Hot and spicy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣟⡾⣏⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠈⢷⡀⠀⠀⠀⠈⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⢿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣷⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⣰⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⡄⢰⣿⣿⣿⣿⣷⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⡆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⣱⡤⠔⠚⠉⠉⠉⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⣠⠔⠋⠁⠀⢀⡤⠔⠂⠉⠉⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠔⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢚⠋⠀⠀⠀⠀⢀⠭⠖⠀⠈⠉⠁⠰⣦⣠⡔⢎⢠⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢁⠽⠂⠁⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠠⠁⠀⠀⠀⠀⣇⠠⠄⢀⡀⠀⡴⠒⠀⠀⠀⢀⠔⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠛⠢⠤⠤⠤⠔⢇⢐⠎⠀⠆⠀⠀⡀⠀⠀⢂⢙⡆⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠁⠐⠦⢄⠀⢀⡜⠈⠆⠈⠀⠀⠒⡄⠀⠀⠂⣈⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢄⣀⣠⠜⠀⠀⠈⠁⠀⠀⠈⠀⠀⣤⣤⠜⠣⡔⠃⠉⠉⡀
⠀⠀⠀⠠⠰⠆⠒⠒⠒⠒⠊⢉⡳⢄⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⢹⠓⠰⠑⠂⠀⠚⠀⠀⠔⢒⠒⠄
⢇⢤⣴⠒⠒⠒⠒⠒⠂⠀⠀⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠁⠸⠠⠜⠀⠘⡆⢀⠀⠀⠀⣸⠀⢸
⠀⠀⠀⢉⠳⢄⣀⠤⣖⡦⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠘⠑⠒⠉⢐⡔⠉⣔⡀⠢⡀
⠀⠀⠀⠀⣰⡫⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠈⢢⣀⡠⠃⠐⠁
⠀⠀⢠⠐⠉⢀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣼⡤⠊⠈⠂⠀⠀⠀⠀⠁⠀⠀⣠⠃
⠀⠀⠈⠀⢯⠀⠀⠀⢀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠔⠒⠒⠊⡅⠀⠀⠀⠀⠀⢀⡜⠁⢠
⠀⠀⠀⠀⠈⢣⠴⠋⠀⠀⠀⠀⣁⣀⣀⠤⠤⠐⢯⠀⠀⠀⠀⠀⠀⡀⠆⠀⠐⡀⢀⡀⠀⠀⠀⠈⠢⠴⢂⢱
⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢀⡔⠉⠁⡀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⡤⠁⠘⢬⠤⠋⠀⡴⠀⠀⠀⠀⠀⠀⠸⠀⠀⠸⡦⡄
⠀⠀⠀⢸⣄⠀⠀⠀⠀⡠⠋⠈⢱⠊⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠈⠤⠀⠔⢡⠘
⠀⠀⠀⢿⠈⠓⠠⠄⠊⠀⠀⢀⡇⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢀⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠈⠣⠀⠀⢰⠀⠀⠀⡃⠀⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠖⠢⡐⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠰⢡
⠀⠀⠀⠀⠀⡆⠀⢰⠀⠀⠀⠑⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠔⢄⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊
⠀⠀⠀⠀⠀⠧⠔⠃⠀⠀⠀⠀⢰⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠼⠀⣀⠠⠀⡀⠈⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⡄⠃⢸
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠤⠤⣄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠛⠷⠤⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡋⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⡄⠀⠀⠀⠀⠀⠀⠀⡖⠉⠨⢽⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠈⠀⠀⠀⡀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠘⢋⡄⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⢀⣿⣿⣷
⠀⠀⠀⠀⢠⣾⠃⠀⢻⣿⣆⠀⠀⠀⠀⠀⣶⣿⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡻⣿⣿⠁
⠀⠀⠀⠠⣿⣯⣦⡄⠀⢿⣿⣆⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠹⢿⣿
⠀⠀⠀⣰⢏⣾⣿⣅⣀⣼⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠠⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⣏⠈⠻⣿⣿⣿⠿⠟⠋⠉⠙⠿⢶⣷⣾⠶⣤⣀⣀⣀⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠙⠳⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠴⠖⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢤⣉⣁⠂⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣉⣁⠃⠰⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣍⣽⢻⣃⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣽⣿⣿⡛⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⣀⠒⠰⢶⡂⢀⡀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⠇⠀⠀⠀⠀⣠⣼⠟⠁
⠀⠀⠀⠀⠈⠻⣦⡤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣦⠀⠉⣥⠀⠀⠉⠐⢺⢆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⠏⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠐⠠⣀⢀⡀⠀⠙⠚⠻⠗⠀⠀⠖⠀⠟⠀⠀⠀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⡄⣀⠀⠀⠀⠀⠀⠀⠀⠈⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣴⣆⣄⠀⠀⠀⠀⠠⣿⡅⠀⠀⠀⠀⠀⠀⣠⣾⠟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣻⣶⣴⣶⣧⠀⠛⠀⠀⠀⠀⠀⢀⡴⠛⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠐⣡⣸⠗⡹⣋⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⡶⠒⠒⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⡄⢘⡅⣩⣀⣌⠛⣾⣏⣸⢿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢤⠷⠈⣯⢁⣽⡧⠤⠵⠷⠾⡵⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁⠀⠀⠀⠀⠱⣧⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⣡⡄⣀⠁⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠘⣤⣀⣀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠑⠆⠀⠃⡀⠀⠀⠀⠀⠀⠐⠀⡾⢷⠀⠀⠀⠹⣯⡀⠀⠁⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠀⠀⠀⠀⠀⢁⣿⡆⠀⠀⠀⢿⣷⣤⣤⡠⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⡞⠁
⠀⠀⠀⠀⠀⠈⠙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠃⠀⠀⠀⣼⡿⠁⢸⣷⡀⠀⠈⠛⠿⠿⠷⢾⠷⡦⠤⡤⠴⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⢘⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⠀⣀
⠀⠀⠀⠰⠀⠀⠸⠟⠀⠀⠈⠳⣤⣤⣀⣀⣀⣴⣴⡴⠟⠋⠙⠿⢿⠿⡿⠋⢹⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⢋⣄⠀⠉⡀⢋⡄
⠀⠀⠀⠀⠂⠀⢰⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠃⠀⠀⠀⠀⣠⣬⣃⠀⠀⠀⠀⠈⠉⠉⠛⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣤⣶⣦⣴⣤⣤⡄⠒⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⡗⣫⣾⠀⠀⠀⠀⠐⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠰⠟⠐⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⢀⠉⠙⠻⠟⠛⠛⠛⠓⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣾⣿⡿⣿⠅⢀⣤⣤⠀⣤⢤⣄⠀⣀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⢀⣴⣿⡿⠢⠀⠀⠀⠸⠏⢫⠤⠁⣠⣿⠀⠈⢣⡀⠀⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⢀⣼⡅⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⠀⠀⠑⠚⠒⠉⠋⠁⠀⠀⠀⠃⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⡆⠀⠀⠀⠀⢀⣰⣶⣾⡿⠁⠀⠀⠘⠁⠀⠀⢀⣀⢠⢐⢋⢭⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠙⠇⣠⣤⣤⣾⡿⠿⠛⠉⠀⠀⠀⠀⡄⠀⠀⠀⠑⠓⠛⠘⠂⠚⠛⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⡂⠬⠩⠳⠦
⠀⠀⠀⡠⣒⡢⣞⠥⠂⠁
⠀⠀⠀⠇⡟⡹⠪⡳⣄⠀⠀⢠⠖⠛⠫⡑⠒⠤⣀⣀
⠀⠀⠀⡴⣴⠁⠀⠘⢎⢢⠀⠀⠀⠀⠀⢸⠀⣀⠤⠤⠊⠑⢦
⠀⠀⢀⠁⡆⠀⠀⠀⠈⡆⢳⠀⠀⠀⠀⡌⡰⠁⠀⠀⠀⠀⠀⠡⡀
⠀⠀⢠⢠⠀⠀⠀⠀⠀⠸⡀⠆⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⢸⠈⡀⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⡜⠀⢇⠀⠀⠀⠀⢠⠃⠀⠈⠲⢄⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⢀⠞⠀⢀⡈⠑⣄⠀⢰⡃⠤⠀⠤⠒⠚⠣⡀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀
⡞⠉⠉⠁⠀⠀⠈⠇⠸⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⡇⠀⠀⠀⠀⠀⠀⠘⡘⡆⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠇⠀⠀⠀⠀⠀⠀⠀⢡⠱⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⢰⠀⠀⠀⠀⠀⠀⠀⠈⢇⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠁⠲⠤⣀⠀⠀⠀⠀⠈⢢
⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠄⡰⠃
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⣀
⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠤⢀⡀
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠘⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠠⢄⣀⣈⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠀⠒⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⢠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⢀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢒⠀⠒⠂⠀⠀⠀⠀⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡒⠀⠐⠀⣀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⡀⠦⣤⡤⠀⠀⣀⣀⣤⡤⠶⠔⡿⠥⠤⠤⠵⠤⠁⠀⠀⠀⠈⠀⠀⠂⠢⠔⡄
⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⡰⠃⠀⠀⠀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠈⠀⣤⠤⠤⢤⣤⣀⠀⠔
⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠳⢤⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⣀⠤⠝⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠈⠀⠀⠌⠦⣷⣄⣀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠄⠀⠀⢸⣷⣄⣸⣿⣷⠀⠈⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠘⢄⠉⠉⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠃⠙⠋⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢠⠞⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⡰⠂⠀⠀⠀⠀⣀⣀⠂⠀⠀⠀⠀⢠⠂⣈⠴⠁⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⠐⡀⠀⠀⠤⣀⠁⠈⠁⠒⠈⠀⠉⢠⣀⣀⣀⡤⠚⢥⢹⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠣⡀⠀⠀⠈⠑⣦⢤⣄⣀⡀⠀⠀⠈⣀⣀⡤⡆⢸⢎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢤⠀⠀⠀⠸⣦⣄⣀⠈⠉⠉⠉⣉⣀⣴⠂⢸⠃⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣌⡀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠣⡀⠀⠹⣿⣿⣿⡿⠋⠉⠉⠉⠙⠢⠄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢆⠀⣀⠀⠈⠻⠿⣤⣴⣶⠶⠄⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠑⠙⡦⡀⠀⠠⠀⠀⠒⠀⠀⠀⠠⡀⠀⠀⠀⠀⠉⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⠘⢌⠲⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠣⡈⢆⡀⢀⠅⠀⠀⠀⠌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⢡⠁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠈⢧⠀⠀⠀⠀⠐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣓⠤⠠⠄⣀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠐⠂⠀⠃
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢉⣉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣙⣿⠛⠛⠿⣦⡀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⣿⡄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣼⣿⠀⠘⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣛⣁⣤⣤⣶⠂⣠⣶⣿⡀⢰⣷⣄⡀⠀⠉⢛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠟⠋⢻⣿⡾⠟⠁⠙⢷⣼⡏⠙⠛⠿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠈⢛⣁⠀⠀⠀⠀⢉⣙⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣠⡶⠛⠻⠆⠀⠀⠰⠟⠙⠷⠤⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣠⣤⣶⣆⡄⠀⠀⢀⣠⣶⣼⣤⣆⡄⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢈⡟⢁⣀⠙⣿⢠⡄⢨⡿⢉⣀⠉⣿⣞⠀⠀⢸⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⣇⣿⣿⣇⣟⣼⠃⢸⣷⣿⣿⡇⣾⠃⠀⠀⢸⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢿⡇⠉⠉⠉⠉⢉⡿⠃⠐⠛⠛⠛⠛⠛⢿⡄⠀⠀⢸⡇⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⠿⡏⠳⣄⢠⣤⣄⡀⠸⣿⡀⠀⠀⢰⣏⣄⣀⣤⣸⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡈⢷⡀⠀⠹⣆⢿⡈⠻⣧⣿⣧⠀⢠⠀⠉⠉⠉⠉⠉⠀⠀⢀⡀⠀⠀⠀⠀⣿⢁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣀⠈⢻⡎⠙⠛⢀⠀⢤⡙⢿⣷⡀⠈⠻⢿⣦⣏⣴⣿⣶⣴⣾⣛⣳⣄⠘⣧⡀⠀⠀⢸⡟⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⠙⠛⠁⠀⠐⠛⠿⣿⣿⣶⣤⣍⣀⣤⣤⣤⣭⠝⠛⢿⣯⣙⠛⠿⣿⣷⡌⢷⠀⢀⡿⢱⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣓⣺⣿⣿⠇⣘⡛⠛⠿⠿⠛⢻⡟⢋⣠⣶⣶⣷⣦⣹⣿⣿⣿⣾⡿⠋⠈⠀⣾⢣⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⠶⠛⣩⠟⢀⣴⠟⣿⠆⢠⣾⡏⠉⣿⡄⣰⠏⠀⢀⣼⡿⠃⠀⠀⠼⢡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠷⠶⠛⣡⣾⣯⣴⠿⢿⣿⠀⢻⣧⣀⣀⣴⣾⡟⠁⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢸⣿⠀⠀⠻⠿⠧⠔⠋⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣤⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠸⣿⣦⠀⣤⡄⢸⡏⣠⡄⠀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣜⣧⠸⢧⡿⢡⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠛⠷⠦⣍⠛⠄⠘⢁⣫⠶⠟⠃⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⣛⣉⠁⠀⠀⢈⣉⣛⠛⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⣩⡴⣢⠀⢱⡌⢦⣍⠻⠇
⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠏⣰⡏⢸⡇⢿⡄⠹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠘⠛⠁⠀⠉⠁⣸⣷⠈⠁⠀⠈⠛⠃
⠀⠀⠀⢀⣤⡀⠀⠀⠀⣿⡇⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⣤⢸⣇⣤⢀⣴⠖⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠈⠻⣷⣄⢿⡄⢿⢰⡿⢀⣾⠿⠋⠀⠀⠀⠀⠀⢰⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡙⢾⡄⠟⠡⢟⣥
⠀⠀⠀⠀⣶⣤⣙⠯⢧⠈⠟⠰⣛⣥⡶⠀⠀⠀⠀⠀⠀⠘⡀⢤⠀⠀⠀⠀⠀⠀⠾⠶⠶⣛⠂⠀⠀⠙⣒⠶⠶⠶⠆
⠀⢴⣶⣶⣶⠼⠽⠓⠀⠀⠀⠛⠯⠵⢶⣶⣶⣶⠀⠀⠀⠀⡷⠘⡆⠀⠀⠀⠀⠀⠀⠀⢛⣭⢣⢢⣦⡜⢯⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣴⡶⢚⡥⣢⠀⣄⢫⣛⠶⣦⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⡀⠀⠀⠀⠀⠀⠰⠟⠁⠛⢸⡇⠛⠈⠻⠗⠀⠀⠀⠀⢀⣠⣴⣿⡟⣏⣷⡄
⠀⠀⠀⠀⣠⣶⠟⣰⡇⣼⢹⣧⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⢀⣤⣾⠿⣻⠟⠸⡇⣿⠸⡇
⠀⠀⠀⠘⠿⠃⠀⠛⠀⣿⡆⠙⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⣀⠧⠔⣚⢂⣩⣭⣭⣭⣷⡄⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠉⢀⣾⠏⠀⠀⢼⡟⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣴⣾⣿⣿⡎⢿⣿⣿⣿⠿⣿⣿⠶⣤⣠⣶⣿⣿⠛⠀⠀⣰⡿⠃⠀⠀⠀⠈⢳⣠⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣿⣿⣿⣼⠟⠋⢁⣤⠿⠁⠀⠀⣙⣿⣟⠈⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠙⣿⣼⠳⣄
⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣿⣿⣿⣿⣉⣀⣀⣬⣷⠾⠛⠁⢀⣤⣾⣿⣿⣿⣿⣧⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⢹
⠀⠀⠀⠀⡞⣿⢹⣟⠻⢛⠛⠛⢿⢿⣮⣤⣤⡟⢠⣿⠿⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡇⢹⠁⣟⠀⠀⠉⠲⢤⡀⠀⠁⣿⠁⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠧⡀
⠀⠀⠀⠀⢸⣿⠇⣹⡄⠀⠀⠀⠀⠉⠓⢴⡿⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠿⠁⠟⣿⢿⣶⣌⡢⢀
⠀⠀⠀⠀⠀⢹⣆⣽⢯⡄⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠁⠘⠱⠟⡿⣷⣬⣀⠄
⠀⠀⠀⠀⠀⠀⠉⣿⢦⠿⣦⠀⠀⠀⠀⠀⢿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣼⣿⣿⣿⣿⣦⣤⣤⣀⣀⡀⠀⠀⠉⠘⠹⢿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢸⠂⠸⡇⠀⠀⠀⠀⠘⣧⣼⡿⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠛⠛⠒⠶⠦⣤⢤⣙⣻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢤⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢩⣿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣀⢳⣌⣷⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡧⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⣤⢹⣾⣿⣿⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣷⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣦⣹⣿⡿⠋⠉⠙⠒⠦⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣸⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣏⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢰⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣷⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⣏⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠘⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣠⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣬⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢀⡴⢋⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠙⢿⣯⣡⡀
⠀⠀⠀⠀⠠⡾⠿⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⣧⣾⣷⣼⣿⣻⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⢿⣯⣓⣶⡤⢤⣤⣤⣤⡴⠶⠛⢉⣠⣤⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣦⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢠⠔⠉⠡⡶⠚
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿
⠀⠀⠀⢺⡛⠛⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⢫⣴⣷⣙⣿⣏⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠙⠓⡶⠦⣤⣤⣤⣤⡴⠶⠛⣋⢡⣤⡼⠧⢓⡳⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠑⢦⣄⣠⣤⣴⣸⣦⣻⣜⡣⣾⢧⢶⣵⡞⠷⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣜⣻⠹⠟⠛⠚⠒⠓⠛⠉⣙⣩⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣯⡓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣻⣫⣵⣶⣿⣿⣿⣿⣷⣤⡉⠢⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣈⣠⣾⣿⣿⣿⡿⠿⣿⣯⡙⢿⣆⠻⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⢻⡿⠟⠋⠩⡀⠠⠀⠙⠿⣦⠻⣇⠹⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡃⠸⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⢿⣤⣦⣤⣀⠁⠉⣀⣤⣤⣤⡇⠀⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠘⣿⣭⣭⣿⠀⠀⢿⣯⣭⣿⠇⢧⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⡀⠈⢸⠉⠁⠀⣴⣶⡀⠈⠩⢙⠀⢸⠘⠀⡆⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠋⣃⠀⠈⡄⠀⢀⣉⣉⣁⠀⠀⠄⠀⢸⡄⠂⢹⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣻⢠⢰⣧⠁⠈⠛⠛⠁⠀⠀⢦⠀⣿⡇⠀⡈⢷⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣿⣦⢏⣆⣾⣿⠀⠀⠀⠐⠀⠀⠀⢸⡆⣿⣿⣄⡐⡈⢷⡇
⠀⠀⠀⠀⠈⠑⠚⡷⢋⡼⢻⢧⣿⣾⢟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣟⡟⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⡈⠰⢋⣴⠟⢿⡏⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣿⡗⣺⣿⡏⡇
⠀⠀⠀⠀⠀⠀⣷⢀⣾⣫⣤⣴⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢛⠓⠩⡁⡇
⠀⠀⠀⠀⠀⠀⣿⡿⣻⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣦⣽⣿⣄
⠀⠀⠀⠀⠀⣾⢿⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⡿⠶⠖
⠀⠀⠀⠀⠐⣿⡈
⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⣀⡤⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⠃
⠰⣏⣴⠶⣤⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡀
⠀⠈⠁⠀⠈⠻⣆⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠈⢧
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠒⢦⡀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠤⣄⣀⣀⣀⣠⡤⠿⡄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⢀⡴⠋⠀⢀⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⢸⣃⣀⡀⢀⣄⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣼⠳⠦⠤⠤⢶⠞⠛⠒⢤⡔⠉⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡤⣄⡿⠀⢀⡞⠁⠀⣭⡏⠀⡌⠁⢹⡆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠃⠀⠶⠀⠀⣷⠘⠃⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡟⠀⠀⠀⠀⠘⣇⠀⠀⠀⣷⣄⣀⣀⠜⠁⣠⠀⢸⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠣⣀⠀⢀⡠⠛⡦⠤⠤⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣾⠀⠀⠀⠀⠀⠀⠈⣳⣶⣏⣁⣤⣤⠴⠒⠋⢹⠀⢸⠐
⠀⠀⠀⠄⠀⠀⠀⠈⢧⡀⠘⡗⠢⠭⢥⣤⣀⣀⡠⠤⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣿⠀⠀⠀⠀⠠⠂⠀⠸⣄⣀⡀⠀⢀⣀⠠⠒⡿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢱⣄⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⢉⡀⠠⠀⠊⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠂⠀⠀⠀⡼⠳⣤⣀⡀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠟⠋⠉⠉⠹⡍⠉⠻⡇⠀⠀⢀⡤⠒⠒⠾⡁⠀⢀⣈⠉⠉⠋⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢷⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⡏⠀⠤⠀⠀⢸⠎⠉⢀⠉⠳⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⡀⠀⠀⠀⣇⠀⠀⠈⠀⠀⡇⠀⣰⠃⠀⠢⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣆⠀⠀⠀⠀⠀⣰⠃⠃⠀⠀⠉⠒⢲⠋⠙⠦⣀⣀⣠⠔⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠀⠀⠀⠈⠑⠒⣲⠖⠊⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⠀⠀⠂⠚⠂⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣤⣤⣴⣶⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠹⠉⠉⠛⣿⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠆⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⢛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⡀⣀⡀⠐⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣴⠼⠶⢦⡀⢈⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣴⠟⠁⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠶⠶⣤⡀⣴⠋⠀⠀⡀⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠶⠶⠶⣞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⣴⠋
⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡴⢄⣀⡠⣄⠀⠀⣠⠟⠀⠀⠀⢀⡾⠷⠶⠶⠶⢤⣀
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠒⠦⣄⡀⠀⠀⢠⡄⠃⠀⠁⠀⢙⠶⠊⠁⠠⢀⠐⠁⢸⡇⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⡤⠆⠈⠲⢴⡇⠀⡀⠀⠀⠁⢠⠀⠈⣷⠒⡲⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡄⠀⠘⠉⠧⣴⡀⢀⡘⠀⡄⠀⠆⣹⡆⠘⠒⠒⣦⠀⠀⠀⠀⠀⠀⢸⡇
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠈⠩⠛⠉⠙⠳⠶⠶⠿⢍⠉⠐⠀⢧⡀⠀⠘⠉⠉⠛⠛⢶⣄⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠶⠶⠶⠲⠶⠶⣤⣀⡏⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼
⢿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡟
⢸⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠈⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⣿⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠻⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⢷⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠉⠙⢿⣦⡀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⣶⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠛⠋⠉
⣿⣿⡿⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠉⠢⡀⣀⡤⠤⠖⠒⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣀⡤⠖⠋⠁⠀⢀⣠⠄⠤⠤⠤⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠤⠊⠁⠀⢀⡠⠔⠚⠁⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠉⠀⠀⢀⠠⣒⣁⠀⡀⠤⠀⣀⠀⠀⢀⠴⠁⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠨⠀⢀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠉⠶⠶⢉⠢⡘⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢀⠄⠐⠋⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⡂⠀⠀⠀⠀⠀⡇⠐⢢⠠⢄⠀⠼⠋⠉⠀⡀⠀⡰⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣎⠉⠒⠒⠒⠒⠘⢣⠀⡇⢸⠘⠀⠀⠠⠀⠀⠈⠇⣹⠆⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠂⠈⠀⠙⠲⠄⣀⢀⠎⠀⢇⠀⠈⠉⠉⠢⠀⠀⠀⣃⣤⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣄⣀⠴⠁⠀⠀⠈⠆⠀⠀⠀⠁⠀⠠⣤⡔⠉⢠⠚⠆⠁⠀⡄
⠀⠀⠠⠀⠤⠐⠒⠒⠒⠒⠒⠛⠉⣙⠦⡀⠀⠀⠌⠀⠀⠀⢀⠀⠀⡀⠈⡟⠓⠊⠑⠒⠒⠊⠀⢀⠔⠛⡴⠦
⠸⢀⣤⡴⠒⠒⠒⠒⠒⠒⠒⠀⠉⠠⢂⡀⠀⠀⠀⠀⠀⠀⠀⠙⢸⠡⠴⠃⠀⢳⠠⡀⠀⠀⠀⢸⠀⠈⡄
⠀⠀⠀⠈⠙⠦⡀⢀⣠⠤⣒⠨⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡇⠊⠲⠖⠊⠱⡢⠊⣁⠒⠢⡄
⠀⠀⠀⠀⠈⡠⣚⠥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠀⠈⢤⠀⢁⡕⠠⠘
⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠎⡠⠂⠐⠀⠀⠀⠀⠈⠉⠉⠀⡀⡰
⠀⠀⠀⠓⠒⡖⠊⠉⠀⠀⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢉⡩⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⢪⠞⠁
⠀⠀⠀⠀⠀⠹⡄⣀⠄⠂⠁⠀⠀⠘⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠉⢀⠄⠀⢬⠁⡄⠀⠀⠀⠀⠀⣇⣀⡠⠳⡄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡜⠓⠒⠋⠉⠀⠀⢙⡄⠀⠀⠀⠀⠀⠘⠰⠀⡄⠴⢀⠔⠉⢙⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢤⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⠞⢤⣀⠔⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠂⠁⠀⠈⠉⠁⠀⠀⠋⠀⢀⡀⠀⠀⠀⠀⢇⠀⠀⣸⠈⢢
⠀⠀⠀⠀⡜⠢⣀⠀⠀⣠⠞⠁⠀⢠⠃⠀⠀⠀⠀⡰⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡅⠈⠂⠀⠀⠀⠀⠑⠂⠁⠂
⠀⠀⠀⠀⢧⡀⠀⠉⠋⠁⠀⠀⢀⠓⢄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠄⢀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⢑⠀⠀⢸⠀⠀⠀⢠⠀⠀⠑⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠙⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣲
⠀⠀⠀⠀⠀⢸⠆⠀⡸⠀⠀⠀⠀⢱⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⠅⠊⢰⡄⡀⠀⠀⠀⠀⠀⢠⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⠀⠇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠄⠠⡀⢂⡠⠄⢁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢰⠘⢀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠀⠀⠀⠀⠀⡄⠀⡇⢀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⢷⠀⠀⠀⢰⡿⣄⠐⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠜⠀⠀⠀⠈⡧⢈⠑⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣧⠀⠀⠀⠀⠀⠘⢖⠁⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⢸⠀⡼⠀⠀⠀⠀⠀⠑⣤⠂⢀⠙⣆⠀⠀⠀⢀⠞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⢀⠨⣸⠀⢇⠳⡀⢤⣀⠀⠀⠈⢢⠀⢐⡘⡆⠀⢠⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⡂⠑⠄⢳⠀⠀⠁⠺⠌⡮⠳⡄⠀⢸⠂⡁⠁⢵⠀⠘⣡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠘⢄⠈⠢⡀⠀⠀⢠⠇⠂⣻⢠⠏⠀⠇⠠⡍⠀⢰⢻⡀
⠀⠀⠀⠀⠀⢯⠢⣄⡀⠀⠀⢳⠅⠘⡀⠑⢄⠙⣆⣠⠋⣠⠀⠿⠉⡠⠁⠂⢠⠃⠀⡼⠐⣇
⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠱⡄⠑⡀⠀⠳⡈⠃⡐⠀⡄⢀⠎⠀⡐⢠⠏⢀⣴⠇⠀⢻⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠰⣤⣅⡚⠀⠀⠀⠘⣦⠐⡀⠀⠱⡐⠀⠀⡖⠁⠀⡐⢠⣮⡴⠛⢁⡀⠀⣼⠀⠀⢀⠜⣹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⡌⠓⢤⠀⠀⢸⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠛⣀⠄⠒⡉⠀⣴⠏⠀⢐⡡⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⡀⢳⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⢀⠎⢀⡤⠁⠀⠀⢈⠔⢫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⡄⢷⠟⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⡜⢁⣴⠏⡴⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠐⠀⢠⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠘⣵⠯⢹⢰⠃⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠃⢦⢘⠟⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠈⢦⡰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠀⣀⣀⠀⠠⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⠐⠈⠐⠠⠀⠠⠀⠤⠂⠄⠂⠀⠠⠀⠠⠤⠆⠑⠀⠀⠁⠒⠒⠐

⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠶⠦⣤⣀⡀⠀⠀⠀⢻⠓⠶⢤⣀
⠀⠀⠀⠀⠀⢹⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠓⠢⣄⣸⠀⠀⠀⠉⠛⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠈⣧⠀⠉⠓⠒⠲⠶⠶⠶⠤⠤⣷⡀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⢴⡶⠚⠛⠋⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠢⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠄⣀⣀⡇⠀⠈⠑⢄⣸⠁⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⠀⣀⣤⠤⠽⠞⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡑⢄⠀⢀⡴⠋⠀⢀⣠⣤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠉⠛⢷⡞⠁⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠐⢼⢀⠞⠙⠳⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣾⠃⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣽⠿⠃⠀⠀⠈⠉⠛⠶⠦⢤⣤⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⣤⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣅⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢋⣙⣫⣿⣩⣮⣛⣭⢻⣳⣶⡦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢱⡷⡟⣛⣫⣭⣥⣾⣷⣾⣧⣬⣭⣽⣛⢷⢾⣏⣉⣭⢙⡳⢦⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠭⠭⠟⠲⣾⣿⣿⣿⠩⠶⠄⠀⠬⡽⣿⣿⣿⡟⠈⠋⠉⠟⢹⡧⢶⠙⢿⣏⠁⣸⡇
⠀⠀⠀⠀⠀⣴⡒⠲⣴⢏⡽⠋⠁⠀⠀⠀⠀⠘⣿⣿⡉⠐⠋⠀⡟⠛⠓⡉⣿⣿⣇⠰⣿⣦⣠⣤⣄⡘⢒⣇⣙⢷⣿⠃
⠀⠀⠀⠀⠘⣿⣿⡶⢣⣾⣇⣀⣤⠤⠴⢤⣤⣶⣿⠿⢷⡠⢮⣤⣡⣴⢀⡾⠟⠛⠙⢦⣽⢿⠂⠀⣤⣝⢿⣉⣿⣤⢳⡀
⠀⠀⠀⠀⠀⣸⢟⣵⣿⣿⡟⠹⣶⠂⣀⣰⠿⣣⡿⠴⠞⢳⣄⠙⠋⣠⠞⠲⣤⣤⠀⣠⣆⣽⣃⠐⠷⠀⢸⣿⣷⡿⡦⢳⡀
⠀⠀⠀⠀⡾⢡⣿⣿⣏⠋⠀⠈⠉⢀⣹⠉⣿⠁⢰⣴⡆⠈⢯⠉⣉⠱⠿⢢⡞⢹⡏⠛⡟⣾⣿⠷⠖⠀⣼⣶⣿⡟⣻⡞⡇
⠀⠀⠀⠘⢧⣾⣿⣿⣿⣇⡀⠀⠛⣾⣟⣱⠏⠀⢸⡏⡿⠀⠀⣴⢿⣷⣶⠟⠀⢀⣽⣤⡱⣮⡥⠴⠖⣻⣿⣿⣿⣏⣼⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠓⠒⠲⠾⠛⠳⢶⡌⠛⠛⠁⠀⠀⠀⣼⠋⠀⠀⠛⠁⠀⠀⠈⠧⣴⠞⠉⠉⠛⠻⠿⠿⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡉⠀⠀⠂⠳⢤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⡀⠄⠠⢀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠂⠀⠠⠈⠀⠀⠀⠀⠁⠀⣹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⢠⡀⠀⠀⠀⠀⠂⠀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠂⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣀⠱⡄⠠⠄⠀⢀⠀⠀⠐⠋⠀⠀⠀⢀⡴⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡠⠔⠚⠛⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣀⣀⣁⠀⠀⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠀⣀⠤⠒⣊⠉⠤⠤⠤⣀⠀⠀⠀⠀⠈⠉⠲⣄⠈⢃
⠀⠀⠀⠀⠀⢀⣠⠤⠄⡀⠀⠀⠀⠀⡔⠁⠀⣀⡴⠊⢁⠔⡁⠀⠀⠀⠐⠀⠀⠒⠄⠀⠀⠀⠀⠀⠈⢳⡘⠂
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠘⢆⠀⠀⡘⠀⢠⠞⠁⠀⠀⢈⠀⠁⠀⠣⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠸⡀⠀⡇⢰⠋⠀⠀⠀⢰⠁⠠⠂⠀⣈⣁⣨⣶⣶⠁⡇⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠱⡄⠀⠀⠀⠀⠀⠒⢠⠐⠺⢿⣿⣿⠿⠁⢀⠃⠀⠀⠀⠀⡠⠞⠁
⠀⠀⣠⠖⢆⢳⠀⠀⠀⠀⢸⠁⡤⢤⡈⠠⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠁⢠⠿⢤⡠⠀⠂⠉
⠀⠀⡇⠀⠀⠘⠇⠀⠀⠀⢸⢸⠀⠀⠹⡆⠀⠀⠀⠂⠒⠶⢦⡤⠛⠢⠀⠀⣠⡄⠀⠁⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⡇⠀⠈⠒⠃⠀⠀⠀⡈⠖⠀⠀⠀⡇⠸⠁⠼⠀⠙⡶⠋⠀⠀⠈⣿⣿⠏⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠐
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡉⠀⢈⡖⠀⡄⠀⠀⢀⡠⠛⡁⠤⠐⠒⠒⠤⢄⣀⠀⢀⡀⠆⠻⡀⢃⠠⣄⢀⣤⠤⠞
⠀⠀⠈⠓⠐⠲⠀⠀⠀⠀⠀⠀⠖⠒⠁⠀⣠⣀⠡⠌⠉⠁⠒⠒⠓⠂⠁⣀⣀⠀⠀⠀⠀⠀⠉⠁⣉⣼⡌⠂⢀⠜⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⣿⣿⠇⠀⠀⠀⠀⣴⡏⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠁⠀⠀⠀⠀⣠⡏⢰⠀⠀⠀⠀⠠⠋⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣤⣴⡾⠋⠀⢘⡀⠀⠀⢠⣾⠀⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⠻⢶⣤⣄⣠⣤⣴⠾⠛⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠃⠀⠸⠁⠘⠆⠀⠀⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⡘⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣦⠀⠀⡆⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣘⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠄⢀⣥⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠁⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠉⠑⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⠶⠞⠒⠠⠄⠀⡉⣩⣁⣉⣉⣉⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣲⣤⣀⣴⣶⣒⣢⣴⠃⠀⡼⠋⠀⠀⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉⠉⠁⠀⠦⣑⣒⣁⣒⣐⣒⡉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠖⠛⠶⠊⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⣀⣤⣤⣀⠀⠀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⡴⠟⣻⣿⠟⢉⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⢠⠋⢀⡴⠟⠀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⠰⣃⣴⠿⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⡟⠃⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠊⠻⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⠀⠀⠀⠙⠷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⠖⠉⠀⠀⠀⠀⢀⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠟⠁⢀⣀⣀⣀⣀⣀⡿⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⢀⣤⣾⣿⣿⣿⡙⡉⠉⠠⠀⢹⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⣾⡇⠀⠀⠸⠃⠐⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠛⠀⠉⢿⣧⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⢀⡏⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡈⠀⠀⠀⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⠀⠀⠀⡀⠀⢠⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠛⡏⠸⠋⠈⠉⠀⠀⠀⠀⠀⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣴⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣄⣠⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⠿⠻⠗⢿⣿⠁⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠋⠋⠀⠀⠀⠀⠀⢈⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠧⣒⡄⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠿⣿⣿⣿⠋⠘⢯⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢈⠁⠋⠀⠀⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠋⠁⠀⣀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⢿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⢇⣀⣤⡿⠛⠁
⠀⠀⠀⠀⠀⠈⢿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠀⠀⠐⣿⣷⠀⠀⠀⠀⠀⠀⡀⣸⣷⠀⠀⠀⠀⠀⣴⣿⣦⡀
⠀⠀⣴⣿⠏⠙⣿⣤⠀⠀⣰⣼⡿⣿⣷⢀⠀⠀⠀⠐⣿⣿⣿⣤⡄⠀⠀⠀⢰⣿⡿⣿⣤⠀⠀⠀⡀⣿⣿⢻⣿⡄
⠀⠀⢿⡏⣰⣆⢹⡿⠀⠀⢻⡿⠃⠘⢿⣿⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠐⣾⠟⣀⠘⣿⠀⠀⠀⣿⡟⠁⠀⢿⡿
⠀⠀⠈⠓⠝⠋⠊⠀⠀⠀⠈⠓⠀⠀⠘⠁⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠙⠐⠿⠂⠋⠀⠀⠀⠘⠃⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⢀⢸⣧⡀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⢠⣼⣿⣿⣿⣇⠀⠀⠀⢀⣼⣿⣿⣧⠀⠀⠀⢠⣄⣿⣿⣿⣀⡀⠀⠀⠀⣼⣿⣿⡄⡀
⠀⠀⢀⣿⣿⣿⣴⡆⠀⠀⣸⣿⠟⠙⢿⣿⠆⠀⠀⣾⣿⠏⢹⣿⡇⠀⠀⢺⣿⡿⠃⢻⣿⡇⠀⠀⣶⣿⣿⣿⣿⣧
⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠘⢿⢠⣷⠈⠟⠀⠀⠀⢻⡇⠀⠘⢻⠃⠀⠀⠘⠯⢀⣶⡄⠿⠁⠀⠀⠻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⣀⣼⣿⣦⠀⠀⠀⠀⡀⢿⣧⣷⠀⠀⠀⠀⠀⣾⣿⣧⣄⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⣾⣿⡇⢀
⠀⠀⣼⣿⡿⠻⣿⣇⠀⠀⢸⣷⣼⣿⠃⣠⠀⠀⠀⣄⣿⠏⠻⣿⡆⠀⠀⠀⢸⣿⣿⡟⢀⠀⠀⠀⣄⣿⣿⣿⣿⣀
⠀⠀⢻⣏⠀⠀⢹⡟⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⢿⡏⣠⣆⢹⠇⠀⠀⠐⣾⡿⠈⣿⣿⠀⠀⠀⣿⣿⡏⠘⣿⡿
⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠈⠃⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣰⠆⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⢆⣎⣀⠀⠀⠀⠀⠀⠀⠠⠠⠁
⠀⠀⠀⠀⢀⠴⠒⠃⠤⣀⣤⣷⢸⣁⡠⠽⠦⢄⠀⠀⠀⠀⠀⠀⠠⢾⣿⠀⠸⣿⠗⣠⣤⡀
⠀⠀⠀⠀⡜⠀⠀⠀⠀⡸⠟⠛⠛⠿⠄⠀⠀⠀⠁⠀⠀⠀⢀⣴⡶⠌⠙⠀⠀⠈⠀⢿⠿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⣯⡀⢡⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⡧⠀⠈⠀⣤⣤⡀⠀⢰⣿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠘⢿⠄⠠⡄⠙⠋⠁⠀⣹⡏
⠀⠀⠀⠀⠀⠈⠀⠀⢀⣆⠀⠀⠀⠀⣠⠤⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⣀⣄⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠐⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋⠋⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣀⣀⣆⣤⣶⣤⣄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠼⡾⠿⠟⠉⠙⠾⣎⠩⡿⣄⠀⠀⠀⠀⠐⢻⠂⠀⠀⡀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⡀⠀⠹⡦⠈⢮⣦⠀⠀⠀⠀⠘⢣⡀⠀⠑⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠄⠀⠀⠈⠳⡀⠀⠉⠈⠀⢻⣧⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠲⢀⡀⠀⠐⠀⠀⠀⠀⠠⠙⣧⡀⠀⠀⠀⠀⠀⠘⠲⢄⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠹⢡⣤⠀⠀⢆⠀⠀⠀⠰⠌⠿⣄⠀⠀⠀⠀⠀⠀⠀⠁⠰⢆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡋⠒⠄⡀⠀⠀⠀⠀⠀⠁⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⡀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⢨⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⢀⠀⠀⠀⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢦⣀⠀⠀⠐⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢹⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠘⠉⠻⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠹⣯⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠻⠶⠶⠤⢄⣒⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢦⣀⠀⠀⢉⠒⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⢄⡑⠂⠤⠤⣀⡀⢄⠀⠀⠀⠀⠀⠀⣤⠜⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠶⠮⣭⢍⣀⡉⠀⡐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⣶⣶⣿⡿⣿⣿⣿⣿⣽⣿⣶⣶⡒⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠿⢛⡉⣟⣋⡽⢿⡿⣿⣻⣿⣟⣿⣿⢿⡿⠧⣉⡻⢁⡙⠷⣄⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠷⢿⡟⢁⣤⡈⢡⡽⣧⡴⢚⣿⣯⣽⣿⣿⣯⣿⡿⡙⢶⣼⢣⡌⣡⣤⡀⠉⠳⢮⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⢯⡀⠀⠀⢸⠸⣿⣿⠀⢴⡌⠲⣯⣞⢩⡏⡏⢻⢹⣆⣳⣿⠟⢠⡦⠀⣿⣾⠃⠀⠀⠀⢨⡿⠟⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⣀⣼⠗⠀⠀⢸⣷⣿⠃⠀⠀⠀⠛⠀⣭⠙⣛⠛⠛⣟⠛⣭⠀⠋⠀⠀⠀⠈⠻⣦⡀⠀⠀⠹⢧⣄⣀⠈⠿⣦⡀
⠀⠀⠀⠀⠀⣠⡾⠋⣴⡟⠺⠿⢶⡄⠚⠛⠉⠉⠀⠀⠀⠀⢀⡴⠲⢌⠀⠛⠀⠀⠉⠀⡤⢖⢲⡄⠀⠀⠀⠀⠀⠉⠛⠛⢡⡤⠬⠽⠿⣦⠙⠿⣆
⠀⠀⠀⠀⣰⣋⣴⠯⠤⣄⠀⠀⢨⣥⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡿⣮⢷⡀⠀⠀⢀⡞⣴⢟⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣬⣅⠀⠀⣠⠔⠿⣦⣙⣦
⠀⠀⠀⢰⣏⣿⡇⠰⣿⠟⢀⡀⢣⣷⡟⢠⣄⠀⠀⠀⠀⡰⣞⣦⢾⢝⣿⣥⠀⠀⣤⣿⡷⡲⣾⣟⢢⠀⠀⠀⠀⣴⡄⢻⣾⡜⢀⣄⠻⠾⠂⠘⣿⣽⡆
⠀⠀⠀⣸⣧⣛⣽⠳⡄⢸⣧⣿⡎⣿⣇⢻⡿⠀⠀⠀⠀⠘⠿⣿⣷⣧⠻⣿⠀⠀⣿⢛⣬⣾⣻⠟⠁⠀⠀⠀⠀⢿⡷⣼⣿⢹⣽⣾⡇⢠⠞⣧⣭⣼⣇
⠀⠀⠀⣿⢾⣟⠁⠦⢹⣎⢹⣿⡇⣿⣼⠟⠁⠀⠀⠀⠀⢀⡤⣬⣙⠿⣷⠟⠀⠀⠻⣿⢟⣫⡴⠦⡄⠀⠀⠀⠀⠀⠻⣧⣿⡸⣿⡏⣷⣿⠶⠉⣹⡷⣿
⠀⠀⠀⣿⢺⣏⡑⠢⣼⡟⣿⣿⠞⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠾⣷⣜⣧⠀⢀⣼⣷⣯⠶⠿⠛⠃⠀⡀⠀⣀⠀⠀⠀⠈⠳⣿⣿⢼⣷⠔⣋⣽⠗⣿
⠀⠀⠀⣿⡷⣧⡈⣹⢟⡴⠛⠁⠀⠀⠀⠰⠀⠛⠀⠙⠀⠶⠆⡐⢿⣎⢻⣟⣆⣸⢻⡏⣰⠿⢂⠐⠗⠈⠋⠀⠉⠀⠃⠀⠀⠀⠈⠙⢿⡛⣧⡀⣼⢻⣿
⠀⠀⠀⣿⣏⣳⡾⣱⠿⣀⡀⢀⣠⣤⣶⣶⣾⣿⣿⣶⣦⣄⠀⠉⠄⢹⣾⣿⣿⣿⣿⣷⡇⠰⠀⠀⣠⣶⣾⣿⣿⣿⣿⣶⣦⣤⡀⠤⣀⠝⣮⠻⣏⠹⣿
⠀⠀⠀⣿⣷⡟⣼⢏⡷⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣼⣯⣽⣿⣿⣯⣽⣷⣄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⡜⣷⣻⣟⣿
⠀⠀⠀⣿⣾⢸⣯⡽⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣻⣝⡋⢉⣻⢽⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣾⣿⣿⣿
⠀⠀⠀⣿⣿⣼⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⡿⣿⠃⠻⣿⠷⣍⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⣿⣹⣏⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢱⣾⣿⣄⣴⣿⣿⣎⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢱⡟⣿
⠀⠀⠀⣿⡁⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⡿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⣿⣝⣿
⠀⠀⣠⣿⣿⡿⢀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣞⢹⣿⣿⣿⣿⣿⣿⣇⢿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡀⠻⣿⣿⣄
⢠⣾⣻⣍⡇⠀⠈⠤⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⠛⠁⣼⠇⣿⣿⣿⣿⣿⣿⣿⣿⡸⣧⠈⢛⣷⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠲⠀⠀⢘⠱⣏⣿
⣿⣬⣭⠾⠁⠀⠀⠀⠃⣼⠿⣿⠿⠿⠟⠛⠉⢁⣈⣽⡿⡇⢰⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡇⢸⢿⣮⣅⣀⠉⠉⠛⠿⠿⢿⠟⢧⠈⠀⠀⠀⠈⠷⠯⣥
⠻⣆⡀⠀⡄⢠⡄⣀⠀⠀⠀⠀⠰⣶⡶⣶⡿⢿⡿⡟⠇⣇⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠙⣼⠼⣹⢿⡿⢻⡿⢶⡶⠆⠀⠀⠀⠀⣄⠰⡄⠰⠀⢀⣴
⠀⠈⢻⣶⡖⠀⠀⣬⣀⠠⠄⠀⠀⢼⣔⢮⣙⡌⢷⡽⣸⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢇⢏⡞⢰⣫⡶⣦⡖⠀⠀⠰⢈⣡⣴⠀⠀⢰⣿⡟⠁
⠀⠀⠘⣏⣿⣶⣾⣿⣿⣷⣿⣟⠁⠈⠋⠸⣿⠀⣼⢣⠇⠀⣷⠸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠇⣾⠀⠘⡎⣧⠐⣿⠇⠙⢁⠈⣻⣿⣿⣿⣿⣿⣿⡿⣸⠇
⠀⠀⠀⠻⣮⣭⣵⣦⣴⣯⣝⠻⣷⣔⣦⣄⡀⢀⣿⡈⠀⠀⢿⣦⣈⠛⢋⣁⡈⢡⣤⡉⢉⣁⣼⡿⠀⠀⢁⣿⣀⢀⣤⢾⣿⡾⢟⣩⣵⣶⣶⣶⣭⡵⠋
⠀⠀⠀⠀⠀⠀⢿⣿⡻⢿⠿⢻⣷⣾⣿⡿⣿⣿⣿⣵⠁⠀⠈⠛⠛⣇⠀⠙⠃⠘⠋⠀⣨⠛⢛⠁⠂⠈⡾⣿⣿⣿⢯⣿⣿⣿⣟⢻⠿⣟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⢻⡾⣿⣿⣿⣿⣴⣿⡿⣭⠆⣼⢠⡇⠀⣸⠀⠀⢸⡇⠀⠀⣇⠀⢸⡄⢧⡐⣵⢻⣿⣷⣿⣿⣿⣿⣿⡏⣰⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣿⣿⣿⣿⣿⡿⡣⣹⢏⡾⠃⡾⠁⢰⡋⣀⣀⠸⢃⣀⣀⠘⡆⠀⠷⠈⢷⡸⣏⢮⣻⣿⣿⣿⣿⣾⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⢻⣿⣿⡟⠘⢛⣥⣈⠶⣤⡾⠛⢷⡼⠋⠉⢳⡞⠉⠙⢷⡾⠛⢷⣴⠶⣤⣬⡉⣓⣹⣿⣿⡟⣹⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⣿⢸⢿⣿⣿⢻⠻⡇⢿⠀⠘⠃⠀⢸⠇⠀⠀⢸⡇⠀⠀⠸⡇⠀⠘⠃⠀⡿⢸⠟⡟⣿⣿⣟⡇⣟⢃⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣸⢸⣿⣿⣶⣤⣷⣼⣤⣴⣦⣤⣼⣤⣤⣤⣼⣧⢠⣤⣴⠧⣤⡾⢧⣤⢷⣾⡴⣶⣿⣿⡿⡇⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣿⣻⢸⣿⡿⣿⣼⢸⠇⢹⠀⠸⡇⠀⢸⡇⠀⢨⡇⠀⢸⡇⠀⢸⠁⠀⡇⠈⡅⢷⣷⢿⣿⣇⣏⣿⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⢸⣿⣇⡘⡷⢿⣤⣾⣄⣠⣿⣤⡼⢷⣤⡼⢷⣤⡾⢷⣤⢿⣄⣠⢿⣤⡿⢾⢡⣸⣿⡇⡇⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣼⠈⠛⢿⡇⢷⢶⡉⢰⡈⠉⣴⡀⠀⣦⠀⢀⡀⢀⣶⠀⢀⣦⠀⢀⡆⢁⡿⡞⢸⡟⠋⠁⣿⢻⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡧⣿⡀⠖⣸⡇⠈⠘⠃⠈⠇⠀⠈⡇⠀⠹⡇⠈⠃⢸⠇⠀⠸⠁⠀⠸⠁⠘⠃⠁⠸⣇⠲⢀⣿⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡻⣧⡈⠈⠀⠀⣤⢠⡄⣀⠀⠀⠀⠀⢀⡀⢰⡇⣀⣀⠀⠀⠀⠀⣀⢠⡄⣤⢀⠀⠀⢁⡾⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⠶⣶⠓⣉⣉⡑⢫⡄⠀⠀⠀⢿⣽⣏⣹⣧⠟⠀⠀⠀⢠⡝⣉⣭⣍⠙⣷⠞⣩⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣳⡿⠉⠹⢆⣚⡀⢠⣚⢦⣹⡟⢻⣯⡴⣿⠆⢀⣃⣸⠏⠀⣻⢟⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣿⣛⣛⣙⠻⣤⣈⠻⢋⣱⣬⣙⠛⢡⡴⢟⣩⣍⣻⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⣠⠐⠄⡄⡀⢤⢻⢰⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣷⣶⣼⡇
⠀⠀⠀⣧⠀⠀⢻⠸⡈⡾⡆⣇⠀⠀⠀⢀⡠⠄⠀⠀⣼⡟⠁⠀⢀⣷⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠻⠗⠊⣘⠀⠁⣁⢤⣠⠐⢄⣎⠂⢣⠀⢀⣾⡟⠀⣀⣠⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⢈⣿⡧
⠀⠀⠀⠀⠈⢋⠈⠃⠐⠼⣀⠇⣷⠘⠈⠂⣀⣴⣾⠏⠀⣲⣿⣿⣿⣿⡿⠋⠀⢀⣀⣠⡀⡀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠈⣧⠈⠃⠀⠛⠀⠉⠀⣠⣾⣿⣿⣿⣶⣾⣿⣿⣿⡿⠛⠁⢀⣾⡿⠛⠻⣷⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⣰⣿⠋⠀⠀⣠⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⡿⠟⠃⠀⠀⠀⢀⣼⡟⠁⣀⣤⣴⣿⣿⣿⣿⠃
⠀⠀⠸⣿⣶⣶⣶⣾⣿⣿⣿⡿⢛⣋⣭⡴⠟⢋⣀⣤⣤⣤⣶⣶⣿⣯⠀⣠⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠈⠙⠛⠒⠛⠛⠛⠛⠛⠉⠉⣁⡴⠞⣛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⡀⠀⠀⠠⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠈⠀⠀⣀⣠⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠓⠀⣯⣝⢱⣿⠟⠉⠀⠀⠀⢀⣩⣿⣿⣿⣙⣻⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠉⠹⣿⣾⣷⣶⣶⣶⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠈⠉⠉⠋⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣸⣧⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⢷⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢀⡏⣽⠀⢸⠸⡆⠀⠀⠀⠀⠀⠀⠸⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⡼⠀⡏⠀⢸⠀⢧⠀⠀⠀⠀⠘⣧⠀⢹⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢰⠃⢠⡇⠀⠘⡇⠘⡆⠀⠀⠀⠀⡏⣇⠀⢳⠀⠘⢦⠀⠀⠀⠀⠀⠀⣼⡇⠘⡇⢧⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⡏⠀⢸⠁⠀⠀⡇⠀⢹⠀⠀⠀⢠⡇⣿⠀⠀⢧⠀⠈⢧⠀⠀⠀⠀⢀⡟⡇⠀⢷⠘⡆⠀⠀⠀⠀⠀⢸⡟⡇
⠀⠀⠀⢰⡇⠀⠸⣆⣀⣠⡇⠀⢸⡇⣤⠀⢸⠁⢸⠀⠀⢸⡆⠀⠈⡇⠀⠀⠀⣸⢰⡇⠀⢸⡄⢹⠀⠀⠀⠀⠀⠈⡇⢷
⠀⢠⡀⢸⡇⠀⠀⠙⠋⠉⠀⠀⢸⢿⣿⡄⠸⡆⠸⣦⢴⢾⡇⠀⠀⡏⣠⠀⠀⡇⠘⣇⣀⣸⠃⠈⡇⠀⢀⠀⠀⠀⢹⡈⢧
⠀⢸⣿⠀⡇⠀⣀⠀⠀⠀⠀⠀⡞⠈⣇⢳⡀⢳⡀⠈⠀⠀⠀⠀⢀⡇⣿⡀⠀⡇⠀⠙⠋⠁⠀⠀⢷⠀⠘⣧⠀⠀⠀⣧⠀⠳⡄
⠀⣸⢸⠆⢹⢠⠟⡆⠀⠀⠀⣸⠁⠀⠸⡄⢧⠀⠳⡄⠀⠀⠀⠀⡼⠀⢧⢳⠀⢻⠀⠀⠀⢀⣀⡀⣸⠀⠀⡏⣧⠀⠀⠸⡄⠀⠘
⡆⡇⢸⠀⠈⢿⠀⢻⠀⠀⢠⠇⠀⠀⠀⣧⠈⣇⠀⠙⣄⠀⠀⠀⡇⠀⢸⡄⣧⠘⣆⠀⠀⣸⠉⢳⡏⠀⠀⡇⢸⡄⠀⠀⢳
⣿⡇⠸⣇⠀⠈⠀⣸⠀⠀⢸⡄⠀⠀⢀⣿⠀⢹⠀⠀⠸⡆⠀⠀⣿⠶⠛⠁⣽⠀⠙⡀⢂⢻⡀⠘⠁⠀⢰⡇⠘⡇⠀⠀⢸⡆⢲
⢻⣷⠀⠹⢶⠶⢾⡟⠀⠀⠀⢿⡶⠶⠿⠃⠀⡞⠀⠀⠀⣿⠀⠀⠈⠂⠀⠀⡏⠀⠀⠩⡌⢀⠹⢤⣤⣴⠟⠠⠂⡇⠀⠀⢸⡇⠸
⠀⠉⠃⠀⠀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⣼⡇⠀⠀⢀⣿⡆⠀⠀⠀⠀⢸⡃⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠻⠶⠶⢾⣿⠁⠀⠀⠀⠀⠸⣷⣤⣤⡴⠟⣷⠀⠀⠀⠀⠀⠀⡞⠉⠋⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠆

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⢀⣤⢖⣻⢿⡿⠂⠀⠀⣀⣤⡀⠀⠀⠀⠀⣠⠿⠛⠉⠉⠁⠀⠀⣠⠔⢶⣄⠀⠀⠉⠉⠓⠦⠤⠀⠀⠀⠀⠀⠀⣤⠀⣀⡀⢀⣤⣶⠟
⠀⠁⠈⠉⡵⠋⠀⢀⣤⣾⣿⣿⣿⣦⡀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣶⠭⣽⢿⣻⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣷⣴⡄⢁⡘⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⡮⢝⣮⣵⢂⠛⣽⢚⣻⢷⠀⠤⠀⠀⠀⢨⣿⣟⣿⣾⠛⣿⡇⣠⡖⣿⣄
⠀⠀⠀⠀⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠄⠀⠀⠀⠙⠚⠓⠴⠯⠹⠤⠛⠘⠉⠀⠀⠂⠀⠠⠾⡵⠋⠐⠉⡌⣿⡀⠙⣿⣿⣿⠂
⠀⠀⠀⢀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⢹⠇⣷⣿⡇⢀⢹⣿⡧
⠀⠀⢀⣞⣇⢀⡴⠂⠀⣀⣀⠀⠀⠀⠀⠀⠀⣽⡽⢢⢆⡄⣀⣠⡇⠐⢩⢭⣆⢀⣀⣀⣀⣀⡀⠀⠀⣤⣀⡀⠘⠃⠀⣿⠀⠸⢹⣿⠇
⠀⠀⠀⢾⠟⠼⠁⠸⡟⠛⠉⠛⠲⡀⠀⠀⠰⠇⠚⠓⠛⠛⠓⠋⠂⠀⠋⠘⢳⠗⡟⠚⠛⠘⠃⠀⠀⠪⠽⠃⢀⣤⡄⠹⠁⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⣤⢸⡄⠀⠻⢧⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈⠀⠁⠀⠀⠀⠀⠀⠀⢠⠢⡄⠈⠽⠟⠀⢠⡶⡄
⠀⠀⠐⠠⣾⢿⣾⣿⣾⡇⣄⠀⠈⡏⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣾⣯⣽⣷⠀⠙⠑⠋⠀⢀⣀⣀⣈⡁⠁⠀⠀⠺⡧
⠀⠀⠀⠀⢤⣾⣿⣿⣿⣷⣿⠀⠀⠀⠀⠸⢀⡀⠀⣤⡒⠿⠛⠋⠀⠀⠾⠁⢲⣆⡄⠀⢈⠀⠉⢁⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠆⠀⠀⢉⣿⠛⢻⡟⢿⠂⠀⠀⠀⢀⡸⠁⠀⢿⡇⠀⠀⠀⠀⠀⠀⠰⠿⢿⡇⠀⠿⠥⠿⠿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠒⠀⠀⢸⡿⠀⣿⡇⣿⠀⠀⠀⠀⢸⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣘⣿⠇⠀⡀⠀⠄⠀⠀⠀⠙⣿⡿⣿⠛⠛⢻⣿⣿⣿⣿⡿⠃
⠀⠀⡄⠀⠀⢸⡇⠀⣿⠁⣿⠀⠀⠀⠀⠀⠄⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠰⣿⡿⠀⠀⣦⣤⣠⣤⡅⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⣀⡏
⠀⠀⠀⠀⠀⠘⠃⠀⣿⠀⠛⠀⠀⣧⣀⣸⣧⡄⠀⠀⠀⠉⠠⠀⠀⠤⠔⠻⡋⠀⠀⠀⣿⣀⣿⣿⡆⠀⠀⠙⠢⢄⡀⠀⢀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠐⠒⠒⠀⡀⠆⠀⠀⠀⠈⠈⢂⠀⢰⣷⣴⡆⢀⠃⠀⢀⣾⡄⠀⢰⠀⠀⢀⠔⠐⠘⠀⠀⠀⣈⠑⠄
⠀⠀⠀⠀⠀⠌⠁⠀⠱⠀⠀⢀⢀⠀⠂⠘⠋⠙⠃⠀⠰⠀⠸⠿⠃⠐⠄⣤⠀⢽⠀⠀⢀⠀⠀⠀⠁⠀⠒
⠀⠀⠀⠀⠀⢄⡁⠀⠰⠀⠀⣈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢄⠀⠀⢁⠀⠀⠀⠘
⠀⠀⢘⣿⠆⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⣴⣷⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⢀⡼⡿⠀⢳⠀⠀⠀⠀⠀⠌⡄⠀⠈⠘⠳⠺⠂
⠀⠀⠈⠛⠁⠀⠀⠀⡀⠀⠐⠀⠀⠀⢿⠿⣿⡿⠀⠀⠀⠸⠁⠀⠀⠸⡇⠀⠀⠀⠘⠀⠀⠂
⠀⠀⢀⠾⠀⠀⠘⢆⠀⠀⠈⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠁⠀⠀⠈⣤⢀⢀⠀⠀⠀⠀⠀⠠
⠀⢀⠂⠘⠂⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠁⠘⠀⠒⠀⠀⠀⠀⠀⢘⣾⣿⣿⣾⡀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⢀⠔⠀⠀⠀⣴⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠉⠙⠁⠈⠁⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⢀⣠⣿⣤⠀⠀⢾⡿⢿⣿⠆⠀⠰⣦⣿⣾⠟⠀⠀⠀⣶⣢⣶⠆⠁⠣⠾⠰⠀⠀⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠛⠃⠙⠀⠀⠈⠓⠈⠁⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠁⢀⣾⠇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠀⡀⢀⠀⠀⠀⢀⡀⠈⠀⠀⠠⠀⠀⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠂⠁⠀⠉⠈⠀⠀⠁⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠆⠙⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⢀⡠⠚⣩⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢋⣡⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠓⢀⢐⡉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠔⠈
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠀⠀⠀⠀⢀⣒⣼⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠇⠿⠇⠟⠁⠀⠀⠀⠀⠀⠀⢠⡠⠀⣐⡤⠤⡂⠀⠀⠀⠀⠀⢀⠀⣀⡀⡀⣀
⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⣐⠀⠀⠀⠁⣈⠀⠀⠀⠀⠰⠈⠀⠁⠀⠁⠀⠁⠐⠦⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠚⣻⠧⡀⠀⠀⣈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⢣⡌⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣯⠏⠁⡀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠁⠨⣯⣆⠹⣆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠹⣧⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠙⢿⣾⣆⢄⢀⠀⠠⠖⠂⠈⠀⠀⠀⠀⠰⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠈⠛⠣⣂⣀
⠀⠀⠀⠀⠀⠄⠐⠁⠁⠐⠆⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠤⣀⠀⠈⠉⡁⡀⣀⣹⠀⠀⠀⠰⠇⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠩⠈⡅⠱⠀⠀⠀⠀⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⠠⠠⠧⠚
⠀⠀⠀⠀⠀⣰⠤⠀⠀⠀⠀⠀⠀⢰⣰⡶⠄⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡈⠒⠄⢶⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡁⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠬⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠮⠘⢁⠾⠛⡩⡤⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢀⣀⣃⠈⠢⠐⠖⢲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠭⡅⠀⢦⣤⠄⣠⣘⡳⠾⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⡆⢀⣄⡐⠀⠠⠾⠿⠛⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠘⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠴⠁⠀⠈⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠄⠂⠀⠈⠉⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣦⣴⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡝
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠈⣱⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣵⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⣷⣴⣦⣴⡟⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠻⡇⠸⣿⣿⣿⣧⣤⡈⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣿⣥⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣆⣼⣿⢹⣿⣿⣿⡀⠉⠙⠿⣿⣷⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠆⠀⣠⣾⣿⡿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢿⣿⣿⣿⣮⣿⣿⣿⣧⣀⣀⠀⠈⠙⠻⠂⣭⣽⣿⣷⡦⢂⣀⣀⣠⣶⣶⠖⣀⣀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢓
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠉⠻⡿⢿⣿⣿⡉⠻⠿⣿⣿⣿⣿⣷⣶⣿⣽⡁⠀⠀⣀⣉⣉⢁⣀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣾⣧⣀⡀⠛⠛⠿⣿⣿⣿⣦⣾⣿⡿⡿⠿⠿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⡀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣷⣶⣤⣤⣥⣀⣿⣿⣿⣿⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⣠⡄⣀⠜⠳⢿⣿⣦⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡘⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠶⠛⠁⠀⠀⡀⢹⣿⣷⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠠⣉⠃⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⠁⠀⢀⣴⠿⢾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣾⣷⣶⣤⣄⣨⣬⠷⡦⢤⣄⣀⣤⡴⠶⠶⠶⠶⠿⠛⠛⠛⣯⣤⣠⣴⣯⣤⣠⣾⣿⣿⡹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣬⣍⣙⣿⡶⢦⣤⡈⠉⠁⠀⠀⠒⠃⣠⣠⣴⣾⡿⠛⠏⢉⣶⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣷⣶⣦⣾⣿⣿⠲⢾⣿⣿⣯⡭⢉⣤⣾⣿⣤⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉



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

⠀⠀⠀⢀⠀⠐⠈⠉⠁⠀⠀⠁⢐⠠⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⡶⠶⠀⠀⠀⠀⠀⠃⠀⠔
⠀⠀⠀⠀⠦⠀⠀⠀⢙⣩⠴⣺⣿⠁⣷⠀⠀⠀⡨⡀
⠀⠀⠀⠀⢴⡶⢻⡏⠀⠙⠃⣿⡟⢸⡿⣭⠁⢄⠈⠎⢂⠀⠀⠀⢀⣬⣭⡗⢀⡶⣯⣰
⠀⠀⠀⢀⠠⠦⣞⣀⠀⠀⢤⢻⡇⣀⠋⠀⠈⠐⠒⠀⠁⠀⠀⠀⠸⣓⣸⡇⠸⢤⢨⡛
⠀⠀⠸⣦⠈⣃⣿⣨⠧⡀⠈⢿⡿⠋⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⢺⡽⠩⢧⣴⢿⡏⠙⡆
⠀⠰⣀⡏⠀⠘⡟⠙⢶⣗⣶⣛⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣿⢾⠅⣀⢴⡏⢚⣹⢀⡇
⠀⠑⢅⠳⣀⠀⠀⠘⠀⢀⣁⣸⣄⣀⣤⣤⣤⣤⣤⣴⡿⡝⣿⣿⠄⢦⣯⢿⣀⣪⠼⠛⠁
⠀⠀⠀⠹⠛⠃⠀⠆⠀⠉⠛⢿⡟⠿⠉⢹⡿⢿⣿⣿⡧⢷⣿⠓⠲⡞⢃⢼⠈⣡⡴⣶⣤⣤⣄⣀⣀
⢀⡠⠶⢶⢤⠰⢦⡶⣶⣶⣦⣾⣄⣀⡾⣿⡇⢸⣿⡟⠀⣼⡉⢸⣠⣇⣨⢸⣿⡇⡿⢡⣿⣿⠀⣿⣠⡄
⠈⠀⠀⢺⢸⣸⢸⣗⣰⠶⡟⣻⡋⣩⣇⣿⡧⢼⠟⠉⡛⣿⡇⠘⢄⠥⢤⣾⠧⠵⠦⠼⣿⣙⣫⣿⡏⢿⡀
⠀⠀⠀⢸⠂⣿⣿⣿⠀⠀⠀⣿⡄⠈⠁⠀⠀⠈⠦⣧⣧⠗⠧⢤⡨⡆⢹⣿⣿⢹⠒⠚⠒⠞⠓⠛⠒⠒⠃
⠀⠀⠀⠈⡄⡅⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡀⢣⡇⠀⣽⡼⣾
⠀⠀⠀⠀⣿⡇⢘⡁⠀⠀⠀⠤⠀⠐⠂⠀⠀⠁⢀⣐⣿⡅⠀⣷⠀⠇⠀⣇⡏⣿
⠀⠀⠀⠤⠟⠃⠘⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠉⠀⠛⠒⠛⠻⣇⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠢⢄⠀⠀⠚⠁⠙⠋⠻⣌⠓⠫⠹⠦⠑⡂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠙⡆⣀⠀⠀⣀⣤⣔⠉⠤⠠⡤⠀⢠⠤⢍⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠛⠉⠉⠉⠁⠈⠁⠉⠀⠉⠉⠉⠙⠒⠦⢤⣐⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠮⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠆⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣦⣤⣄⣀⣠⣠⣄⣠⣀⡘⠂⠀⠀⠀⠀⢀⠞⠁⣀⡀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⢳⠀⠀⠀⠀⠀⡄⠾⣧⠹⠷⠆⣶⠋
⠀⠀⠀⠀⠀⠁⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢝⠱⠾⠛⠀⠁
⠀⠀⠀⠀⠥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠋⣿⢤⡝⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠈⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠠⠚⡽⠀⠀⠀⠀⣴⡏⣹⠿⠀⣶⡞⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠩⠖⠀⠀⠁⣠⠞⠀⠀⠀⠀⢶⣶⠦⠿⠉⢀⣭⠀⠉⠋⠱⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠚⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠘⢦⣄⣀⣀⠄⠊⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⢆⡀⣀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠿⢿⣤⣀⣀
⠀⠀⠀⠉⠉⠀⠘⠿⣦⣀⣀⣀⣀⣤⣴⠴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⣤⣄⣄⣄⣀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠻⣦⡀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠻⣦⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢶⣄⣾⠁⠀⠀⢰⢷⡀⠀⠀⠀⠈⣿⠟⠀⠀⠀⢀⡴⣆⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⣾⠀⠙⢦⡀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠹⡄⠀⠀⠀⠀⠀⠘⣷⡀
⣀⡀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠙⠶⠶⠞⠛⠋⠁⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠘⣧
⠹⣿⡷⣤⣄⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠈⢷⡀⠉⠛⠻⢶⣼⣆⠀⠀⠀⠀⢷⣤⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⡾⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠈⣧⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⣾⠀⠙⢶⡾⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡟⢦⣄⡀⢸⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣇⠀⠈⠙⠛⠲⣿⡀⠀⠀⠀⠀⠀⠀⣴⣯⠶⢷⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡶⣤⣀⠀⠸⡇⠀⠀⠀⠀⠀⣠⣽⣦⣀⠀⠉⠙⢻⡟⠛⠶⣦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢱⠀⠈⠉⠛⠃⠀⠀⠀⣠⠞⠁⠀⠀⠈⠙⠛⢛⣋⣀⡀⠀⠀⠹⣦⠀⠀⠀⠀⡏⠀⠀⠀⢀⣾⠃⠀⢀⣠⣤
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⡶⠛⢦⣄⡾⠋⠉⠉⠙⢦⡀⠀⠘⣇⠀⠀⢠⡇⠀⠀⠀⠈⠛⠛⠛⠋⠉⢸
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢸⠀⠀⢿⡀⠺⠂⢹⡄⠀⠺⠃⠀⠈⣷⠀⠀⢹⡄⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⡄⠀⠈⠳⢦⠴⠟⢧⡀⠀⠀⢀⣴⠃⠀⠀⠘⡇⠀⠀⣠⡿⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠻⣄⠀⠀⠀⠀⢀⠈⠙⠛⠛⠉⢀⣀⠀⠀⠀⣷⠴⠛⠉⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣝⣷⣄⡀⠀⠙⠛⣶⠶⠶⡞⠋⠁⠀⠀⢰⠇⠀⠀⠀⢀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢸⣆⣸⡇⠀⠀⠀⠀⡾⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣤⣀⣀⣀⠀⠀⢀⣘⣧⠀⠀⣯⠸⡇⠀⠀⢀⣾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠉⢹⡇⠀⠸⡆⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢤⣴⠆⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⢀⣀⣙⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⣾⡋⠙⠃⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⡄⠀⣩⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠃⠴⠟⠁⠀⢻⠀⠙⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⢁⡄⠀⢀⡟⣤⠄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⢠⠨⠝⠻⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢀⣀⣡⣤⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣶⣾⣧⣤⣌⣀⡀⠀⠀⡙⢿⣦⡀
⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣿⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠙⠛⠻⠿⣿⣾⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠈⠻⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡠⠍⣹⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣮⣗⣤⣄⣀⣀⣤⣤⣤⣤⣤⠤⡤⡤⣤⣤⣤⣤⣀⣀⣀⣤⣑⣦⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡑⡶⣿⣿⢶⠀⠁⡶⣿⣿⢶⢈⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⠳⣿⣿⠟⣷⣾⠳⣿⣿⠞⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣅⣢⣾⣿⣿⣷⣔⣨⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡟⣿⣿⠿⠿⣿⣿⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠁⠀⠀⠀⠀⠀⢀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⡀⠀⠀⠐⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠄⠢⡀⠀⠈⠦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⢡⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢂⠹⣆⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠚⡀⠹⣦⣤⡶⠾⠛⠛⠋⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣴⢿⡄⠀⠀⡀⠀⢡⠀⠙⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⢀⣴⠾⠛⠋⠉⠉⠀⠀⠈⠉⠉⠉⠀⠈⣷⠀⠀⢁⠀⠀⠑⢌⢘⡳⢶⣤⡄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢀⡿⠀⠀⠘⠀⠠⡀⠀⠉⠑⢦⣀⡷⠔⢉⠘⠂⠀⠀⠀⣠⣤⠶⠾⠿⠶⣶⣤⣀
⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢾⣇⣀⡀⠀⠆⠀⠈⠢⢀⣠⡼⠿⠄⢀⣘⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⣤⣤⣭⣽⣷⣶⣼⣦⣤⡴⠟⢁⣠⠖⠚⠉⠛⠛⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠛⠉⠁⠀⠀⠀⠈⠉⠙⠻⣦⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⢻⣆
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣀⣀⣠⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣤⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢻⢱⠀⢸⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡨⠀⢀⡔⠋⡡⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⡇⠀⢀⡞⣸⠀⢸⡇⠀⠈⣧⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡇⠀⢰⡄⠀⠀⠀⠀⢀⡏⠀⠙⠒⠋⠀⡇⠀⢸⠛⣆⣠⢿⠀⠨⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠈⢷⡀⢸⠀⠀⠀⠈⠓⠋⣷⠀⠀⠀⠀⠸⡇⢠⢄⠀⠀⠀⠙⠒⠋⠀⠀⠁⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢸⠷⠾⠁⠀⠀⣄⠀⠀⡏⢠⡀⠀⠀⢀⣳⠀⠀⢵⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡐⠀⠈⠳⠚⣇⠀⠀⠈⢿⡇⠀⠀⡛⣚⠛⠙⢦⠀⠀⠀⠑⠖⡿⣠⡞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠸⢄⣀⣠⠞⠁⠀⣀⣠⣤⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⠀⠸⡁⠁⠀⠀⠈⣇⠀⠀⠀⡼⢡⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡞⠅⠀⠉⢻⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠇⠀⠀⠱⣄⠀⠀⠀⠹⡄⠀⠀⠳⠞⠙⠲⠚⡽
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⢣⡆⠀⠀⢰⠀⠀⠀⠀⠈⠣⡀⠀⠀⠸⡄⠀⠀⠀⢠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡠⠋⠀⠀⠀⣠⣯⢾⠇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠙⢦⡀⠀⢹⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⢀⣠⠞⠁⠀⣀⠠⠊⠀⠀⠀⠀⠾⢋⡡⠋⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠈⠙⢆⠈⢇⠀⠀⠙⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠿⡉⠉⠭⠤⠔⠒⠉⠀⠀⠀⠀⠀⢀⡴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠑⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠶⠤⣤⠤⠤⠴⠶⠚⠋⠁⠀⠀⣀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢠⣴⠀⠀⠘⣆⠀⠀⢀⢞⡟⢠⡄⠀⠀⠀⠀⠀⠀⣄⠀⢰⡟⠀⠀⠀⣷⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡇⠀⠀⣿⠃⠀⠀⢀⣿⠀⢠⡏⢸⠀⢀⡇⠀⠀⠀⠀⣤⣤⠎⠉⠁⠸⣄⢀⣠⢿⠀⢸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣸⡇⠀⠀⡟⢆⢀⣠⠞⡅⠀⢸⠀⠈⠛⢉⡇⣶⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠉⠁⢢⣀⡼⢼⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⠉⠁⠻⠤⠔⠁⠀⠉⠀⢰⠁⠀⢸⡀⠀⠀⡞⠀⣿⠀⠀⠀⠀⢻⠀⠰⢅⢀⠀⠀⠀⠀⠈⠀⠘⢦⡤⢏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠰⡓⡀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠈⣇⠀⠀⠙⠚⡻⠀⠀⠀⠀⢸⠀⠀⢀⡿⠚⠲⢦⡀⠀⠀⠀⠀⣠⠞⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣷⠞⠶⠶⢤⠀⠀⠀⠘⠦⣀⣠⠏⠀⠀⠀⡼⠃⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠱⡀⠀⠀⡴⠃⢸⣹⠁⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡏⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢴⠀⠀⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⠀⢇⠀⢰⡇⠀⢸⠙⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠹⣄⠀⠀⠀⠀⠈⠑⠤⣄⣀⡀⠀⠀⣀⡙⠶⢾⠁⠀⠀⠀⠑⡀⠀⠀⠀⠳⡄⠀⠀⠸⡄⠀⠓⠶⠊⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⠀⠀⠀⠀⠈⠉⠉⢉⣉⠽⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠹⡄⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠙⢦⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢭⣷⣶⡄

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

⠀⢠⠊⡽
⠀⢸⠀⡇
⠀⠘⡆⢧
⠀⠀⠘⢎⠢⠄⢀⣀⣀⠤⠄⣒⣒⣒⡒⠒⠀⠤⢀⣀
⠀⠀⠀⠀⢱⠀⠉⠛⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⣆⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡒⠝⡻⡸
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠋⠉⠁⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⡒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡯⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⣍⣒⡒⠤⠤⢤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⢄⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠤⠤⢎⣼⣗⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣡⣿⠛⠘⣇
⠀⠀⠀⠀⣠⠖⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠴⠄⠀⠀⠀⠈⢉⣡⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⢴⡆⠀⠙⠀⡀⠘
⠀⠀⠀⠀⠛⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣀⣀⣀⣠⣀⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⣇⠀⠀⠀⣧⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠉⠛⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣦⠀⢸⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠘⡀⠀⠀⠀⠀⠠⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠤⢠⠶⠖⠒⠂⡀⠀⠀⠀⠀⠀⠹⣄⠀⣿⠀⠀⢻⡿
⠀⠀⠀⠀⠁⠀⠀⢀⡇⠀⠀⢀⡾⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡟⠀⠈⠁⣾⡇⠀⠀⠐⠚⠁⠀⣀⠀⠀⠀⠀⢻⡐⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡾⠁⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⣻⣤⣶⣿⡿⠏⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠈⠻⣦⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢿⣿⠷⣛⣭⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⣀⡀⠀⠀⠡⠸⠃
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⠀⠀⠀⠘⠛⠛⠋⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣀⠘⠀⠓⢶⣯⣶⡍⣆
⠀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢰⡛⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣴⣾⣅⠀⢠⣄⣀⣼⣟⡷⢹
⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⠘⠗⠄⠀⣠⣟⣿⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⣿⢿⣗⠸⣿⠻⠇⠸⡇⢸
⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⢢⣘⣿⣿⡿⠋⠛⠦⠀⠤⣄⣀⡚⣶⡀⠀⠀⠀⠀⠘⡀⢳⡘⢿⣄⢹⣄⠀⠘⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣿⡟⣿⠁⠀⠀⠀⠀⠀⠈⠛⠿⠉⢳⡄⠀⠀⠀⠀⠙⢄⠙⠷⣿⣼⣏⢤⣴⠇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⣇⠀⠀⠀⠀⠐⠉⢾⡸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠇⠀⠀⠀⠀⠀⠀⠙⠢⠜⢿⣿⣤⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠂⠀⠀⠀⠀⠐⠾⣷⣍⣙⣂⠀⠀⠐⠒⢞⡻⠻⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⡒⠂⠀⢐⡀⠀⡀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠿⠀⠀⠀⠠⣀⡀⠀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⡇⠀⠀⠹⣅⡈⠁⠀⠀⠀⠳⡀⠰⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⠉⠋⠛⠛⣯⡀
⠀⠀⠀⠀⠀⠀⢰⡀⠘⠶⡖⣯⡀⠀⠰⢏⡷⡶⡤⡆⣤⣤⣝⣦⡄⠀⠀⠂⠛⠀⠀⠀⠀⠀⠀⠐⢶⠤⠤⢤⣼⠂
⠀⠀⠀⠀⠀⠀⠈⢇⠀⣙⣽⣳⢙⡀⠔⢺⣽⢃⣻⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠙⠂⠀⠀⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠈⣄⢬⣖⡛⠠⣧⢨⢞⠶⣫⠨⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⡚⠙
⠀⠀⠀⠰⠀⠀⠀⠀⠈⢱⣾⢄⠀⠠⠐⢾⣹⡟⠀⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣞⡾⠖⠃⠀⠾⢼⣃⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠩⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠃⠀⠀⠜⡹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠍⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠃⠀⠀⠀⠀⣀⣈⡔⢒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣍⠉⠀⠙⠀⠀⠀⢶⢡⡖⠼⣿⠬⣋⢷⣓⡠⠀⠀⠀⠀⠀⣕⠂⠀⠀⠠⠒⠚⡥
⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠀⠀⠠⡽⢖⠠⣛⠭⢏⠼⢋⠾⡁⠀⠀⠀⠀⠀⠛⠉⠒⠒⠒⠚⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⣿⣿⢛⢿⢛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣸⠁⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⡽⡿⠀⠀⠀⠀⡆⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢧⠀⠃⠃⠸⡆⠀⣼⠀⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡚⠟⠀⠀⠐⢀⠀⢸⣤⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠜⡄⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠃⠀⠠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡞⠀⠀⠁⠀⠀⢸⠄⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡰⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣿⠆⠘⠀⠀⠀⢸⠀⠀⢠⡄⢃⠀⠀⠀⢠⣶⠀⡔⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⡈⠃⠀⠂⠀⠀⢾⣄⢀⣾⣿⣄⠀⢠⠀⡬⣿⠊⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⢸⡇⢀⡆⠈⣧⣾⠟⣽⢿⠀⡟⢠⡇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⡇⠀⢠⢸⡇⠀⠀⠀⣿⣧⡞⢁⣾⣠⣵⡟⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡆⠀⠀⠀⠻⠀⠀⣿⠈⡕⠀⠀⢀⡿⠛⠓⢛⣿⣥⣄⣠⣬⣁⣞⣦⣀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣷⣿⡇⠀⠀⠀⣆⠀⠀⣿⠐⠀⠀⠀⣼⣿⣿⣿⡙⠿⣿⣆⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢩⣿⣛⣾⠇⢀⣀⣀⣿⡆⢰⡟⠇⠀⡂⢀⣿⠟⣾⣿⣿⣷⣬⣙⠣⠄⠀⠀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠖⢛⠿⢛⣭⡿⠈⠸⣿⣷⣤⣉⣙⠳⡇⠀⠁⣿⡟⣸⣿⣿⢻⡽⣿⠙⡿⣦⡰⣔⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⣡⠶⠋⣹⠁⡡⢊⣴⣿⣇⣽⣿⣿⠁⢀⣾⣿⢸⣹⣿⣿⡆⠃⢹⡄⣩⠴⠛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠋⠁⠀⡠⣮⠞⣠⣾⡿⣻⣿⢫⣿⣿⣤⣿⠃⢠⠿⠛⣿⠟⢹⡄⡄⢻⡨⡆⠡⠀⠙⠆
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣀⢐⡿⢾⣷⣾⡿⠁⣴⢫⠏⣸⣟⣹⡟⠀⠀⣟⣼⢟⣡⣨⣾⠀⡀⠀⢳⡇⢰⡅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡾⠛⣸⠃⠈⠋⣽⠃⢚⢁⠋⢶⡟⣿⡏⠁⡀⣸⢿⡿⣫⠚⣷⠸⠇⠐⠀⠀⢿⢸⡟⢶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡟⠀⠀⡰⡞⢀⢬⠏⠐⠋⠀⣿⡆⢠⣀⢇⡼⠎⠀⢰⡏⢸⠾⣆⠀⠐⡌⣷⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠘⠀⠀⠀⡻⠁⣌⡟⠀⠀⠀⠀⢹⡗⢉⡵⠏⠀⠀⢀⡼⠁⢸⠃⠘⢧⠀⣱⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠞⠁⠀⠈⠀⠀⠈⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⣀⡏⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⠃⠀⠀⢠⡯⠁⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⠮⢄⣀⠀⢸⢤⡦⣄⡠⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠛⠁⠀⠈⠙⠞⣻⠇⠀⠉⢸⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡝⠀⢦⠂⠀⠀⠀⣿⢤⣀⣄⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢠⡖⢞⡀⡀⢀⡼⠛⡠⠔⣩⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡽⣙⠳⢄⡛⢉⣦⠗⢉⣴⣋⠀⠀⠈⠱⣟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠑⠢⢼⣇⡹⢾⣻⣿⢃⠇⠠⣽⠃⡠⢻⠇⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣳⢤⡄⠀⠀⠀⠈⠙⠦⣌⠓⠦⢤⡞⢁⡞⢠⠏⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⢤⣤⡀⠀⠀⡀⠀⢼⡽⢆⢦⡠⢦⣀⣀⢄⠀⡀⣉⠲⠦⠌⢻⣉⠏⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⣀⠀⠀⠀⢀⣬⣽⠿⠃⠈⠙⢺⣉⣉⣉⣉⣉⠷⠶⣏⡭⠤⠔⠒⠺⣏⠉⠙⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢰⣿⠃⠈⠈⠙⠰⣀⣾⠁⡘⠃⢠⠀⠀⣼⣈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⡤⡴⠾⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⣿⣾⠂⠀⠀⠀⠀⢀⡼⠃⢈⢀⢀⠈⠁⣀⡭⠟⠛⠲⢄⡉⠲⢞⡚⠓⠊⠙⠁⠀⢳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⢖⡛⢦⢀⠀⠀⠀⠘⢧⠀⠑⠉⠺⠆⠸⣭⠀⠀⠀⠀⠀⠉⠓⠤⣈⣀⣤⠤⠖⢻⠮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣾⣽⣌⠉⠓⠚⠲⠐⠻⣄⡀⡀⡀⡀⣨⢯⣤⡄⣤⠀⣠⠀⢠⣀⣠⣭⡽⠛⠋⡉⣭⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⣷⣾⡀⠀⠀⠀⠀⠀⠈⠓⠋⠓⠙⠁⠀⠀⠉⠈⠑⠉⠉⠉⠉⠀⠈⠀⣄⠲⠛⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠟⢹⡷⠷⣄⣄⢠⢄⣀⣠⣄⡶⢤⢤⣠⡴⠤⡴⢀⠀⠀⠀⠀⢀⡀⣬⡧⢀⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢿⣎⡀⠉⠀⠁⠈⣉⡟⠋⠀⡺⠀⠀⠀⠀⣽⠷⠲⠖⠚⠚⠊⠉⣁⡯⢉⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠛⠉⢳⣦⣿⡥⣦⠖⣏⠀⠀⠀⠀⠱⠒⠀⠀⠀⠀⠀⠀⢸⠛⠾⣠⣞⠃
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⡓⢦⠴⠤⠈⡴⠦⠶⠶⠋⠀⠀⠀⠀⢸⡥⠧⠴⠦⠤⡄⠶⠾⠓⠶⡽⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠔⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠴⠲⣠⣄⡤⢰⡾⣤⡤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠦⣆⣄⣄⣠⣪⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠀⣿⣿⣿⡀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⣿⣾⡇⠀⣿⣇⢻⣷⡄⠈⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣋⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣼⣿⣿⣷⢰⣿⣿⡀⢹⣿⣆⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣻⣿⣿⣿⢸⡟⣿⣇⠀⢻⣿⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣷⣸⡿⢻⣿⣿⠇⢻⣿⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠳⣦⣀⣀⣀⣠⡄⠀⠀⠸⣇⢹⣿⡇⠈⢿⣿⠀⢸⣿⢀⡟⢹⣿⣿⠏⠀⠀⢠⣄⣀⣀⣀⣤⠾⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣤⡼⢉⣉⣿⣷⣦⣄⠀⠹⡄⢿⡇⢺⡌⢿⡄⢸⡇⠘⠁⣾⠏⠀⠀⢀⣤⣾⣿⣍⡉⠭⣤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣴⣶⡿⠛⠋⠀⠙⢿⣦⠀⠀⠘⣿⠈⣷⠘⣷⠀⠀⠀⣼⡏⠀⠀⣴⣿⠋⠀⠘⠛⢿⣶⣦⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⢋⣤⠆⠀⠘⢷⡄⠈⢿⣷⠀⠀⢻⣇⢿⡇⣿⡄⠀⣰⡿⠀⠀⣼⡿⠁⢀⡾⠃⠀⠰⣤⡙⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠇⠀⠀⢀⡘⣿⡄⠈⣿⡇⠀⠈⢿⣾⣷⣿⣇⣴⡿⠁⠀⢰⣿⠇⢀⣾⠃⡀⠀⠀⠸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣰⢠⡄⠸⣧⠹⣷⠀⢻⣷⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⣼⡿⠀⣸⡏⣸⡇⢠⡄⣦⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠻⡾⣷⡀⠻⣧⡉⢳⡄⢻⣆⠀⠀⠀⠉⠉⠉⠀⠀⠀⢠⡿⢡⡾⠋⣴⡟⠀⣼⠷⡟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠘⣷⡀⠙⣷⡈⢿⡈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⣾⢁⣾⠋⢀⣾⠏⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡘⢿⣦⠈⢿⣮⣷⡈⢷⡀⠀⠀⠀⠀⠀⢀⡾⢃⣾⣷⡿⠃⣰⣿⠋⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠻⣷⡌⠻⣿⣷⣌⢻⣄⠀⠀⠀⣠⡞⢡⣾⣿⠟⢁⣼⠟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⢻⣦⡈⠻⣿⣦⡙⢿⣶⡾⢋⣴⣿⡿⢋⣴⡿⣡⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣽⣿⣦⣌⣿⡿⠟⣋⣴⡿⠟⢁⣴⣿⣯⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⠿⠟⣋⣭⡴⠞⢛⣩⣴⣾⣿⣿⠿⢿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢛⣋⡥⠶⠞⣛⣩⣥⣶⣿⣿⣿⣿⣿⣏⣛⡳⠶⢬⣉⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣩⠖⣛⣭⠵⠚⣛⣛⣩⣭⠿⠿⠛⠛⠛⠿⠿⣭⣍⣛⣛⡓⠦⣭⣛⠲⢬⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢐⣩⣴⣿⣭⠶⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣬⣝⣶⣭⡂⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣾⣿⡇⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢰⣿⣿⠀⢠⣾⡇⠀⣄⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⢸⣿⣿⠀⢿⣿⡇⠀⣿⡆⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⣿⣿⣿⡇⢸⣿⣷⢀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢿⣿⣿⡇⢸⣿⣿⣾⣿⣿⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢸⣿⠀⠀⢸⣿⣿⣇⣾⣿⡿⣿⣿⡟⢠⣿⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⡆⠀⣾⣿⣿⣿⣿⠟⢰⣿⣿⠁⣼⣿⣿⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣿⡇⢀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣾⣿⣿⡏⠀⢠⣿⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⣿⣷⡀⠀⣿⣧⣼⣿⣿⣿⠿⣿⡆⠈⣿⣿⣿⠹⣿⠁⢀⣾⣿⠇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢹⣿⣷⣤⣿⣧⠹⣿⠏⢹⡇⠈⡇⠀⢹⣿⡟⠀⣽⣶⣾⡿⠋⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⢻⣿⣿⣿⣿⠀⣿⠀⠘⠁⢸⡇⠀⢸⣿⠃⣸⣿⣿⠋⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣼⣿⡿⣿⣿⠀⢹⡄⢠⣄⠈⠁⠀⢸⣿⠀⣿⣿⠻⣷⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⡇⠸⣿⣄⠀⠉⢀⣿⠇⠀⠀⠘⡿⢰⣿⡟⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠘⢿⣄⠸⣿⡀⠀⣾⣯⢰⣶⢀⣄⠀⣸⠟⠀⣼⠏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⣙⠛⠹⡇⠀⣿⣿⣸⣿⣿⡿⢡⣿⠀⢸⠋⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢿⣆⢀⣿⣴⣿⣿⣿⣿⣿⣷⣿⣟⣠⡾⠃⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠏⠀⠀⠀⠉⠙⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⢰⢶⣿⣿⣶⣼⣿⣿⣿⣿⣤⣶⣶⣤⡀⠀⢹⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⠂
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣫⣭⣭⣛⠻⣿⣿⡻⣿⣿⣷⡀⠘⠿⣿⣿⣿⣿⣿⣶⣦⣴⣶⣾⣿⣿⠏
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣧⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⡻⣿⣿⣿⣿⣟⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠹⣿⡇
⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⢀⣿⡇
⠀⠀⠀⣴⣿⡟⠉⠁⠀⠀⠀⠙⠻⣿⣆
⠀⠀⠀⣿⣿⣆⠀⠀⠀⣠⣄⣀⣴⣿⣿
⠀⠀⠀⠹⢿⣿⣷⣶⣾⠿⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡷⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⡀⡀⡀⣷⣻⠮⡩⢨⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠘⠛⢼⡤⢢⣯⣸⣏⡾⣼⢳⣿⣗⢹⡖⢷⣌⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠶⠀⠀⠀⢲⡆⠐⠀⠄⠀⠀⠹⣄⠀⢀⠀⠀⠀⠀⠀⠀⠉⠈⠘⠣⠊⣛⣻⠻⡜⢤⢣⣛⠤⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢘⣻⣓⡥⠀⠀⢀⡷⢶⣀⡀⣀⠀⠀⠈⢦⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠨⠝⠈⠀⠀⠀⠀⠀⠘⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠠⣭⢧⣫⢦⠀⢈⣉⠮⡽⢙⡽⡐⠙⠛⠛⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⠔⢳⢼⠰⡆⢈⣈⡖⣡⠦⢝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠰⢄⠀⠀⠀⠀⢄⠀⠀⠀⢀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠺⢯⠉⠀⠾⠰⠳⣟⣧⡇⠺⣏⠀⠀⠀⠀⠀⡀⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢧⠀⠀⠀⠈⠟⠗⠲⠿⢿⠃
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣼⡝⣀⠈⡑⠀⢻⡬⣟⠁⢘⣯⠄⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⠄⠀⠀⢮⣀⡀⠀⠀⢘⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣝⠼⠂⠁⠀⠹⢲⣏⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⠀⠊⠀⠀⠀⠀⢿⡁⠉⠉⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠄⠃⠀⠀⠘⣉⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⡧⠀⠀⠀⢈⣁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡏⠀⠀⠀⠀⡀⢿⡆⠀⠀⠉⠁⠀⠀⠀⣀⣧⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡗⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⢺⡖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢯⡽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡞⢶⣣⠰⣦⡖⣠⡀⠀⠀⠀⠀⠹⣞⣓⡁
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣦⣤⣤⣤⣤⡤⠶⠆⠀⠀⠀⠀⢉⡅⠾⢭⣛⣯⣙⠣⣞⡤⣥⠂⠀⠀⠀⢐⣣⠿⠁
⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠦⠛⠁⠢⠼⠢⠇⠔⢣⠔⠢⠄⠀⠀⠀⠾⡝⠁⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⣀⣠⠆
⠀⠀⠀⠀⠀⠺⠟⢻⣿⣿⣿⣿⣿⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠀⠀⣠⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⠈⠈⢉⣀⣤⠆
⠀⠀⠀⠀⠀⠄⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠈⠉⠉⠉⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠖⠉⡈⠳⡄⠀⠐⡞⠉⠑⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢻⠀⠀⣯⠁⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⢳
⠀⠀⠀⠀⣼⠀⠂⢀⠀⢠⠸⡦⠒⠁⠂⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣀⢀⠀⡀⢀⢀⡀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡊⠅⠱⢎⣆⣧⣀⣀
⠀⠀⠀⠀⢸⡀⠂⠀⠀⠀⠀⣧⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠅⢠⡄⠀⣤⡂⠐⣌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡅⠀⠀⠌⠳⡞⠋⠀⠀⠀⠈⡹⠁
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⠤⠤⠤⠤⠤⢤⡄⠀⠀⢸⠀⠀⠀⠀⠸⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⣨⣦⣠⢤⠤⠴⢥⣤⡄⠀⠀⠀⠀⠀⠀⠘⣏⠍⠛⣀⣐⢂⡀⠐⠒⣀⠐⠒⠀⠀⠲⠀⠀⠐⢄⣀⡻⠀⠀⠀⠈⡆⠀⠀⠀⠀⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⣄⣉⣈⣁⣉⣉⣁⣉⣉⣉⣈⣁⣀⣈⠴⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡤⠤⣒⣒⡋⠒⠒⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⠀⠀⢀⠇
⠀⠀⠀⠀⠀⣧⡾⣹⣪⠭⠿⡉⢳⠦⣈⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡂⠀⠀⠸⠀⣠⠋
⠀⠀⠀⠀⠀⡜⠉⠉⠉⠲⣄⡸⢨⢖⠫⣿⣄⡏⠀⠀⠀⠀⠀⣠⠔⢊⡭⢤⡖⠉⡹⠥⠽⢏⠙⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡵⠊
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⡷⠚⠋⠉⠙⡏⠀⠀⠀⢀⡴⠋⡰⠉⣍⣸⠥⠞⠛⠛⠉⠉⠛⠓⠿⢤⡎⡓⢤⡀
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢰⠃⠀⠀⡴⢻⠊⣩⣝⠵⠋⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠈⠳⣦⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦
⠀⠀⠀⣸⠁⠀⠀⠰⠀⠀⠀⡇⠀⠐⠀⡜⠀⠀⢰⢡⢞⣉⡹⠋⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⡄⠀⠀⠀⠀⠀⢀⣀⠠⠃⠼⣀
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⠀⡇⢸⡇⣰⠁⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⢻⠁⠀⠀⣠⠜⠛⠓⡰⠀⡀⢀⡈⢑⡢⣤⡄
⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡏⠀⠀⠀⢇⠸⣕⡇⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠒⢀⣀⡨⠀⠀⢰⠁⡆⠀⠈⠉⠉⠹⠎⠉⠊⠷⠈⠳⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠏⡄⠀⡜⠀⠀⠀⠀⠈⠓⠬⢧⣤⣄⣀⣀⣀⣄⣄⣠⣠⠤⠴⠤⠴⠒⠒⠊⠉⠉⠁⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢻⠀⠀⠀⠀⠀⢀⠞⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠃
⠀⠀⠈⢦⡀⠀⠀⡠⠋⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠆⢀⠀⢀⡀⡀⣠⣬⠞⠁
⠀⠀⠀⠀⠉⠙⠛⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⠁⠈⢣⠀⠀⠀⠀⡠⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠀⠂⢸⠀⠀⠀⠀⡇⡀⠀⠈⠙⠢⣄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠈⠙⢦⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠘⡇⠀⠀⢰⡃⠀⠀⠀⠄⠀⣈⠇⠀⠀⠈⣏⢧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠴⠃⠀⠀⠈⡅⠀⢀⣀⠴⠊⠁⠀⠀⠀⢠⠟⢀⠀⡤⢽
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠤⢦⣀⣀⡙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠸⡟⠊⠙⠁⠈⡆
⠀⠀⠀⠀⢠⠏⠀⠀⡤⠶⢤⡀⠀⠀⡇⠀⠀⠀⠀⣾⢭⣝⣛⣒⡒⠐⠶⠶⠶⢒⣛⣒⣬⡽⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠸⣆
⠀⠀⠀⠀⡟⠀⠀⣾⠀⠀⠀⢹⠀⠀⢱⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢀⡇⠀⣀⠔⠋⠱⡄⠀⠀⠀⠳⣄⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀
⠀⠀⠀⠀⢳⡀⠀⠘⠦⠤⠴⠃⠀⢀⡆⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡔⠁⠂⠀⠄⢿⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⡀⢀⢈⣉⠷
⠀⠀⠀⠀⠀⠳⢆⡀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢫⡎⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠒⠢⠤⠄⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢰⢀⠂⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⡀⣠⣿⡄⠀⠀⠙⢶⣶⠏⠁⠛⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠁⠈⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣧⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣺⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⣀⠀⠀⠀⠀⠀⣾⣿⠀⢀⡀⠀⠘⠿⣿⣿⣷⣺⣆⢠⣀⠀⠀⠀⠀⢀⣠⣄⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⠀⠀⠀⠀⣴⣿⠃⢀⣼⣿⡄⠀⠀⠈⠛⠛⠛⠛⠻⠿⠿⠶⠶⠶⠛⠛⠛⠋⠁
⠀⠀⠀⠀⡀⠀⠀⢠⣿⠇⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⢀⣹⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢴⣖⡀⡄
⠀⠀⠀⠀⡗⠀⠀⠼⡿⠀⠀⠀⠘⢶⣤⣄⣀⣀⣀⣠⣶⣶⡴⠿⠋⠙⠻⠿⣿⣿⡿⠛⢸⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⣄⠀⠛⣄⠀⠈⠀⠘⢋⡄
⠀⠀⠀⠀⠰⠀⠀⢨⡶⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⣩⣙⣛⠀⠀⠀⠀⠀⠈⠙⠋⠛⠓⠀⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⣀⡠⡤⠄⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⣼⣽⣿⣿⡇⠀⠀⠀⠀⠠⠤⢤⡀
⠀⠀⠀⠀⠀⠈⠄⠘⣿⡀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠈⣷⠦⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⡀⠀⠃⠀⠀⠀⠀⠁⠀⠈⠙⢒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⠀⠄⢠⠀⠈⠇⠀⠀⠀⠀⢄⠀⠀⠈⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠞⠁⠀⠀⠀⢀⣼⠿⠟⠉⠀⠀⠀⠀⠀⢣⣀⠐⢶⢀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣠⠀⠀⠤⠐⠈⠀⠀⠀⠀⢀⣰⣧⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢻⡆⠀⠀⠀⠀⠀⣀⣲⣾⣾⣿⠃⠀⠀⠀⠈⠀⠀⠀⢀⣀⡀⠠⣐⢉⢾⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠻⠇⣠⣤⣄⣴⣿⣿⡿⠟⠋⠀⠀⠀⠀⣦⠀⠀⠀⠀⠘⠚⠚⠏⠾⠦⠜⠺⡓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣯⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠹⣿⣿⡿⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣿⣿⡇⣀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⢘⣽⣷⢮⠂⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣷⣤⣴⣿⣿⣿⣌⡐⠀⠀⠠⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣾⣥⠷⠷⠦⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠛⠛⠉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢤⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢨⡱⠔⢒⠈⠡⢡⠀⠄⠐⠀⠀⠀⢀⢀⢀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡘⣙⠇⣼⢅⣶⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⢷⢳⢻⠱⡀⠥⠈⡆⣢⣦⣱⣼⣧⣽⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⢾⣞⠯⣻⣺⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣆⠴⢜⡺⣌⡒⢌⣲⣼⣼⣿⣿⣿⣿⠿⢛⣋⣉⠁⠀⠀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⢪⡀⠟⣻⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⡘⢮⣓⠒⣇⣓⣷⣿⣿⢿⣿⣿⣿⡟⠁⠀⠉⣉⣭⣛⠀⠈⣿⣷⠶⡤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣗⢤⡡⣑⡖⣆⣷⣿⣿⣿⣿⣇⠦⡙⢿⣿⣇⠀⠀⠛⠛⠀⠉⠀⠀⠁⡈⠀⠀⠙⣿⣿⡿⡛⠟⠋⠀⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣔⡢⣍⣆⢰⣾⣿⣿⣿⣿⣿⣿⣿⡴⣇⠀⡻⠋⠀⠀⣠⣤⣄⣀⠀⣀⠀⢀⣰⣶⣴⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣦⣧⣿⣽⣿⣿⣿⣿⣿⣿⣷⣈⣄⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⠿⡿⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⠛⠛⢻⣽⡄⠀⠀⠘⠿⠋⠿⣶⣤⣶⡶⣾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⡃⠄⠀⠀⠀⠀⢹⣄⡀⣀⠌⢰⠶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⢿⣈⠉⠻⣾⣦⡀⠀⠀⠈⠳⠦⠄⠀⠈⠙⠒⠂⠤⠉⠉⠛⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⠿⠛⡁⢻⣗⡶⢌⣝⣿⣶⣦⣤⣤⣠⣤⣞⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣶⣴⠷⠂⢶⡶⠶⠤⢬⣙⢣⣌⣟⣠⣿⣯⡓⠛⣿⢿⣿⣿⠿⠋⣿⣾⣿⣿⣭⣭⣉⣒⡒
⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⣶⠾⠟⠛⠛⠛⠛⠛⠛⠶⢿⣧⡹⣿⣿⣿⣿⣿⣾⣝⠛⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⠖⠋⢀⠠⣤⡤⢦⣤⣤⣤⠀⠀⠀⠈⠙⣿⣿⣿⣿⠟⠣⠈⠳⢦⡀⠀⢰⡟⠲⠶⠶⠦⠭⡭⠁⠀⠉⠁
⠀⠀⠀⢀⠀⣿⠟⠁⣢⣾⣯⣿⠃⢀⡾⠋⢀⣭⣷⣤⠄⠀⠀⠀⢻⠟⠉⠀⠀⠀⠀⠀⠹⢆⣾⠁
⠀⠀⠀⠀⢰⡟⢀⠏⣿⣾⠿⠁⠀⠾⠁⠀⡞⠉⣹⣯⣥⣀⣀⢀⡜⠀⠀⠀⠀⠀⠀⣀⣠⣤⣽⣴⡶⣶⣶⠶⡶⣶⣶⡦⢤⠄
⠀⠀⠀⠀⠈⢀⣄⡴⠟⠁⠀⣰⠖⢀⣠⠜⠀⣼⣿⠟⠉⠀⠈⠁⠀⠀⠀⣀⣤⡶⠿⣛⣩⠥⢴⣶⣶⣿⣿⣿⡿⠯⠁
⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠈⠁⠀⠞⠁⠐⠞⣃⠛⠀⠀⠀⢀⣀⡄⠐⠛⠋⠁⠒⣩⠵⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞⠃⠀⠀⠀⠤⠂⠁⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣶⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠸⣿⡀⢁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣴
⠀⠀⠀⠀⠀⣶⠀⠁⠳⠞⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣏⢀⣯⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢡⡸⠶⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⢿⢇⡖
⠀⠀⠀⠰⣤⠘⣔⠛⢰⡆⠛⢋⠀⣆⠀⠀⠀⠀⠀⠀⣆⢹⠘⢿⣾⢻⣾⡇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣼⢧⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢉⠈⢸⡇
⠀⠀⠀⠀⢸⣷⡾⢠⡜⠳⡾⠙⢶⡏⠀⠀⠀⠀⠀⠀⢿⢿⠄⢲⣁⠈⢡⠇⠀⠀⠀⠀⠀⠀⠀⠈⠣⣐⢾⡿⣄⣠⠆⠀⠀⠀⠀⠀⡄⠷⣧⠘⢦⡼⢻⣿
⠀⠀⠀⠀⠈⠳⢄⢶⡿⠀⠻⠏⠊⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⡟⠻⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠐⢮⡹⠀⠀⠀⠀⠀⠀⢿⠿⠟⠀⢷⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠀⠲⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⣠⢾⣿⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢸⡇⢻
⠀⠀⠀⠀⠀⠀⢱⠏⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⡄⣤⠀⢻⠁⣷⣤⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⡿⣿⠸⣇⠀⡀⠀⠀⠀⠀⠀⠰⡇⣿⣶⠏⣥⡇⢀
⠀⠀⠀⠀⠀⠀⢸⡀⡈⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣷⠾⣦⠘⠛⠹⢉⠀⠀⠀⠀⠀⠀⠀⢱⡄⢿⠀⢹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠐⠀⢣⠁⡀⣹⠀⢾
⠀⠀⠀⠀⠀⣧⠈⢷⢧⣤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠾⠂⣸⣶⡄⢻⢿⠀⠀⠀⠀⠀⠀⠀⢺⣿⠻⡇⣬⠃⠿⡿⠀⠀⠀⠀⠀⠀⣀⢻⣬⡇⣷⢹⣤⡆
⠀⠀⠀⠀⠀⠙⠟⠛⢨⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⣤⣿⡎⢧⡆⡜⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣄⣿⡤⠠⠃⠀⠀⠀⠀⠀⠀⠙⠾⡧⠁⠿⣄⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⡇⠹⠓⠁

⠀⠀⠀⠁⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⢸⣿⡿⠀⠀⢰⣾⣿⣿⠀⠂
⠀⠀⠀⠀⡐⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣿⣿⣿⣿⠟⡐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⣆⢸⣿⠀⠀⠀⠀⠙⢿⡟
⠀⢀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢀⠁⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⢿⡀⢻⣆⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⣿⣿⠃⠀⢁⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠿⢷⣄
⠀⠀⠀⠀⠀⢧⠀⠀⠈⠒⢀⠀⠀⡁⠄⠀⠀⠀⠀⠸⡏⠀⠈⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢈⣿⠆⠄⠀⣹⣿⠖
⠀⠀⠀⠀⠀⡌⠀⣀⣀⠀⠘⠀⠀⠆⢠⠀⠀⠤⠦⠠⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠐⢶⡀⠀⠸⠿⠀⠀⢠⣿⡟
⠀⠀⠀⠀⢀⠃⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠁⠀⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠇
⠀⠀⠀⠀⠘⠀⣿⣿⣿⡄⠀⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠘⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠠⠃⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣄⢀⠀⠀⠀⠀⠘⢋⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⢿⣄⣤⣿⡇
⠀⠀⠀⠀⡆⠀⠀⢀⠼⠷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠙⢧
⠁⢀⠀⠀⡁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠨⠀⠀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄
⠀⠘⠙⢀⡁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠱
⠐⠈⠀⠁⠉⠐⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⡁
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀
⠀⠆⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣆⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⡀⠉⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠛⢿⣿⣿⡿⠁⠀⠀⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⠀⣇⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠈⠀⠀⠀⠀⣿⡇⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⡀⠀⠀⠀⢀⣟⡇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⠀⠀⣹⠦⣄⠀⡸⢸⠁⠀⠀⠀⠈⢣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡏⠀⣀⡞⡵⠚⢫⠳⢄⡘⠀⠀⢀⡀⠤⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⢡⠞⠀⣠⠏⠀⠀⠃⠈⠁⠀⣀⠤⠒⣒⠶⠛⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡞⡏⠀⡴⠁⠀⠰⠤⠴⠒⠊⢉⡤⠖⠋⠁⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⡼⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢸⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣸⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡜⠁⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢳⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⣼⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣠⠴⠃⠀⠀⠀⠀⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠀⠀⠀⣸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠿⠀⠸⡇⠀⠀⠀⢀⡴⠚⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣀⡠⠤⡶⠂⢻⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠶⠋⠁⠀⢀⣠⠤⠤⠟⠀⠀⢠⠏⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⢸⠁⠀⢰⡇⠀⠀⢹⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⣀⣤⠤⠤⠤⠖⠋⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠈⣇⠀⠀⡇⠀⢀⡜⠁⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠉⠀⠀⣀⡠⠞⠁⠀⠀⠀⠀⠀⣀⣀⣤⠴⣋⣠⠤⠶⠒⠚⠋⠁
⠀⠀⠀⠀⠀⢰⠁⠀⠀⣸⠀⠀⢇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠉⠉⠁⠀⠀⣀⣀⣀⣠⠴⠋⠁⣠⠴⠋⠁
⠀⠀⠀⠀⠀⣸⠀⣀⣺⣠⣤⠖⠉⠀⠀⣠⠴⠶⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠀⠀⢀⡞⠁
⠀⠀⢀⡖⠋⠙⣆⠹⡄⠀⠘⢆⡀⢠⠞⠁⠀⣠⠖⠊⠁⠈⠉⠙⠛⠛⠛⠋⠉⠀⠀⢀⡠⠔⠂⠀⠀⣰⠋
⠀⠀⠈⢧⠀⠀⠘⢆⠘⠦⣀⠀⢉⡟⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⡤⠤⠤⠶⠖⠚⠉⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠈⢧⠀⠀⠈⠳⣄⠈⠙⣻⠁⢠⠏⠀⢀⡞⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⠓⣶⢃⡴⠋⠀⡐⠋⠀⢀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠞⠁⠁⠀⠀⢸⣷⣀⣀⣾⠀⠈⠓⠶⠶⠶⠶⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠉⠙⡆⠀⠀⠀⠀⢾⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡼⡅⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣳⠟⠀⢁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⡾⠆⢠⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠆⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠛⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣹⠏⢀⣠⣾⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠲⠖⠺⠉⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣹⠚⠁⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠋⠁⢠⣴⠶⠯⣧⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡽⣛⣀⣱⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⢀⣼⣿⢉⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠋⢠⣴⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⣾⡿⣡⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢊⣛⣥⣶⣶⣾⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⣿⣳⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠠⣶⣾⣯⡿⠷⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⣼⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢠⡿⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⡠⠛⠃⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠚⢛⣉⡉⢛⣻⣲⣮⣿⡭⠖⠊⠙⢴⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣭⣤⣶⣿⣾⣿⣿⣾⣿⣷⣿⣿⣷⣶⣦⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⡿⠛⠋⠉⠉⠀⠀⠉⠉⠉⠛⠿⢯⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⠢⠀⠀⣀⠈⠀⠀⢀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡞⠋⠀⠀⡀⠸⣦⠀⠀⠀⡂⠀⠀⢀⣜⠀⢠⠈⠇⠀⠀⡉⠷⢰⣄⠸⠇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠸⡝⠢⡀⢰⣣⡀⠏⠀⠀⠠⠀⠀⡾⣄⠃⠀⢿⠷⢆⠀⠀⣟⡆⢠⣇⢳⠀⣆⠬⠃
⠀⠀⠀⠀⠀⠀⢀⡀⢲⣅⠀⢸⠆⢻⡜⣉⢱⡀⣄⠀⢓⠄⡀⠙⡬⠳⣄⠞⠈⠬⣆⡞⡜⢴⡣⠨⡳⡎⠪⢦
⣄⠢⡀⠀⠀⠀⠈⠳⣰⡛⣤⠊⠈⠌⠀⠈⢠⡇⢸⠵⡄⢠⠿⠀⡏⡦⠉⠌⠀⠌⠛⢛⠄⠀⡈⠄⠀⠁⠮⡋
⠀⠱⣔⢄⠀⠀⠀⠰⡇⠙⠥⠁⠀⢤⠀⢨⣼⣠⠊⢂⢻⢏⠺⡙⠃⠀⠐⠊⠀⡠⠔⠈⠀⠀⠀⢀⠴⠚⠒⠊
⠀⠀⠈⠣⡡⡀⡀⠄⠓⠐⠂⠤⣀⠀⠀⠘⠌⠇⡁⠀⠁⠆⠀⠀⣉⢼⠒⠋⠉⠀⠀⠒⠁⠘⡿⠋
⠀⠀⠀⠀⠙⢼⣳⠌⠐⠀⠀⠀⠀⠉⠑⠲⠄⠂⠐⠒⠂⠘⠁⠉⠀⣀⣄⡀⠉⠁⣀⣀⣈⠛⠁
⠀⠀⠀⠀⠀⠘⣏⣉⡤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠁
⠀⡀⢀⡀⠀⠀⠈⠑⠢⢥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠖⠒⠉
⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠈⠈⠉⠛⠓⠶⠦⠤⢤⡤⡤⠴⠶⠖⠛⠉⠁⠊⠀⠀⠂⠀⠀⠂⠀⠊⠊⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⡀⠤⠤⠀⠂⠀⠐⣌⠓⢄⠀⠀⠀⠀⠀⠀⠀⠏⢀⣀⣐⠒⠬⣑⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣴⠀⠀⠀⠚⠁⠀⠀⡀⢃⠀⠀⠀⠀⠀⠲⠀⠈⠀⢌⠽⡢⢄⡁⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡎⠀⠀⠀⠀⢀⡀⠀⡈⠀⠀⢀⢌⠃⠠⠤⠀⠀⠀⠑⠸⠄⠈⠀⠘⣆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠧⠏⠀⠀⣀⡀⢀⢀⠒⢀⠇⡇⠀⢸⠈⠄⠐⠂⠀⠀⢀⠞⠄⠀⠀⠰⠀⡼
⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠑⠋⠝⠒⠐⠁⠁⠀⢀⠎⠀⠀⠈⡔⠂⠀⠬⠓⡀⡀⠉⠈⠀⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠇⠀⠁⠠⠀⠸⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⣒⣊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠄⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠈⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠆⠖⠤⢀⡀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠀⠀⠀⠁⠘⠃⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⢀⠀⠁⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠄⠈⣁⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠐⠊⠁


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