Softball Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠀⣀⣀⣀⠀⣀⣀⣀⡀⢀⣀⣀⡀⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣌⡃⣼⡏⠈⢻⡆⣸⣇⡉⠀⠉⢹⡏⠉⠀⣿⣤⣿⠀⢀⣽⢻⡀⠀⢸⡏⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣬⠇⠘⢧⣤⠾⠁⣽⣯⠀⠀⠀⣼⣇⠀⢠⣿⣤⡿⠀⣼⡏⢹⣧⡄⣼⣧⣤⡇⢠⣿⣤⡖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠣⡑⢄⠀⠀⠀⠀⠳⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠋⠀⠀⠀⢀⠔⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⢄⠀⠀⠀⠈⠳⣄⠀⡠⠔⠂⠈⠁⠀⠀⠈⠉⠐⠢⢄⡀⠀⢀⡴⠋⠀⠀⠀⢀⢔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢕⠀⠀⠀⢀⡜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡋⠀⠀⠀⡠⢐⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣢⠊⠠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠉⢆⠠⡪⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠰⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢏⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡤⠒⠀⠈⠉⠒⠤⠤⠤⠤⠤⠤⠐⠊⠁⠀⠀⠀⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠠⢊⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠡⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠈⣡⣶⣶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣾⣿⣿⡅⠀⠀⢀⣠⣶⣶⣦⣄⣀⣀⣤⣿⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⠟⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠉⠙⠛⠛⠁⣠⣾⣿⣿⣿⣿⣿⣅⡀
⠀⠀⠀⠀⠀⠠⠚⢉⣅⠈⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠖⣤⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠻⣿⣦⣄⣀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⢳⢢⣀⢀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣀⣤⣀⡀⠀⠙⠛⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠁⠀⠲⠿⣿⡿⠟⠋⠁
⠟⠻⠻⠾⡦⣄⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠛⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠐⠹⡣⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⠀⠀⠀⠉⠛⠿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢚⢔⡀⠀⠀⠀⠀⢀⡼⠟⠿⠿⣿⣿⡟⠋⠛⠻⠿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⡀⠀⣀⣴⡿⠞⠓⠲⠄⠀⢈⣁⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣷⣿⣛⣁⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠚⠉⠉⠉⠉⠉⠉⠛⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⠁⢠⡄⠀⠠⣄⠊⠀⠁⠈⠀⠊⠀⠈⠀⠀⠀⠀⠀⢈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⡀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠃⣴⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⣌⠹⣷⣄
⠀⠀⠀⠀⠀⡴⢉⡟⠲⣿⣀⣸⠀⠀⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⢣⡀⠙⣦⡼⢿⠛⡏⢧⡀
⠀⠀⠀⢀⡞⠀⠛⠀⣶⠋⣈⡿⠲⢦⣗⣀⢠⡇⠀⢺⠀⠀⢴⠀⠀⣠⠀⠀⣀⠀⠀⣠⠀⠀⠠⡄⠀⠐⢆⡀⠈⢧⡄⠀⢻⣤⡤⠿⡏⢩⡇⣸⣇⠳⠀⢳⡀
⠀⠀⢀⡞⠀⢀⠀⠀⠁⠈⠟⢱⣦⡾⣁⢉⣹⠓⠶⣾⢆⣀⣸⣇⠀⢸⡆⠀⠸⣆⠀⠈⢷⡀⠀⢘⣦⣀⣈⣷⡤⠶⣿⠋⢉⡇⠀⣸⡇⠊⠛⠉⠁⠀⠀⠀⢱⡀
⠀⠀⡞⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠝⠟⠁⠶⣶⠋⣀⢀⣽⠋⠉⢉⡟⠛⠛⣻⠛⠛⢛⡟⠉⠉⣽⠁⢀⡿⢀⣰⡏⠰⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠈⠉⠟⠠⠆⣼⠃⠰⠂⣿⠀⠅⠺⠇⠀⠙⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸
⢸⠀⠠
⣼⠀⠈⠁
⡇⢀⠲
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘
⢹⠀⢀
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁
⠈⣇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢤⡀⠀⠀⢶⠀⠀⠰⣆⠀⠀⢤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡟
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠘⢦⡀⠀⠙⣦⡀⣀⣻⣄⣠⣬⣷⣤⣤⣽⣆⣄⣈⣷⣀⡀⢷⡀⠀⢻⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⡼
⠀⠀⠈⢧⠀⢠⠀⠀⠀⢀⡀⠀⠳⣄⠀⠻⣦⣠⡬⢿⠖⠛⢻⠋⠉⢩⡏⠀⢠⡟⠀⠀⣰⠇⠈⢉⣽⠋⠙⣻⠗⠶⢾⣧⣄⣸⡆⠀⢲⠀⢀⡀⠀⠠⠈⠀⡼⠁
⠀⠀⠀⠈⢧⡀⠄⠠⣄⠈⠳⣆⣠⠼⡗⠋⣹⠁⡀⣾⠀⠐⣿⠐⠊⠿⠐⠈⠞⠀⠂⠙⠏⠈⠁⠼⠁⠈⢱⠏⠠⢤⡾⠁⠈⣹⠟⠶⣼⣇⠸⡇⠈⡦⠀⡼⠁
⠀⠀⠀⠀⠀⠻⡿⣧⣨⣷⠚⢻⠀⢸⣇⠀⠿⠂⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠼⠋⠠⣤⡾⠉⢛⡷⣬⣧⠞
⠀⠀⠀⠀⠀⠀⠘⢿⡇⢼⡇⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠾⠋⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠂⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣶⣶⠶⠮⠐⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡀⠰⠃⠒⠩⠛⣄⠂⡀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠛⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠤⣤⣄⣀⣀⣠⣤⠤⠤⠴⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢶⣿⠿⠉⠉⠉⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢭⣙⡾⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣀⠲⣽⣋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠠⣌⣻⠧⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡽⣆
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⠹⣼⣛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⣦⡽⡶⣿⢿⡆
⠀⠀⠀⠀⠀⠀⡿⠀⠀⡐⢦⡿⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣳⣶⢻⡼⠇⠛⠣⠏⢿
⠀⠀⠀⠀⠀⢀⡇⢀⢲⣹⣾⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⢲⣾⠻⡜⠃⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠈⣧⣞⣷⡛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣽⡿⣎⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⣷⠿⣍⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣲⢯⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣟⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣾⢬⡃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣲⠄⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠐⢋⣿⡭⠥⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⣉⣴⣻⣿⣷⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠤⠤⠤⣤⣤⣄⣀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣻⣿⣿⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⣻⡿⢿⡾⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠷⠦⣽⣙⢳⡙⢀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣤⣉⢛⡞⢦⡙⠧⠰⣣⣲⣄⠀⠀⠀⠑⡄
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠦⣄⣽⠉⢛⠈⢳⠀⠀⠈⢿⢟⡐
⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣄⠘⡢⢴⡅⢸⡄⠘⠐⢵⣻⣦⠀⠀⠉⠻⣆
⠀⣰⣿⣃⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⣇⠘⢦⣤⡝⢋⡇⠘⣇⠈⣁⣦⠀⠈⠻⢿⡀⠀⠀⠀⠙⣧⠀⢀⠄⠀⡀
⠀⣻⣿⣿⢳⢦⢠⠆⣴⢰⡇⢸⡀⠙⣦⡝⢳⠃⢸⡇⠘⠇⠀⠀⠀⢻⣿⣧⠀⠀⠀⢻⣆⠀⠀⡀⢸⣦⣢⣿⡄⢹
⠀⣿⣿⢿⢾⣸⣸⣇⢻⡤⢿⠖⣻⠁⢸⠀⢸⠀⠈⠁⣠⣴⣶⣄⠀⠀⠙⣿⣷⣄⠀⠀⢹⣧⣄⣶⡀⣿⣿⣿⡧⠀⡀
⠀⣿⠇⡇⡞⡄⢠⠏⡐⠀⣾⠀⠿⠀⠘⠀⣀⣐⣦⣆⢻⣿⡏⢹⣇⠀⠀⠸⣿⣿⢴⣀⠈⣿⣿⡿⠧⠸⠏⠗⠊⣀⣧
⠀⠉⠀⠁⣃⠃⠘⠃⡩⠀⠈⠐⣶⣤⣷⡆⢻⣿⣿⡿⠀⠈⢿⣿⠿⢷⣆⠀⣻⣶⣻⣿⡷⠈⠙⢊⡁⢰⠀⣆⣷⣿⣿
⠀⠰⠀⢈⣽⣇⠀⠠⣿⣷⣧⠀⢿⣿⣛⡿⠄⠋⠙⣧⠀⠀⠀⢻⣦⣶⣿⡆⠙⠏⠙⠉⡄⠐⡇⢠⡧⢸⢷⡿⣿⣿⡟
⠀⠀⡀⢸⣿⣏⡀⠀⣿⡋⠝⣷⠈⠹⣷⣾⣧⠀⠀⠘⢷⣤⡀⠘⠿⠿⠟⠁⠀⢀⡆⠀⣧⢠⡷⠸⣅⢸⡎⣿⠾⠟⡇
⠀⠀⠐⠘⣿⡒⠉⠀⠸⣧⠀⠘⣧⡀⠙⣿⡛⠃⠀⠀⠘⠿⠟⠆⠀⢀⡀⢸⡇⠀⣧⡼⢿⡉⢻⣆⠙⠋⠁⠀⠀⡐
⠀⠀⠀⢣⠙⣷⣬⣄⡀⠙⣧⡀⠘⢷⣄⠈⣵⣷⣤⠀⠀⠀⠀⢠⡄⢈⡇⢀⡷⣎⠈⢳⡌⠛⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢳⡈⠋⡿⢿⣤⡘⢷⣤⣪⣿⣦⠈⠙⠓⠁⣀⢰⡆⠀⢿⣤⠛⠹⢆⡉⠗⠀⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠑⠄⠴⣂⡸⢻⣤⠉⠙⠻⠛⠼⠀⢠⡄⢻⣄⢷⠴⣯⡈⠻⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠙⠿⣿⣿⣷⢦⡀⠀⣄⢲⣌⢿⣦⣟⡙⠳⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣁⠉⠊⢁⣰⣝⣣⣽⣟⣳⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣿⣿⣿⣿⣟⡟⠉⠀⠀⠀⠀⠀⢀⣀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡆⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣯⣽⡇⣸⡶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠤⢬⡅⡴⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⡷⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⢁⣬⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⠋⠚⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡿⣿⣿⡙⢿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣦⡀
⠀⠀⠀⡆⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⠿⢿⣿⢂⣬⣍⣉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡍⢻⣎⣿⠿
⠀⠀⠀⢘⣆⡧⠀⠀⡭⠹⢿⣿⣿⣿⣿⡌⠁⠞⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⡌⠿⠘⣡⣴⣿
⠀⠐⠒⢠⠟⠁⠀⠀⠀⠀⠀⠉⠣⠈⠟⣰⣿⣿⣶⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠻⢋⣔⢷⡙⣿⣼
⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⠀⠘⡷⣄⠈⠉⣉⣻⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢿⢋⣤⡙⢿⣶⣿⣿⣿
⠀⠀⠀⠲⡿⠻⣅⠒⢦⡀⠀⠀⣖⠻⢿⣷⡈⠙⠿⢯⡻⣿⡿⠿⠿⠿⠷⠿⠿⠿⠿⢿⣿⣿⡟⠻⣿⡟⣠⣙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠉⢹⠯⢳⡄⠉⠒⢤⣀⣀⣤⡤⢠⣤⣤⣼⠿⣿⢋⢗⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡆⠉⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡀⠈⠀⠈⢹⡀⠀⠀⠀⠸⢆⠀⢻⣿⣿⣷⣦⣿⣯⣿⣿⣿⣿⣿⣿⣻⣿⣿⣧⡀⢻⢋⣾⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠁⠈⢹⣿⣿⣿⣿⣧⠤⠻⣿⣿⣿⣿⣿⡿⠿⢿⣿⢀⣬⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠉⠸⡦⣿⠛⣿⡿⣿⣗⣦⡘⠃⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⢴⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⡧⠶⣍⢱⡈⠻⣿⠃⣾⣿⣿⣿⣟⢻⡿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠁⠈⠰⠺⠟⠛⡿⣶⠄⢤⣤⡌⣿⣿⢿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⣤⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠿⢆⣀⡉⠀⣛⡙⣻⣿⣷⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡜⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⠂⠀⠛⠋⢻⣤⣟⣫⣶⣿⣿⠟⠉
⠀⠀⠀⠀⢀⣠⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠈⣹⣿⣿⠟⣹⡿⠟⠁
⢀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣈⡹⠹⠟⠉⠁
⠰⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⠀⠀⠐⠂⠈⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂
⠀⣄⡄⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠐⠾⡞⠁⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠙⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣉⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣬⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣦⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠰⣾⡥⠀⠀⠀⠀⠀⠀⠀⣀⣐⣶⡞⠏⠘⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣌⡿⠷⠀⠀⠀⠀⠀⠀⠀⢤⣴⠻⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⣦⣛⠓⠀⠀⠀⠀⠀⠀⠀⢒⢿⡘⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢠⣄⣼⡼⠻⠄⠁⠀⠀⠀⠀⠀⠀⠀⣉⡟⠆⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠻⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡙⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣌⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠤⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⢀⠾⣖⢂⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⠶⠶⠚⠋⠉





⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⠿⠟⠿⠻⣯⣿⡿⠿⠶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⣤⣿⣿⡋⠙⢻⢦⣀⠀⠀⢀
⠀⠀⠀⠀⣀⣮⡉⠛⠉⠀⠀⠀⣀⣠⠤⡴⠰⡄⠀⠀⢸⠘⠃⠀⢦⣽⣿⠀⠐⢾⡿⠉⢻⡳⡽⣆
⠀⠀⠀⢈⣯⡇⠸⡄⠀⣠⢖⢸⠁⡟⢘⡇⣺⡇⠀⠠⣼⣰⠀⠀⠀⢹⡇⠀⠚⢾⡇⠸⣦⡷⢿⣿⣦
⠀⠀⠀⢸⡁⣇⠀⢹⣼⡖⡧⢼⣉⣿⠩⣆⣼⠷⠒⠛⠉⠉⠛⠒⠦⣿⡇⠀⠀⢸⡇⠻⣾⠄⠀⡏⣿⡇
⠀⠀⠀⠘⡇⢻⡀⢀⢻⡇⢼⢉⠄⢶⣴⣿⡷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣢⡀⣼⠀⠀⣾⠀⠀⡇⣿⣿⠆
⠀⠀⠀⠀⢳⠈⣧⠀⠿⢷⣺⠠⣤⠞⠙⠛⡿⣳⢇⣷⡆⢦⢰⣰⣴⣔⣾⣿⣷⡿⠀⠀⡟⠀⠀⡇⣿⠃
⠀⠀⠀⠀⠈⡇⠸⡆⠐⣆⢻⡦⡟⠀⠀⠀⠈⠁⠋⠸⠁⠟⠸⠁⡇⠿⠸⠙⠉⢱⡀⢸⣿⠇⢸⡇⣿
⠀⠀⠀⠀⠀⠻⡀⢻⡀⢉⡘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⣹⠀⢸⢡⣿⡄
⠀⠀⠀⠀⠀⠀⢣⠈⣧⠀⣷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡇⠀⡾⢸⡟
⠀⠀⠀⠀⠀⢳⣌⢇⠘⣆⠀⢧⠹⣤⣤⣰⢀⡆⡤⢀⡄⣀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣼⠃⢠⠇⣿
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠹⣄⡀⠀⠻⣍⠙⠙⠚⠧⡟⢾⢡⠖⣠⣂⣄⡀⣠⣾⡏⣠⡏⠀⣼⢰⡇
⠀⠀⠀⠀⠀⠀⢸⠹⣷⢰⣧⢾⡀⠀⡜⠳⣦⣀⡀⠀⠀⠈⠘⢻⣿⣭⣷⠟⠉⠠⡿⠀⢰⢇⡿
⠀⠀⠀⠀⠀⠀⠘⣏⣿⢧⣿⣉⣣⠐⠃⡴⠈⠙⠛⠿⢿⠿⠿⣿⣛⠉⠁⠋⠀⠀⠀⠀⣞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⢳⣿⣍⣧⡈⠁⠀⠰⠶⣤⣀⣀⢀⡀⠀⠠⣄⢸⡄⠀⣶⣋⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⢿⠱⣬⠷⢆⣀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠙⠀⣀⡿⣅⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠞⠻⠿⣿⣯⣥⣾⣓⣴⡦⣤⣤⢠⡤⢴⡖⢻⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠙⠻⠿⣷⡾⠓⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣶⣒⡲⠦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠚⠉⠒⣨⣿⠦⢭⡀⠀⠀⠀⠈⠉⠙⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠈⡠⠾⡖⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⢿⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠿⡛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⣽⢟⠛⠛⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⣹⣿⡛⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢹⣿⡛⠉⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⣿⢟⡉⢀⣴⠀⢀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢹⡿⢅⣀⡾⠃⢰⡄⢠⡀⣀⢠⢸⢧
⠀⠀⠀⠀⡼⠁⢀⡄⣴⠄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⡉⠲⡿⢤⣼⢁⣽⣸⣧⣾
⠀⠀⠀⠀⣗⣯⣸⣄⣧⣄⣿⠀⢸⠏⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⡇⠀⡷⠈⡧⢙⠀⣾⠃
⠀⠀⠀⠀⡟⣍⢿⢸⡇⢰⠟⢉⡿⠦⣤⣇⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⢰⠁
⠀⠀⠀⠀⠹⡎⠈⠀⠁⠙⠀⠿⠁⣰⡟⠉⢒⣾⡇⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠋⠁⢈⣳⣿⠀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⢉⣱⣾⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⣉⣳⣾⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣈⣱⣴⠏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣍⣈⣦⣼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣵⣴⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠴⣷⠞⣡⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⢨⣍⣿⣯⡥⢀⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠺⠭⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠒⠒⠒⠒⠒⠂⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⣰⢫⣴⢣⡏⣶⢰⠆⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣜⠛⢦
⠀⠀⢀⣼⣹⢊⢞⠎⡠⢛⡼⠀⣿⠀⡇⠸⡆⠰⣄⠰⣄⠰⣦⡐⢤⡀⢦⡐⢦⡐⢦⡐⠲⡕⢦⢹⡘⢍⠎⠣⡀
⠀⠀⣼⠋⠋⠋⠋⠘⠠⠏⢰⠋⢀⡖⢁⡴⠁⣰⠋⢀⡿⠀⣼⠃⢨⡟⢀⡟⠀⣏⠀⡇⢰⡇⠸⠆⠁⠀⠀⠀⢡
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠐⠋⠀⠺⠁⠸⠇⠀⠾⠁⠸⠇⠀⠃⠀⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⢇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣆⡎
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠀⣾⣸⠰⡋
⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⢠⠇⣼⡀⠿⠙
⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⢘⡇⠸⣧⡘⠳⠈⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡧⠀⠧⡌⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡘⢧⡀⠨⢄⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣀⠀⡀⢄⠲⡈⢣⡀⠳⠄⠉⠀⠀⠀⠀⠀⣀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠓⠮⠥⠤⠤⠤⠤⠤⠐⠒⠈⠉
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⠠⣄⣀⣨⣭⣽⣿⣿⣿⣶⣶⣦⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣩⣶⣿⣿⠿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡔⣡⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠈⠙⠿⣿⣿⣦⡀⠄
⠀⠀⠀⠀⠓⠀⠀⡰⣸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣺⡳⣝⢲⣶⣄⠀⠈⠻⣿⣿⣦⣦⠀⠀⠀⠄
⠀⠀⠀⠀⠐⢆⣾⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣑⡖⢳⠙⣆⡀⠙⠻⣧⠀⠀⠈⢾⣻⣿⡇⣄
⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡙⡦⣏⢹⠈⠁⠻⣿⡀⠀⠈⠳⣄⢀⡀⢻⣿⣿⡄⣀⠉
⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡠⣆⢳⢤⠉⡇⠘⠀⢾⣷⡀⠀⠹⣦⠀⣀⢹⣿⣿⡌⣿⣿⣇⠈⠀⡄
⠀⠀⠀⠀⣺⣿⣿⣿⣦⣤⣤⢠⢰⡸⣄⢳⡎⣏⠘⠃⣠⣤⣄⠀⠙⣿⣦⣀⠈⣷⣿⣇⠻⠟⢃⣿⣿⣿⡀
⠀⠀⠀⢀⣿⣿⣿⡿⡿⣿⢾⣞⡎⡏⢸⠈⠃⢀⣤⣦⡻⢿⣹⣦⣄⢘⣯⣿⣷⠘⠋⢡⢰⣸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢻⣿⣿⣿⡏⠺⠼⠸⠘⡢⡉⢰⣶⣿⡜⠿⢿⡁⠈⠻⣤⣽⡆⠙⠋⢡⡄⣧⣸⠺⣽⣿⠿⣿⣿⡿⠋
⠀⠀⠀⠘⣿⣿⣿⣇⠀⣿⣄⠀⣿⣿⣌⠻⣯⣷⡀⠈⢻⣦⣄⠙⠋⠁⡄⢸⡀⣷⠻⣌⠓⠉⠀⢰⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⡄⣿⡻⡧⠹⣆⠙⣦⡘⢿⣍⡀⠀⠉⠉⢀⠐⡆⠻⣌⠙⡎⠃⠀⠀⠀⢠⣿⣿⣿⠃⠄
⠀⠀⠀⠀⢸⠘⣿⣿⣿⣌⢷⣶⣤⡙⢧⣈⢳⣆⡛⠿⠗⡀⢦⠘⡧⣏⡓⠎⠁⠀⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⣾⠀⠘⣿⣿⣿⣦⡉⢫⠛⢦⣙⠛⠿⠓⠀⢲⡹⡾⢿⡳⠎⠁⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠛⠀⠀⠈⢻⣿⣿⣿⣦⣙⠿⣿⠗⠄⣰⢜⣮⣯⡽⠂⠁⠀⠀⠀⢀⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷⣶⣤⣴⣿⣿⣿⡟⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠏⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠁⠐⠉⠿⠛⢿⣿⣿⣿⡟⠛⠛⠉⠁⣠⠀⢴⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠳⠛⠻⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⣿⣿⠿⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⢲⡌⠙⠛⢿⣿⣿⣿⠿⣿⣿⣷⣶⢶⣶⣶⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠘⠜⣡⣶⣶⣦⣌⠙⢮⡉⠓⠛⠒⠂⠤⠭⠽⣿⣿⡿⠀⠀⠻⠻⢿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣦⡈⠳⣴⡖⠀⠰⢦⣤⣀⣀⢀⣉⡉⠉⠁⠈⠈⠉⠉⠉⠙⠋⠝⣳⣤⡀
⠀⠀⠀⢠⣿⣿⣽⣶⣿⣿⢿⣿⣿⣆⠈⠻⣄⠀⣀⡹⠿⠟⠛⠛⠛⠋⠝⠒⣦⣄⡀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀
⠀⠀⠀⠐⣿⡿⠿⠿⣶⣾⣿⣿⣿⣿⣧⠀⣼⠿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣤⡀⠀⠀⠀⠈⢿⣿⡿⢶⢄⡀
⠀⠀⠀⠀⣿⠤⠰⢏⡁⣉⣿⣛⣯⣿⣿⠀⣿⣀⠀⠀⠉⡀⠀⠀⠀⠐⠀⠂⠤⠄⣈⣪⡛⠻⠷⢢⣀⠀⠈⣧⠤⠀⠁⣨⣆
⠀⠀⠀⠀⢻⡆⠀⠆⠀⠈⢹⡟⢻⣿⡟⠀⣽⡏⠙⠂⠘⠻⢡⣴⣾⣿⣿⣿⣿⣶⣦⣄⡉⠐⠤⢠⣿⣷⣦⡀⠑⠚⠛⠉⠁
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠁⠀⠁⠈⣾⠳⣾⣿⡇⠀⠀⠀⠀⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡉⠛⢿⣿⢷⣤⡀
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠞⢁⣾⡿⠋⠀⠀⠀⠀⣷⣿⡿⣿⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠈⠙⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠾⠿⠾⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⠋⠃⢉⠹⢻⠿⢻⣿⣿⣿⣭⣽⣿⣿⣿⣿⣆⠀⠀⠈⠙⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⠀⠉⠁⡰⠟⢻⣿⡿⣻⣭⣾⣿⣿⣿⣿⠄⠘⠉⠑⣶⣬⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠉⠀⠘⠑⡾⠙⠫⠿⣵⢿⣿⡿⠀⠀⣠⣼⡿⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠺⡛⣋⣿⠃⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢨⣿⢃⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣤⣀⠀⠀⠀⣀⣁⣴⢟⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠾⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⣔⣂⣶⠉⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⣿⡀⠉⠘⠛⣦⠋⣆⡔⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣛⣼⣯⠁⠀⠰⢾⡏⣿⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠣⠤⣈⣙⠒⠰⠼⠿⣯⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠃
⠀⠀⡀⠐⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡘
⠠⠄⡀⢀⠠⠀⠀⢀⡼⠁⠉⠓⢦⣄⣀⡀⡀⠀⣀⡀⣀⣀⠤⢖⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠇
⠀⠀⡨⠄⠀⠂⠀⣾⠁⠀⠀⠀⠈⠉⠁⠉⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡾⠀⡘
⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡇⢠⠇
⠀⠑⡄⠀⣠⣤⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⠃⡘
⠀⠀⠘⣆⠲⠍⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠦⢄⡀⣌⠀⡊⠉⠁⠢
⠀⠀⠀⠈⠑⠒⠒⠛⣆⠀⠀⣀⣠⡠⡴⠴⠖⠆⠆⠤⣄⣀⠀⠀⠀⢀⡞⠈⠑⠦⢽⡙⣾⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣤⠞⠀⠀⠀⠀⠀⠙⢿⡁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣖⠛⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠞⡇⡰⢤⡀⢀⡴⣏⣁⡤⠖⠂⢀⣠⣄⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠛⠁⠀⢹⣯⣿⣦⠄⡤⠀⢠⣿⠵⢋⡟⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠤⠞⠁⠀⠀⠀⠀⠀⣟⣻⣿⣿⢠⠇⠀⠀⠀⢠⡞⠈⠛⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⡶⣿⡁⠀⠀⠀⠀⠀⠀⠀⣘⡳⣾⠯⢍⡞⠀⠀⠀⠀⠚⠀⠀⠢⡘⠦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣴⠃⡇⠙⢶⡀⠀⠀⠀⠀⠰⢮⣿⣛⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠘⠫⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢳⠀⠀⢹⡄⠀⠀⣌⢳⡾⠯⢄⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠲⠦⢤⡀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠈⠳⣤⡼⢁⡐⣦⣹⡟⠓⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠴⠖⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡌⣷⣾⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⣴⣾⣿⣶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢀⣰⣺⣧⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣛⣦⣿⣿⣿⡿⢋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣴⣶⣾⣽⣞⠻⣍⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠶⣦⣽⢿⠹⠚⠉⠀⠀⢸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣯⡳⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⣀⣛⣿⣏⠻⠈⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠌⢠⣤⣽⡟⣧⠉⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠊⠰⢶⡾⣏⠓⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣌⢯⣢⡀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢃⣘⣻⣟⠷⠌⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣙⢮⣢⣄⠀⠀⠀⠀⠀⢠⠏⠶⢾⡿⣭⠃⠀⠀⠀⠀⠀⠀⡠⣪⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣭⡻⢧⣄⣀⠀⠀⡾⠠⣽⣿⣿⡃⠀⠀⠀⠀⣀⢴⣫⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣾⣭⣭⣀⣃⠸⠿⡿⠿⠀⣠⣤⣶⡯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠷⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⢋⣉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⣀⣀⣀⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢁⡾⠤⠒⠒⠚⠛⠛⠓⠒⠒⠤⢄⡀⠉
⠀⠀⠠⢤⡀⠀⠀⠀⠐⠢⠄⣸⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡐
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠉⠒⠤⠤⠒
⠀⠀⠀⠀⠀⢱⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠐⠈⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠒⠁
⠀⠐⠓⠲⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢤⣀⣀⡄
⠀⠀⠀⠀⡠⠔⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠜
⠀⠀⢠⠊⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⠀⠄⠂⠁
⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄
⠀⠸⠈⠁⠉⠐⠠⢀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠔⠂⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡈⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⠤⠚⠩⡀⠀⠀⢀⠞⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⠋⠁⠀⠀⠀⠀⠑⢄⡀⠏⠀⠀⠀⠈⠑⠒⠠⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣀⣠⣤⣬⣥⣤⣄⣀⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠾⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠈⠉⠛⠛⠻⠟⠛⠛⠉⠁⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⡴⣫⣴⣶⣶⣬⣅⡈⢇
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⣫⣾⣿⣿⣿⣿⣿⣿⡻⢶⣕⣄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣾⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠉⠛⠿⠿⢛⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁
⠀⠀⠻⣿⣮⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣠⣤⣤⡀⣶⠖⠶⠯⠽⣿⣿⣛⠻⠭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⣾⣿⣿⣿⣿⣜⡄⠀⠀⠀⠀⢀⣀⣀⣀⡤⠠⠤⠤⠤⠒⠒⠒⣒⣀⣈⣉⣉⣉⣉⣉⣠⣭⣭⣭⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣾⣿⣿⣿⣿⣿⣿⣿⣽⡀⢆⣉⣭⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⡷⠀⠑⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⣝⢿⣷⣮⣽⣿⣿⣿⠟⠀⠀⠀⢿⣿⣿⣿⣿⣿⣟⠅⠀⠀⠈⠪⣻⣿⣿⣿⡿⢿⠛⠏⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣙⣉⣤⣤⣤⣤⣤⡀⠀⠀⠀⣤⣄⠀⣀⣤⣴⣶⣶⣶⣿⣷⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠻⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃⠴⠒⠀⢀⣾⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡼⠀⠀⠀⢸⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⡅⠀⠀⠀⠀⢻⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠀⠀⠀⠀⠀⠈⢿⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠙⠛⠻⠿⠿⢿⣟⣛⠛⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⢠⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠈⠙⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⣦⣴⣶⣶⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣶⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠋⠀⠈⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⡇⠀⢸⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠐⢄⠈⢢⠀⡀⢣⠰⠀⡇⠆⢏⠄⣠⠋⠀⣠⠖⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⡑⢄⠑⠌⠀⠂⠀⡇⠘⠌⡔⡡⡢⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠁⡂⠀⠀⠀⠀⣀⠀⠁⠀⠪⡈⢀⠠⢚⡡⠀⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣀⣀⡉⠑⠀⠀⡐⢡⠊⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣀⣠⣄⣠⣤⡴⠒⠦⠤⠴⠒⠶⠄
⠀⣀⣀⣀⣀⡤⠴⠤⠤⠤⢄⡀⠀⠄⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣁⠀⣀⠴⠉⠉
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⣀⠨⠤⠤⠤⠤⠔⠊⠉⠀⠀⣀⠜⢁⡤⠂⠀⠐⠀⢀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠄⠀⣀⠐⠄⠀⠀⠵⢦⣒⠂⠀⠀⠐⠒⡴⠊⢉⠁⠂⠉⠀⠶⠀⣒⠍⣁⣀
⠀⠀⠒⠐⠒⠐⠀⠐⠂⠐⠒⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⣰⠚⣡⣀⣄⣖⡰⠀⠀⠂⠁⠚⠀⠘⠲⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⢤⣭⠴⠞⠟⠋⣁⣉⠁⣀⣈⠛⠛⠛⠋⠉⠉⠓⠒⠛⠛⠚⠓
⠀⠀⠀⠠⠶⠒⠒⠛⠒⠛⠓⢾⣍⠉⠁⠀⠩⠤⠬⢒⣐⠈⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠰⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡀⣄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡆⢺⠉⠇⠹⠀⠃⠘⠈⠃⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⠹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡘⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣠⡍⣄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⣌⢻⣮⣷⢿⣾⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠙⣦⠼⣯⡻⠞⠃⠁⠉⠉⡙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣙⣧⣝⠻⠎⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣬⠷⢦⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣤⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡒⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡼⠟⠗⠀⠀⠀⠀⠀⣤⠀⠀⡴⠀⢸⣤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⡟⠛⠃⠀⠀⠀⠀⠀⣻⠐⠰⡻⡄⣨⠝⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⡀⠀⠄⣀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠷⣦⣟⣛⠃⠀⠀⠀⠀⠀⢱⠀⠵⠱⡩⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠅⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣻⢶⣟⣋⠀⠀⠀⢠⣶⠄⠈⠀⣴⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠴⠆⠀⡀⢠⠀⠀⢠⣦⢠⣝⢻⣭⣥⠀⠈⠂⢺⡿⡔⢠⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣠⠙⢿⣍⡤⠷⡻⠈⢁⡴⣭⠻⠶⠆⠀⢦⣜⢕⠟⣿⣼⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣦⣄⣀⣀⣤⣤⢿⡽⣮⣛⠋⠀⣃⣼⣿⣷⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣛⢀⣴⣷⣽⣿⣷⣷⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡖⢠⠀⠀⠀⠀⢀⠀⠀⠀⢀⡆⠀⠀⠀⠀⣠⢠⡆⠀⠀⠀⡴⠀⠀⠀⠀⢠⠄⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠓⣯⢠⣞⡅⠀⣏⢠⠿⢇⡿⣴⣠⣞⡄⣰⢃⡞⠀⠀⠀⣼⢱⣄⣾⣡⢠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠴⠋⠈⠙⠛⠣⠊⠙⠂⠛⠰⠋⠙⠙⠋⠓⡟⠀⠀⠀⠘⠑⠟⠈⠉⠓⠛⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⢶⠶⠶⣶⣶⡶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣾⣷⣿⣿⣾⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠰⠤⠖⠲⠶⠿⢿⣷⣶⣶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠤⠤⣄⠀⠀⠠⣤⡄⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣥⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⡘⢧⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣿⣧
⠀⠀⠀⠀⠀⢶⣿⣫⣿⣿⣫⣢⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⠤⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⢿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣶⣿⣿⣿⣌⠀⠀⠀⠀⠀⣀⣀⣀⣤⣽⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡗⠌⣍⣏⣫⣭⣿⣧⠀⠀⠀⠀⠀⠀⣀⣷⡿⠿⠿⠟⠛⠻⢷⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣴⣶⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢰⣞⠿⣿⣿⣿⠿⠿⢿⣿⣿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠹⠁⠤⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣶⣦⡀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠶⠀⠛⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣋⣿⣧⠀⠀⠀⣀⠗⣀⠀⣄⣠⣶⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣷⣤⣶⣾⡻⠶⢦⡄⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣽⣿⣷⣠⣴⣶⣭⣭⣿⣶⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣍⣠⣉⠐⠶⣼⣿⣧⡀⠀⠀⠀⠀⠰⣿⣿⣯⣍⣿⢾⣭⣿⣿⡿⢿⣼⣿⠋⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠙⠿⠿⠟⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠾⠿⣿⣿⣄⠈⣿⣿⠀⠸⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼⣦⣀⣦⣸⣿⣿⡿⠃⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⠀⠀⣤⠙⢿⣤⣼⣿⣦⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡋⣈⠙⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣁⠄⢈⣿⣄⢸⡷⢤⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠃⠀⣿⣿⣿⡇⠀⠀⠈⠻⠻⣿⣿⣿⠁⠐⠛⣿⠃⠉⠁⠈⣿⣿⣟⣛⣉⣯⡅⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠙⣶⡄⣿⣿⣿⣿⣿⣿⣌⠠⣤⠄⠀⠙⠉⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠳⠞⠻⠋⠖⠛⠃⠀⠀⠀⠈⠛⠿⠿⠿⣿⠿⢷⠟⠋⠀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡿⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠀⠈⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢘⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣤⡄⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⡀⠀⢰⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⢳⠀⠀⢀⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢸⠟⠓⡀⢈⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠿⡟⠂⢨⣥⣄⠸⠆⠘⣃⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⡀⢀⠀⠀⠄⠐⠒⠦⠀⠀⠀⠀⡸⣆⡀⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠛⠋⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠤⠤⣀⣀⣀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠯⣄⡀⠀⠀⠀⠀⠀⠠⠤⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡟⠓⢶⣶⣶⣶⣶⣷⠶⣿⣿⣿⣿⣿⡿⠻⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠧⣾⣧⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣉⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡥⢒⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣱⣻⣿⣿⣿⣿⡿⠃⢤⡀⠄⣸⣿⣭⣿⣿⣿⣿⣿⣿⣺⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⡿⠁⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡯⢿⣿⣿⡿⠁⠀⠀⠀⠀⠛⣻⡟⠃⠈⣯⢿⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡟⣋⢿⣿⣿⠇⠀⠀⣀⡀⠀⠊⠉⣁⣴⠈⠁⠘⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣷⣾⣧⣾⢺⣿⣿⣦⣆⣶⣿⡿⠀⠠⠀⢻⣿⡗⣾⢺⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⢾⢸⡿⣿⣿⣿⣿⣿⣿⣷⠒⢺⣿⣿⣽⣷⣿⣿⢇⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣇⣘⣻⣿⣿⣿⣿⣿⣒⣺⣿⣿⣿⣿⣟⣉⣸⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⠿⠿⢿⣟⠛⠛⠛⠛⠛⠛⠛⢛⣻⡿⠿⠷⠶⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⢦⠀⠀⠀⢀⡴⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡧⠖⠢⣾⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣷⡀
⠀⠀⠀⠀⠀⠀⢿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡷⠀⠀⢿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⡟
⠀⠀⠀⠀⠀⠀⢸⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡟⠀⠀⠀⠈⢟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⢯⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠏⠀⠀⠀⠀⠀⠈⠣⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣞⣛⣛⣛⣛⣛⣛⣛⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣛⣛⣛⣛⣛⣛⣛⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⡟⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣀⡴⠞⡻⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⠀⠀⠀⢠⡾⢋⡴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢻⣄⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⣾⠀⡾⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⢹⣄⠻⣄⠉⠛⢶⣤⡀⢸⡇⢸⡇⠸⣇⠀⠀⢻⡄⣷⠈⣷⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠲⣤⡀⠀⠈⠻⢦⣈⠙⠲⣄⠈⢿⣾⠁⢸⡇⠀⢿⡄⢀⣼⠃⣿⠀⢹⡆⠀⠀⣀⡴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⢤⡙⠳⢤⣀⠀⠙⣦⠀⢻⣆⣸⣿⣤⣼⣧⣤⣼⣿⡟⠉⣠⠟⢀⣾⣃⣤⠞⢩⡖⣿⠋
⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠉⠓⢶⣿⡶⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢿⣤⣾⠟⠋⣠⡴⢋⡼⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣶
⠾⠛⠳⢦⡈⣷⣄⠀⠀⠀⠀⢀⣀⣹⣆⠀⠻⣦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⢠⡞⠁⠀⢀⣴⠞⢛⣉⣉⣉⠙⠛⣩⡿
⠀⠀⠀⠈⢷⡹⣍⡛⠿⠿⠛⠋⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⣿⣀⣀⣴⠟⣡⠞⢩⡴⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠈⠻⣦⣉⡛⠚⠛⠛⠛⠲⣦⣼⠏⠀⣠⠶⠚⠋⠻⠿⢷⣴⣿⠀⠀⣾⣆⣤⣶⣟⣿⠓⠶⣦⣄⠹⣏⠉⠉⣁⡴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣼⠇⠀⣀⠁⢠⣶⣿⣿⣿⣿⣿⣿⠂⠀⢻⣿⣿⣿⣿⢿⣿⣦⠈⠙⣆⢻⡟⠛⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⣿⠀⠀⠻⠷⠿⢿⣷⣯⣻⡼⠟⢿⠀⠀⢸⡏⠿⣿⣧⣽⣽⠿⠳⠞⠃⠈⣿⠻⠿⢧⣤⣀⣀
⢀⣀⣠⣤⣶⣒⣚⣛⣉⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣈⣉⣙⣛⣿⣶⣦⣤⣤
⠈⠙⠛⠛⠻⠿⠧⣤⣤⣀⣀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣏⠀⠀⠀⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣀⣀⣤⡤⠴⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⠟⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡶⠚⠛⠛⣷⡀⠀⠀⠀⠀⠀⣠⣀⣀⣠⣴⣤⣤⢶⣤⣄⣤⣄⠀⠀⠀⠀⠀⢠⣿⣄⠀⠻⢤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⣀⣀⣠⣼⠇⣰⠏⣠⡤⠴⣾⠟⠳⣄⠀⠀⠀⠀⠟⠛⠛⢻⣿⠿⠷⠛⢻⡟⠛⠛⠀⠀⠀⠀⣠⣿⡀⠉⠳⠶⠤⠶⠶⠶⢬⣙⠳⣄
⠀⣴⢟⡭⠥⠤⡤⠴⠞⢁⡾⠃⠀⣰⠏⠀⢀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠙⠷⣶⠶⠟⠁⠀⠀⠀⠀⣠⣾⡃⠈⢿⣦⣤⣠⣤⠴⠶⠾⠶⣌⢣⣹⡆
⢸⣷⡶⠟⠛⠓⠶⠶⠟⠋⠀⠀⣰⠏⠀⣴⠟⠁⣀⡽⠷⣤⣀⡀⣂⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣅⠈⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢦⣽⣟⣳⣶
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠞⢁⣴⢾⡟⠀⢠⡟⠉⣙⣿⡿⠿⣶⡶⠶⢾⠟⢻⡇⢀⡟⠉⠻⣦⣀⠙⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣼⣥⠞⠋⠀⠘⣷⠀⣾⠀⣸⠏⠈⣿⠀⢹⠀⠀⣿⡄⠈⢧⠈⢷⣄⠀⠀⠙⠳⢦⣉⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇⢸⡆⣿⠀⠀⢹⡆⢸⠀⢰⡟⢳⣄⠈⠳⢤⡉⢷⡄⠀⠀⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⡼⢀⡿⠀⠀⠘⣧⢸⠀⣾⠁⠀⠉⠛⠷⣦⠹⡄⢿⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡜⢁⡾⠃⠀⠀⠀⢻⣼⢰⡇⠀⠀⠀⠀⠀⢸⡆⠇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣯⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣗⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿

⠀⢠⣤⣤⣤⡀⠀⣠⣤⣤⡄⠀⢠⣤⣤⣤⢠⣤⣤⣤⣤⡄⣤⣤⣤⣤⡀⠀⢠⣤⣤⣤⠀⢠⣤⡄⠀⢠⣤⡄
⠀⣿⣿⠉⠷⡯⢰⣿⣿⢹⣶⡇⢸⣿⡏⠉⠈⠙⣿⡟⠉⠁⣿⣿⠏⣿⣿⠀⣸⣿⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠻⣿⣶⣦⡀⢘⣿⣿⢸⣿⣇⢸⣿⣷⣶⠀⠀⣿⠿⠀⠀⣿⣯⣷⣿⣏⠀⣽⣿⢹⣿⡇⢸⣿⡇⠀⢸⣿⡇
⠀⣤⣬⠙⣿⣿⢸⣿⡿⢸⣿⡇⠘⣿⡏⠀⠀⠀⣿⣿⠀⠀⣿⣿⡇⣿⣿⠠⣿⣤⣼⣿⡇⢸⣿⡇⠀⢸⣿⡇
⠀⢻⣿⡶⣿⡿⠈⢿⣿⣾⣿⠇⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⣿⣿⣷⠿⣿⢰⣿⡿⠹⠟⣗⢀⠾⢿⣷⢸⣿⣋⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⢿⡄⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣟⡤⠚⠉
⠀⠀⠀⣶⣾⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⠿⠶⠿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠙⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠹⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠆⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗
⠀⠀⠘⠋⠠⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣄⣠⣶⣿⠁
⠀⠀⠀⢀⣀⣀⣩⣿⣿⣶⣾⠛⠉⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣿⣿⣿⣿⠷⠶⠂
⠀⠀⠀⠀⠛⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠀⠒

⠀⠀⠀⠀⠀⢀⣠⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣔⣒⣓⠂
⠀⠀⠀⢤⣾⠿⠿⣿⣿⢿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠿⠿⠛⠛⢏⡉⠉⠉⠉⠙⠉⠓⠣⢐⠀⠀⠀⠂⠀⡠⠠
⠀⠀⠀⠀⠁⠀⠀⠈⠉⠛⠿⣟⢾⣷⣾⣷⣾⡿⠛⠉⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣫⣿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⣰⡶⢾⣯⡉⣧⣝⠉⣛⣿⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡄⢸⡿⢁⠻⣽⢋⠉⡙⢟⡿⣯⣽⣶⣦⡀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣠⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢳⠀⢰⣿⣿⢸⣄⣐⣢⡎⡐⠠⠈⡟⣴⠋⠀⠀⠀⠀⠀⠀⠀⠺⣿⢿⣿⣟⡿⠁
⠀⠀⠀⠀⠰⣄⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣽⡇⢸⣛⡫⢚⣉⡉⣉⣉⡪⣭⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⠁
⠀⠀⠀⢆⠀⣸⠟⠀⠘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⠛⠋⠽⠟⢹⣯⣽⠟⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠤⡶⠒⢶⣂⢶⣾⣶⣿⡷⢶⣒⣂⣶
⠀⠀⣀⠈⣹⣿⡆⠐⣄⡇⡰⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠈⠀⠀⣼⠟⠙⣿⣷
⠀⠀⠘⣄⠀⠙⣿⣴⣯⠞⠁⠀⢀⣾⣿⣻⣿⣿⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣎⢀⣀⠀⠀⠀⠀⠀⠀⢸⠟⣠⣶⣿⢿⣇
⠀⡀⠀⠈⣷⣠⣿⣿⠏⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡷⠀⠀⠀⠀⠀⢀⢄⠙⢻⣟⢻⣿⣿⢿⣿⣦⡀⠀⠀⠀⡼⠖⠋⡇⠻⡌⣿⡀
⠀⠀⠀⠐⠘⣏⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠤⡴⠂⠀⠀⠀⣌⣞⣀⠘⢧⢀⣸⡿⠚⢿⣽⣷⠀⠀⠀⠁⠀⠀⡇⠀⠀⠁⠁
⠀⠀⠀⠀⠠⢻⠉⠹⠇⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⣷⣤⣿⣿⣧⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢬⢉⠙⢋⣭⣵⣿⣿⠃⠀⠀⠀⠀⣀⣠⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠀⠉⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⢀⣤⡶⣶⣿⡿⢰⣶⣤⡀⠀⣀⣤⣤⣤⡀⣀
⠀⠀⠀⢀⣴⣿⣿⡷⣈⣿⣶⡿⣿⣟⡻⣮⡙⡗⢿⢫⣽⣿⣷⡄
⠀⠀⠀⣼⣻⣿⣿⣷⣿⣿⡿⢱⡿⣯⣽⠏⣽⣿⣿⣿⢿⠿⣿⣿
⠀⠀⠀⣿⣽⠾⢿⣾⢏⡸⣿⣾⣷⣤⣼⣆⣾⢛⣏⡴⣿⣿⠞⠿
⠀⠀⠀⣿⣽⣿⣿⡠⠛⡷⣹⡿⣩⣶⣶⣟⢿⡹⢿⡋⣴⣏⡞⠁
⠀⣰⣾⡿⣿⣧⣍⣠⣥⣼⣿⣧⣿⣽⡿⠏⠈⣷⢾⡷⣤⣽⣿⣄
⣸⣷⣽⣷⣦⢦⣽⣿⣿⣿⣿⣿⣿⣿⣶⣦⣾⣿⡟⢶⠏⣻⣯⣿⣧
⡉⣉⠻⢶⣿⣮⡞⣉⣿⣿⣿⣿⣿⣿⣿⣿⢶⣿⣧⣤⣹⣿⣾⣯⣼⣧
⠈⣻⣷⡿⢿⣿⣿⠛⢿⣿⣿⣿⣿⠿⣹⣍⣫⠻⣿⣟⣃⣴⣿⣧⡿⢹⣧⡀
⠀⠙⠿⣿⣷⣾⠿⢿⣶⣿⣿⣿⡯⣿⣿⣿⣿⣧⢸⣿⣿⣶⣿⣃⠀⡻⢏⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠿⢻⣇⠿⣿⣿⣿⠷⣼⠿⣿⣿⣍⣉⣀⣤⣾⢫⣻⣆⠀⣰⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣻⣿⣿⣼⣼⣷⣿⣥⣾⢟⣿⠻⣿⣛⣯⡿⣷⡾⢿⣷⣿⠋⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⣯⣥⣴⣆⠀⠙⢿⣿⣍⣴⣟⣽⢿⣻⡟⢠⣼⣿⣿⡄⠐⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣡⣦⡀⣼⣷⣼⣿⡍⡻⣿⣹⣿⡿⣿⣾⣿⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣼⣿⣯⣭⣽⠿⡷⠴⢧⣿⣯⣿⣿⣿⣿⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢻⣿⡟⣫⢴⣾⣿⣿⣾⣿⣿⣿⡱⣻⣿⣿⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⢻⡯⢷⣀⣙⣿⣿⣿⣿⣽⣿⣿⣿⡿⠃⣰⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣼⣷⣾⡟⣿⠻⣿⣿⣮⣙⣛⣛⣉⣤⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠿⣿⢿⣿⣷⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⡟⠉⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢧⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠐⡄⠀⠀⢀⣗⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠽⠿⢿⣷⡄⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⠴⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⢰⣄⠀⠉⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠙⣿⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⣫⡿⣿⣿⡝⢳⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢹⣿⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⡇⣾⣿⣿⢠⣿⣿⣸⣿⣄⠀⠀⢀⣀⡀⠀⠀⠀⠰⣿⡾⢻⡿⣶⣄⠀⠀⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣷⡆⣼⡿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣤⠭⢿⣍⣿⣿⣿⠇⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡿⢶⢛⣼⠟⣰⡟⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⢻⣿⢿⣿⣷⣿⠃⠀⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⡇⢠⣾⣿⢳⣿⣧⣿⡟⠀⠀⠀⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⣛⣯⣭⣿⠀⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡩⣴⣿⣿⣿⡗⢁⣶⠀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⠇⣼⡟⢿⣟⣿⣿⣿⣼⡏⣰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣿⠟⣿⣭⣽⣽⢯⣿⢠⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⢛⣽⡇⢸⣿⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣹⡿⢁⣿⠴⣏⠙⠃⠈⢻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣷⣿⣿⣧⣹⡿⠀⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣭⣤⣤⣤⣤⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣾⣿⠙⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠛⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⢉⣥⣤⣤⣝⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣿⣍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣿⣿⣿⣿⣏⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⢿⣿⡿⠿⠟⣯⣽⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣿⣿⣷⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣄⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠻⠿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⣠⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠶⢶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⢹⣿⣿⣷⡀⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠙⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢿⣿⣿⢿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉














⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠙⣀⡉⣿⡁⢀⣿⠉⣉⣯⡙⢃⣯⡿⢋⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠒⠒⠒⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠿⣍⣿⣩⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣈⠳⣾⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠰⣌⣻⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣀⠳⣼⣋⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣄⠀⣎⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠰⣌⣷⠯⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢦⣬⢷⠚⡟⣿⢻⣷
⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡈⢧⣽⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⡿⣏⢻⠸⠂⠉⠁⠟⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⢀⢶⣹⣾⣭⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣓⣶⡏⢷⠙⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⣰⣸⣾⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣭⡟⢧⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠸⣿⢿⣉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⡿⢦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣽⣛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡥⣿⣍⡃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠾⠅⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⡀⠀⠀⠀⠀⣤⠟⣯⣩⣄⢀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠦⠤⠾⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠙⠉⠙⠿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠈⠻⣿⣿⣿⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⢐⣿⣿⣿⣿⣿⣷⠶⢤⣼⣿⣿⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣼⣽⣿⣦⠖⢻⣿⣿⣷⣿⣿⡟⠀⠀⠀⣽⠛⠝⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣿⣿⣿⡿⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⠀⠃⣠⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢿⣿⠿⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠈⠀⠀⠤⠄⠀⣿⣧⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠂⠀⠍⠠⠀⠀⠀⠀⠂⠀⠩⠭⠭⢍⣉⡒⠭⠀⠉⣁⠂⠌⢐⠠⠭⠭⠍⡐⠀⠀⠀⠀⠀⠀⠀⢒⡀
⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⣒⠀⠀⣀⠀⠀⠀⠀⠈⠡⠀⣒⣒⡒⠒⠒⠂⠨⠥⠀⠀⠌⣐⠒⠨⠭⠀⢐⠢⠀⠀⠀⢀⡀⠀⠀⠠⢒
⠀⠀⠀⠠⡂⠭⢔⠀⠀⠀⠨⣒⠀⢀⡒⠀⠀⠀⠀⠀⢈⠅⣒⠤⠭⠭⠭⠭⠍⠉⢉⣀⠒⠀⠤⠩⠍⠁⡒⠄⠀⠀⠀⡂⢐⠀⠀⠈⠔
⠀⠀⠀⠨⢒⣉⣒⢅⠀⠀⠀⢐⣀⡐⠠⠀⠠⣁⡀⠀⠀⠀⡢⠍⣀⣂⡩⠭⢐⠂⠀⠤⠤⠌⠁⣀⣐⡒⠢⠀⠀⠀⢔⡂⣐⡪⢄⠀⠁
⠀⠀⢄⡪⠤⠒⠢⢕⠠⡀⠀⠀⠒⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢕⡒⣒⠢⠍⡐⠄⠈⠀⠀⠐⠒⠂⠀⠠⠀⠀⠀⡪⢕⣒⣒⡂⠅⡀⠄
⠀⠠⢑⣒⡭⠭⠉⡒⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠁⠐⠒⠂⠭
⣀⢁⣐⣒⣒⣒⣒⠢⠅⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡐⠠⢐⡪⠭⠤⠤⠬⠉⠁
⡀⠀⠒⣒⠂⠤⠤⠬⠅⢒⡈⠄⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠤⠄⣁⠒⠠⠭⠭⠭⠄⡒
⠔⡀⠠⢒⠨⠭⠭⠭⠭⠀⠐⠒⠂⠀⢀⠠⠤⠐⡀⠠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠂⠤⠄⡀⠀⠀⠐⠂⠀⠀⠬⠭⠭⠭⠁⠂⠀⡀
⠭⠐⡪⠄⣂⠉⠭⠉⠉⠉⠉⠉⠉⠀⡐⠭⠭⠥⠀⠐⡡⠀⠐⠈⠤⠤⠁⠂⠠⡐⠀⠨⠭⠭⢀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢑⡂⠌⡒⠨
⠭⢕⡪⢅⣒⠀⠀⢀⣐⠒⠀⠀⠀⠀⠠⠭⠍⠉⠉⠤⠐⠀⠀⠀⠀⠀⠀⢀⠐⠠⠍⠩⠉⠭⠅⠂⠀⠀⠀⠐⢒⣀⡀⠀⣒⡂⠅⣂⠭
⠭⢑⠪⢑⠢⠬⠥⢐⡂⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠤⠄⠀⠀⠈⠄⢒⡂⠠⠐⣒⠭⢒⠩
⠭⠐⠭⡒⠭⠉⠭⢕⠂⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⣀⣒⠢⠀⠀⠀⠀⠀⠠⢒⣀⡀⠤⠀⡀⠐⠀⠀⠀⠀⠀⠈⢐⠨⠭⠭⢔⡪⢐⠨
⠤⡀⢄⠠⢑⣒⡪⢕⠀⠀⠀⠀⠀⠀⠀⠠⠐⡂⠀⣒⠠⠁⠀⠀⠀⠀⠀⠀⠥⢒⡂⠀⣐⠀⢄⠀⠠⠀⠀⠀⠀⢔⡪⠭⠩⠥⣊⢕⠪
⠩⠂⢕⠭⣒⠒⡂⠁⠀⠀⠀⢉⣁⠅⡀⠀⠠⢂⠀⡒⠌⠀⢄⡂⠭⠅⣒⠁⠈⠔⠒⠀⡂⢄⠠⢄⣀⣀⠀⠀⠀⠀⡪⢅⣒⠩⡒⢕⠨
⠀⠀⠐⠅⡂⠄⡂⠀⠀⠀⠠⢐⣒⠨⢐⠀⢊⠄⠄⠠⣐⠨⠕⣒⣈⣐⡒⠬⢐⠠⠀⠀⠨⣒⠨⠅⣒⣒⠨⠀⠀⠀⠀⠅⠒⠨⠂⠁⢀
⣤⣦⣤⣤⣦⣴⣦⣤⣤⣤⣤⣶⣶⣦⣥⣦⣴⣤⣤⣤⣶⣬⣭⣤⣶⣶⣤⣭⣴⣦⣤⣤⣴⣦⣭⣥⣴⣶⣮⣵⣤⣤⣤⣥⣤⣥⣤⣤⣦
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠖⠉⣭⠿⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣎⣀⣼⡠⠃⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣛⠯⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢆⠀⠀⠄⠐⢄⠹⡂⢠⠈⢖⣠⠂⢠⠀⡀⣀⡀⠤⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠦⣸⡄⢄⠻⣇⡜⢣⣸⢁⢂⣾⠟⠋⠁⣀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡀⠰⠈⢻⣬⣶⣿⠞⠳⣿⣿⡏⠀⠰⢀⣼⡉⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛⠻⣦⣴⣯⠝⠉⠀⠀⠀⠀⠉⢋⣷⡶⣋⠥⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣑⠶⠔⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠿⠒⠉⠉⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⠐⠉⠀⠔⠚⢅⣻⣿⠀⠙⠒⠂⠀⠀⠑⠚⠋⢈⣿⢌⡵⠒⢀⣤⠄⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⣠⠔⢲⡿⢋⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣒⠻⢍⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠖⠁⢂⣡⡞⠟⢭⣷⣤⣀⣀⣀⣠⣴⣯⠟⠉⠻⣦⡅⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢋⠘⠂⣡⣿⣿⡟⠷⢻⡟⢯⠻⣅⡘⠀⠘⣷⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄⠉⣀⣠⣴⢾⠏⠌⣸⡏⢶⠋⣿⣄⠣⠹⡏⢦⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠐⠀⡔⠁⢧⠀⠀⠸⣄⠑⠄⢩⠀⠌⠢⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⡶⢯⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠙⠃⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢾⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠙⠶⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠖⠦⢠⡶⢶⡄⢶⠶⠆⠶⣶⠶⠐⡶⢶⡄⠐⣶⡆⠀⢶⠆⠀⢲⡆
⠀⠀⠀⠀⠀⠀⢠⣍⣷⠸⣄⣰⠇⣻⡛⠀⢀⣿⡀⢀⣏⣹⠆⣸⡟⣧⡀⣸⣀⡄⣸⣇⡤

⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⡀
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠠⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠑⠳⣄⠀⠀⠈⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠎⠀⠀⢀⡴⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢕⢄⠀⠀⠙⢦⡠⠒⠈⠀⠀⠀⠀⠈⠐⠢⢄⣠⠞⠁⠀⢀⢔⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⡀⡰⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⡀⠀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠀⠘⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠈⢅⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢂⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡉⠓⠄⣀⠀⠀⠀⠀⠀⡀⠔⠊⠀⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠊⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢔⠄
⠀⠀⠀⠀⠀⢀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠅⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠉⠉⠃⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠀⠈⠀⠒⠢⡀⢀⡠⢀⠀⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠐⠲⠂⠀⠠⠒⠀⠀⠀⠈⠊⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠤⠀⠁⠀⠀⢠⠆⠁⢀⠀⢀⠀⠀⠀⠀⠀⠔⠒⢄⠀⢢⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡐⠀⠀⠀⠀⠀⠉⠀⠀⠋⠗⠛⠀⠀⠀⠀⢁⡐⠀⠀⡦⠚⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣄⠀⡔⠒⠲⠄⠀⠀⠀⣈⠆⡇⢀⡄⡀⠁⠀⠀⠉⣲⠇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠂⠡⡤⢀⠐⠂⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢛⠁⠀⠙⢆⠀⠀⠀⠀⡀⣀⢀⡀⣀⡤⢄⡴⢸⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠢⢦⡀⠀⢧⢤⡀⠀⢱⠀⠠⢀⡸⠀⠾⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢧⠔⠁⠈⠀⣠⠟⠀⠀⠀⠈⠒⠓⠓⠲⠶⠶⣾⠃⠘⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠈⠐⣤⡄⠘⢯⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣳⣀⠀⣀⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠐⠠⣀⡀⠀⢰⠇⠑⣄⠀⠀⠀⠟⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠑⢞⢐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠠⠀⠈⠉⠁⠀⠀⠘⢧⣀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⠐⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⡂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢈⠀⠐⣠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠄⠀⠒⠤⡀⠀⠈⡰⠖⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠂⠀⣲⠈⠉⢉⡆⡇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢆⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠙⠄⢧
⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠘⠀⠀⢀⡀⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⡇⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⢀⣴⣶⣶⣦⠀⢀⣴⣶⣶⣦⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣠⣤⣤⡄
⠀⠀⠀⠸⣿⣿⣿⡿⠀⠘⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⡿⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⠐⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⠀⢰⣾⣿⣿⣿
⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠈⠻⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣤⠀⢠⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠠⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠁⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠄⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠸⠿⡿⠁⠀⠀⠹⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡘⡹⠁⠀⢀⣾⠏⠀⠀⡠⠛⠋⠀⠀⣤⡿⠋⠁⠀⣀⣴⠿⠛⠁⠀⠀⣀⣤⣴⡾⠿⠛⠉⠁⠀⠀⠀⠈⠉⠲⠤⣀⣀⠀⠀⠉⠉⠛⠻⠿⢿⣷⡶⠦⠤
⠀⢰⡷⠁⠀⢠⡿⠃⠀⢀⣾⠄⠀⢀⣴⠟⠉⠀⣀⡴⠞⠋⠁⠀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⣠⣤⣶⣦⡀⠁⠢⢄⡀⠉⠑⠶⢦⣄⣀⠀⠀⠀⠀⠉⠀⠂
⢠⡿⠁⠀⣰⡿⠁⠀⣴⠟⠀⢀⡴⠋⠀⣀⡴⠚⠉⠀⣀⠤⠒⠋⠉⠀⢀⣀⣠⡤⠶⠖⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⠀⠈⠙⠻⠷⣦⢤⣀⡀
⡾⠁⠀⣰⠏⠀⢀⠞⠁⢀⠔⠉⣀⠤⠊⢁⠠⠐⠊⠉⠀⠤⠄⠒⠂⠉⠉⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⡀⠈⠢⡀⠈⠓⢦⣄⡀⠀⠀⠉⠛⠿⣿
⠁⠀⡴⠃⢀⡔⠃⠠⠔⠁⠀⠀⡀⠀⠀⠀⣤⡄⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠳⣄⠀⠈⠛⢷⣤⣀
⠀⡜⠁⡠⠊⠀⠈⢠⠀⠀⠀⣿⡏⠀⠀⣼⡟⠀⠀⣴⠟⠁⠀⣠⡾⠃⠀⠀⣀⡀⠠⠀⣈⠉⠉⠉⠛⠛⠓⠖⠶⠶⢶⢶⡀⠈⢄⠀⠙⢦⣀⠀⠈⠙⠿⣲⣤
⠎⠀⠈⠀⠀⠀⢸⣿⠀⠀⢰⡿⠀⠀⣰⠟⠀⢀⣼⠋⠀⢠⠀⠀⠀⢀⣴⠞⠋⠁⠀⣀⠀⠈⠉⠒⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⢷⣄⡀⠀⠈⠑
⠀⠀⣤⣯⠀⠀⢸⡟⠀⠀⣼⠁⠀⢠⡏⠀⢀⡞⠁⢀⡴⠋⠀⣠⠖⠋⠀⣀⡤⠶⠛⠉⠁⠀⠀⠐⠠⢄⡀⠈⠉⠛⠳⠶⣦⡄⠀⢂⠈⠳⣄⠀⠉⠻⣦⣄
⠀⠀⢹⣿⠀⠀⢸⡇⠀⢀⡟⠀⢀⠟⠀⣰⠋⢀⠴⠃⡠⠔⠉⠀⠤⠒⠉⢁⣀⡠⠤⠴⠖⠒⠂⠀⠠⢄⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⢢⡀⠘⢷⡀⠀⠈⠻⣷
⠀⠀⢸⣿⠀⠀⢸⡇⠀⢸⠃⠀⡞⠀⠔⠁⠀⠁⠐⠈⠀⢀⠀⠀⠀⠀⠈⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠑⠦⣀⠀⠙⠳⢶⣤⠀⠀⠀⠳⡄⠀⠹⣦⡀⠀⠈
⡀⠀⠀⣿⠀⠀⢸⠀⠀⡏⠀⡜⠀⠈⠀⣠⠀⠀⣾⠀⠀⡏⠀⣰⠃⢀⡤⠀⠀⠀⠀⠠⠤⢤⣀⣀⠀⠀⠑⠄⠈⠲⢤⡀⠀⠈⠀⠀⢣⠀⠹⡄⠀⠈⢿⣦⡀
⡆⠀⠀⣧⠀⠀⣿⠀⢰⠀⠘⠀⠀⠀⠀⢿⠀⠀⡇⠀⡸⠁⠰⠁⠠⠈⢀⠔⠁⠀⠀⠐⠠⠄⠀⠈⠁⠀⢂⠈⠣⡀⠈⠙⢷⡄⠀⠀⠈⢆⠀⠹⣧⡀⠀⠙⠛
⢻⠀⠀⢹⠀⠀⡇⠀⠆⠀⠀⠀⠀⡀⠀⢸⠀⠀⡇⢀⠃⠀⠁⠀⠀⠂⠀⠐⠈⠀⠀⠀⠐⠀⠀⠈⠀⠀⡀⢣⡀⠙⢦⡀⠀⠁⠀⠸⠀⠘⣆⠀⠘⣷⡀
⢺⡇⠀⢸⡄⠀⡇⠐⠀⠰⣧⠀⠀⢿⠀⠘⡆⢰⠀⠀⠀⠀⡀⠀⡆⢀⠀⠄⠀⠀⠀⠀⠀⢀⠈⠂⠀⠀⢡⠀⠳⡄⠀⠸⡆⠀⠀⠀⡇⠀⢹⡄⠀⠘⢿⣄
⠈⣷⠀⠘⡇⠀⠀⠀⠀⠀⠹⣇⠀⠘⡆⠀⡇⠨⠀⠀⠰⡀⠸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠆⠀⣇⠀⠹⡄⠀⠀⠀⢠⠀⢹⠀⠀⢻⡄⠀⠈⢿⢧
⠀⢸⡄⠀⡇⠀⠀⠸⣷⡄⠀⠘⣆⠀⠹⠀⠁⠀⠀⢦⡀⠙⡀⠁⠀⠠⠈⠀⠀⠠⠀⠀⢀⠀⡀⢠⠀⢠⠀⠸⡄⠀⠁⠀⢰⠀⢸⠀⠸⣇⠀⠘⣿⡄⠀⠈⠋
⠀⠀⢧⠀⠃⠀⠀⠀⠈⢻⣆⠀⠘⢆⠀⠇⠀⠀⣄⠀⠙⢦⠈⠀⠀⠢⠄⠀⠀⠀⠊⢠⠀⢰⠀⢸⠀⢸⠀⠀⠁⠀⠠⠀⡌⠀⢸⠀⠀⣿⠀⠀⠸⣷⡀
⡇⠀⠈⡀⠀⠀⢿⣦⡀⠀⠙⢧⠀⠈⢆⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠒⠐⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠄⢠⠃⠀⠃⠀⢸⠀⠀⢹⡇⠀⠀⠛⠁⠄
⠘⡆⠀⢃⠀⠀⠀⠙⠻⣦⡀⠀⠳⣄⠈⠄⠀⠘⠓⠦⣄⡀⠁⠀⠀⠀⠒⠂⠈⢉⡀⠄⠂⢀⠤⠂⡠⠊⢠⠂⢀⡎⠀⢸⠀⠀⢸⡆⠀⠘⣿⠀⠀⠀⡐⠀⣸
⠀⠸⡄⠈⠀⢰⣤⡀⠀⠈⠻⢦⡀⠈⠢⡀⠀⢠⣀⣀⠀⠈⠑⠂⠄⠀⠀⠐⠒⠉⢀⡠⠚⠁⣠⠎⠀⣰⠃⠀⡼⠀⠀⣿⠀⠀⣸⡇⠀⠀⠁⢀⠀⢠⠁⢀⡏
⣦⠀⠱⡀⠀⠈⠙⠻⣷⣤⡀⠀⠙⢦⡀⠑⠀⠀⠈⠉⠉⠙⠒⠒⠤⠤⢀⠀⠀⠒⠉⠀⣠⡞⠁⢀⡼⠃⠀⣸⠇⠀⢰⡏⠀⠀⠘⠁⠠⠀⡰⠁⢠⠇⠀⣸⠃
⠈⢦⠀⠡⠀⢀⡀⠀⠀⠉⠛⢶⣄⡀⠉⠢⡀⠀⠰⠶⡶⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠀⠀⠚⠁⠀⠰⠛⠀⠀⠈⠁⠀⠀⢀⠜⠀⡴⠁⢀⡏⠀⢠⡿
⡀⠀⠣⡀⠀⠐⠿⣷⣦⣄⣀⠀⠈⠙⠲⣄⡈⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⠤⠐⠒⠂⠀⠀⠀⠀⠀⢀⡠⠀⢀⡠⠊⢀⡔⠁⢀⡞⠁⢀⡾⠀⠀⣸⠃
⠙⣄⠀⠱⡀⠀⠀⠀⠈⠉⠛⠃⢀⣀⣀⠀⠉⠒⢄⡀⠈⠛⠋⠈⠉⠁⠀⠀⣀⣠⡤⠔⠒⠀⢀⣠⠴⠚⠁⣀⠴⠋⠀⣠⠋⠀⣠⡞⠀⠀⣼⠁⠀⢠⣿
⠀⠈⠳⡄⠀⠀⠠⡤⣤⣀⡀⠀⠀⠀⠉⠙⠒⠤⣀⡈⠀⠀⠀⢤⣴⡶⠜⠛⠉⠀⠀⣀⣤⠞⠋⠀⢀⣤⠞⠁⠀⣠⡞⠁⠀⣴⡟⠀⠀⣼⡏⠀⠀⣼⡟
⠻⣦⡀⠈⠢⡀⠀⠉⠉⠛⠛⠻⠷⠶⣦⣤⣀⣀⠀⠉⠁⠂⠄⠀⠀⠀⠀⣀⡤⣶⠟⠋⠀⠀⣠⡶⠟⠁⠀⣠⡾⠋⠀⢀⣼⠏⠀⠀⣼⡿⠀⠀⢰⣿⠇
⠀⠈⠙⢦⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠢⠤⠄⣀⡀⠈⠙⠉⠀⠀⢀⣴⡾⠋⠁⠀⣠⣾⠟⠁⠀⢀⣾⠏⠀⠀⣰⣿⠁⠀⠀⣾⣿
⠿⡦⣀⠀⠙⠢⡀⠁⠈⠻⠿⠿⠿⠿⢶⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣁⠀⠀⠈⠛⠋⠀⠀⢀⣰⡷⠋⠀⠀⣰⣻⠏⠀⠀⢰⣿⠇⠀⠀⠐⠋⠉⠀⣀⠄
⠀⠀⠈⠱⢤⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠂⢀⠠⠊⠀⡠
⣿⣶⣶⣶⣿⡿⣷⣶⣿⣷⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣾⣿⠷⠶⣿⢿⣶⡾⠿⢷⠶⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣤⣴⣤⣤⣤⣤⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣟⣿⣿⠻⠛⠛⠑⠛⠙⢷⡾⣿⡟⠛⠛⠻⣟⢦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⠿⠛⠃⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡆⠀⢠⣦⣷⣿⣏⠉⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⡚⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⢄⠀⠀⠀⠀⡇⠹⠇⠀⢀⣍⣹⣿⠋⠀⢴⣬⣧⠟⠛⢿⠲⣜⣷
⠀⠀⠀⠀⠠⡶⢻⠏⣶⠀⠀⠀⠀⢀⢠⠔⢚⣏⢠⣇⣀⣼⠀⠀⠀⠀⡇⢀⠀⠀⠈⠙⣿⣿⠁⠀⣤⣉⡟⠁⣤⠘⣇⣿⣽⣷⡀
⠀⠀⠀⠀⢀⡷⣿⠀⠘⣦⠀⡠⠞⡧⢸⡉⣹⣆⣠⡇⠐⢾⠠⠀⢀⣀⣇⣿⣃⠀⠀⠀⣿⡟⠀⠀⠈⢻⡟⢀⡙⢻⡿⠉⢿⣽⣿⡄
⠀⠀⠀⠀⢸⡇⢹⡀⠀⠸⣇⣷⢈⡷⢚⣉⣸⡇⠤⣧⣤⡾⠞⠛⠉⠉⠉⠉⠉⠉⠓⠶⢿⡇⠀⠀⠀⢸⡇⠈⠻⣾⠄⠀⢸⡆⣿⣷
⠀⠀⠀⠀⠘⡇⠸⣇⠀⠀⢹⣿⣁⡞⠈⣇⠠⣇⣴⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⢸⡇⠀⠀⣿⠀⠀⢸⡏⣿⣽⣧
⠀⠀⠀⠀⠀⢿⠀⢻⡄⠈⢦⢿⣇⠶⠊⣇⣠⡿⢟⣿⢟⣾⢣⡆⣤⣤⠀⡄⢀⠀⡀⣀⢠⡰⣾⣿⣦⣾⠁⠀⢠⡿⠀⠀⢸⠇⣿⡟⠁
⠀⠀⠀⠀⠀⠈⣧⠈⣷⠀⠘⡉⢻⣾⡊⢹⡟⠀⠀⠀⠛⢡⠟⣱⠟⣿⠂⣿⢌⡷⢻⡞⣧⢻⡽⣿⡿⢿⠀⠀⢸⣧⡄⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡀⠸⣇⠀⢹⡆⢻⣗⢺⠁⠀⠀⠀⠀⠀⠀⠁⠀⠃⠘⠇⠸⠃⠸⠀⠛⠘⠃⠁⠀⠘⡆⠀⣾⣿⡇⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢧⠀⢻⡀⠀⢃⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡏⣻⠀⠀⣿⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⣷⠀⠘⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⢱⡏⠀⢰⡇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⢠⣄⢿⡄⠸⣇⠀⠋⣄⠹⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣸⠇⠀⣼⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣧⢳⠀⠹⡆⠀⠹⡆⢻⣿⣿⣾⣇⣼⢁⡿⢠⡶⣠⡤⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⠇⢸⡿⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⢻⣄⡀⠀⠈⢻⣦⡀⠉⠙⠘⠇⢿⠹⡟⢠⠞⣡⡖⣠⣄⣀⢀⣴⣿⡇⢀⣼⠁⠀⣼⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⡄⢾⣥⣽⡆⠀⠀⡙⢿⣦⣀⠀⠀⠀⠀⠀⠉⠘⠳⣿⣿⣿⣿⣿⠟⠉⠀⣾⠃⠀⢀⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢿⣷⡘⡿⠶⢿⠀⣰⠏⢀⠉⠻⠿⣶⣶⣶⣤⣶⣶⣶⣿⡿⠿⠋⣡⠖⠀⠀⠉⠀⠀⡼⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣭⣿⡛⣶⠾⣧⠀⢠⡟⠀⠀⠀⠀⠉⠉⠛⠉⠉⠛⠓⠖⠀⠀⢀⡀⠀⠀⢀⣀⡴⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢳⣽⣶⣼⣦⠀⠀⠀⠀⠛⠻⢷⣤⣤⣄⣀⣀⣀⠀⠘⣧⠀⢻⠀⠀⣻⣯⣤⣾⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⢿⡙⢦⣼⠷⣆⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠙⠃⠀⢀⣿⣇⢠⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⣧⣾⣿⣶⣾⣿⣛⣒⣴⡦⢄⣠⣀⠀⢀⠀⢀⡀⣀⣤⠴⣿⣀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠉⠙⠻⠿⠿⠿⣶⣾⣿⣭⣽⣿⣉⣿⣍⣹⣿⣤⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⠋⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠀⢀⣀⡀⠀⠀⠀⣠⣤⣄⡀⠀⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡄⢸⣿⣿⡆⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣧⡀⠉⠉⢀⣿⣿⠛⢛⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⣏⠘⡋⠛⠀⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣾⣿⣶⡯⠼⣷⣿⣶⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠗⠲⠾⢿⣿⣿⣿⣤⣀⣤⣤⣀⣡⣾⡿⠿⠟⠓⢀⠾⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣤⣤⣴⣤⣤⣤⣭⣉⡙⠛⠻⠿⠉⢤⣤⣶⣾⣿⣿⣦⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⠉⣹⠿⠛⠋⣉⣤⣾⣿⣿⣶⣦⣤⣌⣉⠉⠉⡿⠋⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠛⠀⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠐⣿⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠷⠾⠟⠛⠁⠀⣿⣿⠀⠉⠛⠿⢿⣿⣿⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⣠⢤⡀⢤⡤⣤⢤⣤⠄⠠⣤⢤⣤⣤⣤⡤⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠲⢥⡇⠀⡷⢸⢶⠈⠀⡇⢀⠀⡷⠏⡉⢸⡇⠁⣿⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠙⠚⠁⠚⠓⠀⠚⠓⠚⠓⠛⠚⠃⠚⠓⠐⠛⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣧⠀⢸⣦⡀⢀⣾⢂⣼⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣯⣿⣿⣿⣿⣿⣦⠸⢿⣷⣿⡋⢺⣿⣿⣿⣿⣿⣿⣿⠿⠷
⠀⠀⠀⠈⠉⠉⠙⠻⡿⠿⢿⣿⣿⣿⢇⠉⠛⠋⢡⣿⡿⣿⣿⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⣿⣷⣶⣶⠟⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⠇
⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣼⢧⡀⢀⣠⣾⣿⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢘⣿⡀⠈⢃
⠀⠀⠀⠛⠛⠿⢿⣿⣿⣯⣀⡄⡸⠃⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣧⠀⣄⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠀⠀⠰⣩⡿⠛⢀⠀⠀⠀⠀⠀⠀⢸⠿⠿⡟⠋⢻⣿⡿⣿⣷⣹⣿⣾⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠛⠿⡏⠰⠍⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠾⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢒⣒⣿⣻⣿⣋⣽⡿⠟⠛⣉⣁⣠⣄⡀⠠⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠴⠧⠦⠾⢭⣤⣀⣀⡴⠋⢁⡤⠚⣟⠉⠀⠀⣹⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠂⠀⠠⠴⠶⠶⠶⣶⣴⠖⠐⠈⠁⠀⣠⣤⠴⣶⠶⣦⣤⣄⡀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⣿⣽⣷⢸
⠀⠀⠀⠀⠀⠀⠀⡠⢊⡡⠄⠀⠀⠠⢄⣀⣀⡀⠀⢀⣀⣿⣟⠀⠀⢀⣴⠃⠈⠆⠀⢸⠀⠀⣿⣿⣷⠀⠊⢳⡀⣿⡇⠀⢿⠀⠀⠀⢹⣿⡏⡈
⠀⠀⠀⠀⠀⠀⡰⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢱⠀⣾⣿⠀⠐⠀⠀⠈⠀⠀⣿⣿⣿⠨⠀⠀⢧⢹⣷⠀⢨⡀⠀⢰⣿⣿⢇⠃
⠀⠀⠀⠀⠀⣰⠃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠇⠀⠀⠈⣇⢻⡄⢨⣀⣠⣴⣿⢏⠌
⠀⠀⠀⠀⠰⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⣿⡆⠀⠀⠀⠀⢀⣴⣿⡟⠘⠀⠀⠀⠀⠈⠣⣍⣙⠫⠭⠅⠚⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣿⣿⠀⢿⠈⣉⣴⡿⢋⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣌⣛⣂⣘⡛⠋⠩⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⣤⡄⡤⢔⡟⣸⣠⡀⣠⣀⡞⡼⠁⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣖⡼⠻⣧⢿⡟⢸⡵⣿⡼⢻⠟⣿⡴⡧⠂⢠⣿⣿⣯⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣴⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠙⠒⠓⠂⠀⡀⠀⠀⠒⠒⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁⠀⠀⢰
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠘⢿⣾⣿⣿⣿⣿⣿⣯⣿⡿⠁⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣭⣽⣽⣫⣯⣿⣿⣭⣶⡄⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠋
⠀⠀⢀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⣿⣿⡇
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣀⣤⣶⣶⣦⣄
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⠆⠀⠀⠀⠻⣫⣽⣶⣾⣿⣿⣾⣾⣾⣽⡿⡿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢣⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠙⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⣤⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣆⠀⠀⠀⠀⢰⣇⡴⠃⠸⣷⠞⢳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡞⠀⠀⠀⠘⢧⠀⠀⠀⣸⠏⠀⠀⠀⠈⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣰⡌⣧⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠤⠤⠤⣤⡀⠀⠀⠀⠀⢀⠔⠁⠀⢸⣧⡟⠀⠀⠀⠀⠀⣠⠤⠤⠖⠚⠋⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠑⢤⠐⠈⠀⠀⠋⢠⠀⢹⣇⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣰⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢣⠀⠀⠘⠀⢀⡰⠀⠀⠀⠹⣆⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣉⠓⢾⣇⠀⠀⠀⠊⠑⠀⠈⠰⠆⠻⣦⡿⠀⠀⠀⠀⠀⠀⠣⡄⠀⢀⡴⠚⠁⠀⢀⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣾⣤⣄⠀⠙⢷⣗⠀⠀⠄⠈⢀⠀⠀⢀⣽⠇⠀⠀⢸⠀⠀⠀⠀⣧⠀⡎⠀⢀⣴⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠴⠷⠴⣿⠀⠀⠀⠉⢷⡄⠄⠀⠈⠀⣠⡾⠁⠀⠀⠀⠘⠳⢦⠀⠀⢸⡞⣠⣴⡿⢿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠚⠋⠀⠀⠀⠀⠈⢿⠀⠀⢀⡾⠋⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⠀⣀⣀⡷⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣾⠃⠀⠈⠻⣷⡀⢀⣤⣤⡄⠀⣸⣧⠴⠚⠉⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⠀⠀⠀⠀⠸⠟⠁⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠛⢤⣀⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣴⡿⠿⠛⠿⠛⢛⣷⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⢽⣦⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣉⣠⣂⡀⢀⡰⠉⠀⠀⣀⣨⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢸⣿⣏⠉⠘⠁⠉⠀⠰⠋⠀⣠⣶⣤⣿⣽⣇⠘⣟⠶⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠙⢷⣄⠀⢿⣟⣟⡁⠂⠀⠈⠀⠀⢈⡻⣋⣽⣿⣭⣿⡇⠹⠆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣬⣧⣠⣤⠀⠈⠻⣧⡈⢿⣿⣧⠈⡋⠀⠀⠀⠈⢿⣿⣭⣋⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣹⣿⣋⣽⣧⠀⠀⠘⢿⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣛⠚⣫⣿⡿⠀⠀⠀⠀⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣷⣦⣤⣤⣤⣴⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣶⣶⣦⣤⣄⡀

⠀⠀⠀⠀⠀⣠⣤⣶⣺⣛⡛⡙⠛⠓⠶⢤⣀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣄⡀
⠀⠀⢀⡴⠟⠋⠉⠘⠘⠙⠹⢳⢧⡆⡄⡀⠈⠳⣦⣠⣶⡟⠛⠛⠻⢷⣄⡉⠓⠢⠄
⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠷⣷⣦⣀⣈⢻⣿⣷⣤⣀⣠⣤⣤⣭⣭⣄⣀
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⠛⢻⣿⠋⠤⣤⣍⡙⠿⣿⣧⣌⣉⣂⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣄⡈⣿⡿⠷⣄⡉⠉⠩⣭⣉⡓⠂
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⠛⠿⠶⠦⠍⠒⠤⣾⣿⡙⠓⠦⠤
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣶⣶⡶⠶⠶⢚⣒⣒⡶⠾⢭⣀⢲⣶⣤⡀
⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠠⣀⣀⣽⡷⠶⠛⣉⣩⣤⣬⣤⣄⣈⠛⠷⠦⠍⠉
⠀⠘⢿⣦⣆⡄⡀⠀⠀⠀⡀⡀⡀⡀⡀⡀⡀⠀⢠⣾⣿⣷⣤⣤⣤⣤⣶⣾⣿⣍⣙⣛⣻⡷⠒⠒⠒⠄
⠀⠀⠀⠙⢿⣽⢻⠗⡷⡷⡿⡿⠟⠛⠛⢛⣻⣶⣿⠟⠉⠉⠉⠛⠿⢿⣿⣯⣉⣉⣉⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠤⠴⠶⠚⠋⠁⠙⢿⣆⣀⣀⣠⡤⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣦⠴⠶⠶⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⠄⠈⢉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢿⣀⣸⣤⣤⣼⡀⠆⡆⠒⠒⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⢀⠠⢱⡎⠀⠀⠉⠋⠀⡇⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢀⣩⡙⣄⠀⣿⡇⠀⠀⠀⠀⣿⠟⠙⠛⠙⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣻⡄⢹⣿⠉⠹⠿⠋⢹⠠⠃⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠂⠊⠹⣏⢹⣿⡀⠀⣄⣀⣾⠃⠀⠐⠂⠖⠃⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠠⠄⢀⡀⢠⢠⠀⣿⠀⣿⠀⢢⠄⠊⢹⡆⠀⠴⠀⠠⠄⠀⠄
⠀⠀⠀⠐⢨⠀⠀⠀⠀⢜⢷⠀⢈⡤⠠⠾⠷⠾⠳⣾⠀⢸⡀⠀⢸⣇⣀⢈⣄⣏⣠⡉⠁
⠀⠀⠀⠁⠘⢀⡤⡀⢄⠑⢽⠀⢸⡆⢀⣷⣾⡄⠃⢹⠈⢙⣸⠉⢹⡟⠫⢉⡉⠝⢩⠉
⠀⠀⠰⠠⠠⠝⢃⡅⠀⠄⡛⠀⢼⠇⠀⠀⠀⠀⢀⣸⠀⠚⠻⠃⢸⡇⠀⡀⠇⠀⠸⠀⠀⠠⠤⢤⣄⣀⣀⣀⣀⣀
⠀⠀⠆⠿⠞⠘⠞⠳⠴⠄⠊⠀⢸⣛⣛⣙⣛⣛⣿⣟⣛⣖⣓⣷⣞⣿⣶⣷⢲⠖⢲⠀⠀⠀⠀⠀⠁⠉⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣯⡭⢭⢭⣥⣍⣭⣑⣛⣛⢟⢚⢯⢟⢿⢿⣻⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡸⠤⠶⠶⣶⢶⣤⢶⡶⠶⠴⠾⠷⠾⠶⠶⠶⠾⠶⠶⠶⢶⣶⣲⢦⡦⠦⢴⣤⠜⠀⠀⠠⠀⠂⠂⡞⠄⠆⠃⠀⠀⠀⡀
⠀⡃⠀⠀⣴⡗⡩⠤⡉⢻⡄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⢄⠙⣆⢀⡌⣽⡆⠀⡀⠀⢀⢀⠠⢌⠴⠀⠀⠀⢊⣡⠤
⠀⠢⠤⠶⠉⢇⠳⠔⠃⠘⠑⠒⠒⠒⠒⠚⠒⠂⠚⠒⠒⠛⠙⡈⠊⡨⠍⠋⠀⠀⠀⠄⠁⠈⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣦⣤⣤⣴⣾⣿⣿⣿⣷⣶⣶⣤⢀⣀⣠⣀⡀⠀⣠⣦⣄⡀
⡀⣿⣿⣿⣿⣇⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⠇
⣿⣷⣪⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠛⠉
⢻⢿⢕⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣇⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠛⠋
⡄⠀⠀⢀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠋⠋
⣟⠄⢠⢞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠷⠶⠶⣶⢶⢶⢶⣶⣶⣶⣦⣤⣦⣦⣶⣴⣦⣶⣤⣤
⢝⢔⡄⣫⣞⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⠷⣿⠿⠷⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡠⠀⠀⠀⠀⠀⡄⣐⣰⣂⡀
⠀⠀⠀⣤⣿⣿⣿⣿⣤⠀⠀⠀⠠⣿⣿⣿⡧⠀⠀⠀⠀⠬⣿⣿⣿⡎⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⣽⣿⣿⣿⡧⠀⠀⠀⠐⣾⣿⣿⣿⡒
⠀⠀⠀⢠⠿⢿⡿⢯⠀⠀⠀⠀⠀⠹⠿⠟⠃⠀⠀⠀⠀⠈⠽⠿⠿⠓⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠐⢚⠿⡿⢟⠓⠀⠀⠀⠈⠝⢿⠿⠛⠌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣄⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⡄⣂⡐⢀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⢀⢹⣿⣿⣿⡧⠄⠀⠀⠀⠠⣿⣿⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢒⢾⣿⣿⢀⡒
⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠚⡻⢿⠿⡓⠂⠀⠀⠀⠀⠋⠛⠑⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠁⠆⡉⢁⠲⡌⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢤⣰⣦⡆⡀⠀⠀⠀⡀⢦⣴⣤⠆⡀⠀⠀⠀⡀⢆⣶⣠⢄⠀⠀⠀⢀⡄⣴⣇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣅⡂⠄
⠀⠀⠀⣽⣿⣿⣿⣿⡥⠀⠀⠠⣽⣿⣿⣿⣿⠥⠀⠀⢠⢽⣿⣿⣿⡯⡄⠀⠀⣀⣻⠁⠀⠙⡟⠆⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠅⣾⣿⣿⣿⡎⠄
⠀⠀⠀⠚⣻⢿⣿⡟⠂⠀⠀⠀⠊⣻⢿⢿⡛⠂⠀⠀⠀⠊⡻⠿⢿⡓⠂⠀⠀⠈⣙⣷⢤⡞⠻⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠂⠝⠿⠿⠟⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠠⡰⠘⠰⢢⠀⠀⠀⠀⢀⣠⣤⣆⡀⠀⠀⠀⠀⣠⣢⣖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣷⣶⣄⠀⠀⠀⢀⠰⣀⣆⡐⢀
⠀⠀⠠⠅⢰⣿⣷⡄⠩⠀⠀⠀⢺⡏⠀⠈⣿⡂⠀⠀⢸⣹⠁⠀⠈⣳⡅⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠌⣿⣿⣿⣿⡅
⠀⠀⠀⢫⢌⠙⢋⠠⠑⠀⠀⠀⠙⠷⡦⣶⠟⠀⠀⠀⠘⢭⡷⢤⡴⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠀⠀⠀⠐⢙⠿⠿⢟⠐
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣿⣿⣾⣽⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⠤⢤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠘⠛⠻⡷⠚⢁⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠐⠲⠶⢷⠖⠋⡀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠼⣶⠞⢁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⡴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣴⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢶⠋⢠⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡞⣾⠧⣤⣾⠋⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣠⠷⣟⢀⡞⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⡇⠧⠀⡟⠉⡷⠿⣀⣤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⠼⠛⣅⣸⠃⢰⠄⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠐⠇⢰⠏⢱⣤⡖⠛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠉⣶⣿⣀⣼⠀⣶⢸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠀⡾⠁⣵⢶⠋⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⡿⠛⣲⠿⡞
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠸⠁⣠⠷⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣠⠿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠾⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠈⢁⡴⠿⡖⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠈⣠⠴⢷⠶⠦⠄⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⢁⡤⢾⡤⣤⡀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠚⠓⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠉⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠦⢑⡊⠌⠈⠉⠙⠓⢤⣀
⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠍⣉⠂⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡙⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠱⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⢸⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⢀⡇⢊⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠛⠉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⢸⠀⣩⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⢡⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢀⡀
⡇⠀⢁⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠳⠞⣁⡤
⡇⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠾⠉⢀⡤
⣇⠀⠈⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠚⠺⠋⢀⠄
⢸⡀⠀⢁⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠚⢋⣰⠋⢀
⠀⢇⠀⢈⣨⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠋⣁⣠⠟⣠⢀
⠀⠘⣆⠀⣀⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡴⠒⣣⠜⣀⠃⡄⡰⣰⠀⣀⣠
⠀⠀⠈⢆⠠⢬⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡡⠞⡡⢋⠖⡗⡓⠋⡽⠃
⠀⠀⠀⠈⠣⡲⢏⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠘⢯⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠋⠁

⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢿⡟⠉⠉⠉⠉⢻⡶⣶⡏⠉⠉⠉⠉⢻⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⢶⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⣇⠀⠀⠀⠀⡸⠁⠘⣇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣷⢿⣿⣿⣿⣿⠇⠀⢻⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠲⠦⠖⠚⠁⠀⠀⠈⠓⠶⠶⠚⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠙⠛⠻⠛⠛

⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⣠⣿⣿⣿⣿⣿⣿⠆⠰⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⣿⡿⠋⠉⠉⠙⠻⣶⣶⣶⠟⠋⠉⠉⠛⣿⡷
⠀⠀⠘⣿⣿⣿⣿⣿⠟⠀⠀⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠋⠀⠛⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠙⠻⠟⠛⠁⠀⠀⠀⠀⠈⠛⠻⠟⠋

⠀⠀⠀⢀⠔⠂⠀⠒⢄⠀⠀⣠⠖⠀⠐⠢⡀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⣤⢤⣤⣴⣶⣶⣦⣄⡀
⠀⠀⠀⡮⠀⠀⠀⠀⠈⠖⠲⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⢄⡀⠀⣀⠜⠀⠀⠳⣀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠸⣟⣿⣿⣿⡿⠃⠀⠹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⣿⣻⣿⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣾⣿⣿⠇⠀⠀⢸⣿⣿⣿⣽⠟⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡟⠀⠀⠀⣿⣿⣿⣿⠏⠀⣀⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⡋⠉⠀⠀⠀⠈⠀⠀⣠⣼⣿⣿⣿⡏⢀⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶⣄⡀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡏⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢸⣿⡀⢸⣦⡀⣿⡏⠀⣾⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠹⣶⣦⣿⣷⣤⣿⣿⣿⣷⣼⣿⣧⣾⡟⠀⣤⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⣿⡿⠟⠛⠋⠉⠉⠙⠛⠿⣿⣿⣧⣾⣷⣀⣴⣾⠟⠁
⠀⠀⠀⠒⠶⢶⣤⣤⣀⠀⠙⠾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⡿⠀⢀⣀⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣩⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⠤
⠀⠀⠒⠛⠛⠛⠛⠛⠛⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡛⠛⠛⠻⠿⠷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠓⠂
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⢸⣿⠟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠻⢿⣦⣄
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠋⢁⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⢦⠀⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢻⡟⠀⣿⡿⣿⣿⠻⣿⡿⣿⣧⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠀⣸⣿⠁⠉⡏⠀⣿⣇⠀⠘⠂⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⣦⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠴⠚⠛⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠛⠻⠿⠭⠉⠙⠛⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠈⠉⠙⠓⠲⠦⠤⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣄⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⠶⠟
⠻⣦⣤⣆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⢠⡏
⠀⠀⠀⠙⢷⣀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠙⠷⣤⡀⠈⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠞⠋⠉⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⢷⡄⠀⠀⣠⡿⠶⠶⠶⠶⠾⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠋

⠀⠀⢀⡴⠖⠲⠤⠤⠶⠶⠶⠶⠶⠦⢤⣄⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⡤⢦⣄⣤⡤⠶⠶⠶⠶⠶⠶⠶⢤⠴⠲⣦
⠀⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣼⠃⠀⠀⠀⠀⠀⠀⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠏
⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡄⢿
⠀⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣾⠃⣸
⠀⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡾⢀⡿
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡼⠁⣼⠃
⠀⠀⠀⣿⠁⣀⠀⠀⠤⢀⣀⣀⣀⣀⠠⠄⠀⠀⠀⢿⡀⠀⠀⣴⠋⠀⠀⠈⠑⠒⠦⠤⠤⠴⠂⠀⠸⠟⢀⡼⠃
⠀⠀⠀⠙⠛⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠴⠶⠛⠳⠞⠁⠀⠀⠙⠓⠛⠲⠤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣄⣹⡆

⠀⠀⠀⠀⢘⣿⣷⡄⠀⠀⠀⠈⠉⠉⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠙⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⡀⠀⠀⢀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠛
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⡴⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⣰⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠦⣀⡾⠋⠀⠀⠀⣿⠃⠀⠀⠀⢀
⢿⣦⣤⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠁
⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠃⠀⠀⣀⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠃⠀⠀⠙⠛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠿⢿⣷⣦⡀⠀⣸⣿⠟⠉
⠀⠀⠀⠀⣠⣤⣤⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣿⡇
⠀⠀⠀⠀⠁⠈⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⡶⢦⠀⠀⠀⠀⠀⠀⠙⠋
⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⣿⣿⣷⣤⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠁⠀⠈⠉⢿⣷⣄⠀⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠈⠻⣀⣠⣶⣶⡄⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠿⠁⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⢀⣤⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⣼⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣴⣾⣿⠋
⠀⠀⢸⡿⠉⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠻⣿⠿⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣄
⣦⣤⣟⠀⠀⠈⠉⠙⠉⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⣿⣿⢿⣿
⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠀⠀⠀⠘⣿⣷
⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣠⣿⠿⠁⠀⠀⠀⠀⠀⠀⣻⣄⣠⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠛⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠰⣤⣄⠀⠀⠀⠀⠀⢀⣤⣄⡀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠤⠤⣄⠔⠀⢢⡀⠐⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠁⠁⠁⢨⠀⠁⠀⠘⠀⠀⠀⠁⠀⠓⠀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡁⠈⢸⡉⠈⠀⠁⠀⠲⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠘⠃⠀⠀⠇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡸⠿⠀⡀⠀⠀⠀⠀⠀⠤⡄⠈⠀⠄⠘⣀⠀⡆⠀⠀⢸⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⡌⢿⡆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠁⠄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣌⢁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⣛⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣵⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢃⣤⡄⢀⣀⢀⡄⢰⡿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠋⠀⠈⠁⠛⠛⣽⠓⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣀⡀⠀⠀⣸⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⡀⡄⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢠⠋⣺⣰⣦⡀⡏⡴⢧⣎⣦⣶⣄⣞⡜⠀⠀⣴⣦⣼⣱⢶⣰⠀⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠈⠉⠁⠉⠀⠉⠀⠈⠉⠙⠀⠀⠀⠴⠃⠉⠈⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢴⡶⠒⠛⠛⠛⠋⠙⠛⠛⠓⠲⠶⢤⣄⣀⡤⠶⠚⠛⠋⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⠶⠴⠶⠶⠶⠶⠶⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣀⡤⠚⠳⣤⣤⠟⠳⢤⣀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣹⠄⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢸⠁⠀⠀⣴⠖⠋⠀⠀⠀⢠⡄⠀⠀⠀⠙⠲⡆⠀⠀⢸
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢺⠀⠀⠀⣿⠀⠀⠀⠀⠘⢻⡟⠃⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⢀⡞⠁⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⡀⠀⠀⠀⢸⡆⠀⠀⠈⠙⢧⡀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠁⠀⠀⣸
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢻⡀⠀⠀⠀⢾⣁⠀⢀⣀⣀⡀⠀⣨⡷⠀⠀⠀⢠⠏
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠛⠛⠀⠈⠛⠋⠁⠀⠀⢀⣴⢻⡄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠁⠀⣷
⣼⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠻⡆⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠲⠖⠒⠛⠉⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⢹⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⠛⠂⠛⠛⠀⠛⠂⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣓⣲⣶⣶⣤⣤⣤⣴⣶⣶⣒⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠘⢃⣀⣉⠀⠰⠇⠀⠀⠀⠀⠀⠀⣄⠹⣇⠹⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⡀⣄⣴⠟⢯⡓⣄⠀⠀⠀⠀⠀⠀⠙⠁⡄⠀⡀⠀⡀⠀⠀⠶⠂⠀⠀⠀⠲⠀⠁⢈⣤⣤⣄⠐⣡⡴⡆
⠀⠀⠀⠀⠀⢀⠸⡧⣸⣅⣼⣇⠀⠀⠀⠀⠀⠛⠃⠰⠀⠢⣀⣀⡡⠄⠁⠠⡤⠀⠀⠨⠭⠄⠀⣿⣿⣿⣿⡇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠏⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠶⠒⣡⠌⠛⠛⠋⠀⢌⡙⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤⠀⠲⠄⠀⠀⠀⠀⠀⠀⠺⠃⠀⡆⢰⠀⠘⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⠄⠸⡄⠀⢠⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢃⣀⠃⢠⠆
⠀⠀⠀⠐⠶⣄⠢⠄⠐⠈⡔⢀⣤⡄⠀⠀⠀⠀⠀⢠⠿⣟⣛⡛⠿⣷⡀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣦⡀⠤
⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠠⠄⠀⢤⠄⠀⠀⠀⠀⢿⣜⢾⣜⣟⡶⣿⠁⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⡇⠤⠄
⠀⠀⠀⠙⠋⠁⠔⡀⠄⣀⠢⠘⠲⠄⠀⠀⠀⠀⠀⠈⠮⣯⣿⣿⡵⠃⠀⠀⠀⠁⠀⠀⠐⠂⡘⠿⣿⣿⣯⠟⠡⢄
⠀⠀⠀⠀⠚⠁⡸⠁⠀⠘⡄⠐⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣠⠆
⠀⠀⠀⠀⢠⣀⢻⠢⡸⠣⠞⣏⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣤⣀⣹⠟⠀⠀⠀⠀⠑⠧⢤⣤⣤⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠍⠫⣝⢦
⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⠀⠀⠺⣍⣀⠀⠀⠀⠀⠀⢰⠡⠄⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢆⢞⣻⣾⣿
⠀⠀⠀⠒⠛⣡⣟⣀⠀⠀⠀⠀⣏⠈⠁⠀⠀⠀⠀⠀⠈⠢⣈⣋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣂⠥⣕⣵⠏
⠀⠀⠀⠀⠘⠉⠀⡶⠉⣦⠲⡏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠓⠒⠲⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡾⠉⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⣿
⠀⠀⠀⠀⣶⣶⣿⣿⠷⠄⠀⠀⠀⠀⣧⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢿⠓⠦⣤⣀⡀⠀⠀⣀⣠⣴⡾⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⢤⣀⠈⠉⠉⢉⣩⡵⠋⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣤⣤⣤⣤⡤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⡀⢀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠈⠹⠭⠭⠥⠤⠵⠤⠤⠌⠁⠥⠤⠬⠁⠠⠀⡦⠤⠠⠤⠀⠠⠦⠀⠀⠤⠄⠠⠤⢤⣤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠈⠃⠈⠉⠀⠀⠀⠉⠉⠉⠙⠛⠙⠉⠉⠀⠊⠙⠛⠀⠀⠀⠙⠛⣀⣁⣄⣉⣉⣿⣛⡻⢛⣛⠻⠛⠿⢻⣿⣋⡀
⠀⠀⠀⠀⠀⣤⣤⠤⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⡀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠠⠴⢶⠶⠶⠶⠶⠤⠾⠟⠻⠛⠛⠛⠒⠛⠿⠿⢿⣿⣿⣆⣀⣀⢤
⠀⠀⠿⠟⡹⠶⠿⠷⠶⠾⠿⠿⠿⠿⠿⢿⠿⠿⣿⣿⣿⠀⠀⠀⠠⠤⠀⠀⠀⠀⠄⡀⠀⠀⠒⡷⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣤⣤⣄⢀⢀⣄⣠⡄⣠⣄⣀⣀⢰⣷⣬⣃⣠⣤⠖⢒⣒⣒⣒⢂⡀⣐⣋⣀⠛⠀⠀⠀⣀⣩⣸⣟⣛⣛⣛⣻⣟⢀
⠀⠀⠀⠀⠉⠉⢭⠿⢿⡯⡟⣛⡋⠙⠙⠛⠛⡛⡿⢿⣾⣿⣻⣿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣴⢿⢿⣛⣛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠨⣯⣭⢵⢝⣿⡷⠿⣾⣷⣤⣴⣧⣦⡝⣃⣧⣿⣴⢶⣁⣙⡩⠱⣤⡄⠀⠀⣦⠤⠤⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠟⠛⠒⠚⠃⠀⠈⠃⠁⢈⣹⣟⣙⡿⠛⠋⠄⠛⠻⠚⠛⠛⠋⠉
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠛⣿⠀⠉⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠤⠤⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡀⠀⠀⠀⠈⠀⠀⠶⣄⣀⡀⠀⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠘⠋⠀⠀⠀⠀⠈⠰⣄⠀⠀⠙⠻⢿⣶⣶⣄⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠛⢿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠙⠟⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠤⣬⡁⠀⠐⠛⡛⠛⠢⠀⠀⠡⠀⠀⠀⠀⠀⠈⢢⡈⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⠀⠀⠀⠀⠉⠁⠀⠀⠘⠙⠷⠀⠀⠀⠀⢧⠀⠀⠀⠀⠐⠠⣹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣈⣷⡦⠀⠀⢠⠀⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠃⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠛⢻⣇⠀⠀⠀⢣⡙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢗⡦⠀⠉⠈⠀⠀⠀⢠⡳⣌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⡤⠴⢄⡀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠁⠈⡇⠀⠀⠀⠀⠀⠀⢀⢙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⡟⠀⠀⠀⠀⠉⠛⠉⠉⠀⠈⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⢦⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠈⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠄⠐⢀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠀⢀⠄⠠⠐⠊⠁⠀⠀⢀⣀⣀⣤⡾⣛⣭⣾⣿⣯⡉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⡰⠁⠀⠀⠀⣠⣂⣠⢎⡛⣪⣭⣷⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⢄⡤⢰⠁⠀⠀⠀⠈⢿⠿⠋⢁⣶⣿⡿⠟⠉⣉⣭⣭⣝⡛⢿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣿⡿⢡⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣤⣶⣾⣿⣿⣿⣿⣿⣧⡹⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠇⠀⠀⠀⠀⠀⠀⣠⠊⣼⠟⢛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⢷⠏⠀⠀⠀⠀⣠⣴⡾⠁⡾⠁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⠃⠈⠀⠀⠀⠀⢸⣿⠏⠀⡾⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣤⡀⠀⢀⣀⣸⣇⠀⢸⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⡿⣿⡇⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⠿⠿⢿⠟⠀⠃⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣼⣿⡇⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠿⣿⡛⠿⠃⣼⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠛⠋⠉⠉⠛⠲⠶⠶⠶⣿⣿⣿⣾⡷⢰⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡁⢀⣴⣿⣿⣿⣷⣦⣙⢶⣶⣾⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣏⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢿⣿⠃⢘⡿⣷⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠈⢿⡽⠂⠚⠛⠃⠘⠛⠛⠠⢶⣿⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠿⠋⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠈⠁⠲⣏⡀⠀⣀
⠀⡰⠶⢖⡦⢄⡀⠀⠀⠀⠾⢿⣻⠇⠀⠀⢀⣤⣶⡟⢻⣿⣋⣻⣿⠛⣷⣦⣄⠀⠀⠈⠰⣶⠋⠁⠀⠀⠀⣀⠤⣐⠖⠶⡀
⠀⢇⡀⠀⠀⠁⠂⠁⠒⠤⢀⠀⠋⠀⠀⣴⣿⣉⣿⡿⠟⠛⣿⣯⠛⠻⣿⣏⣹⣷⡄⠀⠀⠙⢁⠠⠐⠂⠁⠂⠁⠀⠀⣀⠇
⠀⠀⠈⠑⠲⢤⣀⠀⠀⠀⠐⠈⠐⢠⣾⣏⣹⡿⠁⠀⠀⠀⢘⠟⠀⠀⠈⠻⣿⣉⣿⣦⠐⠈⠐⠀⠀⠀⢀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢴⣀⠀⣼⡟⢻⣟⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⢀⣘⣿⠛⣿⡄⢀⣰⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⣿⣇⣷⡤⠔⠈⠀⠀⠀⠀⢀⡴⠟⢃⣿⠿⣿⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⣧⡈⠀⠀⠀⠀⠀⠀⣤⠎⠁⠀⢰⣿⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣹⣷⣄⠀⠀⠀⢠⣾⠃⠀⢀⣴⣿⣉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣚⡭⠾⣿⣅⣽⡷⢦⣼⣿⣿⣤⠶⣿⣅⣽⡿⠯⣝⡲⢤⣄⡀
⠀⠀⠀⢀⣀⣀⣤⢴⣖⡯⠕⠚⠉⠀⠀⠀⠀⠙⠿⢷⣾⣿⣥⣽⣿⣶⠿⠛⠉⠀⠀⠀⠈⠙⠒⠭⣗⣶⢤⣄⣀⣀
⠀⠀⠀⠸⢿⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣼⠿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈