Kilometre Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠟⠛⠛⣻⣿⣿⡿⠛⠛⢿⡿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡿⠁⠀⢀⣾⣿⣿⣿⡇⠀⠀⠈⢀⣀⣀⡀⠀⠀⠘⠟⠁⣀⣀⣀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠟⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣶⡀⠀⠙⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⣿⣷⡄⠀⠈⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣄⠀⠈⢻⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣦⣤⣤⣿⣿⣷⣤⣤⣼⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡜⠁⡇⢸⠀⢰⠋⠉⣇⢸⢧⠀⡼⡇⣿⣉⠈⢹⠉⢸⣉⡁⢸⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠱⠄⠇⠸⠤⠘⠦⠤⠃⠸⠈⠞⠐⠇⠻⠤⠀⠸⠀⠸⠤⠄⠸⠈⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠉⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⠠⠒⠀⠀⠀⠀⠀⠒⠤⡀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⢀⠠⠒⠈⠀⠀⠀⠀⠒⠄⡀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣸⠁⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢡⠀⠈⣇⠀⠀⠀⠀⣼⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢡⠀⠈⣇
⠀⠀⠀⠀⠀⠀⡟⠀⠠⠄⠀⠀⠀⠀⠀⢠⣴⡏⠀⠀⠀⠀⠀⠠⠄⠀⢻⠀⠀⠀⠀⡟⠀⢠⠀⠀⠀⠀⠀⠀⢠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣧⠀⠈⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢹⡄⠀⢡⠀⠀⡠⠔⠂⠈⠀⠁⠐⠂⢄⠀⠀⠌⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢳⡄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠙⠛⠿⠿⠿⠛⠋⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣀⣀⠀⣀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠋⠉⠉⠉⠉⠉⠙⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠀⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⢀⣠⠴⠚⠛⠛⠛⠓⠦⣄⡀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡠⠄⠒⠊⠉⠓⠒⠦⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⢧⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⢧
⠀⠀⠀⠀⠀⠀⣸⠋⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢷⠀⠚⣇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢷⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣟⠀⢸⠇⠀⠀⠀⠀⠀⢀⣤⡏⠀⠀⠀⠀⠀⠸⡇⠀⣻⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡏⠀⠀⠀⠀⠀⢸⡇⠀⢻
⠀⠀⠀⠀⠀⠀⣯⠀⢸⡆⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⢰⡇⠀⣽⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⢸⡇⠀⣼
⠀⠀⠀⠀⠀⠀⢹⡄⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⡟⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⠁⢀⡞⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⡀⠀⠀⣴⣦⠀⠀⠀⣴⣦⠀⠀⢀⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣙⡋⠀⠀⠀⢉⣋⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣀⡀⠀⠀⠀⢀⣀⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠿⢛⣛⣛⣛⣛⡛⠿⠿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢋⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⡙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣌⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣡⣾⣿⣿⣿⣧⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣼⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣤⣤⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⠟⣡⣦⣍⣛⠛⠛⣛⣩⣶⣬⠛⢛⣛⣟⣿⣿⣿⣿⡟⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡌⢿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠛⣿⣟⣟⣿⡿⣱⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠻⣿⠟⣡⣾⣿⣿⣿⠀⢸⣿⠀⠀⣿⡇⠀⣿⣿⣿⣷⣌⠻⣿⠟⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣤⣼⣿⣤⣤⣿⣧⣤⣿⣿⣿⣿⣿⣷⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⢀
⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⣶⣒⣌⠉⠀⡀⢀⣀⣼
⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣥⣶⣿⣿⣿⣿⡿⠛⠛⠋⠻⠻⠂
⠀⠀⠀⠀⠀⠀⠈⢹⣶⠾⠛⠛⠻⠶⢶⣤⣄⣀⠀⠀⠀⠀⠀⠿⣿⠿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂
⠀⠀⢀⣤⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣄
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣀⣤⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⠿⠛⠉⠉⠉⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣉⡟⠿⣿⣿⡄
⠀⠀⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢧⠀⠈⠙⠛
⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠁⠀⠀⠀⠀⣤⣤⣤⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣷⣶⣤⣄⣀⣀⣤⡤⠆
⠀⠀⠤⠤⣀⣀⣀⠈⠉⢻⣿⣿⠉⠁⠀⢀⣀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣟⡛⠛⠛⠻⢆
⠀⠀⠀⠀⠘⠁⠉⠿⠛⡟⠿⡿⠿⠛⠟⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⠛⠛⠛⠛⠛⠛⠿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⣤⣶⣶⡶⢶⣶⣶⣤⣀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣴⣾⠟⠉⠁⠀⠀⠀⠀⠈⠉⠻⣷⣦⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⢰⣦⣠⣶⡆⠀⣶⣆⠀⣴⣦⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⣿⡇⢸⣿⠀⠀⢸⣿⡟⠁⠀⢰⣿⣿⣤⣿⣿⡄⠀⠀⣿⡇⢸⣿⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⢸
⣿⡇⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⣿⡇⢸⣿⠀⠀⢸⣿⢿⣦⡀⣼⡟⢻⣿⡏⢿⣇⠀⠀⣿⡇⢸⣿⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⢸
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠘⠋⠀⠙⠃⠙⠁⠈⠛⠁⠈⠋⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠻⢿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠟⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠉⠛⠿⠿⠷⠾⠿⠿⠛⠉⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣤⣤⣤⣤⣤⣶⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠟⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⢴⡿⠓⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠉⠉⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣧⠀⠀⠀⠀⢸⣇⣠⣶⠿⠀⢠⣿⣧⣠⣾⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢿⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⢸⡟⢻⡿⠉⣿⡆⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⡄⠀⠀⠀⠸⡇⠀⠀⠙⠃⠘⠃⠀⠀⠀⠛⠃⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⣿⡀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠷⢤⣄⣈⠙⠛⠿⠷⠶⠶⠶⠶⠖⠒⠒⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣤⣤⣭⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠉⢳⠒⠒⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠟⡿⠿⡿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣠⣿⣿⣟⣀⣀⠀⢀⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠾⠿⣿⣿⣿⡿⠁⠀⠘⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⣸⠇⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠋⠀⠀⠀⢀⡴⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⣰⡿⣩⣴⡶⠶⣶⣌⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⠤⠶⠚⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣿⢱⣿⢡⣶⣷⣎⢻⣇⢻⡇⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡌⢿⣮⣿⣿⡟⣸⡿⢸⡇⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣍⣛⣩⣴⣿⠇⣾⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡏⣼⡏⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠶⠤⠴⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣫⢴⡿⠷⠖⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⡶⠾⠷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣧⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⡀⠀⠀⣤⡀⠀⣠⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⠀⠀⠀⠀⢹⣷⣶⣿⡛⠁⠀⢸⣿⣿⣴⡿⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡄⠀⠀⠀⠀⢸⣿⠉⠻⢿⣦⠀⣿⡇⠙⠛⠀⣿⡇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡇⠀⠀⠀⠀⠘⠟⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡖⠒⠛⠻⣦⣄⣹⣆⣀⣀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣦⠾⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠛⠶⠦⣤⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠶⠾⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⡈⠻⢶⣤⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠾⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠾⠇
⢀⡴⠒⠛⠓⠒⠒⠒⠒⠒⠛⠒⠒⠛⠓⠚⠓⠒⠒⠒⠒⠒⠚⠓⠒⠒⠓⠒⠛⠓⠒⠒⠒⠒⠒⠚⠒⠓⢦
⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⣼⣷⠀⠀⠀⠟⠉⢹⡇⠀⠀⢸⣇⠀⣤⠄⢠⣄⣤⣄⣠⣤⣀⠀⠀⢀⣾⣧⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⢹⡏⠁⠀⠀⠀⣴⡟⠁⠀⠀⢸⣷⣾⡃⠀⢸⡟⠉⢻⡏⠈⣿⠀⠀⠈⢻⡏⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣯⣤⡄⠀⠀⢸⣏⠈⢿⡄⢸⡇⠀⣸⡇⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⣿⡇
⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⣸⣧⠀⠀⠀⠘⢻⡇⠀⠀⠀⣿⠀⢀⡀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⢸
⢀⡇⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⢸⡇⠀⠀⠀⣿⣴⡟⠁⠀⣿⠋⢹⣿⠉⢻⡇⠀⠀⠈⢻⡟⠀⠀⠀⠀⢸
⢈⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠉⢻⣆⠀⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠙⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣒⣒⡒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇
⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⣄
⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⢠⣿⡀⠀⠀⠐⢿⣿⠀⠀⠀⣀⣀⡀⠀⠀⢸⣿⠀⣀⡀⢀⣀⣀⡀⢀⣀⡀⠀⠀⠀⢀⣿⡀⠀⢸
⢸⡇⠀⠛⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⣿⣤⡄⠀⠀⢸⣿⣴⡏⠀⢸⣿⠉⢻⡟⠙⣿⠀⠀⠀⠚⣿⠓⠀⢸
⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⣀⠀⣤⣠⡗⠀⠀⢸⣿⠙⣷⡄⢸⣿⠀⢸⡇⠀⣿⡀⠀⠀⠀⣿⠀⠀⢸
⢸⡇⠀⠀⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⠈⠉⠀⠀⠀⠈⠉⠀⠈⠁⠀⠁⠀⠈⠁⠀⠉⠀⠀⠀⠀⠉⠀⠀⢸
⠀⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠤⠍⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠀⠀⢸⣏⣤⠀⣤⢤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢾⢻⣷⢸⣟⣙⡁⠀⢰⣿⣤⠄⣤⣤⣤⣤
⠀⠀⠀⠀⠐⢚⣵⣾⣦⡒⠚⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢸⣿⣿⠀⣿⢸⣿⣼⡇⠀⠀⠀⠀⠀⠀⢠⣾⠃⢈⣍⢹⡇⠀⢘⣿⣿⠀⣿⣿⣿⢸
⠀⠀⠀⠸⠿⠸⣻⣱⣜⣻⡄⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠋⠛⠂⠙⠘⠓⠙⠃⠀⠀⠀⠀⠀⠀⠛⠛⠓⠈⠛⠛⠁⠀⠘⠛⠙⠃⠛⠃⠛⠘
⠀⠀⠀⠾⠿⠎⢿⣫⡝⢿⡄⣾⡷⠆⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⡄⠀⢀⣶⣀⡀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣶⢲⣦⠠⣶⢶⡄⠀⠰⣶⢠⡀⣀⣠⣄⣠
⠀⠀⠀⠿⠿⠆⣿⠟⡻⣷⡁⣴⣾⡧⠀⠀⠀⠀⠀⠀⠀⢀⣐⢿⡇⠀⠘⣿⣿⠀⣿⣽⣿⢸⡇⠀⠀⠀⠀⠀⠀⣀⢛⣧⢐⣿⢼⡇⠀⢘⣿⣾⠀⣼⢿⣿⢸
⠀⠀⠀⠿⠿⠷⢰⡿⠿⣾⠁⣠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠘⠿⠾⠃⠀⠰⠿⠻⠆⠿⠿⠟⠸⠇⠀⠀⠀⠀⠀⠀⠋⠾⠇⠘⠿⠾⠃⠀⠐⠟⠻⠆⠻⠐⠿⠸
⠀⠀⠀⠘⣿⣦⢀⣿⡟⣾⡃⣩⣥⠄⠀⠀⠀⠀⠀⠀⠀⢰⡤⠶⠀⠀⢠⣦⢀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⣶⡦⢠⡖⢦⠀⠀⡴⢲⡄⠀⢰⣆⣀⢀⣀⣀⣀
⢀⣤⠀⢠⣬⡉⠈⠁⣤⡈⠃⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣻⡄⠀⢸⣿⣿⠀⢿⣹⣿⣽⡇⠀⠀⠀⣸⣻⣇⠀⣰⠟⠀⠀⢀⡿⠁⠀⢸⣿⡇⢸⡏⣻⢹
⠙⣿⠀⣿⢰⡿⠀⠀⡿⣶⡇⢸⡟⣿⣺⣷⠀⠀⠀⠀⠀⠸⣧⢶⠇⠀⢸⠿⢿⠄⢻⠽⡷⢿⡇⠀⠀⠀⠈⠙⠋⠘⠛⠒⠰⠂⠛⠓⠂⠀⠘⠛⠓⠘⠃⠛⠘
⠀⣿⠀⣿⣘⡿⠀⠀⣿⣻⡇⢸⡇⣿⡸⣿⠀⠀⠀⣀⡄⠀⣠⣄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠀⣠⡴⣤⠀⠀⢤⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣤⣤⡄⣠⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⢸⣇⢼⠇⠀⢸⣷⣾⠀⣾⣞⣗⢺⡇⠀⠀⠀⠀⠀⠸⠟⣿⡆⣼⡇⣾⠀⠀⢺⣿⡏⢸⣿⣿⡊⣿
⠀⠀⠀⣸⣿⠿⠡⣿⠭⢈⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣹⠇⠀⢸⣿⣿⠄⣿⡱⣇⢾⡇⠀⠀⠀⠀⠀⠸⡧⣽⠇⠿⣧⡟⠀⠀⢺⡟⡷⠸⡿⢾⠇⣿
⠀⠀⠀⠿⠿⢫⣿⣿⡿⣿⣶⠀⠀⠀⠀⠀⠀⣀⣀⠀⠁⡀⠈⠁⢀⡀⠀⢁⡀⠀⠈⠀⠁⠈⠀⠀⠀⠀⢀⣤⠀⣠⣤⡀⢀⣤⣀⠀⠀⢀⣄
⠀⠀⠘⠿⣛⡸⣿⣿⠃⢶⡖⠀⠀⠀⠀⠀⠘⠋⣿⠂⢻⡇⠀⠘⢻⡇⠀⢸⣷⡆⢰⡶⣶⢲⡆⠀⠀⠀⢉⣤⢈⣟⢸⡧⢸⣧⣹⡇⠀⢸⣷⣝⠀⣿⢻⣿⣹
⠀⠀⠀⠀⠂⠀⡀⠈⡙⠓⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⢸⡇⢠⠀⢸⡇⠀⢸⣿⣧⢸⡇⣿⢸⡇⠀⠀⠀⢨⣿⠐⣿⣘⡗⢸⣇⣩⠅⠀⢸⣿⣧⡀⣗⢩⡗⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠛⠛⠓⠒⠲⠶⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⢀⣤⣴⡾⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⠶⢦⣤⣀⠀⠀⡙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣀⣤⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⣾⠟⠋⠀⠀⠀⠀⢀⡀⢤⠐⣧⠈⡇⠸⠇⠸⠁⢸⠀⣴⠂⡄⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠊⠀⢀⣾⠟⠁⠀⠀⠀⢀⡀⠳⡈⠓⠈⠁⠻⠀⠀⠀⠀⠀⠀⠀⠰⠇⠘⠁⠾⢁⡴⠀⡀⠀⠀⠈⠻⣆⠀⠀⠙⣄
⠀⠀⠀⡰⠁⠀⣰⣿⠃⠀⠀⠀⢀⠲⣦⠙⠀⠀⣾⣿⣗⣿⣿⡀⠀⠀⢰⣟⣷⡿⢻⡆⠀⠀⠈⠠⠞⣡⡤⠀⠀⠀⠘⣧⡀⢀⠘⢆
⠀⠀⡰⠁⢀⣼⡿⠁⠀⠀⢀⠰⢄⠑⠈⠓⠀⠀⠸⣤⠟⠷⠾⠃⠀⠀⠸⢯⠿⠿⠾⠇⠀⠀⠀⠀⠾⠋⢀⡴⠀⠀⠀⠈⢷⡀⠀⠈⡆
⠀⢠⠃⠀⣾⣿⠁⠀⠀⠠⣌⡓⠂⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡶⣶⣶⡀⠀⢠⠴⠂⠀⠀⠈⢷⠀⠀⠘⡆
⠀⡞⠀⣸⡿⠃⠀⠀⢀⡲⠶⣥⡀⣾⣿⣼⡿⣿⠀⠀⡀⠀⠀⠰⠾⠂⠀⢀⣀⣀⠀⠀⢸⣿⡿⢯⣽⠇⠀⠀⠠⠖⠂⠀⠀⢸⣧⠈⠀⢳
⢰⠃⢀⣿⡇⠀⠀⠀⢬⣉⠃⠀⠀⠉⠙⠋⠛⠛⠀⠀⠓⠂⠀⠀⠀⠀⢠⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⠀⢻⡆⠀⢸⡆
⣼⠀⢸⣿⠀⠀⠀⠀⠲⠦⠀⠀⣀⡀⣠⣤⡀⠀⢠⣤⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠠⠤⠀⠂⠚⢿⣿⡟⢻⡆⠤⠶⠶⠆⠀⠀⢺⡇⠀⠀⡇
⡇⠀⢸⣿⠀⠀⠀⢈⣉⣁⣀⢸⣿⣿⣿⣸⡇⠀⠀⠈⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠿⠿⠃⠀⠰⠶⠆⠀⠀⠀⣿⠀⠀⣧
⡇⠀⢼⣿⠀⠀⠀⠀⣀⣀⠀⠀⠀⠋⠉⠉⠁⠀⠸⠟⠀⠀⠀⠀⣼⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠄⠀⠀⢀⣿⠀⠀⡟
⣿⠀⠨⣿⠄⠀⠀⠀⣀⣀⠀⠀⢀⣤⣄⣠⣤⡀⠤⠀⠀⠀⠀⣼⡏⠀⠀⢀⠀⠀⠛⠛⠁⠂⠺⣿⢿⣿⣿⠀⠀⠰⠤⠀⠀⠀⢸⡇⠀⠀⡇
⢸⡄⠀⢿⣆⠀⠀⠀⢀⣠⡄⠀⢉⡿⣿⣿⣹⡇⠀⠀⡐⠻⣾⡟⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠰⠏⠸⠷⠾⠀⠐⠦⡄⠀⠀⠀⣿⠁⠀⢸⠃
⠀⣷⠀⠺⣿⡄⠀⠀⠀⢀⣠⠄⠈⡛⠋⠉⠉⠁⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠈⠛⠶⣦⠀⠀⠀⣰⡏⠀⠀⡟
⠀⠸⣇⠀⢹⣷⡀⠀⠀⠀⢀⡴⠟⠁⠀⠰⣟⣿⣿⠛⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣏⢹⡇⠀⠀⠓⠦⠀⠀⠀⣰⡟⠀⠀⡼
⠀⠀⠹⣄⠀⢻⣷⡠⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠙⠛⢁⠈⠳⠆⠀⠀⠀⣴⡟⠀⠀⡼⠁
⠀⠀⠀⠙⣆⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⡏⠀⢀⡀⠀⢀⢀⣠⣄⣀⡀⢠⡀⠲⣆⠀⢦⠈⠓⠀⠀⠀⣠⣾⠋⠀⢀⡞⠁
⠀⠀⠀⠀⠈⢷⡀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠛⢠⡟⠀⠀⠸⠗⣤⠛⠼⠗⠚⠹⠋⠙⠃⠀⠹⠆⠀⠀⠀⠀⣠⣼⠟⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠈⠹⠿⣿⣶⣤⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡈⠉⠉⠿⠿⣿⣶⣷⣦⣦⣤⣤⣤⣴⣶⣶⣾⡿⠿⠛⠁⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣈⡙⠋⠛⠛⠙⠉⠉⠉⠉⠁⢀⣀⣠⡤⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠞⠛⠛⠛⠛⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⣀⣤⣤⠶⢶⡶⠶⣤⣤⣀⠉⠻⢶⣄
⠀⠀⠀⣀⣴⠶⠟⠛⠛⢻⣿⠋⢀⣴⠾⢻⣇⠀⠀⠸⠇⠀⠀⣸⡟⠷⣦⡀⠙⣿⡟⠛⠛⠻⠶⣦⣀
⠀⢠⡾⠋⣠⡴⠶⠀⢀⡾⠁⢠⣾⡁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⣷⡄⠈⢷⡀⠀⠷⢦⣄⠙⢷⡄
⢰⡟⢀⣾⠋⠀⠀⠀⣾⠃⢠⡿⠉⠛⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢿⡄⠘⣷⠀⠀⠀⠙⣷⡀⢻
⣿⠁⣼⠇⢀⣀⣀⣀⣿⠀⢸⣇⣀⠀⠀⠀⠀⠈⣻⠶⠶⡄⠀⠀⠀⠀⠀⣀⣸⡇⠀⣿⣤⡶⠟⠃⠸⣧⠈
⣿⠀⢻⡆⠈⠉⠉⠙⣿⠀⢸⡏⠉⠀⠀⠀⠀⠀⠹⣦⣴⠇⠀⠀⠀⠀⠀⠉⢹⡇⠀⣿⠁⢀⡀⠀⢰⡟
⠸⣧⠈⠁⣀⣀⠀⠀⢿⡀⠘⣧⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣼⠃⢀⡿⠀⣘⣃⠀⠀⠁⣼
⠀⠙⢷⣜⠿⠿⠚⠛⠘⣷⡀⠙⠛⠉⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠉⠛⠋⢀⣾⠃⠀⠿⠿⠂⣠⡾⠋
⠀⠀⠀⠙⠷⢦⣤⣤⣠⣼⣷⣄⠀⠀⠀⣿⣀⣀⣀⣀⣀⣈⣿⣀⣿⠀⠀⠀⢠⣾⣧⣄⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⢷⣄⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣠⡶⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣤⣤⣤⣤⡶⠾⠛⠁
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠒⠀⠀⠀⠀⠈⠀⠈⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠨⠀⠀⢀⣀⡤⣴⣶⣲⣶⣴⡤⣤⣀⣀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠔⠁⠀⢀⣠⣶⣿⠿⠚⠋⠉⠁⣆⠀⠈⠉⢉⡝⠛⠿⢷⣦⣄⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⠟⢉⠀⢻⡄⠀⠀⣀⡛⠀⠀⠀⠸⠃⠀⠂⢀⣭⠛⠿⣦⣀⠀⠀⢌⡀
⠀⠀⠐⠁⠀⠀⣠⣯⠞⠱⣦⡀⠀⠰⠰⠔⠄⠈⠉⠈⠀⠉⠛⠙⣀⣀⣚⠁⠀⠂⢈⡻⣷⣄⠀⠙⢮⠢
⠀⠀⠀⠀⢀⣼⡟⠁⠐⠀⠀⠢⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⢀⠀⠠⠟⠁⠈⠻⣦⠀⠈⠳⡀
⠀⠀⠀⠀⣾⡟⠈⠛⠆⠠⠄⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠙⠃⠀⠀⣀⡴⠙⣷⡄⠀⠹⡄
⠀⠀⠀⢸⡿⢠⣄⡀⣀⣀⢀⠀⠀⠀⠀⠀⠀⢤⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠰⠒⠒⠄⠉⠀⠄⠘⡾⡄⠀⢹⡤
⠀⠀⢀⣿⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⢠⡤⠶⢹⣷⠀⠈⡇
⠀⠀⠐⣿⠇⢠⣤⠀⣄⡠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⣿
⠀⠀⠀⣿⢷⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣴⢛⢓⡼⠧⠤⠤⠤⣀⣀⡀⠀⠛⠒⠂⠰⠶⠶⢀⣿⠀⠀⣿
⠀⠀⠀⢸⣿⡐⠖⠃⠈⠉⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢉⡙⡒⠒⠶⠦⠀⣸⡟⠀⢠⣿
⠀⠀⠀⠈⢷⣇⢈⣤⠄⠘⠙⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠙⠛⠳⢶⣰⣿⠃⢀⣾⡗
⠀⠀⠀⠀⠀⢯⣆⠠⠀⣀⠰⠲⢰⡏⠉⠉⠀⡭⠌⣬⣍⠨⡍⢩⣉⣡⠲⠲⠶⠶⠂⠀⠒⠄⠰⣿⠋⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢻⣦⠚⠉⠀⠀⢸⡇⠀⠀⠀⢩⡇⢧⡼⢐⡇⠀⠏⣹⡏⠀⠀⠀⠉⠳⢄⣾⠟⠁⠀⣾⡿⠁
⠀⠀⠀⠐⢷⡄⠀⠑⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣢⣔⡤⣀⠀⡀⠀⠀⠀⠀⠀⡀⠁⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠄⠀⠀⠣⠄⠀⠀⠀⠀⠀⠈⠃⠙⠂⠀⠀⠗⠀⠀⢀⠄⠈⠀⢀⣶⡻⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣄⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠛⠛⠓⠶⣶⣶⣶⣶⣶⣖⠒⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠈⠀⠉⠉⢱⢸⣷⡦⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⠀⣤⣘⠋⡀⠐⣼⢋⠉⠚⠻⢿⣿⣖⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⣶⣶⣴⣶⣦⣛⣛⣀⡄⡚⠻⣷⠕⢄⠀⠀⠈⠙⠣⠌⡢⡀⠀⠀⠀⢀⠔⠘⠓⠠⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢔⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠺⡿⠂⠑⣕⢄⠀⠀⠀⠀⠀⠉⠂⡤⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⡬⠜⠋⠀⢻⣿⣿⣿⣿⣿⣿⣧⣀⢀⣿⣧⣤⡀⠀⠈⠓⢑⢄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⠛⢧⠀⠀⠀⠀⠑⢕⢀⠔⠁⠀⠀⠀⠀⢀⢔⠁
⠀⠀⠀⠀⢀⡅⠔⠂⢂⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⢀⠔⠁⠀⠂
⢀⣠⠐⠈⠀⠀⣀⣤⣄⣂⡘⣿⣿⣿⣿⠟⠛⢿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⣠⢂⠀⠀⠀⠀⠈⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡝⢿⣇⣀⡀⠀⠀⠀⠀⠀⢀⠔⠀⠠⠔⢁⡀⠀⢀⣤⣾⣿⣷⣕⢄⠀⠀⠀⠈⠄
⡇⠀⠀⣠⢴⠖⡖⢶⢄⠀⠀⣿⡇⠀⠀⠉⠳⢄⣀⣄⣤⣔⡁⠀⢀⢌⣴⠟⡠⠂⠉⠻⠿⠛⢻⡇⠐⢕⢄⠀⠀⠈⠄
⡇⠀⡰⠛⡟⠚⡟⠛⡟⢳⡀⣿⢧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣍⣽⠁⠁⠈⠀⠀⠀⠀⠀⡘⠀⠀⠀⠑⢕⢄⠀⠐
⡇⠀⣟⠒⡗⠚⡗⠒⡗⢚⡇⣿⡆⢢⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠀⢀⠔⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠑⢕⠄
⡇⠀⠘⢶⣻⡒⡗⣺⣳⠞⠀⣿⡟⣀⡧⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⠉⠒⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡇⠀⣀⣀⣉⣉⣋⣉⣁⣀⡀⣿⣏⠕⢂⠡⠀⠀⠀⠀⣿⣿⠟⠋⠀⠀⠀⠀⠀⢀⠔⠁
⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣿⣿⣎⠄⢂⠡⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠄⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⠨⠁⠊⠐⠂⠀⠙⠂⠒⠀⠈
⠃⠈⠉⠉⠉⠉⠉⠉⢉⣉⠉⠡⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⢛⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣠⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢠⠀⠀⠀⠀⠀⠀⠀⠌⡔⠊⠁⡄⠈⠑⡁⠀⠀⠀⠀⠀⠀⣠⠤⣴⠠⣴⠊⠙⢦⡀
⠀⠀⠈⠙⠡⠤⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠉⠁⠉⠒⢄⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠺⣄⡐⢃⣰⢟⣲⡳⠦⢿
⠀⠀⠀⠠⠤⠤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⣠⣄⠀⠲⣀⠀⠀⠀⠀⠀⠐⡌⢆⡀⠀⠀⠀⡠⢂⠂⠀⠀⠀⠀⠀⠀⠙⢭⡑⢋⠥⠵⠒⠋
⠀⠀⢠⣈⣰⣆⣈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠊⠊⠰⠶⠼⠃⠐⠑⠀⠀⠀⠀⠀⠀⠈⠂⠌⠈⠁⠩⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⢀⠄
⢰⣶⢀⠔⢂⡁⠐⢄⠠⢱⠀⠀⠀⠀⠀⠀⡞⠋⠛⠛⠛⠓⢤⠀⠀⠀⠀⠀⠀⠀⣠⣤⣌⠀⠀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢘
⠈⡎⠁⡀⢀⢙⢜⠇⡆⡎⠀⠀⠀⠀⠀⠀⠆⠀⠴⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠸⠭⠇⣔⠊⠑⣢⠤⠝⠀⠀⠀⠀⠀⠀⠀⢠⠀⠨⢀⡁⢨
⠀⣣⠠⡙⠦⠾⠞⠠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⠼⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⣏⡀⣀⢼⢁⠀⡇⠀⠀⠀⠀⠀⠀⠨⠀⠀⠉⠅⠠⠄
⠀⠉⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠙⠃⠀⠙⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⢠⣤⡄⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⣀⠀⠈⣈⣐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠨⢉⠒⠉⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠈⠉⠡⡖⠀⠉⠀⠀⠀⠀⠀⡄⠀⣀⠄⠀⠀⢁⡴⠥⠆⠀⠀⠀⠀⠀⠦⡀⢀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠐⡭⣥⠀⠚⠛⢻⠛⡇
⢐⠀⠀⠀⠀⠀⠀⠅⠀⡀⠀⠀⠀⠀⠀⣘⡈⠀⡄⠒⠒⠫⢄⠀⠀⠀⠀⠀⠀⠀⢆⠈⠘⠂⠀⡄⠈⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣀⡘⠐⠂
⠀⠁⠐⠂⠤⢄⠌⠀⠁⠀⠀⠀⠀⠀⠀⡏⡁⠈⠒⠄⠖⠒⠒⠁⠀⠀⠀⠀⠀⠀⠈⠣⠄⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⡐⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠀⡂
⡄⠐⡒⠒⠒⠛⠛⠁⠠⠬⠀⠀⠀⠀⠀⠀⡠⣤⠈⠆⣈⠀⢙⡾⠀⠀⠀⠀⠀⠀⠎⠙⡉⢹⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣀⠀⢄
⡇⠀⠀⠀⣠⠖⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣿⠀⠀⢠⢤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢛⡉⠱⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⣀⠆
⢁⣀⣀⢀⣃⣐⡉⣀⣀⠄⠀⠀⠀⠀⠀⢀⠙⣯⣹⣿⠲⠤⣶⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠹⣀⣀⣀⣠⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠝⠩⠛⠿⠛⠟⠿⠫⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⠿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠉⠁⠀⠀⠀⠈⠙⠻⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢀⣤⣶⣶⣶⣶⣤⣀⠀⣀⣀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠠⠮⡭⢗⣚⡛⠿⠿⠋⠘⣿⣿⣿⣷⣄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠿⠏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣼⣿⣧⠀⠀⠀⠀⠀⣤⣤⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⡠⠈⠘⠛⠉⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠻⣟⠛⠷⠶⣦⣤⡀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣷⡀⠀⠈⠙⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⡄⠀⠻⣿⣿⣿⣿⡿⢿⣀⡲⠶⠾⠋⣼⡆⢠⣤⣀⣀⣀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⢟⣿⣷⣿⡿⠿⠶⠄⠀⢈⠃⡀⢻⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠰⢯⠈⠀⠈⠁⠁⠀⠀⠀⢀⣴⡾⣧⠀⢻⣿⣿⣧⣼⣇
⠀⠀⠀⠀⠠⡝⣿⡻⢿⣿⣷⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠟⠻⠋⡉⡍
⠀⠀⠀⠀⠀⣹⣯⡫⠬⣻⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣟⡏⠈
⠀⠀⠀⠀⠀⠈⠘⢟⠳⢿⣯⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣯⡋⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠃⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⣰⡖⢥⣃⡂⠄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡁⢀⣴⠾⣧⣼⢟⠘⢍⠀⠉⢳⢌⠢⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠊⠀⠈⠀⠉⠑⣊⡟⡄⣡⢀⠀⢀⡼⠀⠑⠌⢂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠡⠀⠀⠀⠀⠘⢄⡌⠙⢂⠀⠒⠁⠀⠀⠀⢈⡆⢃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⡀⠸⠙⠂⠀⠀⠀⠀⠘⡷⢛⡀⡀⢀
⠀⠀⠀⠀⠀⠠⠂⣀⣀⣐⠂⠄⠀⠐⣄⠀⢀⠤⢤⠘⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠚⠇⠁
⠀⠀⠀⢀⠤⠤⠤⠤⠤⠬⠧⠦⢤⡀⠈⠑⠴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠆⢠
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⠙⠉⠁⠁⠢⠄⡀⠀⠀⠀⠈⠀⡙⠠⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢁⢠⡄⠀⠀⠀⢠⠀⡄⠀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠗⠛⠉⠉⢝⡭⢭⣍⢂⠀⠀⠀⠀⡰
⠀⠀⠸⠀⠀⠸⠇⠀⠀⠀⢹⠋⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣤⠂⣿⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠄⡀⠀⠀⢃⡈⠀⢀⣘⠂⠄⠀⢀⡀⢸⠄⠄⠊⢂⠈⠉⠁⠰⠀⠀⠀⠀⠠⠂
⡔⠐⡇⠀⠊⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠉⢳⠂⠀⠈⠉⠁⠀⠐⠊⠒⠀⠈⠁⠀⠒⡀⠀⠀⠀⠣⠀⠀⡄⠀⢀⡀⠤⠐⡞
⡯⢁⠀⢠⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠍⠁⠀⠒⠀⠀⠒⠀⠉⡀⠀⠀⠀⠀⠀⠛⠀⠀⢀⢔⣑⡀⠃⠀⠀⠀⠀⠀⠐
⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇⢈⢀⠀⣄⠀⠀⣠⢀⠈⢠⠀⠀⠀⠀⠀⠘⠈⡠⠢⠈⢹⡟⢻⠀⠀⠀⠀⠀⠰
⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠠⠀⠀⠀⠀⠀⠀⠃⡀⣄⣀⣀⣠⠀⠐⣋⢀⣀⣀⣤⠤⠼⠤⠤⠤⠤⠤⠴
⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⢄⠖⠌⡀⠀⠀⢀⠂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠀⠊⠂⠀⠠⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠂⠐⠀⠐⠀⠐⠀⠈⠀⠀⠀⡀⠉⠁⠀⠈⢀⠘⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠑⠀⠠⠄⠀⠀⠀⠀⢀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠐⠄⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠒⠑⠄⠀⢀⠀⠀⠀⠀⠀⠂⠛⠋⠁
⠀⠀⠀⠀⠀⠄⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠒⡀⡁⠀⠐
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⡄⢀⠢⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠃⢀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁
⠀⠀⣠⠀⠀⠁⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⢠⠂⣮⠢⠀⠨⠀⢠⠁⠀⠀⠀⠀⠀⢂⠀⠺⠀⠀⠀⣀⠀⢠⠤⠤⠄⢀⢀⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠈⠀⠀⠁⠘⢀⠀⠀⠀⠁⠀⠂⠃⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠆⠀⠀⠂⠐⠐⠀⠀⠄⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠐⠗⠀⠀⠉⠓⠃⠀⠀⠒⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠇⠀⠀⠈⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⢄
⠀⢀⣴⡆⣀⠐⣖⣆⣀⡀⣠⢤⣶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠄⠀⠁⠉⠤⢃⠄⠘⠒⠙⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠄⠀⣀⣀⣀⡄⠁⠠⠠⠀⢀⣀⣀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣈⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⢸⣯⠑⠀⠀⠀⠁⠀⠀⠀⠈⡉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⠨⠟⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠄⠠⠀⠂⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠗⠁⠀⠀⠀⢀⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠦⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡎⠑⣆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠲⠐⢖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠀⡄⠠⠤⢀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠧⠤⢀⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣀⠀⠉⠉⠀⠒⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⣾⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣦⣤⡄⠀⠀⠀⠀⢀⢀⢤⠇⢀⣄⡈⢀⡀
⠀⠀⠀⣰⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣏⠙⠀⠐⢦⣾⣠⣾⣿⠠⣾⣤⡤⠂⡟⠷⣆⡀
⠀⠀⠀⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⡀⠙⠿⣿⣿⣷⣼⣿⣡⡟⣡⠤⠉⢻⡴⣆
⠀⠀⢰⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠉⢌⠙⠛⠷⣥⠰⢚⣒⡀⢿⣿⡃
⠀⠀⡼⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠿⢿⠁⠀⠀⢿⡿⣿⣿⣆⠀⠱⣄⠈⢱⢆⣼⠟⠛⠪⣯⠈⢀
⠀⠀⣶⣿⡏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠷⠾⠿⠇⠸⠶⢾⣿⣮⣿⣿⣧⡄⠈⠃⠈⠨⠟⠁⢈⢂⣿⠀⠆
⠀⠀⣿⣿⣟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⠀⠀⠀⣀⢐⡀⠀⣹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠎⠉⢹⣿⣂
⠀⠀⣴⣿⣿⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣶⣷⣾⢻⣿⠷⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡔⠀⠠⠋⢺
⠀⠀⣿⣿⡟⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣏⠻⣿⣿⣷⣉⣉⣴⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠐⠀⣼⣀
⠀⠀⢸⣿⣿⣦⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣯⠉⠁⠉⠩⢁⡌⠻⢿⣿⠟⢻⡟⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠛⠋
⠀⠀⠘⣍⣿⣿⡅⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⠿⠿⣿⣿⣷⣤⣀⡀⠘⠛⠟⠀⠀⠁⢺⣇⠘⣿⣿
⠀⠀⠀⣿⣿⣾⣧⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⣉⣷⢯⡄⠙⡿⣿⣻⣄⣀⣀⡀⠀⠀⠀⢿⣏⠄⠈⠋⠐⣀⡀
⠀⠀⠀⢹⣿⣿⣿⡇⢘⢷⡀⠀⠀⠀⠀⠀⠀⣀⣼⣥⣤⣶⣿⡷⠀⣸⣴⣾⡏⠻⣯⠁⠉⠳⣤⣤⡴⠿⣧⢤⡄⠀⣀⣼⡴⣦
⠀⠀⠀⠀⢻⣿⣿⣽⣇⠀⡆⠀⢀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣿⡇⠸⣿⢉⣿⣟⠀⠘⢿⡁⠀⣈⠿⣽⡇⠀⠛⠛⠚⢏⣬⡆⣦⡀⢀
⠀⠀⠀⠀⠀⠙⢳⣾⣿⡀⣠⣖⠁⠀⠀⣸⣿⡿⢿⣿⡣⢾⣿⠃⢠⡟⣸⣿⣟⠂⠀⠀⢹⣿⡿⠛⠚⣶⠀⠀⠀⠀⠀⢻⡷⠛⢷⡀⢇
⠀⠀⠀⠀⠀⠀⠚⢛⢘⣿⡾⠿⢷⠎⡛⠥⣠⡄⠘⢿⣷⡾⠛⠂⣸⣷⣿⣿⣾⠲⢤⠀⠀⢩⣷⣾⣾⡿⢧⣀⠀⠀⠀⢸⠇⠀⢸⢿⣦⣑⡀
⠀⠀⠀⠀⠀⠀⢀⣦⣤⢻⣿⣿⣿⢾⣄⠀⣿⠛⠂⢶⣿⣑⣴⣿⠟⣹⣿⣿⣇⡰⣄⠀⠀⠘⠙⠙⣏⠂⠀⢻⣦⠀⢰⣿⡖⠀⡼⠦⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣮⣿⠄⣿⣿⣷⣮⡛⠀⣿⣿⣀⢨⣿⠟⢋⣿⣿⣿⣿⣟⣾⡿⠀⠀⠀⠀⠀⠀⠨⣷⢶⣡⠼⣧⣞⣿⣟⡏⠀⠀⠙⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣽⣦⠈⣟⣿⣇⣴⣧⣿⣿⣯⡙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⡆⡠⠃⠀⠀⠀⣿⣿⣅⣤⡿⣏⣿⡿⢁⣤⢤⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡿⠾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣼⣿⣴⣿⣯⣿⣯⡅⠀⠤⢀⡀⣉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣭⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⡖⠀⠀⠀⢨⣷⡝⣌⠿⢀⣀⠈⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠄⠀⠉⢿⣷⢻⣄⣿⡟⠃⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢀⡇⠈⢿⠀⠀⠀⠈⠉⠉⠙⡟⠉⢿⣿⣿⣿⣧⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢯⠃⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⡟⣿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣞⠂⢀⡀⢣⠀⠀⠀⠀⠀⠀⣿⠂⠀⠈⢿⡘⣇⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⢸⣷⠈⠃⠀⠀⠀⠀⠀⢹⠃⠐⠀⠈⣿⢻⣧⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⠟⠛⢷⣆⠀⠀⠀⠉⠉⠉⠉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⢶⣶⠾⣿⡀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠁⠀⠀⠀⠀⠀⣸⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡉⣹⡇⠀⠀⠀⠀⢐⣿⣩⣿⣉⣙⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣩⡿⢻⡇⠀⠀⠀⠀⣼⡏⠉⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠃⠀⠀⢰⣿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⠛⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⢰⡿⠙⢷⣄⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢿⡇⠀⠀⢹⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢻⡿⠁⠀⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⣼⠏⢠⡿⠁⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠿⣶⡿⠀⠀⠀⠀⠀⠸⠷⠾⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣼⠏



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄⢀⠀⠄⠀⠀⠀⢂⠀⠀⠀⠄⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⡠⢄⠰⢦⣰⠀⣎⣇⠈⡏⢠⢂⡠⢄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢎⡀⠱⡘⠆⠂⠘⠀⠁⠈⠀⠀⠈⠸⡀⡰⢀⠶⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡐⠙⢦⡈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠊⡀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⠨⠶⠄⠀⠀⠀⠀⠀⠀⠀⣰⣲⣄⣰⣦⣄⡀⠀⠀⠀⠀⠉⡱⢃⡠⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣿⡇⠐⠦⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠐⠉⠀⠀⢸⣿⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢿⣿⣦⢸⣿⣷⡀⠀⠀⣠⣶⣄⢀⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣟⣡⣴⣷⠀⣠⣶⣄⠀⠀⢀⣾⣿⡇⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⡏⢸⣷⡄⢰⣄⣼⣿⣿⣿⣿⣿⣿⣿⡿⠷⡀⢹⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣷⣿⡉⠉⠻⣿⣿⣿⣿⡏⠀⠀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠁⠀⠈⢿⣿⣿⣿⣿⡆⠀⢠⣿⣿⣿⣷⡄⠐⠀⠀⠈⠛⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⡟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⢀⣴⣿⣿⣿⣏⠔⠁⠀⡀⠀⠀⠀⠀⠉⢻⡿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠤⢞⢉⡆⠀⠀⠈⠙⠻⠿⠾⠿⠿⠟⠋⠁⠀⠀⢀⣘⣆⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣩⣞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⣤⡀⠉⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⠣⠈⢀⡔⡿⠀⡠⠤⢀⡄⠄⠀⢖⠉⠑⡄⠻⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠁⠈⠇⢸⠀⠀⠀⡖⠂⠀⠈⢦⠴⠃⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡄⠒⢄⡀⠀⠀⠀⢀⡀⠄⠒⢈⣀⠀⡈⠁⠂⣄⡀⠀⠀⠀⠀⢀⠔⠋⠉⡁⠀⠣⡐⠄⠀⠀⣄⠀⠀⢀⠀⠊⠉⠤⠀⠌⠁⠂⠄
⠀⠀⠣⡀⠀⠈⠀⠀⡍⠁⠴⠉⠕⠃⠀⠀⠈⠉⠁⠀⠨⡀⠀⠀⢀⠁⠀⡀⠀⠁⠀⠀⠈⠐⠄⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠠⠘⣀⡤⠃⠀⠀⠁⠔⠀⠐⠲⡄⠀⢀⣈⣀⡠⠔⠁⠀⠀⢸⠀⠂⣀⠈⡀⢀⡔⠀⠂⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠉⠣⠴⠀⠲⠊⠁⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢱⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠓⠒⠓⠈⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠵⡄⢱⠜⠀⣀⠘⠂⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⣀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠀⠀⠀⢀⡠⠤⡠⢊⢋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠠⠄⠀⠀⠈⠲⡄⠀⠀⢀⠔⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⢀⠐⠜⡸⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠄
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠑⡔⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠤⠤⢄⠀⠀⠀⠀⡴⣤⣄
⠀⠀⠀⠀⠀⢀⣞⠂⠉⢑⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⣮⢑⠀⢀⣾⠂⠀⠀⠀⣄⢀⡄
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠉⠑⢢⠀⢰⠀⣀⠠⠤⢀⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⡺⠀⢀⠖⠋⠀⣼⡟⠀⠀⠀⠀⠈⠉
⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⡇⠀⡅⠀⠉⠀⠀⠀⠀⠈⠁⠒⠈⠁⠀⠀⠀⣤⡀⠀⢠⠃⠀⠈⠀⠀⠀⣿⠂⠀⠀⠀⡄⠀⠀⠀⠐⠂
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎⠀⠒⠒⠢⠀⢸⡏⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠃⠈⡇⠀⠀⠈⠁⠀⠀⠀⡀⢰⠤⣀⠀⠀⠀⠀⠀⠀⢤⣀⣰⠀⠀⢀⡾⠀⠀⠀⠀⢠⣤
⠀⠀⡏⠀⠀⠄⠀⠀⢫⠇⢰⠐⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⠳⠀⠀⠀⠀⠀⠘⠃⠘⢰⠀⠀⠀⢀⠊⠀⠀⠀⠀⡔⢳⡟⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢠⣾⣿⣦⠀⠀⢠⠛
⠀⠀⠠⣀⣤⣀⣀⣀⡦⢂⡔⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⠀⠛⠁⠀⠀⠀⠈⠀⠀⠈⠻⠿⠋⠄⠀⣈⡇
⠀⠀⠀⠈⠀⠉⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⢀⠤⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣤⣤⣄⣀⡀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢤⡤⠤⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠁⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠃⠀⢁⠤⠠⢄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠣⡤⡀⣀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠤⢀⠀⠀⠠⠄
⠀⠀⢰⠀⡀⠀⠄⡆⡆⠀⢄⠀⠀⠀⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢹⠉⠉⠓⠃⠀⠀⠀⠀⠀⢀⠀⠕⠁⠀⠀⠀⠀⠑⡀
⠀⠀⠘⠀⠁⠤⠠⠞⠀⠀⠘⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⡆⠀⠀⠤⠀⠀⠀⡀⠀⠀⠈⠐⠀⠀⡀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⢃⠀⣤⡤⠖⠗⠀⢈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢃⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠐⠃⠈⠢⢄
⠀⠀⠀⠸⢄⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠈⠙⠋⠙⠻⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠈⠉⠉⠑⠢⢄⣀⠠⠄⠀⠘

⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣉⡉⠉⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⣿
⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠛⠃⠀⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠃⣠⠶⠲⣦⡈⢻⡄⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠟⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⡀⠀⣸⠇⢸⡇⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣦⠈⠙⠛⠉⢠⡟⠁⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡀⠘⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠿⠛⠛⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡟⠀⣾⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⣿
⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⣿
⣿⣿⠀⠻⠿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⣿
⣿⣿⠷⢶⣶⡶⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⢶⣿⠆
⣿⣿⠀⠘⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⠟⠁
⣿⣿⠀⡄⠸⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠋
⣿⣿⠀⣿⡄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡷⠋
⣿⣿⣴⣿⣷⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠂⠉⢁⣀⣀⣀⣀⣀⣀⡈⠉⠐⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⢁⡠⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠦⢄⡈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⢀⠤⠊⠁⠀⠀⣀⠤⠴⠒⠒⠀⠀⠀⠐⠒⠒⠦⠤⣀⠀⠀⠈⠑⠤⡀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣠⠖⠁⠀⠀⡠⠔⠉⠀⢠⡄⠀⣶⠀⢸⡇⠀⣖⠀⢠⡄⠀⠉⠢⢄⠀⠀⠈⠲⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠜⠁⠀⠀⡠⠊⢀⠀⠸⣆⠀⠿⠀⠙⠂⠘⠃⠐⠋⠀⠿⠀⣰⠇⠀⡀⠑⢤⠀⠀⠈⠣⡀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠃⠀⠀⢠⠊⢀⡀⠘⢷⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡾⠃⢀⡀⠑⡄⠀⠀⠘⢄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡰⠁⡰⠁⠀⠀⡰⠁⣀⠈⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠴⠛⠁⣀⠈⢆⠀⠀⠈⢆⠈⢆
⠀⠀⠀⠀⠀⠀⢰⠃⢰⠃⠀⠀⢰⠁⠀⠙⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠰⠞⠋⠀⠈⡆⠀⠀⠘⡆⠈⡆
⠀⠀⠀⠀⠀⠀⡌⠀⡆⠀⠀⠀⡆⠀⠛⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠰⠖⠛⠀⢰⠀⠀⠀⢱⠀⢡
⠀⠀⠀⠀⠀⢠⠁⢰⠀⠀⠀⢰⠀⠰⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠆⠀⡇⠀⠀⠀⡆⠈⡄
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢸⠀⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⠀⡇⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢸⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡇⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⡀⠸⠀⠀⠀⠀⡆⠈⠉⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠁⢰⠁⠀⠀⠀⠇⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠇⠀⠀⠀⢰⡀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⡆⠀⠀⠀⠸⠀⡸
⠀⠀⠀⠀⠀⠀⠸⡄⠸⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠱⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⡄⠀⠀⠀⠈⠢⡀⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⠀⠀⢀⠔⠁⠀⠀⠀⢠⠊⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢦⡀⠀⠀⠀⠈⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠔⠁⠀⠀⠀⢀⡴⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠙⠦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠋⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠈⠒⢄⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠⠒⠁⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⡰⠶⠶⠶⠶⠆⣶⣶⣶⠶⠄⠀⠀⠀⠀⣠⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⣛⣋⣭⣵⣶⡿⠇⠀⠀⣰⡿⣫⣵⡖⡲⣶⣭⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢟⣛⣭⣵⣶⡾⠿⢟⡻⣿⡅⣶⣾⡇⠀⢸⡟⣼⡟⣭⣶⣶⣮⡻⣷⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢟⣛⣯⣭⡶⠾⠿⠿⣧⣿⣷⢹⣿⡇⠀⢿⡇⡿⢘⣛⣛⣛⣛⣓⢹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣫⣵⣾⠿⠿⢿⣶⣝⢿⡎⣿⡇⠀⠘⣷⡹⣶⣿⣿⣿⣿⣷⠞⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣱⣿⡟⡴⣫⣝⢲⡹⣿⣧⢻⣸⡇⠀⠀⠘⢿⣮⣝⣛⣛⣋⣵⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡏⣧⠻⠿⣹⢇⣿⣿⢸⣧⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡸⣿⣿⣎⢿⡿⣣⣿⣿⡟⣼⣿⡄⠀⠀⠀⠀⠀⠙⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⡻⢿⣷⣴⡿⠟⣫⣾⣧⢻⡇⠀⠀⠀⠀⠀⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣶⣶⢰⣿⣿⣿⣿⡜⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣛⣛⣭⣽⡇⣿⣿⢺⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢟⣛⣭⣽⣿⣦⡱⢿⡿⣿⣿⣿⣿⠇⠀⣀⣀⣀⡀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠿⠿⠶⠶⠆⠀⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⢂⡐⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢈⣌⡠⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠌⡀⢀⠜⠜⠀⠀⠀⠀⠀⠰⠃⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠃⠠⣀⡡⠊⠀⠀⠀⠀⠀⠂⠀⠑⠊⠠⠂⠀⠀⠀⠀⠈⠢⡱⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠘⠀⠤⠂


⠀⠀⠀⠀⠀⢀⡴⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⢀⠥⠀⠆⠠⠅⠀⠀⠀⠀⠀⠰⠌⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⡈⠐⠡⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠃
⠀⠀⠀⠀⢐⡄⢀⣄⠀⡌⠀⠀⠀⠀⠀⢀⣠⣔⣦⡖⠀⠀⠀⠀⠀⠠⢔⠁⠒⡈⠄⠀⠀⠀⠀⠀⠀⣠⠤⢤⡄
⠀⠀⠀⠀⠈⠛⠋⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠃⣶⢲⡄⠂⠀⠀⠀⠀⠀⣾⡆⠉⠁⣿⠆⠀⠀⠀⠀⠀⢗⢄⠠⢪⢆⠀⠀⠀⠀⠀⠀⠀⠛⠘⢆
⠀⠀⠀⠀⠀⠇⠙⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⢰⠐⠙⠀⠀⠀⠀⠀⠀⢀⢕⠕⢈⠁⠀⠀⠀⠀⠀⢖⠑⡢⠀⠚⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⠂⠀⠘⠛⠀⠀⠀⠀⠀⠀⠓⠀⠀⠒⠁⠀⠀⠀⠀⠀⠐⠉⠃⠒⠂⠁


⠀⠀⠀⠀⠀⠠⠒⠘⠢⠀⠀⠀⠀⠀⠀⠒⠀⣀⣠⡀⠀⠀⠀⠀⠀⠠⠽⢄⠀⡸⣀⠀⠀⠀⠀⠀⠠⣄⠓⠚⡦⠄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⠸⠀⢸⠀⠀⠀⠀⠀⠀⠚⡐⠆⠘
⠀⠀⠀⠀⠀⠠⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠃⠙⠛⠇⠀⠀⠀⠀⠀⣄⠀⠈⢰⠊⠀⠀⠀⠀⠀⠤⠚⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⡴⠚⢹⡏⠓⢦⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠙⡏⠉⠀⠀⠀⠀⠉⣹⠋⠉⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢦⣀⣰⡆⣀⡴⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠞⢧⠀⠀⠀⢀⡾⠱⠶⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⣀⣤⠴⠶⠒⠊⢳⡄⣠⠞⠐⠒⠶⠦⣤⣀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠐⢧⣀⡀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⣀⡼⠂⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣈⡉⠙⠛⠛⠒⠒⠒⠒⠛⠛⠋⢉⣁⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⣀⡀
⠀⠀⠀⡤⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣀⠄⠓⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⣀⠀⠀⠲⠆⣈⠁⠀⠁⡠⠁
⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⠤⠀⢂⣀⠩⠓⡄⠀⠀⠼⣀⢤⡀⢇
⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠢⢰⠂⠀⠀⠀⠀⣄⠀⣀⠀⠀⡬⠉⠔⠈⠉⠁⠀⢀⠀⣠⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⢀⡔⠀⠉⠀⢆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠋⢲⠔⠘⠂⠀⠀⠀⣴⡀⢁⣏⠀⠀⢣⡠⠀⠤⡰⠍⠀⠑⢒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠙⠤⡈⣀⣠⣤⠔⠃

⠀⠀⢀⠤⠐⠒⠂⠠⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠔⠁⠀⠠⠒⠈⠉⠈⠁⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠀⠀⠀⣀⡤⡒⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣉⣔⢤⢄⠡⡀
⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡆⡘⠀⠀⠀⢀⠎⢯⠤⠞⠀⠘⠿⠑⡄⠀⠀⠀⠀⠀⡠⠂⣨⠃⢠⠢⠗⡢⠁
⠀⠡⡀⠈⢄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠸⠀⠀⠀⢠⣒⠄⢸⣲⢄⠀⠀⠀⠀⢺⠀⠀⠰⠟⣠⠀⠆⡠⠁
⠀⠀⠈⠂⠀⢢⡀⠀⠋⠀⠀⠰⠂⠀⠀⠀⢠⠀⠀⡰⠀⠸⠀⢀⠀⠀⠸⡼⣂⠀⠀⠀⣀⠀⢁⡎⠀⠀⠀⡠⠁⠀⠀⠀⢀⢀⣄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣼⣡⠃⢠⠇⣴⠏⠀⠀⠀⠙⠢⢄⣐⢣⣴⠷⠋⠀⠀⢀⠌⠀⡀⡤⣀⠔⣿⠂⠈
⠀⢠⢄⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠸⠀⠀⠃⠀⠸⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡃⣬⡨⠊
⠀⣠⠀⢀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢰⠏⠃⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⡠⠋
⠀⠉⠈⠃⠀⠀⢠⣄⡀⣀⠀⠀⠀⢨⠀⢀⢀⣀⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⢜⡀⣁⣏⠀⠀⠀⢸⠋⠀⠀⠀⢣⠀⠎⠀⠀⠀⠀⠀⠀⠀⠰⢀⢠⡀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠤⣄⣴⣖⡉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢘⣄⠀⠀⠀⡜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⢳⡤⣣⠞
⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠈⠪⢁⠀⡩⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠀⣠⢴⣫⡞⠁
⠀⠀⠀⠀⠀⠴⢠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡤⠀⠀⠄⡀⠀⠀⠀⠀⠀⢠⣾⠋⠙⣆⣠⢔⡮⠗⠉⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡎⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢇⠑⢒⡺⠝⠛⠣⠤⠤⠚⠁⠀⠀⡘⠠⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠰⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢀⡀⠤
⠀⠀⠀⠀⠀⠆⠈⢀⠀⠀⠀⠠⢄⡀⠘⠀⠀⠀⠀⠀⢄⠀⢃⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⡘
⠀⠀⠀⠀⠀⠃⠰⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠢⢄⡑⢂⣤⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠔⠂⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠠⠀⠉⠉⠩⠀⠀⠂⠠⠂⠀⢠⢀⡁⠀⠈⡇⠀⢤⣀⡄
⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠈⠲⠚⡈⢀⠀⠀⠀⠆⠐⠈⠂⠀⠄⠀⠐⠿⢄⠀⠀⡇⠀⠀⢁⣠
⠀⠀⠀⠀⠀⠀⠰⢀⡀⠴⠉⢉⡠⠌⠀⡈⠥⠀⠘⠎⠤⠴⠅⠀⠀⠴⢾⡞⣤⠡⠀⠙⠳
⠀⠀⠀⠀⠀⠠⠂⠈⠩⠌⠠⠞⠤⠄⡀⠀⠶⠤⠬⠠⠠⠥⠄⠀⢤⡶⠐⠂⢠⣢⠄⣠⡤⣤⣤⠀⠐⡀
⠀⠀⠀⠀⠁⠀⠁⠈⡠⠤⠄⠀⢈⠃⢩⠈⢈⠀⡀⡉⡀⡀⢉⠀⠘⠁⠘⢠⠯⣎⢇⠩⣅⣤⣄⠀⠸⠀⠠⡅⠘⢶⣥
⠀⠀⠀⠈⡆⠐⠀⠀⡀⠴⡖⠓⠈⠠⣂⠀⢀⠀⠀⣦⡄⠀⠀⠀⠀⢔⡀⡟⢀⡍⣦⠁⠀⠀⢌⠂⠀⢀
⠀⠀⠰⠀⠘⢦⣈⡀⠀⠐⢓⠐⡂⠂⠀⠀⠀⣔⡂⠁⠀⣈⠀⠀⣀⡄⠀⡄⠠⠀⠣⠀⣡⡀⠈⢆⠠⠈⠄⠈⡀
⠀⠀⠀⢐⠀⠄⠀⠠⠸⢉⡹⣿⣉⠀⣨⠉⡍⡅⢠⢈⢻⡉⢀⢀⢉⠉⡭⠠⠆⠀⠀⠐⣰⠔⣸⠆⠉⠁⢀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⢀⢀⠀⠀⠀⡥⠀⠎⠬⠄⢧⠠⡄⠘⠀⠍⠰⡆⠀⠠⢲⠅⠈⢈⣢⣬⡜⠆⡄
⠀⠀⠀⣀⠠⠀⠀⠀⠂⠘⠆⠐⠠⣀⠾⠄⠀⢰⠀⠄⠢⠘⠆⢤⠐⠀⡁⠂⠀⢂⡈⡆⠰⠆⠞⠛⢉⣀⣟⣇
⠀⠀⢰⢣⠈⠆⠀⠘⠀⠀⠀⠀⢀⠇⠀⠀⠀⠛⠓⢆⢀⠀⠀⡸⡤⠈⢀⡠⢤⡈⢷⡀⠠⢨⡐⠴⡉⠀⠉⠉⡆
⠀⠀⡆⠀⠀⣸⠂⢀⢲⠌⠼⡐⢊⠀⠀⠀⠀⠀⠀⠀⠁⠨⠘⠃⠓⢌⣿⡜⠏⡻⠀⢷⡀⠤⠈⠀⠙⠒⣤⠔
⠀⠀⠀⠀⠀⠘⠀⠯⣅⡩⢂⣙⢈⡀⠀⡀⠀⠀⠠⠘⠀⠀⠀⠀⠠⠒⠈⠋⠊⠁⠀⠀⠳⡄⢀⠐⢒⠖⡀⢠
⠀⠀⠀⠐⠀⠀⠀⠘⠆⠈⠼⠏⠙⢃⢶⠗⢻⠂⠀⡀⠢⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣀⠘⠔⡏⡛⢹⢣⠏
⠀⠀⠀⠠⠀⠐⠀⣤⠤⢀⠤⢤⠞⠐⣿⠃⠚⠀⠀⠀⠀⠘⠙⠠⠀⠀⡀⠠⠴⠖⠞⠋⠉⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠉⢧⠴⠃⠈⡀⢄⡠⣏⢠⣀⠀⠀⠀⠠⠀⠒⠑⠀⠀⠀⠠⣄⣘⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠸⡀⠀⠘⠗⠁⠀⠀⠐⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⢠⡆⠁⠀⠒⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠓⠊

⠀⠀⠀⠀⢀⣠⣴⠶⠿⠛⠛⠻⠷⢶⣦⣄
⠀⠀⢀⣴⠟⠉⢀⣠⣤⣤⣶⣤⣤⣀⠀⠙⠿⣦⡀
⠀⣠⡟⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⠙⠳⣦⠀⠙⢷⡄
⢰⡟⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠈⣿⡀
⣿⠃⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢹⡇
⣿⡀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⢹⣇⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⣾⠃
⠈⢿⣄⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⣼⠏
⠀⠀⠻⣧⡀⠀⠈⠙⠛⠿⠶⠿⠛⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠿⠿⠿⠷⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣠⡶⠟⠋⠉⠉⠉⠉⠛⠷⣦⡀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠖⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣶⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⢻⡆⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠈⠛⠷⢶⣦⣤⣤⣶⠾⠛⠉⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆⢠⣤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠛⠐⠶⠶⠆⢠⣄⡈⠻⣦⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢼⡟⠋
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡤⡖⠉⠉⠋⠉⢲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⢰⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢶⣦⣤⣀
⠀⣰⡉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣌⠻⠟⠋⠁⠈⠙⠻⠟⣡⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠈⠙⢿⣦
⢰⠁⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⣠⠖⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠄⠀⠀⢿
⢸⠂⠀⠀⠀⠀⡾⠟⠀⠀⠀⠀⠐⡿⠀⠀⠀⠀⠀⠀⠀⣛⣛⠀⠀⠀⠀⣶⣾⠁⠀⠀⠀⣛⣛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢸
⠘⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠈⠉⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣾
⠀⠈⠢⣀⡄⠀⠀⠀⠀⠠⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠴⠞⠛⠋⠘⠛⠿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢋⡥⠤⠿⠤⢤⡙⢳⣄
⠀⢠⠟⠀⠀⠀⠀⠀⠀⣠⠂⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣀⠄⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⡜⠉⠀⠀⠀⠀⣀⠈⢻⡈
⠀⠏⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣀⣤⡾⠃⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⣇⣼⠀⠀⠀⣴⠶⣾⠏⠀⠀⢧
⠀⠂⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣶⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠸⡏⢻⠀⠀⠀⠻⠶⠟⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢳⡈⢧⠀⠀⠀⠀⠀⠀⠀⡼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠋



⠀⠀⠀⡠⠄⠒⠀⠀⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡙⠟⠋⠁⠈⠙⠻⢫⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆
⢠⠁⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⣠⡴⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⣿⠋⠀⠀⢀
⢀⠀⠀⠀⠀⠀⠿⠿⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢨⣭⣥⠀⠀⠀⠿⠟⠀⠀⠀⣭⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠈
⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠄⣐⠒⠒⠀⠀⢠⡄⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢐⠦⢛⡀⢓⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣘⣁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⢠⢾⡉⠓⡄⢀
⢇⠀⠀⡠⠄⠀⠀⡀⡄⢚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠱⠆⠀⠀⠀⠀⡖⠄⡁⠀⠀⠀⠀⠸⠤⠤⠴⠁⠀⢁
⢸⣷⡞⠀⠀⠂⠑⠁⠀⠀⠀⣈⣉⣀⠖⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡎⠀⠄
⠀⠀⢅⠀⠀⠀⠈⠈⠛⠻⣿⡩⠍⠀⠀⠀⠀⢀⡰⢆⡄⠀⠠⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠒⠒⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⢧⡀
⠀⠀⠀⠀⠀⠀⠃⠤⠖⠒⢦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢹⠀⠀⢠⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣸⠀⠇⠀⠀⠀⠀⠀⠂⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢸⠀⠀⣀⣶⡀⠀⠀⠀⡇
⠀⠨⠀⣠⣤⣤⡁⠀⢀⣤⣤⡀⡀⠀⠀⢀⡴⠀⠀⠙⠃⠀⠈⢀⣄⠀⡀⠀⠀⡀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠉⠿⠉⠀⠀⠀⣇
⠀⠰⠄⠺⣘⡪⠷⠴⢯⣒⡾⠳⠂⠀⠀⠘⠄⠀⠰⣾⡧⠈⠠⠃⣺⡧⠂⠀⠠⠀⠀⠀⠀⠒⠀⡢⣬⡐⡄⠀⣴⣩⡑⡤⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠀⠀⠀⠻⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠓
⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣇⣀
⣄⣀⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⠀⡁⠀⢄⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⡖⠀⠐⠄⣀
⠈⣸⠷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⢌⠁⢈⡇⠀⢀⡎⠉⢱⠄⠀⠆⠀⠀⠀⣠⢐⣛⣛⠁⠀⠖⠁⣀⣈⡀⠈
⠐⠈⠓⠶⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠋⠉⠀⠀⠀⠉⠒⠉⢠⣶⠀⠀⠀⣖⣋⡿⢽⡌⣷⣀⣠⣿⢹⡬⢽⣄
⠀⠀⠀⠀⠀⠀⠑⠠⠀⠉⠉⠈⠉⠗⠂⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠀⠖⠚⢿⡎⠇⠀⠀⠀⠀⠙⠳⠚⠁⠀⠀⠈⠓⠓⠋

⠀⠀⠀⠀⢀⣠⣶⣶⣄⣀⡀⢠⣄⠰⠀⢠⡠⠿⣻⣿⣶⣶⣶⠄
⠀⠀⠀⡾⠿⢿⣿⣿⣿⣿⣿⣮⣿⣵⡭⢻⡆⠀⣸⣿⣿⡿⠋⠀⠀⠀⠁⠀⠀⠠⠂⡀⣠⣾⣷⣤⣤⣤⣴⣤⣤⣤⣶⣦⡤
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠉⣡⡔⠋⠁⢸⠟⠋⠉⠤⠀⠀⢀⣴⣿⣶⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡿⠃
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣗⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠻⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠹
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠛⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠋⠉⠫⠿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢧⢀⡆
⠀⠀⠀⠘⣿⡿⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣤⣤⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠻⢧⣴⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣾⠍⠙⢿⣿⡿⠛⣿⣿⡟⠋⢀
⠀⠀⠀⠀⠀⠀⠙⢇⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣛⠁⠀⠀⠀⢿⠁⠀⠈⡟⠷⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠋⠉⠙⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠁⠀⠀⢶⠀⣴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠌⠈⠀⠀⠙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠦⠴⠶⠾⠿⠷⠦⠶⠾⠿⠿⠶⠶⠶⠦
⠀⡁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠀⢀⣴⡄⣠⣤⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠃⠀⢸⠀⠀⠀⣀⡄⠀⢀⡀⡄⢀⣀⣀⣀⣀⠀⠀⠈⠉⠉⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⢸⠀⠀⠍⡏⢱⣠⣴⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⠄⣠⣄
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⡀⠸⢻⠀⠀⠀⡇⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠀⣠⣶⠷⣾⠿⠻⠿⠄⠘⠛⠛⢻⣿⡿⠿⠿⠿⠿⡿⠿⠶⠼⠷⢤⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠛⢰⡏⠁⠐⣶⣦⣄⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⣠⣤⣤⣌⣛⣛⣀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡧⠴⠶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⣿⢀⣄⠀⢸⡇⠐⠒⣿⣿⡧⠿⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡄⣤⣿⡇⣿⠘⠛⠀⠘⣷⠶⡾⠋⠀⠀⠀⠠⠄⠀⠁⠂⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣿⣧⣻⣶⣶⣶⣶⣶⡞⣰⣿⣿⣷⡀⠀⢰⡟⠛⣿⠛⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣻⡟⣶
⠀⠀⠀⠀⠰⢾⣿⣿⣯⠻⣿⣿⣿⠋⢻⡯⠭⣭⣿⣿⣿⣰⡿⣋⣉⣻⣷⠀⠈⠁⠛⠛⠛⠉⠀⠸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠸⡇⣿⡃⣿⣿⣿⣿⣄⣸⣇⣀⣏⠀⢼⣿⣿⣿⠿⠟⣿⡟⣷⠒⠒⠒⠒⠒⠒⠛⠒⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠻⠷
⠀⠀⠀⠀⠀⠤⠿⠿⣷⣿⣿⣿⣿⣿⣿⣭⣍⣛⣻⣿⣿⣿⣿⣷⣆⠤⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⣿⣿⣿⣿⣿⣿⠿⣡⣦⡇⣾⣤⣤⣤⣤⣤⣤⣤⠄⠀⠐⠀⠒⠒⣶⣶⣶⣶⣶⣶⣿⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠻⢻⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣫⢴⡿⠷⠖⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡶⠾⠷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣧⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⡀⠀⠀⣤⡀⠀⣠⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⠀⠀⠀⠀⢹⣷⣶⣿⡛⠁⠀⢸⣿⣿⣴⡿⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡄⠀⠀⠀⠀⢸⣿⠉⠻⢿⣆⠀⣿⡇⠙⠛⠀⣿⡇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡇⠀⠀⠀⠀⠘⠟⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡖⠒⠛⠻⣦⣄⣹⣆⣀⡀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠾⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠛⠶⠦⣤⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠶⠶⠶⠾⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⠻⢦⣄⡀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡉⠻⢶⣤⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡶⠶⠾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠉⠀⠀⠀⠀⠈⠙⠻⣿⣿⣶⡿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠙⢿⣿⣆⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⢻⣿⣆⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠛⠛⠋⢹⡟⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠋⠉⠙⣿⣿⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⡟⠀⠀⢠⣴⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⠁⠀⠀⣿⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡏⠀⠀⠀⠀⢿⣄⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⡶⣰⠂⣶⠀⣶⢰⠚⠦⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⡿⢇⠀⡿⣶⢻⠈⠳⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠈⡇⠀⠀⠛⠘⠂⠃⠛⠘⠘⠲⠋⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣶⣿⠟⠉⣩⣿⣿⣿⣿
⠀⢀⣠⣴⣾⡿⠟⠋⠁⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉⣠⣴⣿⡿⠛⠁⢸⣿
⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠿⠋⠁⠀⣠⣴⣿⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⢻⣿⠁⠀⣠⣴⣿⡿⠛⠁
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣶⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆⠀⠀⠀⣀⠀⠀⠠⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣍⣘⠲⢶⣁⣟⢃⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣇⡄⠀⡀⠸⢀⣞⢉⣀⠀⠐⠀⢨⣦⠉⢿⣦⣿⣀⣻⠀⢿⡄⢠⠴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣸⣿⡆⠀⠄⠸⣾⣼⣜⣿⣧⠀⠀⠈⢻⠳⠬⢿⣭⠟⠷⠛⠁⡰⠞⣲⡛⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣽⣿⣿⡄⠈⣦⢽⣟⢿⣿⣃⠀⠀⠀⢸⣷⡛⣇⠀⠓⢰⢭⡄⠀⠀⣛⢲⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⡘⠁⠺⠓⣽⣿⣦⡈⠓⠈⠙⠛⠛⠀⣀⠴⠛⠛⠛⢯⡛⠷⣶⢨⡀⢀⠀⢛⡤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠞⣡⠿⠿⠉⢲⠦⠤⠘⠂⠉⠁⠀⠀⠀⠀⠀⠘⠛⠻⣌⠳⡘⡵⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⡷⢶⠞⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠳⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠄⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣦⠀⠀⠀⡄⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣶⠀⠀⣠⠎⠀⠀⠀⡤⣴⡒⢲⣄⠷⠀⠀⠉⠂⠀⢠⡖⣻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠤⣉⡉⠙⠀⠀⠀⠀⠀⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣶⣶⢶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠃⠀⠀⠀⡼⠛⡠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡃⠠⠇⢈⢷⡀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢤⡔⣻⡿⣝⣶⠄⠘⠀⠈⠠⣤⣀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠛⠛⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡁⢨⠌⣯⡙⠒⠠⢶⡄⠀⠈⢻⡁⠰⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⠛⣧⠙⠻⣥⠴⠞⣡⠂⣀⣀⠀⠀⠁⠉⠉⠛⠻⠦⣤⡀⣀⣀⣤⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠃⠀⢀⠁⠖⠒⠎⠠⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡀⢣⠆⣤⡀⠿⠀⠢⠄⠈⠱⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⠆⠁⢠⡏⣰⣶⠦⡴⢤⠦⠀⣤⠀⠀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡄⢸⡇⠠⠄⠈⠙⠄⣀⣈⠟⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⣹⠃⢠⡄⠰⠚⢤⠿⣭⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠰⠦⢀⡀
⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣤⣬⣥⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠀⢉⣳⣿⣷⢦⣄⡤⠤⠤⢤⣀
⠀⠀⠎⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠀⣼⡟⠉⠹⡆⠀⠉⠛⠻⢅⣈⠙⠢⢄⡀
⠀⠀⣦⡏⠀⠀⢀⡟⡿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢠⠀⠈⠐⠢⢤⣀⠀⠈⠉⠒⠶⣤⣷⣦
⠀⠈⠛⠃⠀⠀⢸⡌⡇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢹⣷⡆⠀⠀⠘⡌⠉⠒⡦⠤⣀⠈⠙⠂⠤⡀
⠀⠀⠀⠀⠀⠀⡿⡇⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢸⢿⡇⠀⠀⠀⣧⠀⠀⢸⠀⠈⡇⠑⡖⢤⣬
⠀⠀⠀⠀⠀⢀⣦⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣄⠘⡇⠤⠄⣀⣈⣀⡀⠈⡄⠀⠠⠀⢸⠀⢙⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠙⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⢸⠀⢻⠀⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠈⢤⣿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⣀⠀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠠⠆⣠⡀⠀⠠⠾⣿⡇
⠀⠀⠀⠀⠀⢸⣤⣀⣠⣤⣬⣤⣤⣤⠄⣤⡤⠠⠶⢶⡶⠶⣀⣀⣠⣤⣴⣶⣿⡀⠀⠇⡇⢂⣀⣀⡤⠤⠤⠤⠶⠶⠒⠚⠛⠻⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣔⣏⣿
⠀⠀⠀⠀⠀⢠⣘⣀⣀⠀⠨⣯⣄⠀⠀⠀⠀⠀⣰⣶⣶⠆⠀⢀⣀⣀⣀⣠⡤⢄⠀⠀⣿⣴⣿⣿⡆⠀⠀⢀⡀⠤⠀⠀⣿⡏⠉⠅
⠀⠀⠀⠀⠀⢈⣿⣿⣯⡄⠀⢺⣷⡂⠀⢤⡄⠀⣤⣽⠓⠀⢠⣿⣿⣿⣿⣿⣇⡨⠀⣀⢿⣿⣿⣿⣇⠈⠀⠀⠀⣀⣠⣴⣿⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡅⣀⢒⣶⣲⡶⠶⠶⠶⣶⠶⠶⠂⣐⣒⣒⣒⠒⠒⠒⢂⠀⠁⢸⣿⣿⣿⠏⠀⣤⣤⣼⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠍⠉⣉⣽⣋⣀⣀⣀⠀⠈⠁⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⡏⠀⢀⣵⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣟⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⣀⣀⣦⣀⡠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⣒⡒⠋⠋⣉⣉⣙⣚⠙⣓⣚⠍⣉⣉⣡⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣿⣿⡗⢸⠟⡛⣟⢘⣛⣫⡵⠾⣿⠟⣛⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣿⡇⢼⢸⡇⣿⢸⣿⣿⡄⠭⢾⣇⡿⣿⣦⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢝⣿⣿⣃⣾⣿⣷⣿⠸⡿⢢⣿⣿⣿⣿⡇⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢺⣿⣿⡟⡻⣿⠆⢿⠳⣾⣿⣄⠻⢿⠼⡇⣿⣿⠛⢾⣿⡏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⣿⣿⣯⣧⣿⣤⣾⣔⣻⣼⣷⣼⣴⣶⢇⣿⣿⠷⠞⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣝⣻⣿⡿⠿⠿⠿⠛⠛⠻⠿⠿⢿⣿⣿⣿⣟⠷⢾⣿⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠉⠀⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⢀⠈⠙⠛⢷⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠀⡅⠀⠃⠀⠓⠃⠀⠀⡄⠀⠀⣀⣼⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠠⠈⠀⣀⣀⣀⣀⣀⣀⠀⠀⠐⠀⠀⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣅⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⡟⢿⣿⣿⣿⠏⣄⢻⣿⣿⣿⣿⣿⣭⣟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠉⣾⡿⣛⣿⡉⡇⢘⡧⢻⢹⣿⡿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⢸⣿⣿⠟⣰⣜⢃⣌⡘⢛⡠⣿⠟⣱⡟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⣭⣵⣾⣿⣿⡟⣭⣭⣭⢡⣷⣿⣏⣠⣀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠻⠿⠿⠿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⠀⣀⠀⢀⣔⡁⣸⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⢀⡠⠠⢄⡀
⠀⠀⠀⠀⢠⠉⠈⠀⠀⠀⢀⢍⠱⡀⠈⠉⢳⠀⠀⠀⡋⠀⢀⠤⠬⠤⠤⡀⠈⡇⠀⠀⠀⠀⢠⠀⠀⠀⠈⠑⠢⢈⠆
⠀⠀⠀⣀⠬⢉⡂⠆⠀⢠⠤⣭⠚⠀⡠⠤⠤⠃⠀⠀⣉⠉⢹⠀⣤⠞⢸⠉⠉⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠰⠽⠑⢢
⠀⢘⠉⠁⠀⠀⠘⠦⠔⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡉⠉⠑⠒⠒⠂⠘⠁⠉⡇⠀⠀⠀⡀⢀⡠⠓⠒⠤⣄⣀⠤⠀⠴⣀⣀⡀
⠀⠀⠁⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣠⠤⠀⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠎⡠⠤⠈⠄
⠀⠀⠀⢀⠁⠀⠀⠀⠀⢸⠈⠊⡊⢂⠀⠀⠀⠀⠀⠀⢎⠉⠁⢺⠧⠀⠀⢀⡨⠂⠀⠀⠀⠀⠀⠀⡐⠘⠄⠠⠀⠣⠤⠁⡇
⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⡄⠁⠁⢸⠀⠀⠀⠀⠀⠀⠀⠑⠤⣌⢸⢅⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⡠⢀⠀⡇
⠀⠀⠀⠈⠉⡟⠛⠟⠛⠛⢲⠒⠲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠂⢲⠤⢎⠀⢀⠟
⠀⠀⠀⢂⣀⡈⠋⠀⠀⡀⠂⠛⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡟⠃⡆⠀⡟⠋⠖⠒⡄
⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉


⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⢰⠁⠆⠆⡔⠒⠀⠒⡆⣴⠉⢪⢂⠀⠀⠀⠀⠀⠀⠀⡐⡟⠁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⠿⠓⠢⡰⠒⠒⡕⡄
⠀⠀⢠⠀⠑⠁⠈⠊⠀⠑⠁⠈⠂⠊⢸⠀⠀⠀⠀⠀⠀⣔⡀⢇⣀⣘⠀⠈⡆⠀⠀⠀⠀⠀⠀⡌⠀⡘⠂⠐⠊⠐⠂⣒⠁⡀
⠀⠀⠈⠂⢼⣀⡷⠀⠀⠀⠀⢎⣀⡗⠊⠀⠀⠀⠀⠀⠀⣇⣈⠖⠀⠐⣖⣒⠇⠀⠀⠀⠀⠀⠀⠈⢏⣀⡧⠤⠤⠤⢎⡀⡙
⠀⠀⠈⠭⠉⠍⠉⠭⠭⠭⠉⠩⠭⠭⠉⠀⠀⠀⠀⠈⠩⠭⠉⠉⠭⠭⠍⠉⠭⠍⠀⠀⠀⠀⠈⠩⠭⠍⠉⠩⠭⠉⠉⠭⠭⠍
⠀⠀⠉⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠐⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⢀⠙⠿⠋⠀⠀⠀⠀⠀⣤⣦⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠡⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠐⠠⠄⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⠃⠀⠘⣿⣿⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⠿⣷⣦⡙⢿⣿⣿⣿⣇
⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⡄⢿⣿⣿⣿
⠀⢸⡿⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⡆
⠀⢸⡇⠀⠈⠊⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡆⢹⣿⣿⣷
⢂⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠘⠋⠁⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠐⢴⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠄⠀⠀⠀⠀⠀⡀⠄⡀⠅⠀⠈⠁⠀⠀⠘⢻⣟⢏⠝⣓⡢⡤⠴⢒⡊⠭⣥⣼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠠⠄⡐⠁⠀⠀⠀⣀⠀⡤⠢⡀⠀⠀⢀⣀⢀⣠⣤⣿⢶⢻⠛⡈⡀⣰⡒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⠀⠀⠀⠀⠈⠂⠀⡁⠘⠂⠒⠒⢂⣭⡶⠋⠼⢿⣫⠖⠚⠓⠋⠙⢲⣌⡎⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⡐⠠⠂⠴⡂⠂⡀⠀⠀⠈⢀⣴⠟⠁⢀⡠⠔⠂⢉⣀⣀⣀⣀⡀⠀⠙⢿⣷⣼⡆⠀⠈⠈
⠀⠀⠀⠀⠀⣀⠊⠡⠤⣀⡀⣀⠀⠂⢀⠈⠁⠀⠉⠀⠄⠀⠀⠀⡔⠁⠀⢀⡔⠩⣴⣚⢋⠋⠡⠝⠅⢀⠉⠆⠉⠀⠹⣯⡇
⠀⠀⠀⠀⠈⠀⠤⠤⠤⠤⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⢠⠚⠀⠀⠈⠂⠀⠈⠀⢀⠭⠀⢀⡈⠅⠀⠀⠉⠁⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⢀⠀⠘⠎⠀⢀⡋⠡⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢴⢀⠀⠈⡟⢳⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡦⣾⣉⣉⣑⡄⣧⡀⠡⠀⣐⢑⣀⠈⠶⠰⠤⢄⡀⠀⠀⠀⠀⠀⠘⠠⠀⠉⠀⢆⡑⢘⠁
⠀⠀⠀⠀⡠⢀⠶⡋⠙⠛⠛⠙⠛⠋⠉⠛⠛⠀⠀⠈⠛⠛⠊⠰⢩⡜⠾⢯⠯⢿⠮⣐⢄⣃⠀⠀⠺⠇⠀⠈⠲⠐⠀⠂⠂⡰⣣⢸
⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⡆⡁⠀⠆⠙⠗⠀⠀⠀⠀⠀⠀⠀⠸⢠⡀⡁⡵⡢⠃⠸⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⡨⢸⡄⠀⠀⠀⠀⣠⢤⣠⠻⡇⣰⠅⠀⢡⣶⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢊⢂⠌⠔⠀⢸⠀⡁⠀⢠⢸⡛⠀⠀⠄⡊⢐⠡⢂⠀⡄⠁⢯⡭⠈⡛
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⡡⠡⠀⢀⢂⢈⠈⠄⢸⠀⠇⠀⠂⢨⡗⠀⢂⠅⡂⠅⠪⡂⠁⡘⣏⠘⣥⢉⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⡐⠠⠁⠁⠀⠂⠂⠊⠀⢸⠐⠀⡀⡌⢸⡀⠁⡂⠉⠒⠄⡪⠠⠀⡅⢹⠑⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠐⠄⠈⠔⡐⠁⠁⠀⢀⠂⠀⢄⠄⠈⠘⠂⠃⠃⣰⣤⣉⣁⣉⣀⠃⢊⡘⡡⠃⣸⠞⠁
⠀⠀⠀⠰⣡⡀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠌⡔⡐⠡⠀⠁⠠⢂⠀⠀⠁⡀⣸⠰⡰⢀⠟⠪⣁⣭⠩⣉⠁⠉⣠⣺⢞⡁⠠⠀⢀
⠀⠀⠀⠀⠓⠛⠓⠒⠒⠲⠦⠦⡤⣤⣤⢀⣀⢀⣀⣀⠀⠁⠁⠀⠂⢁⡔⢹⢀⡘⣵⣷⣞⡻⢷⡖⠲⢖⡛⠋⣩⠠⠐⡠⢤⣔⠦⠴⢛⠓⠒⣒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠉⠉⠑⠓⠚⠓⠿⠓⠚⠋⡉⠁⠠⠄⠂⠀⠊⠀⠀⠃⠀⠈⠀⠀⠁⠂⠀⠁⠂⠈⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠋⠉⠈⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⠇⣿⣿⣿⣿⣿⣿⡇⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⣿⣿⣿⣿⣿⣿⡇⢿⣿⣷⡀
⠀⠀⠀⠀⢿⣿⠏⠀⣿⣿⣿⣿⣿⣿⡇⠈⢿⣿⠇⠀⠀⠀⠐⢾⠀⢀⣟⡛⢀⣄⣤⣠⣄⠀⠀⠀⠀⢿⣿⠏⠀⣿⣿⣿⣿⣿⣿⡇⠈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⣩⡿⢸⡇⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢿⣿⠇






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠕⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⡀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡
⠀⠀⠉⠐⣆⠄⠀⠀⠀⠀⠀⠂⠂⠂⠀⠈⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢃
⠀⠀⠀⢠⠇⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠁⠀⡀⠔⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠲⠶⠖⠁⠁⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠁⡀⠀⠊⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠁⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢋⡁⠧⠔⠲⠯⠃⠠⠀⠀⠃⠀⠀⠠⠂⠁⠂⠀⢀⠀⠀⠀⠂⣀⠀⢀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠓⠉⠁⣀⣤⣄⡀⡡⠔⠒⠀⠂⠀⠰⢁⠀⠀⠀⠀⠀⢐⢬⣀⢐⡃⠁⠀⠀⠄⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢦⡞⠉⠀⠀⠙⣧⡀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠀⠀⠠⠄⣈⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣸⠇⠀⠀⢀⡂⠀⠓⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠐⣻⢦⣤⣤⢴⠋⠃⠈⠁⠀⡁⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠑⠒⢺⠉⠉⠉⠐⠊⠀⠀⠀⠒⠊⠁⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⢰⠀⠈⠁⢙⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢀⡠⠖⠒⠓⠓⠚⠢⢄⡀⠀⠁⢀⡤⠛⠒⠚⠓⠒⠢⣤⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⢫⡤⢬⠃⠀⠀⠀⠀⠀⠀⠀⠰⠋⡀⠀⠀⠀⠀⠑⠀⠃
⠀⠀⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⢀⡷⠒⠒⢧⠀⠀⠀⠀⠀⠀⠀⢐⠌⠡⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⡤⠔⠛⠁⠀⠀⠉⠓⠤⣄⣀⣀⣤⠔⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣄⠀⠀⣀⣠⣤⣶⣶⣶⣦⣤⣀⠀⠀⣠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢋⣴⣾⡿⢛⣋⣭⡉⣭⣉⡛⠿⣿⣦⡙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣰⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣷⣬⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⢰⣿⠏⣼⣿⣿⣿⣿⣿⣿⡿⢋⠅⣸⣿⣷⡘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣿⡇⣾⣿⠸⢿⣿⣿⣿⣿⠟⣡⡾⢋⣾⣿⣿⣿⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⢃⣿⡇⣿⣿⢰⣶⣿⣿⣿⣿⣦⣉⣵⣿⣿⣿⣿⣷⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠠⣶⣶⣶⣶⣶⣶⣶⣶⣦⡐⣶⣾⣿⣿⣧⢹⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣿⣿⣿⣿⣿⣆⠻⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣡⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣶⣴⣤⣦⣴⣦⣴⣶⣶⣦⣴⣦⣤⣬⡙⣿⣧⡙⢿⣷⣮⣙⡛⠿⠀⠿⠛⣋⣥⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣶⣌⡛⠻⠿⣿⣿⣿⠿⠿⢛⣩⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣭⣭⣭⣭⣭⡙⢿⣷⣶⣶⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣉⣉⣉⣉⣉⣉⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢸⣿⣿⠟⠁⣨⣿⣿⠁⠘⣿⣿⣿⠃⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠸⠋⠁⣠⣾⣿⣿⣿⠀⠀⢹⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢀⡀⠘⢿⣿⣿⣿⣿⠀⠀⠀⠟⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣦⡀⠙⢿⣿⣿⠀⢸⡄⠀⢰⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣼⣿⣿⣿⣦⣴⣿⣿⣦⣼⣷⣤⣿⣧⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⡄⣀⣀⡀⡠⠤⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠄⠈⢉⠇⡃⠈⠉⠁⠀⠒⢒⢒⡗⠢⠤⠤⣤⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠤⠾⠓⠒⠚⠽⠒⠛⠒⠓⡟⠫⡀⠉⠀⠙⢷⠈⢹⡌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⠀⠉⠉⠤⣠⡽⠟⠂⠀⣀⣀⣀⠀⠠⢤⠀⢢⣦⡶⣿⣶⠀⠶⢾⡟⢻⠇⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢠⢐⠐⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⡕⢺⣿⠀⢠⣠⡀⠴⠟⠛⠹⢻⠂⠀⢸⠇⢺⡠⠊
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠠⠄⣋⠀⠀⠀⠠⣀⣤⡽⠇⣈⡃⠈⠁⠀⠀⠀⠠⠀⢰⢾⠀⢀⣿⡴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣉⣱⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣟⠕⠁
⠀⠀⠀⠀⠀⠀⠸⢶⡼⠂⠀⠀⠀⠀⠀⠀⡆⡉⡦⠀⢀⢀⠀⢀⣦⣾⡿⠿⠛⠟⠃
⠀⠀⠀⠀⠀⠀⡠⠁⡒⠂⢀⣀⠠⠤⠄⠀⠈⠻⠉⠀⢨⣴⣿⣿⣿⠿⡗⠒⠂
⠀⠀⠀⠀⠀⠀⠑⢲⣧⡤⣟⣛⣿⣿⣄⠀⠠⢙⣀⣠⣯⣿⠿⠋⠁⠁
⠀⠀⠀⠀⠀⠀⢀⡩⠗⠋⠉⠉⠉⠒⠚⠒⠻⠿⢟⣭⣛⠁
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊

⠀⠀⠀⣠⣶⠿⠿⠿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⣼⡟⠁⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠶⠶⢦⡀⢸⡇
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠈⡷⢸⡇
⠀⠀⢻⣧⡀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣤⣤⠞⠁⢸⡇
⠀⠀⠀⠙⠻⣷⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⠀⠀⠘⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣿⣛⣛⣿⣛⣛⣿⣛⣻⣿⣛⡃⠀⠀⠀⠀⠀⣿⣤⣤⡤⢤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠷⠾⠿⠿⠿⠃

⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣚⣓
⣶⠓⠓⠛⠚⠛⡻⡟⠛⠚⠛⠛⢻⡆⠀⠀⠀⣿⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣛⣫
⣿⠀⠀⠀⠀⠘⣄⡜⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣠⡴⠶⣦⣄⠀⠀⠀⠀⣠⣴⠶⢦⣄⠀⠀⠀⠀⠀⠘⠃
⣿⠀⣠⠞⠀⠀⢸⡀⠀⠈⢦⡀⢸⡇⠀⠀⢻⣭⡿⠰⠂⠦⣀⣼⠉⠀⠀⠀⢹⡇⠀⠀⢰⡏⣠⣶⣶⠘⣧⠀⠀⠀⠀⢘⡁
⣿⣾⣥⣤⣤⣤⡾⣧⣤⣤⣤⣽⣾⡇⠀⠀⠀⠀⢀⡀⣄⠴⠉⢻⣄⠀⠀⠀⣸⠇⠀⠀⠸⣇⠸⠿⠿⣠⡟⢀⡤⠶⠰⠊
⠿⠿⠿⠿⢿⣿⣦⣾⣿⠿⠿⠿⠿⠇⠀⠀⠀⣒⠊⠁⠀⠀⠀⠀⠙⢻⣶⡟⠉⠀⠀⠀⠀⠈⠻⣶⡟⠋⠀⠀⠓⠶⠰⠆⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠠⠆⠤⠠⠄⠤⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠡⠄⠤⠠⠄⠤⢠⠄⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣒⠲⡄⠀⠀⡰⠒⠲⡀⠀⢠⠒⠒⡄⠀⢠⠖⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣛⣡⠇⠀⠀⠑⢤⠞⠀⠀⠘⢦⡔⠁⠀⠈⠳⡴⠃⠀⠀⢀⡴⠶⠶⣦⡀⠀⠀⠀⠀⣶⣶⡶
⠀⣠⣤⣦⣤⡀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⢠⠄⠀⠀⠀⠠⡄⠀⠀⠀⣾⠁⠀⠀⠈⡷⠀⠀⠀⠀⣿⠛⠛
⣼⢋⣤⣤⣌⢻⡆⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠰⠂⠀⠀⠀⠐⡇⠀⠀⠀⠘⠷⣤⣤⠾⠃⠀⠀⠀⢀⣿⡀
⢿⡘⠿⠿⠿⣸⠇⢠⡤⠆⠶⠉⠀⠀⠀⠀⠀⠘⣤⢀⣀⡞⠉⠉⢳⢀⡀⣬⠃⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⢿⣭⡟
⠀⠙⢷⣶⠞⠋⠀⠰⠆⠦⠴⢄⡀⠀⠀⠀⠀⠀⠀⠈⠈⢧⣀⣀⡼⠈⠁⠀⠀⠀⠀⠀⠀⠀⠴⠦⠐⠐⠂⠒⠐⠠⠶⠆
⠀⠀⠈⡟⣤⣀⣄⣠⣀⣄⣤⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠾⠿⠷⢶⣦⣄⡀
⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣆⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡄
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡜⠁⣀⣀⣀⣀⣀⣀⣴⣶⣆⠀⠀⠀⠀⢀⡀⢘⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⠆⢀⡀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⣀⡀⠰⣿
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠘⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁
⠀⠀⠻⣿⣿⣿⣷⣿⣿⣿⣾⣷⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⣤⣤⣤⣤⣤⣬⣹⣿⣿⣿⣿⣿⣏⣩⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣍⣷⣶⣶⣶⣶⣶⣏⣥⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⢇⣴⣿⣿⣿⡟⣛⣻⣿⢿⣿⡿⠥⠁⠀⠠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⣿⣿⣿⠏⢼⣿⣿⡟⠛⠈⣉⠉⢉⠀⠈⢀⣴⣦⣀⣤⡀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣴⣶⣶⣌⢧⣈⠻⢁⡁⣀⣀⣀⣀⡀⠀⠀⠘⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣿⠿⠋⢠⡿⣀⢺⠛⣿⣿⣿⣿⣿⣿⠀⠀⠠⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠧⢠⡇⣤⣀⣴⣾⠿⠸⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠰⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠙⢿⣷⣶⠶⠚⠻⢿⠿⢬⣽⣧⣀⠀⠀⠀⠹⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠋⡤⢈⡿⠁⠠⣤⢀⠁⠀⠀⢈⢙⠻⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡇⡽⠟⠁⠐⠀⠀⠠⠀⠀⠀⠀⠘⢃⠂⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡎⠃⣀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠐⠀⠀⡀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⡇⠀⣿⡗⠀⠀⠀⠓⢦⣄⣤⠀⢴⣦⡄⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⣿⡧⢦⣠⣀⣀⠀⠀⠀⢰⣾⣿⠙⢀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣴⣿⣷⣶⣿⣯⠀⡷⠶⣾⣿⣶⣿⡖⢀⢾⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡿⢾⣷⠽⠟⣿⡿⣿⠟⠒⠒⣿⣿⣿⣿⣿⠐⠍⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢰⣶⣶⠀⣌⢑⣯⠁⢀⡈⠉⠉⠛⠃⠿⢀⣄⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣀⣉⠁⠜⠉⠈⠁⠀⠈⠉⣗⠀⠀⠀⢀⡽⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣶⣶⡦⠄⠀⠀⠰⢶⣿⠿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾⣮⣿⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⢸⠿⣇⠀⢸⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⡿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣾⣽⣟⡙
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡿⣿⡿⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣟⣿⣟⣿⣯⣯⣿⣯⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣿⣿⣿⣻⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⣿⣿⣷⣿⣿⢿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠷⠦⠸⠿⠾⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡶⢶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣰⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣖⣖⣶⣶⢼⣿⣿⣿⣻⣿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣷⣿⣿⣷⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣟⣿⠿⡻⢿⣇⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿⣿⣿⣿⣿⣿⣷⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣽⣽⣿⣿⣿⣿⣿⣿⣟⡟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣯⣯⣿⣏⣿⣿⣯⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣄⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛



⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣛⣛⣛⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⢀⣴⣿⠟⠁⢸⣿⣧⠀⢠⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣟⠁⠀⠀⢸⣿⣿⣆⣾⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠙⢿⣷⣄⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠻⠟⠀⠀⠙⠿⠃⠘⠿⠋⠿⠏⠻⠟⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢃⠑⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⢠⣾⣿⣿⣿⠾⡄⠰
⠀⠀⠀⠘⢄⣼⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⣿⣿⠟⠉⠁⠀⣷⠒⠓⡄
⠀⠀⠀⣠⣶⣿⠿⠢⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡄⠀⠀⠹⣿⣄⡀⠀⢀⡍⠀⠀⢰
⠀⠀⠀⣿⣿⣇⠀⠀⠀⠱⡄⠉⠒⣄⠀⠀⠀⠀⠀⠀⡆⠰⡀⣀⣀⠈⢿⣿⠏⠀⣉⠏⠒⣼
⠀⠀⠀⠈⠻⣷⣄⣀⡀⠤⠦⠔⠒⠈⠧⣀⠀⠀⠀⠀⠰⡰⠉⠀⠀⠀⢀⠎⠉⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠠⠃⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠣⠤⠄⠒⢩⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠄⠐⢲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠱⢮⠀⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⢹⠤⡤⠔⠓⣄⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠡⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠑⠄⣀⠄⠊⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡠⢦⠀⠀⠀⢰⠀⠀⢠⠃⠀⠀⢀⠔⠲⢄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⡜⠀⠀⡠⠊⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠑⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠊⢉⠂⠀⠀⡘⠀⢠⠞⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡰⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠩⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠲


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⣨⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⡀⢠⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠁⢠⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢉⠂⠠⢬⠀⠀⢠⡾⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠀⠀⠀⠘⠣⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢠⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⢠⠀⠂⢳⣄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠐⣿⣷⣤⣿⣏⣤⣤⣌⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠙⢿⣿⣿⣿⣿⣿⣿⠈⠀⠈⠳⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠈⠉⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠴⠢⣄⡀⠈⠀⠀⠀⢀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⣿⣿⣿⣷⣶⣀⠢⠄⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠂⠸⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢬⠉⠙⠒⠀⠀⠐⡆⠀⠐⠊⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⠀⢀⣀⣀⠀⠀⢇⣀⣀⣀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠒⠊⠉⢀⠚⠛⠉⠀⠀⠀⡈⠋⠋⠉⠉⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢄⣀⠀⠀⢠⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠨⢆⠈⠀⠀⡸⠐⠒⠒⠒⠒⠒⠲⡀⠄⠀⠀⠄⠀⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢰⡐⠄⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠒⠒⠈⣀⠠⠘⠋⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠔⠈⠀⠉⡠⠂⠀⠀⠀⠀⠀⠠⠐⠀⡡⠔⠋⠙⠉⠻⠲⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠀⠀⠌⠀⠀⠀⠄⠈⢢⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⢰⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣠⠀⣤⣤⣶⣆⣀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡲⠄⢀⣀⡀⠉⠁⡒⠒⠒⠂⠤⠀⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠧⡌⣯⠁⣹⠽⢿⣷⣧⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠁⠀⠀⣕⠂⡤⢠⠄⠀⠤⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠃⣿⣷⣸⡦⠀⣿⠁⠀⠀⠁⠐⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⢀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡀⡀⠘⣆⣀⣀⣽⣿⣿⣷⣄⠀⢈⠂⠄⡀⠀⠀⠈⢄⢀⡴⠃⠀⠀⠀⠀⠀⢆⠠⠺⢉⡗⢒⠤⣠⢲
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠠⠀⠀⠒⠁⠀⠀⠐⡄⠰⢊⡴⢁⡞⠀⣠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠃⠈⠀⠀⠀⢠⣄⠀⢀⡜⠵⣋⣴⢏⣠⠞⠀⠏
⠀⠀⠀⠀⠀⠀⡘⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠄⠀⠀⠀⠀⠀⢈⣕⡶⢭⡋⢥⣠⠯⣣⡀
⠀⠀⠀⠀⠀⠀⠠⠰⠄⠤⠠⣤⠚⠓⢲⡴⠒⠒⢦⡀⣀⣀⣀⣀⣀⣇⠘⠁⠀⠀⠀⠀⠀⠀⠉⠐⠠⣈⡑⠪⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠙⠷⠔⠚⠛⠢⠴⠟⠉⠛⠃⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠀⠀⠀⠀⠑⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⡀⠤⠀⠀⠀⠀⠠⠤⠄⠤⠤⠤⠐⠠⠤⠴⠼⠤⣤⣴⣶⣦⣉⡻⠉⠀⠀⢀⢨⣽⣿⣉⣰⣤⡄⠀⠐⠦⠀⠀⠒⠦⣤⠤⢤⣤⣤⣤⡤⣄⡀⢠
⢶⣓⢄⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⣿⣿⣽⣾⡟⣓⣓⣺⣟⣂⢐⢶⡦⠭⠀⠀⠀⠀⠀⣀⣿⠿⣟⠻⠿⠟⠛⠒⠂⠀⢴
⠀⠁⠀⠉⠉⠉⠉⠉⠁⠙⠛⠒⠀⠠⠔⠓⠁⠀⠼⠛⠛⠋⠉⠛⠉⠉⠙⠛⠿⣯⣛⣿⣿⠛⠏⠹⠿⠦⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣠⡀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠙⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣠⣤⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣈⣁⣤⣤⣄⣀⣤⣤⣄⣀⣠⣤⣤⣄⣀⣀⣠⣤⣀⣀⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣠⡴⠋⠁⠀⣠⡴⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀⠀⠈⠉⠓⢦⣀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⣠⠶⠋⠁⠀⠀⠀⣀⣠⣤⡤⠤⠤⢤⣤⣄⣀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣠⠞⠁⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀⠀⠀⠈⠳⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣠⠞⠉⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠙⢶⡄⠀⠀⠘⢦⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢀⡼⢃⡀⠀⠀⠀⠀⠀⠙⠛⠓⠶⢤⡉⠛⢦⡀⠉⠙⢦⠀⠀⠙⢦⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⡞⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣀⣈⣧⣤⣤⣼⣇⠀⠀⠈⢷⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣾⠁⡟⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢉⡽⠋⠀⠀⣠⠴⠛⠗⣿⠀⠀⠀⠈⣧⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢰⡇⠀⠁⠀⠀⠀⣀⣴⣚⣁⣀⣠⣴⣯⡤⠤⢴⢺⠵⠛⠁⠀⢻⠀⠀⠀⠀⢸⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⢠⡇⠀⠀⠀⡏⣭⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠀⠀⠀⠀⢸⠇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣧⠀⢷⠀⠀⠀⢹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠞⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠸⣆⠈⢧⡀⠀⢸⡄⠃⠀⠀⢀⣀⣀⣀⣀⣀⣷⠊⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⣆⠀⠹⣆⡈⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣰⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠳⣄⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣴⠃⣼⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠳⣄⠀⠀⠈⠳⢦⣀⠈⠉⠓⠲⠖⠀⠀⠶⠖⠂⠀⠀⣠⡴⠛⠁⠀⠀⣠⠞⣧⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢷⡀⠀⠙⠳⣄⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣤⠤⠤⠶⠚⠋⠁⠀⠀⢀⣠⠞⠁⢀⡟⠀⢿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢦⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⣠⠾⢇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠳⠄⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠶⠶⠶⠒⠛⠉⠙⠲⠶⠶⢶⡚⠁⠀⠈⢳⡀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⣆⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⢧⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡌⠳⡄⠹⣄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⢧⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⡀⢀⣀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠤⠀⣀⠄⠂⡆⠀⠀⠀⠀⠔⠛⠉⠛⠢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢐⠌⠉⢈⠁⠔⡀⠀⠀⠀⠀⠀⠙⠒⢠⠔⠁⡠⡊⠀⠀⠀⠀⠐⠨⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠩⢉⠩⠋⠍⢂
⠀⠀⠈⠀⠀⠀⠀⠈⠑⠂⣤⠀⠀⠰⢒⡠⠀⡠⠊⠡⠡⠀⠀⠀⠀⠀⠑⣴⣀⣤⠆⠀⠀⠀⠀⣿⠄⠠⠁⠠⠀⠀⠀⡁
⠀⠀⠓⢅⠎⠦⠤⠴⡰⡱⠉⠀⠀⠀⠨⠴⡈⡀⠀⢧⢘⠁⠀⠀⠀⠀⠀⢸⠛⡃⠀⠀⠀⠀⠀⣨⣤⡤⡤⠀⢤⢤⢤⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠈⠑⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⠀⠀⠀⢀⠠⠤⠤⠤⠤⠄⡀⠀⠀⠀⠀⠀⢀⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡋
⠀⠀⠴⠒⠞⢋⠉⢡⠷⠠⡀⠀⠀⠃⠄⠀⢀⠰⠀⡆⠀⠀⠀⠀⢀⠠⠟⠿⠿⠤⣒⠀⡰⠀⠀⢀⢀⡌⠖⡀⢁⠈⡐⢀
⠀⠀⢇⠈⢐⠛⠋⠠⡀⠀⡘⠀⠀⡧⠥⠤⠥⠔⠀⡧⠤⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠛⠤⡘⠊⠓⠉⠚⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⠵⠀⠈⠀⠑⠓⠖⠀⠀⠀⠈⠁⠈⠀⠉⠈⠁⠈⠀⠀⠀⠀⠐⠁⠉⠉⠉⠉

⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡅⠀⢸⠀⠀⠛⢯⠇⠀⠀⠀⠀⠀⣿⠰⠉⠂⣆⠀⠀⠀⠀⢠⣄⢀⣤⡀⢸⢀⠀⠀⠀⠀⣏⠀⠀⠁⠀⠾⠇⠀⡁
⠀⠀⠁⠒⣚⡚⢂⣀⠨⠤⠀⠀⠀⢀⡉⣀⡆⠀⠀⡇⢑⡀⠀⠀⡀⡩⠨⡉⠉⢉⠀⢀⢿⠀⠀⠀⠀⠀⢂⠤⠠⣈⠀⣰
⠀⠀⠀⠃⣒⠐⠐⠂⠃⣛⠀⠀⠀⠈⠁⠙⢦⢀⠤⠋⠉⠀⠀⠀⠈⠁⠁⠉⠛⠉⠈⠐⠀⠀⠀⠀⠀⠀⢈⣓⣄⢠⣘⡡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆
⠀⠀⡄⠀⠀⠀⠀⠀⠒⠒⠒⠂⢄⠀⠀⠀⠀⢠⠤⠤⠤⠾⠭⠽⠦⣖⠲⢤⡀⠀⠀⠀⠀⠛⡉⠉⢉⠛⠀⢶⢲
⠀⠀⣏⠉⣭⠉⣍⠉⣭⡋⢩⡋⡍⡀⠀⠀⠀⠮⠥⠕⠥⠴⠪⠨⠎⠉⠱⡉⣣⠀⠀⠀⠀⢀⣂⣒⣐⣐⣀⣘⣘⣰⠄
⠀⠀⣃⡀⣀⡄⡀⠤⠤⠤⠤⢄⣠⢈⡀⠀⠀⢅⠉⡩⣅⠉⠉⠉⢀⢀⢀⣠⢄⡀⠀⠀⢸⡏⢠⣾⡄⢀⡀⢀⡀⢨⣳⡀
⠀⠀⠀⠁⠑⠑⠁⠀⠀⠀⠀⠈⠛⠚⠁⠀⠀⠈⠈⠓⠛⠁⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠈⠈⠛⠋⠙⠛⠙⠋⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣉⡉⢉⣉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⢋⡴⢦⡙⣧⣀⣀
⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠉⠉⠉⠈⠻⠟⠁⠉⠉⠉⣿⣀⣀⣀⡀
⣿⣿⣉⣁⣈⣉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠉⠉⠉⢹⡆
⣿⣿⠋⢉⣉⣉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⡄⢸⡇
⣿⣿⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⢸⡇
⣿⣿⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣠⠞⢋⣉⣉⠙⢷⡄⠀⠀⠀⠀⣿⢸⡇
⣿⣿⣄⣈⣉⣉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢠⡏⢠⡟⠉⠉⣷⠀⣿⠀⠀⠀⠀⣿⢸⡇
⣿⣿⠋⣉⣉⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠈⢷⡈⠻⢦⡴⠟⢠⡿⠀⠀⠀⠀⣿⢸⡇
⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠈⢷⡄⠀⠀⣠⠟⠀⠀⠀⠀⠀⣿⢸⡇
⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠻⣆⣰⠏⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣿⣿⣄⣉⣉⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣿⡟⠉⢻⣿⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⢸⡇
⣿⡇⠀⠀⢿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣿⢸⡇
⣿⡇⠈⣧⠈⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠓⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠾⠃⢸⡇
⣿⣯⣠⣿⣆⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⠔⠉⠀⢸⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣀⠸⠎⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⣶⠏⠒⠀⠀⠀⠀⣄⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡡⠔⠒⠀⢲⠲⡀⠀⠀⠀⠸⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⠀⢈⠀⠱⡦⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠲⡀⠀⠀⡍⠙⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠤⠴⠒⠈⠁⡧⠤⠊⠀⠀⠀⠀⠑⠤⠤⠁⠀⢡⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⡀⠈⢆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠇⠀⠀⠑⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣂⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠘⡀⠀⠀⠀⠀⣠⠎⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡉⠀⠀⠀⠃⠀⢀⡠⠊⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠓⢄⠀⢰⠤⠄⢀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠱⡀⠀⠀⠸⠖⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢈⡅⡆⠀⠀⠈⠁⠐⠒⠂⠀⡴⠋⠀⠀⠀⠱⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠢⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠈⠛⠓⠀⠀⠠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠒⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠎⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠤⠤⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠳⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡌
⠀⠀⠀⠀⠀⡰⠁⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⡸
⠀⠀⠀⢀⡰⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠘⠛⠹⡀
⠀⠀⣴⢉⣧⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡀⠀⠀⠀⠘⢆
⡴⣏⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⢋⡀⢂⣀⠀⠑⢦
⠑⠪⡓⢄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠳⠶⠋
⠀⠀⠈⠲⣷⣄⠈⢦
⣀⣀⣀⣀⣀⣉⣳⣦⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⣾⠛⠒⠒⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣤⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⠀⠀⠀⠀⣿⣀⣴⡾⠀⢰⣿⣆⣴⣿⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡄⠀⠀⠀⣿⡟⠻⣷⡄⢼⡏⠿⠋⢸⣧⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⡇⠀⠀⠀⠻⠃⠀⠈⠁⠈⠁⠀⠀⠈⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣒⠛⠙⠷⣤⣼⣄⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠟⠓⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠲⢤⡤⠭⠥⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠀⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠄⠀⢀

⠀⠀⠀⠀⠀⠀⣤⠄⣀⣀⣀⣀⡀⣀⠀⣠⠄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠛⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⢸⠀⣿⢦⡾⣇⠀⣧⢀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⢹⠀⠀⢻⡆⣿⠉
⠀⠀⠀⠀⠀⠐⠛⠗⠀⠀⠉⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠘⠃⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠕⠈⠉⠉⠓⢌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡤⠤⡤⠤⡮⠀⠀⠀⠠⠀⠬⣮⣇⣤⡤⢄⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠔⠁⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢮⠥⡽⠀⠀⠀⠀⠀⠀⠀⠀⠮⡅⡽⠂⠈⠢⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⣄⢾⣼⡷⠀⠀⠀⠀⠀⠀⠀⠰⣞⣐⡳⠀⠠⠔⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⡎⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⡰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠛⠛⠒⠒⠒⠂⠀⠀⠀⠐⠓⠓⠒⠓⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⣿⠙⢱⡎⢹⡆⠀⠀⠀⠀⠐⠒⡶⠒⠂⢶⢲⠀⠀⢀⣶⡀⢰⡀⢰⡇⢺⠉⠀⣴
⠀⠀⠀⠀⠀⠀⢿⠀⠸⣇⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⣄⠀⢼⠗⢷⠀⢳⣼⠁⢸⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠈⠀⠛⠀⠈⠀⠀⠋⠀⠘⠋⠁⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣤⣶⣦⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀⣴⣯⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⡀
⠀⠀⠀⠘⣿⣿⣿⣧⣤⣄⣀⠀⠀⠀⠀⣠⣾⢿⣿⣿⣿⣿⢿⣧⡀⠀⠲⠀⠀⣀⣠⣤⣻⣿⣿⣿⠃
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⡿⠋⣨⣿⣿⣿⣿⣅⠙⢿⣦⣴⣶⣿⣿⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢆⠈⠉⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠈⠉⠁⢼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⡿⢻⡿⢟⣯⣶⣿⣧⡀⠀⠉⠉⢩⣭⡍⢩⣭⡍⠉⠉⠀⠀⣸⣿⣾⣝⡿⣿⡟⣿⣷
⠀⠀⠀⠸⢿⠇⠀⢺⣿⣿⡟⠿⣫⣧⡀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⣼⣿⠏⢙⣿⣿⠃⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣦⠀⠘⣿⣷⠀⠀⠸⠿⣧⠾⠿⠇⠈⠀⣾⣿⠃⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠈⠁⠀⣀⠀⣼⠁⣀⣀⡀⠀⠀⠈⠁⠠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢠⣴⣿⡆⠀⢀⡽⢿⣉⣉⣽⠇⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠙⠛⠀⢀⡞⠀⠀⠈⣩⡀⠀⠀⠈⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⣠⡄⠀⢻⣾⣋⣉⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⣼⠁⠉⠉⠉⠀⠤⠀⠘⠿⠂⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠲⣶⣶⣶⣶⣶⣹⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣈⣉⢭⣭⣭⣥⣽⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⡀⣟⠙⠛⠛⠋⠛⠙⠁⠃⣶⣾⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢹⣿⡇⣿⣆⢀⠀⢀⡀⠀⣀⣼⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢇⢹⣿⣶⣟⠃⠛⣿⣿⣿⣧⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣴⣮⣿⣿⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠛⠛⠛⠛⠛⠛⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠳⠜⠙⠏⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡒⣖⣲⢴⢰⣖⡂⣶⣰⣶⠒⢢⠰⣒⠐⣆⣰⢰⢰⡒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠋⠙⠘⠙⠓⢂⠃⠘⠛⠓⠊⠐⠒⠋⠃⠘⠘⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢉⣆⣾⣧⠙⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢋⣉⣉⣉⣉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠶⠛⠉⠉⠉⠉⠉⠙⠳⣤⡙⢷⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠾⠥⠾⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠿⠤⠿⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢀⡾⠋⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠈⠻⣆
⠀⠀⠀⣾⠀⠀⠀⢸⣿⣿⣿⡇⠀⢀⡞⠋⠛⠶⠞⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡆
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠈⢧⡀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠙⢦⡀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⢻⡄⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⣿⣿⣿⣿⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣸⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣿⣿⣿⣿⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣿⣿⣿⣷⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⣿⣿⣿⡿⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣭⣿⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣶⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⣿⣿⣿⣿⣿⣧⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠘⠻⢿⣿⣿⣳⣿⣿⣷⣰⣦⣄⣀⠀⠀⠀⢀⣾⣿⠟⢹⣿⣿⣿⣿⣿⣿⠙⢿⣿⣦⠀⠀⠀⢀⣀⣤⣶⣴⣿⣿⣯⣻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡿⣽⣷⣹⣿⣿⣿⣶⣶⣿⠟⠁⢀⣼⣭⣭⣭⣭⣭⣭⣄⠀⠙⢿⣷⣶⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠙⠛⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠛⠛⠉⢁⣼⣿⣿⣿⣿⣿⣿⣽⣿⣯⡄
⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⠿⣛⣥⣦⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣬⣝⡻⢿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⢰⣿⣿⠇⢻⡿⢟⣫⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢴⣶⣶⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣬⣛⠿⣿⠃⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⠻⠿⢟⣫⣥⡀⠀⠀⠀⠀⢸⣿⡇⠀⠘⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠙⣿⣿⣿⡷⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣿⠿⣛⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⣴⣿⣿⠏⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣧⡀⠀⠈⢿⣿⠷⠀⠀⠀⠀⠛⠁⡀⠀⠙⠃⠀⠀⠀⠀⢿⣿⡿⠁⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡤⠀⠉⠀⠀⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢺⣿⡿⠉⢉⣽⣿⣿⣿⣿⠟⠉⠛⠋⠉⠀⣰⣶⣤⡀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣿⣦⣴⣾⣿⠟⢫⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⣤⣾⣿⣿⣿⡏⠟⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣢⣾⣿⣿⣿⣿⣿⠟⢃⣤⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠹⡿⠿⠶⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠲⡀⠈⠁⢰⣷⣤⠀⢠⣶⣄⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⣴⣿⣿⣿⣧⣸⣿⣿⡶⣾⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠜⢿⣿⣿⣿⣿⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆⢼⡁⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⠿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠌⠿⠿⠿⢿⣿⠿⢿⡄⠀⠀⠀⣴⠟⢰⣿⣿⡿⠿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣤⠀⠰⣿⠀⢸⣿⣆⠀⠀⣿⠀⢸⣿⣿⣷⡀⠙⡿⠃⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠻⣿⣇⣸⣿⣀⢸⣿⣿⡄⡀⣿⡀⣼⣿⣿⣿⣷⣤⣤⡖⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣼⣻⠇⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⠿⣿⣾
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⠉⠉⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠄⡀⠀⠀⠀⠀⢃⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⡉⢑⣶⣷⢰⣶⣇⢀⣾⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠃⠈⠻⠿⠿⠿⠋
⠀⢀⡀
⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘
⠀⠐

⡟⢉⣉⣉⠉⣉⣉⣉⡉⣉⣋⣹⣿⣿⣿⣿⣿⣟⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⢟⠍⠉⠉⣉⡉⠉⠙⠛⠙⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⣿⣿⠃⠀⠈⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⣀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠛⠿⠛⠁⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⡶⠷⠿⠶
⡇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠼⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆
⡇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠈⠹⠆⢠⣖⣀⠀⠀⠀⠀⠀⠀⢠⠀⣂⠀⠀⠰⠒⠀⠈⠙⢛⠃⠀⠀⠀⠀⠠⠀⠠
⡇⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⢟⠔⠀⠀⠀⠀⠀⠁⢨⡁⠀⠀⠀⠀⠀⣠⠀⢸⡄⠀⠄
⡇⠀⠀⠀⠀⢻⠀⠀⠀⢀⡠⠤⠄⠀⠀⠀⠀⣉⠹⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⡀⠀⠀⠸⣿⢶⣋⣡⣤
⣏⣀⣀⣀⣀⡟⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢻⣿⣷⣾⣿⡿⠞⠻⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢤⠖⢒⣶⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠴⠂⢹⠿⢿⣿⣷⣄⠀⢸⣿⣿⣷⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠾⢷⡿⠄⠀⠸⠋⣀⠐⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠀⠀⠀⢸⣯⣿⣿⡿⠃⠀⠀⣿⣿⣿⣦⠂⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈
⡿⠿⠟⠻⠛⢻⡏⠀⠀⠀⠀⠀⠅⠀⠰⠀⢈⡩⠤⠤⣤⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠿⠇⠀⠀⠀⢈⣿⣿⡟⠁⠀⠀⢀⣿⣿⣿⠏⡄⡔⠁⠀⠀⠀⠀⠀⠀⠀⢘
⡗⠀⠀⢰⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠏⠠⠧⣀⡤⠀⠉⢿⣿⠟⠋⢀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⢠⣾⣿⠿⠃⠀⣄⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⣁⡀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠑⠛⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⢿⣿⡄⠀⠀⠀⠀⠀⠀⢐
⣧⠘⠀⠀⠀⠀⠀⠀⡀⠀⢠⣲⡶⢦⠄⣤⠀⠀⠤⠠⠆⠐⢀⠐⠀⠀⠀⣀⡀⠀⠀⡄⡀⠀⠀⠰⠛⠀⠀⠀⡀⠶⣄⠀⣀⡀⠀⠀⠀⠈⠻⠟
⣇⠀⠀⣠⠀⢀⣠⣶⣤⡀⠠⢤⣼⡿⡷⢶⠦⠀⣈⣈⡩⠗⣛⢒⡴⠶⠤⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠘⠉⠤⠶⠿⠖⠈⠙⠐⠀⠘⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡤⠀⠀⠀⠒⠢⢀⣘
⡧⠠⠤⠠⠤⠄⢀⠤⠤⠤⣤⣤⡤⠔⠒⠒⠒⣶⠿⠛⢛⢿⣶⡿⠖⠓⠆⠩⠶⢶⣶⣶⢶⠶⠶⠶⠶⢦⡦⣀⣀⣠⣄⣀⣤⣤⣤⣀⡀⠀⠀⢀⡒⠒⠯⣁⣀⣀⢙
⣧⣈⣿⣀⡀⠀⠀⠶⢶⣀⣀⠉⠀⢀⠻⢷⡶⡄⠠⠆⠀⠒⠀⠀⠂⢀⣭⣴⡀⠂⠀⠉⠀⠒⠀⠠⣤⠤⠤⠛⠋⠉⠀⠈⠁⠉⠛⠀⢐⡍⠁⠀⠀⠠⠄⣀⡀⠉⠘
⡇⠀⠀⠀⠤⢀⡈⠙⠃⠠⠤⢤⡤⠤⠉⠑⠒⠒⠂⠒⠀⠴⠖⠀⠈⣉⡉⠀⠀⠀⠀⠒⠂⠀⢤⣀⣀⠉⠀⠀⠠⢤⣤⠀⠀⠀⠀⠉⠉⠀⠐⠒⠂⠀⠀⣀⣀⡀⢸
⣿⣾⢿⢷⣿⣿⣷⣶⢶⣶⡦⠀⠀⣀⠂⠒⠲⠄⠀⠀⠠⠶⠒⠂⠀⠀⠤⠀⠤⢤⠘⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣶⠆⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣷⣾⣷⣿⣷⣿⣾⣿⣯⣀⣀⣩⣭⣥⣤⣀⣤⣤⣤⣒⣿⣭⣤⣤⣤⣤⣄⣀⣀⣀⣬⣽⣖⣀⣀⣀⣀⣀⣀⣲⣶⣭⣅⣀⣄⣄⣀⣤⣤⣤⣭⣉⣉⣀⣀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⡁⠀⣀⣤⡀⠀⠐⠀⠀⠀⠀⣠⣄⡀⠐⠂⠀⠀⠀⠛⠀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⢧⣀⡿⠀⠀⠀⠀⠀⠘⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠙⠛⠛⠋⠛⠋⠛⠋⠛⠉⠙⠛⠓
⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⣶⡦⠘⠀⠀⠀⢠⣾⣦⣶⡦⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠆⠀⠰⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠰⣛⡖⠐⠒⠂⠐⠐⠂⢞⣳⠰⣛⡆

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⡀⠀⣤⡀⠃
⠀⠀⠀⠁⢰⡞⢳⡆⠀⠓⠓⠛⠛⠋⣴⠋⢳⡄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢉⣥⡀
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠴⠋⠉⠙⠦⠋

⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠓⠒⠛⠚⠚⠛⠛⠓⠀⠀⠀⠀⠐⠛⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠈⠀⠂
⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⢀⠅

⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣴⣿⣿⠿⠟⠻⢿⣿⣿⣦⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠘⣿⣿⣧⢠⠒⠚⢿⡇
⡰⠒⠓⠒⠤⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⢸⠀⠀⠈⢧⡀
⢧⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⣿⣄⡀⠀⠀⣀⣴⣿⣿⡏⠀⢧⠀⠀⠀⠉⠲⢄⡀
⠀⠉⡲⠦⢤⣿⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⢣⠀⠀⠀⠀⠀⠈⠒⢄⡀
⠀⢸⡀⠀⠀⠀⠉⠑⢢⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠉⢒⣦⣤⣀⣀⡼⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢰⠁⠀⠀⠉⠑⠢⡀⠀⠀⠹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⣳⠤⣤⣀⡀⣠⠇⠀⠀⠀⠙⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡀⠀⠀⠈⠉⢢⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀
⠀⠀⠀⠉⠒⣦⣤⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⠧⠄⣀⣀⣀⣀⠤⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣽⡟
⠀⢀⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣉⣀⣈⣁⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⡀
⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣤⣶⣦⡀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠘⠋⠀⣸⣿⠀⠀⠀⣿⡇⢀⣤⠄⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⠀⢹⣟⠁⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⣿⣧⣿⡋⠀⠀⣿⠃⠈⣿⡏⠈⣿⡇⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣾⣯⣤⣄⠀⠀⠀⣿⡏⠙⣷⣄⠀⣿⡀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠁⠀⠈⠉⠀⠉⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⣿⣿
⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣄
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⠠⣿⣿⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⢠⣾⠇⠀⢾⣷⠿⣷⣶⠿⣷⡆⠀⠀⠀⢰⣿⣿⠄⠀⠀⠀⠀⣿
⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⢿⣧⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⡧⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡿⠀⠹⢷⠀⠾⠇⠀⠸⠇⠀⠸⡇⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⣿
⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀
⠀⡶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿
⠀⡇⠀⠀⣸⣧⠀⠀⠀⠀⠟⣿⡇⠀⠀⠀⣠⣤⣤⠀⠀⠀⢸⣿⠀⢀⣀⠀⢀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⢀⣾⣇⠀⠀⣿
⠀⣇⠀⠐⢿⡿⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣤⣄⠀⠀⠀⢸⣿⣠⡿⠋⠀⢸⡿⠛⢻⡿⠋⢻⣷⠀⠀⠀⠘⢿⡟⠂⠀⣿
⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣉⠀⣿⡆⠀⠀⢸⣿⠛⣿⡄⠀⢸⡇⠀⢸⡗⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⡇⠀⠀⠘⠃⠀⠀⠀⠀⠀⠛⠃⠀⠛⠀⠙⠛⠋⠀⠀⠀⠘⠛⠀⠘⠛⠀⠘⠛⠀⠘⠋⠀⠘⠛⠀⠀⠀⠀⠘⠃⠀⠀⣿
⠐⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣭⣭⣭⣭⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠟⠳⢦⣄⣀⠀⠀⠈⠳⣌⡻⣦⡀⠀⠀⠀⣶⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⢀⣀⠉⠙⠛⠒⠶⣾⣿⡾⠳⣄⠀⢰⡏⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⢷⡇⠀⠚⠋⠉⠙⠂⠀⠀⠀⠉⠉⠙⠦⠙⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠳⣶⠞⠛⣧⠀⠀⠀⢸⡇⠀⠀⠸⡿⠂⠀⠀⠀⠀⠸⡿⠆⠀⠀⢸⡟⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣿⣤⡀⠻⣷⡶⠶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠶⠃⠈⠙⢦⡈⠙⠶⣄⡈⠳⣄⠀⠀⠀⠳⠶⠶⠞⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠙⣶⣌⡛⠶⢦⣤⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣞⠁⠈⠛⠛⢻⡇⠀⠀⣺⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣰⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣄⠈⠳⣄⠀⠀⣀⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⣄⠘⢧⣰⡏⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⣴⠖⠓⠿⣇⠀⠙⣧⣀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⠀⠀⢸⣿⡇⠀⠀⠀⣹⠶⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣰⡏⠉⠉⢹⣆⣀⣸⡏⠳⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⡇⠀⠀⢸⡏⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠷⡤⣼⡇⠀⠀⢸⡧⣤⣼⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠾⠇⠀⠀⠸⠷⠶⠶⠴⠾⠷

⡿⢿⠿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠠⠀⠒⠒⠀⠀⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠓⢡⣨⠴⠆⠀⠀⠀⠀⠀⠀⠠⠧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠍⢙⠀⢄⡀
⠀⣦⣠⠈⢀⢀⠤⠂⠑⠂⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢩⠅⠀⠀⠀⠉⠲⠀⠀⠀⠀⠓⠒⠉⠀⢰⠈
⠀⠉⢃⠀⠀⠀⠀⠤⢤⣖⡒⠂⠁⠀⠀⠠⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠠⢀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡏⠉⢂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡇⠀⢈⣤⠀⠀⠀⡆
⠀⠀⠀⢠⡴⣬⠀⣠⠤⣄⠀⠀⠀⣠⠆⠁⠉⢀⠀⢀⢤⢀⠀⠀⠄⠀⠀⠜⠀⠀⡇⠀⠉⠛⠁⠀⠀⡇
⠀⠀⠃⠙⠿⠚⠓⠳⠽⠛⠃⠀⠀⠁⠀⠘⠿⠆⠁⠁⠿⠟⠀⠐⠀⠀⠀⠫⠠⠲⣦⠒⠀⠾⣮⠷⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠧
⠀⠄⣤⠤⠤⠤⠤⠤⠤⠤⠬⠴⠀⠀⠀⠀⠠⠐⣄⠀⠀⡤⠤⡀⠀⠀⠀⠀⠀⠀⣴⠄⠀⣀⠇⠀⠅⢶⠂
⠀⠌⠯⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠐⠣⠔⢁⣄⠀⠀⢠⢞⡖⣛⢦⡀⢡⡴⣚⢲⡀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠐⠿⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⠶⢾⡝⠆⠀⠈⠉⠓⠿⠊⠉⠉⠙⠽⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣻⡻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢀⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠚⠇⡀⠀⠀⠀⠀⠀⠈⢉⡩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡅⢄⠜⠉⠉⡀⡀⠀⠀⠠⡠⠙⠓⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠩⡢⠀⠀⠀⠔⠑⠀⠈⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠢⡤⠀⠀⠈⡂⠲⠙⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠚⢆⠀⠀⠄⠺⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⢀⠀⠄⠃⠀⠀⠀⣀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⣰⠓⠅⠀⠈⠆⡀⠀⠀⠀⠀⢀⢱⣿⣿⣿⣧⠃⠀⠀⠀⢠⡞⠉⢻⡆⠁⠄⡀⠀⢀⢄⠎⠂⠁⢸⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠢⠂⠡⠂⠀⠀⠀⠢⠀⠀⠀⠀⢈⠜⣿⣿⣿⢟⠄⠀⠀⠀⠈⢿⣶⣾⠃⠀⢨⠎⢪⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠋⡆⠀⠀⡌⠀⠀⠀⠀⠐⢄⡀⠔⠐⠁⠁⠐⠈⠉⡀⡀⠀⠀⠀⠘⣿⡏⡠⠑⠁⠀⠈⠑⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⢠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⣀⣤⠾⠛⠁⠀⠀⢀⡠⠀⠁⠂⠢⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⣔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⡁⠀⢀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠢⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠢⠈⠀⠪⣠⡀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⠀⠀⠫⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠄⠠⠖⠀⠢⠂⠀⠀⠀⢸⣷⣄⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⠀⠀⠀⠀⠀⠀⢀⢀⠤⠋⢸⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⢇⠀⠀⠐⠄⠀⠀⣼⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠌⠂⠢⠀⠀⢀⢤⠢⠀⢅⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⢀⠀⠀⠀⠃⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⡠⠊⠃⠀⠀⠀⣈⠆⠊⠀⠀⠀⠀⠁⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠡⠀⠀⠀⠈⠂⢤⢀⠀⣿⠁⠢⡀⢀⠈⠢⣀⡀⡐⠔⠉⠀⠀⠀⠠⡠⠐⠴⡀⠀⠀⠀⠀⠀⠀⠀⢢⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠂⠠⠠⠤⠄⠐⠷⠿⠤⠠⠄⠵⠮⠲⠶⠾⠤⠤⠤⠤⠖⠷⠠⠠⠤⠄⠾⠤⠠⠤⠤⠤⠤⠐⢈⠔⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠔⠁





⠀⠀⣀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀
⣿⡟⣉⣻⠟⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⢢⣤⣄⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠐⠂
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠌⠉⣆⡀⢀⡀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⡀⠀⠀⠀⢠⣤
⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⡈⠄⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁
⢀⠁⠀⠉⢀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡈⠉⠉⢉⡍⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠉⠀⢠⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠁⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀⠀⠀⠄
⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠂⠠⠀⠀⠀⠂⠀⢀⠀⡀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠁⠀⢀⠀⠀⠠⠀⠠⠄⢀⠀⠀⠀⠄⠀⠀⠀⠋⠛⠛⠛⠃⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠐⠠
⠀⠀⠀⢀⠘⠀⠀⠈⠃⠈⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠂⠀⠁⠑⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢡⡀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⠂⠀⠀⠀⠀⠠⡀⠄
⠀⠀⠀⣄⡀⠀⠀⠀⣠⠀⠀⡇⠀⠀⠀⠀⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠄⠃
⠀⠀⠀⠉⠁⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠸⢒⠀⠊⠀⡠⠈⠀⠀⠀⠀⠈⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⠠⠀⠠
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠐
⢀⡗⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⢀⡀⠉⢁⣀⠀⠐⠀⠀⠀⠀⣀
⠘⠀⣀⣀⠈⡉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠃⠀⠀⠘⠋⠀⠈⠊⠀⠀⠀⠀⢠⠇⠀⠀⠈
⠀⠃⡅⠀⠀⡠⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠐⠀⢀⣀⠐⠳⠆⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⣀⣀⣈⣀⡀
⢠⡄⠀⠀⠐⠃⢄⠀⠀⠀⠀⠀⢀⠀⠀⠈⠋⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢶⠶⠀⠀⠀⠀⠀⠐
⠀⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠚⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠊