⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⢇⣿⢇⣽⢆⣶⡍⣖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣿⣿⣿⣿⣿⣧⣠⢽⣟⡸⣇⣸⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢤⠴⣪⣏⣉⣭⣤⠤⠬⢭⡻⣿⣿⣎⠿⣮⣽⡱⢯⣆⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢾⠤⡉⢹⣏⢹⢹⢒⠀⣾⣷⣹⣿⣿⣶⣽⣿⣷⣮⣋⢮⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣾⣮⣝⣿⣿⣿⣾⣟⣿⣿⡿⡿⠉⠉⠉⣿⣯⢿⣾⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠶⠎⠆⢶⣆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡾⣏⣉⣉⣉⣙⣿⡿⠁⠀⠀⠀⠈⠱⢿⣀⡈⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠙⢞⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣯⣿⣿⣿⣻⣼⣗⣇⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣯⣿⣿⣿⣿⠋⠀⠀⠙⠿⠷⢯⣷⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣺⡧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣿⣿⣽⣟⢿⠿⣦⣀⠀⠀⠀⠀⠈⠉⠛⢾⣿⣿⢿⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⠽⠋⠙⠻⣿⣷⣵⡧⣭⡱⢶⣄⠀⠀⠀⠀⠀⠈⠉⠑⣾⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢞⣻⠎⠁⠀⠀⠀⠹⢿⣿⣿⣿⢿⣿⡊⠱⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡵⢆⣀⠀⡀⡀⠀⢀⣀⠄⡤⡞⡧⠜⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⢆⣽⠽⡛⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠽⠽⠯⠿⠿⠿⠾⠙⠉⠀⠀⠀⢀⣠⣾⣿⣿⡿⠿⠟⠋⠉⠸⣿⣼⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣤⣼⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣃⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠈⢳⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⡀⠀⠀⢠⡆⣤⠂⠈⣸⠇⠀⠀⢀⣀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠘⢿⣿⡗⠰⣻⣆⠀⠀⣹⣿⣦⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀
⠀⠀⠀⠀⠀⣴⣾⣿⡿⣿⣹⡿⣗⣿⣿⠿⠿⣿⡷⠗⠃⠀⠀⢸⣏⠀⢀⠎⣿⡆⣴⣿⣿⣿⣷⣿⣿⣿⣎⢶⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀
⠀⠀⢀⣴⣿⣿⢯⣾⣿⣿⣟⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣾⣤⢿⣾⣿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⢠⣿⡿⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀
⠀⢀⣿⣿⣿⣾⡟⡻⢰⣿⣿⣭⣖⣾⣾⣿⡧⢤⡀⠀⢀⣴⣿⡟⢋⣾⣿⣼⣿⣿⣟⣵⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⡿⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣼⣼⣿⣿⣿⣿⣿⣿⢏⣸⣿⠟⠀⠀⠈⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠿⠋⠁⠀⠀⠀⣴⠿⢟⣿⣿⠛⢻⣹⣿⢿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⢀⣀⣤⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⣾⣿⡟⠀⣼⣿⣿⠈⠙⠹⣽⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⠿⠏⠉⠀⠉⠁⠀⠙⠛⠉⠉⠙⠻⣏⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋⠁⣼⣿⠿⠛⠃⠀⠀⠀⠀⠀⠙⠓⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀