Cosy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⢀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠐⠠⠄⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠂⠀⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⠀⠀⠀⠉⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⡀⠀⠐⠀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠉⠉⠈⠈
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢠⢤⣄⠀⠀⠀⠀⢐⠠⠀⠀⠀⠀⠂⠰⠰⠰⠀⠐⠐
⠀⠀⠐⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡄⠀⠀⣤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠄⠄
⠀⠰⠂⠀⠀⠀⣀⠀⠐⢢⠀⡁⢀⣲⠆⠠⠄⠀⠄⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠐
⠀⠤⠀⠈⠁⠀⠁⠃⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠄
⠀⢀⠀⢄⠂⠐⠆⠀⠀⠀⠐⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘
⠀⠘⠓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠄
⠀⠣⠣⠂⠄⠀⠀⠆⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⢐⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⢐⠄⡈⠀⡐⠂⡄⠀⠀⠀⠈⠁⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠁⠈⠗⠄⠉⢀⡡⠌⠁⠠⡀⢀
⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠨⠌⣁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠐⠀⠁⠀⢀⠂⠀⠄⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡐⠄⠁⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⢅⠞⠂⢼⡒⠂⠀⢀⠀⠤⠁⠴⢀⠀⡀⠀⠀⠀⠁⠒⠂
⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠣⠅⢠⠲⠅⡃⢐⠔⠠⠀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠪⠀⡔⠠⠁⣰⡀⠀⡀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⠂⠂⠀⠉⠀⠏⠣⡔⢀⠁⠉⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠀⠔⠄⠀⠤⠖⣂⣀⡐⠢⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⢊⠰⠘⠐⠀⠐
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠳⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢰⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⣯⠀⠉⣆⠀⠀⠀⠘⡃⠈⠀⠀⠃⠃⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠙⠀⠀⠀⠀⠁⣀⠀⠐⠁⠀⠒⣈⡁⠈⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢈⡄⠂⢠⠐⠀⢅⠈⠀⠀⣀⠄⠿⠇⢉⡁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠻⣯⠹⠃⠈⠀⠰⠿⠛⡀⡠⠄⣤⣧⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠃⠀⠀⠙⡇⠒⠷⠽⡀⠍⠓⣀⢸⡉⠁⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡀⡷⢤⣄⣥⣄⣀⠀⠉⡠⠙⠩⢛⡶⠀⠀⡀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⢀⡁⠀⢸⠀⠀⠀⣕⠹⣼⣇⣩⣷⠉⠀⠀⢂⠤⠾⢷⡀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⡀⣉⣀⡁⣢⢉⣻⠀⠔⢋⠽⠳⠸⢝⣏⠻⠖⠊⣁⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠉⠉⠁⠻⠿⠀⡂⠐⡀⠀⠰⣀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠱⠀⣀⠠⠄⠾⠉⠉⠈⠑⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠀⡀⠀⠀⠒⢀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠈⠁⠦⢠⡄⠈⠁⠀⠀⣔⠂⠘⠀⢀⡀⠀⢀⢠⡦⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠲⠀⠀⣸⣝⣘⣦⣶⣿⣴⡿⣻⣮⡇⣾⢧⡡⢄⣱⣦⢹⣿⣾⣷⣷⠄⠀⠀⢆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠻⢶⣾⣷⣤⣿⣿⣿⣿⣿⣷⣽⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡗⠀⣶⣾⣀⠀⣟⣠⢀⠁
⠀⠀⠀⠀⠀⠸⠋⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣻⠿⠿⢿⣿⣿⣿⣧⠀⢿⣿⡏⣴⣿⡿⣿⣤⡲⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⡻⢿⣿⡄⠈⢉⣾⣿⣿⣿⣿⣿⣷⣦⡄⠀⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠛⠛⠛⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠙⠛⠉⠘⣿⣿⣿⣿⣿⣷⠀⠀⢹⡿⠋⠛⠋⠋⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠹⠿⠛⠃⠀⠀⠄⢫⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠘⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⡞⠛⠃⢄⡠⠀⡀⠀⣄⡠⡀⠛⢳⣀⠠⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⣼⠛⣤⠛⣤⢻⡜⣷⡝⣧⡟⢣⣜⢣⣞⡷⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢼⣿⢠⡟⢠⡏⢸⣅⣾⢧⡝⠓⣼⣇⣸⡷⢼⠛⠀⢀⡈⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢙⢺⡀⡘⠂⠀⠧⠈⡍⢻⢸⢤⠀⠈⠀⢹⠖⢻⠀⠀⣸⠇⠀⠀⢀⠸⡆⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠈⠉⠙⠾⠣⠉⠐⠛⠛⠐⠀⠉⠈⠀⠙⠊⠖⠛⠠⠺⠾⠿⠷⠿⡠⠀⠾⣳⠻⠇⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⡃⠀⠀⠀⠀⠀⠠⠞⠀⠀⠰⠀⠀⢀⡀⠾⠞⢢⡀⠀⠀⢀⡀⠤⣈
⠀⠀⠀⠀⠀⠀⢀⠔⡀⠀⠀⠀⣀⠀⠅⡀⠀⠀⠀⠀⠀⠀⢹⠤⠐⠋⡁⠀⠰⡀⠕⣀⠂⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⡀⠀⠊⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⢿⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢉⡝⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠑⠢⢤⣤⣀⣐⣛⠡⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⢰⡒⠖⠒⠒⢲⡒⠖⡒⠒⢒⠖⡒⠶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠻⠿⠗⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢛⣛⣛⣛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡌⢿⣿⠟⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⣴⣿⣿⣿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢹⠟⠉⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣯⣭⣿⣭⣭⣭⣭⡙⣦⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣠⣶⣶⣮⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣿⣿⡇⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⠃⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣀⣀⠀⠀⣿⣿⡇⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠰⠍⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠄⠠⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⣈⠻⣶⣯⣀⠀⠀⠀⠀⢲⠄
⠀⠀⠀⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠞⠋⠉⠀⠈⠉⠙⠻⢶⡤⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠻⠿⠿⠟⠀⠀⢀⡤
⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠀⠰⣿⠀⠀⠀⠀⠠⡀⡄⠀⠀⠀⢠⡀⠀⠀⠀⠘⠋⠀⠄
⠀⠀⠺⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠀⣦⣤⣤⣤⣆⣀⠀⠠⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀⣤⣄⡀⢠⠂⢾⣿⣿⣿⣿⣦⠄⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⠀⠀⠈⢀⡤⠂⠀⠀⠉⢩⠁
⠀⣾⣿⣿⡇⠀⠐⢼⣿⣿⣿⣿⡏⠀⠀⠐⢄⢀⠐⠀⠀⠀⣿⣿⣿⣿⣿⠜⠀⠀⠀⠀⢀⡀⠀⠀⣶⣾⣶⡖⠀⠀⠀⠀⣾⣿⡆
⠀⠈⠛⠿⠇⠀⠈⠸⠿⠿⠿⠿⠀⠀⢰⠄⠀⠉⠀⢠⠀⠀⠈⠉⠉⠉⠁⠀⠀⡄⢰⠊⠉⠀⠀⠀⠹⠿⠿⠁⠠⠄⠀⠀⢿⣿⠟
⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⣤⣠⡀⠀⠀⠀⠀⠑⢎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⣡⠄⠀⣀⣀⣀⣀⣀⡀⠀⢠
⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠐⠀⠀⠀⣠⠀⠉⢠⡟⠛⠛⠛⢻⡄⠁⡀⠀⡀⠀⠈⠉⠉⠁⠀⠀⠀⠈
⠀⠀⠀⣴⣦⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠼⠃⢀⡀⢸⡇⠠⠀⠀⢸⡇⠀⡣⠀⡇⠀⠶⣿⣿⠇⠀⠒⠒⠲⠶⠶⠀⠈
⠀⠴⠀⠉⠉⠀⠀⠃⠃⠀⠀⠸⠧⠀⢀⠀⠀⠀⠀⡈⠁⢸⡇⢀⣀⠀⢸⡇⠀⠈⢉⡤⠤⠤⣄⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠨
⠀⠀⣶⡎⠀⠰⠠⠤⠰⠤⠤⠤⠄⠠⠤⢤⣤⣤⡾⠁⠀⢸⡇⠀⠉⠀⢸⡇⠀⠀⡟⠀⣤⡤⠈⣧⠀⠐⠁⠀⠀⠀⣴⣶⡄⠀⢠
⠀⠄⠻⢷⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄⠀⢸⡇⣀⣀⣀⠸⠃⠀⠀⢧⡀⠛⠁⣠⠏⠀⠁⠀⠀⠀⠀⠉⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⢆⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⡀⠀⡠⢄⣤⡠⠆⠀⠀⠂⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠈⡻⢗⠂⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢧⣼⢱⠋⢠⡏⠁⣿⠁⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⡇⠀⡇⢸⣘⡾⣏⣳⢸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠠⣧⡤⠀⢀⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠴⠛⠛⠙⠛⠣⠀⢠⠖⠒⠉⠙⠛⠛⣲⢆⠀⠀⠂
⠀⡐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⢰⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠈⠀⢤⠀⢠⢤⠀⠀⡀⠐⠀⠀⠀⠀⠀⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡚⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠀⠀⠀⠀⠀⠀⠔⠂⠒⠂⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠐⠂⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠈⠑
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⣀⠀⠀⠀⠄⠆⠐
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠅⠁⢁⢙⠲⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠑⠤⠤⠐⠤⠀⠄⠈⢒⠒⠡⡀⠳⠤⣈⢸⠲⠨⢧⠐⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⠃⠀⠰⠀⣍⠀⡁⡀⠀⠁⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢠⠀⡰⠳⣲⠇⠀⠀⠐⠄⠐⠃⠖⠀⠂⠀⠄⠂⠴⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠄⠀⢈⠏⠀⠀⠘⠶⣌⠰⢸⡀⢒⠄⡤⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠀⠨⠦⣧⠀⠘⣶⡇⡆⡳⣦⠠⠁⡟⣻⡇⢀⠁⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠄⠀⠀⠈⠆⢠⠋⠀⢸⡇⡞⠀⠀⣷⠙⠃⡀⢸⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡴⣦⡈⠀⣳⡏⠀⢢⡜⢱⢅⣤⣄⠁⠀⢀⡅⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠞⢱⠘⠷⢼⢃⡶⠂⠈⢳⠀⡞⣀⢻⡆⠀⠸⣿⣇⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠀⠉⢠⡛⡆⠀⠂⠀⠀⠉⠀⠀⠂⠀⣸⠿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡀⠀⠀⠀⠀⢴⠋⢻⡟⠀⣨⠄⠠⢀⠇⠀⠀⠘⠁⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣸⣵⣖⣤⡈⣀⠈⣿⠀⠀⠀⠐⡌⣀⠀⠁⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠉⢿⠻⠛⠟⠾⢳⡶⡶⣶⡬⢾⠿⠛⠁⠀⠀⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣼⠘⠑⠈⡆⢀⠰⡔⠠⠀⢉⢉⠘⣄⠀⠀⠊⠩⠉⠁⠍⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠂⠀⠀⠀⠀⣩⡄⠠⠬⡁⠀⣻⡺⠀⠁⠀⠦⠠⠀⣐⡄⠀⠂⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠀⠁⠀⠀⠀⠀⠉⠁⠀⡀⠀⠐⠠⠐⠀⢀⠀⠴⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡎⠌⢰⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣄⠀⢀⡀⠀⠂⠀⠀⡴⣶⣬⡁⠀⠀⠐⠄⠤⠄⠠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠛⠀⠘⠂⠀⠀⣤⠈⠁⠀⠉⠃⠀⠀⠠⢠⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⣀⢀⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢥⣆⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠚⠀⠃
⠀⠀⠀⠀⢀⠀⠄⣠⠐⠀⠂⢀⣠⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⢃⡆⠀⠀⠸
⠀⠀⠀⠀⠈⡛⠚⠛⠆⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠈⠚⠗⠷⠇⢀⣇⠄⠐⠒⠂⠀⠀⡠⠀⣢⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠔⠤⠄⠀⠭⢀⠀⠀⠀⠀⠀⠔⠃⠀⠙⠞⠡⠢⠀⠀⠀⠁⡐⠸⠀⣤⣀⣀
⠀⠀⠀⠀⠀⡠⠀⠘⠘⠀⠀⠀⠀⠀⠀⠒⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠞⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠐⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⠀⠀⢀⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠒⠊⠁
⠀⠀⠀⠀⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⠀⠀⠘⠁⠈⠀⠛⢩⠋⠉⠆⠀⠀⡜⡿⠐⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠖⠀⠀⠐⠀⠀⠁⠈⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢦⠍⠀⠀⠰⠶⢶⠆
⠀⠀⠀⢀⠀⠀⠠⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠃⠀⠂⠀⠀⢰⡈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⣾⠆⠀⠀⠄⠀⠀⠈⡀⠛⠀⠀⠄
⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢠⠄⠘⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠃⠀⠠⢀⠀⠀⠀⠀⠀⠀⠛⠀⡆⠀⠀⠀⠁⠀⠀⣰⠈⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⢰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡜⠩⣰⢠⡄⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠃⢀⠈⠀⣶
⠀⠀⣀⠀⠀⠀⠀⠀⡀⣀⢀⣀⠀⢀⡀⢀⣍⠀⠀⠀⠀⠀⠈⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⡀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠈⠀⡄
⠀⠀⠈⠙⠒⠀⢰⠀⢹⡇⢸⣇⠀⡾⢷⠀⣶⠁⠀⠀⡀⠀⠀⠰⠈⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠈⠀⠐⠂⠀⠐⠒⠙⢂⠀⠀⠀⢀
⠀⠀⠀⢰⡆⠀⠏⠀⠀⠀⠀⠉⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢂⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠉
⠀⠸⢀⡈⠀⠀⠀⠀⠑⠲⠤⢴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠠
⠀⠀⠈⠁⠀⠀⠐⠒⠒⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣄⡀⡠⠀⠉⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡆⠀⠀⡄⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⡀⠂⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠐⠀⠀⢓⠐⢀⡀⡀⠀⠀⠠⡜⠤⠀⢠⠀⠀⡦⠀⠃⠀⠀⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⡀⠘
⠀⠀⡅⢀⠀⢀⣀⣰⢀⣀⡅⠀⢠⠁⠀⠀⠨⠀⠀⡤⠐⠀⠒⠂⠀⠠⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⠀⠀⠒⠒⠂⠈⠭⠭⠃⠀⠀⢦⡼⠀⠀⠈⠦⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⣤⡤⠴⢦⣤⡤⠶⣀⣤⢶
⠀⠀⠀⢰⠾⢭⣤⣤⠖⣉⣁⣤⢖⣉⣠⣶⡿⠁⡠⢾
⠀⠀⠀⢸⠀⢰⣯⣿⠀⠂⠀⣇⠀⠀⢸⣿⢣⠄⢠⣿
⠀⠀⠀⡞⠀⠀⡆⢸⠐⠒⠂⡗⠀⠺⢹⡏⠂⠀⠀⢽⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣸⠀⣀⠀⣻⠀⠀⠀⡇⠀⡀⢸⡇⠀⠀⢂⣾⠀⢀⡠⠤⠒⠊⠁⠀⠀⠀⠀⠀⢉⡩⠝⢛⣢
⠀⠀⠀⡇⠀⠈⠀⣽⠀⠛⠀⡇⠈⠛⢸⡏⠘⡣⢻⣗⣈⣁⣀⣀⣀⣀⣀⣀⣀⣠⠔⢊⠁⡤⣾⡿⣋⣀⡀
⣰⢀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠈⢸⡇⠀⣩⣅⣿⣀⣀⣁⣉⣉⣉⠉⠉⠉⣻⣿⣿⣟⣛⠉⣠⣴⣿⠃
⠀⠁⢀⣇⠁⠐⡟⡟⠀⠐⠂⡇⠒⠚⢺⡇⠀⣇⠀⠈⢉⡉⠉⠉⠉⢉⡍⠉⠉⠉⠁⠘⠀⠀⣽⢛⣿⣿⡶
⠀⠀⠒⡇⠈⠈⠟⡏⠀⠀⠆⡇⠀⢐⢸⠁⠀⡞⠙⠓⠛⠛⠓⠒⠒⠒⠒⠒⠒⠚⠗⠾⠷⠶⢿⣿⡿⡏⠀⠀⣀
⠀⠀⠀⣇⠀⠘⣇⡿⠀⠙⡅⡇⠀⠩⢸⣇⣴⣇⣄⣀⣤⣄⣀⣀⣀⣈⣁⣀⣀⣨⣀⣄⣀⣀⣸⣇⣴⣿⡷⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠑⠒⠶⠤⠤⣄⢠⡖⠉⣉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⣀⡤⣬⡇⡞⠁⣀⣀⣠⠞⢷⣀⡾⠉⠓⠦⣄⣼⣾⠉⠉⠉⠑⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⢀⡿⠳⠚⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡴⢋⡤⠴⠒⠒⠲⢤⣀⢀⡴⠚⠉⠉⠉⠉⠓⠢⣄⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⡾⢥⡏⠀⠀⢠⠖⠒⠲⣄⠀⢠⠟⠫⣷⠀⢀⡤⠖⠲⢤⡀⠀⠈⣇⣠⠽⣖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢩⣟⣹⠾⠯⠷⠶⠶⠶⢶⠖⠚⠲⣼⢦⡤⠿⠤⠬⠤⠤⠤⠤⠧⠤⠶⠿⠴⠶⣾⣷⡒⠲⠆⢀⡠⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⡤⠬⢿⠀⠀⠀⡿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡞⡴⠒⠲⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠋⠀⢀⣀⡤⠤⡖⠒⠒⢛⡗⢒⡞⠉⠉⠙⠓⠒⠒⠒⠤⠤⢤⣀⣀⣀⠀⠀⠀⠉⠙⠒⠲⢧⣇⠳⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⡤⣖⠉⠡⡄⠀⠈⠲⣄⡞⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠉⢉⠛⠒⣦⢤⣀⡀⠀⣧⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠋⠀⠈⠳⠤⠞⠀⠀⠀⡾⠀⢀⡷⠤⠤⠤⢤⡤⠤⠴⠖⠚⠉⠀⠀⠀⢰⡃⢀⠜⠁⠀⠀⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠶⠆⠀⠀⠀⠀⠀⠈⠳⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠘⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠒⠢⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠔⠚⠉⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉





⣿⠿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣛⣛⣿⣿⣿⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢁⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⠭⢵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠚⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠞⠋⠉⠀⠀⠀⠀⠀⠀⣀⣤⡷⠆⠀⠀⡀
⠀⠀⠀⠈⡟⠀⠤⣀⡀⠉⠉⠙⠒⠲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠚⠳⣦⣀⣠⣴⠶⠿⠯⠑⠒⠂⠀⠀⠀⢡
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠲⠦⣄⣀⠀⠀⠀⠀⠉⠉⣓⣒⣶⠤⢤⣄⣹⣄⣤⠶⢿⡏⠁
⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣠⣤⣟⡛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠄⠐⢀⣠⣯⣿⠗
⢰⡿⠁⠉⠙⠛⠶⠬⣝⣳⣦⣤⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠔⠒⣒⣈⣩⠥⠤⠔⠒⠒⠉⠉⠉⠀⠀⡟
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣬⣤⣶⠶⣞⣛⣛⣭⣭⡵⠶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠙⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠛⠋
⠀⠀⠀⠀⠀⠉⠛⠳⠶⣦⣠⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡤⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⣤⣄⣴⠄⣀⣀⣀⣤⠤⠶⠗⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⠿⠟⠛⢉⣡⣤⣿⡿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⠿⠛⠛⠉⢁⣠⣤⣶⡿⠿⠛⠉⣁⣤⣾⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⢀⣀⣤⣴⣶⡿⠟⠋⠉⣀⣠⣴⣶⣿⣿⠿⠛⠛⠉⣉⣡⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⠟⠛⠋⠉⠀⢀⣠⣤⣴⣾⠿⠟⠋⠉⣀⣠⣴⣶⡿⠿⠛⠋⠉⢀⣀⣤⣴⣶⣿⣿⣿⣿⠿⠿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⠟⠛⠉⠁⠀⠀⣀⣠⣴⣶⣿⠿⠛⠋⠁⣀⣠⣴⣾⠿⠟⠛⠉⠀⣀⣠⣤⣶⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣠⣴⣶⡿⠟⠛⠉⠁⠀⠀⢀⣀⣤⣶⣾⣿⠿⠛⠋⢁⣀⣤⣴⡾⠿⠛⠉⠁⣀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢰⣿⣿⠉⠁⠀⠀⣀⣠⣤⣶⣿⡿⠿⠛⠋⣁⣀⣤⣶⠾⠟⠛⢉⣁⣠⣤⣶⣾⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⣶⣶⣾⣿⣿⠿⠛⢋⣡⣤⣶⣾⣿⣟⣋⣩⣤⣶⣶⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣾⣿⠋⠉⠛⠿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⣿⡇⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟

⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⣿⣿⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠏
⣿⣿⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠟⠛⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⡿⠟⠋⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⠿⠛⠋⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⠿⠟⠋⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠛⠋⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣠⣴⣶⣿⣿⣿⠿⠟⠋⠉
⠙⢿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠙⢿⣿⣷⣶⣦⣴⣶⣶⣶⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡤⢄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡤⠤⣤⣄⣀⣠⣤⣤⣭⣭⣭⣤⣤⣿⣟⣛⡶⠂⣀⣿⣿⣿⣿⣿⡿⢻⣷⡾
⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢯⣭⣭⣛⣻⣿⣿⣿⢇⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡀⠠⠤⢤⣴⣶⣶⠶⠒⠒⠒⠒⢛⣓⣫⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠗⠃
⠀⠀⠀⠀⠀⠀⣴⣮⣴⠒⣒⣴⣶⣿⣿⣿⣿⣿⣶⣶⢶⠀⠀⠀⣉⣽⣿⣿⣿⣿⣿⡿⠧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⡶⢾⡉⣹⣿⢿⣿⣿⣿⣟⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠛⢛⣻⣿⣯⣽⣯⣽⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡿⢫⣿⠟⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠻⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠸⡀⡀⡘⢆⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣤⠀⢤⡸⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⢿⡿⠿⢿⣿⣷⣶⣶⣶⣶⣶⢶⣶⡶⠿⠤⢤⣭⣭⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣷⣷⣶⢷⣶⣿⣿⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠒
⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠄⠉⠉⠋⠹⠿⠂⠙⢹⡿⠛⠉⠉⠻⠟⠛⠛⠛⠛⠛⠛⣻⣛⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀⣊⣉⣉⣉⣉⣉⣉⣍⢩⡭⣿⣿⣿⣿⡘⣧
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⣤⣛⣂⣠⣿⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⣛⣛⣫⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣽⣯⣭⣭⣿⠿⠿⡟⢻⡿⠟⠋⢉⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠛⠿⢿⣿⡉⠉⢈⣙⣿⣍⣉⣉⠁⠈⠁⠀⠀⠀⡀⣀⣸⣧⠀⠀⠸⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢰⣂⣤⣤⣤⣭⣭⣭⣅⣛⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣦⣀⣐⣿⣿⣷⣄⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠀⡼⠛⣠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⢰⣖⣴⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⡼⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡴⢎⠾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠷⣿⡿⣯⣙⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠘⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡟⣻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⢀⡠⠴⢂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢆⠀⣴⡀⠁⠀⢠⡄⢀⠀⠉⢩⠀⠈⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⣉⠈⠀⠀⡇⠐⠤⠀⠀⠑⠉⠉⠁⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣁⣉⠀⠀⠀⡀⣀⡠⠠⠖⠀⠀⠀⢀⠁⢀⠐⠉⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠑⠆⠀⡴⣀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠃⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣆⠀⠀⠛⠨⠁⠀⠀⠀⡘⠂⠄⠀⠀⠙⠶⢞⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⡠⣁⠶⠂⠀⠉⠀⠠⠂⠀⠀⢄⡠⣔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠀⢙⡗⠒⢲⠶⠚⠛⠛⡖⠒⠓⣿⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠚⠃⢠⡞⠀⠀⣽⠀⠳⢀⣠⢙⠤⠔⠃⠑⠂⡴⢌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣀⢏⠀⠀⡠⠘⠃⠂⢡⠀⡖⠈⠉⠀⠤⠠⡄⠠⡀⠀⠀⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢰⠇⡀⠱⠞⠀⣤⢀⡠⠟⠠⠉⡀⠼⠀⡀⢀⢷⠀⢨⠉⠘⠣⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠟⠢⡀⡘⠀⠀⠀⠈⢠⠁⠸⣀⠀⠀⠀⢄⠀⢀⣄⠈⠠⠄⢠⡇⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡏⠀⠀⠚⠁⠀⢀⠉⠂⢁⠀⠠⠉⠁⡈⠀⢀⠁⠀⢨⠀⢀⠀⠁⠈⡃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡞⠁⠀⠀⡐⠀⠀⠤⣀⢠⠊⠀⡧⠒⠋⠁⠀⠀⠒⠒⢻⠀⠈⠒⠂⢠⡇⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢰⠀⠠⠴⡎⠀⠀⢸⠀⢀⡌⠀⠀⠉⠂⢀⠄⠀⠀⠆⠀⠊⠀⠀⡀⠒⠁⢸⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡼⢆⠀⡰⠁⠀⡀⠀⠈⠉⠀⠀⠀⠀⠂⠉⠀⠀⠸⡀⢀⠾⠀⠀⢑⠦⠀⣼⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡞⠀⠄⠚⠁⠀⠀⠧⠀⠤⠛⠀⠀⠜⠒⠀⡠⠀⠀⢠⠀⠁⢀⠆⠀⢨⠀⠋⠀⡆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⢳⡀⠀⡠⠀⠀⠀⡄⠀⠀⡄⠀⢘⠀⠐⠊⠑⠀⠀⠘⠂⠒⠡⠀⠀⠀⠃⠒⢂⣯⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠉⠛⠲⠦⣤⣄⣄⣀⣀⠀⠀⠀⢸⠢⠄⠀⢀⠂⠀⢀⡀⠀⢀⡀⢀⢀⣃⣠⠮⠜⠂⠀⠡⠀⠡⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⡇⠀⠀⠉⠹⠉⠛⠛⠛⠒⢓⠚⠛⠛⠛⠋⠉⠉⠉⡍⠀⠀⠀⡀⠀⢀⠀⠀⠀⠆⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⢘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸⠀⠀⡇⠀⠀⢘⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢤⠀⣄⢠⣀⣀⣀⣀⣆⣄⠀⠀⠀⠀⣄⢀⣀⠀⠀⠀⠸⠀⠆⠀⠀⠀⢀⣄⣠⠦⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠈⠀⠁⠈⠛⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠒⠂⠉⠉⠩⠐⠒⠒⠢⠤⠤⣤⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣤⠤⠐⠒⠒⠈⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠉⠉⠛⠒⠒⠶⠦⠤⣤⣀⣀⣀
⠀⠀⠀⠀⡎⡀⠀⠉⠉⠒⠒⠲⠤⢤⣄⣀⣀⠀⠀⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠬⢿⡟⠁
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠁⠀⠀⣼
⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⢤⡴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣠⣄
⠀⠀⠀⠴⣴⣖⡛⠛⠉⠓⠒⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣀⣀⣀⠀⠀⠀⠀⢀⣤⣶⠾⠟⣋⣁⡠⠭⠽⠗⠲⡀
⠀⠀⠀⠀⢹⠉⠉⠛⠲⠦⣄⣀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠈⢻⣷⣤⡶⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⣀⣀⣉⣩⡿⠷⠶⠶⣦⣤⣿⣤⡶⠿⠛⡿⠇
⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡞⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⣀⣠⣽⠿⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠖⣲⣿⣓⡶
⣰⡿⠛⠻⠾⢷⣤⣍⣻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⠒⣋⣉⣭⣥⡴⠶⠶⠛⠛⠛⢹⡏
⣿⠃⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣶⣤⣀⡀⢠⠏⣌⠀⠀⢀⣀⣀⣤⣴⠶⠶⠒⢛⣋⣉⣭⣥⡴⠶⠖⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣾⣟⣋⣩⣭⣤⣶⡶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀
⠀⠛⠿⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡶⠶⠞⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣤⣄⣀⠀⣀⣆⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠶⠾⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣯⣵⡶⠶⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣦⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡟⠁⠢⠤⣀⢀⡒⠀⢤⡀⠀⢸⣿⠀⢀⡎⣰⠉⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⡇⠀⠀⠀⠀⡀⠁⠐⠃⠀⢠⣿⡿⠁⣾⠀⣿⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣇⠀⠀⠀⠀⡇⢰⣂⠴⠀⠀⠈⠸⠆⢹⡄⢿⡀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⠀⠀⠀⢀⡳⡀⠀⠀⠀⠠⠃⠀⠀⠘⡇⢸⡇⠀⣾⣿⡿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠓⠒⠈⠉⠦⣼⡄⢀⠤⠓⢤⣄⡀⠀⣷⠸⡇⢰⣿⣿⡇
⠀⠀⠀⡆⣿⢰⢶⣄⠀⠀⠀⠀⡄⠲⣤⠤⠼⠄⡀⢴⡀⠀⠀⡤⠀⢠⡴⠇⠈⠁⢠⠀⠀⠀⠉⠓⢿⢀⡇⣾⣿⣿
⠀⠀⠀⣷⣿⣼⡀⣿⠀⣀⡀⠀⠻⠀⠹⡆⠀⠠⡁⢀⠇⠀⡸⢀⣴⠏⠀⢀⡤⠀⠐⣶⢤⣄⡀⠀⣸⣸⢣⣿⣿⡏
⠀⠀⠀⣿⣿⡇⣃⣿⣎⠳⣄⠉⠙⠒⠒⠒⠒⠰⠤⠠⠴⠒⠂⠈⢠⡀⠀⠾⠀⠀⠀⠟⣀⣠⣭⣼⡏⠁⣼⣿⣿⠃
⢀⠀⠀⠉⠉⠁⠻⠿⠛⠁⠘⠧⢤⡄⠶⢤⣀⣀⣠⣄⠀⠀⠀⢠⡎⢠⠂⣤⣶⠷⠿⠛⠋⠉⠁⠀⠀⠀⣿⣿⣿
⡏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠒⠒⠒⠂⠀⢲⡀⠉⠉⠉⠓⠒⢺⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠳⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⠸⠃⠙⠓⠒⠒⠂⠘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⡇⡎⠀⢘⠰⠀⠈⡔⠋⠈⠉⢍⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁
⠀⢠⠠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡆⢧⠠⣀⣀⣀⡀⠀⠀⠀⢿⢸⡇⠀⠀⠀⢀⣀⣀⣠⡤⠴⠶⠚⠉⠉⠉⠉⠉⠉⠉

⡑⣤⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⠈⢛⢳⣿⣶⣿⣾⡾⣾⣷⣴
⣿⣿⡿⣠⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢯⣑⣸⣧⣽⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣵⣿⡫⢈⣠⣂⠤⠀⣀⡤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿
⣿⣿⣿⣿⣿⣿⣿⡷⢺⣿⣿⣿⣿⣿⣿⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠩⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⠃⠘⠀⠀⢀⣾⣭⡙⠻⣿⣷⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠢⡀⠹⢿⣿⣎⢦⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠝⠉⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡻⡧⠀⠀⠀⠂⠀⠙⠀⠈⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠘⠀⠀⡤⠚⠋⠉⠉⠉⠹⡿⠹⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻
⠈⠈⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠄⡀⠀⠀⣠⠄⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⠘⠁⣼⣿⣇⠀⠛⡴⠽⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠋⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣿⠂⠀⢹⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⣤⠴⠶⠖⠒⢂⠀⠀⠀⠀⠤⠀⠛⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⣀⣀⣀⠌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣯⠷⠿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠠⠄⠐⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⡀⠀⠴⠖⣒⣺⠃
⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠐⡀⠀⠀⠉⠀⢐⡀⠈⣭⣤⣿⣉⣄⣀
⠀⠀⠀⠀⠀⡀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⠿⠿⢿⡄⠘⠟⠿⠟⠋⠁⠀⠘⢁⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠠⠀⠀⠀⡀⠀⡀⣈⠂⠛⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢦⣤⠿⡿⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠈⠘

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣗⠒⠒⠖⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖⠖⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠟⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠋⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠋⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣄
⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣤⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠐⠀⡆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠐⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢈⠐⠦⣍⣛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣉⡓⠦⣌⣙⠒⠌⠉⠛⠷⣦⣄⠀⠀⠀⣀⣠⡤⠴⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⠬⣅⡒⠬⣕⡂⠌⠙⠒⠄⠀⠀⠀⠙⢷⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⠲⢭⣑⠲⠬⣑⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠒⠒⠛⠛⠶⣤⣌⣀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣀⣤⣤⠶⠾⠛⠛⠛⠓⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⠶⠾⠿⠷⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣄⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢩⡿⣍⠿⣯⣉⣉⣉⣉⣉⣁⣤⣤⣤⣤⣴⠶⠶⠟⠛⠛⠋⠉⠁⠀⠀⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⠀⠹⣆⢸⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠇⠀⠀⣀⣀⣘⣿⣧⣶⣶⣶⡶⠶⠶⠶⠚⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⢀⣠⣴⠶⠞⠛⠉⢉⣿⣴⠟⠛⠉⠉⠉⠉⢙⣛⣛⣛⠛⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦
⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠭⠭⠍⠉⢓⣒⣒⣒⣒⣒⣒⣒⡂⠀⠀⠀⠀⠀⢉⡿⣥⣀
⣸⠏⠀⠀⠀⠒⠤⣀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠤⠤⠤⣀⣀⣀⣀⣀⡒⠒⠒⠒⠶⠶⠶⠍⠁⠀⠀⠀⢸⠁⠀⠉⠓⠦⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⢀⠀⣀⣈⡉⠉⠙⠛⠛⠛⠒⠒⠒⠒⠒⠒⠢⠤⠭⠭⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈⣙⣷⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢶⣭⣭⣭⣍⣛⣛⣛⣛⣷⣶⣶⣶⣶⣤⣤⣤⡤⠤⠀⡀⠀⠀⠀⠐⠒⠀⠀⠀⠈⠳⡶⠶⠛⠛⢹⡏⠉
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣍⡉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣛⣛⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⠀⠀⣸
⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣤⠶⠛⠉⠉⢁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠤⠤⠀⠂⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⡀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⠶⠶⠶⢒⣛⣋⣩⣭⣤⠴⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣙⣮⣍⣁⣀⠀⠀⢈⣉⣉⣭⣤⠤⠶⠶⠞⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡮⠶⠒⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠋⠁⣀⠤⣖⡮⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠉⠀⡀⠄⣐⡨⠔⠚⠉⣁⣤⡴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠚⠉⠀⡀⠤⣐⡨⠤⠒⠉⢀⣠⣤⠾⠟⠋⣁⠄⠂
⠀⠀⠠⢔⣀⠦⠀⠂⠈⠀⠀⣀⠠⢐⣈⠥⠒⠋⠁⣀⢤⣰⠼⠚⠉⣀⠤⠖⠉⠰⠀⠀⢀⣀
⣠⠤⠚⠉⠀⠀⢀⠠⠀⣂⠥⠐⠊⠉⠀⡀⢄⣢⠵⠒⠉⢀⠠⠐⠈⠀⠀⠀⠀⡠⠑⣈⠤⢾⠥⢛⡩⠗⣰
⢂⣀⡠⢄⢂⠭⠘⠀⠁⠀⢀⠠⠐⡨⠅⠚⠉⢀⠠⠐⠈⠀⠀⠀⠀⢀⡠⠒⢁⠴⢋⡤⠊⣡⠖⢉⡠⠚⢈
⢠⡤⠖⠋⠀⠀⣀⡤⢐⡨⠔⠂⠁⠀⡀⠔⠊⠁⠀⠀⠀⠀⢀⠠⠂⢁⠤⢊⡡⠚⢁⠴⠋⢀⠔⠁⢀⠔⠁
⠀⣀⡀⠄⢆⠩⠐⠉⠀⠀⣀⠄⠂⠁⠀⠀⠀⠀⠀⢀⠄⠊⢁⠠⠂⣡⠔⠉⡠⠒⠁⡠⠚⠄⠀⠀⢈
⠀⢠⡔⠚⠁⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⡀⠔⠊⢀⠠⠂⢡⠴⠊⢀⠔⠊⡠⠔⠁⠀⠀⠁⠀⠤⠔
⠀⠘⣀⡠⠔⠊⠁⠀⢃⠀⠀⠀⠀⡠⠐⠈⢀⠤⠂⢁⠠⠊⢁⠠⠊⢀⠔⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢣⠤⠒⠁⢀⠄⠊⢀⠠⠂⠁⡠⠐⠁⣀⠔⠁
⠀⠀⠀⠀⠀⢸⡀⠔⠋⠀⡀⠔⠊⠀⡠⠐⠁⡀⠔⠈⢀⠠⠊
⠀⠀⠀⠀⠀⢎⢠⣂⠔⠈⠀⡠⠐⠁⢀⠄⠊⠀⡠⠄⠁
⠀⠀⠀⠀⠀⠄⠀⠑⠂⠵⡃⠄⡠⠂⠁⢀⠔⠈⠀⡈
⠀⠀⠀⠀⠈⠢⢄⣠⠒⠀⠂⢍⠂⠸⠊⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠛⠛⠒⣶⠄⠀⠐⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⢀⠞⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠁⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⠼⢦⣄⠀⠀⠀⠰⠋⠀⠀⢀⣀⣨⡤⠤⠴⠶⠶⢦⡀⣼⡇
⠀⠀⠀⠀⠀⣾⠙⣍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢈⡤⢒⣛⠫⣄⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁
⠀⠀⠀⠀⠀⠸⠀⠹⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡐⠁⠀⢱⠘⣆⠀⠀⠀⠀⣀⡤⠎⢸
⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢠⠂⢀⣿⠤⠐⠊⠉⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⡇⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠡⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠂⣴⡀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣯⡒⠢⣤⠤⣀⠀⠀⠀⠀⠴⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⠙⠷⠊⡬⣹⠲⡁⠀⠀⠀⢠⢀⡇⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣌⡉⠒⠂⠤⠁⠁⠀⠀⠀⡌⡼⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠀⠁⠀⠪⠚⠀⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⢀⡀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⢛⢉⡉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠙⢦⡀⠀⣙⠃⠉⠳⠆
⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠢⡀⠀⢙⡂⠉⠻⠦
⠀⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢮⣄⠈⠛⠶⠄⠻⢶⡤⣮⡀
⠀⠀⠀⠘⣿⠟⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⠛⠦⠶⢦⡄⢶⣄⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡦⣤⣀⠴⣤⠰⣇⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡄⠀⠀⠐⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠉⢳⡄⠀⢹⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⢤⣾
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢳⡆⠀⠘⣿⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣤⣤⣄⣀⡀⠀⢤⣤⣶⣶⣶⣶⣶⣶⣤⠄⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠙⠻⢿⣿⣿⣷⣦⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠂⠀⠰⠀⠀⠈⠃⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠆⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⣆⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⡀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⣴⣿⠂⠀⠀⠀⠀⠂⠀⢀⡈⠀⡉⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⢿⣿⣶⠾⡷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⢺⣾⣷⢚⣏⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣹⣙⣻⣿⣫⡥⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣼⣸⣽⣿⣽⠗⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⢼⣼⣯⠴⣟⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⠺⢿⡿⢚⣯⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⢻⢿⣿⣿⣿⣻⣹⢹⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣝⣹⣺⣿⣿⣿⢼⢬⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⢼⣭⣿⣿⣿⢾⠸⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢺⢶⣿⣿⣿⣻⢸⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣹⣻⣿⣿⣿⣹⢹⣹⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢼⣽⣿⣿⣿⢼⢸⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢶⢺⢺⣿⣿⣿⢲⢰⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⣿⣿⣚⢚⣺⣇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣻⡿⡟⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⢾⡿⡗⣇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡟⣻⣟⣏⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣏⣿⣿⡧⡯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣼⢬⡧⣾⣷⡶⡷⣿⠿⠿⠿⣿⢿⡦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢾⠶⡗⢿⣟⣟⣟⣿⣙⣛⡃⡿⢺⠧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢻⢛⣏⣻⣟⣯⣯⣿⣭⣩⣏⣿⣻⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣹⣩⣥⣽⣷⡷⡶⣿⠶⢾⡦⣯⣼⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣼⠴⡧⢾⡗⡗⣟⣿⢛⢚⡗⠿⢾⡧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢾⠒⡗⣻⣏⣏⣍⣿⣹⣹⡏⣟⣻⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣻⣉⣏⣽⣧⡧⡷⣿⢼⢼⡧⣭⣽⣏⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠶⠷⠾⠷⠗⠗⠟⠚⠲⠶⠶⠾⠷⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣡⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠐⠀⣌⠉⢛⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⣻⣉⡀⠈⠛⠀⠀⢙⣿⣿⣏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢣⠀⠚⣿⣦⣤⠂⠀⣴⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⣶⣿⣶⣶⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠟⠛⠛⢯⣡⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣈⡄⠀⠀⠀⠀⠀⢀⣀⢀⣠⣼⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⣀⣶⣶⣶⢶⡶⣶⡶⠶⠶⠶⢶⠒⠀⠒⠖⠡⠶⠶⠶⠒⠶⠰⠶⠶⡝⢻⣿⣿⣿⢿⠒⠒⠒⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠘⣿⣿⡧⣤⠧⠾⠄⠠⠶⠖⠲⠒⠲⠖⠒⠒⠒⠒⠒⠐⠒⠒⠂⠒⠒⠺⣿⣿⡗⠚⠂⠂⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠀⠀⠀⠀⣿⣿⠇
⠀⠀⣿⣿⣧⡤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⢶⠦⠤⠄⠀⠶⠶⠶⠶⠶⢶⣿⣿⡷⣶⠶⡶⠦⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⡤⠀⢠⣤⣀⣄⣀⠀⣰⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⢀⠀⠀⡀⢀⣀⣠⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣼⣿⣿⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢦⡸⠋⣉⣘⣿⣿⣿
⠀⠀⢻⣿⣿⣟⣀⣃⣼⣿⣌⣉⣀⣁⣤⣀⣠⣤⣿⣤⣾⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣏⣉⣉⣉⣉⣉⣉⣹⣏⣉⣉⣍⣙⣿⣿⣶⣷⣾⣶⣼⣏⣀⣹⣿⣿
⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⡿⢿⡿⠿⢿⡿⠿⠿⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢸⣿⣿⡇⠀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠘⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠚⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢱⣿⡿⣿⠀⠀⠀⠀⢼⣿⣿⣧⠀⠀⢸⣦⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⢸⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡏⣸⣟⢃⣿⡀⢀⡀⣀⣘⣿⣿⣿⡆⠀⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠚⠋⠙⠃⠀⠁⠀⠛⠈⠀⠈⠉⠁⠉⠉⠀⠉⠛⠋⠃⠙⠉⠉⠉⠛⠻⠿⠛⠛⠙⠛⠋⠂⠀⠂⠒⠐⠒⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⡀⠀⣄⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⣀⣤⣤⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠑⣀⡁⠶⡉⣣⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣟⡄⡸⣾⠀⠀⠀⠐⠠⠀⡐⠂⡂⢢⡀⠀⠀⢀⠰⣄⠩⢤⠘⢣⣄⠹⣆⡙⣥⠚⡾⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⠛⠘⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡈⢤⡈⢤⡀⠻⡤⠑⢢⡈⢵⠆⠓⠀⠈⠈⠀⠸⢡
⠀⠀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠦⠉⠠⠚⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢠⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⡤⠧⠀⢀⡤⠀⠀⠀⠀⠀⠀⡤⢀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⡀⠛⠀⣠⡶⠀⠀⠀⠠⠇⠉⠉⠉⠉⠛⠉⠉⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⢀⡤⠖⠀⠀⣀⣠⢠⣤⣤⠶⠶⠴⠁⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀⠰⠁⠀⡀
⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠀⠀⢲⣶⣾⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠘⠲⠆⠒⠊⠀⠀⠀⠀⠈⠉⠠⠈⠀⠁⠉⡗⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠁⣀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠄⢤⠀⠀⠧⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠹⠁⠠⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⡦⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠘⠀⠀⠃⠀⠅⠀⠐⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣐⢁⠺⠁
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⠤⠊
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⢤⠄⠈⠀⠀⠀⠈⡀
⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⢆⠴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⠁⠉⠀⠀⠀⠂⠀⠠⠤⠄⠤⠔⠈
⠀⠀⠀⢀⡀⡧⠀⠴⠆⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠓⡼⠆⠀⣀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠐⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠐⠒⢤⡄⠀⠈⠁⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈
⠀⠀⠀⢀⡀⠀⠀⠟⠀⠀⠴⠀⢁⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠁⠀⠷⠀⢰⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠐⠆⠀⠀⠌⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⡶⢠⡀⢒⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠠⠈⠃⠀⠀⠀⣀⠀⠀⠀⠈⠇⡈⠕⠉⠀⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢁⠀⠃⠠⠄⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣦⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣟⣿⣿⡭⠭⢽⡿
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠙⠛⠚⠳⣮⣽⣿⣯⣭⣿⡭⠽⢭⣭⣉⣉⣙⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠒⠂⠀⠀⠠⠀⠀⠠⡄⡀⠈⠉⡙⠉⢠⠊⠈⠀⠨⠻⣿⡟
⣤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⣈⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣷⣬⣿⣭⢡⣆⣀⣈⣄⣠⣄⣴⣶⣶
⣾⠓⠤⣄⣀⣀⣂⡈⣀⠀⠀⠑⠒⠖⠘⠉⠉⠉⠉⠉⠉⠉⠤⡤⠤⣤⡭⣽⣟⣄⣬⣤⣧⣏⣈⣈⣹⣿⣿
⡉⠉⡒⠀⠐⣦⢤⣬⣾⣿⠟⢻⠇⠚⢛⡟⠛⠛⠙⠉⠉⣽⣿⣿⢻⡉⠻⠉⣯⣌⣁⣉⣘⣩⣭⣿⣿⣿⣿
⣷⣦⣶⡶⠶⠀⠀⠈⠁⢹⠷⡄⢀⣶⡄⠀⠀⠀⠉⠹⣶⣿⣿⡇⢸⣿⡀⢿⣛⣅⣀⣀⣈⣹⣿⣿⣿⣿⣿
⣿⣿⡿⠃⠀⠀⠀⠀⠀⣸⠀⠀⠸⡟⠻⢿⣾⣎⣿⣟⣏⣿⣿⣷⣾⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠉⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠇⠀⠀⠀⢸⣿⣿⣏⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⣸⣷⡀⡰⣾⣦⣲⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⢀⡳⠒⠐⠒⣀⠀⢀⣀⠀⠀⠀⣠⡀⠀⢐⡈⠀⠐⠀⠒⠀⢢⡂⣤⠄⣁⠀⠈⢀⠀⠧
⡀⢎⡠⡐⠀⠀⢠⠁⠈⣖⠀⠀⢂⣦⠈⠠⠈⠤⠀⠤⠈⡒⢀⣀⠀⣡⠂⠬⠡⠈⠩⡀⢠⠑⣇
⠘⠈⣇⡁⠗⠂⢀⠨⠘⠃⠀⣈⣀⣈⠑⠎⠀⢲⠠⠤⠙⢠⠀⠄⠀⢌⢀⢠⢄⡁⠁⠀⡂⢣⠠
⠐⣂⠀⠹⡄⠂⣽⠷⠈⢟⡺⢿⣟⢋⡀⠀⠢⢀⡅⠀⠁⠀⢀⠀⢀⠰⠅⠸⠓⣈⠀⠈⠄⡂
⠀⠈⠓⢄⠜⡿⡻⠘⡅⠈⠄⢠⠳⢥⣔⡆⠀⠆⠤⣉⠆⠀⠌⠄⢚⠒⠀⠀⠈⠁⢯⠀⠦⠀⡄⢀⡀
⠀⠀⠀⠀⠁⢀⠈⢄⣃⠉⢁⣼⠛⠃⠈⠂⠇⠀⠈⠹⠀⠢⠀⠀⠒⢄⠀⠀⠀⠀⠀⠁⠬⢂⠚⠶⠯⡐⣄
⠀⠀⠀⠀⠀⠀⢢⠀⢹⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⢀⡔⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡃⣤
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠨
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠁⢘⡢⠄⠀⠀⠀⠀⠀⠀⠠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡵⠈⣀⠆⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣀⡈⣀⣀⣀⠀⢠⠄⠠⣬⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⣀⣀⣀⡘⡄⠠⡁⠀⠒⠀⠀⢠⢀⣀⠀⠰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠤⠤⡆⢠⠤⠤⠤⠤⠤⠤⠤⠦⠭⠤⡆⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⠂⡦⠤⠰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠐⡇⠰⠀⠀⠤⠤⠤⠤⠤⠤⠄⠐⡇⡴⠠⠤⠤⠀⠀⠀⠀⠀⠀⢺⠀⠷⠶⠲
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡐⠀⣿⢸⠒⠂⠀⠀⠀⠀⠐⠚⠃⠀⡇⢛⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⠀⡗⠒⢺⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠈⠁⠙⠚⡇⠈⠉⠉⠯⡝⠉⠉⢻⠙⠗⠚⢹⠉⠀⠀⠈⠉⠉⠉⠉⠉⡖⠋⠉⠁
⠀⠀⠀⠀⠀⠠⠧⣄⣀⢠⣦⣈⣄⠀⠀⠀⠀⠀⡇⠀⠀⣸⡹⢤⠀⠀⠈⠉⠃⠀⢸⣚⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠇⠀⠀⠉⠉⠉⠀⠀⠀⠀⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⡾⠛⠻⢿⠿⠛⠻⣿⠟⡿⢿⡿⠟⠛⢿⡿⠛⠻⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣤⣤⡀⢀⣤⣤⡀⣠⡀⣄⠀⣠⣤⣄⢀⣤⣤⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⢷⣯⣁⠿⣿⣶⣤⡀⠀⣇⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣈⠑⠒⠁⡀⠉⠛⠻⣿⣧⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡼⠛⠿⣾⣝⡻⢿⡿⢟⣫⣴⣮⣭⣛⡿⠆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⣤⣄⡉⠳⢤⣀⣀⣠⠴⠛⠛⠲⠦⣬⣹⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣾⣥⣤⣤⣤⣭⣿⣷⣾⣿⣯⣥⣤⣭⣽⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠃⠸⠛⠿⠟⠳⠆⠈⠉⠀⠰⠾⠻⠗⠶⠀⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣷⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⣴⣾⣷⣿⣶⣶⣦⣤⣄⣤⣄⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣀⠀⠘⠟⠿⣿⣿⣿⣿⣿⣦⣤⡀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣗⣒⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹⣿⣶⣿⡟⠻⣿⣿⣿⠙⢿⣿⣿⣿⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣈⡉⣁⣀⠤⡽⠿⠿⠀⠈⣿⣿⣿⡆⠀⠀⠀⠈⢿⠿⠟⠛⢿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠡⠀⠀⠀⠁⠉⠉⠙⠿⠟⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠋⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢌⣪⣮⡦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠿⠟⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣷⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠋⠀⠀⠀⠀⠀⢀⢮⣿⣿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢿⢿⢿⢿⡿⠋⠀⠀⠀⠀⠀⠀⡲⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⣾⠾⠿⡾⠀⠀⠀⠀⠀⠀⠀⢔⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣤⣐⡓⠂⠀⡂⠀⠀⠀⠀⠀⠐⢟⣿⣿⣿⣿⣿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥⣤⣄⡀⠀⢀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢪⣪⣾⣻⡿⣻⣿⡯⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣏⣛⡋⡛⢻⣿⣿⣿⣿⣗⢄⢔⣾⡿⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣾⣗⢵⣿⣾⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣵⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠋⠉⠉⠀⠀⠁⠉⠉⠉⠛⠿⠻⣿⣿⣿⣿⣯⣿⣿⢟⣿⣾⣿⣿⣿⢟⣿
⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⣿⡻⣿⢿⣿⡻⣿⣿⢟⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠔⠁⡀⠙⢢⠀⠀⠀⠀⠀⢠⠃⣀⣤⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇
⠀⠀⢨⠀⠠⠋⠓⠀⢣⠤⡄⢠⠒⣼⠘⠋⣤⡀⠀⢹⠀⠀⢀⡀⠤⢤⠐⠒⠒⣦⠀⠠⢠⣀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⡠⠀⠀⠀⠈
⠀⠀⢈⠀⢀⡾⠓⠀⢸⠀⣿⢨⠄⡟⠀⠠⣋⡉⠀⣼⠀⠀⡌⠀⡄⠀⠈⣦⠀⠀⠁⢄⠀⠀⡤⠃⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠘⡀⠀⣰⢦⡀⢸⠀⡇⠀⠇⡇⠀⠴⣫⡓⠀⣿⠀⠀⢰⠀⠈⢢⠂⠀⠑⡄⠁⠀⢣⠊⠈⢴⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⡡⠶⠄⢸⡞⠀⠀⠈⣇⢀⣘⠥⢥⣀⣹⠀⠀⠀⠐⡄⠈⠑⣤⠊⠈⢢⡄⠈⠁⡠⡎⠀⠀⠰⠶⠦⠤⢤⣄⣀⡀⠀⠀⠀⠆
⠀⠀⢰⡗⢚⠃⠉⠉⠙⡇⠀⠀⠀⠸⡅⣠⢠⢠⡆⣸⠀⠀⠀⠃⠈⣦⠂⠀⢡⡀⠀⠈⢂⠈⠑⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠼⠤⡄
⠀⠀⠀⢿⣾⣼⣾⣇⣧⠇⠀⠀⠀⠀⠣⠽⠧⠼⠾⠟⠀⢀⣤⣧⡀⠀⠑⡤⠀⠉⢢⠄⠈⢣⣠⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣩⣀⣸⢧⣀⣀⣀⣀⣀⣀⠀⢠⣾⠉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⠀⠀⢀⠀⠀⠀⠀⢽⠒⢻⡵⠊⠁⠀⠀⠉⠉⠉⠉⠛⣌⡿⠽⡇⠀⠀⠀⣴⣂⡤⢄⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⣜⣛⣉⣉⣉⣉⣛⣛⣒⣒⣒⡒⠒⠾⠷⠴⡇⠀⠀⣸⠇⣏⠑⡞⠉⣴⡂⢀⡀
⠀⠀⠀⢀⠐⠷⠶⠶⢤⣤⣤⣤⣤⣀⣤⠞⢀⣆⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⡏⠀⠀⣿⢠⢎⣩⠓⢠⠧⣸⠁⠲⢇⡴⢠⡄
⠀⠀⡀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠈⢆⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⠟⠲⠤⣍⠸⠁⠶⢥⣝⣉⠖⢀⡟
⠀⠀⢆⡀⠛⠛⠛⠛⠓⠒⠶⠤⠤⠄⠠⣶⣦⡄⠼⡄⠀⢻⢻⡟⠉⠁⠀⠀⠀⠀⠒⠒⠒⠶⢲⠋⠀⢰⡏⠘⠢⣤⣀⠛⢱⠆⣼⠈⡏⢘⡏
⠀⠀⠀⠈⠁⠒⠈⠉⠁⠀⠀⠀⠀⠀⠠⠤⠤⠀⠠⠔⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢀⡴⠋⣤⠋⠰⢠⡍⢉⠓⠲⠤⣤⣁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣟⡶⠒⠂⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢤⣈⠑⠂⠠⠊⡨⠓⠦⢦⢨⠏
⠀⠀⠀⠀⡠⣶⣶⠯⠷⣻⣿⡿⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⠤⢔⢢⠆⠐⠒⠬⡈⠓⠒⠤⠧⣏⡐⠀⡌
⠀⠀⠠⠊⠙⢯⣿⣾⣻⣯⠟⠁⠀⠤⢀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⠌⡀⠴⠋⡀⠀⠀⠀⠈⠆⡀⠀⠀⠀⠙⠛
⠀⠐⠀⠀⠀⣀⠀⠉⠁⡠⢀⡾⡻⣷⣦⣄⣀⠀⠙⠺⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠠⠀⠀⣶⡟⣽⣷⡀⠈⠀⠈⠳⣕⣦⡟⠁⠈⠁⠉⠉⠁⠊⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠱⡀⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⣀⠀⠀⡰⠃⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢄
⠀⠀⠑⣯⠉⠁⠀⠀⢧⠀⢦⡀⠀⠀⠀⠈⣉⠽⠪⣍⣀⠴⠊⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢌⠀⠀⠈⠻⠀⠀⠀⠈⢂
⠀⠀⠀⠙⠀⠀⠀⠀⠈⢧⠀⣈⠥⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠂⠀⠀⠀⠀⠐⢤⠔⠁⠠⡀
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠄⠀⠀⣀⡀⣀⣀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊
⠀⠀⠀⠀⠈⢆⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠶⠤⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⣖⣒⣒⣚⣻⣯⣭⣭⡍⣩⣤⣤⣭⣵⣶⣒⣒⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⣿⠿⡿⢿⡿⣿⡿⣿⢷⡏⠀⠤⠤⠤⠤⠭⢩⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡏⣜⠸⢿⢸⠇⠾⡀⡿⢱⣿⢸⠁⠠⠄⠀⠈⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣷⣾⣤⣾⣤⣦⣴⣧⣴⣮⣿⢸⡄⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⡤⠤⠼⣿⣯⣿⣿⣿⣟⣛⣛⣻⣿⣿⣿⣾⣧⣀⣭⣭⣭⣭⣭⣭⡭⠿⣂⡀
⠀⠀⠀⠀⠀⢠⣟⡭⣭⣥⣀⣒⣒⣀⣀⠤⠤⠤⠈⠉⠉⣙⡋⠉⠙⢛⣉⡉⠭⠭⢭⣀⣀⣒⣦⡤⠽⠿⢷⠟⠃
⠀⠀⠀⠀⠀⢸⢸⣇⡂⠀⠐⠀⠈⠩⠍⠉⠙⠛⠛⣒⡒⠒⢂⡤⠤⠖⠒⠺⠿⠯⠉⠉⠀⠀⠀⠀⠀⡀⢸⡀
⠀⠀⠀⠀⠀⠸⣏⣓⣒⡒⠂⠀⠭⠥⣄⣀⣠⣤⠤⠤⣤⣄⢼⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⣂⣀⣠⣥⣥⣼⣷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣷⣶⣶⡾⢁⠐⣶⢶⣒⡩⠙⣲⢤⣈⣩⣤⣴⣶⣶⣿⣿⣿⣯⣽⠿⠶⠞⠛⠋⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡿⠹⠏⣿⣿⢡⠁⠀⠀⠀⠀⠩⢓⡦⣍⡺⢽⣻⣿⣿⡟⠉⠉⠁⠀⠀⠀⠅⢀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⣼⣧⣾⣿⠃⠂⣠⣴⣦⢠⣠⣀⡀⠈⠑⠫⢗⡾⣝⢮⡟⣦⠀⠀⠒⠀⢀⣀⡠⠤⣖⣚⣿⠆
⠀⠀⠀⠀⣀⣠⣤⣶⠛⠻⠿⠿⣿⠏⠌⠀⠉⠈⠈⠈⠘⠹⠔⡀⠀⠀⠐⢰⠃⡠⢱⣯⡴⣾⣾⣿⣽⠶⠞⢛⣋⣽⣷⣤⣄
⢀⣴⣶⣿⣿⣿⣿⣿⣶⣶⣦⣴⡏⡈⠀⠀⠀⠛⠷⣦⡆⡀⠘⢿⠁⠀⢠⠇⢸⢣⡿⠿⠟⠛⣉⣥⣴⢶⣾⣿⣿⠿⣟⣻⡽
⣾⣿⣷⣶⣦⣬⣍⣙⣛⡻⢿⡟⡐⣥⡁⠀⠈⠈⠓⠂⣉⠛⠀⠀⠀⢀⡏⢠⢇⣾⣥⣶⣾⣿⣿⣿⠾⣟⣫⡽⠶⠛⠩⣸⣧
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⠁⠘⠿⣦⣷⣄⡀⣀⠀⢠⡀⠀⠀⡞⠀⡞⣾⢿⣿⣿⣛⣩⡵⠞⠛⠉⢁⡠⠴⣒⣯⠽⠚
⠀⠀⠉⠉⠙⢛⣿⡿⠿⣿⢃⠃⠀⠀⢠⣄⣹⡟⠿⣿⣤⣄⡀⠀⡜⠀⡜⣼⣿⣭⣽⣿⠍⣀⡠⠴⣒⣩⣵⣾⡛⠉
⠀⠀⠐⢾⣿⣿⣿⣯⣽⠇⠂⠀⠀⠦⠀⠉⠛⠷⠤⡀⠉⠁⠀⣼⠁⣸⣰⣿⣿⣿⣿⣿⣭⣵⡶⠟⣋⠉⢺⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠈⠙⠻⠿⣿⡈⠀⠀⠀⠙⡲⢍⠐⠤⣀⠀⠀⢀⢰⠇⢀⢧⣿⠦⠤⡭⠿⠛⠉⠀⢀⡡⢐⣀⠼⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠉⠃⠀⠁⠀⠀⠀⢢⡟⠀⣞⣾⣏⡀⠀⢫⠀⠀⣀⠬⠖⣊⣭⣶⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣻⣦⡴⠶⠶⢦⣀⠀⣾⣀⣘⣼⣿⣦⣬⣉⡉⢛⣩⣴⡾⠿⠛⠉⠁
⠀⠀⠀⢀⣀⣤⡤⠴⠖⠚⠛⠉⢉⡀⡀⡀⣀⣠⣀⡌⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⢠⡞⠉⣁⠀⠀⢀⠀⣀⣦⡰⠼⠋⠉⠉⠙⠁⡀⢄⣠⣀⠀⠙⠻⢿⣏⡁
⠀⣿⢀⠑⠢⡁⠀⡈⠃⠉⠁⠀⠀⣄⢦⣰⠬⠷⠽⠚⠋⠉⠀⠀⠀⠀⠈⠛⠷⣤⣀
⠀⣿⠸⢞⣠⢈⠐⢄⡀⠀⠀⠸⠒⠋⠁⠀⠀⠀⠤⢄⡲⠤⠴⠖⠒⠋⠀⠀⠀⠀⣙⠳⣄
⠀⠈⠷⣌⡙⢻⣶⠀⠈⠢⢈⠂⠀⠀⠀⠀⠐⠂⠉⠠⠖⠒⢉⠀⡤⠤⢐⣒⣨⡥⢴⡶⠃
⠀⠀⠀⠈⠻⢦⡀⠸⢴⣤⣀⠑⠠⡑⠀⠀⠀⠠⠄⣒⣈⣥⡤⠶⠚⠛⠩⠉⡀⠂⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠈⠳⡏⠀⠈⢠⠃⣴⠚⠋⡉⠡⠄⢀⠀⠀⠀⠀⢈⡡⠤⠞⣹⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠘⠃⡞⣸⡯⠉⠀⠀⣀⠠⠤⠐⣒⣢⣭⡴⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣇⠹⠗⣒⣨⣥⡴⠶⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠀⠀⣠⣤⣄⡀⢠⠔⠆⠀⣀⣀⠰⣇⠆⢰⠀⣦⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⣛⣯⣍⢹⣿⠷⠤⠞⠿⠷⠥⠤⠇⠀⠘⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣰⠛⠂⢀⣄⣜⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠎⠛⠛⠛⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⡀⡀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⣧⣉⣭⣿⣟⣿⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⠉⣉⡉⣙⣻⠀⠀⡁⠀⠀⠀⠀⣁⠀⠀⠀⢰⣶⣶⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⠶⠓⠀⢰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡖⣴⠾⣾⠏⣽⠐⣮⡀⣞⠚⠛⣧⣟⣶⡄⠀⠎⠉⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⣴⣖⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠚⣿⣇⣼⣄⠁⠈⠅⢈⡀⣁⠀⢄⡀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣿⠷⣶⠀⠀⠀⣤⣀⣼⣯⣈⣉⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⡘⣿⣿⣿⣿⣿⣾⣿⣿⣤⣼⣿⣿⣿⣽⣽⣿⣿⣷⣤⣤⣄⣀⣠⣾⣥⣤⣿⣿⣭⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠀⢈⠉⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⠀⠑⠀⠂⠀⠰⠂⠀⡀⣉⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠈⠀⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠠⠤⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢁⣀⢀⣀⡀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠁⠈⠉⠁⠀⠀⠁⠀⠀⠀⠉⠈⠉⠁⠁⠀⠀⠀⠀⢉⠉⠉⠰
⠀⠀⠀⠀⠀⠀⢐⣀⡠⠶⠦⠠⠤⠠⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⢆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⠶⠾⠻⠳⠶⠶⠶⠤⠦⠖⠲⠒⠒⠢⠶⠒⠒⠒⠒⠒⡒⠒⠒⠒⡒⠛⡛⢒⣒⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣿⠁⠀⠈⠉⠈⠀⠉⠉⠉⠁⠈⠉⠀⠁⠀⠉⠉⠉⠁⠀⢸⡟⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠿⡿⡇⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢶⠒⠒⣷⡲⠦⠂⠦⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢆⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣨⢀⣼⠶⠀⠲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢴⠋⢹⠀⣀⢀⡤⢾⢣⠉⢉
⠀⠀⠀⠀⡠⢄⠀⠐⠀⠀⠈⢄⡰⣠⠾⡏⠁⢰⠀⢸⢸⠀⠀⠀⠀⢀⠀⠀⡠⠶⠠⠄⠀⠀⡠⠠⡀
⠀⠀⠀⡈⠀⢈⢀⣀⠀⠐⠞⢻⢁⡇⠀⠁⠀⠀⠁⠸⠸⡀⠀⠀⠀⠈⠀⢬⣲⣶⠀⠀⠐⢳⢶⢈⠇
⠀⠀⠀⡅⢀⠾⠛⠀⠀⠀⠀⢸⠐⡄⠈⠀⠀⠀⠀⠘⡆⣇⢀⣀⠀⠀⠠⢴⣖⡓
⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⠸⠄⡇⠀⠀⠀⠀⠀⠀⢁⣞⣁⣀⡀⠀⠀⠀⣊⡀⣵⠀⠀⠀⣢
⠀⠀⠀⢉⠀⡇⠰⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠈⣆⢻⣿⠤⢤⣀⣑⣀⠡⠜⢀⠀⡤⠟⡅
⠀⠀⠀⠸⡀⠅⠀⠀⠀⠀⠀⠀⢸⡈⡦⠺⠂⠀⠀⠠⠌⣙⠀⠉⠉⠈⠐⠛⠓⠭⢓⣰⣾⣥⣾⠃
⠀⠀⠀⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⢿⡲⢦⣤⠄⢀⠀⠀⠁⠠⣴⠀⠀⠀⠀⣥⣰⢿⡆⠈⠉
⠀⠀⠀⠀⠘⡀⢱⢀⣤⣄⠀⠀⠈⣄⣷⣼⠿⢖⣂⠴⠶⠦⢤⣋⡀⠀⢀⣴⣿⣽⣾
⠀⠀⠀⠀⠀⠹⡄⡏⣴⣀⣀⠀⠀⠀⠁⢬⡆⠀⠉⠉⠉⠩⠑⢦⢱⡴⠿⠿⣽⠏⠁⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠀⡿⡿⠥⢍⣢⣀⡠⠃⠀⡀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠰⠢⠑⠉⠉⠙⠒⠒⠬⢌⠳⠄⠀⣢⣞⡩⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡌⣶⣦⢤⣠⣀⣀⡀⣾⣃⣶⣟⣩⡟⢃⡟⠁
⠀⠀⠀⠀⠀⠀⠔⡾⣱⠋⠁⠉⠀⠉⠛⠛⢶⢷⠀⠀⠉⠉⠡
⠀⠀⠀⠀⠀⠀⠁⠐⠂⠤⠝⡂⠤⣀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠢⠤⠕⠃⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⢀⣠⠀⢀⡀⠶⠘⠉⠐⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢴⡁⢷⠀⣀⠀⢾⡿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣥⠁⠁⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡆⣧⡆⠄⠸⡄⠘⠙⠃⠀⢀⠀⠀⠀⠀⢸⣿⠛⢥⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⡖⡀⢙⡄⠠⠀⠀⠄⠀⠀⠀⠀⠀⣿⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢻⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢿⡌⣇⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠇⢀⣀⣤⣤⣄⣀⣀⢸⡧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⢷⠛⡆⠈⠃⠀⠀⢀⠀⠀⣀⣠⣤⣬⣿⣿⣶⣿⣿⣿⣿⡶⠶⣬⣽⣿⣲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡇⠁⠀⠀⠀⢐⣯⣷⣾⣷⣿⣿⣹⣿⣿⣿⡿⠿⠿⣿⣿⠿⣶⣽⣾⣯⡍⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢿⣄⣀⣲⣿⣿⣿⣿⠿⠛⠉⠈⠳⠭⠽⠿⠃⠀⠀⠀⠈⠉⠛⠒⠯⢽⣟⣻⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⣈⣛⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⣦
⠀⠀⠀⠀⠴⠾⠛⠓⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠺⣶⡬⠿⠿⠿⠵⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⣋⣉⣉⣉⣉⣉⡀
⠀⠀⠀⠀⠘⢷⣤⡤⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⠛⣛⣉⣉⣉⣉⣉⣩⣭⣭⣥⣤
⠀⠀⠀⠀⠀⠐⣧⣤⡶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠋⠉⠉⠉⢉⣉⣉⣉⣀⣤⣤⣤⣤⠆
⠀⠀⠀⠀⠀⠀⢸⣄⣠⣤⣤⡤⠤⠴⠶⠶⠒⠒⠒⠛⠛⠛⠉⣉⣉⣉⣁⣠⣤⠤⠤⠴⠶⠒⢀
⠀⠀⠀⠀⠀⠀⠀⣯⣀⣀⣤⣤⡤⠤⠴⠶⠖⠒⠒⠛⣛⣉⣉⣉⣭⣤⠤⠴⠶⠖⢒⣒⣛⣉⣭⣤⣶⣤⡶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⠤⠶⠶⠖⠒⢛⣛⣉⣉⣭⣭⣤⣤⡶⠶⠶⠿⠛⠛⠛⠉⠉⠉⣉⣀⣠⣤⠤⠤⡖⠲⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⣶⢶⡿⠿⠿⢿⠻⣿⡉⠉⢀⣀⣀⣠⣤⠤⠴⠶⠒⠒⢛⡏⣫⠉⠀⠦⢈⢣⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢈⠃⠹⣻⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢀⡀⠀⠀⠰⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⣰⠁⡰⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠃⢸⠃⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡏⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠀⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠑⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡉⠀⣸⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠑⠕⠙⣬⣇⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠶⡄⠀⠀⠀⠀⠀⠀⠀⠈⢛⣷⣄⠀⠘⣞⡟⠀⡴⠮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⣇⣄⣑⣒⣒⣒⣒⣒⣶⣥⣬⡵⠢⣾⣯⣽⣆⡇⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠅⠘⠒⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠌⠚⠥⠤⠤⠤⠑⠚⠤⠵⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⡠⠠⠤⠠⠄⠀⠠⠤⠄⠤⠤⠤⠀⡀⠠⠠⠤⠤⠤⠤⠤⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⢀⣀⡁⠀⡀⠀⢀⠀⡀⣀⠀⠀⣀⠀⠀⣀⢀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠒⠢⢸⢒⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠰⠒⢢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠉⠀⠀⠀⠀⠀⢣⢱⠀⠀⠀⣧⡀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⠀⠀⠀⢀⡴⠃⣜⡔⠲⠀⡗⠷⠂⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⠤⠀⡔⠁⠀⠀⠻⠀⠰⡀⡇⠈⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⢸⠀⠚⠀⢇⠀⠀⠀⠀⠀⢀⠇⡇⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠤⠜⢸⠀⠀⣠⣊⣑⣖⣉⣑⣖⣉⣢⡇⠀⢀⠐⠄⠚⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠉⠉⡇

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠘⠻⣷⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⡾⠟
⠀⠀⠀⠉⠉⠀⢰⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡄⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠋⢼⣷⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣶⣿⠎⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢸⣿⣷⢆⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣴⣶⠸⣿⣿⠃⠉⠋
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠿⠟⠃⢺⣿⡿⠰⣿⣾⡆⢸⣿⣿⠈⢿⣿⠇⠙⠻⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠋⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠰⠶⣶⣿⡿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠿⣿⣿⠛⠃
⠀⠀⠀⠀⠉⠁⠀⠀⣽⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡖⠀⠉⠛⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠀⠀⠀⠀⣿⠛⠓⢶⣦⠄⣀⣀⣀⣀⣀⡀⠀⣀⣀⣤⣤⣶⡄⠘⠿⡁⠀⢠⡇⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⠞⠁⠀⠈⢿⣿⣿⣿⠃⠀⠈⢿⣽⣿⡟⠀⠀⠀⠈⠳⣿⠇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢀⡀⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠠
⠀⠀⠀⠀⠘⠈⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡠⣯⣿⣿⡗⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢾⣿⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠙⢿⠟⠀⠀⡜⠙⠓⢢⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣠⠀⠀⠞⠙⠈⢥⠀⠈⣽⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠧⢀⣠⡆⠀⠀⣽⣿⡿⡗⠀⠰⠚⠓⠙⡄⠀⢸⣿⣿⣿⡆⠀⠡⣄⠠⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠠⡿⠿⢿⡃⠀⢈⣄⢀⣴⡅⠀⠀⣯⠟⠛⠣⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⡀⣀⣀⡤⠤⠤⠤⠤⠤⠴⠁⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠹⡀⠀⠀⠀⠀⠀⠀⣔⠙⡀⠀⡘⡅⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⢀⡤⠟⠒⠈⠉⠁⠒⠲⡄⠂⠐⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⢍⠑⠂⠤⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠀⠄⡀⢀⠙⢄⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡰⠁⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⢣⠀⠀⠀⠠⠈⠳⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⣱⡄⠀⠈⢄⣁⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠒⠒⢄⠀⠸⠛⠒⠀⣠⡾⠃⠈⡗⠓⠊⠁⠸⠦⡄⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠣⡀⢀⠜⠀⠀⠀⣠⡾⠋⠀⠀⢰⠀⠀⠀⠀⠀⣀⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⣠⠗⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⢀⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠊⠉⠀⠀⠀⠀⣀⡤⠖⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣠⡴⠚⣩⡿⣿⠉⠣⠤⣀⣀⣀⣀⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⢸⣇⣤⠞⡿⠁⣿⡇⠀⠦⠄⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⣀⡠⠔⠚⣇
⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⣰⢿⡇⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠉⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⢸⡷⠀⠉⠁⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠇⢣⡀⠀⠙⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⣄⠠⣠⠀⠈⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠙⠣⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠰⠖⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⡶⠖⠀⢠⡤⠀⠐⠖⡖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠱⡄⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡌⠀⠑⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡈⠧⡏⢇⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣾⣶⣶⣶⣶⣶⣖⣲⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣒⣶⣶⣶⣶⣶⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣟⡿⣿⣿⠿⠿⠛⠛⠻⠟⠛⠛⠛⠛⠻⠿⠻⢿⢟⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⡄⠉⢳⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⢐⡀⠀⠀⣤⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡏⠀⠀⢀⣠⣀⡈⢷⡄⠀⠐⠀⠀⣰⠃⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡃⢉⣿⠀⠙⠷⠶⠶⠚⣡⣏⡹⠈⠛⣦⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⣙⣓⣿⠀⠀⠀⠀⠀⠐⣯⠢⡅⣢⠜⣸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢺⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡤⡤⠴⠋⠀⠠⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣇⡀⢸⣒⣔⣶⣘⣆⡠⡤⠀⠀⠀⠀⣰⣶⣶⣼⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣒⣻⣿⣿⣻⣶⣒⣦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠉⠉⠉⠹⠿⣛⢛⡛⠛⢛⣛⡛⣛⣛⣛⣛⣛⠛⠛⢻⠋⠋⠋⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡄⡇⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⡉⠉⠉⢁⣀⣀⣀⣈⣉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠇
⠀⠀⠀⠀⠐⣶⡀⠀⣹⣦⠀⣴⠁⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅
⠀⠀⠀⠀⠀⢻⣿⡷⣾⠗⣿⡿⡄⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣷⢸⢹⣿⠃⣇⣀⣀⣀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣮⣼⣿⠀⠠⠀⠀⡆⢰⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢻⣠⣤⣦⣤⣥⡼⠿⠻⠛⠻⢳⣶⣶⣶⣶⣶⣶⠶⠿⠾⠿⠿⢦⣷⣶⣦⣬⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢻⣯⣤⣤⣤⣤⣤⠜⠛⠛⠛⠛⠻⠛⢤⣤⣤⣤⣤⣤⡿⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠁⠀⠀⠀⠉⠐⠂⠤⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠐⠠⠄⠀⠒⠠⢀⡀⠀⠈⠑⣤⠔⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⡂⠀⠠⠀⠠⠄⡀⠀⠀⠈⠀⠀⡐⠁⠀⡠⠄⡀⠀⠀⠀⠀⢀⣀⡀⠒⠀⠄⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠄⠀⢀⡈⠐⠂⠩⠒⠥⡂⠅⠢⠀⠀⠌⠀⠀⠐⣀⣀⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠠⠀⠀⠀⡀⠀⢀⠀⠀⡂⠄⡀⠀⠀⠀⢀⠎⠀⢀⠐⠂⠀⠀⠀⠀⠒⠂⠀⣈⠀⠉⠀⠀⠀⠀⢠⢃⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠄⠤⠀⠐⠀⠤⢁⠐⠉⠐⠠⡁⠢⠁⠀⢀⠂⠀⠠⠐⠂⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⡾⣿⡏
⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠨⢐⠂⠤⡈⠐⠠⡑⠠⡀⠈⠀⠀⠠⠁⠀⠐⠠⠁⠀⠀⢀⣀⡀⠀⠈⠉⠁⠂⠀⠀⠀⠀⡰⡳⡿⣽⣿⣀
⠀⠀⠀⠀⠘⡠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠈⠀⠀⡐⠁⠀⠀⠠⠄⠀⠒⠂⠀⠀⠀⣐⡂⠀⠅⠀⠀⠀⢀⡔⡜⡽⣼⣿⢏⣾
⠀⠀⠀⠀⢘⠖⠶⠤⠤⢉⣑⠒⠤⣈⠂⠀⠀⠀⠀⠙⠀⠀⠀⠀⠌⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠎⣜⣞⣼⣯⢫⣻⠃
⠀⠀⠀⠀⣿⠖⠒⠒⠲⠒⢶⢬⢵⣺⢥⣢⣄⡀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣮⢎⡾⡼⢣⡿⠁
⠀⠀⠀⠀⠘⡐⠡⡠⢤⠄⠀⣀⣈⣉⣋⣻⣿⣿⣦⣔⠀⡑⢉⣀⣤⣤⠴⠶⠲⠶⠰⠒⠆⢒⠒⠢⠤⣤⣀⠀⠠⢀⢪⣮⢋⣾⡵⣡⣟⠁
⠀⠀⠀⠀⠀⡴⢼⠲⠟⠭⠉⠈⡩⢍⣙⣷⣮⣮⣷⣿⣿⣿⣭⣯⣽⣶⣿⣿⣻⡿⠿⠟⡑⠒⠍⠃⣅⠂⠌⣙⠓⢪⡻⣣⢿⠟⣱⡟⠁
⠀⠀⠀⠀⠀⡳⡶⣶⡞⢽⡿⠻⠟⠛⠻⢿⣟⡓⣒⣒⠾⠬⠽⢝⣿⣯⣦⣿⣍⣽⠽⠖⣒⠪⠤⣆⣀⠉⠈⠑⡙⠱⣷⢿⠋⣴⡟⠀⠄⠀⠠⢀
⠀⠀⠀⠀⠀⠁⠅⢙⣷⣿⣂⣐⣒⡲⠦⠤⠤⣤⣥⣅⣭⠭⠯⠋⣍⣉⣉⡋⣩⣾⣿⣷⢖⡬⣕⡂⠬⢙⠲⢦⣽⢪⣿⢋⣾⠏⠐⠌⣀⠠⠁⢒⠠
⠀⠀⠀⠀⠀⡇⠘⡤⣻⢸⡃⠖⠂⠙⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢨⠣⣫⣿⡿⡳⡋⠀⢀⠈⠙⡒⢌⡒⢍⡷⣣⡿⠋⡐⠀⠀⢀⠈⠼⠀⠐
⠀⠀⠀⠀⠀⢽⠈⡆⠙⣿⣕⣿⣭⣤⣒⣒⣒⢄⣀⠀⡀⠀⠀⢀⣀⣬⢸⣾⡿⣫⣾⣳⡄⠀⢀⡤⠃⢒⠃⠨⡒⠺⠿⡀⠀⠔⡂⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⣧⠀⢀⣀⣀⣀⣭⣭⣭⣿⡿⠶⠾⠭⣶⢮⣘⣫⠾⠭⢡⠏⠁⠨⠀⡀⠌⠀⣂⠥⡄⠠⠁⠐⠈⣁⣄⠑⠐
⠀⠀⠀⠀⠀⠀⢻⢆⣠⢴⡝⠋⠉⠀⢀⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠲⠉⠀⠀⢸⣄⠀⠀⠐⢀⠐⢉⡠⢒⠥⠒⠋⠰⠋⠊⠁
⠀⠀⠀⠀⠀⠀⠐⢧⠃⣟⡿⠀⠀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢎⣻⣷⢀⠐⠄⠐⠈⠀⡐⣠⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⢇⣊⣉⡥⠤⠤⠠⡂⠰⠶⢒⢒⣂⣠⠔⠴⢀⠴⠽⡂⠔⢒⠩⠀⠀⠀⠡⢒⠉⠉⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠂⠒⠒⠂⠛⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠈⠀⠈⠀⠂⠁⠊

⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠂
⣀⡀⠀⠈⠀⣀⡰⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀
⠀⠠⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠤⠖⠀⢤⠀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠄⠄⠂⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡐⠀⠀⠀⠀⠀⢄⢰⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢄⡋⠠
⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠰⠂⢀⣀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⣷⠈⠻⠀⠀⠀⠤⠀⣁⠤⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠴⠀⠀⠱⠀⠀⠹⠆⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠊
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠁⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠂⠈⠉⠀⠂⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢘⠧⠀⠰
⠀⠀⠀⠀⠀⡄⠀⠀⢀⣁⡐⡇⠀⠀⢸⣆⠈⠃⠈⠀⠀⠀⠀⢀⡀⠀⣼⠇⠀⠀⠀⠀⠂⠀⢎⠁⢰⡀
⠀⠀⠀⠀⠀⠉⠽⠼⠀⠙⢤⢻⠍⠂⠀⠋⣴⣿⣿⣷⠀⠠⣾⣿⣿⣷⠙⠀⠀⠪⠅⢠⠀⣀⠐⡄⠓⠉
⠀⠀⡀⣀⡴⢶⡖⣚⡳⣄⠘⡇⢀⡀⠸⠀⠿⠿⠿⠋⣰⣦⠙⠿⠿⠿⢀⠀⠂⣴⠒⢛⠉⣓⠲⣄⣎⣀⡘⠻⠲
⠀⠀⠀⠋⣇⠀⠘⣫⠁⣼⡰⠓⠻⢆⡀⠑⠲⣖⠄⠀⠉⠙⠀⠠⣶⠒⠁⠀⠀⣋⡀⠸⠿⣭⠗⢠⣏⣅⡐⠀⠁
⠀⠀⠠⣀⡙⠾⡿⠧⢾⠿⠀⠈⠀⠀⠁⠀⠀⡿⠶⣤⣤⣤⣴⠴⡿⠀⠀⡠⡀⣽⠯⢧⡴⢊⠠⠼⠃⠀⢤
⠀⠀⠀⢈⣣⡶⠃⢀⡏⢀⣯⣶⠁⠀⠀⠀⠀⠈⠢⣀⠀⠀⢀⡰⠃⠀⠘⠗⠀⢀⡔⡶⠞⢋⠰⢦⠀⢆⢠⣄
⠀⠀⠀⣤⡴⠽⣄⣬⡃⢀⣺⢷⡧⡠⠶⣠⡦⠶⡶⣾⠏⢽⣦⠶⢖⣠⡴⢦⠀⣈⣷⣤⡄⢼⠁⠠⢀⠯⡋⠁
⠀⠀⠀⣰⠂⠢⠚⣿⡀⢸⡿⠛⠰⠣⠴⣽⣻⣯⢥⣴⠂⢰⣦⢶⣿⡿⠗⠊⠁⢉⠠⢿⣧⣾⡆⢴⡅⠄⢲⡄
⠀⠀⠀⠉⠁⠀⠀⢼⣿⣿⣶⣧⣤⣤⣀⡀⢁⡉⠛⢿⠅⢸⠟⠋⡁⢀⣄⣢⣬⣶⣿⣿⣿⣿⢗⢚⠀⠀⠈⠉
⠀⠀⠀⢀⡀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⠶⠶⠾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢸⠀⠀⠀⣀
⠀⠀⠀⠐⠛⢲⢀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⡀⣧⡰
⠀⠀⠀⠀⢀⣏⡔⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⢡⠀⣠⠤
⠀⠀⠀⠀⠾⡯⡏⠟⢺⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣬⡋⢠⣝⣦⡗⣉⣀
⠀⠀⢈⠶⠀⣇⢶⣛⠤⢾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⡳⢮⣅⡰⢮⣰⣢⠶
⠀⠀⠀⣲⣷⠿⣏⣶⡚⣯⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⣭⡶⠃⢁⣤⡄
⠀⠀⠀⣁⠤⡐⣌⡯⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⠀⠰⣉⢉⡴⢖⡀
⠀⠀⠈⠉⠚⠛⣋⣁⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣿⠿⠟⢠⠜⠀⠁⢈⠀⠙⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠜⢭⣹⠖⢦⠠⠌⢙⠙⠛⠻⢿⣿⣿⣿⠛⠋⠉⠀⠠⠓⢩⡁⢆⠒⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢦⣀⠀⠩⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠁⠀⠀⠐⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢴⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠋⠀⠸⡖⢄⢸⣿⣿⣿⣿⣿⡗⠀⠀⠀⣀⣰⣦⣄⠀⠀⠀⠀⢻⣿⣶⣦
⠰⠖⠀⠀⠀⠀⠈⠛⢠⠀⢀⣠⣦⣤⡀⣠⣿⣿⣶⣶⣿⣿⣧⡀⠀⢸⣿⣿⣿⣿⣷⣶⣣⠀⠀⠈⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣅⣈⠉⠙⠛⠛⠻⢿⣿⠿⠿⠿⠿⠆⠈⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣿⣿⣛⠃⠀⠀⠀⠀⠀⠀⣴⣆⢹⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣷⣶⣦⣤⣶⣶⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠠⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠏⠉⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠙⠻⣿⣧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⣿⠻⠿⠋⠀⠀⠀⠀⠀⠀⠈⠛⢿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠛⠁⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠉⠉⠉⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠙⠉⠉⠉⠉⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⣯⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧⠀⠀⠀⠀⡇⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⢧⡀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠚⠁⠀⠀⠈⢳⣄⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣴⠆⠀⠀⠰⣶⣏⢉⣷⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢰⠟⢱⡆⠙⣯⡉⠀⡿⠀⣿⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⢸⠏⣧⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠈⠳⠛⠁⠀⠈⣻⣤⣟⠀⣿⠀⠀⠀⣼⠹⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣻⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡟⢻⣆⣀⣄⠀⠀⠀⠀⢠⣾⠉⣹⣽⠷⣿⠀⠀⠀⣟⠀⣿
⠀⠀⠀⠀⠀⣤⣬⣽⣥⣤⡀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⣩⣟⣥⡀⠀⠀⠀⠈⠻⣏⠉⣿⠀⣿⠀⠀⣤⣬⣯⣥⣤
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⢰⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⢯⣀⣿⠀⠀⠀⢠⡴⢬⡻⣯⡀⣿⠀⠀⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢶⣞⣉⣀⣀⣀⣀⣩⣤⠶⠶⣤⣼⣦⣼⣃⣀⣉⣻⡶⠴⡇⠀⠀⠀⣿
⠀⠀⣠⠶⠶⡏⠀⠀⠀⢸⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠶⣇⠀⠀⢸⠷⠾⠿⠿⠿⠿⠿⠶⠶⡇⠀⠀⠀⢹⠶⠶⣄
⣠⡞⣁⡀⣀⣙⠂⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣙⣛⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣓⡋⢀⣀⣈⣳
⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣟
⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹
⠀⠙⡟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠙⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⢛⡟⠋⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠈⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠋
⠀⠀⠀⢻⠀⢀⡴⠛⠛⠻⢦⡀⢠⡿⠒⣶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢀⡴⠛⠛⠳⣦⠀⠘⡇
⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠘⡇⢸⣇⣀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢰⡏⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⢶⡄⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⡟⠀⢀⣀⣀⡀⠀⠀⣿⡀⢸⡇⠀⣛⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾⢁⡴⠋⢉⡾⠛⠃⠀⣿⠉⢹⡏⠉⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⢰⡇⣾⠁⠀⠈⣷⠀⠀⠀⣿⠀⠸⣧⣤⣿⠀⣿⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⣠⣤⣼⡇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢷⣏⠀⠀⠀⠸⣇⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⠘⡇
⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢹⡄⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢀⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠙⠛⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢺⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⠸⣆⡀⠀⢀⣼⠃⠸⡇⠀⠀⠀⢸⡇⢀⣴⠖⠛⠻⢦⣄⣠⠶⠛⠛⠶⣄⣠⡴⠞⠛⠲⣦⡀⣿⠀⠀⠀⠀⣿⠀⠹⣄⡀⠀⢀⣼⠃⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠙⠛⠋⠁⠀⢐⡇⠀⠀⠀⢸⣇⣼⣁⣀⣀⣀⣀⣹⣇⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣈⣧⣿⠀⠀⠀⠀⣿⠀⠀⠈⠙⠛⠋⠁⠀⢘⡇
⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣤⣤⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣾⣶⣶⣶⣶⣷⣤
⣿⣿⣿⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢦⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠛⠛⠛⠛⡟⢻⡟⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⢻⡟⢻⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⡄⠀⢀⢀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢠⣤⣤⡤⣠⡄⠀⠘⠃⠀⠀⡁⠀⢀⠀⠀⠺⠿⠂⠀⢀⣄⣀⣄⡀⠀⠀⠀⠀⡄⠀⠈⠀⢠⡄
⠀⠀⠸⢿⣿⡽⠿⠓⠀⠀⠀⠀⠐⠛⠉⠙⠋⢢⡀⠀⠀⢀⣾⣶⣶⣾⣿⡆⠀⠀⠜⠱⢀⠀⢰⡵⡰
⠀⠀⠀⠀⠁⢻⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⠈⠉⣿⣿⣿⣿⡆⠀⠀⠓⡁⢺⠦⠚⠧⠘
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⠁⠀⠀⠀⠀⠙⡿⠀⠘⠇⠀⠀⠀⠈⠀⠀⠀⣤⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣻⣼⣃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⣏⠋⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡾⢀⣀⣀⢷⠀⠀⠀⡀⣤⢄⡐⠊⠉⡉⢿⠄⠀⢀⣄⣠⣤⣴⠀⢀
⠀⠀⢰⡧⠛⢡⢤⠘⠿⣧⠀⠘⢻⣯⣿⣯⡼⠃⠀⠰⠡⢷⣓⢀⣀⠀⠓⠋⠀⠀⠉⢻⣿⡿⣯⠔⠉
⠀⠀⠘⡧⣴⠚⢀⣌⣦⣾⠀⠀⢰⠍⠋⢹⠀⠀⠀⠀⠀⠩⠉⠉⠀⠀⠀⠀⡀⠀⠀⠐⠛⢻⠞⠇
⠀⠀⠀⠑⠈⠓⠿⠿⠛⠁⠀⠀⠐⠶⠿⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠉⠛⠛⠒⠒⠲⠶⠶⠦⠤⠤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠲⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢹
⠀⠀⠀⠀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⡠⠒⠙⣆
⠀⠀⢠⡞⠉⠛⠛⠛⠛⠷⠶⠶⠶⢶⣤⣤⣤⣤⣄⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⠔⠊⠀⢀⡴⢻⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠶⠶⠶⠶⢦⣤⣤⣤⣤⠖⠋⡠⠐⠁⠀⢀⡴⠋⡴⠋⣇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⠊⠀⠀⢀⡴⠋⢠⠞⠀⣠⠋
⠀⠀⠀⡿⠒⠒⠲⠶⠤⠤⢤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋⠀⡴⠃⠀⣴⠁
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠸⡄⢀⡴⠋⠀⢠⠞⠀⢀⠞⠀⠉⠒⣤
⠀⠀⠀⠈⣏⠙⠒⠒⠢⠤⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠒⠒⠷⠿⠶⣤⡴⠁⠀⣠⠋⠀⣀⠴⠋⡇
⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠈⠙⠻⢮⣭⡉⠑⠒⠒⠶⠤⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡴⢁⡤⠚⠁⡠⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠈⠙⠳⢦⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠲⠤⠤⠤⣀⣀⡀⠀⢸⣇⣮⠔⠉⠀⠔⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡽⠋⢀⠠⠊⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⡀⠔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⣀⠔⠋⢀⠄⠊⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠈⢙⠏⠁⡠⠀⠁⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠘⡇⠈⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⣇⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⣶⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣦⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣤⠾⠋⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠈⠻⣿⣍⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⣩⡽⠛⠁⠀⠀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣦⡀⠀⠀⠀⠙⠻⢶⣤⣠⡤⠶⠦⣤⣠⠖⠲⠶⣤⣴⠞⠉⢀⡄⠀⢀⣠⠶⠛⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠉⠻⣄⠀⠀⠀⠀⢀⣀⣉⡙⣷⡦⠄⠙⢷⡄⠀⠀⠙⢷⣶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠙⠛⢿⡄⠀⠀⠻⠀⠀⠀⠀⠹⣦⣤⠴⠆⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣘⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠉⠉⢻⡟⠛⢻⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣾⠃⢀⡿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢰⡏⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⣿⠁⠀⠀⣿⠀⠀⠀⠀⣆⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢻⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢻⡇⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠃⠀⢹⣇⢀⣤⣤⣶⣦⣤⣤⣤⣤⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣠⣤⣶⠶⠿⠿⠿⠿⠷⢶⣦⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣬⣭⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠸⠟⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠛⠻⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣄⣠⣤⣤⣆⣁⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣦⢀⡀⢀⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⠭⠉⠩⣭⣿⣿⣿⣿⣿⣿⠁
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⢛⣋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠛⠿⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⢀⣤⠀⣬⠿⢣⣤⣴⣾⣀
⠀⢹⣿⣿⣿⡿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣹⣿⣾⣿⣾⣿⣿⣿⡟⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡘⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠣⡀⠀⠀⠄⣀⠤⠐⠂⠁⠀⠀⠀⠀⠀⠁⠠⡈
⠀⠀⠀⠀⠈⠀⠀⢀⠊⢠⠤⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⣁⣀⠠⠤⠤⠀⠐⠂⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⣀⡀⠀⠀⠀⢠⠀⠂⠈⠁⢀⡀⣀⡠⠤⠤⠴⠲⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⡀⠸⡀⠂⠘⠉⠓⠋⠀⠀⠀⡀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠈⢂⡒⠒⢚⡛⠉⢉⡉⠭⢁⠒⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣠⠀⠠⠁⢥⣒⣪⣁⣒⣈⣉⣤⢤⣬⣬⡶⣶⠖⠁⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠡⢾⠛⠯⠉⠓⠚⠛⠂⠍⠁⢀⠄⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠸⠀⠀⡃⠀⠇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡼⠄⠠⢤⠤⠤⢄⣀⡀⢀⣀⡀⠀⠅⡔⠁⠑⣌⡀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣠⡀⠀⠒⠀⠀⠈⠉⠉⠁⠉⠁⠋⠀⠁⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠂⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⡤⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠐⠀⠀⠓⠂⠓⠉⠉⡀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠐⠂⠀⠀⠀⠀⠀⠒⠀⢸⠀⢠⠦⠔⠒⠒⠒⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡉⠙⢶⡶⢢⠶⢦⣤⣄⣠⣾⡴⣓⠂⠬⠤⢀⣁⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠻⡎⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠑⠖⠒⠒⠒⠒⠒⠊⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⣀⡀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠔⠒⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠈⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣎⠈⠓⠀⠠⠀⢹⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡀⠀⠀⢰⡆⠈⠀⠀⠀⠈⢑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⣡⠄⠀⣿⣦⣶⠃⠀⠀⡐⠁⡠⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠎⠰⠈⠀⠹⠓⢋⠹⠋⠁⠀⠀⠀⠀⠌⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣃⣀⠄⣀⠀⠀⠀⠀⠀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⡜⠀⠀⡜
⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠈⠉⢺⡄⠠⣐⡂⠈⠒⠩⠀⠀⡠⡀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠰⠀⠀⣸⠁
⠀⠀⢱⣠⣀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠒⠒⠒⠀⠒⠂⢃⠀⠀⠘⠇⢩⠤⠤⠀⢀⠾⣴⣤⡙⠈⢉⡔⠁⠀⣠⢁⡀⠄⠐⠂⢰⢠⠃
⠀⢠⣤⠙⠢⠀⠀⠉⠁⠐⠒⠂⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠁⠐⠀⠀⠦⣄⡈⠲⠄⠀⠐⠀⠀⠀⠀⣉⠙⠲⠎⠒⠈⠉⠁⢀⠀⢠⠄⠐⢡⠏
⠀⠘⣌⠻⡇⠃⠐⠐⠚⠃⠀⠐⠂⢤⣀⣀⣀⣀⣀⣀⣈⣄⠀⠀⠀⠀⠈⠀⡀⠀⠀⠉⠂⠄⠠⢍⠒⠢⡔⠁⢀⣀⣠⠖⠀⠀⠉⠀⣀⠂⠀⢀⡎
⠀⠀⠣⣉⣻⣦⠀⠠⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠐⠖⠐⠦⠄⣀⣣⣀⡐⠈⠑⠄⠀⠀⠀⠀⠀⣄⠘⠀⠃⣨⠁⠀⠀⠀⢀⠄⠠⠁⠀⢀⡞
⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⡉⠁⠲⢤⣄⣀⠀⠀⠀⠈⠃⠀⠀⡇⠀⠀⠀⠀⠁⠜⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠐⠠⣉⠒⠒⠒⠉⠀⡁⡇⠀⠀⢀⠤⠋⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣅⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠉⠁⠂⢀⠀⡃⣇⠀⠒⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠇⠇⠀⠀⠀⢀⡴⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠀⢈⣙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⢀⣠⠔⠉⠉⠀⣉⡱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠄⠀⡠⠖⠉⠁⠀⠀⠈⠙⠒⣤⣤⣀⣀⣀⣠⡤⢔⣚⡭⠴⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⢀⣀⡠⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣠⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠒⠋⠁
⣿⣿⡿⠛⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣴⣿⣶⣦⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠘⡛⠃⣐⣿⣿⣿⡟⠶⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠗⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⠀⠀⢿⣿⣿⣿⣿⣇⢰⡀⠀⢀⣤⠼⠧⠦⣿⣷⣴⣶⡦⠐
⠀⠀⠀⠀⢀⠄⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⠀⠀⢘⣿⣿⣿⣿⣗⣼⣅⠀⠙⣿⣷⣀⣀⡹⣿⣿⣥⣤⣼⡌⣀
⠀⠀⢠⣖⠁⣴⢆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠙⣿⣿⣿⣿⣿⣿⣫⣴⣿⣿⣿⣿⣿⣋⣀⡀⣹⣿⠋⡏⢹⣿⣿⣻⡉⡍⢉⡭⣤⠬⠂
⠀⢠⣾⣟⣓⣈⣤⣴⠄⠛⠋⠋⠉⠁⠒⠒⠂⠙⠓⠿⠿⠿⠿⠓⠿⠿⠟⠻⠿⠟⠛⠿⠿⠿⠿⠛⠛⠀⠂⡛⠛⠛⠿⠅⠰⠈⠉⠉⠈⠁⠈⠀⠀⠀⠠
⠀⠀⠀⠈⠉⠁⠄⠈⡉⠈⡀⠀⠀⠀⣀⠐⢀⠀⠐⢚⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠠⣌⡃⢿⡌⣉⡂⢱⣝⠲⡏⣙⢳⣴⠈⠍⣴⡍⠊⣇⠥⡂⠀⢀⠶⠞⠦⢼⠀⡆⠠⢈⡤⠸⢵⢂⣶⢁⡔⠂⢲⡐⠆⠐⠈⠄⠰⡤⢆
⠀⠀⢀⣿⢙⡆⢸⢫⠁⠔⢿⠇⠘⡸⢻⠛⣤⠗⣾⢈⡝⣺⢶⣧⢡⠙⠤⢠⣐⣦⣾⡅⣶⠻⣍⡆⠞⣀⣽⡆⠶⡏⠋⡍⡉⠁⣀⡛⠃⣷⡤⠖⠣
⠀⠀⢈⣿⢸⣇⣷⡫⠜⢻⣴⡏⠰⣉⢾⣀⣻⡀⢾⠀⠼⢋⢸⡇⡛⠷⢬⡘⢼⡡⡤⢀⣅⠔⢻⡳⢦⡥⠺⢛⠿⠁⠦⣯⡌⠷⡋⠁⠀⣾⣁⣀⠁
⠀⠀⠀⣿⢸⣏⢸⡇⣿⠛⢸⣇⢘⣟⢾⡑⢽⡆⣿⠳⣿⣯⢱⣎⢵⠻⠎⢣⠿⣵⣄⢸⡇⢠⣬⠁⠀⣿⡶⣻⢧⡀⣶⡰⢶⠶⡀⢆⡿⣡⠈⠒
⠀⠀⠉⣿⠘⠷⢸⡦⠯⣀⢸⡇⣀⣉⣻⡓⢺⠇⢹⡘⣿⠣⢸⡅⢒⡼⢦⡡⠾⢼⡇⢸⡏⣈⣭⠗⠀⣿⠒⡯⢿⡁⣽⡵⢿⣷⡅⣬⡝⣡⢌⡉
⠀⠁⠈⠉⠛⠒⠺⠧⠤⠤⢼⣅⣈⣁⣽⡉⠉⠁⢻⠃⠒⠒⢾⠔⠠⠜⢵⣧⠀⠼⣆⢡⡦⠩⡍⠁⠙⣿⠃⠑⢻⡓⣿⡴⣿⣶⠄⢼⡅⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠶⠶⠼⠧⢤⣀⣘⡇⠀⠀⠀⢰⡎⠁⠈⠂⠐⣿⠂⠰⢼⠳⣿⠤⠟⣀⣀⣾⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⠤⣤⣄⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢘⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠓⠛⠒⠤⠤⠀⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣭⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡻⣿⣿⣿⣿⣿⣿⣶⣭⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡻⢿⣿⣿⣿⣿⣿⣷⣦⣍⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣷⣮⣝⡿⢿⣿⣿⣿⣿⣿⣿⣶⣬⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⣍⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣼⣿⠇⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣶⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣽⣿⣿⣿⣿⣿⣿
⢶⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠛
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣷⣦⣝⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠇
⠀⠀⠀⠸⣿⡇⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⡿
⠀⠀⠀⣴⣾⣿⣿⣿⠿⠟⠛⠉⠉⠀⠙⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣰⣿⣿⣿⠃
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠉⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠋⠉⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡤⠴⠶⠶⠒⠒⠂⠀⠀⠀⠀⠀⠐⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢈⠑⠦⣍⣛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣉⡓⠦⣬⣉⠒⠌⠉⠛⠷⣦⣄⠀⠀⠀⣀⣠⡤⠴⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣂⠬⣍⡒⠬⣕⡢⠌⠙⠒⠄⠀⠀⠀⠙⢷⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⠲⢭⣑⠲⠬⣁⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠒⠒⠟⠻⠶⣤⣍⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣀⣤⣤⠶⠞⠛⠛⠛⠓⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⠶⠾⠿⠷⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢉⡿⣍⠿⣯⣉⣉⣉⣉⣉⣀⣤⣤⣤⣤⣴⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⢀⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠸⠀⠹⣆⢸⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠇⠀⠀⣀⣀⣘⣿⣧⣶⣶⣶⡶⠶⠶⠶⠚⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣀⣠⣴⠶⠞⠛⠉⢉⣿⣴⠟⠛⠉⠉⠉⠉⢙⣛⣛⣛⡛⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦
⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠭⠭⠭⠍⣉⢛⣓⣒⣒⣒⣒⣒⣒⡒⠀⠀⠀⠀⠈⢉⡿⣭⡀
⣸⠏⠀⠀⠀⠒⠤⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠥⠤⠤⠤⣀⣀⣀⣀⣐⠒⠒⠒⠒⠶⠶⠦⠀⠀⠀⠀⠀⢸⠁⠀⠉⠓⠦⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⢀⣀⣀⣈⠉⠉⠙⠛⠛⠒⠒⠒⠲⢶⣶⣶⠶⠬⠭⢭⣍⣉⡙⠛⠛⠒⣒⣀⠀⠀⢸⠀⠀⠀⠀⢀⣈⣙⣷⡦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢶⣭⣭⣭⣍⣛⣛⣛⣟⠠⠒⠠⠢⠑⠩⠀⢩⣤⣤⣄⣀⡀⠙⠛⣛⣒⣒⣒⡀⠘⢷⡶⠾⠟⠛⢻⡟⠉⠁
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⣉⡉⠉⠉⠛⠛⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠿⠿⣶⣶⠶⠿⠀⠀⠀⣾⠁
⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⢀⣴⡶⠚⠃⢀⣀⣠⡤⠤⠤⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠀⠀⠀⣿
⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠤⠤⠐⠂⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⡀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⠶⠶⠶⢚⣛⣋⣩⣭⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣙⣮⣍⣁⣀⡀⠀⢈⣉⣉⣭⣤⡤⠶⠶⠖⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠈⠙⠲⡄
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡎⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⡀⠀⠀⠀⠋⢀⢀⢀⠀⠀⠈⡗⠋⠉⠙⢆⠀⠀⡴⠋⠀⠉⢺⠁⠀⠀⡀⠀⡀⠽⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢧⠀⠀⠘⠍⣘⡃⢪⣤⠄⠀⠀⠀⣧⠀⠀⢀⡞⠀⠀⢳⡀⠀⠀⡾⠀⠀⠀⠠⣤⠕⢘⣁⠉⠂⠀⠀⡼
⠀⠀⠀⢸⠀⠀⠀⠐⠀⡔⠝⠆⠮⠦⢀⠀⢸⠀⡠⠊⠀⠀⠀⠀⠙⢄⠀⡇⠀⠀⠴⠴⠰⠋⢦⠈⠤⠀⠀⢀⡇
⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⢝⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠑⠀⢸
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⣀⣀⠤⠤⠤⠴⠒⠓⢤⠀⠀⠀⠀⠀⢀⠴⠛⠒⠦⠤⠤⢤⣀⣀⣀⡀⠀⠀⣸
⠀⠀⠀⠀⡔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣄
⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠤⠖⠒⠋⠀⠀⠀⠀⠀⠀⠑⠒⠲⠤⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⣀⣀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⠲⠶⠦⢞⣁⣀⣠⡴⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⢶⡏⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣟⣉⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢹⠋⢥⣀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡄⠀⠈⠁⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠴⠶⢞⢿⡛⣿⠄⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠈⠛⠻⢦⠐⠘⢛⠁⠋⠳⣄
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢼⠿⠀⠀⠀⠀⠓⢦⣀
⠀⠀⠀⠀⡠⠞⠁⠀⢀⡀⠀⠀⡀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢳⡄
⠀⠀⣠⠎⠀⠒⠒⠤⡄⢹⢀⡄⣴⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠆⠀⣧
⠀⠰⡇⠀⠀⠀⠀⠀⠙⣮⡰⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸
⠀⠀⠱⡄⠀⡄⠀⠀⠀⠘⢦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⢸
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠻⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡠⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠙⢷⣀⠁⠀⣼⠁⠀⠈⠳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⣿⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣤⠀⠁⢠⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣠⣿⠀⡅⠀⠀⠀⠀⠀⡀⠂⠈⡀⠀⠈⡄⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠀⠻⣄⣿⠦⣀⡀⠐⣶⡶⠆⠾⡩⡷⡀⡀⣠⠟
⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⡿⠛⡃⠀⢾⣝⢢⣿⣤⠰⠩⣴⡠⣴⣅⣋⣽⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⣾⠻⠷⠄⠙⢷⣝⢿⣷⡀⠀⠙⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢶⡀⢿⣷⣬⣙⣳⣿⣶⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠈⠛⣺⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠿⣗⣴⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣿⣿⡿⠻⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣒⣒⣒⣒⣀⣀⣀⣒⣀⡀⠒⠒⠒⢒⣒⣒⣚⣓⣒⣒⣒⣒⣖⡒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⡉⠉⠉⠛⠛⠋⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⢹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⣴⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⣿⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢂⡀⡀⣿⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⢸⠟⠳⠁⠞⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡐⠀⠸⠤⠤⠤⠤⠤⠿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⣦⣭⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡄⣴⠆⠀⠐⠹⠃⠉⠀⠀⢀⣠⣀⡀⠻⠁⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⣀⡡⣽⡆⠀⠀⡆⠀⠀⢀⣴⠎⣹⣿⢰⠉⡆⠀⣆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠥⠨⠯⠴⠷⠿⣼⡤⠿⠶⠦⣾⣷⠶⠿⣿⡾⠦⠷⠾⠷⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠉⠙⠛⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⠛⠛⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣀⣀⡀⢀⣈⣉⣉⣉⣁⣉⠉⠉⠉⠀⢀⠀⢀⠀⢠⣤⣼⡄⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠙⣤⡤⠖⠋⠻⠙⠛⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣻⣷⡀⠀⠘⠆⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⣠⠿⠛⠻⢦⣀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣶⠀⣿⣿⣿⢿⡿⣿⣿⣿⡿⣿⣻⣉⣀⣼⣦⣀⣤⡾⠃⠀⠀⠀⢀⣬⡉⠁⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⣀⣀⡿⠀⣿⣿⣽⣉⣇⣶⣿⣌⣠⣷⠉⣿⣿⣿⡿⠿⢿⣱⡀⠀⠀⣀⣼⣿⣿⡄⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣟⣿⣟⣁⡀⠀⠀⠈⠙⠻⣿⣿⡇⣿⣿⣿⠊⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⡤⠿⠷⡾⠓⠒⡶⠶⠶⠶⠒⢶⣿⣿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣿⣷⣿⣿⣿⣴⢿⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠎⠀⠠⠄⠀⠈⠭⠀⠠⠤⠂⠀⢿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣿⣿⣿⣿⣿⣿⡏⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠫⠿⠭⠉⢉⣉⡙⠙⢋⡽⢻⣿⢟⣏⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠈⠉⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠒⠂⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⢀⢶⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⣮⠬⠆⠒⢢
⠀⠀⠀⠀⢸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠂⢉⡠⠤⠒⠊⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⢦⡸⡀⠀⣀⣠⡤⠒⠊⢁⣀⠤⠔⠒⠩⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢏⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⣄⡠⠗⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡱⠔⠊⠉⠀⠀⠉⠉⠐⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⡠⠤⠒⠉⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠘⡌⡆⠀⠀⠀⠀⣀⠤⠐⠊⠉⣀⠤⠒⠉⠀⠀⠂⠘⠁⠀⠀⠀⠀⠀⢀⡠⡔⠊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡤⢴⠊⠙⣄⣀⠤⠒⠉⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡖⠁⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠖⠂⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡎⠁⢸⡉⠑⠢⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠒⠦⢄⡀⠀⠀⠀⠀⠁⠀⠀⣀⠤⠒⠉⠀⠀⠙⠒⠒⠉⠑⠲⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⢄⡀⠀⠉⠒⠠⣀⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⡏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠑⠂⢤⡃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠗⠢⢄⣐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢤⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⣀⠴⠚⢁⡀
⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢠⣼⣿⠿⢶⣄⡀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢸⣧
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠛⠶⣬⣿⣦⣀⣠⡶⣁⣀⣀⣀⣠⣀⣠⠞⠁⠀⠀⠀⠀⣠⠟⠉
⠀⠀⠀⢀⣿⣧⣤⣤⣄⣀⡀⠀⠀⠀⠀⠃⠀⠀⣀⣠⣤⣔⠒⠒⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠿⠿⢿⣛⠛⠲⠤⣄⡀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⢸⣿⣶⣶⣖⣒⣪⣭⣭⣶⣦⣀⣠⣴⣿⣿⣿⣿⣶⣾⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠈⠙⠲⢤⣀⣙⣧⠀⢀⡴⠋
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⡟⠋⠉⠉⠉⠉⠉⠛⠛⠛⠛⠃⠀⠀⠀⠀⠠⠐⠈⠀⠀⢔⠤⣀⠀⠀⠀⠀⠈⠙⠻⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠿⣦⣈⠙⢦⣄⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠁
⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⢞⡀⠀⠈⠙⢷⣤⣈⠓⢦⣔⡢⠔⠊⢉⡠⠔⠊⢁⡴⠚⠒⠒⠲⠤⠤⢄⣀
⠀⠀⠀⠀⠺⡉⡊⠳⡄⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⣔⣥⢟⡩⠔⡀⠀⠀⠀⠈⠛⢷⣾⣋⣦⠶⠋⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⣠⣄⠙⢦⠀⠈⠳⡄⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⢠⣿⣿⢞⣛⣺⣿⣷⡀⠀⠀⠀⠀⠀⠉⠁⠀⠳⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⠇
⠀⠀⠀⠈⠳⡄⠀⠑⢄⠀⠀⢀⠴⠤⠤⠄⠀⠈⠛⣦⣠⣿⣿⣿⡿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠱⠄⣔⣐⣤⣿⡶⢾⣿⣿⡿⢿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⡉⠒⠲⠦⠤⢄⣀⣀⢀⡔⢁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢈⡥⣄⣬⡵⠖⠛⠉⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠉⠓⠒⠦⢤⣀⣀⠉⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⠙⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠶⣤⣀⡀⠀⠉⠙⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣌⡀⠀⠈⠙⠓⠦⣤⣀⠀⠀⠉⠙⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠉⠛⠲⢤⣄⡀⠀⠈⠙⠳⠦⣤⣀⠀⠀⠉⠙⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠦⣤⣀⠀⠈⠉⠓⠦⣤⣀⠀⠀⠉⠙⠲⠦⣄⣀⠀⠀⠉⠙⠓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢾⣄⡀⠉⠛⠲⢤⣄⡀⠀⠉⠙⠳⠦⣄⣀⠀⠀⠉⠛⠲⢤⣄⡀⠀⠀⠉⠙⠓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⣾⣋⠙⠓⠮⣍⡓⠦⣄⣀⠈⠉⠓⠦⣤⣀⠀⠀⠉⠛⠲⢤⣄⡀⠀⠀⠉⠛⠶⢤⣄⡀⠀⠀⠈⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⣟⡉⠝⠲⢦⣀⡉⠙⠲⢤⣀⠉⠓⠶⣭⣛⠲⢤⣀⠀⠉⠙⠲⢤⣄⡀⠀⠈⠉⠓⠦⣤⣀⠀⠀⠈⠉⠓⠶⢤⣀⡘⠀⣿
⠀⠀⠀⠀⠀⠀⣟⠓⠦⣄⡈⠙⠳⢦⣄⡈⠙⠳⢦⣄⡈⠙⠳⢦⣄⡈⠙⠲⢬⣝⡳⢦⣄⡀⠈⠉⠓⠶⣤⣀⠀⠀⠈⠙⠳⠦⣄⣀⠀⠀⠈⢙⠓⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣙⠳⢦⣄⡀⠙⠳⢦⣄⡀⠙⠳⢦⣄⡈⠙⠳⠦⣄⡈⠙⠳⠦⣍⡓⠦⣤⣀⠀⠉⠙⠲⢦⣄⡀⠀⠀⠉⠛⠲⢤⣌⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠶⣬⣙⠳⢦⣀⡈⠙⠳⢦⣄⡀⠉⠓⠦⣄⡀⠉⠓⠦⣄⣀⠉⠓⠶⣭⣙⠲⢤⣀⡀⠈⠉⠓⠦⣤⣀⠀⠀⢈⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠉⠛⠶⣬⡙⠲⢤⣀⠈⠙⠳⢦⣄⡀⠉⠓⠦⣄⡀⠉⠛⠶⢤⣀⠉⠙⠲⢬⣙⡳⠦⣄⡀⠀⠉⠙⠲⢮⣅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣿⠓⠦⣌⡙⠲⣤⣀⠈⠙⠳⢦⣄⡀⠉⠓⠶⣤⣀⠉⠙⠲⢤⣀⡈⠙⠳⠦⣍⡓⠶⣤⣀⢠⠌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠻⢦⣤⠾⠉⠓⠦⣌⡙⠲⣤⣀⠈⠙⠳⢦⣄⡀⠉⠛⠶⣤⣀⠈⠙⠲⢦⣄⡀⠉⠓⠦⣭⣙⠲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣍⡛⠶⣤⣀⠈⠙⠳⢦⣄⡀⠉⠛⠶⣤⣀⠈⢙⠓⠦⣅⣀⣩⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣍⡛⠦⣄⡀⠈⠙⠳⢦⣄⡠⠌⠙⠲⣤⡴⠞⢋⣡⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣉⠓⠦⣄⡁⠈⣙⡷⠖⢋⣡⣴⠚⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣉⢋⣡⡴⠚⠉⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⢹⡇⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠻⢦⣀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠒






⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠻⠛⠉⠉
⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⢦⡴⠖⠒⠲⠿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠘⠃⠐⠂
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣤⣄⣠⣤⣴⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠐⠛⠀⠻⠿⠿⠛⠛⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠐⠀⠀⠒⠬⡑⠢⢀⡀⠀⢀⣀⠠⠤⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡔⠁⠀⠀⠀⠀⠀⠀⠑⠂⠀⠤⠀⠀⠐⠀⠉⠉⠁⢂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠔⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠉⠈⠉⠁⠐⠀⠬⢐⣒⣒⠨⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠄⠄⢀⡀⠀⠀⢀⠔⣊⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠮⡀
⠀⠀⠀⢀⡠⠤⠀⠀⠤⠤⠀⣀⡀⠀⠀⣀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠔⡡⠒⠀⠉⠉⠁⠐⠒⠀⠀⠈⠁⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠀⠤⢀⡀⠀⠀⠀⠀⢀⠆
⠀⡌⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎⠀⠀⠀⠀⠀⠀⠈⠉⠐⡂⠁
⠀⢃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠊
⠀⠈⢄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⡄
⠀⠀⠀⠑⠬⠁⣒⣒⠀⠠⣀⡠⠐⠈⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠈⠣⡂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠐⢺⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⡠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠈⢦⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⣈⠱⠒⠦⠤⣄⣀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠢⡀⠀⠀⠀⠀⠀⡄⠠⡩⢄⢋⡙⠖⠲⡤⢄⣉⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢢⠀⠀⠀⠀⡇⠀⠀⠈⠁⠘⠈⠳⠒⠦⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⠂⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠤⠜⢁⣁⠄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⡰⠠⠋⡉⡰⠒⠃⡤⠴⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠥⠖⡚⣠⠆⠎⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠒⠁⠁⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈