⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡎⠓⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠙⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡓⠢⣄⣀⣾⢹⡄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡇⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⠤⣤⣤⣀⣀⣀⡀⠳⡀⠀⠙⠻⣆⠇⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠉⠓⣦⡀⠀⣀⣠⠤⠖⣚⣋⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢛⡓⠶⣤⣈⠻⣴⣾⣿⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⢰⣿⠀⣠⣿⠿⢛⣡⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣮⣿⣾⠟⣿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⡟⠀⣰⣿⣣⡾⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⣀⠀⠀⠈⠻⣿⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡟⠿⣿⣷⣦⣤⣿⣿⣿⡿⠋⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⣆⠀⠀⠀⠈⠻⣆⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢷⣤⣈⠛⠿⣿⣿⣿⠋⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⣶⣄⡀⠻⣆⠀⠀⠀⠀⠈⢯⠙⡆⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⡆⠀⠀⠀⠀⠀⢳⣇⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⡼⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣼⠀⠀⠀⢠⡀⠀⠀⢸⠙⣿⣿⣿⡄⣷⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀
⠀⠀⠀⣿⣿⣿⣿⠃⠀⣸⠃⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢀⣾⡟⠀⠀⣠⣿⡇⠀⠀⡼⠀⢈⠿⣿⣿⣄⣇⠀⠀⠀⠀⠀⠈⣿⠧⠀⠀⠀
⠀⠀⠀⠿⠿⠿⣿⠀⢠⡟⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠐⠠⣠⣾⣿⡟⠀⢀⡴⢻⣿⠇⢀⣴⣿⣿⡇⠀⣰⢃⠔⠁⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣶⣿⡇⣼⠃⣀⡴⠃⠀⠀⠀⠀⠀⢰⣿⣿⣀⡴⠞⣡⣿⣿⣧⡴⠋⠀⢸⣿⢠⣾⣿⣿⣿⡇⣰⠟⠁⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⣠⠾⠿⠛⠉⠀⠙⠢⠀⢸⣿⣿⣿⡿⢫⣿⠟⠁⢀⣀⣠⣤⣤⣤⣄⡨⣿⠂⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⡀⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡤⠀⣿⣿⡿⠋⠀⠉⠀⠀⢀⣿⣩⡿⢭⢳⡄⠙⣷⣸⣦⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⢸⣇⠀⠀⢿⠀⠒⣲⡿⠛⣉⣹⣿⠳⣌⠳⡼⠛⠁⠀⠀⠀⠀⠀⠀⢸⠀⠈⣄⠐⠀⢹⠀⢸⢻⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣧⠈⣿⣧⠀⢸⡀⣀⣿⠁⠀⡏⠙⣧⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⣸⠀⠈⣼⣿⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣧⣸⣿⣷⡈⣧⠀⠹⠀⠀⢇⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠖⠃⠀⢠⠿⣿⡇⠀⠀⠀⢸⡇⢀⡇⠀⠀
⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣆⠸⣿⣿⣿⣿⣿⣿⣜⣆⠀⠁⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣾⣧⢸⠁⠀⠀
⠀⠀⠀⠀⠀⠀⣿⠈⣿⣿⣿⣿⣿⣿⣆⠻⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⣿⣿⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡏⠀⣿⣿⠻⣿⣿⣿⣿⣆⠙⣿⣿⣿⣿⣿⣿⣯⡙⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⣿⣿⣿⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⣿⣿⣿⣿⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣰⣿⣿⣿⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡇⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣿⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠇⠀⢸⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣯⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠒⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡿⠀⠀⢸⠇⠀⣿⣿⣿⣿⣿⠁⠀⠹⣿⣿⣿⣿⣿⣿⣧⠔⢺⣿⣿⣿⣶⣶⣤⣤⣤⣶⣤⣶⣶⣿⣿⠓⢦⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⠻⢿⣿⣷⣹⠻⢷⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡆⠀⠱⡀⠀⠀⠋⢹⣿⣿⣿⣿⡏⣇⠘⡇⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢙⣿⣿⣷⣦⢸⡏⠁⠀⠀⠀⠉⠓⠈⡿⡏⠁⠀⠀⠈⣷⠀⠀⣹⢶⠀⠀⠈⣿⣿⣿⣿⡇⢸⠀⣇⠀⠀⠀⠀
⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⡠⢊⡤⣿⡍⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⢹⣤⡞⣡⣏⢣⡀⠀⢹⣿⣿⣿⡇⠈⡀⢹⠀