Hai Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⠴⠀⠶⠟⢛⣏⠀⠉⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠓⠢⣄⣴⠋⠀⣠⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠲⢦⣴⣞⡥⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠁⠀⠘⠋⣠⣾⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠩⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢙⡃⠀⠀⠀⠋⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⣄⣀⠐⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡲⠛⠀⠀⠀⠀⠀⢹⠀⠃⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⣴⡼⠫⢥⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡿⠀⠀⢛⣿⣷⡶⢄⠀⠀⡁
⠀⠀⠀⠙⠄⠆⠸⠛⡆⠀⢠⣶⠰⠓⣀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣾⣧⣤⠶⣿⡿⠁⠉⠈⠁⠀⠃
⠀⠀⠀⠀⠈⠉⠛⠒⠻⠋⠈⠉⠀⠈⠉⠛⢿⣧⣀⠀⠑⣄⡀⠀⠀⣆⠀⠀⣀⣀⠂⠀⠀⠀⠀⢠⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠈⠙⠃⣄⠹⢤⣄⡛⠁⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠈⠻⠆⠠⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⢀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣶⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⢀⣴⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁⠀⠀⠉⠛⠛⠋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠛⠻⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢷⣦⣤⣤⣴⡾⠟⠛⠁⠀⠀⢠⣾⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⡇⠀⢠⣤⣴⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠁⢀⣠⣤⣤⣤⣤⣀⠀⠁⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⣀⣤⣤⣤⣭⣅⡀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠟⠛⠋⠁⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁





⣿⣿⡿⠿⣿⠿⢿⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠋⠀⠻⣦⣭⣉⣿⣯⡵⣛⠟⠀⢻⣿⣿⣶⣶⣿⣿⡿⠋⠻⣿⣷⣤⣤⣴⣶⣿⠋⠘⣿⣷⣦⣤⠤⣖⣿⡟⠈⠻⣟⣻⢷⣶⢿⣟⣯⠔⠉⢿⣿⣦⣤⣤⠴⣒⣿⡻
⠀⠀⠀⠀⠉⠓⠺⠷⠂⠁⠀⠀⠀⠙⠒⠿⠷⣿⡟⠁⠀⠀⠈⠛⠻⠟⠛⠋⠀⠀⠀⠘⣿⡿⡿⠿⠿⠋⠀⠀⠀⠀⠉⠛⠛⠋⡉⠰⣿⠀⠀⠙⠺⠽⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠢⠜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠘⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡷⠞⠹⢣
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢠⡶⡀⢨⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⠂⠀⠸⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⡄⠀⠈⠃⠀⠀⠀⠀⠀⣀⡄⣈⣘⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⣩⠟⢉⡁⢀⢀⡍⠄⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣤⠆⠀⠀⠀⠀⠈⠀⠠⠞⢹⡃⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⠻⣂⣺⣿⣿⠀⡷⠀⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢳⡀⠀⠃⠀⠀⠒⠀⠀⠠⣤⣄⠀⠀⣿⣿⠀⠈⢩⣿⡏⠀⢵⡟⠀⡟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠑⠶⠃⢀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠛⠛⠞⠆⠠⠼⠇⠀⢸⠇⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠆
⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠶⠞⠧⠸⠷⠀⠀⠀⠀⠀⠒⠂⠂⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⣿⣀⢘⠀⢀
⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠄⠀⠂⢰⡦
⠀⠀⠀⢸⣦⣤⠀⠀⠂⠀⠀⢠⣄⠀⠀⢠⣄⣤⠷⠀⠘⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⣤
⠀⠀⠀⠸⣯⡿⠄⠀⠠⠀⠀⠀⠤⠀⠀⠘⣿⡿⣷⡆⠀⠀⢀⡀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠛⠃⠀⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⣇⠘⠀⠀⢀⡤⠀⠀⠀⢀⠀⠉⠀⡿⠋⠀⠀⠘⢁⣴⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣬⠁⠁
⠀⠀⠀⠀⠀⠘⠀⢠⣄⡾⠃⡀⠀⠀⣀⠁⠀⠈⠇⠗⣤⣆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⡏⠁
⠀⠀⠄⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⢀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠒⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢰⢆⠄⠀⠀⠸⡿⣅⠀⣠⡄⠀⠀⢀⠀⢠⠄⠀⠀⠀⠆⠀⠀⠈⢠⣧⠀⠀⠀⠀⠀⠈⠛⠃⠸⣕⣷⡄⠴⠂⠀⠀⠀⠀⠀⠀⢀⣶⣠⡆⠀⡀⠘
⠛⠃⠀⠉⠀⠀⠀⠀⣇⡀⠀⠈⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠠⡄⠀⠀⠀⠩⠀⠀⠀⠀⠀⠐⠶⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢤⣤⠀⠁⠙⢸⣦⡄⠀⠀⡀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠉⡄⠀⠀⡇⠻⠀⠀⠀⠈⠻⠧⠀⠄⠉⣶⡆⠀⠀⣀⣀⠀⠀⣀⣠⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⠞⣏⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢤⡄⠀⠀⠐⢲⡀
⠀⢈⡀⣄⡀⠃⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠛⠁⠀⠀⠀⠁⠀⣟⠛⠉⠀⠀⠋⠙⠃⠀⠛⠁⠀⠀⠐⣿⠀⣿⡇⠀⠀⠀⠀⠈⠛⠓⠀⠀⠀⣀⠨⠅⠀⠀⠀⡸⢗
⢀⣹⣿⡿⠀⠀⠀⠀⠰⠀⢀⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⠀⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠂⠂⠀⠀⠀⣿⠃⠀⠙⠁⢀⣴⣿
⠈⠉⢉⡁⠀⠀⠀⣴⡀⠀⢁⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠋
⠀⠁⠌⠀⣀⢰⣄⣶⣤⠀⠈⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠻⠟⠛⠀⣰⣾⠗⠂⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⣤⣤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢯⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢄⠀⢠⣿⠋⠀⢀⣤⡄⢰⣷⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠒⠒⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⣀⡀⠀⠀⠐⠒⢲⣶⠃⠀⠀⡀⠀⠈⠀⠀⣐⣿⣷⠀⠿⠆⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⡶⢾⣭⡑⣮⠉⢳⣒⠶⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣷⠷⡷⣤⠇⢷⣄⣷⡟⢩⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣾⣷⣷⣧⡃⠀⢀⣀⠀⠁⠉⣏⣹⡤⠷⡖⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣈⢻⣿⣿⣠⣿⣴⣿⣽⠀⠀⠀⠀⠀⠣⣴⠃⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠈⢉⣉⣉⣸⣿⣿⠀⠀⠀⠀⠀⠀⠃⡰⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠂⣴⣿⣿⣿⣷⣼⣿⠀⠀⠀⠀⠀⠀⠀⠲⢦⣾⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠮⢿⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠊⢸⣉⣏⡍⠳
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠘⣫⣵⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣀⣿⠐⣨
⠀⠀⠀⠘⡁⡠⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣤⠾⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⡁⡠
⠀⠀⠈⠈⠛⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⡿⣏⠗
⠀⠀⣠⠘⠅⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⡖
⠀⠀⠳⠶⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣼⡿⢹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠿⠎
⠀⠀⣧⣐⠁⡀⠀⠀⠀⢸⣿⣿⣿⣯⢹⣿⣿⣿⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣡⣾⠄⠂
⠀⠀⢈⡿⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢻⣍⠡
⠀⠀⠀⢉⣴⠎⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠃⠦⡤⢄
⠀⠀⠀⠾⡉⢀⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⣴⢾⡀
⠀⠀⠈⢀⡽⠓⠞⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢎⣅⡀
⠀⠀⠀⢨⢉⣉⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣅
⠀⠀⠀⢸⣣⠶⠿⠢⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋
⠀⠀⠀⢸⡟⣦⡀⠀⠀⠀⠀⠸⣿⡏⠛⣻⠿⠿⣿⢻⣿⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠹⣍⣈⠷⣉⡷⠀⠀⠀⢿⣧⢰⠁⠀⠀⠸⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣠⠿⣃⡱⠀⠀⠀⠀⠈⠻⣾⣆⠀⠀⣰⡿⠋
⠀⠀⠀⠀⣧⣶⡳⣶⠒⠒⣤⠖⠀⢲⡄⠈⢷⡞⠁
⠀⠀⠀⠸⣀⢙⡓⣤⡦⠿⠋⣢⣶⠯⠤⠴⠚⠁
⠀⠀⠀⠀⠈⠉⠙⢤⣉⣀⡯⠞⠁

⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⡀⠀⢀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠐⠦⠤⠀⢀⣀⣀⣐⠂⠰⠦⣀⠀⠐⢿⣯⠀⣜⣻⠺⢧⣀⣉⡁⠸⡏⣸⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣭⣭⠤⠤⣬⡙⢶⠀⢹⠀⠀⠀⣿⣘⣻⣷⠐⣿⡟⣻⣻⣦⣰⣿⠇
⠀⠀⠀⠀⠀⢠⣥⠶⠾⢶⠶⢶⣌⢻⡄⠄⣺⠀⠀⠀⠛⠻⢿⠿⢿⣿⣭⣉⣅⡿⠿⢿
⠀⠀⠀⠀⠀⢠⡛⢿⣿⡿⢿⣼⣿⡆⠰⢶⣿⣶⣦⣴⡇⠀⠀⠄⡀⣀⣈⢙⣿⣗⡀
⠀⠀⠀⠀⠀⢰⠟⠀⢈⣤⣤⣽⡿⣿⡿⣿⡟⣿⠟⠛⠀⠛⠧⠀⢻⣿⣧⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣀⣀⢶⣿⣿⣿⠿⣴⣿⣵⡿⠀⣻⣦⣄⢠⣠⣤⡱⣮⣽⣿⡋⣸⣿⠟⠁⠀⠀⠀⣀⠤⠤⡀
⠀⠀⠀⣄⡀⠉⠻⣿⣿⣟⣭⢾⣿⣿⣿⣿⠍⢁⣈⠫⠈⠙⠿⢛⣽⡿⣿⣿⢇⣭⣾⠠⠐⠈
⠀⠀⢸⣿⣷⣦⣄⠈⠛⣨⠀⠸⣿⣿⣿⣷⡞⠋⢉⣉⣶⣶⣾⣿⣿⣿⠻⢫⣾⣿⡿
⠀⠀⠹⠙⣿⢍⠉⠀⠀⡈⣧⠀⣿⣧⣶⣦⠝⠙⠿⢫⣿⣷⣾⡿⣿⣭⡾⣾⣏⡈⠋
⠀⠀⠀⠘⣿⣾⣉⠀⠀⠀⠈⢀⣿⣿⣿⣻⣛⡛⠰⣿⣾⡿⣼⣷⣿⢣⣿⣿⣝⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠿⢻⣧⣄⠘⠁⣴⣿⣛⣿⡿⢿⣷⣿⣿⠋⣿⣷⣿⣣⣿⣿⣿⡿⢛⣀⣶⡀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⢛⡟⢫⣿⢧⠻⣿⣿⠩⠦⠉⠛⠀⣿⣾⣿⣿⣿⡿⣏⣩⣿⡿⢿⣿⣿⣿⡄⠂⠉⠀⡀
⠀⠀⣶⣴⠀⠀⠀⠀⠘⢡⡇⠀⢿⣶⣿⣿⣃⠐⠈⠙⠻⣝⣿⣵⣾⣿⣮⡥⣶⣿⣿⣿⣿⣿⣆⢀⣠⣁⠀⠠⠄
⠀⠀⠛⠛⠱⡶⡲⢃⡖⣰⣾⠟⡈⠋⠛⣻⡟⠀⠀⠀⠐⡈⠙⣿⣇⣾⠁⠐⠋⢹⡿⠿⣻⣿⣿⡀⠃
⠀⠀⠀⠀⠀⠘⡿⢧⠶⢹⡏⣸⡇⠀⠀⠙⠇⢀⣀⡀⠀⠼⠶⣈⠻⠿⣷⣦⡖⣡⠖⢛⢉⣯⣹⣷⡀
⠀⠀⠀⠀⠀⠀⡇⢸⢿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠉⠳⠆⠒⢀⣴⣿⣄⢸⣿⣼⠉⠀⢿⣧⣟⣻⠿⢷⡀⠤⠤⠤⠤
⠀⠀⠀⠀⠀⢾⡇⡄⢠⣾⡿⢋⣾⣿⣿⣶⠄⠀⠀⠀⠀⠀⠻⢿⣿⣿⡇⡿⣽⣟⣶⢿⣎⣻⣿⠃⠀⢻⡄
⠀⠀⠀⢠⣶⡿⣷⢣⣿⣏⣴⣾⣿⣿⣿⣁⠀⠀⠀⠀⢄⡀⠀⠀⠉⠛⠻⣿⣟⣿⣿⣿⣿⣿⣿⣦⣀⡀⢻⡄
⠀⠀⠀⠋⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠶⣶⣾⡗⠦⣀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⠟⠀⡴
⠀⠀⠀⠀⠀⠛⣩⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣧⠀⣼⣿⣿⣿⣶⠿⠑⠢⣄⡀⠀⠀⠈⠙⠻⣿⡿⠿⠛⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠘⠉⣁⣿⣿⣛⣿⣿⡯⠀⠘⣿⡿⣷⣿⣿⣿⣿⠛⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡄
⠀⠀⠀⠀⠀⣼⡿⠿⣿⣉⣿⣿⣿⡇⠀⠀⢘⣿⣿⣿⣿⡟⠃⡤⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⡀⠀⢀⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠴⠟⠀⣀⣩⣿⣿⣿⣿⣧⣤⣺⣿⣿⣿⣿⠏⢠⠟⠃⠀⠀⣠⠾⠛⠁⠀⠀⠙⢿⣿⣦⣾⣿⣿⠏⠀⠹⡇
⠀⠀⠀⠀⣶⣤⣸⣿⣿⣿⣿⣿⣿⠿⠿⢻⣿⣿⣛⠵⢲⠏⡴⢀⡠⠊⢁⣀⣀⣤⡄⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⢠⣾⣿⡿⢿⣿⣿⣿⣿⡷⣹⣿⣟⣋⠽⣟⠋⢢⣏⢬⡴⠛⠛⢛⣉⡌⢀⡴⠀⠀⠀⠀⠀⠀⠀⠙⠿⣆⡀
⠀⠀⠘⠋⠋⠁⠾⠫⠿⣿⣷⣞⣉⣿⣭⣤⣄⣿⣅⢈⣤⡈⡐⢋⣾⡩⣯⠴⠛⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢠⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢸⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⣿⣿⡇⠘⣿⣿⣿⠀⢸⣿⣿⣿⠀⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣿⣿⣿⡇⠸⣿⣿⣿⠀⢸⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣿⣿⣿⡇⠈⣿⣿⣿⣀⣸⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⣿⣿⣿⡇⢀⣿⣿⣿⠛⢻⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⣿⣿⣿⡇⠈⣿⣿⣿⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⣿⣿⡇⠘⣿⣿⣿⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠰⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⢈⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⣿⣿⣿⠀⢸⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡉⢏⠻⠏⠚⠛⠓⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣭⡛⠀⠀⣀⣀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢷⡄⠀⠀⠚⣻⡀⠀⢉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣲⠷⠀⠀⠁⠀⣄⢁⣤⡄⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢙⢥⣂⢠⢄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢎⠖⡲⠋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣠⣠⣤⠔⢠⠁⠴⡋⠰⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡞⠩⠍⠉⢜⢛⡀⠘⢦⠀⠤⠀⠀⠄⠐⠚⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠏⢁⡓⠡⠈⠀⠀⠀⡛⠃⠀⠢⠀⢀⡀⠀⡀⠀⠠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠨⠉⠛⠀⠈⡇⣩⡀⠀⠀⠈⠋⠁⠈⠈⠡⠀⢠⡀⣉⠙⣀⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠓⠂⠉⠂⠈⡠⠏⠀⠠⠤⠉⠉⠋⠀⣦⠄⣉⡀⠭⠠⠣⠄⢈⡀⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢃⡀⡀⠠⡄⠤⠀⠀⠉⠀⠀⠀⠀⢹⠁⢰⡄⠲⠀⠀⠠⠄⠀⠂⠄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡌⢥⡀⠀⠉⡩⢂⡘⠂⠀⠀⠈⠀⢠⡤⠠⠈⠓⠨⠅⠀⠀⠈⠀⠀⣂⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⢿⠂⠉⠀⠀⡤⠀⠀⢀⠁⣠⠀⢀⠀⠀⠀⠠⠀⠀⠀⢒⣠⠀⠤⠶⠚⣉⣁⠀⠀⣀⢠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠅⠀⣾⠄⠀⡤⠂⡀⣠⠇⠀⠀⠁⠀⠀⠈⠉⠉⠡⠌⠋⢈⠞⠀⠀⠀⠀⠀⣠⠘⠃⠡⣎⣴⠴⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡨⠞⣤⢀⠀⣀⠰⠛⠴⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡰⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣼⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡨⢎⡄⠋⠁⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠋
⠀⠀⠀⠀⠀⠀⢀⡠⠾⢏⡛⠄⢁⡀⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇
⠀⠀⠀⠀⣀⠔⣉⢀⠰⠀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣀⠄
⠀⠀⣠⠞⢁⠾⠛⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣠⡾⠁⠀⠀⢀
⢀⡼⠓⡾⠽⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠒⠐⣭⢁⣤⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡏⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠄
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠤⠐⡄⠀⢸⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⣴⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠘⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢻⣷⣽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠾⣿⡯⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣤⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⢹⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⢸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠰⣿⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠠⣿⣛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⣸⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⢀⣿⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣹⡝⠻⣿⡇⣸⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⢠⡟⠀⠠⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡝⠻⡄⢀⣾⡃⠀⢠⣿⣿⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⢿⣿⡇⠀⢸⣿⣧⢙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣿⣧⣀⣾⣿⡏⢻⣿⡿⣿⣿⣿⣧⣄⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠛⣯⣿⣧⣀⣼⣿⡿⠛⣻⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⡘⢻⣿⣿⣿⣁⡀⠚⣹⣿⣃⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⠆⠹⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣏⣿⡇⠀⡁⠀⠚⢹⣿⣿⣿⣟⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠻⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢠⡀⢻⣿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡹⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡧⠀⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣦⠡⢾⣧⣤⣤⣤⣤⣼⣷⣿⣤⣤⣶⣶⣶⣴⣼⣷⣷⣾⣤⣶⣰⣿⣑⣦⣄⣴⣖⣀⣀⣀⣀⣀⡤⣤⡀⢣⣀⣷⡰⢠⡀
⠀⢺⠁⠘⠛⠁⠀⠀⠈⠉⠉⢿⡉⠉⣿⠉⢹⡏⣽⣋⣙⡫⢭⣍⡙⠑⠛⣹⣟⣩⣏⣉⡉⢹⣿⠉⠉⠉⠛⠿⢘⠏⠹⠚⠠⠹⡄
⠀⠀⠀⠀⠠⠄⠁⠚⠛⠂⠀⣤⠄⠀⣿⠀⣼⣯⠀⣿⠉⣿⢸⡏⠙⣦⠀⣸⣿⠁⣿⠉⣿⢸⣿⠀⠀⠀⠀⠀⠈⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⣤⣿⣿⡤⣿⣀⡿⢸⡇⠀⣻⣤⣿⣿⡦⣿⣤⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⣤⣿⡅⠈⣿⢠⡏⢿⠄⣿⠀⣿⣾⡇⠀⢿⣿⡏⢹⡇⣿⠈⣿⣼⣧⠀⠀⠐⠄⠀⠀⠀⠙⠻⡟⠁
⠀⢰⣷⠀⠈⠀⠀⠀⠢⠀⢽⡿⡿⣾⣻⣿⣿⣿⣿⣿⠀⣿⣿⡇⢀⣾⣿⠁⠸⠿⠿⠀⣿⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⣼⡋⠀⠀⠀⢰⡶⠑⠒⢺⣟⢹⣿⣿⣿⡿⢯⣿⣿⣤⣿⡿⢛⣋⣡⡟⠀⠀⣦⡤⣄⣿⣿⣶⠀⠀⠀⣠⢀⣐⣤⣤⣿⡇⠘
⠀⢿⣧⡀⠀⠀⠀⠀⠀⢲⢿⣯⠀⢀⣹⣿⡅⠀⣿⣿⣿⣿⠇⠀⠀⣹⡏⠀⢠⣿⣭⣿⣅⠀⠈⠙⠒⠈⣭⡹⣇⠀⢘⣻⣁⣀
⠀⢸⣿⣷⣶⣶⣴⣶⢿⣄⣀⡨⠇⢻⣿⣿⡿⠀⠈⡿⣿⣭⣎⡚⢲⣟⠀⠀⣿⣿⢲⢯⣿⡝⣿⡾⣝⣶⣤⡀⠹⡏⣿⣿⡟⠁
⠀⠀⠻⡷⠃⠛⢻⣹⢿⢻⠬⣝⡓⠸⠿⠿⠿⣿⡺⠛⣿⡿⣿⣿⣈⣾⡆⣰⣿⣿⣿⣿⣿⣯⡟⣴⣶⣤⣈⣙⣐⣥⣿⣧⡅
⠀⠀⢦⡴⠦⢴⣶⣿⣀⣀⢀⣀⣢⣬⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⣦⣿⣿⣸⣅⣨⡙⠳⣿⣿⣿⣮⠉⠉⡄
⠀⠀⡇⢙⣒⠒⣿⣺⣭⠁⢉⢨⣽⣟⣻⣿⡿⡿⣿⣾⣿⡋⣛⣟⠿⣿⣿⣿⣿⣿⣽⣯⣛⣻⣿⣽⣧⣄⣿⣿⣿⣿⣷⡴⠃
⠀⠀⠑⢤⣽⣿⣟⡛⠿⠙⠛⣷⣍⢙⠹⡤⢹⣁⡿⠈⠉⡁⣩⡴⠟⢛⣩⣾⡿⠁⠈⠉⠩⠍⠓⠚⢻⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠼⠋⠉⠙⠲⠈⠈⣹⠍⠙⠦⢤⡟⠛⠒⠚⠋⠁⠳⠴⠚⠋⠉⠠⠃⠀⠚⠙⠛⠛⠚⠲⠞⠛⠻⣿⣿⣯⣻⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⡴⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠒⠒⠒⠲⠦⠤⢤⣀⣀
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⠈⠳⣄⡠⠔⠊⠉⠀⠀⠀⠀⠀⢀⡤⠟
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣈⠳⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⣿⣧⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡾⠁
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣭⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣼⠃
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢴⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠈⠢⣤⣀⣀⣀⣠⡤⠤⠶⠒⠋⠉⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣠⡀⠀⣠⡀⢠⣄⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡿⠋⠻⡟⠙⠿⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⣀⠤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⠤⠤⠤⠤⠤⠤⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠛⠉⠉⠁⠀⠀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠹⣤⣤⣤⣤⠤⠤⢴⡶⠂⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠉⡽⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣆⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠘⣇⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢸⡄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠖⠛⠀⠀⠀⠀⢳⡀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠸⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⣤⠀⢢⡴⠂⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠇⠀⠀⢀⡄⠑⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠁
⠀⠀⠀⠠⠀⠀⠀⠀⢀⢀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⣀⣉⣿⠾⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡗⠀⠀⣴⣁⢀⣀
⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡶⠴⣮⠤⠀⠀⠹⣥⣎⣡⠀⠀⠴
⠀⠀⠀⠀⠀⢀⢸⠀⡖⣣⡀⠀⢲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⢀⣰⣀⠀⠀⠀⠉⠁⠀⠀⡄
⠀⠀⠀⠀⠀⣄⢸⣷⡏⠻⠹⠶⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠾⡻⣦⠀⠀⢰⣆⠠⢤
⠀⠀⠒⣶⣶⡿⠀⠋⠐⠿⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠆⠀⢹⡅⠸⠷⠀⢀⠉
⠀⠻⠿⢿⣿⠷⠃⠀⢌⠀⡀⠈⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⢸⡇⠀⠀⠈⠋⠀⡀⠠⣏⢁⡀⠉
⠀⢠⣄⣼⠏⠀⠀⠀⢶⣬⣿⣧⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠺⡅⠀⠀⣬⠠⠤⠅⠀⠙⢍⡀
⠀⠀⠿⠃⠀⢦⠀⠐⠄⢹⣯⡿⠋⢫⠀⣀⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢁⠀⠀⣠⡷⢖
⠀⠀⠀⠀⠀⠀⠂⠀⢴⡿⠀⠀⠀⠀⢄⠈⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⠀⣋⠀⠀⣿⣲⣴⡷⠦
⠀⠀⠀⠀⠀⠀⠀⢀⡌⢙⡿⡗⢉⠶⡄⠑⠆⠀⢀⣤⣴⡏⢀⣄⡀⢄⠀⠀⠀⠀⢀⣠⡴⠖⠒⢶⣀⠠⢀⠙⢿⣝⢂⣰⠃⡼⡏⢻⢿⣯⠴⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⢀⣭⣩⡆⢠⣤⣞⢶⠿⣿⣿⠃⠠⢤⣬⣙⣿⠀⣰⣾⣿⡿⠓⢀⡀⣸⣿⣄⡀⠈⢹⣟⣄⡈⢠⡅⢠⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣦⣸⣿⣮⠀⣿⣿⣟⣿⣷⣾⣿⢿⠟⣻⣿⣟⠿⣿⡿⣿⣟⣼⠏⡀⠀⠘⣄⡀⠀⠀⠙⠂⣱⠄
⠀⠀⠀⠀⠀⢀⢀⠂⠀⠁⠀⠀⠀⠘⠉⠉⠻⠛⠇⠈⠉⠛⠋⢉⡿⠛⣾⠀⠉⣯⠙⠀⠈⠉⠉⠉⠹⠟⠉⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠁⠙⠁⠀⠀⠀⠠⣰⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣽⣿⣟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣴⣞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣵⣇⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣄⠀⣤⡈⢻⣇⣻⢋⡀⠀⠀⣠⣄⣤⣄⣐⣒⣰⣶⣶⡟⠛⣿⢭⣥⢴⣤⠄⠀⢸⣿⣿⣿⠀⢰⣶⣦⡀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⣀⢙⣇⠁⠁⣈⡋⣽⠛⠉⠀⠀⣹⢉⡁⢔⣀⣤⣼⣿⣟⣴⣾⣼⠂⣴⠾⡴⢶⠶⢼⣿⣿⣿⠆⢸⣿⣿⠁⠀⡀⠀⢨⠀⠀⠀⠄
⠀⠀⠀⠀⣀⣶⣿⣯⡤⢿⣿⣿⣟⠛⠦⠀⣏⠩⠄⣤⣤⠦⢼⣿⣩⣾⣿⣿⠴⣤⣤⣶⠖⠒⢺⣿⣿⣿⠟⢻⣿⣿⠄⣀⠻⣶⡏⣼⡷⡄⠀⠀⡀
⠀⠀⠀⠠⠚⣻⣿⣣⠀⡈⣿⡏⠻⠀⢀⣤⣷⣶⣾⣿⣿⠟⣾⠋⢉⣀⠈⣹⣯⠿⢿⠟⢉⡀⢁⣻⣿⣿⣷⡘⢿⣿⣦⡉⠙⢾⣿⣿⣇⡹⣷⣄⣵
⠀⠀⠀⠠⣬⡻⠸⢷⢰⠁⣻⡁⢀⣴⣿⣿⣿⣿⣷⢿⡦⣼⣏⡨⠤⠀⢀⣋⣴⡄⠐⢂⣸⡐⠒⠆⢯⣙⣿⣿⣄⠹⣿⣯⣿⡟⠻⣾⣻⣷⢹⣋⣴⡆
⠀⠀⠀⠀⣉⣻⣱⣿⣿⣼⣏⣷⣾⣿⣯⣿⡾⣿⠶⢻⣿⡿⠀⠐⢦⠎⣰⠿⠋⢃⣶⡿⠋⠹⡦⠻⣄⣿⣿⣿⣿⣶⠿⢿⣿⣟⣷⣜⢤⣿⣿⢯⠛⠇
⠀⠀⠀⢾⣉⣼⢿⣿⣿⢿⣿⣿⣿⣿⡿⢻⣾⡛⣿⣿⡿⠀⣤⣤⡬⠉⣿⡟⡟⢻⡽⠐⠉⢒⢿⡤⢠⠼⣿⣿⣿⣿⣦⠀⢹⣿⠿⡿⣿⣟⣿⣌⣷⡀
⠀⠀⠀⠾⢿⣿⣿⣟⠣⢨⡙⣿⢻⣿⣷⣷⡏⠻⢧⣿⠁⠤⠤⠁⠐⢲⣿⣯⣾⡿⠀⢀⣄⢿⣿⣧⣿⡉⢹⣿⣿⣿⣿⣷⠶⣾⣿⣾⣟⣿⣿⣿⣿
⠀⠀⢠⣶⣿⣿⣇⣠⡿⣚⣁⡥⣾⣿⣯⡿⣾⠁⣼⠏⣀⠀⠠⠄⠀⡢⣀⢾⣟⡉⠀⣰⣙⡃⠈⢈⡿⠅⡀⢿⣿⣿⣿⣿⣷⣤⣌⠻⣿⣿⣯⣿⣯⡀
⠀⠀⠀⠛⠫⠾⢠⣿⢿⡂⣌⣽⣮⣼⣿⡷⠇⣼⡿⠀⣥⡤⠐⠤⠐⣧⣴⡿⣷⢾⣭⡶⠇⠠⠀⡄⠠⠄⠡⠈⢻⠟⡿⢟⣶⣿⣿⣟⣙⣿⣿⣾⡋⠃
⠀⠀⢀⣀⣠⣵⠘⠟⠞⣈⣹⡟⣡⣸⣿⣇⣀⡿⢂⡀⣠⣅⡤⠤⢸⣿⡏⣷⢿⣿⣰⣄⣂⡀⢰⣷⡆⢨⠃⠀⢺⣾⣵⣞⣿⣟⣿⣯⣦⣟⣷⣾⣷⡆
⠀⠀⠀⠅⠉⠈⠉⢀⣀⣤⣼⣷⣂⣀⣿⣿⣿⣭⣿⣿⠉⣩⠉⣉⠉⠉⠉⠘⠛⠛⡉⠛⢉⠀⢺⣿⣏⡈⢉⣥⣾⠉⣍⣽⡿⢉⡉⣉⡛⠫⠍⠉⠉⠁
⠀⠀⠀⡠⢶⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣙⠋⣍⣉⣁⡀⢠⢞⡷⠿⣿⣿⣶⣤⣘⢿⣿⡧⢼⡿⣯⣽⣿⣯⣉⣍⡿⢋⣿⣿⣶⠲⠄⠈
⠀⠀⠈⠀⠀⠌⠈⠉⠻⠛⢛⠛⡫⠉⢹⣿⣿⣿⣿⣷⣶⠞⠉⡿⢫⣿⢺⣿⣷⣶⣶⣼⣿⣿⣿⣿⣿⣷⣼⣷⣿⣿⣻⣿⣿⣿⢿⠿⠟⢛⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢘⣳⠖⣾⠛⠿⢿⣿⣿⣿⣿⣿⠗⣶⣷⣿⡭⠽⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠿⠿⣿⣿⣿⣻⣻⡚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡃⣍⢁⡻⢻⣭⣿⣻⣿⣷⢾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⠟⠛⠟⠟⣛⣿⢟⠒⠛⠛⠞⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠋⠈⠉⠙⠛⠛⠉⠋⠙⠀⠙⠉⠙⠙⠋⠁⠁⠀⠀⠉⠋⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣤⢀⡀⠀⠀⣀⣀⣤⣀⡀⠀⢸⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⠉⠻⣿⣤⣴⠿⣿⣿⣿⣿⢿⣧⣨⣿⡅⠀⣰⣶⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢄⠲⣦⠩⢾⣿⣿⣿⣿⣿⠳⡄⠹⣿⡿⠿⣋⣿⣽⠻⡶⢘⣿⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡽⢆⣤⣷⣮⣿⡣⠎⡵⣹⣿⣿⠟⢡⠻⣠⢹⢁⡶⠏⣼⡽⡓⣁⠼⡷⠪⣙⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡝⠻⡷⣼⡌⣾⣿⣷⣶⣾⣿⣷⣿⣭⣷⣯⠾⢿⣿⡧⢠⣟⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠹⣆⠈⣾⣳⣿⣿⣿⡿⠿⠿⢿⣟⣿⣿⠋⠖⠧⢈⢀⣨⢿⣿⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠀⠻⣮⣾⣿⣽⡿⢉⡶⢛⣛⠳⡜⣿⣿⣧⠀⠀⢱⣁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣸⣬⣿⣷⣸⣿⣻⣿⢿⡇⣷⣇⢿⣧⣴⢇⣿⣿⣿⠿⢧⠉⣹⣗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⢿⡿⣿⣹⣛⠿⡿⣮⣇⢛⢟⣶⣤⣴⣿⣿⣿⠣⠹⡰⠞⠻⣿⡇⣠⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠙⢿⢎⡻⣶⣿⠃⡬⣹⣞⡯⠇⠣⢽⣿⡋⢗⠂⣤⡧⣄⣤⣿⣿⣿⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣦⢿⢫⠀⠈⢿⡜⣽⣯⡏⢠⡒⠖⡄⢫⢷⣤⡴⣻⣵⣿⣿⣿⣿⣿⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⢎⠉⠈⣈⣎⢯⠠⡒⣻⣟⣿⣧⣘⣾⣿⣂⢼⡿⣑⣾⡽⣿⡿⣿⣿⣿⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⡂⢁⠵⠋⣁⣤⡾⣿⣿⢿⣿⣿⣿⣾⣿⣿⡷⡿⣟⠟⡵⣚⠝⡿⢿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⡓⠅⣠⠋⣰⡾⣯⠷⣛⣭⣭⣍⡻⣿⣿⣷⣿⣽⣿⢳⣎⡞⡛⣫⣼⡇⠈⠱⠆⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⣾⢅⡀⢱⠂⣼⣿⡿⣡⣾⢿⣿⢡⣽⢇⣿⣟⢾⣿⣷⡉⣻⣼⠞⢱⣯⣀⠑⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⣯⠅⢐⠇⢐⣿⣾⢡⣿⢛⡿⠛⣳⠶⣏⠷⣭⣾⣿⡁⣹⣿⡿⠟⠚⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠸⣿⠶⢴⠀⢸⣿⣿⢸⣿⠿⣿⢿⣍⣓⣤⣿⡟⢫⣀⢝⣵⠁⣴⠄⢠⠀⠡⠀⣿⠄
⠀⠀⠀⠀⠀⠐⣻⡯⢀⡄⠀⣿⣿⡌⡿⣿⣛⠗⣶⣽⣏⠛⠛⠉⠐⢿⣿⣷⣿⡀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠒⣿⠥⡺⡄⢹⣿⣷⣹⡱⢏⡾⠿⢿⣮⢿⣄⠀⠀⣼⣿⢿⡟⠻⠶⠶⠶⠟
⠀⠀⠀⠀⠀⣀⣀⠰⣿⣴⡟⠶⠙⠁⠛⠍⡝⠀⢂⡆⢸⣥⣿⣤⣾⣿⣳⠯
⠀⠀⠀⢠⣾⣿⣹⣿⣝⠃⠁⠀⠀⢠⠻⣼⠷⣤⣍⣀⣼⣿⣿⣿⡫⠾⠁
⠀⠀⠀⠐⣿⣿⣽⣿⢇⡀⠀⠀⠀⢧⠪⠴⠔⣿⡽⣿⠞⠛⠃⠉
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠷⠿⡴⠦⢿⡆⠀⣰⡏⠀⢠⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣷⣶⣿⣿⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡵⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⠄⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠠⠦⣾⣿⣦⠴⠀⠀⠀⠀⠀⠀⢀⣔⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠶⣾⣷⠆⠁
⠀⠀⠀⠴⡶⢦⢴⡶⠶⣶⡤⠶⡶⠂⠀⢠⣔⣦⣄⣴⣦⣤⠀⠀⢀⡐⣛⡛⢛⡛⣀⡀⠀⣀⣀⣀⣳⣀⣀⣌⣀⣀⡀
⠀⠀⠀⠀⣓⡐⣒⣴⣶⣞⣒⡐⡃⠀⠀⢸⣿⠇⣄⠙⢺⡇⠀⠀⠉⢋⣻⢩⠙⣟⠛⠁⠀⠈⠉⠙⣋⠉⠉⡟⢉⠉
⠀⠀⠀⠠⠤⠤⠤⠿⠿⠧⠤⠤⠤⠄⠠⠮⠩⠿⠿⠿⠏⠡⠤⠀⠀⠉⠿⠿⠿⠏⠁⠀⠀⠠⠤⠍⠹⠶⠾⠍⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣛⡃⠀⠀⠀⠀⠀⠀⣀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣃⠘⠁⠀⠀⠀⠀⠀⠉⠛⠋⠛⠛⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣘⣋⣉⣇⠀⠀⠀⠀⠺⠶⣾⣶⣶⣾⠞⠖⠀⠀⠀⠀⠀⣨⡀
⠀⠀⠀⠀⠛⠂⠤⠤⠤⠤⠤⠀⠛⠀⠈⠛⠋⠉⠉⠉⠛⠁⠀⠀⢨⠈⣿⠀⠀⣿⠁⠀⠀⢠⣤⣶⣛⣉⣉⣓⣤⢴
⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣙⠇⠀⠀⠀⡄⠢⠀⠀⡔⢦⠀⠀⠀⠌⠁⠽⠶⠾⠍⠁⠀⠀⠀⠀⠚⠱⠀⠀⠘⠃
⠀⠀⠀⠀⣠⡾⠿⠿⣿⢿⠿⣧⡀⠀⠊⠀⢰⣦⠔⡆⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⠁⠁⠉⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃
⠀⠀⠀⡀⠉⢢⡤⠤⠤⠤⡤⠊⠁⡀⠀⠀⠰⢾⠾⡷⠆⠀⠀⠀⠐⠦⠜⠀⠈⠠⠔⠀⠀⠀⠀⠀⠂⣛⡃⡀
⠀⠀⠀⠙⠢⡤⣤⣤⣤⣤⢤⣤⠚⠁⠀⢠⣶⣿⣾⣷⣶⡄⠀⠀⠲⠤⡀⠀⠀⠀⠴⠂⠀⠀⠀⠀⣀⣽⣁⡄
⠀⠀⠀⠀⢀⠀⢨⣼⣿⡇⠀⡁⠀⠀⠀⠀⠀⣋⣽⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⢿⠤⠄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠦⠀⠀⠀⠰⠂⠾⠿⠷⠤⠶⠀⠰⠶⠶⠾⠶⠶⠶⠶⠦⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢱⠀⢇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢧⢸⢀⣠⠈⠎⢦⡆⢘⡤⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢄⠘⠈⣿⠀⠀⠸⠃⠈⢰⣛⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣈⢀⡀⠀⢀⠀⠀⠀⠀⠘⣻⣯⣽⣯⣿⣷⣦⣀⠀⠀⠀⣠⣶⠾⠿⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡘⣇⣼⡆⣧⡄⡀⠀⠀⢡⣾⣯⣿⢷⣿⠛⠿⣿⣿⣷⣿⣥⣴⣶⣶⣄⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⡰⢀⠉⠈⠉⠙⢧⢇⠀⠠⢼⣿⣾⡿⣿⣿⣶⣤⣀⣠⣔⣫⣥⣤⣤⡌⠻⣷⣄⡙⠻
⠀⠐⠄⠀⠀⠀⠀⠀⠁⢊⡔⢳⡆⠁⠀⠀⠀⠀⢸⣫⠀⠀⢩⣿⣿⢿⣿⣧⣈⣉⣉⢟⣉⣥⣄⣀⡉⢿⣦⣈⠙⠻⠿
⠀⠀⠀⠠⠰⡂⠀⠀⠀⠛⠛⠋⠀⢠⣄⠀⠀⠀⡸⠒⠀⡠⠺⠾⣟⣿⣁⠙⠛⣻⣛⣩⣭⣉⣉⠙⣿⣦⡙⠛⠿⢿⠖
⠀⠀⠀⠂⢁⣁⣀⠤⡀⠀⠀⣀⣴⣶⠝⠀⠀⢸⣍⢀⣚⠻⠿⣶⣬⡙⠻⠿⢿⣋⣉⣉⣉⠛⢿⣧⡈⠙⠻⠿⠻⠂
⠀⠀⠀⠀⠸⢏⢰⡆⣽⣿⠛⠋⣿⠁⠀⢀⣠⣿⠂⣈⡹⠶⠶⣦⡙⠿⠶⠶⠋⢉⣈⣉⠙⢻⣦⡛⠿⣷⣶⡖⠁
⠀⠀⠀⠀⢀⠼⠻⠍⠈⠛⠀⠠⣧⠀⠀⡿⣟⠉⠤⣾⣿⣷⣶⣄⠛⠷⠶⡆⠀⠀⠀⠉⠹⠧⣈⠻⠶⠦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⢀⣿⣾⠉⠀⠚⠛⠛⠈⠙⠻⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⠶⠖⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣰⠏⠉⠙⡥⠀⠀⠀⣛⠛⠛⠛⠷⢤⠦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠁⠃⡀⠰⠀⠀⠀⠈⠉⠀⠀⠀⢀⡈⠀⠂⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠛⠉⠀⠠⠀⠈⠀⠀⠀⢀⡄⠀⠀⠒⠀⠀⠘⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠱⠂⠀⢰⠂⠀⠀⠰⠂⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠈⠀⠍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠚⠀⠀⠞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠀⡀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⢀⣤⢿⠄⣴⠀⠀⠀⣠⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠷⣄
⠀⠀⠀⠀⠀⣴⠋⢠⡟⠀⠰⡂⠀⠀⡉⠁⠀⠀⣀⡀⠀⠀⠀⠈⠃⠀⠀⢠⠀⠀⠀⠋⠀⠀⠀⠀⠀⠰⠄⣿⠀⢸⣧
⠀⠀⠀⠀⠀⣿⡄⠘⢧⣉⠀⠀⠀⠘⠒⠀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⡼⢻
⠀⠀⠀⠀⠀⣇⠙⠶⣄⡉⠳⢦⣀⡁⠀⠘⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠂⠀⠀⠔⡆⠀⠀⣀⣠⡾⠋⣠⠞⠁⣼
⠀⠀⠀⠀⠀⢻⡀⠀⠈⠙⠲⢤⣈⡙⢦⡴⢶⣦⣤⣀⣀⣙⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠾⢉⣠⡴⠚⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⠉⠓⠲⠶⠤⣤⣉⣉⣉⡉⠙⠛⠛⠛⠛⢋⣉⣉⣠⠤⠶⠚⠋⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡈⠉⠙⠛⠓⠒⠒⠶⠶⠶⠶⠶⠶⠒⠒⠛⣉⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠦⠤⠤⠤⠤⠤⠴⠶⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠇⠀⠀⠠⡄⡀⠀⢠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠁⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⢀⢤⣌⡛⢏⡤⢏⣠⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣵⡔⠀⠀⠘⢁⢾⢠⣤⡀⠐⠀⢸⣷⠈⢿⣾⣧⣬⣇⣘⡷⠀⡖⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢮⣿⣷⠀⠠⠈⢿⣿⣿⣿⡷⠀⠀⠀⢋⡑⣚⠷⢾⠋⢋⡉⠠⠛⢩⡏⢹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⢿⣧⠀⠻⡞⢷⣻⣿⣥⠀⠀⠀⣼⣿⣍⣤⣀⠁⠻⠲⠀⠀⢠⡍⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠰⠁⠈⠉⣼⢿⣷⣤⡈⠀⠈⢉⣉⡀⠴⠊⠉⠉⠈⠲⣍⣻⡇⢤⡀⡤⠠⢕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⡋⣙⡠⢸⠛⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⢦⣳⠐⢉⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠀⢀⠈⠀⠆⠀⠆⠀⠀⠀⢀⣀⡀⢀⡆⠀⠸⢧⠀⠀⢐⣀⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠺⠁⠀⠀⠀⢟⠛⠮⠼⢷⠁⠀⠀⠀⠀⠀⢻⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠈⢠⠄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠋⠀⡀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣀⣄⣀⣀⣠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠰⠀⠀⠀⢀⡼⠇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢇⣽⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠻⠀⣼⠀⢦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠽⣷⡞⢦⡄⠰⠋⠑⠀⣀⠀⠀⠀⠀⠘⢿⣶⡶⠶⠶⠶⠾⢿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡝⠈⣋⢿⡛⠧⠀⣤⡀⠀⠙⢻⡆⢠⣦⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠶⢷⡈⠿⣆⣠⡼⢋⡄⠀⠈⠁⠀⠋⠛⠛⠿⢷⣄⣀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡸⠁⢀⠋⡰⠢⣽⠁⠬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠓⡤⢀⢀⠾⡀⢠⡄⠘⢲⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣶⠈⠁⣬⢱⢾⣤⣤⢤⠤⠀⣤⠀⠀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⢈⡧⠀⡤⠈⠓⠄⢀⣈⠷⣀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣨⠇⢠⡄⠰⠚⢠⠽⣭⡴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⡀
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡄
⢩⣕⣗⢨⡗⠀⠀⠤⠒⣴⣿⣿⣿⣿⣿⢾⠇⢀⣀⣤⣤⣀⣨⠄
⣶⡈⠉⣉⡃⠀⡀⢼⣷⣿⣿⣿⣿⣿⡖⠁⣰⣾⣿⣿⣷⣾⣿⡿⡆
⣿⠃⠀⠀⠀⣲⡿⣀⣿⣿⠏⠉⠉⠉⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣴⡄
⠋⠀⠀⠀⠐⣳⣎⣺⡿⠷⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢄
⠀⠀⠀⠀⠀⢸⠟⢀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣷⣾⣿⣷
⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⣼⠟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⣾⡻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⣿⣿⣿⣿⣿
⠀⠀⠀⢰⠇⠀⠀⣤⡐⠀⠀⠀⢰⣿⣾⣿⣿⢃⣸⡿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣄⠀⣤⣿⣿⣿⣿⣿⡀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣸⣟⣡⣄⠀⠀⣾⢿⣿⣿⡇⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢻⣆⡀⠀⠀⠀⠀⠀⢠⣦⣀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⡨⠂⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡽⠟⠒⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠛⠀⠛⢿⠿⡖⢁⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠘⣯⠀⢨⣽⣿⣿⣿⣿⣿⣿⢿⣿⡆⠀⣀⠁⠒⣾⣷⣾⣶⣷⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⣾⢙⣷⣇⣻⣿⣿⣧⠼⠛⠉⠿⣯⡵⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀
⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠳⠲⠟⢛⣿⣿⣿⡉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢙⣷
⣿⠛⠁⣿⠂⠀⣀⣴⡶⠦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⡆⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣤⣶⣶⡶⣦⣤⣀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⡀⠀⡆
⠀⣀⡑⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡔⠀⠀⠛⠇⠀⢀⣿⣧⠿⠿⠛⠳⠿⢾⣿⣃⠀⠀⠀⠀⠀⠀⣤⡿⠯⢻⠆⠃
⠀⠀⠈⠉⠁⠀⢹⣦⣤⣀⣀⠀⠀⣿⡟⠃⠀⠀⠀⠀⡠⠔⠊⢁⠴⠚⡆⠀⠀⠀⠀⢸⠿⠿⠛⢀⡄⠛⠃⣀⠘⠿⠿⠏⠀⠀⠠⡶⣼⣥⣴⣶⣍⣀⡀⡀
⠀⢀⠀⠀⠐⣄⠀⠈⢿⣿⣿⣧⠀⢻⡇⠘⠀⠀⠀⠀⣠⠤⠚⠉⣠⠋⠀⠀⠀⠀⠀⣀⣀⡤⣠⣾⣿⣿⣿⠿⠧⠐⢐⣶⠂⠀⠀⠀⠀⠈⣳⣴⣋⠉⠋⠛
⠀⠈⠀⠀⠀⣻⣿⣦⣄⣘⣿⣿⡆⢸⣿⢘⣤⣶⣦⠀⠉⠑⠒⠛⠁⣤⣶⣶⣧⡄⠀⠈⠉⢻⣭⣥⣿⢿⣿⣷⣿⠃⡡⠎⠀⠠⢶⡇⠀⠀⣿⣿⣸⠆⢀⡀
⠀⠀⠙⠲⠿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣙⣿⡿⠋⢉⡧⢀⣀⠰⠀⢐⣿⣣⣿⣻⣿⡃⢀⣶⣾⣿⣿⣻⣋⣿⣯⡻⣧⣤⡀⠀⣰⡇⢠⡖⢲⣿⡏⠁⠀⠼⡛
⠀⠀⠀⠀⠀⢀⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⡶⠂⢰⠈⠉⠤⠀⠈⠿⣿⣶⡿⠿⠠⠞⢛⢋⣘⡛⣻⠟⠿⠛⠙⠏⠛⠛⠂⠹⣿⣾⣶⣿⣿⣷⣜⣳⣴⡇
⠀⠀⠀⣀⢴⣿⣦⢀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠒⢤⣄⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠸⠻⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡎⠙⠛⠛⠻⡇
⠀⠘⢿⣿⣿⣿⣷⣿⡷⠂⠀⠤⠖⠂⠠⠄⠀⠀⠀⠀⠀⠉⠟⣿⣿⣿⣿⣤⡐⢢⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⢠⣤⣄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⢠⠀⢰⠐⡄⠀⡆⠀⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠸⣿⣿⣴⣶⡄
⠀⠀⠀⠛⠛⠉⠛⠻⠀⠀⠑⢀⡌⠉⢀⠐⠁⠀⢽⣦⡀⠀⠀⠀⣸⣿⣿⣿⡾⠶⠶⡼⢿⡿⠀⠀⠀⢀⣠⣀⣀⡤⠄⡠⠤⠀⠀⠘⣿⡿⠿⠛⠁⠙⠛⠃
⠀⠀⠀⣀⣀⣤⠶⣶⣄⠀⠀⠀⠁⠀⠀⡀⢠⣴⣿⣿⣝⢦⣤⡄⢁⣤⣝⣿⣇⠀⠀⠃⠀⠀⠀⣎⢝⣽⣭⢽⣳⡖⡷⢁⠀⠁⣼⣦⣀⣀⡀
⠀⢠⡾⣽⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠯⣿⡟⣠⣿⣿⣟⣹⣿⣿⠟⠀⠀⠀⠀⠙⢾⣽⣿⣿⣿⠾⠃⠀⠠⣶⡿⣿⣿⡟⠀⢰⣿⣦⠀⠐
⠀⠈⢿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⣷⣤⣴⣦⠀⠀⠀⠀⣴⣵⣿⢿⣷⣿⠿⠟⣁⠀⠀⠀⠀⢀⠀⠀⠙⠻⢿⣋⡀⠀⠀⠀⠈⢿⣿⠻⠿⠀⣿⣿⣿
⠀⠀⠀⠙⠻⠿⠿⠟⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⢠⠞⠛⠉⠀⠀⠀⢠⣶⠿⢻⣷⡀⠀⡀⠀⠀⠀⠐⠿⠿⠿⠛⣀⣀⣤⣤⣌⠀⠀⠀⠀⣿⡿⣿⡄
⠀⠀⢀⣤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⣠⡶⢶⣀⣤⡀⠀⠀⠻⣿⠿⠛⠑⡿⠙⠿⠆⠀⣀⣤⣶⣾⢻⣿⣹⣧⣿⣿⡆⠸⠧⣤⣿⢷⣿⣤⡤⠿
⠀⠀⣦⠘⡇⠈⡆⠀⠀⠀⠀⢰⣄⣀⡄⠀⠀⠀⠀⢻⣷⣾⣿⣬⣿⠀⠀⠀⢀⡴⠃⠀⣿⣿⢿⠿⣿⣯⣫⣻⡿⠟⠋⢹⣟⣂⣿⠆⠀⠀⠈⣿⣏⣽⡏
⠀⠀⠹⡄⢸⠀⣁⠀⠀⠀⠰⢿⣭⣿⣄⠀⠀⠰⠀⠀⠛⡿⠿⠋⠀⢀⠠⢒⣽⡎⠀⠀⠙⠻⠿⠿⠟⠛⡛⠍⡄⠐⠀⠈⠉⠉⠀⠀⠀⠀⣀⠙⣛⠻⠃
⠀⠀⠀⠗⢠⡄⢰⠀⠀⣰⠄⠀⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠎⢡⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡥⢀⣶⡄⠓⠀⠀⠉⠀⠀⠂⠀⠀⢈⠘⠸⢡⠆
⠀⠀⠀⠈⢾⡤⠂⢀⣼⠏⢀⠀⠀⢸⡅⠀⠀⢚⣿⠷⠂⣀⣀⠀⠀⣿⣿⣿⠁⠠⡄⠀⠀⠀⡀⠀⠀⠐⠀⢠⠉⠁⠄⢈⡀⢰⣦⠀⣀⠀⠀⢸⣿⣿⡖⠂
⠀⠀⠀⠀⠋⠀⠀⢸⣿⠁⠿⠀⣿⢼⣷⡄⠀⠈⠉⢠⠼⠛⢿⣿⣾⣿⣿⠃⠀⠀⣀⣤⢞⣭⣤⣤⠄⠀⢀⠀⠐⠂⠀⣘⣿⣾⣿⣾⠟⡀⠀⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣠⣀⣠⣾⡏⠀⠀⠀⠀⠀⢠⣤⡾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠛⠉⠁⣀⠤⠄⠀⠀⠀⠙⠛⣻⣿⣾⣿⡛⠛⠀⠀⢻⣿
⠀⠀⣿⣿⣶⣄⡀⠀⠈⠙⠿⠟⠛⠋⢀⣤⣾⣿⡟⠀⠀⠙⠇⣿⣿⣿⡿⠟⠁⠀⠄⠀⠀⣠⠔⠊⡀⠴⠆⡟⠉⠉⠀⠺⠟⢹⣿⠙⠿⠀⠐⠀⠈⡿⠀⠀⡄
⠀⠀⠸⣿⣿⣿⣿⣦⡀⢀⣀⡀⣠⣶⣿⣿⣿⣿⠁⠀⢠⠊⣰⣿⡏⠀⠀⠀⠀⡀⠀⠀⠈⠁⡠⠊⢁⠔⢂⠄⢠⣄⠀⠀⠀⢈⡁⠄⠐⡇⠄⣀⡀
⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠇⢸⣿⣿⡇⠀⢀⣀⣀⡀⠀⠀⠠⣟⠠⠄⠂⠀⠏⢀⡎⠁⡁⠀⠀⠈⠑⢆⡠⠇⡴⠋
⠀⠀⠘⣿⣿⠿⣿⡿⢟⣩⢿⣛⡿⣿⣿⢿⣿⡿⠀⠀⠈⢾⣿⣿⠀⠀⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⢘⢁⡜⢀⠜⠀⠀⠐⡃⠚⢒⡸⢗⡓⠓⡚⠂
⠀⠀⠀⠈⢋⣴⣿⣶⣾⣛⣟⡋⣶⣶⣿⣮⠉⠁⠀⠀⠀⠈⢿⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠈⡾⠤⠊⠀⠀⠀⠀⢠⡀⠃⠠⠄⠙⢠⡀⠀⢀
⠀⠠⠄⠀⢸⣿⣿⣿⠏⠈⠉⠀⢻⣿⣿⣿⠀⠀⠐⡿⠃⠀⠘⡟⠀⠀⠀⠈⠁⠸⣿⣿⣿⣿⠃⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠈⠈⠀⠢⡖⠀⠈⠁⠀⢜⣸⡁
⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢀⢀⣀⢀⢀⠀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠒⠀⠁⠈⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠛⠛⠉⠻⣷⠶⡄⠀⠀⠀⣱⠄⠀⠳⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⢀⠀⢀⠦⢠⠒⢳⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⣀⣀⡤⠴⠒⠉⢠⠴⠋⠀⡆⣠⠀⠷⠘⣴⣿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡰⡟⠀⢠⡷⠋⠀⢘⡴⢡⠉⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠺⢦⠾⡀⠀⠀⠀⢀⢻⠇⠀⢸⡂⠤⠂⠁⠀⠈⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⢃⣬⡂⣤⡠⠊⡝⠀⠀⢸⡇⠈⠱⠄⠤⡀⡜⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠀⠀⠀⠈⣲⠁⠀⠀⢻⠁⠀⠀⠀⠀⣇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠫⠤⠤⣀⡀⠀⠀⠀⣀⠴⠋⠁⠀⠀⡄⢸⡀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠉⠑⡦⠚⠁⠀⠀⢀⡴⠊⠁⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⡀⠀⠀⠈⡄⡌
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣀⣀⣀⣀⠀⢠⠞⠀⠀⠀⠀⠀⠘⠣⠤⢞⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠘⡇⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⡼⠁⠁⠁⠀⠀⠀⠀⠈⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣇
⠀⠀⢀⡠⠖⢲⠇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸
⠀⢀⠊⠀⠀⢸⠀⠀⡀⡄⣀⠀⠀⣀⣼⠁⠀⠀⠀⣠⠖⠋⠉⠉⠉⠁⠀⠀⠀⠘⠲⣄⠤⠒⠒⠲⣄⣀⢀⢀⡄⠀⠀⠀⠀⠀⠘⡄⠀⡆
⠀⢸⠀⠀⠀⢸⠂⠉⠀⠈⠉⠀⠀⠀⢸⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠐⠀⠠⠴⠁⠀⡇
⠀⠈⢧⣀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠠⣖⣖⣹⠋⣏⢙⣦⣀⠀⠀⠀⢀⡤⠤⠷⠄⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠒⠒⠒⠒⠒⠋⠉⠉
⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠛⠒⠻⠠⢦⣆⣀⣀⡠⠴⠊⠉⠚⠀⠀⠈⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡟⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣾⠠⠺⢶⡀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⢡⠈⠀⠈⣥⠘⣷⣤⡀
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⣠⠼⢋⣼⡁⠀⠀⢠⠀⠘⣆⠈⠙⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢬⣥⣶⠿⠉⠁⠀⠀⠀⡀⠀⠁⡑⠀⠉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣬⢌⡅⠈⠀⠉⠂⠀⠙⠀⠀⠐⠀⠐⠴⠀⠀⠀⣀⣾⠋⢼⡀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠋⠁⣸⢶⣴⠂⢤⡀⣀⡀⡀⠀⣀⡀⢠⣤⠸⡿⢫⣷⣤⣉⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣋⣡⡱⠖⠀⠘⠙⠻⠿⠾⠟⠛⠉⢁⠀⢣⠈⠯⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⡎⠀⠀⠀⠠⠨⠄⠭⣤⠀⢨⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣘⠒⠒⠒⠫⠀⢸⠁⠀⠃⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠀⠀⠛⠛⠉⠉⠉⠛⠀⠤⡉⠲⠂⠀⠀⠀⠀⣀⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⠤⠤⣤⣤⡤⠤⠠⠄⣤⣤⣤⠈⠁⠀⠀⠀⠀⠀⡌⠠⣈⣁⣀⣀
⠀⠀⠀⠀⡤⢤⡦⢄⣀⡀⢀⣤⡤⡾⠛⠯⠟⠙⠛⠙⠋⢹⠉⣮⢹⠋⠛⠙⠓⢤⣀⣀⣀⣀⠐⠛⠒⢏⠉⠈
⠀⠀⠀⠀⠀⠰⠗⠒⠚⢠⠞⢠⠆⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⡀⠀⠈⠹⠁⠀⠀⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⡉⠀⠛⠀⠴⠀⠰⠂⠐⠄⠰⠆⠀⠆⠀⠙⠀⠀⠉⠀⠀⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢿⠿⣷⣲⣶⣤⣠⣀⠀⠀⡀⠀⠀⠀⠀⣀⠰⠶⠶⠾⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⢘⠉⠉⠉⠉⠉⠼⠉⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢐⠀⡁⠀⠀⠈⠉⠁


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣷⣶⣤⣾⣤⣾⣵⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣠⠀⢀⠀⠀⠻⣿⣷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠙⠳⢦⣥⣀⠀⠈⠻⢧⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠠⢤⣀⡀⠀⠀⠀⠉⠐⢶⣄⠀⠁⠰⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⡐⠀⠀⠀⠀⠉⠙⠓⠆⢢⣄⠀⠀⠀⢴⣤⢠⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠿⠿⠿⠷⠶⣶⣤⣤⣀⡀⠈⠁⡲⢦⣜⡲⣿⣶⣷⣶⣬⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢙⣳⣮⣿⣿⣿⣾⣿⣿⠟⠋⢡⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠈⠙⠉⠃⠚⠛⠂⠶⠒⠠⠀⣈⣛⣿⣿⣿⣿⡿⠛⠁⣘⣠⡴⠾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣁⣠⣤⡤⣤⣤⣤⠄⣠⣀⠄⣀⢀⣀⣸⣿⣿⣿⣿⣿⢫⡄⢈⡽⠛⢁⣠⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠈⠀⠀⠀⠀⠀⣀⣀⣈⣻⣿⣿⠿⠿⢯⡽⢤⣶⠋⠀⠀⠼⢿⣿⠙⢆
⠀⠀⠀⠀⠀⠀⢀⡠⠀⡀⡀⠠⠀⠀⠀⠀⡀⡉⠤⠮⠉⠽⢿⣿⡿⠡⠶⣶⣤⡹⡟⣷⡄⠀⠀⠀⠀⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⠒⠛⠁⠐⠀⠐⠀⠀⠀⠀⠀⡀⢀⣴⣴⣾⣿⣧⡴⠿⠿⢉⡼⠃⠘⠷⠀⠀⠀⠀⠀⠀⠄⠈⢇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⡄⢂⣀⡀⠀⠂⣀⠤⠞⠀⠁⠀⢉⣩⣽⣿⣿⡗⢲⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠋⠉⠈⠁⠀⠀⡀⢠⡄⠀⠚⠜⠛⣛⣫⣿⠇⠈⠟⣿⣇⠀⠀⠀⠀⠀⠀⠰⠖⠶⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣶⡯⠓⠋⠀⠀⣠⣴⣾⣿⣿⣿⣦⣄⠀⡁⢹⢦⣄⡀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠐⠊⠀⠀⣤⠚⠉⠁⠁⠀⠀⣠⣖⢾⣽⣿⣿⣿⠟⢛⡿⣿⡿⠋⠀⠀⠀⠚⠉⠓⠶⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠛⠉⠐⠚⣻⣿⣿⡇⢀⠀⠀⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⠿⠀⠀⠀⠀⠀⠈⠓⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⡄⠀⠀⠀⠀⠀⠀⢀⡀⡐⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⠀⣝⠇⡀⠀⠀⠂⠀⠀⠀⢀⡄⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⡈⣿⣆⡆⢀⡀⢀⠁⠀⠠⣤⠮⠙⠡⠀⠞⠃⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⣠⣾⣷⣿⣿⡠⡜⠀⠀⠀⡀⠀⠛⠈⠀⠄⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡄⠈⡇⠀⠀⢠⣿⣋⣮⣽⣷⣭⣽⡳⢤⡤⢿⠀⡠⠘⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⢚⠠⠶⢾⣿⡷⡄⠶⣿⠟⣫⣿⡾⢋⠟⣣⠁⢈⠂⢻⡛⠚⠄⢀⣀⡌⣄⠀⠀⠃⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⠤⠞⠒⠐⢯⡾⣯⡟⣰⣿⣾⣿⣶⣌⠑⠌⡀⠂⠀⡁⠀⠀⠈⠃⠐⠑⠠⢀⠀⢛⡠⠼⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣿⣿⣈⡷⠆⡶⠀⠀⠀⠀⠀⢀⣀⡴⢀⠀⢀⠄⠊⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠠⣄⠄⠸⠳⣈⡀⠀⠀⠀⠀⠉⢷⣈⣻⣾⣿⣿⣿⡇⢁⠀⠀⠀⠀⠀⠀⠈⠋⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⠀⠄⡀⠢⠬⢆⠀⠀⠈⠂⣈⣦⠗⢿⣿⣿⣿⣿⡦⠄⠀⠄⠉⢢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣒⡛⠀⠀⠅⡰⢀⡀⢠⣆⠀⣀⡹⡯⣀⣿⠟⠋⠉⠀⠀⠀⡀⠀⠀⠁⠚⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⣠⠟⣿⠟⠀⠀⠀⠁⠈⠉⠉⠀⠉⠁⠀⣲⣾⣧⣦⣀⠀⠀⠀⠀⠀⢠⠐⠃⢤⠠⠗⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠀⠂⠀⠒⣠⡧⣾⢥⡤⡴⢋⢉⡷⣢⣤⣄⣤⠶⣓⣺⣽⣿⡻⢻⣿⡆⠀⠀⠀⣴⠦⠠⠄⡀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠁⠐⠂⠠⠜⠂⣀⠀⠀⠘⣽⠁⠋⠆⠰⠋⠟⣟⠁⡋⠉⣙⢿⠼⣟⢿⣽⣧⣀⡀⠰⠇⣂⡶⢴⠦⠀⡀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠠⡉⢈⣿⠓⡞⠧⡙⠟⠣⠺⠁⢀⠂⡕⠈⣡⣴⣿⣧⣶⡉⣷⢷⣘⢾⡓⠰⣺⢿⡷⢤⣄⡲⣭⣀⠠⠀⠉⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠀⠒⢓⠂⠉⠉⠤⣼⢷⡇⠀⠀⠀⠀⠀⠀⠋⠀⢸⣟⢸⣽⣿⣿⣿⣷⣽⣿⠋⢉⠋⠘⠓⠻⠄⠐⠀⠁⠈⠉⠁⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢀⢈⡀⠀⠀⠀⡔⠀⠒⢿⣸⡇⡿⠀⠀⡴⠀⠀⢠⠀⣾⣿⣿⣿⣿⣽⣿⣿⡏⠀⠀⢈⣆⣀⠀⠹⣤⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⡀⠄⠂⠛⠠⢎⣹⡿⣡⠦⠞⠳⠗⠒⠂⣤⢝⣿⣿⣿⣿⣿⣿⡿⣀⣸⠀⠀⠈⠄⠙⣈⡀⣻⡏⠀⠂⠀⠤⣀⠤⣀⠀⠄⠀⠀⠀⠀⠀⡀⢀⠁
⠀⠀⠀⠘⠀⢡⡄⠀⣀⠁⣸⣿⡿⠏⠀⠁⠘⡀⠀⠀⠀⠀⠿⢿⣿⣿⣯⣭⣽⣶⣶⣦⣦⣡⣠⡰⠃⣀⠈⠁⢀⡀⠂⠀⠨⡌⠓⠀⠨⠣⠆⣀⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⠋⠛⠀⠀⢀⢤⠋⠥⠄⡲⠒⣀⡀⠘⣿⣿⣿⣿⣿⣿⠿⣇⢌⠙⠻⢿⡒⠻⣧⢤⡰⢄⡆⡰⠔⠉⠉⠁⢔⡀⠀⠀⠉⠂⡞
⠀⠀⠀⠀⢦⣤⠄⠀⠸⠀⠉⠁⢀⠀⠚⠓⠃⢀⠀⢀⣾⢡⣍⣼⣿⣻⣿⣿⣿⣧⠀⢈⠣⠘⢢⡀⠙⢦⠉⠂⢉⠲⡄⠤⠀⠙⡆⠱⢄⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠄⠀⡴⢇⢀⠄⠿⠀⠠⠈⣡⡬⠍⣸⠚⢿⣿⡧⡽⣏⢷⡄⠃⠈⢆⠱⠀⠈⠘⢆⠈⢣⠸⡁⠀⠀⠐⠀⠈⠲⠌⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁⠀⠀⠀⡀⠠⢄⠀⠉⡟⠀⠁⢚⣹⣿⣱⠸⡬⣤⢳⡀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠃⠁⠐⠦⠀⠠⣄⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠁⠠⠀⠀⠀⠓⠀⠁⠃⠔⠀⢞⣽⣹⠁⢳⠹⡄⠁⠀⠀⠸⠀⠀⠁⠀⠘⡈⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⣤⡤⢠⣀⠠⠄⠡⢾⠏⠀⢸⠀⣝⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠉⠀⠁⠀⠀⠀⣸⠋⢠⠀⠘⠀⢸⠄⠆⠆⢀⢰⠀⠀⢸⣆⡟⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⠀⣀⠀⡘⣀⠀⠀⡀⢠⠘⠀⠀⠀⠈⣦⣣⣠⡈⣾⡟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠀⡆⠁⠘⡄⠀⠀⡄⠀⠀⠙⢿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⢸⡇⠀⠀⣧⠀⡄⠈⠀⠀⠀⠀⠈⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡟⢤⡆⡏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢳⠀⡓⣄⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠈⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠲⠠⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⡀⠁⡀⠀⠀⠀⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠐⠀⠂⡀⠀⠘⠂⠀⠀⠀⠀⠀⢀⠘⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠣⠀⠀⠀⠀⠐⠂⠀⠀⠄⠀⠠⠀⢀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠘⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⡀⠈⠀⠀⠀⠀⠀⠠⠀⠠⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢈⠉⠀⠤⠄⠀⠀⠀⠀⢤⡒⣠⠴⢖⠂⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⣦⡀⠀⠉⡆⡄⢀⠉⠀⠀⠀⠉⠥⠀⠀⠂⠀⠀⠀⠀⡴⢀⡐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠖⢀⠀⠤⠀⠓⡀⠄⠈⠅⠀⠀⠁⢅⢂⢠⠄⠈⠄⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⣉⡀⠤⠠⠀⠂⠩⢀⠀⠂⠀⢰⠦⠀⠲⠗⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⠓⠁⠂⠀⠠⠄⠉⣀⢰⢠⡄⢈⠡⠀⣈⠖⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡆⠐⠂⠀⡇⠀⠀⠁⢠⠑⠀⢩⡶⣤⡇⢀⠀⠀⠀⢄⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠐⠀⠐⢀⠀⠉⠀⠀⡀⠘⡀⠀⠀⠐⠀⠀⠀⠉⠀⡀⠀⠀⠀⠂⠀⠀⠀⡤⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠌⠁⢠⢀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⡏⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠆⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠑⠲⠂⠀⠆
⠀⠀⠀⠀⠀⢁⢀⠘⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠣⠄⡀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠠⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣄⣄⣀⣀⣀⠀⠀⢠⡀⢀
⠀⠀⠀⠈⠲⣌⣻⣿⣿⣿⣿⣿⣿⣶⣿⣾⣴⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠄⠀⣀⡠⠄⠀⠀⣀⣀⣀⣀⠄⠀⢀⣠⡤⠖
⠀⠀⠀⠈⠳⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣼⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣿⣤⡶⣻⣵⣾⣿⡵⢶⣿⡟⢋⡹⠯⠭⣤⣴⣞⣋⣁⣠⠖⡊
⠀⠀⠀⠀⠀⠹⣆⢿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣾⣳⢯⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⣠⣿⣾⣿⢟⣽⣿⣿⡿⢛⠟⣃⠔⢪⣤⠋⣿⣧⠄⠈⠛⢛⣿⣿⡟⠛⠉
⠀⠀⠀⠀⢠⡀⢹⣞⣿⣿⣿⣿⣿⣿⡅⠟⢸⣧⣿⣾⣿⣵⣶⣴⠆⠀⢀⠀⣾⣷⣿⣿⡿⣱⣿⡿⢋⣩⣬⠅⣘⣫⣤⣽⣭⣤⣽⣷⣶⣶⣶⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢳⡀⢻⣿⣿⣿⣿⡿⡈⢁⣖⢈⣴⣦⣩⣶⣬⣭⣶⠀⢀⣼⢷⣿⣿⡿⣫⣾⣿⣟⡵⢛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠹⣄⠻⣿⣿⣿⣷⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣀⣼⣿⣿⣿⣿⣧⢹⣿⣿⣁⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠙⢮⡛⠾⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣟⣾⣿⣿⣿⣷⡎⢿⣿⣿⣿⣯⣷⣦⣭⣉⠛⠳⡮⡙⠻⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣷⠲⢦⢌⠙⢿⣷⣄⠈⢪⠑⢄⣉⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢏⣾⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣧⣄⠣⠳⣄⠙⢿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢧⡿⣩⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⠻⣷⣥⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢈⣾⣿⣿⡄⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣣⡿⢫⠈⡄⣿⣿⣿⣶⣤⣤⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⢿⣿⡿⢿⣟⣼⣿⣤⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⢌⢻⣿⣯⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢷⠿⠿⠛⠛⠻⢿⠟⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⡿⠈⢈⠈⣿⣮⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀⣴⣿⣿⣿⣿⡿⠿⣻⣿⣿⣿⣿⣿⡯⠀⣸⡇⢸⠀⣷⣿⣿⠆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⣿⣿⣦⡀⠀⠠⣾⣿⣫⣉⡏⣿⢙⣿⣿⣿⡟⠀⠀⣴⠟⢸⢿⡿⣿⡏⠁⠀⣿⡀⢸⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣻⣿⣿⣿⢸⣿⣿⣿⢿⡀⠀⠘⠛⠉⠉⠈⠀⣀⠀⢈⠈⠁⣠⣾⠏⠀⠈⠘⠁⠃⠀⠀⢸⣿⡇⠀⣧⣘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢿⣿⣊⣻⣿⠟⣿⣦⡀⠀⠀⠀⠀⠀⠤⣽⠿⣿⣿⣿⠿⠿⠶⣦⣶⠖⠀⠀⠀⠀⣸⣿⣿⡄⢻⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣽⣭⣿⣿⣿⣿⡄⠀⠀⠀⠀⠰⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⣿⣿⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣤⢜⣻⣿⣿⣭⢟⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢹⣿⠟⣹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⣡⢾⢿⠊⠉⠉⠈⠉⠉⠁⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⣛⣡⣿⣿⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀⢀⣀⣀⣀⣠⣴⢾⣿⢟⣼⣿⣿⢯⣿⣿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣝⣛⣝⠿⢟⣵⡿⠲⢿⣷⣭⣵⣿⢸⣿⣿⡇⣿⣿⡿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡶⣛⢭⣯⣍⠻⣦⠀⠀⢀⣶⠋⠀⠀⠀⠀⠈⠙⠛⠉⠛⠛⢉⣠⣶⣿⣿⠿⣹⣷⣿⣿⣿⢇⣿⣿⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣿⣿⣣⣤⣤⠁⣽⢃⡦⢸⣿⡆⣀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⢟⣾⡿⣿⣟⣥⣿⣿⡿⢏⡜⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⡶⢦⣜⢷⣩⠷⡽⡏⣢⠿⣈⣁⣘⢿⣿⣏⢿⣿⢿⣿⣿⡿⣿⣿⣶⣶⣿⢟⣽⡿⣹⣿⣿⠟⢡⠊⣸⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣽⣮⠟⠛⠂⠙⣷⢙⡛⠷⢋⡴⠻⡽⠍⠹⢳⡌⠻⠿⠿⢧⣭⣷⣿⡿⢟⡵⣳⣶⣿⣯⣴⣿⠟⠁⠠⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⡛⡷⡸⢿⢧⣠⣏⠺⡍⢀⣿⣩⣷⢠⣴⣦⣡⣷⡄⢻⣽⠦⠬⢉⣁⣀⣬⣾⣿⣿⣯⡿⡟⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣽⣃⣥⣿⠟⠀⣰⡿⠿⣿⣻⢿⡿⡟⣲⣿⠃⠚⠁⣴⠿⢿⣿⢿⣿⡿⢉⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠈⠙⠛⠓⠛⠋⠁⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠈
⣿⣿⡿⠿⣿⠟⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣶⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⡤⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠐⠟⠛⠋⠉⠀⠀⠀⠀⠠⡆⠀⠀⣀⣈⣁⣀⠀⠀⠀⢀⠀⠀⠆⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠄⠱
⠀⠀⠀⠀⠀⢀⣀⠠⡄⠀⠐⠀⠀⠀⠐⠤⢀⠀⣀⢀⠀⠀⠀⣴⡾⣏⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⠠⠌⠁⠡⠘⢄⠀⠀⠀⠀⠀⢀⣁⡚⠁⠀⠑⠢⠎⡟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⡀⠀⠟⠸⢯⠏⠀⠀⠀⠀⠈⠹⣿⡤⠀⣤⠤⢤⡤⠤⣤⠤⠤⢤⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠆⠁⠀⠈⠁⠀⠀⠀⠀⠈

⠀⠀⣄⠀⣀
⠀⠀⠂⠈⠀⠐⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡀
⠀⠀⠌⠀⠠⢂⢀⠀⠀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠓⠀⠀⢐⣆⠀⠀⠀⠀⠀⠀⡀⠐⠠⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠷⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠤⢈
⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⢠⣔⣲⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠣
⠀⠀⠀⠀⠀⣉⡿⣁⡀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣌⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠛⠛⠛⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⢇⣻⠐⢠⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⢁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⣶⣿⣷⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠻⠁⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣟⣿⣋⣓⣶⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠳⣤⡀⡈⣤⠟⠃⢓⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⡶⠒⠀⡘⠃⠀⠠⠻⠷⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢞⣻⣿⡽⢶⡄⠐⠀⠀⡀⠀⢀⠶⠎⠙⢄⠀⠀⠀⠀⢀⠀⠀⣴⡟⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠿⣾⣿⣿⠺⠇⢻⣰⣴⣤⣥⠀⠀⠀⠀⡰⠚⣿⠀⣤⣧⣌⠦⠠⠘⣿⣶⣾⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣶⣿⡼⢿⣟⣻⡶⡟⠛⠁⢉⣿⡁⠀⠃⠘⠙⠞⠉⠛⣧⣴⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⣿⣿⣾⣟⣷⣾⣿⣿⣿⣟⣾⣿⣶⠂⣤⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣠⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣟⣿⡿⠉⠹⣿⣿⣿⣿⣿⣶⡿⢷⠟⣿⣗⠘⡋⣠⣴⣿⡆⠀⠀⠀⢀⣶⢋⡀⣀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡟⠁⠀⠀⠈⢻⣿⣿⡉⡛⠀⢸⡟⠛⣿⠀⠙⠋⠁⣄⣦⠀⠀⠀⠘⢿⣎⠛⠿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠈⠙⢿⠇⠻⠿⠛⠁⢠⡾⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠛⠿⠇⣈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠷⠛⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢳⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠓⠶⣶⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠁⠀⢀⣠⡴⢀⠀⠀⠀⠀⣤⡶⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⣸⣿⣻⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠘⠃⢀⣀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣭⠟⠉⠀⡜⢣⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠟⠉⠁⠀⠀⣼⣿⣿⣿⣿⠇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢾⡾⢇⣶⢟⡙⠛⠷
⠀⠀⠀⠀⠀⠁⠀⣠⣿⣿⣟⠑⠀⠀⠀⠀⣾⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣷⣦⣸⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⣀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠙⢶⣾⠟
⠀⠀⣿⣆⠀⠀⠉⣿⣿⣟⣠⡶⠻⣧⢠⣿⣿⣿⣿⣿⣿⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣦⠀⠉⢠⣄
⠀⠀⢸⣿⣷⣄⣀⠘⠿⣿⡟⠀⠀⠘⢯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠶⠀⠒⠂⠀⠈⠛⠋⢠⡄⢣⣼⡆
⠀⠀⠀⠛⢿⣿⣷⡄⣤⢩⠄⠀⠀⠠⠈⠝⢻⣿⠿⠟⢁⢉⣀⣀⣀⣀⠄⠀⢀⡀⠀⠀⠀⣒⠀⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⡎⠁
⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠚⠻⢿⣿⣋⠛⠉⠁⠈⠉⣀⣸⠋⣙⠛⡁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣉⠉⠓⠓⢦⣤⣴⣶⢶⣤⠊⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⢿⣧⠙⠂⢉⠻⢧⣔⡀⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠖⢰⠆⠐⢿⣆⠀⠀⠀⠀⠈⠻⣷⣄⠀⠁⠀⠉⠛⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠹⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⣀⣠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⡀⠈⠀⠃⠀⠀⠀⠀⠀⠀⢻⡅⠈⠛⠻⣷⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⡸⠿⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣿⠋⠙⢿⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⠀⠀⢸⠇⠀⠀⠀⠃⠈⠑⢶⡞⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣈⠀⠉⠈⠃⢸⠂⠀⠀⠀⠀⠀⠀⡉⠉⣡⡄⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠘⢡⡖⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⣼⢿⣯⠱⡄⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⡀⢳⡤⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢿⣧⠙⠡⠒⡷
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢸⡇⠀⢠⠀⠀⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢀⢼⡏⠘⡶⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢐⠖⠁⣸⠇⠀⠻⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⣠⡎⢡⣴⠀⠀⡇⠀⠀⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡸⠙⢷⠀⢀⣀⠘⢧
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⠀⢸⠀⠀⠀⠀⠀⠷⠀⣠⡄⠀⠀⡀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠦⠀⣀⠀⠀⠀⠂⠀⢉⡄⠘⠳
⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⣒⠋⠭⠒⢂⣈⡠⢤⠄⣀⣀⣀⡤⣤⢤⠄⣀⡀⠉⠉⠛⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢞⣫⡴⠞⠋⠁⣀⠔⣩⣤⠆⣹⢭⣋⡿⢶⣻⣷⣶⣧⣏⠉⠛⠏⡏⠛⡿⡖⣶⣄⠉⢷⣦⣀
⠀⠀⠀⠀⠀⣠⣟⠵⠛⢛⣇⣤⡦⣚⣕⣺⢏⣖⡺⠗⠻⣞⣫⠯⣗⣻⡖⢮⡿⣦⣤⣀⣄⣀⣠⣆⣋⠟⢻⡗⣜⢿⡷⡄
⠀⠀⠀⠀⢠⣿⡏⢀⡰⢯⣿⣤⢙⣖⣩⣿⠿⣿⡾⠿⣷⣿⣻⠿⣟⡼⠿⢏⣸⠯⠘⣿⣿⣿⣶⣒⣻⣍⡽⢫⢸⠘⢉⣧
⠀⠀⠀⠀⢸⡿⣷⢬⡳⣯⣽⡯⣟⠛⢷⡾⣷⠿⣶⡏⡼⢿⣭⡿⢭⣿⣳⣿⣤⠞⠋⠒⣿⡔⣫⠿⠭⣯⣵⣾⣗⡷⣺⣿
⠀⠀⠀⠀⠀⣿⠘⠳⢿⣗⡺⠭⠟⡕⣲⣶⠒⣫⣝⣺⣳⣿⣿⣿⣷⣿⣽⣿⣯⣀⠶⣎⡹⣟⣓⣶⢿⣿⣯⣽⣿⡟⢱⡇
⠀⠀⠀⠀⠀⢸⡆⠀⡈⠀⡏⢛⡲⠤⣝⣚⣣⣏⣸⣟⣿⣙⣍⡁⠻⡩⢟⣿⣞⣯⣶⣾⣼⣷⣾⣿⡿⠿⡟⠋⣧⠇⣾⠃
⠀⠀⠀⠀⠀⠀⢿⠀⢇⠀⠀⠈⠭⣓⣒⠢⠭⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⢋⠩⠭⠁⠀⠀⠈⠁⢸⠋⣸⠏
⠀⠀⠀⠀⠀⠀⠘⣧⠘⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⢚⣩⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⢤⣤⣤⣤⣤⡤⠤⠴⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣋⣀⣂⣐⡀⠩⠙⠟⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠸⠭⡭⣭⡭⠽⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⣴⡿⢋⣴⠓⠲⣦⠄⠀⠀⣠⠴⢀⢔⣉⣁⣔⣂⠓⠶⠀⠙⠄⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢀⡤⣶⣽⣎⣩⣴⣿⣯⣤⣌⢳⡀⠀⡐⢁⡤⠘⠹⡡⣐⣠⢊⠍⣒⢬⡒⡀⠘⠒⠄⠉⢟⡲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠴⢃⠴⠔⠊⡉⣠⣮⡭⠶⠶⣶⠬⠙⢠⣷⠇⠶⡡⣈⣿⣿⡿⠿⠿⠿⣾⣮⡄⢅⢌⡢⠄⠀⠀⠉⠉⠑⣦⠻⣷⣄
⠀⠀⠀⠀⠀⢀⠖⠦⠚⠓⢞⡽⢋⣵⡌⠙⣶⢡⢀⠡⠀⡎⢇⡌⣜⣼⣿⠟⢁⡤⠒⠒⠐⢤⡙⢻⣷⣆⠩⣢⡀⠀⠀⠀⠀⠈⠙⢨⣿⣦
⠀⠀⠀⠀⠀⢘⠒⠘⢋⣩⣵⣿⠍⠑⣡⢰⣧⣿⠟⡀⠟⡼⠎⢰⣾⡿⢁⡔⠁⠀⠀⠀⠀⠀⠘⢆⠙⣿⣶⠐⣤⣄⠀⠀⠀⠀⠀⢫⡨⠟⡇
⠀⠀⠀⠀⠀⠘⠛⡿⣻⡽⢊⡟⢠⠎⣡⠟⠁⠁⣮⠇⡶⡇⠱⣽⣿⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⣿⣇⡌⢲⠆⠀⠀⠀⠀⡆⢡⣖⡇
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⠞⢠⠃⣼⠏⢐⠐⡇⢁⣜⠔⡀⣸⣿⠿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⢿⣷⡄⣯⢏⠄⠒⢲⣣⣾⣿⠇
⠀⠀⠀⢀⢔⣲⣤⣄⠀⠀⢀⢴⠀⣿⡤⢀⢌⣀⣁⣥⣀⡴⠟⣡⣴⢦⡈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣀⠻⣦⣬⣥⣴⡄⢸⡿⡫⠊
⠀⠀⠀⢘⣡⡞⣽⡿⢯⣯⣥⡼⢦⠘⢿⣿⣾⠀⠀⢀⣀⢴⣺⣿⣤⡴⢃⠸⢦⣦⡀⠀⣠⣤⣀⠀⣀⣤⣫⢟⠘⠑⠊⠋⢻⢇⡟⠁
⠀⠀⠀⠈⠀⠑⠃⠀⢹⣿⠀⠀⠀⠙⠀⠤⠭⠿⢿⡿⠿⢫⣖⢟⡵⠛⠛⠋⠁⠀⠁⠀⡿⢞⡿⢿⣞⡻⠞⣱⣴⢞⣷⣶⣯⠞⠙⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠠⠟⠟⠘⠑⠈⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣫⣤⣍⣽⣩⣿⣻⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠤⠀⡀⠀⠀⠀⠀⠤⢄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡀⠒⠐⢂⡉⠉⢄⠉⠐⡆⠤⣀
⠀⠀⠀⠀⠄⣄⣠⣀⣀⣀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢅⡀⠀⣤⠀⠘⣖⠀⣸⣆⡀⣸⠄⢻⠲⡀
⠀⠀⠀⢁⠚⢸⡃⡩⡆⠀⡗⠁⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢰⠎⠉⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡀⠈⠳⡂⠸⠒⠈⠉⠐⠀⠈⠁⠈⠑⢇⣰⠅
⠀⠀⠀⣖⣂⡯⠀⢴⡁⣨⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⢲⡀⢂⡸⠀⢘⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠅⠀⠠⠄⠀⠀⠀⠀⠐⠂⢈⠈⠋
⠀⠀⠀⢠⡠⠕⢞⡷⣿⡿⢾⡀⢨⠀⠀⠀⠀⠀⡆⢃⠡⠴⠀⠘⠾⠇⠀⠉⠛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠐⠂⠀⠀⠐⠀⠀⠄⠆
⠀⠀⢠⠈⠲⠆⠘⠉⠻⠛⠉⠠⢀⠠⠀⠀⠀⣜⠩⠀⠾⣀⣀⡀⠀⠀⠀⠎⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡁⢀⠀⠀⠀⠉⣿⠀⠀⡌⠱⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇⠈⢰⡯⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂⠘⠉⢁⢈⠁⢈⢐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢄⣹⢤⠛⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠜⠁⡌⡄⠘⠈⠢⠈⢸⣄⢀⡸⠀⠀⣸⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⡅⠀⠀⡴⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢦⠀⢇⠀⡨⠤⠄⠉⠀⠀⡘⠉⠙⢀⠙⠷⠺⢍⡔⠠⢤⡀
⠀⠀⠀⠠⡎⠀⠰⠃⠀⡠⠁⠀⡴⠁⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠘⠉⠉⢃⠐⠀⠀⠡⠂⠀⠴⠂⠈⠀⠐⠀⠀⠀⠙⢦⣀⡴⠆
⠀⠀⣿⠀⠉⡒⠚⠁⠒⠓⠄⣠⠃⠀⡞⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣾⡀⢠⠄⡘⢤⣶⢀⡄⢹⢀⠜⢁⡼⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠙⡀⠘⠀⡹⠂⢠⠚⠃⠘⡧⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠐⢤⢱⠈⠰⡤⠠⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⣘⠁⠀⠀⢸⡇⠀⠐⠉⠂⠖⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣌⠁⠀⠀⠀⠀⠀⠀⠐⢄⣀⡀⠀⡇⠀⠀⠀⢸⠤⠬⠿⠻⠃⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠗⠛⠛⠛⠂⠹⢻⠀⠀⠀⢸⠙⠈⠳⠚⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢺⠤⢮⠹⢟⣂⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠀⣄⠀⠨⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⣴⠩⠗⣈⡑⢦⠈⢂⡬⡀⡎⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠂⠈⠱⡀⠀⠀⠀⠀⠀⠅⠈⠚⠤⠝⠘⠏⠹⠀⠎⠐⠑
⠀⠀⢀⢀⠀⣁⣉⠉⠙⣏⣩⡟⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣄⡄⠀⡀⢠⡄⠀⣴⢀⢠⡄⡀⢀⡖⠲⢰⡆⠢
⠀⠀⢸⠀⠀⠉⢠⣷⣋⠉⠁⣀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡹⣿⠞⣋⡉⢶⠄⢉⣴⠒⣁⣤⠉⠀⠄⣊⢨⠟
⠀⠀⠀⠀⠀⠰⢸⡥⠐⠙⠿⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠄⠘⠆⠀⠀⡜⠀⠀⢰⠈⠘⡆⢵⢀⢰⠃⠛
⠀⢀⠀⠀⠠⠀⠉⠀⠈⠀⠀⠀⠀⠰⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠄⠱⠀⡴⠀⠀⠀⠃⠀⣠⠃⠀⠐⡀⠀⢟⠃⢸⠂
⠀⠀⠂⠄⠀⠀⠀⠄⠀⠀⠀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠰⠁⡴⠁⠀⡔⠁⣠⠀⠁⣀⠊⣼⡅⡠⠴⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣀⠔⣠⡿⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢋⣼⡿⣡⣾⠏⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣼⣿⠋⠀⠀⠀⣸⣿⣷⣶⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⣿⣿⡿⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣿⣯⠆⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠜⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣋⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠟⠷⣤⣄⠈⢿⣿⣏⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣀⡀⢀⣙⠻⣶⠛⠛⠻⣿⡟⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣟⠩⢉⣿⣷⡍⠀⠀⠀⠀⠞⣩⣿⠛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠙⠻⠿⠿⠋⠉⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣶⣦⡄⠀⠀⠀⠀⠈⠟⢾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⢯⡝⣿⠻⢿⡷⠀⠀⠀⣠⡄⢀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢘⢳⡶⠾⠻⠷⠀⢀⣀⣹⣢⡈⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣿⣮⡘⡇⠀⠈⢰⣾⣿⣿⣿⡟⢷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢶⣝⣻⡿⢿⠧⣈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣧⡀⢷⡀⡙⠛⠻⣦⡄⣠⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⢳⣿⣦⢸⡀⢾⡏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⠀⠩⢻⣾⡇⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠄⠀⠺⣏⠀⠀⢿⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢂⣶⠀⠀⢹⣧⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠂⠀⣯⡟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡁⠀⢸⣿⡥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣇⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡇⢸⣼⡿⡉⢀⣠⣂⢌⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣿⣸⣇⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣯⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣾⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⢉⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡯⣿⣿⣿⣿⣿⣿⣟⠭⣽⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡙⢽⡿⣿⣿⣿⣿⣷⡿⣻⣿
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠻⡿⢣⣿⣿⣿⣿⣾⣿⠟⠁⠀⠙⣿⣗⣈⣯⣽⣿⣿⣿⣯⣥⡵⢿
⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢐⣻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣰⣿⡿⠥⡝⠟⣼⢿⢿⣿⣳⣳⣟
⠀⢸⣿⣿⣿⣿⣿⡏⠙⠉⠁⠀⠀⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⡟⠓⢲⣁⣧⣟⣞⣿⣿⣧⡼⠸
⠀⠈⠻⢿⣿⡿⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⣟⠋⢹⣿⣧⡦⡷⡷⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣫⣽⣿⣿⣿⡿⠁⠀⠀⢀⣴⣿⣧⡈⠉⠛⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⡿⠋⠀⠀⠀⣴⣿⣿⣭⡉⠛⠻⣶⣾⣟⣯⡯⣿⣿⣿⣿⣿⢿⣾⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠏⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦⣤⣾⡗⡇⣷⣿⣿⣿⣿⣿⣿⢿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣿⣉⡇⣿⣹⣿⣿⣿⣿⣿⣿⣿⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣷⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣴⣶⣫⣷⣿⣿⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡟⠻⠯⠡⠠⠺⠟⢹⣿⣿⣿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⠦⣤⣤⣤⣴⣶⣼⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠉⡉⠃⢁⠪⠛⣧⣥⣬⣭⣽⣿⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣡⣴⣾⣯⢡⣡⣰⣑⢤⢇⣢⣿⣿⢿⣿⣿⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠁⡁⢀⣀⣀⣀⡈⢁⣀⣈⣋⢉⣙⣿⣿⣿⣿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠄⠿⠋⢿⠖⣿⣿⡷⣾⣿⣿⣿⣿⣺⣿⣿⢉⣩⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⢀⠀⠀⡄⠀⡀⠀⢀⠀⠀⣀⠀⣈⠀⣘⡝⣯⣿⡿⡿⢾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⡏⠿⠘⠁⢐⢈⡉⠊⢉⠰⡉⠻⢿⣿⣿⣹⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡉⠁⠄⠈⠀⠛⠉⠙⠉⠙⠘⠃⠁⠂⡄⠛⠻⢷⣯⣿⣿⣟⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡦⠀⣤⣄⣤⣀⡀⣀⣤⣀⣀⠀⣀⠀⠀⠀⠤⢠⡀⠐⣃⣉⣡⣿⣿⣽⣽⣽⣿⣭⣭⣽⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠛⠙⢻⡟⣻⣿⣿⣿⣿⣿⡟⢻⡿⡿⢿⡿⡻⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡾⣠⣿⣿⡟⠰⢿⠟⠛⠠⠟⠀⠸⠘⠋⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⢀⡀⠄⠀⣰⣵⣿⡟⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠚⠉⠉⠡⠈⠩⠀⠁⠈⢡⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⣿⣻⣿⣿⠿⠟⠛⠟⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠋⠧⠀⠦⠒⢤⡄⠀⠀⢀⡀⡱⣼⣿⣟⠇⠛⡉⠀⢄⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠋⠀⠉⠐⠁⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⡇⠀⠀⢀⣤⣤⡀⠀⢀⣤⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⡇⠀⠀⠨⣿⢿⡇⢐⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠰⠒⠲⣶⡄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢲⡄⣤⠆⣸⣧⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢀⡈⡀⠈⠀⢹⣿⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⢸⣿⡏⣉⣠⣿⡥⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⠟⢘⣿⣿⠏⢹⣿⠆⠀⠘⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⢶⣾⡿⠃⣄⠀⠻⠃⢀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⣠⠀⢤⡤⠐⠒⠙⠠⠀⠀⠀⣾⡿⠁⠀⠉⠀⠀⠀⡄⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠘⡄⣀⡑⠀⠈⣧⠄⠘⠀⢀⣿⠀⢔⠣⠀⠀⠀⣴⠀⠆⠀⣠⠖⢀⣤
⠀⠀⠀⠀⠀⠀⢰⡇⠻⢴⠄⠀⢠⠗⢣⡀⠀⠟⠀⠁⠀⠤⠀⠈⣙⠴⠐⠊⠁⠒⠉⠀⠑⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡈⠐⢾⣷⣷⣿⣦⣶⣾⣶⢶⣆⢀⠸⠋⠀⠙⠉⠀⠈⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢸⡽⠃⣞⠱⡀⠉⣻⣿⣿⣿⣶⢾⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣸⠀⠛⠀⠀⠈⠀⣿⣧⣿⣟⣏⣰⣾⣿⣷⡀⠀⠆⢀⠀⠀⡠⣢⠟
⠀⠀⠀⠀⠀⠀⠋⢀⡀⠟⠀⠀⠀⢿⣿⡾⠿⣡⣄⣽⢍⢿⡟⢶⠀⠀⠁⠒⠀⠁
⠀⠀⠀⠀⠀⠀⡏⠰⠁⠁⠀⠀⠀⣿⠈⣵⡾⢿⣇⠿⠿⡑⢳⣾⡆
⠀⠀⠀⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⣯⣁⣈⣿⣿⣽⡟⣿⢷⠼⡷⢻⡄
⠀⠀⠀⠀⠀⢠⠁⠀⢀⠁⠀⠀⠰⣏⠋⡉⢹⣿⣻⣿⣛⡟⠰⢧⣄⢱⡀
⠀⠀⠀⠀⠀⠆⠶⠂⡸⠀⠀⠀⢀⣿⡶⣻⣼⢻⣹⣾⠁⠆⠘⣽⠫⠎⣳⡀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⢰⠀⠀⢠⡇⠀⠀⠀⢸⠟⣿⡿⣧⢈⣹⣁⣼⡇⠀⠸⡦⠀⠈⣧⠀⠀⢀⡎⢨⢧
⠀⠀⠄⣴⠉⠀⠀⠚⠁⠀⠀⠀⠀⠐⠛⢻⣋⣟⢁⡍⠈⠀⠀⠀⢹⡄⠸⠙⣆⠀⡌⠙⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⢳⡄⡄⢻⣞⠃⠀⠀⢠⣤⡴⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⢡⣿⣧⢿⡆⠀⠀⢻⣄⣼⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⠆⣶⠀⠀⠀⠀⠀⠀⠰⣀⡾⡀⠓⢷⠚⢼⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡴⠊⠁⠙⢦⠀⠀⠀⠩⡔⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠬⣁⠀⠁⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠞⠀⠀⠀⠛⢄⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⡶⣾⣶⡆⠙⢄⡀
⠀⠀⠀⠀⠀⠀⢨⡀⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⠔⠀⠈⠲⠀⠈⣇⠀⠉⠛⠁⠀⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠈⠃⠤⠘⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠰⠃⠀⠹⠦⠄⠀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢨⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣠⣤⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⠤⢤⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣻⠂⠀⠘⠓⠀⠀⢲⡄⠀⠉
⠀⠀⠀⢀⣀⠀⠀⠀⠰⣺⣯⡿⣿⢷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠈⠉⠉⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⣘⣷
⠀⠀⠀⠀⠐⠻⣷⣤⣤⡠⠤⠤⠤⠶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⣶⠈⠁⠀⠀⠀⢹⠈⣶⠈
⠀⠀⠀⠀⠀⠠⠟⠐⠛⣻⡷⣦⣤⠤⣼⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⢿⠀⠀⠀⠀⠀⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⠀⠀⠙⢦⠉⠙⠿⠻⠦⠀⠀⠀⠀⠀⠀⠈⠉⣉⠿⠿⠟⠻⠁⠈⠉⠀⠻⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣵⠃⠉⢢⣤⣯⣼⣤⣦⡄⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠛⠋⡕⠀⠀⣀⣋⡁⡉⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⠙⠚⠘⠼⠽⠃⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣤⢐⢀⠢⣀⡀
⠀⠀⠀⠀⠒⢶⡶⠿⠈⢭⣝⣛⣚⠛⣚⢻⡷⠺⡷⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⢉⢉⢉⠻⠇
⠀⠀⠀⠀⠀⣀⣄⣤⣤⣤⢦⣤⣤⣤⣧⣤⣄⣠⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠄⠀⠀⢀⢀⠠⣀
⠀⠀⠀⢸⠛⠉⢳⢴⢨⠛⢇⡹⢎⣩⣭⣿⡷⠾⠁⠈⠀⡀⠀⠀⠀⠀⣆⠀⠈⢯⡝⠱⠾⠾⠴⠤⠦⠗⢪⣿⠷⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠘⠇⠈⠳⠌⠛⠛⣋⠔⠈⠀⠀⠀⠁⠀⠀⠀⠀⠹⡦⢤⣬⣕⡒⠒⢒⣒⣒⡒⠒⠺⠀⣤
⠀⠀⠀⠀⠀⠓⠠⠄⠀⠁⠐⠈⠛⠉⠘⠁⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⠻⣿⠛⠯⠉⠻⠛⠳⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣸⣿⡃⣼⠃⠀⠰⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣻⠏⠀⡀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠋⠰⠏⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠠⣶⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢫⡿⣁⣤⣴⣶⣶⣦⣀⠀⠀⠀⣰⣿⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢸⠃⣿⠉⠈⠻⠟⠈⠛⠇⠀⢰⣿⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢈⠀⠈⠁⠀⠀⠀⢠⠔⠃⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠿⠋⢽⣷⡀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣇⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⢿⠇⢹⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀⣠⠴⠋⠀⢸⣿⡌⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣼⡟⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠠⠞⠋
⠀⠀⠀⠀⣠⣶⣾⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⣿⣟⡻⠿⠖⠂
⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣝⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠸⣿⣿⣟⡁⠄⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠉⠙⠛⢿⣦⡀
⠀⠀⠀⢻⣿⣿⣷⣶⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠈⢦⡀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠈⠛⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣆⠀⠀⠶⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠃⠀⠀⠈⠉⠙⠻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢠⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢛⣿⣿⣿⣿⣶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⣀⣰⣦⣄⡀⠀⠀⠀⣠⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⠿⠛⠀⠀⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠉⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⡀⡴⢒⣄⡴⣆⣀⣤⣄⣤⣤⡤⠀⠀⠀⣠⠀⠀⡠⠀⣠⠀⠀⢀⠀⠤⣤⢠⠄⠀⠀⣠⣴⣄⣴⣿⣿⣿⡀
⠀⠀⢀⣀⣀⢤⣤⠄⠀⠀⠀⠀⠀⠀⠘⣀⡍⣁⣿⣦⣿⣿⡗⠻⢿⣿⣿⣿⣿⣿⣯⣶⠻⠟⠀⣉⠈⢦⣠⣤⠂⠁⠀⠀⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⢁⣄⣴⣘⣒⠂⠀⠀⠀⠀⢠⡝⣡⠔⣶⣿⣿⣿⣿⣶⣌⣠⣿⣿⣶⣿⣿⠿⢿⣿⡰⢿⠶⢶⣯⠛⢪⡏⠳⠄⠀⠀⣀⡰⡴⠫⠈⠻⣻⣿⣿⣿⣿⣿
⠀⣰⠶⠾⣷⣎⣟⠴⠣⠀⠀⠀⠀⠈⠀⠁⠀⠙⣽⣿⠻⣿⣿⣿⣿⡆⠈⠙⠹⠃⢰⣬⣟⠠⠌⠚⠂⢘⡃⠀⠈⠀⠀⠀⢽⢋⣰⠄⢁⢀⣀⣤⢚⠿⢿⣿⣿
⠀⠀⠠⣿⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⠃⠀⠈⠻⠲⣋⠇⠀⠀⠀⠀⠁⠂⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣴⡤⣦⢙⣢⣆⣭⠶⣦⠈⠙⣿
⠀⠀⠀⣿⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠕⣠⣿⠉⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⢳⢮⣿⣗⡼⠋⢉⣿⣶⡶⠄⠁⠀⣿
⠀⢠⣾⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⣡⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣛⡽⢛⣦⠐⣾⣿⠝⣻⡛⠀⠀⠀⣿
⠀⢸⣿⣿⣧⢴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣡⣤⣿⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠥⠾⠽⡇⣈⡣⠛⠟⠝⠀⠃⠐⢿
⠀⢸⣿⠿⠟⠸⣿⣦⠀⠀⠀⠀⠀⡄⢀⡈⠿⠁⣹⣿⣿⣗⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢄⢋⡈⠉⠏⠃⣐⠂⠜⠂⢂⠀⠸
⠀⢈⣷⣤⡤⠠⢶⣾⠄⠀⠀⠀⢐⣧⣯⠃⠘⠀⠁⠁⣠⢤⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠀⢡⣞⣾⠏⢀⠀⡠⠀⠐⠒
⠀⢸⣭⣤⣤⣠⣴⣦⣤⣤⠀⢸⣹⣿⣽⣶⣄⣀⣀⣰⠟⢀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⡷⠿⢷⡿⣽⠁⠋⣤⣀⠀⠀⠀⠀⣀⡠⠆
⠀⢸⣿⢻⣷⣿⣿⠃⢸⣷⢀⣾⣟⣽⣿⣿⣿⣿⣿⡇⠉⢸⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣷⣾⣯⣥⢮⡳⣄⠃⠓⠘⢽⡯⠝⠁
⠀⢸⡟⠀⣿⣿⣿⠀⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠤⣀⠀⡀⡀⢻⣿⣿⣿⣿⣼⣷⣟⠱⣤⣄⡀⠀⠀⠀⠀⠀⠀⡄
⠀⣾⡇⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⡀⠀⠀⠀⠀⠀⢀⠶⠀⡟⠀⠀⠘⡆⣷⠿⠘⣿⣿⣯⠏⡉⠉⠉⠘⠻⠿⠅⣈⣥⢠⡀⣌⠊⠁
⠀⣿⡇⠀⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢠⣿⣷⣄⠀⡀⠀⠀⠘⢿⣯⢴⣻⡛⣻⡤⣀⣠⣤⠀⠀⢀⠀⠹⣿⣿⠒⢇⢄
⠀⣿⡇⠀⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣆⡄⣠⣶⣿⣿⣿⣿⠀⢹⠀⠀⠀⠀⠀⠈⠉⢁⣼⣿⣿⣯⡻⣦⠁⠀⠀⠀⢁⡿⣇⢸⡸⡇
⠀⣿⠀⣰⣿⣿⡇⣴⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣽⣿⡟⢲⣏⡀⠀⠀⠀⠀⢀⡚⠛⢸⣿⣿⣟⣵⣿⠀⠀⠀⠀⢰⠇⢻⠌⢁⠁
⠀⣿⣰⣿⣿⣿⡇⢸⣿⡿⢿⡷⢿⣿⣿⣮⠀⠿⢿⣿⣿⣿⣿⠿⣃⠀⢀⡉⠀⠀⠀⠟⠿⣶⣶⡶⠛⢹⣿⣿⣿⡷⠝⣹⠿⠃⠀⡀⠀⠀⠀⠀⠀⡠⠊⠀⡄
⠀⣿⣿⣿⣿⣿⣧⣿⠟⣧⡈⠄⠛⠟⠛⠁⣀⠀⠈⠛⠛⠋⢀⣀⡩⠜⠋⠁⠳⠶⠤⠶⠖⠛⠉⠀⠀⠸⠿⣻⢪⣾⣻⠋⠀⠀⢈⠀⣦⠀⠀⠀⠀⠀⣠⠂⠁
⠀⠿⠿⠿⠻⢿⣍⢚⣃⣤⠤⠄⠴⣆⠐⠀⠀⠉⠙⢉⣭⣰⣦⡈⠀⠀⠀⠀⣠⠖⢤⣀⣰⢦⠀⠀⠀⠀⠀⠉⠉⠀⣙⠿⣿⣥⣟⣴⠟⣁⡓⠓⠒⢋⡁⢠⡄
⣤⣉⣐⣀⣀⣠⣍⣋⣉⣉⣑⣚⣙⣛⣛⣲⣶⣂⣛⣛⣉⣉⣉⣥⣙⣶⣞⣋⣁⣀⣀⣀⣀⣉⣀⣀⣀⣀⣤⣀⣀⣤⣭⣀⣉⣭⣅⣀⣈⣩⣉⣉⣁⣩⣉⣉⣥
⣿⣻⣛⣟⣿⣹⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣭⣭⣿⣿⣯⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣠⡴⠀⠀⠀⠀⠀⢲⠀⡀⠀⠀⠀⠀⢦⣄⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣟⣿⣿⣿⠿⠿⠓⠂⢀⣴⣾⣷⣾⣇⣧⠀⠀⠀⠐⠚⠿⠿⣿⣿⣿⣻⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⣿⣿⣿⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡿⣤⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣿⣿⣿⣿⡿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⣾⣿⣿⣟⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣹⣿⣿⡀⠋⣿⡟⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣻⣿⣿⣷⣬⡳⣄
⠀⠀⠀⠀⠀⢀⢜⣵⠟⣫⣿⣿⣏⣾⣿⠟⣡⣾⣿⣿⣇⠀⠀⠀⠀⠀⢿⣿⣷⣄⠈⠳⡌⠃⠀⠀⠀⠀⠀⣸⣿⣿⣷⣌⠻⣿⣷⣹⣿⣿⣝⠻⣮⡣⡀
⠀⠀⠀⠀⢀⡵⢋⣤⣾⣿⣿⣿⡸⣋⣵⣿⣿⡿⣿⣿⣿⣄⠀⠀⠀⠀⠘⢻⣿⣿⣵⣤⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢿⣿⣿⣮⣙⢇⣿⣿⣿⣷⣤⡙⢦⡀
⠀⠀⠀⣠⢋⣴⣿⣿⡿⣿⣿⣿⣽⣿⣿⡿⢫⣾⣿⣿⣿⣿⣦⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢠⣴⣿⣿⣿⣿⣷⡝⢿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣦⡙⣄
⠀⠀⡼⢫⣾⡿⠋⢁⣴⣿⣿⣿⣿⡿⠋⣰⣿⠟⣿⣿⣿⣿⣿⣷⣶⣿⠿⠙⢿⣿⣿⣿⣷⠹⢿⣶⣾⣿⣿⣿⣿⣿⠻⣿⣆⠙⢿⣿⣿⣿⣿⣦⡈⠛⢿⣷⡝⢧
⠀⠘⣱⣿⠟⢀⣴⠟⠉⠀⣠⣿⠟⠀⣰⡟⠁⣼⡟⣼⣿⣿⣿⣿⠿⠀⠀⣀⡼⣿⣽⣿⣿⠀⠀⠿⣿⣿⣿⣿⣧⢻⣧⠈⢻⣆⠀⠻⣿⣄⠀⠉⠻⢦⡀⠻⣿⣎⢃
⠀⣰⡿⠃⣠⠏⠀⠀⣀⡴⠟⠁⠀⢠⡿⠃⠀⢿⡇⢿⡇⠈⠻⠇⠀⡞⡟⠋⢳⣿⣿⣿⣿⣀⠀⠀⠘⠟⠁⢸⡟⢸⣿⠀⠘⢿⡄⠀⠈⠻⢦⣄⠀⠀⠹⣄⠘⢿⣆
⢠⣿⠁⢀⣁⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣿⡿⠏⢿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⣈⡀⠈⢿
⣾⠇⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠸
⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⢶⣄⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣿⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⣿⣰⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣰⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣻⣾⣧⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣷⡿⢋⣤⣴⣶⣶⣶⠖⣠⣀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⣀⡈⠁⠟⠁⠁⠀⣠⣿⡟⠉⢿⣿⣿⣇⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣾⠡⠉⠀⠀⠀⠀⣰⣿⡏⠈⢀⣾⡏⠉⣠⢤⡙⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢠⣿⠋⣽⣿⣿⡿⠣⠊⢠⣾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢸⡟⢹⣿⣿⣿⣤⣀⠀⠘⠛⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢀⣤⣿⡿⣿⣿⣟⠀⠘⠻⢷⣦⡀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡟⢡⡖⣍⢻⣿⣶⣤⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⢰⣿⡷⢶⣉⢶⢸⣦⣌⡈⣩⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣿⣧⣘⢙⣂⣾⢿⣯⣭⣁⣰⣿⠀⠀⢀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢿⣿⣿⣿⣁⣾⠀⠀⣿⠟⠁⠀⠀⠀⠀⣿⠃⠀⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡃⡾⠻⢻⣿⢿⡀⣠⠀⠀⢠⡾⡆⠀⠀⢀⠀⠀⠀⣠⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⡟⠟⠀⠀⠀⡏⢘⣧⣝⣵⣿⡟⠀⠀⡀⢀⣾⣷⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣬⣿⣟⡀⠀⠀⠀⢰⣿⣟⡿⣿⣿⣷⣾⣿⣿⣏⠉⣉⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢉⣩⢴⡿⣿⢛⣿⣷⣄⠀⠀⣻⣯⣿⣾⣿⣿⡙⣿⣿⡿⠓⣿⣿⣿⠟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣟⣼⣿⣿⣿⠦⠿⣧⡀⢿⣿⣿⠟⣿⣿⣿⠏⠉⢀⡀⠈⠙⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠀⢨⣷⡄⠀⠼⣷⣾⣿⢟⠀⣉⣏⠡⠀⢴⡋⠉⣧⠀⡙⢿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⢸⣿⣿⣄⣶⣟⡿⠁⠀⣺⣿⣿⣿⠀⠈⠓⠚⠃⢀⣴⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⣿⠃⣠⡼⠛⣿⣋⣽⣿⣷⡐⣠⣶⣾⡻⣿⡿⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⠻⠈⠻⠋⢀⣠⢬⡟⠻⢿⠿⠩⠿⣿⣿⡷⠟⠁
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣇⢀⣽⠀⠀⠀⠀⠀⣸⣷⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⣤⣄⠀⠉⠀⢠⣿⠃⢀⣾⣿⣿⣷⠈⢿⡄
⠀⠀⠀⣿⣿⣿⠃⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡏⠀⠈⠁
⠀⠀⠀⢻⣿⣿⣷⣶⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣹⡿⢿⠁
⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠋⠁⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⢀⣀⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⢦⣀
⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣶⣶⣾⠯⠟⠓⠒⠒⠒⣂⡈⢳
⠀⢸⢇⠀⠀⠀⠀⢀⡠⠔⣒⠭⠕⢒⣉⠽⣿⣿⣿⣿⣍⣉⠩⠭⠭⠤⠤⠌⡷⣄
⠀⠈⢎⢣⡀⠀⠀⣿⡈⠩⠴⠒⣉⣩⠤⠒⢊⣻⣿⣿⣿⡶⠒⠒⢒⡒⣒⠊⢹⠈⠣⡀
⠀⠀⠈⢣⡱⡄⠀⢰⠈⠓⠢⠬⠥⠒⠒⢩⡍⢸⣿⣿⠿⣇⠀⢈⠥⠭⣍⢁⣼⣄⡀⠈⠢⡀
⠀⠀⠀⠀⠱⡜⢆⠸⡀⠐⠒⠒⠀⠀⠀⢀⣠⡚⢻⠀⠀⠀⣟⣇⠀⠀⢈⡇⠀⠀⢸⠉⠙⢿⢦
⠀⠀⠀⠀⠀⠘⢎⢣⣧⡀⠀⠀⠤⠆⢰⠁⢠⠝⠚⠢⣤⡞⠉⠈⣷⠊⠉⠳⣶⠖⠋⠳⣄⡸⠉⡗⢄
⠀⠀⠀⠀⠀⠀⠈⢢⠱⡉⠓⠒⠒⠚⣿⣦⣧⣀⠀⠀⣸⢳⡤⠤⠻⣀⠀⣠⢧⣀⣀⣴⣿⣤⣶⡇⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⢛⡿⣟⠓⠒⠒⠤⢤⣈⢦⣤⣤⡤⢒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠳⡀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢡⠖⢉⡥⣀⠉⠲⠼⠃⢸⠀⠀⠀⠀⠀⠈⣗⣉⠶⣍⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡑⣄⠉⠛⠻⠿⣻⠟⢋⢄⠀⠀⢸⢀⠋⠀⠀⠉⠑⢒⣒⣁⠀⠀⠀⣀⣀⣀⢟⠛⠛⠋⢟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⢆⠀⡴⠋⠀⠀⠸⣌⠲⠤⠞⣸⠀⢀⡠⠔⠊⠉⢀⠀⠀⠀⠀⠀⢀⠀⠘⡄⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢣⣧⠀⠀⠀⠀⠈⠉⣛⡩⠕⠚⠉⠀⠰⠒⠒⠉⠁⠀⠀⠀⢠⠋⠀⢰⠁⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠹⡗⠤⠤⠔⢒⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣠⠔⠁⠀⠀⠀⢀⣠⠤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⢾⡄⠀⠈⠙⠓⠀⠀⠀⠀⣀⣀⣤⠤⠤⠖⠒⠉⠉⠀⠀⠀⣀⡤⠖⠊⠁⣀⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢫⡳⠤⠴⠖⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⢀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⢆⠀⠀⠀⣀⡠⠔⠚⠉⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢳⠒⠉⠁⣀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠤⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠚⢛⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣠⣄⡀⠀⠀⢀⣠⠴⢚⣋⣭⠤⠶⠖⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡛⢭⢼⠖⠗⢋⢻⡶⠞⣫⡵⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⠾⡖⠂⣍⣨⠷⢊⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣞⣉⣭⣥⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠤⠶⠶⠦⠤⠤⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠫⢅⡶⠬⠖⡀⠵⢠⡤⢈⡓⠦⠄⣉⣙⠓⠲⢤⣄
⠀⠀⠀⠀⠀⢀⣠⢞⠚⠣⡼⢑⣩⣄⢘⡀⠱⡶⠀⠆⡈⢁⡤⠤⡄⠨⢙⠊⢂⡉⡳⣆⡀
⠀⠀⢰⡶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠓⠒⠒⠶⠶⠶⠶⠾⠿⢶⡆
⠀⠀⠀⠻⣔⠂⠒⠀⠒⠀⠂⠐⠂⠒⠂⠒⠀⠐⠀⠒⠀⠒⠂⠒⠀⠒⠂⠒⠀⠒⠐⠂⢒⡾⠁
⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣄⠐⣄⠀⠀⣷⣄⡈⢿⣦⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣦⣹⣿⣶⣿⣿⣿⣿⣿⣧⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⣤⠂
⠀⠀⠀⠀⢀⡀⠀⢀⣴⣶⣶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣤
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠁
⠀⠀⠀⠀⠠⣀⠀⣨⣿⣿⣿⣿⣿⡟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⣿⢟⣴⣿⣿⣿⢿⡟⢁⣾⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠃
⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠃⡾⢀⠞⠁⣿⠁⠀⢳⡿⢿⣿⣿⣿⣿⣿⣿⡍⠃⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⣿⣿⣏⣿⣧⣼⣁⡁⠀⠀⠟⠀⠀⢹⡇⠈⢿⣿⣿⣿⣿⠿⡈
⠘⣆⠀⠀⠀⠀⠀⠀⠰⢻⣿⣿⢈⣧⣽⣉⣛⡯⠀⠀⣿⠿⡟⢻⢓⡖⢸⣿⣿⣿⣿⡦
⠀⢿⢧⠀⢠⡇⠀⠀⠀⠈⠙⣿⣰⣿⡏⠀⠀⠀⠀⠀⠀⠉⠈⢋⡉⠀⢸⣿⡛⠿⠀⠁
⠀⠘⣯⣷⠏⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡞⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠤⠤⣿⠏⢠⡷⠀⢀⣀⡀⠀⠀⠀⠁⠀⢤⡄⠀⠐⠐⠀⠀⠀⣸⠀⣴⢟⣇⠀⣀⣤⣀⣀⣀⣤⣠⣤⣾⣿
⣿⢹⠏⣠⣿⣫⣶⣾⣿⣿⣿⣿⣴⢀⣆⠈⠛⠓⠶⠄⢴⠾⠃⣿⣿⢫⣼⣿⣶⣿⣿⣿⣿⡿⢋⢻⣿⣿⣿
⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢸⠹⣷⣄⠀⠀⠀⢀⣴⡞⡹⢣⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣼⣾⣿⣿
⠟⠁⣴⣿⣿⣿⣿⠟⠛⡝⢰⣿⣇⣿⢸⠉⢿⣷⣶⣾⣿⠟⣺⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣾⣿⠿⠛⠉
⡀⣼⣿⣿⠟⠋⠀⢠⣮⣴⣿⣟⣾⡼⡾⠀⢸⡌⠉⢛⣿⣾⣿⣿⣿⣿⣿⣿⠟⢉⣾⣿⣿⣿⢋⢀⡙⢿⡟
⣽⣿⣿⢧⣤⣤⣴⣿⣿⣿⣯⣾⣿⡇⣧⠀⠈⢃⣴⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣼⣿⣿⣿⣿⡀⡖⠂⡸⣳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠿⠖⠿⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠈⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠨⠗⠀⠐⠒⠂⠀⠀⠠⣿⠯⠛⠀⠘⣧⡄⠀⠀⠀⠀⠀⠀⠤⣀⠐⠐⠀⠁⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣼⠀⠀⡀⠀⢠⡌⠉⠀⠀⠀⠀⠈⠠⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡤⣴⠏⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⣤⡄⠀⠀⠀⠀⣸⣟⣉⣽⡿⣶⣀⣀⠀⠀⠀⢼⣻⣿⣷⠀⠀⠀⠀⠀⠠⠀⢠⣤⣾⣿⣿⣿⣿⣿⣟⣉⣳⣦⣤⡀
⠀⠀⠀⢀⣀⡄⠀⣽⣿⣷⡄⠀⠀⠈⠩⣭⡅⠀⠀⢩⣭⡽⠄⠀⠀⢼⣽⣯⡿⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣾⠿⢿⡆⢿⠄⡀⠃⠀⣴⣴⣿⣧⡌⡷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠷⠀⠰⠇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣶⣤⣶⣶⣶⣦⣤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠂⠀⠀⢐⠀⠀⠀⠀⠀⣴⣷⣿⣾⣵⣿⣷⣦⠀⢘⣛⣛⡛⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠋⠋⠛⠉⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⣤⣄⡀⡀
⠀⠀⠀⢠⠶⠶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠻⣿⣿⣶⣦⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣯⣟⠿⡶⣤⡀⠀⠀⠀⠀⠀⣀⣨⣿⣯⠛⢿⡛⢿⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣴⣧⠧⠷⢦⠤⣴⣮⡽⠟⠻⣿⠿⡷⡄⠙⢦⣊⢻⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣯⣿⠏⠁⡴⢂⢲⠙⣿⣁⡰⠀⢰⠁⡠⢠⠋⠢⡀⣑⣜⣻⣿⡱⣦⡀
⠀⠀⠀⠀⠀⠰⣤⣼⢻⣝⢸⢁⣴⣠⣖⣿⣇⠀⣻⢿⡛⣶⣯⣤⣄⡀⡌⢠⣽⡝⣶⣤⢼⣗⡄⣍⢢⡀
⠀⠀⠀⠀⠀⣹⣿⣿⢘⡖⣛⣸⣿⠛⠏⠐⣘⣗⣋⣇⣷⠛⡟⣷⠄⠉⢳⣿⡯⢽⢮⣿⡇⠿⣷⣉⠁⢱⣄
⠀⠀⠀⠀⠀⡏⢹⣴⣿⡿⡟⡏⣽⣷⣼⣿⣷⢿⣩⣿⣿⡄⡕⠘⣖⠀⢸⢸⡇⠀⠉⠉⠀⠀⠀⠓⡔⠁⠉
⠀⠀⠀⠀⠠⣏⣯⣿⡽⠃⡰⣾⠏⡟⣸⣿⣿⣻⡿⣿⡿⣿⠳⣄⠈⠺⠿⢿⠇
⠀⢠⣸⢁⣠⢸⠿⢟⠀⢀⣻⣽⡄⠘⠲⠓⢺⣿⢟⠀⣿⣿⣅⡒⠛⠒⠚⠁
⠀⢽⡿⢟⣿⣵⣯⣞⠀⠘⣩⣯⣀⢀⣴⣾⣿⣿⠿⣿⠛⠋⠺⡿⠀⡀⠀⢤
⠐⣯⡿⣿⣿⣛⣾⣡⠀⠀⠿⠿⠃⢎⠀⠉⠙⠁⠈⠏⠀⠀⠀⠁
⠀⣸⣃⠿⢻⡾⠊⠂⠀⠀⢠⡌⠉⠀⠁⠐⠂⠠⠀⠂
⢸⠿⣿⠽⢿⣡⣏⣀⡄⠀⠀⠑⠒
⠨⢴⣻⡿⠻⡽⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠾⠛⢛⣋⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢟⡛⢛⡯⡽⣧⣄⣀⣤⣴⠶⢟⣛⣋⣭⣥⡷⠾⠟⠛⢛⣉⣉⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠋⣧⠅⠺⣥⢙⡈⢿⣿⠷⠾⣛⣛⣩⣭⣤⡶⠶⣾⣿⣛⣫⣭⣭⡽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠊⣥⠤⠅⣲⣀⣩⡥⠾⢿⣛⣛⣯⣭⣽⠶⠶⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣴⣯⣭⡿⠷⠞⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣼⡟⠛⠿⠛⠉⢁⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠾⠛⣋⡭⢀⠈⠋⣤⡤⠘⠃⠉⣠⡙⠷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⡶⠿⢋⣀⡴⠎⡉⢀⡘⠁⠐⠒⠢⣈⣋⡀⠐⠷⣤⡜⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠆⢉⡤⠿⢉⠀⠛⣀⣥⠀⠻⠿⠋⢙⣉⠉⠷⢤⡃⠈⢛⣀⠈⠻⠶⣦⡀
⠀⠀⠀⠀⣀⣀⣾⣏⣁⣊⣁⣀⣈⣉⣥⣤⣬⣥⣤⣈⣳⣦⣬⣥⣤⣤⣤⣤⣄⣀⣀⣀⣈⣃⣹⣷⣀⣀
⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⣿⣀⣤⠀⢀⣀⠀⣀⡀⢀⣀⠀⣀⡀⢀⣀⡀⢀⣀⠀⢀⠀⠀⢀⡀⠀⢀⣀⠀⢀⣀⢀⣀⣿
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠉⠁⠈⠉⠁⠈⠉⠁⠈⠉⠀⠈⠉⠀⠈⠉⠀⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠻⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠀⠁
⠀⠄

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


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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠒⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢛⣋⣥⣤⣾⣿⣾⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣯⡶⠾⠟⠛⠫⠽⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣴⣿⣿⡭⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣭⣟⣛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣬⣼⣾⣿⣿⣿⣿⣷⣶⣾⣯⣽⣻⢿⣿⣿⣿⠟⠋⠉⠉⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠄
⠀⠀⠀⠀⢀⣠⣢⣦⣾⣤⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣶⣿⠿⠿⠿⠿⠓⠒⠒⠂
⠀⠀⠀⢀⣾⣿⣿⠿⢿⣿⣷⣆⡠⢔⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⡿⠛⠉⠉⠉⠉
⠀⠀⠀⠺⠛⠉⠀⢀⣠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡚⠛⠓⠒⠂⠀⠀⠀⢀⡤⠴⠖⠚⠛⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠟⣿⣿⣿⣿⣿⣿⡿⠟⠛⣻⣿⣿⣿⣿⣯⡻⢷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡏⠀⠀⢀⣿⣿⠟⢿⣿⣿⣿⣷⣭⣛⠿⣤⣶⣀⢀⡀⣀⢀⡀⣀⣀⣠⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠙⣧⠀⠀⠈⠻⣿⡀⠀⠈⠉⣹⣿⠉⠛⠻⠷⠶⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠞⠋⠠⠶⠾⠟⠀⠴⠶⠾⠛⠁⠀⠀⠴⡾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠺⡧⠂⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡾⠟⢛⠻⢷⣤⡦
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠚⠁⠉⠈⡠⠛⣿
⠀⠂⠀⠈⠀⠹⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣚⠀⠀⠀⠀⠉⢳⡋
⠀⠀⠀⠀⠠⠶⠿⠗⠀⠀⠀⠀⠘⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣄⠀⠙⠶⠀⠀⠐⢀⡀
⠀⡂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⣤⠄⠀⠀⠲⠄⠠
⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⣀⢠⣄⠀⡀
⠀⠄⢰⡶⠀⠀⠀⠀⠐⠞⡉⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣿⣶⣤⡠⢄
⠀⠀⠀⠀⠀⠀⢂⡘⠃⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⠈⠉⣽⣿⣜⢳⡀
⠀⠒⠀⠀⢀⣤⠸⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡶⣿
⠀⠀⠀⠰⠋⠁⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣷⣿⡄
⠀⣟⠓⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡯⣿⠃
⠀⡳⠀⠀⡤⠌⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⡤⣤⣄⡀⠀⠀⠀⠀⢀⣸⣯⣹⣷⣿
⠀⡁⠐⠚⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣯⣥⣗⠧⡄⠈⠉⠺⡿⣆⠀⡀⢒⣻⣿⡈⠁⡟⠃⣠⡀
⠀⡆⠀⠀⠀⠠⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠿⠷⣢⢄⠀⠀⣰⡼⠀⢮⣶⣟⠁⠁⣜⡺⠺⠻⢿⡷⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⠛⢿⣿⣿⣶⣶⣔⣯⣺⣿⣷⡿⢿⣧⠾⣧⠤⠋⠀⠀⠀⠀⠁
⠀⣿⠦⠊⠷⣤⣀⠀⠀⠠⠁⠀⠀⠐⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠰⠎⢹⣿⢿⢿⡿⢿⣟⣍⣥⠠⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠀⠀⡀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡀⢀⣿⡀⢀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣷⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣠⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⣠⣇⣄⣶⣾⢛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡂⡄⢨⠇⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⣼⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣧⠀⠀⠀⡏⠉⠉⣻⣿⣿⣿⣿⡿⢿⣿⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⣦⠀⠀⣿⣷⣴⣶⣷⣦⣩⣷⣿⣿⣿⣰⣷⣾⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣲⣿⣿⣿⣿⣿⡟⣻⣿⣿⣟⣻⣿⣿⣿⣛⣻⣿⣿⣿⣌⣹⣿⣿⣿⣿⣿⣿⢿⢿⣹⣿⡍⡁⠀⠀⡀
⠀⢸⣇⠀⣶⡆⢀⣧⠀⣸⣴⣿⣿⣿⡿⢿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣻⣿⣿⣻⣛⣻⣿⣷⣿⣟⣉⣀⣼⣿⣿⣿⣿⣷⣾⣦⣤⣤⣀⡀⢰
⢀⣸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣟⣿⣿⡿⣻⣤⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣹⣿⣿⣷⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣾⣿⣿⣿⢻⣏⣿⢹⣿⣿⠈⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⡽⠿⠛⠟⠻⠛⠛⠿⠾⠟⠿⠿⠿⠟⠛⠛⠻⠛⠛⠋⠁⠀⠁⠀⠀⠭⡉⣌⠍⡙⠭⢴⣶⣼⣿⡇⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣦⣦⣤⣤⣤⣤⣤⣤⣄⣀⣉⣈⣀⣀⣠⣤⣼⣿⣿⣿⣷⣤⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡹⠉⠈⠋⠁⠀⠁⠈⠛⠀⠒⠐⠒⢠⠁⠀⠂⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣼⣷⡄⠀⢈⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠞⢿⠉⢡⠅⢄⡈⣷⡧⠤⢠⣤⣤⣀⣀⡀⠠⢐⣠⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡂⠒⠠⢸⡇⢭⣥⣌⢈⣙⣚⣛⡃⠿⠏⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣥⣴⡶⢈⣰⡶⣶⣶⣤⠤⠤⣭⣅⣈⣀⣀⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠙⠵⠤⠦⢠⣬⣭⢁⣉⣁⣀⠒⠛⠛⠛⠛⠟⠟⠛
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣄⠀⢸⠀⠀⠀⠀⠀⣿⠉⠙⠛⠛⡇⣟
⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣯⠉⠀⠀⠸⠀⠁⠀⠆⠀⢹⠂⠀⠀⠀⣿⠿⠈⠙⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢈⡿⣿⡿⣿⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢸⣷⡩⠀⣷⠙⠆⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠎⡡⠵⠀⡙⠀⢂⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⡗⠠⠈⠠⠀⢸⣦⠀⠠⠴⢾⣧⠀⠀⠃⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠘⠃⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣴⡖⠂⠂⢀⣀⣈⣡⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠃⠀⠀⠲⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠆⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠘⠁⠰⠀⠰⠄⢀⡀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⡀⣤⠀⢀⣤⠠⢆⠨⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠔⠀⣀⠠⣤⣤⢝⣠⣮⠐⣣⣰⣆⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢀⣠⣤⠾⠀⠐⠒⠃⠉⠁⣀⡀⣠⣤⡤⡶⣎⣩⡴⣀⠀⠀⠀⠐⠲⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠠⠖⠋⠉⣀⡤⡤⣴⣶⣶⢿⠿⠟⠻⠍⠃⠴⢲⣄⠁⣉⡉⢂⠲⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠⠂⠀⣀⣤⠔⠚⠋⠅⠒⠂⠉⠁⠈⠨⡷⡤⡄⠀⠐⠂⡔⢈⠭⠀⠈⣯⠀⠁⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠐⠞⢱⡆⠀⠀⠀⠂⠈⢿⣷⣀⣤⣤⢠⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠐⢉⣠⡖⢕⡊⢎⡑⠎⠰⡌⢛⠒⠋⢀⡀⠀⠀⠀⠀⠐⠀⠂⠠⠀⠀⢁⠤⠀⠈⠅⠄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡯⠖⠁⠀⠒⠉⠃⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠛⠏⠿⠉⢿⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠯⡁⠀⠁⠀⠃⠔⠄⠜⠡⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠋⠰⠠⠋⠔⠂⡰⢣⠠⣿⣿
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣧⡨⢀⠐⠃⢠⡚⡉⠻⠔⠠⢈⡖⣤⣿
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠈⡈⠠⠂⣀⠢⡤⠃⠀⠺⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣦⡀⢀⠀⠌⠨⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣿⣶⣷⣾⣿⡿⠟⠁⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡧⡃⡆⣴⠂⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣿⢀⣠⡤⠶⢖⡒⣠⣖⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣼⡇⡔⢿⣏⢸⣧⡈⠉⣱⡏⢾⣽⣻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠔⣚⣉⣥⣽⣌⢿⣷⣿⣷⣔⢿⠀⠀⠈⠛⢿⣮⡻⣮⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣛⣥⣶⣿⣯⣭⣉⡁⠬⢇⣿⣿⣿⣿⣷⡀⠀⠀⠐⢦⡉⢿⣮⣻⣟⣿⣆⢠
⠀⠀⠀⠀⠀⠀⢀⣤⣴⠦⠀⠋⢈⣩⣤⣤⣉⠉⠀⠈⢹⣿⣿⣿⣿⣿⣇⡹⣷⡀⠀⠀⠀⠙⣎⢻⣷⣻⣎⣿⡆⢷⣄⢠
⠀⠀⠀⠀⠀⠀⠈⠅⠙⡠⢀⣴⣿⡟⠻⢦⣈⠓⢦⡐⠾⡿⣿⣿⡿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠸⡎⣿⣿⢿⣾⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⣿⠟⢋⠉⠓⠂⠈⢉⡀⠈⠁⠀⣻⣿⡀⠀⠈⠻⣿⣿⣿⣷⣤⡀⠀⣿⢸⣿⢸⣷⣿⣇⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⡀⠘⠀⠀⠀⠼⠋⠀⠠⣤⣴⣿⣿⠅⠀⠀⠀⠀⠟⠛⠛⠛⠉⢀⡟⣼⣿⡇⣿⢻⣿⣿⣇⣿⡆
⠀⠀⠀⠀⠀⢀⡤⢸⣻⠷⣬⣭⣭⡄⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣅⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣿⣿⢠⣿⣿⣿⣿⢸⣿⣷
⠀⠀⠀⠀⢠⣾⣏⢸⡏⠀⡎⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣖⣯⣾⣿⣿⡏⣼⣿⣿⣿⣿⡆⠻⢹⡀
⠀⠀⠀⠀⠀⠀⣿⠈⡱⠘⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣽⣿⣿⣿⢿⣮⣿⣿⢟⣴⣿⢻⣿⢹⣿⣿⣶⣾⠁
⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣵⣾⡿⢁⣾⠇⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠃⠀⠀⠀⠀⠀⠀⢀⡴⢟⣩⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣟⣥⣴⣾⠟⢰⢇⣿⣿⣿⣿⡶⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⡴⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡭⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣌⣹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠘⣿⢿⣄⠀⠀⠀⠀⠀⠈⣰⣿⡿⠿⠛⠿⢻⣿⣿⣿⣿⣇⠀⢀⣤⣬⣿⣿⡿⢻⣿⣿⣟⡻⣶⠙⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢱⣼⣿⡀⠀⠀⠀⠀⢠⣿⣿⣶⣌⠀⠀⠢⢍⠿⣿⣿⣷⠈⢸⣿⣿⣿⣷⣶⣿⣿⠿⠿⠿⠟⠀⣿⢣⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⠀⠀⠑⠍⠙⠣⠂⠈⢉⣛⣿⣝⠷⣮⡍⠛⠳⠚⠋⣰⡟⡈⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⣿⡀⣄⠀⠀⢦⢻⣿⣿⣿⣿⣿⣧⠈⢷⡄⠀⠀⠀⠀⠸⢿⣿⣿⣷⣦⣶⣦⣠⣤⣾⡿⠹⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠹⣦⣀⠈⢷⣽⣿⣿⣿⣿⣿⣧⡀⠁⠀⠀⢠⡞⣿⣶⣮⣍⣈⣛⣻⡽⠟⣩⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⣌⡻⣿⣿⣿⣿⣿⣿⠙⣦⠀⢀⡀⠱⣌⢰⡿⣻⣿⣍⡁⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣽⣻⣿⣿⣿⣿⠏⣰⣿⣷⡀⣿⡄⣸⡆⡿⣳⣌⣛⣛⣭⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⣻⠏⠘⠻⠿⠿⡇⠿⣿⠉⠛⠃⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠈⣿⣿⣦⠀⠀⣠⡾⣿⡇⠀⠀⠀⢠⣦⣤⣄⡀⠀⠀⢀⣤⣤⣶⠀⠀⠀⣿⢫⠤⡀⠀⠀⠀⠀⡰⣬⢻⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⣠⣴⣿⡗
⠀⠀⠀⠀⠀⠛⡇⠹⣷⣷⡿⢉⡿⠁⠀⠀⠀⠀⠿⢅⣀⡙⣶⣶⣋⣀⠨⠟⠀⠀⠀⠸⣾⣦⣿⣢⣤⣴⣺⣴⣿⠞⠀⠀⠀⢻⣿⣿⣿⣦⣀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣾⠇⣤⠚⠟⢢⡄⣿⡆⠀⠀⠀⠀⢠⣿⠾⢻⡟⢿⣿⠾⣷⠄⠀⠀⠀⠀⣠⣞⣯⢋⣿⢉⣽⣷⣦⠀⠀⠀⠀⠀⣾⣿⡌⠿⠿⠟⣸⣿⣦
⠀⠀⠀⠀⠀⣻⣷⠄⠀⠀⠀⢴⣿⡁⠀⠀⠀⠀⠘⠳⠾⠋⠀⠀⠛⠶⠞⠁⠀⠀⠀⠀⠸⢿⡞⠃⠀⠈⢳⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠸⢿⡿⠃


⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢄⡀⠀⠀⠀⢀⡠⢤⣦
⠀⠀⠀⠀⢹⣷⢿⣱⣄⣄⣴⣹⢷⣿⠁⠀⠀⠀⠇⡀⡀⠀⢀⡀⠀⡀⠀⠁⠀⠀⠀⢸⣇⠈⠁⢂⣠⣠⠊⠉⢸⡿⠀⠀⠀⢀⠄⠲⢶⢄⣤⣠⡤⠄⢀
⠀⠀⠀⠀⠀⣹⣿⣏⣿⣿⣏⣿⣻⡁⠀⠀⠀⠀⠀⢀⡀⠀⢿⡧⠀⢀⠀⠀⠀⠀⠀⠀⣩⡶⠂⣾⣿⣿⠂⠲⣌⠁⠀⠀⠀⠷⣤⣤⣠⣾⣿⣦⣀⣀⣤⠗
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠁⢨⣿⣿⠇⠀⠀⠀⠀⠀⠋⠀⡀⠸⠃⠀⠀⠇⠀⠀⠀⠀⠀⠻⣦⣸⡿⠹⠻⣗⣠⡟⠁⠀⠀⠀⠀⠈⠏⣉⠝⠻⣍⡉⠝
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠉⠙⠁


⠀⠀⠀⠀⣴⡶⢤⣀⠀⠀⣀⡤⢠⢶⠀⠀⠀⣠⠤⠖⠲⣤⣀⣀⣤⠖⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⣢⣄⠀⡀⢀⣤⣆⠐⡆
⠀⠀⠀⠀⠘⣦⠈⠙⣿⣿⡛⠉⣠⡟⠀⠀⠀⠈⠑⢤⡀⠈⢹⡏⠁⣀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠘⢶⠊⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣟⣻⣿⣟⣿⣿⠆
⠀⠀⠀⠀⠀⢸⡙⡇⠛⠛⢲⢋⡿⠀⠀⠀⠀⠀⠀⠀⢻⡾⠃⠹⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠘⣿⣬⣭⢿⣭⣴⡿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣠⠀⡀
⠀⠀⠀⠀⠀⢀⡠⣴⣿⣷⣶⣄⢀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢷⣶⣧⣼⡀⢠⠆
⠀⠀⠀⠀⠔⠃⠀⠀⢸⠋⢹⡿⣿⣷⣄⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠐⣿⠀⣿⢸⡏⢹⡟⢻⣦⡾⠂
⠀⠀⠀⠀⠤⠒⠒⠤⢤⣀⠸⣇⣿⣿⣿⣷⡀⠀⢸⡀⠀⠀⠀⠀⠀⠐⠁⠰⠃⡼⢃⣠⠇⣺⡷⡿⠚⠯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣿⣿⣿⣿⣿⣷⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠿⣄⣁⣦⡀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢻⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠟⠃
⠀⠀⠀⠀⠀⢀⣤⠤⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⣴⣾⣷⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⢤⣀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⢰⣿⢯⣾⢻⣷⣤⣄⣀
⠀⠀⠀⠀⠈⢓⡦⣴⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⣷⣿⣖⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗⣤⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠶⠴⠾⠏⢈⡍⠁⠈⣭⢁⣗⠺⠦⢆⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠮⠟⠀⢀⠀⢴⠀⣘⣋⡾⠐⠀⠀⠉⣀⣀⢀⣌⠓⠲⠦⢤⡀
⠀⠀⠀⠀⡤⢚⣡⡟⠋⠛⢠⠀⠀⠑⠛⠀⡠⠁⠀⢠⢤⡠⠞⠃⢀⣌⠈⠋⠈⢆⣀⡀⠙⢷⡢⡀
⠀⠀⢀⡎⣰⠟⠋⠁⠀⢈⡼⠃⡀⠘⠂⢂⣺⢆⢤⠦⠀⠀⢀⣤⠀⠉⠀⢠⣄⠀⣈⠂⠘⠈⢻⣎⣦
⠀⠀⠸⡇⣿⣄⠺⠤⣀⠬⠀⢈⣉⣀⡥⠈⠉⠛⠀⢆⣴⣀⠨⠈⠀⠀⠒⠋⠀⠀⠀⣬⠀⠀⢰⡟⢸
⠀⠀⠀⣇⠬⡻⢦⣤⠸⠍⠀⠈⣀⠐⠆⠀⠀⠁⢒⠙⣦⡴⠐⢒⠀⡤⠀⠀⠀⢰⠤⢤⣤⡴⢋⡠⣿
⠀⠀⠀⢹⡀⠈⠲⢌⡛⠶⣦⣬⣑⣀⡂⠐⠶⠟⢆⠟⠏⡛⠮⠙⢃⣄⣚⣹⡦⢶⣚⣯⠵⠚⠁⢠⡟
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⠀⠒⠺⠿⠭⠽⠽⣿⣿⣿⣿⣿⣿⠭⠭⠿⠿⠒⠛⠉⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⣄⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣠⠤⣶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠉⠉⠉⠉⠉⠉⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠚⠋⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠟⠋⠘⠀⠠⠆⠀⣰⣿⣿⣿⠃⠀⠀⠀⢀⣤⣶⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⡿⠁⠰⠤⠀⠂⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⢀⣤⡾⠛⠉⣀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⡤⠙⠢⢠⣀⢀⡀⢀⡀⠙⠻⣿⣷⠀⢸⢋⣖⣿⡟⠿⣿⣿⣶⣦⣄⡀⠀⠀⠐⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠏⢀⣤⣶⠿⠿⠿⠋⠋⠀⠀⠀⢹⣿⣷⠘⣿⣿⡟⠈⠈⠁⠘⢻⣿⣿⣿⠶⠄⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠅⠰⠋⠁⠀⡀⠔⡻⣠⣄⣀⣴⣠⢌⣹⣿⣯⠼⡟⠁⠀⠀⠄⠐⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⠀⠀⠂⠐⠀⡐⠐⠙⠿⣿⣿⣿⣿⣿⣿⠀⠂⠀⠀⠀⠂⠀⠀⢂⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠃⠆⠀⠀⠀⢈⣿⣿⣿⣿⠏⠄⠁⠀⢂⣀⠀⢀⣀⠘⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⡙⠄
⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⠋⣁⢀⢀⠰⠶⢘⠁⠈⢀⣿⣿⣄⡀⠀⠀⠀⡃⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⡠⠘⠉⠈⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠀⠰⠱⠸⠡⡅⠻⠇⠈⠀⠀⠀⠀⠀⠀⢿⣿⡦
⠀⠀⠀⠠⠁⣴⠇⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠰⣿⢲⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣾⣿⣷
⠀⠀⠀⠘⣥⣭⣀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠷⠀⠀⠸⢿⡿⠈
⠀⠀⣤⠀⠁⡶⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠨⡇⠀⠀⠀⣼⢿⡂
⠀⠠⣿⡆⠀⣫⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⢀⠀⢰⣽⣯⡁
⢀⣀⣸⣿⡀⣍⢡⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣭⠀⣄⢇⠈⣿⣿⡄
⠘⣏⣹⢿⢝⠂⠀⢀⡘⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠟⢧⡀⢿⣍⠻
⠀⠈⠉⠅⢘⡀⠀⠙⡿⢸⣥⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⡀⡁⢸⡋
⠀⠀⢻⣦⡈⢡⢀⡠⠶⠾⠋⠀⠀⠠⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠾⠋⠀⠀⠀⠀⢠⠶⠀⣽⣷⣧⣟⣯
⠀⣴⠎⢟⡻⣶⣦⡀⠀⠀⠀⠀⢀⣤⢠⣄⣄⡄⠸⢻⣿⣶⣶⣦⣶⠀⢠⣄⡀⢠⣤⣴⣤⣷⣿⠿⠊⠁⠀⠀⠀⠀⠀⠀⢘⢃⣼⣵⣿⡿⣿⠃
⢀⡝⣯⣈⠉⣾⣝⡹⢶⣦⠀⡰⠊⢠⣤⣄⢹⡇⡁⠀⠀⠘⠛⠻⠿⠀⠀⠙⠛⠛⠃⠙⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⣼⣟⣿⣿⣷⠿⣷⡀
⠘⠴⠒⢉⣠⠤⣥⡄⠔⠁⠴⡀⠀⠈⠁⠀⢨⡃⠀⠀⠀⢀⣠⣴⣿⣿⣶⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠼⡉⠣⠽⠿⢿⣷⠄⠈⠁
⠀⠀⠀⣘⠿⠚⠁⠀⢈⡆⠁⠉⡓⢲⣴⡐⣂⡀⠀⠀⢰⡾⠿⢋⣿⣵⣿⡿⣿⣿⡷⣶⠖⠻⡟⣋⠟⣵⡿⢠⠖⠀⣜⠶⣀⣛⠓⡃⠸⣿⠄
⢠⣟⡡⢫⡔⣺⢃⡔⢻⠁⠛⠛⠟⠛⠻⢋⣽⢒⡞⠀⠀⠂⠀⠾⠛⠁⠉⠐⠛⠉⠀⠀⠀⠀⣭⠛⡇⠙⠓⠋⠀⠏⠉⠴⠀⡍⠀⠂⠀⠘⠇
⠀⠉⠀⠘⠚⠁⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠿⠛

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⡐⣦⣦⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⢈⣩⣍⠉⠡⡷⣐⠂⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣦⣿⣵⣿⣿⣿⣿⣿⣿⣤⡈⠀⠂⡅⢀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠎⠉⠉⢽⡟⢼⡿⣫⣿⣿⠛⡀⢐⠊⠄⠁⡳
⠀⠠⢴⣶⠶⠞⠉⠀⠀⠀⣤⣀⣀⡀⢈⠠⡄⢈⣡⣾⠏⠏⠟⠑⠀⠁⠘⠢⠄
⠀⠀⠀⠀⠀⠀⠠⣼⣤⣤⠶⠍⠻⠿⡟⢻⡈⡟⠟⡇⠈⡀
⠀⠀⠀⢠⣴⣦⣤⢮⣿⣟⣃⢀⣱⣦⣄⣽⡅⠀⠀⢳⠀⡀
⠀⠀⣀⣙⡯⢿⠽⣟⣿⠻⠳⠘⣧⠻⣎⢇⠱⡄⠀⠈⢧⠰⡀
⠀⠀⢛⣿⠇⣌⡻⠙⣝⠣⣥⡀⠈⢳⡜⠓⠃⠁⢀⣠⡄⠑⠥⡄
⠀⠀⠨⠻⠘⠏⠳⡄⠈⠀⣀⡙⣂⣥⣠⣾⣟⣤⣈⣯⣶⣶⣾⣶⢦⣀
⠀⠐⣄⣻⡾⠦⣺⡾⣷⣶⠃⠤⢄⠉⠻⣿⣿⢿⣿⣿⣷⣾⣾⣿⣷⡦⢅⡲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠻⣀
⠀⠊⠈⢸⡗⣾⠻⣿⠛⠇⡁⢐⣲⣿⣶⠿⣯⠀⠻⣿⣿⡷⣽⣿⣾⣭⣛⠛⠛⠒⠾⠽⢷⣶⡶⠦⠶⠖⣚⣫⠿⠋⠁
⠀⠀⠀⠘⡄⠹⣄⡆⠀⣾⠵⡊⠿⠳⡹⡄⠘⢧⡀⠙⠻⣶⣤⣈⠙⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⠿⠟⠛⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠧⠐⠻⣿⣷⣿⣥⣅⢰⡀⠙⢿⣄⡀⠙⢶⣤⣀⠉⠙⢻⣿⣶⣶⣶⣶⣾⣿⣭⣤⠄⠠⠀⣀⣀⡠⠄⠀⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣁⣀⣀⣈⣛⣶⣦⣽⣛⣛⣁⣀⣀⣈⣉⣉⣉⣁⣀⣀⣀⣒⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠓⠚⢙⣓⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣸⣇⠀⠀⠀⠀⣸⣀⣽⠿⠩⠭⠩⣄⠀⣆⠀⠀⠀⠀⣸⣏⣀⣌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢿⣭⣭⣬⡿⠆⡀⠀⠀⠓⣮⠻⠤⠤⠤⠤⠃⢴⠚⠀⠀⢀⠰⢿⣥⣭⣬⡿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣞⣼⣿⠉⠉⠉⣿⣶⣳⠀⠀⠾⣶⠛⡆⠀⠀⢐⠐⣾⠷⠀⠀⣞⣶⣿⠉⠉⠉⣿⣷⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣹⢬⣉⣉⣉⡥⣽⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣫⢬⣉⣉⣉⡥⣏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠘⢀⠤⠀⠃⣺⢹⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡏⡧⠘⠀⠤⡀⠃⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢲⠀⠀⢸⠰⢠⡠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⡇⡏⠀⠀⡖⡇⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠘⠚⠶⡆⠺⠘⠀⠀⠀⠐⠐⠂⢉⡙⠚⠃⠂⠀⠀⠀⠃⠿⢰⠴⠗⠃⠆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠⣀⡨⡄⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠐⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⣤⠀⡇⢠⡄⢸⠀⣦⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠉⣷⠐⠃⣶⢸⣌⣩⡇⣿⠀⡇⠀⠀⡼⠀⣧⢸⣍⣩⡇⣶⠘⠀⣶⠉⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣒⠁⠀⣻⢰⢡⢸⠄⣿⠀⡇⢘⠁⢝⠀⡿⠰⡧⣔⠇⣿⠀⠀⠖⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢠⠼⠟⠠⣆⣈⢘⣨⡘⠷⣿⠒⠇⠈⠁⠻⠆⠿⢰⣅⣧⡁⣀⣰⠄⠠⠦⣇⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⣏⠀⢠⡃⠀⣉⡉⢨⣴⣦⠀⡿⠂⢀⠔⠂⠀⠂⠀⢈⣤⣦⠁⢻⣙⠀⠘⠄⣉⣹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢸⣦⡀⡀⠀⢠⡌⢡⡄⣷⠀⡈⢉⠀⢁⠀⣀⢠⡭⢥⡄⣀⠀⢐⣦⡄⡀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠉⣭⠉⠃⣀⢨⠔⠢⡄⣿⠀⡇⠈⠁⣹⠀⣿⢠⠔⠢⡅⣿⠘⠉⣭⠉⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢶⠀⠀⣸⢸⣶⣶⡆⢿⠀⡇⠰⠆⢛⠀⣿⢰⣶⣶⡇⣟⠀⠀⡖⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣰⠼⠾⠤⡄⣾⣸⣿⣿⣇⣻⢠⠧⠰⠄⠝⣄⣿⢸⣿⣿⣇⣷⢠⠴⠷⠧⣆⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠍⠿⠽⠠⠄⠠⠏⠯⠩⠿⠿⠭⠽⠩⠤⠀⠄⠠⠭⠏⠩⠿⠿⠍⠽⠹⠤⠤⠠⠯⠿⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠄⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⢤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢛⠻⢿⣏⣿⡿⣿⣶⣝⢿⣿⣿⣿⣷⡔⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣶⣤⣤⣀⠁⠈⠙⢿⣿⣛⣿⣮⢻⣿⣿⣿⣿⣮⡳⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⣿⣙⣿⣇⣿⣿⣿⣿⣿⣷⡹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⠀⠘⢿⣯⣿⣾⣿⠏⢻⢿⣿⣷⢹⣝⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣟⣿⡏⢾⡷⠞⢛⣿⡏⣯⣷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢧⡼⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣽⣷⣎⠙⣉⠾⣿⣷⢻⣿⣇⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣨⡀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⡿⠬⠿⣿⡺⠿⣿⣟⣿⣿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡌⠁⢠⣤⣶⣶⠒⠀⠉⠛⠻⣿⣿⣿⣿⣿⣧⠀⢻⣶⣶⣤⡁⢰⣦⣿⣿⣿⢧⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⡄⠀⣿⠿⣿⣿⡿⠓⠀⠀⠘⢿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣇⢸⣿⣿⣿⣿⣿⣿⣿⢻⣽⣿⡿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢋⣾⣻⣼⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠠⣿⣿⣿⣿⡿⠋⠀⠀⢫⣾⣿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⢲⡶⣘⣿⡉⠁⠀⠀⠀⠀⣾⣿⢏⢼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⡟⣳⣦⣥⣶⡟⣷⠀⠀⠀⠀⢠⣿⡵⣫⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⡀⠠⢤⣶⣿⣿⠟⠀⢻⣎⣻⣿⣛⣾⠃⠀⠀⠀⠀⠀⣿⣾⢟⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⠙⢿⣾⠿⠛⡄⠀⠀⠀⠀⠀⣟⡵⣫⣸⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣾⣦⡀⠀⠀⠀⢯⣾⢣⣿⡿⣿⣿⣿⠇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⢴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⡤⢞⡯⠋⢀⣿⣦⡀⠀⡿⣡⡿⢿⣾⣿⣿⡟⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢾⣿⠋⢽⢧⡀⠀⣀⣀⠤⢖⣪⠵⠚⠉⢀⣴⣿⣿⢻⣿⣦⠘⣫⣶⣿⣿⣿⢟⡾⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠹⣿⡿⠁⠀⢈⣳⡬⡥⠴⠒⠊⠉⠀⠀⣀⣴⣿⣟⣻⣿⣿⣏⠿⣿⣿⣿⣿⣿⡷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣽⢻⡿⠇⠀⢀⣾⣿⣟⠀⠀⠀⠀⠀⣠⣾⣿⢛⣿⠟⣿⠿⢋⡤⠚⣡⣾⡿⣫⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⡆⠀⠀⠛⠟⠃⠀⠀⣠⣴⠿⣿⣬⣿⣿⡿⢓⡡⠞⣡⣴⣿⣿⣿⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⣿⣿⣦⣤⣄⣠⣴⣾⣽⣿⢿⣿⣿⡽⠟⣩⠴⢋⣴⣾⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠙⠧⣁⠛⠼⠯⢹⣯⡩⠯⠹⠟⢒⣉⣭⠴⢚⣋⣤⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠛⠛⠙⠒⠛⠋⠉⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⣴⡶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⣲⣶⣄⠀⠀⠀⢀⢤⠄⠶⠀⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⢀⣉⡉⢉⡉⠉⢢⠀⠀⡁⠀⠀⠀⠀⠀⢠⡴⣏⣤⣾⣿⢟⠋⠹⠃⠀⠀⣧⣿⣶⡾⠛⠛⣋⡙⢻⣆⠀⠀⠀⠀⠀⣰⡷⠺⣿⣯⡭⣿⣆
⠀⠀⠀⢸⣿⢶⢻⣽⠀⢸⣷⣾⡇⠀⠀⠀⢀⣴⣛⢠⣶⣿⡟⠁⠰⠈⠀⠀⠀⠀⠈⠉⠙⢀⢰⣿⣻⣿⠀⠃⠀⠀⠀⢀⣾⠏⠀⢂⢬⣿⣷⡄⠾⣧
⠀⠀⢠⠐⣯⣿⣷⣿⡆⢼⣿⣿⠇⠀⠀⢠⣾⣓⣮⣿⣽⢟⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠗⠉⠀⠀⡀⠀⠀⢠⣿⠟⠰⠤⠜⡷⠬⣿⣷⢶⡿⠃
⠀⠀⠈⠢⠀⠀⠈⠉⠠⠞⠋⠁⠀⠀⠀⠉⠲⠾⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠁⠠⠒⠃⠀⠀⠈⠛⠒⠒⠂⠀⠀⠀⠼⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢠⠄
⠀⠀⠀⠀⠀⢀⣠⡶⠞⠓⠀⡀⠀⠀⠀⠀⠀⣴⡚⢫⡭⣭⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⣧⣅⣼⠦⡀⠀⠀⠀⠀⣀⢠⠒⢠⠀⠀⢶⣶⢂⡄
⠀⠀⠀⢰⠋⠀⠀⠑⢈⡧⢤⣴⣪⠄⠀⠀⠀⣟⢷⡈⠙⠛⠁⠠⠄⢳⠀⠀⠀⠀⠀⠜⠁⣀⠄⠀⠈⠹⣠⣧⠀⠀⠀⠀⡏⠨⠁⠀⠀⠀⠲⣿⡿⡇
⠀⠀⠀⣅⠡⠶⠒⠀⠘⢃⠘⢁⢼⠂⠀⠀⠀⢃⠈⠿⢶⣆⣴⡴⡶⢤⡆⠀⠀⠀⢀⠀⣸⡭⡳⣆⠀⢸⡏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡗
⠀⠀⠀⢶⡒⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⣈⡉⠈⠃⠀⣸⠇⠀⠀⠀⠘⠀⢿⣿⣎⢯⡇⠈⣷⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣷⡿⠃
⠀⠀⠀⠘⠓⠒⠒⠚⠃⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠓⠛⠃⠁⠀⠀⠀⠀⠀⠦⡀⠉⠉⣩⢅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣄⠂⢀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠑⢂⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠰⣾⠿⠁⡆
⠀⠀⣀⢈⡉⠀⠈⠱⢿⣄⣤⡷⠗⠋⠍⠀⠀⢀⠀⢠⡄⠴⣖⡫⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣶⡞⠁⠀⠀⠀⠀⠀⣾⣁⠎⣁⣹⣥⣿⣾⡇
⠀⠀⠑⠫⠤⠆⠐⠟⣽⠿⠟⠋⠀⠀⠀⠀⠀⠈⠀⠀⠉⠇⠸⠟⠁⠀⡐⠀⠀⠈⠁⠀⠤⠠⠄⣰⣇⢸⡗⠈⠀⠐⠀⠀⠀⡿⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⠃⠐⠈⠀⠀⠀⠀⣇⢸⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣠⠔⠄⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠰⠀⠒⠰⠆⠀⠀⠀⠀⠀⠀⡀⠐⡠⠒
⠀⠀⠀⠀⢠⠀⠄⠄⠀⠄⢰⡆⠆⠀⠀⠀⠀⢰⠤⠀⠒⠐⠒⣀⣈⣀⣀⠀⠀⠀⠀⠀⣤⠈⠀⠀⡀⣀⣤⠽⠞⠛⠀⠀⢰⠁⠀⠑⠀⣄⡀⠀⢀
⠀⠀⠀⠀⢸⣄⡀⣀⣀⣰⠀⣿⣤⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⡜⢁⡠⠴⠞⠋⠁⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢉⡐⢀⠀⢰
⠀⠀⠀⠀⢀⠁⠀⣿⣿⣿⠀⣿⡿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠘⠳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⠐⠁⠐⣈⣴⡿
⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠰⠋⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠼⠿⠿⠟⠃⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠄⠀⠀⠠⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣶⣿⣿⣡⣴⣾⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠛⠋⢉⡙⢀⡉⠋⡙⠿⢾⣿⣿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⣿⡿⠛⡁⢀⠀⠘⢀⡌⢀⣆⣀⣆⢡⡠⠄⠛⢿⣦⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠰⠀⢠⡤⠛⠉⠉⠀⠀⠀⠈⠉⠳⢆⣘⠈⡙⣿⣯⣤⣭⣀⠀⠀⠀⠀⣠⣶⢀⣾⣿⡟⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠡⠀⠂⠀⣳⠋⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣰⠙⠿⣿⣙⣃⣀⡀⢠⣿⣿⣿⣿⣿⣿⣿⡟⣫⡄
⠀⠀⠀⠀⠀⢰⣾⡟⢡⠀⢂⠈⢱⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣬⣙⣛⣿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠠⢸⣿⠀⡀⠒⢀⠀⡇⠓⣀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠻⣿⣿⡷⠶⠈
⠀⠀⠀⠀⠂⠈⣿⠠⠈⣐⠄⡀⢏⡘⢏⣱⡐⡀
⠀⠀⠀⠀⠀⠀⣿⣆⠂⡴⢁⡑⠂⠡⠂⢷⣳⠈⠂⢄
⠀⠀⠀⠰⠀⡀⠹⣟⣧⡀⠂⣴⢈⠒⢉⢺⡝⣅⠐⠤⣄⠃⠈
⠀⠀⠀⠘⠄⠘⢂⢿⣿⣷⡌⡙⣧⢌⡁⢼⠻⠟⢦⢀⣀⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⡠
⠀⠀⠀⠡⣾⣽⡟⠀⢷⣹⣷⠀⡸⣆⢐⡇⠀⠰⣄⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠉⠄⠀⠐⠁
⠀⠀⠀⢠⣿⣟⠀⠀⢸⢿⣿⡁⠄⡏⣛⠁⠀⠀⠀⠀⠀⠀⠀⠸⣤⢤⣤⡀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⠀⠰⠐⠆⠄
⠀⠀⢠⢸⡏⣄⢗⡀⣼⣴⣿⠀⣸⢃⡍⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠿⠿⡿⠆⠀⠀⠀⠀⠀⣊⡆⠀⠸⠄⢀⠈⠀⠃
⡀⠀⠠⠊⢳⠶⠛⣼⢻⣿⢇⢣⣯⠎⠀⠀⠀⠀⠀⠀⠀⠐⠤⣥⣦⣤⣄⣠⣀⡠⠄⠰⡊⣅⠥⠀⢀⣠⡦⢨⠤⠞⠀⠀⡀
⠃⠀⢰⠎⠈⢈⡼⡉⢷⣿⢈⡾⠁⠀⠀⠀⠀⠀⠀⣈⠓⣶⣶⣰⣶⣟⣿⣿⣧⣤⡀⠀⠀⠈⡄⠒⢠⠁⠤⠀⠤⠐⠀⠀⠀⠀⠁⣀⢀⠠
⠀⠀⠈⠀⠀⡘⢠⡀⢺⣿⡾⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣭⣿⣿⡟⣿⣿⢿⣿⣿⣯⣔⡀⣐⣉⠉⠧⡀⠰⢀⣦⠖⠐⡄
⠀⠀⠂⠀⢠⡋⠦⣉⡝⢻⣷⠆⠀⠀⠀⠀⠀⢰⣿⣿⡿⣭⣼⡯⢱⡿⣱⡟⣰⢟⣿⣿⣽⠿⠟⠉⢆⠙⠤⠨⠇⠀⢀⠧
⠀⠀⠂⠘⢨⣁⠒⠢⠪⠌⣿⣦⡇⠀⠀⠀⠀⢊⣽⠯⠤⢧⢤⣷⣿⣿⣿⡴⣿⣾⣿⣿⡁⠀⠀⠀⠈⠹⢠⡄⠴⠄⠈
⠀⠀⠀⠐⠀⡅⢎⡑⠂⠂⢸⣿⡇⣄⠀⠀⠀⣿⣿⠒⣹⢨⡏⠞⠉⢧⠀⠴⠋⣫⠟⣻⠇
⠀⠀⠀⠀⠀⠈⠡⠠⠦⠌⢀⠹⣷⣿⡆⠀⠀⠋⣿⣂⠤⢋⢷⡔⠒⠉⢒⣆⣘⠁⠀⠁
⠀⠀⠀⠀⠁⡄⣀⣎⣅⣱⠌⡄⠹⣿⣇⢳⣄⠀⢿⣿⣌⠏⣭⢹⡳⢾⡁⢀⡩⠧⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠀⣧⠘⡤⠉⢙⠲⢧⡞⢿⣿⣿⣀⡈⠻⠹⣦⡔⢰⠂⣣⡹⢏⠀⣠⠞⣁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡
⠀⠀⠀⠀⠀⠀⠹⡗⠵⢎⠼⠀⠢⠈⡇⠙⢿⣾⣿⣄⢠⠿⣿⣯⠨⢅⠧⣼⠋⠀⣀⣤⠟⣀⣀⣠⡴⢶⠌⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢫⢐⣁⠴⡖⠉⠶⠀⡍⢡⠘⡹⠿⢷⣶⡿⢿⣂⡓⠆⡼⠛⠉⠉⠹⣮⠉⠩⣑⡰⠛⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡬⡽⡍⠳⢄⣥⣈⣧⠔⢆⠴⠀⢠⡁⠢⠠⣅⡔⠓⠒⠒⠒⡟⢛⡌⠖⠋⠀⠀⠄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢑⣏⡟⠒⠆⠛⠺⢈⡛⠚⡶⠼⣤⠬⠷⣞⠉⠙⠲⣦⡤⠼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⢠⠀⢀⡛⠒⠳⡄⡸⣆⣠⡼⠷⢤⠞⠁⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⢊⡴⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣈⣙⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠉⢩⣏⠉⠉⢹⠿⠉⠉⢩⡉⠉⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣭⡭⣭⣭⣭⣝⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠳⣤⠀⠀⠀⠘⡾⠹⠿⠿⠇⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠂⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠈⡃⢤⠀⠀⠀⠀⢒⠴⣝⠄⡔
⠀⠀⠀⠀⠀⠈⢻⡟⢳⣶⣭⣙⠛⠛⠛⠹⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠈⣩⣿⠛⠷⣶⡒⣶⣿⡿⣅⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣀⣛⣇⣀⣀⠀⢀⣀⠘⣀⣻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠈⠳⠀⠈⠛⠞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠛⠛⠉⠛⠒⠂⠀⠈⠙⢿⡿⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠰⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⠅⠀⣫⣠⣿⣱⣄⣠⣤⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠸⠖⢋⡥⠂⠀⠊⠋⡋⠉⠩⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⡔⠶⠿⣹⣓⣖⣖⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠡⢄
⠀⠀⠀⠀⢤⣤⣤⣤⡎⠱⠴⠶⠴⣤⡦⣴⠦⡶⣴⣦⣤⡤⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡎⠉⠚⠒⠖⢿⢗
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠍⡋⠉⢹⠀⠋⠈⡏⠁⠸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠐⠒⠂⠁
⠀⠀⠀⠀⣤⡶⢿⡚⣛⠻⣦⡙⢿⡻⠿⠿⢿⣿⣿⡿⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠠⣤⢒⣡⣠⢀⠘⣄⢮⡓⢤⣄⣠
⠀⠀⠀⠈⠁⠀⠀⠋⠟⢶⡀⠱⣜⠡⣾⣿⣿⠟⠉⢉⠀⠀⡀⠄⠀⠀⠀⠀⢹⣄⠀⠈⠿⢐⡉⢹⠉⣉⡛⢉⡀⢀⠟⠋⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡅⠀⢈⣧⣬⡍⣴⠚⠁⠀⢠⠤⠀⠀⠀⠀⠀⠀⠈⢿⡓⠲⢶⣶⣥⣀⡤⢶⣦⣭⣉⠈⠁⢘⠆
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠈⠀⠀⠀⠈⠁⠠⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢯⠉⠓⠀⠙⠃⠙⠛⠓⠲⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠤⢤
⠀⠤⠒⠀⡁⠐⣀⣠⠄
⠀⠀⠀⠀⠀⠐⠀⠣⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⡀
⠀⣀⣀⣤⣄⣀⢀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣷⣄⡀⠀⡀
⠀⠀⠙⠟⠻⠟⣛⡀⠘⠛⠉⠉⠁⠀⠀⣿⣿⣿⣭⣱⡆⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠙⡋⠃⠹⡀⠀⠛
⠀⠀⠰⠶⢰⣿⢋⣹⢀⠂⠀⠀⠀⠀⠀⠘⠿⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠠⠞⠉⣿⠉⣿⣿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠇⠘⣚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣹⣂⣴⣀⣠⣠⣠⡠⠄
⠀⠀⠀⠀⠒⢸⠏⠉⢿⣿⠀⠒⠒⠒⠒⠒⠛⠛⠛⠒⠒⠛⠛⠛⣿⠃⠙⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠉⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠈⠙⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣤⣤⡶⠶⠶⠶⣤⣴⣦⡾⢿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠞⠋⣠⡙⠒⣠⣄⠐⣃⠀⢀⠀⠰⢦⣀⣉⡁⠙⠛⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡿⢈⡄⠉⣥⠏⠉⣉⠁⠤⠀⠉⠁⠛⠙⠠⢶⠀⠈⠃⠴⠰⠶⣤⣀⢭⡛⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢛⡙⡫⠶⡞⢃⣁⣴⢲⡀⢉⡀⠀⣛⡛⠂⠘⢂⡀⢀⡤⠤⠴⢦⡀⠤⠌⣉⠠⠅⣤⠨⡝⣿⣄
⠀⠀⠀⣤⣤⣤⣤⣿⣯⣥⣭⣤⣥⣬⣥⣤⣤⣤⣬⣦⣾⡦⠶⠾⠷⣶⣴⣾⠶⢶⠶⠶⠶⣾⣶⣶⣦⣥⣤⣤⣤⣤⣤⣤⣽⣷⣤⣤⡄
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣤⣾⣛⣦⠃⠀⠀⠀⠀⢸⡇⢱⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⡇⢸⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣜⣿⡟⣁⡀⢘⣿⣿⡆⠀⠀⠀⠀⠃⠘⡇⠀⠀⠀⡘⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣄⡚⠁⢜⡻⣿⣿⡀⠀⠀⠀⠀⠀⣧⠀⠀⢀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⣿⠀⠲⠍⣼⣿⣷⠀⠀⠀⠀⠀⢸⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⡿⠃⠀⠀⠀⠀⠸⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⣿⣿⣿⣿⣿⡿⠟⡇⠀⠀⠀⠀⠀⠀⠀⢧⡈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠇⠀⠙⢿⣿⣿⣿⠋⠀⠀⢹⣤⣤⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⢸⣿⠀⢄⠀⠀⠉⠁⣿⠀⠀⣄⠀⣻⡿⠏⠀⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⢸⡟⠀⠀⡍⢠⡀⠀⣿⠀⠀⠛⠀⢯⠀⠀⠀⠀⠀⢰⠁⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡅⠀⣿⠀⠀⠀⠀⠘⢶⣶⣦⠀⠀⡞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣿⠁⠀⢸⡇⢸⡇⠀⣿⡄⠀⠀⢴⡀⠸⠿⠋⠀⢰⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠈⠁⢻⡀⠀⢀⡆⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠃⠸⠃⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣈⣁⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢀⣵⣿⣿⣿⣿⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⠻⢿⣿⠟⠁⢀⡄⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⣤⡄⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⡇⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠒⠒⠀⠀⠀⠀⢠⣴⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⡿⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢻⢏⡏⠀⢀⣀⣠⣤⠾⣷⣶⣿⣤⣤⡄⠀⠀⠀⠀⢸⣷⡹⣎⡳⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡽⣁⡿⣸⣇⠀⢀⣿⣻⣿⣿⠿⠿⢿⣿⣿⣷⡦⠀⠀⠀⢸⣿⢿⣹⣷⡝⣖⢟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡫⡼⣱⢹⢳⢸⣿⡰⡿⡛⢹⡿⠋⠀⠀⠀⠑⢻⣿⣷⣄⠀⠀⢸⢿⢸⣇⢷⢻⣞⣎⢿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⡴⣱⣳⢣⡏⣾⢸⡞⣿⡷⡘⠀⢻⣅⣤⣀⠀⠀⡐⣹⣿⡏⠀⠀⢠⠇⣿⢸⣿⡸⡏⣿⡞⣎⢿⣽⣆
⠀⠀⠀⠀⠀⠀⣼⣱⣇⡏⣾⡇⣿⢸⣧⢻⣯⢉⠳⡦⣿⣿⣋⣤⠮⣶⣿⣿⢀⣠⡴⣣⡆⣿⢸⣿⡇⣧⢉⣍⢹⡜⣿⣟⣆
⠀⠀⠀⠀⠀⣸⢣⣿⣸⠃⣋⠃⣿⠘⣛⣃⢻⣎⢳⡘⣿⣿⣿⠓⢒⣿⣿⣿⣿⠋⢾⣿⢹⡏⣨⣤⣤⣹⡟⠙⢷⣷⢉⣉⠘⣆
⠀⠀⠀⠀⢰⡏⠾⠇⣿⣾⠛⣷⣿⡾⠟⠛⠻⢿⣆⢻⣿⣿⣿⠥⠀⢺⣿⡿⠉⠻⣧⠛⣼⡿⠁⠀⠀⠹⡇⠀⠈⢿⠋⠙⠻⣾⡄
⠀⠀⠀⠀⣾⣰⡿⢶⣿⠃⠀⢸⡏⠀⠀⠀⠀⠀⠹⡆⣿⡟⠘⣏⡩⠼⣿⣷⣿⣦⠙⣇⢿⠇⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠘⣷
⠀⠀⠀⢰⣿⠟⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢷⡟⢸⣿⣿⣦⠒⢉⣿⣿⣿⣦⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⡆
⠀⠀⠀⢸⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⣯⣛⣿⡻⣅⠔⢻⡿⣿⡇⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⠛⠙⣯⠋⠀⠑⣏⡜⢻⣁⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⣾⡿⠟⠻⡥⢿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⡯⢼⠂⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⡾⢍⠟⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠑⠘⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⣿⣿⣿⣻⣟⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣭⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣿⣿⣿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠿⠿⢿⠿⠿⢿⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣗⣺⣿⣭⡟⣯⣽⣿⣒⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡭⣭⣽⣿⣿⣫⣿⣯⢻⣿⣻⣭⡭⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣓⣿⣿⣿⠷⢿⣷⣿⠿⢿⣿⣿⣷⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣽⡇⣿⣿⣷⣶⢿⣿⣿⢷⣶⣏⣿⣏⣿⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢾⡿⣿⢟⣿⣿⣿⣿⣿⣟⢿⡿⣿⠿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⡧⡿⣇⣿⣶⣯⣿⣿⣿⣯⣿⣾⣧⡿⠇⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣻⣿⣿⣷⡏⣷⣶⣷⣷⣷⡏⣿⣿⣿⣿⢻⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠿⠟⠟⠿⣿⣯⢿⣿⣶⣶⣿⢯⣿⣿⠿⠿⠺⠿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣽⣿⣭⣿⣼⣿⣿⣿⣿⣿⣿⣼⣯⣿⣿⣽⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠷⢶⣶⣶⣶⣯⢟⣿⣿⣿⣿⣿⣿⣿⣟⡯⣷⣶⣶⣶⡶⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣽⣿⣽⣿⣿⣿⣯⣯⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢼⣷⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⢀⢔⣿⣿⡗⣀⠀⠀⣼⣿⣿⣿⣿⣿⡿⣟⣛⢿⣿⣿⣿⣿⢞⣛⡻⣿⣿⣿⣿⣿⣿⡄⠀⢀⡔⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣽⣿⣸⣿⣿⣿⣟⣸⣿⣵⣿⣿⣿⣿⣿⣿⡁⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⡀⡿⠿⣿⡿⠀⠀⠐⠷⠶⠿⠟⣿⣿⡿⠀⠁⢿⣿⣿⣿⣿⡇⠉⠘⣿⣿⡿⠻⠷⠶⠷⠀⠀⠸⠿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⡧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣯⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣆⡀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⣤⣶⣏⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⣿⣿⣿⢿⣄⠀⠀⠀⣠⡿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⡟⣧⡆⣴⡏⡷⣿⡿⣿⡏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡽⢿⣿⣿⡟⣷⢿⣸⣿⣿⣿⢟⡴⣿⠁⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣙⡾⠿⠿⡇⢹⣷⣿⣿⡭⣿⡟⢰⣟⣠⡴⠞⠋⠉⠀⠀⠉⠛⠛⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⣛⣷⣦⣙⡌⢿⡇⠀⣸⣿⠶⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠠⠄⠀⠀⠉⠳⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣙⣻⣿⣿⣿⠛⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣀⣌⣩⠿⠛⠁⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠀⠀⠀⠐⠂⣉⡿⠿⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠩⡴⠞⠁⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢀⡀⠀⠠⠀⠀⣀⡴⠖⠛⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⢳⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⢀⠐⠂⠀⠀⠐⠂⠀⠁⠀⠀⣠⣴⣛⠁⠀⠀⠀⠀⣀⡴⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣤⣄⣀⣀⣀⠀⢀⣀⣠⡤⠴⠶⠶⠶⠖⠒⠛⠋⠉⠉⠀⠀⢀⣠⠶⢋⡉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠛⠋⠐⠚⠉⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⠛⠶⠦⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠶⠶⠦⠤⠤⡤⣄⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠾⠿⠴⠶⠒⠒⠒⠶⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠶⠶⢤⣤⣤⡤⠴⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡞⠉⠀⠀⠀⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢚⡟⢁⣀⣤⠤⠀⠄⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣿⣿⣗⣹⠋⣁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣮⢹⣾⡟⢡⠎⠀⠀⠀⢢⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢡⠀⡟⢟⠺⣧⠋⢀⠀⠀⠀⢠⣇⣼⣀⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⠜⣸⣵⣯⣵⡏⠀⠀⠀⠀⠈⣾⣿⣽⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣸⡏⠃⢻⣋⠹⣄⡀⢖⣀⣾⠟⠃⠈⠀⠃⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡟⠇⡄⠈⠹⣦⠉⣹⢻⠟⣿⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢼⣌⣶⣄⠀⡉⠧⢙⠣⣰⣿⡄⠀⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠙⢷⠆⠀⠈⠰⢿⣿⡇⣠⠞⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣐⣼⠀⠀⢀⡴⣿⠏⢀⡀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣄⠈⠛⠁⠀⢠⡟⢩⣿⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⣀⣐⣾⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⣇⠀⠁⠀⠀⢴⣄⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣷⣟⡙⠂⣠⣤⣀⡆⠀⡍⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡄
⡿⠿⠻⠿⠋⠀⠀⠘⠀⠘⣿⣤⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠏⢹⣿⢠⡄⠀⠀⠀⠀⠐⢦⣿⣿⡟
⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠿⠒⡱⡄⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⡇⠀⠀⠀⣠⣄⠀⠉⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠁⠈⢶⡄⠀⢐⡇
⠜⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣖⠃⣤⡀
⠑⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⣄⣀⣴⣿⣦⣀⡀⠀⠀⠀⠀⠀⡼⠟⣿⠟⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠛⠋⠉⠒⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠲⢤⣠⣄⣤⣀⣄⣀⣠⣿⣀⣘⣿⣠⣄⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⡏⠁⠈⣿⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⣿⡧⠄⠀⠐⠊⠃
⠀⠀⠀⠠⣒⣁⠰⠃⢀⡠⠄⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣇⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠆⠀⢤⡀
⠀⠀⠀⠀⠋⠈⠉⠁⠲⢄⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⡎⣿⣿⡀⠈⠀⠀⢪⠉⣲⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠻⣿⣧⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⡀⡀⠀⠈⠲⡏⠀⠀⠙⢿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⠠⠈⡀⠀⠀⠀⠐⠂⠈
⠠⢴⠀⢰⠒⠰⠢⣀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⣾⣿⣿⣿⣷⣶⣶⣿⠃
⡲⠚⠛⠉⠐⠒⣄⠉⠒⠀⠀⠀⠈⠉⠉⠉⠁⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠠⠄⠀⠖⠠⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢠⠈⠀⡰⠁⠈⠙⠀⡀
⠀⠀⠀⠀⠀⠀⡾⢭⠹⢸⡁⣊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠋⠉⠁⠐⠓⠠⢆⠀⠠⢐⡀
⠀⠀⠀⠀⠀⣾⢿⣕⣐⣨⣿⣿⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀⢤⣀⠤⠤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢳⣀⠠
⡿⠊⠀⠀⠀⠙⠷⣦⠴⠶⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⡴⡎⣀⣄⣄⣵⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⢿⣷⣤⠤⠀⠀⠀⠀⠀⢀⡽⠛⠉⠉⠉⠙⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢾⡀⢠⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡀⠂⣀⣀⣤⡾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠒⠂⠉⠁⠙⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⡀⢀⠄
⠀⠀⠄⠤⠤⠤⠤⢤⣀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠺⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⠊⡁⠁⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⡷⠂⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⣼⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠠⢾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⢉⡼⢇⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠚⠁⣀⣀⠈⠑⢋⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡟⠉⠀⠀⠀⠀⠈⣿⣿⣧⡄⣸⣝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠀⠀⠀⠀⠀⠉⢻⣿⠻⠏⠙⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣗⣀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⢲⣶⣤⠀⠀⣀⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⠜⠋⢈⣁⠰⢶⣾⣿⣿⣿⣿⡄⠀⠁⠀⠀⠀⠀⠐⠀⠉⠉⠀⠀⠉⠉⠛⠻⢿⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠿⣿⡇⢰⣷⡟⠛⠃⠴⣾⣋⠀⣽⣿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠶⢷⣦⣤⣄⡀⠀⠀⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠰⠀⠻⢿⣶⡿⠃⠀⠀⠀⠈⣻⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡆⠀⠈⢹⡖⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠖⠀⠀⣠⣀⠛⢠⣭⣿⣿⣀⣤⣤⠀⠛⠻⠀⠀⠀⠀⣤⣤⠀⠀⣴⡏⠙⠋⠁⣠⠁⠀⠀⠙⢿⡷⡀⠀⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠘⠳⠆⠀⠀⠀⠀⠸⣿⣷⡀⠀⠘⠋⠀⣤⣶⡄⠀⣿⣻⡆⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠆⠀⡴⠀⢨⣿⣷⠀⢸⣿⣆⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠈⠉⠛⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⢀⣸⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣎⣴⢿⡏⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⡴⠶⢿⣯⣹⣷⣾⡇⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢘⣿⣟⠀⠘⡛⠋⣹⡟⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠀⡀⢾⣁⣼⣿⣷⣴⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢣⣤⣾⣧⣀⢉⡾⢻⣿⠟⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠑⠀⠀⡀⠈⠛⢿⣿⠛⣿⣷⣿⣷⡀⠀⣴⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⡀⠀⠒⣠⡈⠀⠀⠈⠙⠛⠙⠛⠁⠹⠗⢠⣴⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⡀⠀⠀⠀⠀⠀⢂⣤⡾⠛⠁⠀⠀⠈⢑⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠐⢠⠔⠋⠁⠀⠀⢀⡴⠚⠛⠾⡷⣄⠀⠀⠀⠀⠉
⠀⠀⠂⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⢠⣀⣰⣄⣐⠂⣰⣾⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣿⡘⠛⢛⡉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣷⣽⣷⠙⠉⠀⢠⣿⣤⡀⣠⠀⡀⠘⠀⠀⢀⡘⣠⠀⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⡀⠈⠱⠉⣿⡿⣰⣿⠀⠘⣿⡁⢛⠀⠄⠀⠀⠧⠘⣿⣏⠇⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⠿⠁⠀⠀⠀⢸⣿⣿⣿⠇⠀⠈⡀⢉⠃⠀⠀⠀⠀⠘⠉⠉⠀⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣉⠝⠃⠀⢁⠔⠆⠠⠀⠀⠀⠀⠀⠀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠰⡄⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣴⣶⢶⠠⡄⢀⡀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⣨⢙⠡⡃⠆⠀⠾⠏⠁⠀⠎⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⡠⠞⢡⡄⠀⢠⡆⠀⢘⡤⢀⡉⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠸⠁⠀⡈⠀⠸⡈⢀⢤⡀⠰⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠈⢯⠀⣿⣿⠀⠈⠃⠀⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡴⠀⠂⡀⠀⠀⠸⠿⠀⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⣶⠀⠀⠀⠸⠂⠁⣤⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠠⣄⠀⠀⠀⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢾⣿⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣀⠀⣀⠀⠀⣠⣃⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏⢠⡾⠛⠁⠀⠘⠓⠘⠻⠟⠀⠂⠀⢠⣤⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠁⠈⠛⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣴⢤⣴⣤⢦⣄⡀⠀⠀⢀⣠⡿⢧⠈⡷⣠⠟⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣰⣿⡟⣾⣇⣿⢦⣍⣵⠀⢠⣿⢻⣏⠦⣈⠀⣾⣀⣉⣀⡀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠌⠑⠢⡀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢰⠙⠋⠉⠉⠀⠀⠻⣿⣿⡀⠘⣿⣦⡉⠛⠶⢾⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⣷⣄⣑⠂⠄⠀⠀⠃⣠⡇⠀⠀⢸⡿⢿⢻⢛⣿⢿⢿⠅
⠀⠀⠀⠀⠒⠀⠀⠀⣀⠄⢱⠀⢸⣼⠃⠀⡰⠛⠛⠻⠿⣿⣿⡿⠤⠤⠄⠀⠀⠊⠂⠀⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⠀⠀⣾⡗⣚⡂⢼⣿⣼⣇
⠀⠀⠀⠀⢘⠀⠀⡈⢰⠁⡮⠀⢸⣿⠇⠀⡁⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣧⣼⣥⣤⣽⣿⣾
⠀⠀⠀⠀⠘⢀⣀⠀⡈⠈⢀⠠⠾⠋⠀⠀⠁⠠⠄⣀⠀⡈⣿⣿⣧⣀⣤⠀⠄⠊⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣽⡿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠚⣡⠠⡿⢥⣿⣿⣟⡀⠀⡲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠋⠉⠀⠂⠀⠀⠉⢿⣿⡇⠀⠀⢸⠀⠈⠀⢀⠠⣾⣿⣿⠄⠀⠎⠩⢀⡀⠙⡤
⠀⠀⠀⠀⠀⢰⠤⠴⠖⣿⣿⣿⠒⠒⠒⠒⠒⠒⠋⠛⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠘⣿⡇⠀⠀⠘⢶⠞⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠘⡇⠑⢄⠂
⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⢄⠀⠀⢔⢀⠌⣼⡟⠀⠀⠀⠀⠸⡀⠀⠀⠀⠹⣿⣿⣇⠀⠀⠀⠀⡰⠁⠘⢼⡖
⠀⠀⠀⠀⠀⠈⠓⠠⠀⠈⣿⣿⣧⣀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠓⠀⠦⠀⠹⠿⠿⠆⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿