Lizard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣬⣄⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣷⣶⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡟⠉⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⠆⣀⢀⡄⣠⡯⣿⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⢰⣞⢯⡿⢻⢸⣿⢿⣿⢿⡿⣟⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣷⠿⠿⣷⣮⣸⣿⢻⡷⠋⠁⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣷⣻⡶⢟⣛⠿⣿⡿⣵⡿⠁⠀⢠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⡍⣿⣿⡿⣵⠿⠛⠁⠠⣟⢯⣥⣲⣶⣝⣷⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⣝⣃⣿⠙⠁⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠉⡅⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣲⣖⣲⡆⣿⢫⣿⣭⣽⣛⡲⣄
⠀⠀⠀⢀⣤⡶⠖⠓⠲⣄⠀⠀⠀⠀⠀⡾⣾⠛⠻⢥⣞⣿⣶⠶⠛⠛⣿⣥⠎
⠀⢀⣴⠟⠁⠀⠀⡀⠀⣸⠀⢐⢇⡼⣖⡷⢿⠁⠀⡾⠻⢻⡏⠀⠀⢸⣿⡎⣀⡀⡄
⠀⣼⠏⠀⠀⠀⠀⠉⠉⠁⠀⠠⢊⡛⠛⠉⠁⠀⣼⢻⢴⡟⠀⠀⠀⠈⠻⠭⣿⣹⢖⡃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⡟⠀⠀⠀⠀⠀⠀⠀⢨⠍⠦⠁
⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣴⠏
⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠟⠁
⠀⠀⠙⢿⣷⣤⣀⣀⣀⣀⣠⢤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠙⠛⠿⠧⠿⠼⠷⠾⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⣦⣦⠀⣠⣶⣿⣶⡤⠶⠶⠶⠶⠶⠞⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣷⣿⣿⠿⠋⢀⣾⣿⣿⣿⣿⣾⣿⣻⣷⣦⡄⣳⣶⠄⣀⡼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⢠⡇⣼⣿⡿⣟⣛⣛⠛⠻⠽⠟⣁⣨⠭⢖⣫⣵⡾⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⠋⠀⠀⢸⣷⣿⡟⠉⠉⠉⣉⠭⢔⣚⣭⡵⠶⣻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠁⠀⠀⢀⢸⣿⣿⣷⣄⠀⠀⠀⠀⢀⣀⡤⠖⡫⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⢀⣾⣸⣿⠃⠈⠉⠉⠛⠛⠛⠛⠉⣠⡞⣱
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠹⣇⠀⠀⢀⣾⣿⢻⣿⡄⠀⠀⠀⠀⠀⠀⣠⣾⣟⣴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠹⡄⠀⣾⣿⣿⡜⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⣀⣨⣭⣭⣥⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⣿⣿⣿⣛⠛⠛⠊⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠘⠟⠛⠛⢷⣄⠀⢹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠙⠄⠘⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠡⡀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⠀⠀⠀⠀⠠⠊⠉⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⠂⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⢸⠀⠀⠀⡠⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡣⡴⠃⠀⢀⠜⢁⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⠀⢀⠇⠣⡠⢖⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠤⡄⢸⡁⠀⠀⠀⠀⠀⠀⣸⠄⠀⠀⠧⣀⡠⠤⡄
⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⡇⠘⠲⠂⠀⠣⡀⠀⠀⣠⠴⠊⢁⠞⠒⠶⠶⠒⠒⠴⠃
⢀⣀⠀⠀⠉⠒⣌⠢⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⡠⠋
⢏⣈⣉⣒⣒⡤⠞⠀⠘⢦⠤⠴⠚⠀⠀⠰⡉⡱⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢈⠇⢀⡰⠒⠒⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡞⠁⢀⡠⢚⣀
⠀⠀⠀⣖⠉⡷⠒⠉⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣎⢙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⢒⡯⠒⠉
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⠁⠀⠀⠠⠖⡄⠀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⢀⡠⠔⠊⢀⠀⠈⠢⢤⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠘⠤⠜⠀⠀⠀⢎⢉⠆⠀⠀⢀⢤⡀⠀⠙⠋⠀⠀⣰⠃⠉⠉⠉⠉⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠃⠀⠀⢀⡀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡨⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠒⠉⠉⠉⠉⠁⠀⠀⠈⢹⠀⠀⢀⢤⡀⠀⢀⣀⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⠏⠀⠈⢣⡡⠤⠇⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⠔⠋⠀⠘⢆⠀⠀⠀⠱⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⠔⠁⠀⠀⣠⠴⡣⠖⢲⢠⠶⢌⢣⡀⠀⠀⢇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠳⠤⠃⠀⣸⢹⠀⠀⣇⣨⠆⠀⡘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢎⠀⠀⠀⠀⠀⢠⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠀⠀⠀⠀⣠⠋⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠉⠉⠐⠛⠓⠉⠁⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⣀⣀⣀⣀⣀⠤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⡀
⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠚
⠀⢸⣿⣿⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠘⣿⣿⣋⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠸⣿⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣶⣿⣀⣀⣠⣼⣿⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠉⣿⠋⠉⠉⠉⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢀⣴⣾⣿⣄⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠶⣿⣶⠟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠀⠀⠀⠀⠐⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠑⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠿⠃⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣿⠿⠣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣾⣾⢶⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣤⣤⣤⣴⣶⣶⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡾⢀⠀⠀⢀⡠⠰⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠀⣴⡿⠷⣾⣿⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⠀⢸⣿⡠⣶⣿⢱⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⢀⣇⣿⣦⣤⣙⡋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢋⡷⠉⣿⡆⣸⡿⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠤⣹⣧⣰⡟⢠⣶⡁⠀⢀⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⣿⡟⠛⡻⣿⣇⣨⣿⣿⣦⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣬⡀⢹⡿⠟⢈⣿⠋⠀⠈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⡿⠗⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣿⡟⢶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⣿⡇⠠⣿⠀⠀⠀⠀⢀⣶⣶⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠶⣿⡷⠆⢹⣀⣴⣿⣿⣿⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣧⣆⣤⣴⣤⢿⡇⠐⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⡿⢿⣿⣀⠻⣶⡏⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠙⢧⣿⣷⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡷⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⢷⢯⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠰⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⣼⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣄⡀⠀⣀⣠⡾⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⠉⠿⠓⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠮⡄⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⢧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠈⠀⢀⠠⠴⠴⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠿⢧⠀⢀⣀⠄⣴⢛⣟⠟⣻⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣈⣿⡏⡶⢻⣼⣦⡻⠿⢟⡿⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢸⣿⣋⣹⣯⣶⠽⣿⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⡶⠋⣺⠍⠉⠭⣝⠿⣿⠟⠀⢀⣤⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣥⣾⠎⣠⣾⣿⡜⠋⣸⣏⣺⡒⠈⢢⡿⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣋⣼⠉⠈⠟⠉⠀⠀⠀⠙⠛⠋⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⡿⢿⣔⠆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡙⢬⢁⣟⠙⣻⣥⡶⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢩⣲⣦⣹⣧⣾⣌⣡⠞⠉⠉⢻⡿
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀⢻⢹⡀⠀⠙⣿⠁⠈⡟⠀⠀⠀⢸⡧⠴⢮⡉⠤
⠀⠀⢀⣴⠞⠋⠉⠉⠑⡄⠀⡛⢢⡴⣿⢿⡷⠀⠀⠿⣶⣾⠇⠀⠀⠀⠀⠀⠰⡾⢇⠂
⠀⠀⠙⠃⠀⠀⠀⠀⠀⠁⠈⡡⢈⠓⠉⠀⠀⠀⠘⢦⡀⡼⠀⠀⠀⠀⠀⠀⠀⠖⠈⠁
⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⠛⠄⢴⠃
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣴⠃
⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃
⠀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠈⠻⣿⣿⣶⠒⡶⢒⠀⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠂⠓⠂⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⢀⡴⣫⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀
⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣽⣯⡷⣄
⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣉⢿⡧⠑⢤⠤⢀⡐⠒⠖⢂⢤⢀⣀
⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⢿⠃⠀⠀⠀⠀⠀⢀⢠⠤⡒⣘⢊⡽⠦⠤⠄⠑⠚⢫⣢⡄⠀⠐⢻⠻⠿⣎⡛⡳⡀
⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⣴⡂⢀⡠⠐⠅⡐⠐⠑⣈⣨⣤⣴⠲⠆⠀⠀⠀⠠⢭⣄⣤⡺⢷⢾⡶⠿⢿⣿⢿⣷
⠀⠈⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠜⠋⠁⠀⠐⣁⠤⢚⡩⠔⠚⢁⢀⠄⢒⠀⠀⣤⠀⡷⠀⠀⠘⠁⠛⠚⠛⠓⠉⠉⠁
⠀⠀⢿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠄⠀⠀⠼⠛⠋⠁⡀⠀⠀⠀⠔⠑⠁⠀⠀⢸⡀⢱⣄⠀⡀
⠀⠀⠈⢿⢿⣾⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠙⠁⠀⠀⠐⠐⠤⣐⢠⡪⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣎⣩⡷⢯⣀
⠀⠀⠀⠈⠻⣿⣿⣾⣕⡲⠶⠶⠶⣒⣾⣫⣢⣥⣤⣢⡠⣠⣤⡒⣡⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⢣
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣝⣿⣭⢿⣿⣿⣿⡿⠟⠛⠋⢁⡼⢉⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡎⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣮⡶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⡰⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠹⢬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⡠⠶⣮⣔⣀⠀⠀⠀⠀⠀⠀⠧⣄⡉⠲⢤⡀⠀⡠⠖⢒⡚⠛⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠠⡄⠈⡇⠀⠀⠀⠀⢀⣀⡤⠽⠒⣤⠙⢲⢲⠞⢉⣀⡀⠀⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠁⢀⡇⢀⡠⢔⠊⠑⠀⣿⠀⠰⣿⡀⠸⠆⠀⠙⠿⢋⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡸⠟⠛⠀⣾⣦⠀⠐⡛⠲⢀⠈⠉⢀⠟⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠐⠉⠙⣶⠆⠀⠙⠿⡥⢈⣡⠊⢁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣖⡋⢛⡆⠀⠘⠃⢀⣋⡤⠴⠚⠉⠣⣄⡙⢄⡀⠀⣴⠆
⠀⠀⠀⠀⠀⢀⣠⣴⡚⢿⠅⣙⣇⠨⠥⠚⢄⠈⠈⡏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⡉⢳⡚⠁
⠀⠀⠀⢀⡴⠿⣄⡬⠗⠊⠉⠀⠀⠀⠀⠀⠈⢢⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⣏⠙⠿⠂
⠀⠀⢀⠞⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋
⠀⠀⡞⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡤⠊
⠀⢸⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠔⡱⠋
⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⢢⠺⡲⣲
⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠛⠂
⠀⠀⢳⠹⡀
⠀⠀⠀⠳⣕⠤⣀⣀⣠⠔
⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠠⠴⠔⠒⠒⠶⠤⣄⡀
⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢
⠀⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣠⣀⠀⢀⣤⣤⠀⠀⠀⠀⠀⢀⣀⡠⣤⣤⣄⣠⣤⡴⠶⢤⡴⠚⢛⠉⢉⣭⣉⠉⢐⡆
⠀⠘⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢷⣿⣷⢋⣿⡿⠖⠐⠒⠚⠙⠉⠀⠀⢀⠀⠀⠁⢈⡀⠀⠀⠂⢰⣦⣾⣿⣿⣿⣿⡿⠃
⠀⠱⠊⣸⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⠎⠻⠛⢛⢠⢄⣠⣄⣄⠀⠀⢠⠀⠀⢀⣐⢀⣠⢈⠳⣴⣤⣳⣾⠿⣿⠛⠛⠁
⠀⠀⠘⠥⡄⡉⠹⠟⢟⠲⠺⠚⠿⠋⠉⠀⠐⠀⡄⠀⠰⣿⣿⣿⣿⣟⣿⣷⣾⣶⣶⣿⣿⣿⣿⣿⠇⢀⣴⠟⠫⡒⠺⠉⠄
⠀⠀⠀⠀⠈⠉⠙⠷⠶⢶⣤⣆⣀⣄⣀⣀⣄⣄⢴⠦⠾⠟⢻⡟⣛⣛⣯⡵⠿⠿⠿⠟⠛⠛⠉⡁⠠⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠻⢿⣿⠛⡛⠃⢀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠬⠽⣒⡁⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣷⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣤⣷⣣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣯⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢤⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣿⣼⠟⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣙⣻⠻⣷⣦⣤⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⡿⢻⣿⣹⣿⠸⣎⠱⠏⢽⣏⠠⠘⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣿⢾⡀⠀⠀⠀⠀⠀⣀⣤⠶⢾⣿⣛⡿⠖⣛⣻⣌⣶⣍⠺⡶⢦⣰⣼⣿⣿⣀⣀⠠⢍⣿⡇⠘⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣠⢶⡟⡹⢃⣘⣩⣯⠀⠻⣉⣹⣿⣶⣿⢿⣿⣿⡿⠛⠛⠙⠛⠙⠛⠻⠿⢴⣦⣾⠟⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣤⣴⡾⢛⣓⡈⡷⢁⡿⢛⣁⣰⣶⠞⢿⣿⣿⠿⣿⣥⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣯⡿⠛⠄
⠀⠀⠀⠀⠀⣀⡤⠖⣔⡲⣦⢤⣤⣾⡍⢛⡩⠥⠺⠎⢁⣠⡴⠎⣀⠟⠵⣎⣹⣆⣶⣾⠟⣾⣵⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣾
⠀⠀⢀⡴⢎⠱⠒⣀⠆⠭⢸⡏⣿⡷⣻⡄⠴⠚⠚⣋⢈⡅⠤⠶⣤⣜⣿⣿⡿⠞⠉⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⢀⠴⠋⠀⠄⠂⢁⠠⢹⠀⢪⣇⣽⡯⣽⣶⣷⣿⣿⣿⣷⣶⣶⡯⠿⠟⠋⠁⠀⠀⠀⠉⠻⣿⣷⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⣏⡠⠀⠠⠠⠍⠭⠭⠌⣐⡢⣶⡿⠿⣿⣿⣯⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤
⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠋⠀⠀⠈⠙⢿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣻⣿⣿⣿⣝⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢹⣿⠞⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⣽⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠶⠦⠶⠞⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡛⡿⢿⣣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣽⣿⣿⣻⣟⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢿⡿⣿⣞⢹⣿⡟⣿⣿⡄⢃⡰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡋⣩⣿⣿⣿⣷⣹⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⡏⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠐⣹⣿⣿⡄⣻⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢢⣿⠟⢀⣴⣴⡼⠋⣽⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠾⠿⠋⠁⠀⠸⣿⣿⣿⣬⡻⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⣛⣲⢽⣟⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⣟⠿⣿⣿⣾⣿⣾⣿⣿⣿⡿⣿⣿⣿⣿⣦⡀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⢭⣟⣿⡿⣿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣮⣿⣿⣇⡴⣢⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣶⣶⡿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⡯⠔⠒⠓⠒
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⢄⣀⠁⠀⠀⢠⢏⣿⠟⠿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠟⠢
⠀⠀⠀⠀⠀⣴⣿⡿⠟⠛⠉⠙⠛⠻⢿⣿⣶⣆⣀⠐⠒⠒⠒⠒⠤⣽⡿⢦⣔⣻⣿⡟⠀⠀⠀⠈⠉⠁⠙⢿⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣀⠀⠀⠀⠀⠹⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢆⣿⣿⣇⠀⠤⠴⠤
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⢶⣶⣦⣤⣀⠀⠀⠀⠀⠙⢿⣷⣽⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢧⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠈⠻⢿⣿⣿⣽⣲⣤⢤⣤⣤⣠⣤⡴⣾⢏⣳⣿⣿⡟⠃⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠣⠈⠙⠻⠿⣿⣿⣿⣷⣷⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣠⡏⠀⢀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣷⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣷⣄⡀⣀⣠⣖⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠻⠿⠿⠻⢿⢗⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⠀⠀⣤⣤⣀⣴⣠⣤⣀⣀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣤⣴⣾⠿⣟⣿⣿⣿⣿⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣢⣴⣶⣷⣷⣷⡾⢛⡿⠿⣿⣿⠘⢻⣿⣿⣭⣭⣿⣿⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⢒⣿⣟⣿⣿⣯⣭⣁⣀⣀⠙⠛⡄⠐⡶⢾⣿⣾⡟⣻⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣻⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣬⠙⣦⣤⣀⣷⠾⢛⣿⣧⣼⣏
⠀⠀⠀⠀⠀⣠⢾⣿⣿⡧⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⣿⡏⡘⠛⠉⠁⠀⠀⠈⢿⣿⣀⣁
⠀⠀⠀⠀⣴⣥⣮⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣨⡿
⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⢯⣷
⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡐⢆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢝⣾⣾⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⡀
⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣷⢿⡆⠀⠀⠀⠀⠀⢀
⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠰⠴⠾⣴⣯⠞⠙⠁⠀⠀⢠⡈⡆⡜⠀⢀
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣀⣢⣤⡿⣗⢉⠯⣢⡀⠀⠀⠀⠀⣱⣿⡶⢮⠉⠀⣠⡶⣃⡴
⢰⡃⠀⠀⠀⠀⠀⠀⠀⢀⣼⢍⢫⡞⣦⣐⣨⡻⣧⣀⠀⡗⡄⠀⠀⠀⣽⣟⠀⠀⠀⢺⡅⣶⠅⡇
⠰⡅⠀⠀⠀⠀⠀⠀⠀⡈⠀⢰⠎⠀⠘⢻⣷⣛⢛⣿⢟⣮⠻⣦⠀⠀⠈⠟⢯⣤⠤⠬⠛⢷⢿⢣⡀
⠀⡟⣄⠀⠀⠀⠀⠀⣴⡿⡄⠈⠀⠀⠀⡼⣿⢰⣤⡊⠺⣷⢬⣈⠙⣁⣉⡋⠀⠱⣫⣖⠗⠀⢦⢸⢠⠖⣁
⠀⠀⢻⡻⢲⢠⠤⣄⡺⠖⠁⠀⠀⠀⠀⡤⢃⠀⠁⢗⣘⣾⡟⠿⠽⠶⢵⣒⡃⢰⠄⠛⣷⣄⣨⣿⠿⠭⠄
⠀⠀⠀⠈⠛⠈⠺⠗⠋⠑⠦⣄⣀⣀⣔⣶⠟⠀⠀⠈⠀⠾⠽⢶⡶⠶⠞⠓⠂⠁⠀⠈⠑⠫⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣉⡩⢽⡿⡟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢊⣀⢃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠘⠮⠙⠛⠛⠛⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⢉⠵⠂
⠀⠀⠀⢁⣀⠀⠀⠀⠀⠠⣄⣙⡟⢦⣀⡤⠤⠤⠴⠶⠶⠶⢦⣄⢀⣴⠒⠒⠒⠢⠤⠤⢤⣀⡄⣡⣎⣠⠄⠀⠀⠈⠉⣀⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠈⠁⢿⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢶⠿⠿⠀⠀⠀⠐⠲⠇⠘⢶⡄⠀⢀⡀⠀⢀⡵⠂⠠⠖⠛⠂⠀⠘⢿⠿⡶⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⠋⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠙⣄⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⠂⢱⣼⣦⣤⣤⣴⣦⣄⣰⡀⠀⠀⢺⡿⠀⠀⢀⣆⣠⣴⣦⣤⣤⣴⣥⣮⣤⣿⢢⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠾⣷⣠⠿⠟⣿⣷⣤⣀⣼⣇⣴⣤⣾⣿⠉⢻⡇⠾⠷⣖⠂⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠋⢩⡿⠿⠿⢿⡟⠻⠉⠹⣿⠟⠛⢻⡿⠿⠛⠻⢯⡙⠉⠁⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠖⡋⠀⠈⡏⠀⠀⠀⢀⣰⣉⣀⠀⠒⠶⢦⡀⠀⠀⠁⠀⠀⣠⡤⠤⠄⠀⣀⣀⡀⠀⠀⠂⢹⠀⠀⠙⠒
⠀⠀⠀⠀⠀⢰⢁⠀⠀⠇⠀⠀⠀⠋⣤⠞⠛⢛⠋⠈⢻⣷⣄⣀⢀⣐⣊⣁⣶⡶⠒⡛⠛⠳⢬⢻⠀⠀⠸⠀⠀⡈⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⠁⡀⢠⣼⠇⠀⠘⣿⠿⣿⣿⣿⠿⣿⡿⠀⠰⢧⡄⢀⠀⣧⠀⡗⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡸⡖⠁⢸⣧⡁⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⡇⠈⣆⡿⢸
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠛⠀⢻⠀⢉⡿⣹⠟⠒⠶⠶⣿⣿⣿⣶⣶⡖⣾⡟⣿⡟⠀⣿⠃⠀⠑⡜⣦⣳⢀
⠀⠀⠀⠀⠀⠀⠙⠱⡀⠀⠀⠀⠀⠀⠀⠣⠘⣧⣾⠀⣿⡗⢱⠃⠿⠻⣎⢹⡅⠸⣷⣿⢃⡜⡿⠀⠀⠁⠛⢠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠀⠳⣄⠹⣿⡀⣿⣷⡿⠷⣿⠶⢿⣾⣿⢀⣿⢿⣿⠄⠀⠀⣶⡀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠄⢀⠘⠳⡜⡓⢿⣏⠀⢀⠀⠀⠀⣹⡿⣾⢃⠞⡃⢠⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠐⠀⠀⢹⡄⢃⣰⠀⢛⣠⢆⡌⢠⡏⢠⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠦⠍⢁⠀⠒⠉⣚⠻⢾⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠈⢹⠇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢀⣤⠀⠋

⠀⠀⡄
⠀⠀⠀⠀⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠒⠠⢔⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣇⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢁⠠⠀⠤⠤⠤⠤⡀⠀⠄⠀⠤⢈⠀⡄⠀⢀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣾⣧⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢸⡥⠂
⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠼⣛⡽⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⢤⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⣀⡿⠛⠀⠈⠁⠀⠀⠀⣼⠶⠿⠿⣾⣧⡀⠀⢸⣦⠀⠷⠿⣏
⠀⠀⠀⠀⠀⠨⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠙⢇⠼⣿⣿⡏⠁⠀⠀⠀⠉⠙⠂⠄⡀
⠀⠀⠀⠀⠀⠘⡁⠀⠀⡰⡏⠀⠀⠀⢀⣾⣋⠄⠀⠀⠀⢻⡉⠻⠙⠦⠀⠀⠀⠀⠀⠀⣀⠀⠑⢤⣀
⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⡷⠀⢸⠀⢸⢋⠴⢶⣦⣸⡄⠁⠀⢀⠀⠀⠉⠉⠢⣄⣠⣸⣯⣤⡈⣱⠋⡆
⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⡈⡄⣿⡇⢳⣄⣾⠉⢻⡀⠰⡀⠑⡇⠀⠀⠀⠀⠀⠈⠉⠉⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⠘⢾⣿⠊⡦⠩⢅⣒⣟⠷⣶⣤⣹⡄⠀⠀⠀⠀⠀⠀⣠⣼⠁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⢀⡀⠀⠈⠈⠙⢻⠶⠛⠁⠙⣆⢰⠓⠂⠉⡛⠷⠤⠤⠔⠚⠉⣙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣡⣠⣀⣀⢹⡀⠠⠖⠂⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠣⣀⡴⠶⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⣀⠀⠀⡔⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠁
⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡡⠀⠠⠀⢄⡀⢤⡂⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡖⠠⠖⠀⠈⠁
⠀⠐⠁⡀⠀⠀⠀⠐⠀⠀⠙⣴⠀⢃⣀⡈⠁⠀⣷⡀⡏⡀⠀⠀⠀⠀⠀⠀⢀⠀⡞
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠿⠿⠇⠿⣻⣷⣤⠀⠉⠁⠛⡁⠀⡀⠀⠀⠀⠈⠀⡉⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠈⡿⠒⠦⠀⠀⡀⠀⠀⠀⠀⠈⠂⠀⠎⠡⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣰⣤⠀⠁⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⠀⠀⠀⠓⠦⢀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⣾⠚⣡⣶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠑⠛⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢨⠇⢰⣿⡉⣿⣋⣿⣿⠶⠀⠂⠀⡀⠉⠀⠀⠀⠀⣠⡶⠀⢄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢹⢸⣿⣟⣋⣻⣿⣿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⡴⠋⠛⠾⠲⠀⠳⠤⠂⠠⢄⢀⠎
⠀⠀⠀⠀⠀⠰⣆⣼⣿⣿⣿⣏⣿⠀⠀⠀⢰⡞⠀⠐⠂⢄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠿⡥⣝⢿⢿⢻⠄⠀⡀⠀⠃⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠨⠛⠀⠀⠁⠂⠸⣤⡄⠁⠀⠰⣄⡠
⠀⠀⠀⠀⠀⠱⠄⠶⠂
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢝⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢌⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⠯⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠋⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣾⣅⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠂⠀⠀⠀⠀⢀⣤⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠈⠛⠛⢿⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⣠⣏⡜⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡼⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡄⢠⣀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⢀⣠⡎⠀⠀⠀⢠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠶⠿⢷⣤⣄⡀⠀⠀⠈⣻⣿⣿⣿⣿⣾⣄⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣟⣟⣟⣻⣿⣿⣿⣿⣿⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣯⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⢿⣿⣿⢻⣿⣿⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⢿⣷⣿⣾⣷⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢤⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡿⠋⠉⠉⠉⠛⢿⣿⡿⣦⠀⠀⠀⠀⠀⠀⢀⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣮⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⣹⢦⣀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠈⠛⢿⣿⣍⡳⣦⣀⠀⠀⠀⠻⣿⡿⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠀⣴⣻⠟⠛⠛⠿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣧⣿⡓⣦⡀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⢨⢿⡏⠀⠀⠀⠀⠘⢿⢻⣆⠀⠀⠀⠀⠀⠈⠙⢷⣿⣟⣦
⠀⠀⠀⠀⠀⢾⣳⣀⠀⠀⠀⠀⠘⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢻⡿⢽⣇
⠀⠀⠀⠀⠀⠈⠙⢽⣶⣤⡀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣷⡞⠛
⠀⠀⠀⠀⢤⣄⠀⠀⠉⠻⢿⣶⡄⠀⠹⡿⣷⡄⠀⠀⠀⠀⣠⣾⣿⣯⠅
⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠹⣿⡄⠀⠹⣟⣛⢶⢶⣶⢾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣿⠁⠀⠀⠀⠙⠚⠮⠼⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡀⠀⠀⣀⢀⣀⣠⣈⣔⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠉⣡⣼⡿⠟⠛⠁⠀⠉⠙⢿⡇⠀⠐⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠒⠒⡭⡵⣫⢂⡤⠂⠉⢉⡴⠛⠷⡄⠀⢀⡆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠁⢲⢏⡼⣱⡷⠋⠀⠁⠐⢠⠖⠛⠻⠇⠀⠀⣧⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢴⣲⣥⣾⡵⠛⠁⢀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⣧⠴⢊⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣂⣿⢋⣁⣉⠀⠀⠀⡀⠀⠀⢀⣾⣶⣄⠀⢠⡆⠀⠀⣿⣦⢺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⢋⡽⢛⡿⢶⣤⣤⣤⠶⡏⠙⠛⠂⠀⢸⡇⠀⠀⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⢂⣾⠃⣠⠀⡀⠈⠷⣧⠀⠀⠀⠀⢸⡇⠀⠀⠿⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢈⣗⢼⡇⠁⡟⠀⡆⢘⠀⠘⢧⡀⠠⠂⢸⡇⠀⠀⠞⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠜⢀⡼⡇⠀⢰⡇⠀⠈⣧⠀⠀⣿⠦⣄⣸⠇⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠒⠒⣁⡴⠋⠀⢻⣄⣀⢸⣄⠀⠈⠳⣄⣿⠀⠀⢸⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠔⠛⠉⢉⣹⠷⣄⠀⠈⢻⡄⠀⠸⠀⠀⠀⡟⢻⡆⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡀⠀⠀⠀⢡⣤⣽⠀⡆⣸⡇⠀⠀⠀⠀⠀⡇⢨⣳⣄⠠⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡉⠂⠀⠃⠈⠀⢀⡼⡧⣭⡀⣶⣿⠁⠀⢀⠀⠀⠀⣷⡞⠖⠉⣶⣤⣈⡀⠀⠐⠊⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠉⣿⠡⠒⢨⠉⠉⠈⠳⣄⢸⠀⠀⠀⣽⠀⠀⠀⢹⣧⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⡿⠓⣶⡯⢤⠤⣄⡀⠈⢿⠀⠀⠀⣽⠤⠤⣀⠈⣷⣝⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠛⣉⣥⣾⣿⣷⡞⠁⣀⢀⣀⡜⠛⠠⣾⠀⠀⠀⢸⠀⠚⡛⣿⣿⢟⡿⣦⣭⣙⣂⣀⣀⣃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠶⠿⠟⠋⠁⢨⠿⣿⡀⢸⡁⠈⢹⡸⣦⣾⣿⣠⣤⣄⣼⣘⣤⡞⣸⠁⠀⡑⠀⠉⠙⠛⠛⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢠⠏⠀⢻⣧⠀⠙⠒⠫⠝⣻⢯⣯⣀⠀⣀⣸⣭⠶⢻⣭⡷⠊⠙⢄⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⠿⠷⣶⣶⣾⡳⠉⠀⠈⠋⢋⠀⠹⣾⢿⡟⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⣤⠖⠛⠒⠉⠉⠉⠁⠉⠉⠉⠉⠹⠋⠀⠀⠀⠀⠀⠠⠀⣘⠀⢸⠃⠀⠀⠀⠀⢀⣠⠼⠛⠙⠛⠛⠶⠦⣄⣀⡇
⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡴⡫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣄⣤⠤⠴⠞⣿⠁⠀⢸⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⡰⠀⠀⢠⡟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠉⠀⡠⠚⠀⡀⢠⠷⢎⣦⡼⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠱⠀⣠⣾⠓⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⣻⠛⠉⠀⠀⠀⠺⠀⠀⠠⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠒⠊⠉⠉⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣃⠀⠸⡅⠀⠘⡀⢆⣧⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠿⠤⠦⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘⢦⡀⢄⣠⡶⠒⠛⠙⢷⣚⠛⠛⠒⣋⡿⠗⠒⡲⢤⣆⢠⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢷⣄⠉⠲⢿⡦⠀⠀⠉⠛⠶⠂⠀⠀⠶⠚⠉⠀⠀⠐⣿⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡌⣻⣷⣄⠔⠀⠀⠀⠐⢦⡍⠀⠀⠀⠉⠀⢀⡀⠀⠀⠲⢄⠀⣿⣧⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣲⣿⣧⣤⣿⠁⢲⣦⠴⣿⢶⣄⠁⠀⠀⢰⠀⢸⢋⣴⡶⢦⡴⠀⣦⣙⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢤⣬⢿⣿⠀⠋⢹⠀⡈⠃⠓⢿⠾⢻⡆⠀⠁⠀⠈⠱⠻⢬⠷⡺⠁⠀⠈⣤⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠶⢦⡞⣰⡷⠈⣲⠟⢤⡀⢠⠤⣄⣀⣠⣄⠠⠀⠙⢄⠰⣧⢫⡁⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣿⠀⢹⢛⣿⢻⣶⣢⠈⠚⠲⡆⢠⢰⠄⠀⢴⣾⣿⠯⢻⡉⣿⡀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢛⡿⠀⢘⡇⠘⢸⣿⣅⣁⡀⠀⠙⣻⠃⢀⣀⣠⣝⠛⠋⠯⢴⣿⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠂⠹⣦⣾⣁⣀⣼⡿⣿⠉⣙⣛⡛⢛⣛⣋⠙⣿⡏⠙⢻⣶⣶⠏⢩⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠲⢾⠉⣹⠏⡀⣻⡝⡆⡿⠿⢿⡿⠟⣿⠸⣿⣀⢴⣉⢉⡤⠒⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣠⡙⠉⣷⢻⡷⣄⣀⣁⣴⣿⢰⡇⢁⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣴⡟⣾⡇⣛⣛⣛⣃⣿⢷⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢘⠬⣛⠭⠀⠭⢝⣫⡎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢀⡤⠆⠀⠀⠲⠤⠹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣁⡀⣀⣄⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠛⠹⢧⡿⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣭⣿⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠚⢉⣸⣿⣿⣿⣿⡾
⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⣻⠿⣷⣦⣀
⠀⠀⠀⠀⢠⣾⣿⣿⣷⢻⣿⡻⣶⣿⣿⣿⣿⣿⣖⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄
⠀⠀⠀⢠⣿⣿⣿⢿⣿⢸⣿⢏⣻⣿⣿⠛⠛⠛⢻⣿⠉⠉⠁⠈⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣾⡏⣿⡿⣟⣿⣿⣿⣿⡿⣿⡏⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣷⣿⣿⣿⡼⣿⣿⣿⣧⣿⣗⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢈⣿⡟⡟⣿⡇⣷⡽⣿⣿⣿⣿⡀⠀⠀⣰⡿
⠀⠀⠀⣼⣿⢹⡇⣿⣧⢹⣿⣮⡻⣟⡿⣥⢀⣴⡿⠁
⠀⠀⠀⠹⣇⢸⣇⢻⣿⣧⣙⢿⣟⠾⣽⣾⣯⣏⣀
⠀⠀⠀⠀⠙⠘⣿⣦⡙⢿⣟⡷⣾⣿⣶⣭⣟⠻⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⣰⣲⣆⡒⠒⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣹⡝⡋⠭⠰⠦⠰⠄⡌⢀⡃⠀⢮⣙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠺⣏⣷⣎⣤⣉⣀⠀⠀⠀⠀⠐⠈⠅⢙⠂⣈⢢⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣒⣲⠭⠓⢀⡼⣙⣧⡽⣯⡞⠋⠀⡠⠙⢢⠀⠀⠀⠀⠀⠀⠙⢄⡜⣎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⠷⠚⠉⣠⡤⢒⡏⣾⡉⣿⡗⢋⡀⠀⠀⣿⡞⣦⣕⣤⣀⡀⠀⢀⡀⢆⡤⠛⡭⡇
⠀⠀⠀⠀⠀⢠⣾⣍⡗⠁⣰⡶⠋⠁⢀⡾⡰⠿⣿⢯⠻⢷⣨⠃⠀⠰⡻⣧⣼⣿⠒⣿⣽⢙⣳⣆⣾⠟⣽⠃
⠀⠀⠀⠀⢠⡿⣥⡏⢀⡞⠁⠀⠀⠀⣾⢿⢿⣷⣿⡏⠉⠉⠀⠀⠀⠀⠙⠦⣛⣩⡛⡿⣛⣻⡿⣫⣷⠟⠁
⠀⠀⠀⠀⣼⡞⢻⡄⠸⣇⣀⣀⣠⢞⣷⣷⣴⢯⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⢹⡛⢿⢛⣷⣶⠞⢴⣾⣏⣹⠿⢛⡜⠁
⠀⠀⠀⠀⠀⠳⣬⣟⠻⠿⠷⡿⠛⠛⣱⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠿⠛⠛⠋⠁

⣿⠛⠿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⣿⡄⠀⠀⠀⠀⠛⣿⣿⠛⢻⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿
⠛⠁⠀⠀⠀⠀⠀⠈⢻⠀⣿⣿⣿⣿⣿⡏⢠⡆⠸⣿⣿⡟⢹⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢻⣿⣿⣡⡟⠃⠀⢹⣿⠇⠀⢸⣹⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠛⣿⠇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣹⠃⠀⠉⡀⠋⠀⠀⠀⠉⠙⠂⣿⣿⣿⣿⣿⣿⣿⣿⣶⠸⠉⠋⢻⣿⣦⣟⠀⣾⣿⠿⠷⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢾⣿⣿⣧⠀⣿⡇⢠⡄⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢀⣾⡂⠀⠙⠀⠹⡇⠸⣷⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣄⣤⡴⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣧⣼⠇⠀⠀⠀⠈⢠⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣿⠀⠀⠸⣟⣽⣿⣿⣿⣿⣿⢿⠟⠛⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⡏⠀⢀⣠⣿⣿⣿⣿⣿⠿⠃⣸⠀⣴⣈⣿⣿⣿⣟⣿⣿⣿⣿⣷⣤⡅⣿⣿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡾⢪⣇⣠⣼⣿⣿⣿⣿⣿⡏⠀⠀⢻⠀⠻⣼⣿⡏⣿⣩⠓⢿⣿⣿⣿⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣧⣻⣿⣿⡻⣿⣿⣿⣿⣿⣧⣀⣀⡣⣘⣿⣿⣿⣿⣿⣦⣕⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠟⢻⣿⣏⣻⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡯⠀⢸⣯⣿⠟⢳⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢱⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣾⡟⠁⠰⠀⢹⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡀⠀⠀⠀⢾⣟⢿⣿⣿⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡟⠍⣹⣿⣿⡷
⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢇⣿⣿⣿⠇
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢀⣿⢠⠀⠀⣰⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢇⣸⣿⠿⣋⢠⣶
⠀⠘⡟⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡿⢸⠀⣰⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠤⡟⢸⡿
⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⣿⡁⢸⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠃⠘⣿⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠃
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣼⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡄⠀⠀⠀⢸
⡧⠀⠀⠀⣧⢀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢸⣿⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⠇⠀⢀⣧⠀⠀⠀⡘
⡗⠀⠀⠀⣿⡌⠀⠀⠀⠀⠀⣾⣿⡇⠀⢈⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⡿⢸⡧⣸⡿⠀⡄⢸⢧⣆⠰
⡇⠄⠀⠀⠿⣿⠀⠀⠀⠀⠀⢻⣿⡇⠀⡎⠐⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⣾⠸⣿⡇⣼⠁⣼⡇⠀⣇⡜⢸⣿⢸
⣧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⣿⡇⢸⣷⠀⠀⣿⣿⣿⣽⣿⣿⡟⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⠓⢠⣿⠁⠈⠄⢸⡇⣿⡁⠀⣿⡇⠀⣹⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⢸⡟⠀⠀⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣿⠀⠀⠀⣸⡇⢻⡇⢰⣿⡇⠀⢸⣿
⣿⣆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⣿⠀⣼⡧⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⢩⣿⣿⣿⣿⣿⣟⠉⢠⡀⣾⡟⠀⠀⠀⣿⡇⢼⡇⢸⣿⡇⠀⣾⣿
⣿⡟⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⣾⣇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣡⣀⢀⣠⣿⣧⣾⣧⣼⣿⣧⢀⣿⣿
⣿⡇⠸⣿⣴⠆⠀⠀⠀⠀⠀⣧⣿⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣿⣴⠀⢀⣿⢻⣿⣿⣼⣿⠿⣿⣿⣿⣿⣿⣿⣘⣿⣿
⣿⣷⣀⣿⣿⡆⠀⠀⠀⠀⠀⢻⡿⠀⣿⣿⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⠿⠃⠀⠈⠉⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠏⣿⡏⣿
⣿⣿⣿⣿⢿⣇⠀⠀⠀⠀⠀⠘⡇⠀⢿⣿⠀⠀⠀⢸⣿⡟⠏⠀⠀⠀⠙⢻⣿⣿⠏⠀⠀⠀⠀⠈⠻⣿⠛⠿⠿⠿⠼⠛⠿⠿⠛⠀⠟⠁⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠉⠉⠓⠒⠂⠀⠉⠉⣁⣀⡄⣴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠢⠴⠤⣠⣃⡼⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣙⣄⡤⠦⠔⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⢄⠀⠙⡙⠀⠀⠈⠢⠈⢄⠠⠄⡠⠀⠖⠁⠀⠠⠋⠋⠀⣠⣴⠖⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⢀⣣⣴⣠⣄⣠⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⣄⠀⢠⡀⠘⠛⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠛⢛⣦⠀⠀⠹⡏⠀⠀⣴⡛⠛⠀⠀⢿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠍⠁⢺⣿⠓⠦⠴⠳⡖⠙⣿⡷⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠙⢧⠈⠀⠀⠀⣽⠋⠷⠤⡴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡷⢶⣴⡐⠈⠀⠀⠀⢸⠀⠃⠀⠀⠠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠋⡟⢁⢈⡀⠀⠀⢘⡄⠩⣄⡐⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡶⡻⠁⡬⠥⢤⣀⣘⡡⠀⡾⢵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣱⢄⡀⠀⠀⠒⣉⢵⡱⠷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠉⢑⠚⡛⡋⠁⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠤⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣦⣤⣤⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⢿⣻⠛⠿⡿⢿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⠛⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⢿⣿⠃⠀⢀⣸⣿⣷⡖⠋⡀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠋⠹⣷⣄⣿⠀⢀⣼⣿⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⣀⣤⣿⣿⣿⢯⠀⢹⠀⠠⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠦⠤⣊⣉⢱⣿⣛⣿⣿⣧⣈⠇⢀⣼⣿⣿⣶⣶⣶⡶⢶⣶⣶⣴⣄⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⣛⣭⣿⣿⣿⣿⠟⠟⠀⣸⡿⠟⠛⠁⠀⠛⠉⠉⠉⠉⠀⠉⠉⠒⠦⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠾⢿⣯⠄⣀⣴⣾⣿⠿⠿⠛⣩⠄⠈⢠⠞⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⣠⡾⠛⠛⠠⠄⢀⣿⡿⢻⣋⠴⠀⡠⠊⢀⣾⣷⣯⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠄⠀⡄⢀⡀⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⣠⠟⠉⠀⠀⠀⣠⣿⣿⡿⠗⡽⡡⢂⠈⣰⣎⣾⠿⠛⠛⠣⢦⣴⡾⠿⡿⣿⣿⣿⠿⠶⢾⣤⣶⣶⣿⣏⠀⠀⠀⠀⠀⠀⠀⣈⣤⣆⣤⠖
⠀⠀⣿⣾⡿⠓⠀⠀⠼⡟⣷⠹⠞⣼⢡⠟⣰⣿⡟⠣⠀⠀⠀⠀⠀⡁⢉⠰⣾⣿⡯⢿⣶⣤⣤⣿⣿⣿⣿⣿⣀⣀⣤⣄⣠⡤⠈⠛⠉⠉
⠀⠀⠉⠁⠀⠀⠀⠀⠀⣱⣿⡆⠀⢺⢻⢸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠁⠀⠀⠄⠀⠀⡀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠀⡇⣆⣿⣿⣏⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⡄⠀⠤⠒⠂⠀⣀⣠⣴⣾⠋⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⣿⡉⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣡⣤⣤⣶⣶⣿⣿⣿⡿⠉⠁⢻⡆⠀⣠⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣛⣚⣁⣹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⡿⠃⠀⠀⠀⣿⡾⠟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⣸⡇⠬⣿⢫⣿⡿⠟⠋⠁⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⡏⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⡿⣿⣧⠀⡀⠀⣽⣷⡾⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠟⣴⠉⠈⡆⢸⣿⣷⡄⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⢀⣾⣿⣯⣀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠓⠂⠀⠀⠉⠀⠁⢻⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠙⠿⣿⣿⣿⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⠀⠀⠀⠀⠈⠙⢳⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠖⠂⠀⠀⠀⠀⠀⠀⠙⣏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣳⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣦⣀⣀⡀⢀⣰⠛⠀⢀⡀⠀⠀⠀⠀⠀⣀⣴⣋⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣥⣉⠿⡿⢿⠋⠀⠉⠀⠉⠶⠂⢠⣼⣿⠋⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠉⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠾⢿⡠⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢷⡄⡰⢷⣇⠴⠐⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠈⠑⠦⣄⡀⠀⠀⣠⠶⠒⣚⠛⠛⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⣄⣀⣉⡉⠉⠹⠄⠀⠀⠀⠀⠀⠀⠀⢈⣉⡷⠤⢈⠙⢦⣤⣤⡶⠚⠃⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⢠⠇⠀⢰⠀⠀⠀⢀⣀⠤⣖⠋⢹⡄⠘⣠⣿⠀⠀⣖⠙⠀⢰⣿⣿⡇⠐⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣼⣀⡤⠚⠉⣆⠀⠉⠀⣸⣇⡀⠈⠻⣦⠌⢋⣑⡀⠀⠉⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣤⠟⠁⠋⠀⣸⣿⣆⠀⢀⡉⠉⢟⡨⠀⠀⠀⡼⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠁⠘⠳⣶⠂⠀⠈⠻⢿⣄⠈⣡⡔⠉⠀⣈⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠫⢷⡄⠀⠀⠾⠁⠀⠚⢂⣀⡤⠖⠋⠻⣀⠀⠫⣀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⡒⠛⣿⡄⠘⠷⢀⣜⠁⠐⠒⢶⠒⠒⠛⠉⠁⠀⠀⠀⠀⠈⠓⠦⣌⡓⠦⠼⣇⣴⠖
⠀⠀⠀⠀⠀⠀⣠⠶⠋⢿⠄⣈⣣⠤⠞⠒⠊⠉⠁⠈⢳⡀⠠⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣄⠼⢦⣄⡀
⠀⠀⠀⠀⢠⠞⠑⢇⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⢸⡆⠀⠓⠃
⠀⠀⠀⣰⠛⣢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢰⠧⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡠⠊
⠀⠀⣟⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣇⢠⠊⡴⠋
⠀⠀⡏⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡁⢺⡄⠀⡀
⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠹⡍⢯⡳⠟
⠀⠀⢷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣧⠈⠓
⠀⠀⠘⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠘⣆⠣⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠓⠮⠭⡿⠿⠛⠁

⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣾⣷⣶⣊⣽⣷⣄⡀⠀⠀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣖⣛⣏⢻⣷⣿⣿⣷⣀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⠙⡛⣿⣷⣝⠿⣿⣿⣿⣿⣉⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⢁⣿⣟⣾⣝⡺⢿⣿⣿⣿⣍⠻⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠁⠟⣛⣿⣷⣬⣓⣬⣿⡆⠘⡄⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡄⠀⢀⡄⠀⣴⣿⠿⣃⠀⠀⢸⡿⢿⣿⣿⣿⣿⡛⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣟⣠⣾⡟⠁⢹⣁⣴⣨⣩⣿⣯⣿⣿⣿⣿⣿⣬⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⠋⠁⡴⠞⣿⣿⠋⠛⠿⢻⣿⣿⣿⣿⣯⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⣲⣶⣷⣿⣿⣿⣄⠀⣶⡇⣿⣿⣿⣿⢿⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⣸⣿⣿⣿⣿⣿⣿⠟⡦⠀⢻⣿⣿⣿⣿⢈⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⢠⣾⣿⢿⣿⡿⣿⡿⠀⣆⡀⢺⣿⣿⣿⣿⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠈⢻⠾⠛⠀⠘⠛⠐⢣⠱⣿⠻⢸⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠘⠓⠛⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⢀⠓⠒⠲⢶⣶⣤⣽⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠘⠛⢿⣿⣿⣿⣶⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠠⣾⠟⠹⣍⠙⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⣇⠀⢬⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⡀⠒⠐⠶⣶⢦⣄⡀⠀⢻⣿⣷⣌⢻⣿⡾⠅⠃⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⢙⣿⣦⣽⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⢠⡶⢶⣶⣶⣦⣦⣤⣍⠉⢈⣉⣻⣽⠟⠉⠙⠋⠋⡻⠀⣀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⣾⠿⣷⡌⠋⠀⠀⠀⢹⣿⠻⠅⠀⠀⠈⠀⠠⠂⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠊⠁⠀⠀⠀⠋⢰⢿⣷⠂⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠃⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣮⣿⣿⣠⣦⠀⡖⢰⠂⠀⡀⠀⠀⠀⠀⢸⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⡟⢻⣿⣿⣤⣇⠸⣦⠀⣇⣀⠄⠀⣇⣾⠉⢿⣿⣧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣿⣿⣿⠁⠈⣿⣻⠉⢿⣇⢻⣿⣾⣿⢀⣼⢹⡿⢃⠘⢿⣿⣁⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣽⣿⡿⠀⠀⣿⣿⣶⣌⣿⣌⢻⣿⣿⣾⣣⣿⡇⢹⡄⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃⠀⣄⠙⢻⣿⠛⠿⠿⠛⣛⣛⣿⣻⣿⣷⠀⠰⢆⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡏⠀⠘⠈⣠⡾⢃⣴⡿⠶⣟⣿⣿⣿⣿⣵⣿⡄⠀⠹⡄⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⢠⡉⠼⣋⣴⣿⣿⠿⠚⠛⣱⡿⠋⣹⣿⣿⡀⠀⠀⠱⡆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⣝⠀⢈⣵⢾⣿⣿⣿⣷⠀⢠⣾⠟⠀⣼⣿⣿⣿⣁⠀⠀⠀⠛⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠉⢹⣿⠋⣀⣾⠏⡌⢹⣿⣿⠇⢠⡿⠃⠀⣴⣿⣿⣿⢿⡘⠀⠀⠀⠀⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢁⣼⣿⠇⣠⢃⡿⣼⣿⣰⡟⠁⢠⣶⣿⢡⣿⣿⠀⣿⢰⠀⠀⠀⠙⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢁⠞⢰⡟⡘⣷⡟⣱⠋⢀⣴⣿⣿⡏⣸⣿⡇⢰⡟⠀⠁⠀⠀⠀⠈⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢀⡟⠀⣰⠏⠐⢁⣴⣿⣿⡿⣿⠃⢿⣿⡇⢸⠀⢰⠀⢀⠀⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢡⡟⠀⠀⠁⢀⣴⣿⣿⣿⠟⣰⡟⠀⠀⢿⢡⡇⠀⢸⡄⠸⣆⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⢡⡟⢀⣀⣤⣾⣿⣿⣿⡿⠃⣰⣿⡁⢣⠀⠈⠸⡇⠀⠀⢿⡀⢿⡄⠀⠀⠀⠈⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠻⢑⣫⣿⣿⣿⠿⣩⣾⣿⡟⠁⣰⣿⠇⣇⢻⣄⡀⠀⠀⠀⠀⠘⣇⢸⡏⠄⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠉⠁⠀⠚⢹⣿⠿⢿⣶⣿⡟⠀⣇⢸⣿⣇⠀⠀⠘⠂⠀⢸⠀⣄⠂⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣉⣥⣶⣿⣷⣤⣀⣀⣠⣿⠇⢠⠞⢸⣟⠰⠿⣏⠀⠈⠻⣆⠛⠧⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⣤⡾⢛⣡⣴⣿⣿⣿⠃⣹⡇⠈⠉⠛⠛⣿⠆⣀⣠⣿⣿⠀⠀⠈⠀⠀⠀⠹⣦⣀⣰⣤⣀⣀⠀⡤⠚⢀
⠀⠀⢤⣶⠟⢉⣴⠟⢫⣿⣿⣿⠏⢠⡿⠙⠂⠀⠀⣼⠋⠚⠛⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢥⣿⠁⠀⠀⠘⠃
⠀⠀⡀⢀⣴⡿⠃⢀⣾⣿⣿⡟⢠⣿⠃⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠠⣠⡾⠛⠁⠀⣸⣿⣿⣿⠇⠀⠐⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⢀⡾⠋⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⡜⠁⠀⠀⠀⠀⣺⣿⣿⡿⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣿⡟⠛⠻⠷⠶⣶⣤⣄⣠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠿⣿⣿⣇⠀⠀⠀⠘⡏⣿⡇⠀⠀⠀⠀⠀⢸⡟⣿⠀⠈⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⡘⢷⡀⠀⠀⠈⢿⣿⠀⠀⠀⠀⢳⣸⠇⠀⠀⠀⠀⠀⠘⢰⡇⠀⠀⠀⠀⠀⠈⠛⣶⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠈⣷⠆⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⢸⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠜⠁⣰⠁⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠆⠀⢠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢸⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣼⡄⠀⣿⣾⠀⠀⠀⠀⣸⡇⠀⠀⢀⣼⡇⠀⠀⢀⣴⡟⠁⣀⠄⠀⠀⠀⠀⠤⠤⠴⠶⠾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⠶⠿⠿⠶⠶⠶⠾⠿⣶⣤⣤⣾⣿⡁⠀⣰⣿⠏⣴⣾⠅⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣼⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢠⣶⣶⣿⣿⡿⠿⠛⠃⠀⠀⠀⠀⠉⠙⠛⠿⢿⣧⣾⠟⠁⠀⢀⣀⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡾⠟⠋⠉⣀⣀⣤⣤⣤⣶⡶⠶⠶⠶⠶⠶⢤⣤⣌⣙⠻⢶⣟⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣠⣤⣶⣿⡿⠟⠛⣉⣉⣤⣤⣴⣶⣶⣶⣶⣦⣤⣄⡀⠉⠛⠶⣿⣿⣍⠉⠉⠉⠛⠛⠒⠲⠤⠄⡀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠛⠋⠁⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣷⣦⡀⠈⠻⣿⣷⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⢠⡟⠛⠿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⡀⠈⢿⣷⣦⣌⡉⠛⠶⣤⣀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⡉⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠈⢿⣯⣉⠛⢶⣄⡀⠉⠻⢶⣄⡀⠀⢿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣈⡉⠛⠶⣦⣤⣉⡛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠸⣿⡿⣷⣤⡈⠙⢦⡀⠀⠉⠻⢦⣸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣛⠶⣤⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣿⣿⡆⠀⣿⣇⠈⠛⢿⣦⡀⠈⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⢸⡇⠀⠀⣉⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⠃⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡟⢷⡀⠀⠙⢿⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡇⠀⠀⠛⠛⠛⠛⠛⠛⢛⣋⣉⣉⣉⣭⣭⡥⠴⠿⠧⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣿⣿⡄⠹⡄⠀⠀⠙⢧⡀⠀⠀⣿
⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠀⠀⣠⠀⣠⠀⣿⢶⡀⠀⠀⠀⠀⠀⣼⣿⣿⠁⣸⣿⣿⣿⡄⠘⠀⠀⠀⢀⠑⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠛⢉⡉⠁⠀⠀⣿⣿⠀⠀⣿⣄⣿⠀⣿⢸⣷⡄⠀⠀⠀⠀⠻⣿⠇⢀⣿⡟⢿⡿⣿⡀⠀⠀⠀⠀⠳⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣋⣴⣾⠟⠁⠀⠀⠀⢻⡇⠀⠀⣿⣿⣿⠀⣿⠘⣿⣿⡄⠀⠀⠀⠀⠈⠀⣾⣿⡃⠈⢿⡘⣷⡀⠀⠀⠀⠀⢹⣾⡏
⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⠛⠛⠋⠁⠀⠀⠀⠀⣰⡿⠃⠀⢰⣿⣿⣿⠀⢿⡇⢿⣿⣿⣆⠠⣄⠀⠀⣼⣿⣿⣇⠀⠈⠇⠘⣷⠀⠀⠀⠀⢸⠟⠃
⠀⠀⠀⠀⠀⠀⣼⠠⠿⠿⣧⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⢀⣾⣿⣿⣿⠀⢸⡇⠈⣿⣿⣿⣦⠙⣷⣼⣿⡟⠀⢿⡀⠀⠀⠀⠸⣇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢰⣏⠀⠀⢠⡝⢧⠀⠀⠀⣰⠟⠁⠀⢀⣴⣿⣿⣿⣿⣿⡄⢸⣇⠀⠘⣿⣿⣿⣧⠘⣿⣿⣧⠀⠈⠃⠀⠀⠀⠀⠙⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⠀⠈⠃⠀⢸⡇⠀⠀⠀⠛⠉⢉⣴⣿⣿⣿⠆⢸⣿⠀⠀⠸⣿⣿⣿⡇⢹⡇⢿⡆⠀⠀⠀⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠒⠒⠀⠀⠸⣧⠀⠀⠀⠀⣠⣾⡿⠟⠋⠀⣠⣼⠟⠀⢀⣀⣹⣿⣿⣇⢸⣇⠘⣿⠀⠀⠀⣇⠀⠀⠀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣿⡄⠀⠀⠴⠛⠉⢀⣤⣶⣿⡿⢿⣶⣤⡀⠙⠻⣿⣿⣿⠀⣿⠀⢻⣇⠀⠀⢸⡄⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⢀⣸⣧⠀⢀⣠⣴⣾⣿⣿⠟⠁⡀⠀⠉⠛⢿⣷⣤⣀⠉⠻⠆⢹⡆⠈⣿⡀⠀⢸⣧⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⣿⡾⠛⣿⣿⣿⠟⠁⠀⣠⡏⠀⠀⠀⠀⠉⠻⢿⣷⣦⣀⢸⣇⠀⢹⣇⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠿⣿⠀⠈⣿⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣤⣀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣽⣭⣬⣭⣿⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣶⣦⣶⣿⣿⣿⣯⣽⣼⣽⣷⣯⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⡄⣼⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣾⠋⠉⠁⠀⠀⠀⠀⠀⢠⣀⣠⣄⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣧⣤⣤⡴⣴⢦⣏⣿⡄⠀⠠⣽⣿⣀⣠⣿⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣼⡷⣿⢽⣷⣭⢥⣾⣿⡿⠖⠛⠋⠉⠉⠉⠉⠛⠻⢧⣤⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡻⣿⣭⣟⣶⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡿⣏⠻⢍⣼⣿⣿⣯⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣷⢢⣤⠀⠀⠀⠀⢠⣟⣛⣧⡼⢋⣶⡞⢻⣿⡷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⣿⣧⠀⢀⡴⠷⣹⠗⡘⢸⠿⢦⣼⣾⠻⣿⣿⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣶⠟⣳⣟⣠⠾⣃⣍⣜⣿⠟⠁⠀⠈⠉⠛⠻⣿⣿⣶⡶⠆⠀⠀⠀⠀⠀⢀⣄⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡯⡑⢫⡴⢛⣤⣖⣥⣿⠟⠁⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⠻⠆⠀⠀⠀⠠⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⢟⢿⡻⣿⢝⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡑⠁⠂⢂⠈⠇⣺⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠠⣲⡨⣈⣽⡿⠁⠀⠀⢙⣿⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣵⠾⠿⠛⠉⠀⠀⣀⡠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⢿⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠾⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⡀⣀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠋⠟⠻⠉⠇⠯⠄⠵⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⣀⠄⠀⠠⣤⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣤⡄⢤⡄⢤⠄⡠⠤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡗⢁⠞⣉⣀⣀⣀⣀⣀⠠⠤⠕⠺⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠋⠓⠋⠓⠘⠚⠓⠒⠛⠘⠂⠀⠀⠀⠀⠀⢀⡴⢁⡴⠋⠀⣸⠟⠉⠁⠀⠀⠈⠉⠐⠶⣴⣇⡠⠞⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⢋⡠⠛⢀⣴⣿⣛⣋⣀⠀⠀⠀⠀⠡⠀⢘⣯⣌⣁⠒⠒⠒⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢐⠛⠶⠋⠀⣰⡿⠿⠭⢭⣉⡑⣴⠏⣤⠄⣀⡴⣟⠒⠂⠭⣉⠛⠦⢤⣀⣀⠶⠒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠋⢸⠶⠁⠀⣀⡤⠾⠥⢤⣐⡶⠞⠁⠲⣥⣴⡒⢛⡉⡑⠢⣌⠤⡀⡀⠉⠑⠒⠬⠵⢶⡶⠶⠚⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡀⠀⠐⠀⠀⠀⠀⠒⠾⢋⣠⠔⢂⠐⠒⢀⠑⠠⡄⢀⣀⣿⣷⣶⠂⠈⠑⠂⠑⠀⢩⣩⡤⠈⠐⠂⠤⣾⡷⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣷⣾⡀⢰⣾⡷⠒⠊⠀⠀⢀⠙⠓⡀⠈⢠⡴⠓⢄⠂⢀⠀⠀⠀⠀⠈⠑⠢⠤⠤⢤⠐⡞⠩⢉⠉⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⡀⠀⠁⠐⠈⠉⣚⠛⠚⠋⠀⠀⠀⢢⠂⢀⠀⠀⠀⠀⠀⢡⠀⠂⢈⠢⡐⠢⣀⠀⠀⡀⢅
⠀⠀⠀⠀⠀⠀⠀⡀⡈⢳⠄⠀⠀⢠⣔⠃⠉⢩⣼⡏⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⢸⠐⢀⡀⠈⠀⣀⠀⠁⡴⣷⠀⠀⠘⠓⠠⠩⣲⡀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠂⠀⠐⠀⠀⡞⠈⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⢠⠀⠀⠀⠙⢷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣬⠗⠂⠀⡀⠀⠀⢠⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠉⠀⠀⠀⠂⠀⠸⠀⠀⠳⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⡰⠋⠀⠀⠀⠚⢱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⡤⠄⢠⠄⠀⠀⣰⠃⠀⠀⠀⠀⠙⠛⠓
⠀⠀⠀⠀⠠⠂⡁⠀⠀⠤⠊⠁⠐⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⠋⢀⠀⠀⠀⠀⣸⠋⠀⣦
⠀⠀⠀⠀⠒⠁⠀⠀⠀⣠⠄⡥⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠠⠀⡇⠀⠀⠀⠀⠠⠀⡧⡰⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠔⠀⠀⠀⢷⠀⠀⠀⠀⠀⡂⢰⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠋⠀⠠⠀⠀⠀⠀⠀⠀⣈⡷⣆⡀⠀⠀⠰⢀⠇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠩⠀⠀⠀⠀⠀⠀⠠⠀⢀⡤⠖⣩⠀⠀⣑⠢⢄⡀⠈⠒⠤⣤
⠀⠀⠀⠠⢠⣈⡒⠀⠀⠀⠒⠀⠠⠀⠀⠀⢀⣀⡴⢋⠀⠀⠀⠀⢂⠀⢂⣠⠴⠊⠉⠀⠐⠃⢠⠐⠁⢠⠀⠉⠓⠦⣄⠄⠂⠀⠀⢤⡀
⠀⠀⠀⠀⢀⣈⣁⠄⠀⣀⡀⠀⠀⠀⠉⠙⢉⡽⠁⠀⠠⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡄⠀⠀⠹⠁⠀⠿⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠞⢳⡴⠈⠁⠈⠉⣀⡘⠃⠶⣾⠁⠁⠀⠀⠀⡀⢰⡟⠀⠙⠐⠂⠰⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠉⠀⠀⡏⠄⠀⠀⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡠⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⡀⠀⠀⢿⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠒⠀⡀⠀⣀⠀⠔⠁⠀⠀⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠂⠀⠠⠈⠛⠾⢥⣄⡀⢀⣊⢀⣀⠀⢠⠀⠀⠀⠘⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠂⠐⠀⠂⠀⡀⠀⠈⠉⠓⠾⢤⣦⡤⠀⠀⠀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀⠀⠁⠀⢀⠀⠀⠀⠘⠀⠀⡈⠙⠳⠦⣤⠘⠃⠁⠠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣾⠕⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⡋⢉⢿⣿⣦⢠⠀⢀⡀⣰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣽⣿⣷⣶⣦⠿⣿⡀⠻⠟⠋⠙⣿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⣿⣫⣾⣿⡍⣝⡷⡄⠀⠀⠀⣜⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⣯⣼⣿⣿⠟⠍⢿⣯⣷⣷⠢⣄⣤⣧⣄⣤⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⢿⣿⣿⡓⣤⣼⣟⣿⣅⣤⣿⣩⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⣄⠲⣿⢿⣿⣦⢬⣿⣍⡻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢞⣿⡿⠀⠀⠀⠉⣍⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠽⣷⣦⣤⣾⣿⣿⣷⣬⣶⣆⣺⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡟⠛⢹⡿⣏⠀⠈⠙⣿⣿⣿⣿⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣤⣾⣤⠶⠛⠁⠨⡟⢦⣄⣈⣻⣿⣿⣿⡇⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠦⣄⠱⣌⣠⣿⠿⢿⡟⢿⣿⡙⢿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠒⠰⠿⠷⠿⠟⢤⣤⣶⣼⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⡻⠮⣅⣈⣀⣠⣤⣴⣶⡾⠛⠿⠅⠉⣛⣿⣿⡟⠀⠀⠀⠀⠸⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠈⠉⠁⠀⠀⠀⠀⠉⠙⢻⣷⣦⣌⣿⣿⣿⡇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣽⣶⣶⣀⣐⣨⣤⣶⣿⣿⡿⠋⣿⣿⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣿⡿⠁⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠋⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠂⠀⠐⠐⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣿⣿⣿⣿⣿⣾⣿⡗⠄⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⢀⡜⣶⡿⠀⣒⢀⡀⠀⠀⠡⣹⡇⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⠹⣿⣿⢷⣿⣭⣵⣶⣶⠿⠁⠀⢀⢜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⢀⡤⠜⡡⣺⣻⣯⣧⠀⠹⣿⣿⣽⣶⣿⠟⠀⠀⠠⡐⣀⡼⣿⣿⣿⣿⣿⡀⣍⣯⣾⣿⣿⠿⠋⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⠁
⠀⠀⣴⡟⡰⠋⢀⣷⣼⠏⠁⠀⠀⠀⠈⠉⢉⢀⣠⣴⣶⣾⢿⣿⣇⠀⠡⣐⡮⣿⣷⣿⣆⠼⣿⠟⠀⠀⠀⡰⣻⣿⣿⣿⠿⠿⠃⢀⣀⣼⣿⢇⣾⣿⣿⠇
⠀⠀⡷⣀⠇⡀⣞⣿⣟⠂⠀⠀⠀⠀⠀⣀⣴⡿⠿⠯⠁⠀⠈⠙⠉⠇⢢⠸⣥⣿⠟⣿⣷⣶⠿⠖⠀⠀⠀⠁⠒⠋⠉⠀⠀⠀⠀⣋⣉⣹⣵⣿⣿⡿⠋
⠀⠀⠻⣻⡿⣾⣿⠽⠁⠀⣠⣀⣴⢞⣛⣿⣿⣿⣿⣿⣭⣭⣩⡍⢱⣴⣷⡶⠛⠁⠀⠹⢿⣽⣦⣥⣦⡀⠀⠀⣀⣤⠶⠾⠿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠈⠉⠉⠀⢀⣴⣾⣋⣀⣾⣿⣿⣽⣾⣿⠿⣿⣿⣿⣿⡇⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⣁⡠⣤⣤⣠⣤⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠛⣽⣾⣿⣯⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢿⣇⣧⢠⣀⡀⠀⠀⠀⠀⠀⢀⡠⠖⠃⢠⡦⢙⣬⢶⡏⣿⣽⢿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣴⣟⣴⢚⣭⠿⠋⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠛⡟⠛⣾⡄⠀⠀⠀⠔⠊⠁⠀⢀⡶⣿⣷⣞⣿⣿⣿⣯⣿⣿⣷⣿⡍⠉⠉⠁⠓⠾⠿⠿⠟⠋
⠀⠀⠀⡼⣹⣿⣿⠟⠁⠀⠀⡀⣾⣿⣍⠿⣿⢩⣶⣦⡄⠀⠀⠀⠠⠖⠉⠁⠀⠀⠀⠠⠶⢶⡬⠿⠿⣿⣿⣽⣿⣿⣿⣿⡿⠛⠋⠉⠻
⠀⠀⢰⢡⣾⡿⠁⠀⠀⠀⣾⣿⣬⣍⠙⠰⢇⣬⡿⢩⣧⣤⡀⢀⢀⡤⣶⣆⣶⡤⠀⠀⠀⠀⠀⠀⠀⠟⠛⠺⠿⠿⢶⣷
⠀⠀⣿⣿⣿⠁⠀⠀⠀⢰⠻⣿⡿⢡⠃⠀⠀⠋⢰⣾⣿⡯⣾⣿⠾⢷⣏⠝⡸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⡤⠀⡀
⠀⠀⢿⣿⣿⠀⠀⠀⠀⠸⣰⣶⣾⡶⣥⠀⠀⠀⠀⠉⠀⢷⣾⡋⣷⡇⣁⣐⢟⣿⠃⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡴⠀⠀⠀⠈⠃
⠀⠀⠘⡏⢿⡄⠀⠀⠀⠀⣮⡛⢛⣱⣻⣗⢄⣀⢀⡀⣐⣾⣿⠻⣟⣰⡟⣛⣿⣿⣑⣲⠂⡀⠀⠀⠀⣰⡁⠁⠀⠨⠀⠀⡀⠠⢀⡞⣦⣩⣿⡐⡶⠂
⠀⠀⠀⠘⢾⣿⣄⠀⠀⠀⠈⠀⠚⠛⢻⣿⣷⠌⣿⣷⠎⠻⠷⠀⠀⠀⢀⣛⡟⠈⠓⠖⠑⠂⠀⠀⠀⠀⠀⠀⠚⢦⣠⡴⠤⠂⠞⠓⢛⡋⠉⠀⠄
⠀⠀⠀⠀⠀⠙⠻⢷⣦⣀⡀⠀⠀⠀⠀⠹⠄⠀⠙⠛⠿⠀⠀⠀⢠⣶⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠘⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠲⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣱⡀⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⡠⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢤⣷⡆
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⣈⡷⣈⡿⠸⠏⠀⠀⠀⠀⠀⢦⠀⣸⠃
⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠆⢃⠉⣥⣙⡹⢿⡾⡋⠀⠀⠀⠀⠀⠀⠈⢧⢸⣠⠃⣀⡤⠦⠀⠀⠀⢀⡀⢠
⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⡥⣨⠶⣿⣿⠺⣍⠠⠙⢬⣦⠀⠀⠀⠀⠀⠀⣨⡿⠿⠯⠵⠦⠀⢀⣤⠾⢻⣁⣾
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⡙⠯⡖⢳⣬⠐⢨⣙⢿⣷⣈⠄⢰⠓⡀⠀⠀⠀⠀⡝⣯⣋⠀⠀⠀⢀⢺⡏⢰⣾⠀⣿
⢠⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⣩⡼⠉⠈⡿⣿⣯⣝⢮⢷⣿⣶⢦⡳⣝⣆⠀⠀⠀⠑⢟⡿⡗⢀⡀⡀⢎⠻⣿⣥⠶⡇
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠂⡟⠀⠀⠀⠀⢸⢸⠗⠆⠒⡻⣇⣲⣝⢨⠛⠥⣲⡤⠤⣬⡅⠹⠿⢉⡭⢉⢄⠒⠊⠙⠀⡇⠀⢀⡤
⠀⢰⢁⣦⡀⠀⠀⠀⠀⠀⠀⡟⡎⣷⠀⠀⠀⠀⠀⢰⡟⣼⠈⢯⡳⡕⢈⡻⣿⢳⣬⣐⠀⣤⣤⣤⠠⠐⠘⢉⡴⣧⡎⠀⠈⢳⢰⢃⡴⣃⣤
⠀⠀⠀⢧⡻⠓⡆⣠⡤⣤⡀⠷⠖⠃⠀⠀⠀⠀⠀⢠⣤⢏⡀⠀⠁⡿⣀⣙⡮⣝⠻⠟⢽⠷⠶⠶⣚⡉⢄⡧⡘⠛⢧⣦⡀⣰⣿⣿⡯⢥⡄
⠀⠀⠀⠀⠙⠲⠇⣻⣤⣷⡭⠗⠣⣄⡀⠀⠀⠀⠀⢬⣄⣾⠁⠀⠀⠈⠋⢬⣝⣲⠮⢥⣥⣬⣼⠯⠥⠄⠘⠁⠀⠘⠋⣵⢚⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣭⣿⣾⡶⠛⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠖⢊⡽⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡆⠀⠀⢀⠀⠀⠁⠀⠀⠐⠀⡀⠀⠁⠀⢀⠀⠀⠀⢁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠄⢐⡀⢀⡴⠁⠸⠀⠀⠀⠀⠄⠀⠐⠀⠀⠘⡄⠓⣄⠀⠈⠁⢁⠐⠄
⠀⠀⠀⠀⠀⠐⠁⡀⠀⠘⡀⡼⠯⠁⢀⡀⠀⠀⣠⡆⠀⢸⠁⠀⠀⠇⠈⠿⡆⠈⠃⠀⠀⠘⠄
⠀⠀⠀⠀⠀⡄⠀⣁⠂⠀⠁⠧⠈⠀⠀⠉⢢⠜⠘⠀⠀⢨⡤⠚⠁⠀⠀⢂⠿⠘⠀⢂⠘⠀⢀
⠀⠀⠀⠀⠀⡀⠈⡇⢀⠆⡼⡄⠀⠀⡐⢰⡏⠃⠀⠀⠀⠙⠁⢀⠠⠂⠀⠀⣰⡄⢦⠀⠹⠀⠈
⠀⠀⠀⠀⠀⢡⠀⢠⠂⢄⢷⡙⢄⢀⠠⠄⣧⣀⠀⠀⡀⢀⡄⠀⣠⢁⠀⣨⢃⡟⣀⣡⠀⠠⠘
⠀⠀⠀⠀⠀⠸⠀⠈⠇⣈⠹⡎⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠃⠃⠀⠘⠈⠐⣩⠗⠀⠴⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠉⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡐⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⠀⢠⠀⣀⠐⠂⠀⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⣠⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠈⢠⠇⠀⠵⠲⠉⠀⠀⠓⣁⣶⠀⠂⠀⣶⡠⢀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠀⠈⠲⣤⣀⠁⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠀⢁⡄⣹⠁⠘⠀⠒⣆⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠸⠃⠁⠀⠄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠

⣿⡟⡿⢿⠛⢻⠿⠻⡿⠟⢿⠟⠻⡿⠛⢿⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣧⣠⣿⣀⣸⣄⣁⣿⣀⣷⣈⣠⣇⣀⣸⣿⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠉⠉⠉⠉⠋⠉⠉⠛⠉⠛⠛⠉⠙⠛⠋⠙⠉⠀⠀⠀⠀⠇⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠋⢂⣀⣠⠖⠖⠲⢢⣄⣀⡔⡙⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣆⠀⣠⠍⠁⠀⠀⠀⠀⠈⠩⣄⠀⣰⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠅⠀⠄⠠⡄⢠⠄⠀⠀⠨⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⡍⠁⠀⡔⡤⠖⠲⠤⢂⠀⠈⢹⠄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣁⣸⡀⠀⠀⠀⢰⠋⠙⢗⠶⢶⡖⢲⡶⠶⡾⠋⠙⡆⠀⠀⠀⢰⣆⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⢣⠀⠀⠀⣎⠈⠸⠈⠑⠚⡇⢘⠓⠊⠀⠂⢀⣵⠈⠀⠀⡞⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⠿⣶⢼⣇⣛⡦⣬⡢⢒⣧⣼⡒⣐⣤⢴⣻⢹⡦⡾⠿⣷⣿⠠⠀⠐⠀⠀⠠⠀⠀⢠⠋⠁⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡠⠒⠛⠛⠍⢿⡇⠢⠱⣯⣶⣶⣽⠎⠔⢸⡿⠩⠛⠓⠒⢄⣽⡇⡠⣤⢤⢤⣀⡤⠦⠀⡀⠄⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠑⠈⠀⠀⠀⠀⠀⢷⠀⠀⠃⠁⠀⠹⠀⢀⡞⠀⠀⠀⠀⢀⠁⠊⠙⠢⣍⡈⠂⠹⡙⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⢷⠬⣄⠀⠀⠈⠦⣙⣧⡤⡘⢷⡀⠀⠀⠀⠀⢠⡿⢃⣤⣾⣋⠄⠁⠀⠀⣠⠥⡺⠟⠀⠀⡇⠁⠀⠀⢀⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⠈⠀⠀⠐⠀⠒⠉⠈⠓⠐⣟⠊⠙⠳⣆⣰⠞⠋⠑⣻⠂⠛⠁⠁⠒⠀⠊⠀⠀⠁⡇⠀⠀⡇⠀⢠⣄⣸⣿⣄⣀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢺⠀⠀⠀⠀⠀⠀⠀⢰⠖⢈⣁⣙⣢⣤⡿⠷⠆⠒⠋⠈⠁⢴⡊⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⡇⢀⣾⣿⣿⡷⠶⠤⣭⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣆⠀⠀⠀⠀⢀⠀⣺⠛⠁⢡⠉⡈⢰⠁⠀⠀⠀⠀⠀⣀⣾⠧⠀⠀⠀⡠⠀⠀⣸⢰⣷⣰⣧⣾⣿⡿⢿⣷⣿⠛⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡏⢦⠀⠀⠛⠁⠀⠘⠀⠀⡀⠠⠀⡆⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠈⠛⠀⠀⡾⢹⣺⣿⣿⡿⣛⣿⣶⢿⣿⣿⣿⠿⢟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣿⣭⣶⣿⣿⣿⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡄⠀⠀⠀⡘⠀⠀⡈⠠⣁⡬⣔⣽⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⡴⣰⣠⣦⢾⣻⣭⣾⣿⣿⣿⣿⣷⣿⣶⣦⣄⣀⢀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠐⢈⠉⠩⡉⠐⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠤⡈⠢⡀⠱⠀⠃⠀⡇⢠⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠤⢂⠌⠀⠀⠀⠀⠀⠀⠔⠁⠀⢄⡈⠓⢄⠘⠆⠘⠀⢀⡀⠀⢀⠀⠐⠠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠄⠀⠐⠈⠐⢄⠀⠀⠀⠠⢊⠀⢠⡢⡀⠈⢦⠈⠂⠀⠠⠂⠁⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢐⠢⡀⠁⢀⢱⣀⠔⠡⠀⠀⠂⠀⡌⢆⠀⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⣑⠒⠗⠂⠟⠉⠀⠁⠀⠀⠐⠂⠀⠆⠀⢀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⣴⠆⠈⠀⠀⠀⠀⠀⢠⡄⠀⠤⠀⠐⠋⢀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠔⠁⣠⠄⠁⠀⠀⠀⢠⡤⠀⠋⠀⢀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠌⠀⡀⠁⠀⠀⠀⠀⡤⠈⠀⠀⠀⣾⠁⠀⠈⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠰⠀⣸⠃⠀⠀⠀⢠⡄⠀⠀⢀⡞⠀⠁⠀⡀⠂⠀⠄⠃
⠀⠀⠀⠀⠂⠀⠙⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢠⠎⢐⠤⠀⠀⡁
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠋⠀⠀⡠⠊⠀⡁⠸⠂⠄⠐⡀⠀⠁
⠀⠀⠀⠀⡀⠀⣿⣶⡄⠀⠀⠀⠾⠀⠀⡐⠁⠀⠈⠒⢁⠂⠂⢁⢀⠄
⠀⠀⠀⠀⢁⠀⠀⣤⠀⠀⠀⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠆⠸⠿⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠈⡄⠀⠯⠀⠄⠀⠁⣀⠀⠦⠤⠀⠀⠀⢤
⠀⠀⠀⠀⠀⢀⠌⢢⠀⠂⠀⠴⠆⠈⢀⠐⡃⠀⠀⠀⠁⡆
⠀⠀⢀⠄⠂⠁⠰⡀⢡⢸⡂⠀⠁⠀⣨⡅⠀⠀⣰⢀⡀⠧⢂⡆
⠀⠀⠂⠘⢠⠂⠁⠀⢀⠘⠃⣀⠙⠓⢀⡀⠺⠷⠈⠂⠀⠐⠄⡀
⠀⠀⡀⠀⠠⠠⢒⠄⡆⢀⣄⠁⠀⣤⠞⠷⣤⠀⢰⣿⠈⠈⠂⠘
⠎⠀⢀⠀⠀⡘⡐⠂⢁⠘⠻⠆⠴⠰⡄⠀⢥⢠⠀⡌⠈
⠀⢀⠂⡎⠱⣀⠄⠀⠀⠢⠀⡔⠀⣃⢰⡀⡆⠂⣠⠇
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠑⢄⠈⢧⠀⠇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠆⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢈⣿⠞⠋⢉⠀⠀⢀⣠⣤⣨⣅⣤⣄⡀⠀⠀⡉⠙⠳⣿⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⣀⣴⠟⢁⣴⢿⣷⠖⠛⠉⢞⣡⢭⡭⣌⡳⠉⠛⠲⣾⡿⣦⡈⠻⣦⡀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢋⡴⢻⡇⠀⠀⢀⣘⠛⠛⣩⠤⠤⢍⠛⠛⣃⡀⠀⠀⢸⡟⢦⡙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠞⠉⣰⣿⣇⡀⠀⠊⢁⡠⠖⠒⠒⠒⠒⠲⢄⡈⠓⠀⢀⣸⣿⣆⠉⠳⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⡄⠉⣽⡏⠻⡄⠀⠺⠀⠀⣠⠔⠢⣄⠀⠀⠗⠀⢠⠟⢹⣯⠉⢠⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣾⢇⣾⠁⠀⠗⢢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠲⠀⠈⣷⡼⣷⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡏⠈⠙⠲⠄⠀⠙⢺⣿⣏⣀⠀⠀⠀⠀⣀⣹⣿⡗⠋⠀⠠⠖⠋⠁⢹⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣇⣠⡤⡠⢞⣣⣤⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠤⢤⣜⡳⢄⢤⣄⣸⡇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣠⣴⡟⣇⠀⢰⣿⣦⡻⢄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢟⣴⣿⡆⠀⣘⢻⣦⣄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣧⠻⠆⢸⣿⣏⠻⢦⡀⠀⢠⡄⠀⠀⢠⡄⠀⢀⡴⠟⣹⣿⡗⠰⢟⣼⣥⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣍⡛⠛⣻⣿⣓⠦⠀⠀⠈⠻⣤⣤⠟⠁⠀⠀⠴⣚⣿⣟⠛⢛⣩⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠹⣿⣿⠿⠳⠶⡶⢤⣴⣦⡴⢶⠶⠞⠻⣿⣿⠏⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣀⣿⣿⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⣿⣿⣀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣆⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡏⠖⡀⠀⠀⠀⠀⢀⠲⢹⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣟⣉⣉⣉⣙⣩⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡤⠠⣤⠠⣤⠀⠀⣠⠔⠐⢤⠀⠀⣤⠀⠠⡄⠠⣤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢻⡄⠀⠀⢻⠃⠀⣿⠀⠈⠀⠀⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠐⠛⠀⠀⠙⠒⠄⠚⠀⠀⠛⠒⠒⠃⠐⠛⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣠⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⢀⣤⣶⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⣿⠿⠛⠛⢿⣿⣿⣶⣶⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣀⠀⠀⣿⣿⣇⣠⣤⣶⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢻⣿⣿⠉⠁⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⠀⠀⢿⣿⣧⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⢀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠈⠉⠻⣷⡄⠈⢻⣿⣇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⣸⡟⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣷⣦⣤⣾⣯⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⢡⡿⠟⢳⡄⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⢀⣀⡀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠺⠁⠀⠀⠀⡇⠀⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠡⠄⠈⠁⠉⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢐⡏⢀⠀⢄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠊⠁⠐⠄⠀⠈⡋⠀⠇⠀⠃⠀⡀⠀⠀⠀⢀⡀⠋⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢭⣈⡁⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠘⠂⠁⠀⠃⠀⠀⠀⠚⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠰⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠆⠀⠲⠅⢢⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡤⠀⠼⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠁⠀⣿⠃⠸⠅⠀⠀⠀⠀⠆⣠⠂⠃⠈⠡⠄⡤⢤⣿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠃⢹⢇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡘⠒⣠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠸⠄⢪⡀⢀⣀⡤⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⢜⣄⠀⠀⠀⠀⠠⠻⣀⡿⠛⠋⠉⠋⠙⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⣀⢤⡄⠠
⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⣶⢟⣄⠀⠀⠀⠐⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢦⣀⡀⠀⠒⠃⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠓⠾⢡⡀⠀⠀⠀⠀⠉⠉⠙⢫⣥⡄⡀⠀⠀⠀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠑⠀⠀⠀⠞⠣⠂⠀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠑⢴⡠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣰⣷⡏⠹⡷⣦⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠐⠴⠟⠂⠀⠀⢄⡀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⡶⢂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⣘⠯⡓⢈⣐⣻⣆⠀⠠⣀
⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⡀⠀⠀⠀⠀⠋⠻⠆⢈⠁⠀⣠⠤⠶⡀⠀⡀⠀⠍⠀⠀⠙⠀⠉⠀⠀⠀⠚⣯⡏⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⣡⡄⠀⠀⠀⠀⠀⠐⢞⡂⣈⠖⠔⢂⠘⠈⢡⠀⠀⠀⠀⠀⠀⠀⠒⠸⠃⠠⡿⣇⠀⡀⢳⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⢀⠀⠀⡘⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠘⢄⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠏⠀⢐⣈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠛⠀⠀⠛⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⢗⣶⢦⡄⠀⠀⠀⠀⠀⠉⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡅⢿⣿⣼⠏⣴⡖⣤⣀⢀⣀⠐⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⡐⣽⢳⣿⣿⣿⢛⣥⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠻⠘⣯⢛⣟⠁⣿⣏⠻⠃⠉⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠽⠃⡻⢿⣿⣿⣿⣿⣷⣦⣝⢦⡀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⣦⣳⡜⠦⣌⠏⢽⡷⣶⣬⣝⢷⣝⣆⣤⣄⠀⣴⣏⢩⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣻⣴⠀⠀⣨⣿⠁⠀⠈⠙⠲⠿⠿⠿⠟⢨⣿⠙⢷⡿⢸⣿⡟⠁⠠⣉⣦⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣊⠙⠟⠻⠿⠟⠁⠀⠀⠀⠀⠀⢀⣀⣬⣽⣯⣅⡰⠿⠀⠘⠋⠀⠀⠀⠈⡉⢀⡛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣿⠆⠺⣤⡿⢇⣾⣿⣷⡄⠀⠀⠀⠘⡏⢦⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡄⠀⠀⠀⠀⠀⠿⣴⢿⣿⡀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢉⢡⠐⢤⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⣧⠀⠁⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⠘⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠖⠒⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⡀⠀⠻⣿⣿⣶⣦⣀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣦⡹⣿⣿⣿⣿⣦⡻⣷⣄⠙⢦⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢿⣿⣿⣿⣿⣿⣦⣻⣿⣿⣿⣷⡽⣿⣷⣌⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⣬⣻⣿⣿⣿⣿⡏⠙⣿⣿⣿⡻⡼⣿⣿⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠻⣿⣿⡄⠈⢿⣿⠃⠁⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣥⣴⣦⣤⣭⣉⡙⠛⢿⣿⣧⡀⢈⠻⣷⡰⣌⢿⣦⣆⣸⣿⡿⢛⡛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⢿⣿⠿⢿⣿⢿⣷⣤⣨⣽⣿⣎⣿⣾⣿⡿⢿⣛⣿⣿⣿⡼⣫⠽⢦⣻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠁⢀⣹⣿⣿⣭⣍⣀⠈⢭⣭⣿⣿⠿⢛⣩⡶⠟⠛⢉⣩⡍⠉⣶⡇⣾⡆⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣻⣿⣿⣾⠿⣫⡴⢖⡋⠠⠶⣊⣃⡤⠥⢶⣖⣻⣿⣾⣾⣿⣇⠑⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣩⣵⣿⣶⣠⣤⣴⣾⣿⢿⡿⠗⣛⡬⠔⣚⣭⣴⡶⠿⢟⣛⠻⠉⠉⠋⠻⣿⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⣡⣿⣿⣿⣿⣿⢟⣵⡿⢟⣫⣵⠞⢋⣽⣿⠋⣶⣶⡶⣿⣷⣤⣄⣤⣤⡟⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⢡⣿⣿⣿⣿⣿⣯⣾⠟⠿⣥⣾⣿⣿⣿⣆⠈⠻⠉⠘⠉⠋⢹⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢡⣿⡿⢫⣿⣿⣿⣿⣴⣦⠙⡎⡟⠻⣿⡿⡛⢦⡀⠀⠀⠀⠀⠈⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣵⣿⣿⣿⣿⣿⣝⣛⡴⢣⣄⣘⣷⣅⠀⠀⠑⠀⠀⠑⠻⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⣿⣿⣿⣿⣿⡍⠛⠸⣿⣏⣿⣿⣿⣦⡀⠀⠐⠺⣙⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠻⣿⣿⣿⣿⣷⡘⡆⢽⣦⣾⣿⣿⣿⣿⣦⣀⠀⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⠳⢬⡻⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣿⡋⠝⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡢⠄⠀⠈⠻⣿⣿⣿⣿⣎⢿⣿⣿⣷⣼⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠱⠙⣿⣿⣿⣿⣧⣻⣿⣿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⢿⣟⢿⣿⣿⣿⣿⡇⠭⠥⣄⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠂⠙⠾⡻⢿⣿⣶⣶⡌⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢁⣧⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠻⣽⢟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡘⡯⢚⣚⢄⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⠸⢰⢫⢮⡃⡾⣿⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣷⣔⠘⣼⣿⢫⣿⣞⡟⠀⠀⣀⠀⠈⡀⢀⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢌⣿⣱⣌⣛⣛⣻⠟⠁⠀⠀⠁⢲⣴⢷⣮⣀⠄⠸⠂
⠀⠀⠀⡦⡀⠀⢋⠃⠀⣠⠀⠀⠀⠙⠶⢖⣿⢻⢇⣀⠀⠀⠀⠀⠀⣉⣻⣿⢤⡀⣄
⠀⠀⠀⠀⠙⢦⣨⣤⡴⠈⠀⠀⣠⣾⡯⡕⣽⡷⣡⢶⡭⠖⠤⣤⠞⡽⠂⠀⠀⠀⠁
⠀⠘⠃⠐⠒⢸⣽⠛⡇⡀⢀⣴⢋⠸⣧⢟⣽⣵⣳⡯⡭⠽⠂⣦⠞
⠀⠀⠺⠃⠈⠁⠀⠈⠙⣩⣭⣧⠞⢗⣦⡝⣧⣻⣼⡖⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⣏⣺⣽⣿⢻⡳⢯⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡣⢾⣾⣸⣷⡬⠻⢼⢷⡄⠀⠀⠀⠀⠀⠀⠀⢫⢳⠀⣰⠎⢠⡋⠋⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣯⠘⣛⣿⢿⣾⢟⣜⣿⡼⢦⠀⠀⠀⠀⠀⠀⠀⠸⡶⡻⣴⠋⠀⠀⡀⠀⠀⠀⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡱⠭⢯⡝⣣⠣⣬⣽⢗⣣⣯⣷⡀⠀⠀⣠⢴⡶⣦⣆⣩⣿⠒⠒⠀⠛⠁⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢿⠞⢱⣿⢿⣿⣷⣹⣍⢿⣜⣧⠀⢰⢳⣫⡞⠁⠁⠀⠉⠣⣢⡄⠀⠀⠀⡼⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⢹⢮⣴⣯⣹⣮⡻⣟⣷⣩⡽⠆⠾⡿⠿⣧⠀⠀⠀⠀⠀⠈⠀⢀⣤⣾⡾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢬⡓⣾⣏⡯⣽⢿⣿⠷⡷⠴⠶⠲⠖⠋⠀⠀⠀⠠⠴⠶⠻⢻⣿⣿⣷⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⢯⣦⡊⣿⣛⣿⣛⣿⣟⡿⢭⣭⡿⢿⣭⡛⢶⡿⠿⢋⣿⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣨⢝⣮⠭⣭⡉⠻⢿⡔⢿⣽⣩⣝⢾⡝⡆⣯⣛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣻⠿⠶⠿⠚⠛⠀⠀⠀⠉⠒⠚⠿⠷⠚⠃⠐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣢⣤⣀⣈⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣼⡱⢆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠊⢁⣬⠛⣟⠃⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡥⠁⢐⠹⠀⠐⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡀⢀⡀⠠⢤⠤⠤⠤⠴⣤⣤⣀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⢉⡁⠀⠈⠉⠉⠭⡭⢉⣓⣒⠯⢽⣺⠦⣦⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠏⠁⠀⢀⠉⠓⠀⠀⠀⠀⠐⠒⠒⠒⠒⠲⠷⠀⢻⠭⣗⣺⣽⣿⣆⢸
⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⢤⠠⠠⠄⢴⣒⣖⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⣌⡃⠀⠀⣯⡷⣿⡀⣀⣢⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣽⢮⣦⣯⣴⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⢉⣉⣼⠋⠈⠻⢟⣿⡉⠉⠙⢳
⠀⠀⠀⠀⠀⠀⠄⠠⣰⠈⠇⡴⠠⡀⠘⣯⢶⠀⣸⣿⣿⠃⢐⠜⠓⠀⣿⣿⣶⣶⠟
⠀⠀⠀⠀⠀⡀⣨⠡⠀⠴⠀⡄⢆⣿⣾⣿⠆⠘⠛⠋⠁⠀⠠⠂⠀⠈⣿⣿⡟⠁
⠀⠀⠀⠀⠐⠂⠀⠐⠲⠿⠃⠁⠨⢝⣿⣟⣶⠀⠀⠂⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠠⠠⢄⠀⢀⣠⡶⢻⡿⣽⡇⠈⢥⣤⣀⡄⠴⣷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠑⢹⣶⣤⣬⣭⣷⣷⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⣿⠁⠘⢠⠄⣴⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣴⣄⣺⣣⣖⣷⡀⠈⠀⢴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠄⠀⠈⠉⠑⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⣆⣠⣤⢦⡤⠀⠀⠀⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢂⡀⠚⠳⢾⠋⠀⠀⠀⠀⢸⣿⠎⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⢺⡘⠇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣠⠐⠉⠙⣿⣷⡄⠀⠀⡞⠧⣀⠘⢦⠀⠀⠬⠚⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⡹⠀⠀⠈⠀⢻⣷⠀⢀⠀⠀⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠳⠢⠆⢁⠀⢺⡟⢠⡏⠀⠀⠀⠀⣡⠋⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢠⠆⡂⠀⠐⠲⠶⣫⢠⠏⠀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠁⠀⢀⠀⠀⠀⠠⠄⠀⠀⡰⠋⠀⠀⠀⡠⠂⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡔⢀⢇⠀⠀⠀⠘⢀⠎⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠝⠒⠀⠈⠁⠸⠃⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠂⠁⠀⠠⢤⣾⣞⡁⠀⢀⣀⡤⠔⠀⢀⡴⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠛⠛⢅⠀⠀⠀⠀⠀⠒⠋⠉⠙⠛⠛⠯⠥⠤⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⡀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⢀⠘⠋⣉⡝⠁⠤⣤⡤⡄⢤⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠂⠈⠙⠐⠊⠁⠀⠀⠀⠀⣀⠴⠀⣤⠖⠃⢰⣴⣶⣤⠴⡾⣷⠒⠈⣭⣤⠂⠀⠀⢀⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⣀⡠⠶⣿⠉⠉⣹⠛⠙⣿⠦⣀⣉⣳⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⡤⠤⠚⠉⠉⠉⠉⠀⢀⠈⠁⠀⠀⢸⠳⠀⣻⠦⣼⣇⠀⠈⢉⡟⠛⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢨⣿⣧⠀⣿⣤⣈⡏⠁⠀⣼⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠀⠀⠀⠾⠿⠿⠀⠈⠛⠻⠿⠶⠚⠙⠓⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢤⠂⠀⠀⠀⠠⠤⢤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡊⠀⠨⠀⠀⠀⡶⠲⠓⡧⠄⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢩⠈⠂⢢⡼⠒⠂⢴⠨⠛⠚⡓⠀⠀⠠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⡈⢆⢢⠜⠁⠀⠀⠀⠀⠈⠚⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠅⠢⠌⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠐⢈⣉⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠠⠤⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠠⠐⠂⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠂⠬⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠐⠂⠩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣁⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⠠⠈⠩⡄
⠀⠀⠀⠀⠀⣀⢴⠶⣺⡉⢭⣣⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠡⠐⣀⠠⢇
⠀⠀⠀⢀⣼⡋⡐⢉⡄⠋⠁⠀⠀⠈⠘⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠂⢅⠐⢺
⠀⠀⢀⡖⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠔⠲⣈⡛
⠀⠀⢸⣍⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⡟⡐⢊⡑⢈⡇
⠀⠀⢸⡄⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⣰⠢⠌⠤⠑⣼
⠀⠀⠘⠦⢈⡁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢍⠡⡉⢄⡱⠃
⠀⠀⠀⠱⡂⠠⠀⠜⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡚⠋⠠⣈⠂⠠⡜⠁
⠀⠀⠀⠀⠙⢀⠒⢈⠥⢋⠗⠶⢤⡄⢠⣤⣤⡤⠤⡒⢲⡉⠇⡐⠨⣀⡂⠠⣨⠎
⠀⠀⠀⠀⠀⠀⠰⢆⡘⠀⠄⡀⢂⠄⡁⡐⢀⠆⡁⢌⠰⠀⠢⢈⠐⢄⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠆⣡⢈⡄⢣⠘⡜⢀⠀⡆⡐⣀⡧⠬⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠉⠉⠉
⠀⠀⣀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠛⠃⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⠀⠀⠀⠀⢃⠀⠀⠀⣴⣄⣀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⣦⣀⠀⠀⠈⠠⠤⠀⠀⠛⠛⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠣⠀⠀⠀⠀⠀⠀⢀⣶⠟⢛⠛⠃⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠐⠀⠀⠀⠀⠀⣾⣧⡴⠀⠀⠀⣠⣿⡐⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠙⠃⠀⠘⢢⡼⠟⠙⢣⣀⠀⠀⠀⠀⠄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢹⠃⠈⠀⠀⠈⠻⣶⡤⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⠄⠘⢇⢃⣤⣿⣧⣄⠀⣀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣎⡷⢶⡀⠰⣿⡿⠉⠻⢿⣿⣿⣿⣯⠀⠸⡏⠁⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⢇⠀⢈⣿⠁⢤⣤⣤⣄⡙⣉⡣⠀⢠⣿⠟⢠⠄⣰⣿⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⢦⣼⣿⣏⠋⠉⠙⡁⠀⠀⠃⠀⠀⠀⠘⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⣡⡤⠙⢿⡿⣶⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢧⠀⠀⢺⡗⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⢀⠀⠈⢩⢻⣶⣤⣀⣩⣄⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⠀⠀⢸⡤⠂⠀⠁⠈⠡⣼⣯⠉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⡙⠀⠀⠀⠀⠀⢡⣤⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠁⣸⠇⢢⡄⠀⠀⠀⠸⠃⠀⠘⠳⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡾⠁⣾⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠙⠲⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠐⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠲⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣀⣴⠀⠀⠀⠀⠀⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠉⠛⢃⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠐⠀⠀⣀⡀⠀⠀⠀⣣⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⣀⣀⣀⣠⠟⠁⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣦⣴⣶⣾⣶⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠁⢀⣠⣿⣿⣿⣿⣟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣤⣾⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⡀⣀⢀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⣉⠽⢿⣭⣭⣯⠉⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣾⠃⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠙⣦⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣤⡾⠿⠛⠻⣷⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠙⠻⣷⣤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠙⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⠏⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⣦⡀⢀⣆⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⢀⡴
⠀⠀⠀⠀⠠⡀⠀⠀⠸⣦⡀⠄⠀⠀⠈⠣⠀⠀⠀⠀⠹⣿⣅⠀⠀⠀⠀⠈⢿⣴⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠙⢦⡰⣄⠀⠂⠀⣤⡀⠀⠀⠈⠙⢧⡀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠘⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠛⠾⣟⠷⣤⣌⣙⣦⡀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢠⠈⠳⣄⠉⠛⠛⣷⠶⢶⣤⣤⣿⠃⠀⠀⢨⣇⠀⠀⣠⡟⠁⠀⠀⠀⣠⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠓⢾⣄⠀⠈⢷⡄⠀⣿⡀⠀⠙⠻⣿⣦⡆⠀⠈⢿⣤⣾⠋⠀⠀⠀⢀⣼⣁⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡀⠀⠀⠀⠉⠳⡄⠀⠉⠲⣤⡙⢤⠀⠀⠘⢇⢻⡄⠀⢸⡿⠀⠀⠀⣀⣴⣿⠿⠛⠉⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠹⣆⠀⠐⠛⠛⠻⢷⣦⡀⢸⡀⢿⡄⢸⣷⠀⣷⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⣈⠑⠿⠷⣦⣄⡀⠀⠀⠈⠛⢿⣧⠸⣷⠈⣿⣄⣿⣿⠟⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡻⠛⠶⣄⡉⠻⣍⠳⢶⣤⣀⠀⠹⣆⣻⡇⠈⢻⣿⣇⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣀⠀⠈⠀⠀⡠⠌⡙⣷⣮⡑⠂⢤⣙⠿⣷⣽⣿⣷⡀⢀⠻⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠟⠛⠛⠛⠛⠛⠛⢛⠻⠿⣷⣤⣈⣳⣦⣍⣿⣿⣷⣌⣷⣿⣿⣶⣦⣤⣤⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡤⠖⠊⠉⠉⠉⠙⠛⠛⠶⣶⢶⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠛⠋⠉⠉⢁⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠠⠄⠒⠉⠀⠀⢒⣠⣴⣿⣿⣿⣿⡿⠟⢛⠿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠤⠶⠛⣉⣁⣴⣿⣿⣿⣿⣶⡞⠋⠀⠀⣀⣤⣄⣟⠀⣿⡗⢀⡀⠀⠀⠀⠀⠤⠤⠜⠓⠂
⠀⠀⠀⠀⠀⠀⣠⠶⡞⠛⠙⠛⠁⠀⠀⠀⠀⠛⣉⣽⣿⣿⣿⣿⣿⣿⠁⠀⢆⣴⣿⣶⣿⣧⠀⣿⠀⠈⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢀⣤⡤⠐⣂⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣷⡟⠀⠀⠘⠀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠂⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠘⣿⣇⠀⠈⡁⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⣿⣿⣿⣿⣿⣿⠶⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠐⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠿⠛⠃
⠀⠀⠀⠀⠀⡠⠊⢁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣠⠖⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡔⠁⠠⠚⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠀⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⡟⠉⠘⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠉⠀⠀⢠⠟⠠⠋⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠟⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⣤⣤⣶⠾⠉⠙⠒⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣤⣴⣤⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⢿⣿⣿⣧⠀⢰⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣿⣿⣿⣧⣤⡹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⢹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⣿⣿⣿⡇⢻⣿⡜⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⢨⣽⣿⣿⣶⣶⣾⣿⢿⣿⢱⡌⣿⣿⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠸⣿⣿⡇⠈⣿⣿⣦⡙⢷⡄⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣱⣇⠹⢹⣿⣿⣾⣿⣷⣄⠀⣿⣿⣿⣦⢠⣶⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⣌⢻⣿⣿⣿⣿⣿⣷⣿⣿⣯⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣔⣝⠶⣝⡿⣿⣿⣭⣛⡿⠿⣿⣯⣭⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⡿⣅⣨⣭⡝⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢋⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣧⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⡿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⡦⠄

⠀⠀⠀⣀⣀
⠀⠀⢠⣿⣿⣷⢦⣄⡀
⠀⠀⠀⣿⡟⠻⢷⣬⡙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⣿⡇⠀⠀⠈⠙⠷⣯⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠻⣿⣷⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠹⣿⡙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡄⠈⠻⣿⣿⡆
⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣄⠀⢸⣷⡀⢘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣸⡇⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣿⣆⠈⢿⣧⠀⠻⣿⣿⣷⣤⣀⣀⣀⣀⣤⣸⣿⣠⡟⠁⠀⢳⡀⠀⢹⣿
⠀⠀⠀⠀⠀⢻⣦⣄⢠⣼⣿⣿⡄⢈⣿⣆⠀⢹⣿⣿⣏⠻⣿⣿⠿⢿⣿⣿⣿⠿⠿⠷⣾⣷⣤⣀⣿⡀
⠀⠀⠀⠀⠀⠀⢯⡻⢿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠀⠙⠿⢷⡈⢻⣶⡿⠶⡷⠶⢦⣴⣶⣤⣄⣀⢉⡉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⠙⠛⠿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⠉⠛⠛⢦⣠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⣶⠶⠿⠿⠿⢛⣷⣶⠖⠀⠀⠀⠀⠀⣸⡟⠋⠙⠋⠉⠂⠈⠲⣤⣄⣀⠀⠉⠛⢦⣀⠙⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣥⡄⠀⠒⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⢤⣽⣿⢠⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⠀⢻⣧⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⣠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⣀⣀⠈⢉⡀⢿⣶⣄⡀⢠⡀⠀⠀⠀⠈⢻⣿⣆⠀⠛⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡧⠀⠾⢻⡟⠁⠀⠀⠀⢀⣴⣿⠃⠒⠺⢿⣿⣿⡿⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠐⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠋⣿⠟⠁⣠⣾⣿⠋⠀⣠⠄⠀⣼⣿⡋⠀⠀⠀⢀⡈⠀⠀⠀⢻⣿⣿⠉⠛⢿⣷⣄⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⢻⠀⢰
⠀⠀⠀⠀⠀⢀⣼⠏⠀⡰⢫⡿⠁⣠⣾⡏⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠘⣿⣿⡄⢀⣈⣿⣿⣷⣦⣀⣼⣧⣤⣄⣀⣠⡆⢰⠀⣾
⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⡿⠀⠘⣫⣿⡇⠀⢸⡛⣿⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⠋⠙⠻⣿⠄⠀⠉⠉⢳⣿⣤⡆
⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁⠀⠐⣿⣿⡀⠀⠘⠃⠙⠻⣿⣦⣤⣤⣀⣀⠉⠙⠓⠶⠾⠿⠿⠿⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⣃⡇
⠀⠀⠀⠀⠀⠋⠁⠀⢸⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠠⠄⠚⠛⠛⣛⠒⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠠⣄⡀⠠⣀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⠀⠀⠛⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣤⡀⠀⢠⠈⠹⠟⠳⠄⠀⠉⠳⣤⣈⠙⠦⠈⠳⠆⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠘⢻⣇⢸⣿⡆⠀⢀⣿⣧⡖⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⢧⠀⠀⡀⠀⠀⠀⡀⠈⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡎⣿⣷⠀⠸⣿⡟⠀⣰⢃⣾⣿⣿⣿⣿⣿⣿⣏⠙⢿⡎⠙⣿⣿⡇⢸⠁⠀⣀⠀⠻⣆⠀⠟⠀⢰⣶⣦⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠈⠙⣇⢹⣿⡀⠀⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠈⣿⠀⠈⠙⣿⡐⣦⠀⠙⢦⠀⠙⠀⠀⠀⠘⠛⠻⣿⣿⣷⣶⡶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡇⠀⠋⠀⣿⣿⣿⣿⣿⠿⠋⠙⣿⠁⣿⠇⠀⠈⠳⡀⠀⠛⣷⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠰⠃⢀⠟⠀⠀⠀⠀⠣⢄⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⢸⣿⣿⣿⡧⠖⢨⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠛⠛⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢪⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣿⡟⠁⠀⠀⣼⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡀⠐⠋⠉⠛⠿⣷⣶⣶⣾⣿⣧⣄⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢰⡄⠀⢿⣦⠀⠀⢰⣿⣆⢈⡄⠀⢰⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢿⡀⠘⣿⣧⣤⡟⠀⠉⠉⡹⢁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⢷⡀⠈⠻⠿⣿⣷⣶⣾⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠋⠁

⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣶⣤⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣌⠿⣿⡷⠐⣿⣿⠿⠷⢢⡀⠀⠀⢀⣤⡎⣿⣷⣮⣻⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡘⣿⣿⠿⣅⢰⡇⢠⣬⡿⠜⣿⣿⡿⢋⣽⡏⢾⣷⡾⣢⠗⢼⢱⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⡶⡈⠂⠁⠘⠚⠁⣶⢾⣿⠷⣙⢿⣿⣖⣴⡆⠁⠀⢸⢫⡷⢇⣀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⡿⢠⡉⡁⠀⠀⠀⠀⠘⠉⠁⠘⢿⣟⣫⣬⣿⡏⢸⣿⡿⢳⣝⣯⣩⣩⠥⢅⣂
⠀⠀⠀⠀⠀⠀⢸⡇⢍⣉⣼⣥⣬⣡⡀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⠛⣿⡅⠻⠗⣼⣿⣿⠉⠋⠴⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⠇⣿⣾⡶⢤⣤⣄⣤⣶⣒⠿⣿⣟⣬⠏⠁⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣮⡉⢡⣺⣷⡿⢿⣷⣮⡍⣻⣿⣿⢂⡊⢉⠁⠀⠀⣠⣽⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡈⠋⠉⠉⠏⢿⡛⠁⠘⠻⠛⠻⠇⠀⠀⠀⠀⢨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⣠⣆⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠁⡰
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠠⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣂⠀⠀⠀⢀⠔⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁
⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⡀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⢰
⠠⠀⠀⠀⠀⠀⠀⡈⠒⠂⠀⠁⠀⠀⠀⠀⠄⠄⠀⠀⠁⢤⠀⠀⠀⠂⠀⠀⠈⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⢀⠀⡐⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⢊⡠⠊
⠀⠈⠐⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⡀⠠⠌⠃⠀⠈⢁⠀⢀⡡⠠⠤⠀⠀⠀⡀⡀⢀⢀⡀⡀⠠⠤⠠⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⡀⢻⡿⠿⡟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣀⢷⣾⣿⣇⢠⣾⣷⣅⠀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⠾⣿⣢⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣛⣿⣿⣿⣿⣿⣿⣋⣿⣿⠿⠿⣿⣿⣿⣿⣻⡏⠀⣰⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣷⣾⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⣿⣿⣿⠀⠿⣽⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⢋⡷⠊⠽⢿⣿⣿⣿⡿⡟⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⣿⣿⣿⣿⣿⣿⢟⣿⢟⡟⠿⢿⣿⠗⠀⢻⣶⡿⢁⠀⠀⢪⣿⣿⣿⣾⡻⢟⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⠀⡿⠁⠙⠷⠦⣼⣿⣟⣿⣿⢿⣩⣽⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⢔⣣⠞⠁⠀⠀⠀⢰⣿⣿⣿⣿⣯⣶⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣤⣷⠿⠁⠀⠀⠀⠀⢴⣿⠟⢏⡉⠀⢽⡟⠀⠉
⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢟⣉⣡⣶⣶⣦⣤⣶⣿⡟⠁⠀⠀⠀⠉⠉
⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣀⡀⢤⣠⣶⣿⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠋⠘⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠻⣿⡾⡇⠀⠀⠀⣈⣛⡙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⢹⣶⣄⣤⣶⣾⣿⣷⣶⣶⣿⣿⣿⣻⣿⣟⣯⡳⣲⣤⣤⣤⡀
⠀⠀⢀⣀⣤⣾⣿⣿⣿⣷⡿⢷⣿⣿⣷⣽⣿⣯⣥⣉⡙⠻⣿⣷⣟⣿⢿⣿⡿⢳⡄⡀
⠠⠿⠿⢟⣿⡿⣿⣿⠟⢋⠀⠉⠉⠛⠿⠿⣿⣷⣭⡷⣿⠀⠀⠙⠛⠿⡷⡛⠚⡌⠁
⠀⠐⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⠘⠓⠒⠂⠈⠻⢿⡟⡏⠀⠀⠀⠀⠀⠀⠁⠈⠁


⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠟⠋⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡽⠟⠛⠛⣯⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣾⣿⣿⣿⣁⣀⣀⣠⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣤⣶⣶⣶⣦⣠⣾⣿⣿⠿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⣾⡏⠀⠀⢀⣽⣿⣿⡋⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⣘⣧⣠⣶⣿⣿⠟⠹⣿⣶⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣟⢿⡟⠁⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣤⣤⠄⣀⣠⣤⠴⣶⣶⠶⡒⠒⠒⠶⠆⠀⣀⣤⣴⣶⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡷⢖⠛⡩⢶⠃⠿⢟⣫⣽⣶⣶⡜⢟⡴⣛⠽⠻⣟⣭⢾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢏⣴⠇⣠⡴⢖⣒⣛⠉⢷⡁⠀⣹⠗⡞⠀⠁⠠⣾⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢻⣿⡾⢸⢏⡶⢋⣽⣧⠶⠿⢃⡤⢂⣤⣦⣄⠀⢀⣶⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠈⡏⡼⠋⠘⣠⣴⣯⣵⣦⣤⣬⡽⠞⠫⠵⢦⣌⣧⠸⣿⢡⣟⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡷⣰⠃⠀⣸⣿⢻⣤⡥⣋⠕⡊⡲⣶⣦⡴⢣⡿⠀⠀⠃⣸⣿⣆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠄⠁⠀⢰⠋⠀⢀⣤⣬⣤⠄⣽⣇⡴⢂⠀⣯⣜⠏⡀⠀⣿⣿⣹⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡇⠀⠀⠀⠸⢦⣤⠄⠀⠀⢒⣭⢿⣯⡶⣻⠏⠋⣙⣴⠃⠠⢿⣿⠉⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣻⡿⠃⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⠿⠟⢁⠄⠀⠸⣿⣾⠃⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠙⢿⡁⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣲⣶⠶⠞⢉⡀⠂⠆⢻⡏⡇⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠳⣤⣀⣙⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⡿⣾⢁⡀⣴⣃⣾⠀⣾⣿⢸⡿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣼⣷⢾⠋⠀⠈⣇⣿⡆⢻⡇⣿⢋⡼⣻⣫⡏⠀⣿⡟⣾⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠁⠙⠅⠈⠀⠀⢀⣸⢇⡏⣸⣿⣷⠏⣸⣩⠟⢀⣾⣟⣽⣫⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣼⢡⣟⡾⠁⢠⡟⣡⣼⠿⣫⣾⢁⣥⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⡼⣡⠟⣋⣤⣴⠿⣛⣩⣴⠟⠫⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣠⠟⣛⡵⣟⣯⣷⠾⠟⢛⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡡⠂⣴⢃⡾⣫⠾⠛⡉⠀⢀⣶⢛⡍⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⢺⢇⡞⡽⢁⡞⠋⠀⢸⠟⣙⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡌⢸⢸⠣⠇⠈⢧⣀⣀⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⢸⢸⡄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⢀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢠⡿⠖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠞⠋⠩⠀⠐⠂⠂⠏⠀⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⠒⠈⠀⠀⠀⣈⡷
⠀⠀⠀⠀⠀⠀⠐⡒⠉⠀⠀⢀⡈⢁⡐⡤⠄⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠰⠶⠶⠤⡤⢀⣀⢀⣴⢾⠉⢠⣦⡁
⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⢰⠧⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡀⠀⠷⠖⠛⢲⡤⢎⣀⣁⠙⣀⣦⠎⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠐⠁⠂⠀⣀⠄⠑⠃⠀⠘⠩⠋⠁
⠀⠀⠀⠠⠀⠠⡦⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠠⠄⠠⠁⠈⠐⠀⠀⠀⠀⠀⠉⢉⣉⣉⣦⣤⣤⣧⠘⠢⣤⣤⡤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠠⠔⠃⠀⠠⠀⠉⣉⠀⢦⣠⢄
⠀⠀⠀⠀⠰⢄⣀⡀⠐⠀⠀⠀⠀⠀⠰⣟⣯⣍⣀⠘⠏⣙⡦⣄⠿⠏⠄⣏⠹⣷⠶⡆⠀⠀⠀⠛⣢⠄⣀⡀⢁⡀⠀⠐⠂⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠛⠛⠛⠻⠛⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⠈⣩⣍⣁⡷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⢀⣠⣈⡴⠋
⠀⠀⠀⠀⠀⠀⠁⣰⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⢤⠽⠛
⠀⠀⠀⠀⠀⡀⠀⡋⠀⠀⠀⠀⡂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠗⠀⠘⠀⠀⢬⠑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢢⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠐⠚⠀⣼⡋⠓⠢⠤⢦⣤⠤⠤⠶⠖⠋⠀⠀⠀⠀⠀⢸⠣⡀⠈⠀⠄⢠⡁⢐⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠂⠒⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠀⠀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡁⠚⠒⠛⠄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⣀⣀⣤⣄⣄⣀⣀⣀
⠀⠀⠀⢀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⢉⣉⡀⢀⣈⣉⣉⠛⠛⣉⣤⣤⠀⢩⣍⣙⠛⠶⣦⣄
⠀⠀⠀⣾⠃⢀⣀⡈⠉⠛⠛⠶⢶⣦⡴⠞⠋⠁⠀⢀⡖⠋⠉⠀⣴⠏⣉⠉⠉⣽⣛⣋⣴⠖⡲⠀⠀⢹⡇⠀⠀⠙⢿⣦⣀⣤⣴⠶⠶⠟⠛⣷⡄
⠀⠀⠀⢿⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⢠⣀⠀⠀⠀⠊⢀⣠⣶⠟⠀⣿⡁⠀⠀⣼⠀⠘⠃⠞⠇⠀⢀⡌⠳⢦⣴⣇⣀⠙⠏⣀⣤⣶⣿⡷⠂⣿⠁
⠀⠀⠀⠸⡇⠀⠀⢰⣿⣿⣿⣿⡏⠀⠠⣬⠳⣄⠀⠙⠛⣩⡿⠷⢿⡟⠃⠀⣈⣵⣀⣀⢀⣐⣶⣾⣋⣭⡙⣾⡷⣄⠸⡆⢀⡉⡻⠟⠉⠀⣸⡏
⠀⠀⠀⠀⢻⡀⠀⠀⣙⣿⡿⠁⠀⠀⣀⠩⣤⣌⠳⡝⣿⣏⣼⣿⣶⣿⡟⠛⣫⡽⠿⠭⠿⠶⠶⢿⡟⠻⠟⢛⣿⢈⡣⠶⣏⠤⠈⠃⠀⣴⡟
⠀⠀⠀⠀⠀⠙⣦⡀⢉⣿⠍⠉⢛⣒⡭⢤⣀⠙⢷⣝⣮⠻⠿⢿⣶⣤⠁⠘⢁⣠⣤⣤⣤⣤⣤⣤⣉⠃⠐⠫⢖⣩⣴⢺⡅⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⣸⠷⣾⣯⠥⢤⣘⣛⠃⠐⠾⣿⠶⢭⣝⡓⠀⠴⢤⡀⠀⠀⢻⣿⣭⣤⣤⣤⣤⣼⡟⠀⠀⠪⢭⣤⣖⣲⣶⢒⡂⢺⣷⡀
⠀⠀⠀⠀⠀⢠⠏⣞⠟⢻⡧⣤⣙⣛⣉⣻⠷⠒⠒⠦⢬⣽⣓⠂⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⠟⠉⠀⠀⠀⠀⢁⣉⣀⠡⢽⡭⠁⢺⣿⠃
⠀⠀⠀⠀⠀⣿⡈⠐⠀⣸⡧⢬⣭⣥⠤⢽⣯⠭⠭⢍⣛⣒⡦⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⠿⢶⣦⣤⣤⡀⠀⣐⠭⣈⠉⢹⣇⠀⠺⣿⡆
⠀⠀⠀⠀⢠⣯⠔⡢⡀⣸⣿⡀⢻⣿⡀⠘⣿⠀⠀⠄⠶⣖⣻⡅⢀⣤⣶⣋⣉⣁⣠⣤⣤⣤⣤⣀⡤⢍⣻⡾⠋⣑⣈⠁⢸⣇⠀⠙⣮⣷
⠀⠀⠀⠀⣼⠡⡪⢞⠁⣼⣿⡆⠸⣿⡇⠀⠻⣦⣄⠀⠉⢻⣿⣿⣿⣿⠀⠀⣿⣄⣧⣦⣦⣤⣤⣼⠀⠀⣿⠇⠀⣹⠃⠀⣿⡗⠀⠀⠩⣻⣧
⠀⠀⠀⠀⣿⢠⢖⡡⠀⣾⣿⠃⢐⣿⣿⣦⡀⠘⢧⠀⠀⠈⢿⣿⠯⣿⡄⠀⡟⡿⡟⡟⣿⢻⣿⢿⡀⢸⣧⣤⠾⠃⠀⢀⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢿⡶⣫⠖⢀⣿⡇⠀⠈⠸⣿⣿⣷⡀⠈⡇⠀⠀⠼⣿⣢⠙⣿⣄⣿⠉⠧⡇⠟⠈⠸⣸⣷⡏⣼⠁⠀⢀⣦⣿⠿⠃⠀⡀⠀⣜⣿⡇
⠀⠀⠀⠀⢸⡆⠁⠀⢠⣿⡆⣸⡟⡀⠉⢻⣿⡇⠀⢻⡆⠀⢸⣿⣿⣁⢻⣿⣿⠀⠀⠀⠀⠀⢠⢿⡏⣷⠿⢿⣿⣿⡏⠈⠀⠀⢱⣱⣦⣼⣿⠃
⠀⠀⠀⠀⠈⣿⣴⡄⡄⠻⡿⣿⣿⡅⠀⠘⠛⣗⠀⠈⢳⣄⠸⣿⣿⣯⣾⡇⠸⣦⣤⣤⣤⣴⠏⢸⣇⣿⠟⠀⠀⠈⠀⠀⠀⣠⣣⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣧⣷⢠⢠⠡⢻⣦⠀⠀⠀⠀⠀⠀⣈⠛⣶⣾⣿⣿⠏⢧⣀⣉⣥⣤⣭⣏⣀⣸⢻⣿⡆⠀⠀⠀⢠⣀⣰⡟⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣷⣤⣿⣬⢠⢰⠀⢠⢃⢅⣧⣉⢹⡵⠛⣯⣷⣶⠶⠶⣦⣤⣤⡤⣬⣠⢟⡀⢦⢦⢰⣸⢻⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣧⣦⣧⣮⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣼⢸⡞⡄⢾⣸⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠋⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢠⡀⣄⠀⠀⠀⠀⢀⣠⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣦⣼⣶⣴⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠻⠆⠀⣠⣤⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣶⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⢛⣩⣵⣶⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⣿⣿⠟⢉⣤⣾⣿⣿⣿⣿⣿⣿⠟⠉⢀⣄⠀⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⠋⣠⣿⣿⣿⣿⠟⣿⣿⣿⣁⠀⣀⣾⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⢰⣿⣿⣿⠟⠁⠀⠛⠿⠿⠿⠿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⣏⣿⣿⠀⣿⣿⣟⣥⣤⣄⠀⠀⠀⠀⠀⢰⡇⠹⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡙⢷⣬⣿⣿⣿⠀⣿⣿⣿⣿⣿⡟⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣿⣿⡿⢹⣿⠀⣿⠟⠋⣿⣿⣤⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣵⠏⠀⠀⠸⣿⡄⣿⠃⠀⠉⢉⠿⣏⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⠀⠀⠀⠛⠀⠙⠁
⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣴⣿⡿⠿⣿⣿⣦⠀⠀⢠⣿⣿⡟
⠀⠀⠀⢰⣿⠋⠀⠀⠈⠻⣿⣷⣶⣿⣿⡿⠁
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠉⠛⠿⠛⠋
⠀⡀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⠿⠷⠶⠿⠯⣟⡷⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡼⣯⢿⡿⠑⠄⠁⠀⠀⠀⠀⠉⠳⣶⠞⣦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠐⠁⣂⡄⠐⢤⠀⠀⠀⠀⠀⠈⢳⡚⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠁⡉⠀⠀⠀⠀⠀⠋⢻⣤⠜⠲⢤⠀⠀⠀⣠⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠁⠀⠀⠀⠀⠀⡆⣼⣶⡎⠁⢹⠃⠀⠀⠈⡠⡴⠚⣽⠧⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⡰⠁⠀⠀⠀⠀⠀⠈⠁⣾⣤⣿⠀⠘⠀⠀⡠⢊⣼⠁⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⣀⠀⠀⠀⢠⠀⠀⠀⠀⣤⠌⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡺⠇⠀⠞⣠⠞⠁⣬⠻⠶⢤⠶⢒⣲⠖⡜
⠀⠀⠀⠀⡢⠀⢀⣫⠦⢒⢶⣿⣿⣠⣤⡈⠀⠄⠀⠀⠀⠠⠔⣰⡇⢄⡀⠀⠀⠀⠀⠐⠓⠀⢻⣻⡦⣀⠞⠃⠀⠀⠀⠀⠀⣿⠀⠀⠉⣀
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠛⠃⠀⠚⢛⠿⣶⣄⣀⢠⠠⢀⠔⠋⠒⠉⠐⠚⠛⠃⠀⠀⠀⠀⠸⡿⠟⠁⣤⠄⠀⠀⠀⠀⢨⡎⠛⠒⣿⠋⠁⠵
⠀⠀⠀⠐⠀⠀⠠⠄⠰⡆⠀⠀⣠⠀⠦⣤⣶⠟⠛⢿⣤⢣⠂⡀⠀⠀⠀⠀⠀⠀⠀⡠⠌⢠⡄⢷⡀⠈⠁⠀⠀⣠⡦⠀⠀⠀⠀⣔⢻⣍⡀
⠀⠀⠀⠀⠀⢁⠀⢰⣠⣵⣄⣠⢀⣤⡾⠊⣠⣤⣤⡀⠛⢬⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠶⣄⢀⡀⠡⠀⢀⣤⠄⢈⢉⠁⠤⢉⡟⠓⠄
⠀⠀⢠⡐⠀⠢⣪⡿⢿⣿⣿⣥⠟⠉⠀⣼⠛⣠⠿⠷⠀⢢⠈⠁⠀⠀⠀⠀⠀⣴⠄⠂⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⠈⢁⠀⣈⡀⠠⢔⢚⣧⠀⡃
⠀⠀⠀⠑⢀⡞⠁⠞⠁⠒⣿⡟⠀⡀⠄⠑⢾⣟⣆⠰⢷⡀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠑⠲⣤⣉⠁⣀⠀⠀⡈⠑⢦⡃
⠀⠀⡀⠐⠀⠀⠐⢂⣀⡴⢀⡄⠀⢣⡢⣄⠀⣊⣥⠴⠾⠿⣦⣤⣦⡙⠩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠠⠉⡀⠝⠶⣤⠾⠗⠀⣠⣤⢽⡄
⠀⠘⠏⠀⠀⠐⢠⢾⣷⢃⠾⣡⠀⣸⣳⡿⣛⣛⡵⢶⣯⣠⣶⣿⡻⣿⣆⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡅⠐⠂⢠⡄⣗⢉⢧⡀⣽⠾⣽⡀⢿
⠀⠀⠄⠀⠐⠂⠈⢰⣷⣱⣼⣵⠄⡁⣿⣿⠟⡛⡷⣾⢿⡿⠋⣿⠿⢻⣿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠔⠁⠃⢙⡐⠀⠉⣚⢆⢹⡆⡀⠸⢷⡸⡄
⠀⠀⠀⠐⠐⠠⠒⢿⣿⠾⣿⠋⠀⣿⢫⡅⣼⠀⠹⠇⡘⠉⠉⠁⠋⠈⠂⠚⣛⣿⣷⣦⣄⣀⠀⠐⠚⣀⠀⢏⡴⠂⠀⠢⡈⠁⠀⢘⢨⢸⢍⣷⠟⠳⠓⣟
⠀⠀⠀⡀⠀⠀⣠⠘⣿⢄⡇⡀⠀⠙⠳⠺⠉⠣⡐⠀⣠⣶⠶⠾⣿⢿⣷⣶⣔⠀⠻⣶⣿⣿⣟⢶⣦⣤⣙⣦⡤⡼⣶⠶⠾⠧⢴⣭⣀⠉⠘⣿⣈⣷⡀⢸
⠀⠀⠀⠀⢀⠀⠴⢀⡘⢾⠇⠀⠀⠠⣄⢩⠂⠀⢠⣞⡿⣶⠟⢉⡛⢻⣾⣻⣷⢟⣦⡱⡼⣭⢹⣙⣍⣽⡷⠖⠋⠉⠉⠉⡁⢀⣄⣈⣹⡳⣦⣈⡛⡄⢣⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡂⢹⣇⢠⠀⣧⣤⠛⡃⠀⢰⣿⣿⢫⡆⣴⢻⢿⣷⣏⣟⣿⠆⠩⢷⠳⣾⠉⠀⠀⠀⠀⠀⠀⢠⡦⡎⠉⡁⠀⠁⠀⡅⢾⢳⣏⡑
⠀⠀⠀⠘⢄⢩⠄⢁⣸⡹⣄⠀⣪⢻⠷⢠⢈⢸⣿⡛⠈⠀⠻⡾⠶⣋⢾⣟⣿⠙⣰⠋⠻⢿⣷⣄⠀⠀⠀⢀⣼⣟⣿⢀⠑⠀⠀⢣⣀⠀⡹⣆⣏⢳⣇
⠀⠀⠀⢀⠀⠀⠀⠀⠛⠀⠀⢳⣿⣿⡷⣤⠀⠸⣿⣿⣶⣶⠷⠈⠉⡁⢹⣿⣏⣹⠏⠀⠀⠀⠈⠉⠛⣒⣀⣾⣿⣾⡿⠠⠁⠄⠀⠐⠀⠀⡞⣿⣹⢸⣇
⠀⠀⢀⠈⠈⠰⠂⠀⠀⢠⣶⠘⣋⣿⣿⣌⠀⠄⠹⣷⣦⣤⡤⠁⠐⣠⣬⢟⣤⠏⠀⠀⠀⠀⢀⣴⣾⡿⣽⣿⣿⡿⠉⠄⠀⢀⢀⠀⠀⡚⣇⣿⣪⣼⡏
⠀⠀⠈⠁⠀⠀⠀⠀⠀⡟⠛⠘⠛⢿⣿⣧⡝⠂⡀⠙⢿⣯⣤⣴⡷⢛⡥⠚⠁⠠⠀⠀⠐⣴⣿⣿⡳⣻⢧⣿⠟⠁⠨⠘⠌⠌⠀⡤⢻⢁⣾⢯⢇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡜⠃⣀⡂⠙⢿⣿⣱⠖⡀⠀⠙⣻⢟⣴⠻⠁⠂⠀⠀⠀⢀⣼⣿⣳⣯⣾⡿⠟⣁⠠⠨⠠⢀⢂⡀⢛⠟⣲⣿⡿⠋⡤⠚⡸
⠀⠀⠀⠀⠀⠀⠔⠂⠀⠘⢣⣎⢅⠐⠙⠀⢹⣿⣯⡘⣉⡴⣣⠋⠁⠀⠀⠄⠀⢠⡶⢿⡿⡟⡛⠋⠉⠄⡀⠈⠈⡤⣤⢌⣉⠓⣯⣾⣿⡿⠅⣠⢴⡀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣆⢄⠸⣧⠤⢀⡶⣠⣾⣿⣿⣿⠞⠜⠁⠀⠀⠀⣀⣴⡟⢉⣁⡀⠀⣀⠀⣄⣴⣴⠈⣺⠛⠲⠚⣨⣥⣾⠟⢡⠉⣦⠔⣁⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠎⠳⣙⠷⣶⣾⣵⡯⠾⠛⠁⠀⡀⠀⣠⣴⣾⣇⣀⣛⣋⢉⢳⠔⠋⠷⢞⢃⣈⣛⣥⣴⣾⠿⢋⣻⠄⣤⣈⣵⡼⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣼⡓⢬⡑⠒⠂⠲⠛⣄⡚⠉⣀⠿⢿⠿⢿⠿⣿⣿⣿⠟⠻⠟⣛⣿⣿⡿⠟⣉⡀⠰⠆⠈⠉⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣀⣤⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡏⣹⠟⢁⣼⣶⠶⢾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠽⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⠿⠷⠒⡿⠋⠙⢳⣎⠉⠛⣯⣀⡀⠀⠀⠁⠈⠀⠄⠀⢆⣀⣀⠤⣄⣒⣤⣷⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡴⠾⢁⣠⣃⠉⠻⣷⡄⠀⣫⡤⢀⡼⠁⠀⣀⠀⠊⢩⣴⣾⣿⣿⡿⠿⠁⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡿⣿⠟⣿⣿⡀⣸⠀⠀⢸⣷⡁⢂⠔⠁⢀⣶⣿⣿⡿⠛⠋⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢆⣀⡠⢄⣤⠟⠋⠁⣶⡿⠉⢀⠃⠀⠀⣽⠟⡥⠏⣠⣾⣿⣿⣿⣟⣈⣀⣀⣀⣀⣀⠘⠿⢤⡀
⠀⠀⡿⠐⣀⡟⣩⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣹⡭⠿⠋⠀⠀⢈⣿⣿⣦⢸⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠈⠈⠉⡿⠉⠙
⠀⠀⠉⣓⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠉⠈⠁⠀⠀⠀⢾⣾⣿⣟⠙⡟⢻⣿⣿⣿⣿⡤⢀⡀⠀⠀⠀⠀⠀⡌
⠀⣗⣘⠚⠆⠓⠦⣄⠀⠀⠀⠀⠀⣴⠗⠀⠀⠀⠀⠀⢀⣀⡴⠚⠛⠛⠻⠟⣉⣵⣾⣿⣿⡤⢅⡀⠀⠀⠈⠓⠦⣄⣐⣇
⠀⠈⠉⠉⡋⢺⠳⡈⠑⢤⣄⡔⠉⠀⠀⠀⠀⢠⠖⠋⣉⢛⣰⣾⣷⡿⢿⣿⣿⡿⣿⣶⠀⠁⠀⠈⠢⡀⠀⢠⠆⠉⠉⠉⠃
⠀⠀⠀⢀⡷⠜⠀⠈⢶⣴⠋⠀⠀⠀⠀⠀⠔⢣⡾⠟⠉⠀⠁⠀⠘⡁⠈⠀⠀⠁⠀⠈⠂⠠⡀⠀⠀⢁⢐⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣸⣿⣄⠀⠀⠀⠀⢀⠾⠋⢀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⢀⣸⣷⡿⠍⠓⠂⠀⠀⠀⠀⠀⢿⡯⢟⣿⢿⣿⣷⣶⣤⣶⡀⣀⣤⣤⣴⣂⡲⢜⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣴⣾⡆⠀⠀⠀⠀⠀⠀⠛⠀⠀⡈⠀⠀⡈⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣾⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣾⣧⠀⢰⡀⠀⠀⠀⠀⢠⣦⣽⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣵⢿⣇⡧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣾⡉⢻⣟⡿⢿⣣⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡸⠉⠙⠛⠷⣜⡓⣄⣹⠏⠛⠛⠛⠋⠁⠀⠈⠀⠙⣿⡋⢹⡈⢸⡏⡏⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠄⠰⠦⢤⣄⣣⠸⢿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣏⢨⡇⠐⣧⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢀⣀⣀⠀⠀⠀⠉⢆⢈⣦⡈⠁⣀⣀⠀⠈⠉⠐⣄⣸⣉⣩⠄⡀⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⣞⡽⠛⠚⢲⠦⢀⠀⠈⡄⢿⠟⢀⣭⣟⠉⠛⣷⡀⠀⢿⠄⠁⣠⢣⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠀⠀⠀⠛⠒⣦⡥⠟⠀⣴⣾⠋⠁⠀⠉⠉⠛⠳⡆⠘⣷⣾⢣⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢱⠄⠀⠀⢀⣼⣦⣴⠏⢰⣿⣷⣤⠄⢀⠀⣀⠀⢠⠇⢸⡿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠉⠡⢿⣿⣿⣿⡾⠟⠀⢨⡟⠛⠿⠿⢵⠶⢋⡴⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠚⠋⢁⠤⠀⢄⣚⡷⠶⠶⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠴⠷⢶⣾⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣀⣤⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠟⠑⠀⠀⠂⠋⠀⡗⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⡀⢤⣳⣿⠏⢐⣂⣀⢀⡀⣈⢮⣯⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⢹⣿⣿⢷⣯⣭⣷⣿⡿⠏⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠄
⠀⠀⠀⣠⡴⢉⠔⣿⣽⣯⡆⠈⠻⣿⣿⣷⣾⠿⠃⠀⠠⣌⣠⡼⠿⣿⠿⢿⣿⡀⣬⣯⣿⣿⣿⠛⠁⠀⠀⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⡾⠋⡔⠁⢸⡷⣿⠁⠀⠀⠀⠀⠈⠀⠀⣰⣴⣿⡿⢿⣿⣧⠀⠑⣤⡶⣿⣷⣿⣆⠚⣿⠋⠀⠀⢀⠞⣿⣿⣿⡟⠛⠋⢀⣠⣤⣿⢇⣼⣿⣿⠏
⠀⠀⣿⣔⣆⢰⣟⣿⡟⠀⠀⠀⠀⣀⣠⣴⢿⣿⣿⣀⣀⣀⣈⡁⠃⠠⠘⣤⡿⠋⣿⣷⣿⠛⠃⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⣈⣁⣤⣷⣿⣿⠿⠋
⠀⠀⠫⢽⣛⣾⠿⠋⠀⣠⣤⠴⣿⣫⣽⡿⣿⣿⣽⣯⣭⣭⣭⠙⣶⣿⠟⠋⠀⠀⠙⠿⠿⡾⠶⠦⠀⠀⣀⠴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡶⣟⣤⣴⣿⣿⣷⣾⠿⠿⠻⠿⠿⣿⡏⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢊⣁⣤⠰⣶⣦⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⣹⣿⣿⣿⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣴⢦⡤⠀⠀⠀⠀⢀⡠⠔⠋⢀⣴⡆⣵⣼⣿⣸⣷⣿⣿⣿⣿⣿⣶⣦⣀⣀⣀⣠⣤⣤
⠀⠀⠀⢀⣾⣽⣾⣱⠾⠋⠀⠀⣀⣤⣠⣴⣤⠀⠀⠀⠀⠀⠈⠿⠉⣿⡆⠀⠀⠀⠊⠁⠀⢀⣴⡞⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⢀⠏⣼⣿⠟⠁⠀⠀⣠⡸⣿⣿⠞⣻⠇⣾⣿⠦⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠛⠻⠒⠋⢻⣟⡷⢿⣯⡿⢿⡿⠋⠉⠀⠈⠁
⠀⠀⣼⣼⣷⠏⠀⠀⠀⢰⣿⣷⣶⢆⠁⠛⢼⡾⠁⣿⡿⢦⣤⢤⡖⠿⣧⣿⣗⡀⠀⠀⠀⠀⠀⠘⠉⠉⠛⠛⠛⡿⠇
⠀⠀⣿⣿⡿⠀⠀⠀⠀⡿⠙⢟⡁⠦⠀⠀⠀⠀⠿⠿⣃⣻⣿⣜⣛⠳⠒⣌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠄⢠⡀
⠀⠀⢻⣿⣷⠀⠀⠀⠀⠡⣿⣿⣿⢳⣆⡀⠀⠀⠀⠀⣹⣿⣧⣿⡇⣦⣤⣵⣿⣀⢀⡀⠀⠀⠀⠀⢸⢃⠀⠀⡀⠀⠀⠀⠀⠀⡀⠦⣅⡞⡁⠀⢀
⠀⠀⠈⢧⣻⣆⠀⠀⠀⠘⠷⠈⣰⣿⣿⣝⡶⢤⣶⡸⣿⣿⠈⠿⠘⠋⣵⣿⢻⡶⠟⢍⡄⠀⠐⠂⠿⠛⢄⣁⠑⢘⠓⢀⠂⣼⣹⣶⠿⠆⠙⠁
⠀⠀⠀⠀⠻⢿⣦⣄⠀⠀⠀⠀⠁⠀⠫⡿⠋⠸⣿⣃⡘⠋⠁⠀⣀⢠⣶⠍⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⠋⠁⠈⠀⢠⠕⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⡀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢠⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡞⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣴⠒⣄⠀⠀⠀⠀⠀⣠⣤⣤⣤⣶⠶⡶⢖⠒⠤⣤⣄⡀
⠀⠀⠀⠀⣀⣠⣖⣉⠉⠈⠆⣠⣤⢶⡟⠀⠀⢻⣵⢀⡂⡆⠀⠀⠀⠀⠩⠉⠙⢶⡤⣀
⠀⠀⢠⡞⣽⣿⡕⢆⠉⠀⠀⢿⣷⡾⠀⠀⠀⢸⠣⠈⢠⠁⠀⠀⠀⠀⠀⣀⣄⡎⠀⠀⠉⣦⣄
⠀⠀⣾⡁⠧⣍⣠⠎⠀⠀⠀⠸⣿⡽⣄⠤⠤⢚⣚⠓⡞⠀⠀⠀⠀⠀⣾⠇⡄⠉⠑⠺⢳⣾⢩⠷⢄
⠀⢠⠤⠉⠀⠀⢀⡀⣄⣀⠀⠀⢹⣷⣤⡤⠞⣿⢃⡸⡃⡁⠀⠀⠀⡀⠺⠀⣽⠙⠲⣴⣾⣧⡇⠀⠀⠱⡄
⠀⠸⣿⠿⠿⠛⠛⠉⠁⠈⠁⢀⣵⣯⠏⠀⡜⢡⡞⠠⢀⠁⠀⠀⠀⠀⠘⡇⢻⠀⠀⠛⠉⠉⠉⠳⣄⢀⢹⡄
⠀⠀⠀⠉⠛⠒⠲⠖⠒⠚⠛⠛⠉⡉⣒⣾⢀⣞⣤⣤⣌⠀⢀⢀⢄⣞⠭⡇⢸⠁⠀⠀⠀⠀⠀⠀⠈⢿⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣖⣠⡥⠋⠀⠀⠀⠈⠉⠁⠀⢀⠒⠆⠾⢸⠀⠀⠀⢀⣤⡤⣦⢤⠘⣇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⢀⣏⡟⣩⡍⠳⠇⣿⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢾⡷⣫⡜⣷⠏⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣟⣛⣿⡿⠒

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⡤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⣤⠶⠚⠀⠀⠀⠀⠀⠀⠀⢸⣁⣾⣿⡿
⠀⠀⠀⠀⠀⠀⢎⣶⣶⡶⠶⠶⠶⠶⢦⣤⣤⣀⣀⠀⠀⠀⢄⡀⠀⠀⠒⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⣀⡠⠔⠋⠁⢀⣤⠟⢋⠓⠲⠤⣌⡓⢤⡀⠀⠀⠤⣈⠐⢄⠀⠀⠀⠀⡜⢠⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⠀⡀⠀⠤⠐⣪⠅⠀⠈⠙⠣⠀⡀⠀⠉⠓⢿⣦⡀⠀⠈⢣⠀⠃⠀⠀⡼⢁⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡤⣀⠀⠀⢱⢿⡿⠁⠀⠀⠃⢇⡀⠀⠐⠀⢈⢻⣆⠀⠈⠀⠆⠀⡼⢁⡾⡍
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⠿⠿⠀⠀⠁⠀⠀⠀⡴⠃⣌⣷⡆⠀⡀⠁⠘⡿⡘⡔⡀⠀⡼⢁⣾⡝
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣾⣴⡄⠀⡠⠄⠀⠀⠀⠀⠰⠃⣸⠇⠀⢈⠄⠈⣷⣯⣾⣿⣿⣷⣾⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣿⡿⠀⢰⣶⣶⠂⠀⠀⢠⣾⡧⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣭⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡷⠐⣠⡯⠀⡀⠤⠞⠛⠟⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠎⠀⠈⠑⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠀⠚⣻⣧⠀⠀⠀⠠⠃⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⣠⣴⣿⣿⣶⣦⠖⠃⠀⠀⣰⣾⡿⠛⢹⡿⠿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢤⣾⣿⣿⣿⣿⡟⠀⠀⢀⣼⠛⠋⠀⠀⠄⠠⠄⠈⢻⣿⣿⣿⣷⣤⠀⠀⠀⢠⢤⢶⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⣹⣿⣿⣿⡿⣡⢀⣶⣿⠃⠀⠀⠀⠀⠠⢄⡀⠀⠀⠁⣿⣿⡇⠀⠀⠀⢠⣼⣿⣼⠀⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⠏⠌⣩⣿⣿⣿⠀⠀⠀⣀⠀⠀⠀⠸⠃⠀⡄⣿⣿⡇⠀⠀⠀⠀⢹⣧⣤⠀⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣋⡔⠀⢸⣿⣿⣿⡉⢸⠀⠛⠤⢀⡀⠀⠘⠄⢰⣸⡿⠙⠃⠀⠀⠀⠐⡘⣿⣿⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠌⠘⢠⣿⣿⣿⣿⣷⡎⠀⣠⠄⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⢸⡇⠛⣹⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⡀⣠⣿⣿⣿⣿⣿⣿⡇⠈⣀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠙⣛⣫⣤⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠉⠁⠒⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⡭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣅⠽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠐⠂⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠈⠳⠙⠲⡔⠒⠛⠛⠛⠓⠲⠶⠒⠂⠈⠉⠉⠁⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⡷
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢒⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢰⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠐⠮⣄⣀⣀⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠙⢾⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣾⠀⣉⠀⠀⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⡏⠛⣿⣧⡀⠈⠂⠶⠶⣦⣌⡉⠉⠉⠘⠛⠉⢻⠀⣼⣟⠷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠣⣼⣿⣿⣿⣶⣤⠶⡚⠅⠀⠉⠓⠀⠀⠀⠀⣻⡿⠿⢿⡤⢰⡀⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣣⡴⠀⠀⠀⠀⠀⡰⠁⠀⠠⠖⠛⠛⠓⢦⣄⡛⢶⣤⠄⣉⣀⡀⠉⠀⠀⣠⠖⠓⠢⢄⢀⠀⠠⠣⣄⢠⠚⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⢁⡞⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠁⠐⠋⠐⠒⠀⠼⡆⠴⠤⡀⠀⠀⠀⢀⡜⠁⠀⠈⠑⠀⣁
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢢⡀⠀⣠⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢠⡤⠀⠀⠀⠀⠀⠀⢀⠀⠉⣤⡄
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡉⠂⢿⡀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⠂⢸
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠿⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄⡀⠀⠀⠀⠀⣀⣠⠜⠀⢸
⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣰⠃⢰⣿⣿⣿⣿⣷⣦⣀⣭⠙⣿⠻⠋⠙⠿⠛⠯⠉⢯⠭⠭⠓⢿⡟⠙⠋⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢿⠀⢸⣿⡟⠋⠛⢿⣿⣿⣿⣿⣿⠂⠀⣰⠷⣄⠤⠤⠤⠤⣆⠴⡟⠁⠀⠀⢀⡾⢦
⠀⠀⠀⠀⠀⠀⠀⢧⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⣿⢧⡀⠀⠀⠉⠙⢿⣿⣯⣴⣿⠁⠀⠀⡤⠀⠀⠀⠀⣠⠃⢀⣄⡠⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⡆⠘⢻⡄⠱⠾⡤⠲⢀⠀⣿⣿⣿⣿⣀⣀⠢⠀⠀⠀⡠⠀⠄⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠠⢄⠀⢹⡄⠈⢷⣤⣄⣀⣰⣯⠤⠄⠴⠤⠦⣤⡏⣠⠀⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠺⣅⠀⠀⠉⠉⠉⠀⠤⠄⠀⠀⠀⠀⠀⠈⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠞⣧⠚⡧⠊⠑⠸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡾⠁⠀⠠⠀⠀⠀⠀⢉⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⣶⡟⠀⠀⡂⠀⣀⣀⣀⣀⡀⢤⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿⠆⠛⠀⠀⢁⣴⣭⠀⠖⢨⡝⣟⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣾⠇⠀⠀⠀⠀⣈⡼⣆⠀⠀⣹⡇⠈⣦⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣿⡅⠀⠀⣠⡞⠁⠀⠸⣦⢀⣿⠿⣿⣿⣿⣿⣿⡏⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠖⣿⡧⣴⣥⡀⠁⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⣧⡶⠖⣀⠒⠒⠂⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡣⢻⣿⠁⠈⢻⣷⣼⣶⡂⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢩⣽⣷⡆⣴⣿⣿⣿⣿⠉⠙⠒⠒⠛⣻⠞⢡⠿⢿⡇⣿⣿⣿⣿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⡿⣿⡿⡀⠀⠀⠀⡼⠁⣰⡇⢀⣼⣿⣿⠿⡿⠛⣙⡏⢿⣷⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠉⠀⢿⣽⣷⡈⢀⡶⠂⣀⣾⣿⣿⡟⢻⣿⢿⡿⣧⣀⣯⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⣀⣀⠀⠀⠀⠀⠁⡅⢷⣿⣿⣋⡤⠚⡿⣿⣿⣿⡇⢸⣿⣶⣿⣿⣿⣏⣿⣿⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⣀⠘⢿⡄⠀⠀⠀⠀⣻⣿⣿⣏⣀⠀⣿⣛⣿⣁⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⠂⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣷⡀⠃⠀⠀⡠⢊⣽⣿⠧⠀⠀⠀⠉⡉⣉⣠⣴⠿⣻⣽⣿⣿⣿⣿⣿⠏⡀⢀⠂⢸⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⠇⠀⠀⡠⢡⡞⢻⣿⢶⣶⣤⡀⠀⠀⠀⢀⣀⣙⣻⣿⣿⣍⠻⣿⡏⠀⡀⣡⢀⡼⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠀⠋⠄⠀⠨⠀⠁⠀⢸⠁⠀⢻⣿⢃⠀⠀⠉⠀⠉⠹⣿⣿⣿⡌⠀⢨⠀⢊⣴⢣⣿⡃⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢲⣀⣀⠀⠀⠀⠀⠈⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⢀⣴⠋⠉⠡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⣨⢷⣾⣮⡠⠀⢀⣼⣿⡿⠉⠁⠀⡈⠳⢦⣤⣀⣹⣿⣿⣆⣠⣾⣿⠿⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢃⡀⠀⠀⠛⠻⣿⣦⣾⣿⠏⢀⡀⢀⡠⠈⠀⠐⠀⢭⣭⣿⣿⣯⠿⢿⡯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣀⣿⣿⡯⠤⣄⠉⠁⠀⠀⠤⣀⣀⣀⣿⣿⠏⠑⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠛⠒⢦⠀⢀⡞⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⢶⣄⢀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣿⠉⢻⣇⢸⣿⡟⢙⣿⠀⠀⣾⣿⣟⡛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠸⣧⣄⠉⣉⣙⣛⣿⣿⣦⣏⣴⣿⣯⣤⣤⣾⠿⠟⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣾⡟⠋⠉⠉⠛⠷⣶⣄⣤⣶⡿⡟⠋⠩⠭⠭⡉⠫⠍⠍⠁⠀⣀⠀⠀⣀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⡄⠉⣉⠉⠉⠛⠳⠶⣷⣦⣠⣾⠟⠉⡠⠞⠀⠀⠐⠒⠊⠀⣔⡢⠀⠀⣸⣶⢿⡿⠉⣹⡷⠿⢿⣦
⠀⠀⠀⠀⠀⠀⢀⣴⠾⢛⣿⠿⠃⠀⠀⣒⠉⠀⢘⣴⠶⠶⠶⢤⣀⠠⡉⠁⠀⠀⠀⠀⡄⠒⡆⠑⠃⠀⠀⠀⠀⠀⢸⣯⣀⣼⡇⡴⢋⣠⣤⣾⡏
⠀⠀⠀⠀⠀⠀⣾⢃⣾⠿⠓⠁⡀⢀⠈⠒⠁⢠⡟⠁⡔⣆⠀⠀⠙⣆⠹⡄⠀⠐⠵⠀⠑⠈⠀⠀⠀⠀⠀⢀⣦⡀⠀⠙⢛⣉⡀⣼⣟⡋⠉⠉⣻⡇
⠀⠀⠀⠀⠀⢀⣿⠞⠁⢀⣀⠈⠄⠌⠀⠀⠀⢸⡀⠀⢃⢿⡇⡀⠀⣿⠀⠃⢀⠀⢐⣁⠆⠀⠀⠐⠒⠛⠻⣿⡟⠿⠾⠟⠛⠛⢿⡌⠻⣿⡿⠿⠛⠁
⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠙⠀⠀⢠⠒⡄⠀⠈⢷⣀⠀⠂⠕⢀⣴⠃⠀⠀⢸⡆⠀⣀⣠⣤⣤⣤⡄⠀⢰⡿⠁⠀⠀⠀⠀⠀⠸⣷⣀⣸⡷
⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠛⠓⠛⠋⠀⠀⠀⠀⣾⡿⠛⠉⠉⠁⢠⣿⠁⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⢰⣿⠀⢀⣴⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣸⡟⣀⣀⣀⠀⣀⣾⠇⠀⣾⠏
⠀⠀⠸⣿⡀⠀⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⣰⣿⠟⠛⠛⠛⢛⣛⡉⠀⠀⢿⣦
⠀⠀⠀⠙⢿⣦⣀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣞⣡⡴⠟⢀⣠⣾⠟⠻⠷⣶⠶⠿⢛⣿⠇⣤⠰⣦⡙⢷⣤⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣶⣶⣤⣴⣶⣶⠿⠟⠻⠿⣷⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⣴⡿⠋⢸⣿⣧⢻⣿⡆⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣿⠏⣿⠀⢻⣿⣦⣀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡾⠋⠀⣿⣄⣸⣿⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣶⣶⣶⠶⣶⢖⠚⠂
⠀⠀⠀⠀⠀⠀⣈⠀⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⠏⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠁⡠⠲⡇⣀⠦⠄⠀⠀⠀⠀⠀⡤⣶⣒⠤⣄⣀⢀⣀⡤⢒⣿⡷⠿⠲⠦⣀
⠀⠀⠀⠀⠀⢠⣴⠀⠀⢀⢙⠀⠀⠀⠀⠀⠀⡨⠺⠋⠿⠻⠶⣯⣽⠁⣴⣧⠧⠤⣤⣤⡀⠈⢣
⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣾⡁⢰⠂⠀⠈⢿⣌⢄⠀⢀⠀⠐⠦⣵⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣿⠿⠇⠀⠀⠀⠀⠈⠻⢧⡉⢺⣄⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠢⠒⠀⠀⠀⠀⠀⠀⡷⠄⠙⠷⠶⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢇⢦⠀⡇⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⢲⠤⠀⠀⠀⡈⢂⠀⠀⠀⠀⢠⣟⠿⡿⡿⢿⡿⣿⣦⠀⠻⣿⢷⣦⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⣆⣦⣤⣾⠇⠀⠀⠀⡄⠀⣾⣿⣿⣿⣿⠿⣿⣾⣿⠄⠀⠸⡆⠹⣧⠀⠀⠀⠟⠻⡧⠀⠀⠀⢀⣤⣤⣤⣤⣄⠒⠁
⠀⠀⢐⣿⡿⠛⠁⠀⢀⠀⠀⠃⠀⠸⣿⣿⣿⣯⣶⣿⣿⡿⠀⠀⠀⢳⡈⣻⡧⡀⢼⠀⢶⠀⠀⠀⣠⣿⣏⣉⣭⠏⠉⠁
⠀⠀⠘⣿⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠙⢥⠔⣼⡿⠀⡴⡴⠖⠾⡏⠀⢈⡵⠋
⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣄⠉⠁⠒⠸⠦⢮⣉⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⢀⢀⣀⡀⠀⠀⠀⢴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⣀⣀⣀⠍⠑⠲⠶⠤⠂
⠀⠀⠀⠀⠠⠶⣿⡷⢰⡿⠻⢫⣞⠁⠀⢹⣿⠁⠢⠀⠈⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋
⠀⠀⠀⠁⢑⡆⣸⠁⣾⠃⠂⣿⣿⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢴⢶⣦⣄⡀⡀
⠀⠀⠀⣰⣾⠒⣥⣾⡿⠟⣸⣿⠏⠀⠀⠘⠂⠈⠂⠀⠀⠀⠀⠀⠀⣟⠛⠻⣟⠛⠛⠫⠰⡄⠀⠀⠀⠀⣄⣀
⠀⠀⠀⡟⠁⠀⢀⣀⣤⣴⣿⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠄⠈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⢄⡀
⠀⠀⠀⡇⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠦⠄⠀⠴⠀⠀⠰⡏⣷⡘⣷⡆
⠀⠀⠀⠱⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡉⠫⠠⠐
⠀⠀⠀⠀⠐⢀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡏⠀⠐
⠀⠀⠀⠀⢀⡸⣀⠀⣍⡓⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⢰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠿⠂⠀⠀⡀⠀⣄⠀⠀⠀⠀⡰⠇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣷⣶
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣿⡿⠟⢇⡀⠀⢠⡎⠀⠀⢙⣶⣤⣶⣶⣷⣶⣷⡆⠀⠀⢰⣶⣿⣿⣿⣿⣿⡶⠿⠂
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⢽⠋⠀⠀⠀⠀⠀⠀⠙⠒⠛⢛⣹⠟⠛⠋⠉⣦⠁⠀⠀⠀⢨⡿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⢀⣴⣾⣿⣿⣷⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠞⠛⠁
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣷⣽⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣰⣴⠖⢶⡄⠀⠀⠀⠀⠉⠉⣿⠋⠉⠀⠀⠀⠀⠀⠘⠿⣶⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡤
⠀⠀⠀⠀⢸⣿⡶⠴⠗⠀⠀⣾⠋⢙⣻⣿⣶⣦⣄⠀⠀⠀⠀⠚⠃⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢾⠉⠁⠀⠀⣿⠉⠛⠛⠛⠛⠒
⠀⠀⠐⠛⠋⢻⣿⠀⠀⠀⠀⠙⠻⠛⣋⡀⠀⠈⢻⣧⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⣄⣼⣿⠀⠀⠀⠀⠀⠀⢸⡏⢻⣆⠀⣸⡿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠰⠇⣸⣿⠷⢦⠀⠀⠀⣠⣤⠾⠇⠈⠛⠿⠛⠁⠀⠀⠀⠛⠛⠛⠉⠀⠻⣷⣦⣤⣤⡄⠀⠀⢠⣤⣤⣄⡀⢀⣲⣤⣾⡇
⠀⠀⠀⠀⠿⠋⠀⠾⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⡀⠀⠀⠀⠙⠻⠿⣿⠿⠿⠿⠛⠙⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⢀⠤⠂⠀⠀⠀⠀⡄⠀⠀⣀⣄⠀⢰⣿⡿⠋⠉⡛⣿⢿⡄⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠼⠶⠚⠉⠀⢠⣾⣦⡀
⠀⠀⡏⠀⠀⢀⣦⣤⣠⣽⣿⣿⠿⠋⠀⢸⣿⣇⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠳⣄⣀⣀⣴⣿⣿⡿⠟⠛⠲⠀⠀⠈⠿⣯⣟⣻⣽⢟⡿⠁⠀⣤⣄⠀⠰⡆⠀⣠⣭⣽⣽⣿⣷⣤⠀⠀⠀⠀⢿⣧⣶⣤⣄⡀
⠀⠀⠀⠈⠉⠉⠉⠹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣰⠏⠀⠀⡀⠘⣿⣷⡞⠁⠀⠉⠀⠀⠸⣶⣶⣷⣀⠀⠀⢰⠟⠛⣹⡟⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠛⠒⠛⢿⣿⣆⠀⠀⠀⠀⢰⡾⣿⣿⡏⣧⠀⠹⣦⣾⣭⣴⣦⣾⠇
⠀⠀⠀⠀⠸⣄⣤⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣻⣿⡏⢷⡄⠀⠈⠁⢻⣿⣧⠉⠀⠀⠉⠛⠿⠛⠉
⠀⠀⠀⢰⡾⢛⣻⣿⣿⠿⠿⢿⣦⡀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣄⣠⣤⠀⠀⢹⡆⠙⠶⢤⣀⡀⠀⠘⣿⡧⠀⢀⣤⣴⣤⣀⠀⢀⣠⣀⡀
⠀⠀⠀⣘⣻⣿⣿⡍⢿⣧⠀⠀⣿⡧⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠈⠀⢠⣿⣿⣿⣿⣿⣷⣾⠿⠟⠃
⠀⠀⠈⠿⠟⠉⠁⠀⠈⠻⠿⠿⠟⠁⠀⠀⠸⠏⠉⢹⣿⠋⠉⠁⢻⣷⠀⠀⠀⠀⠀⠀⣀⣀⠔⠁⠀⠀⠀⠘⣿⡿⠟⢛⣿⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⣤⣤⣶⠀⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⡴⠶⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⢠⢿⠎⠉⠁⠀⠉⠛⣶⡶⠠⣶⠶⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠰⡖⣦⢀⠀⠀⠀⠀⠛⣁⢀⣀⣀⣀⠀⣠⣄⣀
⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠁⠾⠿⠃⠘⠿⠿⠟⠈⠛⠛⠃⠀⠉⠉⠰⣿
⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠄⣤⡄⣀⡀⣀⡀⢀⣀⢀⣀⢰⠖
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠈⠁⠠⠀⠀⠀⣴⠀⠀⠀⠀⠀⠻⠀⠤⠤⠀⠒⠀⠀⠉⠁⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠕
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠐⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠈⠀⠀⠈⠀⠠⣮⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠗⣰⠃
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣀⡀⠠⠀⠀⠀⢠⡂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣽⠇⠀⠂⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢀⣿⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠂⠀⠁⠀⣉⠁⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠛⠣⠀⢻⣛⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣠⡄⠠⠤⠶⠷⠀⠉⠛⠿⣿⣟⣋⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠙⠀⣿⡙⣫⡍⢩⡍⠹⠉⣟⢹⠟⣭⡟⡍⢹⠉⣿⣿⢹⣶⡏⠍⢻⠟⢭⡟⡉⢹⡍⣩⡏⣿⢹⡏⡍⣻⠻⢽⠛⢽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣼⣴⣧⣽⣧⣿⣧⣼⣧⣼⣤⣿⣼⣦⣹⣧⣦⣼⣤⣽⣷⣘⣻⣧⣷⣼⣤⣿⣥⣦⣼⣧⣽⣧⣭⣼⣧⣧⣽⣌⣿⣝⣼⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⡴⠶⡄
⠀⠀⠀⠀⠀⣰⣗⡈⠁⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉
⠀⠀⠀⠀⢀⣭⣌⣁⠀⠀⠀⠀⠀⠲⠖⠀⢰⣆⣸⣇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠘⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠻⣏⠋⣀⣀⠀⠀⠀⠀⣀⠀⠠⣿⣿⣿⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠀⡹
⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠉⠁⠀⠀⠉⢻⣿⡟⢿⣿⡿⣆⣀⡤⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⡿⠛⠡⡏⢻⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠐⠿⠟

⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⢰⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⠶⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣸⣧⣠⣸⣿⣦⣀⠀⠀⠀⠀⠀⢈⣻⣿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⡿⠈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⣿⢿⠉⣽⢿⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣤⠿⠃
⠀⠀⠀⠀⠀⠸⠃⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠘⠚⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠃

⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠠⠾⠿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠛⣆
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠈⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣀⠋
⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⠙⢠⣴⣷⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡠
⠀⠀⠀⠀⠀⢰⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠻⡟⠛⢿⣿⡈⠇⠀⠀⠀⠀⠀⢾⠛⡿⠛⢻⡿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⡆
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠌⠀⠅⠀⠀⠀⠀⠀⠀⠈⠀⣰⠀⠠⠁⡸⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣴⠶⠤⠤⣤⣄⡀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⢀⡀⣹⣆⠰⡶⠶⢶⡿⢿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⡧⠸⣿⣿⡿⢃⣠⠾⠿⠛⣷⣶⣍⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠋⢰⣟⠁⢠⡞⠋⣡⡾⣶⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⡿⠙⣿⡿⠛⢉⣤⡿⢿⣟⠿⠷⡦⠙⢰⣿⣿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣁⠀⠹⣏⠀⣾⠋⣠⣿⡛⣿⣍⡽⣦⣀⣿⣿⣯⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢟⣿⣿⠟⠀⣴⡟⠁⢸⠇⣰⠿⡟⢧⣙⣻⣷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⡏⠀⣼⣿⡿⠁⠈⢠⣟⠀⠹⣆⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠁⢹⣿⣷⡏⠀⠘⠛⠛⠛⣿⣶⠋⠩⠭⣉⠛⠻⣯⣝⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡀⡈⠟⣿⣇⣶⢀⣠⠄⠀⠹⣿⣦⡀⠵⠒⡨⠀⢈⡿⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣿⣷⣷⡀⠙⣿⣿⣿⣿⣴⣦⣤⠌⠻⣿⣿⣾⠿⠿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⠛⢦⣌⠛⢿⣿⣿⡟⠁⠀⠀⢠⣀⣀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠓⢄⠙⢿⡇⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⠈⠀⠉⠁⠈⢉⠙⢀⠢⠐⢤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡲⢠⡍⠹⠚⣿⣿⠰⠄⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠶⠁⣠⠈⠲⡈⡳⣉⠟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠧⠀⠀⠀⠀⠉⣁⠄⢉⣄⣐⠴⠰⠡⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣠⡄⠀⠀⠀⣣⢉⡆⢹⡭⣗⠒⠀⠀⠀⠀⢀⢀⠀⠀⠸⠂⠦⠀⣤⣤⣶⣶⠿⠋⠀⡈⠀⠠⠁⠀⠄⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣅⢀⡚⣤⡅⢀⡠⢀⣀⣏⠞⣰⠯⢡⡏⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⢣⡀⠀⠉⠉⠁⠀⠀⠀⠘⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠾⢡⠍⢼⣿⣿⢟⡃⠾⣫⠷⣃⣊⣠⠞⡋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠉⠁⠘⠁⠁⠈⠉⠠⡈⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠶⣦⣤⣤⠀⠀⠀⣠⣤⣀⠠⠄⠐⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠘⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⡀
⠀⠀⠀⠨⡻⢐⠀⠄⠀⡀⢀⡀
⠀⠀⠀⠀⠈⠂⠔⢀⡀⡄⠺⠷⣆⡀⢤⠦⠠⢠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠒⠂⠀⠀⠀⢄⠤⠠⠤⠠⢄⣄⣀⡀
⠀⠀⢀⠀⢄⡀⠀⣤⣀⣦⢀⠀⠄⣿⠘⠂⢀⡀⠄⠀⠀⠀⠐⢃⠀⠀⢀⡀⢠⡄⠂⠰⠀⠀⢠⠀⢘⡀⢀⡉⠀⠁⠁⢲⠤⡀
⠀⠀⠀⠈⠉⠐⠚⠙⠒⠉⠙⠛⠂⠉⠀⠀⠀⠈⠡⠄⠀⠤⡀⠀⠐⢀⡀⢼⡏⢠⣼⣦⡄⢀⣉⣠⠼⠧⠨⠆⠠⢃⣠⠟⠐⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠾⣅⣀⠀⠠⡖⢳⣦⡗⣤⠀⠦⡄⠀⢁⡂⣳⡘⣿⣿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡛⠛⣋⣥⣽⠒⠀⠐⠙⠠⠞⠳⣿⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠰⠂⠀⠀⠀⠠⣶⡟⠃⠀⠀⠀⠀⠀⢸⡇⢰⣾⣿⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠈
⠀⠀⠀⢲⣿⣿⣶⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⢉⣒⣙⡛⢷⣿⣅⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠛⠋⠅
⠀⠀⠀⢾⠻⡟⠛⣿⢿⡟⠻⠷⠀⠀⠀⠀⠰⣿⣏⣉⣛⣛⣻⣿⠆⠀⠀⠀⠀⢼⡦⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠓⠉⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠈⠉⠐⠄⢠⠀⠀⢀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣼⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⡄⠀⡀
⠀⠀⠀⠀⠀⢀⠟⣿⣿⠧⣀⠀⠀⠀⠀⠀⣼⡟⣦⣦⣄⣀⣼⣛⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣏⠀⣿⡿⣄
⠀⠀⠀⠀⢀⣾⣿⡿⢿⡆⠘⠀⠀⠀⠀⠤⠿⠃⣹⣿⠻⠿⢿⡿⠳⡄⠀⠘⠋⣰⣾⠿⣿⣿⣿⣿⣷⣿⣗⡀
⠀⠀⠀⠠⠟⢱⠛⠁⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⢏⠀⠀⢸⠀⠐⠇⠀⠀⠀⢯⡸⢰⣏⡘⢯⠙⣻⡋⠙⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⠀⠘⠂

⠀⠀⠀⠀⠀⢰⣿⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⢀⣴⣷
⠀⠀⠀⡀⢠⣾⠟⠋⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠐⠹⠆⠀⠀⠀⠀⠘⣿⣇⣰⣏⢠⣿⡿
⠀⠀⠀⠀⣴⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣲⣶⣤⣬⣧⡈⡀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠟⠋⠁
⠀⠀⠀⠀⢿⣿⠁⣀⡀⢠⠀⠀⠀⠀⠀⢠⣾⣿⡞⣿⠉⠛⢿⡟⠻⠀⠀⠀⠀⠀⠀⠀⠚⢳⣦⣤⣄
⠀⠀⠀⠀⠈⠻⣧⣿⡽⠸⠷⠂⠀⠀⠀⠈⠉⠀⡄⠸⠄⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠂⢀
⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠒⣐⡴⠖⠒⠛⠉⠉⠉⠉⠉⠙⠒⠃⠉⠛⠀⢈⣉⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡎⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠈⠉⠒⠦⣤⣤⣀⣀⡇⠀⠀⠀⠀⠘⡆⠀⢀⡠⠐⠀⣀⡀⠀⠀⣵⡀⢸⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠢⠤⣀⠀⠀⠉⢻⠀⠀⠀⠀⠀⢁⡴⠋⠠⠒⠉⠁⠀⠀⠀⠀⡷⠾⡷⠾⠛⠛⠶⠖⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠙⠢⣼⠂⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠋⠀⠘⠀⠀⠀⢀⣀⣀⣀⡄⠤⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠠⢘⡻⣆⠀⠀⢀⡟⠁⠀⠀⠀⢀⣂⣠⣾⡇⠀⠀⠤⢤⣀⠠⣾⠉⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠒⢩⢴⣿⣦⣴⣾⣇⠀⠀⡀⢀⣾⡏⠉⠹⠋⠀⠀⠀⢠⣨⠛⠦⡉⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠛⠀⠀⠠⡙⢯⠩⠁⢿⣿⣿⡇⡡⠖⠀⠀⠀⢨⣿⡀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣅⠀⠀⠀⠀⠀⣀⣀⠒⣾⠟⠧⠤⠤⠴⠒⠒⠢⠸⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣤⠀⡆⠀⠀⠀⠀⡀⠀⢧⠈⠰⠂⢀⡼⠻⠷⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠟⠑⠁⠀⠀⠀⢾⠉⠻⡧⢀⡀⠴⣯⠔⠀⠛⡿⡖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠏⠀⠀⠀⠀⠸⡃⠐⢹⣉⠷⠐⠛⠒⠒⠟⠁⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡉⣆⠑⠢⠤⠴⠞⠉⠙⠁⠀⡸⠁⠀⢀⣠⠤⠤⠄⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⣒⡶⠖⣂⣀⣤⠶⠚⡅⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠉⠁⠀⠙⢦⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⣤⣤⡤⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣰⣷⠛⠋⠀⠙⠻⣤⡆⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢤⣶⡄⠀⠀⠀⠀⠛⡀⣀⣀⣀⠀⣀⣄⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠿⠏⠘⠿⠿⠃⠙⠛⠋⠀⠉⠉⠰
⠀⠀⠁⠄⡀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣠⡄⠆⠀⠀⠀⠀⠀⠀⠀⢠⠄⠒⠂⠶⠠⠤⢠⣤⢠⣤⢠⡤⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠁⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣆⡈⠁⠈⠀⠀⠀⠠⣎⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⢀⡴⠯⢡⠂
⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣼⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⠀⢧⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠉⠁⣀⣠⣧⡈⠳⠿⣿⣿⠿⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣴⣾⣿⣶⡀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣾⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣼⣿⣴⡂⠈⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⣯⡡⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣝⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡾⠐⠀⠀⢸⣷⣧⣿⣿⣝⣿⣷⣿⣯⣉⡉⠛⠛⠛⠿⠿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠛⠉⠉⠀⢀⣦⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⣾⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⡀⠀⣀⣸⣽⣫⣷⣿⣿⣿⣿⣿⣯⣾⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠇⠉⠛⢀⣠⣿⣿⣿⣿⡟⠻⣷⣴⣿⣿⠟⠛⠛⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡄⠀⢀⡼⢋⡟⣉⡁⠙⣿⠏⣸⣿⡟⠋⠁⠀⠀⠀⠀⠀⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⠛⣿⣷⣿⡇⠸⣿⡿⣿⡂⣿⡇⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠋⠁⣹⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠀⣿⣿⣿⣶⣤⣶⣿⣿⣿⠃⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⣘⣻⣿⣿⣿⣿⡿⣿⣿⠏⠀⠀⠀⠴⣿⣿⠟⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣁⡀⢀⣿⣿⣿⣿⡿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠁⠀⠾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣡⣀⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⣄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠘⣿⣦⡀⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠘⣿⣿⣦⣻⣿⣿⣿⣿⡹⣿⣿⣿⣷⣾⣿⣿⡿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⢿⣿⣿⣿⣧⡀
⢠⣤⣴⣶⣶⣦⣤⣄⡙⢿⣏⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⢞⣿⣿⣿⣿⣷⡄
⠀⠈⠛⠿⢿⣿⣿⣿⣿⣶⣿⣦⣈⣻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣩⡿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⣤⣬⣿⣿⣷⣭⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⣿⣿⠿⠟⠋⠙⢾⣿⣿⣇⢿⣿⣿⣿⡿⢿⣿⣵⢾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠿⠿⣿⣎⣿⡿⠋⣿⣿⣿⣷⣻⣿⡿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢟⣵⣾⣿⣿⣿⣿⣿⣿⣀⣼⣿⣿⣿⣿⣿⣿⠃⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠉⣴⣿⣿⣿⣿⠿⠛⢉⣼⣿⣿⣿⣿⣿⢿⣿⣿⠏⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⢻⣿⡿⠈⠻⠿⢿⣦⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⠃⠀⢸⣿⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠈⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠖⠒⠀⠀⠈⠒⠀⠐⠲⠒
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣤⡤⠾⠷⢄⠀⠠⡀
⠀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⢀⣼⠿⠁⠀⢀⣀⠀⠐⢤⡀⠀⠀⠀⡜⠁⠀⠈⠑⢦⠀⠀⠀⠀⠐
⠀⠐⢯⣀⠀⠀⠀⠀⠀⢀⣠⠞⣇⣀⠀⠀⠀⠈⠓⠦⠤⡙⠂⠀⠀⡄⠀⠀⠀⠀⠀⡑⠠⠤⠤⠐
⠀⠀⠀⠈⠙⠓⠒⠖⠛⠉⠀⠙⠋⠉⠉⠙⢶⡄⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⢰⣆⠠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠉⣽⣦⡀⠀⠀⢀⣀⣤⣤⣀⣠⣯⣈⡆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠂⠀⠀⠤⠒⠛⣉⡿⠷⠾⠛⠉⠙⠛⠉⠻⣏⡁⠀⠈⢢⠀⢆
⠀⠀⠀⢠⣠⠊⠀⠀⠀⠀⠀⢣⣀⡀⠀⠀⣠⠊⢁⡤⠖⠒⠀⠀⠀⠤⠄⠀⠉⢙⠢⡀⠺⡀⠘⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠹⠟⠡⡏⠁⠐⠁⠀⠀⠀⠀⠀⠈⠳⡀⠘⠁⠀⢣⠘⡄⠘⢆⢣
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠈⢿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠑⢶⣒⣏⠹⠷⢦⣀⠀⠀⠀⣰⣀⠀⢀⣠⠆⠀⠀⠀⠈⣇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡴⢻⡏⠙⠒⠒⠋⠁⠀⠀⢀⣀⡤⠍⠉⠉⠉⠉⠛⠿⣅⠀⠀⣆⠀⠀⢸⡄
⠀⠀⣧⠀⠀⠀⠀⠀⢠⠞⢠⡟⠣⡀⠀⠀⠤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⢠⡇⠀⠀⠀⡇
⠀⠀⠸⣾⠀⠀⠀⠀⠏⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡛⡇⠀⠀⠀⡷
⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⣠⣴⣦⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⡄⠀⣀⣤⣾⠶⠶⠛⠉⠉⠉⠉⠉⠀⠀⠀⠈⠲⣤⡄⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠛⠖⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠖⠋