⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⠊⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠂⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠠⢈⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠐⠠⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠄⡈⠐⡀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠂⠄⠡⠐⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠂⡈⠄⢁⠂⠡⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠂⡐⠠⢈⠀⢂⠡⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠐⡀⠂⠄⡈⠄⠠⢁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠐⡀⠂⠄⡁⢂⠐⢈⠐⡀⠂⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠀⠁⠆⠰⢀⠈⠀⠆⡀⠁⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠄⡈⠐⡈⠐⠠⢈⠐⠠⢀⠡⢈⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠠⢀⠡⢀⠁⠂⠄⡈⠐⡀⠂⠄⢂⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠁⠂⠄⢂⠈⡐⠠⢀⠡⢀⠡⢈⠠⠀⠌⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠄⠡⢈⠐⡀⠂⠄⡁⠂⠄⠂⠄⢂⠠⠁⠂⠄⢳⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠒⠶⢤⣤⣷⡌⠐⡀⠂⠄⡁⠂⠄⡁⠂⠡⢈⠠⢀⠡⢈⠐⡀⠸⢼⡇⠀⠀⡔⢫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡾⢄⠀⠀⠀⠀⠀⠘⠻⣿⣿⣷⣠⣌⣄⣤⣡⣦⣼⡷⠷⠿⠿⠿⠿⢿⣿⣿⠷⠿⣿⡄⢠⠛⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣟⠀⠓⠠⡀⢀⣀⣀⣀⣿⣿⣿⣿⣿⠛⠛⠛⠁⢩⡟⠻⠳⣄⠀⢘⡏⠉⠙⠇⠀⠉⣧⣏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣏⢀⡀⠀⠉⢆⠙⢿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢨⡴⠶⡖⢻⡆⢠⡟⣉⣉⢯⢆⠀⢹⣿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢹⠈⠻⣆⠀⠈⠢⢼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣤⣴⣿⡍⢹⠀⣿⣯⡟⠉⠹⡄⠀⢻⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢧⠀⣹⣷⣀⠀⠈⠙⣿⡿⣿⡿⠛⠀⠀⠀⠀⢈⡻⣉⣉⣠⡾⠀⢈⠫⡤⣤⢾⡇⠀⠈⢧⠄⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢣⡌⠀⠸⡄⠀⠀⠘⣧⡏⠁⠀⠀⠀⠀⠀⠈⠙⣶⣤⡞⠁⠀⠈⡳⣦⠴⠁⠀⠀⠀⠘⡆⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⣦⣄⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢴⡇⠀⢿⣀⠀⠀⢰⡄⢰⣶⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡹⡷⠶⠿⠁⠀⠀⣠⣼⣟⣿⠆⠀⠀⠀⠀⢠⣼⡇⠀⠀⠹⣷⣄⠈⠳⠤⣜⢯⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⣿⡞⠙⠁⠀⢸⠇⠀⣤⣿⣿⡇⠀⠀⠀⢹⣿⣿⣦⣀⠙⠳⡄⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⢠⣤⡴⠶⠋⠀⠾⠟⠙⠛⡇⠀⠀⠀ ⠰⢟⠛⠹⠟⠛⠀⢹⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠀⡇⠀⠀⠀ ⡀⢜⣦⠀⣀⡤⢄⣨⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣾⠏⢀⣀⣠⠦⠴⠤⠦⠴⣄⣀⣿⠀⠀⠀ ⠘⡀⠙⡏⠁⠀⠀⡼⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢻⡄⠀⠀⠀⠁⣡⠇⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⠒⠦⠤⣼⣉⣤⠖⠒⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣣⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⣧⣤⣤⣤⡞⠛⠁⠀⠀⠀⡇⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⠖⠁⠙⣟⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀ ⠀⡇⠀⢰⡇⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⠇⢠⠈⡐⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠀⣇⣤⡟⢀⠐⡀⠉⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡏⠀⣾⠀⠰⢀⠈⠹⢶⣆⣀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀ ⢰⣿⠁⢹⡆⣶⡎⠁⡇⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠀⢂⠸⣷⡞⠀⢂⠐⡀⢈⠙⡖⠦⣤⣀⡀⠘⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣸⠀⡐⢸⡇⠘⢻⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠸⡀⠂⠄⡹⠄⡁⢂⠐⡀⠂⠌⢳⡄⠠⢈⠉⡙⠲⠽⣿⣤⡀⢀⡀⠀⠀⠀⠀ ⣀ ⣴⣇⣰⣠⡼⠃⠐⢸⡗⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠲⢾⠃⠐⠠⢀⠂⠄⡁⠂⠌⢳⠀⢂⠐⠠⣐⣴⣶⠚⠟⠿⠓⠶⣶⣶⣶⣶⣶⣿⠁⠉⠉⡁⠠⢀⠁⠂⠙⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⠀⡁⠂⠄⢂⠐⠠⢁⠈⠄⠹⠶⠞⠋⠉⠄⠠⠐⡀⠂⠌⡀⢸⣯⢿⡿⣿⣽⣇⣈⣴⣤⣥⣦⣬⣶⣷⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣶⣦⣥⣌⣠⣈⠐⠠⠈⠄⡁⠄⠠⠁⢌⣠⣁⣂⣤⣵⣤⣦⡿⢿⢯⣽⣭⣿⢿⡽⣏⡿⣽⣫⣿⣷⣻⡾⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⢿⣾⣽⣻⡟⣿⣻⢿⡿⣿⣻⢿⡿⣿⡟⣿⢻⣿⣹⣏⣯⢿⣃⣽⣬⣿⣿⠯⠟⠛⠛⠋⠉⢁⠀⡀⠄⣀⣄⠱⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⣀⢀⠀⠄⠛⢉⠋⢋⠙⡉⠋⢋⠙⡉⠛⢉⠛⠀⡀⠠⠁⡀⠄⡀⡰⠋⣿⣷⣀⣂⣰⣬⣴⡖⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠙⠒⠮⠤⣐⣀⣂⣄⣁⣂⣄⣐⣀⣂⣄⣡⣀⣡⣦⣴⡚⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠉⠙⠛⠛
⠀⠀⠀⠀⠀⣴⣿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀