Anya Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠛⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠘⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣱⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠖⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠙⠛⠻⠀⠀⠀⠀⠀⠀⠈⠛⠙⠃⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠢⠀⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣬⣽⡛⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⣾⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⣼⠛⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣷⣘⣧
⠀⠀⠀⠀⠀⠀⢰⣏⣿⣿⣿⡿⢦⢄⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⡚⠑⠖⢲⠌⢃⠀⠠⠄⠀⠀⠀⠀⠀⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⡆
⠀⠀⠀⠀⠀⠀⡜⢹⣿⣿⣿⣷⣶⣦⣤⡀⠀⣀⣠⡄⠀⠀⠀⠀⠀⢰⠀⠰⠀⠈⡆⠘⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣧
⠀⠀⠀⠀⠀⢰⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠠⠤⠤⠴⠶⠒⠂⠀⠀⠰⠶⠃⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣦⣤⣶⣶⣶⣶⣶⣶⣶⣾⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣧
⠀⠀⠀⠀⣸⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹
⠀⠀⠀⢀⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠻⡆
⠀⠀⠀⣾⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣿⣿⣟⣴⣿⣿⣿⣿⣿⣶⢶⣶⣾⣿⣿⣿⣿⡄⣷
⠀⠀⢸⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⢿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇
⠀⢠⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿
⠀⣼⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠈⠁⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢏⡾⠁⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⣾⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢧⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⠃⠀⢀⠁⠀⠀⠸⢁⠴⠁⠀⠀⢀⢧⢠⠀⡆⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⠀⢸⠀⣣⡴⠷⡔⠀⡠⢴⡧⣢⠴⣯⣢⣠⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶⠁⠀⠀⠀⠨⣿⢱⣿⣥⡀⠀⠀⠀⠀⣱⣯⣤⢹⣿⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⡀⠀⠀⠰⡀⣿⠘⠟⠻⠀⠀⠀⠀⠀⠹⠛⠟⢸⡏⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠑⢤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠁⠀⡎⠆⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⠂⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⣀⡄⠠⢂⠃⠀⠀⠀⠀⢀⡞⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢀⠀⠐⠀⠀⠉⢉⣉⠉⠁⢶⣼⠀⠀⠀⠀⡠⠊⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢢⣄⣠⠐⢩⠤⡙⢦⣸⣿⣠⡄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣼⣿⣥⣤⢋⣤⡅⢤⣀⣿⣿⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣼⣿⣿⣿⣿⣼⣿⣷⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣯⣲⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠈⠉⠲⢄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⢴⣶⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣲⣶⡄⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠘⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡿⠀⠐⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢣⠀⠀⠈⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠸⠀⠀⠀⣤⠤⣀⠀⠀⠀⠀⠀⠠⡠⢄⠀⠀⠀⠈⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⢰⡏⢸⣿⡧⠀⠀⠀⠀⣶⣿⡆⢻⠄⠠⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠁⠠⠁⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠆⣄⡎⠀⠀⠀⠀⠀⠀⠀⠖⠲⠀⠀⠀⠀⠀⠀⠀⠆⡀⢀⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⢿⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⣷⢸⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⣶⣿⣾⣤⢀⠀⠀⢺⠿⠀⠀⠀⠀⡿⡁⠀⠀⠀⣼⣤⣿⣼⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⠒⠀⠀⣥⢼⠐⠒⠚⣻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣄⣤⣼⣿⣼⣧⣄⣴⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⢧⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠞⠛⠒⢿⡅⠀⣿⠟⠋⠉⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⡤⠶⠿⠀⠙⠶⠆⠀⠀⠀⠀⠀⠉⠱⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⡟⣿⡿⠉⣹⠀⣠⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⣿⣟⢿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢋⣿⡟⠀⠀⠻⣶⣿⠃⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣿⡏⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⡟⠀⠀⠀⣴⣿⠃⢴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⣦⠀⠹⣿⡜⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣾⡟⠀⢀⣄⢰⣿⡇⢠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⢷⡄⡿⠣⠄⢻⣷⡾⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣿⣇⣾⣿⣅⣤⠘⠛⣿⡇⠈⠃⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠘⢉⣷⠸⣏⠀⠀⠀⠘⣿⣧⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣽⣿⠇⠀⠀⠀⢸⡏⣿⠀⢀⣿⣿⢰⡆⠀⠀⠀⠀⠀⠀⣻⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⡆⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⢀⣼⠇⣿⠀⢼⠛⣿⠘⣧⡀⠀⠀⠀⠀⠀⠻⠀⢻⣄⠀⠀⠀⠰⣦⠀⠀⣿⠀⠹⣧⠀⣶⡄⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢰⡦⠀⢸⣿⠠⣿⣧⠘⣦⣸⣧⠙⣧⡀⠀⠀⠀⠀⠀⣧⠀⠹⠷⣄⠀⠀⣿⡀⣿⡇⠀⠀⣿⠀⣿⡇⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⢸⡇⠐⣿⡿⣴⡿⠿⣦⣿⠁⠙⠷⣼⣿⣶⡄⠀⠀⠀⢰⡆⠀⠀⠈⠻⣶⣮⣿⠛⣿⠀⠀⣿⠀⣿⡇⣿⠻⣧
⠀⠀⠀⠀⠀⠀⢰⣿⢿⡇⢸⣷⠀⣿⢆⣿⠇⠀⠈⠻⣷⡄⠀⠉⠛⢿⡍⠛⠢⣤⡀⠘⣦⠀⢠⣄⣀⣻⡯⠀⠘⢷⠀⣿⣀⣿⢻⡿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣼⡿⢸⣿⢸⣿⠀⣿⣾⡇⠀⣠⣾⣿⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣱⣿⣿⣿⢿⣾⡀⠘⢣⣿⢹⡿⣼⡇⠀⣿⣧
⠀⠀⠀⠀⠀⢠⣿⠇⠀⣿⡜⣿⡀⣿⡿⠀⢻⣿⡿⢻⣿⣷⣍⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⡀⣿⣿⣦⠹⢿⣆⠈⣿⣼⣇⣿⠀⠀⣿⣻⣇
⠀⠀⠀⠀⠀⣿⣿⠂⠀⣿⡄⢹⡇⣿⠀⢰⣿⡏⢠⣤⣾⣿⣿⡆⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⠆⢸⣿⣀⣿⣿⣿⡏⠀⠀⢸⣿⢹⣇
⠀⠀⠀⠀⣸⣿⣿⠀⠀⢸⣿⢸⣷⣿⡄⢸⣿⠀⢸⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢻⡿⠀⠈⡿⢿⣿⣿⡏⠀⠀⠀⠸⣿⠈⣿
⠀⠀⠀⢰⣿⣿⡇⠀⠣⡄⢻⣇⣿⣻⣧⠘⠟⠀⠈⢿⣧⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠁⠀⠀⢃⣾⣿⡿⠁⡀⠀⠀⢰⣿⠀⠙⣷
⠀⠀⢀⣿⠃⣿⡇⠆⠀⣁⠈⢿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠁⠤⠘⢸⣿⠀⢰⢻⡇
⠀⠀⣾⣿⡇⣿⣷⣴⠀⠀⠀⠈⢿⣿⣿⡇⢠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣦⡀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⡇⠈⠘⢿⡄
⠀⢰⣿⣿⡇⢿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⠋⢹⢿⠿⠛⠛⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⣾⣿⡇⠰⡆⣼⡇
⠀⣾⣿⣿⣿⠸⣿⣇⠼⠀⠰⢆⠀⢀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣛⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢃⣜⠃⣀⣴⣾⢿⣀⣿⣿⣧⠀⢠⣿⣿
⢰⣿⣿⣿⣿⣶⣿⣿⡄⠀⠠⡌⡱⠋⣠⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠈⢹⣿⣿⣿⣿⢆⣿⣿⣿⣿⣧⣾⣿⡟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠘⢀⠘⠇⣀⠋⠀⠈⢿⣧⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡏⢱⠶⣬⣹⣿⣿⢿⣾⣿⣿⣿⢿⣿⣿⣿⠃
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⠣⣄⠙⠞⢳⣇⣼⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⣀⣀⣀⣴⣶⣿⣿⣿⣿⣿⣿⢛⣃⣴⣟⢻⣿⣻⣿⣿⣿⣿⣟⣾⣿⣿⠇
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⣏⡰⢾⣉⡉⢲⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠛⠛⠋⠁⣿⣿⣿⣿⣿⣿⣿⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⡳⠄⣉⡿⣾⣿⣿⣿⣿⡿⠟⠛⣤⣴⣿⣿⣿⡷⠆⠉⠙⠻⣿⣿⣿⣏⠘⣾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣯⠻⢿⣿⣿⣿⣷⣤⠛⣧⣼⣿⣿⡏⢁⠀⢠⣤⡈⠉⠉⠉⠉⠀⢀⣀⢤⣄⠀⢸⣿⡟⢢⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⠈⢻⣿⣿⣿⣷⣤⣾⣿⣿⡷⠞⠷⠲⠶⠿⣷⣶⡀⣶⢶⣾⣹⣄⣩⠿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢠⡶⣶⣶⣿⣼⣷⣿⣾⣏⣉⣈⣉⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠋⣁⣴⡆⣠⢷⣴⡄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠙⠛⠃⠀⠀⠀⣠⠀⠉⠋⠙⠛⠉⠈⠙⠋⠉⠁⠀⠀⠉⠁⢀⠉⠉⠉⠉⠀⠀⠹⠿⠿⢿⣿⡿⠿⠟








⣿⣿⣷⣶⣶⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⢶⣶⣶⣶⣤⣤⣤⣤⠄⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠄⢁⣿⠇⢴⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢰⣼⣿⠀⣸⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⡆⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠀⣯⠀⠀⣦⠀⠐⢠⠠⠈⣧⠀⠀⠀⠀⠀⠰⢸⣿⡇⠸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⣠⠀⠈⠀⣀⠤⠰⠆⠀⠀⠈⢦⠀⠈⠀⠉⠀⠀⠆⠀⢀⣾⠀⠀⣭⠀⠀⠀⠀⠰⠂⠠⢿⣿⣄⢿⡿⠋
⠀⠀⠀⢀⠀⠘⣇⠀⢠⠀⠈⠀⠀⠀⠊⢀⠘⠛⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠍⠰⠂⠩⠃⢤⠀⣽⠀⠀⠀⢀⡄⠈⣴⠋⠿⣿⠟⠀⠄
⠀⠀⠀⠀⠀⠀⢻⡄⠈⢀⠀⠀⠀⠀⠀⡇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢀⠰⠶⡆⠀⠘⠀⠀⣼⠀⠀⠀⠈⠇⣸⠇⠀⠀⣿⠾⠀⠠
⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⡆⠀⠀⠄⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠛⠀⠀⡀⠀⢹⠀⡀⢀⣠⣾⡻⣀⣿⢸⡿⠀⠀⠀⡆
⠀⠀⢰⠀⠀⠀⠀⠘⣇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠰⠀⠀⠀⠀⠀⢸⡰⢠⣦⠟⠑⠛⣿⣿⣧⡇⠀⠀⢸⡇⢻⡀
⠀⠀⢸⣄⠀⠀⠀⠀⠘⣆⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢧⣷⣿⠆⣀⣾⢿⡟⣿⠇⠀⠀⢀⡇⢸⠁
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⢣⡀⢣⠀⠀⠀⠀⠀⠀⣼⠋⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠎⢁⣽⡿⠛⠁⠈⠁⣿⠀⠀⠀⣸⠀⡤
⠀⠀⢸⣷⣤⣄⠀⠀⠀⠀⠀⠱⣼⣿⣦⡀⠀⠀⢘⠁⠈⠃⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢃⡴⠚⠋⠉⠀⠀⠀⠀⣸⠇⠀⠀⣰⡏⢸⠗⠀⢠
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣿⣿⡟⠀⠀⣀⡀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⠿⣰⠏⣴⠂⣼
⠀⠀⢸⠟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⢽⠛⠳⠶⢤⣄⡀⣀⢀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⣶⡶⣿⣾⡟⣴⠏⢠⣿
⠀⠀⠀⠂⠘⣿⣷⡞⣀⢠⠀⠀⠀⠘⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡠⠉⣽⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⣿⣧⢾⣿⣿⣿⣿⣿⣏⢠⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⣠⣵⣼⣿⣿⣫⣵⣿⣿⣿⣿⠟⠀⢡⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠄⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⣠⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣷⣴⣶⣿⡿⠿⠿⠿⠿⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠆⠠⠦⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⢋⡿⠃⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⡾⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣿⠃⠀⢠⠁⠀⠀⢀⠀⢀⡀⠀⠀⠀⢀⠀⠀⢠⠀⠀⢳⢺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠃⠀⠀⡆⠀⢀⠄⠏⠠⡚⠀⠀⢀⢀⠞⢆⣆⠀⡄⠀⠈⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡠⢋⡀⠀⠀⠇⣨⡴⣞⠳⡎⠤⠤⢒⡯⣢⢴⠾⣬⣆⡇⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠞⠠⢀⠁⡀⠀⢾⡏⣼⣭⣴⠀⠀⠀⠀⠀⢱⢯⡤⡌⢿⡇⠀⠄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠠⡀⠈⠀⠐⡄⢹⣇⠹⠟⠟⠀⠀⠀⠀⠀⠘⠞⠳⠃⣼⡇⠈⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡆⠁⠀⡌⠣⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡠⢰⠀⠠⠀⠀⠀⠀⡼⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠁⠀⢰⡄⠈⠐⢄⠀⡏⠻⡛⠛⢛⠿⢄⣀⡇⠀⠀⠀⠀⠀⡼⠁⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠑⠤⡀⠀⡀⠠⠼⡒⠣⣄⠈⣿⡇⠀⣀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠘⠀⠀⠀⠀⣼⣷⠏⠀⡼⠉⠓⠈⠓⢻⣷⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢠⣷⣷⠘⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⢀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿⣿⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠂⠈⠉⠉⠀⠀⠀⠨⢆⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⢰⠃⠀⠀⠀⠀⠀⢀⠀⠠⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠊⡀⠀⣠⠂⠀⡀⠀⢀⠀⠐⠈⠀⠀⡠⠊⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣠⠀⢈⡴⠋⠌⠀⢈⡁⠐⠁⠀⠀⣀⠔⠊⠀⠀⢀⠄⠚⠆⠀⠀⢠⠁⠀⠀⠀⠀⢠⠀⡀⠠⠶⠵⠁
⠀⠀⠀⠔⠀⠀⢄⠤⠊⠀⡠⠋⠀⠐⢠⣾⠟⠃⢲⠀⠐⠊⠤⠤⠄⠒⠈⠁⠀⠘⠀⠀⠀⢊⠀⠀⡐⠀⠀⠈⢢⡱⣄⠇⠔⠁
⠀⠀⠀⠀⠀⠜⠁⠀⢀⠞⢠⢁⠐⠡⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠤⣄⣀⠀⠀⠀⢈⢀⠼⢀⠀⠀⠀⠀⠑⠨⡏
⠀⠀⠀⠀⠈⠀⠀⢀⠂⠠⠃⣆⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⠉⠈⢻⡄⠀⢠⡧⢀⡤⠁⠀⠀⠀⠀⠀⠀⢡
⠀⡘⠆⠀⠀⠀⢀⠂⢀⠂⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢈⡇⢰⣋⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠘⠀⠈⠄⠀⠀⠀⠀⡌⠀⠀⡇⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠶⢋⡵⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⢄
⠇⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⡸⡄⠀⠀⠀⢀⡖⠁⠈⠀⠠⠀⠀⠀⠀⠀⠄⠈⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠨⠀⠈⣇⡴⠪⡄⠈⠦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⣬⢂⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢸⣄⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢂⣠⠁⠀⠀⢀⠀⢄⠇⠴⠤⠤⠀⠀⠀⠠⠎⠀⠴⠁⠠⠿⢷⠀⠀⠁⡈⠁⠠⠒⠈⡄⡄⡀⢀⡀⢸⢀⣡⣀⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠁⠈

⣿⡈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢹⡇
⣿⣷⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⣽
⣿⣿⣧⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⡏
⣿⣿⣿⣆⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣼⠇
⣿⣿⣿⣿⡆⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢻⡟
⣿⣿⣿⣿⣿⡄⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⡆⢈⢠⠀⡀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⣾⠃
⠃⢿⣿⣿⣿⣿⡄⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡿
⠀⠈⣿⣿⣿⣿⣷⡈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡴⠋⢩⣝⠆⠀⠀⠀⠀⠀⠒⢩⣍⠳⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣠⡇
⠀⠀⠚⣿⣿⣿⣿⣧⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢸⣾⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢳⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣆⠸⣿⣇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⣟⣟⡟
⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆⠹⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣨⣽⣹⠃
⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⡄⠹⡇⠀⠀⣶⣸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⣤⣏⡏
⠀⠀⠀⠀⠀⠈⢿⣿⡎⢿⡇⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠐⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⣰⣿⣾⡇⣼⠀⠀⠀⠀⠀⣾⣿⠅
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠻⠀⠀⠀⠘⣿⡿⣿⣧⡀⠀⠀⠀⢹⣿⣿⣷⠒⠶⠶⠖⠛⡿⣿⣿⠀⠀⣰⠺⣷⣿⣿⣽⡏⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠳⡟⡮⠻⣯⣢⣄⠀⢸⣿⠛⠉⠁⠀⠀⠀⠀⠀⠈⢛⡄⣘⣿⣫⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠙⢧⠀⣠⣿⣷⣤⣿⣿⣷⣤⣀⣀⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣷⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⡙⠻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⢻⠟⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠊⠀⠀⠀⠀⠀⠀⡆⠀⠀⠫⠀⠀⠀⠀⣰⡀⣆⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⠘⠉⠀⢀⠀⠣⠀⠀⠀⢈⠑⡌⠂⡀⡄⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠀⡰⠁⣀⣠⣀⣮⠀⠀⠀⠠⠀⣸⣤⣼⣦⣄⣳⢶⡀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠃⠀⠀⠀⠀⡏⢠⡾⠋⠙⣿⡟⠀⠀⠀⠀⠀⠸⡋⢈⣿⡌⢻⡄⠃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠃⣾⠀⣷⡾⢿⠿⠀⠀⠀⠀⠀⠀⣿⠟⠋⠿⠀⡷⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠁⠀⠙⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠒⠀⠁⠀⠀⠂⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡅⡀⡛⠙⠓⠚⠋⠉⣏⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠟⠳⠀⠀⢀⡠⠄⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠸⣄⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⣼⣿⡄⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣠⠂⠔⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢊⣾⣷⣿⣷⣦⣄⠀⢠⠀⠄⠀⠀⠀⣿⣼⣿⢀⡜⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⡜⠻⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⣿⣿⣿⡟⢉⣿⣶⣤⣼⣌⠃⢰⣿⠀⣿⡆⢳⣴⣾⣶⣿⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⠀⣼⣿⣿⣟⠀⠈⠀⣾⡗⠀⢹⣷⠈⠀⢳⣿⣿⡀⣿⣿⣿⣿⣿⣿⣇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⠛⣿⣿⣿⣧⣦⣾⣿⣇⣠⣴⣿⣷⣬⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⠋⠀⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡋⠁⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣮⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠍⠛⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡉⠋⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢏⠄⠀⠀⠀⠀⠀⢀⠀⠀⢶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠈⠄⠀⠀⠀⠻⡆⢣⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⡰⠀⠀⠀⡰⠀⠈⠀⠀⠀⢀⠈⢣⡐⠸⣆⡄⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⢠⠘⣠⡴⠾⣷⣿⡀⠀⠀⠀⠀⢺⠟⢿⣟⠳⣟⢱⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⠀⠀⠀⠀⢸⢨⡏⢠⣠⣽⣷⠀⠀⠀⠀⠀⠈⣦⣾⣿⣇⢹⡆⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠄⠀⠈⠘⠀⠹⡋⢙⡌⠀⠀⠀⠀⠀⠀⠳⣁⡨⠁⠈⢁⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⠲⠲⠶⠴⠖⢲⡟⣴⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡷⢮⠀⠀⠀⣀⠀⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢦⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⣿⡁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡐⢀⠆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢲⣿⣾⣷⣴⣄⠀⢀⡀⠄⠈⠀⠀⣿⣵⣷⠀⡞⠀⠀⠀⠀⠀⢀⣠⣾⣿⣧⠾⠥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠋⣿⣦⣤⣴⣄⡇⢠⣿⠉⣿⡈⣧⣴⣶⣾⡟⢻⣿⣿⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⠃⠈⠁⣼⡗⠀⢹⣷⠉⠘⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠃⠘⢿⣿⣿⣇⣴⣾⣿⣇⣠⣴⣿⣷⣽⣿⣿⠋⠙⠿⣿⣿⣿⣿⣿⣧⢀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⠈⠁⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠉⠀⠈⠀⠈⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣉⢩⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣔⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⢼⠀⣿⠀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣿⣿⣭⣭⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠐⠁⣆⢄⡊⠀⠋⠀⣈⢀⠈⠓⠀⠀⠀⠇⠀⠀⠀⠀⠄⣡⣭⣉⣛⠻⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⢰⢱⠼⡇⠀⠀⠀⡧⢿⠱⢀⠆⠀⠀⠘⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠑⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⢹⣿⣿⠒⠒⠒⣿⣿⡇⠀⠀⢀⠀⢠⠆⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⡀⠀⠈⠡⡈⣳⢤⣄⠀⠈⠃⠀⡀⠾⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⠈⢹⣤⡉⠓⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧
⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⡴⠋⣠⡴⠛⠋⠉⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⢀⠀⠀⠇⠳⡀
⠀⢀⠌⠀⠀⠀⠀⠀⠀⠎⣤⡟⠋⠀⠀⠐⣿⣶⣄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡶⠞⠛⠛⠛⠿⣷⣴⣄⠙⢦⡀
⢠⠋⠀⠀⠀⠀⠀⠀⢸⣸⠏⠀⢠⣆⣠⡾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠐⣶⣄⠈⠹⣦⠀⠀⠙
⠃⠀⠀⠀⠀⠀⠀⠀⣳⠏⠀⠀⠋⠙⢿⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⢀⣴⣿⣿⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠈⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣶⠉⢻⡟⠋⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢰⡶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⠀⢸⠦⠂
⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢵⣈⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡌⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢷⣆⣶⡒⠲⠒⠲⢾⡻⠿⠉⠀⡺
⠁⠢⢄⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⠈⠛⠷⠶⡤⠀⠀⠂⠀⠀⢀⡇

⣷⡘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢹⠃
⣿⣧⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⡼
⣿⣿⣇⢹⣿⡄⠀⠀⠀⠀⠀⢰⣶⡿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣳⡇
⣿⣿⣿⣆⢻⣷⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸⡿⠃⠀⠀⠀⠀⠀⠀⢸⡟⡿
⠻⣿⣿⣿⡌⢿⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡇
⠀⢻⣿⣿⣿⡜⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣝⠂⠀⠀⠀⠐⢹⡕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼
⠀⠀⢻⣿⣿⣷⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠈⠟⠃⠘⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠈⢿⣿⣿⣧⠹⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣚⣻⠇
⠀⠀⠀⠘⣿⣿⣿⡆⠹⠀⠀⣦⡇⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠠⣼⡽
⠀⠀⠀⠀⠸⣿⡎⢷⠀⠀⠀⣿⣿⣄⠀⠀⠈⣷⣦⣤⣀⣀⣠⣤⣴⡆⠀⢀⣴⣻⣿⣶⠀⠀⠀⠸⡿⠃
⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⢻⠿⠻⣦⣀⠀⣿⡟⠛⠀⠀⠀⠈⠙⣣⢐⣷⣪⣿⣿⠇⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣀⣑⣀⣼⣿⣶⣿⣿⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣉⣁⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠛⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠘⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣱⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠖⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠙⠛⠻⠀⠀⠀⠀⠀⠀⠈⠛⠙⠃⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠢⠀⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣬⣽⡛⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⣾⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⣼⠛⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣷⣘⣧
⠀⠀⠀⠀⠀⠀⢰⣏⣿⣿⣿⡿⢦⢄⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⡚⠑⠖⢲⠌⢃⠀⠠⠄⠀⠀⠀⠀⠀⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⡆
⠀⠀⠀⠀⠀⠀⡜⢹⣿⣿⣿⣷⣶⣦⣤⡀⠀⣀⣠⡄⠀⠀⠀⠀⠀⢰⠀⠰⠀⠈⡆⠘⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣧
⠀⠀⠀⠀⠀⢰⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠠⠤⠤⠴⠶⠒⠂⠀⠀⠰⠶⠃⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣦⣤⣶⣶⣶⣶⣶⣶⣶⣾⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣧
⠀⠀⠀⠀⣸⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹
⠀⠀⠀⢀⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠻⡆
⠀⠀⠀⣾⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣿⣿⣟⣴⣿⣿⣿⣿⣿⣶⢶⣶⣾⣿⣿⣿⣿⡄⣷
⠀⠀⢸⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⢿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇
⠀⢠⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿
⠀⣼⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣂⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠷⠹⠦⠶⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣶⣶⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣶⡦⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢹⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣽⠃⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⠇⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠙⠀⡾⠀⠀⠀⡀⠀⠀⠐⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈⣇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠃⠀⠀⢈⡼⢋⣴⣶⡀⠀⠀⠀⠀⢈⣷⣮⠳⣴⠀⠀⠀⢸⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠁⠄⠘⡇⢘⠟⠻⠃⠀⠀⠀⠀⠻⠟⠿⠀⠿⠀⠀⠢⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠂⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⡀⠀⠙⡶⡀
⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⢀⠶⢰⣦⡇⠀⠀⠀⡄⠀⠀⠀⠐⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⡗⣤⠀⣤⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡀⢸⡿⣷⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⣿⡀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣼⣧⣼⣷⣧⣅⢄⠀⠀⠐⣯⡟⠀⠀⠐⠀⢾⠏⡀⠀⠀⠀⣼⣀⣿⣥⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣦⡁⠀⠀⠁⠀⠀⠀⠀⠀⣧⠊⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⡄⡄⠀⠀⣈⣐⠂⣅⣠⡴⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡇⠀⠀⢰⡟⢹⡇⠀⠀⣀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠾⢿⣿⣿⣿⣿⡿⠾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠈⠀⠈⠀⡀⡴⠀⢠⠀⠠⠀⠀⠀⢀⡠⠊⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⡠⠂⠠⠚⢁⠀⣤⣤⢠⠁⢀⡠⠐⠁⠀⡀⠔⠈⠉⠀⠀⠰⠀⠀⡀⠀⠘⡀⣄⠘⠛⡁
⠀⠀⠈⠀⢠⠔⠁⠀⠔⡡⠀⠄⢊⠊⠀⠈⠀⠀⠈⠉⠁⠀⣀⣀⠀⠀⠀⠀⠐⠀⡐⠀⠀⠀⠈⠪⢢⠌
⠀⢀⠀⠀⠀⠀⠀⠊⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠙⠋⠙⣆⠀⠀⡜⠀⡀⠂⠀⠀⠀⠀⠈⡀
⠀⠼⠀⠀⠀⠀⠂⠠⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢸⠂⣞⠵⠊⠀⠀⠀⠀⠀⠀⠀⠐
⠘⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠚⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠁⠀⠀⠀⢠⠀⢸⠀⠀⡠⠣⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⡀⠀⠀⢀⠘⠀⡈⠲⠉⠒⠁⠂⠀⡀⠀⠀⠀⠄⠀⠀⠔⠁⡀⠀⣰⡋⠀⠠⠀⡀⢀⡀⠘⠀⠀⢀⠀⡟⢄⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠔⠀⠂⠁⠉⠉⠀⠀⠀⠈⠀⠈⠁⠈⠉⠃⠄⠀⠀⠠⠎⠀⠐⠒⠂⠘⠀⠃⠒⠒⠒⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⠒⢢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⣿⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⡆⠀⠀⡠⡇⠀⡯⡀⢀⡖⢀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⢀⣠⣤⣭⣿⣶⣤⣬⣭⣽⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠄⠸⠣⡆⠀⠠⠃⠠⠃⠀⠁⠃⠀⠉⢲⠀⠀⠀⠀⡆⠀⠀⠁⠂⠈⢁⢙⣉⡛⠛⠿⠿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣄⢠⢊⣉⣷⠀⠀⠀⠀⣼⣩⣝⢄⠀⡄⠀⠀⠀⠹⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠹⢸⠸⡍⠙⠀⠀⠀⠀⢻⠉⠛⠸⠠⠃⠀⠀⠀⠀⢇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠂⠀⢠⠀⠀⠀⠀⢻⣿⣿⡖⠶⠶⠖⢲⣿⣿⠃⠀⠀⠀⠀⠀⢄⡜⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢣⠀⠀⠀⠙⢏⠁⠳⢄⠀⡠⠈⠉⠻⠆⠀⠀⢀⣏⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠠⠄⠀⠉⠴⡟⠉⡷⣦⡂⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⣠⡠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣀⣀⣀⠀⢀⡤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢠⣿⠏⣡⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⡀⣾⡿⠀⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⡇⠰⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⢰⠀⠀⢀⡆⠀⠀⢰⠀⠀⠀⠘⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡴⠀⠀⣛⠓⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡀⣿⣿⠀⣼⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠋⠀⠀⠘⠃⠀⠀⠈⠀⠄⠀⠀⠀⣀⠀⠀⠀⢲⡀⠀⠤⠀⣿⠀⠀⠛⠈⠀⢰⡆⠁⢹⠀⠀⠀⠀⠀⣀⠉⢺⣿⡄⢹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⡐⠂⠀⠀⡰⠊⣩⣉⠑⠀⠀⠀⠱⠀⠀⠁⠀⠀⢀⢃⡀⠤⡿⠃⠀⢸⡄⠀⠀⠀⠀⠋⡄⢩⣿⣿⣮⡿⠛
⠀⠀⠀⠀⠀⠀⢿⡀⠀⡆⠀⠀⠀⠀⠈⢀⠆⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢁⠀⠃⣈⠁⠐⢁⡘⠄⠸⡇⠀⠀⠀⣼⠄⣸⠏⠈⠻⣿⣁
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢰⡄⠀⠀⠀⠀⠸⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠋⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣰⡟⢀⣀⢸⣏⠋⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⣿⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠉⠀⠀⠀⠀⠀⡇⠠⠀⢦⣾⢿⣷⣼⣧⢸⡇⠀⠀⢠⡇⡄
⠀⠀⢸⠀⠀⠀⠀⠀⢻⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⠀⢸⡇⢁⢾⡞⠁⠈⣹⣿⣿⣿⠇⠀⠀⢸⡇⢹⡄
⠀⠀⢸⣷⠀⠀⠀⠀⠈⠳⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢹⣾⣿⣟⣀⣼⠟⢿⢹⡿⠀⠀⠀⣸⠇⠸
⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠁⢄⠨⣄⠀⠀⠀⠀⠀⢠⡟⢀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡶⠛⣀⣴⡿⠋⠁⠀⠀⢸⡇⠀⠀⠀⡿⢠⡆
⠀⠀⢸⣿⣼⣤⡀⠀⠀⠀⠀⠈⢣⣿⣷⣤⡀⠀⠀⡎⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡁⠴⠛⠉⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃⣾⠃⠀⢰
⠀⠀⢸⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⢠⣄⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⢋⣼⠃⡞⠀⣾
⠀⠀⠸⠟⣿⣿⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⡍⠙⠳⠶⠦⠤⡀⣀⣀⣀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⢰⣶⣿⣽⣿⡋⣾⠃⢰⣿
⠀⠀⠀⠇⠘⢿⣿⣞⢃⠀⡄⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⣽⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣾⣿⡤⢿⣿⣿⣿⣿⣿⡇⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣷⣃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣀⣬⣮⣿⣿⣟⣭⣶⣿⣿⣿⣿⠋⠀⢱⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣶⣾⣿⡿⠿⠿⠿⢿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠱⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡟⡽⠁⠀⠀⡔⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⣏⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣹⢡⠀⠀⡰⠀⡐⠀⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠈⡆⠀⠀⠘⣆⢿⣿
⠀⠀⠀⠀⠀⠀⠹⣯⠇⠀⠀⠀⠇⠀⡇⠀⢿⠀⠀⠀⠀⠘⣆⠀⠘⣄⠀⡇⠸⠀⠀⠀⡿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢸⠀⢀⢇⢰⠤⣦⠀⢠⡀⠀⠸⣱⣄⢹⢦⢠⠀⡇⠀⠀⠃⣏
⠀⠀⠀⠀⠀⠀⠀⣴⠀⢠⠀⢸⢠⢃⡤⠿⢤⡄⠑⠤⣏⠒⠤⡽⣤⠿⢦⡟⠡⣰⠀⢠⢀⠘⡀
⠀⠀⠀⠀⠀⠀⣸⠃⡄⢸⠀⠸⢡⠏⡀⣿⣷⠙⠀⠀⠀⠀⠀⠊⣀⣿⣷⠙⡆⢹⠀⡘⢸⠀⢧
⠀⠀⠀⠀⠀⢠⠛⠀⢱⠀⠆⢰⣺⠀⣿⡿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣗⢸⠠⠁⡆⠀⠸⢇
⠀⠀⠀⠀⠀⠆⡆⠀⠀⢣⠘⣼⠸⡄⠫⠉⠜⠀⠀⠀⠀⠀⠀⠀⠣⡉⠝⢀⠇⠸⠁⡘⠀⠀⠀⡎⢆
⠀⠀⠀⠀⠸⢠⠀⠀⠀⠀⠱⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠔⠀⠀⠀⠀⡇⠘⡄
⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠇⠀⢡
⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠇⠢⠄⣀⣀⠀⠀⠀⠀⠀⣀⣀⠠⠐⠸⠁⠀⠀⠀⠀⠀⡘⠀⠀⠸
⠀⠀⠀⠸⠀⠀⣜⡄⠀⠀⠀⠀⠀⠸⠀⣰⡡⠼⠇⠉⠉⠉⠈⠣⠌⣆⢀⠇⠀⠀⠀⠀⢀⠜⡴⠀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⡺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢄⡔⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⣢⣤⣴⣶⣦⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡙⣆
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⡼⠋⣀⣴⠿⠋⠁⠀⣀⡀⠈⠙⢷⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡀⠒⣀⣀⣍⣀⣄⣀⠀⡇⠈⢧⡀
⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⡘⢠⣶⠟⠁⠀⠀⠀⠈⣿⣿⣷⡄⠀⢷⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⣀⣴⠟⠛⠋⠉⠙⠛⠿⣷⣾⣦⡀⠙⢆⡀
⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢰⢁⣿⠃⠀⢀⣾⠀⣀⣾⣿⣿⣿⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠐⣶⣄⠀⠈⠻⣧⡀⠀⠀⢻
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⠃⠀⠀⠼⠿⣿⣿⢀⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⡀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠙⣷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣽⠇⠀⠀⠀⠀⠀⠈⠾⢿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⣠⣶⣿⣿⣿⠿⠇⠀⠀⢸⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⣿⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣇⣀⡼⡯⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡷⣿⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⣿⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣼⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠉⠓⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠒⠋⡝
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠙⠷⣦⣷⣌⠉⠉⠉⠙⠛⠶⠛⠉⠀⠀⢸⠃
⠀⠑⠢⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢒⡦⠀⠀⠈⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠑⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⢟⡿⠁⠀⠀⢠⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢷⡻⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⡼⠁⠀⠀⢠⠃⢀⠀⢠⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢷⡘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⠁⠃⠀⠀⡎⠀⡘⠀⢸⡄⠀⠀⠀⠀⠘⡄⠀⠀⡀⠀⠘⡄⠸⡀⠀⠀⠈⣷⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⢸⠀⠀⢰⠀⠀⠇⠀⡜⢇⠀⠀⠀⠀⠀⢻⢄⠀⠘⣦⠀⢱⠀⢇⠀⠀⠀⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠘⠀⢠⠻⡀⡗⠼⢆⠀⢰⣄⠀⠀⢣⡱⣄⢹⠳⣸⡄⢸⠀⠀⢠⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡄⠈⡄⠀⡇⢠⢃⣤⠿⠶⣤⠀⠑⠤⣧⠑⠢⢤⣷⡤⠿⢦⣽⠘⡄⡆⠀⢸⢠⠉⡆
⠀⠀⠀⠀⠀⠀⠀⢰⠇⡇⠀⡇⠀⡷⢱⠟⠀⢰⢷⣎⠃⠀⠀⠀⠀⠀⠠⢋⠐⣿⣦⠙⣇⠘⡇⠀⡎⢸⠀⢳
⠀⠀⠀⠀⠀⠀⢀⢿⠀⠸⡀⢱⠀⡇⢼⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⠀⡇⡰⠁⡎⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡌⡆⠀⠀⢣⠀⢆⡇⢻⠀⢿⠫⠝⡟⠀⠀⠀⠀⠀⠀⠀⠸⡛⠯⢿⠃⣸⠀⡷⠃⠰⠁⠀⠀⠇⢧
⠀⠀⠀⠀⠀⠸⢀⠃⠀⠀⠀⠣⡈⢷⠈⠃⠀⠁⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠊⠁⠐⠃⢠⠃⡠⠃⠀⠀⠀⢸⠈⢆
⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠀⠙⢬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⠁⠀⠀⠀⠀⢸⠀⠘⡄
⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢏⠑⠤⢄⣀⣀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⢩⠃⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡇
⠀⠀⠀⠀⠇⠀⢠⡜⡄⠀⠀⠀⠀⠀⠀⠸⡀⢀⣆⡠⠼⠈⠉⠉⠉⠁⠹⠤⣘⣄⠀⡎⠀⠀⠀⠀⠀⠀⡔⣡⠀⠀⡰⠁
⠀⠀⠀⠀⠘⢦⣈⡌⠾⣦⡀⠀⠀⠀⠀⠀⡗⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⣠⠮⠚⡝⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⢳⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠉⣻⡄⢹⣉⠀⠀⠉⠙⠲⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠔⠋⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣤⣄⣀⡀
⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣻⡿⠉⡗⢠⡄⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢿⣟⣿⣿⣿⣿⣶⡆
⠀⠀⠀⢻⣿⣿⣿⣿⣯⣼⡟⠀⠀⣹⣿⢁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢿⡌⣿⣿⣿⣿⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣼⡟⠀⡀⢠⣿⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⢻⣆⡸⣿⣼⣿⣿⠏
⠀⠀⠀⠀⢹⣿⣿⣽⣿⣀⡄⠿⣿⡉⠛⣹⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠛⣧⠿⡊⠀⠀⢻⣧⣽⡟
⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⢸⡏⡇⠀⣿⡇⣰⠀⠀⠀⠀⠀⢸⢳⡄⠀⠀⠀⠀⠀⠀⡇⠸⡇⠀⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠈⣿⡇⢀⡀⠀⣾⢁⣧⠘⡃⢧⡸⣆⠀⠀⠀⠀⠘⡈⠳⣤⡀⠀⢸⡆⣠⡟⠀⢻⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡄⣼⡇⢤⣿⣸⣿⣆⣻⠚⢷⣜⣷⣄⠀⠀⠀⢣⡀⠉⠙⢦⣨⣷⡿⣧⠀⢸⠆⣿⢨⡿⣇
⠀⠀⠀⠀⠀⢸⣿⣇⢸⡇⢸⣇⣿⠀⠈⠻⣦⠀⠉⠻⢿⠑⠂⣤⠈⢠⡄⢀⣄⣙⣿⠁⠹⡆⢸⣄⣿⣸⠇⣿⡄
⠀⠀⠀⠀⢀⣿⠋⣿⢸⡇⢻⣾⠇⢠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⣽⠿⣿⡿⣿⣄⠘⣾⣿⡏⣿⠀⣿⣧
⠀⠀⠀⠀⣼⣿⠀⣿⠘⣧⣿⠏⢀⣿⠟⠙⣿⣿⡌⠗⠀⠀⠀⠀⠀⠀⠾⢡⣤⣿⣿⡌⢻⡆⢸⣿⣷⡇⠀⢹⡿⣇
⠀⠀⠀⢠⣿⣏⠀⢹⡆⢿⣿⡄⣼⡟⢸⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣿⠇⢸⣿⣾⣿⡟⠀⠀⠈⣷⢹⡆
⠀⠀⢀⣾⣿⡇⠰⡈⢻⣾⣟⣧⠸⠇⠈⢿⣅⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠘⢡⣿⡿⠁⠀⠀⠀⣿⠈⢷⡀
⠀⠀⣼⡇⣿⡗⠀⢠⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠇⠈⠁⠈⢠⣿⠀⠊⣧
⠀⢸⣿⡇⣿⣷⠃⠀⠀⠘⢿⣿⣇⠴⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠿⠆⣾⣿⠃⠀⠀⠀⠀⢸⣿⡀⡀⢹⡆
⠀⣿⣿⣷⢹⣿⢀⠀⢠⠀⠈⠻⣿⠀⠉⠉⠀⠀⠀⠀⠀⣤⣶⢶⡄⠀⠀⠀⠀⠀⠀⢠⣿⠃⠄⠀⣀⣤⡄⣿⣿⡇⠁⣾⣇
⢰⣿⣿⣿⣼⣿⣇⠀⠠⡐⠞⢁⠙⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⢀⣽⡇⠙⣷⣾⣿⡷⣹⣿⣿⣷⣼⣿⡏
⢸⣿⣿⣿⣿⣿⣿⡄⣀⠛⢠⢋⡀⠸⣿⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣏⡱⢲⣭⣿⣟⣷⣿⣿⣿⣿⣿⣿⠃
⠘⣿⣿⣿⣿⣿⣿⣿⡌⢲⠌⣋⠽⢔⣿⣿⣿⣿⣿⣿⢷⣶⣶⣶⡶⢻⣿⣿⣿⣿⣿⡩⣵⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣎⢦⡙⠶⣆⣿⣿⣿⣿⣿⠿⢀⣠⣦⣤⣤⠈⠿⢿⣿⣿⣿⠣⣴⣟⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠘⢿⣿⣿⡻⢿⣿⣿⣿⣄⠛⣧⣿⣿⡟⠁⠀⣀⠉⠛⠛⠋⠁⢀⣠⣄⠉⣻⣿⢢⣻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠙⠻⢿⣦⣈⣽⣿⣿⣷⣶⣿⣿⡷⠛⠒⠲⠿⡶⣦⢶⢶⣯⡧⣬⠿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠁⡄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⣿⣿⡟⠀⠀⠞⠛⢿⠿⣿⡿⠿⣤⡶⠦⠀⠸⠿⠻⠿⠿⠿⠟⠡⣴⣿⣴⣿⣷⣤⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠁





⣀⣀
⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣀⣤⣤⣤⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⢠⠤⡀⢠⠙⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣬⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⣠⠚⣣
⡇⠘⡁⠀⢷⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⠷⠁⡰⠉
⠃⡀⠀⠀⠠⠠⣤⣄⣡⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⡇⠀⠛⠀⠼⠁⠀⢀
⠀⢡⠀⠀⠀⠀⠀⠈⠁⢠⠄⠼⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠈⠡⠀⢀⣀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠙⠶⣀⠈⠃⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠳⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠀⠀⠋⠀⠀⠠⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠳⠄⠀⢀⡔⠦⢀
⠀⠀⠀⠈⠛⡦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⣄⡀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢰⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⢴⣆⢀⣀⠠⡤⣤⢤⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⢀⠌⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠘⡇⠀⢹⡆⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⢀⡤⠊⠁⠀⢀⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⠀⠀⣴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣧⣠⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡆⠀⠀⣄⢠⡜⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢨⡏⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⣀⡤⠞⣋⠀⠀⠀⢰⠀⠀⣿⡏⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡘⠁⠀⠀⠀⣤⣄⠤⡤⢤⣤⡤⠚⠛⠉⢀⣴⡿⠀⠀⠀⠘⠓⣤⣿⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣀⣤⡴⠾⢿⣿⠃⠀⠀⢠⣄⢀⡼⠋⠙⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⢋⣀⣀⣀⣼⠃⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢣⠀⠀⠀⠀⠙⢿⣟⠉⠀⠀⣀⡤⠏⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠷⣀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡏⠉⠉⠉⠉⠉⠀⠲⠶⠒⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠉⠠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢛⠉⠉⠀⠘⠒⠒⣶⠶⠤⣤⣤⣤⡤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡆⠀⠀⠀⠀⠀⠀⢀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⡷⠀⠀⠀⠀⣀⡤⠋⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢿⡀⢠⡄⠒⠉⠀⠀⠀⢀⢼⡇
⠀⠐⠋⣀⠀⠁⠙⣀⠀⠀⠀⠀⢀⡼⠶⠊⠙⠂⡀⠀⠀⠀⠀⡰⠃⢠⠃
⠀⠈⠲⠄⣀⣉⣉⣉⣉⣉⠉⣰⠛⠤⢀⡀⠀⠀⢡⡄⢀⣀⠐⠷⢀⡙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣠⣤⣤⣤⣬⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡀⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠃⠀⠀⠀⢧⠀⢳
⠀⠀⠰⣟⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣉⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠓⢄⡙⠁⣠⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣈⡁⠰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠁⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⢄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡀⠈⠻⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⢀⣠⠶⠛⠙⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠏⣹⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠛⠻⠟⠛⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⡥⠄⠀⠀⠀⠀⠀⢤⡀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠘⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⣾⣷⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡐⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡤⢴⠆⠀⠀⠠⠂⠀⠀⡠⠀⠁⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢌⣀⣠⡔⢈⡀⠠⣤⠐⠈⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢹⣧⣼⡷⠀⠀⠀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠠⠐⢻⡀⠀⠀⠀⠙⠈⠁⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⠁⠀⢀⠠⠀⠐⠛⠊⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠑⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣽⣷⠇⠀⠀⠀⣷⣿⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡻⠃⠀⠀⠀⠀⠘⠿⠿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣶⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣼⣿⡏⠀⠀⠀⠉⠉⠈⠙⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⠟⠋⠛⠛⠿⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠋⠉⠈⠁⠀⠈⢻⣿⣿
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠘⣿⡇
⠀⠀⠀⠈⣜⠀⠀⠀⠁⡀⢀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠗⠎⠀⠄⠀⠁⠄⠠⠀⠀⠀⠀⠐
⠀⠀⠀⢠⣿⣦⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠠⡉⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀
⠀⠀⠀⣸⣿⣿⣿⣿⠗⠐⢿⣿⣷⠀⠀⠀⠂⢀⠄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡛⠃⠓
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎
⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⢈⠁⠀⢠⠀⠀⠀⠀⠀⠈⠩⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠦⠲⠀⠀⠀⠀⠸⠲⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢒⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠥⡄⠈⢴⠀⠀⠉⡙⠀⠀⠀⠀⠈⡤⠏⠘⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠠⡄⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠺⠳⢾⡆⠰⠛⡠
⠀⠀⠀⠀⠲⡀⠘⡀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⣴⣀⢀⡀⠀⠰⠈
⠀⢰⠀⠀⡀⠑⠄⠀⠄⣀⠀⠀⠀⢀⠄⢀⠔⠀⠀⠈⠑⠒⠀⡀⠉⠉⠉⠆⠀⡀
⠀⠀⠑⢄⠈⠢⣈⡄⠘⠋⠀⢰⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠶⠈⠀⢠⠄⠉⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢦⡀⠂⡀
⠀⠀⠘⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⢀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢙⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠘⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠉⠀⠀⠊⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢂⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⠶⠀⠉⠉⠉⠛⠻⠶⣿⡄⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢠⠀⠀⢀⡠⠆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠋⠀⣼⣷⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠀⠘⢷⠒⠂⠐⠒⠒⠊⠉⠁⠀⢀⣤⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢻⠀⠀⠀⢧⣤⣄⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⡷⠀⣀⡀⠀⠀⣀⡀⠐⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⡀⠠⠿⢻⡟⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠓⠴⠂⣀⣀⣀⠴⠋⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠐⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠷⣤⣤⣀⣀⣠⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣤⣤⣠⣤⣤⠴⡾⠋⣀⠀⠷⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡅⠀⠀⡇⠀⠈⠁⠶⠂⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁⠀⠘⠀⢆⣤⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠈⠁⠀⠀⠀⠀⠀⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⢛⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⠃⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠀⠀⠀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠛⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠸⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣄⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⣤⡞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠳⠶⠞⠁⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠴⠞⠉⠀⠻⣆⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⡴⢾⠁⢠⣼⣁⣀⣀⣀⣀⠀⡴⢶⡞⠉⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢇⠀⠁⠀⠀⠀⠀⠀⠀⠄⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠡⣀⠀⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⢦⣤⣤⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⠁⣰

⠀⠠⣀⡀⠀⠀⣀⣠⣤⣤⠄⣴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⢀⣀
⠀⠀⠀⠉⠀⠈⣿⣿⡿⢃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠻⣿⣷⡶
⣄⠀⠉⠁⠀⠀⢹⣿⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠸⣿⠁
⠘⡄⠀⠀⠀⠀⠀⢧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠃
⠀⢣⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻
⠀⠘⡄⠀⠀⠀⢠⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡆
⠀⠀⢃⠀⠀⢀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸
⠀⠀⢸⢀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢰⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠈⠀⢱⣃⣸⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⡓⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠰⢱
⠀⠀⠀⠀⠘⠁⠘⢀⣀⠀⠀⣡⠖⠁⠀⠀⠀⠈⢨⡄⠑⢄⣀⠀⠀⣱⠈⠐⠤⠱⠑⠲⢿⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢸⠤⡤⠙⠊⠀⠀⢻⠠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⣀⣠⠤⠄⡼⠀⠀⠀⠀⠀⠀⢹⡀
⠀⢀⠆⡴⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢸⠀⠇⠀⠀⡀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠀⠀⠸
⠀⡆⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡼
⡘⠀⠀⠀⢂⣠⣶⡀⠀⠀⠀⠀⠀⠀⠛⠒⠤⢀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⣾⠁⠀⠀⠀⠀⠀⣰⣧⡀⠀⢸
⠀⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⢀⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡎
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣄⣻⣇⠀⠀⣿⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⠉⠙⠻⣿⣿⠿⢧⢻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣾⣿⣿⣿⠿⠿⠿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⢤⡈⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⢛⡻⠛⠉⠀⠀⠀⠀⠈⠙⢿
⡿⠿⠦⢄⣀⠀⠀⠀⡀⠀⠀⠐⠀⠉⢦⢿⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤
⠀⠀⠀⠀⠀⠈⠉⠉⠑⠢⢤⣌⣂⣬⣼⣮⡦⠀⠀⠀⠀⠀⠀⠻⢶⣾⣁⡔⢀⠀⢀⣀⡠⠔⠛⠛⠫⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⢀⠀⡀⡀⡠⢸⠠⡂⢠⠠⡀⡄⡂⠄⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢖⠂⣆⠗⠁⠇⠀⠣⠸⠀⠚⠘⠀⠃⠃⠁⠀⠀⡏⢀⠇⠄⠐⢰⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠤⠄⠤⢄⠈⠁⠀⠈⢁⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠎⠉⠀⠁⠈⠀⢀⣀⠀⠈⠑⠒⠖⠢⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠔⠉⠁⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⡉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠐⠢⠭⢥⡝⠀⠀⠁⠀⣀⠖⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠄
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠷⣮⡍⠁⠈⢣⠀⢀⣀⠆⠁⠀⠈⠙⠔⠆⠤⠤⠦⠀⠀⢀⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⠆⠋⠁⠀⠀⠀⢹⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢇⠀⢁⠏⡼⠀⠀⠀⠀⠀⠠⡘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃⢣⠀⠀⠀⠀⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠈⢢⣀⠜⠀⠀⠶⠀⠀⠀⡀⡄⢂⠀⠀⠀⠰⡇⠀⠀⠀⠠⠦⠀⠀⠣⣤⢠⡀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠓⢤⣀⠀⠐⠐⠀⠀⠀⠀⢀⡠⠔⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⣿⠐⠒⢲⣙⠷⡀⠈⠑⠀⠓⠋⠁⠀⠒⠒⠂⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⡿⠃⠀⠸⠄⢠⣆⠀⠀⠀⣀⠀⣀⠐⠤⣦⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠀⣈⠀⠛⠂⢸⠀⠉⠉⠂⠴⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⢀⠄⠉⠁⠰⠦⢼⡀⢀⢀⠀⡀⡀⢄⡒⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠃⠴⠻⠦⠐⠛⠒⠀⠀⢘⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠐⠀⠀⢀⠀⡀⠐⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡁⠀⢈⢩⡀⠀⠀⡀⣄⡄⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠰⠠⠔⢀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠂⠃⠘⠁⠈⠠⠃⠀⠀⠢⢂⠠⢦⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠌⠠⠀⠀⠀⠀⠀⠀⠀⠠⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠒⠀⠐⠤⠆⠄⠀⠀⢀⡐⠀⠀⡀⠀⢸⠗⣢⡡⠘⣀
⠀⠀⠀⠀⠒⢀⠠⠈⡀⠀⠀⠀⠚⠧⠰⡞⠀⠀⠀⠀⠱⠪⡄⡢⠀⠐⠐⠂⠀⠀⢈⣱⠈⠊⠤⠀⠭⣲⡠⠠⢴⠀⠀⠀⢬⠇⡡⡈⠈
⠀⠀⠀⠈⠀⠀⠀⢠⣋⣤⢠⣴⠀⢄⠘⠊⠍⠰⢀⣠⡄⠀⠀⠳⠁⠰⠩⠤⢆⡴⠤⠁⡀⠀⠀⢀⣒⠁⠀⡤⡀⣰⠐⠀⡂⠀⠀⢂⠂
⠀⠀⠀⠉⠄⠁⠀⠀⠼⠄⠙⠀⠉⠀⠉⠀⠀⠀⠀⠈⠆⠑⠆⢰⢣⠄⠀⠀⠄⠀⠀⠀⢓⠘⠂⠊⠀⠀⡈⠅⠈⠁⠀⠀⡤⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠄⢀⡤⠈⢐⡀⠀⠀⢀⡀⢠⠀⠔⠲⠀⠈⢠⢠⠉⠈⢁⠀⠀⣈⠰⠀⡐⠤⡍⡒⣱⠄⠒⠤⠑⠠⠶⡀⠋⡀⠉⡀⠠⣉⡧⠐⠂
⠀⠀⠀⠀⠐⠊⢱⠂⠀⠀⠅⠈⡁⠈⠁⠉⠀⣰⡀⠖⠀⠧⡴⠃⠚⠈⠠⠘⠂⡧⠒⠈⠀⠀⢆⠠⠀⠒⠀⠢⠀⠔⠠⡀⣀⠈⠊⢀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠦⢈⠁⠘⠀⠀⠙⠀⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠐⠃⠐⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠨⠒⡠⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠄⠠⠀⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡀⢀⠀⠀⠐⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠢⣄⠀⠂⢄⠀⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⡀⠘⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠄⠀⠁⠀⡐⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⢃⠀⠀⠀⢀⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠁⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠⠀⢢⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠉⠄⡀⠐⠀⢡⠀⠒⢆⠀⢄⠈⠑⠌⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⡄⠀⠃⠀⠀⢠⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠅⠀⠀⠀⠀⡰⠁⠀⢀⣄⣤⡀⠀⠀⠀⠀⢀⣤⣤⣀⠈⢧⡀⠀⠀⠀⢰⠀⠀⠀⡀
⠁⠘⠢⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡜⠁⢰⡷⠁⣠⡌⠁⠀⠀⠀⠀⠉⣠⡌⠻⣇⠀⠷⡄⠀⠀⠸⠁⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠈⠢⡀⠀⠀⢀⣧⠀⠀⡼⠁⠀⠸⡁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⠀⠀⠱⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠑⣄⡞⠉⠈⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡤⠀⠙⢦⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠈⠌⠀⠀⠀⠈⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⡗⣦⣤⣀⢀⣀⣤⣤⢶⡏⠀⠀⠀⠀⢸⠀⠀⢀⡞⢻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠰⡀⠀⠀⠀⠀⠘⢹⠇⠀⠀⠁⠠⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠑⢄⡀⠀⢀⡠⠁⢠⣤⣤⣄⡀⠁⢀⠀⠀⠀⠀⡀⠀⡰⠃⢸⣿⣆⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠣⠀⠀⠀⠀⢻⣷⣄⡀⠶⡶⠞⣀⡴⣫⠏⠀⠐⠋⠀⢘⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠻⢿⣯⡟⠽⣿⣵⣿⠏⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠄
⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠛⣛⣋⣉⣉⣉⣉⡉⠉⠙⠛⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣉⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⡟⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠸⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣠⠞⢱⠏⠀⠀⡿⠋⣿⣿⣿⣟⣁⡿⠁⢹⣿⣿⣿⣿⣿⠋⢻⣿⡿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢿⡀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⡟⠀⠀⠀⠀⠈⠹⣿⠏⠈⢹⡃⠀⠈⠟⠈⢻⣿⡟⠛⠛⢧⣀⣹⡟⠙⢿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇⠀⠀⠀⠀⣤⣰⣷⣶⣤⣽⡇⠀⠀⠀⠀⢒⣿⣀⣀⣀⡈⢧⡙⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠳⠶⢻⡀⠀⠈⢼⣾⠋⠙⣿⢇⠀⠀⠀⠈⣿⠋⢩⡿⠛⠻⣿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠹⣦⣀⣼⣿⠀⠀⣿⠘⠚⠉⠲⠴⠃⠀⣿⠀⠀⠀⣿⣿⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣴⠶⠦⣤⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣴⠟⢷⡴⠻⣼⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡏⠁⠀⠀⠈⠙⢷⣤⣤⡿⠶⠶⠶⠶⢶⣧⡀⠀⠀⠀⠀⠀⠘⠛⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⣯⣶⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⢹⡀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠰⣟⣽⣿⠉⣇⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠓⠾⠃⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⢀⡿⢹⡇⠀⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣄⣶⣶⣶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⢼⡃⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠉⠉⢉⡙⣻⠀⠀⠀⠤⠤⠤⠦⠶⣶⡶⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣦⣤⣤⣥⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠐⢿⠀⠉⢉⣉⣱⣶⣤⣤⣤⣤⣤⣤⡼⣯⣽⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠻⣿⠛⣛⣿⣭⣀⣸⡇⠀⠈⠉⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢛⡟⢹⠁⠉⠉⠀⠀⠀⠀⠀⠀⢀⣽⣯⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣴⠏⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡙⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣷⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣷⣬⢿⣤⣿⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣼⣿⣿⣿⣿⣹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⡀
⢀⢀⡟⠀⠀⠀⠀⢀⣷⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣽⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣭⠇⠘⣷⡀
⣼⢸⡇⠀⠀⠀⢀⣸⣏⣿⢛⣿⡻⢷⣶⣤⣤⣀⣀⣤⣤⡾⣱⣿⣿⣿⣿⣧⣤⣴⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⣷⡀
⢻⠘⡇⠀⠀⠀⢸⣟⣿⣽⣧⡯⢹⣾⣷⣾⣏⢹⣯⢽⣿⣿⠟⠻⣿⣿⣿⣿⡿⢿⣿⣿⡟⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠸⡇⠉⠀⠀⠀⠸⣷⠻⣿⣯⣻⣞⣉⣛⣭⣷⣿⣵⣿⣿⣧⣤⣼⢟⠞⠲⠻⣿⠀⠉⣉⣴⣾⣟⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠙⠀⠀⠀⠀⠀⢻⣷⣉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⠿⠿⠿⠿⠿⠛⠛⠿⢾⣟⢉⣵⢻⣦⡻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣼⣳⣩⣽⠯⠭⢽⢸⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣕⣿⣾⢷⡙⣷⣶⣶⣶⣶⣶⣶⡞⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⣽⣶⠿⠿⠿⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⢷⣻⣤⣥⣼⣷⣒⡴⢏⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣭⣿⣗⣛⣷⣯⡿⣟⣤⡿⣧
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠉⠙⠋⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠈⠙⠻⢶⣯⣭⡾⠟⠉⢀⣿⠀⠛⠛⠛⠛⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣌⡛⠶⣤⣴⠞⣡⣶⣶⣶⡶⠶⠶⣶⡶

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣠⣶⣾⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣶⣤
⠀⠘⣿⣿⣿⡟⠀⣴⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⠀⢻⣿⣿⣿⠃
⠀⠀⠘⣿⣿⠁⣸⣿⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠈⣿⡿⠃
⠀⠀⠀⠈⠋⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠀⠀⠀⠀⠀⠀⠤⡂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠙⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠩⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠖⠀⠀⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠀⠒⣤⣤⣤⣄⣀⡀⠀⡠⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣠⣴⣶⣿⡇⠻⣿⠋⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢀⠸⣿⣿⣿⣿⡄⠁⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣦⡀⠀⠁⠈⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣄⡀
⠀⠀⢀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⢄⣼⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣾⠋⠀⠹⣷⡄⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⡅
⠈⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠴⠿⠁⠀⠀⠀⠘⠿⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡆⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡐⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢫⣤⡈⠛⠀⡀⠁⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠄⠀⠁⡀⠀⠀⠀⠀⠀⠤⠰⠃⣀⡼⠏⠻⡄⠀⠙⡆⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⡆⠀⢠⠶⠒⠄⠠⠀⢀⠀⠀⠀⠀⠈⠙⠇⠘⠀⠁⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠀⢰⠂⡀⣸⡗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠛⣃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣴⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿
⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⣤⣉⣛
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿
⣿⣿⡟⠀⣤⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣷⡄⠀⠀⣿⣿⣿
⣿⣿⠇⣸⣿⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠐⠢⠠⠀⠤⠀⢀⣠⣤⣶⣾⢸⣿⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠐⠰⣿⣿⣿⢀⣼⣿⣿⣿
⣿⣿⠀⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣦⣀⠀⠘⣄⠐⠀⠀⢰⣾⣿⣿⣿⣿⣿⣼⣿⣧⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣆⣿⣿⣿⣷⣦⣄⣀⠀⠈⣉⣽⣿⠷⠦⣼⣷⣤⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⡅⠈⣿⠟⠋⠈⣿⣿⣿⠟⢻⣿⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢠⠀⠁⠀⠀⠀⡏⢀⣤⣄⣼⣿⡿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠄⠀⠀⠀⠀⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠠⠀⠒⠂⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢤⠀⡠⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠷⠦⠄⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠓⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠄⠰⠀⠂⢦⠄⠀⠀⠀⠀⠀⡘⠃⢀⠐⠃⡀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠂⠀⠘⠀⠃⠀⠀⠂⠀⠀⠘⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠄⠠⠀⠀⠀⠚⠤⠀⡄⡄⠀⠀⠀⠛⠐⠁⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⡀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁⠀⠀⠀⠀⡀⡐⠀⠀⠀⠀⠀⢄⠀⢀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠿⠃⠀⠀⠙⠀⢀⠃⠂⡀⠀⠈⢀⠂⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠢⣄⡀⠀⣄⠤⠀⠁⠠⠀⠀⠀⠈⣧⢀⡄⠀⠀⠀⡀⠰⡶⠀⢠⠄⡐
⠀⠀⠀⠀⠀⠀⠐⡀⠃⡄⠃⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⡀⠀⠠⢄⡀⢸⡇⠀⣤⡄⠀⠀⠀⢀⣰⣷⠦⢀⠀⠀⠀⠀⠀⠀⠘⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠌⠀⠘⠀⠈⠒⢷⡀⠙⠁⠀⣠⣴⠟⠁⠀⠐⠃⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣞⡿⠁⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣿⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠚⠛⠛⠋⠉⠉⠉⠙⠛⠛⠒⠒⠒⠤⠄


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⢀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢋⡞⠁⠀⢠⠎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠘⢷⡽⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢅⡾⠀⠀⢀⠃⠀⠀⠀⡎⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠙⡄⠀⠈⢻⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠇⠀⢀⠏⠀⢀⠀⡼⡇⠀⠀⢱⠀⠀⣿⠀⠀⠰⡄⠀⠱⠀⠀⠀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⡜⢰⠀⢸⢀⠃⠱⠀⠀⠀⠣⡀⡇⢣⡀⠀⢳⡄⠀⢇⠀⠀⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠀⢰⢁⠎⢦⡀⣯⡖⠛⠑⢄⣀⠀⣙⣿⠍⠛⠒⠴⠮⠲⢼⡀⠀⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠸⠀⢸⠃⠀⠀⠈⠀⠁⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡇⢈⣠⡴⠶⠶⠶⠶⠯⣥⠀⠀⠀⠀⠐⡯⠶⠶⠶⠶⢦⣄⢣⡜⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢳⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠸⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡸⠀⢻
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢇⠀⠀⠀⠱⡀⠀⠀⠀⠀⠑⠶⣖⣒⣒⣒⡲⠞⠃⠀⠀⠀⢀⡎⠀⠀⠀⢠⠃⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⢆⠀⠀⠀⢱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢀⠎⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣵⣶⣌⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣟⣻⣿⣿⣿⣿⣿⣷⡀⢙⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⣿⣽⣿⣿⡗⡸⡿⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡯⠉⠀⠀⠀⠀⠀⠉⠻⣿⣿⠇⢹⢳⣇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣤⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣠⣤⣤⡀⠘⣿⣯⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣟⡂⠀⠀⠉⠙⠛⠷⠆⠀⢹⣿⣿⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣣⣄⠠
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣯⠳
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⢤⠀⠀⠀⢰⣴⣿⡿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠠⠹⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡠⠐⢹⣿⣿⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢫⡑⠒⠒⠈⠁⠀⠀⢸⣿⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢮⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠨⠂⠠⠀⠬⣔⡀⡀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠍⠳⣦⡄
⠀⠀⢔⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠉⠀⠀⠘⡆⠀⠀⠀⢾⣿⠿⣿⡀⠀⠀⠹⣿⡀
⠀⠀⠀⢄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢸⡟⣀⡘⣿⠠⠴⠂⠙⠣
⠀⠀⢧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠀⠟⠋⠋⠀⠈
⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠘⠀⠀⠀⠰⠶⠄
⣶⣶⡶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢠⠏⠍⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢻⡉⣾⡆⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡟⢰⣶⣮⣿⣥⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣬⣽⣮⡑⣷⠈⣷⠈⣇⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣤⣀⣿⡼⣧⠈⠀⠉⣡⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣌⠉⠀⠀⣤⣴⣟⣽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⡀⣤⣄⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡄⠠⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⠷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡃⠀⢀⣴⣶⣾⣿⣿⣷⣴⣶⣿⣷⣽⣿⣶⣄⡀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⠿⠟⠛⠉⠙⢿⣿⣿⣿⣿⣿⣿⣦⠐⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢁⠀⠀⠀⠀⠶⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢁⡶⠛⣣⡶⣄⠀⠀⠀⠀⣠⢶⣟⠛⢶⣄⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠘⠳⠴⠿⢿⣨⡇⠀⠀⠘⣥⡿⠟⢳⡆⠻⠃⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⢲⡄⠀⠁⠰⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠘⢦⣄⠀⠀⠀⠀⠙⠛⠁⠀⢀⣠⡴⠎⠁⠀⢀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠛⠓⠖⠲⠶⠖⠚⠛⠉⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠘⠷⠶⣶⠶⠊⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡆⠀⢰⣿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣷⢸⣿⣿⣷⢠⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⠿⢻⣿⠋⠘⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⡿⣿⣿⣿⡇⠀⠘⠛⠻⣿⣿⠏⠹⣿⠟⠁⠘⠿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⠛⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢦⠀⠀⠀⠀⡄⠀⢠⠀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣠⣯⣷⠀⢀⣌⠃⠀⠈⢧⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠻⣷⡆⠜⡄⠀⠀⢀⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠄⠰⠆⠐⢫⣧⡀⠈⡆⠀⠈⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡘⠄⠈⢿⣷⡀⠚⣆⠀⡃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠐⠀⣍⢷⡀⢹⢠⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢢⠀⡀⢹⠀⠀⠉⠊⠙⣾⡿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠲⣧⡀⠀⢈⣠⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠚⠛⠦⠌⠉⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⡄⢀⣞⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠈⢸⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠷⠀⠰⡇⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⡄⠀⠀⢙⡾⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⡆⡹⣄⣧⠀⢰⠂⡱⠶⠄⠀⠀⠀⡀⠖⠂⡈⢁⠀⠈⠈⠀⡂⠠⡀⠀⠐
⠀⠀⠀⠀⠤⡁⠘⣇⠺⡏⠷⠴⢩⠎⢴⡄⠀⠀⡠⠉⠀⠀⠀⠀⡄⠀⠠⠘⠀⠀⠀⠀⠀⠂⠀⠒
⠀⠀⠀⢀⠻⣏⣣⣈⡅⠉⠃⠸⠃⠀⠎⠀⠁⢀⠁⠀⠀⢀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠁⠲⠆⢀⠤⠘
⠀⠀⠀⠀⠀⠀⣨⠚⠥⡀⠀⠀⠆⡜⠀⠀⢀⠀⠀⢒⠂⠀⠀⠈⠉⠑⠒⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠁⠈⣏⠀⠀⠀⠀⠠⠀⡐⠃⠁⢒⢠⠈⢀⠉⠠⠄⠀⢀⢀⠠⠀⠅⠀⠀⠀⣨⢃⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠞⠣⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠐⠄⠀⠈⠄⠀⠀⠂⣀⣤⡴⠲⠉⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠐⢤⣔⣲⡍⠁⠀⠁⠉⠁⠁⠀⠂⠂⠂⠉⠉⠀⠈⠀⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠐⠍⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⢀⠂⠀⠀⠠⠈⠒⠠⢀⡀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⡀⠄⠐⠀⠁⠀⠀⠁⠐⠀⠂⠀⢀⠉⠉⢀⠁⠁⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠆⠀⠁⠐⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠠⠄⣀⠀⠀⠐⠌⠀⠀⠀⠐⠠⡀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡶⢂⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢻⣶⣶⣦⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢻⣿⠇⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⠠⡀⠀⠀⠀⠀⠀⠻⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠏⠀⠀⠀⠀⠀⢀⡔
⠀⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠞
⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠘⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰
⠀⠁⠠⠂⠀⠀⠀⠀⣷⡀⠀⢠⠂⠀⠀⠀⠀⠀⡴⠞⠛⠓⢷⡄⠀⠀⠀⠀⠀⠀⢠⡦⠴⣦⣀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠆⠃⠀⠀⠀⠲⡄
⠀⠠⡀⠀⠀⠀⠀⠀⣼⣿⣤⠃⠀⠀⠀⠀⠀⡼⠁⠀⡀⢠⡄⠁⠀⠀⠀⠀⠀⠰⠋⢠⣀⠀⠙⣦⠀⠀⠀⠀⠀⠁⣰⣿⡇⠐
⠀⠀⠐⢄⠠⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠀⠠⡇⠀⢰⣷⣿⡿⠀⠀⠀⠀⠀⠀⠀⣦⣞⡿⠄⠀⠸⠾⠀⠀⠀⠀⢰⣿⣿⣿⣦⡀⠀⡠⠃
⠀⠀⠀⠈⠳⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠨⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⡠⠻⣿⡟⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⣠⠔⠁⠀⠈⠇⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠔
⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣑⠒⠤⣀⡀⡀⠀⠀⠀⣀⡀⠴⠎⢀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠘⠀⠀⠈⠀⠁⠈⠃⠀⠀⢾⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡆⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⣠⣿⣿⣿⢀⠀⠀⠀⠀⠀⡶⣆⠀⠀⠀⠀⠁⢸⣷⡀⣠⣴⣿⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠘⠄⠀⠀⠀⣲⣿⢇⡁⠢⡀⢀⠔⡸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢀⣴⣿⠁⢹⣷⡄⠀⠀⠀⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣀⡀⠀⢀⣚⣛⣛⣀⣘⣿⡿⠆⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣈⢀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⣀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⡞⠀⡇⠀⠀⠀⠀⡶⡄⠀⠀⣼
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠉⠉⢦⠀⠀⠀⢈⡟⠉⠉⠉⠀⠀⠀⠀⣸⠓⠒⢻⠁⠀⠀⣸⠀⢹⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⣤⣄⣀⡀⡇⠀⠀⠀⠀⡜⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⠀⠀⠀⡇⠀⠈⢇⡼⠁
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢀⡽⠳⢤⣀⡤⠞⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠈⠓⠤⠤⡤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢓⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⣾⣋⠽⠛⠉⠉⠉⠉⠒⢤⣀⣀⣀⣀⣀⡀
⠀⢀⣤⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣶⠄⣠⠋
⠀⢸⡏⣇⠀⠀⣼⠹⡀⠀⣷⠀⠳⡀⠀⠀⡤⠀⠀⢠⣏⣠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⢄⣹⡿⠁
⠀⢾⠋⠙⡍⠁⢹⠀⢳⡀⢸⠀⠀⠙⢦⡾⠁⠀⢀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣴⢀⣴⣿⣿⡀⠀⠀⠀⠀⠀⢹⠃
⠀⢸⠀⠀⢱⡀⢸⠀⠀⠱⣼⠀⠀⢠⠏⠀⠀⠀⠜⠀⠀⠀⠃⠀⡠⠒⢳⠋⠓⠦⣄⡀⠀⢀⣾⠀⠀⠀⢠⡟⣿⠛⢯⣩⠳⣄⠀⠀⠀⠀⡿
⠀⢾⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢀⠇⠀⠀⠀⠀⠉⠒⡾⣸⣀⡠⠲⣄⠽⣴⣶⠀⣯⣷⠏⠀⠀⢀⣼⡇
⠀⠀⠀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡉⠀⠀⠀⠀⠀⠈⠀⠈⠉⠑⠲⠬⡗⡿⣿⣀⣰⣃⣀⣠⡴⠟⢁⣇⣀⣀⠤⠴
⢀⡴⠋⠀⠀⠓⢠⠞⠉⠙⡆⠀⣀⡄⡴⣄⠀⢠⠀⠀⠀⣰⡇⠘⡄⢀⡿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⠐⣳⣿⠿⠚⠛⠒⠋⠉⠉
⡞⠀⠀⠀⠀⠀⡇⠀⠀⠀⡷⠘⣄⢸⣁⡜⠀⡞⠀⠀⢀⡧⣧⠄⠘⣼⠃⣸⠀⢸⠛⠲⢤⣀⢀⡀⠀⠻⣀⣹⣷⣏⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀
⢧⡀⠀⠀⠀⠀⢧⡀⠀⣰⢃⡠⠊⢸⠁⠀⠰⡃⢀⠀⣸⠀⢳⠀⢀⠏⠀⠉⠀⡞⠀⠀⠀⠀⠉⠉⣳⠀⣞⣀⣯⣈⡆⠀⠀⠀⠀⠀⢀⣀⣀⣸⣠⠤⠖
⠀⢉⠛⠒⠓⠀⠀⠉⠉⠁⠉⠀⠀⠘⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠈⠀⠀⠀⠘⢇⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠁⠀⢸
⠀⡎⢰⣿⢰⣧⣴⢆⠀⢰⡄⢠⣦⣤⣄⣄⣤⣤⡤⠀⠀⠲⣴⣴⠰⣤⠸⡄⠀⠀⠈⠉⠓⠦⣀⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣽⣦⡀⠹⠦⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⡿⠁⠀⠀⠀⡴⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⡖⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠉⣰⠏⠀⠀⠀⢠⠎⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣦⡸⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⠁⣼⠇⠀⠀⠀⣠⠃⠀⠀⠀⠀⣸⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠐⠀⠙⣷⡜⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣸⡟⠀⠀⠀⣰⠃⠀⠀⠀⠀⢠⡏⠀⠀⠀⠇⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠃⣇⠀⠀⠀⠸⡄⠀⠀⠈⢣⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣹⠀⠀⠀⡎⠀⡄⠀⠘⡀⡘⠀⢸⡀⠀⠀⠀⠘⣄⠀⠀⠈⢆⠀⠀⠀⢿⢄⠀⠀⠀⠀⠀⠀⢰⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠀⢸⠀⡸⢃⠀⠀⢇⢃⠠⠤⠷⣄⠀⠀⠀⠈⠀⢄⡐⠲⠷⣤⣀⠈⢌⠣⣀⠀⠀⠀⠀⢸⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⠀⠀⣛⠔⠁⠈⠓⠤⠜⠲⠀⠀⠀⠀⠙⠒⠦⠤⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠈⡇⢠⠇⣀⣠⢴⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣤⡤⣀⡀⢸⠀⠠⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣿⠀⠀⠀⢹⣸⠋⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⡀⠆⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢰⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⡄⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⡎⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠘⣯⠉⠛⠓⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠤⠶⠒⠛⢉⡟⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⣇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠛⠒⠚⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣸⣇⣀⣠⠴⠊⠁⠀⠀⠀⠀⢠⣯⣾
⠀⠀⠀⠀⠀⠀⢹⡄⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⢾⡆⠙⠶⣄⡀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣀⣠⣴⠾⠛⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡄⢠⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠇⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡄⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢈⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠌
⠀⠀⠀⠀⠀⠰⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⡄⠰⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣆⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠄⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠺⠐⣓⡛⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠉⠀⠀⠀⠀⠀⠉⠀⠀⠈⢣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⢀⡀⡘⡄⠀⠘⣆⢢⡀⠐⡀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⢊⣱⣦⡈⠒⠦⠟⣩⣭⣐⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢰⡞⣽⣯⠳⠀⠀⠈⣿⣷⡙⣿⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢸⠃⠻⠏⠀⠀⠀⠀⠸⠽⡁⢻⠠⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⡈⠁⠂⠀⠀⠀⠀⠀⠈⢁⡟⠁⠀⠀⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢿⡙⣳⣶⣦⣤⡴⣶⠚⡟⠀⠀⠀⠀⢷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣀⠀⠀⠀⢻⠿⣀⠈⠁⢀⠟⢷⠁⠀⠀⢀⠀⠇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⢉⣶⣦⣄⠀⢀⣽⣿⣄⠀⢸⢀⣠⣴⡣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⠂⣩⣿⣻⣯⡵⢉⣹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢲⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⡇⣸⠟⠩⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢻⡄⣿⣆⠀⠀⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠱⣿⣿⣦⡁⠀⢙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠘⢿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠁⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢈⢳⡤⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠈⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠋⠀⠀⢠⡇
⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⢀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠈⠁⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣭⣟⣀⡀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠄⠀⠀⠀⠀⠀⡈⢦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣋⡛⢶⡀⠀⠀⠋⠀⠀⠈⠩⣽⡧
⠀⠀⠀⠀⠀⠀⢠⣶⡯⣷⡴⡀⠀⠀⠠⠀⡤⣶⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⠂⢮⡳⣽⢗⣿⢧⢇⠐⠀⠈⠀⢿⣿⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣪⣉⠉⠀⣼⣽⢾⡯⡿⣻⣸⠀⠀⠀⠀⠸⠃⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠁⠀⢾⣿⣷⣟⢿⣾⣹⠀⠀⠀⠀⠐⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢨⣴⣤⣬⣮⣽⠍⠀⠀⠀⡀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢾⡇
⠀⠀⠀⠠⠤⢴⣽⣤⣤⣥⣬⠃⠀⠀⠈⠁⠀⠀⠀⠁⠀⡷⠀⢰⣶⣶⣲⣶⣶⣖⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠈⢡⠏⡏⠈⠁⠘⠉⠉⠁⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⠀⡇⢨⣿⡏
⡀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢤⠀⠀⢠⠀⠇⠠⠐⠉⠹⠶⡆⠀⠀⠀⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀⠇⠸⢿⡇
⠂⠀⠀⢀⣀⠀⠀⠐⠀⠀⠀⢸⣿⠀⠀⠀⢀⠀⠀⠀⠐⠀⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇
⠀⠤⠀⠤⠤⠀⠄⠈⠀⠀⠀⠈⠻⠁⠀⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠇
⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⢶⣶⠀⠀⠀⣶⠀⠀⠀⣴⣶⣶⣶⠄⠀⠀⠀⠀⠈⣤⡄⣤⡈⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠓⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠉⠶⠄
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠁⠀⢲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡠⠀⠀⠆⣤⡤⠦
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⡁⠀⠀⢠⠄⠀⠀⠰⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡄⠠⠄⣄⡀
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠓⠒⠛⢻⣿⠿⠛⠀⠀⠀⠁⠀⡀⡀⢀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣾⠂⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠰⣀⠀⠶⠆⠀⡐⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠂
⡀⣠⡖⠀⠀⠿⠟⠀⢈⣾⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⢠⣤⠑⠒⠂⠀⠀⠴⣶⣾
⠷⠿⠧⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠀⠀⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢉⣉⣇⣠
⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠠⠐⡄⣄⢤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⡀⠀⠄⠠⠀⢰⡇⠺⠠⠦⠀⠂⠀⠀⠀⠀⠆⠀⠄⠠⠘⠉⠉⠙⢻⠓⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⣄⠀⣠⠀⣀⣤⣶⣿⠗⠈⠃⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠤⢄⣼⣀⣀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠘⣆⠈⠻⣿⣿⣿⣿⣧⠠⠀⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡞⠉⠉⠰⣄⡀⢁⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢹⣿⣿⣿⣿⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⡁⠀⡆⠀⡣⠀⠀⠀⡍⠈⠉⠋⠠⠠⡔⠃⠈
⠀⠀⠀⠀⢃⠀⠀⠦⢿⡀⣿⣿⣿⣿⡟⠁⣶⠀⠀⠀⠀⠀⠀⢶⠀⠀⢀⢷⡎⠀⠈⠉⠀⠀⠀⠠⠀⣀⠀⠀⠀⠄
⠀⣠⣶⡄⠀⠀⠤⡄⠀⠃⢻⣿⣿⣿⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡏⠀⠠⠀⠄⠀⠀⠀⠐⠀⠄⢀⡀⠀⠘
⢸⣿⣿⣿⣆⡀⢀⠀⠀⢰⣿⣿⣿⣿⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⢙⠻⡶⡆
⢸⣿⣿⣿⣿⢁⡜⢤⣦⠾⠿⢿⣿⣿⣿⣁⣀⣄⣤⣤⣀⣀⡀⣀⠀⠀⡀⡄⠀⢠⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⣸⡌⡔
⠀⢀⠄⠀⢱⠶⠻⠇⠀⠀⠀⠀⠈⠘⠿⠿⣿⣿⡿⠿⣿⣿⣿⣿⠃⠁⠘⣧⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡜
⠀⠘⠒⣤⣤⣠⣦⠴⠛⠠⠶⠳⠿⣿⣶⡤⠚⢃⠀⠀⠀⠀⠀⠛⠁⠀⠰⠇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠷⠁
⠀⠀⠘⠙⡿⢿⡟⠃⠀⠠⠴⠄⠒⠂⠀⠉⣁⠀⠀⠀⠲⠀⠐⣒⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⠓⣒
⠀⡤⠘⣂⠐⠋⠈⡈⠙⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠻⠈⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠁⠉⣐⡀
⠀⠀⠀⠻⠒⠲⠤⠀⢰⣶⣤⣀⣀⣀⣤⡝⠙⠛⠚⠛⠃⠄⠄⠀⠀⠀⠉⠛⠺⣿⣦⣄⣀⣴⠾⠯⠤⠔⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡛⠋⠁⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡟⠧⣤⣀⣀⡀⠀⠐⠂⠉⠁
⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢁⠀⠈⠉

⣷⣤⣤⠀⠠⣦⠀⠰⠀⠀⢠⡑⠀⢰⡀⠀⠁⠠⠄⠀⠀⢰⠀⡄⠀⠀⢠⡋⠙⠀⠀⠀⠀⠀⣘⠂⠀⠀⣋⠀⣸
⣿⣿⣿⠀⣼⣿⠀⢸⣗⠲⣿⣿⠀⠠⣃⣴⡠⢿⠀⠀⠀⢰⡶⠇⠀⠀⢈⡇⢠⠀⠀⣿⣿⠙⣿⠃⠀⢸⣿⠀⣿
⣿⣿⣿⡆⢹⣿⠀⠘⣿⡟⣿⡿⠀⠀⣿⣿⣧⣄⣶⠀⠀⣾⣿⡇⠀⠀⢸⡇⣿⠀⠀⢹⣿⢀⣿⡆⠀⢸⣿⠀⣿
⣿⣿⣿⠃⢸⣿⡀⠀⣿⡇⢻⣷⠀⠀⣿⣿⣿⡟⣿⠀⠀⣿⣿⡇⠀⠀⢸⡇⣿⠀⠀⠀⣿⢸⣿⠁⠀⢸⣿⠀⣿
⣿⣿⣿⠀⠸⣿⡇⠀⣿⢧⢸⣿⠀⠀⣿⡿⣿⣇⣿⠀⠀⢾⣿⠇⠀⠀⢸⡇⣿⠀⠀⠀⣿⢸⣿⠀⠀⢸⣿⠀⣿
⣿⣿⣿⠀⠀⣿⡇⠀⣿⣿⢸⣿⡄⠀⣿⣿⣿⡇⣿⠀⠀⣿⣿⡇⠀⠀⢸⡗⣿⠀⠀⠀⠁⢸⣿⠀⠀⢸⣿⡀⣿
⣿⣿⣿⡇⢀⢹⡇⠀⣿⣿⢸⣿⡇⠀⢿⡏⣿⠁⡏⠀⠀⢻⣿⡇⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⡇⢸⠘⣇⠀⢿⣿⢸⣿⡇⠀⢸⡇⣿⠀⡇⠀⠀⢸⣿⡇⠀⠀⠘⡇⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⡇⢸⡇⣿⠀⢠⣿⢸⣿⣿⠀⠘⣯⣿⢸⡇⠀⠀⢸⣿⡇⠀⠀⠀⡇⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⡇⢸⡇⢿⠀⢸⣿⢸⣿⣿⠀⠀⠟⣿⢸⠇⠀⡄⢸⣿⡇⠀⠀⠀⡇⣿⠀⠀⣄⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⠀⠘⠃⠈⠀⠸⠿⢸⡿⠛⠀⠀⠀⠈⠉⠀⠀⡄⠈⠿⡇⠀⠀⠀⣷⣿⠀⠀⣿⠀⢸⣿⠀⠀⢸⣿⡇⣿
⠛⠛⠛⠀⠀⠀⠀⠀⢀⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠈⠉⠀⠀⠘⠿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢘⠀⠀⢐⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄⠀⠰⠀⠀⠀⠠
⠀⠌⠛⠀⠀⠉⠁⠀⠀⠟⠿⠷⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢹⣀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⢻⡄⠀⠱⢀⣀⡀⠀⣀⣀⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⣀⠴⠚⣉⠻⠆⠀⠈⢉⠀⠀⠈⠉⠙⠳⠦⣄⡈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠊⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠑⢢⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⣴⠟⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠙⢯⡈⠍⠉⠉⠉⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⢸⠏⣰⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢰⡏⣰⠁⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠳⡀⢻⡄⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡿⢠⠃⡀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣷⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⡇⠘⠀⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⢣⠘⣇⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⡆⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⢦⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠈⡇⢹⡀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢸⠀⠀⠀⢀⣷⠀⠀⠃⠰⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢹⣄⠳⣄⠀⠈⢠⠀⠀⡀⠀⠀⠀⠀⠀⢘⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠘⠀⠀⠀⢠⠉⣀⠴⠄⣀⠀⠀⠘⣦⣀⠀⠀⢀⡤⠖⠉⠉⣭⡙⠣⣀⠈⠀⠀⠃⠀⠀⠀⠀⠀⣎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡏⠀⠀⢀⠆⠀⠀⠄⡜⠁⠀⠐⢄⠓⠀⠀⠐⠌⠑⠰⠏⢀⡶⠀⠀⠄⠹⡄⠈⣷⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣰⠁⠀⠀⠀⠊⡀⣸⠹⠀⠃⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡁⠀⠀⠀⠀⡇⠀⣸⠀⣄⢀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣰⡇⠀⡀⢠⠀⠀⢘⣷⡝⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠔⠀⢀⡟⢰⠁⠀⢃⡀⠀⢠⠀⠀⢹⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢠⠞⣿⠁⢠⡇⢸⠀⠀⠀⢸⡅⠀⠘⠢⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⡜⠀⠇⠀⠀⢸⡇⠀⢸⠀⡸⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢸⠀⡎⢀⡇⡆⢸⠃⠼⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡄⠀⠀⠸⡇⠀⣇⠘⠀⠀⠀⠌⠓
⠀⠀⠀⠀⠀⠀⢹⠰⠁⠈⣥⠃⢸⠀⠀⣇⡀⠀⠀⢸⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠇⣼⠏⠀⠀⠀⡜⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⢿⡘⣦⡀⢸⢧⡀⠀⠀⠑⢦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡴⠃⠀⠀⠀⠄⠀⠀⠀⠾⠃⠀⠀⠀⣸⡧⣄⡴
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠁⢷⠜⢧⡄⠳⣄⠀⠀⠈⠣⡀⠀⠈⠉⠉⡀⢀⠀⠀⠉⠉⠀⡠⠞⠀⣠⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠘⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡟⠀⠈⠓⢦⣀⠀⢱⠀⠀⠀⠠⠁⠄⠀⠀⢄⠀⠊⠀⠀⠰⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠀⡇⠀⠀⠀⠀⠈⠑⢢⠄⠀⠊⢀⠆⢲⡀⠀⣧⣀⣀⡀⣀⣤⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⢰⡏⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠳⠀⠀⠉⡉⣨⠇⠀⠀⢨⠀⠀⠀⠘⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡰⠁
⠀⠀⠳⠈⠑⠂⠀⢢⡀⢀⣸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠀⠀⠀⢸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡤⠂
⢠⡑⠚⠓⠀⠀⠀⢀⠝⠋⠁⢇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠜⣷⢀⡀⠀⢀⠀⠀⠀⠀⢀⣀⠄
⠀⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢋⠂⢀⠀⠀⠀⠈⡉⣽⠯⠃
⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠣⣄⠉⠒⠚⢉⣭⡞
⠀⠀⠉⠉⠉⠶⠀⠀⠀⠀⢀⠆⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠚⠉⢈⠄
⠀⠀⠀⠀⢀⡆⠀⢀⡠⠴⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠈⠂⠐⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠁⠀⢀⣦⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⢻⡁⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⢶⠶⡖⠛⠉⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⡤⠶⠉⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠤⠄⠤⠐⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟

⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⢢⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡞⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢘⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠤⠄⠶⠶⠶⠶⠠⠄⠀⠀⠀⠀⠀⠄⠠⣶⣶⣦⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢻⣯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣀⣤⡾⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣹⣿⣇⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣿⣿⣿⣶⣤⣴⣶⣤⣼⣿⣿⣿⣦⣄⠀⠀⠀⠙⠉⢛⡟⠓⠲⠖⠒⠲⠚⣟⠋⠉⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡤⠺⡇⠀⠀⠀⠀⠀⠀⠃⠑⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⣴⣶⣄
⠀⠀⠸⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⢀⠆
⠀⠀⠀⠀⢀⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⠀⢠⠃
⠀⠀⠀⠀⢸⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢸⢀⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠴⠚⠉⠉⠉⠉⠉⠒⠢⡄
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠀⠀⠀⠀⠀⣴⣦⡀⠀⠘
⠀⠀⠀⠀⡇⠀⠀⢀⡤⠊⠁⠙⢍⠉⠁⣀⠔⠋⡧⠔⢻⠀⠀⠀⠀⠀⠀⠀⢀⢇⣿⣧⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠇⠀⠀⠉⠀⠀⠀⠀⠈⠓⠚⠲⠔⠉⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠑⠒⠲⠄⡇⠀⠀⠀⠀⠀⠀⠀⠸⡈⠻⠟⠐⠂⠀⠀⠙⠛⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⠀⠒⠲⣦⣤⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠄⠀⠀⠀⠀⠀⠀⠠⣎
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠹⠿⢡⠃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⡁
⠀⠀⠀⠈⣉⣭⣿⣿⣶⣶⣶⣶⣆⠤⠤⠤⠤⠤⢀⣨⢿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠢⡀⠀⠀⠀⣀⠴⠒
⠀⢀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠈⢻⣷⣤⣀⣀⣠⣴⠿⢿⠃⢰⣾⡗⠒⡖⠁
⢰⣿⣿⣿⣿⡿⠋⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣠⣶⣿⣿⣿⣿⣿⡟⠁⠀⡜⠑⠊⠻⠗⠒⠙⠢⠤⠤⠤
⠹⣿⣿⣿⣿⣿⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢠⠃
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢸⠀⠀⠀⢀⡠⠤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠆⠸
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣄⠀⠀⠀⣀⠤⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠘⠉⠁⠒⢄⠀⠀⡀⠀⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠳⠶⠚⠁⠀⣡⣀⠔⠁⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠒⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢠⠟⢉⡍⠀⠀⣇⠀⠉⠉⠓⠛⠒⠒⠒⠒⠒⡟⠀⠀⠰⡟⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢈⡶⢹⠀⠀⠀⡇⠈⠐⠒⠀⠀⠤⠄⠀⠒⡾⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠉⢻⠀⠀⠀⠸⡆⠀⠀⠤⠄⠀⢀⣠⠞⠀⠀⠀⠀⠀⠰⢂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠳⠶⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠃⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⡀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠒
⠐⠂⢉⡆⠰⠒⠈⠀⠀⠀⠀⠁⠒⠂⠤⢄⣉⡑⠢⠤⣀⠈⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⢀⣀⠤⠐⣀⠔⠉⢦⡀⠀⠀⢀⠀⠠⣔
⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣌⡉⠒⠺⠛⣿⠶⠶⠶⣶⠶⠶⠟⠉⠀⢒⣂⣈⣭⠤⠔⠒⠉⠁⠀⠀⠀⠉⠓⠀⢧⠘⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡰⠁⠘⠒⠛⠁⡶⠂⠂⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠐⠉⠑⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⡴⠀⢈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣠⣤⣴⠞⢉⠄⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠢⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣀⠇
⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢰⡏⡎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠀⠀⢸⣷⣷⣾⡇⠀⠀⣰⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡆⠀⢸⣿⣿⣿⡇⠀⢀⣿⣉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣦⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⣿⣿⢹⣿⡇⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⢸⣿⠨⣿⡇⠀⣿⣿⢸⣿⡇⠀⣿⣿⢻⣿⡇⠀⠀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⠀⢸⣿⣶⣿⣿⢸⣿⣿⣿⣿⡇⢠⣿⣿⣸⣿⡇⠀⢀⡇⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⡄⢸⣿⣿⢿⣿⢸⣿⡟⠛⣻⣿⣸⣿⠿⢿⣿⡁⠀⢸⣿⢈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣇⢸⣷⣤⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣧⣾⣿⡇⠀⠛⢻⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⢛⢿⡟⣻⣿⡇⠀⠀⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢿⣿⢹⣿⠘⣷⣆⣾⣇⣿⣧⣿⠸⢻⡟⠿⣿⡇⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡶⣌⢿⠸⣟⣶⢻⣿⣻⣇⢻⡇⣿⢠⣿⡇⣿⣿⣷⠀⣾⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣌⠀⡧⣽⡬⣿⡘⣿⡘⢃⣿⣸⣍⣧⣙⣿⣟⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⡿⢿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣈⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⣿⣛⣉⣿⠟⠀⠀⢙⢶⣶⣶⡾⠋⠻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣃⣀⣈⣻⣿⣿⣿⣷⣾⣿⣿⣿⣻⣷⣦⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⣿⡾⣻⠻⡙⣿⢩⡇⣶⣿⢰⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣇⣿⣧⣷⢿⣾⡇⣻⣿⡀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⢿⡇⣿⡞⣿⡏⠿⣿⣿⣎⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣷⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠌⣿⣿⡫⢻⣿⣿⡏⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⢨⣿⣷⣇⣺⣼⣿⣇⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠬⠿⠷⡵⠾⠻⠿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⣀
⣀⣀⣤⣤⣤⠄⣠⠶⢂⡴⠁⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣍⠉⢲⣢
⣿⣿⣿⣿⣿⠟⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠈
⣿⣿⣿⡿⠋⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⢻⣿⡿⠇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠸⣿⡅⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠐⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠀⠀⡜⢹
⠀⣰⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠁⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄
⢠⡟⠀⠀⠀⠀⠀⠀⡘⠀⣄⠀⡇⠠⠀⠀⠀⢘⠉⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠡⡀⠀⠀⢰⡄⠀⢷⡀⠀⠀⢠
⡜⠀⠀⠀⠀⠀⠀⠀⣇⠞⠙⠢⠇⠀⠀⠁⠂⠀⠆⠀⠀⠀⠀⠀⠀⠑⠠⠠⠠⠀⠼⠄⠀⠀⠀⠀⠀⠠⠀⣁⠂⠀⡌⠢⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⠛⠉⠛⠛⠛⠛⠛⠯⠬⠭⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠩⠽⠿⠿⠿⠿⣿⣍⣙⣲⣤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠙⠓⠲⢤⡤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⠶⠛⠉⡼⢁⠄⠁
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⢀⡎⠀⣠⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡍⣇⢀⠀⠀⡄⠀⣼⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠞⠀⢤⡿⢻⡀⠀⠉⠙⠛⠿⠿⠛⠛⠉⠁⠀⢸⡉⠻⣌⠄⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡠⠋⣠⡾⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠺⣄⠞
⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⣶⣿⠏⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⡩⠄⠀⠀⢀⡴⢏⡨⠁⠀⠀⠀⠊⢄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣃⠀⠀⠀⠀⠀⠀⢠⡮⣕⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠋⠀⠀⠀⠀⠀⢰⠗⠥⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣀⠀⠀⠀⠀⠀⢀⠢⢀⠄⠀⠀⠀⠀⠀⠐⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⢃⠀⠂⠀⠀⠈⢀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠀⠄⠀⡀⠀⠀⣔⢂⣨⠜⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠦⡀⠀⠈⠲⠐⠤⠊⠁⠀⡀⢀⠀⠀⠀⠀⠀⠴⢆⣄⠉⠸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠔⠁⠀⢈⠦⡀⠚⠋⢄⠀⠀⠀⠄⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠄⠄⠀⡠⠀⠀⡀⠐⠁⠀⢀⠰⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠁
⠀⠀⠀⢠⠂⠔⡡⠀⡀⠄⡂⠠
⠀⠀⢠⣏⣀⡨⠔
⠀⠀⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣍⣀⡀⠀⠐⠒⢭⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢟⣫⣭⣭⣭⣭⣇⣀⣀⣀⣐⣾⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⡺⠟⠟⠛⠉⠁⠈⠉⠀⠀⠀⠈⠉⠹⣟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠏⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⢿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⢸⢱⠀⠀⠔⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠘⠙⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⢠⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣠⠔⣣⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⢦⠀⣀⡄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡄⠀⠀⠀⠀⠀⢤⠐⠚⠉⠀⡀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⢂⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠘⢶⠒⠒⣉⢴⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠨⠖⠋⠀⠀⠀⠀⣠⠟⠀⢀⡠⡄⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣄⣀⠀⠀⠀⠀⣀⣠⠴⡿⠥⠒⠒⢉⣀⠟
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢍⠀⣠⠃⡆⠒⠉⠉
⠀⠀⠀⠀⢠⡄⠀⠒⡆⠀⠀⠀⠀⠀⠀⣀⣀⡠⡆⠀⠀⢉⣀⠴⠆⠀⠉⠀⠀⢳
⠀⠀⠀⠀⠀⠤⠀⠁⢁⣀⡠⠜⠒⠒⠒⠒⠒⠚⠳⠒⠊⠉⢀⡞⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡬⢌⣉⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠈⠉⠒⢤⣄⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠼⣄⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣯⠙⡆⠴⠖⠂⠤⠤⠥
⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠊⡠⠚⠉⠹⡦⠴⠖⠒⠒⠒⠤⢀⠉⠉⠛⠒⠲⠶⠶⣂⣀⠀⠀⠀⠀⡏⣸⠁
⠠⡀⡀⠀⠠⢄⡀⢀⠎⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣇⠀⠀⠀⢀⡏⢀⣀⠤⢚⡴⠃
⠀⠑⢌⠲⡀⠀⠈⠙⠦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣼⠢⠤⠒⠊⠁
⠀⠀⠀⠑⢌⡒⢄⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣭⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠐⡆⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢠⢀⡀⠘⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠊⢀⢒⡥⠒⠀⡆⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠌⠾⠠⠈⠀⠉⢶
⠀⠀⠀⠀⠀⠄⠢⠄⣐⣀⣘⡣⠄⠐⢬⣁⠀⢃⢻⡈⠠⣀⡤⠛⡀⠀⠀⢸⢳⠀⢰⣻⠀⠀⢀⢅⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠈⠹⣄⠀⠉⠀⡀⠾⣁⢒⡌⢀⡀⠐⠒⠒⠻⠧⡀⠀⠈⢻⠒⡟⠁⠀⠀⠞⠘⠃⠀⡟⠠⡄
⠀⠀⠀⠀⠀⢰⢣⡆⠀⣬⠄⣤⠀⢠⠎⠀⢸⡴⠉⠉⢣⠐⠑⠲⣠⡆⠀⠀⠰⠀⠃⠀⠀⢰⠒⠐⢦⣀⠀⠺⠄⣤
⠀⠀⠀⠀⠀⠘⡎⠓⢈⡤⠤⠤⠀⣏⠀⠀⠈⣁⡤⠤⢄⠔⠋⠉⠉⠀⡀⠀⠀⠀⠀⠀⢀⠀⢠⠤⠀⠘⠧⠀⠿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠀⠀⣀⣀⣠⡾⢁⡀⢾⠁⠀⠀⠀⠄⣀⡄⠨⠀⠀⠈⠉⠨⠁⠈⠃⠀⣠⡟⠀⣶⠈⠁
⠀⠀⠀⠀⠀⠉⣀⠀⡆⠀⠈⢦⠎⠁⠀⢸⠁⢸⠀⠈⢀⠀⠀⠀⢤⠈⠻⡌⣭⣤⣭⠁⠁⢀⡷⠰⠎⠉⠀⢴⡂⠛
⠀⠀⠀⠀⠐⠉⠉⣁⣇⢰⣃⣀⣀⡀⠀⠸⠀⢀⠔⢉⠀⠀⠀⠀⢈⡇⠀⢇⠈⠙⡅⠀⠀⡼⠁⢠⡔⠀⣯⠐⠋
⠀⠀⠀⠀⠀⠀⡞⠁⢠⠞⠉⡀⠀⠀⠀⠀⠀⡏⠀⢯⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡇⡠⠞⣁⠀⠊⠁
⠀⠀⠀⠀⠀⣈⠤⠀⡟⠀⣼⠁⠒⠉⢉⠃⠀⣇⠀⠘⣆⠈⠲⠤⠄⠀⠀⠀⢀⡼⠁⢠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠈⠒⢠⠊⠉⠉⠙⠚⠆⠀⠘⡆⠀⢰⡀⠀⠀⡂⠉⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠙⠄⠓⠶⠔⠂⠀⠘⡀⠀⢠⠀⢠⡄⠄⠄⠧⠀⠼⠷⠊⠉⠀⠀⠀⠴⠀⠄⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠃⠀⡀⠀⢼⡅⠀⠀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣀⡀⠀⢈⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠘⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠒⠆⠀⠷⠶⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⢲⡾⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⣱⡿⠁⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢰⡟⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⢹⡜⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢥⡿⠃⠀⠀⠈⠀⠀⠀⢀⡔⠀⢠⠄⠀⠀⠀⠀⣧⠀⡀⠀⡆⠀⠀⢻⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⠇⠀⠀⢀⡃⢀⢂⠄⡸⢁⢜⠟⠀⠀⠀⡀⣰⠋⡄⣇⠀⢱⠀⠀⠀⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠋⠀⠀⠀⢸⠃⢜⣥⠶⠿⢴⠁⢀⣤⠴⣺⠕⣡⠴⢾⣌⣧⢸⠀⠀⢀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡜⠁⠁⠃⠀⠀⢸⣲⡟⣱⢟⡃⡄⠀⠀⠀⠀⠁⠘⣱⢟⣀⡈⢿⣿⠀⠀⠄⠀⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠸⠀⠡⡠⠀⣾⡁⢿⣹⣇⡇⠀⠀⠀⠀⠀⠀⢿⣿⣇⡇⢸⣿⠀⡘⠀⠀⠸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠑⠂⠀⠀⠀⠑⣄⢙⠃⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠞⠹⠐⠁⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣃⠀⠀⢠⡀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠀⠀⢀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⡶⠉⡀⢀⡆⢱⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡰⠁⠀⠀⢸⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠐⠀⢰⠀⢡⠀⣀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠃⠀⡈⠀⠀⠀⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠐⢄⢀⠠⠃⠀⠀⣆⠀⠁⠢⣀⠀⠹⡓⢶⠶⢶⠶⢾⣇⠈⠁⠞⠀⠈⠀⠀⠀⠀⢠⡟⢘⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠙⠐⢄⡀⠀⠁⠀⡇⠀⣱⡤⣄⡉⠀⢹⣷⡇⠀⠀⠀⠀⣀⠔⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⢲⣄⠠⡇⠊⢈⡄⣄⠙⢦⣀⣿⣇⣠⡤⠒⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⣿⣟⣀⣠⠞⡁⢸⢡⣀⠀⢸⣿⣿⠃⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⣿⣿⡆⢸⣿⣿⣿⣿⡆⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⢀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⣿⣿⣿⣿⣇⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠂⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠉⠀⠀⠘⠿⠃⠀⠀⠀⣀⣀⠀⠰⠖⠀⠀⠻⣿⠇⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠙⠟⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⢠⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠐⢿⣿⠗⠀⠀⠘⣿⠟⠀⠀⢀⣀⣀⠀⠈⠁⠀⢠⣤⣦⠀⣰⠷⡶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠡⠄⠰⠆⠀⠀⠀⠀⢀⠈⠻⠋⠀⣠⣀⣄⠀⠙⠁⣰⠏⠀⠀⠀⠰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠙⠷⣤⡀⠀⠀⣴⣴⡦⠀⠈⢧⠀⠀⠀⠙⠿⠋⠀⣠⣾⠋⠰⣶⠀⠠⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣤⣤⡄⠈⠙⢷⣄⠈⠛⠁⠀⠀⠘⣧⠀⠀⠀⣠⣴⡿⠛⢁⣀⡀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠀⢠⣤⠄⠹⣧⡀⢰⣶⣶⠀⢻⣇⣴⣿⠟⠉⢀⡀⡈⠛⠁⢿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣷⡀⠉⠁⠀⣸⣿⣿⠃⠀⠀⠙⢿⠏⠀⠀⠀⠙⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣻⣷⣄⠀⠀⣿⣿⠇⠀⢙⣡⣤⡴⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣼⣿⣿⡀⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⣿⣿⣷⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣧⡈⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⠟⠛⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠛⡽⠋⠀⢀⡔⠁⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⣎⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠋⣼⠁⠀⠀⡞⠁⠀⠀⣸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠙⣷⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣸⠃⠀⠀⡜⠀⠀⠀⢠⢿⠀⠀⢰⠀⠀⠸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢿⠀⠀⠸⢀⠀⠀⠀⡆⢸⡀⠀⠀⠣⡀⠀⠑⠀⠀⠸⣆⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢸⠀⠀⠇⡜⢄⠀⣥⡦⠐⠓⢄⡀⠀⠀⠠⠒⠛⠦⠄⠡⠑⠄⡀⠀⠀⠀⠹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠘⠀⢸⠊⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⡇⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠀⡐⠀⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⢻⠀⠀⠱⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣄⠁⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠸⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⢆⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⠆⠀⠀⠀⢠⠁⠀⢸⣿⣿⡀
⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠈⢆⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠆⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⢠⠀⠀⠀⠈⠢⡀⠀⠀⢿⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡾⠀⠀⠀⢀⠎⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⡇⢸⡀⠀⠀⠀⠀⠈⠑⠒⠾⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠠⠖⠁⠀⠀⠀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣜⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠉⠉⠉⠉⠉⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣤⣞⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⢿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠟⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢰⣡⢦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠳⣸⡇⢸⢻⠀⠐⠫⠍⢉⣀⣀⠀⣰⠂⠀⢀⠤⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠘⡄⠁⢇⢁⠌⠀⠀⠀⠀⠀⣷⡮⠊⠁⠀⠀⠈⠁⢰⣶⡤⠜⠁⠀⠀⠀⢆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⢡⡠⠈⠁⡀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠉⠉⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣨⡵⠓⠁⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⡹⠒⡁⢀⣀⠄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⣀⣀⣀⢌⡄⠒⢀⢀⢠⡀⠄⡿⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⢰⣶⣤⣤⣤⣄⡀⠀⠀⠀⡘
⠀⠀⢺⣿⣿⣿⡠⠀⡇⢁⠛⢈⡀⡇⣤⡱⠸⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⢿⣿⣿⡟⠀⠀⠀⢀⠃
⠀⠀⠘⣿⣿⣿⠃⢀⣍⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⢿⣿⠃⠀⠀⠀⠸
⠀⠀⠀⢹⡿⠉⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣸⠏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢁⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠄⠉⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣶⠶⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⠲⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣰⠏⠀⠀⣶⣶⣌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣸⣿⣿⣆⠙⣆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡄⠀⠀⠀⠀⡿⢠⡇⠀⡾⣶⣿⣿⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣿⣷⣿⡙⡆⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠐⠀⠀⡀⠀⠀⠀⠀⡇⡸⠀⠀⠁⠻⡏⠉⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠁⠀⢀⠀⠀⠀⠀⢀⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠣
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠹⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠢⠄⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠄⠉⠒
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⢀⠃⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠆⠀⠀⠀⠠
⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠉⠍⡐⠒⠀⠀⠀⠤⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁