Middle Finger Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⢻⠃⠀⠀⠀⠀⠀⠀⢸⡛⠃⢣
⠀⠀⠀⠀⠀⠀⠀⣰⠒⠶⣤⣼⠁⠂⡞⠀⠀⠀⠀⠀⠀⠀⠈⣇⠊⠈⣦⡖⠛⠙⡆
⠀⠀⠀⠀⠀⠀⢠⢇⠴⠤⢽⠇⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⣀⣸⠀⠀⢹⡅⠒⠢⡹⡄
⠀⠀⠀⠀⠀⠀⣸⡎⠀⠀⠸⠀⡁⠾⠉⠋⢳⡀⠀⠀⢀⠏⠛⠀⠆⢠⡀⠃⠉⠀⢹⣇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠈⢻⡦⠴⣄⠀⠀⢀⠀⢠⢤⡴⠃⠀⠀⠈⢦⠤⣄⠀⣀⡀⠀⡤⠒⣺⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣴⣊⠩⡍⠵⢒⣻⠀⠀⠀⠀⣏⡒⠎⢽⢁⡩⠽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠖⣿⡥⣄⠀⣀⡠⢶⡿⠓⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢼⣯⣨⡿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢩⠏⠁⠀⠈⠉⡝⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠃⠀⠀⠀⠀⠀⠁⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⢉⡀⠀⠀⠀⠀⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠳⣤⣄⣤⡴⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠰⠿⠿⡿⢷⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠈⠉⠩⣭⣿⣿⣿⠀⠀⠐⠒⠒⠒⠓⠒⠂⠀⠘⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠉⠁⠒⠒⠒⠀⠀⠀⣿⡇⠉⠀⠀⠀⠙⣿⣷⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⠿⠟⠛⢿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⢀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆
⠀⢀⣾⣿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣿⡀
⢀⣾⡿⠁⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⢿⣷
⣼⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠸⣿
⣿⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣿
⢻⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⢿⣿⡄⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣿⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿
⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⣒⣒⣒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⡄⠀⠀⠉⢩⣿⡟
⠀⠀⠀⠀⠀⠸⣿⣇⠐⢠⡀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣽⠏⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠙⠲⢦⣤⣤⣤⠀⠂⠀⣤⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣶⣦⣤⣴⣶⣶⣾⣿⡿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡡⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠇⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠂⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠠⠄⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠑⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠲⣾⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠲⢤⣀⠀⢠⡏⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡴⠋⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠁⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢀⡀⠀⠀⣼⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⡁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣼⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢠⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡿⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢶⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⣿⣤⣤⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⣾⣿⣦⣤⣾⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣋⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢩⡭⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣹⡟⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠤⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⡧⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣒⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣶⣒⣒⡀⠀⣿⠀⠀⢀⡤⠤⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠓⢶⣿⣿⣿⣿⠁⠀⠀⣿⠀⢀⣿⣿⠿⠿⠷⢠⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠂⠚⠀⠈⢿⣿⡷⠦⠀⠀⠀⣿⠀⢸⣿⣟⡛⠁⠀⢸⣿⡇⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠓⠀⠀⠀⠀⢸⣿⣿⣗⣀⣀⠀⣿⠀⢸⣿⣛⣒⣀⠀⣸⣿⣷⡿⠥⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣀⣠⠀⣶⡖⢸⣿⣭⣿⠿⣿⡀⢿⡀⣸⣟⣻⣿⡇⠙⣿⢻⣿⢿⣿⠲⢤⡴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⣿⣠⣿⣧⢸⣿⣽⣿⠶⢿⣷⠘⣧⣿⣿⣿⣿⠇⠀⣿⢸⣷⣿⣯⣠⡌⣡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⣿⣸⣿⠀⢸⣧⣠⣱⡿⠗⠀⠘⠛⠃⠙⠿⠓⠾⠿⠄⠀⢹⣿⢋⣿⡇⠈⢠⣿⠏
⠀⠀⠀⠀⠀⠀⣾⡷⢤⡀⠀⠈⠙⢦⣿⢸⠟⠁⠀⠀⠸⠿⢟⣍⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠘⠟⠚⠉⠉⠛⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣝⢦⠀⠀⢠⣿⠭⡂⠀⠀⠀⠀⢻⣿⣟⠃⠀⠀⠀⠀⢡⣉⠀⠀⠀⠀⠀⣉⣉⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣠⣾⣿⣋⣁⡀⠀⠀⠀⢈⢛⣧⡄⠀⠀⠀⢠⣬⣿⣶⣤⡄⠀⠀⣟⡉⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⡇⢿⣿⣷⣎⣀⣤⣴⢾⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⡿⠃⠀⣽⣿⣤⣶⠂⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣻⣿⣿⣆⣻⣿⣿⣿⣿⣿⠟⣿⣻⣿⣿⣷⣾⡛⢻⡿⠿⣿⡇⠀⣼⣿⣿⡏⢡⣼⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣟⣿⣿⣿⣭⣤⡘⢻⣿⣿⣿⣟⠁⠀⠛⢻⣿⠀⠐⠾⠿⣿⠀⠀⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣿⣿⣿⣿⠷⠂⡀⣈⣿⣛⣛⣃⠀⠀⠀⣂⣀⠀⠒⠠⠄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣾⣽⣿⣿⣛⠛⣿⡿⠬⠤⠽⢿⣭⣷⡀⠀⠀⠚⢃⡀⠑⠒⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⢻⣟⡿⣿⣿⣿⣭⣍⣁⣀⣤⠠⠄⠄⠀⠀⣴⣏⣻⠄⠀⠈⠀⠈⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠲⢯⣽⣿⣿⣿⣭⣭⣽⣊⣩⣗⠒⢀⣹⣶⣿⣇⡀⠀⠒⠂⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣾⣯⡛⢿⣶⣶⣿⣿⣦⣼⣿⠄⢨⣍⠛⠿⠤⠭⢭⣍⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣯⣿⣿⣿⣤⡤⠀⠁⠀⠀⠀⠀⠒⢒⡒⠓⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠭⡞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣉⣠⣤⣾⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⣦⠀⠀⠀⠀⠀⠀⡔⠁⠘⠯⠐⠶⠟⢁⡈⠻⣿⣿⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠳⠚⢹⠀⠀⠀⠀⠀⢸⢠⣿⠋⠓⠢⠶⣾⣿⠿⠇⠈⠉⠘⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢸⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⠈⡆⠀⠀⠀⠀⠘⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡇⠀⠀⠀⠀⢀⢿⡇⠀⢀⣠⣤⣤⣄⠀⢀⣤⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⡇⠀⠀⡧⠤⣄⠀⠀⠛⣧⡉⠀⠉⠞⠛⢻⠋⠀⢸⡏⠉⠁⠛⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⡇⠀⠀⠳⠀⠘⣶⠢⡘⡋⠀⣄⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡌⠢⣯⠣⠀⠀⣾⠙⠲⣤⣾⣿⣦⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡏⠀⠀⢀⠀⠀⠀⠀⡀⠀⢰⡅⠀⣸⣆⠀⠈⠻⣶⣒⣚⣩⣽⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⡇⠀⠀⠘⡆⠀⠀⠀⡇⠀⠀⣧⣾⣿⡍⢢⡀⠀⠈⢛⣛⣛⣹⠇⢠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⣿⣿⣿⣷⡄⠉⠢⣀⠀⠉⠉⠀⢀⢾⠅⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣦⠀⠈⠑⠒⠒⣾⣥⣼⣿⣷⣦⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡳⣄⠀⢠⡾⣷⣖⣛⣛⣿⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡄⠀⠀⠀⠀⣠⠟⣿⣿⣿⣿⣿⣿⣷⡝⠿⠋⠀⠀⠙⢯⣽⣿⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⡤⠖⠋⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢸⢟⣭⣷⡀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⣉⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣸⣿⢋⣴⣷⡀⢀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠉⠉⠉⠛⠛⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠟⠉⠁⠀⠙⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣗⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢘⣏⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⠋⠛⣿⣿⡇⠀⠉⣙⡇⢿⡀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣡⣴⡀⠀⢰⡇⠀⠀⡹⢿⡇⠀⠀⢹⣿⣹⣿⣿⡿⠟⣻⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠿⢛⣑⣿⡇⠀⢸⡇⠀⠀⠴⣾⡇⠀⠀⠠⣿⢻⣿⠋⠀⠀⢻⣿⠋⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋⠀⠀⠨⣿⣿⠃⠀⣾⠀⠀⠀⠰⢿⡇⠀⠀⢀⣤⢸⣿⠀⠀⠀⠽⣿⡀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢼⡆⠀⠀⣆⣿⣿⡿⠁⠀⢰⡏⠀⠀⠀⢉⣹⣿⠀⠀⠀⢛⣧⣿⡅⠀⠀⠛⣻⡇⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠘⣧⠰⠤⠍⣽⣿⣇⠀⠀⢸⡇⠀⠀⠀⢚⢨⣿⡄⠀⠀⢉⣿⣿⣧⠀⠀⢨⣿⣿⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠞⢿⣿⣷⣤⣿⡀⠀⠀⠀⠀⠾⡿⠉⠀⠀⠘⠻⢿⡿⠃⠀⠀⠋⢻⣆⠀⢨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢾⣿⣿⣿⡿⠃⠀⠀⠀⠬⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⠁⠀⠒⣄⡀⢀⣲⣶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⣀⠀⠀⣐⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠶⠤⠀⠀⠀⠀⠙⢻⣿⣿⠿⢟⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⢶⣶⣶⣾⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣷⣶⣤⣄⡀⠭⠵⠢⠴⠍⠉⣙⣿⣽⠋⠈⢻⣿⣿⣿⣿⡤⢿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣒⣤⠤⠶⠶⠤⠾⣿⠾⠡⠄⢈⣽⠟⠻⠷⢤⣭⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣽⣿⣿⣿⣿⣯⣶⡶⠄⣀⣹⣿⣉⣹⣶⣼⣿⡶⢄⣠⣭⣽⣿⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣿⣿⣿⣷⣶⣆⣉⣿⣿⣿⣿⣟⡛⣣⣤⣤⣑⣾⡿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⠙⠻⠟⠻⣿⣿⣿⣿⡿⢿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣻⣿⣿⡿⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⡿⠿⣿⣿⣧⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⡁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣒⣚⣷⣿⣏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⣶⡦


⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠴⠟⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⣄⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢦⣀⡇⠀⢤⠀⠀⢸⡟⠁⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢿⡇⠀⠨⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⡾⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢹⡇⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠇⠀⠀⠀⢠⠏⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⠁⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⡀⠀⡀⠀⣄⣀⣀⣀⡀⠘⡇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠘⢿⣿⣿⣷⡾⠁⠠⣿⠿⢿⣿⡧⠀⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣶⣤⣀⣀⠘⣿⠿⣿⣡⣀⣷⡿⣿⣿⡿⠷⣺⣶⠒⣾⣟⣿⠀⠘⡇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡙⣿⣿⣿⣿⡅⠐⠾⠿⠛⠋⣭⣿⡿⠃⢒⣒⣛⣐⠒⠒⢻⣭⣦⡸⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⠀⠿⢾⠇⠀⠐⢾⠅⠀⠸⠿⠧⠀⠀⠤⡴⡦⢐⣻⣿⣭⣷⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠰⠆⠀⠀⠒⠂⠈⠉⠛⠛⠋⠈⠉⠩⠭⣄⣸⣻⣿⠏⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠈⠉⣁⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⢤⡀⠀⠀⢿⣿⡆⣠⣤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢤⡿⣧⠄⢠⣿⣿⡿⠟⠋⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⣧⣶⣿⣿⣿⡿⠋⠁⠀⠀⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⣀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⠿⠋⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⢣
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣇⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⡇⠀⠀⢀⠎⠁⠀⠉⢦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠙⣧⣀⣤⡜⠀⠀⠀⠀⠀⢙⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⣿⠀⠀⠀⠀⡼⠋⠉⠉⠐⠒⠒⠒⠒⠺⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠁⠀⠸⡀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡀⠀⠀⠀⠀⢱⡀⠀⠀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠴⣄⠀⠀⠀⠀⡇⠀⠀⠀⢻⣿⣿⠿⢶⣧⠄⠂⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠏⠀⠀⠀⠸⡆⠀⠀⠀⠀⠸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠋⠳⡀⠀⠀⢃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⢲⡖⠚⠦⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠒⠀⠉⠈⠀⠀⠀⠀⠀⢀⡴⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿
⣿
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⢋⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⠁⢸⠛⡌⠤⠰⣖⡒⠒⠲⢀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡐⠈⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠆⣜⠇⠀⠀⠀⠀⢣⡻⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡀⠄⠘⣧⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠌⠀⠀⠈⠻⢆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠶⠤⢄⠢⠄
⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠹⠐⠀⠁⠀⠀⠙⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⠿⠷⠔⠀⠀⠀⠀⠀⠀⠀⠐⡸⠟⢛⣁⠀⢀⠀⢀⡀⠂⠀⠀⠀⢰⣒⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠻⠛⠛⠿⠛⠛⠛⢛⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠙⢷⠶⠦⠰⠶⠾⠿⠿⢿⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣶⣴⢶⣾⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⣠⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠿⢋⣴⣷⠈⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⣾⣧⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠖⠿⠵⠿⠀⣿⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣽⠉⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡃⢀⠀⠀⠀⠀⠉⣠⣴⣶⣤⣄⣀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠙⠿⠉⡀⣀⠴⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣾⠇⠀⢀⣾⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⡟⠀⣠⣿⣿⣿⣿⠿⠛⠋⠉⠀⠰⣿⣷⣶⣿⣿⣿⣿⣦⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣝⣾⣿⡟⠋⢠⣾⣿⡿⣻⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⡠⠒⠺⠿⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣒⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠙⠛⠿⠿⠿⠇⠀⣧⣄⠈⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢹⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠙⢷⡀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⣠⠟⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⠀⠀⣰⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢧⣦⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣏⠀⠀⠚⠉⠙⠛⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⣠⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⡀⠀⣴⡆⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⢿⣿⣿⠇⠀⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠀⣠⠀⠙⠀⡀⠀⠄⠀⠘⠀⡀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠐⠀⡍⠢⡉⠀⠁⢀⠈⠢⠘⠀⡀⠀⠁⢃⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢄⠂⠐⢦⡐⢄⡰⢸⢢⡀⠆⠃⠤⣀⠡⡀⠀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠔⠢⡉⠀⠒⡈⠢⠀⣌⢀⠱⡠⢮⠞⠉⠳⡘⠈⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠒⠀⠁⣈⣲⡄⢰⡄⢹⡄⠝⢦⡹⠁⠀⠉⢠⠃⠀⠂⡠⠂⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⢁⣻⡈⠉⠻⡌⠋⡹⠀⠈⠇⠈⢀⣴⠁⠁⠠⠦⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⡝⠓⠀⠀⢀⠀⠀⠀⠀⢀⠀⠁⠘⡶⠨⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⠀⢰⡕⠀⠀⠀⣠⡂⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠉⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⠀⠀⠀⠀⠐⠂⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢆⠢⠀⠤⠄⢌⡤⡁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠉⠂⠄⠀⠈⡰⠑⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠨⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢊⠄⠀⠈⠆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠂⠄⡰⢑⠀⢀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠤⠀⢸⠁⡠⢀⠀⠘
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⢜⡉⠀⠈⠄⠀⣀⠀⠀⠤⠷⠃⣴⣄⣠⠀⠀⠀⣾⡵⣶⢗⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠒⢔⠀⡠⣿⠶⠶⠢⠂⡄⠾⠀⠀⠀⠀⠒⠘⠿⢓⡽⡂⢸⠁
⠀⠀⠀⠀⠀⢄⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠵⠚⠁⠀⠳⣀⠓⠄⠀⠀⠀⠀⠐⠘⡗⢊⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠡⡀⠀⡀⠁⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠀⠀⠀⠀⢩⡄⡠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠄⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⡀⠀⠀⠀⠀⠈⡇⠒⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠢⠤⣀⣀⣀⠀⠀⠀⠁⠀⠀⠂⠂⠀⠀⡔⠁⢂⠄⠔⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢂⠀⡠⠀⡀⠀⡠⠈⠀⠐⠂⠀⠐⢊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠈⠔⠁⠠⠀⠠⢊⠔⠀⠔⢁⠔⠠⠁⠌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠐⠁⠠⠀⠄⠀⠊⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⣀⠀⠀⢀⠤⢤⠀⠀⡤⠤⡀⠀⠀⡀⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣹⡇⣿⢎⠀⠀⣱⣿⢸⣏⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣄⠀⣑⣦⣀⣼⡏⣬⠿⣂⣒⠿⣥⢻⣧⣀⣴⣊⠀⣠⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣾⢙⢓⣻⠛⠳⣿⣿⣿⣿⣿⣿⠞⠛⣟⡚⡋⣷⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣴⡿⠘⣝⡧⠬⢼⡐⣿⣿⣿⣿⣋⡧⠧⢼⣫⠃⢿⣦⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣨⣿⣿⢸⡿⠿⣶⠀⣉⢷⣿⣿⡿⣉⠀⣶⠿⢿⡇⣿⣿⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⢿⣯⡴⢉⡝⢣⡼⢾⣤⡌⢡⣤⡷⠧⡘⢫⡉⢦⣽⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⢾⢲⡓⠞⠲⣌⠛⠛⣿⡅⢈⣿⠛⠛⣡⠖⠲⠚⡖⡷⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠷⠆⠀⣑⠉⢿⣾⣿⠁⠈⣿⣷⡿⠉⣋⠀⠸⠾⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣵⣬⣉⠛⠀⠀⠛⣉⣥⣮⣿⣿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠉⣿⢋⣽⣥⣮⣯⠙⣿⠉⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠿⠾⠗⠓⠘⠺⠇⠿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠀⠀⠙⠃⠀⠀⠘⠋⠀⠀⠸⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⣆⡀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠲⠶⣯⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠀⢀⣠⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠛⠛⠓⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⢀⡟⠻⢷⣤⣄⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡀⢸⠇⠀⣀⣬⣍⣉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠸⡌⢧⣿⢠⡞⠉⠀⠀⠈⠁⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠛⠷⠶⣶⠶⢿⡏⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣹⣦⣿⠞⣋⡁⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⠴⠞⠛⠉⠁⠐⠒⠲⢦⡀⠀⢸⡇⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⢁⡟⠁⠉⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⣧⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣾⢁⡼⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠓⠲⢤⣀⠀⠀⠀⠈⣧⠀⠀⢘⡇⠀⠀⠀⣀⣨⣿⣷⣤⣾⣶⡆⣠⡏⣿⠀⣀⢀⠀⠀⠀⠀⣀⣀⣈⠺⣵⡟⣋⡴⠃⣟
⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠛⠁⡀⠀⠀⠀⠀⠤⣍⡉⠒⢤⡀⠘⢷⣴⠋⠀⠀⠀⠸⠣⣄⣾⣻⡿⢿⣿⠟⠋⠀⠹⣾⣿⣿⣧⣤⣶⠿⠛⠉⠉⠑⢹⡟⣯⠀⠀⢿
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⣿⠃⠀⣀⣀⣀⠀⠀⠈⣷⠀⠀⠱⡀⠨⡇⢀⡠⠄⠀⠀⠀⠈⠉⠛⠒⠋⢹⡆⠀⠀⢀⡾⣿⠛⣿⣻⠍⠓⣄⠀⠀⠀⢸⠇⠸⠀⠀⣾
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣰⢏⠅⠒⠈⠁⠀⢢⣸⠀⠀⠀⠈⢢⣿⡏⠀⠀⠀⠀⠀⢀⣀⣀⡙⢶⣼⠃⠀⠀⣼⢁⣀⡙⠛⠛⠒⠋⠋⠀⠀⠀⢯⡀⠀⣠⡼⠃
⠀⢀⠔⠋⠣⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠱⣹⣦⠀⠀⠀⠚⡉⢰⠋⠉⠀⣿⠀⠀⢠⡿⠋⢥⣄⡀⠀⠀⠀⠠⢄⡀⠀⠈⣷⠟⠉
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣰⠇⠀⡠⠀⠀⠀⠀⢻⣇⣀⣠⣤⣾⣷⡦⢼⡳⣼⡇⠀⠀⢸⠃⣀⠀⡏⡙⢦⠀⠀⠀⠀⠈⣦⡼⠃
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣤⣶⣏⣠⠜⠁⠀⠀⠀⠀⠀⠹⡟⠛⢻⡻⣿⣝⢿⣿⣿⠀⠀⠀⣿⣯⣽⣖⠤⣽⡀⠁⠀⢀⣤⡶⣻⡇
⣸⠉⠑⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠙⢷⣀⢠⠆⠀⠀⠀⠀⠀⡇⠀⠀⢡⠹⣏⠿⣿⣿⣀⠀⣰⣿⣿⣋⣭⣿⢷⣿⣦⣤⣹⣟⣴⡿
⡇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⣿⡏⣀⡤⠄⠀⠀⠀⣇⠀⢀⠀⣇⢻⡂⠄⠁⠙⠋⡛⢙⢹⣽⠟⠁⠃⠀⠉⢹⡿⠛⠉
⠰⡄⠀⠀⠠⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⠒⠀⠀⠀⠀⢻⡄⢸⠀⠘⣧⠻⣤⣀⡈⢀⣁⣤⢞⣡⠊⠀⠀⠀⢠⡟⠁
⠀⠙⢆⠀⠀⠹⡄⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⢿⡌⠂⠀⠈⠛⠦⣭⣉⣉⣭⡴⠛⠁⢠⠇⢀⡴⠋
⠀⠀⠈⠳⣄⠀⠹⣄⠀⠀⠑⠦⣤⣤⣂⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢷⡀⠀⠀⠀⠐⠒⠤⠀⠀⠀⠀⠰⢋⡴⠋
⠀⠀⠀⠀⠈⢧⡀⠘⢦⡀⠀⠀⠀⣀⡟⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠚⠁⠀⠘⣧⠀⣴⣋⣽⣿⡝⣦⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⠦⣄⣽⢛⣳⠞⠁⠀⠀⡇⠀⠀⠀⠀⠴⠓⠉⠀⠀⠀⠀⠀⠀⠀⠘⢷⣏⣿⣿⣿⣿⣿⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⡍⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠋⠟⢻⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠛⣻⣿⡛⠁⢠⣤⣤⣉⡛⠿⣷⣤⣄⠈⠉⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⡵⠚⠉⠀⠀⠞⠋⠁⠐⢛⡛⠛⢿⣯⣉⠁⠠⠿⣿⣦⡀⠐⠶⣮⡙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢃⡼⠁⢀⡄⠤⠶⠶⠚⠴⣾⣿⣯⣁⠲⠿⠿⠛⠃⣢⡀⠻⣿⣿⣆⠀⠈⢿⠂⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠈⠛⢢⣿⣾⣶⣶⠶⠖⠀⠀⠠⠾⠟⠛⠃⠀⠀⠈⢻⣷⠦⠀⠙⣿⣷⣦⡀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠒⢦⣤⢈⣫⣿⣿⠟⣇⠀⠀⠀⠈⠉⠛⠳⢦⣀⠀⠀⠀⠀⣿⣦⣀⠚⠻⠿⠟⠃⠀⠀⠀⠀⠠⡘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠉⠁⠾⠷⠦⢍⠒⠾⠃⣠⠀⠀⠀⣀⣀⣀⣀⣙⣦⠀⠀⠀⣿⣿⣿⣧⡀⠘⣷⠞⠁⠀⠀⠀⠀⢷⠸⡆
⠀⠀⠀⠀⠀⢀⡼⣷⣋⣤⠴⠒⣁⣤⣤⣤⡗⢠⣌⣙⠛⠶⢤⣿⡞⠛⠿⠀⢨⣭⡤⢀⣴⠟⢸⣿⡏⠃⠀⢹⠀⠀⢀⡀⠈⢶⣼⡇⠃
⠀⠀⠀⠀⢀⡾⠁⠈⠉⢀⡴⠛⠉⠀⠒⠈⠁⢤⡌⠙⠿⣤⣀⡈⠉⣀⣶⡾⢛⣿⠁⣾⠃⠀⢸⢻⠁⠀⠀⣾⠀⠀⠀⢷⣀⠈⣿⠓
⠀⠀⠀⠀⢸⠻⣶⣶⡶⠛⣠⣴⣾⣿⣿⣿⡆⠈⣁⣀⣀⡀⠉⠛⣿⣿⠁⢀⣸⣿⡄⣿⡀⠀⠀⠸⠀⠀⠀⢸⢰⠀⠀⠈⢿⣷⣽⡆
⠀⠀⠀⠀⠸⣦⣨⣿⣦⣴⣶⣿⣿⣿⣿⣯⣤⣤⣬⣍⣀⣡⡴⠂⠸⣿⠀⢸⣿⣿⡇⢹⣧⠀⠀⠀⣷⢠⠀⠀⠸⣇⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠈⠻⡖⢒⣋⣭⣴⣎⠉⢻⣿⣿⣿⡿⣿⡏⢥⡤⠀⠀⡿⠀⠘⠁⣿⡇⢾⡏⢀⠀⣀⡏⢸⠀⠀⠀⢿⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢄⢫⠉⠻⣿⣆⠘⣿⡇⢻⡇⣿⣷⡄⠁⠀⣰⡇⠀⠀⠀⡟⢀⡾⠁⣼⠀⢹⠇⣾⠀⠀⢰⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⠉⢿⠹⣧⣽⣿⠘⡇⢸⣿⣿⡀⠀⢹⠇⠀⠀⠀⢀⡾⠁⣴⠏⠀⠀⢰⡏⠀⠀⣼⠇⣿⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢱⣿⡝⠿⠀⠁⣼⣿⣿⡇⠀⠀⠀⠀⠀⣠⠞⢀⡼⠋⠀⣠⢀⣿⠁⠀⠚⡟⠀⣿⠇⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡙⣛⣿⣤⣤⣾⣿⣿⣿⢃⡜⠁⠀⠀⠀⠀⠀⡜⠁⠀⣠⣿⡾⠃⠀⠀⠀⠀⣼⣿⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠉⠉⠀⠀⠘⠻⢫⣾⣀⡾⢸⣠⣾⡇⠀⣀⣴⣾⣿⠟⠁⠀⠀⠀⠀⣼⣿⠟⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣤⣀⡀⠀⣀⣴⠏⠈⡿⠁⠸⡿⢿⣷⣾⣿⣿⠟⠁⡀⠀⠀⡀⢀⡾⠋⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠛⠛⢿⡟⠀⠀⠀⠀⠀⠁⠈⠻⣿⠟⠁⣠⣾⣴⡶⠞⠁⡜⠁⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⡴⠂⠀⠀⠀⣠⠀⠀⠉⠀⡼⠋⠀⠁⠀⢠⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣁⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⣽⣿⡞⣠⡾⠀⣠⣶⣿⣾⠟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠿⣿⣿⣿⣥⣾⢻⣿⡿⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⡿⠁⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣄⠀⠀⠀⣀⠀⠀⠀⢀⣿⣿⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⠀⠀⠀⠀⢦⣤⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣿⠀⠀⠀⠀⠉⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠠⣾⠀⠀⠀⣠⣄⣀⣠⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣿⣿⣿⣿⣿⣿⡄⠁⠀⠀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⢸⣿⣿⠉⢻⣿⠿⢿⣿⣿⣿⣿⣿⣟⠉⠿⠃⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⢀⡾⠟⠋⠹⠟
⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⣠⣦⡀⠀⠈⠃⢸⣿⣿⣦⡀⠁⠀⣠⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠉⠛⠳⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡴⠓⠶⠀⠤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠁⠀⢀⡄⢀⠀⣀⡉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⡠⢤⣨⠤⢬⠙⠆⠈⢆⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠃⠀⢀⠎⠀⠈⠀⠀⡰⢀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠉⠁⡀⠀⠀⡀⠀⡀⠀⠀⠀⢀⡴⣝⠶⢤⡄⣐⠢⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠔⡡⢊⣬⣭⡬⠀⢀⣄⡤⠀⠰⡎⠁⠀⡴⠐⠀⠓⠈⠱⣍⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠐⡡⢊⠐⡈⠀⠀⠀⠀⡘⠡⠀⠀⠀⠀⡀⠀⠀⠀⢀⠃⠀⠀⠘⢆⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡆⠘⠀⢃⠀⠈⠔⡡⢊⢷⡀⠀⠀⠀⠇⠈⢀⡤⡴⠶⡶⠶⣤⢀⡞⠀⠀⠀⢀⠎⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⢆⠀⠈⠔⡡⢂⠳⡀⠀⠀⠀⢠⠃⢱⢰⣱⣰⠡⢣⠞⠀⠀⢀⡴⠃⢀⢠⠘⢀⠔⠁⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠐⠡⠂⠙⠆⠀⠃⠀⠙⠜⠋⠁⠀⣰⠋⠀⣀⠔⠁⢠⢰⠘⢀⡴⠋⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣜⠁⠐⠊⠁⡄⡄⠂⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢐⠈⠀⠈⠉⢐⣉⠥⢒⣈⠄⠀⠘⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡩⠔⢘⡤⠖⣋⠩⠖⣈⡭⠖⡃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠠⠐⣈⠤⠒⣉⠤⠒⢉⠠⠒⣉⠤⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣀⠤⠒⡉⠤⠒⢁⡠⠔⢊⠠⠐⡈⠄⠂⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣊⠤⠒⢁⡠⠔⢊⡡⠐⣊⠅⠂⢁⡠⠔⡪⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢈⡡⠄⢊⠥⠐⣈⠤⠒⣉⡠⢄⡊⠅⣂⡡⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⣩⠴⢊⡥⠒⢉⠤⠖⣋⠥⢘⡩⠖⣋⠥⢒⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⠤⠊⡡⠔⢈⣠⡔⠉⠤⠒⡡⠔⢊⠥⠊⡡⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⢉⠤⢒⣡⡇⠸⠀⠀⢉⡤⢊⡠⠒⣉⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠖⡡⢒⡡⡾⠀⢰⠀⠀⠠⠒⡡⠔⢈⡠⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡪⠔⡡⠘⣇⡀⠸⠀⠀⠐⣉⠄⢊⡡⠔⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢈⡴⢋⠴⢂⠴⢉⠿⠀⠀⢈⠤⢊⡡⠔⣊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠔⡡⢞⡡⢂⠡⣺⠀⠀⠴⢊⡡⠔⡋⠔⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠐⡩⢒⡡⢊⢽⡀⠀⣉⠤⢒⡩⠔⡡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠈⠔⠉⠖⣡⠾⠃⠀⠔⢊⡡⢔⡩⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⡤⣚⠁⣠⠆⠀⢊⠤⣒⠥⣪⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠐⠦⠌⢉⢁⡶⠖⠉⠀⠀⠀⠀⠀⠀⢀⣀⠘⠢⠭⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠲⠴⠶⠶⠠⡶⠓⠈⢀⡀⠀⠀⠄⠀⠀⠉⠉⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠁⠀⠀⠛⠀⠀⠐⠈⠃

⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠋⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⣯⣷⣶⣶⣶⣾⣿⣿⠿⠿⠿⠿⠿⠿⠛⠿⢿⣷⣤⠀⠀⠙⢶⣤⡀
⠀⠀⠀⢀⣤⡾⢛⣩⡶⠛⠛⠉⣩⡽⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⠀⠀⣿⢿⣷⣄
⢀⣤⠶⠛⣡⡴⠋⠉⠀⣀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠰⣿⢿⣢⡀
⠟⣁⣴⡞⠁⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣄⠀⠀⠀⠘⢿⣧⣿⡛⣄
⣾⡟⠁⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⣻⣿⣿⣿⣿⣷
⠏⠀⠀⢀⡴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠉⠉⠉⠙⠳⣦⣆⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⡰⣷⡟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠀⠀⣌⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⣿⣿⣿⣿
⣼⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣛⠁⣀⣠⡤⠀⠁⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢙⣷⣿⣿⣇⣠⣴⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣿⣿⣿⣿
⣿⡿⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠛⠘⠲⠿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⣠⢀⠔⣄⣠⠀⠀⢈⣿⣿⣿⣿⡟
⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢼⣮⣅⣾⣿⡟⠉⢻⡆⠀⣠⠄⢀⡼⣱⠏⠎⠀⠀⠀⣀⠈⣼⣿⣿⣿⠇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⣿⠉⣣⣔⣧⣨⣧⣼⣿⣴⣿⡿⡃⠂⠀⠀⠀⠈⠛⠆⣿⣿⣿⡟⠘
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡰⠃⣴⣿⣿⣿⣤⣿⣿⣿⣿⣿⡿⠁⡀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣶⣿⡿⠁⢀⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⣯⠞⠁⠀⡀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⡿⠋⡀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢂⠀⠀⢀⡼⢃⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡿⠛⡟⠀⠈⢙⣦⡙⣿⠃⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⢠⡔⣴⡟⢱⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⠃⠀⠀⠈⠻⣿⣿⠀⠀⠀⣿⣿⠀⠉⠛⠻⣿⣿⣷⣶⣶⠏⠘⣾⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⡍⠨⠃⠀⠀⠀⠈⠛⠀⠀⠀⣙⠘⢄⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠾⠀⠀⣸⣿⠛⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣶⣾⡿⢃⣰⣿⣿⣿⡟⢻⣷⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣴⣿⣿⠏⣹⠏⠀⢨⣿⣾⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠐⢀⠀⣴⣦⠄⠀⠀⢀⣈⣤⣾⣿⣿⣿⣿⣿⡿⣿⣴⠊⠁⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣿⣷⣀⣠⣴⣿⣿⣿⢿⣻⣿⣿⡿⠋⢐⡁⠀⠶⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣼
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣭⣴⣿⣿⠟⣿⣦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣟⣏⣥⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠐
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡿⢟⣩⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣩⣾
⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠟⠀⠀⠀⠀⣀⣤⢸⣿⣿
⣿⣷⣤⡀⣀⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣶⣿⣿⡿⠛⠁⢀⣤⣶⣾⣿⠿⠃⠻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠉⠙⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠉⠉⠉⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⢀⣠⠀⠀⢀⣀⡀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠛⠛⠛⢿⣷⣦⡀⠘⢿⣿⣿⣷⣿⣿⣿⠋⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⠟⠁⠀⢀⣠⣶⣾⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⢀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣦⡀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⣀⣴⣿⣿⡉⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⣀
⠀⠀⢴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠉⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠘⣷⡶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠙⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠋⠉⠁⠀⠀⠉⠙⠻⣦⡀
⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠲⠶⠶⠖⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠲⠤⠄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣴⣶⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣶⣤⣴⣶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠈⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠤⠈⠂⠀⢠⠀⠀⠀⠀⠑⢢⡀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡓⠂⢬⣉⠀⠀⣉⡚⠢⢄⣀⢗⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⢤⣈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢐⡦⠲⠤⣤⠀⣿⢁⠀⠀⠀⠒⠁⡀⡰⠀⠀⠀⠂⠀⢠⠊⣡⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡌⠀⢀⡴⠁⠀⣿⡀⠧⠀⠀⢀⣠⠤⠶⠶⠶⠶⠶⠶⢶⣦⣼⠂⠀⠛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠙⡄⠀⢼⠁⠀⢀⣯⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠉⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣇⠈⡆⣧⠀⢀⠀⢽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠠⢤⡋⢦⡀
⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⡀⠀⠀⠀⠨⢆⡘⢻⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠒⢤⡇⠀⠳⡄
⠀⠀⠀⠀⠈⠀⣀⠴⠀⠀⡏⠀⠀⡀⠀⠈⠷⠤⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠠⠔⠁⢈⠐⠈⠹⢆⠀⢄⠀⠀⡂⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⢴⣧
⠀⠀⠀⠀⠀⠀⠐⠂⠒⠁⠀⠀⠑⠀⠀⠚⠑⠀⠀⣷⠀⠀⠀⠠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠌⠃⢸⡀⠀⠀⢰⡿⣿⠿⡖⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣡⡞⠀⡆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡥⠖⠂⢹⠃⠀⠀⠀⠱⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⣿⣿⡇⢠⣧⠃
⠀⠀⠀⠀⠀⠀⠤⠀⠈⢀⣠⠤⣧⣀⡔⠋⠀⠀⠀⡟⠀⠀⠀⠀⢀⠈⢀⠀⠀⣀⣠⣤⣄⡀⠀⢀⠀⣰⠰⠁⣿⣿⡟⢁⣾⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣒⣁⣠⠟⠀⡽⠀⠀⠀⡠⢞⣻⣾⣷⣶⡶⠣⠀⠀⠉⠉⢿⣛⡽⠄⠉⠓⠁⠠⠃⠀⠈⡋⠁⢰⡿⠯⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣷⡠⢠⡞⡴⠉⢀⡞⠛⠛⠋⠀⠃⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠘⡇⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⠘⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣾⠳⣄⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢄⠀⠀⠀⠀⣀⡀⠀⢀⣤⣤⡼⠁⠀⠀⠀⠀⠀⠀⢸⠆⠀⠈⢣⠑⠌⢷⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⡆⠀⡤⠤⡈⠙⠲⠛⠁⠀⠀⠀⠀⠴⠒⣦⣤⡔⠃⠀⠀⡀⢀⡇⠀⠈⣷⢸⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣦⡐⢄⠀⠀⣧⡼⠓⣦⣀⣀⣤⣴⠒⣶⣿⠀⢸⣿⠃⠀⠀⡆⠀⣷⣾⣅⢠⡄⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠙⣶⢵⣦⣼⣁⣴⣿⣿⣿⣿⣿⣼⣿⡿⠀⠿⠃⢀⠀⢰⢀⢀⡿⢳⡙⣿⠁⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠀⠀⠀⣨⡛⠛⠻⠍⡻⡏⢹⣿⣿⡿⠁⠀⠀⠀⡀⢸⣸⠿⠟⠁⢰⠃⢸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡔⠀⠀⠈⠙⢛⠓⢦⡈⠣⠈⠿⠏⣄⣀⡀⠀⠀⣷⠿⠃⢀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⠰⡇⠀⠀⠀⠀⡀⠀⠁⠀⠰⠒⠛⠉⢁⠄⣿⣿⣁⡤⠖⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡆⠀⠀⠈⠳⣀⠀⠀⠀⢀⠔⠚⠙⢲⣷⡶⣶⣶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡄⠀⠈⠀⠀⣠⠀⠀⠀⠀⠀⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠘⠛⠁⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠶⠶⠖⢀⠀⠀⢸⡆⠀⠀⠰⠀⠀⠄⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⢰⡇⠀⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⢀⠀⢐⡀⠀⢱⠼⣄⠗⠀⡌⢀⡀⣁⠀⢆⣨⣔⠁⡀⠀⠀⣊⣀⡔⢉⣩⠝⠛⢳⢶⡄
⠀⠀⠀⠀⠴⢛⠉⣉⢉⡶⣤⣾⡿⠃⠀⠒⠀⠙⣦⠈⠄⢨⡷⠀⠀⣸⠁⠀⠠⠃⠀⢠⠛⡆⡙⢩⣄⠀⣬⠾⠏⠉
⠀⠀⠀⠀⠀⠻⠲⢪⠄⢀⠄⠀⠊⢳⢀⡇⠀⠛⠻⢷⠐⣿⣟⠀⢸⠟⠓⠐⠀⠛⠷⢊⢀⡇⠘⠂⡁⠀⠀⢠
⠀⠀⠀⠀⠀⠞⢆⠁⢸⠏⠲⠇⢀⣴⡟⡤⠀⠀⠀⠈⠣⣘⣍⡲⠿⡀⠂⠀⣄⡀⢆⡇⠈⠃⡀⠾⠃⡆⠈⠀⠀⠀⣠⡆
⠀⠀⠀⣧⣠⠀⡺⢀⠌⠀⠙⠆⠈⢀⣦⢭⡀⡆⠀⠆⠀⢸⠎⠉⢳⠂⠀⠀⠠⣢⣯⣀⠘⠟⣡⠄⠳⡁⠀⢱⠀⣾⠟⠁
⠀⠀⠀⠛⠃⡜⣷⣻⡆⠀⠘⢾⣟⠉⠈⢀⣘⣥⣤⠞⠋⢻⠖⠦⠼⢦⡶⣤⠂⡹⠎⠉⠀⣤⣤⣾⠧⡀⢀⡈⢆⢁⣴⠇
⠀⠀⠀⠀⣴⠀⢀⠿⠃⣄⠴⠿⢭⣝⢛⣿⠏⣽⡏⢀⣼⣫⣷⣤⣤⣾⣧⠈⢷⣾⣷⠻⠟⢫⣦⣀⠀⠟⠛⠁⠪⡘⠋
⠀⠀⠀⠀⠘⠃⣠⠞⢻⠛⠻⢶⣶⣌⣿⡏⠠⣿⡇⠀⠁⣰⡇⢀⠀⠉⣿⡇⠘⣧⢁⡴⠟⠛⠛⠛⠛⢣⣄⣠⠀⠇⠀⢀
⠀⠀⠀⠀⢇⣴⣤⡾⠛⠻⠛⣶⣦⣽⡿⡇⢀⣿⣿⠀⢤⣼⢧⡀⢀⣠⣿⡇⠸⣷⣋⣤⡶⠿⠋⠛⠚⠻⢿⣏⣤⠸⣾⠏
⠀⠀⠀⣦⡾⠛⣥⣴⣿⣿⠗⢶⣽⣿⡿⢷⠀⣹⣿⣷⣾⠷⣞⣿⡀⢹⣟⠃⣼⢻⣷⢀⣤⣶⣾⣿⣶⣄⡀⠙⣆⠁⡏
⠀⠀⠀⣾⢁⣾⣿⣿⣿⠁⢘⣾⢩⣷⡄⠸⢧⠸⠟⠁⠻⣶⢛⣿⠣⠈⢻⡝⢩⣟⣿⣿⠸⠃⣸⣷⣿⣿⣷⠄⢸⡄⡏
⠀⠀⠀⡇⢸⣿⣻⣿⣿⠀⠙⢹⣿⣿⡇⡀⠀⠀⢀⣤⣴⣟⣸⠁⠄⠤⠆⣰⡾⣿⣿⠟⣩⠿⢻⣿⣿⣿⣿⠇⣼⡇⣜⢠
⠀⠀⠀⣧⠈⣿⣿⣿⣿⣤⡆⠀⣿⠟⣡⣽⣾⡿⣿⣿⡿⠛⠁⢠⣤⣶⣶⣶⣦⡝⠿⣄⣩⡞⢁⣿⣾⣿⡏⠠⣿⠁⡈⢱
⠀⠀⠀⢻⣂⣹⣿⣿⣿⣏⠀⠀⣌⠠⣿⣿⡿⢻⣿⣿⡗⢀⡄⣿⣿⣿⣾⣿⣿⡏⠈⢮⣟⠡⢼⣿⣿⣿⣶⣾⣯⣀⡼⠆
⠀⠀⠀⢿⣿⣀⡀⠙⠛⢻⣄⠰⡙⠀⠙⠓⠞⠛⢋⣍⢰⣿⣷⠈⠉⣛⠿⠟⠛⠁⠁⣾⡝⢃⣾⡿⠟⠛⠛⡹⢿⠁⣴⣆
⠀⠀⠀⣿⠘⣿⣿⣶⣄⠀⠙⢿⡵⢄⣉⣠⡘⠶⠀⢱⣾⣿⣿⡄⠀⠆⠐⢉⣀⣀⣴⡟⣠⠾⠏⠀⡄⣰⣾⢓⣾⢿⢿⡌⠁
⠀⠀⠀⢹⢧⢛⣽⣿⣿⣶⣀⠈⠹⡄⢭⠟⣧⠀⠉⠸⣿⡟⠻⠃⠀⣀⢆⣻⡷⢡⢎⡾⠻⠆⠀⢴⡞⢛⠁⠈⠉⠿⡁⠁
⠀⠀⠀⢠⡈⠛⢛⣿⢻⣿⣆⠀⠂⣷⠀⣺⠸⣷⣶⣤⣦⣶⡄⣄⢢⣽⣾⣧⠁⠰⡿⠓⠀⠀⣵⠏⡹⠋⠀⢰⡀⣦⠄
⠀⠀⠀⢈⣄⠀⠘⢛⣷⣿⣿⡴⣇⢸⡀⢸⠇⣿⣍⣼⢩⣿⣿⡿⠛⣯⡿⢱⠀⡘⣟⠊⠁⣰⡏⣰⡇⠀⠀⣼⡟⢁
⠀⠀⠀⢸⡟⢦⡴⠛⣛⠟⢻⣿⣷⠾⠁⣻⡇⠃⠻⢿⣿⣈⠛⣷⠷⣿⠡⠸⡙⠻⠻⠶⣤⣹⡿⠋⢣⣔⠂⣝⣠⠙⠳⠄
⠀⠀⠀⠉⠀⠸⠀⣸⡉⢆⢠⠹⠿⠒⢡⣿⠋⠲⠀⢄⣈⡉⢁⡀⠀⠀⢠⠔⣱⡶⠛⠛⣯⡁⢀⡁⠻⠞⠓⢧⠄
⠀⠀⠀⠀⠀⡽⢳⢿⠃⢈⣿⣄⠟⠛⠛⢿⣏⠋⠑⠀⠤⠴⠂⠁⠠⠘⢱⣿⠟⠒⢳⡿⡤⠟⡁⠐⠀⠂⠀⠐
⠀⠀⠀⠀⠀⢦⠛⣠⠆⡘⢿⡎⠐⢀⠂⡀⠬⠅⠠⠆⡀⢠⡄⠠⠄⢀⠉⣬⠂⣱⣿⡷⠙⡤⠁⠤⡀⠔⢷⠶⠂
⠀⠀⠀⠀⠀⢴⣗⠣⠀⠀⠈⢷⠀⠀⠲⣂⣀⣠⡁⠀⠀⣾⣽⠂⠂⢈⣡⣀⡚⠙⠋⡀⠀⠀⠈⠰⣿⣿⠞
⠀⠀⠀⢀⠀⢰⠋⣅⣿⡄⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⢻⡆⢳⣶⣿⣿⣿⣿⣿⣏⡉⠀⠀⢰⣶⠻⣷⣤
⠀⠀⠀⠀⠀⠈⢀⠉⠙⣧⣠⠀⠣⠘⠋⠙⣿⣿⣿⣿⣿⣿⡶⣾⣿⣿⣿⣿⣿⠟⠛⠔⠁⣠⣾⢟⠴⠐⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠸⣀⠁⢈⢭⠀⡁⡘⠡⠀⠩⡩⢉⠛⣹⣟⣃⣿⢋⠈⢅⠀⠉⢉⠛⠤⠀⠩⢁⠄⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠰⢀⠀⠠⢤⡄⠀⠀⠀⠀⠀⠳⡙⢿⣿⢿⡍⠀⢀⡀⠀⢠⠄⠒⠂⢠⠀⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠃⠀⢠⡄⠘⠀⠂⠀⠀⢀⡀⢀⠐⣈⠁⠠⠄⣤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠈⠓⠎⠻⠁⠔⡥⠉⣁
⠀⠀⠀⢀⠔⣉⢀⣀⠄⠀⡀⣀⠤⣀⡀⠀⠀⠚⠫
⠀⠀⠰⠟⠉⣤⡊⠤⢒⠁⠀⠀⠀⠀⠠⡄⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⢢⡀⠈⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠲⡆⢠⠤⡀⠀⠙⣄⢼⠀⠀⠀⣀⣀⠀⠀⡠⠉⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡇⠀⠘⠄⠀⠈⢎⢧⢀⠊⠤⠈⠣⠘⡀⠀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠸⡄⠀⢸⠀⠀⠈⠿⠋⠘⠁⢀⡄⠀⡇⠀⠙⠃⢐
⠀⠀⠀⠀⠀⢀⡠⠐⠀⠉⠢⠤⣴⠋⠓⠓⠀⠀⠀⢀⡀⠘⢇⠉⠀⠀⠈⠑⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⢤
⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠈⠄⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡌⠀⠀⠀⠀⠚⢢
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠁⠘⠁⡀⣀⣶⣿⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠀⠄⠀⠀⢤⠀⠀⠀⠐⠂⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢠⡡⠖⠂⠈⠁⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣁⢎⠀⠀⠀⠀⠀⠀⠦⣀⠤⢧⡨
⢇⠀⠀⠀⠀⢀⣀⠤⠔⠂⠁⠀⠀⠈⢁⠐⠈⠉⠉⠈⠉⠉⠛⠇⠀⡆⠀⠀⡠⢄⡀⠀⣐⠩⠃⠀⠈⠢⠕⢀
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⣀⠠⠐⠚⢦⡀⠘⠀⢰⠀⢠⠉⣳⠉⠛⢄⠈⢄⠀⠀⡴⠎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⢰⢦⡸⠷⠤⡇⠀⠒⠈⠁⢀⢧⠀⣀⣁⡦⠑⠢⢄⣠⠀⠃⠒⠈⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠑⢄⣤⠆⠀⢀⣠⢊⣠⡘⢮⡁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠉⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡨⠷⠦⠤⡤⠔⢉⡸⠋⠁⡟⠈⣳⠀⠀⠈⠢⣀⣀⣠⠴⠒⠒⠂⡏⠀⠀⠀⠀⠀⢹⡀⠀⢀⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠘⢊⠝⠃⠌⠀⢸⠺⡇⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⡎⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠘⠃⠐⡄⠀⠐⡄⠀⢐⠂⠒⢸⡀⢹⣍⠀⠀⠀⠀⢀⡀⠤⠐⠒⡖⠀⠀⠀⣀⣤⡞⠥⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠠⠰⡅⢀⡼⠛⠛⠛⠀⠀⡌⢣⡆⢹⡯⡑⠶⠈⠀⠀⠀⠀⣸⠀⡠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡄⢀⡀⢀⢒⠟⠁⠀⠀⠀⠀⠀⢈⠷⡆⠐⠀⠓⠈⠆⠡⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢧⠀⠀⠀⠀⠀⠀⠘⠓⠢⢌⠁⠀⠀⠀⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⠀⠀⠀⣀⣀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⢉⣭⠡⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠤⠤⠐⠲⠈⠙⠛⠛⠉⠃⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡆⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢻⡄⢜⡵⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣄⣈⠳⣯⣬⣿⠀⢧⠀⣺⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢧⣾⣿⣿⣿⣿⣿⡏⣿⡿⠃⢠⣼⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢇⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢸⢹⣿⣿⡿⣼⣿⠃⠀⢰⣽⣿⣿⣿⡿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⡀⠀⠀⠙⠄⠀⠀⠀⡜⣼⡿⢫⣾⣿⣿⠀⠀⣿⣿⣾⣿⣿⣿⣟⢿⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⠀⢀⡰⣾⠀⢀⡼⠰⠯⠴⠛⠉⠀⠈⢀⣠⣿⣿⣿⡟⣿⣿⣿⣯⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠦⡄⠀⠘⢣⡌⠉⠙⠁⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣷⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡦⢄⢀⣏⠀⠀⠀⠐⠂⠀⠈⠉⠀⠀⠙⠿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠠⠤⠤⠤⠄⠀⢀⣠⠔⠺⣿⣷⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣉⠛⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢀⣀⠀⢀⣚⢁⡠⠋⠐⠒⠠⠤⣤⣀⣀⡀⠈⠁⢀⣀⠼⠟⠉⠀⠈⢫⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⡿⣿⠛⠋⣿⠿⣿⣿⣿⣿⣿⡇
⠩⠟⠓⠉⣠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⡜⠡⠾⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣶⣦⣤⣴⣶⣾⣿⣿⣿⣷⣤⡎⠀⢀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠚⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠠⠤⠴⠚⠛⣯⣿⣿⣿⣿⣿⠿⢿⡗⠙⢿⣿⣿⣷⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠠⣴⠶⠒⠀⠀⠀⡀⠤⠚⠉⠁⢈⣿⠉⡿⣅⠀⠀⠸⡀⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠲⠒⠒⠚⠛⠓⠂⠂⠀⠒⠛⠁⠀⠐⠓⢺⡀⠀⢀⢱⡀⠘⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⠓⡄⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⣀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠴⠤⠐⠀⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⡴⡋⠁⠀⠐⠋⠉⠀⠀⠀⠀⠀⣉⠁⢦⠀⠀⠀⠀⡀⠒⠊⣀⡩⢀⣁⡶⠊⠉⠉⠉⠳⠀⣍⠃⠬⣵⢦⡀
⠀⠀⢀⠞⠸⠁⠀⠀⠀⠀⠀⠀⠀⠄⣀⣦⡴⠖⣿⠀⠀⠀⢸⠀⢀⣀⠦⡄⠣⡿⠧⣀⠀⠘⣦⠀⠀⠀⢳⡀⠀⠀⠹⣆
⠀⠀⡖⢈⠁⡀⠀⠀⢀⣀⠴⠓⣻⡯⢟⠝⠀⠀⢻⠀⠀⠀⡸⠁⢀⣰⠚⠀⡘⠀⠀⠈⢷⣦⣘⢀⡀⠀⠘⢧⠀⠀⠀⡎⣇
⠀⣘⡔⠁⠀⠁⠀⡠⠁⣠⢶⣰⣿⠓⠀⠀⢹⣷⢀⠀⠀⠀⠀⠠⢴⡷⠈⠀⠱⣿⠍⠀⠀⠸⣟⣦⣽⣆⠈⠊⡀⠀⠀⠆⢸
⢠⣢⠀⠀⠀⠄⠠⠂⢾⣬⢮⣿⣿⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢫⠸⣭⡋⣦⠀⢷⡀⠀⠇⢸
⢀⢋⠐⣤⠀⠀⡞⠃⠀⢹⣿⡏⡿⠀⠀⡀⠀⢀⢣⠤⠀⠀⠀⠠⠒⠀⠑⢲⣔⡀⠀⠀⠀⠀⠀⠰⣧⡾⣿⠀⠘⢱⠀⢀⡾
⠸⠇⢀⠈⠉⠺⡇⢀⣴⢉⣿⣿⣄⡀⠀⠘⢰⠿⠟⠁⠀⠀⠀⠀⠀⠠⠀⠈⠘⢶⣌⠂⠀⠀⠀⠀⣿⡟⣿⡻⠀⠀⣸⡶⡇
⠀⢁⠘⠀⠷⣄⡀⠂⡹⡦⣽⡿⣿⣷⣷⣶⣉⡂⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠀⠀⠀⠀⣴⣿⡟⣿⣷⢀⣴⣷⣷⡇
⠀⠠⠤⠂⠒⠓⢩⠴⠾⠁⠀⢹⠿⠛⠛⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⢘⣟⣤⣿⣿⣇⣾⣥⢄⣇⣞⣇⡾⠉
⠀⠁⠀⠀⠀⠈⠈⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⠀⠒⠀⠀⢀⠀⠀⠳⠿⠛⠉⠉⠙⢿⠛⣩⢿⡿⣿⣉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠻⣦⡴⣶⠂⢲⢦⣶⠾⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⡠⠤⣀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⢠⠁⣀⠀⠐⠀⠀⠀⠀⠀⠀⣘⣟⡀⠀⠀⠀⠀⢰⡅⠄⣸⡄⠀⠀⠀⡀⣿⠛⣷⡆
⠀⠀⠀⠀⠀⢀⣀⣀⠔⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⠿⠁⠀⠀⠀⠹⣇⠀⣨⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⢋⡀⢀⠱⡄⠀⠀⠀⠐⠁⡀⡀⠀⢰⠀⠀⠀⠰⡶⣶⣶⡆⠀⠀⠀⢠⣾⠿⢿⣿⡆
⠀⠀⠀⠀⠰⡀⠀⠠⢀⠇⠀⠀⠀⠠⢀⢁⡉⠀⠰⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⢻⠟⠻⠟
⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣿⣷⣀⠀⠀⠀⠀⣠⠀⣴⣦⠀⣄⠀⠀⠀⠀⣠⣹⡄⣀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠘⡟⠉⠉⢻⠀⠀⠀⠀⠛⢶⣬⣥⠶⠛⠀⠀⠀⢰⡽⠛⢻⣏⠀⠀⠀⠀⣠⣭⣥⣄
⠀⠀⠀⠀⠀⠑⠄⠠⠂⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠘⠿⠀⠠⠋⠀⠀⠀⠰⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠋⣠⣤⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣆
⠀⠀⠀⠀⠀⣒⠃⢑⣂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠀⣿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠇⠏⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠙⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠟⠷⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣶⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⠲⠶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣁⣤⣤⣤⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠓⠶⢶⣬⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠁⢀⣀⣀⣿⣿⣦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠛⠳⣦⣌⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⢻⣿⣷⡾⠋⢉⣁⣤⣿⣿⣧⣀⣴⣾⣿⣿⠿⠿⠿⠿⠿⣿⣶⣤⡀⠀⠀⣠⣿⣿⡛⠛⠶⣄⠘⣿⣿⡇⠀⠀⢀⣴⣿⠟⢹⣿
⠀⠀⠀⠀⣸⡿⣿⣧⠀⠀⠀⠈⣿⣿⣷⡿⠛⣉⣠⣤⣿⣿⡿⠟⠉⠀⠀⠀⠀⣀⣤⣤⣾⣿⣿⣷⣴⣾⣿⣿⣅⠙⠻⣦⣈⣿⣿⡿⠀⢀⣴⣿⡿⠋⠀⢸⣿
⠀⠀⠀⢀⣿⡇⠸⣿⣧⠀⠀⠀⠘⣿⣿⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⣹⣿⡿⠁⠉⠙⢿⣦⣈⣿⣿⣿⣷⣾⣿⡿⠋⠀⠀⠀⢸⣿
⠀⠀⠀⣼⣿⠁⠀⢹⣿⣦⠀⣠⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣄⡀⠀⠉⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⣸⣿⠇⠀⣤⡀⢻⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉⠉⠉⠉⠙⣛⣿⣦⡀⠀⠉⠈⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⣰⣿⡏⠀⠀⢹⣷⠀⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠛⠛⠛⠛⠀⠀⠀⠀⠙⢿⣿⡄⠀⠀⠀⠀⠀⢸⣿⡆
⢠⣿⡿⠀⠀⠀⣼⣿⣇⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠀⠀⠘⣿⣷
⣼⣿⠃⠀⠀⢠⣿⣿⣿⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠿⢿⣿⣿⣦⣄⠀⠀⢠⣿⣿
⣿⣿⡀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠈⠻⣿⣿⣆⣠⣾⣿⡿
⢿⣿⡇⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠛⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣀⣠⣀⡀⠀⠀⠀⢹⣿⣿⣿⣿⠟
⢸⣿⣧⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠻⣿⣧⠀⠀⠸⣿⣿⡿⠋
⠈⣿⣿⡄⠸⣿⡟⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢹⣿⡄⠀⠀⣿⣿⠁
⠀⠘⣿⣿⡀⠉⠁⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣴⠇⠀⠀⠀⠀⢸⣧⣴⣶⣶⡄⠈⣿⡇⠀⢀⣿⣿
⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠈⠛⠗⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣷⣶⣦⣄⠉⠛⠛⠛⠉⠀⢠⡆⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⣠⣾⣿⡿⠿⠛⠛⠛⠛⠻⣿⣿⣦⡀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⠀⠀⣼⣿⠁
⠀⠀⠀⠀⣿⣿⡟⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠛⣶⣾⡿⠟⠋⢁⣀⣀⣤⣤⣶⡾⠿⠿⠿⠿⠷⠀⠀⠀⢻⣧⠀⢀⣼⡿⠿⠿⠿⣷⣿⡏⠀⣰⣿⠃
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠁⠰⠶⠛⢻⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠈⠀⠀⠀⠀⠀⠀⠉⠀⣴⣿⣧⡀
⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣴⣶⡄⠀⣠⣤⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠃⠀⠀⠀⠀⠀⠸⣿⡄⠀⠿⠿⠰⣿⠇⠀⣼⡿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠛⠳⢶⣤⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣾⣿⠿⣆⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⣿⣿⠃⠀⠿⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡻⣿⠿⣿⡿⢿⡿⠟⣿⠉⣿⣴⣿⠏⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⢻⣿⣿⣶⣿⣦⣼⣷⣤⣿⣶⣿⡿⠃⠀⢠⡀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣴⣿⣧⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⢠⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⢰⡆⠀⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⢸⣷⡀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⣿⣷⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⢿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣤⡻⣿⣦⣤⣤⣾⡿⠿⠛⠻⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣦⡉⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠙⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠈⠻⣿⣿⣿⣶⣶⣶⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣶⣾⣿⡿⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠿⠿⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢠⣴⡶⣦⡄
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⣿⣿⣿⣷⣶⣶⣶⢂⣼⣿⠟⣿⣿⡇⢸⣿⣿⣯⣭⣍⠁⠛⣿⣿⡟⠛⠁⣼⣿⢿⣿⣤⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠿⠿⠿⠁⠀⠀⠀⠠⠿⠿⠛⠛⠿⠿⠇⠶⣾⢿⣿⡿⠟⠀⠰⠿⠿⠀⠀⠀⠙⠛⠛⠙⠛⠃
⠀⠀⣸⣿⣿⣿⣿⣤⣤⣤⣤⣤⡄⣾⣿⠇⢰⣿⡿⢠⣿⣿⠿⢿⣿⡷⢀⣾⣿⢂⣴⣾⡿⠿⣿⣷⡄⣸⣿⡗⢠⣿⣿⢂⣶⣿⣿⠿⡿⠟
⠀⢠⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⢸⣿⣿⣠⣿⣿⠃⣾⣿⠿⣿⣿⣍⠁⣼⣿⡏⢸⣿⣿⣀⣴⣿⡿⢠⣿⣿⣁⣼⣿⠏⣨⣛⣛⣻⣿⣿⠆
⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠀⠈⠉⠉⠀⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉










⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠙⠛⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠀⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠛⠁⠀⢹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠔⠈⠙⠻⣿⣿⣦⣄⠀⢠⣴⣾⣿⡿⠟⠉⠀⠑⢄⠀⠈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⡌⠀⠀⠀⠀⠀⠙⠻⠃⠀⠀⢻⠟⠋⡀⠀⠀⠀⠀⠀⠂⠀⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡠⠔⠂⢿⣿⠇⠀⡆⠀⠀⠀⠀⠶⠀⠀⢸⠀⠀⢸⠀⠀⠀⠺⠀⠃⠀⠀⡐⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢇⠀⢀⡀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⢀⠜⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡠⠃⠀⢸⣿⣿⠋⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠉⠐⠄⣀⠠⠊⠉⠒⠂⠐⠒⠊⠀⠀⠀⠀⠀⠠⣤⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢠⠔⠉⠀⠈⠀⠀⠄⠠⠀⠒⠋⠀⠈⠉⠐⢄⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢰⠁⠂⠈⠉⠐⠂⠀⢀⠀⢀⠠⠆⠀⠰⠀⠀⡀⠐⡀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠤⠄⠐⠂⠠⡀⠀⠘⠀⠀⣀⣇⠀⠠⠀⠈⠃⠀⡇⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠒⠴⠐⠈⠀⠀⠀⠑⠠⣌⣀⡼⠁⠀⡏
⠀⠀⠀⠀⢀⡔⠋⢉⠖⠂⠀⠢⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡤⠒⠒⠢⣄
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠒⠄⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠤⠖⠋⠀⠸⡁⠈⠁⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠓⠤⠄⣀⣀⡠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠠⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⠉⠉⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢰⣶⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣼⣿⣀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠋⠛⠃⠀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠐⠁⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⢀⣠⣴⠿⠛⠛⠛⠛⠛⠛⣿⠟⠀⠀⠀⡶⠻⣿⣶⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⠻⠻⢶⣤⡀
⣤⣶⠟⠉⠀⠀⠀⠀⠀⣤⣤⣤⡏⠀⠀⠀⠀⠖⠀⠈⣿⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⡄⠀⠀⠀⠈⠙⢷⣦
⠈⠛⠷⣦⣀⠀⠀⠀⠀⠈⠉⢹⠇⠀⠀⠀⠀⠈⣄⠀⠀⠀⠉⠉⠉⠀⠉⠙⢶⡄⠀⠀⠀⠀⢫⠉⠉⠁⢀⠀⢀⣠⡶⠟⠋
⠀⠀⠀⠀⠙⠻⠿⠶⠶⠶⠶⣿⠀⠀⠀⠀⢠⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢈⢷⠾⠶⠾⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠠⠄⠀⡠⠂⠀⣀⣬⡆⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠞⢹⣷⣄⣀⠀⠀⠀⠀⠀⣠⡾⡏⡸⣿⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣄⠀⢊⢠⣿⠻⣿⠆⠀⣠⣿⣻⠀⢠⠇⠙⠀⠀⠀⠀⢆⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠘⣆⠃⠀⠈⠒⠐⠊⠏⢀⠀⡙⠁⠉⠁⠞⠀⠀⠀⠀⠀⠀⠸⡆⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠀⠘⣆⠁⠀⠀⠀⢀⠀⠀⠀⢣⢀⡀⠀⢃⡇⠀⠀⠀⠀⠀⠀⣯⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠇⠀⠀⠀⠀⣿⡄⡀⣀⣠⣼⣦⡀⠀⢸⣎⣁⣠⢸⡇⠀⠀⠀⠀⠀⠀⣶⠀⡄⢸⡏⠢⡀
⠀⠀⠀⢀⡤⠂⠁⡌⠙⠀⠀⠀⠀⠀⠀⣰⠟⣹⠇⠈⢷⣮⣭⡿⣶⣾⣫⡟⠉⡃⠃⠀⠀⠀⠀⠀⠀⡏⠀⡀⢸⡇⠀⠘⡆
⠀⠀⠀⡎⢸⠀⠀⢣⠀⠀⠀⠀⠀⠀⣨⣴⣾⣿⠀⠀⠀⠙⠿⢿⣿⠿⠛⡀⣰⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⣷⣿⡁⠀⠇⡇
⠀⠀⠀⠘⢆⠀⠀⠀⢣⡀⠐⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠲⠤⠤⠴⠊⠀⡿⠀⢠⠁⠀⠀⠀⢠⠏⠀⢀⣿⡿⠁⠐⣰⠃
⠀⠀⠀⠀⠀⢧⠀⠀⠀⢷⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣿⣿⣦⣾⠁⠀⡎⠀⠀⠀⢠⠏⠀⠀⢸⠟⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⣇⠀⢠⣻⡛⠿⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡰⠋⠀⠀⡰⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠘⠳⣦⣄⣽⠀⠀⠹⣿⣦⣀⡀⠀⠉⠉⠁⠀⠘⢿⣿⣿⣷⠀⢸⣷⣤⡞⠁⠀⡠⠊⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⣤⣀⣼⣿⣅⠀⠀⠢⢤⣀⠀⠀⠀⠈⠉⠹⠀⢸⣿⡟⣠⠖⢊⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⠭⠑⠊⠛⠉⢻⠶⠔⢺⣦⣄⣤⣠⡶⠀⣾⣿⡧⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣿⠈⠻⣭⡥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠀⠀⠀⠀⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠄⠐⠋⠀⠀⠀⠀⠀⠑⠦⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠊⠀⠀⠀⠀⢠⠚⠐⠢⣴⠓⢦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢠⠞⢩⠾⠃⠁⡂⠈⠁⠀⢷⠈⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠬⠤⠤⠔⠃⠀⠀⠀⠀⣠⣁⠐⠠⠄⢀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠰⠒⠋⠉⣀⠀⡀⣠⢀⢀⢀⠄⡄⡄⠀⠀⠀⡠⠶⠻⠟⠓⠿⡄⢮⡐⠤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⠔⡡⠊⣴⡵⠧⠧⠦⢤⢄⡶⠀⠐⠻⠀⠀⠐⠊⠉⠀⡇⠀⠙⢮⠁⠀⠈⠳⣄
⠀⠀⠀⠀⢠⠀⡇⠀⠐⡡⢊⠄⣇⠀⠀⠀⠀⠀⡼⢜⡉⠀⠀⠀⠠⠀⠀⠀⠀⢰⠁⠀⠀⠈⢧⠠⠀⠀⠈⢧
⠀⠀⠀⡇⠘⠆⢱⠀⠀⠱⢡⢊⡼⢆⠀⠀⠀⢠⡇⠈⢀⡤⡶⠲⡲⢲⠦⡄⢠⠃⠀⠀⠀⢠⠎⠀⠀⢀⠀⢀⠇
⠀⠀⠀⠘⠀⠀⠀⢧⠀⠀⠡⢋⠄⡈⢦⠀⠀⢸⡇⠠⡃⢧⢁⣃⣃⢃⠞⡰⠃⠀⠀⢀⡴⠃⢀⢠⠘⠀⡠⠋⡠⢂⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠁⠊⡤⠈⠳⠄⠠⠃⠀⠻⠿⠞⠉⠉⢀⡜⠁⠀⣀⠔⠉⢀⢰⠸⠀⡠⠚⠀⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡚⠒⠂⠚⠁⡄⠗⡌⢈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠀⠀⠀⠴⢂⡀⠼⢛⡩⠝⣋⠥⠐⢂⠀⢀⠦⠋⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢖⣪⠥⢐⣨⠥⠒⣉⠥⠖⣊⡥⠀⢈⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢚⣉⠤⢊⣩⠔⠒⣉⠤⠒⢉⡤⠒⢁⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠰⢚⣩⠴⢚⡩⠴⠚⢉⡠⠖⢋⠡⠔⣊⡥⢸
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⠴⢊⣡⠔⠊⢉⡠⠔⢊⣡⠔⣊⡥⠔⠉⣀⠤⡎
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡠⠚⢉⡠⠔⢊⡡⠔⢚⡩⠔⠋⣁⠤⢒⡫⠕⣂⡇
⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠴⠚⣁⠴⢊⡡⠴⠊⣁⡤⠔⣪⠔⢊⡭⠖⢋⣠⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡥⠒⢉⡤⠊⣁⠤⠒⣉⠤⠚⣉⠴⠊⣡⠔⣊⠅⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠴⠊⣡⠔⠊⣡⡴⡎⠁⠔⢋⡤⠒⣉⠤⠊⣡⠄⡟
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠊⡠⠔⣉⣼⠁⢠⠀⠈⢁⡴⢋⡠⠔⢉⠤⢚⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⣋⠔⣋⢴⡏⠀⢸⠀⠀⡠⠒⣩⠔⠊⣡⠔⢻
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢚⡥⢊⡥⢾⣀⠀⠸⠀⠀⠔⢋⡤⠒⣉⠤⠒⣳
⠀⠀⠀⠀⠀⠀⢇⠀⠐⣁⠔⢁⠴⢂⠼⢓⣾⠀⠀⢘⡡⠔⢋⡡⠖⣋⡅
⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠔⣡⠞⣡⠖⣡⢼⠀⠀⣩⠴⢚⣡⠖⣋⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠁⠊⡴⢊⠔⣋⠔⣻⠀⠀⠔⢊⡩⠔⣊⠕⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢞⡡⢚⡵⠊⣽⠀⠀⢋⠥⠖⣋⠥⡊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠀⠈⠀⢀⡴⠊⠈⡄⠀⠤⠖⣪⠵⣊⠔⠀⡇
⠀⠀⠀⠀⠠⠤⢀⣰⣏⡀⠠⠴⠞⢛⣀⣩⠾⠁⠀⠐⠉⠒⠋⠔⠁⠐⣇⠠⠄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠊⠉⠉⠀⣶⡋⠁⠀⠀⠀⠀⣀⣀⡤⠤⠖⢒⣒⡀⣉⠩⠭⠠⢤⡐⠀⢀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠁⠀⠀⠀⠀⠎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠹⣄⠡⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡈⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠔⠁
⠠⡄⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣶⠞⠁
⡆⢰⠀⠀⠀⠀⠀⠀⠀⣸⠉⠛⠿⠷⠶⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠑⠈⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠘⣜⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡁
⠀⠀⠘⣮⠢⡀⠀⠀⠀⢿⡀⠀⠀⠀⢠⡥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠎
⠀⠀⠀⠈⢷⡄⠀⠀⠀⢸⡇⡄⠀⠀⡟⠀⠀⢀⣠⣴⠶⠶⠶⣶⣤⡀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣠⠞⢡⡾
⠀⠀⠀⠀⠀⠘⢧⡀⠀⢘⣇⢳⡄⢸⠃⠀⠔⠋⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⡇⠀⠠⠈⣡⠶⠩
⠀⠀⠀⠀⠈⠀⠘⡇⠀⠀⢻⡄⠹⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢻⣿⣶⡆⠁⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⣠⡄⠀⠷⠀⢀⣼⠃
⠀⠀⠀⠀⠀⡀⠀⠁⠀⢹⣮⣷⠀⣿⠀⠀⠀⢀⣠⠔⢋⡀⣀⣀⣀⠉⢢⡀⠉⢿⠁⠀⠀⣀⣀⣀⣀⠀⣴⡶⠟⡉⢀⠀⠀⢰⡄⡎⠀⠀⡰
⠀⠀⠀⠀⠀⠱⡀⠀⠀⢸⣿⠏⢻⡏⠀⠀⠀⠀⠈⠃⠠⠄⢻⣶⣋⣸⡁⠲⠀⠀⠁⠒⠉⠀⠀⠈⢻⠞⢛⢊⡥⣤⠀⠙⠀⣼⢣⠀⠀⣎
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⣿⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢀⡞⠐⠀⠤⠙⠋⠤⠀⢀⡉⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⠓⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠄⠀⠓⠤⠤⠤⠴⠂⠀⣷⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢀⣸⣇⣠⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⢦⠀⠀⠀⠀⠀⠾⠶⣤⡀⠀⠀⠀⠀⠀⠀⢸⠆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠚⠁⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⢀⡆⠐⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢹⣯⣐⠂⠀⠀⠤⠴⠒⠒⠉⠉⠉⠓⢤⣤⡀⠀⠀⠀⠀⢰⣨⠘⠁⣀⣵⣤⣄⣴⡟⠀⠀⡁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣿⠉⠛⠟⠛⠛⠛⢻⣿⡛⠻⣶⣤⣤⣿⣷⣄⣀⣀⣠⡾⢳⣶⣾⡟⠋⠠⠏⠙⠛⠳⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⣄⠉⠐⠎⢛⡋⠈⣿⣿⣿⣿⠋⠀⡀⠙⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡈⠳⣄⠀⠀⠀⠙⣿⣿⣿⠏⠈⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠂⠀⠀⠙⠲⣬⡳⢄⡀⠀⠙⠿⠋⠀⢀⠔⠁⠀⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠀⠀⠀⢦⡀⠀⠙⠲⡍⠛⠀⠀⠃⠀⢀⠌⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠰⢄⡈⠒⠦⠴⠆⠘⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⣀⣤⣴⣾⣿⣿⣷⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡇⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣤⢸⣿⢸⣿⢸⣿⣧⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠻⡇⠀⡇⠀⡇⠀⡟⠉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⢠⡇⠀⠀⠀⠁⠀⠁⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢋⣉⣉⣉⡙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢁⡴⠞⠛⠉⠉⠉⠛⠻⢦⣌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡈⢿
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⣀⣀⣀⣀⣼⠇⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣉⣉⣉⣉⣉⣉⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣀⣀⣀⣿⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣿⣉⣉⠉⣈⠉⠉⢹⡆⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠
⢠⡶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣌⢿⡄⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢠⡾
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠻⣦⡈⠛⠷⣦⣤⣤⣤⣤⠶⠛⢁⣴⠟⠁
⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠈⠙⠳⢦⣤⣤⣄⣤⣤⣴⠾⠋⠁
⠀⠈⠙⠛⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣴⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢦⡀
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠙⠛⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠞⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿
⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶
⠀⢀⣴⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⢶⣄
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⣀⣙⣳⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣆
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣆⠀⠸⠟⢋⣁⣀⣀⣀⣀⡀⠀⠀⣰⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣘⠛⠛⠛⠛⠛⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠋⠀⠀⠉⠻⢷⣦⣄⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣀⣀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⣀⣀⣉⣉⣉⣀⣀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠘⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠚⠛⠛⠋⠉⠉⠛⠛⠛⠓⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠶⠄⣀⠀⠸⠀⠀⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠹⡇⠀⠀⠀⣠⡴⠯⠚⠉⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⢤
⠀⠀⠀⣀⡀⡀⢀⣀⣰⠲⢾⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣷⣰⠶⢤⠴⠒⠲⢤⣀
⢀⣴⣭⣍⡑⠀⠉⠀⢡⠘⣾⠃⠀⠀⠀⢠⣀⣀⡼⢤⣤⠚⠀⠀⠀⠘⣿⠀⠈⠁⢀⠞⠉⠉⠙⡀
⠎⡠⠤⠤⣙⠀⠀⡄⠀⢁⣿⠀⠀⠀⠀⠀⡤⠈⠀⠀⠁⠐⠄⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠄⠠⢵
⠀⡆⠀⠀⠀⣁⣇⡇⠀⠀⠻⣤⣀⡀⠀⢀⡑⠚⠤⠤⠜⢁⡠⠀⢂⣿⠀⠀⠀⢀⣌⠐⠂⠉⠙⡞
⠀⢿⠀⠀⠀⡏⠁⠉⠀⠀⠀⠘⣇⠠⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡞⡿⠀⢠⠔⠉⠈⣧⠀⠀⢠⡇
⠀⢸⡄⠀⠀⣇⠀⠀⠀⠰⡀⠀⠹⣆⠫⠀⡀⠀⠀⣀⠨⠁⠀⠀⣹⠃⠀⣼⠀⠀⢀⡏⠀⠀⢸⠃
⠀⢸⠇⠀⠀⠈⠦⣀⠀⠀⠹⣄⠀⠘⢧⡉⠀⠀⠀⠀⠀⠀⠀⣶⠏⠀⢀⡇⢀⡤⠚⠀⠀⠀⢻⡀
⢠⠇⠀⠀⠀⠀⢀⠈⢣⠀⠀⠙⣦⠀⠀⠙⠚⢷⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁⢀⠀⢄⠀⠀⠀⠀⢱
⠇⠂⠀⠀⢉⡀⠙⣤⠂⠀⠀⠀⠈⢷⣄⠀⠀⠈⠀⠀⠀⠈⠂⠀⢀⡾⠃⠀⠈⡧⠋⠀⠀⠀⠀⠀⡵
⠈⠲⠦⠑⠦⠜⠒⠋⠀⠀⠀⠀⠀⠀⣹⠷⢤⣤⣀⠀⠀⣤⡤⠴⣏⠁⠀⠀⠀⠈⠒⠅⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⢀⡤⠴⠀⠀⢠⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠁⠀⠀⠀⠀⠁⢄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⠀⠀⠀⠀⠀⠈⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⠀⠀⠀⠀⠀⡀⠃⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣽⣵⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣷⣶⣷

⠀⣴⣾⣿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⡄
⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠋⠁⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⢀⣀⣤⣤⣶⣶⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢢⠤⠒⠒⠒⠂⢒⠖⠁⣇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠞⠀⠀⠀⠀⠀⢇⣀⡔⠀⠑⢄⠀⠀⠐⢁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠦⡀⠀⠀⠀⠀⠀⠀⢃⠀⠐⢺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣉⠁⢀⠤⡒⠄⠀⠀⠀⠀⠄⡰⠂⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠟⠀⠎⠀⢻⡇⠀⠀⠀⠀⠀⠁⠀⠀⡤⠑⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡩⠦⠴⢀⠰⢠⡄⠤⠁⠀⠀⠀⠀⢀⠃⠀⡠⡅⠀⡐⠀⢸⠂⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⣏⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⠀⠄⠔⠁⠀⡰⠯⠧⢏⢀⡼⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡑⡀⢌⠒⠀⠚⠻⣇⠀⠀⠀⠀⠀⢠⠖⡦⢠⡌⠀⠀⠀⠀⠁⠀⢀⢂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠄⠳⠄⠀⠤⠎⠁⠂⠀⠀⢀⠃⠎⡠⠢⠊⠀⠀⠀⠀⢀⡰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠐⢳⣶⣶⣦⠀⢀⣠⡤⠐⠁⠜⢰⠚⡄⠀⠀⣠⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠰⣿⡿⣋⠀⠀⠘⡞⠀⠀⠀⢀⡜⠀⠙⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠚⠁⠀⠈⠓⠒⠤⣘⣶⡒⠈⠒⠀⠀⠀⣛⣁⢀⠀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⡀⠠⣀⢀⡀⠀⠀⠀⠀⠒⠒⣒⣛⣆⠀⠀⠀⠀⡵⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⣀⣁⠸⡤⠒⠒⠂⠈⠉⠉⠁⠀⠀⠀⠉⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠠⢘⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⡆⡴⠑⡆⢴⠊⠀⠐⡖⠀⠐⡇⠀⡆⠀⠀⢰⠉⠀⠀⠸⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠏⠉⡇⢧⣀⡇⢈⣉⡆⠀⡇⠀⢀⣇⠀⣇⣀⠀⢸⣉⠁⠀⠀⢀⡴⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠤⠤⡤⠤⠬⠥⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠬⠄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⠀⡇⠸⡇⠀⡇⠀⣀⠤⠤⠒⠒⠢⠤⢄⡀⠀⠀⢀⠔⢦⠀⠉⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣽⠆⠀⠀⠀⢸⠀⠀⠧⠤⠥⢴⡷⢋⠥⠚⠉⠉⠉⠉⠙⠲⢌⡳⣄⠀⠓⠊⠀⡴⠉⡴⡦⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡠⠋⠀⠀⠀⠀⢠⢎⡔⠁⠀⠀⠀⠠⡤⠄⠀⠀⠀⠙⣌⢣⠀⠀⠀⢇⠘⢤⣇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡼⠀⠀⣠⠤⡀⠠⠶⢀⠜⠁⠀⠀⠘⡄⢇⠀⠀⠈⠳⠤⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣀⡠⠶⢦⣀⡠⠖⢶⠀⡇⠀⠀⢧⣰⡞⢀⠔⢡⠖⢦⠀⠀⠀⡇⢸⣀⡠⠖⠦⣀⣠⠔⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠤⠔⠒⠲⠤⠴⠒⠒⡇⢧⠀⠀⠀⢀⠔⠁⡀⠸⢤⠞⠀⠀⢀⠇⡼⠤⠴⠒⠒⠤⠤⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠶⠚⠉⠑⠶⠖⠉⠙⠻⡌⢦⠀⠀⠁⠀⠀⠷⠀⠀⠀⠀⢠⠎⡴⠓⠶⠞⠉⠙⠲⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠤⠤⠤⠤⠤⠤⠀⠀⢀⣤⠀⠘⠦⡑⠦⣁⠀⠀⠀⠀⢀⡠⠖⣡⡾⠤⠤⠤⡄⣠⡞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠖⠲⢤⡀⠀⠀⠻⡤⠃⠀⠀⠈⠓⠢⠭⣭⣭⡭⠥⠖⠋⠀⡇⠀⣱⠀⡇⠀⠉⢹⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡖⠒⠒⠂⠘⡆⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠧⠤⠥⠤⠇⠀⠀⠈⠧⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣏⣉⣉⡁⢀⠇⠀⣖⠒⠀⡖⢢⢠⠖⠢⠐⡖⢰⡒⠂⠒⡖⠂⠒⠒⢰⠒⠂⠀⠀⠀⠀⠈⠉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣅⣩⠴⠋⠀⠀⢈⡽⠀⢧⠼⠸⢤⡤⢠⡇⠸⣍⡁⠀⠇⠀⢠⡄⢸⣍⠀⠀⢎⣉⡱⢾⣉⡹
⠀⠀⠀⠀⠀⠀⢠⠤⠤⡄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄⣤⣀⣄⣠⣀⣄⣠⣀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣀⣀⣀⣀⡠⠄⠒⠒⠛⠓⠢⠤⠄⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠳⠶⣶⣶⠊⢀⣠⣶⠆⢀⣠⣀⡱⡄⠀⡀⡀⠀⡀⣀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡵⡄⠀⢀⡄⠀⠚⠁⠀⡄⠉⢸⡅⢿⠗⠘⠶⠇⠟⠲⣣⣍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡸⠆⠀⣀⡼⠀⠸⣇⠱⣌⠁⠀⣉⣉⠤⠴⠖⣙⢆⣩⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⠁⠀⠀⣨⠗⠀⢸⠃⠠⠃⠀⠜⡽⠉⠁⠀⠀⠉⢿⣿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⡇⠀⠠⠟⢻⠃⠀⢰⣅⠀⠈⠆⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢻⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠰⢲⠀⠈⠳⣄⠀⠀⣠⠏⣰⣿⣷⣶⡖⣶⣶⣶⡾⣤⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡎⢠⡄⠀⠀⠀⠀⢿⣄⠈⣆⡄⠀⠘⠄⠀⣵⢒⢻⣿⢻⢻⣷⣿⡿⠁⣸⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣤⡼⠧⠈⠃⠀⣠⠀⠀⠀⠈⠃⠹⠅⠀⠀⡄⢠⡏⠈⣾⣿⠇⢸⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠴⠶⢶⡄⠀⢀⣿⣀⡀⠀⢀⠀⢀⠀⠀⣼⡁⠸⣧⣦⢹⣿⠸⢿⡉⣿⠇
⠀⠀⠀⠀⠀⠀⣠⠖⠠⠥⠤⠤⠖⠒⠳⡄⠻⢦⣌⠙⠻⣿⠀⠸⡆⠸⣠⡆⠀⢻⡀⠻⡇⢼⣽⣷⣿⣿⣥⣆
⠀⠀⠀⠀⠀⠸⣡⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⡄⢹⡇⠀⢿⣄⠁⠸⣄⡄⠁⢀⠉⢸⣷⣤⠤⠤⠤⡔⡴⠒⠐⠒⠲⠖⠀⠶⠚⣉⣑⣢⡴⠶⠶⣄⡀
⠀⠀⠀⠀⠀⣰⠇⣼⠃⢠⡇⢠⣤⠀⠀⢹⣆⡀⠀⠀⡀⠈⡇⠀⣾⠋⣀⣀⡙⠃⠀⠘⠂⠀⠻⠷⠤⠤⠔⠁⠀⠀⠀⠀⡄⢄⠀⠀⠬⣄⠀⠀⠈⢳⣤⣀⡬
⠀⣠⠶⠦⣾⣿⢀⣿⠀⣾⡀⠸⣿⡆⠀⢿⠉⠉⠙⠲⣣⠀⠃⠀⠁⡴⠋⠉⠉⠁⢀⣤⠖⢀⠼⠀⠀⣀⠀⢴⠀⢀⣷⢠⡗⢸⣷⠷⠶⠙⠿⢶⣄⢠⠦⣹⠟
⡀⡁⣠⠞⠛⢃⣼⣿⢰⣿⡇⠀⡏⡄⠀⠈⡇⠀⠀⠀⠸⣄⣀⣀⣀⣁⠀⣀⡤⠶⠿⠏⠀⠀⠒⠒⠚⠉⠁⠈⢶⣾⣷⢂⢄⠸⠟⣤⡀⢠⡖⢤⠿⠿⣦⠟
⠷⣷⢃⡀⡔⠉⠁⢰⠟⠋⠀⠀⠈⢮⠚⣨⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣸⣥⠾⠛⠿⠿⣇⠎⠀⠘⠉
⠀⠸⠺⢶⠇⣄⠀⢸⣀⠀⠀⣠⣼⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⣶⠿⣶⣾⣏⢿⣻⣛⡵⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⡶⠖⠲⣶⣶⣶⣾⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⡀⠀⠀⣴⣾⡿⠛⠉⠀⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠉⠀⠀⡠⠟⠛⠉⠁⢀⣀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣸⣅⢀⡸⠋⠁⢀⣠⣴⣿⠁⠀⣠⣶⣶⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⣼⣿⣿⣿⣿⡶⠛⠉⠀⠈⢿⣿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢀⣴⠿⠋⠹⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣰⣿⣇⣠⡾⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⡠⠞⠁⠀⠀⠀⠈⠻⣿⣤⣀⡀⠀⠀⠀⣼⣿⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧
⣴⣶⣾⣇⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⣦⣤⣿⣷⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣀⣤⠴⣶⣶⣦⠤⢤
⠈⠉⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠛⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣁⠀⠀⠉
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢇⣠⣤⣶⠟⠙⢿⣶⣶⣦⣤⣄⣤⣶⣿⣿⣿⣅⣀⣀⣀⠀⠀⠉⠁
⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⢦⣤⣾⣿⠟⠁⠀⠀⠀⠙⢿⣿⠟⠛⠻⢿⣿⣿⡟⠉⠻⢿⣿⣿⣿⡇
⠀⠀⢀⣀⣤⣿⣿⣿⣶⣷⣄⣤⣄⠀⠙⣿⡁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣄⡀⠀⠉⠓⠄⠀⠀⠀⠀⠀⠙⠿⣆⡀
⣶⠚⠿⠿⠿⠛⠛⠉⠉⠻⠿⠿⠛⢷⡀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⣦⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⡿⠟⠛⠋⠉⠉⠙⠲⣄⠀⠀⢀⠀⠀⣀⠀⠙⠿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣄⡀⣸
⣿⣿⣿⣶⣤⣤⣀⣀⣀⣠⠔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣾⣿⣷⣤⡀⠈⠻⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠿
⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⣿⣦⣤⣟⠛⠉
⠀⠀⠉⠉⠙⠛⠛⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣶⣶⣶⣶⠾⣿⣿⣿⣿
⠀⠒⢲⣶⣶⣶⣶⣶⣿⣿⣦⣶⣶⣶⣶⣶⣦⣤⣀⣀⠠⢶⣦⣤⣴⣿⣿⣿⣿⠏⠀⠈⠙⢿⣿⣇
⣤⣤⣾⣿⣿⣿⠿⠛⠿⠟⠁⠀⢸⡿⠟⠛⠉⠀⣠⡏⠀⣸⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠈⣻⣦⣤⣤⣤⣀⣀⢀⣠⣶
⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⣾⣿⣿⠁⢀⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠟⠛⠛⠛⠛⠋
⢸⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡟⠉⠙⡇⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠽⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠠⠂⠀⢀
⠀⠀⠀⠀⠀⢀⠔⣦⡴⢦⡄⠀⡄⠀⢸⡇⠀⢀⠀⠱⡄⣀⠀⠀⠀⠀⠀⠀⠀⡐⠴⠞⠉⠑⡇⢀⡠⠀⠃⠒⠛⠘⣇
⠀⠀⠠⠤⢄⡃⠀⠀⠀⣾⡇⠐⢤⠀⢸⡇⠀⡈⠀⠀⠆⠈⠑⠠⠀⠀⠀⠀⢸⣠⣀⠀⠀⠀⠱⡌⠓⠉⢶⠄⠀⢀⠻⡀
⠀⡈⠀⠀⢸⠃⣦⠀⠀⣿⠁⣾⠀⠀⢸⡇⠀⢷⠂⠀⡶⠃⠀⠀⠀⠀⠀⠀⢸⠈⣣⣄⠀⠀⠀⠐⠀⠒⠋⡀⢠⡟⢁⡀⣠⡢⡀
⢀⠆⠀⠀⣾⠀⠙⠀⠀⠀⠀⡿⠀⠀⠈⠁⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⣢⠎⠈⠉⠀⠀⠀⠀⠀⣀⠘⢠⡏⠀⠀⠀⠈⢡⢱
⢸⠋⠁⢠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢾⠟⠁⠀⠀⠀⣱⣆⣀⣹⡏⠆⡜⠀⠀⠀⢀⡴⠟⠙
⡟⠀⢀⡀⠁⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠉⠓⠀⠸⢷⣼⣉⠓⠀⢠⡿⠙⠃⠐
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠁⠀⠀⡤⠀⠀⢠⠀⠀⠈⢙⣩⠄⠀⠀⠀⠀⠀⣇⣸⣁⡴⠯⠁
⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠊⠀⠀⠀⠀⢣⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠀⠀⠀⠀⣇⢰
⠘⣿⠂⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠂⠀⠀⠟⡇⠀⠀⠀⠀⠀⠈⢸
⠀⢹⡇⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢀⠄⠀⠁⠀⠁⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠃⠐⠓⠦⢭⣶⡗⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠘⠛⡛⠈⠉⠀⢀⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢆⠀⠀⠀⡉⠀⠀⠀⠀⡜⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⠓⠲⠤⠀⠀⠀⠀⠀⠇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠷⠤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣤⣤⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢿⠿⠿⠛⣿⣿⣿⡇⣛⣿⣿⡳⠟⠘⠻⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⣿⡿⢟⣣⣶⣾⣿⣷⡾⢿⣿⣿⢻⣿⠿⣿⣦⣿⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣿⣿⣿⣿⣿⣿⡇⠈⠛⠁⢸⣽⡜⣻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣧⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⢿⠁⠀⠀⠀⢸⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠖⣸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡟⠁⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠻⢿⣟⣥⡯⠤⠤⣿⠀⠁⠋⠀⣿⠤⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠁⠀⠀⠀⠀⢠⡏⠀⠈⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⢼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠣⣾⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⢠⣤⣤⠘⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠈⠻⠃⠀⢻⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠛⠉⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⢲⣤⣬⣭⣍⡉⠉⠉⠉⠉⣉⣭⣭⣤⣤⠒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⣇⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⠔⢤⠲⡚⠉⠹⢻⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠘⢧⡀⠀⣤⣀⠀⠀⢸⡆⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠉⠙⡖⢢⠔⢄⣀⡀
⡔⠃⠀⠈⠆⠈⡧⠀⠞⠂⠸⡇⠀⣠⡶⠋⠁⠀⠀⠀⠀⢀⣀⠀⠀⠙⠶⠏⠉⠛⠲⠾⢃⣀⠀⠀⠈⠙⠓⠶⠦⣤⠤⠀⡿⠀⠚⠆⠀⡋⠀⠎
⠧⠤⠄⠤⠈⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣉⠻⠂⠀⠀⠀⠀⠀⠀⠺⢋⣁⣀⡀⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠀⠈⠠⠀⢤
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⡇⠀⠀⠀⠀⠘⠋⠙⠋⠙⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠙⠀⠀⠀⠀⠀⣿⣀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⢹⠀⠉⠓⠲⠤⠤⢤⢀⠀⡞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡆⢀⡀⠤⠤⠴⠒⠛⠁⢸⠂
⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠈⠑⡙⠦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠞⡑⠁⠀⠀⠀⠀⠀⠀⠴⡏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣧⠀⠀⠀⠀⠀⠀⡤⠰⢤⣤⡤⠤⠤⣤⣤⠴⠢⡄⠀⠀⠀⠀⠀⢠⡏⠀⡴⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢄⠘⢧⠀⠀⠀⠀⠀⠹⣠⠎⠀⠀⠀⠀⠀⠈⢦⡸⠁⠀⠀⠀⠀⢠⠟⢀⠜⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠲⣆⣀⣀⣀⡀⢀⣙⡾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣞⡁⠀⣀⣀⣀⣀⡴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⠝⠋⠀⠀⠀⡾⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠋⡀⠀⠀⠈⠛⢅⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠈⠉⠀⢤⠒⠘⢺⠃⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠒⠚⠛⠉⠁⠀⠘⣦⣗⠐⠋⢢⠄⠈⠁⠘⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢂⠟⠂⢠⠜⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⠀⠘⠄⠀⠚⠑⢢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠰⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠛⠉⠙⠋⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠢⠇⠀⠀⠂⠀⠀⠇⠀⠀⠦⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣁⣠⡀⣀⣀⣀⡄⠀⠀⣅⣀⣀⣀⣠⣀⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⡽⠀⠘⣅⣀⣀⣀⣀⣀⣀⡁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⣄⣧⢀⣸⣰⠃⠀⠀⢀⡤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡿⠁⠙⣿⣿⣿⣿⡇⠀⢀⢠⡏⣐⡠⠀⣠⠔⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣦⡖⠈⢿⣿⣿⣿⣶⣾⣾⣷⣿⣾⡟⠁⢀⠴⠊⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⣿⣷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠶⢋⣠⡶⠋⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠇⠀⢹⣿⡗⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣴⡚⣩⣾⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠸⣿⣿⣥⠀⢿⣿⣿⣿⣿⣿⣿⣿⣯⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠙⠀⠀⠀⠹⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡺⠍⠀⣀⢄⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢀⣠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣞⣯⠴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠉⠀⠀⠀⠰⠃⠀⠉⠉⠛⠉⠉⢝⣽⣿⣿⣿⣿⣿⣿⡿⢯⣥⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⣿⣿⡿⠿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡏⢻⠃⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⣼⣿⣿⣿⡿⠟⠛⠛⠖⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠛⠛⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣸⠀⠀⠀⠀⠀⠺⣿⣷⡄⠀⠀⠀⠂⠀⢀⣪⣿⣿⣏⡛⠖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⣤⡶⠖⢃⣀⣴⣿⣿⡇⠀⠀⣠⣴⣿⣿⣿⡿⠿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⢸⠃⣠⣶⣿⣿⢟⣷⠟⠁⢴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠷⠤⣶⣿⣿⠞⠋⠉⠀⠚⠁⠀⣺⡿⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠟⠁⠀⠀⠀⠀⡟⢀⠞⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠁
⠀⠀⠀⠀⡴⠒⣤⠶⠁⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠀⠱⢦⡔⠲⡄
⠀⠀⠀⣼⠇⠞⠁⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠙⠆⢿⡄
⠀⠀⣴⠁⠀⠀⠠⠼⠿⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠸⠿⠤⠀⠀⠀⢱⡄
⠀⠘⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇
⠀⠀⢸⣯⠁⠈⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠈⠀⢩⣿
⠀⠀⠀⡿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡼⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠸⡄
⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢷
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⣇
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄
⣰⣿⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⢸⣷
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣆⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⣍⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣦⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡖⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣷⣷⣍⡤⠀⣻⠀⠀⣤⣒⣒⡒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⢿⣿⣿⠓⠀⠀⢾⠀⢸⣿⢯⠭⠉⢲⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠈⣿⣭⡄⠀⠀⢸⠀⢸⡿⠦⠄⠀⢸⣿⣴⣖⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣀⠀⣤⠄⣿⣭⣼⢿⡇⢸⡇⣸⣿⣿⡏⠱⣿⢻⣿⣶⠦⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡤⣿⡆⣿⢽⡿⠞⣿⡈⣷⣿⠿⣿⡗⠀⣿⣼⣾⣿⣤⠈⣼⡇
⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣀⢸⡇⣿⡀⠸⠇⢤⣟⡋⠀⠉⠁⠈⡋⠉⠛⠃⠀⢹⣟⣼⣇⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠛⢍⡣⡀⠀⠈⢻⡇⠁⠀⠀⠀⠨⣭⣶⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡌⠀⣀⣼⡧⠀⠀⠀⠀⠐⠻⣇⠀⠀⠀⢀⣸⣧⣀⡀⠀⠀⡟⠂⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⠩⣿⣷⣂⣀⣠⣾⣿⣷⣶⣶⣾⣿⣿⠿⣿⠇⠀⣿⣧⣤⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⣷⣹⣿⣿⣿⣿⠟⢿⣿⣿⣿⣶⡉⢿⣿⢿⠇⠠⣾⣿⡏⢠⡼⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣷⣶⡈⠙⢿⣿⣯⡀⠈⠉⠻⠀⠈⡛⠛⠃⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⢻⣿⣿⣿⣿⢭⣤⠂⠚⣷⣶⣆⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢿⡿⣿⣶⣾⣑⠂⠀⣈⠉⠍⠀⠀⢠⡖⢇⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠺⣽⣿⣿⣿⠭⢬⣆⣸⣒⠂⢸⣮⣿⡀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣌⣿⣿⣿⡿⠦⠿⠗⠀⠤⠉⠁⠀⠀⠤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣵⣶⣮⡝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣴⣶⣬⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣿⣿⣿⡤⠧⠤⠤⠤⠤⠤⠤⠤⠤⢼⣼⣿⣿⣿⣿⣷⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢉⣛⠻⢴⡃
⠀⠀⠀⠀⠀⠀⢀⡜⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢄⠊⠈⠇⡙⢆
⠀⠀⠀⠀⠀⠀⡞⠀⠠⡄⠒⡀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠒⠓⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜⢉⣶⣦⣄⡀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⣀⣤⣶⡏⢣⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠁⣾⣿⣧⣈⣹⣶⣤⡀⠀⠀⢀⢀⣠⣶⣛⣩⣿⣿⣿⠀⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢆⠹⣿⣿⣿⣿⣿⡿⡹⢡⣤⠀⢫⢻⣿⣿⣿⣿⣿⠏⢠⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠢⢌⡙⠛⢛⡫⠞⢁⡴⠶⢦⣄⠑⠝⠛⠛⠛⢡⠴⠃⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⢠⣴⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿
⠀⣾⣿⣿⡯⣾⣿⣿⡿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⣿⣿⣿⡦⣿⣿⣿⡆
⠀⢹⣿⣿⣷⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⠁
⠀⢸⣿⣿⣿⡿⠃⠀⠈⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠻⣿⣿⣿⣿
⠀⣠⣝⣻⣿⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣬⣿⣛⣤⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⡟⢛⣛⣛⣛⣛⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣙⣛⣛⣛⣛⣛⣛⣛⡛⠛⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⡇
⠀⠀⢿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠛⠛⠋⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠴⠶⢶⣦⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠈⠙⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣼⣿⣷⣤⣤⣤⣼⡃⠀⠀⣠⡾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⠉⠉⠉⣿⡇⠀⠀⣿⡇⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⣿⡇⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣁⡉⠉⣁⡈⠻⣷⡄⠀⠀⠀⠀⠀⣿⣇⣠⣤⣤⣼⣿⣆⡀⠀⠈⠙⠛⠛⠲⠶⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠉⠛⠉⠉⠙⠻⣿⡷⠀⠀⠀⠀⠀⣿⣿⠟⠉⠉⠉⠛⠛⠛⠻⣦⡄⠀⠀⠀⠀⠀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⡿⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠻⢿⣾⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⣇⠀⠀⠀⠀⢠⣿⡿⢷⡄⠀⠀⠀⠀⠀⢰⣿⣧⣀⠀⠀⢀⣠⡟⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⢀⣼⠏⢹⣿⠀⠀⠀⠀⢸⣿⣷⠈⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠠⣿⠋⠀⠀⠈⠉⠀⠀⢹⡇⠀⠀⠀⠀⢸⣿⣀⣼⣿⠀⠀⠀⠀⢸⣿⣿⠀⠙⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣾⣿⡟⢻⣿⠀⠀⠀⠀⠈⣿⣿⣆⣤⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⢸⣿⣿⡇⢸⡿⠀⠀⠀⠀⠀⢻⣿⣿⠀⢹⣿⡀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⢠⣾⠏⠀⠀⠀⢰⣾⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⢀⣿⡟⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠈⣿
⠀⣰⡿⠋⠁⠀⠀⠀⠀⣿⠟⠀⢀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣄⣀⣀⡴⠶⠶⣄⣀⣀⣼⣿⡇⠀⠀⠀⣀⣀⣀⠀⠀⣿⣿⡟⠀⠀⠀⠠⣿⡀
⣾⠋⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⣈⣉⣙⠿⠿⠶⣶⣾⣿⣿⡟⠛⠻⠷⠶⠶⠿⠟⠻⣿⣿⣿⠶⢶⣶⣏⣁⣽⣤⣾⣿⡿⠃⠀⠀⠀⠀⢻⣧
⣿⣤⠀⢠⣴⣶⣄⣀⣠⣼⣿⠋⠙⠛⠉⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⢤⠉⠙⠉⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠻
⠈⣻⡿⠟⠻⣿⣿⣿⠟⠻⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣷⡶⠶⢿⣦⣤⣤⡀
⠀⣿⡀⠀⠀⠈⠉⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠈⠉⣹⣿⣿
⠀⢿⣤⣶⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⣹⠿
⠀⠀⠙⠿⠿⣶⣤⣀⡀⠀⠀⠀⠸⣿⣿⣿⣦⡀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠈⠻⣯⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀⠀⠀⠀⠛⢿⣿⣷⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⢿⣿⡿⠁⠀⠀⢀⣿⣿⡿⠃⠀⢀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠈⡛⠁⠀⠀⠀⠸⣿⡇⠀⠀⠀⠸⣿⠏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠉⢹⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⢧⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⡀⠀⠀⠀⢀⣨⠿⠓⢤⣤⣤⣿⣦⣤⣤⡤⢀⣴⣶⣤⣀⣠⡄⠿⢧⣤⡄⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⢀⣾⠁⠀⠀⠀⠁⠀⠀⢹⣯⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢀⡸⠇⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⣺⣦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣾⠟⠁⠛⠓⣤⣤⠤⣿⣧⣀⠀⠀⠀⠀⠀⣠⡿⠛⠃⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠐⣶⡄⠀⠀⠀⠀⠀⠀⠘⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠃⠀⠈⢱⣄⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣀⣀⣤⣤⠤⠄⠀⣤⣶⠟⠳⠄⠛⠁⠈⠛⠋⠘⠛⠛⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⠛⢿⣿⣶⣄⣀⣀⣤⣤⣶⣄⡀⢠⣴⡶⢶⣶⣶⣶⣶⡾⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠙⣷⣾⠁⠀⠀⠈⠉⠉⠁⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢻⡆⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢹⡆⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢸⠇⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡄⠁⠀⠀⠀⠀⠢⡅⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⡌⠀⠀⠀⢐⠋⠀⠐⢰⠐⢦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠂⠀⠴⠋⠑⠒⠂⠈⠑⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠘⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣄⢠⠈⠂⠀⢀⠄⠐⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⢈⢁⣼⡇⠈⡀⠈⢀⠀⠈⠁⠀⠀⠐
⠀⠀⠀⠀⠐⠁⢀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠉⠪⢙⡿⠀⠀⡀⠀⠀⠁⠀⠀⠈⡢⠀⠈
⠀⠀⠀⠜⠀⠐⠋⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂⠀⠀⠀⡠⠂⠀⠀⠑
⠀⡀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⢀⢂⡀⡉⠲⡢⠤⡤⠄⠀⢀⠜⢀⡀⡘⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⢀⠀⡤
⡈⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠸⡀⠙⠄⢠⠋⠳⡅⠀⠔⠁⡀⠄
⠁⠀⠀⠀⠀⡰⠂⠰⠀⠀⠀⠀⠀⠀⠁⠢⢕⣄⠴⠋⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠒⠄⠤⠠⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠠⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢏⣁⢀⡀⣀⡀⠀⠀⠀⠂⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠈⠁⢀⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣹⣦⣀⣴⠀⠀⠈⠀⠀⢰⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢙⡛⠛⠴⣶⣶⣵⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⣲⣀⡀⠀⠀⠀⠀⠀⠀⣀⣌⣹⡟⢛⣲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⢸⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠹⣿⡇⠻⣿⣿⣶⣍⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⣿⣁⡾⢻⠀⠀⢀⠴⠒⠊⠙⠒⢄⠀⠀⢹⡎⡻⣮⡻⣿⣿⣿⣷⣮⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣳⣿⣿⣿⣟⠘⠀⢠⣿⠄⠀⠀⠀⠀⠀⣷⠀⢸⠏⣿⣿⣿⣌⢿⣿⣿⣿⣿⣻⣿⡢⡀
⠀⠀⠀⠀⠀⠀⢨⣿⣿⢣⣿⣿⣿⠀⣦⣼⣿⠀⠀⠀⠀⠀⠀⢸⠀⡘⢀⣿⣧⢿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣮⡢⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡟⣾⣿⣿⣿⣇⠸⣦⡉⠀⠀⣠⣀⣄⣀⣰⡴⢁⣾⣿⣿⣾⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣮⣦⡀
⠀⠀⠀⠀⣌⣿⣿⣿⢣⣿⣿⠟⠛⠛⠀⢿⣿⠆⠈⠈⢋⠈⠁⠟⠀⠛⠛⠻⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠰⣼⣿⣿⣿⣿⣿⠃⠀⠈⠻⠄⣿⣿⣗⠲⡌⢻⣤⠛⢀⣠⠀⠀⠀⠘⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣆⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡷⡀
⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⡄⠢⣀⠀⠀⠀⠈⠿⠿⠿⠀⠀⠀⠀⢸⡀⠀⠀⢠⠋⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢄
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠈⢻⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡔⡄
⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⣿⣿⡞⣿⣿⢷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⡀
⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠆⠀⠤⠤⠀⣿⣿⣿⣿⣇⣿⣿⣾⢣⡟⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣽⣷⡀⠀⠀⠀⡄⠀⠀⠀⣒⡛⠛⣶⣶⣿⢸⣿⣿⣿⣿⣿⣿⣼⢳⡏⣿⣼⢿⡏⣽⡿⣿⣿⣿⣿⣇
⡟⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣟⠦⣄⡀⡇⠀⠀⢀⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⢻⣿⡟⣿⣼⣇⣿⣿⣾⣻⣏⣿⠏⣿⣿⣿⣿⡄⢇
⡇⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣟⣷⣿⣿⡇⠀⠀⠸⠛⠿⣿⣿⣿⠿⣿⣿⣿⣿⢸⣿⣇⣿⣿⣼⣿⣿⣿⣿⣼⣯⣼⢻⣿⢻⣿⣷⢸
⡇⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⠙⠉⠀⠀⢿⡇⠀⠀⠃⠀⠀⡠⠈⠁⠆⠘⣦⢹⡿⢸⡿⠋⠉⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⣿⣾⣿⣿⣾
⡇⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⠀⠀⠿⣀⠀⠀⠀⣼⠁⠀⢀⡤⠐⠁⠀⢀⣣⢼⠄⢆⣠⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡯
⠇⣿⣿⣿⣿⣿⣿⣿⣿⣌⣷⠿⠟⠉⢻⣿⣴⡆⠀⠀⠐⠀⢠⡝⡀⠀⡉⠁⢠⣀⠔⠋⢙⠀⠀⠈⣧⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢀⣿⣿⣿⣿⣿⠟⠋⠁⠉⠋⠾⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠑⠄⠁⠘⣥⠃⠀⠀⠈⠒⠒⠋⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢰⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡆⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡟
⠘⢹⣿⣹⡏⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢨⡟⢻
⠀⡼⣿⣿⡃⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡎⠀⣿⣤
⠀⠠⢿⣿⡇⠀⠀⠀⠀⠀⠀⡻⠀⠠⢆⢳⠀⠈⠛
⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢷
⠀⠀⠈⢫⣿⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠛⠛⠳⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⣉⠀⠀⠀⠀⠀⠉⠀⠘⠀⠈⠋⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣠⠔⠁⠀⠀⢀⡠⠀⢀⣴⠀⢀⡴⢁⣴⠟⠹⣆⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⣠⠴⢃⣠⠶⠋⣿⣴⡿⠞⠋⠀⠀⠀⢻⣌⠇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣠⣴⣾⠷⠞⠋⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡙⠉⠁⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠃⠘⣯⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⢀⣌⣙⠳⠆⢀⡆⠀⠀⠶⠞⢉⣀⡀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⣿⣿⠇⠀⣼⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⢰⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠦⠭⠵⠂⢠⡏⠀⠀⠀⠀⠨⣛⣋⡴⠀⠀⣿⣼⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠘⠳⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⠠⠤⠴⠶⠶⠶⠶⠶⠆⠰⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⡿⠓⡖⠒⠒⠒⠛⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⢶⡞⠉⣏⠀⠀⠳⠀⠀⠀⠀⠀⣿⠈⢳⡶⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠁⠀⠀⣼⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⣇⠀⠀⠈⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢠⡀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢦⠀⠀⠀⠀⣠⠟⠀⠀⢀⡟⠀⠀⠀⢠⡆⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢳⡀⠀⠀⠀⢹⡀⣀⣤⡄⠈⠳⣄⣠⡾⠋⢀⣤⣄⣼⠃⠀⠀⠀⠾⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠱⠀⠀⠀⠈⠉⠁⠘⢧⣀⠀⢸⠋⠀⢀⣸⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠎⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣈⠙⠳⠦⣤⣠⡆⠀⢿⡀⠀⠀⠀⢀⣸⠃⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡤⠶⠚⠉⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠷⢶⣤⣼⣷⠀⠀⠉⠛⠶⠚⠉⠀⠀⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⡶⠾⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠲⠶⣴⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣿⢸⠀⠀⢀⣀⣠⣤⡴⠶⠟⠛⠋⠉⠉⠀⠀⠀⢸⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣭⣭⣉⣛⣛⣿⣿⣿⣿⡿⠿⠿⣿⣤⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣶⣶⣶⣶⣶
⠉⠉⠉⠉⠉⠉⠉⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢿⣇⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⣤⣙⢶⣿⣷⣤⠶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢶⣤⣄⣒⠶⣭⣳⣽⣾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡤⠶⠂⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠉⠷⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣠⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠃⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠑⢶⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢰⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠘⣄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡰⠁⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠺⠧⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠀⠚⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣿⣷⣶⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠛⢿⣿⣿⣿⣿⣿⡿⠿⠟⠻⣿⣿⣿⡟⢻⣿⣿⣿⠟⠿⠿⣿⣿⣿⣿⣿⣿⡿⠛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⠨⣽⣿⠥⠄⠀⠀⢀⣰⣿⣿⣿⣏⣹⣿⣿⣿⣦⡀⠀⠀⠠⠬⣿⣯⠅⠀⣠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠄⢀⣀⡀⠀⠀⠐⢾⣿⣿⣿⣝⠇⠸⣫⣿⣿⣿⡷⠂⠀⠀⢀⣀⡀⠠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣝⠣⠜⣻⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⢶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⣠⣿⣿⡿⡷⢤⣼⡟⠈⠛⣿⣿⣇⣸⣿⣿⠛⠉⢺⣿⡿⠿⢿⣿⣿⣄⣹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣭⣴⠷⢶⡶⢬⡁⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⢈⡥⢶⡶⠾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠻⣿⣷⣀⠈⠳⠌⢵⣦⠄⢈⠉⢙⡏⠉⡉⠠⣴⡮⠡⠚⠁⣀⣾⣿⠟⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡏⠀⠐⠿⠟⠛⠒⠀⠢⣨⡏⠐⠠⠀⠂⠐⠀⠀⠂⢹⣅⠔⠀⠞⠛⠻⠿⠒⠀⢹⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⡵⢶⡶⢯⣄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣀⣀⡀⠀⠀⠀⢸⡄⠀⠸⣯⣶⠎⠱⣶⣽⠆⠀⢰⡇⠀⠀⠀⢀⣀⣀⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠂⣠⢖⠘⣅⠀⠆⠈⡛⠿⠿⢛⠁⠰⠀⣸⠃⠲⣄⠐⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢛⣽⠿⠊⠁⠚⠀⢿⣿⠒⠲⢬⣓⣚⠥⠖⠒⣿⣿⠀⠑⠈⠙⠿⣯⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⡀⠈⠓⠘⣿⢀⣦⣶⣶⣶⣶⣴⡄⣿⠃⠚⠁⢀⣦⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣯⠀⠀⣟⣿⡟⠀⠀⠁⠀⢻⣿⣽⠀⠀⣽⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⣇⡷⡇⠀⢀⠀⠀⢸⣷⡨⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢸⣇⣻⡶⣶⣶⢶⣟⣸⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⡏⠻⢲⣦⣴⡖⠟⢹⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢻⣤⣌⣉⣉⣡⣤⡟⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠛⡛⢙⠋⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠋⠁




⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⠿⣿⡿⡿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢡⠀⡀⠀⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢦⣀⠀⢣⣍⡙⠙⡄⠀⠲⠀⢀⠄⣀⢤⠀⠀⠀⠀⠀⠁⠀⠳⢄⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⢈⡇⠛⠢⣈⠖⠒⢧⠒⠀⢆⡠⠊⠀⢠⠀⠀⠀⠈⠋⢉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠛⠐⣀⡀⠀⠛⠀⠉⡐⠢⣀⠀⠰⢇⠀⢼⣿⠇⠁⢺⠈⡱⣞⢱⡆⢀⠚⠁⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⡀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠐⠌⢂⣀⠈⣂⢷⣜⡧⡤⣼⡤⢕⣦⣶⢭⣛⡄⠋⠭⠁⣰⠆⢁⠀⠀⠄⠐⣆
⠀⠀⠀⠀⠀⠀⢀⡀⠁⠠⠀⠄⠈⠀⠂⠄⠀⠄⠀⠀⢌⠁⡚⢈⠻⣯⣭⠝⢣⠬⢈⣧⣁⠀⣈⣶⡧⢿⠊⠑⠈⡋⠤⠉⠓⢑⠄⡀⠈⠊⡀⠀⠃⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠬⠄⠢⢀⠀⢁⠠⠤⠧⠤⠤⢄⢡⠂⠀⠀⠄⠙⣧⠀⠴⠴⠲⢮⣭⣻⣿⢖⣳⠶⢧⠀⣀⠚⠓⠚⠢⠉⠀⠀⠻⣄⡁⢀⠈⠆⢻
⠀⠀⠀⠀⠀⠲⠠⠄⠄⠀⠀⠄⢈⠀⠀⠀⢀⠀⠀⠀⠈⠳⣤⠀⡀⠠⠜⠛⡆⠄⢨⣜⣍⠛⠋⣹⣀⠀⠤⠔⠈⠂⠀⠀⠀⠹⣆⠐⠀⠈⠀⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠾⠀⠀⡀⠉⠀⠀⠈⠳⡄⠐⠈⠣⣄⠀⠀⠀⠒⡮⢴⡡⠄⠰⠛⠛⠂⣹⠇⠀⠀⢺⡀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⢀⠁⠈⢦⡈⢸⡇
⠀⠀⠀⠀⣸⠀⠂⣁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡙⠀⠀⠒⠀⠡⠄⠀⢈⠿⠁⠈⢻⡇⠀⠀⠀⠀⠀⡠⠀⠀⠀⠤⡈⠀⠀⢄⣀⡠⡄⢺⠁
⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠉⠐⡟⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠤⠀⠀⠀⡰⢀⢀⠆⣟⠦⣤⡤⠬⣝⠐⣄⠀⠀⠀⠀⠀⡔⠣⢆⠐⠄⠠⠀⠀⠀⢩⣼
⠀⠀⠀⢸⠠⠂⠀⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠰⠤⠤⠘⣄⢀⠀⠀⢦⠠⡦⠌⠒⠂⡌⠉⠒⢈⡀⠃⠠⡍⠲⠀⠑⠧⠌⠀⠀⡀⠰⣄⣀⡀⢨⡟
⠀⠀⠀⢸⠀⠆⠀⠉⠉⠉⠉⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢦⠰⢦⡆⣄⠀⠀⠴⡇⠁⠐⠆⡀⠚⢄⡆⠀⢈⠀⠀⠐⠀⠀⠀⢀⣾⠁
⠀⠀⠀⢸⡁⡆⠀⠀⠀⠀⠀⡈⢣⡀⠰⢦⣀⠀⠀⢸⡳⣄⡀⠐⢧⡢⠄⠀⠀⠙⢂⠅⠋⢶⠘⢷⣹⠙⠷⣥⠡⠄⣐⡦⢤⠠⠃⠔⠚⢢⡤⣰⠩⡍⢆⢀
⠀⠀⠀⠈⡇⡔⠤⠄⠀⠤⢆⠐⠈⡈⢦⠉⠀⠉⠆⠀⠀⠈⠂⠀⠀⠱⡆⠂⠀⠘⠀⠀⡐⠀⠘⠷⠸⣄⢀⢘⣋⣋⣭⡴⢂⣙⣆⠀⠀⢚⣰⣨⣷⡞⣛⢡⣹⡄
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢈⡀⡦⢬⠤⢕⣤⡀⣡⣹⠂⡲⣄⠀⠀⠀⠈⠙⠲⢤⡀⢂⡈⢀⣄⣀⣏⠙⠉⠋⠉⢁⡤⣼⣧⡘⣿⠳⣤⣨⣿⣿⠟⢿⡿⠞⠷
⠀⠀⠀⠀⠀⢻⣆⡀⣀⡀⠰⢄⠍⣃⠀⠀⠉⠻⠈⠙⠺⠀⡄⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣁⣺⡉⠁⣹⢈⣃⡐⣉⣛⣁⢰⣋⣉⣙⣉⣁⣁⢀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢢⠀⠙⠉⡩⠉⠒⣄⠺⣄⠀⢤⠀⠈⠑⠒⠐⠆⢠⡀⣀⣀⡴⠿⢶⠚⢿⠀⠉⠉⠡⠙⢩⣭⠏⠂⢯⡓⠈⠩⠉⣿⣉
⠀⠀⠀⠀⠀⠀⠀⣳⡄⠀⠀⠀⠈⠁⠲⣄⠒⠒⠴⠚⡄⠉⠑⣇⠲⢀⠁⠸⠄⠒⣀⢠⠀⠀⠀⠐⠀⠁⠀⣈⣂⣀⡤⢬⡦⣬⠙⢂⠈⠣⣢⣸⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢪⡀⠀⠀⠀⢀⠈⠓⠒⠶⣥⠱⣤⣤⠷⠈⠉⠩⠁⠈⢠⣀⠄⣀⠁⠀⢀⡠⠴⠒⠉⠁⠀⠀⣌⢆⠈⠻⢅⣇⡀⢌⡀
⠀⠀⠀⢀⣠⣤⣄⠀⠃⠙⢫⡷⢦⣀⠁⠀⠀⠀⠀⠘⠀⠓⠸⡇⠰⢤⣄⣠⣬⡁⠀⣀⣤⠴⠛⠉⠀⠀⠀⠀⣄⠀⠀⠈⠀⠠⠀⠘⡌⠇⢀⡈⠁
⠀⠀⠐⠓⡿⡕⠚⢷⠀⢀⡴⢉⠓⠠⢛⠿⡒⠶⢅⣀⣀⣀⣀⣀⣀⣀⣡⡤⠤⠗⠊⠉⠀⠀⢀⣠⠄⢀⡀⠀⠐⠀⠁⠀⠀⠀⠀⢀⠇⢀⠈⠉
⠀⠀⠀⠀⠈⠻⢶⣦⣷⣭⣀⣠⠤⠚⢅⠒⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠔⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢽⣃⠒⠺⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡄⢀⡀⠀⢸⡆⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠾⠦⠴⠟⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠛⢻⣿⣿⡇⠀⠀⣀⣴⣾⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⡀⠀⠀⠀⠀⢿⣿⣿⣿⡄⠀⠀⠀⠈⢿⣿⠀⢀⣀⣠⣤⣿⡿⣡⣴⣾⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⣸⠻⣆⣠⢾⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⣶⣶⣶⣝⠷⢿⣿⣿⠿⣋⣴⣿⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠙⠏⠀⠙⠁⢸⣧⣤⡶⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢴⣦⡀⠐⢶⡒⠛⠀⠀⠀⠀⠀⠀⠀⠾⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢀⣶⣶⣦⣄⠀⠙⠇⠀⠀⠙⠀⠀⠀⠀⠀⣀⣀⡀⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠉⠉⠉⠉⠀⠀⠀⣠⢀⣴⣾⣿⣿⣿⣶⣭⡻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣷⣶⣦⣽⣛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠰⠶⠶⠀⠀⣼⢣⣿⣿⣿⣿⠛⢻⣿⣿⣷⣹⣿⣿⣿⣿⣷⣦⣄⡺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣭⣛⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⢸⣿⣯⣼⣿⣷⡿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⡜⣿⣿⡟⠁⣀⣠⣤⣼⡿⣱⣿⣿⣿⡟⠻⠿⠿⣿⠿⠁⠟⠋⠉⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⡻⢷⣿⣿⣿⠿⣛⣵⣿⣿⣿⣿⣷⡘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣮⣭⣛⡻⠿⣿⣿⣿⣿⠟
⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠙⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠀⠀⠈⠉⠁
⠴⢰⠀⠹⣿⣿⠟⠋⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄
⠘⠌⠂⠀⠀⠀⣠⣶⣶⣶⣤⣄⣻⡿⢟⣛⣛⣻⠿⠿⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠀⠀⠀⣀⣀⣀⣒⣒⣒⣒⡒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⡀⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣋⠴⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀
⠀⠀⠀⠀⠀⢀⠔⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⡷⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠢⡀
⠀⠀⠀⠀⣰⡁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⠸⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢈⣆
⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡀⡚⠀⠀⠀⠀⠀⠀⢷⢀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⣦⡀⠀⠀⣠⢎⣰⠞⠁⠀⠀⠀⠀⠀⠀⠈⠲⣄⡱⣄⠀⠀⢀⣴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠇⠀⣀⠀⠀⠀⠀⠀⠀⡀⠠⠾⢛⠥⡖⠉⡰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠉⢲⠌⡛⠷⠄⠀⠀⢀⠀⠀⠀⠀⣀⠀⠸
⠀⠀⠀⠀⣦⣤⣿⠏⠀⠉⠡⠐⠁⠀⣀⡤⠋⣴⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣦⠙⢤⣀⡀⠈⠂⠌⠉⠈⠹⣿⣤⣴
⠀⠀⠀⠀⢿⣿⣁⣀⣴⠴⠛⢛⠉⢉⡤⠖⠋⢉⠡⠊⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠑⠌⡉⠙⠲⢤⡉⢉⡙⠛⠦⣦⣀⣈⣿⡿
⠀⠀⠀⠀⢸⣿⠟⠉⠓⠚⢻⣶⠾⠩⠄⣢⠤⣒⠞⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠳⣒⠥⣔⠠⠍⠷⣶⡟⠓⠚⠉⠻⣿⡇
⠀⠀⠀⠀⢻⣏⠀⢀⡤⠒⣏⠁⡴⠾⠋⢒⣨⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣅⣒⠉⠷⣤⠈⣽⠒⢤⡀⠀⣹⡟
⠀⠀⠀⠀⠸⣿⣿⣿⣷⣤⣿⣋⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⣻⣿⣤⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉

⣿⠿⠿⠿⡿⠟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣛⣛⣛⣿⣟⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠉⠉⢺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠒⠖⠶⣎⠀⡖⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢒⡺⣆⠀⠀⢠⢉⣀⣧⡈⠒⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠈⠀⠉⣿⠀⠀⢸⡏⠉⠉⠉⣳⠀⠀⠀⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣉⣛⡇⠀⠀⡼⣿⠀⠀⣾⡿⣆⠀⠀⣿⣶⣤⠞⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢨⡇⠀⢰⣷⢿⠀⠀⢻⣇⣽⡄⠀⢻⣿⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⡿⠀⠀⣼⣿⣿⠀⠀⢘⣿⣼⡇⠀⢸⣿⣿⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⡿⠁⠀⠀⠙⣿⠁⠀⡀⠈⢻⡿⠁⠀⠀⢻⣿⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⣥⢠⡵⠖⢴⣿⡟⠛⠒⠛⢻⡷⠴⠦⣵⣼⠟⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠿⠛⢻⣄⠀⠀⠰⣿⣷⡄⠀⢠⣾⣷⠀⠀⢰⣿⠆⠶⢤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠈⢻⣷⡀⠀⠹⣿⣇⠀⠘⣿⣿⠀⢠⣿⣿⠆⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠻⣿⣄⠀⠙⣿⠀⠀⣿⡏⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠙⠻⡆⠀⢈⠀⠀⠻⡀⠀⢻⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⡤⠛⠒⠘⢷⠂⠘⠣⠄⠙⠲⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠗⠀⣀⣠⣏⠀⠀⠠⠦⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢈⡓⠈⣄⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠿⠷⣄⣐⠯⣍⡠⠤⣤⡬⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠈⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⠠⠴⠒⠦⣀⠀⣀⣀⠤⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠛⠒⠦⣄⠀⠀⢀⠀⠀⠐⠒⠛⠛⠛⠛⠛⠛⠛⠲⣆
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⠊⠁⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠹⣦⣀⣀⠀⠀⠀⢀⣀⡀⠀⢀
⠀⠠⠓⠂⠉⢉⡧⠤⠔⠒⣨⠕⠉⣀⡺⢤⣶⢖⡠⠄⠊⠁⠀⠈⠀⠀⠀⡀⣀⡀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹⠃⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠉⢁⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠣⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠉⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠤⠤⠤⠤⠤⢤⣄⠀⠀⠀⠀⠀⢀⣠⠤⠴⠾⠿⠳⠷⠴⠾⣤⡀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⢀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡖⠉⠀⢀⣀⣠⣤⢤⣤⣶⠔⠍⠳⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠸⡆⠰⢛⠝⠛⢯⠳⠏⠁⠙⢦⡀⢀⠏⠀⠀⠀⠈⠢⠄⠀⢀⠀⠀⢀⠠⢀⣀⠠⠂⠉
⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⡀⠀⢀⡀⠀⠀⠀⣄⠀⠀⠅⠘
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⢋⠳⠀⠀⠈⠁⢀⡀⠀⠈⠀⠀⢰⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠤⣄⠀⣀⡀
⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠞⠀⠀⠘⣄⣻⠐⠀⠀⢰⠁⠀⠁⠀⠀⠀⢸⣃⠏⠀⠀⠀⢀⠄⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⡀⠀⠀⢀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠒⠉⠉⠉⠙⠲⠄⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣥⣬⣍⣉⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠉⠲⢤⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⣀⠀⠀⠠⠄⠀⠆⠀⡆⠀⠈⠀⠀⠀⢳⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠘⠄⠰⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢠⠂⠀⠒⠀⠃⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠉⠒⠐⠉⠀⠠
⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣦⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣽⣷⣤⣄⣀⣤⣴⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⠀⠀⢾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⡆⣭⣿⣿⣤⣶⣄
⠀⠀⣀⣠⣤⣴⣿⣧⣿⣞⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⡄⣸⠃⣌⣽⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣧⢿⠺⣹⣦⡀⠀⠀⢰⣿⢟⣥⣤⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢴⣷⢄⣟⣌⣿⣼⢫⣿⣿⣿⡟
⢰⣿⢻⣿⣿⣿⣿⣿⡇⣻⣿⢻⣄⠀⢸⣿⣿⣿⣧⡟⠆⣀⠈⠉⢿⣿⣿⠋⣁⢠⣿⣾⣿⠘⣿⣼⡟⢛⣵⡿⣿⡟⠁
⠈⠻⣿⣿⣿⣿⣿⣿⡟⣿⣿⣵⣽⣷⢸⣿⣿⣿⣯⣷⣥⣿⣧⣰⣿⣿⡇⣸⣬⣟⣿⢿⣿⣧⠋⣿⣿⣭⣭⣿⣿⣤⣄⡀
⠀⠀⢈⣿⢿⠻⣿⠟⣷⣾⣿⣿⡿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣿⣟⣽⡿⣿⠏⣚⠿⣿⣆
⢠⡾⢿⣿⣿⣿⣿⣿⣷⡜⢿⡿⣣⣦⣹⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣇⣿⠹⣿⣿⣿⣿⢃⣾⣿⣻⣟⣻⣿⣿⣿⣷⣦⣿
⠀⠀⣨⣿⡿⠿⣿⣿⣿⣟⣈⢄⣿⣿⣧⣽⣿⡿⢿⡿⣟⡻⠮⠝⢿⣿⣿⠰⠞⠿⣟⣿⡇⣸⣯⣽⢹⡟⠛⣭⣻⣿⠛⠛⠉
⠀⢸⢛⣩⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠫⣝⡲⠦⠍⢁⠤⠎⣴⣿⣿⣻⣿⣿⣿⣻⣷⣾⣿⣿⣿⡆
⠀⠻⢿⡿⣿⣿⣿⣿⢿⣿⣭⣍⣿⠻⣿⢿⣿⣿⣿⣿⣿⣧⣀⠄⢭⢤⠵⣂⣼⣿⡟⣱⣿⣿⣿⣿⣿⣌⣿⣷⡋⣿⣧
⠀⠀⠀⠀⣿⣿⣻⡗⠲⣽⢷⣮⣿⣿⣦⣄⣿⡿⣹⣿⣿⣿⣟⠿⠿⠿⠿⢛⣽⣿⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⠷⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⢟⣼⣿⢀⣼⣿⣿⣿⡿⣿⡇⣿⣿⣻⣿⣿⣷⣾⠛⠛⢿⣿⠛⣿⣿⣿⣾⣿⣧⡽⠿⠷⢶⣾⡛⠉
⠀⠀⠈⢿⣿⢿⣷⣄⣹⣡⡾⣿⡿⢛⣿⣿⣿⣳⣿⣿⣟⣷⡄⠈⠁⠀⠀⠀⠀⢰⣯⠿⣿⣿⣿⠿⢷⣶⣿⠿⠋
⠀⠀⠀⠀⢹⣿⣮⡙⠛⠛⣛⣽⣿⠿⠛⣿⣿⣿⣿⠫⣾⣿⣿⡄⠀⠀⠀⠀⠀⣿⡿⢣⣞⢷⣶⣦⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⠟⠻⣿⣷⣘⣿⣿⡇⠀⠀⠀⠀⠠⣿⡴⠿⠋⠀⠈⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠻⡿⡿⠿⠛⠁⠀⠀⠈⠉⠛⠿⠋
⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠠
⢺⠟⠠⣿⣶⣡⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣿⡟⠀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⡔⢀⣤
⣈⣷⣾⣿⣿⣿⣿⣿⣟⣲⣿⡆⠀⠀⠀⠀⠀⣠⣶⣦⠔⢉⣼⣿⣿⢇⣀⣤⣿⡅⠀⣴⡼⣿⣌⣈⣹⣦⣱⣿⣿
⣿⣿⢿⣙⣿⣿⣿⣻⣿⣿⢯⣤⠤⠀⠀⠐⠀⠈⢻⡿⠃⣿⣿⠁⠛⣿⣿⣿⣿⢿⣜⣧⡽⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣅⡀⣸⡿⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣸⣿⠀⠀⠈⣿⣿⣿⣤⣉⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣻⣿⠟⠁⣠⣿⣿⠏⢻⡇⠀⠀⠀⠀⠀⠀⠀⠸⠉⢩⣿⠋⠀⠀⠀⢘⠛⠛⣻⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿
⠘⢯⣏⠀⠰⢿⢯⠀⠀⣸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠀⠀⠸⣿⣿⣿⣏⠅⣿⣿⣿⣿⣿⡧⠏⠹⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⡀⠘⣿⣿⣿⣿⣷⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⡇⠀⠀⠀⠀⠉⠉⢢⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⡖⠉⠉
⡇⠀⠀⠀⢀⣠⣴⣾⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⠀⠀⠀⠀⠀⠀⣷⣤⣄⡀
⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⡇⠀⠀⠀⠍⠻⢿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠩
⠣⣄⣀⢀⡀⠀⠀⠀⠉⠐⠒⠂⠤⠼⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠄⠤⠤⠐⠒⠈⠁⠀⠀⠀⢘⣀⡀⡠
⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠈⠒⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣧⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠢⣿⠀⠀⠀⠀⢸⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠸⣿⣦⣤⣼⡟⠀⠀⠀⢀⠌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠘⣶⣄⠀⣰⣿⡏⠈⣧⠀⠀⢉⣻⣿⡟⠁⠤⠤⡄⠁⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢿⠈⢿⣿⣿⡀⠀⠘⢿⣾⣿⡿⠋⠀⣀⠀⠀⠃⢀⣆⡼⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢸⠀⠈⢿⠈⢳⠀⢀⠀⠈⠁⠀⠀⠀⣸⣿⣶⡶⠟⠋⠀⠀⢘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⠳⢦⡀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠈⡓⠤⡀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⢸⠀⠱⣄
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣾⠀⠀⠀⠈⣇⡾⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠲⠛⠃⠘⣲⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠿⣄⠀⢀⣿⠀⠀⠀⢀⣈⣤⠎⠀⠀⠀⢰⣾⠀⠀⠳⠦⠴⠀⠀⢦⣼⠁⡀⠀⠀⠀⡀⠀⠀⢘⡇⣹⣷⡀⠳
⠀⠀⠀⠀⠀⢸⠀⠈⠙⠛⢁⡤⠶⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⡄⠀⠈⠠⣤⣄⣶⡿⡁⢰⡇⠀⠀⣴⡇⠀⠀⠀⠟⢉⡿⡇
⠀⠀⠀⠀⢠⠆⠀⠀⢠⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠂⠤⣼⣿⣿⢂⠀⠀⠠⣴⠏⣴⣇⠞⠃⢀⠂⣹⠁⠀⠀⠀⣠⠟⠁⡇
⠀⠀⠀⣰⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⠀⢷⣦⣾⠿⣆⠈⢁⣀⣀⠂⢠⠃⠀⠀⢠⠞⠋⠀⠐
⠀⠀⢰⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠘⠉⠇⠀⢀⣾⠟⠛⠃⠀⣮⣄⠀⠀⠀⣠
⠀⠀⢸⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠸⠋⣿⣷⣄⠜⠿
⠀⠀⠀⠑⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⡀⠀⠀⠀⠀⢰⡄⠸⣇⣀⠀⠀⠀⠀⡏⣽⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⣴⡿⣿⣿⣿⣿⡄⠀⢀⣀⣸⣿⣤⠟⠁⠀⠀⠐⠊⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⠀⠀⠘⠁⠀⠀⠉⣻⣿⣿⡿⠿⠟⠛⠋⠁⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⢠⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢺⣻⣿⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠀⠹⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⣿⠟⣿⣦⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡗⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠇⠀⠸⣿⠻⣿⣧⡀⠀⠀⠀⠀⣀⣴⣿⡿⠟⠋⣽⣿⡇⠀⠀⠀⠀⠀⠀⢀⡞⠁⣼⡇⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢻⡆⠈⠛⢿⣦⣤⣴⣾⡿⠿⣷⣦⣤⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠈⠉⣼⠏⠀⠀⠀⢨⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠣⠀⢰⣏⣠⣄⠂⠈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡟⠉⠉⠉⠉⠁⠀⠸⣿⣁⡀⢀⠄⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣭⣥⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⣠⣶⠉⠠⠖⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⠶⣤⣄⣠⣤⡶⠞⠛⠛⠛⠛⠳⢶⣤⣄⣤⣤⠶⢋⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⠀⠁⠉⠑⠣⠀⠀⠀⠀⠀⠀⠞⠋⠉⠈⠀⣰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⡿⢡⡾⠀⠀⠀⠀⠀⠰⣄⣠⠂⠀⠀⠀⠀⠀⢷⡜⢿⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠛⠁⠀⠀⠀⠀⢀⠀⢹⡏⠀⡀⠀⠀⠀⠀⠀⠙⠈⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⡤⠶⢦⡀⠀⠀⠀⢳⣸⢃⡎⠀⠀⠀⢠⡶⠶⢤⡆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⠻⣿⡻⡿⣦⣀⢀⣼⣹⣌⣆⡀⣀⣴⣿⢟⣿⠟⡼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠈⡀⠙⠿⠾⣿⣿⣿⠛⣙⣋⠛⣿⣿⣿⠷⠿⠋⢀⠁⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢹⡄⢀⣾⡟⠁⣠⠭⠭⠭⠭⣄⠉⢻⡃⠀⢠⡏⠀⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⣿⣾⠟⠋⠸⣏⣤⡔⢢⣤⣻⠇⠙⠻⣾⣿⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⣿⠁⠀⠀⠀⠘⢯⣿⣿⠽⠃⠀⠀⠀⠈⣿⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠈⢀⠀⣠⣶⣾⣷⣦⣄⠀⡀⠁⠀⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⣾⣿⣭⣭⣭⣭⣿⣷⡀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡿⢯⡀⠶⠶⠶⠶⢀⡽⢿⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣥⣭⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣹⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣆⣀⡏⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠝⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡒⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣯⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡷⠀⠀⠂⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢸⡇⠐⠃⠀⠀⢸⣇⡴⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠉⠙⣾⠃⠃⠀⠀⠀⢸⣿⠱⠀⠋⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⢻⠇⠀⠀⠀⢀⣻⠃⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⣾⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠂⢸⣷⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡼⠀⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁⠀⣸⡇⠀⠀⠀⢀⠀⣾⠃⠀⠀⠀⠠⠄⢹⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⣠⠏⠀⢀⣀⡀⠀⠼⠃⣀⣀⣀⡀⠀⠀⠀⠈⡓⡀⠀⠀⠀⣾⡀⠀⠀⠀⣂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⡟⣰⠞⠋⠉⠉⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠂⠀⠈⢳⠊⠀⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⡿⣿⣦⣀⣀⡀⠀⠓⠈⢓⣲⣶⣤⠈⠱⠀⢤⣀⣀⠀⠀⠀⠈⠳⢤⡀⠀⢠⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡜⠀⠄⢻⡈⢿⡍⠉⠁⠀⢁⡀⠈⣽⠿⠤⡀⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⣀⠙⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡆⠈⡐⢷⣰⣻⣆⠀⠀⠀⠁⠀⠈⣿⣶⡄⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠦⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠐⠈⢷⡻⡹⣆⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠀⠀⠀⢰⠀⠘⠁⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠄⠸⡇⠀⠹⣧⠀⠀⠀⠀⠀⠃⢣⡀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⢿⠀⣆⠈⢧⠀⠀⠀⠀⠐⣮⡅⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⢠⢨⣿⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠨⠁⠀⠀⠀⠀⠀⠀⢹⡋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⢃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⡖⢦⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡡⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡜⠘⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢉⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣌⠷⣙⡖⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢃⣼⢇⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡷⠿⢬⢯⡱⢆⠂⠀⠀⠒⠀⣦⠐⢀⠄⣠⢞⣵⡾⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⡛⠳⢮⣧⣍⣤⣤⣤⠀⠀⠀⣽⣤⣤⠞⠷⢛⠁⡜⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣝⠅⠀⠀⠉⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠁⣴⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠬⠑⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⢉⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⡶⢿⣿⣿⠿⢿⣟⣛⣛⡁⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢟⣵⣾⠟⣱⡿⡩⢖⣛⣛⣛⣻⣇⣤⢄⡸⣦⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡵⠿⠿⣗⣠⣿⢠⢞⣭⣽⣷⣶⣶⣿⠿⢟⢡⣿⡇⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⣛⣒⣚⣻⣿⣿⣿⣧⣵⣦⣄⣛⡛⠿⢿⣿⢿⣛⡍⣼⡿⠀⣟⣛⣛⠇
⠀⠀⠀⠀⠀⣛⣛⣿⣶⣿⣿⣿⣛⣉⣙⣻⣿⡟⣩⡛⠿⠿⠿⢛⣠⢹⣿⡟⣱⡿⢡⢸⡌⣿⣿⡇⣆
⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣻⣿⣿⣿⣣⣿⢏⣼⠏⣠⡟⣸⠇⣿⣿⢡⣿⡆
⠀⠀⠀⠀⢼⠂⡶⢙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⢋⣾⣿⠟⠡⠚⠵⣾⣿⢣⡿⢸⣿⠇⣾⣿⡇
⠀⠀⠀⠀⠈⠀⠃⠈⣿⡮⢻⣿⣿⣿⣿⣿⣿⣿⡇⣸⡿⠿⢛⡅⠠⢞⡻⡿⣎⠋⣾⢃⣿⡟⣼⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⡏⣿⣿⠻⠏⢿⣿⣿⣇⣿⣷⣾⠏⣴⣾⣿⠿⡇⢈⣄⣿⢸⣿⢸⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⣿⣿⣿⣿⣿⣿⠿⠛⣿⣟⠇⣼⣿⣿⣿⠇⢀⣾⣷⠻⣌⠇⠋⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠘⢿⡿⢏⣼⣿⠘⠉⠉⢵⣒⡶⣾⣿⣿⠀⣿⡟⣼⡿⣰⡿⢟⠋⣁⠬⣠⣾⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⢿⡟⠀⠀⠀⠀⠈⢻⣶⡝⣿⣇⢻⡇⣯⠇⢉⡴⢣⡞⣡⣾⡿⢋⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡷⠈⢿⣎⢿⢹⣧⡾⣱⡏⠼⣛⣩⡶⣣⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⡌⢻⣎⢇⢿⢳⣿⣿⠿⣛⣭⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡙⢧⡈⠛⠵⡾⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣍⡓⠀⠉⢻⣶⣍⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣼⣷⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⡾⠒⢷⠊⡉⠆⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠀⡛⢦⣤⡀⠀⠀⠀⠀⣀⡤⠊⠁⠀⡄⠀⠿⣧
⠀⠀⠀⠀⠀⠀⢸⡗⡫⠊⢀⠳⢙⣟⣉⢦⡀⠀⠀⠀⣠⡇⠀⢸⠀⠀⡃⠈⣆⠙⣆⠀⠀⠸⡇⠀⣢⣤⣟⡟⠁⠀⣈⢧
⠀⠀⠀⠀⠀⠀⡞⠁⠐⠒⢿⣤⡆⣙⢇⠀⠱⠀⠀⢰⡸⠁⣀⠈⠀⢀⠀⠆⠀⠐⢾⠀⠀⠀⠹⡆⠀⠀⠏⠂⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⠑⠿⢞⡃⠀⠇⠀⠀⠑⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⠀⠘⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣯⣽⣭⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡗⠒⣿⣯⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⡞⠛⢯⢿⡇⠀⣸⡇⡎⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⡽⣿⠀⢸⣸⡇⢀⣿⣵⡇⢰⡿⡟⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⣿⠀⢸⣿⣀⣼⣻⣟⠀⣼⡇⢻⡶⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣽⡎⢹⣿⠀⠘⣿⠉⢹⣿⢇⣼⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⣧⠀⢿⡆⠀⣿⠀⢰⡏⡾⠧⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢹⠄⣸⠁⢀⡟⠀⣾⣼⠃⢨⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣀⣿⣀⣿⡁⣴⣿⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⢬⢃⣏⣈⡟⢏⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣽⣟⣻⢿⡾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢻⡟⠋⠉⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⣿⠀⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡏⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣿⢰⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⢀⣈⣷⠀⠀⠀⠀⢀⡿⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣏⢿⣄⠀⢰⠞⣻⣷⢀⠘⣞⢧⣴⠞⠲⣆⠀⢀⣠⡼⢷⣄⣀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠽⣛⡿⠺⠿⢻⡈⠀⠈⢫⡟⢾⢦⣿⣟⡯⢁⣶⠟⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⠀⠀⢹⡇⠀⠸⣀⣐⣹⣃⡠⠀⡴⡰⠋⠈⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⡷⠾⣿⣿⡿⠿⠿⢿⣿⣿⣿⣾⣜⡴⠁⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⢀⡄⣠⠦⠀⠉⠉⠀⠀⠀⠀⠉⠻⢿⣿⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢁⣟⣾⣿⣶⣤⣄⢤⣀⡀⢸⣿⣷⡄⠀⠙⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⣿⡏⣾⣿⣿⣿⣉⣹⡿⠿⠛⠃⠀⠙⠿⠿⠀⠀⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠾⠿⣹⣿⣿⣼⠟⠉⠀⠀⠀⠀⣀⣴⣦⠶⠚⠒⠶⣌⠻⣿⣧⡀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣷⣦⡶⠟⡁⢐⣷⣾⣿⠿⠟⠃⠀⠶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⢀⠘⡇⠀⢿⣷⣴⣾⣏⠻⣝⣆⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣾⣻⡿⠋⠁⠀⠀⠀⣀⣀⣤⣤⣭⣽⢋⠀⠀⠀⠀⠀⠀⠈⣰⣧⠀⢸⣿⣿⣿⣿⡆⣸⣼⢀⣀⣀⣠⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣽⣿⣧⡀⣠⣴⣾⣿⠟⠙⠃⠘⣿⠻⠁⠂⠀⠀⠀⢀⣠⠶⣿⣿⣇⣼⣿⣿⣿⣿⠛⠛⠋⣽⣿⠿⣶⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⣿⣿⠯⠿⠀⢀⣤⣴⡷⣾⣛⣆⠀⢀⣠⣴⠞⠋⠁⠀⠙⢻⠟⣿⣿⣿⣻⣿⣷⡶⠾⣿⣿⣠⡷⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢼⣟⠀⣠⣾⣿⡿⣭⡷⠛⠉⣙⡿⠟⠋⠁⠀⠀⠀⠀⢀⣼⣷⣿⣿⣿⣿⣿⣿⣷⡄⠀⠈⠙⠳⢤⣄⡀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡿⠿⠛⠛⢉⡏⣻⠗⠉⣀⡴⠞⢉⣠⠖⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠉⠀⠀⠀⠀⠀⠆⢸⠃⣠⠞⠉⣠⣶⣿⣏⣀⣀⣠⣤⣾⣿⣿⣿⡿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡋⣠⣾⣿⣿⠟⢻⣿⣏⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡙⢿⣿⣿⣿⣿⡄
⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⣓⠧⣴⣿⣧⠈⠛⢿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣷⡙⠻⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣆
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⠉⠁⢸⣿⣿⠀⠀⠈⠛⢿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣆⠈⠻⣿⣿⣿⣦⡀⠙⢿⣿⣿⣧⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⣟⡹⣇⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⡈⠻⣿⣿⣷⡄⠈⠻⣿⣿⣷⡄⠀⠙⢿⣿⣷⡀
⢿⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠹⡆⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⠈⠻⢿⣿⣆⠀⠀⠙⢿⣿⣦⠀⠀⠙⢿⣿⣄
⠈⢷⣌⠢⣀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⠙⢿⣷⡀⠀⠀⠙⢿⣷⡄⠀⠀⠙⢿⣆
⠀⠀⠉⠛⠷⠿⠷⠛⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄⠀⠙⠻⣆⠀⠀⠀⠙⠿⣆⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡀⠄⡀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣤⣴⠞⠉⢷⣄⡀⠐⣀⣴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡿⣿⢟⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡿⢟⣧⣽⣭⣤⢿⣿⣿⣷⣿⣿⣷⣠⣴⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣥⣾⣿⣿⣿⣿⣿⠈⠿⠋⢸⣿⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⡇⠀⠀⠀⢸⠻⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠃⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⡏⠀⢸⡇⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠉⠉⠙⣿⠒⠛⠒⢺⡇⠀⠈⠀⣿⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠘⡇⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢴⣹⠁⠀⠀⠀⢸⡇⠀⠀⠀⡿⣀⢀⡀⣧⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠁⠀⠀⠀⠇⠹⠿⠃⠸⡄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣠⣤⣤⣄⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⠛⠿⢻⠿⡿⢻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠉⢩⡁⠀⠀⠀⠀⣠⡞⠛⠴⢦⡈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠳⣤⣸⡇⠀⠀⠀⠀⠹⣦⣤⠄⠀⠶⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠳⣬⡙⠷⣄⡀⠀⠀⠀⣾⠁⣠⡷⠦⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢨⣿⢷⣬⣿⠀⠀⠀⠁⠐⠋⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠚⢻⡟⠛⠉⠙⣧⠀⠀⠀⠰⣄⠸⠀⢀⣠⡤⠘⢿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⡟⢠⠶⢦⡀⠘⢷⡀⠰⠶⠀⣠⡾⠋⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠁⠀⠀⠈⠻⣆⠈⢷⡄⠀⠀⣿⠀⠀⠀⠀⠀⢀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⢷⡀⠻⠞⣶⠘⢧⣀⠀⠀⠰⣟⢷⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠉⠀⢀⣽⠗⢀⣤⠟⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣍⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⡴⠟⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿