Slacker Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠠⠐⠀⠂⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠀⡀⠀⡆⠠⠀⠀⠶⠀⠶⠀⠀⢀⠀⠄⠀⠀⢠⡄⢰⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠃⠈⠀⠐⠀⠀⠀⠀⠀⠁⠈⠂⠛⠀⠃⠀⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠂⠈⠑⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⢡⠀⠀⡠⡶⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⢰⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡤⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⣅⣀⠀⠂
⠀⠀⠀⠀⠀⣠⢶⣇⠀⠀⠀⡀⠀⠁⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣨⡀⢀
⠀⠀⠀⢀⠐⠉⡀⠙⣢⣥⣎⠅⠀⠈⠓⡚⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣸⣧⠈⢑⡂⠀⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢤⡀⠀⡸⢀⡉⠁⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣤⣦⡄⠀⣀⢀⣄⠠⠀⡠⢎⢃⠄⠀⠗⠛⠤⠶⠚⠇⡆⠠⡀⠂⢡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠁⠾⠀⠀⣯⠘⣿⠇⠐⠀⠲⠟⠀⠀⠑⠀⠀⠀⢸⡃⢐⡏⠘⠀⣤⠉
⠀⠀⠀⠀⠂⠠⠀⠀⠂⠘⣀⡂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣼⣏⢙⣶⣾⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⡾⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⣿⡿⠛⠋⠙⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠙⡁⠖⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⢃⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠅⠃⣀⠐⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠷⣃⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠤⢤⡒⠂⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢠⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠈⠁⠢⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠃⠀⠀⠀⠀⠀⠀⠙⢠⢂⠤⣈⡉⠉⠉⠉⠀⠈⠉⠉⠉
⠀⠀⢀⣀⡐⠀⠀⠁⠀⠀⣒⡄⠀⠈⠇⡀⢁⣆⣀⠀⢀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠘⠵⡄⠀⠀⣼⠀⠉⠛⠀⠀⢀⠖⡰⠁⠀⠙⠂⠀⠊⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠄⡠⠀⢀⡑⡄⢠⠃⠀⠀⠀⠀⡠⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⢈⢣⡀⠀⠀⠀⠀⠀⠀⠀⡀⠱⢶⠒⠃
⠓⠒⠚⠉⢀⠘⣮⠀⠀⠀⢀⠞⠁⡀⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠉⠀⠑⠀⠀⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠜⠋⣥⣷⣦⡄⠀⠊⠀⠀⣭⠊⠹⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠇⡀⠀⣨⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠙⠛⠚⠁⠀⠀⠀⠆⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠉⠁⠀⠀⡀⠐⠀⠀⠀⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠤⠄⢀⣀⣀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠰⠤⠴⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠙⠃⠊⠛⠀⠀⠀⠐⠄⠄⠸⠰⠀⠀⠖⠄⠀⠄⠚⠚⠘⠡⠐⠔⠀⠆⠀⠐⠒⠀⠀⠐⠐⠃⠂⠘
⠀⠐⠤⠆⠀⠘⠶⠀⠺⠀⢨⠀⠇⠀⠄⠶⠢⠖⠀⠄⠦⠀⠤⢦⡰⠂⠠⠠⠔⠀⠄⠠⠀⠄⠐⠶⠶⠆⠴⠐⠐⠄⠰⠐

⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣿⣿⣿⣦⠀⠀⠀⠀⢸⠀⡇⠀⡤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⣩⣿⣟⠀⠀⠀⠀⢸⠀⡇⠀⡁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣹⠈⠓⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣄⣛⠛⠋⠀⠀⠀⠀⢸⠀⡇⠀⠀⠢⣀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⠛⠀⣀⣀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠈⠁⢁⠠⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠁⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠂⡈⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⢸⡅⠀⠀⠀⠀⢸⠀⡇⠀⠀⣰⡆⢰⡶⢶⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇
⢀⢀⠠⣀⣀⡀⣀⣀⣀⠀⣀⢀⠀⣈⣐⠠⠤⠤⠄⢐⡀⢀⠀⠀⢸⠀⡇⣀⠀⡿⢷⣶⣶⣶⢂⠀⢀⡀⢀⡀⡀⣂⡠⠤⠤⠄⢈⢀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠊⠻⠾⠶⠖⠠⠉⠌⠁⠉⢩⠀⡇⠀⠘⠭⠀⠁⡐⠺⠁⠀⠀⠀⠀⠤⠥⠆⠽⠾⠷⠇⠰⠥⠤⠄⠉
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠀⠇⠀⠀⠐⠒⠚⠟⠛⠚⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⠄⠀⠄⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⣴⠓⢐⡆⠀⢠⣦⢀⡶⠛⠆⣆⡴⠂⡟⠛⠂⡟⠳⡆⣾⢸⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣈⣹⡆⣇⡀⣼⢼⡜⣇⣀⡀⡿⣄⠀⣟⢛⠠⡷⣞⠁⠿⢸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⠁⠀⠉⠁⠁⠀⠁⠈⠉⠁⠡⠈⠡⠭⠭⠥⠥⠈⠁⠋⠸⠘⠃
⠀⠀⣴⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⣄⣀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠄⢀
⠀⢠⡃⠉⠙⠋⢹⡁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⡿⠟⠙⢟⠀⠀⠀⠀⠀⠀⢴⣿⣶⣧⣷⣁⡄
⠀⠘⠃⢠⣤⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⢾⣿⡿⠀⠀⠈⠀⠀⠀⠀⠀⠾⠶⢬⣛⣿⣿⡏
⠀⠀⣀⠀⠙⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⡀⠑⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠂
⠀⠀⠎⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡀⠺⠛⡇⠀⠀⠀⠀⠀⡇⠀⠿⡏⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠁⠀⠀⢠
⠠⠠⠤⡐⢲⠒⠀⣀⡀⣠⣤⣀⠀⠤⣀⠀⣽⠿⢿⢻⣅⠄⣴⠤⠄⠀⡇⠠⠄⠤⢨⡀⠈⠂⠀⢀⠀⢀⠀⢀⠀⠀⠀⠠⠤⠀⠀⢀⣀⡀⢸
⠀⠀⠁⠉⠉⠛⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣨⣀⢁⡀⠀⠀⠀⠀⡧⠤⠤⠶⠖⠃⠦⠤⠄⠠⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠁⠈⠁⡴

⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣄⡀⠀⠆⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢹⣍⣿⣦⡄⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⠟⣩⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣷⣦⢿⣿⡿⠿⣿⡟⣿⣡⣤⣿⣾⣿⣧⢿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢩⣿⣿⣟⣿⣿⣿⢟⡟⣀⣥⣴⣦⢈⣿⣿⣿⣿⡻⣿⣿⡟⣸⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⣿⡿⡤⢿⣿⢿⣿⣿⣿⣵⣿⣿⣜⣻⣿⣿⣿⣿⣿⣙⣿⣿⣞⣿⢃⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⡏⢙⡿⣿⣿⣾⣷⣾⣿⡇⣿⣿⣿⣟⣻⣿⣿⣛⡯⠹⠺⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣯⣿⣭⣿⣯⣿⣿⣿⣿⣿⠻⣿⢿⡿⡿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣲⠢⣴⣶⣄⣀⠀⠀⢿⣗⣀⠠⠀⢀⡄⠀⠀⠠⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣝⢠⡟⠋⠉⠁⠀⠹⠿⠋⡁⠈⠉⠡⠀⠀⠀⠀⠀⠁⠘⢓⣴⣏⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣽⠿⠻⠟⣉⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠈
⠀⠀⠀⠀⠀⢀⡠⠤⠖⠋⠁⠀⠀⠀⠼⢅⣠⡭⣷⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⢠⠐⠇⠿⣦⢸⠚⣧⣶⡦⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠸⠶⢚⣿⡼⠇⠔⠋⢀⣠⣀⠀⠀⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣒⡒⠛⢾⠉⠠⠲⠞⠶⣦⣄⣀⡟⣀⣤⣾⣧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⣻⡟⠶⠊⢀⣼⣻⣿⣷⣿⡿⡏⠙⠉⠛⢛⢹⣷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠉⠀⠘⠛⢠⣿⣿⣿⠟⠈⠁⠃⠀⠀⠀⣺⠈⠏⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠈⣀⡀⠉⠈⠉⠀⠀⢐⠻⢿⡶⡶⣾⢸⠂⣿⡇⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢿⡿⢀⣒⣚⣚⡿⣻⠜⣥⡀⢳⣿⢰⠟⠋⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡷
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⠀⠀⠀⠀⠀⠀⢿⣺⡿⢿⣏⣵⣛⣥⠈⡿⠗⠊⠉⣀⠚⠉⢀⠀⠀⠀⠀⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⡁⠀⠀⠀⠀⠀⠸⢯⣟⣻⢿⠺⣧⠝⠁⠀⠀⠀⠀⠀⠀⢠⢁⣁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡝⠏⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠖⡡⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⠆⠀⠀⠀⠀⠀⠀⠀⠄⠉⠹⠀⠊⠀⠀⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠰⠛⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡆⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠂⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠸⠐⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠻⢵⡅⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣾⣗⣲⣦⣤⣀⠘⠁⠀⢀⣀⠴⠦⠾⣿⣻⡿⠟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠉⠉⠻⢿⣦⠀⠺⠿⠣⣄⡄⠀⠈⠡⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣴⠶⡄⠀⠰⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⣇⣀⠀⠀⠀⡧⠾⣗⢢⣄⡀⠀⠈⠀⠚⡇⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠃⠀⠁⢀⣴⡒⠲⣄⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠒⠀⡞⢉⠀⢀⠎⠀⠀⠈⠛⠿⣯⣙⣒⣶⠤⠤⠤⣶⣶⣾⣭⣭⠼⠛⠁⠀⠀⠀⣴⠮⢧⠈⡖⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⠿⠀⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠛⠛⠛⠛⠛⠉⢉⣤⡀⠀⠀⠀⠀⠀⢿⡶⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⢆⡀⠰⣿⣗⠀⠀⣤⣴⠖⠒⠒⢿⣾⡁⠀⠀⠀⠀⠀⣤⣷⡌⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠾⣶⣟⠊⣯⣯⣯⡛⢓⠀⢧⡉⢹⡟⢋⠫⣍⡇⢀⠄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠚⠓⠛⠿⠿⠃⠉⠁⠀⠘⠻⠗⠋⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁

⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠆⠆⠠⠀⢠⡖⢲⡄⠐⠀⢴⡰⡄
⢀⠀⠸⠟⢋⠁⠀⡤⠀⠀⡀⠀⠰⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠄⡀⠦⢶⠀⠶⠠⠤⢴⢨⡅⠘⠧⠀⠺⠂⣹⣅
⢸⠟⣀⠆⠀⢡⠑⠀⢠⡄⣧⠀⠀⢼⡄⠀⠰⠄⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠆⠐⠀⠼⠀⢠⡖⠀⠐⠀⣀⠀⠂⣷⣤⠖⣼⠏⣀
⢨⠢⠝⠗⣾⣯⣤⣒⢀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠁⠸⡿⠄⠙⠾⠏⠄⢈⠻⡏⠋⢰⢍⣷⣧
⠹⠇⡀⣿⣷⣾⣿⣿⣯⣴⡄⠀⢸⠀⠀⠀⠀⠀⠀⠸⣿⡖⠷⠀⢀⠸⣤⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠘⠂⣀⠀⠀⠠⣶⡆⠀⠈⢻⡇⠒⡤⣭⡿⣿
⢠⣤⡄⢹⣿⣟⣾⣿⡿⣻⡿⡤⢼⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠘⣶⡔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⣀⣾⡡⡄⠈⢻⣥⣦⢫⣧⣿⣿
⠈⠋⣉⢹⣿⣿⣿⣿⣱⣟⣿⣎⣂⡀⠀⠀⠐⠲⠀⡄⠀⠀⠗⣤⠜⠻⢁⠀⠀⢀⠀⠄⠀⣠⡆⢂⢶⡀⠀⠼⠀⠀⠀⠀⠘⢿⣿⣃⡳⣰⣿⠿⣿⣿⡿⢿⣿
⠀⠀⠀⠸⠿⠿⠿⠟⠛⠻⡛⡿⠷⠄⢀⣀⡀⠀⢰⣥⣤⡀⠇⠘⣿⣽⣿⡍⣡⣿⣴⡈⠀⢃⣆⣨⢓⣼⣴⡞⠀⠀⠀⠀⠀⠹⢿⣿⡟⢙⣟⢠⡷⡼⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣼⣿⣿⣦⡀⢸⢱⡟⠐⠄⠀⢈⢿⣾⡇⢿⣿⣿⠁⢄⠉⣿⣿⣿⡿⡿⠟⠲⠃⠀⠀⠀⠀⠀⠁⠐⠋⠉⠋⠀⠀⠁
⠀⠀⠀⢀⡔⢂⠄⠀⠀⠀⠀⠀⠘⠿⢻⣿⣿⠇⣿⣶⠈⣀⠀⠺⡆⠓⣴⣶⣾⣿⡿⠦⢈⠠⠈⠻⣿⣿⣶⡵⢶⣄⠀⠀⠀⠀⠀⠀⠰⠋⢶⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠊⠡⠄⠀⠀⠀⠀⠀⠆⠀⢸⣿⣯⣾⣷⣿⣿⡅⣄⣄⠃⠀⠂⣶⣿⣿⣿⣿⣻⣷⣦⢻⣿⣿⡟⠁⠚⣧⡾⠃⠀⠀⠀⠀⠀⡄⠀⠉⣶⡄⠀⠀⣜
⠀⠀⠀⣄⣀⢀⡠⢦⠀⠀⠠⢒⢑⠠⠈⠻⢿⣷⣾⣿⣿⣿⣏⣿⢠⠂⣼⣯⣿⣿⣿⣿⣿⣯⣿⣿⣿⣏⡀⠶⢨⣿⠁⠀⣀⠀⠀⠀⠀⠰⡄⠀⢸⣇⠀⠃⢻
⠀⠀⢸⡅⣠⠀⠔⠏⠐⠐⠄⠀⣤⡠⠇⠐⠂⠛⠛⠛⠿⣿⣿⣿⡎⢿⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⣿⡟⠀⣈⡏⠀⡄⢨⡟⢿⢿⣭⣤⣄⠀⣿⢷⣀⠀⢸
⠀⠀⣼⣯⡟⣶⡃⡀⠀⢠⠄⠲⠊⠱⠀⠐⣤⣤⠁⠀⠀⠀⠀⠀⠀⠘⢦⣿⠛⠩⣽⣿⡿⠿⠃⢬⣻⡿⢿⢷⠹⠇⢸⢠⠘⠷⠴⢸⣿⠋⣿⢰⡇⢀⣿⠀⢸
⠀⠀⢙⣿⣿⣾⢁⢾⣀⣀⡔⢰⡄⠀⣇⢐⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠠⣤⠤⠄⠉⠉⠀⠀⠄⢰⣿⣼⡆⠡⠨⣾⠿⢤⣿⡈⠓⢾⣿⣤⢸
⠀⠀⠬⣿⣿⣿⠸⢻⠝⠿⠓⠤⠡⢴⡦⢘⣿⣿⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡟⠿⠿⠿⠀⠠⠉⠀⢯⣿⣇⡄⠻⣾⡿⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠛⠋⣉⣉⣉⠙⠛⠛⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣿⡟⢻⣷⡀⠀⣠⣶⡿⠟⢉⣡⡴⠒⠛⢿⡿⠉⠉⠉⣿⠉⠙⠒⣦⢤⣉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣄⠙⣿⣿⠟⢁⡴⢺⣿⡏⠀⣠⠀⠘⡇⠀⢾⣶⣿⠀⢰⣦⡿⠀⣈⠙⣦⣌⠛⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠈⠁⢶⣿⠆⢸⣿⠀⢸⣿⡇⠀⣧⠀⠀⠈⢿⠀⠀⠀⣿⠀⠈⠀⣾⣿⠗⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣾⠟⢁⣾⣷⣄⡙⠢⣌⠙⣆⠈⠛⠀⢠⡟⠿⠆⠀⣸⠀⠘⠛⣿⠀⣷⠔⠋⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⣿⣟⣴⣿⠋⠈⠻⢿⣦⣄⡉⠛⠶⢤⣤⣾⣧⣀⣀⣠⣿⣠⡤⠴⠛⢉⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠉⠛⠿⢿⣶⣦⣤⣤⣤⣉⣉⣥⣤⣤⣶⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⡀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠋⢻⣿⠋⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⠉⠙⣿⠋⠙⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⠀⢈⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢸⣿⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡇⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣻⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠸⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣭⣭⣭⣤⣽⣤⣽⣧⣤⣥⣽⣥⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠺⣿⣿⣿⣿⣿⣿⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣈⠛⠻⠿⠿⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣌⠛⠿⣿⣿⣷⣶⣤⣤⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠆⠀⠸⡿⠟⠛⠿⢿⣭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡶⣤⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⣹⣿⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣠⠞⠉⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢡⣶⣶⣿⠀⢸⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣳⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⢧⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⠀⠀⠘⣇⠀⢀⣿⣿⣿⣿⣿⣿⢛⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠙⣦⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⡄⠀⠉⢉⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠼⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠸⣄⠀⠀⠀⢰⣿⣿⣿⡿⢿⡿⣭⠶⢤⣠⡶⠀⠀⢀⡀⠀⠀⠀⢿⣿⣿⣷⣻⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠹⡆⠀⢠⣿⣿⣿⡿⠁⢸⣿⣿⣾⣿⣿⡇⠀⠀⢾⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢄⣸⣵⣭⡟⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢃⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣸⣧⡂⠀⢂⣀⣀⣿⣿⣿⣿⣿
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠻⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠉⢉⣿⣟⣉⣁⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠍⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⡔⢨⠀⡇⡄⠀⠀⢊⢢⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠠⠙⡀⣤⠈⢄⠘⠀⢃⠆⠀⠔⠋⣼⠄⡻⢀⡀⢀⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠁⠀⠀⠁⠀⠘⠄⠼⠀⠀⠘⠤⠆⠀⠀⠉⢰⡁⡸⠁⠁⣠⠃⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⢂⠀⠒⠄⠀⠀⠉⠀⢀⠄⠐⠂⠉⠉⠉⠁⠈⠀⠀⠂⠠⠌⣀⠀⢸⣀⠞⠋⡰⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠐⠀⠀⠐⠂⠀⠀⠀⢠⣰⣮⠰⠦⠲⣶⡒⠠⡀⠀⠀⠀⠀⠀⠸⠁⢦⣄⡀⠉⡰⠊⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠂⠀⢀⡴⠟⠟⣛⣉⣐⣿⣶⣾⡿⠩⠁⢽⡦⡌⢻⢯⡤⣾⢶⣤⡀⠨⡃⠉⠢⡑⠚⠡⣁⡘⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣿⣿⡿⠍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢨⢻⣷⣱⠀⠀⠈⢆⠀⢀⠔⢂⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⢀⣞⣴⣿⣿⣿⣿⡁⠀⠀⠀⠀⢀⣠⣴⠶⠖⠂⠀⠀⠀⠁⢀⠀⠠⣸⣇⢾⣷⠀⠀⠀⢳⡈⠈⠑⢁⠴⠋
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢠⠃⢹⢿⣿⣿⣿⣿⣟⣷⣶⣤⢰⡯⠿⠶⠛⠁⠀⠀⠀⠀⠀⡉⠀⣰⣿⣿⣧⡏⠀⠀⠀⠀⢣⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠢⠀⡆⠀⠈⠳⡿⣿⡿⣿⣿⣹⡿⣧⣿⣷⣔⣠⣿⡀⠀⠀⠀⠀⠐⡏⠁⣿⡿⠟⢹⣾⢷⡆⠀⠀⠀⠇⠈⢙⣁⢐⣢⡄
⠀⠀⠀⠀⠒⡄⠐⠀⢸⠁⠀⠀⠀⠀⣾⢝⣿⣿⣇⣷⡟⠛⠤⣴⣿⣿⣿⠀⠀⠀⠀⠠⠁⠀⢹⠲⣠⣾⣿⡾⠴⠀⠀⠀⠸⠀⢀⠤⢁⠈
⠀⠈⠁⠀⣀⠠⠀⠀⢸⠀⠀⠀⠀⢸⣿⣼⣤⠘⣯⣽⣿⣷⣶⣿⠉⠉⣿⠇⠀⠀⠀⠀⠀⣠⠏⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠑⠊⠩⣠⠆
⠀⠀⠀⢀⠄⠠⠤⠃⢲⣀⣠⠤⠖⢹⡏⡙⠿⢳⣿⡞⣿⡿⠸⣿⣀⣀⣸⠧⡄⡆⢀⣠⠞⠛⠂⠈⢉⡹⡀⢂⠀⠀⠀⠀⠀⠇⠴⢅⠉⠉
⠀⠀⢀⠀⠤⠃⠀⠀⢸⠤⠭⠿⠶⠿⠷⠿⣎⣹⡿⣦⣿⣻⣴⣿⠿⠿⠽⡇⠐⢷⡏⠁⠀⠀⠀⢠⠟⠠⠀⠈⠀⠀⠀⠀⢠⠁⠠⡒⣛⣀
⠀⠀⠀⠑⠄⠔⢲⠒⠘⣇⠀⠀⠀⠀⠀⢸⣥⠈⢷⣝⣏⢻⡇⢽⡄⠀⠐⣤⠀⠈⡇⠀⠀⢀⡴⠁⠀⢀⠀⠀⠘⠀⠀⠀⡜⠀⣀⣀⡠⠚⠃
⠀⠀⠀⠠⠶⢄⣁⣠⡄⠹⡄⠀⠠⢁⠀⠰⣯⣌⠁⡛⢿⠙⠋⢻⠁⠀⢸⡿⠂⠀⡆⢠⡰⠎⠀⠀⠀⠀⠀⠀⢀⠀⠁⡰⠁⠘⠤⠄⣢⡀
⠀⠀⠀⠀⠀⢀⠤⠜⠁⠀⠹⣄⣤⣤⣤⣴⣷⣄⣈⣉⣻⠀⠀⢸⠀⠀⣸⠁⠀⠀⠸⣋⣀⢀⠈⣓⣀⢀⠀⡀⠀⠀⠰⠁⢠⠒⠒⠊
⠀⠀⠀⠀⠀⢙⣉⣦⠞⢐⡆⠙⢧⡬⡟⢿⠿⢿⣍⠟⣿⠀⠀⢸⠀⢀⣿⠀⠀⠀⢸⣷⡏⢸⣇⠇⢸⢿⠀⡟⣧⠐⢁⠔⢦⠁⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠚⠓⠖⣚⡎⠓⢯⡚⠂⢸⣏⡀⣿⠀⠀⢸⠀⢸⣧⠀⠀⠀⢠⣠⣄⣦⠤⠶⠼⠶⢦⡄⠀⢀⣠⡉⠂⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⢬⢁⡔⣢⠀⠙⠷⣼⡇⠀⢸⡇⠀⢸⢀⡏⢹⠀⠀⠀⣉⠈⣁⠈⠀⣀⡤⠘⢁⡀⢀⡘⠤⣘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠾⠚⠘⣋⠟⢠⠖⡆⠉⠛⣶⠦⣤⣸⣾⣁⣘⣇⢀⣀⣸⡐⠴⠖⠚⠁⠀⠀⠣⡈⠺⠟⠙⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠉⠁⣾⢥⡀⣎⢸⡧⠦⡄⠉⣩⣭⠉⠁⢠⢄⠀⣀⡀⢎⢹⡐⣏⢱⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣜⠥⠞⠈⠃⡜⣡⡄⣇⢺⡴⣆⢣⣸⠸⣅⡹⢠⠃⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠻⠀⣧⠟⠀⠀⢸⡞⠀⠉⠁⠘⠃
⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣴⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢻⣟⣛⡛⠛⠛⡏⠉⢹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⣿⣿⠇⠀⠀⣹⠀⠈⣿⣿⣿⣿⣿⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⢀⣼⣿⣿⠶⢶⣻⣍⣇⠀⢹⣿⣿⣿⣿⣷⣶⢿⣿⠟⠋⠛⠷⢾⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⢰⣿⣷⣦⣤⣿⢿⣿⣿⣿⣷⣶⣆⠙⢿⣤⣀⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠓
⠀⠀⠀⣀⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣤⣄⣀⡀⠀⠘⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⢻⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠉⠉⠉⠉⠀⠸⣿⢸⣿
⠀⠀⠈⠛⠿⠟⠙⢻⣿⣿⡏⠿⠿⠟⠛⠛⠛⠋⠁⠘⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠋

⣿⠀⢠⡀⣸⣿⣿⠇⠀⠀⢠⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⣿⡆⢸⡇⠀⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⣿⠃⠀⠀⢸⣿⣿
⡏⠀⢸⣇⢻⣿⡿⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⣀⣀⠀⠀⠘⣿⣧⠀⢻⡇⢸⡇⢸⣿⠁⠀⣿⣿⠁⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠃⠀⢸⣿⢸⣿⡇⠀⠀⣸⣿⣿⡏⠀⠀⢀⣾⣿⣿⣷⡀⠀⢻⣿⡀⢸⡇⢸⣧⣾⣿⣦⣤⣿⣿⣀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠘⣿⠃⠀⢀⣿⣿⡿⠀⠀⠀⠾⠿⢿⣿⣿⣧⠀⠘⣿⡇⠘⣇⢸⣿⣯⣿⣿⣟⣿⣿⣿⡷⢼⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⣸⣿⡿⠀⣿⠀⠀⢸⣿⣿⣧⣤⣢⠤⠀⠠⢾⠿⠿⠿⡆⠀⣿⣷⠀⠿⡸⡟⠛⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿
⢠⣿⣿⠇⠀⢹⠀⠀⣾⣛⣹⡿⠋⠀⠀⠀⠀⠀⠀⠰⠶⠷⠀⠽⠿⠀⠟⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿
⣿⣿⠏⠀⠀⣾⡇⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣤⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿
⣿⠏⠀⠀⢠⣿⡇⢈⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣰⡎⠀⣿⡇⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⣻⣿⣿⡇⠀⠀⠀⣿⣿
⡏⠀⠀⣠⣿⣿⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣸⣿⣿⣿⠀⠀⠀⣿⣿
⠀⢀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⣶⠀⣾⡆⠉⣿⡿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⣿⣿⣿⠀⠀⠀⣿⣿
⠀⣾⣿⡿⠿⠋⠀⠀⠀⠀⠠⣌⣀⢤⣀⡀⠠⢤⠈⠡⠅⢲⡿⠁⠰⠶⠆⠊⠀⠀⠀⣠⣄⣤⣤⣤⣤⣤⣼⣿⣿⣿⡆⠀⠀⣿⣿
⢠⣿⠃⠀⠀⠀⠐⠀⠸⠒⠈⠈⣙⣓⣻⣿⡛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣋⣩⣤⣄⣈⣹⣿⣯⣾⣿⣇⠀⠀⣿⣿
⢸⣿⠀⣰⣦⣄⣰⠀⠀⠀⠄⠈⠚⢙⣼⢿⡝⠁⠀⠐⠚⠋⠁⠀⠦⠄⠀⠀⠀⠀⡟⠈⠉⠁⠉⠉⠝⠛⠁⢘⣻⣿⣿⡆⠀⢸⣿
⢸⣿⠀⣿⠟⠻⣿⡇⠀⠘⠂⠀⠠⣤⠤⡿⠀⠀⠀⠀⠀⠀⢠⣤⣤⠈⡉⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠀⠀⢸⣿
⢸⣿⠀⣿⠀⢰⣝⠇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠂⡆⠀⠀⠀⠀⠀⠀⢀⣮⡿⣿⡇⠀⠀⠀⣿
⢸⣿⠀⣿⡄⢸⣿⠁⠀⠀⠀⠀⠀⢠⡿⢣⡖⢀⣴⡆⡀⣸⣿⡿⠂⠛⢳⣀⣴⣿⠿⠃⠀⠀⠀⠀⠀⣠⡏⣼⢗⣿⠀⠀⠀⠀⢻
⢸⣿⠀⣿⡇⠸⣿⡇⠀⠀⠀⠀⢀⡿⠰⠾⢇⡘⠿⠓⠀⣭⣭⡍⠀⢀⡼⠿⠛⠁⠀⠀⢀⠀⠀⢤⢰⡿⣽⢿⣿⣿⣆⠀⠀⠀⠘
⢸⣿⠀⣿⡇⠀⣿⣇⠀⢘⡇⠀⢰⣿⠶⠓⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠲⠾⠿⡿⠖⠿⡼⠃⣻⡿⠿⣿⣿⣦⡀
⢸⣿⠀⣿⡇⠀⣿⣿⡀⠈⠀⠀⢸⣧⡀⠀⢹⡇⠠⣤⣤⣤⡜⠛⣿⣴⠀⠠⣴⡶⡶⠞⠃⠀⣤⣾⣼⣰⡟⠉⠹⣿⣷⡄⠀⠙⣿
⢸⣿⠀⢹⣷⠀⢸⣿⣧⠀⠒⣾⣧⣙⣿⡄⠸⣧⠀⠀⠀⠉⣷⣄⠛⢿⡀⠀⠁⠀⠀⠀⠀⢸⣿⣧⣤⣿⣷⠀⠀⢹⣿⣷⠀⠀⢸
⢸⣿⠀⢸⣿⠀⠘⣿⣿⣆⠀⠈⠉⢿⣿⣴⣆⣯⠠⠤⣠⣤⠠⡅⢠⣤⡤⢄⠐⠀⠀⣠⣴⠟⠋⢹⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⢸
⣿⣿⠀⢸⣿⠀⠀⢿⣿⣿⣆⠀⠀⠀⠈⠙⠛⠏⣀⣼⣿⠇⢠⣧⢸⣿⡇⠘⠛⣶⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⢸
⣿⣿⠀⢸⣿⡇⠀⠸⣿⣿⣿⣷⣤⣀⣀⣠⣴⣾⣿⣿⠋⠀⣼⡏⢸⣿⡇⠀⢸⣿⣿⡇⠀⠀⢀⣾⣿⣿⡏⠀⠀⣸⣿⣿⠀⠀⢸
⣿⣿⠀⢸⣿⣇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣰⣿⠇⢸⣿⡇⠀⢸⣿⣿⠁⠀⢠⣿⣿⠟⠋⢀⣤⣴⠿⠿⠇⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⣾⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠙⢦⡀⠀⠀⠀⠀⡯⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣿⢶⣀⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣰⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣔⠂⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⠋⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣁⡴⢃⡼⠀⣠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⣾⡝⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣀⣤⡤⣤⡀⠀⠀⠀⢾⠠⠞⢁⡠⠊⠀⠀⠽⡗⠚
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡾⢛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⣤⢴⡷⠤⠤⢠⣿⣄⡠⡴⠋⠀⠀⠉⢀⡠⢦⡀⠀⣇
⠀⠀⠀⠀⠀⠀⣴⣿⢻⡇⣿⠀⠀⠀⠀⣠⠶⡶⠦⣤⡀⠀⠀⠀⠀⠀⣴⣶⣶⣄⠰⣏⠋⣸⡁⠇⢀⠂⣿⠟⠋⠀⠀⠀⠀⢰⡉⣠⠎⠀⠀⢹
⠀⠀⠀⠀⢀⣾⠟⠁⠈⢧⡙⢲⣶⡤⡼⠁⠠⡄⠀⣈⢻⡀⡀⠀⠰⣾⢿⣛⣻⣿⡀⡞⢯⣿⡝⠶⣄⠜⠁⠀⠀⠀⠀⣠⠔⠋⠙⣁⣀⠀⠀⢸⡀
⠀⣴⠿⣛⢻⡃⠀⠀⠀⠀⠉⠻⢤⣴⣧⠀⠚⢲⠰⠅⣸⡿⣍⢳⡠⣿⣈⣱⡩⠋⠀⢛⣾⣿⣷⡿⠿⣦⣠⡀⣀⡼⢡⢇⢸⠀⡾⡤⣬⠳⠀⢶⠕
⢺⣟⢦⣽⠞⠁⠀⠀⠀⠀⢀⣠⣾⣿⣏⣳⣵⣊⣠⠔⠉⢠⠜⣎⢷⣿⡿⣿⣠⠶⡾⣫⡾⠋⠙⠻⣦⠸⣇⣉⡝⠁⠈⠢⣼⠆⠓⢄⡠⠾⠛⠁
⠀⠈⠛⠋⠀⢠⢤⡀⢀⣴⡿⠟⠛⠛⢿⣏⠀⠀⠀⣤⠤⢮⣼⡟⠀⠈⢷⡏⡻⢊⣾⢏⡤⣒⣒⣄⠹⣧⠀⠈⠻⣶⣄⢠⠇⢦⡴⠊
⠀⠀⠀⠀⠀⣼⠿⠴⠿⠿⠤⡄⢠⡤⠤⡙⣷⡄⠀⣿⣘⢸⣿⣷⣤⣤⣿⣉⣠⣿⠃⠈⠙⢥⠼⡷⠿⢿⣶⠲⠓⢶⢤⠜⢤⠠
⠀⠀⣠⣄⡀⡇⠘⢒⣛⣘⠀⠷⠾⠃⠀⠉⢹⣿⣭⢥⣤⣼⡟⠿⣿⣽⠛⣉⢙⣭⣗⠛⢧⡼⣛⣃⠀⢸⣿⡶⠉⢻⠉⠉⠃⠘⠋⠉⣇⣠⠖⡆
⠀⠋⠁⠀⠙⠃⠈⠉⠉⠉⠀⣠⣸⡉⠉⠉⢹⣿⣯⠉⠉⢀⡀⠀⠀⣀⡷⠛⠒⣆⣘⣶⣦⣞⣋⢘⣓⣺⣿⡇⠀⢸⣦⣵⣿⣦⣀⣀⣐⣒⣒⣛⣒
⠀⠐⠛⠃⠉⣉⣉⣉⠉⠉⣉⣉⡉⣉⡭⠭⢬⣿⣿⢉⡟⠉⢹⣽⠋⣉⢤⠤⢤⣩⠥⠬⣉⣙⣶⠶⠷⠤⣤⢠⠾⠥⠤⠤⠤⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣉⣁⣀⣤⣿⡭⠬⠤⠬⠤⠤⠴⣿⣿⡟⠀⠀⣼⣀⣀⣉⣠⡤⠴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠐⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢁⡴⠋⠀⠀⠀⠀⠭⣉⣓⠒⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣒⡀⠀⠀⠀⣀⣠⡤⠤⠖⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣭⠍⡍⢉⠉⢉⢹⡏⠛⣛⠛⠛⡇⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠾⠃⠗⠲⠃⠳⠰⠛⠘⠎⠸⠃⡇⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⠀⠀⠀⠀⠀⣸⠀⠈⡇⠀⢰⣦⢠⢰⣀⡔⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣒⠰⠢⢿⠀⠈⠉⠉⠈⠉⠈⠀⠀⠉⠁⠀⠀⡿⠒⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡈⠀⠴⠶⠤⠤⠀⠀⠀⠀⠀⠀⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠟⢠⡟⣿⠀⠀⠀⠀⠀⠀⢤⣴⣶⣶⣤⣄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣆⢸⣷⡏⢀⠀⠀⢀⡀⠀⠈⢿⣿⣿⣿⣿⡿⢀⣼⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡗⢸⡿⡇⢸⠀⢀⣿⣿⣦⣤⠞⠋⠙⢻⠏⢀⠋⢉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡆⠀⠶⡇⢸⣷⣧⣼⠀⢸⡿⣿⣿⠟⠢⠄⣀⢠⣿⣿⣿⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠘⢛⡇⢸⣿⣯⣿⣓⣦⢂⣨⡏⠐⣿⡆⢨⡟⣄⢉⢛⣿⠚⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢯⡀⡰⣿⡆⢸⣿⣷⣯⣭⣽⣿⠏⠁⠀⠀⣤⣿⡍⠉⣀⣣⣿⣿⠿⠀⠀⠀⠐⠛⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠃⢀⣺⡇⢸⣿⣿⣿⣿⣿⡏⠀⠺⡿⠀⣯⣼⣇⣼⣿⣿⢯⣼⣄⣀⠀⠀⣠⠼⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⢀⣸⠇⣾⠹⠿⠟⡠⠌⠈⣁⣥⡀⢴⣿⣿⠿⠿⡿⠃⢸⣿⡿⠳⢠⣼⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠳⠿⠽⢤⣿⣿⣥⡴⠶⠾⠿⠿⠿⢿⣿⣿⣿⣤⣶⣶⣿⣿⣝⣤⣔⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⢠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⢾⣿⡆⠀⠐⠿⢀⠘⠀⠉⠀⠀⠠⢰⣴⠶⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⡙⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⢖⠒⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⡧⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⣿⣿⣿⣷⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡶⠶⢶⣶⣶⣶⡄⠀⠀⣦⣶⣾⠟⠳⢶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠆⠀⠘⢿⢦⡀⠀⢀⣙⣧⠀⠀⢠⣿⣤⣇⣀⣤⣼⢿⡿⢿⣿⣏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠉⡀⠀⠀⠠⢿⣟⡋⠉⠹⠙⠀⠀⠘⠇⠛⠙⠿⣛⣿⡟⠃⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣾⡇⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⣠⣶⡄⠀⠀⠀⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⣙⣽⣿⣿⣿⣟⠃⠀⠀⠀⢀⠌⠸⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠂⠀⠀⠀⠉⣭⣥⣠⣿⣿⣷⣦⣀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠿⠿⣻⢿⣷⣶⡆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠁⠀⠀⠀⣀⡀⠀⣀⣤⣴⣿⣾⣿⣧⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠛⠿⠿⠿⠟⠛⢛⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣔⣳⣶⣤⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠟⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠛⠀⢀⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⣈⣤⣤⢿⡇⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⠿⢋⣽⠇⠀⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⡀⠀⠀⠀⠀⠀⡏⠉⢰⣤⠰⡟⠀⣶⠀⢠⣷⡀⡀⢄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡆⠀⢀⣴⣷⡄⠀⠀⠐⣷⣶⠻⠿⠇⠁⠄⢹⡇⣾⣿⣿⠁⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠆⠀⠁⢁⠈⣿⡟⠀⠀⠀⠈⠁⡇⠀⡆⠀⠀⡀⠸⣧⣻⢿⢿⠃⣀
⠀⠀⠀⠀⠐⠒⠂⢀⣁⣌⠘⠆⠀⢀⣀⠀⠆⠀⠈⠄⠀⠀⠀⠀⣡⡄⢁⣄⠀⠠⠰⣼⣏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⢦⡀⠛⠐⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠈⠂⠀⠁
⠀⠀⠀⠀⠀⡀⠠⠓⠠⡄⣾⠇⡀⢠⣄⠀⠈⠀⡀⠀⢀⡶⢄⠀⠀⠀⠀⠀⢩⣿⡗⠲⠷⠶⠃
⠀⠀⠀⠀⠀⠠⢭⡀⠀⢱⠘⠃⠀⠀⠁⠀⢸⣤⢀⠀⠋⠀⠘⠀⠀⠀⠀⠠⣾⡋⠀⠀⢠⣀⠀⠀⣠⣤⣠⡶
⠀⠀⠀⢀⣀⣌⠀⢱⣾⣿⣶⣞⢦⠀⠀⠀⢈⠻⠀⠀⢄⠐⠂⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⣙⣂⡀
⠀⠀⠀⠀⠫⣷⠄⠀⠏⣩⣭⣭⠎⠁⠀⠁⠈⢱⡀⠠⡀⠉⠂⠁⠁⢀⡦
⠀⠀⠀⠀⠀⠀⠈⠆⠉⡁⠀⠉⠳⣦⣄⣤⣦⠀⢀⠔⠈⠀⠠⡄⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠈⢴⣿⣤⣤⠓⠂⣿⣿⣟⡈⢡⡄⢰⠀⣀⠳⣂⡄⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⢻⣶⣰⢿⣟⠹⣇⣀⠀⠤⡏⠓⠀⠈⠙⠹⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢢⡐⠛⢭⣓⠈⠉⠀⠀⠀⠀⠀⢿⡅⠄⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⣧⣀⠘⠦⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⢋⡙⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡀⠁⠀⠀⡅⠀⠈⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠁⠄⣀⡀⡇⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠇
⢀⣴⣶⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠋⠁⠀⠀⠈⠳⣄⠀⠀⠀⡀⠀⠀⣀⡴⠋
⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠷⠒⠛⠉
⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣂⣠⣴⣀
⠀⢸⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣟⢻⡄
⠀⢸⣿⣿⣿⣿⣷⣤⡄⢀⡀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⠏⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⣠⣶⣶⣦⣤⣶⣾⣿⣷
⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣸⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⣰⣿⣿⡄⠀⢀⣄
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠁⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⣿⣿⣿⣀⣴⣿⣿⣿⣶⣶⣶⣷⣷
⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢹⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠹⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢨⠠⣤⣶⣈⡉⠖⠂⠄⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠴⣿⣿⣿⢤⣉⡑⠂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣿⣿⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣏⡡⢭⢑⣾⣿⣿⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⣿⣿⣿⣿⣿⣿⣯⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡧⣴⣶⣾⣿⠛⠻⠧⡇
⠀⠀⠀⠀⠀⠀⢰⣾⠿⠟⠛⠋⠉⠉⠉⠉⠉⠛⠻⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡗⢻⣿⣿⣯⣊⡑⢚⠇⣿⣿⣤
⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠆⠀⠀⠿⠷⠿⠿⠿⠿⠶⠿⠊⠛⠟⠋
⡿⢿⠿⠿⡿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣛⣿⣟⣻⣿⣿⣛⣻⣻

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⢄
⠀⠀⠀⠒⠉⠘⠀⠀⠀⠰⢄⣀⠈⠀⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠
⠃⠀⠁⠐⠤⢀⠁⠀⠀⢀⠠⠏⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⠀⠀⡄⠀⢠⠀⠀⠡⠜⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠃⢂⣴⡀⢄⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⢂⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠀⠈⠉⠉⠀⠒⠠⣤⢀⠀⠀⠐⡶⠣⣤⡄⢫⠀⠀⠀⡠
⠀⠀⠀⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠈⠁⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠃⡶⠋⠈⠙⢷⠀⢈⠶⠂⠤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠒⠈⠁⠁⠶⡀⣸⠑⠁⠀⠀⠀⢆⣁⠀⢑
⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠂⠤⠎⣅⠈⠁⠈⠢⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡖⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣀⡀⠀⠀⠀⢀⣠⡖⠊⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣿⡿⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠓⡁⢀⡉⠂⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⢀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢀⠈⠀⡸⠂⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠉⡉⠹⠿⠿⠋⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⢊⢀⡀⠀⠊⢃⡀⠔⠈⠆⠘⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠀⠀⠠⣄⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣀⠈⢀⠀⠀⢧⠀⡁⠊⠳⡀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠘⠄⠀⠀⠘⡄⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠜⣛⠀⠀⠀⠈⣓⡤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢙⠁⠀⠀⠈⢒⣀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠎⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡿⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡟⠋⠁⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢸⣿⣿⠀⠀⠀⢘⡍⢉⠀⠀⠀⠀⠀⠀⠶⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⡴⠁⠀⠀⢐⠀⠀⠀⠀⣴⡒⢢⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢳⠂⠀⠉⡠⠀⠀⠀⠀⠑⠬⠋⠀⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠿⠀⠀⠀⠐⠈⠀⠀⠀⠀⢳⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠴⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠙⠚⠉⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡴⣿⣧⡄⠀⠀⠀⡴⠋⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠓⠙⠿⠁⠀⡠⠊⠀⠀⠐⠃⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢶⣉⣳⣄⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠂⠀⠁⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠔⠂⠐⠁⠠⢄⣼⢩⣿⡟⠹⣖⡾⠋⣿⣷⣀⣀⣠⣔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠃⠄⡠⠚⠳⡤⢿⣾⡷⠁⡜⠉⡇⠀⢻⡇⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡰⠁⠀⠀⠘⢿⡃⠀⠘⠁⠀⡥⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⣊⠉⠉⠓⠚⠣⠤⣄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡕⢠⡦⣄⠀⠘⣫⠈⣢⣄⡀⠀⠀⠢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⠊⣤⠞⠁⠈⠦⠟⠁⠉⠀⠀⠉⠙⠓⠛⠻⣽⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣉⡛⠛⠋⢉⠗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠾⠦⠯⣭⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠑⠄⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠄⠀⠀⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡂⠀⠡⠂⠀⠄⠀⠀⠀⠀⠠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⠄⠞⠂⡑⢄⠀⠀⠀⠀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠈⡀⢀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⠈⠢⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⠄⠈⠕⠀⠀⠀⠀⠀⣬⡤⠴⠓⠀⡐⠀⠀⠠⠑⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⡑⢄⠂⠀⠀⢈⠢⡀⠢⡑⣌⠢⣀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⠀⠊⠁⢈⣣⡚⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠁⠁⢀⠀⠋⠜⠂⠈⠑⠶⠈⢢⡄⠀⡀⠤⣖⡱⠐⠀⠀⢀⠀⠰⡣⠀⠀⠨⠓⠁⠀⢀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⢌⠢⢃⠂⢄⠈⠀⠀⠒⠲⠦⣤⣤⠖⠋⠁⠰⠂⠁⠘⠛⠛⠃⠁⠀⠀⠁⠘⠁⡤⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢡⣌⠓⡀⠄⠐⣊⣤⠗⠋⡙⠦⢰⢤⠂⡿⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡆⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⢋⡡⠶⡛⠁⠀⠀⣚⣅⢀⡉⡆⠰⡏⠀⠀⠀⠉⠉⠉⠉⠙⢻⠀⢸⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠡⠂⠀⠀⠀⠢⡙⢯⠀⠈⠁⠀⠀⠀⢠⣘⣇⠀⠀⣠⣤⡀⠀⠀⠀⠌⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠄⠀⠀⠀⠰⡏⠀⠀⣹⣿⠁⠀⢙⣷⣤⣤⣤⣤⡞⢠⠍⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠁⠈⠑⠒⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠉⠉⠉⠀⠀⠀⠀⠁⠄⠠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠀⠐⠆⠀⠀⢣⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⡆⠀⠀⠀⠐⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠄⠈
⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠐⠖⠀⠀⣀⡀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠠⠀⠀⡔⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⡠⠤⣀⠂
⠀⠀⡐⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢐⡰⠒⠀⠁⠀⠄⠀⠀⠀⠢⢄⡈⠀⢀⣀⡠⠊⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠐⠒⡒⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⡠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠊⠀⠠⠄⠐⠂⢐⠆⢀⠠⠗⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠄⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠙⠓⠂⠀⠀⠀⢀⡄⠀⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠉⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠤⠤⠤⠌⠀⠀⠤⠄⠀⠤⠤⠀⠂⠂⠩
⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⠶⠒⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢶⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢶⠃⠀⠸⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣄⠀⢶⣄⣀⡸⠛⢢⣤⣠⣴⣛⡦⣤⣤⣼⢳⣦⣤⠟⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⢀⠀⠀⠀⠀⢸⠙⠋⢠⡤⠔⠊⢁⣀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠉⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⡉⠭⢦⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠉⠉⢱⠖⠚⠁⠀⠐⠮⡠⢄⡀⢀⡀⠔⠀⠀⠢⠤⠀⠀⠀⣠⠖⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠈⠁⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣈⣵⡤⠇⠀⠀⠀⠀⠀⣀⡼⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁⣄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⡴⠛⣇⠀⠀⠀⣰⣧⠞⠀⠙⠒⠦⢬⣭⣉⣉⣻⠉⠉⠉⠉⣉⣀⣀⣀⣠⡼⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠈⠛⠛⠛⢡⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⢰⡖⠋⠉⠉⠇⢚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢧⠀⠀⠀⢸⡼⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⣇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣨⣦⡀⠘⠓⢦⠤⠼⠷⢤⡾⢻⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡤⠚⠉⠀⠀⠀⠙⠛⠛⠋⠙⠛⠒⠚⠁⡞⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⣀⢀⣊⣀⠀⢅⠀⣸⠀⡀⠀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⢲⣤⣾⠃⣔⣰⠘⠱⣄⡀⠈⠣⡀⠄⠀⢀⠀⡌⡠⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣏⠎⡞⠏⠀⠀⡷⠎⠀⠘⢿⠆⠀⠀⠂⠙⠀⠀⠊⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠥⠛⣖⠂⠠⢀⠀⠈⠁⠇⠀⠀⠠⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡆⠀⡖⠁⠀⠀⡨⠃⠀⠀⠄⠀⢰⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠃⣰⠇⣀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠑⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⡀⠈⡕⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠘⣄⣧⠖⠂⠀⠀⠀⠀⠈⢁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣠⣤⡼⠀⠀⠀⠀⠀⠀⠀⡄⢈⡄⠥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠚⠟⠠⠀⠉⠛⠀⠀⠀⠶⠶⠆⠀⠁⠀⠀⡀⠠⠄⠂⠉⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠚⠓⠒⠂⠀⠀⠀⠀⠀⠐⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠂⣄⡀⠀⠀⠀⠀⠀⡆⠀⡶⠖⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⣶⠏⣀⢆⡁⠠⡀⠀⠀⠀⠀⠘⠁⠀⢇⠀⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡄⢷⡶⠊⢁⠤⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡖
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡎⢀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⣏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠜⠂⠲⠿⠓⠶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠔⠛⣈⡗⠐⢢⠒⡚⢦
⠀⠀⠀⠀⠀⡀⢒⣿⢤⡅⡀⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠐⢆⠉⠣⡤⢄⠁⢸⠋
⠀⠀⠀⠀⠀⠀⡎⠀⣸⡁⢀⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠈⢿⣀⠀⢡⠀⣁⠸⢂⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠶⠚⠓⠀⠀⠉⠉⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢣⡙⢄⠀⠈⢻⡛⠛⠛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠈⡀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⡃⣇⠀⠀⠘⣧⠀⣀⠄⠀⠀⠀⠀⠀⠀⠁⠑⠀⠤⠤⣤⣤⣴⣶⡿⠉⠀⣹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢌⣿⡅⡇⣠⣔⠒⠛⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⣠⣾⡿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡆⠀⢩⣞⣅⡀⡀⢤⠠⠀⠀⡀⢀⣴⣶⡿⠟⠁⠀⣠⣾⣿⣿⢣
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⠹⣿⠯⠐⠚⠛⠛⠟⠛⡷⠿⢷⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⣿⣿⡟⣾
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⠀⠀⣀⡀⣔⠀⠀⠀⠀⠀⠁⢀⢀⣀⢀⣴⣶⣿⣿⣿⣿⣿⠃⠨
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣟⣒⠀⢀⠀⠀⢠⣯⣷⣿⣾⣿⣿⣿⣿⣿⣿⢏⢄⡇
⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢛⣉⣙⠛⠛⠛⣛⠋⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⢿⡼⠁
⠀⠀⠀⠼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣷⡶⠦⠾⠿⠶⠷⠳⠤⠴⠤⢤⣤⣴⠷⠋⢵⡟
⠀⠀⠀⠘⠻⠻⠿⣿⣿⡟⣿⣿⣿⣿⣷⣄⡀⠹⣶⣤⣴⣤⣤⣄⣄⣠⣤⣤⣤⣤⠤⢤⣀⡀⣼⠛⠓⠚⠓⠒⠛⠛⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣦⣌⠛⠛⣟⠿⠿⠛⠿⠻⠿⢿⣶⣶⣤⣬⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠤⢠⠄⢠⣄⣀⣀⠀⢀⡁⣁⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠉⠁⠈⠉⠁

⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣶⡀
⠀⢰⠆⠀⠀⠀⠀⠀⠹⣿⠀⠀⠄⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⢰⠆⠀⠀⠀⠀⠁⠠⣶⡿⠁
⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟⠀⠀⠈⢃⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⠉⡼⠁⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟
⠀⠀⠈⠙⠺⡤⡴⠚⠁⠀⠀⠀⠀⠈⢳⣴⡿⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠘⢾⢿⡞⠀⠀⠀⠀⠈⠙⠺⠤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⡉⠁⣀⡀⢊⠀⠰⠉⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⢿⡟⠚⠖⠁⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠁⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆
⠀⠀⠈⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⢀⡴⠂⠁⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡀
⠀⠀⠀⠀⠠⠈⠁⢀⠀⠀⠀⠀⠀⠄⠠⡤⠄⢤⡤⢠⡄⠶⠀⠛⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠡⣤⠀⠈⡄⠀⠀⠀⡧⠔⠒⠗⠒⠒⠳⠼⠧⠤⢀⣥⣿⣦⠀⢴⠆⠀⠀⢠⣶⣄⠀⠀⠿⠀⠀⠀⠀⠐⠠⠄⡀
⠀⠀⠀⠀⠀⡩⠋⣑⣷⡀⠀⠀⠁⠤⡴⠃⠂⠂⠁⠀⠀⠀⠀⠀⠉⠫⡀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⢠⣿⣷⠀⣶⠀⠈⠐⢄
⠀⠀⠀⠀⠠⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢶⠄⠀⠀⢰⡦⠀⠀⠈⠛⠋⠀⠀⠀⣠⣤⠀⠐⢄
⠀⠀⠀⠀⠸⠟⠐⣶⣾⠁⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⠃⠀⡀⠁
⠀⠀⠀⠀⣀⣀⡠⠛⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⢀⣠⡜⠀⣿⣿⠂⢀⣄⠀⢠⣾⣦⠀⠀⠛⠁⠀⠀⢀⣄⠀⠀⠘⠋
⠀⢀⣒⠁⣶⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⠈⣥⢀⡈⠁⠀⢈⡉⢀⠈⠿⠏⠀⣀⠀⠀⢀⣤⠀⠉⠐⠀⣤⠈⡀⠀⡀⢀
⠀⢿⣿⡇⠀⠀⠄⠀⠀⠀⠛⡤⠇⣀⣀⣀⠀⣀⣀⣀⡀⢀⠀⠀⢀⠀⠰⡎⠉⠁⠉⠀⠀⡀⠝⠒⠄⠀⠛⠀⠀⢿⣿⠇⣴⠁⠀⢈⡤⠚⢈⣵⣦⠀⣨⡄⢀
⡆⠈⠛⠡⣦⠀⠐⡀⠲⠀⠀⠈⠐⡆⠠⠄⢀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠒⠲⣤⠦⠀⢠⣾⠀⠀⢀⢀⡠⠀⠀⠠⢤⣥⣴⠁⠀⢀⠋⢴⠆⠘⢿⠟⠀⠈
⠇⠿⠂⢰⣿⣧⢠⡀⠀⠁⠀⠀⠀⠈⢠⠀⢠⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⢠⠟⠁⠰⢲⠟⠉⠁⣿⡄⠀⠀⢠⣤⡀⠀⣠⡄⠀⣰⣶⡄
⠀⢄⠀⠈⠛⠃⠈⢁⡀⢀⠀⠀⠀⠀⣨⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣬⠏⠀⠀⠀⣿⣧⢀⠀⢻⣿⠇⠈⠀⡀⠀⠹⠿⠃
⠀⠀⠈⠐⠀⡄⠠⣿⣿⡆⠀⢀⡈⢁⣸⠠⢥⢀⠑⠢⣀⠀⠀⠀⠀⢠⠠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢩⣿⣾⢰⡆⠀⠀⠀⣿⣿⡄⠀⣦⡠
⠀⠀⠀⠀⠀⢘⠀⣉⠉⠙⣦⡙⠁⠀⠊⠀⠉⢚⣴⣶⡄⠱⠆⠀⠀⢲⡗⢦⠅⣀⠀⠀⠀⠀⠀⣀⣀⡠⡧⠴⠐⠂⠈⣿⣿⣍⠀⣠⣤⠀⠀⠙⠋⡶⠈⠉
⠀⠀⠀⠀⠀⢸⠀⠛⠀⢸⣿⡇⠀⣶⢼⣶⠂⠀⠹⠿⠇⢠⣾⣷⣀⣀⡀⣰⣶⣦⠉⠿⠀⠉⠈⢿⡿⣀⠀⢰⣿⣷⠀⠛⠋⣶⠀⢿⣿⠇⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠉⣁⠚⠛⠃⠠⠤⠊⠈⠂⢺⡇⠀⠀⣴⣦⡀⠀⠟⠀⠘⠿⠋⠀⠀⠰⡷⠀⠀⠀⠛⠁⠈⠻⠋⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠄⠒⣁⠀⠀⢸⡏⣿⡀⡄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠆⠜⠛⠛⠿⠥⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢁⣀⠄⠀⠀⠀⠀⠀⠀⣛⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⣶⣾⣿⣿⣿⠶⠀⠀⠈⡘⣻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣍⣻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⠃⠙⠈⠳⣍⠉⠙⠛⠛⢛⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣹⣤⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣯⣽⣿⣿⣯⣿⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⣭⣿⣿⣿⣷⣬⡆⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣠⣴⣿⣶⣶⣶⣤⡄⠀⠀⠀⠀⠈⠉⠁⠀⠙⡿⠿⣿⡇⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⣤⣴⣶⣶⣾⣷⣶⣤⣹⣿⡿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢸⣯⢸⣿⠏⢿⣛⣛⣻⣿⣿⣿⡆⠺⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠈⠛⣾⠛⣿⢿⡟⢹⣿⣿⠁⠀⢸⣿⣏⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠈⠐⠈⢁⡿⠛⡏⠀⠀⣿⣿⣏⡙⠛⠋⣸⣿⣿⣿⣿⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣻⣿⣿⣦⠉⠛⠻⢿⣛⣿⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣏⡀⠁⠀⠀⠀⠀⣹⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠄⡀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣸⣿⡟⣷⡄⠀⢀⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠘⠀⠀⣿⠷⣶⣤⣀⣠⣴⣿⣿⣴⣿⡇⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⣿⡶⠄⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠸⢀⠀⣼⣏⠠⠤⠤⠤⠤⣤⣤⣄⣈⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡼⢿⣭⡟⠁⠀⠂⢤⣤⣤⣤⣄⣠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⢀⣄⠁⠀⠀⠀⠀⢐⠃⠀⠙⠳⣤⢀⣀⠀⠀⣼⣍⢿⣿⣿⣿⣿⣿⠁⢻⣇
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠛⠀⢤⣤⣴⣶⣾⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣾⠈⠄⠀⠀⠉⣺⣿⣿⣿⣿⠇⠀⢈⡏⢷⣄
⠀⠀⠀⠀⠀⠙⠛⠳⠶⠾⢷⣿⣄⣈⣿⣿⢿⣿⠇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⠇⠀⠀⠰⠇⠀⠙⠦⢤⡀
⠀⠀⠀⠀⠲⢦⣤⣄⣀⣠⣇⡀⣈⡉⢹⡇⣸⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⢰⡟⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⡀⠀⠉⢉⡉⣉⣙⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠚⣿⡿⠛⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠙⠛⢷⡄
⠀⠀⠀⠀⣈⠉⠛⠿⣿⣿⣛⠋⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⣸⡏⢀⣦⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠈⠛⠲⠾⠿⠿⢿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⢠⠿⢧⣶⣤⡤⢐⣿⠏⠁⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠿⠟⣠⡾⢁⡄⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⣿⡟⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣩⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣾⡿⠟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣩⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣡⡾⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⢿⠟⠙⠿⢿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣻⣟⣿⣿⣿⣟⣛⣛⣛⣛⣛⣻

⢿⣍⣉⣉⣉⣉⣉⣉⣹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⠋⢹⠉⢹⡟⢾⣝⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣸⣇⣞⣠⠟⢉⣷⣻
⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣛⣉⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⣾⡿⣭⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠜⠁⠈⢿⢦⣈⡽⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡤⠉⠀⠀⠀⠈⠳⣟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⠎⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⡿⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢿⣿⣿⣿⡿⠋⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠛⠛⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠀⢸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠋⠉⠉⠁⠀⢹⣀⣸⠀⠈⠉⠉⠉⠓⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠿⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣭⠒⡋⢱⠀⠀⠀⠀⠘⠁⠀⡙⣥⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠀⢀⠼⠄⠀⠀⠀⠈⢦⡀⠀⠀⣀⣤⠤⠤⠤⣄⣤⠀⠀⠀⢠⠃⠀⢀⣘⠃⡌⢳⠀⠀⠀⡞⠁⠘⠟⠀⢒⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠯⠶⠼⠅⣅⣠⠋⠀⠈⠉⠃⠸⠀⠀⠈⠛⠀⠔⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣛⢙⠉⠀⢻⡰⠂⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠸⠤⠄⠀⣠⠶⠋
⠀⢀⣀⣤⡼⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⡠⠔⡏⠨⠇⠀⠀⡖⢨⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⢡⠁⡄⠁⠀⠀⠀⠀⠀⠀⡀
⢀⣼⠀⠀⠀⠀⠀⢀⣀⣀⠀⠺⢤⡀⠀⣌⣀⣀⡂⠈⠅⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⢣⠀⢨⠆⠀⠀⠀⠀⠀⡼⠁
⠛⠢⠤⠖⠉⠙⠛⠃⠀⡠⠤⢤⠴⠛⠋⠉⠁⠈⠹⡄⢙⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠃⠀⠀⠀⢰⠋
⠀⠀⠀⠀⡀⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠆⠀⠀⠀⡜
⠀⠀⠀⢮⡁⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣠⠤⠤⣶⠫⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢐⣊⠤⡀
⠀⠀⠀⠀⠻⠤⠶⠚⠛⠚⠋⠀⠉⠁⠀⠀⠀⠀⠈⠙⠣⢿⣄⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠙⠚⢢⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠲⠖⠓⠲⠶⠤⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣲⣞⣛⣶⣴⣙⣓⣒⣋⣉⣉⣈⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠂⡄⠰⢄⣴⣯⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢻⣿⣿⣿⣯⣿⣿⣿⣿⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠀⠛⠙⠯⠿⠽⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁

⠀⠀⠀⣠⣤⣾⠃⠀⠀⢀⣰⠟⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠀⢀⣶⣖⣶⣆⢠⡀⠀⠀⠀⠀⢀⡀
⠀⠀⣼⣿⡿⠁⢀⣤⠶⠛⡁⠄⠒⠛⣿⡿⠃⣀⡤⡶⠦⣀⣺⠶⠚⢉⠁⠀⠀⣉⣯⣟⡻⠶⣄⡰⡄
⢀⣼⡿⠋⢀⡶⠋⠀⠀⠁⠁⠀⠀⠈⠙⢀⣢⣭⣿⡐⢛⣋⡄⢐⢎⡴⢒⡽⣚⡵⢋⡥⣒⠾⢛⣻⡄
⣾⡟⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣫⡵⢋⣴⢾⡫⠅⡰⠃⡾⣣⡟⢱⡷⡐⢉⡞⣡⣾⡿⡋⣧⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠏⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠣⣢⠽⣾⣁⡐⣚⠁⡸⢥⣏⢰⡏⣰⠣⢸⢰⠟⢋⠄⢠⡼⣿⣂
⠀⢀⣿⡇⠀⣤⠀⠀⠀⠀⠀⠀⢀⠟⠀⠐⣱⢞⡟⡱⣱⡇⢰⠃⠀⣡⠸⢰⢧⣤⢰⢀⠄⡀⠀⢀⡴⣿⢽⡰
⠀⣼⣿⠁⠀⣿⣆⡀⠀⠀⠀⡄⡛⣀⢂⣰⠷⠊⣴⣣⠟⣡⢇⡀⢰⢇⠀⣎⣟⡿⡿⠈⣴⠃⠂⢠⣆⣿⣷⠹
⢠⣿⠃⠀⢠⣿⣿⠛⣦⡀⠀⠀⢚⣦⠌⡋⠠⡠⠓⣫⣰⢫⠎⠜⣠⡎⢸⣹⠻⣱⠇⠀⣿⣷⠁⢊⢎⣯⣤⠐⠆
⣼⠃⠀⣠⣿⡿⢁⣾⡿⠋⢓⣶⠈⠀⢨⠁⢨⣰⢰⢗⢣⡍⡀⠀⠟⡄⠠⢧⠄⡃⡌⠀⢻⠋⣤⠆⠈⠛
⡏⠀⣴⣿⡟⢀⣾⠏⢀⣴⣿⠏⢠⡰⢸⡇⢎⡏⣼⢊⡴⢳⢡⡄⠾⢁⢖⠄⡐⠸⣵⡴⡼⢚⡄
⢇⣼⣿⣿⠠⠾⡋⠀⣼⡟⠁⠀⠀⠀⠸⢏⡜⠰⢃⣼⢡⢋⢫⠔⠰⠁⢀⡾⢿⢹⢏⠍⠀⠀⡀
⠀⣿⣿⣿⡄⠀⠀⣾⡏⡠⠀⠀⠀⠀⠀⠰⡆⠀⠘⠇⢣⡏⠀⠀⡰⢠⣼⡇⠾⠀⡌⠀⠀⠀⠂
⡀⠈⢻⣿⣷⣀⠀⡿⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢫⡀⠇⠁⣿⠰⡁⢸⠀⠀⠁⡀
⣷⡀⠀⠙⠿⣿⣷⣞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠋⠇⠘⢰⡜⢾⠁⠀⠀⠀⠀⠉
⣿⣿⣦⡀⠀⠀⠉⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠀⠃
⡘⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣌⡛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠻⠀⠀⠀⠀⠀⣀⡩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉⠁⠟⠀⠀⠀⠘⠠⢤⣤⢤⠜⠃⠀⠀⠙⠃⠀⢠⠴⣯⠈
⠀⠀⠀⠀⠀⠈⢆⠀⡬⠁⠀⠀⠀⢰⡦⠤⠀⠀⠄⣴⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⢻⡾⠿⣀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠘⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠐⠺⠉⠀⠀⠀⠀⠧⡀⣈
⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠖⠀⠀⠀⠀⠀⣓⡀⠀⠀⠀⠀⠷⠆⢀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⢩⠙⢢
⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠊⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢏⡇
⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠶⠿⠶⠦⣴⡀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠈⢄⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⢀⣞⢑⡀⠀⠀⣈⣱⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⢻⠃⠀⠀⠈⠀⠄⢀⢠⠀⢸⠀⠀⠀⠀⠁⠲⡄⠊⠁
⠀⠀⠀⠀⠀⣨⡉⠲⠅⠠⠀⢨⣟⠀⠀⠆⠀⠀⠀⠐⣴⠀⡜⠀⠀⠀⠀⠈⠁⢰⡀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢹⣇⣀⣀⡀⠀⢸⣿⠀⠄⠘⠀⠀⠀⠀⠛⠀⠃⠀⠀⠀⠀⠀⠰⠐⠤⠚⠀⠊
⠀⠀⠀⠀⠀⠀⠙⠦⠧⠤⠤⠼⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠚⠀⠀⠀⠀⠀⠲⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠉⠁
⠀⠀⣼⠷⢦⣄⡀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡀⠀⠀⠀⠀⠀⢀⢤⡠⠄⠤⢄⠀⠀⢠⢊⠋
⠀⠀⠀⠀⠀⠈⠛⢗⢄⠀⠀⡇⠀⢱⠀⡀⠤⣀⠀⠀⠀⠀⠀⣏⣉⠉⡇⠀⠀⠀⠀⠰⡅⠀⠑⣴⡄⠀⠑⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡕⢤⡃⡠⢼⠉⠀⠀⠀⢑⠖⠢⢄⣸⡀⣩⡧⠧⠒⠊⠉⠒⡎⠀⠀⢀⣩⠉⣶⠎⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⢌⡐⠪⢅⣢⡎⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠺⠵⣴⣝⣢⣁⠔⠁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠑⠤⣈⡏⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠉⠐⠂⠀⠤⠦⠤⠤⠤⠤⠒⠂⠉⢛⠉⠉⢹⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠤⠄⠐⠂⠉⡠⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣀⣸
⠀⠒⠒⠂⠒⠂⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⣿⣿⣿⠟⠻⠃
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣻⣟⠋⠏⠇
⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠶⣤⠄
⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⠋⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠘⣿⣿⣿⡏⠫⡍⠛⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⢃⢀⣎⡁⡹⡤
⠀⠀⠀⠈⢿⣿⣧⣤⠳⠄⠀⠀⠀⠀⠀⠀⠀⠈⡙⠚⠁⠐⠢⢼⠿⠀⣤⣤⣤⡠⠈⣠⣤⣏⣿⣇
⠀⠀⠀⠀⠀⠙⣿⡟⠀⠐⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢔⠧⠤⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠇⡀⠀⠀⠀⠀⠀⢀⡿⡆⠱⠒⠒⠒⠉⠉⠂
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⣀⠀⠀⠀⢡⠘⡷⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠑⠢⣠⣌⢆⠻⣮⣓⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠛⠶⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠓⠦⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡤⢤⣀⣀
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠁⠉⠳⡄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⡄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀
⠀⠀⠀⣏⢀⣤⣠⣤⣤⣠⣤⣤⣆⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⡸⠁⠀⠀⠀⡀
⠀⠀⠠⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠙⠶⠤⠼⠋⠧⣄⣀⣤⠞⠁⠈⢹⠁⠀⢀⡇
⠀⠀⠀⢙⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠀⢠⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢣⣤⣤⣄⣀
⠀⠀⠀⢠⣿⢠⠓⢦⠠⢀⠀⠒⣆⣀⣠⠐⠂⢲⣄⠀⢀⣀⠤⢤⡤⠔⠢⠐⠆⢸⡇⠄⢸⠁⠀⠈⠉⠳⣦
⠀⠀⠀⢸⣿⢸⠀⢈⠀⠀⠀⠀⠘⠿⠋⢰⡄⠘⠿⠿⠋⠀⠀⠀⡆⠀⠀⠀⠀⢸⡇⢀⡿⠒⠲⢦⡀⠀⠈⠳⣄
⠀⠀⠀⢸⣿⢸⠀⠘⢄⡀⠀⠀⠀⢀⡴⠃⠳⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⢸⡇⣸⠁⠀⠀⠀⠙⢦⡀⠀⠘⢧⡀
⠀⠀⠀⡚⣿⠀⠀⠀⠀⠈⠁⠁⠉⠁⠀⠀⠀⠈⠓⠀⠀⠔⠚⠁⠀⢲⣦⠀⢸⢸⡇⢿⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠳⡄
⠀⠀⡠⠋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⠈⣧⠈⠳⡄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⣆
⠀⣸⠁⠀⡇⠀⢠⡾⠋⠙⢲⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁⠀⠀⠀⠈⠀⣿⠀⠀⢧⡄⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣆
⠀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡆⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢹⡄
⠀⠘⢦⢰⡇⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠃⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣇
⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠛⠛⠁⠀⠀⠈⠑⠢⣄⢠⢸⡇⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡿
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢸⡇
⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⠂⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⢀⣤⠶⠶⣤⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢄⠀⠀⠀⠀⠀⢀⡜⢡⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⢀⡴⠛⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣤⣶⠃⠀⠑⠲⠤⠤⠖⠋⣀⡴⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀⣴⠟
⣼⠁⠀⠀⠀⠀⠈⠑⠂⠤⠄⢀⡀⢀⣀⣀⣀⣀⣉⠉⠁⣀⠀⠀⠤⠄⠒⠊⠉⠀⠀⠀⠀⠀⢹⡆⠀⢀⣠⣴⠾⠋
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠓⠛⠛⠉
⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠉⠻⣷⡶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢦⣬⣭⣭⣍⣭⣍⣭⣭⣭⣭⣤⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣤⡅⣦⡍⠆⠀⠀⠀⢂
⣠⠔⠀⠉⠈⠁⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠉⠒⠀⠀⠀⠀⠈
⠣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⣴⡆⠄⠀⠀⡜⠀⠄
⠀⠑⡀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠙⠁⠀⠀⠈⠃
⠀⠀⠘⡄⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢀⠔⠒⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠉⠓⡀⠀⠀⠸⠀⡀⡇⢄⢀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠄⠁⠈⢑⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠈⠀⠐
⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠤⠀⠀⠠⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⢠⢈⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⣀⣀⡤⣊⡁
⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠈⠉⠉⠁⠈⠐⠂⠤⠤⠼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣶⣦⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢁⣀⣈⡉⣀⣶⢤⡐⠦⡤⠤⠤⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠀⣨⣿⢭⠳⣤⡼⠀⢙⣷⠀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠠⠒⠎⠀⠤⠤⠠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠁⠈⠐⠳⣌⣑⣦⠪⡾⠀⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠂⠀⠀⠀⢠⠀⠄⢸⠀⠫⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠀⠀⠈⡿⠀⡐⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⣁⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⠤⠀⠀⠀⡠⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⢠⣀⣀⣀⣀⣀⣈⢸⠀⠰⢸⡇⠀⠀⠀⠀⢠⡤⣤⣤⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⢀⢀⡣⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣄⣶⡂⠐⠀⠀⠀⠀⠀⠘⠀⢀⣺⡇⠀⣠⠔⢈⢉⣁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣠⠿⠵⠶⠶⠇⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠒⠓⠛⠓⠒⣶⡴⢶⣶⠒⠒⠓⠛⠒⠛⠊⠀⠀⠀⠑⡍⠀⠀⠀⠀⠐⠀⠀⠀⢀⠴⠊⠁⠄⡀⠀⠀⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⢼⢇⣰⣂⣀⣀⣘⣉⣉⣉⣉⣉⢓⣒⣲⣶⣶⣖⡆⠔⠲⡀⠀⡀⠀⠀⠀⠐⠂⠌⢀⡖⠁⠀⣀⡀⠀⠀⠆⠀⠀⡞⢈⡅
⠀⠀⠀⠀⢀⠀⡈⣆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⡇⠙⢞⠁⠠⢣⡀⠀⠀⠀⣀⣂⣈⡁⠀⣊⡀⣱⠀⠀⠰⢀⠈⠀⠸⠁
⠀⠀⠀⠀⠹⡭⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠁⣦⡀⠀⠀⣇⠀⠉⠁⠈⠀⠀⠀⠈⠣⡀⠀⠉⠁⡀⠠⠚⠁⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⡇⠘⠛⠉⠉⠑⠒⠴⠤⠤⠤⠤⠤⠤⠾⠢⠤⠤⡶⠥⠟⠒⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⢊⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡃⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣶⣦⣴⣶⠶⠲⡶⢲⣀⣸⣨⡗⣷⣷⣷⠤⢦⣤⠤⠤⢤⡬⢬⣤⣤⣤⣤⣀⣀⣥⣥⣼⠶⠶⢶⠾⠃⠈⠑⠂⠉⠀⠄⠒⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢺⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣇⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠌⢀⠞⠉⢧⡀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⡎⠉⠑⢦⡀⠀⡿⠀⠘⠀⠈⠀⠀⣴⠛⠶⣄⡀⠀⠁⠒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠈⡇⠀⣠⠒⠉⠙⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⢨⣿⠿⠿⠶⢤⣤⣄⣀⣁⣀⡀⠀⠀⠀⠀⠈⢢⠀⣿⠈⠉⠉⠉⠉
⠀⠀⠀⠀⢴⠉⣉⡵⠂⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠦⣀⣈⣥⣤⣀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⣦⣀⠀⠂⡏⠀⠀⠶⣉⠉⢳
⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠤⣤⠟⠁⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤⣠⠝⣧⣼⡇⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡿⠀⠀⢠⡏⠀⠀⢀⣀⣀⣤⡷⠀⠀⠀⠀⠀⠰⡧⢤⣀⣹⠆⢹⡿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢰⡇⠀⠀⠀⢿⠛⠉⠉⠈⠋⣠⠇⠀⠀⠀⣶⠀⠀⠳⢤⠴⠋⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣼⡇⠀⠀⠀⠈⠳⢤⣤⡤⠞⠃⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⠟⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣉⣧⠐⡄⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⢀⠀⢿⡀⠀⠀⠀⠀⢠⣾⣿⣷⣭⣿⠀⠀⠀⠀⠀⢀⡾⠃⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣅⣄⣀⣘⣧⠀⠀⠀⠀⣿⣘⣻⣿⣿⠏⠀⠀⠀⢀⡴⠋⠀⠀⢿⣐⠏⠉⡷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⠉⠉⢻⡄⠀⠀⠀⠀⠉⠉⠛⠿⠀⠀⠀⣰⣟⣡⣅⣀⣠⡟⠉⠛⢺⠀⢠⠛⣦
⠀⠀⠀⠀⠀⣀⣀⣀⡤⠖⢋⣩⠽⠚⠛⠻⣶⠒⠺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠉⣷⠀⠚⡟⠲⣇⡠⠏
⠀⠀⣠⠖⠋⠉⠉⠁⠀⣴⠋⠀⠀⠀⠀⠀⠛⠶⢶⡞⠻⢦⣄⣀⣀⣀⣠⡴⠛⢧⡀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠄⠐⠉
⠀⠀⣯⣠⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠉⠉⠉⠀⠀⣷⣼⣿⣄⣀⣀⣠⡴⣋⡴⠞⠁
⠀⠀⠀⢿⣠⡔⠀⣴⠤⠟⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠘⠷⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠏⡔⠙⣇⠀⠀⠀⠀⢸⡿⠛⠛⠲⠶⠤⣤⣄⣀⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣠⡟⣡⠊⢠⡴⢻⡆⠀⠀⣠⡼⢷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠏
⠀⠀⢀⣾⢋⡴⠁⣼⣿⢿⡅⢻⣄⣸⣿⡀⠈⢻⣦⣄⡀⠀⠀⠀⠑⠲⢶⣯
⠀⣰⠟⣡⠛⡄⢠⡞⠃⠀⢻⣄⣉⡽⠋⠙⠞⠋⠀⠈⠙⠳⠶⣤⡴⠾⣿⣻⠶⢦⣄⡀
⣾⣋⣞⣁⣠⠿⠋⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠆⣸⡇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⠾⣿⣾⡷⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢤⠤⠤⠍⣁⣀⣀⣰⠋⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠶⠶⢦⣤⣯⣭⣉⣗⣚⣧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢈⣉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠈⠉⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠒⣛⣓⣦⣤⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⠞⠋⢉⡟⢉⣩⡽⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⠒⣶⠟⠋⠉⢁⡴⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣰⠃⣀⣠⠴⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣉⣀⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⡤⠔⠒⠒⠒⠒⠁⠐⠚⠁⠀⠻⠹⠇⢤⠤⠤⠤⠖⠒⢂⡽⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣀⠿⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⢀⡤⠶⠒⠉⠑⢶⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣦⡄
⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠁⠀⠀⠀⠳⣀⠇⣄
⠀⡴⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⣠⠞⢁⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⠘⣆⠀⢀⠀⠀⡀⠀⠀⢀
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣠⠶⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⠄⠀⢹⡄⠀⠘⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣠⠔⠒⠊⢹⠀⣠⠞⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⢰⣘⣇⠀⠈⢧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡰⠃⠀⠀⢠⠇⠊⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠘⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡼⠃⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⢇⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠁⠀⠀⠀⠀⢀⡜⠁⡀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠃⠀⠀⢸⣤⠁
⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢧⡄⠀⠁⠀⠀⢸⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⣼⡤⠆⠀⠇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠘⢆⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠠⡿⠁⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⢷⠈
⠀⠀⠈⢣⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢦⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠁⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠻⣄⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣤⣤⣤⣤⡤⠤⠤⠘⠇⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣦⡀⠀⡼⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡜⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠻⠧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⠻⣍⠉⠑⠶⣴⠲⠶⠖⠒⢫⣉⠁⠉⠑⢦⡀⠀⠀⠉⢲⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠳⡄⠀⠈⢳⣤⠤⠤⠒⠉⠳⣄⠀⠀⠀⠀⠀⠀⡟⠀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠙⠆⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠙⠚⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠘⡆⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⡆⠀⠀⠀⠀⠙⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⠞⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣦⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢄⡀⠀⠀⠀⢸⢻⣴⢈⠈⢻⡄⠀⠐⡇⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠤⠤⠄⣴⠇⢠⡠⠀⠹⠀⠀⠀⢸⣿⠆⠗⢃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢑⣄⠀⠀⠀⠀⠀⣀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠺⣶⣺⡛⠤⡼⠀⠀⠀⠠⣿⡇⣤⣄⢀⠐⢺⠈⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⠤⢀⣈⠈⠁⢹⠋
⠀⠀⠀⠀⠀⠀⢸⡟⡛⠋⢻⠆⠀⠈⠀⠛⣹⢿⠛⠉⠀⠀⠀⠀⠀⣿⡏⣦⠈⣤⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣆⠘⢩⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣠⣖⣧⣤⣀⡄⠺⡏⠀⠀⠀⠀⠀⠀⠀⣽⡉⣄⡿⠀⢀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠁⢂⠎
⠀⠀⠀⠀⠀⠀⠀⢻⣯⡭⠤⠈⢠⡈⠉⢰⢢⠃⠀⠀⠀⠀⠀⠀⠀⣿⢷⡴⠟⠩⢾⡃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⠀⠀⠈⣧⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡿⣤⣀⠀⠶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠯⠀⠀⠀⢖⣠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⢀⣀⣀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡷⢀⡌⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⡀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣌⣄⣴⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠒⠂⣀⠶⠶⠔⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡆⠀⠀⠀⠀⠘⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⠄⠉⠉⢀⠀⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⡄⠐⠐⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡀⢀⡀⠀⣄⠈⠇⣐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⢿⡆⠀⠀⠰⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣀⣶⠒⣳⣦⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢾⣷⣄⣤⣼⢘⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⣩⠋⠁⠀⢨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢟⡉⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⢰⡧⠳⠢⠔⡀⢀⡸⠰⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡛⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⠜⠁⠀⠀⠙⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣀⣄⠄⠈⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠁⠘⠈⠺⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⢀⣴⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠇⠀⠐⢢⡀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠒⠒⠒⠛⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠉⠚⠂⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠂⠀⠈⠀⠠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠠⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⢹⠄⠤⢀⠄⢳⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠿⠄⠤⠀⠀⠓⢃⠀⣸⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠈⢆⠐⠂⠘⠚⢤⣤⠠⠤⠔⠁⠘⣧⠯⠀⢹⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢺⣧⠘⣄⡀⠀⠀⣠⡿⠀⠀⡘⣿⢠
⠀⠀⠀⠀⠀⢀⣴⠻⣷⣄⠈⠻⢷⡘⣿⡼⢋⣭⡆⠀⢰⡇⣿⡄
⠀⠀⠀⠀⠀⣿⡹⣧⡙⣿⡟⠀⠀⠉⠀⠑⠿⣿⡇⠀⢼⣇⢸⡇⠆
⠀⠀⠀⠀⠀⢿⣱⡜⣷⡌⢿⣷⣄⠀⠀⠀⠀⠀⠁⠀⠘⢿⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠈⢿⣽⣮⢿⣆⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡈⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⣾⠷⣻⣧⡙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠆⠀⠀⠀⠈⠀⠀⠈⠀⠀⠒⠀⡖⠠⠠⢴⠐⠂⠈⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡟⣝⢷⣌⢻⡄⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉⠑⠤⣤⣤⣤⠼⢦⡀⠈⣤⣶⣆⣠⣞⣁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣻⣿⣦⠻⡀⠀⢀⡾⠁⠀⠀⢀⣤⣴⣮⣥⣤⣤⣤⣄⣀⡀⠀⠀⠀⠘⠛⠿⠀⣷⣦⣤⡜⣻⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣷⣷⣠⣾⡿⢷⡶⢿⣿⣿⣿⣯⣭⣽⣿⣛⣛⡿⠿⠷⣶⣤⣬⣽⡛⢾⣿⠠⠛⠉⠉⠙⠒⠤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠾⠷⡾⣧⡼⠿⢼⣿⣸⣿⢸⣷⠉⣾⡏⣻⣟⣛⠷⣶⣶⠙⡿⠃⠈⠀⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢰⡶⠂⠀⢰⡿⡵⢞⡪⠓⠋⠁⠀⠀⠀⠀⠀⡀⢉⠉⠓⠛⠣⠿⣇⣽⡏⡸⠟⠁⠀⠀⠀⠠⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⣸⡥⠊⡡⢤⠇⣢⡀⢀⣰⠀⠠⠙⠙⠉⠑⠀⠀⠀⠀⣸⡟⠉⠀⠀⢀⡠⢀⡥⠚⠁⠀⠀⡄⠀⠆
⠀⠀⠀⠀⢀⡸⠀⢀⡀⢶⠸⠟⠏⡾⢠⢗⡀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢌⠟⠀⠀⡀⢔⡡⠞⠁⠀⠀⠀⠀⠀⠊⠀⠁⠶⡄
⠀⠀⠀⢆⠈⠀⠙⠨⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⡶⣀⠐⣞⢀⡠⢊⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠛⠳⡌⠛⠛⠹⢸⡿⣫⣞⢡⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣩⠞⠋⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⣦⣤⣤⣦⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⠘⠛⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⠒⠚⢃⡼⠇⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠻⣤⣼⠀⠀⠀⠀⢠⡶⢆⠀⠀⠀⢀⡴⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠲⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠶⠶⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠙⠋⠀⠀⠀⠛⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣀⡠⠶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣀⠀⠀⠀⠀⢾⡁⠀⣳⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠙⠚⠁⠀⠀⠸⣇⡀⣹⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢾⣭⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠸⣇⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢿⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠀⠲⢤⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠘⠶⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⠟⣶⣤⣀⣿⠀⣰⡾⠛⠛⠛⠳⢦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠛⠛⠛⢉⣽⠇⣸⠏⠠⢤⣀⠀⠀⠀⢱⡄⢸⡇⠀⣴⠞⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠤⠀⠤⠐⠛⠁⠀⣿⠀⠀⢸⣿⡟⣷⣤⣼⠇⣾⠀⣾⠁⡼⠋⣍⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⠀⠐⠒⠒⠒⠒⠲⣤⡀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠘⣇⠀⠀⠙⠛⠋⣠⡟⢠⠟⠀⠻⣄⠻⠆⠜⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠉⠓⠲⠒⠉⠉⠉⠰⠶⠶⠚⠉⢠⡟⠀⠀⠀⠈⠑⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣰⠟
⠀⢀⠀⠀⢀⡄⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠚⠁
⢠⡎⠀⢰⠏⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠋⠁
⠀⠁⠀⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢻⡌⠛⠳⠶⠦⠀⣤⣤⠄⢤⡶⠶⠖⠛⠋
⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣶⠀⣽
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠈⣧⠀⢀⠟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⠳⣾⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡍⠉⠉⣷⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⣠⠾⠛⣻⡷⠶⠾⣧⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠀⠛⠁⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠋
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠋⠓⠒⢦⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣷⡀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠶⣤⣀⣀⣀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⡀⠉⠉⠉⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠰⣆⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢰⡟⠀⠀⠀⡟⠉⠒⠒⠒⠢⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⣆⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣼⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⢷⡀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠈⢷⠀⣼⠃⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠉⠉⠙⠛⠒⠤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⡄⠙⠻⠶⣦⣄⡀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣀⣠⡴⠶⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⣠⣴⠛⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣀⣠⡴⠾⠟⠛⠛⠛⠛⠳⠶⢿⣋⠉⠛⠛⠛⠳⠶⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣼⠋⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢀⡾⠃⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣿⠁⠀⢠⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⡴⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣀⣀
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠉⠻⠶⣦⣀
⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠟⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣤⣶⣶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠋⢹⡟⠋⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣆⠀⠀⠀⣾⠃⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣍⠻⢦⣴⡟⠀⠀⠀⢀⣼⠟⣷⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠹⢦⣀⣿⠀⠀⠀⠀⣾⣿⠀⠘⢷⣄⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⡏⠀⠀⠀⢰⡏⠹⣆⡀⠀⠙⠛⠛⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠐⢷⡀⠉⠳⠶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣀⣠⣤⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⢉⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⣤⣤⣤⣶⣶⠶⠾⠛⠛⠋⠉⠀⠠⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠿⠿⠿⠛⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣿⣿⣿⣾⣿⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠙⠉⠙⠻⠟⢻⡿⠛⣿⠟⠋⠉⠿⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢹⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣮⣤⣤⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⠉⠓⠄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠛⠛⠛⠛⠃⠀⠀⠀⠀⠙⠛⠛⠿⠿⠿⠆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⢖⣶⣾⣶⣶⣶⣶⣦⣤⣄⣀⡤⠤⣤⣤⣀⣀⣐⣒⣠⡤⣀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣊⣿⣿⣿⡿⠿⠿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⣿⣿⣿⣿⡿⠧⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⣿⣿⣿⢏⠴⠶⠄⢹⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⠛⠿⠿⣿⡿⢿⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣿⣿⡏⢸⠶⠄⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠠⠬⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠩⣛⡛⠉⠀⠀⠀⢀⣾⣿⣿⡇⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠉⠉⠉
⠀⠀⠀⠀⠀⣿⣿⡀⠀⢀⣀⣨⣄⣀⣠⣤⣖⣁⡈⡐⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⡷⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣄⣒⠦⡀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⡿⠟⠛⠛⠻⢿⣿⣿⣿⣷⣽⢀⡠⢜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⢳⡴⠄⠀⠀⠀⠈⠿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠸⢿⣿⡿⠋⠀⠀⠀⢫⣀⡀⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠹⠿⣿⣿⣿⡙⠛⠛⠿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠈⡼⠁⠀⠀⠀⠀⠈⠻⠿⠟⠿⠋⠁⠀⠀⠐⣇⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣶⣤⡀
⠀⠀⠀⠀⢺⣿⣷⣶⣶⣶⡆⠀⠀⠀⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⡠⢾⣿⣿⣿⣿⣿⡗⢤⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⡄⠀⠸⣿⣿⣿⣿⣿⡧⠬⣤⣃⢹⠓⡻⠖⠢⠧⠤⡤⠠⡖⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⣘⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠁⠛⠒⠚⠒⠶⠶⠖⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠱⡄
⠀⡏⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⢸⡀
⠀⡛⠀⠀⢻⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⢘⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣀⠀⠀⠀⡇
⢰⠂⠀⠀⢸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⣿⡏⠀⠀⠀⠀⡠⠊⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠈⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠀⢀⢸⣿⣿⣿⡿⠁⠀⠀⢤⡟⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠌⢿⣿⣿⠃⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⠀⠀⢀⣠⠤⠴⠖⠲⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠘⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡁⢀⣔⣲⠄⠀⠀⢠⣖⣢⡀⣈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡄⢸⠈⠁⣀⠀⠀⠀⠀⣀⠈⠁⡇⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠸⣄⣀⣁⠐⠂⠐⠀⣈⣀⣠⠃⣟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⣈⠛⠒⠒⠛⣁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⢏⠝⠉⠉⠉⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠡⠃⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠙⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡀⠀⠀⠀⠉⠓⠲⠶⠶⠖⢚⣉⠀⠀⣠⣄⠀⠈⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⢾⠁⠀⠀⠀⠀⠀⠔⠤⠀⠀⠘⡇⠀⠀⠀⢻⡶⠒⢒⡄⡧
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⡾⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⡇⠀⠀⠀⢸⣯⠭⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⡇⣇⠀⠀⠀⠀⠀⠀⢔⢰⠄⠀⠀⡇⠀⠀⠀⣸⡿⠢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣱⠸⣄⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠉⠉⠉⠉⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠼⠀⢳⠉⠓⠒⠶⠶⠤⠼⠦⠤⠶⠶⠒⠚⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢲⡖⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⡀⠀⢸⡇⠀⢀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠈⠉⠉⢹⡏⠉⠉⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠋
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠟⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠟⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠒⠒⠂⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢂⢀⣀⡈⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠘⠁⠶⠾⣿⣿⣿⡆⠀⢠⣿⣿⣿⡿⠿⠓⠈⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣀⠈⠛⠛⠁⠒⠀⠙⠛⠛⠀⣠⣴⣾⣿⣿⣷⡄⠀⠀⣠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⡀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡇⣠⣞⣩⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⠻⠿⠿⠿⠿⠿⠿⠛⠟⠛⠛⠛⠛⠀⠐⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀⣀⣀⣀⡀⠠⡆⢀⣀⣀⣀⣀⡀⠀⠀⣿⠀⣀⣀⣀⡀⠀⠀⢠⢀⠀⣀⣄⣀⣀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡍⠛⠉⢿⣟⣛⣋⠉⠉⠙⠛⠛⠛⠉⠉⠉⣿⡍⠙⠛⠚⠛⠯⡽⠟⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⠛⠻⠶⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⢷⡐⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡏⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢿⣷⠀⠀⠀⠘⣟⣑⠀⠀⠄⠠⠤⠤⠄⢄⣶⡏⠀⠀⠀⠀⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⡂⠀⠀⢟⣯⠀⣀⢀⡀⢀⡀⢀⣘⡟⠀⠀⠀⠀⣶⣿⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣙⠻⠶⢃⠸⠷⣾⢿⡞⠙⠾⠷⣾⣯⣅⠀⠀⠰⠞⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢂⡨⠾⡀⠕⣸⣸⡇⠀⢐⣾⣿⣷⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇⣀⣚⡃⢀⠈⣤⠘⠋⢠⠀⢀⠁⠀⠠⠌⢉⡉⠃⠸⣿⢋⠐⣤⠀⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠰⡇⢸⠏⡙⢉⠉⠘⠉⠀⠁⡘⠀⠤⠔⢀⡒⣃⡒⠒⠨⡤⠒⠈⠹⣤⠞⠋⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣻⡀⠉⠲⡂⠞⣀⠜⠳⠟⣶⡰⠂⣠⡈⣁⡘⠃⣤⢰⠧⠞⠒⢀⡘⢄⠀⡎⠙⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠁⠈⠟⢰⠆⡾⠇⢠⡰⡶⢯⡁⢻⡭⣷⣯⢽⣗⣠⠞⠲⡶⢦⠀⢻⠀⠀⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠁⠀⠀⢀⠃⠀⠀⠀⠀⣷⢀⣄⠛⠶⢿⣆⠳⡜⢫⡽⣣⠎⢀⣾⠷⠾⢣⡄⢿⠀⠀⠀⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⢰⡆⠀⠤⠁⠀⣀⠀⣷⠀⢡⡖⢀⠀⠉⠀⠸⡄⢰⡏⠀⠀⣃⡀⢺⡄⢱⡇⠀⠀⠠⠀⢀⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣁⣄⠀⠈⠱⠆⠀⣾⡄⠀⠁⠟⠐⡄⠀⠀⢹⡏⠀⠀⠰⠟⠿⠀⣰⢏⡀⠂⠁⠀⡰⢟⡳⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣦⣉⣉⡍⠀⠀⠀⠀⠀⢾⣄⠀⠀⣠⡖⠀⠀⢀⣿⣧⠄⠀⠀⢦⡀⢀⡟⠀⠁⠀⠀⠀⠶⣋⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡉⣉⠉⠀⢀⣿⢀⡀⠀⠈⣧⠀⣸⠏⠀⠀⠀⠀⠿⠇⠀⠠⠀⠘⡇⢸⡇⠀⠀⠀⠀⠀⠘⣋⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠋⢸⣿⢠⡀⢤⠀⠁⢠⡀⢀⣶⣿⣿⣿⣦⡀⣢⣀⡀⢸⡇⠠⠐⠈⠀⠀⠀⢉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣠⣦⡅⠸⠇⣸⠀⣷⡀⠆⢻⣿⣿⣿⣿⣿⡟⡤⢡⣿⠘⣷⠁⠀⠀⠀⠀⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⢠⡀⠈⠋⠁⠀⠐⠿⠀⠙⣷⠂⠈⠻⣿⣿⣿⠿⠁⢃⡼⠃⠈⠟⠀⠠⠾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠁⡄⠁⠀⠈⠓⠢⣤⣴⣷⣤⡴⠶⠋⠀⡀⠘⠀⠀⠀⠉⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢀⣤⣄⢀⣤⣄⠀⡠⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠄⠁⠀⢀⡀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡿⠈⠛⠋⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠿⠁⡀⢺⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⡀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠄⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡿⢿⠋⠀⠀⠀⠀⠙⢿⣯⠀⠐⣿⡞⠀⠀⠐⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠘⠷⠆⣀⠀⠀⠀⠂⠘⠛⣛⠉⠐⠆⢸⡇⠀⠃⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢽⠃⠚⠂⠐⠀⠈⠉⣤⣀⠀⡜⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠃⠀⠀⢀⠄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠃⠀⠁⢾⣧⡄⢠⠀⡏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠙⣷⠀⠸⠦⠘⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣤⠀⠀⠀⢿⠀⠀⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⡠⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⡤⢊⠄⠀⠀⠀⡀⠠⠀⠀⠀⡡
⠀⠄⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⡆⠄
⠀⠠⠀⠀⠀⠀⠀⠈⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⡄⠀⢀⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠁⠀⠀⢿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠃⠢⡤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀
⠀⠈⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠂
⠀⡀
⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣦⠆⠀⠐⠀⠀⠀⠤⠤⠀⡀
⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠈⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡃
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⡈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠄
⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣄⣿⠤⠇⠀⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣬⡀
⠀⠨⠊⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠘⠀⠰⠂⠀⠀⠀⠀⠠⡤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⣰⣶⣀⣄⣤⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⢰⣿⣿⣾⣿⣦⣠⡄⠀⢲⠀⠒⠂⠀⠚⠳⠀⠀⠀⠀⠒⠂⠶⠆⠀⠀⠈⠈⠀⠀⠈⠉⠉⠉⠉⠀⠀⠉⠀⢨⣃⡀⠂
⠀⠀⣿⣿⠻⠁⣾⡿⠁⠀⢹⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠙⣿⡦⠐⡟⠀⠀⠀⢰⠀⠀⠀⠀⢀⣀⡤⠄⠈⡉⠁⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠀⠠⠄
⠀⠀⠀⢟⡛⠛⢛⣳⠶⠶⠾⠆⠀⢀⠔⠁⠀⠀⠈⠉⠁⠀⡧⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠈⠀⢀⠂⠀⠀⠀⠀⠀⠀⠤⠔⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⣿⣷⣦⣶⣤⣤⡖⠰⡟⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠀⢹⢿⣿⠄⠤⠤⠭⠹⣟⠀⠀⠀⠀⠺⣙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡟⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠠⠠⠤⠀⡤⠤⠀⠶⠤⠀⠤⠤⡤⠄⠤⠄⠄⠦⠤⠤⠀⠤⠤⠠⠦⠤⠰⠤⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠰⠲⠀⠂⠀⠆⠰⠂⠐⠂⠂⠰⠆⠀⠀⠖⠔⠀⠐⠀⠃⠤⠲⠂⠐
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣶
⠀⠀⠀⢠⣴⣿⣿⠋⣀⣀⣾⠛⣷⠂⠀⠀⠀⠀⢰⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⡼⢼⣿⣤
⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⠘⠏⠀⠀⠀⠀⠀⠘⡇⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⣿⠶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⡳⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢽⠟⠀⠀⠀⠀⢠⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⣶⣶⣶⠛⠁⠀⠀⠀⠀⢰⣶⣲⣶⣾⣿⣆
⠀⠀⠀⠀⠀⡠⠤⠤⠄⠴⠠⠄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠉⠀⠀⠀⠀⠀⠀⢿⣍⠁⠉⠉⠉⡹
⠀⠀⠀⠀⠀⢸⣧⠴⢿⣦⢴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⠒⠉
⠀⠀⠀⠀⠀⢈⣧⣤⣼⣄⣨⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠠⠶⠿⠷⠦⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⡤⠒⠒⠒⠒⠂⠐⡖⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠖⠂⢩⠶⠿⠙⠉⠀⠀⠘⠟⠛⠙⠉⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢂⡀⡄⠈⠀⠤⠤⢤⡤⠤⠤⣤⢤⡀⡆⠈⠀⣉⠁⠐⠂⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⡄⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠰⢀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡠⠀⠀⠀⠃⠈⠀⠉⠈⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠁⠀⠀⢀⠠⠓⠖⠁⡀⠉⠀⠀⠀⠀⠀⢀⡀⠤⢀⡤⠶⡶⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠊⠀⠀⠀⠀⢀⠄⠈⠀⣠⠞⠁⠀⠂⠀⠀⠀⠙⢦⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⡰⠁⠀⠀⡜⠁⠀⠀⠌⠀⠀⠀⠈⠀⠀⢣⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⡀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠀⠘⠀⠀⠀⠰⠄⢀⣀⡀⣀⠀⠀⠀⡇⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠰⢷⡴⠻⡿⠿⠦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠐⠦⠤⠀⣀⣀⠁⠀⠀⠈⠁⠐⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠼⣦⡀⣿⠉⢹⣿⠗⠀⠢⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠏⠘⣡⠆⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣈⠙⠉⠘⣿⣷⣼⡋⠈⠋⢀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠇⠀⡾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⠀⠙⠶⠶⠶⠛⠿⠿⠷⠦⢀⠀⠁⡀⠀⠀⣤⠀⠑⠀⠞⡈⠀⠀⠀⡀⠀⢘⡄⣪⠃⡜⠁⢙⠃⢆⠅
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⣌⣉⡀⢈⠀⠀⠈⠀⠀⠀⠀⠑⡄⢋⠜⢁⡞⠁⡠⢞⡴⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠁⠀⠀⠀⠀⠐⠶⣀⠀⢞⣐⡭⠞⢟⣤⠞⠁⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠂⠀⠀⠀⠀⡀⠠⢼⢝⠳⢯⡐⠦⣞⣪⠦
⠀⠀⠀⠀⠀⠀⠐⢀⡃⠐⠀⢲⡎⠉⠉⣶⡎⠉⠙⣦⣀⣤⣤⣤⣤⣤⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡬⠑⠂⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠙⠒⠊⠉⠙⠒⠺⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠠⠋⠓⠑⠛⣆⡀⠐⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠂⠈⢳⡀⠀⠐⠯⡘⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⢐⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠈⠀⠀⠉⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠜⢁⠀⠀⠀⠀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠐⢇⠀⠀⠀⠀⠀⢠⠴⡀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠂⠁⡰⠃⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢤⠁⢐⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⡨⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠅⠤⠒⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠐⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⣀⣀⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡇⠀⠈⠢⡀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢀⠀⠀⠀⠀⢱⠀⢀⡠⠦⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢀⠎⠀⠀⠀⠀⠈⠏⠁⠀⣀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠉⠦⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤⢠⠆


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

⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠓⠰⠀⠀⠀⠰⠘⠃⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⡀⠀⠀⠚⠂⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠒⠂⠀⠀⡀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⡀⢀⡀⠀⠀⢤⠀⠠⠀⠀⠄⢠⠄⠀⠀⣀⠀⡀⠀⠀⠀⢈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⡁⢀⠀⠀⠀⢀⡀⠖⢆⣀⠀⠀⠀⢂⠀⡁⠐⠀⠀⠒
⠀⠀⠀⠀⠀⠀⡀⡤⢀⠀⠆⠀⠄⠀⠤⠙⠈⠁⠀⠜⠇⠀⠀⠛⠄⠀⠙⠀⠠⠄⠀⠄⠀⠆⢀⠠⣄⡀
⠀⠀⠀⠀⠀⢄⢀⠤⠎⠒⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠊⠦⢀⢀⠄⢀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠋⠀⠀⡀⠐⠀⠑⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠤
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠦⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠡⡆⠀⠀⠀⠀⠀⠀⠀⢐⣄
⠀⠀⠀⠈⠀⠀⠀⡰⠀⠃⠀⠠⠄⠀⠀⠃⠀⠀⠋⠙⢀⣤⠀⠘⠋⠃⠀⠀⢃⠀⠀⠴⠀⠀⠃⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⢻⣦⡀⠀⠀⠀⠟⠀⠘⠇⠀⠀⠄⣠⣾⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠖⠀⣴⠀⠀⠄⠀⠁⠀⠄⠁⠀⠄⠀⢰⡀⠐⢶⠦⠄⠀⢠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠋⠹⡆⠀⡌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠉⡄⠀⡾⠉⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⢀⣀⠀⠀⡀⠀⠀⡀⠀⢀⣀⡐⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠀⠀⠀⠀⢨⠟⠋⠀⠀⠂⠂⠀⠈⠛⢯⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠀⠀⠀⠀⠰⠄⠙⣤⡜⠁⠴⠀⠀⠀⠀⢀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⠂⠐⠄⠀⠠⠀⠠⠀⠀⠖⠀⠂⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢀⠄⠀⢀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⣷⢸⣿⡇⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⡀⠘⣿⣾⣿⡇⠀⠀⠠⠀⠈⠀⡀⠁⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⣿⣿⣿⡧⠴⡶⠶⠶⠶⠚⠋⠀⠀⣀⢀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠊⠙⠛⠓⠀⠀⠀⠀⢀⣀⠀⠀⡆⣿⣾⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠀⣠⡀⢁⠀⠰⢎⠁⣦⢸⣿⣿⣿⣿⣥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣤⣤⡀⠇⠃⠐⠀⠁⠀⠀⠜⠀⠸⡇⢿⣿⣿⡿⣿⣿⣖⠋
⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣏⢻⣿⣇⠹⣿⣿⠄⠀⠀⠐⠀⠐⠀⠀⠄⢠⡇⠈⠇⢃⣨⣿⣿⠅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣀⠂⠀⣄⠀⠀⠀⢀⣠⠏⠐⡄⢈⣴⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠿⢛⣻⣿⣿⣟⣿⣻⣤⡆⠀⢈⣭⠿⣿⣿⣯⣄⠰⠀⢈⣽⢿⣯⣷⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⢋⣿⣿⢻⣿⣷⡇⣀⠋⠀⠐⠀⠈⠛⣿⡆⠀⠸⢿⣟⡿⢿⣯⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⢏⣼⣿⢩⣏⠉⠢⡀⠀⡳⠀⠂⢸⡇⢀⣿⠾⣿⡗⢶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⡗⠤⢬⢅⣤⣽⡤⠤⠖⢻⠿⠷⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠴⠰⠄⠠⠦⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠐⢋⣁⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⣀⣠⣶⣿⣿⠀⠀⠉⠁
⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣤⣄⣠⣤⣤⣤⣄⡀⠀⠀⠒
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠿⠿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠉⠛⠛⠋⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢿⣿⣿⣿⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇⠀⠀⠀⠻⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠹⣿⣿⣿⣿⡏⠀⠀⠀⠀⢠⣀⠀⠀⠀⢀⣠⠏⣿⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⡿⢿⣿⣦⠤⠀⠀⠀⠈⣫⣷⡶⠽⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡺⠿⠟⠛⠛⠿⠿⣶⣢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⡾⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠃⠌⠩⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⢀⡠⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⡴⡵⠃⠀⡤⠜⠀⠀⠀⠀⠒⠻⣦⠀⠀⠀⢠⣾⣿⣦⣹⣷
⠀⠀⢠⣮⣾⣿⣿⣿⣆⠉⠉⠘⠛⠛⠻⣖⢄⣼⡿⠁⠀⡘⠋⠉⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⣾⣯⣾⢿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⢾⣁⣀⡀⢇⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣄⣇⠏⣌⣄⡑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡩⢂⠀⡂⡷⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⠟⠉⠉⠉⢉⡩⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⣾⠟
⠀⠀⠀⠙⠿⠿⢿⡿⣛⣿⣿⣿⣿⣿⣿⠁⢠⡤⣶⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡯⠟⠊⠁
⠀⠀⠀⠀⢀⣔⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣿⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⠿⠛⠁
⠀⠀⠀⠀⠘⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⣴⣫⣷⣄⠀⠀⠀⠀⠠⣾⣽⠋
⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠐⢀⠤⠲⢌⣯⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢄⠺⣄⠀⢸⢹⣿⣮⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠿⢿⣻⣿⣿⣿⣿⣿⡇⠀⠀⠘⢷⡿⠈⣿⣿⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣇⠀⠸⣿⣿⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⢹⣿⣿⣿⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢝⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣟⡯⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠛⠓⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⠛⢳⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠶⠃⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⣇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠐⣦⡀⣳⢄⣀⠼⠉⣲⣤⣤⣴⣋⡷⠤⠤⠾⢽⣶⣴⠋⠒⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣀⣀⠀⠀⠀⢸⠉⠉⢰⡤⠒⠉⢠⡄⠀⠀⠀⡂⠀⠀⣤⠀⠀⠈⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠸⣁⠺⢧⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠠⠃⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠉⠉⢹⠒⠋⠀⠀⠀⠢⠐⠤⠀⠀⠤⠒⠀⠀⠒⠠⠤⠄⢂⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣤⠽⠶⠃⠀⠀⠀⠀⠀⣠⠞⠁⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠁⣦⣀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⡾⠋⣇⠀⠀⢀⣴⣧⣀⡤⠖⢉⣀⡀⢈⣉⣩⢭⣭⣍⡙⢅⣠⣀⣀⣠⣤⠞⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠛⠉⠁⠀⡾⢡⡞⣉⠔⠋⠉⠀⠀⣀⡬⠗⠛⠂⢹⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠉⠀⣀⣠⠴⠒⠉⠙⠢⣤⣀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠃⠀⠀⠀⠀⠀⠀⢿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⠶⢶⡶⠶⠶⣶⠶⠶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⣾⠁⠀⠀⠘⣶⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀⠀⠘⣇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣡⡼⠀⠀⠀⠀⠀⠀⠸⣄⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⠢⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠖⠀⡀⢠⢀⠅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⢀⣁⠐⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠉⠐⢒⠢⢄⡀
⠀⠀⠀⠠⣖⠐⠆⠘⠀⠁⠀⠐⠀⠀⡔⠁⢹⠀⠸⡏⠍⠢⢀⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠉⣆⠈⢳
⠀⠀⠚⠄⠀⠀⠀⠀⠀⠰⡁⣀⠀⢸⡀⠀⡼⠀⠀⠈⠀⠑⠒⠠⠈⠧⢄⣀⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠒⠂⠀⠀⠀⠀⢀⣀⠤⠒⠉⠘⠦⣔⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠒⠢⢄⣀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰
⠀⠙⠙⠀⢀⡤⠒⠉⠁⠀⠀⠐⢪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⡯⠄⠤⢀⣀⠈⡆⠀⠀⠀⢠⠃
⠣⠤⢠⠤⠬⠵⠶⠤⠶⠾⠭⠬⠭⠉⠁⠒⢶⠤⠤⠄⣀⡁⠀⠀⣇⡤⠈⣿⠀⠀⠀⢀⡧⠤⠤⣁⠀⠈⠉⠴⠒⠴⢇⡀
⠀⠀⢸⠀⠀⠀⠠⢆⣰⠀⠀⠀⠀⠷⡀⠀⠆⠀⠰⣆⡲⠘⡡⠷⡚⠆⠀⠀⢏⠁⠀⢸⠤⠔⠈⠀⢀⠎⠀⡱⠒⠀⠒⠁
⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠟⠀⡬⠤⠤⠤⠤⠤⠤⠥⢦⠼⡟⢦⠤⠘⣆⢠⡋⠀⠀⣀⠴⠋⠀⢸⠁
⠀⠀⢸⠀⠀⠀⢀⡤⢄⠀⠀⠀⠹⣤⣠⠁⠀⠀⠀⠐⡚⢶⡄⠚⢀⠃⠈⠢⠁⡇⢈⣪⢀⠜⢱⠦⠖⠂⠉
⠀⠀⢸⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⡈⢷⠀⠪⡀⠀⢀⡕⠈⠉⠀⢀⡱⠻⠠⠄⠀⢄
⠀⢀⣸⠦⠤⠄⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⠱⡞⢇⠀⢾⣶⡋⠀⠁⠀⢸⢋⠰⠳⠋⠉⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠋⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⡾⠃⢀⠤⠒⠠⢤⡀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡴⠥⠘⢧⠔⠁⠀⠀⢀⡀⣷⣶⢯⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠶⠟⠉⠁⠀⠀⠀⢀⠀⠀⠈⠀⠁⠀⠘⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣾⠁⠀⠀⢀⠀⠀⢠⣾⡁⠆⢰⠁⠀⠀⠀⠐⡮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠨⠉⠉⢀⠀⢰⣿⢦⡀⡾⠉⢙⡸⠀⡀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣞⣱⠀⠸⡀⣿⠁⣈⠻⢧⡀⠀⠀⢡⠘⢾⣇⡤⠖⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡠⠄⡇⢧⠀⠀⠁⡀⠁⠀⢀⣠⣴⣈⠫⠧⠐⠦⢸⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣨⣿⣦⣸⣮⣗⠀⢀⣰⣄⠙⠁⠛⠉⠁⠀⠀⣄⣴⠃⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⡿⣿⢎⠙⡟⠋⠀⢰⠁⠀⠀⠀⠀⣧⠀⠉⠢⡀⠀⢸⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠃⠀⠋⢹⡜⣧⠤⠖⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⣨⢆⢸⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⡘⠀⡎⢰⠎⢀⠇⠀⠀⠀⠀⠄⢼⡇⠛⣰⣟⢆⠀⣞⠃⣄⢿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠠⡔⠀⠐⠀⡐⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡿⢸⠂⠀⠑⠀⢸⡇⠀⠐⠀⠰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡶⣀⠠⣀⠃⠀⠈⠓⢦⣥⢄⡀⠀⠀⢀⡼⠋⡖⢹⡄⠀⠄⡀⢸⡇⢸⠀⢀⢄⠌⢁⡶⠂⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠢⢴⡈⢦⠀⠀⠀⠀⢹⠲⡍⠛⠉⠁⠀⠀⣠⠞⠙⣆⠀⠈⣺⡇⠚⠀⢠⠋⡰⠋⠀⠈⡁⠠⠀⢀⠉⢢⡀
⠀⠀⠀⠀⠀⣀⠤⠒⠚⠧⣀⠀⠀⣀⡿⠀⠀⠀⠀⢀⡼⢀⡱⣆⠦⠖⠚⠉⠀⠀⠀⢸⡄⣸⠃⠁⣹⠂⡟⡴⠁⢀⠔⠁⠀⠀⠀⠀⠈⢀⡇
⠀⠀⠀⠀⡸⢁⠀⠀⠀⠀⠀⢙⡾⠏⠀⠀⠀⢠⡔⡃⢠⡞⢳⠈⠳⢦⣀⠀⠀⠀⠀⢸⣃⣟⣊⡁⠉⢦⣼⠇⢠⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢀⠗⠁⣠⠴⠒⠂⠒⠏⠀⠀⠀⠀⣰⣯⠀⣉⡟⠀⠈⠀⠀⣀⣈⣉⣒⢴⠈⠁⠀⠀⠀⠀⢹⡆⣙⣶⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢠⠃⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⢋⠔⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⢈⡀⠀⠀⠀⠀⣀⡼⠓⠁⠀⠈⠃⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢸⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠟⡠⠖⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⢈⠀⠀⣠⡈⠉⠁⠆⠀⠀⠀⢀⡴⠋
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⡷⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢴⣢⢀⠤⠐⠀⠀⠀⠀⠀⠆⠀⣴⠾⠿⠤⠴⠶⠶⠶
⣀⣀⣀⠙⣷⠀⠀⣀⣔⣩⡴⠾⠯⠤⠚⠓⠛⠛⠛⠋⠉⠉⠉⠛⠛⠂⠀⠉⠉⠉⠉⠀⠹⡆⠀⠀⠀⠂⠈⢀⠀⢀⡞⠁
⠀⠀⠀⠀⠈⠓⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠳⠶⠴⠖⠋⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣤⠌⠀⢂⠀⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡐⠁⠀⠀⠀⠀⠄⠖⠈⠀⢠⠀⠀⠀⡜⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠉⠀⠀⡀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠼⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠒⠈⠡⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉
⠂

⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣂⣠⣄⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⢀⣴⣀⡔⣄⣄⠀⠀⠀⠀⠀⠇⠀⠤⠄⢿⠀⠀⠀⠀⠀⠀⠉⠀⠻⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⠯⠝⠉⢹⣿⠀⠀⠀⠀⠀⡄⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠻⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡆⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡁⢠⣀⣸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣁⠀⠄⣸⠃⠀⠀⠀⠀⠀⣄⠀⠀⢀⣸⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠐⠛⠋⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⢠⡄⠀⠀⣲⡕⣠⠀⠀⠀⠀⠔⠂⠀⠐⠂⠐⡆⠀⠀⢎⠁⠈⠉⠁⢹⡖⠀⠀⢰⠖⠂⣀⢐⡶⠶⡦
⠀⠀⠀⠀⠈⠳⠀⠀⠀⡿⠃⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠃⠀⠀⠀⢣⠀⠀⢠⠋⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠰⠆⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⣷⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣋⣀⠃
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠢⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠌⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢸⠚⠀⠀⠤⠤⢽⡆⠀⠀⠀⠠⠧⠴⠿⠷⣶⡄⠀⠀⠀⠄⠀⠂⠀⣤⠀⠀⠀⠀⢸⠋⠀⠀⢾⡇⠀⠀⠀⠀⣖⠀⣀⠨⡲⢶⡄
⠀⠀⠀⠀⠈⡄⠘⠀⠚⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⡍⠴⢻⡖⣵⠸⡇
⠀⠀⠀⠀⠀⢱⠄⠐⠂⠁⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠯⠂⣑⣁⠞⠐⠇
⠀⠀⠀⠀⠀⠈⣄⠀⠀⣠⠁⠀⠀⠀⠀⠀⢦⠀⢀⣠⡇⠀⠀⠀⠀⠦⠠⠀⣠⡆⠀⠀⠀⠀⠀⠃⠀⠀⣸⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢰⢸⡇⠀⠀⢸⡄⠀⠀⣿⠀⢰⢠⡇⠀⠀⢸⡇⠀⡆⣿⠀⢰⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⢸⡇⠀⡇⢸⡇⠀⠀⣿⠀⢸⠀⡇⠀⡇⢸⡇⠀⡇⣿⠀⢸⠀⣿⠀⡇⢸⡇
⠀⠀⠀⠀⠀⡘⠈⣧⠀⡇⢸⡇⢠⠀⣻⡀⡘⠀⣧⠀⡇⢸⡇⢠⠃⢻⡀⣸⠀⣿⠀⡇⢸⡇
⠀⠀⠀⠀⠀⡇⠀⣿⢰⡇⢸⡇⣼⠀⣿⡇⡇⠀⣿⢰⡇⢸⡧⣼⠀⣿⡇⣿⠀⣿⢰⡇⢸⣷⡄
⠀⠀⠀⠀⠀⡇⠀⣿⢸⠇⢸⡇⣿⠀⣿⡇⡇⠀⣿⢸⡇⢸⡟⣿⠀⣿⡇⣿⠀⣿⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⡇⠀⣿⢸⡆⢸⡇⢿⠀⣿⡇⡇⠀⣿⠸⡇⢸⡟⢿⠀⣿⡇⣿⠀⣿⠸⡇⢸⡿⠇
⠀⠀⠀⠀⠀⡇⠀⣿⠀⡇⢸⡇⢸⠀⣿⡇⡇⠀⣿⠀⡇⢸⡇⢸⠀⣿⡇⣿⠀⣿⠀⡇⢸⡇
⠀⠀⠀⠀⠀⡄⠠⣿⠀⡇⢸⡇⢸⠀⣽⡇⡧⠀⣿⠀⡇⢸⡇⢸⠀⢼⡇⡿⠀⣿⠀⡇⢸⣇

⠀⠀⠀⠀⢠⢠⢠⢠⢀⢀⢠⢰⢠⢀⣤⢠⢠⢠⠀⠀⢀⠀⢠⢠⢠⣀⢤⢰⢠⢠⠀⢠⢰⢠⣠⣀
⠀⠀⠀⠀⠌⠸⠸⠸⠈⠌⠿⠸⠸⠈⠏⠩⠸⠸⠸⠸⠼⠸⠸⠸⠈⠉⠿⠸⠸⠈⠉⠹⠸⠸⠹⠹
⠀⠀⠀⣤⠀⠀⢠⡄⠀⠀⣤⠀⠀⣠⡀⠀⢀⣄⠀⠀⣠⡀⠀⢠⡄⠀⠀⣤
⠀⠀⢰⣿⡆⠀⣿⣷⠀⢸⣿⡀⠀⣿⣇⠀⢸⣿⠀⢀⣿⡇⠀⣼⣿⠀⢰⣿⡇⠀⠀⠀⣶⣶⣶⣶⣶⡂
⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⢰⣿⣿⠀⣾⣿⡇⢸⣿⣇⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡂
⠀⠀⢸⣿⡇⠀⣿⣿⠀⣼⣿⡇⢸⣿⣿⠀⣿⣿⡇⢸⣿⣿⠀⣿⣿⡆⢸⣿⡇⠀⠀⠀⣹⣿⣿⣿⣿⠇
⠀⠀⣸⣿⣇⢠⣿⣿⡀⣿⣿⣇⢸⣿⣿⠀⣿⣿⡇⢸⣿⣿⢀⣿⣿⡇⣸⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⠅
⠀⠀⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣾⣿⣿⣶⣿⣿⣷⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣦⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡯⠅
⠀⠀⣿⣿⣿⢹⣿⣿⡏⣿⣿⣿⢹⣿⣿⠛⣿⣿⡟⣿⣿⣿⢹⣿⣿⡏⣿⣿⣿⠉⠀⠀⣿⣿⣿⣿⡯⠅
⠀⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⢸⣿⣿⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⣿⣿⣿⡯⠭⠅
⠀⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⢸⣿⣿⣀⣿⣿⡇⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠉⠉⠛⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣀⠤⠤⠔⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠃⠄⣀
⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠀⡑⠤⡍⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠳⠽⢢⣀
⠀⠀⣠⠤⠞⠉⠀⠙⢆⡠⠲⠅⢂⠄⠀⡀⢠⠄⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⡃⠀⠑⠐⠉⠢⡀
⠀⠀⢇⣀⠄⠀⠀⠀⢊⢀⡔⠀⡀⠹⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⠀⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠧⠱⣌⡁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢤⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠟⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁
⢀⡰⢄⡀⠀⠀⠀⠀⠄⠀⠀⠙⣦⣀⠀⣀⣀⣠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⢙⣄⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⠂⠘⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⢿⠟⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠃⠐⢂⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠠⠃⠀⠄⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠁⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠢⠖⠆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠒⡄⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣬⣅
⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⠿⠿⠿⠇
⠀⠀⠀⢿⣿⣿⣿⣥⣤⣤⣤⣤⣠⣄⣄⠀⠀⠀⠀⠀⣿⣾⣷⣿⣿⣿⡏⠛⠷⠀⠀⠀⠀⠀⢩⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣍⣉⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠤
⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠤⠤⠤⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠉⠟⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠿⠷⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠉⠁⠀⢀⠎⠀⠀⢀⡄⠀⠀⢀⣆⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠐⠋⠑⠒⠠⣀⠀⠀⠀⣌⢀⠀⢩⢭⣭⠑⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠁⠀⢰⡍⡀⠰⠓⣚⠲⠄⠒⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠞⣳⠃⠈⠚⠁⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⣀⠐⠂⡂⠀⠀⠀⢤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⡀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡆⠱⠀⠀⡼⡝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢷⠀⢆⠀⠑⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⡆⠘⢓⣂⠚⣽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣇⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡖⠻⠀⠇⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⣤⣄⣐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠆⣗⠾⣿⠋⠙⡀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡄⢼⠁⣻⠱⢛⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢚⡀⣠⡕⡏⠑⢉⣑⠢⢤⡑⢄⠀⠀⠀⠀⠀⢀⣀⠽⠦⡀⠀⠀⠀⠀⠀⠀⡠⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠋⡰⠣⠜⠨⢻⠂⠀⠐⠀⠘⠣⡠⡀⠀⣠⣶⠉⠀⣤⠂⣈⣲⠀⠀⠀⠀⠀⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠂⡎⠀⠀⠀⠀⠀⠀⠟⡾⣾⣽⣿⡫⢶⡏⠁⠈⢇⠁⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠻⢿⠓⢌⡰⠀⠀⠀⠙⢌⠢⣀⠈⡓⠢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠑⢌⠣⣇⠀⠀⠀⠀⠑⠬⡑⠇⠀⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣄⠀⠤⠤⡤⠱⣌⠢⡀⠀⠀⠀⠀⠈⢄⢠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⡴⠚⠛⠀⠀⠒⠒⠒⢰⡶⠒⠒⠿⡆⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠒⠒⠘⠇⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣉⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢀⡐⠈⠏⠀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⡰⠁⠀⠀⠀⢀⠘⢝⣂⢴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣶⠂⠍⠀⣵⣀⠀⠀⠀⠈⠿⠁⠀⠂⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠚⢀⣼⣿⣿⣿⣦⣄⣠⠂⠣⡈⡴⠑⢄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⣠⣿⣿⣿⣿⣿⣿⣿⠀⡀⢄⠔⠥⠂⠰⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠉⠑⠢⡀⣴⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡿⢠⠀⠀⠂⠀⠀⠹⠤⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠺⠟⣿⡀⠘⡟⡿⠁⢀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣐⠀⠈⠶⢤⠢⡀⠀⠆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⢸⠇⠤⡉⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠉⠓⢁⣀⣘⡆⠀⠘⢀⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠀⠀⠉⠚⠠⠟⠀⠀⠟⠛⠉⠘⣿⣿⣿⣿⣿⡷⠀⠀⠀⠎⠉⢹⣿⡖⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢂⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⢁⠀⠀⠆⠀⢠⣤⠀⠀⡻⢃⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠁⠀⠀⠀⠈⠀⠀⢢⠀⡇⠀⣠⠀⠀⢀⠠⡀⠈⡄⠀⢦⠀⠀⠐⠶⠔⠁⠸⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⢊⠠⠀⠀⠂⢠⠀⠀⠀⠀⠀⢈⢢⠁⠞⠁⠀⢀⠌⠀⢳⡀⠀⠀⠀⠇⠀⡌⢀⠐⠀⢀⡄⠀⢠⣃⣠
⠀⠀⣀⣤⣤⣤⠃⠀⢠⣼⣀⠇⠀⠀⠀⢀⠀⣮⡜⠀⠀⠀⠀⡌⠀⠀⠀⠱⡄⢀⡀⣀⠐⠀⣰⠀⠀⢀⢙⣈⢉⣴⠟
⠀⠘⠻⠿⠿⣧⣤⣶⣿⣿⣿⠀⠒⣻⣿⠁⢀⣿⡇⠀⠀⠀⣼⣿⣷⣦⡀⠀⣈⠃⡈⠀⡤⠊⠑⢄⣀⣤⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢠⣿⣷⠀⢾⣿⡇⠀⠀⢰⣿⣿⣿⣟⡁⠀⠀⠀⠀⠂⠋⡆⢠⠞⠛⢿⣿⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠻⢷⣾⠛⠇⠀⢀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⠴⠃⠘⠀⣠⠂⠈⡁⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠢⣄⠒⣾⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐⠁⢀⠀⠈⠲⢤⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡁⠰⠃⡡⠄⠀⠀⠀⠀⠑⣄⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣱⠀⠤⠄⠁⠀⠀⠀⠀⠀⠏⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⡼⠃⢀⠤⠐⠠⣄⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠰⠦⠀⠧⠜⠁⠀⠀⢠⢸⣷⡼⢧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠩⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⣷⠃⠀⠀⢀⡄⠀⢠⣮⣅⢂⠘⠀⠀⠀⠀⠱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⣂⠀⠀⡄⢰⣿⠷⣄⣾⢀⠨⠜⠀⢄⢰⡃⠀⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣽⡆⠀⠃⢸⠁⠤⠙⠻⡄⠀⠀⠠⠈⢻⣧⠒⠋⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡂⢹⡸⣄⠀⠀⢀⠀⣀⡠⣴⣾⠬⠛⠓⠘⢃⠗⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠛⣷⣦⣿⣿⣀⣠⣼⣦⠀⠠⡁⠀⠀⠀⢸⠖⠫⡀⠀⠀⠸⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⠿⢹⢵⠤⡇⠀⢀⠌⠀⠀⠀⠀⢸⡇⠀⠈⠣⡄⢰⡇⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⡇⢀⠀⡁⠜⣇⠛⠋⠁⠀⠀⢠⢀⣤⢸⣇⠀⠀⡴⠛⢸⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠐⠀⡀⠀⠁⣰⠇⠀⠀⠀⠀⠁⠙⠃⢠⠿⡎⣧⠘⠢⡘⠊⡇⠀⠀⡘⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢉⡀⠀⠀⠸⠀⠹⢦⣀⠄⠀⠀⠀⠀⠀⢸⠴⡡⡇⠀⠀⠀⠀⣧⠀⠀⠀⡈⠙⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⣝⣂⡑⢆⠀⠀⠀⠈⢿⣑⠢⢤⠄⠖⠁⠘⢀⡾⢄⠀⠀⢤⡇⢸⠆⠀⡱⢁⡴⠉⠠⠀⠘⠒⠀⢤⡀
⠀⠀⠀⠀⠀⠀⣀⡀⣼⡁⠀⠀⠈⢻⠄⠀⠀⠀⠀⣸⠈⢧⣄⣀⡤⠴⠖⠋⠀⠈⣆⠀⡶⠓⠐⡄⣼⢠⠏⠀⡠⠐⠂⠉⠀⠀⠀⢹⡄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠉⠓⢄⡴⠋⠀⠀⠀⣠⡞⠁⣼⠆⠳⢦⣀⠀⠀⠀⠀⠀⡿⢸⠥⠒⠘⢄⣡⡟⠀⠌⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⢀⡥⠋⢀⣠⠤⠤⠴⠋⠀⠀⠀⢀⡼⠁⠲⡼⠁⠐⠀⠀⠉⠑⠢⢄⣠⠒⠉⠉⠉⠉⢐⡄⠻⣅⡎⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⢠⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠠⠉⠁⠀⠀⠖⠒⠋⠉⠉⠁⠃⠀⠀⠀⠀⠀⣀⠷⠛⠉⠙⠢⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢯⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⢂⠤⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⣿⢺⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢸⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⣴⣯⠀⠤⠤⠤⠤
⢀⣀⡈⠛⡆⠀⠀⣠⣐⣡⡴⠾⠥⠤⠟⠒⠚⠓⠒⠒⠒⠚⠟⠓⠒⠂⠀⠈⠉⠃⠘⢧⠠⠄⠀⠒⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠙⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠶⠊⠁⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⣤⡤⠁⠀⠄⠠⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠀⠀⠀⠀⠃⠰⡂⠀⢀⠀⠀⠀⡘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡆⠀⠀⢀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠼⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠉⠉⠈⠢⣢⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣹⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣠⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀
⠀⠎⠉⠑⡆⠀⡇⠀⠀⠹⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠤⣤⢡⣖⣒⣛
⠈⠄⢀⡠⠃⠀⠈⠒⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠄⢀⠃⠀⢈⠷⠠⣤⠤⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡀⠀⢀⡠⠀⢂⡠⠂⠀⠔⢁⢀⣶⠪⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠘⣍⠉⢻⡿⠟⢒⡎⠀⠀⠂⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠻⠒⠋⠀⠀⡤⣜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠀⠀⠀⠀⠀⠐⢋⡤⠤⠈⡀
⠀⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠂⠴⠀⢀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣈⣁⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠶⡎⢹⠴⣴⣩⢈⡇⠀⠬⠭⠥⠄⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠾⢀⠿⠽⣏⠓⠒⠒⠒⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠉⠂⢄⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠦⠙⠦⣀⠀⠀⠀⣠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠉⠀⠀⠀⢈⠆⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣶⣶⣶⠄⠛⠂⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⣀⣴⠂⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠐⠂⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⡶⠟⠻⠃⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⢀⣠⣿⡿⠤⢄⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⡞⡟⢱⠞⣘⡆⠀⢸⠀⣠⡟⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢦⡉⠁⠉⠉⠉⠉⣉⣀⡠⠤⠴⢲⠞⠁
⠀⠀⠀⠀⠀⠀⢁⠈⠈⠁⢸⠅⠀⠒⠋⠈⠁⠈⠀⡼⢩⠖⠒⢒⠀⠠⠄⣀⠀⡼⠁⠀⠀⢨⠗⠒⠀⠉⠁⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠏⠀⠀⢀⠄⠠⠤⠀⠐⠒⠃⠒⠛⠃⠀⠀⠀⠀⣸⠃⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⢠⠀⠀⢀⡞⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⢀⠀⢀⠈⠀⢀⠀⠀⠒⠋⠑⢤⡀⠀⠀⠐⠂⠐⠛⠉⠉⠷⣏⡀⠀⢠⠃⠀⣀⡠⠤⠀⠐⠛⡽⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⠀⠀⢫⣙⠂⠴⠋⠀⠓⠂⢉⣉⠉⠠⠄⠒⠁⣯⠗⠒⠚⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠾⠗⠊⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠠⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢢⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣉⠓⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣳⡄⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢔⣂⠂⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢉⡤⠊⠁⠀⠀⠐⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡯⠤⡀⡼⠀⠘⢄⠀⠀⣀⠀⠀⡘⢦⡈⢳⡀⠀⠀⠀⠀⠘⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠈⠄⠀⠁⠸⢄⣀⣀⠄⠀⠀⠀⠀⢈⠆⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⠦⡤⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠐⠣⠈⡏⠂⣰⠁⣨⠄⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⢐⠿⠀⠀⠉⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠢⢇⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡀⢑⠀⠀⢀⡠⠃⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⡎⠈⠙⠓⠒⠲⠦⠤⠤⠒⠒⠚⡒⠤⠤⠥⠊⠁⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁⠀⠀⠀⠻⠑⣗⠚⣎⣻⠒⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡤⢰⠁⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠘⢅⣈⢸⣉⡉⡑⠒⠢⠤⠤⢤⣄⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠀⠀⠀⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠁⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⢶⣦⣤⣄⡀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠘⣧⣀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠈⡙⠁⠀⢠⠀⣈⢙⠛⠿⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⡀⠼⠞⠒⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣚⡁⠀⠀⠠⠌⠐⢄⠈⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⣀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣼⠀⠨⠀⠀⠉⠃⢤⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠂⠀⠀⢆⡄⠀⢀⣈⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⣅⡼⠃⢠⠐⠃⠀⠀⢀⣿⡖⠁⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⠗⡐⠀⠀⠀⠀⠘⠙⢢⢀⣄⠈⡄⠶⡀⠀⢀⠚⠁⠀⡀⠐⠀⠀⢀⣀⣤⡠⠛⢂⡀⠀⠚⠁⠀⠰⡖⠠⠦⠀⠀⣠⠜⢋⡿⠂
⠀⠀⠀⠀⠀⠀⢀⠆⠭⠀⠈⠀⠀⠠⠔⠀⠀⢁⣤⠾⠯⢤⡨⠀⠉⣵⡆⠀⠀⠘⠿⠂⣶⠠⠛⠁⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠃⠰⠀⡈⣥⠞⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠠⠎⠀⠀⠀⣰⢫⠁⠀⠀⠀⢹⠀⠀⠀⠈⢂⡀⠀⠐⠀⣀⠀⠀⠀⠀⠐⢀⠤⠀⡄⡀⡤⠀⠈⠠⢉⡀⠆⣀⠘⠠⠀⢠⡀⢠⠼⣍
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⠀⠠⣲⠶⡯⠀⠠⡄⣦⣄⣚⠤⠀⠀⠀⠀⠁⢠⠀⠀⠀⢡⠀⠀⢀⣀⢈⠱⠚⠁⠀⠀⢀⠀⠀⠰⠃⠀⠉⢎⣀⡊⣉⠁⠁
⠀⠀⢀⠄⠒⠒⣀⠉⠋⠀⠀⠔⠈⠁⣀⠄⡤⠄⠀⠈⠋⠤⢧⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢠⠇⠸⠡⠀⠒⣃⠀⠃⠉⠀⠪⢀⠈⡀⠀⠀⢴⠶⠁⠀⠀⠆
⠀⣀⠛⠛⠋⠱⠚⠄⠀⠀⡠⠞⢹⠶⠋⠀⢀⣶⡆⠀⣴⣶⠇⠕⠀⠀⠀⠀⠀⠀⠀⠘⠌⠃⠈⠁⠉⠩⡑⠀⢀⠀⠀⠀⠈⠐⠀⠀⠀⠃
⢀⠀⣐⡀⠀⠀⠀⢰⡔⡀⠑⠠⢀⠦⡀⠀⠈⠉⠄⠀⢁⠘⠉⠐⠂⠀⠴⠤⣤⠦⠴⠂⠀⠀⠀⠀⠀⠀⠈⠂⢠⡑⢄⡀⠀⡹⠄⠀⠁⡠⠀⠀⠀⠀⢀⠌
⡔⠀⠀⡥⣀⣌⠉⠆⠀⠀⠀⠀⠀⠉⠈⠐⠠⡄⠀⠆⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣷⣶⣞⠒⣶⣯⣽⣖⣉⣭⡵⡆
⢐⠀⠐⠀⠀⠐⠂⠁⠀⠀⠀⠂⠀⢠⠀⠘⢟⠲⡦⢤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠤⠀⠀⠀⠀⠉⠉⠁⠀⠻⣿⣯⢉⣫⣶⠶⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⣠⠀⠈⠀⠐⠷⠔⠲⠤⣄⡀⠀⠀⡀⠀⠀⠀⠀⡠⠄⠉⠀⠀⣀⣠⣰⡞⠛⢂⠀⠀⠀⠀⠀⠀⠀⠿⡅⠂⠰⠚⠁
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣬⠡⠙⠙⢿⢙⠀⡀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣤⡬⡷⡀⠠⡊⠀⠈⠿⣤⡤⠉⠀⣀⢠⡄⠀⠡⡀⠀⡀⠀⠀⠠⠈
⠀⠀⠀⠈⠀⡀⠀⠀⠀⠓⠬⣅⣀⣉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⡚⠛⢋⡙⠠⡌⢣⠀⠀⠀⠀⠀⢀⡠⠅⢲⡤⠀⣄⠠⠄⠑
⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠙⠧⠘⠃⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢉⡀⠀⠁⢤⡄⢳⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⣌⢠⠀⠩⠄⢰⠅⠈⢹⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠂⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠅⣀⠄⠀⠤⠋⠑⢇⠠⢜⠄⡐⠀⠀⠀⠀⠈⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠂⠽⣶⡧⠤⡠⠂⠀⠸⢋⢱⣌⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⠄⠙⠁⢀⢴⠂⢀⡶⠟⠁⠀⠀⠁⡄⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠑⠤⠴⠒⠁⠀⢀⠴⡿⡡⠒⠉⠉⠢⡀⠐⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⢪⠕⠉⡌⠀⠀⠀⠀⠀⣱⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠦⢀⠀⠀⠀⠀⠀⠀⣀⠄⢺⠀⠀⠀⠀⠀⠀⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠋⠀⡶⠒⠀⠀⠐⠠⠄⠠⠤⠀⠒⠉⠀⠀⠀⢂⣆⠀⠀⠀⢰⠀⠈
⡀⣶⢶⠐⡠⠀⢃⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠉⠐⠆⠀⠀⡊⠘⡦
⢉⠀⢀⡃⠄⠂⠁⣀⠉⠷⠦⢄⣀⡀⠀⠀⠀⠀⡏⠉⠁⡰⠂⠀⢀⠀⠀⠰⡀⢰⡁
⠰⠀⠈⡁⠀⡀⠤⣇⠀⠀⠀⢀⣀⣉⣭⣥⣤⣶⡇⠀⠀⠁⠀⠀⠁⠉⠀⠼⠃⡈
⠀⠐⠉⠓⠀⠀⢀⣈⣂⡀⠰⠆⠛⠧⡄⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢖⡒⢉⠁
⠀⠀⠀⠀⠀⠀⠈⠪⠤⠴⠖⠂⠀⠈⠁⠀⠀⠉⣓⢒⣒⠊⠉⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⣦⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠌⠤⠀⢧⠼⡷⢧⡤⠁⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⣬⡘⣧⡼⣐⣫⣩⠽⢛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣶⣾⣤⠏⣼⠿⣟⣁⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⢸⣿⣿⣿⠿⠛⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉⠐⢺⣿⡗⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣇⡾⠛⠋⠀⠀⠀⣀⠔⠊⠓⢤⣀⠀⠀⠀⣠⡿⢾⣿⠁⠀⢀⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣷⢶⣶⠶⣴⣾⣤⣤⣤⣤⣤⣽⣶⣤⣶⣴⣾⣾⣿⣾⠿⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢻⡀⠀⢀⠁⠀⠀⠉⣿⡯⣉⠉⢉⠀⠍⠤⣾⣿⠟⠃⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣇⡀⠀⠀⠀⠀⠀⠘⠷⠤⠛⠛⠋⣁⣉⣉⣥⣤⠄⢰⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠉⠛⠛⠛⠉⠉⠉⣿⣿⠟⠛⠻⠟⢋⡉⠉⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠂⠀⠀⠀⠻⣿⣿⠆⠀⢀⢲⣿⣆⠀⠀⣸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⢈⢹⡿⠋⠀⣿⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⢻⣿⠁⡀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡹⠋⠀⢠⣾⣷⠀⡃⢀⣿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡆⠀⠄⠀⠀⠀⠀⠀⣌⠏⠀⠀⣠⣾⣿⢿⣴⠇⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣇⠈⠀⠀⠀⠀⠀⢸⡏⠀⢀⣴⡟⠃⣼⠋⣿⠃⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⡀⠀⠀⠀⢸⡧⠾⠋⠁⣠⣾⠿⣌⣽⠆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣇⠀⠈⠳⣄⡀⠀⠀⢀⣠⡾⠻⠋⣸⣿⣿⣄⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢦⣄⣐⡮⠽⡶⢶⣿⡇⠀⢀⣴⣿⣿⣿⡿⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠛⠛⠛⠋⢉⠀⠐⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⣤⣀⣀⣀⣀⣀⣤⣤⠴⠖⠋⠁



⣿⡿⠛⣻⡿⣿⣿⣿⢿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣍⠙⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠳⣄⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣕⠲⠼⣦⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡬⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠐⠒⠒⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡥⠶⠒⠒⠢⣄⢠⡀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⣠⠾⣄⠀⠀⠀⠀⠀⣹⠀⠉⠀⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠸⣆⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠤⠤⣄⠀⡇⠀⠈⢳⡀⠀⠀⢠⣏⣀⣀⣈⠻⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢣⠀⠀⢹⠈⣇⠀⠀⠀⢳⡴⠚⠉⠀⠀⠀⠈⢳⠀⠉⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢰⡀⠈⠃⠀⠀⠀⠈⠉⠒⢲⠏⠙⣄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡇⢀⠀⠙⢳⠒⢦⠀⠀⠀⠀⠀⢸⡀⠀⠘⣆⠀⢀⡴⠋⠀⠀⠀⢀⣀⣀⣀⣹⡄⠀⠀⠀⠀⣼⠀⣠
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⠳⡄⢹⠈⡇⢠⠇⠀⣠⡇⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⣀⡽⠗⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⣽
⠀⠀⠀⠀⠀⠀⠀⡇⢰⣂⣀⠇⠚⠠⣇⣞⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡼⡄⠉⠙⠦⠀⣠⠏⠀⠛⢛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⣏⠙⠒⠒⠒⢺⡳⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⢤⣄⠀⠀⠀⠀⠀⠀⡼⢣⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠉⠐⠶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣲⡄⠈⣇⠀⠀⠀⣠⠞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⡼⠀⠉⠛⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠖⠺⡶⢤⣤⣀⣥⠴⠋⠑⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣯⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣀⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠉⠐⠒⠂⠤⠤⠘⣷⠶⣶⠶⢶⠴⠶⣧⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠙⠦⠴⠓⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣚⡐⣇⣀⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠰⣍⡉⡉⣉⠓⠒⠲⠶⠤⢤⣤⣄⣠⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠓⠚⠂⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠛

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡀⠐
⠀⠀⠀⠀⡌⠀⠀⢀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠠⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⠛⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⡀⣠⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡃⠁⠀⠈⢆⠰⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠌⠂⠀⠀⢆⡈⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠁⠄⢄⢂⠄⠘⠠⠬⣃⡑⣀⣀⠿⢉⣤⢤⠤⢤⠀⢸⡟⠝⠛⠛⡏⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠦⣞⡦⠔⣂⠴⠚⠩⡦⠀⠀⣀⣤⣀⠀⠀⡾⠉⠈⢄⣠⡄⠀⢢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠸⠀⢠⠔⠂⠀⠐⠉⢄⠀⠀⠀⠀⠀⠃⣧⣀⣼⠀⣰⣿⡤⠖⢊⠁⠐⣰⡧⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠀⠀⠀⠘⣿⠠⠓⣀⡥⠖⠋⣿⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠲⣦⣅⣀⡀⢰⡇⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠁⠰⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠟⢩⣭⣭⣭⣭⣭⣭⣭⣭⣽⣷⣶⣾⣿⣿⣿⣿⡿⢉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⢀⣘⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠄⢀⣀⣀⡀⡀⢀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⠉⢉⣠⡀⠀⠀⡀⠈⠉⡟⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠉⠀⢹⠉⢠⠀⣀⣤⡀⠀⠀⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠀⠉⢢⠀⡇⠈⠤⠇⢚⠂⡏⠉⠉⠉⠉⠉⠉⢹⡇⡃⡎⠀⠀⢰⠁⠀⠙⡆⠀⠀⢐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠠⠀⠀⠴⠞⠁⠀⠇⠠⠈⡃⣉⠀⣿⠀⠀⠀⠀⠀⠀⠸⣇⡁⡧⠁⠀⠈⠑⠖⠊⠁⠠⠀⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠈⠀⠀⡀⠀⠀⠈⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠁⠇⠁⢀⠀⠠⠀⠌⠠⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠃⠊⠀⢠⣄⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⢀⠀⠶⠀⡀⢀⠀⠁⠀⠣⠀⠀⢀⡀⠀⠀⠡⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠨⠀⠀⢇⣀⡸⠀⠀⢀⠀⠀⠀⠄⠈⠅⠒⠅⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⢎⢀⡵⠀⠀⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠦⡀⠀⠈⠀⢀⡀⡢⠀⠄⠀⣇⢀⡀⡀⠁⠄⠀⠈⠀⠀⠀⣀⢠⠡⡀⠀⠀⠉⠀⠀⠀⣐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠈⠁⠀⠀⡀⠀⠑⠀⡰⠀⠄⡠⠄⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⢰⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⢀⡀⠧⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⡄⡀⡤⡀⢀⡀⠠⣀⠀⢀⢀⠀⡄⣀⡀⠀⡀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠓⠁⠈⠀⠐⠁⠚⠈⠉⠀⠉⠁⠐⠁⠈⠈⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⡀⢀⡔⡩⠰⠖⠎⣴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⣌⣠⣤⡀⠀⠀⠀⠀⠀⠀⣿⡄⢀⠢⡄⠀⠊⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⣀⣰⣿⡿⠃⣈⣶⠶⢶⣶⠶⢄⣷⠀⠈⠉⢁⣈⠁⠀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣚⢿⣿⣿⣿⣾⣿⡟⣈⣉⡭⠓⠚⡹⣯⡄⠠⢿⡏⠀⠠⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡀⠻⢿⡿⣫⠿⠶⣿⠷⠀⠒⠒⣺⡏⢿⣆⣼⣥⡧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠻⠦⣤⠀⠛⠀⠀⠀⢀⣠⣤⣾⣿⣵⡿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣠⣤⣶⡀⠉⠉⠉⠙⠉⠉⠉⠘⠘⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⠉⠁⠧⡤⠤⠤⠚⠒⠉⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠠⣀⣀⡀⠤⠤⠾⠥⠔⠛⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣖⣿⠈⢿⣶⣶⣶⣶⣒⣲⣶⣶⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣧⠀⠈⠈⠉⠛⠛⠛⠛⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢛⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠒⢶⣄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣄⣙⣧⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡤⠶⠛⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠾⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣠⣤⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣉⣉⠙⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢈⣿⠏⢻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢷⠄⠀⢸⣿⣿⠇⠀⠛⠧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠙⠃⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣸⣿⠏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢳⠆⢠⣿⡏⠀⠀⢿⡙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣿⡾⠃⠀⢈⡙⢿⡄⠀⠈⠻⣄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⠇⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⠈⣉⠙⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠙⠛⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠛⣯⢻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣴⡾⠛⠉⠀⠀⠀⠀⠀⢀⣿⠉⠙⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠁⣄⠀⣠⡿⠁⠉⣿⣄
⠀⠀⠀⠀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⠶⣾⣧⣀⡀⠀⠘⢿⣿⣷⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣴⡾⠛⠁⢸⡇⠀⠉⢿⣏⠀⢀⡴⠋⢙⣷
⠀⠀⠀⠸⣯⣀⡀⠀⠀⠀⠀⠀⠀⢾⡟⠁⠀⠀⠀⢀⣼⠏⠉⠀⠀⠀⠈⠛⠳⢶⣬⣭⣽⣿⠟⠋⠉⠉⠻⣦⡀⠀⢀⣼⠟⠀⠀⠀⠉⠛⢿⡀⣀⣾⠃
⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣸⢿⣶⠟⠁⠀⠀⠀⣠⣤⣠⣤⡿⠛⠁
⠀⠀⠀⠀⠀⠙⢷⣾⠀⠀⣤⣆⣠⡿⠁⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠃⠀⠁⠀⠀⠀⢀⡾⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠟⠛⠋⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠷⣦⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢷⣄⡀⠀⠀⠀⢸⣿⠶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⣠⡾⠋⣠⠎⠙⢷⣄⠀⠀⠀⣿⡄⠀⠀⠉⠉⠛⠛⠷⢶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⢀⣴⠟⣠⠞⠁⠀⢠⡾⠛⢷⣄⡀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣶⣦⣤⣼⠏
⠀⢀⣴⠟⢁⡾⢁⣤⠓⣴⡟⠁⠀⠀⠉⠻⣾⣯⡙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠻⣧⡀
⣠⡿⢃⡴⠋⡶⢪⠉⣧⠾⢷⣀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣹⡿⠻⠷⢶⣤⣤⣀⣠⣿⠁⠀⠀⠀⠘⣷
⠻⣶⣏⣀⣤⣾⠟⠛⠁⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⣀⣼⠟⠀⠀⠀⠀⠀⠉⠉⣹⡟⠀⠀⠀⠀⢸⡟
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣄⡀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠉⠛⣷⠾⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣈⠑⢦⣀⠀⠓⣶⠋⣩⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣌⡑⠢⣄⡝⠓⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡙⠲⢤⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣽⠇
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿