Hut Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠒⠲⣄⠀⠀⠀⠀⢀⡤⠔⠁⠀⠈⠙⠲⣄⠀⣀⣀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⢀⡖⠃⠀⠀⠀⠀⠀⠀⠘⠲⣄⣀⠀⠸⣄⠀⠀⠀⠀⠀⣀⣠⠇⠘⣦⣽⣤⣤⠤⢷⣷⣮⠇
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠈⠉⠑⠒⠊⠉⠁⠀⠀⠺⠥⠞⢸⣿⠀⠸⠝⠀⣀⣀⣀⠴⠒⢦
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⠀⠀⡔⠚⠁⠀⠀⠀⠀⠈⠙⣄
⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⡠⠒⠚⠉⠉⠉⠉⠁⠀⠀⠀⠈⠳⡀⠙⠦⢤⡀⠀⠀⣀⠀⣀⠜
⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢠⠞⠀⡤⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠑⡊⢉⣉⡁
⠀⠀⢸⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⢀⠼⢁⡔⠁⢠⠞⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡔⠋⠈⠁⠀⢹⡄
⠀⠀⠀⠑⠢⠤⣀⣙⡲⡶⣀⣠⡠⠋⠉⣠⠋⠀⡰⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠘⣅⠙⠇⠀⢸⠀⠀⣜⠁⢀⡞⠁⠀⠀⡠⠤⢤⣤⣤⣀⠀⠈⠳⣄⠀⠀⠀⠀⣀⣀⣀⣀⠇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⣸⠀⠀⠀⠉⠁⢳⠀⠀⡞⠀⡴⠁⠀⠀⠈⢦⠀⠀⠀⡏⠉⠉⠉⠉⠀⣹⠀⣀⣶⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⠎⠻⣇⠞⠙⢤⠞⠙⢦⡴⢻⠀⠀⡇⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡤⣤⢤⠀⡿⢹⢘⡟⠓⠚⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠀⠀⢸⠀⠀⢸⠀⠘⡆⠀⡇⠀⡏⢹⠙⡆⠀⢸⠀⠀⠀⡇⠀⡇⢸⢸⠀⡿⠻⡬⢧⡀
⠀⠀⠀⡤⠎⠙⢹⡄⠀⢸⠀⠀⢸⠀⠀⢸⣀⠤⡇⠀⣧⠀⣇⢸⠀⡇⠀⣿⠀⠀⢸⡇⠈⠃⠸⢸⠀⡇⠀⠀⠸⠤⡄
⠀⠀⡴⠃⠀⠀⠀⡇⠀⢸⡀⠀⢸⠀⢠⠔⠛⠀⢳⠀⢸⠀⢿⠽⠟⠃⠀⣿⠀⠀⢸⡇⠐⠧⠾⠚⠀⡇⠀⠀⠀⠘⢧⡀
⠀⠀⢧⡀⠀⠀⠀⡇⠀⠈⡇⠀⣸⢀⠽⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠉⠑⠚⠉⠳⠒⠋⠉⠉⠉⠙⢦⡀⢀⡀⢈⡇⠸⡆⢸⠒⠚⠉⠉⡇⠀⠀⢸⠀⠀⠀⠀⠀⣀⣇⡠⢦⣠⠶⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠓⠒⠛⠉⠀⠀⠀⠀⠓⠦⠤⠼⠤⠖⠒⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢦⡀⠀⠀⠀⠀⣴⠋⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⣀⠀⠙⢦⡀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠈⠳⣄⠀⠙⢦⡀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠙⢿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⢀⡴⢤⡀⠀⠀⠈⠳⣄⠀⠙⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⢰⡋⠀⠀⢙⡆⠀⠀⠀⠈⠳⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢦⡀
⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢦⡀
⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢦⡀
⡴⠋⠀⣠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⠀⠙
⠳⠤⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠤
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⣸⡀⢀⡇⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠉⢹⠉⠉⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡋⠛⡉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⡖⠂⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠙⡯⠉⢉⠩⣽⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⡗⠒⡞⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠓⠒⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠖⠒⠚
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡔⠒⠒⠒⠒⠒⢢⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠂⠀⠀⠀⡀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⡂⠀⠀⠀⠂⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠣⠄⠀⢺
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢮⡱⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠄⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠐⠂⠁⡀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠖⠒⠒⠒⠒⠊⠀⠀⠈⠰
⠀⠀⠀⠈⠂⠄⣀⠀⠀⠈⠉⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠉⠁⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠐⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡞⣤⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠛⠉⠙⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠐⠒⠈⠁⠀⠀⠉⠐⠒⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠝⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠱⠚⠀⠀⠤⠄⠤⠇⠤⠤⠤⢦⠐⢶⠀⠈⠀⠈⠢⡀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⠀⠀⠔⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠈⢆⠀⠠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⡰⠣⢀⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠷⡂⠉⠠⡁
⠀⠀⠀⠀⠀⢀⠌⠀⠀⠔⠁⠀⠀⠁⠀⠀⠰⠖⠀⠠⡧⠠⠤⡤⠤⠀⠀⠐⠒⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠘⢌⢀
⠀⠀⠀⠀⢠⠂⠀⢀⠊⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠀⠀⢀⠀⠈⠆⠀⠀⢂⣡
⠀⠀⠀⠠⠁⠀⠠⠃⠀⠀⠀⠀⡔⠀⠀⠆⠀⠀⠀⠸⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⢃⠈⠄⠀⢁⠀⣈⡤⠐⡈⢡⠱⡀
⠀⠀⠀⠃⠀⡴⠅⠀⠀⠀⢀⡌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠀⣀⠀⠀⠠⠦⠀⠐⠈⠁⠀⠈⡄⠐⡀⠀⠱
⠀⠠⠁⠀⢠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⡛⠉⠉⠉⠉⠉⠉⠀⠈⡄⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠡
⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⢀⣀
⠉⠛⠛⠲⢸⡦⡄⠶⢄⣠⡀⢇⣄⣀⢀⣀⡆⣀⢠⣀⢠⠀⡀⢀⡠⢄⣤⣀⣀⣶⠔⢻⣀⢠⣀⣀⠰⠖⠳⠺⠋⠉⠸⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠋⠀⠀⠈⠁⠘⠉⠉⠀⠁⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣤⡤⠔⠲⠶⠲⠤⠶⠶⠶⠶⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⣇⠀⠀⠀⡄⢀⣀⣀⣀⣀⢀⢀⣇⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⣀⣀⠀⡇⢀⡅⠀⠀⣀⠀⠀⠀⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸
⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⣾⠀⡅⠀⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠘⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⣀⠀⣀⢀⣇⢀⡀⣀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠅⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣐⣀⣀⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣄⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠻⠟⠹⠛⣿⠛⣻⡿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠶⣆⠀⣾⠛⣷⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣌⣻⡄⠹⣼⡇⣸⣿⣿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣽⣷⣿⣿⣿⣿⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠀⣠⣾⡿⠋⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣁⣴⣿⣿⠏⢀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣵⣾⣿⣿⠟⠁⠀⣼⠃⠈⣿⣿⡟⢿⣿⣿⣿⡻⢿⣮⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢻⣿⡿⠃⠀⣰⠃⡿⠀⠀⠸⣿⣿⠈⢿⣿⣿⣷⡀⠙⢷⣌⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⢠⣿⠟⠀⠀⠰⠋⢸⡇⠀⠀⠀⢹⣿⠀⠈⢿⡟⢿⣷⡀⠀⠙⣦⠈⠳
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋⠀⢠⣿⠏⠀⠀⢀⡾⠀⣾⠁⠀⠀⠀⠸⡿⠀⠀⠈⣿⠀⠹⣧⠀⠀⠈⢷⡀⣠
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⢠⡿⠃⠀⠀⢀⣾⠃⢰⣿⠀⠀⠀⠀⠀⠃⢄⠀⠀⠸⣇⠀⠘⣇⠀⠀⠈⢷⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠞⠁⠀⠀⢀⣾⡏⠀⣼⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢻⡄⠀⠀⡀⠀⠀⠈⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣶⣄⣀⣀⣴⣿⣟⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⡀⠀⣳⣀⣀⣠⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣤⣴⣾⣿⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠈⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⣉⣩⣿⡿⠋⠉⢙⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢠⣤⣶⣶⣶⣤⣤⡄⠈⠉⠉⠁⠀⣤⣾⣿⣿⣿⣥⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠋⠉⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢉⣭⡿⠟⠋⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⣶⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣨⣿⠿⠛⠉⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠚⠃⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣄⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣷⣶⣾⣿⣾⣿⣶⣾⣿⣿⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠒⠐⠄⠀⠄⠐⠀⠠⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⢠⠴⣶⣦⣶⡾⠟⡓⠀⠙⠛⠲⠄⠀⠀⠀⡀⢀⣤⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠈⡄⠠⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣀⠀⠤⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠈⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠀⠌⠀⠀⠀⠀⠘⠈⠉⢀⠀⠰⠀⢰⡐
⠀⠀⢀⠠⢀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠖⠒⠒⠀⠄⠀⠀⠀⠦⠔⠀⠀⠀⠐⠊⠉


⣶⣖⡶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⣤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⢀⠈⠓⢦⡒⠲⠶⠾⠭⠭⠷⠶⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⢞⡴⠀⠙⢦⡀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡠⢊⡵⠋⠀⠀⠀⠀⠈⠲⣄⠈⠳⣄⣀⠰⣷⢦⡀⢰⠦⣄⠰⣶⣌⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡠⢊⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠈⠙⢟⠻⠇⠙⠻⠃⠈⠛⠛⠈⠁⣬⣕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠑⣄⢠⡶⢦⣿⠓⢦⣿⠓⠾⠯⠑⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠋⣠⢊⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠛⢧⡀⠉⠀⠀⠀⢀⣀⡤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠲⠞⠁⡁⢠⠀⠀⠀⠀⢀⡤⠀⠒⠒⠒⠒⠒⠂⠠⣄⠀⠀⠀⠀⠀⠙⢦⡀⢁⠀⠖⠚⠋⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡠⠋⣠⠄⠀⠈⠉⠉⠀⠀⢄⠈⠳⡄⠀⠀⠀⠀⢰⡌⢃⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⢀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠈⠂⠀⠀⠀⠈⡇⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠀⠀⠀⡇⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡀⠀⢸⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡇⠀⢸⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⡇⣟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡇⢠⣸⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣈⡃⢿⠋⠙⠒⠲⠦⠤⠄⣀⣀⠀⠀⡇⠀⡇⠀⠀⢸⡇⢸⣿⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣍⠁⠰⠛⠒⠲⠶⠤⠤⢤⣀⣉⠀⠃⠈⠇⠀⠀⠸⡇⠘⣿⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡖⠒⠶⠶⢥⣄⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠛⠃

⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠
⠀⠀⢹⠄⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⢤⠠⢅⡀
⠀⡀⢘⠄⠀⠀⠠⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣄⡀⡺⠛⠻⠂⠠⠶⠤⠴⠤⠂
⠀⢃⢼⢀⠂⠀⠀⠀⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠁⡀⠀⠀⠀⢦⠀⠟⢀⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡤⢶⠀⠂⠂⠂
⠀⠈⢼⡀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⡄⠘⠛⡀⠇⠀⠀⢘⣆⡑⠲⠛⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠄⠀⠀⠁⢀⣀⡀⠀⠀⠂⠐
⠀⠐⠸⠐⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠜⠲⠀⠀⠀⠐⠈⠀⠛⠰⠟⠛⠅⠀⠀⠀⠀⠀⢠⠉⠉⠠⠓⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⢁⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⢐⠀⠠⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣈⡀⢸⠉⠉⠉⡇⢨⢨⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠲⠶⠶⠒⠸⠶⠤⠬⠥⠤⠤⠀⠒⠘⠂
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠
⠀⢸⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠⠄⠠⠀⡄
⠀⡝⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢰⠃⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⡀⡂⢇⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠤⠀⠀⠀⠶⡠⣀⣀⠀⠀⠀⠀⠀⠀⠐⠄⠁⡠⢡⠘⡚⡄⣓⡙⡂⢎⠶⣂
⣎⢉⣀⠀⠤⠉⠄⠈⠀⠀⠃⠑⠚⡤⠀⣀⢁⣉⠀⠀⠀⢠⠆⠂⢁⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⠀⠉⠑⠀⠀⠀⢀⡐⡈⠆⡘⠃⠜⢠⠓⠃⠃⡃⢈⠒⢿⠄
⠀⠀⠴⠀⠀⠃⠀⠀⠔⠒⢌⠉⠉⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠰⠠⡆⠐⠀⠀⠀⠙⠁⠀⠀⠐⠁⡶⠈⠀⠀⠀⠀⠀⠪⣉⣁⢈⠙⢋⣑⠁⠀⠀⠓⣓⠚⡲⡂⠤
⠀⠘⠀⢸⠀⠀⠊⠲⣀⡐⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠒⠂⠀⢀⠀⠀⠀⠀⠠⠤⡠⠤⡃⠄⠀⢀⣠⢈⠀⠀⠤⠄
⠀⠘⠀⠘⠁⠀⠡⢀⠈⢀⠀⠀⠀⠀⠁⠂⠘⠀⠀⠀⠀⠀⠁⠀⠀⠤⠤⠤⠼⠿⠽⠭⠭⠥⠤⠤⠤⠤⠤⠨⠷⠀⠀⢠⡶⢲⣕⠲⡷⠲⢰⣟⣾⡿⣷⠀⡶⢆
⣾⠘⡄⢣⠀⠓⠒⠒⠚⠛⠒⠈⠙⠂⡔⢁⡜⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠈⠉⠉⠉⢸⣷⣾⣷⣿⠀⠉⠁⡀
⢹⠂⣷⢺⠧⠛⠝⠭⠧⠿⠽⠫⠽⠽⡇⣞⠁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠭⠽⠿⠿⠽⠿⠿⠿⠿⠿⠿⠿⠿⠻
⢸⠀⣿⢰⠀⠀⠀⠀⠐⠮⢀⠀⠀⠀⡄⢹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠀⠳⡀
⠈⡉⠁⠉⠀⠀⠀⣀⣀⣀⡀⠄⠀⠀⡋⠀⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠒⡒⢲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⢤⠤⢤⠤⠤⠤⠄
⠙⠷⠷⠀⠀⠀⠈⠀⠰⡶⠀⠈⠀⠀⠇⠸⠶⠟⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⡐⠒⡗⢺⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢒⢒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠷⠲⠥⠬⠆⠀⠠⢄⡀⠀⠀⠀⠀⠤⠤⠠⢠⠠⣠⠂⠀⠐⡀⠀⢠⠀⠀⠀⠠⢥
⠀⠀⠀⠀⠀⠀⠠⠁⠀⣛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⡰⠂⠁⠀⠀⠀⢀⢠⢀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠟⠁⠠⠐⠂⠈⢀⠀⠈⠀⠀⡀⠐⠶⡄⠀⡧
⠀⢠⠄⠀⠀⠀⠀⠌⠀⣙⠀⠀⠄⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡖⠀⠀⠦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⣖⢢⡆⡘⠓⠚⢳⡀⣖⢠⡒⣾⠀⡇
⠀⢀⠀⠀⠀⠀⠠⡆⠨⠿⠉⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠇⣁⠀⠀⢈⠀⠳⠴⠤⠾⠀⣗
⠀⢤⠀⠀⡀⢀⣠⣤⣔⣂⣀⣀⣂⣀⣀⡀⣀⠀⠀⠀⠀⠀⠠⠀⠁⠙⠓⠈⠉⠈⠉⠉⠉⠀⠀⠤⠤⠍⠁⠀⠀⠀⠆⠀⠀⠀⠀⢠⠧⠤⠀⠸⣤⠠⠤⠤⠤⠄⠮

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣤⣤⣴⣶⣾⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣉⣩⣥⡀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⣤⣤⡄⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣿⣿⣿⣿⣧⣤⡀⠀⣠⣶⣿⣿⣿⣧⡄⠀⣿⣿⡆⣠⣾⠟⣉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⡿⢛⣡⣾⣿⣷⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦⡀
⠀⠀⠀⢀⣀⢀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⣴⡏⣩⣍⣭⡍⣿⣿⣿⣿⠋⣭⣉⣭⠙⣷⣌⠻⣦
⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣨⣅⣬⡅⣿⣿⣿⣿⠀⣭⣀⣭⢸⣿⣿
⢀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇⢨⣅⣬⡅⣿⣿⣿⣿⠀⣭⣀⣭⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡏⢹⡟⢹⡟⠙⣿⠀⠀⠀⠀⠀⠀⢸⣿⣧⣬⣥⣬⣥⣿⣿⣿⣿⣤⣭⣤⣭⣼⣿⣿⢀⣄⡀
⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡏⢹⡏⢹⡟⠙⣿⠀⠀⢸⣿⣿⣿⠈⠉⠉⢉⡉⣉⡉⠉⠉⠉⠉⠉⣉⢉⠉⠉⠙⣿⣾⣿⡇⣀
⠀⠀⠀⣾⣿⡿⠿⢿⣿⣶⡄⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⣿⣿⠀⠀⠀⣏⣹⣏⣽⠀⠀⠀⠀⢸⣉⣿⣹⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠙⠋⠁⠀⠛⠛⠋⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠀⠀⠀⠧⠼⠧⠿⠀⠀⠀⠀⠸⠤⠿⠼⠇⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⠏⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢃⡀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠫⠋⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⣋⠁⠋⠠⠒⢂⡄⠰⣤⡿⣛⡓⣶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡎⡡⠐⠟⠁⠀⠄⠰⠆⠀⠸⠀⢠⡈⠀⠈⠉⢁⠈⡻⣷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⡟⢁⠤⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⢀⠀⠆⠟⡲⣹⣷⢤⡀⠀⠀⠀⠀⠀⠀⢀⡄⠨
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠰⡄⠘⠀⠀⣠⣴⣻⣿⣿⠟⠁⣀⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠈⠛⠊⢫⡷⠙⢦⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⣤⠴⠇⠀⠀⠀⠀⠘⢀⣾⣿⡟⡡⢿⣯⢀⣉⣴⣶⣫⠱⡏⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠐⠘⠓⠑⠄⠢⢡⠙⠄⢹⣆⠀⠀⢀⠀⢨
⠀⠀⠀⢰⣃⣀⣐⣀⠤⠄⠂⣠⢿⣿⣯⢰⡟⣱⣿⣿⣯⡿⢿⠇⣰⡄⢀⡔⠀⠀⠀⠀⠀⡀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⡷⠀⠀⠑⢸⢀⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣜⠉⣿⣿⣿⣟⢱⣇⣾⣿⢋⣶⡟⣻⣿⢸⣯⡄⢰⣿⣁⠀⠀⠁⠀⠀⠀⠘⡰⠀⠀⠀⣀⡀⠀⠀⠠⠘⠀⠸⡛⣁⣶⠶⢠⠁⠠
⠀⠀⠀⠐⠒⠀⢲⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⢯⣿⣿⡿⠻⠠⣻⡿⢸⣗⡇⠈⡳⠘⠂⣀⣀⣠⣀⣀⣔⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣧⠀⠀⡄
⠀⠀⠀⠠⠤⠤⣤⠄⣼⣿⣿⢿⣾⣿⣿⣿⢏⣸⣿⣫⣦⣧⠄⢻⢃⣼⣿⡇⠙⠇⠀⠸⣏⠉⠹⠟⢉⠉⣿⠀⠀⠀⠤⡄⠂⠐⠀⠀⠀⠂⣰⣽⣿⡄⠀⡇⡄
⠀⠀⠀⠀⠀⠀⡾⢠⣿⣿⡿⣿⣿⢧⡏⣾⣿⣿⣾⣿⠇⠟⢀⠉⢩⣟⡽⠇⠀⠀⠀⠀⢸⣀⡆⠐⠀⡠⢹⠀⠀⠀⠐⢀⣰⣷⣶⣦⡁⠆⠈⡞⣿⡗⠀⡟
⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣏⣿⣼⣿⡏⣬⡿⣵⣿⡆⠆⣩⠀⢚⠿⡇⠀⠀⢶⠀⠀⢸⡿⠁⠀⠀⡇⢸⠀⠀⢠⣦⣺⣿⣿⢹⣿⣿⣄⠀⠈⣿⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⡯⣼⣿⣿⣿⣿⣿⣳⣿⢽⣧⡸⢜⡷⢻⡧⠃⠀⢰⢰⠂⠀⠈⣷⠀⠀⢠⠁⢸⠀⠀⡀⣽⣿⣿⣿⣿⣻⣿⣟⡄⠀⣿⣿⡃⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢹⡗⠿⡸⣿⣿⣿⣻⣿⡿⢻⣿⣷⣦⠁⣿⠁⠃⠀⠀⠘⠀⢠⠀⢸⠀⠀⡄⠀⢸⠀⣰⣵⡿⣿⣿⣿⣿⢼⢟⣿⠀⢸⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠺⠗⢿⡛⡀⣟⠋⠹⢋⠀⢸⡇⠀⠀⠀⠀⠂⠀⠇⠸⡆⢰⠂⠀⡾⠀⢿⡇⣠⣀⢋⡭⢹⡎⣿⣿⣷⠈⠻⢿⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢇⣠⣤⢿⡾⣿⣷⠶⠖⢃⡾⠀⠀⠁⠀⠈⠻⠗⠖⣀⣠⣤⣭⠬⠩⠽⢆⠥⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣼⣡⣿⣇⠻⣧⡼⠀⣴⡇⠐⠶⠉⠟⠙⠁⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠈⠓
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣁⣈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⣰⡙⣄⡖⡶⢲⣶⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠓⠸⠇⠀⠀⡿⢫⠞⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢐⣛⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠂⠀⠀⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⢠⣎⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣠⣄⡽⠂
⠀⠀⠀⠈⠉⠉⠉⠙⠊⠙⠗⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠢⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⢥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⣒⣒⣚⣉⠁
⣀⡤⠤⠖⠒⠒⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⣒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡠⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠖⠊⠁
⣀⣀⣀⣀⣀⣙⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣒⣒⣒⣒⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⣶⡶⠶⠾⠿⡿⠟⠛⢿⡛⠛⠋⠻⡉⠉⠉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠾⠿⡿⠛⠛⠛⠋⠻⡉⠉⠀⠀⠀⠳⡀⠀⠀⠙⣄⠀⠀⠱⣄⠀⠀⠙⣄⠀⣀⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢻⣧⠀⠀⠹⡄⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⣱⣀⣠⣤⠬⠦⠴⢶⠚⠒⠒⠻⣍⠉⠁⠙⢌⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⣠⠿⣧⠀⢀⣙⣤⡤⠤⡴⠖⠛⠚⠉⢹⡁⠀⠀⠀⠹⡄⠀⠀⠈⢣⡀⠀⠀⠘⢦⠀⠀⠈⢪⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⣴⠋⠀⢻⣯⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢱⡀⠀⠀⠀⠘⢆⠀⣀⣀⣱⠤⢤⠴⠒⠓⠋⠹⣍⠁⠻⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⢀⡾⠃⠀⢀⣼⣻⣧⡀⠀⠀⠀⠀⠘⣆⣀⣀⣠⠤⠷⠶⠲⡖⠊⠉⠉⠙⢆⠀⠀⠀⠱⣄⠀⠀⠀⠘⢦⡀⠹⣷⡀
⠀⠀⠀⠀⠀⣠⡿⠃⠀⢠⡟⠁⠀⢀⡾⠳⠤⠽⣷⡤⠴⠒⢾⠉⠉⠁⠀⠱⡀⠀⠀⠀⠙⢆⠀⠀⠀⠈⢣⡀⠀⠀⣘⣦⡠⠤⢤⠖⠛⠋⠙⣿⣄
⠀⠀⠀⠀⣴⡟⠁⠀⣰⠏⠀⠀⠀⢾⣭⣍⣒⡒⠻⣷⡀⠀⠈⢧⠀⠀⠀⠀⠹⡄⠀⠀⣀⣈⡧⠤⠴⢶⠒⠋⠉⠉⢧⡀⠀⠀⠈⢳⡀⠀⠀⠈⣿⡆
⠀⠀⢀⣾⠏⠀⠀⣴⣋⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣷⡀⢀⣈⣧⠤⠴⣶⠒⠋⠉⠉⠙⣄⠀⠀⠀⠀⠳⡀⠀⠀⠀⢱⣄⣀⣤⣤⣿⣶⠿⠛⠛⠁
⠀⠀⠺⠿⠶⢶⣾⣯⣭⣉⣛⡛⠲⠶⠦⣤⣄⣀⣀⠀⠹⣿⡁⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠈⢣⡀⢀⣀⣠⣽⣶⡶⠿⠟⠛⠋⠉⠁⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠉⠉⠛⠛⠲⠶⢤⣭⣉⣙⡛⠻⣷⡀⠀⠀⠀⠀⢀⣳⣤⣤⣴⣶⠾⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⣷⣤⣶⡾⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⣶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⠿⠷⣶⣦⣤⣄⡀⠀⠀⠀⠀⠈⠁⢸⠀⠀⠀⠀⠀⣀⣀⣤⡤⢶⡄⠀⠀⣿⠉⠁⣽⠀⠀⣿⠀⠀⣾⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠙⠛⢿⣷⠀⠀⠀⠀⢸⠀⠀⠀⣾⠋⠉⢹⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⠀⢀⣿⠀⠀⣿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⣤⣶⣿⣿⠿⠟⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⣿⠀⣀⣸⡴⢖⣻⡇⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡶⠀⣿⠀⠀⠀⠀⢹⠀⠀⠀⣿⣿⡽⠷⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⣺⣿
⠀⠀⠀⠀⠀⢸⣿⠤⣄⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢖⣚⣭⣷⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠘⠿⢷⣶⣿⣿⠿⣿⣳⠶⣤⣀⡀⠀⠀⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⣋⣭⣶⠾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠷⣭⣛⠲⢯⣝⡳⢮⣝⣳⣶⣿⡀⠀⠀⠀⣿⠀⠀⢀⡠⠴⠖⣋⣥⣴⡾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣦⣍⡛⠳⢮⣝⣷⣾⠏⢀⣿⣯⣉⡓⠒⢾⠖⣊⣡⣴⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣼⣥⣶⠾⠟⠋⠉⠙⠛⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣹⢧⡀⠀⠈⣿⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⣠⠖⢉⣬⡳⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠀⠙⢦⡀⠻⣼⠇⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠟⣡⠴⠋⠀⠙⢦⡙⢦⡀
⠀⠀⠀⠀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠃⠀⠀⠀⠀⠀⠙⠲⣝⢦⡀
⠀⠀⣠⣞⣀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣀⣈⣳⡄⠀⠀⠀⣠⠞⢉⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣍⠲⡄
⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢹⠉⠁⠀⠀⠀⠳⢶⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠶⠋
⠀⠀⠀⠀⠀⣿⠀⡴⠶⣄⠀⠀⡤⢤⠤⣤⠀⣿⢀⡴⢦⢸⠀⠀⠀⠀⠀⠀⢸⢸⣧⡞⢹⠉⣦⢸⢩⠭⠭⠭⢽⠀⠀⡇⢸
⠀⠀⠀⠀⠀⣿⢸⡇⠀⣿⠀⠀⡷⢾⠶⣿⠀⣿⢸⡷⢾⢸⠀⠀⠀⠀⠀⠀⢸⢸⣿⡋⢹⠉⢹⢸⢸⠀⠀⢠⣼⠀⠀⡇⢸
⠀⠀⠀⠀⠀⣿⢸⡇⠒⣿⠀⠈⠓⠚⠓⠛⠀⣿⠈⠓⠛⢸⠀⠀⠀⠀⠀⠀⢸⢸⣿⡋⢻⠉⢻⢸⢸⠀⠀⢈⣸⡀⠀⡇⢸
⠀⠀⢀⣠⣦⣿⣸⣇⣀⣿⣀⣀⣀⣀⣀⣀⣀⣿⣀⣰⣒⣞⣲⡄⠀⠀⠀⣀⣼⣸⣏⣻⣛⣟⣯⣸⣼⣄⣀⣘⣹⣥⣄⣧⣸⣀⡀
⠀⠀⠻⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠗⠀⠀⠘⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡶⢶⠶⢦⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡴⠲⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡵⢋⣙⢮⡳⣄⠙⢦⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⠳⣏⡴⠛⢮⡻⣿⢹⡇
⠀⠀⠀⠀⠀⠀⣠⣞⡵⠋⢸⠋⠉⢳⠙⢮⡳⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠉⠳⣌⠳⣄⠀⠙⢮⡻⣧⡀
⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠈⠳⠶⠋⠀⠀⠙⢮⡳⣄⠙⢦⡀⠀⠀⠀⠀⢀⣴⢟⣵⠞⠁⠶⠶⠶⠈⠻⣮⡳⣦⡀⠙⢦⡙⢦⡀
⠀⠀⢰⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣬⣷⣤⣽⡆⠀⠀⠰⠿⢴⡟⠁⠀⠀⠛⠛⠛⠀⠀⠈⠛⠮⠿⠆⠀⢹⡦⠿⠆
⠀⠀⠀⠀⣿⠀⢀⣠⣤⣄⠀⠀⠀⠀⣠⡶⠶⣄⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⣸⠛⢻⡆⢰⡟⠛⡇⠀⢀⡤⠶⢦⡀⢸⡇
⠀⠀⠀⠀⣿⠀⢸⣇⣿⣸⡇⠀⣿⠀⣿⠀⠀⢸⠂⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⠖⢺⡇⢸⡗⠒⡇⠀⢸⠁⠀⠀⣿⢸⡇
⠀⠀⠀⠀⣿⠀⠘⠿⠿⠿⠁⠀⣿⠀⣿⠗⠀⢸⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⠸⠦⠾⠃⠘⠷⠴⠏⠀⢸⠀⣃⠀⣿⢸⡇
⠀⠀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣿⣀⣿⣀⣀⣸⣀⣸⣇⣿⣀⡀⠀⠀⠀⠀⢸⣇⡴⢻⡦⣤⣤⣠⢦⡄⠀⣸⠀⠀⠀⣿⢸⡟⠳⣦
⠀⠀⠻⠿⠾⠷⠿⠿⠿⠿⠿⠿⠷⠿⠶⠿⠿⠾⠿⠿⠿⠿⠿⠏⠀⠀⠰⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⠳⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠛⣆⠘⢷⡄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣰⠏⠀⠀⠈⠳⣄⠙⢦⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠘⢧⡀⠳⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢶⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⡾⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣷⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠖⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⢤⣤⣤⣤⣤⢤⣤⣤⡀⠀⠀⠀⣿⠀⠀⢀⣤⢤⡤⣤⡤⡤⢤⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣴⠒⠒⠒⠒⠒⢦⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣾⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣿⠀⠀⢸⡧⠤⠤⢼⠤⠤⢤⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣿⠀⠀⠀⠰⠆⢸⢸⡇⠀⠀⠀⣿⠀⠀⠘⠷⠶⠶⠾⠶⠶⠶⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣧⣽⣤⣤⣤⣤⣤⣼⣼⣧⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢛⡿⠛⣉⠙⢷⡀⠀⠀⠀⢀⣠⡾⠋⣥⡈⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡃⠻⣾⠇⢸⡇⠀⠀⠀⡾⢻⡇⠸⣾⠟⢀⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⢷⣄⣀⣴⠿⣷⠀⠀⠘⣧⠘⢷⣤⣠⣤⡾⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠈⠉⠀⣰⡟⠀⠀⠀⠹⢦⣀⠈⠉⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠀⠀⢤⡶⠋⠀⠀⠀⠀⠀⣼⠋⠀⠀⢩⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠃⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣟⣠⠴⢦
⠀⠀⡤⢤⡀⠀⠀⣰⠋⣷⠀⠀⠀⣿⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠀⣠⡴⠶⢦⡴⠃⣰⠟⠋⣁⡴⠞
⠀⠀⢳⡀⠙⣦⣤⡟⢠⡏⠀⠀⠀⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠈⢿⡀⠀⠀⠐⣯⣠⣤⣤⡄⠀⠀⢀⡈⠉⠉⢳
⣤⠶⠦⠟⠀⢠⣤⣀⣾⠤⣤⡀⠀⠸⣧⡀⠀⠈⠻⢶⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠈⠁⠀⣿⠀⠀⠀⢸⡟⠛⠛⠋
⠙⠒⠒⣶⠀⠀⠀⢉⣤⠤⢤⡿⠀⠀⠈⠻⣦⣀⠀⠀⠈⠻⣦⠀⠀⠀⣼⠏⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠘⠷⣤⡴⠞⠁
⠀⠀⠀⠙⣦⣀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀⠀⠀⢻⡄⠀⣸⠇⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣼⠇⠀⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⣿⣤⣤⣿⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠉⠉⠉⠃⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⢀⣾⠏⢀⣴⡶⠿⠶⠦⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⣠⣿⠃⠠⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠃⠀⠀⢸⡇⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣾⡏⠀⠀⠀⢸⣷⣄⠈⠙⠳⠶⢦⣤⣤⣤⢴⣶⠋⠀⠀⠀⠘⣷⣀
⠀⠀⠀⢠⣴⠾⠟⠛⠉⠉⠁⠀⣿⡇⠀⠀⠀⠀⢿⣿⣷⣤⣀⡀⠀⢀⣘⣠⣾⣿⠀⠀⠀⠀⠀⢻⡏⠛⢶⣤⣀
⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⡶⠶⣿⡇⠀⠀⠀⠀⠀⠀⠙⢯⣙⠛⠿⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣩⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡏⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣴⡾⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣭⠾⠿⠤⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠈⠶⣶⣦⣶⠶⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠻⣇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠴⠶⠞⠁⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣄⠀⠹⣦⣞⣩⠽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⢿⣶⣤⣄⣀⣀⡀⠀⠀⣀⣀⣠⣤⣾⣿⡁⠀⠀⠻⣦⣀⣉⣠⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠘⣿⡄⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠾⣿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢶⣿⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠀⠈⠉⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⣈⣷
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠾⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠖⡒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠫⠟⣷⣦⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠷⠄⠀⠀⠚⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⡀⣀⣀⣀⣠⣀⣤⡀⢀⣀⣀⣸⣿⡏⢣⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠻⢿⣿⡀⣿⣿⣿⣿⣿⣿⡇⠚⠉⡿⢻⢻⠀⢸⢸⡼
⠀⠀⠀⢀⡀⠀⠒⠴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢠⡇⠀⠀⠷⠾⠾⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠙⠿⠿⢿⣿⣶⣶⡶⣶⣶⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⣴⣾⠿⠋⠁⠀⠀⠈⠉⠉
⠀⠄

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

⠠⣝⡶⠶⠄
⠠⢬⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢐⣺⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⢀⣸⣶⢀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿
⠬⢿⣽⡈⠃⠀⠀⢀⣀⣠⣴⣿⣿⡏⠸⡟⠀⣴⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿
⣞⣻⣿⣍⠓⢤⣤⣍⣉⣉⣠⡀⠈⠃⡖⢫⡀⠸⡖⠘⠧⣶⡄⢀⠀⠀⠀⠀⠀⢀⠀⠄⠠⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡗⣛⣉⣳⡈⠉⢿⣿⡿⣟⣡⡀⠀⣉⡀⠁⡀⠸⠀⢶⢸⢷⣄⡀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⣿⣿⣿⣿⣿⡽⢉⣨
⢹⡇⣯⣟⣿⣏⡀⠀⣯⣤⣤⠈⣑⣀⠈⠒⠀⠈⠀⠺⡘⣌⢲⡙⠻⢀⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿
⣦⠣⣿⣿⣿⣿⠏⠛⣿⣿⣿⣾⣿⢹⡟⠀⣸⣶⣶⣄⢀⣈⠂⠀⠀⢀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿
⣿⣧⡙⢼⡿⢻⣿⢿⡛⣿⣍⠁⣽⣿⠋⠤⠿⠙⠓⠉⠉⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣿⣿⣿⣿⣿
⣿⣿⢷⠀⢿⢎⣿⣿⣶⣷⣾⣷⣮⣷⠰⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣴⣿⣿⣿⣿⡿⢿
⣿⣿⡆⢳⢿⡄⣿⣿⣞⡾⡿⠿⣯⢻⠿⣿⣿⣷⡔⢻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣥⣴⣾⡿⣿⡿⡉⣍⡯⢹⠋
⣿⣿⣷⡾⠿⡟⢿⣯⣽⡉⠓⠒⠶⢎⡳⡭⢿⠿⣿⣶⣿⣽⣿⣛⡛⠛⠛⣿⡿⢿⣹⢿⣾⣸⠯⠷⣯⠞⠛⠃⠀⣀⣀
⣿⣿⡿⢏⣻⣭⣄⣿⣿⣦⣆⣴⣀⠀⠀⠙⠲⠶⠆⠉⠙⠉⠉⠁⠘⠲⠛⠷⠛⢩⠴⡋⣀⡀⢀⡤⢀⡀⡴⠖⠋⠉
⣾⣾⣷⡄⣿⣿⡻⠿⠿⢿⣧⣧⣽⣿⣿⣿⣶⣦⣤⣀⠈⡛⢉⠂⠀⠒⢐⡀⠂⠀⠒⢌⠉⠁⠉⠀⢀⠀⠀⠐
⣽⣭⣿⣽⣻⠟⠻⠿⠿⣿⣿⣷⣇⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣴⣦⣌⢉⣀⢀⡀⡀⢀⠀⠁⠁

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⣀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣰⠟⠉⠀⠀⠘⠳⡄⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⠖⣶⣄⡔⣶⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠳⠀⠀⢸⣄⣛⡿⢇⣉⡽⠀⠀⣀⣸⣧⡶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⢄⠀⠀⠀⠈⠁⠀⠀⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⡽⠓⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⡇⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⢀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⡄⠀⠀⠀⡸⠀⠀⠀⠉⠉⢻⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⣄⡀⠀⠙⠶⠤⠞⠁⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣼⡇⠀⠙⠓⠾⠷⢾⣥⣤⠶⠶⠟⠋⠁⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣡⠶⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡙⢦⡀
⠀⠀⢫⣑⠦⠤⠖⠒⠋⣠⠞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⡙⢦⣀
⡸⠭⠭⠭⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⡉⢳⣤⢤⡔⣨⠇⡆
⠉⣖⣂⣉⠤⠔⠊⢳⠐⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⠝⣀⠀⠀⠀⠐⢛⣒
⠀⠈⠁⠀⠀⠀⠀⠈⠣⣬⠟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠣⠤⠔⠉⠑⢦⣈⠛⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢦⠀⠀⠀⠀⠈⠙⣿⢳⠶⠶⠶⠶⢶⡶⣶⠶⠟⠋⠁⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢷⢤⣤⣤⣤⣤⣽⠸⡇⠀⠀⠀⠀⡇⢸⠀⡀⠀⣀⣠⠞⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠈⠁⢳⡄⠀⠀⡾⠁⠉⠉⠉⠉⠉⠉⢀⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠠⡢⠀⠠⠀⢠⡊⠑⡄⠠⡀⠀⠀⢀⡐⠢⡄⠀⠀⠀⡤⠤⠄⠀⠤⠀⢀⠀⠓⢒⠀⠒⠀⠀⠠⠀⢀⠾⠶⣭⡾⠉⣯⠓⢢⣄
⠀⠀⠀⠠⠊⡱⠊⠁⣀⣠⡑⢢⡔⢠⠀⠰⢈⠐⠁⠀⠀⠀⢗⢭⡉⠀⠛⠀⢅⣤⢀⡄⡁⠀⠈⡁⠀⠀⠿⢶⠦⡂⢩⢀⠁⡠⠋⠂⠂
⠀⠀⠀⡊⢙⠁⠀⠏⢼⡧⣿⠀⢡⠴⡀⠀⠀⠀⠠⠈⢢⠀⢸⢺⠣⠁⠆⠀⡙⠇⠹⠆⠅⠀⠀⠇⠀⣸⢶⠦⠀⠈⣤⠮⡼⠐⣂⡷⠁
⠀⠀⠀⡌⠉⡀⠀⢌⠉⠙⠡⠀⡬⢤⠃⠀⢄⡀⠈⠃⠁⠀⠀⣌⣂⣠⠀⠀⡅⠀⠈⢀⠀⠀⢀⠀⠀⠰⢰⣀⣀⡠⢧⠀⠈⠟⠉
⠀⠀⠀⠈⢊⠑⠦⣀⣈⣁⡠⠚⡤⠈⠀⢤⡘⠮⠂⠀⠀⠀⡀⡀⠀⠠⡢⠀⠀⠀⡄⢄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠃⠀⠨⠀⠀⣼⠃
⠀⠀⠀⠀⣤⣍⡃⠤⠧⠀⠕⡂⠁⡄⠀⠀⠀⠀⠀⢀⡀⠢⡔⠊⠀⠀⠀⣀⠀⠀⠁⠂⠀⠐⢀⠴⠃⠘⢤⠀⡂⠀⠀⠆⠀⠀⠀⠨⠜
⠀⠀⠀⣠⣬⡝⠓⠀⠒⢠⡖⣀⠔⢁⠄⢄⠀⠀⠀⠈⠷⠀⣍⠄⠀⠀⠀⠈⠁⠀⠀⠀⢀⠼⢁⡴⠆⠲⢦⠰⠦⡀⠀⢄⢀⡐
⠀⠀⠐⠛⠊⢦⠠⡄⠘⠁⠀⠀⠀⡬⠰⠠⢧⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡠⠑⠢⠀⠩⢠⠄⠊⠀⠀⢀⣀⠓⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠈⢀⠇⠀⠀⠀⠘⠸⣰⢋⠆⢂⠀⠀⠀⣤⢤⣀⡄⠀⠠⢠⢤⡔⣌⠂⡀⢠⠑⠤⠬⠧⠴⠁⡇⠀⠀⠀⠃⠈⠒⠀⡘⠋
⠀⠀⠀⣀⠄⠀⠠⠊⠀⠐⠅⠀⠠⢀⣭⠄⢦⣬⠀⠀⠀⠷⠘⠧⠁⠀⠀⠀⠘⠃⠉⠀⠃⠀⠀⠀⠑⢊⠀⡀⡐⠈⠉⠐⡀⡀⠈⠂⠂⠃
⠀⠀⠈⠀⠂⠀⠀⢀⠠⠠⡀⠀⠀⠉⠻⠀⢸⠁⠀⢀⡊⠀⠄⡠⠁⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠋⠀⠀⠈⠄
⠀⠀⠀⠀⢀⡔⠪⢙⠀⠅⠨⠢⠀⠀⡆⠀⠰⠀⠀⠈⠉⠀⠀⠀⢀⠴⠘⠃⣛⠰⣀⠀⠀⠈⠀⠀⡡⣀⠀⠀⠈⠔⠐⢌⠀⠀⢀⠠⠁
⠀⠀⠀⠀⣥⠮⢙⣭⣒⡰⢅⣂⡥⠀⠀⠉⠀⠀⠔⠆⠀⠀⠀⠀⠬⠀⢩⣿⢸⣷⢰⠳⣆⠀⢠⠁⠀⠀⠀⢵⢰⡟⠀⡀⢱⠅⠀⠀⠀⢡
⠀⠀⠀⢄⠈⢻⠏⢁⠿⠷⠦⠣⠂⠀⠠⡄⠤⣊⣀⢰⠤⢠⠄⠀⢘⡀⠌⠩⠈⠅⠰⠔⠃⠀⠀⠀⢀⠀⠠⠘⠢⡁⠀⡀⠘⠠⠀⢀⠀⠈
⠀⠀⠀⠁⢀⠂⠀⠆⠀⠀⢀⢀⡀⠄⠀⠈⠂⠡⠿⡾⠗⠁⠀⠀⠠⠃⠃⠈⠀⠁⠣⠀⠀⣄⠀⠀⠊⠀⠀⠀⠀⠁⠉⠐⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠨⡀⡰⠀⠀⠀⠀⠂⠅⠁⠀⢠⡁⠄⠰⠄⣘⡀⠀⠀⠀⠇⠒⠒⠒⠒⠂⠀⠜⠒⠆⠀⠀⠄⠀⠠⡆⠀⠀⠀⠦⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣶⣶⣴⣶⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⢿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡀⠠⢀⡀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡛⣿⣿⣿⠟⠉⠀⠀⠲⣤⠀⠉⠁⢠⡀⢐⠃⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⡙⠿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⠁⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⡇⣿⣿⡀⠀⠀⠀⠀⠀⠀⠐⣠⣍⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣦⣤⣄⢀⣴⠟⠛⠛⠛⠛⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⠇⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⣤⠄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣦⣌⣻⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠿⠟⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠉⢀⣀⡀⢀⡀⢀⣀⠀⣀⠛⣛⣛⣋⣉⡀⣀⣀⢀⡀⢀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⢸⡟⠇⣿⡇⢿⣝⢰⣿⡄⣿⠉⠉⢽⡇⣿⣿⣼⣿⢸⣿⣿⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⢿⡇⢸⣿⡷⡿⣿⢾⣽⢾⠿⡇⣿⣤⣤⣽⡇⣿⠉⡿⢿⣾⠏⢹⣧⢼⣦

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

⣶⣎⠀⠀⠀⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢄⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠐
⠀⢉⠳⠄⠀⠀⠀⠀⠙⢛⡻⡷⠤⣄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠭⠄⠀⠐
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⡆⠀⠠⡀⠁⠂⠀⠀⠀⠀⠤⠀⢀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠁⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠉⠀⠀⠀⠀⠀⠀⡀⠄⠣⠱⠀⠀⠀⠀⠀⢀⠃⠀⠐⠀⠂
⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠀⠀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠄⠀⠀⠀⡘⠀⠂⣁⠘
⠀⠀⠏⠁⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠄⠀⣰⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠰⠦⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⢃⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢀⣠⠄⠀⠀⠈
⠀⠠⠤⠄⠀⠀⠀⠤⠤⠄⠀⠂⠀⠉⠛⡀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣀⡇⠉⠀⠀⠀⢀⡤⠀⠀⠀⠤
⠤⠀⠀⠀⠀⠀⠒⠒⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠂⠀⠔⠀⢀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠜⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⣀⣴⠶⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⡄⠠⠀⠀⠀⠊⠀⠀⠀⠀⢀⠀⠀⠀⠁
⡀⢀⡈⠄⣀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⡁⠚⠡⠂⠀⠀⠀⠀⢀⡴⠁⠀⢠⠀⠀⡨⠀⠂⠀⠀⠀⠀⠀⠀⠀⡤⢃⡜⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠇⠀⠀⠀⡠⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠊⠀⠀⠀⠀⠀⠓⠈
⠀⠀⠄⠂⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⡀
⠤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣠⠴⠐
⠀⠀⠀⠀⠠⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⠄⠀⠁⠀⠄⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠘⠀⠄⠂⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠆⠉⠉⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⡠⠀⠀⠀⡀⠖⠂⠠⠘⠁⠀⠁⠀⠀⠀⠰⠀⠀⢁⠀⠀⠈
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠱⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡀⠀⣀⠐⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠄⠀⠠⢀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠹⡫⠝⠓⠂⠀⠒⠀⠀⣀⡀⠀⠀⠩⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢄⠀⠐⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠒⠄
⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢇
⡀⠇⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⢀⣠⠄⠀⠀⡂⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⡀⠰⠦⠤⠀⠀⢀⠁⠀⠀⠀⠑⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⢀⠃⠐⠰⠂⠇⠘⠀⠃⠘⠀⠃⠚⠘⠀⠃⠘⠀⠃⠘⠰⠃⠖⠘⢀⠃⡘⠀⠃⠚⠐⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⠋⠘⠀⠃⠘⠀⠃⠉⠘⠁⠃⠘⠀⠃⠘⠀⠁⠋⠘⠀⠃⠘⠀⠃⠘⠈⠃⠃⠘⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⠄⠀⠀⠐⠀⠈⠂⠀⠀⠠⠂⠀⠀⠢⠀⠀⠐⠀⠀⠐⢄⠀⠠⠀⠁⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠄⠀⠀⡠⡀⠀⢀⢄⠀⠀⠠⡀⠀⠠⢄⠀⠀⠤⡀⠀⠠⠄⠀⠀⠤⡀⠀⠠⠄⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⠀⠀⠐⡀⠂⠀⠂⠐⠀⠠⠀⠁⠀⢂⠜⠀⠠⠀⠁⠈⢄⠌⠀⠡⡠⠀⠀⢀⠆⠀⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢤⠀⢠⠀⢀⠄⠀⢠⠀⢠⠀⠀⡄⠀⣠⠀⢠⡀⠀⡄⠀⣠⠀⠠⡄⠀⡄⠀⡠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⡀⡄⠀⠀⠰⠀⠀⡀⡄⠀⠀⠰⠀⠀⠀⠀⠀⢀⠰⠀⠁⠀⠀⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⢃⠃⢀⡀⠀⣠⠀⢃⠀⢀⡀⠀⡀⠀⢇⠀⠀⡈⠀⡠⠀⢃⡀⠀⡈⠀⣀⠀⢄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠁⠀⠈⠀⠀⠁⠀⠁⠀⠈⠀⠀⠁⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⠀⠀⡀⠀⢀⠀⢀⠀⠀⡀⠀⢀⠀⢀⠀⠀⡀⠀⢀⠀⢀⠀⠀⡀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠈⠄⠀⠢⠃⠀⠄⠈⠄⠁⠢⠁⠀⠄⠈⠔⠀⠢⠂⠀⠌⠈⠔⠀⠢⠀⠐⠜⠈⠤⠀⠠⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣴⣦⣦⣦⣤⣤⣴⣴⣴⣤⣦⣦⣦⣦⣤⣴⣴⣤⣦⣤⣦⣤⣶⣴⣴⣴⣴⣤⣦⣦⣦⣴⣴⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠂⠀⠑⡀⠀⠀⠐⠁⠀⢂⠀⠀⠀⠌⠀⠐⡄⠀⠀⠠⠁⠀⢁⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠂⠀⠀⠀⠁⠀⡠⠁⠀⠀⠈⠂⡀⠌⠀⠀⠀⠀⢄⠀⠂⠀⠀⠀⠣⣀⠔⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢤⣤⣦⠀⠀⠀⠀⠙⠛⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣄⠀⠀⠀⠉⠰⡦⠀⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⣉⣹⣷⣾⠿⠛⠿⢷⣶⣈⣀⠤⠀⠈⠉⠉⢙⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣾⣉⡁⠀⠈⠉⠱⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠶⠶⣦⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠷⣄⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣸⠋⢱⡄⠀⠀⠛⠛⠒⠒⠒⠀⠀⣰⣿⣿⣦⣤⡜⠃⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⢿⣷⣄⠙⢦⡀⠀⢠⡤⠤⡄⢠⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠙⠻⣷⣄⠹⢄⠀⠀⠀⣀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣦⣀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣶⣶⣤⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⢻⣄⠀⠀⠀⠹⠿⠷⣈⣾⠟⠁⢿⣄⠀⠀⠀⠹⠿⠿⢆⣸⡿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠙⠷⠶⠶⠶⠶⣶⡿⠃⠀⠀⠀⠙⠳⠶⠶⣤⣴⠶⠞⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣰⠟⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠶⣄⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠀⢹⡃
⠀⠀⠀⠀⠀⠘⠿⠶⢿⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣦⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡶⠾⠿⠟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡰⠿⡋⠉⠉⠉⢹⡟⠛⠛⠛⠛⢿⠟⠛⠙⣿⠏⠉⠋⠙⢿⣉⠉⠉⠉⠀⠀⠀⠀⠀⣼⡏⢱⡆
⠀⠀⠀⠀⠀⠀⣸⠇⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⠀⢀⡿⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠟⠀⡸⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⠀⣾⠇⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⣷⡀
⠀⠀⠀⠀⠀⣷⠀⢰⡇⠀⠀⠀⢸⣧⣴⡟⠃⠀⠀⠀⠁⣀⡿⠀⠀⠀⠀⢻⣷⠾⠿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⡄⠁⠀⣈⠁⠀⠀⠀⠘⠃⠹⣇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢻⡇
⠀⠀⠠⢤⣿⣶⠞⠛⣣⣤⡄⠀⠀⠀⠀⠻⣇⡀⠀⠀⣿⡿⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⡇⠀⠀⢸⡇
⠀⢿⣁⡴⠋⠀⢠⡾⠛⠉⣹⡆⠀⠀⠀⠀⠙⢧⡀⠀⣿⡇⠀⠀⣀⣿⣀⣀⣀⣀⣀⣀⣀⡶⠞⠛⠛⣿⡇⠀⠀⠘⣿⡦⠠⠄
⠀⣼⠛⠀⠀⣴⠟⠀⢠⡾⠻⣷⡀⢀⣀⣀⡠⢾⣷⡶⠿⠷⠞⠛⠛⠛⠛⠛⠛⠛⣛⣿⣿⠶⣶⣶⣶⣿⡇⠀⠀⠀⣿⡦⠴⣶⣾⣀
⣶⠉⠀⣠⡾⠃⠀⣀⠏⠀⢀⣼⠟⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣿⡇⠤⣄⣀⣛
⠙⢶⣼⡿⠀⠀⣬⠃⠀⢠⡞⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣾⣿⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⡧⢀⠈⠙⠿⣆
⠀⢈⣿⠀⢰⣾⠉⠀⣴⡿⠿⠶⠶⠶⠶⠿⠟⠋⠉⠉⣻⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⣿⣧⠀⠰⣦⣠⡼
⠀⠈⢿⣦⣈⣿⠷⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢶⠟⠋⠁
⠀⠀⠀⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠰⢀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢾⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡇
⠀⠀⠀⠀⠸⣿⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠛⠛⠋
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⠿⠿⣿⣿⠿⣷⢿⣿⡿⠿⣿⡿⠿⠿⠿⠿⠟⠻⣿⠛⠛⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡄⠻⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠉⠉⠉⠙⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠁⠀⠀⠀⠉⠻⢿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠻⠋⠀⠀⣠⡄⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⢰⣶⣄⠀⠀⠀⠈⠉⠛⠿⢦⣤⣤
⠀⠀⠀⠀⢠⣤⠔⠛⠃⠀⠀⣀⣤⣴⠿⠛⣷⣄⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣾⠁⠙⠛⢷⣤⣤⣄⣀⣀⡀⢀⡈⠹⣿⣿
⣀⢘⣿⣿⣦⣤⣤⣶⣶⣞⠛⠛⠉⣀⣀⣀⡈⠉⠉⠁⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠈⠉⠉⠙⠛⠉⠁⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠁⠘⠛⠉⠋⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣟⣻⣟⣿⣿⣿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢯⣿⣆⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣯⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣩⣼⣉⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣆⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣝⠛⠀⢡⠸⢶
⠀⠀⠀⠀⢼⡿⢟⣿⠀⡏⣿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠉⠉⣡⣤⡄⢈⣱⣤⡀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⢴⣿⣙⢸⣷⣾⡍⠉⠁⢠⠀⠀⣰⠀⠀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣺⣿⣷⣶⠶⣾⠛⠋⣉⣿⣧⣾⣧⣀⢰⡇
⠀⠀⠀⢠⣶⣾⣿⣟⢻⡿⣿⣶⣶⣤⣿⣧⣠⣾⠤⠀⢈⣿⡀⠀⠀⣷⠀⠀⠀⢀⣼⣷⣀⢲⣽⣿⣿⣇⠠⠼⠛⠃⠙⠂⢘⣿⣿⣷⣼⣇⠀⠀⢤
⠂⠀⠐⢿⣿⠟⣽⡿⢷⣿⣿⣇⣙⠿⣿⣿⣿⣿⣦⡴⢺⣿⣀⣀⣈⣿⠅⠀⠀⢬⣿⣿⣷⣠⣿⣿⣿⣧⣤⠀⠀⠀⢠⣤⣈⣿⣿⣿⣿⡧⠀⠀⢾
⡷⣶⣍⣸⡏⠀⠀⠀⠀⡇⢉⠉⠈⢿⣿⣿⣿⣿⣿⣩⣽⣿⣗⠀⢹⣿⣏⣤⣄⣨⣿⣿⣿⣿⣿⣿⣿⢷⠗⠚⠀⡄⠈⣿⣿⣿⣿⣿⣿⣿⡅⠭⣿
⣿⣿⣿⣿⣿⣧⣀⣀⣠⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⡠⢮⢹⣆⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣥⣽⡇⠀⣤⡜⠀⠀⠀⠈⠳⠀⠀⠀⠀⢱⡞⠃⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⡇⡄⠄⡄⢠⠘⣓⣦⡄⠐⠋⠘⣿⠀⠀⠀⣠⣤⣜⠶⠶⢃
⠀⠀⠀⠀⠀⣿⣿⣿⠥⠤⠿⠇⠀⠶⠃⠀⠾⠿⠋⠁⠀⠒⡒⣿⠀⠀⠀⢄⣉⡉⠉⢶⡮⢄⣠⡁⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠛⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠛⠿⣤⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣄⢀⡾⢿⣍⣀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠈⠛⢷⣤⣀
⢀⣀⣤⠴⠞⠋⠁⠀⠉⠛⠷⣤⣈⠉⠛⠶⢤⣀⠀⠀⠀⢀⣴⣚⠛⢿⡷⢦⣄⡀⠈⠙⠛⠷⢶⣦⣤⣤⣤⣶⣶⣤⣤⣀⡀
⣿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠉⠛⠶⣤⡘⢷⣭⡿⠿⠃⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤
⣿⠀⠈⠛⢿⣷⣤⣶⣦⣤⢤⣤⣤⣀⡀⠀⠉⠛⢶⣄⡀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉
⣿⠀⠀⠀⠀⣿⢹⢿⡇⡇⢸⠀⢸⢹⡟⢷⣤⡀⠀⠈⠙⠳⢦⣄⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⢸
⣿⠀⠀⠀⠀⣿⣾⢸⡇⡇⢸⠀⣸⢸⠀⠀⢹⢿⣦⡀⠀⠀⠀⠈⠙⠓⠶⢤⣌⣙⣲⣤⣀⣠⠖⠁⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⣿⢸⠇⡇⢸⠀⣿⢸⠀⠀⢸⠀⡟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⢸⢸⢰⡇⢸⠀⣿⢸⠀⠀⢸⢸⡇⣿⡿⢶⣤⣄⡀⠀⠀⠀⠀⣀⣤⠶⢛⣡⣤⢴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣄⠀⠀⠀⣿⣼⢸⢸⡇⢸⠀⣿⢸⠀⠀⢸⢸⡇⣿⡇⠀⠈⠉⠛⠷⣦⠶⠛⠉⢠⣶⠟⠁⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠻⢿⣶⣼⣿⣿⣸⣸⡇⠀⠀⢸⢸⠀⠀⢸⢸⡇⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠙⠛⠛⠋⠛⠛⠿⣶⣾⣼⠀⠀⢸⢸⡇⣿⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢰⡿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⣺⢸⡇⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⣇⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⢀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣼⢇⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠿⣷⣶⣶⡿⠿⠋⠀⣀⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣤⣀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣸⣿⢀⣤⣴⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡀⣶⣄⣴⣦⣄⣦⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣾⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣭⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣾⣿⣿⠏⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⡤⠖⠛⠋⠙⠳⣦⣄⠀⠀⢻⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠇⢠⣶⣿⠟⠋⠉⠉⠉⠙⠋⠁⠀⢀⣀⣀⡀⠀⠈⣿⣷⠀⠈⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡏⢰⣿⡿⠋⢀⣠⡶⠞⠛⠶⣄⣠⠞⠉⠁⠈⠙⢷⡀⠘⠛⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠋⠀⣰⡟⠁⠀⣠⣄⠀⠘⣯⠀⠀⢴⣦⠀⠈⣷⠀⠀⠀⠀⠀⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠙⠋⠀⠀⣿⣄⠀⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⢻⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣠⡾⠋⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⢿⠁⠀⠀⠈⢿⡿⣿⣒⠒⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡩⠟⠁⣸⠀⠀⠀⠀⠘⣿⡤⣌⠁⠀⠀⠀⠀⠠⠌⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢹⡆⠀⠀⠀⠀⣿⡇⢸⡆⠀⠀⠀⠐⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⣷⣄⣀⣀⣴⡿⢷⠾⠇⠀⠀⠀⠀⢀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠐⠐⠀⠀⠤⠀⠀⠄⠈⠙⣟⢻⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣧⠀⠀⠀⠂⠐⠀⢀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⢄⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⢷⡀⠀⠀⠂⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠤⣤⣤⣤⣤⣄⣀⣀⣈⣃⣸⣧⣶⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠉⠉

⠀⠦⢤⡤⣤⠄⠶⡀⠀⠀⠀⣀⠀⠀⠀⢠⣦⡦⣀⣠⠤⣤⢤⡤⠐⢆⠀⠀⠀⢀⡀⠀⠀⠀⣰⣴⠄⣀
⠀⠐⣠⣾⡇⠒⠲⡌⠃⠀⢸⠹⣿⣷⠀⠉⠉⠁⠛⠃⢂⣴⣿⠐⠒⠦⠙⠀⠀⠎⢿⣿⡇⠀⠉⠉⠘⠛
⠀⠰⣿⣻⡧⠤⠲⣾⣧⠰⣶⢀⣩⣏⣄⠀⡤⢆⡶⠎⢾⣟⣿⠤⠄⢲⣿⡄⢲⡞⣈⣽⣡⡀⢠⠴⢰⠶
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡇⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⣸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⢄⠀⡆⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⢠⢏⢈⡉⠉⠺⣍⣉⡀⠀⠀⠀⠀⠄⠒⡤⠖⠙⠋⠋⠙⠢⣔⠁⠀⠀⠀⠀⠈⠙⠦⠼⠼⠥⣤⣀⡞⣁⡄⢀
⠀⠀⠀⠀⢠⠃⢧⣤⠇⣔⢺⡇⠀⠁⠀⠀⠀⠀⠀⢠⠁⠀⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡗⠀⡆⡀⠀⢠⢲⢿⡗⠒⠄
⠀⠀⠀⠀⠈⢇⠀⣠⡤⢙⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡫⢍⡆⠀⣊⢳⠀⢰⠀⠀⠀⠀⠈⡓⠩⢤⠉⢩⡌⠭⠕⠃
⠀⠀⠀⠀⠀⣠⡇⠀⠀⠙⢲⡖⢤⡀⠀⠀⠀⠀⠀⠈⠲⢭⡉⢀⣄⠉⠁⣴⠋⠀⠀⠀⠀⠀⠙⢢⠏⠀⠀⢧⠔⠋
⠀⠀⠀⠀⠞⢰⠃⠀⠀⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠳⠊⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣠⡞
⠀⠀⠀⠀⠀⠈⢳⡤⠀⠤⢴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁⠒⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢲⣀⣇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠤⣸⠦⠒⡉⢁⣤⡝⢆⠀⠀⠀⠀⠀⠒⣄⣹⣄⣇⡀⠀⠀⠀⠀⠀⠀⢠⠔⠒⡤⠴⠒⠋⠉⠒⠲⢆⠐⠂
⠀⠀⠀⠀⠉⠉⢳⡃⣞⢷⡩⠘⡚⢁⠞⠀⠀⠀⠀⣠⢺⡟⣢⢀⣶⢺⣧⠀⠀⠀⠀⢰⣷⣏⡧⢙⠂⣀⡀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠁⡻⡀⠀⠀⠀⠀⢧⡘⠊⢁⡈⠓⠊⠸⡄⠀⠀⠀⠈⡧⢄⣸⣧⣣⠏⠉⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠇⠠⡄⠀⠀⢸⠁⠘⡄⠀⠀⠀⠀⠈⡆⣄⠀⢠⠀⠀⢳⠀⠀⠀⠀⡃⢸⠉⡏⠀⠀⡔⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠒⠂⠁⠀⡰⠀⠀⠀⠀⠀⠀⢧⠀⠁⠀⠀⢀⡜⠀⠀⠀⠀⠃⠘⠀⠒⠀⣼⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⢦⠖⠒⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡠⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣋⣡⣚⣁⠤⠀⠀⠀⠀⠀⠀⠖⠒⢦⠤⢾⡖⣒⢗⠚⠧⣄⢤⡀⠀⠀⠀⠰⢂⣧⠖⢢⠀⡤⢬⢮⡁
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠙⢶⠲⠀⠀⠀⠀⢀⡤⢴⠋⡘⠞⣃⣑⣜⣃⡀⠘⡀⠀⠀⠀⠀⠐⠁⡏⠯⠼⠀⠮⠥⢹⠀⢠
⠀⠀⠀⠀⠀⣸⠀⣀⠤⣄⠀⢀⣀⣇⠀⠀⠀⠀⠀⠀⣺⠀⠉⠉⠉⠉⣉⠥⠤⠚⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋
⠀⠀⠀⠀⣰⣇⠀⠣⠭⠜⠀⠸⠾⢛⢦⠀⠀⠀⠀⢰⠛⡄⠀⠀⠀⢸⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢁⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠁⠘⢢⣄⣀⣉⣑⣂⡤⠋⠈⠀⠀⠀⠀⠀⠀⠈⣦⢀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⣀⡤⠤⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠤⠤⠤⣀⣀
⠀⠀⡼⠁⠀⢠⡤⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣤⡈⠑⣄
⠀⢰⡇⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡆
⠀⠀⢧⠀⠀⠈⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣸⠃
⠀⠀⠈⠳⣄⡀⠀⠈⣹⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⡞⠋⠁⠀⣀⡼⠋
⠀⠀⠀⠀⢈⣉⠓⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⣿⣤⣴⣞⣋
⠀⡰⠊⠁⠀⠀⠈⠁⢿⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠙⠶⠦⠤⠤⠴⠶⢾⡄⠀⣠⠴⠒⠦⣄⢠⠞⠋⢉⠙⢦⡀⠀⠀⢠⢿⡉⠛⠲⠶⠤⢤⠤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⡇⠀⢰⣾⣹⡇⠀⣼⣿⣴⠄⣷⠀⣦⡱⣄⠙⢦⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡀⠳⣄⣀⣩⠟⢳⣄⠈⠛⠋⣠⠏⣰⠀⣷⡘⢦⠀⠳⡄
⢠⣤⡌⣧⠀⠀⠀⠀⠀⡨⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⢠⠇⠀⣿⢷⡈⢧⠀⠘⣦⢄
⠈⢣⡴⠛⡆⠀⠀⡠⠊⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣸⡏⠘⣧⠈⡇⠀⡗⠈⡇
⠀⠀⢳⡀⠘⣄⠜⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣴⣿⣿⡤⠟⢀⡏⠀⡇⠀⢻⡀
⠀⠀⠀⢳⣠⠋⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣒⠒⠒⠖⠒⠚⠉⠀⢠⠇⢸⠂⢷
⠀⠀⠀⣰⠃⠀⠀⢠⠖⠚⠉⠙⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⡜⠀⣼⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂
⠀⠀⣰⠃⠀⠀⠀⡟⠀⢠⣶⣐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣧⠀⢀⡞⠧⣤⠇⠀⠀⢷⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀
⠀⣰⠃⠀⠀⠀⠀⠹⣄⠀⠉⠁⠀⠀⢀⣀⣀⡤⢤⣀⠀⠈⠛⠁⠀⣰⠇⠀⣬⠴⢆⡜⠀⠀⠀⢸⠀⠠⠂⡠⠰⠂⠊⠳⠦⠴⠋
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠒⠶⠶⠞⠋⠀⠀⠀⠀⠘⠁⠀⠀⠀⢸⡇⠄⢸
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠈⡇
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠂⠀⠈⠉⠁⠀⠐⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⢀⡇
⣧⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠔⢁⠞
⢹⡄⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠖⠁
⠀⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡷⠚⢫⡀
⠀⠀⠀⠀⠀⢀⡼⠁⠉⢹⡟⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠉⠁⠈⣧⠀⠀⢳⡄
⠀⠀⠀⢀⣴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⡄
⠀⢀⡴⠋⠙⠲⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠒⠛⠉⠙⡆
⠀⠻⠶⢶⡆⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⢤⡀⣿
⠀⠀⠀⠸⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠋
⣶⣶⡶⠾⡷⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠
⠀⠀⠀⠀⠀⢰⢄⠃⠀⠀⠀⠀⠀⠀⠀⠙⣠⡆⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⣦⢎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡱⡴⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⣄⢬⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⡥⣠⠀⠀⠀⣼⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⡇
⠀⠀⠀⠈⢳⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢙⠞⠁⠀⠀⠀⢨⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣻⠅
⠀⠀⠐⠶⣭⣼⢃⡆⠀⠀⠀⠀⠀⠀⠀⢰⡘⣯⣭⠶⠂⠀⠀⠀⠹⣽⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⠏
⠀⠀⠀⢀⣨⣭⣬⡇⠀⠀⠀⠀⠀⠀⠀⢸⣥⣭⣄⡀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣞⣷⡤⠤⢤⣾⣻⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠢⠄⠀⠀⠀⠠⠖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⡀⣦⡋⠀⠀⠀⠀⠀⠀⠀⠈⢓⣴⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣇⠶⠂⠀⠀⠲⢆⣷⢠⡄
⠀⠀⠀⠀⣦⣛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣴⠀⠀⠀⠀⢀⡰⣧⣛⠓⠈⠀⠀⠀⠀⠀⠈⠐⢛⣣⡇⡀
⠀⠀⢀⡹⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢃⠀⠀⠀⣈⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢃⡀
⠀⠀⠠⣝⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⣫⠄⠀⠀⣝⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣛⡄
⠀⠀⠀⠮⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢮⠵⠀⠀⠀⢬⣫⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣮⣫⠄
⠀⠀⠀⠙⣚⣱⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣌⣓⠃⠀⠀⠀⠠⢯⡾⣣⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠻⢮⠆
⠀⠀⠀⠀⠨⠵⢟⣴⠀⠀⠀⠀⠀⠀⠀⠀⣦⠳⠮⠀⠀⠀⠀⠀⠀⠀⠛⣛⣽⠗⣷⡀⠀⠀⣰⡷⣿⣙⡛⠃
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠈⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠶⠒⠛⣿⣍⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣋⣄⣀⠴⠒⠲⣍⠃⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⣧⣢⣀⣽⣾⣭⡆⡼⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠼⠛⠉⠉⢉⠈⠈⠉⠁⠀⣦⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⣠⠶⣏⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠡⠄⠀⠀⠀⠉⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢀⡠⢤⠏⢀⡞⠉⠉⠉⡟⠉⡽⢷⠲⢤⣀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⡆⠀⠀⠀⣸⠇
⠀⠀⠀⢀⡤⠞⠑⢰⡏⢀⡟⠉⠙⠲⡀⠱⣐⠃⡎⠀⢀⠽⠳⣦⡀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⣰⡟
⠀⠀⣠⠞⠀⠀⠀⣼⠀⡾⠀⠀⠀⠀⡇⠀⠈⠀⠱⡀⠌⡆⠀⢈⣳⡄⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢦⣀
⠀⣰⠟⠉⠑⢄⢠⡇⢠⡇⠀⠀⠀⠀⣇⠀⠀⣀⠔⠣⠜⢁⡔⠁⠀⠹⡆⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠈⠳⢤⡀
⢠⡏⠀⠀⠀⠀⢺⠁⢸⠃⠀⠀⠀⠀⡼⠂⠈⠀⠀⠀⢀⠊⠀⠀⠀⠀⢹⡤⣀⣠⣤⣾⠟⠛⠶⠶⣤⣤⡴⠶⠞⠋⠹⡄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣼⠁⠀⠀⠀⠀⣾⠀⢸⠒⢤⡀⠀⡼⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⣇⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠠⠙⢦⡀
⣿⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠊⢱⠀⣿⠜⠁⠐⠤⠄⢀⡀⠀⠀⠀⠀⠀⣠⠞⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⣿⠀⠀⠀⠀⠀⢿⠀⢸⡄⡠⠚⠈⠉⠙⠲⣄⠀⢀⠤⠆⠀⠀⠀⠀⢠⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⢻⡀⠀⠀⠀⠀⢸⠀⠘⡏⠀⠀⠀⠀⠀⠀⠈⠙⢄⠀⠀⠀⡠⠂⢠⣼⣤⣏⣀⣀⣀⣀⣠⡤⠖⠛⢫⠑⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠘⣧⠀⠀⠀⠀⠸⡇⠀⣷⠀⠀⠀⣀⠀⠀⠀⠀⠈⡆⠰⠊⠀⠀⠈⣇⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣆⠀⠀⠉⠘⢷⠀⢸⡇⠈⠉⠈⢦⠀⠀⠀⢠⡇⠀⠀⠀⠀⡾⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣽⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⢧⡀⠀⠒⠺⡆⠀⢳⡀⠀⠀⠈⠳⡀⠀⢸⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣿⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠙⢦⣀⠀⢻⡀⠈⢷⡀⠀⠀⠀⠙⢦⡞⠀⠀⢀⡴⠋⡟⠷⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⢠⡇⠀⢸⢄⣀⡀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠛⠲⢷⡀⠈⢷⣀⣀⣀⣀⣠⣤⠶⠚⠉⠀⢀⡇⠀⣿⠀⠀⣀⣀⣀⣠⣤⡴⠞⠋⠁⠀⠀⠀⠀⣸⠁⠀⡆⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠻⣽⠉⣿⠀⠀⠀⠀⠀⠀⠈⡷⡀⠉⠙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠛⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠀⣿⠦⠤⣄⣀⣀⠀⠀⠹⡄⠀⠀⠈⠉⠒⠒⠐⠒⠒⠈⠁⠀⠀⠀⢠⡏⠀⢀⣇⡀⣀⣀⠀⣼⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢻⡀⢸⠉⠒⠲⢤⣍⠉⠙⠒⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡜⠉⠉⠀⠀⠀⡇⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠈⠉⠛⠒⠶⢤⣤⣀⣀⠀⠀⣸⠃⠀⡰⠁⠀⠀⠀⠀⠀⠉⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠦⣴⣅⣀⠀⠀⠀⢀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠤⠴⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢭⣄⣀⡀

⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠠⣖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠁⠀⣠⣤⠄
⠀⠀⠀⠀⠀⢄⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢀⡤⠞⠁
⠀⠀⠀⠀⠛⣶⣿⣯⣼⡔⠠⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠀⢀⠀⠀⠀⠀⠀⠐⢡⡀⢀⣀⡀⠁
⠀⠀⠀⠀⠠⢽⠟⢫⣭⡇⡀⠀⢠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⠓⠀⠀⠀⠀⠠⠞⢈⡙⠀⠁
⠀⠀⠀⠀⠀⠈⣻⣭⣽⡄⠀⠀⠜⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⣄⠀⠼⠬⠄
⠀⠀⠀⠀⠀⠀⠘⢫⣦⣟⠁⢐⣆⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⡀⣀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠀⠘⣖⡂⠈⣿⡈⠄⠀⠀⠀⠀⢠⠄⡆⠀⠀⠀⠀⠂⠀⠀⢠⠎⠂⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢧⠀⠈⠁⠀⠀⠸⠀⠀⢀⠀⡄⢀⠀⢤⣠⡄⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠻⡏⠙⠁⠀⠠⠈⡴⠐⠮⠍⣄⠀⠀⠂⠀⠈⠋⡉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⡀⠔⠊⠁⠀⠀⢉⠞⠖⣀⠀⠢⠙⠠⢌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⢚⠸⠗⢻⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠐⢶⠮⢌⠛⠟⠋⠀⡴⠁⠀⢤⡀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠻⠒⠓⢈⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠒⠒⠚⡩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣆⠀⠠⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠸⠁⠸⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢈⠀⠀⢸⡄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⢸⠀⠀⢰⡁⠀⢸⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠤⠴⠂⠼⢀⠃⠀⠀⣇⠀⠀⠈⡧⠀⠘⡆⠀⠄⠲⠆⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠁⣤⢶⠀⠀⠔⠋⠀⠄⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠠⠀⡀⠀⠰⠈⠧⠀⠀⠀⢤⡆⠀⠀⠈
⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠂⠀⠀⠁⢁⠀⠬⠀⡈
⠀⡂⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⡀⠀⠀⠨⠃⠀⠰⡇⠀⣀⠀⠀⠛⠂⠌⠒⠢⠀⠀⢂⠁⢀⡀⠠⠖⢀⠀⠀⠀⢀⠀⠀⡗⠀⠀⠯
⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠟⢀⡀⠶⡄⠀⠀⠀⠀⠀⠤⠀⠀⠈⠂⠀⠀⡶⠆⣀⠘⠇⠀⠉
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠃
⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⡀⠔⠢⡌⠛⠋⠉⠁⠉⠉⠀⠀⠀⠀⡀⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠸⢆⠜⠀⠀⠐⠟⠀⠀⠀⠀⡠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠐⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠀⡀⠄⠂⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⢀⣀⣀⣀⣄⠀⠐⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⠀⣎⠄⠚⠰⡀⡞⠀⠘⡇⠲⢀⠠⣆⠡⠐⢀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠲⠒⠒⠒⢒⡉⠉⠭⠃⠘⠋⠿⡉⠉⠉⠉⠉⠉⠉⢉⣩⢭⠁⡌⠁⠀⠒⢲⠒⡖⠒
⠀⠀⠀⠀⠀⠀⢀⠠⠁⡀⢈⡁⠠⠤⠀⠠⠉⠁⠃⢀⡀⠘⡆⢀⠢⢈⠙⠒⠒⠇⠀⠘⠂⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠆⠀⠈⠁⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠈⠀⠀⠈⠀⠁⠀⠀⠀⠐⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠈⠀⠀⠯⠀⠄⠈⠘⠀⠀⠀⢨⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠂⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣽⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠒⠒⠒⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠀⣲⠶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⠔⠊⠀⠀⠀⢀⡞⡡⡾⠉⠀⠀⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠔⠛⠉⠉⠑⠐⠦⢄⡀⠀⠀⢀⢠⣖⡽⠎⠁⠀⠀⠈⠒⢒⣶⠣⠥⠊⡰⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡔⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⡿⠛⠛⣻⢥⠤⣤⠞⡀⢀⡠⠚⢀⡀⠠⠖⠀⠀⡠⠀⠀⠀⠠⠀⣀⠠⡀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⢜⣴⠟⠋⠀⠀⠀⠀⠀⠙⣛⣿⡿⠛⠛⠛⠛⠉⠉⠀⢀⡠⠴⣄⠉⣀⡠⠒⠁⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠘⠻⣿⣷⡟
⠀⠀⠀⠀⢠⣴⣴⣾⡧⠌⠀⢀⣀⣀⣴⠾⠟⠋⠀⠀⠈⠁⠂⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠶⢜⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⡤⠶⠒⠀⠐⠐⠛⢛⠖⠀⠀⠀⠀⢤⠀⢀⡀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⡞⠁⢀⣤⠾⠛⠁⢀⠤⠔⠒⠀⠀⠀⠀⠀⠄⠘⡀⠀⠀⠈⢀⠈⡀⠘⠁
⠀⠀⠀⠀⠀⠀⢀⢞⡡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⡀⠀⠃⠠⡅⠀⠖
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⣀⡤⠖⠒⠀⠄⠀⠃⠀⠀⠀⢠⠔⠉⠹⢃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠐⠃⢀⠖⠃⢄⠉⠀⠉⠰⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡀⣄⡀⠀⠀⠀⢠⣋⡤⠔⢈⡀⠀⠀⠘⠃⠀⠀⠀⣠⣿⢶⡀⠀⠀⠀⠄⢀⡿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⢻⣿⣦⣂⡤⠯⠀⠐⠂⠰⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⡀⢸⣼⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠙⡿⠁⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣱⣌⣿⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢈⡂⢀⡀⠀⠶⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡐⠁⠀⠸⣧⣀⣀⡴⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣻⡄⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⣁⠤⠌⠷⠿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣿⠇⠁⠀⠀⢸⣡⣞⣡⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡰⠋⠀⠀⠀⠀⠀⡈⠣⠖⢒⡉⠀⠓⠀⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⡿⠁⢹⠿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠓⠲⣧⣽⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢆⠀⢸⣆⡆⡜⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠤⠶⣿⣿⣿⡇⠀⠀⠃⠀⢰⠃⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣠⠞⠛⠛⠇⠙⠀⠀⠈⢩⡠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣿⡟⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠑⡾⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠂⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛
⣤⣀⣀
⠉⠙⠙⠛⠉⠙⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⣶⣶⣾⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣻
⠿⠿⠿⠻⠋⠹⠿⠋⠙⠛⠙⠛⠻⠙⠏⠉⠛⠍⠋⠛⠋⠋⠉⠉⠉⠛⠉⠉⠀⠁⠙⠉⠉⠀⠀⢀⣩⣼⣿⣶⣿⣿⣿
⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠶⣦⡄⠂⢓⠀⢈⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⡄⠀⢰⡇⣌⠩⠐⢤⡆⣡⠈⣷⡀⠀⠰⣄⣀⢠⢄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⣧⣄⠈⡇⡃⡆⠘⢶⣇⡀⠄⣨⣷⠟⠀⠘⡇⠀⠀⠙⠛⠬⠛⠁⠀⠤⠀⡀⡀⠀⣸⣀
⠀⠀⠀⣦⠀⠀⢀⠀⠀⢠⣤⣻⡔⠀⠄⠀⠀⠈⠛⠶⠳⣷⣌⠲⣤⣝⡻⠾⠿⠟⢀⡔⣶⠇⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠉⠲⣌⠉⠀⠀⠀⠀⠄⠠⡀
⠀⠀⠈⠁⠀⠀⢀⠆⠀⢀⡬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡙⣦⣄⣙⣲⣤⣴⡾⣋⡾⢛⣱⣾⣶⣦⠀⠀⠄⠀⠀⠀⢀⠤⠀⠀⢹⡌⢦⣄⠀⠀⠀⢀⣤⣤⠤
⠀⠀⠀⠀⠄⠀⢁⡠⠖⣥⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠩⠭⢿⡫⠍⠡⣛⣝⠩⠀⠸⡏⠣⠌⠧⠅⠀⠀⠀⠀⠰⠁⠀⠈⠀⣸⠁⣼⣿⡟⠁⠋⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠁⣰⡴⠋⠀⠸⢿⡟⠇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠒⠒⡷⠂⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠒⢄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠁⠀⠒⠀⠀⠀⠀⠀⠘⡄⠀⠀⠧⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠣⢾⡕⠐⠤⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⢀⡤⠀⠘⢦⠀⠀⠀⠀⠇⠀⠀⠀⠀⠔⠀⢠⠀⠀⠀⠀⠀⠀⡴⡁⠓⣺⢷⡹⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠙⠂⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠈⠀⠈⠲⡄⢺⣷⣃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⠛⠀⠙⠃⠀⠀⠀⣐⣸⠃⠀⠤⠴⠊⠀⠀⠀⢠⣀⠜⢳⠄⠀⠀⠀⡜⠱⢿⠅⠦⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡠⠋⠀⠀⠀⠐⣄⣀⠺⠼⣴⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⣤⡤⠎⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠴⣁⡜⠁⠀⠀⠀⠀⠀⠀⠈⢣⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⡒⠒⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⡰⣡⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢃⠘⠦⠍⡡⠤⠞⠁⡄⡇⠀⠀⠀⠀⠀⠜⡴⣣⢏⡞⢰⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠢⡈⣇⠀⢀⡀⠀⢀⠔⠀⣇⡠⠖⢲⡀⢌⠈⠱⢣⢏⠜⠂⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⠎⠛⢉⣙⡋⠁⠀⡠⠋⠀⠀⠀⡇⠈⣷⢄⣀⠞⢴⠋⡓⢠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⡶⠁⠀⠈⢳⡪⠀⠀⠀⠀⢀⡇⢔⠝⢦⢄⡀⢇⣴⣡⣆⡀
⠀⠀⠀⠀⠀⠸⡆⠀⢀⣠⠖⠋⠻⣄⠀⣀⡼⡁⠀⠀⠀⠀⢨⡇⠀⣡⠛⣆⢧⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠋⠐⠀⠀⢀⠈⢉⠁⠀⠘⢦⠀⠀⠀⢸⡇⠀⣇⠀⡆⠘⢦⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠐⠈⠳⢤⣤⠞⠀⠀⢹⡄⢃⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⠀⠀⢠⡎⠀⠀⠠⠀⡀⠀⢠⠇⢸⡆
⠀⠀⠀⠀⠀⠰⡀⢀⠃⠀⠀⠀⢠⠀⢸⠀⠀⠀⡇⠄⠀⠀⢧⠑⠶⠋⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⣷⠀⠀⠀⠸⡗⠤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡆⠒⠒⠒⠂⣿⠈⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢀⣎⣀⣀⣀⣀⡠⠹⠀⠀⠀⠀⠧⠤⠴⠤⠤⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⣄⡀⠀⠀⠀⣀⣠⢶⡋
⡴⠒⠶⣄⠀⠀⠀⠀⠀⠀⢸⢃⠇⠈⠉⠉⠉⠁⢸⡄⣷⠀⠀⠀⢀⡤⠖⠋⠉⠓⣆
⣇⠀⠀⠈⢣⡀⠀⣀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⣸⠃⢻⠀⢀⣰⠋⠀⠀⠀⠀⠀⡿
⠘⣆⠀⠀⠀⠓⠉⠀⠀⠀⠒⠶⠒⢠⠀⠀⢰⠃⠈⠉⠉⠀⠀⠛⠀⠀⠀⠀⣠⠞⠁
⠀⠈⠳⢤⣄⡴⠒⠒⠦⠤⠴⣆⣀⣀⡆⠀⠸⣄⣀⠽⠉⠑⠒⠲⠶⠶⠚⠋⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣛⠛⠙⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⣈⣁⣈⣁⣀⣉⠙⠄⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠚⠋⠉⠁⠀⠀⠉⠉⠙⠓⢦⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡷⢄⡀⠀⠀⠀⠀⣀⣠⡴⢺⡇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⠉⠀⣠⡾⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣿⣷⣶⣦⣴⣶⣶⣿⡿⠃⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡟⠒⠲⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠛⠤⣄⡀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠶⣤⣄⣀⣀⣀⣀⣀⣠⣴⣾⣋⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠈⠉⠉⠉⢉⣭⠟⠁⠀⠉⠳⣄⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⢀⠤⠒⠉⢉⠉⠉⢉⠓⠲⠤⣄⣀⣀⣀⣀⣤⡤⠶⡿⠁⠀⠀⠀⠀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⣆
⣤⢤⣴⠋⣰⡆⠀⢸⣧⡀⣀⣷⠀⠀⠀⣠⣍⣉⣁⣀⣀⣤⣧⡤⠶⠶⠛⢏⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢀⡤⠂⠀⠀⠀⢻⡄⠀⠀⠀⣀⣠⠎⢹⡆
⣇⠀⠉⠉⠙⠷⠶⠾⠃⠛⠉⠛⠓⠶⢤⣽⡄⠀⢁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣴⣛⣁⡤⠴⠊⠁⠀⠀⠀⠀⠀⠀⣷⠀⠆⠀⠉⠀⠀⠀⢿
⠉⠉⠉⠛⠛⠛⠒⠒⠶⠶⠦⠤⢤⡀⢀⠈⠻⣶⡋⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠸⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢹⡓⠢⢽⣳⠛⢧⡀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⡇⠀⠀⠈⠙⠢⣝⠛⢻⠛⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⣿⠀⢳⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⠀⢸⡏⠙⢾⣤⡞⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣺⠀⠀⠚⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣧⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣿⠀⣿⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢺⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢰⡏⠀⡇⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠈⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⢀⣾⠁⢸⠃⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠐⠶⠦⢤⣤⣤⣤⡤⠶⠶⠚⠋⠁⠀⠀⠀⠀⢠⡾⢧⣤⣯⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠛⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠀⠀⠉⠛⠶⠶⠶⢤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡶⠶⠶⠶⠶⠶⠶⠖⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⡿⠀⠀⣀⣀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠾⣏⠙⠛⠛⠓⢻⡶⠋⠉⠉⠻⣇⡀⠀⢠⡀⠀⠀⢠⡀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣴⠸⣦⠀⠀⢀⡞⠀⢠⠤⣄⠀⢻⠉⠛⠛⠷⠶⠦⠾⠷⠶⠚⠛⠓⠶⠶⣤⣤⣤⣀⣠⠴⠒⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠻⠒⠛⡷⠦⣼⠀⠀⡏⢀⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣠⠤⣄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣠⡴⠃⠀⢹⠀⠀⠙⠏⠀⠀⡾⠓⠒⠲⠶⠶⠦⠤⠤⣤⣤⣤⣄⣀⣀⣀⣀⣸⠇⠀⣇⣠⠏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢷⣀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠹⣇⠀⠈⠁⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠿⠷⣤⣤⡴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠄⠂⠀⠀⠒⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⡀⠀⠈⠉⠈⠙⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⠐⠎⣠⡠⠐⠀⠀⠀⠀⠈⢸⡀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⡤⡇⠀⢠⠏⡀⠂⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⡠⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⡎⠣⡀⣧⢴⢣⣾⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠻⢳⣶⣿⠰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⣿⡆⠔⠃⢾⡿⢬⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⣻⣿⣇⣴⣿⣷⣝⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠃⢸⣿⣿⣿⡟⡏⠁⠳⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣼⣿⣿⢿⠁⢀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢸⣿⠇⠸⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠄⠀⠱⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠉⠀⠀⠢⡀
⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⢀⢀⠜⠟⠇⠀⠀⠀⠀⠑
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡠⠞⠋⠁⢹⠎⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡘⠰

⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠟⠁⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡄
⠀⣰⣷⠀⠄⠠⠀⠀⠻⠶⠖⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡘⠅⢴⣶⣄
⠀⠈⠋⠀⠀⠀⠀⠃⠀⠀⠀⠈⠁⠀⠠⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⢼⣿⣻⣋⠉⣀⣀⣰⣀
⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠢⠀⠑⠦⠀⠈⠑⠦⠀⠀⠠⠶⠶⠚⠛⠛⠩⠽⠿⠟⠋⠽⠿⠿⠿⠿⠿⠿

⠀⠀⠠⡤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄
⠀⢠⡌⠫⠤⣤⣤⠸⢧⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣛⢛⣉⡟⣫⣄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⢙⣏⣀⣀⣀⣀⣬⣥⣈⣏⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⣛⣓⣲⣒⣶⣿⣿⣿⣾⣏⣉⣩⣽⣿⣿⠿⡿⠉
⠀⢸⣿⠿⠿⠟⠟⠟⠟⠿⠿⠿⠻⠻⠻⠶⠶⢤⡀⠀⠀⢿⠿⠿⠟⠟⠝⠝⠟⠯⠮⠮⠪⠫⠫⠫⠻⠳⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣶⡄⡀⠀⠀⠤⠈⠁⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣉⡀⠀⠒⢀⡀
⠘⢿⣧⣤⣤⣤⣤⣤⣴⡿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠿⡿⣿⣦⣤⣤⣿⣶⡆
⠰⠿⠿⠿⢿⣿⡟⠛⠛⠓⣼⣿⡄⠀⣀⣀⣀⣤⣤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠙⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢀⢠
⠀⣠⣾⣿⣦⣤⣴⣾⣏⣷⡄⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣤⣦⣦⡄⠘⢺
⠀⠻⢿⣿⢟⣻⣻⣏⣿⣛⡇⣀⣤⣄⢰⣿⣿⡆⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⡶⢤⣤⣋⣼⣿⣿⣿⣿⣉⣿
⢠⣶⣴⣿⣾⣿⣿⣿⣿⠿⣻⣿⣿⣿⣄⣠⣄⣀⠀⠀⠀⠦⣀⣩⣷⢶⢶⢶⢶⢾⢿⢿⢿⢿⢿⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠋⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠉⠙⠙

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠈⠙⠳⠦⣄⣀⠀⠀⠀⢀⣀⡤⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢹⡎⡆⢦⠠⠀⠉⢻⣶⠛⠋⣁⡴⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢻⡜⡞⢖⢳⠀⡸⡌⠉⠉⡉⠩⢭⡃⠙⠦⢤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⢻⡜⣟⣖⢳⡘⡝⣆⠀⣧⠴⡖⣾⢠⢠⣠⣀⡈⢳⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡀⠀⠀⠻⣌⢌⢯⣱⡘⣌⠑⠺⠼⠿⢯⡈⠐⣓⣑⣉⣨⣷⣤⢎
⠀⠀⠀⢀⡼⠃⠀⠀⣿⠉⡏⠈⡄⠀⠙⢦⠢⠳⠵⠌⢆⠀⠀⣠⢞⢭⠭⡩⢍⠉⢀⡴⠃⠘⢧⡀
⠀⠀⠀⠈⠀⠀⠀⠀⢿⠤⡧⠀⡇⠀⠀⠀⠳⣄⣀⣀⣀⠴⠛⢑⠟⡫⢎⠑⣃⡴⠋⠤⠤⠤⡈⠳⣄
⠀⠀⢀⡄⣇⠀⠀⠀⣸⣀⣧⣀⡇⠀⠀⠀⢀⠀⠀⠀⠑⠲⠶⠶⠶⠶⠀⠊⠁⢀⣀⣐⣒⣂⣿⠀⠈⠓
⠀⡠⠞⠀⢹⠀⠀⠀⠈⠋⠉⠉⠀⠀⠀⠀⣾⢐⠒⡆⠀⡆⢰⢐⢲⠀⠀⡆⠀⠀⠂⠠⠄⠀⠀⠀⢰
⣠⠇⠀⠀⣘⡇⠀⠀⠠⡤⢶⠒⢲⠀⠀⠀⡟⢸⢹⡇⠀⡇⢸⢹⢹⠀⠀⡇⠀⣰⠀⢰⠀⠀⡇⠀⡾
⠃⠴⠴⠐⠉⢷⠀⠀⠀⣿⣸⡤⢸⡆⠀⠀⡇⢸⣼⡇⠀⡇⢸⣼⢸⠀⠀⡇⠀⡿⠀⢸⠀⢸⠀⠀⡇
⠀⠀⢀⠰⠀⢺⠀⠀⠀⣿⠀⣇⢸⡇⠀⠀⡇⢸⡶⠇⠀⡇⢴⡾⠾⠀⠀⡇⠀⡇⠀⡉⠀⠸⠀⢰⡗⡦
⠀⠀⠈⠀⠀⠀⡇⠀⠀⠹⠿⠿⠚⠀⠀⠀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⡇⠀⠁⠀⠀⠀⠀⠀⢸⠁⠀⠐
⠀⠀⠀⠀⠀⠀⢀⣀⣒⠂⠬⠭⢉⣉⣉⣉⡉⠡⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⡍⠉⠉⠛⠛⢒⣒⠲⠤⠤
⠀⠀⠀⠀⠀⠡⠀⢒⣒⠈⠭⠥⢀⣀⣀⣠⣤⡤⠤⠤⠤⠶⠶⠶⠖⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠈⣉⡤⠴⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠛
⠀⠀⠀⠀⠠⣏⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣙⣒⣢⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⠀⠸
⠀⠀⢠⣤⣀⠰⣿⣷⣤⣀⣤⣤⣀⢀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⡈⡀⢘
⠀⠀⢈⠛⢿⣷⣤⣈⠉⠉⠙⠛⠿⢶⣿⣮⣧⣉⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠁⠀⢠
⠀⠀⢸⣇⡀⣩⣿⣿⣷⣶⣤⠀⠀⢀⣀⡉⠛⢛⣉⣉⣉⡉⠉⢉⢉⣉⣉⣩⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠸⣿⣿⣿⣿⡍⢛⡻⣷⠒⠛⠛⠛⠟⠛⠛⠉⠉⠉⠉⠐⠛⠛⠛⠛⢒⣂⠚⠓⠀⠀⠀⠀⠀⠀⠀⡤⢃⡣⠂
⠀⠀⢠⣾⣿⣿⣿⣷⣿⣿⠿⠿⠿⠿⠇⠀⢀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢠⡷⠇
⠀⠀⠀⠀⠀⠉⠳⠿⠷⠄⠈⠉⠑⠒⠒⠲⠾⠿⠿⠿⠿⠿⠶⠶⢶⢶⣿⣷⣞⠛⠃⠀⠀⠀⠀⢠⠃⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠐⣶⣤⡀⠀⠈⠛⢢⣄⣁⢀⡠⣴⢷⣷⣦⢠⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠿⠽⠿⠷⠀⢰⣾⡆⣿⢻⣋⣄⣤⣨⣿⣿⣷⣿⣿⠇⠛⠋⠉⠌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢤⣤⣤⣄⣶⣦⣖⠛⠛⠛⠛⠛⠛⠉⠟⠋⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠁⠙⠈⠛⠋⠿⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠒⠻⢄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠝⠈⠀⠀⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠰⠋⠈⠀⠀⠀⠀⠀⢧⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠉⠑⠒⠤⢀⡀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣄⡈⠁⠀⠀⠈⣳⡀⠀⠀⠀⠐⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠻⢶⡄⠀⠀⠀⣿⡀⠈⠀⠀⠀⢸⠠⠐⠁⣀⡄⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠈⠀⠀⠀⣻⣀⡀⠀⠀⠀⢸⠀⠀⡞⢹⡇⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠈⠉⠻⠇⡀⠀⢸⠀⠀⡗⣻⠇⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⢸⠀⠀⡛⣡⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠉⠒⠤⣀⠘⠀⡠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⠁⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣸⢴⣶⡟⣠⡶⢂⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡤⣴⢷⢾⡇⢘⡿⣿⠑⢿⣥⣿⢋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⢣⣽⣄⡾⢟⣻⠿⠟⠒⠶⣾⣿⠟⠉⣁⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠻⢦⣽⣤⡿⣻⣶⡋⠀⠀⠀⠀⠀⣈⠻⣶⣛⣡⠄⠄⠀⠀⠀⠀⠀⠘⣷⠀⣾⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣼⠇⠀⠀⠀⡼⠻⢶⣀⠞⣿⡿⠋⠀⡁⠀⠀⠀⠐⠊⣉⣀⠘⣿⣿⣛⡧⠀⠀⠀⢀⠀⠀⣿⣼⣧⡾⠛⠁
⠀⠀⠀⠀⠈⠻⢦⣄⡀⢻⣾⣿⡄⠀⠀⠀⢛⣉⠹⣏⣾⡟⢀⣾⣦⡀⠁⠀⠀⢠⡞⢉⡉⠣⢸⣿⠁⠀⠀⠀⠀⠀⠙⢺⣿⢛⣿⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣈⣽⣾⡇⢀⣿⡀⠀⠀⠉⠁⣨⣼⡟⠀⠘⢿⡿⠃⠀⠀⠀⠸⡀⢿⠇⡌⢀⣿⡄⠀⠀⠀⠀⠀⠀⣸⡿⠟⠁
⠀⠀⠀⠀⠀⠘⠙⠛⠉⢸⡿⠿⠻⣿⢦⡀⠀⠀⣿⣿⠁⠀⠰⠄⡠⠞⠀⠛⠃⠀⠙⠒⠋⠀⠈⣻⣿⡆⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠙⢆⠻⣆⠀⠸⣿⢃⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⢰⣿⣽⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢳⡌⣷⣤⠹⣎⢳⣄⠀⠀⣿⠿⢃⣌⣹⣿⠇⠀⣰⣿⠟⠁⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠳⣝⢧⣉⠳⣤⣙⠛⠛⠒⠋⢁⣠⣾⠟⢁⣤⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⠀⠸⣿⠛⢷⣿⣯⣝⣳⣦⣴⣿⠋⢀⣴⡟⠉⠙⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠈⢻⣆⠉⢩⡏⠀⠙⢷⣯⠋⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣠⡄⠀⠀⠙⢷⣿⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣧⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣄⣀⣀⣀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⠀⠉⠉⠉⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⠿⣥⣦⣀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠟⠀⠀⣰⡟⣿⣯⠉⠉⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣶⣤⣼⡟⠀⢸⣿⡄⠀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠛⠋⠀⠀⠘⠟⢻⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣷⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠛⠶⠤⣤⣤⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡴⠚⠉⠻⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⠴⠶⠛⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⢶⣄⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠗
⠀⢀

⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⢸⣿
⠈⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⢸⣿
⠀⡃⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⢸⣿
⠀⡇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⠈⣿
⠀⡇⠀⢸⡇⠀⠀⣷⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⢸⢹
⠀⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣘⠀⣼⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⠀⢸
⠀⡇⠀⢠⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣽⢇⠈⠄⠆⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⠀⢸
⠀⡇⠀⢸⡇⠀⠀⡏⠀⠀⠀⠀⠀⢸⣄⠎⢸⡘⠆⠀⣀⣀⡀⠀⣠⣇⣼⣳⣄⠀⠀⠀⡇⠀⠀⣿⣿⠀⢸
⠀⡇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠐⠐⠺⢿⣄⣰⣧⣷⣶⣿⣷⢇⣰⣿⣿⣿⡻⠹⡍⠒⠒⡇⠀⠀⣿⣿⠀⢸
⠀⡇⠀⢸⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢋⡿⢿⣿⣿⣷⣿⠿⢛⣴⣦⣤⣤⣊⡅⠀⠀⣿⣿⠀⢸
⠀⡇⠀⢸⡇⠀⠈⠉⠀⠀⢠⣤⠶⣦⣒⣿⣿⣿⠸⠁⢸⣿⣿⢟⣯⣶⣿⣋⠀⠀⠀⢹⡇⠀⠀⣿⣿⠀⢸
⠀⡇⠀⢸⣗⠀⠀⠀⠀⠀⠀⢈⣷⣶⣿⣿⡿⣿⣷⡆⣼⣿⣿⣿⣿⣿⣯⠀⠄⠀⠀⣺⣇⣀⣤⣿⣿⠀⢸
⠀⡷⠀⢸⡇⠀⠀⢀⡴⠛⢛⣿⣶⣿⣿⣿⡄⣿⢧⡇⣿⣹⣿⣿⣿⣿⣧⡀⠀⠈⠉⠉⠁⠀⠈⠉⣿⠀⢸
⠀⡇⠀⢸⡇⢠⠜⠁⠀⠀⣼⣿⢋⣽⣿⣿⣷⣿⣟⠇⢉⠀⠈⣽⣿⣿⣿⣿⣧⡂⠀⠀⠀⠀⠀⠀⠹⠀⢸
⠀⣷⠀⢸⡷⠃⠀⠀⠀⣠⠿⠁⣡⡞⠉⠙⠻⢿⡟⠢⠘⠀⠈⠉⠁⠀⠈⠙⣿⠋⠓⠢⢄⡀⠀⠀⢰⣠⣾
⠀⣿⠀⠨⠀⠀⠀⢠⡴⠃⢀⣷⡟⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⢠⣼⣧⣿
⠀⣿⠀⢀⠀⠀⠀⢸⡇⠀⣾⠋⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⣿
⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿
⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿
⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡆⣿⣿⣿⣿
⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿
⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣿
⢠⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⢸⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⠀⢸⣿⣿⣿⣿⣿⣿
⢸⣿⠀⢸⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⢸⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⣛⠉⢉⣉⣉⣛⣛⠓⠒⠒⠒⠲⠶⠤⠤⠤⠤⢤⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣔⣡⣶⠶⠖⠒⠒⠀⠀⠈⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠲⠦⠤⠤⠤⠭⣉⡉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡟⠓⠬⠍⣙⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠤⣴⢻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀⣀⣒⣒⣰⠁⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠓⠒⠒⠲⠶⠶⠦⠤⠤⣤⣤⣄⣀⣀⣹⠃⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⣤⣄⠀⠀⢀⠤⠶⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⢉⣉⣳⠀⢰⠁⠀⣠⣴⣬⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⡏⠳⡄⠀⡰⠉⠇⠀⠀⠀⠀⠀⠀⢇⠀⢿⣿⣾⠀⠸⡀⠀⢿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⢯⡉⠓⠿⣄⠙⢄⣳⠘⢦⠀⠀⠀⠀⠀⠀⢸⡟⠒⠋⠁⠀⠀⠙⠲⠶⠶⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⢸
⣔⠛⠓⠲⠀⠀⢸⡁⠀⠸⣆⣀⣀⣠⠤⠔⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸
⠈⠉⡩⠛⣡⣄⣀⣁⣠⠖⠮⠭⠥⠤⠖⠋⢹⡇⠀⢀⠀⠀⠀⠀⠀⠐⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢦⠀⠀⢸⡇⠀⢸⢸
⠀⠀⠓⠋⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢦⣀⠀⠀⠀⣀⡴⢻⠙⠦⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢧⡀⢸⡇⡄⠈⢸⠀⡖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣍⠉⠉⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠚⠧⣥⣸⡼⠋⣀⡼⢀⣀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡼⠦⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡦⣤⠈⠁⠀⠈⡏⠁⠠⠴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣹⣆⠀⠀⠀⠃⠀⠐⠶⢖⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⢸⢻⠒⠒⢲⡀⢿⠲⠤⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⢯⣸⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⡟⠛⡶⠲⠶⠶⠶⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠳⡄⠹⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠙⢄⠀⠀⠀⠀⠀⣼⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠑⠦⣄⣀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠠⣖⡚⠉⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢉⡀⠀⠀⠀⠀⠀⢀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠒⠒⠂⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⢀⡀⢴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣉⣂⣄⠀⠒⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠉⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠁⠀⠈⠉⢹⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⠛⠃⠠⠀⠀⠉⠀⠀⠀⠀⠉⠃⣤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠁⠀⠀⣐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣷⣶⣿⡀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠈⠀⠙⠉⢠⣶⢷⡶⠀⠘⠀⠀⠀⠀⠀⠀⠀⠋⡘⠉⢉⣿⣿⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣹⡿⠃⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠁⠉⠀⠀⠐⠹⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⣤⡀⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠻⣤⡴⢿⡀⠂⠀⠀⠀⠀⣄⣰⠤⠤⠚⠛⠋⢉⣠⡜⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⠂⠀⠀⠀⠠⠾⠉⠀⠀⠉⠐⢠⣀⡤⠟⢋⠡⠀⠚⠁⠀⠁⡠⠀⣤⢤⠀⠀⠀⠒⠰
⠀⠀⠀⡴⣬⠀⠀⠀⠙⡇⠀⠀⠂⠀⠀⠀⠀⢩⠄⠊⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠂⠀⢀⣀⣀
⠀⠀⠀⠁⠈⠀⢰⡿⠟⠁⠀⡄⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡄⠒⠉⢿⣿⣦
⣀⣶⣆⠀⠆⠀⢀⠸⠆⠀⠀⠀⠉⠀⠀⠀⣾⣯⠷⡀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢉⣨⣷⠶⠶⠶⠦⣍⡀
⠁⠀⠈⢻⣇⡀⠈⠁⠀⠀⠈⠀⠀⠀⠀⣴⡿⠈⣤⣷⡄⠀⠀⠀⠀⠀⠠⣀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⢰⣆⣀⡙⢁⣤⠾⠀⠀⠙⠀⠀⠀⠀⣿⠧⠤⠤⠼⢿⡀⠀⠀⠀⠀⣤⠋⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⠈⢳⡆
⠀⠘⢿⣍⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⣴⠿⢦⣀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢠⠊⢸⠃⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠘⠛⠻⣧⠼⠃⢀⣀⡀⠦⡌⠛⡽⠁⠀⢠⡾⠋⠀⠈⠉⠰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠄⣰⠏⠹⠀⠀⠀⣞⣀⡀⢠⡿⠁⠀⠁⠀⠄⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⢰⠉⠉⣠⡟⠀⠀⠀⠐⠀⠀⠀⠛⢢⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡼⢛
⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠁⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠘⣧⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⣀⠴⠆⠈⠉⠀⠀⡞⠉⠀⠈⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⢀⣝⠁⠀⢀⠀⠀⠀⠀⠐⠀⢀⣸⣟⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⣋⠀⢸⠏⠀⠀⠀⠙⠄⠀⠀⠈⠠⠴⠷⢛⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⢿⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⢠⠘⢇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠀⣻⣤⣾⠀⠀⠀⠀⠀⠘⠠⠁⠄⠀⠤⠈⠴⠛⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠒⠂⠈⢁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⣯⣄⠀⠀⠀⠀⠀⠘⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⢻⡀⠀⠀⢰⢧⠀⠇⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣧⡀⣠⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠙⣼⠃⠀⠀⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢈⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⡇⠀⠀⠀⢹⣇⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠈⢠⣡⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀⣈⣷⡀⠀⠀⠀⢻⡆⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢢⣬⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⢿⡇⠀⠀⠀⠀⢿⣤⡾⠁
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⣦⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⠟⠀⠀⠀⠀⠀⠀⠛⠁
⣤⣤⣤⣴⣿⣼⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠟⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣄⣠⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣼
⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⠈⠑⠀⠀⠛⠛⠂⠀⠒⠒⠀⠐⠒⠒⠀⠐⠒⠒⠂⠐⠒⠒⠂⠀⠒⠒⠂⠀⠐⠒⠒⠀⠒⠒⠒⠀⠐⠒⠒⠂⠐⠒⠒⠀⠒⠒⠀⠒⠒⢾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠤⠀⢠⣤⡄⠀⢤⣤⡄⠀⢤⣤⡄⠀⢠⣤⣤⠀⢠⣤⣤⠀⠀⣤⣤⠀⢠⣤⣤⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⣀⣸
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠘⠀⠀⠛⠛⠛⠛⠛⠛⠛⠒⠚⠒⠚⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣀⡀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠿⠿⠿⠿⠿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣶⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⢻⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⢠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠻⣿⡄
⠀⠀⠀⠀⠀⢠⣾⡿⠁⢀⡿⠋⠀⠀⢀⣠⣴⡶⠶⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠳⢦⣄⠈⢿⡆⠹⣿⣆
⠀⠀⠀⠀⢠⣿⡟⠀⠀⠸⠃⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠙⣦⠀⠳⠀⠹⣿⣆
⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣿⣆
⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣻⠇⠀⠀⢸⠀⠀⠀⠀⠰⠄⠀⠀⠀⢠⡿⠀⠀⠀⠀⢻⣿⡆
⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢿⣿⡄
⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠘⣿⣷
⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⣿⡙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠛⠉⠀⡏⠀⠀⠀⠀⠀⠀⠈⢿⣄⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣰⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣻⣆⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣤⣀⣀⣀⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠉⠉⠉⠉⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠤⠴⠾⠷⠾⠷⠶⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃
⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢻⣧⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣯⣤⣤⣴⣶⣿⣿⡿⠟⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢙⠀⠀⠁⠈⠁⠙⢤⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣶⠶⢻⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠀⠀⢀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠂⣠⣀⢀⡀⠀⣐⡂⢡⡄⠉⡈⠁⠀⣶⠶⠄⠀⠰⡆⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠿⠳⣷⣶⠾⣶⣄
⠀⠀⠀⠀⢠⠀⣠⣿⣿⠟⠁⠈⠉⠛⠿⠦⠶⣶⠄⡀⡌⢀⡀⠀⠀⠀⠀⠀⠰⠶⡄⠀⠈⠀⢀⡀⠀⠀⠀⠤⢀⣯⣽⣄⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠛⠀⠀⠀⣴⡶⠀⠀⠀⠋⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⣶⣤⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠙⠒⠲⣶⡄⢠⡔⠀⠈⠀⣀⠰⡟⢀⡀⣷⢶⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⣹⡿⠷⣾⣿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢱⣴⣶⣷⣄⡀⠀⠈⠀⠖⣀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠒⣤⡀⡾⠃⠹⣿⠿⡿⠷⠊⠀⢶⣶⠏⣤⡠⡤⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠠⠃⠀⡀⠀⠘⢻⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠁⠀⢀⣤⠛⢠⡁⠀⠀⣦⠀⢶⡦⠄⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠂⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣹⣿⣶⣾⣟⠀⢀⡙⣀⡙⠃⣀⠈⠦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⡸⠇⠀⣠⠚⣹⣦⣈⠉⠹⠀⠸⣁⣹⣥⣶⣛⡶⠀⠄⢀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠁⣸⡿⠌⠁⠀⠀⠀⢀⢸⣿⣻⣿⠟⠛⠃⠀⠘⠀⠀⠃⠈⠀⠀⠀⠀⠉⠀⠀⠐⡆⠀⠀⠀⢹⣿⣿⣏⠷⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⢀⣶⡀⠀⢠⡦⡜⠹⠉⠉⠀⢾⠓⠛⠈⠁⢠⣄⠀⡀⠀⠀⠀⠀⢀⡀⠀⠙⠒⠀⢦⣴⣿⢿⡼⡇⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢷⣀⠁⡔⠼⢄⠀⠀⣀⣄⡀⠠⣆⢀⣠⡲⣆⠀⢤⣄⢠⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠐⣿⣿⣷⣤⠐⣦
⠀⠀⠀⠀⠀⢰⣤⠀⠀⠀⠸⣿⣛⣦⣀⠀⢠⠄⡼⣿⣇⣜⠛⠿⠏⠁⠉⠀⣨⣽⡄⠀⢄⡀⠀⠀⣀⣄⠀⠀⠀⠀⠙⠛⠿⠿⣏⣻⣗
⠀⠀⠀⠀⠀⢾⣿⣠⣶⣿⣿⣿⣿⣿⡿⢿⣶⡶⣿⠿⣿⣩⣥⣴⣤⣤⣴⣦⣈⣠⣤⠤⠈⠄⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⡿⠁
⠀⠀⠀⠀⢀⡆⣿⣟⢉⣾⣿⠵⣿⢿⡇⠀⢉⣿⣿⣣⣾⣡⣤⠌⡙⠀⢿⡏⠉⢩⣧⣘⠿⡂⠁⠀⠀⢴⠇⠀⠀⠀⠐⠐⣶⣶⠀⠀⠈
⠀⠀⠀⠀⠸⡟⢻⣗⢺⣿⣟⢰⣟⣸⡧⠀⣾⡙⢿⡿⣿⠉⠁⠰⠆⠐⠊⢉⡀⠘⠋⣻⠄⢻⡀⢀⡤⣾⠖⠀⠀⠀⠀⠸⣯⣿⡀
⠀⠀⠀⠀⠀⠛⠸⢿⣿⣿⣿⣬⡟⢿⡗⣸⣯⣵⡘⣿⣿⣿⣧⠿⡀⠀⣠⢾⠋⣤⣼⢻⡇⢰⡇⠘⠛⢿⣟⣰⡄⠀⠀⠀⢻⣿⢇⣀
⠀⠀⠀⠀⠀⠀⠀⢴⣿⣾⣿⣿⡧⢘⠇⢸⣿⣿⣿⣿⣛⡟⢁⣼⠃⠘⣽⠀⠰⣌⢻⣿⠃⢸⣷⢘⡀⠘⠟⠃⠀⠀⠀⣀⢿⣿⢠⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⡮⠃⠀⠀⠀⡹⠿⣿⣿⣶⠟⠈⠛⠱⠊⠚⢂⠙⠟⣿⣿⠂⢴⡟⠈⠛⠿⣶⠀⡆⠀⠀⠈⠐⠋⡙⠀⣛
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡷⠂⠀⠀⠀⠰⢶⣿⣷⠦⢤⣴⣦⣶⣶⣿⣶⣀⣈⣁⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⢿⣿⣿⣷⣷⣀⣶⣾⣿⣿⣿⣦⡼⣿⣿⣿⣿⣿⠻⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠰⣤⡀⠀⢸⣇⢰⣾⣿
⠀⠀⠀⠀⠘⣹⠉⠙⠋⠐⠒⠚⠋⢻⣿⣿⣿⣿⣿⡿⢿⣿⣧⠘⠿⣿⣿⣿⣷⣻⣿⣿⣿⠻⣷⡀⠀⠀⠀⠀⠀⠹⠀⠘⠿⠈⣽⠿⠂
⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⣶⣶⣤⣤⣿⣿⣿⣿⣟⣳⣿⣿⣿⡄⠸⡎⣹⣿⣿⠛⢿⢿⣿⡶⡿⢧⣤⣴⣶⣶⢀⣠⡀⠀⠀⢰⣶⣤⠀⠏
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢤⢀⣹⣿⣟⣿⣿⣿⣿⡟⣿⣎⣿⣿⣿⣿⢸⣿⠀⠀⠀⢠⣄⢳⣆
⠀⠀⠀⠀⠀⠘⢿⣿⡿⣿⣿⣿⠋⠛⠿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣟⣯⣉⣿⡟⡁⢻⣧⢿⡟⣿⣿⣭⣄⠻⣦⠰⡆⠘⠛⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣧⢤⠀⠀⠈⡉⠛⠿⠿⠟⠛⠁⠉⠉⠉⠉⠛⠻⠿⣧⣀⣿⣷⣿⣧⣹⡋⠉⠁⠀⢠⣤⠋⠁⠀⠻⠍⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⠉⠀⠀⠀⠀⠀⠙

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠛⠄
⠠⠁⠲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⡆
⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣿⣿⠆⠀⠀⠀⠀⡀⠀⠈⣄⡄⠀⠀⠀⠀⠀⠀⠸⠿⡿⠃⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠤⢀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠛⠀⠶⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠶⢛
⠐⠂⠀⠀⠀⠀⠀⣀⣤⢦⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠀⠠⣻⠂⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀⣤⢄⡀⢄
⠀⠞⠀⠀⠀⠀⠀⣿⣧⣾⠇⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠰⡴⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⣰⣦⠀⠀⢾⣾⣿⡿⠈⠀⠀⠀⠀⡀⠰
⠀⠀⢰⣒⣦⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⡇⠀⠀⠀⠀⠠⠐⠀⠈⠀⢾⣿⠀⠀⠀⠀⠻⠿⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⣰⠁
⠏⠀⠈⠛⠫⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢰⠀⡀⠀⠀⠀⠀⠀⠋⠀⠘⠟⠃⠀⠀⠀⠀⠈⠇⢀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣼⠀⠀⢷⡀⠀⠀⡋
⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣖⣦⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠁
⠀⠀⢸⣷⠀⠀⡁⠀⢠⣔⡄⠀⠀⠛⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢫⠈
⠰⠀⠈⠛⠀⠈⡅⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠐⢠⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠢⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠆⣤⡙⠀⠀⠀⡄⠠⠀⠀⠀⠀⢠⠡⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠶⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢄
⠀⢰⡄⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠐⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣜⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡻⢛⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡽⡷⠿⢿⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⣿⡿⣅⡀⢀⡞⡈⠀⠉⠛⠚⠉⠀⣈⠱⢀⢀⠤⠤⢤⠠⡀
⠀⠀⠀⠀⠀⠀⢰⠌⠀⠀⣿⠁⠀⢰⡜⠸⣧⠀⠀⠀⠀⠀⠀⡏⠃⢭⢸⠀⠀⠈⠳⡃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⡀⢿⠇⠀⡟⢴⠗⠀⠷⣦⡀⠳⠈⠌⠢⡡⡄⠀⠀⠑⠶⠞
⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⣬⢧⡄⠈⠀⠙⠻⡄⠀⡀⠘⠗⠉⠀⠰⣈⠻⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⣀⣶⠇⠀⠀⣸⣦⡀⠀⠀⠀⢸⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠰⣤⣼⣍⠛⢶⣧⡾⠧⠛⠋⠰⡀⣀⠜⣸⡟⠸⣩⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣷⣖⠿⠟⠋⠛⢶⣲⠒⠀⠀⠐⢻⣿⣿⣶⡏⠀⠐⠛⠞⠂
⠀⠀⠀⠀⢀⡤⢷⠓⢀⢀⣉⣿⡏⠟⢷⣄⠀⠀⠀⠮⠿⢁⡀⠀⠀⣠⡽⠿⣿⣯⣁
⠀⠀⠀⠀⣼⠀⠀⠰⢋⣤⣾⣿⣿⡄⠚⠻⢷⣦⣍⣒⣦⠥⠀⡲⣾⡿⡅⣠⣿⡉⢿⡅⣄⠹⠷⣶⡀
⠀⠀⠀⢀⠟⠀⠀⢢⡙⠋⠉⣻⣿⣿⣖⢌⢻⣮⣿⣿⣿⣿⣿⠾⡋⣾⠜⢣⣿⢰⢮⠔⣼⠀⠀⠈⡇
⠀⠀⡔⠁⠀⠀⠐⠄⣉⡈⠔⣫⣿⣿⣏⢦⣻⣿⣿⣿⣿⣿⣥⣤⠞⣩⢠⣿⣿⣄⢉⣀⠀⠁⠀⠀⢧
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⣷⡟⣿⣿⣿⣿⣿⣿⣷⣞⣷⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠟⣿⣿⣿⣿⣿⣯⡿⢿⣿⣿⣿⠟⣫⣾⣿⣿⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣸⠿⢝⢿⣿⣿⣿⡷⠿⣿⣷⣿⣿⣿⢿⡏⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠠⠿⡘⠁⠘⢿⡳⣦⣤⡴⢻⡦⠈⠙⢣⣾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠉⠀⠀⠀⠙⡗⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠛⠋⠉⠉⠉⠁⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡄⠀⠀⠉⢠⡀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠉⠀⠈⠁⠈⣡⡸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⠀⡴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣆⠈⣀⡀⠀⠀⠀⠀⠰⠾⣿⠿⠶⠀⠀⠠⠀⠋⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢶⣴⡟⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⡇⠀⠀⠀⠀⠱⣤⡀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⠈⠀⠀⠁⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⣷⢲⣤⣤⠖⠁⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡂⠀⣀⣀
⠛⠶⠶⣶⣒⠶⣦⢀⣄⠀⠤⡀⠀⠀⠀⠒⠀⠀⠀⣸⠛⠈⢡⣴⡟⡇⠘⡁⠀⠀⠀⠀⠀⠘⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠓⢀⠐⢶⣺⡾⢙⠉⣉⣉
⣿⣷⣾⣿⣭⣴⣇⢤⣡⠌⠀⠀⠀⠀⡄⠈⠀⠀⡄⠁⠀⢀⠿⠉⠀⣠⣿⣱⡄⠀⠀⠀⠀⠀⠀⠈⣹⠟⠃⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠈⣿⣿⣿⣿⣿⣿
⡤⣧⡌⢁⣀⠉⠙⣎⠹⠀⠀⠆⠀⠆⠀⠀⢀⡜⠃⠀⠰⠋⠀⠀⣴⣿⣿⡟⠓⠀⡀⠀⠀⠀⢀⠜⠁⠀⠀⠈⠉⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠋⠀⢸⣿⣿⣿⣿⣿
⠀⠘⠃⠼⠳⠠⠆⠀⡠⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠃⠀⠠⣾⣿⣿⣿⣷⠀⢀⠀⠀⠀⢀⠎⠀⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠁⢤⣁⣤
⠀⠈⠀⠀⠀⠈⠀⠀⠀⢀⡄⠀⠀⠀⠀⠄⠀⠀⠈⠓⠀⠀⢠⡾⠃⢸⣿⠟⠀⠈⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡀⠈⠃⠉
⠀⠀⠀⠀⠠⡀⠀⠀⠠⠁⠈⠀⠀⠀⢠⡆⠉⠁⠀⠀⢀⣴⣿⠁⢀⣿⡏⠀⠀⠀⠈⠁⠀⠁⢤⠁⠀⠀⠀⠀⠀⠀⠀⠙⠙⠁⠠⠘⠳⠶⠀⠀⠚⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣾⠟⠉⠀⣾⣿⠃⠀⠀⠀⠀⠀⠄⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠚⠻⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠚⣀
⠁⠀⠀⢰⡀⠀⠀⢠⠌⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢸⡇⠀⡠⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⡾⡉
⡀⢰⣘⣇⢉⡄⠀⠸⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠸⣧⠜⠀⠀⠀⠀⣀⢃⠀⠀⠉⢠⡆⠀⠂
⠤⠄⢀⠀⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡰⠋⣤⠀⠠⠚⠳⠜
⠛⠶⠉⣦⣡⠆⠀⢇⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢀⡀⠆⢸⡇
⣁⠑⢤⣀⡅⠀⠀⠘⠂⠀⠀⠀⣠⠞⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠈⠁⡀⠀⠑⠆⠰⠁⢀⣀
⡁⠸⡆⠈⠱⠖⢠⠎⠁⣀⢀⡼⠃⢀⣾⠃⣄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡎⠀⠀⢀⡈⡄⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠆⠀⠰⠀⠀⠀⠀⠈⠉
⠳⠄⠛⠀⠑⠂⠂⠀⠈⣠⢦⡀⢂⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢻⣆⠀⠀⠀⠇⠀⠀⢀⡰⠎⠁⠀⠀⠀⢀⣤⣄⠀⠈⠂⠀⠔⠒⠋⠹
⢄⡀⣤⡀⠰⠀⠀⡤⠀⠀⢀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠉⠄⠋⡄⠀⠀⠀⠀⠀⠸⠤⡟⠀⠀⠀⠀⠠⢤⠴⠁
⣼⡛⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⢀⠀⠄⠙⢠⣶⠀⣢⠀⠀⠀⠠⠀⠀⡀⡀⠀⠀⣀⡀
⡏⢤⡀⠘⠓⠀⠀⠂⠀⣰⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢠⡲⠋⢴⡟⠆⠉⠤⠀⢠⠐⣠⠻⡟⠁⠀⠾⡿⣿⡣
⠀⢀⠃⢐⠬⠀⢀⡀⠀⠁⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣸⠷⡾⠈⡁⠠⢀⣴⡆⢀⣀⣛⠛⠉⠁⠈⠘⠛⡏⠁
⠀⣀⠀⠙⠠⣄⠛⠁⢠⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠄⠀⠄⠀⠈⠛⣧⣼⢿⡉⠀⠀⣴⣦⡄
⣿⡟⠂⠀⠀⣀⠀⣧⡠⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻⢦⡇⢰⣦⠀⣠⠉⠁⢈⣀⠀⠀⠻⠟⠃⠀⢰
⠀⠀⠀⠀⢶⠃⠈⠉⢱⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠛⠛⠳⠂⣄⡠⠆⠴⠀⣄⣠⡀⠄⠛⠘
⣤⣤⡀⠀⠈⡿⠉⣄⢙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡀⠠⠈⠀⠈⠁⠉⠀⠆
⣿⣿⠗⠀⢀⠳⠀⠐⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡶⢤⣿⣀⠀⠀⠐⠂⠀⢠⠆⡘
⠋⣁⠀⠀⣀⡀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠑⢄⠀⠈⠉⠉⠒⠒⠒⠤⢦⣏
⣀⣿⣆⣷⡎⠁⠀⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⢿⡥⠄⠉⠐⢿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠐⠄
⠠⠃⡄⠠⠀⠀⢠⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠲⡄
⠀⠀⢻⡆⢆⠀⡸⢀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⣦⡀⠈⠣⣄⠸⡇⣼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⡈⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢈⣀⣀⡀⠀⠀⠀⠈⠷⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠁⠀⠀⡠⠤⣄⠀⠾⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣠⠀⠀⠀⢀⠎⠀⠀⠀⠑⣄⠒⠺⣄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⢊⠏⠁⠀⠠⠊⠁⠀⠀⠀⠀⠑⠈⢢⠀⢈⣧⠀⠀⠀⢀⠜⢁⡠⡙⢆⡠⢔⠂⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡠⠀⡇⠀⠀⣀⣀⡀⣠⠒⠐⠀⢣⢀⠊⠀⠀⡠⠃⠴⠁⠁⣸⡤⠒⠁⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⠁⠀⢃⢰⠋⠀⠀⠈⠉⠁⠀⠀⠃⢺⣆⠀⡔⢡⠊⠀⣢⠴⡇⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⢄⣈⠉⠑⢄⡄⠄⠀⠀⠠⡶⠄⣀⠠⠃⠀⠓⠊⡔⡁⠖⠊⠁⠀⠈⢳⣈⣁⠀⠀⢀⡄⠈⠄
⠀⠀⠀⠀⠀⣀⣀⡀⡇⠀⠠⠒⠲⠤⣭⣐⡚⠷⠀⠀⠉⡉⠀⠀⠀⠈⠐⢄⣠⠊⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁⢠⠊⠁⢁⠀⠠
⠀⠀⠀⠀⠘⠁⠀⠐⠛⠒⠲⢤⣤⣀⣆⣀⣸⣑⡒⢤⣔⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⢱⠔⠋⠀⠀⠈⡆⠀⠇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣀⠕⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⣀⣠⣴⡿⠿⢿⡃⠀⠀⠀⠀⠀⠀⡘⢈⡠⠒⠁⠈⢣⠜⠀⡰
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣹⠉⢆⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠛⠛⠛⠁⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠃⠉⠀⠊⠉⢁⠆⢀⠜⠁
⠀⠀⠀⠀⠀⠈⡆⠠⡤⠭⡀⠉⢳⡳⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢁⣀⡀⠀⣀⡔⢁⣀⣀⡀⣤⠎⣠⠣⢄⡠⠔⢋⡔⠁
⠀⠀⠀⠀⠀⠀⢡⠀⢇⠀⠘⠤⠜⠣⡈⠢⡀⠀⠀⠀⠑⣄⣀⣤⠚⠁⠀⠉⠙⠛⠡⠭⢀⡠⠖⠁⡰⣁⠔⠉⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠆⠈⡆⢰⠚⢉⢑⣛⠂⠈⠲⢆⣛⣉⡱⠚⢀⡔⠀⠀⠀⠀⣀⠝⠓⠲⡤⠴⠚⢏⢀⡠⠚⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢄⠘⠂⠀⠀⡝⠒⠤⠤⠥⠤⠒⠁⠈⠉⠉⢏⠁⠱⡤⠦⠴⠁⠀⠀⠀⢣⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠅⢂⡀⠚⠢⠤⠤⣀⣀⣠⠒⠒⠢⠤⠖⠋⣄⠠⠋⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⡐⠁⠈⢢⣀⣀⡤⠒⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⡰⠁⠀⠀⢠⡇⣇⢘⠒⢤⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢲⠃⠀⠀⠀⢸⡇⡏⠁⠀⡸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⢸⢽⠁⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⢀⠤⡀⣸⣿⡿⠀⢠⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡤⠤⠔⠒⠊⢫⠀⢆⠀⠀⢀⠔⠁⠀⢳⡜⢿⠃⡠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢣⣈⡦⠒⠁⠀⠀⠀⠈⠧⣏⠔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠑⢄⠀⠀⠀⠀⠀⠐⠚⠉⠱⡀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⢔⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠻⢿⣟⣛⠿⢿⡿⣿⣭⣭⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢘⡏⠒⠀⠈⡉⣉⣉⠛⠺⠿⠿⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠀⠠⠤⠤⠄⠒⠒⠀⠀⠀⠀⠉

⠀⠀⡀⠀⠀⠠
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀
⠀⡄⠀⠀⠳⢤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠠⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠀⢀⡠⠊⠀⠀⠀⢠⠁
⠀⠱⡀⠀⠀⠀⠤⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⢰⠖⠃⠀⠃⠀⠀⢀⠆
⠀⠀⠈⠲⠀⠀⠀⠀⠑⣠⠚⠳⡴⠀⠀⠀⠀⠀⠀⠐⡎⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠔⠂⠈⠁⠀⠀⠀⠀⠀⠘⠢⠜⢳⡀⠀⠀⠀⠀⠀⠀⠛⠶⠚⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠
⠀⠢⢄⡀⣀⢀⢀⠀⠀⠀⠀⠀⠀⠉⠲⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠊
⠀⠀⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡁⡨⠖
⠀⠀⠠⠁⠀⠀⠀⠀⣄⡀⠈⠂⡤⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠢⠀⠀⠀⢠⠖⠒
⠀⠀⠀⣀⣀⠤⠄⠈⡰⠉⠉⠃⠀⠀⠀⠀⢈⠈⠓⠒⢶⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⠒⠉⠀⣀⠔
⠀⠀⠀⢀⣀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⢀⠜⡀⠀⠀⠈⠆⠀⠢⢄⡀⠀⠀⣠⠄⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠠⠤⠄⠀⠀⠘⠀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠠⠂⠀⠢⠉⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠢⣀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠔⠲⠨⠄⣘⠃⠀⢀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⡌⢁⠊⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠑⠒⠐⠀⠐⠀⢰⣆⠀⠀⠀⢀⣼⣿⡆⠀⠀⠀⠀⢀⠘⢁⠀⠠⣰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⡒⢇⠀⠀⠀⠀⠹⣿⡷⠶⠻⠋⠁⠀⠀⠀⠀⠀⠀⡈⠊⠀⠒⢀⠁⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡈⢱⡠⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣀⠔⠀⠠⠤⠔⠠⠃⠤⠐⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠵⠌⠢⣀⠤⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴⠅⢁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠙⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⡔⠀⠀⢀⣷⠓⠃⡙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠀⠀⡀⠀⡄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⢁⠀⠊⠀⠀⠙⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣤⣤⣶⢴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⢸⡂⠋⠑⠚⠛⠿⠿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣶⠀⠀⠀⠀⠀⠀⣿⠿⣤⡾⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⢶⠀⠀⠀⠀⣸⡟⠚⢻⣁⠀⠀⣌⢢⣤⡴⠁
⣤⡄⠀⠈⣦⢰⡤⠀⠂⠀⠀⠀⠀⠀⠀⠾⣿⡄⠙⠀⠦⠠⠖⠀⣴⡾⠀⠁⠨⣯⣄⠀⠀⠀⣀⣜⡜⠁⠉⠃⠀⠀⠀⠺⠿⢤⣀⠐⣋⠤⠄⠀⠀⠉⠩⢕⠀⢀⣴
⠟⠁⠤⠘⠛⡱⠈⠁⠈⠱⠰⠂⠀⠀⠀⠀⡾⢃⠤⢒⡚⣡⡍⠓⠺⣝⢣⣀⣶⠛⠛⠓⠰⠚⣬⣅⢧⠀⠀⠀⢀⡠⢂⡀⠀⣈⠵⣪⠄⠀⠀⠀⠀⠠⠄⠀⠙⡾⠋
⠒⠀⣴⠀⣠⠳⣀⠀⠑⠈⠀⠀⠀⠀⢊⡥⣊⡁⠀⠈⠐⠈⠀⠀⠀⠈⢇⣙⡋⠴⠃⢤⣶⡤⠿⠉⠦⣵⡦⠤⡽⠥⠭⠔⠊⢤⡿⠀⠀⠀⠀⠰⠆⢄⣀⣀⡤⢣⣶
⠀⠀⠉⠺⠟⠀⠨⢕⣂⣒⢀⠐⠒⢋⣥⠘⠙⢑⣶⠆⠀⠀⠀⢤⣀⣀⢮⠝⠁⡉⠁⠈⠙⣿⠂⢀⠀⡿⠟⠀⠖⠐⠷⢠⣄⠹⣿⡆⠀⠀⠀⠀⠀⠀⠐⠂⠻⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠪⠉⠀⠀⢩⣷⢦⣽⡄⠀⠀⠀⠈⠗⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⠿⠹⣶⣷⠀⠀⠀⠀⠀⠰⠿⠟⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⡎⠀⠃
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⡝⠉⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢤⡎⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠈⠳⣄⢸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠌⠀⠄⢀⠄⠀⠀⠀⠀⠀⠈⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣤⣦⣀⣸⣇⢶⠄⡠⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣏⠁⠀⡀⠉⢫⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⡈⠙⠒⠒⠒⠋⠔⠀⣀⡤⠊⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡜⠁⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⡉⠁⠀⠀⠀⠈⠑⢄⠔⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠀⠐⠇⠀⠀⠀⠀⠈⡆⠀⠀⡜⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡌⠡⠀⠄⢕⠀⠀⠀⠀⠀⡇⠀⢸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⡃⠠⣍⠒⠤⢊⡜
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢇⠀⠈⠒⠤⠔⠊⡀⠀⠀⠀⢠⠧⡀⠘⠘⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠉⠐⠭⢴⠃⠀⠀⡰⠓⠤⣀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠙⡀⠽⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢹⣴⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⡗⣦⠤⠤⣄⣀⣀⣀⣀⡠⠤⠔⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⢰⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖⠤⠤⠀⠀⠠⠤⠔⠊⠉⢆⢧⠀⠀⠀⣠⣡⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢲⠙⠂⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⢀⠀⠈⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⡿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣈⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢴⣃⡘⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠄⠈⠛⣇⠀⠙⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈⣆⠀⠀⡀⠘⠀⠀⠀⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠐⠄⠀⢷⣙⣷⣤⡅⠀⠀⠀⠀⣧⠀⡤⠀⠼⢌⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡶⠒⠛⢉⠘⠏⠂⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠶⠶⠋⠲⢧⣀⣀⢘⡧⢀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠳⣄⠀⠀⠁⠉⢯⣀⠀⠂⡢⠀⠈⠉⠙⠒⣦⣄⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣄⣀⠀⠀⠀⡄⠀⠀⠈⠳⣆⠀⠀⠀⢈⣦⠀⠀⠈⠆⠀⠀⠀⠸⡃⠀⠀⠀⢀⡘⣁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠫⠀⠀⠀⠑⠄⠂⠀⠀⠈⢀⠀⠀⠈⠛⢆⠀⢀⠸⢌⣂⠀⠀⠃⠀⠀⠀⣈⡷⡧⠀⠰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠟⡴⠁⠀⠄⠀⠀⢦⡲⠂⠀⠈⢆⡈⠀⠀⠠⠀⠀⠣⡚⢤⡇⠀⣄⠀⠀⠔⠚⠷⠀⠀⠀⠀⠉⠀⠀⣴⡂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠠⠀⠠⠀⠀⠀⠀⠀⠉⠆⠀⠀⠘⣆⠀⠀⠁⠀⠀⠀⢸⠀⠙⢶⡏⠐⠁⠀⠀⢀⠐⠀⠀⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠃⠀⠀⣤⠤⠴⠂⠀⣦⡄⠀⠁⠀⠀⠙⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢱⢄⡀⠀⣤⠁⠀⠀⠀⠐⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠃⠀⠶⠖⠠⠞⣛⣆⠀⠀⠄⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡒⡰⢪⣶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⢠⡴⢮⡟⠉⡀⠸⠂⠀⢀⣀⣑⣤⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⢠⠘⠋⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠨⠃⠀⠀⠂⢀⠤⠴⠖⠛⠉⠉⠉⠉⣙⣧⡄⠀⠀⠀⠀⢽⢫⡿⣦⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠠⠄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡦⠚⠡⠤⠀⠀⠀⡤⠂⡉⠀⠀⠀⠉⢐⠆⠀⠀⠛⠘⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠠⠊⢞⠁⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⣦⠆⠚⣴⠂⠂⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠚⠩⠭⡿⡏⠀⠀⠀⠀⠀⠀⠁⠀⠰⢆⠀⠀⡙⠀⠄⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡵⢲⣾⡏⠂⠦⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠀⠧⠔⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠑⢄⡀⡖⠀⠀⠈⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⣁⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠘⢀⠀⢀⡀⡠⠄⠐⠀⠀⠀⠀⠀⠀⠀⣀⠴⣆⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⢦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⠃⠉⠁⠉⠉⠙⠒⠀⠀⠀⠀⠀⠀⠘⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠶⠦⣤⡞⠉⠉⠙⠒⣤⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠐⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠉⠂⠀⠉⢳⡄
⠀⠀⠀⠀⢠⡖⢤⡀⢀⠰⠛⣷⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⣄
⠀⠀⠀⢸⠉⢀⠔⣹⣿⣶⣶⠟⠀⠀⠀⢀⡴⠛⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢻⣡⡾⠛⠛⠛⣳⢦⣄⡀⠀⢸⣇⡄⠀⠀⠀⢀⣄⡴⣫⠽⠲⢦⡙⢦⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯
⠀⠀⠀⠀⠉⣿⠁⠀⠔⠉⣠⣤⢾⡿⠶⣼⡋⠀⠀⠀⠀⠀⣽⠳⡷⠤⣤⡤⣿⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢰⡏⠀⠀⣠⢾⡇⠨⠐⠀⠒⠀⠙⠷⣤⡀⠀⣰⠃⠀⠙⢦⣽⡵⠋⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴
⠀⠀⠀⠀⣸⣇⡀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡻⣦⠀⠀⢀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣏⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⠰⡂⢤⠖⠁⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⢸⡏⠁⣴⠦⠤⢤⡤⠀⠰⠶⣤⣤⣤⡴⠶⠞⠋⡟⢀⡄⠀⠀⠀⠀⠀⣻⠆⠀⠀⠀⠀⠀⠀⠀⠐⣿⠁
⠀⠀⠀⠀⠸⣇⠀⠘⠷⠶⠊⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⣇⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⣿⣀⣉⣒⣀⣀⣀⣀⣀⣀⣸⣇⣰⣄⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀⠀⠀⠀⣨⡟⠉
⠀⠀⠀⢠⠞⣿⠀⠉⡏⠈⡟⢹⠁⠇⠀⢸⡗⠦⣬⣉⣥⠶⢧⣀⠄⣀⡤⠤⠤⠤⠶⠾⣿⣝⡛⠛⠛⠉
⠀⠀⣰⠋⠀⣿⠀⠈⢧⠀⢇⡜⢈⠇⠀⢸⡷⠒⠒⠛⣧⡀⠈⠛⠋⠁⢀⡄⠀⠀⠀⠀⢸⡆⠉⠓⣦
⠀⣰⡇⠀⠀⣿⠀⠀⠀⠱⣸⢀⠊⠀⠀⢸⡇⠀⠀⠀⢈⣿⠀⠘⠲⠶⠛⠁⠀⠀⣀⠤⠚⢧⣴⠋⠁
⣾⣿⡷⣤⠆⣿⠀⠀⠀⢐⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⣸⠃⠉⠒⠲⠤⠴⠒⠚⠉⠀⠀⠀⠀⢻⡿⠇
⠉⠉⠁⠀⠀⠿⠶⠶⢶⠶⠾⠶⠶⡶⠶⠾⠃⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⠷⠤⠤⠤⢤⢤⡤⠤⣤⠤⠶⠶⠶⠖⠛⠋
⠀⠀⠀⠀⠀⣠⠤⠤⠼⢄⠀⠀⣠⠿⠤⠤⣀⡀⠀⠀⠀⠀⠀⣀⠤⢾⠣⢰⢻⣤⡀
⣀⣀⣀⣀⣘⣓⣒⣒⣒⣚⣃⣀⣓⣒⣶⣶⣒⣛⣀⣀⣀⣀⣃⣐⣊⣙⣆⣘⣎⣒⣬⣵⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀
⠀⠀⢠⣠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠄⠰⠤⠀⡤
⠀⠀⠀⠻⠶⠶⠉⢀⡁⣄⠀⠀⠀⠀⢠⠀⢀⠀⠙⠀⢀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡦⣿⡆⠀⡀⣿⢀⣰⠆⡆⠰⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⡀⠀⢀⣽⣶⢤⣄⣠⠀⠀⠀⠀⣦⣩⣥⡚⣛⣛⣻⡽⢛⠣⠛⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠉⠛⠛⠉⠉⠀⠐⠺⠷⣿⣯⡍⣙⣉⣍⣄⡁⣀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠂⡄⠒⠶⠃⠁⠀⠠⠤⠤⠤⠀⠈⠋⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣌⠀⠉⠶⠉⠛⠾⠟⡨⠄
⠀⠀⠀⠂⠀⠸⠁⠠⠤⠀⠀⠀⠀⠤⠤⠀⣀⡀⠠⢀⢀⣠⣤⣄⡀⡀⣤⣤⣶⣤⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠆⠀⠀⠀⠀⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⡄⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠛⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠒⠂⠐⠀⡁⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡀
⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠀⠰⠦⣤⣤⢤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⡇⢸⠀⠀⣀⢀⣀⢀⢀⠀⡄⢀⠀⡀⢀⢀⡀⣀⠄⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣇⣇⣸⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠀⠂⠂⠀⠀⠀⠀⠀⠒⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⢆⠀⠀⠀⠀⡠⠖⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⣈⣅⣀⣀⠎⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡔⠈⡁⠀⠀⠀⡉⠀⠀⢢⡀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢺⠥⢤⠙⠀⠀⠀⠁⠀⠀⠉⣇⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠎⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣾⣒⡒⢲⡇⠀⠀⠀⠀⢄⠀⠀⢻⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠙⠛⠋⠀⠀⠀⠀⠀⡜⠀⠀⢸⡀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠀⠀⠈⣇⠀⠀⡴⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⣠⠋⠀⠀⠀⠘⠦⠞⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠺⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢧⠀⠀⠀⡴⠋⠁⠀⠀⣀⠀⠀⠱⡄⠀⣀⣀⡤⠴⠖⠚⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⢳⡀
⠀⠀⡇⠣⠀⠀⠱⣄⠀⠀⠀⠙⢆⠀⢀⡟⠉⠁⣀⠤⠚⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠇⢰⡇⠀⠀⢳⡀
⠀⠀⢣⠀⠡⡀⠀⠈⠳⠦⢤⡤⠴⠟⣋⡤⠖⢋⡠⠴⠀⣿⠀⣀⣤⠶⢻⠁⠀⠀⠀⠀⠀⢀⡞⠀⣼⠁⠀⠀⠈⣧
⢰⠉⠛⢦⢀⡑⣄⠀⠀⢀⣠⠴⠒⢉⣡⠔⠊⣁⣀⠀⡇⣿⠉⣁⣠⠴⢾⠀⠀⠀⠀⠀⢀⡞⠁⣰⠃⠀⠀⠀⠀⠸⡆
⡇⠀⢤⣼⠇⠕⠌⣶⠛⢉⡠⠔⠊⠉⠀⠀⡊⠀⠀⡟⠁⣿⣉⡉⠀⠀⡾⠀⠀⠀⠀⣠⠋⠀⣰⠋⠀⠀⠀⠀⠀⠀⢿
⡇⠀⠀⢸⡆⢀⠁⢹⠀⡁⠀⠀⠀⢀⡤⠖⠙⣒⢊⡵⠋⣁⠀⠙⡄⢀⡇⠀⠀⣠⠞⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠸⡄⠈⢻⡇⣄⠑⢸⡀⢇⣀⠴⠊⡁⢂⠐⢉⠠⢈⡷⠛⠉⠁⠀⠘⠾⣄⠴⠊⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠳⢤⠇⠀⠑⢌⡇⠈⠤⢒⠩⠀⠘⡈⢔⡀⢯⣀⡴⠚⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠐⡇⠐⠀⠀⣇⠀⠀⠤⢒⠥⠀⢡⠠⢒⡈⣟⠀⢀⣤⠤⠤⣤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠲⣄⠙⢦⡀
⠀⠀⠀⠀⣇⠀⠑⢄⢸⠀⠒⣡⠴⣊⡥⠀⠀⢁⡠⣞⠷⢻⡇⠀⠀⣇⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢧⠀⠹⣆
⠀⠀⠀⠀⢹⡀⠀⠀⠙⡇⠨⠔⠋⠁⡠⠐⢳⠐⣋⠴⠊⠘⡇⢀⡴⢿⢀⡼⠋⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹
⠀⠀⠀⠀⠘⡅⠀⠀⠀⣷⠀⡠⠖⠉⠀⠀⠀⡄⠀⠀⣠⠞⠁⡾⠀⠈⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⢣⡀⠀⠀⢸⡄⠇⠀⠀⠀⢀⠔⢁⡴⠋⠀⠀⠀⢧⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠸⡇⠰⠀⡠⠂⣁⠴⢋⣀⣀⡀⠀⠀⠘⣆⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⣿⠀⠉⣠⠞⣡⠚⡁⠀⠀⠉⠉⠙⠓⠚⠷⠄⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣰⠋⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠴⠊⠀⠐⠤⣞⣡⠖⣠⠞⠋⠛⠛⠓⠒⠒⠒⠶⠿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠰⡴⠿⣄⣀⣀⣀⣠⡴⠚⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⡃⡰⢀⡆⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠈⠉⠉⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⣤⣀⣾⠉⠙⠛⠛⠒⠒⠲⠶⠶⠶⠦⠤⠤⠤⣤⣤⣼⠏⠙⠛⠛⠛⠛⠉⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⣭⣓⣀⠀⠙⠲⠬⣛⠻⠦⣝⡚⠻⢷⣄⡀⠢⣻⡆⢸⣿⣿⣿⣿⢸⡇⣏⣿⣿⣿⣼⡆⣿⣿⡦⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣁⣈⣙⣓⣄⡉⠻⢦⡙⢿⠀⣿⣿⢻⣿⣿⣿⢰⢿⣿⣿⣿⣇⢿⣿⡇⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠛⠙⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⡎⣿⣿⣿⡎⣽⣿⣟⣿⣏⢻⣿⣯⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠉⠈⣿⣧⢻⣿⣿⣇⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⡏⣿⣿⣿⡾
⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⣿⣿⣿⠀⣿⣿⡸⣿⠸⣿⣿⢨⣿⡇⣻⣿⣿⡇
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢻⣿⣻⠀⠛⣿⡇⣿⡆⣿⣿⠀⣿⡇⢸⣿⣿⡟
⠤⣶⣴⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡟⡈⡇⠰⢹⡇⢹⡇⣿⣿⠆⣿⡇⢸⡯⣿⡃
⠿⣿⢿⣿⠁⠠⣶⠖⠒⠚⠉⠀⠀⠀⠀⠀⠀⣈⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣷⠀⠱⠘⠘⡇⢸⣧⢹⣿⡇⣽⡇⢸⡇⣿⡇
⡀⠀⠀⠉⠀⠀⢹⡆⠀⠀⠀⠀⠀⣀⣤⣄⣀⣁⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⢳⠈⣷⣼⣿⡆⢸⡇⠈⡇⢹⣷
⣿⣆⠀⠀⠀⠀⠀⢧⣤⣤⣴⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠺⠀⢠⡀⠀⠀⠀⡇⠸⡹⡇⢸⣇⠀⡇⢸⣿
⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⢼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣳⢹⢘⡇⠀⠀⠀⠇⠈⡇⡇⠸⠟⠀⠃⠸⠟
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠛⠂⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠸⡌⣷⢰⠀⠡⢐⠀⡇⠿
⣿⣿⣿⠀⠀⠀⠀⠀⠈⠎⢆⠀⠀⠈⠻⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠀⣇⣿⢸⠀⢰⠀⠀⣧⠰⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣿⣹⣿⡇⢸⡇⠀⣿⠐⡄⠀⢀⠀⣿⣛⣿
⣿⣿⣿⡳⣞⢮⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⢸⣿⣇⢸⣧⠀⣿⢠⡇
⡟⣿⣿⣇⠘⢷⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⢻⡞⣟⣿⠀⣿⡀⣿⣸⣿
⣇⣿⣿⣿⠙⣆⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣷⢹⡆⣿⣿⠀⣿⡇⣿⣽⣿
⢿⣿⣿⣿⣇⠨⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠙⣿⣷⡙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⢸⣷⣿⣿⡇⢿⣇⣿⣾⣿
⣷⣿⣿⣿⡇⠙⠢⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⡇⠀⠉⠙⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⢺⡿⣿⢹⡇⢸⣿⣿⣿⣿
⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠃⠀⠀⢀⣄⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⢘⣿⣿⡇⣿⡮⣿⢸⣿⣿⣿⣿
⠸⡟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣩⣵⣤⣶⡛⠛⠛⠷⠮⣍⠻⠾⣿⡿⣷⣦⣤⣄⡀⠀⢸⣿⢨⢿⣿⡇⣿⡇⣿⠘⣿⣿⣿⣿
⠀⢿⡸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠈⠢⡌⠉⠙⠳⠀⠀⠀⠈⠁⣤⡀⠹⠈⠹⣷⡈⠻⣶⣾⡟⠐⢸⣿⡇⢹⡇⢻⡄⣿⣿⡏⣿
⠀⠈⣧⣿⣿⣧⣀⣀⣀⣀⣀⣠⣾⣿⠇⠀⠀⠀⠀⢦⡀⠀⠀⢀⠐⢄⡀⠲⡌⠳⡄⢄⠀⠘⣿⠀⠈⢿⡇⠀⣸⣿⣧⡈⣿⢸⡇⣿⣿⡇⣿
⠀⠀⠸⣿⣿⢿⢿⣿⣛⡟⣿⣿⡇⣿⣆⠀⠄⢀⢠⡈⢁⠘⡆⠀⠉⠂⢀⡀⠀⠀⠈⠈⠂⠀⠁⠀⠀⢸⡇⠀⣿⣿⣇⠁⢹⡌⡇⢻⣿⡇⣿
⠀⠀⠀⣿⢿⣿⢸⣿⣿⣿⣿⣿⣇⣿⣿⡆⠈⠘⠇⢿⠀⠶⡌⠀⡀⠀⠈⢧⡉⠛⠓⠶⠶⣦⡀⠀⠀⢸⠀⢰⣿⡿⠟⠀⠈⣧⢻⢸⢹⣇⢹
⠀⠀⠀⢹⣸⣿⣼⣽⣿⡟⣿⣿⣿⣿⣿⣷⠀⠀⣴⣦⠀⠀⡇⠀⠙⡛⠛⠞⠛⠋⠀⣀⣚⣋⠉⠀⠀⢸⠀⢸⣿⠃⠀⠀⠀⠸⡇⣾⠸⣿
⠀⠀⠀⢸⡇⣿⣻⣷⣿⢶⣿⢹⣿⢹⣿⡏⣠⡾⣵⣿⠀⢰⡏⣳⣾⣷⡤⠀⠀⠠⠸⠛⠋⠙⠓⠂⠀⣾⢀⣿⡇⠀⠀⠀⠀⠀⣇⢻⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⢿⣿⢟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣺⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠀⠉⠁⠀⠁⠀⠈⠁⠀⠈⠁⠀⠉⠁⠈⠁⠀⠉⠀⠉⠁⠀⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⡤⠶⠖⠒⠚⠛⠒⠒⠲⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠶⠒⠚⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⢳⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣆⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠉⠁⠉⠉⠀⠉⠉⠈⠉⠷⣄⡀⠉⠋⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠶⠶⠚⠋⠉




⠀⣀⣀⢀⣀⡀⣀⣀⠀⣀⡀⢀⣀⡀⣀⣀⠀⣀⡀⢀⣀⠀⣀⡀⢀⣀⡀⣀⣀⢀⣀⡀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⠀⣀⡀⣀⣀⠀⣀⡀⢀
⣿⣏⢩⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣴⣦⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣠⡤⠤⠤⠂
⠀⠀⠀⠀⣾⣽⣿⢿⡞⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡾⣁⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⣣⣽⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⠟⡡⠤⠤⢍⡛⢷⣄
⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠺⢷⣦⣀⠸⣿⣿⡎⡴⣚⣐⢤⠙⢎⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣤⣴⣶⡶⠶⠶⠶⣿⣿⠿⣿⣿⣿⣮⣤⣄⣡⡱⡀⠳⡌⠻⣦⡀
⠀⠀⠀⠀⣀⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⣾⠋⠀⠀⣠⣬⢻⡆⠈⠉⠛⢿⣾⡦⡈⠢⡘⢿⣆
⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠙⠂⣸⠇⠀⠀⠀⠀⠈⢻⣯⢆⢱⢸⣿
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠶⢶⣾⡋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⢸⡿⣄⣠
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣜⣧⡈⠁
⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣷⠦⠀⠀⠀⢀⣽⣿⣿⣾⣿⡣
⠀⠀⠙⠿⣶⣤⣄⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣼⠃⢹⡄⠀⠀⠀⠛⠉⠉⢻⣿⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⣿⡟⠛⠉⠉⠀⠀⢀⣠⠞⠁⠀⢸⡇⠀⠀⠀⣤⡀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠲⠶⣶⠛⠋⠁⠀⢀⣤⣿⡇⠀⠀⠀⠙⢻⡿⠋⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣦⣤⣤⣶⠿⠛⣿⠁⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⣀⡀⢀⣼⣃⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠙⠿⣏⣀⣠⠾⠃⣠⠀⠀⠀⠀⣿⠃⠀⠀⠀⠻⠟⢿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⡴⠛⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣠⣾⠟⠿⠛⠉⠹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠲⠶⠤⢤⣤⣤⣄⠀⢀⣀⣀⣀⣀⣀⣤⣤⣾⠇⠀⢀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠛⠓⠒⠶⠶⠾⠷⠾⠥⠤⣤⣤⣤⣤⣴⣿⠶⠟⠛⠉⠁
⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⣰⡏⠉⣿⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⣿⠀⢀⡏⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣠⣿⠄⡼⠁⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢯⡤⠞⠛⠿⠶⣦⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠷⠶⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⣤⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠀⢻⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢨⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠛⠛⠛⠛⣻⣿⣉⣉⡉⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣶⣦⣀⣤⠤⠄⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠛⠓⠶⣶⣶⣶⣶⣶⡖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⢶⣿⣿⣟⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⠶⠶⠖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⣉⣉⣉⣙⣛⣛⣛⣀⣤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠢⠿⠿⠿⢿⣿⣧⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠉⠉⢉⣉⣁⣒⣒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠭⠬⠿⠿⣿⠟⠏⠟⠏⢉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣛⣛⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣀⣤⣤⣤⡤⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⢤⣶⣶⣶⣶⡖⠒⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠾⣿⣿⣿⣿⣿⣿⣏⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠤⣤⡤⠤⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠈⠁⠉⠉⠉⢉⣀⣭⣥⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⣉⣉⣉⣉⡍⠭⠉⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⢶⢶⡶⡖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠶⠶⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠉⠉⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣾⣿⣿⣿⣿⡿⠿⠥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠉⠁⣒⡒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣩⣭⣿⣿⣿⣿⣥⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⠶⠶⠶⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣉⣉⣉⣉⣉⣤⣤⠤⠤⠤⠄
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿