⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋⠠⠞⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⣧⣶⣶⣦⣀⠀⣨⣿⣿⣿⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠻⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣠⠞⠛⣧⠀⠀⠀⠀⠙⣿⢿⣿⣿⣿⡀⠀⠉⠉⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⠁⠀⠀⣠⡾⠁⠀⠀⠙⢷⣄⠀⠀⠀⠈⠸⣿⣿⣿⡇⠀⠀⣄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠹⠟⠃⢼⣿⣿⣿⣿⣿⣀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⢠⣿⣿⣿⠁⠀⠀⠸⣆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠃⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⢃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⡟⠀⠀⠀⠀⣿⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⣸⠟⠉⠙⠇⠀⣿⣿⣿⣿⣯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⡇⠀⣸⣿⣿⠃⣷⣄⠀⠀⣿⣾⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⠴⠏⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⡀⠀⢿⣿⣿⡄⢿⠃⠀⢠⣿⣿⣷⡆⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⢀⣀⠀⠀⢠⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⢻⣿⣷⡀⠀⢀⣾⣿⣿⣿⣃⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣼⣧⢻⣿⣿⣿⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⢙⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡠⠖⠒⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠙⠁⠘⣿⣿⣿⣿⣇⠀⠀⠀⢠⣦⣴⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠸⡿⠋⣿⣿⠀⠀⠀⣿⠟⠋⣼⣿⠏⠉⣿⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠁⠀⢰⡿⠃⠀⣰⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢠⠟⠀⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠊⠀⠀⠀⠛⢩⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⡇⠀⣰⣿⡿⠀⠿⠿⠿⠿⠇⠾⢿⣿⡿⠿⠃⠰⠿⠿⠀⠀⢸⣿⠁⠀⠀⠀ ⣠⣾⠿⠿⠿⠟⠠⠿⠿⠿⠿⠁⠀⠰⠿⠿⠀⠀⠸⠿⠿⠿⣿⡆⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⡇⣼⣿⣿⡇⣰⣿⣿⣿⡿⠀⠀⣸⣿⠀⠀⣴⣶⣶⣶⡄⠀⣿⡟⠀⠀ ⠀⢼⣿⠃⢴⣶⣶⠂⣴⣶⣶⣶⡆⠀⣴⣿⣿⣿⡆⠀⣾⣿⣿⣿⠿⠁⠀⠀⠀
⠀⠀⠀⠀⢠⣿⢹⣿⠏⣼⣿⢀⣿⣯⣤⣤⠄⠀⢀⣿⡇⢀⣼⡟⠁⠀⣿⡇⢸⣿⣧⣤⣤⠀⠘⣿⣧⣤⣾⡟⢠⣿⣧⣤⣤⠀⣼⡟⠀⠀⣿⡇⢰⣿⡇⢻⣿⡄⠀⠀⠀
⠀⠀⠀⠀⣼⡏⣀⣉⣀⣉⣁⣈⣉⣉⣉⣉⣀⣀⣈⡉⢀⣈⣉⣀⡀⠀⣉⣁⣈⡉⢉⣉⣁⣀⡀⣈⣉⣉⡉⠁⣈⣉⣉⣉⣉⣈⣉⣀⣀⣀⣉⣁⣈⣉⣀⡈⣿⣷⠀⠀⠀
⠀⠀⠀⠀⠿⠃⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄⠹⠿⠿⠿⠄⠡⠿⠿⠿⠧⠽⠿⠿⠿⠥⠿⠿⠿⠧⠥⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠘⠿⠇⠀