷⢯⣟⣿⣽⢯⡿⣽⣧⣑⣠⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠐⠊⠀⠌⠠⠝⠈⠃⡝⣲⠹⠂⠁⡀⡣⢐⠂
⠀⠀⠀⠀⠀⣿⢯⣛⠶⣙⣎⢳⡹⢟⡾⢿⠸⣿⣻⣽⡇⢿⡿⡿⠿⢿⣷⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⢁⠂⠄⡈⠐⠠⣈⠇⠠⢁⠒⢀⠱⣋⠶⡘⠠⠑⠀⠀
⠀⠀⠀⠀⠀⢘⡣⢎⡳⡱⢎⢧⡙⢢⠜⡉⠟⢤⠃⡝⠦⣈⠇⡔⢣⠒⡜⡋⠀⠀⠈⠐⠢⠐⡄⢂⠒⢨⠰⢁⠠⢈⠐⠠⢁⡞⢤⣓⠲⠈⣀⠆⠰⢀⠯⡵⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢵⣋⠶⣙⠮⢦⡑⠣⣌⠱⣉⠦⡙⢤⠣⡕⢊⡜⠢⢍⡒⢭⠓⠦⡐⢄⡠⢘⡠⢁⠎⠄⡏⡔⣂⣡⢬⡖⠸⣜⢣⠎⡀⠖⣌⠒⢁⡼⢣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠊⠷⣩⠞⣥⢋⡗⢦⢳⡐⠦⣉⠦⡱⣌⠧⣌⠓⡌⢆⠣⢎⠱⣉⡒⢌⢆⡹⢉⢆⠉⡖⣱⢫⡼⣒⢧⠸⣌⠳⡌⡱⣹⡔⡞⡭⢞⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠒⠯⣜⢣⢧⡙⢶⡡⢖⡳⢌⠳⢌⠚⡌⢬⡑⢪⡕⢁⠣⣎⠒⠁⠘⠈⣧⢛⡵⢪⡽⡜⣎⠷⣎⣱⢂⠱⠱⢺⣱⡙⣮⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠉⠈⠉⢓⠹⣮⠱⡎⢜⡨⢱⡘⢦⡘⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡟⣼⢣⡽⡽⣌⠿⣜⢧⢦⡤⣜⡲⢥⡻⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢣⣛⡜⢦⡓⢧⡹⢲⡱⢣⠃⠀⠀⠀⠀⠀⠠⣁⡞⡽⡸⢧⡻⡼⣭⢻⡜⢯⣞⠵⣊⡷⢫⢗⠧⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡔⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢎⡧⡽⢲⡹⢭⡹⢋⠀⠀⠀⠀⠀⢠⢶⡹⣜⡱⢭⡳⡽⣱⢧⡃⠂⠘⢼⡘⠥⣛⡈⢞⠇⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢐⠡⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣏⠧⡝⠈⠁⠀⠐⠠⠄⠄⢴⣋⠶⣱⢎⡵⢣⢯⡵⢫⣖⡃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠣⡘⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠾⠀⠀⠈⠠⠀⡀⢀⠀⠂⣏⢞⡱⢏⡾⣥⢳⣝⡳⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠆⡱⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢟⠎⠀⠀⠀⠀⠀⠀⠀⢀⡰⢭⡎⡵⣋⠶⣭⢳⢮⣝⡳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢁⠎⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡌⠂⠀⠀⠀⠀⠠⢄⠒⣬⡓⢧⠺⡵⣩⢞⡵⣫⢞⡲⣝⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡰⢈⠔⢣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡜⠠⢍⡀⠀⢐⠨⡑⠌⣼⢣⡟⣎⢷⣹⢣⡟⣼⢣⣏⠷⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡡⢆⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⡀⠀⠄⠢⡁⢆⡑⠢⠁⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠂⠡⠒⠄⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣐⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⠗⢻⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⣾⣝⡀⠑⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣬⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⡛⠁⠂⣹⣾⡍⠶⠡⠀⠞⠱⣎⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣽⣻⡽⢊⠠⠄⠐⠚⡿⢀⠂⠁⠈⠤⢁⠘⡀⠀⠰⠁⡲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣻⢾⣽⣻⠷⠋⢀⣀⣠⣤⣤⡐⠁⠼⣘⠀⢌⠸⣄⢂⡑⠦⠠⡐⠙⠧⢎⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣽⣻⢷⣯⢿⣾⡇⠀⠈⠉⠉⠉⠁⠀⠀⠘⢁⠸⢂⡃⠀⠐⠩⠚⡔⡱⢠⢁⠂⠌⡄⠈⢰⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣟⡷⣯⢿⣽⣻⣗⣯⣟⡿⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠘⠠⠤⣶⣤⡀⠈⠂⢃⢎⠰⢀⠘⠂⢹⡚⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣠⢯⡽⣏⡿⣼⣳⢯⣶⣻⣿⡽⣿⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠠⣋⠀⢂⠁⡞⠀⠋⢐⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠐⡿⣯⣟⣷⣻⢷⣯⣟⣾⣳⣿⡿⣯⢿⣾⡁⠄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⢢⠃⢈⠣⡀⠀⣸⢫⡜⣳⠀⡄⢠⠐⡄⠀
⠀⠀⠀⠀⠀⣿⣳⣿⡷⣯⢟⡾⣟⣾⣳⣿⢻⣽⣻⣞⣷⣨⣔⣠⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠠⠐⠂⠀⠂⢄⠍⠉⠄⠳⣜⡱⠁⢈⠠⣁⠆⠁
⠀⠀⠀⠀⠀⣿⡳⣏⠷⡹⢎⡳⢭⡳⢿⡾⣙⢯⣷⣿⡃⢿⡟⢿⠿⢿⡿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠀⠌⠠⠁⠌⢠⡑⠎⠀⠌⠠⢁⠈⠷⡱⠎⠰⠈⠀⠀
⠀⠀⠀⠀⠀⠘⡵⣩⢞⡱⢫⡜⢣⡑⠦⡘⣍⠚⡔⣘⠣⣄⠓⣌⠒⠦⡘⣍⠀⠀⠈⠁⠊⠰⡐⠌⡂⢅⢚⠀⠌⡠⠁⠌⡐⢎⡜⣡⠛⢈⠐⡤⠡⢂⡽⣹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢱⢣⢎⡵⣓⢮⡑⣌⠲⢡⠜⡘⢤⠃⡖⠬⢣⡘⠬⣑⠱⣌⠳⢒⡰⠠⢄⡘⡠⢑⠨⠄⣏⠲⣉⡤⢞⡖⢸⢣⠞⡅⡰⢄⡫⠐⣁⣼⢲⡱⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠜⣲⢍⠶⣙⢦⡛⢦⡊⢕⡊⡜⣰⢋⠶⣌⠱⢌⠲⢄⡋⢆⢥⡋⡔⢢⢱⠋⠧⡉⣆⢷⠹⣼⢣⢶⡨⡓⠼⡰⡑⣮⢕⢯⡱⢎⢧⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠚⠵⢪⡝⣦⢝⡲⡜⡴⢣⡋⢞⢨⡑⣊⠱⡌⡒⣭⠀⡆⣍⠖⠁⠘⠠⢻⣍⡞⢮⢞⡧⢎⢷⣫⢖⡡⠘⡐⢟⠦⣝⡺⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠠⠀⠀⠄⠈⠁⠉⠒⠽⣼⢣⡓⣌⠲⢰⡡⢇⠲⡑⡬⠀⠘⠁⠀⠀⠀⢠⡝⢨⡽⣚⡞⡾⢭⡞⣵⣫⢖⡤⢖⡫⣞⡼⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣧⡙⢦⡓⢧⣙⢎⡳⢥⡃⠀⠀⠀⠀⠀⢀⠡⣬⠳⡜⣧⢻⢭⡳⣟⠖⣫⢾⢜⡣⣗⢫⢞⡱⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢝⡦⠽⡞⢮⢭⣹⠃⡁⠀⠀⠀⠀⢀⡴⣛⠴⣫⠜⣧⣛⢮⡳⣥⠁⠘⠮⡄⠳⢭⡀⢫⠃⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠆⡡⢉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡹⣜⠣⠊⠁⠀⠀⠢⠄⠄⠰⢏⠶⣍⡞⣥⢛⡴⣫⢞⡵⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠃⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢎⠇⠀⠀⠂⠀⡀⢀⡀⠈⢫⠞⣬⠻⣴⢫⡜⣧⢻⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠎⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡞⢦⡛⣬⢳⡝⣮⢳⡝⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡘⢌⠀⠀⠀⠀⢠⠀⢆⣱⢣⠞⡜⣧⣙⢦⡻⣜⢧⡻⣜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⡄⢣⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡠⠑⡌⣀⠀⢀⠢⡁⢎⡰⣏⢧⡻⣜⣥⣛⢮⡳⣝⢮⡳⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠪⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢠⠀⡒⢌⡐⢢⠁⠆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⡛⠀⠀⠂⡔⠠⠁⠀⠀⠂⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⢏⠹⠆⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⠱⣯⣟⡾⣏⡇⠀⡰⢎⡳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⠟⠁⠂⢻⣞⠯⡗⠐⡐⠁⡙⠼⡉⠀⢂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⣾⣳⣟⡾⠉⠤⠄⠐⢻⡟⢰⠈⠀⠄⣡⠀⠄⢃⡀⠈⢀⠯⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⢿⡽⣾⡽⣞⠷⠋⢀⣀⣤⣤⣤⡐⠐⠚⣄⠡⠐⠰⣌⡐⢢⠉⢂⠡⢀⠋⢳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣳⢿⣯⣟⣷⣿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠁⡚⢤⠃⠀⠀⠉⠧⡘⡌⢡⢀⠂⠌⣄⢀⡜⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢰⢯⡷⣯⢿⣽⣻⠷⣯⢿⡾⣽⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠴⠶⣶⣄⡈⠈⠂⡇⡈⠄⠠⠁⣜⠣⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⡼⣯⣟⣾⣻⣟⣷⡿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠰⡩⢀⠁⢂⡎⠰⠃⠐⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣿⢾⣽⡷⣯⣟⣷⣻⢾⡽⣿⣾⣻⣽⣻⠠⢀⠀⠀⠀⠀⠀⠐⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢈⡰⢀⠘⣄⡀⠀⡞⣥⢛⡇⡄⢠⠠⠰⡈⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣟⣷⢻⣞⣯⢿⡽⣟⣷⣻⢾⣽⣇⣆⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠂⠄⡀⢀⠂⡤⠁⠄⡉⠆⡯⡜⠁⢀⠬⣁⠒⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠾⣜⢣⠞⡼⢻⡽⣿⠸⣿⣯⡿⡍⢿⡟⢿⠿⢿⡿⡗⠤⢀⠄⡀⢀⠀⡀⢀⠠⡐⢠⠜⠀⢂⠈⡐⢀⠢⡜⡀⠐⡠⠐⠠⠑⢭⠲⡍⠠⠁⠀⠀
⠀⠀⠀⠀⠀⠘⡵⣩⠞⣬⢣⡛⡜⣡⠒⣌⠳⢡⠆⣙⠣⣄⢓⡰⢊⠴⢸⡁⠀⠀⠀⠁⠊⢐⠡⠊⠤⠑⣌⠠⢈⠄⢂⠐⢠⠳⣬⠱⠋⢀⠬⣁⠌⢰⢫⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢣⢇⡻⣔⢣⡝⡰⢂⠭⠤⡙⣂⠞⣠⠓⡬⠒⡴⢉⢆⠣⡹⢱⢒⡰⢀⠤⢈⡰⢉⠰⡁⢎⠲⣡⢖⢮⡃⣏⢳⢌⡃⠴⣉⠖⣠⢨⡟⢦⡃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠴⣩⠖⣭⠳⣍⡖⣣⡑⢆⡚⢄⠳⣌⠳⣔⡉⢆⢣⡑⢃⠲⢌⡱⢊⠦⣱⠋⣃⠈⢧⣋⢧⢏⡖⢧⣘⠣⡎⠜⡡⣏⠾⣡⠗⣎⢧⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠒⠽⢢⡝⣦⢝⡲⡜⣬⠳⣌⠛⢤⠚⡌⢆⡜⢡⡛⠀⣇⢣⠜⠁⠘⠠⢓⢧⡚⣦⢾⣙⠮⣝⡧⡼⠄⢁⠚⣹⢆⡻⡜⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠉⠈⠉⠒⠽⡶⣙⠦⣉⠆⡓⣬⢒⠬⡡⡆⠀⠊⠀⠀⠀⠀⣠⠏⡷⣹⠖⣏⢾⣩⢷⣹⢳⢞⡲⣚⠵⣪⠷⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠱⣭⢚⣤⢫⡕⢮⡙⢶⡡⠇⠀⠀⠀⠀⠀⠠⣁⠶⡳⣍⠿⣜⢧⣛⣮⡓⢯⢾⡱⢭⡓⢯⣻⢃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡖⢧⢞⣣⢛⣭⠙⠀⠀⠀⠀⠀⢀⡶⣍⢳⡱⢎⡟⣼⢣⠟⡆⠀⠙⣦⠙⢖⣃⠘⣵⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠜⡠⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣏⠞⡴⠉⠂⠀⠁⠢⠄⠄⢴⣋⠶⣍⠶⣙⢮⡹⣎⣏⠿⡁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠑⡂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡝⢮⠁⠀⠀⠂⠀⡀⢀⠀⠂⡝⢮⡍⢿⡱⣎⠷⣭⢺⣝⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⢹⡎⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛⡦⣹⢲⡙⣮⢻⣜⡳⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡜⣀⠀⠀⠀⠀⢠⠀⢆⣱⢎⡳⠼⣥⢣⡝⣮⢳⢮⡝⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⡄⢣⠑⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⠐⠤⡀⠀⠐⡠⢃⠜⣰⢏⡾⣱⢏⡼⣣⢟⡼⢫⢶⡹⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣄⣠⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡶⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡸⢄⡹⡁⠀⠀⠀⠈⠐⡁⢆⡘⠤⢈⢆⠠⣁⠂⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠲⢡⠎⡰⢍⠳⢢⡐⢄⡠⢐⡂⠔⡨⢘⡌⢳⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⣤⢋⠆⢎⡱⢌⢣⡑⣌⠦⣑⢊⡼⢉⢆⠉⡼⣃⠷⣏⠞⣧⡱⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢊⡱⢊⠲⡐⠎⡔⡖⢡⢃⡆⠓⠀⠘⢈⡷⣱⠭⣖⠻⡞⡴⣏⢷⠬⠄⠂⣃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⡔⢃⠲⣉⠖⡭⡘⠴⡁⠀⠊⠀⠀⠀⠀⣠⠏⡗⣯⡝⣯⠽⣞⡹⣎⢷⣚⢖⡚⡦⣝⢮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⡣⡝⢮⡱⣍⢞⡱⣙⢦⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⣓⢾⡱⣻⢼⡳⠽⣎⢷⢪⡝⡶⣫⢞⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⡦⢷⡚⢮⡹⣹⠚⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢭⢶⣹⢣⣏⡝⠁⢛⣆⠳⡜⡇⠱⣯⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⡷⣙⡖⠉⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣎⢧⡏⢷⡺⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡷⣱⠂⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⠽⣬⢳⡞⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢷⠃⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⠻⣜⡳⡞⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⣖⣱⢻⣜⡳⣝⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⢣⡟⣦⢳⡹⣎⠷⣎⢷⡹⢮⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠠⠀⢂⠱⡈⠆⡡⠊⠐⠁⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠈⠀⠈⠔⢂⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣟⡄⠻⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣝⡞⠀⢨⠖⣭⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡝⢂⠉⡀⠐⡈⠏⠀⠈⡔⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⡧⠀⠐⠨⡔⢀⠘⢤⡀⡀⠜⢲⡹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⢑⡌⢠⠁⠲⢦⡘⡄⢠⠡⠈⠄⠉⡇⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠁⠊⠫⠐⡎⠀⣀⠀⠉⠲⠡⡚⢄⠈⠄⠱⠄⡼⣩⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠘⠛⠻⢶⣄⠈⠐⢊⡐⠈⠄⢂⡇⠑⣏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡅⢈⠐⢠⠃⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢈⠆⢈⠐⢣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⢁⠂⡀⠄⣈⠆⠠⢀⠩⢘⠧⣎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠃⠐⠢⡄⢄⠠⠄⡐⢄⠊⠴⠁⠌⡐⢀⠂⡐⡴⢡⠈⡐⡀⠂⢄⠙⢦⢳⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠁⠊⡔⠡⠌⢌⡃⣌⠰⠀⣂⡐⠼⣡⢣⡝⠠⢀⡕⢂⠌⣸⢣⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣁⠢⡑⠌⢢⡑⢦⡞⡽⣱⡀⡟⢦⢓⡠⢒⡥⢊⢄⡲⣇⠳⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⢖⡨⡱⢉⢎⠁⡽⢣⠹⣣⢇⢷⡸⢡⠣⠜⡱⢮⡝⢮⡕⢮⡝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⡎⠔⠁⠘⢈⡷⣩⢗⣳⢳⡞⡬⣟⢦⠯⠄⠡⢋⡽⢢⡝⡆⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡗⣯⣚⢧⣛⡾⣹⢎⡷⣚⠶⣩⢞⡱⣞⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⢣⣏⠾⣥⢻⡵⢻⡜⣧⢛⣴⠫⡷⡭⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢫⣜⡻⣜⢧⡝⠁⢛⣆⢩⢎⡇⢱⢻⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⣝⢣⡞⣵⢫⡞⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⢭⢶⡹⣎⢷⡹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⢛⣮⢳⡝⣮⢳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⡖⣭⢺⡜⣧⢻⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⢳⡝⣮⢳⡝⣮⢳⡝⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⡀⠠⠀⠂⠘⢠⠃⠜⠀⠃⠐⠀⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠈⠀⠊⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⣋⠀⠠⠑⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠠⠈⡘⡄⠂⠌⠢⣄⠠⠊⢝⢦⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⢬⠡⡀⢆⠡⠈⡀⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡱⢀⣀⡀⠁⠃⠙⡄⢆⠡⠐⠘⠤⣒⢧⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢦⡀⠈⠘⡄⢈⠐⠠⡍⠂⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⠠⢈⢰⠁⢌⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠊⠀⠌⠱⣄⣀⠞⣜⡣⣝⠢⡐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⢂⠁⢂⠠⠀⣌⠒⡀⠠⠘⠰⣓⢎⡀⢠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠁⠉⠂⠅⢢⠐⡄⢢⠘⡰⠰⠁⠌⠠⠈⠄⢂⡽⡐⣂⢄⠡⢈⠐⡈⢮⡹⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠀⢊⠰⡀⢃⠔⡣⠌⠔⣁⣌⡔⣸⢰⡱⠎⡀⢂⠴⣁⠂⣴⡙⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣀⡡⠘⢄⢪⠱⣩⡞⡵⢪⡄⢧⢣⡙⢤⠒⡭⢒⣀⡖⣧⠹⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⡹⢉⢆⠁⡟⣥⠻⣉⢧⣛⣬⢃⠞⡄⢫⠷⣙⠦⣝⢲⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠘⢈⡷⣜⢶⣩⣓⣮⢲⡝⡾⡜⠠⢁⢙⣩⠞⣬⡓⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣠⠏⣧⣛⢶⡹⣎⡷⣹⢳⣝⡳⢎⡝⢦⣛⢶⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⠠⢡⡼⡱⢎⡷⣹⠮⣵⢏⡟⡼⣣⢏⡼⡓⢯⠾⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠞⣧⢳⡙⣎⠷⣭⢳⡭⡍⠀⠱⢧⠸⠱⣇⠘⡻⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣬⢻⡜⣧⢏⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⡝⣦⢻⣜⡳⣎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⠘⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⡼⣱⠺⡵⣫⢶⡹⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⠾⣔⡣⣏⢷⡹⢶⣹⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢐⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⣚⡵⣛⡼⣣⢟⣣⢭⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠂⢂⡘⠄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠈⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⢏⠀⠠⠑⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠀⡈⡘⡄⢈⠈⠦⣀⠠⠚⠼⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⣜⠠⣀⢃⠐⠠⠈⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡕⢀⣀⡀⠈⠑⠌⠆⢎⠀⠡⠐⠦⣪⢕⡺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀⠈⠊⡄⢁⠂⠄⡇⠈⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠂⡈⢰⠁⡘⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠐⠈⢣⣄⡀⢷⡘⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠠⢁⠂⠠⢀⡡⠆⡀⠐⠠⠙⣧⢋⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠃⠈⠐⠡⡂⠔⡠⢂⠜⣀⠲⠁⡈⠐⠠⢈⠐⡼⢰⡀⢄⠁⢂⠡⠈⢧⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠂⡅⢂⠒⣠⢣⠰⠄⣁⣢⠄⣞⡡⢇⠞⠁⣈⠔⡡⠃⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⢄⢡⡐⠡⢊⠤⣃⢧⣛⡥⢏⡄⢧⡹⣊⠴⡘⣤⢋⣠⢖⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⠩⡜⣠⠹⣉⢆⠑⡝⣦⠫⡱⣋⢞⡵⢢⡑⠪⡱⣇⣏⠶⣙⠶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢡⢓⡰⠁⠁⠐⠨⡗⣼⢲⣍⣳⡭⢎⡽⢶⡹⠀⠡⢜⡬⢳⡍⡞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠊⠀⠀⠀⠀⣠⠏⡗⣏⠾⣥⢻⡝⣞⢧⡻⣝⠲⣍⠞⣵⢺⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⠐⣡⢼⡹⣜⡻⣜⢧⣻⢎⡻⣵⢫⡝⣼⡹⢞⡳⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⢗⡹⢆⡳⡜⣧⣛⢮⡵⡍⠀⠸⢧⠸⢒⡇⠩⡻⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⡓⢮⣕⡫⢖⡹⢶⡹⡞⣼⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠀⠂⡟⢦⣙⢞⡧⣏⠷⣭⡝⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢫⡳⡜⢮⡱⣞⡽⡲⣝⠶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⢳⣙⢦⡳⣭⢶⡹⣎⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣎⢧⣛⢮⣓⠷⣎⢷⡹⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⡀⠠⠀⠂⠘⢠⠃⠜⠀⠃⠐⠀⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠈⠀⠊⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⣋⠀⠠⠑⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠠⠈⡘⡄⠂⠌⠢⣄⠠⠊⢝⢦⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⢬⠡⡀⢆⠡⠈⡀⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡱⢀⣀⡀⠁⠃⠙⡄⢆⠡⠐⠘⠤⣒⢧⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠛⠷⢦⡀⠈⠘⡄⢈⠐⠠⡍⠂⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⠠⢈⢰⠁⢌⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠊⠀⠌⠱⣄⣀⠞⣜⡣⣝⠢⡐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠄⡁⢂⠠⠀⣌⠒⡀⠠⠘⠰⣓⢎⡀⢠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠃⠈⠐⠡⡂⠔⡠⢂⠜⡐⡰⠁⠌⡐⢀⠂⢄⡽⣐⠂⣄⠡⢈⠐⡈⢮⡹⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⢠⠑⡄⠒⢤⢣⠰⠐⣀⣢⠌⣲⠰⣥⠛⢀⡐⠤⢃⠐⣰⠝⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣀⡘⢠⠉⡔⢪⡱⢯⣕⢣⡆⢧⡛⡤⢃⠲⣌⣃⣌⠴⣏⠞⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⡹⢉⢆⠉⣗⢣⠻⡨⢇⡻⡴⡑⢎⠥⢓⡧⡞⣬⢛⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠈⠘⣧⠼⣍⢯⣱⣏⢖⣯⢳⠭⢀⠂⢱⣙⢦⣋⠶⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣨⠏⡟⣼⢣⡗⡾⣝⢮⣏⢿⡱⣚⢧⡚⣦⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⠠⢡⡜⢧⢭⡳⡽⣱⢻⡞⡜⣧⠷⣩⢶⡙⢧⡻⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠞⣧⡙⣎⠶⣹⢳⣭⢳⡍⠀⠸⢣⠱⢎⣇⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣴⠹⣌⡳⣭⢳⣎⡗⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣛⢶⡱⣏⡳⣞⢼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⡼⣩⠞⣵⢫⡵⣎⢷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⠾⣔⣣⢻⡜⣧⢻⡜⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⢄⠣⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⣚⡵⣭⢳⡝⣮⢳⡝⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠀⠐⠂⠆⡌⠱⠀⠣⠐⠈⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⡃⠈⠀⠒⢨⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣖⡈⠳⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢷⣃⠀⡸⢜⡬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡷⠀⠡⢀⠂⠙⡃⠀⠡⢂⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⠠⠀⠁⢦⠈⠄⠱⢄⡀⠠⠎⢧⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢰⠁⡌⠀⠳⢬⡡⢂⢡⠂⠌⠂⠱⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠱⠈⢣⠑⠆⠀⣈⠀⠙⠢⢣⠘⡤⠁⠌⠣⠤⣜⡹⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠘⠛⠻⢶⣄⠈⠑⠰⠌⡐⢀⠂⡌⠑⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠄⠂⢌⠃⢌⠁⠈⠔⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢁⠎⠠⠈⠣⣄⡠⡝⣜⢣⡝⡢⠐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠁⢂⠀⠄⢡⠚⠀⡀⠰⠈⡗⣎⡀⣠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠋⠐⠢⢐⠠⢠⠠⡐⢄⠊⠴⠁⡐⠠⠈⠄⡈⡴⢣⣀⢡⠀⡁⢂⠙⢢⡝⡀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠂⠥⡐⠌⣌⠣⡐⠤⠁⣌⡐⣸⢡⢇⡎⠁⡐⡌⡡⠂⣼⣘⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣘⡐⠌⡰⢠⢣⡙⣖⢯⣱⡀⢧⣋⢎⡰⢌⡱⣈⡰⢴⡳⡜⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⠼⠙⢆⠁⣗⢣⢫⡓⢦⢳⢧⡘⢦⠑⡎⢷⣍⢳⢣⢳⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠘⠨⡗⣬⣓⢧⣛⢧⣋⡾⣱⠮⠀⠌⣑⡪⢇⣏⠶⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣠⠏⡷⣹⠮⣝⠾⣜⡳⣭⢻⡝⡲⣍⠞⣵⢺⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⠠⣁⣞⡱⢣⡟⣼⣛⣼⢳⡝⣧⢻⡱⢎⡟⢧⡻⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠶⣙⢦⡙⢧⣛⠶⣭⣚⡅⠀⠸⢣⠙⢎⣇⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⣛⡜⣦⢋⢧⣫⡝⣶⡹⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⡷⢬⣙⢮⣛⢦⡳⣝⢶⡹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⠘⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⡹⢦⡙⣦⢛⣮⢳⡝⣮⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⡓⢧⡝⢦⣫⢖⡯⢞⡵⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢐⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⢶⡹⣎⡽⣣⢏⡾⣜⠯⣞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⠜⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠐⣼⣶⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⡀⠀⠄⠒⢌⠰⡈⠔⠡⠌⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⠃⠀⠂⠡⠒⠄⠀⠀⠠⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡽⣎⠙⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏⢾⡽⣯⣟⢮⠇⠀⡥⣝⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⠏⠈⠄⣿⣳⢹⡏⢈⠂⡐⠈⢱⠃⠀⠆⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣟⡾⣽⣻⠞⡁⠤⠀⠒⢻⠏⢸⠈⠀⠐⢤⠁⠂⢣⡀⠀⠠⢧⣙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠛⠿⠟⠛⠛⢻⣿⣿⢿⣟⣿⣽⣿⣾⢯⣟⣾⣳⣯⣟⠷⠉⣀⣀⣤⣤⣄⡘⡠⢋⢢⠁⡌⠐⠦⣅⠆⡈⢩⠐⡀⢉⠺⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣻⣯⢷⣟⣿⡽⣞⣷⣿⡆⠀⠉⠉⠉⠁⠀⠀⠀⠡⡒⡩⠒⡄⠀⠈⠐⠹⢘⢢⢡⠐⠠⠘⣄⡠⣝⡹⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣳⢯⣷⣻⢷⣯⣟⡾⣽⣻⡽⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠿⢶⣦⡀⠁⠂⠇⡄⠡⢀⢀⡎⠱⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣸⢏⡿⣞⣭⣟⡾⣵⣻⣽⣷⣻⡿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠘⠄⠡⢀⢸⠀⠱⠁⠡⢀⢀⣠⣀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠒⡿⣯⢿⡽⣾⣽⣻⢷⣻⢾⣿⣽⣻⡽⣧⢂⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠐⢀⠎⢀⠉⠦⣀⣀⠞⡴⣩⠇⡤⢀⠆⡘⠀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣻⢷⡳⢯⣿⡽⣯⡟⣷⢯⡿⣽⣆⣜⣤⣀⣄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⢂⠈⢀⠀⢌⠒⡀⢈⠩⢸⢥⡋⠀⡠⠢⡡⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⢗⡫⣝⢣⡛⡽⢷⣯⠹⣿⣻⣿⠉⣿⠿⠿⠿⢿⡿⡇⠒⠄⢂⠄⡠⢀⠄⡠⢐⡐⢢⠁⠌⡐⠠⠈⠄⡼⢨⠐⢀⠂⡐⢀⠣⡝⣥⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⢣⢎⡵⣊⢧⡙⠜⡰⣈⠟⡰⢡⡙⢤⡈⢦⠑⡣⢆⡹⡁⠀⠀⠀⠈⠐⡨⠐⡄⢃⣐⢃⣀⠢⢀⢡⠈⡼⣡⢧⡙⠂⡐⡔⠡⠂⡼⣜⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢜⡣⢏⠶⣩⠖⣌⠣⡑⠆⢎⡡⢃⡜⠤⡙⢤⠋⡔⢢⠱⣍⢓⢢⡐⢄⠠⣁⠱⠈⡔⢨⠒⣔⡦⣏⢽⠐⣧⠓⣎⢠⢒⡱⣈⢡⣜⡳⣜⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠎⡳⢥⡛⣬⢳⡹⡜⡢⢅⠣⠜⡲⣍⢦⡙⣌⠱⠒⡌⢆⠣⣜⢨⠱⣨⡙⠹⢄⠉⡽⣆⠽⣎⢎⡧⣎⠱⣌⢢⠹⣖⡭⢳⡜⣱⡚⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠓⠛⠴⢣⡳⣍⠳⣎⢥⠳⡵⡘⠦⣑⠢⢍⢓⡸⢨⡕⢠⢃⡺⠀⠀⠃⠜⣖⡳⣜⡲⠽⣎⡼⣣⢯⠴⠀⡉⢪⠵⡳⡜⡕⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠄⠉⠈⠉⠚⢮⠷⣱⢃⠞⡠⢍⡲⡌⡔⠣⠅⠀⠋⠀⠀⠀⠀⣬⠋⣵⢫⣝⡻⣜⡾⣱⢏⡞⣶⢒⡏⣞⣱⡛⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢻⡔⣫⢖⡹⢎⡵⡹⣜⡱⠀⠀⠀⠀⠀⠀⡈⢤⣛⠴⣫⢞⡵⣫⢵⣫⠾⣹⢦⢫⡜⠾⣱⠷⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣔⡫⡼⣓⠞⣭⠣⠛⠀⠀⠀⠀⠀⣠⢞⡳⣌⡳⢭⡞⣵⢫⣞⡑⠈⢳⠇⢣⠽⡀⢹⡇⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⢢⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡗⡭⠞⠈⠁⠀⠑⠠⠄⠠⢚⡵⢪⡕⢮⡱⢣⡞⣵⢫⢶⡁⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠼⠀⠀⠀⠄⢀⡀⢀⠀⠍⡞⣥⢛⢧⡻⣥⢻⡜⡯⢞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠔⡨⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⡟⠎⠀⠀⠀⠀⠀⠀⠀⠀⡴⣋⠶⣩⢎⡳⣭⢳⡽⣩⢟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⢡⠊⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡜⡀⠀⠀⠀⠀⢄⠠⢂⡟⡴⣩⢳⡕⢮⡵⣋⢷⠺⣵⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠃⡜⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠜⡠⢡⠀⠀⢂⠜⡠⢃⡽⣺⢵⡣⣏⢞⡧⢯⡝⣮⢻⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠈⠐⡁⢆⡘⠤⢈⢆⠠⣁⠂⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣐⡂⠔⡨⢘⡌⢳⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⢍⠲⡰⡸⢉⢆⠉⡼⣃⠷⣏⠞⣧⡱⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢡⢊⡱⠒⠁⠘⢈⡷⣱⠭⣖⠻⡞⡴⣏⢷⠬⠄⠂⣃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡗⣯⡝⣯⠽⣞⡹⣎⢷⣚⢖⡚⡦⣝⢮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⣓⢾⡱⣻⢼⡳⠽⣎⢷⢪⡝⡶⣫⢞⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢭⢶⣹⢣⣏⡝⠁⢛⣆⠳⡜⡇⠱⣯⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣎⢧⡏⢷⡺⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⠽⣬⢳⡞⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⠻⣜⡳⡞⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⣖⣱⢻⣜⡳⣝⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⢳⡹⣎⠷⣎⢷⡹⢮⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢘⠰⡈⠔⡡⠌⠐⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⡃⠐⠀⠘⠤⡘⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣮⠘⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣌⠵⣋⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⢈⠂⠄⡈⠹⠎⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⠐⡰⢀⠁⢣⡀⠀⢈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡔⠈⠦⡬⡄⠌⡉⡄⠐⡈⠓⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⢫⠑⡄⠀⠘⠐⠉⠎⡴⢠⠑⠠⠈⠧⣀⡜⣏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠒⠻⠶⣦⣄⠈⠡⠙⡄⠡⠐⠀⡞⠘⣎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠈⡒⠠⠁⢌⠃⠰⠁⠂⠤⠀⣠⢄⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⢧⠂⠤⡐⠤⢁⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣥⣂⣄⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⠉⡀⢀⠂⡜⠁⠠⢈⠱⣋⣎⠁⢀⠔⣌⠒⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡶⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠂⡐⠠⠁⢂⠐⣠⢋⠄⡈⡐⢀⠂⡑⢮⡹⠀⠘⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡸⢄⡹⡁⠀⠀⠀⠈⠐⢁⠆⡘⠤⣈⢆⡐⡠⠁⢌⡠⢸⡡⢎⡼⠁⡐⠤⢃⠔⣰⡙⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⠒⢥⢊⠴⣩⠓⢦⡐⢄⡠⣀⡊⠔⡰⠐⡎⣔⡦⣛⢮⠁⣧⡙⣎⡀⢦⡙⠜⣀⡴⣣⠝⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⢣⡝⢂⢎⠒⡤⢋⠴⣘⢢⠑⡤⡹⢉⢆⠉⡶⣃⢷⣋⡎⢷⡰⡑⢦⠙⣢⢯⡝⡱⢞⡥⠻⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⠎⢣⢌⢃⠎⡱⡘⡌⡖⢡⠃⣍⠒⠁⠘⢈⡷⣱⢭⠲⡝⣮⢣⡟⡵⠮⠄⢁⢊⢽⡱⢫⡜⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⠎⡅⢎⡌⢮⡑⠴⡁⡇⠀⠋⠀⠀⠀⠀⣠⠏⡧⣏⢿⡹⢧⡟⣼⣹⢳⣞⠲⣎⢧⡹⢧⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡜⢦⣙⢦⣋⠷⣬⠁⠀⠀⠀⠀⠀⠠⢡⡼⡱⢎⡷⣹⢧⢻⢶⡹⣳⢎⣳⢚⡶⢏⡿⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡝⡮⢞⡲⣍⡻⠘⠀⠀⠀⠀⠀⣠⠞⣧⢳⡙⣎⠷⣳⢎⡯⡕⠁⠹⣆⠱⣋⡆⠙⡾⠈⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣬⢻⣥⢻⡜⡇⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⡝⣦⢻⡴⣫⢞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⡼⣱⠺⡵⣫⢶⡹⣎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⠾⣔⡣⣏⢷⡹⢶⡹⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⢄⠣⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⣚⡵⣛⡼⣣⢟⣣⢟⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢢⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⣢⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠀⠄⠂⡱⢈⠆⠡⠒⠐⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⡗⠈⠀⠈⠐⢢⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣁⣡⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⡾⣄⠙⣂⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡯⠱⣿⢯⡿⡽⣞⠀⢠⠝⣥⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⠟⠀⠡⢻⣯⠟⡽⠆⠈⠄⡐⠈⢏⠀⠈⠔⡠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣽⢯⡷⠋⠤⠄⠐⢛⡟⢨⡇⠠⠈⠰⡄⠡⠈⢦⡀⡀⠜⢵⣊⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣟⣿⣯⣿⣷⢿⣽⣻⢾⣽⣻⠽⠋⢀⣀⣤⣤⣤⡘⠀⠊⡇⡆⢡⠂⠱⢦⡩⢄⠠⣁⠂⠄⡈⡆⠐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣿⣻⣽⢷⣟⣯⢷⣯⢿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠂⠺⠑⣌⠀⢀⠀⠉⠢⡑⠆⡌⡐⢀⠱⠄⡴⣙⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣽⣻⣞⡿⣾⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠛⠻⢶⣄⠈⠈⠰⣁⠀⠂⠄⡇⠃⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⡿⣼⡳⣏⡷⣽⣻⢶⣯⣷⢯⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠡⢈⠐⢠⠃⠰⠁⠂⠤⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⢻⣽⣳⡿⣽⣻⢷⣯⣟⡾⣿⣿⣳⢿⣾⠠⢀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢈⠌⠠⢁⠣⣀⣀⢞⡱⢎⣇⠆⡠⢐⠢⢁⠀
⠀⠀⠀⠀⠀⣾⢷⣻⣟⡷⢯⡟⣷⣯⣟⣿⢳⣯⣟⣾⣗⣤⣢⣄⣀⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠂⠌⠀⠄⢂⡜⠁⡀⢩⠘⡧⢎⠀⣀⠢⡑⠂⠀
⠀⠀⠀⠀⠀⢻⣟⢯⡝⣺⢣⡛⣝⢺⡽⣾⡙⣿⣾⢯⡍⢿⡟⢿⠿⢿⡿⡏⠌⠐⠆⡤⠠⡀⠤⡀⢆⡘⠤⠃⠠⢀⠡⢈⠐⡬⢣⠀⢂⡐⢀⠂⡙⢎⡳⠀⠁⠁⠀⠀
⠀⠀⠀⠀⠀⠘⣭⢲⡙⢦⢣⡝⣌⠣⢔⠢⢝⠢⡅⢛⠢⣄⠳⡐⠎⡴⠸⣁⠀⠀⠀⠀⠁⠘⡐⡐⠢⢄⢣⡈⡔⠂⣐⡠⢚⡥⢣⡝⠂⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢪⢧⡙⣎⠳⡜⡤⢋⢆⡙⣂⠧⣘⢡⠣⠜⡢⢍⡜⣐⠣⢣⢓⠲⣐⠠⢠⢈⡰⠁⡅⢊⠦⣑⡖⣯⢱⡀⡯⣜⢣⡐⠴⣈⠖⣁⡴⣣⢛⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠽⣌⢳⡹⣜⢣⠞⡴⣡⠒⡡⢆⡙⢮⡱⢆⠲⣈⠥⢃⢎⠱⣄⢋⠲⢌⠼⡉⢆⠉⡖⢧⠻⣡⢳⢳⡱⢌⠦⡙⢲⣋⣞⡱⢞⡥⢫⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠓⠮⢣⡻⣔⣣⢛⡴⣡⠞⡥⠛⣌⠱⡂⡍⢎⡰⠓⠀⡏⡜⠜⠁⠘⠈⢧⡝⣎⣳⢖⣏⢎⡟⣮⠶⠄⡁⢃⢧⡝⢮⡜⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠉⠈⠁⠛⠲⣵⢋⡖⡩⢄⢣⡱⡜⢢⢡⡋⠀⠉⠀⠀⠀⠀⣨⠏⡟⣼⢣⡻⡼⢮⣝⢮⡳⣖⢖⡫⢖⡹⣞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣧⢚⡵⣊⢧⠳⣍⠳⣆⠃⠀⠀⠀⠀⠀⠠⣁⢶⡹⡜⣧⣛⣭⢻⣜⡳⣝⢮⢎⣵⠫⣷⣹⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡶⢭⢞⡳⣍⣻⠘⡀⠀⠀⠀⠀⢀⡶⣙⢦⢳⡙⡶⣭⣚⢧⡝⠀⢛⣆⢩⢎⡇⢱⢳⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢏⡞⡴⠉⠀⠀⠁⠂⠤⠀⢴⣋⠶⣍⡎⢧⡹⡵⣎⡳⢏⡆⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡊⠔⡡⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢫⡜⡁⠀⠀⠂⠀⡀⢀⠀⠢⡝⢮⡱⢏⡷⣱⢳⡭⣝⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢹⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛⡦⡝⢮⡱⣏⡳⣝⢮⠞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡘⢄⠀⠀⠀⠀⢠⠀⠆⣵⣊⠷⡸⡝⢦⡳⣭⢳⡝⣮⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠱⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡰⢁⠎⡀⠀⠐⡠⢃⠜⣸⣓⢮⡳⣕⢯⡳⣝⢮⡳⣝⢮⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢢⡁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣢⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⡀⠠⠀⠂⠂⡑⠌⠢⠑⠈⠒⠀⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⣯⠀⠈⠀⠘⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣞⡬⠑⠂⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⡍⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠸⣷⢯⡿⡽⣞⠀⡰⢌⡳⣍⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⠟⠀⠃⢻⣯⠟⡽⠂⢈⠀⠄⡀⢋⠀⠀⠒⡄⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⢾⡽⣯⠷⠋⠤⠄⠐⢛⡏⢘⡧⠁⡀⠘⡆⠐⡈⠦⣄⠠⠘⢣⠞⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⢯⡿⣽⢯⠿⠉⢀⣀⣤⣤⣤⡘⡀⠉⢇⡇⢠⠁⠙⠲⣜⢠⡀⢆⠁⠂⠌⡃⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⣳⢿⣯⢿⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠁⠈⠋⡄⢣⠀⣀⣀⠈⠑⠘⣄⠣⡁⠂⠙⠤⢲⢥⡛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⣽⣳⢯⡿⣞⡿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠛⠳⢦⡀⠀⠑⡠⠁⠌⠠⡍⠂⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⣻⢾⡽⣾⡽⣿⣽⡿⣯⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡑⠈⠄⢡⠃⡐⡁⠁⠆⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⢻⡽⣯⣟⡷⣯⢿⣽⣳⣟⣿⣷⣻⡽⣾⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡈⠆⠐⠠⢑⡤⣀⠞⣜⡣⣝⠢⡐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣯⡿⡽⢯⣷⣻⣞⣿⣳⢯⣟⣷⢧⣑⣦⣠⣀⡀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠃⡈⠐⠠⠀⠄⡡⠎⢀⠠⠘⠠⡗⣎⡀⢠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⢻⣯⠟⡶⣹⢚⢧⣋⠷⣯⢷⡸⢿⣟⣾⡋⢿⡟⢿⠿⢿⡿⡏⠈⠁⠑⠢⡐⢄⠢⡐⠌⠒⠤⠃⡐⠠⢁⠂⡁⡞⡱⣀⢂⠄⡁⢂⠘⢦⡹⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣣⢛⡴⢣⢏⠶⣉⠎⡔⡊⠵⢋⠴⡙⠢⣄⢓⡰⢊⠤⡙⡅⠀⠀⠀⠀⠀⢨⠐⠤⡉⢌⡣⡐⠤⢁⣢⠔⣸⢡⠳⡌⢁⢂⠴⣁⠊⣰⢍⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢣⢏⡜⣣⢎⡳⣌⢒⡡⣙⠸⠌⠦⣑⠣⡜⣂⠖⣉⠲⣁⠯⡱⢒⡰⢀⠤⡈⡌⠒⡈⠤⢣⡹⢞⡝⢦⡃⢵⣋⠳⡠⢎⡜⣂⣄⠶⣙⡎⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠼⡱⢎⡵⣊⢗⡲⢬⡑⢎⡑⢢⠱⣆⢧⡚⠤⢓⡰⢌⡱⠌⢆⡍⠦⡑⢼⠉⢇⠉⠷⣱⠫⡙⡶⣹⢖⡌⢧⠑⢎⡳⣍⢮⡹⣱⠺⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠒⠭⢎⡵⣣⠝⣦⡙⢦⡛⣌⠲⣉⠜⢢⠱⢌⠢⡝⠀⡇⡎⠕⠁⠘⠈⡷⣜⢧⣫⣕⣳⢥⡻⣼⠲⠈⠄⣙⡸⢆⣳⡱⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠈⠉⠉⠒⠽⡶⣙⠦⡱⣈⢎⡱⢎⠬⡑⡅⠀⠋⠀⠀⠀⠀⣠⠏⣧⢛⡶⣹⢞⣮⢳⣭⢻⡝⡲⣍⠞⣭⢶⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠱⣍⢞⡱⢆⠯⡜⣭⠲⣥⠃⠀⠀⠀⠀⠀⠠⢡⡼⣡⠻⣜⡳⣞⡼⣳⠞⣧⢻⡱⢎⡟⢧⡻⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢎⡵⢫⢞⡳⣍⢻⠘⡀⠀⠀⠀⠀⢠⣜⠳⡜⣥⢛⡼⣳⡱⣏⡅⠀⠸⢣⠙⢎⣇⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠄⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣛⢬⡓⠈⠁⠀⠐⠂⠤⠀⠴⣫⠜⣧⣙⢦⣋⢾⡱⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢆⠃⠀⠈⠠⠀⡀⢀⠀⠈⣧⢛⡴⢫⠶⣭⠞⣵⢫⡞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠼⡎⠀⠀⠀⠀⠀⠀⠀⠀⡰⢎⡳⡜⣭⢛⡼⣛⡼⣣⢟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡔⢡⠊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡘⠄⠀⠀⠀⠀⢠⠀⢆⣱⣋⢧⢳⣙⢦⣫⢞⡵⣫⢵⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠆⡱⠰⡀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⡠⠑⡌⠄⠀⠐⡈⢆⡉⢶⣣⢏⡮⣇⡻⡼⣥⢻⡜⣧⢯⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢢⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⣢⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠐⢂⠘⡄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢟⡯⠀⠀⠈⠐⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣁⣡⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣞⢆⠙⠄⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡯⠱⣿⢯⡿⡽⣞⡀⡰⠜⣣⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⠟⠀⠡⢻⣯⠟⡽⠂⠠⢀⠐⠈⢊⠀⠈⠅⢢⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣽⢯⡷⠋⠤⠄⠐⢛⡟⢨⡇⠠⢀⠂⠇⡈⠌⠢⣄⠠⠚⠴⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣟⣿⣯⣿⣷⢿⣽⣻⢾⣽⣻⠽⠋⢀⣀⣤⣤⣤⡘⠀⠊⢇⡆⢰⠂⠙⠲⣌⠥⣀⠒⡀⠂⠄⡃⠈⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣿⣻⣽⢷⣟⣯⢷⣯⢿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠈⠳⣈⠣⢀⣀⡁⠈⠑⢌⠒⣄⠃⠄⠙⠤⡲⢥⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣽⣻⣞⡿⣾⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢦⡀⠈⠰⣈⠐⢈⠠⡍⠈⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢠⡿⣼⡳⣏⡷⣽⣻⢶⣯⣷⢯⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⡈⠄⢲⠁⠌⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⢻⣽⣳⡿⣽⣻⢷⣯⣟⡾⣿⣿⣳⢿⣾⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢐⠊⠀⠌⢓⡄⣀⠷⣘⢧⣓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣾⢷⣻⣟⡷⢯⡟⣷⣯⣟⣿⢳⣯⣟⣾⣗⣤⣢⣄⣀⡀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠁⠄⡁⠂⠄⠠⣈⠆⡐⠠⠘⡘⢦⣃⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⢻⣟⢯⡝⣺⢣⡛⣝⢺⡽⣾⡙⣿⣾⢯⡍⢿⡟⢿⠿⢿⡿⡏⠌⠈⠐⠅⢢⠐⡄⢢⠘⡐⠰⠁⠌⡐⠠⠁⢌⡖⣡⢂⡐⠠⠁⢄⠑⢮⡱⠀⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣭⢲⡙⢦⢣⡝⣌⠣⢔⠢⢝⠢⡅⢛⠢⣄⠳⡐⠎⡴⠸⣁⠀⠀⠀⠀⠀⠒⡈⠔⢂⠡⢇⡌⠔⢠⣡⠌⡴⢪⡔⡣⠁⢂⢍⡠⠂⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢪⢧⡙⣎⠳⡜⡤⢋⢆⡙⣂⠧⣘⢡⠣⠜⡢⢍⡜⣐⠣⢣⢓⠲⣐⠠⢠⢐⡡⠊⡄⢃⡎⡼⡞⣧⠳⡄⣏⢳⡘⡡⢜⣡⢊⣀⢶⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠽⣌⢳⡹⣜⢣⠞⡴⣡⠒⡡⢆⡙⢮⡱⢆⠲⣈⠥⢃⢎⠱⣄⢋⠲⢌⢲⠉⣎⠡⣞⡱⠻⡰⣋⠷⣌⢣⡜⠑⡎⣧⡛⣬⢳⡱⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠓⠮⢣⡻⣔⣣⢛⡴⣡⠞⡥⠛⣌⠱⡂⡍⢎⡰⠓⠀⡏⡜⠜⠁⠘⠈⡗⣬⢳⣋⣖⢯⣚⡽⣲⠼⢀⠐⣑⡹⢆⡧⡝⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠉⠈⠁⠛⠲⣵⢋⡖⡩⢄⢣⡱⡜⢢⢡⡋⠀⠉⠀⠀⠀⠀⣠⠏⡗⣯⢞⡼⣫⢞⡵⣫⢟⡵⢪⠵⣙⡮⣞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣧⢚⡵⣊⢧⠳⣍⠳⣆⠃⠀⠀⠀⠀⠀⠠⣁⡞⣝⠲⣏⢾⣱⢻⣜⡳⣏⠾⣩⢾⡱⢻⡼⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡶⢭⢞⡳⣍⣻⠘⡀⠀⠀⠀⠀⢀⡶⡹⡜⣬⠳⣝⢮⠧⣏⡅⠀⠩⢧⠱⢎⣃⠱⡫⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢏⡞⡴⠉⠀⠀⠁⠂⠤⠀⢴⣋⠶⣣⡝⢦⡛⣼⣋⢾⡱⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡊⠔⡡⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢫⡜⡁⠀⠀⠂⠀⡀⢀⠀⠢⡝⢮⡱⢏⡳⣝⢶⡹⢮⣝⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢹⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛⡦⡝⣎⠳⣝⢮⣝⡳⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡘⢄⠀⠀⠀⠀⢠⠀⠆⣵⣊⠷⡸⣕⢮⡹⣎⠷⢮⣝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠱⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡰⢁⠎⡀⠀⠐⡠⢃⠜⣸⣓⢮⡳⣕⣫⢞⡵⣫⢏⡷⢎⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢢⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⣢⣿⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠐⢂⠘⡄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢟⡯⠀⠀⠈⠐⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣁⣡⣶⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣞⢆⠙⠄⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡯⠱⣿⢯⡿⡽⣞⡀⡰⠜⣣⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⠟⠀⠡⢻⣯⠟⡽⠂⠠⢀⠐⠈⢊⠀⠈⠅⢢⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣽⢯⡷⠋⠤⠄⠐⢛⡟⢨⡇⠠⢀⠂⠇⡈⠌⠢⣄⠠⠚⠴⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣟⣿⣯⣿⣷⢿⣽⣻⢾⣽⣻⠽⠋⢀⣀⣤⣤⣤⡘⠀⠊⢇⡆⢰⠂⠙⠲⣌⠥⣀⠒⡀⠂⠄⡃⠈⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣿⣻⣽⢷⣟⣯⢷⣯⢿⣾⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠈⠳⣈⠣⢀⣀⡁⠈⠑⢌⠒⣄⠃⠄⠙⠤⡲⢥⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣽⣻⣽⣻⣞⡿⣾⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢦⡀⠈⠰⣈⠐⢈⠠⡍⠈⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢠⡿⣼⡳⣏⡷⣽⣻⢶⣯⣷⢯⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⡈⠄⢲⠁⠌⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⢻⣽⣳⡿⣽⣻⢷⣯⣟⡾⣿⣿⣳⢿⣾⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢐⠊⠀⠌⢓⡄⣀⠷⣘⢧⣓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣾⢷⣻⣟⡷⢯⡟⣷⣯⣟⣿⢳⣯⣟⣾⣗⣤⣢⣄⣀⡀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠁⠄⡁⠂⠄⠠⣈⠆⡐⠠⠘⡘⢦⣃⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⢻⣟⢯⡝⣺⢣⡛⣝⢺⡽⣾⡙⣿⣾⢯⡍⢿⡟⢿⠿⢿⡿⡏⠌⠈⠐⠅⢢⠐⡄⢢⠘⡐⠰⠁⠌⡐⠠⠁⢌⡖⣡⢂⡐⠠⠁⢄⠑⢮⡱⠀⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣭⢲⡙⢦⢣⡝⣌⠣⢔⠢⢝⠢⡅⢛⠢⣄⠳⡐⠎⡴⠸⣁⠀⠀⠀⠀⠀⠒⡈⠔⢂⠡⢇⡌⠔⢠⣡⠌⡴⢪⡔⡣⠁⢂⢍⡠⠂⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢪⢧⡙⣎⠳⡜⡤⢋⢆⡙⣂⠧⣘⢡⠣⠜⡢⢍⡜⣐⠣⢣⢓⠲⣐⠠⢠⢐⡡⠊⡄⢃⡎⡼⡞⣧⠳⡄⣏⢳⡘⡡⢜⣡⢊⣀⢶⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠽⣌⢳⡹⣜⢣⠞⡴⣡⠒⡡⢆⡙⢮⡱⢆⠲⣈⠥⢃⢎⠱⣄⢋⠲⢌⢲⠉⢆⠉⡼⣱⠻⠰⣋⠷⣌⢣⡜⠑⡎⣧⡛⣬⢳⡱⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠓⠮⢣⡻⣔⣣⢛⡴⣡⠞⡥⠛⣌⠱⡂⡍⢎⡰⠓⠀⡏⡜⠜⠁⠘⢈⡳⣜⡥⣏⣳⢏⡞⣽⢲⠼⢀⠐⣑⡹⢆⡧⡝⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠉⠈⠁⠛⠲⣵⢋⡖⡩⢄⢣⡱⡜⢢⢡⡋⠀⠉⠀⠀⠀⠀⣠⠏⣧⢻⡜⣧⣛⢾⡱⣏⢿⡱⢎⡵⣙⡮⣞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣧⢚⡵⣊⢧⠳⣍⠳⣆⠃⠀⠀⠀⠀⠀⠠⣁⢶⡙⢮⡝⣶⡹⢾⣱⡛⣮⢳⡭⢲⡝⢾⡱⢆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡶⢭⢞⡳⣍⣻⠘⡀⠀⠀⠀⠀⢀⡶⣙⢦⣋⢧⣛⢶⡹⢧⡅⠀⠱⢧⠸⢙⡆⠩⡽⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢏⡞⡴⠉⠀⠀⠁⠂⠤⠀⢴⣋⠶⣍⠶⣩⠖⡽⢮⣝⡳⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡊⠔⡡⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢫⡜⡁⠀⠀⠂⠀⡀⢀⠀⠢⡝⢮⡍⢿⣡⢯⡝⣾⡸⣭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢹⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛⡦⣙⢦⡛⣼⡹⡖⡽⢎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡌⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡘⢄⠀⠀⠀⠀⢠⠀⠆⣵⣊⠷⡸⣕⢮⡱⢧⣛⡭⣝⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠱⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡰⢁⠎⡀⠀⠐⡠⢃⠜⣸⣓⢮⡳⣕⣫⢞⡝⣧⣛⡼⡭⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢢⡁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣢⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠠⠐⠐⢢⠘⡄⠢⠑⠈⠂⠁⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⡍⠐⠀⠁⠒⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣎⡌⠳⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⡍⢡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠸⣷⢯⣟⢷⣆⠀⡨⢖⡭⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⣎⣴⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⡾⠟⠀⠃⢻⣯⢏⡯⠄⠈⠄⠠⠘⠇⠀⠡⢂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⢾⡽⣯⠷⠋⠤⠄⠐⢛⡏⢰⢃⠀⢈⠸⡀⠡⠘⣄⡀⡀⠞⣥⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⢯⡿⣽⢯⠿⠉⢀⣀⣤⣤⣤⡘⡀⠉⢎⡐⠄⠂⢓⠦⣱⠀⠤⡁⠄⠠⠉⠆⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⢿⣟⣿⡽⣿⣳⢿⣯⢿⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠊⢫⠐⡍⠀⢀⠈⠐⠋⢖⠡⣘⠀⡁⠣⠤⠴⣍⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣷⢯⡿⣯⢿⣽⣳⢯⡿⣞⡿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠛⠻⠶⣤⠀⠑⠌⣐⠠⠁⠄⣫⠐⣭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣭⠿⣭⣻⢾⡽⣾⡽⣿⣽⡿⣯⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⠄⡈⢰⠃⢠⠁⠐⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⢻⡽⣯⣟⡷⣯⢿⣽⣳⣟⣿⣷⣻⡽⣾⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠇⠐⡀⠓⣄⡠⡜⣜⡣⣝⠢⡐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣾⢿⣽⣯⡿⡽⢯⣷⣻⣞⣿⣳⢯⣟⣷⢧⣑⣦⣠⣀⡀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡈⠠⠀⠄⢠⡙⠀⡀⠰⠈⡗⣎⡀⢠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⢻⣯⠟⡶⣹⢚⢧⣋⠷⣯⢷⡸⢿⣟⣾⡋⢿⡟⢿⠿⢿⡿⡏⠈⠐⠢⢐⠄⡠⠄⡠⢂⠜⣠⠃⢀⠂⠄⠡⠈⡴⢣⣀⢂⠄⡁⢂⠉⠶⣹⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣣⢛⡴⢣⢏⠶⣉⠎⡔⡊⠵⢋⠴⡙⠢⣄⢓⡰⢊⠤⡙⡅⠀⠀⠀⠀⠈⢰⠈⡔⢂⠢⣅⡐⠤⠊⣠⡡⢸⡡⢧⡜⠂⡐⡔⢡⠊⢴⢣⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢣⢏⡜⣣⢎⡳⣌⢒⡡⣙⠸⠌⠦⣑⠣⡜⣂⠖⣉⠲⣁⠯⡱⢒⡰⢀⠤⣈⡰⠐⡌⠰⢢⡍⡶⣫⢕⡂⢧⡝⢲⢠⢒⡱⣈⣄⠖⣯⠲⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠼⡱⢎⡵⣊⢗⡲⢬⡑⢎⡑⢢⠱⣆⢧⡚⠤⢓⡰⢌⡱⠌⢆⡍⠲⢄⢳⠉⣆⠉⢧⢣⠳⣋⡎⢷⣎⠜⡥⠊⠖⡯⣕⢮⡹⢲⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠒⠭⢎⡵⣣⠝⣦⡙⢦⡛⣌⠲⣉⠜⢢⠱⢌⠢⡝⠀⡧⣉⠎⠁⠘⠠⣳⢹⣎⣳⢣⡟⢦⣛⠾⡴⠁⠌⣑⣩⠶⣩⠳⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠈⠉⠉⠒⠽⡶⣙⠦⡱⣈⢎⡱⢎⠬⡑⡅⠀⠓⠀⠀⠀⠀⣠⠏⡳⣎⡳⣏⡞⣧⢏⡿⣱⢻⠲⣍⠶⣭⢳⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠱⣍⢞⡱⢆⠯⡜⣭⠲⣥⠃⠀⠀⠀⠀⠀⠠⢡⡜⢧⡹⡵⣣⢟⡼⣫⠞⣵⣫⣓⢮⡝⢮⡻⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢎⡵⢫⢞⡳⣍⢻⠘⡀⠀⠀⠀⠀⢠⡜⢧⡙⢶⡱⢯⢵⣫⢞⡍⠀⠸⢥⠘⢎⡇⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠄⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣛⢬⡓⠈⠁⠀⠐⠂⠤⠀⠴⣫⠜⣧⡹⢦⣙⢮⡳⢮⡝⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢆⠃⠀⠈⠠⠀⡀⢀⠀⠈⣧⢛⡴⢫⡗⣮⢳⡝⣧⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠤⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠼⡎⠀⠀⠀⠀⠀⠀⠀⠀⡰⢎⡳⣜⢣⡝⢮⡳⣝⢮⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡔⢡⠊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡘⠄⠀⠀⠀⠀⢠⠀⢆⣱⣋⢧⢳⣬⠳⣜⢧⡻⣜⢧⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠆⡱⠰⡀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⡠⠑⡌⠄⠀⠐⡈⢆⡉⢶⣣⢏⣮⠳⣜⡻⣜⢧⡻⣜⣫⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⣿⡋⠂⠂⢤⠀⡄⠠⡀⢄⠰⡀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⣌⠒⡬⡁⠀⠀⠀⠈⠐⢡⠘⡠⢂⠅⣆⠈⡤⠁⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⡨⢜⠰⣍⠳⢢⡐⢄⡠⣀⠣⡐⢡⠘⡤⢋⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠱⡌⡱⢄⠫⡔⣘⢢⡑⡌⡱⠙⢆⠉⡶⢇⢟⣧⢋⣧⣓⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⢡⢃⠖⡡⢎⠱⡔⢡⠣⣰⠑⠁⠉⠘⡧⣝⢮⠲⡽⣎⠶⣹⢶⠬⠐⠠⢃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢆⡩⢲⡅⢎⡡⡃⠀⠋⠀⠀⠀⠀⣨⠏⣏⢾⡹⣝⢮⣝⢧⣏⣟⣲⢓⢮⠳⣜⢧⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⣣⢞⡱⢆⠃⠀⠀⠀⠀⠀⠠⢡⣜⢣⢏⡳⣝⠾⣼⢳⠾⣜⢧⣋⢾⡱⢯⡞⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⡬⢗⡣⢏⡽⢋⠀⠀⠀⠀⠀⣠⢗⡫⣜⢣⠞⣵⢫⡝⣮⠝⠈⠹⣆⢩⠞⡅⢣⢟⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣥⡛⠈⠁⠀⠐⠂⠤⠀⢴⢣⠮⣕⡎⢧⡛⣼⢣⡟⣼⠃⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡼⢭⡳⣝⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⣜⢣⡝⣮⢳⡝⣮⢳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⣌⡳⣜⢧⡻⣜⢧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⢣⡟⣦⡝⣮⢳⡝⣮⢳⡝⣮⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠈⠐⡁⢆⡘⠤⢈⢆⠠⣁⠂⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣐⡂⠔⡨⢘⡌⢳⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⢍⠲⡰⡸⢉⢆⠉⡼⣃⠷⣏⠞⣧⡱⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢡⢊⡱⠒⠁⠘⢈⡷⣱⠭⣖⠻⡞⡴⣏⢷⠬⠄⠂⣃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡗⣯⡝⣯⠽⣞⡹⣎⢷⣚⢖⡚⡦⣝⢮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⣓⢾⡱⣻⢼⡳⠽⣎⢷⢪⡝⡶⣫⢞⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢭⢶⣹⢣⣏⡝⠁⢛⣆⠳⡜⡇⠱⣯⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣎⢧⡏⢷⡺⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⠽⣬⢳⡞⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⠻⣜⡳⡞⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⣖⣱⢻⣜⡳⣝⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⢳⡹⣎⠷⣎⢷⡹⢮⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠈⠐⡁⢆⡘⠤⢈⢆⠠⣁⠂⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣐⡂⠔⡨⢘⡌⢳⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⢍⠲⡰⡸⢉⢆⠉⡼⣃⠷⣏⠞⣧⡱⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢡⢊⡱⠒⠁⠘⢈⡷⣱⠭⣖⠻⡞⡴⣏⢷⠬⠄⠂⣃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡗⣯⡝⣯⠽⣞⡹⣎⢷⣚⢖⡚⡦⣝⢮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⣓⢾⡱⣻⢼⡳⠽⣎⢷⢪⡝⡶⣫⢞⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢭⢶⣹⢣⣏⡝⠁⢛⣆⠳⡜⡇⠱⣯⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣎⢧⡏⢷⡺⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⠽⣬⢳⡞⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⠻⣜⡳⡞⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⣖⣱⢻⣜⡳⣝⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⢳⡹⣎⠷⣎⢷⡹⢮⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⠜⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠐⣼⣶⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⡀⠀⠄⠒⢌⠰⡈⠔⠡⠌⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⠃⠀⠂⠡⠒⠄⠀⠀⠠⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡽⣎⠙⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏⢾⡽⣯⣟⢮⠇⠀⡥⣝⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⠏⠈⠄⣿⣳⢹⡏⢈⠂⡐⠈⢱⠃⠀⠆⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣟⡾⣽⣻⠞⡁⠤⠀⠒⢻⠏⢸⠈⠀⠐⢤⠁⠂⢣⡀⠀⠠⢧⣙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠛⠿⠟⠛⠛⢻⣿⣿⢿⣟⣿⣽⣿⣾⢯⣟⣾⣳⣯⣟⠷⠉⣀⣀⣤⣤⣄⡘⡠⢋⢢⠁⡌⠐⠦⣅⠆⡈⢩⠐⡀⢉⠺⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣻⣯⢷⣟⣿⡽⣞⣷⣿⡆⠀⠉⠉⠉⠁⠀⠀⠀⠡⡒⡩⠒⡄⠀⠈⠐⠹⢘⢢⢡⠐⠠⠘⣄⡠⣝⡹⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣳⢯⣷⣻⢷⣯⣟⡾⣽⣻⡽⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠿⢶⣦⡀⠁⠂⠇⡄⠡⢀⢀⡎⠱⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣸⢏⡿⣞⣭⣟⡾⣵⣻⣽⣷⣻⡿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠘⠄⠡⢀⢸⠀⠱⠁⠡⢀⢀⣠⣀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠒⡿⣯⢿⡽⣾⣽⣻⢷⣻⢾⣿⣽⣻⡽⣧⢂⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠀⠠⠎⢀⠉⠦⣀⣀⠞⡴⣩⠇⡤⢀⠆⡘⠀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣻⢷⡳⢯⣿⡽⣯⡟⣷⢯⡿⣽⣆⣜⣠⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⢂⠁⠠⠀⢌⠒⡀⢈⠩⢸⢥⡋⠀⡠⠢⡡⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⢗⡫⣝⢣⡛⡽⢷⣯⠹⣿⣻⣿⠉⣿⠿⢿⠿⢿⡷⡇⠂⠤⡀⢄⠠⢀⠄⡠⢐⡐⢢⠁⠌⡐⠠⠈⢄⡱⢊⠐⣀⠂⡐⢀⠣⡝⣥⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⢣⢎⡵⣊⢧⡙⠜⡰⣈⠟⡰⢡⡙⢤⡈⢖⡰⢊⢆⡹⡁⠀⠀⠀⠈⠐⡡⠌⡐⢡⢐⡃⣀⠢⢀⢡⠈⡼⡰⣍⠞⢀⠐⡔⠡⠂⡼⣜⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢜⡣⢏⠶⣩⠖⣌⠣⡑⠆⢎⡡⢃⡜⠤⣙⢂⠖⡡⢎⠰⣍⢓⢢⡐⠤⣀⢡⡘⠠⢃⢸⠰⣡⡖⣫⢽⠐⣧⠳⡜⢠⢆⡹⢈⡡⣜⡳⣜⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠎⡳⢥⡛⣬⢳⡹⡜⡢⢅⠣⠜⡲⢥⢎⡜⢡⢊⠵⢠⠋⡔⣬⠑⡢⢅⡞⠹⢄⠉⡞⡥⠞⣵⢪⣇⡎⡱⢌⠣⣚⡵⣫⠕⣮⢱⡚⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠓⠛⠴⢣⡳⣍⠳⣎⢥⢳⡹⣘⢊⠬⣑⠪⠔⢣⣉⡖⢠⢣⡱⠊⠀⠑⠌⡶⣝⡹⢲⣛⠶⣸⠽⡼⠬⢀⠡⠚⣥⡛⣴⠃⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠄⠉⠈⠉⠚⢮⡗⡱⢆⡩⢒⣌⠲⣍⠒⡤⠃⠀⠋⠀⠀⠀⠀⣜⢋⡼⣹⢯⡹⣏⡾⣹⣝⡳⣖⠶⣛⢴⡹⠞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢹⡱⢫⡔⣣⢎⡳⢎⡳⣜⠀⠀⠀⠀⠀⠀⡈⣔⢮⡱⡝⣶⡹⣎⢷⡳⢮⢳⣎⠳⣭⠾⣹⠷⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣕⡮⡵⢺⡱⢫⡝⠪⠀⠀⠀⠀⠀⣠⢞⡱⢎⡵⡹⢶⡹⣎⢯⡑⠈⢳⠇⢹⢲⡀⢹⠇⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⢢⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡏⢧⠝⠉⠀⠀⠑⠠⠄⠠⢚⡵⢪⣕⡫⢖⡭⣳⡝⣮⢳⠂⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡼⠀⠀⠀⠄⢀⡀⢀⠀⠍⣞⡱⢮⡝⡾⣴⢣⡟⣜⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠔⡨⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⣟⠒⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⡝⢦⡹⣱⢎⡷⣹⢎⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⢡⠊⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡌⠂⠀⠀⠀⠀⢄⠠⢂⡟⣜⢣⡝⡶⣱⡝⣮⢳⡝⣮⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠃⡜⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠜⡠⢃⠄⠀⠂⡜⢠⢃⡽⡺⡵⣣⢞⡵⡳⣞⡱⣏⢾⡱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢆⠌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣿⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠠⠀⠒⢠⢊⠱⠐⠌⠂⠘⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣟⡇⠈⠀⠈⠔⢂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣞⡄⠻⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡍⠄⡀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡛⢺⣽⣳⡿⣝⡞⠀⢨⠖⣭⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⡾⠏⠐⢁⢻⣷⢻⡝⢂⠉⡀⠐⡈⠏⠀⠈⡔⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣟⡾⣽⣻⡞⡁⠤⠄⠐⢻⡏⢸⡧⠀⠐⠨⡔⢀⠘⢤⡀⡀⠜⢲⡹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠻⠿⠟⠛⠛⢻⣿⣿⢿⣟⣿⣽⣿⣾⢯⣟⣾⣳⢿⣽⠳⠃⢀⣀⣤⣤⣤⠘⡀⠋⢃⡌⢠⠁⠲⢦⡘⡄⢠⠡⠈⠄⠉⡇⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣻⣯⢷⣟⡿⣽⣞⣯⣿⡆⠀⠉⠉⠉⠁⠁⠀⠀⠁⠊⢫⠘⡔⠀⣀⠀⠉⠲⠡⡚⢄⠈⠄⠱⠄⡼⣩⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡷⣯⣷⣻⢷⣯⣟⡾⣽⡷⣯⣿⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠘⠛⠻⢶⣄⠈⠐⢊⡐⠈⠄⢂⡇⠑⣏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣸⣝⣧⡟⣭⣟⡾⣵⣻⣽⣟⣷⡿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡅⢈⠐⢠⠃⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠒⣿⣞⣷⣻⡽⣾⣽⣳⣟⡾⣿⣾⣻⡽⣧⠡⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢈⠆⢈⠐⢣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠠⣟⡾⣿⣳⡟⡷⢯⣷⢯⣟⡿⣷⢯⡿⣽⣇⣥⣢⣄⣀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⢁⠂⡀⠄⣈⠆⠠⢀⠩⢘⠧⣎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡟⡯⢇⡻⣍⢳⡹⢻⢾⢷⠹⣿⣻⣿⠉⣿⠟⢿⠿⢿⣿⡋⠃⠒⠠⡂⢄⠠⠄⡐⢄⠊⠴⠁⠌⡐⢀⠂⡐⡴⢡⠈⡐⡀⠂⢄⠙⢦⢳⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡹⣜⢣⠳⣬⢣⠝⣡⢊⡌⡛⠴⢡⡙⠢⣌⠲⣁⠎⡴⢨⠅⠀⠀⠀⠀⠁⠎⡐⠡⠌⣌⢃⡌⠰⢀⣂⡐⠼⣡⢣⡝⠠⢀⡕⢂⠌⣸⢣⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢖⡭⢎⡳⣌⢳⡘⢤⠒⣌⠱⢊⠥⣘⡑⢆⡓⢌⠲⣐⠃⢯⠱⡒⢤⠀⠤⡰⡁⠣⢌⠰⢣⡸⡕⡯⣜⡀⡟⡴⢣⡐⡔⣣⠜⣀⡴⢧⡓⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠏⡶⣩⢖⡹⢆⡻⢤⡙⡌⣒⠡⡜⢦⡚⡬⢑⢢⡙⢢⠱⣘⢢⡙⢢⠥⡹⢉⢆⠉⣗⢣⠻⡱⢎⣳⡜⡡⢇⠜⣰⢇⡯⡱⢞⡥⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠊⠷⣩⢳⡣⣝⠲⣥⢣⡝⢢⡙⠴⡉⢦⢘⡡⢃⡖⢡⢃⡣⠚⠁⠈⡘⡷⢬⣓⣳⣛⢧⢎⡽⢦⠯⠄⠁⣋⢵⡙⣎⡖⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠄⠂⠀⠀⠉⠁⠉⠓⠮⡷⣘⢇⡘⢆⡍⡲⣌⠔⠣⡅⠀⠋⠀⠀⠀⠀⣠⠏⡷⣹⠶⣹⢮⡞⣽⢫⡞⣶⠳⡜⢮⡱⣞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠳⣍⠞⡜⢦⡹⡱⢎⡽⣱⠂⠀⠀⠀⠀⠀⢠⢁⡞⡵⡹⣞⡵⣫⢾⣱⢻⡜⣧⢛⡼⠳⣝⢮⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢼⡹⠶⣳⠹⣭⢹⠚⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⣓⡳⡞⣵⢫⡕⠉⢺⣅⢳⢚⡇⢹⢺⠀⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠜⡠⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⡿⣰⠛⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⣣⠳⣥⣛⡽⣜⡳⠆⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡱⢀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣡⠃⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡼⣱⢻⡴⣹⠶⣭⡝⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠔⡡⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢟⠆⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⢶⡩⢧⣛⠶⣛⢦⡻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⢃⠒⡀⠀⠀⠀⠀⠀⠀⠀⠂⢁⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⢦⣙⠶⣭⣛⡽⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⢌⠢⢡⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⢣⡟⣦⡝⣧⢏⡟⡶⢭⡞⡽⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⡀⠠⠀⠂⠘⢠⠃⠜⠀⠃⠐⠀⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠈⠀⠊⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⣋⠀⠠⠑⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠠⠈⡘⡄⠂⠌⠢⣄⠠⠊⢝⢦⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⢬⠡⡀⢆⠡⠈⡀⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡱⢀⣀⡀⠁⠃⠙⡄⢆⠡⠐⠘⠤⣒⢧⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢦⡀⠈⠘⡄⢈⠐⠠⡍⠂⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⠠⢈⢰⠁⢌⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠊⠀⠌⠱⣄⣀⠞⣜⡣⣝⠢⡐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⢂⠁⢂⠠⠀⣌⠒⡀⠠⠘⠰⣓⢎⡀⢠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠁⠉⠂⠅⢢⠐⡄⢢⠘⡰⠰⠁⠌⠠⠈⠄⢂⡽⡐⣂⢄⠡⢈⠐⡈⢮⡹⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠀⢊⠰⡀⢃⠔⡣⠌⠔⣁⣌⡔⣸⢰⡱⠎⡀⢂⠴⣁⠂⣴⡙⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣀⡃⡘⢄⠪⡱⣩⡞⡵⢪⡄⢧⢣⡙⢤⠒⡭⢒⣀⡖⣧⠹⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⡹⢉⢆⠉⡵⢣⡹⡙⣣⢝⡮⡑⢎⠆⡹⢧⣛⡴⣙⢦⠻⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠈⠘⡗⣼⢣⣝⣱⢧⣚⡵⣻⢌⠂⠡⢑⣣⢞⡱⡎⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣨⠏⣧⢻⡜⣧⢻⣜⡳⣝⢮⡛⣖⢫⠖⣭⣞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⠠⢡⡼⢥⣛⡼⣣⢟⡼⣳⡝⣮⠽⣌⣳⡛⢗⡾⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠞⣧⡙⢦⢣⢯⡵⢫⡞⡅⠀⠸⢧⠘⢖⣃⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣴⡩⢇⡏⣶⡹⢧⡻⡅⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡴⢫⡗⣮⢳⡹⢧⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⡜⣣⠝⣮⢳⡝⣧⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⠾⡜⣥⢻⡜⣧⣛⠶⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⢄⠣⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⣚⡽⢎⡷⣹⠶⣭⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠂⢂⡘⠄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠈⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⢏⠀⠠⠑⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠀⡈⡘⡄⢈⠈⠦⣀⠠⠚⠼⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⣜⠠⣀⢃⠐⠠⠈⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡕⢀⣀⡀⠈⠑⠌⠆⢎⠀⠡⠐⠦⣪⢕⡺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀⠈⠊⡄⢁⠂⠄⡇⠈⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠂⡈⢰⠁⡘⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠐⠈⢣⣄⡀⢷⡘⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠠⢁⠂⠠⢀⡡⠆⡀⠐⠠⠙⣧⢋⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠃⠈⠐⠡⡂⠔⡠⢂⠜⣀⠲⠁⡈⠐⠠⢈⠐⡼⢰⡀⢄⠁⢂⠡⠈⢧⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠂⡅⢂⠒⣠⢣⠰⠄⣁⣢⠄⣞⡡⢇⠞⠁⣈⠔⡡⠃⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣁⠢⡁⠎⣠⢃⡧⣛⡥⢏⡄⢧⡹⣊⠴⡘⣤⢋⣠⢖⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⢖⡨⡱⠙⢆⠁⡟⢦⠻⡱⣋⢞⣕⠢⡕⠊⡵⣣⡏⢶⡙⢶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⡎⠔⠁⠉⠘⡗⣼⢣⣏⣵⣋⢞⡼⣳⠭⠐⠠⢑⡹⣆⣛⡖⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣨⠏⡗⣯⡜⡶⣛⣮⢳⣭⢻⡝⢮⡹⡱⣞⡼⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢡⡜⢧⢳⣹⠳⡽⣼⢳⡚⣧⡛⡦⢽⡱⢏⡾⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⢣⢏⡶⣫⢗⡮⡅⠀⠱⢧⠙⢎⡇⠙⡾⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣴⡩⢞⡬⣳⡝⣮⢳⡅⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡴⢫⠷⣜⡳⢞⡱⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣩⠻⣼⣙⡏⡷⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⡖⣥⣛⢶⣙⢾⡱⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⢳⡝⣶⡹⢮⡝⣮⣝⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠀⠐⠂⠆⡌⠱⠀⠣⠐⠈⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⡃⠈⠀⠒⢨⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣖⡈⠳⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢷⣃⠀⡸⢜⡬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡷⠀⠡⢀⠂⠙⡃⠀⠡⢂⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⠠⠀⠁⢦⠈⠄⠱⢄⡀⠠⠎⢧⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢰⠁⡌⠀⠳⢬⡡⢂⢡⠂⠌⠂⠱⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠱⠈⢣⠑⠆⠀⣈⠀⠙⠢⢣⠘⡤⠁⠌⠣⠤⣜⡹⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠘⠛⠻⢶⣄⠈⠑⠰⠌⡐⢀⠂⡌⠑⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠄⠂⢌⠃⢌⠁⠈⠔⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢁⠎⠠⠈⠣⣄⡠⡝⣜⢣⡝⡢⠐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠁⢂⠀⠄⢡⠚⠀⡀⠰⠈⡗⣎⡀⣠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠋⠐⠢⢐⠠⢠⠠⡐⢄⠊⠴⠁⡐⠠⠈⠄⡈⡴⢣⣀⢡⠀⡁⢂⠙⢢⡝⡀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠂⠥⡐⠌⣌⠣⡐⠤⠁⣌⡐⣸⢡⢇⡎⠁⡐⡌⡡⠂⣼⣘⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣘⡐⠌⡰⢠⢣⡙⣖⢯⣱⡀⢧⣋⢎⡰⢌⡱⣈⡰⢴⡳⡜⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⠼⠙⢆⢡⡳⢣⢫⡓⢦⢳⢧⡘⢦⠑⡎⢷⣍⢳⢣⢳⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠘⠘⡧⣝⣣⢳⣹⢇⣏⢾⣱⠎⠄⡈⢑⣪⢇⡏⡶⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣠⠏⡷⣩⢷⡹⣞⡼⣣⢏⡿⣱⠞⣭⠲⣭⢞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⢠⢁⣞⡱⣹⠶⣹⢎⣵⣛⡞⡵⢧⡛⣴⡛⢧⡻⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⣏⠶⣙⢦⡻⢵⣫⠶⡍⠀⠙⢧⠙⠦⣇⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣌⡳⣍⠶⣹⢳⢮⡝⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⢎⡷⣭⢳⣋⠾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⠘⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⡼⣡⠻⣜⢮⡳⣭⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⠾⣔⡣⣟⡼⣣⢟⡼⣣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢐⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⣚⡵⣛⣬⢳⡝⣮⢳⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣄⣠⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡶⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡸⢄⡹⡁⠀⠀⠀⠈⠐⡁⢆⡘⠤⢈⢆⠠⣁⠂⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠲⢡⠎⡰⢍⠳⢢⡐⢄⡠⢐⡂⠔⡨⢘⡌⢳⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⣤⢋⠆⢎⡱⢌⢣⡑⣌⠦⣑⢊⡼⢉⢆⢩⡜⣇⠷⣏⠞⣧⡱⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢊⡱⢊⠲⡐⠎⡔⡖⢡⢃⡆⠓⠀⠘⠘⡧⢽⣘⠶⣹⠞⡴⣏⢷⠬⠄⠂⣃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⡔⢃⠲⣉⠖⡭⡘⠴⡁⠀⠊⠀⠀⠀⠀⣠⠏⡟⣼⢫⣝⡻⣼⡹⣎⢷⣚⢖⡚⡦⣝⢮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⡣⡝⢮⡱⣍⢞⡱⣙⢦⠃⠀⠀⠀⠀⠀⢠⢁⡞⡽⣸⢳⢎⡷⣳⡝⢾⣣⢏⡮⢵⡳⡽⣺⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⡦⢷⡚⢮⡹⣹⠚⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢣⡟⢮⣓⢧⡝⠈⢣⢇⢸⡍⡇⢱⡳⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⡷⣙⡖⠉⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⣝⢣⡝⣧⣛⠾⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡷⣱⠂⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⢭⢶⡹⠶⣭⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢷⠃⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⢛⡮⣝⡻⣜⡳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⡖⣭⢺⢵⡫⣵⢫⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⢣⡟⣦⢳⡝⣮⢛⡶⣹⢖⢯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠠⠀⠂⡑⠢⢌⠱⠀⠃⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⡇⠈⠀⠈⠄⠣⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣞⣄⠻⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣝⠶⠀⢠⠗⡼⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡝⡊⠁⠄⠠⠑⡍⠀⠨⠐⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠀⠈⡰⡁⢂⠘⢤⡀⡀⠜⢣⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⢒⡌⢠⠁⠳⢤⣊⠤⡀⠥⠈⠄⡈⠇⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠊⠓⡘⡔⠀⣀⠀⠘⠢⠙⡄⣃⠐⡀⠳⠄⡼⣩⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠘⠛⠻⢶⣄⠈⠐⠨⡐⠠⠐⡀⡇⠑⣫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡑⠠⢁⠰⠃⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠠⠍⠠⢁⠣⣀⣀⢞⡱⢎⢧⠂⠤⡐⠤⢁⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⢈⠐⡀⠄⢂⡜⠁⠠⢈⠱⣋⣎⠁⢀⠔⣌⠒⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠃⠐⠂⡤⢀⠤⢀⠰⡀⠎⡰⠁⠠⠐⡀⢂⠐⡴⢣⠀⠌⡐⢀⠂⡑⢮⡹⠀⠘⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠁⠢⡁⢆⠡⢌⡅⣂⠥⠂⣐⡀⢮⡑⢮⡹⠈⡐⠤⢃⠔⣰⡙⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣀⡱⠀⢎⠰⡘⡴⡲⣏⢳⡀⣧⠛⢦⡁⠴⣉⠖⣁⣤⢳⡙⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⡹⢉⢆⠩⣓⢧⠳⣋⢎⠷⣬⠙⢦⠙⡲⣭⢞⡱⢎⢧⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠘⠨⣗⢭⣎⣳⣓⢯⣚⡵⣫⠶⠄⠡⠚⣥⢛⡜⡮⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣠⠏⡗⣮⢇⡟⣮⢞⡵⣫⢗⣞⢲⢫⡜⣣⢾⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⠠⣁⡞⡵⣩⢞⡽⣜⡻⣜⡳⣏⢮⣣⢳⡹⢧⡻⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠶⡹⣜⡱⢣⡟⣼⢣⢟⡅⠁⠺⣃⢨⣓⠇⠘⣷⠈⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠯⣕⢮⡱⢣⡟⣼⣋⠾⡅⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡬⣳⡝⣧⢞⡵⣎⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⠘⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⠶⣱⠺⡵⣎⠷⣎⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⢮⡳⢥⣻⡱⢯⡝⣮⣛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢐⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⢖⡽⣣⠷⣹⢣⣟⡲⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠂⢂⡘⠄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠈⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⢏⠀⠠⠑⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠀⡈⡘⡄⢈⠈⠦⣀⠠⠚⠼⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⣜⠠⣀⢃⠐⠠⠈⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡕⢀⣀⡀⠈⠑⠌⠆⢎⠀⠡⠐⠦⣪⢕⡺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀⠈⠊⡄⢁⠂⠄⡇⠈⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠂⡈⢰⠁⡘⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠐⠈⢣⣄⡀⢷⡘⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠠⢁⠂⠠⢀⡡⠆⡀⠐⠠⠙⣧⢋⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠁⠉⠂⠅⢢⠐⡄⢢⠘⢠⠒⠁⠌⡐⠠⢈⠐⡼⢰⡀⢄⠁⢂⠡⠈⢧⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠀⢊⠰⣀⠃⢆⠣⡌⠔⣀⣡⠆⡼⣡⢇⠞⠁⣈⠔⡡⠃⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣈⡔⠠⢊⢌⠳⡬⡗⡽⣸⡀⢷⡱⣊⠴⡘⣤⢋⣠⢖⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⢖⡰⡸⠙⢆⠸⡹⢥⠻⡱⢣⣛⡴⠱⣌⠲⡙⣦⠟⡴⣋⠶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⡎⠔⠁⠉⡘⡷⣙⢮⣝⣣⢷⣡⢟⣳⠴⠀⠡⢑⡻⢴⡙⡞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡟⣞⡼⢣⡟⣼⢫⡞⣽⡹⢎⣳⡙⣶⣹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⡹⣜⢧⡻⣼⢣⡟⡶⣝⠺⣴⡹⣖⡳⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⡹⣎⢷⡹⣎⡅⠀⠹⢦⠹⢆⡇⠸⡹⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⣣⢳⡱⢏⡾⣱⢏⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡼⣱⠧⣝⣫⢞⡵⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⢶⡡⢟⡼⣣⢟⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⢦⡝⣮⢳⡝⣮⢳⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⡝⣧⢻⡜⣧⢻⣜⡳⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠂⢂⡘⠄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠈⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⢏⠀⠠⠑⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠀⡈⡘⡄⢈⠈⠦⣀⠠⠚⠼⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⣜⠠⣀⢃⠐⠠⠈⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡕⢀⣀⡀⠈⠑⠌⠆⢎⠀⠡⠐⠦⣪⢕⡺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀⠈⠊⡄⢁⠂⠄⡇⠈⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠂⡈⢰⠁⡘⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠐⠈⢣⣄⡀⢷⡘⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠠⢁⠂⠠⢀⡡⠆⡀⠐⠠⠙⣧⢋⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠃⠈⠐⠡⡂⠔⡠⢂⠜⣀⠲⠁⡈⠐⠠⢈⠐⡼⢰⡀⢄⠁⢂⠡⠈⢧⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠂⡅⢂⠒⣠⢣⠰⠄⣁⣢⠄⣞⡡⢇⠞⠁⣈⠔⡡⠃⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣁⠢⡁⠎⣠⢃⡧⣛⡥⢏⡄⢧⡹⣊⠴⡘⣤⢋⣠⢖⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⢖⡨⡱⠙⢆⠡⣛⢦⠻⡱⣋⢞⣕⠢⡕⠊⡵⣣⡏⢶⡙⢶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⡎⠔⠁⠉⡘⡷⣙⢮⣹⣱⢏⡞⣼⢳⠭⠐⠠⢑⡹⣆⣛⡖⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡟⣎⠷⣭⢞⡽⣎⢯⡻⡝⢮⡹⡱⣞⡼⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⣩⢟⡼⣎⢷⣹⠺⣵⢫⠶⣹⡵⢫⠾⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢣⡟⡼⣎⢷⡉⠀⠱⢧⠹⢆⡇⠩⡟⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⣝⢣⣝⡳⣝⢮⡅⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⢭⢶⣙⢮⡳⣭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⢛⡮⣝⢮⡳⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⡖⣭⢺⢵⢫⡞⣵⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⢳⡝⣮⢛⣮⢳⡝⡮⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠀⠐⢈⠆⡘⢄⠢⠑⠂⠁⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠈⠀⠂⢅⠂⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⢳⡈⠣⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣯⢷⣃⠀⡜⢥⣋⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡷⠀⠌⡀⠄⠙⡊⠀⠡⢂⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⠠⠀⠐⠬⠐⠠⠱⢄⡀⠠⢋⠶⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢫⢰⠁⡌⠈⢓⢦⡱⢀⠡⠆⡀⠃⠱⠃⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠂⢍⠲⡁⠀⣈⠀⠊⠣⣉⠒⢤⠁⢂⠣⡠⡜⡭⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠘⠛⠻⢶⣄⠈⠑⠊⡄⢂⠐⠠⡍⠘⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡒⠀⠌⢰⠃⢨⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢀⠇⠐⡀⠣⣄⣀⠗⣎⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⢁⠂⡀⠄⢠⠋⢀⠠⠘⡘⢦⣋⠀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠋⠐⠆⠰⡀⠄⠤⢐⠠⢊⠴⠁⠌⡐⢀⠂⡐⡜⣡⢈⢄⠂⡁⠄⡑⢮⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠜⣀⠣⢘⢠⢃⢌⠰⠀⣂⡐⣜⡱⢆⠏⡀⢂⠜⡠⠂⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣐⡠⠊⠔⡨⢎⡜⣖⡻⣱⡀⢧⡙⣎⠰⣌⢣⣊⡰⢴⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⠖⡤⡹⢉⢎⠱⣎⢇⠳⣋⢖⣳⢣⡑⢎⠱⢬⡳⣜⡱⣋⠶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⣍⠒⠁⠘⠘⡷⣸⢎⣳⣛⣎⢖⡯⢾⡱⠂⠠⢑⡫⢖⡭⡓⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡗⣯⠶⣹⢎⣾⡹⢧⡳⡽⣚⢭⠳⣭⢞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⣞⡹⢲⣏⡳⣏⢶⣝⡫⣗⢧⣙⢮⡝⡮⠯⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⣏⠶⣩⢓⣮⢳⡝⣮⠅⠀⠹⢦⠩⢎⢇⠙⡻⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣌⡳⢥⣋⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⡧⢏⡵⢫⡗⣮⢳⡝⣮⢳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⡝⢮⡜⣣⠝⣮⢳⡝⣮⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⠳⣍⢳⡜⣥⢻⡜⣧⢻⡜⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣬⢇⡿⡜⣧⢻⡜⣧⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠄⠢⢑⠢⡁⢆⠱⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠐⠀⠡⠒⠌⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⡘⠇⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣟⢧⠇⠀⣑⢭⡛⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡟⠈⠂⠄⡈⠱⡋⠀⠡⢄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⢈⠀⢁⢒⠠⠁⢡⡀⠀⡈⠞⣥⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢋⢠⠈⡤⠘⢢⢥⡂⠌⡉⡄⠂⠃⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠒⣋⠱⠄⠀⠈⠐⠉⠎⡴⢠⢑⠈⡐⢣⡀⣜⢏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠻⠷⣦⣀⠈⠂⡍⡄⠐⡀⠄⡏⠚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠐⡂⠡⠀⢬⠁⠰⠁⠂⢄⠀⣠⢄⡀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠎⠠⢁⠣⣀⣀⢞⡱⢎⡇⡆⢠⠐⢢⠁⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣄⣠⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢂⠉⡀⠐⢠⠜⢀⠠⢉⠜⣣⢎⠀⣀⠢⣁⠚⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡶⡇⠒⠄⡠⢀⠄⡠⢀⠄⡰⢀⢆⠃⠠⠐⡀⠂⠄⡱⢊⠄⢂⡐⢀⠂⡙⢮⡱⠀⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡸⢄⡹⡁⠀⠀⠀⠈⠐⡁⢆⡘⠤⢈⢆⠠⣁⠂⢄⡡⢸⡡⢏⡼⠁⣀⠦⢁⠆⣸⠱⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠲⢡⠎⡰⢍⠳⢢⡐⢄⡠⢐⡂⠔⡨⢘⡌⢳⡴⡺⢭⠀⡷⡙⣎⠠⡔⣌⠒⣁⡴⣇⠻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⣤⢋⠆⢎⡱⢌⢣⡑⣌⠦⣑⢊⡼⢉⢆⠉⡼⣃⠷⣏⠞⣧⡱⠱⣌⠓⣜⣣⢟⡱⢞⡬⡛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢊⡱⢊⠲⡐⠎⡔⡖⢡⢃⡆⠓⠀⠘⢈⡷⣱⠭⣖⠻⡞⡴⣏⢷⠬⠄⠂⣃⢯⡜⣣⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⡔⢃⠲⣉⠖⡭⡘⠴⡁⠀⠊⠀⠀⠀⠀⣠⠏⡗⣯⡝⣯⠽⣞⡹⣎⢷⣚⢖⡚⡦⣝⢮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⡣⡝⢮⡱⣍⢞⡱⣙⢦⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⣓⢾⡱⣻⢼⡳⠽⣎⢷⢪⡝⡶⣫⢞⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⡦⢷⡚⢮⡹⣹⠚⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⢭⢶⣹⢣⣏⡝⠁⢛⣆⠳⡜⡇⠱⣯⠈⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⡷⣙⡖⠉⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⡣⡝⣎⢧⡏⢷⡺⡄⠀⠀⠈⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡷⣱⠂⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡜⣳⠽⣬⢳⡞⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢷⠃⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⡼⣡⠻⣜⡳⡞⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⣖⣱⢻⣜⡳⣝⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⢣⡟⣦⢳⡹⣎⠷⣎⢷⡹⢮⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠪⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⡀⠀⠀⠂⡱⢀⠣⡐⠌⠒⠈⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⣛⠀⠁⠈⠄⢣⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡟⣤⠙⠄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⠱⣯⣟⡾⡽⣞⠀⢠⠳⣥⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⠟⠁⠂⢻⣞⠿⣽⠂⠌⠠⠐⠈⢓⠀⠈⠔⡠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⣾⣳⣟⡾⠉⠤⠄⠐⢻⡟⢨⡇⢀⠀⠱⡌⠐⡈⠦⣀⠀⠜⢥⣋⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⢿⡽⣾⡽⣞⠷⠋⢀⣀⣤⣤⣤⡐⠐⠊⠇⡆⢈⠆⠱⠦⣜⠠⡈⡔⢈⠐⠈⡋⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣳⢿⣯⣟⣷⣿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠈⠚⡉⢜⠀⢀⠁⠈⠓⡘⡔⢂⡌⠠⠙⢄⢤⣋⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⢰⢯⡷⣯⢿⣽⣻⠷⣯⢿⡾⣽⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠘⠛⠻⠶⣤⠀⠈⠒⣀⠂⠌⡀⣚⠌⢛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⡼⣯⣟⣾⣻⣟⣷⡿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠁⢂⠐⢠⠃⡀⠋⠐⠠⠀⣠⢄⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠐⣿⢾⣽⡷⣯⣟⣷⣻⢾⡽⣿⣾⣻⣽⣻⠠⢀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠐⡌⠐⢀⠣⣄⣀⠼⡱⢎⡽⢀⠤⡐⢄⠃⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣟⣷⢻⣞⣯⢿⡽⣟⣷⣻⢾⣽⣇⣆⣦⣠⣀⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⢂⠡⠀⡀⢂⠴⠁⡀⢌⠑⡯⣜⠁⢀⠔⢌⠌⠁
⠀⠀⠀⠀⠀⣿⡳⣏⠾⣜⢣⠞⡼⢻⡽⣿⠸⣿⣯⡿⡍⢿⡟⢿⠿⢿⡿⡏⠌⠐⠢⡄⠤⢀⠤⢀⠆⡘⢰⠊⠠⠈⠄⠠⠁⡴⢋⡄⠂⡐⢀⠂⠑⢮⡹⡄⠘⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡵⣩⠞⣬⢣⡛⡜⣡⠒⣌⠳⢡⠆⣙⠣⣄⢓⡰⢊⠴⢸⡁⠀⠀⠀⠀⠈⢂⠢⢡⠘⢠⢣⡀⠥⠊⣠⡡⢸⢅⣣⡜⢉⢀⠢⢊⠌⢰⢣⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢣⢇⡻⣔⢣⡝⡰⢂⠭⠤⡙⣂⠞⣠⠓⡬⠒⡴⢉⢆⠣⡹⢱⠒⡰⢀⠤⣀⡘⢄⠊⡄⢧⣘⠶⣫⢕⡃⣭⢚⠴⣁⠔⣪⠕⣁⢢⡟⣬⢃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠴⣩⠖⣭⠳⣍⡖⣣⡑⢆⡚⢄⠳⣌⠳⣔⡉⢆⢣⡑⢣⡉⢖⣉⠲⠤⣹⠉⢧⠈⣧⢣⢛⢣⡎⢷⣌⠣⡞⠄⣋⢶⡹⢬⢇⡻⣔⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠒⠽⢢⡝⣦⢝⡲⡜⣬⠳⣌⠛⢤⠚⡌⢆⡜⢡⡜⠁⢇⡪⠕⠁⠘⠨⢱⡜⣧⣋⡶⣫⠖⣯⢳⠼⠄⠁⡓⣹⢎⢮⡱⠊⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠉⠈⠉⠒⠽⡶⣙⠦⣉⠆⡓⣬⢒⡌⠲⡄⠀⠋⠀⠀⠀⠀⣠⠏⣝⢶⡹⣜⢧⣟⡼⣫⢟⣲⢓⠮⣕⡺⣎⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠱⣭⢚⣤⢫⡕⣎⠳⣜⢣⠃⠀⠀⠀⠀⠀⠠⣁⡞⡱⢎⡷⣹⠞⣼⢳⡽⢎⣷⡩⢞⡥⢟⣼⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡖⢧⠞⣎⢯⡹⠙⡀⠀⠀⠀⠀⢀⡶⢳⡜⡹⢎⡵⣫⡝⣮⠇⠁⠛⣦⠙⣎⢆⠘⣞⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠜⡠⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣏⠞⡜⠈⠁⠀⠁⠂⠤⠀⢴⢫⡜⢧⡚⡵⢫⡜⣧⢻⣜⡃⠀⠀⠈⠁⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡝⢮⠁⠀⠈⠠⠀⡀⢀⠀⠂⢷⡘⢧⢻⢵⣣⢟⡼⣓⢮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠆⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⢹⡎⠀⠀⠀⠀⠀⠀⠀⠀⣤⢣⣛⠬⣇⢏⡞⣮⢳⡝⣮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠓⡀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠀⠀⠀⡜⣀⠀⠀⠀⠀⢠⠀⢆⣱⢣⠳⣜⡳⡜⣮⡝⣮⢳⡝⡮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⢄⠣⡑⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⠐⠤⡀⠀⠐⡠⢃⠜⣰⠯⣝⡳⣬⢳⡝⣶⡹⢎⡷⣹⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⡀⢀⠀⠠⠀⠂⢂⡘⠄⠣⠘⠀⠃⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠈⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⢏⠀⠠⠑⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠀⡈⡘⡄⢈⠈⠦⣀⠠⠚⠼⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⣜⠠⣀⢃⠐⠠⠈⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡕⢀⣀⡀⠈⠑⠌⠆⢎⠀⠡⠐⠦⣪⢕⡺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀⠈⠊⡄⢁⠂⠄⡇⠈⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠂⡈⢰⠁⡘⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠐⠈⢣⣄⡀⢷⡘⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠠⢁⠂⠠⢀⡡⠆⡀⠐⠠⠙⣧⢋⡀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠁⠉⠂⠅⢢⠐⡄⢢⠘⢠⠒⠁⠌⡐⠠⢈⠐⡼⢰⡀⢄⠁⢂⠡⠈⢧⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠀⢊⠰⣀⠃⢆⠣⡌⠔⣀⣡⠆⡼⣡⢇⠞⠁⣈⠔⡡⠃⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣈⡔⠠⢊⢌⠳⡬⡗⡽⣸⡀⢷⡱⣊⠴⡘⣤⢋⣠⢖⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⢖⡰⡸⠙⢆⠸⡹⢥⠻⡱⢣⣛⡴⠱⣌⠲⡙⣦⠟⡴⣋⠶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⡎⠔⠁⠉⡘⡷⣙⢮⣝⣣⢷⣡⢟⣳⠴⠀⠡⢑⡻⢴⡙⡞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡟⣞⡼⢣⡟⣼⢫⡞⣽⡹⢎⣳⡙⣶⣹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⡞⡵⡹⣜⢧⡻⣼⢣⡟⡶⣝⠺⣴⡹⣖⡳⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⡱⡹⣎⢷⡹⣎⡅⠀⠹⢦⠹⢆⡇⠸⡹⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣔⣣⢳⡱⢏⡾⣱⢏⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡼⣱⠧⣝⣫⢞⡵⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⡏⢶⡡⢟⡼⣣⢟⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⢓⢧⡹⢦⡝⣮⢳⡝⣮⢳⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣦⡝⣧⢻⡜⣧⢻⣜⡳⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢢⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⣦⣷⣾⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⡀⠠⠀⠂⠘⢠⠃⠜⠀⠃⠐⠀⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠈⠀⠊⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣏⢆⠙⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⡿⣽⠞⡀⠰⠜⣦⡙⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢻⡧⠃⠈⠄⢂⠀⣋⠀⠠⠑⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⣇⠠⠈⡘⡄⠂⠌⠢⣄⠠⠊⢝⢦⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⠋⣂⡆⠰⠀⠙⡲⢬⠡⡀⢆⠡⠈⡀⡇⠀⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠡⠈⢓⡈⡱⢀⣀⡀⠁⠃⠙⡄⢆⠡⠐⠘⠤⣒⢧⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢦⡀⠈⠘⡄⢈⠐⠠⡍⠂⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢮⡽⣏⡿⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⠠⢈⢰⠁⢌⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠐⡿⣯⢿⣽⣻⢷⣻⣞⣷⣻⣿⡿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠊⠀⠌⠱⣄⣀⠞⣜⡣⣝⠢⡐⠄⡓⡀⠀
⠀⠀⠀⠀⠀⣿⡽⣿⣳⣯⠟⣷⣟⣾⣳⡟⣿⣳⢯⣿⣇⣱⣢⣄⣀⡀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⢂⠁⢂⠠⠀⣌⠒⡀⠠⠘⠰⣓⢎⡀⢠⠒⠤⠉⠀
⠀⠀⠀⠀⠀⣿⡽⣫⠗⣎⡛⢶⡙⡾⢷⢿⠸⣿⣟⣷⠉⣿⠟⢿⠿⢿⡿⡏⠃⠈⠐⠡⡂⠔⡠⢂⠜⡐⡰⠁⠌⠠⠈⠄⢂⡽⡐⣂⢄⠡⢈⠐⡈⢮⡹⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣱⢣⡛⡴⣙⢦⠛⡰⢡⠎⡝⢢⠥⣙⠢⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⢠⠑⡄⠒⢤⢣⡐⠌⣁⣌⡔⣸⢰⡱⠎⡀⢂⠴⣁⠂⣴⡙⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢡⢧⡙⢶⡩⢎⡱⢡⠣⠜⡘⡔⡒⣌⠱⢆⡩⢒⠜⢢⠡⢏⠵⢒⡰⢀⠤⣀⡊⠤⢉⠰⣢⡹⡞⡵⢪⡄⢧⢣⡙⢤⠒⡭⢒⣀⡖⣧⠹⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⡙⢧⣙⢦⠳⣍⡒⡡⢜⡠⣙⢦⠳⡌⢎⡡⠓⡌⠎⢆⣱⠊⡔⢢⡹⢉⢎⡑⢧⢳⡹⠱⣋⠾⣌⢣⠚⠤⣋⢷⡹⢦⡹⢲⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠚⠼⣊⠷⣌⡳⡱⢎⡴⣍⢊⡓⣉⠦⢡⢋⠜⣘⠖⢡⠓⣌⠃⠁⠈⠘⡷⣸⢧⣙⣳⡭⢞⡽⣲⠽⠀⠄⢱⡹⢦⣙⠧⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠮⣶⢩⢆⡱⢂⢎⡥⢎⡜⠤⡃⠀⠋⠀⠀⠀⠀⣠⠏⣧⢻⡜⣧⣛⣮⢳⣭⢻⡝⣎⢧⡹⢶⣹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⣋⠞⡴⣋⠞⣜⢣⠞⣦⠃⠀⠀⠀⠀⠀⢠⢁⡞⣥⠻⣼⡱⣞⡼⣳⠞⣧⢻⡜⣲⡝⢧⡻⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡵⡭⢞⠮⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⡯⣜⢲⡹⢶⡹⢶⡹⡅⠀⠸⢣⠸⠥⣇⠘⡽⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡟⣬⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣴⡩⢖⡹⢮⡝⣧⢻⡅⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⠴⠁⠀⠈⠠⠀⡀⢀⠠⠈⣧⢛⡴⢫⠷⣭⢳⡝⣮⢳⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢏⡼⣩⢛⡼⣣⢟⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⢂⠀⠀⠀⠀⠠⢄⠢⣝⢥⣋⠾⣔⣣⢻⡜⣧⢻⣜⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⢄⠣⠱⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠆⡀⠀⢐⠨⡑⠌⣼⡹⠶⣭⣚⡵⣋⢧⡻⣜⡳⣎⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢊⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣮⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⠀⠀⠐⢈⠆⡘⢄⠢⠑⠂⠁⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣟⡇⠈⠀⠂⢅⠂⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠠⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⢳⡈⠣⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢺⣽⣳⣯⢷⣃⠀⡜⢥⣋⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣌⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⠏⠐⠠⢻⣷⢫⡷⠀⠌⡀⠄⠙⡊⠀⠡⢂⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⢷⣻⣞⣯⠷⠃⠤⠄⠐⢻⡏⢸⠠⠀⠐⠬⠐⠠⠱⢄⡀⠠⢋⠶⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⣟⡾⣽⣻⢷⠻⠊⢀⣀⣤⣤⣤⠘⡀⢫⢰⠁⡌⠈⢓⢦⡱⢀⠡⠆⡀⠃⠱⠃⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣟⣷⣻⣽⣻⣽⡿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠑⠂⢍⠲⡁⠀⣈⠀⠊⠣⣉⠒⢤⠁⢂⠣⡠⡜⡭⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣞⡷⣯⢿⣽⣻⡽⣾⣽⣞⡷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠘⠛⠻⢶⣄⠈⠑⠊⡄⢂⠐⠠⡍⠘⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⣼⣳⣟⣶⣻⣯⢿⣟⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡒⠀⠌⢰⠃⢨⠁⠈⠤⢀⡤⢤⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⣿⢾⣽⣳⣟⣾⣳⣟⡾⣽⡿⣿⣽⣻⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢀⠇⠐⡀⠣⣄⣀⠗⣎⢧⡓⡆⡐⠢⡑⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣷⣻⢞⡷⣯⣟⡷⡿⣷⢯⣟⣯⣇⣱⣢⣄⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⢁⠂⡀⠄⢠⠋⢀⠠⠘⡘⢦⣋⠀⣠⠒⡤⠉⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠶⣹⢚⡼⣙⠾⣽⣟⠸⣿⣯⣿⠉⣿⠟⢿⠿⢿⡿⡏⠋⠐⠆⠰⡀⠄⠤⢐⠠⢊⠴⠁⠌⡐⢀⠂⡐⡜⣡⢈⢄⠂⡁⠄⡑⢮⡹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡱⣍⠞⣥⢋⠶⣉⠞⡠⠜⡙⠦⡑⡜⠣⣌⢃⠖⡌⢆⡹⡁⠀⠀⠀⠀⠈⠜⣀⠣⢘⢠⢃⢌⠰⠀⣂⡐⣜⡱⢆⠏⡀⢂⠜⡠⠂⡼⡱⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⣌⠻⣔⢫⠳⣌⠒⡥⡙⣌⠱⣘⠰⢃⠦⣉⠖⣘⠢⢅⠫⡕⣒⠰⡀⠤⣐⡠⠊⠔⡨⢎⡜⣖⡻⣱⡀⢧⡙⣎⠰⣌⢣⣊⡰⢴⡳⣍⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣬⢣⡛⣬⢓⠶⣡⢆⠓⡬⢘⢥⡚⡴⣊⠴⣉⢌⠣⢜⡠⢣⡉⠖⡤⡹⢉⢎⠱⣎⢇⠳⣋⢖⣳⢣⡑⢎⠱⢬⡳⣜⡱⣋⠶⡩⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑⠛⠴⣋⠷⣱⢎⡳⢬⡱⢎⡱⢣⠑⡒⡌⠦⡙⢢⡜⢁⠣⣍⠒⠁⠘⠘⡷⣸⢎⣳⣛⣎⢖⡯⢾⡱⠂⠠⢑⡫⢖⡭⡓⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠧⣝⡺⣄⠣⢍⡒⡱⣌⠱⡡⠆⠀⠋⠀⠀⠀⠀⣠⠏⡗⣯⠶⣹⢎⣾⡹⢧⡳⡽⣚⢭⠳⣭⢞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠳⣌⠳⣎⠵⡹⣌⠳⣥⠃⠀⠀⠀⠀⠀⢠⢁⣞⡹⢲⣏⡳⣏⢶⣝⡫⣗⢧⣙⢮⡝⡮⠯⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡳⢮⣓⠳⣍⠿⠘⠀⠀⠀⠀⠀⣠⠖⣏⠶⣩⢓⣮⢳⡝⣮⠅⠀⠹⢦⠩⢎⢇⠙⡻⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡷⣩⡓⠈⠁⠀⠑⠠⠄⠄⢴⢣⠻⣌⡳⢥⣋⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠢⡑⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⢡⠁⠀⠈⠠⠀⡀⢀⠠⠈⡧⢏⡵⢫⡗⣮⢳⡝⣮⢳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢗⡋⠀⠀⠀⠀⠀⠀⠀⢀⡰⡝⢮⡜⣣⠝⣮⢳⡝⣮⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢠⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡌⠄⠀⠀⠀⠀⠠⢄⠒⣬⠳⣍⢳⡜⣥⢻⡜⣧⢻⡜⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⡱⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⢈⠜⡀⠀⠐⡨⠑⡌⣼⢣⡟⣬⢇⡿⡜⣧⢻⡜⣧⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠪⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢠⠀⡒⢌⡐⢢⠁⠆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⡛⠀⠀⠂⡔⠠⠁⠀⠀⠂⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣈⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⢏⠹⠆⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⠱⣯⣟⡾⣏⡇⠀⡰⢎⡳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣼⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⠟⠁⠂⢻⣞⠯⣗⠒⠐⠁⡙⠼⡉⠀⢂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⣾⣳⣟⡾⠉⠤⠄⠐⢻⡟⢘⠠⠀⠌⣐⠠⢀⢃⡀⠈⢀⠯⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⣯⢿⡽⣾⡽⣞⠷⠋⢀⣀⣤⣤⣤⡐⡀⠛⢄⠂⡐⠨⣄⡂⢆⡉⠢⡁⠄⢋⢳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣳⢿⣯⣟⣷⣿⡇⠀⠈⠉⠉⠁⠁⠀⠀⠰⠈⡞⠤⠃⠀⠀⠙⠦⡐⢣⢐⠠⠂⡐⢆⠀⣜⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⢯⡷⣯⢿⣽⣻⠷⣯⢿⡾⣽⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠁⠴⠶⣶⣄⡈⠂⠊⡆⡡⠀⠌⠁⡼⠱⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⣨⣏⡷⣯⣻⡼⣯⣟⣾⣻⣟⣷⡿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠰⡱⠈⠠⢁⠞⢠⠃⠂⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣿⢾⣽⡷⣯⣟⣷⣻⢾⡽⣿⣾⣻⣽⣻⠠⢀⠀⠀⠀⠀⠀⠐⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠡⡘⠠⠘⢄⡀⠈⣮⢱⣋⢇⡄⢠⠠⠰⡈⠀
⠀⠀⠀⠀⠀⣿⣻⣾⣟⣷⢻⣞⣯⢿⡽⣟⣷⣻⢾⣽⣇⣆⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠄⠠⠁⠀⠌⡠⢁⠐⣈⠑⣎⠎⠁⢀⠬⣁⠒⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠾⣜⢣⠞⡼⢻⡽⣿⠸⣿⣯⡿⡍⢿⡟⢿⠿⢿⡿⡗⠤⢀⠄⡀⢀⠀⡀⢀⠠⡐⢠⠜⠀⠡⠈⠄⠡⣈⠜⡁⠄⡂⠄⢂⠘⢝⡲⡍⠠⠁⠀⠀
⠀⠀⠀⠀⠀⠘⡵⣩⠞⣬⢣⡛⡜⣡⠒⣌⠳⢡⠆⣙⠣⣄⢓⡰⢊⠴⢸⡁⠀⠀⠀⠁⠊⢐⠡⠊⠤⠑⣌⠠⢈⠄⠡⢈⢰⢃⢮⡱⠊⢀⠲⢈⠌⢰⢣⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢱⢣⡛⡴⢣⡝⡰⢂⠭⠤⡙⣂⠞⣠⠓⡬⠒⡴⢉⢆⠣⡹⢱⢒⡰⢀⠤⢈⡰⢉⠰⡁⢎⠲⣡⡜⣞⡃⢮⣙⠦⣁⠦⣉⠎⣁⡤⢯⢣⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠽⣘⢧⡚⡵⣋⢖⣣⡑⢆⡚⢄⠳⣌⠳⣔⡉⢆⢣⡑⢃⠲⢌⡱⢊⠦⣱⠋⣃⠈⢧⣋⢧⢟⡴⢣⡳⢌⡖⡡⠎⣷⢹⡱⢎⡳⢭⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠛⠴⢭⡺⢴⡙⢶⡸⣌⡳⢌⡛⠤⠚⡌⢆⡜⢡⡛⠀⣇⢣⠜⠁⠘⠠⢓⢧⡚⡦⢾⣜⢣⡟⣦⠼⠄⡁⠚⣥⢛⣬⢓⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠓⠳⣼⢱⢣⡘⡌⡓⡼⣰⠨⢅⡆⠀⠊⠀⠀⠀⠀⣠⠏⡷⣹⡝⣳⢎⣧⢻⡜⣯⢖⡖⡻⣌⣳⡜⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢱⣋⠶⡱⣜⡱⢣⢇⡛⢦⠃⠀⠀⠀⠀⠀⠠⣁⠶⣳⢱⣫⢧⡻⣜⣧⠻⣜⣳⠼⣱⠣⢷⣹⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢎⡵⢎⡗⣫⢞⡹⢋⠀⠀⠀⠀⠀⢀⡶⣍⠳⣥⢣⢗⣧⢻⣜⡃⠁⠫⣖⠹⣌⢇⠘⣳⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠜⡠⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡯⣜⡱⠈⠁⠀⠁⠢⠄⠄⢴⣋⠶⣍⡳⢬⡓⡾⣜⡳⣎⠇⠀⠀⠈⠁⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠑⡂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣳⢌⠃⠀⠈⠠⠀⡀⢀⠀⠂⡝⢮⡱⢏⡷⣹⡜⣧⢻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⢱⡞⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛⡦⡝⢮⡱⢧⡻⣜⡳⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⣁⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡜⣀⠀⠀⠀⠀⢠⠀⢆⣱⢎⡳⢼⣙⢦⡹⢧⡻⣜⡳⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⡄⢣⠑⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡰⠐⠤⡀⠀⠐⡠⢃⠜⣰⢏⡾⣱⢇⡯⢞⡹⢧⣛⡼⢳⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡡⢆⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⡀⠀⠄⠒⢌⠰⡈⠔⡡⠌⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠂⠀⠡⠒⠄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣐⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⠗⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⣾⣝⡀⠁⡂⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣬⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⡛⠁⠂⣹⣾⡍⡾⠤⢀⠓⢭⢲⡹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣽⣻⡽⢊⠠⠄⠐⠚⡿⢀⠩⠀⠂⢰⠀⠱⡀⠀⠐⠡⡲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠛⠿⠿⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣷⡿⣽⣻⢾⣽⣻⠷⠋⢀⣀⣠⣤⣤⡐⠁⠜⢠⠁⠄⠣⣌⣀⡑⠦⢠⠐⠙⠶⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣻⣽⣿⢯⡿⣯⣟⡷⣯⢿⣾⡇⠀⠈⠉⠉⠉⠁⠀⠀⠘⣬⢘⠢⡉⠀⠐⠡⠺⡐⠦⢡⠈⠄⡈⡄⠈⠰⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⣿⣳⢿⡽⣯⢿⣽⣳⢯⡿⣽⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠐⠡⠤⣶⣤⣀⠈⠊⠆⡹⢠⠐⠈⠒⢩⠳⣍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⢷⣭⢿⣹⣭⣟⣾⣳⣟⣿⣳⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠙⠳⢄⠀⠁⡖⠠⠁⠌⡸⠁⠎⠐⡠⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⣿⣻⣞⣷⣻⢾⡽⣞⡷⣯⣿⣯⢷⣻⣾⡐⢀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⢂⠇⡀⠧⡀⠀⡸⣍⢞⡹⢠⠀⡄⠰⣀⠀
⠀⠀⠀⠀⠀⣯⢷⣿⡷⣯⠿⣽⡿⣽⣳⣿⢯⣟⡷⣯⢷⣠⣆⣤⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠐⠊⠀⠠⢐⠌⠉⠑⡘⢮⡹⠁⠈⡠⢅⠢⠀
⠀⠀⠀⠀⠀⣿⢯⣏⠷⣹⢚⡥⣛⢳⠿⣾⠸⢿⣻⣟⡏⢿⡟⢿⠿⠿⣿⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠀⠄⡁⢂⠁⢂⡡⠎⢀⠁⢂⠁⠌⢳⡱⢎⠁⠊⠀⠀
⠀⠀⠀⠀⠀⠘⢧⢎⡳⢥⣋⠶⣉⠎⡒⡌⢏⢣⠱⡘⠣⢤⡑⢆⡚⢔⡩⣍⠀⠀⠈⠁⠊⠰⡐⠌⡂⢅⢪⠁⡐⠠⠐⠠⢈⡖⣡⢋⠖⠉⢠⠜⢠⠂⡝⢮⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢣⢮⡱⢣⢎⡳⡌⡜⢡⠜⡌⢆⢣⢩⠑⢦⡑⣊⠔⣊⠴⢨⢓⠲⡐⠤⢠⢘⡀⠎⡐⠌⣆⠳⣈⣥⡚⡇⢸⡜⡥⢋⡠⢜⡡⠊⢄⡼⣙⢞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠭⣣⢏⠶⡹⣜⢣⠞⡌⡜⡰⢊⢍⠶⡸⣄⢋⡔⢊⠥⣊⠱⣌⡃⠇⢦⢉⠏⡥⠈⢢⢗⢹⡲⣍⣳⠸⡜⡸⢡⠚⣬⣓⢏⡳⢞⡭⠎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠚⠵⣊⠷⣹⢜⡲⢥⡍⢮⣃⠳⢌⡒⢌⡃⠞⣠⢳⠀⡬⣙⠔⠉⠐⠠⢻⢺⡬⣣⠽⣖⢭⣛⢶⣙⠆⢉⠒⠝⣮⢱⢫⠜⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠉⠁⠉⠘⠧⣞⢣⡆⣍⠒⡌⢦⣍⠲⣁⠎⠀⠑⠁⠀⠀⠀⢠⡽⢱⡹⣖⡻⣞⢦⣛⢮⡳⣎⡤⢞⡹⢆⡯⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣧⢚⣤⢛⡜⣣⢎⡳⣌⡃⠀⠀⠀⠀⠀⢀⠡⡴⣋⠶⣭⢳⡝⣮⢟⡮⢷⣹⡜⣣⢟⠾⣱⢧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡨⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡶⢭⢞⡣⢏⡽⠘⡁⠀⠀⠀⠀⢀⡴⣛⠴⣍⢞⡱⣏⠾⣥⢯⠂⠙⢶⡈⢧⢻⠀⢻⠆⠁⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣙⢮⡱⠉⢀⠀⠀⠢⠄⠄⠰⢏⠶⣍⡞⡜⢮⡱⢯⡝⣞⢾⠀⠀⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡂⢍⢂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠇⠀⠀⠠⠀⡀⢀⡀⠈⢫⠞⣬⢫⡝⣶⡹⢧⢻⡜⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠆⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡝⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡞⢦⢣⡝⢶⣹⣋⢧⡻⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡘⢄⠀⠀⠀⠀⢠⠀⢆⣲⡱⢎⡼⣣⣓⢮⡳⢇⡻⣎⠷⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢃⠆⡩⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡠⠑⡌⣀⠀⠐⡠⢃⠜⣠⢷⡹⣞⡴⣣⠟⣮⡝⣏⠷⣭⣛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡡⢆⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣡⣷⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⡀⠀⠄⠒⢌⠰⡈⠔⠡⠌⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⡉⠄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠂⠡⠒⠄⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣐⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⠗⢻⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⢄⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠱⣯⣟⣾⣝⡀⠑⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣇⣬⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⡛⠁⠂⣹⣾⡍⡾⠤⢀⠓⢧⡚⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣟⡾⣽⣻⡽⢊⠠⠄⠐⠚⡿⢀⠩⠀⠂⢤⠀⠙⡀⠀⠰⠁⡲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣽⣻⢾⣽⣻⠷⠋⢀⣀⣠⣤⣤⡐⠁⠜⢠⠁⢄⠪⣄⠡⡑⠦⠠⡐⠙⠧⢎⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⣽⣻⢷⣯⢿⣾⡇⠀⠈⠉⠉⠉⠁⠀⠀⠐⡜⣘⢢⡁⠀⠈⠙⠮⡐⡱⢠⢁⠂⠌⡄⠈⢰⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣟⡷⣯⢿⣽⣻⣗⣯⣟⡿⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠐⠂⠤⣶⣤⡀⠈⠒⠡⣊⠰⢀⠘⠂⢹⡚⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣠⢯⡽⣏⡿⣼⣳⢯⣶⣻⣿⡽⣿⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠁⡞⠀⠄⡁⠞⢀⠋⠐⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠐⡿⣯⣟⣷⣻⢷⣯⣟⣾⣳⣿⡿⣯⢿⣾⡁⠄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠂⢰⠋⠀⢦⡀⠀⣸⡹⢬⢳⡀⢄⠠⡐⢄⠀
⠀⠀⠀⠀⠀⣿⣳⣿⡷⣯⢟⡾⣟⣾⣳⣿⢻⣽⣻⣞⣷⣨⣔⣠⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠄⡐⠂⠀⠡⢀⠎⠉⠄⡙⣎⠳⠈⠀⠤⣁⠆⠀
⠀⠀⠀⠀⠀⣿⡳⣏⠷⡹⢎⡳⢭⡳⢿⡾⣙⢯⣷⣿⡃⢿⡟⢿⠿⢿⡿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⡈⠐⡀⢂⠁⣆⠃⡐⢈⠐⠠⠘⢝⡲⡍⠰⠈⠀⠀
⠀⠀⠀⠀⠀⠘⡵⣩⢞⡱⢫⡜⢣⡑⠦⡘⣍⠚⡔⣘⠣⣄⠓⣌⠒⠦⡘⣍⠀⠀⠈⠁⠊⠰⡐⠌⡂⢅⢚⠀⡐⠠⢁⠐⡀⡞⣤⢫⠔⠁⣈⠔⡡⢂⠷⣡⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢱⢣⢎⡵⣓⢮⡑⣌⠲⢡⠜⡘⢤⠃⡖⠬⢣⡘⠬⣑⠱⣌⠳⢒⡰⠠⢄⡘⡠⢑⠨⠄⣏⢒⡩⢴⡢⡏⢰⡝⢦⠃⣄⠚⡔⠊⣀⣼⢣⢳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠜⣲⢍⠶⣙⢦⡛⢦⡊⢕⡊⡜⣰⢋⠶⣌⠱⢌⠲⢄⡋⢆⠥⢋⡔⢢⠱⡍⣆⠉⢆⣳⢙⣮⢕⣣⡸⠜⣌⠓⡬⣝⢎⢯⡱⢎⢧⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠚⠵⢪⡝⣦⢝⡲⡜⡴⢣⡋⢞⢨⡑⣊⠱⡌⡒⣍⠖⢣⡘⢆⠝⢠⠉⢶⣫⡜⢮⢞⣎⠖⣯⢳⣌⡃⠐⡉⢞⡣⣝⢺⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠄⠈⠁⠉⠒⠽⣼⢣⡓⣌⠲⢰⡡⢇⠲⡑⠼⠀⢧⠘⠁⠀⠀⠈⣼⢦⢛⡞⣞⢮⣛⡼⣣⢗⡦⡥⢖⡫⣕⢮⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣧⡙⢦⡓⢧⣙⢎⡳⣍⡆⠀⠀⠀⠀⠀⠀⠜⣂⡼⣩⢞⡽⢮⡝⣾⡱⣏⢾⡱⢫⡵⢞⡧⢇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢝⡦⠽⡞⢮⢭⡹⣜⠆⠀⠀⠀⠀⠀⣐⡼⢣⠞⣥⢻⡜⣧⣛⢦⠃⠙⣞⡈⡧⣛⠈⣞⠇⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠒⠤⡑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡹⣜⢣⠮⠕⠀⠂⣀⠀⠀⢀⡴⣍⡞⣥⢛⡴⣋⢾⡱⣎⠷⠀⠀⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡉⢆⠡⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡎⡍⠀⠐⠀⠀⠀⠈⠁⠡⢗⡼⣘⢧⣛⢶⡹⣎⠷⣭⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠎⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢸⣣⠁⠀⠀⠀⠁⠀⠀⠂⠀⡞⡴⣍⠶⣩⢞⡵⣫⡝⡶⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⢡⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠡⠀⠀⠀⠀⠀⠀⡀⢄⡞⣱⡹⣌⡳⣥⢻⡜⣧⡝⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⢢⢁⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠆⡡⠀⠀⢀⠀⡌⠱⣈⢾⡹⣥⢳⢭⡳⣭⢳⡝⣶⡹⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⠠⢐⠰⡈⠔⢢⠁⠆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⠓⠀⠐⢠⠐⡄⠁⠀⠠⠈⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⡝⣯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢳⣯⢷⣟⡴⠈⠃⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡟⠁⠂⢿⣻⢞⡽⢄⡀⠷⡸⣌⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⡾⣽⢯⣟⡊⢀⠠⠤⢺⡿⢈⠣⠀⠠⡀⠄⢹⠀⠈⠔⠢⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⣿⣿⡿⣿⣿⣻⣿⣽⣾⢿⡽⣯⣟⣷⣻⡽⠋⠀⢀⣀⣤⣄⠘⠠⠜⠄⢂⠠⢱⡈⢀⠳⢄⡀⠔⠹⢆⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡿⣯⣿⣿⣽⣿⢯⣿⢯⣿⣻⢷⣻⣾⡇⠀⠐⠛⠛⠉⠉⠈⠁⢡⡈⡼⣀⠎⠀⢉⠳⢅⡢⡘⢄⠂⠄⡑⡀⠐⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢹⡽⣞⣯⣟⣾⣻⣞⡿⣾⡽⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠣⠐⠆⣠⣤⣄⠀⠁⢉⠆⣚⠠⢀⠑⠒⢪⢖⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣸⡝⣯⣗⣻⢾⡵⣯⢿⣽⣟⣷⡿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠀⠡⣒⠠⢈⠐⡼⠀⠏⢐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠔⣯⣟⣷⢯⣟⣯⢿⣽⣻⢾⣿⣾⣻⣽⢯⢀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠂⢤⠃⠈⢆⠀⠁⣰⢫⡝⣣⠄⡀⢠⠀⡔⠁
⠀⠀⠀⠀⢠⣟⡾⣿⣻⠾⡽⣻⡾⣽⢯⡿⣷⣻⣞⡿⣇⣐⣠⣀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡀⠐⠌⠀⠡⢈⠍⡈⠑⡣⢞⡱⠂⠁⡀⡣⢐⠂
⠀⠀⠀⠀⠘⣯⢟⡵⢫⡝⣳⢣⠟⣯⢿⣳⠹⣷⣟⣿⠙⣿⠿⡿⠿⢿⣷⡦⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⠂⠄⡁⠂⠄⡡⠎⠀⠄⠡⢀⠹⡜⢦⠎⠐⠁⠀⠀
⠀⠀⠀⠀⠀⢫⠞⣜⢣⠞⣥⢋⠞⡠⢆⡱⢋⠖⣨⠙⢦⡘⠦⡑⢎⢢⡙⡆⠀⠀⠈⠒⠠⡂⠔⡠⠒⡈⠴⠁⠠⢁⠂⠄⡁⡞⣡⢓⠼⠈⡐⡔⢠⠈⣧⢣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⡺⣌⢧⡛⡴⣋⠴⡑⣊⠴⡉⢦⢡⡙⢤⠣⡱⡘⡌⢆⡱⢍⠖⣢⠐⡄⢠⢘⡠⢁⠆⡡⢏⡔⡃⣤⡜⡆⢼⡱⢣⠏⡀⠴⣡⠒⢁⣼⠲⣭⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠚⢦⣙⠶⣩⠖⡵⣒⢦⡉⢆⠲⢌⡲⣱⢱⢢⢉⠖⡰⢊⡜⠤⡙⡌⢥⠢⢍⠲⣈⠁⣎⡴⢛⡦⣝⢣⠲⣍⠳⡨⢱⡹⣔⠞⣭⢖⡹⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠚⠡⠟⣜⢣⣎⠵⣪⠕⣮⢱⢡⠋⡆⡍⡒⠥⣃⠜⡦⠱⣘⠢⣑⠪⡑⣜⢲⢳⡜⢯⡽⣌⢏⡷⣌⣣⠅⠡⠹⣚⡝⢦⢫⠜⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠈⠉⠈⠙⠲⢽⡆⣏⠦⡙⡰⢌⣱⡑⢢⠍⡜⢣⠆⡓⡌⣱⠸⣌⢧⣋⢾⡱⢮⣝⢎⡾⣱⢧⡬⡤⢵⢢⡝⣮⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢻⣌⠳⣌⠳⣍⠶⣩⠳⡜⣌⠲⣌⡱⢸⢥⡛⡜⢦⡙⢮⣝⡳⣎⠿⣜⠧⣟⣼⡱⣋⢶⡻⣜⣃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⣳⠭⡳⢞⡹⣍⡟⢉⢮⡳⠜⠁⠢⠘⣧⢛⡬⣝⠺⣬⢳⣭⣛⡍⠇⠸⣞⠱⣍⡞⠐⡯⡎⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠷⣩⠖⣥⠃⠈⠇⠁⠀⠀⠀⢡⠟⣎⠶⣩⠳⣭⢳⠶⣙⡆⠀⠀⠈⠠⠀⠉⠁⠘⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠃⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣡⡛⡼⠀⠀⠀⠀⠀⠀⢀⠋⣴⢋⢿⡱⣏⠾⣍⢿⡱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣟⡴⡹⠜⠀⠀⠀⠀⠀⢠⡜⣫⡜⣭⢲⡹⣎⠿⣜⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡇⠓⠁⠈⠰⠀⡄⡀⢴⢫⡜⣱⢺⣔⣣⢳⣭⢻⡜⣧⣛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢐⢢⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢠⠀⠀⠁⢀⠀⠀⠀⠀⢊⡷⣹⢦⣓⢮⣓⢯⢖⠯⣞⡵⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⠠⢐⠰⡈⠔⢢⠁⠆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⠓⠀⠐⢠⠐⡄⠁⠀⠠⠈⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⡝⣯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢳⣯⢷⣟⡴⠈⠃⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡟⠁⠂⢿⣻⢞⡽⢄⡀⠷⡸⣌⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⡾⣽⢯⣟⡊⢀⠠⠤⢺⡿⢈⠣⠀⠠⡀⠄⢹⠀⠈⠔⠢⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠛⠿⠿⠛⠛⣿⣿⡿⣿⣿⣻⣿⣽⣾⢿⡽⣯⣟⣷⣻⡽⢋⣠⣤⢴⣶⣤⢘⠠⠜⠄⢂⠠⢱⡈⢀⠳⢄⡀⠔⠹⢆⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡿⣯⣿⣿⣽⣿⢯⣿⢯⣿⣻⢷⣻⣾⡇⠀⠉⠁⠀⠈⠉⠀⠀⢡⡈⡼⣀⠎⠀⢉⠳⢅⡢⡘⢄⠂⠄⡑⡀⠐⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢹⡽⣞⣯⣟⣾⣻⣞⡿⣾⡽⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠣⠐⠆⣠⣤⣄⠀⠁⢉⠆⣚⠠⢀⠑⠒⢪⢖⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣸⡝⣯⣗⣻⢾⡵⣯⢿⣽⣟⣷⡿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠀⠡⣒⠠⢈⠐⡼⠀⠏⢐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠔⣯⣟⣷⢯⣟⣯⢿⣽⣻⢾⣿⣾⣻⣽⢯⢀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠂⢤⠃⠈⢆⠀⠁⣰⢫⡝⣣⠄⡀⢠⠀⡔⠁
⠀⠀⠀⠀⢠⣟⡾⣿⣻⠾⡽⣻⡾⣽⢯⡿⣷⣻⣞⡿⣇⣐⣠⣀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡀⠐⠌⠀⠡⢈⠍⡈⠑⡣⢞⡱⠂⠁⡀⡣⢐⠂
⠀⠀⠀⠀⠘⣯⢟⡵⢫⡝⣳⢣⠟⣯⢿⣳⠹⣷⣟⣿⠙⣿⠿⡿⠿⢿⣷⡦⠄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⠁⠂⠄⡁⠂⠄⡡⠎⠀⠄⠡⢀⠹⡜⢦⠎⠐⠁⠀⠀
⠀⠀⠀⠀⠀⢫⠞⣜⢣⠞⣥⢋⠞⡠⢆⡱⢋⠖⣨⠙⢦⡘⠦⡑⢎⢢⡙⡆⠀⠀⠀⠁⠢⡐⢄⠢⢡⠘⡰⠁⠠⢁⠂⠄⡁⡞⣡⢓⠼⠈⡐⡔⢠⠈⣧⢣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⡺⣌⢧⡛⡴⣋⠴⡑⣊⠴⡉⢦⢡⡙⢤⠣⡱⡘⡌⢆⡱⢍⠖⣢⠐⡄⢠⢈⡰⠈⡔⢂⡳⢌⡒⣡⠴⡎⢰⡝⢦⠋⣀⠦⡑⠆⢁⣼⠲⣭⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠚⢦⣙⠶⣩⠖⡵⣒⢦⡉⢆⠲⢌⡲⣱⢱⢢⢉⠖⡰⢊⡜⠤⡙⡌⢥⢊⡌⢳⡀⠁⣇⢮⣙⣮⢓⢧⠱⢎⡱⢊⡔⣧⢓⡞⡭⢖⡝⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠚⠡⠟⣜⢣⣎⠵⣪⠕⣮⢱⢡⠋⡆⡍⡒⠥⣃⠜⡦⠱⣘⢂⠎⡔⢣⢸⡱⢞⡼⣜⢮⡝⣎⡟⣦⡱⡃⠨⠱⢻⡜⣱⢫⠜⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠈⠉⠈⠙⠲⢽⡆⣏⠦⡙⡰⢌⣱⡑⢢⠍⡜⢣⢆⢩⠒⡬⣑⢮⡱⣋⠶⣭⢮⡝⢦⣛⢶⣣⢥⡤⢭⢲⣙⢦⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢻⣌⠳⣌⠳⣍⠶⣩⠳⡜⣌⠲⣌⠲⣙⠶⣩⢖⡱⣍⢻⡜⣧⢻⡽⣹⢎⡷⣫⡜⣣⠗⣮⢏⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⣳⠭⡳⢞⡹⣍⠯⡝⣮⢳⢬⠳⣍⢞⡱⢎⡵⣊⢧⡻⣜⡳⡞⡕⠃⢸⢳⠙⣦⠻⠈⣯⠆⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢃⠜⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠷⣩⢖⠝⢊⠙⡸⣝⢎⠷⣘⢮⡱⣏⠶⣩⠖⡽⣎⢷⡹⡅⠀⠀⠉⠀⠀⠉⠁⠘⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠃⠆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠠⠁⠀⠀⠉⡰⣏⢮⡹⡜⢦⠳⣍⢷⣣⢏⡷⣹⢎⡷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⢃⢮⢣⢳⡙⢮⠳⡜⢦⡛⣼⢳⡹⣎⢷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠃⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⢫⢎⢧⣣⠝⣮⢹⣜⡣⣝⢮⡳⣝⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⠒⠁⠀⠀⠀⢀⣰⢣⢏⡞⣮⠵⣛⢦⣏⢶⡹⣎⢷⡹⣎⡗⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠠⠀⢅⠒⡌⠰⡈⠆⠡⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠁⠂⠀⠣⢄⠃⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⠗⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢳⣯⢷⡿⣜⠀⠁⣂⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡟⠁⠂⢿⣯⢟⡭⢂⠠⠓⢎⠶⡙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣽⡾⣽⢯⣟⡊⢀⠠⠤⢺⡿⢨⠑⠀⢂⠰⠀⠹⡀⠀⠊⠔⣢⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠛⠿⠿⠛⠛⣿⣿⡿⣿⣿⣻⣿⣽⣾⢿⡽⣯⣟⣷⣻⡽⢋⣠⣤⢴⣶⣤⢘⠠⠜⣀⠂⢄⠩⣄⡁⡱⠤⠠⠌⠑⠏⡶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⡿⣯⣿⣿⣽⣿⢯⣿⢯⣿⣻⢷⣻⣾⡇⠀⠉⠁⠀⠈⠉⠀⠀⢡⠌⡜⢄⠎⠀⠰⠙⠦⡑⣌⢃⢈⠐⠠⡄⠈⢰⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⡽⣞⣯⣟⣾⣻⣞⡿⣾⡽⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠓⠘⠂⢤⣶⣤⡀⠁⠂⢎⡘⢠⠁⠘⠂⢹⢎⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣸⡝⣯⣗⣻⢾⡵⣯⢿⣽⣟⣷⡿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠐⣣⠈⠄⡁⡞⢀⠋⠐⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠔⣯⣟⣷⢯⣟⣯⢿⣽⣻⢾⣿⣾⣻⣽⢯⢀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⢰⠁⠂⢆⡀⠀⣸⡹⢬⢳⡀⢄⠠⡐⢄⠀
⠀⠀⠀⠀⢠⣟⡾⣿⣻⠾⡽⣻⡾⣽⢯⡿⣷⣻⣞⡿⣇⣐⣠⣀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠠⢀⠃⠀⠌⢠⠎⠉⠄⡙⣎⡓⠈⢀⠤⣁⠆⠀
⠀⠀⠀⠀⠘⣯⢟⡵⢫⡝⣳⢣⠟⣯⢿⣳⠹⣷⣟⣿⠙⣿⠿⡿⠿⢿⡷⡗⠠⠄⡀⠀⠀⠀⠀⠀⢀⠠⠄⡰⠀⢂⠐⡀⠂⢌⡰⠃⠠⢈⠐⠠⠘⢹⠲⡍⠰⠈⠀⠀
⠀⠀⠀⠀⠀⢫⠞⣜⢣⠞⣥⢋⠞⡠⢆⡱⢋⠖⣨⠙⢦⡘⠦⡑⢎⠢⣝⡁⠀⠀⠀⠈⠂⠅⢒⠨⡐⠌⣒⠁⡐⠠⠂⠄⣁⢮⡰⣙⠖⠁⣈⠔⡡⢂⡟⣤⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⡺⣌⢧⡛⡴⣋⠴⡑⣊⠴⡉⢦⢡⡙⢤⠣⡱⡘⣌⠱⡄⡛⢖⢢⡐⢄⠠⣈⠂⡅⢢⠁⣇⠲⣉⡴⡜⡖⢸⢆⡳⢅⡠⢚⡔⠊⣀⡼⣜⢲⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠚⢦⣙⠶⣩⠖⡵⣒⢦⡉⢆⠲⢌⡲⣱⢱⡒⠌⢦⢑⡩⢒⢢⠩⢌⡃⢦⢙⡸⡀⠉⢦⡝⣱⣞⡱⣣⡸⢎⡜⢢⠱⣭⢚⡭⢣⣛⢬⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠚⠡⠟⣜⢣⣎⠵⣪⠕⣮⢱⢡⠓⢬⡉⠖⡨⠔⢣⢎⠜⢢⡑⣊⠔⣢⢙⠶⣍⡞⡴⣖⢧⡍⣟⢦⣜⡡⠘⠰⢛⣜⡣⣝⠊⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠈⠉⠈⠙⠲⢽⡆⣏⠦⡙⢤⢊⡵⣡⠋⠦⡙⢎⡡⢒⡡⢚⡤⣋⠞⣜⣳⢳⡞⣾⡸⣝⡞⣶⢲⡌⣖⢏⢶⣱⡋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢟⡬⢳⡜⣎⠳⡜⣥⢛⠦⣍⢦⡑⢣⢚⡵⢪⡕⣫⢖⡹⢮⡵⣣⠿⣼⡹⢎⡷⡜⣎⡞⢮⡳⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠾⣬⠳⡝⠭⠏⡛⡼⢦⡙⢧⢫⡜⣣⢞⡱⢎⡵⢫⡞⣵⢫⡅⠃⠙⣞⠘⣦⢛⠈⢷⡃⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠣⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠧⠋⠀⠀⠘⡐⡞⣧⠻⣌⡳⣜⡱⢮⣱⢫⡜⢧⡻⣜⢧⢏⠀⠀⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡑⠄⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠔⡼⡱⢭⡳⢥⠳⣌⡳⣙⢮⡳⢞⣣⢟⡼⣋⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡜⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠆⠀⠀⠀⠀⣞⡱⣙⠦⡝⣎⠳⣥⠳⣍⠶⣙⣏⢮⡝⣞⠽⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠌⠀⠀⠀⠀⡜⢦⡝⣬⢳⡹⣌⠷⣌⠻⣌⣳⡱⣎⢷⡹⣎⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠔⡡⠀⠀⠀⠀⠀⠀⠀⠀⡦⠒⠉⠀⠈⠰⠄⠠⠻⣜⣣⠞⡴⢣⠷⣭⢳⢮⡝⣮⢧⡝⣮⢳⡝⣞⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠋⢀⠀⡀⠀⠄⠀⠂⡑⠌⠢⠑⠈⠂⠂⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠁⢢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢈⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢇⠹⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⡡⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⢿⣽⢮⡇⠀⡒⣰⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⢀⣤⣇⣴⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⠿⠃⠤⢹⣟⡞⡟⡔⠀⠋⠑⢯⠜⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⣟⡿⢞⠁⡀⠤⠔⣿⠁⣇⠀⢈⠰⢁⠀⢆⠀⠈⠐⣱⢋⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀
⠚⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣷⡿⣯⣟⣷⣻⣽⡻⠞⣁⣤⡴⣶⣦⢄⢃⠘⠦⠐⡠⠉⠦⣌⢨⠑⠒⡌⠀⠏⠲⡅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⡿⣯⣿⡿⣿⣽⢿⣯⣟⣷⣻⢾⣷⣏⠁⠉⠉⠀⠀⠉⠈⠀⠈⡌⢚⠣⡐⠀⠈⠐⠩⢎⠒⠤⣁⠂⠁⢆⠀⣰⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⡽⣾⣻⢯⣟⣟⡾⣽⡾⣽⣻⣯⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠱⠰⢶⣶⣄⠀⠉⠢⠱⢨⠐⠠⠉⢰⡙⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⣹⢷⣫⣟⡾⣭⣟⡷⣿⣳⣿⢿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⡀⠀⢇⠂⠡⢈⠲⢀⡃⠈⠔⢀⡤⢤⣀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠠⢾⣻⡽⣷⣻⢾⡽⣷⢯⣟⣿⣷⢯⣟⡿⡄⡀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⢂⠃⠠⠑⢤⠀⠼⣘⠧⣎⠡⡐⠄⢣⠁⡀
⠀⠀⠀⠀⠀⣼⣷⣻⣷⣯⠿⡽⣯⣟⡾⣽⢿⣻⢾⣽⣳⣐⣤⣀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠄⡀⠃⠈⠐⣈⠆⠐⠠⠌⠸⣜⡀⣀⠒⢤⠃⠀
⠀⠀⠀⠀⠀⢻⣞⢯⣓⠮⣝⠳⣍⡻⡽⣿⠸⣿⣻⣾⡏⢻⡿⢿⠿⠿⣿⠿⠰⠠⠀⡀⢀⠀⠀⠀⡀⢄⠢⡐⠠⢁⠂⠄⠡⢈⠴⣡⢐⡨⠐⢈⡐⠈⢞⡱⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣭⠲⣍⠞⣬⠳⣌⠱⠒⣌⠳⣡⠣⣘⠣⣄⠳⣈⠎⡱⢌⡛⠀⠀⠀⠀⠈⢢⠑⢌⠰⡈⠴⠁⡐⡠⠈⠄⢡⢎⡳⢆⠋⢀⡰⢃⠔⢁⣮⢳⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⡛⣬⢛⡴⢫⡔⡩⣑⠢⡑⢆⡱⢢⠱⢌⠲⣁⢎⡑⠦⡙⢭⠒⡤⢂⠄⣀⣊⠐⣂⠡⣋⠵⣡⡔⣏⠏⢰⢫⠜⡩⢔⢣⡜⣡⠴⣪⢇⠯⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠦⢫⡜⣣⢞⡱⢆⠧⣍⠒⡴⢡⠚⣬⠳⡴⣈⢆⢣⢉⠦⡙⠤⢋⡜⠤⡌⠳⢄⡁⢱⢞⡱⢾⡸⣹⣌⣇⠺⣑⠨⢺⡜⢧⢫⡕⢮⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠙⠒⠭⢞⡹⢞⡬⣓⢦⢣⣛⠤⢛⡰⢂⠎⡴⢉⠆⣭⡘⠥⣘⠢⡱⢉⢆⡳⢺⣌⠳⡵⣣⢓⠾⣜⣳⡒⠠⢁⣜⢫⢖⡹⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠈⠑⠎⣷⢊⢧⢃⡒⣉⠖⡴⣉⠜⡄⡛⠴⣁⠎⢥⣉⠶⣩⢓⢮⡻⢵⣫⢏⡾⣣⢷⡹⢏⡽⣌⣳⢮⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⣏⠞⣦⠱⣍⢞⡱⣍⠶⣬⢱⡑⣌⠺⡱⢎⡳⢥⣋⠮⣝⢧⡝⣮⢳⡭⠱⣏⡟⡴⣹⠘⡧⡏⢆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣲⠽⠜⠊⠉⠉⠉⢶⣣⡝⣬⢳⡙⢮⡱⢣⢎⡳⢎⡳⣝⢮⣳⡈⠀⠑⠆⠑⠣⠆⢱⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡑⢄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠠⢱⢎⢶⣙⠦⢧⡹⢲⠭⣇⠯⣜⢣⡟⣼⢣⢧⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⢰⣋⠞⡼⣩⠞⣥⢋⢧⡛⣜⢻⡜⣧⡝⣮⣛⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⠒⠂⠀⠀⠀⣠⠳⣎⡝⡲⢥⡛⡴⣋⠶⡹⣌⢧⡹⢶⡹⢶⡹⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⡱⠀⠀⠀⠀⠀⠀⠀⠀⠂⡔⠉⠁⠀⠈⠒⠤⠚⢡⡟⡴⣩⢓⢧⡙⣶⢩⢞⡱⣎⠶⣹⢣⡟⣣⢟⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⠑⡄⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠂⠄⠀⠀⠀⠂⣟⡴⢣⢏⢶⡹⣎⠷⣮⡱⢯⡹⢧⡻⣜⢧⡻⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⡀⢀⡀⣀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⡘⠄⠒⠀⠃⠈⠀⠁⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⢿⡽⣾⠀⠀⢖⡝⣲⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡿⠃⠤⢹⣯⡟⡧⠑⠊⡀⠙⢦⠋⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠜⣿⠱⡀⠀⠁⡔⢁⠈⣄⠀⠀⢠⢻⡙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣾⡿⣽⢯⣷⣻⢯⡿⠝⣁⣤⡴⣶⣦⢄⢃⢢⢡⠈⡔⠘⠦⣔⡨⠑⢊⡁⠠⠙⠲⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⣯⣟⣾⣽⣏⠀⠉⠉⠀⠀⠉⠈⠀⠘⡂⡙⠢⠆⠀⠘⠀⠣⠝⢢⢐⢡⠂⠁⢦⡀⡜⣏⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣻⣞⣷⣻⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠈⠁⠒⠿⣶⣤⡀⠈⠢⠅⣌⠐⡀⠄⡸⠌⡳⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⡵⣯⣷⢯⣷⣿⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠈⡖⠠⠐⡀⡏⠠⡁⠀⠂⢠⡔⢦⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⢷⣿⣯⣟⡾⣽⡄⠀⠀⠀⠀⠀⠀⠀⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⡰⠁⠄⡑⠢⠄⠺⢜⢣⢞⡄⠣⠘⠰⣀⠀
⠀⠀⠀⠀⠀⣯⢿⣽⣷⢯⠿⣽⣷⢯⣟⣾⢿⡾⣽⣳⣧⣉⡤⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠂⠄⡁⢀⠐⡠⠃⡈⠄⠌⠸⢦⣀⡠⢊⠲⠀⠀
⠀⠀⠀⠀⠀⣿⢯⡻⢬⡛⡞⣥⢛⠿⣞⣿⠸⣿⣻⣽⡏⢿⡿⣿⠿⢷⣶⣶⢂⣀⠀⡀⠀⠀⢀⠀⢠⠐⠢⠜⠀⠄⡁⢂⠐⣠⢎⡱⢄⠔⡈⢐⠄⠌⣲⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⢮⡱⢣⡝⡜⢦⢋⠖⡰⢌⡛⠴⢡⡙⠣⣌⠣⢔⡊⠴⣨⠓⠀⠀⠀⠈⠀⢉⠢⠘⡄⠃⡭⠐⣈⠔⢀⡂⢰⢣⠞⡜⢁⠠⡔⠣⠌⢠⣇⢯⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢶⣩⠳⣜⡹⢎⡒⠬⣑⢢⠑⡎⡡⠜⡑⢦⡉⢖⡨⡑⢆⢫⢓⠲⢄⠢⢄⡀⣡⠑⡠⠃⢧⡙⣤⢞⡭⠇⢪⢇⣋⠖⣌⢳⠬⡥⢖⣫⡜⢮⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⠏⡶⣙⢮⡙⢶⡱⣊⡱⢢⠱⣉⠜⣦⢚⡤⠱⡘⡌⠦⣉⠦⣉⠎⢆⡱⢂⡝⢢⠁⣣⢞⢬⠳⣜⢻⡜⣦⠜⡘⠰⢪⠯⣕⢫⢖⡹⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠑⠊⠽⢆⡳⣥⢛⢦⠳⣌⢻⡐⡋⡔⢣⠱⢌⡑⠦⣸⡐⢎⢢⡑⣊⠴⢡⠳⡜⣎⡳⡜⣧⢣⡟⣼⢫⡕⣀⣁⢮⢣⢏⡮⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠉⠁⠉⠚⠳⣎⡳⢆⡱⢌⢃⡞⣤⢉⠖⣡⠙⢦⢡⢒⠡⢎⡲⡹⣜⢣⡟⣝⢧⣳⢻⡜⣧⣛⢧⡚⣬⢳⣏⣦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠹⣜⢣⠖⣎⠳⡜⢦⣋⠶⣤⢋⣆⠱⣊⡝⢮⡱⢳⣌⢳⠺⣭⢞⡱⣏⡞⠱⣭⠳⣙⢮⠉⡾⡅⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢮⡽⢬⢗⡻⠉⠉⠉⠲⣏⡜⣣⠳⣜⢣⡝⣣⢎⢧⡛⡼⣭⢳⡝⣆⠀⠑⠣⠉⠊⠂⠸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢄⠣⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢺⠘⡁⠀⠀⠀⢀⡼⣙⠶⡹⣌⠳⡜⣵⢪⢖⡹⣜⢧⡻⣜⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠊⡔⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠊⠅⠀⠁⠢⠄⠒⢫⠼⣹⢎⡵⢪⡝⡼⣌⢻⣎⠷⣜⢧⡻⣜⡥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⠀⠠⠀⡀⢀⠀⠈⣳⠱⣎⡜⢧⡚⡵⣊⠷⣌⠿⣜⢧⡻⣜⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢌⢢⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⡻⢴⣙⠦⡝⡼⣥⢛⣬⢻⡜⣧⢻⡜⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠚⡀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡜⣀⠀⠀⠀⠀⢠⠀⢆⣱⢓⢮⡱⢧⡏⢿⡼⣱⢎⡟⣬⢳⡝⣮⢳⡝⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⡀⢀⡀⣀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⢋⠀⡀⠀⠄⠀⠂⠁⡘⠄⠒⠀⠃⠈⠀⠁⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠳⣯⢿⡽⣾⠀⠀⢖⡝⣲⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⡿⠃⠤⢹⣯⡟⡧⠑⠊⡀⠙⢦⠋⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣾⢯⡿⣽⠞⡁⠀⠤⠜⣿⠱⡀⠀⠁⡔⢁⠈⣄⠀⠀⢠⢻⡙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣾⡿⣽⢯⣷⣻⢯⡿⠝⣁⣤⡴⣶⣦⢄⢃⢢⢡⠈⡔⠘⠦⣔⡨⠑⢊⡁⠠⠙⠲⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⣯⣟⣾⣽⣏⠀⠉⠉⠀⠀⠉⠈⠀⠘⡂⡙⠢⠆⠀⠘⠀⠣⠝⢢⢐⢡⠂⠁⢦⡀⡜⣏⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣻⣞⣷⣻⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠈⠁⠒⠿⣶⣤⡀⠈⠢⠅⣌⠐⡀⠄⡸⠌⡳⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⡵⣯⣷⢯⣷⣿⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠈⡖⠠⠐⡀⡏⠠⡁⠀⠂⢠⡔⢦⣄⠀⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⢷⣿⣯⣟⡾⣽⡄⠀⠀⠀⠀⠀⠀⠀⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⡰⠁⠄⡑⠢⠄⠺⢜⢣⢞⡄⠣⠘⠰⣀⠀
⠀⠀⠀⠀⠀⣯⢿⣽⣷⢯⠿⣽⣷⢯⣟⣾⢿⢾⣽⣳⣧⣉⡤⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠂⠄⡁⢀⠐⡠⠃⡈⠄⠌⠸⢦⣀⡠⢊⠲⠀⠀
⠀⠀⠀⠀⠀⣿⢏⡷⢎⡛⣞⡱⣛⠿⣞⣿⠸⣿⣾⢿⡍⢿⡿⡿⠿⣷⣶⣶⢂⣀⠀⡀⠀⠀⢀⠀⢠⠐⠢⠜⠀⠄⡁⢂⠐⣠⢎⡱⢄⠔⡈⢐⠄⠌⣲⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣚⠼⣘⢧⡙⢦⠳⣉⠲⠰⣌⠛⡔⡌⢛⠧⣈⠣⠔⣣⠰⢹⡌⠀⠀⠀⠈⠀⢉⠢⠘⡄⠃⡭⠐⣈⠔⢀⡂⢰⢣⠞⡜⢁⠠⡔⠣⠌⢠⣇⢯⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⣞⡱⢎⡝⣎⢳⡈⢥⠓⠤⢋⠴⣈⠇⡜⠴⣉⠚⣄⠣⢃⢮⡑⠲⡐⠤⡀⢄⣡⠑⡠⠃⢧⡙⣤⢞⡭⠇⢪⢇⣋⠖⣌⢳⠬⡥⢖⣫⡜⢮⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠪⢳⡜⣬⢣⡝⢦⡛⣌⢣⢒⡡⠎⣜⠲⣥⠩⣄⠫⡘⢤⢊⡕⡉⢖⡉⢆⢆⡙⢢⠁⣣⢞⢬⠳⣜⢻⡜⣦⠜⡘⠰⢪⠯⣕⢫⢖⡹⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠒⠳⢜⣣⠽⣌⢳⠲⡴⡹⣌⠳⢌⠣⡔⢣⢉⠖⢢⡜⡸⢰⡘⢌⠢⠍⡆⣏⠶⣩⠶⡭⣎⢧⢻⡜⣯⣑⡠⣁⢮⢣⢏⡮⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠉⠈⠉⠓⢧⡗⢮⡑⣊⠱⣘⢤⡋⢜⠢⢜⡱⠥⡘⠬⣑⡩⢴⡩⢞⡱⢯⡽⣹⢎⣷⡹⣖⢧⡳⣍⢮⢳⣏⣦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢟⢦⠳⣌⡳⣍⠶⣙⠮⡜⣆⢆⣣⢙⠲⢥⡛⢦⡝⢮⡱⢫⡞⣵⢫⠶⡝⠸⣣⠳⢎⢧⠉⡾⡅⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⣳⠥⢳⡚⣭⠋⠉⠉⠚⣮⠲⣍⡛⢦⡝⢦⡹⢦⡙⢧⡹⣎⢯⣝⣣⠀⠉⠣⠉⠚⠠⢘⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢣⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠶⠉⠐⡁⢀⠀⡴⢭⣛⠦⣝⢲⡙⡦⣝⡲⣍⢧⡹⣎⠷⢮⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⠂⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠁⠀⠄⠀⠀⠁⠈⠁⣝⢮⠳⣬⢣⡝⡲⣍⢳⡝⣦⡝⣮⣛⢧⡳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡌⠀⠀⠀⠁⠂⠀⠀⠂⢠⢏⡳⣌⡳⡜⡵⣊⠷⣘⢧⣛⢶⡹⣎⠷⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⡠⢏⢮⡱⢎⡵⡹⡜⣥⣛⣬⢳⡝⣮⢳⣭⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⢁⠆⠀⠀⡀⠐⡌⡘⣤⢛⡼⢢⣟⡹⡞⣵⣙⠶⣭⢖⡯⢞⡵⣫⢖⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠱⡂⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢣⣷⣾⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠋⢀⠀⡀⠀⠄⠐⠐⠠⠃⡜⠠⠃⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡉⠄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⠀⠈⠀⠐⠡⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⢐⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢅⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠐⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢯⣷⣻⣽⢮⠀⠀⡕⡲⢦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⣀⣤⣆⣭⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠋⠄⣹⢷⣛⠧⠃⡐⠉⠱⣋⠎⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢾⣻⣟⣯⠗⢁⡀⡀⠬⣿⢋⡐⠀⠄⣡⠂⠘⡄⠀⠁⢊⡳⣍⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀
⠛⠿⠿⠛⠛⢻⣿⣿⢿⣿⣻⣿⣽⣾⡿⣽⢯⣟⡾⣯⢷⡛⠈⠉⢀⣀⣀⠀⠏⠰⣀⠂⡰⠀⢦⣁⠜⡒⠒⡀⢁⠚⢦⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⢸⣿⣻⣿⣟⣿⡽⣿⢯⣟⣯⣟⣾⣻⡭⠁⠐⠒⠛⠛⠋⠉⠁⠘⠁⡙⢆⠅⠀⠁⠊⠵⡐⢭⠐⣠⠂⠀⣆⠀⣸⢒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⣷⣻⢾⡽⣯⢿⣽⣻⣞⣷⣻⢾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠘⠀⠂⠴⢶⣦⣄⠀⠑⠊⠴⣀⠡⢀⠡⢸⠍⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣏⣯⢿⣹⣏⡿⣾⡵⣯⣷⢯⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠉⠓⠆⠀⠐⡬⢀⠂⠄⡏⠠⠁⠈⠔⢀⡤⢤⡀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠄⣿⢾⡽⣷⣻⢾⡽⣷⣻⢷⣿⣯⢿⣽⣻⡀⡀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⡰⠁⠄⠓⢄⠠⠼⣘⢧⡹⢄⠢⠐⡡⠡⡀
⠀⠀⠀⠀⠀⣯⢿⣽⡷⣯⢿⡽⣷⢯⣟⣾⢿⣻⣞⡷⣧⠐⡠⢀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠄⠠⠁⠀⠌⣠⠃⡀⠂⠌⠲⣍⡀⣠⠒⢤⠃⠀
⠀⠀⠀⠀⠀⣿⢯⡻⣝⠧⢯⡝⠿⣿⣞⣿⢸⣷⡿⣽⡟⣿⣿⣿⢶⣶⣦⣴⠄⠀⠀⠀⠀⠀⠀⠀⡀⢄⠂⡜⠀⡐⠈⠄⠡⢈⡴⢡⡀⡤⠁⢂⠄⡁⢻⠴⠈⠀⠀⠀
⠀⠀⠀⠀⠀⣜⢣⡝⣬⢛⢦⡹⣙⠢⢍⣚⠳⢎⡙⠻⣇⡈⢇⡔⢢⠌⡽⠚⠀⠈⠈⠑⠐⢂⠅⡊⠔⡨⠜⡀⢐⠠⠁⢌⢠⢏⡴⢣⠉⡀⡜⣈⠔⢠⡏⣞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⢊⡳⡜⢦⣋⠶⣑⠢⢍⡒⢌⠲⣁⢎⡱⢄⠳⡌⡔⢃⡚⢌⡳⢲⡐⢄⡠⢠⠀⣄⠱⢈⠄⡏⠴⣡⢖⡏⡇⢸⢎⡜⢣⠔⣱⠜⣤⠴⣪⠳⣜⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⢝⢦⡙⢮⡱⢫⠴⡸⣌⡑⢆⠲⡐⢮⡱⣒⡌⢣⠜⢢⡑⢆⡱⢊⡔⢣⠱⣈⢓⢪⠀⣙⢮⢱⠯⣜⣱⢭⡒⠼⣁⠚⡼⣙⢦⢫⡕⣫⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⠥⢏⡧⣛⢵⣊⠶⣉⢲⡩⣖⠱⠣⢜⢡⢊⠥⣘⠢⣒⠥⣘⢂⡓⠬⣘⢂⢯⡹⢎⠧⡼⣎⡜⣧⣛⢷⡒⠠⢐⣩⢎⡳⣜⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠑⠚⠓⠹⠬⢧⡗⣬⠃⡝⠢⡅⣎⡒⣡⠣⢌⠳⡌⢆⠥⢓⢤⣋⠶⣩⢏⢯⡳⣽⡸⢧⣛⡮⣝⢯⡙⣦⢫⡞⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢞⡴⢫⡔⢧⣙⠦⡝⣆⠳⣬⢱⣈⠎⡜⣭⠲⣍⢞⡱⢎⡳⣝⢶⡹⢧⡯⠱⣏⠶⣙⠶⠉⣟⡏⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣳⣜⠧⣞⢣⠏⠉⠉⠲⢧⡎⡽⣘⢦⡛⡜⢮⡱⣋⠶⣹⠮⣝⡳⣆⠀⠙⠆⠙⠺⠄⢱⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡁⢎⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢢⠸⠊⠰⡀⢀⠀⣴⢫⢞⡱⢭⢲⡹⡜⢧⡳⣍⢞⣱⢻⣜⡳⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠜⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠁⠀⠄⠀⠀⠁⠈⠀⢽⢪⡝⣎⢧⢳⡙⣎⢳⡞⢮⣜⡳⣎⢷⢣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡌⠀⠀⠀⠁⠂⠀⠀⠈⢠⢗⡹⡜⣎⠶⡹⣌⠳⣜⡳⣎⢷⡹⣎⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠠⣰⢋⢮⡱⣙⢦⢫⡕⣮⠝⣦⢻⡜⣧⢻⡜⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⢁⠆⠀⠀⡀⠐⡌⢌⣱⢣⢏⠶⣹⡝⣮⢳⡼⣱⢻⡜⣧⢻⡜⣧⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠂⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⢂⣷⣾⣾⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⡀⠀⠄⠂⡘⠰⢠⠉⠆⠘⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠁⠘⢄⠂⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠌⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡉⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢋⠍⠁⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⢯⡷⣿⣳⡝⠀⠈⡤⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⢀⣠⣌⣸⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⠏⠄⢻⣷⣻⠝⠢⠔⠊⢱⢣⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⢿⣽⣻⡽⠂⠀⣈⡸⣷⢫⠘⠀⠐⡈⠄⠸⡀⠀⠁⠣⣜⡲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠾⠿⠿⠿⠟⢻⣿⣿⣿⢿⣿⣻⣿⣽⣷⢯⣟⡿⣾⣽⣻⡞⠃⣀⣨⣥⣀⡀⠏⠰⢂⠐⢠⠙⣄⢂⡙⠢⠄⡊⠐⠹⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⢸⣿⢿⣾⣿⢿⡿⣽⡿⣽⣟⡾⣽⣳⣏⠃⠒⠛⠉⠉⠛⠙⠉⠘⠁⠫⢄⡃⠀⠨⠉⠶⡑⢬⠁⢌⠐⡈⡄⠈⣰⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⡾⣟⣾⣽⣻⡽⣯⣟⡷⣯⢿⣽⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠌⠒⠁⢤⣶⣤⡀⠈⠐⢍⡊⡐⠀⠌⢉⢰⠛⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⣹⣞⡳⣯⣽⣳⢯⣟⣯⣟⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢄⠀⠰⠱⢈⠐⡀⡞⠠⠉⠐⢠⠀⣠⢄⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠄⣿⣞⡿⣾⣽⣳⢯⣟⡿⣞⣿⣞⡿⣽⣻⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⢰⠁⢂⠣⣀⣀⠼⡱⢎⣭⠂⠤⡐⢄⠃⠀
⠀⠀⠀⠀⠀⣿⢾⣽⡷⣯⣟⣯⣿⡽⣯⢿⣿⣽⣳⣟⣇⠢⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠠⠀⠄⠃⠀⢂⢐⠂⠠⢀⡑⢩⢎⠁⢀⠔⠌⠌⠁
⠀⠀⠀⠀⠀⣿⢯⡿⣛⢷⣻⢾⣽⡿⣽⣿⢸⣷⣟⣾⡟⣿⣿⣶⣷⣤⣤⣤⡁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡰⠁⠠⠁⠌⡐⢈⠤⣃⡈⣐⠠⢀⠂⡙⢮⡹⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣯⢳⡝⣮⢳⣻⣯⡹⢻⢾⣇⡙⠿⠽⠇⠘⠧⠉⠿⣽⣿⠳⠈⠉⠐⠐⠂⠔⡰⢈⠢⢘⡰⠁⠠⠡⢈⠐⣀⢮⡱⢦⠙⠀⡤⢊⡰⠀⡧⣝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢧⢻⡜⣧⢏⡟⢆⠣⢌⡙⢌⠳⢟⣆⠢⣄⠣⢃⠦⢹⡇⠀⠀⠀⠀⠀⢀⡂⠥⢘⡀⠗⣌⢣⡴⣒⡇⢸⢲⡙⢆⡌⣚⡔⣃⣠⢞⡳⣜⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡜⢮⠳⡙⢌⠎⡵⢋⠖⣒⢚⠲⢡⠒⣎⠱⡄⣋⠜⡌⠥⣚⠱⣉⠦⡙⡘⢆⡩⡑⢦⠈⡱⣌⢳⢞⡥⢳⣨⠇⡼⢢⠘⡼⣜⢣⠞⣮⠱⠎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡙⢮⠱⡘⡌⠎⡔⢃⠎⡔⢃⠎⡥⣉⢆⡓⡬⡐⢎⠬⣑⢂⠳⡐⠦⣑⢩⠢⡑⡜⢂⢯⡱⡭⢮⡜⣮⣙⢮⣛⢶⠃⡈⠤⣉⢧⡛⣴⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡝⢮⣅⠓⡬⢱⡌⣃⠞⣈⠎⡜⡐⢆⢮⠱⢣⠍⡒⡜⡠⢍⡒⣉⠖⡡⢆⢣⠱⢌⡳⢎⡵⡹⣖⡻⣞⡬⢷⡹⣎⡟⡖⢮⣱⢣⡟⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⡁⠚⠹⠜⣧⣚⠵⣚⡤⣍⠲⣍⠞⣬⢃⢇⢪⠑⡴⢑⡪⣔⡡⢎⠱⠌⢦⡛⣬⠳⣍⠶⡱⢭⡳⢮⡝⣧⡟⢱⢫⠟⣲⢣⠙⣞⡏⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠆⠤⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠚⠧⣋⠶⣨⡑⢦⣋⠶⠥⠛⢮⡝⣚⢦⡙⢦⡛⣬⢳⡙⢶⡹⢧⣛⠶⣅⠀⠛⠆⠃⠫⠄⢱⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢎⠰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢥⣛⠦⢋⠁⡀⣰⢎⡽⣚⢦⣋⢧⡙⢦⢧⡹⢦⡹⢧⣫⢟⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠎⡄⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠁⡸⣱⢭⠲⣍⠶⣩⢓⠮⣝⢶⡹⢧⡳⣎⢷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢨⠀⠀⠀⠈⠀⠀⠀⠀⠐⣧⢋⠷⣌⢳⢣⢏⡞⡜⣧⣛⢧⡻⣜⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠰⡁⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⢄⡞⢦⣋⠾⣌⢧⣋⡞⡼⣱⢎⣧⢻⡜⣧⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠡⢣⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠃⢆⠀⠀⢀⠐⠤⡉⢆⡞⣜⢣⡞⣽⣚⢦⢧⣹⢳⡝⡾⣸⢧⢻⣜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢪⡐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢂⣵⣷⣾⣿⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⡀⠠⠠⢁⠒⡌⠰⡈⠆⠡⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡉⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡟⠀⠐⠀⢡⢂⠁⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠁⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠙⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡙⡉⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡿⣏⣳⠈⠰⡀⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣠⣔⣡⣬⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⠟⡘⢷⣻⣽⠣⠆⠰⠙⢦⣋⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⢿⡿⣿⢯⠋⠀⠀⣹⣷⢻⠀⠐⡀⠂⠄⢩⠀⠀⠊⠔⡦⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀
⠾⠿⠿⠿⠟⣻⣿⣿⣿⢿⣿⣻⣿⣽⣿⣳⢿⣻⢾⡽⣯⢿⠝⢀⣒⣂⠉⢉⡏⡸⡁⠄⠰⢩⡐⢀⠳⠤⢠⠈⠓⢧⡚⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⣻⢿⣽⣾⣿⢿⡿⣽⣻⣾⡽⣯⣟⣯⡿⠙⠱⠚⢛⡛⡿⠟⠃⠐⠂⣳⡈⡄⠀⠩⠙⢦⠡⣂⠅⡈⠄⢘⡀⠈⢰⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣟⡾⣷⣻⢯⡿⣽⣳⢿⣽⣳⢿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠠⠔⡁⣠⣤⣄⠀⠁⠲⠘⡔⢈⠀⡉⢂⢫⢖⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡼⢮⣽⡳⣏⣯⣟⡷⣯⢿⡾⣽⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠙⠻⠦⡀⠈⢠⢣⠐⢀⠂⡼⢀⠋⠐⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠚⡿⣯⢷⣻⡽⣾⣽⣻⣽⣻⣿⣽⣷⣻⢷⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠂⢬⠁⡐⢆⡀⠀⣜⢫⡜⣣⠄⡠⢠⠐⢂⠀
⠀⠀⠀⠀⢠⣿⡽⣿⣯⣟⡷⣯⣷⢯⡷⣿⣟⡾⣽⣻⡔⠠⢀⠀⠀⠀⠀⢈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠀⠌⠂⠀⡐⢀⠎⠀⠌⡑⢪⡕⠁⢀⠤⢉⠢⠀
⠀⠀⠀⠀⠈⡷⠛⡙⠳⣯⣟⣷⣿⣯⢿⣷⢻⣽⣳⡿⠷⣷⣦⣦⣤⣠⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡐⠁⠄⡈⠐⠠⠁⡤⢋⠀⢌⠐⡀⢂⠹⢲⡹⠠⠉⠀⠀
⠀⠀⠀⠀⢠⡔⢯⡹⡹⣌⢻⣽⣞⡹⢛⢾⣌⠻⢿⣻⠇⠹⡏⠙⣟⣿⣯⠏⠐⠒⠐⠠⠄⡄⠢⠄⣃⠱⡐⠁⠌⡐⠠⢁⠢⣝⡰⣍⠞⠁⡠⠜⡠⠂⣧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⣦⢹⢲⡱⢣⡝⠦⡍⢛⠌⡅⢊⠜⡩⠶⢳⡆⠠⢄⡀⢡⣿⣿⡆⠀⠀⠀⠀⠀⠈⣄⠃⡄⢂⠳⣌⢒⣠⢥⡆⡜⢦⠳⡌⣠⠒⡥⢃⣐⢼⢣⢏⠄⠀⠀⠀
⠀⠀⠀⠀⠰⣋⠶⣩⠓⢬⡑⢣⠎⡰⢌⡒⢌⠰⡈⢆⡉⢣⠖⣉⠖⣨⠛⣤⠠⣄⠒⡤⢩⠜⣈⠓⢮⡀⣓⡜⣫⡜⢦⢇⣸⢃⢳⡘⢤⠻⣜⢫⡜⣎⠳⠎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠱⣋⠦⡙⢤⠊⡕⡌⠐⠢⠘⡌⢢⡑⠢⢌⢣⠚⢤⠚⣄⠳⣈⠕⠢⡍⡔⢣⡘⢤⠋⣆⠹⡞⡼⣱⢼⣋⠞⣵⢻⡤⠧⠀⡙⠪⡗⡼⣸⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢩⢖⡩⢢⢙⡰⢘⠢⡀⠀⠘⡤⢌⡓⠺⣄⢋⠦⣑⠢⡱⢌⢪⠑⠦⡑⢆⡩⢂⡓⣌⠳⣍⢳⢧⢮⣝⣚⢧⡻⣜⡳⡖⣖⣫⢵⣫⣅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡄⢫⠖⣥⢊⡴⢉⠲⢩⠍⡣⠜⢢⢩⡑⢢⢏⠶⣄⢣⢑⢪⠰⣉⠖⣉⢦⡑⣥⢚⡬⢳⡍⢮⡹⣞⡼⣹⢞⡵⢩⣗⠫⡖⢧⠙⣶⡋⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡔⢂⢹⢆⡏⡰⢩⡘⠅⣎⠑⡎⠥⡒⢬⣋⢎⡳⢐⡊⡌⢆⡓⣬⢛⡼⢦⡛⡴⢫⡜⣣⠞⣥⢓⢮⡳⣝⢮⣅⠀⠚⠆⠙⠪⠄⢱⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⢌⠢⠄⢫⢖⡡⠣⢌⡓⠤⢋⡔⣣⢕⡣⢎⢧⡑⢣⠜⣈⠖⡜⢦⣣⠞⣧⠹⣜⢣⠞⣥⢛⡴⣋⢾⡱⣏⠾⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢢⠁⠊⠀⠫⣜⠳⣎⢶⣩⠷⠼⠓⡊⢍⠩⡑⢭⡓⡞⣬⢛⠜⠁⠂⡹⢖⡹⣌⢧⡛⡴⢫⣜⡳⣎⠷⣭⢻⡼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠃⢆⠀⠀⠀⠈⠉⠀⠀⠀⠀⣘⠀⠁⠈⠀⢩⠀⠈⠑⠊⠁⠂⠘⠀⠁⣏⠶⣩⢖⡹⣜⢣⢎⡵⣫⡝⣮⢳⡭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⢀⠆⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣄⠾⣌⡳⢧⢎⡵⣋⡞⣬⢳⢧⢻⡜⣧⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢌⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⢃⠆⠀⠀⢀⠐⠤⣉⢒⣎⢳⢮⣝⣫⢞⡴⣫⢞⡵⣫⢞⢧⣛⢶⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⡰⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠠⢱⣌⣦⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠠⡀⠤⠠⢄⡐⢠⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡟⠯⠁⢠⢀⡐⠀⠈⠀⠠⠘⠀⠃⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠠⠡⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡙⢋⠉⠉⠁⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡳⡌⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⢀⣠⣕⣨⣐⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⡟⣿⣻⢷⣃⠀⢰⡍⣏⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀
⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⣿⡿⣟⡯⢁⡘⣯⡗⠈⠄⡁⢈⠊⡗⠀⠰⡀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣻⣽⡾⣟⣯⣟⡿⣯⣿⣍⣀⠀⠤⢽⡇⠁⠐⢠⠂⠄⠱⡀⠀⠈⣰⢫⡝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀
⠀⠀⠀⠀⠀⢸⣿⣯⢿⣾⣿⣿⣻⡿⣽⣻⢾⣽⣻⢾⣽⣻⣷⣥⣤⣤⣀⠀⣸⢱⡈⠄⡂⢣⡌⡰⢉⠡⢡⠀⡘⢈⠳⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⢷⣯⢷⣟⣯⣟⡿⣾⣽⣻⠊⠁⠈⠀⠂⠛⠿⠛⠁⠰⠣⣘⠀⠀⠓⠡⣃⠲⡁⢆⡐⠠⠘⣄⡠⣛⠭⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣟⡻⣾⢽⣛⣾⣛⣮⢷⣯⣟⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠒⢀⣠⣤⣀⠀⠀⠑⢌⠆⡂⠡⢀⠀⡙⢺⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⢞⣷⣻⣧⢿⡽⣞⣯⣟⣯⣿⢾⣽⡷⣿⡀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠉⠛⠻⠶⠤⠄⠸⣡⠁⢂⠐⡜⢱⠀⠁⠆⢠⠴⣤⣀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⣺⢯⣷⣟⣯⢿⣽⣻⣞⣷⣻⣿⣯⣟⣷⡇⠀⠀⠀⠀⠀⠠⢢⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠂⢠⠃⢈⠲⢄⠠⠋⠘⢆⣳⠂⠔⠨⡘⠠⡀
⠀⠀⠀⠀⠀⢿⣯⣿⣿⣞⡿⣾⣽⣾⣳⢿⣻⡾⣽⢾⣽⢆⠄⡀⠀⠀⠀⢀⣏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠂⠄⡁⢀⠠⢠⠎⠀⠌⡐⠠⢈⡀⣠⠒⢤⠃⠀
⠀⠀⠀⠀⠀⢸⠓⡩⢄⡉⠻⣷⣿⠹⣿⣻⣇⢿⣟⣯⣿⢹⣿⣷⣶⣤⣤⣾⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠀⠄⡈⠐⡀⢂⡴⢃⡖⡱⠊⢀⡱⠠⢈⢧⣛⠀⠀⠀
⠀⠀⠀⠀⠀⢄⠊⠔⡠⢂⠥⢈⠻⡧⠦⣩⡙⡎⣍⣻⠓⠈⠣⠀⢫⢿⣽⡞⠀⠒⠀⠆⡰⢀⠢⡐⠄⣃⠲⡀⢌⡐⠠⢁⡐⢮⡜⣣⠞⢀⡔⢣⠔⠁⣼⢢⡝⠀⠀⠀
⠀⠀⠀⠀⠐⡈⢌⠂⡅⢂⠆⡥⢒⠭⡑⠥⡒⣍⠐⡋⠯⡌⠒⢆⠀⣾⣿⣿⠀⠀⠀⠀⠀⠨⠶⣄⣣⠄⢧⡙⣤⢖⡞⡍⢰⠳⣌⡱⡘⠦⣜⢣⡖⡻⣜⢣⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠊⠰⢈⢦⢏⡰⢉⠦⡙⡰⣑⠸⢄⡑⢢⢡⣿⣎⠒⠌⢿⡇⠀⣠⢐⠢⢍⢣⠱⢌⠳⣆⣢⢽⠸⡾⣸⢹⣌⠿⣄⢣⠅⠣⢋⡷⣜⡱⢎⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠦⣥⠋⠴⡡⣑⠢⡙⣆⡘⢤⣻⢽⢚⢣⠛⣌⢫⡑⠦⣉⠜⡌⠦⣉⠆⡓⢬⠳⣎⢳⢳⢧⡓⢮⡻⣭⢏⣤⣁⢦⣚⢴⣹⡋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠘⢳⣌⢣⠓⡰⣁⢣⡑⡆⡑⠆⠌⣫⢂⠇⢎⡔⢢⡑⠎⡤⢋⠔⣃⠦⡙⣘⢎⡳⣍⠷⣫⢞⣭⢏⣷⡹⣎⢷⡸⢆⡯⢞⡵⣃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢞⢢⡙⢤⡑⢆⠲⢡⠈⠀⠀⡇⠎⡜⠢⠜⢢⠜⣑⠢⡍⡜⡐⢦⡱⣍⢮⡱⢎⡳⡝⡾⣜⣫⢶⠁⠸⣏⠸⣣⢝⠈⡷⡃⠑⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⢆⠀⠀⠀⢸⠢⠜⢢⠜⡨⢅⢫⠀⠀⠀⣏⠼⡰⣉⢎⠥⠚⣄⠳⠰⢌⡳⢣⢳⡜⢦⡝⢮⡱⣙⠾⣥⢳⣫⡄⠀⠈⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⠜⡠⢃⠀⠀⠀⢧⡙⣂⢎⠱⡌⡒⡄⢀⠴⣊⠷⣡⢏⣎⢣⠙⣄⢣⡙⣬⢓⣏⠳⣜⢣⠞⣥⠳⣍⢿⡸⢧⡳⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠨⠄⠈⠀⠀⠀⢺⠔⣌⢢⢃⠖⣡⠓⣌⠺⣥⢛⡴⢫⡔⣊⢳⡐⢦⡚⣴⢫⡜⡳⢬⢣⡛⡴⢻⡜⣮⡝⣧⢻⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠣⡘⠀⠀⠀⠀⠈⡻⢤⣂⢧⡚⣤⢛⡬⢳⣌⠳⣜⢣⠞⠭⢖⡹⢦⠝⢠⡳⣜⡱⣋⠶⡹⣜⢣⣟⡲⣝⢮⡳⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⡁⠇⠀⠀⠀⠀⢏⡳⡜⢦⣙⢦⢫⡜⣣⢎⡽⠌⠁⡀⠄⡀⢠⠀⣦⢳⡱⢮⡱⣍⢾⡱⣎⠷⣎⢷⡹⣎⢷⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⢚⠀⠀⠀⠀⢺⡱⣙⠮⡜⣎⠳⣜⡱⢊⠄⠐⠠⠀⠒⡌⢢⠱⣎⢧⣛⢧⡻⣜⣎⠷⣭⢻⡜⣧⢻⡜⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⢿⣟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⡒⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⢢⢱⣢⣤⣶⣶⣶⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢄⡀⢂⠤⠠⠄⠤⡐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠿⠏⡀⢄⡐⠈⠁⠠⠘⠄⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠎⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠑⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡡⠂⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣴⣵⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⡷⣟⣧⡇⠀⠰⣢⢳⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣿⣾⣿⣿⡻⢁⠌⣷⡏⠒⡈⠁⢏⡶⠁⠄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⢿⣿⣻⣿⣽⡿⣷⣻⢯⡿⣽⣻⢯⣷⣛⣀⡀⢠⣼⠆⡁⠠⢁⡂⢤⠀⠀⠀⠳⣍⡛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀
⠀⠀⠀⠀⠀⢸⡿⣷⢿⣻⣿⣟⡿⣽⣳⣟⡷⣯⢿⣽⣳⢯⣿⢥⣤⣤⣀⠀⢸⢂⠄⠠⡀⢇⡀⠙⡒⢂⠡⠈⠕⢪⠀⢂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⡽⣯⢿⣻⡽⣾⣽⣻⡽⣯⢿⣽⣻⣾⠉⠁⠀⠀⠀⠸⠟⠛⠂⢹⠂⡱⠀⠓⠌⣣⢑⢢⡘⠄⣈⠐⠦⡀⡴⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⢯⣝⣯⢳⣻⣵⣻⢶⣻⣯⣟⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠱⠀⠀⣀⠀⠈⠂⠵⡘⡄⢂⠐⠠⠘⢂⢹⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣻⣞⣷⣻⢷⣯⣟⣯⢷⣿⣞⣿⢿⣻⡀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠁⠀⠉⠛⠻⢶⣦⣤⣀⢃⠎⡘⠠⠁⠂⡷⠀⠀⠃⣠⠖⡶⢲⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠀⠀⣿⣳⣿⡷⣯⣟⡾⣿⣞⡿⣾⢿⣯⣟⣯⣇⠀⠀⠀⠀⠀⠰⢲⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠁⠈⡴⠁⢉⠒⢠⠊⠉⠘⢣⠞⠀⠑⡈⢌⠡
⠀⠀⠀⠀⠈⣿⣳⣿⣿⣳⣯⣟⣿⣾⣽⣻⢿⡾⣽⢾⡽⣆⠀⠀⠀⠀⠀⢡⣿⠥⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡀⢂⠐⡀⢀⠂⢤⠃⠠⠈⠌⡀⠢⢤⣎⡘⠂⠀
⠀⠀⠀⠀⠀⢹⡿⣟⣿⠟⢓⡉⠛⡼⡻⢯⣧⢿⣟⣯⣿⢹⣿⣶⣦⣤⣠⣿⡓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡐⢀⠂⡐⢀⠢⢜⢢⡚⠔⢁⠢⠔⡡⢐⢮⡱⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠋⢄⡘⠤⠘⡰⠠⡙⢦⡙⢦⠩⣏⡓⠈⠳⠀⢹⢯⣷⡏⠁⠸⢀⡀⠀⠀⢀⠀⡀⢄⢠⠋⠀⣄⠂⡐⢀⢂⡟⡬⢧⠁⡌⡔⠣⢊⢀⣾⢲⡹⠀⠀
⠀⠀⠀⠀⠀⠀⠠⡉⢄⠒⡈⠥⢐⠡⡐⠢⢜⠨⣁⠛⠵⣈⠒⢆⡀⣾⣿⡿⠀⠀⠀⠀⠈⣘⠳⣤⠵⣌⡰⣉⠧⣄⢶⡚⡇⢺⡜⡱⢊⡜⡰⣏⢶⢣⢏⡖⣣⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠡⢈⠆⡑⡈⠆⢡⠢⡕⣊⠖⡤⣉⠒⣌⣾⣏⡔⠠⢻⡧⠀⠀⠀⢠⡘⠤⠓⡌⠻⣜⡳⢬⡎⣏⢶⡙⣦⠳⣣⣕⢣⠈⡑⢮⢇⡏⡞⡼⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡐⠈⠈⠁⠒⢩⠎⡑⠦⢡⠚⣄⠣⢓⡾⡽⢾⡀⠣⠌⣛⠂⣄⠎⢆⠥⢋⡜⠬⠱⣌⢻⣕⡚⢮⡻⣜⡜⣯⢳⡞⡭⣄⣐⡴⣚⡼⣱⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⡇⢎⡑⢎⡡⢃⠦⡙⠼⠈⠻⣻⢆⠳⠞⡬⢓⠬⡘⢬⡘⠆⡜⡌⢣⢌⡳⣌⢻⣳⢳⡝⣮⢵⣫⠾⣹⢞⡴⢳⡥⢟⡵⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠱⡨⠜⣄⢣⡉⢦⠩⢜⠀⠀⠸⣈⠞⡰⢑⡊⠦⡙⢤⢊⡱⠌⣜⣱⢎⡵⣊⠷⣌⢷⡹⣎⠷⡓⠈⢳⣃⢸⡓⣆⠘⡷⠈⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡠⢁⠀⠀⠀⠀⠀⢃⢣⠒⣢⠑⠦⣉⠎⡄⠀⢰⢡⠚⣐⢣⡘⢅⡚⠤⢃⠦⡙⢦⡓⣎⠶⣩⠞⡜⣎⠷⣭⢻⡥⠀⠀⠈⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡑⠌⣂⠀⠀⠀⠀⠈⡦⠙⡤⢙⠢⣅⠚⣄⠀⢸⢢⡝⣤⠳⣜⠢⣡⠋⡜⡰⢡⣛⠼⣌⡳⢥⡛⡼⢭⡻⣜⢧⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠑⡂⠐⠀⠀⠀⠀⠀⠱⡩⢔⡩⢒⠤⢋⡼⣄⢎⡳⡜⢦⡛⣬⠲⡅⠞⡰⢡⢣⢞⡱⢎⡵⢣⡝⡞⣧⢻⡜⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠃⡔⠀⠀⠀⠀⠀⠀⠀⡟⣔⠢⣍⠲⣍⠞⡜⢮⡱⣙⠦⡝⢦⠳⣌⢧⡱⣋⢞⡼⣱⢫⡜⣣⠞⡽⣜⢧⡻⣜⡳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠠⡅⠀⠀⠀⠀⠀⠀⢸⡜⡳⣌⠗⣮⡙⣎⢧⠳⣍⢞⡹⠊⢉⡈⢆⡷⣭⢻⡔⣣⠳⡼⣥⢻⡵⢫⡞⣵⢫⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡃⢦⠀⠀⠀⠀⠀⠀⠘⡼⡱⢎⡝⢦⡹⡜⣎⠳⡜⠎⢀⡑⢢⢘⡼⣣⡝⣮⡝⡶⣫⢗⣭⢳⠞⣧⢻⡜⣧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⣠⣴⣾⣿⡿⣿⣟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⠡⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠠⠘⣬⣼⣴⣶⣶⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⢀⠀⠄⡠⢀⠄⡠⢀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣟⠁⠀⡀⠠⠐⠈⢂⡘⠤⠑⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠿⠛⠛⠋⠉⠀⠀⠀⢃⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡁⠈⠀⠈⠆⡑⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡌⢡⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⣠⣴⣮⣤⣧⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⡞⣿⣜⡆⠁⢡⢄⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣽⣿⣿⢿⠏⢠⠹⣞⣧⢀⠇⠫⣖⡱⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀
⠛⠿⠿⠿⠿⢿⣿⣿⣿⡿⣿⣿⣻⣯⣿⢿⣳⡿⣯⣟⡾⣿⣻⣞⣋⣀⠀⠴⣿⡇⠐⢀⠂⡈⡅⠀⠈⠑⣢⢖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀
⠀⠀⠀⠀⠀⢼⣿⡷⣿⣿⣿⡽⣿⣻⡽⣯⢷⣻⢷⣯⣟⡷⣯⣤⣤⣄⡀⠀⡻⠅⡊⠀⢀⠇⠘⠢⡄⠤⢈⠊⠳⠚⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢸⡾⣽⣷⣻⢷⣻⢷⣯⣟⣯⢿⣽⣻⡾⠉⠀⠈⠀⠀⠿⠻⠃⠴⡜⡅⢠⠃⠮⡑⢦⡘⢄⠢⠌⡀⠹⣀⡈⡽⣩⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⡝⣧⡟⣭⢿⣹⢯⣶⣻⣞⡿⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢃⠆⢀⣈⣄⠙⠄⣣⠒⠤⢁⠠⢀⠑⢃⠻⡜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣯⣟⣷⣻⡽⣯⣟⡿⣞⡷⣿⡽⣿⢷⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠌⠀⠉⠩⠟⠳⣶⣄⡙⠌⡂⠜⡀⠌⡀⠏⡄⠀⠉⢠⠖⡶⣒⠖⡠⢀⠄⡒⡀
⠀⠀⠀⠀⢠⣟⡾⣽⣷⣻⢷⣯⣿⣽⣻⣟⣿⢯⣟⡿⡄⠀⠀⠀⠀⢀⠲⣶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠈⣥⠃⠀⠐⡨⢁⠘⠒⢠⠊⠉⠰⢩⠋⠀⢁⠌⡰⠐
⠀⠀⠀⠀⠐⣯⢿⣿⣷⣯⣟⣾⣽⣾⣳⣿⢻⣟⡾⣽⣷⠠⠀⠀⠀⠀⠐⣈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⢀⠂⠁⠀⠂⢌⠂⠄⡁⢊⠀⠳⣒⠯⡔⠁⠀
⠀⠀⠀⠀⠀⢹⣿⣟⣿⢿⣾⢯⣿⠹⡳⣿⡛⣯⣿⣟⡏⣿⣷⣧⣤⣄⣴⣿⠙⢡⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡁⠂⠌⠠⢁⡜⣡⠞⡐⠁⣄⠊⡐⢨⠳⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣄⠀⢉⠛⠿⠶⠱⣂⠟⣌⢋⣎⡃⠘⠇⠈⣫⣿⣾⡙⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⡜⠀⡐⠠⠁⠌⢠⢏⡼⣡⠃⡰⡘⠤⠊⢄⣯⠳⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡘⢤⡉⠖⡤⠞⡡⢍⠛⢦⡁⢒⡀⢠⣿⣿⡇⠀⠀⠑⠒⠐⠠⢦⣜⢀⠣⢘⡔⢣⢌⡤⡗⡎⡜⣎⠲⡡⢎⡱⣝⢲⣋⢗⣎⠳⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠡⣜⡢⠌⠵⢈⡑⠢⢡⠙⡤⣸⣾⣴⠹⢿⣷⡇⠀⠀⠀⠀⠀⢠⠳⣬⢣⣂⣱⢌⡇⡹⣎⠵⣣⢸⢧⣣⠕⡀⠣⢏⠷⣘⠮⡼⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠂⠒⠤⡈⠁⢆⠡⠒⣯⠷⡯⢐⠂⡌⢓⠀⠀⠀⠀⠂⠁⡱⢊⠥⠻⣜⠯⣞⣍⡘⣧⡙⣯⢞⡵⣫⢄⣡⢖⡫⣵⢻⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠐⠀⠀⠀⠘⡀⠀⠈⠐⠘⢻⡽⣂⠙⡠⢁⠂⠀⠐⠀⠀⠰⣅⢋⡔⠣⡜⠻⣜⢮⡽⢶⡹⣎⠿⢼⣣⡟⡴⢫⠶⣫⢽⡂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⡠⠀⢌⡁⠾⣥⢊⠐⡡⠊⠄⡄⠀⠀⡝⠲⡌⡔⠣⡜⢡⢊⡱⢏⢾⡱⣏⡻⣝⢶⣃⠹⣍⡇⠘⣷⠈⠐⠀⠀⠀
⠀⠀⠀⠀⠀⢈⠔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠌⢦⢩⠱⣉⠆⠑⢄⣩⡘⠀⠀⡸⣌⠣⡔⡌⡓⠬⡑⢬⡘⡜⢦⠳⣍⢷⡹⢮⡅⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⡈⠔⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣃⠚⡤⢃⠎⡔⡩⠦⠌⠀⠡⡀⣔⠳⢌⡑⠦⡑⣌⠓⡸⢰⡸⣙⢎⡳⣜⢧⣛⢧⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⢁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⠡⡍⡴⣩⢒⡡⢱⣋⠠⠀⡠⢘⠤⢋⠴⡘⢢⡑⢢⡙⡰⢡⠓⡭⢎⡵⣫⢞⡭⣞⡱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡃⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢷⡸⡱⠡⢆⡱⢣⢖⡀⢸⡁⢇⠎⡱⢢⠙⡤⢣⢑⢢⡑⢃⡚⢌⠳⣜⢧⡻⣜⢧⡻⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢳⡱⣍⡒⣔⢫⠞⡄⢸⡜⣌⠳⣅⢎⡽⣘⠇⣌⢒⡌⢣⠜⣨⡳⣹⢎⡷⣹⢎⡷⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠆⢣⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⡵⢪⡕⣎⢧⣋⢷⠀⡜⢎⡳⡜⣎⠶⣩⠞⣬⢣⡜⡱⢮⡕⣣⠟⣎⢷⡹⣎⢷⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⢿⣿⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⡐⢌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠠⣘⣦⣷⣶⣶⣶⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠀⢄⠠⡀⢄⠠⡀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⡉⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⢀⠠⠐⠐⠈⠱⡈⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⣌⣰⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡏⠈⠈⠀⠣⠌⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢄⠢⠄⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⡷⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣴⣾⣷⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠹⣯⣗⡃⠈⠰⢄⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣷⣿⣿⣿⢿⡯⠁⠂⢹⡾⣷⠠⠜⠊⢶⣩⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀
⠈⠉⠉⠉⠉⣻⣿⡿⣟⣿⣿⣽⣿⣾⢿⣯⣟⡿⣞⡷⣯⢷⡯⠏⠤⠐⠒⠓⣿⢉⠠⠁⠌⠠⢁⠀⠀⠉⢦⡓⢶⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⢹⣟⡿⣿⢿⡾⣯⢷⣯⢿⡾⣽⣻⣽⣻⡽⠛⠟⠛⠷⣶⣤⡀⠃⡠⠂⠄⢈⠱⡈⠓⡤⠔⠂⡉⢊⠳⠀⢂⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⢯⡟⣯⢿⡽⢯⣟⣾⣻⣽⣳⣿⣾⡇⠀⠀⠀⠀⠈⠉⠁⠀⠴⡁⡟⡀⠎⠒⢕⡲⣐⠢⣐⠡⢀⠘⠦⠄⣏⠶⣩⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡾⣧⢿⡽⣾⣽⣻⣞⣷⣻⡷⣯⣷⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢑⡁⠨⣤⣤⣅⠃⠜⡰⢌⢀⠢⠀⠌⡁⣱⠋⠐⢠⠀⣀⣀⣀⡀⠀⠀⠀⡀⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣽⣳⢯⡷⣟⡾⣽⣿⣳⡿⣽⣷⠀⠀⠀⠀⠀⠀⣀⡀⠀⡤⠠⠐⠀⠀⠐⠋⠙⠛⢷⣌⡉⢆⡜⢀⠂⠄⡇⠣⣀⠀⡰⠣⠞⡴⡣⠘⠄⠣⢘⠠
⠀⠀⠀⠀⢸⣯⣟⣿⣾⡽⣯⣟⣿⣽⣳⣿⡿⣽⣳⣟⣆⠀⠀⠀⠀⠈⡉⠿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢣⠂⠀⠀⡜⠀⡐⠠⡘⠠⠐⡈⠐⡥⣀⣆⠑⠎⠀
⠀⠀⠀⠀⠀⢷⣻⣿⣿⣽⡷⣿⣻⢷⣿⣳⢻⣷⣻⣾⠿⣴⣤⣀⠀⠀⢰⡾⠿⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠄⠡⢈⠀⠄⢠⠱⣡⠠⠁⠄⡡⢈⠱⣎⠇⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣧⡛⠙⢻⢿⣳⡌⡎⡝⢯⢹⠻⡽⠇⢻⡉⠻⣿⣿⣟⡇⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⢈⠐⠠⢈⡰⢣⠳⠅⢂⡡⢚⠤⠁⡼⣌⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⡐⠤⣈⠓⢌⡉⢒⠣⠽⣳⡡⢠⢀⠀⢩⣿⣿⡄⠀⠀⠰⠀⠀⢀⠀⡀⢠⠀⡄⢧⡐⠤⠊⢠⣁⢢⡝⡭⡙⡔⢪⣔⢣⠴⣚⡵⣊⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠒⠤⢩⠂⡜⢠⢃⠒⠤⡑⢢⣜⠆⣾⣿⣯⠁⠀⠀⠀⠀⠀⠀⣟⡱⡆⢡⠘⢦⡘⢶⢫⡓⡇⣸⣜⠰⡱⢈⡓⡮⣏⢞⡱⢎⠅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠉⠀⠃⠜⢨⠑⣌⡘⠤⣱⡟⣿⡇⡌⢿⡿⠆⠀⠀⠀⠀⠀⡰⣎⠵⡹⢄⣊⢧⢳⡸⢣⡝⣹⢦⣏⢿⣑⠂⡐⣨⡜⢮⡵⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠂⠀⠄⠀⠀⠘⢠⠘⠴⣉⠺⣷⢧⢃⠆⡌⡐⢀⠀⠀⠀⠉⣱⠮⡛⠹⢯⡹⣞⣭⢳⡦⣽⢢⡟⣼⢣⣏⣟⡹⢦⢫⢷⣹⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠀⠀⢀⠄⠈⢆⠢⢐⠱⡈⠔⠀⠀⡞⡥⠘⣌⠓⣢⠙⢤⠋⢇⢻⡜⣧⢻⣜⡳⣞⡌⢣⡛⡇⢣⢟⠐⠠⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠌⠀⠀⠒⠣⡂⢅⠊⡔⠀⢻⠁⠸⣤⠙⡤⢙⢢⡙⢌⡎⠼⣌⠳⣎⢷⣹⠘⠈⠈⠓⠀⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⡟⠠⠊⢄⠣⡘⠀⠸⣄⢋⠴⡉⢆⠜⡢⠜⡡⢏⡳⣝⠮⢧⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢡⠘⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠌⡇⠠⠑⠊⠀⠡⣀⠸⡜⡌⢦⢩⢘⢢⡑⠣⣅⠫⠜⢧⣛⢧⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⢤⢀⠒⢌⢠⠃⠀⠐⢀⠀⠀⢘⡱⡉⢆⠦⣉⠦⡘⢥⠢⣙⡘⢢⠹⣎⢷⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⢎⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠢⢉⡄⠦⠥⢄⡀⠀⠀⠀⡎⡔⣉⠖⡌⡒⡴⡉⠦⣑⠢⡜⢡⣒⢯⡞⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢂⢃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠧⡘⣅⠫⡰⢌⡆⠀⡐⡜⠤⢣⠜⡰⣑⢧⣙⢦⢣⠳⣜⢣⣞⡳⢞⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⣿⢿⣻⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⡔⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠐⣬⣷⣷⣿⣿⣯⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠠⡀⢄⡐⠠⢄⡐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠡⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⡀⢀⠠⡐⠀⠉⠠⠑⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⢀⣠⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣾⡏⠀⠀⠈⠠⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⣀⢣⢘⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣯⣷⣿⣻⣟⡈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀
⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⡟⢈⢷⣯⢷⣻⣜⠀⢘⡢⣖⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⣯⢷⡿⣿⢿⡿⣯⠗⠈⠀⣺⡽⠇⣿⢾⠰⠎⠐⠩⡖⠁⢄⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀
⠀⠀⠀⠀⠀⣿⡿⣯⣿⣷⣿⣻⡿⣯⣟⡷⣯⢿⡽⣯⢿⣝⣁⣈⡉⠈⠁⡟⢠⣿⢯⠐⠀⡁⢂⠱⡀⠀⠈⠞⡴⣩⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⢹⡽⣯⢷⣟⡷⣯⢿⣳⡿⣽⢯⡿⣽⠏⠁⠉⠈⠩⠿⠷⠆⢀⠎⣹⠇⣨⠀⢄⡩⢄⠈⡥⠉⢠⠐⢀⢣⠀⢂⠶⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⡻⣽⣛⡾⣽⢯⣯⢷⡿⣽⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠁⡍⢄⠎⠈⠡⠫⢔⡡⢆⡁⠂⠄⠓⠈⠷⠱⢎⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⠴⣿⣽⣳⢯⡿⣽⣻⣞⣯⣿⢷⣿⣟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⠂⠐⠻⣶⢦⣍⡐⢪⠆⡌⡄⢁⠂⢰⢋⠀⠐⠁⡴⣒⢖⡒⡄⢠⠠⠰⡀
⠀⠀⠀⠀⢰⣟⣾⣿⢯⣟⣷⣻⣞⣷⣻⣿⢷⣯⣟⡆⠀⠀⠀⠀⠀⠶⣦⣰⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠈⠟⠓⠰⣁⠂⠠⠈⡜⠀⡑⢂⠒⢁⠈⠎⡕⠁⢀⠬⡁⠆
⠀⠀⠀⠀⠘⣿⣾⣿⣿⣞⣷⣻⣿⢾⣽⣻⣟⡾⣽⣻⣀⠄⠀⠀⠈⢐⣨⣿⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠠⢀⠒⠀⠐⠠⡜⢀⡂⠌⢠⠘⠲⣍⡆⠁⠀
⠀⠀⠀⠀⠀⢻⣿⡻⢟⢾⣿⣽⣃⢟⠻⣧⢻⣿⣻⡇⢿⡿⢾⣶⣤⡾⡏⠉⠀⠀⠀⠑⡂⠄⠀⠀⠀⠀⠀⠀⢀⠎⢀⠁⢂⠐⡈⢤⢫⠴⠋⠀⡔⢊⠄⢱⢎⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠛⠓⠦⠀⡌⡙⠻⣈⠳⢌⠣⣆⣧⡅⢈⠃⠀⠟⣾⣯⡁⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢀⢀⠎⢀⠂⠌⡀⠂⣜⢣⢏⠆⡴⢩⠜⣁⣤⠻⣎⠖⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠦⣁⠣⡄⠣⢌⠒⠤⣈⠓⢤⢊⣄⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠂⡶⢬⡂⢅⠪⡜⢬⣡⠶⣫⠑⣮⢃⠎⡜⠰⢯⣹⠱⣎⢳⡜⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠘⠐⠀⠣⢌⡑⢊⢜⠰⢠⢉⡶⣿⡖⡘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡝⣲⠹⡄⢚⡬⠇⡮⢗⡭⢦⢏⡟⣎⠆⢉⠨⣚⠵⣪⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⡀⠀⠈⠐⡌⡑⢆⠞⡹⣧⣏⡐⡛⠍⠀⠀⠀⠀⠀⠀⠀⠐⢧⡚⣥⠟⣳⢧⣛⠿⣴⣙⡼⣙⣮⣝⣮⢻⡌⡟⣬⡟⣵⡂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠆⠈⠀⠑⡽⠆⢁⠀⠀⠀⠀⠀⠀⠀⠀⡰⢎⠉⢆⡚⠤⢋⡚⢝⠲⣝⢮⡳⣓⠾⣜⡻⢸⡱⣃⠹⢮⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⠀⠀⢀⠄⡀⠀⠀⠀⠀⠽⡎⠀⣎⠴⡉⢦⢉⡜⠒⡼⢨⠳⣍⠿⣼⡹⠂⠁⠓⠂⠝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡡⢊⠔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢀⠆⢂⠈⠄⠀⠀⠀⢈⠁⠀⣎⠱⣘⠢⣅⠪⢱⢡⠃⡟⣬⢛⡶⣹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠔⡡⢊⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠈⠀⠀⠀⠈⠠⠀⢊⠀⡀⣎⠳⡄⡓⢤⡙⠢⠥⣙⠸⡱⢏⡾⣱⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⢁⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⢡⠀⠀⠈⠀⡭⢓⡘⣌⠒⣌⢃⡓⡌⡱⢂⢣⠹⣣⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠎⡰⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⣯⠗⡀⠀⠀⡄⢢⠀⠐⠀⢠⡑⠎⡔⢢⠍⣔⡊⠴⢡⢱⠉⣆⢣⣟⡺⣜⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠑⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣼⢏⡇⠎⠪⢁⠢⡁⢆⠀⡀⢦⠩⡜⣘⠤⢋⠶⣙⢎⢧⢣⡝⣬⢳⣎⠷⣭⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣻⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⣰⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣦⣾⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠤⡐⡐⠂⠆⠁⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠉⠄⡂⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⠀⠄⢠⢀⠂⠈⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠛⡉⠉⠉⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⡿⣽⡆⠐⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀
⠀⠀⠀⣀⣴⣨⣰⣥⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⣾⢿⣳⢯⣟⡳⡌⠸⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣷⣿⣿⠛⢄⢹⣯⠿⣯⢾⡽⣧⠀⣘⠵⣙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⠛⠛⠛⠛⠻⣿⣿⣿⡿⣿⣿⣻⣯⣿⣟⣷⣻⣽⣻⣟⣾⣳⣋⡀⠤⠤⣿⠃⣿⢯⣟⡿⠉⠄⠠⠙⡂⠀⠡⢂⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀
⠀⠀⠀⠀⠀⣿⢿⣾⢿⣿⣽⣻⣟⣾⡽⣞⡷⣯⢷⡻⢾⠶⠦⣤⣄⡀⠏⠰⢿⣯⠏⠵⡈⠀⠁⡆⠑⢤⡀⠄⡙⠒⠮⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣯⣟⡾⣯⢷⣯⢷⡻⣾⡽⣯⣿⣽⡋⠀⠀⠀⠀⠛⠉⠁⠀⡁⠜⠁⢀⠦⢥⠈⡐⠮⡐⡤⢂⡐⠤⢈⠐⢣⡠⢝⣫⢓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣼⣻⡽⣞⡾⣯⣟⣷⣟⣷⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠁⣀⣤⣙⠒⠬⣑⠂⡌⠀⠄⡈⢂⡹⠈⠐⣀⠀⣀⣀⣀⠀⠀⠀⢀⠀
⠀⠀⠀⠈⢹⣻⣞⣷⣻⣽⣻⣷⣻⣞⣿⣾⢯⣷⡇⠀⠀⠀⠀⠀⣀⣀⠀⠐⠀⠀⠀⠀⠠⠈⠂⠀⠹⠋⠛⠶⣦⡙⣠⠙⠠⠐⡀⠇⠦⣀⢀⠎⠱⢣⢶⠃⠜⠨⠌⡂
⠀⠀⠀⠀⢸⣷⣻⣿⣳⢯⡷⣿⣳⢯⣿⣯⣟⡾⣽⡀⡀⠀⠀⠀⠌⠻⣼⡆⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠊⠀⠀⡜⠀⡐⢀⠆⠈⠄⠃⠰⣀⡴⡁⠖⠁
⠀⠀⠀⠀⠘⣷⣿⢿⣿⣻⣽⣿⢻⢿⣳⣹⣾⣟⣿⢳⣦⣤⣀⡀⣠⣷⠻⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡈⠐⡀⠄⡰⣊⡤⠃⠌⡠⢃⠈⡶⡱⠀⠀
⠀⠀⠀⠀⠀⠘⠿⣏⡁⠛⠻⢾⡆⢣⣙⢣⠝⣺⠛⠌⠯⠈⢿⣻⣽⠧⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠂⠄⠡⠐⡼⣱⠣⠐⡤⢋⠔⢁⣼⢱⣓⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠒⠤⢨⠁⢆⠩⡘⢩⠻⡄⢂⡄⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢄⣀⠠⣄⡠⢂⠌⡘⢆⢆⣡⠬⡖⢩⠖⣡⢣⠙⣴⣋⠞⣭⢎⡳⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠑⠌⢩⠁⠞⡠⢃⠜⡠⢃⢜⣶⣏⠙⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⡱⣣⠌⡸⣬⢚⡶⣍⢧⣸⢻⣤⢣⠘⡐⢭⡛⡴⡫⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠈⠀⠱⢈⠒⣅⢊⢾⡽⣺⢁⠻⡿⠑⠀⠀⠀⠀⠀⠀⠀⠀⣄⡞⢦⢳⡭⢦⡵⢧⢮⣑⢮⡚⣵⢫⡞⣖⡦⢴⢫⡼⣳⠅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠒⡄⠃⠂⠙⢟⡆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⢉⠖⣨⠓⡝⢫⠞⡽⣎⠷⣭⢳⣝⢮⠳⣍⢧⠙⣧⡓⠄⠀⠀⠀
⠀⠀⠀⠀⠀⡔⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⡆⠸⡘⣄⠳⣈⠇⡚⠤⡏⡝⣎⠷⣎⡟⠦⠙⠪⠄⠳⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⡁⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢘⠀⠀⡝⢢⡑⠆⣎⢡⠓⡜⢸⣌⡳⣝⠾⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡑⢌⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠠⠀⠀⠀⠀⠀⡐⠠⠀⠃⣀⡝⡆⣌⠣⡔⢃⡚⢌⡱⢎⢵⣫⢞⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠡⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡱⡇⠀⠀⠀⠀⠀⢐⠡⠁⠀⠀⢀⡞⢡⠆⡓⠬⡑⣌⠲⡐⠎⡤⠓⣏⢾⣱⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠢⢌⢡⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⣱⢧⠀⠀⠀⠌⠀⠀⢃⠀⠀⠀⠸⣐⠣⠜⣌⠱⣘⠤⢃⠭⡘⡔⣩⢼⣣⠷⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⠊⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣰⢯⢯⠀⠀⠌⠀⠀⠀⠈⢢⠀⡀⡗⣠⢋⢲⡈⢵⡩⢞⡥⣚⠵⡸⢥⡳⣝⠾⡅⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣻⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢢⣿⣾⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⡒⡐⠢⠉⠂⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⡄⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⣷⠀⠀⠀⡀⠀⠐⢠⠂⠅⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡛⡉⠉⠁⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣿⢿⣧⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⠀⣠⣬⣐⣡⣬⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⡿⣿⢯⣟⡾⣫⢿⡝⡆⠉⢄⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣿⣿⣾⣿⠿⡁⠜⣯⢿⢮⣷⣻⢿⡇⢠⣀⠞⢦⡓⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠛⠛⠟⠛⠻⣿⣿⣿⡿⣿⣟⣿⣯⡿⣟⣾⣻⡽⣟⣾⣻⢞⣁⠠⠤⢿⡏⢼⣳⢯⣿⠃⠄⡠⠀⠄⠹⠀⠈⠐⢡⡖⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠘⣿⣟⣾⣿⢿⣻⢯⣷⣻⡽⣞⡷⣿⡽⠾⡷⠶⣤⣄⡀⡘⢠⢾⡿⠟⡘⠀⡠⠅⢀⡘⠆⠙⢢⠐⢁⠈⡙⡂⠐⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣷⣻⢾⡽⣯⣟⡿⢾⣽⣻⣽⣻⡷⠁⠀⠀⠀⠘⠋⠉⠀⠆⠸⠉⠀⠀⠐⠨⠣⢰⠈⠚⢱⠢⡅⢢⠃⡀⠙⠤⠚⡴⣩⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⡷⣽⣏⣾⣳⢯⣟⡿⣾⣳⢿⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠥⠀⠶⣶⣦⣅⠣⢜⡄⡁⠂⠄⠡⢰⠋⠀⠆⢀⡤⢤⢤⠀⡀⢀⢠⠀
⠀⠀⠀⠈⢸⣟⣷⣟⡾⣽⣻⡾⣽⣻⣯⣿⣻⣽⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠁⠀⠉⠻⠦⠰⣁⠃⠌⠀⡏⠘⠤⠤⠘⠘⢣⠞⠁⠘⣀⠢⡁
⠀⠀⠀⠀⢽⡾⣽⣯⣟⣷⣻⣟⣷⣻⣿⣳⢯⣟⣇⢀⠀⠀⠀⠈⡙⢷⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠠⠘⠀⠐⣀⠎⠠⢁⠂⡘⢦⣚⡄⠃⠀
⠀⠀⠀⠀⠘⣿⡿⣿⣿⣞⣯⡟⣯⣷⣏⣿⣟⣾⡟⣶⣤⣄⣀⢀⣾⢟⠃⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠂⡐⠠⢈⡰⣌⠲⠊⡀⠴⢀⠂⣳⢲⠀⠀
⠀⠀⠀⠀⠀⠙⢿⣎⠈⠛⠻⢷⠌⣬⡙⡬⢛⣋⠧⠹⠌⠹⣟⣯⡿⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠠⡜⠀⢂⠐⠠⠁⡼⡱⢎⠡⢤⡙⢂⢡⠼⣥⠳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠱⣈⠱⠠⢌⠡⢋⠹⢆⡰⢠⠀⣼⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠲⡴⣈⠌⡑⡜⢬⣡⡜⢾⢰⢳⠡⢏⡘⢦⡝⣭⢣⠟⡴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠎⢡⢊⠣⢌⠢⢡⠊⡔⣰⣷⡎⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⠹⡄⠢⡝⣎⢧⡛⡼⣌⢯⡳⠮⠄⠡⡚⡥⢏⡾⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡀⠀⠀⠑⠨⢌⠣⡘⡴⣻⡳⡇⡙⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⡕⣣⣏⢧⡥⢿⣘⡦⣙⠶⣹⢮⣝⡳⣞⢲⢭⢳⣏⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢣⠘⠐⠈⢻⢳⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠍⢥⣈⠳⡙⢧⠻⣜⢯⡞⣵⢻⡌⢷⡋⡞⡼⠉⣞⡕⠄⡀⠀⠀
⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡆⠘⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠘⡴⢠⢣⢉⢆⠓⣌⢇⡻⡜⢧⣏⣇⠓⠈⠓⠂⠸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠐⢆⠀⠀⠀⠀⠀⠀⠀⠠⢀⠁⠀⣣⢃⠦⣉⠦⢩⡘⠤⢳⡙⣞⢦⣏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡊⠔⡡⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡄⠀⢆⠀⠀⠀⠀⠀⡠⡐⠀⠡⣀⢇⡎⢲⠐⡎⡡⠜⣡⠣⢝⢮⡳⣎⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠑⡌⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠸⣥⠀⠸⡄⠀⠀⡀⠀⡑⠀⠀⠀⠀⡞⢌⠥⢚⡰⢡⢋⠤⠓⡌⠦⣙⢮⡻⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡰⠡⢌⠡⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸⡼⠀⡜⠼⣀⡐⠀⠀⢡⠀⠀⠀⠸⣈⠎⡜⢢⡑⡪⢔⡉⢞⡈⢖⡡⣯⢵⢫⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠣⢌⠢⡑⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⡾⣽⠰⣉⢧⠓⠀⠀⠀⠀⢃⡀⠀⣇⠆⢳⠨⡅⢼⡱⢣⡜⣎⡜⢮⣱⢳⣎⢯⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⢁⠣⠌⠑⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠁⠉⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠠⡔⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠃⠀⣘⠼⣋⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣿⣾⡿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⢈⠃⠡⢈⠣⠃⠀⢆⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⣽⣻⢷⣻⣞⣷⡛⠃⠐⠒⠚⡿⠁⣿⣯⠿⡿⠀⠀⠎⠀⠄⡆⠣⣀⡀⠌⠳⣌⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⢯⣟⣯⠷⠏⠉⠋⠙⣻⣶⠄⠃⠘⢹⡞⢁⠃⠀⠜⠦⠁⡌⠔⡢⢔⡠⢂⠅⢈⢠⡀⢡⠖⡦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⣾⣳⢿⣽⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠃⢱⠀⣠⣥⣅⡒⢩⡘⠤⠂⡈⠌⠛⡘⢧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣞⣷⢯⣟⣾⡽⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠚⠉⠛⢷⣌⡑⠆⠆⠠⢁⢸⢁⠈⠐⢠⢴⠲⣖⠠⡀⢄⠰⡀
⠀⠀⠀⠀⣼⣻⣽⣯⢿⡽⣾⣟⡾⣽⣿⣟⡾⣽⡀⠀⠀⠀⠀⠒⢶⣤⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠁⢠⠃⠈⡑⠢⠈⠌⠱⢊⠁⢈⠤⡁⠆
⠀⠀⠀⠀⢹⡷⣿⣿⣯⣿⡽⣿⣽⣟⣳⣯⢿⣽⣧⣐⡀⠀⠀⠈⣌⣿⠃⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⢁⠂⡀⠡⡰⠁⢌⠠⠁⡌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢾⣿⣇⢫⠛⣧⠻⣿⣻⡖⢿⠻⢿⢶⣾⣻⠈⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠠⢁⠂⡐⢠⡣⣕⡛⢀⢢⠑⡄⢂⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⠉⢇⡉⢒⠱⠦⣗⡠⠄⡀⠘⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⠐⢄⠒⠦⣌⠒⣁⣤⢂⢧⠳⡘⢤⢊⡖⣩⠤⣞⡱⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠢⡙⠤⣘⠄⣃⠒⠤⡑⢪⣔⠦⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⢯⠐⡈⡗⡔⡯⡗⡼⡠⢯⡱⢍⠢⡙⡮⣇⠻⣔⡓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠁⠒⠠⢊⠤⡉⢆⣱⢿⣻⢃⢹⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡼⣙⢮⣑⣤⣛⠼⣝⡱⣣⢛⣧⣛⢮⣁⣐⠬⣎⡳⡍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠢⢑⠲⠀⠛⡽⣆⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⡔⢫⠑⠯⠶⣭⣛⢶⡳⣵⢫⠶⡽⣎⢷⢪⠽⠼⣳⢧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⠸⡄⢫⢌⡱⢂⡌⢣⡝⢮⢏⡻⡵⡜⢣⠩⢞⡀⢛⠆⠁⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⠩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡝⢆⠲⢌⡒⡌⢓⡌⢳⢎⡵⣫⡝⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⠣⢌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠀⠰⡀⠀⠀⠀⠀⠀⢀⠢⠈⢂⠀⡳⣌⠱⢢⠱⢌⠣⢌⠣⣏⢾⡱⣝⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢌⡑⠢⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢹⡃⠀⡱⡀⠀⠀⡀⠀⡃⠂⠀⠀⠈⡳⠌⡥⢃⡚⢌⡜⠬⣑⠢⢎⠳⣭⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡌⠱⡈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢰⢧⠀⠵⡱⡀⠐⠀⠀⢃⠀⠀⠀⠸⣁⠳⣈⠕⣌⠲⣈⠖⣡⠚⢤⢣⡷⣹⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠑⡌⢢⠑⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢠⣟⣯⡘⡕⣣⠝⠀⠀⠀⠀⢢⡀⠀⣗⡈⢖⡡⢎⡸⢣⢇⡞⣤⢋⣎⡳⣞⢵⣫⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠲⢶⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠁⣎⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣌⣳⠹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡩⡙⠾⣜⣣⢟⡵⣳⡝⣮⢳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡱⢑⠲⣐⠢⡩⢜⠣⣝⢮⡳⣖⠽⢢⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡳⢍⠲⢄⢣⡑⢎⡱⢪⢖⡹⣎⡟⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡌⢎⠜⡰⢌⠲⡰⢩⠞⣵⢫⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⣉⠲⢌⡑⣊⠱⣐⠣⢜⢨⠳⣝⢮⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⠔⣃⠎⠴⣁⠳⣈⠜⡌⢆⡣⣟⠮⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⢦⡙⢜⡐⢎⡱⣍⠳⣌⠞⡼⣌⡳⣭⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣨⣳⠹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡩⡙⢮⠳⣭⣛⢶⡳⣝⢮⡳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡱⢑⠢⢓⡰⢌⢃⡟⡼⢣⡟⣼⢫⢦⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡳⢎⢡⢃⠖⡌⡚⢤⢋⠷⣸⢧⣛⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡌⠦⣉⠲⢌⡱⠢⣍⠻⣴⢫⡞⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⣉⠒⠥⡚⡐⢆⡓⠤⠓⣌⠳⣝⡳⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⡔⣩⠒⢥⡉⢖⢨⠱⡩⢄⡣⣯⢵⣫⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⣆⠓⡬⢄⠫⢴⡙⣎⠶⣩⠖⣭⡲⣝⠶⣳⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠲⢶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠁⣎⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⡑⠪⢔⡡⢊⠔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⠑⠢⢘⠢⡘⢄⣳⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣌⣳⠹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠱⣈⠲⠀⠛⡽⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡩⡙⠾⣜⣣⢟⡵⣳⡝⣮⢳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡱⢑⠲⣐⠢⡩⢜⠣⣝⢮⡳⣖⠽⢢⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡳⢍⠲⢄⢣⡑⢎⡱⢪⢖⡹⣎⡟⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡌⢎⠜⡰⢌⠲⡰⢩⠞⣵⢫⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⣉⠲⢌⡑⣊⠱⣐⠣⢜⢨⠳⣝⢮⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⠔⣃⠎⠴⣁⠳⣈⠜⡌⢆⡣⣟⠮⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⢦⡙⢜⡐⢎⡱⣍⠳⣌⠞⡼⣌⡳⣭⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠁⠃⠜⢠⠑⡌⢢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣌⣳⠹⣜⡸⣣⢛⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠣⢘⠒⠄⠛⡽⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡩⡙⠾⣜⣣⢟⡵⣳⢧⣛⢾⡱⢯⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡵⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡱⢑⠲⣐⠢⡩⢜⠣⡟⡼⣣⢷⡹⢦⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡳⢍⠲⢄⢣⡑⢎⡱⣙⠶⣙⢶⡹⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡌⢎⠜⡰⢌⠲⡐⢭⢚⡽⣎⢷⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⣉⠲⢌⡑⣊⠱⣘⢂⠣⢜⢪⠷⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⠔⣃⠎⠴⣁⠓⡬⢌⡓⡌⢦⡟⣵⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⢦⡙⢜⡐⢎⡱⣍⠳⣔⢣⠖⣜⣣⢟⢶⣹⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠃⠜⢠⠑⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⢴⣋⣔⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠐⠈⠀⠀⠀⠀⠀⠣⢄⠳⠀⠻⡽⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡉⢇⠻⡜⣶⡹⣖⡻⢶⡹⣞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡹⢈⠖⡡⢆⡱⢉⡝⢧⢻⡜⣮⠵⢣⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠧⣋⠴⡑⢢⠜⡱⢌⢣⡓⢾⡱⢯⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡔⢢⡙⠤⠓⡬⢌⠣⣝⢮⣝⡳⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⡉⢖⢨⠱⡩⠔⣊⠱⢌⡘⢮⡵⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⠜⡌⢆⢣⡑⢎⠴⣉⠦⣑⢢⡟⢧⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⢦⡙⢌⣒⣉⢲⡙⢮⡔⣣⠖⣬⡳⣝⢧⣻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠊⢄⠓⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⢴⣋⣔⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡠⠓⠄⠛⡽⣆⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡉⢇⠻⡜⣶⡹⣖⡻⢶⡹⣞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡵⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡹⢈⠖⡡⢆⡱⢉⡝⢧⢻⡜⣮⠗⣣⠹⢚⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠧⣋⠴⡑⢢⠜⡱⢌⢣⡓⢾⡱⢯⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⢂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡔⢢⡙⠤⠓⡬⢌⠣⣝⢮⣝⡳⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢱⡆⠀⢣⠄⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⡉⢖⢨⠱⡩⠔⣊⠱⢌⡘⢮⡵⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠊⡔⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣸⢦⠀⣍⠞⡀⠠⠁⠀⢡⠀⠀⠀⢠⠣⠜⡌⢆⢣⡑⢎⠴⣉⠦⣑⢢⡟⢧⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡐⢡⠄⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣰⢯⢯⠜⣌⠞⡱⠁⠀⠀⠀⠡⡀⠀⢦⡙⢌⣒⣉⢲⡙⢮⡔⣣⠖⣬⡳⣝⢧⣻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⣀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⡀⢠⠀⠒⢀⠡⠘⠐⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⢆⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠉⡀⠰⢍⢯⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⢉⠁⠌⠒⡁⠈⠔⢠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢨⠀⡀⠌⢆⠱⢄⡠⠌⠛⠴⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠂⡳⠀⠆⠪⢆⠲⡠⣀⢃⠈⠰⣀⢠⠞⡴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⢹⠀⠥⣶⣬⣑⠰⢌⡪⠐⠠⠈⡙⠘⣭⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠡⠙⣻⡦⠁⡇⡑⠠⢀⢹⢠⠀⠀⢡⢖⡚⢦⠂⡄⠰⢠⠁
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠀⢠⠃⠀⡉⡰⠁⠠⢉⢹⠁⣠⠒⡅⠈
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠈⡒⠠⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⡁⢂⠠⠐⡔⣁⢄⠁⠂⡄⠛⡴⣃⠀⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⠐⡀⢢⢏⡴⠍⠠⣈⠎⠔⢠⡳⢭⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠰⣄⡂⠔⡡⢎⡔⣂⣡⢖⠂⣏⠞⣰⢡⢓⡜⣤⡜⢾⡱⡋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠽⡐⠠⡧⡜⣧⢏⡞⣄⠿⣌⢣⠂⠝⣎⠷⣘⢧⠓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⠊⠄⡓⢌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣊⢧⣋⣇⣅⡹⣜⡬⣸⡜⣭⢻⡜⣏⢥⣀⢖⣫⠵⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠳⠀⠻⡽⣆⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⡘⡎⠜⡚⠼⢳⡝⣞⡵⣛⡴⢯⡞⡝⡾⣌⢻⠸⣏⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⣍⠚⣌⠱⢃⡜⢤⡋⢷⡹⣳⢮⡙⢥⠸⠩⢆⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⠐⡠⢉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠱⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡙⢎⠤⢋⠴⣈⠖⣉⠶⣱⢣⢯⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡄⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⣛⡌⢲⢉⠲⡐⢎⡰⠙⡖⢯⡞⣹⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⢃⠆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢱⡆⠀⡱⠄⠀⠀⠀⠀⡦⠁⠀⠀⠉⡖⡩⢆⢩⢒⡉⢖⣈⠓⡌⠦⡹⢧⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠊⡔⡀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⣸⢦⠀⢣⢏⡄⠀⠁⠀⢡⠀⠀⠀⠰⢡⡑⣊⠦⢡⡚⢤⢊⡱⢌⡱⢢⡟⣼⢃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡐⠩⢄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⣰⢯⢯⣘⢣⢎⡜⠀⠀⠀⠀⠡⡀⢀⢣⢃⠖⣡⢊⡵⣙⢦⢣⡓⢮⡜⣳⡝⣮⢻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⣀⠠⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠂⢡⠘⠰⠀⠃⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⠈⠀⠈⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⣟⡿⡆⠀⠴⠂⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡻⣞⡽⣷⠀⠈⡁⢀⠮⡹⣍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣽⣳⢿⣽⠀⠠⢁⠂⠄⠑⢊⠀⠨⠐⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠈⠉⠉⠉⢹⣿⡿⣟⣿⣿⣽⡿⣞⡿⣽⢯⣟⡿⣾⡽⡞⠁⠒⠒⠛⣯⠁⣿⡽⣯⡇⠀⢀⢌⠀⡈⠲⠈⠦⣀⠤⠙⢣⢚⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⢨⢿⡽⣟⡿⣞⣷⣻⢯⣿⣽⣻⢾⡽⠗⠙⠉⠛⣷⡦⠀⠁⠐⢹⡟⠱⠁⠀⠂⡚⠄⡜⠂⠗⡲⢄⠔⢢⠀⠂⢆⡀⣲⢣⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢨⣏⡿⣝⣻⡽⣾⡽⣯⣷⢯⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠸⠄⠴⢶⣦⣍⡊⠥⣊⠰⠀⠄⠑⢋⣎⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⣻⢾⣽⢯⣷⣻⢷⣻⢷⣿⣻⣾⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠈⠛⠶⠄⡞⡈⠄⠡⢸⢠⠀⠀⢡⠖⡶⢲⠄⢀⡀⢄⠀
⠀⠀⠀⠀⣿⣻⣾⣟⡾⣽⣻⣽⣻⢾⣷⣯⣟⣾⡀⠀⠀⠀⢐⠳⣶⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡐⠃⢀⠉⡐⢁⠈⠱⢩⠂⢁⠬⡐⠂
⠀⠀⠀⠀⢿⣳⣿⣿⣽⡷⣿⢯⣿⣻⣹⡾⣽⡾⣧⣀⠀⠀⠀⢢⣹⡟⠀⠀⠀⠀⠀⠀⡈⠓⠄⠀⠀⠀⠀⠀⠀⠀⡠⢁⠐⠠⠐⠠⢀⠖⣁⠤⢈⠐⡀⠓⣎⢆⠁⠀
⠀⠀⠀⠀⠈⢿⣿⠻⠻⠽⣿⣜⣊⢛⢧⠻⣟⣿⠿⡟⠿⡷⣶⣟⡇⠁⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⢀⠰⠁⠠⠈⠄⡁⢢⢏⡼⡘⢀⢢⢊⠔⢠⢏⡞⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⡆⢄⠃⠆⡌⢡⠓⠼⣖⠠⠄⡀⢫⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⡃⢌⠂⡏⡔⣃⢬⢴⠂⣏⠞⣰⠰⣉⢦⢥⡒⢯⡚⡌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠩⠌⡙⠒⡌⢆⡉⢒⠨⡑⣼⡴⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⢺⡀⠎⡵⡜⣧⡏⡞⣜⡼⣌⢣⠑⠜⣣⢧⡙⣧⠙⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⢂⠱⢌⠢⣝⡿⣽⠀⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⣩⢶⣡⢂⡹⣜⣤⣸⢧⡹⣞⡭⣏⢤⡨⣔⢧⡝⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢊⠜⠈⠳⣻⠆⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢓⡋⣌⠳⣋⠷⣹⢖⡳⢧⡝⣮⡓⡽⠮⣱⢞⠺⣹⠧⡀⠀⠀⠀
⠀⠀⠀⠀⢀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣳⠈⢜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠸⡐⠦⢱⡈⠖⡡⢎⡝⢧⣛⠶⣭⡙⠧⠘⠺⡄⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡧⠈⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡙⢎⠢⣅⠫⢔⡩⡐⢧⢎⡽⣲⡝⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠢⡑⢂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠀⠫⠄⠀⠀⠀⠀⠀⢀⠢⠈⢂⠀⡝⣆⠣⡔⢩⢂⠖⣉⠞⡜⣞⡵⣺⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠱⡈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢯⡄⠈⡝⡀⠀⠀⡀⠀⡃⠂⠀⠀⢈⡳⢌⡒⢌⠣⡜⢌⡒⣌⠲⣈⠾⣱⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢃⠜⡠⢓⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣧⠇⠠⣍⠳⡄⠐⠀⠀⢃⠀⠀⠀⢘⡰⢂⠭⡘⡒⣌⠲⢌⢤⠓⡰⣌⢷⣫⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡃⠜⡠⢃⠜⣀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⣰⢯⣇⡚⡴⢋⡜⠀⠀⠀⠀⢢⡀⠀⣇⠒⡍⢦⡑⡹⣌⠳⣎⢦⡛⣴⡹⣞⡲⢯⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠑⡄⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣢⣿⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⣀⠠⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠀⡡⠘⠰⠀⠃⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠐⠈⠀⠈⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⣟⣿⡀⠀⢦⠈⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⣻⢞⡵⣇⠀⠈⡀⢠⠍⣏⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠌⣿⡻⣽⣞⡿⣇⠀⢰⢁⠂⡐⠈⠏⠀⠨⠐⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠈⠉⠉⠉⢹⣿⡿⣟⣿⣿⣽⢿⣳⡿⣽⣻⡽⣯⢷⣻⡞⠃⠒⠒⠚⢫⠃⣿⢾⣽⠇⠀⠨⢀⠀⢠⢃⠘⠦⡀⠔⡑⢦⢙⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⢨⢷⣻⣯⣟⡾⣽⣯⢿⣽⣳⢯⡿⣽⠏⠙⠉⢛⣶⡤⠀⠃⠐⢹⡿⠙⠀⠀⠂⡓⡈⡔⠊⠖⣢⠑⣄⠢⢀⠂⢆⡀⣖⢣⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣨⣏⣷⣛⣮⣟⡷⣞⣯⣷⢯⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠐⠸⠀⠴⢶⣦⣍⠂⢇⡪⢀⠂⠄⠘⢣⠞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠒⣻⣞⣷⣻⢾⣽⣻⡽⣞⣿⣻⣾⣟⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠉⠀⠈⠻⠦⠄⣋⠄⡈⠐⡬⢃⠀⠈⢠⢖⡳⣒⠄⡀⢀⢄⠀
⠀⠀⠀⠀⣷⣻⣾⣯⣟⡾⣽⡿⣽⢾⣿⢷⣻⣾⡀⠀⠀⠀⠚⢶⣆⡌⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠰⠁⠠⢉⡐⢁⠠⠑⢩⠂⢈⠤⡊⠐
⠀⠀⠀⠀⢻⣽⣿⣷⡿⣽⣟⡻⣽⣿⡹⣯⡷⣯⣷⡀⠀⠀⠈⣢⣿⠇⠀⠀⠀⠀⠀⠀⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⢀⠐⡴⢁⡀⠆⢁⠂⡑⡞⣆⠁⠀
⠀⠀⠀⠀⠀⢻⣿⡋⠻⡛⢯⣧⢣⡙⡳⠻⡟⣿⣽⠻⢿⢶⣾⣻⠈⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⢀⡰⠁⡐⠠⢈⠐⣠⢏⡴⠋⠐⡠⠎⠔⢠⡛⡼⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⢀⢡⠂⡌⠢⢌⠱⡙⠞⣦⢀⠁⠘⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⣦⡑⡈⠆⡱⢰⣈⣰⢴⢂⡗⡎⡜⡰⣉⠶⣡⡜⢧⠽⡌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠘⠨⠑⢌⠓⡌⡒⢄⠣⢄⠣⣼⣥⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢦⢳⢀⠣⡕⣇⡧⣗⠺⣄⢿⣐⢣⠑⠜⣫⠗⡼⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠐⠡⢊⡔⣈⣾⣛⡷⢈⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡝⣎⣳⣨⡐⣹⡜⣬⣸⢣⣛⢮⡽⣭⢄⣨⠴⣫⢵⡃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠢⠌⠻⢾⡀⢗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⠜⢎⡰⠣⠟⡶⣹⢖⡳⣏⡼⣣⡟⡼⡫⢖⣹⠚⣧⠗⡀⠀⠀⠀
⠀⠀⠀⠀⢀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢫⢥⠘⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⢘⡢⠥⡙⢢⡑⢢⢩⠓⡽⣚⡵⢮⡑⠧⠹⠸⡄⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⢠⠃⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣆⠀⢣⠄⠀⠀⠀⠀⠀⠀⠀⡀⡀⠁⠀⡓⢎⡡⢣⡘⢅⢎⡱⢣⡓⣞⢧⣛⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⠱⡈⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠳⡀⠀⠀⠀⠀⠀⠠⢠⠈⢀⠀⡟⡤⢡⢃⡜⠌⣆⠲⣉⠷⣎⢷⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢢⠑⡌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⢷⠀⢈⠵⡀⠀⠀⠀⠀⠣⠁⠀⠀⢉⠞⣡⠣⢌⠢⡍⢤⠓⡰⠒⣌⠳⣝⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢡⠊⡔⢡⠡⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣯⠆⢠⣋⢖⡀⠠⠀⠀⢃⠀⠀⠀⢘⠬⡰⢑⡊⡕⣘⢢⢙⡰⢩⢄⡳⣏⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢎⠰⡁⢆⠱⣀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣼⡳⣇⠦⢣⠞⡜⠀⠀⠀⠈⢢⠀⡀⣏⠰⣡⠣⡜⡸⢥⢣⢎⡵⢣⢎⣵⢫⡻⣼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⣀⠠⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠂⢡⠘⠰⠀⠃⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⠈⠀⠈⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⣟⡿⡆⠀⠴⠂⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡻⣞⡽⣷⠀⠈⡁⢀⠮⡹⣍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣽⣳⢿⣽⠀⠠⢁⠂⠄⠑⢊⠀⠨⠐⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠈⠉⠉⠉⢹⣿⡿⣟⣿⣿⣽⡿⣞⡿⣽⢯⣟⡿⣾⡽⡞⠁⠒⠒⠛⣯⠁⣿⡽⣯⡇⠀⢀⢌⠀⡈⠲⠈⠦⣀⠤⠙⢣⢚⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⢨⢿⡽⣟⡿⣞⣷⣻⢯⣿⣽⣻⢾⡽⠗⠙⠉⠛⣷⡦⠀⠁⠐⢹⡟⠱⠁⠀⠂⡚⠄⡜⠂⠗⡲⢄⠔⢢⠀⠂⢆⡀⣲⢣⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢨⣏⡿⣝⣻⡽⣾⡽⣯⣷⢯⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠸⠄⠴⢶⣦⣍⡊⠥⣊⠰⠀⠄⠑⢋⣎⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⣻⢾⣽⢯⣷⣻⢷⣻⢷⣿⣻⣾⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠈⠛⠶⠄⡞⡈⠄⠡⢸⢠⠀⠀⢡⠖⡶⢲⠄⢀⡀⢄⠀
⠀⠀⠀⠀⣿⣻⣾⣟⡾⣽⣻⣽⣻⢾⣷⣯⣟⣾⡀⠀⠀⠀⢐⠳⣶⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡐⠃⢀⠉⡐⢁⠈⠱⢩⠂⢁⠬⡐⠂
⠀⠀⠀⠀⢿⣳⣿⣿⣽⡷⣿⢯⣿⣻⣹⡾⣽⡾⣧⣀⠀⠀⠀⢢⣹⡟⠀⠀⠀⠀⠀⠀⡈⠓⠄⠀⠀⠀⠀⠀⠀⠀⡠⢁⠐⠠⠐⠠⢀⠖⣁⠤⢈⠐⡀⠓⣎⢆⠁⠀
⠀⠀⠀⠀⠈⢿⣷⠫⠛⠿⣽⣎⢎⡛⢧⠻⣟⣿⠿⡟⠿⡷⣶⣟⡇⠁⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⢀⠰⠁⠠⠈⠄⡁⢢⢏⡼⡘⢀⢢⢊⠔⢠⢏⡞⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⡆⢢⠉⠆⡌⢡⠛⡸⢖⡠⠄⡀⢫⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⡃⢌⠂⡏⡔⣃⢬⢴⠂⣏⠞⣰⠰⣉⢦⢥⡒⢯⡚⡌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠩⠡⠙⡒⠌⣆⠱⡐⢌⡑⣦⡵⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⢺⡀⠎⡵⡜⣧⡏⡞⣜⡼⣌⢣⠑⠜⣣⢧⡙⣧⠙⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⠠⠓⣌⠢⣼⡻⣽⠀⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⣩⢶⣡⢂⡹⣜⣤⣸⢧⡹⣞⡭⣏⢤⡨⣔⢧⡝⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡉⠌⠳⡟⡆⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢓⡋⣌⠳⣋⠷⣹⢖⡳⢧⡝⣮⡓⡽⠮⣱⢞⠺⣹⠧⡀⠀⠀⠀
⠀⠀⠀⠀⢀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⣳⠈⠜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠸⡐⠦⢱⡈⠖⡡⢎⡝⢧⣛⠶⣭⡙⠧⠘⠺⡄⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡧⠈⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡙⢎⠢⣅⠫⢔⡩⡐⢧⢎⡽⣲⡝⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠢⡑⢂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠀⠫⠄⠀⠀⠀⠀⠀⢀⠢⠈⢂⠀⡝⣆⠣⡔⢩⢂⠖⣉⠞⡜⣞⡵⣺⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠱⡈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢯⡄⠈⡝⡀⠀⠀⡀⠀⡃⠂⠀⠀⢈⡳⢌⡒⢌⠣⡜⢌⡒⣌⠲⣈⠾⣱⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢃⠜⡠⢓⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣧⠇⠠⣍⠳⡄⠐⠀⠀⢃⠀⠀⠀⢘⡰⢂⠭⡘⡒⣌⠲⢌⢤⠓⡰⣌⢷⣫⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡃⠜⡠⢃⠜⣀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⣰⢯⣇⡚⡴⢋⡜⠀⠀⠀⠀⢢⡀⠀⣇⠒⡍⢦⡑⡹⣌⠳⣎⢦⡛⣴⡹⣞⡲⢯⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⣀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⡀⢠⠀⠒⢀⠡⠘⠐⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⢆⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠉⡀⠰⢍⢯⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⢉⠁⠌⠒⡁⠈⠔⢠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢨⠀⡀⠌⢆⠱⢄⡠⠌⠛⠴⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠂⡳⠀⠆⠪⢆⠲⡠⣀⢃⠈⠰⣀⢠⠞⡴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⢹⠀⠥⣶⣬⣑⠰⢌⡪⠐⠠⠈⡙⠘⣭⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠡⠉⣻⡦⠁⡇⡑⠠⢀⢹⢠⠀⠀⢡⢖⡚⢦⠂⡄⠰⢠⠁
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠀⢠⠃⠀⡉⡰⠁⠠⢉⢹⠁⣠⠒⡅⠈
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠈⡒⠠⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⡁⢂⠠⠐⡔⣁⢄⠁⠂⡄⠛⡴⣃⠀⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⠐⡀⢢⢏⡴⠍⠠⣈⠎⠔⢠⡳⢭⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠰⣄⡂⠔⡡⢎⡔⣂⣡⢖⠂⣏⠞⣰⢡⢓⡜⣤⡜⢾⡱⡋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠽⡐⠠⡧⡜⣧⢏⡞⣄⠿⣌⢣⠂⠝⣎⠷⣘⢧⠓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⠊⠄⡓⢌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣊⢧⣋⣇⣁⡹⣜⡬⣸⡜⣭⢻⡜⣏⢥⣀⢖⣫⠵⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠳⠀⠻⡽⣆⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⡘⡎⠜⣊⠷⡹⣎⢷⡳⡽⣜⢧⡛⣼⠫⣜⣺⠸⣏⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⣍⠚⢤⠒⡥⡘⢤⠏⡳⡝⣮⢽⡘⠧⠸⠸⡄⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⠐⡠⢉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠱⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡎⡍⠦⡩⠔⣑⠪⠜⣱⡹⣜⢮⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡄⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⣌⠱⢢⡙⠤⣙⠸⣐⠳⣎⢷⡹⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⢃⠆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢱⡆⠀⡱⠄⠀⠀⠀⠀⡦⠁⠀⠀⠉⡕⣊⠥⢃⠬⣑⠢⠓⣌⠲⣈⠳⣝⢮⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠊⡔⡀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⣸⢦⠀⢣⢏⡄⠀⠁⠀⢡⠀⠀⠀⠰⣑⠢⢎⡑⢎⡰⢩⡑⢢⠱⡌⣱⢮⣛⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡐⠩⢄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⣰⢯⢯⣘⢣⢎⡜⠀⠀⠀⠀⠡⡀⢀⢣⠆⠳⣌⢢⡹⣜⢣⡜⣣⠳⡜⣵⢫⡞⣼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠃⡇⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⡑⠄⠣⠐⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⡔⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣿⠀⠙⠀⢘⡼⣙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⡀⠋⠆⠐⠠⢄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡏⠀⠐⢈⠀⢀⠇⠱⢄⣀⠌⠳⢜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠘⡾⣯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠙⣻⣶⠄⠃⠜⣹⠗⠡⠁⠀⠌⠦⠈⡄⠞⡤⢢⠄⣀⠃⠠⢰⡀⢁⠶⣒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣟⣳⢯⣟⣻⣞⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠡⢑⠂⣠⣤⣅⡚⠤⣉⠖⡀⠌⢈⠳⠉⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⢾⣽⣳⢾⣳⢿⡾⣽⢷⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠘⠉⠛⠳⣤⡉⡔⡌⠠⢀⢹⢠⠈⠐⢠⡔⢶⢲⢠⠀⡄⠰⠂
⠀⠀⠀⠀⣼⣻⣾⣯⢿⣽⣻⡽⣯⢿⣷⣟⡾⣽⡀⠀⠀⠀⠀⠲⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠁⢀⠎⠀⡑⢂⠂⠉⢌⢣⠁⢈⠤⡑⠌
⠀⠀⠀⠀⢹⣷⣿⣿⣻⣾⣽⡿⣯⣿⣹⣾⣽⣻⣧⣀⡀⠀⠀⠁⣎⣿⠃⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⢈⠀⡐⣠⠃⣈⠐⡀⢂⠹⣍⠖⠁⠀
⠀⠀⠀⠀⠈⢳⣯⠻⠹⢷⣟⣇⠯⡛⢧⠻⣷⣿⠞⣿⠻⢷⣶⢾⣻⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⠠⠁⢂⢰⡸⢤⡛⢀⢂⠜⡠⢂⡼⣹⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⢌⡙⠤⡉⠎⡓⠴⣧⡐⡈⡀⠘⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠂⡌⢎⠴⡐⣁⡬⠄⣮⠱⢣⠔⡪⡜⣌⡤⢞⡱⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⢢⠘⢢⠱⣈⠔⡡⢂⡑⣢⡗⡰⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠈⡄⢏⡲⡝⣧⠺⣄⢷⡉⣇⠊⡕⢯⠞⡼⣩⡓⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠃⢄⠓⣄⠣⣴⡟⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢿⡘⣦⢇⣌⢳⡹⣜⡘⣧⢛⡮⣝⢮⣁⣀⢮⡙⣶⠓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡠⠓⠤⠙⡟⣦⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡦⠛⡌⣋⠞⢧⣻⣱⣛⢶⡹⣞⡽⣎⢷⡩⢶⡹⢧⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡵⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠠⣙⠰⡡⢚⡰⠰⣄⡛⢮⢳⡭⡞⡽⢦⠙⠞⣄⠹⡃⠈⠀⠀⠀
⠀⠀⠀⠀⠠⢌⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢎⢣⠱⢡⢒⡱⢌⠒⡭⣖⡱⢯⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⡈⢆⡁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠹⠄⠀⠀⠀⠀⠀⠀⡰⠀⠢⠀⣫⢆⡙⢢⢅⠲⢌⠓⡜⠲⣝⢧⣻⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠢⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⢫⠄⠀⠀⠀⠀⠧⠀⠀⠀⠈⡵⢊⠬⣡⢊⠜⡌⡚⢌⡱⢌⠳⢮⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠊⠥⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣸⢦⠀⣣⠞⡀⢀⠀⠀⢡⠀⠀⠀⠰⣑⢊⠦⣁⠎⣜⠰⣉⢲⡐⠎⡜⣧⢻⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡁⢎⠰⠡⢄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⣰⣏⣯⠜⡴⢩⡓⠀⠀⠀⠀⠡⡀⢀⡱⢂⠣⡜⢤⡹⣌⡳⣌⠧⣜⡹⣜⢧⡻⣼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠃⠜⢠⠑⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⢴⣋⣔⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠐⠈⠀⠀⠀⠀⠀⠣⢄⠳⠀⠻⡽⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡉⢇⠻⡜⣶⡹⣖⡻⢶⡹⣞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡹⢈⠖⡡⢆⡱⢉⡝⢧⢻⡜⣮⠗⣣⠹⢚⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠧⣋⠴⡑⢢⠜⡱⢌⢣⡓⢾⡱⢯⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⡳⡔⢢⡙⠤⠓⡬⢌⠣⣝⢮⣝⡳⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠉⡵⡉⢖⢨⠱⡩⠔⣊⠱⢌⡘⢮⡵⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⠜⡌⢆⢣⡑⢎⠴⣉⠦⣑⢢⡟⢧⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⢦⡙⢌⣒⣉⢲⡙⢮⡔⣣⠖⣬⡳⣝⢧⣻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠃⡇⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⡑⠄⠣⠐⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⡔⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣿⠀⠙⠀⢘⡼⣙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⡀⠋⠆⠐⠠⢄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡏⠀⠐⢈⠀⢀⠇⠱⢄⣀⠌⠳⢜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠘⡾⣯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠙⣻⣶⠄⠃⠜⣹⠗⠡⠁⠀⠌⠦⠈⡄⠞⡤⢢⠄⣀⠃⠠⢰⡀⢁⠶⣒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣟⣳⢯⣟⣻⣞⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠡⢑⠂⣠⣤⣅⡚⠤⣉⠖⡀⠌⢈⠳⠉⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⢾⣽⣳⢾⣳⢿⡾⣽⢷⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠘⠉⠛⠳⣤⡉⡔⡌⠠⢀⢹⢠⠈⠐⢠⡔⢶⢲⢠⠀⡄⠰⠂
⠀⠀⠀⠀⣼⣻⣾⣯⢿⣽⣻⡽⣯⢿⣷⣟⡾⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠁⢀⠎⠀⡑⢂⠂⠉⢌⢣⠁⢈⠤⡑⠌
⠀⠀⠀⠀⢹⣷⣿⣿⣻⣾⣽⡿⣯⣿⣹⣾⣽⣻⣧⣀⡀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⢈⠀⡐⣠⠃⣈⠐⡀⢂⠹⣍⠖⠁⠀
⠀⠀⠀⠀⠈⢳⣯⠻⠹⢷⣟⣇⠯⡛⢧⠻⣷⣿⠞⣿⠻⢷⣶⢾⡻⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⠠⠁⢂⢰⡸⢤⡛⢀⢂⠜⡠⢂⡼⣹⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⢌⡙⠤⡉⠎⡓⠴⣧⡐⡈⡀⠘⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⡠⢄⠂⡌⢎⠴⡐⣁⡬⠄⣮⠱⢣⠔⡪⡜⣌⡤⢞⡱⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⢢⠘⢢⠱⣈⠔⡡⢂⡑⣢⡗⡰⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⣣⠌⡐⡫⣔⡏⡷⡜⣆⡳⣍⢣⠊⢕⡫⡞⡼⣩⡓⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠣⢄⠓⡄⠣⣴⡟⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢷⡸⣱⣎⣐⣱⢍⡞⡱⡹⣜⡳⣭⢧⣁⣂⡬⣕⣣⠗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠐⠈⠀⠀⠀⠀⠀⠊⡌⠱⠠⠙⣟⡆⢲⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠣⡋⢕⠪⢗⣮⠻⣜⢷⡻⣴⢻⡜⣧⢻⡔⣳⠼⣣⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠠⡙⡌⡱⢊⡄⢫⠌⡷⢹⠖⣏⠾⡜⢧⠸⠱⣆⠩⡇⠈⠀⠀⠀
⠀⠀⠀⠀⠠⢌⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡵⢊⡑⠦⣘⢡⠚⣡⢋⡞⣼⢫⡽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⡈⢆⡉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠹⡀⠀⠀⠀⠀⠀⠀⡰⠀⠢⠀⣞⡡⠜⡡⠜⠢⡍⢤⠫⡜⣮⢳⢧⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠢⡘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⠧⠀⠀⠀⠈⠶⡡⢋⠴⣉⠣⠜⡢⢱⠘⡤⢛⡮⣝⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⢓⡀⠀⠀⠀⢡⠀⠀⠀⢨⠱⣁⠇⢎⡰⣉⠖⣡⢣⠩⠔⣣⡝⡾⣔⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡁⢎⠰⡑⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡴⣩⠒⠁⠀⠀⠀⠡⡀⠀⣆⠓⡬⣘⢢⠱⣍⠶⣡⢎⡵⣋⢶⣹⠳⣞⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠃⡇⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⡑⠄⠣⠐⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⡔⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣿⠀⠙⠀⢘⡼⣙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⡀⠋⠆⠐⠠⢄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡏⠀⠐⢈⠀⢀⠇⠱⢄⣀⠌⠳⢜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠘⡾⣯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠙⣻⣶⠄⠃⠜⣹⠗⠡⠁⠀⠌⠦⠈⡄⠞⡤⢢⠄⣀⠃⠠⢰⡀⢁⠶⣒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣟⣳⢯⣟⣻⣞⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠡⢑⠂⣠⣤⣅⡚⠤⣉⠖⡀⠌⢈⠳⠉⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⢾⣽⣳⢾⣳⢿⡾⣽⢷⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠘⠉⠛⠳⣤⡉⡔⡌⠠⢀⢹⢠⠈⠐⢠⡔⢶⢲⢠⠀⡄⠰⠂
⠀⠀⠀⠀⣼⣻⣾⣯⢿⣽⣻⡽⣯⢿⣷⣟⡾⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠁⢀⠎⠀⡑⢂⠂⠉⢌⢣⠁⢈⠤⡑⠌
⠀⠀⠀⠀⢹⣷⣿⣿⣻⣾⣽⡿⣯⣿⣹⣾⣽⣻⣧⣀⡀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⢈⠀⡐⣠⠃⣈⠐⡀⢂⠹⣍⠖⠁⠀
⠀⠀⠀⠀⠈⢳⣯⠻⠹⢷⣟⣇⠯⡛⢧⠻⣷⣿⠞⣿⠻⢷⣶⢾⡻⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⠠⠁⢂⢰⡸⢤⡛⢀⢂⠜⡠⢂⡼⣹⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⢌⡙⠤⡉⠎⡓⠴⣧⡐⡈⡀⠘⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⠤⠘⡬⢰⠐⣁⡬⠄⣮⠱⢣⠔⡪⡜⣌⡤⢞⡱⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⡘⢢⠑⣌⠰⡁⢆⡑⣢⡗⡰⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣳⢈⠡⢇⣣⠟⣵⠹⡄⣷⡉⢇⠊⡕⢯⠞⡼⣩⡓⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⠂⠍⡄⢣⡘⢄⣲⠿⣽⠁⢿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢷⡚⣴⣃⣌⢓⠮⣝⡸⢭⡹⣖⢯⣏⣐⣀⡪⣝⣲⡓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢡⠘⡒⠄⠛⡽⡎⢴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡦⢛⢡⠛⣬⢏⡟⣶⢛⣧⢳⢯⡞⣼⢣⡭⢳⠼⣣⡗⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡽⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢀⢣⢊⡱⢂⠎⡜⣨⠛⣬⢛⡶⢳⡼⢣⠸⢙⡆⠱⡃⠈⠀⠀⠀
⠀⠀⠀⠀⠠⢌⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡓⢎⡔⢩⢢⠱⢌⡓⢬⠳⣜⢧⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⡈⢆⡉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⠀⡰⠀⠢⠀⣝⠢⣌⢃⠦⣉⠲⢌⠺⣙⢮⡳⣭⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠢⡘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⠧⠀⠀⠀⠈⣎⠳⣐⠪⢔⡡⠚⣌⠱⢌⢢⢛⠶⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⢓⡀⠀⠀⠀⢡⠀⠀⠀⢠⢃⠖⣡⠚⡰⢌⡓⠬⡘⣌⠲⡌⣿⡱⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡁⢎⠰⡑⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡴⣩⠒⠁⠀⠀⠀⠡⡀⠀⣜⠢⢍⡤⢃⡝⢮⡜⣥⢫⡔⣣⢽⡲⣝⢯⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠙⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣈⢳⡹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡩⡙⠮⡝⣧⣛⢶⡳⣝⢮⡳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡱⢡⠓⢬⡐⡌⢣⡝⢮⠳⣝⢶⡹⢦⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡵⢊⡜⢂⠖⣉⠖⣌⢣⡛⣼⢣⡟⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⣇⠣⣌⢃⠞⣠⠚⣄⢣⢛⡼⣣⢟⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠈⣎⢓⡰⢊⠦⡑⡜⡠⢎⠒⣌⠳⣏⡞⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⢠⠣⠜⣠⢃⠖⡡⢎⠱⣈⢇⢢⢣⡝⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⢦⡙⢜⡰⢌⢺⡱⣋⠶⣉⠖⣎⡳⣝⢶⣹⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠈⡱⢀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⣛⡾⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⣷⣿⡿⣟⠇⠐⠸⣯⡟⣳⢯⡿⣽⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⡿⣞⡷⣯⢷⠛⠁⠐⠒⠛⡷⢁⣿⢯⡿⡏⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⢿⡷⣯⢿⣽⡻⠝⠙⠉⠛⣻⣶⠄⠃⠘⣽⠏⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢤⡀⢁⢶⡒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣭⣟⡿⣽⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⢀⠈⠌⠣⠙⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣻⣞⣿⣳⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠙⠷⣬⡑⠢⡜⠀⠡⢸⢁⠀⠁⢢⡔⢶⢲⠀⡄⢠⠐⡂
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣷⣻⢾⣿⣳⣟⣾⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⢀⠇⠈⡑⠢⢈⠈⡡⢋⠀⢈⠄⣡⠂
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣯⣿⣹⣷⣻⢾⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠂⠌⢀⠐⣠⠃⢄⠂⡁⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠹⢷⣟⣧⢋⡛⢧⠻⡿⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⡐⠠⠁⠌⢠⣚⡤⡛⢀⠢⢜⠠⢁⡻⣜⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⡈⠍⠥⣉⠚⡱⠼⣣⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⠄⣏⠶⡱⣁⠎⡵⣈⡴⢺⡱⠎⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡘⡘⠴⣀⠣⡐⢡⢂⡙⣤⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⡛⡄⣯⡒⠵⠈⠞⡵⣫⢜⡣⡝⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⠒⠠⢃⠱⡂⠆⣴⡟⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣈⣳⠹⣜⡱⣍⢻⢶⡹⣏⣡⣈⢦⡓⣮⡓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠰⢉⠜⠠⠙⣟⡆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡲⡩⡙⢮⠳⣭⣛⢶⡳⣝⢮⡳⡽⢎⣷⢸⢲⠽⣲⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠼⢀⡱⢑⠢⢓⠤⢩⢌⠷⡹⢎⡷⢭⡻⢦⠩⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡵⢊⡱⡉⡜⣂⠞⡰⣙⠮⣜⢧⣛⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢀⠢⡀⠢⠀⣇⢣⠒⡑⠦⢡⠎⡱⢌⠻⣜⢧⣏⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡦⠁⠀⠀⠈⣎⠣⢍⡜⠬⣡⠚⢤⡉⠖⣌⠳⣮⢽⣁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠀⠁⠀⢡⠀⠀⠀⠰⢡⠍⠦⡘⠆⡥⢚⡰⢌⡃⢆⡣⣗⢮⡝⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠁⠀⠀⠀⠡⡀⠀⣎⠥⣊⡕⡡⢛⡜⣣⠖⡭⡜⢮⡱⣏⢾⣹⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠈⡱⢀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⣛⡾⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⣷⣿⡿⣟⠇⠐⠸⣯⡟⣳⢯⡿⣽⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⡿⣞⡷⣯⢷⠛⠁⠐⠒⠛⡷⢁⣿⢯⡿⡏⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⢿⡷⣯⢿⣽⡻⠝⠙⠉⠛⣻⣶⠄⠃⠘⣽⠏⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢤⡀⢁⢶⡒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣭⣟⡿⣽⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⢀⠈⠌⠣⠙⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣻⣞⣿⣳⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠙⠷⣬⡑⠢⡜⠀⠡⢸⢁⠀⠁⢢⡔⢶⢲⠀⡄⢠⠐⡂
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣷⣻⢾⣿⣳⣟⣾⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⢀⠇⠈⡑⠢⢈⠈⡡⢋⠀⢈⠄⣡⠂
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣯⣿⣹⣷⣻⢾⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠂⠌⢀⠐⣠⠃⢄⠂⡁⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠹⢷⣟⣧⢋⡛⢧⠻⡿⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⡐⠠⠁⠌⢠⣚⡤⡛⢀⠢⢜⠠⢁⡻⣜⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⡈⠍⠥⣉⠚⡱⠼⣣⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⠄⣏⠶⡱⣁⠎⡵⣈⡴⢺⡱⠎⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡘⡘⠴⣀⠣⡐⢡⢂⡙⣤⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⡛⡄⣯⡒⠵⠈⠞⡵⣫⢜⡣⡝⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⠒⠠⢃⠱⡂⠆⣴⡟⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣈⣳⠹⣜⡱⣍⢻⢶⡹⣏⣡⣈⢦⡓⣮⡓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠰⢉⠜⠠⠙⣟⡆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠖⡩⡙⢮⠳⣭⣛⢶⡳⣝⢮⡳⡽⢎⣷⢸⢲⠽⣲⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠐⣡⠑⡆⡓⢤⠩⣌⠷⡹⢎⡷⢭⡻⢦⠩⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠧⡙⢤⡙⠤⢓⠸⡰⣙⠮⣜⢧⣛⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⠠⡐⠀⠡⣀⢳⡉⠦⢌⠓⣌⢃⠖⣩⠳⣎⠷⣭⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⡓⠀⠀⠀⠀⢧⡙⢢⠍⡪⠔⣊⠜⡄⡓⡌⢻⡜⣮⢃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⡀⠀⠀⢡⠀⠀⠀⢨⠒⢬⠡⢎⠱⣘⠤⢓⡘⡔⣉⠦⣟⡼⢫⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠖⠀⠀⠀⠀⠑⣀⠀⢦⡙⢢⢍⢢⠹⣌⡳⡜⡼⡰⣍⢾⡱⣏⡟⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠙⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠲⢶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠁⣎⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⡑⠪⢔⡡⢊⠔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⠑⠢⢘⠢⡘⢄⣳⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⠴⣇⣈⢳⡹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠱⣈⠲⠀⠛⡽⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠒⣅⠫⠝⣮⢳⡝⣶⢳⡝⣮⢳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠈⡤⢋⡜⠤⢢⠙⡌⠷⡹⢎⡷⣲⠝⢦⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠐⠐⠄⠀⡳⠥⡘⢬⠡⣓⠸⡡⢛⡼⣸⢧⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢠⠡⠃⠀⠂⠷⡡⢍⢢⠓⢤⠣⢱⠩⢖⡧⣏⢷⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⠀⣡⠀⠀⠀⠀⢯⡑⠪⡔⢩⢂⠇⣣⢉⠖⣨⠹⣶⡹⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡀⠌⠀⠀⠀⢆⠀⠀⠸⣠⢃⠓⡬⢑⡊⡜⣐⠪⢜⡠⢣⡷⣹⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⣋⠎⠀⠀⠀⠀⠀⢡⠂⣇⠒⣌⡓⢌⢮⡱⢣⡜⣣⢎⡜⣳⢵⣣⢟⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢤⢁⠃⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢈⣴⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⡌⠱⠠⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣆⠀⠀⠀⠠⠀⠈⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢀⡤⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣼⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠘⠀⢰⡜⣫⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⢈⠱⠊⠀⠢⢄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠈⠉⠉⠉⢩⣿⣿⣟⣿⣯⣿⢿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⠯⡇⠀⠐⡈⠀⠂⢆⠱⢄⣀⠌⠳⢜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠐⣯⣟⣾⢿⣯⣟⣯⢿⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠌⣽⠏⠁⠁⠀⠌⡲⠀⡉⠬⣂⠴⡠⢀⠃⡈⢰⡀⢁⢶⡒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⠾⣽⣛⡾⣽⣞⣯⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⢡⠃⣠⣤⣅⡣⢉⠖⡰⠀⠌⡈⠣⠙⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢾⢯⣟⡷⣯⣟⣷⣻⣞⣷⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠘⠉⠙⠷⣬⡑⡌⡒⢀⠡⢸⢁⠀⠁⢢⡔⢶⢲⠀⡄⢠⠐⡀
⠀⠀⠀⠀⣼⣻⣾⣟⣷⣻⢾⣷⣻⢾⣷⣟⡾⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠁⠀⢠⠃⠈⡑⠢⢈⠈⡡⢋⠀⢈⠄⡡⠂
⠀⠀⠀⠀⢹⡷⣿⣿⣾⣽⣻⣷⣯⣿⣻⢾⣽⣻⣧⣀⡀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠂⠄⡀⠡⡰⢁⠂⡄⢁⡈⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠀⢿⣯⠻⠳⢿⣯⣇⢫⠳⢯⠻⣯⣷⠚⣿⠛⢷⣶⢾⡏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⡐⠠⢁⠂⢄⣫⢔⡋⠡⣀⠖⡠⢁⡻⡜⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠑⠀⡐⢨⠙⠤⣉⠒⠣⠶⣣⠄⡠⠀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡤⢐⠠⠒⡬⢰⠐⣁⣤⠌⡞⡴⢋⡔⢢⡵⣈⡤⢾⡱⠍⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠓⢢⠙⠢⠤⣉⠆⡱⢠⠙⣤⡗⡰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣱⠈⡅⡇⣧⡛⣵⢪⡄⣿⡐⢧⠈⠧⡳⢧⡙⢧⡙⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠐⠡⢂⠱⡄⠣⣜⡿⣽⢁⢻⣿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣏⠞⣬⣇⣀⣛⠴⣝⡘⣧⢹⢶⡹⣎⣡⣐⢬⢣⡽⠎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠐⠀⠀⠈⠰⣈⠓⠤⠙⡟⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⡱⠑⢮⠳⣎⡟⣶⣛⢶⡹⢮⡳⡝⣶⢩⢞⡱⢯⠗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠐⣡⠋⢆⡓⠤⡙⢤⡛⢮⡝⣧⢳⡼⢥⠸⠚⡄⢫⠇⠁⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡈⡈⠂⠀⡣⠝⢢⠜⡱⢘⢢⡑⣣⠞⡼⣓⡞⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡈⢆⡑⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⡐⠰⠁⠀⠁⡳⡍⢲⠘⡴⢉⢆⠲⣡⠻⣜⢧⡻⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢡⠂⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⡱⠀⠀⠀⠀⠐⠨⠀⠀⠀⠀⡳⣉⠦⢩⠔⢣⡘⢢⢅⡚⢌⠳⣽⡱⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡉⢆⡑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣱⢣⡀⠀⠀⠀⠄⠣⡀⠀⢰⢑⠢⢎⡡⢎⡡⠜⣡⠒⡌⢎⡱⣮⡝⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⢢⠘⡄⢓⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⣢⠧⠁⠀⠀⠀⠀⢀⠀⢇⢎⠬⡑⢦⠑⣬⠳⣍⢦⢫⡜⢮⣱⢳⢞⣱⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠃⠜⢠⠑⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡷⣚⢴⣋⣔⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠐⠈⠀⠀⠀⠀⠀⠣⢄⠳⠀⠻⡽⣆⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠊⡰⡉⠷⢎⡳⣝⢶⣛⡶⣹⢞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠀⠐⣡⠑⡎⠦⢱⣈⢣⠞⡵⢫⡞⣼⡱⢣⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠣⠄⠀⠀⠀⠀⠀⠀⢀⠄⢂⠁⡀⠧⣋⠴⡉⢆⠦⡙⡰⣩⠳⣜⢧⣛⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡀⠀⠀⠀⠀⠀⢄⠘⠀⠀⠀⢷⡐⠦⣙⠰⢢⠱⡑⢬⠳⣝⢮⡳⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠤⡑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⠵⡀⠀⠀⠀⢈⠐⡀⠀⠀⠀⢧⡙⠴⠨⣅⠣⡱⢘⡂⠧⢌⠳⣝⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢆⠱⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣸⢦⠀⣣⠕⡠⠀⠀⠀⠀⠑⠄⡀⢨⠒⡌⢎⡱⢄⡣⣑⠣⡘⡜⡨⢥⡟⣼⢃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⠢⡑⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣏⣯⠜⡔⡛⠀⠀⠀⠀⠀⢀⠠⣡⠇⡭⢘⠆⢦⡙⢶⡡⢞⡱⢬⡱⣳⡝⣮⢻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠃⡇⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⡑⠄⠣⠐⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⡔⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣿⠀⠙⠀⢘⡼⣙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⡀⠋⠆⠐⠠⢄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡏⠀⠐⢈⠀⢀⠇⠱⢄⣀⠌⠳⢜⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠘⡾⣯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠜⣹⠗⠡⠁⠀⠌⠦⠈⡄⠞⡤⢢⠄⣀⠃⠠⢰⡀⢁⠶⣒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣟⣳⢯⣟⣻⣞⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠡⢑⠂⣠⣤⣅⡚⠤⣉⠖⡀⠌⢈⠳⠉⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⢾⣽⣳⢾⣳⢿⡾⣽⢷⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠘⠉⠛⠷⣤⡉⡔⡌⠠⢀⢹⢠⠈⠐⢠⡔⢶⢲⢠⠀⡄⠰⠂
⠀⠀⠀⠀⣼⣻⣾⣯⢿⣽⣻⡽⣯⢿⣷⣟⡾⣽⡀⠀⠀⠀⠐⠚⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠁⢀⠎⠀⡑⢂⠂⠉⢌⢣⠁⢈⠤⡑⠌
⠀⠀⠀⠀⢹⣷⣿⣿⣻⣾⣽⡿⣯⣿⣹⣾⣽⣻⣧⣀⡀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⢈⠀⡐⣠⠃⣈⠐⡀⢂⠹⣍⠖⠁⠀
⠀⠀⠀⠀⠈⢳⣯⠻⠹⢷⣟⣇⠯⡛⢧⠻⣷⣿⠞⣿⠻⢷⣶⢾⣛⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⠠⠁⢂⢰⡸⢤⡛⢀⢂⠜⡠⢂⡼⣹⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⢌⡙⠤⡉⠎⡓⠴⣧⡐⡈⡀⠘⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠂⡌⢎⠴⡐⣁⡬⠄⣮⠱⢣⠔⡪⡜⣌⡤⢞⡱⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⢢⠘⢢⠱⣈⠔⡡⢂⡑⣢⡗⡰⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠈⡄⢏⡲⡝⣧⠺⣄⢷⡉⣇⠊⡕⢯⠞⡼⣩⡓⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠃⢄⠓⣄⠣⣴⡟⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢿⣘⠦⣇⣌⣳⡹⣜⡘⣧⢛⡮⣝⢮⣁⣀⢮⡙⣶⠓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡠⠓⠤⠙⡟⣦⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⡰⣉⠻⡜⣧⡝⣶⣛⢶⡹⣞⡽⣎⢷⡩⢶⡹⢧⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡵⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⢂⠈⡴⢁⠞⡰⠰⢌⡱⠎⡷⡹⢞⡜⡯⢦⠙⠞⣄⠹⡃⠈⠀⠀⠀
⠀⠀⠀⠀⠠⢌⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠱⠄⠀⠀⠀⠀⠀⠀⠐⡠⠁⠀⠁⡳⢍⢢⠱⣉⠲⡘⠥⢳⡙⢮⡝⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⡈⢆⡁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠹⠄⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⡳⡌⢆⡓⢌⡱⢌⢃⠧⣛⢧⡻⢵⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠢⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢱⡆⠀⢫⠄⠀⠀⠀⠠⠀⠣⠀⠀⢀⢳⣉⠲⠌⠦⣑⢊⡌⢲⡐⣊⠽⣣⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠊⠥⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣸⢦⠀⣣⠞⡄⠀⠀⠀⠀⠐⠑⢌⠬⢃⠤⢣⢙⠢⡅⢎⠔⣣⠰⡡⢲⡝⣮⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡁⢎⠰⠡⢄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⣰⣏⣯⠜⡴⠍⠀⠀⠀⠀⢀⠀⡄⡬⢎⡱⢊⡥⢊⡱⣙⠮⡜⢦⡱⢣⡳⣝⡞⢾⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⣀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⡀⢠⠀⠒⢀⠡⠘⠐⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⢆⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠉⡀⠰⢍⢯⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⢉⠁⠌⠒⡁⠈⠔⢠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢨⠀⡀⠌⢆⠱⢄⡠⠌⠛⠴⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠂⡳⠀⠆⠪⢆⠲⡠⣀⢃⠈⠰⣀⢠⠞⡴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⢹⠀⠥⣶⣬⣑⠰⢌⡪⠐⠠⠈⡙⠘⣭⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠡⠙⣻⡦⠁⡇⡑⠠⢀⢹⢠⠀⠀⢡⢖⡚⢦⠂⡄⠰⢠⠁
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠀⢠⠃⠀⡉⡰⠁⠠⢉⢹⠁⣠⠒⡅⠈
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠈⡒⠠⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⡁⢂⠠⠐⡔⣁⢄⠁⠂⡄⠛⡴⣃⠀⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⠐⡀⢢⢏⡴⠍⠠⣈⠎⠔⢠⡳⢭⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠰⣄⡂⠔⡡⢎⡔⣂⣡⢖⠂⣏⠞⣰⢡⢓⡜⣤⡜⢾⡱⡋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠽⡐⠠⡧⡜⣧⢏⡞⣄⠿⣌⢣⠂⠝⣎⠷⣘⢧⠓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⠊⠄⡓⢌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡊⠧⢏⣧⣁⣹⠼⣬⡸⡜⣭⢻⡜⣏⢥⣀⢖⣫⠵⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠳⠀⠻⡽⣆⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠀⡘⡍⠶⢫⣳⢛⡶⣳⠽⣜⢧⡛⢮⠯⣜⢺⠸⣏⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡀⠠⡈⠵⡘⠬⡑⠤⢋⡔⢏⡻⡜⣧⢯⡙⠧⠸⠩⢆⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⠐⡠⢉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠱⡂⠀⠀⠀⠀⠀⠀⡢⠁⠀⠀⠀⡻⢌⡱⢌⢃⠧⡘⢆⢳⡙⢶⡭⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠱⡄⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⡳⡌⡔⢪⢌⢒⡩⢌⠲⡝⣮⢳⣝⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⢃⠆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢱⡆⠀⡱⠄⢀⠀⠀⠄⠀⠑⣀⠀⡄⣳⠱⠌⡥⢊⡒⠴⡉⢦⠱⣈⢳⢮⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠊⡔⡀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⣸⢦⠀⢣⠯⠀⠀⠀⠀⠈⠐⠈⢒⢨⠅⡎⡱⢂⠧⡘⢆⢍⢢⠱⢌⡢⣟⡼⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡐⠩⢄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⣰⢯⢯⣘⡣⠁⠀⠀⠀⢀⡀⢄⡢⢎⢇⡚⢔⡡⢃⢞⡹⢎⡜⣎⠵⣪⢵⣫⢶⣹⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠑⡂⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⢀⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⡄⠐⠂⡈⠄⠃⠂⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠉⠀⠀⢂⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀
⠀⠀⠀⠀⠈⡔⡀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⢿⡽⡆⠀⠴⡀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⢀⣠⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⡞⢿⣻⢷⡯⣟⡽⣷⠀⠈⣁⠀⠮⣹⡙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣿⡿⣟⠏⠐⢈⡿⡯⣟⡾⣽⣻⠀⠀⠁⠌⠐⡀⢋⠀⠨⠐⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠀⠈⠀⠁⢸⣿⣿⣻⣯⣿⢿⣾⣟⡿⣽⣻⣞⡷⣯⢷⡻⠋⠐⠒⠚⢻⠃⣿⡽⣷⡏⠀⢀⢸⠀⠐⠬⠈⠦⢄⠠⠙⠲⢌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠈⣾⣳⢿⣻⢯⣟⣾⣽⣻⢷⣻⢾⡽⠏⠛⠉⠙⣷⡦⠄⠈⠐⢹⡿⠁⠁⠀⠊⠜⢄⡃⠊⠕⡲⢌⡐⠦⠁⠂⢆⡈⡴⣓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡯⣏⡿⣹⣻⣞⣧⣟⣯⢿⣽⣿⢿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈⢰⠂⠱⢶⣦⣅⡊⠥⣣⢁⠂⡐⠈⢇⣏⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢺⢿⡽⣷⣻⢷⣻⢾⣽⣻⣟⣾⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠻⠶⠄⢎⡐⠠⢁⢸⢠⠀⠀⢡⠖⡶⢲⠄⢀⠠⣀⠂
⠀⠀⠀⠀⣾⢯⣿⡷⣯⢿⣽⣟⡾⣽⣿⡽⣞⣯⡄⠀⠀⠀⠐⠳⣶⣰⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠⠃⠀⡉⡰⠁⢈⠱⢩⠂⢁⠤⡂⠔
⠀⠀⠀⠀⢹⣯⣿⣿⣯⣿⢾⡿⣽⣿⡹⣿⡽⣞⣷⣀⡀⠀⠀⢡⣼⡿⠀⠀⠀⠀⠀⠀⡈⠒⠤⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⡐⠠⢀⠐⡴⢁⡐⠄⠂⡄⠓⣎⣆⠁⠀
⠀⠀⠀⠀⠀⢻⣿⠿⠹⢾⢿⣇⣍⠳⠷⡹⢿⣻⡿⡟⠻⣷⢶⣾⡃⠁⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⢤⠃⢀⠂⠄⠡⢠⢏⡴⢋⠀⡢⢍⡐⢠⠳⣜⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⡈⡔⢨⠂⠥⡙⡐⠳⣖⡠⢈⡀⢈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢶⡈⠔⡉⣆⠲⣈⡰⣬⠂⣏⠞⣰⠡⢎⡵⢤⣔⡫⠽⡌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠌⢡⠚⠤⡑⢌⠱⢠⠑⣢⡖⡸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⡻⡀⠆⣇⢳⣍⡷⣘⣆⡽⣌⢣⠑⢊⠷⣣⢎⡝⡓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠈⠀⠱⡈⢦⠁⣾⡽⣻⠁⢿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠭⠶⣇⣌⣹⢎⣬⡸⡕⢮⡳⣭⢏⡥⣈⡴⣓⢮⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠙⠠⠙⣯⢇⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠡⡀⠁⡜⣙⠺⡜⣧⣛⢶⡳⣝⢧⡻⡜⣫⠾⣱⢺⠹⣎⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⡀⢫⠄⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠐⠚⡰⢌⡒⣡⠒⠬⣡⠟⡼⢣⣟⡹⡜⢣⠰⠫⡄⢹⠃⠈⠀⠀⠀
⠀⠀⠀⠀⠠⡑⢌⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠀⠹⣀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⡻⠔⡘⢤⡙⢢⡑⢎⣱⠳⣜⡳⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⢌⠢⢡⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠱⣂⠀⠀⢀⠀⠘⡐⡄⠀⠀⠀⡷⣉⠜⢢⠜⡡⠜⢢⠜⡳⣎⢷⡹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢌⠢⡁⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢱⡆⠀⡱⡂⠀⠀⡀⠀⠀⠈⢢⠘⢄⡳⢡⠎⢥⢊⡑⠎⡥⢊⡱⣈⠳⣝⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠣⡘⠤⡉⢄⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣸⢦⠀⡕⠀⠀⠀⠀⠁⠐⠂⢀⠭⣰⢡⢃⡚⠤⣃⡜⡑⢢⢃⠦⢡⢣⡟⣬⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢂⡑⢢⠑⡌⠢⡀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣰⣏⣯⠜⠀⠀⠀⠀⡀⢠⠀⡦⢋⠞⣰⠂⠧⣌⠣⢴⡩⢇⡳⢎⡜⣣⡳⣝⢮⣛⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⡌⢡⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣡⣿⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⣀⠠⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠂⢡⠘⠰⠀⠃⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠉⠀⠀⢂⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠀⠈⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀
⠀⠀⠀⠀⠠⠡⠄⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣷⢿⣻⡆⠀⠴⠂⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⢀⣠⣶⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⡷⢟⣿⣳⢯⡟⣵⢷⠀⠈⡁⢀⠮⡹⣍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣿⣾⡿⣿⣟⡟⠀⠌⣷⣻⢫⡿⣽⣻⠀⠠⢁⠂⠄⠑⢊⠀⠨⠐⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠀⠈⠀⠁⢸⢿⣟⣿⣿⣽⣿⡾⣿⡽⣯⣟⣾⣳⢯⣟⡷⠋⠐⠒⠚⢹⠃⣼⣻⣷⡇⠀⢀⢌⠀⡈⠲⠈⠦⣀⠤⠙⢣⢚⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠈⣯⣟⡾⣽⣻⢾⣽⣳⣿⣳⣟⡾⣽⠏⠙⠉⢛⣶⡦⠀⢈⠐⢹⡿⠸⠁⠀⠂⡚⠄⡜⠂⠗⡲⢄⠔⢢⠀⠂⢆⡀⣲⢣⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡟⣼⣛⡷⣽⣻⢶⣯⣷⣻⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠸⠄⠴⢶⣦⣍⡊⠥⣊⠰⠀⠄⠑⢋⣎⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⢹⡿⣽⣯⣟⣷⣻⣟⡾⣽⣯⢿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠉⠻⠶⠄⡞⡈⠄⠡⢸⢠⠀⠀⢡⠖⡶⢲⠄⢀⡀⢄⠀
⠀⠀⠀⠀⣾⡽⣿⣷⣻⢾⣽⣾⣽⣻⣿⢿⣽⣻⡄⠀⠀⠀⠚⢶⣆⡌⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡐⠃⢀⠉⡐⢁⠈⠱⢩⠂⢁⠬⡐⠂
⠀⠀⠀⠀⠸⣟⣿⣿⣽⡿⣞⡟⣾⣽⡝⣿⢾⡽⣷⡀⠀⠀⠈⣠⣿⡇⠀⠀⠀⠀⠀⠀⡈⠓⠄⠀⠀⠀⠀⠀⠀⠀⡠⢁⠐⠠⠐⠠⢀⠖⣁⠤⢈⠐⡀⠓⣎⢆⠁⠀
⠀⠀⠀⠀⠀⠻⣿⡫⠛⠛⠿⣧⢣⢍⢻⡙⠿⣿⣻⠛⢿⡶⣶⡿⠐⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⢀⠰⠁⠠⠈⠄⡁⢢⢏⡼⡘⢀⢢⢊⠔⢠⢏⡞⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡆⠱⡈⢅⠊⠥⡘⠹⣖⠠⢁⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⡃⢌⠂⡏⡔⣃⢬⢴⠂⣏⠞⣰⠰⣉⢦⢥⡒⢯⡚⡌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠩⠡⢑⠊⡜⢤⢁⠣⢌⡑⣮⡖⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⢺⡀⠎⡵⡜⣧⡏⡞⣜⡼⣌⢣⠑⠜⣣⢧⡙⣧⠙⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⠢⡑⣌⠢⣼⡻⣽⠠⢿⡷⠄⠀⠀⠀⠀⠀⠀⢀⠀⠰⠊⢅⣉⢮⢳⣔⣈⣳⠹⣤⣸⠵⣹⢞⡭⣏⢤⡨⣔⢧⡝⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠉⠌⠳⡟⡆⣹⠄⠀⠀⠀⠀⠀⠀⢀⠤⠉⠀⠑⠰⣎⠏⡱⠮⣝⢮⣛⢶⡳⣛⡼⣫⠞⣝⠮⣕⢺⠺⣹⠧⡀⠀⠀⠀
⠀⠀⠀⠀⢀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⣳⠈⠞⡄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⡸⢌⡱⠒⣌⢢⢉⠖⡝⢧⠯⣵⡛⡜⢣⠸⠩⡄⢳⠃⠈⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣣⠈⠸⠄⠀⠀⠀⢤⠁⠠⡀⠀⠀⠠⡁⡳⠔⡩⠔⣊⢌⡚⢌⢧⡛⣴⢫⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠢⡑⢂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣃⠀⠫⡄⠐⠀⠀⠠⠔⠰⠔⡠⠃⢌⡳⣉⠒⡭⢐⡊⠴⡉⠶⣙⢮⡳⡽⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠱⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢯⡄⠰⠀⠀⠐⠀⢀⠀⡀⠘⢆⡩⢒⡵⢃⠭⣐⠣⡘⢥⠊⡕⠬⢌⢳⣳⢣⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢃⠜⡠⠣⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣧⢇⠂⠀⠀⠀⠀⠀⠀⢀⡰⡋⠴⢹⠠⡍⡒⡌⢥⡉⠦⡙⢤⢋⡌⢦⡗⡯⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡃⠜⡠⢃⠔⢄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⣰⢯⠃⠆⠀⠀⠀⢠⠐⡌⢦⢡⠹⣌⢇⡚⢔⡱⠘⢦⡙⢧⡜⣣⠖⣜⣣⢟⣭⢻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠑⡂⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⢀⡀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⡄⠐⠂⡈⠄⠃⠂⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠉⠀⠀⢂⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀
⠀⠀⠀⠀⠈⡔⡀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣷⢿⡽⡆⠀⠴⡀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⢀⣠⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⡞⢿⣻⢷⡯⣟⡽⣷⠀⠈⣁⠀⠮⣹⡙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣿⡿⣟⠏⠐⢈⡿⡯⣟⡾⣽⣻⠀⠀⠁⠌⠐⡀⢋⠀⠨⠐⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⠀⠈⠀⠁⢸⣿⣿⣻⣯⣿⢿⣾⣟⡿⣽⣻⣞⡷⣯⢷⡻⠋⠐⠒⠚⢻⠃⣿⡽⣷⡏⠀⢀⢸⠀⠐⠬⠈⠦⢄⠠⠙⠲⢌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠈⣾⣳⢿⣻⢯⣟⣾⣽⣻⢷⣻⢾⡽⠏⠛⠉⠙⣷⡦⠄⠈⠐⢹⡿⠁⠁⠀⠊⠜⢄⡃⠊⠕⡲⢌⡐⠦⠁⠂⢆⡈⡴⣓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡯⣏⡿⣹⣻⣞⣧⣟⣯⢿⣽⣿⢿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈⢰⠂⠱⢶⣦⣅⡊⠥⣣⢁⠂⡐⠈⢇⣏⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢺⢿⡽⣷⣻⢷⣻⢾⣽⣻⣟⣾⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠻⠶⠄⢎⡐⠠⢁⢸⢠⠀⠀⢡⠖⡶⢲⠄⢀⠠⣀⠂
⠀⠀⠀⠀⣾⢯⣿⡷⣯⢿⣽⣟⡾⣽⣿⡽⣞⣯⡄⠀⠀⠀⠐⠳⣶⣰⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠⠃⠀⡉⡰⠁⢈⠱⢩⠂⢁⠤⡂⠔
⠀⠀⠀⠀⢹⣯⣿⣿⣯⣿⢾⡿⣽⣿⡹⣿⡽⣞⣷⣀⡀⠀⠀⢡⣼⡿⠀⠀⠀⠀⠀⠀⡈⠒⠤⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⡐⠠⢀⠐⡴⢁⡐⠄⠂⡄⠓⣎⣆⠁⠀
⠀⠀⠀⠀⠀⢻⣷⡛⠙⢯⢿⣇⣍⠳⠷⡹⢿⣻⡿⡟⠻⣷⢶⣾⡃⠁⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⢤⠃⢀⠂⠄⠡⢠⢏⡴⢋⠀⡢⢍⡐⢠⠳⣜⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⡈⡔⢨⠂⠥⡙⡐⠳⣖⡠⢈⡀⢈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢶⡈⠔⡉⣆⠲⣈⡰⣬⠂⣏⠞⣰⠡⢎⡵⢤⣔⡫⠽⡌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠌⢡⠚⠤⡑⢌⠱⢠⠑⣢⡖⡸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⡀⢄⣠⢇⡻⡀⠆⣇⢳⣍⡷⣘⣆⡽⣌⢣⠑⢊⠷⣣⢎⡝⡓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠈⠀⠱⡈⢦⠁⣾⡽⣻⠁⢿⣿⠤⠀⠀⠀⠀⠀⡌⠄⠐⣁⠨⢈⠳⣎⠵⣇⣌⣹⢎⣬⡸⡕⢮⡳⣭⢏⡥⣈⡴⣓⢮⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠙⠠⠙⣯⢇⢊⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⠈⣎⡳⡜⣙⠺⡜⣧⣛⢶⡳⣝⢧⡻⡜⣫⠾⣱⢺⠹⣎⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⡀⠄⠀⠀⠀⠀⢀⠘⡀⠀⠀⠀⡐⡉⠱⡘⠤⠣⢜⡠⢎⢡⠟⡼⢣⣟⡹⡜⢣⠰⠫⡄⢹⠃⠈⠀⠀⠀
⠀⠀⠀⠀⠠⡑⢌⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠀⠀⠀⠤⠀⠨⡀⢀⣆⠠⠐⡄⢇⠀⡝⢎⡱⢢⠱⣈⠞⡰⣩⠳⣜⡳⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⢌⠢⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⠁⠀⢀⠀⠀⠈⠀⢡⠃⡔⠬⢆⡹⣂⠖⡡⢃⠖⠬⡑⢬⠳⣭⢳⡭⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢌⠢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢱⠎⠀⠀⠀⠈⠀⠀⠀⣠⢋⡜⢡⢆⡳⢡⠎⡡⢍⡜⠢⢍⢢⠓⡌⢳⡝⣮⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡑⢄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡘⡀⠀⠀⠀⢀⠠⣀⠒⠴⣨⢔⡣⢞⢡⠣⠜⣡⠒⣌⠓⡌⢦⢩⡘⣡⢾⡱⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⠢⢌⢂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡜⢠⠡⠀⠀⡠⢊⠔⡠⢏⢒⡱⢎⡵⣋⠤⢋⡜⠤⡹⣌⠳⡜⣎⠴⣪⢵⡫⢗⡾⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⣀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⡀⢠⠀⠒⢀⠡⠘⠐⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⢆⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠉⡀⠰⢍⢯⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⢉⠁⠌⠒⡁⠈⠔⢠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢨⠀⡀⠌⢆⠱⢄⡠⠌⠛⠴⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠂⡳⠀⠆⠪⢆⠲⡠⣀⢃⠈⠰⣀⢠⠞⡴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⢹⠀⠥⣶⣬⣑⠰⢌⡪⠐⠠⠈⡙⠘⣭⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠡⠙⣻⡦⠁⡇⡑⠠⢀⢹⢠⠀⠀⢡⢖⡚⢦⠂⡄⠰⢠⠁
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠐⠚⣶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠀⢠⠃⠀⡉⡰⠁⠠⢉⢹⠁⣠⠒⡅⠈
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠈⡒⠠⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⡁⢂⠠⠐⡔⣁⢄⠁⠂⡄⠛⡴⣃⠀⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⠐⡀⢢⢏⡴⠍⠠⣈⠎⠔⢠⡳⢭⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠰⣄⡂⠔⡡⢎⡔⣂⣡⢖⠂⣏⠞⣰⢡⢓⡜⣤⡜⢾⡱⡋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⠀⢀⠄⠂⠤⢀⠤⡀⣔⢣⠽⡐⠠⡧⡜⣧⢏⡞⣄⠿⣌⢣⠂⠝⣎⠷⣘⢧⠓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠁⠊⠄⠓⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⡌⠒⠀⠑⢀⡌⠓⢮⡙⣎⣇⣅⣹⡜⣬⣸⡜⣭⢻⡜⣏⢥⣀⢖⣫⠵⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠱⢀⠳⠀⠻⡽⣆⢂⠁⠀⠀⠀⠀⠐⡌⠁⠀⠀⠀⢰⡹⢲⡝⡑⠺⡜⣧⣛⢶⡳⣝⢶⣫⠞⣝⠾⣌⢻⠸⣏⠷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡼⡀⠄⠀⠀⠀⢀⡀⠑⡀⠀⠀⠀⢜⠹⠁⢦⠩⡱⡘⡄⢎⡡⠟⡼⣓⢮⣛⡜⢣⠸⠩⢆⢩⠇⠈⠀⠀⠀
⠀⠀⠀⠀⠐⡠⢉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠠⠠⠐⠀⠠⣀⠐⣤⠀⡌⢂⣇⠀⠧⢣⡑⢆⡩⢆⠹⢌⠳⣍⠾⣱⢞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⠀⠀⢀⠀⠀⠀⠀⢣⢐⠥⡌⠤⣛⡔⣘⠢⢱⡈⢇⢪⠱⢭⡞⣵⢫⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠀⡜⡡⠎⣌⢳⡱⢊⠴⣉⠒⡜⡨⠆⣍⠒⣌⠳⣏⡞⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⠩⠄⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠀⢠⠀⢆⠁⠴⣱⢚⡬⢣⠱⣉⠒⡬⢑⡬⢑⡜⣠⢋⠤⣣⣏⠾⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣁⠚⡄⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⢀⠎⡰⢁⠀⡐⢠⠃⡜⠂⣌⢣⢱⢫⡜⣣⠣⢌⡓⣌⠺⣜⢣⡔⣣⠜⣮⡱⣎⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠲⢶⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠁⣎⣿⠃⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⢀⠘⣀⠁⠄⢢⢐⠄⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠊⢄⠓⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠐⡠⢃⠌⠂⠈⢂⠘⠊⢷⡚⡴⣏⣐⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡠⠓⠄⠛⡽⣆⢂⠁⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⢡⡜⢦⡝⣑⢫⠳⢧⣛⢶⣛⡶⣹⢞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡵⡀⠄⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⢸⠚⠡⡒⡌⠦⡙⢤⡉⢎⡞⢵⢫⡞⡼⠧⢧⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⢠⠐⠂⠈⠠⠠⡸⣄⠤⠘⡠⣇⠀⡳⢌⡱⢘⠢⢜⠸⢄⡫⢖⡹⣝⣫⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠄⠀⠀⠀⡀⠀⠀⠀⣱⢁⠦⠜⡄⡳⡌⢆⡍⢚⠌⢎⢢⡙⢧⡻⣜⢧⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠠⣄⢋⡜⣘⠼⣱⢉⠦⣘⢡⠚⣌⠒⡌⢆⡙⢼⣣⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢩⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⢃⠀⠀⠀⠠⡀⠆⠬⢁⠰⣌⠶⣩⠞⣡⢊⠴⢡⢊⣑⠢⡍⢜⠢⣉⠦⣗⢯⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡁⢎⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡠⠓⡌⢄⡀⢠⢂⠱⠨⠁⢤⢃⡝⢮⡱⣋⠔⡌⢎⢆⡹⣌⠳⡜⣎⠵⣊⢷⡹⢮⡽⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⡄⢂⠆⠤⠐⠢⠔⣨⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠃⠜⢠⠑⣌⠢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠐⢁⠈⠀⠀⠀⢒⠁⠊⢷⡚⡴⣏⣐⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠁⠀⠂⠀⠀⠀⠣⢄⠳⠀⠻⡽⣆⢂⠁⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⢄⡜⣣⠝⣑⢫⠳⢧⣛⢶⣛⡶⣹⢞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⠀⠀⠤⠀⠈⠄⠀⠀⠀⣸⠚⠡⡚⣄⠣⣉⠦⣉⠜⠮⡵⢫⡞⡼⠧⢧⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡀⠂⠁⠀⠢⠔⠸⠣⡐⠰⡀⣇⠀⡳⢄⠳⣈⠖⡌⡚⠥⣙⢧⢺⡝⣯⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠄⠀⠀⠀⠈⡕⣁⠶⡘⡄⡳⡌⣃⠖⡌⣒⡉⢖⠩⣎⢷⡹⡖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⢀⡐⢠⢡⢒⠱⣜⡱⢃⠖⣌⢒⡡⢜⡨⢱⡐⢊⢳⡝⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡜⣀⠀⠀⠀⢀⠆⡘⠄⢀⠰⣪⢜⡳⡜⡡⠎⡜⡠⠎⣔⢢⡑⢆⡩⠎⡴⣛⡼⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡜⢠⠂⠤⡀⢅⠊⡔⠉⠀⢤⢃⠳⣎⠵⣣⢡⠓⣌⡑⢺⡌⢧⡜⣎⠴⣋⡶⣝⡞⣽⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⡀⠰⠐⡄⠠⠁⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⡒⡑⠪⠔⣡⠊⡔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⡄⠁⠀⠈⢩⠊⠕⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠁⠃⠜⢠⠑⡌⢢⣱⠿⣽⢁⢻⣿⠧⠀⠀⠀⢘⡄⠀⠀⠀⢁⠐⠊⢷⡚⣴⣋⣔⣣⡝⣬⡙⣦⢛⡮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠣⢘⠒⠄⠛⡽⣆⢂⠁⠀⠀⠀⠀⠂⠈⠄⠀⠀⠈⢀⡜⡦⡝⢡⠻⡜⣶⡹⣖⡻⢶⡹⣞⡵⣫⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⡵⡀⠄⠀⠀⠀⡰⠀⠀⠰⠀⠀⠀⢺⡘⠱⣈⠇⡒⠥⣂⠕⡩⡝⢧⢻⡜⣮⠵⢣⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠆⠉⠀⠀⠂⠜⠪⢳⡀⠎⡐⣆⠀⡳⢌⡱⢃⠬⣘⠱⢌⣣⢓⢾⡱⢯⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠠⠀⠀⠀⠠⠷⣠⠥⠜⣄⠳⡌⠴⡉⠦⣑⢊⠖⢬⢫⡞⣭⢳⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⠞⡔⡡⢎⠱⣌⡳⣉⠖⣉⠖⣡⢊⠜⡢⢅⣊⠳⣏⡞⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠆⡂⠀⠀⠀⠢⢌⡑⠎⠀⢜⡡⢎⡳⣜⢡⠒⡜⡰⢊⡴⢈⠞⣠⢃⠦⡱⣏⠾⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠂⡜⠰⢠⠐⡌⡑⢂⠞⠀⢜⠢⣙⢧⡓⣎⠴⢩⡰⢡⠳⣜⢣⠞⣤⢣⢞⡱⢯⡝⣾⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⠀⡀⢔⣂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⢂⠍⡘⢃⠶⢧⡐⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⢆⣰⠁⠀⢠⠆⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⡑⠪⢔⡡⢊⠔⢢⠉⣦⡖⡸⣿⣿⡁⠀⠀⠀⢀⠕⢂⠀⠀⠀⠂⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⠑⠢⢘⠢⡘⢄⣳⠿⣽⢁⢻⣿⠧⠀⠀⠀⠀⠀⠈⡄⠀⠀⠸⠈⡷⣚⠴⣇⣌⣳⠹⣜⡸⣣⢛⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠱⣈⠲⠀⠛⡽⣆⢂⠁⠀⠀⠀⠀⠆⠀⠀⠄⠀⠀⣄⠺⡴⣍⡙⠾⣜⣣⢟⡵⣳⢧⣛⢾⡱⢯⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⡵⡀⠄⠀⠀⠀⡰⢀⠀⢨⡄⢀⠢⢹⡣⠙⡤⡘⠦⡐⢆⡩⢜⠣⡟⡼⣣⢷⡹⢦⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠆⠉⠀⠀⠃⠘⠎⢲⡈⠔⡉⣆⠀⡳⢌⠥⠓⣌⠲⡘⠥⣛⠴⣹⠶⣹⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠠⠀⠀⠀⠂⡗⢨⠴⡘⣄⠳⡌⢎⡱⢢⢑⡩⢒⡩⢳⡝⡾⣱⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⠞⠰⣃⠎⡱⣌⢳⡉⠖⡰⢡⠎⡰⢃⠬⣁⠎⣳⡝⣮⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠆⡂⠀⠀⠀⠢⢌⡑⠎⠀⠴⣨⢞⡱⢎⠥⢊⡱⢡⠣⣘⠅⡎⣒⢡⠚⡤⣟⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠂⡜⠰⢠⠐⡌⡑⢂⠞⠀⣌⠱⢣⢎⡵⣋⠔⡣⣌⡑⢮⡱⣋⢴⡩⢆⡻⣴⢫⡝⣯⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⡐⠀⣐⢢⣠⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠆⢀⣱⠈⠀⢈⠠⢀⣄⠰⢀⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⠀⠰⣎⢳⠈⡔⡣⢇⡿⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠘⠀⠀⢰⡁⠀⠀⠰⠊⡷⢪⣍⣇⣰⣫⠝⣜⡘⣧⢹⢶⡹⣏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡄⠀⢐⠃⠀⠀⣆⠼⣘⠧⡙⢎⠷⣎⢿⡱⣟⡴⣫⢾⡱⣏⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⠀⡀⢰⠐⡄⢸⡀⢀⠆⣹⠏⠘⣢⠑⡎⠴⣈⠦⣙⠎⡷⡹⣎⠷⡜⢣⠙⠞⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡐⠈⡀⠀⠑⠨⠴⢑⡨⢐⡈⣇⠀⠧⡓⠬⣑⠢⣒⢩⢒⡱⢳⡜⣯⠽⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠀⡳⢀⠶⡘⣄⢳⡡⠓⡤⠓⡤⢃⠦⡙⢧⡞⣵⡛⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡔⢩⢃⠖⡱⣌⠧⢣⠙⡤⢋⡔⠣⡜⡘⢤⠊⢧⡻⢵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⢆⡙⠀⡔⣪⢜⡱⢎⡱⢡⠋⠴⡡⣌⠓⡰⣉⠦⡙⡤⣟⣣⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⡜⠀⢆⡱⢣⢎⡳⢣⠜⢢⡙⢢⠱⣎⢧⡱⢎⡴⣙⢶⡹⢮⡽⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⡐⠀⣐⢢⣠⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠆⢀⣱⠈⢀⠰⠠⢀⣄⠰⢀⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰⠀⢰⣊⢷⠈⡔⡣⢇⡿⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠘⠀⠀⢰⡄⠀⠀⡐⠊⢷⡘⣎⣇⣰⣫⠝⣜⡘⣧⢹⢶⡹⣏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡄⠀⠰⠀⠀⢀⣁⠼⣲⡙⠱⢎⠷⣎⢿⡱⣟⡴⣫⢾⡱⣏⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⠀⡀⢰⠐⡄⢘⠀⢄⠂⢿⡣⠓⡌⡓⣌⠲⢌⢢⢙⠮⡵⢫⡞⣵⢫⢦⠹⢹⡀⢫⠆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡐⠈⡀⠀⠑⠨⠜⢃⠌⢒⠰⣴⠀⡳⠅⡆⡓⡌⠦⣙⢢⡙⢧⢺⠵⣫⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠈⣱⢈⠦⡙⢦⡱⡍⡰⢱⢨⡑⢢⠆⡹⣙⢮⣻⡱⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡴⠍⣆⠣⣉⠶⡱⢃⡕⡊⠴⡨⢅⠎⡱⠌⣌⢣⡟⣵⢃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⡵⠀⠱⣤⠳⣍⢞⡡⢃⠦⡙⡰⣑⢊⠬⣑⠪⣄⢣⡞⣵⢫⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⡜⠀⢐⠣⣎⢳⡜⢦⢡⢃⠞⡰⢱⡍⣎⠶⣉⠶⣌⡳⣝⢮⣛⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⡐⡀⠰⣂⡄⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠆⢀⠡⠀⡇⡁⠠⢀⣄⠰⢀⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠐⠀⡄⠁⠀⠀⠀⠰⣎⢳⠈⡔⡣⢇⡿⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠘⠀⠀⢰⠀⠀⠀⡐⠊⡷⡜⣭⣇⣰⣫⠝⣜⡘⣧⢹⢶⡹⣏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡄⠀⢨⠁⠀⢀⢂⣜⠲⡝⣐⠫⠶⣭⣛⣞⡳⣵⢫⢾⡱⣏⢶⡩⢞⠼⣳⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⠀⡀⢰⠐⡄⢰⠀⢄⠂⢿⡬⠛⣄⠣⣉⠖⠤⡱⣈⠷⣍⠟⣎⢷⣫⢦⠹⢹⡀⢫⠆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡐⠈⡀⠀⠑⠨⠲⢁⠎⢨⠰⣴⠀⡺⠔⡡⢎⡑⢆⡹⢐⢮⡹⡜⣧⢳⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠂⢭⢐⠦⡙⢦⡱⡍⣒⠡⢎⠒⡬⢡⠎⡵⣹⢎⣟⡺⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡴⢋⡌⣒⢩⢖⡱⢣⠌⡱⢊⡱⢂⠣⡜⢰⠉⠾⣜⡳⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⡵⠀⢡⠖⣥⢋⡮⢑⠦⡙⣐⠣⡔⣩⠒⣡⠣⣙⢢⡯⣵⢫⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⡜⠀⢠⢃⡟⡴⣋⢖⡉⢦⠱⢌⡚⡵⢣⡜⢦⠳⣌⣳⢳⣣⢟⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⢒⠄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠈⣼⣾⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠑⠢⡑⠌⠂⠑⠀⠈⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⡿⠿⠻⠛⠉⠁⠀⠆⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⠉⠀⠁⠢⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡍⡁⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⣻⡆⢀⡔⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⢀⣤⣶⣴⣵⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⣻⢷⡯⣟⡽⣷⠀⠚⠀⢘⣢⡓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣿⣿⢿⣟⠏⠐⠸⣯⡟⣾⣽⣻⣽⠀⢠⠃⠃⡀⢫⠃⠀⢂⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⠉⠉⠉⠉⢹⣿⡿⣿⣟⣿⣯⣿⡷⣿⡽⣯⣟⡾⣽⢾⡟⠠⠐⠒⠚⣷⠁⣷⣯⠷⡏⠀⢘⠈⠀⠐⣂⠡⣄⢀⡘⠣⢭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠈⣷⣟⣿⣻⢯⣟⣷⣻⣷⣻⢷⣯⢿⠝⠛⠙⠛⣳⣶⠄⠃⠜⣻⡝⠡⠁⠀⠄⡮⢀⡁⢜⡤⣐⢂⠀⡃⠈⢤⠀⢁⠶⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⣾⣳⢯⡷⣯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠁⢢⡁⣀⣤⣑⠪⢱⡘⢔⠀⡉⠌⠳⠑⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢼⣻⡽⣯⣟⣾⣳⢯⣟⣿⣳⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠙⢉⠛⢶⣌⡁⢎⡰⠀⠡⢘⢂⠈⠐⢠⡔⣲⢒⠄⣀⠠⣀⠂
⠀⠀⠀⠀⣼⣳⣿⣷⣻⢾⣽⣟⡾⣽⣷⣯⢿⣽⡀⠀⠀⠀⠀⠴⢦⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡣⠄⠈⢀⠏⠈⠒⢠⠊⢈⠡⢋⠂⢀⠤⡂⠌
⠀⠀⠀⠀⢹⣷⣿⣷⣯⣿⢾⣿⣽⢿⣹⣞⡿⣞⣧⣀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⡀⠂⠌⢀⠠⣡⠃⢀⠂⠄⡉⠳⢭⠆⠁⠀
⠀⠀⠀⠀⠈⢷⣯⠻⠻⢾⣿⣇⢫⠻⢧⢻⡿⣟⡞⣿⠻⢷⣶⢾⣏⠀⠀⠀⠀⠀⡐⡀⠲⡢⡄⠀⠀⠀⠀⠀⠀⢀⠂⡐⠠⠁⠌⢠⡚⣤⠛⡈⢐⠜⡠⢁⡏⣞⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⠘⡠⢙⠌⢣⠙⠆⢶⣳⡀⠌⡀⠘⣽⣿⡄⠀⠀⠀⠀⠆⠀⢱⠀⠇⡅⠠⢀⡄⢠⠐⡄⡏⢤⠐⣁⡬⢄⡳⡙⢦⠡⡜⣡⣊⡤⢞⡼⡘⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡱⢈⡜⢤⠉⡜⢠⢂⠱⣌⡖⢬⣿⣿⡁⠀⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⢠⢏⢧⠒⡀⡏⣦⡛⣵⠺⡄⣷⡉⢦⠓⡜⣣⠗⣜⡣⠞⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠁⠐⠈⠤⢉⠦⡁⢎⣰⣟⢿⠂⣽⣿⠦⠀⠀⠘⠀⠀⢰⠁⠀⠀⡐⠊⢷⣊⢮⣇⡰⣛⢴⣙⡱⣫⢝⡶⣹⢞⣀⣐⣨⢛⡴⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠂⠀⠀⠀⢢⢉⠲⠈⠚⡿⡎⠄⠁⠀⠀⠀⠀⡄⠀⢨⠁⠀⢀⢃⡜⢦⡝⡘⢮⢳⣭⣛⢶⣫⠵⣎⢷⡹⣎⢷⡜⣼⠹⣞⡗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡽⠈⠄⠀⠀⡀⢰⠀⠆⢰⠀⡀⢆⢻⡎⠛⡄⣓⠢⣅⢢⢉⢆⡏⠿⡜⣧⣛⢮⢧⠸⢜⡂⢣⠇⠁⠂⠀⠀
⠀⠀⠀⠀⠠⢌⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⡐⠈⡀⠀⠉⠸⠘⠥⢘⠠⠒⣴⠀⡳⢆⡱⠢⢜⡨⠜⡰⢫⡜⣲⡝⣮⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡌⢒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠀⠀⠀⠈⣣⠬⣑⡙⢦⡱⢆⡒⠍⣆⠲⣉⠲⣡⠻⣴⣛⣬⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⡁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡔⡋⡔⢢⡜⢦⡙⢦⡘⡱⢄⠳⡠⢓⠤⠓⡤⢛⣬⢳⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⢑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⡐⢌⠢⡙⠀⠰⣙⢦⡙⢮⡑⢆⠲⢡⠎⡱⠌⡥⢊⡕⢢⢣⣞⡳⢮⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢡⢂⠱⡀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠜⡠⢑⠂⡄⢢⠘⠤⡙⠀⢨⠱⣍⠶⣩⢇⠜⢬⡑⠦⢹⡱⢫⡔⣫⢔⡫⢶⡹⢞⣹⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⡐⢀⡐⠒⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠂⠀⠈⠰⠀⡀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠀⠈⠀⠁⠀⠀⡁⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⡀⠀⢠⠂⠀⠀⡃⠊⢷⡚⣴⣃⣬⢳⡹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡄⠀⠰⠀⠀⠄⣃⣜⠲⣍⠱⣋⢮⡳⣝⢶⡳⣝⢮⡳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⠀⡀⢰⠐⡄⢘⡀⠰⣈⢹⡎⠛⡄⣓⠰⢢⡑⡌⢎⡳⡝⢮⡳⣝⢮⢧⠸⠹⡄⢛⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡐⠈⡀⠀⠑⠨⠜⢡⠁⢆⡘⣴⠀⡳⠌⡥⢃⠴⡉⡜⠤⣛⠦⣝⢮⢏⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠂⣹⢠⠒⣌⢣⠵⡩⢔⡉⠦⠱⢌⠣⡜⣹⡜⢯⡺⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡔⠣⣌⢃⢦⣋⢞⡑⠦⣉⠜⡱⢊⡱⠌⡴⢈⠷⣹⠶⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⢆⡙⠀⠰⣆⢏⠶⣩⢒⠸⡰⠡⢎⡱⢌⢢⡙⠤⢋⡴⣏⠷⣭⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⡜⠀⢨⡑⢎⣎⠳⣅⠎⣱⢢⢙⢢⡝⢦⢣⢎⡵⣋⠶⣭⣛⢶⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠒⣦⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣠⠀⠀⠀⠈⣴⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⡶⣾⣏⠀⠀⠀⠀⠀⡀⠄⣐⢺⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠆⠁⠸⠸⠰⠀⠠⢀⡄⢄⠢⡐⡎⢤⠒⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⢏⣳⠐⠄⣏⡲⣏⠷⣸⡄⢿⡠⡓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⡀⠀⢠⠂⠀⢠⡐⠊⡽⣊⢖⣏⣠⣳⢹⣌⡹⣆⢻⢮⡽⣝⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡄⠀⠰⠀⠀⡸⢁⣜⠲⣍⡙⠮⢳⣭⢳⢞⡳⣝⢮⡳⡽⢎⡷⣌⡳⠾⣭⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⠀⡀⢰⠐⡄⢘⡀⠰⠠⣽⠎⠛⡤⣘⢡⠣⢄⡋⣌⠷⡹⢎⡷⢭⣻⢦⠱⠹⡄⢫⠆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡐⠈⡀⠀⠑⠨⠜⢁⠃⠥⡐⣶⠀⡓⠦⣑⢊⠦⡑⡜⠢⡝⢮⡜⣧⢳⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠈⢳⡠⠓⣌⢣⡝⡒⡌⠦⢱⡘⡰⢃⡹⢣⡞⣵⢫⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡔⢣⠜⣑⢢⠳⣜⠱⢌⡱⢃⠴⢡⢃⠖⡡⢎⢳⣫⠼⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⢆⡙⠀⠰⣚⢬⢣⡛⠤⢋⡔⠢⣍⢢⢃⠎⡬⡑⡌⢦⡝⣏⡞⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⡜⠀⢨⠱⣍⢎⢧⡙⢬⢡⡊⡕⢬⢣⢏⡲⡱⡜⣜⣣⢟⡼⣹⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠐⣤⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣐⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⣶⢶⡟⠀⠀⠀⠀⠀⡀⢴⢐⡂⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⡀⠐⠃⠀⠂⢃⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠌⠀⠀⠠⠀⢸⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⢀⠄⠀⠠⡐⠊⣗⠺⣔⣇⣬⢳⡹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡃⠀⠈⠀⠀⢡⢂⡼⣌⠗⣩⠺⡜⣧⣛⢶⡳⣝⢮⡳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⢀⠀⢢⠑⡄⢘⡀⢌⡘⢼⠳⠘⡸⣀⠧⡘⡄⢎⡡⠟⡼⢣⡟⣼⢫⢦⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡀⠉⡀⠀⠑⠨⠜⢁⠆⡰⠘⣴⠀⡳⢆⠲⣑⡘⢢⡙⡌⢳⢣⢞⡵⣫⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠈⢵⡠⢓⡌⢣⡕⡎⢒⠤⣉⠖⡰⢉⠮⣙⢮⡳⣭⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡴⢋⠔⠣⣌⠳⣜⢡⠋⡴⢁⢎⡡⢃⠞⡠⢍⢳⣣⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⡕⠀⢨⡜⡳⣜⡹⢄⢣⠩⡔⣉⠦⣘⠡⢎⡑⢪⡔⣧⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⡜⠀⢈⠦⢝⡱⢎⡕⣊⢆⡓⢬⡘⢧⢣⡝⣎⠼⣱⢺⡱⣏⢾⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠠⢤⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣐⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⣶⢶⡟⠀⠀⠀⠀⠀⡄⢒⡐⡆⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠀⠚⠁⠀⠑⠥⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠘⠀⠀⠠⠐⠠⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⢀⠄⠀⡇⠔⠊⡷⣚⠴⣇⣨⣳⠹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡃⠀⠈⠀⠀⢌⢤⡚⡴⣍⡙⢮⠳⣭⣛⢶⡳⣝⢮⡳⡽⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⡀⡠⢲⠁⢆⢘⡀⠌⡌⣼⠣⠛⢤⡘⢢⡑⢆⡌⢣⠟⡼⢣⡟⣼⢫⢦⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡀⠉⡀⠀⠉⠠⠞⢡⠘⢠⠐⣶⠀⡳⢌⠥⡘⢆⡜⠱⡊⢵⠳⣜⢧⣛⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠀⢯⡠⠓⣌⢣⡕⡎⢒⡉⠦⢌⢣⡑⢎⢻⣬⢳⡝⡾⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠔⢢⠱⣉⢤⠳⣜⢡⠣⠜⣑⠪⢔⡸⢈⠦⢌⠳⣝⡳⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⠍⠀⢠⡳⡜⢮⡹⢄⠣⢎⠱⡌⣚⡐⠦⣉⠦⣉⠶⣭⢳⡭⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⠜⠀⢈⠆⢷⡙⢦⡓⣌⢣⢊⡱⢸⢥⠳⡜⣬⠲⣥⢻⡜⣧⢻⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣠⣤⣅⡪⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠛⠷⣬⡑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠀⢤⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣐⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢂⠐⡈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⣶⢶⡟⠀⠀⠀⠀⢄⢂⡐⠒⡆⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡀⠂⠄⢡⡎⡵⡩⠁⡠⢍⡰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠐⠌⠀⠈⠑⣥⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠣⣕⣊⢤⣚⠶⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠘⠀⠀⠠⠀⣿⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⡹⠶⣍⠶⣩⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⢀⠔⠀⡇⠐⠨⢗⡺⢴⣃⣬⢳⡹⣜⡱⣋⢞⡶⣹⢏⣠⣐⢪⡕⣫⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡇⠀⠈⠀⠀⣐⢃⡜⢮⡕⡩⠳⢎⡷⣹⢖⡷⣹⢎⡷⡹⣎⢷⣘⡳⠼⣫⢗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⡀⠦⢓⠰⣀⢘⡀⠰⡈⣼⠹⠊⡔⣡⢙⡂⠖⡡⢍⡞⡵⢫⡞⣵⢫⢦⠱⢙⣂⠹⡆⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡈⠉⡀⠀⠑⠠⠚⢅⠑⡰⠐⣶⠀⡽⠐⣆⠩⡜⢡⠚⣄⢫⠳⣜⢧⣛⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠄⠈⢳⢠⠓⣌⢣⡜⡱⢨⡑⡌⢣⠱⣈⠮⠽⣜⠧⣏⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠔⢊⠆⡭⢤⠳⡜⠥⣃⠦⡙⠤⢓⡰⢊⡱⢌⠻⣜⢮⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⠅⠀⢨⠼⣘⢧⡛⠬⣑⠢⣒⠩⡜⢡⠒⢥⠒⡬⡱⣏⠾⣜⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⠌⠀⢈⠆⢯⡱⢎⡥⠓⡤⢓⠤⢛⡜⣣⠝⣎⡜⡲⣵⢫⡻⣼⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⣀⣤⣁⠊⠵⣈⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⠉⠻⠶⣄⡉⡖⡈⠄⠡⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠀⢤⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢁⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣐⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠂⠌⠠⠀⠌⡰⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⣶⢶⡟⠀⠀⠀⠀⢄⢂⡐⠂⡅⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⡐⠈⠄⢡⡎⡵⣉⠡⢠⢍⠰⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠐⠌⠀⠈⠓⣥⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⢄⡳⢎⡱⢄⠎⡵⣈⡤⣞⢲⡑⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠘⠀⠀⠠⠀⣿⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣿⡠⢓⠌⠺⡵⣋⢶⡩⡖⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⢀⠄⠀⡇⠐⠨⢗⡺⢴⣃⣬⢳⡹⣜⡱⣋⢞⡶⣹⢏⣠⣁⡬⣓⢮⡓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡇⠀⠈⠀⠀⣐⢃⡜⢮⡱⣉⠾⡜⣧⢻⡱⢷⡹⣎⢷⡹⣎⢷⡸⢼⠹⣎⠷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⡀⠦⢓⠰⣀⢘⠠⡐⠌⣼⠹⠊⡱⣐⠲⢨⠔⢣⢉⡷⢹⢎⡷⣹⢮⢧⠙⢮⡁⢫⠇⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡈⠉⡀⠀⠑⠠⠚⢅⠢⢁⠒⣴⠀⠱⠦⣉⠖⡌⣃⠦⣡⢋⡞⣜⢧⣏⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠄⠂⢳⡨⢒⡌⣓⢮⡑⢆⡚⢤⡑⢢⢅⡚⠼⣜⢧⡞⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠔⢢⡑⢎⡰⢣⢎⡝⠰⡌⡒⠬⣡⠒⣌⠣⠜⡪⣝⡳⣅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⠅⠀⠰⡜⢮⡱⣋⠖⣈⢇⢒⡉⢖⡡⠚⣄⠣⢣⡑⢮⡳⣭⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⠌⠀⢨⠱⡝⢦⠳⣭⠘⡴⣈⠦⡙⢮⡱⢣⢎⡵⢣⢽⣣⢟⢶⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠱⣀⠣⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠈⠀⠉⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⢠⠔⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠙⠀⢰⣍⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⠋⠄⠈⠓⡂⠈⡐⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠈⠀⢁⠆⠱⣀⣀⠌⠳⢬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡮⠐⡈⠼⡠⢔⡠⢀⠃⡈⢠⡀⢁⡖⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢑⡁⠀⠀⠁⠊⠱⡌⠴⠀⠌⡈⠙⠎⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠚⠻⢶⣤⣄⣈⢑⡊⡔⠠⢁⢸⢀⠈⠐⢠⠴⣲⢲⠀⡄⢠⠰⡀
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠀⢤⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠐⢀⠎⠈⠑⠢⠊⢁⠡⢋⠀⢈⠄⡒⠄
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣐⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡐⠠⢈⠀⠄⡱⡁⢂⠄⠂⠌⠫⡝⡆⠁⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⣶⢶⡟⠀⠀⠀⠀⢄⢂⡐⠢⡅⠀⠀⠀⠀⠀⠀⠀⣀⠂⡐⠠⠁⢂⢈⢮⡱⡍⠁⢤⢉⡔⢁⡽⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠐⠌⠀⠈⠒⣥⠀⠠⢄⣠⠐⢄⠒⡬⢰⠐⣁⡬⠄⣞⢣⠓⡤⢚⡴⣡⠤⣞⠲⡕⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠘⠀⠀⠠⠀⣿⠀⠀⢠⢇⣻⢀⠊⣕⡣⡟⣥⢛⡄⣯⡂⢏⠰⡙⢶⣣⢛⡬⡛⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⢀⠔⠀⡇⠐⠨⢗⡺⢴⣃⣬⢳⡹⣜⡱⣋⢞⡶⣫⢏⣄⣈⢦⢣⣏⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡇⠀⠈⠀⠀⣐⢃⡜⢮⡱⣉⠾⡜⣧⢻⡱⢷⡹⣎⢷⡹⡞⣼⡱⢎⠷⣎⠷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⡀⠦⢓⠰⣀⢘⡀⠰⡈⣼⠹⠊⡱⣐⠲⢨⠔⢣⢉⡷⢹⢎⡷⢭⣻⢦⠙⢞⣀⢫⠇⠁⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡈⠉⡀⠀⠑⠠⠚⢅⠑⡰⠐⣶⠀⠱⠦⣉⠖⡌⣃⠦⣡⢋⡞⡼⣣⢗⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠄⠈⢳⢠⠓⣌⢣⠞⡥⢒⡸⣐⢡⢒⡡⢚⠼⣱⢏⡾⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠔⢊⠆⡭⢤⡓⢮⡑⢣⠔⡌⢆⠣⡔⡩⢢⠅⡻⣜⢧⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⠅⠀⢨⠼⣘⢧⣙⠦⡉⢖⡘⠬⣌⠱⣂⠵⡡⠚⣄⡻⣎⢯⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⠌⠀⢈⠆⢯⡱⢎⠶⣡⢉⡖⣨⠱⣎⠳⡔⢮⡱⣙⢦⡻⣜⣫⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣄⠣⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢀⣶⣷⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⣀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⡀⢠⠀⠒⢀⠡⠘⠐⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠂⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠠⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡷⣿⢿⣽⡆⠠⢆⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡞⣿⢯⡿⣽⢫⣞⣷⠀⠉⡀⠰⢍⢯⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣷⣿⡿⣟⠇⠐⠸⣯⡟⣽⣻⣞⣿⠀⢠⢉⠡⠈⠒⡁⠈⠔⢠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢹⣿⢿⣟⣿⣿⣽⣿⡾⣯⣟⣯⢷⣻⣞⣷⠛⠁⠐⠒⠛⡷⢁⣯⣷⡻⡇⠀⢘⠀⡀⢈⠅⠱⢄⡠⠌⠛⠴⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⣾⢯⣿⣻⢾⣽⣳⣟⣷⣻⣞⣯⠷⠏⠙⠉⠛⣻⣶⠄⠃⠘⣽⠗⠡⠁⠀⠄⡶⠀⠆⠺⢔⠢⡔⣀⢃⠈⠰⣀⢠⠞⡴⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣯⣛⡷⢯⣟⡾⣧⣟⣷⣻⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠠⢙⠄⠁⡀⠈⠐⠌⢆⡚⢠⠀⠌⡙⠘⣭⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢾⣶⣻⣽⣻⢾⣽⣳⢯⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⠻⠶⣤⣄⣀⠉⡆⡜⠀⠄⢹⢠⠀⠀⢡⢖⡚⢦⠂⡄⠰⢠⠁
⠀⠀⠀⠀⣼⢾⣽⡷⣯⣟⣾⣯⣟⣾⣿⢯⡿⣽⡀⠀⠀⠀⠀⠀⢤⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠈⢀⠃⢀⠉⡰⠉⢈⠹⢩⠁⣠⠒⡅⠈
⠀⠀⠀⠀⢹⣯⣿⣿⣷⣻⣾⡷⣿⣾⣹⣯⣟⣷⣧⣐⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⡁⠂⠄⡀⢖⠁⡐⡀⠂⠌⠻⣔⡃⠀⠀
⠀⠀⠀⠀⠈⢻⣯⠻⠳⢿⣷⣇⢏⡛⢧⠻⣯⣿⡜⣿⠻⢿⣶⢶⡟⠀⠀⠀⠀⢄⠂⡄⢂⡅⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠠⢀⠁⢢⡝⣬⠳⠈⢠⡑⡡⢁⢮⡹⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡈⠤⡙⠌⡌⢓⠱⠦⣗⡠⢈⡀⠘⣽⣿⡄⠀⠀⠀⠐⠈⠁⠈⠒⣵⠂⠀⠢⣔⡠⠘⡄⣇⠲⣀⡥⢦⠊⢷⡘⢦⠡⡜⣡⢂⡰⣼⢲⡙⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡒⠱⠌⡔⡡⢊⠔⡠⢑⣢⡖⡸⣿⣿⡁⠀⠀⠀⠘⠀⠀⠠⠀⣿⠀⠀⢰⢣⢳⠁⡄⣇⠷⣍⡞⡭⣆⢧⡙⣆⠓⡜⡧⣏⢵⣊⠧⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠘⠠⡑⢪⡐⢡⡺⣝⣿⢁⢻⣿⠧⠀⠀⠀⠀⠀⢀⠆⠀⡇⠐⠨⡗⡭⢎⣇⣰⣩⠞⣬⣸⠵⣹⢎⡷⣚⢂⠠⡱⢎⡶⡹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢡⠘⡒⠄⠛⡾⣆⢂⠁⠀⠀⠀⠀⡇⠀⠠⠀⠀⣐⢃⡴⡹⡜⣩⠚⡵⣫⣝⡳⣬⡝⣶⢫⡾⡹⣞⢭⢳⢽⡺⣥⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⡼⡀⠄⠀⡀⠦⢓⠰⢠⢐⠠⡐⠌⣼⠣⠉⡖⢡⠚⢤⢡⠢⡙⡖⠽⣎⢷⡚⣷⣃⢪⠽⡀⢛⡆⠑⠀⠀⠀
⠀⠀⠀⠀⠐⡈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⡈⠉⡀⠀⠑⠢⠜⢃⠰⠡⡐⣶⠀⢘⢇⡚⠤⡃⢥⠣⢍⠺⣜⢲⡝⣶⠊⠀⠀⠉⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠑⡌⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠂⠠⠀⠀⠈⢣⡑⢲⡈⢇⠞⡦⢌⢓⡘⢢⠍⣢⠹⡜⣣⢟⡼⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠔⡊⢬⠡⡜⢮⡙⠖⣉⠦⡘⡅⡚⣄⠣⢒⡡⢛⡼⣱⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠜⡠⠀⠀⢀⠰⡈⠆⠅⠀⠸⣔⢫⡜⣣⠙⡜⡰⢢⠱⣌⠱⢄⡋⠦⣑⢊⣼⢣⣏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠜⡠⢑⠂⡄⢢⠑⡌⠌⠀⢨⡑⢎⡳⡜⣥⠋⡴⣑⠢⣓⢬⠳⣌⢞⡱⣌⢧⣏⢷⣚⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⢀⠄⡠⢀⠄⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⢀⡀⢄⠐⠈⠁⡘⠄⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢶⠐⢂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠈⣀⡰⠼⣌⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⢀⠃⠄⠠⠈⡇⠀⠑⠢⡔⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⠀⣘⠀⡀⢣⠘⠢⠤⠐⠙⠲⢜⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠐⡚⠄⡜⠂⠕⣊⠖⣈⠖⠠⠘⢄⡠⢏⡳⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠐⠸⠄⢀⣀⠀⠉⠂⢎⡡⢂⠄⠠⠉⢌⢧⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠉⠛⠳⠦⣤⠀⠐⡎⡄⠡⢈⠸⢠⠀⠀⢡⢖⣚⢦⠂⡄⠰⢠⠁
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⢤⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⢀⠉⡰⠋⠚⢘⢎⡁⢠⠒⡅⠈
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠄⡁⢂⠠⢀⠖⡁⠄⠡⢈⠘⢼⡩⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⣟⠈⠀⠀⠀⢄⠂⡄⠂⡆⠀⠀⠀⠀⠀⠀⢀⠰⠁⡈⠐⡀⢂⢰⢋⡜⡴⠊⢁⡔⢂⠀⣏⢳⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠑⡱⠀⠐⠂⣤⡃⠜⠠⡏⠴⣈⣰⢴⠂⣏⠞⡜⢡⡐⣣⠘⣠⠼⣍⠾⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣓⠀⠀⡐⡦⡝⡄⢃⣎⡅⣏⡶⣍⡖⣭⠚⣌⠣⡜⢧⡏⡵⢫⡜⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠄⠈⡳⣱⠹⣆⣡⣺⡜⣬⣰⢧⡹⣎⢷⣪⠐⠈⣕⡹⣜⡳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠵⣣⠋⡝⠶⣣⢻⡵⣬⢵⡹⣝⢮⡳⣏⠻⣴⢳⡝⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⢣⠱⣘⠢⢅⠣⣌⡱⢫⠳⣝⢮⡳⣍⢛⡴⠂⢟⡆⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠣⢆⢍⣊⠱⡄⢣⠍⡗⣎⢷⡹⣭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡳⢌⠢⠜⢢⡉⠦⡙⡜⢮⡳⣝⢶⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⢹⢨⠓⣌⠣⣙⠢⢅⢣⠱⣘⠢⡙⣮⢳⡵⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⣋⠧⢩⢄⠳⣠⢙⡌⢆⠓⡤⢣⠱⣼⣓⠾⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⣍⠲⡡⢎⡒⡡⢏⡼⣌⠞⣥⢲⡹⠶⣭⢻⡅⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⠤⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⢿⠈⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⢀⠠⣤⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⣟⠀⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠁⠆⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣲⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⢄⠠⠄⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢀⠠⣀⠂⠁⠀⠌⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠐⠡⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣼⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⢀⠡⠀⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀
⠈⠉⠉⠉⢩⣿⡿⣟⣿⣯⣿⢿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡟⠀⢀⢘⠀⡀⢓⡈⠓⠄⠂⠌⠛⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⣿⣻⣟⡿⣽⢯⣟⣿⣳⣟⡷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⠟⢁⠃⠀⠂⡙⢄⡉⠑⠢⢙⡌⢢⠍⡐⠈⠦⠄⣞⡹⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣷⣛⡾⣽⣫⣟⡾⣵⣟⡾⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠸⠀⣠⣄⠀⠈⠐⠣⢱⢠⠂⠐⡈⢡⠏⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢾⣳⢯⡷⣯⢷⣻⣽⣳⣿⡽⣿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠖⡌⠐⠠⡸⢠⡀⠈⢰⢺⡹⣹⠄⢢⠐⡌⠄
⠀⠀⠀⠀⣼⢯⣿⡿⣽⢯⣿⣞⣷⣻⣿⢯⣟⣿⡀⠀⠀⠀⠀⢀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡀⠰⠁⡀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡔⠁
⠀⠀⠀⠀⢸⣯⣿⣿⣽⣟⣾⡿⣾⣽⡻⣟⣾⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⠠⠁⠄⣐⠬⢁⠐⡀⢂⠐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠹⢾⣯⣗⡩⢛⢷⠻⣯⣿⠆⢿⠛⢿⣶⢶⡟⠀⠀⠀⠀⢄⠠⠄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠁⠌⡰⢍⡚⡴⠊⢀⠆⡨⠀⣏⢖⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠑⠀⡘⢠⠩⣉⠑⡊⠕⠦⢷⡀⠄⡀⠘⣼⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⠬⠁⠈⠂⣥⡊⠔⡈⡖⢢⣉⡴⢺⠠⣏⢧⢹⡀⠴⣉⠎⣁⣼⢚⡬⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⢂⠓⠤⢃⡜⣈⠒⠤⡙⢢⡵⡤⣿⣿⡅⠀⠀⠀⠐⠀⠀⠀⠄⣋⠀⠀⡰⢦⡹⡄⠡⡝⡆⢧⢏⡝⣦⣙⠆⡧⢘⡱⢏⣞⡱⢎⡳⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠂⠥⢘⠠⡉⢆⣱⢿⣻⠅⢺⡿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠄⢈⢳⢣⡓⣇⣡⣹⡜⣫⠜⡼⣱⢫⠷⣜⠠⠈⣕⢮⡱⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠱⢈⠲⠀⠛⡽⣎⠰⠁⠀⠀⠀⠀⡄⠀⠈⠀⠀⢸⢠⡎⢧⡣⢍⡋⠷⣣⢟⡭⡽⡼⢭⡻⣽⢎⣟⡹⣜⣲⡝⣇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡼⡄⠀⠀⠠⡰⢊⠕⡠⠐⡂⠌⡂⢖⢫⠃⢗⡨⢌⡱⢂⠎⡜⣱⢫⠷⣹⢎⡿⣌⢱⢺⠀⢻⡌⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⡀⡈⠉⠁⠀⠐⠡⠱⠕⡌⠰⡈⣼⡀⢘⠦⢃⡒⡩⢆⠱⡃⣜⢣⢇⡿⣸⢣⠀⠈⠁⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢎⠰⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠂⠠⠀⠀⠀⠱⣡⠒⡬⣁⢎⠶⣁⠎⡱⢨⠱⡘⢤⠛⣼⢣⣏⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠌⢢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠐⢃⠦⡙⢤⡹⢨⡓⢬⡘⢡⠣⢥⢉⠦⡙⡄⡛⡼⢮⣕⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢢⠡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⠘⠀⢈⠶⣙⠶⣩⠇⡜⢢⡘⢅⡚⢤⠃⡎⠴⢡⠱⣼⢳⢮⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢁⠆⡱⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⡈⠆⠀⢠⠊⢷⡩⢞⡱⡘⢬⡡⠜⢢⡝⢮⡱⢎⡵⢣⣛⡼⢫⡖⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⡀⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⣟⠈⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⢀⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⢿⠈⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⢀⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⢿⠈⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⣟⠈⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⢁⡃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣥⣷⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠤⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠠⠀⠀⡄⠰⠀⠁⢈⠰⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⠢⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠩⠄⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⢿⡽⣇⠀⢧⠘⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⣀⣴⣶⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⢿⣻⢷⡯⣟⡽⣿⠀⠀⣀⡰⠚⣬⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢿⣿⣾⣿⣟⠏⠐⠘⣯⡟⣷⢯⡿⣽⠀⠠⠁⠄⢂⠈⠇⠀⠐⠡⡖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⣿⡾⣿⡽⣯⣟⡿⣾⡽⡞⠁⠐⠒⠚⡿⠁⣿⢯⡿⡿⠀⢀⠸⠀⣀⠓⡈⠓⠄⠂⠌⠛⢸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⣯⣟⣿⣻⢯⣟⣾⡽⣷⣻⢷⣯⡟⠗⠙⠉⠙⣻⣶⠄⠃⠘⢹⡟⢁⠃⠀⠂⠛⢄⡌⠑⠡⢋⡜⢠⠍⡐⠈⠦⠄⣯⢱⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡟⣞⡷⢯⣟⡾⣧⣟⣷⢯⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠸⠀⣠⣄⠀⠈⠐⠣⢱⠰⢀⠂⢈⠡⠧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢺⡽⣯⣟⡿⣾⣽⣳⣟⣾⣯⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⠞⡄⠈⠄⡸⢡⡀⠈⢰⢺⡹⣹⠄⢢⠐⡘⠄
⠀⠀⠀⠀⣼⣻⣽⣯⣟⡾⣽⣷⣻⢾⣷⣟⡾⣿⡀⠀⠀⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠰⠁⠀⠌⡩⠃⢃⠳⣡⣀⣠⠂⡜⠀
⠀⠀⠀⠀⢹⡷⣿⣷⣯⣿⡽⣷⣯⣿⡻⣾⣽⣳⣷⣠⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠄⠡⠀⢌⡜⠁⡐⢀⠂⡐⠧⣎⠇⠀⠀
⠀⠀⠀⠀⠀⢻⣯⠻⠽⢷⣿⣇⢫⢓⢯⠻⡷⣿⡖⢿⠻⢿⡶⣶⣟⠈⠀⠀⠀⢄⠂⡄⢲⠀⠀⠀⠀⠀⠀⢀⢀⢢⠁⡐⠠⠈⠄⣱⢋⣌⠳⠈⢄⡒⢠⠂⣝⢲⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⡘⢠⢉⠋⡌⢊⠵⠬⣗⡠⢀⡀⠘⣽⣿⡆⠀⠀⠀⠐⠌⠡⠈⠒⡜⠁⠈⠂⡵⣈⠢⠌⡖⢢⣉⡤⢗⠢⣇⠯⡜⢡⡘⢦⢈⣁⡼⢎⡳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⠒⡌⠖⣌⠢⡘⠰⢠⠑⣪⡴⡴⣿⡿⡇⠀⠀⠀⠘⠀⠀⠀⠄⣚⠀⠀⣰⢣⢳⡐⠌⣣⢇⠧⢟⡺⣄⣏⠲⡍⠦⢹⢎⡏⢶⡙⢮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠐⠄⢣⠡⢃⠆⣹⣽⢻⠆⣹⣿⠧⠀⠀⠀⠀⠐⠀⠔⠀⠆⠌⢈⢇⡏⡖⣇⣬⣱⡎⢯⡜⡵⣹⢎⡷⣜⠠⠁⣙⡜⣣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠱⣈⠲⠁⠙⡿⣆⠒⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠡⣎⠞⡼⡉⠞⡖⢧⡻⣭⢭⡜⣧⢻⣼⢣⣟⡹⣱⢺⡵⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡽⡀⢂⠀⣀⠢⢓⠡⢂⠄⡃⠌⢂⢇⢫⠎⡓⠬⡱⢘⢢⠱⣈⢳⠹⣎⠷⣎⠷⣎⢱⡹⠀⢟⡅⠂⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⡀⡐⠈⠁⠀⠑⠨⠔⠓⡌⠢⡘⢴⡀⢘⠧⣘⠡⢎⢒⡡⢣⠹⣌⡳⣭⢻⢭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡄⢣⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁⠀⠈⠀⠠⠀⠀⠀⡱⣡⠒⡭⣁⢎⡖⡡⢍⢢⢃⠬⣑⠪⡕⣳⢭⣳⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡐⠦⢡⠓⣤⠹⣌⠞⡡⢎⠢⢍⡒⡌⡱⠰⣁⢋⡶⣫⢖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡔⢡⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢌⠒⠀⠀⢀⠐⡄⢣⡐⠀⢨⢲⡙⢦⡛⣌⠲⡑⠬⣑⢊⠴⢡⡑⢣⠜⢢⢼⣣⢟⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡄⢒⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢌⠢⡉⢆⠠⢂⠱⣈⠒⠀⠐⡌⢧⡹⢲⡹⢄⠳⣨⠱⢌⡏⣞⡡⢞⡥⣚⣥⢟⡼⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣴⣾⣿⣟⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠑⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣡⣿⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⢂⠄⢢⠐⡠⠂⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠄⠠⢄⠢⠈⠀⠠⠈⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠛⠛⠉⠁⠀⠆⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀
⠀⠀⠀⠀⢈⡉⡁⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⢿⡽⣇⠘⢣⠈⢂⠤⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⣀⣴⣶⣤⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⡗⢿⣻⢷⣯⢟⡽⣿⠄⠈⡄⡜⠚⡴⠫⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⢯⣿⡿⣿⢿⠛⠀⠎⣿⡻⣞⣯⢿⣽⠂⢰⠐⠀⡐⠈⢇⠀⠈⠱⣎⢳⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀
⠈⠉⠉⠉⢩⣿⣿⢿⣻⣿⣽⡿⣷⡿⣽⢯⣟⡿⣽⣻⡽⠋⠀⠒⠒⣻⠃⣿⣞⣯⠿⠁⠰⢈⠀⢤⠩⡈⢑⠒⡀⠌⠃⢫⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠐⣯⣟⡿⣿⣻⣽⣻⢷⡿⣽⣻⢾⣽⠓⠋⠏⠛⣻⣶⣄⠂⡐⢹⡯⢁⠊⠀⠂⠏⢄⠎⠁⠣⢍⡒⢤⠃⢌⠈⠦⠐⣧⡙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⠾⣽⢳⣟⣾⣹⢯⣟⣷⣻⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠀⠀⠀⠀⠀⠈⡘⢀⣠⣄⡀⠈⠐⠩⠖⠰⢀⠂⠄⢩⠎⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢼⣯⣟⣷⣻⣞⣷⢯⣟⣾⡷⣿⣻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠉⠛⠓⠦⠄⠈⢇⠃⠠⠈⡜⢢⡀⠁⣰⢫⡝⣣⠅⠢⠐⡱⢀
⠀⠀⠀⠀⣼⣳⢿⣷⣻⢾⡽⣟⡾⣽⣿⢯⣟⡾⡄⠀⠀⠀⠀⠀⡀⢀⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡀⠱⠀⠠⠀⡽⠀⠓⢸⢡⣀⡤⡑⡄⠃
⠀⠀⠀⠀⢺⣽⣿⣿⡽⣯⣿⣿⣽⣯⢿⣻⢾⣽⣷⣀⠄⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠡⠐⡀⠂⣄⢏⠀⢂⠁⢂⠘⢦⡓⡇⠀⠀
⠀⠀⠀⠀⠀⢿⣯⠻⠻⢿⣞⣧⢛⠺⢧⠻⣿⣻⡆⢿⠻⢷⣶⡶⣿⠘⠀⠀⠀⢄⡐⡄⢺⠀⠀⠀⠀⠀⠀⡀⢠⡘⠀⠌⡠⠁⠄⡱⢎⡜⡜⠊⢐⠤⡁⢂⡝⣲⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠓⠀⡘⢨⠙⡌⢩⠘⡓⠴⢯⡑⠨⢁⠘⣺⣿⡇⠀⠀⠀⠨⠄⠐⠈⠑⣼⠀⠁⠚⣤⡃⡘⣀⠳⡌⣥⠴⣺⠰⣙⠮⡜⡠⢜⣡⢊⣐⡼⣜⡱⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠓⣂⠓⡌⢆⡱⢈⠒⠤⡉⢦⣏⡄⣿⣿⡇⠀⠀⠀⠌⠀⠀⠀⠄⡿⠀⢀⣘⢦⢳⡐⠄⣳⠎⡧⠟⡼⣌⢧⡃⢧⠑⠮⣇⡏⣖⠳⡬⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠁⠈⠢⠌⡱⢌⠢⣱⣟⢯⠇⣸⣿⠧⠀⠀⠀⡀⠀⢀⠆⠀⡇⠀⢉⡜⣎⢧⣇⣌⣣⣛⠦⡝⡶⣹⢎⡷⣚⠠⢁⣑⠯⣜⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⡨⠑⠤⠙⢯⣗⠨⠁⠀⠀⠀⠠⡅⠀⠀⠀⠀⢌⣡⠞⡼⡘⢆⠫⠞⡵⣭⢻⢭⡴⣫⢟⣼⢳⣏⠯⣜⣳⡝⡦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢞⡄⠂⠀⢄⡢⢳⠐⠤⢘⠠⡈⠌⡜⢫⠊⡕⡊⢥⠋⡔⢢⠍⡺⢱⢏⠾⣜⡳⣎⢳⠼⡀⢟⡌⠃⠀⠀⠀
⠀⠀⠀⠀⠀⡱⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡀⡘⠁⡀⠀⠘⠠⠙⢅⡘⠰⠘⣶⠀⢘⠵⢊⡜⠌⡥⢊⢇⡹⢎⡝⣮⢳⡭⠀⠈⠁⠘⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡅⢣⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠂⠠⠀⠄⠂⢱⡡⠓⣌⠓⣎⢖⡡⠜⡘⠴⡉⢦⠘⡭⢞⡵⣫⢞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⡀⠔⢊⠴⠩⣤⢛⢬⠚⣄⢋⠬⡱⢘⠤⢋⡔⢣⠙⣵⢫⣖⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠔⡩⡀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⢀⠊⡄⠀⠀⢀⠰⡈⠆⠅⠀⠨⣖⡹⢆⡏⢎⡑⢢⠎⣒⢡⠍⡜⢢⠜⠢⣍⢼⡳⣎⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠃⡔⡑⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠢⡑⢌⠂⡄⢢⠑⡌⠌⠀⢨⡑⢮⡱⢫⡜⢢⢍⡲⡘⢤⣋⠞⡼⡸⣌⢳⢬⣳⢳⡭⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣟⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢊⡔⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢢⣾⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⢀⠢⠐⡌⡐⢢⠡⠌⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠛⠀⠀⡀⠄⠀⠂⡔⠈⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠀⡐⠠⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⢿⣷⡀⢀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠉⡍⠁⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⢿⣯⣷⡈⠇⠈⠤⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⢀⣤⣶⣥⣶⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡽⢾⣻⢷⣯⡟⣮⢿⡆⢠⢤⠃⠋⡶⡙⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣯⣿⣷⣿⢿⡛⠡⠌⢿⣻⢲⡿⣽⢯⡇⠈⡠⠀⠡⡈⣄⠀⠁⠲⣭⢓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀
⠉⠉⠉⠉⠙⣿⣿⡿⣟⣿⣯⣿⢿⣞⣯⣟⡷⣯⢿⡽⡾⠋⠤⠂⠒⢻⠏⢸⣽⣯⢿⠅⡀⡇⢀⠡⢅⡀⢓⠂⡁⠄⠹⢘⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⣿⢯⣿⢿⡿⣽⢯⣟⣾⢷⣯⣟⣯⡿⠙⠛⠛⣳⣶⣤⠘⢠⠻⣟⠎⠘⠐⠈⡑⠌⠌⠑⠬⢃⠎⠴⣁⠢⠘⠢⠰⣫⢝⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣯⢟⡾⢯⢿⣹⣟⡾⣽⣻⢾⣽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠀⠀⠀⠋⢄⣤⣄⡀⠈⠐⠢⡙⡀⠆⢁⠠⠸⡕⠡⢀⠀⢀⡀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢼⣳⢯⣟⣯⣟⡷⣯⢿⣽⣯⢿⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⠓⠶⠀⠐⢥⠃⢀⠂⡝⢠⡀⠀⣸⢣⡝⣣⠅⠢⠘⠰⣀
⠀⠀⠀⠀⣸⢯⣿⣟⡾⣽⣻⣽⣻⣞⣿⣟⡾⣿⡄⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠘⠀⠠⠐⡜⠁⡑⠪⢥⣀⡴⡉⠖⠀
⠀⠀⠀⠀⢸⣟⣿⣿⣽⡷⣯⣿⣳⣯⣟⣾⣽⣳⣿⡀⠄⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠡⢀⠁⠂⣤⢋⠄⠂⡄⠡⠘⠦⡳⡅⠀⠀
⠀⠀⠀⠀⠀⢿⣿⠿⡻⣽⣟⣇⢫⠻⢮⡻⣾⣟⡇⢿⠿⢷⣶⣤⣾⠉⠁⠀⠠⡀⣌⢁⢢⡀⠀⠀⠀⠀⢀⠀⡄⡍⠀⢌⠐⠠⠈⣜⢦⡩⡜⠡⢀⠖⡡⠐⣧⢹⠀⠀
⠀⠀⠀⠀⠀⠀⠛⠓⠀⡢⢌⠛⠤⢋⠅⡓⠴⢯⡑⢨⠁⠈⢳⣿⣏⠀⠀⠀⠊⠀⠀⠉⠐⡄⠈⠁⠸⣥⠊⡰⢠⠣⣍⡤⣞⠺⢱⡩⢖⡱⢠⠒⣍⣂⡰⢞⡬⡓⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢅⠚⡘⠢⡅⡊⠔⡡⢂⠍⣢⣯⡄⣾⣿⡟⠀⠀⠀⠃⠈⠀⠀⠀⠆⠀⣀⢞⡴⢳⠠⢡⠳⣆⡳⢍⡻⣤⣛⠤⣃⠣⡹⢎⡶⣩⠳⡜⠁⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠁⠑⠠⢉⠒⣅⠊⣼⣛⣷⠃⢼⣿⡷⠀⠀⠀⠀⠀⢠⠁⠀⠓⠀⣉⠞⣜⢣⣃⣆⡻⣜⠦⣝⠶⣱⢏⡷⣓⠀⢡⣙⠶⣱⡛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠣⢄⡙⠄⠙⠾⣇⠪⠙⠀⠀⠀⠤⡀⠀⠠⠀⠀⢏⢤⣋⢞⡜⣡⠛⡼⢳⣭⢻⡤⢯⡝⣏⢾⣱⡻⣍⢞⣹⢧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡽⡄⠂⢀⢂⠜⡚⠄⣂⢸⠀⡡⠚⡘⠧⠊⡔⠢⡍⡔⠣⣄⢋⡜⢧⠻⣜⢧⡳⡏⢜⢮⠀⢯⠎⠁⠀⠀⠀
⠀⠀⠀⠀⠠⣁⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣀⠉⠈⠀⠈⠐⠠⠊⢣⠐⡡⠱⣇⠀⣍⠳⡰⢌⠣⡔⢪⠜⣸⢓⢮⡳⣝⡣⠀⠈⠁⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡄⠣⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠂⠄⠠⠀⠈⢧⠔⡣⠜⡲⣌⠳⢄⠣⣃⠜⡡⠎⡔⢫⢞⡵⣫⢵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢌⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢠⠘⠆⣍⠒⣥⠳⡜⣍⠊⡵⢠⢋⡔⠣⡜⢡⢊⡜⢧⡻⣔⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠃⠴⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⢃⠆⠀⠀⠀⠢⢌⡑⠂⢀⠱⣌⠻⣔⢫⠔⡬⢑⢢⠃⣖⢨⠱⢌⡃⢖⢨⡳⣝⢮⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡜⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠑⡌⠢⢄⠰⣈⠑⣂⠂⠀⢤⢋⡜⡳⢬⢣⠚⢤⢋⠤⢛⡬⢳⡜⣎⠼⣌⢧⡻⣜⢧⡇⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣻⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⣡⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠐⣠⣿⣶⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⣀⠐⠤⡐⢠⠂⡄⠢⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠛⢀⠠⠀⠄⣀⠢⠈⠀⠀⠌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠡⡀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⣧⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⢛⠛⠋⠉⠈⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣿⣻⢬⠑⠂⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣨⣢⣁⣆⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣾⣟⣯⢷⡯⣽⢿⣀⡰⠜⣥⢛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣟⣾⣿⠛⠠⢹⡾⡽⣳⣟⣾⣻⠁⡁⠄⡈⢋⠀⠘⠰⣠⠖⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀
⠚⠿⠿⠿⠿⣿⣿⣿⣿⡿⣿⣟⣿⣟⣯⣷⢯⡿⣽⣻⢿⡽⢊⠠⠤⠴⣿⠃⣿⣞⡷⣿⠰⠀⠠⢘⡈⠢⢄⠄⡁⠛⡴⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⡿⣟⣷⢿⣿⣟⣿⣻⣞⡷⣯⢿⣽⣳⡟⢿⠶⠶⣤⣄⡀⠏⡰⢿⡾⠏⠅⡺⡀⣃⠢⢕⠲⡌⢄⡘⠠⢀⢣⣀⢨⡳⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡽⣾⢯⣷⣻⣞⢷⣻⡽⣯⣿⣾⡃⠀⠀⠀⠀⠙⠉⠁⠀⡀⠜⠁⠀⠀⠁⢰⠁⠀⡀⠈⠘⠢⢅⠳⡀⡄⠠⢈⠓⢩⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣼⣻⣞⣶⣻⣞⣯⢿⣽⣳⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛⠻⠶⣤⣄⡀⠈⢱⢠⠁⠂⢄⡓⡀⠉⠐⡠⢴⡒⡖⡄⢀⠠⢠⠂
⠀⠀⠀⠈⢹⡾⣽⣷⣻⣞⡷⣟⣾⣻⣯⣿⢯⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠆⠈⠀⢢⠁⠑⠂⣤⠋⢧⡹⡜⠁⢈⠤⡃⠐
⠀⠀⠀⠀⢸⡿⣽⣷⣻⢾⡽⣿⣞⡷⣿⣯⣟⡾⣽⡀⡀⠀⠀⠀⠀⢂⣾⡄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠂⠌⠁⡀⠌⡴⠁⡐⠀⠆⢹⠲⣍⡆⠁⠀
⠀⠀⠀⠀⠘⣿⣿⢿⣿⣯⣿⣟⢫⣿⣷⢻⣾⣽⢿⢳⣦⣴⣀⣀⢀⣾⠻⠁⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠄⡈⠄⠡⡰⢎⠴⣀⠤⠁⢌⠄⡙⢦⣃⠀⠀
⠀⠀⠀⠀⠀⠙⡾⣏⠂⢛⠳⢿⡦⠥⣙⠧⡙⣟⠻⠘⠯⠉⢿⣻⣯⠧⠁⢀⠱⠄⠣⣘⠤⠄⠀⣄⠠⢀⠰⡀⢳⡀⡌⠔⢠⣈⠰⣍⢞⡲⠉⢠⠜⡢⠌⢰⡳⣌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⡘⠤⡘⠰⢠⠉⡍⠛⠦⡁⢆⡀⢀⣿⣿⡇⠀⢐⠄⠈⠀⠀⢒⠀⠀⠀⣛⢦⠁⢆⠱⣘⠴⣚⢏⡆⢹⡜⢪⠔⡩⢦⢫⢤⡚⣝⡲⠍⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠘⠠⠘⠤⡑⠓⠤⡃⢌⡑⠢⣽⡾⡔⢺⣿⣿⡀⠀⠂⠀⠀⠀⠁⢊⠀⠠⣜⠵⣪⠇⡌⢸⢣⡼⠙⣎⢞⡮⢮⡑⡎⠰⢩⡳⢣⡝⢮⠑⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠈⠀⠀⢀⠀⠀⠈⠑⠂⡍⠢⡌⠵⣯⣛⡇⠜⠻⡿⠓⠀⡄⠀⢀⠃⠀⢁⠃⢠⢎⡳⢥⡟⡴⣭⢳⣜⣳⡹⢎⣽⢣⡟⣭⣀⣢⢜⡳⣼⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠈⠒⠈⠹⣽⠈⡄⠀⠀⡐⠆⡀⠀⠀⢀⠚⡐⣏⢮⡱⢃⡜⠱⢣⠟⣬⢳⡜⣯⣚⠷⣛⠶⣇⠞⣮⠝⡶⢥⠀⠀⠀⠀
⠀⠀⠀⠀⠠⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢣⢇⠐⣄⠃⢹⠐⠤⡘⡄⢢⢁⢣⢙⠂⢸⠰⣈⠇⢣⠜⣐⢪⡙⢶⡩⢟⣭⢻⡌⠻⡜⡈⢽⠃⠁⠀⠀⠀
⠀⠀⠀⠀⢀⠃⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠄⠀⠉⠐⠘⢈⡆⠢⠌⢺⡄⢨⠓⡬⣘⢡⠚⣄⠋⡔⣣⡝⣺⣜⡳⣌⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⢉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠀⡸⡔⢫⢡⢃⣎⢳⡰⢁⢎⠲⣈⠵⢨⡑⠞⡵⣎⢷⣣⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡊⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⢀⠀⠆⠱⢨⠅⣦⢫⣌⠳⡰⢉⢆⠓⣌⠲⢡⢌⠓⣌⠣⣗⢮⣃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠑⡸⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢠⠃⡅⠀⠀⠀⡰⢈⠆⡉⠀⡈⢶⡙⢦⠳⣌⠱⢢⠍⡢⢍⡢⣑⠣⣘⠜⣠⢣⡟⢮⡝⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠡⠆⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠠⡁⢎⠰⢠⠐⡌⠰⡁⠎⠀⢠⡉⢧⣙⢎⡳⡌⠎⣅⢎⡑⢮⡱⣃⠞⣥⢚⡴⣣⢟⣣⢟⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡨⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⣡⣷⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣀⠐⠄⢢⢁⠒⡌⠤⠁⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠿⠋⢀⠠⠀⠂⡄⠢⠁⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⡿⠟⠛⠋⠉⠠⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⣷⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠛⡉⠉⠉⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⡿⣧⠣⠈⢂⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀
⠀⠀⠀⣀⣴⣬⣴⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⡗⢿⣻⣽⢾⡽⣫⣷⣀⡌⠲⣥⢛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⣯⣷⣿⢿⠏⠐⠌⣿⢾⢻⣞⡷⣏⠠⠐⡀⠈⠇⠀⠐⠡⣖⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀
⠛⠛⠛⠛⠛⣿⣿⡿⣿⣿⣻⣯⣿⣟⡿⣞⡿⣾⡽⣯⢿⡾⠩⠄⠒⠚⢻⠇⢺⣽⣻⡇⡐⢀⠠⢣⠘⠢⠄⠒⢈⠙⠦⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⣟⣯⢿⡿⣽⢯⣟⡷⣯⡿⣽⣻⢷⣻⠽⠛⠛⠛⢶⣦⣄⠉⢠⢻⣿⠽⢥⡓⠠⡘⠂⠇⡎⣍⠰⡌⡀⠂⠧⡀⣜⢏⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⢾⣻⡽⢯⣟⡾⣝⣷⣻⢷⣯⣿⡇⠀⠀⠀⠀⠈⠉⠀⠀⠆⠘⠁⠀⠀⠠⠱⡁⢀⡀⠈⠐⠡⢆⣡⠃⡐⢀⠘⠊⣱⡛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣰⣯⣞⡷⣽⣻⣞⡿⣽⣞⣯⣟⣿⡽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠈⠙⠻⠷⣤⣄⣀⠀⠵⠨⢀⠂⣁⠇⡀⠐⠂⣠⠖⣖⡒⡄⢠⠀⠴⡀
⠀⠀⠀⠀⢸⡷⣯⣟⣷⣻⣞⣿⣳⢯⣿⣾⢯⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠠⡘⢀⠉⠂⡴⠣⢛⡴⡋⠁⢀⠌⡒⠄
⠀⠀⠀⠀⢸⡿⣽⣿⣞⣷⢯⣿⣽⢯⣿⢯⣟⡾⣽⡆⢀⠀⠀⠀⠀⢁⣶⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠂⡁⢀⠀⢌⡜⠠⠐⡀⠂⢝⡲⣋⡆⠁⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣾⣟⣿⠸⡿⣯⡽⣾⣟⣿⣿⣶⣬⣄⣀⣀⡾⠿⠁⠀⠠⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⢀⠂⡐⢀⠂⡜⢆⠦⣁⠆⠠⣁⠂⠑⡧⢆⠀⠀
⠀⠀⠀⠀⠀⠈⠟⢯⡁⡙⡙⠻⢇⠶⣡⠛⡬⣏⠙⠙⢏⠙⢟⣿⣻⠇⠀⡈⠦⠓⠌⣘⢠⠠⠀⣄⠠⢠⠐⡰⡘⣄⠢⠐⣀⡢⢸⡍⡞⡜⠁⣠⠑⡤⠊⣸⡱⣋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⣌⠱⣈⠒⠤⡉⠜⡙⠦⡁⢆⡀⢈⣿⣿⡆⠀⡨⠀⠀⠀⠀⣯⠀⠀⢀⡛⢦⠡⡐⢣⠜⠶⣫⢓⡇⢳⡜⣡⠓⡜⣤⢫⡔⣺⡱⢣⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠤⢉⠄⡋⠴⡁⢎⠰⣁⣾⡷⡇⢺⣿⣿⠀⠀⠀⠀⠀⡈⠀⡗⠀⠤⣎⡝⣎⡇⠐⢯⣚⡼⢡⢏⡞⣧⢞⡰⣉⠰⠩⡾⡸⢥⣋⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡀⠀⠈⠐⠡⡘⢢⡑⠼⣣⢿⡅⢚⢿⡿⡃⠀⠀⠀⠰⠀⠀⠇⡄⣠⠞⡜⡦⢧⡽⢤⣳⣜⢣⠯⡜⣧⢯⡽⣡⣄⡡⢞⣱⢧⡃⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠘⠀⠉⢳⢫⠐⠄⠁⠠⣐⠂⠀⢘⠀⢀⢳⢰⣇⠻⡜⡡⢎⡙⢣⠗⢮⣳⢻⡜⣧⣛⢾⡱⣎⠵⣏⠞⣧⢇⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢫⢇⠐⣡⠋⠅⢊⠔⣨⡐⢄⠂⢎⠱⠁⢣⡑⢆⡩⠒⡜⠢⣜⠣⣛⠶⣹⢎⡷⡌⠛⢼⡈⢧⠇⠁⠀⠀⠀
⠀⠀⠀⠀⡱⢈⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠁⠘⠐⠱⣈⠘⡈⢾⡀⢈⠳⢌⡒⢍⠲⡑⣊⠴⣩⢞⡱⣏⡞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠆⡡⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠐⠀⠀⠀⠁⢳⠜⣉⠦⣑⢎⡳⢌⠢⡍⣒⠱⣈⠖⡡⢏⢷⣹⠺⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⡀⢄⠚⢡⠚⢤⡚⡼⣨⠓⡬⢑⠢⣅⠚⡤⢊⡕⠌⠦⣙⡻⣬⢇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠔⣁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡘⢌⡀⠀⢀⠀⡔⢡⠘⠀⠀⣰⡙⡦⣝⢲⠃⡵⢈⢣⡑⢢⡍⡔⢣⡘⣌⠓⡤⣟⠶⣋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠴⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡐⢌⠢⡐⠤⡀⢎⠰⣁⠊⠀⢒⠤⣛⢴⡩⢎⡱⠢⣍⠆⡜⣣⠞⡜⢦⢳⡰⢫⣜⢧⡻⣭⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡓⡌⢢⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⣰⣾⣷⣿⣿⣿⡿⠟⠀⠀⠀⠀⢀⢀⠢⠐⡌⠰⡈⠔⠨⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⠛⠁⠠⠐⢠⠐⠌⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠡⠄⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠷⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡛⠛⠋⠉⠈⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣷⣟⡲⠈⠃⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⢀⣰⣁⣣⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⡽⢺⡿⣽⣞⡿⡽⡄⢀⠷⣌⢳⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⡿⣷⣿⡿⡯⠑⠄⢿⣳⣯⣟⣵⠋⠡⢀⠈⢓⠀⠈⠒⢤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀
⠚⠛⠛⠛⠻⣿⣿⣿⢿⣿⣟⣿⣯⡿⣟⣷⣻⣟⣾⣽⣻⢷⠟⠠⠤⠠⢾⡟⢸⡾⣽⢠⠁⠠⡘⡈⠦⣀⠠⠌⠓⢦⡝⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠈⣿⢿⣽⢿⣯⣟⡿⣽⣻⣽⣞⣷⣻⣞⡷⠛⢳⠲⠶⣶⣤⠘⠁⢸⣿⣝⡰⡄⢂⠥⡱⣀⠖⢂⠄⡃⠄⠸⡀⠐⡲⢦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣻⢾⣯⢷⣯⢿⡽⣳⣟⣾⣳⣿⡎⠀⠀⠀⠀⠈⠙⠋⠑⢠⠁⡿⠃⠘⠑⡬⠀⠀⠁⠊⢥⢚⠰⣈⠄⢉⠁⠻⠜⣎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⡷⣽⢾⣼⣳⢾⣯⣟⡷⣟⡾⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠈⠐⠴⣶⣦⣄⠀⠀⠑⠊⡌⡠⢈⠐⢠⠏⠀⠃⠄⢀⣠⢄⣀⠀⠀⢀⠀⠄
⠀⠀⠀⠈⢹⡽⣯⣟⡾⣽⢯⣷⢯⣟⣿⣽⣟⣯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⠈⢒⠁⠄⠈⡜⠘⢤⣀⣠⢛⡴⣋⠦⠃⠜⠠⡉⠤
⠀⠀⠀⠀⣾⣻⣽⣿⡽⣯⣟⣿⣻⣞⣿⣟⡾⣽⣞⡆⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢀⠀⡘⠀⠐⡀⡰⠈⠡⠈⡕⢦⣀⢮⡐⠎⠀
⠀⠀⠀⠀⠸⣷⣿⣿⣿⣳⣯⣿⢳⣿⣽⣺⣟⣷⣻⣽⣐⠂⠄⠀⠀⢀⣾⡧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠄⢂⠐⠠⢁⡴⡁⠌⢠⠁⠄⡑⢎⡳⠆⠀⠀
⠀⠀⠀⠀⠀⠹⣿⡿⡛⣿⢳⣿⡌⣎⠹⢧⠛⠯⠿⣿⣿⣿⣿⣶⣶⡿⠛⢡⠀⡌⢰⠁⠀⠀⠀⠀⠀⠀⡀⠄⡎⠠⢈⡐⠠⢈⢰⡋⡴⡱⠎⠁⡰⠌⡐⢨⡓⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⢰⠂⠆⡌⡑⢊⠓⡬⢹⡶⡀⠄⠳⠀⢹⢽⣾⡛⢀⠉⠡⠈⠂⢥⠃⠈⠐⢨⢆⡑⠤⢙⡔⢢⣁⣤⢶⠁⣧⡙⠶⣁⢢⠱⡑⠊⣠⢾⡱⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠜⡘⠴⠤⣁⠣⡘⢄⠃⡌⡑⢎⣕⡀⠀⣾⣿⡏⡑⠀⠠⠀⠀⠠⠇⠀⢀⡞⡜⣆⠌⠲⣌⠇⣳⡎⡞⣄⢧⡙⡜⠤⣃⣯⢹⡍⢧⣣⠳⠁⠀⠀
⠀⠀⠀⠀⠀⢀⠐⠈⠀⠁⠊⠐⢡⢉⠒⣌⠒⠤⣱⢿⣻⠇⣹⣿⣿⡇⡀⠀⠀⡠⠀⢨⠡⠐⢫⡜⡵⢪⡐⡉⢮⠳⣅⡸⣕⢫⡞⡵⣜⡡⠐⡘⢳⡜⣣⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠄⠀⡀⠀⠱⠠⡙⠴⡉⢷⣻⡆⠜⠹⠿⠍⡇⠀⠐⠀⠀⢰⠰⣐⢧⡚⠵⢫⡽⣹⢫⠷⣍⣓⡹⢮⣝⣳⣳⣒⠶⣘⠧⣞⡵⡂⠀⠀⠀⠀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘⠀⠀⠀⠱⣹⠈⢄⢀⣢⠃⠄⠀⡄⠠⣈⡃⣯⠲⠝⢬⡁⢎⡑⢫⡙⢮⢳⣝⡳⣎⠷⣧⢻⢎⡭⢏⠳⣝⠣⡀⠀⠀⠀
⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡇⢬⠂⠘⡄⠣⣘⢤⠡⢐⠸⣰⠁⢘⢦⢘⠢⢍⡒⢌⠦⠇⢮⡱⢭⡻⣜⠷⠈⠞⠥⠌⡹⠀⠀⠀⠀⠀
⠀⠀⠀⢘⠠⢍⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⠀⠁⠈⠠⢃⠌⣂⡹⢆⠨⡜⣨⠑⠦⣉⠲⢌⡹⢰⣋⢾⡱⣏⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⡒⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠐⠀⠂⠀⠙⡖⢡⢒⡡⣝⢲⠡⢎⡑⠦⣉⠲⣐⠣⢎⠳⣝⢮⣋⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⡀⠠⠌⠓⠬⣑⢪⢴⡩⢎⡱⢊⡔⠣⣌⡑⢢⡑⣊⠱⠌⢷⣭⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⢌⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡔⢄⠃⠀⠀⠀⡌⢢⠡⠉⠀⣐⡳⢬⡓⢮⡑⠦⣑⢊⡔⢣⡔⡌⢣⠔⣡⠋⣜⡺⣴⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡐⢌⠢⡉⠤⡐⢌⠰⡁⠎⠀⢒⠤⣙⠶⣩⠳⣌⠱⢢⢅⡊⢶⠹⣌⢳⡚⣤⢛⡴⣻⠼⣭⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⡍⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢂⣼⣶⣿⣿⣿⣿⡿⠂⠀⢀⠀⡂⠔⡐⢢⢁⠒⠰⠈⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⢀⣠⣶⣿⣿⣿⡿⠛⠛⠩⢄⠰⠈⠀⠀⠠⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⡉⠄⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⠿⣗⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠿⠟⠛⠋⠉⠉⠀⠀⠀⢀⣠⣬⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⢦⠙⢂⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⢘⡌⢒⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⢷⣿⣻⣾⠀⢀⡫⢖⡹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣯⣷⣿⢿⠃⡌⣷⣟⡾⡏⠃⠠⠉⢞⠁⠈⠔⡠⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⢿⣽⢾⣻⡽⣿⣟⣯⡟⠁⣀⣀⣼⣯⢙⠦⠁⠠⢡⠈⢆⡀⢀⡐⠳⣎⠽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠀⠀⠀⠀⣿⣿⡽⣿⣷⡿⣿⣟⣯⣟⡿⣞⣯⢷⣟⡷⡯⣧⣬⣤⣀⡀⢸⠃⠸⢸⠀⠂⣅⢣⡀⢭⢀⠐⡄⠈⢌⡀⠐⠤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣵⢯⣟⣯⢿⣽⣳⢯⣷⣻⣽⣻⢾⣻⡞⠉⠀⠈⠀⠈⢽⢿⠂⠀⠟⣏⣧⢘⠠⠁⠚⠤⣃⠲⡈⡔⡀⠑⠒⢪⡕⣎⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⢻⡼⣏⣟⣮⣟⡿⣾⣽⢾⡽⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢀⠩⣀⣤⣄⠀⠀⠁⠓⠴⢁⠌⡐⠠⠀⣜⠓⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⢼⣯⢿⣽⢾⡽⣾⣽⣻⢧⣿⣯⢿⡿⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠉⠙⠛⠶⠦⠤⠀⢎⡘⠀⠄⡁⡎⢆⠀⠈⢠⢏⡝⣫⡍⠰⡀⢆⠱⢀
⠀⠀⠀⠀⢸⣞⣯⣿⢯⣟⣷⣻⣽⣻⣞⣿⡿⣽⣳⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠠⡘⠀⠄⡉⡰⠋⠚⠼⣡⡁⠀⡄⠢⡜⠀
⠀⠀⠀⠀⢸⣟⣾⣿⣿⣞⣷⣻⣷⣟⣾⡟⣿⣳⢯⣟⡆⢄⠀⠀⠀⠀⠀⣙⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⢁⠂⠄⠠⠐⡴⠁⢂⠡⠐⡀⢟⡭⣛⡂⠀⠀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣾⢿⣽⡇⣟⢯⣿⢹⣿⣻⡿⣿⣦⣼⣀⡄⠀⣴⣿⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠄⡁⠂⠌⠠⣡⠏⡴⢨⡄⠂⢡⠐⠠⢑⢧⡃⠀⠀
⠀⠀⠀⠀⠀⠈⠿⠿⣑⠝⣉⠻⠷⣌⠲⡌⢧⢳⠉⠙⠻⡟⠻⣿⢿⣿⠿⠸⢄⡙⢂⠤⠀⢄⠠⡀⢄⠢⢁⠧⣈⠰⢀⢡⣈⢰⢣⠞⣜⠃⢠⡘⢢⠜⠁⣼⠲⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⢄⠣⡘⠤⡉⠜⣂⠻⢏⡰⢀⠍⠀⠘⣻⡗⠂⠐⠀⠈⢱⠂⠀⠀⢸⡱⣂⠡⠊⡖⣡⢞⢮⢣⡇⢸⢣⠛⣌⠲⡡⣞⣡⢤⣛⢎⡳⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⡍⢢⠑⠪⠔⣡⠊⡄⢃⠆⡑⣮⡞⡄⣰⣿⠋⠀⠀⠀⠐⢸⠈⣀⡠⢧⢳⡱⠌⡁⣷⣡⢯⢺⡱⢺⡜⣧⠙⡤⠃⠵⣣⠞⣦⢋⣮⠁⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠈⠀⠀⠀⠉⠐⠩⠄⡓⢌⠢⣘⡼⡿⣽⢂⢹⣿⡁⠀⠀⡌⠀⢸⠰⠀⡨⢇⡧⣹⣆⣡⣐⢧⡓⢦⣏⢧⢻⣜⡻⣜⡀⠒⣨⡝⢦⡻⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠐⠀⠀⠀⠀⠘⡄⠓⡌⠌⠻⡽⣆⠚⠘⡇⠀⠀⠄⠀⠠⡃⢧⣙⢮⠜⡱⢎⡷⣹⢮⠽⣥⢦⣏⠷⣎⣷⢣⡟⣭⢳⢼⡳⣽⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠹⡼⡈⠵⡁⢂⠀⡆⠀⡅⢣⠎⡓⢪⢘⡡⢒⡌⣑⠪⣙⢚⠶⣭⢻⡜⣞⡧⠏⢲⡹⡆⠻⣜⠑⠠⠀⠀⠀
⠀⠀⠀⠀⡉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠁⠘⢄⡃⢌⠧⡘⢠⠊⢾⠀⠘⣦⢡⠣⡘⠤⢓⠌⡏⡜⢦⣋⢾⣱⢫⠝⠈⠁⠒⠈⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠱⡈⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡘⠀⠀⠀⡀⠀⠀⠀⢀⢣⠂⣜⡈⣆⠸⣄⠣⡜⢡⡙⠌⡎⠴⣈⠷⣌⢷⣩⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣈⠳⠤⣑⢢⡹⣌⠧⣘⠡⣌⠓⢬⡑⠬⡑⢎⠷⣭⣛⢶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⢀⠰⠈⢁⢣⠣⡜⢦⠳⣌⠱⢂⢇⢢⡙⢤⠊⡕⡘⢆⠚⠴⣫⢞⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⠀⠀⢀⠎⡰⠀⠀⢀⠀⡡⢊⠤⡍⠀⢤⢣⡛⣜⢣⠏⣄⠫⢌⠆⡣⣜⠢⣉⠖⡩⢌⡍⡲⣏⠾⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡌⢢⢁⠣⢄⠰⡈⠔⡡⠞⠀⡜⢢⠱⡹⣌⡳⡜⡠⢝⢢⠩⣔⢣⡛⡴⢪⡕⢮⡜⣵⢫⡻⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣾⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⣘⠤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠐⣼⣶⣷⣿⣿⣿⡿⠃⠀⢀⡀⠢⢄⠡⢒⡐⠂⠆⠁⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⣠⣴⣿⣿⣿⡟⠛⠛⠙⡐⠤⠁⠀⠀⡐⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢉⡁⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠿⠿⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠆⠙⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⢘⡄⢣⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⢷⣟⣯⢧⠀⢰⠱⣎⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣿⣾⡿⠏⣈⢻⣞⣿⠏⡁⠂⠑⠞⠀⠈⡔⢠⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⢿⣽⣞⡿⣽⣻⢿⣯⠷⠋⢀⣀⣼⣻⠎⡇⠀⠡⣘⠘⣄⡀⢀⠜⠱⣎⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀
⠀⠀⠀⠀⠀⣿⣿⡽⣿⣷⡿⣿⣟⡿⣽⣻⢾⣽⣻⡽⣯⠿⣮⣤⣥⣀⡀⠀⡟⢀⣃⠄⡰⡐⢢⡀⡜⠠⢀⠃⡈⢸⠀⠁⢢⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢱⣯⣟⣯⢿⣽⣳⢯⡿⣽⢯⣟⣾⣳⡟⠋⠀⠈⠀⠈⠽⡿⠧⠀⡝⢩⡆⡅⠈⠑⠘⠬⡑⢎⠰⣀⠂⠓⠂⢏⡳⢬⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣹⢞⡽⣞⣻⢮⡿⣽⣻⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⡘⢀⣠⣤⣀⠀⠈⠈⠢⠕⢰⢈⠐⠠⠐⡬⠏⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠼⣯⢿⣽⢾⡽⣯⢿⣳⣟⣾⣷⣻⡿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠛⠻⠶⠤⠄⠈⠖⡌⠠⢁⢂⠇⢆⠀⠁⢰⢫⡝⣫⡕⢠⠂⢆⠱⢀
⠀⠀⠀⠀⢸⣯⣟⣿⢯⡿⣽⣻⣷⣻⣞⣿⣽⣻⣽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠎⢀⠈⠡⣘⠃⠳⠜⣥⡁⠀⡄⠢⡜⠀
⠀⠀⠀⠀⢸⣷⣻⣿⣯⣟⣷⣻⣷⣟⣾⡟⣷⣻⢾⣽⣇⠄⡀⠀⠀⠀⠀⢉⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠠⠁⠌⡀⠠⢈⠴⠁⡐⠠⠘⠀⢯⡹⣙⡂⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣯⣿⣳⡇⣟⢯⣿⡹⣿⣻⣾⣿⣶⣤⣥⣀⠀⣰⣿⣟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⡀⢂⠁⠂⠄⣱⢋⡜⢤⡄⠃⢄⠡⠂⢱⡍⡇⠀⠀
⠀⠀⠀⠀⠀⠈⠿⠿⡕⠝⣉⠻⢿⣠⠣⡜⢳⢸⡉⠑⠛⣿⠻⢿⢿⡿⠟⠧⢅⡚⠠⡄⠠⣀⠄⡠⠐⡄⢣⢃⡐⠤⠂⣈⡰⢨⢇⡞⣜⠃⢀⡌⣊⠔⠁⣞⠼⡁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡐⢂⠆⡄⠣⢌⢃⠻⠽⣐⠐⡌⠂⠘⣻⠓⠀⠂⠀⠈⠡⡇⠀⠀⢘⢦⡑⠰⣈⠦⣙⡖⣫⡕⡃⣏⢞⡸⣄⠚⡴⡜⣤⢦⣛⢎⡳⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⢊⠍⡘⢡⠚⠤⣃⠌⢢⠑⠢⢌⣲⣧⠇⣀⣿⠁⠀⠀⠐⠀⠃⡇⢀⡠⣝⠦⢧⢁⠢⣗⢣⢟⢣⠞⡵⣜⣎⠴⣨⠘⡜⡳⣎⠶⣩⢎⠅⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠁⠘⠰⠠⡙⢢⢉⠒⣼⡻⣽⠃⠼⣟⠀⠀⢀⠆⠀⠃⡔⠀⡨⢎⡝⣺⣠⣑⣨⢏⡞⡬⣟⡜⣳⢎⡷⣃⡐⢠⡩⣜⢣⡗⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠂⠀⠀⠀⠑⢢⠉⠎⠌⠻⡽⣇⢊⣈⠀⠀⠨⠀⠀⢸⠈⣴⡙⢮⢊⠳⢧⢻⡜⣏⢧⡵⡼⣜⡳⣏⣾⡱⣏⢧⢳⢭⡳⣽⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠘⣞⠤⡹⢠⠀⢰⠀⣀⠃⡕⢸⠙⡇⢪⠱⢌⢢⡙⢌⠳⣙⠾⣱⣛⡼⣲⡽⡉⢮⡹⡆⢛⡶⠑⠀⠀⠀⠀
⠀⠀⠀⠀⢣⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡊⠀⠃⢤⠉⡜⢆⠄⡃⠜⣡⠀⢱⣃⠎⡜⡰⠌⢎⡒⢭⢸⡱⢎⡳⣇⡟⠑⠈⠁⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢢⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠀⠄⠀⠀⠀⠀⡐⢎⠄⣣⢘⡄⢰⡌⡜⡰⢑⡊⠦⣉⠲⢌⡳⣍⢷⣙⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠆⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⢀⠚⢎⠴⠡⡜⢦⢳⠰⡡⢣⡘⠥⢢⠍⠦⡑⢎⠷⣭⢞⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠰⡁⠀⠀⠀⠀⠀⡀⢠⢐⠊⢑⢊⢆⡳⣜⣣⠋⢦⠑⡥⡘⢬⠡⠎⡥⣉⠖⡸⡘⣮⢳⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡠⢃⠔⡀⠀⠠⢀⠱⡈⢆⠢⠠⡘⣎⢮⡱⢎⢆⡹⢠⢋⠔⣉⢦⡉⢖⢡⡒⠬⣑⡜⣧⣛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠑⡌⠒⠤⡀⢆⠡⢆⣑⡎⠠⣃⠵⣘⠦⡝⣎⠲⣐⢃⡎⢜⡸⢦⡹⣌⠶⣩⢖⡱⣞⡵⣭⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠌⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣣⣾⣶⣿⣿⣿⣿⠗⠀⡀⡀⢆⡐⢂⠒⡐⠂⠆⠁⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⠟⠛⠋⠋⡅⢂⠂⠀⠀⠠⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢉⡁⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⢿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠿⠿⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡱⡈⠃⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀
⠀⠀⠀⠀⢀⢣⠘⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⡯⣟⣷⡇⠀⡜⡱⢎⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⣾⡿⡟⢠⢹⣷⣻⠏⠐⠈⠣⡍⠀⠡⢂⠄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣟⣯⣷⣻⢯⡿⣿⣻⡽⠋⢀⣀⣠⣿⡽⡀⠈⠄⡅⠱⣀⠀⢀⠙⢦⣍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠀⠀⠀⠀⠀⣿⢿⡽⣿⣻⣾⣟⡿⣿⣽⣻⣞⡷⣯⢿⣽⡳⢿⣤⣥⣄⣀⠀⡾⠑⡄⢂⢸⠰⡄⣈⠍⡠⢈⠄⡈⢡⠀⠂⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⡿⣽⣟⡿⣽⢾⡽⣷⣻⢷⣯⢿⣽⣛⠊⠁⠈⠀⠀⠭⢿⡗⠄⠱⢹⡄⠆⠈⠐⠡⢞⡐⢎⠰⢠⠈⠓⠂⢗⠦⣝⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⢻⡵⣏⢿⣭⢿⣽⣳⢿⣻⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⡋⢄⣤⣤⡀⠀⠈⠘⠢⢇⠰⢈⠐⡈⠀⣮⠍⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠴⣟⣯⢿⣞⡿⣞⡿⣾⣽⣻⣷⢯⣿⣾⣝⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠈⠉⠛⠻⠶⠤⠄⠐⡣⠆⠐⠠⢱⠂⣄⠀⠁⡰⢏⢯⡹⣱⠀⠆⢢⠑⠄
⠀⠀⠀⠀⢨⣟⣾⣿⣽⣻⣽⣻⣷⣳⢯⣿⣿⣳⣟⣾⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠀⠈⢀⠎⠀⠄⡉⡴⠙⠚⠦⣝⡁⠀⡔⠢⡜⠀
⠀⠀⠀⠀⠸⣟⣾⣿⣷⣻⣞⣷⣿⡽⣯⣿⢷⣻⣞⡷⣧⠠⠀⠀⠀⠀⠀⠘⣿⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠌⡐⠠⠀⠄⢂⡜⢀⠂⡐⠌⠀⢯⣙⢞⡁⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣯⣿⡾⣇⢻⠿⣽⡺⣟⣾⢿⣿⣶⣍⣔⡀⠀⣠⣾⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⡈⠐⡀⠡⠈⡴⢋⠴⣠⠒⠠⢈⠔⡠⢙⠮⡅⠀⠀
⠀⠀⠀⠀⠀⠈⠻⠿⣕⠝⡡⠛⠿⣆⠳⣌⠳⢬⡉⠙⠛⣿⠛⢿⡿⡿⠟⠧⣂⠜⠤⠄⠠⣀⠄⡠⠄⡂⢦⣁⡐⠄⢃⢠⡁⣜⢣⢏⡞⢁⠠⣑⢪⠐⠁⣎⡳⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⣉⠒⡌⢢⠡⡙⠚⠽⣌⠐⣈⠁⠀⢛⠂⠐⠀⠀⠈⡖⠀⠀⠀⡸⣅⠢⢁⠲⣄⢳⡼⣍⣳⡀⣏⠞⡬⢜⣂⠳⡜⣤⢒⣏⢧⢳⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⢌⠩⡑⢂⠓⡌⢆⠱⣈⠱⡈⠤⣹⣶⠡⢄⡞⠀⠀⠀⠐⠀⡇⠀⣀⣰⢱⢎⠧⡈⢼⡜⣇⢞⢧⡒⢷⡸⣍⠜⣢⠈⡝⡳⣎⠳⡜⣎⠃⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠁⠘⠐⠌⡑⢢⡑⢌⣲⡟⣯⡗⢨⡇⠀⠀⠰⠀⠀⡇⠆⢀⣨⢓⢮⣳⣈⣔⡺⣜⠲⢮⡝⣎⡷⣭⣛⣒⠠⢐⡩⣎⢽⣱⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠂⠀⠀⠀⠑⢢⠘⢢⠄⠛⡷⢯⡀⡧⠀⠀⠁⠀⠀⣒⠬⣔⡣⡏⢎⠳⣹⠶⣹⢎⡯⣥⠾⣜⡳⢧⡻⣜⡯⣙⢶⣩⢾⣱⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠈⡷⠅⡖⢠⠀⢢⠀⠒⡘⡄⠟⠒⡍⢢⠓⡤⡙⡌⢍⡚⡵⢻⡼⣙⢷⡺⣵⠋⡜⠾⡄⢛⡶⠁⠄⠀⠀⠀
⠀⠀⠀⠀⢣⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠈⠄⣃⠘⡴⡌⡑⠤⢑⡇⠀⢧⣃⠞⣠⠱⣘⠢⢱⢃⠳⡜⡭⢶⡹⣎⠓⠈⠉⠒⠈⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢢⠑⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠂⡀⠀⠀⠀⢀⠸⡐⠌⡤⢱⡀⢶⡈⠖⣡⠚⢤⠙⣢⢉⢞⡱⣭⢳⡝⣮⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠆⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⢀⡛⢪⠔⢣⡘⢧⡹⡘⢤⡉⠦⡙⢤⠊⢦⠙⡜⢧⡻⣜⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⡜⠀⠀⠀⠀⠀⠀⣀⠠⡌⠈⡒⠥⣊⠵⣊⠧⡱⢑⠢⣅⠣⢍⢢⡙⠤⢋⡔⡊⢵⣫⢞⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡜⢠⠃⠀⠀⠠⢈⠒⡄⡣⢁⠤⣙⢮⡱⢫⡜⡱⠌⡥⢃⠦⣉⢆⠣⣌⢃⠧⡰⢉⣖⢧⡻⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡜⢠⠃⡜⢠⠐⡌⢢⠑⣰⡏⢜⡰⢡⢮⡱⢣⠳⢌⡚⣤⢉⠦⣙⢎⡳⣔⢫⢲⡱⢫⡜⣧⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠣⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢢⣽⣶⣿⣿⣿⣿⡿⠂⣀⠐⡄⢂⠒⡐⢂⠒⠤⠁⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣤⣾⣿⣿⠛⠛⠋⡙⢩⠐⠀⠀⠀⠄⠃⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢉⠡⢀⠀⢀⣠⣴⣾⣿⣿⣿⠿⣷⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⣀⣤⣵⣶⣿⣿⣿⣿⣿⣿⣿⣟⢦⠘⠃⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀
⠀⠀⠀⠀⢀⡣⢘⡠⢀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡽⣻⣾⡀⢀⡫⢖⡭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⣾⣿⠿⢁⡙⣾⡽⠃⠠⢈⠺⠁⠀⢡⠂⣄⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⡿⣯⣟⣾⣻⣟⡿⣟⡿⠎⠁⣀⣀⣿⡧⠁⠐⢠⠂⢣⣀⠀⣀⠫⢖⡭⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠀⠀⠀⠀⣿⢿⡽⣿⣻⣾⣟⡿⣿⣽⣻⢷⣻⣞⡷⣯⢿⢽⣥⣬⣄⣀⠀⢸⢗⡀⢂⣄⠳⡀⢤⡉⢀⠰⠀⠌⣁⠀⠢⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣯⣟⣯⣟⡿⣾⡽⣷⢯⣟⣯⢷⣻⣽⠋⠁⠀⠁⠐⠉⢿⡿⠎⢁⠷⡰⠀⠉⠘⠢⢱⡈⠆⣅⠂⠌⠒⠂⣏⠶⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⡳⣭⢷⣫⣟⣵⣻⣭⢿⡾⣽⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢈⠑⣀⣤⣄⡀⠀⠁⠙⠤⡊⡄⡁⠂⠄⢂⡗⠊⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠴⣿⡽⣯⢷⣻⣞⣷⣻⣞⣯⣿⣳⣿⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠈⠙⠛⠷⠦⠤⠀⠰⡡⠁⠌⡐⢨⠸⣀⠀⠁⡴⣋⠯⣝⣱⠀⠆⢢⠑⠄
⠀⠀⠀⠀⢠⡿⣽⣿⢯⣷⣻⣞⣷⣻⣞⣿⣷⢯⣟⣾⡅⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⣀⠃⠄⢈⠁⡼⠑⠩⢞⢲⡁⢀⡔⠢⡘⠀
⠀⠀⠀⠀⠘⣿⣳⣿⣿⣞⡷⣯⣿⣷⣻⣾⢻⣟⣾⣳⣿⡀⠄⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠄⠡⠀⠄⡀⢂⠞⠀⠄⠡⢈⠰⢍⡳⣚⡁⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣯⣿⣟⣧⠹⡷⣿⡏⣿⣾⢯⣷⣷⣬⣆⡠⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠀⠌⠠⠁⠌⡐⡴⢋⡜⢤⠘⡀⢂⠔⡈⢱⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠈⠹⠿⣝⢸⠋⡹⠿⣎⡔⢦⡙⢆⡏⠉⠛⢿⠛⠿⣿⠿⠿⠿⢘⠂⠤⠄⢠⡀⢄⠠⢂⠜⡸⣀⠡⠌⢠⣁⠆⡼⣡⠳⡜⢁⠂⡴⡉⠔⡀⣏⢎⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢊⠔⡡⢂⠜⡠⢙⠚⠽⣃⠔⠢⠑⠀⠛⡂⠐⠀⠀⠉⡖⠀⠀⠀⡳⣌⠂⢆⠱⣌⢳⡜⣧⢹⠀⡷⡱⢋⠖⣌⠳⡔⣥⠴⣚⡭⣚⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⡉⢅⠚⡰⠡⢎⠰⡁⢎⠰⢠⢹⣼⣳⢀⡘⠀⠀⠀⠐⠀⡇⢀⣀⠼⣱⢚⡌⠤⢹⡜⡼⠞⣥⣋⢷⡱⣇⢹⡘⢠⠛⡧⣇⢯⡱⢎⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠁⠐⠁⠎⠱⢨⡂⡑⣢⣟⡯⡷⢈⠇⠀⠀⡰⠀⠀⡇⠆⢀⡝⣥⢫⣖⣈⣆⣹⡜⡳⢼⡜⣎⠷⣭⢗⣊⠀⡑⡬⢞⢦⡽⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠀⠂⠀⠀⠀⠁⠆⡌⢣⠡⠙⢾⣳⠌⢆⠀⠀⠄⠀⠀⣒⢘⡦⣙⢦⠋⠞⡵⣎⢷⡹⣭⢦⡼⣎⠿⣜⡯⣞⡽⡹⣜⣹⢮⣝⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁⠀⠀⠀⢏⠧⠎⡠⠀⠤⢀⠒⡘⠬⢱⠙⢆⡹⢘⡰⢌⡊⠵⣉⠾⣱⢫⡝⣮⢳⣭⠑⡳⢼⠀⢻⢶⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠱⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠢⣡⠁⡷⡀⢎⠐⡱⡇⠀⢣⢖⡡⢒⠬⡘⢆⣡⠓⣬⢣⡝⢮⡳⢞⡃⠈⠉⠓⠈⠊⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠀⠀⠂⡀⠀⠀⠀⢀⠸⣀⠣⣐⠱⣀⢣⢎⡰⡉⢦⢑⡊⢤⠋⢴⠳⡜⣧⢻⣭⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⢀⡛⡸⢄⠳⣈⠗⣮⢐⡱⢌⠢⢍⢢⡙⢢⡙⠹⣜⡳⣎⢯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢎⠀⠀⠀⠀⠀⠀⣀⠠⡌⠉⡔⡡⢎⡱⢎⡝⠢⢎⠰⣊⠱⢊⠴⢨⠥⡘⢥⠢⣙⢮⢏⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⠎⠢⠄⠀⢀⠠⢈⠒⡄⣳⠀⠤⣱⡙⢮⡱⢫⢌⠓⣌⠣⡔⡩⢎⡘⢆⠲⣉⠦⢱⡌⣟⡞⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠌⡌⠱⡈⢆⠠⢂⢅⠊⣴⣻⢈⠖⡡⣛⢦⡙⢧⡘⠼⣀⠳⣈⠵⣫⠜⣬⠳⣥⢚⡵⣺⠵⣋⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠖⡅⢊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠒⣸⣥⣶⣿⣿⣿⣿⠖⢀⠠⢄⢂⠒⡄⢒⠰⠠⠡⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢀⣴⣾⣿⠿⠟⠛⡛⢩⠡⠈⠀⢀⠰⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⡇⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢿⠿⠟⠛⠛⠉⠁⠀⠀⠀⢀⣢⣤⣶⣾⣿⣿⣿⣿⣿⣏⡄⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣳⣎⠀⢀⣇⢏⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠟⡘⣿⡖⡊⢈⠪⣕⠊⠠⣀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡽⣷⢿⣿⣿⢿⣻⠊⠀⠐⢹⡇⠀⠂⢄⠘⡀⠀⠀⢈⠞⣭⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀
⠉⠉⠉⠉⠉⣿⣿⢿⣟⣿⣯⣿⣷⡿⣷⢯⣟⣯⢿⣽⣻⢾⡽⠟⠓⠒⠂⠉⢩⠇⠠⠈⠌⡆⢉⠒⡒⢈⢈⠐⠛⡄⠐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢻⣞⣯⣿⣟⣿⣳⢯⣿⡽⣿⡽⣞⣿⢾⡽⠏⠉⠻⠛⠛⣿⣶⡔⢸⡄⢃⠓⠸⢢⡙⡔⢄⠎⡠⠁⠣⣄⠤⣋⢯⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡽⣞⣷⣻⢾⣽⡻⣞⣽⣷⣻⣽⣾⣿⠀⠀⠀⠀⠀⠀⠉⠙⠈⢰⢸⠅⠀⢀⠀⠈⠘⠨⢆⠱⡁⡂⠠⢀⠙⠂⡷⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣼⣻⢶⡽⣞⡷⣿⡽⣞⣷⣻⣾⢿⣻⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠊⠀⠚⠿⠷⣦⣄⣀⠀⠉⣰⢁⠃⠄⡈⢸⢡⠀⠑⠂⢠⡔⢦⡔⢄⠀⡀⢄⠠⠂
⠀⠀⠀⠈⠁⣿⢾⣽⢯⡿⣽⣻⣷⣻⣽⣻⣷⣿⣻⣟⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣋⠀⠂⡀⠏⢀⠲⠄⣠⢧⡹⢆⣛⠞⠀⢁⡈⢆⠡
⠀⠀⠀⠀⢸⣯⣟⣿⣯⣟⡷⣯⣷⣟⡾⣽⣷⢯⣷⣻⢷⡀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⠄⠘⠀⢀⠂⡰⠁⠄⡈⢩⠘⡦⣔⢣⠌⠂⠀
⠀⠀⠀⠀⠀⣿⣾⣿⣿⣾⣟⣷⢿⢯⣿⣻⡽⣻⣾⡽⣯⣇⠰⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠐⠠⠈⠄⡁⢂⡜⡡⢈⠐⡀⠂⠌⠱⢎⡽⡀⠀⠀
⠀⠀⠀⠀⠀⠙⣾⣿⣿⢷⡿⣯⣿⡜⢪⢛⢷⠛⡷⢿⣿⣿⣿⣷⣯⣶⣤⣾⢟⠀⠀⠀⠀⠀⠀⠀⠀⣀⢢⠁⠂⢌⠠⢁⠂⢴⡋⡴⡱⠎⡁⢄⡡⢃⠆⣸⠲⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠙⠓⡌⠡⢌⢋⠣⠇⢮⣘⣹⢦⠀⡀⠘⡇⠉⢿⠋⠉⠉⠊⢕⢂⠒⠂⠰⣁⢊⠡⡐⡎⡔⣃⢄⣢⠶⢈⡶⣙⠶⡉⣐⠰⣊⠔⢁⣠⠷⣙⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣉⠆⡌⢢⠉⡔⠠⢌⡉⠳⣈⢒⡄⠀⢨⠄⠀⡀⠀⠄⡇⠀⠀⢠⡝⣆⠡⢂⡗⣸⢪⢳⢣⢻⡠⢳⢍⠲⡱⢌⢳⡱⡞⣭⢲⢫⠵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠃⠰⠐⢌⠡⢃⠎⡔⡉⠆⡌⠱⣼⡿⣜⠠⡀⠀⠀⠀⠈⠀⡇⠄⠖⣧⢚⡜⡆⠡⢞⡭⢞⡋⢧⡓⢯⡻⣌⡣⡅⠘⡸⢕⡫⢖⡭⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠈⠀⠀⢀⠀⠀⠈⠀⠊⠔⡱⢌⠰⠽⣳⣛⣏⡐⡃⠀⠀⠜⠀⠀⠇⢃⠴⣌⡳⢎⣧⠵⣦⠾⣥⡛⠮⣝⢺⡵⣫⢷⣡⣂⡰⢖⡹⣎⣞⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠂⠀⠀⠈⠀⠀⠈⠀⠀⠀⠈⠰⣈⠙⠐⠈⢻⢞⡄⡃⠀⠀⠂⠀⠀⢭⢸⡳⢬⠳⣡⠚⡹⢖⡻⢶⡹⣖⢯⡳⣝⢧⡟⣶⡣⡝⣎⠷⣹⢮⡀⠀⠀⠀⠀
⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠏⣒⠅⡃⠄⡡⢀⠱⡈⠞⡱⠋⡱⠤⡙⢤⠣⠜⢢⠙⡼⢣⠟⣼⢣⡟⣶⡈⡳⢭⡀⢟⢶⠉⠂⠀⠀⠀
⠀⠀⠀⠠⡐⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠀⠈⠂⠜⡠⠝⢆⡁⠆⡩⡇⠀⠱⣎⡑⢆⢣⢙⠢⢍⡃⠮⡝⢦⡻⣜⢧⠇⠀⠁⠁⠈⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢁⠆⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠜⠀⠀⠠⢀⠀⠀⠀⠀⢊⡔⢨⡰⠜⡤⡱⢆⡜⠌⡆⢣⡉⢦⢉⠖⣹⢣⢷⡹⣎⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⠀⠈⠀⢠⢩⡑⢢⡙⢤⡛⡼⢠⢋⠬⡡⠜⢢⠩⡌⠥⢋⠞⣵⢫⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢠⢊⠀⠀⠀⠀⠀⢀⠀⠤⡘⠁⡒⢬⢡⡜⣣⠝⣢⠃⢎⡒⣡⢙⡂⠧⡘⠥⣃⠚⡌⡟⡶⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠦⡈⠄⠀⠀⠄⡘⠤⣉⣲⠁⢆⣙⢎⡳⡜⣥⠓⡤⢙⠢⡜⡐⢦⡉⠖⣉⠖⣌⠱⣌⡟⡵⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃⢆⠱⡈⠤⡐⢌⠰⣁⢲⣽⢈⠲⣈⠮⡵⣙⢦⠱⢌⢣⡑⠦⣙⠶⡹⡜⣬⠳⣌⢳⡼⣹⢭⡳⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂⠄⡉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣁⢂⠄⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢂⠐⠠⢁⢂⡎⡅⢂⢡⠀⡐⠠⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣶⣶⣿⡿⡀⠀⠀⠀⠀⠀⢀⠀⠄⢲⠁⠐⡀⠆⢈⠐⣰⢋⡴⣩⠎⠁⢠⠔⡡⠊⡴⣩⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠉⡁⠋⠤⡉⡇⠘⠂⣍⠰⡈⢌⢣⠚⣌⣡⢴⢺⢀⢧⣋⠶⣁⠰⣌⢣⠚⢀⡴⡳⢥⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⠀⠀⠀⡇⡇⠀⢠⢞⡣⠌⢄⣣⡹⢼⣲⢭⣚⡌⢶⡉⠶⣡⠓⣬⡣⢟⡭⢶⡙⡇⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⢎⡵⡊⢄⠣⣝⡣⡇⢷⣌⢻⣎⢧⡳⣄⠘⠰⠭⡳⣜⢣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⡚⣴⣫⠴⣦⢵⣣⡽⠇⣎⡳⣎⢷⡳⢦⡤⡵⣚⠵⣎⢷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠽⡠⢍⠻⡜⢮⣓⡞⣾⣱⠻⣜⢧⡟⣧⢗⣱⢫⠺⣝⠮⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠃⡇⡱⢊⡱⢘⡂⠦⡙⡴⢫⡝⢮⡳⣝⢾⡌⠲⣹⡀⠩⡟⠈⠁⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⠸⡱⢌⢢⠣⠜⣑⠢⢇⠣⡞⣥⢻⣜⡳⢎⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠜⡤⣘⢥⢊⢆⠣⢍⢢⡙⠤⢣⡝⢦⣛⢶⡹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⠦⣑⠩⡔⣎⢞⡰⢊⡑⠎⡴⢈⠣⡅⢎⡱⠫⣞⣭⢳⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢨⠱⣨⢕⡺⡜⡊⡔⠣⡌⢓⠤⢋⢆⡱⢊⠴⡑⡜⡼⣣⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⡘⢧⡓⢮⡱⢃⡱⢌⠓⣌⢃⢮⡡⢎⠰⣉⠦⡑⢦⡝⣧⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⣌⠱⢧⡙⢶⣙⢢⡑⢎⡑⢢⡙⢦⠳⣌⡳⢥⡲⣙⡮⣝⡞⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂⠄⡉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣁⢂⠄⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢂⠐⠠⢁⢂⡎⡅⢂⢡⠀⡐⠠⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣶⣶⣿⡿⡀⠀⠀⠀⠀⠀⢀⠀⠄⢲⠁⠐⡀⠆⢈⠐⣰⢋⡴⣩⠎⠁⢠⠔⡡⠊⡴⣩⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠉⡁⠋⠤⡉⡇⠘⠂⣍⠰⡈⢌⢣⠚⣌⣡⢴⢺⢀⢧⣋⠶⣁⠰⣌⢣⠚⢀⡴⡳⢥⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⠀⠀⠀⡇⡇⠀⢠⢞⡣⠌⢄⣣⡹⢼⣲⢭⣚⡌⢶⡉⠶⣡⠓⣬⡣⢟⡭⢶⡙⡇⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⢎⡵⡊⢄⠣⣝⡣⡇⢷⣌⢻⣎⢧⡳⣄⠘⠰⠭⡳⣜⢣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⡚⣴⣫⠴⣦⢵⣣⡽⠇⣎⡳⣎⢷⡳⢦⡤⡵⣚⠵⣎⢷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠽⡠⢍⠻⡜⢮⣓⡞⣾⣱⠻⣜⢧⡟⣧⢗⣱⢫⠺⣝⠮⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠃⡇⡱⢊⡱⢘⡂⠦⡙⡴⢫⡝⢮⡳⣝⢾⡌⠲⣹⡀⠩⡟⠈⠁⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⠸⡱⢌⢢⠣⠜⣑⠢⢇⠣⡞⣥⢻⣜⡳⢎⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠜⡤⣘⢥⢊⢆⠣⢍⢢⡙⠤⢣⡝⢦⣛⢶⡹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⠦⣑⠩⡔⣎⢞⡰⢊⡑⠎⡴⢈⠣⡅⢎⡱⠫⣞⣭⢳⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢨⠱⣨⢕⡺⡜⡊⡔⠣⡌⢓⠤⢋⢆⡱⢊⠴⡑⡜⡼⣣⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⡘⢧⡓⢮⡱⢃⡱⢌⠓⣌⢃⢮⡡⢎⠰⣉⠦⡑⢦⡝⣧⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⣌⠱⢧⡙⢶⣙⢢⡑⢎⡑⢢⡙⢦⠳⣌⡳⢥⡲⣙⡮⣝⡞⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠐⠠⠉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣅⡂⠄⡀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠄⠡⠈⢤⠚⡅⠠⢁⠂⠄⢂⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣷⣶⣾⡿⣀⠀⠀⠀⠀⢀⠀⡠⢀⠲⠁⠠⢁⠌⠠⢁⡸⢣⣙⢬⠓⢁⢈⠴⢁⠂⡼⣘⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠋⠉⠊⠤⡉⡇⠉⠊⣄⠣⡐⡁⢏⠲⣉⢤⢦⢳⠠⣏⠳⡜⡊⡄⢦⡩⢂⢁⡲⣏⠵⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⡀⠀⠀⡇⡇⠀⢠⠞⣥⠐⠡⢎⣱⢻⢼⢣⡛⣄⢏⡳⣘⠱⣘⢲⡍⣯⡙⢶⡩⠞⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⠞⡜⡎⡁⠻⣌⠯⡎⢧⡝⣺⢎⡷⣌⣃⠄⢋⠜⣣⡝⣲⠍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⣚⢭⡧⡼⣤⢽⣪⡽⠇⡻⣬⡻⣜⡳⢦⡤⢦⡛⣵⢺⡵⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠞⡡⠜⢳⠹⢮⡵⣚⣞⡳⢮⡝⣧⡟⣧⠟⣦⢛⠼⣣⠟⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠁⡏⠴⣉⠦⣙⠰⢂⠭⣸⠹⣇⢟⡶⣹⢞⡌⠖⣫⡀⠹⡫⠈⠁⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⢘⢧⡐⠎⡤⢋⠜⢢⠇⡹⡜⢮⡜⣧⢻⡌⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠼⡠⣘⠆⡜⣑⠢⢍⡊⢥⢊⡑⣎⠷⣜⢧⡻⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⢦⢡⠣⣑⢮⡹⠰⢌⠚⡤⣉⠖⡌⡒⠬⡙⠮⣳⡝⣮⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢘⠢⢆⢧⡙⣦⢉⠇⢎⡑⢲⠐⡎⢔⡩⢒⡡⢓⡘⢾⡱⢮⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⢪⡝⣎⠶⣹⠐⢎⡘⢆⡩⢆⣍⠲⢌⡔⢣⢌⡱⣌⡷⣹⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⡌⠥⢞⡬⢳⢥⠋⢦⣉⠖⡰⢪⡜⡹⢆⡞⣥⢲⡱⣎⠷⣭⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂⠄⡉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣁⢂⠄⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢂⠐⠠⢁⢂⡎⡅⢂⢡⠀⡐⠠⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣶⣶⣿⡿⡀⠀⠀⠀⠀⠀⢀⠀⠄⢲⠁⠐⡀⠆⢈⠐⣰⢋⡴⣩⠎⠁⢠⠔⡡⠊⡴⣩⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠉⡁⠋⠤⡉⡇⠘⠂⣍⠰⡈⢌⢣⠚⣌⣡⢴⢺⢀⢧⣋⠶⣁⠰⣌⢣⠚⢀⡴⡳⢥⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⠀⠀⠀⡇⡇⠀⢠⢞⡣⠌⢄⣣⡹⢼⣲⢭⣚⡌⢶⡉⠶⣡⠓⣬⡣⢟⡭⢶⡙⡇⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⢎⡵⡊⢄⠣⣝⡣⡇⢷⣌⢻⣎⢧⡳⣄⠘⠰⠭⡳⣜⢣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⡚⣴⣫⠴⣦⢵⣣⡽⠇⣎⡳⣎⢷⡳⢦⡤⡵⣚⠵⣎⢷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠽⡠⢍⠻⡜⢮⣓⡞⣾⣱⠻⣜⢧⡟⣧⢗⣱⢫⠺⣝⠮⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠃⡇⡱⢊⡱⢘⡂⠦⡙⡴⢫⡝⢮⡳⣝⢾⡌⠲⣹⡀⠩⡟⠈⠁⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⠸⡱⢌⢢⠣⠜⣑⠢⢇⠣⡞⣥⢻⣜⡳⢎⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠜⡤⣘⢥⢊⢆⠣⢍⢢⡙⠤⢣⡝⢦⣛⢶⡹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⠦⣑⠩⡔⣎⢞⡰⢊⡑⠎⡴⢈⠣⡅⢎⡱⠫⣞⣭⢳⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢨⠱⣨⢕⡺⡜⡊⡔⠣⡌⢓⠤⢋⢆⡱⢊⠴⡑⡜⡼⣣⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⡘⢧⡓⢮⡱⢃⡱⢌⠓⣌⢃⢮⡡⢎⠰⣉⠦⡑⢦⡝⣧⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⣌⠱⢧⡙⢶⣙⢢⡑⢎⡑⢢⡙⢦⠳⣌⡳⢥⡲⣙⡮⣝⡞⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠐⠠⠉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣁⢆⠀⡀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠄⠡⠈⢤⠚⡅⠠⢁⠂⠄⢂⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣶⣶⣿⡿⡀⠀⠀⠀⠀⠀⡀⠠⣀⠲⠁⠠⢁⠌⠠⢁⡸⢣⣙⢬⠓⢁⢈⠴⢁⠂⡼⣘⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠋⠁⠩⠴⡉⠇⠁⠊⣅⠢⢑⡀⢏⠲⣉⢤⢦⢳⠠⣏⠳⡜⡊⡄⢦⡩⢂⢁⡲⣏⠵⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⡀⠀⠀⡇⡇⠀⢠⢞⣣⠂⠌⣎⣱⢻⢼⢣⡛⣄⢏⡳⣘⠱⣘⢲⡍⣯⡙⢶⡩⠞⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⠞⡴⡃⠜⠲⣍⢮⡣⢷⣩⢛⣮⢳⣌⣃⠄⢋⠜⣣⡝⣲⠍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⡚⣵⣣⡼⣤⢽⣢⡽⠇⢮⡹⣎⢷⡳⢦⡤⢦⡛⣵⢺⡵⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠽⣀⠇⠻⡜⢧⡳⣝⣞⡳⣝⢮⡳⣏⠷⣝⡲⣝⠺⢧⡻⢄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠃⠧⡘⣌⠓⣘⢢⠱⢌⡲⢹⢎⢷⡹⣎⢿⡌⠱⣚⡀⠫⡽⠈⠀⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⠸⢥⢢⢙⡰⢌⠲⢡⠏⡜⣎⠶⣹⢎⡷⣊⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠜⡤⣘⠦⡑⣊⠔⣊⠥⠣⠜⢤⣋⡞⣵⢫⡞⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⠦⣑⠩⡔⣎⢳⡑⠬⡘⡔⡊⠵⣉⠦⣘⡘⢣⡟⣼⣓⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢨⠱⣨⢕⡺⣌⠣⣌⠣⡑⠦⡙⡔⢢⢒⡡⠜⡡⠜⣳⢎⡯⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⡘⢧⡓⢮⡱⢊⡱⢄⠣⢍⡒⣥⡘⡡⢆⡱⢊⡅⣣⡽⣎⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⣌⠱⢧⡙⢶⡹⢄⠣⢎⣑⢊⠴⣣⢳⡱⢎⡵⢪⡜⣱⡳⣭⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠐⠠⠉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣅⡂⠄⡀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠄⠡⠈⢤⠚⡅⠠⢁⠂⠄⢂⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣷⣶⣾⡿⣀⠀⠀⠀⠀⢀⠀⡠⢀⠲⠁⠠⢁⠌⠠⢁⡸⢣⣙⢬⠓⢁⢈⠴⢁⠂⡼⣘⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠋⠉⠊⠤⡉⡇⠉⠊⣄⠣⡐⡁⢏⠲⣉⢤⢦⢳⠠⣏⠳⡜⡊⡄⢦⡩⢂⢁⡲⣏⠵⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⡀⠀⠀⡇⡇⠀⢠⠞⣥⠐⠡⢎⣱⢻⢼⢣⡛⣄⢏⡳⣘⠱⣘⢲⡍⣯⡙⢶⡩⠞⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⠞⡜⡎⡁⠻⣌⠯⡎⢧⡝⣺⢎⡷⣌⣃⠄⢋⠜⣣⡝⣲⠍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⣚⢭⡧⡼⣤⢽⣪⡽⠇⡻⣬⡻⣜⡳⢦⡤⢦⡛⣵⢺⡵⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠞⡡⠜⢳⠹⢮⡵⣚⣞⡳⢮⡝⣧⡟⣧⠟⣦⢛⠼⣣⠟⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠁⡏⠴⣉⠦⣙⠰⢂⠭⣸⠹⣇⢟⡶⣹⢞⡌⠖⣫⡀⠹⡫⠈⠁⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⢘⢧⡐⠎⡤⢋⠜⢢⠇⡹⡜⢮⡜⣧⢻⡌⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠼⡠⣘⠆⡜⣑⠢⢍⡊⢥⢊⡑⣎⠷⣜⢧⡻⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⢦⢡⠣⣑⢮⡹⠰⢌⠚⡤⣉⠖⡌⡒⠬⡙⠮⣳⡝⣮⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢘⠢⢆⢧⡙⣦⢉⠇⢎⡑⢲⠐⡎⢔⡩⢒⡡⢓⡘⢾⡱⢮⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⢪⡝⣎⠶⣹⠐⢎⡘⢆⡩⢆⣍⠲⢌⡔⢣⢌⡱⣌⡷⣹⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⡌⠥⢞⡬⢳⢥⠋⢦⣉⠖⡰⢪⡜⡹⢆⡞⣥⢲⡱⣎⠷⣭⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⣿⠆⣀⠰⡀⢎⡐⢢⠁⠆⠡⠒⠈⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⡿⠛⠛⠋⡙⠩⠄⠂⠀⢀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⢿⣗⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠋⠁⠀⠀⠀⢂⣢⣴⣾⣿⣿⣿⣿⣿⣏⢆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢾⠀⢀⡪⢽⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣬⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⠟⡈⢿⡳⠎⢀⠑⢝⠂⠐⡠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣟⣾⢿⣻⢿⡽⠊⠀⠐⣸⡇⠀⠂⣈⠌⢆⠀⠀⢈⠷⢬⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀
⠙⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⡾⣯⣟⡿⣾⡽⣞⣯⣟⣛⣁⣀⠉⠉⠁⣟⠀⠄⡂⢆⡈⡉⢆⠁⢠⠈⠊⡋⠀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣻⢿⣽⢿⡿⣽⣻⢾⡽⣷⢯⣟⣷⣻⡽⠚⠁⠙⠉⠝⣻⣷⣞⢘⣆⠰⠉⠒⠥⣙⠢⢌⠆⡌⠠⠑⠤⠔⣣⢝⡺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣯⣟⡾⣯⣟⡷⣯⠿⣽⣯⣟⣾⣷⣿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⢒⠫⠀⣀⡀⠀⠉⠂⠣⣘⡄⡁⠂⠄⡁⠊⣵⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣳⢾⣝⣳⢾⡽⣯⣟⡷⣿⢾⣽⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⡔⠂⠉⠀⠉⠛⠻⢶⣤⣄⣀⠀⢰⡁⠇⠐⠠⢙⢰⠀⠘⠀⣡⢖⡲⢖⢢⠀⡄⢠⠐⡀
⠀⠀⠀⠀⠈⣿⣽⣻⣞⣯⢿⡽⣷⢯⡿⣽⣿⢾⣯⣟⡇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡝⠀⠁⠐⠎⡀⠓⠂⣤⢓⢮⡱⢫⠋⠀⢈⢄⠣⡐
⠀⠀⠀⠀⠀⣷⢯⣿⣟⡾⣯⣟⣿⣯⢿⡽⣿⣻⢾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂⠄⡉⠀⢀⠂⡱⠂⠠⢀⡘⠱⣆⠖⣧⠢⠁⠀
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣟⣷⣻⣾⢯⣿⣽⡏⣿⣻⡾⣽⣧⣁⢂⠄⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢂⠐⠠⢁⢂⡎⡅⢂⢡⠀⡐⠠⠙⢺⡔⠃⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣯⣷⡿⣌⡛⠾⢯⣙⠿⣽⢿⣿⣿⣿⣿⣶⣶⣿⡿⡀⠀⠀⠀⠀⠀⢀⠀⠄⢲⠁⠐⡀⠆⢈⠐⣰⢋⡴⣩⠎⠁⢠⠔⡡⠊⡴⣩⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠳⠹⡉⢌⡙⠷⡌⠧⣉⢇⣧⠄⣀⠘⢧⠈⢻⠏⠉⡁⠋⠤⡉⡇⠘⠂⣍⠰⡈⢌⢣⠚⣌⣡⢴⢺⢀⢧⣋⠶⣁⠰⣌⢣⠚⢀⡴⡳⢥⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣑⢢⠘⡰⢈⠆⡡⢊⡉⠳⢄⢒⠄⠀⠈⠔⠀⠀⠀⠀⡇⡇⠀⢠⢞⡣⠌⢄⣣⡹⢼⣲⢭⣚⡌⢶⡉⠶⣡⠓⣬⡣⢟⡭⢶⡙⡇⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠆⠐⠂⢆⠩⠔⢣⠘⡄⠣⢌⠱⣸⣾⢿⠐⣈⠀⠀⠀⡈⠀⡇⠃⠒⢭⢎⡵⡊⢄⠣⣝⡣⡇⢷⣌⢻⣎⢧⡳⣄⠘⠰⠭⡳⣜⢣⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠀⠀⠉⠀⢣⠘⣅⠊⡴⢯⣏⡿⢀⢣⠀⠀⠰⠀⠀⢇⣁⡰⢭⡚⣴⣫⠴⣦⢵⣣⡽⠇⣎⡳⣎⢷⡳⢦⡤⡵⣚⠵⣎⢷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢡⢂⡙⠐⠈⠫⡽⡎⢰⠀⠀⠐⠀⢀⢸⢸⡕⣣⠽⡠⢍⠻⡜⢮⣓⡞⣾⣱⠻⣜⢧⡟⣧⢗⣱⢫⠺⣝⠮⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⢙⡃⠔⡠⢘⡀⢄⠊⠞⡘⠃⡇⡱⢊⡱⢘⡂⠦⡙⡴⢫⡝⢮⡳⣝⢾⡌⠲⣹⡀⠩⡟⠈⠁⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠀⠂⠱⠠⡝⢢⠌⡘⡘⣥⠀⠸⡱⢌⢢⠣⠜⣑⠢⢇⠣⡞⣥⢻⣜⡳⢎⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⡀⠀⠀⠀⠌⣖⢠⡡⠜⡤⣘⢥⢊⢆⠣⢍⢢⡙⠤⢣⡝⢦⣛⢶⡹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢠⡈⠦⣑⠩⡔⣎⢞⡰⢊⡑⠎⡴⢈⠣⡅⢎⡱⠫⣞⣭⢳⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠤⡈⠃⢨⠱⣨⢕⡺⡜⡊⡔⠣⡌⢓⠤⢋⢆⡱⢊⠴⡑⡜⡼⣣⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠄⡈⢆⡉⢦⠁⡔⡘⢧⡓⢮⡱⢃⡱⢌⠓⣌⢃⢮⡡⢎⠰⣉⠦⡑⢦⡝⣧⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⢢⠑⢢⢸⣻⠰⣌⠱⢧⡙⢶⣙⢢⡑⢎⡑⢢⡙⢦⠳⣌⡳⢥⡲⣙⡮⣝⡞⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠣⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢢⣽⣶⣿⣿⣿⣿⡿⠂⡠⠐⠌⡰⢉⠜⠨⠑⠈⠐⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⠋⠋⠉⠉⡅⠂⠁⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⡉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⣷⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠿⠟⠛⠉⠉⠁⠀⠀⠀⣐⣨⣵⣾⣿⣿⣿⣿⣿⣿⣟⡲⠈⠃⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢯⡿⣄⢀⠦⢫⡜⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⣠⣴⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⠿⢁⠂⢿⡝⢊⠀⢂⠹⠁⠀⠑⠢⢤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⣽⣞⡿⣽⣻⢿⣽⣞⣁⡀⠈⢼⣏⠀⠈⠄⡆⠓⣄⣀⠠⠋⠶⣌⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢸⣿⡿⣯⣿⢿⣾⡿⣿⣻⢯⣷⣻⢾⡽⣯⣟⠯⣥⢤⣤⣤⣀⢨⠏⡔⢈⠤⣘⠤⡰⣀⠂⡘⡀⠄⣡⠀⠁⡦⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢘⣯⢿⣽⣻⣟⣷⣻⢷⣻⣟⡾⣽⣯⣿⠓⠀⠀⠈⠀⠐⠻⠿⠓⢈⠓⡬⠐⠈⠂⠓⠬⡘⣅⢢⠐⡀⠓⢈⠳⡍⢮⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⣯⣛⣾⢳⣻⢾⣝⣯⢷⣯⣟⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠤⣶⣤⣀⠀⠀⠈⠒⡅⡔⡈⠄⠂⢄⡳⠁⢢⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠠⣿⢶⣯⡷⣯⣟⡿⣾⡽⣯⣿⡾⣽⡷⣯⡅⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠉⠙⠛⠲⠆⠀⠘⡤⠁⠄⠡⠸⠘⣄⠀⠈⡼⡹⣌⡳⡜⠰⠈⠆⡡⢀
⠀⠀⠀⠀⠀⣿⣻⢾⣟⡷⣯⣟⣷⣻⢷⣻⣿⢿⡽⣯⢧⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⠀⠰⠃⠠⢀⢉⡜⠑⠃⢣⠹⡄⣀⡔⢒⠡⠂
⠀⠀⠀⠀⠠⣿⡽⣿⣿⣽⣳⢯⣿⣯⢿⣽⣻⣯⢿⣽⣻⡆⢄⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡀⠂⠌⡐⠠⠐⣠⠎⢀⠂⠡⠈⠌⢳⡱⣚⠌⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⣿⢿⣽⠹⡿⣯⣇⢿⡿⣾⣟⣿⣶⣯⣴⣠⣀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⠠⢀⠁⠂⠄⢡⠞⣡⠚⣤⠊⠁⣌⠂⡄⢱⠭⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠻⢿⣟⠍⠟⠻⠿⣧⠱⡌⡝⠦⣍⠉⠛⢻⡟⠻⡿⣿⠿⠿⢐⡒⢂⡤⠄⠄⡰⢀⠆⡑⡸⢄⡡⢂⢡⣨⠔⣎⠳⣥⠛⡀⢄⠳⡄⢊⢀⣏⢞⡁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢉⠆⡱⢈⠆⡩⠌⡓⠫⢧⡁⢂⠌⠀⠸⠒⠀⠀⠀⠈⠡⡆⠀⠀⡝⡆⢌⡐⢱⢊⡴⢏⣳⢺⡀⣞⡹⢤⠓⡜⢬⠳⡴⣊⢧⣋⠮⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢩⢁⠚⡐⠣⠜⡠⢃⠌⡑⢢⢉⣶⡏⠤⣰⠁⠀⠀⠐⠀⠃⡇⠠⣜⠵⣋⡆⠌⣜⢫⠞⡭⣖⢣⠷⣼⣑⢎⡱⠘⠜⣫⠗⣭⠲⣍⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⠀⠁⠀⠀⠈⠀⠃⠜⡠⠓⣌⠢⢡⣞⡿⣽⢂⡱⠀⠀⢀⠂⠀⠱⠠⢠⡜⣣⠧⣇⣌⣄⣫⢞⣱⣎⢧⣛⢶⡹⢮⣅⣈⣐⢦⡛⣴⣛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠂⠀⠄⠀⠀⠣⢄⠓⡅⠊⢻⡞⡆⢼⠀⠀⢘⠀⠀⢸⠱⣣⠞⣥⠋⡵⢺⡼⣱⢏⡶⣤⠾⣜⠯⣝⡷⣎⣧⢹⢲⢽⣣⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⢹⠽⡈⠔⡀⢨⠀⡐⢂⡝⢡⠟⢢⡙⠤⣃⠴⡉⣌⠓⣭⠻⣜⡻⣜⡳⣽⡘⢣⢯⠀⢳⣳⠘⠂⠀⠀⠀
⠀⠀⠀⠀⡈⠔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡈⠀⠣⢘⠠⣙⢆⡘⠄⡸⣄⠀⠱⣜⠢⢅⠲⡑⢬⡘⠦⡙⡦⡝⣼⢓⡧⡕⠀⠉⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⡐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠀⢀⠀⠀⠀⠀⠐⢆⠱⣠⢘⡄⢸⡄⣋⠬⡑⡜⡰⢌⡱⢡⠳⡝⣬⢻⡼⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠢⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠈⢀⠎⠳⡄⢣⠜⣦⢹⡰⢌⡱⠰⡡⢎⡐⢣⡙⠜⣣⠟⡶⣭⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⡘⠄⠀⠀⠀⠀⠀⡀⢠⢐⠊⢘⡱⢌⡥⢺⡔⢣⠒⡬⣐⠣⡑⢆⡩⢒⠬⣑⢢⠙⣗⡳⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⡔⢡⢂⠀⠀⠠⢀⠱⡈⢆⡇⠰⡘⢶⡩⢞⡥⢋⠆⡭⡐⢆⠣⣍⢢⡑⠎⡴⣈⠦⡹⣜⡳⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡰⢈⠆⡌⠰⡀⢆⠡⢒⢡⣟⠶⣁⠇⣣⠝⡮⣜⡡⢚⢤⡑⢪⡑⣎⠳⣌⡳⡱⢆⢯⡱⢯⡵⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢪⠡⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣷⣾⣿⣿⣿⠟⢂⠠⠠⠌⢢⠉⢆⠩⠘⠐⠀⠁⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣤⣾⣿⣿⠋⠋⠉⠉⡅⠊⠀⠀⠀⠌⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⢉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⣿⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⣀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⡱⠀⠣⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⢀⠢⡐⡀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣞⡿⣅⢀⠦⢚⡬⣛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⣠⣴⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡟⠡⡈⢿⣽⠊⢀⠂⡑⡃⠀⠐⠢⢤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣽⢾⣽⣻⢿⣻⣽⣞⣀⡀⠠⢽⡾⠀⠀⢂⠆⠱⢄⣀⠠⠙⠲⣌⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣻⣽⣿⣾⢿⡿⣯⢷⣻⣞⡿⣾⡽⣯⠷⣥⢤⣤⣤⣀⢸⡏⡄⢡⠬⣘⠄⢦⡐⠠⢑⠠⢀⢡⠀⠁⡦⢦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⢯⡷⣿⣻⣟⡾⣯⣟⣯⣿⣳⢯⣟⣷⠋⠀⠀⠀⠀⠐⠻⠿⠢⢀⠛⡤⠂⠀⠉⠂⠝⡘⢆⢡⢂⠈⡑⢈⠳⡍⣎⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣏⣟⣷⢻⣞⣽⣳⢯⣞⡷⣯⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠋⠠⣴⣦⣀⠀⠀⠉⠒⡌⡐⠄⠂⠄⠠⣏⠊⠐⣀⠀⢀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠠⣟⡾⣽⣮⢷⣻⣞⣯⣟⡾⣿⡽⣿⣟⣾⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠛⠳⠆⠀⠐⡥⠃⠈⠄⡱⠈⢆⡀⠀⡸⣍⢮⡱⡜⠰⠈⠆⡡⢀
⠀⠀⠀⠀⠀⣿⣽⣳⣯⣟⡷⣯⣷⢯⡿⣽⣿⣟⡾⣯⣇⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⠀⠤⠃⠈⠄⣈⠍⠓⠘⢢⠝⣄⢀⡔⢒⠡⠂
⠀⠀⠀⠀⠐⣿⣞⣿⣷⣯⣟⡷⣿⣯⢿⣽⣻⢾⣽⣳⡿⡄⢂⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠂⠌⠠⢀⠂⣰⢊⠐⡈⠐⡈⠘⢬⢣⡛⡄⠀⠀
⠀⠀⠀⠀⠀⢹⣾⣿⣿⣷⣿⣻⣽⠹⣿⣻⣎⢿⣾⡷⣿⣿⣶⣬⣄⡄⣀⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠀⠄⡈⠄⡁⢂⡼⢡⠎⡴⠐⡁⢄⠃⡄⢣⡝⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠻⢿⣟⠍⡟⠛⢯⡗⢢⠥⡛⢦⣹⠉⠛⢻⡟⠻⣿⢿⠿⠻⠜⡑⢂⠤⠄⠤⡐⢠⠂⠥⢘⢢⠡⠆⣁⣰⠄⣮⢱⢣⡛⢀⠡⡔⠣⠌⣀⡞⣜⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⡉⠆⡌⠣⢌⡉⢒⠫⢟⡠⢂⠅⠀⠘⡓⠀⠂⠀⠈⢱⠂⠀⠀⡜⢦⠑⡨⢘⢦⡹⢎⣧⢹⡀⣎⠧⢣⡑⢎⡱⣎⠵⡲⣍⡳⢬⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡅⢓⡘⢢⠑⣢⠘⡄⢃⠆⡘⣦⣟⡆⢴⠁⠀⠀⠐⠀⢸⠀⢠⡲⣍⢏⡆⢡⠺⣌⢗⡫⢖⡭⢳⡜⣇⠳⡘⠠⡙⡮⢗⡣⢧⣙⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠈⠀⠍⡄⠓⣌⠢⣘⣼⡻⣽⠆⢺⠀⠀⢀⠆⠀⠸⠰⢀⡴⣍⢎⣇⣢⣄⣫⢎⣵⣎⢷⣩⢟⡼⣫⣍⣐⣠⢜⡣⣝⢮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠁⠀⠄⠀⠀⠱⣀⠓⡌⠌⠻⡽⣎⠰⠀⠀⠠⠀⠀⡘⡁⣏⠶⣩⢎⠱⢳⣎⡳⢏⡶⣤⢞⡶⣛⠾⡵⣎⡷⣘⢮⢵⣫⢞⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠱⡹⢜⠡⡀⠰⠀⡐⢄⢣⠙⡇⢣⠊⡥⢃⠴⡉⢭⡘⢣⢟⡲⣭⢻⡝⣮⠓⢭⠺⡄⢫⢶⠙⠂⠀⠀⠀
⠀⠀⠀⠀⢀⠃⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠃⢆⡑⢸⢃⠔⡨⢈⢮⠀⢨⡓⡴⢉⠆⡍⠦⡘⡍⢆⠷⣘⢧⡻⣜⠇⠈⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⡉⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠐⢀⠀⠀⠀⠀⡀⢇⡰⢁⡜⢆⢰⡱⠌⢎⠜⡰⢃⡱⡘⡌⡻⡜⣎⢷⡹⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⢡⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⡜⢃⠣⡜⢨⢲⡹⡜⣈⠎⡱⠌⢆⡱⢌⡱⠙⢮⡳⣝⢮⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⡁⠀⠀⠀⠀⠀⢀⡀⢤⠉⢰⢉⠖⣡⢏⠶⡱⡘⢤⡙⠤⢋⡒⢌⡒⡌⡍⢆⡙⢮⡳⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡰⠑⠤⠀⠀⡀⠄⡉⢆⣘⡆⠰⣈⢮⡙⢦⣋⠖⡱⡘⢤⢊⠥⣣⠜⢢⡑⣌⠲⢡⢎⡳⣝⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠡⡉⢆⠱⡀⠔⡨⠔⣂⡾⡅⡣⠜⣢⢝⡣⢞⡌⡱⢜⡂⢎⠲⣥⢛⢦⡚⣤⢛⡜⣮⢳⣭⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢪⠡⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣷⣾⣿⣿⣿⠟⠂⡀⡐⢠⢂⠒⡐⢂⠒⠤⠁⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣤⣾⣿⣿⠟⠛⠋⢋⠍⠢⠁⠀⠀⠄⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⢉⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⣀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⡰⠈⠣⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀
⠀⠀⠀⠀⢀⠢⡐⡀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡯⣟⣿⡄⠀⡭⢎⡵⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣴⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡟⠡⢹⡾⣽⠋⠀⠌⠚⡅⠀⠡⢂⢄⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⡽⣾⡽⣿⢿⣻⣟⠎⢀⣀⣠⣿⣻⠀⠈⠐⡆⢡⣀⠀⣀⠋⢶⣩⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀
⠀⠀⠀⠀⠀⣿⡿⣟⣿⣽⣿⢾⡿⣿⡽⣞⣷⣻⢷⣻⡽⣯⢷⣥⣬⣄⣀⠀⡸⡑⡄⢂⢡⢚⡀⢤⠩⢀⠰⡀⠐⢡⠀⠂⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⡿⣽⣯⣟⣯⢿⣽⣳⢿⣻⢾⡽⣯⡷⠋⠁⠀⠁⠐⠩⢿⡷⠌⢡⢳⡌⠠⠉⠘⠢⢓⡌⢆⢡⠈⠄⠓⠂⢗⢮⠹⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣝⣳⡟⡾⣭⣟⡾⣽⢯⣟⣯⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠏⣀⣤⣤⡀⠀⠈⠘⠢⡍⡐⡈⠐⡈⢀⡟⠌⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠴⣟⣾⣳⢿⣽⣳⢯⡿⣽⢯⣿⣞⣿⢾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠛⠳⠶⠤⠄⠐⡥⠁⢂⠐⣠⠋⣄⠀⠁⢴⣋⠯⣝⣱⠀⠆⢢⠑⠄
⠀⠀⠀⠀⢰⣯⢷⣿⣻⢾⡽⣯⣿⡽⣯⣿⣾⢯⣟⣾⡄⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⢐⠂⠄⠠⢁⡜⠃⠊⠗⢮⡁⠀⡔⠢⡜⠀
⠀⠀⠀⠀⠘⣯⣿⣿⣯⡿⣽⣳⣿⣽⣳⣿⢯⣟⣾⣽⣳⠠⠀⠀⠀⠀⠀⠈⢿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠡⠈⠄⠠⢀⢡⠎⠐⡈⠐⠌⠰⢍⢯⣙⡂⠀⠀
⠀⠀⠀⠀⠈⢻⣽⣿⣿⣿⣻⣽⡧⢻⠻⣾⡝⣿⣾⣽⣯⣷⣥⣒⡀⡀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⢁⠂⠡⢈⠐⡴⢃⠎⡔⠤⠁⢌⡐⠈⢲⢥⡃⠀⠀
⠀⠀⠀⠀⠀⠈⠻⠿⣝⠜⢋⡙⠿⡆⣏⠔⡛⣌⣋⠘⠛⣿⠿⢿⣿⢿⠿⠿⢌⡡⠤⡄⠠⣀⠄⡠⠐⠤⢙⡄⢂⠄⢊⣐⡀⢞⡥⣋⡞⢁⠂⡜⢢⠔⢁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠤⡘⠰⡈⠤⣉⠑⣊⠻⢇⠢⢌⡂⠈⢾⠂⠀⠀⠀⠈⡆⠇⠀⠀⡸⢦⠉⡔⠸⣌⢣⡞⡽⣸⠀⣏⠶⡑⢎⠦⣙⡬⢥⠴⣊⡟⢦⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢩⠑⡌⠱⡘⠴⣀⠣⢄⠣⢌⣑⣶⡼⠠⣘⠁⠀⠀⠐⠀⡇⢇⡀⡴⣙⠞⡔⢨⢸⡬⣳⠜⣧⢣⠷⣜⢧⠹⡌⠰⡙⡮⣇⢯⡱⢎⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡐⠀⠁⠀⠀⠈⠁⠘⠰⣈⠱⢌⠒⢤⣞⡿⣽⠂⠧⠀⠀⢠⠁⠀⡇⠂⢀⡍⢧⣋⣇⣢⣌⣲⡍⡶⢬⣇⣛⢮⣏⢷⢚⡀⢁⡼⡜⢦⡽⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠂⠀⠄⠀⠀⠱⡈⠥⠃⠜⢳⣻⡌⢼⠀⠀⠠⠀⠀⣘⠙⡴⣹⢲⡉⠞⡵⣎⢷⡹⡽⣤⠾⣜⡳⢮⣏⠷⣍⡏⢶⣹⢧⣛⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⢣⠽⡌⢠⠀⢰⠀⡐⠌⡆⠻⠕⡣⢌⢓⡰⢌⡊⡕⡙⢮⡝⣮⡝⣧⢻⣝⡃⢞⡳⠂⢻⡼⠁⠄⠀⠀⠀
⠀⠀⠀⠀⡈⠔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠡⣂⠉⡼⠔⡈⠆⠱⡇⠀⢱⢎⠢⡜⢄⠣⡜⢨⠇⡜⣦⡙⣮⢳⢞⡃⠈⠑⠓⠈⠊⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⡐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠁⠀⠠⠀⠀⠀⠀⠀⡘⡤⢉⣂⠹⢄⡰⣊⠱⢌⡊⠵⣈⠣⡜⡰⢣⡝⣮⡝⣮⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠢⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠁⠀⠀⠀⢀⡙⢎⡰⡉⢦⡱⢣⢍⠢⢍⡒⡡⢃⠖⣡⢋⠜⣳⢺⢵⣚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⡀⢆⠋⠘⡔⢢⡙⢦⡝⢃⠎⡱⢢⢡⢃⡍⢲⡐⠪⢜⡠⢛⢮⡝⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡜⢠⠃⠀⠀⡀⢨⠐⡡⢬⠁⣠⠚⣜⡣⡝⢦⡙⡌⢎⡑⢆⠣⣌⡒⠥⣘⡱⢊⠴⣩⢞⡼⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⡘⠤⡐⢄⠣⡘⢰⡿⢰⠡⠎⢧⢳⡙⣦⠱⡘⣤⡉⢦⡙⢦⡝⣲⠱⡜⡜⢮⡵⣫⢞⡵⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⡿⠆⢀⠠⡐⢄⠒⡰⢐⠢⠐⠡⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⠿⠻⠛⡛⢋⡑⠈⠀⠀⠰⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⡧⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣀⣢⣤⣶⣿⣿⣿⣿⣿⣿⡟⣄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣼⠀⠀⢖⡽⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠟⢹⣾⡳⠎⢁⠚⣥⠋⠀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢷⣿⣿⡿⣿⣻⠍⠐⠂⢻⡇⠈⠠⢀⠰⡄⠀⠈⢀⢯⡹⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀
⠉⠉⠉⠉⠉⣿⣿⡿⣟⣿⣯⣿⣷⢿⣳⡿⣯⣟⡾⣯⢷⣯⣟⠗⠡⠀⠒⠒⢻⡗⠠⠐⠌⢆⠈⢓⠒⡈⢠⠑⠃⡍⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣿⣽⣻⣟⡿⣽⢾⡽⣯⣿⣽⣳⢯⡿⣽⣻⠞⠹⢛⠻⠷⣶⣄⠼⠸⣄⢡⠙⠨⣚⢌⡒⢄⠆⡠⢁⠲⢄⢠⡏⡵⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣞⣳⢯⠿⣽⢯⣻⣗⣿⣳⢯⣿⣽⣷⠁⠀⠀⠀⠀⠉⠛⠋⠁⡐⠳⢌⠀⢀⠀⠈⠘⠈⢖⠡⠆⡠⢈⠠⠘⡑⣳⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣯⢾⣽⣫⢿⣽⣻⢷⣻⢾⣯⣟⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠀⠈⠘⠻⠷⣦⣄⣀⠀⠉⢢⠅⠆⠠⢁⢰⢣⠀⠐⠂⣀⢴⢢⢴⢀⡀⠀⡄⢠⠂
⠀⠀⠀⠀⢸⣽⣻⣾⣽⣻⢾⣽⣯⣟⣯⣿⣞⣿⣻⡽⡄⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⡞⠀⠁⠂⡜⠀⠲⠄⣠⢏⣎⠳⣎⠗⠈⢁⡘⠄⡒
⠀⠀⠀⠀⣻⢾⣽⣿⡾⣽⣻⢾⣷⣻⢾⣽⣿⣳⢯⣟⣧⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠠⠘⠀⠈⡐⣠⠃⠠⢀⠹⢨⢆⡤⢧⡈⠂⠀
⠀⠀⠀⠀⠘⣯⣿⣿⣿⣷⢿⣯⡟⣿⣯⣯⢹⣯⣟⣾⢿⡄⢂⠀⠀⠀⠀⣈⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⠠⠁⠌⠠⢁⡴⡁⠄⡁⢂⠐⡀⠫⣜⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠹⣿⣿⢟⢿⠟⣯⣷⡘⢬⠛⣯⢻⡾⣟⣿⣾⣷⣮⣄⣄⣰⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⡀⢤⠃⠈⠄⠡⠈⠄⣡⠏⡴⣉⠶⠉⢀⠦⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⠒⡥⢈⠔⣋⠚⠦⣙⠤⣳⢦⠈⡙⢿⠛⢻⡟⠛⠉⠁⠪⢑⠲⠐⠂⠂⣜⠠⢃⠔⡪⢔⡱⣈⡤⣵⠂⣧⢛⡴⢋⢠⠰⣉⠆⠡⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⢣⡘⢄⡉⠒⡄⢊⠌⡙⠶⡐⢬⡁⠀⢫⠀⢀⠀⠀⠀⡝⠀⠀⢀⡭⢣⠂⢆⡱⢪⢲⢧⠳⡼⡠⢇⡏⢴⡉⢆⢳⡵⣚⢭⣣⠳⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠃⠌⠢⢌⠡⡘⠥⡘⢄⠚⡄⢣⣼⣷⣑⠢⡁⠀⠀⠀⠁⠀⠞⠀⠰⢧⣚⠽⡈⢄⢫⡝⣎⠏⢷⡱⣛⠾⣜⡢⡍⠘⡌⢷⡩⢖⣥⡛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠢⢡⠉⢖⠨⢼⡳⣏⡧⠘⡅⠀⠀⡌⠀⠀⡍⡁⢤⣚⢬⣛⣤⢮⡤⢿⣌⡻⠮⢵⣩⡟⣼⢣⣥⣂⢤⢖⡹⣎⢶⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⢍⠢⡙⠀⠉⢿⡹⡌⡇⠀⠀⡂⠀⠀⢧⢰⡳⡜⡎⠴⡙⠮⣝⢳⣎⢷⣚⢧⡳⣝⢮⡟⡶⣍⠞⡮⢵⣫⢞⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⢂⠔⡡⠀⡅⠠⡑⠸⡘⡱⠉⡜⢢⡑⠎⡤⢃⡌⢎⡝⢮⠳⣝⢮⡝⣷⡈⢻⠼⡀⢳⢯⠈⠂⠀⠀⠀
⠀⠀⠀⠠⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠈⠢⠔⡡⢛⠤⢁⠃⠥⡆⠀⡝⢦⡘⣑⢢⠣⡘⢦⢉⢞⡱⢎⡷⣹⢎⡑⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢈⠅⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠂⣀⠀⠀⠀⠀⠚⡤⢉⡴⢙⠄⣳⢂⠵⡈⠦⣑⠱⢌⡊⢼⡱⣫⡜⣧⢏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠁⠀⣈⢍⡱⢄⡋⢼⡡⢏⠴⣉⠲⣁⠞⡠⢍⢢⡑⢣⠻⣼⡹⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⢂⠀⠀⠀⠀⠀⢀⠠⡐⠜⠈⢆⠲⣌⡜⣣⠝⣌⠲⣈⠕⣂⢎⠱⡌⡒⠬⡑⢬⠱⣳⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⢠⠒⡌⡀⠀⠀⠄⡱⢈⠆⣹⠀⢆⢫⠞⡴⡹⡜⠬⡰⢑⡌⠲⣁⢎⠦⣑⢡⠣⢍⣂⢗⡯⡞⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠣⡘⠤⡁⢆⠰⡈⠔⡡⣺⡽⢊⡜⢢⡛⡴⢳⣍⠒⡥⣃⡌⡓⡜⣎⠳⣔⢫⡜⢦⣙⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠐⣸⣦⣷⣾⣿⣿⣿⠗⠀⢠⠀⠤⠠⠄⠤⢠⠐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣴⣾⣿⠿⠿⠟⠻⢛⡉⠀⠉⠀⡘⠄⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠇⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠉⠀⠀⠈⢂⣠⣴⣾⣿⣿⣿⣿⣿⣏⡘⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡶⡅⠀⢂⢒⡖⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⡟⠙⣿⡤⠎⢉⠺⣱⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣾⣿⣻⡿⣽⠞⠀⠑⣸⠥⠀⠂⢄⠡⡀⠀⠀⢩⢞⡹⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀
⠙⠛⠙⠋⠛⣿⣿⡿⣿⣻⣿⣯⣿⣷⣟⡿⣽⢿⡽⣞⡷⣯⢿⡽⠃⠀⠒⠒⢙⡆⢀⠈⠜⡀⠙⠢⠔⠂⡈⠉⠧⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡿⣯⣟⡿⣿⢷⣯⣟⣾⣽⣻⡽⣯⢿⣽⣻⠝⠛⠟⠛⠳⣶⣦⡌⢲⡀⠎⠣⠝⣢⠙⡔⣠⠱⠈⡀⢧⣀⢠⢟⡹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⢷⢯⡿⣽⣻⢾⡽⣳⢯⡿⣽⢯⣿⣾⠁⠀⠀⠀⠀⠀⠙⠙⠀⢩⢱⠃⠀⠀⠀⠉⠒⠄⣣⢡⠃⡀⠄⡈⢊⠑⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣮⣟⣳⢷⣯⢿⡽⣯⣟⣿⣽⣻⣿⢿⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠂⠐⠚⠿⣶⣤⣀⠀⠀⠁⣣⢐⠂⡐⠠⢸⠌⠀⠱⠠⢀⡤⡤⢤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠈⢹⣻⢾⣽⢯⣟⡾⣯⣟⡷⣯⣿⣞⣿⡽⣿⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠰⡉⠀⠄⠁⡎⠘⠢⠄⣀⡏⢶⣩⢓⠞⠀⠑⡈⢌⠡
⠀⠀⠀⠀⢼⣯⢿⣿⣻⢾⣽⣳⣯⣟⣷⣻⣿⢯⣟⣷⣳⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢀⠚⠀⠠⢁⢢⠃⠈⠄⢡⢫⡄⣤⢎⡘⠂⠀
⠀⠀⠀⠀⠸⣯⣿⣿⣿⣻⣾⡽⡟⣿⣞⣿⡹⣟⣾⣳⣯⣇⠂⠄⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⢀⠂⡐⢀⢂⡜⡂⠌⠐⡈⠄⡀⠻⣔⢫⠄⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⡷⣿⣯⠜⡝⠯⣗⢻⣯⣷⡿⣿⣿⣾⣧⣴⣤⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠁⡐⢀⠂⡐⠀⢦⡋⡴⣑⠎⠡⠐⡤⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠩⠛⠮⢋⠡⢊⠛⠧⢎⡱⣊⡵⣤⠈⡉⢿⡉⠛⡿⢋⠉⠉⠪⢔⠲⠐⠠⠂⣌⠰⠡⠌⡖⡰⡐⢡⡰⠴⢡⢧⡙⢶⡉⢐⠤⣋⠔⢁⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢎⡡⢊⠔⢢⠐⡡⢘⠓⢎⡰⢐⡀⠀⢋⠀⠀⠀⠀⠀⡹⠀⠀⢠⢏⡆⡑⢂⠧⣱⢻⣱⢋⢷⡘⢦⡛⠴⣡⠋⢶⡱⣚⢮⡱⣣⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠢⠘⠤⣈⠑⡚⠤⡃⠴⡁⢎⠰⣼⡿⡔⢂⡅⠀⠀⠀⠉⠀⠵⠠⢔⣫⠞⣼⢀⠃⢿⣘⠶⠍⣏⠶⣹⢳⣌⡳⠄⡙⡸⣕⢫⢖⣱⡃⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠈⠐⠡⢌⠣⡘⢄⢿⣹⡽⡇⡘⡆⠀⠀⡜⠀⠀⣃⠃⡴⣌⠻⣔⢧⡼⡬⢮⣹⣙⠮⣳⣹⡳⣎⢯⢥⣀⡤⢖⡫⣞⡖⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⢣⠘⡌⠂⠙⢞⡷⡁⡇⠀⠀⠆⠀⠀⢇⢺⡱⢎⡝⡌⢓⠞⣹⢧⣳⠳⣖⡳⢧⣛⢾⣫⢞⡵⣊⡟⢼⣣⣟⡀⠀⠀⠀⠀
⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡏⢣⠐⡡⠀⡇⠠⡘⠸⡈⠷⠉⡒⣌⢃⡚⠤⡒⢤⠋⡞⡝⢧⣛⢮⡝⣾⡐⣣⠽⠀⢗⡶⠁⠂⠀⠀⠀
⠀⠀⠀⠈⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠈⠢⢡⢡⠛⢄⠡⢡⠩⡆⠀⡳⢤⠣⢌⢓⡸⢄⢫⠱⣘⠧⣎⠷⣹⢖⡇⠀⠉⠁⠐⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠐⡌⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠈⠠⠀⠀⠀⠀⠀⡚⡄⢃⡴⠹⢄⡳⡌⡱⢊⢆⠲⠌⣆⠓⣌⢳⣬⠻⣵⢫⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠁⠀⣠⠙⠥⣒⡉⢦⢳⡱⡑⢪⠰⢩⠜⡠⢍⡂⢇⠎⡻⣜⡳⣞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠑⠀⠀⠀⠀⠀⢀⠠⡐⠜⠀⡙⢆⢥⡚⣥⠧⠱⢌⠣⣑⠣⣌⠱⢢⢉⠖⠬⡑⡜⣳⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⠀⠠⢃⠜⡀⠀⠀⠄⡱⢈⠆⣹⠀⠦⣙⢎⠶⣙⠦⡱⢉⢆⡓⢌⠲⣌⠥⢃⠎⣌⠓⡬⡔⣯⢞⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠃⢎⠰⡐⢄⠒⡌⠰⡁⣾⡝⢎⡒⢩⣎⠳⣍⠶⣁⠫⣔⠸⣈⠳⣬⠳⣍⠞⣤⢛⡴⣹⢮⡝⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣶⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠈⢟⠻⠉⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠆⠒⠵⣎⠼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⢂⣰⠹⣌⡳⣧⢭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⣎⢳⡱⠱⢌⡛⢮⡳⢮⡝⣶⢣⡟⣼⢣⡿⣜⡧⣚⡵⢹⣎⣯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡘⠃⡇⡹⠰⡘⢤⡑⢣⡘⣱⢫⠞⣵⢫⣞⣳⡈⢷⡸⡀⢳⠶⠁⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⠸⣅⠣⢍⠢⡜⢡⠒⠧⢌⡻⣔⣫⢖⣧⡓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠤⢋⢆⠓⡬⢑⢪⢑⡊⡵⢎⡼⣋⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡡⠎⡜⡰⡉⢆⠎⡔⡩⢎⠳⣝⢮⡳⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠔⢣⡘⡡⠜⣌⠲⢡⢒⡡⢚⠌⣷⣹⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⡙⢢⢡⡑⣣⠢⢍⠆⡣⠜⢢⡙⣶⢣⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⡘⢥⠢⣱⢣⡛⡤⢏⡵⣩⢧⡹⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢢⣵⣾⣶⣿⣿⣿⡿⠂⠀⡀⢄⠢⡐⠢⢌⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⣠⣴⣿⣿⡟⠿⠛⢛⠛⡀⠆⠁⠀⠠⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠀⠀⠀⠠⢑⣠⣶⣾⣿⣿⣿⣿⣿⣿⢣⡘⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡩⢁⡀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣧⡇⠀⣩⢚⡝⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⢀⣠⣴⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⡟⢉⢿⣳⠒⠁⡘⠼⡑⠁⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣯⡿⣾⢿⣿⢿⣯⠗⠈⠀⢊⣧⠈⠐⠠⠄⢣⡀⠀⠈⡰⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢛⣿⣿⢿⣻⣿⣽⡿⣷⣿⢿⣻⣽⣳⢿⣽⣻⣞⡟⣒⠒⠊⠉⠉⡷⢀⠠⡁⢇⡀⢩⠒⠊⢠⠀⡙⢣⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠸⣯⢿⣻⣿⣟⣯⣟⣯⣟⣯⡷⣯⣟⡿⣞⠷⠉⠛⠉⢛⣿⣶⣔⠃⣇⠰⠉⠒⡌⡣⢎⠰⡄⡡⠐⠘⣄⡀⡼⣋⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢿⡽⣾⡽⣾⡽⡾⣽⢾⣽⣳⣯⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⢠⠁⠏⠀⣀⠀⠀⠉⠢⠕⣰⠁⡂⠄⠠⠑⡉⣞⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣛⣮⢿⣱⢯⣷⣻⡽⣯⢿⣞⣷⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠒⠈⠘⠛⠿⢶⣤⣀⡀⠀⢣⠅⡊⠄⠡⢐⢧⠀⠉⠰⢀⡤⢤⠤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠀⠐⢹⣟⡾⣯⣟⡿⣞⣷⣻⣽⣻⣟⣾⣿⣳⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡞⠀⠄⠡⡘⠀⠦⠄⣀⠷⡸⣍⠾⠍⠂⠑⡈⢌⠡
⠀⠀⠀⠀⠀⣾⣽⣻⣷⢯⡿⣽⢾⡷⣯⢷⣿⣟⡾⣽⣻⡀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠠⠐⠁⠈⠄⢢⠍⠠⠁⢌⠳⡆⡤⣎⠘⠂⠀
⠀⠀⠀⠀⠀⢹⣞⣿⣿⣯⣿⣽⣻⣿⣽⣯⡟⣯⣟⣷⣻⣧⣁⠂⠄⠀⠀⢀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⡀⢂⠁⠂⢌⡔⠣⢀⠂⡁⠂⠄⠻⢴⡩⠆⠀⠀
⠀⠀⠀⠀⠀⠘⢯⣿⣿⡿⣷⣿⣻⡆⢯⠳⣿⡹⣿⢾⣟⣿⣿⣿⣷⣶⣤⣿⢟⡀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠆⢈⠐⡀⠂⠌⢰⠎⡜⡱⠆⠋⡀⣌⠂⡔⢨⡓⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⡙⢌⡙⠻⠦⡙⣤⢓⣬⡌⢈⠙⢇⠉⢟⠟⠋⠉⠊⠴⡉⡇⠘⠂⣕⠈⢆⠡⡘⢤⠒⣈⣤⡱⠌⣮⠹⣜⠣⢁⠦⣘⠄⢃⣠⡏⡵⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢈⠆⡌⡑⢢⠑⠤⡉⠜⡱⢌⢂⢆⠀⠘⡉⠀⠀⠀⢀⠂⡇⠀⢀⣎⣳⠈⡔⢩⢆⢻⣱⢆⣻⡀⢧⡛⢤⢋⠜⣲⡱⣚⢮⡱⢎⡵⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⠈⠒⡌⡑⠢⢍⠢⡑⢌⠰⣨⣿⣎⠒⢼⠀⠀⠀⠈⠀⠌⠇⠐⢧⡚⡴⡃⢄⢫⢞⣣⠏⢞⡴⣛⢮⣳⡘⡌⠘⠴⣙⠧⣎⠵⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠁⠃⢌⠂⢧⡈⠲⣽⣣⢿⡉⢜⠀⠀⢠⠁⠀⢊⡘⣠⠖⣭⢓⡧⣬⢦⡽⣆⣛⡞⢶⣩⡗⣧⢯⡍⣄⣢⠴⣛⣬⣛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⡔⡈⠇⠀⠻⡽⣖⢸⠀⠀⢨⠀⠀⠸⣘⣥⠻⡴⢋⠜⢣⢏⡶⣹⢞⡲⣎⢷⣚⢧⡟⡼⣇⠮⣝⢶⣣⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⠘⡄⢊⠄⣰⠀⢌⠡⢎⡐⠋⡕⣊⢌⠣⣒⠰⡡⢎⣱⢫⢞⡼⣣⢟⣳⡉⢞⡼⡀⢳⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠉⠔⣨⢘⠎⡰⠈⡔⢴⠀⢘⢦⢊⡱⢄⢣⡑⠆⢧⡘⣎⠶⣹⢎⣗⣓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠰⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢀⠀⠀⠀⠀⠘⢢⠑⡤⢎⠦⡘⢦⠑⡲⢌⡒⠬⣉⢆⠒⡭⢞⣥⢻⣬⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁⠈⠀⢀⣉⠫⢔⡊⠴⣙⢮⡑⢆⠣⣌⠱⣂⠎⡱⢌⠫⡜⣳⢎⡷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡔⡀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢢⢉⠖⣬⠳⣍⠦⣉⠆⡓⠤⢓⠤⡙⢤⢃⠣⢜⢨⣛⠶⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⢢⢁⠀⠀⠠⢈⠆⡱⢈⡇⠰⡘⣮⡙⢦⡛⡔⢢⡑⣊⠥⡙⣌⡒⣉⠆⣎⠱⢊⢦⡹⣝⢶⡁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⢌⠰⡀⢆⠡⢊⢔⡿⢲⡡⢃⢶⡙⢶⡹⢌⡱⢰⡁⠞⣰⢣⡝⣤⢛⡤⣛⡜⣮⢳⡝⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⠠⣀⠢⢄⠡⠒⡌⠰⠀⠆⠰⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⡟⠛⠛⠉⠣⢄⠃⠀⠀⠠⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠡⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⢿⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠛⠟⠛⠉⠉⠀⠀⠀⠀⣀⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⡳⡄⠛⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠰⡀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢯⣷⣇⠀⡘⢵⡩⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⣠⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣟⠏⢠⢻⣾⠉⠐⡀⠙⠎⠀⠈⡔⢠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣯⣿⢾⣟⣿⣻⢿⣯⡷⠃⣀⣀⡼⣟⠀⠐⡀⠇⡘⣄⡀⢀⠜⠳⣬⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣯⣿⣷⡿⣿⣟⣿⡽⣾⣽⣻⣞⡷⣯⡿⣮⣤⣤⣀⡀⠀⣏⡔⢠⢨⡱⣀⠤⡘⠠⢐⠠⢀⠸⠀⠐⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⣽⣻⢷⣻⡾⣽⢷⣯⢷⣯⡟⠁⠀⠀⠀⠀⠿⢿⠊⢀⠳⡰⠀⠀⠑⠪⢱⡑⡊⢔⠠⢈⠓⡄⠻⡴⣩⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢟⣾⢳⢯⣟⣯⢷⣻⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠋⢠⣤⣤⣀⠀⠀⠙⠢⠍⡄⢂⠠⠁⠄⣧⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⢷⣻⣞⣯⣟⡾⣽⢯⣷⣻⣷⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠙⠛⠶⠦⠀⠘⡤⠃⠠⢁⢸⠒⡄⠀⠉⢰⢺⡙⢯⡱⢠⠐⡄⢊⠁
⠀⠀⠀⠀⠀⣸⣯⢷⡿⣾⣽⣻⣽⣻⢾⣽⣷⣻⢷⣻⣇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠁⢀⠎⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠀⢿⣞⣿⣿⣳⢯⣷⣻⣯⣟⣾⠿⣯⢿⡽⣞⡇⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠄⠡⢈⠀⡈⢄⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⠸⣯⣿⣿⣿⣿⣳⣿⡝⡿⣾⣟⢿⣯⣿⣟⣿⣦⣥⣂⡀⢀⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⢀⠂⠌⡐⢀⠂⡴⢣⠜⣠⠢⢀⢁⠒⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⡿⠛⢷⣧⢚⠬⣙⠮⣹⠚⠻⢿⡟⠿⣿⢿⠿⠿⢜⡑⠤⠤⠠⡀⡰⠀⢆⢂⠱⣁⢂⠌⡐⣀⡂⢼⡡⢗⠮⢁⠐⡤⢊⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢡⠊⡔⡈⢆⢊⡑⢊⠳⢯⡐⠠⠅⠀⠙⢃⠀⢂⠀⠈⢰⠁⠀⠀⣸⢃⠆⡌⢸⢄⣣⠶⣹⣱⠀⡷⣙⢎⠣⢆⡹⢴⣡⠜⣎⢏⠧⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡅⢓⠰⠡⢎⡰⢈⠆⡑⢢⠉⣶⣏⡆⣰⠃⠀⠀⠀⠀⢸⡀⢠⣔⡣⢏⡖⢈⢜⡎⣖⠻⡕⣎⢧⡳⣅⠮⡱⠨⡜⢧⡮⡝⣎⠞⠅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠀⠉⠆⡐⠣⢜⠠⢃⣾⢻⡽⢆⠹⠀⠀⢀⠆⠀⢰⠰⢀⠴⣩⡓⣎⣄⣊⢞⡭⢖⡽⡜⣎⢷⡹⣖⡓⡀⢐⣩⢖⡹⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠂⠀⠄⠀⠈⠁⢎⡑⠎⠌⠻⣽⡆⢹⠀⠀⠠⠀⠀⢰⡈⣏⠞⣥⢋⠵⢮⡝⣮⣝⡻⢬⡼⣺⢭⠷⣭⢳⡝⣫⠶⣩⢷⣹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠱⡺⠴⠂⠄⢐⡀⠐⡄⢣⠙⡝⢊⡌⡒⠦⣘⠡⢎⡙⢣⢯⡵⣎⢿⡱⢯⡝⢦⣛⠌⢳⣭⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡒⠀⠃⢩⡘⢰⢆⠱⢈⠌⢮⠀⢸⡰⡡⠓⣌⠱⣂⠍⡞⠤⡳⢬⢳⡝⢯⡖⠈⠈⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠒⢌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠄⠀⠀⠀⠀⡀⢇⠌⢢⡘⢆⢰⣡⠱⢩⢄⡓⠬⡘⡔⠣⣝⡣⣏⣞⡳⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢭⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⠀⠀⠀⢀⡘⣋⠕⡸⣈⠶⣡⢇⢃⠲⠌⡥⠓⣌⠱⢌⠳⣩⢶⡹⣎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⠁⠀⠀⠀⠀⠀⢀⡀⢰⠈⢱⠐⢎⡱⣌⡳⣑⠪⢌⠣⡍⠴⣉⠤⢋⢆⢣⠒⡌⢷⣙⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡰⢁⠣⠀⠀⡀⠄⡑⢢⢘⡆⠰⣈⡞⣥⠳⣌⠳⢄⡋⡔⠣⢜⢢⡅⢎⡱⢌⠆⡭⣘⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠑⡌⢢⠑⡠⠐⡌⠰⣁⡾⡇⡱⢂⡹⢆⡻⣜⡡⠎⢴⡉⡜⣌⠳⡜⢦⡚⣥⢚⡴⣭⢳⡝⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⢀⢀⡐⢄⠢⡐⢄⠢⡐⠄⠢⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⠟⠟⠛⢛⠡⢂⠑⠈⠀⡐⠌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢉⠡⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⣿⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠻⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢡⡘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡱⢈⣤⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡾⣷⡀⠀⣍⡼⣙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣽⣾⣿⢿⠃⡙⣯⣟⠖⠁⢈⠣⠏⠀⠢⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀
⢴⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣯⣷⢿⣻⢿⣟⡿⣽⠯⠁⠀⢐⣹⡾⠀⠈⠄⠆⢣⡀⠀⢀⠚⡬⣝⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠀⠀⠀⠀⢸⣿⣯⣷⣿⣻⣿⣯⢿⣻⣞⡷⣯⢿⣽⣻⢾⣽⣋⣉⣉⡁⠀⢸⠟⡀⢂⢌⠣⢄⡈⡍⡀⢢⠀⡉⢌⠀⠂⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣳⣟⣾⣻⣟⣧⣟⣯⢷⡿⣽⢯⡿⣾⠝⠉⠀⠃⠉⠽⣿⡷⠌⠁⠷⣨⠀⠍⠒⠕⣢⠑⡆⢡⠐⡈⠦⠐⢧⡚⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣳⣛⣾⢳⣛⡾⣽⣞⣯⡿⣽⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠇⣀⣠⣄⡀⠀⠁⠚⠤⣃⡐⠠⠈⠄⡁⣞⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣳⣟⣮⢿⣽⣻⣷⣻⣞⣿⣳⣿⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠉⠙⠛⠶⢦⡤⠀⠰⣁⠃⠌⠐⢠⠓⡄⠀⠉⢰⢺⡙⢯⡱⢀⠆⡰⢈⠁
⠀⠀⠀⠀⠀⣿⣳⢿⣯⣟⣾⣳⣯⢷⣻⢾⣷⣿⣻⣞⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢀⠏⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠈⣷⢿⣿⣷⣻⣞⡷⣿⣯⣟⡿⣿⣞⡷⣯⣿⡀⠀⠀⠀⠀⠀⠘⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠌⡐⠈⡀⢀⢂⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣽⣿⣻⠼⡻⣿⣞⣯⣿⣳⣿⣧⣕⣂⡄⠀⢀⣾⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠌⡐⢀⠡⠐⣤⠣⡜⣠⠢⢀⠁⠆⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣟⠪⠏⡜⠿⣇⡱⢢⠹⣌⣓⠛⠷⣿⠿⢿⣿⡿⠿⠯⢍⠣⠄⡠⢀⡀⢠⠀⡄⢂⠱⡄⣈⠔⡀⢂⡐⡜⢦⠳⣜⠡⢀⠦⣉⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠰⣈⠒⠤⢡⢉⢃⠒⠯⣇⠰⣈⠂⠈⢾⠐⠀⠂⠀⠉⡆⠇⠀⠀⡸⢦⠘⡠⢙⠴⣡⠶⣙⢯⠀⡟⣬⠓⡌⢆⢣⠞⣤⠴⣋⠟⡦⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⡉⢅⠢⡙⠤⢃⡌⢢⢉⠒⡨⢑⣴⣏⠠⡸⠁⠀⠀⠀⠀⡇⡇⢀⡤⣛⢼⡠⠑⣸⢎⡵⣛⠭⣎⢧⡹⣆⢹⡘⢨⠺⣹⢆⡻⣜⡹⠆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠁⠈⠢⠡⡘⠢⣌⠢⢡⣟⢿⣹⠐⠧⠀⠀⢠⠂⠀⡇⠄⠀⣍⠳⣎⣇⣒⢌⣺⠜⡦⢽⡎⡞⣵⢫⣗⡚⢀⠐⣡⢞⡱⢮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠣⢄⠣⢃⠙⢾⡻⡌⢼⠀⠀⠠⠀⠀⢸⠘⣼⣘⡳⢊⠳⣎⡟⣼⢫⡽⣡⡼⣹⢳⣏⡾⣹⡍⣏⢳⢮⡻⣝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢹⠧⡌⢄⡀⢰⠀⡈⠜⡄⠷⠜⢣⡉⢖⡰⠌⡥⢋⡜⣣⡝⣧⢻⡴⢻⡵⡋⡼⣹⠂⢻⣬⠑⠄⠀⠀⠀
⠀⠀⠀⠀⡐⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠄⠂⠀⠣⡐⠤⡹⠔⡈⠆⠱⣃⠀⢣⡜⢂⠦⡙⢤⢃⡒⠇⣜⡲⣍⡞⣧⢻⡅⠈⠁⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣁⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⠀⠀⠀⠀⡘⡤⢉⣂⠹⢄⠰⣌⠣⡜⡘⡔⢊⡌⢓⠤⣳⡌⣟⡼⢳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢃⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠁⠀⠀⠀⢀⡙⢎⡰⣉⢆⡻⣔⢃⠲⡑⣌⠣⠜⣌⠲⡑⡚⢵⣫⢻⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡘⠄⠀⠀⠀⠀⠀⡀⠠⢄⠋⠘⡔⣒⠴⣪⢕⠪⣌⠱⡘⣄⠣⡍⠤⠓⣌⠱⣂⠭⣳⡝⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡜⢠⠃⠀⠀⠠⣀⠣⢌⣡⠁⣠⠚⡼⣌⢳⡱⢊⡱⢄⢣⡑⢢⡱⢌⢣⡙⡤⠓⡤⣓⢧⣛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⡘⠤⡐⠢⢄⠃⢦⣻⠰⡡⠎⣵⢊⢧⢳⠡⡒⣌⠲⢌⢣⡝⣎⠶⡸⢥⢳⡱⣝⢮⣭⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⡿⠆⢀⠠⡐⢄⠒⡰⢐⠢⠐⠡⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⠿⠻⠛⡛⢋⡑⠈⠀⠀⠰⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⡧⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣀⣢⣤⣶⣿⣿⣿⣿⣿⣿⡟⣄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣼⠀⠀⢖⡽⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠟⢹⣾⡳⠎⢁⠚⣥⠋⠀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢷⣿⣿⡿⣿⣻⠍⠐⠂⢻⡇⠈⠠⢀⠰⡄⠀⠈⢀⢯⡹⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀
⠉⠉⠉⠉⠉⣿⣿⡿⣟⣿⣯⣿⣷⢿⣳⡿⣯⣟⡾⣯⢷⣯⣟⠗⠡⠀⠒⠒⢻⡗⠠⠐⠌⢆⠈⢓⠒⡈⢠⠑⠃⡍⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣿⣽⣻⣟⡿⣽⢾⡽⣯⣿⣽⣳⢯⡿⣽⣻⠞⠹⢛⠻⠷⣶⣄⠼⠸⣄⢡⠙⠨⣚⢌⡒⢄⠆⡠⢁⠲⢄⢠⡏⡵⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣞⣳⢯⠿⣽⢯⣻⣗⣿⣳⢯⣿⣽⣷⠁⠀⠀⠀⠀⠉⠛⠋⠁⡐⠳⢌⠀⢀⠀⠈⠘⠈⢖⠡⠆⡠⢈⠠⠘⡑⣳⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣯⢾⣽⣫⢿⣽⣻⢷⣻⢾⣯⣟⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠀⠈⠘⠻⠷⣦⣄⣀⠀⠉⢢⠅⠆⠠⢁⢰⢣⠀⠐⠂⣀⢴⢢⢴⢀⡀⠀⡄⢠⠂
⠀⠀⠀⠀⢸⣽⣻⣾⣽⣻⢾⣽⣯⣟⣯⣿⣞⣿⣻⡽⡄⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⡞⠀⠁⠂⡜⠀⠲⠄⣠⢏⣎⠳⣎⠗⠈⢁⡘⠄⡒
⠀⠀⠀⠀⣻⢾⣽⣿⡾⣽⣻⢾⣷⣻⢾⣽⣿⣳⢯⣟⣧⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠠⠘⠀⠈⡐⣠⠃⠠⢀⠹⢨⢆⡤⢧⡈⠂⠀
⠀⠀⠀⠀⠘⣯⣿⣿⣿⣷⢿⣯⡟⣿⣯⣯⢹⣯⣟⣾⢿⡄⢂⠀⠀⠀⠀⣈⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⠠⠁⠌⠠⢁⡴⡁⠄⡁⢂⠐⡀⠫⣜⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠹⣿⣿⢟⢿⠟⣯⣷⡘⢬⠛⣯⢻⡾⣟⣿⣾⣷⣮⣄⣄⣰⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⡀⢤⠃⠈⠄⠡⠈⠄⣡⠏⡴⣉⠶⠉⢀⠦⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⠒⡥⢈⠔⣋⠚⠦⣙⠤⣳⢦⠈⡙⢿⠛⢻⡟⠛⠉⠁⠪⢑⠲⠐⠂⠂⣜⠠⢃⠔⡪⢔⡱⣈⡤⣵⠂⣧⢛⡴⢋⢠⠰⣉⠆⠡⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⢣⡘⢄⡉⠒⡄⢊⠌⡙⠶⡐⢬⡁⠀⢫⠀⢀⠀⠀⠀⡝⠀⠀⢀⡭⢣⠂⢆⡱⢪⢲⢧⠳⡼⡠⢇⡏⢴⡉⢆⢳⡵⣚⢭⣣⠳⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠃⠌⠢⢌⠡⡘⠥⡘⢄⠚⡄⢣⣼⣷⣑⠢⡁⠀⠀⠀⠁⠀⠞⠀⠰⢧⣚⠽⡈⢄⢫⡝⣎⠏⢷⡱⣛⠾⣜⡢⡍⠘⡌⢷⡩⢖⣥⡛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠢⢡⠉⢖⠨⢼⡳⣏⡧⠘⡅⠀⠀⡌⠀⠀⡍⡁⢤⣚⢬⣛⣤⢮⡤⢿⣌⡻⠮⢵⣩⡟⣼⢣⣥⣂⢤⢖⡹⣎⢶⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⢍⠢⡙⠀⠉⢿⡹⡌⡇⠀⠀⡂⠀⠀⢧⢰⡳⡜⡎⠴⡙⠮⣝⢳⣎⢷⣚⢧⡳⣝⢮⡟⡶⣍⠞⡮⢵⣫⢞⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⢂⠔⡡⠀⡅⠠⡑⠸⡘⡱⠉⡜⢢⡑⠎⡤⢃⡌⢎⡝⢮⠳⣝⢮⡝⣷⡈⢻⠼⡀⢳⢯⠈⠂⠀⠀⠀
⠀⠀⠀⠠⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠈⠢⠔⡡⢛⠤⢁⠃⠥⡆⠀⡝⢦⡘⣑⢢⠣⡘⢦⢉⢞⡱⢎⡷⣹⢎⡑⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢈⠅⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠂⣀⠀⠀⠀⠀⠚⡤⢉⡴⢙⠄⣳⢂⠵⡈⠦⣑⠱⢌⡊⢼⡱⣫⡜⣧⢏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠁⠀⣈⢍⡱⢄⡋⢼⡡⢏⠴⣉⠲⣁⠞⡠⢍⢢⡑⢣⠻⣼⡹⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⢂⠀⠀⠀⠀⠀⢀⠠⡐⠜⠈⢆⠲⣌⡜⣣⠝⣌⠲⣈⠕⣂⢎⠱⡌⡒⠬⡑⢬⠱⣳⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⢠⠒⡌⡀⠀⠀⠄⡱⢈⠆⣹⠀⢆⢫⠞⡴⡹⡜⠬⡰⢑⡌⠲⣁⢎⠦⣑⢡⠣⢍⣂⢗⡯⡞⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠣⡘⠤⡁⢆⠰⡈⠔⡡⣺⡽⢊⡜⢢⡛⡴⢳⣍⠒⡥⣃⡌⡓⡜⣎⠳⣔⢫⡜⢦⣙⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠐⣸⣦⣷⣾⣿⣿⣿⠗⠀⢠⠀⠤⠠⠄⠤⢠⠐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣴⣾⣿⠿⠿⠟⠻⢛⡉⠀⠉⠀⡘⠄⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠇⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠉⠀⠀⠈⢂⣠⣴⣾⣿⣿⣿⣿⣿⣏⡘⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡶⡅⠀⢂⢒⡖⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⡟⠙⣿⡤⠎⢉⠺⣱⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣾⣿⣻⡿⣽⠞⠀⠑⣸⠥⠀⠂⢄⠡⡀⠀⠀⢩⢞⡹⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀
⠙⠛⠙⠋⠛⣿⣿⡿⣿⣻⣿⣯⣿⣷⣟⡿⣽⢿⡽⣞⡷⣯⢿⡽⠃⠀⠒⠒⢙⡆⢀⠈⠜⡀⠙⠢⠔⠂⡈⠉⠧⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡿⣯⣟⡿⣿⢷⣯⣟⣾⣽⣻⡽⣯⢿⣽⣻⠝⠛⠟⠛⠳⣶⣦⡌⢲⡀⠎⠣⠝⣢⠙⡔⣠⠱⠈⡀⢧⣀⢠⢟⡹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⢷⢯⡿⣽⣻⢾⡽⣳⢯⡿⣽⢯⣿⣾⠁⠀⠀⠀⠀⠀⠙⠙⠀⢩⢱⠃⠀⠀⠀⠉⠒⠄⣣⢡⠃⡀⠄⡈⢊⠑⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣮⣟⣳⢷⣯⢿⡽⣯⣟⣿⣽⣻⣿⢿⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠂⠐⠚⠿⣶⣤⣀⠀⠀⠁⣣⢐⠂⡐⠠⢸⠌⠀⠱⠠⢀⡤⡤⢤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠈⢹⣻⢾⣽⢯⣟⡾⣯⣟⡷⣯⣿⣞⣿⡽⣿⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠰⡉⠀⠄⠁⡎⠘⠢⠄⣀⡏⢶⣩⢓⠞⠀⠑⡈⢌⠡
⠀⠀⠀⠀⢼⣯⢿⣿⣻⢾⣽⣳⣯⣟⣷⣻⣿⢯⣟⣷⣳⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢀⠚⠀⠠⢁⢢⠃⠈⠄⢡⢫⡄⣤⢎⡘⠂⠀
⠀⠀⠀⠀⠸⣯⣿⣿⣿⣻⣾⡽⡟⣿⣞⣿⡹⣟⣾⣳⣯⣇⠂⠄⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⢀⠂⡐⢀⢂⡜⡂⠌⠐⡈⠄⡀⠻⣔⢫⠄⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⡷⣿⣯⠜⡝⠯⣗⢻⣯⣷⡿⣿⣿⣾⣧⣴⣤⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠁⡐⢀⠂⡐⠀⢦⡋⡴⣑⠎⠡⠐⡤⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠩⠛⠮⢋⠡⢊⠛⠧⢎⡱⣊⡵⣤⠈⡉⢿⡉⠛⡿⢋⠉⠉⠪⢔⠲⠐⠠⠂⣌⠰⠡⠌⡖⡰⡐⢡⡰⠴⢡⢧⡙⢶⡉⢐⠤⣋⠔⢁⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢎⡡⢊⠔⢢⠐⡡⢘⠓⢎⡰⢐⡀⠀⢋⠀⠀⠀⠀⠀⡹⠀⠀⢠⢏⡆⡑⢂⠧⣱⢻⣱⢋⢷⡘⢦⡛⠴⣡⠋⢶⡱⣚⢮⡱⣣⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠢⠘⠤⣈⠑⡚⠤⡃⠴⡁⢎⠰⣼⡿⡔⢂⡅⠀⠀⠀⠉⠀⠵⠠⢔⣫⠞⣼⢀⠃⢿⣘⠶⠍⣏⠶⣹⢳⣌⡳⠄⡙⡸⣕⢫⢖⣱⡃⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠈⠐⠡⢌⠣⡘⢄⢿⣹⡽⡇⡘⡆⠀⠀⡜⠀⠀⣃⠃⡴⣌⠻⣔⢧⡼⡬⢮⣹⣙⠮⣳⣹⡳⣎⢯⢥⣀⡤⢖⡫⣞⡖⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⢣⠘⡌⠂⠙⢞⡷⡁⡇⠀⠀⠆⠀⠀⢇⢺⡱⢎⡝⡌⢓⠞⣹⢧⣳⠳⣖⡳⢧⣛⢾⣫⢞⡵⣊⡟⢼⣣⣟⡀⠀⠀⠀⠀
⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡏⢣⠐⡡⠀⡇⠠⡘⠸⡈⠷⠉⡒⣌⢃⡚⠤⡒⢤⠋⡞⡝⢧⣛⢮⡝⣾⡐⣣⠽⠀⢗⡶⠁⠂⠀⠀⠀
⠀⠀⠀⠈⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠈⠢⢡⢡⠛⢄⠡⢡⠩⡆⠀⡳⢤⠣⢌⢓⡸⢄⢫⠱⣘⠧⣎⠷⣹⢖⡇⠀⠉⠁⠐⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠐⡌⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠈⠠⠀⠀⠀⠀⠀⡚⡄⢃⡴⠹⢄⡳⡌⡱⢊⢆⠲⠌⣆⠓⣌⢳⣬⠻⣵⢫⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠁⠀⣠⠙⠥⣒⡉⢦⢳⡱⡑⢪⠰⢩⠜⡠⢍⡂⢇⠎⡻⣜⡳⣞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠑⠀⠀⠀⠀⠀⢀⠠⡐⠜⠀⡙⢆⢥⡚⣥⠧⠱⢌⠣⣑⠣⣌⠱⢢⢉⠖⠬⡑⡜⣳⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⠀⠠⢃⠜⡀⠀⠀⠄⡱⢈⠆⣹⠀⠦⣙⢎⠶⣙⠦⡱⢉⢆⡓⢌⠲⣌⠥⢃⠎⣌⠓⡬⡔⣯⢞⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠃⢎⠰⡐⢄⠒⡌⠰⡁⣾⡝⢎⡒⢩⣎⠳⣍⠶⣁⠫⣔⠸⣈⠳⣬⠳⣍⠞⣤⢛⡴⣹⢮⡝⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣶⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠈⢟⠻⠉⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠆⠒⠵⣎⠼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⢂⣰⠹⣌⡳⣧⢭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⣎⢳⡱⠱⢌⡛⢮⡳⢮⡝⣶⢣⡟⣼⢣⡿⣜⡧⣚⡵⢹⣎⣯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡘⠃⡇⡹⠰⡘⢤⡑⢣⡘⣱⢫⠞⣵⢫⣞⣳⡈⢷⡸⡀⢳⠶⠁⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⠸⣅⠣⢍⠢⡜⢡⠒⠧⢌⡻⣔⣫⢖⣧⡓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠤⢋⢆⠓⡬⢑⢪⢑⡊⡵⢎⡼⣋⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡡⠎⡜⡰⡉⢆⠎⡔⡩⢎⠳⣝⢮⡳⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠔⢣⡘⡡⠜⣌⠲⢡⢒⡡⢚⠌⣷⣹⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⡙⢢⢡⡑⣣⠢⢍⠆⡣⠜⢢⡙⣶⢣⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⡘⢥⠢⣱⢣⡛⡤⢏⡵⣩⢧⡹⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢢⣵⣾⣶⣿⣿⣿⡿⠂⠀⡀⢄⠢⡐⠢⢌⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⣠⣴⣿⣿⡟⠿⠛⢛⠛⡀⠆⠁⠀⠠⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠀⠀⠀⠠⢑⣠⣶⣾⣿⣿⣿⣿⣿⣿⢣⡘⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡩⢁⡀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣧⡇⠀⣩⢚⡝⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⢀⣠⣴⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⡟⢉⢿⣳⠒⠁⡘⠼⡑⠁⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣯⡿⣾⢿⣿⢿⣯⠗⠈⠀⢊⣧⠈⠐⠠⠄⢣⡀⠀⠈⡰⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢛⣿⣿⢿⣻⣿⣽⡿⣷⣿⢿⣻⣽⣳⢿⣽⣻⣞⡟⣒⠒⠊⠉⠉⡷⢀⠠⡁⢇⡀⢩⠒⠊⢠⠀⡙⢣⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠸⣯⢿⣻⣿⣟⣯⣟⣯⣟⣯⡷⣯⣟⡿⣞⠷⠉⠛⠉⢛⣿⣶⣔⠃⣇⠰⠉⠒⡌⡣⢎⠰⡄⡡⠐⠘⣄⡀⡼⣋⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢿⡽⣾⡽⣾⡽⡾⣽⢾⣽⣳⣯⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⢠⠁⠏⠀⣀⠀⠀⠉⠢⠕⣰⠁⡂⠄⠠⠑⡉⣞⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣛⣮⢿⣱⢯⣷⣻⡽⣯⢿⣞⣷⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠒⠈⠘⠛⠿⢶⣤⣀⡀⠀⢣⠅⡊⠄⠡⢐⢧⠀⠉⠰⢀⡤⢤⠤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠀⠐⢹⣟⡾⣯⣟⡿⣞⣷⣻⣽⣻⣟⣾⣿⣳⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡞⠀⠄⠡⡘⠀⠦⠄⣀⠷⡸⣍⠾⠍⠂⠑⡈⢌⠡
⠀⠀⠀⠀⠀⣾⣽⣻⣷⢯⡿⣽⢾⡷⣯⢷⣿⣟⡾⣽⣻⡀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠠⠐⠁⠈⠄⢢⠍⠠⠁⢌⠳⡆⡤⣎⠘⠂⠀
⠀⠀⠀⠀⠀⢹⣞⣿⣿⣯⣿⣽⣻⣿⣽⣯⡟⣯⣟⣷⣻⣧⣁⠂⠄⠀⠀⢀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⡀⢂⠁⠂⢌⡔⠣⢀⠂⡁⠂⠄⠻⢴⡩⠆⠀⠀
⠀⠀⠀⠀⠀⠘⢯⣿⣿⡿⣷⣿⣻⡆⢯⠳⣿⡹⣿⢾⣟⣿⣿⣿⣷⣶⣤⣿⢟⡀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠆⢈⠐⡀⠂⠌⢰⠎⡜⡱⠆⠋⡀⣌⠂⡔⢨⡓⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⡙⢌⡙⠻⠦⡙⣤⢓⣬⡌⢈⠙⢇⠉⢟⠟⠋⠉⠊⠴⡉⡇⠘⠂⣕⠈⢆⠡⡘⢤⠒⣈⣤⡱⠌⣮⠹⣜⠣⢁⠦⣘⠄⢃⣠⡏⡵⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢈⠆⡌⡑⢢⠑⠤⡉⠜⡱⢌⢂⢆⠀⠘⡉⠀⠀⠀⢀⠂⡇⠀⢀⣎⣳⠈⡔⢩⢆⢻⣱⢆⣻⡀⢧⡛⢤⢋⠜⣲⡱⣚⢮⡱⢎⡵⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⠈⠒⡌⡑⠢⢍⠢⡑⢌⠰⣨⣿⣎⠒⢼⠀⠀⠀⠈⠀⠌⠇⠐⢧⡚⡴⡃⢄⢫⢞⣣⠏⢞⡴⣛⢮⣳⡘⡌⠘⠴⣙⠧⣎⠵⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠁⠃⢌⠂⢧⡈⠲⣽⣣⢿⡉⢜⠀⠀⢠⠁⠀⢊⡘⣠⠖⣭⢓⡧⣬⢦⡽⣆⣛⡞⢶⣩⡗⣧⢯⡍⣄⣢⠴⣛⣬⣛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⡔⡈⠇⠀⠻⡽⣖⢸⠀⠀⢨⠀⠀⠸⣘⣥⠻⡴⢋⠜⢣⢏⡶⣹⢞⡲⣎⢷⣚⢧⡟⡼⣇⠮⣝⢶⣣⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⠘⡄⢊⠄⣰⠀⢌⠡⢎⡐⠋⡕⣊⢌⠣⣒⠰⡡⢎⣱⢫⢞⡼⣣⢟⣳⡉⢞⡼⡀⢳⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠉⠔⣨⢘⠎⡰⠈⡔⢴⠀⢘⢦⢊⡱⢄⢣⡑⠆⢧⡘⣎⠶⣹⢎⣗⣓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠰⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢀⠀⠀⠀⠀⠘⢢⠑⡤⢎⠦⡘⢦⠑⡲⢌⡒⠬⣉⢆⠒⡭⢞⣥⢻⣬⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁⠈⠀⢀⣉⠫⢔⡊⠴⣙⢮⡑⢆⠣⣌⠱⣂⠎⡱⢌⠫⡜⣳⢎⡷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡔⡀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢢⢉⠖⣬⠳⣍⠦⣉⠆⡓⠤⢓⠤⡙⢤⢃⠣⢜⢨⣛⠶⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⢢⢁⠀⠀⠠⢈⠆⡱⢈⡇⠰⡘⣮⡙⢦⡛⡔⢢⡑⣊⠥⡙⣌⡒⣉⠆⣎⠱⢊⢦⡹⣝⢶⡁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⢌⠰⡀⢆⠡⢊⢔⡿⢲⡡⢃⢶⡙⢶⡹⢌⡱⢰⡁⠞⣰⢣⡝⣤⢛⡤⣛⡜⣮⢳⡝⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⠠⣀⠢⢄⠡⠒⡌⠰⠀⠆⠰⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⡟⠛⠛⠉⠣⢄⠃⠀⠀⠠⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠡⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⢿⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠛⠟⠛⠉⠉⠀⠀⠀⠀⣀⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⡳⡄⠛⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠰⡀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢯⣷⣇⠀⡘⢵⡩⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⣠⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣟⠏⢠⢻⣾⠉⠐⡀⠙⠎⠀⠈⡔⢠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣯⣿⢾⣟⣿⣻⢿⣯⡷⠃⣀⣀⡼⣟⠀⠐⡀⠇⡘⣄⡀⢀⠜⠳⣬⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣯⣿⣷⡿⣿⣟⣿⡽⣾⣽⣻⣞⡷⣯⡿⣮⣤⣤⣀⡀⠀⣏⡔⢠⢨⡱⣀⠤⡘⠠⢐⠠⢀⠸⠀⠐⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⣽⣻⢷⣻⡾⣽⢷⣯⢷⣯⡟⠁⠀⠀⠀⠀⠿⢿⠊⢀⠳⡰⠀⠀⠑⠪⢱⡑⡊⢔⠠⢈⠓⡄⠻⡴⣩⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢟⣾⢳⢯⣟⣯⢷⣻⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠋⢠⣤⣤⣀⠀⠀⠙⠢⠍⡄⢂⠠⠁⠄⣧⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⢷⣻⣞⣯⣟⡾⣽⢯⣷⣻⣷⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠙⠛⠶⠦⠀⠘⡤⠃⠠⢁⢸⠒⡄⠀⠉⢰⢺⡙⢯⡱⢠⠐⡄⢊⠁
⠀⠀⠀⠀⠀⣸⣯⢷⡿⣾⣽⣻⣽⣻⢾⣽⣷⣻⢷⣻⣇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠁⢀⠎⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠀⢿⣞⣿⣿⣳⢯⣷⣻⣯⣟⣾⠿⣯⢿⡽⣞⡇⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠄⠡⢈⠀⡈⢄⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⠸⣯⣿⣿⣿⣿⣳⣿⡝⡿⣾⣟⢿⣯⣿⣟⣿⣦⣥⣂⡀⢀⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⢀⠂⠌⡐⢀⠂⡴⢣⠜⣠⠢⢀⢁⠒⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⡿⠛⢷⣧⢚⠬⣙⠮⣹⠚⠻⢿⡟⠿⣿⢿⠿⠿⢜⡑⠤⠤⠠⡀⡰⠀⢆⢂⠱⣁⢂⠌⡐⣀⡂⢼⡡⢗⠮⢁⠐⡤⢊⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢡⠊⡔⡈⢆⢊⡑⢊⠳⢯⡐⠠⠅⠀⠙⢃⠀⢂⠀⠈⢰⠁⠀⠀⣸⢃⠆⡌⢸⢄⣣⠶⣹⣱⠀⡷⣙⢎⠣⢆⡹⢴⣡⠜⣎⢏⠧⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡅⢓⠰⠡⢎⡰⢈⠆⡑⢢⠉⣶⣏⡆⣰⠃⠀⠀⠀⠀⢸⡀⢠⣔⡣⢏⡖⢈⢜⡎⣖⠻⡕⣎⢧⡳⣅⠮⡱⠨⡜⢧⡮⡝⣎⠞⠅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠀⠉⠆⡐⠣⢜⠠⢃⣾⢻⡽⢆⠹⠀⠀⢀⠆⠀⢰⠰⢀⠴⣩⡓⣎⣄⣊⢞⡭⢖⡽⡜⣎⢷⡹⣖⡓⡀⢐⣩⢖⡹⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠂⠀⠄⠀⠈⠁⢎⡑⠎⠌⠻⣽⡆⢹⠀⠀⠠⠀⠀⢰⡈⣏⠞⣥⢋⠵⢮⡝⣮⣝⡻⢬⡼⣺⢭⠷⣭⢳⡝⣫⠶⣩⢷⣹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠱⡺⠴⠂⠄⢐⡀⠐⡄⢣⠙⡝⢊⡌⡒⠦⣘⠡⢎⡙⢣⢯⡵⣎⢿⡱⢯⡝⢦⣛⠌⢳⣭⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡒⠀⠃⢩⡘⢰⢆⠱⢈⠌⢮⠀⢸⡰⡡⠓⣌⠱⣂⠍⡞⠤⡳⢬⢳⡝⢯⡖⠈⠈⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠒⢌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠄⠀⠀⠀⠀⡀⢇⠌⢢⡘⢆⢰⣡⠱⢩⢄⡓⠬⡘⡔⠣⣝⡣⣏⣞⡳⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢭⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⠀⠀⠀⢀⡘⣋⠕⡸⣈⠶⣡⢇⢃⠲⠌⡥⠓⣌⠱⢌⠳⣩⢶⡹⣎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⠁⠀⠀⠀⠀⠀⢀⡀⢰⠈⢱⠐⢎⡱⣌⡳⣑⠪⢌⠣⡍⠴⣉⠤⢋⢆⢣⠒⡌⢷⣙⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡰⢁⠣⠀⠀⡀⠄⡑⢢⢘⡆⠰⣈⡞⣥⠳⣌⠳⢄⡋⡔⠣⢜⢢⡅⢎⡱⢌⠆⡭⣘⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠑⡌⢢⠑⡠⠐⡌⠰⣁⡾⡇⡱⢂⡹⢆⡻⣜⡡⠎⢴⡉⡜⣌⠳⡜⢦⡚⣥⢚⡴⣭⢳⡝⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⢀⢀⡐⢄⠢⡐⢄⠢⡐⠄⠢⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⠟⠟⠛⢛⠡⢂⠑⠈⠀⡐⠌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢉⠡⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⣿⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠻⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢡⡘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡱⢈⣤⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡾⣷⡀⠀⣍⡼⣙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣽⣾⣿⢿⠃⡙⣯⣟⠖⠁⢈⠣⠏⠀⠢⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀
⢴⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣯⣷⢿⣻⢿⣟⡿⣽⠯⠁⠀⢐⣹⡾⠀⠈⠄⠆⢣⡀⠀⢀⠚⡬⣝⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠀⠀⠀⠀⢸⣿⣯⣷⣿⣻⣿⣯⢿⣻⣞⡷⣯⢿⣽⣻⢾⣽⣋⣉⣉⡁⠀⢸⠟⡀⢂⢌⠣⢄⡈⡍⡀⢢⠀⡉⢌⠀⠂⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣳⣟⣾⣻⣟⣧⣟⣯⢷⡿⣽⢯⡿⣾⠝⠉⠀⠃⠉⠽⣿⡷⠌⠁⠷⣨⠀⠍⠒⠕⣢⠑⡆⢡⠐⡈⠦⠐⢧⡚⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣳⣛⣾⢳⣛⡾⣽⣞⣯⡿⣽⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠇⣀⣠⣄⡀⠀⠁⠚⠤⣃⡐⠠⠈⠄⡁⣞⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣳⣟⣮⢿⣽⣻⣷⣻⣞⣿⣳⣿⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠉⠙⠛⠶⢦⡤⠀⠰⣁⠃⠌⠐⢠⠓⡄⠀⠉⢰⢺⡙⢯⡱⢀⠆⡰⢈⠁
⠀⠀⠀⠀⠀⣿⣳⢿⣯⣟⣾⣳⣯⢷⣻⢾⣷⣿⣻⣞⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢀⠏⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠈⣷⢿⣿⣷⣻⣞⡷⣿⣯⣟⡿⣿⣞⡷⣯⣿⡀⠀⠀⠀⠀⠀⠘⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠌⡐⠈⡀⢀⢂⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣽⣿⣻⠼⡻⣿⣞⣯⣿⣳⣿⣧⣕⣂⡄⠀⢀⣾⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠌⡐⢀⠡⠐⣤⠣⡜⣠⠢⢀⠁⠆⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣟⠪⠏⡜⠿⣇⡱⢢⠹⣌⣓⠛⠷⣿⠿⢿⣿⡿⠿⠯⢍⠣⠄⡠⢀⡀⢠⠀⡄⢂⠱⡄⣈⠔⡀⢂⡐⡜⢦⠳⣜⠡⢀⠦⣉⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠰⣈⠒⠤⢡⢉⢃⠒⠯⣇⠰⣈⠂⠈⢾⠐⠀⠂⠀⠉⡆⠇⠀⠀⡸⢦⠘⡠⢙⠴⣡⠶⣙⢯⠀⡟⣬⠓⡌⢆⢣⠞⣤⠴⣋⠟⡦⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⡉⢅⠢⡙⠤⢃⡌⢢⢉⠒⡨⢑⣴⣏⠠⡸⠁⠀⠀⠀⠀⡇⡇⢀⡤⣛⢼⡠⠑⣸⢎⡵⣛⠭⣎⢧⡹⣆⢹⡘⢨⠺⣹⢆⡻⣜⡹⠆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠁⠈⠢⠡⡘⠢⣌⠢⢡⣟⢿⣹⠐⠧⠀⠀⢠⠂⠀⡇⠄⠀⣍⠳⣎⣇⣒⢌⣺⠜⡦⢽⡎⡞⣵⢫⣗⡚⢀⠐⣡⢞⡱⢮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠣⢄⠣⢃⠙⢾⡻⡌⢼⠀⠀⠠⠀⠀⢸⠘⣼⣘⡳⢊⠳⣎⡟⣼⢫⡽⣡⡼⣹⢳⣏⡾⣹⡍⣏⢳⢮⡻⣝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢹⠧⡌⢄⡀⢰⠀⡈⠜⡄⠷⠜⢣⡉⢖⡰⠌⡥⢋⡜⣣⡝⣧⢻⡴⢻⡵⡋⡼⣹⠂⢻⣬⠑⠄⠀⠀⠀
⠀⠀⠀⠀⡐⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠄⠂⠀⠣⡐⠤⡹⠔⡈⠆⠱⣃⠀⢣⡜⢂⠦⡙⢤⢃⡒⠇⣜⡲⣍⡞⣧⢻⡅⠈⠁⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣁⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⠀⠀⠀⠀⡘⡤⢉⣂⠹⢄⠰⣌⠣⡜⡘⡔⢊⡌⢓⠤⣳⡌⣟⡼⢳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢃⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠁⠀⠀⠀⢀⡙⢎⡰⣉⢆⡻⣔⢃⠲⡑⣌⠣⠜⣌⠲⡑⡚⢵⣫⢻⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡘⠄⠀⠀⠀⠀⠀⡀⠠⢄⠋⠘⡔⣒⠴⣪⢕⠪⣌⠱⡘⣄⠣⡍⠤⠓⣌⠱⣂⠭⣳⡝⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡜⢠⠃⠀⠀⠠⣀⠣⢌⣡⠁⣠⠚⡼⣌⢳⡱⢊⡱⢄⢣⡑⢢⡱⢌⢣⡙⡤⠓⡤⣓⢧⣛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⡘⠤⡐⠢⢄⠃⢦⣻⠰⡡⠎⣵⢊⢧⢳⠡⡒⣌⠲⢌⢣⡝⣎⠶⡸⢥⢳⡱⣝⢮⣭⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⡿⠆⢀⠠⡐⢄⠒⡰⢐⠢⠐⠡⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⠿⠻⠛⡛⢋⡑⠈⠀⠀⠰⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⡧⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣀⣢⣤⣶⣿⣿⣿⣿⣿⣿⡟⣄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣼⠀⠀⢖⡽⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠟⢹⣾⡳⠎⢁⠚⣥⠋⠀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢷⣿⣿⡿⣿⣻⠍⠐⠂⢻⡇⠈⠠⢀⠰⡄⠀⠈⢀⢯⡹⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀
⠉⠉⠉⠉⠉⣿⣿⡿⣟⣿⣯⣿⣷⢿⣳⡿⣯⣟⡾⣯⢷⣯⣟⠗⠡⠀⠒⠒⢻⡗⠠⠐⠌⢆⠈⢓⠒⡈⢠⠑⠃⡍⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣿⣽⣻⣟⡿⣽⢾⡽⣯⣿⣽⣳⢯⡿⣽⣻⠞⠹⢛⠻⠷⣶⣄⠼⠸⣄⢡⠙⠨⣚⢌⡒⢄⠆⡠⢁⠲⢄⢠⡏⡵⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣞⣳⢯⠿⣽⢯⣻⣗⣿⣳⢯⣿⣽⣷⠁⠀⠀⠀⠀⠉⠛⠋⠁⡐⠳⢌⠀⢀⠀⠈⠘⠈⢖⠡⠆⡠⢈⠠⠘⡑⣳⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣯⢾⣽⣫⢿⣽⣻⢷⣻⢾⣯⣟⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠀⠈⠘⠻⠷⣦⣄⣀⠀⠉⢢⠅⠆⠠⢁⢰⢣⠀⠐⠂⣀⢴⢢⢴⢀⡀⠀⡄⢠⠂
⠀⠀⠀⠀⢸⣽⣻⣾⣽⣻⢾⣽⣯⣟⣯⣿⣞⣿⣻⡽⡄⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⡞⠀⠁⠂⡜⠀⠲⠄⣠⢏⣎⠳⣎⠗⠈⢁⡘⠄⡒
⠀⠀⠀⠀⣻⢾⣽⣿⡾⣽⣻⢾⣷⣻⢾⣽⣿⣳⢯⣟⣧⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠠⠘⠀⠈⡐⣠⠃⠠⢀⠹⢨⢆⡤⢧⡈⠂⠀
⠀⠀⠀⠀⠘⣯⣿⣿⣿⣷⢿⣯⡟⣿⣯⣯⢹⣯⣟⣾⢿⡄⢂⠀⠀⠀⠀⣈⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⠠⠁⠌⠠⢁⡴⡁⠄⡁⢂⠐⡀⠫⣜⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠹⣿⣿⢟⢿⠟⣯⣷⡘⢬⠛⣯⢻⡾⣟⣿⣾⣷⣮⣄⣄⣰⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⡀⢤⠃⠈⠄⠡⠈⠄⣡⠏⡴⣉⠶⠉⢀⠦⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⠒⡥⢈⠔⣋⠚⠦⣙⠤⣳⢦⠈⡙⢿⠛⢻⡟⠛⠉⠁⠪⢑⠲⠐⠂⠂⣜⠠⢃⠔⡪⢔⡱⣈⡤⣵⠂⣧⢛⡴⢋⢠⠰⣉⠆⠡⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⢣⡘⢄⡉⠒⡄⢊⠌⡙⠶⡐⢬⡁⠀⢫⠀⢀⠀⠀⠀⡝⠀⠀⢀⡭⢣⠂⢆⡱⢪⢲⢧⠳⡼⡠⢇⡏⢴⡉⢆⢳⡵⣚⢭⣣⠳⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠃⠌⠢⢌⠡⡘⠥⡘⢄⠚⡄⢣⣼⣷⣑⠢⡁⠀⠀⠀⠁⠀⠞⠀⠰⢧⣚⠽⡈⢄⢫⡝⣎⠏⢷⡱⣛⠾⣜⡢⡍⠘⡌⢷⡩⢖⣥⡛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠢⢡⠉⢖⠨⢼⡳⣏⡧⠘⡅⠀⠀⡌⠀⠀⡍⡁⢤⣚⢬⣛⣤⢮⡤⢿⣌⡻⠮⢵⣩⡟⣼⢣⣥⣂⢤⢖⡹⣎⢶⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⢍⠢⡙⠀⠉⢿⡹⡌⡇⠀⠀⡂⠀⠀⢧⢰⡳⡜⡎⠴⡙⠮⣝⢳⣎⢷⣚⢧⡳⣝⢮⡟⡶⣍⠞⡮⢵⣫⢞⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⢂⠔⡡⠀⡅⠠⡑⠸⡘⡱⠉⡜⢢⡑⠎⡤⢃⡌⢎⡝⢮⠳⣝⢮⡝⣷⡈⢻⠼⡀⢳⢯⠈⠂⠀⠀⠀
⠀⠀⠀⠠⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠈⠢⠔⡡⢛⠤⢁⠃⠥⡆⠀⡝⢦⡘⣑⢢⠣⡘⢦⢉⢞⡱⢎⡷⣹⢎⡑⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢈⠅⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠂⣀⠀⠀⠀⠀⠚⡤⢉⡴⢙⠄⣳⢂⠵⡈⠦⣑⠱⢌⡊⢼⡱⣫⡜⣧⢏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠁⠀⣈⢍⡱⢄⡋⢼⡡⢏⠴⣉⠲⣁⠞⡠⢍⢢⡑⢣⠻⣼⡹⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⢂⠀⠀⠀⠀⠀⢀⠠⡐⠜⠈⢆⠲⣌⡜⣣⠝⣌⠲⣈⠕⣂⢎⠱⡌⡒⠬⡑⢬⠱⣳⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⢠⠒⡌⡀⠀⠀⠄⡱⢈⠆⣹⠀⢆⢫⠞⡴⡹⡜⠬⡰⢑⡌⠲⣁⢎⠦⣑⢡⠣⢍⣂⢗⡯⡞⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠣⡘⠤⡁⢆⠰⡈⠔⡡⣺⡽⢊⡜⢢⡛⡴⢳⣍⠒⡥⣃⡌⡓⡜⣎⠳⣔⢫⡜⢦⣙⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠐⣸⣦⣷⣾⣿⣿⣿⠗⠀⢠⠀⠤⠠⠄⠤⢠⠐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣴⣾⣿⠿⠿⠟⠻⢛⡉⠀⠉⠀⡘⠄⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠇⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠉⠀⠀⠈⢂⣠⣴⣾⣿⣿⣿⣿⣿⣏⡘⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡶⡅⠀⢂⢒⡖⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⡟⠙⣿⡤⠎⢉⠺⣱⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣾⣿⣻⡿⣽⠞⠀⠑⣸⠥⠀⠂⢄⠡⡀⠀⠀⢩⢞⡹⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀
⠙⠛⠙⠋⠛⣿⣿⡿⣿⣻⣿⣯⣿⣷⣟⡿⣽⢿⡽⣞⡷⣯⢿⡽⠃⠀⠒⠒⢙⡆⢀⠈⠜⡀⠙⠢⠔⠂⡈⠉⠧⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡿⣯⣟⡿⣿⢷⣯⣟⣾⣽⣻⡽⣯⢿⣽⣻⠝⠛⠟⠛⠳⣶⣦⡌⢲⡀⠎⠣⠝⣢⠙⡔⣠⠱⠈⡀⢧⣀⢠⢟⡹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⢷⢯⡿⣽⣻⢾⡽⣳⢯⡿⣽⢯⣿⣾⠁⠀⠀⠀⠀⠀⠙⠙⠀⢩⢱⠃⠀⠀⠀⠉⠒⠄⣣⢡⠃⡀⠄⡈⢊⠑⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣮⣟⣳⢷⣯⢿⡽⣯⣟⣿⣽⣻⣿⢿⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠂⠐⠚⠿⣶⣤⣀⠀⠀⠁⣣⢐⠂⡐⠠⢸⠌⠀⠱⠠⢀⡤⡤⢤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠈⢹⣻⢾⣽⢯⣟⡾⣯⣟⡷⣯⣿⣞⣿⡽⣿⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠰⡉⠀⠄⠁⡎⠘⠢⠄⣀⡏⢶⣩⢓⠞⠀⠑⡈⢌⠡
⠀⠀⠀⠀⢼⣯⢿⣿⣻⢾⣽⣳⣯⣟⣷⣻⣿⢯⣟⣷⣳⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢀⠚⠀⠠⢁⢢⠃⠈⠄⢡⢫⡄⣤⢎⡘⠂⠀
⠀⠀⠀⠀⠸⣯⣿⣿⣿⣻⣾⡽⡟⣿⣞⣿⡹⣟⣾⣳⣯⣇⠂⠄⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⢀⠂⡐⢀⢂⡜⡂⠌⠐⡈⠄⡀⠻⣔⢫⠄⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⡷⣿⣯⠜⡝⠯⣗⢻⣯⣷⡿⣿⣿⣾⣧⣴⣤⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠁⡐⢀⠂⡐⠀⢦⡋⡴⣑⠎⠡⠐⡤⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠩⠛⠮⢋⠡⢊⠛⠧⢎⡱⣊⡵⣤⠈⡉⢿⡉⠛⡿⢋⠉⠉⠪⢔⠲⠐⠠⠂⣌⠰⠡⠌⡖⡰⡐⢡⡰⠴⢡⢧⡙⢶⡉⢐⠤⣋⠔⢁⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢎⡡⢊⠔⢢⠐⡡⢘⠓⢎⡰⢐⡀⠀⢋⠀⠀⠀⠀⠀⡹⠀⠀⢠⢏⡆⡑⢂⠧⣱⢻⣱⢋⢷⡘⢦⡛⠴⣡⠋⢶⡱⣚⢮⡱⣣⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠢⠘⠤⣈⠑⡚⠤⡃⠴⡁⢎⠰⣼⡿⡔⢂⡅⠀⠀⠀⠉⠀⠵⠠⢔⣫⠞⣼⢀⠃⢿⣘⠶⠍⣏⠶⣹⢳⣌⡳⠄⡙⡸⣕⢫⢖⣱⡃⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠈⠐⠡⢌⠣⡘⢄⢿⣹⡽⡇⡘⡆⠀⠀⡜⠀⠀⣃⠃⡴⣌⠻⣔⢧⡼⡬⢮⣹⣙⠮⣳⣹⡳⣎⢯⢥⣀⡤⢖⡫⣞⡖⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⢣⠘⡌⠂⠙⢞⡷⡁⡇⠀⠀⠆⠀⠀⢇⢺⡱⢎⡝⡌⢓⠞⣹⢧⣳⠳⣖⡳⢧⣛⢾⣫⢞⡵⣊⡟⢼⣣⣟⡀⠀⠀⠀⠀
⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡏⢣⠐⡡⠀⡇⠠⡘⠸⡈⠷⠉⡒⣌⢃⡚⠤⡒⢤⠋⡞⡝⢧⣛⢮⡝⣾⡐⣣⠽⠀⢗⡶⠁⠂⠀⠀⠀
⠀⠀⠀⠈⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠈⠢⢡⢡⠛⢄⠡⢡⠩⡆⠀⡳⢤⠣⢌⢓⡸⢄⢫⠱⣘⠧⣎⠷⣹⢖⡇⠀⠉⠁⠐⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠐⡌⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠈⠠⠀⠀⠀⠀⠀⡚⡄⢃⡴⠹⢄⡳⡌⡱⢊⢆⠲⠌⣆⠓⣌⢳⣬⠻⣵⢫⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠁⠀⣠⠙⠥⣒⡉⢦⢳⡱⡑⢪⠰⢩⠜⡠⢍⡂⢇⠎⡻⣜⡳⣞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠑⠀⠀⠀⠀⠀⢀⠠⡐⠜⠀⡙⢆⢥⡚⣥⠧⠱⢌⠣⣑⠣⣌⠱⢢⢉⠖⠬⡑⡜⣳⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⠀⠠⢃⠜⡀⠀⠀⠄⡱⢈⠆⣹⠀⠦⣙⢎⠶⣙⠦⡱⢉⢆⡓⢌⠲⣌⠥⢃⠎⣌⠓⡬⡔⣯⢞⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠃⢎⠰⡐⢄⠒⡌⠰⡁⣾⡝⢎⡒⢩⣎⠳⣍⠶⣁⠫⣔⠸⣈⠳⣬⠳⣍⠞⣤⢛⡴⣹⢮⡝⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣶⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠈⢟⠻⠉⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠆⠒⠵⣎⠼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⢂⣰⠹⣌⡳⣧⢭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⣎⢳⡱⠱⢌⡛⢮⡳⢮⡝⣶⢣⡟⣼⢣⡿⣜⡧⣚⡵⢹⣎⣯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡘⠃⡇⡹⠰⡘⢤⡑⢣⡘⣱⢫⠞⣵⢫⣞⣳⡈⢷⡸⡀⢳⠶⠁⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⠸⣅⠣⢍⠢⡜⢡⠒⠧⢌⡻⣔⣫⢖⣧⡓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠤⢋⢆⠓⡬⢑⢪⢑⡊⡵⢎⡼⣋⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡡⠎⡜⡰⡉⢆⠎⡔⡩⢎⠳⣝⢮⡳⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠔⢣⡘⡡⠜⣌⠲⢡⢒⡡⢚⠌⣷⣹⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⡙⢢⢡⡑⣣⠢⢍⠆⡣⠜⢢⡙⣶⢣⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⡘⢥⠢⣱⢣⡛⡤⢏⡵⣩⢧⡹⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠖⡤⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢘⣴⣧⣶⣶⣾⣶⣶⠆⢀⠠⢀⠔⡠⢂⠔⡠⠄⠂⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣿⣿⠿⠿⠟⡛⢛⡁⠂⠀⠀⠡⠈⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣋⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠆⣂⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⠡⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢧⡂⠀⣐⢮⣒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⣀⣴⣧⣮⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣷⣿⡿⠉⢿⣳⠔⠉⡐⢭⡚⠀⢂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣿⣾⡿⣿⢿⣻⡗⠁⠘⢈⡇⠀⢂⠐⡀⢃⠀⠀⠈⢴⢫⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀
⠛⠛⠛⠛⠛⢻⣿⡿⣿⣿⣻⣿⣽⣿⣾⢿⣻⣽⣻⣞⣷⣻⣽⡻⠓⠐⠒⠀⠊⡽⠀⡀⠜⡄⢈⠓⠆⢊⠠⢁⠹⢪⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢸⣟⣿⣻⣽⣿⢯⣟⡾⣽⣯⡷⣯⢷⣻⣞⠷⠋⠻⠛⠛⢿⣶⣔⠱⣆⠐⠓⠨⣃⠞⡌⢤⡘⡀⠐⠰⡄⢀⡞⣹⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣻⢾⡽⣾⣽⣻⣞⡿⣽⣾⡽⣯⣿⣷⡏⠀⠀⠀⠀⠀⠉⠉⠈⠠⠣⠝⠀⠀⠀⠈⠘⠤⣑⠬⡁⢂⠈⡁⠜⠡⣏⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣏⣷⡻⣵⢯⡶⣯⣟⣷⣻⣽⣳⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⡠⠤⠓⠈⠚⠻⠷⣦⣄⣀⠀⠈⢡⠆⡃⠠⢈⠰⢭⠀⠁⢂⢀⣠⢤⡤⢄⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠒⣿⣞⡷⣿⡽⣯⣟⣷⣻⢾⣽⣷⣻⣯⣟⣧⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⡖⠁⠐⠠⡜⠈⠦⣀⣀⠞⡜⣦⠹⠎⠐⠁⠌⡱⠠
⠀⠀⠀⠀⠀⣷⢯⣿⣷⣻⢷⣻⣾⣽⣻⢾⣟⣿⣳⣟⣾⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠀⠒⠁⢀⠂⢤⠊⢉⠘⠰⢛⡤⣠⢎⡘⠄⠁
⠀⠀⠀⠀⠀⣿⢯⣿⣿⣽⣯⣷⣻⢷⣯⣿⡏⣷⣟⣾⣳⣟⣠⢀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⠠⢈⠐⡀⢂⡌⡃⠐⠠⢈⠐⡈⠲⣥⢫⠄⠀⠀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣷⣿⡘⣟⠾⣟⡸⣿⡾⣟⣿⣿⣿⣾⣶⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠆⠈⠄⡁⢂⠐⢠⠏⡴⣑⠎⠊⠄⢢⢂⠁⣜⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠻⠯⡸⢋⠜⠛⠷⡌⡲⣉⢇⣧⡈⠉⠙⢏⠘⢻⠟⠛⠉⠣⢅⡑⡖⠐⠂⣜⠠⠃⡔⡌⢦⠘⣂⡰⣤⠊⢧⡛⡴⠍⢂⡡⢎⠅⠊⣠⣎⢧⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⡑⢢⢐⠡⠌⡌⢓⠣⣉⠱⣀⠀⠘⡁⠈⠀⠀⠀⡇⡆⠀⢀⡎⢧⢁⠢⣙⢤⢻⢬⠳⡼⡈⢧⡹⢰⢋⠦⣱⢎⡶⢫⠵⣊⠶⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠎⠰⢀⠣⡘⠒⠬⣂⠱⡈⢆⠱⣨⣾⣞⠐⡈⠀⠀⠀⠈⠀⡇⠇⠐⢧⢞⡹⡄⠡⢎⣛⣎⠏⡷⡱⢏⡷⣅⣋⠆⡘⠬⡳⣍⢧⢫⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠈⠑⠢⢐⠱⢌⠢⡱⣯⣛⡾⢁⢇⠀⠀⠰⠁⠀⢇⢂⡰⣍⢮⡱⣧⣱⣊⢧⣎⣳⡺⢕⣫⢞⡵⣫⢍⣠⣐⡴⣋⣮⢳⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠰⣈⠱⠡⠌⠻⡽⡆⢸⠀⠀⠡⠀⠀⣸⢸⡳⣜⡲⢡⢃⠷⢎⡷⢎⡗⣦⢻⡜⣯⠾⣧⣛⡦⢳⠼⣱⢮⣏⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠳⠙⡆⢢⢀⠰⢀⠰⡀⠞⡱⠊⢇⡱⢊⡔⠣⣌⠣⡙⢼⢣⢟⣲⢻⡜⣧⡙⢣⢻⠀⢳⢶⠉⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠁⠈⠠⠁⡆⡝⢆⠡⡘⡘⣥⠀⠸⢦⡑⠬⢱⢠⠣⣉⢇⡚⡜⢦⢻⡜⣧⡓⠀⠉⠁⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⢀⠀⠀⠀⠀⠌⣆⢡⡰⠜⠤⣘⠆⡥⢋⢆⡱⡘⢔⠢⡜⡹⢎⡳⣝⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠌⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠁⠈⠀⢠⡉⢦⠘⣌⠳⣌⠻⣄⠣⡌⠴⡑⣊⠱⡰⠩⢍⠳⣝⢮⡳⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢎⠀⠀⠀⠀⠀⠀⡀⢄⡂⠃⢘⠢⣍⢦⢛⡬⠓⡤⠓⢬⡑⠜⣠⠣⢱⢩⠘⣆⠙⣮⢳⡭⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠜⡠⢃⠀⠀⠠⡈⢆⡑⢢⠁⡔⣊⠷⣘⢎⡳⢌⢓⡰⣉⠦⣘⡱⣂⠝⢢⡡⠓⡌⡼⣜⢧⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢢⠑⠢⢄⡐⢢⠑⢢⢸⣻⠰⡘⢤⢛⡜⢮⡱⢊⠦⣑⡔⢊⠴⣣⠝⣬⢣⠵⡹⣜⡱⣏⠾⣥⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢢⣵⣾⣶⣿⣿⣿⡿⠂⠀⡀⢄⠢⡐⠢⢌⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⣠⣴⣿⣿⡟⠿⠛⢛⠛⡀⠆⠁⠀⠠⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠀⠀⠀⠠⢑⣠⣶⣾⣿⣿⣿⣿⣿⣿⢣⡘⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡩⢁⡀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣧⡇⠀⣩⢚⡝⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⢀⣠⣴⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⡟⢉⢿⣳⠒⠁⡘⠼⡑⠁⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣯⡿⣾⢿⣿⢿⣯⠗⠈⠀⢊⣧⠈⠐⠠⠄⢣⡀⠀⠈⡰⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢛⣿⣿⢿⣻⣿⣽⡿⣷⣿⢿⣻⣽⣳⢿⣽⣻⣞⡟⣒⠒⠊⠉⠉⡷⢀⠠⡁⢇⡀⢩⠒⠊⢠⠀⡙⢣⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠸⣯⢿⣻⣿⣟⣯⣟⣯⣟⣯⡷⣯⣟⡿⣞⠷⠉⠛⠉⢛⣿⣶⣔⠃⣇⠰⠉⠒⡌⡣⢎⠰⡄⡡⠐⠘⣄⡀⡼⣋⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢿⡽⣾⡽⣾⡽⡾⣽⢾⣽⣳⣯⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⢠⠁⠏⠀⣀⠀⠀⠉⠢⠕⣰⠁⡂⠄⠠⠑⡉⣞⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣛⣮⢿⣱⢯⣷⣻⡽⣯⢿⣞⣷⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠒⠈⠘⠛⠿⢶⣤⣀⡀⠀⢣⠅⡊⠄⠡⢐⢧⠀⠉⠰⢀⡤⢤⠤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠀⠐⢹⣟⡾⣯⣟⡿⣞⣷⣻⣽⣻⣟⣾⣿⣳⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡞⠀⠄⠡⡘⠀⠦⠄⣀⠷⡸⣍⠾⠍⠂⠑⡈⢌⠡
⠀⠀⠀⠀⠀⣾⣽⣻⣷⢯⡿⣽⢾⡷⣯⢷⣿⣟⡾⣽⣻⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠠⠐⠁⠈⠄⢢⠍⠠⠁⢌⠳⡆⡤⣎⠘⠂⠀
⠀⠀⠀⠀⠀⢹⣞⣿⣿⣯⣿⣽⣻⣿⣽⣯⡟⣯⣟⣷⣻⣧⣂⠄⡀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⡀⢂⠁⠂⢌⡔⠣⢀⠂⡁⠂⠄⠻⢴⡩⠆⠀⠀
⠀⠀⠀⠀⠀⠘⢯⣿⣿⡿⣷⣿⣻⡆⢯⠳⣿⡹⣿⢾⣟⣷⣿⣿⣷⣾⣤⣿⠟⡀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠆⢈⠐⡀⠂⠌⢰⠎⡜⡱⠆⠋⡀⣌⠂⡔⢨⡓⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⡙⢌⡙⠻⠦⡙⣤⢓⣬⡌⢈⠙⢏⠉⢟⠻⠋⠉⠚⠥⡉⡇⠘⠂⣕⠈⢆⠡⡘⢤⠒⣈⣤⡱⠌⣮⠹⣜⠣⢁⠦⣘⠄⢃⣠⡏⡵⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢈⠆⡌⡑⢢⠑⠤⡉⠜⡱⢌⢂⢆⠀⠘⡡⠀⠀⠀⠀⠔⡃⠀⢀⣎⣳⠈⡔⢩⢆⢻⣱⢆⣻⡀⢧⡛⢤⢋⠜⣲⡱⣚⢮⡱⢎⡵⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⠈⠒⡌⡑⠢⢍⠢⡑⢌⠰⣨⣿⣎⠒⢼⠀⠀⠀⠈⠀⡡⠇⠐⢧⡚⡴⡃⢄⢫⢞⣣⠏⢞⡴⣛⢮⣳⡘⡌⠘⠴⣙⠧⣎⠵⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠁⠃⢌⠂⢧⡈⠲⣽⣣⢿⡉⢜⠀⠀⢠⠁⠀⢰⢈⡰⢲⣍⢳⡧⣬⢦⡽⣆⣛⡞⢶⣩⡗⣧⢯⡍⣄⣢⠴⣛⣬⣛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⡔⡈⠇⠀⠻⡽⣖⢸⠀⠀⢨⠀⠀⠸⣨⡗⣣⢞⡡⠜⡣⢗⢾⡱⣏⠶⣎⢷⣚⢧⡟⡼⣇⠮⣝⢶⣣⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⠘⡄⢊⠄⣰⠀⢌⠡⢖⠘⠅⡣⠜⢢⡑⠎⡤⠓⢬⠹⢎⠷⣭⢞⣹⢳⡉⢞⡼⡀⢳⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠉⠔⣨⢘⠎⡰⢈⡘⢤⠀⢘⢧⠱⣈⠞⣠⠋⢆⡛⢌⡳⡜⢮⣳⢫⢗⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠰⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢀⠀⠀⠀⠀⠘⢢⠂⡴⠍⠦⡘⣆⠓⡌⠦⡑⡜⢢⡉⢦⡙⣎⢷⡣⡟⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁⠈⠀⢀⣉⠫⢔⡩⢒⡽⣌⡓⠬⡑⡜⡰⢡⠚⠤⡙⣌⠳⣝⡳⡵⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡔⡀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢢⢉⠖⣰⢋⡶⢌⠒⡥⢓⡰⢡⢃⢎⡱⠡⢆⠣⠜⣳⡝⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⢢⢁⠀⠀⠠⢈⠆⡱⢈⡇⠰⡘⣮⠹⣌⢧⢃⠎⡑⢆⠣⡔⣣⢊⡔⢢⡙⢤⢋⡼⣱⢞⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⢌⠰⡀⢆⠡⢊⢔⡿⢲⡡⢣⢜⡳⣜⠺⣌⠸⣡⢎⠱⣘⢥⢫⡔⣣⠝⣦⢋⣶⡹⣎⢷⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⠠⣀⠢⢄⠡⠒⡌⠰⠀⠆⠰⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⡟⠛⠛⠉⠣⢄⠃⠀⠀⠠⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠡⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⢿⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠛⠟⠛⠉⠉⠀⠀⠀⠀⣀⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⡳⡄⠛⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠰⡀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢯⣷⣇⠀⡘⢵⡩⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⣠⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣟⠏⢠⢻⣾⠉⠐⡀⠙⠎⠀⠈⡔⢠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣯⣿⢾⣟⣿⣻⢿⣯⡷⠃⣀⣀⡼⣟⠀⠐⡀⠇⡘⣄⡀⢀⠜⠳⣬⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣯⣿⣷⡿⣿⣟⣿⡽⣾⣽⣻⣞⡷⣯⡿⣮⣤⣤⣀⡀⠀⣏⡔⢠⢨⡱⣀⠤⡘⠠⢐⠠⢀⠸⠀⠐⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⣽⣻⢷⣻⡾⣽⢷⣯⢷⣯⡟⠁⠀⠀⠀⠀⠿⢿⠊⢀⠳⡰⠀⠀⠑⠪⢱⡑⡊⢔⠠⢈⠓⡄⠻⡴⣩⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢟⣾⢳⢯⣟⣯⢷⣻⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠋⢠⣤⣤⣀⠀⠀⠙⠢⠍⡄⢂⠠⠁⠄⣧⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⢷⣻⣞⣯⣟⡾⣽⢯⣷⣻⣷⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠙⠛⠶⠦⠀⠘⡤⠃⠠⢁⢸⠒⡄⠀⠉⢰⢺⡙⢯⡱⢠⠐⡄⢊⠁
⠀⠀⠀⠀⠀⣸⣯⢷⡿⣾⣽⣻⣽⣻⢾⣽⣷⣻⢷⣻⣇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠁⢀⠎⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠀⢿⣞⣿⣿⣳⢯⣷⣻⣯⣟⣾⠿⣯⢿⡽⣞⡇⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠄⠡⢈⠀⡈⢄⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⠸⣯⣿⣿⣿⣿⣳⣿⡝⡿⣾⣟⢿⣯⣿⣟⣿⣦⣥⣂⡀⢀⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⢀⠂⠌⡐⢀⠂⡴⢣⠜⣠⠢⢀⢁⠒⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⡿⠛⢷⣧⢚⠬⣙⠮⣹⠚⠻⢿⡟⠿⣿⢿⠿⠿⢜⡑⠤⠤⠠⡀⡰⠀⢆⢂⠱⣁⢂⠌⡐⣀⡂⢼⡡⢗⠮⢁⠐⡤⢊⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢡⠊⡔⡈⢆⢊⡑⢊⠳⢯⡐⠠⠅⠀⠙⢃⠀⢂⠀⠈⢰⠁⠀⠀⣸⢃⠆⡌⢸⢄⣣⠶⣹⣱⠀⡷⣙⢎⠣⢆⡹⢴⣡⠜⣎⢏⠧⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡅⢓⠰⠡⢎⡰⢈⠆⡑⢢⠉⣶⣏⡆⣰⠃⠀⠀⠀⠀⢸⡀⢠⣔⡣⢏⡖⢈⢜⡎⣖⠻⡕⣎⢧⡳⣅⠮⡱⠨⡜⢧⡮⡝⣎⠞⠅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠀⠉⠆⡐⠣⢜⠠⢃⣾⢻⡽⢆⠹⠀⠀⢀⠆⠀⢰⠰⢀⠴⣩⡓⣎⣄⣊⢞⡭⢖⡽⡜⣎⢷⡹⣖⡓⡀⢐⣩⢖⡹⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠂⠀⠄⠀⠈⠁⢎⡑⠎⠌⠻⣽⡆⢹⠀⠀⠠⠀⠀⢰⡈⣏⠞⣥⢋⠵⢮⡝⣮⣝⡻⢬⡼⣺⢭⠷⣭⢳⡝⣫⠶⣩⢷⣹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠱⡺⠴⠂⠄⢐⡀⠐⡄⢣⠙⡝⢊⡌⡒⠦⣘⠡⢎⡙⢣⢯⡵⣎⢿⡱⢯⡝⢦⣛⠌⢳⣭⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡒⠀⠃⢩⡘⢰⢆⠱⢈⠌⢮⠀⢸⡰⡡⠓⣌⠱⣂⠍⡞⠤⡳⢬⢳⡝⢯⡖⠈⠈⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠒⢌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠄⠀⠀⠀⠀⡀⢇⠌⢢⡘⢆⢰⣡⠱⢩⢄⡓⠬⡘⡔⠣⣝⡣⣏⣞⡳⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢭⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⠀⠀⠀⢀⡘⣋⠕⡸⣈⠶⣡⢇⢃⠲⠌⡥⠓⣌⠱⢌⠳⣩⢶⡹⣎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⠁⠀⠀⠀⠀⠀⢀⡀⢰⠈⢱⠐⢎⡱⣌⡳⣑⠪⢌⠣⡍⠴⣉⠤⢋⢆⢣⠒⡌⢷⣙⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡰⢁⠣⠀⠀⡀⠄⡑⢢⢘⡆⠰⣈⡞⣥⠳⣌⠳⢄⡋⡔⠣⢜⢢⡅⢎⡱⢌⠆⡭⣘⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠑⡌⢢⠑⡠⠐⡌⠰⣁⡾⡇⡱⢂⡹⢆⡻⣜⡡⠎⢴⡉⡜⣌⠳⡜⢦⡚⣥⢚⡴⣭⢳⡝⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢪⡑⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⠀⡀⢄⠰⡀⢆⠰⡀⠆⠰⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⠟⠟⠛⢛⠡⢂⠘⠀⠀⠠⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢉⠡⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⣿⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠻⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢡⡘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡎⡘⣄⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡾⣷⡀⠀⣍⡼⣙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣽⣾⣿⢿⠃⡙⣯⣟⠖⠁⢈⠣⠏⠀⠢⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢴⣿⢿⡿⣿⢿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣯⣷⢿⣻⢿⣟⡿⣽⠯⠁⠀⢐⣹⡾⠀⠈⠄⠆⢣⡀⠀⢀⠚⡬⣝⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀
⠀⠀⠀⠀⠀⢸⣿⣿⣽⣾⣿⡿⣯⣿⣻⣞⡷⣯⢿⣽⣻⢾⣽⣋⣉⣉⡁⠀⢸⠟⡀⢂⢌⠣⢄⡈⡍⡀⢢⠀⡉⢌⠀⠂⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢘⣷⣻⢾⣻⢷⣻⢷⣯⢷⡿⣽⢯⡿⣾⠝⠉⠀⠃⠉⠽⣿⡷⠌⠁⠷⢨⠀⠍⠒⠕⣢⠑⡆⢡⠐⡈⠦⠐⢧⡚⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⡷⣫⢿⡝⣯⣻⢿⡼⣯⡿⣽⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠏⢀⣠⣄⡀⠀⠁⠚⠤⣃⡐⠠⠈⠄⡁⣞⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⡽⣯⣷⣻⢷⣯⡿⣽⣳⣿⣳⣿⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠉⠙⠛⠶⢦⡤⠀⠰⣁⠃⠌⠐⢠⠓⡄⠀⠉⢰⢺⡙⢯⡱⢀⠆⡰⢈⡀
⠀⠀⠀⠀⠀⣷⣻⣽⡷⣯⣟⡾⣟⣷⣻⢾⣷⣿⣻⣞⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢀⠏⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠂
⠀⠀⠀⠀⠐⣿⣳⣿⣿⣳⢯⣟⣿⡷⣯⣿⢿⣞⡷⣯⣿⡀⠀⠀⠀⠀⠀⠘⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠌⡐⠈⡀⢀⢂⡜⠁⠌⡀⠉⠌⡳⢭⢳⡀⠀⠀
⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣟⣯⣿⠸⡿⣽⣞⣯⣿⢷⣻⣧⣕⣂⡄⠀⢀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠌⡐⢀⠡⠐⣤⠣⡜⣠⠢⢀⠁⠆⡈⢣⡳⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣟⠨⠏⡙⠿⣧⢜⡰⡙⡬⣋⠛⠻⣿⠿⢿⣿⡿⠿⠯⣍⠡⠤⡀⡀⡀⠄⡠⢀⠆⢱⠄⣈⠔⡀⢂⡐⡜⢦⠳⣜⠡⢀⠦⣉⠔⠁⡮⡕⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠤⡑⢢⠐⢢⢁⢃⠓⠯⣇⠰⡈⠂⠈⠶⢁⠐⠀⠀⠉⡆⡅⠀⠀⡸⢆⠡⡘⢨⠚⣴⠲⣝⢫⡀⡟⣬⠓⡌⢆⢣⠞⣤⠴⣋⠷⡹⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉⠱⡘⠢⠍⢆⡌⢢⢉⠒⡄⢓⣼⡵⢀⡼⠁⠀⠀⠀⠀⡇⠆⢀⡤⣝⢺⡁⠔⣩⠞⣖⠻⣥⢋⡷⣸⢆⡹⡘⢨⠺⣹⢆⡻⢬⠳⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠁⠊⠔⣈⠲⣈⠒⣌⢿⡻⣽⠂⢇⠀⠀⢠⠂⠀⡇⠂⢀⡩⢎⡳⣎⣐⣌⢻⡌⠷⡜⣧⢚⣧⣛⣖⡓⡀⢂⣍⠮⣕⠯⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠐⠀⠠⠀⠀⠀⠱⣈⠲⠌⡙⢷⡻⣌⢸⠀⠀⠠⠀⠀⢸⠉⣴⡙⣮⢃⠻⣜⢮⡳⣝⢯⡭⣼⡹⢶⣝⣮⢳⡽⣩⢎⣳⢏⣟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⢹⠶⠌⡄⡀⢰⠀⡈⠜⡄⠷⠙⡆⣌⠣⡔⢊⡱⢉⠞⣲⢧⣛⡵⣎⢷⣫⠕⣣⢞⠄⢻⡬⠑⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠄⠂⠀⠣⣐⢁⡺⢄⠱⢈⠴⣃⠀⠱⣆⠱⢌⣃⠲⣉⠒⣇⠸⢦⣙⢮⡳⣝⠆⠈⠘⠒⠀⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠰⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⠀⠀⠀⠀⡘⣄⠣⣠⠙⣄⠰⣌⠣⠎⣄⠓⣌⠱⡌⡱⢳⡬⣳⡝⣮⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠨⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠁⠀⠀⠀⢀⢉⠖⡡⢍⡰⣙⢦⢃⡓⠬⡘⠤⠓⣌⡑⢣⠚⣵⢺⠵⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡘⠄⠀⠀⠀⠀⠀⡀⠠⢄⠋⢈⢎⡱⣌⠶⣍⠎⡒⢬⢑⡩⠜⣑⠢⠜⣡⢃⠦⡙⢯⡳⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⡈⠀⠀⡜⢠⠃⠀⠀⠠⣀⠣⢌⣡⠁⣠⠚⣦⠳⣌⡳⢌⡒⡩⠆⢎⡰⣉⠦⡙⡘⡤⢊⠴⣩⢷⡹⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡘⢄⠣⡘⠤⡐⠢⢄⠃⢦⣻⠰⣡⠚⡴⣋⠶⣩⠆⡱⢡⡚⠤⣓⢭⢲⡱⡹⢴⡩⢞⡱⢧⡻⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠌⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣣⣾⣶⣿⣿⣿⣿⠟⠀⠀⣀⠐⡠⢂⠔⡠⢂⠔⠠⠂⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⣀⣴⣾⣿⣿⡟⠿⠛⢛⢁⠢⠐⠁⠀⠠⠑⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⡄⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠻⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢥⡈⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡣⢘⣄⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡾⣷⡃⠀⡔⣭⢫⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣽⣾⣿⠿⢃⠹⣷⣻⠖⢁⠀⠫⡞⠀⢁⠂⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀
⢴⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣯⣷⢿⣻⣟⣿⢿⡽⡏⠁⠀⣐⣫⣿⠀⠀⠌⡄⢡⡀⠀⠈⡜⡱⢎⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀
⠀⠀⠀⠀⠀⢸⣿⣯⣿⣾⡿⣿⣯⢿⣯⢷⣻⣞⣯⢷⣻⣞⣯⣋⣉⣉⡁⠀⢸⠋⡆⠐⢨⠰⣀⢌⡩⢁⠠⡀⠩⢑⠀⠡⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢸⣷⣻⢾⣻⣟⡷⣯⣟⡾⣟⣷⣻⢾⣯⠗⠉⠈⠁⠈⠽⣿⡷⠈⠁⠷⣌⠂⠉⠒⠌⣲⠁⡖⢁⢢⠈⠢⠔⢣⢎⡵⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡞⣭⣟⢷⣫⣟⡷⣭⣟⡿⣾⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠗⣀⣠⣄⡀⠀⠁⠚⠬⢒⠠⢁⠂⠌⡀⡞⠍⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣞⡿⣮⣟⡷⣯⣟⡷⣯⣿⣳⣿⣷⣻⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠀⠉⠙⠛⠷⢦⡤⠄⠁⡇⠆⡈⠄⢰⠉⡄⠀⠈⢰⢞⡹⢫⡕⢠⠐⡄⢊⠁
⠀⠀⠀⠀⢀⡿⣾⣽⡷⣯⣟⡷⣯⣟⣷⣻⣷⣟⡾⣯⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠍⠠⢈⠑⡰⠋⠮⢱⢫⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠠⣿⣳⣿⣿⣳⢯⣟⣿⣾⣳⡿⣟⡾⣽⣳⣷⠀⡀⠀⠀⠀⠀⠘⣷⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠌⡐⠠⢀⠀⢂⡜⢁⠐⡀⠒⠈⢷⣩⢳⠂⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⣯⣿⢸⡻⣽⣞⣿⡷⣿⣽⣧⣔⣠⢀⠀⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠐⡀⠡⠀⣜⠣⡜⣠⠰⠀⠡⠌⡀⢣⢏⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣟⠪⠏⡹⠟⣧⡘⢦⠹⡌⡟⠋⠿⣿⠿⣿⣿⡿⠿⠯⡍⡘⠤⠀⡀⣀⠠⢀⠄⠢⢌⡅⡐⠄⢃⢠⡁⢼⢪⡕⢮⠁⢂⡡⢋⠔⠁⡞⡜⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠤⡑⢌⠰⡈⢅⠓⡘⠟⣣⠐⡸⠂⠈⠾⠀⠐⡀⠀⠉⡖⠀⠀⠀⣘⢦⠈⡅⢚⠴⣱⠞⣎⣳⠀⣯⠲⡍⣆⠚⣤⢓⡥⢴⢺⣍⠳⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢩⠑⡘⢢⠱⢌⠢⡑⢌⠢⣁⢓⣴⣧⠠⡜⠁⠀⠀⠀⠀⡇⠀⣀⡠⢞⡼⡡⠌⣸⢎⣥⠟⢧⢎⢷⡸⣇⠼⡠⢙⡸⣣⡞⣣⢏⡜⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠁⠂⠱⠈⡜⠰⣈⠒⢤⣟⡯⣟⠠⠇⠐⠀⠠⠂⠀⡇⠆⠀⣩⠳⡜⣇⡰⣄⢫⡜⡺⠼⣎⢎⡷⣹⢞⣑⠠⢐⣡⢞⡱⢮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠂⠐⠀⠠⠀⠀⠈⠑⡌⢪⠡⠙⢾⣳⡜⢰⠀⠀⠁⠀⠀⡒⢌⡴⢣⠯⡙⢮⢳⣭⢳⢏⡯⣭⡜⣞⡳⣝⣮⢳⡏⡝⢦⣫⢟⡽⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠁⠀⠀⠀⢧⠧⠎⡠⢀⠰⠀⠤⠙⡄⢟⠓⡡⢓⡌⠦⠌⡍⢎⡱⢧⢻⣜⡳⣭⢶⣫⠏⣜⠳⡎⢹⢮⠑⠄⠀⠀⠀
⠀⠀⠀⠀⡐⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠈⠔⣁⠢⡱⡌⢢⠡⡑⡇⠀⢇⣣⠘⡴⢉⡜⡰⢌⢣⠺⡴⣙⡞⣶⡹⠂⠈⠙⠒⠈⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⠀⠀⠂⡀⠀⠀⠀⠀⢸⡀⢆⣡⢱⡀⢧⢌⡱⢂⠇⠦⣑⢊⠦⣙⠶⣡⢟⡲⣝⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠊⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⢀⡛⢊⡔⢢⢜⡣⢞⡠⢍⢚⡰⠡⢎⠒⣌⠳⡙⢮⡳⣝⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠠⡌⠉⡔⢣⡘⢦⢎⡵⢋⠔⣊⠆⢥⠋⡴⠩⣄⢓⡘⠦⢹⣎⢷⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⡌⠀⠀⡜⢠⠃⠀⠀⠠⢈⠆⡡⢳⠁⢤⡘⢧⣙⢎⡞⠤⣃⢚⡐⢎⢢⡍⡔⣃⠲⡌⣒⢩⢖⡽⣪⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡜⢠⠃⡜⢠⠐⢢⠁⢎⣰⢿⢈⡒⡌⢷⡘⢮⡜⡡⢆⡣⠜⣌⢲⡙⢦⡱⢫⡔⣎⢳⣎⠷⣭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⡿⠆⢀⠠⡐⢄⠒⡰⢐⠢⠐⠡⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⠿⠻⠛⡛⢋⡑⠈⠀⠀⠰⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⡧⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣀⣢⣤⣶⣿⣿⣿⣿⣿⣿⡟⣄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣼⠀⠀⢖⡽⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠟⢹⣾⡳⠎⢁⠚⣥⠋⠀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢷⣿⣿⡿⣿⣻⠍⠐⠂⢻⡇⠈⠠⢀⠰⡄⠀⠈⢀⢯⡹⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀
⠉⠉⠉⠉⠉⣿⣿⡿⣟⣿⣯⣿⣷⢿⣳⡿⣯⣟⡾⣯⢷⣯⣟⠗⠡⠀⠒⠒⢻⡗⠠⠐⠌⢆⠈⢓⠒⡈⢠⠑⠃⡍⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣿⣽⣻⣟⡿⣽⢾⡽⣯⣿⣽⣳⢯⡿⣽⣻⠞⠹⢛⠻⠷⣶⣄⠼⠸⣄⢡⠙⠨⣚⢌⡒⢄⠆⡠⢁⠲⢄⢠⡏⡵⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣞⣳⢯⠿⣽⢯⣻⣗⣿⣳⢯⣿⣽⣷⠁⠀⠀⠀⠀⠉⠛⠋⠁⡐⠳⢌⠀⢀⠀⠈⠘⠈⢖⠡⠆⡠⢈⠠⠘⡑⣳⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣯⢾⣽⣫⢿⣽⣻⢷⣻⢾⣯⣟⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠀⠈⠘⠻⠷⣦⣄⣀⠀⠉⢢⠅⠆⠠⢁⢰⢣⠀⠐⠂⣀⢴⢢⢴⢀⡀⠀⡄⢠⠂
⠀⠀⠀⠀⢸⣽⣻⣾⣽⣻⢾⣽⣯⣟⣯⣿⣞⣿⣻⡽⡄⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⡞⠀⠁⠂⡜⠀⠲⠄⣠⢏⣎⠳⣎⠗⠈⢁⡘⠄⡒
⠀⠀⠀⠀⣻⢾⣽⣿⡾⣽⣻⢾⣷⣻⢾⣽⣿⣳⢯⣟⣧⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠠⠘⠀⠈⡐⣠⠃⠠⢀⠹⢨⢆⡤⢧⡈⠂⠀
⠀⠀⠀⠀⠘⣯⣿⣿⣿⣷⢿⣯⡟⣿⣯⣯⢹⣯⣟⣾⢿⡄⢂⠀⠀⠀⠀⣈⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⠠⠁⠌⠠⢁⡴⡁⠄⡁⢂⠐⡀⠫⣜⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠹⣿⣿⢟⢿⠟⣯⣷⡘⢬⠛⣯⢻⡾⣟⣿⣾⣷⣮⣄⣄⣰⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⡀⢤⠃⠈⠄⠡⠈⠄⣡⠏⡴⣉⠶⠉⢀⠦⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⠒⡥⢈⠔⣋⠚⠦⣙⠤⣳⢦⠈⡙⢿⠛⢻⡟⠛⠉⠁⠪⢑⠲⠐⠂⠂⣜⠠⢃⠔⡪⢔⡱⣈⡤⣵⠂⣧⢛⡴⢋⢠⠰⣉⠆⠡⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⢣⡘⢄⡉⠒⡄⢊⠌⡙⠶⡐⢬⡁⠀⢫⠀⢀⠀⠀⠀⡝⠀⠀⢀⡭⢣⠂⢆⡱⢪⢲⢧⠳⡼⡠⢇⡏⢴⡉⢆⢳⡵⣚⢭⣣⠳⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠃⠌⠢⢌⠡⡘⠥⡘⢄⠚⡄⢣⣼⣷⣑⠢⡁⠀⠀⠀⠁⠀⠞⠀⠰⢧⣚⠽⡈⢄⢫⡝⣎⠏⢷⡱⣛⠾⣜⡢⡍⠘⡌⢷⡩⢖⣥⡛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠢⢡⠉⢖⠨⢼⡳⣏⡧⠘⡅⠀⠀⡌⠀⠀⡍⡁⢤⣚⢬⣛⣤⢮⡤⢿⣌⡻⠮⢵⣩⡟⣼⢣⣥⣂⢤⢖⡹⣎⢶⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⢍⠢⡙⠀⠉⢿⡹⡌⡇⠀⠀⡂⠀⠀⢧⢰⡳⡜⡎⠴⡙⠮⣝⢳⣎⢷⣚⢧⡳⣝⢮⡟⡶⣍⠞⡮⢵⣫⢞⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⢂⠔⡡⠀⡅⠠⡑⠸⡘⡱⠉⡜⢢⡑⠎⡤⢃⡌⢎⡝⢮⠳⣝⢮⡝⣷⡈⢻⠼⡀⢳⢯⠈⠂⠀⠀⠀
⠀⠀⠀⠠⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠈⠢⠔⡡⢛⠤⢁⠃⠥⡆⠀⡝⢦⡘⣑⢢⠣⡘⢦⢉⢞⡱⢎⡷⣹⢎⡑⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢈⠅⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠂⣀⠀⠀⠀⠀⠚⡤⢉⡴⢙⠄⣳⢂⠵⡈⠦⣑⠱⢌⡊⢼⡱⣫⡜⣧⢏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠁⠀⣈⢍⡱⢄⡋⢼⡡⢏⠴⣉⠲⣁⠞⡠⢍⢢⡑⢣⠻⣼⡹⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⢂⠀⠀⠀⠀⠀⢀⠠⡐⠜⠈⢆⠲⣌⡜⣣⠝⣌⠲⣈⠕⣂⢎⠱⡌⡒⠬⡑⢬⠱⣳⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⢠⠒⡌⡀⠀⠀⠄⡱⢈⠆⣹⠀⢆⢫⠞⡴⡹⡜⠬⡰⢑⡌⠲⣁⢎⠦⣑⢡⠣⢍⣂⢗⡯⡞⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠣⡘⠤⡁⢆⠰⡈⠔⡡⣺⡽⢊⡜⢢⡛⡴⢳⣍⠒⡥⣃⡌⡓⡜⣎⠳⣔⢫⡜⢦⣙⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠐⣸⣦⣷⣾⣿⣿⣿⠗⠀⢠⠀⠤⠠⠄⠤⢠⠐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣴⣾⣿⠿⠿⠟⠻⢛⡉⠀⠉⠀⡘⠄⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠇⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠉⠀⠀⠈⢂⣠⣴⣾⣿⣿⣿⣿⣿⣏⡘⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡶⡅⠀⢂⢒⡖⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⡟⠙⣿⡤⠎⢉⠺⣱⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣾⣿⣻⡿⣽⠞⠀⠑⣸⠥⠀⠂⢄⠡⡀⠀⠀⢩⢞⡹⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀
⠙⠛⠙⠋⠛⣿⣿⡿⣿⣻⣿⣯⣿⣷⣟⡿⣽⢿⡽⣞⡷⣯⢿⡽⠃⠀⠒⠒⢙⡆⢀⠈⠜⡀⠙⠢⠔⠂⡈⠉⠧⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡿⣯⣟⡿⣿⢷⣯⣟⣾⣽⣻⡽⣯⢿⣽⣻⠝⠛⠟⠛⠳⣶⣦⡌⢲⡀⠎⠣⠝⣢⠙⡔⣠⠱⠈⡀⢧⣀⢠⢟⡹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⢷⢯⡿⣽⣻⢾⡽⣳⢯⡿⣽⢯⣿⣾⠁⠀⠀⠀⠀⠀⠙⠙⠀⢩⢱⠃⠀⠀⠀⠉⠒⠄⣣⢡⠃⡀⠄⡈⢊⠑⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣮⣟⣳⢷⣯⢿⡽⣯⣟⣿⣽⣻⣿⢿⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠂⠐⠚⠿⣶⣤⣀⠀⠀⠁⣣⢐⠂⡐⠠⢸⠌⠀⠱⠠⢀⡤⡤⢤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠈⢹⣻⢾⣽⢯⣟⡾⣯⣟⡷⣯⣿⣞⣿⡽⣿⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠰⡉⠀⠄⠁⡎⠘⠢⠄⣀⡏⢶⣩⢓⠞⠀⠑⡈⢌⠡
⠀⠀⠀⠀⢼⣯⢿⣿⣻⢾⣽⣳⣯⣟⣷⣻⣿⢯⣟⣷⣳⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢀⠚⠀⠠⢁⢢⠃⠈⠄⢡⢫⡄⣤⢎⡘⠂⠀
⠀⠀⠀⠀⠸⣯⣿⣿⣿⣻⣾⡽⡟⣿⣞⣿⡹⣟⣾⣳⣯⣇⠂⠄⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⢀⠂⡐⢀⢂⡜⡂⠌⠐⡈⠄⡀⠻⣔⢫⠄⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⡷⣿⣯⠜⡝⠯⣗⢻⣯⣷⡿⣿⣿⣾⣧⣴⣤⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠁⡐⢀⠂⡐⠀⢦⡋⡴⣑⠎⠡⠐⡤⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠩⠛⠮⢋⠡⢊⠛⠧⢎⡱⣊⡵⣤⠈⡉⢿⡉⠛⡿⢋⠉⠉⠪⢔⠲⠐⠠⠂⣌⠰⠡⠌⡖⡰⡐⢡⡰⠴⢡⢧⡙⢶⡉⢐⠤⣋⠔⢁⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢎⡡⢊⠔⢢⠐⡡⢘⠓⢎⡰⢐⡀⠀⢋⠀⠀⠀⠀⠀⡹⠀⠀⢠⢏⡆⡑⢂⠧⣱⢻⣱⢋⢷⡘⢦⡛⠴⣡⠋⢶⡱⣚⢮⡱⣣⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠢⠘⠤⣈⠑⡚⠤⡃⠴⡁⢎⠰⣼⡿⡔⢂⡅⠀⠀⠀⠉⠀⠵⠠⢔⣫⠞⣼⢀⠃⢿⣘⠶⠍⣏⠶⣹⢳⣌⡳⠄⡙⡸⣕⢫⢖⣱⡃⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠈⠐⠡⢌⠣⡘⢄⢿⣹⡽⡇⡘⡆⠀⠀⡜⠀⠀⣃⠃⡴⣌⠻⣔⢧⡼⡬⢮⣹⣙⠮⣳⣹⡳⣎⢯⢥⣀⡤⢖⡫⣞⡖⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⢣⠘⡌⠂⠙⢞⡷⡁⡇⠀⠀⠆⠀⠀⢇⢺⡱⢎⡝⡌⢓⠞⣹⢧⣳⠳⣖⡳⢧⣛⢾⣫⢞⡵⣊⡟⢼⣣⣟⡀⠀⠀⠀⠀
⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡏⢣⠐⡡⠀⡇⠠⡘⠸⡈⠷⠉⡒⣌⢃⡚⠤⡒⢤⠋⡞⡝⢧⣛⢮⡝⣾⡐⣣⠽⠀⢗⡶⠁⠂⠀⠀⠀
⠀⠀⠀⠈⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠈⠢⢡⢡⠛⢄⠡⢡⠩⡆⠀⡳⢤⠣⢌⢓⡸⢄⢫⠱⣘⠧⣎⠷⣹⢖⡇⠀⠉⠁⠐⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠐⡌⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠈⠠⠀⠀⠀⠀⠀⡚⡄⢃⡴⠹⢄⡳⡌⡱⢊⢆⠲⠌⣆⠓⣌⢳⣬⠻⣵⢫⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠁⠀⣠⠙⠥⣒⡉⢦⢳⡱⡑⢪⠰⢩⠜⡠⢍⡂⢇⠎⡻⣜⡳⣞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠑⠀⠀⠀⠀⠀⢀⠠⡐⠜⠀⡙⢆⢥⡚⣥⠧⠱⢌⠣⣑⠣⣌⠱⢢⢉⠖⠬⡑⡜⣳⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⠀⠠⢃⠜⡀⠀⠀⠄⡱⢈⠆⣹⠀⠦⣙⢎⠶⣙⠦⡱⢉⢆⡓⢌⠲⣌⠥⢃⠎⣌⠓⡬⡔⣯⢞⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠃⢎⠰⡐⢄⠒⡌⠰⡁⣾⡝⢎⡒⢩⣎⠳⣍⠶⣁⠫⣔⠸⣈⠳⣬⠳⣍⠞⣤⢛⡴⣹⢮⡝⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣶⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠈⢟⠻⠉⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠆⠒⠵⣎⠼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⢂⣰⠹⣌⡳⣧⢭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⣎⢳⡱⠱⢌⡛⢮⡳⢮⡝⣶⢣⡟⣼⢣⡿⣜⡧⣚⡵⢹⣎⣯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡘⠃⡇⡹⠰⡘⢤⡑⢣⡘⣱⢫⠞⣵⢫⣞⣳⡈⢷⡸⡀⢳⠶⠁⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⠸⣅⠣⢍⠢⡜⢡⠒⠧⢌⡻⣔⣫⢖⣧⡓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠤⢋⢆⠓⡬⢑⢪⢑⡊⡵⢎⡼⣋⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡡⠎⡜⡰⡉⢆⠎⡔⡩⢎⠳⣝⢮⡳⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠔⢣⡘⡡⠜⣌⠲⢡⢒⡡⢚⠌⣷⣹⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⡙⢢⢡⡑⣣⠢⢍⠆⡣⠜⢢⡙⣶⢣⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⡘⢥⠢⣱⢣⡛⡤⢏⡵⣩⢧⡹⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣷⣦⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠉⢻⠏⠋⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠃⠰⢧⡚⡼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⣁⡰⢣⡝⣲⢧⡭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⡕⣫⠼⣁⠎⡽⢹⡞⣼⡹⣖⢧⡻⣜⢧⡟⣼⡣⢞⡱⢽⣎⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡸⠁⡗⣈⠖⡌⣅⠲⠤⡙⣬⠳⣝⢮⡳⣝⢶⡙⢮⡹⠀⢞⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⢘⢦⢊⡔⢢⠣⢱⠡⢇⡚⡜⢦⢻⡜⣧⡓⠀⠉⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠆⡥⢊⢆⡙⢢⠍⡒⢬⣙⡎⣷⡹⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡰⡉⢦⠘⡅⢎⠱⢢⠜⡸⢡⣟⣣⢟⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠴⡑⡌⢣⠜⣈⠇⢎⠴⢡⢃⡜⢞⡺⣜⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⠒⡅⠎⣅⡚⡔⢪⠜⣢⠃⢎⡴⣫⠷⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⢩⠜⣘⢰⡹⡜⣥⢚⡥⣚⠵⣺⠵⣛⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢢⣵⣾⣶⣿⣿⣿⡿⠂⠀⡀⢄⠢⡐⠢⢌⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⣠⣴⣿⣿⡟⠿⠛⢛⠛⡀⠆⠁⠀⠠⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠀⠀⠀⠠⢑⣠⣶⣾⣿⣿⣿⣿⣿⣿⢣⡘⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡩⢁⡀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣧⡇⠀⣩⢚⡝⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⢀⣠⣴⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⡟⢉⢿⣳⠒⠁⡘⠼⡑⠁⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣯⡿⣾⢿⣿⢿⣯⠗⠈⠀⢊⣧⠈⠐⠠⠄⢣⡀⠀⠈⡰⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢛⣿⣿⢿⣻⣿⣽⡿⣷⣿⢿⣻⣽⣳⢿⣽⣻⣞⡟⣒⠒⠊⠉⠉⡷⢀⠠⡁⢇⡀⢩⠒⠊⢠⠀⡙⢣⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠸⣯⢿⣻⣿⣟⣯⣟⣯⣟⣯⡷⣯⣟⡿⣞⠷⠉⠛⠉⢛⣿⣶⣔⠃⣇⠰⠉⠒⡌⡣⢎⠰⡄⡡⠐⠘⣄⡀⡼⣋⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢿⡽⣾⡽⣾⡽⡾⣽⢾⣽⣳⣯⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⢠⠁⠏⠀⣀⠀⠀⠉⠢⠕⣰⠁⡂⠄⠠⠑⡉⣞⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣛⣮⢿⣱⢯⣷⣻⡽⣯⢿⣞⣷⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠒⠈⠘⠛⠿⢶⣤⣀⡀⠀⢣⠅⡊⠄⠡⢐⢧⠀⠉⠰⢀⡤⢤⠤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠀⠐⢹⣟⡾⣯⣟⡿⣞⣷⣻⣽⣻⣟⣾⣿⣳⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡞⠀⠄⠡⡘⠀⠦⠄⣀⠷⡸⣍⠾⠍⠂⠑⡈⢌⠡
⠀⠀⠀⠀⠀⣾⣽⣻⣷⢯⡿⣽⢾⡷⣯⢷⣿⣟⡾⣽⣻⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠠⠐⠁⠈⠄⢢⠍⠠⠁⢌⠳⡆⡤⣎⠘⠂⠀
⠀⠀⠀⠀⠀⢹⣞⣿⣿⣯⣿⣽⣻⣿⣽⣯⡟⣯⣟⣷⣻⣧⣂⠄⡀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⡀⢂⠁⠂⢌⡔⠣⢀⠂⡁⠂⠄⠻⢴⡩⠆⠀⠀
⠀⠀⠀⠀⠀⠘⢯⣿⣿⡿⣷⣿⣻⡆⢯⠳⣿⡹⣿⢾⣟⣷⣿⣿⣷⣾⣤⣿⠟⡀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠆⢈⠐⡀⠂⠌⢰⠎⡜⡱⠆⠋⡀⣌⠂⡔⢨⡓⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⡙⢌⡙⠻⠦⡙⣤⢓⣬⡌⢈⠙⢏⠉⢟⠻⠋⠉⠚⠥⡉⡇⠘⠂⣕⠈⢆⠡⡘⢤⠒⣈⣤⡱⠌⣮⠹⣜⠣⢁⠦⣘⠄⢃⣠⡏⡵⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢈⠆⡌⡑⢢⠑⠤⡉⠜⡱⢌⢂⢆⠀⠘⡡⠀⠀⠀⠀⠔⡃⠀⢀⣎⣳⠈⡔⢩⢆⢻⣱⢆⣻⡀⢧⡛⢤⢋⠜⣲⡱⣚⢮⡱⢎⡵⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⠈⠒⡌⡑⠢⢍⠢⡑⢌⠰⣨⣿⣎⠒⢼⠀⠀⠀⠈⠀⡡⠇⠐⢧⡚⡴⡃⢄⢫⢞⣣⠏⢞⡴⣛⢮⣳⡘⡌⠘⠴⣙⠧⣎⠵⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠁⠃⢌⠂⢧⡈⠲⣽⣣⢿⡉⢜⠀⠀⢠⠁⠀⢰⢈⡰⢲⣍⢳⡧⣬⢦⡽⣆⣛⡞⢶⣩⡗⣧⢯⡍⣄⣢⠴⣛⣬⣛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⡔⡈⠇⠀⠻⡽⣖⢸⠀⠀⢨⠀⠀⠸⣨⡗⣣⢞⡡⠜⡣⢗⢾⡱⣏⠶⣎⢷⣚⢧⡟⡼⣇⠮⣝⢶⣣⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⠘⡄⢊⠄⣰⠀⢌⠡⢖⠘⠅⡣⠜⢢⡑⠎⡤⠓⢬⠹⢎⠷⣭⢞⣹⢳⡉⢞⡼⡀⢳⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠉⠔⣨⢘⠎⡰⢈⡘⢤⠀⢘⢧⠱⣈⠞⣠⠋⢆⡛⢌⡳⡜⢮⣳⢫⢗⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠰⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢀⠀⠀⠀⠀⠘⢢⠂⡴⠍⠦⡘⣆⠓⡌⠦⡑⡜⢢⡉⢦⡙⣎⢷⡣⡟⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁⠈⠀⢀⣉⠫⢔⡩⢒⡽⣌⡓⠬⡑⡜⡰⢡⠚⠤⡙⣌⠳⣝⡳⡵⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡔⡀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢢⢉⠖⣰⢋⡶⢌⠒⡥⢓⡰⢡⢃⢎⡱⠡⢆⠣⠜⣳⡝⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⢢⢁⠀⠀⠠⢈⠆⡱⢈⡇⠰⡘⣮⠹⣌⢧⢃⠎⡑⢆⠣⡔⣣⢊⡔⢢⡙⢤⢋⡼⣱⢞⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⢌⠰⡀⢆⠡⢊⢔⡿⢲⡡⢣⢜⡳⣜⠺⣌⠸⣡⢎⠱⣘⢥⢫⡔⣣⠝⣦⢋⣶⡹⣎⢷⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⠠⣀⠢⢄⠡⠒⡌⠰⠀⠆⠰⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⡟⠛⠛⠉⠣⢄⠃⠀⠀⠠⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠡⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⢿⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠛⠟⠛⠉⠉⠀⠀⠀⠀⣀⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⡳⡄⠛⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠰⡀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢯⣷⣇⠀⡘⢵⡩⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⣠⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣟⠏⢠⢻⣾⠉⠐⡀⠙⠎⠀⠈⡔⢠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣯⣿⢾⣟⣿⣻⢿⣯⡷⠃⣀⣀⡼⣟⠀⠐⡀⠇⡘⣄⡀⢀⠜⠳⣬⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣯⣿⣷⡿⣿⣟⣿⡽⣾⣽⣻⣞⡷⣯⡿⣮⣤⣤⣀⡀⠀⣏⡔⢠⢨⡱⣀⠤⡘⠠⢐⠠⢀⠸⠀⠐⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⣽⣻⢷⣻⡾⣽⢷⣯⢷⣯⡟⠁⠀⠀⠀⠀⠿⢿⠊⢀⠳⡰⠀⠀⠑⠪⢱⡑⡊⢔⠠⢈⠓⡄⠻⡴⣩⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢟⣾⢳⢯⣟⣯⢷⣻⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠋⢠⣤⣤⣀⠀⠀⠙⠢⠍⡄⢂⠠⠁⠄⣧⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⢷⣻⣞⣯⣟⡾⣽⢯⣷⣻⣷⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠙⠛⠶⠦⠀⠘⡤⠃⠠⢁⢸⠒⡄⠀⠉⢰⢺⡙⢯⡱⢠⠐⡄⢊⠁
⠀⠀⠀⠀⠀⣸⣯⢷⡿⣾⣽⣻⣽⣻⢾⣽⣷⣻⢷⣻⣇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠁⢀⠎⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠀⢿⣞⣿⣿⣳⢯⣷⣻⣯⣟⣾⠿⣯⢿⡽⣞⡇⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠄⠡⢈⠀⡈⢄⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⠸⣯⣿⣿⣿⣿⣳⣿⡝⡿⣾⣟⢿⣯⣿⣟⣿⣦⣥⣂⡀⢀⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⢀⠂⠌⡐⢀⠂⡴⢣⠜⣠⠢⢀⢁⠒⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⡿⠛⢷⣧⢚⠬⣙⠮⣹⠚⠻⢿⡟⠿⣿⢿⠿⠿⢜⡑⠤⠤⠠⡀⡰⠀⢆⢂⠱⣁⢂⠌⡐⣀⡂⢼⡡⢗⠮⢁⠐⡤⢊⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢡⠊⡔⡈⢆⢊⡑⢊⠳⢯⡐⠠⠅⠀⠙⢃⠀⢂⠀⠈⢰⠁⠀⠀⣸⢃⠆⡌⢸⢄⣣⠶⣹⣱⠀⡷⣙⢎⠣⢆⡹⢴⣡⠜⣎⢏⠧⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡅⢓⠰⠡⢎⡰⢈⠆⡑⢢⠉⣶⣏⡆⣰⠃⠀⠀⠀⠀⢸⡀⢠⣔⡣⢏⡖⢈⢜⡎⣖⠻⡕⣎⢧⡳⣅⠮⡱⠨⡜⢧⡮⡝⣎⠞⠅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠀⠉⠆⡐⠣⢜⠠⢃⣾⢻⡽⢆⠹⠀⠀⢀⠆⠀⢰⠰⢀⠴⣩⡓⣎⣄⣊⢞⡭⢖⡽⡜⣎⢷⡹⣖⡓⡀⢐⣩⢖⡹⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠂⠀⠄⠀⠈⠁⢎⡑⠎⠌⠻⣽⡆⢹⠀⠀⠠⠀⠀⢰⡈⣏⠞⣥⢋⠵⢮⡝⣮⣝⡻⢬⡼⣺⢭⠷⣭⢳⡝⣫⠶⣩⢷⣹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠱⡺⠴⠂⠄⢐⡀⠐⡄⢣⠙⡝⢊⡌⡒⠦⣘⠡⢎⡙⢣⢯⡵⣎⢿⡱⢯⡝⢦⣛⠌⢳⣭⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡒⠀⠃⢩⡘⢰⢆⠱⢈⠌⢮⠀⢸⡰⡡⠓⣌⠱⣂⠍⡞⠤⡳⢬⢳⡝⢯⡖⠈⠈⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠒⢌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠄⠀⠀⠀⠀⡀⢇⠌⢢⡘⢆⢰⣡⠱⢩⢄⡓⠬⡘⡔⠣⣝⡣⣏⣞⡳⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢭⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⠀⠀⠀⢀⡘⣋⠕⡸⣈⠶⣡⢇⢃⠲⠌⡥⠓⣌⠱⢌⠳⣩⢶⡹⣎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⠁⠀⠀⠀⠀⠀⢀⡀⢰⠈⢱⠐⢎⡱⣌⡳⣑⠪⢌⠣⡍⠴⣉⠤⢋⢆⢣⠒⡌⢷⣙⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡰⢁⠣⠀⠀⡀⠄⡑⢢⢘⡆⠰⣈⡞⣥⠳⣌⠳⢄⡋⡔⠣⢜⢢⡅⢎⡱⢌⠆⡭⣘⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠑⡌⢢⠑⡠⠐⡌⠰⣁⡾⡇⡱⢂⡹⢆⡻⣜⡡⠎⢴⡉⡜⣌⠳⡜⢦⡚⣥⢚⡴⣭⢳⡝⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⢀⢀⡐⢄⠢⡐⢄⠢⡐⠄⠢⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⠟⠟⠛⢛⠡⢂⠑⠈⠀⡐⠌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢉⠡⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⣿⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠻⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢡⡘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡱⢈⣤⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡾⣷⡀⠀⣍⡼⣙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣽⣾⣿⢿⠃⡙⣯⣟⠖⠁⢈⠣⠏⠀⠢⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀
⢴⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣯⣷⢿⣻⢿⣟⡿⣽⠯⠁⠀⢐⣹⡾⠀⠈⠄⠆⢣⡀⠀⢀⠚⡬⣝⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠀⠀⠀⠀⢸⣿⣯⣷⣿⣻⣿⣯⢿⣻⣞⡷⣯⢿⣽⣻⢾⣽⣋⣉⣉⡁⠀⢸⠟⡀⢂⢌⠣⢄⡈⡍⡀⢢⠀⡉⢌⠀⠂⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⣳⣟⣾⣻⣟⣧⣟⣯⢷⡿⣽⢯⡿⣾⠝⠉⠀⠃⠉⠽⣿⡷⠌⠁⠷⣨⠀⠍⠒⠕⣢⠑⡆⢡⠐⡈⠦⠐⢧⡚⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣳⣛⣾⢳⣛⡾⣽⣞⣯⡿⣽⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠇⣀⣠⣄⡀⠀⠁⠚⠤⣃⡐⠠⠈⠄⡁⣞⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣳⣟⣮⢿⣽⣻⣷⣻⣞⣿⣳⣿⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠉⠙⠛⠶⢦⡤⠀⠰⣁⠃⠌⠐⢠⠓⡄⠀⠉⢰⢺⡙⢯⡱⢀⠆⡰⢈⠁
⠀⠀⠀⠀⠀⣿⣳⢿⣯⣟⣾⣳⣯⢷⣻⢾⣷⣿⣻⣞⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢀⠏⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠈⣷⢿⣿⣷⣻⣞⡷⣿⣯⣟⡿⣿⣞⡷⣯⣿⡀⠀⠀⠀⠀⠀⠘⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠌⡐⠈⡀⢀⢂⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣽⣿⣻⠼⡻⣿⣞⣯⣿⣳⣿⣧⣕⣂⡄⠀⢀⣾⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠌⡐⢀⠡⠐⣤⠣⡜⣠⠢⢀⠁⠆⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣟⠪⠏⡜⠿⣇⡱⢢⠹⣌⣓⠛⠷⣿⠿⢿⣿⡿⠿⠯⢍⠣⠄⡠⢀⡀⢠⠀⡄⢂⠱⡄⣈⠔⡀⢂⡐⡜⢦⠳⣜⠡⢀⠦⣉⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠰⣈⠒⠤⢡⢉⢃⠒⠯⣇⠰⣈⠂⠈⢾⠐⠀⠂⠀⠉⡆⠇⠀⠀⡸⢦⠘⡠⢙⠴⣡⠶⣙⢯⠀⡟⣬⠓⡌⢆⢣⠞⣤⠴⣋⠟⡦⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⡉⢅⠢⡙⠤⢃⡌⢢⢉⠒⡨⢑⣴⣏⠠⡸⠁⠀⠀⠀⠀⡇⡇⢀⡤⣛⢼⡠⠑⣸⢎⡵⣛⠭⣎⢧⡹⣆⢹⡘⢨⠺⣹⢆⡻⣜⡹⠆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠁⠈⠢⠡⡘⠢⣌⠢⢡⣟⢿⣹⠐⠧⠀⠀⢠⠂⠀⡇⠄⠀⣍⠳⣎⣇⣒⢌⣺⠜⡦⢽⡎⡞⣵⢫⣗⡚⢀⠐⣡⢞⡱⢮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠣⢄⠣⢃⠙⢾⡻⡌⢼⠀⠀⠠⠀⠀⢸⠘⣼⣘⡳⢊⠳⣎⡟⣼⢫⡽⣡⡼⣹⢳⣏⡾⣹⡍⣏⢳⢮⡻⣝⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢹⠧⡌⢄⡀⢰⠀⡈⠜⡄⠷⠜⢣⡉⢖⡰⠌⡥⢋⡜⣣⡝⣧⢻⡴⢻⡵⡋⡼⣹⠂⢻⣬⠑⠄⠀⠀⠀
⠀⠀⠀⠀⡐⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠄⠂⠀⠣⡐⠤⡹⠔⡈⠆⠱⣃⠀⢣⡜⢂⠦⡙⢤⢃⡒⠇⣜⡲⣍⡞⣧⢻⡅⠈⠁⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣁⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠀⠀⠀⠀⠀⡘⡤⢉⣂⠹⢄⠰⣌⠣⡜⡘⡔⢊⡌⢓⠤⣳⡌⣟⡼⢳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢃⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠁⠀⠀⠀⢀⡙⢎⡰⣉⢆⡻⣔⢃⠲⡑⣌⠣⠜⣌⠲⡑⡚⢵⣫⢻⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡘⠄⠀⠀⠀⠀⠀⡀⠠⢄⠋⠘⡔⣒⠴⣪⢕⠪⣌⠱⡘⣄⠣⡍⠤⠓⣌⠱⣂⠭⣳⡝⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡜⢠⠃⠀⠀⠠⣀⠣⢌⣡⠁⣠⠚⡼⣌⢳⡱⢊⡱⢄⢣⡑⢢⡱⢌⢣⡙⡤⠓⡤⣓⢧⣛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⡘⠤⡐⠢⢄⠃⢦⣻⠰⡡⠎⣵⢊⢧⢳⠡⡒⣌⠲⢌⢣⡝⣎⠶⡸⢥⢳⡱⣝⢮⣭⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢢⠌⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣣⣾⣶⣿⣿⣿⣿⠟⠀⠀⣀⠐⡠⢂⠔⡠⢂⠔⠠⠂⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⣀⣴⣾⣿⣿⡟⠿⠛⢛⢁⠢⠐⠁⠀⠠⠑⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⡄⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠻⠛⠋⠉⠉⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢥⡈⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡣⢘⣄⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡾⣷⡃⠀⡔⣭⢫⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣽⣾⣿⠿⢃⠹⣷⣻⠖⢁⠀⠫⡞⠀⢁⠂⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀
⢴⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣯⣷⢿⣻⣟⣿⢿⡽⡏⠁⠀⣐⣫⣿⠀⠀⠌⡄⢡⡀⠀⠈⡜⡱⢎⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀
⠀⠀⠀⠀⠀⢸⣿⣯⣿⣾⡿⣿⣯⢿⣯⢷⣻⣞⣯⢷⣻⣞⣯⣋⣉⣉⡁⠀⢸⠋⡆⠐⢨⠰⣀⢌⡩⢁⠠⡀⠩⢑⠀⠡⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢸⣷⣻⢾⣻⣟⡷⣯⣟⡾⣟⣷⣻⢾⣯⠗⠉⠈⠁⠈⠽⣿⡷⠈⠁⠷⣌⠂⠉⠒⠌⣲⠁⡖⢁⢢⠈⠢⠔⢣⢎⡵⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡞⣭⣟⢷⣫⣟⡷⣭⣟⡿⣾⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠗⣀⣠⣄⡀⠀⠁⠚⠬⢒⠠⢁⠂⠌⡀⡞⠍⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣞⡿⣮⣟⡷⣯⣟⡷⣯⣿⣳⣿⣷⣻⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠀⠉⠙⠛⠷⢦⡤⠄⠁⡇⠆⡈⠄⢰⠉⡄⠀⠈⢰⢞⡹⢫⡕⢠⠐⡄⢊⠁
⠀⠀⠀⠀⢀⡿⣾⣽⡷⣯⣟⡷⣯⣟⣷⣻⣷⣟⡾⣯⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠍⠠⢈⠑⡰⠋⠮⢱⢫⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠠⣿⣳⣿⣿⣳⢯⣟⣿⣾⣳⡿⣟⡾⣽⣳⣷⠀⡀⠀⠀⠀⠀⠘⣷⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠌⡐⠠⢀⠀⢂⡜⢁⠐⡀⠒⠈⢷⣩⢳⠂⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⣯⣿⢸⡻⣽⣞⣿⡷⣿⣽⣧⣔⣠⢀⠀⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡈⠐⡀⠡⠀⣜⠣⡜⣠⠰⠀⠡⠌⡀⢣⢏⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠹⠿⣟⠪⠏⡹⠟⣧⡘⢦⠹⡌⡟⠋⠿⣿⠿⣿⣿⡿⠿⠯⡍⡘⠤⠀⡀⣀⠠⢀⠄⠢⢌⡅⡐⠄⢃⢠⡁⢼⢪⡕⢮⠁⢂⡡⢋⠔⠁⡞⡜⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠤⡑⢌⠰⡈⢅⠓⡘⠟⣣⠐⡸⠂⠈⠾⠀⠐⡀⠀⠉⡖⠀⠀⠀⣘⢦⠈⡅⢚⠴⣱⠞⣎⣳⠀⣯⠲⡍⣆⠚⣤⢓⡥⢴⢺⣍⠳⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢩⠑⡘⢢⠱⢌⠢⡑⢌⠢⣁⢓⣴⣧⠠⡜⠁⠀⠀⠀⠀⡇⠀⣀⡠⢞⡼⡡⠌⣸⢎⣥⠟⢧⢎⢷⡸⣇⠼⡠⢙⡸⣣⡞⣣⢏⡜⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠁⠂⠱⠈⡜⠰⣈⠒⢤⣟⡯⣟⠠⠇⠐⠀⠠⠂⠀⡇⠆⠀⣩⠳⡜⣇⡰⣄⢫⡜⡺⠼⣎⢎⡷⣹⢞⣑⠠⢐⣡⢞⡱⢮⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠂⠐⠀⠠⠀⠀⠈⠑⡌⢪⠡⠙⢾⣳⡜⢰⠀⠀⠁⠀⠀⡒⢌⡴⢣⠯⡙⢮⢳⣭⢳⢏⡯⣭⡜⣞⡳⣝⣮⢳⡏⡝⢦⣫⢟⡽⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠁⠀⠀⠀⢧⠧⠎⡠⢀⠰⠀⠤⠙⡄⢟⠓⡡⢓⡌⠦⠌⡍⢎⡱⢧⢻⣜⡳⣭⢶⣫⠏⣜⠳⡎⢹⢮⠑⠄⠀⠀⠀
⠀⠀⠀⠀⡐⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠈⠔⣁⠢⡱⡌⢢⠡⡑⡇⠀⢇⣣⠘⡴⢉⡜⡰⢌⢣⠺⡴⣙⡞⣶⡹⠂⠈⠙⠒⠈⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⠀⠀⠂⡀⠀⠀⠀⠀⢸⡀⢆⣡⢱⡀⢧⢌⡱⢂⠇⠦⣑⢊⠦⣙⠶⣡⢟⡲⣝⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠊⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⢀⡛⢊⡔⢢⢜⡣⢞⡠⢍⢚⡰⠡⢎⠒⣌⠳⡙⢮⡳⣝⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠠⡌⠉⡔⢣⡘⢦⢎⡵⢋⠔⣊⠆⢥⠋⡴⠩⣄⢓⡘⠦⢹⣎⢷⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⡌⠀⠀⡜⢠⠃⠀⠀⠠⢈⠆⡡⢳⠁⢤⡘⢧⣙⢎⡞⠤⣃⢚⡐⢎⢢⡍⡔⣃⠲⡌⣒⢩⢖⡽⣪⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡜⢠⠃⡜⢠⠐⢢⠁⢎⣰⢿⢈⡒⡌⢷⡘⢮⡜⡡⢆⡣⠜⣌⢲⡙⢦⡱⢫⡔⣎⢳⣎⠷⣭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠲⣉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢱⣴⣧⣶⣿⣿⣿⡿⠆⢀⠠⡐⢄⠒⡰⢐⠢⠐⠡⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⠿⠻⠛⡛⢋⡑⠈⠀⠀⠰⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⡧⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⢿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣀⣢⣤⣶⣿⣿⣿⣿⣿⣿⡟⣄⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢢⠐⡀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣼⠀⠀⢖⡽⣱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⢀⣤⣾⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⠟⢹⣾⡳⠎⢁⠚⣥⠋⠀⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢷⣿⣿⡿⣿⣻⠍⠐⠂⢻⡇⠈⠠⢀⠰⡄⠀⠈⢀⢯⡹⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀
⠉⠉⠉⠉⠉⣿⣿⡿⣟⣿⣯⣿⣷⢿⣳⡿⣯⣟⡾⣯⢷⣯⣟⠗⠡⠀⠒⠒⢻⡗⠠⠐⠌⢆⠈⢓⠒⡈⢠⠑⠃⡍⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣿⣽⣻⣟⡿⣽⢾⡽⣯⣿⣽⣳⢯⡿⣽⣻⠞⠹⢛⠻⠷⣶⣄⠼⠸⣄⢡⠙⠨⣚⢌⡒⢄⠆⡠⢁⠲⢄⢠⡏⡵⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⣞⣳⢯⠿⣽⢯⣻⣗⣿⣳⢯⣿⣽⣷⠁⠀⠀⠀⠀⠉⠛⠋⠁⡐⠳⢌⠀⢀⠀⠈⠘⠈⢖⠡⠆⡠⢈⠠⠘⡑⣳⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⣯⢾⣽⣫⢿⣽⣻⢷⣻⢾⣯⣟⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠀⠈⠘⠻⠷⣦⣄⣀⠀⠉⢢⠅⠆⠠⢁⢰⢣⠀⠐⠂⣀⢴⢢⢴⢀⡀⠀⡄⢠⠂
⠀⠀⠀⠀⢸⣽⣻⣾⣽⣻⢾⣽⣯⣟⣯⣿⣞⣿⣻⡽⡄⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⡞⠀⠁⠂⡜⠀⠲⠄⣠⢏⣎⠳⣎⠗⠈⢁⡘⠄⡒
⠀⠀⠀⠀⣻⢾⣽⣿⡾⣽⣻⢾⣷⣻⢾⣽⣿⣳⢯⣟⣧⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠠⠘⠀⠈⡐⣠⠃⠠⢀⠹⢨⢆⡤⢧⡈⠂⠀
⠀⠀⠀⠀⠘⣯⣿⣿⣿⣷⢿⣯⡟⣿⣯⣯⢹⣯⣟⣾⢿⡄⢂⠀⠀⠀⠀⣈⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⠠⠁⠌⠠⢁⡴⡁⠄⡁⢂⠐⡀⠫⣜⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠹⣿⣿⢟⢿⠟⣯⣷⡘⢬⠛⣯⢻⡾⣟⣿⣾⣷⣮⣄⣄⣰⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⡀⢤⠃⠈⠄⠡⠈⠄⣡⠏⡴⣉⠶⠉⢀⠦⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⠒⡥⢈⠔⣋⠚⠦⣙⠤⣳⢦⠈⡙⢿⠛⢻⡟⠛⠉⠁⠪⢑⠲⠐⠂⠂⣜⠠⢃⠔⡪⢔⡱⣈⡤⣵⠂⣧⢛⡴⢋⢠⠰⣉⠆⠡⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⢣⡘⢄⡉⠒⡄⢊⠌⡙⠶⡐⢬⡁⠀⢫⠀⢀⠀⠀⠀⡝⠀⠀⢀⡭⢣⠂⢆⡱⢪⢲⢧⠳⡼⡠⢇⡏⢴⡉⢆⢳⡵⣚⢭⣣⠳⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠃⠌⠢⢌⠡⡘⠥⡘⢄⠚⡄⢣⣼⣷⣑⠢⡁⠀⠀⠀⠁⠀⠞⠀⠰⢧⣚⠽⡈⢄⢫⡝⣎⠏⢷⡱⣛⠾⣜⡢⡍⠘⡌⢷⡩⢖⣥⡛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠢⢡⠉⢖⠨⢼⡳⣏⡧⠘⡅⠀⠀⡌⠀⠀⡍⡁⢤⣚⢬⣛⣤⢮⡤⢿⣌⡻⠮⢵⣩⡟⣼⢣⣥⣂⢤⢖⡹⣎⢶⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⢍⠢⡙⠀⠉⢿⡹⡌⡇⠀⠀⡂⠀⠀⢧⢰⡳⡜⡎⠴⡙⠮⣝⢳⣎⢷⣚⢧⡳⣝⢮⡟⡶⣍⠞⡮⢵⣫⢞⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⢂⠔⡡⠀⡅⠠⡑⠸⡘⡱⠉⡜⢢⡑⠎⡤⢃⡌⢎⡝⢮⠳⣝⢮⡝⣷⡈⢻⠼⡀⢳⢯⠈⠂⠀⠀⠀
⠀⠀⠀⠠⡑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠈⠢⠔⡡⢛⠤⢁⠃⠥⡆⠀⡝⢦⡘⣑⢢⠣⡘⢦⢉⢞⡱⢎⡷⣹⢎⡑⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢈⠅⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠂⣀⠀⠀⠀⠀⠚⡤⢉⡴⢙⠄⣳⢂⠵⡈⠦⣑⠱⢌⡊⢼⡱⣫⡜⣧⢏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠁⠀⣈⢍⡱⢄⡋⢼⡡⢏⠴⣉⠲⣁⠞⡠⢍⢢⡑⢣⠻⣼⡹⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⢂⠀⠀⠀⠀⠀⢀⠠⡐⠜⠈⢆⠲⣌⡜⣣⠝⣌⠲⣈⠕⣂⢎⠱⡌⡒⠬⡑⢬⠱⣳⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⢠⠒⡌⡀⠀⠀⠄⡱⢈⠆⣹⠀⢆⢫⠞⡴⡹⡜⠬⡰⢑⡌⠲⣁⢎⠦⣑⢡⠣⢍⣂⢗⡯⡞⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠣⡘⠤⡁⢆⠰⡈⠔⡡⣺⡽⢊⡜⢢⡛⡴⢳⣍⠒⡥⣃⡌⡓⡜⣎⠳⣔⢫⡜⢦⣙⢮⡳⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠐⣸⣦⣷⣾⣿⣿⣿⠗⠀⢠⠀⠤⠠⠄⠤⢠⠐⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣴⣾⣿⠿⠿⠟⠻⢛⡉⠀⠉⠀⡘⠄⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⠇⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠉⠀⠀⠈⢂⣠⣴⣾⣿⣿⣿⣿⣿⣏⡘⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡶⡅⠀⢂⢒⡖⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀
⠀⠀⠀⣠⣴⣧⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⡟⠙⣿⡤⠎⢉⠺⣱⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣾⣿⣻⡿⣽⠞⠀⠑⣸⠥⠀⠂⢄⠡⡀⠀⠀⢩⢞⡹⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀
⠙⠛⠙⠋⠛⣿⣿⡿⣿⣻⣿⣯⣿⣷⣟⡿⣽⢿⡽⣞⡷⣯⢿⡽⠃⠀⠒⠒⢙⡆⢀⠈⠜⡀⠙⠢⠔⠂⡈⠉⠧⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡿⣯⣟⡿⣿⢷⣯⣟⣾⣽⣻⡽⣯⢿⣽⣻⠝⠛⠟⠛⠳⣶⣦⡌⢲⡀⠎⠣⠝⣢⠙⡔⣠⠱⠈⡀⢧⣀⢠⢟⡹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⢷⢯⡿⣽⣻⢾⡽⣳⢯⡿⣽⢯⣿⣾⠁⠀⠀⠀⠀⠀⠙⠙⠀⢩⢱⠃⠀⠀⠀⠉⠒⠄⣣⢡⠃⡀⠄⡈⢊⠑⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣟⣮⣟⣳⢷⣯⢿⡽⣯⣟⣿⣽⣻⣿⢿⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠂⠐⠚⠿⣶⣤⣀⠀⠀⠁⣣⢐⠂⡐⠠⢸⠌⠀⠱⠠⢀⡤⡤⢤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠈⢹⣻⢾⣽⢯⣟⡾⣯⣟⡷⣯⣿⣞⣿⡽⣿⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠰⡉⠀⠄⠁⡎⠘⠢⠄⣀⡏⢶⣩⢓⠞⠀⠑⡈⢌⠡
⠀⠀⠀⠀⢼⣯⢿⣿⣻⢾⣽⣳⣯⣟⣷⣻⣿⢯⣟⣷⣳⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢀⠚⠀⠠⢁⢢⠃⠈⠄⢡⢫⡄⣤⢎⡘⠂⠀
⠀⠀⠀⠀⠸⣯⣿⣿⣿⣻⣾⡽⡟⣿⣞⣿⡹⣟⣾⣳⣯⣇⠂⠄⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⢀⠂⡐⢀⢂⡜⡂⠌⠐⡈⠄⡀⠻⣔⢫⠄⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⡷⣿⣯⠜⡝⠯⣗⢻⣯⣷⡿⣿⣿⣾⣧⣴⣤⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠁⡐⢀⠂⡐⠀⢦⡋⡴⣑⠎⠡⠐⡤⢁⠆⣸⢣⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠩⠛⠮⢋⠡⢊⠛⠧⢎⡱⣊⡵⣤⠈⡉⢿⡉⠛⡿⢋⠉⠉⠪⢔⠲⠐⠠⠂⣌⠰⠡⠌⡖⡰⡐⢡⡰⠴⢡⢧⡙⢶⡉⢐⠤⣋⠔⢁⣠⢧⣋⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢎⡡⢊⠔⢢⠐⡡⢘⠓⢎⡰⢐⡀⠀⢋⠀⠀⠀⠀⠀⡹⠀⠀⢠⢏⡆⡑⢂⠧⣱⢻⣱⢋⢷⡘⢦⡛⠴⣡⠋⢶⡱⣚⢮⡱⣣⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⠢⠘⠤⣈⠑⡚⠤⡃⠴⡁⢎⠰⣼⡿⡔⢂⡅⠀⠀⠀⠉⠀⠵⠠⢔⣫⠞⣼⢀⠃⢿⣘⠶⠍⣏⠶⣹⢳⣌⡳⠄⡙⡸⣕⢫⢖⣱⡃⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠈⠐⠡⢌⠣⡘⢄⢿⣹⡽⡇⡘⡆⠀⠀⡜⠀⠀⣃⠃⡴⣌⠻⣔⢧⡼⡬⢮⣹⣙⠮⣳⣹⡳⣎⢯⢥⣀⡤⢖⡫⣞⡖⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⢣⠘⡌⠂⠙⢞⡷⡁⡇⠀⠀⠆⠀⠀⢇⢺⡱⢎⡝⡌⢓⠞⣹⢧⣳⠳⣖⡳⢧⣛⢾⣫⢞⡵⣊⡟⢼⣣⣟⡀⠀⠀⠀⠀
⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡏⢣⠐⡡⠀⡇⠠⡘⠸⡈⠷⠉⡒⣌⢃⡚⠤⡒⢤⠋⡞⡝⢧⣛⢮⡝⣾⡐⣣⠽⠀⢗⡶⠁⠂⠀⠀⠀
⠀⠀⠀⠈⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠈⠢⢡⢡⠛⢄⠡⢡⠩⡆⠀⡳⢤⠣⢌⢓⡸⢄⢫⠱⣘⠧⣎⠷⣹⢖⡇⠀⠉⠁⠐⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠐⡌⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠈⠠⠀⠀⠀⠀⠀⡚⡄⢃⡴⠹⢄⡳⡌⡱⢊⢆⠲⠌⣆⠓⣌⢳⣬⠻⣵⢫⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠁⠀⣠⠙⠥⣒⡉⢦⢳⡱⡑⢪⠰⢩⠜⡠⢍⡂⢇⠎⡻⣜⡳⣞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠑⠀⠀⠀⠀⠀⢀⠠⡐⠜⠀⡙⢆⢥⡚⣥⠧⠱⢌⠣⣑⠣⣌⠱⢢⢉⠖⠬⡑⡜⣳⢞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⠀⠠⢃⠜⡀⠀⠀⠄⡱⢈⠆⣹⠀⠦⣙⢎⠶⣙⠦⡱⢉⢆⡓⢌⠲⣌⠥⢃⠎⣌⠓⡬⡔⣯⢞⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠃⢎⠰⡐⢄⠒⡌⠰⡁⣾⡝⢎⡒⢩⣎⠳⣍⠶⣁⠫⣔⠸⣈⠳⣬⠳⣍⠞⣤⢛⡴⣹⢮⡝⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣶⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠈⢟⠻⠉⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠆⠒⠵⣎⠼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⢂⣰⠹⣌⡳⣧⢭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⣎⢳⡱⠱⢌⡛⢮⡳⢮⡝⣶⢣⡟⣼⢣⡿⣜⡧⣚⡵⢹⣎⣯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡘⠃⡇⡹⠰⡘⢤⡑⢣⡘⣱⢫⠞⣵⢫⣞⣳⡈⢷⡸⡀⢳⠶⠁⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⠸⣅⠣⢍⠢⡜⢡⠒⠧⢌⡻⣔⣫⢖⣧⡓⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠤⢋⢆⠓⡬⢑⢪⢑⡊⡵⢎⡼⣋⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡡⠎⡜⡰⡉⢆⠎⡔⡩⢎⠳⣝⢮⡳⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠔⢣⡘⡡⠜⣌⠲⢡⢒⡡⢚⠌⣷⣹⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⡙⢢⢡⡑⣣⠢⢍⠆⡣⠜⢢⡙⣶⢣⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⡘⢥⠢⣱⢣⡛⡤⢏⡵⣩⢧⡹⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡜⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢘⣴⣧⣷⣿⣿⣿⡿⠆⢀⠀⡄⢂⠒⡰⢈⠤⡐⠠⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣠⣴⣿⣿⠿⠛⠟⠛⡋⠔⠁⠈⠀⡀⠣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣴⣾⣿⣿⡿⣿⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠁⠀⠀⠠⢑⣢⣴⣾⣿⣿⣿⣿⣿⣿⢡⠙⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢩⢁⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⢯⡄⠀⣡⠽⣙⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⢀⣠⣴⣧⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⡿⢁⢻⡷⠒⠁⡙⠜⡎⠀⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣽⡷⣿⣿⡿⣿⣻⠗⠀⠈⠌⣇⠁⠂⢄⠂⢃⡀⠈⠀⠳⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢻⣿⡿⣿⣻⣿⣯⣿⣷⡿⣿⣻⡽⣯⣟⡷⣯⣟⡷⢋⡒⠊⠉⠉⡵⠀⠄⡊⢆⠠⢑⠒⡈⢐⠈⠹⢊⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠸⣟⡿⣽⣿⣷⢿⡽⣾⣽⣳⣯⣟⡷⣯⣟⠗⠋⠹⠉⢛⣻⣶⣔⠺⣄⢢⠉⠊⣕⡊⢖⢠⠃⡌⠠⠘⣄⡠⡜⣏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣽⣻⡽⣾⡽⣯⢿⡽⣞⣿⣳⢯⣿⣷⡇⠀⠀⠀⠀⠀⠈⠉⠀⠠⠓⠦⠀⣀⠀⠀⠉⠢⢣⡘⡄⠡⠀⠄⠙⠌⣳⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⢧⣷⡻⣵⣻⣞⣯⢿⡽⣯⣟⣯⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⠈⠘⠛⠿⣶⣤⣀⠀⠀⢡⢃⡃⢈⠐⢨⢓⠀⠁⠆⢀⡤⡤⢤⢄⠀⡀⡀⢄⠀
⠀⠀⠀⠀⠈⣿⣻⣞⣿⣳⣟⡾⣽⢯⡿⣽⣿⢾⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⡖⠀⠂⠈⡜⠈⠦⠄⣀⢏⠶⣩⣓⠞⠀⠑⡈⠆⠢
⠀⠀⠀⠀⠀⣿⣳⣿⣷⣻⢾⡽⣿⢯⡿⣽⣿⡿⣽⣻⢿⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠠⠑⠀⠂⡐⢠⠋⠈⠌⠱⢘⡆⡤⣎⠘⠈⠁
⠀⠀⠀⠀⠀⣿⣳⣿⣿⣽⣯⢿⣽⢿⣽⣷⡏⣿⣳⢯⣿⣳⣀⠆⡀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⢂⠐⡈⠐⢠⡜⡁⢂⠡⢈⠐⡈⠼⡱⢎⠆⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⣾⣿⣻⡌⢟⠾⣿⡸⢿⣟⣷⣿⣿⣿⣷⣷⣦⣾⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⢌⠃⡐⢀⠂⡐⠠⢡⠳⣌⡱⠆⢊⠀⡆⢂⠄⣹⢚⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠛⠯⡹⢉⠌⡛⠿⡌⠖⣌⢣⡧⡌⢈⠙⢏⠉⢻⠏⠋⡁⠫⠌⡉⡇⠘⠀⣇⠘⢄⠃⡎⠴⡰⢈⣰⢤⠃⣏⢳⣌⡓⢈⠤⣚⠔⠁⣢⡇⢯⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡡⢊⠔⡡⢘⠰⡈⠥⡙⠱⣂⠒⣄⠀⠈⠅⠀⠀⠀⠀⡇⡇⠀⢠⢞⣣⠌⢂⡝⢢⢽⢫⢖⣹⡠⢏⠶⡰⢡⢋⢶⡱⣎⢯⣱⢚⡵⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠍⠐⠄⢣⢉⠒⡡⢆⡑⢢⢁⠣⣰⣿⢮⠐⣈⠀⠀⠀⡈⠁⡇⠃⠰⢧⡚⡼⡄⠃⢞⡹⢎⠏⢾⡰⢻⡜⣧⡱⢃⠈⠮⡕⡞⢦⢣⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠒⠤⡙⢄⡊⡴⣻⣭⢟⡈⢃⠀⠀⠰⠀⠀⢇⣁⡰⢣⡝⣲⢧⡭⢦⢯⣍⣛⠮⣳⢹⡞⣵⢫⡥⣈⡤⢶⡙⣧⢏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠀⠐⠀⠀⠀⠘⠤⡑⠰⠈⠫⣟⡆⢹⠀⠀⠡⠀⢀⢸⢸⡕⣫⠼⣁⠎⡽⢹⡞⣼⡹⣖⢧⡻⣜⢧⡟⣼⡣⢞⡱⢽⣎⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠣⠛⢆⠂⡄⢰⠀⡄⢊⠞⡸⠁⡗⣈⠖⡌⣅⠲⠤⡙⣬⠳⣝⢮⡳⣝⢶⡙⢮⡹⠀⢞⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡑⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠁⠈⠂⠥⡘⡜⢆⡘⠄⠚⣥⠀⢘⢦⢊⡔⢢⠣⢱⠡⢇⡚⡜⢦⢻⡜⣧⡓⠀⠉⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢌⠒⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠁⠄⠀⠀⠀⠀⠌⣆⢌⡱⠬⠤⣘⠆⡥⢊⢆⡙⢢⠍⡒⢬⣙⡎⣷⡹⢶⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢂⡅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠀⢠⡈⢎⠴⣉⠲⣍⢞⡰⡉⢦⠘⡅⢎⠱⢢⠜⡸⢡⣟⣣⢟⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢘⢢⢒⡬⢳⡜⣊⠴⡑⡌⢣⠜⣈⠇⢎⠴⢡⢃⡜⢞⡺⣜⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠌⡌⣁⠀⠀⠠⣈⠱⡈⢦⠁⡔⢚⡬⢳⡜⣣⠚⢤⠒⡅⠎⣅⡚⡔⢪⠜⣢⠃⢎⡴⣫⠷⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡌⢒⠰⡀⢆⡐⠢⢄⠣⢸⣻⠰⡌⣃⢞⡱⢎⡵⡉⢦⢩⠜⣘⢰⡹⡜⣥⢚⡥⣚⠵⣺⠵⣛⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠃⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢢⣵⣾⣶⣿⣿⣿⡿⠂⠀⡀⢄⠢⡐⠢⢌⡐⢠⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⣠⣴⣿⣿⡟⠿⠛⢛⠛⡀⠆⠁⠀⠠⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡋⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠛⠋⠋⠀⠀⠀⠠⢑⣠⣶⣾⣿⣿⣿⣿⣿⣿⢣⡘⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡩⢁⡀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣧⡇⠀⣩⢚⡝⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⢀⣠⣴⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⡟⢉⢿⣳⠒⠁⡘⠼⡑⠁⢂⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣯⡿⣾⢿⣿⢿⣯⠗⠈⠀⢊⣧⠈⠐⠠⠄⢣⡀⠀⠈⡰⣍⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀
⠙⠋⠛⠙⠙⢛⣿⣿⢿⣻⣿⣽⡿⣷⣿⢿⣻⣽⣳⢿⣽⣻⣞⡟⣒⠒⠊⠉⠉⡷⢀⠠⡁⢇⡀⢩⠒⠊⢠⠀⡙⢣⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠸⣯⢿⣻⣿⣟⣯⣟⣯⣟⣯⡷⣯⣟⡿⣞⠷⠉⠛⠉⢛⣿⣶⣔⠃⣇⠰⠉⠒⡌⡣⢎⠰⡄⡡⠐⠘⣄⡀⡼⣋⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢿⡽⣾⡽⣾⡽⡾⣽⢾⣽⣳⣯⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⢠⠁⠏⠀⣀⠀⠀⠉⠢⠕⣰⠁⡂⠄⠠⠑⡉⣞⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣛⣮⢿⣱⢯⣷⣻⡽⣯⢿⣞⣷⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠒⠈⠘⠛⠿⢶⣤⣀⡀⠀⢣⠅⡊⠄⠡⢐⢧⠀⠉⠰⢀⡤⢤⠤⢄⠀⡀⢀⠄⠀
⠀⠀⠀⠀⠐⢹⣟⡾⣯⣟⡿⣞⣷⣻⣽⣻⣟⣾⣿⣳⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡞⠀⠄⠡⡘⠀⠦⠄⣀⠷⡸⣍⠾⠍⠂⠑⡈⢌⠡
⠀⠀⠀⠀⠀⣾⣽⣻⣷⢯⡿⣽⢾⡷⣯⢷⣿⣟⡾⣽⣻⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠠⠐⠁⠈⠄⢢⠍⠠⠁⢌⠳⡆⡤⣎⠘⠂⠀
⠀⠀⠀⠀⠀⢹⣞⣿⣿⣯⣿⣽⣻⣿⣽⣯⡟⣯⣟⣷⣻⣧⣂⠄⡀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⡀⢂⠁⠂⢌⡔⠣⢀⠂⡁⠂⠄⠻⢴⡩⠆⠀⠀
⠀⠀⠀⠀⠀⠘⢯⣿⣿⡿⣷⣿⣻⡆⢯⠳⣿⡹⣿⢾⣟⣷⣿⣿⣷⣾⣤⣿⠟⡀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠆⢈⠐⡀⠂⠌⢰⠎⡜⡱⠆⠋⡀⣌⠂⡔⢨⡓⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⡙⢌⡙⠻⠦⡙⣤⢓⣬⡌⢈⠙⢏⠉⢟⠻⠋⠉⠚⠥⡉⡇⠘⠂⣕⠈⢆⠡⡘⢤⠒⣈⣤⡱⠌⣮⠹⣜⠣⢁⠦⣘⠄⢃⣠⡏⡵⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢈⠆⡌⡑⢢⠑⠤⡉⠜⡱⢌⢂⢆⠀⠘⡡⠀⠀⠀⠀⠔⡃⠀⢀⣎⣳⠈⡔⢩⢆⢻⣱⢆⣻⡀⢧⡛⢤⢋⠜⣲⡱⣚⢮⡱⢎⡵⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⠈⠒⡌⡑⠢⢍⠢⡑⢌⠰⣨⣿⣎⠒⢼⠀⠀⠀⠈⠀⡡⠇⠐⢧⡚⡴⡃⢄⢫⢞⣣⠏⢞⡴⣛⢮⣳⡘⡌⠘⠴⣙⠧⣎⠵⡋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠁⠃⢌⠂⢧⡈⠲⣽⣣⢿⡉⢜⠀⠀⢠⠁⠀⢰⢈⡰⢲⣍⢳⡧⣬⢦⡽⣆⣛⡞⢶⣩⡗⣧⢯⡍⣄⣢⠴⣛⣬⣛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⡔⡈⠇⠀⠻⡽⣖⢸⠀⠀⢨⠀⠀⠸⣨⡗⣣⢞⡡⠜⡣⢗⢾⡱⣏⠶⣎⢷⣚⢧⡟⡼⣇⠮⣝⢶⣣⢯⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⠘⡄⢊⠄⣰⠀⢌⠡⢖⠘⠅⡣⠜⢢⡑⠎⡤⠓⢬⠹⢎⠷⣭⢞⣹⢳⡉⢞⡼⡀⢳⣳⠈⠂⠀⠀⠀
⠀⠀⠀⠀⡀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠉⠔⣨⢘⠎⡰⢈⡘⢤⠀⢘⢧⠱⣈⠞⣠⠋⢆⡛⢌⡳⡜⢮⣳⢫⢗⠀⠁⠉⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡘⠰⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢀⠀⠀⠀⠀⠘⢢⠂⡴⠍⠦⡘⣆⠓⡌⠦⡑⡜⢢⡉⢦⡙⣎⢷⡣⡟⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢨⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁⠈⠀⢀⣉⠫⢔⡩⢒⡽⣌⡓⠬⡑⡜⡰⢡⠚⠤⡙⣌⠳⣝⡳⡵⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡔⡀⠀⠀⠀⠀⠀⡀⢄⠢⠉⢢⢉⠖⣰⢋⡶⢌⠒⡥⢓⡰⢡⢃⢎⡱⠡⢆⠣⠜⣳⡝⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⢢⢁⠀⠀⠠⢈⠆⡱⢈⡇⠰⡘⣮⠹⣌⢧⢃⠎⡑⢆⠣⡔⣣⢊⡔⢢⡙⢤⢋⡼⣱⢞⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡘⢄⠣⢌⠰⡀⢆⠡⢊⢔⡿⢲⡡⢣⢜⡳⣜⠺⣌⠸⣡⢎⠱⣘⢥⢫⡔⣣⠝⣦⢋⣶⡹⣎⢷⡃⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢢⡑⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣢⣷⣾⣿⣿⣿⣿⠟⠀⠠⣀⠢⢄⠡⠒⡌⠰⠀⠆⠰⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣤⣾⣿⣿⡟⠛⠛⠉⠣⢄⠃⠀⠀⠠⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠡⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⢿⡄⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠛⠟⠛⠉⠉⠀⠀⠀⠀⣀⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⡳⡄⠛⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢆⠰⡀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢯⣷⣇⠀⡘⢵⡩⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀
⠀⠀⣠⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣯⣷⣿⣟⠏⢠⢻⣾⠉⠐⡀⠙⠎⠀⠈⡔⢠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣯⣿⢾⣟⣿⣻⢿⣯⡷⠃⣀⣀⡼⣟⠀⠐⡀⠇⡘⣄⡀⢀⠜⠳⣬⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀
⠀⠀⠀⠀⠀⢸⣿⣟⣯⣿⣷⡿⣿⣟⣿⡽⣾⣽⣻⣞⡷⣯⡿⣮⣤⣤⣀⡀⠀⣏⡔⢠⢨⡱⣀⠤⡘⠠⢐⠠⢀⠸⠀⠐⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⣷⢯⣿⣻⣽⣻⢷⣻⡾⣽⢷⣯⢷⣯⡟⠁⠀⠀⠀⠀⠿⢿⠊⢀⠳⡰⠀⠀⠑⠪⢱⡑⡊⢔⠠⢈⠓⡄⠻⡴⣩⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣯⢟⣾⢳⢯⣟⣯⢷⣻⣽⣻⣞⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠋⢠⣤⣤⣀⠀⠀⠙⠢⠍⡄⢂⠠⠁⠄⣧⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⢷⣻⣞⣯⣟⡾⣽⢯⣷⣻⣷⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠙⠛⠶⠦⠀⠘⡤⠃⠠⢁⢸⠒⡄⠀⠉⢰⢺⡙⢯⡱⢠⠐⡄⢊⠁
⠀⠀⠀⠀⠀⣸⣯⢷⡿⣾⣽⣻⣽⣻⢾⣽⣷⣻⢷⣻⣇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠁⢀⠎⠀⠌⡁⠶⠉⠧⠹⢎⡁⠀⡠⢢⡁⠎
⠀⠀⠀⠀⠀⢿⣞⣿⣿⣳⢯⣷⣻⣯⣟⣾⠿⣯⢿⡽⣞⡇⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠄⠡⢈⠀⡈⢄⡜⠁⠌⡀⠉⠌⡳⢭⣓⠂⠀⠀
⠀⠀⠀⠀⠀⠸⣯⣿⣿⣿⣿⣳⣿⡝⡿⣾⣟⢿⣯⣿⣟⣿⣦⣥⣂⡀⢀⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⢀⠂⠌⡐⢀⠂⡴⢣⠜⣠⠢⢀⢁⠒⡈⢳⢬⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⡿⠛⢷⣧⢚⠬⣙⠮⣹⠚⠻⢿⡟⠿⣿⢿⠿⠿⢜⡑⠤⠤⠠⡀⡰⠀⢆⢂⠱⣁⢂⠌⡐⣀⡂⢼⡡⢗⠮⢁⠐⡤⢊⠔⠁⡞⢦⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢡⠊⡔⡈⢆⢊⡑⢊⠳⢯⡐⠠⠅⠀⠙⢃⠀⢂⠀⠈⢰⠁⠀⠀⣸⢃⠆⡌⢸⢄⣣⠶⣹⣱⠀⡷⣙⢎⠣⢆⡹⢴⣡⠜⣎⢏⠧⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡅⢓⠰⠡⢎⡰⢈⠆⡑⢢⠉⣶⣏⡆⣰⠃⠀⠀⠀⠀⢸⡀⢠⣔⡣⢏⡖⢈⢜⡎⣖⠻⡕⣎⢧⡳⣅⠮⡱⠨⡜⢧⡮⡝⣎⠞⠅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠀⠉⠆⡐⠣⢜⠠⢃⣾⢻⡽⢆⠹⠀⠀⢀⠆⠀⢰⠰⢀⠴⣩⡓⣎⣄⣊⢞⡭⢖⡽⡜⣎⢷⡹⣖⡓⡀⢐⣩⢖⡹⣜⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠂⠀⠄⠀⠈⠁⢎⡑⠎⠌⠻⣽⡆⢹⠀⠀⠠⠀⠀⢰⡈⣏⠞⣥⢋⠵⢮⡝⣮⣝⡻⢬⡼⣺⢭⠷⣭⢳⡝⣫⠶⣩⢷⣹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠀⠀⠱⡺⠴⠂⠄⢐⡀⠐⡄⢣⠙⡝⢊⡌⡒⠦⣘⠡⢎⡙⢣⢯⡵⣎⢿⡱⢯⡝⢦⣛⠌⢳⣭⠑⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡒⠀⠃⢩⡘⢰⢆⠱⢈⠌⢮⠀⢸⡰⡡⠓⣌⠱⣂⠍⡞⠤⡳⢬⢳⡝⢯⡖⠈⠈⠓⠈⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠒⢌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠄⠀⠀⠀⠀⡀⢇⠌⢢⡘⢆⢰⣡⠱⢩⢄⡓⠬⡘⡔⠣⣝⡣⣏⣞⡳⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢭⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⠀⠀⠀⢀⡘⣋⠕⡸⣈⠶⣡⢇⢃⠲⠌⡥⠓⣌⠱⢌⠳⣩⢶⡹⣎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⠁⠀⠀⠀⠀⠀⢀⡀⢰⠈⢱⠐⢎⡱⣌⡳⣑⠪⢌⠣⡍⠴⣉⠤⢋⢆⢣⠒⡌⢷⣙⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡰⢁⠣⠀⠀⡀⠄⡑⢢⢘⡆⠰⣈⡞⣥⠳⣌⠳⢄⡋⡔⠣⢜⢢⡅⢎⡱⢌⠆⡭⣘⢧⡻⣜⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠑⡌⢢⠑⡠⠐⡌⠰⣁⡾⡇⡱⢂⡹⢆⡻⣜⡡⠎⢴⡉⡜⣌⠳⡜⢦⡚⣥⢚⡴⣭⢳⡝⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣨⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠐⣤⣷⣾⣿⣿⣿⣿⠗⠀⢀⠀⡂⠔⡐⢂⠆⡡⠂⠔⠠⠂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⣀⣴⣾⣿⣿⡿⠛⠛⠙⢡⠐⡐⠈⠀⠠⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢋⠡⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⢿⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡿⠿⠿⠟⠛⠉⠉⠀⠀⠀⢀⣀⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⢆⠙⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡄⢢⠀⠀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⢾⡿⣞⠀⢠⠧⣎⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀
⠀⠀⣠⣴⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣾⣿⠿⢃⡘⣿⡽⠏⡁⠄⠘⢜⠁⠈⠔⡠⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⣿⣽⡾⣟⡿⣿⣻⡽⡞⠁⣀⣀⣼⣿⠄⠀⠌⣐⠈⢆⡀⢀⡐⠳⣎⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀
⠀⠀⠀⠀⠀⢸⣿⡿⣯⣿⢿⣾⡿⣿⣟⡿⣞⣷⣻⣽⣻⢷⢯⣧⣬⣄⣀⠀⢸⠣⣃⠄⢒⡨⠔⣠⢉⠄⡐⠄⠠⢈⡁⠠⢄⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠘⣯⢿⣽⣻⡿⣯⣟⣷⢯⣿⡽⣞⡷⣯⠛⠉⠀⠁⠐⠩⢿⡷⠌⢁⠽⡄⠆⠈⠑⠢⢍⡒⢜⠠⣁⠂⠱⠄⢪⠖⡭⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣟⡟⣾⡳⣟⢷⣻⣞⣯⣷⣻⣽⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡋⢂⣤⣤⣀⠀⠈⠘⠢⠕⣨⢀⠂⠌⠉⡴⠍⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣼⢷⣻⢷⡽⣯⣟⣷⣻⣞⣷⣻⢾⣿⣽⠂⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠈⠉⠙⠻⠶⠤⠄⠐⢆⠆⡈⠐⢠⠏⡄⠀⠈⢰⢎⡽⣹⡡⠠⢄⠢⡐⠁
⠀⠀⠀⠀⠀⣸⣯⣟⣯⣟⣷⣻⢾⣳⣟⣾⣿⣻⢷⣯⡇⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢁⡜⠀⠌⠒⡲⠍⠞⠴⣣⠃⠀⡠⠤⡡⠉
⠀⠀⠀⠀⠀⣿⢾⣽⣿⣞⡷⣯⢿⣷⣻⢾⣿⣽⣻⢾⣽⡠⢀⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠐⡀⢂⠀⡀⠌⡴⠁⡐⢈⡐⠡⣛⠼⣱⠂⠁⠀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣯⣿⣻⣿⠹⣿⣯⡗⣯⣟⣿⣾⣷⣤⣆⡠⠀⢠⣾⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⡀⢂⠐⡀⢂⡰⢚⠴⡁⡤⠂⠠⡐⢈⠳⣥⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⠿⠻⢾⣧⠱⣌⠻⡜⢻⡳⠿⣿⠟⢿⣿⣿⠿⠿⢌⡘⠤⡄⠠⢀⠄⡠⠄⣂⢑⠆⡐⡠⠂⡐⡀⢲⡹⢬⢖⡉⠄⣠⠓⡄⠃⣜⢲⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠈⢁⠣⡑⢢⢉⠒⡌⢡⠙⡋⢷⡠⠘⠦⠀⠗⡓⠀⠀⠀⠈⢱⠀⠀⠀⢸⡱⡐⠄⢺⢌⡱⢦⡳⢭⡁⡧⡝⢎⡬⡐⢎⡴⣃⡤⢞⡭⢖⠅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠥⡑⢂⠎⠴⣈⠆⡱⢈⠆⡌⣣⣯⡆⢼⠁⠈⠀⠀⠂⢸⠀⣀⡠⢧⡓⡇⡘⢬⡲⢏⢧⢟⡲⢧⣱⣍⠚⡔⢉⠾⣱⡞⡼⣩⠞⠍⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠂⠈⠀⠀⠀⠁⠘⠰⢀⠓⢤⠃⡜⣼⡻⢷⡃⢺⠀⠀⢀⠆⠀⠸⠰⠀⡨⢧⡹⢼⣐⣄⣻⢜⡲⣌⠷⣩⢶⡹⣞⡱⠂⠐⣅⡺⣱⢣⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠂⠐⠀⠀⠀⠀⠉⢢⠙⡌⠄⠻⣯⢗⢠⠂⠀⠠⠀⠀⠸⡐⣼⢱⢣⢋⡝⠾⣜⣣⢯⡝⣭⣜⢧⡻⣵⣫⠷⣭⢛⡴⢣⣏⣟⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠊⠀⠀⠀⠘⡾⠴⠡⡀⢐⡀⢠⠁⢇⠏⡎⠣⢎⡰⠱⣈⠦⡉⠞⣱⢎⡷⣹⠶⣹⢏⡗⣭⢚⡏⠼⢮⠱⢀⠀⠀⠀
⠀⠀⠀⠀⢀⠱⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠡⢑⡈⢲⢆⢂⠩⡈⢖⠀⠸⣆⡡⠓⡔⢢⡙⢌⠧⡘⢶⡩⢏⣗⣫⠦⠘⠑⠢⠈⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠎⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠐⢀⠀⠀⠀⠀⡀⢇⠂⣅⠚⡆⢸⠤⡑⢭⠨⢥⡘⢌⡒⡩⢖⡭⣛⣬⢳⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢣⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⡜⢓⠬⢱⡘⣬⢳⡉⠦⣉⠦⣘⠢⢅⠣⡍⢚⠵⣎⡟⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠃⠀⠀⠀⠀⠀⢀⡀⢤⠉⢰⠉⣆⢣⡜⢦⠓⡬⢑⢢⢒⡡⢚⢌⡱⢌⢃⠎⡜⡽⣣⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢠⠃⡜⠀⠀⠀⢄⡑⢢⢘⡆⠠⣌⡳⢬⠳⣜⠣⡩⢔⢩⢂⠎⡴⡉⢆⠲⣌⠪⢜⡰⢯⡵⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢠⠃⡜⢠⠃⡄⠒⠤⡘⢄⡾⡇⡱⠤⣙⢎⡳⣜⡡⡑⢎⣂⠇⣚⠵⡹⣌⠳⢦⡙⢮⣱⢏⡶⡳⡇⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣟⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢣⠃⡜⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⢢⣿⣶⣿⣿⣿⣿⠟⠁⠀⢀⠠⠐⠌⡰⢁⠎⠰⠡⠈⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⣤⣾⣿⣿⣿⣿⠋⠋⠃⠤⢈⠒⠀⠀⠠⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠿⠛⠛⠉⠡⢀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡟⢿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠿⠛⠛⠉⠉⠁⠀⠀⠀⢀⣠⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡴⡀⠃⢤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⠀⠀⠀⢀⣨⣐⣡⣆⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⡞⣿⣻⣽⣆⣀⠖⢫⡜⡱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⣿⣾⣿⣟⠏⠐⠸⣷⣻⡎⠀⡐⢀⠙⡁⠀⠘⠰⡰⢦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀
⠺⠿⠿⠟⠿⢿⣿⣿⣿⡿⣿⣟⣿⣽⡿⣟⡷⣟⡿⣽⢯⣟⣷⡻⠂⠠⠤⠴⢿⠇⢸⠀⠀⠬⡀⠱⠤⡀⠔⠙⠪⠖⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢸⢿⣷⣻⣿⣿⣻⢿⣯⢿⣽⣻⣽⣻⣽⣻⡞⠳⠶⠶⣦⣤⡀⠏⠠⡷⡈⠐⠦⢑⢆⠲⡡⢀⠣⢀⠁⢇⠀⢂⠶⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⢾⡽⣞⡷⣯⣟⡾⣯⣷⣻⢾⣽⣾⠁⠀⠀⠀⠀⠘⠛⠋⢀⠌⢰⡽⡘⠀⠀⠈⠃⠥⢉⠖⣂⠌⢈⠒⠉⡞⢱⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣏⣷⡻⣭⢿⡵⣯⢷⣯⡷⣯⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠄⠑⠲⠿⣦⣄⣀⠀⠈⠒⣨⢐⠠⢁⠐⡜⡊⠐⡠⠀⣀⣀⣀⡀⠀⠀⠀⡀⠀
⠀⠀⠀⠀⠒⣻⡽⣞⣿⡽⣯⣟⣯⣟⡾⣿⣳⣿⣻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⢡⠎⠀⠂⢄⠃⠣⣀⠀⣸⠱⣎⠶⡱⠈⠆⠱⢈⠂
⠀⠀⠀⠀⠀⣯⢿⣽⣷⣻⢷⣯⣿⢾⣽⣻⣿⣳⢯⣟⣦⠀⠀⠀⠀⠀⠀⢀⡠⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠀⡐⠌⠀⠡⢀⠞⠁⠡⢈⠳⡤⣀⣆⠑⠎⠀
⠀⠀⠀⠀⠀⢽⣯⣿⣿⣯⣿⢾⣽⢿⣾⣻⣹⣯⣟⣾⣽⣄⠂⠄⠀⠀⠀⣴⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⢂⠐⡀⢂⡡⢎⠐⡈⡐⢀⠂⠱⢍⣎⡇⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⣾⢿⣻⣌⠻⡹⢧⠻⣯⣿⣻⣿⣿⣷⣾⣤⣾⡟⡟⢀⢆⠀⠀⠀⠀⠀⠀⢀⢀⡰⠁⡐⢀⠂⡐⢀⢮⡑⢮⡱⠊⢀⠆⢊⠔⢨⠖⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠙⠓⢁⠣⢌⡉⠧⣁⠓⡩⠇⣷⣂⠥⠿⠈⠻⣏⡏⠁⠀⠈⠑⢂⡏⠀⠁⠊⢦⡉⡐⢂⡕⢢⢒⡡⡴⡒⢸⢆⡛⢦⢁⠴⣉⠎⣁⡤⡟⢮⡅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡒⠢⠜⣰⢀⠣⡐⢡⠂⡌⢒⣡⣓⢈⣼⠂⠀⠁⠀⡀⢃⡆⠀⠀⣨⢇⣳⠈⡔⡪⣕⣫⡼⡱⢇⡹⣎⠱⣃⠎⡲⢇⣻⠱⣎⡝⡲⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠁⠑⠈⠔⢨⠑⠬⡁⢎⠰⣽⡻⢿⠀⣿⠀⠀⠀⡄⠀⢂⡁⠈⢉⠲⣍⠮⡅⣐⢳⡙⣆⠜⡳⢭⡳⣎⢷⠬⠄⠡⢋⢧⡛⣴⡙⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠠⠀⠀⠀⠘⢠⠉⠦⣉⠓⢿⣻⡔⢸⠀⠀⠘⠀⠀⢰⠰⣀⠮⡝⡜⢫⡽⣭⢻⡜⣧⣻⣙⢧⡽⣎⡟⣞⣲⢓⡎⣧⡝⣶⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠆⠀⠀⠈⢱⢧⡸⢀⡀⢘⠀⢀⠊⡥⢸⠳⠙⡬⢡⢒⡉⢣⠛⢲⢣⢟⡮⣵⢫⢾⣣⠗⣭⢺⡌⡻⣜⠣⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠃⡢⢐⢸⡔⠈⢆⠱⣲⠀⠸⣔⠃⢦⠙⡤⢋⠆⡏⠼⡸⢥⣛⢮⡝⡶⠘⠡⠇⠘⡭⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⠠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠉⠀⠠⣍⠂⢆⣙⢆⠘⡌⡍⠦⢩⠔⣃⠚⣌⠱⡹⣆⡟⣮⢝⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠂⠀⠂⠀⠜⣊⠵⡈⢦⡹⡱⢎⡘⣅⠚⡤⣉⠆⢣⢓⡱⢞⣥⢻⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠀⢀⠣⠀⠀⠀⠀⠀⠀⡀⢄⠎⠑⢆⡒⢭⢲⡱⣙⠢⡱⢄⡋⡔⢢⢙⠢⡅⠲⢌⡚⢧⣳⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢠⢃⠒⡀⠀⠀⠄⡌⠱⣈⠂⢀⠜⣦⡙⣎⢧⠓⡤⢣⠑⠦⡑⣌⡃⢎⡱⢌⠓⡬⣘⡧⢯⡝⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠦⡈⢆⠡⢄⠢⡑⢌⡑⣾⠀⢇⡚⢴⡹⡜⢮⡑⢆⢣⢍⡒⣙⠦⡝⢦⡚⣥⢛⡴⣹⢞⣣⠟⡅⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢢⠂⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⣢⣽⣾⣾⣿⣿⣿⠷⠂⠀⠀⠀⡀⡐⠠⢄⠂⡄⢂⠰⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠻⠋⢀⡀⢂⠄⠁⠀⡀⠣⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⡉⠄⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⢯⢁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⠟⠛⠛⠋⠉⠈⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡯⢆⠱⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⠀⠀⢀⣘⡌⣄⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣟⡿⣯⠀⢠⠽⣩⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣯⣷⣿⣿⠃⢻⡾⣽⣻⣽⠈⠁⠄⡑⠏⠀⠈⡔⠠⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀
⢴⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⣽⡾⣟⣿⢿⣿⣻⡽⠂⠁⢂⣿⣳⢹⡮⡐⠈⠀⡆⠘⣄⡀⢀⠝⠲⣬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠀⠀⠀⠀⣿⣿⣽⡿⣷⣿⡿⣟⣿⡽⣞⣷⣻⣽⣞⡿⣾⣍⣍⣉⡁⠀⢸⠃⣸⣧⠅⠂⢬⡘⡄⢤⡘⠠⢐⠠⠐⢨⠀⠂⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢹⣞⡷⣟⡿⣯⢿⣽⢾⣻⣟⣾⣳⣟⠾⠉⠀⠁⠈⠭⠿⠗⠌⠰⢉⡿⠊⡜⡄⠀⠉⠂⢝⡐⡃⢆⠡⠈⠒⠂⠷⣌⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣝⣻⡽⣻⣝⡿⣮⣟⣷⣻⣞⣷⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢀⠘⠤⣴⣦⣄⠀⠀⠉⠒⡅⡃⠌⠠⢁⢨⠏⠡⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠴⣟⣾⣳⣽⣳⢯⣿⣳⣟⣾⣷⣻⣿⣯⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠉⠛⠳⠒⠀⠘⡴⠁⠂⠄⡞⢠⡀⠈⢠⢞⡹⣋⡎⠔⡠⠊⡔⢀
⠀⠀⠀⠀⢀⣿⢾⣽⡾⣽⣻⢾⣷⣻⣞⣷⣿⣳⢯⣧⠀⠀⠀⠀⠀⠀⡀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⡰⠁⠠⢈⠡⠋⠈⡁⠧⣅⢀⡐⠒⡌⠂
⠀⠀⠀⠀⠸⣯⢿⣿⣟⡷⣯⢿⣷⣻⢾⡟⣷⢯⣟⣾⡄⢂⠀⠀⠀⠀⢀⣾⡇⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠠⢈⠐⡀⠄⡐⡔⠃⠠⢁⠐⠠⠑⢯⡜⡇⠀⠀
⠀⠀⠀⠀⠀⢻⣿⡿⡿⣿⣻⡿⡇⣟⢿⣟⢻⣯⣿⣻⣿⣿⣶⣧⣤⣠⣿⢻⠃⠄⢄⠂⠀⠀⠀⠀⠀⠀⠀⠠⠌⠠⢁⠂⡐⢀⢢⢓⡬⣙⠎⠀⣌⠂⡍⢰⢎⡅⠀⠀
⠀⠀⠀⠀⠀⠈⠹⠷⣌⢨⢉⠛⠿⣈⠒⡘⠧⢹⣳⠛⠜⣇⠉⢿⣽⡋⠁⠀⠑⠊⡤⠁⠂⠐⠠⢆⡒⢨⠐⡩⢆⡡⢂⣐⡤⢂⡏⡞⢴⢃⢠⢚⠤⢉⡠⣞⢎⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⠍⠦⣁⠎⡰⢁⠆⡰⢙⢢⣴⢂⢠⣿⠀⠀⠄⠀⠀⡇⠀⠀⠀⡠⢏⠾⣀⠱⡘⢦⢵⢫⠞⣴⠸⣱⡉⡖⡌⢆⣯⠺⣍⠶⣍⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠃⠈⠁⠘⠐⠤⣉⠒⣅⠊⡔⢡⡾⣟⡦⢙⣧⠀⠀⢀⠄⠀⡇⡄⠀⠈⡳⣋⢞⡅⢂⢝⣚⢎⠏⡞⣜⡳⢧⢧⡱⠄⠣⡘⢽⣌⢳⠞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠘⢠⠓⢬⢉⠻⣞⣧⠈⣷⠀⠀⠂⠀⠀⡅⢂⠀⣬⠳⡭⠺⣝⠯⣞⡵⢮⣛⡼⣖⡹⣏⣞⡳⣖⠲⣜⡣⣞⡽⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠂⠀⠀⠉⡞⣅⡚⠠⠀⢈⠀⡐⢸⠈⡄⠲⢻⢐⠣⣌⠹⡘⢚⠣⢟⡼⣣⢟⡼⢮⡳⡽⣙⠶⡉⢳⡽⠡⡀⠀⠀⠀
⠀⠀⠀⠀⢀⡐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠠⠍⠀⢃⠱⡈⡼⡐⡈⠆⢣⠘⡀⠈⣆⠓⡤⢃⠭⢌⠓⡮⠸⣥⢛⡼⣣⢿⠱⡈⠳⠥⠈⡷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠘⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⠁⠁⡘⡄⠣⢌⡻⣅⠀⣎⠱⣌⠱⢊⡜⠸⣐⠣⣇⢏⡾⣱⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⢈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠂⠐⠀⢀⠛⠬⡑⠴⣨⠳⣌⠧⡰⡉⢖⣈⠓⡤⠓⣌⠳⣝⢮⡝⣞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠠⠃⠀⠀⠀⠀⠀⢀⢀⠰⠈⡍⣒⢩⠲⣥⢛⡌⡓⡔⣉⠦⢌⡱⢢⡙⠤⢣⢘⠪⣝⡞⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢠⠃⠥⠀⠀⠀⢠⠑⡌⠢⠁⠀⢰⢎⢧⡛⡴⣋⠴⡑⣌⠒⣌⢲⠰⡡⢌⠳⣈⢎⡱⣏⢾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠃⡜⢢⠡⢄⠢⡁⢎⢰⡇⠂⢘⡌⡞⢦⣙⠶⣡⠒⢥⠢⡍⣔⢫⢳⡱⢎⡳⡜⣬⢳⡝⣮⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⣿⣟⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢐⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢍⣦⣵⣶⣶⣶⣶⡦⠀⠀⠀⠀⢀⠠⡀⠤⠠⠄⠤⡐⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⣿⣿⡿⢿⠃⣀⠠⢀⠐⠈⠀⢡⠘⠄⠃⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢋⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⢀⣡⣬⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣧⠘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⢢⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⡿⣷⡃⠀⢱⢪⡝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⢀⣠⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣷⣿⠋⢿⣽⣻⢷⡻⠒⠁⠌⠺⡅⠀⠆⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⡷⣿⣿⢿⣻⡽⠂⠁⠚⣷⡻⢯⡇⠌⠐⠈⡄⢣⡀⠀⢁⠯⣜⡹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀
⠀⠉⠉⠉⠉⣿⣿⣟⣿⣿⣽⣿⣾⣟⡿⣽⣻⢾⡽⣷⢯⣟⠟⠀⠒⠂⠛⣿⠁⣿⢁⡇⠠⠘⡰⡀⢌⠣⢀⠰⠀⠃⡇⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢻⣳⢯⣿⡾⣟⣾⣳⣯⢿⣳⢯⣿⡽⡯⠟⠹⠛⠛⣷⣦⠄⠃⠰⢹⣏⡒⡅⠆⠉⠑⠎⣱⠂⡎⢄⠡⠘⠄⢪⠖⣥⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⢯⣟⡷⣻⢽⣞⣷⣫⡿⣽⣻⣾⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⢃⠘⠀⠀⠸⠄⠚⢻⡿⢶⣬⡘⠬⠆⡂⡌⢀⠉⢠⡓⠂⡄⠀⠀⣀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⡿⣞⣾⣱⣯⣟⣾⣳⢯⣿⣳⢿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠈⠁⠀⠈⠛⠓⠸⣁⠂⠄⢈⠲⠰⣀⠀⢁⠟⡜⢦⣋⠔⠠⠃⡜⢀
⠀⠀⠀⠀⢀⣿⡽⣾⣟⡾⣽⢾⣽⣻⢾⣿⣻⡽⣞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠀⢀⠢⠁⡐⢀⠩⢀⠂⠄⠃⣅⢀⣔⠰⠌⠂
⠀⠀⠀⠀⠸⣷⣻⣿⣾⡽⣯⢿⣷⢯⡿⣟⣷⣻⡽⣷⢀⠀⠀⠀⠀⠀⢀⣾⠆⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡁⠂⠄⠠⢀⢆⡃⡄⠌⠠⢁⠈⠳⣬⠃⠀⠀
⠀⠀⠀⠀⠀⢿⣿⢿⣿⣽⣿⣻⡝⡿⣟⡯⣷⣯⢿⣿⣷⣮⣴⣀⣀⢀⡞⡟⠡⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠠⠁⠌⣠⢏⢦⡓⠡⢈⠴⣁⠊⣰⢣⠇⠀⠀
⠀⠀⠀⠀⠀⠘⢿⣯⡑⢙⠚⢿⣧⠉⡌⢛⠬⢻⡟⠾⡘⣆⠉⡿⣽⠉⠐⠀⠑⢪⢸⠂⠀⠠⢄⣠⠐⡠⢂⠹⡄⢆⢂⣡⢬⢂⡳⢎⢲⢡⠒⣭⢒⡤⢴⣋⠧⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠜⡊⠴⢠⣃⠜⡠⢊⠔⣈⠓⣔⣤⠆⣰⡷⠀⠈⠀⠀⢸⢰⠀⠀⠀⢠⢣⠷⣀⠣⢸⣘⠼⣺⢜⡺⡠⣟⡌⢣⠎⠱⢧⣫⢜⡣⣝⠚⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠌⠂⠁⠈⠐⠁⠢⢌⠱⢌⠒⡄⢳⡾⣟⡇⢹⡇⠀⠀⢠⠀⢸⠠⠀⠀⠦⣏⢎⡳⡅⢢⠱⣎⢏⠎⢧⡓⢷⡹⣞⣓⠂⡁⢪⡴⣋⠶⠍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠄⠠⠀⠀⠈⠐⡌⠱⢌⠍⢷⡻⣆⠡⠇⠀⠀⠃⠀⠀⡖⠆⠀⠴⣌⠧⡓⠿⣭⢻⡵⣺⡝⣶⡹⢺⡵⣫⢞⡽⣍⡳⣜⣭⢻⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠈⠀⠀⠹⡼⣁⠏⣀⠀⡃⢀⠐⢣⠐⢸⡓⠮⢃⡍⣒⡉⢣⢚⡱⢺⢵⢫⢷⣹⢣⢫⣞⢡⠳⡅⢸⢯⠐⠄⠀⠀⠀
⠀⠀⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠘⢐⠄⡂⣧⠂⠩⠄⢏⠰⡅⠀⢧⠢⡅⢎⡑⢢⢅⠫⡌⣏⢞⡲⢯⣭⠘⠀⠉⠓⠂⠛⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⠔⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡀⠀⠀⠁⠀⠭⢂⠩⣌⢻⡵⠀⣋⠱⡘⢤⠙⣂⠎⡱⡘⡜⣮⢹⣓⢮⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⢌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠐⠀⠐⠀⠡⡓⣩⠢⢅⡺⢥⡍⢧⡉⠦⡙⢤⠙⡤⢃⠝⡒⢯⢞⣣⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⠀⠰⡀⠀⠀⠀⠀⠀⢀⠠⡐⠱⡘⡄⢳⣌⢳⢣⠞⡡⠜⣡⠚⣄⢋⠴⡉⢦⢉⠦⡙⣮⢳⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⡈⠀⠀⡰⢁⠆⠀⠀⡀⢠⠉⢆⡑⡀⠀⢒⡭⢳⡌⢧⢫⢌⡱⠩⣄⠓⡬⣌⢒⡱⢌⢢⠓⣬⢳⢏⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠑⡌⡘⠤⢠⠐⡡⢊⡴⡇⠀⠰⡸⣜⢣⠞⣭⠲⡌⡔⣃⠦⡙⡖⣭⠲⣜⢣⢎⡝⣲⢏⡾⣍⡃⠀⠀⠀⠀⠀⠀⠀⠀