Sprightly Emojis & Symbols

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠈⠉⠉⠉⠉⠉⠡⠶⠒⠀⣀
⠀⠀⠀⠀⠙⣢⣴⠊⠉⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⡆⠀⠀⠀⠒⠶⣄⣠⠖⠋
⣀⣀⡀⢀⡼⠁⠈⠑⢦⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⣀⠴⠋⠙⣆
⠀⠈⠉⣿⠓⠲⠤⠄⠀⠀⠀⣸⠁⠀⢀⠞⡠⠒⡧⠐⠈⠹⠐⢀⠃⡀⠴⡆⠄⠢⠐⠁⡀⠀⠈⠄⠀⠁⢀⣀⣠⡬⣧⠖⠒⠋
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠂⠋⠀⠔⠀⠠⠀⠀⠀⠈⠁⠀⠀⠔⠀⠢⠀⠀⢠⠀⠀⢰⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠀⠀⠀⠀⠀⠀⠈
⠀⠠⠠⣅⠀⢄⢠⠂⠂⠀⠀⠞⠛⢤⠀⠀⠀⠔⠒⠄⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢⠀⠀⠀⠗⠓⢂⠀⠀⠀⠐⠂⠀⠔⠁⡤⠠
⠠⠰⢤⡈⠃⠀⠘⠀⠀⠀⠐⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠊⠀⠚⢁⡴
⢈⠦⠀⠀⠀⠀⠀⠀⢘⠄⡀⠈⠐⢂⠰⠉⡂⠀⠀⠀⢠⠀⠀⠀⠀⠠⡄⠀⠀⠀⠎⢁⠀⠜⠊⠁⠀⢀⠴⠀⠀⠀⠀⠀⠀⠠⠔
⠈⠂⠤⣀⠀⠀⠀⢀⠄⠀⠈⠐⢄⠀⠢⡀⠀⠀⠀⠀⠀⠢⣀⠀⣀⠔⠀⠀⠀⠀⠀⣠⠊⠀⢀⠀⠊⠀⠀⠢⠀⠀⠀⠀⡀⠄⠒
⠀⠀⠀⠀⠉⠀⠈⠡⢄⠀⠀⠀⠀⠈⠀⠌⡐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⢁⠤⠂⠁⠀⠀⠀⠀⡠⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠈⠰⠄⠘⡃⠀⠐⢲⠀⠀⢸⠁⠠⡄⠈⠀⠀⠀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠐⠀⠀⠀⠉⠀⠚⠚⠒⠈⠁⠀⠀⠈⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠗⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⠀⢀⢆⠀⢀⡀⠀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⡄⠀⠠⠓⠒⠣⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣬⣤⣤⣤⣦⣶⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠉⠉⠉⠉⠉⠉⠉⠀⠒⠂⢠⣤⣄
⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢑⣤⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶
⠀⢠⠋⠀⠀⢀⠞⣀⠴⢲⡀⠤⠰⢲⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⡞⠃⠀⡰⠁⠋⠀⠀⠈⠀⠀⠀⠈⠀⠠⡆⠤⠐⢺⠧⠄⢺⠤⠒⢆⠀⠀⠈⠆
⠀⡇⠀⢠⠁⠀⠀⠀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈
⢠⠟⠷⣈⠀⠀⠀⠀⢡⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⡇⠀⠀⡾
⡇⠀⠁⠈⠀⠀⠀⠀⠐⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡆⠀⠀⠀⠀⡇⣀⠀⡄⠀⠀⠀⠀⢀⠤⠐⠂⠉⠉⠉⠉⠐⠒⠤⣄⡀
⠑⢄⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠠⠞⡍⠻⡀⠀⠀⡠⠊⠀⠀⠀⠀⢠⠊⢀⡠⠭⡐⢄⡀⠉⢢⡀
⠀⠀⠘⡄⠣⣀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠁⢠⠃⠀⠌⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠑⡄⠀⠑
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⢎⣉⠖⡰⠗⠚⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡉⠤⠄⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢒⣦⠄⣀⡀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡀⡘⠚
⠀⠀⠀⠀⠀⢀⡔⠋⠀⠸⢄⣀⢈⣉⡏⢉⡁⠍⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠸⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⣀⠈⠉⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁
⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠐⢍⠇⠀⠀⠀⠀⠀⠀⠘⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠗⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⡇⠀⢸⠀⣠⠔⠒⠛⠙⠒⢮⠶⠏⠈⢙⣀⢢⡀⠀⠀⢰
⠀⠀⠀⣃⡠⢘⡿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠑⠤⣼
⠀⠀⠀⡋⠀⠈⠶⣉⡁⠀⣀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠐⠠⢤⡀⠤⠄⠒⠉⠉⠁⠀⠀⠀⠉⠐⠂⠤⠄⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠀⠤⠤⠠⠞⠄⠠⠄⠀⠖⠒⠁⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠒⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠈⠧⢀⣀⠀⣀⡰⠀⢠⣀⣀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠓⠂⠛⠀⠀⠀⠞⠓⠂⠳⠤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⣀⣴⡖⣀⠀⣬⣷⠿⠿⣷⣄⣒⠶⡞⣠⣀⣸⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠟⢿⣿⣿⣶⣿⣛⡋⠴⢃⣺⣟⣳⠾⡞⠋⠽⠿⢷⣭⢭⠿⢿⣟⠿⠛⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣾⣿⣿⡿⠋⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣷⣮⣌⡠⣤⣶⣶⡶⣶⣴⣤⡜⣉⣮⣡⣷⣄⣡⣤⣤
⠀⠀⠀⠀⠀⣠⣤⣻⣟⠛⠟⣉⣩⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡽⣿⣿⢯⡽⡻⣿⣿⢿⣧⠾⢝⣻
⠀⣠⢴⣾⣿⡿⠋⠋⣉⡉⣶⣭⣭⣽⣿⣿⣿⣿⡄⠀⠻⣿⣿⣿⣿⣿⣷⣶⣾⣷⣿⣿⣿⣿⣿⣿⢙⣿⣿⣿⣾⣶⣿⣿⣿⣿⣷⣎⠺
⠉⢿⣯⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⡀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣿⡟⠻⢟⠿⠟⠿⠛⠛⠣⣴
⠀⠀⠀⢛⣿⣿⣿⡿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⣀⠀⠐⢈⡉⣉⠩⣽⠿⠛⡻⠿⠻⣷⣞⣱⣲⠂⠀⠝⠋⠙
⠀⠈⠀⠀⣠⣴⢒⣈⣥⡠⢤⣶⣤⡀⢉⣥⣍⠉⢹⣻⠿⠿⠿⠒⠀⠶⠈⠛⣻⠟⠛⣷⢛⠙⢿⠿⠤⣧⣜⣒⠚⠒⠚⠁⠀⠀⢀⣵⣶
⠉⠉⠉⠤⠉⠻⠿⣿⣿⣷⠈⠽⠿⠹⠈⢛⡿⠿⠟⠀⠀⠐⢢⣐⣮⣁⣠⣾⠃⠀⣼⠃⠀⠀⢠⡶⠤⡁⠋⠈⠙⠓⢦⣄⣀⡰⠷⠿⣿
⠈⠀⠒⠂⠀⠤⣤⡤⣤⣤⠴⠶⡶⠖⠚⠁⢀⠀⠀⠐⠀⡀⠀⠈⠛⣛⠉⠉⢙⡿⠁⢉⣀⡀⠠⠀⠀⠉⠀⠘⠉⢤⣻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢠⡀⠀⢂⡥⠈⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠐⠁⢓⣮⣿⠾⠟⠛⠓⠛⢀⢂⠀⠀⠀⠈⠁⠀⣈⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⢡⠋⠀⠀⠀⠀⠀⠐⠀⠎⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⢤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣠⣠⣶⣷⣶⣾⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢀⡋⠛⠀⠐⠠⠄⢨⡍⠿⠗⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣫⣭⣽⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⠛⠉
⣶⣶⡶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⣶⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡇⠀⠀⠀⠀⠀⣿⣹⠿⢧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡰⣞⠀⠀⠀⠀⠀⢀⣼⠖⠛⠛⠁
⠀⠀⣠⠶⢦⡀⠀⢀⠀⢀⡴⠶⠓⢢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣗⣦⣀⠀⠀⣸⠁⢀⡴⠃
⠀⣾⢵⢠⣀⢷⡼⠟⣷⢿⣦⢄⡀⠈⣿⣇⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣷⣼⠁⠀⡾
⠀⣿⠀⠎⠛⢂⡱⠀⢻⡾⠅⠈⠀⠀⠙⢉⣰⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⢬⣧⣿⠀⠀⡇
⠀⢹⠀⠀⠀⠈⠁⣠⡿⠶⠶⡞⠉⠛⣶⠛⠻⣿⣶⣭⡙⠛⢿⣶⣤⡀⠀⣰⡷⠋⣿⣿⣿⠀⢠⣇⣀⣠⡀⠀⢀⣠⠄
⠀⢸⠆⠀⠀⢀⣼⣿⠀⠫⠶⠛⣄⠀⢸⣿⠿⠛⠛⠻⣿⣷⣾⠙⢻⣿⣾⣽⣃⠄⣿⣿⣿⣶⣟⣻⡿⡠⠉⠙⠉
⠀⠘⠦⠿⠞⠋⠁⢹⣄⠀⠀⣠⣄⠀⢸⠃⠀⢠⣴⣞⡽⣟⣷⡛⠋⠉⢁⢀⣿⣟⣔⢟⡛⢽⣿⣿⠟⠓⠒⢂⣠⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⡤⡈⢁⣴⠋⠀⠀⠘⠉⢉⡟⡿⢿⣾⣦⣤⠶⠛⠙⣿⣧⠉⠉⢒⣈⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠛⠁⠀⠀⠀⠀⠀⠀⠙⠓⠲⣦⠬⢭⣭⡝⠟⢛⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣰⢯⡝⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣿⣾⠇⠀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣥⡬⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳

⠀⠀⠀⢀⠀⠀⢸⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⣀⡀
⠀⠀⠀⠀⣁⠴⠒⠛⠓⠦⣄⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⠆
⠀⠉⠂⡴⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣦⡄
⢠⡤⠘⡇⠀⠀⢀⣤⡾⣿⢆⠘⡄⠙⠁⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠆
⠀⠀⠈⢧⠀⠀⢿⡷⠛⠀⡸⠀⢣⡀⠀⠀⠀⠀⠀⢠⠿⠀⠀⠀⢀⡴⠃⢀⡠⡶⠀⢀⣀⠠⡤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠐⠋⠀⠑⠦⢄⠙⠒⠳⡑⠄⣠⡗⠀⠀⠀⠀⠀⡘⠆⠀⠀⣰⢋⡧⠊⠁⠀⠓⠉⠀⠀⠐⠒⠒⠈⡝⠀⢀⡀⣤⠀⢀⢠⡄⢀⡠⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠘⠀⠀⠱⣆⣠⣴⠿⣉⢿⣄⠀⠀⠀⠀⣧⠀⠀⠰⠁⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠘⠃⠋⠀⠀⠛⠉⠁⠘⠋⠁⠀⢡⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣹⡾⠋⠀⠀⠀⠀⣪⠀⠀⠇⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡷⣄⠀⠀⠀⠀⠀⢀⠔⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠈⠀⠀⠀⠀⠀⠸⣄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⡄⠀⠀⠀⠀⠀⡇⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠁⠀⠀⠀⠀⢀⡥⠞⠳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡀⠰⠊⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⡠⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⠀⠀⡜⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠣⠄⠼⢀⡜⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⢸⠉⠁⠒⠂⠤⠤⠄⢀⣠⠖⢲⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⢙⡁⠀⠉⠛⢷⠖⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⠚⠂⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⡏⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡈⠲⢄⠀⠀⠀⠀⠀⡰⠣⡌⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠃⠀⠀⠀⠀⠀⠀⠀⠘⠤⠕⠀⠉⠓⡖⠘⠋⠀⠀⠙⢌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠒⠀⠉⠉⠢⣙⠄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠤⠐⠒⠀⠀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⡈⠢⣬⠳⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠦⣠⣤⡀⠐⣄⣱⡶⠋⠀⠈⢦⣄⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⠤⠤⠐⠒⠉⠁⠀⠀⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠒⠂⠤⠄⠀⢀⣀⡞⢅⣀⡀⠀⠤⠤⠒⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⣤⠤⠤⡤⠼⠀⠰⠤⣤⠤⠤⣤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠇⠀⠀⠷⠀⠀⠀⢰⠧⠀⠀⠷⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢰⣴⡄⠀⠀⠀⡤⣦⣠⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣾⣿⣿⣾⣂⡅⣸⣒⣱⣿⢷⣤⣿⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⢛⢳⣿⢿⢺⣷⣿⣝⢿⢓⣻⠇⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣄⣌⠻⣾⣷⣿⣿⣿⣿⣿⠮⠙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⢀⠀⠻⠀⢿⣿⣿⣿⣿⡛⢀⣴⣾⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣿⣿⣿⣬⢿⣿⣿⠟⠋⠁⢀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⠶⠢⢀⠳⠀⠀⠠⡀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣧⣻⣏⡟⠀⠀⠉⢛⣀⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣄⠲⡎⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⡀⠋⠈⠁⠀⠉⢹⠘⣿⠠⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⠄
⠀⠀⢀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠈⠀⠋⣠⣾⣿⣿⣿⣿⣿⣛⠋⠙⠛⢻⣿⣿⣿⣿⣿⣿⠃⠀⠂
⠀⠀⠀⡀⠀⠀⢀⣠⣄⣀⣤⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠢⠄⣀⠀⠈⠙⣻⣿⣿⣿⠃⠠⣄⢀⡀⣀
⠀⠀⠀⠋⠀⠀⠠⠀⠸⣋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠅⠀⠈⢈⣁⣐⣿⣿⠟⠁⠀⠀⣁⣿⣥⣇⣠⠄
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠟⠋⠩⠿⠟⠉⢉⡛⣻⣿⣽⣶⣶⡾⠿⣿⡟⠛⠄⠀⠀⢴⣄⣼⢿⣽⠁⠀⣀
⠀⠀⠀⡀⠐⣿⣿⠿⠟⠉⠀⠀⠀⠀⠘⠀⠀⠻⠋⢻⠷⣌⣰⡅⢠⡿⠿⠷⠆⠀⠀⢀⣸⣿⠏⣿⠟⡀⠘⠻⠂
⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠟⣦⠀⣀⠀⡄⠠⢀⣰⡤⡷⡶⠝⠓⢶⠶⡆⠀⢀⣦⣼⣿⣇⠾⠟⣉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣴⡆⡺⠯⡾⠿⠀⠀⠀⠰⠀⢰⣤⡿⢿⣷⣿⢋⣴⣿⠏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠻⢖⠄⠛⠀⠀⠀⠀⠀⢠⣤⠞⠉⣰⣿⠿⡃⢸⣿⡷⣏⠉⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣋⣀⣀⣀⣀⣀⣀⣚⣙⣁⣠⣚⣉⣀⣈⣁⣁⣀⣀⣉⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⠉⠉⠉⠁⠐⠀⠄⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠠⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⣿⡗⣿
⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢖⠀⠀⠀⡜⠀⠀⢀⠜⠡⠀⢤⠑⠀⢀⠄⡀⠀⡀⠀⠈⡄⣿⡗⣿
⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢰⠀⠀⠀⡌⢀⠁⠀⢀⠀⠀⡌⢠⠠⢊⠔⠃⠀⢰⣿⣏⣿
⢠⠏⠀⠀⢠⠆⡠⠐⡆⠄⠒⠨⠥⠀⡚⢀⡀⢴⣀⠄⣦⠐⠒⡄⠀⠀⢑⠘⠀⠀⠀⠠⡀⠦⠤⠂⡠⠱⠀⡬⠰⠁⠀⠀⠀⢸⣿⡟⣿
⢸⠀⠀⢠⠱⠃⢀⠤⠠⠀⠀⠀⠀⠀⠋⠁⠀⠈⠤⠠⢄⠀⠀⠰⠀⠀⠀⠀⠡⡀⠀⠀⠈⠁⠀⠁⠀⢇⡘⠀⠄⠆⠀⠀⠀⡘⣿⡇⣿
⣿⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⢀⠔⠀⣿⣧⣿
⢹⠷⣄⡀⠀⠀⠀⢠⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠀⠀⠀⠀⢀⡴⢤⠀⠀⠀⢀⡌⡠⠠⠄⠀⠀⠀⠀⡀⠠⠄⠂⠁⠀⠀⣿⣷⣿
⡆⠰⠀⠁⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠉⠸⠀⡆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠱⠄⣂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠉⢂⠓⠐⠃⠀⠀⠀⠀⠒⠤⠀⠠⠄⠂⠀⠀⠀⠀⠓⠀⢃⠎⠁⠀⠀⠀⡆⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡀⠠⠠⢤⠃⣸⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠁⠀⠀⡀⠖⠒⠆⠀⠀⢁⡴⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠠⠒⠘⡁⠉⠉⠋⡇⠉⠁⡏⠐⠠⠀⠀⠀⠀⢃⠈⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⡦⢀⠀⠀⠳⢄⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣐⡆⠀⠀⠀⠀⠀⠀⡇⠀⠑⠠⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⢂⠆⠀⠀⠀⠀⠀⠀⡏⠐⠄⣀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠌⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡇⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠙⠀⠃⠀⠁⠐⠒⠠⠤⠊⠢⠄⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹
⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⡂⡂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣺
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢾
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢾
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣺
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣉⠽⠅⠹⠍⠀⠈⡹⠍⠨⢍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣴⡶⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠶⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣠⠀⠀⢀⡠⠀⠀⣀⠠⠶⠀⠀⣀⡤⠀⠀⠀⢀⠀⠀⢀⡀⠀⣠⣄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠁⠀⠀⠀⡼⣇⡠⠚⠁⡷⠒⠉⠀⠀⠛⠊⠉⢀⣃⣠⠔⠊⣽⠤⠒⠉⠷⠊⠁⠘⡆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⡞⠰⠏⠀⣀⠤⠤⠤⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠤⠤⠤⣄⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣤⠖⠘⢳⠀⠀⠀⠀⠀⢈⠾⠻⢦⣅⠀⠀⠀⠀⠀⢰⠁⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⡃⠀⠀⠀⠀⠀⢸⡠⡟⠰
⠀⠀⠀⣀⠔⠳⡀⠘⢦⡀⢸⠀⠀⠀⠀⠀⠘⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠘⢇⡀⠀⠱⡀⠀⠱⣼⡀⠀⠀⠀⠀⠰⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀
⠀⡴⢤⡀⠈⠙⠒⠷⠀⠀⠀⣠⠶⢦⡀⠀⠀⠉⠲⠾⡄⢠⡞⠉⢢⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡴⠉⠱⡄⢀⡾⠖⠋
⡜⠁⠀⠙⠢⣀⠀⠀⠀⠀⢰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠙⣄⠑⠒⠊⠀⠀⠀⠀⠀⠀⠉⠓⠠⠤⠤⠤⠚⠉⠀⠀⠀⠀⠀⠀⠙⠒⠚⢠⠞
⠉⠙⠛⠛⠁⠙⠀⠀⠀⠀⠘⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠀⠀⠡⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⣀⣀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⣤⡤⢀⣀⣀⣀⣀⣀⣀⣀⢠⣤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠉⠲⣀⡘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⡄⠀⠀⠀⡇⠘⠀⠀⠀⢀⡇⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠤⣸⡇⠼⠆⠀⠀⢸⡀⠀⠀⠀⠀⢀⢼⠏⠀⠀⠀⠀⠉⠑⠒⠒⠓⠚⠒⠒⠊⠉⠀⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠊⠀⠀⠀⠀⢀⠎⠱⡄⠀⠀⢠⠎⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡖⠒⠂⠁⠀⠀⠘⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⡀⠀⢀⡠⠞⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠠⣎⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⠤⢄⣀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⣀⣠⣤⢶⣿⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⠀⠀⠀⠄⠤⠄⠹⠤⠤⠀⠘⠛⠉⠉⠀⢸⡏⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⣠⣠⣤⣤⣤⣤⠇⠀⣇⣤⣤⣤⣤⣤⡠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠓⠒⠒⠻⡄⠀⠀⠀⡜⠒⠒⠚⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣉⣀⣀⣀⡀⠰⠇⠀⠀⠀⠦⢀⣀⣀⣀⣈⣹

⠀⠀⠀⢀⠀⠀⠸⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠄⢀⣀⡀
⠀⠀⠀⠀⣁⠴⠒⠛⠃⠶⣄⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⠆
⠀⠉⠂⡴⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣦⡄
⢠⡤⠘⡇⠀⠀⢀⣤⣾⣿⢆⠘⠀⠉⠁⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠄
⠀⠀⠀⢧⠀⠀⢿⡷⠛⠀⡸⠀⢧⡀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⡴⠁⢀⡠⠦⠀⢀⣀⠠⡤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠐⠋⠀⠑⠢⣄⠙⠒⠳⡙⠄⣠⡟⠀⠀⠀⠀⠀⡜⠇⠀⠀⣰⢋⡧⠊⠁⠀⠓⠉⠀⠀⠘⠓⠒⠈⡝⠀⢀⡠⣤⠀⢀⣠⡄⢀⣠⡀⠀⠀⠀⢧
⠀⠀⠀⠀⠘⠀⠀⠑⣆⣠⣴⠿⣉⢿⣄⠀⠀⠀⠀⣧⠀⠀⠰⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠛⠉⠀⠘⠃⠁⠀⢡⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣿⡿⠋⠀⠀⠀⠀⣨⠀⠀⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⡷⣤⠀⠀⠀⠀⠀⢀⠔⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠈⠀⠀⠀⠀⠀⠘⣄⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⡄⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⢀⡥⠞⠳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡀⢰⠊⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⠀⠀⡜⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠣⠀⠼⢀⡜⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⢸⡉⠁⠒⠂⠤⠤⠄⢀⣠⠖⢲⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠙⡁⠀⠉⠛⢿⠖⠀⠀⠀⠀⠀⠁⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠉⠒⠂⠐⠿⠀⠀⠀⠀⠀⠀⠀⠀⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⠲⣄⠀⠀⠀⠀⠀⡰⠣⡌⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠃⠀⠀⠀⠀⠀⠀⠀⠘⠤⠗⠀⠈⠛⡖⠘⠋⠀⠀⠙⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠒⠀⠉⠉⠢⣙⠄⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡥⠤⠐⠒⠃⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠚⠁⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠠⡈⢢⣬⠳⡄⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠦⢀⣀⡀⠐⣄⣱⡾⠋⠀⠈⢦⣀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⠤⠤⠐⠒⠉⠁⠀⠀⠀⠀⠈⠒⠋⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠒⠂⠤⠄⠀⢀⣀⡞⢀⡀⠀⠀⠤⠄⠒⠂⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣤⠤⠤⡤⠼⠀⠰⠤⡤⠤⠤⡤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠇⠀⠀⠷⠀⠀⠀⠰⠇⠀⠀⠷⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⠐⠒⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⣥⡶⠶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡴⠀⣀⠴⠆⣠⠄⠒⢙⡀⠤⠐⡆⠀⠀⢠⡄⢀⡠⣴⡀⠔⠢⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⢎⡥⠊⠀⠀⠋⠀⠀⠀⠈⠀⠀⠸⠤⠒⠉⠐⠛⠈⠀⠈⠁⠀⠀⢣⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡌⠈⠀⠀⠊⠁⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠑⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠠⠄⢲⠀⠀⠀⠀⠀⢸⠷⣆⠀⠀⠀⠀⠀⠀⠰⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠳⠀⠀⠀⠀⠀⠨⣠⡦⣴
⠀⠀⠀⢀⠴⡀⠈⢦⠀⢸⠀⠀⠀⠀⢠⡎⠀⡄⠘⠂⠀⠀⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠀⠀⠀⠀⠀⠘⢡⠀⠈
⠀⠀⠰⢁⠀⠘⢆⠀⠑⣨⡀⠀⠀⠀⠘⢇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀
⠀⠤⢄⠀⠙⠂⠼⠆⠀⠈⣀⠤⢤⠀⠀⠈⠓⠼⣆⠀⡔⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠲⡄⢀⠷⠔⠋
⣜⠀⠀⠑⠤⡀⠀⠀⠀⢰⠁⠀⠈⠀⠀⠀⠀⠀⠘⢆⠁⠂⠉⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠒⠈⢠⠂
⠈⠁⠀⠀⠒⠃⠀⠀⠀⠈⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⢠⠤⠐⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⡐⠶⠀⠤⠤⢤⢤⠤⠤⠄⢲⠖⣈
⠀⠀⠀⠀⠀⠀⠀⢘⣄⠈⠲⠆⢠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠰⠤⠀⡀⣈⣸⡀⡀⢠⠼⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⡙⣺⠓⠋⠀⠀⠀⣧⠀⠀⠀⠀⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⡀⠠⡤⠊⠀⠱⡀⢀⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠤⣀⡀⠀⠀⠀⠀⢠⢆⠀⠀⠀⠀⢀⣀⠠⠐⣏⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉⠁⠀⢉⡈⠂⠈⠉⠉⠀⠀⠀⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⢙⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣀⣀⣀⣀⣀⠆⠀⣄⣀⣀⣀⣀⡀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠼⠒⠒⠚⣄⠀⠀⢀⠖⠒⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⡠⠤⠋⠀⠀⠈⠤⢤⣀⣀⣀⣨⡇
⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿

⠀⠀⠀⠀⡀⠄⠂⠀⠀⠉⠉⠉⠉⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡜⠁⠀⡠⢂⠠⡆⠀⠤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠠⠀⠀⠈⠈⠀⠀⠀⡀⠀⠀⠀⠆⠀⠸⠀⠐⠂⠘⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⢠⣧⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗
⡆⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠸⢀⢀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢸⣿
⠐⠤⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠍⠁⠀⢀⠔⠁⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⡀⢸⣿
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠀⡀⣰⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠐⣸⣿
⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡈⠤⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣀⠡⡆⠠⢄⣀⠀⢀⡀⢤⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠠⡊⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⡎⠀⠀⠀⠂⠐⠐⠀⠂⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠐⠀⠁⠀⠀⠀⠀⠀⡠⠂⢸⣿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠁⠐⠀⠀⠀⠂⠈⠀⠀⠀⢸⣿
⠀⠀⠈⠀⡇⠀⠀⣀⢀⣀⢀⣠⢠⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⢧⡖⠀⠀⠀⠉⠉⠁⠀⠐⠐⠦⢄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠚⠀⠩⢂⠀⢀⡀⢠⡄⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠀⠦⢐⠀⠀⠀⠀⡀⠀⠀⠁⢐⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⢀⠰⠂⠲⠀⠀⠰⠀⠲⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉

⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣠⣤
⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢀
⠀⡰⠃⠀⢀⡄⢀⢠⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠓⠀⠠⠹⠊⠀⠈⠀⠀⠈⠈⢠⠡⠄⢰⠤⠰⠤⠐⠄⠀⠘⡀
⠀⣅⠀⠁⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⡃
⡜⠉⠲⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⢑⠂⠀⠀⢠⠀⠀⠇⠀⠀⠀⠀⢀⠀⠠⠤⠤⠀⡀
⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠨⠜⠢⠀⠀⢀⠔⠊⠀⠀⢠⠀⠀⣀⠀⡈⠒⢄
⠀⠈⢂⠪⠩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠠⠁⠀⠀⠀⠀⠈⠀⠁⠀⠱⠈⠂⠀⠡
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠘⠤⢂⠚⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⢀⠄
⠀⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢐⡁⠈
⠀⠀⠀⠀⢀⠄⠂⠹⠁⠀⠂⡆⠒⠰⠂⠂⢅⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠂⠐
⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠈⠈⠀⠀⠀⠀⢀⠠⠐⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠚⠈
⠀⠀⢠⠀⠀⠀⢀⣠⠔⢤⣀⣤⠄⠤⠄⡀
⠀⠀⠈⡀⢰⡞⠁⠀⠀⠀⠈⠀⠀⠀⠉⠘⠒⠄⣀
⠀⠀⢘⠀⠘⠆⡉⠀⣀⠀⢤⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠑⠲⢐⠂⠈⠀⠀⢀⠀⠀⠈⠁⢒⢠⠒⠈
⠀⠀⠀⠀⢰⠀⠈⠁⠀⠀⢁⡁⠀⠀⠉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡤⢠⡤⠂⠠⣤⠀⡤⠐
⠀⠀⠀⠀⠀⢐⣂⣁⡈⠁⠀⠀⠁⢀⣈⣐⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠀⠴⢄⣀⡴⠙⠛⠉⠦⣀⡠⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠘⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣦
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠃
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠈⠳⣄
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠉⠀⢀⡀⠀⠙⣦
⠀⣸⣿⡀⠐⡀⠀⢀⣰⠆⠀⢀⡤⣾⠀⣀⣤⠄⠒⢻⡃⠀⣤⠄⡿⠀⠀⠀⠀⣤⡄⠀⠀⣤⣄⣿⡤⠞⣆⠈⢥⠄⠀⠘⢇
⠀⢿⣄⠟⠀⠉⠈⢻⡏⢠⠞⠉⠀⢾⠎⠉⠀⠀⠀⠘⠉⠉⠀⢰⡀⣀⠰⠆⠉⣦⠶⠎⠁⠸⠟⠉⠠⠎⠙⣇⠀⠀⠀⠀⢸
⡆⠸⣋⢠⡄⢀⡾⠻⡟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡄
⡇⠸⠋⠀⠀⢸⠛⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠇
⡇⢰⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⠳⢄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡠⠞⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⣼⣧⡀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡐⠚⡍⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢩⠓⢆⡀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡄⢀
⡠⠟⠙⠛⢶⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡶⠛⠋⠻
⣷⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠
⠏⡀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀
⠀⠓⠤⢠⣇⠀⠀⣠⠄⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⣄⠀⠀⣸⡄⡤⠊
⠀⠀⠀⠈⠙⢆⠀⢧⣄⣠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠂⠒⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢀⡼⠀⡰⠋⠁
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠐⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠀⢹⠁⠉⠉⠉⠙⡟⢹⠋⠉⠉⠉⠈⡟⠠⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠘⢤⠄⠀⠀⣀⣇⣸⡀⠀⠀⠠⡤⠃⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⠀⢀⣠⣷
⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠰⣎⠀⠀⠀⠀⠀⠀⠀⢀⡛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⢀⣀⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⣀⡀⠀⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠉⠛⠂⣤⣤⠀⠀⠀⠀⠀⢀⡼⢇⡀⠀⠀⠀⠀⢀⣤⡤⠐⠚⠉⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠁⠈⠁⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠄⢀⣀⠀⠀⣀⡀⠴⠀⠀⠦⢀⣀⠀⠀⢀⡀⠠⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠹⣇⠀⠀⠀⠀⣸⠇⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠉⠑⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⢰⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⣧⠀⠀⠀⢹⠀⠀⠀⢸⢀⠀⢸⠀⠀⠀⡏⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠘⡆⠀⢆⠈⠆⠀⡇⠈⢸⠀⡈⢠⠃⠸⠀⢀⠀⣰⠋⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡀⠳⡄⠘⢆⠘⠀⠘⡀⠀⠀⠇⠀⢸⠀⠀⡘⢠⣶⢦⡉⡰⠃⢀⠄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢦⡀⠘⢦⠘⠀⠈⢆⠀⠀⠁⠀⠀⢠⠀⠘⠀⠀⠁⣾⠃⣸⡇⠁⣠⠋⡠⠋⠀⡀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠑⢤⡙⠀⠀⠓⡀⠀⠈⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢸⣿⡴⠛⠀⠐⠁⠀⠀⠰⠊⣠⢖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠄⡀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⢸⣿⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠈⠛⠲⢤⣀⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⣰⠏⠠⠒⠉⢸⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣧⠀⠀⢀⣼⠏⠀⠀⠀⡀⢸⣿⠐⢁⣀⠤⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⣏⢸⣿⣧⡿⠈⠳⠶⠋⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⢀⣤⣴⠆⠀⠀⣰⣿⡶⡟⢰⣿⡇⠀⣿⠀⠙⠛⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣉⣁⣀⣤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣼⣧⠀⢀⣿⣿⠀⣿⠈⠻⡷⠼⠃⠀⠀⠀⠀⢹⡇⠀⣤⡀⠀⠀⠀⠀⢀⢸⡿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⢰⣿⡇⢀⣿⣿⣆⡾⢿⣿⢀⡿⠀⠀⠀⠀⠀⠀⣀⡀⠀⣸⣧⣼⣻⠇⠀⠀⠀⠰⡄⢠⡄⠀⢀⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⠃⣿⣿⡃⠈⣙⠛⠁⠀⠀⠀⠀⠀⣸⣿⢻⡄⣿⣿⣿⠟⠀⠀⠀⠀⢸⡇⠘⠷⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠙⠛⠁⠀⣼⣿⣇⠀⣿⣧⠀⢀⣀⠀⣤⠀⣿⡯⠀⢰⡿⣿⡟⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⠀⣸⠏⣿⣿⠀⢸⣿⡀⢸⣿⡀⣾⠀⢹⣿⣠⡞⠁⢿⣧⠘⣿⡄⠀⣸⠇⠀⠉⠒⠦⠄⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⠄⢀⡠⢠⡟⠀⣿⣿⠀⢸⣿⡇⠸⣿⡇⣿⣇⡾⠁⠀⠀⠀⠀⠀⠀⠈⠻⠶⠏⠠⢄⡀⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣡⠖⠉⠀⣾⢀⠄⢸⣿⠀⠸⣿⣇⣼⡿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⢄⠙⠂⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣀⠔⢰⡏⠀⠀⣼⡏⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠑⢆⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⡔⢠⡟⠁⡀⢠⠈⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⢀⠘⣆⠀⠱⣄⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣈⠳⠴⠋⠀⡜⢀⠏⠀⡄⢸⠀⢀⠀⡄⠀⠀⢃⠀⢀⠈⣆⠀⢳⡈⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⡼⠁⠈⠀⢰⠀⠟⠀⢸⠀⡇⠀⣆⠘⡀⠀⢆⠈⠀⠀⢳⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡏⠀⠀⠀⣾⠀⠀⠀⢹⠀⠀⠀⠸⡄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠃⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠐⠂⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⡎⠀⠤⣤⡤⢠⡄⣀⢤⠟⢰⢁⣀⠀⢠⣏⡇⢀⢄⣤⡠⢤⡄⡠⢀⢀⣄⢰⠗⢠⡄⣠⡇⠰⠁
⠀⠀⠈⠒⠋⠈⠀⠈⠃⠉⠉⠒⠈⠉⢹⠀⠘⠀⠁⡘⢹⠋⠁⠈⠃⠉⠉⠉⠉⠉⠉⠈⠁⠉⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢀⠇⠀⠀⠀⠀⠇⠘⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠬⠥⠒⠐⠒⠂⠀⠉⠉⠉
⠀⠀⠀⠀⠴⠒⠊⠉⠁



⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢟⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣯⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣛⣹⣿⡿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣯⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣭⣷⠿⠻⢍⠀⠈⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣼⣿⡉⠀⢦⡀⢱⣾⣿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣻⣾⣿⡁⠀⠙⠆⠈⣿⡿⠿⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⡟⠛⠃⢠⢀⡘⣦⣾⣇⠁⠀⢠⢻⡿⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣿⣿⣿⡟⠻⣖⠀⠀⠊⢷⣈⣿⣿⠀⠀⢸⣿⣧⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⣿⣾⣿⡟⠛⠋⠃⠈⣃⠀⠐⣾⣿⣿⣏⢠⠞⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⢣⣤⡀⠀⠀⠘⢦⣿⣿⣿⣿⡿⣁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠶⠾⠛⢒⠀⠀⡎⢯⢋⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⠀⠀⢛⠁⠠⢻⠀⣿⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣞⣀⡤⢜⠳⣠⡆⢹⣷⡿⣿⠿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⢠⣾⣿⣿⣿⣷⣿⡿⢿⣿⡟⠹⣇⣸⠀⢸⣿⣸⠟
⠀⠀⠀⠀⠠⣄⡀⡀⠠⢿⢲⠀⣠⣿⠏⢻⣿⣿⣭⣤⠀⢿⣛⡿⠀⢬⣽⠷⣸⣿⠟
⠀⠀⠀⠀⠀⣿⣿⣶⣿⡇⠀⢰⡿⠁⢀⣿⣿⣿⣏⣿⠲⢌⠛⢣⡝⣰⢻⣾⣿⡟
⠀⠀⠀⠀⠀⢹⣿⣿⣟⠁⣰⠟⠁⠀⣾⣿⣿⣿⣹⣷⣤⣿⠂⢹⣧⡽⠟⢿⣿⣷
⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⢀⡆⠀⣸⣿⡟⠻⣿⣿⣷⣿⡿⣀⣀⠈⣿⣴⣿⣿⠋
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⠟⠀⢠⣿⢙⡁⢰⣿⡟⠋⠙⠳⠼⢥⣀⣿⣿⣿⡏
⠀⠀⢀⣾⣿⡿⠃⠘⣿⡇⠐⣠⣼⡟⠀⡄⢸⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⠏
⠀⠀⣾⣿⡿⠃⠀⣰⢋⣷⣾⠃⡿⠁⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⢩⡟⠁
⠀⣼⣿⣿⠃⠀⢰⡟⢀⣿⣿⣴⠁⠀⠀⡾⠇⠀⠀⠀⠀⠀⠀⢰⡟
⣼⣿⣿⠃⠀⢠⣿⢁⣾⣿⣿⣿⣦⠄⡸⠀⠀⠀⠀⠀⠀⠀⠀⡼
⣿⣿⠃⠀⢀⣾⠇⣾⡿⣿⠽⣿⣏⡰⠃
⣿⠇⠀⠀⣾⠃⠈⠉⠀⠀⠀⠈⠉⠁
⠋⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡀

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣚⣛⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣗⡉⣤⢿⡌⢿⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣙⣉⡍⠠⠚⠁⡈⣸⢟⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣠⡖⠀⠀⢋⣼⣧⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⢣⠀⢸⡉⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⡀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣧⣌⣻⣿⣿⣿⠄
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⣴⣿⣿⣿⣿⢟⣹⣿⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⢺⣅⣀⣠⣴⣿⣿⣿⣿⣇⣿⢋⡇⠹⣯⣿⡁⢿⣿⣟⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠋⢉⣿⣿⣿⣿⣿⣿⣿⠎⠀⢀⡨⢿⠁⣤⣟⣿⣾⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⠏⠐⡂⠸⣅⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⢹⣃⣠⣴⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡼⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⣠⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦
⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿

⠀⠈⠳⠿⣦⣄⣀⡀⠀⠈⣈⣻⣷⣤⣤⣹⣶⣤⣀⣀⡀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⣼⢃⠀⠀⣠⣿⠁⠐⢛⡕⠉⠀⠀⢀⣀⣠⡤⠶⠒
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠉⢉⡉⣛⣿⡿⠿⠿⠿⢿⣷⣦⣤⠄⠀⠀⠀⠀⢀⣤⣾⣷⣤⣾⣿⣧⣤⠶⠾⠛⠿⠶⠶⠚⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⡀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⡇⠀⠀⠀⠀⠀⠛⠛⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⠀⠀⠀⠀⠀⠀⢺⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠉⠢⢄⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣀⡀⠀⠀⠄⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠽⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⣿⣿⡀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣠⠞⠁⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠘⣻⣽⣠⣤⣼⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣉⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⣛⠗⣿⠆⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣮⣤⣀⡀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠑⠦⡀⢀⣤⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⡾⡿⠛⢷⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣤⠌⣼⣿⣯⣜⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣍⣿⣾⣢⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⡹⠖⢲⣿⠋⠉⠉⣵⠂⠙⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⢟⣷⣶⣍⣿⣿⣿⡇⠀⢹⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠟⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⢿⡄⢀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣥⣠⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⣸⡯⣿⣿⣿⣙⣿⣟⣿⣿⣿⠀⣿⣿⣟⣩⡿⢿⢻⣿⣿⣿⣿⣿⡿⠙⣧⣿⡸⠏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢤⣽⣿⣧⡿⣧⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣦⣝⡟⢿⣷⣿⣿⣿⣿⡿⠁⠀⠘⣿⠇⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⢸⣿⣧⣿⣿⣿⣷⣝⣿⣿⣷⣿⣿⣏⣽⢿⣿⣶⣿⣿⣿⣿⡿⠁⠀⠀⠀⢹⡄⠀⡌⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⡀⣿⣿⡆⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢹⣿⣿⣿⣿⣿⣿⣿⣆⡸⢿⣿⣿⣿⣷⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢳⣽⠿⢋⣸⣿⣿⠆⢀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⡿⢿⢿⣿⣿⣿⣿⣿⣿⣶⣝⠻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣰⠟⢿⡟⠁⠀⣿⣶⡿⠛

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣽⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⢰⣶⡶⠿⢿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⣆⠘⠁⠀⠀⠀⠈⠳⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣽⣏⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⠀⡄⢀⡤⣄⠀⣄⣤⠠⣄⣠⠄⠀⣠⢤⡄⢀⣤⡀⠠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀⢿⣤⠇⠸⣧⡤⠀⡟⢧⠀⢹⡏⠀⠀⢿⣤⡄⣼⠿⣧⠀⣿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣶⣿⣿⡿⣿⣿⣶⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⣿⣿⣾⣿⠋⠁⠀⠀⠀⣠⣿⣿⠿⠟⠛⠿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⢸⣿⠃⠀⡀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⠿⢶⣄⠀⠸⣿⡄⠀⣿⠀⠀⢸⣿⡀⠀⣴⣶⣦⡀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠁⠀⠀⠀⠀⠀⢀⡀⠙⣷⡀⠹⣷⣀⠀⠻⣧⣀⢹⣿⣿⡇⠀⢰⡄⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⢀⣠⣤⣤⣤⣤⣄⡀⠙⢶⣬⣙⠶⣬⣿⣷⣶⣾⣿⣿⣿⣿⣧⣴⣿⣿⣿⠟⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣾⣿⠿⠛⠛⠛⢻⣿⣿⣷⣦⣽⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⢿⣿⣿⠛⠁⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠁⠠⣤⣤⣶⣿⣿⠃⠈⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣀⣀⣀⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢤⡈⠛⢿⣿⠏⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠸⣿⣿⣗⡀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠈⠻⣿⣿⡟⠀⢐⣫⣭⠭⣭⣍⡓⠄⠀⠀⠀⠀⠰⣋⣭⣭⢭⣭⣕⡀⠀⠹⣿⣿⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⣴⣟⣠⣾⠟⠁⣠⣤⣤⣌⠻⣦⣀⠀⢀⣠⡾⢋⣤⣤⣤⣀⠉⢻⣦⣄⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣉⣛⢿⣿⡇⠀⠀⠀⣸⣿⡟⠃⢹⣿⠿⢿⣿⠃⠚⣻⣿⡀⠀⠀⠀⣿⣿⠟⣋⣩⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⢹⣧⡀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢠⣿⢃⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⣙⢷⣦⣄⣀⣠⣴⡿⠋⠀⠀⠀⠈⠻⣷⣤⣀⣀⣤⣴⠿⡃⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⣦⠈⠉⠉⠉⢁⡀⠀⢶⣤⣤⠖⠀⢀⠉⠉⠉⠉⢁⣾⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣅⢻⡆⢀⣀⣤⡾⠡⣤⣄⣈⣁⣠⣤⠜⢷⣄⣀⠀⣸⡇⣬⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠘⢿⣄⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⣰⡿⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⠈⢿⣦⣤⣀⣀⣤⣶⣿⠃⠰⣾⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡿⠟⠛⠉⠁⠀⠀⠀⢻⡟⠛⠛⠋⣽⠏⠀⠀⠀⠈⠉⠛⠻⣿⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠏⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠒⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠔⠈⠉⢁⠀⠖⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠀⡀⠈⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢐⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠢⠤⢤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⣿⡿⣿⣶⣤⣤⣀⣀⠀⠀⢀⠀⠀⠈⠉⠛⠛⣿⣆⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠸⣿⣷⣿⡿⢿⣿⡿⣿⣷⣆⣈⣄⠀⠀⠀⠀⣰⠿⣿⡆⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣾⠿⠃⠀⠉⠓⢦⡀⣼⠃⢀⡿⠈⠢⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣠⡾⠛⢿⡆⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣼⣿⣿⣶⣿⣿⠇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠘⣿⣿⡀⠀⡾⠁⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠘⢰⡟⠋⠀⠈⠻⣿⣶⣇⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⢀⠀⠑⠂⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⡿⢦⣤⣶⣶⣾⣿⣿⣷⣄
⠀⠀⠀⠠⠁⠀⢀⠀⣀⡀⠀⠀⠊⠀⣀⠔⠡⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣽⣿⣿⠂⠀⠃⠀⢤⡀⠀⡤⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣡⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣼⣿⣿⣿⣧⣽⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣿⣿⣿⠟⠋⠉⠁⠈⠉⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣕⣦⡄⠉⢀⣀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣭⣾⣿⣦⣙⠂⠘⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⢻⡞⣹⠇⢦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣿⣿⣿⣻⣿⠁⣸⣿⡿⠟⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣜⣇⣽⣿⣿⣿⣶⣿⣿⣾⣟⠃⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣟⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠿⣛⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣞⡿⢿⣿⣉⠲⢄⠀⣰⡆⠀⣼⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣷⣾⣍⡛⠛⢧⠙⠇⣰⣿⣬⡁⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣷⣤⣀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠐⠀⠀⠀⠀⠀⠀⢀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣓⣤⢲⡇⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣦⣀⠀⠀⠀⠀⠀⠙⢱⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⠃⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⠉⠛⠛⠶⠷⡈⠀⠐⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠲⠶⠿⠦⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⠈⠉⢷⠀⣰⠈⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠀⠈⡳⠋⡠⣿⣿⣿⣿⣿⠆⣘⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⣰⣧⠸⡇⣌⣙⣛⡭⠵⠞⢉⡥⠄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠖⠒⡉⠉⠁⠀⢀⣤⠥⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣊⣁⡀⠀⠀⠀⠉⠤⡔⠛⠋⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⢽⠂⢀⣀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣶⡀⣐⣰⡷⡦⠦⢵⠾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢆⡲⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⣸⣿⣧⢀⢈⡯⢐⣦⡞⠉⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢬⣭⠄⠀⠀⠀⠀⠀⠈⠶⢦⠐
⠀⠀⠀⣿⣿⢻⣸⣯⣶⠰⣿⣬⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠖⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣏⠸⠿⢿⠟⢧⣬⡷⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡂⠀⠀⠀⠀⢂⠠⣅⠀⠀⣐⠉
⠀⠀⠀⠀⣹⣤⡀⠙⠟⠁⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⠀⠀⠀⢨⣴⠀⣷⠀⡁⣼
⠀⠀⣠⣾⣟⠻⢍⡟⣿⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⢴⣿⣿⣏⢉⣼⢗⡡⠄
⠀⠀⣿⣿⣿⣧⡀⣿⣿⢢⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡤⠶⠂⠀⠈⠉⢁⣘⠄⠀⢀
⠀⠀⣻⣿⣿⣿⣷⣸⣿⣆⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣴⣮⣛⠛⠻⣿⠧⠄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠞⠖⠀⠀⠙⠿⠿⠛⠓⠒⠀⠀⠀⠓
⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


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

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⠖⠊⠉⠙⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠉⠑⠲⡀
⠀⠀⠀⠀⡾⠀⢀⣠⣤⣤⣄⡈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⣤⣶⣤⣄⣀⢠⣷
⠀⠀⠀⢸⠁⢠⣾⣿⣿⣿⣿⣿⣆⠀⢢⡀⠀⠀⠀⣀⣀⣠⣤⣤⣬⣝⢻⣟⣛⣥⣤⣤⣤⣀⣀⡀⠀⠀⢀⡴⠁⣴⣿⣿⣿⣿⣿⣷⡄⠘⡇
⠀⠀⠀⢸⠀⢘⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⠓⠻⠃⢉⣤⣴⣶⠦⠤⣬⡭⢭⣥⠤⠴⣶⣦⣬⡉⠻⠟⡻⠏⠀⣸⣿⣿⣿⣿⣿⣿⣿⡃⠀⡇
⠀⠀⠀⠀⠡⡸⣿⣿⣿⣿⣿⣟⠙⠙⢀⠴⠋⢀⠔⠋⣉⣠⣤⣴⡶⠶⡖⢲⠶⠶⠦⣤⣄⣉⠙⠦⡀⠙⠦⡀⠋⠋⣽⣿⣿⣿⣿⣿⢇⠌
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡋⠁⢠⠐⢠⠄⠀⠀⣠⠾⠿⠋⠉⢀⣠⠄⠰⣧⣦⣄⡀⠈⠛⠿⢷⣄⠉⠀⠠⡄⠢⡄⠈⢙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣽⣿⠿⠟⢁⡔⠁⣰⠋⠀⢀⠼⠁⣠⡴⠖⢂⣉⣁⡀⠇⠆⣶⣌⣉⡐⠲⣦⣄⠈⠧⡀⠀⠹⣆⠈⢦⡘⠻⠿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⣾⡁⠀⣴⡟⠁⢰⡇⠀⠀⠀⠀⠜⠁⡄⠀⠉⣸⠏⡤⡀⢠⠸⠹⣏⠙⠀⣦⠙⠣⠀⠁⠀⠀⢹⡆⠈⢿⣦⠀⢈⣷
⠀⠀⠀⠀⠀⠀⢹⠏⢸⡟⠁⢰⣿⣧⠀⣠⣀⣤⡄⢸⡇⠀⢰⡷⠀⠇⠇⠈⠃⠐⠞⡆⠀⢽⠇⣠⣤⣀⣄⠀⣾⣿⡆⠈⢿⡆⠹⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣤⠘⠛⣯⠀⠉⠉⣉⣙⠀⠻⡄⠀⠀⠚⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⣋⣉⠉⠉⠁⣿⠛⠃⣧⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⣿⠀⢠⣿⠷⢶⣿⢿⣿⣻⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⣿⠶⠺⣿⡀⠀⣿⠀⢿⣷
⠀⠀⠀⠀⠀⡆⢸⣿⡇⠀⣿⠀⢺⠻⢷⡄⠻⢮⣍⣥⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣬⣩⡵⠋⣠⡾⠛⡇⠀⣿⠀⢸⣿⡆⢰
⠀⠀⠀⠀⡐⠀⣾⣿⡇⠀⣹⠀⢸⡄⢀⡙⠶⠦⠄⠀⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢿⠀⠠⠶⠾⢋⡀⢠⡇⠀⡏⠀⢸⣿⣷⠀⢄
⠀⠀⠀⠀⠁⣠⣿⣿⠃⠀⢻⠀⡘⢿⣄⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠛⠋⣠⡿⢀⠀⡟⠀⠸⣿⣿⣄⠈
⠀⠀⣠⠀⠀⣿⣿⣿⠀⠀⣸⡂⡇⢀⡹⠶⡣⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣔⠜⠶⢏⡀⢸⢘⡇⠀⢀⣿⣿⣿⠀⠀⣄
⠀⢐⠁⠀⠀⣻⣿⣯⡄⠀⣿⣧⢰⣜⢿⡴⠃⢀⡔⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠑⠒⠬⣞⢦⡿⣣⡆⣼⣿⠀⢨⣿⣿⣟⠀⠀⠈⡀
⠀⣾⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⡏⠻⠄⠀⣠⠏⡴⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠠⣄⠀⢠⠟⢸⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⣷
⠈⣏⠆⠀⠀⠀⢿⣿⣿⡄⠈⠿⣧⠀⠰⣼⡏⢀⡤⠀⠀⢠⣄⣀⣀⣀⠀⠀⣀⣀⣀⣸⣿⢷⠀⢤⡀⢹⣧⠇⢀⣾⡿⠁⣠⣾⣿⡟⠀⠀⠀⠐⣹
⠐⢿⣏⠀⠀⠀⠸⢿⣿⣿⣤⣀⠙⢷⣄⠙⢳⣿⡃⠐⠀⠘⣯⣤⣀⠉⠙⠋⠉⣀⣤⡽⠻⠚⠓⠚⢿⣾⣿⣶⡾⢋⣤⣶⣿⣿⣿⠇⠀⠀⣠⣿⣿
⠀⠙⡿⠂⠀⠀⠀⠉⠙⣿⣿⣿⣶⣄⠙⠧⣀⡙⠛⠛⠓⠤⠈⠛⢿⣷⠀⠀⣾⡿⠛⠃⠴⠖⠒⠒⠊⠉⠾⠋⠴⣿⣿⣿⣿⠟⠙⠀⠀⠀⣿⣿⠟
⠀⠀⠐⣶⣦⡀⠀⠀⠀⠀⠈⠁⠸⣿⣿⣶⣿⡏⠛⠛⠲⠶⠤⠄⠀⠉⢻⡏⠁⠀⠠⠰⠖⠖⠛⠛⢻⣿⣾⣿⣿⠟⠉⠉⠉⠀⠉⠀⢠⣿⣿⠋
⠀⠀⠀⠙⢿⣷⡆⢀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣛⠐⠂⠃⠁⠀⢀⣤⣾⣷⣄⡀⠀⠈⠘⢐⡂⣛⣿⣿⡿⠿⢁⡄⠀⠀⠀⠀⡀⢰⣾⡿⠃
⠀⠀⠀⠀⠀⠙⠗⣼⣧⣠⣀⡆⠀⠀⠀⠈⠉⢹⣿⣿⣿⣶⡖⠺⠋⠉⢻⠏⠈⢙⠗⢶⣶⣿⣿⣿⡿⠛⠁⣶⠾⠃⢰⣀⣠⣴⣧⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣧⣤⠀⢀⣴⣀⢈⣙⣿⣿⣿⣇⢀⣟⣶⣶⣶⣶⣻⡀⣸⣿⣿⣿⣿⣿⣼⣶⣄⠀⣤⣼⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠋⠙⠛⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠏⠙⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣛⡿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣛⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣼⡁⠘⠉⠃⠁⠚⠚⠉⠃⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⡀⠀⠀⣀⠀⡈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠳⠞⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠄⠀⠀⢸⡴⠊⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠠⠀⠀⠀⢀⡞⠀⠈⠀⠀⢸⡀⠀⠀⠠⡀⠀⠀⠀⡦⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣶⣾⣤⢤⣾⣄⣀⠀⠀⠀⣸⠁⠀⠀⠀⠱⡄⠀⠀⠇⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⡋⠰⠇⠞⠚⠙⠉⡙⢷⣎⡁⠀⠀⠢⢄⠀⢈⡠⠄⠒⠒⠢⢄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡦⡀⠠⠀⢠⠋⢠⠀⠀⠂⠀⠀⢱⠀⠤
⠀⠀⠀⠀⠀⠀⠺⢣⢊⡇⠶⠀⠀⠈⠀⠀⠄⠀⠈⠁⠀⠠⠦⡿⣿⣜⡄⠀⢸⡀⢫⡀⠀⢀⡀⠀⡸⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠋⠹⠘⠅⠀⠀⣹⣦⢍⣉⣉⠤⠚⢁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠓⠤⣄⣀⡠⠴⠊⢀⠴⠃⠀⠀⠀⠀⠀⣀⠜⠃⠀⡀⠀⢄⠀⠈⠑
⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣉⡶⠢⠤⠤⠤⠒⠊⢷⠤⣀⠀⢀⠀⡠⠊⠁⠀⠀⠀⡇⠀⠸⡄
⠀⡴⠉⡩⢅⠈⣇⠀⠀⢤⣴⠊⢁⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⢹⠿⠋⠀⠀⠀⠀⠀⣈⠥⢄
⠀⢣⣼⣁⣄⡇⠈⡇⠀⠈⠉⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣹⠚⠁⠀⠀⠀⠀⢀⠤⢺⠔⠈⠚⡍⢱
⠀⢮⡤⢽⣉⡜⠉⣀⠀⠀⠀⠀⣀⣀⠀⡇⠠⠤⠄⡀⠀⣤⣄⣀⣀⡀⠀⠀⠘⠤⣌⠪⣶⡷⠙⡇
⠀⠀⠀⠰⣒⣾⣾⠽⠒⠊⠉⠉⠀⢠⠶⠃⠀⠀⠀⠳⠦⠀⠀⠀⠀⠈⠉⠁⠒⠦⢼⣷⣋⠑⠊⠁
⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡄
⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠐⠁⠐⠱⢠⠂⠀⠀⠀⠠⠀⠐⠀⠆⠀⠤⠆⠈⠇⢣⠂⠧⠀⠀⠖⠀⠀⠀⠴⠠⠀⠀⠏⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠓⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠑⠦⣀⡀⣸⡇⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣰⠋⠀⠈⢻⡆⠀⠀⠀⠉⣿⠓⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⡠⠃⢀⠔⠒⢦⡀⠀⣿⠀⠀⠀⠀⣷⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠉⠀⠈⠁⠀⠀⢺⣴⠀⠘⡇⡠⢻⣿⡀⠀⠀⡟⠀⣼⠃⢀⣿⡆⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣴⡆⠀⠀⠀⠀⣸⣿⣒⡺⠋⠀⢸⣏⠙⠓⣾⠓⠋⢿⣧⠞⠙⣿⣄⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⠔⠁⢉⣉⡀⠀⠀⠈⠻⠶⠞⠁⠀⠀⠀⠀⢀⠀⠈⠛⠷⠦⠤⠔⠁
⠀⠀⠀⢀⠴⠒⠛⠛⠛⠛⠛⠛⠓⢻⡟⠋⠉⠀⢠⡞⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⡏⢀⣤⡀
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢠⡿⠀⠀⠀⢀⠏⠀⠀⠀⠀⣰⠏⠉⢹⠀⣾⡗⠁⣾⠁⠀⠀⠀⠀⣀⣀⡀
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠑⣾⡧⠤⠄⠒⣡⠀⠀⣾⠀⠀⣿⠀⠀⠀⢠⣿⣷⣞⣁⡀⠀⢀⠔⠋⠁⠈⢻⡆
⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⠀⢰⣿⠀⢰⢿⣦⣤⠴⠃⠸⠇⠀⠈⢙⠿⣧⣤⣀⣀⣠⠞⠁
⠀⠀⠀⠙⠷⣶⣤⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⡸⣿⣤⠞⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠔⠒⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠔⠀⢠⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀
⡆⣧⠀⣾⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣇⠔⢩⠆
⠇⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠰⠊⣰⠁
⠈⡄⠀⠀⠀⠉⠛⢠⠄⠷⠀⠀⠀⠀⠀⣤⣀⠀⠉⠂⠀⣀⣀⡀⣀⠀⠀⠀⠀⠐⠧⣀⠙⠀⠀⠀⠀⠠⢀
⠀⠘⠦⣀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⠃⠀⠉⠀⠠⠋⠀⢓⠀⠀⠀⠀⠀⢸⠀⠉⠦⠀⢠⠒⠂
⠀⠀⠀⠈⠃⠘⣇⠀⠀⠀⠀⠀⠀⢀⠰⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⡦⠀⠀⠈⠳⠀⢀⡠⠋
⠀⠀⠀⠀⢸⠀⠹⡄⠀⠀⠀⠀⢀⡹⠖⠒⠉⠉⠉⠁⠒⠦⣀⠀⠀⠀⠰⡁⠀⠀⠀⡔⠁⢠⡎
⠀⠀⠀⠀⠀⣧⠀⢳⠀⠀⣀⠔⠉⠀⣀⠀⠀⠀⢀⣀⠀⠀⠈⠲⡀⠀⠀⠘⠀⢠⠞⠀⢠⠟
⠀⠀⠀⠀⠀⢀⣄⣠⣧⡴⠁⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⡄⠀⠱⠀⠀⢤⣠⠋⠀⢀⠎
⠀⠀⠀⠀⠀⢯⠁⠀⢸⠁⠀⠀⠀⠀⠈⠀⠀⢀⣀⠤⠚⢩⡆⠀⠀⣛⡀⣸⠃⠀⢠⠎
⠀⠀⠀⠀⠀⠘⣇⠀⡆⠀⠀⠀⣿⡏⠉⠉⠉⢁⣀⣤⣶⡿⠁⠀⢀⠀⢙⠟⢦⣴⠋
⠀⠀⠀⠀⠀⠀⢹⡀⢡⠀⠀⠀⢻⣿⣶⣾⣿⢋⣉⢀⡼⠃⠀⠀⢨⡟⠁⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⢧⠈⣄⠀⠀⠀⠙⢿⠁⢀⢠⡤⠊⠀⠀⠀⣠⠏⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠓⠶⠶⠶⠶⠖⠋⠉⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠡⡤⠒⠒⠒⠒⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠉⠙⠓⠒⠲⠤⡶⠤⠼⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠤⠤⠄⢾
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠉⠻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠣⣀⣀⡠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡖⠒⠲⠶⠖⠊⠻⣤⣤⣀⣀⣀⣀⣀⣠⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠁⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠧⠴⡇⠀⠀⠀⠀⢀⠞⠀⢊
⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⡇⠀⠀⠤⠒⠁⠀⠀⣸
⠀⠰⢂⠀⠀⢋⠀⠀⠀⠀⣀⠦⠚⠣⠁⠀⠀⠀⠀⠆⢨
⠀⠉⠂⠠⣭⣭⡥⠍⠁⠎⠑⠀⠀⠀⠨⡆⠠⠄⠚⠠⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣠⣤⣤⣈⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢊⣡⣤⣤⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠑⢄⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣿⣿⠏⠐⠈⠉⠁⠀⠀⠉⠉⠐⠈⢿⣿⣿⣿⠇⡘⠀⠀⠀⠠⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡫⠋⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠌⠫⡛⢀⠃⠀⠀⡆⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠑⡋⣿⢿⡅⠄⠀⠀⠀⠀⡊⡿⣿⡍⡗⠁⠐⡆⠀⠀⡠⠃⢄⠆⠄⠄⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠈⠉⠈⢀⣤⣤⣤⣄⠈⠈⠉⠈⠀⠀⠀⢠⠀⡐⠀⠀⠀⠉⠘⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢁⣀⠈⠥⢰⠁⠙⢿⠟⠁⢳⠨⠍⢁⡈⠉⠀⠸⠠⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⢁⠠⠒⠉⠹⠀⢀⠜⢄⠀⡸⠉⠐⠢⡀⠉⢀⠃⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡁⠀⠀⠀⠀⠙⠢⠤⠤⠚⠀⠀⠀⠀⠀⣰⡇⠘⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡟⡷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⠃⠃⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠑⠻⠾⠭⠿⢿⣿⣿⠿⠿⠿⠟⠛⠉⠢⡀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢄⠀⠀⠀⠀⠀⠈⢦⣀⣠⣮⣐⢄⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠂⠀⠑⠄⣠⡀⣀⣠⣿⣿⣿⣿⣿⣷⡑⡀⠀⠀⠠⠂⠉⠑⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⢂⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⠊⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⡠⠄⠂⠀⠐⠠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⠠⡀⠠
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣨⠻⣿⣿⣿⣿⣿⣿⣿⣿⢫⡁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡂⣂⣂⡣⠗⠓⠱⢍⠻⠿⣿⡿⠿⢛⠵⠁⠒⠦⣆⡆⡄⣄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢀⠔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣾⣧⠀⣴⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⣷⠉⣻⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠷⠀⠀⠐⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⠛⠛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⠛⠛⡉⠉⠁⠀⠀⣀⣀⣤⣴⣶⣿⣡⠀⠀⠀⣰⠀⡀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⢠⡄⠀⣠⠀⠀⠀⠀⣠⠀⣠⣤⣤⣷⣴⣶⣾⣿⣿⣿⣿⣿⡟⠡⣿⡄⡂⠀⣿⡂⠠⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢀⣸⡄⡠⠲⢾⡇⡀⠀⢀⡷⢤⡴⠒⠛⡦⠌⣿⣿⡿⠻⠙⠟⠛⡿⢏⢁⠀⠘⠿⠇⣈⢰⣿⣇⠲⠇⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡿⠛⠋⠀⣠⠚⠉⠒⠣⠀⠂⠀⢠⣆⢁⠔⠀⠀⠀⠀⢄⠈⢰⡆⠀⠁⠋⠳⠴⢄⣆⡀⠁⠉⠑⣄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠾⠋⢀⠤⠀⠀⢊⠉⠁⣷⢠⠀⢢⠤⠀⠀⠀⠀⡀⠠⠒⠉⠀⢠⡆⣼⣇⠀⠐⠂⠀⠠⠐⠩⠙⡄⠀⠀⡇
⠀⠀⡀⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠡⠿⠜⠂⢀⡀⠀⠀⠀⢰⡇⣀⣐⡀⠀⣼⣧⡿⠿⠖⠒⢂⠀⠀⠀⠠⠀⡇⠀⠀⠇
⠀⠀⠇⠠⠄⠀⠠⠄⠁⠒⠂⠀⠶⣿⣿⣽⣷⠀⠀⠉⠁⠀⠚⠃⠼⣷⡐⢉⡁⠨⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠀⠈⠠⠆⠀⠆⢴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠂⠀⠠⠤⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠣⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡆⠙⡖⠐⠤⠐⠒⠀⢒⠂⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠁⠁⠀⠀⠙⠃⠀⠉⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡠⠀⠃⠤⠒⢤⡄⡀⣀⠠⠔⠢⢄⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠔⠸⠁⠀⠀⠀⢢⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⡐⠲⠂⠀⠐⠖⢰⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⡁⠀⠀⡐⠀⠀⡆⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠁⠀⡀⠀⠀⠎⠀⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⢤⡀⠠⠒⠀⠀⠀⠀⠀⠰⡄⢀⠀⢬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠑⢆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⢹⠀⠀⢠⠂⠀⠀⠀⠈⠁⠒⠒⠒⠀⠁⠀⠀⠀⠰⡄⠀⠈⣋⣉⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠘⠒⠈⠐⠀⠠⠼⠀⠀⢀⠀⠀⠀⣀⠀⠀⣦⡀⢀⠝⠉⠉⣅⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⡄⠀⠀⣶⠀⠠⠄⠊⠱⢄⠀⠉⠀⠐⠀⠀⢩⠀⠀⣀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠐⠀⡎⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⠤⠁⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡐⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢣⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠰⢧⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠄⠠⠀⠀⢀⠂⠀⡀⠴⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⢄⠉⢤⡀⠘⢆
⠀⠀⣀⠠⠴⠊⢁⡴⠋⠀⠀⠀⢠⠀⠤⠊⠀⡀⠀⠌⠘⡀⠀⠀⡄⠈⢆⠀⠶⠒⠤⢄⠱⠀⠈⠓⢄⠑⠒⠤⠤⣤⣤
⣴⢟⣯⡁⢠⠶⠊⠀⠀⠀⠀⢀⡧⠀⠀⠀⢀⠁⡸⠀⠀⠃⠀⠀⠇⠀⠀⠳⡀⣥⠀⠀⠁⠷⠀⠀⠀⠙⠠⣐⢃⠘⠷
⠀⠈⣡⠎⠁⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠸⢀⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⣾⠀⠀⠀⠀⠀⠠⣀⠀⠀⠙⢌⠁
⠀⣴⠃⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠀⠀⠀⠸⠆⠸⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠓⢒⠒⠈⣁
⢠⡇⠀⠀⠀⡰⡆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢹⡄
⢸⠁⠀⠀⣠⠁⠈⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣀⠀⢀⣀⠤⠀⠀⠀⠀⠀⣸⠇
⠘⢧⠤⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍
⠀⠀⠀⠀⠘⢧⣀⡀⠀⠀⠀⠠⠂⠑⣄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡴⠦⡄⣀⣀⠀⣀⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢄⣀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣋⣀⡠⠜⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⡖⠒⠂⠔⠒⠂⠈⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠆⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠈⠀⠀⠀⠀⠇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡜

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤⣤⣤⠤⣤⣤⣤⣀
⠀⢀⣀⣀⣾⡷⣶⣀⣩⣈⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣦⣴
⣦⢿⣿⣻⠄⣍⠈⠀⠨⣯⠿⣯⣁⣀⠀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿
⠀⠀⠈⠘⠿⠎⠀⠀⢀⠉⣫⡭⣉⠁⢘⣽⣿⣿⣟⣩⣏⢟⣙⣋⡛⢙⢉⣏⡝⣽⢻⣿⣟⡂⠀⠀⠀⠀⠉
⡀⣦⠀⣦⠐⣦⠀⠀⢠⣿⣿⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⡾⠠⠄⠥⠸⠀⠇⠀⡏⠸⠀⠅⣤⡀⠀⢀⡀⠀⠀⠀⠟
⣧⣭⣤⣥⣤⡿⣄⠀⣸⣏⣀⡒⠀⠈⢿⣻⢛⡿⢿⢿⡿⣯⠀⠹⠿⠘⢀⠃⠀⠂⠰⠀⠂⠿⢳⡄⠀⠁⣠⣦⣤⣦
⣿⣤⣬⣿⡯⠠⣽⢿⣿⣿⣿⣷⠀⠀⠀⠉⠈⠑⠲⠅⠈⠉⠀⠀⠠⡆⠘⠤⣤⢰⡄⢀⠀⠀⢿⣿⠀⠘⣿⣟⣛⡛
⢹⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠂⠀⠀⠒⠀⠀⠀⠀⢀⣙⡀⢠⣷⡀⠀⠘⣬⣷⣦⣬⡅⠀⠀⠉⠉⠉⠉
⠈⠹⠟⠋⠉⠁⠈⠀⣠⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣭⣿⡿⢸⡮⣧⠄⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠐⠄
⡀⠆⠀⠀⠀⢀⣀⡀⡛⢻⣿⣦⣤⡦⠀⠀⣀⠀⠙⢟⣿⣄⣦⢰⣴⠂⠄⠙⠚⠀⠀⠛⠛⠓⠈⠀⠀⢀
⡀⠀⠴⡆⠀⣌⠉⠛⠛⢿⣿⣿⣅⣀⠀⢰⣿⡆⠀⡿⡇⠈⠛⠸⣿⠀⠀⡆⠀⠀⠀⠀⠀⢈
⠀⠛⠀⢰⣷⠈⢀⡀⠀⠘⢛⣻⣿⣿⣿⣌⠿⠃⢀⣧⡇⠀⠀⠸⠿⠃⠠⠤⠄⠀⣀⡀⠈⠀⠀⠀⠀⠀⣠⣷⠆⣰
⣄⠀⠀⠘⢫⣿⡄⠀⠀⢠⣿⡿⣿⡷⠟⠿⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠐⠒⠒⣺⣿⣄⠀⠀⡄⠀⠀⠀⠑⠉⣰⣿
⢴⠄⠀⠀⢸⣿⡇⠀⠀⢈⣿⠁⠹⣷⣄⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⢤⣅⣠⣭⡛⠗⠂⡰⠁⠀⠀⠀⠀⠀⣸⣿
⠌⠋⠀⠀⠈⢏⠀⠀⠀⢸⣷⠀⠀⠙⣿⣧⠀⠼⠛⠁⠀⠈⠉⠀⠀⠀⠐⠋⠉⢸⣿⣷⠚⠁⠄⠀⠀⠀⠀⠀⣸⣿
⠀⠠⠀⠀⠀⠛⠑⠠⠄⠘⣟⠀⠀⠀⠘⢻⠀⢀⣀⣴⠆⠀⠤⠔⠀⠀⠀⠄⠀⢸⣿⣿⡄⠒⠀⠀⠀⠀⠀⠀⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡀⠀⠈⠀⠋⠐⠎⠀⢠⠔⠀⠀⠀⣀⠀⢀⣸⠿⣿⠇⠀⡀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠠⠉⠆⠀⠀⠐⠋⠀⠀⠃⠀⠐⠓⠒⠀⣀⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠬⠁⠀⠀⠀⠀⠀⠨⠬⠥⠾⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⣴⠂⠀⠀⠀⠠⠀⢀⡀⠀⠂⣀⣼⡇⣀⡖⠃⣼⠛⠻⣶⣬⡛⠒⠀⠀⠻⠄⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⣿⢶⢫⠟⢈⡡⠶⠀⠀⠀⠀⣠⠐⣼⡟⣿⣿⣿⣿⣶⢾⡾⡛⠀⠐⠻⣿⣏⠐⠀⢀⠠⠉⠂⠀⣈
⠀⠀⠀⠀⠀⠀⡀⠈⠀⠚⣼⣦⣿⠾⢻⣷⠶⢀⠀⠀⠠⠚⢀⣸⣿⣿⠋⣿⣿⣿⣿⠦⠤⠄⠠⣄⣻⣿⡇⢠⠴⠣⣰⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⠳⠔⣦⣿⣿⣵⠈⠐⠋⠀⡔⠓⣾⣴⣾⡿⢧⠃⣸⣿⣿⡏⣠⠀⠐⢂⠀⠀⠀⠈⢿⣧⣖⠁⠀⢠⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡕⢦⣘⣿⡿⣿⠞⠅⠛⠠⡼⠓⢿⣿⣿⣿⠀⣼⣿⢫⣅⣴⣾⣟⣿⣻⣿⣲⣆⠐⢾⣿⣯⣵⠀⠘⢀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠿⢷⡙⣋⣿⡷⠀⠀⢀⠘⢨⣿⣿⣿⣿⠧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⠀⠀⣾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠙⣾⣷⣿⣿⡇⠀⠐⠁⢠⣿⣿⣿⣿⣧⣾⣿⣶⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⣩⣿⣷⣆⢠⣏
⠀⠀⠀⠀⠀⠀⠐⠺⣽⡟⣿⣿⣿⡏⠀⠐⠲⣾⣿⣿⠵⣿⣿⣿⣯⣁⣾⡿⠛⣟⣿⣿⣿⣿⣿⣟⣳⣿⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⠙⢿⡜⢠⣤⣾⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣩⣿⣿⣿⣿⣛⣿⣻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣾⣿⣾⡧⠈⣹⣿⡿⠎⠸⣿⣿⣿⣿⣿⠿⢍⣉⣉⡹⠿⢾⣷⣿⣿⣿⠂⠀⠋⡇⣨⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⣾⣿⣿⣾⠟⢿⣷⠃⠐⠀⣿⣿⣿⣿⣿⡧⣄⠈⠛⣉⠠⠀⠄⠀⡀⠀⠀⢠⢈⡏⢠⣿⠷
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⣿⣿⣿⣟⡀⠘⣫⡴⠊⠁⢸⣿⣿⣿⣿⣷⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣘⣿⣿⠋⡷
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠹⣿⣿⡿⢦⡞⣉⠀⢀⣵⣿⣿⣿⣿⣿⡍⠛⠖⠀⠀⠀⠀⠀⡀⠀⢐⠟⢰⣾⠟⡏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⣿⡞⠀⠂⢠⡆⢋⣹⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠀⠀⢠⢩⡴⣻⢯⠜⡋⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠐⣥⡾⠀⠄⠀⠰⠛⣉⣲⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⠀⢀⣠⢷⣾⣿⡟⢁⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⢱⣃⡄⢸⡂⠀⣌⣹⣿⣿⣿⣽⣿⠀⠀⠀⠀⠀⡼⣯⡻⢿⣿⣿⣻⠟⠀⠁⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠻⠉⢤⠉⠁⠀⠉⣷⣽⣿⣿⣿⣼⣦⡀⣀⠀⠠⠈⠐⣦⠲⠿⡀⠁⠀⠄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⡀⢠⠄⣨⣿⣿⣿⣿⣿⣿⣷⣼⣦⣷⢀⣶⣿⡀⠀⠰⡦⠀⠀⠀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠞⠀⠀⠐⠂⢀⠀⡻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⡀⠀⠀⠄⠲⠖⠋⠉⢀⣙⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⠂⠁⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⣀⠀⠀⢀⣄⠰⠄⠀⠀⠘⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣧⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⡀⠈⠁⠠⠀⠀⠀⠀⡀⣄⣰⣾⣿⣿⣿⡟⣻⣿⣧⢋⠶⣀⠘⣿⣿⣿⡠⠄⠀⠀⠀⣰⡾⠓⠀⠀⠐
⠀⠀⠀⠨⣶⣮⣄⠃⠀⠀⠀⠀⣀⣌⣶⣿⠟⠿⣙⣿⣿⣿⣿⡿⠟⣛⠠⠞⠀⠙⠉⣿⣟⣠⠀⠀⣠⣴⢏⠗⠀⢠
⠀⠀⠀⣻⣿⣿⣿⣽⣿⣶⣶⣬⣯⣥⣯⣴⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠒⠆⠀⠀⢤⣾⠿⠧⢀⣾⠛⡁⠀⠀⢀
⠀⠀⠀⣽⠋⠉⠻⠿⠿⢿⢿⢿⣿⡿⠟⣉⠉⢈⣹⣧⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⣾⢿⠏⠀⠀⠀⠠⠉
⠀⠀⠐⢿⣿⣿⣿⣷⣄⢤⣦⠰⡿⠣⠶⣧⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠐⠀⣼⠿⠶⣽⠇⠀⠘⠀⠐⠃
⠀⠀⠀⠈⠀⣰⣿⢿⣿⣷⣶⣾⣿⣶⣾⣿⡿⢿⡟⠋⠛⢥⡀⠀⠀⠀⠀⠀⠀⢾⡷⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁⠀⠈⣤⣿⣿⣿⣿⣿⡿⠛⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡿⠀⠀⠈⠁⠂⠀⠄⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡗⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠈⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡆⠀⠀⠀⢀⠀⠈⠐⠀⠒⠆⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠈⠀⠸⣆⠀⠀⠀⠉⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢠⡐⠂⢰⠐⠄⢠⠀⡆⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣐⠀⡀⠀⢱⣎⣈⠐⠡⣀⡨⡅⠀⠉⠠⡀⢀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠃⠠⡆⠀⡂⠙⣏⡥⢀⠛⢿⢏⠀⠃⣤⡀⣿⡏⡄⠧⢀⢀⠁⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢠⠀⣜⣠⢾⠛⡏⢫⠿⣦⣼⠒⡿⣠⡁⠀⠷⣷⠂⠜⡠⠁⠖⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣠⢉⣽⡈⢛⣶⣬⣏⠘⢁⣍⠿⢻⣿⡴⠛⢻⣿⡅⡲⢎⣽⣇⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢘⡟⢃⣞⣱⣧⣿⣵⣾⣷⣞⣿⣿⣽⡤⣿⣬⣤⣤⠜⢃⡈⡛⠂⠊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠚⢆⠈⣋⣀⡍⢫⣯⣽⡍⠥⣿⣿⣿⣿⣿⣷⣿⣞⣷⣾⣿⢂⣻⣶⣎⡹⡇⢀⢦⡆⠁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠹⠀⢠⣿⠈⡙⠘⣬⣿⣷⣶⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⣿⣿⢻⣏⣻⡓⢻⠮⣨⣁⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡾⢗⣁⣈⢓⣻⣿⣿⣿⣯⠿⣾⣷⣾⣯⣈⣿⣿⡶⣟⣻⡿⣿⠛⢫⣶⡄⢞⠣⢙⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡈⢠⡏⠉⣴⣡⣽⣯⣷⣮⣿⣴⣯⣿⣿⣿⣿⣿⣟⣛⣿⣿⠟⡿⣋⣿⣵⣦⢎⠟⣈⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⡴⢣⡍⢟⣲⠼⣿⠿⣿⣯⣾⣿⣿⣿⣟⣺⣿⣿⣇⡟⠸⣧⡟⢬⠉⢁⡀⠤⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⢊⠋⠉⠷⣬⣿⢶⣸⣻⡿⣿⣿⠻⣿⣿⢛⣽⣿⣛⣿⡅⠿⣞⢃⠁⣸⣻⠃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢂⢃⡀⢽⣯⣀⢾⣽⢮⣤⡤⣿⣿⡟⣿⢹⣿⡷⣼⣿⣟⠻⠟⢟⡰⠅⠱⠒⠊⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌⡙⠳⠆⡸⠚⠫⠙⢻⣾⢗⣾⠟⣟⠱⢦⡽⢻⢷⣬⡸⢋⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⠠⠓⠃⠒⠉⣼⣠⣥⡴⢦⠐⣀⣼⢿⡐⢻⣯⡴⣤⠐⠴⠻⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⠠⠒⡀⠉⢻⠳⠛⠊⢁⡱⠥⠚⠙⠂⡁⠀⠀⢀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠋⠁⡀⠁⡀⠈⠴⢆⢉⡆⠹⠄⠋⠒⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠤⠄⠂⠐⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀



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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠊⢀⣄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠀⠿⠀⠀⢹⢹⠀⠀⠈⡇⣀⣀⣀⠀⠀⢀⣀⣀⣀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢸⠁⠈⠃
⠀⠀⠁⠸⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⢨⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⣿⠈⠀⠀⠀⠈⠀⠀⠀⡇⠸⠀⠀⠁
⠀⠀⠀⠀⡆⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⡀⠀⠀⠀⢀⠀⠄⡆
⠀⠀⠀⠀⢁⠀⠀⢈⣀⠁⠀⠉⠰⠀⠂⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⡀⢠⡇⡃⠀⠀⠀⠀⠀⠀⠄⠀⠇
⠀⠀⠀⠀⠸⠀⠀⢘⢠⠀⠐⠤⢄⠐⡂⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⢸⡇⡁⠀⠐⠀⠀⠀⠂⠼⢠
⠀⠀⠀⠀⠀⡆⢀⠘⠀⠁⠀⠉⠈⠀⠀⠀⣸⠀⠀⠸⠀⠀⢸⢀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢀⡸⠆⢣⠀⠀⠀⠀⠀⢀⡄⢸
⠀⠀⠀⠀⠀⠳⠢⢤⠴⠶⣲⡒⠀⠈⣉⠍⠤⠀⠂⠚⠂⠤⠼⣸⡀⠀⠀⡇⣸⣀⠀⠀⠀⠀⠑⢲⠋⠁⠀⠀⠀⠀⠀⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠰⠁⠀⣉⠥⢭⡖⠉⠀⡦⢄⠀⠀⠀⠀⠀⠀⠀⠉⠢⣴⠛⠋⠋⠙⢆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠸⢀⠜⠀⢠⠎⢠⠃⢀⡀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠹⡇⢲⢄⠀⢨⠀⠀⠀⡿⠠⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⡌⠀⠆⠀⠨⠑⢄⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⢰⣼⠀⠆⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠃⠀⠀⠑⠤⣀⠀⠀⠀⠈⠁⠀⠀⠘⣿⠀⠸⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⢰
⠀⠀⠀⠀⠀⠀⡔⠂⠈⢹⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠒⠐⢆⠀⠉⠁⠲⣤⣤⣤⣼⡃⢐⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⡁⢰⠒⣺⣀⠋⠀⠀⣢⡆⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠈⣤⣄⠀⠀⡇⠀⠀⡀⢀⣀⣈⣉⣉⣉⣉⣁⣀⡄
⠀⠀⠀⠀⠀⠀⢁⠀⣿⠁⠹⠆⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠁⢨⠀⠀⠇⠀⢠⢖⡒⢀⡲⡆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠰⣄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠙⡇⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⣤⠤⡀⠀⠈⠓⢤⡀⠀⠀⠈⠂⠂⠀⠀⠀⣀⡤⠎⠀⠀⣡⢄⣸⠀⣠⠆⣄⠀⣀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣧⡤⠝⠋⠒⠒⠀⠠⠭⢶⡖⠒⠒⣶⠶⠫⠄⠒⠒⠒⢹⢥⡤⢻⢸⡇⠑⣦⣠⣤⡔⠊⠉⢦
⠀⠀⡂⠀⠀⠀⠀⢠⡇⠸⢄⣄⣀⣠⡄⡄⠀⠉⠉⠉⠁⠀⣤⣤⣤⣤⣤⡴⠀⣣⡈⠈⠷⠀⠈⠛⠋⡤⠠⠤⠤⠷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡤⠤⠤⠤⠤⢼⡟⠁⠀⠀⠀⠀⢸⡧⠔⠶⠒⠂⠤⣀⣽⠀⠀⠀⠀⠀⠀⢹⡕⣺⡀⣀⣀⠀⢀⠁⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠂⠀⠀⠀⠀⣰⠤⠄⠒⠒⠒⠒⢋⠤⠒⠒⠒⠂⠤⢄⣉⡐⠒⠒⠚⠀⠂⠀⠛⣋⢡⠀⠀⠘⢿⠏⠈⠉⣿⣇⣀⣀⡀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢠⡧⠒⠶⠒⡞⠃⣈⠁⠀⠐⣠⢰⠡⡆⠀⠄⠉⠛⠓⣛⣛⣛⠓⠀⢿⣁⠀⠀⠀⠁⣴⣤⣤⡿⠀⠀⠈⣷⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡴⡔⢐⣄⣀⠀⡚⣉⡀⠀⠈⠐⠚⠓⠜⠇⣀⣀⣢⠘⢤⠉⡏⣹⠀⡀⢻⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣸⠃⡤⠈⢦⠀⡔⠁⠁⡮⠤⣄⣀⣂⣈⣀⡠⠹⠆⠀⠱⣬⢀⡑⣀⠖⠻⠴⢧⠀⠀⣔⡉⢉⠉⣉⠆⠀⠀⠀⠰⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠓⠒⠂⢉⣁⣀⣡⣤⠎⠀⠀⠤⠤⠤⠤⠤⠤⠤⠬⠦⠴⠀⠀⠀⢀⣉⣉⣈⣉⢣⠀⢀⠔⠛⠋⠁⠒⢢⠀⠀⠀⢃⠀⠀⠂
⠀⠀⠂⡖⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠑⠒⠚⠒⠊⠁⠀⠀⠀⠈⠆⠀⡀
⠀⠀⠄⢰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠻⡀⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡥⡀
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠂⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠎
⠀⠀⠙⠂⠃⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⣿⡿⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢺⣿⣿⣿⣷⣤⣶⣶⡖⢲⣶⣶⣤⣼⣿⣿⠃⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⣻⣿⣿⣿⣿⣧⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡟⣿⣿⠧⠤⠄⠘⢿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⣠⣶⣿⣷⣶⡄⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠘⢿⣿⣿⠟⠁⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣩⣄⠀⠀⠀⢀⣼⣅⡀⠀⠀⠀⢀⣤⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡟⠉⢸⠃⠉⢻⣒⣾⣿⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⢿⣿⣧⣀⣀⣀⣠⣾⠿⣻⠟⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⡙⢯⣝⣛⣛⣉⣭⠎⠁⠀⢀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠠⡀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢸⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣿⣿⣷⣤⡠
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⢶⡦⢀⠁⠀⠀⠈⠁⠉⠊⠢⠻⢿⣿⣭⣤⣴⣄⡢⣤⣤⣀⢠⣦⠀⣀⡀⠀⠀⢀⣀⣀⣀⢀⣀⣀⣀⢀⣀⣠⣤⣶
⣶⣴⣦⣶⣿⣿⣿⣶⣭⣉⢴⣤⣀⣨⣤⣿⣦⣿⣤⣦⣤⣤⣤⣶⣶⣤⣤⣔⣶⡬⠻⣿⣿⣿⣿⢧⣞⣷⣳⣷⣿⣷⣦⣷⣾⣿⣷⣿⣿⣾⣾⣿⣿⣿⣾⣯⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣥⣿⣿⣿⣿⣯⣿⣿⣿⣿⣇⣿⣿⣿⣟⣿⣿⣿⣿⣾⣿⣿⣌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣉⣸⣿⣿⣿⣿⣿⡿⠹⣿⣿⣿⣿⣿⣿⠿⢿⡿⢟⣫⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣹⣿⣿⣍⣵⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⡿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠙⢋⣴⣟⢫⢻⠀⡏⣸⠋⣽⣦⣀⡁⠁⠙⠋⢉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠋⠁⠀⠀⣿⣷⢰⡿⣿⣸⢧⠏⣸⠏⢘⣿⡄⠀⠀⠀⠉⠀⠉⠛⢿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣋⡁⣠⡶⠿⠛⠁⠀⠀⠀⠄⠀⣿⣿⣾⢳⣷⣿⠟⣴⠋⣴⠟⣽⣿⠂⠀⠀⠀⢄⡀⠀⠀⠚⠛⠷⣦⠐⣉⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⠟⠄⠀⠀⢄⠁⠀⠀⢀⣿⣿⣿⣿⡿⣫⡞⣱⢞⣥⣼⡿⢿⣄⠀⠀⠀⠀⠀⠲⠄⠀⠀⠞⢯⡟⠛⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠏⠀⣀⣈⢈⣀⣀⣀⣀⣀⡑⠲⣶⣾⣟⠿⣽⣿⣿⣿⣿⣡⠾⣛⣽⣶⣾⡟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢸⣦⢀⠘⠿⣿⣿⣿⣿
⣿⣿⣿⣍⢀⣴⡟⠀⠀⠀⡀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣾⣷⠀⣶⣿⣦⣄⠀⠀⠀⠑⠀⠀⠀⠀⠈⣷⡄⢈⣽⣿⣿
⣿⣿⠏⠊⡍⠋⠉⠀⠠⠂⠀⠀⠀⠀⠀⠀⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡏⠀⣠⡿⠸⣿⣿⣶⣄⡀⠀⠀⠀⠢⠀⠈⠉⠫⡈⠋⢿⣿
⠿⢿⣤⣾⡀⣀⡀⠊⠀⠰⣤⣴⣄⠴⣵⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡿⠉⠀⠀⣿⡇⢰⣿⣯⣙⡿⣿⣦⡀⠀⠀⠈⢂⣀⡀⣸⣦⣼⠿
⠐⠉⠉⠉⠚⠛⠑⠀⠀⠀⢸⠿⠏⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⡇⢀⣤⣿⣿⢁⣾⣿⣛⣿⣷⣶⣽⣿⣶⣄⠀⠀⠉⠛⠋⠉⠉⠰
⡀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠆⢀⣠⣶⣾⣿⣇⢿⡿⠿⢿⣿⣿⣿⣷⣿⣿⣟⠀⠀⣿⣿⠟⣾⣿⣿⡿⠛⠁⠀⠈⠉⣿⣼⣷⣤⡀
⠀⠉⠒⠠⠀⠀⣀⣴⣟⣯⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣿⣿⣻⣿⣤⣀⣿⠇⣠⣾⣿⣇⣀⣀⡦⣶⣶⣧⣿⣿⢛⣙⢻⣄⠀⠐⠚⠁
⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⢟⣉⣨⣿⡄
⠀⠰⢾⣻⣉⣁⣀⡀⡀⠀⠀⠀⣿⣁⣀⡀⠉⠙⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣧⣼⡿⠋⣙⡇⠀⠐
⠀⢰⠌⣉⣥⣤⣤⠼⣤⠤⠀⢀⠹⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢿⣻⣿⡟⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⠟⣿⣿⣿⣧
⠀⠘⣷⣬⣼⣶⠿⢿⣿⣿⣯⣠⣴⡖⠉⠉⢸⣷⡀⠀⠀⢀⣠⣾⣿⣿⡃⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣥⡾⣯⣶⣿⣿⣿⣿⣬⣦⣀
⠀⣨⡿⣽⣿⣁⣀⣀⣾⣟⣃⣼⣿⡇⠀⠀⠀⢿⢷⡀⠰⣟⣿⣿⣿⣿⢃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣡⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⠿⠟
⠀⠹⣿⣓⣛⣚⣛⣛⣛⣚⣓⣳⣷⣶⠀⠀⠀⠈⠃⠁⠀⠀⠉⠉⣡⡿⣯⣯⣝⣿⣿⠀⠀⠀⠈⠹⣿⣿⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣴⡿⢷⣦⠈⣿⡀⢀⣴⠾⠟⠷⢶⣄⠀⢸⡏⢠⣶⠿⣶⡄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢿⣷⣾⡟⢀⣿⢁⣿⢁⣴⡾⣶⣄⢻⡇⢸⣇⠘⣿⣴⣿⠇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⣀⠀⣿⠃⠘⣿⡈⣿⣤⣾⡏⣸⡏⠀⢻⡇⢀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⣶⣄⠉⠙⠻⠟⠀⠀⠙⠷⣦⡀⣡⡶⠟⠀⠀⠘⠿⠛⠉⢁⣴⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣀⣴⣦⣀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠀⠀⠀⢰⡿⢁⣴⣶⣦⡈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⣿⣅⣹⡿⠀⣿⠀⠀⠀⠀⠀⣴⠾⠿⠶⢶⣤⣤⣾⣧⠀⠀⠀⢸⡇⠘⣿⣤⣾⠏⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣈⡉⠉⠁⢼⣯⠀⠀⠀⣰⡟⠛⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⢈⣿⠄⠈⠉⣉⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠟⠁⠀⠀⣿⡄⣤⣤⣤⣠⣤⣤⣦⡄⣿⡆⠀⠀⠘⠿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢀⡈⠉⠉⣀⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠉⣀⠈⠉⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠛⢁⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣿⠛⠛⠛⢻⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠻⣦⡀⣠⡿⠃⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⣧⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣿⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠁




⡟⠉⢹⡟⠿⠿⠿⠟⠿⡟⠿⠻⠿⠿⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢛⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠤⠶⠄⠀⠀⡠⢊⣵⣿⣿⠷⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⣱⣶⣯⡉⠳⢦⣀⣠⣤⠴⠶⠖⠚⠛⠓⠻⠷⠿⣿⣿⣋⢸⢸⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⢙⠹⣿⣿⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡇⠀⣠⠄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡰⠂⠘⠛⠓⠶⣍⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢤⡀⠀⠀⠀⠈⢷⡶⡟⠛⠲⣤⡈⠻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⡴⠞⠛⠓⢦⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠉⠃⠀⠀⠀⠀⣷⢰⠀⠀⢀⢹⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠇⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡷⠷⢸⣶⡇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸⡿⡷⠿⠆⢺⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣾⠀⠀⠀⠀⢸⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠓⣶⠆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣴⠟⣹⣿⡄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢮⣛⢶⣬⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣋⣥⣪⢕⣿⠇⠀⠀⠀⠀⢰⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣼⢫⡑⠮⣷⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣾⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⠦⣭⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⡿⢴⣶⣿⡟⠛⢫⡍⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⡷⣻⣿⢧⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⠖⠀⠀⠀⠀⠀⠀⠀⢸⡀⠰⢛⠿⣿⣻⣿⠿⠁⣩⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠿⠃⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡏⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⣾⣷⣀⠀⠀⠀⠀⠀⠐⠏⢀⡾⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢋⢿⡆⠀⠀⠀⠀⠀⠈⢿⡄⣼⠟⠉⠉⠉⠉⠀⡀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠈⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⣦⣰⠀⠀⡀⠐⣀⣼⠻⣆⡀⠀⠀⣒⣢⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠈⠉⠛⠋⠉⠁

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡑⠢⣄⡇⠘⢀⡠⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡴⠛⡁⠈⣇⠛⠁⠐⠳⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠰⠂⠁⠀⠀⠀⠑⢠⠈⠉⠲⠂⡀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⣈⡠⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠞⡡⠆⠀⠀⠀⠀⢠⠖⠂⠙⠲⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⢰⣮⣷⠀⠀⠀⠁⢰⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠉⠟⠀⠀⠀⠂⠀⠩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠹⡀⠀⠀⠀⠀⠀⠈⠰⡒⠂⢀⣠⣒⡲⠒⡩⠃⠀⠀⠀⠀⠀⠀⠀⢠⠏⢷
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⢧⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠈⠤⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⡞⠀⡠⠃⠈⠳⡀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⠁⢸⠃⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⣇⠀⣧
⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠸⡄⣻
⠀⠀⠀⠀⠀⢀⠟⠸⡀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⢠⠏⠉⠉⢹⡄⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡴⠁⠸⡀
⠀⠀⠀⠀⠀⡎⠀⠀⡘⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠀⠀⠀⠈⣧⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠣⣄⠀⡏⠋⠀⠀⠀⠀⠀⠀⠘⣇⠀⢾⠀⠀⠀⠀⠀⣼⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⢸⡄⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣧⣸⡄⠀⠀⠀⠀⢨⣇⣸⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠿⠿⠇⠀⠀⠀⠀⠺⠿⢿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣟⠉⣻⣛⣛⣛⣛⣻⣛⣛⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣛⣿⣟⣛⣛⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⣰⡾⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠓⠦
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⡿⢀⠄⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠂
⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⢼⠠⠀⠀⠀⣠⣶⠟⠳⡄⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠈⣿⣇⠀⠀⠸⡆⠀⢀⠴⣿⡿⠤⠒⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡀⢻⣿⠀⠀⠀⣿⣿⠀⠀⢀
⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠘⣿⣇⣀⠔⠁⠀⠹⣿⡄⢀⡔⠉⠈⠁⠀⠻⣷⣄⣀⣀⠔⠂⠀⢸⣿⡆⠀⠀⢀⠔⠁⠘⢿⣦⠔⠀⣿⣿⠀⠀⠁
⠀⣀⡀⣠⣿⣏⣀⣀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠖⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⢏⣥⡴⠿⠋⠀⠀⠉⠉⠛⠒⠲⠤⠤⣤⣤⣤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣤⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⠄⡀⣄⠍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠒⠉⣂⡬⣁⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡴⢊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠀⠀⡴⡫⠊⢁⠄⠠⢄⠀⠀⢀⠔⠉⠲⠀⠀⠀⠀⠀⠀⣎⡉⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡜⡁⢠⡞⡔⠀⢀⡤⠀⠀⠀⢣⠀⡎⠀⠀⠐⠥⢀⠀⠀⠀⠀⠰⠀⢭⠤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡾⡸⢠⡓⡜⠀⠐⠀⠀⠈⠢⣀⠀⣣⠅⢠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⡐⠀⡼⠳⠁⡞⠜⠀⠀⠐⢄⡀⠀⡀⠘⢣⣽⣷⢃⣀⣢⠔⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠁⢰⡷⠃⢰⢍⠂⠀⠀⠀⠀⠀⡉⢭⠓⢿⢿⡿⢟⡉⡉⠉⠑⢆⠀⠀⠀⣘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⣼⣠⠀⢸⠪⠀⠀⠀⠀⢠⠊⠀⠈⠶⠉⢸⡗⠆⠙⠑⠀⢀⡼⠀⠀⣰⠃⢠⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠆⠀⢯⣼⠀⢰⢽⠀⠀⠀⠀⠈⠢⠤⠄⠀⠀⡘⢃⠀⠀⠀⠀⡅⠀⠀⣰⠓⢀⠿⡔⠀⡜⠁
⠀⠀⠀⠀⠀⠰⡆⠀⢸⠙⡆⠈⣟⡆⠀⠀⠀⠀⠀⠀⠘⠂⠒⠁⠈⢳⣤⠔⠊⠀⢀⡜⠃⣠⠟⡩⠀⣼⡙
⠀⠀⠀⠀⠀⠀⣧⠀⠘⣷⠚⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡑⢄⡠⠋⢀⣴⢋⠟⢀⡜⡬⠁
⠀⠀⠀⠀⠀⠀⢈⡀⠀⣈⠀⠈⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠊⠀⣴⣿⡷⠃⢠⣾⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠗⠾⡀⠐⢤⡀⠀⠀⠀⠈⠲⠄⡀⠀⣀⣀⡠⠔⠀⢁⣠⣶⠚⡲⠋⢀⡴⣿⣾⣷⡶⠶⣲
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠢⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠾⠛⣉⠕⠋⠀⡠⡛⡥⠋⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠒⢄⡈⢙⠲⠦⣀⡀⠀⠀⠀⠀⠀⠓⠉⠀⣀⠰⢫⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣠⠈⠑⠢⠤⢀⣨⣇⡄⠀⢀⣀⡤⠔⣋⣥⣖⠉⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠓⠮⢦⣠⣠⡻⠲⠗⣚⣙⡡⠜⠚⠁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠂⠡⠀⠨⠐⡠⢀⠄⡉⢍⠡⡀⠄⡀⢔⠠⣐⠄⡂⠄⡂⠅⡂⢅⡪⠑⠌⠂⠅⠪⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⢶⡆⡀⠐⠠⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠛⡛⠣⢛⡈⢎⡂⠋⠁⢠⣻⣧⢮⡔⣄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⢹⣶⣤⡔⣾⢠⡶⣼⣳⣤⡃⠀⠌⠪⠥⠞⣱⣸⡇⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⢠⢾⣷⣌⡞⣿⡿⣹⣿⣶⢿⣮⡻⣿⣿⣷⡀⠀⠀⠀⣽⡟⠉⠀⢀⡀⠀⠁⠳⣀
⠀⠀⠀⠀⠀⡴⢣⢷⣙⢿⡆⡿⣱⣿⣿⣿⣏⢿⣿⣮⡳⣿⡇⠀⠀⠀⠏⠻⢻⠿⢳⠷⠀⠀⠀⠈⢦
⠀⠀⠀⠀⣼⡷⢽⢼⣿⡷⡙⡤⣽⣍⡹⠶⡶⣴⣿⣛⠷⢾⠀⠀⠀⢄⣀⠀⠻⢮⣥⠴⠂⢀⠀⠀⠘⣧
⠀⠀⠀⣸⣷⣶⣇⣮⣻⣿⡏⣿⣮⣟⣿⡾⣰⣿⡿⠿⠿⠿⢳⠀⠀⠺⡽⢟⣀⣶⡀⠀⠀⠀⠀⡘⢠⣽⣇
⠀⠀⢀⣏⢿⢹⡿⠻⣿⢷⡋⣽⡿⢿⣷⣶⣶⣤⣟⣿⣿⡥⠏⠀⠀⠀⠀⠘⠯⡿⠆⠀⠀⠀⠀⣠⣷⣹⣿⡀
⠀⠀⢸⠁⠻⢓⣥⣴⣾⢟⣾⣷⣭⠖⠋⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢰⣠⣤⠀⠀⢀⣠⢴⣮⣝⣛⡿⢸⡇
⠀⠀⢸⣤⣶⣾⢮⣟⠳⣛⣫⡽⠁⠀⣠⠀⡄⢀⢔⣤⣉⠦⢈⣠⠈⠀⠘⡋⠀⠀⣴⣿⢯⣿⣿⠿⣫⡇⣼⡇
⠀⠀⢸⠝⢉⣾⣿⢿⣾⣿⣶⣧⠀⠀⠀⢴⣒⢺⣼⠿⣧⠎⠃⠈⠀⢸⣿⢳⡞⢸⢿⠻⣟⣯⠒⣫⡾⢰⣿⡇
⠀⠀⠈⡇⠿⡟⢯⣿⣻⣯⣽⣾⠀⠀⠀⢶⣒⠃⠾⢴⠖⣶⡶⠀⢀⠈⡁⠅⣠⢿⣾⣳⣝⢿⣷⣾⢣⣿⣿⠁
⠀⠀⠀⢹⣿⠂⣜⢿⢟⠻⢍⣽⠀⠀⠀⠀⠀⣱⢶⣭⣞⡟⠛⠴⠨⣥⣶⡿⣿⣾⢣⣿⣿⢯⣿⠏⣛⣭⡏
⠀⠀⠀⠀⢫⢸⢹⣦⣠⡤⠛⠁⢀⣀⣤⠀⠀⠹⠸⣿⡿⡈⠀⠂⢸⣿⣿⣧⣿⢧⣿⣿⠟⣪⣵⡾⣶⡝
⠀⠀⠀⠀⠀⠳⣿⣿⣮⡀⠀⠠⠛⣡⣿⢀⠀⡀⢑⠚⠉⠀⠐⠀⠸⣟⣛⡧⠛⠞⠯⣶⣿⣤⣽⣞⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣟⠂⢀⣶⠟⠙⠁⢎⢁⡑⠈⠀⢀⡀⠀⠀⠀⠙⢯⣾⢱⣎⢿⣶⢬⣍⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠘⠙⠓⠈⠠⠀⠄⠽⠀⣀⣛⠋⣺⣷⡀⠀⠈⢻⣸⡿⡎⢿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⡛⢺⡵⠜⣥⣽⡁⠙⠉⠀⠀⠀⠀⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠐⠉⠀⠀⠀⠀⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠆⠰⠈
⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠛⡘⠛⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠰⡇⠀⠀⠠⠤⠖⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠠⠄⠀⠀⠠⠠⠄⡄⠀⠀⠀⠀⡂⠆⠅⠀⠀⠀⠰⠰⠀⠀⠀⠀⡀⠀⠀⣠⢭⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠰⠠⠀⣃⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢇⣃⠸⡤⡀⣧⣅⠈⣆⠀⠀⠀⠁⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⣩⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⢀⡄⡄⠀⣄⠀⢠⠀⡀⡀⢠⡀⠀⡄⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢀⣿⡇⠀⠈⠫⣶⡽⠃
⠀⠀⡡⢖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⡀⠐⠇⠧⠀⠷⠄⠸⠀⠬⠄⠀⠧⠀⠧⠉⠆⠮⠐⠦⠰⠥⠀⠀⠠⠻⠛⠃⠀⠀⠀⣿
⠀⠀⠖⠉⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢄⠐⠀⣠⠆⠀⠀⠀⡆⠀⡆⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⠀⣄⠀⠀⠀⢰⡿⠁⠀⢈⠀⠄⠀⠀⠁⢈⠄
⠀⠀⠀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⣠⣴⣾⠟⠋⠀⠙⠉⠢⣴⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡀⢸⣾⢀⡷⠀⣠⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠀⡈⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣆⠀⠁⠀⠉⣞⣴⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠹⣄⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢀⡟⠀⠀⠀⠈⢻⣿⡿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣀⣤⣶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⢀⠀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣼⣿⣠⣴⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠗⠁⢀⣴⣿⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠛⠓⠀⠀⠀⣺⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⢠⡿⠛⠁⢀⣴⣿⣿⣿⣦⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠨⠚⣭⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⣻⣿⣿⣿⣿⠟⢻⡇⣀⠀⠀⠀⠀⠀⣴⡰⠁⠀⠁⠀⠀⠀⠀⠀⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠟⠻⣿⠟⠁⠀⡞⠘⡿⠀⠀⢀⣄⡾⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠐⠀⠜⠀⠀⣴⠟⠛⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡇⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢁⡾⠃⠸⣡⠞⠋⠀⣠⠔⣢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡄⢠⡟⠀⣸⠃⠀⣰⠏⣞⣠⠞⢁⡞⠁⡤⠶⠖⠛⠛⠛⠓⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣿⠀⢰⡇⢸⠁⡀⡿⠀⢠⣿⠘⠛⣡⣶⡯⠖⠛⠛⠛⠉⠉⠒⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠘⢿⣆⠈⢧⡸⢗⣱⣣⣴⣿⣭⣤⣶⣿⣥⣤⣤⡴⠞⠉⢑⡐⠶⠶⠿⠷⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⣿⣄⠀⠀⠈⠓⠀⠙⢋⣉⣉⣈⣉⡙⠻⢿⣿⣦⣤⣬⣴⠶⢛⣭⣥⣴⣶⣚⣛⠛⠛⠛⠒⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⠬⠍⠉⠁⠀⣠⣠⠞⠉⠀⠀⠀⠀⠹⡟⠦⡙⠻⣧⣤⣤⣶⠶⠖⠒⣒⣒⣒⠈⠱⣶⡚⠓⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣿⠗⠀⣺⣿⡿⠛⠿⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠿⣦⣘⠻⢿⣿⣯⣭⣙⡓⠬⡙⠦⣄⡀
⠀⠀⠀⠀⠀⠤⣤⣤⠴⢚⡁⠠⠊⠁⠀⠀⢤⣳⣤⢄⣤⡶⢆⣀⡀⠀⠀⠀⠒⢿⣿⠶⢦⣍⡛⠶⣤⡤⣤⡌⠉⠛⢶⣄⣀⡈⠁
⠀⠀⠀⠀⠀⠔⠉⠀⠚⣉⣁⠀⠀⠰⣄⠀⠢⢍⣵⡿⠣⠀⠀⠉⠀⠀⠀⠰⢶⣤⡹⣦⣶⠦⢭⣓⠦⡀⢦⣄⣀⣀⣀⣈⠉
⠀⠀⠀⠀⠀⠀⣶⡌⠉⢠⣾⣷⣀⡄⠹⣦⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡌⢙⠷⢤⣈⡙⢮⡑⢌⡛⠿⣯⣍⣉
⠀⠀⠀⠀⠀⣤⠛⢿⣷⣿⠿⠿⠛⠁⠀⠈⠙⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢷⣌⠓⠶⣤⡀⢀⡉⠂⠙⢷⣤⣍⡛⠿⣿⡛⠛⠋⠁
⠀⠀⠀⠀⢸⣟⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠐⠠⣘⣷⠙⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠉⢷⣠⠀⠹⣷⡝⢷⣤⡐⠬⡛⠻⢶⣄⡉⠓⠤⡀
⠀⠀⠀⢀⣿⠃⠀⠀⢨⣄⠀⠀⣄⠀⠀⢈⣁⠦⣝⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⣿⠦⣤⣀⣙⠳⣌⢏⠀⠻⣿⣷⣌⠳⣦⡈⠙⠦⡀
⠀⠀⠀⠉⠸⣆⠀⣰⢿⣿⣷⣦⣨⠓⠦⣤⣌⣱⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⡙⠷⣤⣉⠛⢷⡘⣎⠆⢀⠹⣧⠙⢷⣌⠹⣦⣄⠈⠢⡀
⠀⠀⠀⠀⠀⣿⡇⠻⣈⣿⣭⣭⣥⣦⠀⠀⠈⠙⠻⣿⠀⠀⠀⢀⠀⢒⣤⠾⣿⡏⠀⢹⣷⣶⣯⣭⡂⠀⠘⣦⠘⣧⠘⣇⢨⡻⣷⣌⠻⣿⢦⡈⠂
⠀⠀⠀⠀⠀⡁⢻⡤⠉⠉⠉⠉⠙⠛⠳⢶⣀⡀⢤⡭⠀⢀⣠⡤⠾⠍⢓⣴⢿⡜⡄⠀⠻⣟⢦⡉⠛⢶⣄⠹⣧⠸⣧⠸⡌⢷⡈⢿⣧⡙⢦⠉⠒
⠀⠀⠀⢀⣼⡇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣧⣾⣋⠄⢀⡴⢁⡾⢡⡾⡇⣿⡄⠀⠹⣦⠙⠷⠄⠈⠂⠘⢷⡹⣆⠃⠘⢧⠀⠙⣷⡀⢣
⠀⠀⠀⣼⠉⣿⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠛⢋⠁⣴⣿⠁⠸⢁⣿⠀⢿⠸⣿⣦⡀⠙⢷⣄⠀⠀⠙⢦⡀⠙⢿⣦⡀⠈⠳⡄⠘⣷⠀⠆
⠀⠀⠀⡟⠀⢸⡇⠀⣿⠀⢀⣿⡄⠀⣶⡀⠁⠀⠀⣴⠃⣸⠋⡇⣼⢀⢸⣿⡀⠈⠣⠻⡌⠻⣦⡀⠙⢷⣦⡤⠄⠙⢶⣄⡉⠛⢦⣄⡈⠂⠈⢃
⠀⠀⠀⠱⠀⣾⠁⣷⢸⠀⢸⣿⠇⢸⠿⠁⠰⠀⠀⣿⠀⡇⡀⢻⣌⣾⢸⣿⠷⣄⠄⠀⠀⢳⣌⠛⢶⣄⠈⠛⢦⣀⠀⠈⠙⠳⠦⣀⣉⡁
⠀⠀⠀⠀⠀⠟⠀⢿⢸⠀⠀⣿⠀⡟⠀⠀⣄⠀⠀⢻⡀⢃⢸⣦⡙⢯⠘⣿⡄⠙⡾⣦⣥⠈⣿⣿⣦⣌⡙⠲⢤⣈⣉⠒⠠⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠸⡆⢸⢸⠀⠀⣿⠀⣧⠀⠀⣿⡘⡀⣤⠳⡀⠀⢿⡙⢆⠁⠹⣧⡀⠸⠙⢿⣧⡘⢷⣌⡙⠛⠷⠶⣬⣭⣛⣛⠛⠛⣛⣋⡁
⠀⠀⠀⠀⠀⠀⠈⠀⢸⡀⠀⢹⡀⠘⢦⠀⠸⣧⠹⣜⣧⡙⢦⡈⢣⢠⣀⠀⠈⠳⣄⠀⠈⠻⣗⠂⢉⡙⠛⠓⠒⠂⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⣇⠘⣦⡀⠀⠃⠳⡘⢿⣷⣄⠙⠦⣁⠙⢿⣶⣦⣬⣙⡒⠀⠈⠳⣌⡙⢿⣿⡛⠛⠛⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣆⠘⢿⣦⡀⠲⣌⠂⠙⢯⠳⣄⠈⠑⠂⠙⢿⣍⠛⠛⠿⠶⣤⣈⠛⠶⣌⡙⠲⢭⣙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠈⠻⡟⢷⣾⣷⣄⠀⠀⠈⠙⠶⣤⣀⠀⠉⠳⢦⣤⣀⣀⠈⠙⠂⠀⠉⠓⠲⠤⠭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⢦⡙⢿⣟⠿⣶⣄⡀⠀⠀⠉⠛⠲⠦⠤⣄⣀⣉⣁⣀⣀⣀⣀⡀⠠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠉⠀⠈⠓⢤⠉⠛⠿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠙⠛⠛⠒⠲⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢍⠁⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠴⠃⠀⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡶⠦⠾⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⣄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⢠⡤⠠⣄⡜⠋⠹⡀⠀⠀⢠⡆⢀⠦⣄⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⢸⠷⡄⠀⠀⢀⡇⠀⢳⡀⠀⣰⠃⢸⡀⠀⠀⠀⣀⠀⠀⠀⢀⡖⠲⡆
⢀⡠⠊⠁⠀⠀⠀⠀⠁⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠙⢦⡀⡼⠀⠀⠀⠻⡴⠃⠀⠈⡇⠀⣠⠞⡻⠰⡖⠂⠋⠀⠀⠻⢠⣤
⠉⠒⠠⣤⠀⠀⢀⢀⣀⡀⠠⠖⠋⠀⠀⠀⠀⠀⣟⠒⠦⣄⡀⢸⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⡇⠀⢈⠆⠀⠀⠀⠂⣾
⠀⠀⠀⢸⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣄⣀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣸⣤⣴⡷⠤⡤⠟
⠀⠀⠀⠀⣧⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠃⠀⢈⠻⡀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠈⣹⠛⢦⡀⠀⠀⠀⠘⠛⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⡴⠤⣄⠀⠀⠀⢰⡖⠒⠛⠳⡆⢠⡤⠐⠋⠀⠀⡠⢶⣉⠁⠀⠀⠀⢀⠔⠁⠀⠀⠈⠲⢄⠀⠀⠀⠀⣴⣃
⠀⠀⠀⠀⠀⡜⠀⠀⠽⠦⢤⣤⡀⠙⠢⣄⠀⠛⠚⠁⠀⠀⠀⡜⠁⠀⠈⠉⠒⠤⣤⠋⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠉⠉⠒⠒⢤⣤
⠀⠀⢶⡐⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣈⣳⡄⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⡀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⡦⠀⠀⢀⡀⠀⠀⠸⣞⠉⠉⠉⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⡸⠁⠰⣎⠁⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠓⣻
⠀⠀⠀⠀⣇⣀⡠⠞⠉⠉⠘⠚⠉⠑⢤⡀⠀⠀⠀⢀⠇⠒⠀⠀⠤⠤⠀⣀⠀⠀⠀⣃⠀⠠⠔⠊⠁⠀⠀⠀⠀⠰⠁⠀⠀⠈⠉⠲⢄⠀⠀⠀⠀⠛⠢⡄⢀⣀⣨
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣠⠶⠒⠒⠋⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣀⣉⡶⠀⠀⠀⠀⣧⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⢛⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⡴⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠎⠋⠀⣠⠔⠉⢇⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⡰⢻⣄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣀⡞⠉⠳⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⠉⠓⢄⠑⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⡶⠦⠿⢦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣀⣀⠀⠉⠂⢵⣼⡆⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠙⡄⠀⣸⣠⡟⠲⠛⠁⠀⠀⠀⠀⡴⠋⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠁⢈⠉⠉⠉⠁⠀⢠⣦⠀⠀⠀⠀⠀⠀⠑⡄⠀⣷⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⢍⡁⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠉⣇⠀⠀⢀⡴⠋⠸⡆⠀⠀⢀⣶⡀⠀⠈⢦⣿⣮⣀⠀⠀⠀⢸⡟⠒⠦⢤⣀⠹⡄⢀⢀⠀⠀⠀⠀⠀⠀⢸⣀⡶⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣰⠏⠀⠀⠀⢳⡀⢀⡞⠀⢳⡀⠀⣼⠋⠀⠈⠛⠲⢤⣘⡆⠀⠀⠀⠈⠙⢋⠏⠈⢳⡀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠳⠟⠀⠀⠀⢳⡀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠠⠤⠄⠚⠀⠀⠀⠛⠒⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠙⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢠⣀⠀⢁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠊⠀⠀⠀⠀⠀⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⣀⡔⠁⠉⠳⠾⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠟⠒⠤⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⣤⣤⣀⠀⣀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⣀⣀⡠⠀⢤⣤⣤
⠀⠀⢹⣿⠋⠀⠹⣿⡏⠀⠹⣿⡯⠉⢽⣿⡷⠀⠲⣿⡶⠂⠰⣾⡶⠂⠰⣟⡶⠃⠈⢻⡿⠏⠈⢻⡿⠇⠀⠛⣿⠇⠀⠐⢿⠏
⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠛⠀⠀⠈⠛⠀⠀⠀⠻⠃⠀⠀⠸⠃⠀⠀⠸⠇⠀⠀⠈⠏⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⡴⠄
⠀⠀⠀⠀⢛⣼⡺⠀⠀⠀⠀⠀⣻⣿⣿⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣿⣿⡟⠁⠀⠀⠀⠀⢠⣆⣀⠀⢿⣿⡘
⠀⠀⠀⠀⠀⠉⠁⠐⣶⣾⣇⠀⠈⠀⠉⠀⣾⣿⣿⠀⠀⠀⠀⠀⢴⣿⣿⡃⠈⠁⠈⠁⠀⠀⣀⠀⠘⢻⣿⣿
⠀⠀⠰⠖⠀⠀⠀⠰⠿⢿⡏⠁⠀⠀⠀⠀⠈⠉⠉⠀⢠⣾⣷⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⡞⡹⠇⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠘⠿⠾⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⠁⠁
⠀⠀⠀⢀⡾⠿⠞⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣦⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠄⠉⠃
⠀⠀⠀⣿⣧⠀⠀⢈⣿⡇⠀⠀⣀⣀⡀⠀⠀⡀⠀⡀⠀⣀⡀⢘⢋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡀⢀⡀⣀
⠀⠀⠀⠻⣿⠀⠔⢺⡿⠃⠀⠨⢛⣧⣧⠀⠀⡟⠶⢻⠀⡧⠄⢸⢴⠃⣯⢼⠁⢱⠎⠀⠀⠈⠁⠁⠀⠀⢛⠋⣉⣡⣟⡃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠈⢀⠃⠃⠈⠀⡁⠁⠈⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠊⠀⠀⠀⠀⠙⠁⠘⠃⠑⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠈⣠⣤⡀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣂⠀⠈
⠀⠀⠀⠰⣾⣷⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠐⠽⠿⠯⠛⠄
⠀⠀⠀⢀⣝⣛⡿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⡖⠀⠀⠠⣄⣀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠹⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠠⡾⠃⠀⠀⠀⡞⠛⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠤⠄⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⠿⠿⠶⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣀⡀⢮⣷⡄⠀⠀⠀⠀⢀⡴⣆⣠⣄⣤⣀⡄⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢮⠅⠘⠛⠀⠀⠀⠈⠀⠠⢤⣀⠀⢠⣀⣈⡀⠀⠁⠠⠄⢀⣠⡼⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠀⠀⠀⠁
⠀⠐⠀⠀⠄⠤⠀⠀⠀⢀⣠⡝⠁⡈⣿⠇⠀⠀⠀⠀⠀⠀⢠⣦⣴⣦⣴⣦⣤⠀⠀⠀⠀⢿⡏⠋⠁⠀⠀⠀⠀⠒⠒⠲⠆⠐
⠀⠀⠀⠀⡀⠀⠙⠿⠛⠛⣿⣇⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠒⠒⠒⠘⠋⠙⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣄⢠⠀⠀⠀⠀⠀⠀⠀⢠⡿⠓⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠞⢾⡿⠿⣛⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡋⠀⠀⠀⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⣆⠄⠠⠠⢀⠒⠟⠛⠫⢛⠾⠉⠀⠀⠀⢀⣤⣺⣷⣶⣾⣿⣶⣾⡆⠀⠀⣿⡿⠠⠆⢲⣾⣿⣿⣇⠀⠘⢿⣇⣌⢣⣌⢻⣿⡇⠀⢠⣿⣧⣾⡇⣠⣾⣿⠇
⠀⠀⠀⡠⠴⠀⠠⠆⠔⠓⠀⠀⠀⠀⠀⠀⢠⣿⣿⡗⠙⣿⣿⣿⣿⣿⠃⠀⠀⣿⡿⢰⠲⣾⣿⣿⣿⣿⣷⣄⠀⠈⣉⡉⡉⠙⣿⡇⢠⣾⡿⣿⢿⣿⣿⠟⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣿⣿⣿⣿⡷⠀⠀⠀⣿⣿⡀⣹⣿⣿⣿⣿⡟⠛⠛⠷⣄⠙⠋⠁⣰⣿⡇⢸⣿⣷⣿⣾⣿⣥⣤⣶⡆
⣶⣶⣶⠶⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠛⢛⣓⣿⡿⢿⣿⣿⣿⣷⣶⣶⠹⣷⣖⠚⣹⣿⡧⠀⢻⣿⠟⠁⠉⠉⠉⠩
⣿⠟⠁⢤⣿⣿⣧⣀⠈⠻⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⠰⠀⠀⢸⣿⣿⣷⠀⣼⣿⡙⡟⢿⣿⣤⣿⣿⢾⣿⣿⣏⠁⠀⢉⠀⠀⠀⠀⢰⣦⣴
⠁⣀⣿⣿⢻⣿⣿⣿⣷⣄⠹⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢸⡄⢇⢀⣿⠟⣿⠟⡀⣿⣿⣿⣧⠈⠻⠿⠿⠋⣸⣿⣿⡧⠉⠀⣾⣧⣄⣀⣤⣾⠟
⢨⣷⡜⠃⢠⣿⡿⣿⣿⠿⠤⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣐⣆⣧⡸⠈⢡⡤⠿⠷⣿⣿⣿⣿⣿⠛⣶⣤⣴⣾⡇⢻⣿⡯⠅⢸⣿⢸⡟⠟⣿⠁
⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣿⣿⢀⣿⣿⠀⠓⠖⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⠉⠉⠉⠀⢀⠀⢹⡗⣸⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣁⠛⣭⣭⣉⠍⠏⣭⡉⣍⡋⠍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢰⣤⣿⣽⡆⠀⢳⣿⣯⠛⠛⠛⣻⠿⠁
⣿⣿⣿⣿⠛⡟⡿⠿⠿⢿⠿⣿⣿⠿⠷⠶⢿⠟⢿⣿⢻⠻⢟⠻⢿⠟⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠷⠝⠿⣿⣿⡿⢻⡍⠁⠀⣠⣾⣁⣄⣀⣀⡀
⣿⣿⣿⣿⣤⣧⣦⣵⣮⣿⣤⣿⣿⣮⣴⣾⣴⣥⣾⣿⣧⣦⣾⣼⣴⣶⣵⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⠃⢸⣷⠀⠀⠀⠀⠀⠀⠈⠛⠃
⣿⣿⣿⣿⠁⠉⡹⢛⠉⡇⠏⣝⠍⣏⡽⡋⢝⣫⣹⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣇⢸⠏⠀⠀⠀⠀⠀⠀⠴⠿⠗

⠀⠀⠀⠀⠀⠀⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡴⠚⠓⣶⡼⢿⣾⣷⣾⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣶⠏⠀⠀⠀⡰⣿⠃⢋⣴⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⣠⠋⠀⠀⢺⡂⠁⠀⠐⠛⠛⠛⠿⠛⠶⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢽⠀⠀⡁⠀⠠⢔⣾⡅⠀⢀⣻⣛⣻⡟⡟⢛⡟⠉⠉⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢀⡾⠏⠀⡇⠀⢀⡼⣻⢸⣷⣦⣽⣏⢙⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⣇⠐⠡⣰⡇⠻⠛⣧⡈⣻⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⢀⠀⠀⠀⢀⣠⣤⣆⣀⣨⣿⣿⣧⠶⢻⣿⣿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠲⢶⡿⠟⠛⢉⣽⠟⠉⠁⢈⣿⡿⠛⣧⣠⣼⡿⠿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠋⠌⣿⠂⢤⣾⡟⠊⠁⢉⣿⢻⡗⢂⡾⣻⡟⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⢀⣁⣈⣆⠔⠋⠀⠀⢠⣿⣿⣿⣴⠪⣭⣬⡙⢦⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢚⣿⣿⣏⣀⠀⣠⣟⣯⡿⠋⠁⢚⣃⣨⣤⣤⣤⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣥⣽⣿⣿⣿⡿⠿⠿⠿⠿⠿⠻⠛⠛⠋⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⡀
⣇⡀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠓⣤⣾
⣿⣿⠙⣆⣄⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⡀⢱⣾⣿⣿
⣿⣿⣿⣿⠿⡉⣀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⢏⢢⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠺⣥⣷⣿⣁⢀⡴⠂⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⢡⢈⡢⣄⣾⣆⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡾⣿⣂⣼⣤⣸⡎⡹⠀⢰⡄⣠⠄⠀⣄⢄⠰⡀⣦⠀⡀⢲⣀⠙⢄⣠⣷⣏⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣤⣷⣿⣴⣿⣿⣴⣄⣼⢸⣾⣶⣧⢿⣴⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿