Real number Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡇⠀⠀⠀⠀⠀⢠⡦⢞⣥⣠⣽⣾⢦⠀⠀⠀⠀⠀⠀⢀⣾⠛⣠⠟⣷⠙⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡟
⠀⠀⠀⠀⠀⠀⢀⡘⢂⡼⠃⠀⠀⠀⠀⠀⠿⠿⠁⠀⠀⠀⣽⡇⡇⠀⠀⠀⠀⠀⠬⡿⠿⠁⠀⠀⢰⣯⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣼⢃
⠀⠀⠀⠀⠀⠀⠀⢺⣍⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢛⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡏⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣥⡾⢛⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡧⢔⡁⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⣷⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⡯⠀⢽⣾
⠀⠀⠀⠀⠀⠀⠀⢈⡷⢋⡆⠀⠀⠀⠀⠀⢀⡀⢧⣈⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣭⡀⠀⠀⠀⠀⠀⢀⢀⡽⢶⣟⠀⢾⣼⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⡇⠀⠀⠀⠀⠀⣈⡙⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡠⣶⠄⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⣠⣿⡿⣛⣆⣹⣯⣽⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⣫⣽⡆⠀⠀⠀⠀⠀⡙⣫⣼⡟⣿⣡⣭⣦⡇⠀⠀⠀⠀⠀⠀⠸⠋⣠⣏⣀⠛⠰⠋⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⣻⣿⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠉⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙


⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⢤⣤⠀⡦⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣦⣤⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣶⢠⣤⣤⠀⠆⠀⠀⠀⠀⠀⠀⠀⠲⡴⢆⢀⡀
⠀⠀⠀⠀⠀⠀⣠⠚⢹⠀⠹⠷⠔⠃⠀⠀⠀⠀⠀⠀⣠⠞⢹⠈⠹⠧⠴⣁⡄⠀⠀⠀⠀⠀⠈⠀⠞⠻⠁⠸⣅⡀⠀⠀⠀⠀⢀⡌⣯⣭⠆⠿⠉⢩⣧⡀
⠀⠀⠀⠀⠀⠀⢹⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡌⠀⠀⠀⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⡥⠀⠀⠀⠀⠀⠘⡧⠙⡃⠀⠀⠀⠟⣉⡃
⠀⠀⠀⠀⠀⠀⠼⣟⠛⣠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣻⠆⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢞⡿⠀⠀⠀⠀⠀⠀⠀⣧⣶⣥⣦⠀⣰⢁⣠⠁
⠀⠀⠀⠀⠀⠀⢠⢞⠛⢿⡮⣻⣽⣦⡀⠀⠀⠀⠀⠸⣠⡜⠻⣿⠌⣛⣼⣧⡀⠀⠀⠀⠀⠀⠀⢀⡜⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢅⡈⢿⡏⠀⠧
⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⢰⣶⢾⡇⠀⠀⠀⠀⠀⣟⣥⡷⠀⠀⠀⣭⢿⡟⠀⠀⠀⠀⠀⠀⣸⣉⣥⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣷⠋⠀⠀⠀⣠⠶⡦
⠀⠀⠀⠀⠀⠸⡟⢻⡦⢀⡀⢼⣿⣾⠁⠀⠀⠀⠀⠀⠿⠮⠴⢄⡀⢤⣿⣾⡷⠀⠀⠀⠀⠀⢰⡿⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣿⣤⣄⡠⣼⣯⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠾⢥⡎⠉⠋⠁⠋⠀⠀⠀⠀⠀⠀⠀⠘⢥⠊⠐⠋⠡⠊⠀⠀⠀⠀⠀⢠⡟⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣵⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡐⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣁⡛⠯⢴⣮⠷⠆⠀⠀⠀⠀⠀⢠⣾⣧⣽⠸⠡⠓⣚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡹⠃⠀⠀⠀⢱⣞⢂⠀⠀⠀⠀⠀⢚⠛⠻⠀⠀⠀⠸⣐⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢀⡤⢶⠑⢿⣃⠀⠀⠀⠀⠀⢫⣵⠆⠀⠀⠀⢠⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢯⡄⠹⣠⣞⣥⠀⠀⠀⠀⠀⢙⡧⠤⠀⠀⠀⠈⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⠅⠀⠀⠀⠀⠀⣾⡶⣞⠀⠀⠀⠰⣠⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡙⢻⢀⡀⢴⣿⡙⡿⠀⠀⠀⠀⠀⠻⣐⡨⣦⠀⠀⠴⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣟⣛⣠⠹⠇⠀⠀⠀⠀⠀⠀⠀⠛⢷⣯⣠⣴⠾⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠀⢀⡄⢀⡀⠈⠉⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡇⠰⡇⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⣀⣀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⡔⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠎⠀⢀⡄⢠⣄⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡌⠀⠀⢸⣇⢸⣿⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠰⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡃⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠀⡠⠈⠀⠀⠈⠢⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡜⠀⠠⠆⣠⠆⠀⠐⡀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⠀⠀⣴⢫⡤⠀⠀⢡⠀⠀⠀⠀⠰⠁⢠⡄⢠⡄⢄⠀⢠⡄⢠⡄⠱
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢸⡇⢸⡆⠈⡆⢸⡇⣸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⢀⠂⠁⠀⠑⢄⠀⠀⠇⡄⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠜⠀⠀⠘
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄⠃⢀⠃⠀⣀⠀⠀⠀⡆⠀⡄⠃⠀⠀⠀⠀⠑⠠⠀⣀⣀⣀⣀⠤⠒⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠠⠂⢠⠀⠈⠻⣧⡅⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢺⡀⡀⠀⠀⠘⠃⠀⠄⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠕⡐⢄⠀⠀⣀⠜⢔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠪⡢⠌⠩⠔⡢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢀
⠀⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠀⠠⡖
⠚⠃⠙⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⠀⠁⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⡄⠀⠰⠐⠀⠀⡀⠀⠠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣀⣠⡀⠈
⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠀⠀⠀⣀⢀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢧⢀⡀
⠀⠀⠀⠰⠿⠀⠸⣀⣀⣄⣀⠽⠛⠉⠁⠀⠀⠀⠂⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⣠⠶⠂⣠
⠀⠀⠀⡄⠀⠀⠐⠇⠈⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠞⠒⠁
⠀⠀⠀⢸⡀
⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡑⡄⠀⠀⠀⠀⠀⠀⠀⢀⢎⠞⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⢐⢠⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡠⢣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡘⠀⠀⠀⠀⠀⠀⠐⠓⠣⡀⠀⠀⠀⠀⠀⢀⣴⠀⠀⣀⡇
⠀⠀⠀⠀⠴⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⢀⠜⠐⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠈⡜⡄⠀⠀⠀⠀⠁⠀⠀⢉⣹⢽
⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⡴⢥⠋⢀⣀⠀⠀⠀⠀⠐⠴⠄⠀⠀⢠⢢⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣀⠀⣸⣼⡀⠀⣀⠀⠀⣀⡀⠀⠀⢘⣼⡀⠀⠀⢀⡀⠀⠀⣀⣐⡑⠁⠀⠀⠀⠀⠀⢀⠀⣐⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⠀⠁⡅⠆⠀⠀⠀⠀⠀⠀⠀⡄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⠁⠀⠀⠀⢀⡀⠀⠐⡵⡄
⠀⠀⠀⠀⠀⠁⠃⠒⠤⠀⠀⠀⠀⠀⠀⠀⢀⡌⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠃⠀⠀⠀⠠⣪⡀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠡⠀⠀⠀⠀⠀⠈⠉⠁⠢⠀⡀⠀⠀⠀⠀⠀⠀⢀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⢴⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡖⠀⠀⠀⠈⡌⠄⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⠀⠀⡴⡲⠂⠀⠀⠘⡌⡄
⠀⠀⠀⠒⠲⠤⠀⠀⠀⡊⡌⠀⠀⠀⠀⢣⢡⣀⠀⠀⠀⡃⠁⠀⠀⠀⠀⠌⡄⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⡊⡄
⠀⠀⠀⠀⢀⢀⣀⣠⣊⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠴⠁⠀⠀⠀⡸⣰⠀⠀⠀⠀⠀⠀⣀⡀⠉⠀⠀⣀⡐⠑
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠀⠀⠈⠈⠉⠉⢉⣀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠁⠀⠀⠀⠈⠘⠀⠀⠀⠀⢀⢔⠟⠁⠀⠀⠀⠀⠀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠄⠀⠀⡰⠰⠀⠀⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⠁⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⠁⠀⠀⠀⠀⠀⠑⢴⣀⠀⠀⠀⠀⠀⠀⢀⢑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⢠⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠈⣿⣷⠀⠀⠀⠀⣿⣿⡇⠀⠀⠠⣿⡷⢈⣿⣷⠀⠀⠀⣿⡷⢸⣿⡷⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠂⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⡼⠟⠁⠀⠀⠀⠀⠐⠺⣿⣤⠀⠀⠀⢀⠎⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⡄⢀⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣤⣮⣤⣤⣾⠀⠀⠠⣾⡗⠀⣿⡿⠀⠀⠀⠛⠒⢺⣿⣿⠺
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢰⡷⣶⣶⡾⠀⠀⠀⢠⣶⠏⢉⣷⡆⠀⠀⣾⣿⣿⣿⣿⠀⠀⢀⣶⣿⠉⣿⣷⡄⠀⠀⣤⣾⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠸⠇⠤⣤⣄⠀⠀⠀⣿⣿⠄⢾⣯⡀⠀⠀⠉⠀⢀⡴⠁⠀⠀⠈⠻⣿⣤⣿⡛⠁⠀⠀⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣤⡄⣿⣿⡇⠀⠀⣿⣿⠀⢸⣿⡗⠀⠀⠀⣰⣿⠁⠀⠀⠀⢰⣿⡿⠀⣿⣿⡆⠀⠀⣨⣿⠁⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠿⠤⠟⠋⠀⠀⠀⠙⠻⠦⠜⠛⠁⠀⠀⠀⠻⠿⠀⠀⠀⠀⠈⠻⠿⠤⠿⠛⠁⠀⠀⠙⠿⠤⠟⠋

⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢲⣤
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢸⣿⡆⠀⠀⠀⠀⣿⠉⣿⢠⠎⠀⠀⠀⠀⠀⠀⢿⣷⣼⡋⢤⠀⠀⠀⠀⢀⡔⢁⡄⢠⡈⢳
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⡿⠀⠀⠀⠀⠉⠈⣡⢋⡤⣄⠀⠀⠀⠀⢰⣿⠙⠻⣿⣿⡀⠀⠀⠀⢸⠀⣿⠇⣾⡇⢸
⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⡴⠃⢿⣇⡿⠀⠀⠀⠀⠈⠛⠓⣂⡈⣿⠇⠀⠀⠀⠈⠣⣈⠁⠉⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠉

⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠿⠀⠀⠾⣷⠀⢀⣴⡄⠀⠐⠒⠒⠒⠀⠀⠉⢡⣾⠟⠀⠀⣀⣠⣅⣀⠀⠀⢶⣾⣴⠆⠀⠀⢻⠟⢻⠛
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠈⠉⠉⠉⠀⠀⠀⣸⡁⠀⠀⠀⠀⠰⠄⠀⠀⠀⠋⠿⠙⠁⠀⠘⢻⠛⡿⠃
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⢤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠳⢤⡀
⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⢀⡞⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⡼⠀⠀⠀⠀⢀⡼⠁⠀⢠⠀⠀⠀⢀⡤⠒⠊⠁⠠⠀⠀⠀⠀⡀⢠⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢳
⢠⡇⠀⠀⠀⢀⡞⠀⠀⠀⠂⠀⡠⠚⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠕⢦⡀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠸
⢸⡇⠀⠀⠀⣸⠁⠀⠀⠀⢀⠞⠀⠈⠀⢈⠀⣇⣀⠤⢤⠒⠒⠒⠒⠢⠤⣀⠀⠀⠀⠐⠁⠀⠙⡄⠀⠀⠀⠀⠸⡆
⠘⡇⠀⠀⠀⣿⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⠔⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⢳⠀⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⣇⠀⠀⠀⢀⡇⠀⠀⠀⠀⡼
⠀⠈⢧⠀⠀⠘⣆⠀⠀⠸⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢈⠂⠀⠀⠀⠘⡄⠀⠀⠀⡏⠀⠀⠀⡼⠁⠀⠀⠀⣸⠃
⠀⠀⠈⢧⡀⠀⠙⣆⠀⠀⢣⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⣰⠁⠀⠀⡴⠃⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠙⢦⡀⠈⠣⡀⠀⠳⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⢀⠞⠀⢀⡴⠁⠀⣠⠞⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠲⣄⡈⠢⣄⠙⠦⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠴⠋⣀⠴⠋⢀⡠⠞⠁⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡙⠲⠤⣉⡒⠬⢕⣒⣠⠤⠤⠤⠤⢤⣒⡊⠥⠒⣊⣡⠴⠚⠉⠀⣀⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣌⣉⡓⠒⠒⠾⠿⠯⠿⠖⠒⠒⠋⢉⣁⣀⠤⠴⠚⠉⠁

⢀⣤⡄⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⠄⢠⣤⡀⢠⣤⡄⣠⣤⣤⣤⣤⣤⣤⣄⠀⣠⡤⠤⠤⣤⡀
⢸⣿⣇⣀⣿⣿⣛⠛⣛⣛⣛⣿⣿⠀⠙⢿⣿⣄⠉⠛⠁⠨⣿⠇⠙⠛⠻⠿⠿⠿⠏⠁⣼⡿⠀⠀⠀⢸⣿
⠘⠿⠿⠿⠿⠿⠟⠀⠻⠿⠿⠟⠃⠀⠺⣿⣿⡀⣾⡷⠂⢰⡶⠄⠀⣠⠖⠒⢤⡀⠀⠀⠸⣿⣀⠀⢀⣼⡟
⠠⠖⠂⠶⣶⣤⡶⣦⣴⣶⣆⡠⠶⡄⢠⣤⣤⣤⣤⣤⡀⠈⠽⠃⠀⣇⠀⢘⡆⣷⠀⣿⠀⣴⠉⠉⠉⣁
⠈⠓⢦⢠⣿⢸⡿⣯⠻⢿⣿⡇⠀⡇⢸⣿⣿⣿⣿⣿⡇⢰⣷⠆⠀⠈⠙⠉⢠⠇⠀⢿⣤⣿⠄⠀⠈⣿
⠀⠒⠁⠘⠃⠘⠛⠋⠙⠛⠋⠙⠚⠁⣈⠉⠉⢹⣿⣿⠇⠘⠛⠀⠀⠠⠴⠚⠁⠀⠀⠀⠀⠿⠀⠀⠀⣿
⠀⠀⡀⠀⠀⣴⡞⠛⠛⠛⢶⣦⠈⣍⠭⠁⠀⣿⣿⣿⠀⠲⢾⣭⡇⠀⣰⣤⡀⠀⠀⠀⠀⠢⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠛⠷⣤⣤⣤⠾⠋⠈⠇⠻⠁⢠⣿⣿⡿⠀⢠⣤⣤⡄⠀⢿⡆⠈⠑⠢⢤⡤⠴⠟⠀⠀⣿
⠀⠀⠨⠀⠀⢰⣿⠁⠀⠉⣿⡆⠈⢻⣙⡁⢸⣿⣿⡇⠀⢸⣿⣿⡇⢀⡤⣄⢤⡄⡤⣄⣠⠤⣄⠀⠀⣿
⠀⢀⣈⡀⠀⠀⠉⠛⠛⠛⠉⠀⠈⠉⠁⠀⣾⣿⣿⠃⠀⣿⣿⣿⠁⠠⡟⡧⣾⡄⣱⡄⢧⣀⡼⠀⠠⠿⠄
⠀⠀⣿⠀⢀⠖⠑⢆⠒⣂⢚⡷⢶⣶⠀⠀⣿⣿⣿⠀⢀⣿⣿⣿⠀⡀⣀⡀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⣿⠀⠀⠓⠔⠃⠓⠛⠚⠃⠛⠛⠀⢸⣿⣿⡏⠀⢸⣿⣿⡇⠀⣩⣰⡄⠀⢸⣿⡁⠀⢈⣿⡇⠀⠈
⠀⠀⣿⠀⠀⣴⠖⠛⠓⠶⢄⡀⠈⣷⠀⠘⠛⠛⠃⠀⣼⣿⣿⠇⠀⣳⡐⡆⠀⣠⡼⠟⠛⠻⢧⣄⠀⠀⠄
⠀⠀⣿⠀⠀⠙⠢⠄⠀⠀⠀⠈⠓⠿⠀⠰⣏⡷⠦⠀⣿⣿⣿⠀⠀⣐⣈⡃⠀⠻⢧⣄⣀⣠⣼⠟
⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⠴⠆⠀⢀⣶⠀⢰⣿⣿⣏⣀⣀⡈⢀⣤⡀⣠⣤⡀⣠⣤⡀⣠⠀⣀⠤
⠀⠀⣿⠖⠀⠈⣿⠉⣿⠀⣰⠃⣀⣀⡀⠀⠘⠿⠀⢸⣿⣿⣿⣿⣿⡇⡏⠀⣷⣿⢾⡍⣿⣼⢧⣿⠀⠧⣀
⠀⠀⣀⣀⣀⣀⠛⠀⠿⠀⢿⢸⡁⠀⢹⠀⠸⢿⡀⠈⠛⠛⠛⡛⠛⠃⠻⠤⠋⠿⠼⠇⠿⠿⠻⠷⠆⠤⠼
⢀⣾⠏⠁⠀⠙⣷⡄⠀⠀⠈⠲⠤⠴⠋⠀⠘⠞⠁⠀⠾⠿⠀⣹⣿⡶⠀⣠⣴⣶⣶⡆⠀⣶⣶⣶⣶⣶⣶
⠸⣿⡀⠀⠀⢀⣿⠇⣀⣸⣿⣲⣶⣦⣤⡄⢸⢿⡀⢀⣤⣄⠈⢻⣿⣦⠸⣿⣯⣭⣭⣥⣤⣭⣿⣯⠉⣿⣿
⣀⣈⣛⣒⣒⣛⣉⣀⣛⣛⣛⣛⣛⣛⣛⣃⣈⣛⣀⣈⣛⣛⣐⣚⣛⣉⣀⣉⣛⣛⣛⣛⣛⣛⣛⣁⣀⣙⣛
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢸⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢠⡴⠾⢿⣤⡀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣾⣧⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠈⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⣀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠐⣊⡿⢿⠀⠀⠀⠀⠀⣨⠥⢤⡅⠡⠀⠀⠀⠀⢨⠭⠤⡌⠑⡀⠀⠀⠀⠀⠀⡴⣷⠋⡄⠀⠀⠀⠀⣸⣽⠯⠿⠽
⠀⠀⠀⠀⠈⠀⡯⢽⠀⠀⠀⠀⠀⠀⠀⢀⣗⣸⠃⠀⠀⠀⠀⢠⠤⢞⡤⠁⠀⠀⠀⢀⡾⡝⢹⠤⡇⠀⠀⠀⠀⣗⣞⠂⠤⢄
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⡤⢎⡴⠃⠀⠀⠀⠀⠀⠉⠉⢳⡌⢆⠀⠀⠀⣞⣿⣶⡞⡟⢓⠀⠀⠀⠀⠀⠈⠙⣧⠀⡆
⠀⠀⠀⠀⠀⠀⣗⣺⠀⠀⠀⠀⢰⣛⣺⣛⣒⣒⡂⠀⠀⠀⣞⡖⠒⢚⠠⠂⠀⠀⠀⠉⠉⠀⢹⣙⡏⠀⠀⠀⢠⣲⣦⣖⣁⠜


⠀⠀⠀⠀⠀⣀⠤⢤⠀⠀⠀⠀⢠⣤⡤⡄⡤⠤⠄⠀⠀⠀⢀⠤⣐⡢⢄⠀⠀⠀⠀⠀⠄⠀⠀⠠⡀⠀⠀⠀⠀⢀⠤⠂⠢⡀
⠀⠀⠀⢠⢛⣔⣋⡉⠁⠀⠀⠀⠈⠉⠉⢉⡿⡿⠁⠀⠀⠀⡇⢾⡉⢹⠄⡅⠀⠀⠀⡊⢠⠋⠙⣆⢸⠀⠀⠀⢠⠇⡞⠉⢳⠈⡄
⠀⠀⠀⣧⣿⠝⠢⣌⠱⠀⠀⠀⠀⠀⢀⣞⡺⠁⠀⠀⠀⠀⡸⢖⡿⣗⠺⠄⠀⠀⠀⠢⠈⠲⢐⣷⢠⠂⠀⠀⢸⢒⡇⠀⢸⠀⡇
⠀⠀⠀⢣⠹⣄⢀⡗⠠⠀⠀⠀⠀⢀⡾⡷⠃⠀⠀⠀⠀⠨⡁⢿⡀⣸⠇⢈⠀⠀⠀⠀⣀⣈⠜⠁⠜⠀⠀⠀⠸⡀⢧⣀⡼⢰⠁
⠀⠀⠀⠀⠑⠪⠥⠒⠁⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠑⠂⠥⠇⠒⠁⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠙⠀⠠⠀⠂


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢒⠀⠀⠀⠀⠀⠀⠐⠟⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠩⠥⠤⠆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡰⠒⠒⠀⠈⠀⠀⠀⠒⢒⣀⣘⠛⠁⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡄⠲⡔⢂⢰⡇⡀⡀⢀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⠀⠀⠠⠧⠮⠹
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠃⠋⣛⣛⣙⠀⣤⣤⣤⡄⠀⠾⠆⠀⠈⢁⠀⠀⠘⠛⠋⠃⠀⢈
⠀⠀⠀⠀⠉⠀⢠⣤⣤⡦⡄⢀⣀⡂⠚⢂⣛⡀⢀⣀⣀⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⠀⣀⣀⣀⣀⡀⠀⠈⡶⣶⣶⡶⡆⠀⣻⣿⡿⣿⡯⣿⣟⣿⣿⠏⣿⡇⢠⣤⠬⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠛⠛⠻⠝⠁⠀⠴⠤⢠⠄⠒⠂⠀⠿⠿⠶⠻⠿⠿⠻⠿⠷⠆⠿⠇⠀⣀⡀⢀⣀⡀⣀⡀
⠀⢈⠁⠀⠀⠲⠖⠀⠀⠤⢼⡇⠀⣖⠘⠁⠀⠀⠀⠰⣔⠀⠀⠀⠠⡤⠍⠀⠉⣁⣛⣽⠛⠋⠃⠸⠀⣖⢀
⠀⢀⠀⠀⠀⠀⢭⠅⠀⢀⣈⡁⢐⣟⠀⠀⣶⣒⠲⠼⠿⠶⠰⠀⠤⡀⠀⠀⠀⢈⣸⣿⠀⠔⠒⠲⠄⠛⠈
⠀⠀⣾⣶⣴⣤⢽⠁⠘⠈⠀⠒⠲⠂⠀⠀⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠀⣎
⠀⠀⠠⣶⡶⣲⠀⠲⠀⠀⠀⠉⠀⠀⠀⠈⠉⠡⢀⣀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠠
⠀⠀⠀⠚⠓⠁⠈⠁⠀⠀⠀⠐⠒⠂⠀⠀⠈⣠⣤⣄⣦⣀⠠⠤⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣴⠈⡁⠀⠀⠀⢫⣹⣻⣏⢘⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⡤⠄⠀⠄
⠀⠀⠀⠀⠀⠤⠤⠀⠀⠴⠆⠿⠀⠀⠀⠀⡴⠀⠀⣿⢰⣶⣷⣶⢶⡆⡀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠒⠼⠙⠛⠉⠓⠐⠒⠀⣀⢤⡖⣀⣟⠀⠀⠶⠶⠂⠰⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠋⠻⠄⠀⠀⠀⠈⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡤⢤⠀⠀⠀⠀⠀⠀⣀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⠀⣭⣥⣍⣬⢠⡄⠀⠀⠈⠀⠈⠁⠀⢀⡀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⠛⠛⠙⠈⠉⠈⣉⡉⠈⠁⠀⠀⠀⠐⠒⠀⠀⡄⠡⠀⠄⠠⠀⠀⠠⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠘⠀⠀⠁⠀⢏⡅⠌⡁⠄⠀⠀⢀⣙⢛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠿⠾⢸⠄⠐⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠠⣰⡀⣀⡚⠂⠂⠋⠓⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⣀⣀⠀⣤⣤⡤⣤⡀⠠⠶⠂⠀⠀⠒⠒⠀⠀⠁⠙⢠⣥⣠⣤⣠⣤⣠⡀⣀⣈⣡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠁⢘⣢⡄⠃⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠹⠿⠻⠯⠸⠇⠊⠋⠓⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠤⠀⢨⣭⠀⢸⢞⡃⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣂⢐⠲⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠆⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠜⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠀⠁⠁⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣙⣛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠠⠤⠽⢥⠹⠷⠶⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣶⣶⢶⠄⢦⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠒⠇⠀⠀⠀⠀⠀⠤⠄⠀⠤⠍⢠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣿⣿⣵⢴⡄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⣿⠐⠀⠀⠀⠀⠀⠄⠨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠾⠯⠛⠀⠀⠀⠀⠐⠓⠶⢷⠖⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⣠⣀⣄⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⠙⠛⠛⠃⠛⠀⠀⡤⣄⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡆⣤⢄⢤⠀⠀⠀⠀⠀⠀⠀⠀⣥⣬⡄⠀⠖⠲⢶⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠠⠀⠄⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⡂⡩⠭⠀⠀⠀⠀⠀⠀⣤⣤⠁⠀⠀⠀⠀⠘⠛⠛⠃⣸⡿⣾⣿⣺⡿⢺⡆⠀⠐⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠓⠇⠷⠀⣀⠐⠓⠂⠉⠀⡿⡦⠀⠀⡀⢀⢀⠀⡀⠀⠀⠠⠤⠬⠁⠈⠀⠈⠀⠒⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣤⣤⣤⡅⣠⣤⣤⣤⡀⣿⠿⠀⢀⡌⢄⢠⠁⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠛⠋⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⠈⠀⠀⠀⠀⠐⠒⠊⠉⠄⠀⠛⠛⢀⡀⠈⠗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢏⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠰⠾⠔⠒⠀⡰⠟⠿⠾⢸⣀⠀⠼⠻⠯⠙⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠄

⣽⣿⣏⡿⣤⣴⢸⡹⣦⣼⣾⡁⣖⣦⣈⡝⢮⣁⣞⠯⢟⣖⣛⠃
⣬⡿⢸⡷⠿⡗⠒⠈⠉⠋⡴⢣⢔⣭⣖⣋⣛⡹⠾⡄⠨⣀⠴⠁⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⠀⠠⢀⠀⠀⢠⢄⠄⠐⠀⡀⠠⠄
⣧⡔⠉⣷⢲⣿⡾⣟⠛⡋⠉⠉⠽⡟⠑⣾⠋⡹⠉⠛⢉⣉⣸⣻⡴⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⢨⣶⣶⢻⣝⣻⣄⣻⠞⣡⣬⠓⠤⠟⣒⣨⠯⣀⣒⠭⣉⢡⡉⣀⢹⠟⢭⡆
⠛⢏⣙⣟⣹⠋⣈⠉⠿⠒⣉⠓⠀⢸⠁⢰⡛⠉⣽⠟⠷⠚⢫⠀⠈⢒⡉⣠⢦⡀
⣳⠞⢙⡽⠋⣩⣽⠀⣴⡛⣎⠓⢤⣿⠚⣦⣹⠛⢧⡔⢒⣋⣭⣴⣊⢭⠗⣓⡮⢿
⠈⣶⠛⣧⣐⣛⣿⡏⠫⠖⣡⡶⠤⣉⠉⢸⠛⠛⠋⢱⡯⠤⣤⣀⣉⠈⠀⡏⠀⠂
⠴⢚⣯⠗⠋⠱⠒⢳⠀⠣⣀⣝⡷⢢⢡⣏⡴⣂⠙⠪⢝⠲⠤⣬⢏⣙⣶⠷⠶⢶⡟
⠖⢛⣟⣉⡇⠶⠲⠭⣤⣒⡛⠓⢲⣧⠙⢺⢣⢮⠙⠺⣛⠲⣄⣴⠛⣡⠟⡀⢀⡟⠀⠀⢀
⢉⣙⡾⠥⣠⡞⠉⣱⠙⠒⠚⠒⠛⢿⠁⠘⠈⠙⠷⣄⣙⢯⡉⢈⡟⠳⣬⠉⠛⠓⢦⡖⣾⣷⣄
⢯⣈⠻⢷⡛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁⠀⠀⠠⣤⣶⣿⣿⣿⣿⣦⡀
⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣶⢶⣦⡶⣰⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡤⠀⠀⠀⠀⠀⡤⠀⠀⠀⠐⠂⠀⣤⠀⠤⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠒⣒⢓⣀⣀⡀⢸⠀⣽⠇⣄⡀⠡⡤⠀⠉⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠻⣿⢿⡟⢹⡇⢈⣀⠉⠁⡀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣒⠂⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⢸⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡁⠀⣾⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠤⠖⠀⠀⠈⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠆⡉⠀⠀⠀⠀⠸⠻⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠠⠀⠀⠀⠀⢀⣲⣶⣐⢲⠶⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠨⠉⢨⣭⣭⣬⣥⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠶⠂⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀⠀⠙⠫⠿⠉⢛⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠀⠀⠶⡖⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠒⠀⡄⠉⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢐⡒⠂⠀⠀⠶⠦⠂
⠀⠀⠀⠀⠀⠀⠀⢶⠖⣃⠀⠃⡀⠀⠀⠀⠂⡀⢡⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣤⡄⣤⢠⡀⡀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢤⣤⣤⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⢛⡛⡛⡘⠃⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡲⡔⡄⡶⡏⠨⠅⠩⠈⣤⣆⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠇⠓⠠⠤⠠⠶⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣥⡁⣅⢀⡁⠀⢀⣟⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⢴⡟⢻⣆⠛⣿⠶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⢿⣛⢯⣸⠇⠀⠉⠉⠉⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⣃⢸⣇⣸⡇⣴⣋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠛⠏⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠉⠉⠁⢉⣉⣄⣉⡉⠡⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡞⠐⠀⢄⡀⢤⠀⠀⠈⡉⠁⠀⠠⠄⢀⠀⠀⠀⠀⠀⢀⠐⠀⠋⠛⠛⠙⠿⠈⠉⠉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⣽⡟⣿⣿⣽⡟⣿⠀⣀⠘⠀⠀⠀⠀⠀⠀⠁⠠⠀⢈⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣄⣤⡄⣽⡠⡏⠋⠁⢠⣗⣶⣆⣷⣲⡀⠈⠉⠍⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠿⠈⠛⠛⠛⠐⠓⠀⠶⠶⠶⠆⠀⠀⠀⣀⣀⡀⠀⠀⠃⠀⠀⠐⠓⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠙⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⣠⣤⠀⠀⠤

⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠤⢄⡀⠀⠀⠀⠀⠀⣠⠄⠒⣀⠀⠰⠄⠀⠀⢠⠂⢲⠀⠀⠀⢰⢊⠱⡄⠀⠀⢰⡫⠙⠩⠙⠛⠁⢦
⠀⡠⠶⢡⡬⠦⡌⢀⡆⠀⠀⡜⢁⡱⢤⣸⠀⠱⡄⠀⠀⠸⡥⠋⠀⠈⡗⢁⡄⠀⠀⢰⠊⣸⠀⠀⠀⠘⣃⠄⡇⠀⠀⢸⠔⢁⠋⠉⠉⠈⠁
⡔⢒⣨⠍⠀⠀⡷⠁⡀⠀⠘⣄⢚⠄⠀⠈⢲⠀⣰⠀⠀⠀⠀⢀⡠⠞⡡⠋⠀⠀⠀⢸⠂⡅⠀⠀⠀⢘⡁⢴⠇⠀⠀⠘⡖⠉⡢⠤
⠙⠐⠁⠀⠀⠀⡁⠐⡇⠀⠀⠀⠀⠀⠀⢀⠖⢊⠏⠀⠀⠀⡔⢋⠘⠋⢉⠶⡀⠀⠀⣸⠘⣧⢠⠤⡤⠎⠀⠼⠀⠀⠀⠀⠉⠈⠉⢢⠂⠉⠄
⠀⠀⠀⠀⠀⢸⠁⡠⡇⠀⠀⠀⠀⢀⡴⠃⡰⠋⠀⠀⠀⠀⠑⠉⠁⠈⢃⠀⠲⠀⠀⠣⠴⠠⠄⠴⠐⢳⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠹
⠀⠀⠀⠀⠀⢸⠊⠀⢇⠀⠀⠀⣠⠋⠰⠆⣤⠀⠤⣄⠀⠀⠀⠀⠀⢀⡎⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡗⢈⡇⠀⠀⠀⣀⣀⣀⡀⠜⠁⠁⡙
⠀⠀⠀⠀⠀⠘⠣⠤⠃⠀⠀⠀⠑⠒⠒⠀⠒⠐⠒⠛⠀⠐⣍⠒⣈⣁⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠨⠇⠀⠀⢸⠀⠂⠀⠑⣄⡨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠈⠁
⠀⠀⠀⠀⠀⡀⠀⡀⣀⠀⠀⠀⢀⣠⠀⠤⠤⠤⢤⣀⠀⠀⢀⠴⢂⠉⣏⣉⠋⢢⡀⠀⠀⢀⢤⠀⠀⠉⠀⣱⠀⠀⠀⠀⢀⠴⡂⢑⠒⢄
⠀⠀⠀⡄⠁⢠⠴⢤⠀⢳⠀⠀⠀⠠⠘⠂⠓⣤⠁⡸⠀⠀⡮⠠⢤⠋⠀⠙⡆⠀⡇⠀⢀⠆⢀⡶⠊⣶⡀⢀⡀⠀⠀⠠⠓⠄⢡⣸⠀⠌⠣
⠀⠀⣄⠀⠰⠀⠀⠸⡖⡹⠀⠀⠀⠀⠀⠀⢠⠗⣚⣁⠀⠀⣆⠀⠜⠦⠐⠚⢁⠘⠀⠀⠘⡄⠐⠓⠊⢃⢠⡠⠇⠀⠀⠓⢀⠈⠋⠀⠛⢀⠰
⠀⠠⠀⢈⠃⠀⠀⠀⠈⠀⠀⠀⠀⢔⢈⣁⠭⢜⠁⠄⠀⠀⠈⠲⡂⠀⣀⣴⡋⠀⠀⠀⠀⠘⠦⣴⡀⠸⢚⡀⡄⠀⢸⠀⠀⠛⠀⠀⠘⠁⣀
⠀⢀⠀⠺⠖⠂⠒⠢⡀⠀⠀⠀⠀⠀⠀⠞⢀⠁⠀⠀⠀⠀⠠⠚⠍⡢⡈⠀⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⠀⠈⠀⠈⡃⠀⠀⣘⠉
⠀⠠⡀⠀⢰⠓⡦⠀⢸⠀⠀⠀⠀⠀⠞⢀⠃⠀⠀⠀⠀⢀⠃⠢⡔⠉⠉⢹⡇⠔⠃⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⠂⠈⠙⠦⠖⠩⡉⢈
⠀⠀⠙⢄⡀⢈⣁⠔⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠘⡄⠈⢱⠤⠔⠢⢀⡰⠃⠀⢀⠤⡤⢤⠤⠎⡁⡸⠁⠀⠈⢆⡀⠁⠀⠐⡀⢈⠏
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⠘⠒⠉⠀⠀⠀⠘⠠⠤⠀⠦⠐⠉⠀⠀⠀⠀⠀⠑⠒⠦⠀⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣭⢳⠀⠀⢠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡉⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠙⠚⠁⠀⡰⠃⠺⠀⠀⠀⠀⠀⠠⠤⡤⠤⢤⠤⡄
⠀⠀⠀⠀⡤⢤⠤⠞⡡⠚⢒⡚⡢⠀⠀⢌⣔⣁⣎⣠⠉⠜⠁⠀⠀⠀⠀⠀⠀⡜⠃⡙⠁⠀⠀⠀⠀⠲⠤⠮⠤⠴⠥⠐⠁
⠀⠀⠀⠀⠈⠉⠉⠛⣆⢘⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⠎⠀⠀⠀⠀⠀⠀⠔⡒⠀⠋⠉⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡰⠃⠀⢀⢴⣠⡀⠀⠀⠚⠒⠓⠒⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡤⠀⢀⣤⡀⠀⣠⣤⡀⢠⡄⢠⡄⢀⣤⣤⠀⠀⣀⣤⡀⢠⣤⣤⡄⠀⢀⣤⡀⠀⣀⣤⡀⠀⢀⣠⣄⠀⠀⢸
⠀⠀⠘⣿⡇⠈⣡⡿⠀⠡⢼⡃⠸⠧⢼⡇⠐⠟⢲⡄⢠⣿⣤⡄⠀⢠⣾⡇⢀⣼⣿⣧⠀⢿⣼⣿⠀⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠸⠃⠸⠷⠶⠀⠲⠞⠃⠀⠀⠸⠇⠀⠶⠞⠁⠈⠻⠿⠃⠀⠸⠟⠀⠀⠻⠿⠋⠀⠲⠖⠃⠀⠙⠿⠟⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⡆⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠉⠉⠀⡀⠀⠀⠀⠀⠀⢀⣤⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⠠⢀⠌⠀⠀⠀⠀⠀⠐⠊⠑⢢⠀⠰⡀⠀⠰⠟⠂⢀⡷⠀⠀⠀⠀⢣⣤⢠⠄⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠀⠀⠀⠀⠠⠤⠄⠻⠹⠀⠂⠀⠀⢦⡤⠄⣐⡡⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠀⠀⠀⠀⡀⢊⡀⣀⡄⠀⠀⠀⠀⡀⢀⢤⡀⠀⢠⠈⢦⡄⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⢸⠀⠂⠀⠀⠀⠀⠀⢸⠁⠛⢁⠴⠃⠀⠀⠀⠀⠫⠝⠋⠁⡠⢸
⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠈⠓⠈⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠉⡔⠊⠀⠀⢀⠠⠴⠆
⠀⠀⠀⠀⠠⠁⠐⡶⠈⡄⠀⠀⠀⠼⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡗⢠⠂⠁⣀⡀⠀⠆
⠀⠀⠀⠀⢰⡈⠃⠂⠀⠢⡀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠁⢠⢢⠀⠀⠀⠀⠀⠈⠐⠃⠄⠐⠋⠀⠚⠃⠈⠙⠃⢐
⠀⠀⠀⠀⠈⢩⠁⠊⠿⠂⠀⠀⠀⠀⠀⠀⡀⠘⠉⡀⢤⡀⢀⠽⡀⠀⠀⠀⠀⠀⠠⠀⢂⡈⢣⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠠⠈⠡⠄⠀⡼⠀⠀⠀⠀⠀⠈⠲⣈⠤⠊⠁⠲⠊⠁⠀⠀⠀⠀⠀⠂⠀⡸⠓⡀⠀⠀⠀⠀⢈⡤⠞
⠀⠀⠀⠀⠀⠈⠓⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢰⡄⠀⠝⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠁⠐

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠈⠛⠛⠻⠋⢀⡴⡾⠃⠀⠀⠀⠀⠀⢤⣀⣼⡋⠁⠀⣰⡇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⢫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠚⢋⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠱⠆⠀⠀⢀⡟⠀⠀⣠⣼⠄⠀⠀⣼⠀⠀⠀⠀⠀⢀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠀⠀⣵⠀⠘⠀⠀⢀
⠀⠀⠀⠘⠢⢤⣤⣠⡴⠎⠁⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡘⡈⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⢤⣀⠀⠀⢀⣴⠇⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠐⢲⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠶⠿⠃⠀⠀⠀⡠⡦⡄⠀⠹⡆⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠶⠶⠿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠀⢸⡇⡀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠖⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠾⠉⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠘⠙⠓⠚⠀⠀⢸⠞⠛⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠀⠀⢤⣤⣤⣤⣀⣤⣼⡇⠀⠀⠀⠒⠀⠀⠑⠚⠋⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠉⢂⠀⠀⠀⠀⢿⡉⠉⠂⠀⣷⠀⠀⠀⠀⠀⠰⡟⠙⠂⠀⢸⠆⠀⠀⠀⠈⠓⠦⠤⠤⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠛⠀⠀⠀⣻⠀⠀⠰⣄⡀⠀⠀⡀⢔⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠶⠆⠀⡿⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⡪⠒⠆⠀⠰⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠘⠦⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⢀⣀⢤⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⠤⠦⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⢀⠄⠒⠒⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⢘⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⠀⠀⠠⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⣀⣀⣉⣀⣀⣉⣀⣀⣀⣀⣀⣈⣁⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠐


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠛⠻⣷⣄⠀⠀⠀⢀⣴⠿⠛⠋⠙⠛⠿⣦⡀⠀⠀⠀⣠⣶⠟⠛⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢀⣤⣾⠀⠀⠈⢻⣦⠀⢠⣿⠁⠀⣴⠛⠛⣦⠀⠘⢿⡄⠀⣼⡟⠁⠀⠾⠋⢳⡄⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⣿⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⣠⠟⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠰⢾⡅⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⣿⠀⠀⠀⣸⡟⠀⠸⣷⡀⠀⣴⣟⣁⣀⠀⢀⣾⠇⠀⢻⣇⠀⠀⢦⣀⣠⠟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣈⣀⣤⡾⠟⠀⠀⠀⠙⢿⣦⣍⣉⣉⣩⣴⡿⠋⠀⠀⠀⠻⢷⣤⣀⣉⣁⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁


⠀⠀⠀⣠⣴⡶⠶⠶⢶⣦⣄⠀⠀⠀⠀⠀⣠⣶⠶⠶⠶⢶⣤⡀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⣶⣄⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⢶⣦⣄
⠀⠀⣾⠟⠁⠀⢀⣤⠀⠈⠻⣷⡀⠀⢠⣾⠋⠀⣴⠶⠶⠄⠙⢿⣦⠀⠀⣴⡿⠉⢀⣤⠶⣄⠀⠙⣷⡄⠀⢀⣾⠟⠁⠤⠤⢤⣤⠈⠻⣷⡀
⠀⢸⡏⠀⠀⣠⠞⣿⠀⠀⠀⢹⡇⠀⣾⡇⠀⠀⣿⠴⣤⡀⠀⠀⣿⡆⢰⡿⠀⠀⣼⣡⢤⣌⠀⠀⢸⣷⠀⢸⡏⠀⠀⠀⢀⡾⠁⠀⠀⢹⡇
⠀⢸⣇⠀⠰⠧⠤⣿⠤⠀⠀⣸⡏⠀⢿⡇⠀⢠⡄⠀⢈⡇⠀⠀⣿⠇⠸⣿⡀⠀⢹⡁⠀⢸⠆⠀⢸⡿⠀⢹⣇⠀⠀⠀⣾⠁⠀⠀⠀⣸⡇
⠀⠀⠻⣦⡀⠀⠀⠛⠀⢀⣴⡟⠀⠀⠘⢿⣆⡀⠙⠒⠋⠀⣠⣾⠏⠀⠀⠹⣷⣄⠈⠛⠒⠋⢀⣠⡿⠃⠀⠀⢻⣦⡀⠀⠋⠀⠀⢀⣴⠟
⠀⠀⠀⠈⠛⠷⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠿⠛⠁⠀⠀⠀⠀⠈⠛⠿⠶⠶⠾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢉⣉⠉⠛⢿⣦⠀⠀⠀⣠⡾⠟⠉⣉⡉⠉⠻⣷⣄⠀⠀⠀⣴⡿⠋⠉⣉⡉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢰⡏⠉⣳⠀⠀⢹⣧⠀⢰⡿⠁⠀⣾⠉⠉⣧⠀⠈⢿⡆⠀⣼⡏⠀⢠⡞⠉⠹⡆⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⠟⠛⢷⡀⠀⢀⣿⠀⢸⡇⠀⠀⠻⢦⠴⣿⠀⠀⢸⡇⠀⣿⡀⠀⢸⡇⠀⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠹⢦⣤⡼⠃⢀⣼⠟⠀⠘⣷⡄⠀⠲⣤⣤⠏⠀⢠⣾⠃⠀⠻⣧⡀⠈⠷⣤⡼⠃⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣠⣤⣴⠿⠋⠀⠀⠀⠈⠻⣶⣤⣄⣠⣤⣶⠟⠁⠀⠀⠀⠙⠿⣦⣤⣄⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⡷⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠠⣤
⠀⠀⠀⣸⡟⢻⣟⣿⡏⢻⡆⠀⠀⠀⢂⡀⠀⠂⠀⠰⠀⠁⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢀⣴⣶⣦⣀⡿⠁⠀⠀⠀⠀⠀⠓⠙⠀⠀⠀⠀⠀⠀⠄⠘⠀⠡⢀⠁
⠀⠀⠀⠘⣻⠛⣿⣿⠛⡋⠀⠀⢀⡀⡀⡰⣨⡄⢩⠀⠠⠀⡄⠀⠀⠀⠙⠛⠋⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⢀⣻⣻⣆⣾⣹⣟⣖⠀⠀⠀⢀⡄⠁⢀⡈
⠀⠀⠀⠰⠿⣦⡉⣁⣼⠿⠄⠀⠀⠀⠠⠐⠠⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣧⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣍⣷⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⣤⣼⠟⠿⣤⣤⡀⠀⠀⠉⠀⢀⡂⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣿⣏⣹⡿⠛⣿⣿⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠛⠁⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠙⣷⢸⡿⠁⠀⢻⢸⡇⠀⠀⠀⠼⠛⢻⠟⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⡟⣸⠃⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣷⠖⠀⠀⠀⢻⣼⠇
⠀⠀⠀⢰⢻⠆⠀⠀⣿⣿⣧⣽⣿⣿⣥⣿⣿⠇⠀⣠⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣀⡀⠀⢰⣦⣿⡆⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠛⠻⠛⢉⣰⠏⠀⠀⣿⣿⣟⠛⠀⠀⠀⠈⢛⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⣿⣦⣨⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣤⡄
⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁⠀⠀⠀⠘⠻⣿⣿⣦⠄⠀⢴⣿⣿⡟⠁⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⣴⣾⣿⣦⡄
⠀⠀⠀⢸⣿⡇⠀⠀⣠⠄⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣴⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠉⠉⠙⠛⠃⠀⠀⣀⡀⠀⠀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠉⠁⠀⠀⣿⣁⡀⣘⣹⣇⣶⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡀⣀⣠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢸⣿⣅⣽⣿⣉⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⡿⣿⣻⣿⣿⠟⠃⠀⣴⣷⣤⣄⠀⠀⠀⢀⣠⢀⣦⢰⣷⢾⡿⢙⡏⠀⠀⠀⢹⣦⣀⡧⠀⣾⠀⠀⡀⠀⠀⠀⢸⣧⠙⠻⠟⠟⣸⣿
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣽⣿⣿⣿⡟⠉⠀⠀⠀⠀⣿⣿⣿⡧⠀⠀⢻⣿⣼⠿⠾⠁⠘⠀⣀⡁⠀⠀⠦⣤⣽⣿⣷⡀⢹⣷⠾⣡⠖⠀⠀⠀⠻⠿⠶⠶⠾⠿⠃
⠀⠀⠀⣠⣄⡀⠀⠀⠈⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠛⠛⣿⠆⠀⢈⣉⢀⣤⢰⡆⣿⢸⣿⠇⠀⠀⢲⣦⣄⣸⣿⣷⣾⡇⢀⣿
⠀⠀⠈⠹⠛⠆⠀⠀⠀⠀⠀⠀⠉⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⢿⠸⠤⠻⠊⡋⠀⠈⠋⠉⠉⠙⣿⣿⣿⣿⡿⠛⠁⠀⠰⣒⣦⣤⣞⠁⠂
⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣦⠀⠀⠀⠀⠘⠋⣈⣤⣤⣶⢶⣿⡿⠀⠀⠀⠀⠀⠀⣹⣿⡿⢾⡇⠀⠀⠀⠀⠋⠉⠉⠉⠛⢷⣄⣀⡀
⠀⠀⠀⠀⠀⢯⣽⣿⡽⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢿⣿⡏⣿⡀⣿⠄⠃⠃⠀⠀⠀⠀⣀⣠⣿⣦⣤⣼⣧⣀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣿⣿⠗
⠀⠀⠀⢀⣤⠤⣽⡿⠤⢤⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠋⠁⠉⠀⠀⠀⠀⠀⢄⣤⡀⠀⠉⠛⠻⠻⠟⠙⠏⠉⠁⠀⠀⡿⣦⣤⠀⣤⡾⣿⠻⢶⡄
⠀⠀⠀⢸⡏⣿⣿⣿⢾⢽⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡏⢛⣹⣏⡛⢹⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⢸⣧⡿⣿⣿⢯⣼⡇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⡿⢿⣿⣧⠀⠀⠈⠛⢿⣷⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⢿⡿⡇
⠀⠀⠀⢸⡿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⣾⣾⣿⣧⡀⠀⠀⠀⠘⠷⣿⣾⢧⣼⣿⡟⠀⠀⠀⠀⠀⣾⣿⣿⣛⣡⣤⣶⣾⡁⠀⠀⠀⢸⣷⣾⣿⣶⣿
⠀⠀⠀⠀⢸⣧⠀⠀⣾⡇⠀⠀⠀⠀⠀⢀⣾⣿⠟⣈⢱⣿⡿⣱⠀⠀⠀⠀⠀⠛⠛⠀⠉⠁⠀⠀⠀⠀⣀⣴⡿⣿⣿⣿⡿⣿⣿⠿⠇⣠⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣟⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⣿⣴⣿⣿⡿⠿⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⢦⡜⣛⣥⣶⣿⣿⣿⣗⣠⡤⣶⣿⣿⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣷⡙⣟⣛⣭⣵⣶⣾⡿⠋⠀⠀⠀⠀⣀⠀⢢⠀⡦⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠃⠈⠀⠀⠀⠁⠀⠀⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠸⠟⠀⠀⠻⠇⠀⠀⠀⠈⢻⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠥⠅⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠴⠤⠤⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠒⠐⠀⠂⠂⠀⠀⠀⠀⠐⠺⠿⠸⠿⠒⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⣴⣶⣦⣄⣀⡀⠀⠀⠀⠀⢀⡄⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⣿⡿⠿⢿⠿⣍⠃⠀⠀⠀⠁⢸⡿⠿⠛⢿⠀⠀⠀⠀⠈⠃⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠉⠈⠁⠈⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢸⡧⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢷⠿⢧⠀⠀⠀⠀⠀⡇⠱⠿⣧⠇⠀⠀⠀⠀⠀⢀⣉⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠒⠚⠒⠚⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠰⣶⣾⣶⣿⣶⠆⠀⠀⠀⢠⣼⣏⣙⣷⣶⠀⠀⠀⠀⠀⣖⡖⢰⣖⠄
⠀⠀⠀⠀⣿⣿⣶⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣷⣿⠀⠀⠀⠀⠰⠦⠤⣀⡀
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣶⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣮⣕⢄
⠀⠀⠀⠀⠀⠀⠀⢠⢣⣿⣿⢟⡁⣾⣿⣿⣮⢇⠀⢠⢯⣾⣿⣡⣶⣦⢹⣿⣮⢧⠀⢀⢯⣾⣿⣡⣶⡦⢹⣿⣧⢣
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⡇⣿⣿⣿⣿⢸⠀⢸⢸⣿⣿⣿⠟⣡⣾⣿⣿⢸⠀⢸⢸⣿⣿⣿⣯⣤⠻⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢎⢿⣿⣿⣇⣿⣿⣿⢫⠇⠀⠈⢏⢿⣿⣁⣚⣛⣻⣿⢏⠏⠀⠀⢯⢻⣿⣌⣛⣋⣴⣿⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣟⣛⣛⡯⠖⠁⠀⠀⠀⠀⠑⠯⣛⣛⣛⡫⠕⠁⠀⠀⠀⠀⠑⠯⣟⣛⣛⡯⠕⠁


⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⢔⣭⣷⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣯⣷⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣯⣗⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣯⣕⢆
⠀⢠⢣⣿⣿⡿⢃⢸⣿⣿⣮⢇⠀⢠⢫⣿⣿⠁⣶⣶⣾⣿⣮⢇⠀⢀⢯⣾⣿⢣⣶⣦⣽⣿⣧⢧⠀⢀⢏⣾⣿⣶⣶⠆⣼⣿⣧⢣
⠀⢸⢸⣿⡟⠱⠿⠸⣿⣿⣿⢸⠀⢸⢸⣿⣿⣴⣶⣦⢹⣿⣿⢸⠀⢸⢸⣿⣿⢠⣶⣦⢹⣿⣿⢸⠀⢸⢸⣿⣿⣿⠇⣾⣿⣿⣿⢸
⠀⠈⢎⢿⣷⣶⣶⣰⣿⣿⢣⠇⠀⠈⢏⢿⣿⣌⣛⣋⣼⣿⢯⠇⠀⠀⢯⢿⣿⣌⡛⣋⣼⣿⢏⠎⠀⠀⢧⢻⣿⣿⣸⣿⣿⣿⢏⠎
⠀⠀⠈⠓⠭⣛⣛⣛⡫⠖⠁⠀⠀⠀⠀⠑⠭⣛⣛⣛⡫⠕⠁⠀⠀⠀⠀⠑⠯⣛⣛⣛⡫⠕⠁⠀⠀⠀⠀⠑⠮⣛⣛⣛⡫⠕⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣫⣽⣶⣾⣭⡲⣄⠀⠀⠀⢀⠴⣫⣽⣶⣾⣭⡳⢄⠀⠀⠀⢀⡴⣪⣽⣶⣾⣭⡓⢄
⠀⠀⠀⠀⠀⠀⠀⢠⢫⣾⣿⠡⣶⡎⢻⣿⣎⢇⠀⢠⢏⣾⣿⢡⣶⣮⢻⣿⣎⢧⠀⢀⢎⣾⣿⢋⣶⣦⢻⣿⣮⢣
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⢃⣬⣄⢻⣿⣿⢸⠀⢸⢸⣿⣿⣌⠛⣋⢸⣿⣿⢸⠀⢸⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢎⢿⣿⣜⠻⢋⣼⣿⢏⠇⠀⠈⢏⢿⣿⣝⠿⣋⣼⣿⢏⠏⠀⠀⢯⢿⣿⣌⠻⢋⣼⣿⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢽⣛⣛⣛⡫⠕⠋⠀⠀⠀⠈⠓⠭⣛⣛⣛⡫⠕⠃⠀⠀⠀⠈⠓⠭⣛⣛⣛⣫⠕⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠐⠶⠖⠀⠰⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⢀⣠⠴⣾⣷⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠞⠁⠀⠈⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠓⢦⣀⣠⠞⠀⡆⠀⠀⣶⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣠⠶⢤⣀⢀⡼⠁⢸⠀⠀⠈⡁⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⡼⠁⣶⠀⠈⢉⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠿⠀⣠⡤⠶⠒⠲⠦⢥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠋⠀⠀⠛⠀⠀⠘⠀⠀⠘⠁⠀⠈⠃⠀⠀⠃⠀⠀⣰⠋⠁⠀⣀⣰⣆⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⡼⠁⠀⠀⢸⠁⠀⠀⠈⠇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢡⡇⠀⠀⠀⠸⣄⣀⣀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠹⡄⠀⠀⠸⣄⣀⣀⣠⠇⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠙⠦⣄⡀⠀⠘⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉






⣿⠿⠿⠿⣿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠠⡒⠀⠀⠀⠀⠀⠀⠀⢀⠬⡀⠀⠀⠀⢀⢊⡠⠀⢳⠀⠀⠀⠀⠀⠀⠠⠀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠿⡶⠏⡆
⠀⠀⠀⢀⡴⠢⠟⠫⣯⠶⣄⠀⠀⠀⠀⢀⡤⠧⣘⡆⠀⣠⠞⠉⠙⠛⠾⠻⣄⠀⠀⢠⠆⠰⢕⣺⣰⡾⢦⡄⠀⠀⠀⢠⠟⠚⠑⢲⡇
⠀⠀⢠⠋⠀⡴⠚⠛⢢⡀⠈⣧⠀⣠⠴⠋⠀⠀⡀⡇⠘⢧⣀⡴⠒⢦⠀⠀⢹⠀⠀⠸⠦⠤⢤⠄⠀⣠⠏⠀⠀⠀⣰⠃⢀⣼⠀⢸⡇
⠀⠀⡏⠀⣸⠀⠀⠀⠀⢷⠀⢸⡄⢻⣠⠴⡆⠈⠀⡇⠀⠀⠈⠀⢀⡞⠀⢀⡟⠀⠀⠀⢠⠖⠁⠀⠰⣇⠀⠀⠀⡰⠃⢠⣎⣸⠀⢸⡇
⠀⠸⡇⠀⢻⡀⠀⠀⠀⡞⠀⢸⡇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣠⠎⠀⣠⠞⠀⠀⠀⠀⠸⠤⠤⣄⠀⠀⢱⡀⡴⠁⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⢷⡀⠀⠓⠶⠶⠚⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⡇⠀⠀⡎⠁⠀⠈⠉⠉⠉⢹⠀⢀⣀⣀⣠⠟⠀⠀⢀⡇⠉⠉⠉⠙⠛⣻⠀⠀⡏⠉
⠀⠀⠀⠳⢤⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⣇⣀⣀⡇⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼⠀⣼⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⢿⣀⣠⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⡄⠀⠀⠀⠀⠀⠀⢀⠊⣠⠚⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢴
⠀⠀⠀⠀⡤⠆⣤⡣⣌⠤⢤⠀⠀⠀⠀⢀⣤⣬⣤⡸⢤⠀⠀⣾⠛⠛⠛⠚⠳⣍⠴⠛⡇⠀⠀⠰⠃⡔⠛⡄⠀⠀⠀⢀⡤⢆⣰⣷⣀⢧⡀
⠀⠀⠀⠀⡇⠀⣠⠭⠭⠤⠼⠀⠀⢀⡔⠉⠀⣀⡬⢤⡜⠀⠀⣯⣀⣀⣠⣀⠀⠀⠈⣠⡇⢀⠔⠛⠛⠛⠟⠳⡄⠀⢠⡏⠀⣠⠔⠲⣄⠀⣹⡄
⠀⠀⠀⠀⡇⠀⠿⢤⣀⠀⠀⠀⠀⡞⠀⠀⠺⠓⠶⢤⡀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠞⠁⠀⣾⠀⠰⣍⣈⠆⠀⣿⠀⢸⠀⠀⢧⣀⣀⡟⠠⠟⡇
⠀⠀⠀⠀⠷⠶⢤⡀⠈⠱⣄⠀⢰⠇⠀⢀⡤⠦⣄⠀⠹⡄⠀⠀⢠⠞⠀⢀⡼⠁⠀⠀⠀⢘⡦⠀⠈⠁⠠⣔⡁⠀⠈⢧⣀⠀⠈⠁⠀⠀⢀⡇
⠀⠀⠀⢀⣀⠀⣀⡇⠀⠀⢹⠀⠸⡆⠀⢸⡀⠀⣸⠀⠀⡇⠀⢀⡞⠀⠀⣼⠁⠀⠀⠀⣰⠋⠀⣠⠖⠲⣄⠀⠙⡆⠀⠀⠈⢉⣹⠇⠀⠀⡼⠁
⠀⠀⠀⡜⠉⠉⠁⠀⠀⣠⠏⠀⠀⠳⣄⠀⠉⠉⠁⢀⡼⠁⠀⢸⣁⣀⣰⡇⠀⠀⠀⠀⢧⠀⠀⠳⣤⡤⠏⠀⢀⡇⠀⡏⠉⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠉⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠈⠓⠒⠒⠚⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⡤⠞⠀⠀⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢀⡀⢀⠀⣀⠀⡀⢀⠀⣀⡀⡀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢸⣻⡿⣿⣸⣿⣄⡟⣾⢺⣉⡁⣇⣾⣷⣸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣤⡀⣠⣄⢀⣤⡀⣀⢀⡄⣄⢠⢠⣤⢄⣀⡄⢠⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠼⢤⠜⠧⠤⠻⢤⠟⠻⡼⠇⠟⠿⠀⠇⠸⠿⠟⠿⠻⢼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡍⣴⡌⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠃⠻⠶⣍⡵⠾⠃⠚⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠤⠤⠤⠤⠍⠁⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠦⠶⢐⠀⣛⣃⣈⣈⡁⢀⣤⡴⠻⠁⢸⠷⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⣿⠀⠖⠒⠲⠂⠀⠛⠶⠖⠒⠖⢸⡇⠀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⢰⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣳⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣃⣀⣀⣀⣀⠀⠉⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣀⠀⠀⢀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⡭⠭⠤⢤⡉⠉⢹⠀⠀⠀⠀⠀⠀⡏⣭⠭⠭⠭⠤⠥⠬⣭⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠌⢛⣛⣛⡛⠁⣠⠞⣿⠳⣄⠀⠀⠀⡇⢿⣀⣘⣛⣀⣀⣃⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣛⡦⠐⣶⣶⠖⠀⣇⠀⠹⣖⣿⠀⠀⠀⡇⠠⠤⠀⠠⠤⠀⠠⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠬⠤⠬⠭⠭⠤⠤⠌⢳⠶⠞⠁⠀⠀⠀⡇⣶⣲⣲⣶⣖⡆⣶⢲⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡛⢛⣛⣛⣛⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⡇⣿⢿⢿⣿⡿⡇⣿⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⠐⠖⠖⡀⡀⠶⠖⢸⠀⠀⠀⠀⠀⡀⡇⣩⣭⡍⣩⣭⡅⣿⢻⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣁⣈⣉⣉⣉⣁⣀⣀⣸⣝⠲⠶⠶⠚⠃⣇⣛⣒⣃⣙⣒⣃⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣈⡁⠢⠀⠀⠀⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠛⠂⢀⡼⠃⢀⠂⠀⢰⣄⠀⠑⠞⠁⠀⡀
⠀⠀⠀⠀⣠⠤⠼⠃⠀⠈⠘⠒⠢⢲⠀⠀⠀⣇⠀⠀⢀⡀⠀⠀⠸⠀⠀⠀⢀⣤⠊⢀⠔⠁⠀⠀⠀⣙⠗⡀⠀⠀⢎
⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⣀⣀⠼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢈⡧⠐⠁⠀⠀⠀⠀⠺⣅⡀⠀⠲⣄⠀⡣
⠀⠀⠀⠀⠙⠒⠂⢻⡆⠄⢸⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊

⠀⠀⠀⠀⢀⡤⣀⠀⢀⣴⠞⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⢀⡀⠀⠀⠀⢀⡖⠈⢡
⠀⠀⠀⠀⠘⢅⡡⢣⡿⠃⡀⡠⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⡀⠀⣶⠒⠚⠁⠘⠋⡩⠀⠀⠀⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⡔⠁⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠖⠀⡀⠸⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢀⣾⠎⠀⢁⠊⠀⡏⠍⡂⠀⠀⠀⠉⠂⠒⠒⠀⠀⠉⠈⠉⠁⠀⠀⠈⠒⠋⠙⠒⠁⠀⠀⠀⠀⠈⣓⣒
⠀⠀⠀⠀⠈⠣⢄⠒⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⣄⠀⠀⢠⡤⠄⣀
⠀⠀⠴⠟⠉⣀⠑⡄⠀⠀⢰⡆⠀⠀⠈⠛⣦⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⡠⠊⠀⠀⠈⠓⠠⢀⡀⠀⠂⢀
⠀⢐⡘⠀⢀⣾⠇⢸⠀⠀⣼⠃⠌⠀⠀⠀⣭⡀⢆⠀⠀⠀⠰⣿⠋⠀⠰⣋⡀⠀⠀⠀⠀⢀⣠⡶⠟⢁⠔⠁
⠀⠘⣧⠈⠚⠋⢀⠇⠀⠀⢰⡄⢸⠀⠀⠀⠉⡁⡆⠀⠀⠀⠀⠀⠉⠒⠤⣀⡈⠙⡲⠀⠰⡟⠁⢀⠔⠁
⠀⠀⠀⠑⠒⠊⠁⠀⠀⠀⠘⠧⢈⠀⠀⠻⢋⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠑⠚⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⢀⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠚⠀⠀⠀⠀⢀⠀⠀⠀⢰⠎⢇⣠
⠀⠀⢩⣈⣁⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠠⠬⠃⠀⠀⠀⠒⠲⣼⠀⠀⠀⠀⠀⠪⢉⡤⠆
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠒⠀⣰⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠠⠬⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⣤⢦⠈⠆⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⢿⠄⠠⠀⠀⠠⣤⣌⣋⢀⡆
⠀⠀⠀⠀⠂⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⣀⢉⡀⠐⠂⠀⠐⠂⠀⠂⠀⢠⣿⡛⢛⣼⠃
⠀⠀⠀⠀⡆⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⠂⠀⠈⠍⠉⠋⠀⠁⠀⠈⠉⢰⠟⠁⠈⠈⠿⠛⠁
⠀⠀⠀⠀⢀⡀⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⠛⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣀⣤⠤⠤⠤⣤⣀⡀⠀⠈⠻⣦
⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠋⠁⠀⠀⣀⣀⡶⠈⠙⢶⡄⠀⠈⢳⡀
⠀⠀⠀⢰⠏⠀⢀⡾⠁⠀⠀⠀⢠⡞⠉⠉⠳⡄⠀⠀⠹⡆⠀⠀⣷
⠀⠀⠀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢀⡏⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⣉⣉⣠⣤⣤⣤⣤⣤⣄⣉⣉⠙⠓⠦⣄
⠀⠀⠀⢿⡀⠀⠸⣆⠀⠀⠘⢷⣤⣤⠾⠃⠀⠀⠀⣠⡞⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⠖⢋⣉⣥⠀⠀⢀⣤⠶⠲⠶⣬⣭⡝⠲⡄⠈⢳
⠀⠀⠀⠘⣷⡀⠀⠈⠳⣤⣀⠚⠁⠀⠀⠀⣀⣤⠞⠉⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠘⠦⣍⣉⡛⠶⠶⠛⠁⠀⠀⠀⢋⣀⣤⠴⠃⠀⣸
⠀⠀⠀⠀⠘⣿⣦⣀⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⣀⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⣀⣠⣴⣾⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣟⠿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⣫⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡻⣿⣷⣮⣭⣽⣛⣛⣛⣛⣛⣛⣛⣯⣭⣵⣾⣿⢟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠒⠚⠛⠛⠒⠒⠶⠶⢤⣄⣸⣟⢿⣷⣯⣽⣛⣻⣿⠿⠿⠿⠿⠿⣿⣟⣛⣯⣽⣾⡿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠋⢁⣤⠴⠶⠒⠚⠛⠛⣛⣛⣒⡲⠶⢤⣄⠉⠛⢷⣮⣽⣛⡿⠿⠿⣿⣿⣿⣿⣿⡿⠿⢿⣛⣯⣵⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠰⣏⠠⠴⣾⣁⣀⣠⠞⠉⠉⠉⣽⠟⠓⢈⡷⠀⠈⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣵⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠈⠛⠲⠦⢭⣭⣥⣤⣤⣤⣤⠤⠶⠞⠋⠁⣠⣼⣿⢿⣿⣶⣿⣿⣭⣭⣭⣭⣭⣽⣿⣶⣿⣿⢿⣫⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⢟⣿⣷⣾⣭⣭⣿⣛⣛⣛⣛⣛⣿⣭⣭⣷⣾⡿⣟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⣯⣟⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣟⣫⣷⡿⣿⣶⣭⣽⠿⠟⠛⠛⠛⠿⢿⣿⣛⣯⣭⣷⣾⠿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣝⡻⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⣟⣭⣾⣿⠟⠉⠀⠀⢀⣀⣀⣀⡀⠀⠈⠙⠻⣿⣭⣷⣿⢿
⠀⠀⠀⠀⢀⣀⣠⣤⣤⡤⠤⠤⢤⣤⣤⣄⣀⡀⢸⣷⣽⣻⢿⣿⣷⣶⣿⣭⣭⣭⣭⣭⣭⣿⣶⣾⣿⠿⣿⣿⠃⠀⢀⡴⠞⠉⠉⠀⠉⠉⠛⠶⣄⠀⠈⠻⣯⣶⣿
⠀⣠⠴⠛⢉⣁⣠⣤⡤⠤⠴⠴⠤⠤⣤⣄⣈⠙⠛⢿⣿⣿⣿⣶⣯⣭⣽⣿⣛⣛⣛⣻⣿⣯⣭⣷⣾⣿⣿⠇⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠹⣯⣾
⣾⠁⠀⡶⢋⣁⣰⠖⠀⠀⣠⠖⠛⠛⢶⡤⠬⠙⢦⠀⠈⢷⣯⣽⣛⣿⡿⠿⠿⠿⠿⠿⠿⢿⣟⣻⣽⣿⣿⠀⠀⣼⠀⠀⣰⠆⠀⠀⠀⣀⣤⣤⡀⠀⢹⡄⠀⢹⡟
⣿⣄⠀⠙⠦⣭⣉⣛⣛⡛⠁⠀⢀⣀⣉⣠⣤⠶⠋⠀⣠⣿⣿⣻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣻⣿⣿⠀⠀⢿⠈⠛⢻⡄⠀⠀⣴⠏⠀⠈⣷⣄⡀⣧⠀⠀⣧
⣿⣿⣷⣦⣤⣀⣀⠉⠉⠉⠉⠉⠉⠉⠉⣀⣀⣤⣴⣾⣿⣿⣛⡿⢿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⡿⢿⣿⣿⣇⠀⠘⣇⠀⠀⠛⠶⠞⠁⠀⠀⢰⠏⠈⠀⡟⠀⠀⣿
⣿⣮⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣵⣿⣿⣿⣿⣶⣶⣿⣯⣭⣭⣿⣷⣶⣾⣿⣿⣿⣿⣿⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⡇
⣿⣝⡻⣿⣷⣶⣯⣭⣭⣿⣿⣿⣯⣭⣭⣽⣶⣾⣿⢟⣯⣿⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠹⣿⣿⣦⡀⠀⠙⢦⣄⣀⠀⠀⠀⣀⣤⠞⠁⠀⢠⡟
⠙⢿⣿⣷⣾⣭⣿⣛⣛⣛⣛⣛⣛⣛⣛⣿⣭⣷⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⠀⠉⠙⠛⠉⠉⠀⠀⢀⡴⠋
⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣦⣤⣤⣤⣤⣴⠞⠋

⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠲⠦⠀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠁⠀⣄⠀⢶⡄⠀⠀⠞⠐⠂⠀⠒⠠⠑⠀⠀⠀⠈⠀⢰⡀⠰⣦⠀⠀⠀⠐⠈⡤
⠀⠀⠀⠰⣶⠆⠀⠀⠻⠆⠸⠃⠀⠀⢀⠂⠀⠀⠀⠀⡀⡀⢴⡾⠀⠀⠘⠿⠀⠟⠀⢀⠀⢊⣄⢀⡀
⠀⠀⠀⠀⠀⠀⣶⡆⣶⢳⡆⠀⠀⠀⢸⠀⠀⠀⠀⠐⠉⠑⠀⠀⢰⣲⢠⡞⣶⠀⠀⠈⠀⠙⢿⠙⠇
⠀⠀⠀⠀⠀⠀⣠⠄⠈⠉⠀⠀⠀⠀⣴⡄⠀⠀⠀⠐⠱⠄⠀⠀⢀⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⢉⠁⢀⣠⠀⠀⠀⠀⢠⡀⠀⠠⠔⠀⠀⠀⠀⡀⠈⡉⠀⣀⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠌⠯⠝⠓⠚⡊⠀⢱⠀⠀⠀⠀⣰⢶⡖⠁⠁⠀⠀⡹⠭⠋⠒⠳⠁⠈⡆⠀⠀⣠⡴⠀⢀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠀⠁⠁
⠀⠀⠀⠀⢀⡴⢶⣤⣤⣰⡦⢄⠀⠀⠀⣀⣠⣖⣐⡦⡀⠀⢀⡴⢦⣤⣄⢤⠤⡀⠀⢀⠄⣀⣴⡄
⠀⠀⠀⢠⣿⠛⢻⡋⠿⣿⡅⠞⢣⣠⣼⠛⡽⠏⡙⢳⣇⢠⣿⠛⢻⠉⠿⣿⡑⡞⠆⠂⢠⣲⣛⢧⣄
⠀⠀⠀⠸⠭⠀⠘⣏⣢⣹⡷⢚⡟⣿⡿⠄⠈⠒⢓⠍⣿⠨⠩⠀⠸⡋⣄⣿⡧⢓⠇⣠⡾⢻⣷⢱⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠋⡌⠛⠉⠀⣿⠆⠀⠀⠀⡃⣶⣿⢱⠀⠀⠀⢸⢫⡍⡟⠋⠀⠉⠓⣘⣿⡗⡖⢸
⠀⠀⠀⠀⡠⣖⠉⢦⡔⠀⠁⠀⠀⣿⠛⣶⠀⣀⣑⣴⣾⠁⡠⡖⡒⣦⡕⠂⠀⠀⠀⠀⠈⢓⣸⣋⡴⠋
⠀⠀⠀⠀⢷⣴⡷⢛⠇⢀⣀⡀⠀⢿⣟⠋⢸⡑⣷⣽⣿⠀⢧⣼⡖⠻⠇⢀⣀⠀⠀⠀⣾⣽⣿⡇
⠀⠀⠀⠀⠸⣽⣶⣦⣤⣿⣿⣿⠀⠈⠿⣶⢊⡑⣶⡷⠃⠀⢸⣻⣵⣧⣤⣿⣿⣷⠀⢀⡞⣢⣿⣷⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠉⠉⠉⠉⠉
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⢹⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣬⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣝⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣦⣭⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⣜⣸⣿⣍⣉⣉⣉⣉⣉⣩⣭⣽⣿⣿⣿⣯⣩⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢻⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⢱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠮⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠿⢸⣿⣭⣭⣭⣭⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⢿⣿⡿⠿⠿⣿⣿⠿⠿⣿⣿⡿⠿⢿⣿⡇⣿⣿⣛⣛⣛⣛⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠾⠿⠆⣿⡰⠿⠷⢸⡇⠿⠿⢸⣿⠸⠿⠇⣿⡇⣿⣿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢛⣛⡻⣿⠟⣛⣛⢿⡟⣛⣛⢻⣿⢛⣛⡛⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣛⣛⣃⣿⣜⣛⣛⣸⣇⣛⣛⣸⣿⢸⣿⡇⣿⡇⣿⣿⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣩⣭⡙⣿⢋⣭⣍⢻⡟⣭⣭⢻⣿⢸⣿⡇⣿⡇⠻⠿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣭⣭⣥⣿⣮⣭⣭⣼⣧⣭⣭⣼⣿⣬⣭⣥⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠃
⠀⠀⠀⠀⠀⠀⣤⠀⢠⠀⠀⠀⢀⠚⠃⠸⠀⠀⠀⠀⡶⠈⢆⠀⠀⠀⠀⣤⠤⠇⠀⠀⠀⠒⢀⡇⢀⣛⣀⠇
⠀⠀⠀⠀⠀⢄⣁⣠⠇⠀⠀⠀⠈⠉⠃⢸⠀⠀⠀⠐⠠⠔⠃⠀⠀⠀⠤⠤⠀⢳⠀⠀⠀⣤⠌⢀⠀⠉⠁
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠲⠖⠃⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠠⢤⠴⠃⠀⠀⠠⣀⡠⠌⠀⠀⠐
⠀⠀⣀⣉⠀⣉⡄⠀⠀⣤⠈⡷⠀⠀⠀⠈⣡⠃⠀⠀⠐⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠸
⠀⠀⠀⠈⠲⠯⠀⠀⠈⢁⠆⠀⡦⢄⠉⢁⠄⠀⠀⠉⠙⣀⣶⠛⠀⠀⠀⣀⣀⠀⢰⠀⠀⠀⠀⠀⢀⠃⠈
⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠒⠚⠁⠀⠉⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠁⠀⠀⠀⠀⠐⠞
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠦⠁⠀⠀⠀⠐⠾⠀⡇⠀⠀⠀⠀⣠⡵⠀⠀⠠⠰⡤⡷⠀⠄⣴⣄⢰⠀⡄⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠒⠗⡠⠇⣄⠀⣀⠀⢀⠄⢤⡇⠀⠀⠀⠰⠆⢀⠀⡄⠀⠐⢦⠀⠀⠀⠃⠰⠞⠀⢳⡶⢌⣷
⠀⠀⠀⠀⠠⠀⠀⠠⠤⠖⠋⠀⠈⢠⠋⠀⠀⠀⠀⠀⠈⠐⠒⠋⠀⠑⠟⣁⠟⠀⠀⠉⠙⣉⠀⡀⠙⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠤⠛⠀⠀⠢⢤⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠰⠁

⠀⠰⣤⣀⣀⣀⣤⠾⠃⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠘⡉⠀⠀⠀⢀⡀⠀⠀⠈⠙⠛⠛⢋⣡⠖⠀⠐⠞⠛⠷⣦⠀⠀⢀⣀⣀⣀⠚⠋⠁
⠀⢀⣀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢠⡶⠟⠁⠀⠀⠀⠀⠒⠚⠉⠁⠀⠲⣤⣤⣀⣠⣤⠶⠂⠀⠉⠉⣉⠙⠛⠛
⠟⠛⠉⠁⣠⠄⠀⠀⢀⣠⣤⠆⠀⠐⠛⠋⠉⠉⠉⠀⠀⠰⠶⠶⠶⠶⠀⠰⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣬⡥
⠀⣀⣴⠾⠋⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠚⠶⠶⠂⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠐
⠛⠋⠀⠀⠀⠀⠠⡾⠛⠛⠷⠀⢀⠀⠀⣠⡶⠶⠶⠶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⢈⣁⡀⠀⠀⠀⠀⠀⠀⠈⠛⣀⠀⢀⣤⡀⢶
⠀⠀⠀⠀⠀⢤⣀⡀⢀⣀⣠⡴⠋⠀⠀⠀⣀⣀⣀⡀⠒⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠛⢷⠀⠀⠀⠻⠛⠛⠛⠛⣃⣀⣡⣤⡌
⠿⠶⠀⢀⡀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⣀⠀⠐⠟⠛⠻⠀⠀⠀⠀⠀⢶⣄⠀⣀⣀⣤⠴⠁⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠰⠟⠀⣀⣤⠶⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠐⠶⠛⠛⢿⣆
⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⢤⣤⣴⠶⠶⠶⠶⠆⠀⠉⠀⠀⠀⠀⠀⠁⠀⡠⢤⡀⠀⠀⠀⠀⠀⠻
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠋⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠠⣀⣀⡁
⣀⣀⣠⣤⣤⡴⠶⠶⠛⠛⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣰⠿⣧
⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠟⠛⠁⢠⠇⠀⠀⠀⡤⠔⠁⠀⠀⠀⠰⠇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠟⠿⣶⣄
⠀⢀⡀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣶⣶⣦⠀⠀⠿⠷⠀⢠⠀⠀⠀⠀⠀⠀⢻
⡶⠛⠁⠀⢀⣰⠏⠀⠀⠀⠉⠀⠀⣠⠀⠀⢠⣤⡴⠞⠋⠀⠀⠀⢠⡶⠛⠛⠻⣦⠀⠀⠀⠀⠀⢀⣀⣉⣀⣀⣀⣠⣄⠸⣧⡀
⠀⠀⡠⠀⠈⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠈⠙⠷⠶⣴⠶⠾
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⠀⠀⠀⣀⣤⡶⢶⣦⡀⠀⠀⠀⠀⠀⠤⠴⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⠀⠀⣀⣠
⠀⠀⠀⡠⠊⠀⠀⠈⠉⠀⢀⣀⣀⣠⣤⣤⠀⠀⠀⠠⠤⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠀⠠⠀⠀⠀⠀⢳⡀⠀⠀⠉⠛⠋⠉
⠀⠀⠜⠁⢠⡴⠶⠶⠛⠛⠛⠉⠉⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣀⣤⠞⠁⠀⠃⠠⠀⠀⠀⠀⣀⣁⣠⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣠⣶⣤⣄⠀⠀⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠠
⠶⠶⠶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠁⠀⠀⠀⡟⠁⠀⠀⠙⣧⠀⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⣠⡾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⢶⣤⡀⠀⠀⠀⠀⠀⣧⠀⢸⠟⠛⠛⠋⠉⢹⡟
⠀⠀⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠈⠓⠶⠶⣶⠆⠸⠏⠁⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⢹⣀⡀⠀⠀⠀⠀⠀⠸
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⣀⣀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁⠀⠀⠚⠛⠳⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠛⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣿
⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⣴⠟⠛⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢆⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠘⡇⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣀⣀⣀⡀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢀⠇⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣄⠊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⣴⣏⠀⠀⠀⠀⠀⠀⠙⠲⡿⠛
⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣆⣀
⣿⠿⡿⢿⠿⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⠛⣿⣿⡟⢿⡿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⠻⣿⣿⣿⡇⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠟⠿⠿⢿⡇⣿⡇⣿⠿⠿⠃⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⡇⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢩⣤⢹⣿⡇⣿⡇⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⠈⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⡇⢿⠇⣿⣿⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⠟⣈⢿⣿⣿⣿⣿⣿⣿⣘⠛⣸⣿⣿⡆⣾⣿⣿⣿⡅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣧⣿⣿⣿⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⢀⣿⣿⡏⢿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⢰⣿⣿⡏⣴⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢸⣿⣿⣧⡍⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣴⣿⣸⣿⣿⣿⣷⣿⣿⠿⠿⠛⠛⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⠾⢛⣛⣛⡛⢷⣦⡙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣯⠸⢿⣿⣿⠿⢂⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⠟⣋⣭⣤⣤⣬⣭⣁⠠⣙⠛⠶⠶⠶⠶⢛⣩⡄⠸⠟⢛⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣾⠏⣵⣶⣶⣶⣬⢻⣷⠘⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉⣉⣉⣿⢀⠻⣷⣌⣙⣛⣋⣡⡾⠟⠀⣬⣉⣉⣉⣩⣥⢀⣿⣿⣿⢡⣤⣌⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣶⣬⣭⣭⣭⣥⣶⣿⠀⠿⢿⣿⣿⡿⠿⠸⣿⣿⣿⠲⣶⠌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣌⡛⠿⠿⠿⠿⠿⢛⣡⠀⣷⣶⣶⣶⣶⣿⣧⠹⣿⣿⣷⠶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣶⣶⣶⣿⣿⠟⣠⣙⣛⡛⣛⣛⣉⣵⣶⣮⣙⣛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠈⠉⠛⠛⠛⠉⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⣶⣲⠀⡴⡆⢰⠒⣦⣰⢰⣆⣰⢠⠖⡂⠀⢠⢶⢀⡶⠒⣰⠒⢠⠖⢦⢰⡆⣴⢰⣆⣰⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠟⠹⠘⠋⠻⠸⠶⠋⠸⠸⠋⠿⠘⠦⠟⠀⠞⠙⠎⠳⠄⠙⠶⠘⠧⠞⠘⠧⠟⠸⠈⠿⠀⠇


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

⠀⣄⠀⢦⠀⠀⡤⡄⢀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠂⠒⠒⠒⠒⠒⠂⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⠀⠀⠀⠀⡀
⠸⣼⠁⠸⡇⠀⢰⠆⠈⠃⠂⠀⠆⡄⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡆⣠⡾⠂⠀⠁⠀⠰⠁⣼⡿⠁⠄⢉⣼⠟
⠀⢿⠀⠀⠰⠦⠀⠀⠆⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⡟⣡⣤⡄⠀⡐⠀⠀⡿⠀⢠⠚⣿⠏
⠀⢀⣠⠞⠀⠀⠀⡄⣠⠞⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠙⠊⠁⡐⠁⣠⢀⠁⠀⠸⠛⣉⡴⡆
⢀⡏⠀⠀⠀⠀⠀⡯⠁⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠰⣀⡀⠀⠻⣄⣼
⠈⠁⠀⠀⠀⠀⡘⠁⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠦⣤⠆⣈⡱⠀⠶⠘⠉⠀⠀⠈⠀⠢⠀⢿⠿
⠀⢦⣄⣄⣀⣤⣤⡶⠾⠋⠁⠀⠂⠒⠀⠈⠈⣈⣴⡀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠈⣉⣭⣍⠀⠀⠀⠀⠠⠠⠇⠀⠀⢀⠈⢿⣀⡇⠀⢀⠀⠤⠄⠂⠂⠁⠀⠈⠃⠀⠀⠀⠀⠲⠶⠶⠶⠆⢠⡀⢆⠀⠀⠁
⢠⣶⣿⣿⡝⣿⢦⡀⠡⣤⠠⠄⢐⣈⠀⠐⠟⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠧⠄⠀⠀⠀⠀⠀⠀⠀⠨⢻⡄⠀⡀
⠀⢿⣷⣻⣿⡈⢻⣿⣖⠬⣑⣚⠿⠛⠀⠀⠀⠀⣷⣶⠀⠀⣀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁
⠀⠈⣿⣷⢿⣧⠀⢙⠋⢲⠈⠀⠀⢠⣦⠀⠀⠀⠉⠀⠀⠀⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⠟⠊⠩⣂⣥⣾⠈⠀⠀⠀⣞⣘⡇⢀⡀⢀⡄⢀⣤⡀⡀⢠⣠⣠⡔⢻⢳⢠⣠⡄⣠⢦⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⠰
⠀⠀⡀⣀⣴⣾⣿⣿⣿⢸⠀⠀⠸⠋⠉⠹⠘⠴⠛⠶⠘⠶⠋⠳⠞⠘⠇⠻⠸⠘⠺⠃⠟⠳⣾⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⡄⣿⣿⣿⣿⣿⠟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠾⠵⠃⠀⠀⠀⣀⡀⠀⠐⣤⠌⡞
⠀⠀⠀⣿⣿⠟⠋⠀⠀⠉⡱⠂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠦⠤⠃⠀⢀⡔⠀⠈⢂⠀⠀⠊⠀⠀⡴⠁⠀⠐⣈⠤⠲⡇
⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢀⡀⠀⡀⢢⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⠀⠀⢘⡀⠀⠀⠀⡀⠀⠀⢠⠘⢀⡠⠔⣁⢀
⠀⠀⠁⠬⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣊⡡⢦⡼⢻⣥⣀⡀⠀⠈⠀⠚⠋⠀⠈⠓⠤⠔⠚⠁⠀⠀⠀⠣⠀⢐⠊⢠⡀⡀⠀⢀⣴
⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⡤⠖⢉⠈⠁⠀⠀⠁⢾⣿⣿⡏⠊⠓⠒⢲⡆⠀⢠⡀⠀⠀⠀⡿⠈⠀⠀⠀⡄⠈⠉⠉⡀⠰⣾⣿⣿
⠀⠠⠆⠘⠀⢀⠀⠄⠚⠉⠀⠀⠀⠁⠀⢶⣄⠀⠸⣿⣷⣗⢄⣔⢲⣾⣟⡀⣼⡟⠀⢀⣼⣿⠀⠀⠀⠀⠔⠀⠀⠂⠀⢠⣿⠛⠿
⠀⠀⠀⣠⡴⠏⠃⠀⠀⠀⠈⠀⠄⠢⠀⠀⢹⠷⡄⣿⣿⣮⢿⣻⣿⡫⣞⢳⣾⡇⠀⢮⣨⠏⠐⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠋⠁⣺⡶⠆⠀⠀⠀⠀⠀⠈⢻⣦⡀⠓⠃⢹⡿⡁⡀⠀⠉⠙⠋⡛⠇⠑⠀⠘⠁⢠⠀⠀⠀⠀⣀⠀⠀⠀⡿⠋⣤⡤⣶
⠀⠀⠀⠀⠸⣉⣧⢤⣄⠀⠀⠀⠀⢀⣀⣀⣉⡁⣀⢘⢙⡂⡑⣀⣈⣉⣉⡀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⣤⡞
⠀⠀⠀⠀⠀⣤⣹⡤⠋⠀⠀⠀⠄⢸⠐⡇⢸⠀⡇⢰⠂⡟⢰⡆⢸⡇⠀⡇⢸⣿⡆⠀⠀⠸⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⠿⡿
⠀⠀⠀⠀⠀⠈⠨⠁⠀⠀⠐⠀⠀⠸⠶⠷⠞⠐⠳⠼⠦⠗⠘⠧⠘⠦⠴⠇⠈⠉⠀⠀⠐⠀⠀⠀⠰⠂⠀⠀⠀⠀⠠⠿⠟

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠾⣿⡇⠀⠀⠀⠀⠀⠸⠋⠉⠙⣿⡆⠀⠀⠀⠋⠀⠈⣿⠆⠀⠀⠀⠀⠀⣼⢻⣿⡅⠀⠀⠀⠀⣸⠉⠉⠉⠋
⠀⠀⢀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣁⡼⠟⢡⡀⠀⠀⠉⠳⠺⣧⣄⠀⠀⠀⠀⡼⢋⢸⣿⠀⣤⠀⠀⠀⠛⠛⠒⢲⢦⡠⠄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣶⣷⣟⠁⠀⢂⡄⠃⠀⢀⡄⠀⠄⢸⡍⡄⠀⠀⢘⣷⡿⢺⣿⡟⠃⠀⠀⢀⣤⠀⠀⢀⣯⠏
⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠘⠙⠛⠛⠛⠛⠃⠀⠀⠈⠉⠒⠒⠉⠐⠇⠀⠀⠈⠉⠉⠚⠛⠛⠁⠀⠀⠈⠉⠓⠒⠛⠁

⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣠⣠⣀⣄⡤⠀⠀⠀⠀⢀⡀⢤⣀⠀⠀⠀⠀⠀⣀⡠⢄⡀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡀
⠀⢠⣿⣟⣀⣈⠧⠄⠀⠀⠀⠁⠀⢀⡼⠁⠀⠀⠀⣀⣿⣤⣀⣿⠆⠀⠀⠀⡼⡏⠀⠂⢻⢆⠀⠀⠀⢄⣾⡇⠀⠘⣿⡄
⠀⢸⣿⣯⣄⠙⣷⡄⠀⠀⠀⣀⣀⣾⣡⠀⠀⠀⠀⢹⣾⢟⢿⣿⣄⠀⠈⢋⠻⣧⣀⣀⣼⣽⠀⠀⠀⢸⣿⡠⣤⣠⣿⡇
⠠⣼⣿⡅⠔⢀⡿⠇⠀⠀⠀⠦⣿⣿⣁⠀⠀⠀⢀⣿⣇⠉⠄⣿⣿⡁⠀⠈⠀⣄⠉⢹⣿⣟⠿⠁⠀⠈⢿⣇⠨⢹⡿⢛⠓
⠀⠣⠈⠉⠉⠉⠑⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠄⠉⠙⠉⠉⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⢀⣠⣴⣦⣤⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⡿⠛⠀⠙⣿⣦⣾⣶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠉⠀⠀⠀⣐⡬⠞⠉⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡄⠀⠀⠐⣯⠶⣚⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⣠⣄⠾⠟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠘⢦⠀⢸⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡶⢶⣶⣶⣭⣳⣄⠀⠀⠀⠀⠘⢿⣷⣀⣈⣷⣬⣏⣉
⠀⠀⣀⣀⡀⢤⣀⡤⠴⠛⠋⠉⠀⠀⢀⣠⢬⡙⠻⣿⣧⡀⠀⠀⠀⠀⣻⣻⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⣿⣿⣿⣿⠀⣸⠁⠀⠀⠀⠀⣀⣴⢶⣏⣴⣶⣽⣒⣼⣿⣿⣤⡀⠀⢀⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⣿⣿⣿⡏⠀⢯⠀⠀⠀⠀⠙⢩⣼⣯⡻⢶⣾⣿⣿⣮⣿⣿⣋⣿⣆⠸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⣶⣶⣤⡀
⣿⣿⣿⣿⠀⢸⡄⣠⣤⣤⡀⠈⠻⠿⠿⣿⣿⡏⣴⢾⣿⣿⣿⣿⣿⣆⠈⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⡿⣻⣿⣿⣿⣶⣶
⣿⣿⣿⣿⣆⠀⠹⣿⠉⠙⠛⠻⠾⣶⣶⣿⣝⡿⢿⣿⣿⣿⣿⣿⡿⣿⠀⠈⠩⣿⢯⣾⠟⣫⣭⣭⣴⠓⠈⢩⣋⠛⣏⢰⣿⡟⣰⣿⣿⣿⣿⣿⣿
⣿⣷⠉⠻⣿⣖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠃⣼⣿⣿⣿⢟⠀⡹⠀⠀⠀⠈⠻⣷⣬⣿⣿⠟⢌⢧⠀⠀⠀⠉⢻⠘⣿⣇⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣦⣄⡙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢟⣵⣟⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠈⡆⢣⠀⠀⠀⠈⠣⣹⣿⣿⠿⡿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣄⡙⠢⣄⠀⠀⠀⠀⠀⢠⣸⡿⣟⡵⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣠⠇⠀⠀⢀⣀⣨⣽⣿⣶⣤⣌⣈⡈⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠋⠳⢦⣀⠀⠀⠘⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⠞⠉⠛⠲⢤⣤⣴⣶⣶⣶⣶⣶⣤⣄⣀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣷⡙⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⡴⠃⠀⠀⠀⢠⣿⣿⣿⢿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠿⣿⣿⣿⣿⡃⠀⣠⢾⣴⣶⡀⠀⠀⠘⢻⣿⣿⡿⠋⠉⠀⢈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡏⠀⠸⡿⢻⠻⠿⠿⠿⠛⠻⠿⠻⠻⢿
⠀⠀⠀⠉⠛⠃⠈⠛⠁⠀⠈⣿⣧⡀⠀⠹⣯⡿⠏⠀⠀⠀⠀⠀⠙⢶⣦⣤⣿⣿⣿⣏⣭⠭⠶⠶⠟⠛⢧⡀⣀⡷⠦⠤⠤⠤⠤⠤⠤⠶⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡈⢻⡛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣆⠀⠀⠀⠀⢄⣠⡈⠻⢟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣠⣄⠀⣒⣉⣻⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣴⠛⣷⠀⣰⡿⠀⠰⠟⣿⠀⠀⢶⣿⠀⠀⣼⠃⣾⠁⣼⠋⠉⠁⣴⠋⠁⠈⢛⣿⠋⠰⣟⣽⠇⢰⡟⢹⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⣰⠏⠀⣽⠃⢀⣤⣾⣡⡶⠀⠛⢉⣷⠘⠛⣿⠃⠀⠋⣹⡇⢸⡿⣻⡇⠐⣾⠛⠀⣴⠟⣳⠀⠘⢛⡿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠈⠁⠀⠉⠀⠀⠉⠉⠁⠀⠀⠉⠀⠉⠉⠉⠀⠀⠉⠉⠀⠀⠃⠀⠀⠈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⠄⠙⡆⠀⠀⠘⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⣄⠙⣦⠀⠀⠀⠠
⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠈⡆⠀⠀⡈⠓⠀⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢄⣈⣁⠀⢸⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠣⡄⠀⠉⠛⠛⠊⠀⠀⠀⠀⠀⠠⣤⣤⠄⢨⡇⠀⠀⠀⠀⠀⠠⣠⣶⠆⢈⡇⠀⠀⠀⠀⠈⠉⠁⠀⣿⠀⠀⠀⣀⡀⠲⡄
⠀⠀⠀⠀⠀⠀⢦⣤⣤⠀⠠⣤⡷⠀⢀⡁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢥⡀⠙⣅⠀⠀⠀⠙⢶⢀⡄⡢⠃⠀⠀⠓⠿⠀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣄⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⡠⡎⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠨⡄
⠀⠀⠀⠀⢠⣄⠀⢱⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠀⠈⠉⠁⠀⢰⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⡀⢰⡦⠈⣳⠀⠀⠀⠀⠀⢰⡄⠲⠶⠲⠖⠿
⠀⠀⠀⠈⢻⣉⠐⢾⠃⠀⠀⠀⠀⠀⠀⢀⡔⡆⠀⠇⡄⠀⠀⠃⠀⠀⠸⠟⠫⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⡀⠰⡏⠀⠀⠠⠟⠻⢛⠃⠀⠀⠄
⠀⠀⠀⢀⠀⠙⠀⡸⠇⠀⠀⠀⠆⠀⠀⠀⠉⠁⠀⢹⠇⠀⠀⠀⠀⠐⣶⠷⠀⠽⠀⠀⠀⠀⣠⡀⠱⡄⠀⠀⠸⠟⠀⢸⠀⠹⢝⢻⠆⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠛⠓⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣤⣿⡇⠀⠀⠀⠀⠐⢤⣦⡺⠂⠀⠀⠑⠾⠟⠁⢰⠇⠀⠐⠶⠔⠄⠊⠀⠐⣤⢤⢤⡺⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠼⠋⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢹⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⣉⠉⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠘⢦⠀⠀⠀⠰⢤⣤⣴⡴⠻⠀⠈⠛⠛⢀⠒⠒⠲⠀⠀⢠⠎⠴⡿⠇⠸⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠉⢠⢔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⢴⡕⠀⠀⢀⠀⠀⠀⢈⠥⠴⠶⠆⠀⠠⠆⠴⠄⠰⠇⠀⠸⠿⢶⣶⡄⢰⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠁⠀⢤⣀⡄⢀⣽⠀⠀⠀⠀⠀⢀⡤⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠖⠀⠀⠀⠒⠆⢴⡆⠀⠤⠤⠀⠉⠻⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⢠⠻⠋⠀⠀⠀⠀⢄⠀⠁⠀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⢸⡆⠀⠀⠀⠳⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠙⠪⠯⠚⠀⠀⠀⠀⠀⠀⠀⠑⠦⣲⡊⠝⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⡇⠀⠀⠀⠀⠈⢯⡉⠉⠹⣤⡞⠉⠉⡽⠃⠀⠀⠀⠀⠀⢾⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠿⠀⠀⠷⠶⠶⣆⠀⠀⠀⠻⣄⠀⠉⠀⢠⡞⠁⠀⠀⢀⡤⠤⠤⠬⠿⠿⠥⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⠀⠀⣀⣀⣀⡿⠀⠀⠀⢀⡼⠀⠀⠀⠻⡄⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏⠀⠀⠀⠀⠀⣠⠟⠀⢀⡶⡄⠀⠙⣦⠀⠀⠀⠀⠀⠀⡴⠋⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠒⠃⠀⠀⠀⠀⠰⠧⠤⠴⠏⠀⠙⠦⠤⠼⠗⠀⠀⠀⠀⠀⠙⠦⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠉⠉⠉⣿⠀⠀⡞⠛⠛⠛⠛⠛⠛⠛⠛⡆⠀⡏⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢷⣤⣤⣤⣤⣤⣤⣤⣴⠇⠀⡇⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⢠⡴⠶⢦⡄⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⡇⠀⡇⠀⢀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡼⠀⢸⣇⡀⢸⡇⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠷⠤⠇⣰⠉⣤⡈⢳⠀⡼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠿⠃⣸⡾⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⣿⠀⠈⠓⠒⠒⡿⢁⣾⠞⣛⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠉⡇⠀⢸⡏⠙⠲⠦⣄⣀⠀⠀⠀⠀⢸⠃⢠⡇⠀⠀⠀⠀⡼⢁⡞⣯⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⢁⣀⡤⠶⠚⠁⠀⠈⠛⠲⢤⣄⡀⠉⠙⢶⠀⠀⡟⠀⣸⠁⠀⠀⠀⢼⣥⠞⠀⠙⢦⣭⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠈⠙⠳⠦⣤⡀⠀⠀⣠⡴⠖⠛⠉⢀⣠⡼⠀⢸⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⣀⡇⠀⢸⡇⣀⡤⠶⠛⠉⠀⠀⠀⡿⠀⣼⠁⠀⠀⠀⠀⠀⡖⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⠇⢀⡏⠀⠀⠀⠀⠀⠀⠧⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠀⢠⢶⠆⠀⠀⠀⠀⡠⠔⢦⠀⠀⢀⠶⡆⠀⠀⠀⠀⠀⡤⢒⠲⢄⠀⠀⣠⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⡇⠀⣰⢣⠎⠀⠀⠀⠀⠀⠙⡇⢸⠀⢠⢎⠞⠀⠀⠀⠀⠀⠀⢩⡖⠃⣜⠀⡰⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡰⢣⢋⡴⠋⡇⠀⠀⠀⠀⡇⢸⢠⢋⢾⣊⣍⠓⡄⠀⠀⢀⡤⠭⠗⣸⡞⡱⣣⠎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡽⣱⢣⠎⣼⠇⢧⡀⠀⠀⠀⠙⣹⢣⠎⠀⢀⡼⣡⠇⠀⠀⠀⠉⠒⢋⡝⣴⡟⢡⡿⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡱⠁⠙⠒⠒⣆⡞⠃⠀⠀⠀⣰⣡⠇⠀⢰⣋⣘⣓⡦⠀⠀⠀⠀⢀⣞⡼⠉⠓⠒⢲⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠄⢀⢀⣀⢀⠀⠀⣀⠀⢀⠀⡠⠀⣀⠀⡀⠀⣀⢀⣀⡀⠀⡆⢄⣀⣀⡀⠀⣀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠉⠚⠉⠀⠑⠀⠁⠀⠉⠸⠀⠑⠀⠁⠊⠃⠏⠑⠈⠁⠘⠀⠃⠊⠁⠀⠁⠉⠠⠋⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣿⡇⠀⠠⢆⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⡠⢚⣭⠀⠀⣯⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢱⠁⢸⣿⠀⠀⣿⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⢸⡀⠸⠿⠀⠀⠿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⢞⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⢈⣵⢟⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣛⣛⣛⣛⣛⣛⣛⣛⢻⠀⠀⠀⠈⠁⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣛⣛⣛⣛⣛⣓⣒⣛⣸⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣭⡍⣭⡍⢭⡍⢩⣯⢹⠀⣠⣴⢶⣤⡀⠀⠀⠙⢿⣿⣿⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣽⡅⣼⣧⣼⣧⢸⣧⠘⣼⣿⠰⠶⢿⢿⣀⣀⡀⠀⠙⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣶⡆⣶⡦⣴⣦⢺⣿⢰⠹⣿⣿⡋⣰⣿⢋⣙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⠶⠶⠶⠶⣶⣶⣾⠀⠈⠉⠁⣿⣿⠾⠯⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠀⠀⠀⠀⠈⠻⠷⠾⠟⠁




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

⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠸⢿⣿⡇⠀⠀⠀⠀⠀⣴⣿⡿⠿⣿⣷⡄⠀⠀⠀⠸⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠘⢿⣿⣏⠀⠀⠀⠀⠀⣰⣿⡿⠀⢠⣤⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡧⠀⠀⠀⠀⣿⣿⣧⣤⣼⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣯⣥⣤⡄⠀⠀⠀⢿⣿⣦⣤⣼⣿⡗⠀⠀⠀⠀⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠘⠛⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣷⡀
⠀⢸⣿⣷⣴⣶⣤⡀⠀⠀⠀⠀⣰⣿⡿⣵⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⣸⣿⣷⣿⣿⡃
⠀⠘⠛⠛⠛⠙⢿⣿⡆⠀⠀⢠⣿⡿⠉⠙⢿⣿⡆⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⢰⣿⡿⠋⠙⢿⣿⡀
⠀⠐⣶⣶⣄⣠⣾⣿⠇⠀⠀⠸⣿⣷⣄⣠⣾⣿⠇⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠸⣿⣷⣄⣠⣾⣿⠃
⠀⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠛⢿⣿⣆⠀⠀⠀⣼⣿⡟⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣸⣿⡟⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣽⣿⡿⠀⠀⠀⠀⣿⣿⡀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠹⣿⣷⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢒⣒⣒⣒⣒⣒⣒⣒⣒⠂⠀⠀⠀⠈⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣩⣭⣭⣭⣭⣭⣭⣭⣭⡁⢰⠶⡗⠒⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⢈⣹⣏⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⢛⣛⣛⣛⣛⣛⠛⣷⠀⠀⠈⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡉⠉⠉⠉⢹⡇⣿⠐⠒⠒⠒⠒⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢈⣉⠉⣉⠉⣉⡁⣿⢈⣉⣉⣉⣉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠤⠀⠤⠀⠤⠄⣿⠠⠤⠤⠤⠤⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠐⡒⠀⡒⠀⢒⠂⣿⠐⡒⠒⠒⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣬⣭⣤⣭⣤⣭⣥⡿⢈⣭⣭⣭⣍⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣠⡄⠈⠻⣦⠀⢀⣾⠋⢁⡤⢤⡈⠙⣷⡀⠀⣴⠟⠁⡤⠤⡄⠙⢷⡄
⠀⠀⠀⠀⠀⠀⢼⠁⠀⠈⢹⡇⠀⠀⢹⡆⢸⡇⠀⠈⠀⣠⠇⠀⢸⡇⢰⡏⠀⠀⠁⠴⣏⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢸⠇⠀⢀⣾⠁⠘⣧⠀⢠⣞⣁⡀⠀⣼⠃⠈⢷⡀⠐⢦⣀⠟⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡶⠟⠁⠀⠀⠘⠳⣦⣤⣤⣴⠞⠁⠀⠀⠈⠻⢶⣤⣤⣴⠾⠋


⠀⠀⢀⣤⠶⠶⠶⣦⣄⠀⠀⠀⢀⣠⡴⠶⠶⢦⣄⠀⠀⠀⠀⣠⡴⠶⠶⢶⣄⡀⠀⠀⠀⣠⣴⠶⠶⠶⣤⡀
⠀⣰⡟⠁⠀⣰⡆⠀⠙⣧⠀⢠⡿⠁⢠⠖⠒⠂⠙⢷⡀⢀⡾⠋⢀⡔⠲⡄⠈⢿⡄⠀⣼⠋⠐⠶⢖⡶⠈⢻⡆
⠀⣿⠀⢀⣜⢹⡇⠀⠀⣻⡇⢸⡁⠀⠘⠛⠳⡄⠀⢸⡇⢸⡇⠀⢸⠖⠓⣆⠀⢈⡇⢸⣟⠀⠀⢀⡞⠀⠀⠀⣿
⠀⠹⣦⠈⠉⠹⠏⠀⣠⡟⠀⠘⣧⡀⠸⠤⠼⠃⢀⡾⠁⠈⢷⡀⠘⠦⠴⠃⢀⣼⠃⠀⢻⣄⠀⠸⠁⠀⠀⣴⠏
⠀⠀⠈⠻⠶⠦⠶⠞⠋⠀⠀⠀⠈⠛⠶⠶⠴⠞⠋⠁⠀⠀⠈⠙⠷⠶⠶⠶⠛⠁⠀⠀⠀⠙⠳⠶⠶⠶⠟⠁


⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠳⢶⣤⡀⠀⠀⢀⣴⠶⠛⠛⠶⣦⡀⠀⠀⢀⣤⡶⠟⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢰⠚⢲⠀⠘⣷⡀⢠⡟⠁⢰⠞⠳⡄⠈⢻⡄⢀⣾⠃⢀⡖⠓⣆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢺⡀⠀⣼⠛⢻⡄⠀⣸⡇⢸⡇⠀⠘⠦⠴⡇⠀⢸⡇⢸⣇⠀⢸⡇⠀⣿⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠦⠞⠁⣠⡟⠀⠈⢷⣄⠘⠶⠞⠁⣠⡾⠁⠀⢻⣄⠀⠳⠶⠃⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠖⠛⠉⠀⠀⠀⠀⠙⠛⠶⠖⠛⠋⠀⠀⠀⠀⠉⠛⠲⠶⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⢄⡀⠀⠀⠀⠀⢀⡠⣤⣤⢄⡀⠀⠀⠀⠀⢀⡠⢤⣤⢤⣀
⠀⠀⠀⠀⠀⠀⢀⢞⣷⣿⠟⣿⣷⣿⢆⠀⠀⣔⣵⡿⢛⣛⢿⣮⣢⠀⠀⡰⣿⣾⢟⣛⢿⣾⡷⡀
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣷⡇⣿⣿⣿⣿⡆⢸⣹⣿⣷⣿⠟⣼⣿⡗⡇⢰⣿⣿⣿⣿⣋⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡽⣿⣿⣇⣿⣿⡿⡻⠀⠘⣟⣿⡟⠱⠿⢿⣿⣻⠁⠀⢟⢿⣿⡙⠟⣡⣿⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠪⠽⠛⠻⠭⠊⠀⠀⠀⠈⠛⠽⠟⠻⠯⠛⠁⠀⠀⠀⠑⠽⠟⠛⠯⠕⠁


⠀⠀⢀⢤⣲⣶⣖⡢⣄⠀⠀⠀⢀⡠⣒⣶⣶⣢⢄⠀⠀⠀⠀⡠⣔⣶⣶⣒⢄⡀⠀⠀⠀⣠⢔⣲⣶⣖⡤⡀
⠀⢰⣳⣿⣿⢏⢹⣿⣯⢣⠀⢠⢮⣾⡏⣭⣭⣿⣷⣱⠀⠀⣎⣾⡿⣩⣭⣻⣷⡵⡄⠀⡼⣽⣿⣭⡭⢩⣿⣞⡆
⠀⢀⣿⡿⠡⠿⠸⣿⣿⣿⠀⢸⢿⣿⣧⣶⣦⢻⣿⡇⠇⠸⢸⣿⡇⣵⣦⢹⣿⣿⡇⠀⡿⣿⣿⡿⢱⣿⣿⣿⡠
⠀⠘⢜⢿⣿⣿⣼⣿⢟⠎⠀⠈⢾⢿⣧⣝⣥⣾⢟⠝⠀⠀⠫⡻⣷⣜⣫⣼⡿⡷⠁⠀⠱⡻⣿⣧⣿⣿⡿⡣⠃
⠀⠀⠀⠙⠚⠋⠛⠚⠁⠀⠀⠀⠀⠙⠚⠋⠛⠓⠁⠀⠀⠀⠀⠈⠒⠛⠙⠓⠉⠀⠀⠀⠀⠈⠓⠛⠙⠓⠋


⠀⠀⠀⠀⠀⠀⠀⣠⣖⣭⣿⣿⣕⣦⡀⠀⠀⢀⢔⣮⣿⣿⣵⡢⡀⠀⠀⢀⣴⣪⣽⣿⣥⣢⣄
⠀⠀⠀⠀⠀⠀⢰⣵⣿⡏⢶⡎⣿⣷⣵⡀⢠⢳⣿⡏⣴⣦⢹⣿⡞⡄⢀⣮⣿⡿⢱⣶⠹⣿⣮⡆
⠀⠀⠀⠀⠀⠀⢸⢿⣿⠫⣶⡦⢻⣿⢿⠃⢸⢸⣿⡷⣭⣥⢸⣿⣇⡇⠘⡿⣿⣇⢻⣿⢁⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠈⠫⡻⢷⣭⣵⡿⡫⠋⠀⠀⠳⣿⢷⣮⣵⡿⣿⠞⠀⠀⠙⢝⢿⣮⣥⡾⢟⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠂⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡤⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠒⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⢤⡀
⠀⠀⢠⠋⠀⣰⡄⠀⠀⠀⢳⡀⠀⠀⠀⣰⠁⠀⠀⠀⣰⣦⠈⢣⠀⠀⠀⠀⡼⠁⠀⢠⣄⢐⡀⠈⣆⠀⠀⠀⢀⡞⠀⢰⡆⠀⣖⠀⠙⡄
⠀⠀⢸⠀⠈⠛⠉⠒⠢⠄⠀⡇⠀⠀⠀⡇⠀⠀⠀⣀⣾⡏⠀⢸⠀⠀⠀⠀⡇⠀⠷⠈⣷⠀⢅⠀⢸⠀⠀⠀⢸⠀⠐⣿⡟⠞⣿⠇⠀⡇
⠀⠀⠘⢆⠺⠛⠗⠀⠀⢀⡼⠁⠀⠀⠀⠹⣄⠰⠿⠿⠿⠇⢀⠞⠀⠀⠀⠀⠳⡀⠠⠈⠿⠀⠄⣠⠏⠀⠀⠀⠈⢧⡀⠻⠇⠐⠿⠀⡰⠃
⠀⠀⠀⠀⠙⠲⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠖⠋⠁

⠀⠀⠀⠀⢀⣠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀
⠀⠀⢀⡔⠁⠀⡀⢀⠀⠙⢦⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⣠⠊⠀⠀⣀⠀⠈⠳⡄⠀⠀⠀⠀⡴⠋⠀⢀⡄⠀⠈⢢⡀
⠀⠀⢸⠀⣠⢙⢐⠸⠷⡆⠈⡇⠀⠀⠀⡏⠀⢨⣤⢠⣤⡤⠀⢸⠀⠀⠀⠀⡇⢀⡄⠀⣄⠀⣀⠀⢹⠀⠀⠀⢸⠁⠀⢀⡞⠁⠀⠀⠀⡇
⠀⠀⠸⡀⠘⠻⣒⡀⠂⠀⢰⠃⠀⠀⠀⢳⠀⠼⠿⠛⠿⠿⠄⡸⠀⠀⠀⠀⢇⢀⣄⠀⣄⠀⣤⢀⡞⠀⠀⠀⠘⡆⠀⠞⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠙⠦⣄⣀⣀⣠⠔⠃⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠈⠳⢄⣀⣀⣀⡤⠊⠀⠀⠀⠀⠀⠘⠢⣄⣀⣀⣀⠴⠋


⠀⠀⠀⣠⠖⠋⢉⡉⠙⠢⡄⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠑⢦⡀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⢠⠔⠋⠉⠉⠙⠲⣄
⠀⠀⢰⠁⢀⡷⢻⡗⢶⠀⠸⡄⠀⠀⠀⡼⠀⣸⣇⣽⣤⣯⡀⢱⠀⠀⠀⠀⡎⠀⠀⠀⡲⠀⠀⠈⢧⠀⠀⠀⢠⠇⠀⢰⣾⢳⣶⠀⠘⡆
⠀⠀⢸⠀⠙⣧⣾⣶⣼⠃⢀⡇⠀⠀⠀⣇⠀⢻⡟⣿⢻⣿⠁⢸⠀⠀⠀⠀⡇⠀⠛⢀⣀⠘⠛⠀⣸⠀⠀⠀⢸⡀⠀⢸⠧⠤⣿⠀⠀⡇
⠀⠀⠈⠣⣀⠉⠙⠋⠉⣠⠞⠀⠀⠀⠀⠘⢦⡈⠁⠛⠈⠉⡠⠋⠀⠀⠀⠀⠙⢄⡀⠉⠉⠁⢀⡴⠃⠀⠀⠀⠀⠳⣄⠘⠓⠚⠛⢀⠜⠁
⠀⠀⠀⠀⠈⠙⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁

⠀⠀⠀⠀⣠⠴⠒⠲⠦⣄⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠦⣄
⠀⠀⢠⠎⢀⣴⣶⡖⠠⠈⢳⡀⠀⠀⠀⣰⠋⣀⣀⣀⣀⣀⡈⢦⠀⠀⠀⠀⡴⠁⣤⣤⣤⣤⡄⠙⣆⠀⠀⠀⢀⡞⢁⡄⠀⢦⣠⡄⠱⡄
⠀⠀⢸⠀⣾⣿⢿⣧⣀⡄⠀⡇⠀⠀⠀⡏⠀⡤⢤⣤⣤⢤⡄⢸⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣧⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⡇⠀⡇
⠀⠀⠘⣄⠈⠻⣿⡿⠟⠁⡸⠁⠀⠀⠀⠹⡀⠛⠛⠛⠛⠛⢁⡜⠀⠀⠀⠀⢣⡀⠛⠛⠛⠛⠃⢀⠏⠀⠀⠀⠈⢧⠀⠡⠤⠔⠈⠁⣠⠃
⠀⠀⠀⠈⠓⠦⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣶⡍⣿⡏⣿⡇⣾⣍⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣛⡡⣿⡇⣿⡇⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠿⢇⣿⡇⣿⡇⠿⣛⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣤⣤⣤⡄⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠁⠀⠀⢸⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢀⣤⣤⣤⣤⣬⣥⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⡇⠛⠛⠛⠃⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡈⠛⠛⠛⠛⢛⡛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠻⠟

⠀⠀⠀⠀⢀⣴⣿⢰⠆⠀⠀⠀⣠⡾⣫⠝⢷⡄⠀⠀⠀⠽⠿⠾⠷⢍⡿⣫⠖⠀⠀⠀⠀⢀⣼⡇⡶⠀⠀⠀⢸⡏⡽⠿⠿⠯⠶⠂
⠀⠀⠀⣠⡾⡁⣿⣸⠀⠀⠀⢀⣿⣼⠁⠀⢸⡇⡇⠀⠀⠀⠀⢀⣴⠋⠼⠋⠀⠀⠀⠀⣠⡟⢹⡇⡇⠀⠀⠀⢸⣇⣥⢴⣤⣀
⠀⠀⠈⢫⡾⢹⡏⣿⠀⠀⠀⠘⣿⢹⠀⠀⣼⢣⡇⠀⠀⠀⠀⢻⡵⠎⢷⡄⠀⠀⠀⣴⢏⡶⢸⣿⡇⠀⠀⠀⢼⡟⣵⠋⠁⢹⡆⡀
⠀⠀⠀⠀⠀⢸⡇⡏⠀⠀⠀⠀⠐⠛⢀⡼⣣⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢀⡾⢣⠟⠀⣼⡿⠃⠀⠀⠀⠐⠚⠁⠀⠀⢸⡇⣷
⠀⠀⠀⠀⠀⣼⢷⡇⠀⠀⠀⠀⢀⣴⣋⡾⠋⠀⠀⠀⢠⣄⠀⠀⠀⢀⡾⣳⠇⠨⠭⠭⠭⠏⣿⢯⡿⠷⠂⢶⡀⠀⠀⠀⢀⣾⢳⠇
⠀⠀⠰⣶⠶⢿⣎⣀⣀⡀⠀⢰⣟⣰⣯⣦⣤⣄⣤⡄⠀⠙⠶⠴⣶⣫⡵⠋⠀⠀⠀⠀⠀⠀⠻⢸⡇⠀⠀⠀⠙⠶⣠⡶⣟⡵⠋
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⢀⣤⢖⡔⢶⣄⠀⠀⠀⢶⣶⣶⣲⣶⢲⡶⣠⠀⠀⢀⣴⣲⡖⢶⡀⠀⠀⠀⢀⡴⣒⠶⣄⠀⠀⠀⠀⢀⣴⢖⡒⢶⡄
⠀⠀⠀⣾⣳⠏⠀⠀⠹⢇⡄⠀⠀⠀⠀⠀⢀⣾⣱⠏⠀⠀⣼⣷⡏⠀⢸⡧⡆⠀⠀⣿⣼⠁⠀⢹⡇⡀⠀⢀⣾⢳⠏⠀⠀⢿⣤
⠀⠀⢸⡏⣩⢖⣲⡒⢦⡉⠁⠀⠀⠀⠀⢀⣾⢳⠏⠀⠀⠀⠹⣧⣁⣤⡞⣵⠃⠀⠀⣿⡹⠀⠀⢸⣷⣇⠀⢸⡇⡏⠀⠀⠀⢸⡇⡇
⠀⠀⢸⣿⢯⡟⠁⠀⠘⣷⢡⠀⠀⠀⢀⣾⢳⠏⠀⠀⠀⢀⣾⢫⠞⠃⢹⡆⡀⠀⠀⠈⠻⣦⣴⢿⡿⣿⠀⢸⡇⡇⠀⠀⠀⢸⢧⡇
⠀⠀⠘⣿⠸⠀⠀⠀⢀⡿⣸⠀⠀⢀⣾⢣⡟⠀⠀⠀⠀⢸⣧⡟⠀⠀⣸⡇⡧⠀⠀⣄⠀⠈⠉⣸⣇⡟⠀⢸⣇⡇⠀⠀⢀⡟⣸
⠀⠀⠀⠙⠷⣄⣀⡤⢞⣵⠃⠀⠀⣼⢣⡟⠀⠀⠀⠀⠀⠈⠷⣅⣠⡴⢟⡼⠃⠀⠀⠙⢧⣀⡴⣋⡾⠁⠀⠀⠻⣤⣀⡤⢞⣵⠋
⠀⠀⠀⠀⠀⠐⠒⠛⠉⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠉⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠁⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⢀⠀⠀⠀⢀⡀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣼⣇⣣⠤⡄⢀⠀⣤⣤⣤⣤⣤⣤⠀⡀⠙⢷⣤⢟⣵⠏⠀⠀⠀⠻⠟⢲⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠘⠟⠃⢹⣇⡟⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠁⣠⡾⣛⢷⣄⠀⠀⠻⠏⣡⣬⢰⠷⠾⠂⠀⠀⠀⢀⡽⢟⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠞⠁⠀⠩⠵⠆⠀⠀⠉⠭⠟⠀⠀⠀⢀⣠⢖⣫⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢾⣿⢰⠀⣠⠞⣵⠟⢀⣼⢫⡵⠒⠉⣷⡄⡀⠀⣶⢤⠀⠀⠀⠀⠀⣠⡴⣞⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⣫⠞⣡⠞⠁⠀⠈⢻⠮⠀⣀⣴⠿⣹⠃⠀⣿⢸⠀⢀⣤⢶⣫⡵⠞⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⢟⣥⡄⡀⠀⠀⠀⢰⡟⡽⠶⠚⠁⠀⠀⣿⣾⠀⠙⢦⣋⠁⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⢒⠠⠆
⠀⠀⠀⠀⠀⢠⠾⣫⠞⠁⠈⢛⣵⠇⠀⠀⠀⣬⣅⠇⠀⠀⠀⠀⠐⡛⡿⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠨⠭⠿⠿⠶⠖⠖⠃
⠀⠀⠀⠀⠀⠀⠉⠁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠙⣥⠏⠀⠀⠀⠀⠘⢟⣴⠀⠀⠀⠀⠀⠀⠉⢣⡤
⠀⠀⠀⠀⠀⢠⡴⠋⣩⣟⣃⣽⢷⣆⠀⠀⠀⠀⠀⣴⢦⣄⣾⣲⠆⠈⠀⠀⣤⢶⣄⢀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⢰⡿⢠⡿⣱⠆⣿⢣⡞⣿⢡⡀⣀⣀⣰⣏⢚⣼⡇⢏⠀⠀⠀⢀⣿⡟⡵⠋⠀⠀⠰⢦⣸⣧⣴⢗⡴⠀⠠⣶⢆⡄
⠀⠀⠀⠀⢿⡇⢻⡇⡇⣸⡏⡞⢰⡿⣼⢁⣘⣢⣿⡐⣸⡏⠞⠛⠛⠁⣴⢟⡍⢻⡄⣠⠀⡐⡖⢲⣿⣿⡖⢲⣆⣤⠀⣠⡛⠁
⠀⠀⠀⠀⠸⣷⡈⠻⠶⡿⠷⠶⣛⡵⠃⠐⠃⣿⢣⢸⡟⡽⠯⠶⠂⠸⣏⠟⠀⣀⡿⣯⡚⠁⠸⣫⢸⡇⡝⠲⣠⠀⠀⢹⡟⡷
⠀⠀⠀⠀⠀⠈⠛⠶⣭⢭⣖⣬⠍⠀⠀⠀⠘⣧⡇⢟⣼⠃⠀⠀⠀⠀⠈⠗⣾⡭⠖⠈⢣⡴⠀⠉⠈⠡⠇⠀⠁⠀⠀⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣄⠀⢀⣄⠀⠀⣀⣀⠀⣀⠀⣀⠀⢀⣄⡄⠀⠀⣀⣀⠀⣠⣠⣄⠀⠀⣠⡀⠀⠀⣀⣀⠀⠀⣀⣄⡀
⠀⠀⠛⣿⠀⢋⣸⠇⠈⠥⣾⠀⣿⢤⣿⠀⠸⠲⣦⠀⣾⣥⣄⠀⠀⣤⣿⠀⣠⣿⣿⡀⠸⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠿⠀⠿⠷⠆⠐⠶⠟⠀⠀⠀⠿⠀⠰⠶⠛⠀⠙⠿⠟⠀⠀⠿⠁⠀⠙⠿⠟⠁⠰⠶⠟⠀⠈⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠀⠀⣦⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢟⣽⠛⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣴⣿⣶⣿⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠀⠀⠂⠸⠀⣠⣤⣤⣀⡾⠛⠿⠟⣋⣻⣷⣀⠛⢿⣿⣷⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣯⣃⡈⣿⣿⠿⢿⣿⣿⣬⣿⠿⠿⠿⢿⣿⣿⣿⣷⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⡾⠛⣛⣉⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠀⠲⣄⠀⠀⠀⠀⣤⣄⣤⡀
⠀⠀⠀⠀⢀⢳⣤⠀⠀⠀⠀⠀⡄⢠⠀⠀⢨⣿⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⣅⢀⡘⠛⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢰⠀⣸⠠⠀⣿⡏⠻⠿⠯⠥⠭⠟⠛⠛⠛⠛⠋⠉⠉⣹⣿⣟⣿⡗
⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⣀⣨⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣟⠋⠀⠀⠀⠀⠀⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠉⠋⢾⢿⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣶⣶⣦⡄⠀⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣾⣿⡇⣿⡄⢸⣷⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣿⣿⠘⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣿⡟⠹⣧⢸⣿⣤⣄⣓⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣿⣟⣻⠿⠋⠈⠀⠀⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢻⡿⠟⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠃⠉⠀⠀⠙⠛⠛⠻⠿⠾⠿⠿⢶⣤⡶⠿⠃

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠛⠒⢤⣀
⠀⠀⠀⠀⠀⡴⠋⠀⢀⡠⠴⠖⠒⠲⠤⣄⠀⠈⠳⣄
⠀⠀⠀⢀⡞⠀⢀⡴⠋⠀⢸⣄⣄⠀⠀⠈⠑⣄⠀⠘⣆
⠀⠀⠀⣸⠀⠀⡞⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡄
⠀⠀⠀⡇⠀⠀⡇⠀⠀⠈⠓⠶⠤⢤⡀⠀⠀⠀⡿⠀⠀⣇
⠀⠀⠀⢹⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⢠⠇⠀⢸⡏
⠀⠀⠀⠈⣧⠀⠈⠳⣄⠀⠀⠈⠉⢻⠀⢀⡴⠋⠀⣠⣿⠃
⠀⠀⠀⠀⠘⢷⣄⡀⠈⠙⠒⠲⠶⠒⠚⠁⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣀⣀⣤⣤⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⢁⣀⣀⡤⠴⠒⠚⠉⠉⠉⠉⠉⠉⠙⠒⠦⢤⡤⠤⣶⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⣿⣿⣿⣿⣿
⡴⠒⠒⠒⠦⠤⣀⣀⠀⠀⠀⢾⣀⣀⣠⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣿⣿⣿⣿⣿⣿
⠨⡟⠉⠉⠉⠓⠒⠚⠻⠿⠷⠶⠶⠶⠖⠒⠚⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠒⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠚⠛⠒⢿⣀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⠤⣄⣀⣀⣀⡤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣴⡉⣰⡋⠃⣞⢳⢼⣀⡗⡷⣼⢹⡏⡇⡷⣼⢴⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠉⠁⠈⠁⠀⠉⠀⠁⠈⠀⠁⠁⠁⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠚⠙⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠦⠀⠀⠀⠐⠂⠀⠘⠀⠀⠀⠀⠀⠀⠈⢩⠀⠀⠀⠀⢀⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⡠⣀⡀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⣶⢸⠁⠀⠈⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣡⡀⠀⠀⢤⣤⣤⣤⣤⡤⠀⠀⠀⢀⠠⣀
⠀⠀⠀⠀⠛⠼⠀⠀⠀⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠟⠀⠀⢀⢀⢀⡀⠀⡀⠀⠠⠁⢇⠀⣨⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⠀⡄⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠁⢰⣦⣰⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⡀⠀⠀⠉⠉⠉⢉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡄⢀⣇⣀⣀
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠘⢧⢀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠴⠲⠄⠀⠀⠀⠱⡁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠈⠉⠉⢀⡀⠀⠀⠀⢀⠀⢠⣠⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠘⠷⠿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠀⠈⠉⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠶⠿⠷⡀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⢀⠀⠠⠀⡀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠄⠚⢃⠀⠀⠀⠀⢀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠉⠉⠝⠀⠀⠀⠀⡀⢀⣤⡀⠀⠀⠀⢀⠀⢀⡠⠀⠀⠀⠀⠀⢡⡶⣶⡀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠿⠾⠾⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠃⠘⠉⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠘⠷⠿⠁⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⢿⣿⣿⣿⠿⠃⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣵⡟⣭⣷⣹⣿⠁⣴⠛⠛⠛⠛⠛⠛⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⣿⣧⡻⣟⣣⡿⣹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣤⣤⡄⣿⣿⣿⠀⣿⣿⣿⣶⣶⣾⣿⣿⠀⣿⠟⠻⡟⠛⡿⠛⢿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⣶⣦⢸⣿⣿⡇⣿⣿⣿⠀⣿⣷⣒⣒⣒⣒⣲⣿⠀⣿⠦⠴⡧⠤⡷⠤⢾⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⠀⣿⣯⣭⣭⣭⣭⣭⣿⠀⣿⡤⢤⡧⠤⡧⠤⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠸⠿⠿⠇⠿⠿⠿⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⢿⣤⣤⣧⣤⣧⣤⣼⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣤⡄⢠⣤⠀⠀⢠⣤⠀⣤⡄
⠀⠀⢠⣴⡾⠟⠋⣁⣈⠀⠀⠀⣉⣉⣉⣉⣉⡉⠻⣷⡀⠀⢈⣉⣉⣉⣉⣉⣉⣉⠙⠇⠀⣿⡇⢸⣿⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⢁⣤⣶⣿⠛⣿⡇⠀⠈⠛⠛⠛⢛⣿⡿⠀⣿⡇⠀⠈⠛⠛⢻⡿⠛⣿⣿⡀⠀⠀⢿⣇⠘⠿⣶⣶⣼⣿⣀⣿⡇
⠀⠀⠀⠀⠉⢸⣿⠀⣿⡇⠀⠀⠀⣠⣶⠟⠋⣠⣾⠟⠀⠀⠀⠀⠀⠹⢿⣦⣀⠙⢿⣆⠀⠈⠻⣶⣤⣤⣄⣈⣉⠙⠻⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⣾⠛⢁⣴⣾⣿⣷⣶⠄⠀⢰⣶⣶⣶⣶⣾⡿⠇⢸⣿⠀⠀⠀⠀⠈⠉⠉⢹⣿⢿⣦
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⢰⣿⣯⣤⣤⣤⣤⡀⠀⢠⣤⣤⣤⣤⣤⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⡶⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⠿⠋⣁⣠⣤⣤⡄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⢻⡆
⠀⠀⠀⠀⠀⢠⣾⠛⢿⣿⣿⣿⣟⡛⠛⠀⠀⠀⠀⢀⣾⠏⢠⣾⠿⠻⠿⣷⣅⠀⠀⠀⠀⠘⠛⠛⠛⠛⣻⡿⠛⣷⠀⠁
⠀⠀⠀⠀⠀⠘⣿⣤⣤⣀⣈⣉⠉⠻⣷⡀⠀⠀⠀⢸⣿⠀⣿⣿⠿⠿⣦⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⣼⡟
⠀⠀⠀⠀⠀⢠⣤⣭⣭⣽⣿⣿⡿⠀⣽⡇⠀⠀⠀⠸⣿⠀⢻⣧⣀⣠⣿⠃⣸⡟⠀⠀⠀⠀⠀⢀⣼⡟⢀⣾⠟
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣴⡿⠁⠀⠀⠀⠀⠻⣷⣄⠙⠛⠋⢁⣴⡿⠁⠀⠀⠀⠀⢀⣾⠏⢀⣾⠏
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠚⠋⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣀⠉⢻⣆⠀⠀⠀⠀⠀⢀⣾⠟⢉⣀⣤⣤⣀⡙⠂⠀⠀⠀⢠⣾⠟⢉⣻⣷⣦⣄⠉⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣄⠙⠻⢿⣧⣼⡟⠀⠀⠀⠀⠀⣼⡏⢠⣿⠋⠉⢻⣯⠛⣧⠀⠀⠀⣾⡇⢠⣿⠋⠉⠉⠹⣿⣦⡈⠁
⠀⠀⠀⠀⠀⢠⣾⠛⢿⣶⣤⣀⠉⠻⣦⠀⠀⠀⠀⢻⣧⠈⠿⣶⣶⣾⣿⠀⣿⡇⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⣿⡟⢿⣦
⠀⠀⠀⠀⠀⣾⡇⠰⣿⡉⠉⣻⣿⣦⡈⠃⠀⠀⠀⠀⠛⢷⣶⣤⣴⡿⠃⣠⣿⠁⠀⠀⣿⡆⠸⣿⣀⡀⣀⣠⣿⠁⣸⡟
⠀⠀⠀⠀⠀⠹⣷⣀⠙⠛⠛⠋⢁⣹⡗⠀⠀⠀⠀⠀⠚⠛⠛⠛⢉⣠⣴⡿⠁⠀⠀⠀⠘⢿⣄⡈⠛⠛⠛⠋⢁⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⠾⠟⠋⠀⠀⠀⠀⠀⠀⠶⠶⠶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡟⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⢤⣤⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠾⠟⢩⡈⠀⠀⠀⠈⠉⠁⠀⠀⠖⠲⣦⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠁⠀⠀⠀⠀⢷⡀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠙⣿⣆
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⣸⣿⠀⠀⠀⠀⢸⡟⢀⡴⠃⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢀⣿⠇⣾⡇⠀⠀⠀⣿⣧⡶⠶⠶⠶⣤⣶⣾⠛⠋⠉⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿⠀⣿⣇⣠⡶⠟⢿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠶⣦⣤⣀⣀⣾⡿⠋⠙⢻
⠀⠀⢠⣿⠁⠀⢀⣠⣤⣤⣽⣇⠀⠀⠀⠀⠀⠀⢃⣀⣠⣤⡴⠶⠶⠃⢻⣿⣿⡀⣤⡀⠹⣦⠀⠀⠀⠀⢀⡴⢁⡠⣀⡼⢩⣿⠋
⠀⠀⣼⣿⠀⠀⠉⠀⠀⠉⠙⠿⣷⣦⣴⣶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⢸⡇⢸⣿⡀⢸⣷⠀⣠⡶⢋⡴⢫⡾⢋⣴⡿⠁⠀⠀⠀⢀⡾
⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⡿⠿⠷⢶⣤⡀⢀⣠⡤⠶⣦⣄⠀⠈⠁⠀⠛⠷⠾⠟⠈⢉⡴⠋⠀⣈⣴⣿⠟⠀⠀⠀⠀⣠⡿⠁
⠀⠀⣀⣽⣿⠿⢿⣷⣶⣶⣶⣾⣿⠁⢰⡇⠀⠀⠀⢮⣻⣮⠁⢠⡶⢈⡙⠻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠙⢷⡄⠀⢀⣼⡟
⢤⣀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣬⡿⣦⣼⣿⣿⣿⣿⠀⣀⡀⠛⠃⢶⠀⢙⣿⣿⣟⠶⣄⡀⠀⠀⣿⡁⠸⣷⡀⢸⡇⢠⣾⠏
⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠠⠿⠃⢠⣶⠄⢛⣥⠀⣀⣴⠾⠛⠉⠀⣙⡿⢦⣉⠲⢤⣙⣿⣾⠿⠣⠟⣡⣿⠋
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠈⣷⠈⠳⣄⠐⠋⣠⡀⠀⠈⠉⠀⠉⠀⠀⢀⣴⣟⣉⠀⠀⠙⣷⡄⠈⠛⠻⠶⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠿⣦⡀⠈⠳⣄⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠹⣇⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠳⣄⠀⠀⠀⢀⣠⡶⠟⠻⣶⣄⣠⣿⡀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⠀⠀⠘⢿⣦⡀⠈⢷⣤⡾⠋⠁⠀⣠⡴⠟⠋⠉⠈⣷⠈⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠙⢿⣦⣄⣿⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⢻⣧⣀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⠀⠀⠙⢿⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠛⢸⣟⢻⣦⠀⣾⢿⡄⢸⡟⠛⣶⡀⣿⢸⣿⣆⢸⡇⣴⠟⣻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡟⢻⡅⣼⡷⠾⣷⢸⣧⣤⡾⠃⣿⢸⡏⠹⣿⡇⢿⣬⣽⡿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⡛⠛⠛⠛⠛⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⢀⣼⣇⡀⠀⢠⣿⠀⠀⣤⣾⣧⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢀⣼⣧⡀⢸⡏⢹⡇⠀⢈⣿⡀⠀⣿⡇⣸⡇⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡏⢹⡇⢸⡇⢸⡇⠀⣿⠛⣿⠀⢿⣇⣽⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣿⠶⠀⣿⢀⣼⣧⡀⢸⣧⣼⡇⢸⣧⣼⡇⢠⣿⣀⣿⠀⠀⣿⡇⠀⢠⣦⠰⢶⣷⣄⣤
⠀⠀⠀⠀⠀⣤⡶⠟⠋⠁⠀⠀⠀⣿⢸⡇⢹⡇⠀⢹⡇⠀⠀⢹⡇⠀⠀⢙⣿⠉⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠉⠙⠷⣶⡆
⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⠿⠃⠀⢸⠇⠀⠀⠸⠇⠀⠀⠀⡿⠀⠀⠀⢻⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⡶⠤⠤⠤⠤⢶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢶⣶⠶⢶⡶⢶⡶⣶⡶⣶⠶⣶⡶⠶⣶⠶⡶⠶⢶⡶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠠⡟⠁
⣤⣄⠀⠈⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣷⡶⢶⡾⢷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣤⡀
⠀⠙⠻⣦⡀⠋⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⠻⣦⡀
⠀⠀⠀⠈⢿⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡇⠀⣠⣶⣤⠀⢨⣿
⠀⠀⠀⠀⠈⢿⡄⠀⣠⣤⣤⠀⣤⣤⣤⠀⣤⠀⠀⢠⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⣷⢶⣄⠀⢸⡇⠀⣿⣤⣄⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⣄⠀⣿⣤⣤⠀⣿⠀⠀⢸⡇⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⣾⣁⣀⣀⣴⢿⣄⣿⠀⢸⡇⠀⣤⣤⣿⠂⢸⡇
⠀⠀⠀⠀⠀⢸⣇⠀⣤⣤⡿⠀⣿⣤⣤⠀⣿⣤⡄⢸⣧⣤⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠉⠁⠀⢹⡟⠀⢸⡇⠀⠀⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣿⣄⣀⣀⣠⣿⣁⣀⣈⡁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢠⣤⣼⡿⠀⠀⠀⠀⠸⣧⣠⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⣀⣼⣧⣤⣤⣄⣠⣤⡄⠀⠙⠋⠻⣦⡶⣦⠀⢠⣤⣴⡟⠀⠀⠀⣴⠟⣷⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠻⣧⡄⠀⠸⠟⠋⠀⠀⣤⣤⣤⣤⣤⣤⣼⣷⡄⠀⠀⠀⠀⠀⢰⡏⠀⣿⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠚⠈⣷
⢀⣀⣀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣄⣀⡀⢸⡇⢀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡄⢹⣧⣤⣄⡀
⣾⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠋⠁⣿⠁⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢸⣯⣤⣬⣿
⣿⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⡇⠀⠀⠀⠸⣼⣧⡤⠤⢤⣄
⣤⠤⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠋⠉⢿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢈⡇⠀⠀⠀⠀⠀⠀⠀⣷⣤⠶⠶⠶⢶⡄⠀⠀⠀⢰⠛⢛⣛⣻⣯⣤
⠸⣷⠶⠶⠚⠛⠛⠛⠛⠛⠛⣛⣉⣉⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⡇⠀⠀⠀⢸⠀⣿⠉⠉⠉⠁
⠀⠈⠻⠶⠶⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⣤⣴⠶⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠻⢦⣀⠀⣠⡞⠁⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠉⠻⠋⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣷⣤⣞⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣼⠇
⠀⠀⠀⠀⣀⣀⣀⣈⣉⣭⣥⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⠀⠀⠀⠀⠀⠈⠳⢿⣁
⠀⢰⡟⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⣦
⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣠⠎⣙⠶⣄⡀⠀⠀⣰⣻
⠀⠘⢷⣖⣛⣻⣭⣭⣭⣭⣭⣤⣤⣤⡴⠶⠟⠀⠀⠀⠀⠀⠀⠀⢰⣯⡀⠀⠀⢀⡴⢣⡾⠛⠻⣦⡙⠲⠾⣱⠟
⠀⠀⠈⠉⠉⠁⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡳⠴⠟⣰⠟⠀⠀⠀⠈⠙⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣭⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⣐⡤⠖⠋⠉⠙⠲⢯⡢⠀⠀⠀⠀⠀⠀⡀⠠⠔⠀⠉⠙⠒⢄
⠀⠀⠀⠀⠀⡜⠀⠀⠀⣿⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠸⠉⡇⠀⠀⢳⣀⠀⠀⠀⠀⠀⡐⠀⠀⠘⠉⡧⠀⠀⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⡁⠀⠀⢠⠞⠃⠀⠀⠀⡇⠀⠀⠀⠀⠃⡅⠀⠀⠀⠒⣇⠀⠀⠀⡄
⠀⠀⠀⠀⠠⢣⡀⠀⠀⢿⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠉⢧⡀⠀⢸⣄⡀⠀⠀⡼⡁⠀⠀⠀⠀⠀⠠⡀⠀⠰⣤⠗⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⡤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⣤⣴⡻⠊⠀⠀⠀⠀⠀⠀⠑⠈⠢⢤⣤⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣨⠖⠋⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠙⠳⣤⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠛⢶⡄
⠀⠀⠀⠀⠀⡼⠁⠀⢸⠀⣿⠀⠀⠘⣆⠀⠀⠀⠀⠀⣺⠃⠀⠀⣿⠉⠁⠀⠈⢷⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⠉⠇⠀⠀⠹⡆
⠀⠀⠀⠀⠠⡇⠀⠀⠸⣤⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⢳⡄⠀⠀⢸⡇⠀⠀⠀⠀⢾⡇⠀⠀⢸⠓⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣿⠀⠀⣰⠃⠀⠀⠀⠀⠀⠽⣆⠀⠀⢶⣼⠃⠀⢀⡞⠀⠀⠀⠀⠀⠈⢳⡀⠀⠸⣤⠇⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠪⠓⠤⢤⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⢀⡔⠋⢁⣀⡈⠉⠳⣅⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⢁⡀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⣔⡵⠊⠉⣉⡉⠙⠺⡄
⠀⠀⠀⠀⢠⡏⠀⠀⠈⢩⠇⠀⠀⠘⡆⠀⠀⠀⠀⠐⣼⠁⠀⢐⡏⢹⠀⠀⠈⢆⠀⠀⠀⠀⠀⣸⡏⠀⠀⢸⡏⣿⠀⠀⠹⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⡾⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⢧⠀⠀⢠⡗⢿⠀⠀⠀⣸⠀⠀⠀⠀⠀⢻⡇⠀⠀⠈⠳⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡳⣄⠀⠠⠇⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⢆⠀⠘⠧⠾⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⣳⣄⠀⠐⠦⠟⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠈⠛⠷⡶⣤⡶⠶⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⣤⣴⢶⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡤⠒⠋⠉⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏⠀⠀⢸⠋⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣆⠀⠀⢸⣀⡇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣷⣤⣀⣀⣀⡤⢞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⠀⣿⠁⣿⢧⣿⢀⣞⣧⠀⡿⣦⡇⣼⠋⠉⠈⡏⠀⣾⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠛⠃⠛⠈⠋⠘⠉⠙⠃⠃⠈⠃⠈⠛⠋⠘⠛⠘⠉⠙⠃⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⡤⠦⢀⡤⠦⢀⡤⢦⡄⣤⠀⣤⢠⣤⢠⡄⠤⡤⠄⢤⡄⢠⣄⢠⠀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢷⠸⣧⣠⠸⣧⣠⠘⢧⣠⠇⢻⣄⡿⠸⡏⢿⡇⠀⡇⠀⣼⡇⢸⠙⢿⠘⢧⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢸⣿⣅⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⢺⢸⣿⢨⣗⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⣿⢘⡏⡷⡶⣤⣄⠀⠠⠶⢶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⣿⢪⡇⡇⡇⡇⡏⡇⠀⠀⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⠀⢸⣸⣟⣙⣗⣇⣇⣇⣇⡇⢐⣖⡲⠀⢨⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⢸⣄⣀⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣉⣁⣀⣨⡇⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣰⠃⠀⠀⠀⠀⢙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⣀⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣓⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⠞⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠦⠀⢐⠖⣒⣒⠀⢹⠀⠀⠀⢠⣦⠀⠀⣒⣒⣂⠰⢀⣀⣀⠀⡼⠁⠀⠀⠰⢒⣂⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠨⡽⠀⠨⠶⠯⠿⠀⢸⠀⡴⠛⠋⠀⠀⠀⠉⠉⢩⡿⣏⠁⢸⢳⠖⠦⣄⠀⢀⠼⠯⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠤⠤⠤⠤⠬⠭⠉⠀⡤⠾⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⢙⡦⣄⢸⢸⠀⠀⢨⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠭⠭⠥⠬⠤⠭⠀⠀⡇⣟⡉⠉⠉⣹⢸⡇⠀⠀⠀⠀⣾⠀⠈⠻⣼⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣞⣺⢲⣓⣖⣶⢺⡆⡇⠉⠉⠉⡉⣩⢸⡇⠀⠀⣠⠆⢹⡀⢀⡴⠋⠀⠀⠀⠀⣼⡹⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡯⣿⢽⡯⡯⣿⢽⡇⡇⠂⠂⠀⠂⠒⢸⡇⢀⡞⠁⠀⠀⣟⠉⠀⠀⠀⠀⢀⡴⣣⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⢽⡇⡇⠁⠁⠁⠁⢩⢸⡅⠋⠀⠀⠀⠀⠈⠻⢶⣶⣶⣚⡭⠖⢁⣀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠷⠾⠼⠶⠧⠾⠼⠇⠓⣲⠒⠒⠒⠒⠚⠁⠀⠄⠠⠄⠤⠤⠄⠀⠀⠀⣤⠀⠀⠒⠒⠒⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣵⢲⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⣿⣾⣿⢽⣯⣿⣯⣽⣿⣿⣿⣿⣿⣿⡇⣾⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣴⣦⣄⠙⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⢸⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣯⣯⣿⣿⡏⠏⠛⣿⣿⣿⡇⣿⢸⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣿⡷⢾⣿⣿⣿⣿⣿⣶⡾⣿⣿⣿⡇⣿⠉⠀⠀⠀⣬⣤⣤⣤⠄⢀⣿⠂⣶⣶⠄
⠀⠀⠀⠀⠀⠀⣿⣾⣷⣿⣿⣿⣿⣿⡿⡇⠀⠙⠋⠀⣀⠿⣿⣿⣿⣿⣿⡝⢛⣿⣿⣿⣿⣠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠀⠀⣸⣰⣷⣮⡽⠿⠛⠓⠒⢛⣩⡛⢛⣽⣭⣿⣿⣿⡿⢿⡟
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⡤⠒⠿⣿⣿⣿⣯⣤⠄⠀⠀⠀⠉⢁⣤⣬⣿⣿⣿⣭⣿⣿⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉⠀⠐⠚⠛⠉⠀⠀⠀⠀⠠⠄⠄⠀⠀⠈⠁⠘⠛⠿⡿⠿⠿⠿⠿⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⡾⠖⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣟⠋⠀⣉⡷⠀⢛⡇⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣰⠃⠘⢦⣄⣙⣁⣤⠞⠀⢳⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢰⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢷⢸⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⢸⡄⠀⣀⡴⣶⢶⠲⡶⣤⡀⠀⣸⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠃⣼⠛⠶⢧⣼⣤⣿⠾⢻⡄⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢰⣏⠓⢦⣤⣼⣤⡼⠶⠋⣷⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠲⣼⣇⣸⣀⣠⡤⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣿⣹⣡⣯⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉






⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠉⢉⠉⠛⢷⣄⠀⠀⢀⣴⠟⠋⢉⣉⠉⠻⣶⡀⠀⠀⣠⡾⠛⠉⣉⡉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠠⠶⣿⠀⠀⠀⢿⡆⠀⣿⠁⠀⠸⠉⠉⣷⠀⠈⣿⠀⢰⡿⠀⠀⠚⠁⣹⠆⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⣿⠀⠀⠀⣠⠞⠁⠀⠀⣿⠃⢸⣇⠀⠀⡀⠈⠙⡆⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠿⠀⠀⣰⡿⠁⠀⠻⣧⡀⠾⠷⠶⠶⢀⣼⠟⠀⠈⢿⣆⡀⠙⠶⠞⠃⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⡶⠶⠟⠋⠀⠀⠀⠀⠈⠛⠷⢶⠶⠾⠟⠁⠀⠀⠀⠀⠙⠻⠶⠶⠶⠾⠛⠁


⠀⠀⠀⢀⣤⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣀
⠀⢀⣼⠟⠁⠀⣠⡄⠈⠻⣧⠀⠀⢠⡾⠋⢁⣤⣤⡄⠉⢻⣦⠀⠀⣰⡿⠋⢀⡤⣤⡈⠙⢷⡄⠀⠀⣴⠟⠁⣤⣤⣤⡌⠻⣷⡀
⠀⣸⡇⠀⠀⡴⢻⡇⠀⠀⢸⣇⠀⣿⠁⠀⢸⡧⢤⣄⠀⠀⢻⡆⢰⡿⠀⠀⣿⡤⣬⡁⠀⠈⣿⠀⢸⡏⠀⠀⠀⣰⠏⠀⠀⢸⣧
⠀⢹⣇⠀⠾⠷⢾⡷⠀⠀⣸⡇⠀⣿⡀⠀⢠⡀⠀⣿⠀⠀⣼⠇⠸⣷⠀⠀⣿⠀⢈⡗⠀⢀⣿⠀⢸⣇⠀⠀⢠⡏⠀⠀⠀⢸⡟
⠀⠈⠻⣦⣀⠀⠈⠁⣀⣴⠟⠀⠀⠘⢷⣄⡈⠙⠋⢁⣠⣾⠏⠀⠀⠙⢷⣄⡈⠛⠋⢁⣠⡾⠃⠀⠀⠻⣦⣀⠈⠁⠀⣀⣴⠿⠁
⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠶⣦⣄⠀⠀⠀⠀⢀⣤⡶⠾⠷⠶⣦⣀⠀⠀⠀⠀⣠⣴⠶⠾⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢀⡴⠶⣄⠀⠙⣷⡀⠀⣴⡟⠁⣠⠶⠶⡄⠈⠻⣦⠀⢀⣾⠏⠀⣠⠶⢦⡀⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢈⣷⠤⣟⠀⠀⢹⡇⠀⣿⠀⠀⢿⣀⣀⣿⠀⠀⣿⡄⢸⡏⠀⠀⡏⠀⠘⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠸⣇⣀⣼⠀⢀⣾⠇⠀⣿⡀⠀⢠⣉⣁⡟⠀⢀⣿⠀⠸⣷⡀⠀⢷⣀⣸⠇⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣈⣉⣀⣤⡾⠋⠀⠀⠈⠻⣦⣄⣉⣁⣠⣴⠟⠁⠀⠀⠙⢷⣦⣀⣉⣁⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⠀⠀⠀⠀⢠⠊⡇
⠀⠀⡀⠀⣀⡬⢶⣖⠤⠀⡴⠲⢶⣅⠤⠤⢀⣐⣗⡀⠀⣿⠀⠐⠲⢄⡀⠴⠓⢲⡧⢀⡐⢶⡂
⠀⠀⠉⠉⠩⠌⣻⠲⢇⠤⣥⠄⢰⡟⠀⠀⣐⣓⢀⠀⠤⠷⠄⣈⠉⠁⠤⠀⣶⡏⠁⠰⠀⣩⣤⡤
⠀⠀⠈⠛⠋⠉⡡⠤⠄⡉⠀⣠⠋⠀⠂⠐⡚⠋⠕⣒⡂⠔⢺⣧⢰⡄⠠⣤⣤⠎⣰⡖⠚⠒⠒⠓
⠀⠀⠀⠀⠀⣘⣁⡠⠜⢁⣬⡙⠋⠿⠃⠤⠤⠠⠤⠤⠤⠒⠀⠀⠤⠤⠒⠚⠛⢋⣋⠐⢶⠦⡐⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠒⢐⣄⡄⢩⣶⡆⢘⢒⡀⠀⡴⠶⠶⣦⡦⠀⠀⠦⣍⣙⣓⡨⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣠⠞⠁⠰⣠⣬⣉⠛⠉⠀⠀⠀⢀⠔⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠭⠤⠴⠚⠁⠀⠀⡘⠛⠛⣋⡴⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁





⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣄⠐⣦⠀⠠⢦⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⣠⡄⠀⠀⠀⡀
⢸⣼⠀⠸⡇⠀⡶⠀⠉⠑⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⢂⣴⠟⠀⠀⠀⠀⠀⢰⡷⠉⠀⢈⣼⠟
⠀⠿⠀⠀⠐⠂⠀⢰⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⡴⣦⠀⠀⠀⠀⢻⠁⢠⡞⣻⠏
⠀⢠⡤⠂⠀⠀⢠⣈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠛⠁⠀⠀⠠⢆⡀⠀⠘⠋⣡⡾⢇
⢠⠇⠀⠀⠀⠀⢸⠇⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣀⠠⠃⠀⡀⡈⠐⢦⡄⠀⠘⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠾⠠⢬⠇⠘⠃⠁⠀⠀⠀⠀⠘⠂⢹⣛
⠀⠳⢶⣶⡶⠶⠾⠛⠋⠁⠀⠈⠀⠁⠀⠀⣠⠾⡆⠀⠀⠀⠀⠀⠀⢀⡀⠘⢄⠀⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠸⠟
⠀⠀⣠⣤⣶⣆⠀⠀⠐⠀⠀⠈⠀⠀⠂⣤⠹⣦⣾⠀⠰⠀⢂⠁⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠛⠛⠉⠀⣦⠀⠄
⠸⣿⣿⣿⡎⢿⣷⡄⡐⢦⣘⣠⣤⡄⠀⠉⠀⢻⣛⠀⠀⠀⠀⠀⠀⢰⡄⠀⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⢧⡀⢆
⠀⢹⣿⣽⣿⡄⠙⣿⡏⡓⠦⠒⠉⠁⠀⠀⠀⠼⠟⠀⠀⢴⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⢻⣿⠻⠗⡠⢀⡀⠇⠀⠀⢀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⢐⣩⣶⣿⡇⠀⠀⠀⣼⠶⢿⣰⣖⣰⣗⣰⣟⡦⣆⣰⢲⡞⡆⢿⢲⣶⢺⣶⣟⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠁⣾⣿⣿⣿⣿⡧⡀⠀⠀⠁⠀⠈⠁⠉⠁⠉⠀⠉⠀⠈⠁⠀⠀⠁⠈⠀⠀⠀⢀⡼⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⣿⣿⣿⠟⠉⠐⠦⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⢐⡄⠀⠀⠀⠀⢠⡀⠉⡁⠀⠀⢀⡴⠃⠀⠄⢊⣁⡇
⠀⠀⠀⠟⠋⠀⠀⢀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⢀⣀⡆⠉⡀⠇
⠀⠀⡂⣀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠠⠀⡠⣰⡿⣞⠀⠀⠀⠐⠀⠤⠖⠀⠹⢄⡀⠠⠶⠀⠀⠀⠸⠀⠀⠤⢀⠈⠉⠀⠀⣠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⠔⠛⠒⠉⠀⠋⢰⣿⣿⡷⠲⠦⠤⢤⠀⠀⢀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠂⠐⠁⢠⣤⣾⣿
⠀⢀⠄⠐⠀⠈⠀⡀⠆⠉⠀⠀⠐⠀⢠⣄⠀⠘⣿⣯⣇⠄⡀⢠⣼⣷⠀⣾⡗⠀⢀⣼⡇⠀⠀⠀⡀⠄⠀⠄⠀⢀⣿⡿⢿
⠀⠀⠀⣀⣴⣺⠆⠀⠀⠀⠀⢀⠀⠠⠀⠹⡷⡄⢿⣿⣿⣷⣻⣿⢣⣏⢷⣿⡃⠠⣎⣹⠇⠒⠀⠀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠈⠛⠁⣾⣤⠀⠀⠀⠀⠀⠀⠲⣦⡀⠓⠺⢸⣿⠋⠀⠁⠙⠛⠛⡻⡎⠃⠀⠙⠁⡀⠀⠀⠀⢀⠀⠀⠀⢸⠟⢠⣤⣤
⠀⠀⠀⠀⢺⣹⣤⣤⡀⠀⠀⠀⠀⣀⣈⣉⣃⣀⣘⣋⣀⣁⣀⣀⣀⣈⠁⠀⠀⠂⠀⠂⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣅⡼
⠀⠀⠀⠀⢀⣄⣧⡼⠁⠀⠀⠀⢈⡇⢺⢘⡇⣖⢰⡆⣿⠰⡆⢸⠃⢸⡇⢸⣿⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⢀⡿⡿⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⠷⠾⠼⠧⠿⠜⠧⠿⠀⠧⠸⠤⠼⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⠉⢹⡇⠀⢰⡞⠉⢁⠀⠉⠓⣦⠀⠀⢰⠉⠁⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣤⠀⠀⠘⡇⠀⠨⠟⠉⠉⠙⣧⠀⢸⡄⠀⠘⠉⠁⣨⠆⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⠀⠀⡇⠀⠀⠀⠀⣀⣼⠃⠀⡼⠁⠀⠀⢰⡉⠁⠀⠐⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣷⠀⢀⣤⠾⠋⣀⡴⢊⣁⡀⠀⠀⠀⠉⠙⣦⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⢸⡇⠀⠀⠉⠉⠁⠀⡇⠀⣾⠑⠒⠒⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠒⠒⠋⠀⠈⠧⠒⠒⠒⠚⠋⠈⠁⠀⠈⠓⠲⠖⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠛⡏⠀⠀⠀⡖⠛⠛⠛⠛⢳⠀⠀⠀⠀⢀⡤⠶⠒⠒⢲⡆⠀⢸⠟⠛⠛⠛⠛⠉⠛⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⡇⠀⠀⠀⡇⠀⢰⡖⠒⠚⠀⠀⠀⡴⠋⠀⢀⡴⠶⠾⠇⠀⠸⠶⠖⠲⢶⣶⠆⢀⡏
⠀⠀⠀⠀⠀⠀⡼⠁⢠⢿⡆⠀⣷⠀⠀⠀⣗⠀⠈⠉⠛⠢⣄⠀⢰⡇⠀⠀⠘⠛⠛⢶⣄⠀⠀⠀⠀⢠⡞⠁⢠⡟⠁
⠀⠀⠀⠀⢀⡾⠁⠀⡯⠼⠃⠀⠙⣷⠀⠀⠉⠉⠙⢳⡄⠀⢹⡆⢸⠃⠀⢰⠛⠓⡄⠀⢿⠀⠀⠀⢀⡟⠀⢠⠟
⠀⠀⠀⠀⢸⣄⣀⣤⣤⡄⠀⠀⡶⠞⠀⢰⠶⠶⠶⠚⠁⠀⣸⠇⠈⢧⠀⠘⠲⠚⠁⢠⡟⠀⠀⠀⣼⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠼⡇⠀⠀⠘⠦⠤⢤⠴⠶⠚⠁⠀⠀⠈⠳⠤⠤⠤⠞⠋⠀⠀⠀⠀⣿⣶⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠶⠲⢦⡀⠀⠀⠀⣀⡤⠤⠤⣤⡀⠀⠀⠀⢀⡤⠖⠒⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⡴⢲⡄⠀⣱⠀⢀⡞⠁⢀⣤⡀⠀⠘⣆⠀⢀⡟⠀⢠⣄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⠓⠚⠁⣰⠃⠀⢸⠀⠀⢻⣠⡿⠀⠀⢿⠀⣼⠀⠀⡟⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢀⣤⢶⡄⠈⢧⠀⠈⢷⣀⠀⠀⣀⡀⠀⡿⠀⣿⠀⢸⡇⢠⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣿⣧⡼⠃⠀⢸⠀⠀⣤⣨⣭⡽⠛⠁⣠⠇⠀⢻⡀⠀⠧⠼⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠁⠀⣠⣴⠏⠀⠰⣇⣀⣀⣀⡤⠖⠉⠀⠀⠈⢷⣄⣀⣀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⠞⠛⠛⠳⢦⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠭⠭⠭⠥⠤⠄





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

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢪⠀⠀⠀⠀⠁⠀⣀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣠⣤⣤⣼
⠀⠀⠀⠀⠈⠉⠁⠀⣯⠀⠀⠀⠨⠤⠼⠛⠆⠀⢀⡟⠀⠀⠀⠶⠺⠭⠁⠀⣠⠇⠀⠀⠀⣠⠆⠀⢸⡀⡀⠀⠀⠀⠛⠐⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⣤⠀⠈⠙⢂⠀⠀⠀⠉⠁⠀⠈⠀⣿⠀⠓⠒⠲⠦⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠻⣠⠀⠀⠀⡏⠀⠰⠛⠂⠀⣠⠀⠀⠀⡀⠐⠒⠀⠀⢀⡞⠐⠛⠛⠓⠂⠀⢰⡞⠋⠀⠉⠉⠲⠀⠀⣸
⠀⠀⠀⠀⠀⠤⢤⣤⠶⡿⠀⠀⠀⠣⠤⠤⠤⠤⠦⠟⠀⠀⠀⠑⠦⠤⠤⠴⠋⠀⠀⠀⠀⠀⠴⠴⠾⠀⠀⠀⠑⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠈⠈⣳⠀⠀⢸⠀⢀⣀⣀⡀⠀⣸⠀⠔⠉⠀⠀⠀⠁⡄⠀⠀⠊⠁⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⣠⣶⠋⢁⠀⠀⠈⠉⠉⡁⠃⢀⣰⠏⠀⠀⠐⢏⠋⠀⣠⠇⠀⠀⠀⠸⡹⠂⠀⢸⡇⠀⠀⠀⣴⠛⠂⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢳⡀⠀⢠⠊⠀⣰⠟⠁⠀⢠⠗⠀⠀⠀⠘⠳⠄⠀⠳⣄⣀⣠⡄⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠐⡀⠀⠻⠨⠃⠀⣸⠀⢀⠇⠀⢰⠃⠀⠀⠀⡇⠀⢴⠋⠳⠀⠀⣠⠀⡀⢀⣀⠀⠀⠀⡼⠀⢀⠀⠀⢧⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠱⢤⣀⣠⡤⠞⠁⠀⢀⣀⣀⣸⠀⠀⠀⠀⠳⣄⡀⠉⠁⢀⡴⠃⠀⠍⠁⠀⢀⣠⡴⠃⠀⠘⢄⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣱⠀⠀⠀⠀⡰⠊⠳⣄⢀⠔⠓⢀
⠀⠀⠀⠀⠐⠒⠐⠀⠀⠙⠒⠒⣤⠀⠀⠒⠒⠒⠒⠒⢰⠀⠀⠰⠂⠑⢦⠀⠀⠒⠒⠋⠋⠉⠐⠲⡆⠀⠈⢢⡀⠈⠁⢀⡴⠋
⠀⠀⠀⠀⠴⠤⠴⠀⠀⢰⠶⠷⠏⠀⠀⠶⠶⠶⠴⠶⠾⠀⠀⠱⣤⣠⠞⠀⠐⠶⠶⡆⠤⣤⠴⠼⠇⠀⡠⠔⠁⣠⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⡼⠀⠀⠀⠈⠳⣤⠞⠁⠈⠲⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠆⠤⢄⣤⠈⠉⠉⠉⠙⠛⠒⠒⠶⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢰⡇⠀⠀⠈⠁⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠘⢀⡈⠀⠁⢠⠀⣀⡀⠀⠀⠠⠈⣉⠁⠒⠒⠂⠠⠠⡄⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠒⢂⡀⠀⠡⠀⠀⡀⠑⢀⡀⠄⠀⠀⠒⠀⠀⠌⢡⠀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠻⠶⢤⣤⣄⣀⣀⣀⠀⠀⠐⠀⠠⠄⠁⠰⠐⡀⠄⠐⡀⢠⠃⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠢⠤⠄⠀⠀⠀⠁⠉⠉⠉⠛⠛⠓⠶⠶⣦⣤⣤⣄⣀⣉⣀⡘⠀⢰⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⣀⡀⠀⠈⠀⠂⠁⠐⠒⠀⠤⢄⣀⡀⠀⠀⠉⠉⠉⠙⠋⠀⢿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⢡⡀⠸⠉⢹⠛⠲⠶⢤⢠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣒⣀⠀⠈⢘⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡤⠁⠀⢈⡀⠓⠂⠧⠤⠃⠊⠀⠀⢠⠂⢐⠆⢠⣤⢤⣀⡀⠘⠿⠇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡘⠁⡴⠋⣸⠃⡞⣡⠖⢀⡄⠀⡀⠀⠈⠑⠚⠠⠾⠅⡤⠀⠄⠀⠀⡸⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠇⠒⠲⢠⡏⡄⡘⠘⠀⣸⠃⡜⠐⣫⠃⣼⠀⡴⠀⡂⠀⣄⠀⠀⢠⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠂⡐⡂⢠⠀⡀⠉⠉⠑⠚⠇⠸⢶⠬⡤⣤⠏⣐⠐⠋⠀⢰⠇⠀⠀⡞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⡁⠈⠳⢁⡾⡐⠡⢎⠃⢸⠆⢠⠒⡀⠀⣄⠉⠀⠉⠁⠒⠰⠏⠀⠀⣸⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡈⠋⠙⠋⠾⠁⣆⣘⡫⢀⡟⢠⠃⢏⠣⢠⡟⢀⠃⣠⠀⢰⠆⠀⠀⣰⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠜⠁⡨⠀⣶⢀⡄⢀⠀⢈⠉⠁⠘⠙⠂⠆⡾⠀⣆⡀⠀⢀⡾⠀⠀⢀⣿⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⣀⠀⠁⣰⠃⠎⢠⡜⠀⣾⠀⡔⢉⡤⠀⣦⠀⡀⠀⠀⢉⠙⠁⠀⢀⣎⢯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢠⠄⠀⠈⡁⠛⠘⠿⠔⢖⣶⠃⣜⢀⠒⠃⣸⠃⡘⠁⠀⠀⣸⠃⠀⠀⣞⢸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠠⠋⣔⣹⢀⡏⡰⠁⠀⠐⣦⠀⢀⢈⠉⠉⠒⠋⠀⠷⠶⣒⣠⡏⠀⠀⠸⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠻⠦⢮⣀⡾⢰⣁⠀⠃⢰⡏⠠⠃⢀⡀⢰⡇⢠⠂⠀⠂⢠⠈⠀⠀⢀⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣍⣉⣐⣒⠀⠀⠀⠈⠁⠉⠓⠟⠀⠧⣄⣠⢠⡟⢀⡀⠐⠃⢀⡿⠀⠀⠀⡌⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡯⠿⢍⡉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠉⠁⠈⠃⠉⠉⠾⠁⠀⠀⢰⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⢦⣤⣀⣀⣀⠀⠒⠀⠀⠀⠉⠉⠙⠒⠒⠦⠤⣤⣬⣉⣐⣂⣿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠢⠤⠤⢤⣤⣀⣀⣤⣥⣚⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛