⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣄⣤⡄⠤⠤⣤⠤⢤⣄⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣷⣟⠤⠘⡉⠀⠐⠃⠀⠀⠎⣈⠙⡻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠕⢼⠜⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷⡁⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠬⠀⠀⣠⠀⠀⠀⠀⠀⠁⠀⠐⣆⠀⠀⢌⢂⢸⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠨⠵⣽⣽⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠇⠀⠀⠁⡿⠛⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⡆⠀⠀⠀⣶⠀⠀⠀⠐⢳⣦⡸⢧⠀⢴⠘⡀⠈⢈⣤⠋⠈⢽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡀⠀⠀⡇⢀⢰⠰⠩⢄⠀⠀⠀⣸⢯⠀⠀⣣⠈⠀⠁⢀⡟⠀⠀⠀⣸⣿⣿⡙⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠀⠀⠀⡏⠉⡀⡄⠀⠀⢦⠀⠀⢺⣾⣿⣷⣶⣶⣤⠀⣶⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣇⢀⢷⣾⣿⣿⣶⡦⠘⡀⠀⠸⠈⢹⡏⣩⢁⡏⢀⣹⡄⠀⠀⢠⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⠀⠽⡻⡍⠪⣍⠭⠀⠀⠀⠀⠂⠀⠀⠀⡘⢠⡧⣼⡏⠄⠀⢸⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⠘⣷⣦⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⢑⣿⠁⣿⣿⡄⢰⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡝⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢤⡇⣿⡾⠄⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢸⣦⠈⢿⣿⣶⣤⣀⠀⠀⠀⣠⣴⣿⢿⠀⠀⢰⣿⢧⠐⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠊⠛⢃⢹⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠀⣆⣾⣷⡀⢢⡠⣻⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢺⠀⢻⣿⣿⣿⣛⡛⠛⠋⠹⠈⢷⡿⠋⠛⣿⣶⣥⡄⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡁⠀⠈⠀⠂⠀⠁⠀⠀⣿⠙⠛⠻⢾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢸⣿⣿⠯⠉⠶⢤⣄⣀⡀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⠘⣿⡋⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡇⠀⠀⠀⠀⠀⢸⣇⠀⠀⢳⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣶⣶⣾⣿⠧⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣇⠀⠀⠀⢠⣸⣾⣿⣷⣤⣺⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡿⠟⠳⢶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢲⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⠀⣿⡿⣬⣧⣹⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⣟⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⠋⠉⠙⢻⣿⣿⡏⠈⠿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣮⣿⠀⠀⠀⢻⣿⢿⣿⡿⠋⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢞⣛⣼⠟⠑⠠⠀⠀⢹⣾⡟⢁⣠⣾⡿⠛⠛⡻⠿⠇⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣿⡇⠀⠀⠀⡀⠀⣀⣾⡿⠟⠉⢻⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡿⡇⠀⣠⠄⠛⠙⢻⣿⡓⠂⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣥⣶⣿⠟⠁⠀⠀⠀⠀⠛⣏⠃⠀⢀⣤⣃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣦⣴⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⡟⠀⣸⣿⣿⣿⣿⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⢈⣤⣶⣿⣿⣿⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣿⣿⣿⡏⡼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⠇⡗⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢈⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠙⢿⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⢧⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣤⣤⣤⣾⣿⡟⣿⣇⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀