Counting Emojis & Symbols

⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⠃⠀⣶⡆
⠀⠀⠀⠨⠛⠇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠘⢃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠿⠂⠀⢠⣄⠀⠀⠀⢰⡀⠒
⠀⠀⠡⠒⡂⠨⠉⠄⠀⠀⠀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢥⡀⠀⣀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠈⠠⠒⠒⠂⠂⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠶⠀⠛⠆⠀⠀⠀⠰⠂⠀⠖⠀⠐⠂
⠀⠀⢀⢤⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠚⠁⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⣴⣦
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⢺⣜⣻⢯⣄⣆⢄⠦⠀⠀⠇⠂⠀⠀⠤⠀⡄⠀⡤⢨⡍⢠⡀⢀
⠀⠀⠀⠀⠀⣠⣄⠀⣀⠀⠴⠀⠀⠀⢀⠄⢀⠂⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠂⠐⠁⠄⡡⢄⡡⠜⢁⢘⡀
⠀⠀⠀⠀⠀⠙⠫⠤⠀⠀⠀⠉⠁⠀⠌⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⡀⠐⠒
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠀⠀⠀⢀⣀⠀⠘⡆⠠⠀⠀⠉⠀⠔⡠

⠉⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠁⠁⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢄⡀⢀
⠀⠀⠀⠤⠤⠄⢄⠀⡎⠀⠀⠀⠈⠆⠀⠀⢀⠀⠀⠀⠀⠀⠆⠀⠀⠔⠀⣠⣶⡄⠀⠀⠀⠀⢗⠻⠣⠜⠛⠻⢀⡀⡀
⠀⠀⠀⢀⡄⠤⠄⢠⠠⣀⠀⢀⡀⣡⣀⣀⡀⠄⠀⠀⠀⡠⠃⠈⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⢀⠷⢻⡷⡜⢾⠸⢑
⠀⠀⠀⣾⠀⠀⠀⢀⢈⠦⠀⢴⠟⠉⠙⠛⠿⣇⠌⠁⠁⠀⠉⠀⠤⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⡇⢀⠀⠀⢨⠈⢐⠀⢢⠀⠀⠀⠀⠀⠂⢤⠀⠀⠀⠀⠀⢠⢠⡾⠉⠀⠋⠈⣿⠀⠀⠀⠀⠀⡀
⠀⠐⠖⠀⠁⠀⠀⠁⠲⠆⡀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⢊⣿⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⠀⠐⡿⠀⠀⠀⠔⠀⠐⠐⠀⠐⠲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠉⠀⠀⠀⢀⠠⠐⠀⠀⠈⢁⠄⠀⠐⠀⠀⠁
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠚
⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⢨⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⡀⠔⠂⠄⠈⠈⡀⠀⢠⡕⠀⢀⠠⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⡁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠡
⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠉⠱⠀⠂⠉⠀⠈⠀⠂⠒⠀⠁⠊⠁⠈⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠨⠄⠀⠒⠀⠀⠄⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠠⠜⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠑⠀⠀⢀⣋⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠺⣭⡽⠳⡧⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢂⠎⠀⠀⠀⣀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⠸⠶⠖⠁⢱⣧⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⠆⢈⣠⡀⠀⠠⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⠇⠁⠀⠀⢸⡄⠈⠁⠀⠀⠃⠘⠀⠀⠀⠀⠀⢖⠔⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⠀⠀⠀⡸⠉⠀⠀⠀⠀⢀⠀⠀⠀⠐⡗⠶⠘⠴⣐⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⢯⣀⠀⢈⠶⢤⠀⠀⠀⠀⠀⠀⣄⠀⠂⠀⡴⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡷⠦⢤⣀⠈⠑⢾⣦⣄⡀⠀⠀⠀⠀⢀⠟⠂⢰⠾⡦⡞⠈⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠰⠿⠛⠲⣝⠆⠈⡆⠀⠀⠠⠂⠀⠀⡞⢰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣀⠈⡶⣇⢀⡜⠁⠀⠀⣸⣠⠞⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⢀⠆⠀⡨⠓⠢⢄⣛⢀⡠⠔⠊⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠓⠀⠀⢸⠀⢀⠃⠀⠀⠀⠈⠋⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠑⠢⢤⣼⠶⠼⠷⠦⢤⣄⣀⣀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⡶⠦⣤⣤⣠⡤⠔⠺⡉⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⢥⣄⣀⣀⢁⡀⠀⠀⠂⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡆⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⡉⠀⢀⠁⠁⠀⠈
⠀⠀⠀⣀⡰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡬⡦⡇⢸⢷⣦⡜
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠈⠆⢃⡍⣭⣨⠀⡅⠤⢄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢸⡗⠳⠏⠿⠜⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠠⠄⠤⠀⠤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣷⢸⠀⢹⢧⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠇⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠈⡇⠘⢈⢓⡅⣜⠀⡃⢠⢤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡿⠎⣇⡯⢆⡟⢾⢀⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡇⠀⠀⠀⠛⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⠰⠀⠀⠐⣀⠂⡀⠀⠀⠀⢀⢂⡄⢂⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠃⣩⡄⠃⠀⠃⣤⠘⢁⠀⡄⢡⡈⠠⠤⠘⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒
⠀⠀⠀⠀⠀⣀⠆⠀⢀⠀⢈⠠⠶⠈⠀⡀⢀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢸⡃⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠘⢀⣥⠘⠁⠆⠰⠂⡶⠀⠀⢠⠐⠃⡶⠀⠀⠠⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⡄⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠐⠀⠀⠀⠘⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⡆⠖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠃⠓⠁⠀⠀⡀⠀⠀⠀⠬⠀⠀⠀⠉⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢈⡀⠀⠀⠀⠁⠀⠀⠀⢲⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⣛⠆⠀⠀⠀⠀⠮⠅⠀⠀⠠⠽

⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣾⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣷⣶⣶⣶⣶⣶⣶⣾⣶⣿⣧⣀⣀⣸
⣿⣿⡏⠙⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠒⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠘⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣰⠒⣴⣀⢀⠆
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢤⠀⠀⠀⢠⡄⠀⠀⢀⢄⠀⠀⠐⢏⠹⠘⠤⠃⠋⠻⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠾⠄⠀⠀⠢⠋⠀⠀⠈⠙⠀⠀⠀⠀⠀⡿⠞⣽⣳⡹⢼⢈⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⠀⠀⠚⡖⠀⠀⢀⣓⠄⠀⠀⢞⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢥⠀⠀⠀⡹⠀⠀⠀⡮⡆⠀⠀⠐⢽⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠁⠀⠀⠀⠁⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢔⣄⠀⠀⢠⡄⠔⡄⠀⣠⢠⡄⠀⠀⠀⠀⠀⠀⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠀⠀⠂⠋⠀⠀⠀⠃⠂⠃⠀⠘⠀⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⡞⠉⢳⠀⠀⠀⠀⠀⠀⠀⢠⠴⢦⡀⠀⢠⠴⢤⠀⠀⠀⠀⠀⣀⣀⠀⢰⠚⢲⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⣇⠀⡟⠀⢸⠁⠀⠀⠀⢸⡅⠈⡇⢸⠀⢸⢀⡏
⠀⠀⢀⡤⣄⡴⠶⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢹⣸⠃⠀⡏⠀⠀⠀⠀⠀⢧⠀⢹⣾⠀⢸⣾⠀⢰
⡞⠉⣻⠀⢸⡇⣠⢧⣀⣸⠀⠀⠀⠀⠀⠀⢀⡴⢾⣄⣈⣋⡀⢸⠃⠀⠀⠀⠀⠀⠘⡆⠀⠿⠀⠸⠃⠀⡞
⡇⠀⣸⡀⣸⣇⣧⣀⠀⠉⢳⡀⠀⠀⢠⡞⠙⣇⠰⣇⡀⠉⠉⠙⣇⠀⠀⠀⠀⠀⡤⠷⣄⡴⠒⠚⠋⠛⢷
⢹⠛⠉⠉⠁⠀⣠⠟⠁⠀⠀⡇⠀⠀⠀⢧⠀⣹⠤⠼⣹⠃⠀⠀⢸⠀⠀⠀⠀⠈⡧⣄⠈⣳⠴⣶⠂
⠸⡆⠀⠀⠀⠀⠁⠀⠀⢀⡼⠁⠀⠀⠀⠘⣏⠁⠀⠀⠙⠀⠀⢀⠞⠀⠀⠀⠀⠀⢷⡈⠉⠁⠀⠃⠀⠀⣰
⠀⠙⠲⠤⠤⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠶⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠶⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣤⠶⣄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢾⠁⢹⡀⡇⠀⣿⠀⡼⠁⣹⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢻⠀⡇⠀⣿⠀⡞⠀⣽
⠀⠀⢀⡤⣄⠸⡆⠈⣇⣯⠀⣿⢰⠃⢀⡏⠀⠀⠀⠀⠀⢠⠤⣄⠸⡄⠘⡇⡇⠀⣿⢸⠃⢠⠇
⠀⠀⠸⡄⠈⢦⢷⠀⢹⡿⠀⢿⡞⠀⣼⠀⠀⠀⠀⠀⠀⠹⡄⠘⢆⢧⠀⠻⡇⠀⢿⠏⠀⡞⠀⣀⣀
⠀⠀⠀⠙⣆⠈⠛⠀⠀⣴⠛⠓⠒⠲⣷⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠛⠀⠀⠀⠀⠀⠀⢸⡧⠞⠁⣨⠇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠙⠒⡶⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣠⠴⠒⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠸⠇⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⠃⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⣤⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⢤⣤⣤⠴⠋

⠀⠀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡀⢠⠚⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢰⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣼⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡇⠀⠈⢣
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢻⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢣⡇⠀⠀⠸
⠀⢸⠀⠀⠀⢸⠒⠲⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣧⣀⠀⢸⠃⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣷⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠸⣀⡀⢸⠉⠑⡄⠀⠀⠀⡇⢀⡤⠊⠀⠈⢣⠖⠁⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢹⠀⠀⠀⡗⠒⣆⣀⡀
⠀⢸⠀⡠⠔⠋⠀⠉⠲⢄⡀⡛⢦⠀⢠⠜⠁⠀⠀⠀⠀⣸⠀⠀⠀⡰⠁⠈⢳⠀⡠⠒⠉⠉⠙⢳⡀⠀⠀⠀⢀⡜⠁⠀⢸⠀⠙
⠀⢸⠀⠀⠀⠠⡀⠀⠀⠀⢱⡇⢸⠀⡄⠀⠀⠀⡀⠤⠖⠣⣍⢲⠉⠀⠀⢀⠌⠀⣇⢀⠀⠀⠀⠀⡡⠖⠒⠂⡞⠀⠀⠀⡜
⠀⢸⡀⠀⠀⠀⠈⠑⠲⠚⢏⣀⡜⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠒⠒⡏⠀⠀⠈⠉⠉⠳⡀⠀⠀⠀⠀⠀⠙⡤⠴⠺⣀⣠⠞
⠀⠀⢣⡀⠀⠀⠀⠀⠠⠁⠀⢠⠃⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠎
⠀⠀⠀⠉⡲⠤⠤⠴⢄⣀⠤⠃⠀⠀⠀⠀⠑⠤⣀⣀⣀⠤⠒⠂⠰⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢤⡤⠒⠒⠒⢾⠁
⠀⠀⠀⠈⠳⠤⣀⣀⡠⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⢀⡀⣀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠂⠉⢢⡈⠀⠈⢲⠀⠀⠃⢀⢀⠀⠀⠀⠀⢀⠔⠢⡀⠇⠀⢱⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⣸⠁⢈⠀⠀⠀⠀⢸⠀⠀⢹⡇⠀⢸⡏⠀⢡⡠⣄⠀⠀⠀⣀⣠⡋⡀⠉⠉⠙⡖⠢⢄
⠀⠀⢃⠀⠀⡸⠤⡀⠸⠀⠀⠇⠀⡸⠀⠀⡀⠀⢸⠀⠀⠀⡇⠀⢸⡇⠀⡾⠁⢸⠀⢠⠞⠁⠀⠈⠙⠲⢄⡀⡇⠀⠀⡇
⠀⠀⢈⠄⠊⠀⠀⢰⠀⠀⠘⠃⢠⠁⡎⠉⠈⠉⠲⣆⠀⠀⠓⠀⠘⠂⠠⠇⠀⡇⢠⠃⠀⠀⠀⡀⠀⠀⠀⠉⠑⣆⠀⡟⠲⡄
⠀⠀⠁⠀⠀⣀⠠⠦⡀⠀⠀⠀⡄⠀⠑⠢⢄⠀⠀⢈⡖⠉⠉⠑⢦⠀⠀⠀⡸⠀⣾⠀⠀⠀⠀⠹⣄⡀⠀⢀⣴⢃⡰⠁⢠⠋⢦
⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⡇⠀⠘⣄⠀⠀⠀⠀⠀⠉⠉⢧⣈⠉⢀⡠⠋⣠⠃
⠀⠀⠈⢄⠀⠀⠀⠀⣀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠰⠀⠀⠀⢰⠃⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⠀⠙⠓⠒⣺⠁
⠀⠀⠀⠀⠈⠱⠌⠉⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣤⡀⠀⣠⠎⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⣠⣄⣀⡀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠤⠤⠭⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾⣀⠀⠀⠀⠀⢁⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠀⠈⠉⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡝⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠈⠢⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠆⠁⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⡀⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡂⢂⠌⠛⠛⠲⢝⠝⢦⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣷⠼⠇⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡰⠔⠁⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠤⠴⠥⠊⠀⠀⠀⠀⡌
⠄⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣤⠖⠊⠁⠀⠀⠀⠉⠁⠒⢄⠀⠀⠀⠀⠀⠰⠈⢀⡌⠀⠀⠀⠀⠀⣀⣤⡞⠁
⠈⠀⡅⢀⣇⢂⣀⣤⡄⠀⠀⠀⠀⠀⡆⠀⠀⢀⠔⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⣇⠄⠻⢦⣄⡀⠀⠀⠛⢿⠋
⠀⠀⢁⣸⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠁⠀⠀⠎⠀⠁⡄⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠘⣄⠠⢤⠈⠙⠲⢮⡁⢻⡄⠀⠀⠀⠡
⠀⠀⠈⠁⠀⠀⢹⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⣛⣠⣤⣤⠟⠳⡮⡺⠄⠀⠀⠀⠀⡿⠈⠁⢧⠀⠀⠀⡿⢁⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣾⣯⠽⠯⠞⠁⠀⢀⣃⠜⠀⠀⠀⠀⢀⠏⠀⠀⠀⠑⠒⠒⠓⠉⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠁⠀⠀⠀⠀⣠⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠀⠀⠀⠀⠀⠉⠉⠇⠀⠀⢀⡠⠊⠀⠀⠀⠀⣀⡤⣶⠟⢷⡀⡪⠲⣄⡀⠀⠀⠀⠀⠠⢜⣡⡾⠃
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⢰⢀⠔⡙⠀⠀⠀⠀⣠⠞⠛⠉⠀⠀⠀⠙⢾⠖⠔⠹⠳⠶⠲⠶⣾⡿⢋⣤⡀
⠀⠀⠀⠀⠀⠘⡀⢠⢆⠀⣀⣀⣠⡤⠤⢶⣶⠞⠃⠀⠄⠀⠀⠀⠀⠓⠒⠒⠂⠀⠉⠉⠩⣽⠈⠛⠓⠾⠾⠒⠚⠁⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⣆⣨⣞⣙⣡⡥⠬⠦⠖⠚⠁⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⢐⠨⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣧⣁⣀⣤⡤⠤⠤⡴⡲⡲⠲⢖⡶⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠤⠬⠾⠦⠾⠦⠖⠂⠒⠒⠚⠚⠛⠉
⠀⠀⠀⠀⠀⠀⡀⠄⠠⡀⠄⠢⢐⠄⡂⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⢄⠄⠀⠀⡠⡠
⠀⠀⠀⠀⠀⠀⠀⠁⠊⠔⠡⠊⠔⠠⢂⢕⠐⠡⠈⡢⢀⠀⠀⠀⠀⠀⡠⠀⠄⠄⠄⡠⢀⢀⠀⢐⢔⠁⠁⠈⠂⠀⢀⠄⠀⠁⠀⠀⠠⠂⢀⢔⢐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠔⠀⡀⠀⢀⠁⡈⠀⠐⠄⠀⠊⢀⢕⠁⠐⠁⠁⠈⠈⠐⠀⠁⠁⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣶⣿⣿⣿⣿⣷⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢹⣿⣯⣭⣭⣙⠻⣿⣿⣥⡉⠒⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣿⣿⠿⣿⣿⣿⣿⣶⡏⠌⠀⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⣇⠀⠀⠉⠛⠻⠿⣅⢀⡤⢸⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣷⠄⠀⠀⣁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⠿⠀⠀⣤⠀⠀⢀⣤⣼⡴⠁⢹⣿⣿⠿⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠀⠀⡀⠀⠐⡈⠙⠂⡈⠀⠀⠀⣉⣵⠒⠂⠉⠻⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠁⠀⠀⢱⡴⢰⠀⠀⢠⠎⠀⠀⠀⠄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡀⠐⠀⠠⡀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⠼⣌⠈⠁⠐⠛⠃⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡏
⠀⠀⠀⡇⠀⠀⠀⡀⠀⢀⠖⠉⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡷⠖⣀⡠⢀⡑⠠⡀
⠀⠀⠆⠀⠈⠂⢘⠥⠔⠉⠀⠀⢀⡷⠂⠛⠂⢤⠘⠂⠀⠀⠀⡀⡰⠀⢨⠁⠂⠁⠀⠀⠸
⠀⠈⠉⠀⠰⡄⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢺⠂⠀⠀⠀⠀⢠⠋⠁⠀⠀⢀⠀⠀⠀⠠⠁
⠀⠀⠀⢀⡀⠆⠀⠀⠀⠀⡀⠁⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⢀⠇⠀⢀⠃⠀⠈⠀⠀⢀⡈⠘⠢⡄
⠄⣄⣠⠀⠀⠈⠀⠀⠀⣴⣤⣗⣤⣀⣦⣄⣀⠰⠀⠀⢀⡔⡋⠀⠀⣸⣀⣠⣦⠀⠈⠁⠈⠑⠀⠘⢄⡀
⠀⠀⠙⡀⠀⠀⠈⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⣀⣀⡀⣾⠴⠃⠀⢀⣿⣿⣿⣿⣦⣤⣤⣄⠀⠀⠑⡄⠙⢆
⠀⠀⠀⠁⠀⠐⠂⠀⠀⠀⠙⣿⣿⣿⣿⠿⢿⠀⠀⢸⣿⣿⣿⣶⣾⣿⣿⣿⠿⠛⢿⠛⠛⠓⠮⣵⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠈⠉⠁⠀⠀⠎⠀⠀⢸⣿⣿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠶⠊
⠀⠀⠀⠀⠀⠈⠐⠀⠠⠒⠈⠈⠀⡂⠀⠁⠐⢀⠐⢈⠉
⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⡀⠁

⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⣠⠶⢄⠀⠀⣠⠶⢤⠀⠀⠀⠀⠀⣀⣀⠀⢠⡖⠳⡄⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠸⡆⢀⡏⠀⣸⠀⠀⠀⠀⠰⡇⠈⢧⢸⡇⠀⡇⣸⠁⢠
⠀⠀⠀⣀⣀⡤⠦⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢷⣸⠁⠀⡏⠀⠀⠀⠀⠀⢳⠀⠸⣼⡇⠀⣧⡇⠀⡼
⡞⠉⢻⠀⠈⡇⢀⣽⣀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠤⢷⠀⠘⠏⠀⢸⠃⠀⠀⠀⠀⠀⠘⡇⠀⠻⠇⠀⠿⠀⢰⠃
⡇⠀⢸⠀⢀⣇⣾⡀⠈⠙⠳⡄⠀⠀⠀⣰⠒⢻⡀⢸⡉⠉⠉⠙⠻⣆⠀⠀⠀⠀⠀⣠⣽⣀⣠⠤⠶⠒⠒⠾⣆
⢻⠶⠋⠛⠋⠉⢉⡽⠃⠀⠀⢹⠀⠀⠀⠸⡄⠀⣧⣀⡽⣻⠂⠀⠀⢹⠀⠀⠀⠀⠀⣧⡀⠈⢳⣤⢤⡤⠀⠀⢸
⠸⡄⠀⠀⠀⠀⠚⠀⠀⠀⢀⡞⠀⠀⠀⠀⢿⠚⠁⠀⠀⠇⠀⠀⢀⡞⠀⠀⠀⠀⠀⢳⠉⠓⠋⠀⠾⠀⠀⠀⡼
⠀⠹⢤⣀⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⠤⢤⣤⠴⠊⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣠⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⢰⠚⢳⡄⢸⠁⢸⡆⢀⡞⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢦⠀⣿⠀⢹⠀⢰⠋⢳⡄
⠀⠀⠀⢀⡀⠘⡆⠀⣧⢸⠀⢸⡇⣼⠁⢰⠇⠀⠀⠀⠀⠀⠀⣀⠀⢹⡀⠸⡆⣿⠀⢸⢀⡏⠀⡼
⠀⠀⢸⡉⠙⣆⢳⠀⢸⣼⠀⢸⣧⠇⠀⡞⠀⠀⠀⠀⠀⠀⢾⠉⠳⡈⣇⠀⢷⣿⠀⢸⣸⠁⢰⠇
⠀⠀⠀⢳⡀⠘⢾⡆⠀⢉⡤⢬⣯⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⢧⠀⠹⣿⠀⠈⠁⠀⠈⠁⠀⡞⣠⠔⠚⣦
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠸⣤⣀⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢀⣀⠘⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⡞⠉⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⢾⢉⣙⡿⢶⣯⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠌⣹⠋⠠⠈⠉⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠸⢰⠃⢠⢀⡇⡉⢶⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⢀⡠⠷⣾⣿⠿⠋⠄⠨⣶⣶⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⡴⡏⣟⢱⡁⠀⠀⢀⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⡇⠂⠀⠀⠀⢸⡅⡙⠞⠛⠉⠀⠠⣤⠠⡏⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠋⠀⠀⠀⠀⠀⠈⡷⣥⠀⠀⠀⠀⠀⠀⠀⡹⠂⠀⠀⢀⡀⠀⠏⡼⢀⠕⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⠄⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⣤⡼⠃⠀⠀⠀⠹⡍⠾⠀⠁⠞⣴⠵⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠷⢞⣋⠀⠀⠀⠈⣶⣄⣀⢀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠑⠀⠀⢠⠎⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⠀⠀⠈⠙⠲⣤⡾⣅⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡷⠄⠀⣀⣴⣃⢠⡇⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠙⠓⠶⣤⣀⣀⠙⢾⡟⠳⢦⡀⠀⠀⠀⠀⢀⠞⠀⠀⢀⡟⠙⡏⠛⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠐⠒⠛⠻⠍⠉⠙⢦⢻⡄⠈⣷⠀⠀⠀⣠⠎⠀⠀⠀⡼⠁⣸⠁⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⡴⣿⠀⢀⡾⠁⠀⠀⠀⢰⢃⣠⠿⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⠀⠀⠀⠀⠀⣰⠔⠉⠛⣷⣮⡗⠉⢳⠏⠀⠀⣀⣤⠴⠛⠋⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣆⠀⠀⠀⣠⡏⠀⠀⡾⠁⠀⠉⠑⠶⢤⡴⠚⠉⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠸⡏⠃⠀⠈⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠙⠦⢤⣀⣈⣷⣤⣼⣷⢦⣤⣀⣀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠹⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⠿⠿⠶⢶⣤⡶⠶⠞⠉⠹⡄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣬⣅⣀⣀⣀⣈⡓⢀⣀⠀⠀⠐⠉⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡇⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠜⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠦⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠀⠀⠀⠀⢀⡜⠉⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠐⠒⠉⠀⠀⠀⠀⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠋⢁⣊⡉⡁⢀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠘⠉⠉⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣹⣏⣛⣟⣛⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠉⠙⠦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⠀⠀⠀⠈⠢⡑⢤⡀
⠀⠀⠀⠀⠀⢀⠆⠀⢴⠆⠀⡶⠀⢰⡆⠀⢶⢄⡴⣋⠔⠁⠀⣠⣴⢶⣶⢦⣤⡀⠈⠢⡙⢦⡠⡶⠀⢰⡆⠀⢶⠀⠰⡦⠀⠰
⠀⠀⠀⠀⠀⢸⣀⣀⣟⣀⣰⣇⣀⣸⣁⣀⠟⣩⠞⠁⠀⢀⣾⠋⠀⢸⡇⠀⠈⢿⣆⠀⠈⠢⣝⠻⣀⣈⣇⣀⣸⣆⣀⣻⣀⣀⡇
⠀⠀⠀⠀⠀⡏⠉⢹⠏⠉⣽⠉⠉⣿⠉⠹⡞⢹⠀⠀⠀⢸⣧⣀⣀⣼⣧⣀⣀⣨⣞⠀⠀⠀⢘⢳⠍⠉⢿⠉⠉⣯⠉⠹⡏⠉⢱
⠀⠀⠀⠀⢰⠓⠒⠛⠒⠒⠛⠒⠒⠓⠒⠚⠓⣺⠀⠀⠀⢸⡏⠉⠉⢹⡏⠉⠉⠉⣟⠀⠀⠀⢸⠚⠓⠒⠚⠒⠒⠛⠒⠒⠛⠒⠚⡆
⠀⠀⠀⠀⠀⠫⡭⠭⠬⠭⠭⠭⠭⠭⠭⠭⠭⢽⠀⠀⠀⢸⣷⣤⣤⣼⣷⣤⣤⣴⣯⠀⠀⠀⢸⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠝
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⢸⡟⡟⢻⠀⠀⡟⣿⢻⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⣹⢹⡇⠀⣿⢹⡋⡇
⠀⠀⠀⠀⠀⠀⡇⢸⡗⡷⢾⠀⠀⡗⣿⢺⡇⢸⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⢸⠀⡏⢻⢙⡇⠀⣿⢹⡋⡇
⠀⠀⠀⠀⠀⠀⡇⢸⣧⣧⣼⡄⢠⣧⣿⣼⡇⢸⠀⠀⠀⢐⣿⠟⠟⢿⡽⠛⠿⣿⡇⠀⠀⠀⢸⠰⠷⠿⠾⠗⠀⠿⠾⠷⠷
⠀⠀⠀⠀⠀⠀⡇⢠⣤⣤⣤⠀⠀⣤⣤⣤⡄⢸⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⠀⡶⢶⢶⡆⠀⣴⢶⠶⡆
⠀⠀⠀⠀⠀⠀⡇⢸⣇⣇⣸⠀⠀⣇⣾⣸⡇⢸⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⠀⡧⣾⢼⡇⠀⣿⢼⡦⡇
⠀⠀⠀⠀⠀⠀⡇⢸⡇⣏⢹⠀⠀⡇⣿⢸⡇⢸⠀⠀⠀⠸⡟⠒⠒⠚⠓⠒⠒⢻⡇⠀⠀⠀⢸⢀⣇⣼⣸⣇⠀⣿⣸⣅⣇
⠀⠀⠀⠀⠀⠀⡇⠙⠛⠛⠛⠁⠘⠛⠛⠛⠃⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠈⠉⠉⠉⠁⠀⠁⠉⠁⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⢸⡧⠤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⢀⣀⣀⢰⣇⢀⣀⣀⣀⣀⣀⣸⣇⢀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟
⠀⠀⢣⠀⠀⠾⠟⠁⢀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣧⣀⠀⠀⠀⠀⡆⠀⠀⠀⠸⠿⠋⢰
⠀⢀⠠⠱⣖⣒⣒⡩⢒⠀⠅⠂⠀⠀⠀⣨⣭⣭⣭⣭⡀⠀⠀⠀⠁⠀⠀⢠⡄⠀⠀⠈⠀⠀⠀⣿⣷⣶⣦⣶⣤
⠀⠀⠢⠑⠒⠀⠒⠚⠁⠀⠀⠀⠀⠀⠰⠟⠋⣻⡛⠻⠧⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣗⣤⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⠼⢶⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠄⠀⠀⠀⠀⠸⠿⠯⠙⠟⢸⣏⠝
⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣠⣴⣤⡀⣿⠀⡀⠀⠀⠀⠀⠀⠀⢰⣦⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣿⣿⣿⡧⠟⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⠀⣛⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⣠⣤⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⣥
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢲⠛⠟⠒⡟⢻⠖⢺⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣍⣙⣉⣥⣴
⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣘⣇⣛⣈⡏⣸⢁⠟⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣯⣿⣯⣵⣾
⠈⠀⠀⢹⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣼⣼⡽⠥⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠿⠿⠿⠟⠛
⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⡴⠊⠉⣉⠉⠑⢄⠀⠀⠀⠀⠀⠀⢠⡴⠊⢉⣉⡉⠛⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⣉⡉⠑⠆
⠀⠀⠀⠀⡼⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⢰⡿⠀⠀⠈⢀⡇⠀⠈⣧⠄⠀⠀⠀⠠⠁⠀⠀⢁⡷
⠀⠀⠀⠀⢳⠀⠀⠀⣿⠀⠀⠀⡞⠀⠀⠀⠀⠈⢧⠀⠀⢸⠋⠀⠀⠀⡿⠀⠀⠀⠀⠐⠀⠀⢀⡈⡷⠀⠀⠘
⠀⠀⠀⠀⠀⠳⣄⣀⠉⣀⡠⠞⠀⠀⠀⠀⠀⠀⠘⣷⣄⣘⠛⣃⣠⡞⠁⠀⠀⠀⠀⠀⠔⢄⣀⠉⢁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⠴⠚⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠛⠳⢤⡀
⠀⠀⠀⠀⣰⠃⠀⢸⠀⡇⠀⠈⢳⠀⠀⠀⠀⢠⡟⠀⠀⣾⠉⠀⠀⠹⡆⠀⠀⠀⠀⣰⠃⠀⢰⡏⠃⠀⠀⢳
⠀⠀⠀⠀⢿⠀⠀⠸⠶⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠙⢹⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⢸⡟⡆⠀⠀⢸⠃
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠇⢀⡠⠋⠀⠀⠀⠀⠀⠳⣄⠀⠙⠞⠀⣀⡜⠁⠀⠀⠀⠀⠈⢧⣀⠘⠷⠃⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠋
⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠲⢦⣦
⠀⠀⠀⠀⣰⠃⠀⠐⢲⠂⠀⠘⡆⠀⠀⠀⠀⠀⡜⠀⠀⡖⣦⠀⠀⢣⠀⠀⠀⠀⠀⣼⠏⠀⠀⡞⣦⠀⠈⢦
⠀⠀⠀⠀⣯⠀⠀⠀⡾⠀⠀⠀⣹⠀⠀⠀⠀⠸⡅⠀⠀⡷⣿⠀⠀⢀⡆⠀⠀⠀⠀⢻⠀⠀⠀⠧⣿⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠀⠰⠇⠀⠀⡠⠃⠀⠀⠀⠀⠀⢣⡀⠀⠧⠟⠀⢀⡜⠀⠀⠀⠀⠀⠘⣧⡀⠀⠦⠟⠀⣠⠏
⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⢀⣤⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣈⠉⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡉⡤⢤⠀⠀⠀⠀⠀⡠⢔⣬⢖⡐⡀⠀⢨⢙⡟⠂⡇
⠀⠀⠀⠀⢀⣠⣄⣤⣖⡔⡔⠀⠀⡠⠨⡠⡞⠁⠀⠈⠙⠾⢶⠀⠀⠀⠀⢠⠇⠀⡥⣕⣋⠉⣷⠃⢁⠈⢹
⠀⠀⢀⡴⠋⢀⡤⠤⡑⠗⢣⢀⡬⠣⢽⡁⠘⠚⠉⢹⠀⠀⢸⠀⠀⠀⠀⣼⠀⠘⠧⣀⠀⡹⠁⢠⢺⠀⢸⡄
⠀⠀⡞⠀⣰⠃⠀⠀⠘⣆⠼⠋⠀⢀⠼⡇⠀⣀⠴⠋⠀⣠⢏⠀⢀⡀⠀⠉⠉⢢⠀⠈⢻⠁⠐⠓⠚⠀⠘⠓⣦
⠀⢸⠃⠀⣇⠀⠀⠀⢰⠛⢦⡖⢻⠀⠀⡗⠉⠁⠀⣤⡚⢁⣮⣴⣥⣵⠖⠲⠤⠞⠀⠀⢸⠦⠤⠤⢤⠀⢠⡖⠟
⠀⠘⣇⠀⠘⠦⠤⠖⠃⠀⡼⠁⢸⠀⠀⣧⣄⣀⠀⠀⡸⠉⢈⠥⢍⠉⢳⣄⣀⣀⣠⠴⠃⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠈⠲⣄⣀⣀⣀⣠⠞⠁⠀⢸⠀⠀⢿⠀⠈⠉⠓⢧⡀⠘⠲⠚⢀⡼⠀⠀⠀⠀⡀⢦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢳⡖⠚⠀⠀⠀⢠⠖⠉⢀⣀⣀⠉⠲⣄⡠⠶⠶⣤⡸⣤⡀⢠⠚⢄
⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠲⠴⢞⠥⢹⣀⠀⠀⠀⣏⠀⠐⣇⣀⣸⠆⡰⠋⠀⣠⠴⠦⠤⣧⢆⡺⠫⠎⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣤⣀⣀⡀⠈⠉⠁⣸⠤⢤⢡⡛⣐⣇⠈⠉⠁⣰⠁⠀⠀⠉⠉⠓⢤⡇⠀⡔⠉⠙⡆⡸⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠔⠿⠤⣀⡉⠑⠚⡩⠋⠉⠉⣿⠀⢠⠋⠉⢱⠀⠘⣇⠀⠙⠶⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠋⠀⠀⢠⠔⠊⠀⣠⠊⠀⠀⠀⠀⢹⡀⠈⠳⠴⠊⠀⣸⠉⠲⠤⡤⠴⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡞⠁⠀⠀⠀⠘⠶⢤⡀⠈⠱⡄⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠁⠀⠀⣀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠻⠤⢤⡞⠀⠀⠀⠀⡴⠤⣤⠼⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⢠⢰⣶⠄⠀⠀⣰⣿⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣤⣘⠁⠀⡐⠋⠰⣿⣿⣷⣍⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⣿⣷⣾⣷⣯⣭⣴⡌⢙⣦⠀⢸⣿⣿⡿⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣿⣿⣿⣿⣿⡏⠁⢡⣌⠋⠀⠸⠿⠉⠁⠀⠀⠀⠀⡈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⣿⡿⠿⡄⠀⠀⠀⠀⠀⠀⣄⠘⣷⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣈⡾⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⣶⣶⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠟⣒⡀⠤⠤⠐⠒⠒⠀⠘⠂⠀⠒⠒⠒⠤⠤⢄⣀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠤⠔⠒⢒⣉⣉⣉⠀⠀⠐⠀⡉⣉⣉⡐⠒⠆⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢚⣉⡉⠤⠄⠀⠄⠀⢈⠠⠀⠠⠤⢌⣉⢉⠒⠆⠰
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⡥⠤⠔⠒⠂⠀⠂⠀⠠⠀⠀⠐⠒⠂⠴⢨⡅⣈⡁
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠠⢈⠀⠀⠀⠀⠅⠀⠀⠀⠤⠤⠔⠒⢊⣉⠀⠀⠀⠀⠐⠀⠀⠀⢉⣡⣼⣤⣦⣤⣥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⡆⠀⠀⠀⠀⠒⢋⣉⠤⠤⠐⠂⠀⠀⢈⠀⠐⠒⠠⢼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣉⠤⠤⠒⠂⣁⡀⠀⠀⠠⠀⠀⣈⡐⠒⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠤⠔⠒⠉⠁⠀⠀⠀⠐⢐⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠾⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠶⠶⠦



⡶⠶⢶⠶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⡶⢶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⢀⠀⣁⠆⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠒⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡤⣌⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣤⣤⣄⠠⣤⣀⣠⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠤⠶⠂


⠀⠀⠀⢀⣀⡀⢠⣀
⠀⠀⠀⠘⠘⣀⣄⡍⣁⠠⡀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠉⠸⠉⠋⠃⠈⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⢚⠷⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⡀⠀⣠⠴⠦⡶⣥⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠓⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣧⣛⣇⡞⠁⠀⣀⣀⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⡻⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢹⡉⠲⠋⠁⢸⡆⠀⢠⡇⠀⠀⠀⣀⠊⣰⢊⡙⢆⠀⠀⠀⢀⠔⠁⠀⠐⠣⠤⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡖⠁⢀⡄⠘⠉⣇⠀⠀⢀⠞⠀⢀⡾⠀⣾⠉⠉⠉⠑⠉⠮⠴⠋⢹⠀⠀⠈⠛⠲⠦⢤⣀⠀⠀⠀⠉⡶
⠀⠀⠀⠀⠀⠀⠀⠹⠶⠋⢹⡀⠀⢸⣄⠴⠃⠀⣠⣾⣤⣀⣿⠤⠤⠤⣤⠄⠀⠐⢁⡼⠀⠀⢀⠀⡠⠀⠀⡾⠀⢠⡟⠒⠃
⠀⠀⠀⠀⠀⢀⠄⣀⠴⡆⠈⣇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠃⠀⢀⡖⠉⣠⣅⣶⢇⠷⣸⡀⠘⠧⣤⡼
⠀⠀⠀⠀⠀⣢⣤⢥⣄⡰⣤⢹⡄⠀⢸⡗⠒⠒⠒⠒⠒⠒⠃⢀⡞⠀⠀⣰⠋⢠⠞⠁⢀⡤⠤⣍⠀⠙⢦
⠀⠀⠀⣰⠋⠀⠀⣀⣡⣀⡏⠀⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⢀⡏⠀⢰⠃⠀⠀⠀⢳⠀⠘⡇
⠀⠀⣼⠁⠀⠠⠯⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⢸⠃⡀⣿⡐⠆⠀⠀⢸⠀⠀⣿
⠀⠀⡇⠀⠀⢀⣠⣄⠀⠈⢳⠀⢠⠀⢀⣰⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⡼⣇⢤⡋⣌⡦⢤⣠⠞⠀⠀⡟⠀⠀⠀⢀
⠀⠘⡇⠀⠠⡏⠀⠀⣷⢀⡬⠧⡄⢮⡡⢮⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢁⣭⣬⣀⣸⠁⠀⢀⡼⣃⡤⠶⢶⠡⣰⠐⢶
⠀⠀⢷⡀⠀⠓⠦⠖⠃⢸⣇⣀⣈⡁⠈⠉⡴⠁⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⡇⠀⢸⣳⣦⣤⣤⠤⠶⠋⡼⠁⠀⣀⢬⣉⠫⠥⣎
⠀⠀⠀⠳⢤⣀⣀⣀⣤⠖⠃⡠⠊⠁⢀⡞⠀⠀⠀⣰⣤⣭⣤⣵⠀⠀⠀⠀⠧⠤⣄⡀⠈⠑⣆⠀⠀⢸⡃⠀⢸⡁⠀⢈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢺⣀⣀⡀⠀⠉⠳⣤⠊⠁⣉⠭⡉⠉⢳⡀⠀⠀⠀⠀⠀⡷⠀⠀⢸⠂⠀⠈⢧⡀⠀⠙⠒⠋⠠⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣿⠀⠀⠧⡤⠟⠀⣸⠃⠀⡸⠉⠙⠉⠀⠀⢀⡼⢠⠶⠤⣤⣩⠷⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠁⠀⠀⣠⣯⠷⠀⠀⠀⠀⠾⢅⠀⠀⠙⠶⠤⠤⠤⠖⠋⠀⢿⣀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⠤⠴⠚⣽⠁⠀⣰⠚⠛⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⠤⠤⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡤⠞⠁

⣿⣿⠟⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⠘⢿⣿⣿⠟⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣉⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⢉⡉⠙⡟⠉⣉⠉⢻⠋⢉⡉⠙⡟⠉⣉⠉⢻⣿⡇⠀⣛⣒⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠰⠇⠀⡇⠀⠶⠀⢸⠀⠠⠂⠀⡇⠀⠷⠀⢸⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⢠⡂⢲⡀⣴⣶⣦⡀⣶⣒⠂⢰⠶⢶⡖⠀⠀⣰⠐⢲⡴⠶⣶⡖⢠⠰⣆⠆⡐⣄⠀⣴⠖⠒⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠈⢁⣼⠹⡿⠀⣽⡇⢬⣉⡗⢰⠀⣾⠁⠀⠀⣿⢠⣬⠀⠾⢿⣄⢐⢸⣿⠇⠒⢸⣧⣹⡗⠒⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⣿⣿⢀⡀⢹⡄⢀⣾⠟⠀⣠⣷⣦⡀⢸⣿⠀⠀⠰⣿⠀⢶⠐⢶⡾⠇⢨⢘⠿⡀⠍⣸⣿⠘⢓⣒⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⢸⣿⠈⠃⠚⠰⠿⠿⠿⠇⣿⡇⡘⠃⢨⠭⡄⠀⠀⢸⠀⣿⢸⡇⢸⡇⠑⢾⣶⣷⣾⣿⠏⠸⠿⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⠟⢰⣋⣵⠐⠒⠂⠈⠀⣿⡟⢻⡇⢸⡀⡿⠀⠠⢖⡦⡾⡌⢛⢻⡗⠷⣶⢪⣍⢹⠓⠀⠐⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⡀⣰⠞⢻⠀⡆⠀⠀⠰⠂⣿⣇⣼⡇⢸⠛⣧⠀⢀⡘⣧⠙⠁⡀⡨⡁⢰⡟⠈⠛⣈⠙⡆⢸⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡽⠙⣒⡋⢀⣥⣝⡀⣾⢸⡎⣉⣋⣀⢸⡀⣿⠀⠀⢩⠱⣢⢀⡇⢀⡇⣿⠃⣾⠃⠤⠴⠃⣐⣦⢠⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⢸⣉⡁⣿⠋⣹⣷⠈⠼⠃⢉⣿⠃⢈⣛⡁⠀⠀⠸⠴⠥⠨⡌⠉⠧⣖⠂⣿⠻⣦⡆⣷⠀⣿⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣸⠇⣿⠿⢻⣿⢸⢿⡂⠈⢹⡇⠸⢤⡄⠀⠀⣺⣼⠃⡜⢱⡾⠀⣿⠃⣿⠀⣿⠤⠟⠉⠁⠘⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠅⠀⠻⠿⠿⠏⠸⠼⠑⠦⠾⠃⠯⠼⠃⠀⠐⠧⠼⠁⠐⠸⠯⠘⠤⠏⠻⠾⠟⠰⠂⠀⠀⠄⠅⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⢀⣀⠀⣀⠀⣀⣀⢀⣀⠀⣀⣀⢀⡀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡤⠘⢧⣜⠸⣿⡌⣑⡆⡏⢹⢺⣿⢨⡕⣺⣭⠃⠀⢰⡋⠀⢹⠤⣿⡇⠈⠀⠐⠛⠸⠟⠛⣿⡆⣭⣍⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠣⢰⡟⢹⣧⠈⣴⠒⠀⣥⠾⢀⠋⠰⠲⠶⣽⠧⠀⠀⡑⣲⠋⠀⢺⢶⡶⠶⢸⣻⣴⡟⠻⠿⠳⠆⠹⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠄⠠⣼⡟⢰⣿⢹⡆⣧⣴⠸⠠⠇⢁⠎⣿⣷⠀⢀⡴⣧⠀⣦⡉⠸⠟⣷⠰⡿⠻⠿⡿⠿⠇⢈⡟⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣦⣄⢸⣿⠀⠻⠞⢡⣶⠀⡄⢦⢀⣿⢀⣭⡍⠀⠘⠧⣿⠸⠷⡷⠰⣦⣿⠸⣿⢰⣿⠅⢀⠘⠋⢸⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⠻⠿⠋⢸⠀⡇⢻⣿⣦⣅⡜⠀⠋⠞⣶⡄⠀⠀⡉⠵⢚⣶⢴⠶⠶⠶⣶⣴⣆⠀⠀⠔⠀⠤⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣶⠑⢹⡴⣿⢈⡐⡇⣴⠙⠻⢿⣿⣆⠺⠴⠟⠀⠀⢀⡅⢘⣿⢅⡆⠈⣱⢞⣁⣿⣟⣩⣭⠙⢬⠋⣈⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡶⡾⢿⡃⣿⢈⣌⡈⢿⡐⠇⠀⣿⠇⠀⡖⠒⣇⠀⠈⠲⠛⠋⢈⠋⢺⡇⢸⢲⣬⠁⢋⣿⠀⢢⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇⠿⣿⣸⡇⠀⢁⣀⠈⣥⡄⠀⣇⣀⡄⠀⠐⠟⣿⠈⢸⠰⣿⣿⡏⣭⣐⢠⣆⢻⡗⡃⠸⢷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⢠⡞⠙⡆⣿⢻⡇⠀⠅⠄⠄⠰⡇⠀⡇⠀⡇⠀⢠⡾⢋⠀⢸⢼⣏⢹⡧⡾⢠⡏⡛⡋⢸⢾⠰⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠛⠁⠈⠛⠁⠀⠂⠈⠀⠛⠃⠘⠓⠚⠃⠀⠈⠛⠛⠀⠛⠈⠛⠛⠁⠃⠘⠁⠁⠈⠈⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⣟⠲⠦⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠐⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠁⡔⠂⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⣀⠴⠚⠁⣀⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢊⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠋⠀⣠⡴⡚⣕⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡄⠀⠠⡐⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠑⠁⢀⢤⢶⢯⠾⠛⠈⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⢈⠂⠀⠀⠘⠂⠂⠀⠀⠀⠠⠀⠐⠒⠆⠈⠀⡠⣠⣩⠵⠻⠃⠁⣀⡤⢞⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⣼⠥⣤⣤⠒⠆⠈⠉⠐⠁⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⢄⣢⡦⠞⠉⠀⢀⣠⢶⣫⡻⠊⠋⢩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠠⠯⠘⠋⠐⠀⠀⠀⠀⣀⢠⡄⠖⠚⠩⠉⠛⠛⠛⠓⠲⠶⠶⠛⠉⠀⣀⢄⣖⡽⠷⠑⠁⢀⣤⠶⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠔⠀⠀⠀⠀⡀⢀⣄⣴⠼⠓⠋⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣂⣮⠾⠏⠋⠀⣠⡤⠺⠋⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢀⢠⣄⠰⠄⠗⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠒⢒⡒⠶⠶⠶⠤⠤⢤⡴⠞⠋⠁⢀⢀⠔⠞⠉
⠀⠀⠀⡠⢀⠀⡄⡰⠄⠃⠈⠀⠀⠀⢤⡞⠠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡠⡢⠚⠁
⠀⠀⠄⠀⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠒⠒⠀⠀⠄⠀⠀⠀⠔⠋





⡿⠿⠿⠿⡿⠻⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣟⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠒⠂⠤⠤⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠈⠉⠁⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠼⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠞⠀⠀⠉⠀⠀⠹⠀⠀⠀⡅⠀⠈⠁⠀⠈⡇⠀⠀⠜⠀⠀⠉⠀⠀⠁⠀⠀⠀⡁⠀⠈⠇⠀⠈⡄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢘⠒⠒⠚⠒⠒⡇⠀⠀⠀⡗⠒⠒⠓⠒⠚⠂⠀⠀⠘⠒⠒⠓⠒⠒⡇⠀⠀⠀⡗⠒⠒⡗⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠤⠀⠀⡇⠀⠀⠀⡧⠀⠠⠇⠀⢤⡀⠀⠀⢨⠄⠀⠤⠀⠀⡇⠀⠀⠀⡧⠀⠠⠇⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠃⠀⠀⠀⠑⠒⠒⠒⠒⠊⠀⠀⠀⠈⠒⠒⠒⠒⠒⠃⠀⠀⠀⠑⠐⠒⠒⠒⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠜⠉⠉⠛⠋⠉⠳⠀⠀⠀⡛⠉⠙⠋⠉⠙⡆⠀⠀⡔⠉⠋⠛⠉⠙⠃⠀⠀⠀⡛⠉⠙⡋⠛⠙⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠴⠀⠀⡆⠀⠀⠀⡇⠀⠠⡆⠀⠰⠀⠀⠀⢰⠀⠀⠦⠀⠀⡆⠀⠀⠀⡇⠀⠠⡆⠀⠰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣀⣀⣀⣀⣀⡇⠀⠀⠀⣇⣀⣀⣄⣀⣠⠀⠀⠀⢠⣀⣀⣀⣀⣀⡇⠀⠀⠀⣇⣀⣀⣃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠀⠀⠀⠓⠒⠒⠒⠒⠚⠀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠀⠀⠀⠓⠒⠒⠒⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠒⢢⠀⠀⠀⠖⠒⠒⠖⠒⠒⡄⠀⠀⡴⠒⠒⠖⠒⠒⠢⠀⠀⠀⠖⠒⠒⠖⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⠀⠀⡄⠀⠀⠀⡇⢀⢀⡄⠀⢠⡀⠀⠀⢡⡀⠀⣤⠀⣀⡄⠀⠀⠀⡇⠀⢀⡅⠀⢠⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡃⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠺⠁⠀⠈⠀⠀⠇⠀⠀⠀⠇⠀⠈⠁⠀⠸⠂⠀⠀⠺⠀⠀⠉⠀⠀⠇⠀⠀⠀⠇⠀⠀⠁⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠈⠀⠉⠀⠒⠤⢄⡀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠰⠂⠀⠑⢬⡀⠱⠾⢅⡀⢇⡀⢇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⡞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠀⢠⠖⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠈⠉⠙⠁⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡀⢸⡄⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡂⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣇⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠃⠜⢙⣻⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣟⡉⠃⠘⠋⠛⠙⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣁⡖⢸⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢣⡸⠂⠀⠀⠘⠁
⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢫⠏⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠂⠀⠈⠉⠁⠀⠀⠀⡞⡸⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠠⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⢢⠀⡀⠀⠀⠀⠀⠀⣴⡇⠀⠀⡆⣥⣀⡀
⠀⠀⠀⠀⠁⠒⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠐⠁⠀⠀⠀⠤⠄⢠⡤⡀⠀⠀⠀⠀⢱⢙⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢀⣴⣡⠊⠀⢀⡀⠀⠀⠀⠀⠀⠢⠷⠄⠀⠀⠀⡖⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠉⠈⠈⠉⠁⠀⠁⠀⠉⠉⠀⠀⠀⠀⠉⠐⠐⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠊⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠖⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢒⠆⠀⠀⠀⠀⠀⠀⠒⢀⡤⡀
⠀⠀⠀⠀⠀⢀⡤⣠⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡌⠀⠀⠀⠀⡠⡴⠀⠀⠀⢱⢹
⠀⢀⠄⠀⠀⠀⠀⠀⠉⠙⢢⠢⡀⠀⠀⠀⠀⠀⠀⢀⣎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠜⠀⠀⠀⠀⠀⠢⠧⠄⠀⠀⡨⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠱⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⡖⡶⠀⠀⠀⠀⠱⡈⡄⠀⠀⠀⠀⠀⠀⡜⠐⠁⠀⠀⠀⠀⠀⢀⠔⡶⠀⠀⠀⠈⢢⠱
⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠎⡐⠀⠀⠀⠀⠀⡇⢇⠀⢀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡸⠠⠁⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⢨⠆⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⢜⡴⠁⠀⠀⠀⠀⠀⠉⠐⠓⠁⠀⠀⣰⢁⠆⠀⠀⠀⠀⣰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠀⠀⠀⢀⠔⡸
⠀⠀⠀⠀⠀⠀⠴⠶⠾⠷⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣠⡤⠾⠵⠁⠀⠠⠤⠤⠤⠯⠾⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠤⠤⠤⠴⠶⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠑⢬⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⡠⡺⠋⠀⠀⠀⠀⠀⠀⠀⠱⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣱⣄⠀⠀⠀⢀⢇⡇⠀⠀⠀⠀⡘⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⡘⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠎⠀⠀⠀⠀⠀⠀⢡⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠤⠤⡄⠀⠀⠀⠀⢀⡌⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⢀⣠⣣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡾⠋⠉⠙⠃⠀⡏⢹⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⣿⣿⣿⣿⣿⠉⠀⠀⣼⠀⢸⠁⢳⡸⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⡰⢶⣿⣿⣿⣿⢇⡞⠀⣰⠇⣰⠏⢀⢧⣧⣎⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢠⠟⣰⣿⣿⣿⣿⣿⡿⢁⡴⢋⣴⢏⣤⣏⣮⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⡏⣰⣿⡿⣿⣿⣿⣫⣴⣯⣴⠿⣷⠿⠧⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡿⢠⡿⣿⢣⣿⣿⣿⣿⡏⠁⠀⠀⣻⣶⠄⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⠇⣼⢇⣿⢸⣿⣿⡿⠿⠇⠀⠀⠀⠉⠀⠀⠀⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠀⣿⢸⡇⢺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⡿⢸⡇⢸⡇⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⠀⡇⠘⡇⠈⢣⠙⣿⡟⠀⠀⠀⣤⣄⣀⣀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿⡀⢱⠀⢻⡄⠈⡆⠈⠻⢷⣤⡀⣿⣿⣿⡟⠉⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢗⠆⢆⠈⢿⡄⠐⡀⠀⣸⣿⠇⠘⢿⡿⠃⢀⡼⠃⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣌⠀⠘⢧⣈⣿⣶⣿⡿⠛⠁⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢻⡿⠋⠉⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠋⠋⣼⡋⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠉⠳⣞⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠙⣇⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠠⣟⣷⡤⣤⠀⠀⠘⠂⠴⢱⡠⢤⣄⠀⠀⣼⢃⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠻⣧⡀⠀⠙⠲⢤⣀⡀⢀⣨⠏⢻⣦⣄⣀⡀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⣏⣹⣠⣄⠀⠈⠁⠿⠁⠀⣼⢃⣾⣿⠇
⠀⠀⠀⠀⢀⣤⣶⡟⠋⣤⣻⡝⣿⣶⣤⡀⠀⠈⠁⠉⠀⠀⣼⡇⢽⢿⣿⣶⡄⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠓⠀⠀⠀⠀⠀⣼⢃⣾⣿⠇
⠀⠀⢠⣾⠟⠉⢻⠻⢿⣾⣿⣿⣿⣿⠻⣏⠉⠐⠒⠒⠒⠛⠉⣷⣸⢿⣶⣿⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⣽⠏
⠀⠀⣿⠃⢠⡷⣈⠲⣈⣿⣿⣿⣿⡽⢣⠙⣧⠀⠀⠀⠀⠀⠀⢹⣇⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡟⣽⠏
⠀⢸⡏⢰⡏⢰⡟⣷⣽⣈⣿⣿⣿⣷⣌⣦⢹⣧⠀⠀⠀⠀⠀⠘⣿⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⢏⣾⠋
⠀⣿⡇⣾⠧⣾⣿⢿⡿⢁⣿⣿⣿⣿⣿⣷⠈⢻⡄⠀⠀⠀⠀⠀⢿⡆⢸⣷⠴⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⣴⡟⡜⣸⠏
⠀⣿⣇⣿⢄⣿⣷⣿⣷⣸⠻⣿⣿⣿⣿⣟⠁⠸⣇⠀⠀⠀⠀⠀⢸⡇⣠⣽⡤⠖⢻⡟⢶⣤⣀⠀⠀⠀⠀⢀⡴⠲⣾⠉⢾⡍⢳⣴⡟⡐⢰⠏
⠀⢸⣿⣽⡏⠉⣬⡍⡉⢹⠸⣿⣿⢻⣿⣿⢺⣧⣿⡀⠀⠀⠀⠀⢸⠟⠉⠀⢀⣴⣿⣏⣤⠶⢛⣿⣶⣤⣀⡏⢹⡀⠹⡄⠈⡇⠈⢿⡴⢡⠏
⠀⠀⣿⡿⣿⣦⣼⣯⠙⡟⢰⣿⣿⣸⣿⣿⣾⣿⢻⡇⠀⠀⠀⢠⡏⠀⠀⠀⠈⠉⠉⢀⣠⠴⢊⣿⣀⠹⢿⣿⡀⣧⠀⠣⠀⠀⠀⢸⣧⠊
⠀⢠⣿⢰⣿⣿⣿⡄⢈⣤⣾⣿⣿⣿⣿⢿⡼⢟⣈⣿⡤⠖⢶⡏⠀⠀⠀⠀⠀⠘⠋⠉⣁⠴⢋⣯⣽⡿⣦⣽⡇⠈⠀⠀⠀⠀⠀⠀⡟
⠀⢸⡏⢾⣿⣿⣹⡟⠀⣍⣱⣿⣿⣿⣷⣾⣶⠟⠛⠋⠻⣦⠀⢳⡀⠀⠀⠀⠀⠐⠊⣉⡠⢴⣯⣿⣿⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠸⣧⠸⡏⢋⠹⢟⣴⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⠀⠀⢀⣀⣈⣉⣥⣴⡿⠙⠾⡟⣁⠹⡟⠲⣤⣀⣀⣀⣤⠞⠁
⠀⠀⠹⣷⣷⣆⢱⣾⢿⣿⣿⡟⠡⠀⠀⠀⠀⠀⠀⣀⣀⠀⢹⡄⢸⡿⠛⠛⢉⢹⣿⣷⣿⣿⠠⢞⡓⣌⢦⠹⢦⣀⣌⡩⠏⠁
⠀⠀⠀⣿⣽⣿⣧⣴⣿⣿⣋⢀⣀⣀⣀⣤⢶⣷⣿⣿⣿⣿⢸⣷⣾⡇⠈⠀⢿⡌⣿⣿⣧⣝⣀⣀⣁⣬⣤⡶⠛⠉⠁
⠀⠀⠀⢻⡎⢻⢿⡾⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⡟⢋⣁⣾⡟⣹⡇⢀⠠⡄⠂⣿⡇⠈⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠙⠄⠑⠹⢿⣿⣿⣿⣿⣀⣿⣿⣿⣿⠿⠿⠿⠻⣟⠀⣿⡇⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠘⠇


⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡂⠀⠀⠀⠀⣜⡁⠔⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠱⢶⡏⢣
⠀⠀⠀⠀⣠⠦⠿⠯⣛⡸⢄⠀⠀⠀⠀⠀⢠⣐⢇⣚⡇⠀⣠⠖⠋⠛⠛⠾⠟⣆⠀⠀⠀⣤⠤⠴⣰⣎⠶⡼⢤⠀⠀⠀⠀⢠⠞⠒⠓⠲⡎
⠀⠀⢀⡞⠁⢠⠴⠒⠦⡀⠀⢳⡀⠀⣀⡴⠋⠀⠀⢸⡇⠰⢇⡀⣠⠶⢤⠀⠀⢸⡆⠀⠀⢧⣀⣀⣈⠀⠉⣰⠏⠀⠀⠀⢠⠏⠀⣰⠂⠀⡇
⠀⠀⡾⠀⢰⠇⠀⠀⠀⢸⡄⠀⣧⠀⣇⣀⣤⡄⠐⢹⡇⠀⠀⠉⠁⠀⣸⠀⠀⣸⠁⠀⠀⠀⣠⠔⠁⠀⣴⠃⠀⠀⠀⣰⠃⢀⡜⢸⠀⠀⡇
⠀⠀⡇⠀⢸⡀⠀⠀⠀⢸⠇⠀⣹⠀⠉⠁⢸⡇⠀⢸⡇⠀⠀⠀⢀⡴⠁⠀⣰⠃⠀⠀⠀⠈⣧⣠⣀⠀⠀⠙⢦⠀⣰⠃⠀⠈⠉⠉⠀⠀⠙⢻⡆
⠀⠀⢷⠀⠀⠳⣤⣠⡴⠋⠀⢀⡟⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⡔⠋⠀⠀⠚⠓⠒⢲⡆⠀⠀⠀⠀⢈⡇⠀⠀⢸⡖⠳⠲⠶⠶⠶⢶⠀⠀⡖⠚⠃
⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⢸⡇⠀⠐⣇⣀⣀⣀⣀⣀⣀⣸⡇⢀⡏⠙⠛⠉⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠒⡤⠒⠄⠀⠀⠀⠀⠀⠀⠠⠁⠐⠈⢣⡀⠀⠀⣠⠤⠤⠴⣐⣾⠱⠄⡧⣤⠀⠀⠀⢠⠀⡠⢦⠀⠀⠀⠀⠀⠀⡐⢁⣶⡉⢃
⠀⠀⠀⠀⡏⠑⠙⠣⠭⠏⠉⡇⠀⠀⠀⣠⠞⠛⠛⠓⠊⣹⠀⠀⣿⠀⠀⠀⠀⠀⠙⠋⠅⢸⠀⠀⣠⣧⣼⣶⣴⣄⠀⠀⠀⡴⠋⠙⢒⣙⠒⠋⠳⡀
⠀⠀⠀⠀⡇⠀⢰⠋⠉⠉⠉⠁⠀⠀⡼⠁⠀⣠⣒⡛⠙⠃⠀⠀⠛⠒⠚⢛⡷⠃⠀⢀⡴⠛⠀⡞⠀⢠⠴⢬⡀⠘⣇⠀⢸⠃⠀⣸⠉⠈⢱⠀⢰⢷
⠀⠀⠀⠀⡇⠀⠈⠑⠒⢦⡀⠀⠀⢸⠁⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⣠⠏⠀⢀⡴⠋⠀⠀⠘⣇⠀⠙⠦⠞⠁⢠⠏⠀⠸⡄⠀⠘⠲⠶⠋⠀⠉⢸
⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⠹⡄⠀⣿⠀⠀⡴⠉⠉⢣⠀⠘⡇⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠀⣠⠞⠃⠀⠀⠀⠘⠳⢄⠀⠀⠙⠦⢤⣤⣄⠀⠀⠀⡟
⠀⠀⠀⢠⠤⠤⠤⠞⠀⠀⢠⠇⠀⢸⡀⠀⠳⣄⣠⠞⠀⢠⡇⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⢠⡎⠉⠙⡆⠀⠈⡇⠀⣠⣤⠤⠖⠁⠀⢀⡼⠁
⠀⠀⠀⢾⣀⡀⠀⠀⣀⡤⠋⠀⠀⠀⠳⣄⣀⠀⠀⣀⡤⠏⠀⠀⠸⣥⣤⣤⠇⠀⠀⠀⠀⠘⣇⠀⠀⠓⠒⠚⠁⠀⣠⠇⠀⣷⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⡁⢠⡏⠀⠀⠀⠀⠀⡠⠀⠄
⠀⠀⣤⡶⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡿⠷⠀⠀⠀⠀⡖⢃⠀⠎⢀⠏
⠀⠀⢹⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⠀⠀⠀⠀⠀⠘⡈⠀⠁⠞⠀⠐⠒⠠⠄⡀⠀⠀⢠⡴⠞⠛⣳⠄
⠀⠀⠀⠻⠋⠉⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡡⠂⣜⠀⣿⣿⠆⢀⣴⣴⠇⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡾⠛⣷⠀⠀⠀⠀⠀⠈⡗⡼⡳⡀⠀⢀⣀⡀⠛⠋⠀⣤⡖⠋⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣼⠋⠀⠀⠀⠀⠀⠀⠸⠑⣜⡁⠀⢺⣿⡗⠀⠀⠀⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⣦⠀⠀⠀⠀⠀⡠⠛⠛⠿⣀⣀⠀⢠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠾⠃⠀⠀⠀⠀⠘⠓⠒⠲⠛⠋⠉⢈⠉⠁⠀⠠⡠⠄⠀⠰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⣀⢈⡀⣅⠀⠀⢈⠂⠀⠐⢤⡐⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⢀⡀⢣⡀⠀⢜⠁⠀⠀⠀⢑⠄⢸⢀⡀⣐⢼⠈⠀⠀⠀⠀⢿⡀⠀⢀⠀⣤⠀⣶
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⠄⠐⠋⠀⣾⣯⠄⣀⡒⢄⠀⠀⠀⠀⢸⣷⠄⠙⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠁⠈⠉⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠤⠀⠀⠀⠤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⡦⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡐⠀⢢⠀⣠⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⢠⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⡧⣀⡈⠀⡇⡀⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣐⠳⠤⠘⡏⠀⠀⠀⠈⠆⢠⠄⡀⠀⠀⠀⠀⠀⡇⠈⢫⠖⠺⣌⣑⠂⠄⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢠⢂⣏⠑⠀⠀⠀⠡⠀⠀⠀⠀⠈⢇⢻⠑⣄⠀⠀⠀⠀⠈⢰⠀⠀⠀⠃⠈⠑⠲⣄⡀⣀⠀⠀⠠⠉⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠩⢜⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡋⠀⢣⠜⢦⡀⢀⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⢠⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡨⠔⠈⠐⠦⣝⡆⠀⠀⠀⠷⡀⠀⠀⢀⣄⡀⢸⠀⠂⠤⣄⣉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠘⣁⣀⣤⡤⠀⠠⠜⠁⠀⠀⢸⠖⠶⣄⣠⠈⠄⠈⠛⠈⠂⠀⠘⠢⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⡁⠀⡟⢆⠄⠠⣄⠀⠀⡐⢡⠆⣷⡄⣣⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠈⢉⠯⠆⠀⠞⠟⢉⠀⠨⡀⣿⠏⠀⠀⠀⠀⠑⡀⠀⠀⠀⠠⠄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢉⡴⠂⠐⠁⠀⠀⠀⠀⠀⠆⡿⢋⠉⠉⠋⠋⠀⠙⠶⠾⠶⠶⠾⠿⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⢐⠾⠛⠒⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠀⠒⢶⣫⣄⣀⣀⡵⠄⣀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠘⠭⠄⠀⠀⠭⠌⠁⠒⠦⠤⠤⠤⠀⠀⠈⠓

⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠈⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡌⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠤⡄⠀⠀⠢⠀⠀⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⡀⣀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠆⠀⠀⠘⠃⠀⠉⠀⠀⠀⠴⠀⠘⠃⠀⠀⠀⠉
⠀⠀⠀⠀⢀⠲⠶⢓⠉⢉⠙⡠⡤⠤⢄⠒⠒⢒⠤⣤⡤⢄⠤⠦⠦⡒⠒⠓⢢⣤⣠⡀⠲⠶⠖⡲⠚⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠁⠰⡀⠁⠀⠀⠀⠁⢴⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠈⠀⠤⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁


⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢲⢀⠆
⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠔⠀⠀⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠀⠀⠀⢈⠀⠀⠀⠰⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠠⠐⠀⠀⠈
⠀⠀⠀⢲⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⡄⠀⢦⠄⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠠⢢
⠀⠁⠀⠈⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠁⠄⠠⠄⠀⠒⠀⠀⠄⠈⠀⠀⠀⡀⠊⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠈⠳⠋⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠄⡐⠠⢆⢀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠊⠔⠁⠀⠐⠉⢉⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠐⠀⠁⢀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠁⠀⠀⠀⠁⡠⣀⡀⠀⠀⠀⣠⣶⣙⠀⡾⢋⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠐⠠⠀⠀⢹⣿⣿⡆⠀⠀⠛⠿⠋⠀⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠛⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⣀⣀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐⣨⣄⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄⠀⠀⠀⠀⠀⠀⢸⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠑⠶⠀⢟⡀⠀⠙⢓⡀⠀⠀⠀⡘⠁⠙⠌⠆⠀⣠⡀⠀⠀⢸⠴⠾⢯⡀⠀⠀⠀⠀⠀⣿⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠲⡤⣥⠆⠈⡁⠁⠀⠐⠈⠀⠰⠀⠀⠀⣰⡟⣧⠀⠀⠈⠠⠤⠞⠁⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣅⠠⡀⡷⡾⠶⠛⢔⠠⠄⠊⠀⠀⠀⢰⠋⠉⠉⣧⠀⠀⠀⠀⠀⣀⠠⠖⠶⠀⠀⣿⠀⡇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠶⠓⠻⣿⠀⠉⠀⡠⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⣿⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡌⠒⢤⡤⠀⠀⠉⢀⣤⠀⠃⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠲⠃⠀⣿⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠚⠁⢠⠄⠐⠋⠀⢀⡀⠰⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⣄⡒⠀⣀⠀⠐⠂⢀⡀⠀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠃⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡉⠉⠛⠛⠛⠛⠛⠙⠋⠁⠀⠀⠈⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⢸⡀
⠀⠀⠀⠀⠘⠃⠀⠇⠀⡀⠀⠀⠁⠀⠀⠉⠉⠁⠀⠀⠀⠘⠈⠁⠀⠀⠈⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠐⠀⠀⠄⠄⠀⠃⠀⠈⠀⠀⠸⡂⠠⢄⠀⠀⠀⠠⡄⠀⠂⣄⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠁⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄⠀⠀⠀⣠⠤⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⠖⢲⡄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢹⡄⠀⢰⡇⠀⢨⡇⠀⠀⠀⠀⠀⣼⠋⠙⣆⠀⣿⠀⠀⡇⢀⡟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⣧⠀⣾⠀⠀⣼⠁⠀⠀⠀⠀⠀⢹⡄⠀⢹⡄⣿⠀⠀⡇⣼⠁⠀⣼
⠀⣀⣀⣠⠶⢶⣴⠋⠛⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⣴⠇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣷⣿⠀⠀⣷⡇⠀⢰⡇
⡾⠉⠉⣏⠀⠀⣿⠀⣠⣿⣄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣧⣀⣀⠛⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠸⠇⠀⠀⠟⠀⠀⡾
⡇⠀⠀⣷⠀⠀⣿⠰⣇⠀⠈⠉⠛⢦⡀⠀⠀⠀⠀⣴⠖⢾⡆⠀⣾⠉⠉⠉⠙⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⢀⣠⡤⠴⠶⠒⠶⣷⡀
⢻⣤⣴⠟⠶⠶⠋⠛⠛⣳⠦⠀⠀⠈⣧⠀⠀⠀⠀⢷⠀⠀⢿⡀⠙⣷⢶⡆⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡃⠈⠙⢿⡁⠀⣀⣀⠀⠀⠈⢷
⢸⡆⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⣧⣠⡼⠛⠛⠁⣾⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⡟⠶⣄⣠⠟⠉⣽⠋⠀⠀⠀⣸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠉⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠏
⠀⠈⠳⢤⣤⣤⣤⣤⣤⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠴⠶⠦⠤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣤⣤⣤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⠶⢦⠀⢠⡏⠉⢹⡄⠀⢠⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡄⠀⣾⠁⠉⡇⠀⢀⡴⠲⣦
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⡇⢈⡇⠀⢸⡇⢀⡟⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢷⠀⣿⠀⠀⡇⠀⣼⠁⠀⡾
⠀⠀⠀⢀⡤⣄⠀⢹⡀⠀⢿⠀⡇⠀⢸⡇⣼⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀⠘⣇⠀⠸⡆⣿⠀⠀⡇⢰⡇⠀⢸⠇
⠀⠀⠀⢿⡀⠈⢷⡈⣇⠀⠸⣧⡇⠀⢸⣧⡏⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⣄⢻⡀⠀⢷⣿⠀⠀⣇⡾⠀⢀⡟
⠀⠀⠀⠈⢷⡀⠈⢳⣽⡄⠀⠙⣡⣤⣬⣋⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⣾⡇⠀⠈⠉⠀⠀⠉⠁⠀⣼⢁⣠⠖⠛⢳⡀
⠀⠀⠀⠀⠀⢳⡄⠀⠉⠀⠀⠐⣏⠀⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠙⣿⠂⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⡴⠚⠋⠁⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⢀⣠⠏

⠱⠄⠈⢁⣀⣵⡙⠳⡿⠄⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⢀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢕⣤⠀⠀⣠⠊
⢀⣴⣕⠄⠛⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠉⠀⠛⠀⢀⡀⠀⠀⠀⡔⢠⠀⠀⠀⠘⠁⠀⠜⠁⠸⣆
⣝⠂⠙⢀⣴⠄⠓⠀⠀⠀⣠⠰⠲⠲⠆⠁⠀⠀⠴⠀⠒⠠⠀⠀⠈⠀⠀⠀⠀⠀⠁⠤⡄⠀⠀⠀⠀⠠⠀⠀⠈
⡙⠂⠀⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡚⠶⠄⠀⠀⢀⠥⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠘⠉⠁⢢⠀⠀⠀⠈⠀⠀⢸⠒⠒⢀⡀⠀⠀⠀⠁⠀⠔⠀⠀⠀⡠⠚⠁
⠀⠀⢠⠀⠤⠰⠇⠀⠀⠀⢰⠃⠚⠁⠠⡄⠀⠀⢁⠇⠰⠀⠀⠀⢰⠤⠐⣀⠸⠂⠒⢀⠰⡄⠀⢀⡀⠀⠀⠀⠠
⡰⠈⠙⠂⠀⠈⡱⠔⠶⠤⠌⢠⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠸⡀⠀⠉⣠
⠳⠦⠀⠀⠀⠀⣁⠀⡤⠬⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠄⠀⠠⠅⠀⠠⠒
⣦⠀⠀⠀⢰⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⠂⠀⠀⠉⠉
⠀⠈⡁⠀⠘⠠⠂⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⡤⠤⠄
⣤⡦⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣴⡶⠖⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣼⡷⠖⣋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⢴⡶⠟⣋⠄⠀⠀⣠⠔⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠢⣌⠙⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⠲⡀⢄⠂⠀⢠⢲⣤⡟⢇
⠀⠀⠀⠀⠀⡀⠀⣀⢀⢄⠀⠀⠀⠠⠀⣠⠴⠒⠺⢶⣜⡍⡇⠀⠀⠀⠀⢠⠏⠑⠛⢔⡦⠧⣄⡜⠲⠓⠤⡎
⠀⠀⣠⠴⠚⠺⠛⣵⣆⡇⢠⢠⡎⢬⡎⠣⣀⡤⢤⡀⠉⠙⡆⠀⠀⠀⠀⡼⠀⢰⠛⠒⠲⢤⡟⠀⣰⡆⠀⡇
⢀⡜⠁⢠⠔⠚⠓⢎⠀⢹⡤⠋⠉⠹⡇⠀⠀⠀⣠⠃⠀⢠⠇⠀⠀⠀⠀⠧⣄⡈⠑⠦⣀⠞⠀⡼⢡⡇⠀⡇
⣼⠀⢠⠇⠀⠀⠀⢸⢶⠉⢀⣀⠠⠂⣇⣀⠤⠚⠁⢀⡴⢋⠤⣀⠶⡀⣀⠀⠀⣱⠀⠀⢹⠀⠀⠉⠉⠀⠀⠉⢹
⡇⠀⠸⡄⠀⠀⢀⡼⠘⢻⠏⢹⠀⠀⣏⠀⠀⠀⠲⠯⣄⠼⡶⢷⣶⢿⡉⠉⠛⠁⠀⠀⡼⠛⠛⠛⢻⠀⠀⡏⠉
⠹⣄⠀⠈⠓⠒⠉⠀⢀⡞⠀⢸⠀⠀⣿⠲⠤⣄⣀⢰⠁⠀⡖⠒⡄⠀⢳⠦⠤⠤⠴⠊⠀⠀⠀⠀⢸⣄⣠⡇
⠀⠈⠳⠤⣤⣤⣤⠴⠋⠀⠀⢸⠀⠀⢸⠀⠀⠀⠈⠙⢆⠀⠙⠋⢁⣠⠋⠀⠀⡔⠠⠄⢷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⣏⠉⠀⠀⠀⢀⠖⠁⢀⡤⠤⡀⠈⠳⣤⠔⠛⠓⠶⠜⣦⡤⢠⣊⢣⣀
⠀⠀⠀⠀⠀⠀⣼⠉⠓⠚⠼⢏⠶⢨⣄⡀⠀⢀⢸⠀⢀⢻⣀⣀⡿⢀⠞⠁⢀⣤⡒⠒⠶⣣⠦⠽⢛⡊⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠧⢤⣤⣀⡀⠀⠉⠑⣸⠦⠤⡠⣜⢩⣻⡀⠀⠀⣀⡞⠀⠀⢀⣀⠉⠑⢤⡇⠀⡜⠉⠉⡇⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣠⠖⠻⠤⢄⣉⠉⠒⣩⠟⠉⠉⢩⡇⠀⢰⠉⠈⢳⠀⢸⣇⠀⠙⠲⠖⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠀⢀⠴⠊⠁⢀⡔⠁⠀⠀⠀⠀⢧⠀⠘⠢⠔⠃⠀⣸⠉⠲⢤⣤⠤⡦
⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞⠁⠀⠀⠀⠸⠤⢤⡀⠀⠙⣆⠀⠀⠀⠀⠈⠓⠦⣤⣤⠤⠞⠁⠀⠀⢀⡠⠔⠁⠀⣠
⠀⠀⠀⠀⠰⠧⣤⣤⡞⠀⠀⠀⠀⡤⣤⣀⣠⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠐⠈⠉⠉⠑⠢⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠙⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠂⠀⠀⠤⠄⠀⠀⡄⠀⠀⠀⠑⠖⠖⢀⣀⠄⠀⠀⢀⣄⣀⠄⠀⠀⠈⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠒⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⢀⡉⢣⠀⠀⠀⠀⠐⠦⠤⠄⢀⣀⡀⠀⠢⠤⠔⠊⠒⠃⠀⣠⢀⠄⠀⠀⢠⠀⠀⠀⠈⢢
⠀⠀⠀⠰⠊⠁⠀⢀⠀⠀⣈⣧⣀⠉⡉⠀⠙⠓⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡜⣠⠀⠀⠀⠀⠁⢳
⠀⢀⡤⠂⠀⠐⠊⠹⣄⡀⢀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⡀⠀⢀⡜⠀⠀⡀⠈⠹⠄⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠀⡴⠀⠀⠈⠀⢤
⠀⠊⠀⠀⠒⣆⠀⠀⣀⡍⠀⠀⠀⠑⢪⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠀⠠⣄⡀⠐⠆⠀⠒⠁⠒⠒⠋⠀⢀⠦⠔⠄⠈⠇
⡀⠀⠀⠀⢤⡈⠁⠀⠰⡁⠀⠸⣄⠀⡀⠑⠂⢄⠀⠰⣄⡀⠄⠀⠀⣀⠀⠀⠣⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢄⣰⠤⠂⠈⠋⠁⢀⠄⠀⠀⡀
⢇⠱⠄⠀⠀⣉⠁⣠⠀⠉⠀⠀⢀⠉⡀⠀⠀⠀⠉⠁⠀⢀⠀⣉⡉⢀⣀⡠⢄⣈⠉⣉⣁⡀⠠⣄⣠⡄⠀⠀⢤⡀⠀⠀⠈⠉⠁⠠⠖⢊⡁
⠈⠢⠀⡄⣄⠈⠀⠻⠀⠀⠙⠤⠈⠉⣁⡜⠐⠋⠈⡐⠊⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⣀⣜⠃⠀⠀⠀⠠⠴⠒⠊⠀⠄⠈⡄⠊
⠀⠀⠀⠈⠢⡁⠀⠤⡀⠄⠐⠂⠀⠀⠀⠈⠋⠉⠉⠀⣿⠀⠀⠀⢰⠀⡇⠰⠀⠀⠀⠀⡼⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⣀⠀⠚
⠀⠀⠀⠀⠀⠉⠒⠂⠐⠤⡍⠉⢀⣐⡊⡐⠒⠀⠀⠖⡿⠀⠀⠀⢸⠀⢇⡇⠀⡈⠠⢸⠃⠒⠓⠒⠂⠀⠀⠉⡈⣁⣀⡠⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠑⠄⠀⢺⡀⢈⠸⠈⡇⠸⠀⢃⠇⡆⡿⠖⠤⠒⠐⠊⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡆⡀⡇⡄⠠⣸⢰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⡇⠁⠀⡇⡏⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡼⢇⡠⠴⢧⣇⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠌⠉⠁⠀⠀⠀⠀⠀⠁⠘⠋⠳⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⢠⣴⠦⣴⠠⢤⣤⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠲⠄⠠⣿⡇⣿⠀⢸⠀⢸⢣⠔⠢⠤⠆⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⡿⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⡇⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢳⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠴⢦⠀⠀⢈⠤⢤⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠣⠀⠀⠀⠀⠋⠑⡀⠀⠀⠉⠀⠀⠁⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠀⠀⠔⢢⡤⡈⠂⢀⡀⢩⠤⠒⢤⢤⡀⠙⠀⠐⠳⡄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⠠⠄⠀⢤⡀⠀⠀⠀⠀⠀⠈⠀⠐⠆⠀⠀⠑⠲⠀⠈⠓⠀⠈⠈⠋⠂
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠔
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⡠⠀⠀⡤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢃⡀⡄⠀⠀⠈⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣛⣛⣻⣛⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠉⠐⠀⠉⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠐⠀⠤⠀⠀⠤⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⣀⠀⢀⠀⠀⠀⡀⠀⢀⡀⢀⠀⠀⠀⢀⠀⢀⡀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄⠀⣀⠀⢀⡂⠀⠀⡇⠀⢀⡀⢀⡄⠀⠀⣃⠀⢀⡀⠀⡀⠀⠀⢃⠀⠀⡂⠀⣘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠃⠀⠀⠇⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⠃⠀⠀⠛⠀⠊⠐⠐⠃⠀⠀⠚⠀⠂⠑⠀⠃⠀⠀⠚⠑⠀⠁⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠤⠀⠠⡄⠀⠀⡇⠀⠠⠀⠀⡆⠀⠀⠤⠀⠠⠄⠀⡄⠀⠀⠠⠀⠀⡆⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣀⣀⣀⡀⠀⠀⣃⢀⣀⣀⣀⡇⠀⠀⣀⣀⣀⣀⣀⡆⠀⠀⢠⣀⣀⡆⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠘⠀⠐⠆⠀⠀⠇⠀⠘⠀⠀⡇⠀⠀⠑⠀⠘⠂⠀⡃⠀⠀⠐⠀⠀⡃⠀⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣗⣀⣒⣂⣰⠆⠀⠀⣖⣂⣐⣂⣒⡇⠀⠀⢔⣂⣐⣂⣒⡃⠀⠀⢰⣂⣐⣂⣀⡲⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠘⠀⠈⠐⠄⠘⠒⠀⠐⠀⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠰⠀⠠⡄⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠠⠄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢠⢠⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠈⠁⡏⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⢹⠘⠁⠁⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠧⠏⠡⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠌⠹⠜⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⡀⡂⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠠⣴⠂⠀⠀⠀⠀⠀⠠
⠀⠀⡀⠔⠁⡃⠀⠀⠀⡴⠉⣉⠙⠱⠀⠀⠀⠀⠸⠀⡈⠉⡝⠀⠀⠀⠀⡐⢁⠁⡇⠀⠀⠀⠀⠈⢈⠽⠅⠅
⠀⠀⠦⢲⠀⠁⠀⠀⠀⠀⠉⢀⠃⢰⠁⠀⠀⠀⠀⣊⡀⠘⢄⠀⠀⠀⡜⠠⠨⠀⠇⡀⠀⠀⠀⢤⣈⠐⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠔⠁⠐⠃⢤⠀⠀⠀⢠⠤⠜⠀⢸⠀⠀⠘⠀⠀⢤⠀⡔⠃⠀⠀⠀⡤⠤⠃⠀⡇
⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠚⠀⠀⠀⠓⠂⠐⠒⠁⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠈⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡠⠤⡌⢄⠀⠀⠀⡤⠤⠠⠮⡡⡄⠀⠀⠀⣀⣠⣩⡀⠀⠀⠀⠀⡠⢠⠌⢄⠀⠀⠀⠀⢀⡤⢤⣬⡀
⠀⢀⠎⠀⠔⣀⠂⠀⠀⠀⠦⠤⡤⠀⣠⠇⠀⠀⢘⠀⣆⠄⢹⠀⠀⠀⡎⢠⠂⢲⢀⡇⠀⠀⡰⢁⠔⠒⢄⠑⡀
⠀⢸⠀⠠⠀⠄⠱⠀⠀⠀⢀⠞⢀⠞⠁⠀⠀⠀⠜⢃⠀⡐⠣⡀⠀⠀⠱⢄⣉⠁⠀⡆⠀⠀⡇⢸⠀⠀⢸⠀⡇
⠀⠘⣄⠐⠒⠃⡸⠀⠀⠀⡞⠀⡞⠀⠀⠀⠀⠘⡄⠘⠤⠂⢀⠇⠀⠀⢰⠒⠊⢀⠔⠁⠀⠀⠳⡀⠑⠒⠁⡠⠃
⠀⠀⠀⠉⠂⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣛⣶
⠀⠀⢀⡄⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⣿⣿⢻
⠀⠀⠀⠳⡀⠀⠘⠛⠋⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠟⠋⠀⢘⡆⠀⠀⠀⢀⡀
⠀⠠⡈⢆⠱⠴⣒⣒⡚⢊⠴⢒⡈⠥⠴⠀⠀⠀⠀⠀⣀⣤⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠠⣄⠀⠀⠒⠉⠀⠀⠀⠀⣸⣿⣷⣶⣶⣤⢗⣤
⠀⠀⠱⣈⣻⠭⠥⠤⠭⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⡿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⢯⠅⣠⡄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⠛⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣘⠷⢟⡻⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⣿⢛⢛⣯⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡧⠀⠀⠀⠀⠀⠀⠸⡿⠿⠧⠉⠿⠃⣿⣋⡬⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⠀⢀⣥⡀⠀⢸⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡀⢺⢻⣿⡿⡗⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠋⢻⢸⣿⣇⡇⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣩⣔⣚⣿⣭⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠉⠉⠉⠉⠉⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠁⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⡷⠶
⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⢟⣛⣛⠿⢛⣛⡻⠿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⡇⠸⠄
⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡟⠛⡟⠛⢻⠛⢻⠛⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣦⣖⣤⣭⣉⣥⣴⣶⣷
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⢳⠒⡗⠒⢺⠒⢺⠓⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣯⣽⣿⣋⣭⣴⣿
⠐⠐⠒⠒⢲⣿⣿⠒⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡚⢗⢻⠒⡟⡲⢛⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠋⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢻⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢚⠋⠁⠈⠑⠦⢷⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢽⣯⣤⣤⣀⡀⠀⠀⠀⠈⠙⠓⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠛⠶⢶⡶⢶⠖⠀⠲⢆⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢞⣋⡉⠀⠀⡿⢀⡟⠀⠀⠴⣦⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⣀⡴⠚⠉⠉⢹⡆⣼⠃⢸⡇⢀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣁⠀⠀⢠⡾⠋⠀⠀⠀⠀⢰⡿⠁⠀⣸⣡⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⠉⠉⠻⣶⠋⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⣤⣀⡾⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⡷⣿⡴⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠛⢚⣛⣻⣧⣾⣆⣀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠿⢋⡴⣶⠆⠀⣿
⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠻⣯⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢀⣴⠏
⠀⠀⠀⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠶⠶⠛⠁
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠙⠷⣤⣀⠀⠘⠉⠔⠒⠤⠆⢀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⣶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⢻⡀⣀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠄⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣴⣎⣠⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣀⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⢀⣠⠞⠙⢦⡀
⠀⠀⠀⢠⡴⠶⢤⣄⠀⠁⣴⠛⠀⢀⣀⣀⣀⣻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⣀⡴⠋⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠈⠳⣄⠀⠈⣧⠀⠀⢰⡏⠉⠉⣈⣁⣄⡈⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣄
⣾⠋⠙⠳⢦⣌⢢⡀⠸⡇⠀⠀⠉⠉⠉⣵⠖⠚⠓⠂⠀⠘⣆⠀⠀⣠⠖⢊⠉⠁⠀⠀⠀⠀⣠⠴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢤⡀
⠙⠳⠦⠤⣤⡈⠛⣷⡴⠛⠉⣷⠀⠀⠀⠙⣾⠋⠉⠉⠀⠀⠸⣆⡴⠋⠀⠈⢦⣀⠀⣀⡴⠋⠁⢀⣠⠞⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⣴⠖⠛⠉⠛⣿⡀⣰⢇⡿⠀⠀⠀⠀⣹⣄⠘⣶⣤⣀⣀⡈⠧⣄⠀⠀⠀⠉⠛⠉⢀⣠⠶⠋⠀⠀⠀⠀⢠⡾⢯⡥⠔⠀⠀⠀⠀⣠⡴⠻⡄
⠀⠀⠻⢦⣤⣐⣋⣉⡙⠉⣾⠁⠀⠀⠸⡛⢧⡉⠛⠁⡜⢸⡍⠙⢦⡈⠳⣄⡀⠀⠀⠀⠸⣝⣶⣄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢬⡑⢦⣄⣀⣽⣾⣿⣆⣼⠃⠈⠀⠀⠀⠙⢦⣀⠙⠲⣤⣤⠾⢿⡿⣏⠳⣄⣰⠋⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢬⡁⣨⠟⠧⣼⣧⣤⣤⣀⣀⣀⣀⣀⣈⣳⣤⣀⣠⣴⣋⣀⣈⣛⣿⣯⣤⡶⢞⣭⣤⡄⠀⠀⠀⠀⣀⣤⠶⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⡛⠛⠓⠋⠓⣄⣿⣬⣯⡭⠥⠾⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⢀⣀⣠⡤⠶⠞⠛⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠐⠤⣭⣛⡒⠶⢴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣞⣛⣉⣭⣤⣄⣠⣤⣤⣤⠤⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠉⠉⠉⠙⣦⠀⠀⠈⠉⠙⣛⣛⣛⣷⠶⠶⠿⠛⠛⢯⡉⠀⠀⣽⠁⢹⡄⠀⠀⠀⢻⡾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⡀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣄⣿⠀⠈⣧⣀⣀⣀⣘⣇⠀⢷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣎⣉⣉⣯⣉⣙⣛⣲⣶⣤⡤⣬⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣈⣿⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣵⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠎⣱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠆⡀⡀⠂⣠⡾⠣⠄⣀⠀⠀⠀⠀⠀⠐⣆⠀⠃⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠈⢀⣤⣀⠀⠀⠀⣎⣀⣀⠀⠈⢡⢰⣶⡆⠀⠀⠀⠉⠁⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⢹⠶⡏⠀⠈⡷⠀⣼⣿⣿⣿⣿⣶⠟⠈⠋⠀⠀⠉⠉⠀⠀⠀⠀⠙⠷⠘⠿⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠓⠖⠋⠀⠙⠲⠚⠷⠀⠸⣿⣿⣿⣿⡏⠀⠀⡀⠀⢀⡙⠳⣤⡀⢰⠇⠴⠰⠆⠄⠨⠀⠀⡠⠾⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡰⠶⢤⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣔⣅⡈⠛⢁⣸⠇⠈⠻⣆⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⣀⣛⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢾⡄⠀⠀⣠⡄⠀⠀⢶⣲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠹⣆⠀⠀⠀⠀⠀⢀⣤⡶⠛⡋⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠰⠤⠄⠹⠀⣀⣈⣁⣀⣀⣸⣿⣧⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⢻⡄⢺⡗⢀⣴⠟⠁⢰⣿⡇⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⣦⡆⠀⢘⡷⠶⣤⠀⠘⠿⣿⣿⣿⣿⣇⣿⣿⡆⠀⣀⡀⠀⠘⠿⠿⠃⠀⠀⢸⡇⠀⣰⡟⠁⠀⠀⠈⠉⠁⠛⣩⣀⣈⣀⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠾⠡⠤⠌⠧⠀⠺⠿⠿⠿⠿⠿⠸⠿⠷⠀⠛⠃⠀⢀⠐⠃⠐⠃⡀⢸⣯⢠⣿⠃⢸⣾⣾⠀⠀⢀⠀⠉⠀⠀⠈⠉⢄
⠀⠀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⣄⡀⠘⠁⠀⠀⡀⠁⣼⡇⣼⡟⠀⢸⣿⣿⠀⣀⠻⠀⠀⠀⠀⠀⠀⠸
⠀⣴⣧⣄⣀⠘⠛⠀⡄⠰⠀⡔⢀⠀⠀⠘⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠙⣦⡈⠃⠈⠁⢠⣿⢡⣿⠃⠀⠀⣉⣣⣄⣿⡆⠀⣤⡀⠀⢀⡄
⠀⠀⠀⣼⠋⢀⣈⠃⠀⠀⠀⠀⢋⡀⠀⠀⣿⠀⣴⣤⣴⡀⠀⠀⠀⡂⠀⠀⠈⣿⣄⠀⢀⣾⡏⣼⡿⠀⠀⠀⢽⣿⣿⡾⡷⠀⠉⠁⠀⠈⠁
⠀⠀⠘⠁⠀⠀⡤⠀⠀⠀⠀⠀⠡⠤⠀⠀⣿⠸⠿⠿⠿⠗⠀⠀⠀⠁⠂⠀⠀⢸⣿⣤⣾⡿⣱⣿⠇⣼⣇⡤⡄⠀⠀⣠⣤⣶⣿⣿⣿⣿⠶⠒
⠀⠀⣠⣤⠀⠀⠀⠀⢀⣤⣄⠀⠰⢛⡆⢰⣿⠀⣀⣤⣴⣶⣶⣦⣄⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠏⠈⠋⠉⡱⢣⠀⠀⠡⠀⠉⠉⠉⠉⡀⠂⠁
⠀⠸⠟⠁⠀⣷⣾⡄⠀⠻⠋⠀⠀⠈⠀⢸⣿⠀⠉⢳⣿⣿⣿⣿⠙⠉⠀⠀⢠⣿⣿⣿⣿⠟⠃⠀⠘⢲⣞⠁⠿⢦⡄
⠀⠁⣤⣦⣀⠈⡿⠀⠀⠀⠀⢰⣟⣠⢄⠈⣿⣆⠀⠀⠉⠙⢉⣵⡂⠀⠀⢠⣿⣿⡿⠟⢁⣠⠀⠀⠀⠉⠉⠁⠀⠎⠀⢀⡀⠐⠀⠀⠀⢀⡞
⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠛⠉⢹⠎⠀⠘⢿⣷⣤⣀⡀⠻⡿⠃⠀⣠⣿⣿⠋⠀⠈⠉⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠄⠚⠀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠲⣶⠋⠀⠀⠀⡀⠉⠻⣿⣿⣷⣦⣀⣴⣿⡿⠁⠐⠀⠠⠆⠀⡜⠀⡀⠙⠛⠀⣀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠉⠋⣀⡀⠀⠀⣀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠮⠉⢭⡩⠨⠏⠀⠀⣷⠀⠀⠀⣍⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣰⣶⣶⡄⠀⠀⠈⡆⠀⢀⣼⣿⣿⣿⣿⣷⣄⠀⠼⠧⠀⠀⠀⣸⡟⢠⡄⠈⠉⠑⢦⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢀⣾⣿⣿⣿⣿⣿⣿⡟⠁⠀⣀⣠⠤⠤⠖⠒⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠏⣀⣤⡤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠶⠶⢤⣤⣀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠴⠶⠦⠤⠤⣤⣤⣀⣀⡀⠘⢧
⠀⠀⠀⠀⣼⣿⣋⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡉⠛⠶⣼⣀⣤⣖⣚⣋⣩⣤⣤⣤⣤⣤⡤⠤⠤⠤⢤⣤⣤⣭⣭⣿⣾⣧
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣼⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀
⠀⠘⠛⠟⠻⠛⠻⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⠛⠻⠿⠿⠿⠟⠻⠿⠿⠿⠟⠛⠛⠛⠻⠛⠟⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣿⣟⡛⣿⣿⣿⠿⠻⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⢶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢟⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢍⠉⠃⠀⠀⠀⢀⠄⠒⠒⠂⢀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣺⣯⣶⠄⢀⠤⠄⠂⠀⣀⠀⠀⠔⠃⠀⠐⠤⡀⠐⠄⠀⢀⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡝⠟⠉⡀⠎⠉⣶⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⢀⡈⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣫⡞⠀⢼⠀⢀⡀⣷⡀⠀⢄⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡇⠊⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣾⣦⣼⣶⣤⣽⣿⣿⣶⣾⣷⣤⣿⣿⣦⣄⣀⣀⣤⣄⡀⢀⣀⣁⡀⠀⠀⠁⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣝⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⡿⠿⠶⣶⣦⣴⢀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠳⣭⣿⣥⣝⣿⣷⣶⣿⣿⣿⣿⣁⡀⠀⣀⡀⢠⣿⣿⣿⣾⣿⡿⠿⣿⣫⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⢸⠿⢿⣿⣷⣿⣏⠉⢻⡟⣿⣇⠉⠋⠀⠀⡟⠃⣼⠟⠉⣹⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣬⡉⢿⡙⢿⣆⠀⠁⣿⣿⠀⠀⠀⢠⡇⢰⠏⢀⣾⣿⠟⣡⣾⣿⠁⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠲⣝⠆⠙⢧⠀⣿⣿⠀⠀⠀⠈⡇⠀⣠⡿⢋⡴⡚⠱⠂⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣭⣙⡛⠂⠁⠀⠀⢰⣿⣿⠀⠀⠀⠀⢱⡐⠋⠀⠑⠊⠁⠔⢛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣭⣽⣛⣛⠟⠃⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⢧⠀⠀⠀⠴⠒⢛⣛⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣾⣿⣿⣿⡄⠔⠒⢂⣀⠜⠆⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠁⡀⠀⠀⠈⠺⣿⠟⠛⠻⣿⣦⣴⣈⡀⠀⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣖⣁⣀⣀⣀⣈⣙⢿⠶⣶⣀⣀⣠⡤⠈⠛⠉⠙⠛⢿⣿⣁⣙⣒⣲⣷⣶⣶⣀⣾⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⣟⣹⢛⣟⣻⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠛⠛⠃⠘⠛⠛⠛⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡇⠀⠀⣺⡅⠀⠰⣿⠄⠀⢘⣧⠀⢰⡟⣧⠀⠈⣹⠃⠀⢸⣿⡅⠀⢻⣽⠀⢈⣿⣏⡇
⢀⣶⠶⢤⠀⡠⠶⣄⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠌⠀⠀⡈⠁⠀⠐⠉⢁⠀⠀⠁⢐⠐⠂⢉⡒⢐⠨⠁⡀⢀⠡⢙⡀
⠈⠙⠶⠎⠉⠱⠶⠋⠸⠇⠺⠇⠻⠇⠿⠃⠿⠘⠿⠘⠿⠸⠗⠸⠟⠸⠇⠺⠇⠿⠇⠿⠘⠿⠸⠿⠸⠟⠸⠏⠹⠇⠸⠇⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠛⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠤⣤⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠁⠀⡴⢦⡀⠈⠛⠛⠛⠛⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣠⠎⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⣠⠞⠁⢀⡤⢤⡀⠈⠳⣄⣀⣀⣀⣀⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⢀⣀⡀⢠⡏⠀⠀⢹⡄⢀⣀⡀⢀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢸⠀⡇⠀⠳⠤⠤⠞⠀⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⣛⡁⢈⣛⡃⠀⠀⠀⠀⠀⠀⢘⣛⡁⢈⣛⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢸⠁⡇⠀⣾⠓⠛⣷⠀⢸⠈⡇⢸⠀⡇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠇⠘⠛⠃⠈⠛⠃⠀⣿⠀⠀⣿⠀⠘⠛⠁⠘⠛⠃⠸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁




⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠉⠁⠐⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠁⠢⢬⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡔⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡜⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣈⣕⣪⢅⡀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢰⢱⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢁⡴⠉⠀⠀⠈⢷⠈⣷⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡎⠀⠀⠀⠀⢀⠀⠴⢚⣩⡤⢀⡞⠀⠀⠀⠀⠀⠘⡇⢸⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡤⠞⠓⠢⠀⠀⠈⠀⠀⠀⢀⡴⠒⠺⣄⠀⠀⠀⠀⠀⠀⠇⠨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⡀⠙⠍⠁⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠈⡆⠀⠀⠀⠀⡸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠒⠀⠀⠀⠀⠐⡀⠀⠁⡁⠀⡀⠀⡀⡕⣮⡇⢀⡔⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⡀⠁⠁⠀⠒⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡘⣄⡀⣀⠈⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠀⢀⠖⠣⡈⢳⠠⠼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠘⡆⠓⠋⠇⢄⠀⡨⠉⠒⠤⢄⣀⣀⣠⠤⠖⠋⢀⣀⣀⣀⡀
⠀⠒⠒⠒⠒⠂⠀⠀⠐⠒⠒⠂⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠛⠒⠒⠶⠶⠚⠒⠒⠒⠒⠒⠚⠃⠛⠒⠒⠒⠃⠀⠡⠈⠌⡳⠒⠖⠀⠀⠐⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⢀⣀⣃⠀⢀⡴⠠⠈⠉⡐⠀⠀⠀⠠⠀⢀⠀⠀⢠⢀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢟⣛⣛⠀⠹⣇⢀⣾⣿⡄⠀⠸⢯⣿⡄⡘⠀⠀⢹⣩⣃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠛⠐⠛⠛⠋⡀⠈⠉⠉⠛⠈⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠭⠤⠽⠀⠀⠣⠶⠶⠶⠷⠀⠀⠀⠘⠒⠒⠒⠋




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠐⠀⠂⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠰⠋⡀⠘⢶⠀⠀⢤⠶⠎⡐⢆⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡠⠖⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⠀⠀⡆⠀⠀⠈⢻⡆⠀⠀⠀⠀⢀⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⢡⠴⠒⠲⢦⡀⠀⣬⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡇⠀⣴⠶⢄⣆⢸⣠⣶⣄⠀⢀⡇⠀⠀⠀⠀⡏⠀⠀⠀⢹⠀⠀⠠⠊⠁⠉⢢⠀⠀⢐⣴⡏⠀⠀⠀⠀⢳⣤⣾⠀⢠⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠱⣀⣹⣦⠞⠉⠉⢧⣼⠟⢀⡞⠀⢀⣠⠴⠚⠁⣠⣾⣦⠸⠦⡴⠁⠀⠀⠀⢀⡇⠀⠀⠈⢳⡀⠀⣼⡿⢶⠀⠀⣣⡬⠿⣤⠁⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢉⠉⢻⢀⡴⠋⠀⠀⠀⢣⣹⣇⡽⠃⠀⠀⠀⣾⠿⡇⡸⠀⠀⠀⠀⢠⠟⠢⢘⠿⠚⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢀⠤⠊⠀⠀⠙⣄⡀⠀⠀⣠⡏⠁⣼⠏⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠄⠿⠤⢿⡁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘⠳⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣴⠒⠋⠁⠀⠀⠀⠀⠀⠘⡈⢉⢉⡜⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠀⠁⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⡄⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠒⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⠀⣀⣀⠽⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠋
⢸⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀
⠸⣤⣤⡤⠶⠚⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠀⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣀⡤⠤⠿⣄⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠴⠒⠋⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠦⣤⣤⡤⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠛⠉⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣠⠤⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣀⣠⡤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠆⠀⠀⠆

⠀⠀⠀⢀⣠⣾⣿⡏⠀⠀⣠⣴⣶⣦⣄⠀⣀⣤⣄⠄⠀⢀⠘⢣⣼⠂⠀⠀⣠⣤⡄⢀⣴⣿⢿⣷⣦⠀⢠⢄⣀
⠀⢀⣠⣤⣌⠻⢩⣴⣤⢸⣿⣿⣿⣷⣿⡆⠃⣩⣥⣤⣤⡘⢷⣍⣷⠄⠀⠀⠛⠿⠃⢸⣿⣿⣿⣿⣿⡇⢀⠙⠛⠢
⠀⣾⣿⣿⣷⣷⠘⠿⠟⠘⢧⣬⣿⣥⡽⠁⣼⣿⣡⣶⣿⣿⡀⣬⡁⣠⣶⣿⣿⣷⣦⡈⠻⣷⣶⡿⠟⠀⣳⢚⡶⡄
⠀⠻⣿⣷⣿⢟⣴⡾⢿⣶⡄⠉⠩⢅⣠⡄⠻⡟⠙⠋⠙⡿⠁⣩⢰⣿⠉⠈⠉⠈⢹⣿⠀⢀⣴⡾⣶⣄⠐⠋⠁⠀⣿
⢀⣤⣶⣭⡀⢾⣦⢀⢀⣰⣿⠀⣧⡞⣯⡀⠀⠉⠛⣛⣋⠀⣿⡿⠘⣿⡀⠉⠀⠁⣸⡿⠀⢸⣧⡀⠀⣿⠀⣠⣶⠛⢢
⣿⡕⠒⢫⣿⠈⠻⣾⣿⠿⠃⣀⣈⣉⠉⠀⢀⣴⣿⠻⠟⣿⣶⡄⠀⠈⠛⠿⠿⠿⠋⡴⠞⡳⢿⡛⢋⣥⡆⢻⣟⣀⣸
⠙⢶⣶⠶⠋⣠⣶⣄⠀⢠⣾⣿⠿⢫⣷⡀⣼⣿⡏⢣⠐⣿⣿⣿⠀⣠⣴⣶⣦⣄⠸⡿⠻⠟⢿⡇⢾⣿⠇⠀⠈⠉
⠀⠀⠀⠀⠀⠛⢿⡿⠃⠸⣿⣑⣤⣿⣿⠃⠸⣿⡿⠇⠶⣿⣿⡟⢰⣿⢿⣿⡿⣿⡆⢈⣛⡛⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢛⡟⠁⣾⠆⠈⠛⠶⠶⢟⠋⢀⠈⠳⣼⣿⣧⠟⠀⣿⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡧⢠⣶⣿⣿⣶⡄⠠⢸⡚⡶⠄⢠⣶⣄⠀⣤⡆⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠁⠀⠀⠈⣿⠀⠐⠛⠈⣀⣘⠛⠋⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠘⠂⠀⣾⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣷⣶⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣇⣉⣙⣋⣉⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡒⠋⠙⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠦⠴⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣧⣴⣦⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠠⢤⠂⠀⠀⠀⠀⠀⢄
⠀⠀⠠⠊⠁⡇⠀⠀⠠⡊⡈⡉⠉⡄⠀⠀⠀⠫⢤⠍⢩⠃⠀⠀⠀⠠⠋⡄⠃⠀⠀⠀⠀⡇⠩⠽⠨
⠀⠀⠒⠃⠀⡇⠀⠀⠀⠈⢀⠆⢠⠃⠀⠀⠀⠐⠅⡀⠑⠄⠀⠀⡰⠁⠦⠇⠠⡄⠀⠀⠀⠧⢌⠁⢢
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠔⠁⠐⠓⢲⠀⠀⠀⡴⠐⠊⢀⡔⠀⠀⠓⠒⠒⠂⠐⠁⠀⠀⢀⠂⠐⠀⡰
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉⠁⠉



⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡰⠊⡉⠩⠃⠀⠀⢈⠉⠉⠉⠈⡇⠀⠀⠠⢚⠛⠛⡄⠀⠀⢀⠎⠁⢉⠐⡄⠀⠀⢀⠖⡉⢝⠓⡄
⠀⢸⠀⢈⢉⠐⡄⠀⠀⠀⢠⠋⢠⠎⠁⠀⠀⡰⠈⠀⠠⡃⠀⠀⠸⡀⠃⠐⠀⠹⠀⠀⡇⠘⠀⠀⠃⢸
⠀⢸⡀⠣⡠⠀⡀⠀⠀⢠⠃⢰⠃⠀⠀⠀⢰⠁⢨⣉⠆⢸⠀⠀⠀⡈⠽⠃⢠⠃⠀⠀⢇⠘⠦⠴⠃⡸
⠀⠀⠑⠂⠔⠚⠀⠀⠀⠘⠒⠊⠀⠀⠀⠀⠀⠑⠢⠤⠐⠁⠀⠀⠀⠂⠒⠊⠁⠀⠀⠀⠈⠑⠢⠔⠚⠁

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠛⠒⢶⠖⠀⣠⡶⠇⠀⣀⠀⠀⠀⢀⡀⣠⡟⠑⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⢟⠀⠀⡰⠀⠀⠀⢀⢎⠝⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠾⡊
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣸⠀⠀⠀⡀⣖⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠳⠀⠀⢈⡄⠀⠐
⠀⠀⠀⠐⢄⢀⣀⠀⣀⣄⠉⠀⠀⠈⠚⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⡸⠐⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠈⠉⠉⠉⠀⠀⠠⡄⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠦⣤⣧⡇⠀⠀⠀⢀⣀⡀⠀⠹⡆⠀⠀⠙⠒⠲⠴⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠤⠆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣦⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⠀⣯⠀⠀⠀⠚⠚⠙⠆⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⡤⢀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠋⠁⠀⢤⡀
⠀⠀⠀⠀⠀⠠⠤⣤⣤⡄⠀⠀⣤⣼⡇⠀⠀⠘⠀⠀⠀⠀⣀⣤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡀⠀⠰⢤⣄⠶⠦⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠛⠛⠛⡻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠉⠁⠀⠀⢄⠀⠀⠀⠀⠀⡶⠶⠤⡀⠈⣆⠀⠀⠀⠀⠀⠀⣐⡆⡀⠀⠐⣆⠀⠀⠀⠈⢦⣂⣀⣀⣠⣼⠋
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⠟⠒⠀⠀⡆⠀⠀⡀⠀⠑⠤⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⡠⡏⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢀⣀⠀⠀⡇⠀⠀⠘⠲⠤⣤⣤⠦⠔⠁⠀⠀⠀⠀⠀⠀⢀⢀⡀⠈⠫⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠉⠉⠉⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⡕⠉⠉⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣁⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠉⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠴⠔⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⢀
⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⣩⠿⠥⣄⣠⣤⡄⢀⡎⠁⠀⠀⠙⡆⠀⠀⠀⠀⡀⠀⡀⠀⠁⠈⠀⠀⠀⢀⠀⠈⠢⠱⡀⠀⠈⠲⡀⠀⠡
⠀⠀⢺⡻⣦⠀⠀⠀⠀⡞⠁⠀⠘⠀⢹⡙⠁⠘⣇⠀⠀⠀⢀⠇⠀⠀⠀⣢⣴⣦⣤⡴⠁⠀⢄⠀⠀⠀⠡⠀⠀⠑⡈⢆⠀⠀⡇⠇⠀⠡⡀
⠀⠀⢸⡇⠘⢷⡆⠀⠀⣇⠈⠈⠁⠀⣸⠃⠀⠀⠈⠓⠒⠒⠙⢷⣦⡀⠀⠹⣿⣿⡿⣵⣶⣶⣦⡺⠁⠰⠀⠀⠀⠀⠈⢄⠢⠔⢀⠀⠀⠀⠑
⠀⠀⠈⣧⠀⠀⠁⡀⠀⠉⠑⠶⣴⠚⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢿⡆⠀⠈⠉⠍⠛⠛⠛⢻⣇⠀⠁⠀⠀⠈⢀⠀⠈⢢⡀⠸⢀⠠⠐⠈
⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡿⣃⠀⠀⠀⠀⢠⣄⣀⣀⡀⠀⠀⠀⠘⡿⡃⠀⠀⠀⠀⠀⠠⡀⠀⠑⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⠈⢠⣶⣿⣿⣿⣷⠀⢀⣴⣿⣟⣻⣿⡿⠃⠀⠀⠈⠈
⠀⢰⢲⣤⣄⣀⡀⠀⠈⣧⣀⣨⣇⣀⡀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠘⢛⣿⣿⣿⣿⣷⡀⢰⠀⢀⡀⠀⠘⢃⡀⠀⠀⠀⢀⠤⠀⠒⠂⠄⡀
⠀⠀⠙⢿⣿⡿⠯⠃⠀⠀⠉⣹⡟⠉⠙⣷⠀⠀⠈⠙⠛⠉⠀⠀⠀⠸⠿⠟⠛⠛⠋⠁⠘⠁⠙⠋⠀⠀⠈⠁⠀⠀⠠⠃⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠘⣧⣄⣤⠿⠁⠀⡠⠄⠘⠩⢗⢦⡀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⢾⣿⡄⠀⢰⡶⠛⢧⡀⠀⠀⠉⠈⠀⢀⣾⣿⣶⣦⡀⠀⠔⡱⡀⠠⡄⠠⢠⡃⢸⠠⠀⢠⠀⠀⠀⠀⠐⠡⠄⠐⢄⠀⠀⠐⠀⠂⡰⠃
⠀⠀⠀⠀⠀⠀⠸⠷⣴⡞⣡⡄⠀⠀⠀⠀⢀⠸⣿⣿⣿⠀⣤⣇⡀⡇⠀⢈⡢⢴⣿⣟⡦⡔⠁⠀⠀⡴⠀⠀⡀⠀⠀⠀⠉⠐⠂⠒⠈
⠀⢰⣶⣄⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠢⡈⠻⢧⠀⢠⢟⡝⠁⠀⣊⠡⠸⠛⢻⢤⣌⣢⠀⠀⡁⠀⠰⠁⠀⠀⠀⠀⣰⣦⣄⡀
⠀⠀⠻⡿⠂⠀⠀⠐⢶⣿⣿⠁⠀⠀⠀⠀⡀⠀⠈⠵⠴⢤⠗⠋⠀⠀⠀⠀⠀⠀⢆⠌⠀⠀⠀⠀⢀⠀⠀⣀⠤⠀⠀⠀⠀⠈⠙⠻⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾⠶⠆⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠂⠁⠀⠺⢷⣦⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡤⣤⡆⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠠⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠓⠋⠀⣟⣹⠄⠠⠀⠀⠀⠀⠈⠁⢠⣿⣿⠛⣇⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⡆
⠀⠀⡜⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠜⠁⠀⠀⠀⠀⠘⢿⠿⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⣆⠀⠀⢴⠆⠈⠃⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⢧⣤⣤⣤⣤⣠⣤⣤⣤⣤⡇
⠀⠚⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢶⣶⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠟⠀⢀⠀⣭⣀⢀⠸⣿⣦⠐⠛⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡟⢺⣿⣿⠀⠀⠙⠋⠇⠀⠀⣀⠾⠋⠀⠀⠀⠺⠇⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⢀⡴⠋⣀⠜⣁
⠀⠀⠀⠀⠤⠒⠀⠀⣤⡀⢸⡢⠀⠀⢣⣰⠁⠀⢱⣶⣶⣀⠄⠀⠼⠏⠀⠀⠈⠀⠀⣘⠟⠀⠀⠐⢿⣿⡗⠀⠀⠒⠀⠻⣤⢞⣡⣾⣿⣷⡄
⠀⠀⠀⠀⠀⣄⣆⠀⠁⠀⠀⠀⡈⠃⠀⠙⠷⠀⠘⠻⠏⠂⠀⠀⠀⠀⠀⠀⠶⠠⡦⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠛⠁⠀⠀⠀⠉⠀⠀⠀⠀⠂⢨⡏⠀⣴⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⡿⠈⢁⡼⠉⣡⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠲⠀⡸⠃⢠⠞⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⢨⠖⠀⠀⠀⠀⠈⠁⢆⠀⢀
⠀⠀⠀⠀⠀⠐⢆⣀⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠺⢆⠛⡀⠀⠁⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠠⢠⠄⠈⢀⠀⠀⠀⣶⣂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⠘⠀⠀⠈⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠒⢤
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠁⢤⣀⢀⡀⠀⠀⠀⠀⣠⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣷⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠉⠉⠁⢶⡟⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠴⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠈⠀⠀⠀⠀⠈⠒⠲⠄⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠛⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠠⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠠⠄⣠⢄⠀⠄⣀⠴⠖⠒⠣⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡈⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠉⠉
⠀⠀⠀⠀⠀⣶⣡⣼⣤⣧⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⢀⣀
⠀⠀⠀⠀⠀⠁⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⢀⠀⠁⠀⠀⠀⠀⠀⣷⣾⣿⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⢻⢈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡄⠐⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⣤⠀⡄⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠀⠀⠁
⠀⣀⠀⠀⠀⠀⠑⠀⡀⢂⡀⠀⢀⠂⣀⣄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⢀⠀⡀⢀⣠⠤
⠘⢋⡙⠉⠉⢀⠀⠘⠻⠟⡹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠁⠀⠀⠁⠂⠀⣀⠀⠀⠀⠀⠀⠐⠚⠀⠖⠈⠁⠀⠀⠀⠀⠀⠙⠰
⠀⠼⢃⠠⠁⠀⠀⣠⠁⠈⠃⠀⠀⠄⠀⠒⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⡀⠀⢈⠀⣴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀⠀⠀⠀⣄⡀⠀⡀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠛⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⠀⠘⠋⠀⠀⠀⠈⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢠⠀⣴⡆
⡆⠀⠀⠀⠀⡷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠤⢈⣡⠔⠘⠋⠀⠉⠓⠤⠴⣀⢄⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠞⠉⠁⢠⣶⡄⠀⠀⠀⢠⡄⠈⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢀
⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠚⠀⠀⠀⠠⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⢢⣄⢠⠀⠀⡄⠶⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠢
⣤⣤⣤⣤⣤⣤⣤⣤⣥⣬⣷⣶⣶⣦⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠰⣄⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠠⠀⣦⣠⠔⠀⡀⠀⠘⠆⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠀⠼⠀⠀⠤⠀⠸⠛⢀⡀⠀⠁⠀⠀⠔⣈⠁⠀⠀⠀⠠⢖⣀⣰⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⡀⠀⠉⠈⢱⣌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠒⠰⢿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⣛⣓⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡄⣸⣷⢸⣇⠷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⣬⠍⠀⠀⠈⠀⠀⠀⠆⠒⠋⠀⠂⠰⡤⠴⢚⡁⢰⠀⠩⢾⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠐⣏⠀⠀⠀⠓⣚⠁⠙⠂⢑⣂⠀⢁⠀⢤⠴⢖⣼⣌⡐⠆⢠⣿⡿⠿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠁⠠⣿⡈⢉⡳⢬⢟⣻⢿⡿⢿⡿⣶⢶⡿⢿⣿⣿⣶⠀⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣨⣿⠖⠙⣿⣷⠟⠷⠾⣰⠄⠸⣿⣿⠛⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡧⠒⠒⣧⡇⢤⣌⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⢳⢶⡦⠁⣀⡌⠙⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡗⡎⢈⣷⠘⡄⣠⣿⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢢⣿⣿⢹⠀⣅⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠻⠃⠺⠂⠀⠀⠉⠉⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⢀⠀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠘⣠⣤⣀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠤⠤⠴⠤⠴⠾⠿⠧⠴⠌⠛⠓⠲⠷⠾⣿⣟⡡⠞⠒⢚⣾⣿⣿⣶⣆⣀⣛⣛⣺⣷⣶⣶⣶⣶⣤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿