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