⣠⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉⠁⠀

⣠⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉⠁⠀

⣠⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉⠁⠀

⣠⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉⠁⠀

it got me

⣠⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉⠁⠀⠀

⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉

⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⡟⠉⣻⣿⣿⡏⠰⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠙⣻⣿⣿⣄⣠⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣭⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠞⢡⣽⣿⣿⠿⢻⣿⣿⣿⣏⣿⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣴⡨⠛⠋⠁⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣯⢈⣿⣿⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣿⣷⡜⣿⣧⡉⠉⠙⠋⠁⠈⠉