Alphabetical order Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡀
⠀⠀⣤⡄⢠⣀⣀⣀⣤⡀⢀⣀⣤⣀⢀⣀⢀⡄⠀⢠⡀⣀⠀⠀⣀⡀⢀⢀⠀⡀⠀⠀⠀⠀⠟⠻⠶⣽⠆⠀⠀⠀⠀⠀⢸⡿⠸⣽⠇
⠀⠘⠛⠛⠚⠻⠟⠛⠛⠛⠘⠛⠛⠛⠙⠛⠘⠓⠀⠊⠃⠋⠘⠃⠙⠐⠛⠛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⢴⠠⣦⣤⣠⠄⢀⠔⠢⣄⢠⣠⠄⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠀⠈⠉⠈⠀⠀⠉⠁⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠴⠀⠂⠰⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⡤⠄⣀⡀⠀⠀⠀⠀⠀⢠⡤⢄⢀⣤⠀⠀⠀⠀⠀⠀⣤⠄⢀⣀⠀⠀⠀⠀⠀⠀⢠⡤⢠⣖⠀⠀⠀⠀⠀⠀⣠⠴⠀⣀⡀
⠀⠀⠀⠘⠦⠖⠷⠀⠀⠀⠀⠀⠀⠘⠧⠎⠳⠾⠀⠀⠀⠀⠀⠀⠻⠁⠻⠟⠀⠀⠀⠀⠀⠀⠘⠃⠘⠁⠀⠀⠀⠀⠀⠀⠻⠼⠛⣥⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠘⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢶⣀⡖⣦⡄⠀⠀⠀⠀⠀⠀⠀⡖⢠⠀⠀⠀⠀⠀⠀⠀⠐⢲⠀⣄⠀⠀⠀⠀⠀⠀⢴⡔⢸⡤⠀⠀⠀⠀⠀⠀⠀⡆⠀⡆
⠀⠀⠀⠙⠈⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠀⠘⠛⢠⠏⠀⠀⠀⠀⠀⠀⠉⠙⠈⠙⠀⠀⠀⠀⠀⠀⡀⠓⠂⠃
⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠆
⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄⠀⠀⠀⢄⠀⠀⠀⠈⠐⠄⠀⠀⠀⣠⢄⢠⢁⣀⠀⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠁⢄⠀⠀⠀⠀⠀⠀⠉⠈⠂⠉⠀⠀⠀⠀⠀⠀⠈⠣⢴⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠀⠀⠀⠀⠀⠀⠈⠁⣀⠉
⠀⠀⢸⣷⡼⣧⢶⢶⡆⠀⠀⠀⠀⢰⡷⣼⡶⢦⠀⠀⠀⠀⠀⠠⡎⢙⣦⢶⡄⠀⠀⠀⠀⠀⣿⡽⢶⢶⡄⠀⠀⠀⠀⠀⣞⣉⡇⡴⣦
⠀⠀⠈⠀⠁⠉⠀⠈⠁⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠸⠋⠀⠀⠀⠀⠀⠀⠈⠉⠃⠉⠿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢒⠀⠠
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠸⡶⡃⡾⠃⠀⠀⠀⠀⠀⠀⣙⣣⣺⣇⠀⠀⠀⠀⠀⠀⠈⡇⠘⡗⠀⠀⠀⠀⠀⠀⣿⣠⢻⣰⡆⠀⠀⠀⠀⠀⠸⣶⠁⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⠄⠀⠀⠀⠀⡀⠔⢄⣐⣂⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠃⠀⠀⠀⠀⠈⠀⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⢠⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣀
⠀⠀⠈⠿⠹⠇⠹⠿⠏⠀⠀⠀⠀⠠⠜⢣⠼⠿⠀⠀⠀⠀⠀⠀⠘⠇⢙⡞⠀⠀⠀⠀⠀⠀⠴⣥⢤⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠬⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢀⡤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛
⢸⠂⠁⣘⣉⠘⢸⠒⠈⠁⡗⠈⠁⠀⠀⠀⢀⢻⡄⠀⠀⢀⣀⡀⢀⡀⠀⣀⡀⡇⣀⠀⣀⣀⣈⡠⠀⠀⠀⡐⠉⠀⠀⣀⡀⠀⣀⠀⣀⣀
⢸⠟⠁⢿⢧⣄⢸⡟⠉⠀⡟⠉⡀⠀⠀⠀⣸⠇⠷⣾⢻⢁⡟⣿⠃⢸⡎⢸⣽⡏⣩⢾⠎⢉⣽⠁⠀⠀⢸⡇⢀⣴⣾⠁⢸⡎⣸⢸⠏⠈
⣘⣀⣀⣘⣀⣈⣙⣓⣊⣁⣛⣊⣉⣀⣀⣀⣋⣀⣀⣛⣻⣋⣀⣛⣀⣘⣛⣋⣘⣛⣁⣀⣛⣋⣘⣀⣀⣀⣈⣛⣋⣀⣃⣀⣙⣛⣉⣚⣀⣀

⡉⠁⠀⢀⡀⠀⠅⠌⠀⠁⠀⠄⠏⠀⠁⠀⠁⠠⠹⠈⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈
⠆⢠⡀⡆⢐⠀⢠⠠⢴⣤⡄⠀⡤⠀⠀⢠⡆⠀⣦⣄⠀⢠⡤⠠⢌⠒⠀⠀⠀⠀⠀⠀⠀⣴⣦⢙⡃⠀⠀⡀⠀⣀⢀⣄
⠁⣤⡵⠉⢰⠀⣧⡀⢘⣁⡀⠀⡇⢠⣀⢸⡍⢀⡏⠁⠘⠐⣻⡀⢸⣤⠀⠀⠀⠀⠀⠀⠈⡃⢀⡸⡋⡅⢸⡿⠉⡟⢻⣄
⠂⠈⠀⠀⠈⠁⠉⢁⣾⠀⠀⠈⠉⠈⠁⠈⠁⠀⠀⠁⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠈⠁⠈⠁⠀⠉
⡀⠀⠀⠀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢠⢠⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠠⡀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠘⠀⠁⠐⠤⠇⠀⠀⠸⠸⠀⠧⠖⠀⠀⠐⠤⠄⠠⠄⠀⠀⠠⠠⠃⠀⠼⠀⠀⠐⠄⠀⠸⠌⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⣄⢀⡄⢀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠆⠀⠀⠑⠀⠈⠀⠁⡇⠀⠀⠀⠃⠈⠁⠈⠐⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢢⠘⠁⠐⠲⠀⠀⠀⠀⠁⢤⠀⠆⠄⠀⠀⠀⢇⠀⡐⠠⢐⠀⠀⠀⠀⠸⠀⠀⠨⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂

⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⢀⢤⠀⢊⡆⠀⠀⠀⠈⠒⠁⠖⠀⠀⠀⠀⠀⠁⠆⠐⡄⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢠⠀⡀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣄⠀⠀⡀⠀⠀⠀⢁⡌⢣⠈⠀⣤⡀⠀⠀⠀⡱⠀⠀⡄⠀⠀⠀⠀⠈⡎⠀⢄⠀⠀⠀⠀⠀⡐⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⠉⡟⠽
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠋⠙⠢⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⢀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡁⠹⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⡇⣏⣫⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠗⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰


⠀⠀⠀⠀⡀
⠀⠀⠋⠀⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣷⣀⣁⣈⣜⣀⣀⣡⣀⣔⣤⣀⣳⣶⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⡤⡲⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠒⠒⠒⣲⣤⣤⣤⣄⠀⠀⠤⠄⡏⠉⠉⠁⢀⠐⠐⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⠀⢀⠐⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠃⠀⠀⣷⠀⠀⠠⠇⠀⠀⠀⠀⠀⢀⡴⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⠀⢸⡀⠀⠸⠆⢀⠄⠀⠀⣠⠟⠉⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠦⠤⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠐⠁⢈⣥⣿⠇
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⣦⣤⣤⡏⠀⠀⢀⠀⠀⢹⡄⢘⠀⠀⠀⢠⣢⣴⣿⠿⠛⠉⠀⠀⠀⣀
⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠦⠤⡍⠉⣉⣚⡛⠛⠻⠇⢘⣀⣤⣾⠿⠛⠉⠀⠀⠀⢀⡤⠶⠋⢩
⠀⠀⡖⠒⣶⠶⠶⠶⠶⠶⢖⣒⠒⠛⡇⠀⠈⢩⣅⣉⡉⠉⠙⠋⠉⠉⠀⠀⠀⣀⡠⠴⠊⡟⠀⠀⠀⢰
⠀⠀⡅⠀⡇⠀⠀⣀⣀⣀⠀⠈⢳⠀⠃⠀⠀⢸⡇⠀⠀⠌⠛⠛⠓⢒⡒⠒⠆⢻⠠⠁⠘⠁⠐⠀⠀⢻
⠀⠀⡇⠀⡄⠀⠀⠧⠤⠼⠂⠀⣰⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⣀⣘⣀⣀⢀⡄⣀⣨
⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠐⠺⢅⠀⠀⠀⠀⢸⣇⠀⠀⢠⡖⠒⢦⣤⠤⠉⠠⢨⡭⡿⣿⠿⣿⣿⣿⣿
⠀⠀⡃⠀⠀⠀⠀⡎⠉⡦⠀⠀⠀⡇⠀⠀⠀⢸⡿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⢰⠒⠒⠒⠒⠻⠛⣻⣿
⠀⠀⡇⠀⡇⠀⠀⠉⠉⠀⠀⠀⣠⠇⠀⢠⣀⢸⡇⠀⠀⠈⠓⠂⠐⠢⡀⠀⠀⢰⡄⠀⠀⡠⠂⣴⢋⣿⠒
⠀⡀⠦⠤⠕⠖⠲⠒⠖⠒⠒⠈⠁⠤⠾⢸⢸⢸⣇⡈⢄⡀⠀⠀⠀⢀⣸⠇⠀⢸⠀⠀⡐⢁⣠⠷⠋⠁
⠀⠉⠉⠛⠛⠓⠶⠋⠉⠝⠋⠉⠲⠍⠉⠛⠛⠓⠦⢤⣍⣉⣛⡒⠮⠭⢄⣀⣀⢰⠇⣠⠶⠋⠁⠤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠩⠭⣁⣒⣒⡦⠘⠛⠝⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡔⠛⠛⠛⠛⡟⢹⠀⠀⠀⠰⠚⠋⠉⠉⠉⠐⠓⢤⠉⢦⠀⠀⠀⠀⢀⠤⠚⠋⠉⠉⠓⢤⡉⢢⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣇⠠⡆⠀⠀⢸⠀⠀⢠⠄⣤⡀⠀⠀⡇⢸⠀⠀⠀⢠⠏⠀⠀⣀⢤⢄⠀⠀⢹⠀⣧
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢣⠀⡇⠀⠀⢸⠀⠀⠸⠤⠞⠁⠀⣰⡧⠊⠀⠀⠀⠸⠀⠀⢰⠁⡇⠈⠲⠒⠊⠉
⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⢸⠀⢃⠀⠀⢸⠀⠀⠀⠀⠀⠐⠚⠛⠭⡓⢢⡀⠀⠀⠀⠀⡗⠘⡇
⠀⠀⠀⠀⡇⠀⢰⠹⠀⠀⠀⠘⡂⢹⠀⠀⢸⠀⠀⠀⣀⣀⣀⠀⠀⠀⠘⡄⠸⡀⠀⠀⠀⢟⠈⡇
⠀⠀⠀⢸⠀⠀⠿⠾⠂⠀⠀⠀⡇⢸⠀⠀⢸⠀⠀⢸⠃⣹⠈⡇⠀⠀⠀⢉⢀⡇⢸⠀⠀⠘⡄⠸⣄⡖⠻⢯⠝⠳
⠀⠀⠀⡇⠀⢀⣀⣀⡀⠀⠀⠀⣇⠀⡇⠀⠈⠀⠀⠘⠷⠟⠚⠁⠀⠀⢀⣏⡜⠀⠘⡄⠀⠀⠈⠒⠉⠀⠀⡸⢀⠇
⠀⠀⠸⠀⠀⣏⡸⠀⠇⠀⠀⠀⢧⢀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠋⠀⠀⠀⠘⢄⡀⠀⠀⠀⣀⣴⠵⠃
⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀⣀⠀⣀⣠⠀⣀⢠⡀⢠⠄⣀⢀⣀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠋⠹⠛⠉⠋⠛⠙⠉⠛⠉⠀⠘⠂⠙⠙⠛⠃⠙
⠀⠀⠀⢀⣶⡄⢀⣤⣄⠀⠀⠀⣄⠀⡀⠀⠀⠀⢰⣟⡧⢸⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠂⣤⡄
⠀⠀⠀⠸⠛⠷⠘⠷⠟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠸⠯⠟⠸⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠊⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞
⠀⠀⠀⣴⢦⡄⣀⣴⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡖⠂⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⢀⣶⠂
⠀⠀⠀⢿⡴⠃⢿⣼⠇⠀⠀⠀⠘⠷⠞⠀⠀⠀⠸⣯⡅⢻⣿⠃⠀⠀⠀⣤⣀⡀⣄⠀⠀⠀⠿⠉⠸⠏⠁⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⠴⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⢠
⠀⠀⠀⢿⣸⡇⢿⣽⡇⢀⠀⠀⠂⠂⡠⠡⠀⠀⢸⡟⢻⢸⠉⡇⠀⠀⢤⣀⣀⡀⠀⡀⢀⠀⣸⣇⢸⠂
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠁
⠀⠀⠀⠠⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣀⡄⢠⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⣤⠀⢠⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⢀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸
⠀⠀⠀⠈⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⣸⢷⡾⣇⣿⢳⠛⡆⠀⠀⠀⣀⠀⠀⠀⢸⡟⣾⡇⡟⣷⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⠀⣽⢰⡛⡧
⠀⠀⠀⠉⠈⠁⠉⠁⠈⠀⠁⢀⣿⣰⣿⣿⡆⠀⠈⠁⠈⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠁
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠀⠂⣀⣀⠀⠀⠀⠀⠀⠈⠒⠚⠀⠀⠀⠀⠒⢀⣀
⠀⠀⠀⣿⣼⠇⣾⢳⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⡋⣹⡇⡶⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠃⡶⠒
⠀⠀⠀⠙⠀⠀⠟⠋⠀⠀⠀⠆⣠⣾⠟⠀⠀⠀⠀⠙⠛⠃⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠻⣍⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠁⢾⠆⠀⠀⠀⠀⠀⠶⠀⠰⠀⠀⠀⣾⢸⣧⡄⣤⠀⠀⠀⡠⠛⢆
⠀⠀⠀⠘⠛⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠀⠀⠀⠀⠀⠀⠠⢔⢾⠀⠀⠀⠻⠾⠁⠷⠿⠀⠀⠘⠳⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃
⠀⠀⠀⢸⣄⡾⢠⡄⣠⠀⠀⣀⣰⣆⣀⣤⡄⠀⠘⣆⣶⣼⢠⢠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠺⣴⠃⢤⣠⠄
⠀⠀⠀⠀⠻⠁⠀⠻⠃⠀⠖⠀⢩⡍⠉⢀⡆⠀⠀⠻⠙⠏⠘⠟⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠺⠙⠆⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠲⣤⠆⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠆⢠⣤⠀⠀⠀⣾⠳⣤⣤⢤⡀
⠀⠀⠀⠀⠿⠀⢙⡟⠀⠀⠀⠠⠜⠁⠀⠀⠀⠀⠰⠿⠄⠼⠧⠀⠀⠀⠀⠀⠡⠁⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠆

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢰⠃⢀⡀⠀⠀⠀⠀⠀⡅⠀⠢⠆⢀⠀⠀⠀⣰⠁⢀⠐⣢⠄⠀⠀⢈⠂⠀⠠⠐⡀⠀⠀⠀⢀⡆⠀⠨⠥⠵⠁⠀⠀⠀⠀⠀⠲⠶⠆
⠀⡬⠀⠈⠃⠀⠀⠀⠀⠐⡇⠀⢀⡀⠁⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢐⠇⠀⠀⠀⢸⠀⠀⠀⢀⠄⠀⠀⠀⢸⠀⠀⣄⠰
⢐⠃⠀⠰⡐⡀⠀⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠨⠀⠀⠀⠈⢀⠄⠀⠨⠀⠀⠀⠈⠀⡠⠀⠀⢸⢀⠀⠈⠈⠀⠀⠀⠀⢨
⠞⠱⠋⠀⠘⠄⠈⠀⠀⠀⠏⠀⠀⠘⠋⠀⠀⠀⠀⠐⠱⠀⠐⠃⠀⠀⠐⠀⠀⠄⠒⠊⠀⠀⠀⠐⠊⠘⠉⠏⠙⠀⠀⠀⠰⠱⠊

⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⢠⠃⠀⠐⠂⠒⠀⠀⠀⢸⠀⠀⠀⠓⠀⠀⠀⠀⣞⡥⢤⠀⢀⡄⠀⠀⠀⡓⣀⠀⡄⠀⠀⠀⠀⠁⠀⢰⠋⠀⠀⠀⠀⠀⠐
⣾⠀⠀⡴⠀⡀⠀⠀⠀⢸⡀⠀⠴⣰⠀⠀⠀⠀⠀⠈⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⢠
⠁⠂⠀⠙⠛⠁⢀⠀⠀⢸⠁⠀⠀⣿⠀⠀⠀⠀⢀⡠⠗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠀⠈⠃⠀⢸⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠁⠀⠀⠀⠐⠁⠀⠀⠸⠡⠒⠀⠏⠤⠂⠀⠀⠾⠶⠀⠤⠠⠆⠀⠀⠀⠿⠆⠶⠖⠁⠀⠀⠀⠤⠤⠀⠐⠈⠄⠀⠀⠀⠨⠵⠂⠤⠴⠂

⣠⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⢀⢀⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⡾⠀⢀⠀⠹⡂⠀⠀⠀⡘⠀⠀⠊⢬⠂⠀⠀⠀⠀⡂⠀⠈⠙⠁⠀⠀⢀⡞⠀⠀⠢⡂⡀⠀⠀⠀⠈⠁⠀⠐⠚
⠋⡄⠀⣹⠀⡴⠃⠀⠀⠀⣘⠀⠈⠢⠀⠁⠀⠀⠀⡇⠀⠀⠀⢸⠆⠀⠀⠀⢘⡇⠀⢠⠠⠞⠁⠀⠈⠀⠀⠀⢀⠑⠁⠀⠀⠀⢰⠠⠀⠰⡀⠀⠃
⢀⡃⠀⠈⠀⠐⠀⠀⠀⠀⣿⠀⠀⠑⠣⡀⠀⠀⠀⠠⠢⠀⠀⠈⠀⡀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠈⠄⠀⡀⠀⠀⡀⠀⠀⠀⢰⣗⠀⠘⠴
⠸⠟⠋⠀⠀⠸⠌⠉⠀⠀⠺⠁⠁⠀⠑⠏⠈⠀⠀⠀⠁⠂⠤⠒⠋⠀⠀⠀⠘⠊⠛⠁⠀⠀⠀⠀⠀⠀⠁⠒⠑⠛⠛⠡⠋⠀⠐⠛⠉⠀⠈⠊⠟

⢠⠄⠀⣀⡀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠀⠀⠀⡠⠄⠀⠀⢀⠄⠀⠀⢠⡅⠀⠀⢠⠆⠀⠀⠀⢰⠁
⠪⡀⠀⠉⠳⠀⠀⠐⠚⢩⠃⠀⠂⠁⠀⠀⣰⠀⠀⠀⠇⠀⠀⠀⠰⢠⠀⢀⡞⠀⠀⠀⠈⣢⠀⠀⡞⠀⢀⠀⠀⡏
⢀⠈⠠⣅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⡷⠀⠀⠀⠈⡄⡀⠀⠁⠀⠀⠀⠀⠋⠂⠀⠁⠀⡜⠀⠀⠁
⢋⠀⠀⠀⢀⠄⠀⠀⠀⢸⡀⢀⠀⠀⠀⠀⢐⠀⠀⠀⢀⡠⠀⠀⠀⠰⡡⣀⡀⠀⠀⠀⠀⠘⣧⢀⠀⠀⠀⠣⢀⣀
⠀⠉⠀⠀⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠎⠀⠀⠀⠀⢌⠀⠀⠠⡎⠀⠀⠀⢠⡇⠀⡀⣠
⠀⢣⠂⠀⠀⠀⠀⠀⠀⠀⢢⠢⡀⠀⠀⠀⠀⠀⠁⠀⡸⠁
⠀⣠⠃⠀⢄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⡔
⣠⠇⠤⠈⠂⠀⠠⠄⠀⠀⠀⢠⡤⠠⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⠄

⠀⠀⢀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⡀⢀
⠀⠀⠤⠇⡀⠀⠂⠀⠀⢀⣰⣶⣶⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⢀
⠀⠀⠀⣠⠇⢇⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⠀⠐⠀⠀⠀⣀⠄⠒⢀⠐⠀⢤⡀⠀⠀⠀⠀⡠⠊⡄⠰⠠⠀⡄⡆⠀⡑⢄
⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⢀⣶⣶⡆⠃⠀⠀⡁⠠⠊⠀⢣⠀⢸⠀⠠⠃⠈⢂⠀⠀⠜⠢⡀⠃⠀⢋⠓⠹⠀⠓⡠⠚⠂
⠀⠀⠈⠧⣀⠈⠂⢀⣀⣠⡔⠻⠿⠟⠀⠀⠀⠠⢁⠁⠸⠒⠇⢣⠀⠰⠁⣈⠁⠀⡀⠠⠀⠰⢌⠑⢤⠜⠂⠃⠔⠋⣄⢆⡎
⠀⠀⠀⠀⠈⠉⠉⠋⠉⢫⡀⠀⠀⠀⠀⠀⠔⠁⠈⢉⡰⠂⠤⣠⠃⠣⠤⠐⠒⠂⠁⠰⠀⢨⡥⠄⡄⠀⠀⠀⠰⠐⠚⢚⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣩⡁⠒⠴⠀⠀⠀⣃⠘⠬⠂⠀⡄⠀⠂⠀⠶⡁⠀⠁⠈⠀⠱⢮⣀⣠⠀⠀⠀⣆⣀⡜⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣋⠙⠀⠆⠀⠀⠀⠁⠈⡭⠉⠁⠃⢘⠓⢒⠂⠐⢒⠀⠀⠓⠈⠟⡂⠀⠁⠀⠄⠀⣄⠄⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠌⠉⡀⠀⠀⠠⠦⠄⣃⡀⠀⠀⢈⠀⠘⢄⡀⠼⠀⠀⠠⠤⠍⠭⠄⠆⠰⠂⠤⠤⠄⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡩⠕⠊⠑⡤⠐⢆⠀⣖⠀⠈⠹⠀⠰⠂⣯⣠⡄⢠⠓⠖⠳⢄⡧⠜⠀⠀⡆⠀⡸⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⣎⡁⠀⠇⡄⢌⢢⠁⠀⠠⠂⠀⠀⠂⠁⢁⠕⣥⣠⠀⠐⢢⠑⢄⠐⠀⢠⡠⠔⣫⣽⣤⡾⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⡘⠀⠓⠁⠀⠡⠀⠀⠀⠀⠀⠀⠑⠋⡀⠁⠙⢰⠈⢈⠀⠊⠀⠀⠀⠑⢊⠔⣱⢹⡬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⠟⠛⠀⠀⠀⠀⠀⣠⣴⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⡄⢀⣀⡀⠀⠆⠀⣀⠉⠉⠛⣿⠿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⣀⠀⢰⠛⣾⣖⠥⠄⠀⠀⠹⠗⡄⣀⣀⣄⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠤⠖⠊⢉⣡⠶⣁⢿⣭⡟⣿⣭⣭⠁⠄⠀⡀⠈⠉⠁⢀⠂⠒⣄⠲⣔⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣾⡿⢿⣾⣾⣿⣿⡽⣡⡞⠀⢟⠀⢤⣁⢀⢠⠘⠀⠀⠙⢘⡴⢋⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⢠⣟⣷⣦⣤⠠⣌⣿⡾⢷⡯⡞⣿⡉⣿⠀⠈⢰⠣⣌⡙⢿⠦⠀⠓⣴⠮⠄⠔⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢏⣟⣻⣿⣰⣿⡾⡲⢾⠎⠿⡿⠗⢂⣼⣷⠘⢦⡄⠁⠀⠘⣀⠔⠁⠀⠐⠂⣰⡾⠛⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡻⣯⡿⣟⣿⣟⡇⢼⣻⢁⠊⢸⣷⣚⡁⠘⢌⠁⢀⠿⣉⡷⡽⣯⡀⠀⢀⠘⣷⣉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣾⢸⣿⣾⣷⣏⣼⠟⣉⠔⠛⠃⣠⣤⣀⣈⠳⣬⠃⢡⢐⠦⣿⣧⣗⠀⢧⣾⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡯⢽⣙⡾⣾⣿⣭⣿⣣⡞⣁⣴⠊⠉⠀⠀⠀⠈⠻⣯⡀⣰⢚⢷⡾⢋⡨⣑⣾⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣫⠷⣧⢿⣼⢻⣿⣼⣿⣿⣿⠀⣀⠀⠀⠀⠀⠀⢻⡿⢟⠧⣀⡿⢛⣴⣼⠷⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⣻⣿⣯⣹⣮⣝⣽⣞⢽⣿⠿⠋⠀⠄⠀⠀⠀⠀⠀⣸⠁⣆⢦⣉⣧⢭⣤⣏⡱⠊⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣿⣿⣿⡝⣿⣻⣭⠏⠃⠀⠀⠀⠀⠀⠀⠠⣴⡿⣧⡈⠳⡏⠙⣯⡠⠤⢰⣦⠔⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⠛⠲⣾⡏⢾⣿⡆⠀⠀⠀⠀⠀⠀⢴⣾⠟⢀⡜⢡⣼⠸⢷⣾⠀⠆⠈⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢠⠈⠉⡍⣿⡇⠀⠀⠀⠀⠀⠢⡀⠀⢀⣠⣾⣧⡹⣿⣳⢾⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠛⠾⣝⠲⣧⣿⣇⠀⠀⠀⠀⢀⠈⠁⠀⠀⠉⣛⢿⣿⣷⡱⠁⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣆⡜⡉⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢻⣿⠟⡩⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣣⣸⣤⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⢠⣧⡾⠤⠶⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡝⡍⢳⣦⢤⣀⣀⡀⣠⣿⣦⣤⠎⢠⢳⠶⠚⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣌⣮⣷⡤⡭⣿⣬⢟⢩⢭⣿⠒⠓⢺⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡏⢁⡴⠚⢻⠧⣤⣈⠈⢷⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡾⠀⣂⠷⡀⠀⠙⢆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠉⠁⠀⠉⠈⠃⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣬⡓⢤⡀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡛⢿⣶⣌⡳⢤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠀⣠⠞⣡⣄⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣍⡻⢿⣦⣍⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⡿⢋⡴⠀⠀⠀⠀⠀⠀⠛⠛⢀⠀⠀⠀⠨⠀⠀⢀⠔⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⠻⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠳⠄⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢀⡤⠀⠀⢀⠖⠋⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢩⠛⠉⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡀⢂⠘⢆⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⣀⠔⠓⠢⠀⠀⠀⡄⠀⠀⢠⠔⠈⠀⠉⠃⡀⠀⠁⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⡀⢀⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢣⠀⠀⠀⠀⠈⡆⠰⡇⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⠁⠀⠀⠀⢀⠀⠀⣀⡀⠀⠾⢥⡤⠚⠀⢴⡇⢤⠀⠀⠀⢀⠇⠀⠙⠂⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⢀⣀⡀⠀⠀⠈⠀⠀⢀⡇⠀⠁⢀⡴⠟⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠚⠉⠉⠁⠀⠀⠀⠀⠀⠄⠊⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡂⣒⣾⠟⠀⠀⢤⣄⣀⠀⣀⣠⠴⠿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢠⡐⢄⠂⠭⣛⡭⡭⣒⠵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠓⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂




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

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣰⣷⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠐⢶⣦⣴⣶⣶⣦⣄⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡄⠀⣰⠆⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⠀⠀⠀⠛⠀⣽⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣬⣀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠘⢿⡿⣿⣿⣿⣿⣿⣿⣋⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⡿⣿⣟⣛⣙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠟⠛⠉⢻⣿⢿⡇⢸⣿⣿⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠡⠜⠟⠻⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣟⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⡄⠀⠀⠀⠀⠈⢀⣿⣿⣿⣿⡟⠁⠀⢶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠃⠀⠸⣿⣶⣧⣀⣀⣴⣿⣿⣿⣿⣿⣴⣿⣦⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣄⠀⠘⠄⠀⠀⠀⠈⠻⠋⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⡀⠀⠀⣠⣴⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⠟⣼⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⡀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⡿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠈⣯⣩⣬⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⡶⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠄⢰⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣾⣿⣇⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠟⣉⣿⣽⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠁⠀⠀⡿⠿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⠀⣿⣿⢻⣿⡇⠀⢀⣾⣿⡿⢿⣿⠆⠀⣿⣿⣿⣿⣦⠀⠀⣿⣿⠿⠿⠟⠀⢸⣷⡿⠿⠿⠃
⠀⠀⠀⠀⣼⣯⣼⣷⡀⠀⠀⣿⣿⣿⣯⡀⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⡇⠀⣿⣿⣶⣶⠀⠀⢸⣿⣷⣲⡄
⠀⠀⢀⣾⣿⡟⠹⣿⣷⡀⠀⣿⣿⣨⣿⣿⠀⠸⣿⣷⣦⣶⡷⠀⠀⣿⣿⣶⣿⡿⠃⠀⣿⣿⣤⣤⣤⠀⢸⣿⡏
⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠁
⠀⠀⢀⣴⣶⣶⣶⣤⠀⢰⣶⡦⠀⠀⣶⣶⠀⢰⣶⣶⣶⣶⣶⠀⢰⣶⣲⣲⣲⣖⠀⢰⣶⡆⢀⣶⡖⠀⢰⣲⡆
⠀⠀⣾⣿⠟⠉⠙⠟⠁⢸⣿⣷⣤⣤⣿⣿⠀⠈⠉⣿⣿⠉⠁⠀⠈⠉⢹⣿⡏⠉⠀⢸⣿⣷⣿⠿⠁⠀⢸⣯⡇
⠀⠀⣿⣿⣄⣀⣻⣷⡆⢸⣿⣿⠛⠻⣿⣿⠀⢀⣀⣿⣿⣀⡀⠀⢀⣄⣼⣿⡇⠀⠀⢸⣿⡟⢿⣦⡀⠀⢸⣿⣧⣀⣀⡀
⠀⠀⠈⠻⠿⠿⠿⠋⠀⠸⠿⠿⠀⠀⠿⠿⠀⠸⠿⠿⠿⠿⠷⠀⠙⠿⠿⠋⠀⠀⠀⠸⠿⠇⠘⠿⠇⠀⠸⠿⠿⠿⠿⠇
⠀⠀⣀⣀⡀⠀⠀⣀⣀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣀⣀⡀
⠀⠀⣿⣿⣷⣀⣼⣿⣿⡄⢸⣿⣿⡄⢸⣿⣧⠀⣴⣿⡿⢿⣿⡆⠀⢸⣿⡿⣿⣦⠀⣰⣟⣿⣿⣿⣆⠀⢸⣿⡟⣿⣷
⠀⠀⣿⣿⣿⣿⡿⣿⣿⡇⢸⣿⣿⣿⣾⣿⣿⠀⣿⣏⠀⠀⣿⣿⠀⢸⣿⣷⣿⡟⠀⣿⣿⠀⣠⣿⣿⠀⢸⣿⣿⣿⠏
⠀⠀⣿⣿⠀⠉⠀⣿⣿⡇⢸⣿⡇⠹⣿⣿⡿⠀⢻⣿⣶⣾⣿⠇⠀⢸⣿⡏⠁⠀⠀⠹⣾⣷⣿⣿⡏⠀⢸⣿⡿⣿⣷
⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠉⠛⠋⠁⠀⠀⠈⠙⠁⠀⠀⠀⠀⠈⠉⠉⠙⠁⠀⠘⠛⠁⠙⠙
⠀⠀⠀⠀⠀⣠⣶⣶⣤⠀⢰⢶⡶⣶⢶⣶⡆⠀⣶⡆⠀⠀⣶⣶⠀⢰⣶⣦⠀⣴⣶⡆⠀⢰⣶⣆⠀⠀⠀⢀⣶⣶
⠀⠀⠀⠀⠐⣿⣿⣉⠁⠀⠈⠉⢻⣿⡏⠉⠁⠀⣿⣧⠀⢀⣿⡟⠀⠀⣿⣿⣀⣿⣿⠀⠀⠈⢿⣿⡄⢀⠀⣼⣽⠇
⠀⠀⠀⠀⣀⠈⠙⣿⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣄⣼⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠘⣿⣷⣯⣷⣿⡟
⠀⠀⠀⠘⠿⣿⣿⡿⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠿⣿⠿⠋⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠹⠟⠁⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⣀⣀⠀⣀⢀⡀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣾⡿⠃⠀⢻⣿⣆⣿⡿⠁⠀⠛⠛⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢿⣷⡄⠀⠀⢸⣿⡇⠀⠀⠀⣾⣿⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣯⣻⡙⠿⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣶⣯⡿⡟⠿⢿⣿⣿⣯⠙⡏⠻⡙⣏⠐⠀⠡⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣼⣿⣿⣿⣿⣿⣿⡔⣆⠀⡘⠛⣿⠀⠀⢀⡗⢨⠀⠀⡀⠀⠒⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠰⣿⣟⣿⢿⣿⣿⣿⣿⣿⣾⣷⣠⣿⣃⣤⣴⠇⣎⣄⣠⣁⡤⠀⠘⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣿⣿⣿⡿⣫⡄⠳⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⡿⠟⢹⣧⡝⠀⠀⠀⢻⣿⡿⢿⣟⡛⠻⠿⠿⣿⣿⣿⣿⣯⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠘⠾⣻⡿⣟⢳⣤⣀⠀⠂⠈⢻⡟⢷⣤⣄⡀⠀⠀⠀⠀⠉⠛⢿⣷⣕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠈⠀⢴⣤⣬⣉⣉⠓⢉⠳⣄⠨⢀⣿⣦⣉⠉⠓⡀⠀⠀⠀⠀⠀⠀⠉⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠄⣴⣶⣦⡍⠙⢷⡀⠀⢘⣧⣼⣿⠻⣆⠀⠀⠀⠀⠀⠠⢤⣀⠀⠀⠈⠻⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠦⠞⣷⠀⠀⠘⣿⣇⠈⣻⣿⣷⣿⣶⡐⢺⣿⡗⠀⠈⠳⣄⠀⠠⠀⠀⠀⠈⠛⠶⠄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠳⠀⠐⠈⣧⡀⠀⠈⠛⠷⠿⠿⠾⢻⠎⠀⢷⡇⠀⠀⡀⠀⡀⠀⠳⣦⡀⠀⠀⠀⠀⠀⠘⢦⡿
⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⢄⡀⠒⢠⡇⡛⠀⠀⣠⣤⣤⡀⠀⢠⣤⣾⣷⠀⠀⠙⣆⠀⠀⠀⢮⣿⣦⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣥⣙⢷⡌⠱⠆⢳⣠⡇⠀⠘⣿⣿⣿⣿⣖⣻⡍⠉⢻⡀⠄⠀⠈⠣⡄⠀⠈⢻⣿⣿⣷⣄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢠⣞⣿⠉⠀⠀⠀⠀⠈⠛⡄⠲⠆⠹⠻⠿⠛⠉⣻⣷⣄⡀⠻⣄⠀⠀⠀⠈⠀⠀⠈⣿⠹⣿⠎⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠘⢧⡏⠀⠀⠀⣬⣾⣷⣶⣤⠠⠔⠀⠀⠄⢴⣆⣦⡉⢛⣣⣤⣬⡢⡄⠀⠀⠀⠀⡄⣿⡇⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣴⣿⣿⣿⣿⣿⠇⡀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⡝⢧⡀⢄⢶⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢮⣻⠟⢻⡟⠟⠿⡿⢿⣿⣿⣿⣿⡿⣿⣿⣭⣌⣿⣛⣿⢻⣎⢿⡎⡞⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣋⡀⠀⠈⣿⠀⡀⢀⠐⢂⠈⣽⣿⣷⡿⣿⣷⣿⣿⢱⡏⠀⠋⠘⠟⣿⡏⢠⠃⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣷⣷⢸⣿⣧⣕⠠⡍⣬⣹⡷⠋⠀⣀⠞⣹⢯⡟⢸⠇⠀⠀⠀⣠⡟⠀⠘⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⢿⣿⣿⣿⡞⡽⣣⢫⡟⡠⡀⠁⠀⠹⠏⢠⡿⢠⠀⠀⢰⣿⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹⠋⠁⢸⡟⢿⠻⣾⡵⣣⢏⡼⡃⠂⠀⠀⣰⣶⡿⠁⠃⣄⡴⣿⠇⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠧⠀⠀⠀⣀⢿⡄⠀⢸⣷⡀⠀⢻⣷⣇⣾⢸⣿⣤⠀⢠⣿⡟⣡⢏⣾⠋⢠⡟⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢦⣰⣿⣿⡿⣧⠀⢸⣧⢡⠀⠀⠙⢙⣿⣛⣿⣿⡄⠷⢟⠰⣱⣿⣿⠀⣼⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠈⢠⣾⣇⢸⠀⠀⠀⣸⢿⣏⣿⣿⣷⠀⡌⠀⣽⣿⣿⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⡜⠀⠀⠀⢼⣷⣿⠻⣽⣿⡀⠁⢰⡏⢻⢿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠠⣀⡄⢠⠉⡴⣭⢱⣾⣿⣿⡀⠀⠀⠀⠈⣧⡇⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣠⣄⣠⡁⠀⠤⠀⢐⣊⣶⣻⣿⣿⣅⠀⠀⠀⠀⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⢖⡨⢃⡤⢀⠀⠀⢀⠀⠀⢀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⠠⠤⢦⠆⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⡿⠰⣣⡞⠀⢀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡛⠠⣤⣿⠁⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⢀⡠⠀⢀⠄⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣋⠉⠳⣄⠑⢤⠀⢶⣈⣉⢡⣻⣿⣿⣷⣀⣾⣏⣴⠊⢀⡔⢁⣴⣉⣉⣵⠖⣊⣭⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣥⢹⣶⣾⣿⣯⣼⣧⢸⣧⡼⠯⠉⠀⠈⠛⠁⠀⠉⢻⡿⣿⣇⣟⢠⣿⣭⣭⣽⡿⢟⠋⡉⠅⠠⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠍⠿⣿⣿⣿⣿⣿⣿⡂⠀⠐⠀⠈⡏⠀⠀⠤⠀⣆⢻⣿⣿⣼⣿⣤⣿⣿⠘⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣁⣀⣐⢻⣿⣿⡟⣷⣿⠃⡀⣁⡶⢿⣿⡷⣤⡠⠎⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿⢛⣡⡴⠟⣉⣤⣿⠆
⠀⠀⠀⠀⠀⢠⣏⣈⠹⠷⠓⢙⢙⢻⣻⣿⣿⡵⠟⠡⠜⠛⠁⣀⣀⣀⣀⡈⠙⠾⣡⠟⢿⣿⣿⣿⣿⣿⠯⠉⠆⠰⠀⠆⢠⠄⢠⠈⡁⠐⠃
⠀⠀⠀⠀⢠⡶⠈⢁⠰⠄⢃⠉⠄⠒⣃⣽⣿⣡⠝⣀⢴⡂⣀⣀⣩⣍⣁⣉⡓⣲⣬⣿⣷⣏⣿⣿⣿⣿⣄⣨⣅⠈⠀⠀⠀⠀⠈⠀⠁⠘⠂⡄
⠀⠀⠀⠀⠊⢀⠐⠈⠀⠈⢀⠐⢦⡼⢛⣻⣿⣷⣮⢕⣀⣿⠤⠤⠼⣶⠦⠭⣽⣟⠂⠙⣿⣿⣿⣿⣿⣿⠃⡍⢘⠛⠛⢋⣃⣀⣀⣀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⢡⡶⢃⡐⠂⠼⣿⣷⣽⡯⠀⣸⣇⡒⢡⡘⣛⣒⡿⠄⣠⢾⣿⣼⣿⣿⣿⣌⠑⠀⠁⠚⣠⠄⠉⠋⢥⣭⣥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡞⠀⠖⠀⠈⣴⣿⣿⣿⠇⢉⢨⡯⣠⣾⡵⣩⣽⣛⣣⣿⣿⣿⣿⣿⡿⢉⠷⠶⣤⠈⠀⠀⠈⢁⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠃⣾⣿⠱⣿⣿⡾⢋⠀⠳⠾⢁⣰⡷⠚⢠⣿⢿⡟⣿⣿⣿⣿⡩⠄⠳⡶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡆⢩⣿⣻⣐⡧⠘⠃⠀⠀⠋⠑⠛⢦⣜⣯⡿⣎⣿⣿⣏⠣⠨⠄⠻⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣾⡟⣳⣎⡑⠀⠒⠀⠂⠴⠆⢴⣼⣿⣿⣿⣿⣿⣿⣿⣄⣥⣵⣞⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣿⣿⣧⣶⢶⠰⠂⠆⢶⣾⣿⣿⣿⣿⣷⣿⣿⣿⡟⣿⠟⣉⣰⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠚⠁⢙⢁⠎⣟⢻⢺⣿⣿⣿⣿⣿⣿⣿⣷⢛⣌⣷⣶⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣿⢧⣶⡖⡾⣮⡶⣥⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣤⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⣿⠋⢸⡏⡇⢠⠉⣟⣛⣿⣿⣿⣿⣿⣿⣿⡏⠻⣿⣀⠐⣾⣿⣿⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣾⠃⠀⠀⠄⠀⠀⣀⣁⣈⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣏⡑⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⠀⠀⠀⣙⡀⠀⣤⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣏⠛⣿⣿⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⢠⡀⠈⠁⠳⣿⣿⣿⢿⣿⣿⣿⡟⠛⣿⠀⢨⣿⡡⣽⣟⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠂⠩⣱⣿⣿⣿⣿⣻⣿⣿⣿⠀⠀⠛⢠⣿⣿⣷⠚⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⡄⠀⠑⢤⡀⠀⠀⣄⡳⢦⣻⣿⣿⣿⡿⢿⣿⣿⣿⣄⣀⣤⣿⣟⠉⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠷⠀⢀⡀⢉⣲⣅⡠⣸⣿⣿⣿⠿⣹⠿⣾⣿⣿⣯⡏⠉⢸⣛⡉⣸⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⣴⣾⡏⣸⣿⣿⣿⣿⣿⠿⢿⣶⢿⣶⣞⣿⣿⣿⡇⠀⠐⣿⠞⣯⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⠿⠿⠿⠛⠁⠀⣹⣛⣿⡷⣶⣿⣿⣿⣿⠀⢠⣷⠞⣿⣾⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⠿⠟⣺⣿⣿⣿⢿⣇⠘⡿⣿⣿⣯⡷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣒⡈⠿⣛⣿⣿⣿⣿⠀⣧⣜⣓⡚⣺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣬⡴⠾⠛⢻⣿⠿⢿⣧⣀⠋⢙⣛⣿⣟⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣻⡛⠀⢲⡖⠛⣶⡝⠓⠚⣿⣛⣿⣉⣀⣾⣉⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠹⠿⠿⠿⠀⠀⠀⣿⣿⣿⣿⣾⠟⠀⠀⣾⣿⠛⠗⠘⠛⠀⠀⠛⠃⢀⣿⣿⣿⡿⢀⣤⠶
⢶⡄⠀⠙⠿⠿⠛⠷⠂⠀⠀⠀⣀⡀⠀⠀⠀⢿⡟⢸⣿⡟⠀⠀⠀⠈⠛⠀⠀⠀⢀⣤⡴⠶⣤⠈⠙⠛⠉⠀⠘⣿⣾
⣾⠏⠀⠀⠀⠀⣠⡴⢦⡀⠀⠀⢹⡇⣴⠟⠀⠀⠀⠀⠉⠀⠀⢀⣶⡄⠀⠀⠀⢰⣿⣿⣧⣤⣿⡇⠀⠀⠀⡀⠀⠈⢿
⠁⢠⣤⡄⠀⠘⢿⣷⣾⡗⠀⠀⢸⣿⠋⠀⠀⢰⣿⠿⣷⡄⢀⣾⣿⣧⣼⣿⠇⠈⢿⣿⣿⠿⠛⠁⠀⠀⠹⣿⣆⠀⠈
⢠⣿⣻⣷⠀⠀⠈⢻⣯⡀⠀⠀⠸⠟⠀⢰⣶⣄⣻⡄⠈⠀⣾⡟⣿⡿⣻⣿⠀⠀⠘⠃⠀⠀⣀⣠⣴⡄⠀⠘⣿⣆
⣾⡿⠿⣿⣇⠀⠀⠀⠙⠁⠠⣴⡆⠀⠀⠀⠙⠻⠟⠁⠀⠀⠀⠀⠉⠀⠿⠿⠀⠀⠀⠀⠀⢻⠟⢿⣏⠀⠀⠶⠿⠋
⠛⠁⠀⠛⠋⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⢀⣶⡄⠀⣤⣤⡀⢰⣶⠀⠀⠀⢠⣦⠀⠀⠀⠈⢿⣇⠀⠀⠀⢀⡀
⣦⡀⠀⠀⠀⠀⣴⡿⠃⣀⠀⠀⢻⣷⡀⠀⢀⣴⣿⠟⠁⠀⢸⣿⢿⣜⣿⡀⠀⠀⣾⡟⣠⣾⡦⠀⠘⠋⠀⠀⣰⣿⠗
⣽⣿⠀⠀⣠⣾⠿⣷⣾⡿⠃⠀⠀⠁⠀⠐⢿⣿⣁⠀⠀⠀⢸⣿⡆⢻⣿⡇⠀⣸⣿⣿⠟⠉⠀⣀⣤⠶⣤⠀⢻⣿⣄
⠟⠁⠀⠀⠙⠁⣴⡿⠋⠀⠀⢀⣤⣀⡀⠀⠀⠙⠋⠀⠀⠀⠈⠛⠁⠀⠉⠁⠀⠈⠛⠁⠀⠀⠀⠹⣿⣶⣿⠇⠀⠙⠻
⠀⢠⣤⠀⠀⠀⠈⠀⠀⠀⢀⣾⣟⠛⠃⠀⣀⣠⣀⡀⠀⠀⢠⣾⡧⠀⠀⠀⠀⢠⣶⣦⣄⠀⠀⠀⢻⣿⡉
⣷⣌⣿⡆⠀⢀⣤⡀⠀⠀⣾⣟⡉⠀⠀⣼⣿⣀⣼⣿⠆⠀⢸⣿⡄⠀⢸⣷⠀⠈⢿⣯⣿⣿⣶⡄⠀⠛⠃⠀⣤⡀⢹
⡏⠻⣿⣷⠀⠈⢻⣷⡄⠀⠉⠙⠛⠀⠸⣿⣿⣿⡿⠋⠀⠀⠀⠙⠿⣿⡿⠋⠀⠀⠘⣿⣇⠈⠋⠀⠀⠰⣶⣄⣿⡇⠈
⠿⠀⠘⠛⠀⠀⢀⣿⣿⠀⠀⢀⣀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠘⠋⠀⢰⣶⠒⣦⠈⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠘⣿⣿⣷⣶⣤⠀⠀⢰⣿⣡⣶⠆⠀⢠⣿⡿⠛⠻⣿⠆⠀⠀⣼⣿⣿⣟⠀⠀⠈⠛⠃
⠂⠀⠀⠀⠐⢿⣦⡀⠀⠠⣶⣶⣾⣿⣷⡍⠉⠀⠀⣾⣿⣿⡁⠀⠀⠸⣿⣦⣼⡶⠀⠀⠀⢠⣿⣧⣤⡟⠀⠀⢀⣤⣾
⡄⠀⠺⣿⣄⠀⠹⣷⡀⠀⠈⠙⠿⣿⣭⠁⠀⠀⠠⢿⠃⢻⣧⠀⠀⠀⠈⠙⠋⠀⣠⣶⡦⡄⠀⠈⠁⢀⣤⠀⠹⠋⠻
⣿⠆⠀⠙⢿⣶⣶⡿⠁⢀⣤⡀⠀⠈⠁⠀⠀⣶⡄⠀⠀⠀⠈⣀⣀⣀⣠⠀⠀⣴⣿⣿⣦⠏⠀⢠⣦⡈⣿⣷⡀
⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠻⠂⠀⠀⢰⣶⣼⣿⣶⣤⣄⠀⠙⠋⣽⡟⠀⠺⠏⣸⡇⠀⠀⠀⠀⠹⣿⣟⠙⣿⡆
⡟⢀⣤⡀⠀⠀⠠⣾⠟⠙⠃⠀⠀⠀⠀⠀⠀⠈⠹⣷⡉⠁⠀⢀⣼⡟⣀⡀⠀⠀⠉⠁⣤⡄⠀⠀⠀⠘⢿⠆⠈⠀⢀
⣶⡿⠋⢁⣠⣤⣀⠀⠀⠀⣠⣾⠿⢿⠂⠀⠀⠀⠀⠙⠁⠀⠀⠸⠿⠟⠛⠃⠀⠀⠀⠀⢿⣧⠀⠀⢠⣦⣤⡀⠀⢀⣾
⠁⠀⢠⡏⠈⣿⣿⡆⠀⠀⣿⡏⠀⢀⠀⠀⠀⣿⡇⣠⣤⢀⣄⡀⠀⠀⢀⣠⣴⣾⡄⠀⠸⣿⡆⠀⣾⣿⣀⣽⡄⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣳⣶⣾⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡄⠀⠀⠀⢠⡾⣻⠏⣿⢻⣇
⠀⠀⠀⠀⢀⣸⢃⣿⠀⣿⠦⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⢏⠹⡇⡹⡆⠋⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠋⠸⠟⠀⣴⣿⣿⠛⢿⣿⣦⡀⠀⠀⠀⠀⡄⡆⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡟⠀⠈⢻⣿⣿⣦⡀⠀⢀⡡⠐⠊⠁⠀⠀⢀⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣷⠀⠀⣀⣹⣿⣿⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⠀⠀⢹⠀⠀⠀⢸⣿⣿⣿⠀⣠⣄⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⡄⠀⠀⣀⠀⠀⢸⣿⣿⠿⢐⣇⣈⡄⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⢿⣿⣿⠦⣀⠀⢀⣠⠞⣿⡿⠀⠘⠉⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠛⠉⠁⠀⠀⠀⠃⠀⡀⠠⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⣿⠀⠠⠄⠀⠄⠄⠤⠂⠈⠀⠀⠀⠑⣄⢚⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡀⠀⠔
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣇⡀
⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠋⠘
⣿⣶⠀⡀⢀⠀⣾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⣠⠞⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣷⣿⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⡆
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠹⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⢯⣅⣀⣤⣤⣤⠀⢰⢿⡿⣟⠻⣿⣿⡿⠿⣿⣿⣿⣿⡇
⣿⣿⣿⣦⣬⣾⣿⠏⣉⣿⣶⣿⣯⣭⣠⣾⣿⣿⣦⣤⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣿⣿⠋⣙⣿⣬⣿⣽⡏⣉⢹⣿⣿⣿⡿⢷⣥⡈⢹⠿⢿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠿⠿⠛⠛⢿⣿⡿⠛⠓
⣿⣿⣿⣾⠿⣾⡍⠉⢞⣿⣿⣧⣴⣿⣮⣵⣾⣿⢿⢿⣶⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⠃⠀⠈
⣿⣿⣿⣦⢔⣾⣿⣦⣿⣿⡋⠉⠿⠛⣿⡿⠿⢷⣄⣤⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⣿⣿⣿⣿⣿⣏⡑⠀⣩⡿⠿⢿⡾⠿⣿⣠⣴⢸⡟⠉⣿⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⠶⢆⣤⣤⣤⡿⠀⠀⠀⠀⠀⠸⡧
⣿⣿⣍⠀⢎⣻⡿⣿⣿⣇⢈⣷⣶⡀⣸⡿⢿⡟⡛⢶⣽⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⡀⢀⣤
⣿⣿⣿⣷⣿⣏⣈⣃⣼⡟⠹⡿⣿⣿⣿⣿⣴⣿⣿⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀
⣿⣿⣁⠘⠛⢿⣿⠛⣿⣿⣶⠿⢿⣿⡿⠳⣠⣿⣇⡙⢠⣿⣿⣿⣯⠀⠀⠀⠀⣿⣿⠛⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣧⣿⠟⢻⣿⣯⣤⣿⣿⣬⣴⣾⣿⡟⠿⢿⡟⠻⣿⠙⣻⣿⣿⣿⠀⠀⠁⠹⣿⡿⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣟⠿⠿⢿⡗⠐⠒⠒
⣿⣿⣿⣷⣮⡿⠿⣿⣏⠉⡻⡿⠟⠉⠰⠶⣽⣷⣿⢷⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⢸⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⣁
⣿⣿⠏⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⢿⣿⣿⣿⡄⠀⠀⠈⠉⠱⠎⠉⠉⠆⢿⡏⠉⠉⠉⠁⠉⠈⠉⠁⠈⠀⠀⠈
⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⣿⣿⣿⣶⣦⣤⣤⣤⣤⣦⣤⣤⣤⣀⣀⣴⣶⣶⣶⣾⣿⣿⣿⣿⡿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡠⠒⢄⠀⠀⠀⢄⡤⠒⠒⢤⣠⠄⠀⠀⣠⠀⠐⢦⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀
⠀⢸⠀⠈⢻⣧⠀⠀⠀⠀⠀⢸⡟⣦⡜⠀⠀⢰⡟⠀⠀⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⢹⡆⠀⠀⣿⣀⡀⣄⠀⠀⠀⣿⠀⠀⠈
⠀⠘⣿⠗⢠⣿⣧⠀⠀⠀⠀⢸⡧⠋⢹⡆⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⣿⠀⠀⠁⠀⠀⠀⣾⠒⠲⠄
⠀⠀⠈⠀⣼⠘⣿⡆⠀⠀⠀⠘⠇⠀⠘⠁⠀⠀⠛⠳⠤⠤⠔⠀⠂⠀⠀⠿⠆⠤⠞⠋⠀⠀⠀⠟⠀⠠⠴⠀⠀⠀⠿⠄
⠀⠀⠀⢰⠇⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⡾⠶⠒⠒⢻⣧⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⡆⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⣏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⣸⠃⠀⠀⠀⠈⣿⣆⠀⠀⠻⣄⠀⢈⡇⠀⠀⣟⠰⠛⠚⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⠀⠀⣿⠀⠑⡄⠀⠀⠀⣄⠀⠈⡀
⠀⣰⣯⠀⠀⠀⠀⠀⢸⣹⣆⡀⠀⠰⠛⠀⠁⠀⠠⠿⠀⠀⠀⠳⠀⠀⠀⠴⠀⠀⠘⠇⠁⠀⠀⠀⠿⠀⠀⠙⠀⠀⠀⠃⠐⠒⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⡄⠀⠀⠀⠒⢲⡀⠀⠀⠀⠀⠀⣿⡁⠐⢺⣟⠀⠀⠀⢀⠔⠁⠀⠈⠑⢦⡁⠀⠀⢹⡿⠀⠉⣷⠀⠀⣴⠋⠀⠀⠀⠙⢦⡀
⠀⠘⠁⠀⠀⠀⠀⠘⣾⡀⠀⠀⠀⠀⣿⢷⡀⢸⡇⠀⠀⠀⢺⠀⠀⠀⠀⠀⠘⡷⠀⠀⢸⣷⠲⠤⠛⠀⢸⣯⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡄⠀⢀⠀⠀⠘⠧⠀⠀⠀⠀⣿⠀⠻⣼⣷⠀⠀⠀⠹⡄⠀⠀⠀⠀⣸⠃⠀⠀⢸⣇⠀⠀⡠⠀⠀⠻⢄⠀⠀⢀⣀⡾⠁
⠀⠐⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠉⠀⠀⠀⠀⠀⠑⠒⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠉
⠀⢠⣤⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⢤⡄⠀⠀⢤⡄⠀⠀⠀⢠⡤
⠀⠀⣿⡇⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢉⣴⠋⠁⡀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢸⡃⠀⠀⠘⣷⡀⠀⠀⡜
⠀⠀⣿⡧⢤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠳⢗⣆⠀⠀⠀⠀⣿⡃⠀⣞⠿⠤⡀⠀⠀⣸⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠹⣧⠀⡰
⠀⠀⣿⡇⠀⠙⣂⡄⠀⣤⠀⠀⢀⡀⠀⠀⣹⠀⠀⠀⠀⢻⣇⠀⠈⠉⢀⡄⠀⠀⢹⣧⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢿⡦⠁
⠀⠐⠓⠃⠀⠀⠈⠙⠒⠛⠀⠀⠘⠙⠒⠒⠋⠀⠀⠀⠀⠀⠉⠛⠒⠒⠋⠀⠀⠀⠀⠉⠛⠛⠋⠀⠛⠋⠀⠀⠀⠀⠀⠁
⠀⠀⣠⣀⣀⣀⣠⣄⢀⣀⠀⢠⡄⢠⠀⠀⢤⡤⡄⠀⠀⠀⣠⣤⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⢤⢀⠀⠀⣄⠀⠠⣤⣀⣀⣀⣤⡄
⠀⠈⠃⢉⡤⠀⠁⠀⡽⠁⠀⠀⣧⣝⠀⠀⠀⠹⣷⣄⠀⢀⡈⠀⠀⠀⠀⢸⠇⢸⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄⠀⠀⢀⣾⠟
⠀⠀⠀⣾⠃⠀⠀⣴⡇⠀⠀⠀⢸⣯⠀⠀⠀⠀⠙⢿⢢⡏⠀⠀⠀⠀⠀⠁⠀⢸⣻⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣞⠏
⠀⠀⢸⣿⡄⠀⠀⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⢫⣵⣄⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⢀⣼⣿⠀⠀⠀⠀⠀⢀⣭⠂
⠀⠀⠀⢻⣧⣄⠀⢹⣷⣄⣀⣀⣼⣿⠀⠀⢠⡤⢰⣃⠀⠙⣿⣦⡀⠀⠀⣀⡀⠀⠀⠉⠉⠉⢸⢿⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠁⠉⠉⠁⠀⠀⠘⠛⠃⠀⠀⠘⠒⠛⠀⠀⠛⠃⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠐⠖⠓⠛⠛⠲⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⢀⡀⢀⠀⡔⠀⠢⠀⠀⡀⣀⠀⠀⣀⠀⠀⠀⢰⠄⢰⠂⢀⠆⠀⢲⠀⣀⠀⠀⠔⠒⠒⡄⠀⣀⠀⠀⡠⠄⠤⡀
⠘⠁⠃⠀⠀⠁⠀⠀⠀⠄⠘⠀⠀⠑⠂⣤⡈⢿⣵⣮⠣⡈⣄⠾⠲⣤⣘⡠⢃⣀⠀⠀⠀⠀⠘⠨⠁⠀⠈⠀⠀⠀⠸⠀⠀⠀⠉
⠀⠤⢄⠀⠆⠉⠁⠀⠀⠀⢀⠀⠰⠃⠀⢻⣝⣶⣇⣉⢦⢋⣤⣶⣶⣄⣈⡶⣋⠏⣄⠀⠴⠤⠀⠔⠀⠀⠈⢢⠠⠄⠀⠂⠈⠁⠂
⠀⠤⠄⠀⠀⢂⠀⠘⠁⠀⠀⠀⠠⡔⢂⢦⣽⣿⣿⣄⣶⠸⣿⣟⠛⠛⣇⡶⠧⠾⣥⣄⣀⠀⠀⡀⢐⠈⠉⠀⠀⠀⠀⠉⡀⠀⠁
⠀⠀⣠⠴⣶⣿⣿⣷⣶⣤⡀⠀⠀⠙⠣⠿⢿⠛⠛⣏⣉⠭⣿⣶⣾⠟⠃⣸⠒⢾⣿⣿⣿⣷⠄⠀⠀⠱⣠⣶⣶⣾⣿⣳⣶⣤⡀
⢠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣛⣀⡤⠞⣹⠉⡸⣧⢴⣇⣴⡦⠻⣎⢙⠉⠙⠈⠛⢯⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⡽
⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠁⠐⠃⠚⠅⡾⢳⡈⠁⠈⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣶⣄⠀⠀⠀⠀⢠⠞⠀⠀⠙⣄⠀⠀⠀⠀⢠⣴⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⠋⠀⠀⠀⠀⠈⢦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁⠸⡻⠟⠋⠹⡄⡴⠁⠀⠀⠀⠀⠀⠀⠈⢧⢀⠟⠉⠛⠛⡏⠉⠛⠛⠛⠛⠛⠋⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀⣳⡀⠀⡇⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⡀⠀⠲⣭⣯
⠠⣶⣦⢠⣤⣄⢀⣄⡀⠙⠿⠋⢀⡀⣧⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⢾⠁⠀⠀⠀⡏
⠀⢹⠇⠂⠻⠃⠘⠿⠃⠒⠚⠒⢻⠟⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸⣇⣀⣀⣠⣤⣤⠴⢻⠲⠤⠒⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⢾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠉⠉⠉⠉⠉⠉⠉⠀⠐⠀⠐⠒
⢠⣀⣀⣄⡄⣀⢠⣀⣀⣀⣤⣤⠤⠖⣺⡆⠀⢨⡴⣖⣋⠉⠉⠉⠉⠉⠉⣓⣲⢤⢰⠀⠀⣼⠆⣄⡀⡀⣀⣀⣀⠸⠴⡇⠀⠤⠤
⣼⣿⣿⠿⠿⠟⠛⣋⡉⠀⠢⠃⠫⣯⢽⢳⠀⡿⢸⣏⡛⠻⠿⠿⠿⠿⠛⣋⣿⡜⣿⠀⢠⣿⠈⣐⡪⣳⠐⢚⡀⢀⠐⠀⢀⠊⢰
⢀⢢⢤⠤⣷⡉⠉⢤⠐⣾⢫⡶⡂⢠⢸⡘⣷⠇⢸⣿⣿⡆⠀⠀⠀⢠⣾⣿⣿⡇⢻⡄⡼⢸⢰⢢⠢⠄⠡⠠⣤⠳⣞⣀⡸⠱⣅
⠐⢽⡉⢡⣈⡟⣉⣉⣇⣼⠐⠀⠅⣀⠘⣷⡇⠀⠘⠿⢿⣧⣶⣤⣦⣼⣿⣿⡿⠁⠀⢻⡇⣾⢀⠝⢁⡂⣸⡀⠨⠤⠏⠋⣠⣀⣤
⠰⣄⡈⠣⡜⡷⣨⠃⡦⢤⠼⠲⠠⠁⠄⣟⠧⣴⣶⣾⡿⠿⠿⠿⠿⠿⠿⣿⣷⣶⡦⠞⢡⡇⡜⠋⢯⢔⡹⠈⠞⠆⢒⠫⣥⡽⠥
⠀⡁⢈⠀⢿⢻⡁⡈⢤⡾⣤⢺⡜⡼⠀⡜⢦⡀⠉⢪⡻⠭⠭⠭⠭⠭⠭⢋⡞⠁⠀⣠⠏⠄⢧⠆⣸⣁⡄⠁⡘⠧⣾⠤⢃⠰⢰
⠀⡧⢸⠎⠉⢸⡈⢣⡘⠿⠛⡯⡠⠃⡂⠹⣄⠳⢄⡀⠳⣄⣀⣀⣀⣀⣠⠞⠀⡠⢚⡁⠐⣞⡎⢉⠓⢴⠘⣒⢡⣥⡌⣢⢌⠄⢛
⠀⠎⡆⢼⠫⣳⠙⡼⣗⠓⢊⡅⡀⠼⣡⠀⡏⠳⣄⠉⠓⠦⣍⠉⣉⠭⠖⠒⠚⠱⡱⠀⠆⢻⠘⢤⡾⠀⡣⣠⠸⣍⡆⠣⡜⢢⠐
⠈⡄⡱⠼⣥⡥⡜⢫⣘⢾⠄⣹⠁⡄⠀⡴⡁⠸⡈⣉⠙⢲⢬⡛⢁⡴⡞⠛⡩⢠⠁⣶⠀⣟⢉⡌⠩⢹⢴⠈⢏⢸⠰⣃⡨⢶⢸
⠀⠻⠆⠲⠑⠃⠣⠇⠋⠖⠈⠻⠄⠫⠁⠠⠅⠸⠰⠇⠎⠀⠰⠟⠻⠴⠁⠘⠓⠚⠷⠠⠆⠨⠉⠃⠰⠚⠖⠖⠉⠾⠰⠡⠰⠟⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢲⠁⠐⠋⠀⣀⡀⠉⢷⠉⡴⢟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⢀⣎⣀⢠⠔⠀⠀⠀⠀⡀⠀⠀⢁⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠓⠉⠎⡉⠣⠀⡀⣤⠔⠒⠉⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠘⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠚⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠰⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⣀⣤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⣇⢸⣻⣿⡇⠙⢶⡀⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠈⣏⠟⠓⠲⣾⣿⣷⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⡀⢐⡶⢿⡗⢠⡄⠀⠙⢻⡟⣉⣉⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣨⣃⡈⠙⠊⢱⠀⢠⡄⢀⡬⡟⢿⡟⠷⠀⠀⢠⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠬⢭⣀⠀⠀⢢⠀⠀⣈⣿⡟⡮⢁⣄⠐⣜⣳⣿⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣿⢽⣛⡟⠡⠴⠷⣾⣿⣿⣎⠝⠋⠀⢀⡜⠿⢽⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣾⣧⣞⠉⠀⢤⢞⠀⠀⠈⢉⣀⡤⡄⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠈⠋⠤⠴⠿⠿⢟⢵⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣽⣿⣷⣦⠀⢀⡀⠀⣀⠜⠫⠄⠂⢈⣷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⡯⢾⡟⠉⠀⠀⡐⢈⢂⣾⣿⣮⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠙⢳⣸⠃⣍⣷⣤⣤⠾⠛⠁⠈⣻⠟⠉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⠟⠛⠋⠉⠀⠀⠀⠀⠼⠣⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠀⠰⠄⠀⠀⠀⠀⠈⠻⣿⠿⠀⠐⠾⣻⢓⣧⠀⠀⠀⢠⠤⢀⠀⠀⠀⠀⣀⠒⢀⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢤⡀⠰⠦⠐⠓⠸⠃⠶⢰⠦⠒⠘⠛⢶⣶⣿⡄⡀⡍⡉⡁⠰⠃⠀⠀⠀⣴⣦⡤⠀⠀⠙⣉⣿⣿⠏⠁
⠠⢊⣠⠀⠀⠀⠤⠀⠆⠐⠂⠛⡀⠛⢠⣮⣤⠆⠀⠀⠀⠈⢹⠇⠆⠻⠃⠓⠀⠀⠀⠀⠀⣿⡟⠁⠀⠠⠾⠟⠛⠁
⠀⠈⠀⠀⠰⢠⡄⠰⢂⠐⠃⣈⠁⠶⠂⣈⠙⠀⠀⠀⠀⠀⣾⠀⠀⣀⣀⠀⣵⣦⡤⣴⢶⣿⠈⠠⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡄⠗⠈⢁⣄⠋⡀⠒⠀⠉⣀⠉⠁⢠⡤⠚⠽⣏⣻⠟⣡⣼⣿⣿⣿⡇⣓⠿⡻⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠋⡀⠛⣀⠁⠘⠋⣠⣏⣭⣽⢛⣛⠭⠥⢲⣿⠿⣿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠛⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⠢⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢉⡇⡷⠀⢀⡀⠀⡰⢩⢤⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠇⣠⠟⢿⠀⡇⠰⠬⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡼⠋⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠙⢅⠀⠀⠔⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⠙⠲⣄⠀⢸⠀⠀⠀⠀⠉⢀⣿⠀⠋⠀⠀⢠⠇⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠑⣼⠳⢄⣀⣀⡤⠞⠀⠓⠦⠤⣶⠋⢀⡤⠚⠉⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⣂⠀⢠⡾⠃⢀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢹⡖⠁⠀⠀⠀⠈⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠟⠀⠀⠀⠑⠠⠤⠤⠤⠴⠚⠁⠀⠘⣯⣐⣀⣀⣂⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡋⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠁⠀⠀⠀⠀⠀⢠⡾⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣸⣥⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢳⡀
⠀⠀⠀⠀⠰⠏⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⠁
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠀⠀⠐⡄⠀⠈⣇⠀⡇
⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠔⢠⠉⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⠒⠊⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣶⣶⠾⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠔⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣾⣿⡀⢰⢁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠀⣰⣾⣿⣳⣿⣿⣿⣿⣿⣿⣻⣿⣉⣻⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣚⡂⠠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⠉⢹⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⣈⡉⠁⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⢡⡄⠀⢀⣀⣠⣤⣶⣾⣿⡿⢱⠄⢀⠀⣾⠠⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⢀⡇⠀⠰⠀⣀⠀⠀⠒⣀⠲⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⡇⠀⢠⣿⡀⠚⡍⠛⠢
⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⣴⣷⠔⠉⢛⡥⡤⣀⠀⣛⠒⠋⠉⡜⠛⣒⠿⠛⠋⠉⠌⡁⠩⠩⣠⠄⡈⠉⡓⠴⠸⢿⡇⠛⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠧⠘⠉⠁⢀⠔⡅⢀⠈⠁⠀⠁⠀⠸⠓⠒⠉⠀⢀⠄⠘⠆⢀⣿⠀⠀⡀⠉⠈⠑⡷⢤⠀⠀⠈⡖⠉
⠀⠀⠀⠀⠀⠄⠀⠙⠤⠊⠁⠀⡀⠂⠀⣼⢰⡄⠘⠃⠀⠀⠀⢠⡀⠂⢀⠀⠀⣿⢸⣿⡄⣀⣀⠀⢀⠀⠂⠀⡆⠀⠀⠇
⠀⠀⡆⠀⠂⠀⠀⠠⢀⣀⠀⢀⣤⣤⡤⣭⡁⠀⠰⠆⠀⣠⡄⣾⡇⠲⠵⠄⠜⠛⠛⠉⠉⠀⠄⠈⠀⠀⠈⠀⡇
⠀⠀⠁⠨⠁⠠⢈⠀⢀⠀⣀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠀⠒⠠⠅⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠿⡷⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠃⢂⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⠀⢠⠞⢉⢤⡈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⢠⡏⠹⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⡑⠀⢀⠉⠀⠀⠀⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣠⡄⢰⣅⢀⡀⠀⢀⠙⠀⠀⠀⣠⠀⠀⠀⠀⠂⢹⠃⢀⡾⠁⠀⠛⠍⠀⠀⣀⣷
⠀⠀⠀⠠⢀⡸⠀⡠⠆⡠⠼⠿⠨⠦⣤⣄⡙⠲⣼⡄⠀⠂⣤⣤⣿⠶⠯⠴⠿⣿⠶⢤⣤⣤⣭⠟
⠀⠀⠀⠀⠙⠀⠀⠇⣀⠃⢀⣨⣤⡶⠶⣿⣿⣷⡘⣿⣆⣼⡿⠛⠉⠐⠀⠈⡀⢈⣳⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠛⡸⠋⠀⠁⡀⣀⡀⠈⢿⣿⣾⣿⢏⣴⠶⠛⠛⠓⢒⡀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠻⠆⡮⠚⢿⡰⣶⠀⠙⣿⣿⡟⢀⣠⠘⡄⠁⢈⣉⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠋⡀⠉⠁⠂⠀⢘⣿⠇⠐⠂⠁⠀⠀⠉⠘⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡿⣹⡆⠀⠉⠀⠈⠀⢨⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⠄⣸⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠟⡁⠀⣼⠞⣧⡤⢸⣦⠀⣳⡶⠶⠀⣬⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁⡸⡝⠀⣈⣵⡄⠈⢬⠃⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠘⣤⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⡀⠐⠄⠀⡄⠀⣤⠀⠾⢿⡏⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠨⠀⣁⣤⢀⡀⢹⠀⠈⡀⢻⡆⣿⠀⢰⡄⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐⣚⠿⢋⡴⣾⣿⢼⠊⠰⠧⣿⣇⣹⣶⣾⣇⣻⣿⣦⡀
⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⡇⠉⠁⡀⢸⠀⠙⡟⠸⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠒⠿⠿⠿⣿⣿⣿⣿⣿⠟⢃⣷⣦⣤⣼⣼⣧⠀⠀⠛⠀⠸⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⣉⠑⠢⡀⠀⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠂⣇⠀⢀⡀⠤⠀⠘⠀⠀⠇⠀⠠⠌⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠇⠀⠃⠘⠆⠘⠄⠀⡄⠠⡄⢰⡄⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠆⠀⢠⠂⠀⡄⠉⠀⠀⡇⠀⣀⠣⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠁⠀⡀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠋⠐⠒⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀⠀⢄⠀⠠⡄⠀⠀⣑⠀⠀⠀⠀⠂⣠⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡇⠀⡌⠳⠀⣧⠀⣀⠈⡇⠀⠜⠀⠀⠈⠀⠰⠀⢠⠄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⢁⠀⠀⠁⠀⠁⠀⠃⠀⠀⠀⠆⠰⠄⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆⠀⠰⠀⢸⠀⠀⢣⠀⡄⠀⠀⡄⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⡘⠀⠀⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠂⠀⠀⠀⠂⠀⢀⠀⠀⠀⢀⠁⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢻⠿⠻⣿⣿⣷⣶⣶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠻⠿⠿⠿⠿⠛⠋⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣟⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⢾⣛⣿⣭⠽⠿⡟⠚⢟⢷⣄⠄⠊⠁⠀⡀⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢽⡿⠛⣿⠉⣷⣿⣀⣁⢹⡤⠈⢧⢿⣧⠤⣤⡀⠀⣤⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⣁⢠⢿⢈⣿⣿⣧⡁⢣⣿⣰⣸⣬⣿⣄⣾⣧⠴⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢸⣇⢸⣙⣿⣿⡿⣿⡟⣿⣻⣯⣵⣶⣷⣿⠿⠘⡘⢸⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⣾⣟⣿⣯⣽⣿⣶⣿⣿⢿⢿⣿⣿⡿⡟⣿⡀⡆⠃⢁⣗⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡟⣿⣧⣘⢻⡏⣿⣿⣿⣾⣿⣿⡟⠁⢧⢹⣧⣰⡰⠐⢸⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢳⠈⢻⠿⣿⢿⠛⢿⣿⡏⢀⣀⣷⣤⣼⣼⢿⣿⡷⠷⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⡆⠘⣆⣠⣼⣦⣾⣿⡿⢿⣿⣛⣿⣿⡷⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⢻⣿⣛⣿⣩⣿⣤⠿⠳⢻⡟⠛⠛⠁⠀⢸⣧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠙⠛⠛⠉⠈⢶⣶⣾⣿⣥⣀⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣦⠀⠀⠀⣴⣾⣿⣿⣿⣿⡿⠿⠂⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠛⠛⠿⣿⣷⠟⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣄⣠⣄⣤⣤⣴⣿⣿⣿⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⡀⢰⡀⠀⢢⣌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡄⠀⠚⠛⠒⠴⣯⡜⠂⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⢛⠀⠐⠁⠀⠀⠈⠛⢦⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢠⠄⢠⡀⠰⠋⢀⡌⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠁⠀⠄⠀⠈⠀⠀⠑⣆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠘⠀⠀⠀⡀⠀⠀⠈⣿⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⡸⠀⠀⠀⠀⠠⠀⠀⠀⡅⠰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠁⢀⠀⠀⠇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠆⠀⠀⠠⠀⠀⠀⠰⢌⣄⡀⢈⠀⢸⠎⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠨⡀⣀⡀⠀⠀⠀⠀⠖⠒⠀⢀⣈⢙⣧⣸⡀⠋⠄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀⠀⠀⠀⠀⠀⢂⡀⠀⠐⠈⠀⡠⠀⠘⢙⢷⣅⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⢸⠉⠓⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈⠀⠀⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢃⢸⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠈⠀⠀⠀⠀⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢘⠀⠀⠀⠀⠠⠀⠠⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢘⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢘⠀⠀⠀⠀⠉⠀⠈⠠⠄⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⢰⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠘⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⣿⡇⠀⠀⠂⠀⠀⠠⠀⢀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿
⣿⡇⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⣿⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠐⠀⠐⠘⠀⠀⠄⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈⠐⠀⢀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠆⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠄⠉⠀⠀⡀⠀⠒⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⣿
⣿⡇⠀⡀⠁⠀⠀⠘⠀⠂⡀⠁⠀⠃⠐⠀⠄⠀⠀⡄⠀⠄⢀⠀⠀⠀⠄⢀⠀⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⡇⠁⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿
⣿⡇⠀⡏⠁⠀⠁⠃⠀⠐⠀⠂⠀⠁⠀⠀⠁⠀⠐⠠⠀⡁⠂⠀⠀⠀⠂⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠄⠀⠄⣿
⣿⡇⠀⢀⠁⢀⡀⢈⠀⠀⠀⠀⢠⠀⡀⢀⡀⠀⡀⠀⢀⠄⠀⢀⠀⠀⠀⡀⠀⠀⠀⡁⠀⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⡀⠀⠂⣿
⣿⡇⠀⠈⠀⠀⠀⠀⠀⠁⡀⠐⢀⠀⠁⠀⠈⠀⠁⠀⠀⠀⠁⠈⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣿
⣿⡇⠀⢰⠠⢠⠀⠆⣀⠀⠈⡀⠠⠀⠀⠠⢀⠀⠄⠀⠒⢀⠀⠄⡀⠀⠀⠀⠈⠠⠘⠀⠄⠐⠀⠄⠂⠀⠀⠈⡀⠀⠀⠀⠄⠈⠀⠃⣿
⣿⡇⠀⢸⣀⣀⣀⣀⢂⣐⣄⣀⣀⣐⣀⣀⣀⠀⣀⡀⢂⠀⡀⣀⠀⠀⡀⢀⠁⠀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣥⣿

⠀⠀⠀⠀⠀⣰⢣⠃⠀⠀⢀⣸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⡇⡇⠀⡌⠀⠀⣛⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠃⢰⠀⠀⠀⠸⣿⣿⡗⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⣽⡄
⠄⠀⠀⠀⠀⢻⡀⢰⠀⠀⠀⠀⣻⣿⣿⣄⠀⠀⠀⣿⣦⡀⡀⡀⣾⡟⠀⠀⣠⣼⣿⣿⣧
⣧⠀⠀⠀⠀⠀⢳⡀⢆⠀⢀⣶⣿⣿⣿⣿⣤⣤⣤⣻⣿⡿⣷⣿⣿⣅⣶⣿⣿⡟⣿⣿⢯⢃⡀
⣿⣧⡀⠀⠀⠀⠀⠙⢌⢲⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⡿⡌⠛⢸⣿⡿⠉⠙⠜⢋⣵⣿⣿⣷
⣿⣿⣷⣶⣦⣄⣀⢀⣼⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣾⣤⣼⠿⢤⣤⡴⣚⠛⢛⣛⡿⠟
⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠛⠀⠀⠈⣿⣿⣯⣿⠿⣿⣿⠁⠀⠀⠉⢅
⠀⠀⠀⠀⠅⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⡀⠀⣠⣿⣿⣟⢃⣶⣤⣭⣔⢦⣄⠀⠀⠱⡀
⠀⠀⢠⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⢫⣿⣿⣿⢿⣿⣷⡄⠀⠐⡀
⠀⠀⢼⢷⣿⣿⣿⣿⣿⣿⡿⠙⠛⠉⠙⣿⣿⣿⣿⣿⣟⣿⣿⣿⣦⣾⣿⣿⢹⣿⣿⣿⠛⠀⠀⠱⠀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⢈⣿⣿⣿⠿⠛⠉⠻⢧⣅⣈⣿⣿⣤⣿⣿⠃⠀⢀⣴⣶⣿⠇
⠀⢠⠷⣿⣿⣻⣿⣿⠟⡃⠀⠀⠀⠀⠀⡿⠛⠃⣥⣿⣧⡄⠀⠀⡀⢰⣼⣿⣿⣿⣹⣾⣟⠞⣻⡿⡏
⠀⠀⢠⠟⢿⣿⣿⣧⣦⢉⠁⢀⠆⢸⣾⣃⠀⠀⠉⠿⠏⠁⠀⣼⠇⣸⣿⣿⣿⣿⣶⠁⠈⠉⠁⠀⡁
⠀⢰⣧⣦⣶⣿⣿⣻⠇⠘⠐⣿⠀⠀⢻⣿⢷⢦⡀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠋⣠⡖⠀⠀⠀⠀⠀⠃
⠀⢰⣿⣿⣿⣿⣿⡁⠀⣆⢤⠉⠀⠠⣾⣿⣅⣁⢡⠀⠀⢀⣤⣆⢀⣙⠋⠉⠀⠁⠀⠀⠀⠀⠀⠰
⠀⠈⠇⣿⣹⣿⣿⣿⣦⡘⣤⡀⠀⣼⣊⣿⣝⠈⠳⠀⠀⣿⡻⣿⠇⠀⠀⠀⠠⣀⢠⣀⠀⠀⢠⠁
⠀⠀⠀⢻⣷⣿⣿⣿⣿⣿⢛⣿⠛⠿⣿⣿⣿⣆⠀⠀⠐⣿⡿⠯⠀⠢⠀⠀⠀⢽⣿⣿⣷⣶⣧⣤⣀⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠆⢤⡤⢘⠻⣷⣄⡀⠀⠀⠙⣷⢂⠀⠈⠢⡀⠀⠀⢋⠟⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⢰⢿⢿⠿⣿⣫⣿⣿⣿⣿⣆⣿⣿⣾⣿⣿⣦⣈⢿⣿⣿⣆⡀⠀⠀⡡⠖⡁⠀⠀⠈⠈⠙⢿⣿
⠀⠀⠀⠀⢸⣆⡴⣷⡿⣿⣿⣿⣟⢿⣿⣿⡿⣿⡝⡿⣿⣿⣿⣿⣿⣿⣶⡶⠉⠀⠈⢊⠲⡀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠟⢰⢻⢹⢻⢉⣎⡾⢿⣿⣿⠻⢿⡿⡸⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢃⠙⣆
⠀⢤⠄⠀⠀⠀⠘⡆⡸⣼⠞⠋⠀⢠⢏⡿⠀⠀⠙⣧⡼⠏⠉⠀⠈⠹⣿⣿⡇⠀⠀⠀⠘⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢼⠃⠀⠀⠀⢸⠞⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⢰⠀⢠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⣿⣿⣯⡅⠀⠀⠂⠀⡌⡞

⠀⠀⠠⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡴⠦⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⢤⡄⠀⠀⠀⠠⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡷⠈⠄
⠀⠀⠀⠁⠀⠀⠀⢮⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⣼⣯⣀
⠀⠀⢠⣄⠐⠀⠀⢶⢉⡀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢀⡄⠀⡀⢀⡀⠀⠒⢠⠂⠂⠀⠀⠀⠘⢁⣽⣅
⠀⠀⠀⠙⠢⣀⠀⠈⢏⢧⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠉⠁⠀⠀⢀⠈⠁⠀⠀⠀⡤⠀⠈⡁⠰⠆
⠀⠀⠀⠀⠀⠈⠓⠀⠈⠯⠢⠀⠀⠀⠈⠋⠀⠀⠂⠀⠀⠀⣧⡆⠀⠂⢶⠀⠀⠀⠀⢀⡀⠀⠀⠙⠃⠶⣄⠀⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⡰⠂⢠⡄⠀⠀⠠⠰⣆⠀⠀⠀⡧⠀⠀⠈⣧⠀⠀⠀⠈⠡⠋⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠦⠋⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⢆⣀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠙⢂⡀⠀⠀⠀⠀⠉
⠀⠀⢰⠃⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡂⠂⠀⢀⡤⠶⠟⢀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⣠⠀⠀⢀⡤⠖⠂
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡄⣸⢃⣤⡦⠀⠀⠂⠀⠀⠀⠔⠀⠠⠄⠀⠀⠂⠀⠐⠠⢶⠀⠀⠀⡾⠀⠀⠀⠂⡀⣠⠄⠀⠀⠘⡆⢨⡤⠖⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠙⠛⠙⡆⠐⠂⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠐⠳⠀⠓⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠁
⠀⠀⠢⢂⡀⠀⠠⠀⠀⠀⠈⢀⠀⠀⡀⠀⠠⠀⡀⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠋⠉⣉⣉⡓⠦
⠀⠀⠀⡈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣼⠁⢰⡇⢄⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠠⠀⠀⠀⠘⠇⠀⠈⠙⠻⢛⢫⡢⡀
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⠆⠐⠐⠈⠘⠃⠀⠾⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠑⠒⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠿⠗⠈⠢⢑⡇
⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⡟⠀⠀⠀⠀⠀⢰⠀⠀⢀⣀⠴⠒⢦⣀⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⠠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢢⡄⢀⣠⠴⠺⣁⠄⢈⢀⠄⠀⠀⣀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠁⠈⠀⠀⠆⠀⠀⠀⠀⠈⠂⠀⠒⠒⠁⠀⠀⠸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠸⠁⠌⠁⠀⠀⠘⠀⠘⡄⠀⣦⣠⣟⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣶⣦⣤⣄⡀⠀⠀⠘⡆⢀⣴⣶⣤⣤⣤⡀⠀⠀⠀⣠⠀⣠⣤⣄⣠⣤⡀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣄⡀⠀⠀⠀⠀⠀⢀⣤⣦⣀⣤⡀
⠘⠓⠾⠻⢿⣿⣿⣿⣿⡆⠀⠀⠙⠛⠛⢿⣿⣿⣿⣿⠀⠀⠀⠷⣴⣿⣿⣿⣿⣿⡗⠀⠀⠀⢮⣴⡿⢿⣿⣿⣿⣷⠄⠀⠀⢠⣀⣾⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠋⠻⡿⠛⠷⠀⠀⠀⠀⠀⠾⠛⠻⠿⠛⠓⠀⠀⠀⠀⠀⢸⡿⠻⠟⠛⠀⠀⠀⠀⠀⠀⠻⢿⠟⢻⠇⠀⠀⠀⠈⠉⠀⠟⠙⡿⠛⢿

⠀⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀
⠀⠀⢰⡿⢿⣿⣿⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷
⠀⠀⠾⠀⡼⠛⣿⣿⠿⡄⠀⠀⠀⠈⠣⣼⠿⢿⡿⠷⠆⠀⠀⠀⠹⢏⢹⣿⠿⣿⠿⠀⠀⠀⠘⠒⠀⠙⣿⠻⣿⡁⠀⠀⠐⠯⠁⠀⠸⠿⡿⠿⢿
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠻⣧⡀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⢿⣄⠀⠀⠀⠀⠀⡉⢠⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⢻⣦⠀⠀⠀⡰⠁⣾⠃⠀⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠛⢀⡠⠀⠈⢿⡄⠀⢻⣧⠀⢰⠇⣸⡏⠀⢸⣿⠀⠀⠀⢂⠄⠉
⠀⠀⠀⠀⠻⣿⣛⠛⠻⠷⢶⣤⣄⡒⠉⠲⣄⠘⣿⡀⡄⢻⣆⠈⢰⡿⢀⠆⣼⡇⠀⣠⠔⠁⠀⣀⣤⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠈⠉⠻⢷⣤⡈⠳⣼⣧⠰⠈⣿⣧⣿⢁⡞⣠⡿⢀⠜⣁⣤⡶⠟⠋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠠⣀⠉⠻⣦⡙⣿⡆⠃⢸⣿⡇⠎⢠⣿⣷⡷⠟⢋⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡙⢦⡈⠻⣿⣷⣿⣿⣿⣿⣶⣿⠟⢋⡠⠖⣁⣴⠿⠋⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⢲⣿⣿⠏⣁⠉⠻⣿⣿⡎⣁⣤⣾⠿⠥⠖⠚⠉⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⢂⣴⣿⣿⣿⡏⢀⣭⠖⢀⣿⣿⣿⠿⠿⠿⠶⠶⠶⠶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠁⠀⢀⣤⣾⠟⠋⡩⣿⣿⣷⣤⣥⣤⣾⣿⡿⠉⠈⠛⠒⠀⠀⠂⠀⠀⠀⠀⣉⣉⣛⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣶⠿⠋⠔⣪⣵⠞⣻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⠶⠶⠶⠶⠶⠿⠿⠿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⣤⡶⠛⠉⢀⣴⣿⢃⠏⣸⣿⣿⢰⠀⢿⣿⣌⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠞⠋⠁⠀⠀⡠⠊⣼⡏⡎⢰⡟⢸⣿⡀⠆⠸⣿⢻⣆⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⢀⠜⠁⢰⡿⡘⢠⡿⠀⡟⢻⣧⠘⡀⢿⡄⢿⡄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠑⠀⠀⣾⠃⢡⡿⠁⢠⠇⠈⣿⣆⠀⢸⡇⠸⣷⠀⠀⠳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡿⢠⡿⠁⠀⡞⠀⠀⠘⣿⡄⢸⡇⠀⣿⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢇⣾⠁⠀⠰⠃⠀⠀⠀⠹⣿⣸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠃⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣼⡇⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢷⢀⣿⣡⠞⠫⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢸⣿⡿⠋⠆⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⣸⣿⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣿⣧⡾⠋
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠉
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀
⠀⢈
⠀⠈

⠀⠀⠀⠀⢀⢻⡀⠀⢸⡇⢸⠆⢠⡆⠀⠘⠀⣼⠁⢲⡆⠀⣏⡌⠃⠀⣾⣉⠛⢠⡞⠁⠘⠇⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⣄⠀⠀⢰⠀⠀⢀⠀⢰⠍⢀⠀⡀
⠀⠀⠀⡀⡂⠂⣷⡀⠘⣇⣠⡆⠈⣇⣀⣤⠀⣿⡀⠸⠇⢀⣏⢁⡤⠀⣿⡙⠀⠘⠧⠀⣰⠇⠀⠀⢸⢰⡀⢸⠀⠸⠀⢧⣀⠐⠇⢸⡀⢻⣈⠀⢸⠀⢉⣐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠙⠃
⠀⠀⠀⠀⢀⡄⠀⡄⠀⢀⡄⠀⢀⡀⠀⢀⠀⣀⠄⠀⣀⠀⠀⠀⣠⡀⠀⣤⡄⠀⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⢸⠦⠀⡇⠀⢸⠀⠀⢸⡇⠀⣿⢾⡁⠀⠀⡇⠀⠀⠀⠉⢧⡀⢹⡇⠀⠉⢣⣼⠀⠀⠀⠐⠠⡀⠀⠂⢠⠀⢸⣠⠂⠀⣥⠀⠴⠒⡔⢢⠀⠰⠲⡆
⠀⠀⠀⠀⠘⠆⠀⠳⠀⠸⠆⠙⠘⠁⠄⠷⠀⠸⠦⠀⠓⠠⠏⠐⠆⠈⠀⠸⠇⠀⠄⠀⠻⠀⠀⠀⠀⠄⠃⠈⠀⢸⠄⠸⠌⠦⠀⠛⠀⠣⠀⠇⠺⠀⠐⠔⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡴⠀⠲⣄⠀⠰⠀⡄⢀⡄⠀⢠⡀⠀⡆⠀⡆⠀⢄⠐⠆⠀⠀⡖⠲⠀⡔⠀⠰⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⢀⡀⠀⡀⠀⣄
⠀⠀⠀⠀⢃⠀⢀⠟⠀⢾⠈⠁⠘⡇⠀⢰⠇⠀⡁⠸⡅⠀⡀⠙⡆⠀⠀⡇⠀⠀⢇⠀⣠⠀⠀⠀⠄⢨⡅⢸⠀⣳⠀⡃⢨⠀⢸⠉⠁⡐⣍⠀⣎⠀⣆⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⠥⠀⠀⠉⠀⠉⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠀⢸⡉⠁⠀⠉⢩⡄⠈⠁⠀⠉⠁⠀⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢥⠀⠊⠀⠀⢧⢀⠿⣀⠎⠀⠈⢶⠀⠀⠈⢣⡠⠃⠐⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⢀⡄⢤⠠⠀⢠⡤⠀⢠⠀⡄⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠋⠀⠟⠀⠠⡄⠀⢷⡄⠀⠀⣇⠀⠀⢼⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠟⠈⠏⠀⡄⢣⡀⠈⡟⠀⢰⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡄⢱⠀⠀⠀⡇⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠠⣍⠀⠀⢠⡎⡀⠀⠀⠉⠹⠁⠀⠘⢤⡇⠀⠀⡄⢸⠀⠀⢀⠈⡇
⠀⠀⠀⠀⠀⠀⠰⡇⠐⠀⠀⠀⣇⠀⠀⠀⣮⣤⠀⠀⣀⡹⠀⠀⠐⢺⡗⠀⠀⣀⡸⠃⠀⠸⣀⡗⠀⠀⢠⠇⠀⠀⢰⣈⡗⠀⢀⣩⠞⠀⠀⠸⠀⠃

⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠄⣷⠀⠀⡇⠀⠀⣠⠀⠀⠀⠀⠀⠀⢠⣰⠆⠀⠀⠠⠂⢈⣀⠢⠀⠠⣾⠲⠄⠀⣆⡆⢠⠆⠀⠀⠀⣴⡦⠀⢲⡄
⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⢀⡀⠀⣀⠁⠀⢀⡁⠀⠉⣿⠙⠀⠴⠶⠀⢰⡈⢳⣼⡁⠀⡇⢰⣡⡟⣰⠃⢀⢨⣿⠄⠀⠈⡡⠃⣰⣲⠀⠀⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠦⠀⠉⠀⠸⠁⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠙⠤⢉⣉⠕⠁⠈⠙⠋⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣹⣷⢤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⡇⣧⣿⣿⣼⡯⣶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣷⣿⣇⣿⣿⣿⣿⣿⣿⣶⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡦⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢸⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡏⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟⡇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢻⣿⡟⣿⣿⣿⣿⣿⣿⣿⢼⢸⣿⡇⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠟⠃⣷⡜⡏⣿⣿⣿⣿⣿⢸⠘⡇⡇⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠙⠃⡇⣿⣻⡿⢹⠹⢸⠀⠇⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡇⣿⠃⠈⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡙⠂⠙⠢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣯⡑⡄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣷⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⡆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⢠⣴⣦⣴⣶⣶⡤⠤⠄⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠉⠛⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢿⣿⣿⣿⣿⡟⣼⠋⠀⣿⣿⣿⣿⣿⣿⣿⣯⡚⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠏⠀⠀⢠⣽⣶⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⢹⣧⡿⠇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠋⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⢀⣠⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣾⡟⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣯⡿⠋⡀⠀⣿⣿⣿⣿⣿⣿⣟⡿⠛⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣶⡿⠁⢠⢸⣿⡟⠻⣏⡁⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠟⠀⠀⠄⢸⣿⠁⠀⠈⢍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣠⠴⢾⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⠁⣸⣿⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⡘⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⠋⠀⠀⠀⠀⠀⠁⢀⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⣰⠏⢷⡀⠀⠀⠀⠀⠀⠀⡘⠋⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⡀⢠⡶⠛⠛⠻⠶⣦⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⠙⡿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⢀⣿⡁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣴⠟⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⣙⣉⣁⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢰⠞⠉⠈⠙⢷⡀⢀⣤⣤⣤⣄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣟⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣧⠀⣰⣾⢲⣀⡿⠀⠀⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⢦⣿⣿⣿⣿⡇⠀⣾⣿⣋⣥⢸⡇⠀⠀⠀⠀⠀⣦⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠉⠉⠉⠀⠹⣆⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣰⣿⣆⠀⠀⠘⠓⣲⡶
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⢰⡏⠈⠻⠿⣶⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣗⡀⠀⠀⠀⣰⣏⣀⣀⣀⣀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣦⠀⢰⡟⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⣇⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⡿⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘
⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⣰⠋⠛⠀⠀⢠⣏⠉⠉⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⢀⣤⠾⢷⣄⠀⠀⠀⠀⢻⡄⠀⢀⣴⠟⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⡾
⠀⠀⠀⢀⡴⠋⠁⠀⠈⠛⢷⣦⣤⣤⣤⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢀⣿⠁
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢇⠀⠀⠀⣾⠁
⢠⡾⢀⣤⣾⠇⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⡏⢸⡆⠀⠀⣿
⣮⣴⠟⢡⡟⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⠀⢸⡇⢸⡇⠀⠀⢹
⠛⠁⠀⢸⡇⢀⡄⠀⠀⣤⠀⠀⣿⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⡿⢀⡿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⢸⡇⣼⢷⡀⠀⣿⡄⢠⡟⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⣀⡈⣃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠀⠀⠀⢠⠇⣼⠃⠀⠀⣿⠀⢀⡟
⠀⠀⠀⠘⣷⡿⠘⣧⠀⡏⢳⣀⡇⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠉⠛⢻⡄⠀⣿⡿⠶⠶⠶⠾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡏⣼⠏⠀⠀⢰⡏⠀⣸⠇
⠀⠀⠀⠀⠙⠗⠀⠘⢷⣧⠀⠛⠓⠀⠀⠀⠀⠀⠀⣸⠇⢸⡏⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠛⣧⠀⠀⠀⣸⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣰⠏⢠⡿⠀⠀⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠴⣶⣠⠏⢠⡶⠾⠷⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠛⠶⣿⡁⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠀⢸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⣷⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⣧⡞⠁⠀⢀⣀⣀⣸⡃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡏⠉⠛⠻⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣄⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⢻⣄⡾⠁
⣀⣀⣀⣄⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣈⣡⣀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⡿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣼⣧⣠⣧⣠⣶⣦⣥⣼⣿⣱⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣦⡄⣠⠃
⠀⠀⠀⣿⡿⠃⠀⠀⠠⣆⡀⠀⠀⠀⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⢠⡄⠀⠀⣰⣆
⠀⠀⠀⢿⣧⡀⠀⠀⣠⢿⣿⠀⠂⠤⠾⣿⠉⠀⠀⠀⠀⠘⣷⡄⠀⠀⠙⣶⡏⠛⠈⢿⣶⠢⡄⢸⣷⣠⣄⣠⣿⡆⠈⢻⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠺⣿⠀⠀⠈⢀⣼⡟⠀⠀⠀⢀⡿⣧⣀⣀⡀⠀⠴⣻⣷⡀⠀⠀⣼⡿⠀⠀⣾⣿⡀⠘⠟⣿⠇⠀⠀⣼⡏⠀⢸⡏⠀⢠⡄⣨⡉
⢀⡀⠀⠀⠻⠀⠀⠀⠈⠹⡗⠀⠀⠀⠸⣇⠈⢉⡈⠁⠀⢻⡉⠉⠳⠤⢼⠻⣦⠀⣸⠃⠘⠧⣄⠘⣿⢦⡄⢸⡇⠻⡄⢸⡄⠀⠰⠆⠶⠶⠆
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠉⠀⠉⠀⠀⠉⠁⠈⠁⠈⠁
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄
⠀⠀⠀⠀⠁⠀⠀⠙⠋⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣸⡿⢋⣀⠇
⠀⠀⢀⣶⣿⣿⣦⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡇⠀⠀⠀⠀⣄⣀⠘⠿⠿⠟⠁⠀⠀⢀⣤⡔⠁⠀⢀⡾⠁
⠀⠀⠘⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣤⣴⡄⠀⠀⠹⣻⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⢀⡴⣋⣴⡆
⠀⠀⠀⠀⠉⡙⠀⠀⠀⠀⠀⢠⣤⣤⡈⠍⠀⠸⠤⡄⠀⠀⡀⠑⢀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠙⢿⣿⠿⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⣠⣶⣦⣤⠄⣤⣤⣀⠀⠀⣤⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⢠⣄⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣿⣿⠿⠿⠣⣍⣭⣯⣴⠂⠉⢸⣿⡻⣿⣿⣿⣿⣟⣿⣿⣿⣿⡇⢀⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠽⠁⠀⠀⠀⢿⣿⣿⣿⣿⡂⢰⣿⣿⡎⠉⢉⠙⢉⠙⡛⡿⣿⡇⢺⡇⠀⠀⢀⢆⣀⡝⢻⣿⣿⡇⠸⠿⠿⠿⠇⢀
⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⢸⣿⣿⣿⣿⡇⠘⣿⣿⡇⢲⡀⢰⠆⠀⠃⠷⣸⡇⢸⣧⡀⠀⠻⠟⠙⢨⢁⣿⣿⡇⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢈⣵⣤⣶⣶⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣦⡉⠻⠄⠀⠀⠀⣸⡇⢸⣿⣷⣌⠒⠀⠐⢢⣿⣿⣿⣇⢀⣀⣀⣀⣀⣐⡀⠄⡠⠀⣠⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⣀⠄⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⠏⠤⠊⠀⠠⠈⠁
⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⢠⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⡄⢨⣭⣤⣤⣤⡬⢭⣭⣭⣭⣭⡅⢠⣭⡉⠉⠉⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣿⣿⣿⡟⢡⣶⣶⣝⣿⣿⣿⢸⣿⡿⠛⠉⠉⠙⠛⠛⠿⣿⣿⡅⢸⣿⣿⠟⠋⠀⠀⠹⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⣼⣿⣿⣿⢸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⡇⢸⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⡗⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣜⣿⣿⡟⣼⠟⢻⠘⣿⣷⣄⠀⠠⡀⠀⠁⢀⣼⣿⡇⢸⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡧⠀⠈⡇⡀⠀⠀⣿⣿⣿⣷⣤⣀⡀⣴⣿⣿⣿⡇⣾⣿⣿⣄⡀⠀⣠⣾⣿⣿⣿⡇⢸⣿⣿⣄⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⠀⠀⠐⠃⠀⠀⠿⠿⠿⠿⠿⠿⡧⠸⣿⠿⢿⠇⠿⠿⠿⠿⠿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣟⠀⠀⠀⣰⣿⣶⡿⠃⣠⣶⠀⡶⣶⣶⣶⣶⣶⠀⣶⣶⣶⣶⡆⠰⣶⣶⠶⢶⣶⡆⢶⣶⣶⣶⡆⠀⠰⣶⣶⣶⣶⣤⠠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣿⡀⠀⢀⣿⣿⡟⢀⣴⣿⣿⠀⣿⣮⣻⠉⢹⣿⢰⣿⣿⣿⣿⡇⢺⣿⣻⠇⠀⢻⡇⣼⣿⢿⣿⡇⢠⣆⠈⢿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡷⠀⢿⣿⠿⠁⣼⣿⣿⣿⠀⣿⣿⣷⢀⣴⣷⣾⣿⣿⣿⣿⡇⢸⣿⡟⢀⣤⣾⣿⣿⢃⣾⣿⣷⠸⣿⣷⣦⣙⢿⣿⣿⢫⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣷⣤⣻⠷⠀⢠⣿⣿⣿⣿⠀⣿⣿⡃⠈⢻⣿⣿⣿⣿⣿⣿⡇⠚⠋⠀⠉⣉⠻⣿⡏⣾⣿⣿⣟⢸⣿⣿⣿⣿⣦⣈⢻⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⢰⣿⣿⣿⣿⣿⠀⡟⡾⢀⠾⠷⠽⠿⠿⠿⠿⠿⠇⠀⠓⠄⢠⣾⣿⣯⣼⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠿⠿⠿⠷⠀⠸⢿⣿⡿⠿⠿⠂⠿⠻⠉⠒⠒⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠛⠋⠀⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⣶⣶⣶⣖⣶⣶⣶⣶⣶⣶⣶⠀⢶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⡄⠰⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⢠⣦⣤⣤⡤⡆⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⣽⡻⣿⠿⠿⣿⣿⣿⣿⣿⣿⠀⢮⣝⡛⠉⠙⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⢿⣿⡗⠀⠀⠀⠹⠛⠛
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⠀⣾⣿⣷⠀⢀⣿⡟⢿⣿⣿⣿⡇⣴⣾⡗⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡼⢩⠜⢿⣀⠀⠀⠂
⠀⠀⠀⠀⠀⣿⣿⠟⢑⢿⣿⣽⣿⣿⣿⣿⠀⣿⣿⡿⠀⢛⣿⣿⡞⣿⣿⣿⡇⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡲⢦⠀⢹⣴⠀⣠⣦⣂
⠀⠀⠀⠀⠀⣿⣿⢦⢠⣦⣟⣿⣿⣿⣿⣿⠇⣿⡃⣄⢀⢀⣤⣉⣁⣿⣿⣿⡇⢹⣿⣿⣿⣦⣴⣤⣶⣦⣶⠌⠀⠀⠀⠘⠁⠄⣱⡾⣍⡼⠻⠛⠋
⠀⠀⠀⠀⠀⣿⣟⠣⠈⠉⠉⠉⠉⠉⠉⠉⠀⣿⣧⣡⠀⣾⣿⣿⣿⣿⣿⣿⡇⢸⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣃⠀⠀⠀⠀⠀⠙⠋⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠘⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠙⠀⢀⠈⠁⠀⠺⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢛⢫⡀
⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⢀⡀⠀⠀⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠉⠁⠛⠑⢯
⣶⣼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⡀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢣⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⡶⠖⠒⠀⠀⠀⠸⠆⢀⡀⠤⠀⠒⠐⠒⠂⠠⢸⠁
⣎⣿⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⠇⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣍⣩⡆⠀⠀⠀⠀⠀⠴⡀⠣⡀⢠⡄⠀⠀⠀⢴⡄⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠠⣀⣤⡀⠀⠀⠀⠸
⠉⠀⠀⠀⢀⠀⢀⡀⠈⠅⠀⠀⢀⠀⢠⠼⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠋⠁⠀⠀⠀⢀⡀⢐
⠀⠀⠐⠇⢀⡠⠀⠀⠀⠀⢀⡀⠛⠛⠛⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣸⢒⣶⠶⠆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⡀⡠⠀⠠⢰⡀⣴⠶⠶⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠪⠟⠷⢆⣤⠦⣀⣤⠂⠀⢦
⠀⢀⣤⠀⠀⠀⠓⠀⢤⠀⣌⡁⠀⠀⢊⣡⡍⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣴⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⣄⠀⠀⡀
⠀⠈⠙⣄⢀⠀⠈⠁⠀⠀⡏⠉⠀⠀⠨⠉⠹⠶⣤⣿⣤⣴⠆⠠⠲⢲⠚⠧⠤⠤⢀⣀⠀⣿⢀⢀⡄⣀⡀⠢⣠⡏
⠀⠀⠀⠍⣧⢀⠀⡈⠁⣁⣁⠀⠀⠀⠀⠀⠁⠀⠈⡈⠁⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⢈⡧⠒⠻⠙⣷⠉⠻⣏⠑⠂
⠀⠀⢠⠆⠁⠘⠉⠉⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠚⠁⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠡⡀⠀⢈⠁⡈⠐⡾⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⡟⢿⡄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⢾⠋⠩
⠀⠀⠀⠈⠉⠀⠀⢉⠖⠤⠀⢃⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⠋⠀⠀⠹⡄⠀⠀⢀⠶⠦⠐⡀
⠀⠀⠀⠀⠀⠙⠀⠀⢟⣿⠶⣾⡄⠀⠰⡖⠋⢹⠀⢀⣴⠁⣠⡄⢀⡀⢰⡀⢠⠿⠀⠂⡐⠉⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⠒⠒⠉⣿⣾⡟⢼⡧⠤⠧⠀⠺⣤⠟⠁⠚⠛⣣⠾⠿⠈⠛⠋⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣴⣤⠈
⡀⣀⣠⠀⠀⣤⡄⠀⠀⠙⠛⠻⠦⠄⠰⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠈⠛⠷⠚⠀⠀⠀⠀⠀⠸⠋⠠⠤⠀⠂
⠅⠈⠀⠂⠀⠼⠤⠊⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠖⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡠⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⠂
⠀⣶⠄⠁⠀⣀⣥⠰⠃⠀⠈⢡⠀⠌⠀⠀⠀⠀⢀⣀⢀⢀⣠⣴⠶⢶⡀⢀⡀⠀⠀⠀⠀⠀⠰⠀⠶⠂⠀⠀⠀⠀⡀⠀⠀⠓
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠾⠟⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⣴⠦⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⣀⣀⣴⣇⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⠀⠀⠐⡀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠦⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠰⠂⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⣈⠘⠁⠀⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠒⠂
⠀⠀⠈⠂⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠠⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠅⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠷⠀⠀⠀⢠⣄
⠀⠀⠀⠤⢤⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⡀⠉
⠀⠀⡀⠀⠀⠈⡁⠀⠀⠙⡀⠀⠀⠀⠀⠛⠛⢶⣦⣤⣤⣤⡶⠚⠛⠋⠀⠀⠀⠀⠀⠀⢾⣟⡶⠈⠰⣿⣟⠑
⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⡀⠀⠀⠠⡄⠈⠀⠀⠀⠔⠉⠢⠀⠀⠔⠁⠀⠀⠀⠤⠀⠐⠛⠿⢋⠔⠜⠛⠋
⠐⡆⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠈⠂⠄⠀⡀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠈⠇⢲⠀⠀⠐⠂⠀⠁
⠀⠀⠘⢤⠄⠠⠀⠠⠀⢂⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠘⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢼⡄⣀
⠀⠀⠀⠀⠁⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠠⠴⢤⣤⣀⡀⡀⣀⣠⡴⠶⠆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠶⠲
⠀⠠⡀⠈⠉⡙⠛⡉⠃⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⡄⠈⢠⠀⠀⠀⢀⡀⠀⠀⠣⠀⠈⠋⠀⠶⠀⠉⠁⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠠⠀⠄
⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠀⠀⠀⠀⢰⡔⠀⠀⠀⠀⠀⢠⡎⠁⢦⠀⠀⠀⠀⢠⠀⠀⠈⢟⢄⠀⠀⠀⢀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⣾⠀⠀⣔⡟
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠾⣦⠀⠀⢀⠅⠀⠤⠀⠀⠀⠘⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⠋⠀⠀⠢⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⡉⠀⠀⠀⠀⠰⠄⠀⠀⢀⡄⠀⢴⠋⠀⠀⠆⢠⠜⠀⡄
⠀⠀⠀⠀⠀⠈⣟⠂⠀⠀⣾⡅⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⢀⣙⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠶⢀⣏⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢈⡁⠀⠀⠆⠀⠨⡌⠀⠀⠀⠰⠀⣸⡇⠀⠀⣀⠀⠚
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡰⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡀⠀⠀⠀⣤⠔⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⠁⢨⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⣇⠀⠈⠁⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⡀⠀⠀⠀⢀⡃⠀⠀⠀⢀⠀⠰⣷⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⡀⡨⠂⠁
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠄⠀⠀⠀⠀⠈⠹⠆⠀⠀⠀⠀⠘⠶⠄⠀⠀⠀⠀⠈⠕⠊
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣦⠉⠀⠀⠀⠀⠀⠀⠉⠀⠟⠀⠀⠀⠀⠀⠀⢿⣇⠂⠀⠀⠀⠀⠀⠲⢠⡿⠀⠀⠀⠀⠂⠀⢠
⠀⠀⠀⠀⠀⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠀⠀⠀⠀⠀⢀⡀⠙⢷⡄⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠈⢡⣄⠀⠀⠀⠀⠀⠀⢃⡐⠎⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠐⠃⠰⠟⠀⠀⠀⠀⠀⠀⠨⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡄⠸⠁⠀⠀⠀⠀⢤⠌⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⣓⡀⠀⠀⠀⠀⠀⠒⠀⣘⢝⠁⠀⠀⢠⠄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠆⠀⠀⠀⠀⠀⠀⠀⣅⡌⠀⠀⠀⠀⠀⡰⢯
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢩⣅⠀⠀⠀⠀⠀⠒⣠⡟⠀⠀⠀⠀⣠⣼⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣆⣰⣿⣿⣦⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⢊⣉⣁⣤⣤⣄⣀⣉⠙⠻⣿⣶⣄
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣻⣿⡆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠟⠛⠁⠉⠛⠿⠟⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⣿⣿⠏⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣫⢁
⠀⠀⠀⠀⣘⢍⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠃⠈⠀⠀⢀
⠀⠀⠀⠀⠉⢘⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠸⠂⠀⢰⠀⠀⠀⠀⠀⠀⠰⢒⠶⠶⠀⠀⠀⠀⠀⠀⢀⣿⡆⠘⠂⠛⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⣖⠂⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠱⣶⣤⣤⡄⠀⢠⣴⣶⣾⠿⠛⢁⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠉⠛⠁⠄⠊⠛⠉⠀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⡄⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠊⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠈⠂⠁⠐⠂⠀⠀⠀⠀⠀⠂⠊⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⣄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠑⠂⠄⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⢀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⡄⠀⠸⠠⢠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡰⠋⠀⠀⠘⠒⠒⠐⠀⠅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠰⡿⣣⣦⡀⠀⣴⣜⢿⠷⠿⣟⣶⡀⠀⢠⣦⡻⡷⠾⢿⣴⣄⠀⢀⣶⣻⠿⠸⠿⣣⣦⠀⢀⣴⣜⠿⠆
⠀⠀⠈⣻⠿⣻⣇⢸⣏⡿⣟⠋⢻⡿⣏⣷⢠⣯⡻⢿⠋⠙⡿⢟⣿⡄⣾⣹⢿⡛⠙⣻⢿⣹⡇⣸⣟⠿⣟⠁
⠀⠀⠀⠈⠛⣛⣿⣿⣟⠛⠁⠀⠈⠙⢛⣿⣿⣟⡛⠉⠀⠀⠉⢛⣻⣿⣿⡛⠋⠁⠀⠈⠛⣻⣿⣿⣛⠛⠁
⠀⠀⠀⣐⣟⢻⡿⢻⡿⣻⣳⠀⢰⣻⡻⣿⠹⣿⢛⣗⠀⠀⣚⡛⣿⠏⣿⢟⣟⡆⠀⣚⣟⢿⡟⢿⡟⣻⣂
⠀⠀⢘⣿⣻⣿⠇⠈⢿⡿⣻⣗⣿⡻⣿⠏⠀⢿⡿⣻⣟⣻⣟⢿⡟⠀⠹⣿⢟⣿⣺⣟⢿⡿⠁⠸⣿⢟⣿⡃
⠀⠀⢈⣿⡇⡁⠀⠀⢈⢸⣿⡁⣹⣏⢈⠀⠀⠀⡁⣿⣏⣹⣿⢈⠀⠀⠀⡁⣹⣏⢈⣿⡃⡁⠀⠀⢈⢸⣿⡁
⠀⠀⢸⣿⣵⣷⡄⢀⣾⣷⣿⡯⣿⣵⣿⣆⠀⣼⣷⣽⣯⣽⣯⣾⣧⠀⣰⣿⣮⣿⢽⣿⣾⣷⡀⢠⣾⣮⣿⡇
⠀⠀⠀⠩⣯⣼⣧⣼⣷⣽⡯⠀⠨⣿⣵⣿⣰⣿⣭⡯⠁⠈⢽⣭⣿⣆⣿⣮⣿⠅⠀⢽⣯⣾⣧⣼⣧⣽⠯
⠀⠀⠀⢀⣤⣭⣿⣿⣭⣤⡀⠀⠀⣠⣭⣿⣿⣯⣥⣀⠀⠀⣀⣬⣽⣿⣿⣥⣄⠀⠀⢀⣤⣭⣿⣿⣭⣤⡀
⠀⠀⢀⣽⣿⣽⡏⢸⣏⣷⣯⣄⣬⣿⣍⡿⠘⣿⣽⣿⣄⣨⣿⣯⣿⠃⢿⣩⣿⣥⣠⣽⣿⣹⡇⢹⣯⣷⣯⡀
⠀⠀⠰⣶⡝⠟⠁⠀⢻⢫⣶⡦⣶⣯⠿⠁⠀⠙⠟⣷⡶⢶⣾⠻⠋⠀⠘⠿⣽⣷⢴⣶⡝⡟⠀⠈⠻⢫⣶⠆
⠀⠀⢠⣿⢇⣆⠀⠀⣰⡰⣿⡦⣾⡗⣤⠀⠀⢠⣄⢿⣦⣴⡿⣠⡄⠀⢀⣤⢾⣷⢴⣿⢆⣆⠀⠀⣠⡸⣿⡄
⠀⠀⠐⢶⣿⢿⡇⢰⡟⣿⡶⠖⠶⣾⠟⣧⢀⣿⢿⣷⠖⠲⣾⡿⣿⡀⣼⠻⣿⠶⠲⢶⣿⢻⡆⢸⡿⣿⡶⠂
⠀⠀⠀⠙⠻⠿⣿⣾⠿⠿⠋⠀⠈⠻⠿⣿⣾⡿⠿⠛⠁⠀⠛⠿⢿⣷⣿⠿⠟⠁⠀⠙⠿⠿⣷⣿⠿⠟⠋
⠀⠀⠀⠤⡶⢿⡿⢿⣿⢶⡦⠀⠠⢶⢿⣿⢻⣿⠷⡦⠀⠀⢴⠾⣿⡟⣿⡿⣶⠄⠀⢴⡾⣿⡿⢿⡿⢶⠦
⠀⠀⢰⣿⢿⣾⠇⠘⣷⣿⣿⡦⣾⣿⣷⡟⠀⢯⣾⢿⡷⢾⡿⣿⡿⠀⢻⣾⣿⣓⢴⣿⣿⣾⠃⠸⣷⡿⣿⡆
⠀⠀⠘⣿⡇⠃⠀⠀⠘⢑⣿⠋⢻⡗⠙⠀⠀⠈⠃⣿⡟⢻⣿⠘⠁⠀⠀⠋⢺⡟⠙⣿⡊⠃⠀⠀⠘⢸⣿⠃
⠀⠀⢘⣿⣣⣷⡄⢀⣼⣮⣻⣗⣿⣫⣾⡄⠀⣴⣧⣻⣟⣻⣟⣼⣦⠀⢠⣷⣝⣿⣺⣟⣵⣧⡀⢠⣾⣜⣿⡃
⠀⠀⠈⢛⣟⣽⣇⣸⣯⣻⣛⠁⢹⣻⣫⣿⢠⣿⣙⣿⠉⠉⣻⣋⣿⡄⣿⣝⣟⡏⠈⣛⣟⣽⣇⣼⣏⣻⡟⠁
⠀⠀⠀⠀⣉⣛⣿⣿⣟⣉⠁⠀⠀⢉⣙⣿⣿⣟⣋⡁⠀⠀⢈⣉⣻⣿⣿⣋⡉⠀⠀⠈⣉⣻⣿⣿⣛⣉
⠀⠀⠀⣺⣭⣻⡟⢹⣟⣯⣗⡀⣸⣯⡟⣿⠹⣿⣫⣿⡀⢀⣿⣝⣿⠏⣿⢻⣽⣇⢀⣺⣭⣻⡏⢻⣟⣭⣗
⠀⠀⢨⣭⡻⡿⠃⠈⢿⢟⣽⣍⣭⣝⣿⠃⠀⠻⡿⣯⣯⣭⣽⢿⠟⠀⠘⣿⣫⣯⣩⣭⡻⡿⠁⠘⢿⢟⣽⡅
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⢀⣀⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣿⠟⠀⠀⡼⠿⣿⢋⡠⠄⠀⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠙⢿⡏⠀⠀⠀⠀⠀⢰⣩⢆⠗⠚⠑⠒⢂⣀⠄⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣅⠤⣴⡎⡇⠀⠀⠀⠀⠀⠁⠀⣀⡀⠐⠒⠂⠣⠀⠀⠀⢀⠀⢸⣿⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⠱⡌⠳⡸⣷⡟⠀⣇⠀⠀⠀⠢⣜⠀⣀⣀⠠⠎⠀⠀⣀⠐⢸⣇⠠⡟⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⣷⣤⡐⢌⢰⢠⡇⠀⠀⡐⠈⠀⠈⠀⠉⠀⠘⠁⠀⠀⠄⠈⣿⠀⢡⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠿⠿⠿⠷⠿⡈⢻⣿⣾⠀⠀⠀⠉⠢⠤⠀⢀⣀⣀⡠⠔⠂⠀⠸⡄⠹⣧⡾⠁⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠹⣭⠀⠀⠀⠀⢀⠀⡜⠈⣿⣏⢄⠀⠀⠀⠠⣧⠀⡏⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢰⠃⠀⠀⠀⢹⡳⣦⣠⡄⣼⠀⡇⢰⡿⠋⠄⠀⢼⣆⠀⢿⣰⠟⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣣⠃⠀⠀⠀⠀⠘⣧⠀⠈⠑⠪⠒⠷⡾⠀⠀⡄⠀⠈⢿⡆⢘⣡⣤⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡀⠀⠀⠀⡀⢀⣿⣷⣄⡀⠀⠀⣸⠃⠀⢀⣿⡄⠀⠘⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡿⣄⠀⡀⠀⣸⣿⣿⣿⣿⣿⣶⣿⣶⣶⣿⣿⡗⠀⠀⠀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢀⣀⣌⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡸⢣⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠴⠛⠉⠈⠻⡿⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⣡⡆⠀⠀⠀⠉⢄⡰⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣴⡴⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⡿⠛⠻⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⡤⢤⠊⠀⠀⠀⠲⠰⢾⣧⣎⣶⣃⢀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⡇⠐⣤⠠⢠⠴⠆⢪⣧⠀⠠⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢉⣰⡏⢸⡄⠸⠇⠀⠀⠰⠘⠇⣠⠂⣀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣁⡏⣧⠀⢿⡆⠀⠀⠀⠀⢧⢰⡿⠀⠀⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠉⡐⣧⣿⠆⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢹⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⡉⡇⢼⡇⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣥⢜⡅⣿⢈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠏⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⣤⠀⣠⣄⣀⣠⣤⣄⣀⣠⣄⡀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣄⠀⢠⡿⣦⣄⣼⠗⠦⢾⡃⢈⣻⣆⠀⠀⢙⣷⣆⡈⠛⢶⡎⠹⢧⣤⣢⣄⡀
⠀⠀⠀⢀⡀⠀⠰⣦⣶⣤⠄⢀⣻⣥⣡⡾⠃⠙⣠⡿⠀⢠⡾⠇⢸⡟⠛⠀⠀⠘⠛⠛⠁⠀⠘⠿⠿⠿⠛⠿⣭⣿⣦⡤⣰⣦⣄⡀⠀⠀⣀⡀
⠀⠀⠀⢈⣹⢿⣶⡶⢿⣿⠗⢻⡉⢩⡏⠀⣠⡾⠋⢀⣴⠟⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣧⣤⣉⠉⣿⡗⠀⣈⡽⣿⣶⡆
⠀⠀⠀⠀⠀⠈⠉⠛⠂⠹⠃⠸⠃⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠉⠙⠛⠁⠀⠉⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⢀⣄⠀⣤⣀⣠⣤⣄⣠⣤⣤⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣤⡄⢸⣧⣄⣈⡷⠄⣽⠁⢛⣿⡀⠀⢹⣷⡀⠈⢻⣏⣙⣷⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣘⡳⣾⣷⣀⣨⣷⣩⡾⠁⢛⡼⠃⢠⡾⠃⣻⠉⠁⠀⠉⠉⠁⠀⠀⠉⠉⠁⠙⠻⣯⠻⣶⠬⣿⢦⣤⡀⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠻⣿⣟⢻⣿⢙⣯⢠⡏⠀⠶⠟⠁⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⢷⠆⠠⣿⣥⡤⠻⣿⣟
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠁⠈⠁⠀⠀⠀⠀⠀⢈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢰⣄⡀⢳⣖⢾⣷⣎⠛⠛⠻⣷⣖⠶⠷⣷⣤⣹⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⠀⠀⣸⡿⣿⠭⣾⡏⠙⣻⣧⢸⡿⣿⡆⠀⠾⠿⠿⠇⠀⠀⢹⣿⣀⣹⡿⣷⣼⣦⣦⡀
⠀⡀⠀⠀⢠⣤⣶⣄⣀⣠⣿⣿⢳⡾⠟⠁⣫⣾⠟⢁⣼⠟⠉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠙⠿⣷⣝⣿⡷⣴⣦⣤⣀⠀⢀⡀
⠀⣛⣿⣷⣶⡽⢿⣿⡞⣿⠉⣡⣿⠀⢲⣾⠟⠁⠒⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠻⠷⣾⣏⢩⣿⣗⢘⣛⣿⣷⣶
⣀⣉⣉⣙⣛⣃⣐⣛⣀⣛⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣉⣉⣙⣛⣋⣉⣉⣛⣛
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣢⣭⡭⣭⣭⣫⢭⢭⣭⣅⡓⢄⡀
⠀⠀⠀⢀⣤⣶⣶⣿⣶⣦⣀⠀⠀⠀⠀⣀⡴⣿⣷⣿⣿⡙⠋⢟⣱⡮⡽⠟⡕⣿⣾⣾⣿⢦⣄⡀⠀⠀⠀⢀⣤⣶⣾⣶⣶⣤⣀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⣫⣷⣿⣿⠟⠛⣿⣷⢀⡄⠚⡛⠠⢀⣷⣿⠋⠛⣿⣷⣾⣭⣋⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⠏⢘⠽⣦⣫⣽⣿⣿⣟⣩⣿⢶⠂⠀⡻⠿⣶⣦⣾⣾⠿⣟⠀⠐⣶⣿⣭⣿⣿⣿⣯⣭⣅⡟⠓⣈⣻⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣷⣼⣷⣛⣽⣻⣴⠿⢯⡁⠀⠙⠏⢿⡀⠁⡀⠘⢻⡿⠡⢐⠉⢀⡿⠝⠉⠀⢈⡽⠿⢶⣟⣯⡟⣿⣯⣷⣿⣿⣿⣿⣿
⠀⠀⠛⢿⣿⣿⣿⣿⣯⣵⣮⣾⣷⣖⣢⡈⢂⠀⠈⡇⠹⣟⡛⢷⡃⢨⡞⡋⢟⡋⣬⠀⠀⠐⢁⣔⣲⣾⣷⣽⣧⣭⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⡿⣿⠻⣿⣿⣿⡄⡇⠀⣷⠀⢀⣟⣼⣁⣈⣧⣯⡜⠀⣾⠀⢸⢠⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⠀⣿⡿⢿⡿⣧⡀⠟⡠⡾⠁⠀⠀⠀⠀⠉⢳⢄⠁⢀⣾⢻⡿⢿⣿⠀⠀⠀⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠸⣷⡘⣇⠠⢀⠤⣒⠋⠙⠛⠑⣒⢌⡀⠀⡾⢡⡾⣇⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⢣⣸⢳⢹⡶⣭⠀⠉⠀⣆⣡⠄⠉⠑⣝⢦⡇⡾⣷⡼⠀⢸⢄
⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⠏⣩⠗⣻⡄⠈⣿⣼⣼⢱⡀⢰⡖⠈⠀⠀⠉⠰⡄⢀⡟⣷⡇⣿⠁⢀⣟⢹⣉⡝⣷⡦⢄
⠀⠀⠀⠀⠀⠀⣸⡴⠟⠛⠋⢿⢸⠁⠻⣄⣿⣿⢻⠁⣈⠃⠱⠤⠄⠤⠤⠌⢸⡁⣁⡇⣿⣿⣠⠞⠀⣇⡿⠙⠛⠳⢦⣡
⠀⠀⠀⢀⠋⠀⡟⢀⠀⠀⠀⢸⡾⡄⠀⠈⣿⢹⣼⣫⠥⠖⠓⠋⠉⠉⠙⠛⠺⠮⣟⣣⣿⣿⠁⠀⢠⣿⠇⠀⠀⠀⡈⢷⡀⠑⡆
⠀⠀⠀⠁⠀⠘⢷⡀⣸⡆⢀⡼⠃⢽⠂⠒⣾⣿⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⣿⣿⠒⢐⡏⠼⠦⣀⠠⠾⣀⡾⠃⠈⠉⡄
⠀⠀⠀⣆⠀⠀⢨⠈⠀⠀⢈⣧⡟⠷⠃⣼⣹⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⣧⠘⠾⢻⡼⣇⠀⠀⠀⢕⠀⠀⣀⠁
⠀⠀⠀⡏⠑⠓⡻⠂⠶⣀⠉⠀⢨⠎⠉⢹⡿⠀⣸⣿⣿⣿⣷⡙⣿⣿⢫⣿⣿⣿⣾⣧⠀⢻⠋⠉⢻⡁⠀⠉⢩⠖⠂⢞⠓⠪⢹⠂
⠀⠀⠀⢹⣤⢰⡀⠀⢀⠘⡶⠖⠻⢇⠠⣼⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⢸⡇⠄⡸⠗⠲⢶⠁⡀⠀⢀⣅⣠⡄
⠀⠀⠀⡇⠀⢀⡵⢤⡏⠩⠰⠐⠀⣄⡷⣽⢻⡀⠀⠙⡻⢿⡿⠿⠿⠿⠿⠿⡿⢟⠋⠀⢀⡞⣿⢿⡤⠀⠀⠈⡉⠰⡤⣮⡁⠀⢸
⠀⠀⠀⢳⠀⢸⠀⢈⣷⠀⠀⡀⠀⠐⠛⢸⡷⠿⠿⣿⣿⣦⣄⡀⠀⠀⠀⣀⣤⣿⣿⠿⠿⢾⠃⠿⠀⣀⣈⠁⢀⣿⡀⠀⡇⠀⡚
⠀⠀⠀⡴⠓⠛⠻⢹⠀⠱⡶⠀⠀⡓⠐⣸⠃⠀⣴⠳⢼⣾⡍⠉⠉⠉⠉⢉⣷⠋⠲⡆⠀⠘⡆⠀⢎⠀⠠⢷⠀⢀⡟⡟⠓⠛⢳
⠀⠀⠀⣧⡀⣰⠞⢸⡗⠀⠀⠠⢎⠄⡰⠙⣄⢀⡿⠀⠚⢇⠙⠶⣤⣤⠔⠋⡸⠃⠁⢦⣀⣰⠹⣄⠨⠓⠠⠈⠠⣶⠗⠚⡄⠀⣸
⠀⠀⠞⠉⠉⡻⣄⣠⣝⣚⣓⡠⠄⠈⣇⣀⡟⠈⠀⠀⠀⠀⠉⢲⡇⢸⡗⠉⠀⠀⠀⠀⠉⢻⣄⣸⢁⠬⡀⣐⣛⢫⣄⡴⡟⠉⠁⢁
⠀⠀⡇⠀⠀⣐⠀⠀⠈⡸⠛⠙⠷⣋⡽⢅⣸⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⣧⣠⢿⣈⠻⠛⠙⣇⠀⠀⠁⣃⠀⠀⢈
⠀⠀⠈⠀⠈⠿⡦⡀⢾⠁⢀⠀⣸⡝⠈⠀⣀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠀⡀⠩⣇⠀⠀⠘⡷⠀⣴⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠑⠀⣛⣒⠪⠊⠈⠁⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠁⠀⠀⠃⠙⠗⡒⠃⠀⠈

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡤⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠀⠀⠀⠀⠠⠤⠀⠀⠄⠁⠀⡄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠡⠀⠀⠀⠠⠀⡀⠀⠀⠀⢨⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠀⠀⠈⡶⠀⠀⠀⠀⢀⣃⢴⠏⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠴⣰⡎⠉⠑⢤⠀⠀⠿⠋⠀⢀⣴⣶⠢⡀⠀⠰⢀⣤⡠⠖⢫⢆⠎⠾⠿⠋⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⢇⠀⠀⣀⠀⠈⠛⠁⠀⢸⠀⢠⠜⠉⠀⠀⢠⠎⢀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡄⠀⠈⠢⣙⡀⠀⠀⠀⠀⣼⠴⠉⠀⠀⠀⣡⠟⠀⠀⢀⡈⠱⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⠲⣾⣿⡿⠃⠀⠀⠀⣠⠚⠁⠀⠤⠬⠊⢀⣀⡐⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢁⠔⠈⠀⠀⠀⢀⣉⣛⣆⠀⠀⠀⠈⠋⠀⠀⠀⢠⠞⠁⠀⠠⠄⠀⣾⠟⠉⠉⠙⠳⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠄⢰⠃⢀⠤⠤⠤⠾⠋⢩⠉⠛⢳⡄⠀⠀⠀⠀⠀⣴⠠⠽⠀⠀⠀⠐⠀⠙⠀⠸⡞⠆⠀⢻⠀⠀⠀⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⠀⠀⢠⡄⠀⠀⢰⠀⠘⡴⠋⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⡀⠀⠆⠀⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠰⠆⠀⠀⠄⡡⠋⠀⠀⠀⢀⡴⠙⢦⡀⠀⠀⠈⠢⣀⠀⠀⠊⠀⠱⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣵⢦⡀⠀⠀⠀⢠⡰⠊⠀⠀⠀⢀⡴⣿⣿⣄⢰⠟⠢⣄⠀⠀⠈⠳⡄⠀⠀⠀⠀⣿⡏⠁⠉⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣦⣁⠿⣷⡤⣽⣡⠞⠁⠀⠀⢀⢴⡿⠀⠃⠀⠙⠳⠀⠀⢸⣳⣄⠀⠀⠈⠣⡀⠀⢀⠘⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠋⠰⢿⠁⠀⠀⢀⣴⠏⠈⠀⣀⠈⠀⠀⠀⠀⠀⠘⣽⢻⣷⢄⡀⠀⠘⠀⠀⠠⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⢠⣾⡏⠢⠤⠔⠈⠄⠀⢠⡀⠀⠀⣴⡀⠀⠀⠀⠀⣿⣷⡙⠆⢻⣷⣎⠐⢠⣀
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⢻⠿⠁⣼⡗⠀⠀⢠⣶⠁⠀⠀⠀⠁⠠⣄⠈⠡⣆⠀⠀⠰⠛⢿⣇⢦⠠⠛⢿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢾⡟⠀⠀⠀⠀⠀⠀⠁⠀⠠⢬⡶⠄⠁⠀⠈⡯⠀⢁⠀⠀⠀⠀⠀⠠⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠌⠃⠘⠁⠀⠀⡀⠀⠀⠀⠀⠀⠐⠻⠁⠎⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⣦⡆⠀⠂⠀⠀⠈⢀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣲⣄
⠀⠀⠀⠀⠀⢠⡎⢲⡀⠀⠀⠀⠄⠀⠂⠀⠏⠋⠋⢉⠍⠁⠀⠀⠑⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⡀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠘⠁⠀⠋⣠
⠀⠀⠀⠀⠀⠁⠀⠀⣳⠀⠇⠀⠐⠆⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠃⠰⠄⠠⠶⠀⠐⠅⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠈⡀
⠀⠀⠀⠀⠀⠇⢃⠈⠁⠀⠈⠤⠠⡀⢃⠀⠬⠁⠀⠄⠐⠕⠀⠀⠄⢈⠁⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⡄⠓⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠉⠘⢁⣀
⠀⠀⠀⠀⢀⠀⠀⢄⣀⡀⠐⡄⠠⠁⠀⠌⠀⠀⢢⠴⠁⠓⠀⠀⠀⠀⠀⠀⠌⢠⡨⠀⠀⠀⠀⠀⠀⠀⡏⠳⠅⠠⠰⠆⠀⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠠⠤
⠀⠀⠀⠀⠈⠀⢨⡞⠡⠰⠀⠭⠣⠞⠀⠀⠃⠀⠐⠤⠀⠀⠀⠂⠀⠂⠀⠑⠰⠂⠀⠐⠀⠀⠀⠒⠀⠀⡶⡆⠀⠀⡠⡠⠀⠀⠀⠂⠄⠀⠠⠀⠀⠈⢖⠒
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠠⠈⠄⠀⠠⠀⠀⠤⠠⢠⠤⣄⠀⠀⡠⢂⠐⠀⠀⡐⠀⠀⢀⠀⠀⠀⠀⠢⠀⢁⡀⠀⠐⢠⠁
⠀⠀⠀⠀⡠⠠⣅⠄⠀⠀⠀⠀⠀⠀⠠⢸⠉⠁⠀⠈⢸⠀⠐⠀⠁⠀⢀⢀⠁⠀⠀⡄⠠⠄⠀⠠⠀⠀⠀⠀⠄⠠⠄⠐⠀⢸⠀⠀⠆
⠀⠀⠀⠀⡀⣆⠀⠘⢂⡦⠀⠀⠘⠉⠉⢉⠀⠀⠠⠄⠀⠀⠁⢠⡀⢂⠸⠁⠀⠐⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⢀⠂⠔⠀⠀⠈⠀⠐⠀⠀⠆⣠⠎⠁
⠀⠀⠀⠀⠔⢋⠑⢆⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⠂⠀⠁⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⠈⠛⣠⡀⠠⠂
⠀⠀⠀⠀⠀⢨⠒⠸⠂⠠⠀⠀⠀⠀⠼⠂⡄⠀⠐⠀⠀⠀⠄⠠⠀⠀⢞⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡄⠀⠂⠀⠀⠈⠀⠒⠀⠀⠀⠀⠊
⠀⠀⠀⠀⢸⢛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠊⠀⠀⠁⠄⡀⠀⠀⠁⠳⡌⠘⢦⢀⣀⣂⢀⠅⠀⠀⠀⠀⠀⠀⠀⠇⢠⠁⠀⠀⢡⠀⠀⠀⢀⢾⡷⠊⠀⠖
⠀⠀⠀⠀⠐⠈⠃⠐⠂⠂⠀⠀⠀⠀⡀⢁⡀⣀⡠⠀⠀⢤⠀⠀⠇⠀⠈⢢⡁⠳⡀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠄⡏⡁⠀⠀⠀⠃⠀⠀⠀⠀⠄⠠⠄⠂
⠀⠀⠀⠀⡇⠠⠀⠀⠀⠚⡆⠀⠂⢀⠀⠀⠁⠁⠀⡉⠀⠈⠖⠰⠀⠀⣠⠄⠑⡄⠹⡄⢀⠂⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⡐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠁⢰⠀⠀⠈⠂⠀⠙⠈⠉⠙⠁⠐⠀⠀⠀⠀⠀⢐⡀⠠⢸⠀⠈⠉⠄⠲⡁⠀⠀⠀⠀⣄⣀⠀⠐⠂⠾⢿⡄⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣤⡀⠀⠀⠀⠀⢠⡀⠀⠀⠄⠈⠀⠨⠀⡃⠀⠀⠀⠄⠀⠀⠈⠉⡀⣀⠼⠀⠀⠀⠀⠴⠘⡄⠀⠈⠀⡀⠈⠀⣦⡶⠸⡏⠀⠀⡄⢒⠂⠐⡆⠀⠰
⠀⠀⠀⠀⠠⢄⣀⡀⢄⠈⡻⠀⢀⠀⠄⠀⢳⠀⠃⡀⠀⠠⠀⡀⠀⠀⠀⠈⠀⠠⡞⢳⠀⡀⢐⡃⠀⠀⠀⠨⠠⠀⠀⠀⠀⢨⡗⠀⠀⠀⠉⠃⣠⠁⠀⢀
⠀⠀⠀⠀⠀⠘⡐⠢⡄⠀⠁⠀⠀⠐⠈⣠⢲⠀⠀⠀⠀⠠⡄⠀⠀⠄⠀⠀⢠⠀⠉⠉⠀⠀⠀⠵⠈⠀⠀⣀⣀⡇⠘⠀⠃⢸⡇⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠑⢀⠀⠀⡡⢇⠀⠀⠀⠀⠈⠁⠙⠿⠶⠤⠀⠀⠀⢀⣄⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⣄⠘⠀⠀⣀⡄⠤⠓⠂⠈⠉⠙⢳⠄⣐⣴⣀⣀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠪⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠀⠁⠀⠀⠀⠊⠀⠒⠈⠓⠘⠃⠛⠈⠒⠁⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠢⢄⣀⠉⢀⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣫⣝⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣽⣿⣯⣭⣭⣭⣭⣭⣽