⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣴⣶⣿⠿⠟⠛⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⢿⣿⣶⣦⣤⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀
⣼⣿⣏⣁⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡈⠉⠛⠿⣿⣶⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀
⠈⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠋⢉⣉⡉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⡿⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠈⠉⠉⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠋⠁⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⣠⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⡀⠀⠀⠀⣿⡿⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀
⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⣰⣿⡏⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣶⣼⣿⠃⢀⡀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀
⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⢠⣿⡟⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣿⣿⡇⠈⢻⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀
⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⣸⣿⠁⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣿⡏⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⢸⣿⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀
⠀⠀⢸⣿⡀⣠⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀
⠀⠀⠘⣿⣧⣿⣿⣆⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣰⣶⡄⢀⣾⡟⠀⠀⠀
⠀⠀⠀⠘⢿⣿⠟⢿⣷⣄⣘⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡇⠀⣠⣿⡿⣀⣀⣀⣀⣤⣶⣿⣿⣷⡿⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣾⡿⠿⠿⠿⢿⣿⣧⣾⡿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣯⣭⣄⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⡿⠿⠛⠛⠋⠉⢱⣿⡏⠀⠉⠉⠉⠉⠛⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⣿⣧⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⣿⣷⣦⣄⡀
⠀⠀⠀⢠⣶⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠉⢻⣿
⠀⠀⠀⣿⡟⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⣼⣿
⠀⠀⠀⢿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣄⡀⠀⠀⠀⠀⢸⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠙⠻⠿⢿⣶⣦⣤⣿⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠿⠿⠛⠋⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⣁⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀