Dog Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠂⠀⠁⠉⠉⠉⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣈⠁⠀⠀⠀⠓⠀⠀⠄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠘⠛⡁⠀⠀⠈⠭⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢸⠀⠀⠐⠄⢀⠀⠀⠀⢀⠀⢠⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠜⣄⢣⢀⡠⠤⠤⡤⢴⠋⡠⡞⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠢⡹⠀⠀⠀⡖⡡⠊⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡇⠀⠀⡇⠀⠀⠀⡋⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⡀⠀⠱⢄⡠⠔⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⡞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡆⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⢰⠀⠀⠀⢛⠀⠀⠀⠀⠀⡜⢇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠄⠑⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢠⠞⠀⠈⢆⠀⠀⠘⡀
⡔⠋⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⡰⢻⠀⠀⠀⠈⢣⠀⠀⢃
⡇⢄⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⡟⠀⠀⠀⠀⠀⠘⡄⠀⠀⢣⠐⠁⣸⠀⠀⠀⠀⠀⢣⠀⠸⡀
⢣⠈⢦⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⡀⠀⠘⡤⢖⣁⠀⠀⠀⠀⠀⠈⢇⠀⢣
⠀⠑⢤⣈⠉⠛⠉⠉⢳⡄⠀⠀⠢⣇⣀⠀⠀⠀⠀⠀⢀⣱⠀⠀⢸⡁⠀⢙⡆⠀⠀⠀⠀⠀⢣⠈⠓⠤⢄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠼⠤⠴⠼⠀⠉⠉⠉⠉⠉⠁⠸⣀⢀⠀⢈⢻⠉⠁⠀⠀⠀⠀⠀⠀⠣⣀⣰⣀
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠲⢄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡄
⠀⠀⠀⠀⢰⠀⠀⠀⡴⠁⠀⠀⠀⠈⢃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡀
⠀⠀⠀⠀⢠⠀⢀⡞⠀⠁⣀⡤⠤⠶⠖⠒⠺⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⡇
⠀⠀⠀⠀⠈⠦⠊⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⠞⠛⠛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⡠⠴⠒⠒⠒⠒⠒⠤⠄⠀⠘⠀⡟
⠀⠀⠀⣀⠄⠈⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠐⣌⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⣦⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢰⠁⣠⣤⡀⠀⠀⠀⠀⠀⠀⠧⠞⠀⠐⠀⠀⢀⡠⠴⠒⠒⠓⠲⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢃⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠈⠀⠘⡟⠛⠃⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⢀⣀⠀⠀⢢⠜⠙⠛⢛⠟⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⡜⢁⡄⠀⢰⠃⠀⠀⡴⢃⣀⣀⡤⠖⠀⠤⢄⣀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠡⡎⠀⠀⠸⣤⡤⠞⠁⠀⠈⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠑⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠳⢄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠆⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠳⣄⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⠔⠒⠈⠉⠉⠉⠉⠉⠉⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⣤⡀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠆⠀⢀⣄⡀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠆⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢰⠀⠀⠳⠟⠉⠀⠀⠀⠀⠀⣙⣃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡎⠀⠀⠀⠀⠞⠁⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⠀⡇⠀⡀⠀⠈⠳⡀⢀⡤⠂⠀⠀⠀⢠⠀⠀⢰⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⢿⡀⢣⡂⡀⠀⣨⣅⣀⣀⣀⣠⢴⠏⠀⣠⡿⢦⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡜⠳⣄⠹⣌⡏⠀⠀⠀⢹⣉⡴⠃⡠⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⢧⣸⠀⠀⠀⠀⢸⡁⣠⠎⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⡄⠀⠀⠀⡇⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠀⢳⡀⠀⠀⠘⠲⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢸⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⡇⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⣆⠇⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡜⢳⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⢧⠀⠀⠀⠀⠈⣇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠉⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⢧⡀⠀⠀⠀⢻⡀
⢠⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⡇⠀⠀⠀⡼⢻⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠘⡇
⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢿⠀⢀⡜⠁⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢳
⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢀⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠸⡆⠈⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣇
⢳⡀⠘⢆⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢻⡀
⠀⠳⣄⠈⠙⠲⠤⣤⣤⡤⠶⠚⣆⠀⠀⠀⢀⠀⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⣷⠖⠛⠓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⢷⣀
⠀⠀⠈⠳⢦⣄⣀⠀⠀⠀⠀⢀⣘⣶⠀⠀⡀⠀⠉⠻⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡇⠀⠀⠀⠸⣦⣀⡀⠑⣎⡧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠳⣤⣤⣧⣀⣸⣄⡼⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⣧⠀⠀⢰⠀⠠⡍⢻⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⡦⠀⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣶⣛⣤⣤⣷⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣊⣉
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⠉⣀⠀⠉⠐⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢠⡙⣿⠁⣼⠟⣀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠉⠁⠃⠄⠹⠀⠋⢈⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢈⣿⣷⠀⠰⢲⢳⠢⠀⢠⣿⢽⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠃⣿⡀⣤⢠⣇⢠⠀⢸⣿⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢧⣥⢟⡹⢦⣀⣸⣏⣷⡍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣽⠃⢲⣿⠿⠊⠑⣫⣾⣿⠇⠘⡆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⢘⡇⠀⠀⢠⣿⣿⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣾⣷⣆⣴⣿⣿⣿⠁⠀⠀⠀⣾⠁⢠⠀⢸⣄⣤⣤⠀⠀⠀⠻⠦⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⣰⣿⠀⣾⠀⢸⣿⣿⣿⣤⡀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠄⠀⠀⠙⠻⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣾⡟⢀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⣸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣀⡀⠸⣧⣤⣀⠀⢀⣠⣤⣀⣠⣶⠃⠀⢰⣿⣿⣿⡿⠛⠛⠻⢟⡁⠙⠻⡷⣷⡀⠰⠁⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣠⡝⠁⠀⠀⠈⠛⠿⣿⣿⡿⠿⠟⠋⠁⢀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣼⣿⡤⠔⣃⡼⠃⠀⠁⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡰⠋⠁⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⣤⠞⠋⠁⠀⠀⠀⠀⠀⠈⢌⡉⠟⠋⠀⢸⡋⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿
⠀⠀⠄⠒⠉⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒
⡤⢀⢀⡄⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⠀⠁
⠓⠛⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⠀⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠤⣤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣽⠀⠀⢠⠄⠀⠘⠇⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡝⠀⠀⠈⠛⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠠⣇⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠓⠀⠀⠄⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡗⠲⠛⠦⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠙⡄⠀⠘⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⠤⠶⢤⡀⠀⢢⠀⠀⠀⡔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⠀⠀⠀⠀⠹⡆⠈⡄⠀⠀⡇⠀⣠⠀⠸⡇
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢧⠀⠀⣧⠞⠹⡄⠀⡇
⠀⠀⠀⠀⢸⡷⡄⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠀⠀⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠰⣧⠙⢦⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡠⠚⡇⠀⡇⠀⠀⢇⠀⠇
⠀⠀⠀⠀⠀⠈⣷⣄⠈⠓⠦⢄⣸⣄⣀⠀⠀⠀⢀⡤⠚⢁⣠⣖⠉⠉⠙⡇⠀⢧⠀⠀⠈⢣⠘⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠄⠠⠤⠾⣅⣀⣀⣀⣈⣉⣉⡀⣠⡠⠏⠉⠀⠙⢄⠀⡁⣠⠀⠀⠁⠠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⡤⠀⢒⣯⣻⣿⣿⣯⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⡄
⠀⠀⠀⠀⠸⡀⠂⠈⢟⣿⣽⣿⣿⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⡄
⠀⠀⠀⠀⠀⠈⠑⢤⣼⣿⣿⣿⣿⡟⠛⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⠏⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣋⡉⠛⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣷⣿⡿⠟⠷⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣷⣿⣿⣯⡉⣸⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⡇⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠘⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣩⣭⣦⣦⣄⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⠀⢘⡟⠛⠛⠿⠗⠒⠒⠛⠉⠉⠀⠀⠀⠘⢿⣧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠆⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡳⣄⡐⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢨⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠏⢻⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣥⣀⡼⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣁⠼⠀⢸⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣄⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣶⣶⣾⣿⣷⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⡿⠛⠟⠋⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣶⣶⣦⢹⣿⣿⣿⣿⠿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣆⠀⠘⣿⣿⣿⠏⣸⣿⡿⠋⠀⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣽⣤⣴⣿⠟⠁⣀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢹⣮⣉⣠⣾⠂⠀⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣹⡇⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⡀⠀⠀⠀⢀⣼⣿⠃⠙⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠰⣿⣄⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟⠋⠙⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠋⠱⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⢀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠘⠀⠉⠙⠉⢻⡄⠀⠷⠀⠀⠑⠀⠀⠐⠀⠀⠀⠁⠀⠈⠀⠉⠉⠁⣶⣦⣤⣤⣴⣶⣤⣤⣤⣤⣤⣶⣤⣶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠔⢁⠀⠀⠀⡇⠈⠃⠂⡄
⠀⠀⢀⡀⠀⠀⠺⠄⠀⠀⡇⠀⢀⡾⠀⠢⠄
⠀⠀⢽⠀⠀⠀⢀⣀⢄⡀⠳⡴⠋⣼⠀⠀⠀⢦⡤⣀
⠀⠀⠀⠀⢐⠓⡩⠗⠋⠀⠀⠀⠀⣾⠀⠀⢠⡾⠁⣹⣧⠘⢶⣶⣶⡶⠶⠶⠶⠶⠖⠲⠶⠦⢤⠄⡀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠄⠀⠈⣿⠀⠀⠋⠀⠰⣿⣿⠀⠀⣿⠟⠙⢲⣶⠶⡟⠿⠓⠒⠀⠀⢀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⡄⢸⡟⠀⠀⠛⠀⠀⠟⠉⠀⠁⠀⠀⠀⠀⣰⠃⠀⠀⢣⠄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠁⠀⠀⠀⠈⠀⡄⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢲⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢌⠳⠀⠀⢀⣯⠄⠀⠀⠀⠀⠀⠄⠀⠀⢿⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠟⠶⠾⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠉⢳⡄⠀⠀⣾⠤⠤⠤⠀⠀⠀⠀⠀⠀⠸⣆⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⢿⣿⣶⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣌⢿⣿⠷⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠐⡀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⠀⡴⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡏⠀⠀⠃⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄⡨⢃⢤⣤⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣑⣾⠁⠀⠠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠘⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⠀⠀⠀⠀⠀⣰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠈⡇⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣾⠀⢱⣤⣤⣿⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⡇⠀⣿⣿⣿⣿⡻⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡇⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣆⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠉⣭⣽⣆⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⢹⡿⠃⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢻⣿⣧⡀⠠⣄⡀⠀⠀⠀⠀⢹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠙⢿⣿⣶⣤⣭⣷⣶⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣯⣍⠉⠛⠉⠉⠁⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⡄⠀⠀⠀⢠⣤⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⠀⠀⠀⠉⠛⠿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣛⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠸⡇⠀⠀⣠⣒⠲⣦⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣇⠀⠀⠀⠁⠀⠀⢿⣿⣿⣿⠃⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣿⠈⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠙⣿⣿⣿⣿⣿⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⢦⣄⠈⠄⢰⡏⠀⠈⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠛⠁⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣶⣶⣶⣾⣿⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⡟⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⡆⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⢀⣤⠴⠚⠛⠉⠉⠓⣦⣤⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠈⢽⢷⣶⣎⣩⣤⣤⣤⣶⣶⠾⠋⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢣⣄⠀⠀⠀⠀⠀⠀⠈⠙⠻⡿⢿⡿⠛⠛⠁⠀⢠⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣾⠟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠶⣤⣴⣶⣤⣤⣶⡶⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡺⠑⠤⢲⡞⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⠒⢤⣐⡊⠁⠠⠀⠀⠀⠉⢽⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣤⣽⠁⠞⠀⠀⠀⠀⠀⢠⢜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⠝⠛⠛⠛⠋⠀⠀⠀⢠⠄⠐⠂⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠸⡅⡜⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠼⢶⠤⠔⠋⠙⠁⠀⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⡘⠀⠀⠈⠙⢦⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣏⣉⠉⠒⠒⣚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠐⠣⠋⠉⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠀⠀⠘⠳⢤⡀⠀⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠸⠞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⠀⠀⣦⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠸⡄⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⢸⠀⠀⡇⣠⠖⠚⠛⠓⠲⢄⡠⠁⠹⢄⠀⠀⠀⢳⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣇⠀⢻⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢳⡀⢠⠏⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠸⡄⢸⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡄⣀⡬⠽⢧⣸⡆⢀⡵⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡰⠂⠀⠀⢀⣾⠀⢀⠀⠀⣠⣀⡀⠀⠀⢀⣀⣀⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉⠈⠙⠛⠒⠚⠁⠀⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣶⣶⣾⣿⣷⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⡿⠛⠟⠋⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣶⣶⣦⢹⣿⣿⣿⣿⠿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣆⠀⠘⣿⣿⣿⠏⣸⣿⡿⠋⠀⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣽⣤⣴⣿⠟⠁⣀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢹⣮⣉⣠⣾⠂⠀⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣹⡇⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⡀⠀⠀⠀⢀⣼⣿⠃⠙⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠰⣿⣄⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟⠋⠙⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠋⠱⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⢀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠘⠀⠉⠙⠉⢻⡄⠀⠷⠀⠀⠑⠀⠀⠐⠀⠀⠀⠁⠀⠈⠀⠉⠉⠁⣶⣦⣤⣤⣴⣶⣤⣤⣤⣤⣤⣶⣤⣶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠂⠀⢾⠋⠀⢀⣀⠈⠒⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢳⡄⢀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠄⠀⠀⠀⣶⡎⠀⠀⢠⣴⣌⠀⡀⠀⠀⠀⠀⠀⠀⣤⠦⣄⠀⠀⣿⣦⡀⠀⠀⠈⢧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⡇⢠⠀⠻⠿⠛⠐⠀⠀⠀⠀⠀⠰⡎⠻⠿⠟⠁⠀⢻⣿⡇⠀⡆⠀⠀⠑
⠀⠀⠀⢀⡞⣤⡆⠀⠀⣿⡇⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠋⠀⢿⣇⠀⠃⠀⠀⠀⠀⠆
⠀⠀⠀⣰⣿⡁⠀⠀⠀⢻⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠨⡁⣻⣿⡄⣀⠀⠀⠀⡈⠋⠣
⠀⠀⠀⢻⢿⣿⣾⣄⣄⢸⡇⡏⠔⡠⠀⠀⢠⣿⣿⣿⣿⣿⣷⡄⠰⠀⠀⣦⣤⢸⣏⣿⣇⢿⣿⣄⣸⡟⣾⡟
⠀⠀⠀⢸⣿⢿⡏⣽⣽⡇⣏⢧⡀⢿⠢⠀⠈⠛⢿⣿⣿⣿⠛⠀⠀⠁⣴⠟⢁⣾⡷⣿⣿⢸⣿⣯⠁⣴⡏
⠀⠀⠀⠀⠙⢿⣿⣿⣏⢰⣿⠀⠁⠈⣇⠀⠀⠀⢀⣈⠡⠀⠀⠀⠀⣠⠃⠀⢀⣈⠀⣸⣿⡸⣿⣫⡸⢢⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣏⣞⣤⠗⠀⣀⠻⣷⣤⣶⡶⠿⣿⣶⣶⣤⡶⡫⠟⠀⠁⠈⠀⢻⣻⣇⢿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢚⠇⠀⡀⠀⠙⠃⠘⢻⣧⠀⠀⠁⠀⣨⣿⠎⠀⠀⠀⠀⠢⠀⢀⣡⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣶⠟⠁⠀⠀⠀⡀⠀⢆⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢷⡄⢃⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢯⠀⢨⣶⣾⢿⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⠋⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡀⠉⠈⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠳⣶⣄⢰⡆⣀⠀⣄⣠⣅⠀⠿⠂⠁⠀⠀⠀⠙⠂⢈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠙⠛⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣭⣿⣻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⢻⠋⠉⠀⢮⣳⣞⡿⣻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠱⡀
⠀⠀⠀⠀⠀⠘⢄⡁⠀⠌⣹⣿⣾⣿⣿⣿⣿⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⣿⣿⣿⣿⣿⣿⠋⠉⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⠞⠛⠲⣶⣶⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣽⣷⣿⣽⣿⣿⣟⣻⣿⣷⣶⣦⡘⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⡟⣹⣿⣿⡿⠯⣅⡈⡛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡗⣆⠀⠀⠀⣸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⢔⢶⡟⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⠀⠀⠀⣿⣻⣿⣿⡽⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠉⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣉⣡⣵⣤⣦⣀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣆⠀⠈⡾⠻⠿⢿⣿⡯⠤⠵⠶⠖⠛⠋⠉⠀⠀⠀⠹⣿⣦⡀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠈⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣀⠀⢘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢹⢷⡎⠋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠀⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠉⡘⠀⢳⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢴⣆⣠⠟⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠠⠤⠃⠀⣸⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⣆⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢂⣄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠖⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣀⣀⣀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣿⣿⣿⣿⣷⡄⠀⢱⡜⠒⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡞⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡀⠀⢀⠜
⠀⠀⠀⣇⠀⠀⠙⠒⠦⠤⣤⣀⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣷⠀⠙⠿⣿⣿⣿⠿⠓⠒⠒⠔⠚⠚⠁
⠀⠀⠀⠈⠓⠦⣀⣀⠀⢰⣿⣿⣿⣿⣿⣿⣯⠙⠻⠿⠿⠿⠋⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠛⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠔⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⡠⠊⡨⠋⠒⠒⠂⠤⠒⠒⡆⠀⠀⡔⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢠⠊⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣷⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠘⢤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢘⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠲⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠄⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⢠⣤⡆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢊⠏⠀⠀⢠⣤⡄⡈⠁⠈⠀⠀⠀⠀⠀⠀⢤⠠⠀⠀⠀⠀⠹⣦⡀⢄⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣾⣾⠀⠀⠀⣠⣶⢦⡀⢀⠀⠀⠀⠀⠀⢠⠀⠀⣠⡴⢦⣄⠂⠀⢸⣿⣄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣽⡇⠀⡀⠐⢻⣿⡿⠃⠌⠀⠀⠀⠀⠀⠀⢠⡖⠻⣿⣷⠟⠃⢡⠸⣿⣿⠀⠀⣴⠀⠀⠈⠢
⠀⠀⠀⠀⠀⣾⠀⣠⠀⠀⠘⣿⡇⠀⠁⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠴⠀⢻⣿⡆⠀⢿⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢸⣥⠷⠋⠀⠀⠀⣿⡇⠀⣐⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠆⢠⣿⣷⠀⠀⠀⠀⠀⠀⠠⣬⣤
⠀⠀⠀⠀⣿⣿⣦⠀⡀⠀⠀⢸⣧⢠⡈⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠐⠦⣨⡿⣿⡆⣱⣄⡀⠀⢀⣄⠀⣤⡑
⠀⠀⠀⠀⣿⢿⣿⣿⣥⣆⣦⠀⣿⢸⠡⢂⡤⠀⡀⠀⣼⣭⣿⣿⣿⣿⣿⣿⡆⠈⠂⠀⠀⣶⣵⠀⣿⣧⣿⣿⢹⡯⣷⣠⣼⡏⢹⣿⠛
⠀⠀⠀⠀⢹⣾⡿⣿⢩⣯⣼⡇⣏⡻⣄⠘⣿⢆⠄⠀⠙⠛⣿⣿⣿⣿⣿⠟⠀⠀⠈⠀⣶⠟⢉⣸⣿⣿⣿⣿⠸⣷⣿⣏⠁⣰⡟
⠀⠀⠀⠀⠈⠻⣼⣯⣾⣿⠟⠁⣿⠁⠈⠃⠘⡇⠀⠀⠀⠀⠀⠙⠟⡉⠀⠀⠀⠀⠀⣸⢃⠀⠈⠹⠋⠀⢹⣿⡀⣿⡿⡏⢸⢇⠎
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⣸⠏⣱⡀⠀⠀⢿⣦⡀⢀⣀⣀⣷⣶⣥⣀⣀⠀⢀⣰⢯⡤⠂⠐⠚⠂⠹⣿⣿⣇⢹⣷⣿⣊⡼
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣳⡯⠿⠟⠀⠀⣤⡈⠻⣿⣿⡟⠋⠉⠹⠻⠛⢻⣿⢏⡵⠋⠀⠀⠀⡀⠀⠀⠑⢜⢿⡜⠓⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⡄⠀⠀⠀⠀⠀⠉⢻⣷⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠈⠂⠀⢠⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⣶⣾⠟⠡⠀⠀⠀⡄⣀⠀⢲⡀⠀⠸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⢻⣆⠀⠇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⣵⣦⣾⣿⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡏⠀⠻⠱⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠁⠀⠘⠋⠛⠋⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠛⢷⣀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠢⠀⢱⣌⠀⠲⠀⠀⠀⠀⢤⣀⠀⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡸⣥⢰⣤⣀⣶⣼⡶⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⠉⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⣀⣤⣄⡀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡴⠋⠁⠀⠀⠉⢢⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⢳⡄
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⡾⢁⠀⠀⠀⠀⠀⠀⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⢠⡞⠀⠀⢑⣠⡤⠴⠶⠒⠚⠛⠚⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣻
⠀⠀⠀⠀⠀⠈⢧⡴⠋⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣤⡶⠞⠳⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢰⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡼⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣠⠴⠚⠋⠉⠉⠁⠉⠉⠉⠓⠒⠀⠀⠊⠀⢸
⠀⠀⠀⠀⣀⠤⠐⢡⡟⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠴⡚⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢰⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣿⠀⠀⡜⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⠘⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢀⡇⠀⣼⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠤⠒⠉⠉⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠈⣇⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠒⠀⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢂⠘⡷⢶⣤⣤⣤⣤⠶⢞⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⡠⠤⠤⠄⠀⣠⠎⠀⠀⠀⢠⠞⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⡞⠀⡠⠂⠀⢰⡏⠀⠀⠀⡴⠃⣀⣀⣠⠴⠚⠓⠒⠒⠦⢤⣤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠃⡜⠁⠀⠀⠸⣧⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡆
⠀⠹⣄⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠒⠂⠠⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠂⠈⢇⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠚⠉⠁⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡆⠀⠀⢀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢴⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠸⣄⠀⠀⣆⠀⠀⠀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⠉⢁⡀⠀⠉⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⢀⠛⢿⡎⠃⣼⠶⠂⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠾⠃⡹⠁⠀⣟⠠⠿⠂⣿⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣧⡇⠀⠀⣀⣈⣀⡈⠀⠀⣴⡟⣧⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣄⣼⢿⣿⠀⠘⠑⢃⡃⠁⠀⢸⣿⠞⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣸⣿⡄⣼⢀⣾⡄⠇⠀⢸⣿⡀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠷⣤⠞⣭⡻⢤⣠⣸⡏⣾⣷⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠃⠲⣿⣿⠷⠊⠐⣫⣾⣿⣿⠀⢹⠀⣦⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣿⠁⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣀⣠⣿⣄⠀⣠⣾⣿⣿⠟⠀⠀⠀⠀⣸⠇⠀⡀⠀⣶⠀⣀⢀⠀⠀⠀⠘⣷⡌⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠀⠀⠀⠀⢠⣿⠀⢠⡇⠀⣼⣿⣿⣾⡀⠀⠀⠀⠈⠉⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⣿⣿⣀⣾⠃⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠟⠃⠀⠀⣀⠀⠉⠿⡿⠟⠛⠋⠁⠀⠀⠀⣀⠀⠀⢰⣿⣿⣿⣿⣀⣼⣿⣿⣿⣿⢿⣿⣇⢀⠀⠀⡀⢰⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣀⣠⡤⠤⠿⣦⣤⣄⡀⣀⣠⣤⣤⣤⣴⣾⠏⠀⠀⢸⣿⣿⣿⡿⠛⠉⠙⠛⢛⣁⠀⠙⢿⠿⣷⡀⠐⠇⠈⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⣠⡽⠁⠀⠀⠀⠈⠻⠿⣿⣿⣿⡿⠿⠟⠋⠁⠀⡀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⣼⣿⣦⠤⠚⣠⡿⠃⠀⠈⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⣠⣀⡴⠟⠓⠀⠀⠀⠀⠀⠀⠀⢃⡘⠛⡿⠋⠀⠀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿
⠀⠀⢀⡠⠴⠚⠁⠀⣴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⣠⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡤⠠⠃
⠀⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠰⠋
⣟⣰⣧⣾⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⡄⠀⠂
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢼⡄⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡶⣶⡷⢶⠞⢻⡿⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢀⠀⠀⣤⡐⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣮⡴⠛⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠛⠛⠻⢶⣄⡀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢸⣿⡵⠂⠀⠀⢀⣠⡴⢶⣶⣦⣄⠀⠀⠀⠀⣠⣶⠿⣿⣶⣦⣄⠀⠀⠀⢻⣷⡄⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠚⣿⣁⡀⠀⣠⢿⣿⣷⣾⣽⣿⣿⢀⣀⢠⣠⣿⣿⣶⣿⣿⣿⡿⢷⠀⣸⣿⣿⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠿⣿⡿⠷⠁⠘⣿⡿⠿⠛⠛⠃⠁⠀⠀⠀⠉⠉⠉⠻⠛⢿⣁⠘⠛⠛⢃⡈⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣏⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠣⡄⠀⢀⡾⠁⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⣟⠀⠀⠀⠁⠀⠀⠀⠀⣴⣿⣿⠛⠛⣻⣿⣶⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠻⣿⠖⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣾⣿⠿⣷⡆⠀⠀⠀⠀⠀⠀⢀⢀⡤⠋⠙⣆⣀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣦⣀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠻⣄⡀⢳⣤⡤⠖⠛⠋⠀⠀⠀⠈⠋⠙⠾⣦⣀⣠⣞⣀⡴⠃⠈⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⢤⣤⣠⣀⠀⢀⢀⣠⡤⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠶⣶⠶⠞⠿⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡀⣠⠶⠌⠀⠐⠤⡀⠀⠀⠀⢀⠠⠔⠒⢲⣤⣄⡀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣼⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⡀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢰⣿⠅⠀⠀⢀⣤⢶⣶⣤⡀⠀⠀⢠⣴⠶⣶⣦⣄⠀⠀⣹⣷⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⣿⣤⣀⡔⢿⣿⣾⣿⣿⠧⠤⠠⠼⠿⠿⣿⣿⡟⢳⣠⣿⠿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⡟⠉⠀⢸⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠹⢖⠀⠀⢠⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⢰⡇⠀⠀⠁⠀⠀⢀⣴⣶⠾⠛⣷⣦⡀⠀⠀⠀⠁⢠⡏⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠈⢻⡗⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠹⡆⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠠⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⢀⡔⢣⡀⠀⠀⠀⠀⠀⠀⢀⣇⡀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠹⢄⣘⣦⡤⠔⠓⠁⠀⠀⠙⠑⠶⣄⣀⣜⣠⠞⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠲⠦⠦⣄⣤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡌⠊⠒⠹⠅⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣍⠉⢲⡭⢀⡐⠀⠀⠀⠀⠻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⣿⣿⣿⣶⠇⠈⠀⠀⠀⣀⠀⢖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠠⠁⠀⠀⠀⠀⠀⠀⢸⡅⠸⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡧⣅⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⣠⣄⣀⠴⠲⠊⠀⡇⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⢠⠀⠀⠑⠢⣀⡀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣼⡉⠒⠒⠒⢒⡿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠃⠛⠉⠉⠉⠁⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⠑⠦⡀⠀⠁⢱⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠘⡆⠀⠀⠋⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣆⠀⢹⠀⠀⡇⠀⣀⣀⣀⣀⠀⠙⠶⣄⠀⠀⠀⠸⡀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠬⡇⠀⣧⠞⠁⠀⠀⠈⠙⠦⠀⠈⠳⣄⠀⢠⠇⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢹⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡎⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠙⣲⣷⢸⡆⠈⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢴⣋⡤⠈⠉⠁⣼⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠔⠊⠙⠳⠛⠶⠚⠉⠉⠛⠛⠛⠋⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⣶⣿⣿⣶⣤⡀⠁⠀⣌⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⣿⣿⣿⣷⡄⠠⠙⠂⠈⠉⠉⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢰⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠃⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⡀⠀⢀⡰⠃
⠀⠀⠀⢸⠀⠀⠀⠙⠲⠤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣷⡄⠈⠻⣿⣿⣿⣿⡿⠟⠲⠴⠦⠤⠜⠒⠉
⠀⠀⠀⠀⠳⢤⣀⠈⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⣿⣿⠿⠃⠀⠀⠀⢨⠏⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠑⡆⠸⣿⠿⠋⠉⠙⠛⠿⠟⠃⠀⠀⠀⠀⠀⠀⢀⡐⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⡴⢤⣀⣀⠈⠀⠀⠀⠀⠀⡉⠀⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢊⡴⠋⣠⠞⠁⠈⠉⠉⠉⠉⠉⠉⡇⠀⠀⣸⢹
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡎⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠣⣀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⠳⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⢀⣴⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⣿⣿⣿⣿⣿⣿⣇⠀⣀⣀⣀⣠⡿⠁⠀⠀⠀⠀⢠⡄⠛⢠⣤⡌⠉⠀⠀⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠘⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠃⠀⠈⣿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠳⣦⣤⣤⣶⣶⣶⣿⣿⣿⠟⠇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢤⠀⠉⢠⡞⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⢿⣿⠋⠈⢿⣇⠻⣧⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠈⠁⠀⠀⠀⠀⠀⢀⣼⡿⢛⣿⡟⠉⠛⠛⣿⢿⣿
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣸⡇⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⢹⣇⠀⠀⠀⠀⣿⠘⣏
⠀⠀⠀⠀⠰⠿⢠⣿⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠙⠃⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⢠⡐⣀⡆⣆⠂⡇⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠙⠛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⢠⡿⠀⠀⢀⣽⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠀⠘⠛⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣆⠀⠀⠀⢀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣤⠸⠀⠀⣠⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢠⣿⣿⣿⣿⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣾⣿⣿⣣⣾⣿⣿⣿⣦⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠋⢭⣷⡄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⢹⠏⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢶⣶⣆⠀⠀⠀⠀⠀⠀⠀⢶⣺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠻⣿⣦⣌⣓⢦⣄⣀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⠻⢿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣭⡉⠛⠋⠉⠁⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣄⠀⠀⠀⢶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀⠀⠉⠙⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⡀⠀⣀⣀⣠⣤⣤⣠⣤⣤⣤⣴⢿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⠄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣼⠁⠀⢀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠀⠀⠘⠀⠀⣷⣾⣿⠈⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠿⠟⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⢀⣀⣤⣤⣀⠘⠀⢸⠃⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣥⣶⣶⣾⡇⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣿⡗⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠰⣄⠀⠀⢀⣠⠴⠒⠛⠓⢢⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠸⢻⣿⣧⣤⣶⣶⡿⠛⠉⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠧⢀⡀⠀⠀⠀⠈⠙⠉⠋⠁⢀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠶⣄⣤⣦⣤⣤⢴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠔⢁⠀⠀⠀⡇⠈⠃⠂⡄
⠀⠀⢀⡀⠀⠀⠺⠄⠀⠀⡇⠀⢀⡾⠀⠢⠄
⠀⠀⢽⠀⠀⠀⢀⣀⢄⡀⠳⡴⠋⣼⠀⠀⠀⢦⡤⣀
⠀⠀⠀⠀⢐⠓⡩⠗⠋⠀⠀⠀⠀⣾⠀⠀⢠⡾⠁⣹⣧⠘⢶⣶⣶⡶⠶⠶⠶⠶⠖⠲⠶⠦⢤⠄⡀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠄⠀⠈⣿⠀⠀⠋⠀⠰⣿⣿⠀⠀⣿⠟⠙⢲⣶⠶⡟⠿⠓⠒⠀⠀⢀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⡄⢸⡟⠀⠀⠛⠀⠀⠟⠉⠀⠁⠀⠀⠀⠀⣰⠃⠀⠀⢣⠄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠁⠀⠀⠀⠈⠀⡄⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢲⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢌⠳⠀⠀⢀⣯⠄⠀⠀⠀⠀⠀⠄⠀⠀⢿⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠟⠶⠾⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠉⢳⡄⠀⠀⣾⠤⠤⠤⠀⠀⠀⠀⠀⠀⠸⣆⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⢿⣿⣶⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣌⢿⣿⠷⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠐⡀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⠀⡴⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡏⠀⠀⠃⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄⡨⢃⢤⣤⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣑⣾⠁⠀⠠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠘⠒⠁

⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢀⡀⠿⠇⣀⡀⠛⠃⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷
⠀⠹⡿⠻⣿⣿⣿⣿⣷⣶⣿⣿⣷⣿⡁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠙⠻⠷⢦⣄⣀⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠆⢀⡔⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠻⣿⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⡿⢹⣿⠀⠀⠙⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠡⣾⠟⠁⠀⠀⠈⣿⠻⣿
⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠿⠄⠀⠀⠀⠀⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠛⠃⠸⠿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠹⠸⡈⠠⠹⢸⢘⠸⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣼⠇⠀⠀⢠⣾⡿⠿⢿⠿⠿⠿⠶⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠻⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠼⣿⣿⠃⠀⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠋⣄⡀⠀⠀⣠⡴⠾⠛⠛⠛⠳⠶⢤⣤⣤⡶⠿⠛⠛⠛⢷⣮⡻⠀⣠⣬⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⢿⣇⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡌⠉⠉⠈⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠘⠃⠁⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⠀⣿⣷⠀⠀⠀⠀⠙⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⣤⣈⠹⣿⠀⠀⠀⠀⠀⠀⣠⣤⣀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠈⠻⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣽⡿⠆⠏⠀⠀⠀⠀⠀⠸⣿⣽⣏⠷⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⣁⣀⣠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠈⠉⠛⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⣤⣿⣿⣧⠚⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⢀⣴⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣤⣤⣤⣶⣿⣿⣿⣿⡿⢶⣶⣶⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⣰⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⢤⣶⣶⡦⡤⢀
⠀⠀⠀⣴⣿⣿⠟⣱⠇⠀⠉⡽⠡⣩⣍⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⣤⣤⣼⣶⣬⡑⢂⢹⣦⡀
⠀⠀⠘⡟⢿⣿⠂⣟⠀⠀⢸⡗⢰⣿⡟⠃⣘⣦⠤⣶⢾⢿⢿⠿⠻⣖⠂⢉⡀⠶⢶⣭⡏⠉⠉⠛⢦⡻⣿⣿⡄
⠀⠀⠀⢣⠘⣿⣀⡡⠂⠀⣾⠃⣸⣿⡷⠾⠋⠁⠴⢁⡴⠀⣸⠇⡐⠈⠓⢦⡌⠓⢄⡙⢇⠀⠀⠀⠀⠹⡝⣿⣿
⠀⠀⠀⠀⠁⢸⣿⠀⠀⠸⠏⢠⣿⠟⠀⠀⠀⠀⠠⠏⢠⡾⠋⢠⡷⡄⠐⣦⡉⠓⢤⡙⢌⢦⠀⠀⠀⠀⠸⣽⣿
⠀⠀⠀⠀⠀⡈⣿⠃⠀⡁⠀⣾⡟⠀⠀⠄⠀⠠⠁⢠⠟⢀⣶⠟⢀⣷⠰⠘⡟⠁⠀⣀⣀⠘⢧⠀⠀⠀⠀⠙⠿⠂
⠀⠀⠀⠀⠀⢀⠉⠑⢸⠁⣸⣟⠀⠀⠀⠤⠀⠀⠀⠁⠀⢁⣠⡤⣄⠙⡆⡆⢷⡄⠘⢩⣿⣫⣍⣦
⠀⠀⠀⠀⠀⠀⢗⣻⣿⢠⣿⣿⣦⣐⠀⠀⠀⠀⣠⠖⣛⣛⣻⣷⣤⣤⡗⠁⢸⣷⣴⣾⣟⡷⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠘⣿⣿⣿⣿⣿⣟⠀⠈⣡⣾⣿⢿⡝⠻⣿⣿⣷⠀⣾⣿⣿⣿⡿⠴⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⢹⣿⡿⠀⠍⠉⣶⣄⠁⠱⢽⡯⠴⢃⣿⣿⣿⡆⠈⠻⣿⣭⡳⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠰⠂⢀⢻⣦⡀⣀⣶⡄⠈⡼⡫⣿⠇⠠⠧⠃⠀⠉⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠨⡄⠻⣿⣿⡿⢁⠎⠜⡴⠁⢀⠄⢀⡀⡀⠫⣤⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣆⠀⠀⣷⠀⢹⡟⢠⠎⢀⡾⣂⡀⠚⠀⢸⡧⢸⢀⠻⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡏⣿⣿⣦⡀⠈⣇⢸⢁⡞⣠⣟⡴⢛⣷⡀⠉⠑⢤⣸⢨⣞⡡⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢱⢹⣿⣿⣷⠰⣹⣸⡸⢠⣿⡯⠶⢋⣹⡇⢰⠖⠁⢸⡂⠀⠛⠂⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠂⠁⠉⢻⡀⣿⢻⠇⣼⣯⠤⠶⠟⢻⡆⠈⠀⠀⡀⠷⢦⡘⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠄⠀⠀⠈⡇⠟⣼⢀⣿⣷⡦⠴⠖⣻⣇⢠⠄⠃⠐⠂⠈⢷⡆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⠐⠐⠆⠀⠀⢳⣾⣿⢸⣿⣿⣿⣄⣀⣿⡏⠈⡄⠖⠒⠛⢋⠸⣧⣸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⠀⠈⠀⠀⠀⠀⠔⠀⢻⡇⣼⣿⣿⠗⠊⠙⡻⠀⣀⣤⣤⡀⠀⠀⠀⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠛⢧⣸⣶⠀⠇⠀⠌⡖⠀⡤⠀⠳⣽⣟⣧⣴⡶⢏⣵⠃⣿⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢴⣄⣀⠈⠄⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠀⣹⣿⡿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠀⠀⠀⠀⠀⠰⠟⠛⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠄⠐⠒⠂⠠⠄⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠃⠀⠀⢸⣯⠆⡠⠤⠤⡀⢯⣿⠀⠀⠀⠇⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠸⠒⠒⠒⠐⠒⢊⣤⣒⣠⣌⠑⠒⠒⠒⠐⠺⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠇⠀⠀⠀⠀⠀⣀⣉⡿⣏⣁⡀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠫⣀⡨⠁⠀⠀⠀⠀⠀⡰⡖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠢⠤⢀⣀⣀⣀⣀⠀⠤⠄⠲⡋⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⠃⠉⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢣⠀⠈⠢⣀⠀⠀⠀⢀⡠⠊⠀⢠⠃⢱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆⠀⠀⠈⠉⢲⠊⠁⠀⠀⠀⡜⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠂⡜⠢⠤⠤⠀⢸⠀⠀⢀⡀⠜⣄⠚⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣰⣘⣴⡀⢰⠀⣀⠼⢄⡀⢐⠀⢰⣼⣰⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠒⠲⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢋⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⡙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢰⠁⠀⠀⠀⣼⡏⣳⠀⠀⣀⣀⡀⠀⣞⢹⣇⠀⠀⠀⢸⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣼⣀⣀⣀⠀⠘⠛⣋⠴⠋⣀⣀⠉⠲⣝⡛⢃⣀⣀⣀⣀⣇⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⡏⠀⠀⠈⠉⠉⠉⠀⢼⣿⣶⣶⣿⡆⠀⠉⠉⠀⠀⠀⠀⠹⡄⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⢀⡀⢉⣽⢿⣁⣀⡠⠀⠀⠀⠀⠀⠀⢀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠉⣏⠁⠀⡹⠉⠀⠀⠀⠀⠀⠀⠀⣸⡵⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⣀⡤⠞⢹⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠉⠑⠒⠲⠶⠤⠤⠶⠖⠒⠒⠉⠉⠀⣇⢀⡼⣄⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡎⠀⠀⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢸⡀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⡇⠘⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢇⠀⠀⠀⠈⠓⠦⢤⡤⠴⠚⠁⠀⠀⠀⣸⠁⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⣀⠘⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠃⢀⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠉⡰⠋⠓⠦⠶⠂⠀⢸⠀⠀⠠⠤⠤⠖⠛⢦⠉⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡇⢰⠀⢧⣶⠀⢰⠀⠀⢀⣸⢆⡀⠀⠀⡆⠀⣷⣸⠀⡆⢰⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠚⠓⠋⠉⠀⠀⠈⠙⠚⠓⠚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢀⡴⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠰⠤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠐⠊⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠘⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠉⠓⢦⡶⠒⠓⠒⠂⠀⠁⠀⠋⣀⢤⠀⠀⠀⠘⣤⣤⣤⣤⡤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⡀⠀⢈⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⠀⠀⡴⠛⠂⠘⠀⠀⣤⣬⡁
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠙
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⠏⢸⠀⠀⠀⠀⠀⢀⢠⠀⢀⣄⠀⠀⣀⡀
⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⠰⠋⠀⣴⠀⠀⠐⡻⠛⠊⠀⠀⠈⠁⠀⠔⠻⢁⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⣲⣾⣿⡿⠿⠿⢿⣷⡄
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣤⣤⣾⣿⣿
⠹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⠞⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⠿⣿⣿⣿⣿⣷⣄⡀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⢢⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠋
⠀⠀⠀⠈⠙⠻⠿⢿⣿⠿⠋⠁⠀⢹⣿⣿⣶⣶⣶⣶⣶⡿⢻⣧⠀⠀⠀⠡⡀⠀⠀⠀⠈⠑⠒⢀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠙⢷⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⠰⡆⠦⠀⣀⡄⠀⠀⠀⠀⣀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡞⠀⢀⣴⣿⣿⣿⣶⣄⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢸⠁⠀⢾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣶⣿⣿⣿⣶⡄⠀⠀⠉⠢⢄⠀⠀⠀⠀⠈⠐⠂⠼⠷⢦⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠈⠿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠒⠄⣀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠐⣆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠼⠿⡶⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠾⠿⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡘⠀⠀⢄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠈⠆⠐⠀⠀⠀⠀⠁⠀⠀⠂⠠⢀⠀⠀⢀⠀⠤⣀
⠀⠀⠀⠀⠈⠓⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠑⠄⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡈⢢⡀⠐⣀⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⣠⣄⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣞⠀⠀⠀⢠⠀⠀⠈⠁⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠙⠓⠒⠒⠒⠒⠒⠚⠋⠁⠀⠉⠀⣀⡤⠋⠀⠀⢀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠢⢤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⢢⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠹⠶⠆⢀⣀⠤⣿⠀⠀⠀⠀⠀⠀⡼⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿
⠀⠀⠀⠀⠉⠁⠀⡏⠀⠀⠀⠀⣠⡞⠉⠉⣠⠆⠀⠲⢤⠠⢤⡤⠜⠀⠀⠀⢠⢧⡏
⠀⠀⠀⠀⠀⣠⠞⠀⢀⡠⠚⠉⠹⢧⡀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⢠⡏⠀⢰⠛⠁⠀⠀⠀⠀⠳⠀⠈⠢⣄⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠈⠓⢤⡀
⠀⠀⠀⠀⠀⡇⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠢⠴⠦⠃⠀⠀⠀⠀⠈⠙⢆⠀⠀⠓⢶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠤⠄⠤⠤⠀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⢠⠆⠀⠀⠀⠀⠀⠀⣀⢰⡀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠰⠂⠀⡏⢰⣿⡆⠀⠀⠀⣾⣷⠈⡇⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡄⢀⡁⠀⠀⢀⣾⢲⣦⠀⠀⠀⣿⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠔⣼⠁⠀⠀⢀⣿⣿⣏⠀⢀⠀⢸⣄⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣸⡀⡾⣦⡜⠛⠟⠻⣤⣾⠀⢰⢩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠦⢐⢈⡻⢦⣤⠾⢋⢈⡤⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣤⣀⣨⠀⣉⠁⠀⠈⠁⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠺⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣾⡏⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠙⣿⣿⣿⢿⣿⣿⣿⡿⣻⣿⠋⠶⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢰⡄⠘⣿⣿⣿⣿⣿⣿⡟⠱⣿⠈⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠘⣘⡧⠀⠈⣿⢿⣿⡿⠃⠀⠀⣏⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡙⠁⠀⠀⠀⠀⠙⠃⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣲⠞⣇⠀⠉⠀⠀⣃⢀⠀⠀⣤⠀⠀⠀⠠⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⢳⣄⠞⠈⠁⠛⠂⠴⠸⠀⠀⡔⠓⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠐⣆⠀⠀⠀⢠⡞⠀⠀⠀⡇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⠂⠀⠘⢣⡦⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⡴⢹⢇⠀⠀⠀⠀⣸⠿⠁⠀⡅⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⢸⠀⠙⠒⠒⠋⢩⠀⠀⠀⡇⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠖⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⠳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⡠⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡻⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣶⣶⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣎⠁⠈⠋⢹⠟⢘⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡈⠀⠀⠀⢸⣏⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢺⣿⡿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⢹⠿⢿⣿⠏⠀⠀⠀⠀⢠⣄⣀⣀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣏⠁⠀⠀⠁⠀⠀⠀⠀⢢⣶⣿⣿⣿⡇⠀⠻⢿⠿⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⠿⠁⠀⠀⠀⢀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠿⣅⠀⠀⠀⠐⠂⠺⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⢀⣉⠁⣀⣤⡗⠀⠀⠀⠀⢀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢐⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠿⠿⠿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⢀⣤⣴⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠻⢷⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣙⣷⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠻⣷⣄
⠀⠀⠀⣀⣴⣿⠟⠁⣀⣤⣶⣾⣿⡷⠀⠀⠀⢀⣼⠏⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⢷⡄⠀⠀⢰⣿⣷⣤⣀⡀⠈⠻⣷⣦⡀
⢀⣴⣾⣿⣿⣷⡿⠿⠛⠋⠉⠀⣿⡇⠀⠀⠀⣼⣇⣀⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⠈⣿⠀⠀⢸⣿⡍⠉⠛⠿⣷⣶⣬⣽⣿⣷⣤⡀
⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⣿⣏⣽⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣩⡇⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿
⠀⠙⠛⠿⣿⣶⣶⣤⣤⣤⣤⣾⣿⠁⠀⠀⠀⠙⠿⢿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡷⠟⠀⠀⠀⣿⣧⣀⣀⡀⢀⣀⣠⣤⣶⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣯⣍⣙⢲⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⠛⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⣀⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠈⣽⡿⠛⠻⠿⠿⠿⠿⠿⢿⣿⣥⣬⣿⠿⠿⠿⠟⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⣧⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠈⢿⣷⣤⣀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠛⠻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⢀⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠁⠛⠿⣷⣶⣶⣶⣶⣶⣶⣿⠿⠿⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠂⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠀⢀⣠⣴⣯⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣭⣨⣸⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣼⣿⠿⠻⠧⢘⣿⡟⣹⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠘⠀⠀⠛⠀⠀⠘⢛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⣤⣀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠘⠿⡟⢿⣿⣿⣿⣿⣿⡿⢿⠿⢿⣾⡿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⢈⣿⣿⣿⣿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⢏⣿⡿⠤⢄⣠⠶⠊⠉⠀⠘⢿⣿⣿⣿⣷⣬⡛⣾⣿⣿⣿⣷⣄
⠀⢰⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠘⣇⠀⠀⠀⠀⢀⣠⣤⣴⣆⢿⣾⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣤⣤⣄
⠀⠰⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣥⣶⡆⣼⣿⣿⣿⡟⠻⠿⠟⠿⢿⣾⣿⣿⠿⢣⣿⠿⠿⠟⠛⠿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠪⠿⠛⠋⠁⠈⠀⠀⠀⠰⠿⠟⠛⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⣠⣤⡄⣠⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣶⣿⣿⣷⣤⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡟⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠤⠻⠿⠿⠿⢻⣿⣍⣿⣿⣿⢿⠿⠟⠋⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠉⢹⣿⣻⣟⣿⣿⣿⣿⣿⣟⣟⣿⢟⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠴⠖⠛⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣠⠚⠉⢉⡤⠟⠛⠛⠛⠛⠛⠛⠲⢶⣤⣄⡀⠀⠀⣀⣤⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢰⣇⢀⡔⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠈⠙⠳⣟⠉⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⠟⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢸⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⡀⠀⡄⠀⠀⠀⣀⣀⠀⠀⢧⣼⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⢞⣉⠉⠲⡀⠀⠀⠀⠀⡃⠀⠀⣀⣈⡉⠁⠀⢸⡟⣧⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⣿⣿⣶⢀⠇⠀⠄⠀⠀⠀⠀⣜⣴⡆⠈⡆⠀⠀⣧⠈⠛⠋⠁
⠶⣷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠛⠓⠁⠠⢪⣤⡶⠤⢄⣀⠹⣿⣟⡡⠃⠀⠀⣿
⠀⠘⣇⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠄⠂⠀⠉⡀⠀⠈⠻⣿⠿⠟⠁⠑⠀⠄⠀⠀⠀⠀⣿
⠀⠀⣿⠈⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠈⠒⠤⠴⠺⢄⡀⠔⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⢻⡀⠀⠈⣧⠀⣀⣤⡤⠶⠖⠛⠛⠋⠊⢙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟
⠀⠀⠈⠻⣄⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩⢙⠒⠲⠶⠶⢤⡤⠤⠤⣶⠶⠖⠛⠉
⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠀⠚⠀⢘⢰⡟⠁
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠐⠂⣼⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⢰⢣⡟
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⢇⠀⠀⠀⠀⠀⢠⡇⠀⠀⣠⠋⣾⠁
⠀⠀⣠⠟⠀⠀⠀⠀⣀⣤⠚⠣⢄⣀⠀⠈⢆⠀⠀⠀⠀⠸⡇⣠⠞⠁⢰⡇
⠀⣼⠃⠀⠀⢀⡴⠟⠉⢹⡄⠀⠀⢸⡟⠓⠾⣆⠀⠀⠀⠀⡟⡇⠀⠀⢸⡇
⠀⡟⠀⡤⢤⣼⡁⠀⠀⠘⣧⠀⠀⣼⣤⣄⠀⠹⣆⠀⠀⠀⢻⣷⠀⠀⢠⠷⢤⣄
⠸⣇⠀⠀⠀⠈⣷⠀⠀⠀⢹⡄⠀⠀⠀⣹⡇⠀⠹⣆⠀⠀⠉⠉⠙⡆⠀⠀⠄⢸⡇
⠀⠈⠙⠛⠛⠚⠋⠀⠀⠀⠀⠙⠛⠓⠛⠛⠁⠀⠀⠘⢷⣤⣤⣤⣴⠿⠛⠛⠛⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣌⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢟⣥⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⣷⠄⢦⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⡔⠠⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣗⠀⠻⣦⡙⠷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣴⠏⠀⣺⣿⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠦⠀⠙⣿⣦⡈⠻⣦⡶⣴⣴⢶⣴⠟⢁⣴⣿⠋⠀⠴⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⡍⠀⠀⠘⣿⣷⣴⠂⠀⠀⠀⠀⠐⣦⣾⣿⠁⠀⠀⢩⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣗⡀⠀⢀⣼⠟⠁⠀⡐⠀⠀⢄⠀⠘⠻⣇⡀⠀⢀⣻⣿⣇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⣿⠛⠁⠀⠉⠀⠀⠀⣰⠇⠀⠀⠸⣄⠀⠀⠀⠉⠀⠈⠛⣿⣿⡄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣀⣀⣀⣈⠳⣄⡀⢰⠀⠀⡄⣀⣠⢞⣁⣀⣀⣀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣥⠀⠀⠀⠀⠀⠀⠉⠉⢹⡿⣿⣌⢱⡏⠀⠀⢻⡌⣡⣿⢿⡏⠉⠉⠀⠀⠀⠀⠀⢀⣬⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⠄⠀⠀⢽⡆⠀⠀⠀⠀⠓⠒⠻⠀⠀⠀⠀⠀⠐⠟⠒⠋⠀⠀⠀⠀⣸⠇⠀⠀⠠⣬⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣁⠀⣠⣄⡀⢻⣄⠀⠘⠶⣄⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣠⠶⠃⠀⣰⡟⢀⣠⡀⠀⣈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣥⠖⠀⠈⠛⢦⠻⠦⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠴⠧⡴⠋⠁⠀⢲⣬⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣤⣤⣤⣀⠀⠐⣶⣤⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢻⡄⠀⢀⣤⡶⠂⠀⣀⣤⣤⣤⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡟⠓⠀⠈⢿⣷⡀⢺⠁⠀⠀⠀⠀⠀⠀⠘⡇⢠⣿⡿⠁⠀⠛⢻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣯⣤⡀⠈⣿⣧⡀⠀⣠⣤⣶⣦⣤⣄⠀⢀⣾⡿⠁⢠⣤⣽⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣶⣦⣼⣏⢳⣄⠹⣭⡍⢩⣭⠏⣠⡞⣹⣧⣶⣾⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠽⣿⣿⣆⠻⣷⣤⣭⣭⣴⣾⠏⣰⣿⣿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⣜⠛⠛⣡⠋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣉⣩⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠭⠑⡄
⠀⡘⠩⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰⠌⠈⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠈⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⢀⣄⡀⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣠⣤⠐⠂⠀⢀⣄⣀⣄
⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⡤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠐⠃⠉⠀⠀⠀⠀⠀⠀⠔⣒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⠤⣀⠢⠙⠆⠀⠀⠀⠀⣀⠀⡎⣿⠿⠀⢀⡀
⠀⠀⠘⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⡄⠻⠿⠁⠀⡌⠀⠀⠀⠰⠁⠀⠀⠈⠀⣶⡋
⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠄⢰⠍⠂⠀⠀⠀⠙⠢⠤⠈⠀⠀⢂⠀⠀⠀⠀⢀⣾⢠⠀⡀⠉⠂
⠀⠀⠀⣦⣥⣧⡄⠀⠀⠀⠀⠰⠄⠂⠐⠄⠀⠀⠀⠀⠉⠠⠜⠅⠉⠀⠀⠀⠀⠶⠀⠅⠇⠈⠉⠀⠀⠀⠀⠹⠈⠑⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢤⢄
⠀⠀⠐⡋⠟⠤⠀⠀⠀⠀⠐⢀⡉⠀⠀⠀⠀⠀⠀⠀⢀⣩⣍⢃⠀⠀⠀⠀⠀⠒⠂⢊⡆⠀⠀⠀⠀⢀⠄⠠⠂
⠀⠀⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡈⢀⠀⠀⠀⠀⠀⠠⠠⡷⠀⢀⠀⠀⠉⠀⠀⠀⣀⠘⠂
⠀⠀⠀⠀⠛⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡐⡀⢂⢔⡄⠀⠀⠀⠀⠁⠐⡀⠐⠀⠀⠀⠀⣞⠉
⠀⠀⠀⠐⠁⠰⠁⠀⠀⠀⠈⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡦⡤⠀⠀⠀⠀⠀⠀⠐⣢⠢
⠀⠀⠀⠁⠿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡤⠄⠀⣀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠁⠀⠀⠐⠉⠉⠀⠀⠈⠁⠀⠀⠐⠀⠒⠒⠂⠶⡶⢄⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢀⣿⡏⠠⠂⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢀
⠀⠀⠀⠀⡀⢀⠀⠀⠀⢰⡆⠀⠀⣤⠏⡤⠀⠀⠀⠐⠂⡰⠦⠀⠀⠀⠄⠂⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⠛⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⢀⣴⣏⣜⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢳⢦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣡⡄⠀⠀⣀⣤⣔⢄⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡆⢿⠘⢠⠁⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⣼⣟⣵⠶⠓⢠⣿⣿⣧⣿⡌⠹⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⢷⣶⡀⠀⠈⠛⢾⡇⠀⡇⠀⠀⠀⢉⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠉⣿⣿⡿⣀⠀⠀⠑⠘⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠺⠻⣟⡽⠳⠀⣸⠸⣿⠁⠀⠀⠀⠀⠀⠈⢸⠃
⠀⠀⠀⠀⠰⡟⡀⠀⠀⠀⡟⢰⣿⣿⠀⠈⠹⣿⣶⠀⠀⠀⠀⢀⣤⡄⠠⠀⡀⣀⠀⠀⠀⠣⡀⠀⢀⣤⠀⠘⡓⢿⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠈⠻⣆⠀⢀⠁⣾⠏⡿⠀⠀⣀⢹⣿⡀⠀⠀⣰⡿⣋⡀⠀⠌⠁⠈⠻⣄⠀⠀⠐⣶⣿⡏⠀⠀⣿⣿⢸⣇⠀⠀⠀⢰⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢇⡜⢴⣿⢸⡇⠀⠀⠛⡿⡭⠁⠀⢠⣿⣶⣿⣿⠁⡀⢺⣿⣷⣼⡄⠀⠀⠻⣿⠀⠀⠀⣽⣿⡘⢿⡀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣼⡇⣸⠗⠋⠁⠈⠀⠃⠀⠀⠀⠑⣧⡈⢡⣼⣗⣠⠉⣻⠿⠇⠀⠀⠀⠉⠲⣀⢀⡿⣹⣇⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢃⣿⢀⡴⣶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢠⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⡿⣿⠐⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣻⡯⠸⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠈⣧⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣩⣿⠚⣧⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣤⣿⣦⣡⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⡼⢏⣼⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠸⡇⢻⣿⣿⣤⣄⣀⣠⣴⡿⣿⣟⣙⣧⣀⠀⠀⠀⠀⢠⠎⠁⠔⠃⠛⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣇⣿⣿⣿⣿⠗⠒⠂⢺⣿⡋⠀⠿⣷⣶⣶⣶⡿⠐⠀⠀⠛⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡷⠞⣨⠛⠁⠀⠀⠀⠙⠋⠁⠀⠀⠸⣯⣽⣟⠓⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠹⠿⣿⡿⠿⠛⣷⡶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠖⠛⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠙⠒⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⡀⠀⠀⠈⢧⠀⠰⡄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⡠⠀⠀⢀⣐⣾⠛⠀⠀⠀⠘⠀⠀⠃⠀⠀⢠⡌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠐⠀⣰⣾⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣦⡘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣻⣿⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⡟⣳⡘⣿⡀
⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠉⠉⠉⠁⠘⠁⠀⠀⠀⠀⠀⠄⠀⠀⠤⠍⠻⢿⣇⣿⣧
⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢙⠶⢤⣄⡀
⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣧⡀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣷⣤⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⠟⠋⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣆⠡⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡜⠄⠀⠀⠀⠀⢠⣿⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⡀⠀⠀⢀⣾⡯⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⣿⣿⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣿⡏⠁⠐⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣪⣴⣿⣿⣿⣿⣿⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠠⡪⠀⠀⠀⢀⣴⣶⣶⣶⣦⣤⣴⣶⣤⣶⡶⢿⣿⣿⣿⣿⣿⣿⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⠃⠀⠸⣇⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠰⣬⣻⣿⣿⣿⣻⡿⠿⢛⣯⣷⣿⣿⣿⣿⣿⣿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠄⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣛⡟⠉⠁⡠⡾⡿⣿⡟⠁⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⢹⡄⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠊⠀⠊⡜⡑⢑⣿⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡆⢳⡀⠀⠀⠀⠀⠑⡀⠀⠠⡐⢄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⠁⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢳⡀⠀⠀⠀⠀⠈⠀⠀⠘⢮⠣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠜⢀⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⢀⠎⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠿⣿⡛⠛⠿⣛⠛⠛⠛⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣠⣤⠴⠶⠒⠛⠋⠉⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⣠⢞⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠞⢡⡾⠁
⣠⠖⠒⠒⠒⠒⠚⠋⣁⣠⣴⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⣀⠀⠀⣿⠛⠶⣤⣤⡤⠿⠃⠀⣴⠃⢀⡿⠁
⠉⠓⠶⢤⣄⡉⠉⠛⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡏⠉⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣼⠇
⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣤⣤⣉⣷⠀⢻⡏⠉⠙⠛⠒⠲⠞⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣾⡏⠀⣠⣤⣴⣶⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢛⣿⣿⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠙⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠛⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⢀⣤⣤⣤⣤⣶⣦⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠙⠿⠿⣿⠟⠁⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣀⣰⣇⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠤⠤⣤⣿⣦⡶⠞⢠⡞⠁⠀⠀⠀⠀⠀⠀⢸⡇⣀⡠⠋⠀⠀⠹⣆⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⢀⣴⠶⠦⣤⣄⣀⣀⡤⢶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⠁⠀⠀⠀⠀⠘⠲⢤⣄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣠⠞⠁⠀⢰⡇⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⠃⠈⣧⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠃⠀⠈⣻⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠈⢀⣀⣤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⣟⣠⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⠤⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣰⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠿⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⣠⡞⠉⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣶⣄⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠀⢰⡄⠀⠀⠈⠙⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠉⢉⠉⠛⢷⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠛⠉⠁⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡙⠻⣿⣆⠀⣿⣷⡄⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣰⡿⣧⡉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠹⡆⠘⣿⡀⢹⣿⣿⣄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⠁⠈⣿⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣸⣷⠀⣿⡇⠀⠙⣿⠿⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⣿⣷⣤⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⢀⣿⠁⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⣿⡀⠀⠀⢿⣿⣿⡿⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣤⠾⢃⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⡾⠿⣷⣤⡀⠈⠻⣿⣤⡾⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠙⢧⣴⠶⠒⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⢿⣷⣤⣠⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠿⠿⢿⣷⠀⠀⠀⠀⢘⡏⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠑⠛⠻⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⡀⠀⢀⣰⣿⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠙⢿⣿⠿⠿⠿⠟⠛⠁⠀⣠⣴⣿⠏⠀⠀⠀⠀⠠⠾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠛⠿⣶⣤⣤⣄⣠⣤⣤⣴⠾⠿⠗⠶⠾⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣾⠿⠿⠿⠿⣷⣄⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠿⢿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣶⠞
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣤⣶⣿⣿⡇⠀⠀⢀⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢠⡇⣆⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏⢸⠀⠀⠀⢸⣧⠀⠘⣿
⠀⠀⠀⠀⠈⣇⠘⢦⡀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⡿⠀⠀⠀⢸⣿⠀⠀⠻⣧⣤⡀
⠀⠀⠀⠀⠀⠹⣄⠀⠙⠶⣤⣤⣿⣿⣧⠀⠀⠀⠀⠀⣸⣿⠿⠟⠛⠛⢦⡇⠀⠀⠀⠘⠿⠛⢦⡀⣄⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣉⣉⣁⣀⢀⠀⠀⢀⣉⣄⡀⠀⣾⣇⠘⣧⡀⠀⠀⣤⠀⢰⠘⣧⣼⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⡿⠿⠛⠙⠛⠿⠿⠟⠻⠿⠛⠿⠿⠿⠿⠷⠟⠻⠟⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⢒⣐⣖⣀⡀⠒⢒⠒⠒⡒⠒⠶⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢒⣡⣄⠊⠁⣠⠀⠀⠀⠀⠀⠉⠛⢿⣿⠁⠀⠀⠀⠈⠀⠀⠙⣆⠀⠀⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠚⢋⠉⠀⠈⠀⠀⠀⠀⢀⠄⠀⠀⠀⠹⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⡽⢃⡾⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢯⠀⠀⠁⠀⠀⠀⠀⠀⠸⡎⣇⢸⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡇⠀⢠⣴⣾⡇⠀⡞⠀⠀⠀⠀⢠⣄⠀⠀⣤⣴⢂⣼⠃⠀⠀⠀⠈⠘⣆⣴⣿⣀⡲⠄⠀⠀⣿⣿⠈⠁⠀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⠸⣿⡟⠀⢿⣿⣿⡇⠀⡇⠀⠀⠀⣠⡿⣿⣽⣿⣿⣿⡜⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣽⣿⠦⡄⠀⢸⣼⡆⠀⠀⠀⠸⠿⣇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⢻⣇⠀⠘⣿⣿⡇⠀⣷⠀⠈⠉⠉⠁⣽⣿⠋⢻⣿⠁⠀⠀⠀⠀⣿⡀⠀⠀⢎⠻⠧⠀⠀⡠⠀⢸⡏⡇⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⠀⠀⠘⣿⣧⠀⣿⡆⠀⠀⣄⠀⠀⠀⠐⣻⣯⠀⠀⠀⠀⢀⡿⠷⠀⠀⠀⠁⢰⣶⠟⠁⠀⢸⣿⡇⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢻⣿⠀⣿⠈⠀⠀⠘⠳⣶⡆⠀⣻⠃⠀⣠⡶⠛⠉⣉⢀⠀⣤⣤⡈⠻⣅⠀⠀⠀⢸⣿⡗⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⢸⣿⡇⣿⠀⠀⠀⠀⠀⡼⠃⠀⡙⠀⡾⠃⠀⠀⠾⢿⡛⠃⠀⣻⡷⠀⢹⡄⠀⠀⢸⣿⣷⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣸⡟⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠸⠁⠀⠀⠀⠳⠿⠇⣠⣦⣙⣿⠃⠀⠃⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣴⣿⣿⡇⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⠿⡿⠛⠁⠀⠀⢀⢺⠄⠀⡟⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⢿⣿⣷⠀⠘⠁⠘⣾⣧⠀⠀⠀⠀⢀⡁⡅⠨⢽⣿⡇⣸⣿⡒⠂⠀⢸⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⣿⢿⣆⠀⠀⠀⠻⣿⣧⡀⠀⢠⢠⡄⣴⠶⢿⠟⣿⣿⣿⣧⣉⣀⡌⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠸⡆⠻⣆⠀⠀⠀⠈⡿⢿⣶⣤⣀⣤⡤⢺⠛⢷⠹⠉⠈⢿⠻⠋⡴⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢹⡄⠀⢷⠀⠈⠓⢦⣄⠀⠘⠺⠿⠻⣿⡏⣱⠹⣆⠀⠀⠀⠀⠸⣳⠟⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠁⠀⠈⢧⠀⠀⠀⠹⠦⣀⣀⣀⠀⠈⠿⣿⣷⣤⣉⣁⣠⣤⣤⡟⠀⠀⠀⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⣀⣉⣡⠀⠤⠞⠋⠀⠀⠀⠀⠈⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⢰⣰⠃⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡟⡀⠀⠀⠀⠀⡰⠟⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠆⠀⡿⠀⢸⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠈⢰⡇⢀⠀⠀⠀⠀⠀⠈⠻⣿⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⢿⣇⣴⡄⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⠀⠀⠀⠈⠻⡿⣧⠀⠀⠀⠀⠀⠈⠃⢻⣶⣿⡄⡄⠀⢀⣤⣄⠀⢀⠀⠀⠀⠀⠀⡈⣧⠀⠀⠀⠀⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠙⠏⠻⢿⣴⡿⠁⠙⠻⣾⠃⠙⠂⠀⠀⠹⡿⠀⠀⠀⣸⠇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠋⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⠋⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠧⠀⣤⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⡀⠀⠀⠀⠀⠈⠛⢿⣾⣷⣤⣘⣦⡀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⡷⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⣠⣴⣶⣶⣶⣶⣿⣿⣿⠛⠉⠁⠉⠀⠀⠈⠹⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⡴⢋⣵⣿⣿⣿⢏⣽⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⠳⣄⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣴⡏⠀⣼⣿⣿⣿⡟⣸⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠒⠲⢿⣿⣿⣿⣿⡇⢻⣷⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⣿⡇⠀⣼⣿⣿⣿⡇⣿⣿⣿⣫⣭⣭⣄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡗⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢿⣿⣿⡄⢹⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠃⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠻⣿⣿⣆⣿⣿⣿⡇⣿⣿⢿⣿⣿⣽⡿⠃⠀⠀⠀⠘⢿⣿⣿⣻⠛⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇⣿⣿⣷⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⢧⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣤⣤⣄⠀⠸⠃⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠙⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣤⣾⠷⠤⠤⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢸⣿⣿⡟⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠈⣿⣿⣿⣌⠻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡙⠻⢿⣦⣌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣉⣙⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⣤⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⢀⢀⣾⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⠷⠊⣴⡾⢣⡮⠟⣡⣿⡦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⣶⣷⠊⠀⠁⢰⣼⢃⣤⠂⢈⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣸⢋⣿⣴⣶⡀⢺⡆⣸⠋⣄⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣱⡟⡁⠀⠈⣾⣿⣈⣡⣾⡿⡿⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣥⣶⣿⡾⣷⢀⣐⣁⠙⠿⣟⡙⠇⡀⣇⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⣄⣀⣼⣣⡶⣛⣉⡙⢷⡜⣶⣝⠆⢹⣿⡆⣿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⣱⠋⣾⡿⠛⡿⣸⠇⣿⣿⡄⢚⣳⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠁⠀⣿⣷⡀⠠⠴⠾⣿⣿⠇⠞⠛⠢⡜⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⣀⣈⠻⢿⣶⣶⣾⣿⣿⣷⣦⡈⢻⡟⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣦⣾⣿⣿⠁⠀⢘⣿⠀⠃⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠶⠾⠏⣸⣧⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄⠙⢿⡉⣷⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠉⠉⢿⣎⢷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢠⠀⠀⠀⣿⣿⣿⡇⠀⢲⡀⠈⢿⡜⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢸⡄⠀⢀⣿⣿⣿⣿⣶⣾⣿⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠘⣿⣷⣿⡿⢻⣿⡿⠛⠛⢿⣿⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠹⣷⡀⠀⢸⣿⡇⠀⠀⠈⣿⠇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠙⣿⣦⡈⣿⣇⠀⠀⢀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠈⠻⣷⣾⣿⡄⠀⠘⠁⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣶⣶⣦⣴⠀⠀⠀⠀⠀⠀⣠⡄⠄⠀⠀⣤⠀⠀⠀⠀⠀⢀⡀⣶⣶⠿⢦⣤⣦
⢶⣆⠈⠀⠈⢀⣶⠀⠀⠀⠀⠀⠀⣼⠃⠀⠋⠁⢷⡀⠀⠀⠀⠀⠀⠈⣴⡆⢠⣶⡌⠀⠀⠀⠀⠀⣨⣿⠉⠀⠀⠀⠈⢻⣶
⠘⣒⠀⠲⠆⢐⡚⠀⠀⠀⠀⠀⠀⢿⣶⣾⣿⣶⣾⠃⠀⠀⠀⠀⠀⢜⠋⠡⣤⠉⢋⠀⠀⠀⠀⠀⣹⣧⠀⠀⠶⠂⠀⣸⣇
⠶⠈⠆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⠁⠀⠀⠀⠀⠀⠉⢹⣷⣦⣤⣴⡾⠏
⠸⣼⠁⡄⣤⢸⡀⠀⠀⠀⠀⠀⠀⢠⠯⢭⡭⡭⡄⠀⠀⠀⠀⠀⠀⣠⡀⣴⢰⡆⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡠⠂
⠈⠛⠠⠇⠧⠘⠛⠀⠀⠀⠀⠀⠀⠾⠭⣭⢩⡍⠷⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠘⠛⠻⡿⡿⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⣦⠀⣶⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⠉⠀⠉⠙⠟⠁⠀⠀⠀⢀⢤⣤⣀⣤⣀
⡀⠄⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⠄⠀⠀⠀⢀⣧⣴⠟⠛⠛⠛⢷⣤⣄
⠈⠀⢠⣦⣦⠀⠈⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠙⢻⢿⠟⠋⠀⠀⠀⠀⠘⣷⠯⠀⠀⠀⠀⠈⠷⣟
⠈⠂⡀⠉⢁⡀⠀⠀⠀⠀⠀⢼⣿⣧⣀⠀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠨⢌⡀⠀⠀⠀⠀⠀⠀⠘⠳⢾⠖⠒⢶⣄⢼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⢨⠡⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⣸⣏⡞⠈
⢔⣅⣀⡀⣀⣀⡲⠀⠀⠀⠀⠀⠀⢸⣟⣡⣿⣿⡆⠀⠀⠀⠀⠀⠀⠐⣃⡅⠦⠦⣻⠁⠀⠀⠀⠀⠀⠻⠿⣷⢶⡿⠿⣿
⠀⠀⠑⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⠛⠛⠲⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠈⠣⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⢀⠴⠒⠒⠢⢄⠀⠁⠀⠺⡀⠀⠀⠀⠉⢢⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡤⠋⠀⠀⠀⠀⠀⠁⠀⣰⠶⢸⠀⠰⢶⡄⠀⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠠⡦⠀⠀⡶⠰⠄⡀⠀⠀⣰⣿⣿⣟⠛⠛⠛⢿⣷
⠀⠀⢀⠀⠀⠀⢀⡔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠐⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⡴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣼⠁⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠃
⠀⢀⠀⠠⡇⠀⢷⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡄
⠀⠀⣄⠀⢧⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠇⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠓⢤⣀⠙⢦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⠀⠀⢠⣔⡉⠉⠑⠒⠢⠤⠄⢀⣀⣀⣀⡠⠔⠂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⠏⠁⠀⠀⢻⣿⣿⣿⣿⣶⣾⡿⠀⠘⢄⠀⠀⠀⠐⠺⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠿⠏⠓⠒⠶⠤⠤⠤⢦⠀⠀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠄⣀⠂⠄⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡝⠂⠉⠀⠀⠀⠀⠈⠀⠐⠠⣄⠒⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠘⠚⢄⣀⠀⠀⠀⠀⠀⠀⠀⠢⡐⢂⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢀⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣼⠿⣤⣾⡿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠂⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⡀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⢝⣲⡤⣤⣤⠼⠉⢹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⡆⠀⠀⡆⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣅⣠⠀⠀⢸⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣴⣧⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣦⢻⣿⣿⣿⣿⣿⣿⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢿⣟⢛⣱⠋⢈⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡆⣫⡁⠀⠀⢈⣧⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⣦⣈⢿⣦⢶⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢾⣿⠻⣿⣾⡆⠀⢻⣿⣿⣿⣿⠋⡟⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⣿⣆⠘⢻⣷⡀⠀⠹⡝⢿⡇⢰⠁⠈⢿⣿⣿⣿⣿⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⣿⡄⠘⢿⣷⡀⠀⠀⠈⢳⠃⠀⠀⠈⠉⠛⠻⢷⣤⠄⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⡀⢸⡇⠀⡀⢀⢸⠀⠀⠀⠉⠀⣴⡾⠶⠧⣤⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡺⠛⣿⡇⠙⢿⠀⢡⣘⣮⡀⠀⠀⢀⡴⠁⠀⢠⣴⣶⣾⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣃⠹⣷⠀⠀⡇⠠⢹⣻⡁⠈⡆⣼⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣇⣿⡄⠀⡇⠹⣼⣿⣿⣷⣼⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⣿⡇⠀⢈⠘⣿⡯⠙⣿⣿⣿⣇⠀⠸⣿⣿⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡛⢿⠇⠀⠘⣤⣿⣇⣄⣈⣻⣿⣿⣦⠀⠈⠻⠿⣿⡞⠁⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⢶⡯⢆⣤⣿⠀⠀⡾⢠⢻⣿⣿⣿⣿⣿⣿⠾⣷⣄⠀⠰⢽⣿⣶⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠿⠤⠟⠓⠁⢠⣞⡴⣡⠋⣿⠀⢰⣣⢋⡵⢞⣻⣶⣶⠾⠿⠟⠛⠛⠛⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠘⠛⠛⠷⠾⠛⠧⠀⠀⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⢀⡼⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣆⠈⣦⠲⠾⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⡾⠛⠅⠀⠀⣆⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⢻⣿⡿⠵⠀⠀⠀⡀⠀⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⡀⠁⠀⠀⠀⠻⠆⠙⠊⠟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⢯⠵⠍⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠋⠹⢏⣰⣦⣶⠄⠀⠙⠿⣿⡏⠓⠤⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡶⠲⠶⠖⠚⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢴⣾⣷⣶⣴⠀⢀⢀⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣻⡗⣮⣿⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠍⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣶⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣿⣿⣠⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠠⣷⣾⡯⠀⢈⣧⠀⠀⠀⠀⠀⠀⣼⠆⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⣽⣷⣾⣿⡿⢿⡿⣷⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡰⠀⠈⣿⡷⠀⠀⠀⠀⠐⣿⠇⣠⣴⣤⣄⠻⠔⢀⣤⣦⣾⣿⣿⣿⣿⡇⢼⠐⣻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠰⣻⣧⠀⠀⠀⠀⣾⠟⣼⠟⠛⠛⠛⠿⣾⡿⠿⠛⠛⠛⢿⡟⣿⡃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣣⠆⣉⢻⣿⣿⡀⣠⡜⠃⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣿⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣸⣿⣿⡟⠃⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⢃⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣍⡤⣾⣿⣿⠁⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠀⠄
⠀⠀⠀⠀⢀⡠⠀⣿⣿⣿⠿⣿⣿⡿⣡⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⢠⡏⡀⣼⣿⣿⢧⣺⣿⣿⠀⠃⡔⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣸⠓⣿⣿⣿⣿⣿⣽⣿⣿⡷⠆⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠠⠀⠀⠀⢀⡀⣿⠿⣿⠉⠉⣻⣿⡿⠿
⠀⠀⢠⡟⢰⣿⣿⣿⣿⣿⣿⣿⠿⣇⠐⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣤⢄⣶⣶⣶⣶⣾⡾⠍⠉⠙⠐⠂⠐⠂⡠⠴⢲⡾⠇⠐⠁⢔⣿⣿⡿⠄⠒
⠀⢠⣿⣃⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠀⢃⠤⣀⣀⣤⣤⡔⠺⠿⠈⠋⠁⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⢀⣤⣿⡿⠛⠁
⠀⢸⣿⣛⣻⣿⣿⣿⣿⡟⢛⣧⠀⢷⣄⠠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⡀⠀⠀⠀⢠⣿⠟⠋⠁
⠀⠈⠉⠉⠀⠻⣿⢿⡿⢁⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⠀⠀⠀⠀⠈⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡃⠀⠈⠀⠀⠀⠀⢀⠀⣶⣠⣄⢀⡀⢒⣐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠉⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⡠⢤⡤⠚⠉⠀⠈⠉⠐⠢⣀⠀⠀⣀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡠⠞⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣀⠴⠊⢀⠌⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢻⠢⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠉⠀⠀⣠⡊⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⢄⠀⠀⡄⢡⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⣿⡷⠈⢢⠀⢠⠃⠀⣀⡀⡀⠀⠘⡆⠀⡇⠀⢢⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⢿⣿⠆⠈⢀⠃⠀⣼⣿⡷⠈⢻⠀⢰⠀⡇⠀⠀⠑⠢⢀⡀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⣤⠶⠦⠤⢼⣀⠀⠙⢿⣿⠧⠊⠀⢸⠀⠇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣶⣦⣤⣤⣼⡇⠀⠀⠀⠀⠀⢠⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠙⡿⠿⠛⠉⠈⠀⠐⠒⠒⠊⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠉⠉⠙⠉⠉⠁⠀⠀⣀⠤⢺⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⡤⠀⠤⠀⠐⠂⠂⣉⠬⠖⢊⠔⠊⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢣⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠷⠤⠠⠤⠒⠊⠁⢀⠤⠊⠁⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠀⢃⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⣀⠀⡀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠘⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⡹⠀⠀⠀⢱⡀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡜⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣤⠾⢳⠀⠀⠀⠀⢇⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣎⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣎⣀⣠⢃⠄⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⢇⡠⠔⠃⠙⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃


⣀⣀⣀⠀⡀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⡀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠛⣿⠿⡆⠀⢠⣶⠾⠓⠶⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⢰⣄⡀⠁⠀⠃⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣼⣿⡟⡛⠿⣿⣿⣿⣿⡷⠀⠀⠀⢿⡟⠉⠙⠻⢿⣿⡿⠟⢫⣴⣶⣶⣾⠏⠁⠀⠴⠏⢹⣿⣿⡝⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠸⣿⣿⠇⡏⠀⣸⡏⠈⢻⡇⠂⣀⡀⠸⣇⠀⠀⢀⡄⢿⡀⠀⣿⣿⣿⠋⢹⠀⠀⠀⠀⢠⣼⣿⣿⠷⣄
⠀⠀⠀⠀⠀⠀⣴⣀⣭⣆⠀⢸⠛⠁⣧⣀⣛⢁⣄⠘⣯⠄⠻⠀⠀⠀⠀⠀⢈⡷⢀⣀⣀⡟⠙⢿⣆⣀⡀⠀⢀⡐⠋⠘⠟⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠉⠻⠋⢉⣥⡴⠁⠀⡟⢿⠇⠘⢿⣿⡷⠀⠈⠰⣦⣤⣶⠔⠊⠁⢠⣴⣶⣤⣤⣤⣤⡀⠀⠰⣿⣿⣿⣶⣄⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠰⠒⢺⡿⠋⠀⢀⣠⣴⡶⠀⢀⣨⣭⣁⣀⣀⣀⣼⣿⣧⠀⠀⠀⣿⡿⠿⠿⢿⡟⠛⠿⠀⣼⠟⠻⣿⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣤⣴⣿⠿⠛⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠿⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠁⢹⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⠀⠀⠀⠈⣿⣿⡿⢿⣿⣿⣿⣿⡟⠁⠀⠀⢶⣶⣦⠀⢀⣴⣆⣀⣶⣄⠀⠀⠀⢸⠙⠂⠀⠀⢀⢷
⠀⠀⠀⠀⠀⠀⢠⡾⠿⠋⠀⠀⠀⢀⡀⠀⢸⢿⠋⠀⠀⠀⠘⡇⢸⡇⠀⠀⠀⠀⢿⣿⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠞⠠⠃⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⣶⠶⢟⠂⠀⠸⠸⠄⠀⠀⠀⠀⠅⠀⠇⠀⣀⢠⣤⠈⠛⠛⢿⣿⣿⣿⣿⢿⣿⣷⠀⢀⣤⣤⣄⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣠⣴⣶⣶⣦⣤⣶⣾⣷⣦⣀⣄⠀⠀⠀⠈⢹⣿⢋⣀⠀⠀⣿⠀⠀⠀⢘⡏⣿⠀⠋⠟⠉⠉⠃⢈⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡠⠀⢲⡚⠛⠿⠃⠀⠀⣷⠆⠀⢠⠜⠁⡏⠀⣶⣶⣶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢰⡿⠋⠉⢻⣯⡙⠛⣿⢟⣿⠁⠈⠛⠛⠀⠴⠂⠁⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⢠⡿⢋⡛⠋⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣘⠃⠀⠀⠀⠉⠁⠀⠛⠂⠉⢀⣴⣿⡆⠀⠀⢸⣀⡠⠚⠀⠀⠀⢿⣿⣿⣿⣷⣶⣄⢸⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣀⣀⠀⠀⠀⢠⣴⣾⣿⣿⠿⣧⡀⠀⠈⢿⣷⣤⣦⣴⣤⠀⢿⣿⣿⣿⣿⣿⠀⠀⢤⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣆⠀⣻⡿⣿⣽⣿⠀⠹⢿⡄⠀⠈⠙⣿⠛⢿⣿⠀⠈⣿⣿⠉⢿⡟⠀⠋⠀⠉⠈⠻⠯⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⣿⣿⠏⠰⠿⠁⠘⠛⠃⠀⠀⠀⠁⠀⠀⠀⠽⠀⠜⢈⠀⠀⠿⠇⠀⠾⢇⣀⣠⣤⡊⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠘⣿⣴⢀⣴⣶⣶⣶⣶⣾⣿⣶⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣶⣄⡈⠉⠉⠙⠿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⡀⠀⠀⢿⠁⠘⠟⢻⣿⣿⣿⡿⢻⡿⣿⡦⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠘⠧⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⣸⢸⡄⠀⠀⣼⠋⢻⠀⠀⣸⠇⠀⠓⠀⠀⢸⡟⣿⠛⠻⠛⣻⣿⠋⠛⢿⣿⡄⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡉⠀⠃⠚⠀⠀⠘⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢧⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡄⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠄⢿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣸⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⣿⡿⣿⠛⣷⠀⠀⣼⠃⠉⠛⡿⢿⠿⢿⡇⠀⢀⣈⢁⣠⣤⡀⠙⣿⠙⠛⠟⠛⠁⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡇⠀⠀⠰⠃⠀⠸⠀⠀⠻⡆⠀⢠⠃⢸⠀⠈⣡⡾⠛⠿⠿⣃⣨⡁⠀⣿⠀⢶⣶⣶⡄⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠋⠀⠈⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠀⠀⠀⠀⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡇⠙⢀⡘⠃⣧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠛⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⣈⣡⠔⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⠀⢳⣰⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⢠⣄⠀⠀⢠⣤⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡇⠀⢸⠀⠈⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣼⠀⠈⠁⠀⠀⠈⠁⠀⣿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠧⠤⠿⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠛⠛⠀⠀⣰⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠙⠲⠶⠶⠶⠛⠁⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⡆⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⢤⣤⠤⠆⠀⢸⠀⠀⠀⢻⣼⠃⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠘⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⡇⠰⠀⠀⠆⢸⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⣰⣧⡀⠀⢀⣼⡀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⡘⢃⣠⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠈⠙⠛⠋⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⢀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢲⡆⠀⢸⠀⠈⠟⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣀⣸⣧⣀⣼⣀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠄⠀⠀⠘⠐⠀⠀⠀⡀⠀⠀⠀⢀⠀⢈⠀⠁⠀⠍⠁⠈⠁⡉⠁⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡐
⠀⠀⢠⠁⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠈⡆⠀⠀⠀⠈⠀⣠⠀⠀⠀⡄⠀⠁⠀⠀⠀⠈⠉⡅⠀⡄⢀⣖⠀⢠⠀⠈⠉⠀⠀⠀⠛⡛⠀⠈⠃⠀⠀⠉⠀⠘⡛⠃
⠀⠀⠀⠂⠀⡄⠀⠀⢴⠆⠀⠀⣘⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠀⠀⠀⠛⠀⠀⠀⠀⠃⠀⠀⠀⠀⣁⠀⠀⠀⢙⠃⠀⠀⢀⡁
⠀⠀⠀⠀⠈⠉⠢⢠⠀⠀⡄⠄⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡀⠀⠀⡘⠀⠀⠀⠀⠠⣀⠀⠀⠜⠙⠡⠀⠀⣄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠁⡍⠀⢀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠠⠀⠄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠒⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠔⡠⠈⠐⠀⠐⠂⠐⠁⢈⠐⡀⠀⠀⠀⠄⠈⠉⠀⠉⠉⠉⠁⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠑⣀⢀⡐⠀⠈⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠐⠁⠆⠀⠀⠀⠀⠀⠑⡎⠑⠂⠀⠀⠆⠀⠠⠀⣀⠀⠀⠀⣀⠀⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠠⠀⢋⠀⣤⡤⡘⠁⡀⠀⠀⠀⠀⠀⠢⠆⠀⠈⠀⣤⡀⠁⠀⠘⠔⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠐⠂⠀⠈⠀⠀⠀⣀⠀⠀⠀⠠⡦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢡⠆⠀⣩⡀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠐⠐⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⢀⠤⠀⠵⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⣀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠑⢄⠀⠂⠀⠂⢀⠄⠁⠀⠀⠀⣿⢀⠀⢀⣀⢄⣀⠀⠀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠈⠻⢤⢌⠉⠀⠉⣠⡆⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⡎⠀⠀⠐⠖⠀⠀⠀⠶⢐⠀⠀⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠠⠊⠀⠂⡀⠀⠀⢀⠘⠀⠈⢀⠀⠀⠀⠀⠄⠀⠛⠀⠙⠀⠠
⠀⠀⠐⡀⠀⡄⠀⢀⣄⠀⠀⡬⠀⡐⠀⠀⠀⢀⠀⠀⠀⠲⣄⠀⢠⠖⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠁⢀⡀⠈⠁⠀⠀⢨⠀⠀⠀⢶⡀⢠⡀⠀⢀⡄⠀⣦
⠀⠀⠀⠈⠐⠀⢄⡈⠯⣀⡄⠈⠊⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⣶⠀⠈⠦⠄⠀⠀⠀⠀⠀⢄⠀⡦⢀⡘⢏⣀⠰⠀⢀⠀⠀⠀⠀⠘⢦⠀⠀⣤⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠛⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠂⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⢸⡤⠀⣉⡀⢠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⡐⠈⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠈⠀⠈⢂⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢖⠀⠀⠀⣲⣀
⠀⠀⡀⠀⠀⠀⠘⠂⣤⠐⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⣤⠀⠖⠈⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠀⠉⡀
⠀⠀⠉⠰⠀⢤⡄⠀⠲⠆⢀⣦⠀⠐⠈⠀⠀⠀⠀⢀⠀⠀⢀⠸⢃⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡅⠀⠀⢀⠃⠀⠀⠀⠀⠀⡎⠀⠀⣛⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠠⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠂⠄⠒⠁⠀⠀⠀⠀⠀⠀⠳⠦⣄⡌⣠⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡄⠐⡖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⢁⢀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⣨⠛⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⡈⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠤⠤⠤⢴⣿⣿⣯⣾⠄⠘⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢦⡀⠀⠐⠄⢱⡾⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⣿⠏⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣿⠀⠀⢰⡾⠁⢰⣿⠀⠀⠀⢰⠀⠂⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⣿⣆⢀⠀⣀⡔⠃⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⡀⢰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠟⠺⡕⠤⠄⠀⢤⡆⢹⣿⣿⣷⠏⠀⠀⠀⠀⠠⠀⠀⠸⠀⡘⠃⠒⢿
⠀⠀⠀⠀⢀⡀⢀⠛⣦⣄⢹⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠈⢇⡾⠀⣿⢿⡀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠼⠆⣰⠀⣸⣧⡠⠳⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠐⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠖⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷⡃⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⣠⡤⠀⠀⠀⠀⢀⣤⢤⠶⡖⠔⢾⠟⠉⠋⡀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⠈⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⢨⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢰⣿⣶⡎⢻⠁⠀⠀⠀⠀⠀⠠⢀⡄⠀⡀⢦⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣼⠀⠠⣾⡄⠀⠀⢠⠆
⠀⠀⠀⢀⠄⠀⡞⠉⠇⠈⠉⠉⠁⠈⢇⠀⣇⡖⠀⠀⠀⣰⣣⡾⢹⠀⠀⠈⠀⠈⠉⠉⢀⠻⠁⠀⠀⠀⠀⠟⢀⡞⠁⠉⠀⠀⠀⠀⠠
⠀⠀⠀⡌⢀⠾⡇⠀⠀⠀⠀⠀⠀⠀⢹⡤⣿⠃⠀⠀⣴⡿⠋⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣿⣿⠀⠷⠼⠶⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠚⠒⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣶⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⢉⣠⣤⣤⣤⣤⣀⡀⠀⣀⣤⣶⠶⢶⣮⣝⢿⠁⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁⠀⠻⣶⣄⡴⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⢻⣦⠐⠛⠛⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠻⠛⠁⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⣧⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⢠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⣉⡉⠻⣿⠀⠀⠀⠀⠀⢀⣀⡉⠙⢶⣿⣿⡆⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⣏⣿⠆⠟⠀⠀⠀⠀⠀⣾⣏⣿⡷⡌⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣀⣀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⣾⢿⣿⣿⡿⣦⠀⠀⠀⠉⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙⣴⣿⣿⡦⠋⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⣠⣾⣷⣄⡀⠀⠀⠀⣀⣴⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠾⠻⠿⠿⠿⠿⢛⣻⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠿⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⣴⠶⠶⠶⠶⠶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⢦⣤⣤⣾⠟⠁⠘⠋⠹⠿⢿⣿⣿⡾⠋⢉⣤⠴⠒⠒⠒⠛⠋⠉⠉⠉⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠟⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠶⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⡀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣋⡵⠒⠋⢹⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢽⣶⡄⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⣴⢿⡿⠋⠁⠀⢀⣴⣯⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣀⣴⠿⠃
⠀⠀⠀⠀⠀⣸⡏⣼⣀⣀⠀⠀⣨⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠻⣿⠋⠁
⠀⠀⠀⠀⠀⣿⣵⠟⠉⠙⠷⣦⣹⠟⠀⠀⢀⡴⣫⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢨⡿⠃⠀⠀⢸⢸⣿⣿⢛⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⡏⣧⠀⠀⠀⠀⠀⠀⠘⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠈⠿⣿⣿⣿⠟⢁⣄⡀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣣⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⣠⣤⣴⣤⣄⣀⣸⣿⠀⠀⠀⠀⠀⢠⣤⠴⠿⠉⠳⠿⠱⡄⢀⡴⣾⣦⠬⠭⢍⡉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣦⡀
⠀⠀⣠⡾⠛⢛⠁⣬⣿⡿⠟⠻⣄⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⢿⣇⠀⠀⠀⠀⠀⢀⡴⠋⢠⣤⣌⠻⣶⣄
⠀⣸⡟⠀⢰⣿⣷⡸⣿⢇⣶⣆⠘⡆⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠛⠛⢶⣦⠰⠆⠀⠀⠈⠉⠛⠢⣄⠀⠀⠀⡏⣰⣦⡸⣿⡏⣠⣾⣿⣆
⠀⣿⣧⡀⣶⣮⡍⣴⣶⣮⣛⠋⠀⣿⠀⠀⠀⠀⠀⠈⠅⠸⣿⣶⣶⣶⣿⠟⠀⠒⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⡇⢙⣫⣥⣶⣜⠿⡿⢸⣿⡀
⠀⠹⣿⣅⠈⠋⣴⣿⣿⣿⠿⠃⠀⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠃⠛⠿⢿⣿⣿⣴⣶⡦⣿⡇
⠀⠀⠈⠻⣿⣀⠉⠉⠀⠀⠀⠀⠀⠀⠋⣧⠀⠀⠀⠀⠀⠀⠠⠚⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠿⠏⢋⣼⡟
⠀⠀⠀⠀⠈⠙⢿⡶⢤⡄⠀⠀⠀⠀⠀⣿⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⡤⢖⡿⣹⣆⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⣿⡄⣤⣀⠀⠀⠀⠀⢻⡈⠉⠉⠛⠻⠶⠦⠖⠳⣤⠦⠤⢀⣈⣻⠶⠛⠃⠉⠉⠀⡼⠁⢻⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠉⠉⠉⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠘⠂⠀⠀⠀⣸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠸⠟



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⠀⣿⡿⣲⠀⠀⢀⣀⠀⠀⠀⢀⢴⣥⣃⠁⢀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣹⣿⡿⠶⡞⢻⣟⠩⢰⣿⠸⢿⣠⣷⡂⣼⢿⣯⣄⣀⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣍⣅⡄⠔⡶⠙⢄⡘⠋⣞⠺⣿⣿⡉⢻⠚⡏⠰⢾⣿⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠙⠆⠁⠀⠺⠀⡄⢠⢸⣿⣿⢦⡏⡁⢠⣯⠛⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢌⡒⠾⠘⣷⣿⡟⢾⠳⡇⢸⢋⡐⢛⠛⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠛⣫⣀⣷⣘⠙⣷⠀⠀⢸⢣⡞⣉⡥⡶⢲⡶⣶⣿⣌⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⠀⠀⠛⠿⣿⣏⣹⡆⠀⢈⣾⣧⣿⢶⣴⣯⣶⠷⠎⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠑⠀⠀⠀⠙⢻⣿⢿⡇⠀⠀⣾⣿⣿⡿⢿⣿⣿⡗⠾⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠦⠀⠀⠀⠋⠁⠉⠋⠴⣶⡿⠿⠿⣿⣆⡙⠹⠶⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⡀⠀⠀⠀⠀⠀⣈⡋⠀⠀⠀⠛⠋⣹⣷⣿⣿⣟⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠢⣄⣀⠀⠀⠀⠉⠉⠓⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣟⠻⣶⣤⣤⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢷⣾⣿⣿⣿⣿⣾⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣱⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢠⠎⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠿⣿⢟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⠁⢈⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣸⣿⡏⣿⣿⡿⢿⣿⣿⣭⣿⣿⢿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠛⠻⢻⣥⡾⣿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⢀⡈⠆⠐⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠑⡆⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡠⠔⠁⡐⠁⠀⠀⠀⠀⢀⠀⢀⠀⠀⠘⡒⡀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⣠⢎⡀⠀⠀⠀⢀⠔⠁⠀⠀⠑⡄⠀⠀⠰⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢱⣿⣟⡀⠃⢀⠊⠀⣠⣄⢤⡀⠐⠀⠀⠀⠡⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠛⠻⠃⠀⠀⠀⢸⣿⣯⡄⡏⠀⡇⠀⠀⠀⠀⠑⠠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢾⣍⣉⠉⠙⣦⠀⠉⠛⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⠛⠀⠤⠀⠤⠔⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⡀⠔⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡶⠀⠂⠒⠂⠀⡈⠅⠂⡩⠂⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢛⡖⠂⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠁⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠻⢶⡒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢰⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡴⢇⠀⠀⠀⢣⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⠀⠀⠀⢸⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠸⠀⠀⠀⠘⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠀⢀⠔⠁⠀⠀⠀⣘⠤⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠣⠰⣥⠂⡄⠀⢀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿