/* ⡀⠎⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⠃⠈⣶⡛⠿⠭⣉⠛⠿⡿⠛⠉⣀⣠⣤⣭⡏⠴⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣱⣬⠛⠉⠀⠀⢠⠀⠀⠀⢀⣀⠀⠉⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠳⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠋⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⣿⠟⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠻⣿⣿⣿⣿⣷⡀⠘⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣄⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡄⠈⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⠁⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠸⠀⠀⠻⣿⣿⣿⣆⢦⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⠀⠀⠀⠀⠐⣇⠀⠀⠙⣿⣿⣿⡄⠙⣄⠀⣴⣿⣿⣿⣿⠏⠀⢸⠀⠀⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣈⣦⠀⠀⠀⠀⢹⠀⠀⠀⠸⣿⣿⣿⠀⠀⠳⣀⠋⣸⣿⣿⣿⡟⠀⠀⠀⡆⠀⠀⠀⠀⠀⡏⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⢧⠀⠀⠀⠀⡇⠀⠀⠀⠘⣿⣿⣷⠀⠀⠘⠀⣿⣿⣿⢩⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠀⢱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢀⣴⣶⣿⣿⡀⠀⠀⢻⠀⠀⠀⠀⠹⣿⣿⡄⢸⣿⣿⠃⠈⠀⠀⢸⠀⣿⣆⠀⠀⠀⠀⣿⣿⣿⠷⠘⡀⠀⠀⠀⠀⠀⠀⢠⢹⡀⠈⡿⠻⣿⣛⢿⣿⣷⡀⠈⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⠀⡇⣼⣄⠀⠀⠀⢻⣿⡄⠑⠑⣿⡀⠀⠀⠀⢀⠀⠂⠇⠀⠀⠖⠛⢿⣿⣿⣌⢿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⡀⣿⣿⡇⠀⠀⠀⠀⢸⠀⣾⣿⣿⡷⠿⣷⣤⣿⣿⡄⠀⠀⠀⠑⠤⡀⠀⠃⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣆⠙⣿⣧⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠁⠀⠀⠀⠀⠘⣾⣿⣿⠁⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡏⠙⣿⠉⠻⣿⠀⠀⣿⠀⠀⠀⣄⠀⣿⢸⣷⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠁⠀⣿⣿⠋⣿⠏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⢻⠀⠀⢀⡟⢀⣿⣸⢃⠟⣿⣿⣿⠀⡄⠀⠀⠀⠘⠻⡄⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣿⠃⣿⣿⡗⠁⣧⣿⣿⣧⢹⡀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⣿⣾⣿⣿⣿⢿⠘⣿⣿⣿⣿⣤⠀⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠉⣿⣿⣿⡿⣿⠻⣷⣬⣓⣬⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠉⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠼⢉⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠀⢸⠀⢻⢿⣿⣿⡏⣿⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⢸⣿⣿⠘⡀⢦⡿⣿⣿⣿⢿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⡄⠀⠈⣿⣿⡄⠱⣴⠛⣾⣿⣿⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⣯⠛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣇⠀⠀⣿⣿⣿⠿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠸⠀⠀⠸⣿⣿⣷⠆⢳⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠿⠿⢿⡟⠀⠀⠉⠦⣀⡤⢶⠀⠖⠲⠶⠊⠀⠀⠀⢻⡛⠛⠛⣿⣿⠀⠀⠀⠀⠃⠀⠀⢿⣿⣿*//*⡀⠎⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⠃⠈⣶⡛⠿⠭⣉⠛⠿⡿⠛⠉⣀⣠⣤⣭⡏⠴⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣱⣬⠛⠉⠀⠀⢠⠀⠀⠀⢀⣀⠀⠉⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠳⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠋⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⣿⠟⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠻⣿⣿⣿⣿⣷⡀⠘⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣄⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡄⠈⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⠁⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠸⠀⠀⠻⣿⣿⣿⣆⢦⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⠀⠀⠀⠀⠐⣇⠀⠀⠙⣿⣿⣿⡄⠙⣄⠀⣴⣿⣿⣿⣿⠏⠀⢸⠀⠀⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣈⣦⠀⠀⠀⠀⢹⠀⠀⠀⠸⣿⣿⣿⠀⠀⠳⣀⠋⣸⣿⣿⣿⡟⠀⠀⠀⡆⠀⠀⠀⠀⠀⡏⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⢧⠀⠀⠀⠀⡇⠀⠀⠀⠘⣿⣿⣷⠀⠀⠘⠀⣿⣿⣿⢩⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠀⢱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢀⣴⣶⣿⣿⡀⠀⠀⢻⠀⠀⠀⠀⠹⣿⣿⡄⢸⣿⣿⠃⠈⠀⠀⢸⠀⣿⣆⠀⠀⠀⠀⣿⣿⣿⠷⠘⡀⠀⠀⠀⠀⠀⠀⢠⢹⡀⠈⡿⠻⣿⣛⢿⣿⣷⡀⠈⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⠀⡇⣼⣄⠀⠀⠀⢻⣿⡄⠑⠑⣿⡀⠀⠀⠀⢀⠀⠂⠇⠀⠀⠖⠛⢿⣿⣿⣌⢿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⡀⣿⣿⡇⠀⠀⠀⠀⢸⠀⣾⣿⣿⡷⠿⣷⣤⣿⣿⡄⠀⠀⠀⠑⠤⡀⠀⠃⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣆⠙⣿⣧⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠁⠀⠀⠀⠀⠘⣾⣿⣿⠁⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡏⠙⣿⠉⠻⣿⠀⠀⣿⠀⠀⠀⣄⠀⣿⢸⣷⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠁⠀⣿⣿⠋⣿⠏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⢻⠀⠀⢀⡟⢀⣿⣸⢃⠟⣿⣿⣿⠀⡄⠀⠀⠀⠘⠻⡄⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣿⠃⣿⣿⡗⠁⣧⣿⣿⣧⢹⡀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⣿⣾⣿⣿⣿⢿⠘⣿⣿⣿⣿⣤⠀⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠉⣿⣿⣿⡿⣿⠻⣷⣬⣓⣬⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠉⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠼⢉⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠀⢸⠀⢻⢿⣿⣿⡏⣿⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⢸⣿⣿⠘⡀⢦⡿⣿⣿⣿⢿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⡄⠀⠈⣿⣿⡄⠱⣴⠛⣾⣿⣿⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⣯⠛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣇⠀⠀⣿⣿⣿⠿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⢻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠸⠀⠀⠸⣿⣿⣷⠆⢳⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠿⠿⢿⡟⠀⠀⠉⠦⣀⡤⢶⠀⠖⠲⠶⠊⠀⠀⠀⢻⡛⠛⠛⣿⣿⠀⠀⠀⠀⠃⠀⠀⢿⣿⣿*/
