Wtf Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣦⡀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⢀⡀⠘⠁
⠀⠀⠀⠀⠀⠠⢤⡘⠿⢷⣿⣿⠿⣿⡿⢶⣾⠟⢠⡄⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣰⣿⣦⣴⣿⣇⣀⣠⡄
⠀⠀⠀⠀⠀⠂⢨⡇⢠⡄⢩⠀⣤⠀⠀⠀⠃⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠂⢿⠉⠟⠛⠛⠿⠛⡟⠙⡟⢱⡎⢹⠆
⠀⠀⠀⠀⠀⠁⠠⣦⣈⣠⣴⣄⣁⣤⣆⣰⣴⣀⡸⠇⠿⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠆⠀⡀⠀⡀⠀⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣿⣿⠛⢿⣿⠉⠉⠉⠀⠀⠀⠀⠀⢠⣄⡀⢰⣷⣄⣴⣿⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢠⣾⣷⣤⣤⣾⣷⣶⠷⠤⠾⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢹⠏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠒⠲⣿⠛⠻⣿⠛⢻⡟⠻⠟⠛⠓⣦⠐⣶⠀⠀⠀⠀⠼⠟⠉⠸⣿⠏⠙⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠐⠏⠀⠀⢻⠀⡀⠁⠀⠀⢎⠉⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣶⣦⣼⣤⣿⣤⣼⣦⣤⣤⠙⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⢹⣿⠋⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⣤⣀⣠⣿⣦⣴⣿⡇⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣦⡀⣠⣾⡆
⠀⠀⠀⠀⢀⣀⠴⣿⠛⡟⠛⠛⡿⠛⠛⣿⠛⢳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣾⣿⣿⠿⠿⢷⣾⠿⠧⡤⢤⡄
⠀⠀⠀⠀⠀⠉⣂⠀⣼⡇⠀⢽⣧⣄⠉⢻⡀⣸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠁⣤⠀⠹⠀⠶⢀⡀⠀⠲⡇⠀⡇
⠀⠀⠀⠀⠀⠚⠻⢀⣿⣧⣤⣤⣧⣤⣴⡿⣇⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣦⡀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢀⣼⣄⣉⣠⣞⠀⣶⣿⣏⣁⣠⢿⠘⠄
⠀⠀⠀⠀⠀⠐⠀⣸⠿⠋⢹⣿⠏⠙⠿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢌⡿⢶⡿⣿⠿⠿⠿⣷⠶⠿⠀⢤⡀⠀⠀⠀⠀⠀⢸⣿⠟⢻⣿⡟⠻⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⠀⠃⠈⠃⢠⣄⠀⣴⡇⠐⢲⡆⢸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⡟⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢀⣀⣆⣀⣾⣿⣀⣿⣇⣘⣯⡑⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠟⣿⣿⠟⢿⣿⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠏⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣤⣼⣿⣷⣾⣿⣇⣠⣤⡆
⠀⠀⠀⠀⠀⠀⢲⣦⣤⣿⣿⣶⣿⣷⣠⣤⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡆⠉⢻⠋⢹⠛⠉⠙⢿⡏⢹⣆⣀
⠀⠀⠀⠀⠆⠉⢿⠉⢹⡏⠉⢬⣯⠉⠃⣼⠀⣿⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⣄⢀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⡀⠀⢨⠀⠻⠂⢸⡇⣼⠛⠉
⠀⠀⠀⠀⢀⠀⢠⠀⢸⡇⠐⢺⣿⡄⢸⣿⠤⣟⠀⠀⠀⠀⠀⠀⠠⢽⣿⠾⢿⡿⠿⣿⡿⠶⠿⣿⠶⡦⠀⠀⠀⠀⠉⠙⢺⣷⣶⣾⣷⣶⣶⣿⡇⠻⠷⠄
⠀⠀⠀⠀⠈⠀⠈⣷⣾⣿⣿⣿⣿⣿⡟⠛⠲⠃⠀⠀⠀⠀⢀⠀⣤⡄⠈⠀⠘⠃⠀⡟⠀⡴⠒⣿⢀⡇⠀⠀⠀⠀⠀⠀⠼⠋⡁⠸⣿⠃⠈⠙⠧
⠀⠀⠀⠀⠀⠀⠸⠛⠁⠈⣿⡟⠈⠉⠻⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⢀⣴⣀⣦⣤⣀⣿⣀⣉⣀⣿⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠻⣿⡿⠻⢿⣏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠿⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠛⠛⠋⢹⣇⣠⣴⠶⠛⠉⠉⠀⠀⢸⡏⠉⠉⠙⠛⠛⠛⠛⠛⣧⡴⣶⠾⠿⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠄⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣸⢻⡇⣿⣀⣀⡀⠀⠈⢻⡆
⠀⠀⠸⡅⠀⠉⠉⠉⠉⠙⠄⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠛⠛⣅⢹⡇⠀⠀⠈⣿
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢰⡀⣠⣄⠀⣸⠃⠀⠀⠀⠀⠀⠀⣾⣀⡀⠀⠀⢠⣤⡴⠶⣶⠶⠛⠀⠀⣠⣤⣀⣀⣀⣸⣇⠔⣶⠾⠷⠛⠀⠀⠀⣴⠏
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠛⠁⠈⠙⠃⠀⠀⠀⠀⠀⠀⣰⣏⣩⡇⠀⠀⢹⠖⢒⠒⢿⠀⠀⠀⠰⠿⢤⣬⣉⡩⠟⢸⠀⡿⠀⢠⣴⣶⠶⠟⠋
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⣠⣤⡼⢫⠋⠀⠀⠈⣿⠿⢟⣻⠇
⠀⠀⠀⠀⠀⠘⣦⣠⣤⣤⣤⡴⠉⢣⣀⠀⠀⠀⣀⣀⣀⣾⠁⠀⢸⣇⣀⣠⣿⠀⠀⠀⢸⣇⣀⣠⡏⡱⠊⠁⠀⠀⠀⠀⢼⢹⡋⠉⣷⠱
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠘⣤⠃⠫⣉⠉⠉⠉⠉⣩⠇⠀⠀⠘⢏⣉⣀⠬⠂⠀⠀⠘⢏⣉⣩⠽⢃⡀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠝⠛⠛⠉⠁⠀⠀⠈⠉⠉⠉⠉⠀⠚⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⢛⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠛⣩⣭⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣽⣶⣧⣶⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⠋⣿⠀⠀⣿⡇⠀⢸⣿⡄⠀⢻⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⡇⢿⡏⠀⢻⠀⠀⣿⡇⠀⠘⣿⣇⠀⠠⣶⣿⣻⠃
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠀⠀⠈⠻⣿⣿⣿⣦⠀⠀⠀⢿⣿⣷⢸⡇⣇⠘⠀⠀⣿⣧⠀⠀⣿⣿⣀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠹⣿⣿⣿⣧⠀⠀⠘⢿⣿⡌⡇⣿⣄⢀⣠⣿⣿⣷⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠤⠴⠤⣤⡀⠀⠀⠴⠋⣀⣀⣀⣿⣿⠛⢻⣧⠀⠀⠀⠉⢷⣤⣿⣿⣿⣿⠿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⢀⡠⠴⠦⡀⠀⠀⠀⠰⠁⢰⡮⢻⣿⠇⢈⣿⡄⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠈⠃⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⢈⡏⡀⣸⣿⡇⠀⠀⠀⢸⣿⠟⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠉⠁
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⣼⣿⣿⣿⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢀⡴⠶⡦⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠔⠻⣿⣿⠃
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠂⢄
⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣿⠟⠫⣄⡀⠀⠀⠀⠀⠀⡀⢀⡤⠊⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠰⠛⠁⠈⣿⣿⣿⡿⠁⠀⠀⠀⠉⠒⠂⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁


⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣏⢽⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣿⣿⣆⣀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠀⠀⠀⠀⠀⠀⣴⣿⠿⠛⠛⠋⠉⠉⡏⠸⠇⠠⠀⣿⡿⠛⠛⢋⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠻⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣀⣀⣤⣤⣄⠀⠃⠀⣀⣤⣤⣿⡷⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⠀⠀⢸⡏⣿⣿⡄⣀⠀⠀⣴⣿⡿⠿⠛⠛⢻⣿⡆⠀⠈⣿⡿⠉⠁⠀⢴⣾⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡇⠸⣿⣇⠀⢀⡀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠅⠀⢻⣿⣤⣿⣿⣾⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⣿⣇⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠘⣿⣿⣿⠿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⢿⣿⠀⠀⠀⢸⣿⣿⣿⡿⠟⣛⣛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠙⠻⠋⠀⠉⠛⠀⠀⠀⠀⢸⣿⣷⣀⣤⣼⣿⡀⠀⠀⠘⣿⣿⣿⡄⠺⠿⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣧⣤⣶⣾⣿⣿⣿⡇⢠⣶⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⠗⠀⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⠀⢹⠛⠉⡏⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣆⣠⣦⣄⣤⣤⣴⣦⣤⣀⣤⣴⣶⣦⣶⣶⣶⣿⣿⣾⣶⣦⣤⣀⣦⣶⡄⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⣿⣿⣿⣿⡟⠉⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣶⠶⢶⣶⣶⣦⠀⠀⣾⣿⣿⣿⠉⠁⠈⠉⠛⠓⠦⠶⣶⣤⣤⣀⡀⠀⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠉⢿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⡇⢈⣿⣿⠀⠀⣴⣤⣀⡀⠀⠀⢸⣿⣿⣿⡟⠀⢸⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠘⣿⣿⠇⠀⠀⢸⣿⣿⡇⠀⢸⣼⣿⡏⠀⠀⣿⣿⣯⣀⣸⣷⣾⣿⣿⣿⡇⠀⢸⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠀⠸⣿⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⣿⡇⠀⠀⠛⠛⠻⠿⠿⣿⣿⣿⣿⡿⠃⣠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠁⠀⢀⡀⠀⠹⠃⠀⠀⢸⣿⣿⡇⠀⢠⣦⣤⣤⣄⣀⣾⣿⡿⠋⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⡆⠀⠀⠀⣼⣧⠀⠀⠀⢰⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⡀⠀⢀⣿⣿⡀⠀⢀⣾⠀⠈⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣄⣤⣿⣿⣿⣿⣿⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⠿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠛⠿⣿⣿⣿⣿⠟⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠙⠂⠈⠁⠀⠋⠁⠀⠀⠈⠀⠀⠁⠀⠙⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⢀⣀⡀⢀⣀⣀⣤⡀⡀⠀⣀⣠⣄⣄⣀⣤⣤⣶⣿⣯⣧⣤⣂⡀⢀⣀⣤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⡟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⡀⠀⠀⣤⣤⣤⣤⠤⠟⠻⠿⢿⣿⣀⢀⣀⡀⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⣿⣿⣿⣿⠛⠛⢻⣿⣿⡇⠀⠀⡟⠻⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣶⣦⠀⠀⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢻⣿⣿⡟⠀⠀⠸⣿⣿⣷⠀⠀⡇⣰⣿⣿⠀⠀⢸⣿⣶⠦⠄⣀⣀⣾⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠈⢿⣿⡇⠀⠀⠀⢿⣿⣿⠀⠀⣷⣿⣿⣿⠀⠀⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡀⠀⠘⣿⠀⠀⠀⠀⢸⣿⠟⠀⠀⢿⣿⣿⣟⠀⠀⠈⠉⠉⠛⠛⠻⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⢀⣄⠀⠀⠏⠀⡇⠀⢸⣿⣿⡇⠀⢀⣶⣶⣶⣤⣤⣤⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⡄⠀⠀⠀⣸⣿⡀⠀⠀⠀⡇⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣷⡀⠀⢀⣿⣿⡇⠀⠀⢠⣷⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡛⢛⣿⣿⣿⣿⠿⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⠿⠛⠟⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⠋⠁⠙⠗⠈⠛⠁⠉⠛⠉⠁⠉⠙⠁⠀⠉⠁⠙⠗⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⡰⠃⠠⡇⠀⠀⠀⠀⠀⠀⠀⣼⢁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠃⠀⠁⠸⡄⠀⠀⠀⠀⠀⣠⠋⡀⣿⢸⠁⠘⠀⢧⠀⠀⢀⡄⠀⠀⢀⠇⠈⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢤⣼⠀⠀⠀⠀⢹⠄⠀⠀⡠⣶⠃⠀⠁⣿⡏⠀⠀⠀⠘⡇⢀⡏⢳⡀⢀⠞⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡴⢿⡀⠄⠉⠀⠀⠀⠀⢸⠀⣠⠞⠁⠉⠀⠀⠀⠉⠁⠀⡀⠀⠀⡇⣸⠱⠀⠳⠃⠀⠀⡀⢀⣧⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⢡⠸⠆⠀⠀⠀⠀⠐⠀⣯⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡿⠃⠀⠀⠀⠀⠐⠀⠀⢈⡄⢸⠾⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⣧
⠀⠀⠀⠀⠀⠀⢣⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡟
⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⣠⠞⢃⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢧
⠀⠀⠀⡰⠁⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢦⡀
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⡞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠀⠐⠢⠄⡀⠀⠀⠀⠀⠀⠈⠁⠀⢧
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⡤⠞⠑⠠⢄⡀⠀⠀⠘⡆
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠒⠒⠂⠈⠉⠉⢣⣀⠀⠤⠤⠤⠤⢤⣼⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠐⠀⢹
⡇⠀⠀⠀⠀⣀⡠⢤⠀⠀⢀⡀⢀⡤⠼⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣉⠽⢠⡤⢤⠔⠀⠁⠀⠀⡰⠁⠀⠀⠀⠂⢸
⡇⠀⠀⢰⠋⠁⠀⠈⠾⠉⠁⢹⡎⠀⠀⠠⠤⠤⡀⠀⠀⠀⠀⢚⠀⠀⠀⠤⢀⠀⡸⣰⠋⠀⢸⠀⡜⠀⠄⠀⠀⡔
⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⠀⠀⠀⢶⠶⢶⠀⠀⠀⠀⢸⠋⠁⡧⠇⢀⡾⢀⠇⠌⠀⢀⡜
⣿⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠈⠆⠘⠀⠀⠀⡌⠁⠀⠀⣳⠖⠻⡄⠘⣲⠤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⡄⡀⠀⠀⠸⣀⠀⠠⠁⣀⣀⡀⠀⢠⡇⠰⡇⠀⠀⠀⠀⠟⡇⠀⠀⢀⡇⠀⠀⠀⡏⢂⡞⠁⣜⠑⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼
⠈⣧⠀⠤⠀⠀⠑⢤⡀⠔⠦⠤⠤⠄⣸⠀⠀⠓⠂⠐⠒⠚⠀⠧⠤⢄⠼⠁⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠸⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢹⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠈⠳⣄⡈⠂⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣤⣀⠈⠁⠒⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣉⣤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣤⡤⠤⠤⠴⠶⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡀⡇⠀⠀⠀⠀⠀⠀⢀⠔⣻⢀⡜⢁⢸⠀⠀⠀⠀⠀⠀⠀⡾⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠁⠀⠀⠹⡄⠀⠀⠀⠀⠠⠋⡄⣿⡜⠀⠀⠘⡆⠀⢠⡄⠀⠀⢰⠃⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡞⠓⢾⠀⠀⠀⠀⢹⠀⢀⡴⢻⠃⠀⠀⣿⡇⠀⠀⠀⢹⠀⡏⠘⣆⡰⠃⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⡔⢹⡇⠐⠀⠀⠀⣀⠀⡎⡴⠋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠁⠃⠈⠁⢀⠀⠀⠼⠒⡄⣴
⠀⠀⠀⠀⠀⢠⠁⠀⠃⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠋⢹
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⢀⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞
⠀⠀⠀⠀⣠⠾⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⡀
⠀⠀⠀⡼⠁⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠣⡀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠁⠀⢳
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡌⣀⠀⠀⠀⠀⠈⠖⠋⠑⠂⢤⠀⢀⠀⢇
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠉⠉⠀⠀⠀⠱⠤⠀⠒⠒⠒⠒⡶⢇⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⢸
⡇⠀⠀⢀⡤⠤⠒⢆⡤⠤⢤⢠⠒⠃⠐⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⡟⢠⠚⠁⠖⢲⠁⠐⠀⠀⠠⠃⠀⠀⠀⠀⠘
⡇⠀⠀⢹⠀⠀⠀⠉⠀⠀⠘⠃⠀⠀⠐⠀⠒⠀⠀⢃⠀⣐⠀⠀⠀⠁⣠⠤⢱⠃⠀⢸⠀⡎⠠⠀⠀⡰⠁
⣧⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠃⠀⠀⠘⡍⢹⠀⠀⠀⠤⠃⠀⠨⣁⣠⡚⠠⣁⡁⣠⠔⠁
⢻⠀⠀⠀⠀⡄⠀⠀⡤⠀⠀⠀⠀⢀⠃⢸⠀⠀⠀⠀⣴⡄⠀⠀⢀⠃⠀⠀⢰⠃⣀⠇⢠⠚⣁⡹⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡇⢂⡀⠀⠘⢄⡈⣠⣄⣀⣀⠀⣹⠀⠘⢄⠀⠀⡤⠃⣏⣀⠀⡸⠀⠀⠀⠘⠒⠋⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⢻⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠃
⠀⠀⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠈⢳⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠙⢦⣄⠁⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣤⣀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣀⣤⠴⠶⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣒⣀⠖⣢⣤⡀⢀⡠⠄⢤⡠⢊⠕⠊⠁⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡀⠀⠀⡔⣪⡒⢭⡲⣀⡠⢄⢊⠭⠒⠒⠉⠹⡡⠃⠀⠀⢀⣀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡊⠥⠒⠈⢉⡉⢝⡛⠭⠓⠊⠁⠘⢅⢲⠁⠀⠀⠀⠀⢀⡀⣁⠀⠀⡔⠁⢨⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠊⠐⡈⡯⠙⠒⠚⠁⠀⠀⠀⠀⢠⠤⣚⠈⡄⠀⠀⢆⠉⠀⠀⠀⠁⠒⠀⢀⠇⠀⠀⣸⠃
⠀⠀⢀⣀⠤⣄⢔⣢⣰⡱⢥⠤⣠⡇⢡⠀⠀⢠⣀⡠⢤⠀⠀⠰⣏⠈⡇⢂⠀⠀⠸⠀⠀⣀⣀⠀⠀⠀⡎⠀⠀⣰⡃
⠀⠀⠘⡄⢫⠀⠀⠘⣷⠐⡉⠀⠹⡧⢸⠀⠀⠈⡖⢣⠸⡀⠀⠀⢿⠀⢱⠘⡄⠀⠀⠓⠉⠀⠀⢡⠀⠰⠀⠀⢀⡿⠦⣄
⠀⠀⠀⠰⡈⢆⠀⠀⢻⠀⠇⠀⠀⢣⠘⠀⠀⠀⣗⠈⡆⢡⠀⠀⠸⡇⠀⡇⢡⠀⠀⠀⣀⠤⢒⣈⠡⡄⠣⠤⣤⢓⡄⠈⢷⡀
⠀⠀⠀⠀⠱⡈⡆⠀⠀⢇⢱⠀⠀⠀⠄⡃⠀⠀⣿⠀⠑⠈⡀⠀⠀⢻⡀⠐⠈⡀⠀⠀⢑⠈⠁⠀⠀⢱⠘⡉⠀⠈⢷⠀⢈⡇
⠀⠀⠀⠀⠀⢣⠘⡄⠀⠈⣶⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⢆⢡⠀⠀⠈⣧⠀⢇⢱⠀⠀⠈⣧⠀⠀⠀⠀⡇⢁⠀⠀⢸⣧⢊⠃
⠀⠀⠀⠀⠀⠀⢣⠘⡄⠀⠘⠀⠀⣤⠀⠀⠀⠀⢸⡆⠀⠘⡄⢇⠀⠀⠹⡄⠸⡀⢆⠀⠀⠸⡆⠀⠀⠀⠱⡬⠭⢭⢭⡷⠊
⠀⠀⠀⠀⠀⠀⠸⣇⠰⡀⠀⠀⠀⠌⢣⠀⠀⠀⢨⡇⠀⠀⢡⠸⡀⠀⠀⣷⠀⢁⠘⣒⣒⡊⠽⠀⠀⠀⠀⣷⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⡀⠀⠀⢠⡄⢢⠀⠀⢸⡇⠀⠀⠀⣇⣙⣒⠉⠭⠃⠈⠁⠀⠀⠀⠀⠀⠀⢀⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢡⠀⠀⢸⡟⢄⡑⠨⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⠤⠤⠤⠒⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠩⠭⠵⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠜⡔⠉⠑⠒⠒⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⢄⠀⠀⠀⠀⠀⠀⢀⡠⢋⣀⣉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⠑⠒⠤⠤⠔⠒⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠐⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣽⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠋⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢰⣿⠀⠀⠀⠀⠀⢀⡾⠛⠛⢷⡀⠀⠒⠤⣄⣀⠠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠇⠀⠀⢸⡷⠶⠶⢤⣤⣉⣉⠛⠒⠾⠯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⣧⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⢤⣤⣀⡀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣿⣇⣤⣴⡶⠶⠶⣿⠄⡀⢠⠃⠹⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡿⠇
⠀⠀⠀⣠⡀⢿⡄⠀⠀⢠⣿⠁⢀⣠⣴⠾⣿⡋⠁⢀⣠⣶⣶⣿⠛⣁⣠⣤⠶⠟⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⣈⡻⣌⢷⡀⢀⣾⠃⣴⠟⠋⠀⠀⣿⡶⠞⣿⣭⣴⠶⣿⠛⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠈⠻⠿⠿⠁⣼⠇⠀⠀⠀⢀⣠⡤⣿⡟⢻⡏⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠖
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⡀⠀⠀⠀⢿⡇⢸⠇⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠰⠃
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠈⢦⠀⠀⢸⡇⣿⠀⠀⠀⠀⣿⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢰⡿⠁⣰⣶⠀⠀⠀⠈⣧⠀⢸⣧⣿⠀⠀⠀⠀⣿⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⣿⢁⣼⠟⢿⡄⠀⠀⠀⢸⡆⠘⣿⣿⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⢀⣀
⠀⠀⠀⢀⣿⢣⡾⠃⠀⠈⢿⣄⣀⣤⡾⠁⠀⣿⣿⡇⠀⠀⠀⣿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣾⠟⠁
⠀⠀⢀⣾⣷⡿⠁⠀⠀⠀⠀⠉⢩⢥⡄⠀⠀⢸⣿⡇⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠅
⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⡰⢫⡟⠀⣤⣧⡈⣿⡇⠀⠀⠀⢻⡇⣷
⠀⠘⠟⠁⠀⠀⠀⠀⢀⡤⢊⣴⠋⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠘⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠠⠒⠉⠰⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀⣤⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⣄⡀⠀⡼⠁⠳⣄⢠⡟⠀⠹⣦⠀⢠⡞⠙⣆⠀⠀⠀⠀⣠⠴⡄
⠀⠀⠀⠀⠀⢠⣤⣀⡀⢠⡏⠀⠈⠙⠚⠁⠀⠀⠈⠋⠀⠀⠀⠈⠷⠏⠀⠀⠈⢧⡀⣠⠞⠁⠀⠹⣄⠀⢀⡤⢾⡆
⠀⠀⠀⠀⠀⢸⡇⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡏⠁⠀⠐⠻⢟⠉⠀⣸⠁
⠀⠀⠀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠀⠱⠚⠉⠒⠤⣄⣀⡀
⠀⠀⠈⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠉⠉⠉⠉⠁⠘⠤⠤⠄⠐⠒⠒⠲⢮⣀⡀⠀⠁⠀⠀⠀⠁⠀⠀⡰⠁⣹⠃
⠀⠀⠀⠀⠙⣆⡠⠤⠒⡆⣀⡠⢤⢀⠖⠚⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⠀⡤⠚⠰⠐⡖⠁⠄⠀⠀⡰⠁⣴⠃
⠀⠀⠀⣠⠞⢫⠀⠀⠀⠙⠁⠀⠘⠇⠀⠀⠐⠐⠒⠀⠀⠈⠀⠈⡀⠀⠀⠘⢠⣀⠌⡜⠁⠀⡇⢠⠁⡀⠀⠀⡔⠀⠀⠈⣻⠆
⠀⠀⠘⠧⣤⡀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⡎⠀⠀⠈⠏⠙⠇⠀⠀⢨⠽⠀⠀⡗⢂⡴⠃⣜⠐⠂⣠⠜⠀⠀⣴⡛⠁
⠀⠀⠀⠀⠀⢹⡟⡄⠀⠀⢠⠀⠀⠀⠀⠀⡇⢠⡇⠀⡀⠀⢸⣴⠀⠀⠀⡈⠀⠀⠀⡾⠁⣸⠀⡠⠟⠉⡇⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⢠⡟⠀⠱⡀⠀⢁⡐⠀⠀⠀⢸⠁⠘⣄⣀⣀⣀⡌⢘⠀⠀⠐⠇⠀⠀⠀⠣⠜⠁⠀⠣⠜⠋⠀⠀⠀⣴⠖⠚⠛⠁
⠀⠀⠀⠀⠉⠛⠛⢻⡟⠉⠉⠈⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣤⠴⠖⠛⣿⡇⠀⠀⢀⣤⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⣿⠉⠛⠲⠦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣸⠋⠳⣄⣠⠟⠉⠙⣦⡀⢀⡶⠛⢦⣀⣼⠃⠈⠙⠲⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣿⠀⠀⢰⡏⠀⠀⠙⠉⠀⠀⠀⠈⠻⠋⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠳⣿⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡴⢶⣄⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣾⢁⡾⠁⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⣸⣷⠟
⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⢦⡀⠀⢀⡶⣄⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠲⣄⣰⠏⠀⠳⣤⠟⠀⠈⢳⡀⣴⠃⠙⢆⠀⠀⢀⡴⠚⢧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣶⠲⠤⣤⡇⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠙⠁⠀⠀⠈⠳⡴⢯⠄⠀⢘⣧⣠⠴⠛⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠑⣄⡤⢇⡀
⠀⠀⢰⡖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠀⠐⠒⠲⡀⢀⢀⣀⣀⣀⣀⠁⠂⠀⠀⠀⠀⠉⠀⠀⠀⡝⠛⡷
⠀⠀⠀⠙⢦⠀⣀⣠⡄⠀⠀⡀⢀⠤⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡎⢉⡱⢀⡀⡤⠀⠀⠀⠀⡜⢀⡾⠁
⠀⠀⠀⣠⠾⡋⠀⠀⠸⠋⠁⠸⠎⠀⠀⠀⠠⠄⠀⠰⠀⠘⠀⠀⠀⠂⢀⢠⢣⠎⠀⠈⢀⠃⠀⠀⢀⠌⠀⠘⢳⡤
⠀⠀⠺⣥⣀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢤⠀⠀⠀⢖⠒⠇⠀⠀⣀⠏⠁⣜⠂⣠⠆⢘⠐⠀⣠⠎⠀⢠⡶⠋
⠀⠀⠀⠀⠈⣿⢇⠀⠀⢠⠀⠀⠀⠀⢰⠀⣼⠀⠀⠀⢸⣴⠀⠀⠀⡃⠀⠀⠸⠋⢈⠀⡨⠟⢙⡅⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣼⠃⠘⢆⠀⢁⣔⠀⠀⠀⢛⠀⢣⣀⣀⣀⡜⢸⡀⠀⢸⠁⠀⠀⠑⠔⠃⠀⠓⠜⠋⠀⠀⠠⡖⠚⠛⠁
⠀⠀⠀⠀⠉⠉⢉⡟⠉⠁⣈⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠿⠶⠖⠛⠉⣿⠇⠀⠀⣰⣄⠀⠀⢀⣤⣄⠀⠀⠀⢀⡀⠀⠀⣴⠦⣄⡀⠀⡏⠈⠉⠓⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡏⠈⢳⣴⠏⠀⠈⠳⣄⡴⠋⠉⠳⣼⠇⠀⠀⠙⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣀⣿⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣈⠛⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢠⡟⣠⠟⠉⠳⠾⠃
⠀⠀⠀⠀⠀⠀⣾⣵⠋
⠀⠀⠀⠀⠀⠸⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣒⣀⠖⣢⣤⡀⢀⡠⠄⢤⡠⢊⠕⠊⠁⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡀⠀⠀⡔⣪⡒⢭⡲⣀⡠⢄⢊⠭⠒⠒⠉⠹⡡⠃⠀⠀⢀⣀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡊⠥⠒⠈⢉⡉⢝⡛⠭⠓⠊⠁⠘⢅⢲⠁⠀⠀⠀⠀⢀⡀⣁⠀⠀⡔⠁⢨⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠊⠐⡈⡯⠙⠒⠚⠁⠀⠀⠀⠀⢠⠤⣚⠈⡄⠀⠀⢆⠉⠀⠀⠀⠁⠒⠀⢀⠇⠀⠀⣸⠃
⠀⠀⢀⣀⠤⣄⢔⣢⣰⡱⢥⠤⣠⡇⢡⠀⠀⢠⣀⡠⢤⠀⠀⠰⣏⠈⡇⢂⠀⠀⠸⠀⠀⣀⣀⠀⠀⠀⡎⠀⠀⣰⡃
⠀⠀⠘⡄⢫⠀⠀⠘⣷⠐⡉⠀⠹⡧⢸⠀⠀⠈⡖⢣⠸⡀⠀⠀⢿⠀⢱⠘⡄⠀⠀⠓⠉⠀⠀⢡⠀⠰⠀⠀⢀⡿⠦⣄
⠀⠀⠀⠰⡈⢆⠀⠀⢻⠀⠇⠀⠀⢣⠘⠀⠀⠀⣗⠈⡆⢡⠀⠀⠸⡇⠀⡇⢡⠀⠀⠀⣀⠤⢒⣈⠡⡄⠣⠤⣤⢓⡄⠈⢷⡀
⠀⠀⠀⠀⠱⡈⡆⠀⠀⢇⢱⠀⠀⠀⠄⡃⠀⠀⣿⠀⠑⠈⡀⠀⠀⢻⡀⠐⠈⡀⠀⠀⢑⠈⠁⠀⠀⢱⠘⡉⠀⠈⢷⠀⢈⡇
⠀⠀⠀⠀⠀⢣⠘⡄⠀⠈⣶⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⢆⢡⠀⠀⠈⣧⠀⢇⢱⠀⠀⠈⣧⠀⠀⠀⠀⡇⢁⠀⠀⢸⣧⢊⠃
⠀⠀⠀⠀⠀⠀⢣⠘⡄⠀⠘⠀⠀⣤⠀⠀⠀⠀⢸⡆⠀⠘⡄⢇⠀⠀⠹⡄⠸⡀⢆⠀⠀⠸⡆⠀⠀⠀⠱⡬⠭⢭⢭⡷⠊
⠀⠀⠀⠀⠀⠀⠸⣇⠰⡀⠀⠀⠀⠌⢣⠀⠀⠀⢨⡇⠀⠀⢡⠸⡀⠀⠀⣷⠀⢁⠘⣒⣒⡊⠽⠀⠀⠀⠀⣷⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⡀⠀⠀⢠⡄⢢⠀⠀⢸⡇⠀⠀⠀⣇⣙⣒⠉⠭⠃⠈⠁⠀⠀⠀⠀⠀⠀⢀⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢡⠀⠀⢸⡟⢄⡑⠨⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⠤⠤⠤⠒⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠩⠭⠵⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠜⡔⠉⠑⠒⠒⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⢄⠀⠀⠀⠀⠀⠀⢀⡠⢋⣀⣉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⠑⠒⠤⠤⠔⠒⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠐⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠉⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣆⢰⡆⢠⠆⢶⡶⠆⣶⠶⢠⡆⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡾⢷⡟⠀⢸⡇⠀⣿⠛⢈⡃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⠀⢠⠤⠤⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣶⡀⠀⠀⠀⢀⣴⣧⠀⠀⠁⠀⢀⣴⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⣀⣙⠻⢷⣾⠉⠉⣿⡿⠉⠉⣷⡶⠟⣋⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡬⠀⠀⠘⠁⠀⢹⡇⠀⠈⠋⠀⠀⢭⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠛⠉⠀⠀⡏⣝⡷⠀⠀⠀⠀⢻⣡⢸⠀⠀⠉⠙⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣶⣤⣤⡤⠀⠀⠉⢙⡁⠀⠐⠒⠀⠈⡋⠉⠀⠀⢤⣤⣤⣴⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⠤⠒⠒⠒⠢⢄⡸⠀⠀⠀⠀⠀⠀⢇⡠⠔⠒⠒⠒⠤⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⢰⠁⠀⠀⠀⠀⠐⠀⠹⣿⣶⣶⣶⣶⣿⠏⠠⠀⡀⠀⠀⠀⠈⡆⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠸⡀⠀⠀⠀⢢⠠⠀⠐⠈⠙⠻⠛⠋⠁⠀⠀⠀⡕⠀⠀⠀⢀⠇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡑⢄⠀⠀⠀⠓⢤⣠⣀⣠⣼⣧⣄⣀⣀⡬⠚⠀⠀⠀⣠⢊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣀⠀⢸⠻⣿⣉⡉⢉⣉⣿⠿⡇⠀⣀⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⡀⠀⠀⠀⠀⠀⠀⢀⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⡤⢄⣀⣀⡠⢤⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠋⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⡄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⡄⠀⠀⠀⠀⠀⣶⡿⠿⠿⠛⠛⢻⡏⡿⠉⠁⣿⣿⠿⠛⠛⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢻⣿⣧⠀⠀⠀⠀⠀⣿⣇⣀⣠⣤⣄⠘⠃⢀⣠⣤⣼⡷⠀⠀⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠀⢸⢹⣿⡆⠀⠀⠀⣾⣿⠿⠛⠋⢻⣿⠀⠀⢹⣿⠋⠁⠀⠶⡿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢸⠀⢿⣷⠀⣠⣄⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⠀⠀⠀⢸⣿⡆⠀⢸⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢻⣿⡿⠙⢿⠟⠀⠀⠀⢸⣿⡇⠀⢘⣿⡀⠀⠀⢸⣿⣿⡟⢉⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣇⣀⣀⣼⣿⣿⣿⠀⣩⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⢀⣀⣼⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡈⢹⠟⢻⡏⠉⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠏⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢸⠀⠘⠃⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠀⢀⣀⠀⢀⣀⢀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⢸⣿⣿⡿⠿⣿⡿⠿⣿⡿⠿⡿⠿⠿⠿⢿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⣿⡇⣼⣿⡇⣼⡟⠈⠉⣿⡏⠁⣿⣏⣉⠁⠀⠀⢸⣿⣿⣿⠀⢻⠃⡀⢻⠃⢸⣷⣶⠀⣶⣾⠀⠰⠶⣿⣿⣿
⠀⠀⠀⢸⣷⣿⠘⣷⣿⠃⠀⠀⣿⡇⠀⣿⡟⠉⠁⠀⠀⢸⣿⣿⣿⡇⠈⢀⣇⠈⠀⣿⣿⣿⠀⣿⣿⠀⢰⣶⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣖⣺⡿⡶⣾⣿⣿⣿⣶⣿⣿⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⡄⣀⣠⠤⠖⠒⠋⡧⠤⠤⠤⠤⠤⢤⣀⣠⣤⣤⣖⡀
⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣸⠋⠉⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣼⢇⣿⣠⡀⠈⠻⣆
⠀⠀⡆⠀⠀⠀⠈⢁⠀⠀⠀⡸⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣀⣀⣸⠃⠀⠀⠀⠀⢠⡇⣈⣉⣤⠏⠀⠀⡿
⠀⠀⠸⡄⠀⠀⠀⠈⠖⠉⠲⠃⠀⠀⠀⠀⣼⣓⡆⠀⢸⠭⠭⢿⠀⠀⢸⣯⣝⢛⡾⢹⣸⠉⣡⣤⡤⠞⠁
⠀⠀⠀⠹⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢰⠃⢰⡇⠀⢸⠀⠀⢸⠀⠀⣤⢴⡻⠁⠀⠘⣟⣻⣿⡀
⠀⠀⠀⠀⠹⡒⠒⠒⢇⡘⠶⠤⠤⠶⠶⡏⠀⠸⡶⠒⢛⠀⠀⢸⠶⠾⣧⠁⠀⠀⠀⠀⠳⢧⠼⠃
⠀⠀⠀⠀⠀⠨⠒⠒⠊⠁⠀⠁⠈⠀⠉⠐⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⡐⣶⣴⣿⣷⣿⣧⣴⠶⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣶⣄⣴⣇⢀⣀
⠀⠀⠀⠀⡀⠏⢠⠌⠅⣤⠈⠀⠉⠀⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠟⠹⠛⠛⠏⠻⠙⠋⣶⢰⠂
⠀⠀⠀⠀⠀⠚⣦⣶⣷⣦⣾⣤⠶⠦⠟⠘⠀⠀⠀⠈⠀⣠⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠘⠃⣠⢀⣄⢀⡇⠼
⠀⠀⠀⠀⠀⠀⠛⠉⢹⡟⠀⠙⠃⠀⠀⠀⠀⠠⢬⣷⠶⣿⠿⢿⠿⣶⠶⢤⡠⣄⠀⠀⠀⣼⠿⠻⣿⠟⢿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠸⡄⡈⠀⠀⠖⢻⡆⡿⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⣿⣶⣷⣿⣶⡗⠶⠎⠁⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⢠⣄⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠚⠡⠀⠿⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄
⠀⠀⠀⢀⡠⢽⠿⡿⠛⣿⠛⢲⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣶⣿⡷⠿⢷⣶⠶⢀⢠⡀
⠀⠀⠀⠀⣩⡀⣼⡇⠨⡷⠄⢹⡇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⢀⠠⠄⠉⠠⠄⠁⠐⣀⣄⡐⢺⢀⠃
⠀⠀⠀⠀⡉⢁⣿⠿⣿⡿⢶⣟⠓⠛⠀⠀⠀⠀⠠⣤⣀⣿⣦⣾⣧⣀⣤⠀⠀⠀⠀⠀⠐⢻⣶⣶⣧⣼⣿⡷⠴⠎⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠹⠁⠀⠉⠀⠀⠀⠀⠀⠲⠉⠏⠹⠉⣍⠉⣽⠉⢥⠀⡇⠀⠀⠀⠀⠛⠁⠘⡟⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣠⣄⣸⣿⣠⣿⣠⣯⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠹⣿⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣷⣤⣾⣇⣈⡀
⠀⠀⠀⠀⠐⣦⣤⣿⣶⣾⣧⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠹⡟⢻⠛⠙⢿⠙⣧⣀
⠀⠀⠀⠂⠘⠃⢸⠃⣰⣿⠈⣰⡇⣿⠀⠀⠀⠀⠀⢤⣀⣰⣦⣤⣿⣀⣀⣄⠀⠀⠀⠀⠠⣆⣠⡀⣤⡈⢃⣼⢤⣏⠁
⠀⠀⠀⠀⠀⢧⣼⣦⣤⣿⣦⡿⣧⡇⠀⠀⠀⠀⢀⡉⠋⠉⠏⠉⡟⢉⣽⡏⣽⠀⠀⠀⠀⠀⡿⠟⢻⡿⠛⠿⡄⠉⠁
⠀⠀⠀⠀⠀⠟⠋⠹⡿⠉⠛⠇⠀⠂⠀⠀⠀⠀⠈⢁⣄⣠⣤⣀⣆⣀⣠⣇⠏⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⣿⡟⠻⢿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⡛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⢾⣥⣄⣀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣾⣭⣥⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣤⣬⣭⣭⣭⣭⣥⡶⣼⣿⣿⣁⣀⣀⣀⣀⣀⡀
⠀⠀⣀⣴⣶⣿⣷⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢠⣶⣿⠟⢩⣍⡉⠉⠉⠙⠛⠿⠿⠿⣿⣷⣆⠀⠸⣿⣿⣿⠿⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⣾⡿⠉⠉⠉⠉⠉⣉⠉⣽⢿⣿⣿⣷
⣿⣿⣿⣶⣶⣮⣭⣀⣀⠀⠀⠀⢀⣴⣿⣿⣿⠀⢰⣿⣿⡏⠀⠀⠀⠀⢀⢸⣿⡏⠉⠉⠉⠙⠛⢻⡿⠿⠿⠿⠿⢶⣾⣿⣷⣶⣶⣶⣦⣤⣭⣽⣯⣤⣿⣿⣿
⢻⣿⡇⣠⣬⣭⣛⡛⠿⠿⠿⣿⣿⠟⢡⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⢸⣿⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⣿⣿⣿
⢸⣿⣷⢹⣿⠟⠛⠛⠛⠁⠀⣿⣿⠀⢸⣿⡏⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢸⣿⣿⡌⣿⡆⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⡇⠀⠀⠀⠀⡏⢸⣿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⣿⣿⣧⢸⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣶⣾⣿⣿⡇⠀⠀⠀⢠⡇⢸⣿⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⠇
⠀⢻⣿⣿⡀⠃⠀⠀⠀⠀⢠⣿⣿⠀⢸⣿⠉⠉⠉⣽⣿⣇⠀⠀⠀⢸⡇⣾⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣾⣿⣿⠛⠋⠉⠉⠙⣿⣟⠻⠿⠿⠿⠿⠟
⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⡇⠀⣿⣿⣿⣿⣿⡏⣿⣿⠀⠀⠀⠈⡇⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣽⣿⣿⠀⣀⣀⣀⣀⣹⣿⣆
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣇⠀⠀⣿⡇⣿⣿⠀⠀⠀⠀⡇⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⠿⠿⠿⠛⢛⣛⣛⢉⣿⣿
⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⢹⣷⠀⣿⣿⠀⠀⣿⡇⣿⣿⡀⠀⠀⠀⠃⣿⣿⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⢸⣿⣿
⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣄⣿⡏⠀⠀⣿⣧⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸⣿⠁⣿⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣿⣿⢘⡏⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣀⣀⣀⣀⣿⣿⠘⠃⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⡼⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⠀⠀⠀⣠⠏⢷
⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⢸⠀⠙⢆⠀⣀⢠⠇⠀⠀⠙⣆⠀⠀⣀⠀⠀⡼⠁⠀⠀⢧⠀⠀⡴⠃⠀⠸⡆
⠀⠀⠀⠀⠈⡇⠙⠦⣄⠀⠀⡾⠀⠀⠈⠻⠋⠛⠀⠀⠀⠀⠈⢧⣸⠋⢣⠜⠁⠀⠀⠀⠘⣧⠞⠁⠀⠀⠀⢷⠀⠀⠀⣠⣶
⢶⣤⡀⠀⢰⡇⠀⠀⠈⠳⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣇⡴⠋⠁⡏
⠘⣆⠙⠦⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠇
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠛⢻⠇
⠀⠘⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡇⠀⠀⠀⠀⡞
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⡜⠀⠀⠈⢧⠀⠀⠀⠀⠀⡴⡄⠀⢠⠎⠀⢀⠃⠈⠀⠀⢸⠃
⠀⠀⠀⠘⢆⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⢦⣀⡶⢄⡰⠁⠀⠀⠀⠀⠳⣀⠖⢆⡜⠀⠸⣤⠃⠀⠀⢸⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠨⠳⣄⢠⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢃⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢣⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡴⠾⠧⢤⡄⠀⠀⣰⡖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠘⡀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠃⠀⠀⠒⠚⠲⡶⠲⠀⠀⠀⡴⣠⠞⢁⠧⠤⡀⠀⠈⠳
⠀⡴⠖⠒⠒⠒⠒⠉⡆⠀⠘⣆⠀⡜⠀⠀⠉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢠⢷⠁⠐⠁⠀⠀⠄
⠀⠹⡀⠀⠀⠀⠀⠀⠸⠜⠉⠉⠶⠀⠀⠀⠀⣹⠁⠀⣀⡁⠀⠀⠘⡀⠀⣈⠀⠀⠀⠀⠄⢀⠀⠀⠀⣴⠀⠀⠀⠀⡎⠈⠉⡹⠃⠀⠀⠀⠀⢀⠎
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⢉⠃⠀⠀⠀⢫⠀⢸⠀⠀⠀⠀⠐⡲⠒⠒⠚⢱⠀⠤⠀⣠⠁⠀⠈⡄⠄⣀⡏⠉⠉⠁
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡼⠀⠀⠀⡘⠀⠀⠀⠀⠈⡆⢸⠀⠀⠀⠀⡏⠁⠀⠀⠀⠘⠺⣤⠞⢢⠀⠀⠀⣉⡴⠋⢦
⠀⠀⠀⠈⢦⠤⠀⠀⠀⠐⢀⡈⠀⠀⠀⠉⡇⠀⠀⠀⣇⠀⠀⠀⠀⢸⠃⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣏⢀⡤⠜⠀⠀⠰⣁⣐⠤⠊
⠀⠀⠀⠀⠀⠙⠀⠒⠒⠒⠋⠈⠉⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠐⠁⢠⣃⠂⠀⠒⡲⠃⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢴⣦⠀⢠⣶⡄⠀⣰⣦⢠⣶⣶⣶⣶⡄⣴⣶⣶⣦⠀⠀⠀⢸⣿⣿⣿⡟⠛⣿⣿⠛⠻⣿⡟⠛⣿⠛⠛⠛⠛⢻⡟⠛⠛⠛⣿⣿⣿
⠀⠀⠀⠸⣿⡆⣼⣿⣿⢀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⡏⢀⠀⢿⠁⢠⣿⣿⡇⠀⣾⣿⡇⠀⠻⠿⣿⣿⣿
⠀⠀⠀⠀⢻⣷⣿⠃⣿⣾⣿⠀⠀⠀⣿⣿⠀⠀⣿⣿⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⡇⠈⠁⣸⡆⠈⠀⣾⣿⣿⡇⠀⣿⣿⡇⠀⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⠛⠛⠀⠘⠛⠃⠀⠀⠀⠙⠛⠀⠀⠛⠛⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡤⠤⣿⣷⢤⣴⣿⣿⣿⣧⣤⣿⣿⣧⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣷⣶⣶⣶⡿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢟⣕⡉⠉⠙⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣿⣿⣿⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠉⠉⠉⠙⠛⢛⣛⠿⠿⠿⠿⠿⡟⣿⣿⠇
⠀⠀⣠⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣹⢯⣿⣿⣶⣿⣿⣶⣶⣦⡀
⣰⣾⣛⠹⣍⠉⠉⠙⠛⠻⣿⣷⡀⢸⣿⣿⠛⠿⠿⠶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣀⣀⣾⡏⠉⠉⠉⣩⣩⡿⣿⣿
⣿⣿⢛⡻⠷⢶⣤⣤⣤⣾⢿⣿⠁⢸⣿⡇⠀⠀⠀⡆⣿⡇⠀⠀⠀⢸⡏⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠷⣿⣿
⢹⣿⢸⣿⠛⠛⠂⠀⣿⡇⢸⣿⠀⢸⣿⡇⠀⠀⢠⡇⣿⡇⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⣿⠀⠀⠀⠀⣿⡇⢸⣿⣀⣿⣿⡇⠀⠀⢸⡇⣿⠁⠀⠀⠀⣿⡇⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⣀⣠⣿⣿
⠈⣿⣿⠸⠀⠀⠀⠀⣿⡇⢸⡟⠛⠛⣿⣇⠀⠀⢸⣧⣿⠀⠀⠀⢠⣿⠁⠀⠀⣾⣿⡿⠟⠛⢻⣿⠿⣿⣿⣿⠏
⠀⢻⣿⡆⠀⠀⠀⠈⣿⠀⣾⡿⠿⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣀⣀⣀⣤⣿⣧
⠀⢸⣿⡇⠀⠀⠀⠀⣿⡀⣿⡇⠀⣿⢸⣿⠀⠀⠘⢻⣿⠀⠀⠀⢸⣿⠀⠀⠀⠙⠋⠉⠩⠽⡧⣼⣿
⠀⠀⢿⣿⠀⠀⠀⠀⠸⣇⣿⡇⠀⣿⣼⣿⡄⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣀⣀⣁⣿⣿
⠀⠀⠸⣿⡆⠀⠀⠀⠀⠉⠛⠁⠀⠉⠋⠉⠁⠀⠀⢸⣿⠀⠀⠀⣼⡇⡟⠀⠀⠀⣾⣿⣿⡿⠿⠛⠋
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⡇⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠸⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣘⣿⣀⣀⣀⣿⣛⠃⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣚⡩⠭⠭⢗⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⠤⠤⠦⠤⠬⠵⢦⣄⡀⠤⠖⣒⣉⣧⠞⣡⠊⠁⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⠔⣡⣖⡩⠤⢒⡄⢀⣀⠤⢒⣉⠥⠒⠊⠉⠀⠈⣇⡴⠁⠀⠀⠀⢀⣀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢊⡡⠤⠒⠒⠒⠒⣻⠶⠟⣚⡩⠔⠒⠋⠉⢏⠉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡴⠁⠈⡆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡥⢊⡁⠤⠤⠤⢒⣒⡩⠄⠚⠉⠀⠀⠀⠀⠀⠀⠈⡆⢸⠀⠀⠀⠀⣀⠤⠄⠒⠉⠉⠲⢄⣀⡜⠀⠀⢸⠁⠀⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠊⠀⠀⡇⢰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢒⣉⢅⠀⡇⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠃⠀⠀⠀⡜⡌
⣀⣀⠤⠤⠴⢖⣒⡄⣼⣠⠧⠤⠄⠦⣷⠀⡇⠀⠀⠀⢠⡀⣀⠤⠆⠀⠀⠀⢸⣧⠀⠸⡄⢸⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⠀⠀⠀⠀⡎⠀⠀⠀⡸⡜
⢱⠀⢺⠉⠉⠁⠘⠿⡇⠀⡎⠉⠉⠹⣿⠀⢡⠀⠀⠀⢸⣩⠴⡄⢸⠀⠀⠀⠀⢿⡀⠀⢇⠀⡆⠀⠀⠀⠸⠔⠊⠉⠀⠈⢆⠀⠀⠸⠀⠀⠀⢰⣱⣇⡀
⠀⢣⠈⢇⠀⠀⠀⠐⣥⠀⢁⠀⠀⠀⢻⠀⢸⠀⠀⠀⠈⣿⠀⢣⠀⢇⠀⠀⠀⠸⣧⠀⠘⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠆⠀⠠⡀⠀⠀⠸⣇⠀⠉⠳⣄
⠀⠈⢆⠈⣆⠀⠀⠀⢳⠀⢸⠀⠀⠀⠈⢇⠰⡆⠀⠀⠀⣿⠀⠘⡄⠸⡀⠀⠀⠀⢻⡆⠀⢣⠀⢇⠀⠀⠀⠀⡤⢒⣉⠥⠔⠒⢳⠀⠉⣒⣒⡨⢭⣣⠀⠀⠹
⠀⠀⠘⡆⠘⡄⠀⠀⠀⢧⠸⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⣿⡇⠀⢱⠀⢃⠀⠀⠀⠘⢷⠀⠈⡄⠸⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢇⠈⡇⠀⠀⠀⢻⡆⠀⢀
⠀⠀⠀⠘⡄⠸⡄⠀⠀⠈⣆⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⡄⠀⠈⡆⠈⡄⠀⠀⠀⢸⡇⠀⢱⠀⢃⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠘⡄⢸⠀⠀⠀⠈⣿⢀⠜
⠀⠀⠀⠀⠘⡄⠹⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡇⠀⠀⠸⡀⢱⠀⠀⠀⠈⣿⡀⠀⢇⠈⡆⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⡇⠈⠦⠤⠤⠤⢻⢋⡴
⠀⠀⠀⠀⠀⢱⡀⠱⡀⠀⠀⠉⠀⠀⢸⣆⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢣⠀⣇⠀⠀⠀⠸⣇⠀⠸⡀⢱⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠙⢶⢒⣒⣒⡨⠟⠋
⠀⠀⠀⠀⠀⠸⡳⡀⠱⡀⠀⠀⠀⠀⢸⠈⢆⠀⠀⠀⠀⠸⡆⠀⠀⠀⠘⡆⠘⡄⠀⠀⠀⢿⡄⠀⢣⠀⠓⢒⣈⣉⡡⠷⠀⠀⠀⠀⠀⡘⡇
⠀⠀⠀⠀⠀⠀⠱⣵⡀⢳⡀⠀⠀⠀⠈⡄⠈⢆⠀⠀⠀⠀⣷⠀⠀⠀⠀⠹⡀⠱⠄⠒⢒⣊⣹⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢳⠀⠀⠀⠀⣿⣧⠈⢧⡀⠤⠴⢻⠀⠀⠀⠀⠀⠓⠒⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡠⠞⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢣⣀⣀⣠⢿⡇⠳⠤⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣀⣀⠀⠉⠁⠈⠈⢁⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⠤⠄⢲⠚⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⠞⡸⠁⠀⠉⠙⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠉⠉⢉⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠓⢤⣀⠀⠀⠀⠀⣀⡠⠔⢋⡴⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠉⠉⠉⠉⣀⡠⠔⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⡩⠭⠭⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣊⠥⠄⠒⠒⠒⠋⠷⣤⠤⠔⣒⡨⢽⠴⢁⠔⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⣀⠀⢀⢞⣵⠮⠄⣒⣱⣄⡠⠔⣊⠥⠐⠊⠉⠀⠀⠘⣤⠃⠀⠀⠀⣀⢤⡀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⠥⠒⠊⠉⠉⣉⠭⠛⣋⠭⠔⠊⠉⠀⠈⢆⠰⡋⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⠀⠀⡔⠁⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠞⠡⠖⢒⡨⠭⠭⠔⠊⠉⠀⠀⠀⠀⠀⠀⣀⠼⠀⢇⠀⠀⠀⢠⠒⠉⠉⠀⠈⠑⠢⠜⠀⠀⡰⠁⠀⠀⠰⡍
⠀⠀⠀⠀⠀⣀⡀⠀⡜⡰⠁⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠔⠚⡆⠸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⢻⠁
⢖⠂⡭⠭⠭⠒⣷⣼⠒⡩⠭⠭⢽⣧⠀⡇⠀⠀⠀⡧⢒⡉⢱⠀⠀⠀⢸⣇⠀⢸⠀⢇⠀⠀⠀⢡⣀⡠⠄⠒⢆⠀⠀⠀⡇⠀⠀⢀⢮⠃
⠘⡄⠱⡀⠀⠀⠈⢹⠀⠁⠀⠀⠈⡿⠀⠇⠀⠀⠀⣿⠁⢣⠈⡆⠀⠀⠀⢿⡀⠈⡆⠸⠀⠀⠀⠈⠁⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⡟⠓⠦⣄
⠀⠸⡀⢣⠀⠀⠀⢹⡁⢸⠀⠀⠀⠸⡄⢰⠀⠀⠀⢹⡂⠘⡄⢰⠀⠀⠀⠸⣧⠀⢸⠀⢇⠀⠀⠀⠀⣀⠤⠔⣂⡩⠤⡀⠑⠤⠤⠼⠛⡄⠀⠘⣧
⠀⠀⠱⡀⢣⠀⠀⠀⢣⠀⡆⠀⠀⠀⠱⢘⠀⠀⠀⢸⡇⠀⢡⠀⠆⠀⠀⠀⢻⡄⠀⠇⠘⡀⠀⠀⠈⣷⠊⠉⠀⠀⠀⢣⠀⡎⠉⠉⠉⣿⡀⠀⢸
⠀⠀⠀⢣⠀⢆⠀⠀⠀⢆⡇⠀⠀⠀⠀⢻⡁⠀⠀⢸⡇⠀⠈⡆⠨⠀⠀⠀⠀⣷⠀⠸⡀⢡⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⡆⠸⡀⠀⠀⠸⡇⢀⠇
⠀⠀⠀⠀⢣⠈⢆⠀⠀⠈⣧⠀⠀⠀⠀⠈⠃⠀⠀⠈⣷⠀⠀⢸⠀⢣⠀⠀⠀⢹⡆⠀⢣⠈⡆⠀⠀⠀⢿⡄⠀⠀⠀⠀⢡⠀⠧⠤⠤⠤⠻⢃⠜
⠀⠀⠀⠀⠀⣆⠈⡆⠀⠀⠈⠀⠀⢹⡄⠀⠀⠀⠀⠀⡿⠀⠀⠀⢃⠘⡄⠀⠀⠀⢿⡀⠘⡄⢹⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⢲⢒⣒⣒⠠⠛⠉
⠀⠀⠀⠀⠀⢟⢆⠘⡄⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⠀⣷⠀⠀⠀⠘⡄⢰⠀⠀⠀⠸⣧⠀⢱⠀⣓⣂⣉⡉⠥⠇⠀⠀⠀⠀⡘⡇
⠀⠀⠀⠀⠀⠈⠪⣆⠘⡄⠀⠀⠀⠸⣀⠘⣄⠀⠀⠀⣿⠀⠀⠀⠀⢱⠀⠓⢒⣒⣉⠩⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠘⡄⠀⠀⠈⣻⢆⠈⠦⠔⠒⣻⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⢀⠀⠀⣀⠤⠚⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⠤⠤⠐⢻⡄⠓⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠤⠤⣀⣀⣀⣀⡀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋⢹⡁⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢀⡠⠔⢋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠤⠀⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠉⠒⠢⠤⠤⠤⠒⠊⣁⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣶⣤⣤⣤⣴⣖⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⡾⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⣇⠀⠀⠀⠀⣰⠋⣇
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡏⠈⢳⡄⠀⡀⢰⠇⠀⠀⠹⣆⠀⠀⢀⠀⠀⢠⠏⠀⠀⢹⡄⠀⢀⡼⠁⠀⢹⡀
⠀⠀⠀⠀⠀⡇⠙⠦⣄⠀⠀⢠⡇⠀⠀⠙⢾⠛⠟⠀⠀⠀⠀⠘⢧⣠⠏⢳⣴⠋⠀⠀⠀⠀⢻⣤⠞⠀⠀⠀⠈⣧⠀⠀⠀⣀⣴
⣤⣄⠀⠀⠀⡇⠀⠀⠈⠳⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⣆⣠⠞⠁⡾
⠘⡌⠳⢦⣀⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢠⡇
⠀⠹⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⢴⡆
⠀⠠⣬⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⠀⠀⣼⠁
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⣄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⢆⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡔⠁⢸⠀⠠⠀⠀⢠⡇
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠀⢀⠇⠀⠀⠀⠈⢦⠀⢀⣄⢀⡜⠘⣆⢠⠎⠀⠀⢸⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠙⠶⠁⠙⠎⠀⠀⠀⠀⠀⠀⠳⠏⠈⠋⠀⠀⠘⠃⠀⠀⠀⣼⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢡⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⡇⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⣀⣀⣀⣀⡀⠀⠀⠀⣀⠚⠀⠛⠛⠓⠒⠒⠀⢀⡼⡏⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⡄⠀⠀⣰⠄⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⡞⡰⠋⡠⠓⠒⠢⠀⠀⠈
⠀⣞⠋⠉⠉⠉⠉⠁⢹⠀⢀⣹⡄⢠⠃⠀⠀⠀⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢰⢻⣀⡀⣀⠀⠀⠐
⠀⠘⡄⠀⠀⠀⠀⠀⠀⠳⠁⠀⠘⠇⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⣄⣀⣀⡅⠀⠀⠀⠐⠀⠄⠀⠀⢀⠿⠀⠀⠀⠀⡇⠀⠉⡜⠃⠀⢀⡀⢀⣀⠜
⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢠⠁⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⢀⢡⠏⠉⠉⠋⢸⢠⠆⢀⡴⠀⠀⠀⣇⠤⢠⡜⠈⠈
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡏⠀⠀⠀⡜⠀⠀⠀⠀⠀⣃⢀⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠘⢓⠶⠛⢲⠀⠀⠀⣉⠽⠛⠲
⠀⠀⠀⠀⠣⡄⠀⠐⠒⠀⠀⣀⡁⠀⠈⠉⢹⠇⠀⠀⠀⣇⠀⠀⠀⠀⢩⠇⢸⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⣁⣀⠤⠚⠀⠀⠰⣅⣈⠶⠚
⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠚⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⣇⡁⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠘⠃⣰⣿⣿⣿⡿⣿⣿⣷⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢠⣿⣿⣿⡏⠀⣸⣿⣿⠃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⣸⣿⣿⣿⣦⣤⣾⣿⣿⣿⠃⠀⠻⠿⠋⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⠀⢀⣤⣤⡀⠘⢿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⣸⣿⣿⣷⡀⠘⣿⣿⡿⠛⠻⣿⣿⣿⠁⠀⣻⣿⣿⣿⣤⣤⣀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣧⠀⢸⣿⣷⡀⠀⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⡇⠈⣿⣿⣧⠀⣿⣿⣿⠀⠙⠻⣿⣿⣿⠁⠀⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⡀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠈⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣿⣿⠀⠀⠀⠘⢿⠋⠀⠀⠀⠀⠀⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠈⠙⢿⡿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠁⠀⠀⠀⠀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠀⠿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣧⢸⡆⡾⠘⣿⠃⢸⣟⠁⡇⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠸⠏⠻⠃⠀⠻⠀⠸⠁⠀⠆⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣠⡄⠀⠘⠁⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠿⣦⣤⡖⠒⣿⣿⠒⢲⣤⣴⠿⠋⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣭⡁⠀⠙⠃⠀⢹⡇⠀⠘⠋⠀⢈⣭⣥⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠁⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠁⠀⢙⣩⡿⠀⠀⠀⠀⣟⣅⡃⠀⠈⠛⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠷⣶⣤⠤⠄⠀⠈⠉⡁⠀⠀⠁⠀⢈⠉⠁⠀⠠⠤⣤⣶⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⡠⠒⠈⠉⠑⠢⢼⣀⠀⠀⠀⠀⣀⡧⠔⠊⠁⠁⠒⢄⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⢘⠀⠀⠀⠀⠀⠁⠈⠿⣿⣿⣿⣿⠿⠁⠂⠠⠀⠀⠀⠀⡃⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⡄⠀⠀⠈⢆⠀⠀⠀⠀⢉⡉⠀⠀⠀⠄⡰⠁⠀⠀⢠⠃⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣦⡀⠀⠀⠉⣶⣷⡶⠿⠿⠶⣶⣶⠉⠀⠀⢀⣴⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡉⠙⠒⠂⠀⠒⠛⢉⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣮⡉⣉⣵⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣷⣾⣿⣷⣶⡦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⢡⣶⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⡀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠛⠛⣋⣍⠀⠀⣿⣷⠀⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠻⣿⡏⠀⣿⠀⠀⢸⣿⠀⠀⣿⣿⡄⠀⠘⠟⢿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⠀⣿⡇⡀⢻⠀⠀⢸⣿⠀⠀⢸⣿⣧⠀⠀⣿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠛⠉⠙⠛⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⠀⠀⠀⢸⣿⣿⡇⣿⡇⡇⠘⠀⠀⣸⣿⣆⠀⢸⣿⣿⣄⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⢿⣿⣿⣿⣧⠀⠀⠀⠻⣿⣧⠸⡇⣿⡄⠀⣠⣿⣿⣿⣷⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠠⠤⠤⣀⣀⠀⠀⠀⢤⠟⠀⢀⡀⣸⣿⣿⠟⢿⣧⠀⠀⠀⠈⠻⣧⣴⣿⣿⣿⣿⡿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⣀⣤⣉⠀⠀⠀⠀⢀⠏⠉⣭⡹⣿⣿⠀⠘⣿⡄⠀⠀⠀⠀⢻⣿⣿⣿⠋
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠈⠏⠀⢀⣬⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡟⠀⣸⣿⣧⠀⠀⠀⠀⣸⣿⡿⠋
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣶⣿⣿⣿⠀⠀⠀⠀⠿⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣣⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠠⠟⠛⠻⠂⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠠⠚⢻⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀
⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⣋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠕⠁⠀⠀⠁⢦
⠀⠀⠀⠀⠀⣰⠿⠃⢸⣿⣿⣿⣿⡿⠁⠀⠈⠑⠫⢤⣒⣀⣒⣀⣨⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣽⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠋⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢰⣿⠀⠀⠀⠀⠀⢀⡾⠛⠛⢷⡀⠀⠒⠤⣄⣀⠠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠇⠀⠀⢸⡷⠶⠶⢤⣤⣉⣉⠛⠒⠾⠯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⣧⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⢤⣤⣀⡀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣿⣇⣤⣴⡶⠶⠶⣿⠄⡀⢠⠃⠹⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡿⠇
⠀⠀⠀⣠⡀⢿⡄⠀⠀⢠⣿⠁⢀⣠⣴⠾⣿⡋⠁⢀⣠⣶⣶⣿⠛⣁⣠⣤⠶⠟⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⣈⡻⣌⢷⡀⢀⣾⠃⣴⠟⠋⠀⠀⣿⡶⠞⣿⣭⣴⠶⣿⠛⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠈⠻⠿⠿⠁⣼⠇⠀⠀⠀⢀⣠⡤⣿⡟⢻⡏⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠖
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⡀⠀⠀⠀⢿⡇⢸⠇⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠰⠃
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠈⢦⠀⠀⢸⡇⣿⠀⠀⠀⠀⣿⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢰⡿⠁⣰⣶⠀⠀⠀⠈⣧⠀⢸⣧⣿⠀⠀⠀⠀⣿⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⣿⢁⣼⠟⢿⡄⠀⠀⠀⢸⡆⠘⣿⣿⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⢀⣀
⠀⠀⠀⢀⣿⢣⡾⠃⠀⠈⢿⣄⣀⣤⡾⠁⠀⣿⣿⡇⠀⠀⠀⣿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣾⠟⠁
⠀⠀⢀⣾⣷⡿⠁⠀⠀⠀⠀⠉⢩⢥⡄⠀⠀⢸⣿⡇⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠅
⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⡰⢫⡟⠀⣤⣧⡈⣿⡇⠀⠀⠀⢻⡇⣷
⠀⠘⠟⠁⠀⠀⠀⠀⢀⡤⢊⣴⠋⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠘⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠠⠒⠉⠰⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠐⠂⣼⣿⣿⡿⣿⣿⣇⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢰⣿⣿⡿⠀⣸⣿⡿⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠂⠀⠀⠀⢿⣷⡀⣀⣠⣼⣿⣿⣿⣾⣿⣿⣿⡇⠀⡈⠋⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⣾⣿⣦⠀⢿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣇⠀⠁⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⣆⠈⣿⣿⠀⢸⣿⣿⡇⣠⣾⣿⣿⣿⣿⣿⣦⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⡀⢹⣿⣧⠸⣿⣿⡇⠙⢻⣿⣿⡏⠉⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣧⣸⣿⣿⡀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⢿⣿⡇⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⠟⠀⠘⠻⠃⠀⠀⠀⡏⠀⠀⠀⠀⠘⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠛⠃⠀⠀⠀⣿⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠀⠀⢻⠀⡼⠁⣠⡄
⠀⢀⣄⠀⠀⢴⣦⠀⠀⠀⠘⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠢⣄⡀⠀⠀⣀⡴⠋⠘⣇⠀⠈⠀⠠⠞⠉⠀⠀⠀⠀⢀⡀
⠀⠈⠙⠛⠶⢤⡈⠑⠀⠀⠀⠀⠳⡌⠓⠦⣄⣀⠀⢀⣀⠜⠁⠀⠀⠈⠉⠙⠉⠁⠀⠀⠀⠈⠲⣄⣀⣀⣀⣀⣠⠤⠖⣺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣆⠀⠀⣨⠭⢭⠁⠀⣠⠤⠤⣤⡤⠖⠚⠛⠓⠲⢤⡀⢀⣀⣀⠉⠁⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠸⡄⢠⡇⠀⠈⢧⠀⡏⠀⢀⡇⠀⢀⣠⣤⣤⡀⠀⠈⣿⠀⠉⡇⠀⢀⡤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡒⠒⢷⠀⠀⢳⡼⠀⠀⠀⠘⣾⠇⠀⢸⠀⢠⠏⠀⠀⠈⢻⡄⠀⡟⠀⠀⡇⡰⠋⠀⠀⣧⣄⣠⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠘⣇⠀⠘⠃⠀⣰⡆⠀⠘⠀⠀⡟⠀⡏⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠟⠁⠀⠀⠀⡇⡸⠃⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⢹⡄⠀⠀⢀⡏⢻⡄⠀⠀⢠⡇⠀⡇⠀⠀⠀⠀⢠⠇⢰⡇⠀⠀⠀⣠⣶⠀⠀⡿⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⢴⠶⠭⠥⠤⣤⣀⣀⣧⠀⠀⣸⠁⠀⢷⡀⠀⢸⠀⠀⠹⣄⠀⢀⡠⠋⠀⢸⠀⠀⢀⡴⠃⣿⠀⠀⠀⢀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡤⠇⠀⠀⠈⠷⠶⠛⢷⣄⠀⠈⠉⠉⠀⠀⣠⢾⣀⣠⠞⠀⠀⣿⠀⠀⢠⠞⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⠻⠶⠶⠶⠖⠋⠁⠈⠉⠁⠀⠀⢀⣿⣀⣠⠛⠒⠒⠒⠒⠺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢠⠏⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⠞⠉⠀⠀⠀⠉⠓⠦⣀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠈⠳⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣧⠀⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠂⣘⢸⣶⠄⠀⠀⠀⢀⡄⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠷⣶⡶⠂⠈⠁⠀⡔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢴⣿⡏⣻⢻⣾⠃⠀⠀⠀⠀⠘⠛⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⣤⣴⣍⡿⢳⠼⢼⣧⢠⢀⣀⠀⠀⠀⠀⠰⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⢻⠋⠁⠈⠹⠀⠀⠀⠀⠈⠉⣻⠆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠛⣂⡄⡀⠀⢀⣄⣀⠀⠀⠀⠀⣴⢿⡆⠀⠀⢀⣠⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠷⣾⡉⠁⢀⣀⠤⣅⠀⠀⠁⠀⠹⣇⣴⠆⠈⢗⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠘⣿⣀⡞⢰⠀⠘⠓⠦⣄⣀⣰⠌⠙⠶⣄⣼⣇⢈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠸⣧⣀⡷⢋⣿⣗⢼⠀⢰⣂⣠⣿⣿⡁⠀⠀⠀⢻⣶⠝⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⢿⡇⡞⠻⡏⠀⠀⠘⣋⠀⣀⣽⣆⠀⠀⠀⠸⡄⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢲⣾⣞⣻⢾⠉⢀⠀⠰⣍⣯⠥⠭⠝⠂⠀⠀⠀⡃⠀⠀⠀⢀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠙⠈⠉⠁⠀⠘⠀⠀⢃⣴⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⢀⡿⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⢰⣆⠀⢰⣿⢸⣀⠀⠀⠀⡀⡐⠈⡁⠙⠀⢻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡟⢀⠀⠀⠀⠀⡈⠀⠸⠁⡬⠉⠀⠀⠀⠋⢈⣇⠀⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠈⢉⠄⠀⠀⣀⣿⠀⠀⢀⡇⠤⠄⠀⠀⠀⢻⣷⠛⠛⠋⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠤⢤⣈⣹⠁⠟⠃⠀⠀⠀⢀⡅⠀⠸⠛⢣⣀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣅⡿⠆⠀⠀⠀⠀⠻⢷⣾⣇⠀⢈⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠕⠁⠀⠀⠠⡌⠀⠀⠀⠀⠙⢿⣶⡦⠐⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠰⡿⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢰⣟⡀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠙⢿⣷⡿⠙⠇⠀⠀⠀⠀⠀⠀⠀⢸⠇⢐⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠱⣄⠀⠀⢢⠀⢀⡴⠻⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡶⠦⣤⣀⡘⠦⠀⣈⡷⠋⠀⠀⠙⢿⣄⣠⡏⠀⠀⣠⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡄⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠾⠷⠖⠚⡿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣃⣤⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡤⠶⠋⣿⡆⢀⡄⢀⣾⡿⠀⢀⡀⠀⠀⣤⣤⡄⠀⠀⠀⠀⣀⣿⡞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣿⣧⣾⣧⣾⡿⢡⣾⡿⣿⣷⡀⢸⣿⡇⠀⠀⠀⣰⣿⠛⣿⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⢻⣿⣿⠿⣿⡿⠁⠸⣿⣷⣿⡿⠃⠈⣿⣇⣴⡆⣴⣿⠃⠀⠈⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⢀⣾⠏⠉⠀⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠿⠿⠻⣿⣿⠃⣀⣤⣾⣟⣫⡉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⠁⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣹⠛⠁
⠀⠀⠀⠀⠀⠐⠚⠉⠁⠚⠋⠉⠉⠉⠉⠉⣩⡛⢷⣦⡀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠻⠛⠳⣤⡀⠀⠀⠀⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣴⠏⣰⠗⠙⣷⣠⠞⣋⠭⠡⣶⠈⠹⣍⠃⣍⣉⠀⠀⠉⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠃⠀⠀⢸⢣⣿⠇⠐⠳⠸⡇⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠻⡀⠀⠀⠀⠐⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⣠⡴⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠚⠁⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠀⢠⣿⠟⣿⡄⢸⣿⣆⢀⣿⡇⢀⣾⠟⢿⡄⣾⡆⠀⠰⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡀⠀⠀⢸⣿⠀⢸⡇⢸⡟⣿⣾⢻⡇⢸⡿⠀⠈⠁⣿⡇⠀⠀⣦⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠱⠀⠀⠸⣿⡀⢸⡇⢸⡇⠙⠋⢸⡇⢸⣿⠰⣶⠀⢿⠇⠀⠀⠘⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⠀⠀⠀⠹⢷⡾⠃⢸⡇⠀⠀⠸⡇⠈⢿⣶⡟⠀⢴⠄⠀⢤⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⣁⡀⢀⣀⡴⠞⠦⠴⠳⢤⣀⣀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡄⠀⢰⠀⢸⠀⡌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⡀⢹⢸⡇⡿⠸⡠⠖⠛⠛⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⢰⡇⢸⢸⡇⡇⢠⠇⠀⠀⠀⠉⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠈⠷⣦⣤⣥⠶⠋⠀⠀⠀⠀⠀⢮⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠀⡄⠀⢧⡀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⢎⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠄⠠⣅⠀⠀⠀⠀⠀⠀⢀⡴⠲⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠟⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠙⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣿⣿⡿⠿⠋⠉⠉⠙⠛⠉⠁⢠⠀⢀⡀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⡴⠀⠀⠀⠲⢄⠀⡀⡀⠒⠦⣤⣄⠈⢰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⡄⠀⠀⠀⠀⠘⡿⣯⣛⢳⣄⠈⠙⠻⣄⣿⣿⣿⣷⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠙⢲⡝⣷⡀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⠼⣿⡀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣰⢇⡴⠃⠀⠀⠀⢻⣆⣹⡇⠀⠀⠀⠻⣿⡿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⡇⠀⠀⠀⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⢼⣿⣿⠇⠀⠀⣀⣀⣜⣃⠹⡇⠀⢀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣆⢹⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⠾⠟⠻⠿⠿⠛⢻⡿⢿⣿⣿⣿⣿⡿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣯⣤⣤⣤⠀⠀⠀⠀⠀⠀⣾⠁⣾⡟⠉⠈⠀⠀⠠⠁⢀⣠⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣴⡟⢱⡟⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⢐⣿⢘⣿⠃⠀⠀⠀⠠⣿⠟⠛⠫⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠺⣿⠀⠀⠀⠀⠀⢸⡷⢊⣿⠀⠀⠀⠀⢸⡟⠀⠀⠀⠱⠀⠰⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⡀⠀⠀⠀⠀⠸⣿⢈⣿⠀⠀⠀⠀⠸⣷⠀⠀⠀⣼⣿⣿⣿⣿⡋⠉⠀⠀⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⠀⢀⢹⣧⠀⠀⠀⠀⠀⠿⣎⠻⣆⠀⠀⠀⠀⢿⣆⠀⠰⠿⠿⢿⣿⣿⣿⣿⣶⣦⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣧⣼⣶⣿⣿⣷⣶⣶⣄⣀⣐⣻⣷⣿⣶⣶⣦⣄⠘⣿⣿⣶⣶⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣛⣙⡛⠛⠛⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡿⠿⡿⠌⠙⠛⠛⠋⠙⣭⠋⡍⠻⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢴⡞⣧⢺⡛⣿⣿⣷⣄⡀⠀⠀⠈⢫⡈⣷⡀⣠⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⢋⢷⣛⡙⠋⠉⠉⠉⠁⠀⠀⠀⠀⢳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⠴⣭⠆⠀⠀⠀⣀⣠⣴⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣴⣿⣿⠟⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢨⠀⢀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠐⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠤⠤⠐⠒⡠⠀⠠⡀⠀⡀⠀⠀⠀⠘
⠀⠀⠀⣀⣀⡤⠤⠤⠒⠒⠀⠉⠉⠀⠀⠀⠀⡠⠄⠀⡌⠀⢀⠠⠁⠀⠇⠀⠀⠀⠀⡄
⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢂⡉⠀⠈⡀⠀⠀⠃⡠⠊⢡⠠⠤⠂⠀⠀⠃
⢸⣷⠀⠀⠀⠀⠀⢀⠔⠁⠈⠳⡘⠀⠠⡘⡇⢀⠀⢸⡀⠀⠈⠄⡖⢸⢀⠂⢐⠀⠀⢸⠀⠀⣀⣀⠀⠀⣀
⠀⢻⣧⠀⠀⠀⠀⡌⠀⢀⠤⡀⠸⡆⠀⣷⠀⢸⡄⠀⢧⠀⠁⠊⢁⠆⠘⠐⠈⠀⠀⠀⣶⣿⣿⣿⡿⠷⠾⠛⣛⣛⠻⢝⡳⢆⡀
⠀⠈⣿⣆⠀⠀⠀⠇⠀⠀⠀⠐⠀⡇⠀⢠⠆⠘⠡⣀⠬⠃⠂⠐⠁⠀⠀⠀⠀⣠⣖⣾⠛⢿⣿⠷⠒⠛⠿⠛⠉⠉⠿⣷⣿⣎⢿⣤⣀
⠀⠀⠸⣿⡄⠀⠀⠆⠀⢆⢀⡀⢠⢱⠀⣸⠘⡄⠂⠁⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⢿⣿⡞⣡⣤⣶⣖⣶⡿⣟⣻⣄⠀⠈⠻⣿⣿⣽⣧
⠀⠀⠀⢹⣿⡀⠀⠈⢄⠀⠀⣠⠂⠘⠁⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠟⠛⣻⣟⣯⠁⠸⠟⠚⠛⠛⠻⠿⠻⠟⠿⠿⠟⠛⢆⠀⢸⡏⣿⣿⡄
⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠹⡁⠀⠀⠀⣿⣿⠟⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⣀⣤⡴⠾⠛⠉⠙⠿⣿⣷⣦⢄⣙⣄⠀⣰⠏⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⢿⡷⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠀⠈⠁⢻⡷⢛⣵⡀⠀⠚⠉⠛⠳⠆⠀⠀⠀⠀⠶⠛⠛⠳⠼⡴⣽⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣼⢿⣿⣿⠀⠀⣴⠶⣶⠦⠄⠀⠀⢠⠆⠘⣿⠟⣶⠄⢸⣿⣿⣾⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠉⣿⣄⣰⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡛⠊⣽⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣼⣿⣮⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⡁⣀⡉⠀⠀⠀⠠⠏⣾⣿⣿⣿⣿⣿⣇⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡻⣿⣿⣧⠀⢀⠀⠀⠀⢾⡭⠭⡭⢭⡷⠀⠀⠀⢰⣿⣿⣯⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡄⠀⠀⠢⠀⠀⠀⠀⡠⠀⠀⠀⣼⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠷⠀⠀⠀⠁⠀⠀⠐⠁⠀⠀⠐⠛⠧⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠉⠀⠈⠀⠀⠀⠌⠀⠀⢀⡆⠀⠀⠀⠄⠀⠀⠐⡀⠀⠀⠀⠈⠙⠛⠶⢶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡌⠡⡀⠀⢀⠛⡄⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡸⠁⠀⠘⠀⡘⠀⢱⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡾⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠘⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣌⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣐⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠠⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠳⢄⠀⠀⠌⠀⠘⢷⡀⠀⡀⠈⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⡀⠄⠀⠀⠙⠎⠀⠀⠀⠈⠷⠈⠀⠀⠀⠀⣿⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠂⠀⣾⠇
⠀⠀⠀⠀⠀⠈⢉⠀⠀⠄⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⢤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢶⠶⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠹⡄⠀⠀⠀⠸⣆⠀⠀⠀⡼⠁⠀⢀⣠⣀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠞⠃⠀⠂⠀⠀⠿⠆⠀⠀⠁⠀⢀⡀⠀⠋⠀⠀⠀⠀⠀⣰⡿⠋⠉⢀⣀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢀⡀⠀⠀⠀⣿⠛⠁⠀⠀⠀⢿⠁⠀⠀⠀⢠⡀⠀⠈⠃⠀⠀⠀⢀⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⣀⣀⣠⣿⠀⠀⢤⣤⣶⠾⠆⠀⠛⠛⠻⠃⢤⣀⡀⢀⣀⣴⡿⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣶⠟⠋⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠛⠉⠉⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠁⠒⠘⠷
⠀⠀⠀⠀⠀⠀⠘⠚⠛⠉⠉⠀⣀⣴⠾⠛⠀⠀⢀⠀⠀⣰⡄⠀⠀⠀⣶⠄⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠴⡆⠀⢠⣾⠏⠀⠚⠄⢰⣿⠀⠀⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠚⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣯⣽⣿⣯⣽⣯⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⠉⠉⠐⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣡⠄⠀⢀⠜⠁⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠃⠀⠀⢀⠔⠂⠁⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠋⢀⠏⠀⠘⠃⠉⠀⠄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⡄⠀⠀⢰⠁⠀⡠⠂⠀⠀⠰⠋⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠄⢀⡤⠐⠒⠂⠸⠀⠀⠀⠊⡠⠀⠀⠰⠄⠄⢀⠸⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⡠⠁⢠⠃⠀⠀⠀⠀⢠⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠊⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢀⡔⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠘⠀⠴⠃⢀⡄⠀⢀⡔⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡦⠴⠃⠀⡫⠀⠀⢀⠂⠠⠠⠖⠉⠀⠛⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢊⢁⠔⠁⡠⠃⠀⠈⠈⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢀⠠⠐⠂⠀⠀⠀⢄⠀⠀⠀⢠⠂⠀⠈⡀⠔⠊⠀⢠⠤⠄⠐⠪⠁
⠀⠀⠀⢀⡠⡄⠄⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠸⠁⠌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⢀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠘⠂⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡟
⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠂⠀⢀⠔⠀⠀⠀⠀⡠⠊⠀⡾
⠀⠀⠀⢠⡀⠀⠎⠀⡞⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠂⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠁⣀⠞⠀⢀⡾⠁
⠀⠀⠀⠀⡀⢠⡀⡜⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠰⠀⠀⠇⢀⠞⠃⠀⠀⢠⠊⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠂⠀⠀⣄⠀⠈⢀
⠀⠀⠀⠀⠀⠁⢀⠔⠁⠀⠀⠀⠀⢸⠀⠀⠀⢘⠀⠀⢀⠀⠀⡠⠀⠠⠀⠀⠀⠀⠀⢸⠁⣀⡀⠀⠀⠀⠀⠀⢻⡀⢀⠀⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣀⣤⡆⠀⠀⠘⠀⠀⠀⢆⡰⠀⠁⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠚⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⡀⠀⠀⡇⠀⠀⠀⠀⡗⡀⠀⠀⠀⢸⡄⠀⠀⠀⢂
⠀⠀⠀⢀⠀⠀⠸⠀⡄⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⡃⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⡈⡅⡠⠃⡔⠁⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠰⣤⠤⠤⠶⠋⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⡰⠁⠉⠴⠉⡀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣼⠏
⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⢂⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠡⡀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡦⠻⣿⣿⠂⠀⠀⠀⠀⢦⡑⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⢻⣿⣦⣖⣮⠁⠀⠀⠀⠀⠀⢨⡟⠀⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⣿⣿⡿⠛⠀⠀⠀⠀⠦⠀⣶⣿⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣷⠀⢀⣾⠿⣍⢔⣲⣒⣀⣀⣀⡘⠻⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢿⣷⣼⡇⠀⣈⠉⠛⠛⠛⢛⠿⠿⣦⢙⢀⣾⣿⣷⢲⡎
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠙⢿⣥⠶⣚⠀⠀⠀⠀⠔⣲⢶⣼⡿⠿⠛⠁⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⡀⠀⢀⣸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠷⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠷⣷⣿⣿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⠀⠈⢉⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⢠⣤⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠁⠸⠿⠟⠀⠰⡿⠿⠶⠂⠀⠀⣿⠿⠟⠿⣿⣿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⡦⠀⣀⣾⣄⠀⠀⡀⠀⠀⢀⣼⡷⣤⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡏⠀⠀⠀⠤⠖⠋⠉⣷⢄⣠⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⠀⠀⠀⠘⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠒⠦⢄⣀⣀⡀⢀⣀
⠀⠀⠀⠀⡠⠄⠠⠴⠲⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠔⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⣒⠒⠒⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄⠀⠀⠀⣶⣶⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣿⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣷⡄⠀⣠⣿⣿⣇⠀⠀⢠⣿⡏⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⢸⣿⡏⢻⣷⣴⣿⠛⣿⣿⠀⠀⢸⣿⡇⢠⣤⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇⠀⢸⣿⡇⠀⠻⡟⠁⠀⢹⣿⡇⠀⠘⣿⡇⠈⢹⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠈⠿⡧⠀⠀⠘⠿⠶⠿⠃

⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⣿⣿⠀⠀⠀⢸⣿⣿⡄⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⣰⣿⠛⠉⠛⣿⡇⠀⣼⣿⣿⣿⣦
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⢻⣿⡆⠀⠀⢸⣿⢿⣿⡄⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠛⠃⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⣿⡇⠀⠀⢸⣿⠀⠹⣿⣿⠟⠈⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣆⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⣿⣷⠀⠀⢸⣿⠀⠀⠙⠃⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⢲⣶⣀⣾⡟⠀⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠘⠛⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⣶⣶⣶⣶⣶⠀⢠⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠙⠿⠇⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠅⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⣤⡄⠀⢰⣶⣿⣿⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣴⡿⠃⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠙⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣠⢦⡀⠀⠀⠀⢸⡉⠁⡠⠄⠀⠐⠂⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⢀⡟⠓⢦⡀⠀⢸⡹⠙⢦⡀⠀⢀⡾⠁⠀⠹⣦⣀⡴⣋⣳⠀⠁⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⣾⠈⠙⢦⣼⠃⠀⠀⠙⠶⠟⠁⠀⠀⠙⢳⣾⣀⣄⠀⠀⡹⠿⠊⠀⢈⠀⠀⠀⠀⠀⠀⡄⠸⣀
⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣸⠉⡹⠷⣶⠃⠀⡠⢄⣀⡤⣄⠀⠀⣠⠤⣄⠼⠢⣀⡎⠀⠀⠀⠱⠞⢀⣀⡀⡰⠹⡀⢀⠀⠀⠀⠀⢠⠀⢘⡄
⠫⣵⠀⠀⠀⡰⠋⢉⡁⠀⠀⠈⡉⣀⠀⠀⠑⠖⠁⠀⣀⡀⠀⠑⢴⠁⣀⠠⠀⠀⠈⠁⠆⠐⡄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡀⠸⡀
⠀⠀⡠⠏⢉⢁⡊⠁⠀⠀⠀⠀⠒⠀⠐⠠⡀⠀⠉⠁⠀⠈⠢⣀⠀⠀⠁⠀⠀⠐⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠚⠀⠀⠀⠀⠁⡡⠤⡹⡄
⠀⠀⢳⠀⢡⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠈⢆⠀⠐⠀⠀⠀⠈⠉⠙⢄⠀⠀⠀⢆⠀⠀⠊⢰⠂⠀⠀⠒⠒⠀⢀⠅⠀⢁⠘
⠀⠀⠀⢆⠈⡇⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠈⠀⠀⢰⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⠀⠀⠃⠀⠀⠀⠀⢀⠀⠂⠀⠠⠈
⠀⠀⠀⠸⡄⠁⠀⠀⠀⠀⠀⠀⠀⠠⠮⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠀⡇⠀⠀⠓⣄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⢘⠀⠀⣀⢀⣀⣠⠖
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⣀⠀⠀⠀⠑⠈⠆⠀⠀⠸⠓⢄⠀⠀⠀⢰⠐⢄⣀⠠⠠⠋⠉⠈⠀⠀⡏⢩⣤
⣠⢄⡀⠀⠘⡆⡄⠀⠀⠀⠰⠝⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠀⠑⡀⠀⢀⠄⠐⠤⠘⠐⠀⢀⡑⠈⠉⠀⢀⣄⠀⠀⣤⠤⠴⠾⠿⠾⠀⠀⠯⠗
⠙⠈⠁⠀⠀⠇⢰⠀⠀⢀⡀⠀⠀⠀⠀⡈⢀⡤⡠⠄⣀⠄⠀⠀⠈⠀⠁⠀⠀⢀⠶⠤⠴⠋⠙⠚⠛⣲⠧⣤⣙⡊⢹
⠀⠀⠀⠀⠀⢸⠈⡄⢀⡄⠡⣀⠤⠔⠋⠀⣀⡀⢀⣠⢄⣀⣸⠉⠲⠒⠉⢉⣉⡉⠀⠀⠀⣰⠦⣄⣀⣿⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠘⠦⣀⠤⠴⣄⣠⣤⡤⠤⣴⡷⠟⣿⡄⠀⠈⢹⡷⣄⡀⢠⡟⠉⠙⠶⣄⣼⠃⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠉⠀⠀⢹⡇⠀⠀⢸⠇⠀⠙⠟⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠗⠀⠀⢀⡀⠈⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠢⣿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡶⣤⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣼⠃⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⣿⣿⡾⠿⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⠂⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⡰⠋⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠂⠲⠃⢠⠎⡄⢀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⢀⠊⠀⡠⠄⠀⢀⠖⠀⠀⠀⢀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢈⠀⠈⠀⠀⡄⢀⡠⠄⠀⠠⠅⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠈⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠀⠀⠀⠂⠀⠀⠀⠀⡰⠀⠀⠀⠐⠁⠔⠁⣠⠀⠀⠄⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡇⠒⠀⠪⠁⠀⠠⠂⠀⠰⠈⠀⠘⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠔⢀⠔⠁⠀⠁⡁⡤
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠐⠁⠐⠀⠁⠀⠀⠀⠀⡀⠀⠰⠁⠀⠤⠂⠁⠀⠐⠀⠀⠁
⠀⠀⠈⡴⠐⡉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠜⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢀⢴⠄
⠀⠀⠀⠀⠀⠀⡔⠁⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⢀⠔⠈⠀⠁⠀⠀⡠⠂⠀⠂⠀⠀⠤⠀⠀⠀⢀⠔⢁⡞
⠀⠀⠀⡄⠀⠈⠀⠎⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠊⠀⠀⢀⡔⠁⢀⡾
⠀⠀⠀⠀⠀⡅⡌⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⢃⠽⠉⠀⠀⠰⠁⠀⠀⡠⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢰⠁⠀⢲
⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⣇⢠⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠛⠃⠀⠀⢁⠀⠀⠰⠂⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⢹⡆⠀⡀⠀⠀⢸⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠂⠀⠘⠀⠀⠈⠀⠀⠁⠀⠀⢸⡄⠀⠀⠅
⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠸⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠠⢁⡠⢁⠜⠀⠀⠀⢠⠟⠀⠀⠀⠀⠈⠦⠖⠒⠚⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠋
⠀⠀⠆⠀⠈⠠⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠁⢀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁

⡶⣶⢶⠶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣶⣦⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⣠⣤⠨⣷⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟⠛⢿⡿⠀⠘⣇⠘⣯⣤⠄⢹⡖⢺⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠋⠉⠙⢻⡀⢀⡀⣰⡄⠹⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⣾⣿⣷⠀⣷⡀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠈⠉⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⠿⠿⠿⠿⡿⠿⠟⠋⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⣁⣀⣉⠙⢿⣦⡀
⠀⠀⠀⠀⢀⣴⣾⠿⠟⠛⠛⠻⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣾⠋⡉⢻⣧⠈⣿⣷⡀
⠀⠀⠀⣴⡿⠋⣁⣴⡾⠿⠿⢶⣦⣄⠙⢿⣦⡀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⣠⣤⣶⣶⣶⣼⣿⠁⣼⡏⣸⡇⢸⡏⢀⣿⣿⡇
⠀⠀⣼⡿⠁⣼⡟⣡⡶⠖⠒⢶⣌⠻⣷⡀⢻⣷⣄⣴⣾⠟⠛⠛⠛⠛⢿⣿⠟⢉⣁⣀⣈⡉⠋⠀⣿⠁⣿⠃⣿⡇⢸⣿⣿⡇
⠀⢠⣿⡇⢸⡟⢰⡟⢠⣾⣿⣦⠙⣆⢹⣷⠀⠿⠿⠟⢁⣴⠿⠛⠻⢷⣄⢀⣴⠟⢋⣉⡙⠻⠿⢿⣿⢰⣿⠀⣿⠀⢸⣿⣿
⠀⠘⣿⡇⢸⣷⠘⣷⡈⠻⠿⠿⠇⢻⠀⣿⣦⣤⣶⣶⡿⢃⣴⠟⣷⡄⣿⣿⠏⣴⡿⢩⡙⢦⣾⠀⣿⢸⣿⠀⣿⠀⢸⣿⣿
⠀⠀⢻⣷⡀⢻⣷⠈⡛⠿⠷⠶⠿⢿⣷⣤⡙⢋⣴⢦⡀⣾⠃⡀⣸⡇⣽⡏⣸⣿⢁⡿⢁⣾⣿⠀⣿⡌⠛⠠⣿⡀⠸⣿⣿
⠀⠀⢸⣿⠀⣾⡏⣼⠇⣾⣶⣶⠆⣾⠀⢹⣧⣾⠁⠘⣿⡏⢸⠁⣿⠁⡿⢀⣿⠇⡼⢁⣾⠃⣿⡀⠛⢀⣿⣦⠈⣿⡄⢹⣿⣧
⠀⠀⣿⡇⠀⣿⠀⣿⠀⣿⣿⡟⢠⡟⢰⡀⢿⡇⣸⡀⠿⠁⡟⢸⡿⠀⣠⣾⣿⠀⣠⣾⠃⡀⣿⡇⣷⠦⠤⢤⣴⠟⢀⣼⣿⣿
⠀⠀⢿⣷⠀⣿⡀⣿⣇⠹⠟⣠⡿⢡⣿⢷⣤⣴⡿⠿⣶⣾⡇⣼⣇⣼⠏⠸⣿⣶⡿⢁⡾⠃⣿⣤⠶⠖⠛⣠⣿⠀⣸⣿⣿⠏
⠀⠀⠸⣿⡄⠹⣷⡈⠻⠶⠾⠛⣠⡿⠃⣠⣀⣀⣤⣦⡀⢻⣇⠸⠿⢋⣴⣦⣌⣩⠄⣡⡴⠛⣿⠀⣶⠿⠛⠛⢁⣴⣿⣿⡿
⠀⠀⠀⠹⣿⣆⡈⠻⠷⣶⡶⠿⠋⣁⣼⣿⣿⣿⣿⣿⣷⡀⠛⠿⠶⠿⠋⢨⣿⢃⡼⢁⡄⢸⡿⢰⣿⠀⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣤⣶⣾⣿⣿⣿⠟⠀⠀⠀⠙⠿⣷⣶⣶⣶⡆⢸⣿⢸⣇⠺⢃⣾⠃⣼⡇⢰⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣧⠈⢿⣆⡙⠻⠟⢃⣼⠟⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠙⠻⠿⠿⠛⢉⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣶⣶⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⣿⢸⠉⠙⠛⠲⢤⡀
⠀⠀⠀⠀⣀⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀⡟⠀⢢⣤⠞⠸⠶⠶⣦⣀⡾⠀⢀⣤⣄⣀⡀
⠀⠀⣠⠞⠁⠀⠀⠀⢹⠀⠀⠀⣀⣤⣄⠀⠀⢹⠈⢳⡀⠀⡏⠉⠉⢡⠀⡾⠁⠀⠀⠀⠀⠀⠉⠁⠀⢸⠃⠀⠈⠉⢳⡆
⠀⠀⢻⡄⢰⣒⡄⠀⣾⠀⠀⡟⠁⠀⠘⢧⡀⠘⡇⠀⠻⡄⣇⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡟
⠀⠀⠀⢷⠀⠉⢀⣴⠏⠀⢀⡇⠾⠓⢦⡈⢳⡄⣧⠀⠀⠹⣿⠋⠀⢸⢸⡄⢀⡶⠲⠶⣤⣀⡀⠀⢠⡇⠀⢀⣈⡿
⠀⠀⠀⢸⡶⠛⠛⠛⠳⡶⠾⢧⣄⣀⡤⠽⠂⣹⣿⠀⠀⠀⠙⠀⠐⢺⠀⢳⡘⠷⣤⠀⠀⠈⢹⠆⣸⠦⠀⠀⣼⠁
⠀⠀⠀⢸⣧⠀⢀⣠⣄⣀⠀⠀⠈⢧⢠⣄⡀⠁⠘⢆⠀⣠⠀⠀⠀⢸⡀⠀⠉⠹⠉⠀⣴⢤⡾⠀⣿⣀⠀⣰⠃
⠀⠀⠀⠈⡗⠀⢸⠀⠀⠈⠳⡀⠀⢸⢹⠈⠛⢦⣀⣸⠁⢸⣦⠆⠀⢸⠋⠛⠛⣿⠀⢸⠃⠀⠀⢀⣠⣍⡙⠋
⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⣰⠁⠀⣸⢸⡄⠀⣸⣉⣁⣠⡿⠹⠶⠶⠚⠀⠀⠀⠛⠒⠛⠀⠀⠀⡟⠃⠈⣷
⠀⠀⠀⠀⢻⠀⠘⠓⠚⠋⠁⠀⣠⡧⠼⠃⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠁
⠀⠀⠀⠀⢸⣄⣴⣤⣤⣤⡷⠞⠋
⠀⠀⠀⠀⠀⠈⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⠊⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⢳⠤⣄⣀⣀⣤⠞⠀⠳⢤⣀⣀⣠⠴⢛⡇⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠙⠢⠀⢸⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣄⠀⠀⢰⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢠⡄⠀⠀⢸⡇⠀⡏⠈⢳⠀⠹⣆⣶⣬⡇⠀⢠⡟
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠘⡆⠻⡄⣶⣼⡇⠀⢷⣀⣼⢀⠀⠹⠇⠙⠃⠁⠸⡇⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠀⠖⠂⠀⠀⡇⠀⠹⡏⠻⠃⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀
⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⡀⠀⠈⠳⣄⠀⠀⢀⣠⡤⠴⠦⢤⣰⠃⠀⠰⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠎⠀⠀⢸⣆⡴⠋⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢳⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢳⡀⠀⠀⠀⢰⣇⠀⠀⠀⠀⡴⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢳⡀⠀⠀⣼⢻⠀⠀⢠⡞⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢳⡀⢀⡏⠸⡇⣰⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢳⣼⠇⠀⡿⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠀⢿⡆⠀⠀⢻⠀⠀⠀⠀⠀⣸⠁⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠛⠓⠶⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠟⠚⢉⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠶⠶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢲⡶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣴⠾⡿⠀⠀⣠⡀⠀⠀⣿⠷⣦⣄⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⢸⡇⣠⡾⠛⣷⡄⠀⣿⠀⠀⠉⠙⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠈⠻⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠒⠒⠒⠒⠒⠀⠀⠄⡀⠀⠀⠀⠀⡠⢄⣀⡉⠐⠈⠀⠤⣤⣍⠙⠀⠀⠑⢄
⠀⠀⢀⠀⠒⠉⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠀⢀⠴⠛⠁⡀⠈⢙⡁⠀⠒⠶⠌⢻⡀⠀⣀⠀⠀⠀⠀⡀
⠀⠐⠁⠘⣿⡀⣄⢻⠀⣴⣦⢸⡇⡄⣼⢀⣿⠃⠀⠀⠀⢠⠤⡂⠚⠀⠀⠈⢿⣶⣦⠤⠈⠂⠹⣦⡘⢿⣶⣶⡀⠈⢄
⠀⠀⠀⠀⠘⢷⣿⣾⢸⡇⢸⡏⣿⣿⠇⡼⠃⠀⠀⠀⢀⡊⢰⣿⠶⢔⢦⠶⡆⢻⡛⠁⠀⠇⢀⡹⣿⡛⢊⠻⠓⣤⣈
⠀⠀⠀⠀⠀⠈⠿⠹⠀⠛⠛⠀⠉⠘⢠⠁⠀⠀⢀⣴⡶⠀⠈⠁⠀⠀⠳⣤⣄⣤⢙⡤⣶⣷⣼⣟⣛⠛⠦⣑⢤⣚⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠠⠀⡆⠺⣻⣿⡀⠀⠀⠐⠁⠀⢀⣠⠴⠶⠶⠤⠙⣝⢯⡉⠉⣛⣻⣶⣬⡛⣯⣻⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡵⠛⠒⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠳⠯⠛⠉⠋⠑⣦⣙⣎⠛⠷⠿⠁⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏⠘⢿⡹⣿⣴⣆⠐⠀⢸⡆⢍⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡥⠄⡀⠀⠀⠀⠀⠂⣉⠉⠻⠆⠀⠀⠀⠀⠘⠘⢸⡇⠘⣿⣍⣁⡠⢞⣣⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡿⠥⠴⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠖⠆⣼⡃⡀⣿⣧⣄⡻⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣄⣄⣼⣧⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⢳⢧⣧⡿⠋⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣁⠘⣽⣿⣦⠀⠀⠐⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣼⣇⢠⠟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠾⢣⠶⢿⠛⠹⣷⣦⡀⢶⣀⣠⡤⡾⢩⠀⠀⠀⠀⠀⣴⠃⣸⣟⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣽⣴⠉⢀⣼⡦⣳⣼⣿⣿⣦⠹⣞⡇⣀⠘⠀⠀⠀⢀⡾⠁⢈⣿⣿⠘⠂⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠋⡿⣷⡄⠉⢠⣜⡊⣙⣼⢟⣷⡄⠬⠥⠚⠀⠀⡐⠛⠁⠀⠈⠛⠛⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠓⠀⠀⢬⠿⠞⠉⠞⠹⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡓⣄⣀⣈⣴⣦⡒⠼⢻⡄⡀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⡿⠋⠓⠋⠑⢈⣭⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠘⣇⡀⣿⣿⢸⡇⢶⣀⣀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⠄⢀⠠⠀⠀⣿⣿⢸⡿⠋⠁⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣳⠦⠡⠲⠠⠉⠀⡩⠀⢼⣿⠜⠃⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠋⠀⡀⡄⠀⢄⠀⠘⠀⡹⠘⠃⠀⢠⡠⠂⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠤⠀⢀⠐⠄⠀⠐⠄⢀⡀⠵⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠢⣈⠀⠀⠀⠐⣀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠉⠁
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢆⣽⣿⣿⣿⣳⡎⠤⣐⠊⢔⢦⠁⠄⠉⠡⢒⡹⢿⣾⡿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣯⢶⣱⠃
⢃⣌⢻⣿⣿⣽⣷⡿⡐⡚⣬⡐⠬⠀⡀⠀⢀⢫⢽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⡶⢶⣶⡄⠀⢰⣆⠀⠀⢀⣠
⣏⡞⡭⢗⣿⢵⣳⣿⣷⣻⣾⣷⣫⡜⣀⣄⠤⠞⠋⢛⠉⠙⠉⠀⡀⠉⠛⠙⠻⢿⣿⣿⡇⠀⠀⢸⡏⠀⠀⣿⠃⣰⣿⣿⣠⣾⣿⠏⢀⣴⠾⢷⣦⡄⠀⢠⣤
⠩⡖⠬⠀⢊⠯⢗⣯⢿⣿⣿⣿⣿⣛⡉⠀⣠⣠⡶⠧⠶⢾⣟⡁⠀⠀⠀⠀⣀⡤⠽⠿⣷⡀⠀⠘⠷⣶⠾⠋⢰⡿⠀⣿⠟⣸⡿⠀⣿⡁⢀⣄⡀⠀⣰⣿⠋
⠱⠀⡀⠀⠀⠈⣜⣞⣿⣿⡿⠋⠁⢸⣧⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠈⠀⠀⠀⠀⠘⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠷⠾⠋⢁⣘⠛⠁
⢁⠀⠀⠠⢐⣸⡽⣺⣿⣿⠀⣤⣽⣿⣿⣿⣶⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠉⠑⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⢁⢎⣣⢇⣳⣭⣷⣿⠟⠁⣀⣀⢼⣣⡿⠋⠀⠀⠀⠀⠈⠲⠤⣀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡐⠏⠾⣽⣿⣾⣿⣿⣿⠟⠉⢡⡬⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠿⠿⠿⠿⠿⣿⣿⣿⣦⣿⣿⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠚⠋
⢐⡤⣽⣾⡽⣿⣿⣿⡏⠀⠀⣨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠯⡻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣴⣿⣿⣿⡵⢂
⢖⣻⢿⣳⣿⣿⣿⡿⡀⢀⡞⣹⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢶⣤⣀⣀⠀⠀⠀⠀⠀⢡⡀⠀⠘⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⣷⣯⣗⡢⢡⣰
⣞⣮⣿⣿⣿⣿⣿⣷⣷⡟⢰⠟⠀⠀⠀⢀⡀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠐⠄⠀⢸⣿⣄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣷
⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⡏⢰⡄⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡴⠶⣤⣀⠀⠀⠘⢻⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⠀⠀⢠⡴⢋⣷⡌⠑⠀⠆⠀⠀⠀⠁⠸⣿⠇⡼⠋⠀⠀⢣⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⢸⠀⠀⠈⠙⠂⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⣯⠙⠂⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣎⢯⣿⣿⣿⣿⣿⣿⠟⢁⣼⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠢⠹⣾⣦⡀⠀⠀⠒⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⠿⢛⣥⡴⠛⠁⠘⡿⢁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⠘⣟⠍⠢⣄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⡿⠋⠰⠋⠋⠀⠀⠀⡜⠃⠘⠀⡁⠲⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣧⠀⠆⠘⠀⣿⣄⠀⠀⠑⣄⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⡅⠀⢢⢀⠀⣤⣾⣶⡏⠀⠀⠀⠀⢸⡶⠈⠇⢀⣠⡔⠒⠒⠚⠉⡷⠶⠆⠀⡧⠀⣿⠀⡄⠀⠇⢹⣧⠑⢦⡀⢾⣦⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶
⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⠿⣷⠀⡇⠘⠀⠘⣄⠀⠀⠀⠀⢄⠒⠀⠀⣀⡴⠁⠀⠀⣇⠀⠟⠀⠁⠀⠀⢸⣿⣷⣄⣱⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣵⣿⣿⠛⢉⠟⣿⣿⡟⠁⢀⣿⠀⡇⠀⠇⠀⢿⡀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠃⢸⣿⣿⠿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⣿⣿⣟⣿⣎
⣽⣿⣷⣠⢨⣼⡏⠈⡇⢸⣾⣿⡄⠠⠀⠸⡀⠈⠇⠀⢳⣤⣀⡀⠀⠀⠀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠐⠀⣼⣿⠙⡄⠸⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡮⠽
⣯⢷⣻⣿⣿⣿⣷⣼⣧⣼⡟⣸⣷⠀⠣⠀⠐⠀⠈⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣼⡏⣡⡀⣴⢀⣿⡷⠛⠛⠛⠻⠿⣿⣿⣿⣿⢵⣚⠩
⣞⠯⠂⠀⢙⣿⣿⠿⣿⣿⣇⡇⢿⣆⠀⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⢀⣼⣥⣶⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣯⢇⠣
⠈⠠⣀⠾⠋⠁⠀⠀⠀⠉⠛⢿⣼⣮⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⢸⠀⠀⠀⠀⠀⠀⢀⠞⠛⠉⠉⢀⡿⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠙⣿⡀
⡤⠞⠁⠠⣄⣀⡀⠀⠀⠀⠀⠀⠹⣏⢛⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣇⠀⠀⠀⠀⢠⡞⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠹⣄⠈⠐⠘⠷⣄⠀⠀⠀⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠻⣧⠀⣠⡴⠋⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠈⠓⠤⣀⠀⠙⣷⡆⠀⠀⠀⠀⢿⠀⠘⡄⠀⠀⠀⠠⠿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠹⣆⠀⠀⠀⠀⠀⠈⠙⢺⣿⠀⠀⠀⠀⠀⢸⣀⣠⡷⠊⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⢸⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢽⣆⠀⠀⠀⠀⠀⢰⠟⠋⠀⠀⠀⠀⠑⠾⡍⢹⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢰⠉⠠⠖⠂⠀⠀⠀⠀⠀⠀⠀⣿⣾
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣾⣶⡶⣶⢶⡶⠶⣶⣶⠶⡾⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣀⠀⠐⠢⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠈⠉⢢⡠⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠒⠐⠢⣠⠊⣠⢀⣈
⠀⠀⠀⠀⠀⠄⠀⠂⠈⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⢀⠴⠊⢉⠭⠍⠐⠀⠀⠀⢀⠤⠤⠤⠤⠄⠤⠔⢂⠉⠩⢁⡒⠤⠤⠤⠤⠤⣀⠀⠀⠀⠢⠤⢄⠠⠒⠂⠀⠐⠲⢄
⠀⠀⠀⠀⢀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⢹⡷⠊⠀⠀⠀⠀⠈⠻⡀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡠⠜⢀⡠⢚⣍⠢⣄⠀⠀⠀⠀⡇⠀⠀⠀⢀⠊⠀⠀⣀⣤⣤⣄⠀⠀⠘⡀⠀⢸⢸⠀⠀⠀⠀⣀⠔⣊⣑⠦⡀⠀⠀⢀⠆
⠀⠀⢸⠀⢐⢡⠒⠁⠘⣿⣆⢃⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⣴⣿⠟⠉⠙⠆⠀⠀⣷⠀⠈⢸⠀⠀⠀⢰⢡⣾⠏⠀⠑⡌⠓⢐⠪⡐
⠀⢀⡬⠓⠈⢦⢣⠀⠀⠸⣿⡌⢒⣐⠢⠇⡀⠀⢠⠀⠀⠀⣿⠁⠀⠀⢠⠂⠀⠀⣿⠀⠀⠘⠤⢒⡐⢃⣿⡟⠀⠀⡰⣱⣿⠟⠑⡘⢆
⠀⡞⠀⠀⠀⠀⢣⠣⡀⠀⢹⣿⠃⠀⠑⢦⡅⠀⠘⠀⠀⠀⠙⠄⠀⠔⠁⠀⠀⢀⣿⠀⠀⣦⠊⠀⠈⣾⡟⠀⠀⡐⣴⣿⠋⠀⢀⢊⠜
⠀⡇⠀⠀⠀⠀⠀⢣⡱⡀⠀⠻⠀⠀⡀⠈⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠁⠀⡀⠀⠸⠁⠀⢄⣼⡿⠁⢀⠔⡱⠋⡄
⠀⠙⢤⣀⣀⠀⠀⠀⠱⡑⡀⠀⠀⢸⣷⡄⠀⠀⢠⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⢀⣾⡇⠀⠀⠀⢌⣾⡟⠁⡠⣡⠎⢀⠜
⠀⠀⠀⠀⠈⢣⠤⠀⠀⠱⡐⢄⣀⣸⣿⡿⠲⣶⣾⣿⡿⠻⣷⣶⣶⣾⣿⠟⢻⣿⣿⣶⠶⢻⣿⣧⣀⣀⣈⣹⠟⠟⢋⠜⠁⠀⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⠪⣙⠛⣛⣃⠞⠢⠭⠤⠒⠒⠤⢭⣭⠥⠤⠒⠒⠠⠭⠥⠺⣜⣛⡻⢛⡁⣿⡷⠤⣂⠜⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠒⠒⠉⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠃⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⢑⡦⠤⠤⠀⢱⡀⠀⠀⠀⠀⠀⠒⠒⠄⠀⠀⠀⠀⡐⠤⠤⠂⠀⠑⠲⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠠⠆⠀⠀⠀⠀⠙⠢⠄⣀⣀⠠⠄⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⠁



⡶⣶⢶⠶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⡶⢶⡶⢶⣶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢦⠀⠀⢀⡠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡆⠀⠀⠱⡤⠋⠀⠈⡆⠀⣀⡤⠔⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⠐⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠓⠉⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⢀⣀⡀⠀⣀⣀⡀⢀⣤⣤⠀⣀⣤⣤⠀⣤⣌⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣧⣿⣿⣇⣼⣿⡟⣼⣿⢛⣉⡀⣿⡇⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⣿⣤⣿⡟⣿⣿⢿⡟⣿⡇⢿⣿⣬⣿⡇⣛⡃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠀⠀⠋⠁⠀⠉⠉⠉⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠘⠃⠀⠀⡀⠒⡀⠀⠀⡃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠞⠀⠀⠈⠢⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣷⣶⣟⡛⣿⣿⣿⣇⠀⢰⣤⣴⡶⢠⣤⣤⡄
⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⡿⠀⠘⠉⣿⠁⣿⣚⣿⠃⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣿⣿⣿⣤⡖⣟⣟⠋⠀⠀⠀⠏⠘⠱⠿⠏⠀⡿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣦⣦⣵⣦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⡉⢸⣿⡁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠉⢹⠟⣿⣾⣿⣟⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣧⣸⣿⣗⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣱⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣼⣿⣷⡇⣿⢢⣿⣿⣿⣿⡗
⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸⠇⠀⠀⠀⠀⢺⡗⢰⢿⡅⣋⠀⠸⣿⡄⣀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⠀⠈⢹⣿⣿⣿⣗⠈⢻⣿⣿⣿⣿⣿⣷⠀⠀⣿⣍⠀⠀⠀⠀⠀⠈⠙⠿⠟⢀⡀⠙⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢁⡀⠀⠀⢰⠛⠻⠿⠿⠛⠀⢠⡏⠈⣿⣿⣿⣿⡆⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⠛⠂⣀⠀⡀⠀⠀⢰⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣀⣴⣿⣿⣷⡄⢸⣼⣿⣦⣶⣾⣶⣬⣅⡴⢻⣿⣿⣿⣿⠟⠛⠛⠂⠀⠀⠀⠀⠀⠀⠁⠀⣇⣄⠚⠁⠻⣿⡆⠀⣼⣧
⠀⠀⠀⠐⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣯⠀⣴⣁⡽⢛⣩⣠⡠⠆⠀⢢⠀⠀⠀⢰⡆⠀⠐⠉⢉⣭⣤⣶⣿⡿⠇⢰⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣇⣤⡄⠀⠀⠉⠙⠛⠿⣯⡇⣿⣙⣯⠘⢃⣴⣶⣤⡐⠒⢦⠀⡁⣀⢀⠀⠈⢥⠀⢻⣿⠁⢠⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃⠒⣽⣼⣿⣿⣿⡿⠀⠀⣀⣀⣀⣶⣿⠛⠉⠉⠈⡃⣿⣿⣿⡟⢀⡀⠀⠀⣧⠙⣿⣦⣾⡏⠀⣾⠃⢀⣾⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠋⠉⢰⣵⣿⣻⣿⣿⣿⣇⢠⣿⣿⣿⣿⣧⣤⣷⣶⣾⡿⢿⠍⠛⠋⠉⠛⠀⣀⣤⣿⣼⣿⣿⣿⠃⣸⣃⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠿⠿⠿⢿⣿⣟⠀⣦⡀⣼⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⢧⣿⠃⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣲⣶⣶⣶⣶⠆⠀⠀⠏⠐⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠈⠿⣿⣿⣿⣿⣾⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠋⢉⣩⣥⠀⠀⠀⠀⣶⣤⣽⣿⢿⣿⣿⡇⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣴⣿⡿⢿⠿⢿⣿⣿⣿⣿⣿
⠀⣤⣿⣿⡇⠀⠀⠀⠀⠹⣿⣧⠁⢾⣿⣿⡇⣿⣿⣿⣿⣿⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠟⠟⠿⠿⢫⣿⡟⠰⢞⣿⣿⣿⣿⡿
⢸⣿⢿⣿⣷⣶⣶⣴⣶⣶⣿⣿⣦⣾⡿⠛⢁⣿⣇⣼⣿⣿⠅⣿⠀⢻⣯⢹⣿⣿⡿⠿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣧⣤⠶⠶⠶⣿⡟⢳⣾⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⠉⠀⠘⠙⠛⠛⠁⠀⠀⠁⠉⠛⠛⠓⢾⣿⣿⣿⣧⣾⣷⣄⠙⣿⣿⣿⣿⣿⣿⣿⠷⠆⠐⣉⡤⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠿⠿⠷⠼⠟⠉⠁⠀⠀⠀⠀⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠈⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢹⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠘⣿⣄⠀⠀⠈⣿⡆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠁⠀⠀⠀⠘⣿⣆⠀⠀⢹⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇⠀⢸⣿⠀⠀⢀⣴⡿⠿⢿⣦⣠⣄⡀⠀⠘⠿⠿⠿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⠈⠀⠀⠀⠁⠀⠀⠸⠿⠀⠀⠀⠻⠋⠙⣿⣶⣶⣦⣤⣤⣀⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣴⡶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⡿⠿⢿⣶⣦⠀⢀⣤⣿⠋⠀⠀⠹⠋⠀⠀⠙⠿⢿⠿⣷⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠉⠀⣀⣤⣴⣤⣄⣀⣀⡀⠀⠀⣼⡟⠉⠉⢉⣩⣿⠿⠋⠁⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠀⠀⠀⢀⣀⣿⡏⠉⠀⠀⣼⡟⠉⠁⠉⠛⠛⠛⠛⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⢀⣠⣴⡾⠿⠛⠙⠻⠿⠀⠀⠉⠁⠀⢀⣀⣀⣤⣤⣶⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠐⠿⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⠾⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⠿⠿⠛⢋⣉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⠟⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⢀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣤⡄⠀⠀⠀⠀⠈⠻⢷⠿⣷⣶⡀⠀⢀⣴⡄⠀⢀⣼⡟⠋⠁⠀⠀⠀⠀⠀⠀⢴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠻⠿⠿⠋⠀⢠⣤⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⣄⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⡇⠀⠘⣿⡄⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣿⡇⠀⠀⢻⣧⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⢀⡴⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠉⠤⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠙⢚⠇⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⠆⠃⠀⠀⠀⢸⣿⠻⣿⣦⣀⠀⢸⠀⡈⠀⣠⡆⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠈⠻⢿⣿⣶⣶⣶⡾⢿⣿⣤⣤⣴⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⢠⣿⡇⠀⢰⡖⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠻⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠴⠒⠚⠛⠷⢤⣀⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢹⣷⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡞⠉⢷⡾⡁⢀⣤⣤⣄⡀⢠⣟⢹⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣁⣀⣠⡾⠿⠶⠶⢤⣸⣿⣝⢦⡀⠀⢠⡇⠀⠈⢿⣢⡟⢩⡴⠛⠛⠉⠉⣿⣇⠀⢸⣄⣀
⠀⠀⣀⣀⣄⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⢀⣀⠀⠀⠈⢻⣿⠀⠹⣆⣸⠁⠀⠀⠘⣿⣧⡀⣷⣠⡤⠀⠀⡿⣿⠖⠚⣇⡀
⠀⠀⢸⣃⢈⠄⠀⠀⠀⠀⢀⣿⣿⢤⡾⠛⠛⠻⣦⠀⠀⢻⣔⡀⠈⠛⠀⢠⡆⠀⠸⣎⠉⠛⠉⠀⢀⡼⢃⣽⣀⣤⣽⠛⠛⠓⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣼⣿⣿⣿⡁⠀⠀⠀⠘⡇⠀⢸⡯⣿⣄⠀⠀⢸⢿⡄⠀⢹⣶⣶⣶⠾⠛⣽⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⢷⣄⣀⣠⠾⠁⠀⣸⣿⣿⠙⢷⣤⡿⠀⠿⠛⠛⠁⠀⢄⠀⠀⠹⣄⠀⠀⠀⠀⡐⠀⠆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡴⠛⣿⣿⣷⠈⠉⠁⠀⢀⣴⣿⠿⠟⠃⠈⠉⠀⠀⠀⠀⠀⢀⠵⣎⠻⣍⠉⠉⠁⠀⠀⠀⠀⠶⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣠⠞⠁⠙⢻⡿⠶⠶⠿⠛⠙⢿⣆⠀⠀⠀⠀⣠⡤⠤⠤⣄⢸⠀⠘⢦⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⠞⠁⣀⣠⣄⡀⠙⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠖⠀⠀⠀⠀⠀⠀⣿⢰⠏⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⠝⠀⠀⠀⠀⠀⢹⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠻⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠉⠙⠳⣤⠴⠚⠛⠛⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠶⠛⠉⠉⠙⠓⣦⣶⡿⠟⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡶⠿⠟⠛⢻⣧⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⡶⠿⠟⢿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡶⣾⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⢀⣤⣤⣶⣶⣿⣇⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣀⣠⣤⣤⣄⠀⣾⡏⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣤⣤⣴⡿⣿⡀⠀⠀⠈⠁⠀⠀⢿⣿⣿⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣴⡶⠿⠿⠻⣷⡄⢠⣿⠛⠉⠉⠈⢿⣦⣿⡇⠀⠀⢀⣿⢿⣷⣶⢾⣧⠀⠀⠀⢹⣯⠀⠀⢻⣇⠀⠀⠀⠀⣀⣀⣸⣿⢿⣇⢀⣀⣀⣿⡇
⠀⠀⠀⠀⠻⣷⡀⠀⠀⠘⢿⣼⣿⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⢸⣿⠀⠀⠀⠈⣿⡄⠀⠀⠈⣿⡀⠀⠸⣿⠀⠀⠀⠸⣿⠛⠉⠁⠘⣿⡟⠉⠉⠻⣷
⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢸⡿⠀⠀⠀⠀⢻⣇⠀⠀⠀⢻⣇⠀⠀⢿⡇⠀⠀⠀⢿⡇⠀⠀⠀⢿⣇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠈⠃⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠘⣿⠀⠀⠀⠸⣿⠀⠀⠸⣿⣤⣤⣶⣾⠟⠀⠀⠀⣨⡿⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⣧⣤⣶⣶⠿⠃⠀⠀⠉⠉⠉⠀⠀⡆⢠⡖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⢸⣿⣿⣧⡀⣀⣀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣀⣤⣤⣼⡟⠙⠻⠿⣿⡿⠿⣥⡀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣀⡾⠲⢿⡄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠒⠛⠋⠁⠀⠙⠛⠛⠛⠉⠀⠀⠀⢻⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⡿⠿⠿⣿⣿⠿⠿⢿⣿⣿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⢰⣿⠟⠙⢿⣷⠀⣿⣿⣧⠀⣼⣿⣿⠀⣾⡿⠋⠙⠋⠀⠀⠀⠀⣿⣿⣿⣿⠁⢠⣶⣆⠀⢻⡇⠀⠀⢿⡟⠀⠀⢸⡟⠀⣠⣶⣦⣿⣿⣿
⠀⠀⠀⢸⣿⡀⠀⣼⣿⠀⣿⡏⣿⣴⡿⢸⣿⠀⣿⣇⠘⢻⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠸⣿⡿⠀⢸⡇⠀⣇⠘⠃⣼⠀⢸⡇⠀⢿⣤⠀⢸⣿⣿
⠀⠀⠀⠈⠛⠿⠿⠟⠁⠀⠿⠇⠸⠿⠃⠸⠿⠀⠈⠻⠿⠿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⣀⣠⣴⣿⣧⣀⣿⣄⣠⣿⣀⣼⣿⣦⣄⣀⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣭⣥⣼⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣠⡤⠶⠦⣤⡀⣠⠶⠚⠉⠉⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠀⠉⠉⠛⢦⣠⠞⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠙⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡤⠖⠚⠛⠒⠦⣄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡤⠶⠦⡄⢠⡞⠉⠉⠙⣦⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⣴⠞⠉⠉⠉⠙⠳⣄⠀⢀⡼⠁⠀⠀⠀⢹⡟⠀⠀⠀⠀⠸⡏⠀⠀⣠⠴⠦⢤⣤⡴⠏⠀⠀⢸⣇⣀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢧⣼⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⢀⣤⠤⣤⡀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⡾⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠠⣏⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣠⠾⠃⠀⢸⡇⠀⠀⢀⡏⠁⠹⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢷⣄⣉⣙⡷⠀⢠⡇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⢀⡼⠃⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢰⡀⠀⠀⢠⡀⠀⠀⢀⣿⣆⠀⠀⠈⠉⠁⠀⣠⠞⠀⢠⣤⡤⠶⠋
⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⡄⠀⠀⠘⣇⠀⢀⡟⠀⠀⠀⢸⡇⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⣼⠃⠈⠓⠶⠤⠴⠶⠛⠁⠀⠀⠘⣇
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⠶⠞⠁⠀⠀⠀⣸⠁⠀⢀⡿⣧⠀⢀⡾⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣀⣀⡾⠁⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢀⣴⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙⠉⠉⠁
⠀⠀⠀⠀⠈⠳⠤⣤⣀⣠⣄⠀⠈⠙⠶⠦⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⡴⠻⣆⡀⠀⠀⠀⢀⣠⠞⠻⣆⡀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⡤⠖⠋⠀⠀⠈⠙⠓⠒⠚⠋⠁⠀⠀⠀⠙⠓⠒⠒⠛⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⢻⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⡀⠀⠀⠀⣀⠠⠐⢉⠔
⠀⠀⠀⣀⡀⠀⠀⠀⠉⠀⠀⠀⢠⠑⠠⣀⠀⠀⣀⠂⠀⠈⠉⠉⠁⠀⣀⣠⠃⠀⠀⠀⣀⡤⠆⠀⢀⡀
⠀⠀⠀⠀⠉⠳⢖⠀⠤⠀⠤⠔⠁⠀⠀⠀⣉⡉⠀⠀⡄⢠⠀⠔⠊⢁⣀⣈⣁⠶⢈⠝⠁⠀⠀⠈⠉⠁
⠀⠀⠠⣤⠀⠀⠀⠉⠢⡠⠊⢁⣀⡀⠉⢲⠁⠘⠄⢠⠁⠀⡟⠀⠰⠁⣀⣀⡀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⢰⠁⠀⠉⢦⢸⠀⢀⠈⠆⠀⣄⢃⠀⢃⠀⢥⣄⠀⠈⠇⢀⣈⡁⠴⠦
⠀⠀⠀⠀⠉⠙⠢⡀⢸⡀⠘⢄⠀⠀⢘⢸⠀⠘⢦⠀⢠⠙⡎⢂⡀⠓⠐⠊⢀⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣄⠈⠑⠒⠁⣸⣤⣆⠈⠓⠊⠀⠈⠀⠉⠓⠒⡎⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⢊⠠⠔⠒⠂⡍⠓⡲⠋⠁⠀⠀⠑⡀⡠⠊⠁⠈⠁⠢⡀⡄⠀⠀⠀⠑⠜⠻⣦⡀
⠀⠀⠀⠠⠖⠋⠁⠀⠀⣠⠀⠰⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢏⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢸⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡇⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣿⣦⣄⣘⠀⣂⣀⣴⣿⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣆⣼⠿⠷⠺⣦⣿⡇⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⢀⣀⣀⣀⡀⢀⣤⠖⠛⠋⠛⠺⢿⣧⠀⠘⣿⣿⠀⠀⠀⢹⣿⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣘⣿⣤⣿⣟⠉⠉⠉⣷⠟⠀⠀⣀⣤⣄⡀⠀⠙⣦⠀⣿⡿⠀⢀⡀⠀⢿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠿⣷⣾⡟⠉⠉⠙⣿⣿⠀⠀⠀⣿⠀⢀⡼⠛⠻⣿⣿⡄⠀⠸⡇⢸⡇⠀⣸⣷⡀⠈⠇⠀⢸⡧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⢾⣿⣯⠀⠀⠸⣿⡇⠀⠀⠀⠘⢿⡆⠀⠀⣿⠀⣸⠃⠀⠀⠈⢿⣿⠀⠀⣿⠀⠁⢀⡿⣿⣧⡀⠀⠀⢸⣄⡀
⠀⠀⢈⡏⠀⣣⠀⠈⢿⣿⣆⠀⠀⢻⡇⠀⠀⡀⠀⠈⠇⠀⠀⢹⡂⢿⠀⠀⠀⠀⢸⣿⠀⢀⣿⡀⠀⢸⠇⠹⣿⣧⠀⠀⣼⠿⠿⠛⠓⠒⠂
⠀⠀⠈⠉⠙⠃⠀⠀⠈⢿⣿⡆⠀⠘⠇⠀⢠⣷⡄⠀⠀⠀⠀⢸⡇⠘⣧⠀⠀⠀⣼⠟⠀⣼⣿⣇⣀⡿⠀⠀⣹⣿⣷⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡄⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⢸⣿⣄⠈⠛⠲⠚⠋⣠⡾⠁⠻⠛⢟⠁⠀⠀⢻⡉⠉⠀⠀⠀⠀⡤⠤⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠙⣿⣷⡀⠀⠀⢸⡁⠻⣿⣧⡀⠀⢈⡿⣿⣿⣶⣶⣶⠿⠋⠀⠀⠀⠙⢂⠳⣶⡶⠶⠿⠦⠀⠀⠀⠙⢷⡼⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠙⣿⣷⠀⡀⣾⣷⠿⠿⣿⣷⣾⣿⡁⠀⠉⠉⠉⠀⢀⣀⣀⡀⠀⣉⢉⢣⣈⠫⣄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠹⡿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠲⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⠀⠀⣀⡀⠀⢀⡀⢀⣀⣀⣀⡀⢀⣀⣀⡀⠀⠀⢸⣿⣿⣿⡿⣿⣿⡿⢿⣿⡿⢿⣿⢿⣿⣿⢿⡿⣿⡿⣿⣿⣿
⠀⠀⠀⢿⡇⢠⣿⣧⢀⣿⠇⠙⢻⣿⠛⠃⣿⡟⠛⠃⠀⠀⢸⣿⣿⣿⡆⠸⣿⠀⠈⣿⠁⢸⣧⣤⠀⣠⣼⡇⠠⣤⣼⣿⣿
⠀⠀⠀⠸⣿⣼⡏⣿⣼⡿⠀⠀⢸⣿⠀⠀⣿⡿⠛⠂⠀⠀⢸⣿⣿⣿⣷⠀⠃⢰⡄⠛⠀⣿⣿⣿⠀⢹⣿⡇⢀⣤⣼⣿⣿
⠀⠀⠀⠀⠻⠟⠀⠹⠿⠃⠀⠀⠘⠿⠀⠀⠻⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⣾⣧⣤⣼⣿⣿⣿⣤⣼⣿⣧⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣶⣶⣖⣒⣻⣭⣝⣒⡲⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢼⠉⠙⠛⠓⠒⠒⠒⠒⠒⣯⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡜⠀⡠⠞⠀⠀⠀⠳⠤⠀⢣⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⠁⢀⡤⠤⡄⠀⢠⠤⢤⡀⠈⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠸⣀⣂⡽⠀⢯⣐⣀⠇⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⡇⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⢸⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢳⣀⡼⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⡜⠀⢳⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⣀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠿⠿⢛⣛⡋⠀⢶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠛⣿⠁⠉⢶⣆⠀⢸⣿⣇⠀⠸⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣾⣿⣷⣿⣶⣄⠀⠀⠀⠀⠀⣾⣿⡇⣿⡇⠀⣿⠀⠀⣾⣿⠀⠀⣿⣿⠀⠀⢋⣹⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢰⣿⣿⡇⣿⡇⡆⢫⠀⠀⣿⣇⠀⠀⣿⣿⡄⠀⣿⣿⣯⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠛⠛⠛⠛⠉⠉⠙⢻⣿⣿⣆⠀⠀⢿⣿⣇⢹⣣⣧⠀⠀⢠⣿⣿⣦⣀⣿⣿⣷⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠙⢿⣌⣸⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠁⠉⠱⣄⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⣆⠀⠀⠀⣿⣿⣿⡿⠋⠋⠈⠉
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⣠⠤⢤⠈⠋⠀⠀⠐⠶⢒⣉⡉⠈⣿⣿⣿⡄⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⢀⣿⣿⠟⢻⣿⢻⡀⢠⡤⠁⠀⠀⠀⠀⠐⢫⡍⠙⠃⢹⣿⣿⣷⠀⠰⠟⠋
⠀⠀⠀⠀⢸⣿⣿⡆⠸⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⡀⠀⠀⠀⣰⠶⠶⡄⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆⠀⠰⠿⠶⠶⠟⠀⠀⣠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢀⣤⣤⣀⣀⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣠⡴⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⠖⠒⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠒⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣤⣀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⣀⣤⣤⣤⣄⡀⠙⠳⣾⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠈⠙⠻⣦⣿⠀⠀⠀⠹⣦⠀⠀⣴⠟⠀⠀⠀⣿⣴⠟⠉⠀⠀⠀⠉⠛⢷⡄⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⣴⣶⣄⠀⠀⠘⣿⠀⠀⠀⠀⠙⣷⣼⠋⠀⠀⠀⠀⣿⠃⠀⢀⣤⣶⣦⣤⣤⣬⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡏⠀⣹⡇⠀⠀⣿⠀⠀⠠⡄⠀⠈⠁⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠛⠛⠋⠀⠀⣰⣿⠀⠀⠀⣿⣆⠀⠀⣠⣿⠀⠀⠀⣿⡄⠀⠀⠐⠒⠒⠀⠀⢀⣾⣿⣤⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠈⠷⣦⣀⣀⣀⣀⣠⡾⠋⣿⡀⣀⢀⣿⠹⣦⣰⠟⣽⡀⣀⢀⣿⠻⢦⣄⣀⡀⣀⣠⣴⡟⣡⣿⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡇⠀⠀⠉⠛⠛⠛⠁⠀⠉⠉⠀⠈⠛⠛⠛⠉⠀⠀⠉⠉⠛⠋⠉⠈⢿⡋⠙⠷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣬⠟⢶⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⡴⠞⠲⠶⠖⠛⠉⠈⠉⠉⠉⠀⠀⠀⠹⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠉⠉⠉⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠀⠀⢰⡇⢠⣇⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣄⠀⠙⠛⠋⠀⠀⠀⣾⡇⢸⣿⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠁⠀⠀⠀⠉⠛⠿⣿⣦⡀⠀⠀⠀⠀⢿⡇⢸⣿⣿⠏⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⢿⣷⡄⠀⠀⠀⠀⠀⠀⠛⠁⠀⣠⣴⡾⠟⠛⠉⠉⠉⢉⣛⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣴⠿⣿⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⣼⣿⣿⣋⣉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠹⣷⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⡀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⢃⣸⠇⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠿⢷⣦⣄⡀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠁⠀⠀⠀⠀⠈⠙⠻⢷⣄⠀⠉⠛⠲⣤⣤⣤⣤⡤⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣀⣀⣀⣠⣤⣤⠶⠋⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠋⠁⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣠⡴⠟⠛⠋⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠈⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣴⣶⣶⣶⣶⠶⠾⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⡀
⠀⣤⣤⣤⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣼⡇⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⣀⡀⠀⠀⠀⢀⣸⣟⣀⣤⣼⣿⡀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⣠⣤⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿
⣿⠀⠀⠀⠀⠀⣦⢀⣴⠟⠛⠷⣦⡀⠉⣿⣷⠟⠛⢻⣦⠀⠀⠛⢻⡟⠛⠁⢹⣿⠀⠀⠀⢸⣿⠀⠀⠀⢠⣾⠋⠉⠉⢻⣿⠁⣠⡶⠶⢶⣤⠀⢀⣴⠶⠶⢾⣿
⣿⡀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢹⣷⠀⣿⡇⠀⠀⠈⣿⡆⠀⠀⢸⣷⠀⠀⢸⣿⠀⠀⠀⣸⣿⠀⠀⠀⢀⣿⠻⠿⠷⢾⣯⣰⣿⣀⣀⣀⣹⣷⣾⠏⠀⠀⠀⣿⡄
⠘⢷⣤⣤⣤⡾⠋⢿⣧⡀⠀⢀⣼⡟⣠⣿⣧⣀⠀⠀⣻⣧⣀⠀⢸⣿⠀⠀⠘⣿⡀⠀⢀⣿⣿⡇⠀⠀⢨⣿⣧⣤⣤⣤⡿⠹⣿⠋⠉⠙⠛⢻⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠋⠀⠉⠉⠉⠉⠀⠸⠛⠛⠋⠻⠿⠿⠿⠟⠀⠙⠻⠶⠟⠋⠻⠿⠗⠀⠘⠋⠀⠀⠀⠀⠀⠀⠹⢷⣶⣶⠶⠂⠹⠷⣶⠶⠶⢿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠑⢬⢦⡀⠀⠀⠈⠐⠀⡢⠄⡀⠀⣀⣐⣠⡒⠚⡁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠐⢂⣠⠄⠀⠱⢤⠂⠄⠄⠠⠄⡤⠚⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠢⠖⠉⠀⠀⠀⠈⠙⢅⠐⠂
⠀⠀⠀⠀⠰⠒⡊⠐⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠄⠀⠀⠀⠀⠁⠂
⠀⠀⠀⣠⢈⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠀⠀⠀⡖⢲⠙⠉⠀⠀⣸⢙⠀⠀⢐⡄⠐⠈⠉⠀⠀⠉⠻⡄⠂⠀⠀⠀⠀⠉⢲⣄
⠀⢀⡀⠴⠈⠠⠄⡀⠀⠁⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠻⡄⠄⠀⠨⠃⠀⠀⢀⣀⣀⣀⠸⠃⠆⠀⠀⠀⠀⠀⣘⢃
⠀⠀⠱⠰⠃⠀⢦⣤⠀⠀⠀⡦⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⠀⠀⠀⠂⣀⣀⣀⣀⠤⣄⢠⠀⠀⠀⠀⠀⠀⢣⡞
⠀⠐⢢⠠⡀⠀⠀⡄⠇⠀⠀⠋⠀⠀⠀⡇⢸⠀⠀⣛⡀⠀⠀⢫⠀⠀⢸⢆⠀⠀⠀⠀⠀⢸⡉⡍⠀⠀⠀⠀⠈⡛⢢⠶⠀⠀⠀⢀⠜⣡
⠀⠀⠰⠀⣉⠂⠀⠇⢠⠀⠀⠀⠀⠈⠉⠳⣴⠀⠀⠀⠀⠀⠀⠈⠆⠀⠨⡄⢂⠀⠀⠀⡀⠈⠛⣓⠂⡀⠀⠀⠀⡔⠀⠀⠀⠀⡀⢈⢨⡀
⠀⠀⢄⠀⠸⠃⠀⢰⠸⠀⠀⢀⣄⡀⠀⠀⠘⠀⠀⠠⢤⢄⠀⠀⠀⡀⠀⡙⣄⢂⡀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣄⠡⡀⠄⠀⠠⠁⠀⢳⡆
⠀⠀⠐⣠⠁⠀⠀⢸⣤⡄⠀⠀⠿⠆⠀⠀⢀⡀⠀⠀⢨⡈⣄⠀⢀⠗⠀⠁⠈⠉⠉⠉⠑⠚⠒⠒⠒⠂⠀⠀⠈⠉⠉⠛⠈⠀⠀⠀⢐⠜
⠀⠀⠀⠋⣄⠀⠀⢸⣬⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠂
⠀⠀⠉⠣⠈⠢⢀⠀⠁⠁⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡠⠊⠈⠀⠈⠀⡠⠆
⠀⠀⠀⠀⠁⠉⠀⠉⢣⢲⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⢀⠐⠈⠹⠲⡂⢤⠄⢄⡀⠀⢀⠠⠀⢀⠀⠁⠒⠲⠅⠀⠒⠢⠈⠈⠑
⠀⠀⠀⠀⠀⠀⠂⠀⢀⠑⠁⠒⠒⠀⡂⠜⠠⢤⠑⠀⠲⡤⠀⠀⠀⡀⠠⠉⠈⢃⠬⡀⡀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠸⠃⠉⠐⠐⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠢⠀⢀⣄⣤
⠀⠀⠀⠀⠀⡖⠀⠒⠠⠀⠀⠀⠀⠀⠇⠀⢠⠀⠀⠈⠻⠟⠀⠀⠀⣀⠠⠄⠐⡄⠀⢀⢠⠤⠀⠀⢀⠀⠀⠀⠀⠑⡄⢑⠀⠀⢀⢀⢴
⠀⠀⠀⠀⠀⠡⠀⠀⢂⢣⠀⠀⠀⢠⠀⠀⢸⠰⠀⠀⠀⠀⠀⠀⡊⢸⠀⠀⠀⡇⠀⡇⢠⠀⠀⠀⢸⠀⠀⠀⢠⠞⠒⠊⠀⠀⢈⣁⠷
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠆⢳⠱⡑⡼⠀⠀⢸⠀⡤⠀⠈⠁⠀⠢⢇⢸⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⡌⠀⠀⠠⣢
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠆⠀⠐⠆⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠅⡜⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⢀⠄⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⢠⡏⠀⠈⠳⠀⠀⢠⠀⢸⡌⢡⡄⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡔⠁⠘
⠀⠀⠀⢈⠀⢸⡄⠀⠑⡀⠀⠀⢸⡄⠀⠀⢸⠀⠃⠰⡘⢄⣀⡔⠀⠀⢸⠀⠘⠁⠸⠇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⣀⠤
⠀⠀⠀⠈⡄⢸⡇⠀⠀⠀⠀⠀⡠⠉⣠⢄⡀⢤⠂⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠱⠎⠤⠊⠀⠀⠣⣀⠑⠄⠀⠀⢀⣀⡄⠸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⢠⠠⢲⠀⠀⠀⢁⠔⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠱⠀⠀⠀⠀⡀⠀⠈⠒⠊⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠄⠐⡄⠀⠀⠀⡇⠀⠂⢀⠞⢡⣀⣉⣀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⢡⠇⠀⠀⠀⡃⠈⠀⠀⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⡡⠃⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠐⠉⠳⠄⠀⠀⠈⠀⠐⠊⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⣠⡾⠃⠀⠀⠀⠀⠀⣠⠞⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠰⠠⠴⠁⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠈⣻⢯⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠤⠊⠀⠀⢀⡜⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⢀⠈⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⡠⠊⢀⠂
⠀⠀⠀⡜⠀⡀⠀⠀⠀⠀⠈⠀⠀⡠⡊⡀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠠⠂⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠾⠊⠀⠀⠀⠀⠀⢁⡔⠴⠊⠀⠀⠀⠀⠀⡰⢻⠇⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⢀⠔⠀⠀⠀⣀⠜⡱
⠀⠀⠀⠀⣡⣀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⡀⢀⠊⡦⢁⡾⠊⢉⠽⠀⠀⠀⠀⠀⠐⠠⠔⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠊⠑⠊
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⡠⢄⢀⠎⠸⠞⣸⣷⠿⠒⠋⢁⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⢸⠘⢀⢇⣡⣿⠷⠶⠶⠖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⠀⡇⡘⣃⣮⣾⡏⠀⠀⠀⡀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠐⣦⠀⠀⠀⠀⠱⠀⡇⣷⣿⡿⠛⠡⡀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠠⠔⠂⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣱⡿⠿⢁⠀⠀⠀⠈⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐⠒⠚⠛⠛⠚⠙⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠀⠀⢂⠁⠀⠠⠃⠀⠀⠀⠀⠀⠐⢢⢠⡄⠀⣔⠀⣰⡆⢀⣖⠀⣰⠀⢀⡂⠀⣐⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠉⠉⠉⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢠⡄⠀⢀⠘⡀⠀⠠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠫⠄⠀⠀⠀⠀⠈⠀⡐⢄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⢠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⢀⠂⠀⠢⡠⠃⠀⠀⠀⠘⡀⠴⠀⢀⠄⠊⡐⠀⢀⠶⠉⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠰⡐⠂⠠⠴⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⢀⠁⠀⠁⠀⠀⠀⠀⠠⠗⠊
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠐⡄⠀⠀⢀⠀⠀⠀⠀⡀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠤
⠀⠀⠀⠀⠀⠀⠤⡐⠀⠈⠁⠀⠀⠀⢸⡇⣰⣆⢸⡇⣾⠋⢻⡄⣿⣠⡄⢰⡟⠀⠀⠀⠁⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠁⠢⢄⠀⠀⠀⠸⣷⡟⢻⣿⠛⣿⣀⣿⢱⣿⠟⣿⣾⠁⠀⠀⡀⠀⠠⠄⠒⠀⠉
⠀⢀⢀⡀⣀⣀⡈⠀⠠⠂⠁⠀⡀⠀⠀⠿⠁⠀⠛⠀⠉⠋⠁⠘⠋⠀⠹⠃⠀⠀⠀⣐⡀⠀⠀⠀⡀
⠀⠀⠉⠈⠀⠀⢀⠀⠀⠀⠀⠴⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠅⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠝⠂⠈⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠤⠀⠀⡠⠀⠀⡠⢀⡔⢄⠀⠀⠀⠆⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⡐⢀⠤⠊⠀⠀⠄⠊⠀⡅⠠⠁⠈⠀⠀⠑⢄⠀⢀⠀⠀⠀⠉⠒⠤
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⢀⠀⠀⠢⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⣔⠀⠀⠀⠀⡄⠀⠀⠀⠈⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⢶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠒⠛⣹⢟⣀⣀⣀⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⣠⡴⠚⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠓⠦⣄⠉⠓⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠀⠀⠀⠀⠀⠈⠉⣹⠃⠀⠀⣠⠖⠶⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢉⣑⠀⠈⠳⣄⠀⠀⠀⠀⢠⡏⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢠⡴⠛⠙⠻⣟⠦⠀⠈⣧⠀⢀⡴⣋⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠛⠀⠀⢆⠀⡀⠢⢀⣀⠀⠀⠐⠋⢁⡴⠋⠀⠀⠀⠀⠙⣆⠀⢾⠛⠀⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠢⣀⣌⣤⡿⢦⣄⣀⣁⣀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢹⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⡾⠛⣩⣴⣿⣧⠉⠉⠉⠀⠀⠀⠀⠀⣠⣤⣄⣀⡀⢸⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣧⣾⡿⠛⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠙⣛⡻⢿⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠉⣰⠛⠉⠉⠛⠳⣄⠀⠀⠀⢀⡶⠋⠉⠙⠳⡄⢸⡇⢸⡇
⠀⠀⠀⠀⠀⣠⠞⠛⠿⣆⡸⣼⠃⠀⠀⠀⢠⡄⢹⡆⠀⠀⢸⠃⠀⠰⠶⠀⢹⣸⡇⢸⡇
⠀⠀⠀⠀⢰⡏⠀⢄⠀⠈⠛⡇⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣠⠾⠛⠛⢦⡀
⠀⠀⠀⠀⠘⣇⠀⡴⢳⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠸⣆⠀⠀⠀⠀⡼⠉⠋⣤⠒⠂⠀⣿
⠀⠀⠀⠀⠀⠹⣞⢧⣈⠀⠀⠈⠧⣄⣀⣀⣀⣠⠞⢠⠖⢲⣦⠈⠙⠒⠒⠋⠀⠀⣸⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡇⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣄⠀⠘⠁⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣀⣀⣀⣀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠤⠀⢀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠈⠀⠀⠀⠐⠊⠀⠀⠀⠈⠂⠀⠀⠀⣐⠤⠤⠄⠠⠄⠱⣀⡀⠀⠀⢧⡋⠸⢰
⠀⠀⠠⣔⣛⣈⣡⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠆⠶⠐⠀⠄⠀⠈⠃⢠⣶⣶⣴⣦⠀⠍⠃⠀⢀⣀⣢⣔⡃⢀
⠀⠀⠄⠙⠛⠛⠋⠀⠀⠀⠀⠓⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠐⠀⠐⢀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠋⠀⠀⠀⠀⣆⠐⠷⠶⠂
⡀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢄
⠀⡀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡢⠀⠁
⠀⠀⢰⣶⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣬⣤⣵⠆
⠀⠀⠄⠉⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠔
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠗⢺⣯⡿⠃
⠀⠀⠀⣴⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠏
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠄⠀⠄⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠁⠀⡤⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣀⠀⣀⠐⠄⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠡⠞⠶⠁⠀⠀⠀⠀⠀⠀⠴⠲⠆⠀⠀⠀⠀⠈⠀⠀⠳⠞⠉⠀⠁⠀⠀⠀⠒⠒⠐⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠒
⠀⠰⠆⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠐⡀⠀⢠⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣧⠀⠈⠺⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣀⣦⣀⣀⠁⡀⠀⢿⣿⣰⣇⣶⣿⣿⣫⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠄⠠⠢⣀⠀⠀⠘⠉⠰⣧⡦⡄⢀⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡄⣦⡏⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠥⣦⠀⢀⠀⠀⠀⠱⣀⡂⡄⢾⠻⡟⣬⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡉⡐⠀⠀⠆⡹⣤⠉⣪⣼⣶⣺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣚⣟⢁⣠⡃⠀⠀⠠⢿⣍⡗⣶⣿⡿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠹⣮⣀⣀⣀⣮⢻⣛⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠹⠝⡁⠹⣿⣸⣿⣿⣿⡿⣿⣷⣿⣛⢽⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣍⣭⡀
⠀⠀⠸⠍⣿⣷⣤⠀⠀⠀⠀⢀⣀⠠⣀⠈⠉⠀⢀⡿⣠⣽⠿⣻⣿⢿⣿⣶⣚⣻⢺⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⠁
⠀⢀⣀⠀⢠⣾⣻⠀⣠⣤⣠⡾⣿⣻⣿⡑⣶⢒⣿⣷⠏⠙⠁⠈⠈⡏⠻⡿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⡦
⠀⣶⣾⣿⣾⣽⣿⣾⡿⣿⡿⣧⣽⠿⢻⣷⣡⣿⣽⠟⢠⠀⠈⠁⠤⠴⠊⢄⣸⣋⣽⣿⣿⣿⣏⣽⣿⣿⣿⠶⠃
⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡧⣂⢸⠸⣷⣿⣿⡦⠀⠻⠶⣤⡄⠀⠈⠈⠻⠟⠡⠴⠏⣰⠻⢿⠏
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⢷⡈⠀⠈⢻⠿⡦⢁⣠⠀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣽⣿⡾⣿⠎⠀⠀⣰⡞⠀⠈⡧⣼⣳⡜⣷⠇
⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⡟⠁⠉⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡄⠀⠀⠀⡀⣐⠣⠔⠻⠟⠀⢠⣤
⠀⠀⠉⠿⣻⣿⣿⣿⣿⣍⠉⠀⣠⣴⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣬⡇⠀⠰⣼⡿⢉⢧⣠⣤⠀⠀⢘⣣⡤
⠀⠀⠀⠀⠉⣿⣿⣿⡿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣟⣿⣿⣿⣟⡿⠀⠀⠰⠏⠑⠈⣾⣟⣥⢠⠀⢟
⠀⠀⠀⠀⠀⠁⠻⡿⠃⠀⠀⠀⢻⣿⣧⣽⣿⣿⣟⠳⠀⠀⠙⣿⠿⠟⣉⠀⠀⢠⣶⣀⣤⡼⣯⣤⣿⣧⠀⠛⠡⠂⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⡿⣯⣿⣿⣿⡘⠝⠶⣤⢀⡘⢯⣃⣭⣥⡷⢯⣽⣿⣿⣶⣿⣾⣞⠀⠐⠳⠐⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸⣠⡝⠂⠘⣿⢱⣎⡦⣬⡆⠹⣧⣄⠠⠉⢿⡚⣃⣀⠛⢷⡌⠻⣟⣌⠁⣾⡵⠀⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡿⠲⢈⣆⣄⣿⣿⣷⡋⣁⣶⣧⠿⣷⡤⢑⣦⡉⢳⣶⠾⡿⢤⣤⣾⠿⠿⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠒⠺⣿⠛⠩⣿⣾⣑⣿⢁⣿⣟⣽⣿⣽⡳⣷⣆⡤⣠⣴⡎⠉⠻⠇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠤⠀⠻⠳⢦⣅⣾⣿⡏⠽⣌⡙⣿⠧⠜⢻⡀⠚⠅⠉⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠆⠀⠹⢧⠸⣷⢴⠛⠓⠀⠘⠻⢤⡀⠀⠁

⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠎⠀⠴⣾⣿⣿⣇⠀⠑⠈⠛⠒⠀⠀⠐⠀⠀⠀⠀⠞⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠤⠔⠚⣱⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⢧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⠏⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰
⠀⢀⡠⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠄⠀⠀⠉⠉⠈⠁⠀⠀⠤⢀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⣾⡿
⣠⠋⠀⠀⠀⠀⠀⠓⢀⠀⠀⠀⠀⠛⠢⢀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠁⠒⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣡⣾⣿⠗
⡄⣠⡤⠤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠉⠉⠉⠉⠀⠋⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠁⡄⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⣀⣢⣤⣤⣶⣶⣦⣀⣶⣿⣿⣿⡥⠐
⠀⠀⠀⠀⢈⠀⠀⠀⢀⠀⢀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠓⢦⡄⠊⠋⠘⠁⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⢿⢿⣿⣿⣿⣯⣤⡤
⣄⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⣤⢦⡀⠀⠀⠀⠀⢀⡀⠀⢀⣩⣿⣿⣏⠻⢿⣿⣿
⠉⠁⢄⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡄⠠⠒⠉⠀⠀⠀⠀⣾⡿⠉⠀⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠑⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⣷⣆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠠⠓⣄⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⢀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⢀
⠀⠀⠀⠀⠀⢿⠙⠒⠒⠒⠀⠀⠳⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⣸⡇⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠁⠈⠙⠷⠀⠄⠀⠀⠀⠐⠲⠤⢬⣿⣿⣿⡇⢀⣿⣇⣠⣤⠤⡀⠊⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠒⠂⠀⣠⣤⣿⣿⣿⣷⣸⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⣠⠃⠀⠀⠺⠆
⠀⠰⠀⢠⡀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⡿⠿⣿⡆⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣰⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⡀
⠀⠐⠠⠼⢷⣠⠀⠀⢀⠠⠀⠘⠁⠀⠀⠀⡀⠀⠀⠀⠀⢉⡉⣤⠶⠟⣡⠂⠀⠈⠻⡄⠀⠀⠀⠀⠉⠉⠓⠀⠀⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚
⠀⠀⠀⠀⠈⠋⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠩⠄⡔⠁⠀⠘⠁⠀⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⠀⠀⠀⢀⣠⣰⣶⣤⣄⣇⠀⠀⠀⠀⠀⠄
⢀⡴⠀⠀⠀⠀⠀⠀⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠈⠻⠙⠻⠰⠦⢤⡄⠀⠀⠀⠀⠈
⠉⠀⠀⠀⠀⢀⠀⠀⢀⠄⠀⠈⠓⠊⠧⣀⠠⠤⠴⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣈⠀⠀⣤⠀⡀⠀⠀⣠⣮
⠀⠀⠀⠀⢶⠠⠤⡀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⢿⣆⢀⣻⣦⣵⣶⣿⠟⠉
⣀⡀⠀⠀⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠹⣦⣷⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠛⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⡎⠀⠀⠀⢠⣴⣿⡷⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢟⠿⣠⢀⠀⠁⠈⠁
⠀⠀⢀⣠⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠊⠀⠀⠐⠦⢄⡀⠀⠀⠀⢽⣿⣿⣦⣄⡇⠀⣰⣿⣿⡿⢁⠤⠀⠂⠀⠀⠀⠐⠂⠀⠞⠀⠈⠀⠘⠳⢌⣀⠒⠂
⠙⠀⢄⠀⠀⠀⡀⠀⠀⠐⠄⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣀⠀⠙⠻⣿⡏⠁⠀⣿⣿⣫⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣿⣿⣿⣶⡄⢤⠚⠒⣨⣿⣯⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠘⠀⠀⠀⠀⠒⠛⠉⠉⠀⠐⠚⠛⠙⠉⣻⡿⢿⠇⣀⣶⠚⣵⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠓⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⣄⣀⣀⣀⠀⠀⠀⠀⠀⡗⢹⠁⠈⠙⠒⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠒⠒⠚⠛⠋⠉⠉⠉⠉⢹⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⡏⠀⠉⠀⠀⠘⡆⠀⠀⠀⡘⢠⠃⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠙⠒⠒⣦⠀⡰⠁⠀⢀⠔⠁⠎⠀⠀⠀⠀⡸⠁
⠀⣀⣤⠤⠖⠛⡇⢀⣀⣠⣤⣤⠀⣼⠋⠉⠁⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⠒⠋⢹⠖⠒⣻⠁⠀⡜⠀⠀⠀⠀⡰⠁
⢾⠁⠀⠀⠀⠀⠹⡞⠀⠀⠀⢸⡴⠁⠀⠀⠀⢠⣀⣀⢤⡀⠀⠀⢰⡒⠀⠉⢹⡀⠀⠀⠀⠀⣏⣉⡉⠉⢹⠃⡼⡏⠀⠀⢸⠀⢀⡏⠀⡰⠁⠀⠀⠀⡼⠁
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡘⠀⠀⢰⠁⠀⠀⠀⢇⣀⣀⣀⡀⠀⠀⠀⠀⢠⣤⠷⠶⠋⢸⠷⠇⠀⠀⣿⠀⣸⠇⣰⠇⠀⠀⢀⡾⠁
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠛⡏⠀⠀⠀⠀⠘⣇⠀⢸⠇⠀⠀⠀⠀⣾⣼⠄⠀⠀⢴⠀⠐⢓⡶⠟⠀⡿⠀⠛⠲⢶⣤⡟⠁
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⠃⠀⠀⠀⠀⠀⠸⡄⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣿⠖⠀⢶⡄⠀⢀⣰⠶⠶⢿⠁
⠀⠀⠈⣇⠀⠀⠀⢠⠓⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠤⠤⠤⠤⠖⠒⢒⡗⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠙⠄⣠⡞⠀⢠⡏⠹⣀⣀⡼
⠀⠀⠀⠸⡖⠂⠉⠉⢀⡑⠒⠒⠂⠉⠉⠉⣟⠀⠀⢻⡀⠀⠀⠀⠀⢀⡾⠀⡟⠉⠉⠉⠐⢺⠀⠀⠀⠀⠀⠘⠦⠴⠋⠀⠀⠘⠶⠴⠞⠉

⠀⠀⠀⠀⠀⢀⠀⣴⢦⣰⢷⣦⣟⣆⣀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢯⣧
⠀⠀⠀⢠⠴⢿⡟⣿⣹⣟⣶⣯⠿⢿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⣁⠀⣈
⠀⠀⣆⣡⡁⠹⢿⠟⠿⠋⢛⣦⢑⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⣨⡞⠋
⠀⠠⠽⠏⠖⠻⠦⠀⠻⠦⡴⠦⠈⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢰⠀⠶⠓⠂⠀⢠⡶⣿⣿⣆⠀⠀⠀⠀⡂⢀
⠀⣀⡐⢲⣶⠒⣲⣶⣶⠶⡿⣷⣶⣖⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣰⡃⠢⡀⠀⡚⠈⠀⠀⠀⠄⠀⠀⣿⢻⡟⣿⣿⡄⠀⠀⠀⢻⣄⣀
⠀⠈⠛⡛⢛⡰⢮⣝⣀⠀⣇⣊⣈⣉⠀⠈⠈⠀⠀⠀⠀⠀⠀⣨⡽⣯⡟⠸⡄⣈⡀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠸⡟⣠⡀⠀⠀⠀⢻⣷⠄
⠠⣿⣿⣿⣿⠁⠸⣿⡙⠦⡘⠀⢭⣿⠀⠛⠃⠀⠀⠀⠀⠀⢐⠋⠉⡙⢿⠀⠷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣿⠋⢡⡜⠀⢀⡀⢀⣀⣀
⠀⢸⣿⣿⡿⠀⢰⣿⣷⡀⠈⠂⣄⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⣶⠖⢽⣶⣶⠾⢿⣆⠀⠀⠀⠀⠀⠀⢹⢣⡼⢿⠿⠟⠁⣬⣿⣟
⢸⣿⣿⣿⣯⣀⢸⣿⣿⣧⣾⣿⣾⠛⠛⢻⡀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣦⣤⠀⠉⠈⢿⣦⣾⡟⠀⠀⠀⠀⠀⠀⠙⣹⡧⠁⢀⡀⣠⣬⣿⣟
⢨⣿⢻⣾⢿⣿⣿⣿⡿⠟⢋⠿⠻⡷⣦⡈⣧⠀⠀⠀⢠⠎⠀⠀⢀⠀⠈⣿⣷⣀⣶⣿⣯⡍⣿⠀⠀⠀⠀⠀⠀⠀⣿⣡⡀⠈⠉⠉⠉⠉⠁
⢸⣿⣿⣷⣿⡷⠛⡏⣤⣠⣿⣉⠀⣾⠻⣇⣿⣶⣴⣶⣧⢃⣤⣬⣿⡄⠀⠉⠉⠉⠛⠛⠛⠛⠿⢦⡀⠀⠀⠀⠀⠀⢿⡩⠽⠆
⠙⣿⣿⣯⣿⣃⡾⢹⣿⣿⡿⣿⣳⡿⢰⣿⡏⠉⠙⢯⠕⠉⠀⠈⠛⡇⠀⠐⢰⡖⠒⠻⠟⠛⢻⡏⠀⠀⠀⠀⠀⠀⠸⠙⣆
⠀⣿⣽⣽⣿⡿⢉⠈⠀⠀⠀⢈⢹⣷⣾⣿⠀⠀⢀⣤⣿⡀⠀⠀⠀⣷⠀⠈⠈⣿⡀⠀⠀⠀⣼⣧⡀⠀⢀⣀⣀⠀⣀⡆⠘⣷
⠀⠛⢿⡟⣿⡃⣴⣄⠀⠀⠀⠀⣿⣿⣿⠗⠀⢠⣿⣿⣿⣷⣦⣄⠀⠘⣇⣀⣰⣿⣷⣟⣿⣶⣿⡌⠷⠀⡀⠈⠉⠙⠛⠓⠄⠈⠣⡀
⠀⠀⢛⣾⣯⡇⣿⣿⣿⣄⣴⢷⣿⣿⡇⠀⠀⣾⣿⣿⠯⠍⠉⠙⡄⠀⠙⠃⠈⠃⠀⠙⠁⠈⠹⣧⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠷⠒
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⣾⣿⠟⠀⠀⢀⣿⣿⣿⣶⠂⠀⠀⠈⣶⣦⣲⡶⠂⣤⣄⣀⣀⢰⡿
⠠⣾⡷⣶⣿⡿⠛⠛⠋⣰⣶⡿⠇⢀⣤⣾⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠿⠲⣤⣤⠈⢉⡉⠙⠛⠛⠓
⠀⠈⠙⠛⠉⠑⠓⣿⣾⠿⠛⢳⠘⠟⠿⠟⠻⣟⣿⣿⣿⡀⢀⠖⠋⠀⠀⣠⡼⠁⠉⠀⠸⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣧⡀⣀⣦⣶⣿⣡⡴⠾⠓⠒⠛⠒⠛⠚⠶⠆⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⣀⣀⣀⣀⢀⡀⡀⠀⠀⠀⢺⣆⣿⣼⣿⣧⣸⣇⣾⣿⣼⣿⣴⣿⢿⡗
⠀⠀⠀⠀⠀⣾⠛⣷⢹⣤⣿⠹⢀⡰⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣯⡿⢿⣿⣿⣻⠃⠀⠀⠀⠈⠿⠏⠿⠟⠟⠙⠛⠋⠛⠻⠋⠃⠿⡭⠁
⠀⠀⠀⠀⠀⠙⠛⠃⠘⠁⠉⠈⠘⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠠⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠔⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⡄⣠⣤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⣠⡤⢰⠆⠀⠀⠀⠀⠀⠀⣰⡟⢻⣾⣿⣶⢿⣿⣏⣲⡆⠿⠀⡄⠀⠀⠀⠀⣴⢶⡄⣿⢸⡇⣾⠋⢸⣇⣿⢹⠇
⠀⠀⠀⠀⢾⣻⣺⣿⣾⠟⠿⢿⣼⠯⠀⠀⠀⠀⠀⠀⠠⡘⠷⠾⠛⠛⠉⠘⠃⠙⠋⠑⠃⡠⠃⠀⠀⠀⠀⠻⠾⠃⠙⠛⠁⠙⠛⠘⠁⠻⠐
⠀⠀⠀⠀⠘⠛⠉⠉⠉⠀⠀⢀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠀⠀⠐⠒⠂⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠾⠷⠞⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣡⣠⡄⠀⠀⠀⠀⣠⡦⣠⡶⣦⣴⠶⣦⣴⠆⢠⡄
⠀⠀⢀⣤⡤⣶⣖⡦⢰⣿⡄⣿⣛⠀⣧⣸⣿⡇⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣾⢻⣾⣷⢿⣿⡃⠀⠀⠀⠀⢿⣤⠿⠦⠞⠛⠶⠟⠿⢶⡛⢠⠃
⠀⠀⢻⣄⣤⢹⡟⠷⠼⠗⠿⠰⠾⠛⠋⠈⠋⠂⠀⠀⠀⠀⠀⢺⣿⣿⠷⠾⠉⠛⠁⠁⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠔⠁
⠀⠀⠀⠉⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡀⠀⠀⠀⢀⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⡶⠆⣶
⠀⠀⠀⠐⣿⣻⢾⣏⣿⣿⡾⣷⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠲⣴⢶⣶⢃⣶⣆⣶⢸⣾⡇⠀⠀⠀⠀⠀⠠⢀⣤⣄⣾⠛⣿⣿⠞⠉⣛⣷⣏
⠀⠀⠀⠀⠹⠃⠀⠉⠉⠈⠁⠉⠛⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠷⠾⠛⠿⠟⠹⠯⠁⢲⠀⠀⠀⠀⠀⢿⣁⣽⠻⠶⠋⠛⠀⠀⠀⡈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠠⠤⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠙⠁⠀⠂⢀⢤⡄⠰⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠶⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⣀⣀⡠⠤⠄⠀⢀⣀⡀⠀⠀⠀⠉⠁⠶⠆⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢾⠏⠉⠉⠉⠳⣄⠀⠀⠈⠉⠉⠶⢦⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠶⠶⠀⣀⣴
⢀⣠⣤⣀⣀⠀⡀⠐⠒⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡈⠿⣦⠀⠀⠀⠀⠘⠻⣶⠶⠶⠄⣀⡀⠀⠀⠀⠀⣀⠴⠋⠉⠰⠆⣀
⠈⠻⠃⠈⠁⣀⡀⠐⠲⠶⠖⠀⣼⣆⡀⠀⠀⠀⠈⠀⠀⠈⠶⡀⠀⠀⠀⠀⠘⠧⠀⠀⠈⢩⣶⣄⠰⠂⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠿⢧⣀⠀⠀⠀⠀⠀⢹⣤⣀⣠⣀⠀⠸⡿⠒⠂⠀⠈⢹⡆
⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠈⠀⠀⠉⢳⣤⡀⠚⠓⠀⠉⠻⣄⠀⠀⠀⢰⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⣦⡀⠀⢀⡄⠀⢻⡇⠀⡄⣼⠗⠚⠀⠺⠋⠙⣿⠟⠢⣤⡀⠀⠈⢳⣄⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠂⢀⡌⠀⠀⢸⠀⠀⠀⠈⣶⡄⠘⠿⠃⠛⠋⠙⠻⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣷⠀⠀⠀⠀⢸⡄⠀⠀⠘⠋⠀⠀⠀⠆⠀⠀⠀⠁⠀⠘⠛⣦⣤⠤⠄⠀⠀⡀⠀⠤⠤⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠈⠻⡆⠀⠀⠀⠀⠙⣷⣄⣀⠀⣴⡄⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⢤⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢋⡄⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠉⠙⢿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⢶
⠀⣴⣇⡀⠉⠀⠀⠀⠈⢣⡀⠀⠀⠘⢇⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠉⠁⠁⠀⠀⠀⠀⠀⠘⢣⡀⠀⠀⠈⠁⠀⠀⠀⠶⠶⢾⣉⠉⢉⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣀⣀⣀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠀⢀⠀⠀⠂⠈⠉⠁⠈⠉⠉⠁⠲⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⢾⡷⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢀⣀⡀⠀⠀⠀⣀⡀⠀⠶⠦⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡆⠀⠉⠉⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠒⢀⣀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢂⣀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣄⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠘⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣭⣤⣤⣦⣤⣤⣄⣀⣀
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠉⠁⠀⠀⠀⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣴⣦⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣯⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠒⣒⡶⠴⠶⠒⠒⠒⠒⠒⠲⠶⠶⣶⣒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿⣿⣿⣿⣿⣿
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠾⣭⠩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣤⢉⠭⠭⠭⠭⠭⠭⠭⢭⣍⣭⣭⣭⣭⣍⣩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⣭⢽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣤⣬⢷⣒⣒⣒⣒⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣿⣿⣿⣿
⢰⣿⡇⣿⡇⣾⡿⣰⣾⣿⣷⣸⣿⣴⣿⣇⣿⡏⣿⡏⠘⣗⣒⣒⣾⣿⣿⡟⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣒⣒⣒⣒⣒⣒⣒⣒⣺⣿⣿⣿
⠈⣿⣷⣿⣷⣿⡇⣿⡇⣸⣿⠛⣿⣿⢿⣿⣿⠁⣛⠁⢀⣧⡾⠋⠙⢿⣿⡇⢻⢡⡏⣸⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠤⠦⠶⠶⠦⢿⣿⣿
⠀⣿⣿⡏⣿⣿⠃⠻⠿⠿⠋⠀⠟⠛⠘⠛⠋⠈⠛⠀⣼⣿⣷⣫⣤⡴⠞⠿⣆⠀⠀⠠⠶⠶⣷⣶⣤⡙⢿⣿⣿⣿⣟⢿⣿⣿⡭⠭⠭⠭⢽⣿⣿
⠀⠙⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠋⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⢻⣷⣌⠙⣧⢻⣿⣿⣹⣽⣫⣩⣿⣿
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣚⣹⡇⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⡆⠘⠆⢿⣿⣗⣒⣒⣒⣿⣿
⣛⣛⣒⡦⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⢳⣒⣒⣿⠃⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣒⣒⣂⣿⣿
⠭⠤⠤⠵⠤⠤⠤⠤⠤⠤⠤⠭⠷⠦⣤⢀⠹⡤⢼⣾⣾⣿⠁⠀⠀⠀⠐⠦⣤⡀⠀⠀⠀⢀⣀⣤⠖⠒⢾⣿⣿⣿⣿⡆⠀⢸⣿⣿⠤⠤⠤⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠿⣽⣿⣿⠇⠀⢀⣴⣞⣓⣀⠀⠁⠀⠀⠀⠉⠉⢀⣐⣲⣾⣿⣿⣿⠉⠁⠀⢸⣿⣿⠬⠭⠭⣿⣿
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⣿⣿⣿⣿⠀⠀⠀⠻⠍⠻⠟⠁⠀⠀⠀⠀⠀⠘⠉⢽⣟⣽⣿⡿⢁⠀⠀⠀⣼⣿⣇⣙⣉⣁⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠁⣼⠀⠀⢠⣿⣿⣖⣒⣒⣺⣿⣿
⠶⠶⠶⠶⠶⠶⠒⠶⠶⠖⠖⠶⠲⠲⠖⣿⣿⣿⣿⣇⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣰⡿⠀⠀⣼⠏⣿⣿⣷⡖⣾⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣿⣿⣿⢳⠹⣆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⣿⣷⣼⣿⡇⠀⢠⡟⠸⡇⠙⢿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⣿⣎⢧⡁⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⣿⣿⣿⢿⠃⠀⢸⠃⠀⠁⠀⠘⣿⣿⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣿⣦⡁⠀⢻⡆⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡀⠀⠀⣿⣿⠏⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣛⣷⡄⠈⣿⣆⠀⠀⠀⠀⢀⡈⠛⠛⠛⢉⠄⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠤⠤⠤⠶⠶⠶⠤⠤⠤⠴⠶⠶⠦⠤⠤⠤⠤⠤⠼⠽⣿⠀⢹⡟⣧⡀⠀⠀⠀⠈⠑⠒⠚⠁⠀⠀⣰⣿⠁⠀⠀⠀⠀⣸⡄⠀⣴⣿⣿⣿⣿⣿⣿
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⡿⠀⢸⡇⢿⣿⣦⣄⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠀⠀⠀⠀⣠⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿
⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠍⠋⠀⠀⢸⡇⢸⣿⠿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⠂⠀⠀⠀⠀⠈⠀⠘⡟⠀⠈⠻⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣒⣒⣒⣒⠒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⠃⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡭⠭⠭⠤⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣉⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣮⣧⣤⣥⣭⣭⣭⣭⡭⠭⠭⠭⠭⠅⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿