⠀⣀⣤⣤⣤⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠉⠛⠛⠛⠿⣿⠿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⡀⢀⣽⣷⣆⡀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠶⠋⠀⠀⣠⣤⣤⣉⣉⣿⠙⣿⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠴⡟⣻⣿⣿⣿⣿⣿⣶⣿⣦⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⡿⠻⣿⠃⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠏⢹⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⡶⣿⣄⠀⠀⠀⠀⠀⢉⣿⣿⣿⡿⠀⠸⣿⣿⡿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣦⢀⣿⣿⣄⡀⣀⣰⠾⠛⣻⣿⣿⣟⣲⡀⢸⡿⡟⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣾⣿⡛⣿⣿⣿⣿⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣍⣿⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣽⣿⣷⣙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⠿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣠⣤⠶⠚⠉⠉⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣿⣿⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠞⠋⠁⠀⠀⠀⠀⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣾⣿⠋⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⢀⣷⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⣿⠁⢠⠃⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡛⠛⠿⠿⠿⠿⠿⢷⣦⣤⣤⣤⣦⣄⣀⣀⠀⢀⣿⣿⠻⣿⣰⠻⠀⠸⣧⡀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⣦⣼⡏⢻⣿⣿⠇⠀⠁⠀⠻⣿⠙⣶⣄⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣐⠀⠀⠀⠈⠳⡘⣿⡟⣀⡠⠿⠶⠒⠟⠓⠀⠹⡄⢴⣬⣍⣑⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣀⠐⠲⠤⠁⢘⣠⣿⣷⣦⠀⠀⠀⠀⠀⠀⠙⢿⣿⣏⠉⠉⠂⠉⠉⠓⠒⠦⣄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⣿⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣷⡀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢹⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⢶⣄⠀⠀⠀⠀⠀
⠯⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠈⠳⠀⠀⠀⠀
⠀⠀⢀⣀⣀⡀⣼⣤⡟⣬⣿⣷⣤⣀⣄