Thought Bubble Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠄⢀⣀⠀⠄⠀⠢⠤⠄⠤⠄⠐⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠈⠉⠈⠉⠲⢄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⡀⠀⡁⠂⠀⠑⡄
⢀⡂⠀⠀⠀⠘⠟⠀⠏⠀⠀⠀⢱
⢠⠑⡄⠢⠄⣀⠀⠀⠀⡀⠀⠀⠘
⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⡌⠑⠢⢄⣀⢀⠀⡄⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠢⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⢄
⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠁
⣶⣶⡶⡶⠶⢶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠀⠐⠂⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠁⣤⠀⠀⡀⠔⠀⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠀⡄⠄⠐⠑⠀⡃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⡘⠀⢈⡄⠀⠀⠀⠀⠀⠸⣁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠄⠊⠀⠀⠠⠈⡀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠂⠐⠒⠠⠂⠀⠈
⠀⠀⠀⠳⠀⠀⠀⠀⢀⠎⠈⠀⠀⠉⠘⡄⠀⠠⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠎⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⣀⠄⠄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡀⠤⣠⠊⠁⠀⠉⠦⠠⠤⡀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⢠⠊⠁⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢄⠀⡀⠀⠰⠉⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⠢⡀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⢲⢀⡔⢦⢠⡖⡄⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⢄⣀⡀⡀⡀⢀⠝
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⠈⠣⠞⠘⠋⠐⠆⠀⠀⠀⠘⠀⠀⢀⠀⠄⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣀⢀⡀⡸⠀⠀⠀⠀⠀⠀⢠⠛⠀⠠⡆⠰⠗⠰⢒⠀⣂⠀⣖⠀⠠⠀⠆⠀⡜⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠤⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⡀⠀⠀⠀⠀⠀⠔⠀⠀⠘⠀⠁
⠀⠀⠀⠀⢠⣀⡀⠀⠈⠀⠀⠀⠀⠀⠈⣄⠀⠑⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠒⠘⠁⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠈⠒⠀⠒⢄⣀⣀⠔⠓⠂⠂⠀⠀⠀⠀⡰⠤⠤⡀⠀⠀⠈⠂⠀⠉⠀⠉⠀⠩
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡤⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠀⢀⠂⠈⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄⡀⠀⠀⢠⠋⠀⠀⠀⠀⠉⠠⠀⠀⠀⠐⠂⠁⠀⠠⠄⠄⡀⠀⠀⠀⣰⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠞⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠀⠱⡀⠀⠀⠀⠀⢀⡪⠀⢀⠄⠐⠐⠀⠂⠠⡀⠀⠀⠀⠉
⠀⠀⠂⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠉⠓⠒⠢⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠑⠂⠅⠉⠀⠀⠀⣴⠀⠀⠀⠀⠈⠢⠀⠀⣀⠄⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠁⢀⠀⠀⠀⠀⡀⠈⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⣒⠄⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⣛⣟⣙⣛⣟⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠉⠩⠙⢋⡀⡈⠉⠉⠩⠰⡒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢄⣠⠀⣠⠖⠋⠉⠀⠀⠀⠀⠉⠙⠲⢌⠀⠣⡀⠉⢙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡤⠊⢀⠞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠱⡀⠈⡗⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃⢁⠎⠀⠀⠀⣠⣴⠒⡶⣋⠶⢆⠀⠀⠀⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠖⠂⢸⠀⠀⠀⠘⡄⠓⠀⠓⠉⢁⠎⠀⢀⠀⠂⢈⠀⢀⠕⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣔⠁⡀⢸⠀⠀⠀⡀⢸⡀⠀⠀⠀⡎⠀⠀⠀⠀⠁⡸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⡄⠈⢆⠀⠀⠠⢀⡇⠀⠀⢸⠀⠀⠀⢠⠐⢠⠃⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠴⠎⣀⠀⠈⢦⡀⠀⠀⠛⠀⠀⡃⡀⡄⡆⠀⡰⠃⠀⠀⡔⠋⠁⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠤⠤⢄⠀⠸⠀⠀⠀⠀⠀⠀⠀⢱⣠⡄⡉⠢⢄⣸⡀⢑⣇⣠⠤⠔⠋⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠑⠀⢇⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠰⣎⡀⢘⡽⢟⡆⠈⠑⢦⡜⠁⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠀⠠⠔⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣉⣉⣤⡴⢞⡧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡘
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⢠⣀⣴⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡌⠀⡜
⠀⠀⠀⠀⠀⠰⡁⠀⢀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠀⠀⠀⠀⠈⠻⣙⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠘⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠰⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡔⠉⠉⠲⡀⠀⠀⠀⢻⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢥⣌⣊⠔⠰⠀⠆⡔⡠⢠⢀⠎⡞⠠⠃⢋⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠄⠊⠀⠀⠀⠀⢸⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⠤⠤⠤⠤⠤⠔⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣱⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢣⡸⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⢈⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡴⠆⣨⢔⠵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣷⠆⢐⠵⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠕⡪⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣩⣂⠅⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣩⣿⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢘⣻⢹⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢕⡏⢸⠀⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡠⢀⠴⣂⢔⡢⢊⠤⣂⠔⡪⠔⡡⢐⡠⢂⠴⣂⠔⣢⢔⡠⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠥⢻⡰⣛⢕⡾⢳⣄⣿⢔⡠⢂⠄⡢⡤⠔⠔⠠⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠶⠀⠠⠄⠠⣤⠀⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠴⠀⠠⡄⠀⣤⠀⢠⡄⠀⣈⠀⠀⠁⠀⠘⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠶⠀⠰⠆⠀⣤⠀⣠⡄⢀⣀⠀⣈⠁⠀⠛⠀⠘⠃⠀⠖⠀⠰⠄⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠠⡄⠀⣤⠀⢠⣄⣀⣈⣁⣈⡋⠀⠛⠃⠘⠟⠀⠶⠆⢰⡦⠀⣤⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠰⠆⠀⣤⠀⣠⡄⢀⣤⠖⠉⠀⠀⠀⠀⠀⠉⠛⢿⡀⢾⡦⢠⣦⠀⣤⡄⢀⣀⠀⢈⠁⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⣤⠀⢀⡄⠀⣈⠀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⠖⠁⠀⠀⠀⠀⠘⣏⠀⠛⠁⠘⠇⠀⠶⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠤⠀⢠⡀⢀⣀⠀⢈⡁⠀⣻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡶⠿⠀⢶⠆⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⡩⠊⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠀⠀⠄⣠⡀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠈⠁⠀⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠖⠀⠠
⠀⠀⠀⠀⠀⠛⠀⠰⠆⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠄⢠⡄⠀⣀
⠀⠀⠀⠀⠰⠦⠀⣤⠄⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⠀⢀⡀⠀⡀
⠀⠀⠀⠀⢠⡄⠀⣀⠀⣈⡉⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⢀⡁⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠀⠁⠀⠙⠁⠘⠟⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢉⣉⠀⢈⠁⠀⠛⠀⠘⠂
⠀⠀⠀⠀⠉⠀⠘⠃⣴⠟⠋⠙⠣⢰⣶⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠛⠀⠛⠃⠐⠗⠀⠰
⠀⠀⠀⠀⠀⠀⠺⢸⣏⠀⠀⠀⠀⣤⣤⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠇⠐⠷⠀⢰⠄⠀⡀
⠀⠀⠀⠀⠀⠀⣦⠀⢻⣦⣤⣤⠖⣉⡁⢈⡛⠻⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⠆⢰⡆⠀⣤
⠀⠀⠀⠀⠀⣾⠁⢀⣄⠀⢀⡁⠀⠙⠁⠘⠛⠐⠿⠛⢿⡿⢿⣿⡿⣿⣿⢩⣭⡁⣙⡻⣿⣶⣤⣤⣤⣤⣴⣶⣿⡟⢠⣦⠀⣠⡀⢀⡀
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠘⠃⠘⠛⠀⠾⠇⢰⣶⠀⣴⡄⣠⣭⢀⣭⡁⢈⣛⠀⠛⠃⠸⠿⠛⢿⠟⢻⣿⠁⣤⠀⢠⡄⠀⣈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠀⠰⠆⠀⣶⠀⣠⡄⢀⣭⠀⣈⡉⠈⡛⠁⠻⠟⠘⠿⠃⢾⡷⢠⣶⠀⣤⡄⢀⣌⠀⢈⠁⠀⠉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣄⠀⣈⠁⢈⡋⠀⠛⠃⠘⠿⠀⠿⠇⢰⡶⠀⣴⡄⢠⣄⠀⣉⠁⠈⠉⠀⠛⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠘⠋⠐⠿⠂⠲⠖⠠⣶⠀⣤⡄⢀⣌⠀⢈⡁⠀⠉⠀⠘⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠰⡶⠀⣤⡄⢠⣄⠀⣈⠀⠀⠉⠀⠙⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠋⠉⠉⠛⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢚⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡸⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⢣⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣔⢟⢯⣎⡿⠾⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⢀⡾⣝⢮⣮⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣜⣓⠦⢝⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⣯⠬⠭⠭⣭⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠸⣍⡭⠭⠭⠛⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠈⢙⣶⡯⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠓⠶⢤⡀
⠀⠀⠀⠀⠀⣴⠭⣒⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢰⣛⡭⢖⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣾⢒⡪⠵⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢹⠥⢒⡩⠽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠈⢯⠥⣒⠭⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠲⠦⣄⣀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠈⠫⡖⣋⠥⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⢋⣤⠶⠶⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⡥⣚⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠮⣼⣛⡶⣦⣤⣀⣀⣀⣀⣠⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⡤⠤⠊⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⡟⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣟⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠏⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣴⣶⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣾⣿⡿⢉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣸⣿⣿⠁⡈⠀⠏⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣤⡀⠀⠀⠀⠸⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⡆⢰⡁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡇⢠⣼⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣦⠠
⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⣷⣤⣀
⡌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠺⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠘⣿⣿⣿⣿⣿⣿⣍⣳⣦⣄⡀
⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣾⠿⠛⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠙⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⡳⡀
⠀⠀⠙⢯⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⡀⣀⣤⣴⣿⣿⣿⠏⠀⣰⣮⡟⢋⠝⠫⠟⠉⠛⣋⠀⢂
⠀⠀⠀⣨⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣍⠋⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⠷⠏⠀⠀⢀⡰⣟⠁⠚⢋
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠋⠕⠛⠻⢿⣿⣿⢻⣿⣿⣿⢻⠇⢀⣠⠴⣂⡛⠰⠀⠀⠰⣿
⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⢻⡿⢿⠛⠋⠞⢰⣿⠃⣾⣿⠿⢿⣷⣦⡀⣽
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⢻⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣀⣼⣇⡼⠁⠿⠁⠂⠀⠀⠈⢻⣿⠈
⠀⠀⠀⠀⣌⣉⣉⣥⣤⣶⠒⠻⣿⡛⠿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⡴⠂⠀⠈⡐⠀⠀⠀⣸⡏⡄
⠀⠀⠀⠀⣿⡘⣿⣿⣿⣿⣷⡀⠹⣿⠃⠀⢙⡄⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⠿⠋⣀⣀⣀⣀⣴⣤⣶⣶⣿⣾⣷
⠀⠀⠀⠀⣿⡿⣾⣥⣌⠉⠛⢿⡆⢭⣄⠺⣿⣿⡄⠀⠀⠙⣧⣶⣎⢻⣶⣦⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢃⠔⠋⡍⠉⠛⠋⠉⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣧⣴⡤⠀⠀⠉⠁⠛⠓⠋⠓⠋⠉⠁⠀⠀⠀⠈⠻⠟⢸⣿⣿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⢣⡴⣣⣤⣤⡄⠤⣀⡀⠻⣯⡙⣿⣿
⠀⠀⠀⢰⣿⣿⣯⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⣤⢸⣿⣿⣿⣿⣿⣄⠙⢿⣦⡉⢻⠟⠁⠰⡾⣟⣷⣾⣷⣌⢻
⠀⠀⠀⢸⡿⠛⠛⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢿⣧⠙⣿⣸⣿⣻⣿⣿⢿⣿⣇⡾⣿⣿⣆⡀⢀⡀⢈⢯⣼⡏⢿⣿⣿
⠀⠀⣰⣾⢿⡟⣫⠄⢀⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⡧⢈⣿⡏⡽⣿⣤⣈⣿⡿⠀⢸⣿⣿⣷⡈⣷⢸⣷⡽⣇⢸⣿⣿
⠀⢀⣿⣷⣟⡽⠧⢀⡌⢀⣀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠⣟⢻⡟⠰⡄⣿⣿⣿⣿⠃⠀⠀⠻⣿⣿⣷⡀⠘⡇⢻⣽⣎⣿⣿
⠀⢸⣿⣿⣿⠂⣠⣾⣷⣿⡿⠛⠁⠀⣠⣴⣶⣾⣿⣿⢿⣇⣀⠠⣶⣶⣿⣶⣤⣼⣟⣫⣩⣿⣿⣿⣿⡟⠀⣠⣤⣄⣉⣿⣿⣇⣠⣿⡀⢘⣿⣿⡏
⠀⠀⢿⣿⣯⡄⢹⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣉⣁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠃
⠀⠀⠈⢿⣿⣿⡜⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠻⣿⣷⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢛⣿⣿⣿⠿⢻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣖⠒⠲⠤⣄⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⣙⢦⡀⠀⠙⠪⣧⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡉⠛⠂⠀⠀⠈⠃⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣂⡀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠎⠀⠀⠀⠀⡀⠀⠀⡀⠀⠙⠦⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢦⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠣⡀⠀⢠⡠⣄⡠⠴⠃⠀⢠⠖⠉⠀⢀⠲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠆⠀⠀⠀⠀⠈⠒⢢⠀⠀⣘⡚⢣⠉⠋⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠄⠒⠂⢊⠔⠤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣨⡘⡻⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠒⠀⠀⠂⠬⣐⠒⢒⣛⢆⢳⡎⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢉⠆⠐⠎⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣯⠋⠉⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⡏⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣰⡟⡹⢽⡲⢒⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠒⠊⠉

⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣤⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣿⣿⣿⣿⣿⢟⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡶⢿⣿⣿⣿⣏⢸⡄⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠀⠈⠹⣿⣿⣿⠸⠁⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⢻⡄⢰⠀⢸⣤⢀⢸⣻⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣧⣀⠸⠟⣿⣿⡇⠀⡟⣿⣿⣿⣿⣿⣿⠏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠁⠹⣿⣿⣿⣿⡿⢠⡄⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣏⠀⡃⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⡿⠿⠇⢀⡁⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠏⠀⠀⠀⠀⡇⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠂⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠨⠭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠂⢀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠿⣿⣛⡛⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢯⣉⠙⠛⠳⠦⣤⡀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠳⣦⡀⠀⠀⠙⠳⣜⢦⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⡓⠶⢽⣦⡀⠀⠀⠈⠙⢧⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠥⠆⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠖⠋⠙⠲⢄⠀⠀⠀⠤⠄⠈⠙⠦⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣰⠁⠀⠀⠀⠀⠈⢳⡤⠚⠑⠲⡀⠀⠀⠈⠑⢦⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⣄⠀⠀⠀⢀⡀⡇⡀⠀⠀⣰⡃⠀⠀⠀⣠⠴⠒⠈⠁⢠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢈⡒⠤⣤⡼⠟⠛⣿⠞⠉⠁⠀⠀⠀⣜⣀⡤⠤⣄⣸⢦⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⣄⠀⠠⢀⠂⠀⠀⡔⠊⠙⢢⠀⠀⢀⣈⣉⠙⢧⠀⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠀⠀⣠⠖⢤⠤⠧⠦⠴⢃⡴⢄⡀⠉⠈⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠉⠑⠋⣠⠋⠀⠀⠀⣠⠏⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⠑⡄⠀⠀⠀⠀⠀⢹⣀⣀⠀⣧⡀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠙⠁⣄⠈⠀⠘⠲⣄⠀⠀⠀⠀⠀⠈⡆⢣⠘⣆⠀⡼⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠦⣤⢤⡤⣄⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⠘⡴⡍⠋⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣃⣸⡀⣘⡂⠉⣉⠑⠢⢄⣀⠀⢀⣘⣯⠇⢱⢄⡴⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢬⣉⣁⠠⣴⣸⡈⠿⢆⠀⡐⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡷⠐⠚⠷⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢸⠙⠉⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣸⠗⠀⠀⠀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢺⣤⣤⣧⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠰⢭⡃⡴⠧⠭⡿⣉⣉⡩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠥⠭⠭⠥⠤⠕⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠂⠠⠀⡀⠀⠀⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⣘⡿⠃⠀⠠⠀⠉⣀⢀⣤⣄⠠
⠀⠀⠀⠀⠀⠀⠄⢰⠾⣷⠀⠀⠀⠠⠀⠀⣸⡋⡿⢀⣿⠂⠂⠸⣉⠛⠁⡧⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠀⠈⠂⢌⣉⠁⠿⡿⠀⠘⠄⠈⠛⠞⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠢⠄⡀⠀⠽⠄⢀⣦⣄⡀⣠⠺⠏⠀⠀⠀⠂⠀⠘⡀⡀⢤⣤⠂⠈⠀⠀⠀⡀⠤⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠀⠉⢱⡶⠛⠁⠀⠀⠢⡀⠀⠀⠀⠀⠈⠩⠀⠀⠰⠀⠀⠀⠰⠆⠉⣆⡡
⠀⠠⠄⠀⠀⠀⠂⠢⠀⠀⠬⠀⠀⠀⠈⠦⡂⠤⡀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢘⡠⠴⡏⠉⠓⡢⠤⢀⣀
⠀⠀⠀⢸⣶⣧⣤⡄⣤⢦⣘⠶⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠜⡴⠷⡀⠀⢀⠀⠰⠿⡀⠀⠀⣀⠀⠐⢀⣄⢡⠀⢠⠁
⠀⠀⠀⠘⢿⣿⣿⠀⠋⠁⠀⠀⢂⣣⢀⡀⣠⠊⡀⠀⠀⠀⠀⡀⠇⠀⠀⠐⢠⣦⡀⠀⠉⠉⠀⡌⠀⠀⣦⡴⠊⠀⠄
⠀⠀⠰⠀⠀⠀⠀⠒⠁⠀⠀⡀⢈⡴⠆⢀⠁⣜⡛⡽⣩⣤⣭⡝⢎⠭⢜⡶⡀⢰⢡⣀⠀⠀⠐⠄⣀⡀⠛⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠇⢈⠁⠊⠘⡀⢏⣱⣌⡁⣴⣃⠾⠷⠴⢋⠙⢀⠏⡘⠉⠀⡂⠄⠀⡸⠋⠀⠀⠀⠁
⠀⠀⡀⠔⠈⠉⢀⢀⡄⡈⠉⡐⢶⡶⣦⠲⣴⠦⣭⣥⠃⠈⠀⡿⣖⣘⣿⣶⡵⢾⢃⠒⠀⠈⠉⠁⢠⠀⠐⡒⠂⠀⠉⠉⠂
⢔⠁⠀⡀⠠⡀⠀⢻⠁⠀⠀⠈⠐⠍⣚⣀⢃⠴⠂⣡⢃⠤⣐⣡⡌⢒⠉⠙⢟⢉⣬⢊⣤⠀⢀⣠⣾⠀⠶⠀⠇⣠⠂⣠⠄
⠀⢂⠈⠀⠀⠀⠀⠀⠣⠀⠀⢀⡀⢀⣾⣻⡗⠻⡷⣡⣾⣷⣬⠻⣷⣾⠳⣿⣽⢀⣿⣯⡉⢒⠀⠀⠀⡀⠀⠀⢮⠐⠓
⠀⠀⢀⠀⡀⠄⠂⠁⠀⠀⠀⡈⠉⠒⠩⠙⡛⠻⠶⠧⢿⣥⣾⣴⣿⣼⡧⠿⠧⠾⠛⠋⠀⡈⠀⠀⠀⢇⠀⠰⠾⠤⠄⠀⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢠⠆⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢕⡄⠃⡷⠊⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠁⠀⠀⡠⠦⣀⣬⢵⢄⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣠⣄⠈⠀⣧⣚⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢰⠋⠀⠀⠀⠑⠌⡉⠑⡀⠔⠁⢀⠀⠀⡄⠘⠡⡉⢛⡛⢈⠝⠀⠀⡇⠘⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠜⣇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠺⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠃⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⢤⠶⠞⠥⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠂⣁⣢⠔⠂⠒⢄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠯⢥⣤⠤⣠⣀⣉⣁⣚⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⠢⡈⠓⠒⠒⠚⠚⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠈⠐⠤⠤⡤⠒⠀⠀⠀⠀⠀⢂⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠴⠑⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠳⢄⣀⠀⠀⠀⠀⡠⠠⢀⡾⠁⠄⠐⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠒⠒⠒⠉⠀⠀⠀⢀⢜⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢
⠀⠀⠀⠀⡀⢠⡄⢀⠄⡠⢥⣤⣠⢄⣤⣤⣤⡤⣠⢤⡤⣤⣤⣤⡤⣤⢠⣤⡤⣤⣤⣤⡤⡤⣤⣷⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠗⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣤⣤⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⣿⣿⣿⠿⠛⠋⠉⠙⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠘⢿⣿⣦⣄⣀⣀⣀⣤⣶⣿⡟⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣶⣶⣿⣿⠿⠿⠿⠿⠿⠟⠛⠁⠰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢱⡼⣿⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣾⣿⢟⣡⠟⠁⠀⠈⠛⢳⣤⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠴⠳⠞⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⡴⠟⡏⢈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡾⠋⢀⣄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣠⣾⡿⠛⠻⣦⡀⠀⠀⠀⢀⡴⠦⣦⠞⠉⠀⣠⠀⠈⠃⠈⠉⢰⣷⠀⠀⠀⠀⠀⠀⣀⡜⠋⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠈⠻⣷⡀⠒⠠⡀⢾⣡⠤⢀⠀⠁⠈⣠⣦⠀⠀⣼⣿⡇⠀⠀⠀⣠⣴⡿⠞⠊⠀⠀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⢶⡄⠀⢀⣠⠔⠀⣀⡀⠀⣻⣿⡇⢠⣿⣿⣷⣠⠴⠊⠴⠊⠀⠐⠁⣠⣏⠐⣻⣿⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠎⢀⡴⢀⠄⠀⠀⠀⠀⠀⠀⢠⣌⣿⡷⠊⠀⠀⠜⠉⠀⠀⣼⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠤⠁⣁⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⣴⣿⣤⣀⣀⠐⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⢀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⣡⣤⣾⠟⢁⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣦⣜⣱⣿⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⣫⡄⠀⠀⠀⠘⠄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣷⠀⣠⣦⡀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠛⠿⣿⠟⠉⠉⠛⣷⣄⠀⡀⣴⣿⡿⠋⠀⠀⣴⣧⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⡄⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⠀⠀⠀⠀⠀⠿⠿⣿⣿⡋⣀⡀⢠⣄⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠛⠲⣤⡄⠀⠀⠀⠈⠉⠉⠉⠻⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁⠉⠀⡀⠄⠐⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⡏⣀⠀⠀⠀⠘⠳⣄⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣶⣶⠶⠊⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈⠿⠷⢤⣀⡀⠀⠀⠉⢻⣷⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠀⠋⠁⠀⠀⠀⠀⠀⢀⣴⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠄⠈⠳⢾⣦⣠⣦⠈⡛⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠯⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠉⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠋⠀⠈⠁⠀⠀⠀⢀⣠⣾⣧⣤⡶⠃⠀⠀⢀⠜⠋⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢁⣾⣿⣿⡿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠐⠀⠀⣴⠿⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠦⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣴⣤⣿⣧⣤⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⣾⡏⠀⠈⣿⣶⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢽⢿⣶⡾⣿⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠉⠁⢲⠞⢳⡆⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠘⠳⠞⠃⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⣉⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⠋⠉⠩⠍⠉⣳⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡡⠖⠉⠀⠀⠀⣀⠔⠁⠹⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠞⠀⠀⢀⣠⠴⠊⠁⠠⢄⣀⡱⡄⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡤⠴⠒⠾⠷⠄⠀⠀⠀⠀⠀⠀⠁⠈⠢⣀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡏⠀⠀⢤⡦⢷⣶⣄⠀⠀⠀⣠⣶⡾⣾⡄⠀⠈⠛⣧⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⢃⢤⣴⡇⠀⠀⠀⣇⠈⠋⣸⠀⠀⠀⣇⠙⠁⣸⠀⠀⠀⠀⣷⣤⢄⢱
⠀⠀⠀⠀⠀⠀⠘⡆⠘⢾⡇⠀⠀⠀⠈⠑⠊⢹⠀⠀⠀⠈⠓⠊⠁⠀⠀⠀⠀⣿⠊⠀⣼
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠦⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠠⢤⣄⣀⣉⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡁⠀⠀⠀⠀⠀⠀⠈⠛⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢀⡤⠶⠀⠀⠰⣒⡒⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣀⡀⠘⠉⣰⣀⡀⠉⠉⢀⣿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢞⠝⣩⠧⡄⢈⣉⡀⢀⠔⢁⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠁⠋⠀⢀⣹⠀⠶⠀⡨⠛⠉⠁⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⢣⠀⠀⠀⢸⣯⣴⠶⠾⡇⠀⠀⢀⡞⠳⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⢀⠜⠳⢤⣀⡾⢿⡁⠀⣠⠷⡄⢀⡸⢷⡀⠀⠀⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⡶⠛⢷⣤⡀⣩⡃⢀⡿⢯⡀⢀⣽⣅⣠⠾⠓⠶⣄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣠⡾⠋⠀⠀⠀⠈⢻⠉⢷⡾⢧⣠⡷⠛⠰⡏⠀⠀⠀⠀⠈⢧⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡎⢠⡾⠋⠀⠀⠀⠀⠀⠀⢠⡇⠀⠁⡼⢹⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠳⣄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⡼⣴⠏⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠇⠈⣧⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⢿⣶⣧
⠀⠀⠀⠀⠀⡸⠹⡿⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⠀⠀⣿⡄⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠈⢷⠹⣇
⠀⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠀⢀⡤⠞⢹⡆⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠙⢦⣀⠀⠀⠀⠀⠈⠇⢹
⠀⠒⠒⠒⠚⢧⡀⠀⠀⢀⣠⣖⣉⣀⣀⣬⣧⣤⣤⠴⠧⠤⠤⠾⠷⠤⠤⠤⠷⠤⠤⠽⢷⣤⡀⠀⠀⠀⣸⣧⣤
⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⠀⠀⠀⠀⠀⠀⠛⠀⣸⠁⠀⠀⠀⠀⠀⠘⠁⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀⠀⠳⡄⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⡴⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⡏⠀⢈⣿⠀⠀⠁⠀⠀⠀⣀⣠⣴⠶⠶⠶⠞⣿⢷⣶⣄⠀⢠⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣰⣾⣯⣍⣵⡶⠶⠿⠶⠟⠋⠉⢹⡆⠈⠉⠀⠀⢀⡠⠀⠀⠀⠀⠀⣀⡤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⠀⠀⣴⢏⣿⠏⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢻⡄⠀⠀⠀⠈⠀⠀⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠘⠿⠃⢠⣿⢸⣿⠀⠀⠀⠀⠅⠀⠀⠀⡀⠀⣀⣀⣧⠀⠀⠀⠀⠴⠚⠉
⠀⠰⢦⣄⠀⠀⠈⠳⣌⠀⠀⠀⠀⠀⣿⣤⣼⣧⠀⢀⣤⠶⣶⠶⠀⠀⠙⣯⣭⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠂
⠀⠀⠀⠉⠓⢦⣀⠀⠈⠑⢄⠀⠀⠀⣿⣿⣷⡉⠀⠈⠡⢞⣛⠃⠂⠀⢎⠑⠒⠘⡆⠀⠀⠀⠠⠤⠐⠒⠚⠉⠉⣀⣀⣀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⢸⣛⡛⠋⠀⠀⠀⠈⠀⠀⠠⢂⢈⡷⠀⠀⡇⠀⠀⠀⠀⠠⠤⠤⠜⠟⠛⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⡆⢧⠈⠀⠀⠀⠀⠀⡠⠈⠉⠈⣀⠙⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡄⠀⠀⣄⠰⠃⠒⠉⡭⠵⢆⡁⡇⠀⠐⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣸⡰⣦⡀⠘⠦⠀⢀⡞⣁⠴⠒⢼⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣤⠤⠖⠒⠚⠉⠁⠀⠀⠀⢀⣼⣿⡍⢌⡻⢷⣦⣤⡟⠀⠀⠴⠒⠂⢱⡄⠀⠀⠀⢈⣁⠉⠁⠁⠘⠓⠒⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⣇⠀⠈⠒⢬⣹⡇⠀⠀⠚⠀⠒⠉⣳⠀⠀⠀⠀⠒⠈⠉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⡇⢹⡀⠀⠀⠀⣸⠁⠀⠀⠀⡠⠊⡼⢻⡿⢶⣦⣄⣀
⠀⠀⠀⠀⢀⣠⣤⡶⠟⠋⠉⠛⠛⠛⡛⠛⠋⠁⠀⢻⣄⠀⣼⡿⠀⠀⠀⠀⠀⢰⡇⠈⠀⠀⠀⠉⠉⠛⠿⠶⣤⣀
⠀⠀⢀⣶⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣏⣾⠿⣁⠀⠀⠀⠀⡠⠋⣿⠀⠀⢀⡆⠀⠀⠀⠀⠀⠸⣟⢦
⠀⠀⣼⡟⠈⢳⡄⠀⠀⢠⠀⠀⠀⠀⠻⣦⡀⠀⢀⣠⠞⠁⠀⠈⠑⣄⢀⣾⣷⠀⢻⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿⠘⣧
⠀⢸⡿⠀⠀⠀⢻⡄⠀⢸⠀⠀⠀⠀⠀⠈⣡⣴⣿⠟⠉⠲⡄⠀⠀⠈⣿⣿⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢹⣇
⠀⣿⠃⠀⠀⠀⠈⢷⠀⡾⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠘⢆⢀⡠⢘⣿⣿⣿⣸⡇⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠃⢀⣤⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⢦⣴⣿⣿⣿⣿⣿⡇⠀⠀⢠⡏⠀⠀⣇⠀⠀⠀⠀⠀⠘⣷
⣾⠇⠀⠀⠀⠀⠀⢀⣤⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣃⣼⠼⠟⢻⣿⣿⣿⡇⠀⢀⡟⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⣿⠓
⠀⠀⠀⠀⠀⠀⢀⣀⡴⢛⡓⠈⠙⢷⡀⢠⣤⢄⣀⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣠⣤⣶⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠙⠶⡉⠉⠙⢦⢤⣾⠿⠷⢤⣄⡠⠛⠻⠆⡀
⠀⠀⠈⠉⣃⣙⠃⣴⣿⡿⠂⠀⠀⣠⣶⣾⣧⢦⣀⠀⠀⠶⢤⡄⠠⠔⠛⣿⠁⠀⠀⢀⠉⠒⠛⠘⠢
⠀⠀⠀⠀⠉⠉⠃⠈⠉⠦⣤⣴⣾⣿⡿⠿⢿⣤⣤⡀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⢀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠉⠁⠀⠘⡽⠿⠻⠀⠀⠀⠀⢴⣿⣏⠀⠀⠀⣸⣿⡇⢀⡀⡆
⠀⠀⣿⣷⣦⣤⣀⠀⠀⠀⠀⣠⠀⠀⠁⠄⠊⣀⣶⣦⣶⣄⠀⠀⠀⠈⠿⢶⠂⢀⣿⣿⣿⡃
⠀⠀⠈⠁⠀⣿⣿⣷⣿⣴⣆⣿⣇⣴⡄⠀⠀⢘⠛⠿⣍⡁⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣯
⠀⠀⠰⢴⣷⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢀⣿⣶⡆⠀⠉⠂⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠛⠛⠿⠻⠿⠿⠛⠿⠿⢿⣄⣼⡆⠀⠀⣠⣠⣿⠿⠿⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠉⠉⠉⠲⢄⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⣿⠒⠲⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⢴⡋⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢾⣧⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣤⣤⣿⣤⣤⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣀⣀⠀⠤⠔⠊⠙⠒⠒⠒⠉⠓⠒⠒⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠂⠀⠘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣈⣢⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡉⠀⠀⠀⠀⠀⠉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠛⠃⣠⣤⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠓⠦⠤⠀⠄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⢀⠊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠙⠢⣄⣀⣀⣀⣀⣠⠴⠊⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡘⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠄⠀⠀⠈⠉⠉⠁⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠔⠀⠘⢆⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⢆⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⢆⢡⠀⠀⠀⠀⠀⢀⡀⠄⠂⠁
⡀⠀⠀⢀⣠⠊⡘⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡀⠤⠐⠊⠁
⠈⢁⣀⣉⠀⢠⠁
⣰⣤⣤⣤⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⢀⣴⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠅⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⡄⠀⢿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣻⠀⠀⠀⠀⠈⠇
⠀⠀⠀⣰⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡏
⠀⠀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠴⠖⠉
⠀⢠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢾⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣤⣤⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣻⣿⣿⣿⣿⢀⣼⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣟⣹⣿⣿⣿⣿⣿⣿⡧
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠷⣼⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⢿⣿⣿⣟⡿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣯⣿⣿⣿⣿⣿⠁⠈⣻⣿⣄⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣴⣿⡿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡟⢀⣴⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠸⣿⣿⣿⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣴⠶⠶⣤⡴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣾⣿⣶⣶⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠋⠁⠈⢙⡋⣉⣩⣿⣿⠛⠛⠀⠀⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠔⠒⣋⣉⣉⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⡴⠞⠛⠛⠛⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⣁⣠⠤⠞⠋⣀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠛⢋⣁⣤⣴⡾⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⢋⣩⣷⣶⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣫⣥⣶⣿⣿⠿⠛⠋⣱⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⢻⡟⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⢀⡴⠟⣫⣭⣿⣿⠁⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⠞⠛⠛⢻⡇⠀⠈⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡌⢿⡷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣤⣤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢛⣻⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⡿⣿⣿⡿⡿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⢿⣿⣟⡽⠋⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠛⢶⣤⠀⠀⠸⣯⡤⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⡫⢑⠢⡈⠀⠀⠙⣷⡄⠀⠀⠀⠀⠈⠻⣷⣤⣂⠴⢂⢔⠛⢶⣤⣀⣠⣿⣿⣧⠀⠹⣧⣀⣤⡿⠃⠀⠀⠹⠛⢶⡆
⠀⠀⠀⠀⠀⢀⣼⣿⣿⢿⡙⠪⡑⠌⠪⠁⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⡑⡍⣲⣿⣿⣿⣿⠿⣿⣷⣀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣾⣿⠛⢬⡓⢌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢪⡟⣿⣿⣿⣿⠉⠀⠀⠀⢸⣿⣿⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣾⣟⢧⡙⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣄⠀⠀⠀⠀⠔⣡⣾⡿⣛⣿⣿⠀⠀⠀⠀⢸⣿⣿⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣼⡟⠈⠓⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣀⢀⣼⣿⣿⣋⣥⢶⣿⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣩⣦⣯⣛⠿⣿⣿⣿⣿⣯⣶⣿⡇⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠋⠀⠀⠀⠉⣿⣿⡿⠛⢙⣧⠀⠀⠀⢀⣶⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⣰⠏
⠐⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡤⠖⠋⢿⡀⠀⢀⣾⣿⣿⡿⠿⠿⠿⠿⠶⢦⣤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠈⠧⠠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⣿⠿⠿⠿⡿⡟⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠲⣤⣤⠒⠚⠉⠉⠈⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠛⠦⠤⠤⠤⠤⠤⠮⠽⠯⠥⢤⠭⠽⠤⠤⠤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡴⠟⡋⠈⠉⠉⠈⠃⠒⠶⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠐⣿⡇⠀⢀⠴⠦⠀⠀⠀⠀⠤⣠⣭⠉⠙⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢷⣄⣰⢋⡙⠢⠀⠀⠰⢋⣭⢱⡀⡠⡿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠀⠳⣝⣣⠜⡏⠈⠛⢬⠭⠞⡁⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠞⠃⠀⠘⠦⠖⠃⠀⠀⠒⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠠⢤⣄⣀⡤⠔⠂⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠈⠉⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⠉⠉⠓⠒⠉⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡇⠘⢦⡀⠀⠀⡠⠃⣸⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠘⡄⠀⣹⢖⠚⡅⢰⠃⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠦⠀⠀⠀⠀⠀⠘⣤⣯⡙⢻⣧⡎⠀⠀⠀⠠⠆⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢾⣏⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡐⠄⢹⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠐⢼⣈⡀⠀⠀⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⢦⠈⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡷⢄⠑⣼⡇⠀⠀⠀⠀⢹⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣇⠀⠑⣼⡇⠀⠀⠀⠀⠈⡄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣭⠑⢦⣸⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠶⠶⠶⠶⠦⠾⠦⠶⢤⣤⣤⣷⣤⣤⣤⣤⣤⣿⣵⡶⠿⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠿⠶⠶⠖⠒⠒⠒⠒⠛⠺⠶⣤⡄
⠀⠀⠀⠀⠴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⢰⡶⢶⣦⠀⣀⡀⠀⢀⣀⣀⠀⢀⣀⣶⠀⢀⣀⠀⢀⣀⠀⠀⠀⣴⡶⣶⠄⢀⡀⣀⠀⠀⢀⣶⣶⣶⣶⢰⣶⣀⠀⢠⡶⠀⢀⡀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⣾⣧⡾⠏⣾⠋⣿⠀⣿⠋⣿⢲⡟⢹⡟⢰⣿⣽⡇⢸⡟⠁⠀⢸⡿⢀⣿⠀⣿⠋⣿⠃⠀⠀⢁⣿⠁⠁⣸⡏⣹⠇⣸⡇⠸⣯⣙⠠⠟
⠀⠀⠀⠀⠀⠰⠿⠂⠀⠀⠿⠾⠋⠰⠿⠀⠿⠸⠷⠾⠇⠘⠷⠞⠀⠿⠇⠀⠀⠘⠷⠾⠋⠰⠿⠐⠿⠀⠀⠀⠼⠿⠀⠠⠿⠂⠿⠀⠿⠀⠷⠾⠟⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⠶⠶⠶⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⢀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠾⠟⠛⠉⠉⠉⠉⠉⠉⠛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⡿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢹⣇⠀⠀⠀⢀⣿⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠈⣿⡀⠀⢀⣾⠃⠀⠀⣿⣇⣤⣤⣶⡶⠿⠟⠛⠋⠉⠉⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠹⣧⣠⡿⠃⠀⠀⠈⠛⠉⠉⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀⣀⠀⢀⣀⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⣶⣶⣤⣴⣶⣶⠶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠢⢄⣠⠴⠒⠋⠉⠉⠉⠉⠒⠢⢄⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠋⠁⠈⠉⠉⠉⠉⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⣀⡤⢄⡀⠀⢰⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣀⠠⣈⣲⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠘⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠬⠷⢤⠔⠒⠒⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠦⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠒⠚⠋⠉⠉⠉⠉⠛⠛⠒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠀⠉⢩⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠈⠁⠀⠀⠀⠉⠉⢛⣳⣒⠀⠀⠀⠀⠀⠀⠐⢶⡚⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠢⣄⠀⠉⠉⡿⠉⢉⡟⠀⣀⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣠⡤⣷⣤⠴⠶⢦⣄⠘⠀⠀⡼⠁⢠⡏⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢉⣿⡋⠀⠀⠀⠀⠸⡆⢀⡾⠁⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠛⠛⠲⠶⢤⣤⣄⣀⠀⠉⠛⠛⠋⠉⡟⠃⠀⠀⠀⠀⢀⣷⠏⠀⠀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠒⢿⣄⠀⠀⠀⢀⡼⠁⠀⠀⣰⠟⣉⣁⢹⡄
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠈⠫⡓⠶⠶⠛⠁⠀⠀⠀⠁⠸⠽⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡁⠐⠒⠠⠤⠀⠄⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⠶⠶⠶⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠴⣚⡉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠟⡱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡙⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢉⣉⣙⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠀⢀⡾⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⢀⣉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣠⠟⠀⠀⠀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡌⠉⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡼⠁⠀⠙⢦⡀⠒⠦⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣠⡞⠁⠀⠀⠀⢤⡀⠀⠀⠀⣸⣿⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⢀⣤⠤⠤⢶⡟⠳⣄⠀⠀⡴⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠙⢦⣀⡴⠿⣏⢀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⠓⢺⣿⡆⠀⠺⠞⠉⣾⢕⡾⠃⠈⠑⢦⣀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⢾⣿⠇⠀⠀⣀⡤⣿⠘⣇⣾⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠤⠖⠋⠁⠀⣿⠀⡇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣶⠖⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠴⠶⣶⣶⠶⣶⣒⣂⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⡦⠬⣀⣸⣦⣀⣀⣀⣀⣀⣀⣤⣤⣴⡖⣻⢛⡙⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠷⠾⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡰⠀⡴⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⡇⡗⠒⠂⠤⠀⠀⠀⠐⢈⣹⢪⡧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⠇⡿⡿⢻⣿⠦⠀⠀⠀⠘⠻⠟⠐⢄⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠊⠉⠉⠑⢄⠈⠇⠰⡀⠀⠀⠀⠐⢾⣿⠗⠀⢀⠊⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⡜⠀⠀⠀⠀⠀⠘⡄⡆⠀⡇⠀⠐⢄⡀⠀⠀⡠⠀⠘⡄⠀⠀⠀⠀⢱⡦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡇⠀⣠⡇⠀⠀⠀⠀⢉⠏⠀⠀⠡⠈⠲⢄⠀⠀⢸⡟⣿⠤⣶⡄
⠀⠀⠀⡜⠁⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣇⡴⣹⠀⠀⠀⠀⠀⡎⠀⠀⠐⠢⠤⢄⡀⢱⠢⣼⡇⣿⠀⢹⡇
⠀⠀⠸⠀⠀⠀⠀⠀⡴⠁⠀⣀⠀⠀⠀⠀⠀⠀⡟⠐⠁⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠈⠙⠚⢿⣼⣿⢦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡰⠇
⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⡠⢎⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⢠⡤⠮⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠋⢻⠿⠋⠀⠀⠉⠻⢿⣟⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⠛⢻⠉⠉⢹⠤⢼⡉⠉⢹⠖⠉⠉⣴⠋
⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀⠈⢿⠉⠙⠀⠀⠉⠉⠉⠀⢠⠄⣯⡀
⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⠳⣄⠠⠤⠄⠀⢀⡴⠋⠀⢿⡎⡆
⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁⠀⠀⡠⠜⠛⠀⢭⡥⠟⢀⡤⠔⠊⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠢⡇⠀⠀⠀⡔⡛⠀⠀⠀⢀⣀⡿⠆⡈⢸⠀⢀⡠⠂⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠣⢄⣰⣶⠚⢬⠀⠸⡇⠸⠖⠉⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣆⠀⠀⡀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠁⠉⠊⠳⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⡿⠀⠀⠀⠠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡇⡔⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⠀⡖⠤⢤⣃⣀⣀⠀⠀⢀⣆⣈⣀⣀⠀⠀⢨⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⠬⠦⠀⠘⡆⠐⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⢤⢤⣃⣀⡸⠀⠀⢀⠤⣀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠀⠉⠉⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⣀⣀⣀⣀⣼⣀⣀⣀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⣤⠥⡴⣚⣡⡆⣉⣿⣿⣿⣿⣿⣶⣿⣿⣶⣤⣤⢰⡏⠀⠛⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠠⡏⣴⣤⣷⡟⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡿⠿⢿⣿⣭⣤⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣶⣾⣯⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣡⠐⡻⠻⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣋⡻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⠉⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢉⣿⣿⣿⢛⠂⠘⠂
⠀⠀⠀⠀⠀⠀⠠⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣟⠉⠉⠉⠉⠛⣧⣍⠻⢶⠒⠦⠶⠴
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠉⠀⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣤⣝⣿⣿⣿⣟⠿⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠻⠋⠀⣠⡀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠁⠀⡴⢾⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠁⠀⠀⢶⠀⣾⡀⠀⠐⠀⠀⠀⢀⣾⣇⠀⢸⣿⣿⣶⡶⠞⡁⢀⡔⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠳⣤⣀⡺⣐⡇⢹⣄⣠⣌⠀⢰⣿⢻⣭⣶⣾⣿⠿⠿⣀⡞⢻⡿⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⣼⣿⡙⢯⢻⣄⣿⣿⢚⡆⣾⣁⣼⣿⡛⣿⡀⢀⣀⡿⣷⣸⡧⢀⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣬⣿⣿⣿⣟⣾⠗⣮⣿⢻⣿⣿⣿⡇⢸⡆⠀⣾⡏⣱⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣯⣷⣋⣤⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣸⡏⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⠉⠀⠉⠀⠄⡄⠀⠀⠀⠀⠀⠔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⢋⢨⡟⠃⠉⣹⡷⠦⣤⣾⡇⠀⣠⠴⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⣿⠃⠀⡰⠊⠀⠀⠻⣿⡿⣿⣷⣾⠿⠤⠼⠆⠙⢁⡀⠀⠹⡧⠘⠓⡞⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠲⡆⣾⡏⢀⡜⠁⠀⠀⠰⠮⢉⣰⣺⣟⣒⡒⠀⠀⠀⠀⠀⣧⠀⠀⢰⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⣷⣾⠀⠀⠀⠀⠀⠓⠀⡤⠼⠷⠄⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠈⢋⣿⣿⣿⣽⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣯⣿⡇⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢼⣿⠿⣿⣿⣿⠋⠻⠇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠤⠠⠄⠤⠤⠤⠤⠤⠤⠤⠀⠀⢀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠖⠂⠀⠀⠀⠀⠐⠒⠂⠒⠀⠀⠀⢇⠆⡀⢠⠀⠀⠀⠀⠀⠀⢡⠀⢀⡀⠀⠈⠀⠀⡡⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⡠⠀⠀⠀⠀⠀⠀⠀⣀⢸⠃⠀⠀⠀⢤⣬⣀⡀⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢄⠀⢾⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠑⠐⠂⡤⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣠⣄⠀⠈⠳⣅⠈⠀⠀⠀⠀⠀⠀⠀⠰⠃⠐⡋⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠁⠠⠀⠨⡒⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⠈⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⣉⡁⢋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢁⠀⠀⠀⠀⡀
⠀⠀⠀⡬⠹⡄⠀⠀⠀⠀⠀⠀⠂⢀⡡⠀⠀⠡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣃⡈⠂⠐⠓⠠⠀⠂⢠⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢱⡀⠀⡠⠂⠐
⠀⠀⠈⠀⠀⠈⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣶⣾⢋⣈⡯⠥⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠁⢀⣀⠤⠟⠛⠫⢀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⠋⠀⢀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡄⠚⠁⠀⠀⠀⠀⠐⠑⠀⡂⠀⠀⠀⡅⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠄⠁⠀⢤⡀⢘⣁⣀⠈⡅⠂⠐⠋⠀⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠐⠒⡺⠋⠀⠀⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⡁⠀⠀⡀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠾⣿⢿⣿⣷⣄⣤⣤⣤⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⠋⠁⠀⣠⠀⠀⢠⡟⠋⠉⠹⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠁⡀⠀⠀⠀⠀⠸⢷⣾⣿⣆⢠⠀⠻⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⡿⢛⣃⣉⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣄⣀⣄⢰⣄⠀⠀⠀⠀⠀⢸⣿⣿⠶⠭⣤⣄⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠙⢿⣿⣿⣼⣇⣤⣿⡱⣶⡞⠻⣿⣄⠀⠀⣹⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠞⢷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠘⣿⣧⠴⠇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡑⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣦⠀⠋⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠙⠷⣦⣾⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣷⣄⡀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⡏⠀⠀⣉⣿⣿⣿⣷⣿⣿⣿⣯⠄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⠟⠉⠉⢉⠁⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠳⠀⣀⠀⠀⠘⢶⣶⣤⣤⣀⣀⡀⠀⠀⢀⠀⢸⠖⣦⣤⣼⣾⡿⣐⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀⡁⣡⣴⣶⣶⣾⣿⣿⣿⣿⠏⠹⢖⠷⣬⣶⣾⣤⣌⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⠀⠀⠈⢳⣿⡿⣛⣟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⡼⢹⣿⣟⣿⡷⢚⣿⣿⣿⣿⣿⣷⠃⣴⡇⣸⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢰⠁⢻⣿⣿⣥⡀⠈⣿⣿⣿⣿⣿⢡⣶⣟⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠸⡇⠘⡿⠀⠉⡁⠀⠙⢿⣿⣿⣿⣇⣿⣿⣿⣿⠻⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠐⠄⣀⡉⠀⠐⣛⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⢏⡀⠰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣵⡤⠄⠛⢛⣿⠶⠆⠀⢹⣿⠻⢿⣿⣿⣯⣌⠙⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠁⣷⣾⣿⣿⣿⣿⣿⣄⠁
⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠂⠀⠀⠀⠂⢤⡏⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣰⣧⣿⠆⠀⣰⣦⠀⢀⠀⠀⠀⠻⣄⠀⠀⠈⠛⢿⣿⣿⡷⠻⣦⣤⣀⡀
⠀⠀⠀⢺⣩⠉⠉⠈⡁⣈⠀⠈⠉⠉⠁⠀⠀⠀⠐⢌⠓⢶⣤⣀⡼⠟⠛⠁⠀⠙⣿⣿⣿⣷⣦⣤⣤⣄
⠀⠀⠀⢹⣧⣴⣶⣥⣤⣽⣓⣻⣶⣶⣶⢀⣰⣶⣿⡻⣧⣤⣤⠽⢤⣴⢶⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣤⣶⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⠿⠋⠉⠉⠉⠉⠛⠿⠛⠉⠉⠉⠉⠉⠛⢿⣿⣷⣄⣀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⡿⠿⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣆
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⢿⣿⣧⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⡿⠋
⠀⠀⠀⢿⣿⣆⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉
⠀⠀⠀⠈⠻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣄⣀⠀⠀⢀⣀⣤⣆⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⣤⣤⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⣾⣿⠟⠉⠙⢻⣿⣿⣿⣿⠿⠿⣿⣿⣶⣶⣾⣿⡿⠟⠉⠙⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⢘⣿⣿⡀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢰⣿⡟⠙⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠿⣷⣶⡿⠏⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠶⢦⠠⠀⣤⠀⣤⣤⢤⢤⢤⣤⡄⣠⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠆
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢨⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⡀⢀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⡞⠁⠀⢹
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠘⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⢠⠀⣸⡁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠀⠐⠂⠀⠀⢀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⠂⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠟⠃
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣦⢠⢸⢯⡄⡇⢌⠶⡀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⡡⠃⣇⠟⠀⢸⠸⢿⡷⢎⣶⡳⣿⠦⠆⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡗⡤⡆⠘⠃⡧⠈⠀⢸⡁⣉⠀⠊⢁⡚⠓⠛⡋⠀⡀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⠉⠛⠒⠶⣓⡄⢠⠛⡇⠲⠸⠇⠘⣇⠰⠄⣷⠀⣭⠥⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠄⢁⠐⡤⢒⡁⢀⠀⡠⡀⠀⡀⠸⡀⢹⠄⠠⠉⠲⠶⠊⡒⠀⠠⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⣸⠂⠬⠤⠙⠂⠀⠀⢄⡁⢀⠨⠀⠀⠀⠃⠀⠀⠀⠀⣀⢋⡀⢀⣠⠃⣟⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡈⢋⠄⠠⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠐⣠⠉⠄⠀⠆⡨⡄⠂⠘⡁⢺⡬⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⠉⠛⢋⠀⠀⠀⠀⠘⡀⢰⠁⠰⢂⠚⡈⠀⠑⠀⢉⣒⡌⠰⢱⠷⡽⣤⠂⡾⣝⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⢆⡑⠤⢂⠐⠈⠁⢐⣈⣡⡤⣀⣤⢨⣵⡚⣠⡿⠀⣘⣷⠟⣿⣇⢲⢸⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⣡⣀⢄⢀⠲⡎⢻⠇⢛⣷⣃⢾⣹⣜⣻⣅⡈⠀⡥⢆⢋⣈⡑⠟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢻⡄⢾⠘⣣⣹⠶⣸⣞⢾⣳⠬⢞⣏⣸⢧⡄⠘⡏⠀⣹⢲⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠟⢈⡻⠤⠗⣾⡛⢭⡵⠶⠟⠛⠙⡅⠾⢳⣀⣻⡴⢽⡙⠛⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠝⠀⠀⠀⠞⠒⠻⢏⠬⠠⡶⢄⢃⣴⣼⠖⢛⣻⢯⡀⢁⠀⠠⠀⠀⠀⠈⠒⠕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢰⡾⣟⣙⠓⢲⠃⠘⢃⠬⢎⢟⣩⡭⣶⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⡳⡖⠳⣒⣶⢒⡝⡏⢯⢥⣩⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣺⠯⣻⠏⠄⠃⠊⣘⢩⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣉⣜⡳⢥⣋⣙⡾⠧⢅⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠉⣉⣉⣭⠿⠔⠂⢁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠉⠉⠉⣀⣀⣤⠶⠚⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢉⣀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣁⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠘⣿⣿⣿⡿⢣⣾⣿⣦⢰⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠻⠿⠋⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠐⠂⠀⠉⠉⠉⠉⠉⠉⠁⠐⠒⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠉⢆⠀⢀⡀⠤⠤⠤⢠⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠈⠱⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠛⠉⠉⠙⠲⡄⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢻⠀⠀⠀⠀⠀⠀⡃
⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠈⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠁⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⠠⠐⠉⡘
⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠒⠉⠀⠀⠀⡔
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠂⠉
⣶⣶⡶⢶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⢀⠬⠅⠀⠀⠠⠬⢘⠉⢀⠒⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠔⢡⠎⠁⠀⠀⠀⠀⠀⠀⠀⠑⠄⠑⡄⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠜⢠⠃⠀⢀⢤⠖⢲⣺⠶⡀⠀⠀⠈⡀⢢⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢨⠉⢸⠀⠀⠈⠂⠁⠈⠀⡜⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠄⠘⡀⠀⠀⢐⠀⠀⢰⠁⠀⢀⠌⡐⠀⠘⡀⠀⠀⠀⠀⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠴⠁⡀⠐⣄⠀⠘⠃⠀⠃⡠⠆⢀⠔⠀⠀⠔⠉⠀⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠂⢄⢸⠀⠀⠀⠀⠀⠰⠔⠀⠑⡠⠦⣬⣧⠒⣊⠁⣀⠄⣰⠀⠀⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⢨⠗⢐⣋⡵⡁⠀⠙⠁⠀⠈⠀⠀⠀⠀⠀⠠⠃⡈
⠀⠀⠀⠀⢀⠠⠄⠤⡀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⢨⡋⣉⣥⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠁⡐
⠀⠀⠀⠀⠨⣀⡀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⠽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊
⠀⠀⠀⠀⢀⠠⠄⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠠⠀⠀⡀⡀⠀⠀⡀⠀⠀⡀⠄⠂⡠⠔⠁
⠀⠀⠀⠀⠘⠤⠤⠊⠀⠀⠀⠋⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠄⠀⠀⠨⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠐⠥⠆⣀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⣴
⠀⠀⠀⠀⠀⠀⢠⢪⠶⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠘⢼⠛⠌⡪⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⣠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠆⣩⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠰⢀⡽⢸⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡇⣼⣇⣸⡈⢣⡀⢀⢀⢀⣀⣀⠀⠀⠐⠑⠁⠊⠊⠐⠁⠀⠂⠑⠀⠂⠐⠑⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠉⠉⠉⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠒⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡰⠠⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠘⡆⠀⠀⠀⠀⠲⡀⢸⡰⠃⡠⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠈⠉⠉⢳⠁⠞⠁⠔⢀⡤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣆⠔⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⢲⣥⣤⠤⠖⠊⠀⠀⠀⠀⢈⢀⡀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢀⢀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠑⡄⠀⠀⠀⠀⠁⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠫⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢰⠁⠀⠀⣀⠄⠈⠀⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠗⠊⠁⡀⠀⠊
⠀⠙⣍⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠈⠀⠀⠀⡀
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠐⢀⡖⠱⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠧⡀
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣼⠀⠀⡀⠀⠱⣴⣥⠀⠢⣀⣠⠴⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠰⢎⣉⠉⠑⡆⢿⠄⡞⠀⠀⢺⠝⠊⠀⠀⠈⣡⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠊⢡⠤⠂⠀⠃⠀⡜⠀⠀⠀⠁⠀⠀⠀⠀⠈⣓⡄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠀⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠒⠀⠠⣤⠄⠨⣭⠀⢀⣒⣀⣐⡀⣀⣉⣩⠤⠘⣆⣠⠀⠀⠀⠄⠐⠂⠈⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠚⠀⠀⢳⠀⠀⠀⠀⣻⠀⡎⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠚⢻⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠼⣤⣀⣀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠻⠭⠤⠬⠿⠯⠭⠭⠭⠿⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂

⠬⠐⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠨⠀⠀⠹⡀⠙⡏⢻⣮⠹⣣⣼⣾⡟⠉⡽⠁⡸⢫⠑⠁⠎⡠⠛⠙⠁⣡⠞⠁
⣥⣀⠀⠀⠁⠀⠀⠀⠀⠘⢆⣤⢤⣤⣄⣀⡀⠀⠄⠀⠀⠀⢠⡴⠲⢧⣁⠰⡎⡶⠛⡋⠀⠈⣷⡼⢁⣀⣀⣈⠀⢀⣀⣀⣀⡀⠊⠀⠀⠀⠀⢀⣠
⢦⢈⠁⠢⠀⠀⠀⠀⠀⢀⡿⠉⠀⠀⣀⣈⡿⢳⠶⢄⢈⣤⡿⠈⠀⠘⠉⠡⠿⢥⡄⠀⠀⠒⠟⠓⠛⠀⠉⠉⣹⠉⠁⢹⠉⠉⢷⠀⢀⡠⠂⠉⠈
⠽⠐⠀⣀⠀⠀⠀⠀⠀⠀⣻⠅⠀⠘⠧⠈⠁⠈⠀⠀⠈⠀⠘⠓⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠁⠀⠀⠛⠀⠠⣼⠇⠁
⠶⠤⡀⠀⠉⠁⠀⠀⠀⠀⠳⣤⣀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠜⢋⣷⠶⠒⢦⠀⠀⠀⠀⢲⡁⠀⠀⠀⠀⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⠠⠄⣂⢀⠀⠀⠀⠀⠀⠀⠔⢁⡠⠂⠀⠆⠀⠀⠙⠀⢠⡼⠀⠀⢀⣤⠼⠁
⠉⣁⠀⠂⠀⠀⠀⣠⢤⣯⠙⢻⠘⡏⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠀⠀⠘⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠀⣾⠥⢤⣀⢀⡀
⢉⠉⠉⠓⠂⠀⣾⣥⠀⠀⠀⣈⠀⢸⡁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⢀⠀⠋⠻⠉⣷⣠
⣭⣋⠁⠀⣀⣠⠿⠀⢀⡴⠋⠁⠀⠀⠉⠀⠀⠀⢈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠀⠀⠀⠀⠀⠴⠋⠤
⣉⣣⡶⠟⠀⠘⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠠⠚
⠀⢾⣡⡀⠀⠀⡀⠀⢀⡀⠀⢠⠀⠒⠂⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠀⠲⣶
⡒⣘⠹⠷⠶⣾⡃⠀⠘⠶⢴⣋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠆⠀⠀⠀⢀⠀⠀⠤⢀⠈⠈⠀⠀⠀⠀⠀⠈⠀⠰⠀⢰⠀⠀⠀⣀⣀
⠀⢀⡀⠀⠀⠀⠻⢤⣠⣀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠂⠀⡀⠀⠀⠁⠀⠀⠀⠈⠐⠐⠄⠀⠀⠀⠀⠀⠀⣤⡿⠀⣀⠚⠛⠉⠂⠉⣑
⣿⠏⠁⠀⠀⠀⠀⠀⠨⠍⠁⣷⠀⠀⠀⠀⠀⠑⠛⠙⠂⠀⠀⠀⠀⠈⠁⠀⢠⠀⠀⠀⠘⠀⠀⠀⠘⢁⢀⣠⣄⣀⣤⠖⠃⢠⡄⠀⠀⠐⠤⠄
⢐⠀⠀⠀⠀⡠⠐⠉⠀⢀⣀⣿⡀⠀⠀⣀⡄⠀⠀⣨⡤⠾⠁⠀⠈⢹⡋⠆⢠⣁⠀⢀⣠⠶⣄⣠⣆⠀⠉⢄⡼⠻⠤⣤⡴⠟⠁⠀⠐⠀⣀⠀⢅
⣡⢄⠤⠒⠉⠀⢀⣤⠖⡿⠋⠀⠉⠛⠛⠁⠙⠶⠖⣻⠃⢰⠖⢐⢠⠞⣇⡄⠀⠉⠛⢫⡇⠀⠈⠀⠌⠛⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠰
⠞⠁⡀⠀⡤⠚⣉⠤⠂⠀⠀⡠⣀⠀⠠⠆⡠⠀⣰⠁⣄⢉⢻⣏⢏⣆⡎⠳⣦⢠⡶⠏⠀⠀⠀⠤⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠻⣿⡿⠀⢿⣿⠟⢿⣿⣿⣿⠟⠃⢁⣈⠀⠛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡁⠰⣿⣿⡇⠀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣄⠀⠀⣰⣾⣿⣷⣆⠀⠀⣠⣼⣿⣧⡄⡈⢉⠀⣤⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⢻⣿⣿⣿⡿⠀⠀⠻⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣦⠀⠀⠉⠉⠉⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣏⣠⣶⣦⠀⣴⣶⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣥⣬⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣛⢺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⣛⣛⣿⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷⠾⠭⠭⣭⣽⠛⠛⣉⣉⣥⣭⣉⣡⣴⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣯⣍⣡⣄⣀⠀⠀⢚⡔⢩⣿⣿⣿⣿⣹⣗⢜⢿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣷⣯⣐⠈⣟⠿⣿⣿⣶⣄⠀⡞⢀⣿⣿⣿⣿⣿⣿⣿⣦⣩⠉⢱⡛⠙⠋⢿⣿⡿⣛⢿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡿⡿⡇⣧⣌⣿⣶⡄⠈⠁⠀⡇⢼⣛⣿⣿⣿⣿⣿⣿⡿⣌⠒⣫⣶⣿⣿⣿⣿⢹⣿⣿⣿⡇
⠀⠀⠀⢸⠿⣿⣿⣿⣿⣷⣧⣿⣿⣽⣿⣿⣾⣷⠇⡼⢿⣿⣿⣿⣿⣿⡷⠆⢺⡾⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠻⣾⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣀⠀⠀⠺⠀⢠⡄⠁⠈⢹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠐⠀⠀⠀⣠⣾⡇⠀⠀⢹⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠀⠈⠂⢰⣿⡿⠙⡀⠀⠀⢃⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠔⢡⣾⠿⣇⡀⢠⡀⠀⠈⢆⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⢸⡛⠁⠀⢻⣿⣿⣷⣶⣶⣾⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠂⠀⠀⣤⡂⠐⢤⠀⣃⠀⠀⠈⡇⢹⠋⠉⠙⡻⠛⢻⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠈⡿⠀⠀⠀⠑⡉⣦⠀⠐⠾⠾⠒⠒⢦⡃⠀⠀⠠⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⢻⡁⠀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⢻⠀⡀⡰⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣶⠀⠀⠀⢠⠒⠈⠉⠲⣑⣶⠀⠀⠀⠀⠀⠐⠲⡆⠀⠀⠀⢣⣭⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⠆⠸⠥⢠⡇⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠉⣦⠀⠀⠀⢻⡍⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢡⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢆⠘⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⢿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠰⠑⢛⣛⢧⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣶⣤⠀⠀⣀⣰⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠨⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢙⣿⣷⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣈⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣾⣿⡯⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⠿⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠛⠛⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣦⣤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠘⠉⠀⠀⠀⠈⠛⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡀⢹⡀⠀⠀⠀⠀⠀⣠⠀⠙⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠉⠉⢻⣿⣎⣳⡈⠃⠀⡔⠋⠁⠀⢠⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢌⣙⣿⣿⣶⣬⣟⢠⡀⠀⠂⠆⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠿⣽⣯⣹⣾⠟⠁⠀⢰⣆⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⣰⠃⣽⠣⠀⠀⠀⠙⢻⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⣂⣤⡀⠀⠀⠁⠀⡇⠀⢀⣦⢒⣤⠋⠀⣯⡀⠀⠀⠁⠐⠀⠀⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣛⣁⣒⣈⢉⣛⡿⠷⠦⠀⠀⣈⣤⣾⣿⣿⣯⣤⢤⣾⣿⣦⣤⣤⣤⣴⣶⣼⣿⣶⣦⣤⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⢀⠁⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡇
⠀⠀⠀⠀⠀⠈⠀⡌⢞⠀⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⡴⠀⠀⠀⣀⢉⣈⣹⣉⣉⣹⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⡿⠧⠼⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠈⠲⠊⡉⠁⠈⠐⠦⢼⣄⣠⣤⣴⣾⠋⠀⠀⠀⠈⣨⡟⡉⣹⣿⣿⣻⣿⣩⣿⣿⣿⣩⣿⠉⠉
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠁⠀⠛⠒⡶⠤⠙⠛⡛⣿⡏⠀⠀⠀⣠⣾⣿⠂⠀⣿⣿⣷⣿⣷⡋⣽⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠈⠉⠰⠠⠤⠤⣤⣄⣀⠀⠀⣀⢴⣿⣿⡿⠁⠀⣠⣼⠿⠿⠟⡡⣾⣿⡭⣿⣟⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⣿⣿⣿⠋⠁⠀⠸⢿⣧⣴⣴⣾⣧⣿⣭⣠⡬⠻⣿⠏⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣤⣧⠿⡝⠿⢶⠤⣤⣌⣉⠿⠿⢿⣿⣿⡷⠀⠉⠁⠰⣿⣄⣀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠘⠉⠀⠀⠀⠀⠀⠈⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠀⠀⠒⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂



⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⣾⣿⣷⣦⣀⣤⣴⣶⣿⣿⣿⡇
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⣠⣴⣶⣿⢿⣿⣿⣿⡖⢠
⠀⠀⠀⢰⡁⠈⠛⠁⣼⣿⡟⠉⠀⢸⡇
⠀⠀⠀⠈⢷⣤⣤⣾⣏⣁⡀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠻⢿⣿⣿⡇
⣀⣀⣀⣀⣀⣛⣛⣉⣀⣀⣈⣻⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⠀⣲⡾⢁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⣠⣾⠋⠀⠘⠀⠀⣹⡇⠀⠀⣡⡀⡰⢀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡴⢿⡇⠀⠀⣠⠞⣿⡇⠀⠀⠀⡆⠀⣿⠀⠀⢠⣿⠁⠳⠀⠀⢶⠻⣭⠟⠀⠘⠀⡀⢠⣤⠀⠠⠄
⠀⠀⠀⠀⠄⡀⢋⣁⣴⠛⢷⡤⠾⠷⠀⣸⡞⠃⠀⠀⠁⠂⠏⠀⣡⣾⡟⠠⡄⠀⡄⠀⠘⢻⡆⠀⠀⠀⠀⠀⠘⠦
⠀⠀⠀⢠⠤⠞⠛⠛⣿⡀⠀⠀⠈⠀⠀⠿⢠⣤⡀⠀⠀⣴⢃⡼⣿⡏⠀⠀⠁⢀⣀⠀⠡⠀⢹⡆⠰⣇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡀⣠⡀⣀⠈⣲⣆⠀⠀⠀⠀⠘⢻⡇⠀⠘⠛⢫⣾⡿⢢⡀⠀⠀⠀⠙⠦⣀⠀⠀⠱⠀⠻⣄⢀⣶⣸⣷⠴
⠀⠀⠀⠀⠘⠋⠈⠉⠉⠈⠈⠙⠓⢠⣾⡆⠰⢿⡇⠀⠀⠀⠀⢻⣅⠀⠀⢠⡀⠀⠀⠀⠈⠃⠀⠀⠀⣀⢨⡿⢛⡟⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣄⡰⠖⠉⠀⠀⠀⠺⢈⡙⠓⠆⢤⠠⣽⣇⠀⠀⠉⠀⠀⠀⠀⠀⢀⡼⠗⢋⡤⠟⠈
⠀⠀⠀⠀⠀⣠⠿⠒⠈⠉⠀⠀⢠⣦⢠⣦⢠⡄⠀⠑⠂⢀⡀⠀⠀⣭⣢⠀⠀⣠⠀⣠⠴⢚⣿⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⣠⣼⣿⣿⣿⣿⣿⡀⣄⠀⠈⠉⠓⠀⠀⠀⡈⠁⢒⣛⡥⠾⠋⢠⣾⡄⠀⣀⣴⣆⣰⡷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⠿⠿⠿⠻⠿⠿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠩⢴⠂⠉⠀⠀⢀⣿⣿⡿⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⡠⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⠿⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡟⢦⡈⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢶⣿⣿⣯⣿⣿⣿⣿⣿⣜⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣼⣿⣿⣦⣿⡿⢿⡟⢿⣿⣿⣿⣿⠿⣌⠙⠋⡿⠘⣿⣿⣿⢹⡿⢿⠏⠀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠻⣿⣿⣏⡉⢃⠀⢿⠀⠉⠙⠟⠙⠂⠙⠃⠘⠀⠀⠘⠿⠇⠸⠃⠀⣿⣿⡿⣿⣀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣾⣿⣤⡤⢄⣀⢙⣧⣄⣹⣶⣾⣷⠶⣶⡶⣴⣷⣶⣶⣾⣶⣤⣆⣴⠄⢿⡏⠀⠘⢿⣿⣷⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣯⣥⣥⣤⢀⣉⣾⣿⣿⣿⠃⠈⠿⡷⠿⣷⣹⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠿⣿⣤⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⣩⢷⡆⠰⠂⠈⠛⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠙⠿⠻⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡝⢷⡀
⠀⠀⠀⣿⣿⣿⣿⡟⢛⣿⣩⡾⠟⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣷
⠀⠀⠀⣿⣿⣿⣿⣷⡿⢿⡁⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣠⠼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣫⡅⠘⣉⡴⠋⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠸⣿⡿⣿⣿⣿⣷⣿⣿⣿⠟⠉⣰⢞⣵⠂⠀⠀⠀⡰⠚⠀⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣾⡿⠛⢻⡇
⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⡿⣷⣾⣽⠟⠁⠀⠀⠀⠀⢀⣴⡾⠿⣿⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣉⡀⠀⠻⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠙⢛⣿⣿⣤⣤⣀⠀⠀⠀⣴⠟⣻⠞⠻⣿⣿⡇⢰⡆
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⣠⣶⣿⣿⣿⡍⠛⠻⠉⢻⡆⠀⠀⠀⠈⣁⠀⠀⠉⠁⠀⠘⡇
⠀⠀⢠⡶⠊⠉⠉⠛⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣿⣁⡀⠹⠛⠃⠀⠀⢀⠂⠈⠀⠀⠀⠀⡀⠉⠓⠶⠀⠀⠀⠀⢹
⠀⢠⠈⣠⣶⣶⣤⣀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣀⠀⠀⠄⠁⠀⠀⠀⠀⢠⣄⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠸
⠀⠸⠇⠿⢿⡀⠉⠛⢿⡀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠐⣦⠈⠇⠀⢼⣿⡇⠀⠀⠘⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠱⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠆⣴⣀⠀⠀⢿⣦⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠁⠀⠀⠀⠀⣨⣦⡄⠀⠀⣸
⠀⠀⠀⠀⠈⢧⠛⢷⣄⠀⠈⠁⠰⣀⠰⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⢿⣤⣴⠶⠶⢶⣤⣀⣴⠋⠀⠹⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠓⠤⣍⡀⠀⠀⠀⢻⡄⠰⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣿⣿⣿⡄⠀⠀⠀⠈⠓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠉⠉⢠⣀⣠⡤⠀⠀⢤⣤⣄⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡿⠋⠀⠀⠀⠀⠉⠛⠦⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣰⣷⣤⣤⣤⡄⠀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⠃⠘⢿⡉⠙⢿⣿⣿⣷⣦⣤⣄⠀⠀⣠⡾⠋⠀⠀⠰⠶⠶⠾⠛⠉⠉⠀⠀⠀⠀⠀⠉⡆⠹
⠀⠀⠀⣀⣀⣀⣠⣾⣿⠟⠁⠺⣿⣿⣿⡆⠤⠀⠈⠳⣤⡙⠛⢿⣿⣿⣿⣤⣰⣿⠃⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣷⣀
⢀⡴⠚⠛⠛⣿⣿⠛⠁⠀⠀⠀⠙⠻⡟⢳⠄⠀⠀⠀⠀⠙⢶⣄⠈⠻⣿⣿⣿⡇⢀⣶⠆⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠁⠘⠛⠿⠿⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡦⠀⠀⠀⠀⠀⠀⠈⠻⠦⣈⣿⣿⣧⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢄⡀⠀⠀⠀⣴⣿⣿⣿⡟⢿⠇⠀⣠⠁⠀⠀⠀⠀⠀⠉⠛⠛⠳⢤⣤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡄⠀⠘⣷⣀⣼⣿⣿⣿⣿⡏⠀⠄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠀⠀⠀⠀⠙⢶⣄⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⢀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⢧⣤⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠿⢶⠶⠶⠛⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡿⠧⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠁⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢣⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠶⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣿⢱⣿⣀⣴⠏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠿⣿⣿⠛⣿⠇⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠚⠋⢿⣶⡇⣿⣿⣸⡿⣴⠋⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣼⠀⣿⡅⣿⢻⡿⠸⡿⠀⡎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠘⣤⠿⣿⡇⠘⠷⠀⠁⠄⢡⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠶⠶⠶⠶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠣⡀⠀⢀⠄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⣸⠆⠑⢤⣀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠥⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠴⣃⡀⠀⠀⠀⠀⠈⠂⠠⠤⠴⠚⢣⡀⠀⠀⠀⢠⡀⢠⣄⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠔⠋⠉⠉⠀⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢿⠒⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠠⠒⢋⠁⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠵⠯⠖⠀⣙⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠐⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⢧⠹⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠳⠘⡷⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⡤⡈⠓⠀⠀⠀⣀⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⠀⠀⠀⠀⠘⠁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⣩⠀⠀⠀⠀⠀⢂⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣸⣯⠀⡠⠐⠰⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣶⡖⢦⠀⠀⣠⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⡰⢋⡟⠐⠁⠆⠀⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢤⣿⣿⣦⠀⠀⠀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠑⠒⠒⠈⠀⠀⣰⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠹⣿⣿⣥⣤⣾⣿⡿⠻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣷⣶⡀⠀⠃⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⢿⣿⣿⣿⡿⠁⠀⠹⡏⠙⠻⠿⠿⠶⠶⢚⣿⡀⠀⠀⠹⣿⣿⣿⣶⣦⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⣿⣿⠟⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠘⠛⢿⣄⠀⠀⠈⠻⣿⣿⣿⣿⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⣠⣶⣭⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠠⠀⣀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢹⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠂⠈⠳⢄⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠠⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠂⠐⠀⠀⠰⠀⠆⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠁⠀⠛⠀⠐⠀⠰⠆⠀⠦⠀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠀⠘⠁⠐⠃⠀⠶⠀⢶⠄⢠⡄⢀⣄⠀⣈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠈⠁⠀⠋⠀⠛⠀⠲⠆⢰⡦⢠⣤⠀⣤⡀⢀⡁⠀⡉⠀⠘⠀⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⢀⡀⠀⠉⠀⠘⠁⠘⢃⡰⠗⠒⠀⠒⠒⠤⣠⣄⢀⣍⠀⣈⠁⠘⠋⠀⠛⠀⠰⠀⠠⠄⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠋⠀⠛⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⣋⠄⠛⠃⠛⠃⢴⠗⠀⡦⠀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠙⠀⠘⠃⠰⣷⠴⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⢳⣠⣤⢀⣄⠀⣈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢛⠄⠊⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠉⠀⠉⠀⠘⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠘⠃⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠚⠀⠐
⠀⠀⠀⠀⠀⠛⠀⠲⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠖⠀⠤
⠀⠀⠀⠀⠐⠃⠀⠦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⡄⠀⡀
⠀⠀⠀⠀⠰⠆⢠⡄⢀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠤⠀⢠⡀⢀⡉⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠁⣤⠀⢀⠀⠀⠁
⠀⠀⠀⠀⡄⠀⣀⠀⢈⣁⣈⠛⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣠⡄⢀⡁⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡉⣰⠏⠁⠈⠁⠸⠿⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⠁⢈⠁⠀⠋⠀⠐
⠀⠀⠀⠀⠀⠈⠁⢿⡀⠀⠀⢀⢾⠆⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⠀⠙⠁⠘⠃⠀⠄
⠀⠀⠀⠀⠀⡸⠄⠘⠛⠳⠶⠋⣴⠄⣤⡄⢹⣷⣦⣤⣀⣀⣀⣀⣤⡾⠿⠛⣿⣦⣀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠛⠀⠛⠀⠲⠂
⠀⠀⠀⠀⠘⢧⣠⠶⠀⠠⡆⢠⣤⠀⣨⡁⣈⡉⠈⡛⠛⠿⠋⠿⠟⢸⡷⢰⣶⠟⣿⣿⣿⣿⣿⣿⠟⠛⠁⠘⠃⠐⠇⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⠀⢀⡀⠀⣉⠀⢙⠃⠘⠟⠸⠿⠀⣿⠆⣶⡦⢰⣦⢀⣭⠀⣙⡁⠘⠋⠀⠛⠀⠾⠂⠰⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⠋⠀⠛⠁⠺⠇⠰⡷⢠⣶⢀⣤⡄⣨⡅⢈⡉⠀⡙⠁⠛⠃⠰⠗⠀⠶⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠐⠗⠀⠶⠀⢴⠄⢠⡄⢀⣍⠀⣉⠁⠘⠋⠘⠛⠀⠿⠂⠰⠆⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⣤⠀⣠⡁⢈⡉⠀⡛⠁⠛⠃⠘⠇⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠘⠛


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠒⠀⠤⠀⠀⠀⠀⠀⠀⠠⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⣾⣿⠀⠀⠀⡈⠰⠒⠈⠀⢀⡀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣠⣤⠉⠒⠀⠀⠀⠀⠀⠘⢠⡾⢹⡆⢀⣤⡀⠆⠀⣠⠶⣦⡟⢹⡆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⣍⣷⠇⠀⠀⠀⡘⠀⢠⠀⣾⣄⣼⢁⣴⣿⠁⠘⡄⠈⣖⡁⠀⡘⠃⠀⢡
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⠀⠀⠀⠠⠁⠀⠈⠑⢢⣌⡁⠘⠷⠏⠀⡜⢡⠀⠀⠉⠛⠁⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠶⣀⡀⠀⠀⢸⡆⠀⢠⣆⣀⠀⠀⣠⢪⠟⠀⠀⠒⠀⠖⠀⠀⠧⠀⠀⢀⣀⡤⠐⠃⠁⠀⠀⠀⠀⣀⣀⠠⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⢤⠈⠉⠒⠋⣌⡡⠷⠖⠒⠂⠀⢄⠀⠀⠀⠀⠁⠉⠀⡟⢹⠀⠀⠀⠖⠈⠉⠁⢀⣠⠒⠢⡀⢡
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡠⠀⠠⡀⠀⠀⠐⢟⠃⡀⠀⠀⠀⠐⡄⠱⡀⠀⠀⠀⠠⢈⠴⠁⠀⠀⠀⢠⠀⡠⢤⡅⠈⣠⠴⠾⣮⡄
⠀⠀⠋⠉⣉⠁⡀⠀⠀⠁⠀⠇⢊⣁⠈⠀⠀⠀⠈⠑⠂⢉⠂⡀⠀⠈⠀⢈⠢⣤⢀⡐⠂⠀⠀⠀⠀⠀⣀⡄⠇⠠⠬⠙⠺⠀⠐⠠⠬⢉⠑⡐⠒⡄
⠀⠀⠀⠀⢻⣶⣿⢶⣶⠀⣦⠞⣦⣙⠣⢄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠑⢁⠾⠿⣀⠂⠄⣀⠔⡀⠟⠳⠀⠀⠀⣀⡄⠀⢈⠀⡤⡄⢰⠁⢠
⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠋⠁⠀⠀⠑⢢⢡⡀⠀⠀⣀⠞⣠⠁⠀⠀⠀⠀⠀⡘⠁⠀⠀⠉⠒⣵⣧⠀⠀⠀⠉⠉⠁⢠⠁⠀⠀⣶⣠⠖⠋⠀⡆
⠀⠀⠀⣄⣀⣈⠀⠀⠠⠴⠃⠀⠀⠀⢠⠘⣡⡤⠉⢩⠃⣰⠿⢒⡶⢂⣀⢐⡘⢧⡔⣀⡀⢦⡰⠀⣡⠂⡀⠀⠀⠀⠀⣌⠀⠀⠀⠿⠿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⡅⢸⠓⢀⠔⣾⠀⣿⢦⡈⠲⠉⢛⣹⢋⣦⡙⢄⣼⠛⡇⠉⡼⠀⠿⠗⠠⡀⠀⠀⡀⢿⡶⠒⠀⠠⠤⠄
⠀⠀⠀⠀⠀⣀⠠⡀⠀⠀⠀⠀⣧⡈⢀⠰⠀⠁⢸⠈⢆⠑⠴⡿⠶⡀⢻⠤⢬⡙⠛⢋⣡⡎⢀⡶⣡⣄⠀⠈⠐⠬⣀⣈⠤⠊⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⡠⠔⠉⠀⠀⠘⣄⣠⡆⠅⠀⠠⣉⠳⣟⠻⡦⠛⡿⠿⢢⣶⡎⡉⠀⠀⢸⠻⣏⠴⣽⣿⣣⡾⢻⢋⡈⡕⡄⠀⠀⠀⠀⠰⠀⠈⣩⠉⠁⠀⠀⠀⢸
⢶⠁⠀⢀⡠⠤⢀⠀⡈⣿⠁⠀⠀⠀⠈⠑⠌⣙⣛⡡⢃⠴⠶⢘⣤⠃⠴⣀⠮⣱⡌⢲⠈⠈⠓⣟⢌⣠⡔⢁⣄⣀⣀⣀⣤⣾⠀⠼⡄⢸⠃⣀⠶⢄⡸
⠀⠆⠀⠘⠄⠀⠔⠠⠀⠈⢦⠀⠀⠀⠀⡀⠐⢨⡿⢳⠣⠾⣯⡖⣡⣾⣳⣌⡙⢶⣧⣼⣶⣾⣖⣻⠀⣿⣿⢛⠛⠤⠒⠈⠉⠉⠀⠀⠀⢰⡎⡔⠶⠂⢸
⠀⠀⢣⠀⠂⠀⠀⣀⠠⠐⠂⠀⠀⠀⠘⠣⣀⠻⠿⢮⣷⣶⣆⣴⡋⣯⠟⣩⡗⢈⡿⢫⡇⢙⣿⡏⣴⣿⣴⠾⢁⠰⠆⠀⠀⠀⡆⠀⢀⣀⣱⠀⠀⠀⢈⡄
⠀⠀⠀⢣⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠁⠐⠀⠈⠉⠉⠛⠛⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠂⠈⢀⠄⠀⠀⠀⠀⠓⠂⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢠⠐⠒⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢀⠀⠀⢀⡴⠂⠉⠉⠉⠁⠢⣧⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠈⠐⠂⠨⣱⡀⠃⣼⠊⢁⠍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⠀⠀⠈⢁⣤⣀⠀⣠⣓⣤⡀⠀⠀⠀⠀⠀⠀⠀⢁⡠⣀⣀⣀⡠⢀⠈⢉⠄⣪⠰⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⣼⠆⠀⠀⠀⠻⡁⠰⠛⠟⠥⠁⣹⠄⠀⠀⠀⠀⠀⠠⡞⠙⢇⣷⣿⡋⠛⡇⠀⠁⢾⠉⡾⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢇⠠⡀⠀⠀⠀⠈⠐⠂⠀⠐⠊⠁⠀⠀⡇⠀⢘⡇⠀⠀⠑⠀⠤⠤⠄⠊⠀⠀⢀⡜⠈⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡷⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠺⠗⠆⠀⠀⠀⠀⠀⠀⠀⢰⣿⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠗⠀⢠⡆⠀⢠⡄⠀⠀⠀⠀⠀⣸⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢤⣄⢀⢀⣀⣠⣄⣅⠀⠀⠀⠀⠀⣀⡁⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣈⠋⣛⠡⣉⠉⠁⠀⠀⠀⠀⠀⠋⠃⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⡊⠤⠤⠬⠂⢀⣀⣉⠳⣄⠀⠀⠀⠀⠀⢄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢿⠲⣤⣤⠠⡄⣤⣤⣤⣄⣉⣉⣳⡄⠀⠀⠀⠕⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠉⠓⠾⠦⠶⠶⠜⠙⢉⠝⠁⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⡀⣀⣀⣤⠂⠁⠀⠀⠀⠀⠈⢈⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠖⠙⠃⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠙⠦⣄⣀⠀⠀⠀⢀⠠⡤⡐⠔⣰⠏⡌⠄⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠡⠉⡙⠃⠲⠖⠶⠲⠈⠁⠀⠌⠌⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠡⠁⠀⠈⠘⠀⠀⠀⠋⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠠⠄⠒⠒⠀⠂⠙⠲⠛⠶⠚⠶⠃⠾⠺⠶⠻⠾⠱⢛⠸⢃⠾⡚⢶⠻⢛⠾⡃⢿⡚⢛⠗⠛⠛⠚⡛⢛⠻⠟⠛⠈⠢⠐⠈⠀⠀⠀⠀⠄⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠿⠟⠛⠛⠻⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣦⡀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠛⠛⠛⠃⠀⠀⠸⣧⡀⠀⣴⡿⠋⠉⠉⠉⠙⢿⣦⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⠿⠶⠶⠶⠶⠶⠶⠶⠿⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⢶⣶⣶⠶⠶⠾⠟⠛⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢴⢰⠀⠀⢔⢲⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣞⣲⠀⢠⣞⡦⠈
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠤⠀⣁⠀⢀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⢈⣥⣤⣤⡉⠁
⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠘⠴⠾⠷⠜⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠂⠻⠶⠿⠶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⡀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⢀⣀⣠⠴⠋
⣶⣶⣶⠾⣿⢿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⡀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧
⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣤⣤⣤⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⢿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠿⠻⣿⣿⣿⣶⣤⣄⣀⣀⣠⣤⣴⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⣠⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠉⠙⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⢀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⣠⣾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠶⠶⣦⣤⡶⠿⠿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡖⠁⢀⣼⣿⣧⣀
⠀⠀⠀⠀⠀⠀⢀⡛⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣀⣴⠿⠛⠁⠀⠉⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠛⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠘⣧
⠀⠀⠀⢸⡏⢠⡾⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⢠⣤⣀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠘⠞⠉⢹⡆
⠀⠀⠀⢸⡇⣶⡀⠀⢰⡾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣀⣸⣷⠀⢀⣼⠃
⠀⠀⠀⢸⡇⠈⠛⠾⠛⠁⠀⠀⢀⣴⠶⠺⠷⢶⡶⠀⠀⠀⠀⠉⢰⠟⠿⣇⠹⣆⠺⣇
⠀⠀⠀⠈⣿⠀⠀⣀⣠⡄⠀⠀⠻⠁⠀⠀⣠⣤⣄⡀⠀⣀⠀⣷⠘⠃⠘⠃⠀⠘⣧⠙⣦⡀
⠀⠀⠀⠀⢿⡄⠀⠙⢿⡿⠃⠀⠀⠀⢴⡾⢉⣹⡟⠃⠀⠈⠁⠘⢷⣄⡀⠀⠀⠀⠘⣧⠈⣷
⠀⠀⠀⠀⠸⣧⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⣿⠀⣿⡀
⠀⠀⠀⠀⠀⣿⠀⣹⠟⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⣿⠀⢻⣆
⠀⠀⠀⠀⠀⢻⡆⢹⣆⢺⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣷⠀⠀⣿⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢸⣧⠀⠹⢦⣍⣁⣴⣄⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠸⢿⣿⣿⡿⠀⠀⢿⡀⠀⠀⠀⠹⣷
⠀⠀⣠⠾⠿⠿⣿⠀⠀⠀⠉⠙⠉⠹⣦⡀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿
⠀⣼⡏⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⣿⡶⣦⣄⣼⠀⣿⡀⠀⠀⠀⠀⣀⣀⣤⣾⣏⡄⠀⠀⠀⢀⣿
⠀⣿⡀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠉⢹⡀⢸⡟⠛⠛⢛⡿⠛⢋⣉⡠⠮⠀⠀⠀⠀⣼⠋
⠀⠹⣧⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡇⢸⣇⠀⣴⠟⢁⣼⣋⣥⡄⠀⠀⢀⣤⣾⠃
⠀⠀⠙⠳⣦⣤⣤⣾⠟⠃⠀⠀⠀⠀⠀⣤⠀⣿⡀⠸⣇⠀⠻⠟⠃⣴⡿⠛⢉⣠⡶⠶⣤⣬⣭⡿⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣄⠀⣴⣦⣤⡟⢀⣿⡇⠀⢻⡄⠀⠀⠀⠀⠀⠰⠟⢉⡴⠶⢦⣿⠶⠶⠶⠶⠞⠛⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣏⣀⣽⠟⠛⢻⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⢰⡟⢀⡴⢿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡼⠟⠛⢿⡄⠀⠘⢿⣀⡀⠘⢇⠀⠀⠀⠀⠀⠀⡇⠈⠀⢸⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣀⣤⣤⣴⣶⣿⣿⣏⠁⠀⠀⠀⠘⣷⠀⠀⠀⠉⠛⠛⠿⡆⠀⠀⠀⠀⠀⢰⣄⣴⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣴⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⠿⢻⡿⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣴⠋⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠛⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉
⢸⣿⣿⣿⣿⡿⠿⠿⠛⠿⠿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢸⡏⢹⡏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⠀⢀⣾⣿
⠀⠙⠻⣧⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠠⣶⣤⡶⣶⣶⡶⠶⠛⠁⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠀⣿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣀⠴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠶⠟⠛⠛⠛
⠀⠀⠀⣿⡀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣀⣠⣤⡤⢶⣤⠶⠶⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠋⠉⠀⠘⢧⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠾⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⣴⣾⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠖⣶⡟⠁⠀⠀⠀⠀⣠⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣀⣴⡞⣋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣀⣴⣾⡛⣿⠛⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠘⠿⠁⢾⣅⣷⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣴⡟⠿⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠈⠛⠋⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠏⢉⣉
⡇⠀⠀⠀⠑⠠
⡇⠀⠀⠀⢰
⡇⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖
⡇⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠴⠃⠂
⡇⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠲⣀⡀⠀⣀⣀⢄⠀⠀⠀⠀⠀⢀⣠⡀⠀⢀⠤⠄⠂⣤⡠⠤⠖⠋⢻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⢲⣿⢧⡀⠈⠉⣙⠒⡲⠊⡀⠀⠀⠀⠀⠘⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠀⠀⠀⠀⠠⡜⠽⡆⠀⠀⠀⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣰⠠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠠⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢼⢃⡄⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢼⣏⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠂⠀⠀⠠⠤⣄⠀⢴
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⡑⠀⠀⠀⠀⢿⡄⢀⣤⣶⡶⠶⠶⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⡀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⢴⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠙⣆⠀⢤⣾⣿⠫⠃⠀⠀⠀⠀⡌
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⢿⠀⠀⠀⠈⠀⠀⠀⠀⠀⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢈⣁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠇⠀⣀⡴⠟⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠀⠀⢀⣠⢄⡴⠀⠀⢸⣠⣼⣫⢔⣠⠞
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠭⣭⠉⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⠽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠤⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢹⡧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡩⠖⠛⡰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⢶⢹⡾⣤⣇⡇
⠀⠀⠀⠀⠀⠰⢫⠖⠂⢠⡃⠀⠀⠘⣄⢘⠥⢤⡀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⣉⠤⠤⡄⠁
⠀⠀⠀⠀⠀⢰⡧⡆⢀⠀⠙⠢⣄⠀⠈⠺⡄⠀⠈⣳⡀⣠⠖⠊⠉⠉⡹⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⢼⠀⡟⣄⡴⠲⢜⣧⠀⠀⠹⣄⠀⠀⠹⣍⣀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠈⢁⡀⠤⠿⣥⠀⠀⠈⠳⢤⣀⣽⢩⠑⠒⠒⠒⠒⠶⠦⠤⢄⠀⠀⠀⣠⢽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠚⠛⠓⠈⠁⠀⠀⠀⠀⠒⠁⠈⠑⠦⣄⣀⣀⣀⣀⡠⠞⠀⠀⠀⠳⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠊⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⢸⡛⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⠛⠲⠤⣄⣀⠀⠀⠀⠈⠀⠉⠳⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠦⢄⡀⠈⠉⠑⠒⠤⢄⡀⠀⠀⠀⣸⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⣙⠒⠤⣄⡀⠀⠀⠉⢱⠿⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠃⢀⡆⠀⠀⢿⠤⠀⠀⠉⠓⠦⣀⡏⠀⢸⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠼⢤⣤⣤⡼⠀⠀⠀⠀⠀⠀⠨⠥⢤⣸⢇⡡⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠒⠤⣤⣀⡿⡇⠀⠀⠀⠀⠀⠀⠒⠒⠒⠋⣻⠉⣠⠞⡡⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣀⠀⢱⡏⠙⢒⡒⠒⠒⠒⠒⠒⢲⢶⠞⠙⡷⣮⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢦⣵⢋⣈⠮⠤⠋⠀⠀⠀⠀⡠⢫⡎⣀⣴⡧⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠠⣖⣚⡷⣄⣈⠓⠚⣟⣩⠆⣀⣀⡤⠴⠖⢺⠟⠛⠛⠛⣆⡛⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠓⠒⠢⠤⠤⠤⠤⠤⠤⠛⠒⠒⠺⠯⠯⠭⠗