⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣖⣀⣀⣠⣴⣿⣿⣷⣤⣀⣐⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡞⣛⣉⣽⡿⠟⣉⢛⡛⡡⢄⠤⢤⢠⣀⣉⡉⡙⠛⠷⣬⡙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣟⣿⣿⣿⠿⠿⢿⢿⡿⣿⣿⣿⣧⣟⡜⢇⡇⡤⣄⠧⣃⠟⡤⣀⠛⢻⣿⠿⣤⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⣏⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣜⠲⣌⠳⣍⡞⣱⣌⠳⣌⡛⢷⣬⣿⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣩⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⠳⡬⣝⠲⣬⢳⡘⣬⢣⠌⠛⣿⣷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣋⠶⣩⢓⡎⢧⡹⢤⢣⢋⡆⠹⣏⣷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⡞⡱⢎⡜⣧⡙⢦⢋⡼⣘⢣⠻⣿⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣭⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢬⡱⢎⡜⢶⡹⢌⡳⡜⣌⠧⣃⣿⢿⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⢢⡓⠮⣜⢣⡝⣪⠔⡯⡜⢲⣩⣿⡿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣛⣛⠁⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠳⣍⠳⣌⢳⡙⢦⠛⡴⡩⢇⠲⣿⡇⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠛⠋⠉⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢌⡻⢜⠳⣌⢳⡛⣬⠛⡴⣙⢬⠓⣿⣷⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⠿⣾⠃⠀⠀⢰⣿⣿⣇⣠⣤⣴⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣎⡜⣣⢫⢔⣣⠹⢤⠛⡴⣩⠲⣍⠤⣏⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⠀⠀⠀⠈⠿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣶⣄⡀⢀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⣌⠶⣡⢓⣮⣴⣭⣮⣝⡲⣱⢋⢦⣙⡇⠀⠀
⡀⠀⠀⠀⣠⣿⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⡱⢊⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⡭⠲⡭⠇⣿⠀⠀
⠁⠀⠀⢰⣻⣟⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⡙⠆⢻⡇⠀
⠀⠀⠀⣽⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⠹⣩⣼⡟⡗
⠀⠀⢰⣿⡿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡬⢓⡥⢿⡇⣿
⠀⠀⢸⣿⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣬⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⡼⢱⡩⢖⡩⣿⡿
⠀⠀⠈⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠛⣬⠙⢻⡟⣿⠛⡟⢹⡌⢳⡌⣧⠙⡎⣴⣿⡇
⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⢧⡩⢛⡴⢓⡬⢓⡭⢳⢜⡣⡝⣬⠳⡍⣆⣾⡇
⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠦⡹⢥⢚⡵⡘⠧⣜⢣⢎⡵⢚⡤⢳⠱⣎⣿⣷
⠀⠀⢸⡇⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠓⣭⢲⢮⡹⣱⠪⣜⢊⠶⣉⠶⣩⣳⣾⣧⣿
⠀⠀⠸⣧⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⠴⣩⠲⡅⣇⠳⠬⣍⣲⣭⣰⣿⢻⣿⡿⣷
⠀⠀⠀⠀⠉⠉⢿⣯⣉⣛⡛⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣴⣣⣵⣼⣦⡿⢟⣛⣿⠿⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⠷⠶⣶⣭⣭⣭⣿⣷⣶⣟⣛⣛⣛⣛⣿⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢚⣛⣛⣛⣿⣿⣭⣽⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀