⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⢒⣛⣻⣿⣿⣿⣶⣒⡲⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠑⠦⣄⣈⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠙⡆⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣠⠇⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⠞⠃⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀
⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡤⠴⠖⠚⠛⠛⠛⠛⠒⠶⢤⣀⠀⠀⠀⢀⣴⣿⠶⠋⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⣸⣿⣿⣿⣿⠟⠡⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣾⣯⣉⣀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠀
⣿⣿⣿⣿⡟⠀⢠⣾⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠙⠳⢦⣸⣿⣿⣿⣿⣿⣿⣯⣤⣤⣄⡀⠀⠈⠙⢿⣿⣿⣿⣿⠀
⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⠀⠀⠀⢻⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣇⠀⠉⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⡏⠀⡄⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣆⠀⠀⠀⠀⠀⠀⣄⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⢸⣿⣿⣿⣿
⠀⠈⢿⣿⡟⣿⠀⢰⡇⠀⠀⠀⣠⢀⣠⣿⣴⢶⡾⡷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⡇⠈⣧⠀⠀⠀⠀⠀⣸⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡎⠇⠀⠀⢸⣿⣿⣿
⠀⠀⢀⡟⠀⣿⠀⢸⡇⠀⠀⢀⡿⠋⠉⣿⢀⡾⠀⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣰⣏⣸⡇⠀⠸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿
⠀⠀⡾⣠⠀⣿⣰⢿⡇⠀⣤⣼⣇⣀⣀⣿⣼⠁⠀⢿⡏⣇⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⡴⠃⣏⡿⠉⠉⠓⢿⣦⡀⠀⢰⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⣴⣿⣿⣿
⠀⣸⢣⣿⠀⢿⡟⠘⣟⠛⠛⠛⠉⠛⣿⣿⣿⣷⣶⡞⢧⠸⣄⠀⠀⠀⠀⠀⣼⠃⣠⠞⠁⢰⡿⠁⠀⠀⠀⢸⡄⠙⢢⡿⠀⠀⠀⠀⠸⣎⣿⣿⣿⣿⣿⣟⣠⣾⣿⣿⣿
⠀⣿⠏⢹⡀⠘⠁⣸⢿⡄⠀⣰⣄⣀⣿⣿⣿⣿⣿⡇⠈⠀⠙⣆⠀⠀⠀⣸⣇⠞⠉⠙⠒⠛⠶⣤⣤⣀⣀⠸⡇⣠⠏⣿⠀⠀⠀⠀⠀⢿⣿⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⡏⠀⠈⣧⠀⢀⡿⠈⢷⡀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠳⢄⢠⡟⠁⠀⠀⠀⣦⣀⣠⣿⣿⣿⣿⣿⡟⠷⢤⣿⣀⡀⠀⠀⠀⢸⣿⠀⠈⠙⠻⠿⢿⠿⠛⠀⠀
⠈⠀⠀⠀⠈⣧⣸⡇⠀⢨⠛⢼⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡏⠛⠛⠃⠀⠀⠸⠃⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡞⠁⠙⠇⢀⡜⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢷⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣞⡄⠀⠀⣴⠋⠀⠀⠀⠀⠙⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠃⠀⡼⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⣯⢳⣀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⠃⠀⢀⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠁⢀⡞⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢰⡆⠀⢸⡛⠛⠃⠀⠀⠀⠀
⠀⠀⢸⠹⡇⠀⣸⢳⡈⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣀⣀⣤⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀
⠀⠀⠘⠀⠻⡄⡏⠀⢳⡀⠀⠀⠉⢻⡲⢦⣄⡀⠀⠀⠀⠀⢧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⠏⣷⠀⡾⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⠇⠀⠀⠙⢦⣀⠀⠸⣷⠄⡀⠉⠓⠦⣄⣀⠘⠄⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡀⢠⡟⢀⣾⠁⠀⠀⣴⠋⠀⣿⠞⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢽⣆⠙⠒⠛⠓⠲⠻⠉⠀⠀⠀⠀⠀⠀⠛⠛⠛⣛⠛⠛⠋⠉⠉⣿⠁⠀⠀⠀⢠⣷⢿⢡⣾⡏⠀⣠⠞⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣿⠀⠀⠀⢀⡞⠁⡾⢫⣿⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠛⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀