Win Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠁
⠀⢀⠔⠂⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⢄
⠐⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱
⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠰⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰
⠀⠂⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠈
⠀⠀⠀⢀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠄
⠀⠀⠀⠀⠢⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⠠⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠠⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⣀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠤⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⢈⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⢀
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡺⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠶⠖⠘⠋⠲⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⢁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⣔⢢⠤⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠰⢤⡕⣶⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⢀⣀⡤⡀⠀⢠⣀⣀⠠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⣤⠤⡇
⠀⠀⠀⠀⠀⠘⡞⡾⠁⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⡜⠀⠀⠈⠖⠁⠀⠀⠈⡕⠀⣀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⡽⣎⢓⡃
⠀⠀⠀⠀⠀⣸⠼⠱⠤⡀⢈⠜⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⣈⣙⠂⠀⠐⣛⠂⠀⡇⡎⡆⢃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢓⠃⠀⠀⠙⠄⡠⠃
⠀⠀⠀⠀⢀⣿⠀⢠⠀⠩⣀⡠⠊⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠀⠀⠃⠀⠂⠀⠇⠃⢊⣶⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠘⣄⠀⠓⢤⠒⠂⠉
⠀⠀⠀⠀⢸⣇⢀⡸⠀⠀⠙⠄⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠰⣘⠀⠀⠀⡀⠀⠑⢊⡷⠋⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢈⠵⠒⠁
⠀⠀⠀⢀⢨⡿⣎⠆⠀⠀⠀⠈⠂⡀⠀⠀⢈⠢⡀⠀⠀⠀⠀⠀⠘⢄⠱⣈⣀⣠⠞⠁⢀⠔⠊⠀⠀⠀⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⢀⠔⠁
⠀⡀⣮⡼⠎⠳⠿⣵⣄⠀⠀⠀⠀⠑⢆⡀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠈⠢⣄⢀⣀⠤⠊⠁⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠰⣹⠑⠰⠠⠀⡆⠈⡶⢅⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠉⠒⡤⢠⠔⠀⠀⠀⠀⠈⡤⠢⡄⠄⠂⠁⠀⠀⠀⠀⠀⡠⠊
⠐⢭⢶⣄⠀⠠⠀⣰⢷⠊⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡘⠇⡃⠀⠀⠀⠀⠀⡰⠁⡜⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠈⠚⠫⠿⡻⠯⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡐⠀⡆⢃⠀⠀⠀⢀⠜⢠⠺⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠁⠀⠐⢌⠓⠶⠒⣁⠔⠁⠸⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠑⠤⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡅⠤⠥⠄⡁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠠⠤⠀⢀⠀⠀⠀⠠⠤⠐⠒⠈⠉

⠀⠀⠀⠀⠀⡠⠒⢄⠀⠀⠀⠀⢀⠔⠢⡀⠀⠀⠀⠀⡠⢤⢄⡀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⡰⢂⡀
⠀⠀⠀⠀⡐⠑⡀⠎⢣⠀⠀⠀⢀⡀⢀⡁⠀⠀⢀⠾⡔⠋⠢⡝⡄⠀⠀⢸⡀⠀⣘⠀⠀⠀⣿⣳⣺⣿
⠀⠀⠀⠀⡇⠄⣿⠠⠄⡆⠀⠀⠈⠁⠈⠁⠀⠀⠸⠤⠤⠄⠤⠧⠇⠀⠀⠸⠁⠀⠉⠀⠀⠀⣧⠀⠀⢸
⠀⠀⠀⢠⠄⠀⢤⠀⠠⡇⠀⠀⢠⡤⢠⡄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⡤⣤⣬⠀⠀⠀⣧⠤⠤⢼
⠀⠀⠀⢠⠤⠀⠤⠄⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢧⠚⢮⠲⡀⠀⠀⢸⠀⠀⢐⠀⠀⠀⡧⠀⠀⢼
⠀⠀⠀⠨⣄⣀⣤⣀⣀⡇⠀⠀⢰⡆⢰⡆⠀⠀⢸⣀⣸⣀⣇⣀⡇⠀⠀⢸⡆⠢⣶⠀⠀⠀⣷⠀⠀⣼
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣌⠁⠀⠀⢱⡄⠀⠀⢀⣤⢊⡩⠧⣉⠢⣄⠀⠀⠀⢀⠴⢊⠴⠣⣉⢢⣄⠀⠀⠀⡴⡎⢱⠦
⠀⠀⠀⢀⣸⠍⠉⠉⣇⡇⠀⠀⣬⢺⠁⠀⠁⠀⠙⠢⡇⠀⠀⣇⡶⢺⠀⠀⢸⢺⣤⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⡀⠀⠀⡇⡃⠀⠀⡇⢸⠀⢀⣅⠀⢸⠀⡇⠀⠀⣇⣄⣸⠀⠀⢸⣸⣈⠀⠀⠀⡇⡇⠘
⠀⠀⠀⠀⠀⠂⠀⠀⡇⠆⠀⠀⠇⢸⠀⠀⠁⠀⠘⠀⡇⠀⠀⡇⠀⢻⠉⠁⢻⠀⠘⠀⠀⠀⡇⡃⢈
⠀⠀⠀⢈⣻⣅⣀⣀⣏⡇⠀⠀⣥⣸⢀⢀⣄⣀⢠⣀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡇⡧⢸
⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠒⠚⠒⠒⠓⠒⠒⠒⠃⠀⠀⠓⠒⠛⠒⠒⠚⠒⠚⠀⠀⠀⠓⠓⠚⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠋⣁⠙⣉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠉⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⣲⢤⠀⠀⠀⠂⠀⠟⠸⣿⣧⡀⡀⣀⡄⢀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠐⠄⠀⠀⠀⠘⠉⠟⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠃⠀⠘⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠘⠀⠀⠀⠀⢿⠿⠟⠀⠈⠀⠀⢀⡄⠀⠀⠀⠒⠀⠒⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢶⡋⣿⡥
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣴⣴⣶⡞⣉⣾⣤⣴⣴⠇⠁⣀⠃⢀⠀⣀⠀⣀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡠⣾⢶⣯
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠤⠄⠤⠀⠀⠤⠤⠀⠀⠀⠀⣠⠀⠀⠦⢽⡿⠀⠿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣷⡟⠣⢀⣮⡉⠀⢨⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⣂⢰⣿⣿⣦⡈⠙⣯⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⣠⣴⣴⣿⣯⣥⡶⣴⣂⠘⣻⡷⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠁⠀⠀⠀⠀⠔⠃⠁⠙⠻⠋⠀⠁⠘⠚⣿⢯⠆⠚⠁⡀⠠

⠀⠀⠀⢀⡠⣄⠀⠀⠀⣀⣤⣤⣴⣶⣶⡾⠿⠟⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⢀⣤⣄⡀
⠀⢀⠔⣡⣦⡘⣆⠀⣾⣿⣛⡛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣛⣿⣽⡦⠀⣼⣷⣤⡉⠢⡀
⢠⠏⢰⡿⠉⠙⢮⣳⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣿⠿⠿⢿⣟⣋⣻⣿⢥⣾⠟⠁⠈⢻⡵⠙
⣾⠀⣿⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠞⠛⠋⠉⣸⣇⢾⡇⠀⠀⠀⠘⣷⡇
⣿⠀⣿⠀⠀⠀⠀⠀⠁⢹⣿⣿⣿⣿⣲⣶⣶⣦⠄⠀⠀⣠⡴⠶⠀⡀⠀⠀⢀⣶⢞⣦⠀⡿⣿⣿⠉⠉⠀⠀⠀⠀⠀⣿
⢹⡆⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷⣿⣭⠅⠀⠀⠀⢰⣿⠉⠀⠀⠀⠀⠀⡟⠀⠀⠈⢰⣯⣿⡟⠀⠀⠀⠀⠀⠀⢰⣧⢣
⠈⢷⡸⣧⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⣾⣇⠂⠀⠀⠘⠀⢰⠃⠀⠀⠀⡸⢉⣿⠃⠀⠀⠀⠀⠀⢀⣿⣾⡟
⠀⠈⢷⡜⣧⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡅⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⡄⠀⠀⠀⢃⡏⣼⡟⠀⠀⠀⠀⠀⢠⡾⣣⠟
⠀⠀⠀⠙⣮⡳⣄⠀⠀⠀⠈⣿⣿⣯⢻⣿⣷⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠃⠀⠀⠌⢸⠐⣿⠃⠀⠀⠀⢀⡴⣫⡴⠃
⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⢹⣿⣿⣎⣿⣿⡆⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠠⢡⡟⣲⡏⠀⠀⢀⣴⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣻⣿⣿⣹⣿⡷⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⠀⠀⡼⢤⣿⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢿⣿⣿⣧⢻⣿⡂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⣰⣣⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⡿⢻⣿⡏⢿⣷⠀⠀⢹⠀⠀⠀⠀⠀⠀⢡⣥⣿⠻⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣿⣧⠀⠈⡂⠀⠀⠀⠀⢠⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣞⣷⡄⠀⣧⠀⢀⠇⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆⣈⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣋⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣯⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⡏⣷⠹⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢿⣿⣭⣛⣙⣗⣘⣖⣉⣙⣩⣴⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣄⣈⣩⣉⣛⣻⣿⣟⣉⣙⣛⣋⣩⣥⣤⣤⣬⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠏⠀⠉⣿⠙⠋⠛⠛⠛⠛⠓⠙⠉⠉⠷⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⢰⣧⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣴⣴⣼⣧⣭⣯⣭⣭⣯⣭⣭⡭⠭⠭⣭⠬⠤⢤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣆⣀⣑⠶⠶⠤⠤⠤⠤⠤⠶⠶⠶⠶⠦⠤⠤⠴⡒⣆⣀⣴⣷
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⣠⠎⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠏⠀⠄⠠⠀⠈⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠈⣶⠀⠀⠂⢤⠊⠀⢠⠇⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠘⠚⠃⡀⠀⠀⠀⠀⣀⣀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⠄⠤⠰⣯⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⡄⢠⠀⠄⠁⠃⠀⠀⠀⠀⠀⢀⡔⠩⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢰⠀⠀⠰⡀⠀⡀⢂⠀⠐⠁⠀⠀⠀⡊⢁⠻⢆⣌⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠁⠀⠀⢀⠸⠀⠀⠀⠐⣄⡀⢈⡠⢆⠀⠀⠀⠀⠁⠈⢙⠒⠜⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠠⠴⠈⠀⠠⠀⠐⠈⠁⢈⠉⠉⠀⡈⡃⠐⠂⠀⢀⠀⠀⠙⠒⠛⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢠⠠⠂⠁⠀⠀⠀⠀⠐⠊⠁⠠⠈⠑⠃⠀⠀⠀⠀⠁⠢⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠂⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣩⠭⠍⠋⢁⣉⠓⢦
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢰⠃⠚⢧⠼⣲⣶⠿⢟⢯⡙⠞⡇
⠀⠀⢠⡿⡙⣼⠛⠳⡄⠀⠀⢸⠀⠀⡇⡴⠲⡄⢀⣉⠒⡽⠐⠃
⠀⠀⢺⣑⣾⠋⠠⣴⠃⠀⠀⠘⣶⠛⣾⡷⢌⠵⠀⡸⠉⠁⠀⡀
⠀⠀⠀⠉⠙⠳⣄⠈⠙⠢⣄⠀⠈⠳⡥⡀⠀⠀⣀⢀⣠⠂⡸
⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠘⠶⣶⣄⣀⣵⠀⠠⣤⣀⣀⠔⡁⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⣀⡞⠀⠙⣿⣀⢀⣿⠟⢿⣉⣉⣳⣤⡤⢤⣴⢞⣯⡉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠘⠿⣏⣁⡀⠀⠀⠀⢀⣙⣳⣾⠷⠞⢛⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠉⠙⢻⡿⣏⣉⡵⢿⡀⠀⣾⡸⡇⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣷⠈⠉⠑⠢⢗⣦⢇⢣⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣸⡅⠀⠀⠘⡛⠌⢂⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⠶⠶⠛⠛⠋⠉⢹⠀⠀⠀⠀⠘⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⢀⡴⠋⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⢀⣸⡧⠤⠶⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠋⠉⣿⢷⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠰⡇⢨⡇⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠀⠈⣷⠾⣇⠀⣗⠒⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠺⣄⣀⡿⡆⡋⠉⠣⢙⡲⠤⡈⣧⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⠶⣾⠃⣆⠸⠽⢀⡇⠉⠒⠒⠤⢥⠴⠃⢻⡋⠉⠉⠉⠉⠉⠉⠹⣆
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢀⣿⣀⣈⣛⣚⣋⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⣩⣭⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⢀⡾⠥⠤⠤⠤⠤⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⣠⡾⢹⣿⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⢛⣛⣛⡛⠛⠛⠛⣷
⠀⠸⡇⠀⠀⣰⠞⠛⠳⣦⠀⠀⢸⡇⠀⠀⠀⠾⠋⠀⠸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿⠉⠉⠹⡆⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠐⠂⠑⠂⠌⠈⠂⠤⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⢷⡀⠀⢀⣀⡀⠀⠀⠀⢀⣤⣠⠁⠠⠃⠀⠁⠂⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠁⠀⠀⠀⠀⠀⠈⠡⠏⠀⣸⢠⠁⢀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡘⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣨⡴⠊⠁⡄⠀⣈⠉⠂⠀⠀⠀⠀⣀⣠⣀⠀⢃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⠀⠀⠀⢀⠔⠻⠀⠀⢀⠇⢪⣬⣿⠀⠀⠀⠀⢀⢤⣄⠉⠀⠈⡂⠀⠀⠀⠀⡸⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠛⠥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠰⣶⣿⠇⠀⠀⢃⠀⠀⢀⠔⠁⠀⡐⠉⡉⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠧⣖⠀⠣⢀⠀⠀⠀⡀⠀⠀⠀⠀⢤⢀⣄⣠⢤⠀⠀⠀⠀⠀⠀⡡⠀⢩⠀⢀⠔⠁⠠⠛⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⢇⣀⠀⠀⠁⠢⣀⠈⠢⢄⠀⠀⠘⠦⠀⠤⠂⠀⠀⠀⠀⡀⠤⠀⠠⡂⠔⠁⠀⢀⢔⡀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣁⣀⣀⣂⡁⠢⡀⠀⠀⠁⠊⠒⠊⠑⠖⢢⡆⠀⠐⢲⠂⠩⠤⠀⠒⠊⢊⠀⠀⢀⠔⠁⡸⠿⠿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠒⠠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⡤⠐⠁⠀⠰⠁⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠐⠚⠀⠀⠀⠀⠀⠑⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠒⠀⠀⠄⠤⠀⠀⠀⠀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠋⡄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢀⣀⣀⣀⣠⠀⢇⣀⣀⡀⠀⢤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⠁⠀⠃⠀⠀⠀⠀⠃⠀⠉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠪⡙⢖⠒⠒⡆⠀⠀⠀⠀⠖⠒⠒⡞⡩⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠁⠀⠄⡦⠀⠀⠀⠠⡆⠄⠀⠉⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⠊⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠀⠀⠂⠁
⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⠉⢱⣀⣀⣀⣤⠤⠤⠤⠤⢴⣶⣶⣶⣶⣶⣶⡖⠒⠒⠒⠒⢢⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀⢸⠉⢳
⠀⠀⠀⢹⣿⣿⣿⣯⣧⠀⠀⣀⣀⣨⣿⡿⠿⠿⠿⠿⣧⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢉⣿⠁
⠀⠀⠀⢸⣿⠿⠛⠛⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢹⣿⣷⣶⣶⣦⣼⡿
⠀⠀⠀⠘⣿⠀⠀⠀⣘⣿⣿⠿⠿⠿⢿⣤⣶⣶⣶⣶⡟⠛⠛⠛⠛⣿⣤⣤⣤⣄⣼⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⡟⣶⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⡇⠀⠀⠉⠉⠛⡇⡇
⠀⠀⠀⠀⡇⣿⡿⠿⠿⠛⠚⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠦⠤⣄⣀⡇⡇
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢀⢠⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠇
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⢱⢸⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⠴⠒⠛⠲⢄⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠃⠉⠁⠀⢈⡄⢣⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⢸⢸⡄⠀⠀⠀⠀⠀⠀⠀⠙⣾⡈⠳⡒⣺⠟⡻⣸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡶⠯⠟⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡠⢤⣀⣀⡇⢐⠆⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠘⢄⣀⣀⡠⠔⠋⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡸⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⣀⣠⠴⣧⡞⠁⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠸⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡆
⠤⠀⠠⠠⠇⠸⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡔⢹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣸⣠⣧⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣴⣶⣿⠛⣭⡉⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠛⠛⠉⢁⣴⠲⣶⣤⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⡇⠘⠷⣶⡄⢸⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⢸⣧⣈⡉⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⢷⢻⣿⣿⣿⣿⣦⣈⣙⣤⡿⠿⠛⠛⠛⠛⠀⠀⠀⢸⣿⣿⣿⡘⡄⠀⠀⠀⠈⠻⠷⠶⠛⢀⣀⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⠈⣧⠻⠿⠿⠿⠿⠛⣋⣡⣴⣶⣿⠿⠿⠿⠀⠀⠀⢸⣿⣿⣿⣷⡹⣄⣀⡀⣀⣠⣤⡶⠿⠛⠉⣁⣀⣀⣈⣿⣿⣿
⠀⠀⠀⠀⠘⣷⣶⣶⣶⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠐⠀⠉⠉⣁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣈⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⣠⣤⣶⣾⠿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⣁⣤⣴⣶⠿⠟⠛⠉⠁⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⣁⣠⣤⣶⣿⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠂⡉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⣀⣠⣤⣶⣾⡿⢛⠩⣀⠀⠀⠐⢨⠙⠿⠖⠐⠀⣀⣀⣠⣤⣴⣶⣶⠿⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⣀⣤⣶⣾⠿⠛⠛⢉⡼⠿⠴⠀⠀⠉⣀⣀⣤⣤⣴⣶⣶⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠉⠀⢴⣶⣿⠿⠟⠉⠁⠀⢀⣈⣀⣤⣤⣶⣶⣾⣿⠿⠛⠉⠉⠉⠉⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⣶⣶⣾⡿⠟⠛⠛⠉⠉⢀⣰⡟⣁⠊⢀⣨⣧⣤⣄⠀⠈⢂⣻⣷⡀⠀⣀⢊⣠⣤⡈⣀⢸⣿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⡏⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡿⠟⠋⠩⢉⠭⠙⠛⢿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠀⠁⢀⣤⣿⣤⡀⠀⠐⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⣿⠀⠀⢀⣀⠀⠄⠀⠀⣿⠇⠀⠀⠀⣾⡁⣿⠸⠿⠀⠀⠈⠈⣿⡄⠀⢀⠐⣀⡀⠢⠀⢸⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣿⣿⢰⠀⣿⣿⡧⢈⠀⢸⣿⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠆⢸⣿⣿⠆⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⠀⠀⠈⠉⠠⠂⠀⠀⣿⡄⠀⠀⠀⣾⡆⣿⠈⣿⠀⠀⢀⢀⣿⠃⠀⠈⠀⠉⠁⠔⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠀⡀⠈⠛⣿⠛⠁⠀⠠⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣷⣦⣄⣒⣈⣐⣀⣤⣾⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡄⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣿⠋⠀⠀⣷⠀⠀⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠀⠘⠀⠀⠀⢸⡄⠀⠂⠀⠀⠀⢻⠘⠳⣤⣶⠶⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠀⠀⠀⣀⡈⢢⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣼⠇⠀⠲⠒⠠⣤⣀⢀⣀⡤⠴⠦⠴⠼⢏⡁⡀⠘⣿⣿⡄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⡿⠄⢠⡤⠖⠋⠙⠉⠉⠁⠀⠀⠀⠐⠀⠀⠈⠱⢶⡙⣿⠃⠖⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⢿⡿⢾⣇⡰⠊⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠈⢻⣄⠠⡃⡀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠴⠁⠀⠀⡾⢻⡏⠻⣦⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢻⣿⣯⡐⡀⠀⠄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡀⠸⣇⣸⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠿⢿⡁⠒⢀⠅
⠀⠀⢨⠀⠀⠀⠁⠉⠀⠀⣈⡩⠁⠀⠙⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣾⠟⠂⠉
⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠂⠀⠄⢀⡀⢹⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠠⠘⠋⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠐⢀⣐⠀⠀⠀⠀⠀⠀⠀⡳⣤⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢀⡆⢀⣴⡏⡁⠒⠍⣿⣷⠆
⠀⠀⠀⠀⠰⠀⣒⡎⠄⣀⡀⠀⠀⠀⢳⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣠⣾⣿⣿⣿⣕⡬⣭⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠯⢀⠒⠒⠀⣨⣄⣈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠢⣿⡿⠋⣨⡿⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⡇⠀⠀⠀⠀⠀⠞⢁⣾⡯⢇⡀⠀⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⢰⣿⡏⢄⣛⣮⢶⣿⡟⠁⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣭⣵⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡭⠋⠉⠉⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠦⢤⣤⣀⠀⠀⠀⣿⢮⣅⣀⣀⣀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡫⡞⠉⠀⠀⠀⠀⢙⡇⠀⢠⡿⣦⣄⣉⣉⣉⣁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡛⠶⢤⣤⣤⡤⢶⣿⡇⠀⢸⡟⠶⣬⣭⣭⣭⡭⣾⣿⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠟⠶⣤⣤⣤⣤⣾⡟⠁⠀⠀⠛⠦⣤⣤⣤⣤⡤⠿⠋⢀⣀⣤⠴⠞⠋⡡⠴⠞⠛⠦⣉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠓⠶⠶⠶⠖⢻⣿⠇⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⣛⣉⣤⣤⠶⠶⠚⠛⠋⠉⢀⡙⢿⣿⣿⣿⣷⣄⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⢿⣿⣿⣿⣷⣄⠈⢻⡀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠙⢿⣿⣿⣿⣷⣬⡃⠀⠀⠀⣀⡤⠞⣋⣥⠶⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡛⠷⣄⠀⠙⢷⣄⠠⢄⡀⠀⠀⠀⠈⠛⠿⠶⠤⠌⠻⣿⣿⣿⣿⣦⢖⣛⣭⢶⣛⣭⠶⢚⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣛⢦⣌⠳⣄⠀⠉⠳⣄⠙⢶⣤⡤⠤⠤⠀⢀⣀⣤⡴⣾⣿⣿⣿⣿⡿⠿⣛⣩⡥⠶⣛⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢷⣝⢷⣌⠳⣦⡀⠈⠻⠦⠶⠶⢖⣛⣻⡭⢿⣞⣿⠿⣿⣿⣿⡷⠖⣛⣭⡵⠖⠛⢉⡴⢖⡛⠛⠛⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣝⠷⣌⠻⣦⠴⠶⠚⣛⣩⡭⠶⠚⣋⣥⠴⢾⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⣾⣄⠋⠀⠀⠀⠀⢀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣌⠳⣤⡴⠖⢛⣉⣥⠶⢚⣋⣭⠶⠚⠋⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⢿⣍⠙⠛⠒⠒⠛⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⣤⠶⢛⣫⡥⠶⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣬⣉⡙⠛⠒⣚⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠶⣻⠝⠉⠉⠉⠛⢶⡄⠀⠻⣄⣉⠉⠉⠉⢉⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣁⣀⠀⠀⣀⣀⣴⡇⠀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⡉⠉⠉⢉⣩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣾⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠿⠿⢿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠸⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⢸⣿⣇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⢿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢫⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠘⠃⠀⠙⠋⠀⠘⠃⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢰⡖⠒⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠒⢲⡦⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⣀⠸⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⠃⣀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠀⠹⣧⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⠏⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣌⠟⣷⠀⠈⠻⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠟⠁⠀⣾⠻⣡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⡧⣿⢠⡀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢀⡄⣿⢼⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣮⣌⣿⣦⣤⡀⠀⠈⠙⣿⣿⠋⠁⠀⢀⣤⣴⣿⣡⣵⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⣬⢿⠘⢿⣦⠀⢠⣿⣿⡄⠀⣴⡿⠃⡿⣥⣿⣷⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⣭⣽⣶⣿⣿⣿⣿⣶⣯⣬⣿⣿⡿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣖⣖⣺⣻⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠀⠀⠛⠁⠉⠉⠛⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠠⢄⣀⠀⠀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣓⠄⠰⡱⠙⠦⠠⢀⣩⠤⢀⠝⡦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠁⠀⠀⠉⠛⠒⠀⠒⠛⠋⠉⠁
⠀⠀⠀⣀⠤⣆⡒⠺⡺⢻⡅⡀⠲⣦⠴⠀⠀⠁⢐⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠰⢤⣀⡀⠀⢀
⠀⠠⠎⠘⠒⠛⠒⠬⠠⣈⠽⠛⠚⠁⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⠀⠀⢀⠖⠀⠀⠠⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⠔⢠⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠿⣥⠭⡅⠂⠀⢀⢄⠞⠁⠀⠀⢀⠀⡂⠤⣀⠠⡀⢀⣼⠇⠒⠒⠀⣲⣤⣤⢀⣀⣠⡷⢶⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠲⡀⠀⢀⠄⠠⠂⠁⠀⠀⠀⠐⠃⢈⡠⠤⢍⡿⢀⠾⢇⡀⠀⠀⠈⢚⣈⣡⠁⡠⢄⡀⠹⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠒⠒⣩⠔⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠁⠀⠀⠀⠢⠀⠀⠀⠉⠀⠀⡀⠞⠩⠝⡟⡳⠁⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢆⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠸⠀⠀⠀⠀⠀⠊⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠢⢧⣀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠲⠥⠠⠄⠀⠀⠀⠐⠒⠦⠒⠂⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠕⣒⠀⠀⢰⠽⠒⠰⡦⠤⣔⢃⢷⣮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠖⠒⠒⠒⢀⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⠚⠉⠁⠊⠀⠑⠫⣢⡰⠂
⠀⠀⠀⠀⠀⠀⣀⠀⡀⣀⣀⡠⣄⠀⠺⠄⢲⣆⣀⠨⣟⠐⠯⠕⠶⠤⠤⢤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢥⣀⠐⣀⠀⠀⠈⠑
⠀⠀⠀⠠⡊⠁⠀⠏⠃⠁⢠⣄⡜⢢⠀⠈⠀⠀⠀⠈⠁⠛⠬⠤⠤⣥⡤⠤⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠉⠀⠉⠐
⠀⠀⠀⢠⡁⠂⢅⠱⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⡖⠚⠒⠲⠄⡀⠄
⠀⠀⠀⠈⢏⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠯⠷⡷⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠉⠊⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠆⢁⣠⠬⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡴⠶⣏
⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣠⣄⢹⡄⠀⢠⠔⠀⠐⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⣤⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡏⢠⠏⠀⣏⣠⣗⠀⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢸⣇⣻⠆⠘⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠘⡇⢸⡀⠀⠀⠁⣿⠀⢠⠄⢄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡀⢧⡀⠀⠀⢹⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠳⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡴⢃⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢧⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣠⠎⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢿⡌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⠻⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⢸⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢋⡾⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠹⣜⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠓⠦⣤⠀⣤⡴⠞⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡎⣝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⢈⣁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢍⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠁⠤⠈⠩⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠙⠂⠀⠈⠠⠀⠂⠂⠒⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⠴⠿⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠶⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢉⡠⠖⣏⣉⣉⣠⣤⣴⡯⠦⣌⢳⡄
⠀⠀⠀⠀⣠⣤⢶⣄⠀⠀⠀⠀⠀⠀⣼⠀⠀⢉⡗⠒⠚⠛⠛⠋⣽⠒⢦⡸⢸⡇
⠀⠀⠀⣴⡿⣍⢶⣟⡉⠹⡆⠀⠀⠀⢻⠀⠀⠸⡄⡞⢙⢦⠀⣠⣅⠙⢸⠃⠛⠁
⠀⠀⠀⣿⠑⢮⠟⠁⣡⡴⠃⠀⠀⠀⠘⣆⡞⠳⣷⡷⣌⣹⠦⠁⣸⠏⠉⠂⢠
⠀⠀⠀⠙⠛⠻⠶⣄⠀⠙⠳⢤⡀⠀⠀⠘⢷⡝⢈⠁⠈⠁⠀⠀⠀⠀⡄⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠙⢦⣀⣀⠀⠉⠁⢆⠀⠀⠈⠒⠒⠚⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠈⠙⢿⡟⢦⣴⣿⠀⠀⢹⣶⣦⣤⣎⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣠⡾⠀⠀⠹⣿⣄⣀⣠⡿⠉⠻⣤⣤⣤⣽⣦⡴⠶⢶⣾⠟⣯⣤⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⠙⠿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⡷⠾⠛⢻⣆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠈⠉⠛⠛⠻⠶⣿⣾⡛⠛⢋⣽⣿⠀⠀⢠⡏⢻⡄⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠛⠿⢿⣁⠹⣆⡀⢊⠹⡌⣃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢹⣫⠿⡛⠆⠛⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠄⠀⠀⠀⠘⡌⠃⢉⣠⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣤⣴⠶⠾⠛⠛⠋⠉⠉⣯⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⣠⡖⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢸⡇⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣀⣀⣀⣤⢴⣿⡏⠉⠉⠁⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠉⠁⠀⢸⡏⢿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢾⡁⢨⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⠀⠀⢺⣤⣼⣇⠀⢸⡧⠤⠴⠶⠶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡀⠀⢨⣷⡀⡿⠶⢾⠓⠦⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡈⢳⠤⡾⠉⡇⣧⣀⡀⠉⠒⠆⠉⢳⠈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⢧⡘⠛⠓⣸⠃⠀⠈⠉⠙⠲⠶⠶⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣀⣀⣙⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠋⠉⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⢀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣆
⠀⠀⣿⠋⠉⠙⠉⢉⣩⣭⣍⡉⠉⠉⠉⠉⣿⠀⠀⠀⠀⢀⣰⡿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡶⠶⠶⣦⡀⠀⠀⠀⣿
⠀⠀⠿⠀⠀⠀⠰⠿⠋⠉⠙⠿⠆⠀⠀⠀⠿⠀⠀⠀⠀⠛⠋⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠸⠿⠀⠀⠀⠘⠿⠀⠀⠀⠿

⠄⠀⠤⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣠⢤⠄⠤⠀⠀⠀⠀⠀⠀⢠⡤⢠⣄
⠀⢏⠐⠁⢴⠊⠀⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠌⠁⢎⡱⠂⠠⠆⠀⠀⠞⢠⠋⡸⠉⡝
⠀⠀⣆⠰⡀⠜⠸⣇⠀⠀⠀⠀⠈⡆⠀⠀⡤⠊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢈⠀⡀⠠⠃⠀⠀⠀⠀⡄⠠⢁⠂⠰⣶⠁
⠀⠀⠈⠂⠀⢀⠞⠹⡄⠀⠀⡀⠀⠘⢆⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠣⠘⡌⠐⠉⢂⠀⣸⠄⢀⢀⠂⣠⠃
⠀⠀⠀⢈⡴⠊⠐⠌⠱⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠀⡀⠀⠂⠀⠈⠀⠁⣤⠋⡰⠁⠎⢱⠏
⠀⠀⠀⠀⠸⡀⠈⠒⠆⠑⡀⠁⠈⠂⠀⠰⡄⠠⠂⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠡⠀⠀⢀⡞⠀⠀⠀⠀⠠⢀⠚⠠⠂⣐⠇
⠀⠀⠀⠀⠀⢹⡂⢈⠢⠜⢣⠀⠀⠀⠀⠀⠫⡀⢀⠀⡠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⠀⠀⠀⡀⠀⠀⠀⠀⢀⡰⠀⡰⢠⢠⡜
⠀⠀⠀⠀⠀⠈⠷⠠⡀⠀⡉⠆⠀⠀⠀⠀⠀⢴⡀⠀⠔⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢈⠁⢄⠐⢁⠐⣇⠀⠀⡾⠄⢀⣠⠈⡩
⠀⠀⠀⠀⠀⠀⠈⢃⠀⡁⠀⠘⣆⠀⠀⠀⠀⠀⢧⠀⠀⠁⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠊⠁⠈⠰⠁⠈⠢⠀⢀⢾⣁⡄⠀⣥⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠂⠎⠑⡄⠀⠀⠀⠀⠈⠶⠀⠀⠀⠱⠃⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡰⠆⠀⠀⠀⠀⣼⢃⠐⠁⠄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠁⠀⠀⣱⡀⠀⠀⠀⠀⠐⠇⠀⠀⠂⠘⠆⠀⠀⠀⠀⠀⣘⠄⢀⠀⢠⠜⠀⠀⠀⠀⢰⠁⠀⠋⣝⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⡔⢁⠲⡀⠀⠜⠀⠀⠘⡀⠀⠀⠀⠈⢀⠀⠀⠀⢴⡁⠀⠀⠑⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠂⠀⢷⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⠈⡄⠀⢠⠘⢦⠀⠄⠐⠀⡄⡈⠀⢠⠀⢈⡐⠂⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠘⡀⠀⠀⠐⢩⣀⡧⠀⠀⠘⡶⠁⠀⢀⠀⢀⣎⠠⠀⠛⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠠⠀⠘⡆⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠙⢷⡀⠐⠐⠁⠀⠀⠀⢀⠞⠠⠛⠶⢣⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠁⠸⡄⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⡇⠀⠀⢀⠀⠀⠀⣼⡀⠁⢤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⢹⣶⡶⣦⠂⠀⡠⠀⠒⠖⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡐⠀⢳⡀⠀⠀⠀⠀⠀⠆⠐⠀⠀⡀⢿⣯⠿⡀⣨⠁⠠⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠠⠈⠇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⢯⣼⣷⡉⢆⠡⠀⢄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⡄⠀⠀⢁⠀⠀⠀⠀⠀⠐⡄⠀⠀⡄⠘⣿⡟⢿⡄⠂⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂⠀⣠⠋⠀⠀⡀⠠⣤⡤⠼⠀⠒⠐⠀⠈⣷⣤⢮⡁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠙⠋⠀⢠⡀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠘⠿⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠀⣈⣰⡶⣾⢷⠴⣿⠿⠿⠾⣷⣿⣷⣶⣂⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠽⠀⠋⠉⠉⠀⢀⣀⣀⣀⣀⣀⣀⣈⠉⠙⠛⠻⠿⣶⣟⡢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣀⡠⠴⠓⠚⠉⠉⠉⠉⠉⠉⠙⠛⠻⠦⣀⡀⠉⠙⠻⣾⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⢤⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣢⡄⠀⠀⠀⠈⠙⠢⣤⡀⠈⠩⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣱⡟⠁⠀⡠⡶⠋⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢿⢼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣸⠎⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⠛⠛⠻⠿⠿⠿⠷⣦⠈⠳⣄⠰⣦⠈⢿⣷⡌⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⡻⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣦⡄⠈⢳⡌⠀⠀⢻⣿⡄⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⢼⠒⣾⠁⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠹⣾⣿⠀⢹⣿⡀⢿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠊⣿⠃⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠈⢹⠇⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣏⢹⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠠⡆⠸⡅⣸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡆⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠂⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⡀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢙⣆⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣌⠃⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢻⡆⠀⠀⠐⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢠⡛⠀⢈⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⢿⡄⠀⠀⠀⠁⠀⠀⢰⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡔⠀⢜⡎⢠⠒⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡀⠀⠈⠻⣧⡀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠋⠠⣼⠎⠀⠁⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⣤⡄⠈⠿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⣰⠞⠁⢰⡄⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢿⣄⡀⠈⠙⠷⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡕⠊⠁⢀⡠⡿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣴⡆⠀⠈⠙⠛⠳⠶⠶⠒⠀⡀⠀⠒⠦⠤⠙⠒⠈⠁⠀⣰⠎⠲⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠝⢓⠶⠶⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣦⣌⠓⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⣉⣉⣋⣉⡙⠋⠙⠿⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⢀⣀⠀⠈⢻⣿⣾⡿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⣴⠟⠳⠶⠶⠛⢉⣷⣾⡿⠟⠉⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⣠⠞⠁⠀⠀⢀⣠⣶⣿⠟⠉⠀⡀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢠⡾⠋⠀⠀⣠⣶⣿⠟⠋⠀⣠⡴⠟⠙⠶⣤⣤⠶⣦⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠴⠏⠀⣠⣴⣿⠿⠋⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⠰⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢰⣶⣾⡿⠛⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⣴⣾⡿⠟⠁⢀⣀⠀⠈⢿⣦⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⣠⣾⡿⠟⠉⢀⣠⡶⠋⠁⠀⠀⣾⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⠟⠉⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⢻⣷⣤⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⢀⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⣿⣧⠀⠀⠀⠀⣠⡴⠟⠁⢀⣤⣾⡿⠛⠁
⠀⠀⠀⢀⣾⡿⠋⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⣤⣿⠇⠀⠀⠰⠞⠉⢀⣠⣾⡿⠟⠉
⠀⠀⠀⢸⣿⡁⠀⠙⣿⣦⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⢿⡆⠀⠀⢀⣠⣶⣿⠟⠉
⠀⠀⠀⠈⢿⣷⣦⡀⠈⢿⣷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠻⣿⣆⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣜⣿⣷⡀⠈⢻⣤⣤⣤⣤⣀⢀⣠⡶⠋⠁⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣄⠀⠁⠀⠀⠀⠙⠛⠁⢀⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀⠀⠀⠀⢀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣤⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⢸⣦⡙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢋⣴⡆⣸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⠟⠛⢶⡄⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣖⣬⣿⠿⠛⢻⣿⣷⣇⣶⡇⠀⠀⠀⠀⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⣿⡾⡗⠀⣀⠁⢀⣀⠀⠀⠉⠉⠉⠁⢻⡆
⠀⠀⠀⠀⠀⠐⠓⠒⠒⠒⠂⠀⠀⠀⠀⣿⣿⣷⣜⣿⡆⣠⣤⣬⣍⠉⠁⠀⠀⠠⠄⠠⢸⡇⠀⠀⠀⠀⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡙⣿⠟⠃⠀⠀⠈⣿⠀⠀⠀⡄⠠⢠⡀⠀⣰⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⠁⢀⣠⠉⠉⠒⠒⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠁⠀⠀⠈⠁⠉⢠⠀⠀⠃⠀⠀⠀⠀⠲⢤⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠙⠻⠿⠿⣤⠞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠿⠀⠀⠂⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢁⡀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠆⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠤⠤⠤⠤⠄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣏⠲⣄⠐⠦⢤⣤⡤⢴⠀⠀⠀⠠⣀⠀⠉⠲⣤⣄⣄⣀⠀⠀⢹⡓⠶⢦⣤⣤⠀⣀⣇⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠙⠲⢤⠤⠤⣅⢀⠀⠀⠀⠀⢹⡄⠀⠈⠃⡀⠈⠙⢉⡾⡟⣷⠆⢄⡀⢠⠝⣿⡇⠹⣆⡀
⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠀⢀⡆⠀⠀⠀⠘⣧⠀⠀⠀⢾⡀⠀⠁⠈⢹⣆⠀⠉⠀⣁⣉⡁⢘⡶⠋⠀⣿⣿⠀⠈⠋⢢
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠀⠣⡋⠉⠓⡆⠉⠑⢼⡆⠀⠀⠀⠉⠓⠄⠈⢂⢻⣇⠀⠀⠉⢹⣿⣯⠀⠀⢀⠟⣿⢇
⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⡀⢸⡇⠀⠀⠈⢷⢄⠀⠀⠀⠀⠀⠀⠀⠡⣻⣧⢰⣦⡀⢿⣿⣶⣄⠊⢰⢿⢸⡆⠀⣄
⠀⣰⠋⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣟⡄⠀⠀⠀⠀⠂⠀⠀⢻⣿⣆⢿⣷⡈⣿⢧⡙⠿⣄⡟⠀⠘⠂⢸
⢠⠇⠀⠀⠀⠈⠳⣤⡀⠀⠀⠱⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢸⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⠑⠀⠀⠀⠀⠀⠀⢉
⣸⠀⠀⠀⠀⠀⠀⠘⡿⣆⠀⠀⠸⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠸⡇⠀⠀⢻⡿⠿⠿⠿⠷⠶⠶⠤⣄⡀⠀⠀⣠⠊
⡇⠀⠀⢸⠀⠀⠀⠀⠘⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠁
⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢺⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⣴⡷⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⢻⣷⡌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢋⣼⡿⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠹⣿⡌⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠙⣿⣆⢻⣷⡀⠀⠀⠀⠀⠀⣼⡿⢡⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠹⣿⡄⠀⠀⢀⣾⡟⢡⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠹⣿⣆⢠⣾⡟⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡘⣿⣿⠏⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⠏⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠁⣰⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣤⣶⣶⣿⣶⣶⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠱⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢥⠆⡈⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⢓⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡺⣹⣷⡄
⠀⠀⠀⠀⠀⠀⢮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⡴⠀⠀⠀⠀⠀⠀⢠⣿⡓⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⢞⣿⡄
⠀⠀⠀⠀⠀⢌⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡥⢰⠀⠀⠀⠀⣿⡷⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⢾⣿
⠀⠀⠀⠀⠀⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠖⢨⠀⠀⠀⠀⣿⣟⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⣻⣿
⠀⠀⠀⠀⠀⠩⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠯⠍⠘⠀⠀⠀⠀⣿⣶⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡲⣶⣿
⠀⠀⠀⠀⠀⠀⢚⡵⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⡒⢠⠃⠀⠀⠀⠀⠘⣿⡭⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣭⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠫⢇⡄⡀⠀⠀⠀⠀⠀⢀⢠⡲⣝⠂⡰⠃⠀⠀⠀⠀⠀⠀⠘⢿⣯⡼⣹⢿⣿⣿⣿⣿⣿⣿⡿⡟⣏⢮⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠽⠃⠀⠀⠀⠸⠝⠂⢁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣯⣞⣼⣿⣿⣿⣇⣣⣵⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣾⣶⡤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣉⣤⣶⣾⠿⠿⠿⠿⣷⣶⣤⣉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡘⠿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣌⡙⠻⠿⢿⣿⣿⡿⠿⠟⢋⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠦⠴⠶⠶⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⢄⡀⠀⠀⠒⠤⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⡀⠀⡤⠤⡀⠀⠤⠤⡀⠄⠈⠁⠈⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠐⠀⠰⠁⠀⠐⠂⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠨⠀⠰⠀⠛⠶⢤⣤⣀⣀⣼⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢷⣤⣀⠀⠀⠀⠀⠀⡀⢀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡱⢲⠂⠘⣄⠀⠀⠀⠀⡰⠁⢰⠂⠀⠀⠀⠈⠉⠙⠛⠛⠛⠁⠀⠀⣹⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠐⠁⠂⠀⠀⢨⠗⠀⢲⠸⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⠎⡆⠀⠈⠀⠃⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠘⠀⢧⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠡⠤⠤⠤⠀⠒⣆⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡆⠀⠀⢸⠀⠀⢀⠴⠓⠲⠤⢄⣀⣱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⣻⣿⢿⡿⠷⠖⠒⠐⠶⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⣎⠻⡀⠀⢀⣴⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠻⣿⣿⡄⣷⠀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣍⣭⣿⣿⣿⠿⠿⠆⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢡⣾⣿⣿⣿⢿⠋⢿⣧⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠃⠞⣀⢈⡉⠃⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⢀⣤⣿⣾⣧⣄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⡇⠀⣾⣿⣻⢸⡟⠛⠃⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠙⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠿⠄⠀⢀⢸⡟⣿⣿⣆⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠐⣷⣶⣶⣦⡤⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣇⣧⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠛⠿⣿⣷⣴⣭⣿⣿⣿⣿⣷⣆⣠⠤⢲⣿⡏⢻⡿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⣸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡽⢛⣫⢤⣤⣼⣿⣷⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆
⠀⠀⠀⠀⢈⣿⣷⣶⣶⡦⣤⣤⣉⡙⠃⠀⠀⣀⠤⠊⣹⣿⣦⠀⣸⣦⣤⡄⠀⢀⣀⣤⠖⠋
⠀⠀⠀⢠⣤⣀⣉⢉⠉⠛⠛⠻⠿⢿⣿⣟⣉⡤⠔⣲⣿⣿⣿⣿⣶⣺⣾⡆⣲⠿⠉
⠀⠀⠀⠘⠿⠿⢯⣿⣿⣶⣶⣤⣤⣀⡤⠟⠓⢊⡉⠭⠃⠀⠈⡉⠉⠉⢹⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠒⠉⠁⠀⠀⠀⠀⠀⣓⡿⠀⣚⣅⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣯⡀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⡴⠦⠌⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣀⠐⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣾⣷⡦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⣀⣬⣅⣈⠙⠲⣦⡀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠁⠄
⠀⠀⠀⡄⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠐⢿⡿⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⢠
⠀⢀⡒⢰⠀⠀⣀⡀⣀⣀⣀⡈⣀⡀⠀⢐⠆⣆⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢽⡿⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⢹⡯
⠀⠀⢧⢨⣆⠀⠑⢜⣿⣿⣿⣿⠡⠃⢀⣼⢁⠧⠀⠀⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⣼⡟
⠀⠀⠉⠷⣘⠳⠆⠀⠀⣽⣏⠀⠀⠸⠚⣡⠮⠁⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠏⠀⠀⠀⠀⠐⣿⢧⡀⠈⠀⠀⠀⠀⠀⠀⠁⠀⡼⣿⠂
⠀⠀⠀⠈⠩⠟⠀⠠⠾⠿⠿⠶⠀⠀⠳⠉⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠫⢟⡤⣀⡀⠀⠀⢀⣠⢤⡻⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠑⠫⠃⠘⠟⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⣶⣶⡆
⠀⠀⠀⠀⡀⠠⠀⠂⠈⠀⠀⠁⠃⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠑⢿⡿⠂⠀⠀⠀⣀
⠀⠀⢀⡞⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡔⠁⠀⠀⣠⣾⣷⣄⠀⠀⠈⢲⡱⣆
⠀⠀⣸⠂⡅⠀⠀⣀⣀⣤⣀⣀⡈⠀⠀⢀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢼⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠧⢱⡂
⠀⠀⢰⡆⠱⡀⠀⠄⢿⣿⣿⡇⠘⠀⣠⠏⣰⠂⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠧⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠨⠅
⠀⠀⠀⠛⣤⠉⠓⠀⠈⢿⡏⠀⠀⠟⢁⣔⠃⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠹⢤⡙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠶⢛⡤⠏
⠀⠀⠀⠀⠀⠋⠀⢀⣴⣿⣷⣦⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠋⠀⠀⠀⠀⠀⠈⠱⠃⠐⠂⠤⢠⡄⠤⠐⠂⠐⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⠀⢀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣾⣷⠴
⠀⠀⠀⠀⠀⢀⣀⡀⠸⣿⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠸⣿⣿⠃⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠈⢿⡿⠁⠀⡤⡀
⠀⠀⠀⠀⢠⣊⠏⠀⢀⣼⣄⠀⠈⢶⣣⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⡜⡰⠂⠀⠀⣸⣇⡀⠀⠑⢆⢣
⠀⠀⠀⣰⢱⠆⠀⠀⠈⢉⠉⠀⠀⠀⢱⡻⡄⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣝⢨⠁⠀⠀⢈⢉⠉⠁⠀⠀⠈⡅⢫
⠀⠀⠀⠯⢼⡂⢀⠀⠀⠀⠀⠀⠀⡀⢸⠥⠇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠂⠀⠀⠀⢐⡂⠰⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠦⢐⡂
⠀⠀⠀⣹⡘⣥⠀⠀⠀⠀⠀⠀⠀⠁⡜⢰⡇⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⢀⠀⠀⠀⢀⠀⢠⣿⡿⠀⠀⠀⠀⠀⢭⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⣴⢆⡭
⠀⠀⠀⠀⢻⡄⠷⣄⠀⠀⠀⢀⡤⠿⣼⡓⠀⠀⠀⠀⠀⠀⠈⠿⣿⣦⣄⣁⠀⠀⣀⣠⢴⡿⠟⠁⠀⠀⠀⠀⠀⠈⢳⣌⠲⢠⡀⠀⠀⢀⡄⠶⣡⡞⠁
⠀⠀⠀⠀⠀⠘⠱⠎⠀⠀⠀⠈⠳⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⠷⢾⣷⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠤⠟⢀⡀⠻⠤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠡⡀

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⢩⣍⠯⣆⠀⠀⠀⠀⠀⠸⢏⣭⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⠒⢦⡏⣿⣿⣿⣿⣿⣿⣦⠖⢲
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣸⣿⣷⢿⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⢸
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⢻⣿⢸⠀⠀⠀⠀⠀⣸⣿⢣⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢧⡀⢸⣼⣿⣿⣿⣿⣿⠀⢠⠏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣭⣋⠆⠀⠀⠀⢀⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⢟⣿⣿⣿⣿⡧⠖⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣶⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊⠛⢿⣿⣿⣿⣿⡿⠛⠑⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣄⣀⡤⣶⠶⠶⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠶⠶⢶⡦⣄⣀⣤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢹⡛⠛⠻⠷⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠿⠛⠛⣻⠁⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⠖⠚⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠒⠒⢦⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣏⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢦⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣠⠎⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠉⠳⠤⣈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠴⠋⠁⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡡⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠦⠤⠴⠆⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣤⡤⠤⠾⠦⠀⠀⠸⠦⠤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠒⠒⠛⠒⠒⠒⠒⠒⠒⠛⠒⠒⠚⠓⠒⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠗

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

⣶⡶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⠀⠀⠀⠀⠀⣀⡴⠛⢷⡄⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⠉⠉⠉⠉⠉⠡⠀⠀⠈⢡⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠶⠷⠶⠶⠷⠶⢶⣶⠟⠀⢀⡀⠀⠀⠀⣀⣠⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⣀⡀⢀⣀⣠⡾⠋⠠⠀⠀⠈⡙⠉⠙⢉⣰⠟⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⡟⠛⢿⣿⣿⣿⣶⣶⣷⢶⢾⠶⣶⣶⡿⠿⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣥⠶⠛⠋⠉⠉⠁⠀⠀⠀⠈⠉⠉⠙⠛⠶⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠛⢦⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⣀⡼⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠒⠒⠶⠶⢬⣝⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠶⠶⢶⠶⠄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⠁⠀⠀⢀⡏⠀⠀⢀⡄⠀⠳⣤⣤⣤⣤⣤⡀⠀⠀⠰⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⢠⠏⠙⢦⣀⣀⣬⣤⠤⠾⠃⠀⠀⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⢠⡶⣄⡀⠀⠀⠀⣠⡼⠇⠀⠈⢻⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠄⠀⠀⠈⠳⣄⠙⠷⠶⠟⢋⣀⣼⢷⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⠢⡀⠀⠀⠈⠛⠶⠶⠞⠛⠉⠀⠀⠙⣆⠀⢱⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⡇⠀⠀⠀⢀⡠⠊⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⠈⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠘⢦⡇⣀⡠⠖⠉⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣴⣀⠀⠈⠉⡙⠛⠓⠒⠒⠖⠒⠛⡉⠉⠠⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣡⣁⣂⡂⡈⡀⡈⣠⣁⣤⣴⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠋⠉⠉⠀⠀⠀⠈⠉⠙⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⡠⣀⣵⡾⠟⠋⠉⠉⠉⠉⠉⠙⠛⠷⣮⣔⢄⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⢀⢈⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣔⠄⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠄⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠊⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠁⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠊⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡡⠁⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠑⢝⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⡋⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠁⠊⠙⠻⠷⣶⣶⣶⣶⣶⠶⠟⠛⠅⠊⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⣤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⣶⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡈⢿⣧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠄⠀⢀⠀⠀⢄⠀⠀⠀⠀⠈⢻⣦⠀⠀⠘⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⣠
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣀⣀⠐⠐⠒⠒⠒⠒⠒⠂⠀⠀⠉⠁⠀⠂⠀⣀⠀⠦⠀⠀⠻⣶⠀⣀⠀⡀⠉⠓⠲⠖⠂⣼⣾⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠻⣦⡁⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠑⣈⢥⡾⠻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⡤⣰⢾⡟⠃⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⠀⠀⠐⠂⠀⠀⠀⡄⠀⠀⠀⠰⠀⠀⠀⣯⡴⠋⠁⠀⡄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠄⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣠⣻⠋⠀⣠⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢤⠀⠀⠈⠳⣦⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⢀⠀⡜⠀⠀⠀⠀⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠂⠀⡀⠹⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⠂⠐⠀⠙⠀⢀⠒⠀⣠⣄⣶⡤⣤⣄⠁⠀⠐⠆⠀⡈⠊⡀⠘⢫⡆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢄⠂⠀⢀⣠⠀⡀⠘⠏⠙⠉⠀⢀⠈⠙⠛⢷⠴⠈⣀⠑⠠⠁⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣜⣦⣶⠿⠛⠉⠀⠇⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⣤⣀⠈⢿⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠈⠀⠀⠀⠈⡄⠀⠀⠠⠤⣀⠀⠀⠀⢀⢰⠀⠘⠃⠀⠈⠙⠛⠾⢿⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠈⢃⠀⠈⠀⠀⠈⡄⠀⠀⠈⠄⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠆⠈⢠⡀⠖⡐⠀⣁⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⠀⠈⠀⢀⠀⠀⠐⠂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢤⠀⠀⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠤⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⢉⠀⠆⠀⠀⠀⠰⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡐⢆⣠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢣⠌⢂⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⣜⣁⣶⡉⣆⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⣈⢀⠈⠁⠄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢩⠀⡃⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢹⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⢘⡀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠛⠋⠉⠀⠓⠐⠳⠾⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠌⠁⠐⠒⠂⠀⡄⢠⠐⠖⠒⠲⠒⢆⡉⢶⢀⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⢀⠀⠆⠀⠀⣀⣀⣡⣼⣷⢏⣭⡤⣤⡥⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠘⣽⣃⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠉⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⡄⠤⠰⠲⠀⠈⢂⠃⢀⡈⡶⠋⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⣡⡀⢤⣮⡾⠪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠓⠳⠗⠸⠿⠃⡈⠠⠤⠀⠉⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢰⣤⡤⣤⡴⠀⠀⠀⣴⣶⠄⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠰⣶⣶⡶⠀⠀⠀⠀⢶⣶⣶⡖⠀⠀⠀⠀⡒⣶⣶⠖⠂
⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠈⢘⣿⣿⣿⡀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠀⠀⠑⣿⣿⠊
⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠻⣿⠋⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣇

⠀⠀⠀⠀⠘⡉⣷⣶⣾⢉⠃⠀⠀⢀⣠⣦⣀⠀⠀⠠⡖⣿⡿⢿⣿⢲⠀⠀⠘⢿⣿⡿⠟⠀⠀⠲⠀⠀⢰⠀⠀⣖⣿⣿⣿⣗⠆
⠀⠀⠀⠀⠀⠘⠿⣿⡿⠃⠀⠀⠀⠀⠿⡿⠇⠀⠀⠀⠈⠿⣿⣿⠿⠁⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠓⠖⠋⠀⠀⠈⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⢤⠾⠿⠦⠀⠀⠀⠀⠰⠾⠷⠄⠀⠀⠀⠀⠴⠿⠷⠀⠀⠀⠠⠿⠿⡇⠀⠀⠀⠴⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⢒⣉⠉⣶⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠔⠒⣂⡉⠤⠄⠒⠂⠉⠀⠈⢄⡚⢃⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠐⢂⣉⣡⠤⠒⠂⠉⠉⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⡀
⠀⠀⠀⠀⡀⠤⠐⠒⣈⡁⠤⠐⠒⠈⠁⢀⠔⠁⠀⣠⣿⣶⣦⡀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠈⡈⡇
⠀⠀⠀⠀⢡⠀⢣⠀⡃⠀⠀⠀⠀⠀⢀⠂⠀⠀⢸⣏⠸⣷⠈⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢣⢹
⠀⠀⠀⠀⠘⡄⡤⠒⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠷⣿⣶⣤⣄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⡈⣆⡀
⠀⠀⠀⠀⠀⢡⢱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣿⠈⢻⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⡔⢩⠁⢠⡄
⠀⠀⠀⠀⠀⠈⡄⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢰⣦⣤⣿⣷⡿⠃⠀⢠⠃⠀⠀⠀⠀⢀⣀⠇⠀⠇⢀⡇
⠀⠀⠀⠀⠀⠀⢣⠰⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠉⠉⠙⠛⠀⣀⣔⡁⠤⠄⠒⣊⡉⠥⠄⠒⠊⢹⠉⡇
⠀⠀⠀⠀⠀⠀⠸⡆⢇⠤⢀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠲⠶⠒⢒⣉⣥⠄⠒⢒⠏⠁⠀⠀⠀⠀⢠⠊⡤⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠲⡀⢡⠠⠤⠐⠂⣈⢩⡤⠀⠒⠚⠛⣿⠟⠁⠀⡠⠊⠀⠀⠀⣀⡀⠠⠼⠀⠘⠂⣸
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣈⠡⡄⠒⠂⠉⠁⠀⠀⠈⠐⣀⣤⡤⠤⠤⠖⠛⢒⣈⣉⠡⠤⠀⠒⠒⠂⠉⠉⡇⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⡄⠸⠤⠤⠐⠒⢀⣈⢉⡤⠤⠀⠒⣶⠞⠋⠉⢀⠔⠁⠀⠀⠀⠀⠀⢀⠖⣀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⡏⠦⠤⠄⠶⠦⠭⠭⠭⠤⠤⠤⠬⠴⠦⠤⠤⠤⠴⠮⠥⠤⠤⠤⠤⠤⠤⠤⠼⠀⠵⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⣀⣀
⠀⠀⠐⢿⡿
⠀⢀⣴⡶⠶⠶
⢰⣿⣯⣇
⠀⠈⢹⡿⠃
⠀⠀⢸⣧⣤⡄
⠀⠀⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦
⠀⠠⣿⡇⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡉⢩⢍⣤⠀⣄⢠⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⠈⣁⣉⡉⣉⣉⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠛⠙⠛⠛⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠙⠘⠚⠉⠊⠁⠛⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈⣿⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢽⡄⣦⣶⡞⣰⢷⢰⡞⣴⢛⡇⠀⡇⣴⠃⣾⢻⡔⡟⣷⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠈⠁⠈⠉⠈⠁⠈⠉⠁⠀⠁⠈⠉⠈⠉⠀⠁⠉⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⡦⠀⠀⠀⠀⠀⠀⢰⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⢾⡜⢳⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣼⣟⠄⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠿⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠁⡴⠋⠀⠀⠀⠀⢰⣮⠉⠈⢯⡆⠀⠀⠀⠀⠀⠠⣶⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡄⠀⠀⠀⠀⠀⠀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⠀⠀⠀⠀⢶⣿⣿⡶⠂⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⡜⡵⠟⠙⣮⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠰⢟⣟⠇⠀⠀⠀⠀⠀⠀⣬⡍⠋⣥⠀⠀⠀⠀⠀⠳⡻⢶⣶⢟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⡆⠀⠀⠀⠀⠀⠀⣌⣭⣭⡗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢶⣤⠀⠀⠀⠀⠀⠀⢙⠻⠿⣇⠀⠀⠀⠀⠀⢰⣶⣿⣿⣷⡶⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠘⡜⣿⡿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⣶⠷⠁⠀⠀⠀⠀⠀⠻⠇⠈⠟⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⣠⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⡾⠇⠀⠀⠀⠀⠀⠀⠘⠻⡿⠛⠀⠀⠀⠀⠀⠀⠈⡿⢿⡅⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⣘⣑⡄⠀⠀⠀⠀⠀⠀⠈⣨⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣆⡀⠀⠀⠀⠀⠐⠲⣷⣶⡗⠂⠀⠀⠀⠀⠀⠹⣷⣸⡟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⡘⠿⠿⡋⣀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣩⣥⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠟⠀⠀⠘⠟⠂⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠚⠒

⠀⠀⢀⡤⠖⠒⠲⣤⠀⣠⡞⠛⠛⠓⠒⠶⣤⣤⣀⡀
⢀⡜⣡⠾⠋⠉⠳⣬⣿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⡼⢰⡇⠀⠀⠀⠀⢸⠃⠀⠀⠈⠙⠻⠶⣦⣄⣀⠀⠀⠀⠀⠀⠉⠳⡄
⠇⢸⡄⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣀⣀⠈⠙⠛⠳⠶⣶⣶⣤⣴⣇⣀⣀
⠰⣄⠳⣀⠀⠀⠀⡇⠀⠀⠀⠀⢰⡟⠉⠉⣻⠂⠀⠀⠀⠀⠀⠀⢰⣿⣥⣬⡙⢶⡀
⠀⠘⠳⣝⠳⣄⣸⡇⠀⠀⠀⠀⠘⢋⡏⢰⡇⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠙⡆⢻
⠀⠀⠀⠈⣷⠈⢻⣧⠀⠀⠀⣤⢀⣸⠃⣼⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡯⢸
⠀⠀⠀⣰⠟⠀⢸⠟⢧⡀⠰⣧⣄⡀⠀⠓⢦⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡴⢃⠎
⠀⠀⠾⣧⣤⠖⠋⠀⠀⠹⣦⡀⠀⠉⠙⠓⠋⠀⠀⣀⣴⣿⠤⢤⠤⠴⠖⣋⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣠⡶⠶⠖⠛⠛⡋⠀⡶⠋⠉⠉⠋⠉⠁⠀⠀⢀⣤⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠋⠀⠀⠀⠀⠀⡇⠀⢳⠀⠀⠀⠀⠀⣀⣀⣠⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢸⣷⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⠀⠀⠀⠀⠀⠀⠙⠆⠼⠃⣠⣴⠾⠛⠛⠉⠹⠛⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠼⠋⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠂⠈⠙⢿⣷⡀⠀⠀⠀⠀⣰⠏⠀⠀⣤⡀⠀⠀⠀⠀⡴⠛⠶⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠼⠈⠻⣦⡀⠀⢠⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡂⠀⠀⢸⡇⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢹⡇⠀⠀⣸⣷⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠰⠷⠶⣶⡾⠛⠋⠀⢀⣼⠃⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⡀⠀⠀⠈⠛⠶⠶⠞⠛⠁⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠛⢷⣦⣤⣀⣀⣀⣀⣤⣴⡾⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⡇⠀⣉⡽⠛⣿⣍⣿⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⠞⠋⠳⣴⠏⠈⠋⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⢀⡏⠹⡆⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢻⡄⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢀⡼⠋⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⠛⠛⠿⠿⠿⠿⠿⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣦⣤⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣤⡴⣦⣄⡀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣦⣤⣤⣤⣬⣽⣧⣀⣀⣀⣀⣀⣠⣿⣯⣤⣤⣤⣤⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢚⣿⣿⡟⠂⠘⣶⣶⠇⠈⢸⣶⣿⠈⠀⢺⡗⠀⢉⣿⣏⠃⠀⣾⣿⣷⠀⠘⢿⡿⠃⠀⢻⣿⡟⠀⠀⢹⣿⡿⠀⠀⢸
⠀⠀⠀⠙⡟⠁⠀⠀⢸⡇⠀⠀⠈⣻⡋⠀⠀⢸⡇⠀⠀⢻⡃⠀⠀⠹⣿⠏⠀⠀⢸⡇⠀⠀⠈⣹⠁⠀⠀⠀⣿⠁⠀⠀⢸
⠀⠀⠀⠴⠷⠄⠀⠀⠾⠷⠀⠀⠸⠿⠿⠀⠀⠼⠧⠀⠰⠿⠷⠀⠀⠰⠿⠆⠀⠀⠾⠷⠀⠀⠼⠿⠷⠀⠀⠰⠿⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⣯⣀⣿⣭⣭⣭⣭⣯⣯⣭⣽⣭⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣽⣿⣿⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⢠⣄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠁⠀⠐⠀⠀⠈⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢂⠀⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⠤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠈⡄⠀⠀⡇⠀⠁⠀⡤⠤⡀⠀⠀⠇⠀⡘⠀⡌⠐⡷⠀⠀⠴⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⢠⠀⠈⡈⠀⠀⠘⡀⠰⠀⢀⠁⠀⡁⠀⠁⠀⠀⠠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⢸⠀⠀⠈⡄⠀⢡⡰⠜⠀⡡⠃⠀⡄⠀⠀⡀⠀⢠⠀⢢⠀⡄⢸
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⢱⠠⠀⠘⠀⠀⠀⠁⡀⠸⡷⢠⡞⢃⠀⡜⠀⠀⠀⢃⠠⡇⡠⡀⡶⠀⡄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠘⣇⢠⠁⠀⠀⠀⠀⢱⠀⢌⡙⠀⠈⠚⠀⠀⠀⠀⠘⡆⣷⣉⣠⡄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠶⣆⢼⡁⠃⠀⠀⠀⠀⠀⠀⡀⠀⠐⢀⡀⡄⠀⠀⠀⠀⠀⠘⠀⣛⠏⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠴⠐⠓⡂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠁⡃⠀⠀⠀⠀⠀⢘⠀⠴⠂⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢢⡉⠩⢥⠒⡀⠀⠀⠀⠀⠀⠀⡖⠲⢖⣲⡆⢡⠀⠀⠀⠀⠀⡄⠀⠀⠈⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⣇⠀⠀⠀⠀⠀⠀⢿⡲⣸⣄⢅⣿⠀⠀⠀⠀⢠⡒⡀⢀⠒⠂⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⡂⢑⠄⠸⡄⠀⠀⠀⠀⢸⠛⠚⠟⠳⠋⠅⠃⠀⠀⠀⠈⣹⠀⣁⣤⢖⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣨⠂⣔⢕⣠⡇⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⢰⠀⠀⠀⢸⣗⣬⢻⡄⢃⠮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⢳⢩⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠁⢸⠀⠀⠸⠀⠀⠀⢸⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⢸⠀⠀⠀⠀⠀⡀⠀⠤⣄⣀⠠⢼⣀⡤⢧⠄⡀⠀⠈⠀⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡀⠂⢸⠀⢀⠀⣀⢀⠁⠀⠾⠤⠍⠺⠜⠦⠥⠾⠃⢰⠀⠀⡀⣀⣘⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠈⢡⡕⣋⡧⣬⣘⣿⠀⠀⢸⠉⠀⠈⠁⠀⠈⠀⠉⠀⠈⠉⠉⠀⠸⣄⣹⠞⠭⢮⣉⣼⢭⠱
⠀⠀⠀⠀⠀⠀⠰⠄⠠⠠⠤⠤⠥⠀⠉⠈⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠠⠤⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠂⠑
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠐⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠐⠒⠒⠒⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠉⠙⠛⠓⠶⢤⡀
⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⠴⠖⠒⠒⠶⢤⣀⠀⠙⢦⡀
⠀⠀⠀⣠⠏⠀⣠⠞⠉⠀⠀⣠⡤⢴⣇⠀⠉⢳⡀⠀⢹⡄
⠀⠀⢠⡏⠀⢰⠏⠀⠀⠀⠀⢯⡀⠀⠙⠂⠀⠀⣷⠀⠀⣷
⠀⠀⢸⠀⠀⢾⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡿⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⣋⣉⣉⣉⣉⣙⠛⠒⠶⢤⣀
⠀⠀⠸⡆⠀⠸⣆⠀⠀⠳⣤⣤⡴⠃⠀⠀⢀⡼⠃⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣤⠖⠛⣩⠉⠉⢉⡭⠿⠯⣍⣛⡲⣄⠈⢳
⠀⠀⠀⢻⣄⠀⠈⠳⣄⣐⠋⠀⠀⣀⣠⠴⠋⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠳⢬⣉⣛⡲⠚⠋⠀⠀⢀⣋⣠⡤⠞⠀⣰
⠀⠀⠀⠀⠻⣷⣤⣀⠀⠉⠉⠉⠉⠉⠀⣀⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⣦⣤⣀⣉⡉⠉⠉⠉⠉⢉⣁⣀⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣶⣶⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣽⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⢸⣷⣽⣻⠿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⢿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣩⣤⠤⠤⠤⠤⠤⢤⣬⣉⡙⠻⢷⣽⣻⠿⣿⣷⣾⣿⣿⣿⣿⣷⣾⣿⠿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⡞⢩⣤⣖⣀⣀⡴⠛⠉⢛⡶⠾⠙⣦⠀⢹⡿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣶⣿⣿⢿⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠉⠓⠶⠭⣭⣭⣤⣤⣤⡤⠤⠖⠛⠁⣠⣾⡿⣷⣾⣿⣭⣭⣭⣭⣭⣭⣿⣷⣾⠿⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡿⣻⡿⣷⣾⣿⣭⣽⣿⣿⣯⣭⣿⣿⣾⠿⣛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣛⣿⣿⡿⠿⠿⣿⣿⣟⣻⣯⣷⡿⣻⣿⣷⠾⠛⠉⠉⠉⠙⠛⠿⣽⣿⣾⡿⣟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣻⣿⡿⠿⢿⠿⠿⣿⣿⣻⣯⣷⣾⣿⠏⠀⢀⣠⠴⠶⠶⠦⢤⣀⠀⠙⢷⣿⢟⣿
⠀⢀⣠⠴⠖⠛⣋⣉⣉⣉⣉⣉⡙⠛⠲⠦⣿⣿⣶⣭⣟⣻⣿⠿⠿⠿⠿⠿⠿⣿⣛⣯⣽⣿⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⣿⣿
⣴⠋⢠⠴⠛⣉⡍⠉⠀⣠⠶⠿⢭⣍⣟⢦⡀⠙⢯⣟⣻⡿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣻⣿⡟⠀⢠⡇⠀⢠⡄⠀⠀⠀⣀⣀⠀⠘⣦⠀⠹⡿
⣷⡀⠘⠲⢭⣙⣓⡒⠛⠁⠀⣀⣘⣁⡤⠞⠃⢀⣾⣟⣻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧⠀⢸⡇⠓⢿⡀⠀⢀⡞⠁⠈⣧⣀⢸⡄⠀⣷
⣿⣿⣷⣦⣤⣀⣉⡉⠉⠉⠉⢉⣉⣀⣠⣴⣾⣿⢿⣟⣿⠿⣿⣿⣷⣶⣶⣿⣿⡿⢿⣟⣿⣿⡄⠀⢷⡀⠀⠙⠶⠋⠀⠀⠰⠏⠈⢸⠃⠀⣿
⣿⣿⣽⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⣿⠿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡿⠿⣿⣷⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⡇
⢿⣾⣽⣻⠿⣿⣿⣶⣶⣶⣶⣶⣿⣿⠿⣟⣫⣷⡿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⢿⣿⣦⡀⠈⠓⠶⢤⣤⣤⠴⠚⠁⠀⣠⠞
⠀⠙⠿⣿⣿⣿⣶⣾⣾⣿⣷⣷⣶⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⣀⠀⠀⣀⣀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⠀⠀⢠⠄⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢂⠀⠀⠀⢡⠀⠀⠀⠈⠄⠠⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠘⡇⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠐⡀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠆⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠌⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢡⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⡌⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢡⠀⠀⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠤⠤⠤⠤⠦⠤⠤⠤⠦⠤⠤⠤⠼⠶⠶⠶⠿⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⣀⣀⣀⣀⣀⣠⠐⢢⣀⣀⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⡉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣧⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⣀⣤⣤⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠈

⢀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⡀⣀⠀⠀⢀⡀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠈⠿⣿⣿⠿⠁⠀⠀⠸⡙⢻⣿⠛⣿⣿⠋⡕⠀⠀⠀⢆⢸⣿⣿⣿⠀⠎⠀⠀⠀⠹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠈⣾⣿⣿⣿⢧⠊⠀⠀⠀⠀⠈⠺⢿⣿⠿⠊⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣶⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠼⠿⠿⠧⠀⠀⠀⠀⠀⠰⠾⠿⠿⠆⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠄⠀⠀⠀⠀⠠⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢩⢹⣿⣿⡏⡝⠀⠀⠀⠀⢰⣿⣿⣿⡦⠀⠀⠀⠀⠀⠰⣥⣤⢤⣤⡼⠀⠀⠀⠃⢹⣿⣿⣿⡇⡐
⠀⠀⠀⠈⠺⣿⣿⠗⠁⠀⠀⠀⢃⢸⣿⣿⣿⡇⠨⠀⠀⠀⠀⠀⠘⠿⡾⠟⠀⠀⠀⠀⠀⠘⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⣀⣾⣷⣄⠀⠀⠀⠀⠀⠀⢀⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⣿⣦⣴⣿⡀⠀⠀⠀⠀⢀⣿⣤⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⢀⣾⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣠⣤⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠚⣉⣉⠑⠲⣄⣠⣴⢶⢻⡙⣙⠳⢷⡀⠉⢉⠁⠈⠁⠉⣧⠀⠛⠿⠟⢉⢻⡏⣶⠦⣤⢴⡒⢋⣩⠿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣅⡷⠞⠛⠛⠲⠤⣽⡀⠻⠭⠞⠁⠀⠀⠀⠉⡑⡆⠀⠀⠀⠀⠨⢱⠄⠀⠀⠊⠈⡿⠁⣴⣟⡦⠿⠛⠓⠚⠶⡈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡿⠀⠀⠀⠀⠀⠀⠈⡷⢦⡤⠬⣄⠀⠀⣀⠠⣳⣧⣤⣴⣦⣤⣼⣟⢀⢀⠀⠀⢀⢳⡀⠙⡏⠀⠀⠀⠀⠀⠀⠹⡷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠁⣾⣤⠾⣿⡏⡉⠦⠀⠘⠃⠀⣋⡙⠙⣿⡧⣦⣦⠞⠀⢈⡇⠀⠀⠀⠀⠀⠀⢀⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢻⡀⠀⠀⠀⠀⠀⠀⢸⡀⣤⡬⢿⠋⡐⣘⠧⡌⡷⠁⠀⠀⠈⢧⡴⢴⡋⣀⢭⠛⢴⣤⢺⠁⠀⠀⠀⠀⠀⠀⣸⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢣⡀⠀⠀⠀⠀⠀⠈⣯⠟⠶⠀⠘⠒⠚⠀⠀⡌⢠⣾⣷⡄⡄⠀⠈⡤⠗⠀⠄⢠⠿⡟⠀⠀⠀⠀⠀⠀⣰⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣳⣄⠀⠀⠀⠀⠀⢻⡄⢀⠈⠀⣤⣤⣄⢑⣿⣸⣿⣿⣧⣵⡞⣀⣤⣤⠀⠁⣀⣰⠃⠀⠀⠀⠀⢀⣼⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣷⣄⠀⠀⠀⠈⣟⣚⠄⠀⠻⣿⡿⡿⠊⠁⠀⣀⠈⠑⢿⣿⣿⡟⠠⢠⢡⡏⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠸⡍⠀⠴⠶⣾⡏⠀⠛⣀⠤⠤⡀⢤⠀⠻⣯⣴⣈⠀⡾⠁⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⢻⡀⣴⣶⣏⡄⠰⠆⢇⠺⠇⡰⢀⠀⠀⣯⣤⣄⣸⢁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⡿⠿⠿⣥⡀⠀⡄⠉⢩⠀⠈⢀⣾⢿⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠄⢿⢓⣿⣢⣄⣀⣀⣠⣴⡟⢻⡖⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢸⣿⣿⠏⣺⡎⢿⣿⣿⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠁⢐⠁⠀⠀⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠡⢣⠴⢉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣣⣠⣿⣄⣀⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⠉⠋⣽⠯⠉⠙⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⠳⣅⠀⠀⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡐⠂⣤⠤⣄⠈⣷⣄⠰⠀⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⠟⠡⠈⡛⠸⡇⠯⠼⠀⡿⢈⠘⣁⠻⣬⣵⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠀⣴⡫⢍⢾⡟⠒⠶⡛⡇⠈⣞⡭⡙⢷⠀⢺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣙⢦⣈⣛⣡⠞⣀⣉⡀⠘⣷⣄⠙⠚⣁⡞⢰⣫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠧⠴⠥⠦⠼⠤⠿⠼⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠛⠛⠛⠁⠀⠘⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃⢀⣠⣀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣓⣒⣒⣒⣒⣒⣶⠾⠿⠯⠭⠭⢭⡿⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠘⢄⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢁⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⢄⠀⠀⠈⢂⠹⣷⣒⣒⣂⣈⣭⣭⢭⣭⠟⠀⠂⠀⠀⠀⢠⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠑⠄⠀⠀⠁⠘⢿⡉⠁⠀⠀⠀⢠⠊⡠⠀⠀⠀⡠⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠈⠢⠀⠀⠀⠈⢷⠀⠀⠀⣰⠃⡴⠁⠀⠀⣔⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠑⠀⠀⠀⡌⢧⠀⣰⠃⡜⠀⠀⢀⣾⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢆⠀⠘⡌⣷⠃⠎⠀⠀⡠⣣⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣠⣤⠴⠶⠷⠶⠶⢦⣴⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣀⡤⠴⠲⠖⠲⠒⠦⣌⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡠⢊⡥⠀⠀⠀⠀⠀⠀⠀⠠⡉⠳⣌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠞⣁⢹⠶⠃⡀⠀⠀⢀⡐⠶⣆⣐⡳⠘⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢠⡏⢠⣛⠛⠁⠀⢇⡩⠉⡆⣳⠀⠙⣮⣑⢝⣘⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⣆⣩⠀⠀⠀⠀⠀⠀⡀⢣⡀⠀⠶⠧⣽⣡⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡌⣯⠰⠂⠀⢢⡀⠀⠀⠀⣰⠇⢀⡻⢶⣎⣸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠱⡈⢀⠿⣠⠀⠙⠲⠶⠚⠉⠀⣬⢻⡎⣱⠃⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣳⡀⠘⢮⡻⠿⣾⣳⣆⠀⢰⣠⣷⠽⢷⡴⠁⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢝⣦⣀⠙⠲⠭⣽⣟⣓⣺⣏⡥⠖⠋⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢟⡶⣦⣤⣤⣬⣭⣤⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠛⠛⠋⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠿⣟⢿⣿⡿⡷⣶⡤⠤⠄⠤⣤⣤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣿⣷⡽⢷⡄⠀⠀⠀⡠⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀⣷⡀⠀⣼⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢦⣼⣿⣿⣧⣿⣇⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣾⣿⣿⣿⣛⣛⣙⣠⢄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⢀⣴⣿⣿⣿⣿⢻⡏⠘⣿⣦⠐⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⠃⠾⢁⡀⣈⠊⠃⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠀⣠⣼⣷⣿⣤⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣟⣿⡁⠀⣼⣿⡿⡏⣿⠿⠿⠆⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⢀⠀⢻⣿⣿⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠀⠀⠙⢻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠀⠠⣤⣷⢸⡟⣿⣷⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⠓⢶⣦⣤⣤⣀⣀⡀⠀⠀⠄⠀⠀⣀⣠⣿⣿⣾⣷⣿⡿⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣀⣙⣛⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣤⠴⠚⢻⣿⣧⢙⣿⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣷⡾⢟⣫⣥⡤⣴⣾⣿⣿⡗⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⣹⣿⣶⣶⣤⣦⣤⣄⣀⡉⠙⠓⠁⠀⠀⣀⠤⠈⣹⣿⣷⡄⢢⣽⣤⣤⡄⠐⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⣀⣈⠉⠉⠛⠛⠻⠾⠾⣿⣿⣿⣷⣶⠖⢀⣀⡤⢴⣾⣿⣿⣿⣯⣬⣭⣽⡭⢠⣾⡟⠉
⠀⠀⠀⠀⣿⣿⣟⣓⣾⣶⣤⣄⣀⣀⠀⠈⠉⣻⡛⠭⠖⠒⢛⡏⠉⠙⠛⠛⠛⠛⢻⡆⣘⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⢿⣿⣿⣿⣏⣁⡤⠔⠚⠉⠁⠀⠀⠀⣳⣢⡄⠀⣸⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣑⢼⠇⠠⡾⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣏⡬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⡀⠀⢠⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⣤⣤⣄⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⢿⣿⣧⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠁⠀⠉⠉⠁⠀⠀⠀⠉⠉⠀⠀⠛⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⠏⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠉⠉⠁⠀⢀⣠⣴⣾⡿⢻⠟⣿⢫⡟⣹⢋⡿⣹⢟⡿⣿⣷⣦⣄⠀⠀⠀⠀⠈⣿⣿⣄⣠⣤⣴⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡿⠃⠀⠀⣠⣶⡿⣿⢋⡾⣱⢏⡾⢡⠏⣼⢣⡞⣴⢋⡞⣱⢏⡼⢹⣿⣿⣦⣤⣶⣾⣿⣿⣿⠿⠿⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠀⠀⣠⣾⣿⠟⡼⢣⡟⣴⢃⡞⣱⢏⡼⣡⠏⣼⣣⣿⣴⣷⣾⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠇⠀⠀⣴⡿⣻⢏⡾⣡⠏⡼⢣⣟⣴⣧⣾⣷⣿⣿⣿⣿⠿⠿⠛⠛⠉⠁⠀⠀⣀⣀⠀⢠⣶⣶⣿⣿⣿⡆⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠁⠀⢀⣾⡟⣴⣣⣾⣵⣿⣾⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⣀⣀⠀⢰⣶⣾⣿⡿⠿⠂⠸⣿⣯⠀⠈⣿⣿⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣤⣤⣾⣿⣿⣿⣿⠿⠿⠛⠛⠉⠁⠀⠀⢀⣀⡀⠀⣶⣶⡀⠀⢿⣿⡄⠘⣿⣿⠀⠀⠀⠀⠀⢻⣿⡆⠀⣹⣿⡇⠀⢻⣿⣇
⠀⠀⣠⣤⣴⣶⣾⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⣀⣀⠀⢴⣶⣆⠀⠸⣿⣷⠀⢻⣿⣿⣆⠘⣿⣷⠀⢻⣿⣦⣴⣶⣆⠀⠘⣿⣿⣿⣿⣿⡃⠀⠘⣿⣿⡀
⠀⠀⢻⣿⣟⠛⠉⠁⠀⠀⠀⢀⣄⠀⠀⣶⣿⠀⢻⣿⡄⠘⣿⣿⣷⡀⢻⣿⡆⠘⣿⣿⣿⣧⣻⣿⡆⠈⣿⣿⠋⠉⠁⠀⠀⢻⣿⡇⠘⣿⣷⡀⠀⢻⣿⣇
⠀⠀⠘⣿⣿⡀⠀⢰⣿⣇⠀⢸⣿⣧⠀⢻⣿⡄⠘⣿⣷⠀⢹⣿⣿⣿⣜⣿⣷⠀⢹⣿⡏⢻⣿⣿⣿⠀⢹⣿⣇⣀⣠⣤⡄⠈⣿⣿⠀⠈⠛⠛⠀⠈⣿⣿⡄
⠀⠀⠀⢻⣿⣧⠀⠀⢻⣿⡆⢸⣿⣿⣧⠸⣿⡇⠀⢻⣿⡆⠈⣿⣿⠙⣿⣿⣿⡆⠈⣿⣿⠀⠙⣿⣿⡇⠈⣿⣿⠿⠟⠛⠋⠀⠀⠀⠀⣀⣀⣤⣤⣶⣿⣿⣧
⠀⠀⠀⠈⣿⣿⡄⠀⠈⢿⣿⣸⣿⡟⣿⣧⣿⣷⠀⠈⣿⣿⠀⢹⣿⣇⠈⢻⣿⣿⠀⢸⣿⡇⠀⠈⠛⠉⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⠛⠛⠉⠉
⠀⠀⠀⠀⢹⣿⣧⠀⠀⠘⣿⣿⣿⡇⠘⣿⣿⣿⡀⠀⢹⣿⡇⠈⣿⣿⠀⠀⠙⠋⠁⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⢿⣿⠟⠉⠁⠘⣿⣿⡀
⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠈⢿⣿⡇⠀⠈⠛⠉⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⡿⣿⢛⡿⣩⢏⡼⢡⣿⠏⠀⠀⣴⣾⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠻⠟⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⢿⡿⣻⢏⡽⢣⠟⣼⢣⡞⣰⢋⡾⣱⣿⠋⠀⠀⢸⣿⣿⠉
⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⡿⣿⢟⡿⢩⠟⣼⢣⡞⣰⢋⡾⣱⢏⡼⢡⠟⣼⢣⣾⣿⠟⠁⠀⣀⣤⣼⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⠛⠋⠉⠻⣿⣾⣴⣋⡾⣱⢏⡼⣡⠟⣼⢣⡞⣴⢋⡞⣱⢏⣼⣷⡿⠟⠁⠀⠀⢰⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⣿⣿⣦⣤⣄⡀⠀⠉⠛⠿⣿⣷⣾⣵⣯⣼⣡⣯⣼⣧⣿⣶⡿⠿⠛⠉⠀⠀⣠⣴⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⡄⠀⣀⡀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⣀⣀⣀⣼⣿⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣄⣠⣤⣤⣄⠀⢀⣤⣤⣤⣄⣠⣿⣿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⢿⣿⣿⠿⠻⣿⣿⣿⣿⠟⠿⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠞⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠦⠈⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣂⣄⡯⠉⠀⠀⠀⠰⠙⡄⠀⠀⠖⠙⢴⠉⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠂⠠⣉⡭⠋⠀⠀⢀⡀⠀⠈⠁⠀⠙⠀⠠⡄⠐⠂⠀⣐⠀⢠⡄⠀⠀⠐⢖⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡠⠬⣳⡀⠲⠐⢊⠰⣄⠀⠰⡆⠺⢀⠀⠈⠢⠀⡠⢼⢥⠉⠀⠀⠀⢀⣈⡠⡎⢶⠛⠁⠀⠀⡀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⡥⠮⣒⠁⠀⠰⠕⢵⠋⠁⠀⢸⡠⠚⠑⢦⡇⠀⠀⠀⠈⠁⠀⠈⠺⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⣒⡒⢤⡀⠀⠈
⠀⠀⠀⠁⠀⠀⡞⡵⠒⠢⡙⣄⣀⠈⡉⣉⠉⠉⠈⠀⠀⠀⠁⠀⠀⡈⠉⣀⠉⢠⠠⢠⠀⠅⢁⣀⣠⠷⠃⠀⠀⠈⢆⢳⠀⠀⠉
⠀⠀⠄⠀⠀⠘⣸⠁⠀⠀⠓⠛⠛⠓⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⢒⠒⠒⠐⠂⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⢸⢸⠁⠀⠒
⠀⠀⠄⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⡆⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠡⢊⠔⡡⢐⠄⡡⣊⢸⠀⠀⠀⠀⠀⠀⠀⡌⢸⠀⠀⠄
⠀⠀⠀⠀⠀⠘⢸⡀⠀⠀⠀⠀⠀⠃⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢊⠔⡡⢊⠔⡁⠂⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠡⢇⠀⠀⠀⠀⠀⠸⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢚⠔⢈⠔⡡⢊⠔⠃⠀⠀⠀⠀⠀⠜⡠⠏
⠀⠀⠀⠀⠀⠀⠐⢟⣆⠀⠀⠀⠀⠀⠆⠀⡪⠀⠢⡀⠀⠀⠀⠀⠀⠀⠐⡡⢒⡡⢊⠔⡡⠆⠀⠀⠀⠀⠠⢂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢎⡷⡀⠀⠀⠀⠈⢆⠐⠄⠀⠐⠀⠀⠀⠀⠀⠀⣪⠔⡡⢊⠄⡊⡐⠀⠀⠀⢀⠔⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡢⣄⠀⠀⠈⢦⠀⠠⠀⠀⢀⠀⠀⠀⠠⢔⡡⢂⠔⢪⠊⠀⢀⡠⢐⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢵⣢⢤⣀⡑⣄⠀⠂⠀⠀⠀⠀⢐⣡⡖⣻⣭⡥⣴⣶⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⠦⣄⠀⢀⠨⢛⡡⠚⠛⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠑⢐⡺⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⡠⠖⣝⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠢⠒⠔⡈⣹⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⢀⠕⡊⠢⢛⢛⣞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠺⠻⠯⠤⠞⠁⠭⠤⠿⠿⢿⠿⠿⠿⢿⠿⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡁⡟⠛⠛⠒⠒⠒⠒⠒⠒⠚⠚⠚⠚⠓⢺⠌⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣼⣶⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⣀⠸⡀⠀⠈⠀⠀⠀⠐⡛⠛⠛⠓⢒⠂⠖⠖⡶⢋⡞⣸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠙⠉⠛⠐⠛⠓⠓⠓⠓⠓⠟⠺⠞⠗⠻⠛⠟⠩⠑⠨⠃⠩⠂⠝⠊⠐⠚⠊⠓⠉⠊⠂⠒⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆
⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠁⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⠟⠛⠋⠁⠀⠀⠈⠙⠛⠻⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⣶⣶⣤⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠉⠀⠀⠉⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⡿⠷⠾⢿⡿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣀⣠⣀⣠⣄⣤⣀⣀⣀⣀⣀⣀⣠⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠓⠚⠉⠀⢀⡎⠙⢀⡼⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⠺⠷⠤⣤⣤⣀⠀⠀⠀⠉⣨⣦⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⠀⠈⠀⠀⠀⠀⠈⠉⠓⠲⠤⣾⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡢⠌⠋⢸⠈⠙⠛⡲⠦⢤⣄⣀⠀⠀⡀⠀⣿⣿⣿⣿⢿⠉⣩⣳
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢒⠭⣦⠉⠀⠀⠀⠈⠳⢄⣠⣌⠦⠅⣺⣫⣽⠐⡕⡰⣿⡿⠿⢉⣼⣾⣿⡿⡄
⠀⠀⠀⠀⠀⢰⣎⡀⠰⠚⣡⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣀⢔⣩⣾⣿⣿⠄⣼⣿⠿⠭⠴⣿⡷⣝⣟⡼⠃
⠀⠀⠀⠀⠀⢸⠚⢭⣙⡓⠮⣭⣒⠠⣠⠄⢒⣲⡖⣐⡬⣖⣿⣿⣿⣿⣟⡤⠿⠉⠀⠀⢠⢸⣯⣫⡞⠢⣄⣀
⠀⠀⠀⠀⠀⣸⠂⠀⠙⠪⢕⡒⣉⢿⣷⡲⢭⡴⣾⣶⢿⣿⣿⣿⣿⡛⠫⠥⣤⣀⣠⣼⢏⣸⣿⠟⠓⣤⠀⢘⠏⢒⡢⢄
⠀⠀⠀⠀⠀⠈⠑⠒⠠⢍⣒⠦⢭⣈⡈⠍⣛⣚⣯⣽⣿⣝⡋⠃⣀⠤⠤⢲⣫⣿⠗⡟⢊⡿⠟⠛⢉⠹⢖⠦⣐⣉⣵⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠲⠮⢍⡓⢃⣾⢿⡟⠩⠬⠀⠀⠀⢀⢔⣵⢿⣿⠙⢈⡞⠁⠀⡠⠂⠀⠀⠉⣋⠛⠿⣿⠿⣂
⠀⠀⠀⠀⠀⠀⢀⡔⠊⡡⠉⢠⡀⠐⠠⢍⣙⡉⠈⠀⢀⣀⣀⣀⢔⣩⣿⣷⣿⣿⣼⠯⠀⠐⠊⠀⠀⠀⠀⠤⠀⠀⠀⡃⠀⣆⡍⢁⣶
⠀⠀⠀⠀⠀⠀⣺⠦⠥⠌⠉⣀⢒⣂⠉⠱⠖⠦⠤⠂⣁⣰⣋⣴⡟⣫⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢂⡱⣀⠤⢚⡭⢞⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠍⠭⠄⠒⣒⣀⡀⠤⠍⠡⣤⠆⢤⣄⣐⣩⣮⣿⣿⣧⣀⣀⠠⣤⡀⣀⠀⠀⠀⢀⠀⠀⡐⡨⢔⣫⣴⣮⣵⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢋⣭⠩⣤⣔⣒⣒⣒⣒⣉⡩⣥⢭⣈⣭⡹⢫⣿⠟⠏⠉⠛⠷⡦⢌⡑⠎⣑⣲⣭⣐⠾⣽⣶⣿⣿⡿⠿⢛⡩⢔⡎
⠀⠀⠀⠀⠀⠀⠀⢸⡓⠑⠪⠢⢌⠐⠁⠩⠊⢔⡅⡩⢈⠑⡡⡩⠛⢶⣵⠀⠀⠀⠙⢳⠪⢹⣿⣧⣽⣷⡿⠿⢛⡋⠥⣒⣮⡾⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠼⡃⠀⠀⠀⠀⠉⠒⢷⡤⠄⠒⢒⣤⢄⣤⣶⣷⣿⣽⣷⣄⡀⠀⠈⠀⢽⢿⣟⠯⣭⣖⡾⡅⠯⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠈⢮⢷⠬⣩⣴⣾⣻⣿⡿⣿⡻⠕⠁⠈⠐⠤⣀⠽⠟⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠈⢶⠀⣩⣷⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⢄⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⡠⢀⠄⡀⠐⡀⡠⠐⠠⢊⠄⡨⠀⠔⡡⢂⠄⠠⢂⠄⠀⡀⡀⢂⠔⠡⢂⠔⡡⢊⢄⠄⡠⢊⢔⠕⡵⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠊⠐⠁⠂⠁⠀⠂⠐⠨⢀⠄⠀⠔⠁⠪⢊⠀⠐⠡⠂⠔⠁⠊⠐⠁⠂⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠛⡓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⣀⣤⠤⣤⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣤⣤⣄⡀
⠀⠀⢠⡞⢩⡴⠲⢦⢻⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠱⡄⡔⠉⠈⠑⣆⠀⠀⠀⠀⠀⠀⢸⢡⠖⠲⣌⠻⣆
⠀⠀⡾⠀⡟⢀⡤⠾⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣲⡟⣴⣢⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⠸⠒⣆⢸⡀⢹⡄
⠀⠀⣇⠀⡇⠘⢦⣤⢾⠀⠀⠀⠀⠀⢠⡄⠑⠢⠤⠚⠁⠈⠓⠒⠊⠀⡶⠀⠀⠀⠀⠀⣾⠦⠴⠋⢸⡇⢸⡇
⠀⠀⢿⡀⢷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢿⡦⣀⠀⠀⠀⠀⢀⣀⠤⣾⠃⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠇⣸⠃
⠀⠀⠀⠳⣌⢧⡀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢷⣀⠉⠙⠙⠉⠉⣀⣴⠏⠀⠀⠀⠀⠀⠀⡇⠀⣠⡴⢋⣴⠏
⠀⠀⠀⠀⠈⠳⣝⡲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠙⠣⠤⣤⡤⠬⠛⠁⠀⠀⠀⠀⠀⠀⢸⡧⢚⣥⠶⠛⠁
⠀⠖⠒⠚⠉⣉⡭⢿⡎⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⣿⢻⡟⠓⠒⠲⠤⠄
⠀⠶⠖⠛⠋⠉⢧⣈⣡⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣤⡾⠛⠛⠛⠲⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡉⠉⠉⠉⠉⠉⠉⠉⠉⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⠶⠶⠶⠶⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠄⠀⠀⠀⠛⢶⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣡⣤⣤⣶⣾⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⠀⠀⣠⣀⣠⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣶⠦⠴⠴⠤⠶⠶⠶⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠦⠀⢿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣼⠁
⠀⠀⠀⣾⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣞⡉⢉⣉⡉⠉⠉⣿
⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⠀⡴⠞⠛⠛⢦⠀⠈⢻⡀⢻⡄⠀⠀⠀⠀⠀⣀⣀⣼⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠁⠈⠉⠉⠀⠀⣿⠤⢤⡄
⠀⠀⠀⠸⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⢷⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⣤⠶⠶⢦⡀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠹⣄⣿⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠈⢳⡄⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣠⡾⠁⠀⠀⠀⣿⠀⣾
⠀⠀⠀⠀⠀⠘⢿⢰⡆⠀⠀⣴⣾⣭⣤⣤⠄⠀⠀⠀⣼⠛⠷⣄⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠘⢧⡻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠙⠳⣶⣤⣤⣤⣤⣤⣴⠖⠛⠉⠘⣧⢷⡀⠀⠙⠓⠒⠛⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣙⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⢯⡳⢤⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠛⠛⠛⢻⡍⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠙⠳⣦⣤⣤⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣰⠇⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣤⣤⣤⣤⣤⣼⣧⡄⠀⠀⠀⣴⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠲⠷⣦⡀⠀⣠⣯⣤⣤⣤⣤⣬⣧⡀
⣿⣿⣿⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⢀⣤⡴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣤⡀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣏⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣹⡇⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⢃⣀⣉⠉⠉⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠉⣉⣀⡈⣿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠛⠋⠁⠀⠀⢰⣿⢫⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⣿⡆⠀⠀⠈⠙⠛⠁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡄⠻⠾⠿⠋⠀⠀⠀⠀⣼⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣧⠀⠀⠀⠀⠙⠻⠷⠟⠁⣿⡀
⠀⠀⠀⠀⢰⡆⢻⣇⣴⡾⠃⠀⠀⠀⠀⠀⠹⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⠟⠀⠀⠀⠀⠀⠘⢷⣶⣌⡿⢠⡆
⠀⠀⠀⢀⢸⣿⠀⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⡀⣾⡏⢀
⠀⠀⠀⣿⣆⠹⠛⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠛⠋⢰⣿
⠀⠀⣤⡘⢿⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣆⣿⠏⣠
⠀⠀⢹⣷⡀⢁⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⡄⠀⣾⡿
⠀⢰⣦⡛⢷⠿⣋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠹⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠻⠾⠟⣡⡆
⠀⣀⢿⣷⣈⣼⡿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣛⣛⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⡀⣼⡿⣁
⠀⠸⣷⣍⢻⢩⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣍⠛⢩⣾⡏
⠀⠀⠙⠿⢮⡿⠋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡙⠿⣷⠿⠛
⠀⠈⢿⣦⡈⣇⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢰⢃⣴⣾⠃
⠀⠀⠈⠻⠿⠾⡟⢁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣿⣿⣿⣿⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡌⠻⡿⠿⠟⠁
⠀⠀⠈⢿⣶⣤⣱⣸⣿⢃⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡌⢿⣟⡜⣠⣶⡾⠃
⠀⠀⠀⠀⠙⠻⠿⠻⡅⢸⣿⡆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣿⣿⢈⠟⠿⠟⠋
⠀⠀⠀⠀⠘⢿⣿⣶⣜⣾⡿⢰⣿⡇⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡿⢿⡿⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣄⢰⣿⡇⢻⣷⣫⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⣙⣛⣉⣙⢦⣻⡏⢸⣿⡆⣶⣄⠀⠀⠀⢸⣿⠿⠿⣿⡿⠿⠿⣿⠿⢿⠿⠀⢀⠀⣠⣶⢠⣿⡿⢸⣟⡵⢋⣉⣙⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠿⠟⠳⢬⣻⣿⠹⣿⣷⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣼⣿⡟⢸⣟⡥⠞⠻⠿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⣿⣿⣷⡿⠛⠲⢬⣿⣇⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⢁⣿⡯⠶⠛⠿⣶⣿⣿⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣴⣾⣿⣿⡿⢏⣉⣝⣛⡿⠶⢦⣤⣤⣤⣤⡤⠶⢿⣛⣛⣉⡙⢿⣿⣿⣷⣶⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠾⢿⡿⠿⠛⠁⠀⠘⠉⠉⠉⠉⠃⠀⠀⠙⠿⠿⠿⠷⠦⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⢰⣿⣿⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣸⣿⣟⣴⣿⡟⠀⠀⣴⡆⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⠃⠀⠀⠀⠀⢹⣿⣿⠃
⠀⣼⡇⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⢿⣧⣤⣤⣤⣤⣼⣿⣿⣀⡤⢿⢦
⢠⡟⠁⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣿⣿⣿⣿⣧⣼⣧
⢿⣧⣤⣤⣀⣤⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠉⠙⠛
⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⠸⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⢻⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⢸⠇⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢹⣿⡟
⠀⠀⠀⠀⢹⣿⡟⢻⣿⡟⠈⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠘⣿⡇
⠀⠀⠀⠀⢸⣿⠇⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣷
⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⡟
⠀⠀⠀⠀⢸⣿⠀⠀⢹⣿⠀⠀⠀⢠⣶⣶⣿⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⠀⠀⠀⣿⡁⠀⠀⠀⣸⡃
⠀⠀⠀⠀⢸⡟⠀⠀⢸⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣼⣿⣇⣀⣀⣀⣿⣷⣀⣀⡀
⠀⠀⠀⢠⣾⣿⠀⠀⢸⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣶⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⢩⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⠋⠉⠩⠀⠀⠀⠋⠀⠀⣤⣀⣉⣉⡀⠀⠂⠉⠀⠉⠂⠘⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢀⡐⠤⠄⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠘⠓⠂⠤⣀⡀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⣀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⣿⣿⣿⣿⢏⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⢃⠀⣠⣤⣤⣄⡀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⢠⣤⠀⢴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⡏⠎⠀⢻⡟⠛⠿⣿⣦⡀⢠⣿⡿⠀⣠⣴⣶⣶⣶⣦⣿⣿⠀⢸⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣿
⣿⡿⠇⠀⠀⡇⠀⠀⠀⠈⠻⣿⣾⣿⠃⢰⣿⡏⠀⢻⣷⠈⢹⣿⡀⢸⣿⣇⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠐⢻
⣿⣷⡄⠀⣴⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠸⣿⣇⠀⢸⣿⠀⠸⣿⣷⠟⠙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⣿
⣿⣗⠂⢠⡏⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠙⠿⡿⠿⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣿
⣿⣦⡄⢸⡇⠀⠀⢀⣀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣼
⣿⣿⠀⢸⡇⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⢰⣿⣷⠀⠀⠀⢹⣿⡇⠀⠈⢛⡁⠀⣠⣦⢀⣾⣿⡄⠀⠀⠀⠀⢸⡇⠀⣿
⣿⡗⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠀⢰⣿⡇⠀⢸⣿⣾⡿⣿⡇⠀⢀⣤⠀⢸⠁⠀⢸
⣿⡏⡠⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⡇⢿⣷⣴⡿⠋⠀⣸⠀⠀⣿
⣿⣿⠃⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿⣿⣧⠀⠀⢀⣿⣿⣦⣼⣿⣧⣴⡿⢿⡿⠀⠈⠛⠉⠀⠀⢐⠍⠀⠘⣿
⣿⣧⠰⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⣸⡿⢻⣿⣆⢀⣼⡿⠋⠙⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠜⣼
⣿⣿⣗⠦⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⣴⣿⠇⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠘⣸⣿
⣿⣿⣿⣆⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠉⠑⠒⠠⢤⡤⣀⣀⣀⣀⣀⣀⢀⣀⣤⡤⠄⠒⠊⠉⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣕⣢⢤⠀⢀⣀⠀⠀⠀⠉⠉⠉⡁⠀⠀⠀⢀⠀⠀⣤⣔⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣽⣶⣿⣿⣿⣿⣮⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣀⣤⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣤⣀
⣴⣿⠋⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠙⣿
⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣸
⠸⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿
⠀⢻⣷⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢾⣿⣿⣿⠿⠿⠿⠿⠀⠀⠿⠿⠿⠿⣿⣿⣿⡷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠻⣿⣆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠈⠻⣷⣄⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡇⠀⠀⠀⠀⢸⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⣴⣾⣷⣦⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⠁⠈⠙⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⡄⠀⠀⠀⠀⢠⣶⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣅⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣩⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⢷⣾⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⣷⡾⠷⢶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠘⠻⠛⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠟⠃⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣽⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣯⡶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣯⠉⠉⢻⡀⠀⠀⢀⡟⠉⠉⣽⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢷⠀⠀⡾⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠇⠘⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢰⣿⠋⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣇⡀
⠀⠀⠀⠀⢸⣿⠀⢠⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⡀
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠁⠀⣶⡄⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⢿⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⣿⡇
⠀⠀⠀⠀⠈⢿⣷⣼⣿⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣂⣀⣙⣁⣀⣀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⢀⡴⠚⠉⠙⠲⣄⠀⠀⣠⣤⠤⠤⠤⠤⠴⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⣤⡀⠀⢀⡴⠚⠉⠙⠲⣄
⢰⠋⠀⡠⠒⠲⢤⠈⠳⣼⣁⠀⣠⣤⠤⠴⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣄⡀⢀⣽⡴⠋⣠⡶⠒⢦⡀⠈⢷
⡏⠀⢰⡇⠀⠀⠈⢳⡀⣿⠈⠉⠛⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠉⢸⠀⡰⠋⠀⠀⠀⢷⠀⠘
⡇⠀⢸⡇⠀⠀⠀⠈⠳⣽⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⣿⡞⠁⠀⠀⠀⠀⣼⠀⢀
⢹⡀⠈⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠉⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢠⡏⠀⡸
⠀⢧⡀⠘⢦⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠀⣰⠃
⠀⠀⠳⣄⠀⠑⢦⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠈⣧⡀⢀⣀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠁⢀⡴⠁
⠀⠀⠀⠈⠳⢤⡀⠙⠦⣸⡇⠀⠀⠰⣟⡉⠉⠉⠀⠀⠀⠈⠉⠉⣩⠟⠀⠀⠀⡇⡴⠚⠁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⢩⠷⠀⠘⣇⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⢸⠟⠀⠰⣏⠁
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣼⡀⠀⠀⠀⢠⠃⣠⠔⠋⠲⣄⠀⢧⠀⠀⠀⠀⣾⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠁⢧⠀⠀⠀⠘⠚⠁⠀⠀⠀⠈⠓⠾⠀⠀⠀⡼⠁⠙⠲⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠦⣤⣀⣀⡀⣀⣀⣀⡤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣉⣉⣉⣡⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⠛⠧⢤⣤⣤⠼⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠉⠛⠛⠛⠛⠛⠋⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠉⠉⠛⠛⠛⠛⠛⠛⠉⠹⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠻⢤⣄⣀⡀⠀⠀⠀⣀⣀⣠⣤⠟⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠉⠉⠉⠉⠉⠉⠉⠁⣀⣠⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠰⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠂⠈⠓⢆⣼⣶⣶⣿⣍⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠜⠉⠉⠈⠙⣿⠦⢬⡀
⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⣤⡤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⢠⣿⡿⠿⠋⠛⠛⠿⠅⠀⠀⠀⠀⠀⠀⠈⡤⡤⠀⠀⠒⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⣠⣞⡉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠀⠀⢴⣖⠛⠛⠛⠛⢿⣷
⠀⠀⠈⠘⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠔⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⢉⣦⡀
⠀⠀⠀⠀⠊⠀⠀⠋⠸⠀⠀⠀⠀⠀⠀⠀⠐⠦⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⡴⡀⡀⠀⣀⠹⠹⠁
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡱⠁⠅⡁⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠄⠀⠀⠀⡤⠤⠼⠤⠤⠄⠀⠀⠌⠀⠄⣀⠀⠀⠀⠐⢈⠐⠀⣧
⠀⠀⠀⠀⠐⠴⠒⠈⠁⠘⡄⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠀⠀⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠈⢹⣀⠀⠀⠀⠀⢸⠀⠀⡰⠁⠉⠐⠒⡴
⠀⢀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢘⠀⣰⠁⠀⠀⠀⠀⠇
⠀⠈⠀⠀⠀⠀⠀⠠⠔⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢡
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢧⠀⠀⠀⠀⢸
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠐⠒⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠈⠙⢷⣆
⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠏⠀⠀⠀⠀⢀⣴⡶⠀⢻⡆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⣦⣴⡿⠋⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢠⡞⠉⠳⢠⡏⠉⢷⠀⡏⠉⣷⢸⡏⠉⣷⢀⡿⡆⠀⠹⣧⡀⠀⠙⠋⠀⠀⠀⢀⣾⠃
⠀⠀⠀⣿⠀⠘⣧⣀⡴⠸⣇⣀⡼⠀⡏⠉⠁⢸⡏⠉⠁⣼⠓⢻⡀⠀⠘⠻⣦⣤⣀⣀⣤⣶⡟⠁
⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⡇
⠀⠀⠀⣿⠀⢠⡏⠙⢢⡏⠙⡆⣿⡄⣼⡇⣏⣙⠆⡇⠀⢸⡇⢰⢧⠀⡿⣄⡇⠉⡏⠁⠀⢸⡇
⠀⠀⠀⣿⠀⠈⠳⠔⠈⠳⠜⠁⠇⠹⠙⠃⠏⠀⠀⠷⠤⠘⠃⠏⠙⠇⠇⠘⠇⠀⠇⠀⠀⢸⡇
⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣭⣭⣭⣭⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣼⠃⠀⠀⠘⣧⣤⣄⣀⣀⣀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⢿⣏⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣹⠿⠃⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⣿⢳⣄⠀⠀⣀⣀⡀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣀⣀⠀⠀⣠⡞⣿⡀
⠀⠀⣀⣸⡇⠀⠙⠛⠛⢻⡿⠁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⢿⡟⠛⠛⠋⠀⢸⣇⣀
⢶⣿⣋⠁⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣿⠀⢀⣠⣴⣦⣄⡀⠀⣿⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⣙⣿
⠀⠈⠙⢳⡆⠀⠀⣀⡀⢻⣆⠀⠀⠀⢸⣿⠾⠛⠉⠀⠀⠉⠛⠷⣿⡇⠀⠀⠀⣰⡟⢀⣀⠀⠀⢰⡟⠋⠁
⠀⠀⠀⢸⣇⣴⠟⠉⠛⠛⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠛⠉⠻⣦⣸⡇
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⢀⣴⠟⠛⢶⡄⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠻⢶⡄⠀⠀⠀⠀⠀⢠⡶⠛⠻⣦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣧⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⣼⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢻⡄⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣄⠀⠀⠀⠻⣦⣀⠀⠀⣀⣴⠟⠀⠀⠀⣠⡟⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠙⢷⣤⡀⠀⠈⠉⠛⠛⠉⠁⠀⢀⣤⡾⠋⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠁⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢶⣤⠀⠀⠀⠀⠀⠀⣤⡴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠂⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⠀⠀⠉⠉⠑⠐⠂⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⣖⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠴⣏⣀⡀
⠀⠀⠀⠀⣄⣀⣀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠚⠻⢥⣀⠀⠀⠀⠀⠉⠒⠩⠽⣳⡆⠀⠀⠀⢀⣠⣴⢯⣥⣶⣶⣛⣫⡀
⠀⠀⠀⠀⢠⣌⣙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠿⣾⣯⠓⠦⢄⡀⠂⢤⣀⡠⠬⠉⠑⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣧⣈⠙⠒⠦⣍⡙⠲⢤⣤⡤⠤⠤⣶⣶⣶⣶⣶⣾⣿⣿⡇⠀⠀⠀⢀⠀⣷⣿⣿⣿⣿⣿⣟⣛⣛⣉⠁
⠀⠀⠀⠀⠀⢹⣌⠉⠂⠀⠀⠉⠙⠲⣶⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣀⣤⠼⠶⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠈⣱⠦⢄⡀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⣝⠲⢬⣑⠒⠤⢀⣉⠻⣛⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢍⣙⡒⠲⠤⠴⠚⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠐⠚⣏⡉⠀⠉⠐⠲⠶⡄⠀⠀⠀⢀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠾⠁⠀⠀⠀⠀⠀⠀⢘⠣⠤⣀⠀⠀⠀⠀⠀⠙⠛⢋⣀⡄⣦⠄⠒⠀⢸⣿⣿⣿⡿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣤⣤⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠂⠀⠉⠒⠴⢶⣾⣿⣿⣿⣿⡇⣯⣀⣀⣤⣾⣿⣯⣷⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢩⣑⣲⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠤⢀⡀⠈⣿⣿⣿⡿⠟⢛⣹⣾⡞⠉⢀⣼⣿⣿⠗
⠀⠀⠀⠀⠀⢭⣐⣒⣶⣾⣓⠲⠬⣿⣿⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣔⠚⠣⢤⣀⠉⠈⠉⠀⣤⣾⣿⣿⣿⠓⠛⠛⠋⠉
⠀⠀⠀⠀⠀⢐⠂⠭⢽⣯⠤⢍⣛⣿⣿⣿⢄⣒⣶⣾⣿⠀⠀⠀⠀⠈⣇⠉⠉⠒⠢⠤⣀⡀⢐⣻⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠠⢌⣙⣻⣿⣑⡒⠶⣿⡿⣿⡀⠬⢿⣭⣽⡗⠀⠀⠀⠀⠉⠒⠠⠀⢀⡀⠀⠀⠽⣺⠟⠋
⠀⠀⠀⠀⠀⢘⠒⠲⣿⣿⠦⢈⣉⣛⣛⣿⠍⣙⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣓⣒⣺⣿⣿⣿⠃⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠻⡀⠀⠀⠀⠀⠀⠀⢀⠟⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠒⠂⠠⠤⠥⠀⠀⠀⠠⠭⠤⠄⠒⠊⠱⡀
⡴⠮⠭⠭⠭⠭⠭⠛⠀⢀⢤⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡤⠀⠈⠻⠤⠭⠭⠭⠭⠵
⡇⠠⣶⠒⠐⡆⠀⣀⠤⡗⢄⣈⠀⠒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠂⢉⠠⢺⠤⣀⠀⠐⠐⠐⣶
⡇⠀⣿⠀⠀⢆⣀⣁⣀⡇⠀⠀⠉⠀⠃⠀⠒⣒⡂⠀⠀⠀⠀⠀⠒⠒⠀⠘⠀⠉⠀⠀⢸⣀⣉⣀⡀⠀⠀⣿
⠇⠀⣿⡆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⣿
⢰⠀⢻⣇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠄⢠⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣸⡏
⠀⡄⠘⣿⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢠⣤⡄⠤⢀⠀⣤⣄⡀⠐⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣿⠃⢠
⠀⢠⠀⢹⣧⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠘⢄⠈⠛⡡⠂⠘⢄⠻⠟⡠⠃⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⣼⡏⠀⡄
⠀⠀⠆⠀⢿⣆⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⠀⠁⠀⠀⠀⠀⠀⠁⢀⣠⠂⠀⠀⢀⠂⠀⠀⠀⠀⣰⡟⠀⠐
⠀⠀⠀⠀⡀⠻⣦⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠑⠦⡤⠄⠤⠄⠀⠀⠉⠁⠀⠀⠀⡌⠀⠀⠀⢀⣴⠟⢀⠄
⠀⠀⠀⠀⠑⢄⠈⠳⣤⡀⠀⠀⢣⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⠃⠀⠀⠀⡜⠀⠀⢀⣤⠞⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠉⠢⢌⡙⠛⠶⠶⠷⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⢀⠾⠶⠶⠛⢉⡡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠐⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠂⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠶⢤⣤⣤⠤⠶⡪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⠊⠳⠶⠤⠶⠶⠞⠓⢄⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠠⠤⠤⠄⠀⠤⠐⠃⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣇⣈⣉⣉⣉⣉⣉⣉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⡀
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡘⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⠀⣷⣄⣤⣼⣭⣴⡴⣟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣽⣿⢿⣿⠯⠉⠀⠀⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⢀⣟⡛⠛⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡋⠙⠛⠻⠿⣶⣀⣴⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣶⣶⣦⣶⠉⢿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⡟⢛⡛⡿⠋⢿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⡘⣷⡄⡸⡷⠋⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⡀⠡⢤⡀⣉⠷⣴⠲⣿⡛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢧⡴⢺⡏⠸⣀⣽⠀⠁⠀⢤⣀⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡔⠋⠸⣟⠁⠀⠀⢠⣾⣿⣤⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠃⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡄⠀⠀⠀⠀⠮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠆⠀⠀⠀⠀⣓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢑⣣⡀⠀⠀⠀⠀⠀⠀⠀⣬⡛⠁⠀⠀⠀⠀⠈⠔⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡋
⠀⠀⠀⠀⠀⠀⠁⠾⣠⣀⡀⠀⣀⣤⠺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠤⡀⠀⠀⠀⠐⠦⠈⠀⠀⠀⠀⠀⠀⠀⠉⠿⡤⡄⡀⠀⢀⣄⣴⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠋⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠀⠘⠊⠁


⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣾⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡗
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠛⢷⡦⣄⠀⠀⠀⡠⣖⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡄⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣑⡆⠀⠀⠀⠀⠀⠀⠀⢸⣻⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⢲⢰⡀⠀⠀⠀⠀⠀⠀⠀⢐⡆⡆
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣍⢧⠀⠀⠀⠀⠀⠀⢀⡖⡭⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⢣⡽⢲⠀⠀⠀⠖⢧⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣭⣻⣉⣍⣻⣟⣉⣿⣫⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡿⠿⠟⠛⠛⠛⠿⢿⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣡⣴⣾⣿⣿⣿⣿⣷⣦⣌⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣠⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣦⣉⠛⠛⠿⠿⠟⢛⣉⣴⣾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡉⠻⠟⢿⡿⢿⡿⠻⠟⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣿⣄⣴⣦⠀⣴⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣿⡏⠀⠸⣿⣿⡿⠁⠀⠉⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢹⡟⠁



⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⣶⣶⡄⠀⢠⣶⣆⠀⠀⢠⣶⡄⠀⠀⣴⣶⡀⠀⢠⣶⡄⠀⠀⢠⣶⡄⠀⢀⣴⣦⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⠀⣽⢿⠀⠀⠀⠿⠇⠀⠀⠚⠛⠗⠀⠀⠟⠻⠂⠀⠚⠛⠛⠀⠀⢸⣿⠁⠀⠀⣿⣟⠀⠀⢨⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⡰⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠃⠀⠄⠢⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣀⡇⠀⠀⠑⠊⠀⠠⠟⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠄⠀⠀⠀⣠⠶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⡠⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⢀⠄⡔⠈⠀⠹⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⢅⠀⠂⠠⠈⠀⠠⠀⠀⠀⠠⠄⣴⡁⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⡆⠀⠀⢂⠀⠂⠀⢀⠂⠀⠀⠀⠇⡐⠬⣉⡲⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠅⢀⡀⠂⠀⠀⠀⠠⠈⠛⠒⠊⠀⠧⠀⠀⠀⠀⠀⠀⠦⠥⠔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡀⠀⠁⠀⠀⠀⠠⠘⠂⡠⠚⠴⠀⠀⠀⠈⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣀⠤⠤⠤⣄⠀⢠⡊⠉⠁⣁⣀⣀⣀⣀⡠⢤⠤⣤⣤⣀⣀⣀⣀⣀⠀⢈⡷⢀⠴⠒⣊⠑⠲⣄
⠀⢠⡾⢡⠖⠒⠲⢄⠙⣼⠉⣿⠿⠿⠒⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⢙⡶⣡⠞⠋⠉⠉⢢⠘⡆
⠀⣿⠃⡇⠀⠀⠀⠀⢱⣸⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣏⠀⠀⠀⠀⣸⠇⠃
⠀⢹⡆⢧⠀⠀⠀⠀⡸⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠧⣙⠂⠀⠀⣠⠏⡴
⠀⠈⢻⣌⢳⡀⠀⠨⠴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⡴⢃⠞
⠀⠀⠀⠙⢧⣉⠢⣄⠀⠀⠈⣇⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣠⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠈⠳⢤⣑⠦⡀⠸⡜⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⢦⣱⡸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣻⡳⡙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡇⠙⢎⠻⢦⣀⠀⠀⠀⠀⠀⢀⠎⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡱⠃⠀⠀⢹⣦⣍⣙⡛⠂⠠⢖⠁⠀⠘⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⠶⢶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⣁⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣤⣤⣤⡶⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠘⠿⣷⣤⣄⣀⣀⣀⣤⠴⠃⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠡⠤⠤⠤⠤⠤⠭⠽⠽⠴⠶⠶⠶⠶⠶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣷⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣁⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣄⢀⠄⠀⡀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡘⠀⠀⠀⠀⠀⢣⡜⠉⢀⠀⠀⣠⡄⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠆⠀⠀⠀⢰⣥⡀⠀⢀⠈⠉⠉⠀⡔⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂⢪⠆⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠭⠤⠍⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡴⠞⠋⠛⠳⢦⣄⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠶⠦⠴⠞⠉⠀⠀⠉⠳⠶⠶⠶⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⣠⠞⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⣆⠀⠀⠀⠀⠀⢳⣄
⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠘⣇⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⠀⠀⠀⠀⠀⣸⠃
⠀⠀⣿⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿
⠀⢠⡏⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣼⠃⠀⠸⣆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢻⡄
⢰⠟⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠹⣆⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣰
⠀⠹⣇⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣠⡴⠶⣦⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣸⠏
⠀⠀⣿⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠙⠷⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿
⠀⢠⡏⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⢻⡄
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠹⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠏⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⡴⠶⠶⣤⡀⠀⠀⢀⣤⠶⠶⢦⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣄⣀⣀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣀⣀⣤⠞⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⣹⠇⠀⠉⠙⠳⣦⣄⣀⡴⠞⠋⠁⠹⣇⠉⠉⠉⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠘⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⢘⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠈⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⢀⠀⣠⡾⠋⣀⣀⣀⣠⣤⡴⠾⠛⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⠶⠶⢦⣤⣤⣀⣀⡀⠈⢷⡄
⠀⢀⠞⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢠⡿
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⢀⡿⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣾⠃

⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⠟⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣄
⠀⠀⠀⠀⠀⣿⣿⡏⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⡟⢰⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⡆⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢋⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⣄⡈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡙⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⡇⣿⣿⣿⣶⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡉⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠃⣿⣿⣿⣿⣿⣿⣘⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣇⣸⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣁⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⣤⠤⣤⣤⢤⣤⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢈⣭⠿⠋⠉⠀⢀⣴⣿⠂⠀⠀⠀⠙⠻⣏⠁⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣰⠟⠁⣠⠴⠦⣄⠸⣿⣿⣶⡆⠀⣀⣀⠀⠈⢳⡄⣧
⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣾⣁⣤⠤⠼⠆⠘⣿⠟⠰⢯⣥⣈⡙⣆⠀⠹⡏⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⠁⠀⠀⠈⣁⣤⣿⠃⠈⠀⠀⠈⠙⠋⠀⠀⣧⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⣘⡷⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡄⠀⢠⣿⡟
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠆⢰⣿⣟⡆
⠀⠀⠀⠀⠀⠈⠙⢿⡇⠀⣸⣛⢷⣤⣀⣸⣇⣤⣤⣤⣄⣀⠀⢈⣀⣤⣤⣤⣤⣤⣤⣞⣻⢿⠀⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡿⢶⣤⣬⣿⡝⠲⢦⡼⢿⣿⣿⠻⡶⢞⣿⣧⣴⣶⣿⣿⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⠿⢟⢿⣷⣿⣿⣆⠀⠀⢸⣿⣷⠀⣠⣿⣿⣿⣿⢻⣭⠿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠷⣽⣿⣿⣿⣷⣦⣾⣿⣿⡿⢻⣿⣿⢟⣞⡋⠙⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣠⡤⠀⢸⣻⣿⡏⢹⣿⡿⠿⣷⡟⢻⣿⡏⠋⠁⢀⡠⠹⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⡉⠀⠀⡈⡿⣿⡇⢸⣏⡃⠀⣭⡁⠘⣿⠀⠀⠀⠀⠻⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢭⣽⣿⣶⣾⣿⣷⣶⣿⣶⣿⣿⣀⣀⣤⡤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⢦⣐⠒⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠻⣾⣋⠛⠲⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⢀⣸⢁⣤⣂⣀⣀⣄⠀⠀⠀⠀⠀⠀⢠⡇⢀⠄⠂⠀⠀⠀⠘⢏⠛⠑⠘⡉⠓⠤⡅
⠀⠀⠀⠀⠀⠀⠀⠨⠻⠟⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠁⣼⠅⠎⠀⢦⡀⠀⠀⠀⡼⢶⣤⡴⠧⠨⡄⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠛⢀⠀⣀⢈⢅⠀⠀⠀⢘⡏⠙⣲⡈⠀⡅⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠘⠿⣿⣿⣿⣿⣿⣆⢠⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣼⣡⣲⡧⣍⣈⣀
⠀⠀⠀⠀⠈⠆⠀⠐⠠⠀⠀⠀⠀⠉⠭⠉⠉⠀⠈⢀⠄⠰⠄⠈⠙⠒⠒⠂⠀⠀⣾⣎⣿⣛⣟⣻⣛⡁
⠀⠀⠀⠀⠀⠤⡉⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠬⠤⠶⠂⠷⠒⠒⠀⠈⠁⡉⣩⠙⡻⡁⠉⠹⠀⠃
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠁⠘⠃⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⣇⠀⢀⡏⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠸⡄⡸⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢷⠃⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠘⡆⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢻⡀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠶⠶⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⡤⠖⠒⠒⠦⣄⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⢦⡀⠀⠀⠀⠀⣠⠞⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠉⠛⠒⠚⠉⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠀⠠⠤⠔⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢋⣙⡛⣛⣩⣌⠛⣫⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣡⣿⣿⣿⠟⣋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⢣⣾⣿⠿⢋⣵⣾⣿⡿⢛⣥⣦⣍⣛⣋⣍⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣴⡿⢋⣥⣾⣿⡿⢛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢛⣡⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢟⣡⣶⣿⣿⣿⣟⠹⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣩⣴⣿⣿⠟⣋⣴⣿⣿⠃⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣠⣾⠟⣫⣴⣿⣿⠿⣋⣴⣾⣿⣿⣿⣿⡀⠙⠛⠋⠉⠙⢻⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⡿⠛⠁
⠀⠀⠀⠀⢀⣾⡟⣴⣾⣿⣿⣿⡁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⡿⢛⣡⣶⣿⡿⠟⠁
⠀⠀⠀⣰⣿⣿⡇⢿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠛⠁⢠⣿⣿⣿⣶⣿⡿⠟⠉
⠀⠀⠀⢻⣿⣿⣿⣆⠻⣿⣿⣅⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠙⠿⣿⣿⣧⡙⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣌⢻⣿⣷⡘⠿⠟⠿⢿⣿⠿⣋⣵⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠹⣿⣿⣾⣿⣿⣶⣤⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡜⢿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠄⠀⠀⠀⠀⠀⢠⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣔⣶⢿⠟⠃⠉⠀⠀⠀⠀⠀⠀⠈⠁⠛⠻⡽⣺⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠸⠿⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠾⢇⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣧⣯⡝⠋⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠙⢩⣭⡾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡷⡶⠖⢰⣶⠶⠶⣾⡟⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠶⣶⡄⠳⣾⣾⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣻⣭⠍⠀⠸⣿⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠇⠀⠹⣯⣏⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⣶⠟⠃⠀⠀⠀⢻⣆⠀⠀⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣼⠏⠀⠀⠀⠛⠿⡴⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡛⣾⠿⠃⠀⠀⠀⠀⠹⣷⣄⠘⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⡾⠋⠀⠀⠀⠀⠘⠿⣿⢋⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⣷⠞⡃⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⢘⠳⣿⢋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣷⢟⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡳⣟⣩⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣻⡟⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⢻⣛⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⢳⠟⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⢻⡟⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣻⡋⣴⣆⡀⠀⠀⠀⠀⠀⢀⡴⢏⣼⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⢶⣆⢹⣟⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣭⣽⣅⣾⢿⣤⠄⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⡀⢠⣤⠿⣇⣼⣯⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⣦⣿⣬⣿⢺⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⡇⣿⣴⣯⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠟⠒⢿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⡿⠚⠿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣶⢿⣿⣈⣷⣴⠭⠄⠏⠁⡒⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠆⠙⠂⡠⣘⠉⣿⣿⢋⡉⠑⡀⡁⡬⡃⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠯⢠⠺⢦⠘⠂⢸⡜⣿⡿⠺⠀⢀⠇⣟⠄⠃⠀⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠈⠀⡞⠡⠀⠼⠉⣿⣿⣙⠉⠉⢈⣠⢘⢰⠈⠰⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡂⠐⣦⠀⠚⣶⠃⠤⢸⣿⠛⠒⠠⡼⠉⠸⡌⠓⡀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡉⠉⠯⢰⣍⠒⢀⡌⠑⠚⣿⣧⣮⠗⠦⣶⠎⠐⠸⣣⣌⠀⣛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⡓⡹⢄⡤⢄⢹⠀⢰⡉⢻⠞⢹⡟⣻⢀⠀⢹⣄⣴⣆⠀⠘⠃⠀⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠵⠇⠈⠀⢩⡸⡦⠄⢋⡅⠀⣿⣧⡸⣍⠀⠘⡹⠃⠺⠂⡄⠠⠀⠄⢳
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⠂⢸⡀⣸⠄⣽⠀⠘⠃⣈⣿⣷⣤⡞⢁⡈⠀⡠⣔⣉⠁⡀⠀⠤⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣏⢅⠘⠁⠀⡰⢿⠀⠘⡆⠀⠈⢳⣼⡟⠋⠉⠩⠥⠄⠁⡻⠀⠀⢃⣶⠒⣻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡈⠟⢒⠳⡀⠸⠒⠀⠻⢖⠀⢰⣼⣿⣶⡒⠎⢠⢄⡀⢀⠠⣄⣤⣀⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⠃⡱⡬⠤⢖⡶⢰⣿⠀⠀⣿⣿⡳⢣⢂⡵⡾⣅⣀⡤⠛⠠⠀⠹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠄⠼⠛⡰⠆⢀⡷⢀⣙⣿⣹⢿⣿⣷⣌⡀⠀⣰⣧⠛⠁⠀⢀⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⢀⢀⣦⠠⠔⣁⢈⡋⠉⢁⣿⣿⣯⣤⠀⠀⠚⢁⠀⣢⠀⢱⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣷⣄⣑⡶⢯⣶⣭⠙⠻⣿⣿⣤⠤⠄⠀⠼⢏⡔⠉⢱⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣫⡑⠚⢉⡁⣱⢶⣿⣟⠣⠀⠀⣀⡶⣲⢱⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣯⠀⣨⠷⣿⢻⣟⣖⠘⡁⢻⡯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠒⠁⠀⠉⠉⠚⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣯⠉⠉⠛⠣⢖⣒⣒⡒⠒⠒⠒⠒⠒⣒⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣡⡟⣠⠊⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠈⠄⠙⣶⣒⣒⣒⡒⠂⠠⠤⠤⢤⣾⠟⡰⠁⠀⠀⢀⠌⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⡀⠀⠀⠈⠀⠘⢯⠉⠉⠉⠉⠛⠛⢳⡿⠃⠜⠀⠀⠀⠐⠁⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⢄⠀⠀⠀⢢⡈⢳⡀⠀⠀⠀⢀⡿⠁⠀⠀⠀⣠⠊⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠱⡀⢷⡀⠀⢀⡾⠡⠀⠀⠀⡴⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡹⡷⡀⠀⠀⠘⣄⢳⣀⡞⣡⠃⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣎⢂⠀⠀⠘⢂⣻⡀⠃⠀⠐⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠿⠟⠛⠉⠉⠉⠛⠓⠲⠦⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣤⠴⠚⠉⠉⠉⠉⠉⠒⠶⣄⠀⠉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢡⡾⡋⢰⠒⠀⠀⠀⠀⠀⠀⠀⠐⢦⡙⢦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢰⣏⡐⣅⣀⣿⠛⠂⠀⠀⠀⠀⠀⠙⢾⢇⡄⠳⣄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡞⡘⢍⣩⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣋⡆⢸⡆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣹⣭⢭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣰⠀⡅⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⠙⣧⡴⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⣽⢣⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⢧⠛⣴⡿⣃⠀⠀⠀⠀⠀⠀⠀⢀⡠⣷⠀⠋⡜⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠈⢳⡉⣽⢿⣶⡀⣄⠀⠀⣤⣶⣼⣿⠞⣡⠎⠀⣰⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠙⢦⣌⡛⢛⣷⡠⠾⣿⣛⣃⡵⠞⠁⣠⡞⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣈⠉⠛⠓⠒⠓⠉⣉⣁⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢯⣿⣿⠿⠿⠟⠋⠑⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣭⣥⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⢲⠂⠀⠠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢥⡀⣠⣘⣦⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠙⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢀⣠⣴⠶⠞⠛⠛⠛⠛⠛⠛⠶⢶⣤⡀⠉⠶⠋⠃⠀⠸⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⢀⡤⠶⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣫⣍⡓⢾⡏⠛⠻⠶⠶⣶⣶⣶⣶⠶⠶⠟⠛⠋⠉⣿⠏⡔⠋⠉⢲⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠋⠀⠀⠛⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠃⡀⠀⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠳⠶⠊⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠁⠀⡼⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⣠⠴⢋⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⣆⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠗⣋⣭⠴⠖⠋⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡑⠲⠶⠴⠖⣚⣉⣉⢻⣦⡀⠀⠀⠀⠀⣠⢾⠁⠚⢳⠀⠀⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠉⠡⣄⡴⠃⠹⣦⠀⠀⣼⠃⠈⠓⠖⠋⠠⣄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣀⣠⡄⠀⠀⠀⠀⢀⣸⡇⠀⣧⣀⡀⠀⠀⢠⣴⠂⠉⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢰⡁⠀⠀⠀⡼⣿⠸⡣⠴⠟⣜⡿⡆⠀⠀⠈⢣⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣆⡴⠖⠓⠀⠀⢸⣧⡙⠻⠷⠿⠟⢋⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢠⣶⡄⢸⡇⠙⠛⠲⠖⠚⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡼⢧⣀⠀⠀⠀⢀⣠⡾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣀⠉⠛⠛⠛⠋⢁⡴⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡉⠓⠒⠚⢉⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⡀⠀⠈⠛⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠷⠿⠿⠿⠷⠿⠿⠿⠿⠷⠟⠋⠀⠀⣠⣶⣿⣿⣿⣷⣶⣤⡀
⠀⣴⣿⣿⠟⣋⣭⡝⠻⣿⣿⣆⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⡿⠋⣩⣍⡛⢿⣿⣿⡄
⢸⣿⣿⠏⣼⡿⠋⠀⠀⢹⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠁⠀⠈⠻⣿⡆⢿⣿⣿
⣿⣿⣿⢰⡿⠁⠀⠀⣠⣾⣿⡿⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠸⣿⣿⣦⡀⠀⠀⠸⣿⢸⣿⣿
⣿⣿⣿⠸⡇⠀⢺⣿⣿⣿⠿⠁⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⠆⠀⡿⢸⣿⣿
⢻⣿⣿⡄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣼⣿⣿
⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣄⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⢻⣿⡽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠹⣿⣾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⡿⠁
⠀⠀⠀⠀⠈⠻⣷⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⡿⠀⠀⠀⠀⠀⠀⢀⣤⣞⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⠃⠀⠀⠀⢀⣠⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⣄⡀⠀⠘⣿⣿⣇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⠃⠀⢀⣤⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⡀⠘⣿⣿⣧⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣴⣿⠋⣠⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠘⢿⣿⣷⣄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⢃⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠈⠻⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⣉⣉⣛⣛⣋⣉⠀⠀⠀⠀⠀⠈⠛⠿⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⢰⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢙⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠋⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡉⢹⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⢀⣄
⠀⠀⠀⠀⠀⢰⣿⡇⠟⣋⡀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⡀⠀⠀⢀⣙⠿⢸⣿⡆
⠀⠀⠀⠀⢀⢸⣿⢡⣿⡿⠃⠀⠀⢸⣿⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⣿⡇⠀⠀⠈⢿⣿⡌⣿⡇⡀
⠀⠀⠀⠀⣼⣷⠉⢛⣫⠁⠀⠀⠀⠸⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇⠀⠀⠀⠈⣙⡛⠉⣼⣷
⠀⠀⠀⠀⢿⣿⢠⣿⡿⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⢿⣿⡆⣿⣿
⠀⠀⠀⢠⣎⠻⠸⢟⡁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢈⡻⠇⠿⣣⡄
⠀⠀⠀⠸⣿⣧⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣸⣿⡇
⠀⠀⠀⠀⠻⣿⣸⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⡟
⠀⠀⠀⠀⢷⣮⡁⢱⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡎⠉⣥⣶
⠀⠀⠀⠀⠘⣿⣷⡸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⠇
⠀⠀⠀⠀⠀⠈⣛⠷⠛⢱⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠛⠼⢟⠁
⠀⠀⠀⠀⠀⠀⢻⣷⣦⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣝⠏⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠸⣫⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⡘⢿⣿⣀⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⡀⣾⡿⢋⣤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⠝⠁⢿⣿⣦⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣴⣾⡿⠁⢫⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⣶⣶⣭⣛⠇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠸⣛⣫⣴⣶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣿⣿⣿⣿⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣤⠄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠠⣤⣤⣤⣾⣿⡇
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⠿⠛⠁⠀⠀⠈⠛⠿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⢸⡿⠋⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣦⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⣴⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠉⠻⠿⠿⠛⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⢠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⣿⠟⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⣴⣄⠀⣿⣿⢁⣠⣤⣀⣾⣇⣀⣴⣿⣿⣿⡟⠋⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⣿⣉⣉⣉⣿⣿⣉⣹⣿⣿⣿⣉⣿⣿⣿⣯⣉⣉⣉⣉⣉⣉⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⡿⠽⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠏⠃⠀⠈⠉⠉⠉⢹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⡙⠻⣿⢦⣿⣿⢿⣿⣹⣿⣥⣭⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⢾⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⣷⣾⣿⠇⣼⣿⣿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣼⣤⣿⣾⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⣤⣶⣄⣠⠖⠉⠀⠀⡀⠄⢀⣄⣀⣀⠀⣀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠉⠉⠉⠉⠙⠓⠠⠄
⠀⠀⠀⢀⣿⣾⣿⣿⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣤⣄⡀⠀⣀⣀⣀⣠⣴⣿⣷⣤
⠀⠰⠛⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⣠⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡾⢿⣿⣿⣿⣿⣻⣿⡿⢹⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠈⣿⣿⣿⡿⠛⠛⡇⢸⡇
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣷⠀⠀⡟⠛⠉⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⣿⡿⠟⠈⠙⠛⠓⡀⠀⣿⣸⡿⠋⠀⠀⠹⣿⣿⣿⣿⡇⢠⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⠀⠀⠀⠙⣿⣿⣿⡄⠀⠀⠈⠿⠷⠷⡀⣿⡋⠀⢀⣤⣾⠀⢹⣿⣿⣿⣇⡌⠀⠀⠀⠀⠀⠀⡇⢸⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⠃⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠰⠷⣿⠟⢠⣾⡿⠃⠀⠈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⡇⠀⠀⣤⡀⠀⢻⣿⣿⣷⡀⠀⠀⠀⣦⣿⣤⡿⠋⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⣿⡇⠀⠀⢻⣿⡀⠘⠿⢿⣭⡿⠀⢿⣦⣌⢹⡏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⠀⠀⠀⢘⡇⣿
⠀⠀⠀⠀⠀⠀⠻⣯⣽⡾⣿⣏⠀⠀⠀⠙⣃⡀⢰⣄⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⣀⣤⣶⣶⠶⢶⣶⣿⠿⠿⣿⣦⣤⣄⣸⠀⣿
⠀⠀⠀⠀⣠⣴⣾⣿⣿⣇⢹⣿⡆⠀⠀⠀⢙⣁⣬⣿⣧⠀⢀⣠⣴⣶⣞⣻⣭⠹⣿⣿⠉⢙⢆⣿⣻⣿⡆⠀⠀⠀⠀⠙⠻⢶⣏
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠈⠉⠁⣈⣿⣗⡛⠉⢾⡿⡏⠀⣨⢳⣿⡷⠐⠋⢨⣿⡏⠉⢡⠀⠀⠀⠀⠀⠀⠐⠈⣳
⠀⠀⠻⣿⣿⣿⣿⣿⡿⠻⡄⢿⣽⢄⣀⣠⣴⠶⠛⢡⣿⡟⠙⡆⢸⣷⡟⠉⠀⠨⠿⢓⣀⣠⣬⣥⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠘⢯⡛⠛⠻⠶⠶⠷⠾⠟⠟⣻⣿⠟⠗⠀⢼⣾⡃⠀⡇⠸⠿⢋⣠⡴⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⢶⣄⢰⡏⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢠⣿⣿⠂⢀⠀⠈⠛⠷⣚⣤⡶⠻⣿⣿⣾⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⣿⣿⣿⡆⠀⠀⣾⠋⠋⣿
⠀⠀⠀⠀⠀⢸⠹⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣯⣤⠶⠞⠛⠁⠀⢲⣶⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⣼⣿⣿⣿⣿⡁⠀⣰⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠈⠓⠒⠠⠤⠤⠐⠒⠚⠻⣿⡻⣿⡀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢋⣴⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡱⢦⡀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⣠⠔⢁⡴⠟⠙⢦⡀⠐⠌⠙⠻⠿⢿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡙⢶⣄⠀⠀⠰⢻⡏⠀⠀⠀⠀⣠⠞⣡⡶⠋⠀⠀⠀⠀⠙⠲⢀⣤⣤⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⠻⢦⡀⠀⠀⠀⡠⠖⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠓⠚⣉⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣲⡨⢭⣽⣦⣦⢭⣽⣿⣄⢰⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠗⢉⡱⠿⠗⡾⠩⠽⠋⠉⠉⢉⡈⢹⠟⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣿⣫⢖⣤⣄⡶⢀⣽⡗⢀⠀⢀⡾⣿⣿⣷⡄⡰⠄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣥⣞⡛⣉⣽⢾⣿⣷⣾⣦⣿⡟⠽⣿⡿⢽⣷⡄⢀⠠⣈
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⡟⠻⠞⠛⠋⢛⣿⣟⢻⠉⠁⠀⠀⠀⠀⠈⢹⢸⣦⣀⠻⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣇⣠⣀⡤⣾⣿⣿⢟⣿⢠⡀⠀⠀⠀⠀⠀⠀⢿⠘⢿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠋⣩⡬⠁⢈⣵⡿⠃⠚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠿⣿⣨⣡⣴⣶⣯⣬⣀⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣾⠿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣀⣀⣠⣴⣦⣤⣤⢿⣾⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡫⡞⢷⣏⡱⣿⠛⢙⡁⠀⠒⠀⠽⠛⠟⣷⡿⢻⣿⡏⠻⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣧⠀⠻⢿⠇⠻⡗⢸⣇⠀⠀⠀⠀⠀⠚⢙⡷⠟⠤⠂⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⡃⠀⡠⠀⣳⣦⠛⣦⣀⡀⠀⠀⠀⠁⢀⡾⠷⣶⣦⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠻⠢⢤⡖⢻⣿⠐⡛⠫⣇⠀⠀⠀⢀⠞⠀⢠⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠛⠀⠀⠈⢿⣆⠙⠛⠁⡈⣻⡤⠀⠀⢠⡜⠋⠉⢍⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠆⠀⠀⠀⠀⠙⠛⠿⣶⡿⢡⡄⠀⠀⠉⠀⠂⠈⠈⢻
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⡿⠷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠃⠁⠤⠄⣀⣀⠀⠀⠠⠜
⠀⠀⢀⠀⢤⡶⣫⣭⣏⠛⢎⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠐⠈⠁⡡⣼⣿⣿⣷⡶⣤⣀
⠀⠀⠩⠭⡴⡁⣻⡎⣏⣿⠾⣽⡽⣻⠆⠀⠀⠈⠑⢄⠀⠀⣀⣄⣀⣠⠞⠁⣿⣿⣟⣿⣞⣿⣝⣙⣦⡄
⠀⠀⢰⢞⡴⣷⡾⣲⣽⣦⣹⠵⢫⢸⣿⣦⠀⠀⠀⠀⠑⠢⣀⣋⡭⠋⠀⢸⣿⣟⣻⣟⣿⡷⢲⣥⠁⠁⠠
⠀⢀⣀⣽⣤⣿⣽⣷⣾⣧⣧⣦⣩⣶⣿⣷⣧⡀⠀⠀⠀⠀⣨⣶⣦⣄⠀⣿⣿⣿⣿⡏⠷⢱⡃⡸⢰⢦⠄
⠀⠀⠀⠹⠿⠉⠟⢋⠫⠛⠉⠩⠩⠉⡭⠹⣿⣷⣄⠀⣠⡾⣿⣿⣿⣿⣿⡿⢯⣿⢿⣧⡏⠮⣭⣢⣍⠷⡆
⠂⠁⠀⢘⣛⣛⣙⣿⣿⣟⣿⣯⣟⣛⣛⣿⣿⣽⠻⣿⣿⣿⣿⣿⣿⣿⣙⣛⡿⣿⣟⡽⠷⡽⠭⣛⠟⡛
⠉⠉⠉⠈⠙⢉⡛⢿⣿⠿⣏⠟⢿⡽⠿⠟⠛⢯⢶⢻⡟⣙⢿⣿⣿⠿⣿⣿⣞⣿⣭⡧⢴⣓⡢⠀⣴⠅
⠀⠀⠀⠀⠀⠈⠨⡯⠚⠷⣞⠨⡺⠄⠄⠘⣲⠷⠋⡢⢬⣿⣿⣿⣿⢺⡿⠻⡟⢛⡟⠓⡀⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠈⠁⠀⠂⠛⠙⠀⠛⠿⠿⠿⠃⠈⠀⠊