Tortilla Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠛⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢰⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠤⢔⡳⠶⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⠀⡤⠚⠁
⠀⠀⠀⢀⢶⠂⠀⠀⠀⠀⠙⠂⡹⠋⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠒⠂⠸⢤⠴⠀⠀⠀⠀⠀⠁⠀⠀⠋⠁⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠷⠃⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⢰⣮⠃⠀⠀⠀⠀⠀⢀⢾⠁⢆⣀⡀⠀⠀⣠⠅⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠒⠦⠠⡔⠀⠀⡰⢀⡈⠆⠃⠀⠀⠀⠀⠀⠀⠐⣦⠎⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠓⠀⠀⠀⠈⠀⠔⠑⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠐⠊⡅⠀⠠⡤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠠⡆⠀⠠⠴⡅⠀⠀⠀⠀⢀⠁⡉⠒⠢⠄⢧⢴⠂⠕⠁⣃⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠁⠀⠀⠀⠀⠠⢶⡀⠀⠙⠄⠁⠂⠀⠶⠊⡙⡆⠀⠀⢀⠄⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⢰⠊⠀⠀⠨⠁⠀⠀⠀⠢⡄⠀⠀⠀⣈⠩⠀⢦⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢀⡟⢀⠄⠀⠊⠢⠄⠀⠰⠄⠘⠀⢒⡎⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡄⠀⣆⣤⠋⡀⠀⡀⠠⠨⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣭⣤⣍⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⡉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠤⠤⠤⠀⠀⠀⠠⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠒⠤⣄
⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⡄
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡇
⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠈⢳⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣠⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠻⣦⣌⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⣲⡺⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠤⣤⣶⣒⡪⠭⠽⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠊⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢀⣠⣄⣀⣀⡀⠀⠀⠘⣷⠀⠀⠀⠀⢈⠉⠉⠛⠛⠳⠶⢶⣤⣤⣀⣀
⠀⠀⠀⠀⢸⣏⠉⠉⠉⠛⠛⠛⠻⢿⡆⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⣦⣤⡀
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠰⡶⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠒⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠠⡦⠀⠀⠀⠀⠙⠋⠀⠀⣠⡾⣯⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠻⠂⠀⠀⠓⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠈⠉⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠐⠆⠀⠀⢀⣴⠟⠁⠀⠀⠓⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠰⣿⠆⠀⠀⣸⣧⠀⠀⠀⢀⣴⠟⠁⠶⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢀⣴⠏⢻⡆⢀⣴⠟⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣴⣿⡁⠀⠀⠛⠛⠁⠀⢤⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠟⠁⠸⣧⠀⠀⡀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⠁


⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠒⠀⠃⠀⠠⠘⠃⠒⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢠⡀⠀⠀⠒⢄⡀⠀⠐⢂⡀⢀⣀⠀⠉⠉⠂⠠⣄⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠀⡀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠈⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠈⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠈⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠀⠈⠁⠀⠀⠀⣠⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠀⠀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠂⠀⢄⣰⣣⢿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠙⢄⠀⠢⣀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠁⠀⢀⡀⠀⣀⣴⡞⠺⣹⢗⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡙⠂⢌⣛⣒⠠⠤⠤⠤⢄⣀⣀⣀⡠⠤⣴⢾⣋⣁⣴⡞⣻⣥⡿⠉⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠐⠦⠬⣍⣙⣓⡒⠲⠦⠵⠴⠤⠤⠤⠴⢾⣿⣭⠿⠿⠋⠉⠁⣠⣶⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡱⠖⡀⠤⠀⠀⢀⠉⠉⠉⢉⣀⡀⠈⠛⠋⠉⠁⣀⣀⣠⡤⠶⠋⣩⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢶⡆⢄⣀⣈⣉⣙⡠⠉⢉⠘⠛⠒⠋⠁⠡⠁⠂⣡⠞⢃⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠅⣀⣛⠚⠐⣲⠶⠤⠤⠶⠶⣞⣋⠥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⡄⠀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⢐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠰⠞⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣻⠃⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠢⡀
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠈⠁⠐⢄
⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠂⠁⠠⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢦⡀
⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠑⢦⣄⡂⠤⠀⠀⠠⠄⢀⢀⣀⣀⣠⡤⢤⢖⡶⢌⠆
⠀⠀⠈⢄⠠⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠽⡹⢿⣿⣿⡯⠸⠳⠉⠚⠹⡘⡸
⠀⠀⠀⠀⠀⠨⡀⢒⠢⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⠀⠀⠀⠠⠄⣠⣄⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢠⣤⠀⡈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⣤⡮⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⣠⣾⡫⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠦⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⢔⠃⠁⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁⠼⢀⠈⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉⢢⡐⠈⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠎⢲⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠆⡏⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⡅⣄⠡⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣀⠤⢂⢅⠟⠁
⠀⠀⠀⠀⠀⠀⡀⡗⡘⢆⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠚⠅⢚⠔⠏
⠀⠀⠀⢀⠀⡨⠳⣄⣌⠢⠀⠀⢉⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⠤⡀⠆⡑⠨⠁⡒⠌⡤⠊
⠀⠀⠸⠷⣤⣤⡄⠀⠽⠛⢦⣈⠑⡠⢅⣂⠈⠐⠦⠦⠤⠤⠤⡤⠠⠀⠀⠀⠐⠬⠭⠉⠙⡫⠉⢁⣀⣢⣤⣴⣶⣶⠖⠒⠁
⠀⠀⠀⠇⡈⠇⠉⠩⠉⠲⡒⠬⡬⡏⣛⠒⠂⠂⢈⠉⠀⠀⠉⠉⠉⠙⢈⣀⣀⣬⣥⠤⠴⠖⠟⢛⡽⢋⣷⢿⠏⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠡⣀⠈⠫⢯⠲⠒⠤⣉⡈⠁⠀⠉⠀⠠⠨⠐⠈⠀⢀⠀⢂⡄⣡⢾⠟⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢓⡐⠒⠤⠄⣀⣀⣀⣀⣴⣇⣵⡃⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠠⢈⢀⡌⠉⠹⣿⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠠⠀⡀⡀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣶⠿⢶⣤⡴⠟⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠻⣿⡿⣿⣶⣄⣠⡶⠟⣉⣉⣉⣉⠛⠿⣿⠙⠳⣤⡴⠛⠲⠟⠁⠀⠀⠹⠁⠀⠀⣴⣋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠐⣿⡟⠛⠛⠋⠀⠀⢹⣿⠏⣰⡾⠛⠉⠉⠉⠳⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣾⣿⣶
⠀⠀⠀⠀⠀⠀⣾⣟⠉⠉⠉⠛⠛⠻⠆⠀⠀⠀⠀⣼⠏⣼⣿⠟⠛⠛⢷⣶⣄⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠋⠁⠀⠀⢠⣿
⠀⠀⠀⢀⣠⣤⣌⣹⡿⠀⠀⠀⣀⣤⣶⣦⣤⡀⢸⣿⣠⣿⣷⣤⣤⠶⠶⠿⣍⡀⠀⠈⢻⣷⠀⠀⠀⢀⡾⠁⠀⠈⢻⣶⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⢰⣿⣏⡀⠈⠀⠀⠀⠀⣾⣏⡀⠀⠀⠈⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡤⠾⠛⠒⠒⠒⠾⠷⣤⡀⢀⣼⡿⠦⡴⢿⣷⠀⢰⣟⠉
⠀⠀⠀⠉⠉⢻⡄⠀⠀⠀⠀⠈⠙⢿⣷⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⣤⣴⣶⡾⠿⠛⠛⠳⠖⠀⠙⣿⡿⠃⠀⢀⡼⠃⠀⣀⠿
⠀⢀⣴⣾⠓⠉⠀⠀⠀⠀⢠⣴⣤⣄⣹⠟⢁⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣤⣾⣿⠟⢋⣁⠀⠀⢀⣤⣄⣀⡤⢤⡟⠀⠀⠀⠘⠣⣤⡄⠙⣦
⠀⣠⡬⠛⠁⠀⠀⢠⣴⣤⣀⣿⠁⠙⠿⡴⠻⢷⡼⠿⣧⠀⠀⢠⡞⠁⣾⣿⡟⠁⢰⣿⠛⠛⠢⠾⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⡅⢰⡇
⠀⠛⠿⣶⠊⢀⣤⣾⣿⣁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢷⣤⣤⣾⣅⣼⣿⣿⠀⠀⠈⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠿⠾⣿⡿⠃
⠀⣤⠞⠁⠀⠈⣉⣿⢷⣦⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⣉⣁⣤⣤⣄⡀⠈⠙⠢⣄⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⣸⡇⣠⣄⣀⣾⠃⠀⣼⣿⣆⠀⠀⠀⠀⢠⡞⢁⣴⠾⠛⠋⢹⣿⠛⣿⣷⣶⣤⣤⡿⠿⠂⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⣿
⠿⠟⠛⠻⣿⡿⠀⠀⣿⣿⢿⣷⡀⠀⠀⢿⣴⠟⠁⠀⠀⠀⢸⣿⠀⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣿⠿⠟⠶⠄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⢻⣿⣦⣄⠀⠀⠀⠀⢀⣤⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠿⣿⣿⣦⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⡀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣤⣀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡄⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠰⠆⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠘⠃⠀⠀⠀⠀⠀⣰⣿⣧⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣶⣶⣿⣿⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢈⡙⣿⣿⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠒⠒⠦⠤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠙⠻⣟⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠈⢀⠀⠙⠲⣌⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢵⡄⠀⠈⠙⠶⣤⡙⢷⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠄⠀⠀⠈⠻⣦⣿⣄
⠀⠀⠀⠀⡴⠃⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠙⢿⣷⡄
⠀⠀⠀⡼⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢳⡀
⠀⢰⠃⠀⣾⣿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢾⣷⡉⣷
⠀⣿⠀⡄⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠉⢿⣼⡇
⢀⡇⢰⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⢹⣷
⢸⡇⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⢘⡄⠃⣿
⢸⡃⢁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⢈⢇⠀⣿
⢸⡇⢰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡿⠀⡟
⠈⣇⢘⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠀⠀⠀⢠⠀⠀⠀⠀⠌⢠⢶⡇⢸⡇
⠀⢹⣎⢷⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡦⡞⠁⣼⠁
⠀⠈⢿⣮⣎⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠄⠀⠀⠀⠀⣤⡄⡀⢠⠏
⠀⠀⠈⢿⣆⠺⡯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣾⠏⠀⢠⠏
⠀⠀⠀⠈⢿⣦⡉⠻⣼⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠱⠀⣰⡏
⠀⠀⠀⠀⠀⠹⣿⢦⡸⣧⣶⠀⣀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠘⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣏⡷⢿⠟⢧⡀⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠠⠂⠐⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡛⠷⢯⣄⣶⣷⣾⣿⢦⡀⣅⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⢠⠆⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⢬⣍⣛⠻⢭⣂⣓⡬⠅⠁⣠⡀⢠⡒⠁⠊⠀⠉⢁⣠⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠽⠷⣤⣤⣬⠤⠤⠴⠶⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠉⠉⠁⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠉⡠⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢧⡀⠀⠀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠓⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⠙⠒⠲⠤⠤⣄⣑⡢⢤
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⠀⠀⢹⡖⢦⣄
⠀⠀⠀⠀⠐⡇⠀⠀⠈⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣰⠋⢄⠠⠂⣱⠂⠀⠀⡧⠹⢦⡀
⠀⠀⠀⠀⠀⣧⠀⠀⢀⡀⡀⠐⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⡀⠂⠀⡀⠐⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢀⡇⡤⠗⠀⢀⠯⡤⠆⠀⢣⠔⢄⡃
⠀⠀⠀⠀⠀⢸⡀⠂⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠸⡇⠀⠄⠈⠀⠃⠀⡆⠲⡁⠀⠀⢇
⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠆⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢧⡀⠀⠠⠀⠁⠀⢀⠀⠘⠳⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⡄⣀⠨⡄⠺⠄⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣁⡀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠻⣄⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⡤⣪⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡈⠒⠶⠤⣤⣤⣀⣀⣀⣀⣠⣤⣤⡤⠤⠴⠚⣡⠖⠉⣚⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣬⣉⠁⠒⠂⠤⠤⢤⣤⣤⣤⡤⠶⢶⣖⣲⡿⠧⠔⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠳⠛⠻⠿⠿⠷⠒⠒⠚⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⡤⠀⠀⣀⢰⣄⠀⠀⣼⡦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠠⡦⣸⠷⡘⠈⠘⣇⠨⠟⢡⠟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣠⡆⠑⠛⠹⠾⠗⠚⠓⠛⠳⢦⣇⡶⣏⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠈⠑⠠⡰⠋⢀⠀⠀⢀⠀⡉⠀⠒⠒⠂⠀⠤⠤⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⢀⠀⠀⠈⠑⠲⠵⣤⣄⣀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⢪⠟⢦
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠉⠉⠒⡒⠶⣦⡄⠀⣠⣢⠏⠀⣘
⠀⠀⠀⠀⠀⠀⡴⢄⠄⡀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠁⢀⡼⣪⠋⠙⡆⣠⠝⠁⣠⠞⠉⣧
⠀⠀⠀⠀⠀⢰⠖⡌⠟⠚⠧⢄⣀⡠⠤⠤⠴⠦⠤⠶⠒⠚⠋⠋⠁⢀⣴⠩⠴⠖⠊⠁⢀⠀⠏⡇
⠀⠀⠀⠀⠀⠈⢌⢜⠢⣄⣀⣀⣒⠤⠤⠄⠒⠦⠥⢤⣤⠤⠤⠔⠊⠁⠀⠀⠀⣈⡡⠴⡃⢤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣌⠩⠭⢍⡒⠒⠒⠶⠦⠀⠀⠀⠠⠤⠤⠤⠴⢓⣋⢥⣢⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠉⠭⠓⢒⣒⣒⣒⣒⣒⣲⠾⠿⠓




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⢀⣀⣀⣼⣿⣟⠀⠀⢀⣤⠀⠀⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠟⠉⠫⡴⢻⣿⣿⠭⡽⡿⠋⢹⣤⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣴⠿⠉⣰⣤⠊⣤⣦⣆⣸⣿⡿⠥⠶⠿⠶⢮⣤⣬⣽⣿⡷⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠹⣿⣿⣍⢹⣿⡬⣿⡟⠉⢱⡤⠆⠀⠈⠁⠂⢨⡉⠹⢿⣿⣻⢷⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣃⣳⣠⢫⣿⣿⢟⠉⠀⠀⡄⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁⠁⠀⠉⢊⠓⢝⢦⡀
⠀⠀⠀⠀⣠⣤⠀⣵⡀⠉⢉⢹⣾⣿⠇⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⣤⡄⢀⣄⠈⠓⢳⡀
⠀⠀⠀⠀⠻⣍⣼⠷⣿⣉⣹⠋⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠐⠛⢛⣀⡤⠗
⠀⠀⠀⠀⢀⡼⢉⣈⣻⣿⡟⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⡀⠀⢄⣠⠼⠚⠋⠁
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⣂⣠⠴⠚⠉
⠀⢠⣿⣿⠟⢫⣷⣾⡿⢋⡀⠀⠀⠀⠀⠂⠀⠀⠀⠠⣀⡠⣴⣾⡥⠖⠋⠉
⠀⢀⣹⡋⢳⠖⣉⡿⣇⠈⢀⠃⠀⠀⢰⣲⡀⠡⢖⣫⡥⠖⠋⠁
⠀⠸⣝⣋⡼⡏⢠⡇⡿⠀⠀⡀⢀⠀⢛⣫⠴⠚⠉⠁
⠀⠀⠀⠀⢠⣇⣿⣷⣧⣠⣤⣴⠶⠚⠉
⠀⠀⠀⠀⠈⠛⠛⠛⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠄⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡀⠀⠀⣄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢧⣀⣠⠞⠙⠒⠒⠁⠙⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣠⣤⠄⠀⢀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠄⠂⠂⠄⠀⡈⢉⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⠀⠀⠀⠈⡎⠀⠀⢡⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠰⡀⠀⠈⡸⢇⠉⠀⡸⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⣀⠤⠴⠦⠤⢤⡜⠁⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⣤⠤⠤⠶⠤⢄⡀
⠀⠀⠀⢠⠎⢁⡴⠒⠉⢑⠞⠀⠀⠀⠀⠀⠀⢀⠤⠒⠠⠂⠀⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⠀⠘⢖⠉⠑⠲⣄⠉⢦
⢀⠔⠉⠁⠀⢺⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢸⠂⠀⠉⠓⢄
⢇⣠⢺⠳⡀⢳⠀⢠⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢰⠃⡰⢻⢢⣀
⠀⠈⠁⠀⠈⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠀⠀⠉
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⡏⡏⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⢸⡇⡇⡇⣿⢀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠉⠁⣇⡇⠉⠁⠀⠀⠀⠀⠈⢳
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠈⡀⢀⠀⡄⠂⠀⠀⠀⠀⠀⠀⢠⠀⢼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⣿⣿⠇⠀⠠⣾⣿⣿⠀⠂
⠀⠀⠀⠀⡈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⣿⣿⣿⣿⡟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣿⡇⠀⠀⠀⠈⢿⡟
⠀⠀⠄⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⡿⠠⡡⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⣧⠘⣧⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠀⠀⢻⣿⠃⢀⠠⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠙⠿⣦⣀
⠀⠀⠀⠀⠀⡧⠀⠀⠈⠑⠄⠀⠠⠐⠀⠀⠀⠀⠈⠗⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠇⢀⣤⣄⠀⠁⠀⠐⠠⠄⢀⠐⠓⠀⠂⠀⡈⠀⠀⠀⠀⠂⠈⣠⡀⠀⠀⠙⠧⠀⠀⠛⠃⠀⢀⣿⡿
⠀⠀⠀⠀⢰⢀⣿⣿⣟⠀⠀⠀⠠⢠⡀⠀⠀⠀⠀⠘⢂⡠⠀⠠⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠄⠀⠈⢻⡗⠂⢠⣤⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠓⠚⢿⣿
⠄⠀⠀⠀⠀⠀⢠⠚⠉⠁⠄⠀⠀⠀⠀⠂⡠⠀⠀⠄⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠙⠃
⠀⠰⠤⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠀⠻⠏
⠀⠠⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠂
⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⣛⣿⣍⣉⣿⠛⢛⣻⣿⣭⣿⣷⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣡⣶⡿⠛⣿⠟⢉⣵⠾⠛⠉⠉⠁⠀⠀⠉⠉⠛⠻⣧⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠋⣤⣾⠿⣶⣴⠶⢟⣴⠟⠁⠀⠀⡄⠀⠀⠀⠀⠰⠆⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢠⡿⢁⣾⠟⠀⢀⣸⡏⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠠⡄⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣾⠃⣰⡿⠶⢾⡟⠉⣰⡟⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⢰⡟⢸⣏⠀⢠⣼⠇⣴⠟⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠰⠂⠀⠀⢹⡇
⠀⠀⠀⠀⢸⡇⣿⡅⠀⢻⡇⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⣿⡏⠁⣴⠟⣽⠏⠀⠀⠘⠃⠀⠀⠐⠇⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣇⣀⣿⣾⠋⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠾⠏
⠀⠀⠀⠀⠀⠈⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣧⣶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠁⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡾⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠖⢶⡀⠰⢖⡓⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠉⠉⠀⠀⠈⠁⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣀⠙⠃⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢄⠀⠈⠙⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣾⠐⣤⣀⣀⡠⠀⢸⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠤⠞⢿⣟⠢⡐⠉⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠈⠛⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠒⠉⠻⣇⢰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣖⣈⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢿⣂⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⠟⠋⠁⠑⢆⢻⣷⡂
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢻⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⢈⣠⡏⠉⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡄⢀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠉⠉⠛⠶⣄⠄⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⣱⠞⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣼⡏⠀⠀⠀⠀⠀⠀⢀⡠⠰⠿⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠄⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠛⠉⣹⣿⠿⠋⠉⠀⠘⢦⣴⣿⣿⣿⣿⡛⠉⠛⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠙⣿⣭⡀⠁⠀⠀⡠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⣿⣿⣿⣿⠗⠀⠰⠟⣋⣉⡁⠀⠀⢯⣤⣾⣶⣷⣾⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠘⣿⣿⡿⣿⠆⠀⠘⠻⠛⠋⠀⠀⠀⢻⢫⡟⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⣀⠂⣁⠀⠀⣸⣿⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⣠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣾⡿⢿⣧⣤⣶⣿⣿⡆⢙⢉⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⢀⡇⠀⠙⠿⣾⣿⣿⣿⣿⣿⣿⡻⣭⡽⠁⠀⠀⠀⠀⠀⢀⠜⣿⣿⣿⣿⢿⣿⣿⣿⡀⠀⠀⠀⠀⢸⡀⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡜⡇⣠⢀⣴⣯⣥⣉⠛⢿⣿⣿⣧⡀⠀⠀⢀⣀⣠⣤⠖⠁⠀⣿⡿⠛⠁⣼⡟⠋⢻⠱⡀⠀⠀⢀⢸⢣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢶⣂⢼⣿⠇⣇⣿⣿⣿⣿⣿⣿⡷⢈⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣧⣰⣶⣿⠀⣠⣾⣇⠹⠀⠀⣼⣼⠈⣿⡧⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠄⢻⡆⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⢿⠿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶⠃⢠⢧⡇⢸⡏⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⡾⣷⣼⣿⠏⠈⢹⣿⣿⣾⣿⣶⣦⣤⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⡴⠛⣻⡿⠿⣿⣿⣿⣿⡆⢸⣾⠁⣾⢿⣿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⡻⣿⣏⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁⠀⠀⣀⣤⣴⣿⣵⠶⠋⠁⠀⠠⣴⠿⣿⣿⣿⣾⡿⣰⢛⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣜⣷⣿⠿⢻⣿⠟⠿⢿⣿⡿⣿⠟⢻⣿⣧⡀⢀⣠⡴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣼⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢻⣷⣿⣿⣦⣶⣦⣬⣼⠏⠀⠀⢻⣿⣿⢫⣡⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⣡⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⣿⡿⢛⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⢏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡙⢶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⢛⣉⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣗⠁⠀⢉⡟⣳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⠗⣺⣿⣄⠹⣷⣀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠉⠁⠀⢠⣴⠟⣡⣾⣗⠺⣯⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢯⣍⣶⣦⣝⣻⣦⣬⣄⡀⠀⠀⣀⣠⣤⣴⣞⣯⣶⡶⣩⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⣻⢶⣿⣯⣿⣿⠿⠿⣿⣿⠿⠟⣿⣿⣿⣿⡶⣟⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠫⠽⢿⣆⢠⣤⣧⡄⣰⣿⠯⠉⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣋⣙⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠋⢻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⠏⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣄⢸⠃⠀⠀⣴⡀⠀⠀⠀⢹⡾⠃⠀⠀⢠⣦⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⡇⠘⢦⣀⠀⠈⠀⣀⣠⠔⠃⠈⣆⠀⠀⠀⠀⠈⢷⡀
⣤⣄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠙⠓⠛⠋⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧
⠈⢿⡛⠶⢦⣄⡈⣇⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣴⠒⠒⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢳⡀⠀⠈⠙⠻⠆⠀⠀⠀⡏⠳⢄⣸⠁⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⢀⡀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠲⣄⡀⣸⠀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠉⣙⣻⣧⣀⡀⠀⠀⠀⠰⠿⠞⣷⠀⠀⠀⠀⢰⠇
⠀⠀⢰⡇⠀⠀⠀⠸⠓⠦⣤⣸⠀⠀⠀⠀⣠⠖⠛⠶⢤⣄⣠⠟⠉⠙⠳⣄⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡜⠁⢀⣤⣄⠀⣠⠶⠖⠶⣄⠀⠈⢧⠀⠀⠀⡇⠀⠀⠰⢿⣤⠴⠶⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠁⠀⡏⠠⠎⢻⠁⠀⡶⠀⠈⡇⠀⠘⡆⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡆⠀⠙⠦⠤⠾⣧⡀⠀⢀⣰⠇⠀⠀⣷⠀⠈⣩⠗⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠢⣀⠀⠹⣄⠀⠀⠀⣀⡈⠉⠛⠉⣀⡄⠀⠀⣽⠖⠋⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⠒⠾⠷⢦⡀⠈⠉⣟⠛⡟⠉⠀⠀⠀⡇⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡖⣇⠀⠀⠀⣼⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠖⢻⡄⠈⣧⢹⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠤⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠲⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣶⠓⠟⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢦⣤⣤⣤⣤⡤⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⣤⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞
⠘⢧⠾⠻⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷
⠀⠀⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢧
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠓⠒⠒⠒⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠤⠀⢀⡴⠃⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⣠⣤⠶⠭⣀⡠⠒⠀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣉⢳⠀⠀⡸⠓⠒⣲⡞⢑⡒⠐⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⣤⠀⠈⡇⠰⠄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠋⠈⠉⠀⠀⡇⠀⠀⠉⣇⡈⠀⢀⠞⠀⡀⠸⡇⠠
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡀⠀⠀⣠⠗⡤⡀⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡏⠀⠀⠀⠀⠀⠘⢲⣴⣚⣉⣉⣉⠥⠔⠊⡇⢠⡇⠄
⠀⠀⠀⠄⠀⠀⠀⠻⡄⠐⢷⠬⢥⣉⣁⣀⣀⡀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⢀⠄⠀⠸⣀⢈⠀⠀⢀⣀⠠⢾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣄⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⢉⠀⠀⠐⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡟⠉⠉⠉⢻⠈⠉⢻⠀⠀⢀⡤⠒⠒⠺⣅⠀⣀⣈⣉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠐⠀⠀⠸⠋⠀⢀⠈⠱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⢸⢀⠀⠘⢄⡀⠀⣀⣧⠀⠀⠀⠀⢠⠃⢠⠏⠀⠑⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠣⣀⠀⠀⠀⡠⠎⠈⠀⠀⠀⠈⣹⡁⠀⠑⠦⠤⠒⣁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠉⠉⢉⣀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⡿⠿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠤⣀⠀⢠⣄⣰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠳⣀⢂⣝⡅⠀⠠⢚⣩⣥⠶⣶⠶⠾⠶⠶⣦⠌⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣧⠂⢉⢃⠮⢎⡤⣲⠜⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢌⠁⠲⠛⠒⠠⣠⢆⣲⢞⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡈⠄⠒⠒⡆⢤⣨⢞⡕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡐⠀⡠⠔⠠⠀⢔⠔⡱⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢰⠒⠰⠀⠀⣇⡤⠂⢜⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠏⡀⠃⡴⠋⡇⢀⢪⠞⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡠⢤⣄⠂⠀⠄⣀⣀⣠⣤⠤⠤⠖⠚⠁
⠀⠀⠀⠀⠀⡸⠅⠠⠎⠀⠥⡠⣣⠋⠀⠀⠀⠀⠀⢀⡠⠄⣒⠢⠍⣒⣂⡭⠥⠖⠒⠛⠉⠉
⠀⠀⠀⠀⢀⠗⠀⢄⣀⠚⡴⡵⠃⠀⠀⠀⠀⠀⠀⣀⣨⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠈⣆⠈⣀⣊⢌⠞⠁⠀⣀⣠⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠘⣝⣒⣵⡥⠴⠚⠋⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡐⠉⠁⠀⠀⠀⠀⠀⠱⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢀⠄⠒⠀⠚⡆⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⠄⠀⠀⢀⡄⢠⠤⠄⠒⠀⠘⠛⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⡰⢋⡄⠃⠀⡐⠀⠀⢠⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡀⠐⠈⠔⢣⡘⡀⠀⠸⠇⠀⠀⠘⢞⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠀⠀⡇⠉⠀⢀⠀⠀⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠀⠀⠈⠲⣆⠀⠐⠂⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠄⡀⠀⠐⠡⠅⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠑⠤⡀⠀⣠⢔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠈⠀⠀⠀⢠⣄⠀⠐⠤⢌⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢀⠀⠀⡟⠀⡴⣇⣀⠘⡄⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣀⠤⠒⠂⠉⠉⠀⠀⠉⠉⠙⠒⠠⢄⡃⠀⠀⠀⠃⠀⠈⠑⠢⡀⠀⠀⠀⠘⠀⠀⡐⠊⠀⡠⠚⡀
⠀⠀⠀⠀⠀⡠⠐⠂⢒⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡾⠀⠀⠀⠐⠉⢀⣄⠠⠀⠀⠀⢐⠒⠈⣀⠤⢑
⠀⠀⠀⢀⠌⡀⠈⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡇⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡊⣉⡀⠄⠊⠄
⠀⠀⠀⢀⠈⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣤⠀⠄⠊
⠀⠀⢀⡼⠞⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠚⢆⠀⠀⠀⠀⠁
⠀⠀⢎⠀⠀⢪⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠐⠂⠀⡀⠔⣸⠃⠀⢠⠃
⠀⠀⠈⠳⢤⣀⠑⠢⠤⣀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⣐⣠⠥⠖⠋⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠭⢭⣽⣿⣟⣛⣛⣛⣛⣻⣿⡭⠭⠭⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠋⠀⠬⠘⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠁⠀⠁⠈⠈⠀⢈⠉⠀⠀⢮⣭⡷⠂⠀⠀⠈
⠏⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠨⡀⠆⠀⠀⠀⠀⠤⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⣀⡔⢒⠙⠛⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⣠⠂⡸⠂⠀⠰
⠁⠀⠈⢁⡉⠀⢬⠀⡀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠁⠀⠀⠁⠀⠀⠸⠁⠐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⣑⢶⣄⡀⠀⠀⠀⠀⠘⠁⠀⣀⡀⠀⠀⠀⠀⠘⡅
⠃⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠂⠀⠀⠀⠀⠀⢀⠠⠝⠛⠖⠀⣀⠀⠀⠃⢀⠀⢰⠒⢰⡄⠀⠲
⠁⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠁⠀⠀⠀⠉⣈⠁⠠⡌⠁⠀⠀⠀⠂⠙⠀⠀⠁⠩⠤⠀⠀⠁
⠀⢀⡀⠀⠀⠀⡆⠀⠉⠠⠄⠈⠇⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠠⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰
⠰⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⢱⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠋⠙⢠⠁⢐⣀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢠⠴⣶⣧⠀⠀⠨⠁⠀⠀⠀⠂⠀⠀⠑⠘⠾⡌⠁⠙⠀⠀⠀⠋⠠⡐⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠖⠐⠀⠀⠠⠀⠀⠈⠙⠉⠁⠀⠀⠀⢠⠄⠀⠐⠀⠤⠄⠲⠄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠡
⠲⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣛⡂⡀⠀⠀⠀⣀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠈⡀⠀⠈
⠄⠀⠠⠤⠒⠂⠀⡀⠘⠠⠐⠂⡀⠈⠁⠀⠻⢡⡗⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠈⠀⠀⠀⢃⠀⡅⠀⠀⠀⠀⠶
⣄⠀⠳⠀⠀⠀⠙⠉⠁⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡁⠀⠀⠢⠀⠀⠀⠀⠖⠆⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠳⠀⠀⠀⢠⠁⡸⡃⠀⠀⠀⠀⠀⢠⡄⠘⠁⠀⠀⠐⠀⠀⠀⢀⣀
⠷⠃⠀⠀⣀⣤⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢲⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢲⡀⠀⠀⠀⠀⡠⡐⢰
⠀⠀⠀⠯⠽⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠀⠀⢀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠈⢄⠸⠀⠀⢱⠈⠁
⠀⣶⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠈⠩⠡⢤⠄⡀⠀⠀⠲⠊⠁⠀⠀⠀⠉⠁⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁
⢠⠋⠠⠂⠀⠀⠸⠸⢰⠋⠁⠀⠄⠀⠀⠰⠂⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⢠⡄⡀⠀⡄
⡮⢄⠢⠀⣤⡴⢒⢉⣢⠖⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠎⠸⠇⠀⠈⠠⠀⡀⠐⠚⢀⣠⠀⠀⠀⠠⡌⠀⠀⠀⠁⠈⠐⠆
⠁⠁⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠀⠈⢃⣘⠆⠀⠦⠄⠀⠀⠀⠁⠀⠀⠄
⢀⠀⠀⠀⠀⢰⠂⣢⡀⠀⠆⠀⠀⠀⠀⠀⡀⠀⡒⡒⡄⢂⠀⠀⠀⠀⠀⠀⠴⡿⠞⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠁⠀⢀⠀⠀⢈⢉⡅⠁⠀⠁⠀⡇⠀⠀⠟⢋⠄⢁⠪⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⣠⠐⠂⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⢄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠇⠡⢃⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡀⠀⡀⠂⠄⣁⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠂⢄⣀⢌⢂⠀⠀⠈⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⡀⢜⣀⠜⡘⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢀⠀⠈⠀⠂⠈⠀⢀⠀⠤⠄⠀⠀⠠⡄⠀⠀⢀⡀⢀⣀⠀⢀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠰⠉⢈⠲⠂⢠⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⣀⣀⣀⠠⠄⡤⠀⠰⢆⠀⡝⠅⠝⠃⡨⠑⣠⠉⢂⡔⢑⡠⠉⠄⠔⢡⢈⠕⠄
⠀⠀⠀⠀⠀⠙⡃⠤⠆⢁⣆⠔⠙⡀⠜⢣⢀⠛⡌⠘⠣⠀⠛⠄⡌⠡⣀⠂⢑⣴⠀⠱⠦⠤⠑⠒⠚⠃⠈⠁⠀⠉⠀⠈⠈⠈⠁⠀⠉⠀⠘⢃⠌
⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠉⠀⠙⠂⠐⠓⠀⠘⠃⠀⠛⠐⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⢀⣀⠈⠉⠭⠉⠒⠃⠈




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢈⢀⡃⠀⠀⠀⠀⠸⢤⡀⠀⠐⡄⢿⠂⠀⠀⠞⠭⠇⠀⠀⢐⢠
⠀⠀⠀⠀⠀⠐⠞⣮⢫⢧⡀⠀⠀⠀⢂⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠢⢀⡀⠀⠠⣄⠠⠤
⠀⠀⠀⠀⠀⠀⢠⡹⣇⣩⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡄⠠⣤⠀⠀⠀⠁⠲⠦⠤⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠶⠇⢠⠀⠀⡀⠀⠀⠀⠀⠉⠰⠬⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠸⠒⠒⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠆⠐⢢⡀⠀⡀⠈⢲⠑⢲⣀⡐⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⡀⣠⠔⠀⠀⢠⠞⠁⠀⠀⠀⣀⠔⠚⠓⢲⣷⡀⡉⠫⢿⣤⡧⠉⣹
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⢠⠟⠘⠁⠀⠀⠆⠀⠀⣰⣦⣆⣿⣿⣿⣠⣴⣿⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠴⣴⣻⠖⠁⠀⠀⢟⣃⠈⠀⠀⠀⠀⢶⣶⣿⠿⠛⠋⠉⠙⠛⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠠⢀⣈⣥⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠚⠻⣦⣀⡀⠀⠀⢀⣄⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠚⠛⠛⠛⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠉⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠠⠤⠒⠊⠉⠀⠀⠉⠉⠓⠒⠒⠒⠒⠘⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠤⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠄⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣔⣂⡀⠀⠀⠀⢀⠤⠈⠉⠑⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡡⠀⠀⡞⠁⠙⡀⠀⠀⠐⢋⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣍⠤⡃⠀⠀⢣⡔⡄⠀⠀⠀⠀⠋⠂⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀⢀⡇⠀⢀⠞⠁⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢣⠀⠀⣀⣀⠀⣀⡠⠀⠈⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⡀⢠⠓⠤⣀⠀⠣⡀⠣⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣧⠱⡼⣄⠀⠈⠳⡄⠙⠦⡀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠒⠓⠢⡈⢳⡀⢀⠋⠀⢀⡜⡖⠀⣀⡴⠊⢣⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⢀⣠⡤⠚⢦⡴⢀⡌⠀⠀⠈⢧⠘⢾⡉⢀⡠⠃⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠴⢮⠠⡜⠀⠀⠀⠀⠀⠉⠀⠉⢟⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠤⠤⢤⠴⠚⣄⢣⠀⠀⠀⠀⠀⢀⣀⣠⠦⢖⠴⠂⣸⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⣸⠋⡆⢣⣀⡤⠖⠚⠉⢁⡠⠖⠉⠀⠀⢻⣇⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⠞⡄⠁⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⢸⡟⢦⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⡴⠒⣬⣇⡼⠋⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠎⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠋⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠇⠀⠈⠳⡄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠚⠋⠉⠀⠀⠀⠀⣠⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡈⠙⠒⠶⠦⠤⠴⠿⠶⢶⣶⠶⢒⣀⣠⠄⣠⠞⠉⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠉⠉⣉⣉⡳⠿⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣷⣶⣶⣾⣶⡿⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣾⣷⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⡾⢿⠿⠿⠿⠾⣷⡶⠶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⣿⡟⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠿⢿⣿⣿⣶⣦⣄⣈⡇⣿⢹⣿⠃⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣟⢿⣿⣿⣧⢿⣸⣿⣿⡿⣟⣻⣭⣭⣭⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⠿⢿⣶⣝⡿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣟⣾⣿⣟⣽⣾⣷⢻⣿⣿⣿⣶⣽⡻⣮⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣼⢟⣽⣿⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣮⡋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡇⡿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⢠⡀⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⠀⠀⢀
⠀⠀⠀⠲⡄⠈⢧⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⡠⢀⠀⠀⡠⢦⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⡠⠋⠀⣀
⠀⠀⣄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⢄⡠⠁⠈⠳⠔⠁⠀⠣⠐⠉⠀⠸⠀⢀⡠⢦⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⢀
⠀⡀⠈⠃⠀⠀⠀⠀⢰⠆⠠⢄⣠⡀⢄⠀⠀⠀⠀⠀⡠⠄⠒⠀⠀⡂⠀⠀⠃⢁⡀⠼⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠈⠁⠀⣀⡀⠤⢤⠀⠀⠊⠁⠀⠀⠀⢂⠀⠀⢀⠎⠀⠀⢀⡀⠀⠇⢀⡴⠋⠀⠀⠀⠀⠈⢫⡇⠀⠀⡤⠠⢀⠀⠚
⡠⠄⠂⠉⠀⠀⠀⠈⡜⡀⠀⠀⠀⣇⠀⠈⠆⠀⣼⠀⠀⠀⣿⠀⡴⢀⢺⠁⠀⢠⠎⠑⢆⠀⠀⢣⣀⡾⠀⠀⠀⡹⠈
⠘⡀⠀⠀⠀⠀⢰⠂⠑⣴⡄⠀⠀⠉⠀⠀⠘⣄⠿⡀⠀⠀⠓⠀⢰⡇⢸⠀⠀⠈⠈⢂⡘⠀⠀⣸⣱⠁⠀⠀⡐⠁
⠀⠑⠮⣉⡄⠀⠈⡇⡆⠀⡇⠀⠀⢲⠳⡄⠀⠘⣇⠑⢄⡀⠀⠀⠈⡇⠘⡄⠀⠈⠒⠋⠀⠀⣠⢧⠃⠀⢀⡜
⠀⠀⣀⠠⠴⠀⠀⠸⡰⠀⠇⠀⢀⣸⠦⢗⠔⠂⢹⣄⣀⣈⣉⠉⣹⢣⡀⠈⠢⢤⣤⣤⢤⠞⣿⠵⠬⡕⢀⡽⠃
⠀⠉⠢⡀⠀⢧⢀⣀⠵⢇⠘⢉⡁⣠⣶⢾⡓⣛⠿⢉⠟⣻⣅⣛⢿⣽⣍⡳⢤⡄⠐⠀⠀⡼⠙⢤⡤⠾⢉⡀
⠀⠀⠀⡈⠓⠈⠉⠂⠐⡈⣠⣮⣵⠷⠾⠷⠶⠿⠶⢾⣮⣜⡡⡯⠹⠿⣶⢻⣶⣬⠷⣄⡀⠈⠀⠈⠀⠀⠀⢈⡳⠂
⠀⠤⣊⠀⠀⠀⠀⢀⡾⠛⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠩⢈⠙⠶⣄⣀⠻⡐⢻⠰⣇⠘⣿⢦⡀⠀⠀⢠⠖⠉
⠀⠀⠀⠈⡑⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠪⡳⣌⠺⢟⡓⢾⠦⡛⢛⣿⣦⠀⠀⠉⠢⡀
⠀⠀⢠⠎⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠳⣍⢙⣧⢴⣽⣈⣆⣻⣳⣀⠤⠒⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢊⢷⡇⣤⢜⣽⠋⣃⣿⣷⢄
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡼⢑⠶⠞⠉⢿⡎⡏⠁
⠀⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣆⠀⠉⢀⣰⢇⡇
⠀⣀⠀⠀⠉⠓⠲⠦⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠩⢳⡾⢟⣫⡾⠁⠀⠀⠀⣄
⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠤⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⡖⠒⠲⠶⠞⠛⠉⠁⠀⠀⠀⠰⣄
⠀⠉⠀⣠⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢇⠀⡠⠖⢀⠀⠀⡰⠀⡀⠀⢰⠁⠢⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈⠁
⠀⠀⠘⠁⢀⡔⠀⢀⡄⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠢⠊⠀⠀⠙⢦⠃⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠸⣄⠀⠳
⠀⠀⠀⠀⠉⠀⠀⠛⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠓⢶⣤⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣡⡀⠀⠀⠘⠁⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠄⠀⢻⡌⠦⡰⠀⢼⣟⣀⣀⣀⣀⣀⡀⣀⣀⣠⣤⣤⡤⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠻⣧⣀⠀⢿⣴⡇⢨⣾⣷⣡⡤⠴⠚⢛⣩⣵⣖⣻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣀⣠⣤⡤⠤⣤⣌⣻⣷⢽⣿⡀⣾⣿⣴⣦⣶⣿⣿⣟⡛⠋⠉⠉⠉⠛⠂⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉⠉⠉⠉⠙⣻⠷⠶⣾⣯⣿⣯⢽⣿⣿⣿⣾⣷⣿⣿⣷⣾⣿⣷⣶⣶⣷⣴⣠⣌⢠
⠀⠀⠀⠀⠀⠠⠊⠜⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠛⠉⠁⠀⠀⠀⠙⣿⣷⡄⠀⠉⠙⠈⠀⠈⠉⠉⠙⠛⠻⣿⣿⣿⣷⡶⡤⠄
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠹⢷⣷⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢧
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣷⣃
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣿⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣯⣿⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣿⠃
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢠⢀⣷⣿⡄
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⢳⠁
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣿⡟⡞
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣩⡿⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⣘⣿⣿⣟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣾⣽⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⣠⣔⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢬⣀⡀⠁⠀⠀⠀⡀⠀⡀⠀⠐⠀⡀⠁⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠶⠴⠶⠶⠶⠷⠶⠦⠷⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⢠⡄⠠⠄⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⢀⠀⢀⠀⠀⡀⠀⡀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠿⠇⠿⠇⠺⠿⠻⠿⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⣠⣤⠀⠠⡄⠀⠀⣤⠀⢠⣤⡀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⠿⠿⠟⠛⠛⡛⠉⠉⠉⠉⣉⠉⠉⠉⠉⣍⠉⠉⠉⠉⣉⠉⠉⠙⠛⠛⠛⠻⠿⠿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⣀⣴⣾⡿⠿⠛⠉⠉⠀⢀⣶⣄⠀⠀⣼⣿⣦⡀⢀⣼⣿⣷⡄⢀⣾⣿⣷⡀⢀⣼⣿⣷⡀⠀⣠⣾⣦⠀⠀⠀⣠⣄⠉⠉⠛⠿⢿⣷⣦⡀
⢀⣾⣿⠟⢡⣄⡀⠀⣼⣶⣄⣾⣿⢿⣷⣼⣿⠋⠻⣿⣾⡿⠁⠙⣿⣿⡟⠈⢻⣿⣿⠟⠉⢿⣷⣾⡿⠛⣿⣧⣠⣾⡿⣿⣆⠀⢀⣤⣦⠈⠻⣿⣧
⣾⣿⣁⠀⣾⣿⢿⣶⣿⣛⣿⣿⣧⣤⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣾⣿⣿⣶⣶⣾⣿⣿⣯⣄⣹⣿⣾⡿⠻⣿⣆⠀⣨⣿
⣿⣿⣿⣿⣿⣥⣴⣿⣿⡿⠿⠛⠋⣹⠉⣹⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⢿⣍⠛⠛⠿⢿⣿⣶⣤⣽⣿⣿⣿⣿
⠹⣿⣿⣿⡿⠟⠋⠉⢀⡀⠀⣶⡆⠙⢷⣿⣿⣿⣿⠐⠿⠿⠿⠿⠿⠿⠇⠀⠸⠿⠿⠿⠿⠿⠷⠸⣿⣿⣿⣶⠟⠠⣶⠆⢀⡀⠉⠛⠿⣿⣿⣿⣿
⠀⠹⣿⣏⠀⢠⣦⠀⠙⠛⠀⠀⣤⡔⢛⣿⣿⣿⣿⠀⠀⣴⣶⣦⠀⢸⣧⠀⣿⠀⢠⣶⣶⡄⠀⠀⣿⣿⣿⣟⠳⣦⡀⠀⠙⠛⠀⣴⡄⢀⣽⣿⠏
⠀⠀⠈⠻⣿⣦⣅⡀⠀⠺⠷⠈⠋⣱⣿⡿⢿⣿⡏⠀⠀⠙⠛⠁⠀⢸⣿⢸⣿⠀⠀⠙⠛⠁⠀⠀⣿⣿⡿⢿⣷⣍⠁⠺⠷⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣤⣀⣻⣿⣿⡟⡧⢻⡇⠀⠀⠀⠀⠀⠀⣸⡟⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⢿⡟⣾⣿⣟⣀⣤⣴⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣀⣸⣇⠀⠀⣤⣤⣤⣾⣿⣷⣶⣿⣿⣦⣤⣤⡄⠀⢀⣿⣀⣰⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⡈⠻⠿⣿⣿⣤⣴⣿⣿⣿⣿⡿⠋⠉⠻⢿⣿⣿⣿⣿⡄⣼⣿⡿⠿⠋⢁⣀⣀⠀⣀⣀⠀⣀⣀⣀⣀⡀
⠀⠀⠸⣿⠇⠀⢿⡿⠀⠸⣿⠇⠀⠺⣿⠆⠀⠘⣿⣿⣍⠛⠛⠋⠁⢠⣶⣶⣶⡄⠉⠛⠛⠉⣰⣿⡟⠀⠀⢰⣿⡿⢿⣿⡿⢿⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⣠⣾⣿⣷⡿⠟⣠⣿⡟⢠⣿⠃⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣤⣤⣤⣀⣠⣴⣿⠟⠁⠀⢀⣼⣿⠟⠁⠀⢀⣴⣿⠋⢠⣿⡇⢸⣿⡇
⠀⠀⠀⠀⢻⣿⡌⢿⣍⠻⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣿⣿⣀⣀⣤⣴⣿⣿⣿⡿⠿⠋⠁⠀⣠⣴⡿⠋⠀⠀⢀⣴⣿⡿⠁⣰⣿⣿⢃⣿⣿
⠀⠀⠀⠀⠈⢿⣿⡄⠻⣷⣌⠛⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⣀⣠⣴⡶⠟⠋⠁⠀⢀⣠⣶⣿⡿⠋⣠⣾⡟⣽⢏⣾⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠈⠻⣿⣦⣌⡙⠛⠷⠶⠶⣶⣶⣶⣶⣶⣶⠶⠶⠿⠛⠛⠉⠁⠀⢀⣠⣤⣶⣿⣿⠟⠉⣠⣾⡿⢫⣾⢋⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠈⠙⠻⢿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⠿⠛⠉⣀⣴⣾⡿⢋⣴⡟⣡⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠢⣀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⣀⣤⣶⣿⠿⠛⣁⣴⡿⣫⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣬⡳⢦⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣿⡿⠿⠛⢋⣡⣴⡾⢟⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣭⣛⠶⢦⣤⣄⣉⣉⣉⡉⠉⢉⣉⣉⣉⣁⣤⣤⣶⠾⢟⣋⣥⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣭⣭⣙⣛⣛⣛⣛⣛⣛⣋⣩⣭⣥⣶⣾⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⠿⢿⣶⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢤⣬⣭⠤⠶⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⢠⣤⣀⣤⡄⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣄⣀⣉⣠⣄⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⠀⠀⠀⠀⣈⣉⣩⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠟⠀⠀⠀⠀⠈⠉⣉⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡀⠀⠀⠀⠀⠀⠒⠛⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠃⠀⠀⠀⠀⣘⣋⣉⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⠀⠀⢀⣀⣩⣭⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⡏⠀⠀⠀⠀⢈⣉⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠀⠀⠀⠀⠀⢬⠍⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠⠏⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠰⠋⠀⠀⠀⠛⠓⠚⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⣀⣘⣛⣛⣓⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠤⣤⣉⡉⣩⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠶⢦⣤⣤⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢀⣀⣀⠙⠛⠶⠦⣴⣾⡿⠋
⠀⠀⠀⠀⣀⣠⣶⠿⠋⢀⡀⠴⢤⣀⣉⣉⣙⣛⣿⣿⠟⠁
⣀⣤⣾⡿⠟⠋⠁⠠⣤⣀⠙⢻⣶⣤⣽⣿⠿⠛⠋⠁
⠛⠻⠿⣶⣶⣴⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠒⠂⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⡾⢺⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠒⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⣟⡾⣿⡀⠀⠀⠀⠀⠀⣠⡾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠀⠀⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⠈⠁⠀⠠⢀⠀⠀⠀⠀⠈⢺⣯⠈⠌⠛⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢂
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠄⠀⡀⠀⠀⠀⠀⠐⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡤
⠀⠛⠤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠒⠉⠱⠀⠀⠀⠀⠉⠀⠀⣤⢺⠈⠲⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠐⠀⠀⠀⠰⠐⠋
⠀⠀⠀⠀⠈⠈⠙⢛⠲⣴⠢⣄⡤⢀⡀⠀⣀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠀⠐⠘⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠄⢀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠓⠈⡍⢀⠐⠟⡙⠛⠛⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢬⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠓⠀⢿⠗⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠈⠄⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠁⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⣓⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠄⠀⠀⠐⠉⣇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢹⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡐⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠊
⣿⣿⡟⠻⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣴⡠⢀⡀
⠀⠀⠀⠀⠀⣠⠶⢛⣽⣳⢄⣾⢿⡛⢶⡀⠀⠀⠀⣴⣾⢿⣟⣣⣀⣿⣿⣿⣦⡀⠀⠀⠀⣴⠶⣶⣶⣿⣯⣭⣽⣶⣦⡄
⠀⠀⠀⠀⢠⡏⣴⠈⠁⠀⠀⠛⠀⠀⢢⣿⠀⠀⣸⡏⠀⠊⠁⠀⠘⣿⣿⣿⣿⢷⠀⠀⢰⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⢻⣧⠀⠀⠀⠀⣠⣿⣿⣿⡿⠘⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣜⠆⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⢿⣿⣶⣶⣾⣿⣿⣿⠟⢁⠎⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣯⠥⠒⠁⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣜⣂⣿⣚⣠⡴⠁⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⣤⠄
⠀⠀⠀⠀⠀⢠⡴⠿⢹⡿⣿⣿⣭⠶⣄⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠿⢿⣷⣦⡀⠀⠀⠀⢀⠀⢠⣞⠧⢴⡿⠦⠀⢀
⠀⠀⠀⠀⢰⡏⠀⠀⠘⠀⠈⠃⠈⠃⠘⣧⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⢠⡏⠀⠈⠀⠀⠀⠃⠀⠀⠈⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠙⠳⠦⢤⡤⠴⠞⠋⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣤⠤⡴⢲⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣄⡇⡄⢀
⠀⠀⠀⠀⢠⣶⢿⡿⠗⣀⠀⣙⢲⣶⣦⡀⠀⠀⠀⠀⡀⠀⢰⠟⢿⠻⣄⠐⣦⣄⠀⠀⠀⠀⣀⡤⠙⠷⣾⣿⠶⠋⢤⣄
⠀⠀⠀⢠⣿⢤⣼⣷⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣶⡹⡆
⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢀⣾⣿⡗⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢹⣿⣿⣦⣤⣤⣤⣶⣿⣿⣿⠇⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⡾⣱⠇
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣀⣬⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⢶⣤⠙⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣶⣬⣿⠟⠦⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣼⠋⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⠆⣠⡅⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠐⣋⣽⡇⠀⢾⣭⡉⢸⡿⢤⠖⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⠀⠉⠀⠀⠀⠀⠸⠿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⠀⠓⠒⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⢓⠂⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⢀⣴⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⢿⣿⣿⣿⠋⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣍⡋⠁⠀⣠⠎⠀⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣌⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⣤⠴⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣢⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⡰⡁⠀⠀⠀⡀⠐⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠁⢀⡸⠇⢀⠀⠀⠀⠸⡆
⠀⠀⡼⠁⠈⠀⠀⡌⠀⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣤⣾⣿⣿⡟⢻⡗⣦⠙⢷⣌⠀⡀⠀⠀⣿
⠀⢠⡇⠤⠒⠘⣼⡃⠀⠀⠀⢥⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠋⣾⣿⣿⣿⣧⣸⣧⠀⠂⠀⠻⣷⡘⣆⠀⢸⡀
⠀⢸⡇⠀⠀⠀⠹⣁⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠃⠠⣿⣿⣿⣿⣿⣷⠟⠂⠀⠀⠀⠻⣖⠘⡆⢸⡇
⠀⣜⠁⠀⠀⠀⢀⣻⣴⣶⡞⠁⣇⠀⠀⠀⠀⠀⠀⢸⡇⢰⣴⣿⣿⣿⣿⣿⣿⣶⡶⠀⠀⠀⠀⠀⢹⡇⣷⢸⡇
⠀⠘⣦⡀⠀⠀⠸⣴⣿⡟⠀⠀⠏⠀⠀⠀⠀⠀⠀⠘⠣⣾⣿⣿⣿⣿⣿⣿⣿⡉⢻⠀⠀⠀⢔⣀⢴⠂⠀⠸
⠀⠀⠈⠻⣄⣀⣀⣻⠟⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠦⠶⣄⡐⠀⠒⠀⢁⡾⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠷⠲⠾⣽⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢸⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡴⠒⣒⠒⣆⣀⣴⢲⣟⣿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣳⡯⣽⢋⣭⠭⣟⣾⡋⣥⣴⣿⣶⣮⣹⢯⢵⡚⠳⣶⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣚⠽⠖⠻⠵⠋⠠⠾⠽⠛⠧⣑⠦⣽⣟⡚⠻⢿⡇⢠⣠⢰⣶⡇⣠⢜⣴⡫⠿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣞⣽⣷⡶⠬⣟⢿⡼⡿⢚⣠⣴⣿⡇⠀⠀⡈⣽⣂⣤
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⣿⢿⣩⢤⡖⢲⡛⠀⠹⡍⠉⢭⣏⣿⢻⡚⠘⢩⢟⡿⡗⡶⠂
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣀⣀⣀⣹⣦⡙⠻⣖⡓⠚⠾⠶⠿⠮⡭⣝⢯⣱⠹⣽⣳⣆⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠛⠋⠉⢿⡶⣤⣍⡛⢿⣖⣒⡚⣅⣞⢿⡛⠃⣮⠔⢻⡽⠻⣦⡀
⠀⠀⠀⠀⡼⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢧⣷⣏⡛⠛⢷⡦⣏⠘⣎⢦⡑⢄⠪⣵⡌⢩⣭⠴⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠕⠛⢿⡯⣗⣌⠢⡱⣿⣞⠙⢲⣧⣮⡛⡶⣾⢵⡆
⠀⠀⠀⢰⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠷⢻⠳⣭⣒⣿⣮⡪⣴⣷⡛⣿⡙⣟⡟⣆
⠀⠀⠀⠸⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣝⣒⣶⠀⢩⠍⣤⠣⣬⣟⡻⣷⢿⡀
⠀⠀⠀⢠⠇⡆⡆⡀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣹⣜⡶⢃⣜⣁⠹⢼⡼⣩⠏⣩⡛
⠀⠀⠀⢸⡄⡇⡇⡇⢸⢸⢸⠰⡄⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⢹⠘⠲⣟⢮⡛⣟⣫⡞⢧⣀⡞⡇
⠀⠀⠀⢸⠛⢇⠇⡇⣸⢸⢸⠀⡗⢇⢸⠘⡆⢰⢰⢠⠀⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠙⠚⣧⣳⣬⣿⡠⠄⠊⢹⣸⣍⠦⣹
⠀⠀⠀⠈⢧⡀⠀⠁⠙⠈⠘⠃⢧⢸⣸⠀⡇⢸⠘⣬⡆⡇⢸⠀⡏⠘⡆⡆⢠⠀⡄⡀⢀⠀⠀⠀⠀⠀⢸⡉⠙⢯⡷⢀⣤⣎⠙⢺⢳⢺⠃
⠀⠀⠀⠀⠀⠉⠓⠲⠮⠤⢤⣀⣀⠀⠉⠃⠳⠈⠇⢧⢧⢻⠘⡆⢳⠀⢇⢹⠸⡄⡇⢳⠸⡄⡇⡄⠀⠀⠀⢿⢳⠦⠽⢤⡞⣽⣇⣨⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⢤⣀⡀⠀⠈⠘⠂⠳⠘⡆⢸⡸⡄⣇⢱⢸⡀⡇⢹⢹⠈⡆⢧⠘⡿⣰⣆⠈⢷⣖⠥⣌⠻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣀⠀⠀⡀⠁⠓⠸⠜⡆⡇⢹⡈⡇⣇⢻⢸⡀⢳⣈⢻⠀⢳⡙⢄⠨⣝⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⢤⣀⡀⠀⠃⠀⠃⠹⠘⡌⣆⣇⠀⢷⡀⢾⢨⣇⢸⡀⠹⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣀⣀⠀⠀⠀⢘⠀⠸⣷⡎⡃⢹⣾⣟⡸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠬⣄⠀⠹⣧⣿⡦⣽⡿⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣉⡭⠥⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢶⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢋⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⡀⡏⢸⣉⡉⠛⠛⢳⡶⠖⢛⣛⣿⣿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣯⢭⡤⣄⣸⡇⢡⠞⢑⠏⣠⠶⢃⡤⠚⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢉⡥⠤⠤⠴⠶⢍⠲⢍⣠⠃⠀⡾⢞⡅⣠⣏⡀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⣑⡄⠀⠀⢰⡏⠀⠉⠀⠠⢍⡓⢤⡀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠉⠀⠀⠀⠀⠀⡠⠎⢁⣀⣤⡀⠡⠐⠂⠋⠓⠒⠶⣌⡂⢹⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡞⠄⡼⠉⠀⠸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡜⠀⠀⣀⣤⠶⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣷⡜⠁⠀⠀⠀⢱⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣀⢀⠞⠁⠀⡀⠀⠠⠤⢤⡀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣀⡠⠤⠤⣿⣊⠜⠀⠀⠈⠀⠐⠀⠀⠀⠙⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠞⠁⢀⠀⠄⢀⠀⠀⠀⠔⠀⡔⢅⠶⡀⠒⠄⡘⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡜⠁⢠⠖⠁⢀⠐⡂⠑⢄⠀⣄⠀⡄⠀⢙⣃⠔⢊⡀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡼⠀⢠⠗⢉⠁⢠⠜⠉⠱⢬⠇⠈⠈⠀⠀⠈⢄⡔⠉⠁⢤⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣝⠋⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇⠀⢸⠄⣂⠐⠓⢠⡄⠀⠀⡠⢄⠀⡖⠐⠄⢀⣼⠢⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣂⣈⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢣⠀⠀⢧⣀⡈⠵⢊⣠⡄⠀⣠⢼⡆⡻⡤⡈⠟⠈⠠⢠⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠩⣤⢭⣓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣹⡆⠀⠈⣆⠀⠀⠀⡴⠈⠈⠊⠱⠒⠣⠀⡣⠁⠘⠦⠴⠒⠓⣉⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢶⡌⠉⠛⢛⠃⠈⣙⠛⠃⠈⢉⠩⠝⠛⠉⠉⠉⠀⢹⣆⠀⠘⣆⠀⠀⠳⡜⠠⠠⠀⡅⣠⠖⠁⢀⣠⣤⣴⣶⣟⣉⣙⣛⠛⡷
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠞⠋⠁⠀⠀⠀⣀⠀⠀⠙⠒⠋⠀⠀⢀⠄⣀⡤⠶⠒⣾⠈⠳⣤⡈⠣⡀⠀⠉⠒⠢⠴⠖⠁⢀⣴⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⣠⠞⢉⡀⠤⠒⠒⠈⠉⠉⠙⠁⠀⠀⠀⠀⠌⣠⠄⡰⣡⠞⠁⠀⠀⣠⠏⡀⠔⢹⠙⢳⡾⣷⣤⣄⣀⣤⡤⠶⠋⠁
⠀⠀⠀⣴⣧⣬⡥⠤⠤⢤⣤⣤⣀⣀⣀⣀⣤⠖⣃⣠⠞⣯⣼⠞⠋⠀⢀⡴⣛⣥⣤⠀⠀⡦⠀⡀⠳⣄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢼⠷⠛⠉⠀⠀⠉⠀⠀⠀⠀⠘⢋⡵⡚⣁⠀⠩⠁⡀⣿⠳⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣹⠏⣠⢋⣼⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡷⠛⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁

⢫⣵⢤⣤⣤⣤⣤⠀⣠⣬⡙⠿⢀⠙⠿⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠀⠀⣀⠐⠟⠁⠀⢠⠊⠁⠀⠀⠡⣀⣀⣸⡿⠀⢠⣤⠈⠉⠀⠀⠀⠈⠨⢀⢁
⠀⡝⢟⣿⢵⣿⣿⣿⣿⣷⡶⠀⠀⢰⣄⠘⠃⠍⠑⡀⠈⠇⠀⠀⠀⠀⢤⠀⠁⢀⠇⠀⡩⠤⠄⠄⠀⢀⠠⠀⠸⢦⠀⠈⣉⠉⠀⢠⣶⠿
⢦⣀⣀⠈⠙⠇⢻⣷⠈⣿⣿⡦⠀⠀⢑⡙⠄⠸⡀⠀⠀⠀⠀⢀⠀⠀⡈⠃⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡐⢒⠉⠀⣀
⠻⠿⠟⠁⠀⣀⠀⠈⠀⠙⢋⠀⠶⠀⠈⢴⣤⠄⣀⣈⠀⠐⣤⡀⠀⡼⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠘⢇⣴⡀
⢀⡆⠰⡄⠺⠿⢣⠀⠀⠀⠘⠁⠀⠀⢴⡄⠙⣀⣴⠋⠀⠀⠹⣤⣶⣤⣮⠤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠑⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣡⠆⠚⠛⣦⢾⣿⣿⡛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢯⠿⣦⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣾
⣶⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠞⠫⠰⠀⠚⠋⠈⣿⡿⠁⠲⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣽⡿⢷⣿⢋⣠⡤⡠⠀⢤⡶⠄⣀⣴⣿⢋
⡧⣷⣞⡷⠀⠀⠀⠀⠀⠠⠀⣸⣿⡆⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠂⠀⠁⠀⠀⡀⡈⠙⠋⠁⠈⠛⠀⠀⠘⠉⠀⠈⠉⠉⠀⢸
⠛⠚⠋⢀⣀⣠⣤⠶⠆⠁⠀⢈⢉⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣾⡇⠀⣤⡀
⠚⠛⠛⠛⠉⠁⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠑⠂⠀⠀⠀⠈⠉⠀⢠⣽⠗
⠀⠰⠂⣀⡴⠟⠉⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⠃⠀⠀⠀⠉⠢⠀⠀⠀⠈⠁
⠀⠒⠛⠉⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠈⢻⠓⢄⡛⠁⠐⠒
⠰⠵⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⢀⡐⣷⠄⠒⠲⠈⡆⠀⠉⢀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⠉⠀⠀⠀⠀⠀⠀⠔⠂⢀⣈⣿⣿⣤⣿⣿⢷⣟⢷⢛⠁⠀⢠⠃⡀
⠀⠀⠒⠰⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠚⠀⡀⠀⣄⠀⣠⣽⣿⢋⣽⡿⢾⠏⠋⠁⠘⣠⢀⠠⠂⠀⠀⠁⠀⠰⠆⠐
⠀⠂⠀⠀⠘⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⡈⠆⠙⠻⠟⡘⠋⠁⠀⠀⠐⢾⣶⡷
⠒⠀⠀⠀⠀⠀⠰⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⠖⠂⠀⠀⠷⠀⡰⣧⠀⠀⢀⡀⠀⠀⠉⠀⠀⠠
⠀⠀⣉⣀⡀⢀⣤⡄⠈⠁⠀⠀⠀⠲⠶⠤⠰⢂⠀⠀⠠⡤⠀⠀⠠⣤⠆⠀⠀⠀⣤⣀⠀⠀⠀⠐⢶⠶⢤⣇⣤⣔⣨⠟⠀⢆⡀⢧
⣿⣿⢿⣿⣷⣿⣽⣶⡖⠀⠧⡀⠇⢀⡈⠑⠚⠁⠀⠚⠀⠀⠐⠈⠢⡀⠀⠀⠀⠢⠛⠛⠀⡯⢸⡆⢀⠔⠚⣿⣿⣿⣁⣤⣄⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠
⠚⠙⠽⣟⢿⣷⡌⣿⣿⡗⠀⠀⠀⠈⠘⠐⣶⣄⣠⠞⠀⠀⡄⠈⢢⢑⡄⠀⢠⡄⠐⠂⠀⢤⡍⠐⠛⠓⠈⣿⣟⠏⢿⡌⠁⠐⡦⠀⠀⠀⠐⠦⢤⣤⡴⠶⠚⣁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠛⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡸⣿⠃⠀⠀⠀⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡘⡼⠋⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⣩⣟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣟⢯⣉⢀⠀⣀⠀⠀⡤⡤⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣓⣉⣙⣓⣛⣛⣛⣋⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣸⡿⢀⣀⣦⣄⣠⡀⢶⢆⡈⢿⣿⣿⣿⠿⠟⢩
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⢼⣿⣿⣿⣿⡿⠶⣾⣿⣿⣾⠋⠁⠀⠀⠀⠸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠻⣯⣿⣿⣿⢏⣉⡛⠛⠋⠀⠀⠸⣿⡟⠽⠉⠁⠐⣬⡿⠃⡟⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⣿⣿⣿⣦⠈⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⣠⠇⠀⠀⠀⡇⠀⠀⠆⠀⠀⡆
⠀⠀⠀⠀⠀⢀⠈⡇⢰⠁⠀⠀⠀⠀⠃⠀⠀⠀⠈⠛⠛⢻⡆⠀⠀⠀⣿⣿⣿⣿⣶⣭⣶⣼⣿⣾⣿⣅⠀⠀⠀⠀⣇⠀⡸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⣇⣎⠀⠀⠀⠀⠈⠘⢾⡇⠀⠀⠀⠋⢸⣿⠛⠿⢿⣿⡿⢿⢿⡿⠃⠀⠀⠀⠀⠹⣦⡄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠘⡄⠁⣸⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠺⠦⣌⣣⠀⠀⠀⠀⠀⠀⠪⢛⡶⣦⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠙⣷⠸⠂⠀⠀⠀⠀⢸⡆⠀⠀⠀⡤⢀⣈⠛⢦⣤⣀⠀⠀⠀⠀⠀⠈⠉⠁⢼⠒⠂⠀⠐⠒⠂⠀⠂⠀⢸⡆⠀⠛⠐
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠤⠴⠺⠆⢀⣄⢰⣿⣛⡈⠿⠿⢿⣿⣷⠢⢀⡀⠀⠀⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣠⣶⡿⠿⠋⠘⢿⡿⠙⢳⣿⡶⠿⢿⣿⣶⣭⣙⣛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠒⠚⠋⢀⣀⣤⣴⡿⠛⠁⠀⠀⠀⠀⠈⡳⣶⣤⣴⡶⢷⣿⠀⠉⠛⠛⠿⠿⠿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠁⠀⠀⢠⣴⣶⣶⣿⣿⣿⣋⠙⢿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠈⠙⠷⠤⠀⠀⠀⢀⣠⣼⣿⢀⡁⣠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⣻⠽⠛⠉⠉⠁⠀⠀⠻⢿⡿⠈⠙⠛⠛⠻⠿⣿⣿⣿⣶⡄⠀⠀⠀⠈⠉⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣟⡵⠊⠁⠀⠀⠀⢀⣠⣴⣶⣿⡌⠣⣤⣤⣤⣤⣀⣀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠁⠀⠀⠀⢀⣠⣶⣿⣿⠟⠛⠋⠁⠀⠘⠿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁

⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢿⡿⠿⣿⡿⠿⠿⠷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠲⢦⣄⣀⡀⠀⠀⣀⣠⡴⠶⠛⠻⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢀⠀⠁⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣠⡤⣀⠈⠶⢤⣀⡀⠀⠀⠀⢀⣀⣤⡶⠂⠶⠶⢁⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⠖⠋⠘⠃⠀⠀⡀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠈⢦⠊⠀⢢⡝⠢⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣠⣾⣿⣦⣤⣦⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⣄⣨⣦⡀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣤⣤⣤⣤⣴⣦⣤⣤⣤⣴⣾⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢻⣿⣿⡿⠋⠉⠀⠀⠀⠉⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠻⣿⣿⡟⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⡏⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠤⠤⠤⠶⠖⠚⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠆⠙⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⢀⡠⠚⣩⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢋⣡⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠓⢀⢐⡉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠔⠈
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠀⠀⠀⠀⢀⣒⣼⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠇⠿⠇⠟⠁⠀⠀⠀⠀⠀⠀⢠⡠⠀⣐⡤⠤⡂⠀⠀⠀⠀⠀⢀⠀⣀⡀⡀⣀
⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⣐⠀⠀⠀⠁⣈⠀⠀⠀⠀⠰⠈⠀⠁⠀⠁⠀⠁⠐⠦⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠚⣻⠧⡀⠀⠀⣈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⢣⡌⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣯⠏⠁⡀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠁⠨⣯⣆⠹⣆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠹⣧⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠙⢿⣾⣆⢄⢀⠀⠠⠖⠂⠈⠀⠀⠀⠀⠰⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠈⠛⠣⣂⣀
⠀⠀⠀⠀⠀⠄⠐⠁⠁⠐⠆⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠤⣀⠀⠈⠉⡁⡀⣀⣹⠀⠀⠀⠰⠇⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠩⠈⡅⠱⠀⠀⠀⠀⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⠠⠠⠧⠚
⠀⠀⠀⠀⠀⣰⠤⠀⠀⠀⠀⠀⠀⢰⣰⡶⠄⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡈⠒⠄⢶⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡁⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠬⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠜⠁⠁⠀⠀⠀⠀⠆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⣃⠠
⠀⠀⠀⢀⠀⠀⠀⠌⠀⢀⠔⠱⠋⠅⠀⠀⠀⠈⡁
⠀⠀⠀⠈⠢⢀⠀⠄⠠⢀⠘⠀⠰⠇⠀⠀⠰⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠈⠧⠠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠤⡀⠁⠁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠔⠀⢢⠑⠀⠀⢀⡀⠁⢈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡔⠀⠸⠤⡄⠄⠀⠀⠈⢄⠀⠀⢀⣤⠄⠀⠐⠀⠢⣞⠆⠀⠀⡠⡄
⠀⡀⠀⠀⠀⠀⡀⠀⠀⣆⠄⢨⣣⠀⠈⡄⠀⠀⠀⢀⠈⠀⠨⣄⠈⡁⠀⠄⠐⠀⢀⡀⡄⠀⠀⠈⠡⣄⠀⡆⠈
⡌⡀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⢸⠂⢀⠃⠀⠐⠀⠾⣄⠀⠛⢁⠓⠀⠂⠀⠀⢀⡀⣄⠀⠂⡄⣊⠀⠐⠋⢁⣠
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠊⠁⠀⠀⠀⠀⠀⠾⡙⡶⡤⢤⣈⣉⣉⣉⡀⠉⣉⣉⣉⣀⣠⣤⣴⠾⠛⠉
⠘⠤⠀⠠⠀⣄⠀⠀⠀⠀⢀⢠⠉⠉⠉⠉⠉⠉⠈⠉⠀⠈⠉⠉⠉⠉⠉⠋⠙⠛⠻⠛⠟⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣉⣢⣤⣤⣔⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠉⠛⠦⣤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺⠗⠛⠲⣄⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢀⣠⡴⠚⠉⢰⠏⠀⡀⠀⠘⢿⡏⠁⠀⠀⠀⠀⢀⡏⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠳⢤⣄⣀⡟⠀⠀⢻⠀⠀⠘⢿⣶⣤⣤⣄⣀⣼⡁⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠸⡆⠀⠈⠻⣇⠀⠀⠈⢧⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣿⣿⣄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⠳⢦⣄⡈⢧⡀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⣿⣿⣿⣦⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⠙⠳⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣄⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣹⣿⡿⠿⠒⠒⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣿⣦⣶⣀⣀⡄⠀⠀⠀⠀⣠⣞⣓⣻⠉⢙⣓⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣦⣤⡀
⠀⠀⠹⣶⣾⣿⣿⠿⠿⠿⢿⣿⣦⡆⠀⠀⣿⢏⣭⡍⣿⢏⣭⣍⢾⡇⠀⢀⣤⣠⣄⠀⠀⠐⠆⠾⠹⠭⠧⠖⠃
⠀⢀⣀⣼⣿⣿⣶⣶⣶⣴⣶⣾⣿⣧⣀⠀⢿⡘⠿⣣⠿⢇⠻⢟⣼⠇⠐⠿⠟⠙⠿⠗⠀⢤⣰⣤⣀⣠⡴⢦⣄
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠂⠀⠙⣿⢟⣺⣺⢿⡟⠁⣤⣤⣤⣤⡀⠀⠀⣰⣿⣴⣏⣿⢳⣜⣷⣽⡄
⠀⠀⠰⠿⠿⠿⠟⠛⠛⠿⠿⠿⠏⠉⠁⠀⠀⠀⢻⣐⣒⣒⡸⠀⢸⡏⠁⠈⢹⡇⠀⢀⡽⠿⣿⣿⣿⣿⣿⠿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠶⠶⠀⠀⠀⣩⣅⠀⠀⢸⠁⠀⠀⠈⡇⢀⣿⣧⣿⣸⢈⣆⣱⡆⣷⣸⣿⡀
⠀⠀⠀⠀⠀⢀⣤⣰⠻⣿⠀⠀⠀⠀⢩⣋⡊⠂⠀⠀⠘⣷⣅⠀⠈⠀⠀⠀⠀⠁⠠⣟⡿⣿⣿⣿⣿⣿⣿⣿⡟⣯⠆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⢹⡄⣠⣤⡀⢀⣒⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢡⠙⠾⣇⣾⢩⡏⣱⣅⡷⠟⢁⠐
⠀⣼⣶⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⠇⠸⠿⠀⢀⠲⡙⢷⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠁⠄⠈⠙⠻⠗⠛⢉⠠⠘
⠀⣿⣿⡄⢸⣿⣿⣿⣿⣿⣇⣿⡟⠀⠀⣀⣄⠀⠳⠉⠂⠀⠀⠀⠀⢻⣦⣴⣶⣄⠀⠀⠀⠀⠐⠠⠀⠂⠈
⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⠉⠁⠀⠀⢄⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣝⢻⣆⡀⠀⠀⠀⠀⡤⢴⣶⣶⡤⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⠀⠻⢿⣌⣉⣼⣿⣿⣷⣆⠀⢰⣿⣿⣷⣿⣿⣷
⠀⠀⠀⠀⠀⢹⣿⣏⣿⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡆⠀⠘⣿⡿⢋⣥⣾⣿⡿⠀⠈⢿⣿⣿⣿⣿⠟
⠀⠀⢠⣴⣶⣾⡿⢻⣿⣶⣶⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠣⣄⡤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠿⠿⠿⠿⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣋⣡⣶⣷⠾⠻⣾⠿⣦⣤⣤⣌⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⣿⣋⣁⣈⣛⡋⠉⠻⣦⣤⣤⣾⠋⠈⠉⠻⢿⣦⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣶⠿⠛⠛⠉⠉⠙⠛⠻⠷⣾⣥⡀⠹⣷⣴⣶⣴⠾⠻⠿⣶⡌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠁⠈⠁⠀⠀⠈⢁⣀⣀⣀⣂⡀⠉⠛⢷⣦⡀⠈⢿⣤⣄⠀⠘⠿⣦⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⣀⣤⣴⠾⡟⢛⣛⠛⣛⠛⠛⠿⢶⣤⣈⠙⢷⣦⣉⠹⣷⣀⠀⠛⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢀⣣⣴⡿⢿⣭⣴⣿⣿⠛⠛⠿⢛⣷⢶⣶⣦⣈⠛⢷⣦⡈⠛⢷⣿⡛⢿⣄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢻⣿⣥⣤⣤⣭⣥⣀⡙⢷⣶⠶⣿⡁⠀⠀⠙⢻⣷⣀⠙⢿⣦⢀⡍⠻⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⠿⠛⠉⠉⠀⠀⠈⠉⠉⠛⠷⣦⣄⡘⠿⠶⢷⣾⠟⠛⠛⢿⣆⠙⣷⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠋⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⡀⠻⣶⣤⡀⠈⠛⣷⣘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡘⣿⣄⡀⠉⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠉⠻⣾⣍⣻⣧⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣴⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠙⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠆⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠶⠂⠀⠀⠀⢀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⡶⠟⠋⠉









⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⣿⣿⣿⢿⣿⠿⢻⠿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠙⠶⠦⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⢀⣰⡩⠂⠀⠀⠀⠀⢠⣞⣠⣴⡂⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢧⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⣇⡀⠀⠀⠐⠒⣞⠉⠁⠀⠀⠀⠀⢀⣠⠤⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⢹⠁⠀⠀⠀⠊⢈⢦⡀⠀⠀⣠⠖⢋⠀⠀⠀⣂⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠸⣄⠀⠀⠀⠀⠁⣠⣷⣄⣼⣅⠀⠀⠶⠀⠀⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠈⣷⣶⣦⣴⣶⣿⣿⣿⠟⠿⠂⠀⠀⠀⠄⠚⠀⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣌⠓⠀⠀⠸⢆⡀⠨⠄⠺⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣛⣛⣟⣿⣛⣿⣿⣛⣦⣤⣤⡀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠀⣾⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠾⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠟
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠙⠲⠶⠶⠚⣹⣤⠼⠷⠛⢻⠈⠉⠙⠛⠿⣿⣿⣿⣶⣶⣶⡀⠐⠶⠶⢶⡶⠶⠒⠒⢒⡶⠒⠊
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠐⢧⣀⡀⠀⠀⣠⠶⢿⣁⠤⠀⣠⠟⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠠⣴⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠉⠉⢹⠉⠶⣄⣹⠆⠉⣀⠴⠊⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⡏⠛⠷⣦⣤⣤⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠑⠒⠲⢶⠶⣶⠺⡘⢆⢸⢹⠀⠀⠈⠢⡀⠀⠀⠀⣈⡆⠀⠀⢱⡀⠙⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢰⠙⣆⠰⡑⢷⡀⡸⣺⠀⠀⠀⢀⡜⠀⠀⠀⠘⢷⡆⠀⠀⠇⢠⡏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡄⠀⡜⠠⡿⢤⣀⠈⠛⣥⠇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠸⣿⣿⣿⣿⡿⢁⡴⠀⠀⠀⠈⡿⠚⠧⠰⠧⡀⠈⠉⠒⣾⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⢿⣿⣿⡿⣡⠏⢀⣤⠤⢶⡞⠉⠀⠀⠦⢀⢹⠀⠀⠀⣿⠊⣀⡤⠤⠀⠀⠀⣤⣄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏
⠀⠈⢿⣿⣷⠃⠀⠁⠀⣠⠾⠀⠁⠀⠠⢀⢈⡇⠀⠀⠀⡸⢿⡁⠰⡶⠀⠀⠀⠈⠡⠀⠀⠀⠀⢹⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠈⢿⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠢⢤⠟⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠘⡇⠀⠢⣼⣷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣴⡄⠀⠀⠀⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠱⣄⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣶⡆⠀⠀⠈⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣿⣷⣶⣶⣶⣾⣶⣶⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠹⡄⢸⡆⠀⠀⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡆⠀⡇⠘⡇⠀⠀⡇⡇⡠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣏⡛⠒⠲⠶⣻⠄⡟⠉⡇⠹⠛⠓⠶⢁⡏⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠁⠉⠢⣁⢿⡰⡡⢠⠀⡦⠀⠈⣡⠾⡛⠍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡼⡁⠀⠀⢠⠜⠊⠉⠉⠈⠉⠛⠻⠚⠉⠉⠉⠉⠛⢣⢤⠀⠘⢛⢦
⠀⠀⠀⠀⠀⠀⣼⠓⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⡀⠪⣮⣧
⠀⠀⠀⠀⠀⢀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠀⠂⡀⡔⢔⢅⠀⣹⣿⡇
⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⠀⠀⠀⠀⠺⠱⠰⢈⣎⠃⢿⣿⣷⠀⠀⢀⣀⣠⠤⢦
⠀⠀⠀⠀⠀⢸⡏⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣠⣀⡤⠼⠶⠶⠴⠶⠗⠒⠚⠛⠛⠋⠉⠉⠐⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠈⣇⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⣼⣿⠋⠁⠀⠀⠀⠒⠒⠀⠄⠀⠀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠹⣤⠀⠘⠀⠀⢀⣤⠀⠸⠀⢰⠏⢀⡀⠠⠀⠐⠒⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣶⡷⠄⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠐⢄⠃⡜⣿⡀⡶⠄⣿⠀⠠⠰⡀⠀⢀⡀⣟⢛⡬⠇⣴⢦⡄⣤⢄⠠⡰⣶⣦⠏⢰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡉⠂⢷⢟⢽⡇⠂⣿⠀⠀⠣⠳⣄⠀⠙⢮⡚⠲⠊⠊⢸⡁⣿⣾⠘⣷⢘⡷⢠⠊⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⣘⠒⠁⢈⡾⣇⠀⢑⠵⡜⢮⡓⢀⠙⠲⢦⣜⣁⠙⣉⣠⣛⣉⠥⠚⠁⠰⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣬⣧⣀⢈⠚⣷⣿⣦⣀⠰⢦⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢭⣛⣿⣿⣿⣽⡶⣤⠶⠀⠴⠦⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠛⠛⠚⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣀⣀⡀⣀⣤⠤⠤⠤⠶⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠉⠀⠤⠠⠤⠤⢀⠀⡀⠤⠐⠒⠒⠀⠐⠂⠷⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠊⠁⠀⡸⠀⣀⣠⡎⣁⣀⠀⣀⡀⡠⠋⠀⣀⠤⠒⡒⠶⠶⢦⣄⠀⠙⢕⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡡⠔⠋⠀⠂⢛⠣⠚⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⢤⡖⠀⠒⠾⠀⠀⠀⠀⠈⠥⣤⣀⣳⠉⠉⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⡨⣠⠔⠊⢁⠞⠁⠀⡤⣖⣉⡍⠄⠠⡜⠋⠐⠂⠔⠊⠀⢀⣠⣀⠀⠄⠄⠤⠤⠤⠈⣁⣀⡀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢀⠔⡅⠂⠁⠘⠡⡡⠀⢀⠂⠀⠠⠺⣜⠁⠀⠁⠲⢞⠀⠀⢀⣠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠐⠠⡄⠘⢣⡀
⠀⠀⠀⠀⠀⠀⢠⠋⢸⠀⠀⠀⣠⠐⣇⠀⠆⠀⠀⠛⣆⠉⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣱
⠀⠀⠀⠀⠀⢀⠃⢀⡸⢄⡀⠴⠃⠀⢛⢸⠀⠀⠀⡎⠈⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⢀⠆⠀⡙⠀⡎⠀⠀⠀⠀⠈⠾⡄⠀⠀⡀⠀⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⡜⠀⣺⠄⠒⠛⠒⠒⠀⠀⢠⣀⠇⣀⡧⠂⢀⠎⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠱
⠀⠀⠀⢰⠁⠀⠃⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⠔⠁⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⡎⠀⠀⡄⠀⠀⠀⠀⡠⠤⠷⠀⠤⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⢠⡔⠁⠀⠀⠀⠀⡇⠀⠀⠀⣀⠞⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢱⠀⢆⠀⠀⠀⠀⢠⠄⠀⣀⠞⠁⢀⡴⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠘⣆⠈⠳⢄⣀⣤⡩⠴⠛⠁⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡼⠁
⠀⠀⠀⠀⢸⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠳⣄⠁⠉⠉⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠴⠖⠒⠒⠒⠚⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠖⠒⠒⠊⠋⠉






⣿⣿⡿⠿⣿⡿⢿⠿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠙⠓⠲⠦⠤⠴⠞⠋⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⢶⡶⠶⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣤⡖⠋⠉⠙⡇⠀⠀⠀⠘⡇⠀⠀⠀⢸⠋⠉⠙⢲⣤⣀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣴⠋⠁⠀⣇⠀⠀⠀⣧⣀⣀⣀⣴⣧⣀⣀⣀⣼⠀⠀⠀⡸⠀⠈⠙⣦⡀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⢀⣴⠋⠈⡄⠀⣠⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣧⣀⠀⢠⠁⠙⣦⡀⠀⠹⡄
⠀⠀⠀⠀⡾⠀⢠⢏⠀⢆⣠⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣷⣄⡰⠀⡿⡄⠀⢷
⠀⠀⠀⠀⡇⢠⣏⢦⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣄⣹⡄⢸
⠀⠀⠀⠀⣇⠈⠻⣿⠻⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⡟⣿⠟⠀⣸
⠀⠀⠀⠀⢹⡀⠀⠈⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠛⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠳⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠦⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀
⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⣀⣠⡾⠋⠥⣯⣜⣿⣿⣿⣿⣿⣿⣓⣽⠡⠑⠢⣦⣄⡀
⠀⠀⠠⠀⠠⠄⠀⢀⡴⠿⡍⠀⢱⣿⡆⠩⢱⣾⣻⣛⣛⣏⡿⡎⠥⠱⣿⠂⠀⢩⡿⠲⣄
⠀⠀⠠⠀⠰⢆⡴⣫⠤⣞⡁⠤⠜⠛⠀⠀⢈⣐⠋⠏⠹⠏⣃⢄⠀⠀⠈⠓⠤⢌⡓⠦⣍⠳⣄
⠀⠀⠠⠐⢰⣯⠞⣡⡀⢀⣥⠀⠀⠀⠀⠀⡳⣿⣧⠀⠀⣼⡾⢛⡀⠀⠀⠀⠀⣬⡅⠀⣤⡳⣬⣆
⠀⠀⠰⠀⣿⣏⡦⡉⣶⡦⣿⢷⡗⠀⠀⠸⣿⣿⣿⡆⢀⣿⣵⡿⠟⠀⠀⠸⣇⣿⢷⣷⢉⢴⣾⣿
⠀⠀⠐⠀⣿⡏⠐⣻⣼⣿⠿⠋⠀⠀⠀⢰⣒⣬⣻⡁⢈⣝⣣⣲⡶⠀⠀⠀⠙⠺⣼⣿⣾⣂⢸⣿
⠀⠀⠐⠒⣿⣯⣰⡿⠞⠁⠀⠐⢀⡂⠂⠰⢀⢯⣻⣧⣼⣟⡽⢂⠐⠘⣀⣃⡂⠀⠈⠛⢿⣆⣽⣿
⠀⠀⠐⠂⣿⣯⣿⣵⣢⣶⣾⣿⣿⣿⣿⣦⡀⢀⣿⣿⣿⣿⣆⢀⣴⣿⣿⣿⣿⣿⣶⣴⣮⣿⣿⣾
⠀⠀⠘⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⢩⣿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠁⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⣽⣧⣼⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡀⡀
⠀⠀⠈⢁⣿⡧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡂
⠀⠀⠈⢸⣟⡃⠡⡻⣿⣿⣿⣿⣿⠿⣿⡟⣰⢻⣿⣿⣿⣿⣟⣧⢙⣯⡿⢿⣿⣿⣿⣿⡟⡈⠀⣹⣯
⠀⠀⠀⢈⢉⣁⢀⠘⠉⠋⣭⣥⣤⣶⣿⣇⢣⣿⣿⣿⣿⣿⣿⡎⣌⣿⣷⣦⣤⣭⠍⠉⠁⡀⡀⠈⠙
⠀⠀⠀⢸⣿⡁⣨⣤⣀⡀⢸⣶⣷⠱⡿⠀⢸⣿⣿⣿⣿⣿⣿⡇⡀⢸⡞⢸⡷⡖⠀⣔⣤⣦⣀⣾⡟
⠀⠀⠀⠈⢿⣿⣯⣽⣟⣷⣤⣄⡁⣹⠀⠈⣦⣛⢛⣙⢫⡙⢛⣵⠇⠀⣿⣈⣠⣶⣶⣻⣯⣽⣿⠿⠁
⠀⠀⠀⠀⢈⡁⠻⣟⢫⣽⣿⣿⣻⣿⡁⣀⠩⠉⡄⢉⡌⠀⡌⢩⢠⢈⣿⣿⣿⣿⣿⡝⣻⠟⠁
⠀⠀⠀⡀⢀⡁⠀⠈⣎⣿⣿⣿⣿⡽⠷⠣⣃⡾⣠⣘⣣⣤⡱⣈⡎⠳⠳⣿⣿⣿⣿⣧⠁
⠀⠀⢀⠀⢀⡄⠀⢠⡿⣿⣿⣿⠶⡗⡎⠻⠁⠿⠁⠈⠃⠀⢻⠀⠟⢹⢻⢶⣾⣿⢷⡿⡇
⠀⠀⢀⠀⢀⡄⠀⢸⣿⣿⢻⣿⣶⡶⢶⠞⡦⠲⡤⠰⠖⢲⠖⢶⠓⠾⠶⣾⣿⡟⣷⣿⡿
⠀⠀⢀⡀⢠⡄⠀⢨⣾⣿⣼⣯⡻⠿⣼⣄⡴⣤⠧⡰⠧⠼⠦⢾⣠⢧⡿⡟⣽⣧⣿⣿⡇
⠀⠀⢠⠄⢠⡄⠀⠘⣿⢻⣏⢙⡟⠻⠐⡆⠰⡄⢲⠀⠄⡞⢀⠆⠠⠃⠞⠹⡏⡀⣿⣿⠏
⠀⠀⠠⠄⢠⡄⠀⠀⠘⢿⡧⡈⢀⣤⣤⡀⠀⠀⣠⣠⣆⣄⠀⠀⢠⣠⢤⣀⢈⣼⡿⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠄⢠⡄⠀⠀⠀⠀⠙⢮⣻⡶⠲⣜⡀⢠⣌⣷⢾⣧⡤⢀⣣⡞⢳⣜⡵⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠭⠙⠦⠙⠩⠮⠋⠴⠋⠽⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐⣤⣤⠀⡎⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⣹⣷⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⡀⢽⡄⢀⠆⠀⠀⡤⠄⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢀⣂⣫⡾⣶⡋⠀⠀⠁⢠⠀⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⢹⡿⠟⠿⣟⠉⠀⠀⣾⣷⣎⠁⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠈⠂⢠⠀⢈⢹⠧⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⡀⢸⡇⢀⣽⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⣀⣴⠏⠀⢧⡾⠉⡍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⣉⣰⣼⣾⣡⡦⢈⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⣨⠠⠵⢔⣢⡀⠀⠀⠀⠉⠛⠻⢿⣷⡼⠇⠀⠰⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢂⡠⠠⠔⠀⣿⢿⡃⠀⠀⠀⠀⠀⠀⠀⠉⡷⠹⣶⡀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠙⠛⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠞⣉⠻⣟⠲⢶⣒⣲⣦⠄⠀⢀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣤⣿⣿⣭⣭⣝⣛⣳⠮⠤⠬⢭⠷⠞⣛⠉⠉⠛⢦⡇⣸⠗⢦⡀⠀⠀⠀⢀⡀
⠀⠀⠀⢤⢤⣀⣀⣠⡿⢒⡭⢕⣚⡽⢖⣛⡭⣤⢶⡶⠖⣫⣴⠞⢉⣍⡛⣯⣡⡬⢇⡿⢋⣝⣿⣶⣶⠿⡟⣀⡀⣠⡤
⠀⠀⠀⠘⠒⣦⣬⠭⠿⠟⣛⣓⣛⡚⡓⠢⠬⣴⣉⣴⣞⡓⣻⠟⠁⠳⡙⠿⡏⢸⢹⡶⣿⠿⠟⣛⣻⣤⣭⢁⡖⠁
⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⠍⣛⠻⢅⡀⠀⠀⢧⠀⡏⡞⢸⡼⠃⡼⢋⡡⠒⠉⣼⣛⣓⣒⣒⡒⢲⠆
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢍⡓⢦⣼⣷⡿⠁⠛⠳⠦⠿⣟⣲⣿⣿⣿⣿⡷⠼⠬⣍⣁⣀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⠻⣥⡠⠤⠄⠒⠂⢠⣿⣿⣷⣶⣶⣶⣶⣦⢤⡤⡜
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡙⢦⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢣
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⡆
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿
⠀⠈⠓⠲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⠦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⢿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⠢⠻⣿⣿⣿⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⢦⣤⣄⣈⣙⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⠶⢶⣤⣤⣀⣠⣤⣴⣶⠶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠍⠉⠉⠉⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⢶⡿⠛⠻⠶⢶⣦⣤⣤⣤⡶⠶⠶⠶⣶⢶⣤⡀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣁⣐⣀⣀⣤⣤⣴⣶⣿⣯⣄⣀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣯⣀⣈⣉⣳⣦⣤⣀⡀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣠⣿⣤⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⡿⠋⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣿⣿⣿⣿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠋



⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣤⡀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣷⠆⢻⡉⢻⡥⢤⡄
⠀⠀⠀⠀⠀⢀⣀⣠⣿⠂⠚⢹⢈⡟⠚⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣏⣸⡗⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣤⣶⣦⣼⡦
⠀⠀⠀⠀⠐⢉⡏⠉⢻⣰⣾⡏⣢⣽⠆⢰⣋⣭⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣹⣿⡈⢇
⠀⠀⠀⣀⠀⢿⣿⣿⡿⠟⣻⣟⡽⢋⡄⢺⡿⢿⣿⣿⣿⣿⣿⡿⣿⠻⠛⡻⢽⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⡿⠃⡜⠂
⠀⠐⢲⣿⣶⣾⣿⡏⣠⠺⠟⢋⣠⣾⣧⣾⠀⣼⡇⢀⣤⡉⠓⠉⠀⠸⠷⠁⠀⠈⠁⠀⠒⠦⠌⢻⣿⣿⣿⣿⣾⣿⣿⣿⣦⣿⣿⣟⠛
⠀⠊⣋⣿⣿⣿⣿⣿⣿⡶⠾⢋⣵⡏⣩⣿⣄⡟⠠⠐⠂⠀⠀⠠⢀⡐⠖⢀⠄⠀⠀⣀⡀⣆⡐⣀⢻⣿⣿⣿⣿⣿⠙⣿⣿⣿⡟⣼⠇
⠀⠐⠌⢛⡛⣿⣯⣶⣾⣷⣶⠟⣫⡾⣿⡿⣿⢀⣤⣶⣾⣶⣶⣤⣄⠀⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⣌⣿⡿⣿⣿⣿⣇⣩⣴⣿⡅
⠀⠤⠴⠚⣋⣽⣼⠟⠑⣷⣦⣶⣿⡀⠈⣿⢻⣿⣿⣿⣛⣿⣿⣿⣿⠀⢀⠰⠸⣿⣟⠛⠛⠛⢿⣿⡿⢸⣧⣿⣯⣿⢻⡿⣿⣿⠇
⠀⠀⠑⠛⠋⡿⠁⣤⡾⢛⣥⣋⣵⣿⣆⣸⢮⣽⣿⣥⣤⣴⣿⣿⠇⣠⣼⣤⡀⠻⠿⣿⣿⣿⣿⣿⢇⠸⣿⣿⣿⡟⣹⣿⡎⢿
⠀⠀⠀⠀⢀⣇⡠⢥⣾⣿⣿⣿⣿⣿⣿⣿⡙⠿⠿⠿⡿⠉⠀⠀⣸⣿⣿⣿⣿⣦⡀⠀⠀⢙⡯⠁⠀⢰⣿⣿⣿⡇⣽⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠳⣌⠁⠚⠁⢠⡄⠘⠛⠋⢻⠉⠙⠋⠁⠀⠴⠀⠈⠁⠠⠞⢿⡿⢿⣿⣿⣿⣧⡈
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⢿⣿⠀⠘⠂⠁⠀⠀⠁⢠⠐⢃⣉⣡⣬⣟⠦⡀⠀⠀⠀⠀⠀⠀⣾⣇⡼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⡾⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⢀⣠⣮⣶⠛⠂⠋⠘⣛⣷⠿⠲⠂⠤⡀⠀⠀⣿⣿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣴⢟⣵⣿⣷⣾⣧⡿⢿⠿⣷⠀⡑⠒⠺⠉⠁⠀⠘⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⡰⢰⣿⡟⠿⣿⣿⣇⢸⡀⣠⣀
⠀⠀⠀⢀⣴⣛⣵⣿⣿⣿⣿⣿⣿⣜⢢⣶⣿⣧⡑⠦⢤⡤⢄⡀⡀⣀⣶⠶⢮⣵⣀⠀⣠⡒⠰⠊⢁⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⡄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣾⠟⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣀⣈⣉⣯⣽⣴⣿⣷⣾⣯⣄⣀⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⡟⢏
⠀⠀⠀⠀⣿⢀⣾⣿⡟⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⢰⠏⣼⣿⣿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⠂⢠⡔⣩⣾⣿⡈⣿⣿⣿⣧⠀⢸⣿⣦⢦⣄
⠀⠀⠀⢼⣄⣿⣿⡏⢷⣄⡘⣧⣻⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⢿⠃⠀⠀⠈⢠⡏⢠⣻⠞⣽⢿⣿⣷⣿⣿⣿⣿⡤⢸⣿⣿⣦⣈⣙⣤
⠀⠀⠀⠸⣯⣟⣿⣿⣼⣿⣿⣼⣿⣿⣿⣿⣿⣛⣻⠿⣿⡀⢿⣿⣿⣿⠃⠀⠀⠀⠔⠉⠀⢸⡏⢸⣧⡾⢿⣿⣿⣿⣿⣿⡵⢸⣿⣿⣿⣿⡿⣙
⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢌⣷⡈⢿⡿⡂⠀⠀⠀⠀⠊⠁⠀⠘⣇⢈⣿⡀⣿⣿⣿⣿⣋⣼⢧⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣦⣌⣸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣷⣦⣤⣿⣿⠟⢣⣾⣿⣿⣿⣿⡸⣿⠁
⠀⠀⠀⣾⢩⣤⣾⣿⣿⣿⣝⣻⣿⣿⣿⣿⣟⣿⣿⣿⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⡘⠁⢈⣍⣻⣿⣭⣤⣤⣤⡄⠈⣿⣿⣿⣷⡇
⠀⠀⠀⢿⣮⣻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣮⣁⣾⡿⠻⠆⡇⠀⠀⠀⠀⠀⣼⠇⠀⠀⠁⠀⠈⠛⢿⣯⣛⠿⣿⣿⡏⠀⢻⣿⣿⡿⠃
⠀⠀⠀⠈⢻⣿⣷⣬⢉⣿⣯⡙⢿⣿⣿⣿⣿⣿⡿⣿⣻⣏⣙⣚⠃⠀⠀⢀⣀⣼⡿⠀⠀⡶⠆⢀⡄⠀⠀⣀⣙⣷⣝⢿⣇⠀⣈⣭⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢻⣿⣆⠹⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⣆⠉⢛⠿⣿⡷⠀⣠⠶⠛⢉⣶⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣽⠃
⠀⠀⠀⠀⠘⢿⣿⡌⢻⣷⠀⠻⣿⣶⡟⡛⠻⣿⣿⣿⠿⠟⠋⣠⣶⣿⠀⠈⠳⠶⠟⠠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣙⠻⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⢻⡇⠀⠹⠀⠀⠉⠹⠿⢿⣷⣿⣿⣿⣄⣴⣿⣿⢧⡟⠀⠈⢁⣄⠀⠀⠘⠛⢱⣿⣿⣿⣿⣿⣿⣿⠟⠛⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡾⣻⣶⣭⣿⣿⣿⣿⣿⣿⣯⡞⠀⠀⠀⠨⠵⠂⣠⢰⣦⣼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣾⣿⣿⣿⣿⣿⣹⣿⡿⢿⣧⣄⣱⣾⣙⣲⣞⣙⣿⡶⠋⠉⢿⡻⣿⣿⣿⣿⡻⣦⣸⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠾⠿⠟⠃⠀⠀⠀⠉⠉⠈⠙⠋⠉⠉⠉⠁⠀⠀⠀⠀⠉⠈⠉⢿⣯⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⡀⠀⠘⣧⠀⠀⠻⠀⠀⠓⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡾⠃⠀⠀⣰⠗
⠀⠀⠀⠀⠈⢻⡄⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⠓⢄⠀⠀⢀⠞⠙⣆⠀⠀⢀⡠⠞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠞⠁⠀⢀⡶⠄
⠀⠀⢠⣄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠑⠦⣄⢠⠁⠀⠀⠑⠶⠁⠀⠀⠘⠦⠐⠁⠀⠀⠸⡆⠀⢀⣠⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⡀
⠀⠀⠀⠙⠣⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠤⠄⠐⠂⢠⠀⠀⠀⠷⠒⠉⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠽⠓⠋⠉⠉⢆⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠑⠢⣤⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⢀⣠⠤⠖⠒⠙⢦⢀⡇⠏⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢠⠃⠀⠀⠀⢀⡤⠒⠒⠎⠀⣠⡾⠁⠀⠀⠀⡀⠀⠀⠀⠈⢻⠁⠀⠀⢠⡞⠙⠲⢤⡀⠀⢀
⢀⠠⠐⠈⠁⠀⠀⠀⠀⠀⠘⡌⠃⠀⠀⠀⠀⢸⣇⠀⠀⠘⡆⠀⠀⡎⠀⠀⠀⢠⠇⡀⠀⡸⠀⡌⣽⠁⠀⠀⢠⠞⠉⠑⡆⠀⠀⠀⢧⢀⡴⡟⠀⠀⠀⠀⢨⠃⠉
⠘⡄⠀⠀⠀⠀⠀⠀⢀⡤⠔⠳⣸⢴⠀⠀⠀⠘⠛⠀⠀⠀⠘⡄⢘⣧⠀⠀⠀⠈⠯⣁⣀⣁⡜⠁⡎⠀⠀⠀⡏⠘⢢⠀⣸⠀⠀⠀⣸⠋⡼⠁⠀⠀⠀⡰⠃
⠀⢹⡀⣀⠤⠄⠀⠀⠈⡷⢄⡀⠜⢻⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⣼⠘⢆⠀⠀⠀⠀⠀⠀⠸⡇⠀⣥⠀⠀⠀⢇⠀⣰⡧⠃⠀⠀⢀⡟⡸⠁⠀⠀⠀⡰⠁
⠀⠀⠙⠠⢤⢸⡀⠀⠀⠸⡀⢅⠀⠸⡀⠀⠀⠀⣻⠈⢧⠀⠀⠀⠙⣄⠈⠳⣤⣀⠀⠀⠀⢀⡇⠀⠘⣆⠀⠀⠀⠉⠁⠀⠀⠀⢀⡞⣱⣃⠀⠀⢀⣜⡀
⠀⠀⠀⠀⣀⡠⢵⠀⠀⠀⢳⠘⡀⠀⡂⠀⠀⠀⣙⢣⣸⡧⠐⠂⠚⣻⡀⠀⠀⢉⠙⢙⠉⢩⢣⡀⠀⠈⠳⢄⣀⣀⣀⣀⣤⡶⢻⣽⠤⠬⣉⠗⠁⢈⠝
⠀⠐⢎⡉⠀⠀⠈⡆⠀⠀⠈⢧⢁⠀⠧⠄⠒⠊⢡⣠⣀⣻⣤⢤⣤⢟⠿⢟⣻⣿⡎⢠⢽⣿⠶⠗⠦⣤⠀⠀⠀⠀⣁⠕⠈⢠⠛⢇⠀⠀⢈⣦⠞⠁
⠀⠀⠀⠈⠲⣄⠀⢹⣤⠴⠒⠉⠀⡦⠈⢢⣴⠀⣾⢿⣋⠷⠀⢵⠖⠋⢲⡈⡠⢾⠿⠭⣦⣽⡿⠿⠚⣍⢛⠳⣬⡉⠀⠀⠀⠺⣀⠈⢙⠟⠉⠙⠒⠦⣄
⠀⠀⠀⠀⣀⠜⠁⠀⠀⠉⠐⠀⠡⢒⣲⣥⣶⡿⠿⠿⠛⠛⠛⠛⠻⠛⢺⠿⣤⣉⡀⢾⠧⠉⠾⢟⡛⢙⠷⢶⣤⠛⢶⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣈⠽⠃
⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⢀⣦⡿⠟⠉⠈⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠈⠐⠨⠛⠶⣤⠐⠂⠚⣧⠲⠞⡅⢠⡃⡀⢿⡗⢦⡀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠁⠙⠒⠢⠄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡔⠈⡻⢦⡃⢤⣦⣤⣧⣄⢹⡄⢘⣿⡲⡙⣆⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⡠⠂⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠦⡙⢧⡀⢹⣤⡈⣯⠛⢣⡂⢄⡼⢮⣷⡄⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢠⠞⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⡢⠹⣧⡀⢈⣧⡤⣮⣽⣀⣘⣂⣷⡹⣄⣠⠴⠚⠁
⠀⠀⠀⠉⠐⠒⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠑⢌⢻⣍⠏⢀⣀⠹⣋⡷⠛⡏⢳⢹⣦⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠹⣧⡨⠥⢖⢊⣃⣴⠲⡿⣇⢻⡝⠦
⠀⠀⠀⠀⠀⡟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⣧⡃⢨⠏⢛⡠⠀⠸⢿⢸⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠱⡘⣿⣄⡀⢡⡀⠄⣀⣿⢸⡇
⠀⠘⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢌⡻⣤⣤⣰⣼⠿⢣⣾⠁
⠀⠰⠖⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠙⠛⠛⠓⠒⠒⠲⠶⠶⠶⠶⠴⠤⢤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣈⣐⣪⣭⣴⡾⠟⠁⠀⠀⠀⠀⠀⠙⠶
⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠤⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠉⣹⠉⠉⠙⠛⠻⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀
⠀⠈⠋⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠓⠈⠀⠀⠀⢧⠀⠀⢀⡴⠂⡀⠀⠀⠀⡴⢄⠀⠀⠀⠀⡌⠑⢄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠛
⠀⠀⠀⢴⠟⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠔⠉⠀⠀⠑⡄⢀⠞⠀⠀⠑⢄⠀⡸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢻⣄
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⣴⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢰⡄⠀⢱⡄⠀⠸⣧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣷⣾⣷⣶⣾⣷⢷⣿⣶⣿⣾⣷⣷⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡷⡷⢾⡷⠾⢶⣶⠶⠿⠶⠾⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⣙⠻⢦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣶⣟⣉⣈⣭⣿⣿⣿⣿⣯⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠉⢭⣽⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠐⣺⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠁⠀⠀⢤⣼⣿⣿⣿⣿⡭⠪⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⢒⠺⢿⣿⣿⡋⠐⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⡀⣸⣿⣿⠿⡉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⣙⣿⡿⠓⠚⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣂⣈⡉⠉⠁⠀⠀⢴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣿⡿⠇⣠⠄⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⢀⣤⡾⠃⠀⣠⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣀⣤⣶⡿⠟⣡⣶⣿⣿⠏⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⣾⡿⠛⢉⣐⣿⣿⣿⠿⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠉⠀⠀⢀⣤⠞⠋⠀⣀⣼⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠖⠒⠋⠉⠀⠀⠀⢀⣠⣤⡤⠈⠉⠀⠠⢀⣨⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣤⣾⡿⠋⠉⣀⣠⣤⣶⣶⣾⣿⣿⣿⣟⢛⣋⣀⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠰⣿⣏⠠⠚⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠀⡰⠚⠉⠉⠩⢭⣉⡙⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠁⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢰⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠑⠆⠀⠀⠰⠚⠉⠉⠁⠀⠀⠀⠀⠀⡎⠀⠠⠴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⢰⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣇⠀⠐⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠶⠒⠛⠛⠉⠙⠛⠛⢻⡇⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁⢀⡤⠤⠀⠀⠠⠖⠲⠄⠘⡇⠀⠀⠀⠘⢦⠀⠀⠀⢰⠑⢮⠱⢄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⠀⠈⠁⣠⣤⣤⣷⡌⡦⢀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡄⢹⡀⣠⣴⣾⣿⣶⣿⣿⣿⣶⣾⣧⣴⡗⠀⠀⠀⣴⣋⣀⣄⠈⢡⠃⢰⠟⠀⣱
⠀⠀⠀⠀⠀⢰⠀⠀⠀⣾⠀⠀⠘⢷⣾⣿⣿⡿⠟⢉⣉⣉⠉⠙⠛⣿⠛⠋⠀⠀⠀⢰⠛⡇⠀⢸⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠁⠀⣀⡼⠃⠠⡄⠀⠀⠀⢸⡀⠙⠤⠎⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⡆⠀⠀⠀⠀⡎⠀⠈⠙⠓⠶⠚⠛⠉⣀⠀⢀⡇⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳⡀⠀⠀⠀⣹⠖⢋⣩⣤⣷⠀⢸⣧⠤⠈⠻⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠱⡄⠀⢰⠇⠀⠈⣈⠅⠛⠒⠋⠑⠢⢄⣀⣈⡷⢰⠚⢲⣀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠈⠢⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡶⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣄⣀⣀⣀⣀⣀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣻⣉⣟⣻⣋⣿⣫⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠸⣇⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢠⡶⠀⠀⠀⣠
⠀⠀⠀⠀⢾⣄⠀⠈⢷⡀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⢶⡀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠞⠁⠀⢀⡾⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠂⢄⠀⠀⠀⠎⠀⠑⢄⣀⡔⠁⠀⢳⡀⢀⠴⠊⠁⢰⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⣴⠟⠁
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠉⠢⠜⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣇⢀⡤⠚⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⡤
⠀⢀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢐⡗⠒⠢⢤⣀⡧⡤⢤⡀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠉⠉⠉⢸⠀⠀⠀⠈⠁⣀⢠⡤⠧⢤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠈⠓⠀⠀⠀⠀⢀⣀⣤⡀⠀⠁⡔⠊⠉⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣏⡇⠀⠀⠀⢀⢀⣀⠀⠀⠀⠚
⠀⠀⢀⡠⠴⠒⠉⠉⠀⠀⢧⢠⡄⡇⠀⠀⠀⢠⡀⠀⠀⢣⠀⠀⠀⣼⠀⠀⠀⠀⡴⠛⡉⢉⠁⢀⢼⡏⠀⠀⠀⣠⠤⢄⡀⠀⠀⠈⢇⠀⠀⣰⠏⠀⠀⠉⠲⣄⠠
⢖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢅⠁⠀⠀⠀⢸⣧⠀⠀⠀⢧⠀⠀⡇⠀⠀⠀⠀⣇⠧⠤⠞⢀⡎⣾⠀⠀⠀⣰⢃⠀⠀⢱⠀⠀⠀⢸⣤⢞⡟⠀⠀⠀⠀⣠⠃
⠘⡄⠀⠀⠀⠀⠀⠀⢰⡃⠁⠘⢴⣰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢰⢿⡀⠀⠀⠀⠑⠒⠒⢲⡎⠀⢿⠀⠀⠀⠂⠀⠘⡆⡸⠀⠀⠀⢸⠁⡼⠀⠀⠀⠀⡰⠁
⠀⠻⡤⠖⠚⡀⠀⠀⠈⣯⠢⠄⠊⢸⠀⠀⠀⠀⣤⢶⡄⠀⠀⠈⢾⠀⠳⣄⠀⠀⠀⠀⠀⠈⡇⠀⢹⡀⠀⠀⠰⠄⠴⠟⠁⠀⠀⢀⡏⡼⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠈⠓⠸⡀⠀⠀⠘⡄⢃⠀⢸⠀⠀⠀⠀⢫⠂⢻⡀⠀⠀⣈⣧⠀⠈⠓⠦⢤⣀⣀⣠⡗⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⣠⣟⠔⠣⠤⣀⡠⠾⢤⡀
⠀⢀⣀⡤⠤⠒⢣⠀⠀⠀⢳⠸⡀⠈⡄⠀⣀⣠⡤⠳⠘⣅⠉⠁⠀⣼⣤⣠⣤⡰⠀⡠⢄⣰⡁⣄⠀⠀⠈⠒⠦⠤⠤⠴⣒⠟⢹⡾⠚⠓⠲⡌⢀⣠⠊
⠀⠈⠓⢄⠀⠀⠈⡆⠀⠀⢀⣷⢠⠀⠣⠋⠁⠀⢠⣼⣤⡼⠶⠞⠻⣯⡋⠱⠞⢛⡏⠹⣛⣿⣛⣉⡉⢶⣇⣀⠀⠠⠔⠊⠁⢠⠇⠳⢄⣀⣠⣿⡋
⠀⠀⠀⠀⠑⢦⡀⠸⠖⢊⡁⢀⠌⠃⢀⡵⠺⣬⣟⣻⣬⣩⣥⣹⣃⣀⣸⣄⠔⣹⠛⢛⡿⢶⣞⣻⣭⣦⣄⠈⠒⢦⡀⠀⠀⠘⠢⠠⠤⠊⠀⠀⠈⠙⠲⣤⣀
⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⣀⣼⣭⣷⠞⠟⠛⠋⠛⠉⡉⢉⠉⠉⠉⠹⣟⠳⢦⣌⡙⠛⢒⡋⢻⡌⢈⢟⣿⠿⣍⠹⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠰⢚⣁⠀⠀⠀⠀⠀⠀⣸⡾⠛⠋⠀⠄⠀⠀⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⠄⠁⠪⠙⢶⣍⠁⡈⠷⡙⢫⡵⠀⢧⡤⠘⣿⡻⣆⠀⠀⠀⠀⢠⡞⠋
⠀⠀⠀⠀⠉⠒⡂⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠎⢰⢌⠳⣄⡐⠴⣾⠷⢶⣤⣧⡀⠿⢝⢮⡳⣄⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⣠⠎⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠑⢈⠳⣆⡠⠖⢦⡰⠋⠑⣦⠠⢦⡙⣟⣦⠀⠀⠀⠀⣈⠦
⠀⠀⠀⡔⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⢈⢳⣄⣠⣣⠖⠿⢼⣤⣬⢇⣸⣎⢷⡤⠒⠉
⠀⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢⡙⣧⡟⢀⡤⠜⢣⡿⠉⠳⢸⡎⣷⣄
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⡌⢿⡄⢖⢈⣅⣧⡼⠙⢟⣻⡘⡧⠓
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡌⢿⣝⠀⠊⢩⡄⠀⠀⢻⡇⣿
⠀⣀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⢄⢻⣇⠄⠐⠄⠀⣤⣿⢃⡟
⠀⠉⠀⠀⠀⠈⠛⠲⢦⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢝⠳⢤⣮⣷⠟⣡⣾⠃⠀⠀⠀⠀⢀
⠀⠚⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⢹⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠶⠶⠶⠶⠶⠶⠤⠤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣉⣺⣦⡶⠾⠛⠁⠀⠀⠀⢀⠀⠈⠛
⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡠⠔⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⠁⠀⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡀
⠀⠈⠁⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢷⠀⠀⣠⠔⠐⡄⠀⠀⢀⡴⠢⡀⠀⠀⠀⡎⠉⠢⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠁
⠀⠀⠀⠾⠋⠀⢀⡴⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠚⠁⠀⠀⠘⣄⣠⠎⠀⠀⠈⢦⡀⡸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠘⣧⡀
⠀⠀⠀⠀⠀⢰⡟⠁⠀⢀⣾⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡦⠀⠘⣆⠀⠈⢿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣾⣿⡿⠿⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠇⠀⠀⠀⢀⣠⣴⣶⣶⣿⣯⢉⣾⣿⣿⣿⣿⠶⠌⣤⠁⠴⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣶⠏⠈⠙⣻⣿⢶⡷⠹⣿⠿⠻⠛⠉⠀⢰⡤⣴⠀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣾⣷⣖⠆⠀⠔⠋⠄⠘⠛⠄⠈⠀⠀⠀⠀⠀⠀⠰⠿⣶⣄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡍⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣠⣼⡟⠙⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢹⢿⠀⢀⣠⠀⠀⡀⠠⠀⡀⢰⣄⠀⠀⣤⣄⠀⢿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠾⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣷⠀⠙⠋⠀⠈⠉⠀⠀⠈⠹⠿⠿⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⣤⣠⣴⡾⣷⣦⠀⠀⣿⠀⣰⣶⡶⣶⣄⡀⠀⠀⠀⠀⠐⡇⠀⢈⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⣠⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣼⣿⣷⣿⣿⡥⢉⣤⠀⠀⣿⠇⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢷⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⡀⠀⢸⣷⢼⣷⣿⣾⣿⣿⠄⣤⡟⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠻⠿⣿⣿⣿⠏⠀⣸⣿⡄⠀⢳⣿⣩⣿⠿⠛⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠇⠁⠀⠀⡄⠀⠀⠈⠛⠀⠘⣿⣿⡟⡆⠘⠋⠁⠀⢀⡄⠄⠐⢿⣶⣶⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣶⠠⠀⠀⠀⠀⠀⢠⠀⢸⡐⣿⣿⡿⣸⠀⢠⡄⢂⠠⠤⠂⡌⢸⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣼⢷⣶⣥⣈⣉⠀⠃⠀⠀⠛⠋⡈⠛⠋⠀⠈⠛⢈⣋⣠⣵⣶⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠆⡛⠘⣏⡛⣽⡇⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⢷⣻⣿⡗⢿⣟⣹⡿⡀⠻⠟⠓⠚⠆⠒⠉⠛⠳⠒⠞⠻⠿⣿⠃⠀⠿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⢴⣿⡶⠀⠘⣡⠀⠀⠃⠀⠀⠀⠤⠤⡧⠤⢤⠓⠗⣺⡇⠂⢰⣄⣄⣉⣀⠉⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡋⠀⠉⢁⣀⠈⠛⠀⠀⣀⠁⠒⠲⢶⢷⣶⢷⡾⡶⠟⢫⡀⠂⣸⣿⣿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠉⠁⠀⠋⠠⠸⡄⠀⠀⠀⠐⠗⠀⣠⣦⡠⣜⣷⡌⠁⠀⣰⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢃⠴⠄⣀⣶⣄⠀⠀⠀⢼⣿⡿⠿⠀⠀⠀⣀⣴⡿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠉⠈⠙⠷⣶⣤⣿⣷⣣⣴⣴⣶⣿⣿⣄⠙⢿⣾⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢠⠀⣿⣛⠘⠻⣻⣾⡿⣿⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⡾⢸⣿⠇⠀⠀⠈⢻⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⣤⣠⣿⣄⠀⠀⢠⠒⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣁⣈⡃⡰⠈⠒⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠉⠀⠀⣀⣤⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣶⣶⡆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⠒⢲⢀⡆
⠀⠀⠀⠀⠀⣀⣤⣾⣿⣶⣿⣿⣷⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢯⢳⣺⠀⢸⢊⠧⣀⣀⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⠶⣖⣚⡀⢸⢀⡇⠮⢔⠽⠛⢦⡀
⠀⠀⣰⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠎⡀⠀⠀⠀⣀⢎⠤⡉⣄⠐⠥⣀⣀⡀⢳⡀
⠀⢀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡸⢰⠁⠀⠐⠛⠛⠁⠀⣷⠀⠉⠑⠒⠋⠁⠀⣇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢳⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠉⠙⠒⠤⣀⠀⢀⡠⠔⠒⠉⠉⠑⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀
⣠⣖⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠔⠒⠉⠀⢸⠀⠉⠑⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣈⣑⣦
⠘⣟⢿⣿⣿⣿⣿⣭⣭⣉⣹⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣸⣭⣭⣽⣿⣿⣿⣿⠟⣡⠋
⠀⠈⢆⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣶⣤⣤⣤⣤⣼⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡴⠁
⠀⠀⠈⢆⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⠞
⠀⠀⠀⠈⢧⠀⠀⠈⠻⣿⣿⣿⠿⠛⠋⠉⠉⠙⠻⣿⠿⠛⠋⠉⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⡿⠋⠀⠀⣠⠋
⠀⠀⠀⠀⠈⢣⠀⠀⠀⢈⠟⠁⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣽⠟⠁⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⢣⡀⢠⠏⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⢁⠔⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠳⡎⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡴⠋⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠋⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠔⠚⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠘⠦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣶⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡤⠤⢄⡀⢀⡠⢎⠉⠳⣄⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡴⠁⠈⢳⢤⡤⠃⡠⠴⡌⠭⠀⠈⠂⠀⢈⠭⣓⣂⣚⢳⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠉⠉⢆⠀⠀⢦⠃⠀⠘⡀⣀⣱⠰⡶⠧⠴⢊⡵⠮⢭⢤⣍⣿⢀⠎⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠲⠤⣀⠀⢦⡱⣀⠮⠁⠀⣊⣉⠭⢵⣿⣭⠥⠤⠽⢊⣉⣠⠤⢾⠋⠉⠑⢤⣀⣠⢷⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣇⠀⠀⢲⠤⠞⣓⡩⠭⠭⠝⣒⣦⣀⣹⡀⣁⣠⠤⠔⠚⠛⢛⣛⣛⣛⣛⣒⠲⠦⢄⡀⠀⢃⠄⢀⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠤⠤⢩⠖⣡⣶⠋⣠⡂⠤⠔⡺⠉⢉⡥⠖⠉⢁⣠⠴⠒⠚⠉⠉⠁⠀⠀⠀⠉⠉⠙⠒⠮⣕⡺⡀⠁⠈⠉⡧
⠀⠀⠀⠀⢀⡖⠉⢱⠀⠙⠲⢡⢞⣥⣶⠦⢤⣛⡏⢈⡱⠚⠁⣠⠴⣚⠥⠂⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠲⢍⠻⣦⡀⠘⢑
⠀⠀⠀⠀⠀⡇⢀⡴⣒⣤⢣⣣⠟⠋⠘⡄⠀⢜⡵⠋⢀⡴⢋⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⢷⡔⠋
⠀⠀⠀⣠⠚⠉⡰⡪⠃⣟⢿⣧⠀⠀⡠⣇⡴⠋⢀⠔⢩⠞⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷
⠀⠀⠀⢇⠀⡰⠃⠀⣸⡟⢀⡾⣖⠊⢠⠎⢀⡴⠃⠔⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆
⠀⠀⠀⣸⠙⠁⢅⠮⠊⡠⢪⡄⠀⡴⠃⣠⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢰⠃⢠⡌⡆⠠⡪⡾⠛⠃⡜⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⢀⡏⠀⢾⡀⠀⡇⡇⠳⢄⡞⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤
⠀⡼⠀⠖⠀⢨⠂⢄⠀⠀⡜⢀⡞⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠋⠉⠉⠁
⢠⠇⡆⠠⣀⡴⠒⢲⠁⡼⢀⡞⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠤⢒⣊⠭⠐⣂⡥⠴⠒⠉⠁
⣸⢰⠉⡀⢀⠀⠐⠊⡼⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢒⣊⠥⠖⠊⠉
⡇⠘⢀⠀⠘⢰⠄⡰⠁⣰⠁⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀
⡇⡇⠈⠓⢚⣜⡼⠁⣰⠃⠀⠀⢀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠉⢻⡏⠉⠀⣸⣧⠀⠀⣴⠟⠋⠙⠁⣴⠟⠉⠛⢷⡄⢼⡏⠉⠋
⢻⡿⣌⣩⠥⠊⢀⣜⣡⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⣯⣼⣇⠀⣿⡀⠀⠀⠐⣿⠀⠀⠀⢸⡇⠈⠙⠻⣦
⠀⠙⠒⠶⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠠⠟⠀⠀⠹⠆⠘⠻⠶⠶⠆⠙⠳⠶⠶⠋⠀⠲⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⠀⣐⠀⠀⠀⢀⠀⢠⠂⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠈⠻⣦⠀⣠⡶⠛⠉⠁⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣠⡴⠞⠛⠛⠳⢦⣀⠀⠀⢀⣠⣤⠶⠞⠛⠶⢦⣀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⣀⣴⣿⣏⣀⣠⡄⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⣀⡴⠟⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣠⣶⣄⠀⠀⠀⢀⣴⠶⣤⣀⣹⣆⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠈⠻⣄⠀⢺⣿⣿⣿⣿⣿⠿⠋⠁⠀⠈⠉⢉⣿⠟⠋⠉⠛⠛⠛⠋⠁⠀⠀⠉⠉⠙⠳⢦⣤⡄⠀⠀⠀⣹⡗
⠀⠀⠀⠀⠀⠙⢧⡀⠹⣿⣿⠟⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢷⣤⠟⠁⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣰⠏
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⠏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠋
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣠⡞⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣀⣴⠞⠁
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠚⠉
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⡀⠀⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡷⣯⢉⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⡄⠀⠀⣿⣿⠐⠀⠀⠀⠐⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⡀⠀⢩⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⡇⠂⢸⠀⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠶⠀⣿⣿⣿⡇⣨⣴⣿⣿⠁
⠀⠀⠀⢀⠀⠐⠀⠀⡀⠀⠀⣿⣿⠨⠀⢸⣿⣷⡇⣹⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠃⠀⠀⠠⡄⠀⠀⠀⣿⣿⣨⡀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⡆⠈⣿⣷⣧⢈⡆
⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣿⣿⣿⣨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣇⠠⠐⠀⠀⠀⠐⠠⢄⡀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢾⣿⣿⠟⣻⢶⡟⣙⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⡀⠀⢹⣿⠃⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠃⣾⠏⡧⣷⠱⣎⢓⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⣗⡻⣿⣶⠿⠃⣘⡿⢆⣰⣿⣷⠀⠀⢀⠀⣤⣦⡄⠤⢐⡈⢿⣾⡿⣽⣿⣸⣧⣿⡎⢿⠆
⠀⠀⠀⠀⠀⠀⡰⠉⠽⣟⣧⡙⣮⡻⣿⣿⢾⣿⣿⡿⠎⡙⣿⣿⣿⣷⣮⡃⣿⣿⣿⣿⣟⣿⡈⣿⡿⡿⣛⣻⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢧⣦⣿⢿⡟⣷⡜⣷⡾⠃⢿⣿⣿⣷⡋⠸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢿⣶⢸⣿⣾⣧⠈⡏⠪⡑⠙⣿
⠀⠀⠀⠀⠀⠀⢸⡧⣿⡸⣿⣽⣿⣽⡇⠀⠘⢹⣿⣿⡇⠀⠈⡙⠛⠛⠋⠀⠸⣿⣿⣿⣾⡧⢸⣿⣿⣇⣏⣽⡀⣷⢸⡗⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡹⣿⣿⣿⣿⣿⣏⣈⣉⢉⠀⢠⣤⣤⡀⠀⠀⢀⣀⣀⠙⣿⣿⣿⣾⣿⣿⣿⢿⣿⣷⣿⣿⠟⡮⠞⠥⠂
⠀⠀⠀⠀⠐⢴⣴⣬⣿⣿⣿⢿⣟⠻⣿⣏⣉⣠⣀⠁⢉⣿⢽⣿⣿⣶⣶⣿⣿⣿⣷⣾⣿⣿⣿⢯⠟⠁⠘⠋⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁⠀⠈⠓⠈⠙⠉⠉⠿⠶⣿⣷⣦⣉⣹⣿⡿⢿⣿⣿⠿⠿⡿⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢱⠃⠀⠀⠀⢀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡙⠘⠦⠶⢮⣽⠟⠋⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠙⠳⡦⠤⣴⣾⠉⣠⢰⡦⣼⡧⠤⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠒⣶⣒⣒⡚⠛⠉⠉⠛⠋⠛⢶⣄⡴⣻⡥⠼⠟⡠⠀⠈⠀⠀⠀⣞⢠⢿⠀⡇⠀⠈⠳⢤⣶⠆⠀⠘⣦
⠀⠀⢀⠔⠁⠀⠀⠈⠉⠛⠓⠶⣶⣶⡷⢢⠋⠸⣤⡤⢤⣴⡃⠀⠀⠀⠀⠀⢯⠊⠈⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⢀⡤⠋⢳⠀⠈⠹⠷⣌⡻⡄⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠰⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡤⠚⠉⠀⠀⠘⡆⢰⠀⠀⠘⢧⣻⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⠑⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠈⠀⠀⠁
⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋

⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣠⣀⣀⣄⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⢾⠁⢾⠇⠀⠀⣠⠴⠛⠛⠻⣦⣄⠀⠸⠿⠃⢹⡄⠀⠀⢿⣿⣿⡿⠀⠀⢠⡏⠘⠿⠇⠀⣠⣴⠛⠛⠛⠦⣄⠀⠀⠸⡷⠈⡷
⠀⠈⠳⠶⠶⠖⠋⠁⠀⢴⣶⣀⡼⠉⠓⠦⣤⡴⠟⠀⣤⠀⠀⠙⠋⠀⠀⣤⠀⠻⢦⣤⠴⠚⠉⢧⣀⣶⡦⠀⠈⠙⠲⠶⠶⠞⠁
⠀⢠⣤⣀⣤⡀⠀⢠⣶⠀⢉⠉⢠⣦⠀⠀⠀⠀⠘⠃⠀⠀⠀⣰⣆⠀⠀⠀⠘⠃⠀⠀⠀⠀⣶⡄⢉⡉⠀⣶⡄⠀⢀⣤⣀⣤⡄
⠀⠹⣿⣿⡿⠃⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣠⣦⡀⢀⡀⠀⠀⣿⣿⠀⠀⠀⡀⢀⣴⣄⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠘⢿⣿⣿⠏
⠀⠀⠈⠛⠀⠀⠀⢠⣤⢰⣶⢀⣤⡄⠀⠀⠈⠃⠀⠈⢿⣦⡀⠹⠏⢀⣴⡿⠁⠀⠘⠁⠀⠀⢠⣤⡀⣶⡆⣤⡄⠀⠀⠀⠛⠁
⠀⢘⠀⠀⠀⠰⣿⣎⢻⡆⡿⢸⠟⣴⡿⠀⢀⣠⣤⣤⣀⠉⠃⣠⣄⠘⠉⣠⣤⣤⣄⡀⠀⢿⣦⠻⡇⢿⢰⡟⣱⣿⠆⠀⠀⠀⠃
⠀⢙⠁⠀⠀⠀⠀⠙⢃⣠⣶⣄⠘⠉⠀⠀⠈⠛⠛⠛⠋⢁⡀⠙⠋⢀⡈⠙⠛⠛⠛⠁⠀⠀⠉⠃⣠⣶⣌⡘⠋⠀⠀⠀⠀⠈⡋
⠀⠈⠀⠀⢀⣤⡀⠀⠈⠙⠿⠉⢀⠀⠀⠀⢀⡀⠀⢀⣴⡿⠁⣠⣆⠈⢿⣦⡀⠀⢀⡀⠀⠀⠀⡀⠉⠿⠋⠁⠀⢀⣤⡀⠀⠀⠁
⠀⠀⠺⢿⡌⠿⢠⡿⠇⠀⠀⠨⡣⢎⠄⠀⠻⠟⠀⠈⠁⠀⠀⣿⣿⠀⠀⠈⠁⠀⠻⠟⠀⠠⡁⠄⠥⠀⠀⠸⢿⡄⠿⢡⡿⠗
⠀⢰⣷⠶⠰⣿⠆⠶⣾⡆⠀⠀⡁⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠹⠏⠀⠀⠀⠤⠀⢀⠀⠀⠀⠁⠃⠁⠀⢰⣷⠶⠰⣿⠆⠶⣾⡦
⠀⠀⢴⣾⠃⣶⠘⣷⡆⠀⢰⡟⠿⠇⣠⠞⣋⡙⣧⠀⢀⡀⠀⢀⡀⠀⢀⡀⠀⣾⠻⠟⢀⡴⢛⣉⢻⡄⠀⢰⣾⠃⣶⡘⣷⡦
⠀⢀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠙⠒⠛⠁⠀⠛⠛⠋⠀⠘⠛⠀⠙⠋⠀⠛⠃⠀⠘⠓⠚⠋⠀⠘⠻⠛⠁⠀⠀⠀⠈⠛⠃⠀⠀⡀
⠀⢈⡁⠀⠰⣶⡜⣴⡶⠀⠰⣶⡜⣱⡶⠀⠀⡀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢶⣎⢣⣶⠆⠀⢶⣦⢣⣶⠆⠀⢈⡁
⠀⣨⡀⠀⠀⠙⠉⠙⠁⢀⠀⠙⠉⠙⠁⠀⠲⡿⠂⠸⣿⣷⡄⢿⡿⢀⣾⣿⠇⠐⢿⠖⠀⠈⠋⠈⠋⠀⡀⠈⠋⠉⠋⠀⠀⢀⡅
⠀⠀⠀⢀⡔⢤⠀⠀⠰⣝⣽⠀⠀⢠⢔⢤⠀⠀⣀⣤⣈⡙⠃⣠⣄⠘⢋⣁⣤⣄⠀⠀⣤⡢⡄⠀⠀⣯⣫⠆⠀⠀⡤⣢⡀
⠀⢀⣄⠈⠳⠚⢀⢀⡀⡌⣄⣀⢀⠘⠲⠛⠀⠀⠙⠿⠛⢋⡀⠛⠛⢀⡙⠛⠿⠛⠀⠀⠓⠖⠃⡀⣀⢠⢡⢀⡀⡀⠓⠞⠁
⠀⠈⠉⠀⠠⡜⠎⠇⢃⢃⠃⡃⠏⠞⡤⠀⠀⠀⠀⢠⣾⣿⠃⣼⣦⠘⣿⣷⡄⠀⠀⠀⠀⢤⠳⠹⢘⠘⡘⡘⠘⠱⢣⠄⠀⠛⠃
⠀⠀⠀⠦⡝⠈⢤⡳⠸⠘⠸⠱⢣⠌⠐⡭⢆⠀⠀⠈⠉⠁⠀⣿⣿⠀⠈⠉⠁⠀⠀⠰⢭⠂⠡⡜⠎⠇⠃⠇⢞⡤⠁⢪⠴
⠀⠀⠬⢝⠠⢬⡓⡑⢦⠇⢏⡴⡁⣪⠥⢸⡭⠄⠀⠀⢠⢖⡲⣄⢠⢔⣲⣄⠀⠀⠠⢭⡃⠬⣕⢈⢦⡹⠸⡴⢊⢚⡥⠄⣫⠥
⠀⢀⣛⣃⠘⠛⠀⠉⠁⠐⠀⠉⠉⠐⠛⣀⣋⡁⠀⠀⠹⠭⠟⠘⠏⠸⠭⠞⠀⠀⢘⣙⣀⠛⠂⠉⠉⠀⠂⠈⠉⠀⠛⠃⣘⣛⡀
⠀⢾⠁⣿⠇⠀⢀⣤⠖⠛⠛⢶⣤⡀⠘⠿⠃⢹⡄⠀⠀⠀⣴⣶⣴⣶⠄⠀⠀⢠⡏⠘⠿⠃⢀⣤⡶⠛⠛⠲⣤⡀⠀⠸⣿⠈⡷
⠀⠈⠳⠶⠶⠚⠋⠀⠀⣶⣆⣸⠇⠙⠶⠤⠴⠞⠀⢠⣄⠀⠘⢿⣿⠏⠀⢠⣤⠀⠳⠦⠤⠖⠋⠸⣇⣰⣷⠀⠀⠙⠓⠶⠶⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣁⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠉⠀⠉⠓⢤⡴⠋⠁⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣠⡤⠤⣄⣀⣀⣤⠤⣤⣀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣠⠴⠋⢹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡏⠓⢦⣀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⢀⣠⣴⣋⣡⣤⣶⣾⣯⣀⣀⣀⣼⡀⠀⠀⠀⣸⡇⠀⠀⢀⣀⣧⣀⣀⣈⣽⣶⣦⣤⣈⣓⣤⣀⡀
⠀⠀⠀⠀⠀⠻⣍⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⣽⡟
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠙⠿⣿⣿⠿⠟⠛⠛⠛⠛⠻⢿⡿⠛⠛⠛⠛⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⠉⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣶⣶⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣤⠤⠴⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⡀⠀⠒⠒⠒⠒⠢⢄⡀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆⠠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠭⠍⠉⢩⠭⠑⢦⣸⡧⠯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠀⠘⠃⠀⠸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣟⣀⡀⠀⠀⠀⠀⠀⡇⢜⣁⣀⠒⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠉⢗⢦⡀⠀⠀⢣⠈⠻⠛⠋⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⡄⠀⢠⡗⠢⠄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢗⡦⣟⠣⣄⣀⠤⠖⠋⢡⠷⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠙⠦⢭⣙⣒⣒⡒⠒⣲⣶⣶⣾⣿⡿⠟⠃⠘⢦⡈⡇⢀⣠⠔⠁⠀⠀⠀⠈⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⡏⢿⡉⣹⢹⠃⠀⠀⢰⠃⣬⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠁⠘⢦⣄⠀⡞⢀⠁⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣦⣄⠤⠞⠉⠳⣅⠘⠃⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠛⠟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡔⠀⣼⣃⠀⢀⠈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠛⢅⣺⣤⣴⡿⢿⣿⠭⠭⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⣄⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡏⠀⢠⣤⣤⣄⠀⠑⢆⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠋⠉⠙⠳⣤⡴⠄⣇⠀⠈⣿⣿⣿⣷⣆⠀⠑⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢰⡈⠛⢿⣷⠀⠀⠙⠙⠛⢿⣿⣷⣄⠀⠉⠠⣀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢘⣇⠸⠁⠀⠀⠀⠀⠀⠀⢻⣿⠿⣳⣤⡀⠈⠙⠾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡘⠎⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠈⠈⠻⣿⣷⣤⡀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣀⠀⠀⠀⠀⢀⣾⣿⣷⣦⣄⡀⠉⠑⠻⣷⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣴⣾⣿⣿⣿⣶⠾⣿⣽⣿⣿⣿⣿⣟⣿⣿⣷⣤⣿⠂⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠛⠿⠿⢿⣿⣦⡀⠀⠀⠸⠙⣿⣿⣿⣾⣽⡿⠏⠀⣴⣿⣇⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⢀⡀⠉⠙⠷⣦⣴⣄⣫⣿⣿⣿⣿⠀⠀⠞⠁⠉⠉⠉⠀⠁⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⠉⠙⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⠿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣭⣶⣖⣒⣀⣀⣀⣚⣒⡒⠲⠂⠨⡍⠙⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠟⠛⣉⣉⣉⣉⣉⣉⡙⠛⠻⠿⣿⣟⡛⠛⠿⠶⣦⣤⣈⠳⢦⣍⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣩⣴⡶⠿⠛⠛⠉⠉⠉⠉⠛⠛⠻⠶⣦⣤⣉⠛⠷⣦⣀⠀⢉⡙⠻⢶⣬⡙⠦⣍⠻⣷⣦⡀
⠀⠀⠀⠀⠀⣰⡿⣫⣴⠟⠋⠁⢀⣠⣤⣤⣶⠶⠶⣶⣤⣤⣄⣀⠀⠉⠛⠷⣦⣈⠙⢿⣟⠛⢻⣆⠉⠻⢷⣌⣷⠈⠻⣿⣆
⠀⠀⠀⠀⣼⣿⡿⠋⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣄⡀⠈⠙⢷⣄⠙⢷⣬⣿⣀⡀⠀⠙⢿⠳⠦⣌⠻⣷⡄
⠀⠀⠀⣼⣿⠟⠀⣠⡾⠋⠁⠀⠀⠀⠈⠀⠀⠀⠞⠋⠀⠀⠀⠀⢠⡄⠉⠻⣶⣄⠀⠙⢷⣄⠙⣿⡍⠻⣧⠀⢸⡄⣼⣟⡇⠙⣿⣆
⠀⠀⣰⣿⠋⢠⣾⠋⠀⠀⢀⡄⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠈⠙⣷⣄⠀⠹⣷⡈⢻⣆⣿⣦⣤⡙⢦⣽⣇⢠⡈⢿⣧
⠀⢠⣿⠇⢠⡿⠁⠀⠀⠀⠈⠓⠀⠀⠀⠈⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠻⣦⡀⠘⢿⡄⠹⣯⠁⠈⠻⣦⠈⢻⣆⠻⡌⢿⣧
⠀⣿⡏⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⣷⡄⠈⢿⣆⠹⣧⣀⠀⣿⠀⠀⢻⣇⢻⡌⢿⣧
⢸⣿⠁⣾⠃⠐⠖⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠈⠋⠀⠀⠠⠄⠀⠈⢿⡄⠈⢿⡄⢹⣿⢿⣿⡀⠀⠀⢻⣆⠁⠘⣿⡆
⣾⡟⢸⡟⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⣿⡄⢻⣆⢹⣿⣆⣀⡀⢿⡄⠇⢸⣿⡀
⣿⡇⢸⡇⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠳⠀⠘⣷⡀⠘⣷⡈⢿⡾⠿⠛⢻⣧⠸⣷⠀⡄⢿⣇
⣿⡇⢸⡇⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡆⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠹⣧⠘⣷⣖⣛⣩⣿⠆⢻⡇⣷⢸⣿
⣿⣇⢸⡇⠀⠋⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣇⠀⢻⣆⠸⣿⠿⢹⡇⠀⢸⣧⢸⡈⣿
⠘⣿⣾⣿⠀⠤⣤⣄⠀⢀⡀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠄⠀⠛⠋⠀⠀⡄⠀⠀⢿⡆⠀⢿⡄⢻⣿⠽⣷⠀⠈⣿⠈⠁⣿
⠀⠈⠛⠿⣿⣶⣶⣶⣤⣬⣥⣀⣘⣛⡒⠲⠶⠦⠄⠀⠀⣀⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠘⣿⡀⢿⣞⣿⠀⢰⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠿⠿⣷⣶⣶⣦⣤⣭⣭⣁⣈⣓⡀⠀⠶⠤⢤⣤⣄⣀⣀⡀⠀⠀⠘⣷⡀⠘⢷⣄⠻⢿⣴⠿⠁⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⣷⣶⣶⣶⣤⣭⣭⣉⣁⣀⡈⢿⣦⡈⠻⢷⣤⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⣿⣷⣶⣾⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⠀⠒⠈⠉⠐⠲⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡊⢠⠄⠠⣄⠀⢰⡄⡄⢠⠂⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠣⣜⢀⢀⡙⠧⠀⢁⣷⣃⠀⠀⠴⣾⡧⡠⠶⠆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡇⢀⠜⠋⠀⠀⠀⠀⠀⠀⠁⠘⠃⠾⠋⣨⣔⣥⣤⡄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠯⢷⣶⣦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⣤⣻⠶⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⠴⠒⢛⣛⠩⠉⠉⠉⠛⠒⢦⣄⠀⠀⠀⠀⣼⣀⣒⣋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⠛⣡⠶⠶⢦⠀⠀⠀⠀⣿⣿⣦⣀⣰⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⣽⠛⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠶⠄⠀⠀⡀⠠⢒⠤⠀⠐⠂⠀⠀⠀⠀⣷⢈⡇⠹⢻⡄
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣇⠀⠒⡧⠀⠓⠐⠁⠂⠀⠈⠁⠀⠀⠀⠀⠸⡌⠰⣡⡏
⢀⡀⠀⡎⠠⠄⠄⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠸⠞⢻⠃⠀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡶⣿
⢀⠹⡀⡇⠀⣠⠀⠐⣋⡔⠒⠶⢄⡀⠀⠀⠀⡜⠀⢸⣄⡰⣶⣬⣜⠩⠉⠀⡇⠀⠀⠀⠀⠀⣸⠁⣯⡀
⠈⠄⠡⣡⡀⠙⣆⡀⠛⠀⣰⠂⠀⣉⠀⠀⠀⢧⠠⡇⢾⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⢃⠔⣵⠟⠉⢹⡆
⠀⠐⣄⠘⠓⠂⠀⠀⠁⠀⠘⡆⠀⣯⠀⢧⠀⠘⣆⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠑⢱⠞⣩⡤⠀⠈⣿⠲⠖
⠀⠀⠀⠢⡀⠙⠄⡀⠉⠀⠀⠁⠀⣼⠀⡼⠀⠀⠈⢢⠸⡇⠀⠀⠀⡠⠈⠀⠀⣠⠎⢂⠄⠈⠀⠀⠀⠀⠰⠾⠿
⠀⠀⠀⠀⢧⠀⠀⠈⢦⠀⠀⠀⠀⠘⢦⠃⠀⠀⠀⠀⠑⣌⠐⠒⠈⠀⢀⣠⡞⡡⠂⠄⢒⡁
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⡀⠀⣠⡧⢄⡀⠀⠀⠀⠀⢘⣶⣤⣴⠚⣉⢂⣪⠔⠒⠚⠉
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡠⠚⠓⠒⠓⢌⠒⠃⠉⢩⡶⢚⠿⡠⣞⠻⠛⠄
⠀⠀⠀⡔⢿⠁⠀⠀⢀⠀⡌⠀⠀⠀⠀⠀⠀⠑⠦⠤⠗⠋⠁⠀⠀⠈
⠀⠀⢰⠀⠈⠀⠀⠈⠀⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⠠⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠈⠑⢤⡀⠀⠀⠀⣀⠔⠈⣀⠔⠊
⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠔⠀⠀⠀⠀⠀⣇⡠⠔⠉⣀⠔⠊
⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⡠⠔⠁⡠⠔⠉⠑⠖⠊
⡠⠂⠀⠀⠠⠶⠀⠀⠀⡠⠔⠉⢀⠔⠊
⡇⠑⠤⡀⠀⠀⠀⠐⠉⢀⠤⠊⠁
⠑⣄⠀⠈⡆⠀⢀⠤⠊⠁
⣤⣬⣵⣶⣶⣪⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡔⠀⠀⣠⣶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢠⡘⢿⣦⡀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢰⢸⡸⣽⡶⢄⠈⣐⠩⣻⣦⡀⠀⠘⢿⣷⣶⣄⣤⣠⣴⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⠁⡀⠀⢀⣴⣾⢨⣿⠁⠀⣸⠿⣷⣌⢸⣿⡄⢲⣞⣿⣍⡛⢿⣭⡿⢶⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠗⠋⡡⠈⠘⣛⡻⠷⣿⣾⡟⡟⢠⣼⣌⣺⣿⣿⠐⡋⣿⡛⠿⣆⡽⢷⣄⡙⢷⡢⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡔⣽⠁⣾⣟⢫⠾⠃⠰⣨⡿⣇⣸⢿⣏⣿⡿⡋⠐⣥⣿⡟⢧⣘⣿⣦⡙⢷⣄⠹⣦⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢫⣷⡄⠀⢋⡷⠟⡸⡇⠈⠹⣿⠐⡡⣂⣿⡛⢿⡄⢿⣦⡙⣷⣌⠻⣧⠘⣷⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣑⡾⠁⡴⣾⣡⣏⣧⣷⣷⣿⣿⣟⣣⣿⠙⢿⡰⣿⣦⠻⣿⡌⢻⡆⢹⡇⢹⣷⡈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣴⣦⡀⠀⣠⢫⢸⠐⠁⠀⠠⠐⡞⢹⠏⢏⢃⠈⢹⡇⠙⣿⣿⣧⣾⣷⠈⣿⣇⣽⣷⣾⣷⡾⢿⡛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⢠⡔⣲⠁⠀⠀⠀⠀⡄⣧⢠⣤⡈⢆⣦⣼⠟⣷⠟⢿⡇⢸⢿⠶⣿⢿⣟⠻⣝⢿⣧⠸⣷⡈⣿⠉⣃
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢨⠽⡂⠐⠎⢼⣀⡀⠀⠀⠀⠸⣿⣾⣿⣿⣾⣇⠙⣧⠈⢷⣬⡿⢷⠀⠙⠦⠳⠬⠳⡜⣦⠹⣧⠙⣇⢹⡇⢸
⠀⠀⢠⣞⣁⠀⠀⠐⢥⡄⢀⠘⢫⡝⣦⠈⠊⢿⣽⣵⣶⣄⡄⠘⠙⣷⡹⣆⢹⣆⢸⣷⣼⣏⠁⠘⣇⠀⠀⠀⢂⠀⠁⠨⠀⢸⡇⢹⡿⣇⢸
⠖⠀⢸⣿⡋⠀⠀⠀⡜⠁⠀⡀⠀⠈⢧⢇⠀⠀⠠⣠⣨⣿⣿⣄⢸⡏⢿⣾⣦⣿⣿⣿⠛⢻⡆⠀⢻⣷⣤⡀⠈⠁⠈⠀⡀⠀⡇⠘⡇⣿⢸⠀⠀⠀⢀⠀⣰⠆
⠐⢂⠸⡟⢃⡀⠀⠀⠀⢢⡀⠀⠀⠀⠰⡞⣰⣬⡄⢹⣶⢻⢻⣴⣾⣷⡼⣿⠳⣌⢆⢹⣦⠀⢿⡀⣸⡧⡉⠻⣶⣄⠀⠀⠙⠰⣧⢠⣇⢹⣿⠀⠀⠀⢸⡁⣿
⠐⢦⣶⣇⡿⠃⠀⠀⠀⠀⠙⣿⣿⣶⣄⠀⠃⠾⠛⠛⠉⠁⠉⠟⠒⣼⣃⣹⣆⠉⠣⣷⣍⣷⣤⣿⣿⠻⣮⡳⣌⠻⢷⣦⡀⠀⢻⡈⣿⢸⣿⠀⠀⠀⢸⡇⣿⢸
⠀⠀⠀⠹⡁⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠆⢀⡀⢀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢣⢹⣽⡏⢷⣌⠻⣮⡳⣤⡙⢿⣶⢄⠁⠙⠀⡟⠀⠀⠀⣤⠻⣿⠟
⠀⠀⠀⠠⠹⣿⣴⡄⠀⠀⠀⠈⠪⠛⠿⠿⣿⣾⣿⣿⣤⣕⠄⣤⠰⠄⠀⢀⡀⠀⠀⢀⠀⠸⣿⢛⣇⢀⠙⠷⠌⡻⠮⠿⠦⠽⠶⠇⠀⠀⠀⠀⠀⠀⡀⢹⡇⠙
⠀⠀⠀⠓⠀⠹⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠙⠙⡿⣿⣿⡿⣷⣷⣿⣿⣿⣿⣦⣔⢾⣦⡞⣪⣥⣴⣼⣿⣁⣠⡀⠀⠀⡀⠀⢀⠆⠀⡾⠀⠀⠀⢠⡞⣽⠟⢁⣤
⠀⠀⠀⠀⠀⠀⡘⢿⢰⣷⠀⠀⠀⢲⡐⢄⡀⠀⠈⠉⢉⣀⠈⢿⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠏⠙⢁⣸⣶⣶⣾⣇⠠⠐⠊⠀⠀⠀⣿⢸⡏⣠⠟
⠀⠀⠀⠀⠀⠠⣤⣾⣿⣅⣀⠀⠀⠀⠙⠙⠻⣶⣤⣀⠻⣼⡁⠀⠉⡚⠉⠉⠉⠀⣾⣿⣿⣿⠿⣻⡛⠥⠿⣿⣿⣿⣿⡿⠟⠅⠀⠀⠀⠀⣀⠀⢹⣜⡴⠫
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠻⣷⣷⣠⠀⢀⡴⣤⣽⣿⣿⣿⣷⣀⣞⢽⠗⢖⠲⡖⠿⠿⡻⠶⣖⢲⡀⠀⠀⠀⢉⡐⠂⢈⠀⠀⠀⠀⡀⠀⠈⢡⣼⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠻⣿⡇⠀⢙⠋⠀⠀⠉⠙⣿⣿⣟⠀⠀⠈⢧⠹⡌⢯⠀⠀⠉⠘⣷⣶⣾⠶⠟⠓⠒⠁⠀⠀⣠⢞⣧⡶⠞⢋⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠈⠛⢷⡾⡗⠀⠀⣁⠼⠛⠛⢿⢰⣤⡀⠈⣆⢻⡈⣇⠀⣠⢤⣿⣿⠿⠿⣽⠆⠀⣀⠀⡾⣱⠟⢋⣠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣾⠽⢷⣤⣴⣶⣴⣶⣄⡈⠉⠙⠻⠿⠿⠿⢿⣿⡿⠷⠾⠋⢀⣀⣤⠄⣠⠀⢈⣠⠿⠳⢾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠙⠛⠛⠿⠿⠿⠛⡳⠖⠒⣢⣶⠀⣶⠤⢄⣀⣤⡶⢟⣩⣴⠾⠛⡳⠛⠉⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠐⠚⣩⡴⢟⣿⣿⣑⠦⣈⠛⠿⢿⣭⣶⠶⠾⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⢴⡿⠻⣦⣙⠟⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣋⠁⠀⠀⠀⢢⡀⠀⠀⠀⠀⡀⠂⠐⢀⠀⠀⠀⠠⣠⠀⠀⠀⠀⠀⠉⢠⡈⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠐
⠀⠀⠀⠀⢰⢄⠀⢀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠑⢤⠀⠈⠃⠐⠄⠀⠀⠀⠀⠀⠑⠠⠐⠐
⠀⠀⠀⠀⣀⠀⠀⠘⣃⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠰⢢⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣷⡾⠛⠀⠀⠀⠈
⠀⠀⢰⢔⣽⠄⢀⣼⡿⣶⡄⠀⠹⠛⠿⣷⣦⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠋⠁⠀⠀⠀⠀⠱⠒⠀⠀⠀⠈⢿⡽⠷⠛⠉⠀⠀⠀⠀⠀⠀⢶⣢⠀⠀⠉
⠀⠀⠀⠉⠁⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠁⠀⠘⠀⠢⠄⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⡠⠖⠀⠀⠀⠀⣿⣿⣻⣿⡆⠀⠀⢀⣰⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⣤⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⡄⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠈⠃⠀⠀⠼⠂
⠀⠀⠀⡀⠻⣄⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠬⣤⢁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁
⠤⢄⠀⠀⠀⠐⠀⠀⠀⣠⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠰⠀⠀⠀⠀⡀⡤⢦⡄
⣰⠀⠌⠁⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣆⠰⠂⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀
⠀⠖⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠀⠀⠊⠂⠀⠠⠀⢀⡀⠑⠕⠢⢀
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⣲⣿⣾⢁⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⡴⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠋⠚⠁⣿⣀⣠⠟⢚⡷⠶⢀⡤⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠐⠋⠤⣀⠐⠀⠀⠀⠀⣀⠠⠖⠨
⠀⠀⡄⠀⠀⠀⢀⡀⢀⠀⠀⠀⢰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠁⠠⠎⠀⣰⠟⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⡁⠀⠂⠀⠀⠀⠀⠐⠆
⠔⠈⠀⠀⠀⠀⢸⣥⡛⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⠈⠀⠀⢀
⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣤⣤⣤⣤⡄⠞⠃⠘⠀⠀⠀⠀⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠀⠀⢀⣾⣄⣀⠀⠀⠀⠀⠸⠷⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾
⣿⣿⣯⣿⣿⡇⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡀⡖⠀⠀⢀⣾⣿⣿
⣿⣿⣿⣿⡿⠁⠀⣀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⣉⣡⠔⠂⠀⠀⠹⣿⣿
⠛⢉⡉⠉⠀⠀⠀⠜⢀⠂⠀⡄⠀⠤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⡠⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠉⠀⠐⠇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠲⡄⠐⠀⠀⠄⠀⠀⠀⣠⠁⠀⠀⠠⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⢠⠀⢒⠀⠀⠀⠀⠐⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠈
⠀⠡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠠⠀⠀⠇⠀⠌⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠞⢀⠈⡁⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠂⠸⠀⠀⠀⠄⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⢀⡶⣆⣀⠐⠃⠀⢀⢰⣯⡴⢇⡴⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⠀⠀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠈⠁⠓⠆⢹⠄⠀⣴⣟⣸⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⣤⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣏⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠄⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠉⠉⠀⠀⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣂⡤⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠭⢥⣴⠞⣣⠈⠢⡀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠀⣀⡴⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⡊⠀⠧⠀⠈⢂⠀⠀⠀⠀⠈⠑⣶⠀⠈⠉⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠿⠻⣷⡆⠀⢧⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⣀⠀⠐⢲⡍⠻⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⢣⡔⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠙⢶⣄⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠁⠀⠀⠀⠀⠀⠀⢠⡖⡾
⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⠐⢶⣶⠶⣎⠈⠛⠻⣿⣿⣿⣿⠿⠒⠛⣛⠃⠈⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⢀⣴⡟⣸⠃
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣂⣸⡿⠇⠿⠧⠀⠘⢛⣩⣥⣦⡀⠸⣿⡹⣶⡗⠀⠄⠀⣼⢸⣿⣦⡀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⣴⡿⠏⢠⡏
⠀⠀⠀⠀⠀⠀⢰⠘⣿⣿⣿⣿⣿⣿⣟⠁⠐⡆⢀⣿⡇⠘⠁⠀⠈⠻⣧⣞⣶⠀⠀⡿⠀⠉⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠿⣿⣿⣿⣿⣿⡒⣠⠀⠸⠟⠃⠀⠀⠀⠀⠀⢸⣿⡇⣠⣴⡇⠀⠀⠸⣿⣿⣷⣆⣀⣠⣶⣿⣿⡿⠏⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣴⣿⣿⠁⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣛⠿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠐⠂⠀⠀⠁⠀⠄⠸⠂⠀⠀⠀⠶⠰⠀⠀⠀⠀⠠⠰⠀⠄⠈⠐⠀⠀⠀⠐⠢⠒⠀⠄

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠉⢿⢰⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣟⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⢳⢾⣿⡟⠰⠀⠀⠀⡴⠃⠀⠀⠙⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⣠⢰⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠡⠟⠸⡛⣛⣯⡿⠁
⠀⠀⠀⠀⢀⣸⣿⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⡿⠸⠀⠀⠀⠀⠀⠀⠀⣬⣿⣷⢿⠰⣾⡳⢶⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⣘⣛⢣⣬⣤⣂⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⡹⠟⣼⠁⢹⣆⠙⣯⠘⡿⠇⠀⠀⠀⠀⠀⠀⠈⠈⠙⠽⢿⣿⣿⣿⣿⡿⠿⠋⣀⣀⠔⠁
⠀⠀⠀⠀⠘⢿⡇⡄⠀⠀⢨⣭⣶⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠹⠀⠀⣿⠂⡸⣄⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⣠⣴⣶⣾⣭⡏
⠀⠀⠀⠀⠀⢸⠃⠁⠀⠀⠠⣿⣿⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠘⢀⢙⡝⡃⠀⡈⢹⠀⠀⠠⠤⢴⡗⠀⠀⠀⠀⢠⣾⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣸⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠈⠙⠻⠅⢈⡉⠉⠌⠋⠓⠊⠣⡜⠀⠀⠀⠀⠾⠇⠀⠀⠀⣠⣿⣿⠿⢿⣿⣬⣿⠆
⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣾⡄⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢸⢹⠇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠰⠾⠒⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉
⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⡇⠀⠀⠀⠀⣶⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣤⢸⠁⠀⠀⠀⢠⠀⠀⠀⠀⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⣀⠀⢿⣿⠷⠶⠶⠀⠀⠀⠐⠺⣷⣿⣿⣄
⠀⠀⠀⠀⢀⣿⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢠⡿⢟⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢿⣿⣿⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡧⢼⣿⣢⣾⡁⠈⣿⡃⠀⠀⠀⠀⠀⠠⢶⣯⣿⠁⠀⠉⠁
⠀⠀⠀⠀⠸⠁⣿⠀⠀⢀⣤⣿⣧⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢹⡇⣶⣯⡘⠛⠟⠛⠻⠏⠁⠀⠀⠀⠾⣴⣿⠃⠀⢰⣦
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠿⠏⣼⡿⠀⠀⠀⠸⠛⣻⡄⠀⠀⠀⠀⠠⠿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠉⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢁⣾⣿⣷⡀
⠀⠀⠀⠠⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢠⣤⣤⣠⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠈⠉⠀⠉⠀⣤⠀⠀⠀⠀⠀⢸⣿⣯⠏
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⢀⠃⠀⢠⠄⠀⠀⠀⣶⡟⠀⠀⠀⠸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣾
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢾⣦⣠⣸⣀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠐⢀⠀⢀⣀⣸⣦⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣧⣬⣿⣧⣼⣿⣵⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠶⠦⢄⡓⠚⠻⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⢰⣶⣯⣀⣀⣀⣀⠗⠻⠿⣿⠷⣤⡀⠀⠀⢲⣦⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⡼
⠀⠀⠀⠀⠀⠉⠁⠀⠐⣲⢇⡦⣶⡙⡀⡉⠙⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠊⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣄⣀⣀⣸⠀⠴⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡰⠋⠸⠀⢿⠓⠈⢄⣈⠄⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠀⠀⡠⣿⠟⠁⠀⢠⣀⣼⠀⠀⠐⠉⠘⠀⠀⠀⠙⠛⠁⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⢀⣀⠀⣾⣻⡆⢠⣤⣿⠟⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⠷⠂⠀⠀⠀⠀⠀⠠⢄⣀⣄⠀⠀⢠⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠈⢹⣯⡿⠷⠏⠀⠀⠰⣿⠆⠀⠀⠀⠀⠁⠀⠀⠀⠈⠳⠶⠶⣄⢀⣀⡽⠉⠁⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠂⠨⠂⠀⣤⠀⣀⣼⣿⣿⣿⠀⠀⠀⠀⠐⢲⠖⠀⠀⠰⢠⠞⠁⠀⠢⠀⢀⠐⡄⠂⠀⠀⠀⠀⠀⠀⠠⡚⠑⡄⠀⠑⠁⠀⣔⣞⣿⣽⣟⣿⣿⣦
⠀⠀⠠⠀⠀⠀⣠⣾⣟⣽⣾⡟⡁⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠴⠀⠈⠋⠀⢌⡁⠁⡸⠆⠀⠘⠤⣴⣀⠂⠐⠓⠀⠀⠀⠘⠛⠋⠉⠈⠉⠉⠉⠻⣧
⠀⡀⠀⢐⣠⣾⣿⣿⠟⠋⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠘⠓⠋⠀⠀⠈⠀⣀⡔
⠀⠀⢶⠀⠙⠱⠌⠂⠉⠁⡀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠤⠞⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⡀⠂⠄⡀⢂⣜⣀⡖⠊⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠙⢋⠓⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⠈⠁⠉⠁⠀⠤⡰⢶⠁⣷⣯⢆⡍
⠀⠀⢀⣄⢢⢀⠀⠀⠀⠀⠀⡀⢈⠇⠀⠀⠀⠀⠘⠊⠠⠀⠄⠈⠀⠀⠸⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣰⣤⠵⣁⢼⠏⢻⣎⠗
⠀⠀⢀⠀⢠⠯⢀⠂⠀⠂⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠄⠐⠅⣿⠟⣿⢷⣈⠪⠁⣿⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏⢏⣀⣤⣶⣾⠣⠄
⠀⠤⠀⠀⠄⠀⠀⠠⠀⠄⡀⠀⢀⡔⣦⢖⣴⡤⣀⠀⠀⢈⠅⠀⠀⠰⠀⣰⣻⣿⣿⠷⠀⠀⠀⠠⡐⢺⠱⠈⠋⢣⡔⠂⡀⠀⠀⠀⠀⠈⠙⠃⠉⠉⠀⠐
⠀⠃⠐⠀⠃⠀⠚⠀⠀⠀⢠⣴⡣⠄⠂⠛⠋⠙⠿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠳⠋⠁⠀⠀⠀⠀⢂⠔⠚⣤⠀⡐⡦⠙⠥⢘⠀⠀⠐⠀⠀⠀⣠⣀⣄⡀⠠
⠀⠢⠀⠀⠀⠀⠀⠈⠀⠀⣾⡛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⠐⠃⠀⠈⠁⠀⠀⡔⠀⠉⠛⠿⠿⢿⠃⢂⡄
⠀⢠⡀⠲⡴⢡⠄⡄⠀⠐⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡁⠅⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⢀⠀⡀⠀⠀⡀⠠⠀⠀⢀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠂⠠⠋⠀⠀⠑⠒⠂⠂⠁⠀⠀⠀⠀⠤⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣶⣶⣶⣶⣶⣶⣶⡖⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⣿⣿⣿⣿⣿⣿⣾⣽⣿⣽⣿⣿⠿⠟⠛
⠀⠀⠀⠁⠰⠄⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠿⣿⠿⠿⠿⠿⠿⠛⢛⠛⠂
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⣀⠀⠠⠀⠀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡦⠤⠴⠦⠤⠤⠤⣧⣀⡘⣷⠠⠟⠁⠀⠀⠳⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠺⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⡐⢂⣇⣒⣹⣴⡎⠑⠀⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⣄⠰⣄⣭⡷⢴⡟⠿⡔⣢⢀⠁⠒⣄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⣤⠀⣉⢀⢷⣆⡁⡸⡿⣡⢣⢶⡐⢄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠌⠳⣌⣾⢃⣾⣷⣧⠴⠃⠎⠀⡧⣘⡆⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⠀⠫⣥⣚⡠⢀⡜⣱⢃⡀⠙⠯
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣤⣀⠙⣿⠿⠓⢉⠙⠳⣷⣀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⢃⠈⣿⠿⣈⠀⣶⣤⡁⠺⠷⢀
⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣶⡦⠀⠉⢳⠀⠨⣃⡘⠓⢀⡀⢀⠂
⠀⠀⠑⠦⢤⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣷⣲⣤⠖⣄⣶⠛⢻⡄⠸⢃⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⢻⣏⠙⠛⢳⠾⡟⢿⣻⡟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⣧⣉⠁⠠⠟⣮⢱⣎⢹⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣬⣖⣄⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠈⠀⢿⣮⣉⣷⣡⡼⢧⣜⠳⣞⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⢦⡙⣿⣟⠡⠆⢼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣀⣀⠀⠀⠠⠙⢧⣌⣿⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠓⠒⠒⠚⠉⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠈⠳⡒⢄⠰⡎⣢⠄⡲⠋⠀⠉⠙⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⢤⣀⣸⠈⠢⠃⡇⢰⡥⠤⠐⢒⡩⠃⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠰⠯⠀⠠⠤⣀⡀⠠⠤⠔⠀⢠⠔⠊⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⡠⠔⠢⠄⣉⡐⢄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⠔⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⡀⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣖⣲⣾⣿⣿⣿⣿⣿⣿⣶⣖⣠⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⠋⠠⠜⣯⢖⣻⣿⣿⣿⣯⣿⣿⣿⣓⣾⠡⠎⠙⠦⣔⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡛⠉⠙⡤⣾⣦⠙⣙⢫⣾⡟⡿⠿⠿⢛⢻⣷⠝⣡⡋⣾⣿⠀⠀⠉⢛⣿⢦⣀
⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⡿⠀⠀⣳⣿⠃⠀⠁⠘⢉⡝⢳⠷⠿⡟⢻⠉⠃⠈⠀⠘⠿⣄⠀⠀⠻⣄⣀⠙⢷⣄
⠀⠀⠀⠀⡴⢛⡵⠟⠛⠲⡦⠉⠉⠉⠀⠀⠀⠀⡔⣒⢄⠈⠁⠀⠁⡰⣊⣲⠀⠀⠀⠀⠀⠉⠉⢴⠒⠛⠻⢦⡙⢧⡀
⠀⠀⠀⡼⣵⠏⣰⣦⠀⢠⣾⡆⠀⠀⠀⠀⠀⣀⣙⣿⣿⡇⠀⠀⢸⣾⣻⣋⣄⠀⠀⠀⠀⡀⢴⣶⡆⠀⣴⣭⠻⣶⢷
⠀⠀⢰⣟⢟⡢⡌⢡⣾⣦⣿⡇⣾⡇⠀⠀⠀⢽⣿⣿⡸⣿⠀⠀⣿⢇⣿⣿⠗⠀⠀⠀⢸⣿⣸⣿⣶⣿⡎⣡⢶⣛⣻⡆
⠀⠀⢸⣿⣟⠡⢸⣎⣹⣷⣹⣽⠏⠀⠀⠀⠀⡠⣬⣙⢿⡞⠃⠘⢻⠿⣫⡤⢤⠀⠀⠀⠀⠙⢧⣇⢸⣏⣵⡧⠌⣹⣿⡇
⠀⠀⢸⣟⡧⢁⣾⣏⣿⠟⠁⠀⠀⢀⡀⣀⠀⡙⠓⣺⢷⣿⡄⢠⣿⡾⣛⠛⢉⠀⣄⢀⡄⣀⠀⠈⠻⣿⣺⣗⠩⢼⣻⡇
⠀⠀⢸⡯⢗⡴⣯⠋⠁⠀⢀⣀⣩⣤⣤⣀⡈⠁⠤⡙⣇⣿⣷⣾⣿⡼⢋⠀⠈⣀⣠⣤⣤⣤⣀⣀⠀⠀⠙⣯⢶⡾⢼⡇
⠀⠀⢸⣷⣟⡽⣣⢞⣵⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⢁⣾⣻⣿⣿⣿⣷⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣆⢷⣻⣾⡇
⠀⠀⢸⣿⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⡋⢙⡿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢜⣻⠃⢙⣿⡣⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡇
⠀⠀⢸⡾⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣷⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣷⡇
⠀⠀⣼⣿⡟⣀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡄⣿⣿⣿⣿⣿⣿⢰⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⠻⣿⣧⡀
⣰⣟⡯⡋⠀⠉⣘⢿⣿⣿⣿⣿⣿⣿⠟⣿⣟⠃⣸⢧⣿⣿⣿⣿⣿⣿⡜⣧⠉⣻⣯⠻⣿⣿⣿⣿⣿⣿⡿⣁⠂⠀⢘⢿⣽
⣷⠖⠚⠁⠀⠀⠉⠞⠻⠛⡛⣋⣉⣠⣴⣾⣿⢠⢋⣾⣿⣿⣿⣿⣿⣿⣷⡘⡄⣿⣷⣦⣤⣀⣉⣉⠛⠋⠓⠀⠀⠀⠈⠛⠲
⠘⢷⣤⡥⠘⢐⠂⠄⡀⠀⢙⣋⣻⣍⢯⣻⡙⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⢉⡾⡽⣩⣿⣙⡏⠀⢀⠀⢐⡂⠁⢠⣤⠞
⠀⠀⢿⣷⣦⣾⣿⣶⣶⡂⠈⠟⢿⡏⢠⢣⠇⢰⠸⣿⣿⣿⣿⡿⣿⣿⣿⠇⡆⠘⡞⡄⢹⡟⠙⠁⢐⣶⣾⣿⣷⣶⡿⡿
⠀⠀⠘⠾⢽⣶⣶⣿⣝⣻⢶⣦⣤⣀⣼⡀⠀⠸⣧⣌⡉⣥⡌⢡⡬⢉⣥⣾⠇⠀⢀⣧⣀⣤⣾⣿⣟⣩⣷⣶⣶⡿⠿⠃
⠀⠀⠀⠀⠀⠻⣯⡛⣏⣿⣿⣿⣟⣿⣿⠿⠀⡄⡉⠉⡄⠀⢡⡌⠀⣰⠈⢡⢠⠀⣟⣿⣿⣻⣿⣿⣿⣽⢋⡽⠟
⠀⠀⠀⠀⠀⠀⠀⢳⣸⣿⣿⣿⣿⡿⢣⣟⡼⢣⠇⢠⠇⠀⢸⠇⠀⠘⡆⠸⡌⢧⣻⡾⣿⣿⣿⣿⣿⣇⡞
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡏⠛⣋⣨⢤⣨⠖⢮⡾⠛⢳⡞⠉⢳⡴⠒⣦⠤⣅⣝⢛⣹⣿⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣟⣧⣿⣿⣟⠟⢇⢻⠀⠋⠀⢸⠁⠀⠠⡆⠀⠀⡇⠀⢉⠀⡟⡸⠹⣻⣿⣿⢸⣋⢿⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠣⣿⣿⣷⡾⡟⡶⠛⡶⠚⣶⠒⠺⡗⠒⣶⠛⢶⠛⢳⠻⢻⣛⣿⣿⢻⣼⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣾⡟⣄⣿⡍⢿⣴⣀⣧⠀⣷⣀⣿⣀⣰⣇⣠⣿⣀⣾⠀⣼⣀⣧⡿⣉⣿⣹⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣿⢿⡏⠛⢿⠸⣶⡉⣭⠉⢡⡉⢠⡍⠁⡀⢡⡄⢉⡌⠉⣥⢉⣶⠃⡿⠛⢹⡟⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢿⢺⣇⢂⠼⠀⠀⠃⠘⠀⠈⠇⠀⠛⠀⡁⠛⠀⠘⠀⠀⠃⠈⠀⠀⠧⡘⣸⡇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠦⣀⣠⠴⠴⢠⠄⠀⠀⢠⣦⡼⢧⣲⡄⠀⠀⣠⡄⠧⠷⣤⣠⠴⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡣⣾⠛⢦⠹⠄⢠⢤⡹⡷⢿⣏⣴⡄⢀⣉⣶⠙⣷⢞⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣛⣟⡳⢤⡙⢟⣱⣬⡛⠋⡴⣟⣭⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠳⢶⣄⣀⣤⠶⠛⠛⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀
⠀⠀⠀⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣄⡀
⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⣤
⢿⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠾⠷⠶⣴⠾⠿⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠖⢻
⠸⣇⠀⠀⠉⠻⣿⣍⡉⡉⡉⣹⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠉⠉⣿⡛⠛⠛⠛⢛⣿⣭⣭⣭⣽⢇⣴⠟⠁⠀⠀⣾
⠀⢻⡄⠀⠀⠀⠈⣻⡽⠟⠋⠉⠛⠶⠶⠶⠶⣿⠟⠳⠶⠶⠶⠶⠾⠋⠙⠛⠻⢿⣋⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⢰⡏
⠀⠀⢿⡀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⣤⠞⠋⠀⠀⠀⠀⢀⡿
⠀⠀⠈⢷⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠁⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⢷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠞⠋
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠟⠋
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⠳⢶⡖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠰⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢀⡀⠀⢻⣷⣄⡀⠘⣷⣤⣀⠀⠀⢀⣀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠘⢉⣛⡋⠀⠉⣛⣋⡛⠈⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢺⡇⠀⠙⠛⣷⠀⢸⡟⠛⠉⠈⢿⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡗⠀⠀⠀⡼⠀⠸⡄⠀⠀⠰⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢶⡶⣿⠶⢶⡷⡶⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⡀⠀⠀⣀⡐⠀⠀⣠⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡉⠁⠀⠈⢲⣄⣀⣀⣤⠊⠁⠀⠈⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠉⠳⠤⣄⣭⡃⠀⠀⣸⣅⡤⠴⠋⠁⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⡔⠁⠘⢁⡼⣄⠙⠀⠙⡶⢤⡀⠀⠀⠀⠀⠈⢹⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢹⡀⠀⠀⠀⠀⠀⢀⣜⣀⡤⠖⠋⠀⠈⠓⠤⣄⣘⣄⢈⡗⠀⠀⠀⠀⡼⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠲⡀⠀⠀⠀⠀⣀⣀⡞⠀⠀⠀⠀⢠⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣇⣀⠀⠀⢀⠉⡞⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠀⠀⠈⠉⠳⣄⠋⡼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⣐⣼⠿⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠸⠀⣿⣸⠃⠀⠀⣀⣠⡞⠉⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⣠⣀⣠⡄⠀⢠⠿⠛⠲⠶⣿⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠉⠉⠓⠚⠋⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢻⡀⠀⠀⠀⢀⡀⣀⡤⠞⠓⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣹⣦⣤⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⣈⣷⡶⠦⠤⠶⠿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣶⠷⣿⣍⣩⣿⠾⣶⢖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢒⣈⣐⣀⣂⣉⠣⠴⣏⡼⢻⣘⠆⠈⠁⣐⣀⣒⣉⣲⣤⡀
⠀⠀⠀⠀⠀⢀⣴⢖⣏⣉⣤⣤⢀⡀⠀⠨⣁⠲⣼⠟⠻⣧⠔⣈⠅⠀⢀⢠⣤⣤⣉⣻⣷⣦⡀
⠀⠀⠀⢀⣐⣋⣼⠋⠉⠉⣡⡀⠈⠓⠠⠄⠛⠓⢮⡀⠀⡡⠚⠛⠠⠄⠋⠀⠀⡍⠉⠉⠙⣷⣟⣆⡀
⠀⠀⢀⠞⣯⡽⣟⠂⠀⠘⠊⢀⠥⢠⡄⠀⠀⠀⢨⠟⠻⣃⠀⠀⠀⠀⡤⠼⡀⠁⠉⠀⠘⣻⢯⣿⠳⡄
⠀⢀⠂⠺⠄⠀⠈⠒⠂⠀⠀⠄⠀⢠⠟⠉⠙⣻⣿⣀⣀⣿⣟⠋⠉⠻⠄⠀⠈⠂⠀⠀⠒⠠⠀⠠⠇⠀⡀
⢀⣾⡗⠰⡂⠀⠀⠄⠲⠿⣦⣆⠙⠀⠀⣠⠞⠉⢹⣿⣿⣯⠉⠳⣄⠀⠀⠋⣰⡴⠟⠂⠤⠀⠀⠸⠆⢾⣷
⠘⢁⠔⠘⠁⠀⠠⠀⢀⠀⠀⠩⠷⣶⡞⠉⠙⠿⣿⠀⠈⣿⠛⠛⠉⢻⡶⠾⠍⠀⠀⡀⠀⠄⢀⠈⠁⠢⡉
⣷⣆⠈⢀⣬⣄⣀⡀⠀⠀⠀⠀⡔⠹⡇⠀⠀⠀⡿⢆⡠⠿⠀⠀⠀⣸⠇⢰⠆⠀⠀⠀⠀⠀⢠⣤⡀⠃⣴
⣿⣟⣤⣾⣿⣿⣿⠃⠀⠀⠀⠀⠰⠀⠀⠂⠀⠄⢀⡼⢧⡀⠠⠀⠐⠀⠀⠄⠀⠀⢀⠀⠀⣼⣿⣿⣷⢦⣻
⠟⢠⣾⣿⣿⡿⠃⠀⠀⠀⠀⡀⢀⠀⠉⠛⠒⠲⡄⠠⠄⢠⠖⠓⠛⠉⠀⡀⢀⠀⠀⠀⠀⠘⣿⣽⣿⣷⡄
⠰⢻⣿⣿⣻⠃⢀⣐⣤⣾⣷⡷⢾⣷⣾⣄⡤⠀⢡⡀⢀⡌⠀⠴⣤⣶⣶⡶⢷⣶⣷⣔⣆⠀⠘⣾⣿⣿⡟
⠀⢻⣿⣿⠇⢠⣷⣿⣏⠛⠏⠀⡼⠛⢹⣿⣷⠃⠘⢷⡾⠃⢘⣾⣿⡏⠛⢧⠀⠹⠛⣻⣿⣾⡄⢹⣿⣿⡟
⠀⠘⣿⣿⠠⢾⣿⣏⣉⠑⣷⣻⡷⠂⣉⣹⣿⡗⠀⠠⠄⠀⠺⣿⣏⣉⠐⣾⣗⣾⠎⣉⣹⣿⡷⠄⣿⣿⠁
⠀⠀⠘⣿⠰⢸⣿⣯⡄⠘⠛⠛⠁⣀⣀⣿⡿⠓⠀⣼⣧⠀⠒⢿⣿⣁⣠⠉⠛⠛⠂⢠⣽⣿⡇⠂⣿⠃
⠀⠀⣠⠏⠀⠨⠺⣿⣾⣿⣴⣷⣴⣾⡿⠿⠁⠀⣼⣿⣿⣧⡀⠈⠻⣿⣷⣶⣾⣦⣿⣷⣿⠟⡇⠀⠸⣆
⠀⢠⠁⢀⣄⣀⣨⠀⢝⠛⠻⡏⠉⠁⠘⣇⠒⣼⣿⣿⣿⣿⣧⠖⢸⠃⠉⠉⠹⠛⠛⠫⠀⣥⣄⣄⡀⠈⡄
⠀⢰⣾⣛⣙⢿⣿⣀⠀⠀⢀⠀⠐⠦⣠⡿⢰⣿⣿⣿⢿⣿⣿⡇⢹⣄⠰⠂⡀⠀⠀⠀⣀⣿⡿⣉⣛⣿⡆
⠀⠈⢻⣟⣯⣴⠛⣿⣴⡆⠀⡈⠀⠀⠀⠐⡘⠿⠿⣳⣜⠿⠿⢅⠂⠀⠀⠀⠀⠀⢰⣦⣿⠛⣦⣹⣿⡟⠁
⠀⠀⠀⠉⠳⣦⣶⣿⣷⣶⣶⡌⠷⠀⠂⠀⠀⠀⠜⠟⢻⠃⠀⠀⠀⠐⠂⠾⢃⣾⣷⣿⣭⣤⣴⠟⠉
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⢻⣧⣄⡀⢀⠈⢒⠀⡀⠀⡀⢀⠀⠘⠃⢀⢀⣀⣼⡟⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣈⠛⠿⡧⣟⢔⡟⢓⣟⠺⢗⡺⡞⢻⡢⣻⢽⠿⠛⣁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⣿⣿⠇⡸⢶⣤⣄⣉⠀⡃⠀⠇⠀⠃⠈⠁⠈⡁⢈⣈⣤⡶⢯⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣃⣉⠀⡀⢿⣦⣇⣉⠙⠋⠟⠘⠇⠰⠇⠰⠋⠺⠉⠋⢉⢨⣴⡿⢀⠀⣩⣰
⠀⠀⠀⠀⠀⠀⠧⠿⠏⠀⠀⠻⢿⡷⣾⣦⣧⡤⢦⣴⢧⡴⢤⣼⣤⣶⢿⡛⠟⠁⢀⠹⠳⠼
⠀⠀⠀⠀⠀⠀⠀⣥⠸⣯⣤⣄⡀⠀⢛⡩⠟⢛⠟⠚⠛⢺⣛⠻⢉⡛⠀⠀⣠⣴⣾⠇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡼⣿⡿⠳⣄⠀⠀⣀⡚⢄⡠⢐⣂⠀⠀⣠⠞⣿⣿⢧⣽⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠘⠟⢇⢸⡎⣥⢤⣬⢹⡟⢁⡴⣤⢰⡇⡊⠘⠃⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠛⢳⣿⣦⣿⣻⡟⢿⣤⡿⠟⠛⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢛⣚⣋⣙⣓⡛⠋⠐
⣿⣿⡿⠿⣿⡿⢿⠿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⣿⣧⡶⠟⢋⣁⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡶⢶⣄⠘⢿⣛⣥⣤⣾⣻⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣯⡿⢟⣽⣷⡄⢿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀
⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⣤⠖⣊⡤⢚⣩⡶⢟⣫⣵⣾⣿⢿⣿⠟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢪⣤⡾⢋⢔⣷⡿
⠀⠀⠀⠀⣀⡘⣷⠚⣛⢉⡥⢞⣵⠶⠋⣡⣶⣿⣿⢿⣯⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣊⣾⣿⢿⡿⣯⡾⠋
⠀⠀⠀⠀⢿⠉⢿⣽⠻⡧⠞⠋⣁⣴⣿⡿⢿⣽⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⣾⡿⢏⡴⢫⣾⠟
⠀⠀⠀⠀⢀⣄⣀⠻⡆⣻⣶⣿⣿⠟⣯⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠐⠀⠀⠀⠀⠀⠀⠀⢀⠔⣱⣾⡿⢛⣷⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠃⠁⠀⠛⠻⠿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠁⠀⠀⠀⠀⣀⠠⣄⣾⠟⠋⣸⡿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠘⠏⢿⠿⢿⣶⣾⡿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢠⣿⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⢹⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⡇
⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠰⡆⠀⠌⣩⡼⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠤⠌⠳⡀⣿⢀⣴⠟⠋⠀⢒⣶⡶⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠈⠱⣶⣶⣶⣽⡉⢻⡿⠷⠾⠿⠟⠋⠀⠀⠀⠀⠀⠀⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⣾⡿⢿⣯⢿⢯⣀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⡟⠛
⠀⠀⠀⠘⠃⠈⠀⠀⠀⠤⠶⠟⠛⠉⠐⢁⡏⢿⡀⠙⠙⠛⠷⠞⠓⠀⠒⠀⠀⠀⠀⠀⠘⠾⢿⠿⠿⢿⠿⠟⠟⢿⣿⣿⣿⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠃⠘⢧⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⠀⠚⠒⠈⠻⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠓⣼