Chemical structure Emojis & Symbols

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈⢃⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⠂⠒⠀⠀⢀⠞
⠀⠀⠀⠀⠊⠀⠙⡄⢠⡌⠉⠉⠀⠀⠀⠀⠀⢤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠕⠐⠃⠀⠀⠐⠀⠀⠀⠀⣄⠀⡀⠀⠉⢀⠀⣄
⠀⠀⠀⠀⠠⠀⠈⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⢀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠄⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠁
⠀⠀⠀⠘⠀⠀⠀⠃⠀⢀⠀⢀⠀⠀⠀⠀⠀⠙⢀⢀⡄⠁⠙⢃⠀⠀⠀⠀⠀⠀⠄⠴⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⢀⠆⠈⠣⡀⣠⡀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠐⠲⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠌⠀⠈⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡀⡠⠋⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠘⠂⠐⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠈⠀⠈⡄⠠⡄⠀⠀⠀⠈⢀⢀⣄⠀⠈⢁⠀⣠⠀⠀⠀⠀⠈⠢⠀⠄⠀⠀⡡⠄⢀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⠈⠐⠐⣀⠀⠀⠀⢂⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠠⠀⠀⠈⠄⠀⠎⠀⠀⡂⠀⠀⠀⠐⡀⠀⠀⠖⠀⠅⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠐⠐⢂⡀⠀⠐⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤
⠀⠀⠀⠈⠀⠀⡘⠉⠈⠀⠀⠠⠋⠀⠀⠀⠀⠀⢀⡀⣀⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠆⠰⠆
⠀⠀⠀⠀⠀⠀⢰⠄⠤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠛⠀⠃⠀⠀⠘⠀⠂⡀⠀⠀⠀⠀⠀⠂⠂⡀⠀⡀⠂⠚
⠀⠀⠀⠀⠖⠀⢂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⠀⠚⠁⠀⠀⠀⠀⣤⣄⣴⠋⠉⢳⡄
⠀⠀⠀⠈⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⣀⡀⠋⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠀⠀⠀⠀⠀⠀⠠⠆⠈⢃⡀⢀⠈⠉⠁⡄⠀⠀⠀⠀⠀⢦⠀⠀⠰⠀⠠
⠀⠀⠀⠰⡄⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠈⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠓⠐⠂⢀⡀⠄
⠀⠀⠀⠀⠐⠀⠺⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⡤⠠⠈⠀⠈⡠⠠⡄⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⢀⡈⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢴⠀⠀⢰⠄⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠙⠄⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴⣶⣾⣿⣿⣿⣤⡇⢀⠐⠺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⢦⣴⣿⠀⠀⠀⠀⠉⡟⠉⢾⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠊⠙⡉⠀⠁⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠈⠀⠀⠀⠰⠞⠀⠀⠀⠀⢀⠀⣀⣀⣠⣴⣾⡇⠀⣤⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣠⠶⠶⣟⣃⣀⡀⠀⡀⠱⠃⠀⠀⠀⠀⣀⠄⠀⠀⠄⣀⠉⠀⠌⠛⠿⠟⠉⠉⠂⠉⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠐⠆⠐⠺⠟⠠⣖⣶⢷⣷⣾⣍⣭⣋⣈⣻⡇⠰⠠⢀⣠⣴⣶⣿⠀⢲⡶⠀⠀⠀⠀⠂⠀⡀⠒⠐⢶⣤⣶⣾⡗⠒⠒⢷⣶⣶⣾⡿⠰⢶⠀⠀⠴⣴⠇
⠀⠀⠀⠀⠀⠡⠋⠝⠻⠀⠛⠋⠁⠡⠐⠂⠙⠚⠖⠙⠻⣿⠟⠋⠲⠁⠀⠀⠀⣤⣄⣀⣴⣿⣧⠀⠌⠉⠛⠙⢆⠀⢀⠎⠙⡟⠉⠙⠲⣤⣠⡄⠀⣿
⠀⠀⠀⠀⠀⠛⠀⠀⣬⠴⠉⠛⠀⠒⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠉⣤⠛⢿⢿⠟⠻⣀⣀⣰⠄⡄⠘⣶⣇⣀⡀⢁⡤⠤⠔⠚⠋⠉⢲⣴⣄⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣯⡄⣀⠀⣤⣤⣴⣾⣿⠦⡄⠠⠀⠴⣿⣿⡿⠒⠺⣿⣿⡟⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠶⠺⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⠿⢯⠀⠀⣀⠀⠀⠀⠀⠀⠰⡾⠉⠁⠈⠒⡄⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⠁⠀⠀⠀⢀⠄⠩⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣴⣿⡄⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠿⣿⡿⢿⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠃⣠⠀⠴⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢀⣠⣿⣢⣦⣤⣦⠀⠀⠀⠈⠛⣤⡎⠁
⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠉⢤⡀⠀⠀⠈⠙⠛⠁⠙⠛⠁⠀⠀⠀⠀⠆⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢠⡄⠁⠀⠀⠀⠀⠀⠀⠂⠐⡀⠀⠀⠀⠀⠀⠀⡛⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠀⠀⠀⠄⠀⠀⣀⢀⠀⠀⠀⠀⡀⠠⠈⠀⠆⠀⠀⠀⠀⣠⡠⠴⡿⡉⠁
⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠄⡀⠂⠈⠀⠀⠀⠀⠀⡈⠀⠀⠀⠘⠃
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠴⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠤⣀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⡰⡛⢄
⠀⠀⠀⣀⡸⣿⠦⢤⡄⠚⠃⠀⠀⠈⢃⠀⠀⠀⡜⠀⠀⠀⠀⢀⡀⠐⠆⠀⠀⠀⠀⠀⢾⠠⡇⢸⡶
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠲⠆⠀⠀⠀⠀⠀⢀⠜⠂⠤⠀⠀⠀⠀⠀⢳⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠠⠀⠁⠀⠲⠀⠒⠀⡀⠀⠀⠀⠁⠶⡀⠀⠄⠀⠀⠀⠘⣇⠀⠀⣠⠟⠁⠀⠀⠀⢠⠼⠡⡄⠃
⠀⠀⠀⠠⠄⠠⠆⠐⠂⣤⠀⠀⠀⠀⠀⠀⠸⡦⠠⠄⠀⠀⠀⠛⠋⢉⠛⠂⣀⠀⠀⠀⢠⡘⢂⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⡠⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡠⢄⠀⠀⠀⠀⢀⡀⠀⢀⠓⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⢠⠂⠐⡀
⠀⠀⠀⢀⣀⠶⠤⠒⢄⠄⡠⠆⠆⠀⠀⠀⠀⠀⢰⠁⠀⠑⠚⠄⠠⠂⠀⠀⠀⠑⠚⠈⠳⠇⠀⠀⢀⡐⠊⠀⠀⠀⠀⢍⢈⠃⠉⠂⢞
⠀⠀⠀⠥⠔⠀⠀⣀⡰⠐⠣⣠⠤⠀⠀⠀⠀⠀⠈⡲⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⠒⠓⢇⠀⠀⠀⠀⠀⠀⣃⠀⠀⡠⠊⠐⠢
⠀⠀⠀⠀⠀⠀⢸⠀⡨⠀⠀⢅⣀⡸⠀⠀⠀⢀⠔⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠄⠆⡇⠀⠀⠀⠐⠝⠀⠀⠀⠀⡎⠀⢸⠂⠣⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡁⠈⡆⠀⠀⠀⠀⠀⠀⠰⡰⠦⢄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⡦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠬
⠀⠀⠀⠀⠀⠀⡔⠉⠩⠑⢄⠀⠀⠀⠀⠀⠀⠣⠀⠀⡱⠪⡀⠀⡠⠀⠀⠀⠀⠸⡝⠣⠄⣀⠦⣀⡠⠝⢤⠂⠀⠀⠀⠀⠀⢀⠠⡆⠀⠈⠇
⠀⠀⠀⡠⠠⣎⡀⠠⠄⡦⠺⡨⠂⠀⠀⠀⠀⠀⡱⢎⠰⡱⢈⠶⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⣶⠁⠀⠀⢸⠀⠀⢰⠀⡦⠀⠪⣀⣸⠈⢨⠃⠉⠁
⠀⠀⠀⠢⠤⠒⠀⠤⢀⡔⠲⡀⠀⠀⠀⠀⠀⡐⠁⡀⠱⢔⠁⠀⠱⠀⠀⠀⠀⠠⢧⠄⠒⠙⢻⠐⠂⢠⠶⠀⠀⠀⠈⠀⠀⠀⠀⠈⡢⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠂⠐⠒⠁⠀⠉⠂⠀⠀⠀⠀⠀⠈⠈⠁⠂⠰⣉⠦⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠈⢲⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠢
⠀⠀⠀⠀⠀⡠⡀⠘⠀⠀⠀⡀⠜⠤⠂⠀⠀⠀⠀⢅⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢐⠂⠉⠑⠘⠢⣀⠀⠀⠀⠀⢔⢑⠂⠁⢈⡃⠢
⠀⠀⢀⡀⠀⠉⠑⡖⠓⢖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠑⢤⠀⢄⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⣅⠐⠀⠀⠀⠀⢘⢁⠆⠁⢈⡃⢰
⠀⠀⠰⠼⠂⠉⣠⠛⡖⢣⠈⠙⠁⠀⠀⠀⠀⠀⢠⠐⠢⡀⠄⠦⣀⡠⠀⠀⠀⠀⢎⢙⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⢘⢁⠂⠁⢈⡋⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢕⡄⠀⠀⠀⠀⠀⠀⠘⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠠⡀⠤⠧⠜⠀⠀⠀⠀⠀⠀⠀⢸⣉⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠪⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠰⠉⠁⠢⡀⠀⣀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄
⠀⠀⠀⠘⠤⠓⡤⠄⢤⠧⠤⠆⠀⠀⠀⠀⠀⠀⢡⣀⠀⣠⠾⣀⢀⣌⠀⠀⠀⠀⠀⠀⢎⣹⡂⠀⠂⠺⡀⢀⠇⠀⠀⠀⠀⠀⢀⠔⠶⠣
⠀⠀⠀⠀⠀⠀⡡⣀⢤⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠱⡔⢀⢒⠈⣶⠁⢉⡢⠀⠀⠀⠀⢀⡁⠈⠢⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠤⡀⠑⡀
⠀⠀⠀⢜⠁⢙⠀⠀⠀⠋⠀⠈⠢⠀⠀⠀⠀⠉⠐⠞⠘⠲⣤⠞⠐⡖⠙⠁⠀⠀⠀⠰⠁⠈⡆⠀⠘⠀⠀⠃⠀⠀⠀⢀⠠⣃⠤⠚⠂⠓⠄⣘⠄⡀
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠢⠀⠠⠎⠀⠀⠀⠀⠀⢰⠀⣤⠔⠉⠑⠀⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠂⠐⠃⠀⠀⠀⠢⠀⠗⠒⠐⠒⠐⠂⠣⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣸⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡾⠋⠙⠲⢤⣠⠖⣋⡥⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⡭⠓⠦⣍⠓⠀⣀⡤⠞⠓⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠙⠋





⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⠀⠀⢀⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣾⣿⣿⣿⣻⣿⣿⣿⠀⠀⠈⠉⣹⡋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⡌⢋⠙⠋⣙⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣧⣽⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠀⠒⠢⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣶⣒⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠤⠤⢄⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣑⣲⣶⣤⣤
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠿⠟⠛⠛⠛⠛⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠟⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢶⣤⣶⣿⣿⠟⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠉⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡴⠶⢶⣒⣶⡚⠋
⠀⠀⠀⠀⠀⠀⢀⢼⣓⠢⢄⡀⠀⠀⠀⠉⠛⠲⢤⣄⡀⠀⣿⣧⣀⣀⣀⣀⣀⣤⡤⠶⠖⣚⣋⣭⣭⣶⡾⠶⠶⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠙⠂⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠓⠺⢿⣿⣿⣿⣵⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠐⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠒⠤⣄⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⣖⣛⣻⡟
⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡉⠓⠤⣀⠀⢸⣿⡤⠀⠀⠀⢀⣀⣠⣤⠴⣖⣾⣿⣭⣷⠶⠿⠛⠛⠋⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠛⠿⠿⣛⣛⣯⣿⡿⠿⠾⠛⠛⠉⠉⠁⠀⠀⡀⠠⠄⠐⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣶⣿⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⢸⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣸⣿⣿⡅⣀⣀⣤⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢀⣀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠏⣠⡗⠻⣍⡱⠀⡀⠀⠀⠀⢠⣧⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠
⠀⠀⠀⠀⠘⢈⡶⣦⣾⣗⠟⢉⡝⠀⠀⠎⢻⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠌⠀⢀⣿
⠀⠀⠀⠀⠀⠹⢆⣽⠉⠀⢦⣠⠀⠀⠀⠀⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠄⠁⢀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠈⠻⡄⠠⠁⠆⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠊⠀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠚⣧⠀⢀⣷⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠉⡇⠙⠻⠜⠉⠻⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠀⣀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣀⠀⢀⠀⣀⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠄⠀⠀⠀⣠⠲⠀⠀⠀⠄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⠀⡀
⠀⣴⣔⡱⣄⠘⣄⡹⢿⡛⠓⠓⠛⠛⡇⠀⢘⡀⢀⢤⣰⠞⠛⢧⠀⠠⠀⠀⡠⠀⢚⠀⠀⡀⠠⠻⠀⠀⠀⢘⠀⠀⠀⣸⡏⠀⡀
⠀⣿⣿⡟⢸⡆⣼⠙⠲⣝⠢⡀⠀⠀⢹⣾⢋⠐⠁⠆⢷⢠⣴⣶⣶⣦⡀⠀⣗⠀⠀⢡⠈⠀⠀⠘⣶⠁⠀⠀⢱⠠⣿⣿⠙⠑⢳
⠀⣿⣿⣧⣌⣀⡏⠀⠀⠈⡷⠇⠀⠀⠘⡿⣧⠀⠀⢸⠈⢺⣿⣿⣿⣿⡇⠀⡿⠀⠀⠀⣶⠻⡀⠀⡿⡋⠀⠀⡙⢸⣻⣿⡦⠄⠀⡇
⠀⢸⡿⠃⡀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣁⣠⣞⠀⠀⠙⠿⢿⣛⢀⡔⢻⡀⠐⢇⠀⢥⡀⠀⣴⠘⠒⠀⢠⠀⠀⠛⠛⢦⣼
⠀⠀⡧⣤⣭⠿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡭⠭⠭⣭⣽⡇⣼⡁⠀⠀⠀⠈⢧⠀⠉⠀⠈⡇⠀⠀⠈⠙⠀⡀⠀⠉⢰⡆
⠀⠐⠀⠀⢸⡄⢇⠀⠀⠀⠀⠀⠀⠀⢸⢯⣽⣿⡯⣛⣿⣿⠀⠀⠀⠘⣄⡴⠀⠀⠀⠀⣇⠀⠀⠀⠀⠄⠄⠀⠀⢞⡤⢀⠀⠀⠀⡄
⠀⠀⡀⠀⠈⠁⠰⠦⣤⣀⣀⣀⣠⠤⠚⣼⠀⢸⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠄⡠⠠⠀⣠⣀⣀⣴⠎⣀⠀⢀⠠⣤
⠀⠸⠤⣤⣤⡤⠂⠀⢨⡉⠀⠋⡅⠀⢠⣿⣦⣼⣿⣿⡿⠉⠏⠉⠉⠁⢿⡼⠦⣤⡴⡏⠀⠀⠃⠀⠀⠉⠈⠁⠀⠈⠉
⠀⠀⠀⣻⠀⣇⢀⡴⢾⣧⢰⣤⣇⠀⠈⠙⡟⡟⢻⠋⠁⠀⢁⠀⠇⠀⢰⢀⣀⢠⡤⠃⢀⠀⠀⢀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠁
⠀⠀⠈⠿⠟⠚⠓⠀⠾⢿⣿⣟⠿⠿⠦⠀⣿⣿⣽⣀⠀⠀⢠⠀⡄⠀⢤⣄⡄⠸⢤⡀⢋⣿⣷⣬⣑⣢⠜⠳⠃⠻⠛⠘⠀⠪⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⢿⢊⡛⠚⡳⠄⠛⠖⢿⡤⠚⠉⠀⠀⠉
⠀⢀⠔⠁
⠔⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣧⣤⣴⣶⣶⠶⣿⣿⣶⢰⣶⣶⣶⣶⣤⣤⣤⣄⠈⣿⡆
⠀⠀⠀⢰⣾⣿⡏⣽⡿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠻⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⣤⣬⣤⠀⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⠀⠀⠉⣡⣝⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷
⠀⠀⠀⠿⣦⣧⠀⠀⠀⠀⠈⠩⣗⣀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⢦⣘⣼⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢹⣿
⠀⠀⠀⣾⠿⣟⣄⠀⠀⠑⢀⡼⢯⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠤⡮⣸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣽⣿⡇⢸⣿
⠀⠀⠀⣿⠀⡋⣏⠀⠁⠠⡽⡷⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣶⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⢰⠈⠀⠀⠸⡇⢂⠀⠀⠨⣽⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣴⣶⣶⣶⣶⢰⣶⣶⣶⣶⣶⣶⣤⣬⣭⣅⠸⣿
⠀⠀⢸⣇⠀⠀⠀⣿⡀⢶⡖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢦⣿⣼⣿⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⡼⠿⣦⠀⠘⣿⣷⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣤⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⣷⠘⠿⠠⠊⢉⣥⣤⣤⣦⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿
⠀⠀⣿⣤⣶⠤⠄⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣏⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹
⠀⢰⣟⣉⣐⣶⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠧⣼⣿⡛⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠇⡿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⢀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡷⣧⣤⣝⣶⠃⠀⠀⠀⠀⠀⠀⠀⣀⠛⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠰⠀⣰⣁
⢀⡜⠁⢰⣿⣞⠈⢦⡀⠀⠀⠀⠀⢐⣶⡿⠁⠀⠀⠀⠀⠈⠉⠙⢿⣷⣼⣿⡆⠀⠀⠀⠀⠀⠀⡼⡿⢿⠃⠀⠆
⠮⢴⣎⣀⣉⣀⣱⣆⣳⠀⠀⠀⢦⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢴⠦⠄
⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⠀⠀⣄⠀⠀⠀⡠⠋⢹⣿⡟⠑⢄⠀⠀⠀⠀⠀⢀
⠘⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠁⠀⠀⡇⠀⡱⠀⠀⣀⠀⠀⠀⢁⣤⣄⣽⣡⣤⣌⡀⠀⠀⠀⠀⠸⣷⠆
⠀⠈⠻⣿⣿⣿⣿⡏⠀⠀⠀⠈⠑⠁⠈⠲⠀⠀⠁⠉⠀⠀⠀⠺⡿⠋⠀⠙⠿⡟⠀⠀⠀⠀⢀⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠊⠁⠀⠀⠀⠀⠸⠿⣿⡿⠗
⠀⠀⠀⠀⡄
⠀⠀⣶⠀⡄⠀⠔⡀⠀⠀⠀⢀⣘⠁⣤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣷⠀⡇⠀⠰⠇⠀⠀⠀⠘⠛⢄⣀⠶⠀⡶⠀⡀⠀⠀⠀⠀⠰⠿⠿⢆⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣶⠦
⠀⠐⠛⠂⡃⠀⢰⡆⠀⠀⠀⢀⣤⠜⢻⠂⠘⠛⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣰⣄⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⢀⡘⠗⠀⠀⠁⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⡆
⠉⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢶⣤⣴⣧⣄⠀⠀⠀⠀⠀⠀⢀⠉⢁⡘⠙⣀⠛⢃⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣆⠀⠀⠀⢀⢞⡟⠉⠉⢙⠲
⠀⢰⣏⣠⣟⣀⡀⠀⠀⠀⠀⠸⠉⠉⠉⠈⠉⠁⠘⡃⠀⠀⠀⠠⣾⣷⡇⠀⣸⣿⠀⠀⠀⣤⠞⠳⠄⠰⠚⠳⡇
⠀⠀⢉⣿⡋⢉⢹⠆⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠂⠂⢿⣿⣿⠿⠃⠀⠀⠀⠠⡀⡀⢸⡀⣠⠀⠄
⠀⠀⠀⠙⠛⢻⡟⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣠⣶⣾⣿⣿⣶⣶⠀⠀⠀⠀⠈⠛⠋⠉⠑⠈

⠀⠀⠀⠀⠀⣠⣴⣿⣿⡋⠀⠀⢀⣠⣴⣶⣤⣄⠀⢀⢀⣀⣀⣀⡀⠀⠀⠐⠷⣤⡶⠀⠀⠀⢀⣤⣄⡀⠀⣠⣶⣿⣿⣷⣦⡀⠀⢀⠔⡀
⠀⠀⠀⠀⠈⠙⢿⡿⣋⣀⡀⢠⣿⣿⣿⣿⣯⣿⣆⠸⡸⠟⣋⣀⣀⠀⠸⣦⡞⢿⣇⠀⠀⠀⠸⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣷⠀⠊⠺⠿⢦⡀
⠀⢠⣾⢻⣟⡛⣮⠑⣿⣿⣿⠸⣿⠻⠿⣿⠯⢻⡿⠀⣰⣾⡟⢉⡻⣿⣦⠙⠓⠟⠛⠀⣀⣤⣤⣤⣄⡀⠘⣿⣿⣙⣉⣿⣿⡏⠀⣎⣤⢒⠀⠉
⠀⣿⣿⣼⡟⣽⣿⡇⢉⣉⣁⡀⠙⠿⣿⣿⡿⠟⠁⠀⣿⣿⣾⣿⣷⣿⣿⡆⢸⠷⢠⣾⣿⠿⣿⡿⢿⣿⣆⠈⠛⠿⠿⢛⡋⠀⠢⠯⣢⣏⠳⢀⣠
⠀⠘⢿⣿⣿⡿⢟⣴⡿⠿⠿⢿⣦⠀⠀⠐⢶⣠⡦⠀⠻⣋⣀⣉⣀⣉⡿⠀⣠⠄⣾⣿⠀⢀⠀⣀⠀⢸⣿⡆⠀⣰⡿⠿⠿⣷⡄⠚⠁⠀⠀⠸⣿
⠀⣠⣴⣦⣄⡀⢸⣷⡄⠀⠀⣰⣾⡇⠸⣦⡞⣿⡁⠀⠀⠈⠛⠛⠛⠉⠀⣿⣿⡇⢹⣿⠀⠈⠀⠈⠀⢸⣿⠁⠀⢿⣆⠀⠀⢸⡟⠀⣠⣶⠛⠓⣦
⣾⡟⠷⠶⢻⣷⠈⢿⣧⣿⣿⣿⡟⠁⠀⠙⠓⠛⠃⠀⠀⣠⣾⡿⣿⡿⣷⣮⡉⠀⠀⠙⠿⣿⣿⣿⡿⠟⣡⣴⠶⢾⣟⠿⠿⢛⣠⠀⣿⣸⠀⠀⣿
⢻⣇⣠⣄⣸⡿⠀⣀⡈⠉⠉⠁⣠⣾⣿⣿⡿⢦⡀⠀⣼⣿⡿⣷⠈⢰⣿⣿⣿⡄⠀⠀⣀⣀⣀⡀⠀⢰⣿⣤⣶⣦⣽⡇⢰⣿⣿⠇⠈⠛⠻⠛⠁
⠀⠉⠛⠛⠉⠀⣶⣿⣿⣶⡀⢰⣿⣿⡛⠉⣰⣿⣷⠀⣿⣿⣷⣌⠃⢈⣻⣿⣿⡇⢠⣾⣿⣿⣿⣿⣦⠈⢿⣁⣉⣈⡹⠃⠋⠛⠯
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⠀⠈⢿⣧⣝⣾⣿⣿⠏⡀⠹⣿⣿⠛⠈⠛⣿⣿⡟⠀⢾⡿⣻⣿⣿⣻⣿⠀⢀⣬⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡛⠁⢼⡿⠂⠈⠙⠻⠿⠿⢛⠉⢀⡀⠈⠷⣶⣶⣶⡶⠏⠀⣿⡋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣧⠀⣠⣶⣾⣷⣶⣄⠀⢄⢸⢞⢣⢤⡀⢠⣤⣍⠁⢀⣰⡆⠙⠳⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣸⡟⠉⠉⠉⠉⢿⡆⠀⢍⡷⢛⠆⠁⠻⣿⡿⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣤⣄⣠⣤⣸⠃⣠⠀⡠⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⡄⠀⢸⣿⣎⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣧⠀⠈⠙⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⠿⣿⣿⣿⣿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢸⣿⣿⣶⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣼⣄⣈⣉⣋⣉⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠶⠟⠙⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⢉⡉⠀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣄⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⠼⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠛⠛⠛⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⡀⠀⠂⠀⠀⠠⠶⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠤⡄⠉⠻⣿⣷⣾⣀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠶⢴⠞⠁⢰⠾⡇⠀⠀⢤⡄⠀⠤⢼⠟⠛⠛⠓⠀⢀⡀⠀⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠰⠃⢻⣤⣾⣧⠆⠈⠁⠀⡀⣠⣴⡆⠈⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⡀⠐⠀⢿⣶⡦⡤⠀⠖⠀⣶⢀⡄⠈⣿⠿⠋⠷⠖⠀⠀⡙⠛⠛⠀⠀⠐⠳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠉⠁⠚⠁⢄⡀⠀⢀⣤⣼⣿⡿⡘⢀⢠⣀⣶⣤⠀⣤⠉⠀⠀⢠⠤⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⡀⠀⠀⠀⠀⡀⠉⠩⠉⠓⢷⣿⣦⣿⣟⣛⣿⣿⣾⣆⡄⠐⠀⠐⠀⠠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠖⠀⠋⠉⠂⠠⣄⢀⣼⠟⠉⠁⠉⢻⣿⣿⣿⡿⣓⠂⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠸⣿⡇⠀⠀⠀⠀⠘⡿⢉⣿⣿⡋⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⢦⠄⠀⠞⠋⡀⠚⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠐⢀⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣄⠀⠉⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢫⠀⡤⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⡄⠶⠇⢀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠋⠚⡃⢀⡀⠈⠃⠀⠀⡀⠀⠐⠃⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠙⢎⠁⣿⠁⡀⠀⠁⠉⠀⠀⡄⠉⣭⡀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠷⠰⠋⠑⢧⡀⠋⠀⠀⠀⠀⢠⣄⢰⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠂⠀⡀⠀⢀⠄⠀⠘⠢⠤⠀⠁⢦⠀⠀⠀⠐⣯⡉⣿⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠦⠈⡃⠀⠸⠃⢠⡓⠀⠀⠆⠀⢀⡃⠀⣻⡄⠈⣧⠉⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢠⡤⠁⠀⠈⣁⠀⡟⠆⠀⠛⠀⠀⢻⠃⠈⠛⢿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣙⠀⠀⠰⠎⠁⠘⠋⠁⠴⢀⡙⠀⠶⠀⢻⠤⣀⡀⢠⡤⠴⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠀⠠⠠⠆⠐⢀⠀⠠⠙⠰⠆⣀⣼⠀⠉⠷⠀⠀⠀⠀⡠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣀⠰⣌⣀⣧⡄⠀⠀⠀⠁⠈⠱⠋⠀⠀⠀⠈⢻⣆⡀⠶⠛⠚
⠀⠀⠀⠀⠀⠀⠈⠁⠋⠙⠿⢁⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡏⠠⠆⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠒⣰⠃⣠⣄⠀⠀⠀⠀⠐⡃⠀⠀⢸⣬⡅⠈⠓⢀⠀⢰⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠶⣧⣄⣻⠀⠀⢠⡶⠀⠈⠁⠀⠀⣸⠉⠀⢮⠁⣄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡏⠶⠚⡻⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠁⠸⠐⢶⣋
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠶⢄⠶⣀⠤⢨⠗⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⠀⠀⣿⠛⢁⡀
⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠃⠒⠀⠀⠟⠀⠀⠀⠹⠥⡶⠄⢀⣖⣠⠟⠛⠁
⠀⠀⠰⣿⡃⠀⠀⠀⠀⠀⠀⢒⡀⠠⡐⠆⠀⡀⣾⠃⠀⠠⠀⠲⠀⠀⠠⠈⢙⣁⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠘⢣⡄⠀⠀⠀⠀⠀⠀⠀⢂⣤⠘⠢⣤⠀⠀⠀⠄⠀⢀⠀⠀⠲⡶⠆⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⢀⣄⠙⠀⠀⠀⠀⠀⠀⠐⣿⣿⢀⣤⠘⣀⠰⠂⡄⠀⣼⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠐⢶⠀⢠⣦⠀⠠⣌⠛⣤⡿⣀⡽⢃⡀⠀⢰⠇⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣅⠀⠀⠙⠻⣿⣾⡏⠀⠁⢄⠀⡜⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠀⣴⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⣠⡀⠘⡿⠀⠈⠀⠀⠰⡇⢀⠃⠀⠀⠀⣀⣦⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢺⣷⠀⠀⠀⠀⢰⣦⠀⠠⠐⠚⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣴⠀⠀⠀⢈⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠘⠻⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠈⢰⡞⠀⠀⠀⠐⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢄⢸⡇⢀⣾⠇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡆⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⢀⣞⢿⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢾⣽⣶⣶⢳⣶⣿⣿⡇⠀⡀⠀⠀⠀⠀⢀⣤⢾⠉⠁⣠⣄⣨⢽⡶⢄⠀⠀⠀⠀⠸⣟⣩⠇⠀⠀⠀⠀⠓⠈⢸⣀⣀⣀⣀⣀⣴⣭⠥⠚
⠀⠀⠀⢰⠀⠀⠀⣷⣭⠉⠉⠀⣘⣿⡇⠀⠉⠀⠀⠀⢠⢾⡵⡏⠀⠸⣿⣿⣿⢿⣧⡈⢳⠀⣄⡀⠀⠈⠁⠀⠀⠀⠀⣠⠶⠛⠋⠿⣌⠀⠀⡤⠀⣠⡀
⠀⠀⠀⠈⠀⠀⠀⡏⠙⠀⠀⠀⢸⣚⡇⠀⠀⠀⠀⠀⣞⣋⣤⡽⠀⠀⠿⢯⣔⣓⣠⠟⠀⡇⠈⠹⠀⠀⠀⢀⣀⡀⣀⠹⡀⠂⠈⠀⢈⠑⢄⠑⠀⠉⠰⠇
⠀⠀⠀⠀⠀⠀⡼⠉⢳⠀⠀⠀⠚⠃⣇⠀⠀⠀⠀⠀⢏⠁⢀⣴⡶⣤⡀⠀⠈⠉⢁⣴⣶⠇⠀⠀⠀⠀⠀⠘⠉⣁⣹⣅⣘⢦⡀⠀⡀⠠⡈⢧
⠀⠀⠀⠀⠀⢸⣡⣶⣷⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⡜⢆⠻⣅⣴⣆⡇⠀⣠⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠀⣀⠭⠛⢷⣤⣤⡤⠎⠲⠄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠂⠈⠓⢬⣻⡛⣃⣀⣿⡿⠟⠁⠀⠀⠀⠀⣀⣠⢠⠄⣷⣄⡀⠉⠄⠀⠨⠀⢱
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⢖⣛⡲⡄⠀⠀⣀⣀⠉⠉⠉⠀⢀⠔⢤⠀⠀⠀⠀⠉⠁⣨⠷⠛⡉⠍⡻⣍⠙⠛⠛⠉⠀⢀
⠀⠀⠀⠀⠀⠻⣿⣛⣛⣻⣿⣿⣿⡿⠿⠋⣾⢼⠦⢇⡇⠀⠘⢥⡾⠃⠀⠀⠀⠈⠲⠛⠀⢠⡚⣦⠀⠊⠁⠂⠁⠀⠀⠀⢈⡇⠀⠀⠀⠟⠻⠰⠆⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⢀⣀⡀⠃⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⡔⠉⠀⠀⣆⣀⡇⠀⣏⣉⠁⣰⡄⣴⡀⠀⡆⠀⢯⡉⠀⠉⣿⠉⢸⣠⠞⠀⠘⠞⠁⠀⠒⠒⢿⣷⣾⠆
⠀⠀⠀⠀⢠⣀⡄⢠⡀⣤⢀⠀⡀⢀⢉⠉⠀⠸⣀⢀⡀⠐⠇⠀⠸⠀⠻⠤⠀⠃⠛⠁⠃⠀⠇⠀⠰⠛⠀⠀⠙⠀⠘⠀⠉⠀⠀⠂⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠸⠁⠇⠹⠀⠛⠘⠀⠉⣿⠻⠸⡇⣀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠐⠃⢠⡟⠛⡇⠀⠀⢸⠿⢷⠀⠃⠀⢸⠛⠛⡇⠙⠀⠀⠀⠀⠴⠀⣶⢻⡿⠿⠿⠟⠻⢷⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠁⠭⢉⠳⣴⡒⢲⠄⠀⠀⠀⢸⡇⠀⡃⠀⠀⢸⠀⢸⠀⠀⠀⢸⣤⣤⡇⢀⠀⠀⠀⠀⠀⠀⡟⠉⡇⠀⠀⠠⡣⢼
⠀⠀⠀⠀⠀⢊⠘⢀⡎⠀⠀⠀⠀⠀⠀⣡⣴⣸⡖⠋⠀⠀⠀⡇⢰⣟⣿⡇⠀⠀⢸⣀⣸⠀⠀⠀⢨⣿⣿⡇⢸⡀⠀⢠⣄⡄⠀⣧⣄⡗⠀⠀⢀⡧⢸
⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢀⠠⠄⣂⡥⢖⢛⠽⡇⠀⠀⠀⠛⡟⠛⣿⣿⡿⠛⠛⠛⣿⣿⠛⠛⠛⢻⣿⣿⡟⢻⠋⠀⠈⠿⠁⠀⣇⣤⣧⣤⣶⣾⣶⣾
⠀⠀⠀⠀⠀⣰⢯⢻⣄⡬⠴⠒⠉⠁⠀⢰⢎⡼⠁⠀⠀⠀⠀⡇⠀⣿⣷⣇⠀⠀⠀⣿⣿⠀⠀⣀⣨⣿⣿⣇⣼⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠉⠙⠦⣄⣀⣀⣀⣀⠴⠋⣠⣤⣤⡀⠀⠤⣿⠽⠿⠛⠛⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠀⠀⠙⠀⠀⣴⡄⠀⠀⠉⠛⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣉⣩⡀⠀⠀⣇⠻⣪⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣹⠃⠀⢀⣀⡋⠉⣟⡆⠀⠀⡄⢀⡤⠴⠧⣄⣀⣀
⠀⠀⠀⠀⢀⡐⠀⠀⣤⠤⠶⣿⣛⣛⡟⠉⣻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢹⠀⠀⠀⠈⢁⡴⠞⠋⠤⣹⢦⢸⠀⠀⠠⢰⠋⣀⠀⡈⠜⣿⠤
⠀⠀⠀⠀⠀⠀⣠⢴⠷⠾⢷⠖⢾⠉⠀⠄⢹⠀⠀⠀⣸⠀⡤⠖⢦⣀⣤⣤⣴⡁⠀⣹⠃⠀⠋⠊⡼⠛⢀⡴⠚⢹⠁⢊⡇⠀⠀⠤⢼⠀⣏⡉⣹⢆⣼⠧
⠀⠀⠀⠀⠀⠀⣧⠘⠒⠖⠋⠀⢸⠀⠄⠀⢸⠀⠀⠀⠘⠲⣇⠀⢠⠟⠉⠁⠀⠙⠋⠁⠀⠀⠀⡼⠁⡰⠃⠀⣀⡇⠠⣿⠀⠀⠀⠐⠘⣆⢹⢣⣽⣼⠟⠄
⠀⠀⠀⠀⠀⠀⡗⠐⠂⠀⠀⠀⢸⠀⠁⡂⢸⠀⠘⣤⡆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⢸⠃⠀⠀⢿⡇⢸⡟⡿⠁⠀⠀⠁⠀⢻⣇⣿⠉⠂
⠀⠀⠀⠀⢀⠀⡇⠀⠒⠀⢀⡀⢸⡀⠀⠤⠈⡇⠀⠙⠀⢀⣴⣾⠇⠀⣠⣶⡆⠀⣀⣀⠀⠀⢸⡇⠈⢳⠤⣀⣀⣹⣋⠉⠀⠀⠀⠀⠀⠀⣼⣼⣿
⠀⠀⠀⠀⠚⠁⡇⠀⠀⠀⠀⠀⠸⡧⠴⠖⠚⠁⠀⢀⣤⣞⠛⠛⠒⠲⣿⡿⠤⠿⠾⠼⠁⠀⠘⣧⠸⣯⡇⠀⠀⠠⣭⢉⡇⠀⠀⠀⡀⣀⣿⣿⣿⠀⠄⠄
⠀⠀⠀⠀⠀⠀⣣⡤⠖⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠁⠹⣿⣷⡀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠷⢶⣦⣤⣤⣤⣼⠟⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣒⣚⣉⣉⣉⣍⣉⣉⣙⣒⣒⣶⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠰⢞⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣷⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠓⠶⠤⠭⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠤⠴⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⡶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⣴⡶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡀⢠⠀⠀⠠⠄⠠⠀⠄⠠⠄⠠⠀⣉⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡾⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠎⠑⠠⢤⣤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠤⠄⠀⠀⠈⢧⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠋⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡼⠁⠀⠀⠀⠀⢯⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡜⢦
⠀⠀⠀⠀⠀⠀⣠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢧⡀
⠀⠀⠀⠀⠀⡰⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢳⡀
⠀⠀⠀⠀⡼⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⡄
⠀⠀⢀⠞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⣄
⠀⢠⢎⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣆
⠠⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠃
⣦⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠
⢯⠳⢭⣓⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⣋⡵⢫
⠀⠑⠦⣄⣉⠙⠒⠲⠮⠭⢭⣥⣒⣒⣒⣒⣒⣢⣤⠤⠤⠤⠤⠤⠤⠤⠒⠒⣒⣒⣒⣒⣒⣋⣩⣭⡭⠤⠶⠒⠚⣉⣠⠴⠋
⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⠤⣤⣀⣀⣈⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣀⣠⡤⠤⠤⠶⠒⠚⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠘⠁⠀⢈⠂⠀⠀⠀⠀⠺⣿⣿⣿⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠿⠿⠇⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠣⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣷⣷⡄
⠀⠀⠀⣀⣀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⣧⣦⣤⣤⡄⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⠀⠀⠈⣿⣿⣼⣿⣾⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⡟⣾⣿⡇⠀⠠⣶⣶⣶⣶⣶⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣠⣿⣿⣿⣄⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠿⣿⣿⣧⣿⡿⠃⠀⠀⠙⠿⢿⠿⠟⠁⠀⠸⢿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⢤⣤⣤⠀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠈⠉⢱⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣀⣸⣄⣼⣀⣸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⠀⢰⣤⣼⠀⢰⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢙⣉⣻⠋⢻⣉⣻⠋⡟⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⠿⢿⣿⣿⠿⢿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡂⢸⣿⣿⠀⢸⣿⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣶⣾⣿⣿⣶⣷⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣶⣶⣿⣷⣶⣾⣿⣶⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢡⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⢾⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⣲⢤⡀⠹⣿⣧⣿⣿⢃⣤⡖⣲⣶⣔⣦⣄⡀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣾⣷⣤⣿⣆⣿⡦⡈⠻⠛⢡⣾⣻⣷⣿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⣿⣧⣤⣴⣶⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠓⢦⣤⣝⠙⡻⣿⣿⣷⠀⠂⢸⣾⡟⣡⠇⠀⠀⠀⠀⠀⠀⠐⡄⢀⡆⢾⣧⣴⠛⠋⠀⠙⠻⠿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠷⠤⠽⢿⠀⡔⠐⡯⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⡴⠏⠓⠀⠀⠀⠀⠀⢀⣠⠆⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⢀⣀⠀⠀⠀⣀⢈⣡⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⣄⣀⣀⡀⠀⢄⠀⠀⢀⠂⠀⠀⢀⣸⢉⣉⣠⣼⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣲⣦⣽⣦⡐⣤⣄⣠⣤⣄⠀⠀⠀⠈⣷⡿⣿⣿⣿⣷⡶⠦⢱⣦⣼⣿⣍⣶⡾⢿⣿⣿⣿⣗⣾⠗⠲⣤⡀
⠀⠀⠀⠀⠀⢀⢐⣿⠟⠩⠤⠤⠈⣉⡛⢿⣼⣿⣿⣿⣿⣿⣆⡀⠀⣼⣿⠟⠻⣿⣿⣧⡀⡟⠓⡛⢉⣠⣤⣴⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣦⡤⠄
⠀⠀⠀⠠⠖⠋⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡉⠋⢀⣰⣾⣼⠏⠀⠀⠉⢹⣿⣾⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠉⠉⠁⠀⠀⠈⢹⣽⠏⠁⠀⠀⠀⠀⠀⢻⡿⠟⠋⠋⠩⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠷⣄⠀⠀⠢⢤⣬⣵⣏⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣄⣀⠉⠉⢉⢉⣁⣀⣹⣧⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠿⠛⠛⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠙⠿⡿⢻⡿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣿⡇⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣱⣶⣿⣿⣿⣿⡇⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣽⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡠⠀⣀⢀⣀⠀⣀⣠⠔⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠛⠿⣿⣿⣿⣿⣇⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣿⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡠⠄⠒⠀⠀⠈⠁⠀⠀⣠⣔⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠖⠉⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⠜⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡴⠃⣰⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣉⣀⢀⠀⠀⡀⡀⠐⠨⡢⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣼⠇⢰⠃
⠀⠀⠀⠀⠀⠀⢠⡲⢿⣿⣿⣿⣟⠳⢖⣒⡚⠵⡲⣦⣄⢱⡀⠀⠀⠀⡠⠊⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣠⣿⠏⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠿⢿⣶⡾⢿⡟⠛⠛⠂⠉⠫⣷⣿⣆⣠⢊⣠⣴⠶⠛⠉⠀⠀⠀⠀⠠⠤⢀⣠⠞⠁⢀⣼⣿⠏⢠⡞
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠒⢁⡴⠊⠙⠒⠦⢄⣒⠤⣄⣈⣿⣿⣿⣿⣭⣤⣤⣤⣀⣀⠀⠀⠠⠄⠠⢤⣀⣀⢠⣾⣿⡏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⢀⡙⢿⣿⣿⡿⣿⣿⣯⣭⣭⡭⠭⠍⠉⠒⠂⠒⠈⠉⠹⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠋⠉⠙⠛⠛⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠤⠀⠀⢽⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠀⡨⠋⢵⣶⣶⣝⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣶⣭⣵⡌⢙⣆⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠞⠛⣿⣿⣿⣿⣿⡁⢰⣮⡅⠀⠛⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⠙⠂⠀⠀⠀⠀⠐⢆⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣶⣦⣄⡀⠀⠀⠀⠀⠈⠉⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⠛⠛⠋⠤⠀⢒⣀⡉⠀⠈⠀⠀⠈⣀⡒⠂⠤⠄⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡩⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣁⠒⠀⠠
⠀⠀⠀⠀⠀⠀⣠⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢒⡀⠉⠀⠀⠀⠀⠀⠀⠈⡅⣒⠐⠌⠤
⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠩⠤⠐⠂⡀⠀⠀⠀⠀⠀⡐⢒⣰⣸⣀⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⣒⣉⠭⠄⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠠⠄⠒⣒⠈⠉⠀⠀⠀⠀⠁⠉⣐⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⣀⣀⣀⠐⠈⠉⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢹⣿⣿⣿⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿

⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡤⢠⣀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⣷⡀⠀⠀⠀⠀⠀⡨⠟⠓⢤⣰⣶⠀⢰⠁⠀⠀⠈⢃⠀⠀⠀⢀⠄⠠⡀⢈⠀⠁⠀⠀⠠⠀⠀⠢⡑⡀⠀⠈⢂⢀⠐⡄
⠀⠀⢿⠹⣆⠀⠀⠀⡜⢀⡀⡈⠀⢹⠉⠀⠘⣄⠀⠀⢀⡆⠀⠀⠀⣵⣶⣶⣶⠃⠀⠐⠀⡀⠀⠐⠀⠀⠑⡘⢄⠀⠨⠈⠀⠐⠄
⠀⠀⢸⡄⠘⢿⠀⠀⣹⣄⠀⠀⢀⡜⠀⠀⠀⠈⠉⠉⠁⠹⣷⣄⠀⠙⡿⢿⣯⣿⣿⣿⣾⠀⠘⠀⠀⠀⠀⠈⢄⠒⠁⡄⠀⠀⢈
⠀⠀⠸⠇⠀⠀⠰⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⡄⢠⡶⠀⠀⠈⠛⠃⠀⠀⠀⠂⠉⠉⠉⢣⣄⠀⠀⠀⠀⠀⠀⠀⠢⡀⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⢀⣀⣼⣥⣤⠀⠀⢀⣴⣶⣶⣦⣤⠄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⢀⣀⠀⠀⠀⠀⠀⣼⠋⠙⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠰⢿⣿⣷⣿⣿⡁⠀⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⢾⣿⣿⣶⣲⠀⠘⠳⢾⡷⠾⢶⣄⠀⠈⠻⠿⠿⠋⠀⠀⢰⣿⣿⣿⠿⠗⠀⠧⠰⡷⠀⠀⠰⠆⠀⠀⠠⠊⠀⠀⠀⠀⠑
⠀⠀⠀⡽⠉⠉⠀⠀⠀⠀⠸⣇⡀⣠⡟⠀⠀⢀⠀⠖⡦⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⢠⣴⣧⠀⠀⣤⠴⢶⡀⠀⠉⠛⠉⠀⢠⣾⣦⣤⡀⠀⠐⢳⡀⢀⠠⢀⣸⠀⣆⡀⠤⡀⠀⠀⠀⠠⢢⠀⢃⠀⠀⠀⠂⠢⢀⠆
⠀⠀⠀⠉⠀⠀⠿⣤⡼⢋⡀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣴⡀⡇⠀⠑⢤⣼⣿⣧⣤⠊⠀⠀⠠⠀⠀⠀⠀⠀⠁⠂⠄⠠⠐⠁
⠀⢤⣦⡀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠈⢄⠙⢿⡄⠀⡿⡱⠁⠀⣊⠀⠼⠓⡧⣤⣑⠄⠀⠁⠀⠸⠁⠀⠀⠀⢀⣴⣤⡀
⠀⠈⠻⡟⠀⠀⠀⢲⣿⣿⡇⠀⠀⠀⢀⠀⠀⠉⠦⢦⡔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠂⠀⡠⠤⠀⠀⠀⠀⠉⠛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠶⠷⠒⠀⠀⠶⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠂⠂⠁⠀⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⡖⣶⠆⣦⡀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠉⠛
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠯⠽⠀⠒⠀⠀⠀⠀⠀⠰⢿⡟⢩⠧⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⡞⠛⠛⠛⠛⠛⠛⠉⠛⣷
⠀⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠠⠀⢁⡀⠠⡄⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣛⣻⣟⣛⠛⠛⠋
⠀⠀⠀⠉⠉⠙⠓⠒⠦⠤⢄⣼⠁⠀⠀⡒⠉⠙⣶⣄⠀⣤⠘⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣌⠉⠉⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣼⣿⡖⠁⠙⠟⢣⠀⠀⠀⣴⠟⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⡴⠋⣹⠆
⠀⠀⠀⠀⡠⠄⠀⢀⡀⢰⠡⠀⠀⢇⡘⠉⠑⣤⣤⡄⡘⠀⢰⠟⠀⠀⠀⠀⠀⠶⠆⠀⠀⣾⣿⡧⠀⠀⠄⠐⣏⣠⠞⡡⣪⣶⣄
⠀⠀⠀⠀⠠⣰⡀⠘⠀⠀⠀⡀⠂⠈⠻⣦⠀⠛⡿⠋⠀⠀⠀⠀⠀⠠⡶⢠⡜⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⢻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⡀⠀⠀⠀⠀⠠⡆⠀⠚⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁
⠀⠀⠀⠀⠐⠁⣠⠀⠠⡌⠁⠀⠀⠀⠀⢀⣦⠈⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠠⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡂⠉⢂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠐⡐⢀⣀⡀
⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠸⠅⠀⢨⡀⠀⢠⠀⠀⠀⠀⠈⠉⡅⠒⢦
⠀⠀⢀⡄⠴⠥⠤⠤⠤⠤⠴⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⠶⠃⠀⠀⠀⠀⠀⠘⠀⠀⠸⢧⡆
⠀⠀⠂⢵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⢸⡀⠀⠀⠀⣀⡴⠁⠀⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠈⠉⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠨⠟⠀⠀⠈⠋⠀⠀⠀⠀⠴⠀⠀⠀⠀⠰⠀⢡⠀⠀⠀⠀⠀⠈⠁⠃⠙⠿⠶⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠘⠀⠀⠀⠀⠀⠀⠀⠰⢸⠀⠁⠀⠀⠰⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣤⣤⡤⠀⠀⠀⠀⢰⠁⣇⠀⠄⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢧⠀⠀⠀⠀⠀⠁⠁⣆⡆⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠲⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⡅⢶⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣇⠀⠀⠟⠁⣸⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣿⣿⣿⠀⣿⠀⣆⣀⡤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢿⣿⣧⢰⠦⠟⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⠆⢸⣿⣧⠈⣠⠚⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⡟⣋⣨⡽⠿⠛⣿⣿⠗⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣏⣻⠟⣿⠀⠶⠂⠁⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠈⢿⣷⡶⡄⣄⠒⣘⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⣏⣤⣤⢰⣂⣭⣿⣿⣿⣷⠶
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠏⠀⠈⠙⠋⠛⠛⠋⠉⠁
⢀⠙⠉⣠⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀⠀⣿⣰⣿⠄⠀⠀⢀
⠀⢱⡶⢼⡧⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣦⢙⣟⣁⡤⠂⣠⡎
⠀⠈⠃⡤⠛⠀⠈⠁⢠⣤⠀⠀⠀⠀⠀⠀⠀⢀⡘⢿⣿⣿⡿⠛⠁⡤⣨⠏
⠀⠀⠀⠀⠀⣤⡄⢳⢀⡟⠀⠀⠀⠀⠀⠀⢠⡌⢻⣄⠙⣿⣿⣧⡾⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡈⠃⠎⠀⢠⠀⠀⠀⠀⠀⣿⣿⣄⢿⠀⡞⠉⠀⣀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⡞⠀⠀⠀⠀⠘⣿⣿⣿⣧⣷⢁⣶⠈⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⢀⡰⣤⡀⣿⡿⠙⢿⡿⠈⣳⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⣠⠛⢱⡍⢿⣾⠃⡀⠀⢀⡾⣥⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡆⣿⣀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣯⣄⠀⣤⡈⣿⡀⢲⠀⢸⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⢿⡍⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣉⢈⡉⠀⠘⠿⠻⣷⢸⣦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢿⣿⣦⡀⠀⢿⣿⠋⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣧⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡇⢸⠟⣿⣿⠇⠀⡏⣀⢃⡼⠃⠀⠀⠀⠠⣀⠀⠀⠀⠐⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠊⠀⠈⠻⠀⠆⠘⣿⠚⠁⠀⠀⠀⠀⠀⠙⢷⡌⠀⠰⡀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⢠⡇⠀⢺⣇⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⣤⣹⣦⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠁⠀⠀⣿⡄⠄⠀⠀⠀⠀⠀⠀⠈⢻⡉⣿⣿⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢰⠀⢠⡆⢸⣇⠀⠀⠀⠀⠀⢀⡴⠀⢾⣇⣡⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢲⣤⣄⡀⠀⠙⠷⣦⠸⡇⠘⡟⠀⠀⢀⡦⢀⠈⠁⠀⣬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣌⡳⢦⡄⠀⠘⠇⠀⠀⠀⢐⣈⢰⣷⠐⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠻⢿⣿⣷⡆⢰⡇⠀⠀⣴⠿⣿⠀⢻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⢿⣇⣘⡁⢉⡼⢁⣤⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢀⠙⢿⠁⢩⡄⢸⣏⢉⢡⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠈⡀⣼⡇⢸⣿⡤⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⢘⣷⠘⡇⢠⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⣿⣿⢀⡟⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡆⠘⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢎⣿⡄⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢀⣿⡄⠀⠀⠐⣄
⠀⠀⠀⠄⡀⠀⠀⠂⣤⡀⠘⣿⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠫⢝⣶⣄⠀⠀⠀⠀⠀⡖⠀⢹⣿⠀⢠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⢀⡀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠙⣆⠀⢀⠀⣱⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠖⠀⠀⣠⠀⠀⠀⠀⠁⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⠙⠇⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⣀⣤⣤⣀⡀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢈⣉⡱⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⡇⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⢃⠀⠀⠀⡠⠐⠉⠀⡀⠀⠀⢤⠀⠉⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢆⡔⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢀⠀⠠⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⢈⠀⠀⠘⡄⠁⠀⠒⠀⠀⠀⠀⠀⠆⠈⡂⡽⠒⢄⠐⠕⡠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡸⢃⡀⠅⠹⣄⡔⣉⣂⣄⣢⣠⣠⣄⣱⣤⣇⣁⣀⢑⢿⣋⣡⣴
⠀⠀⠀⠀⠀⠀⡐⠠⠊⠬⠾⠤⠲⢵⢬⣔⠷⢦⢹⡉⠑⢁⡬⠆⠄⠀⠀⠸⡬⠐⠒⢔⠀⠀⣊⡴⠇⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⢀⡴⢻⠜⠣⡪⡧⢯⡁⢕⣙⠒⠲⣶⠀⢐⣌⡪⣐⣗⡡⠪⠂⢕⢇⠁
⠀⠀⠀⠀⡰⠀⠀⠀⠈⠂⠊⡤⣨⡂⢕⣕⣄⡪⢐⣎⡗⠻⣕⠀⠀⣀⠀⡋⠀⠉⠖⡡⢈⢔⢕⡽⠸
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢍⠪⣺⣇⢠⡿⡊⣽⡿⣤⣷⡑⠉⢊⠤⡱⢂⢖⠭⡪⡱⢗⠵⡡⠁
⠀⠀⢠⠁⠀⠀⠀⠀⢅⠌⠀⠸⣋⣌⠞⡊⣕⠄⡪⢐⢡⣱⣹⡶⡋⢕⠅⡪⢚⢕⠈⣊⡔⢁⠔⠁
⠀⠀⠈⢔⠄⡀⠀⠀⠀⠀⠀⢨⠢⠐⠍⡪⠐⢅⢨⣲⡋⢇⡗⣬⣪⣕⡥⢪⠄⠵⠊⣉⠔⠁
⠀⠀⠀⠀⠈⠉⠀⠘⠁⠰⠶⠷⠾⠶⠿⠾⠿⠥⠉⠐⠊⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠠⡀⢀⠄⡀⢄⢠⠠⢀⠀⡀⡀⠄⠠⡀⢀⢀⠀⠀⠄⡀⠄⠀⡀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈⠐⠁⠊⠀⠑⠁⠊⠂⠅⠊⠐⠀⠐⠑⠨⠀⠅⠊⠔⡩⠂⠕⠨⢂⠕⠪⠐⠡⡪⠠⠃⠔⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠰⡀⠀⠀⠀⠀⠀⣠⢎⡕⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠳⡀⠀⠀⠀⡴⢡⠎⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠻⡲⢄⠀⠀⠀⡠⠜⠀⡿⠀⠀⠀⢱⠖⠑⠋⣰⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠈⠢⡱⣄⠀⡇⠀⠀⣇⠀⠀⠀⢘⠀⠀⣰⠏⢀⣄⠀⣙⠀⠀⠀⠀⣀⣤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠙⢮⡑⠋⠳⠀⣿⠀⠀⣀⠜⠀⢀⡟⡰⡟⢠⠀⢧⠀⢀⢔⡫⠋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠳⣄⠀⢇⢹⡀⢸⠀⠀⠀⣸⠏⡜⢀⡜⠀⡰⢟⡵⠋⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢦⣀⠀⠙⣆⠈⠛⠧⡀⣇⠀⢀⡏⡼⠀⡜⡔⠉⣴⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠻⡳⡄⠘⣧⠀⠀⢃⡸⠓⠊⡼⠁⠀⢧⣹⡾⠁⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⡀⠀⠀⠀⠘⢆⠘⢮⢢⣘⣧⠀⢘⡂⠀⣸⠃⠀⠀⣸⡟⠁⠀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠛⢕⠢⣀⠀⣸⠀⠈⢳⡄⠈⣧⣿⠇⢀⡏⠀⠀⠀⢻⠀⠀⠰⣋⡠⢄⡶⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠙⢮⡉⠉⢄⠀⠀⠹⡄⠸⡉⠀⣸⠀⠀⠀⢀⡼⠄⠒⣋⡵⠚⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠙⢦⡎⠀⠀⠀⠹⡄⠈⠱⡇⠀⠀⣰⠛⢁⡴⠞⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⢻⡖⢤⡀⠀⠹⡄⠀⡄⠀⠀⢱⣴⠋⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠆⠀⠀⠹⣤⡃⠀⠀⢳⣸⠃⢀⣴⠟⠁⠀⠀⡔⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠒⠀⠛⠘⠒⠀⠘⠛⠚⠃⠛⠂⠐⠒⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⡀⠀⠀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⣀⠤⠔⠂⢈⣁⠨⣧⣀⢠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⢔⡠⠖⠉⠑⠠⢀
⢀⢴⡚⠉⣁⡤⣔⣲⠭⠕⠒⢉⣀⡄⠀⠀⠀⠀⢀⠄⢀⣀⣀⠤⠤⠒⠊⠁⠀⠑⢅⠀⠀⠀⠀⠀⠉⠒⢄⡀
⡇⠃⢹⡩⠾⣍⢁⡠⠴⠒⠋⠉⢚⢤⠀⠀⡀⠐⠁⡞⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀
⡇⠀⣼⠃⠀⢸⠀⠀⠀⢀⠔⣋⣑⣤⣤⠎⠀⠀⠀⠳⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢈⡵⠂
⡇⠀⣿⡄⠀⣾⠀⠀⢐⣤⠚⣀⢨⣯⣕⠐⠢⠁⠀⠱⡘⢆⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⣠⠴⠊⡡⢧⡄
⣇⣆⡇⡇⠀⡇⠀⢶⡉⠙⠈⠀⠀⠉⠉⠙⠲⢄⡢⢹⡀⢈⣳⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣶⠉⣀⢔⡪⠚⠉⠓⢦⣀
⠀⠈⠷⣇⣀⣧⠶⣟⢍⡓⠦⣄⡀⠀⠀⠀⠀⢀⡹⢦⠿⣉⠀⠘⢄⠀⠳⣄⣀⣠⠤⠴⠖⠒⢋⣉⢿⣔⡪⠖⠉⠀⠀⠀⣠⢴⡟
⠀⠀⠀⠁⠸⣿⣗⠦⣕⡪⡂⢄⠉⠒⢄⡠⠖⠉⣠⣜⣄⡈⠓⢤⡈⠳⣠⣏⡂⢀⠤⢄⣒⠭⠕⠚⠉⢈⣱⠆⣀⡤⠖⠋⣁⠜⣳
⠀⠀⠀⠀⠀⠈⠪⢑⢄⡉⠛⢷⣬⣀⢠⣧⠴⣫⡞⠣⣀⠉⠢⣀⠉⠲⣌⠓⠒⠒⠉⠁⠀⠀⣀⡤⠞⣉⢿⣞⠉⢀⠴⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣐⠀⢈⣯⣀⠴⠊⠀⠣⣀⠈⠳⢄⡈⠑⢤⡀⠙⠢⣄⢀⡠⠖⠋⢁⠤⣪⢕⣩⡽⢚⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠁⠀⠀⠀⠀⠈⠑⢤⡀⠙⠲⣄⡈⠳⣄⣠⣏⠂⠀⣊⣵⡯⠖⢋⡨⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠙⠢⣄⠉⢓⣲⠿⠋⠁⣠⢔⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣹⠫⠒⣀⠔⣪⠔⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣻⣆⣤⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⢀⣀⠀⠀⣀⣀⣀⠀⠀⢰⢴⠾⢿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠁⢢⠄⠀⠀⠀⠈⡿⢉⣻⠁⠤⠀⠒⢒⣦⡤⠟⠀⠀⠀⠢⣄⣀⢀⡀⣀⣀⣂⣀⢀⣀⡀⠄⠀⠀⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⡶⠲⠤⠜⣉⡐⠶⢶⣯⣤⢴⣶⣶⠖⠉⢉⠠⠀⠀⠀⠀⣀⠎⠈⠋⠉⠩⠥⠄⠀⣙⣟⠁⠒⢂⣠⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠉⠉⠑⠚⠳⡞⠉⠤⡧⢂⣠⠴⠓⠀⠀⠀⠙⠛⠓⠒⠒⠒⠒⠒⠚⠁⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⢄⣀⡀⠀⠀⠀⡏⣀⡤⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠁⠀⠀⠀⠀⠀⠀⠀⣴⠶⡂⠀⠐⠌⠉⠁⠈⠉⠏⠓⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⡾⠷⠒⠚⠛⠿⠿⡶⠦⣄⣈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⢣⣤⣤⣤⣤⣤⣴⣧⣄⣀⠈⠙⠲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠭⠁⠀⠰⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠠⠶⠖⠶⠶⠶⠶⠶⢿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⡀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣧⡴⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠛⠛⠛⠛⠙⡿⣟⡛⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⣶⣇⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⣄⡀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⠆⠀⡁⠀⢀⣤⡼⠔⡈⠀⠺⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠓⣦⣤⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠊⡍⠁⡀⠀⡿⣚⢉⡀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠑⠠⢀⠀⠀⠀⠈⠘⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆⠀⠀⠀⡇⠀⢇⠀⠃⠀⠸⡇⠀⡁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠢⢄⣠⣶⣿⣷⡀⠐⡀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠀⠀⡇⠀⢻⠀⠁⠒⠒⠁⠀⡷⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⢸⠇⣹⠿⠋⢁⠰⢅⢀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⢀⠀⢸⠀⠀⠄⠤⣆⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠠⠈⠱⣄⣴⠾⠋⠀⠀⠀⠀⠀⣦⣼⠀⢸⠀⣍⠀⠀⣿⠁⡄⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠉⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⠀⠀⠐⠔⣤⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠓⠞⠋⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠀⢀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣅⣈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⢉⣉⣉⣉⡁⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠤⠤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⠟⠛⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢡⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠐⢖⠚⡞⡒⠒⠒⠒⠒⠒⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠃⠀⠀⠀⠀⡂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⣸⣀⣀⣀⣙⣇⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⡤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡏⡇⠀⠉⣉⡉⢹⡇⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣤⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⢡⠀⠀⠀⠆⢸⡇⠀⠀⠀⠀⠀⡼⠁⠀⠀⣤⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠘⡄⠈⣁⡁⢸⡇⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⣇⠀⠀⡃⢸⡇⠀⠀⠀⠀⠀⢿⢆⠀⠘⠝⣀⠺⢠⣿⠇⠀⠀⠀⠀⠀⠀⣿⣇⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣣⣤⣼⣧⣬⣭⣴⣪⠇⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣯⣶⠵⠃⠀⠀⠀⠀⠀⢀⣴⣻⣧⣤⣼⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠛⠛⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⠒⠒⠒⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠒⠒⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠠⠬⣏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠚⢻⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠐⠒⢷⣻⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠉⠩⣇⣿⡄⠀⠀⠀⠀⠀⠀⣿⡏⠉⢹⢿⡉⢩⣍⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢀⣺⡾⣷⠀⠀⠀⠀⠀⠀⠸⡿⡄⠈⠉⢀⣬⠁⣤⠀⠀⠀⠀⠀⢠⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠹⣝⢄⣀⣀⣉⣀⣉⣀⣀⣀⣀⡴⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⡟


⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⡏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣄⠀⠀⢻⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣦⣾⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣦⣤⣴⣶⣤⣄⣠⣴⡾⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⠉⠀⠀⠈⠉⠉⠁⠈⠛⢷⣤⣶⣶⣶⣶⣄⣤⣻⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣽⣿⡏⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠘⠻⢿⣿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⠿⠃⠀⢿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⡆⡄
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣻⢸⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⡇⣟⣤⢀
⠀⠀⠀⠀⠀⢀⣴⠇⡇⡟⣸⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣇⢻⢸⠸⣧⡀
⠀⠀⠀⢠⣤⣾⣿⠀⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⣿⣷⣤⡄
⠀⠀⠀⢸⣿⡿⢿⠀⡇⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⢸⠀⡿⢿⣿⡇
⠀⠀⠀⣿⡇⢧⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⡇⡾⢹⣿
⠀⠀⠀⣿⡇⢸⠘⢸⢡⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡌⡇⠃⡇⠸⣿
⠀⠀⢠⣿⠀⠀⡇⢸⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⡇⢸⠀⠀⣿⡄
⠀⠀⢸⣿⠀⠀⡇⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⢸⠀⠀⣿⡇
⠀⠀⢸⡏⠀⠀⠃⡌⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢡⠘⠀⠀⢹⡇
⠀⠀⣾⡇⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⢸⣷
⠀⠀⣿⡇⠀⠀⠀⠃⠀⠌⣐⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣂⠠⠀⠸⠀⠀⠀⢸⣿
⠀⠀⣿⠁⠀⠀⠀⠂⠠⠤⣀⣉⠓⠺⣵⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣮⠕⢚⣉⣀⠠⠤⠀⠀⠀⠀⠘⣿
⠀⢀⣿⠀⠀⠔⠀⠀⠒⠒⠀⠤⠭⠕⣲⣭⣛⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣛⣭⣖⠫⠭⠤⡀⠐⠒⠀⠀⠢⠀⠀⣿
⠀⢸⣿⠀⠂⢀⡀⠤⠄⣀⣒⠢⠬⣍⡒⠮⢭⣛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣛⡭⠕⣒⣩⠥⠔⣒⣀⠤⠄⢀⡀⠐⠀⣿⡆
⠀⢸⣿⣍⣐⡒⠤⢭⣑⡒⠤⢍⣓⠒⠬⣙⡒⠬⣙⠢⢝⠻⣷⡀⠀⠀⢠⣾⠟⡩⠔⣋⠥⢒⣋⠥⠖⣚⡭⠴⢒⣊⡭⠤⢒⣒⣨⣿⡇
⠀⠈⠉⠛⠻⠿⢿⣶⣦⣭⣉⣒⠂⠍⠑⠂⠈⠁⠀⠉⠀⠁⠀⠙⢆⡴⠋⠀⠈⠀⠉⠀⠈⠉⠐⠊⠩⠐⣒⣉⣭⣴⣶⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣤⣤⣀⡀⠀⠀⠀⣀⣴⣾⣷⣦⣀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⡿⠿⠿⠟⠿⢿⣶⡶⠿⠛⠛⠉⠁

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⡤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⠀⣀⣀⡀⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⠇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠐⠂⠣⢄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⡇⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣷⣿⡿⠫⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣟⠉⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢿⣿⣿⣿⣏⣿⣏⣿⣿⣿⣿⣿⣿⢯⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⡣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠠⣉⡐⢬⣳⣄⠀⠀⣴⣫⣴⣾⡟⠍⠉⢉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠉⠛⠿⣷⣌⢻⣆⣼⣿⣟⠫⠭⠭⠻⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⠘⠁⠀⠀⠀⠀⠀⠉⠛⢿⠿⠿⡋⠄⠀⠀⠀⠄⠈⠉⠛⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡤⠤⠄⠀⠒⠂⠁⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠧⣀⡀⠴⠂⠒⠒⠒⠒⠒⠒⠒⠚⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠲⠒⢲⠖⢉⣰⣄
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠸⠒⢇⣠⡤⠵⠶⠒⠛⠉⠁
⠀⠀⠀⣼⣇⣀⣀⣀⠀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⢈⣧⠀⠀⠈⢦⡀⠀⠀⠀⠀⣀⠔⠉⣀⡴⠗⠒⠋
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠾⠿⠡⠹⠛⠛⠶⠚⠛⠿⠿⠾⠿⠿⠿⠭⠀⠀⠀⠀⠱⡄⢤⣴⡾⠵⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠁⠉⠀⠐⠂⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡔⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⢟⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠳⠄⠀⠀⠀⠈⠁⠒⠀⢤⠄⢒⢍⣢⡷⠖⠛⡁⠐⡖⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠉⠙⠛⠒⠂⠠⠼⠶⠿⠋⣁⠄⠂⣁⣤⠖⣇⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⡜⠙⠆⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣠⡽⣾⠚⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢃⡀⠀⠀⠀⠈⠉⠁⠂⠤⢈⡉⠀⠀⡀⠶⠂⠛⡋⡕⣈⡴⠄⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠠⠴⠒⠲⠤⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠈⠁⠒⠐⠠⠄⣀⠀⠀⠀⠀⠙⡁⠀⣀⣶⠷⠟⠋⠁⠀⠀⠀⠀⠀⠸⠥⠾⠤⠶⠶⠶⠶⠈⣉⣉⣑⡖⠀⠈⢢⣄⣀⣀⣠⣤⣀⣀⣈⣁⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠴⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢠⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢀⢈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠥⠂⠈⠁⠐⠰⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡔⠂⢸⠞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢠⠀⠈⣿⣷⠆⠀⠈⠻⠟⠁⠘⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣦⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣄⠀⠄⣤⠄⠠⣀⠀⠀⠀⠀⢀⠀⡼⡄⠀⠀⠀⠀⠀⢰⠒⣲⡄⠀⠀⠀⠀⠀⡟⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡀⠑⢄⡀⠨⠤⠀⠀⠀⠈⢀⠀⠘⠀⣰⣆⠀⠀⢹⠛⢛⡆⠀⠀⠀⠀⠀⡇⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠫⣁⣄⣠⡀⠀⠀⡀⠀⠀⠀⡰⠧⠼⡄⠀⠀⠉⠉⠀⠀⣀⣀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⣷⣬⣦⣝⠩⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣴⠋⠁⠀⠀⢰⡇⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠥⣀⣨⠉⠐⠂⢈⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠤⠄⢸⡇⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠴⠐⠂⢀⠀⠐⠂⠀⣁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣖⡀⣀⠀⠒⠀⣈⢀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢈⣉⠋⠋⠙⠙⠉⢉⠁⠀⢀⢈⡉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠁⣧
⠀⠀⠂⠀⠁⠀⠃⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣡⠀⠀⠀⠻⠀⠁⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⠀⠉⡀⠠⢀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡱⠃⠀⠀⠀⠀⠀⠀⠔⡟⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠤⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠠⠀⢐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠠
⠀⠀⠀⠀⠀⠐⠄⠠⠆⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⢂⠀⠀⠀⡀⢀⠀⠀⠀⠀⠄⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢄
⠀⠀⠀⠀⢀⠁⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠐⠀⠀⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠠⢁⠂⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⡀⠔⠁⠈⠒⢀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⡄⢀⢀⠁⢠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡨⠀⡀⠀⢐
⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⡀⠀⠀⠀⢈⠀⠀⢃⠐⠆⡁⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠡⠀⠀⠀⠀⠌⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠠⡀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⢀⣠⣤⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣦⣄⠀⠀⠀⠀⠀⢹⣇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠛⠛⠛⠛⠛⠛⠻⣶
⣤⣾⠋⢡⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠠⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⢀⣀⣤⣶⡶⠶⠞⣩⡿⠋
⠈⠉⠐⢦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠻⠶⣶⣄⡀⠀⠀⡙⢿⣄⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⣰⣤⣴⣾⠿⠉⠉⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⠀⢻⣆⠀⠀⠀⢿⣷⠀⠀⠀⣰⡿⠋⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⣶⣄⢻⡄⠀⠀⢸⣿⠀⠀⣼⠟⠀⠀⣠⣾⢛⣁⣀⡤⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⢸⣿⠀⣼⠏⠀⣰⣿⠟⠓⠉⠉⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠺⢿⣦⡀⢸⡇⢠⡟⢠⣾⠟⠁⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⠘⣷⣿⣃⣤⣴⣶⡶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣰⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⡶⠶⠶⢾⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠟⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢶⣦⣤⣀
⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡀
⠀⠀⢸⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢻⡇
⠀⠀⢸⡇⠉⠛⠻⠷⢶⣶⣤⣤⣤⣤⣴⣶⣾⣛⣛⣛⠛⠉⠉⠉⠉⠉⠉⢹⣏⣿⠉⠉⠉⠉⢛⣛⣛⣛⣻⣷⣶⣦⣤⣤⣶⣶⠾⠟⠋⠀⠀⢸⡇
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠻⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠏
⠀⠀⠀⠀⠀⠉⠉⣿⠳⠶⢶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠶⠶⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⣶⠶⢶⡶⠶⢶⣶⢶
⠀⣿⣏⢩⡅⢸⠛⢝⣿⢠⠄⠀⠀⠀⠀⡀⣤⠀⠀⢠⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡤⠀⠀⠀⢠⡄
⠐⡦⣟⡃⠎⠘⣿⠋⠻⠸⣽⠻⠟⠻⠀⠇⢯⡷⡇⢸⣧⠜⢋⠢⠶⠩⠾⠰⠅⣿⠐⠷⠀⡇⢾⠸⠿⠦⠞⢸⠸⢷⠀⡇⢾⠸⡮⡽⣗
⢰⠓⠀⡱⣆⠨⠉⡶⢽⠀⡄⡀⢀⣀⠀⣠⡀⡀⠀⠬⠁⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠈⠋⠀⠈⠛⠉⠋⠀⠋⠀⠁⠀⠋⠉⠃⠈⠋⠃⠘⠃⠘⠃⠋⠈⠘⠀⠋⠃⠘⠘⠉⠋⠁⠛⠃⠘⠉⠀⠚⠁⠘⠐⠃⠃⠃⠛⠍⠃⠛

⠀⠉⠛⣶⢌⣿⠟⡶⢬⣍⠙⢿⣯⠋⡏⠀⣻⠉⣤⡹⣿⣏⣩⣽⣿⣍⣉⣹⣿⡿⠿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⡈⠋⠉⣉⣻⣶⣿⢟⢳⣋⢂⣦⣯⡀⠙⠉⣁⠂⠉⠛⢻⣅⡉⠉⠀⠉⠁⠒⠇⠐⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠱⠀⠀⠀⠄
⠀⠢⠋⠉⠀⠀⠀⠁⠹⠿⣷⣮⣝⣿⠙⠛⢿⡀⡄⠀⡄⠀⠀⠀⠸⠀⠀⠉⠉⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠀⠀⠀⠀⠀⠐⠟⠿⣷⣽⡟⠋⠡⣤⠀⢻⣧⠁⢀⡃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠀⣐⣶⣿⣿⣿⣿⣿⡿⢿⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⠻⢬⠎⠃⣙⣻⢌⣛⠀⠀⠙⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠛⢋⣭⣍⡉⠉⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠘⢀⠸⠁⠀⠐⠀⠀⢀⣭⡿⣘⣯⠸⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⡍⢻⡄⠀⠀⠀⠀⡃
⠰⠄⢀⠀⠀⠠⡤⠄⠐⣦⠢⣤⠀⠀⣹⣧⠋⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠁
⠀⠠⠁⢀⠀⣀⣀⠀⢀⡁⠀⠀⠀⠀⠻⢿⡆⠀⠀⠀⣠
⢀⠠⠀⠁⡜⠋⠉⠗⢀⠁⠀⠀⣰⣿⣟⣀⠀⠐⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠄⠀⠷⠀⠀⡸⠳⠂⠀⢠⡟⢻⣿⣿⣷⠂⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡟⢧⣄⢀
⢠⠀⠀⠀⠀⠠⠾⠃⠌⠀⠀⠈⠀⠸⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠛⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠦
⢘⢲⡀⠀⠸⣦⣄⣾⠁⡆⠘⣦⡀⠀⠻⣿⣿⣿⣿⠓⢄⠀⠀⠀⠶⠞⣁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣾⡿⣳⣦⢀
⢘⡨⠁⣀⣀⣈⣛⣛⢀⡇⠀⣿⣷⣄⠠⠙⠯⠒⠉⠃⢰⠀⠀⠈⠐⠆⣹⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠸⣇⠻⣷⠾⠃⠀⠀⡀⠀⢀
⠸⠧⡆⢈⠿⠻⠃⣿⣌⠀⠀⣿⣿⣿⡿⢦⣴⣤⣴⡄⠾⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⣿⣶⣶⣷⣽⣿⣿⣻⣿⡃
⢘⠻⠇⡈⢒⡀⠒⢿⡇⠀⢸⣧⣹⡁⠀⠀⠙⠉⠈⠁⠀⢠⡦⢀⠀⢀⠸⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠉⠙⠛⠉⠉⠁
⢘⣻⣛⡙⠻⣧⣗⠺⠇⠀⢸⣿⣿⣧⠀⠀⠘⠀⠀⠀⣆⠀⣇⠟⠀⣴⠃⠀⠀⠀⠈⢟⠃
⠘⠁⢿⣏⣵⣿⣾⣾⠃⠀⣸⣿⣿⣽⠀⠀⠀⠂⠀⠀⣿⡀⠘⠀⠘⠏⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠠⠞⣿⣿⣿⠀⠠⡿⢻⣿⣿⡆⠀⠀⠒⠀⠀⣿⣿⠷⠒⢴⡄⠀⠀⠀⠀⢰⣆
⠀⠀⠸⢃⣬⡿⣿⡏⠀⢸⡇⣾⣏⣽⡟⠀⠀⠀⠀⠀⣿⣿⣇⣐⣈⠉⠉⠀⠀⠀⠈⠙⠓⠓⠒⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠐⠲
⠀⠀⠀⢴⠁⠡⡭⠀⣠⣺⠇⣿⡏⠉⡇⠀⠀⠀⣶⣿⣿⣇⢿⠋⠻⣟⡀⢀⣀
⡀⠀⠀⢸⣴⣶⠇⢰⡟⣿⠀⢸⣧⠀⠃⣄⠀⠀⢫⣤⣿⡟⠀⠀⠀⠈⠁
⠄⠀⠦⠿⠿⠋⣠⠟⠀⢽⠀⠈⠻⠚⢺⣿⣷⣾⢛⡿⠉⢇⠀⠀⣠⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠄⠄⠀⠀⠀⡠⠄⠘⠉⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⣶⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⣀⣀⣀⣀⣀⣀
⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⣀⣤⣄⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠚⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠞⠒⠤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣾⣿⣿⣶⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢁⡴⠛⠁⠀⣆⣇⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠁⠀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠃⢀⠴⠊⢁⡤⠛⠁⠀⠀⠀⠀⠈⠁⠀⢴⢿⠆⠀⠀⣠⠀⡀⠀⠱⡀⠀⠀⠀⠀⢀⠜⠀⠀⠠⣳
⠀⠀⠀⠐⠈⠀⢿⣅⣠⠖⠉⠀⠀⣤⡄⠀⢠⠆⠀⡇⠀⠈⠉⠉⠀⠀⠁⠀⠇⠀⠀⠈⠒⣤⣴⠶⠁⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡄⠈⠀⠀⢺⣀⣞⣀⣀⠀⠀⡄⢀⡀⠀⢠⠀⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀
⠀⠀⠀⠀⠀⠘⠁⢰⣖⠤⣀⠀⢀⢀⠀⠀⠀⡟⢉⣁⣹⠃⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠈⡕⠂⠉⠰⠦⠀⣀⢠⡯⠚
⠀⠀⠀⠀⠀⠀⢀⡮⠉⣻⠚⢁⠀⠂⠂⠀⠀⠙⠋⡼⢉⣹⠂⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⡎
⠀⠀⠀⢀⡤⠒⠊⠁⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⡹⠋⠉⢱⠄⠀⠀⠑⢄⠀⠀⠀⣀⠇⠀⠀⠀⢠⠀⣽⢒⡲
⠀⠀⠀⣿⣿⣶⣦⣄⠘⡆⠀⢰⡠⡇⡀⢰⠆⠀⠀⠀⢧⠄⠀⠉⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⣀⡀⠀⠸⡟⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣷⠇⠀⠘⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠘⣄⡜⠀⠊⠀⢀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⢀⠤⣶⣶⣶⣐⡦⠀⠟⠳⠄⠇⠀⠀⠀⢹⡀⠀⢀⣴⠷⣴⠁
⠀⠀⠀⢀⠠⡞⢶⠀⠀⠀⠀⢾⣁⣀⣀⡤⠆⠀⠀⠈⢺⢻⠟⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣴⠛⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠚⠋⠿⣅⠀⠀⠀⠀⠀⠀⠀⡞⣷⡏⢹⡄⠀⠀⠀⠀⠀⣤⠀⠀⠘⠂⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠠⠚⢧⠀⠀⠈⢛⣦⣄⣼⠆⠀⡀⠀⠀⠀⠀⢹⢻⣷⠘⣇⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢾⣉⣁⣹⠀⠀⠘⠉⠈⠀⠀⠠⠈⡼⣿⡄⢿⠀⠋⠉⠹⡇⢱⠆⠀⠀⠀⠀⠀⠀⠘⠬⠜
⠀⠀⠀⢠⣄⠀⠀⢠⠖⠊⠛⢿⡉⠀⠀⠀⡤⠄⢤⠀⡀⠀⢣⢿⣷⣸⡇⠀⠀⠀⡸⢧⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠈⣁⠀⠀⠙⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠯⠚⠀⠀⠀⠘⠃⠀⢱⡄⠀⠰⣮⣭⣭⣵⠂
⠀⠀⠀⠘⠖⢃⣘⡛⠀⠀⠈⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠇⡇⣀⡤⠒⠁⣠⣀⠀⢸⠀⠉⡇
⠀⠀⠀⠀⢐⠛⠉⠉⢹⠛⠉⡇⠀⠀⠀⠀⠀⠀⣀⡀⠤⠶⠶⠤⠤⣀⣀⣀⣀⣩⣤⡄⠀⠑⠋⣠⠜⠀⠀⠡⡀
⠀⠀⠀⠀⠈⡄⠀⠀⢸⠇⠀⢰⠀⠀⠛⠋⠉⠉⠙⠓⠤⠤⠤⠤⠜⠋⠉⠁⠀⠀⠀⠠⣶⠀⠀⣷⣶⣿⣿⣿⡟⢄
⠀⠀⠀⠌⠀⠉⠉⠹⠥⠤⠔⠚⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠠⠉⠀⠅⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⡄⠀⢀⠄⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠈⠠⢎⡀⠀⠀⠀⠀⠀⡤⠊⠘⠄⠀⠀⠀⠀⡀⢃⢀⠂⠀⠀⠀⠀⢊⠀⢀⡂⠀⠀⠀⠀⠀⠔⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡀⣤⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⢀⠀⣶⠄⠀⠀⠀⠀⠀⡋⢶⠎⡃⠀⠀⠀⠀⣤⡅⢸⡾⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠉⠐⠊⠁⠀⠀⠀⠀⠈⠛⠀⠁⠀⠀⠀⠀⠀⠑⠿⠋

⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⢹⣤⢩⡬⠀⠀⠀⠀⢀⣵⠆⢾⡄⠀⠀⠀⠀⢑⡋⢡⠆⠀⠀⠀⠀⡚⠈⠘⡿⠃⠀⠀⠀⢸⡟⣿⡅
⠀⠀⠀⠼⠁⠘⠱⠀⠀⠀⠀⠈⠓⠦⠈⠀⠀⠀⠀⠀⠟⠁⠉⠦⠀⠀⠀⠀⠉⠰⠞⠁⠀⠀⠀⠀⠈⠺⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣿⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢠⠄⠀⠄⠠⠄⠀⠄⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡴⣟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣥⠾⢋⡄
⠀⠀⢀⣀⣀⣀⣀⣀⡄⡠⠚⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣡⠾⣛⠴
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⣄⣀⣤⣤⣤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡥⢚⣥⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠋⠥⠴⠄

⠀⠀⢠⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢶⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⣤⣤⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠩⠖⠂⠀⠀⣚⡧⠄⠀⠀⠀⠀⠈⠀⠲⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⠦⢴⠀⠀⠀⠀⠀⠀⠈⢨⣖⡀⠀⠀⠠⠄⢀⠀⠤
⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⡀⠈⠡⠀⠀⠀⠀⠀⠈⠀⠀⡄⡄⣀⡠
⠀⠀⢧⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⢿⠀⢄⢢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠤⡀⠀⠀⠒⢝⠄⠁
⠀⠀⠀⠸⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢠⠀⠈⡀⠤⠀⡐⠀⠀⢀⠀⠈⡀⠈⢀⠀⡀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠈⢀⢂⠀⠃
⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠶⠀⣦⠀⠀⠀⠀⠀⠀⠾⣸⡜⣯⣗⢯⠵⡆⠀⠀⠀⠀⠠⠒⠤⡀⠀⠀⠀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⡄⣤⠡⣶⣦⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠼⣠⡎⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⢿⣿⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡁⢹⡂⡈⢀⠎⠀⠀⡁⠀⠂⠀⠐⠀⠀⠐⡘⡿⢰⠀⢀⠂⠀⠀⡄
⠀⠐⠲⣄⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠰⠁⠀⠀⠐⠀⠀⠀⢀⡙⢦⣾⡣⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢈⢳⡁⠀⠀⠀⠀⢈⡗⠈
⠀⠀⠀⠴⠆⠋⠀⠀⠀⠀⠀⠀⠄⠠⠀⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠡⠒⠻⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠒⠂⠈⠉⠉⠁⠒⠒⠒⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⡠⠤⣖⡒⠂⠉⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠓⠒⠲⠶⠤⢤⣤⣀⣀
⠀⠀⠀⢸⢡⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⣄⣀⣀⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⡟⠁
⠀⠀⠀⠸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠸⡇
⠀⠀⠀⣀⣑⣲⣤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠷⠓⠤⢤⣄⡀
⠀⠀⠀⠙⡟⠲⠤⣄⣀⠀⠈⠉⠙⠒⠲⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠐⠚⢷⣦⣀⣀⣤⡶⠿⠿⠗⠒⠒⠊⠉⠉⠀⠸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠈⠉⠉⠛⠓⣶⣦⣤⣄⣀⣈⣇⣠⣤⡶⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡶⠶⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⣈⡿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⢶⣿⣶
⣰⡿⠛⠳⠷⣦⣭⣛⣿⣶⣤⣀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠔⠒⢒⣋⣩⣭⡥⠶⠶⠛⠛⠋⢹⠃
⣿⠁⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣶⣤⣀⣠⠏⡄⢀⣀⣀⣤⣤⠶⠶⠒⣚⣋⣩⣭⡤⠶⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣯⣭⣴⡶⠶⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⣦
⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣖⣷⣀⣀⣤⣤⡶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠒⠒⠒⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠠⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣄⡀
⠀⠀⠀⠀⡄⠁⢀⠄⢄⠀⠉⠀⠀⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠉
⠀⠀⠀⡗⡁⠀⢈⠀⢈⠀⠀⡁⢈⠀⠀⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⡌
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠐⠐⡁⢸⢀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⣇⣂⠿⡀
⠀⠀⠀⡇⠐⠄⢠⠀⠀⠀⠀⡇⢰⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣈⠛⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃⠂⠘⠈
⠀⠀⠠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠄⠘⠚⠀⠀⠒⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⠄⠈⠁⠠⠭⠬⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⡀⢀⣤⠀⣠
⠀⠀⠀⠀⢰⠋⠙⠦⠤⠤⠤⠤⠜⠋⢱⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⠖⠶⣧⠀⢀⡇⠠⡟⣋⡛
⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡎⠑⠒⠢⠤⠄⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠨⢐⡂⢸⠁⢙⡀⢠⢣⣛⣸⠆
⠀⠀⠀⠀⢸⢀⣒⣒⣒⣒⣒⣒⣒⣒⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡁⢸⠀⠐⢅⢨⠀⡆⠒⡀
⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠒⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⢘⢨⡅⢺⠀⠸⡤⠰⠀⠊⠯⠇
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠄⢺⠀⠀⡇⢸⠀⢸⢻⡼
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣿⠀⢘⡇⢘⠀⢸⣶⣾⡆
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠉⠉⠿⠆⠾⠇⠠⠀⠀⠯⠤⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠝⢻⡍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠄⠀⠠⠀⣠⡿⠤⠀⡀⠀⠀⠀⠀⠀⠈⢆⠀⠋⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⣁⣀⡀⠀⠀⢠⡋⠀⠀⠀⠉⡀⢤⣤⡄⠀⠀⠈⠁⠀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠉⣦⣾⠁⠀⢹⡆⠀⣼⣿⣿⣿⣷⣶⡗⠘⠛⠁⠀⠒⠒⠂⠀⠀⠀⠙⢷⠄⢷⠇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠤⠶⠁⠘⠦⠴⠾⠄⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠓⢦⣄⠀⠀⣆⢠⠄⢦⢠⠄⠂⠀⠀⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣲⡒⣰⠆⠀⠀⠀⠀⠈⠉⣿⣿⣿⣧⣰⡋⢉⠶⡉⢉⡆⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠙⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣏⠀⠀⠀⠀⠀⠀⢀⣨⣭⠉⠉⠉⠁⠀⠀⠀⠈⢁⣀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣤⣤⠹⡆⠀⡈⠛⠁⠀⠀⢿⣻⡆⠀⠀⠀⠀⠀⢀⣶⣾⣅⠀⠀⢹⡆⢠⣴⠀⢀⣴⠟⠉⣠⣾⡁⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⢦⣀⣀⠀⠀⢺⣿⣿⣿⣿⣿⠹⣿⣷⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠘⣿⠀⠉⣴⡟⠁⠀⠀⠻⠟⠡⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⡞⣀⣉⢳⡀⠠⣾⣿⣿⣿⣿⡇⢻⣿⡇⠀⢿⠇⠀⠀⠈⡄⠀⡤⠀⠠⣿⠀⣸⡟⠀⣤⣤⡄⠀⠀⠀⡰⠟⠁⠀⠙⠷
⠀⠀⠀⡀⠀⢠⣤⡄⠀⠈⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⢉⠁⠈⢉⣁⣀⡀⠀⠀⠰⠆⠀⠀⠀⠶⢸⣿⢀⣿⠃⠀⣿⣿⡇⠀⢰⡖⠀⠀⠀⠀⠀⠀⢰
⠀⢀⡞⠀⠀⠈⣿⠅⡀⢠⡄⢀⠀⠀⠀⠙⢶⡀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⡆⠀⠆⠀⣼⡏⣸⡿⠀⠀⠻⠿⠇⢰⣦⠈⢆⠀⠀⠀⠀⠠⠊
⠀⠙⠛⢻⡶⠀⠠⢄⠙⠀⠁⠈⠠⠆⠀⠀⢸⡇⠀⣘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢰⣿⢃⣿⡇⠀⠀⠺⣷⣿⣞⣿⡆⠀⠿⠂⠀⠺⠇
⠀⠀⢀⡞⠁⠀⠒⠂⠀⠀⠀⠀⠀⠚⠁⠀⢰⡇⣸⣿⣿⣿⣆⠀⠈⠐⠆⡀⠀⠀⢻⣷⠀⣠⣿⡏⣼⣿⢀⣦⠀⠈⠛⠛⠛⠙⠃⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠈⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣙⣀⠀⢸⡇⠈⠉⠉⠉⣁⡀⠀⠂⠀⠀⠀⠀⢸⣿⣾⣿⣿⣼⣿⠇⣼⠿⢶⣁⠄⠀⠀⠚⠻⠿⠿⠿⠿⠛⢉⡅
⠀⢠⣶⠿⠀⣬⣐⡀⠀⣿⣾⠇⠀⢡⡞⠁⣾⡇⠰⣶⢿⣿⣿⣿⣿⣶⣤⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠀⣀⣀⡼⢣⣆⠀⠀⠐⠒⠂⠀⠤⠐⠊⠁
⠀⡘⠁⠀⠀⠻⣿⡇⠀⠈⠉⠀⣠⡈⠀⠀⢻⣇⠀⠀⠸⢿⣿⡿⠏⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⢨⣿⣦⠈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠾⣿⣦⠀⠛⠁⠀⠀⠀⢠⣿⣶⡖⡆⠘⣿⣦⡀⠀⠀⠀⣴⣾⡀⠀⠀⣰⣿⣿⠟⠉⣠⠔⠀⠄⠀⠀⠀⡤⠀⠐⠀⠀⠤⠀⣉⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠀⢠⡟⠀⠀⠈⠻⣿⣷⣤⣄⡙⠛⠀⢀⣼⣿⡿⠁⡀⡀⢀⠀⣀⡀⠀⢰⣿⡇⠀⠀⠀⡀⠐⡀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠉⣿⣯⡀⠀⠀⠀⠂⠀⡙⢿⣿⣿⣷⣤⣾⣿⡟⢀⡄⠀⡰⠀⢀⡔⠁⢠⠀⠉⠀⢤⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠉⠉⠁⡴⠄⠀⠀⢠⠁⠀⠈⢿⣿⣿⣿⣿⡇⠢⠤⢴⡤⠤⠋⠀⠀⣼⡆⠀⠀⣰⣄⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⢠⣿⣿⣷⠀⠀⠀⢸⡄⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠚⠓⠀⠀⠀⣠⡿⠀⠶⠁⠀⠀⠑⠖⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠶⠶⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⣰⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣤⡤⠶⠶⠒⠒⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢠⡞⢁⣀⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠹⣆
⠀⠀⠀⠀⣠⣿⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠛⠳⢦⣼⢀⣠⣴⢖⣛⣋⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣉⣛⣶⣿⣆
⠀⠀⠀⢠⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡄
⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡞⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⢻
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠛⠿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⡴⠋⠀⢰⡆⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⣠⣾⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⢆
⠀⢸⣁⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢤⣀⣠⣤⣄⡀⢈⣧
⠀⢸⣿⣿⡛⢳⣶⣤⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡀⠈⠗⠀⢠⣬⡁⠈⢧
⠀⠈⢻⣿⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠙⠋⠀⠀⠀⠹⠿⠃⠀⢈
⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⡄
⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⢿⣿⠇⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣒⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣀⡀⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢒⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠻⡇⠄
⠀⠀⠈⢩⣶⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡾⠇⠅
⠀⠀⣴⣾⣿⣷⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠭⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠿⡿⢣⠆
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⡏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲⣮⡀⣸⡖⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⢈⣆
⠈⠢⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢦⢴⡷⠛⠳⣷⢤⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢈⣧⠤⢹⡇⠤
⠀⠀⠈⣇⢰⠁⠹⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠫⣀⣨⢻⡄⠚⠂⡼⢧⣀⡸⠗⠀⠀⠀⠀⠀⠀⠠⠤⡷⠤⢵⡁⠀⣠⡧
⠀⠀⠀⢡⢸⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⣏⠿⣏⡏⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⡵⠒⠛⠁
⠀⠀⠀⠸⠇⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠊⠱⡀⡰⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡤
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰
⠀⠀⠀⠀⢰⡀⠃⣴⠀⠀⠀⠀⠀⠀⠏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣥⠁⠀⠀⠀⠀⠀⠈⠓⠺⠃
⠀⠀⠀⠀⢸⢻⡾⠸⡠⢲⠃⠀⠀⣀⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⠓
⠀⢀⠀⠀⠸⠂⡟⡺⠁⡏⠀⠀⠀⡆⠉⠱⡇⠀⢸⠀⡀⢰⠀⠀⠀⠀⠀⢸⣏⡙⢹⡆
⠀⠀⣷⢄⠀⢠⡇⠃⠞⠀⠀⠀⠀⠙⠀⠚⠀⠀⠀⣼⡇⣾⣧⠀⠀⠀⠀⠸⣏⠀⣨⠃
⠀⠀⠸⣌⢳⡀⢡⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡗⣿⣼⡶⢠
⠀⠀⠀⠈⠑⠀⡄⢰⣦⣀⡈⢇⠀⠀⠀⠀⠀⠀⠓⣆⣿⣾⣿⢇⢈
⠀⠀⡠⠒⢀⠐⡇⠁⠀⡨⠙⠾⠀⠀⠀⠀⠀⠀⠸⣯⣿⡻⣿⣷⡏⠀⠀⢀
⠀⠴⠁⣴⠏⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⣓⡻⣿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠻⡜⠁⠀⠈⣧⠏⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣧⢸⣅⡿⠀⠀⠸⣄⢀⠇⠀⠀⠀⠀⠀⠀⣆⣁
⠀⠀⠁⠀⣤⣴⡿⠦⢤⠀⠀⠀⠀⢂⡀⠐⠀⠀⠀⢀⢽⣿⣿⢧⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⢠⡄⣾⣿⣤⡀⠀⢀
⠀⠀⠀⠀⢿⠋⠀⠂⠒⠀⠀⠀⢠⠸⠷⢤⣄⠀⠀⠀⠀⠉⠉⢁⠀⠀⣠⣧⣤⡀⠀⠀⠀⠀⢸⡃⠀⠀⠈⠇⠀⠈⠃
⠀⠀⠀⠀⠸⣧⡀⣦⠆⠀⠀⠀⠸⣄⠀⣀⠌⠀⠀⠀⠀⠀⠀⠜⠀⠀⣿⣸⣽⠇⠀⠀⠀⠀⠈⢧⣀⣀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡴⠖⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠄⠐⠀⡀⠀⢀⡀⠀⣀⣀⣀⣀⠀⠀⡤⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛
⠀⠀⠀⠀⠀⠀⢡⣬⣭⠥⠤⠤⠤⠄⠀⠀⠐⠒⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⠻⡄⠀⢀⢀⡴⠛⠉⠀⠀⡀
⠀⠀⢀⠀⣈⠀⠀⠀⠀⣄⣙⣘⠀⢀⡠⠤⠀⠀⠀⢠⣐⣲⠶⣾⠀⠀⠘⢿⡵⠋⠀⠀⡠⠀⠀⠀⠀⠂
⠀⠀⡁⠀⡇⠀⠰⣆⣀⡏⢹⢨⠖⢰⣧⣤⣄⠀⠀⠜⠿⠋⠀⠸⠲⡒⠒⣦⠀⠀⠠⠪⣀⠀⠀⠀⠀⠀⠀⠈⠐⡤
⠀⠀⠃⠈⠁⠀⠀⠉⠀⠁⠈⠉⠀⠈⠀⠀⠐⡋⠀⡇⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⢈⣲⣤⡁⠂⢀⠀⠀⢀⠔⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠂⠀⠀⠀⠈⠈⠈⠀⠀⠀⡊⠳⢤⣉⡓⠦⣌⣒⣤⡞⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⢀⡀⡧⠄⡀⠀⠀⢐⠙⠳⢄⡈⠉⠓⠢⢍⣑⡴⠃⠀⠂
⠀⠐⡷⢾⡄⠠⢤⣀⣀⣾⡇⢰⣶⣶⣧⣤⡇⠀⠀⡄⡀⢹⣻⠇⠇⠉⠁⡀⠀⠀⠑⢄⠀⢝⠲⢄⡀⠰⢈⡴⠚⡠
⠀⠀⠀⠐⠃⠄⠜⠀⠐⠋⠓⠘⠂⠚⠃⠂⠓⠐⠒⠋⠀⠋⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠑⠀⠑⠠⡉⣛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠕
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠓⠤⠀⠈⠁⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⡴⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢸⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠇⠀⠀⠿⠴⠶⠶⠾⠿⠛⠓⠶⠶⠶⠖⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⠀⠀⠀⠀⡆⠀⠀⢸⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣌⣉⣉⣉⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢸⡸⠀⢸⡁⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠟⠛⠛⠻⠛⠇⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⢱⡇
⠀⠀⠀⠀⠠⠂⢠⡖⠒⠲⡄⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠃⠐⡠⢃⡄⠀⠀⠀⠀⠰⠪⢿⣄⣀⣴⠟
⠀⠀⠀⠀⠠⡀⢀⠻⠻⢟⠃⣀⠄⠀⠀⠀⣧⣀⣀⣀⣠⣄⣀⣀⣀⣠⡇⠀⠀⠀⠛⠛⢏⠔⠩⠬⠤⡄⠀⠀⢀⣤⣤⣤⣭⣥⣤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣶⣶⣶⣀⣀⣨⣶⣶⡆⠀⠀⠀⠀⠁⠀⣤⣶⣶⣦⡀⠂⠀⠀⠀⢀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⠉⢻⣄
⠀⠀⠀⠀⠩⡭⡭⡭⡭⠭⠭⠭⠁⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠘⠟⢂⣠⣤⣶⣦⣐⠚⠟
⠀⠀⠀⠀⢀⣃⣓⡓⣓⣒⣒⣒⠀⠀⠀⠀⠀⢀⠈⢻⢸⡟⡼⠃⠀⠀⠀⠀⠀⢣⡀⠀⡐⠀⢀⡠⠀⠀⠀⠀⡏⢌⡟⠛⠛⠛⣿⠩⡃
⠀⠀⠀⠀⠠⠆⠦⠆⠁⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠷⠷⠇⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠔⠃⠀⠀⠀⠀⣛⣚⠛⣛⣛⣛⢛⣓⡳
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠒⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠊⠛⠗⠀⠠⠀⠀⠀⠀⠀⠀⡠⠀⡴⢁⠀⠲⠄⢤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢰⣾⠖⠚⠛⠓⢾⣶⡆
⠀⠀⠀⠀⠀⢁⠀⠠⠠⠄⢀⠀⠀⠀⠀⠀⠘⣇⠀⡏⠀⢢⡀⠀⢀⠏⠀⠀⠀⠀⠀⠲⣶⣶⠆⠀⠀⠀⠀⠀⢨⠃⠀⠀⣀⡤⠂⢹⡀
⠀⠀⠀⠀⠀⣀⣤⣀⣀⣠⣆⠀⠀⠀⠀⠀⢠⡜⠓⣧⡙⠛⡡⠔⠋⣆⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠁⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠒⠛⠛⠛⠛⠓⢐⡄⠀⠀⠀⠀⠙⠒⢻⡉⠉⠓⠃⠜⠁⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⢘⡦⢤⣤⠤⢾⠁
⠀⠀⠀⠀⠀⠃⠙⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⣶⠿⠛⠛⠿⣷⣄⠀⠀⠀⢰⠿⠟⠃⠀⠙⠿⢷⡆
⠀⠀⠀⠠⠤⣶⣶⣿⣿⣿⣿⣶⣶⠤⠄⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡙⢿⣵⢆⡀⠀⠀⠘⣿⡆⠀⠀⢸⠀⣤⣤⣤⡄⠀⢸⡇
⠀⠀⠀⢀⠀⣿⣶⣮⣭⣭⣵⣶⣿⠀⠀⠀⠀⣿⡇⠐⠂⣃⡀⣠⡄⠀⠀⠀⠡⠐⢝⢿⣵⣆⡀⢀⣿⠇⠀⠀⢸⠀⠶⠶⠶⠂⠶⢸⡇
⠀⠀⠀⣸⠀⠻⠿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠿⠧⠿⢧⠀⠀⠀⠀⠈⠢⡁⢝⢿⣯⡾⠋⠀⠀⠀⢸⠀⣏⣉⣉⡁⣋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⠙⠟⠁⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢔⡉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⠃⠀⠀⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⣠⡀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣹⠀⠀⠀⠀⠀⢀⡄⠀⠀⠢⠀⠙⠀⠀⠀⠀⠀⠉⢴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢤⡔⠠⠤
⠀⠀⠻⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⢀⣀⠀⣀⣠⣀⢀⡘⠃⠀⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⢠⡄⠀⠁⠶⠂
⠀⠀⠸⡀⠀⢠⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⣿⣿⠿⠿⠛⠻⢿⣷⣶⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡀⠃⠠⠈
⣦⠀⠴⠆⠀⠈⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣾⠉⠙⠀⢠⣄⣀⠇⠀⠙⠋⠀⠀⠀⠀⢃⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⣿⠙⠒⢦⠀⡈⠀⠀⣶⠀⠈⠂⠀⠀⠠⠀⠀⠀⣀⠀⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠞⠧⠀⢀⠀⠀⠀⠃⠀⠄⠀⠀⠀⠀⢜⢀⡔⠀⠆
⠻⢀⠀⠈⠀⠁⠀⠀⣿⠀⣦⡀⠀⠀⠀⠀⠀⠀⠽⣶⣿⣞⠁⠀⠀⠀⠀⠀⡠⠀⠀⠐⠀⠘⠀⠀⡀⠠⠀⠀⡤⠀⠀⠲⠶⢈⠁⡀
⠀⣿⣤⡀⠈⠀⠀⠠⣿⡀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠘⣿⠛⢀⠴⡇⡄⠐⠭⠃⠀⢀⠠⡄⠀⣆⣰⢧⠀⠀⠀⠀⠶⡀⠼⢂⣰⡀⠉
⠉⠁⣀⠘⠆⠰⠅⠀⣻⣄⣽⠀⠀⠀⠀⠀⠀⠀⠦⣄⠙⡧⠃⠎⠁⠛⡀⠀⠀⣰⠛⡠⠓⠃⢸⣝⣤⠀⠀⠀⠀⠀⠘⢦⣌⠀⠀⠠⠦
⠀⣸⠏⢳⡀⠀⠇⢀⡽⠉⣿⣧⢀⡄⠠⠀⠀⠀⣀⠋⢀⠁⠐⠛⢁⣽⠁⣶⢦⣛⢉⣇⠐⡆⢾⡟⠈⠳⡄⠀⠀⠀⠀⠂⠘⠉⡈
⠀⠁⢀⡀⣷⣀⣾⠦⣯⢠⣏⣹⣯⠀⠀⠀⢀⢄⣿⠀⢿⠀⠀⠀⠥⢸⠋⠉⠢⣬⠐⢻⣤⡙⠢⠙⠀⠀⠈⠶⢠⠀⢀⡄⢤⡄⠀⠓⠘
⠼⢆⠀⠛⠚⡉⠖⣂⣞⠒⣉⠙⣟⠳⡄⠠⠀⢸⠏⠀⠎⢓⣦⣇⡄⢠⡀⡀⠀⠀⢰⣷⣾⣾⢤⡄⠐⢃⡤⠖⡴⢖⣘⠢⢂⠉⠀⠀⠰
⠀⢺⡶⣥⠤⡵⡟⠟⣿⠞⠁⠸⣏⠀⢿⠀⣾⢿⡷⢀⡆⠾⠴⢛⡍⢻⠐⠉⣀⡄⠀⢐⣿⡦⣈⠄⠀⣬⣄⢠⣷⣋⠉⡀⠀⠀⠐⢦
⠒⢈⣴⣤⣈⢿⣯⣭⣿⣄⠘⠰⢄⢠⠘⠚⢷⣏⠀⠸⠛⢋⡀⠀⠰⣏⡠⣶⠋⡳⢻⢰⠉⢛⢃⡾⢀⠙⠁⠀⠁⠘⠳⣼⢳⣠⣶⠄⠻
⠻⢧⣟⣿⣱⣮⣦⡜⣿⠫⠀⠀⠹⢀⡿⣿⣿⣿⣿⣷⠶⠆⠠⠰⢾⣋⣾⡛⣼⣯⣣⣬⣃⠉⠡⠇⣼⣀⠄⠀⣴⣀⡰⠏⢀⡗⠙⠺⠃
⣟⠻⣿⠽⠛⠃⢽⣿⢁⣀⣀⠴⠆⣼⡦⣷⠟⠛⣳⡏⠀⠙⠲⠐⢹⣿⠋⡡⠈⣻⣻⣿⠊⢁⣼⠉⣀⠡⣛⣠⣿⡟⣷⢶⣿⣿⣶⡄⢀
⠠⠻⠿⠵⠷⠄⠠⠴⠫⠿⠿⠟⠿⠹⠥⠀⠿⠿⠳⠯⠄⠾⠃⠴⠼⠯⠏⠄⠄⠁⠹⠌⠁⠇⠛⠀⠛⠳⠈⠛⠾⠡⠽⠿⠿⠯⠝⠃⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠤⢠⠀⠀⡒⡆⠀⠠
⠐⡆⠀⡄⠀⢀⡠⠄⠀⠀⡀⢠⣾⣿⣛⣛⣛⣛⡛⣿⣷⡄⢀⢁⠠⣀⠔⠀⠁⠐⠀⠊⠀⠀⠀⡁⠀⠀⢀⠃
⠀⠹⡀⠀⠀⠠⠄⢀⡀⠠⠄⢸⣛⡛⡿⠿⣻⠻⠟⣻⢻⡇⠘⡀⠀⢀⠀⢀⠀⠀⠀⠒⠚⠀⠆⠂⠀⠃⠱⠰⠒
⠀⠨⢷⠀⠀⢀⣈⠂⢒⡁⠀⠘⣿⣷⣾⣷⣶⣿⣷⣾⣾⠃⠀⢈⡀⠒⠀⠁⡀⠂⠀⠀⠈⠃
⠀⣄⠂⠁⡀⠀⠸⠒⠐⠣⠀⠀⠈⠙⠛⠛⠿⠛⠛⠋⠁⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠆⠀⠀⠀⢀⢀
⠀⢃⠀⠀⠈⠰⠸⠀⠰⠀⠂⠁⠀⠘⠃⠀⣀⠀⠒⠬⠁⠀⠀⠀⠀⠀⠀⠀⠉⠤⠂⠢⠶⠠⠥⠅⠠⠄⠤⠠⠤
⠀⡤⠁⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⡠⡀
⠀⠀⠀⢡⡀⠠⠤⠀⠀⠀⠔⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⣋⠀⠴⠰⠤⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠬⠄⠀⠀⢀⠀⠀⠀⠀⠀⢠⣠⠃⢘
⢀⡀⠀⠀⣰⠀⠁⢀⠀⠀⠘⠐⡀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠸⠀⠀⠀⠤⠔⢨⣸⢰⣈
⠈⠁⠀⠀⠈⠀⠀⢉⢼⠄⠀⠀⠁⠀⠐⠅⠀⠁⢀⡤⠤⠤⠀⠀⢀⣀⡀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠐⠉⠉⠉⠉⠁⠐⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⡀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⠈⠋⠉⠁⢰⡿⠙⠛⢻⡏
⠀⠀⢠⡤⠤⠤⠤⣤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⢀⠀⠀⠰⡀⠀⢣⠀⠀⠂⠀⡀⠀⢆⣼⣅⣸⠐
⠀⠀⡂⠀⠀⠀⠁⠀⠀⠘⠒⠒⠀⠀⠒⠂⠀⠐⠲⠂⠀⠀⠘⠄⠀⠀⢃⠀⠸⢷⡀⠠⡄⠁⢀⡈⠛⠛⡁
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠀⠪⠂⠀⠀⠀⢠⠐⠂⡄⢀⠀⠠⠐⠈⠁⠛⠑⡄⠲⠓⠄⠀⠄⠀⠐⠀⠀⠈⠁⠄
⠀⢀⡁⠐⠀⠀⠀⠀⠁⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠐⢀⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠂⣀⡖⠤⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠃⣨⡄⠀⠀⠁⠦⠐⠠⠰⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠿⢖⠋⠀⠀⠁⠁⠠⠀⠀⠀⠀⠀⠀⢀⠱⠴
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⠿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⠿⢷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⠍⢙⡛⠻⠿⢷⣶⣦⣤⣤⣤⣤⣤⣀
⣴⣾⣟⠿⠿⣿⣿⣟⣣⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠖⠛⠉⠡⠶⠚⠉⢁⣠⣴⣶⣿⠿⠿⠟⠛⠛⠛⠛⢿⣦⡀
⠈⠛⠿⣿⣶⣦⣬⣉⠙⠛⠿⠿⣷⣦⣶⣿⣁⣤⣀⣤⡤⠖⠀⠀⣀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠛⠉⢉⣀⣤⣤⡶⠶⠶⢶⣦⣄⠀⢹⣷
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣶⣤⣄⡈⠉⠛⠻⠿⣿⣿⣶⣿⣶⣋⣤⣤⣤⠶⠞⠛⠉⢁⣀⣤⣴⣶⠿⠟⠋⠉⠀⠀⠀⠀⠀⢻⣿⡀⠈⣿⡇
⠀⠀⠀⢀⣸⣿⣿⣿⣷⣦⣉⠛⠛⠿⢿⣶⣦⣤⣀⡈⠉⢻⣿⠋⠉⢁⣀⣤⣴⣶⣿⠿⠟⠛⣉⣠⣤⣶⣶⣾⡁⠀⠀⠄⠀⣼⣿⠀⢰⣿⠇
⣠⠶⠞⢻⣿⣿⣿⡿⢭⣒⠩⠙⠲⠤⣄⡀⠉⠙⠛⠿⢿⣿⣿⣶⣿⡿⠿⠛⣋⣩⣤⣶⣾⠿⢟⡛⢯⡱⣶⡬⡙⠳⣤⣴⡿⠟⠁⠀⣾⡿
⣿⣦⣄⡀⠈⠛⠿⣿⣶⣬⣍⡒⠦⣄⣀⠉⠙⠲⠦⣤⣠⣤⣽⣿⣷⡀⠀⠉⠉⣀⠀⠈⠻⣷⡈⠙⠆⠙⡌⢳⡌⠂⠙⣿⣀⣀⣠⣾⡿⠁
⠀⠉⠛⠻⢷⣦⣄⣀⠉⠛⠻⢿⣷⣦⣌⣙⠒⠦⢤⣀⣈⠙⠛⣿⣿⡷⠀⠀⠈⠁⢀⣀⣤⡼⢿⣦⡀⠀⠀⠀⢷⠀⠀⢹⣿⠻⠿⢿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⢈⣿⡿⣷⣦⣤⣀⠈⠙⠛⠿⣿⣶⣤⣈⠉⠛⣿⣿⣿⣃⣤⣴⡶⠿⠛⠉⠁⣀⣠⣽⣷⡀⠀⠀⠈⠀⠀⠈⣿⣦⢀⣀⠀⠈⠉⠙⢳⣦⡄
⠀⠀⢸⣿⡛⠛⣿⡇⢁⠏⣿⣿⣷⣦⣤⣀⠀⠉⠙⠻⢿⣾⣿⠿⠟⠛⠉⣁⣠⣤⣶⡶⠿⠛⠛⠉⠙⣷⠀⠀⠀⠀⠀⢀⣿⣿⣭⣤⣶⣾⣿⡿⠿⠟⠃
⠀⠀⠈⢿⣷⡀⢿⣇⠀⠀⠋⢴⠋⡿⢿⣿⣿⣦⣤⣀⠀⢀⣤⣴⣶⣾⣿⠿⠛⠉⠁⠐⠉⠀⠀⣀⣰⣿⡆⠀⠀⢀⣠⣼⣿⡿⣿⣿⣿⣏
⠀⠀⠀⠈⢿⣿⡌⠙⢷⣦⣄⠀⠈⠁⠾⢡⡟⣿⣿⣿⣷⣿⣿⣿⠟⡋⢀⠀⠀⢀⣠⣤⣶⣾⡿⠿⣻⣿⣷⣶⡿⠿⠻⢟⣑⣬⣿⣿⣿⡙⠛⠷⢶⣤⣄
⠀⠀⣀⣠⣼⣿⣿⣆⠀⠙⠻⣿⣦⣄⠀⢸⠁⡟⣼⡏⢸⣿⠟⣿⣿⣶⢋⣶⣿⠿⢛⣫⠥⢐⣚⡥⠿⠛⠉⠁⠀⠲⠾⠟⠛⢋⣩⣴⣿⠿⠃⠀⣀⣩⣿
⠀⢿⣏⠀⣶⣟⢻⣿⣧⡀⠀⠈⠙⠿⣿⣦⣀⢡⣿⠀⣿⡇⠀⢿⣿⡇⢸⡟⠁⠀⠀⠐⠊⣉⣤⣴⡶⠞⠉⢀⣀⣤⣴⠶⠿⠛⣋⣁⣤⣴⣶⡿⠿⠛⠉
⠀⠈⢿⣆⠘⣿⣆⠫⣿⣷⡄⠀⠀⠀⠈⠙⠿⣿⣿⣤⣿⡇⠀⢸⣿⡁⢸⣷⣄⣤⡴⠾⢛⣋⣩⣤⠶⠾⠛⢋⣉⣥⣴⣶⣾⠿⢿⣿⡟⠋⠁
⠀⠀⠘⣿⣆⠹⣿⣧⠈⠻⣿⣆⠀⢤⡀⠀⢀⠀⠨⠉⢿⣧⣄⡀⢿⣇⠀⠻⠿⢷⡶⠿⠛⢉⣡⣤⣶⣾⣿⠿⠟⢛⡉⠁⠀⠀⣶⣍⠻⣷⣄
⠀⠀⠀⠘⣿⡆⠹⣿⣷⡀⠹⣿⣷⠈⠃⠀⠀⠁⠀⠀⠀⣉⡙⠻⢿⣿⣷⣦⣤⣤⣴⣶⡾⠿⠟⠛⠉⠁⠐⠊⢉⣁⣀⣀⣀⣠⣭⣤⡀⠙⣿⣧
⠀⠀⠀⠀⠘⣿⡄⠘⣿⣇⢀⠙⣿⣷⡀⠀⠒⠀⠀⠀⣈⣉⣉⣀⣀⣈⣭⣭⣭⣭⣤⣶⣶⣶⣶⣶⣿⡿⠿⠿⠿⠿⠿⢛⡛⠛⠛⢿⣿⡄⢸⣿⡄
⠀⠀⠀⠀⠀⠘⣿⡄⠘⣿⣎⠃⠈⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⣛⣛⣛⣋⣭⣭⣭⣭⣴⡶⠶⠶⠒⠚⠛⠋⠉⠭⠤⠴⠶⠂⢀⣿⡇⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⢿⣆⠀⠈⣿⣿⣿⣦⣒⢂⡀⠈⠉⠉⠉⠉⠉⠋⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣭⣭⣤⣾⣿⠁⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠈⢿⣦⣼⣿⣿⣿⣯⣍⣀⣐⣒⣒⣠⣭⣭⣭⣥⣤⣶⣶⣶⣶⣶⣾⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢮⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⣦⣆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠢⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⢙⣧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠰⣅⣩⣥⡴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠛⠙⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⣴⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡟⣀⢀⣄⣀⣀⠉⠛⠁⠀⣤⠀⠀⠀⣤⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠟⠋⠙⠿⣦⠀⠀⠀⢼⣦⠀⣸⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⢴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢎⡄⠀⠀⠀⠀⠘⢿⡆⠀⠀⡹⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣄⣤⡿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠈⠁⢀⣠⡴⠟⠀⠀⠓⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⠉⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⡷⠆⠚⢍⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡤⣾⣷⡦⣶⠶⢿⡾⠿⢿⣦⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣾⠟⠋⠀⠃⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⢦⡀⣀⡄⣠⠂⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡄
⠀⠀⢀⠱⣀⡀⠀⣤⠔⣂⠥⠄⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⠈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣄⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣳⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠐⠒⠓⠒⠈⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡍⣟⣛⣯⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠠⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡇⣀⣤⣶⣤⡀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⣽⣿⢽⠥⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠾⠾⠾⠿⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⣛⣯⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣹⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣷⣶⣦⣶⡾⠿⡇
⢀⠀⠀⣀⣾⣿⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣥⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣾⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠐⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠠⣉⠒⠬⢍⡛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠒⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠠⢍⡓⠦⢌⣑⠂⠈⠙⠛⠶⣤⡀⠀⢀⣀⡤⠴⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣐⠨⢍⡒⠮⢍⡒⠌⠉⠒⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠙⠳⢮⣍⠒⠤⢁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠛⠛⠛⠷⢦⣤⣀⡀⠀⠀⠀⣠⡾⠁⠀⣀⣠⣤⡴⠶⠞⠛⠛⠋⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠓⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⢶⣴⣦⣦⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡸⠱⣞⠻⣦⠤⠤⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⡠⠴⢶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠁⠀⠹⣦⡿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢶⡿⢀⣠⣤⣤⣤⣽⣿⣿⣿⣿⣿⣛⣋⣉⣉⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⣀⣴⠶⠛⠋⠉⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠭⣭⣭⣍⣉⣉⣉⣛⣛⣛⣛⠛⠓⠶⠶⠶⠶⠶⠶⠶⣶⠿⠇
⢀⡾⠋⠀⢀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠚⠓⠒⠒⠒⣒⡒⠒⠶⠶⠾⠏⠭⠭⠁⠀⠀⠀⢰⠛⠲⢤⣀
⣾⠁⠀⠀⠀⠈⠙⠒⠄⠀⢸⡇⠀⠀⠀⠀⠠⠤⠄⢀⣀⣀⣀⣉⣛⣓⣒⣒⣒⠒⠢⠤⠤⠭⢭⣭⣍⣉⣁⣀⠀⠀⠀⣿⠀⠀⠀⠈⠛⢶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣂⠤⠤⠤⠤⣤⣤⣤⣤⡀⢀⠈⡉⢽⠿⠍⠭⠉⠛⠓⠒⠂⣤⣄⣠⣒⣂⣀⠀⢿⣀⣀⣤⣤⣴⣶⡾⠿⠃
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠶⠶⠶⠿⢤⣤⢦⣥⣥⣬⣥⣼⣿⣿⣷⣶⣦⣄⣀⣒⣲⣶⣶⣀⣈⣹⣏⠉⠀⠀⣾
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠁⠀⠀⢰⡏
⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠁⠀⠀⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠐⠒⠛⠉⠁⠀⠀⢸⡇
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣤⣤⠤⠶⠶⢒⣒⣀⣤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣈⡻⣝⡒⠀⠀⠀⠀⠐⠒⠚⢛⣋⣉⣭⣭⡥⠶⠶⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⠟⠉⢀⠉⣀⢸⠸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⣿⣿⣿⣿⢁⠀⢀⡏⢀⡇⢨⣶⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⢎⡽⣿⣿⣿⣧⡟⢀⡞⣠⠞⣠⢎⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⢃⣾⣿⣿⣿⣿⣫⣴⣫⣴⣯⣾⡋⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⣾⢠⡏⣼⣿⢧⣿⣿⣿⡟⠉⠁⠀⣿⡦⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⡏⣼⢡⡟⣿⢸⣿⣿⠿⠗⠀⠀⠀⠁⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠃⣿⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⢸⠁⡇⢸⠀⣿⠸⡇⣿⣦⡄⠀⠀⠀⠀⠀⠀⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡐⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢻⠀⡇⠸⡀⢡⠀⠸⡈⢿⣤⡀⠀⣾⣶⣶⡶⠆⢰⠇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢸⡆⠱⢆⢣⠈⣷⡀⢣⠀⠈⣿⡇⢹⣿⣿⠁⢠⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢳⣄⡃⠀⢆⡘⣷⣤⣷⡾⠛⠀⠀⠈⢀⡴⠋⠀⠀⠀⠀⠀⠀⡜⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⡿⠛⠉⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠋⠊⣞⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠈⠙⠾⠀⢠⠆⢀⠀⠀⠀⠀⠀⢠⡟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠈⠳⣄⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠽⢷⢲⣄⢄⠈⠀⠃⠷⢢⡦⠀⢠⡟⣴⣿⠋
⠀⠀⠀⠀⠀⣀⣤⡾⢫⣽⣷⣄⠀⠉⠒⠤⠤⠔⠁⢸⡛⢶⣦⣄⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠜⠾⡖⠀⠀⠈⠂⢠⠟⣴⣿⠃
⠀⠀⢀⣴⠟⢿⣷⣮⣳⣿⣾⣿⢿⡖⠄⠀⠀⢠⠤⠻⣇⢿⣾⣿⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⠏
⠀⢀⡿⠁⣤⠈⠠⠙⣿⣿⣿⣿⢆⢻⣆⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⣻⠋
⠀⣸⠇⣮⠁⡟⣷⣽⣹⣿⣿⣿⣜⡆⢻⡄⠀⠀⠀⠀⠸⣏⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢞⣿⠃
⠀⣿⢰⡟⢼⡿⣿⠟⢹⣿⣿⣿⣿⣷⠈⣷⠀⠀⠀⠀⠀⣿⠀⣿⡶⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⢠⡿⠱⣱⠇
⠀⣿⣼⣇⡾⢿⣿⣧⡏⢹⣿⣿⣿⣏⢀⢻⡄⠀⠀⠀⠀⣿⣠⡼⠷⠚⣻⡝⠳⢦⣄⡀⠀⠀⣤⡞⢺⡄⢻⡆⢳⡿⠡⢠⠏
⠀⢸⣿⣿⣀⠾⡦⢤⡄⢹⣿⢛⣿⣿⣹⣾⣧⠀⠀⠀⢀⡞⠁⠀⠠⠾⠿⠷⠛⣉⣽⠟⢷⣶⡆⢣⠀⢇⠀⠃⠈⣷⠁⠏
⠀⢸⣿⣹⣿⣿⡛⠈⢁⣿⣿⣿⣿⣿⣻⠏⢿⡀⡠⣄⡜⠀⠀⠀⠀⠠⠤⠒⢊⡡⢾⢶⣬⣙⣿⠘⠃⠈⠀⠀⠀⢸⠇
⠀⣸⠇⣿⣿⢻⡧⠘⠛⣿⣿⣿⣯⣬⣵⡾⠿⠻⣄⠈⢆⠀⠀⠀⠀⠠⠔⠒⣉⣴⣻⣿⡿⠿⡟⠀⠀⠀⠀⠀⠀⡼
⠀⢹⡆⢿⠛⠛⢗⣴⣿⣿⣿⠟⠋⠁⠉⠀⠀⠀⠘⡇⠈⠆⠀⠀⠀⣐⣂⣩⣴⠟⠿⡿⡂⢰⡛⢄⣀⣀⢀⣠⠞⠁
⠀⠀⢻⣾⣶⡰⣾⢿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣀⡀⢹⡀⢸⠞⡛⢋⢹⣿⣷⣿⡇⠖⡓⣬⢦⠳⣤⣬⡭⠋⠁
⠀⠀⢸⡯⣿⣷⣴⣿⣿⣁⣄⣠⣤⣶⢿⣿⣿⣿⣿⢸⣿⣿⡀⠀⠸⠌⣿⡿⣶⣥⣤⣤⣬⡶⠟⠉
⠀⠀⠀⢻⡘⢟⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣯⣭⣴⣿⠋⣿⠃⠂⠂⠀⣿⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠉⠀⠈⠙⠛⠟⠻⠶⠟⠛⠛⠛⠉⠉⠉⠗⠀⠿⠀⠀⠀⠀⢺⡇


⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠈⠉⠉⠉⢻⠉⠉⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠉⠓⠂⠒⠀⠤⠀⣀⣰⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠴⠶⠶⠆⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠒⠀⠀⠠⠀⠀⠐⢶⠐⠒⠒⠒⠒⠂⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠤⠂⠑⠂⠀⠀⠄⠠⠀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠐⠠⠤⠤⠄⠀⠰⠖⠂⠒⠂⢀⣀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⠠⠤⠄⠀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠊⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠐⠒⠀⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠬⠛⠉⠁⠀⣀⠤⠶⠊
⠀⠀⠀⠀⠀⣀⠀⢠⣄⡀⠀⠀⠀⠠⠄⠉⠀⢈⠙⠛⠲⠶⠶⠤⠤⣄⣀⡀⢀⣄⡄⠐⠂⠁⠀⢀⠀⠀⠀⠀⠀⣀⣤⡀⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠙⠫⣝⣓⡒⠢⠤⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡄⠈⠀⢀⠀⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠨⠁⠐⠈⠭⢙⡒⠢⠄⢀⠀⠉⠉⠉⠛⠒⠒⠲⠶⠭⢄⣀⠄⡡⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⣼⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐⠀⠄⠀⠀⠀⠛⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠒⠋⠁
⠀⠀⠀⠀⠀⢠⡀⠈⡉⠛⠒⠂⢤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊
⠀⠀⠀⠀⠀⠈⠑⢳⣠⢕⠂⠀⠀⠭⣓⠠⡂⠄⣀⠠⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠠⠂⠁⢀⠀⠁⠪⢀⠀⠀⠀⢈⠀⠀⠠⡌⠐⠂⠈⠀⠀⡈⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⡐⠚⠉⠁⠀⠀⢀⠀⠀⠄⠀⠈⠙⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠡⢀⡀⠀⠐⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠀⣀⡤⠖⠉⠀⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠑⠡⡀⠄⡈⢷⣶⡮⣤⣀⠐⠱⠦⣀⣸⣃⡀⠠⠐⠒⠉⠁⠀⠀⢀⣠⠔⠛⠁⠀⠀⢀⣠⠴⠋⠈⠀⠀⠀⠀⠀⠀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠈⠙⠓⠮⢍⡓⠲⠤⣭⣑⠊⠴⢂⠄⠀⠀⠴⠚⠉⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠔⣠⡑⠨⢙⠲⡠⢀⡀⠀⣄⢀⠄⠊⠀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⡑⠤⡉⠙⢁⠀⠁⠉⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠟⠄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⠢⢍⣛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠑⠦⣌⣉⠒⠤⡉⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣑⠢⢭⣑⠢⢍⡙⠒⠤⣀⠀⠀⠈⠙⢷⣄⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠓⠮⣍⣒⠦⢍⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣭⣷⣦⣌⣁⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣯⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣄⣠⣤⣴⣟⣁⣤⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⣿⢿⣿⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡜⠘⣯⠉⣿⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠖⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠁⠀⠘⢧⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡟⣀⣠⣤⣴⣦⣬⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⣼⣿⠏⠉⠀⠀⠀⠀⠂⠠⠤⠤⠤⠤⢤⣭⣉⣉⣉⣉⣉⣉⣛⣛⣛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⢶⡾⠿
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠒⠒⠤⠬⠭⠭⠭⠭⠭⠍⠉⠁⠀⠀⠀⢠⡟⠳⢤⣀
⣼⠃⠀⠀⠀⠉⠑⠢⠄⠀⠀⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣒⡒⠒⠒⠢⠤⠤⠤⠤⢬⣭⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠄⢀⣀⣀⣀⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⠉⣛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⣀⣤⣽⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣭⣭⣭⣭⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⡦⠄⠤⠄⠀⠀⠀⠉⠁⣀⣀⠀⠀⠙⣟⠛⠋⠉⠉⡟
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣩⣭⡿⠿⠟⠛⠛⠛⠛⠛⠒⠶⠶⠾⠿⠛⠀⠀⠀⢸⠃
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠞⠛⠉⠁⢀⡀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⠶⠒⣛⣉⣉⣤⡤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣈⡳⣭⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡶⠶⠖⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢿⣿⠿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣀⠀⣀⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣤⣤⣤⠶⠋⠁⠀⠀⣿⠀⠀⠈⠛⠦⣤⣴⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⠛⠛⣩⣴⣾⣿⣿⢸⣿⣿⣷⣮⣙⠛⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⢷⣄⡀⠀⠀⠀⣿⠀⠀⠀⢀⣠⠿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⣘⠻⣿⣿⣿⣿⢸⣿⣿⣿⡿⠟⣁⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠲⣤⣠⣥⣄⡴⠞⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣷⣦⣍⡛⠟⠚⠿⢛⣩⣶⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡘⢿⣿⡿⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡿⢿⡀⠀⢀⡿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣸⣇⣀⡤⠞⠋⠁⠀⣿⠀⠈⠙⠶⣤⣀⣼⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠸⠿⢛⣩⣶⣿⣿⢰⣿⣷⣦⣍⡻⠿⠃⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠸⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠂⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠙⠳⢦⣀⣀⣿⣀⣠⡴⠚⠉⠀⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣾⣿⣶⣭⡛⠿⡿⠘⢿⠿⣋⣥⣶⣿⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣭⡇
⠀⠀⢠⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾⡿⠋⢡
⠀⠀⢀⠳⣀⠤⠤⠤⣂⠬⠄⣂⡄⠀⠀⠀⠀⣛⣛⢛⣛⡛⠀⠀⠀⠀⠘⠠⠀⠀⣀⠀⠀⠠⠘⠀⠀⠀⢰⣦⣤⣤⣀⡄⡀
⠀⠈⢄⣒⡪⠭⠭⢙⠵⠊⠁⠀⠀⠀⠀⠀⣾⡿⠿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢺⣷⢀⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣙⣺⣹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡏⣿⡭⢿⠺
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠰⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠈⠋⠉⠁⠉⠀⣿⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣴⣿⣷⡦⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣷⡆⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣯⣿⣿⡇⠛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣨⣵⣒⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢩⡄
⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⡶⠿⠷⠶⡿⢶⠾⢾⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣙⣛⢛⣉⣤⣤
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⢳⠒⡗⠒⠗⢺⠒⡺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣫⣭⣿
⠐⠀⠀⠐⣿⣿⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣫⣛⣹⡹⣋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⠿⠟
⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⢒⣀⠑⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⡊⠉⠀⣠⡶⠞⠋⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢸⣿⣿⣯⣭⣥⣵⣚⣃⡢⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡴⣖⡿⠟⠛⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⢸⣾⣿⠟⠉⠀⠀⠀⠀⠉⠉⠀⠒⠒⠒⠶⢤⣭⣶⢶⣾⢿⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⢠⡎⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠈⢿⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠶⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠈⠛⢦⣄⣀⠀⢠⠟⠉⢻⣦⠀⠀⠀⠀⠀⠀⣾⣯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⢀⣼⣿⣶⣷⠿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠈⠻⣦⣀⣀⣀⣠⡾⠛⠁⠀⠀⠀⡴⠒⢶⣄⣀⣀⡎⠉⠙⣇
⠀⠀⠀⣠⣶⠟⠱⠁⠘⣧⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⡨⠋⠀⠈⢳⣦⠼⠿⠄⠴⠲⣦
⠀⠀⢀⣿⣏⣿⡓⣲⣶⡾⠂⠒⠛⣻⣿⠀⠀⠀⠀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⡁⠀⠀⠘⣧⣀⣼⠇
⠀⠀⠘⣿⣾⣿⣏⣿⣿⣹⣟⠀⣉⣿⡾⠃⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⡇⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⠛⠻⠿⠷⠾⠷⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⢶⣦⡄
⠀⠀⠀⠀⠀⣠⢤⣤⢤⡄⣠⠦⢤⣤⠀⣤⠤⡤⢤⠀⣤⠤⢤⣄⢠⡤⢴⣦⡄⠀⠀⠀⠀⠈⠐⠌⢰⡄⠀⠀⠀⠀⠀⠀⡿⠉⠉⢹⠁
⠀⠀⠀⠀⠀⢘⠟⠉⢻⠀⢸⠋⠙⢻⠀⢹⣉⣻⡏⠀⡙⠉⠉⡇⠀⣏⣯⣩⡅⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠶⣤⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⣀⣼⢸⣿⣶⣿⠀⢸⡀⠀⣼⠀⢸⣿⡇⡇⠀⣷⢤⢠⡇⠀⡇⠀⠁⠀⣷⣄⠀⠀⠀⠀⠀⢹⠟⠟⠛⡏⠀⠀⠀⡇⠀⠀⠸⡀
⠀⠀⠀⣭⡿⡿⣿⠿⠿⠁⣸⣦⣠⡿⠀⢽⣿⣧⠧⠀⣿⣤⣵⠧⠀⣧⣤⣴⠇⢿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠀⢁
⠀⠀⠀⣈⣿⢡⣽⣽⣯⠭⢭⣯⣥⣽⠻⢭⣥⡴⠤⠤⣤⣼⢤⡷⠤⣼⡿⢿⡿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⢠⠇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⣁⣿⢸⡟⠃⣿⠀⢸⠋⠉⢹⠀⢸⣿⠃⠀⠀⣿⠁⠀⡇⠀⡇⠀⠈⠁⣿⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⣘⣀⣀⣀⣀⣀⣀⣱
⠀⠀⠀⠀⣿⢸⡇⠀⣹⠀⢸⠀⠀⢸⠀⢸⣿⡀⠀⠀⡿⠀⠀⡇⠀⡇⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⣸⠀⠀⠀⢿⣰⣿⠻⠿⠧⠤⠶⠾⠛⣇
⠀⠀⠀⠀⣯⠘⠁⠀⣿⠀⢸⠀⠀⢸⠀⢸⣿⠁⠀⠀⣿⠀⠀⡇⠀⡇⠀⠀⡄⣿⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠙⠳⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠄⣿⠐⡆⠀⣿⠀⢸⠀⠀⣸⠀⢸⣿⠀⠀⠀⣿⠀⠀⡇⠀⡇⠀⢸⠇⣿⠆⠀⠀⣾⣖⣒⠒⠒⠀⠀⠀⠐⢒⣶⡄⠀⠀⠀⠀⣠⣿⣇
⠀⠀⠀⠖⣿⢸⡇⣠⣿⠀⢸⣀⣠⣿⡀⣸⣉⠀⣄⣀⣿⠀⣠⣇⢀⣇⠀⣰⣄⣿⠂⠀⠀⣿⣿⡏⢉⠉⠉⠉⠉⠉⢿⣿⣧⣶⣶⣴⣊⣻⣿⠟
⠀⠀⠀⡤⣟⠘⡿⠿⣿⣈⣈⡛⠛⠛⢓⡈⠻⣿⣁⣀⡘⠷⠿⣓⡒⢛⣿⣿⣆⣻⣆⠀⠀⢻⣿⣇⠀⡀⠀⠀⠀⣆⣼⡿⠉⠉⠉⠉⠉
⠀⠀⠀⡄⡿⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠙⠒⠛⠛⠛⠛⠛⠛⠛⢿⡷⠀⠀⠀⠙⠿⣿⣷⣦⣴⣾⡿⠋
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡿⠒⠀⠀⠀⢡⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡀⠊⠀⢀⡤⢴⠕⢀⢀⡆⢠⡖⠊⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡈⠑⢦⠀⠘⢧⣄⠒⠉⠀⣸⣶⣾⣾⢧⡸⢗⣋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⣀⠄⠀⠀⠳⡀⠀⡢⠟⠛⠉⠀⢠⣿⣏⢈⡤⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⢢⡀⠀⠀⣯⠆⠀⠀⠀⠀⠀⣼⠟⠋⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣺⣟⣢⣴⣯⡀⣀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⣀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡿⠛⢿⣯⣤⡦⠦⣠⡀⠀⠚⣁⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣙⡀⠀⠠⡄⢰⠀⠀⠸⡇⢀⢄⣀⠀⢀⠄⠀⠀⠀⡸⠋⠀⠀⠀⠈⠙⠻⢶⢤⣄⣀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⣑⠤⣀⣼⣧⣄⡀⢤⡵⠋⢫⠀⠀⡌⠀⠀⠀⠚⠁⠀⠀⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⠀⡀⠀⢠⡀⠤⣼⣒⢿⠙⣯⣿⠋⠙⣿⡟⠁⠀⠠⣠⡚⠀⠀⣰⠇⠀⠀⡀⣶⠂⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠙⡄⠀⢹⠀⠀⠸⠀⠀⢠⣿⠋⠁⠀⢠⠏⠀⠀⣼⠁⢻⡶⢤⠈⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠈⠒⠀⠀⣡⡐⣸⠀⠀⣸⣠⡔⠋⠀⠀⠀⠈⡏⠀⠀⠀⣿⢀⡾⡀⠀⠀⢰⡇
⠀⠀⠀⠀⢀⣠⠤⠴⠲⠻⠦⠤⠠⢀⣀⣉⣩⣿⠶⢾⡷⠦⠤⠤⠀⠀⠚⠛⠁⠀⠀⠀⠺⣿⣽⣗⣠⣖⡁⠀⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⠟⠉⠀⠀⠸⣷⡀⠀⠀⠀⠀⠠⠇⠀⠀⠰⠀⢸⡿⠉⠉⠛⠉⠤⠀⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⡼⠀⠀⠣⠀⢠⡟⢁⠀⣄⠀⠠⠀⢀⡀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠⡇⠀⠀⠀⢰⡟⠁⠈⢡⡜⣶⠖⣤⣯⣤⣀⠀⡤⠄⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣼⠁⠄⠈⢠⡟⠀⠐⠠⠖⠾⡌⣽⣛⠋⢉⡀⠀⣄⣤⣄⡀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣆⠘⢦⢛⠀⣠⡅⠀⠀⠀⠀⠀⠉⠙⠦⠠⠀⠀⠀⠀⠉⠑⠢⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢹⡄⣸⠻⣅⡂⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢻⡏⠀⠈⠙⢲⣾⣧⣄⣀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠀⠐⢠⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣇⠀⠀⠀⡾⠃⠀⠀⠉⠉⠉⠙⠓⠦⠤⢰⣦⣀⣦⡄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡆⠈⡇⢀⡾⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠳⠦⣴
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⣀⡤⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⢁⣠⣴⡺⠕⠓⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠑⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⣀⣲⠿⠋⠉⣀⣤⢶⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡴⣯⢤⠤⠒⠀⠀⠀⠀⠀⡀⣢⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁⢀⡠⣔⡮⠟⠋⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠈⠉⠀⠀⠀⣀⡤⠐⠐⠈⠁⠉⠉⠉⠉⠛⠓⠛⠁⠀⣀⣲⠷⠛⠉⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⣄⡤⠴⠶⠿⠥⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⠀⣰⡴⠞⠋⠁⣀⡴⠚⠉
⠀⠀⠀⠀⠀⡀⣄⠤⠐⠘⠁⠁⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢀⣄⠶⠋⠁
⠀⠠⠀⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠦⡀⠀⠀⠀⠀⠀⣠⠔⢲⢰⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⡆⡇⠀⠀⠀⠀⠀⣼⠕⠂⢀⠀⢠⠂⠀⠈⢆⠀⠀⠀⡜⠁⠀⢸⢸⠴⣚⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢸⠀⡇⡇⠀⠀⠀⠀⡸⠁⠀⢠⢇⡴⠃⠲⡀⠀⠘⡄⠀⠀⡇⠀⠀⢸⠔⠊⠁⢀⡇
⠀⠀⠀⠀⠀⠀⡞⢳⢸⠀⢸⠀⡇⡇⠀⠀⠀⢀⠇⢀⡴⣷⢍⠀⠀⡠⣈⢦⠀⠱⡄⠀⢹⠉⡣⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⢀⣧⣸⣼⣠⣼⣄⣧⣇⣤⡀⠀⢺⠊⣏⠀⠐⠆⠀⠀⠰⠀⠀⣳⢄⣹⢀⡼⠃⠯⡖⠒⠊⠁⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣷⡀⠀⣐⣂⠀⢀⠟⡏⢀⡞⠈⠉⠉⠉⠁⠀⡤⡄⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢹⢦⡀⠀⠀⠀⠹⣄⣇⠙⡦⢍⡡⡴⠋⢠⢃⡟⠀⠀⠀⠀⠀⢀⡴⠋⠙⢢⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠭⠬⠉⡇⠀⠀⠀⠀⠈⢁⣹⡇⠀⠀⣸⣾⡓⠋⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡴⠚⠉⠓⡇⠀⣛⣛⠀⡇⠀⢀⠔⢺⠉⠉⢣⡈⢲⡞⢁⠇⠈⠙⡶⠦⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠰⡁⠀⠀⡀⠳⠤⢤⠤⠄⠇⠀⠆⠀⣸⠀⠀⢘⠑⢻⡟⠋⠃⠀⠀⡇⠀⢹⠀⠀⢸⠖⠖⠖⠒⠲⠚⠳⠲⡇
⠀⠀⠀⢀⠔⠁⠀⢄⣇⡠⠀⠘⢦⠀⠀⢰⠀⠀⢻⠀⠀⠘⠖⠑⣞⠒⠂⠀⠀⡷⠀⠈⡄⠀⢸⠀⠀⠰⣉⣱⠀⠀⠀⡇
⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠀⠀⢠⠇⠀⢸⠀⠀⣸⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣧⠀⠀⡇⠀⢸⠀⠤⠤⠤⠬⠤⠄⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣇⣀⡀⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡸⠀⠀⢳⠀⢸⣀⣉⣉⣉⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠒⠬⣷⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣟⡩⠭⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡞⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣇⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢼⣄
⠀⠀⠀⠀⠀⠀⠤⢤⡴⣫⡿⠓⠒⢾⢶⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣳⣯⣻⣦⡤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠺⠛⡏⠀⠀⣀⠜⠓⠚⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠭⣓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢕⡢⠭⣐⠈⠙⠓⢦⣄⠀⠀⠤⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⢴⣪⠥⣂⠅⠀⠈⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠠⠬⠿⠶⣥⣀⡀⠀⠀⠀⠀⣼⠀⠀⢀⣀⣠⡤⠶⠶⠿⠯⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠿⠗⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⣶⢶⣶⠶⠶⠶⠶⠖⠒⢒⣒⣒⣒⣶⣶⠶⠶⠓⠛⠛⠛⠛⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠁⢳⣸⡟⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⢀⠀⠀⠀⠀⠐⠊⠙⢦
⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃⣀⣠⣤⣿⣷⣶⣶⣶⣖⣒⡚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⢀⡴⠖⠛⠉⠁⠀⣿⠟⠉⠀⠀⠀⠀⠐⠋⠭⠭⠭⠭⠭⠍⢉⣛⣛⢛⣛⡒⠒⠒⠒⠲⢶⡾⠓
⣰⠋⠀⠀⠤⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⠉⠡⠭⠭⠭⠤⠄⠀⠀⠀⣼⠉⠓⠦⣄⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⣀⣀⣈⣉⣉⡀⠐⠒⠒⠒⠠⠭⠍⠉⠉⣀⣀⣀⠀⠀⣇⠀⣀⣀⣈⣹⣷
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠾⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⣀⣭⣽⣯⣤⣈⣿⠋⠉⢹⡇
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⣠⠴⠖⠒⠛⠉⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡘⢷⡠⠀⠀⠀⠀⠤⠤⠶⢒⣒⣛⣫⡭⠭⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⣼⢇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢠⣼⢇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⢿⣅⣀⣤⣀⡀⠀⠀⠀⠀⣀⣠⣀⣸⡿⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠉⠛⢷⡀⠀⣼⠋⠉⠉⠻⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⡈⣿⠾⣿⡀⢠⠾⣡⡟⠉⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⣠⡾⢣⡿⠀⠈⠛⠷⠾⣏⣀⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣬⡻⣦⣤⣙⣭⡴⠟⠁⠀⠀⠀⠀⣴⠟⠋⠛⢷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡌⣿⣿⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣾⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣐⣛⣵⠟⠈⣿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠙⠷⢶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣯⣁⣀⣴⠿⢦⡾⠛⠉⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢸⣇⠀⣤⡗⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠉⢸⠤⠤⡇⡇⠄⠸⠄⢀⠏⠈⠔⠁⠈⠀⣰⣣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠘⠀⠀⠃⠒⠀⠀⠘⠂⠀⠘⠈⠂⠒⠐⠁⠀⠑⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⣀⠀⡀⠀⢀⡀⡀⣀⠀⡀⢀⡀⡀⠀⠀⢀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⢎⠀⠀⡇⡙⣀⠜⡁⠆⠜⢸⠀⠀⡱⡇⠀⠀⠈⠱⣀⢀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠁⠉⠀⠁⠈⠀⠀⠀⠀⠀⠉⠈⠀⠈⠀⠁⠀⠀⠈⠀⠁⠉



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

⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠂⠀⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⠀⠐⠋⠋
⠀⠂⠀⠀⠀⢣⢀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠤⡉⡰⢀
⠀⠀⠀⠈⠀⠀⠸⠊⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠤⢀⠞⢠
⠀⠀⠀⠀⢤⠂⠀⠆⠨⠉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠰⠃⢀
⠀⠀⢀⠀⣦⠁⠃⡌⠐⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃
⠀⠠⠄⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⡂⠀⠐⢊⠤
⠀⠀⢳⠂⠀⠀⠀⠀⣀⠀⠀⠀⠠⠤⠐⠀⠀⠈⠁⢰⠠⠀⠉⠀⠀⠀⠀⠀⡀⠐⠑⠂⠀⠀⠀⡀⠃
⠀⠀⢸⡀⠀⠀⠀⣸⡃⠀⠀⠀⠐⠒⠈⡀⠀⠀⢀⣀⡓⠘⢀⡄⠀⠂⠾⠃⠀⠀⠀⠀⠀⠀⠐⢌⠍
⠀⠀⢸⠀⠀⠠⡐⢕⣀⠀⠀⠐⠛⠆⠂⠀⡀⠀⠀⠀⠠⠄⠋⠀⢦⡀⣤⣄⡀⠈⠀⠀⢀⠈⠀⠤⠂
⠀⠀⡞⠀⠀⢤⣈⣩⠢⠂⠀⠢⣀⠀⢀⠀⠁⠀⠀⠀⠀⠈⠉⠱⡆⢢⠀⡀⠀⠀⠀⠀⠀⠁⠲⠘⠋
⠀⠀⠀⠀⠀⠢⣭⣠⠶⠖⠂⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠤⣀
⠀⠀⠀⠀⣰⠀⠀⠀⠀⢀⡤⢄⡀⣀⡴⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡩⠾⠖⠒⠒⠀⢴⠄⡤⡄⠠⠶⠶⠆⠀⠷⠇⠤⠼⠃⠐⠂⠸⠥⠿⠢⠆
⠀⠀⢠⣎⣿⡄⠀⢀⠂⢇⠀⠀⠈⡿⠧⡀⠀⢣⠀⠀⠀⠀⢠⣦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⣠⣤⣤⣄
⠀⠀⠈⠉⠸⣽⠀⠈⠀⠈⣆⡤⢴⠗⠒⠛⠳⣻⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢠⡀⠸⠾⠿⠟⠁⠀⢹⠛⠷⡞⠀⣄
⠀⠀⠀⢠⢿⠀⠀⠀⡰⠊⠈⠣⣼⠀⠠⡶⠀⢸⢦⠀⠀⡇⠀⠀⢂⢀⠀⠀⣯⡤⠴⣿⠀⠀⠈⠁⠀⡶⠀⢸⠀⠀⢸⠀⠠⡇⠀⠈⠀⠀⠘⡷⠋
⠀⠀⠀⠳⢟⡀⠀⠀⢧⣀⡀⠀⠙⠢⣀⠀⠀⣜⡠⢷⠋⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⡇⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⣿⣢⠀⠀⠘⠋⠐⢻⠒⠋⠛⢯⣇⣀⡸⠀⠛⠀⠀⠀⠀⠀⠀⣠⠁⡀⢇⠀⠀⠀⠀⠀⣿⣤⣸⠀⠀⢸⣠⣼⡅⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢱⠉⠀⠀⠀⠀⢀⠸⡄⠀⢠⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠆⠀⠘⢦⡀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣚⣧⠀⠀⠀⠀⠚⠁⠙⠒⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣰⠃⠀⠁⠀⠀⠀⠱⡀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠘⠛⢸⣿⠁
⠀⠀⠀⠀⠀⠉⠈⡷⡀⠀⠀⠀⠀⣀⣤⡴⠼⠞⠚⠉⠀⢃⠀⠀⠀⠀⢧⣀⣀⣂⣀⣀⠠⠴⠁⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠽⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠸⣿⡿⠃⠀⠸⠔⢻⢽⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠶⠖⠛⢇⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠐⠁⠄⠠⡞⠋⠉⠉⠀⠀⢀⠀⠘⡀⠀⠈⠀⠈⡆⠀⠀⡀⢀⢀⠀⠀⢀⠀⣦⣀⠀⠠⠶⠀⢦⡀⡶⡇⣤⠐⠷⠀⠀⠀⢺⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠈⠀⠀⠀⠀⢡⠀⠀⠀⠀⢱⠀⠀⠓⠚⠙⠐⠆⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠁
⠀⠀⠀⢠⣦⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠘⡆⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠭⢠⠀⠀⠀⠀⠀⣇⠟⠧⠀⡋
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢱⠀⠀⠈⡀⠀⠀⠀⠀⢳⠀⢀⡠⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⣀⢧⡀⠀⠀⠉
⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠔⠋⠀⠉⠢⣀⠀⣠⠜⠁⠀⠙⠢⡀
⠀⠀⢀⡠⡼⠶⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠀⡠⠠⠀⠀⠀⠤⠤⡔⠁⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⢸⠀⠀⠈⢦⠀⠀⠀⢣⢀⡤⠖⠊⠉⠉⠁⠀⠀⢀⡁⠉⠲⠀⠀⣇⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠏⢇⠀⠀⠀⠀⠀⠀⢀⠁⠄
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠑⢄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠴⠃⠈⢣⠉⢈⠆⠀⠀⠀⠱⡀⠀⠀⠀⠀⡴⠀⠘⡄⠀⢀⣀⣀
⠀⠀⠠⠖⢸⠒⠂⠀⠀⠀⠀⠉⠓⠒⠶⠤⣴⠄⠀⠀⠀⢀⠊⠀⢆⠀⠃⣸⠦⠂⠀⠀⠀⠀⢀⠗⠒⢒⡒⠒⠃⠀⢠⠋⠉⠁⢠⡶⣦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠒⠆⢚⠓⣍⠁⠀⠀⢀⢀⠀⠠⠋⠀⢰⢵⣌⡇⠀⠐⠃⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⢸⠀⡀⡀⣀⣀⠀⣇⡀⢀⣘⠁⠀⠀⠀⠀⠄⠉⠁⠀⠀⠀⠂⠀⠀⠀⠘⠽⣛⠁⠀⠀⠀⠘⠃
⠀⠀⠀⢉⠽⠉⠉⠉⠉⠉⣩⡍⠉⠉⠉⠉⢩⢡⡄⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦
⠀⠀⠀⠀⠀⠐⡏⠉⠉⠉⠉⠛⠛⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⢰⠉⣷⠀⢣⠮⡀⣴⠃
⠀⠀⠀⠀⠀⢀⡇⢀⢀⣀⣀⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠈⠁⠘⠚⠀⠘⠒
⠀⠀⠀⠀⠀⣏⢛⣉⣉⣉⣉⣉⣉⣹⣝⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⢀⣉⣁⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠈⠉⠉⣿⢸⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠤⠤⠤⠀⢸⠀⠀⠀⠀⠀⢻⡍⠉⢩⣭⠉⠉⠉⢹⠏⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⢰⡶⠚⠚⠛⠒⠚⠛⠛⠛⠛⠉⢹⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠘⠛⠀⠀⠀⢸⠂⠀⠀⠀⣼⣁⣣
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠓⠒⠒⠚⠿⠿⠿⠿⠇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⣠⣄⣤⣄⡀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⣠⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢧⢤⡼
⠀⠀⠀⠀⢠⡏⠘⡆⠀⠀⠀⠀⠌⣿⣿⡏⠻⣿⡆⠀⢰⣿⣿⡟⠙⣿⣦⠀⡐⣿⣿⡟⠛⣿⣆⠀⢘⣿⣿⣠⣶⣿⡿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⣾⡶⠶⢿⠀⠀⠀⠀⠀⣿⣿⣷⣶⣿⠃⠀⢸⣿⣿⠀⠀⢸⣿⠀⢸⣿⣿⠀⠀⢸⣿⠀⢸⣿⣿⣿⡛⠉
⠀⠀⠀⠀⣿⠀⠀⣼⠀⠀⠀⠀⠀⣿⣿⠋⠙⢻⣷⠀⣸⣿⣿⠀⠀⣸⣿⠀⢸⣿⣿⠀⠀⣸⣿⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢰⣿⣿⣀⣠⣼⡿⠀⠸⣿⣿⣄⣴⣿⠃⠀⠂⣿⣿⣄⣴⣿⠏⠀⢼⣿⡇⠘⢿⣿⣧⠀⠀⠰⣯⣤⣤⡴⣾⢤⡀
⠀⠀⠀⠀⢹⣤⣤⡼⠀⠀⠀⠀⠈⠛⠙⠛⠛⠉⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠘⠛⠀⠀⠀⠙⠛⠀⠀⠀⠻⣖⡒⣃⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠋⡇⠀⠉⠙⠒⠒⠤⠤⢄⣀⣠⠤⠴⠤⠤⠤⠤⠤⠤⠴⡆⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⡀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠒⠀⠀⠀⠀⣿⠁⠀⠀⠒⠀⠀⠤⠀⠀⡼⡂⠀⠀⠀⠀⣠⣴⢫⠖⠀⠀⢹⡄⣀⡀⣴⣫⠖⠀⠙⣆
⠀⠀⠀⠀⠀⠈⢸⡁⠁⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡐⠀⠤⢤⡷⠀⠀⠀⠀⠀⠀⢿⠉⠀⠀⠀⣸⠋⠉⢹⡇⠃⠀⠀⠀⡿⠶
⠀⠀⠀⠀⠀⠐⠂⠭⠭⠍⢑⡲⠤⣀⠀⠀⣿⠀⠀⢀⣀⣠⣤⣅⣈⣹⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠞⠁⠀⠀⠀⠳⢤⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠙⠛⠛⠛⠋⠉⠀⠒⠒⠒⠒⠐⠁