Joking Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⡤⠖⠉⠉⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠴⢅⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠲⣄⠂⠉⠉⠀⠉⠉⠒⢄⡔⠋⠀⠀⠀⢀⠤⠒⠑⠬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠈⢁⡢⠤⠄⣀⠀⠀⡀⢀⡤⠒⠂⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⣱⠊⠉⠫⣠⠤⠤⣄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠠⠤⢰⣍⠉⠉⠁⠀⠀⠀⠀⠀⣼⡻⠭⢯⡉⣏⢩⡆⠀⣀⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣽⡋⢭⣭⣺⣀⠀⠀⠀⠀⣀⣀⣤⠯⣤⠶⢖⣛⣻⡄⢉⡩⠥⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢿⣉⣉⣁⣤⠭⣽⠋⠉⠉⢫⡏⠀⠀⠈⢏⠉⠀⢰⡇⠸⣄⠀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⢮⠽⠖⠒⠛⡇⠀⠀⠀⣀⠀⠀⠤⠐⢺⠀⠀⢸⡇⢠⠤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣽⠀⠀⠀⠀⡗⠀⠁⠁⢀⢆⠀⠀⢀⠃⠀⠀⠘⡇⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠈⠢⠤⠔⠃⠀⠉⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢘⡧⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠉⠒⠤⠤⠀⠀⠤⠤⠐⠉⠀⠀⠀⢨⡗⠒⠒⢦⡀⠙⢄
⠀⠀⠀⠀⠀⣀⠤⠤⠼⠧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢹⠀⠈⠓⢤⡀
⠀⠀⠀⡠⠊⢀⠔⠒⢰⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡜⠀⡰⡦⡀⠑
⠀⠀⢰⠁⢰⠃⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠑⠊⠁⠳⠝⠒
⠀⡴⠋⠀⠈⡆⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⡜⠀⡴⡦⣀⣙⡆⠀⢸⡇⠀⣿⠀⠀⠀⢀⡀⢀⠀⡄⡆⠀⠄⠪⠀⢒⣂⣠⡼⠇
⠳⠴⠵⠃⠀⠀⠀⠀⠈⢧⠀⣿⠀⠀⠀⠐⢃⣘⣪⡥⠤⠶⠒⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣿⣶⣶⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⠻⠛⡛⡛⠛⠛⠛⢻⡟⠛⡟⡟⠟⡛⣿⠻⣿⠛⠛⠛⠛⢻⡟⡟⡟⠛⡛⣛⢿⠛⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠙⢿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠚⠛⠓⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠍⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⡇
⠀⠀⡇⠀⠀⠀⠈⢂⠀⠀⠀⡠⠁⠀⠂⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢀⠈⠁⡃⢠⠎⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠑⠄⡀⠀⠂⡴⣶⠐⡀⠀⠀⠔⡭⣅⠢⠀⡤⠁⠀⠠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠛⠛⠄⠁⠀⠀⠢⡳⢛⡴⠁⠁⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡜⠀⠰⢓⠆⠀⢰⠦⠀⠀⠀⡾⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⡀⠂⠋⠀⠀⠋⠀⠀⣀⡞⠁⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡌⡀⠈⠉⣁⡀⠤⠚⠁⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠁⡄⢂⠀⠀⠀⡀⡀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠈⠁⠀⠈⠁⠉⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⣀⣀⡠⠖⠉⠀⠀⠀⠀⠑⠎⠁⠀⠀⠀⠈⠓⢤⣀⣀
⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⢸⠒⠤⢀⣀⠤⠔⠊⠁⠀⠙⠢⠤⢀⣀⠤⠔⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡒⢤⠀⠀⢀⠔⠈⠉⠒⠈⠉⠒⢄⠀⠀⢠⠄⠠⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎⠮⠜⣀⠔⠁⠀⠀⠀⣠⠀⠀⠀⠀⠑⢄⠐⠼⠌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠀⠉⠀⠀⠀⠀⢀⠔⠁⠳⡄⠀⠀⠀⠀⠉⠒⠊⡰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⡀⠤⠒⠁⠀⠀⠀⠈⠑⠤⢀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⢦⠔⠒⠢⣄
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡇⡇⠀⠀⡎⡔⠃⠀⡠⠃⠀⠀⠀⢠⠏⢆⠀⠀⠀⠀⢣⡀⠐⠓⡌⡆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⢇⠉⠒⠊⠀⠀⠀⠀⡰⠃⠀⠈⠣⡀⠀⠀⠀⠈⠒⠊⢁⠃
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠑⠠⠤⠠⠤⠔⠋⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠤⠔⠁⠀⠀⣀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠂⠀⠀⠢⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠈⠄⢸
⠀⠀⠀⠀⠀⣄⣀⣀⡠⠔⠒⠁⠀⠀⠀⠀⠈⠑⠢⠄⣀⣀⣀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠘⠦⣀⡀⠀⢀⣀⠤⠚⠑⠦⣄⣀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠀⡒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠆⠵⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢦⡽⠉⠑⠰⠤⡀⠀⠔⠃⣀⡤⠀⠀⠀⡘⢀⠞⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⢠⠀⠠⠃⠉⢠⡶⢀⠀⠀⠠⢲⠌⣠⡁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠛⠒⠏⠠⡘⢀⠀⠀⣀⢶⣡⠧⠄⠎⡡⢡⢰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠈⣴⣴⣶⣿⣇⡸⣷⠀⠠⢀⣀⠀⠀⠂⡞⣨⠞⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠔⠂⠀⠉⠁⡜⠿⠴⢟⠇⠀⠃⠀⠀⢀⠈⠉⠃⢌⢁
⠀⠀⠀⠀⠀⠀⢠⠀⠇⡴⠗⠴⠶⠀⢀⠀⠁⠀⠈⠁⠀⢀⠖⠀⠀⠈⣗⠀⠖⠀⡈⠑⠠⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠈⠘⢠⠤⠤⠄⢠⡰⠁⠀⠐⡦⢰⡀⠄⠋⠀⠀⠀⢠⡷⠠⠄⠀⢁⠀⣼⠐⣧⠗⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠘⠌⢤⡀⢀⠄⢪⠌⢔⠆⠀⠘⠃⡏⠀⠀⢀⡀⠶⠀⠈⠐⣧⠔⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⡴⢀⠀⣩⠀⠀⠨⣿⢍⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⡒⠋⠑⠒⠪⠷⠈⠀⠀⢀⠃⠀⠀⠀⠀⠀⠦⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠠⠀⠀⠘⢦⠀⠀⠀⢀⡜⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡀⢢⠀⠀⡅⠀⠀⠀⢀⠾⢷⡖⠀⠀⠘⡆⠀⢀⠰⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠰⠂⠀⠀⡰⠃⠀⠀⠘⡇⠀⠀⡀⠀⠈⠁⠀⠀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⡀⠄⠀⡀⢀⠃⠀⠀⠀⠀⢸⠔⠋⠉⠑⠢⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠚⠝⠂⠉⣱⡄⠀⠀⠀⠸⠤⠄⢀⣄⡀⠀⠠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡈⠀⢀⡴⠊⠉⠁⡈⠀⠀⠀⡀⢀⠀⠀⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠔⠉⠁⠀⠀⠀⠫⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⡀⠔⠃

⠀⠀⣿⣿⣿⣿⣿⣿⣟⣏⣋⣍⣏⣋⣋⣻⣻⣿⣛⣿⣏⣿⣋⣋⣉⣿⣟⣟⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⡟⠉⠉

⠀⠈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⡀⠀⢀⣀⡀⠤⠂⠁⠀⠀⠀⠁⠒⠀⠤⠤⠤⠀⢒⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⡹⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣎⠦⠤⡀⠀⠚⠑⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣄⠀⠀⠀⠀⢀⡀
⠀⠰⠀⠤⠤⠒⢒⠯⠭⠭⡀⠀⣠⠀⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡄⠀⢀⠁⠐⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠒⡖⠁⠀⡌⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⠒⠁⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠀⠀⣧⠀⠴⠀⡀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡐⠋⠣⣤⠴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⠰⠀⠀⠀⠀⣀⣣⠜
⠀⢀⠀⠀⡀⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⡁⠉⠁⠑⠊⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣯⡍⠉⠉⠉⠙⠂⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣴⣮⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⣨⠟⠁⠀⣠⠖⢻⠉⠉⠙⠒⠲⠤⠤⠤⠤⢤⡀⠀⣼
⠀⠀⠀⢀⣾⣵⠆⠀⡞⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃
⠀⠀⠀⢘⡝⠁⠀⠰⡇⠀⠈⣠⡶⠶⠶⠴⡄⠀⠀⢠⣤⣤⠀⣇
⠀⠀⢠⣾⣴⠂⠀⢀⡷⠀⠀⠋⡠⠔⠢⢄⠀⠀⠀⣈⠤⣈⠓⡿
⠀⠀⠘⣱⡇⢀⡀⢸⠀⠀⠀⡞⠀⠀⠀⠀⢣⢀⡞⠀⠀⠈⢷⡇
⠀⠀⡞⡡⠒⠂⣼⡎⣇⠀⠀⣇⠈⣀⡤⢀⡼⢸⣀⡤⣈⠁⡼⠓⠲⢤⣀⡤⠤⢄⡀
⠀⠰⡏⢷⠁⣎⠉⠓⠛⠀⠀⠈⠙⠛⠒⠊⠀⠹⡀⠀⢢⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠦⣄
⠀⠀⢳⣄⣈⣃⠀⠐⣫⠔⣒⣢⠀⠀⠀⡀⠀⠀⡇⢀⣮⠤⢤⡀⠀⢠⠇⠀⠀⠀⢀⠀⠀⠀⠘⠦⠤⢤⡀
⠀⠀⢸⣤⠀⣸⠀⠀⢠⡾⠓⣌⠑⡄⠸⣀⡀⢠⠃⡞⠓⠒⢤⣸⢀⠎⠀⠀⠀⢠⠋⠀⠀⠀⡴⠀⠀⠀⣧
⠀⠀⠈⠹⡄⡏⠀⠀⣾⣤⢼⠋⠉⢻⣦⣤⠬⣭⡤⢷⣤⣀⣀⣥⡟⠀⠀⠀⠀⡏⠀⠀⠀⡸⠁⠀⢀⠀⡟
⠀⠀⠀⠀⢷⡇⠀⠀⢸⠀⠀⠓⠶⢯⡀⢈⣟⠉⠈⣯⠃⠀⠀⡼⠀⠀⠀⠀⡸⠀⠀⠀⢰⠃⠀⢀⣏⣰⠇
⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡖⠢⣄⣠⢍⠉⢉⠛⠛⡏⠀⠀⠀⣧⡀⠀⠀⣰⠃⠀⠀⢠⠏⠀⢠⠟⢩⠏
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠑⠶⣿⣄⣠⢿⣉⣿⠷⠳⡄⠀⠀⠈⠛⢍⡉⠉⠒⠒⠲⣅⣀⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠈⠉⠉⠉⢁⣠⠔⠛⢦⠀⠀⠀⠀⣹⢤⣀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠒⠚⠉⠀⠀⠀⠈⠳⣄⣀⣠⠃⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⡄⠢⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠒⠉⣁⠀⠀⠀⠀⠉⢢⠃⠀⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⠊⠉⠲⡄⠀⠀⠀⠈⡀⠀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡀⠀⢀⠀⠀⠀⠀⢀⢠⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠒⠤⡀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠤⠴⠁⠤⠌⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⠑⠢⡤⠐⠀⠀⠀⠀⠀⠀⠀⠇⢁⣠⣞⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠃⠠⢄⡉⠑⠂⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⠙⠲⢤⣀⡀⠀⠀⠀⠰⠿⡏⠐⠤⣀⣀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠈⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⢦⠀⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡀⠀⠀⠀⢀⠔⠉⠀⠈⠀⠠⢄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠀⢀⣀⠄⠀⠀⠀⠡⡑⢀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡈⠁⠛⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡄⠀⣀⠄⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠘⡰⠐⡀⠀⠀⠀⠈⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠤⠄⠀⠀⠀⠀⠀⢡⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠂⠀⠠⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠘⢢⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⣄⠀⠀⠀⠃⠀⡤⠤⠀⠰⡀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⠀⠘⣿⢷⣄⡘⠀⠰⠀⠀⠑⣄⠙⢄⠀⠀⠀⠀⣿⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⢨⠊⡝⢧⣘⣀⣀⣀⣈⣿⡷⣌⣡⣄⠀⠀⠏⠀⠀⠀⠀⠙⠛⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠒⠂⠸⢧⠙⢿⠟⣷⣾⡛⠙⢋⡉⢩⡍⣃⠀⢸⣧⣿⢤⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⠊⠀⡴⠛⠒⠂⠠⢼⣷⠟⠻⠋⡀⠴⠂⢚⠁⠠⠤⠇⠀⠀⠉⠁⠘⡗⠁⢑⠀⠀⠀⠠⠒⢢⡄⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⡄⠀⡄⠀⠀⣴⡟⠁⢄⠴⢊⣠⡴⢿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢏⠀⢻⠀⠀⢸⡷⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣷⣼⠏⣈⠶⢃⠔⠈⠻⣷⣟⣽⣧⡀⠀⠀⠀⠀⠀⢈⡇⠀⠁⠉⠀⠀⠀⠀⠀⠊⠀⠀⢀⣼⣿⡟⠃⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠿⠩⠤⣍⠰⠁⠀⠀⠀⠈⢿⣦⡙⣷⡄⠀⠀⠀⠀⠈⢃⠀⠀⢰⠶⣄⠀⠀⠀⠀⠀⠀⠈⠁⡌⠓⠀⣠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣷⢾⣷⠀⠀⠀⠀⠀⠈⢤⠀⠀⠦⢀⠆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣹⣧⠀⠀⠀⠀⠀⠀⢱⣤⣀⠀⠀⠀⠀⢀⠀⠤⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣙⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠛⢻⠏⠁⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠖⠒⠒⠊⠉⠉⣿⡟⢿⡗⠢⣤⣄⣀⡀⠸⣿⡟⠀⠀⠰⠎⠀⠀⠠⠤⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢫⡝⠣⠶⢶⣤⣄⣀⡀⠀⠀⣿⣧⣼⡇⠀⠀⠈⠉⠙⠲⠿⣥⣀⡀⠀⠀⠉⠁⠀⠀⠀⠃⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢸⡇⠀⠀⠈⠁⠘⠛⠩⠶⢶⣿⣧⣼⡇⢀⣀⣀⣠⣤⡤⠔⠒⠛⠋⠑⢄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⢻⡅⠀⠀⠀⠀⠰⡄⠀⠀⠈⠁⠐⢋⢿⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⢀⠀⠀⢠⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⡖⠒⠊⠀⢡⡄⠀⠀⠀⠀⣾⠀⠹⣆⠀⠀⠀⢀⣀⣠⣤⣤⣴⠒⠚⠃⠀⠀⠀⠀⠈⢸⡏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠑⠀⠀⠀⠀⠈⠐⢦⠀⠀⣿⠂⠀⠙⠷⠚⠉⠙⣿⣿⡏⠉⢡⣄⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⢰⡀⠀⠀⠀⠠⠊⠁⠀⠀⢸⠇⠀⠀⠀⠀⢀⣬⢻⣿⠟⢻⡁⠉⠙⠚⣤⡉⠉⠉⠁⠘⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠈⠉⠉⠉⠲⣤⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣿⢰⣯⣼⠀⢀⠉⠁⠂⢰⡄⡸⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢦⣄⣀⡀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠿⡌⠛⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢦⣄⣀⣸⡅⠀⠀⠀⠀⢀⣀⣠⠄⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⠖⠈⠉⠁⠀⠀⠈⠦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⢤⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⠠⡤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⢩⠀⠀⠀⠀⠈⢉⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣸⠀⠀⠀⠀⠀⠀⠀⣇⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠁⠙⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣸⢿⣶⠟⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠛⡈⠉⣿⣿⣦⣤⣼⠀⠀⠀⠀⠀⣤⡊⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⡈⠀⠈⢺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⣿⡿⣿⣿
⣿⣤⣶⣴⣴⣦⣾⣤⣼⣧⣤⣼⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣾

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠻⠟⠟⢛⣻⠛⠛⠛⢻⡟⣻⢻⣛⣛⣿⠛⣿⠟⠛⠛⠟⣿⣻⢻⣛⡛⣻⣻⢻⣟⡿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⠆⠀⠐⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡠⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠀⠉⠁⠀⠀⠈⠁⠀⠐⠠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⢨⡠⢠⠂⠁⠄⡄⠀⠸⠢⣲⠐⢇⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⢾⠀⣡⠤⢀⠐⠀⠀⣇⠀⠌⡆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⡎⠈⠛⠀⣇⠾⢆⣧⡄⢃⢫⠈⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⣹⠖⠋⠀⡬⠋⠉⠘⠒⠻⢌⢉⠉⣦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢸⠀⠀⠀⠀⠀⢰⠁⣠⡆⠀⠙⠲⠤⠤⠤⠤⠚⠀⢻⡁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢸⠀⠀⠀⠀⠀⠘⣄⠀⡷⢄⡀⠀⠀⠀⠀⢀⡼⠛⠀⡇⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢦⢸⣈⠉⢱⠒⠒⠏⣹⠁⠀⡼⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⣿⣾⣶⣾⣿⠃⢀⡜⠁⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⠘⢟⠻⠿⠻⡏⣰⠋⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣆⠨⢕⣓⠉⢀⡇⠀⠀⠀⠀⢄⠀⢀⡀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣤⠤⠞⠀⠀⠀⠀⠀⠀⠉⠒⠒⠐⠒⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠙⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢀⡠⠴⠒⠉⢯⠀⡧⢌⡡⡄⢈⡟⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣇⠀⠀⡎⠀⠀⠀⠀⠇⠀⠀⠈⢀⡇⠀⡡⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⢞⣁⠔⡄⠀⠀⠀⠀⠀⢣⢈⡴⠎⠀⠙⣆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⠱⣡⠩⢬⡃⠀⢀⡀⢠⠀⠈⠌⠑⠄⠀⠀⠙⢣⢆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⣼⡁⠀⠠⢀⠀⠀⢀⣀⠤⣴⡶⡄⡀⣀⠀⠀⠀⢴⢤⡈⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠼⡂⣿⡿⢂⣤⡤⢖⠮⠡⢰⠀⡀⣈⣨⣩⢹⡉⠻⠟⢷⣦⠁⡆⠑⢌⠂⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⣸⠇⡞⣴⣿⢹⣉⣸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣯⠱⢃⣤⠮⠴⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⡿⣄⢘⠊⣿⣭⣿⣿⣿⣟⠉⠙⢻⡟⠉⢽⡿⢛⡩⣽⡏⢩⡎⣼⠀⠀⢺⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡖⢁⣧⠟⣿⢹⡄⠙⢧⢏⡉⡿⡷⠀⢸⠁⠀⢸⣼⣚⠴⠿⠁⣼⡘⣧⢸⠀⡞⠧⡵⠕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣴⣯⡞⡗⣄⠀⠻⣶⢷⡇⠀⠈⠀⠀⠀⢿⠁⠀⣠⣾⡿⣠⡄⣿⡀⠃⢸⡆⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⢢⠋⠀⣿⡟⢷⡷⣯⡳⣄⣀⣸⠇⠀⠀⠀⠀⠀⠈⢧⡾⡽⣿⣿⣏⢈⢻⣷⣠⠿⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⢳⣤⣾⣼⣼⣷⡿⢛⡏⣹⡟⡆⠀⠀⠀⠀⠀⠀⠀⣳⡀⠈⢻⣿⠦⣿⡙⡅⡑⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⢨⡿⣃⠽⡿⠛⠁⢸⡇⣿⡇⠣⣄⣀⣀⣀⡤⢂⣼⣿⡷⢰⠌⢘⣷⡄⠉⣢⣥⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣏⠀⠀⢾⣤⡴⠏⠈⠁⣿⣧⣀⡀⠈⠁⣀⣴⣿⣿⣿⡇⡈⠿⣿⣿⣿⡀⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠈⡿⠀⠀⢾⠁⠀⢀⣦⢀⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣿⣿⣿⡀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⢸⣼⣆⠐⣼⣿⣞⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⢃⣄⣾⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⢸⣾⣿⣿⡄⢻⣿⣿⣿⣿⡇⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠋⣼⣿⣿⣿⣿⣞⣿⣿⣿⣿⡇⠀⠀⢸⣿⣳⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⡾⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣞⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋⣛⣶⣿⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡿⣁⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣿⡟⠉⠛⠻⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⢿⣿⣿⣷⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠿⠁⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣧⢠⣄⣀⣀⣀⡄⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⣦⡙⠛⠛⠋⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣿⣦⣤⣶⣾⣟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠛⠻⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⣠⡆⠀⠀⡀⣠⣤⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠉⣿⠙⠁⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠶⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢼⣿⣿⡷⠀⣿⣠⣤⣽⣿⣿⣿⣿⣿⣿⢋⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢺⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⠃⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢻⡛⠛⣿⠋⣿⣿⣷⢸⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⡇⠀⠋⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢿⣿⡿⠸⣿⣿⠇⠀⠀⠀⠀⠀⠾⠗⠀⠀⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢤⣀⡀⠀⠀⠀⣠⠔⠒⠉⠉⠉⠉⠑⢲⡞⢉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡉⢲⣊⣁⠀⠀⠀⠀⠀⠀⠉⠓⢤⠊⠀⠀⠀⠀⠀⣠⠔⠋⠁⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠉⣲⡤⠤⢄⣀⠀⠀⢸⠀⠀⡠⠔⠒⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠑⡤⠚⠒⢾⠀⣀⣀⣀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⡤⠴⠒⠒⠞⠁⠀⠀⠀⠉⠁⠀⣠⡷⠶⣶⠖⠢⡼⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠧⣖⣋⣩⡭⠽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢚⠉⠉⢳⣄⣣⠼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣤⡭⠴⣖⣒⠭⢿⣄⣀⡀⢀⣀⣀⣀⣤⣴⠾⠳⢶⣚⣛⣩⡭⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣏⣉⠁⣐⣠⣤⡬⢽⠟⠉⠀⠉⢫⡜⠁⠀⠀⠀⢫⠁⠀⠀⢸⡇⠀⠀⠀⢠⠒⢢⠞⠉⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⢬⣉⣩⡭⠥⠶⢲⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⢠⠄⠾⡀⢠⣀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⢸⡄⠀⠀⢠⠄⠀⡀⠀⠒⠀⠀⡸⠀⠀⠀⢸⡇⠀⠙⢄⠀⠈⠐⠄⠈⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⣿⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡰⠛⢤⣀⣀⠔⠁⠀⠀⠀⢸⡇⠀⠀⠀⠣⡄⠀⢀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⠀⠀⢰⠃⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⢸⡇⢀⣠⠴⠃⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠈⠒⠤⢄⣀⣀⠀⠀⣀⣀⡠⠄⠊⠁⠀⠀⠀⢸⣏⣉⣀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠴⠚⠉⣉⣉⣉⡁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢀⡞⠁⣠⠖⠉⠁⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⡞⠀⢠⠇⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⠴⠃⠀⠘⣦⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⢰⠋⠀⣠⣄⠀⠻⢄⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⢀⢰⠠⢸⣀⠆⠀⢸⡇
⣏⠀⡸⢹⠎⠲⠶⠚⠀⠀⠀⣏⠀⢀⣿⠀⠀⠀⠀⢒⡆⢄⣇⡸⠁⠇⠄⢘⣀⣨⣤⠤⠶⠖⠛⠋⠁
⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⣋⣊⣠⣤⠤⠶⠖⠛⠋⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣶⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣭⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠂⠐⠉⠐⠀⠀⠀⠀⠐⢤⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⢀⡀⠂⠄⠀⡀⠀⠀⠠⡀⠀⠣⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢦⣄⠁⠒⠀⠃⣀⣁⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⡇⠀⠀⠁⠀⠁⠀⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠰⠋⠀⠀⠙⠦⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠐⠀⢄⠀⠀⠀⠀⢠⠄⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⠤⠄⠘⠂⠀⠇⠀⠀⠀⢲⡤⢤⠔⠁⣠⡞⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠙⢆⠀⠀⠐⣀⠀⠘⡀⠀⠀⠀⠀⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡥⠀⠀⠀⠀⠀⠀⠙⣄⡀⠸⡇⠀⠀⠀⠙⠀⠀⠘⠧⠀⠀⣉⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠛⢠⠀⠍
⠀⠀⠀⠀⠀⠀⠀⢀⡛⢠⠀⠒⠀⢠⠀⠰⠀⠀⣀⠀⠀⠀⠀⠀⠑⠂⢀⡴⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣈⠇⠸⠀⠀⠀⠂⠀⠀⠉⠢⣀⠔⠈⠡⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠟⠠⢤⣀⠴⠈⠀⠀⠀⢃⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡤⠂⢀⣷⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⡈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⣁⢰⠎⠀⠆⠀⠀⣐⡠⠔⠊⠀⠀⠀⠀⠆⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⢀⠀⠀⣠⡄⠈⠁⠀⠸⠀⠀⠰⡀⠊⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠠⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⠤⠤⢄⠀⠀⠀⢀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠓⡀⠀⢦⠀⠀⠀⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠀⡀⠀⠘⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠐⠇⠀⠑⢀⡀⢀⡴⠂⠀⠀⠀⠀⡀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠅⠀⠀⠀⠠⠗⠠⠤⢀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢤⣀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠈⠃⣲⡆⠘⠀⢹⠀⡠⠈⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠈⠐⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⣅⠀⢀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢟⠛⣻⡿⢿⣿⣿⡿⠿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣎⣩⣿⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠛⠉⠉⠉⣉⡵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠂⠀⠀⠀⣀⠠⠤⠤⠤⠦⠤⠤⠤⢤⣀⣀⠀⠀⠀⢀⣀⣀⣤⠬⠭⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠔⢩⠃⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⢄⠀⠀⣷⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⡆⢀⡇⠀⠀⣀⣀⣀⣈⣙⡛⠀⠀⠒⠋⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠤⠬⣍⠓⠾⠀⠀⠀⢸⠁⠀⠛⠀⢸⠍⣹⠈⠉⠉⠉⢣⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠚⠲⢤⡀⣠⠞⠦⣅⣀⠀⢀⣠⠎⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠙⡇⠈⣹⢧⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⣀⠤⠤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⢱⢤⡀⠀⠀⠀⠀⠀⠁⠀⢤⣀⣀⣀⡼⠀⣸⠁
⠀⠀⠀⠀⠀⢀⡠⠖⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡴⠁⢀⠏⠓⡦⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢏⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⣠⣋⣀⣘⣁⣰⡭⠭⠟⠒⠂⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⢦⣀⡤⡔⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣹⢃⣀⢿⡄⠀⠀⠀⠉⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡴⡊⠁⢀⡔⠐⠒⠒⠙⢷⠋⠀⠀⠀⢸⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠥⠞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣤⢤⡤⠴⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣸⣸⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡲⣾⣤⣤⣽⡤⢤⣷⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠭⠟⠚⠉⠑⠺⠿⠾⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠐⠂⠀⡄⢳⢄⠀⠀⠀⠀⠀⢼⠂⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⡴⠣⠁⣴⡇⠀⠀⠐⠀⠀⠠⣈⠞⠈⣆⠀⢸⢂⣼⣿⣿⣬⣀⣀
⠀⢀⠀⢰⡅⠘⡇⠈⠀⠀⠀⠀⣸⠀⢠⣾⡿⣿⡇⠀⠀⠀⠀⠀⡼⠃⠀⡰⢁⠃⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠀⣸⠜⣽⠟⣿⠻⣿⣿
⠀⢸⣧⡘⠀⢀⠙⡀⠀⠀⠀⠀⡏⣰⣿⠟⠁⣿⣇⣤⣶⣾⣿⣿⡷⣤⡐⠀⣸⢒⠠⠀⠀⠀⡔⡇⠀⠀⢀⡄⠀⣿⣼⡿⢀⡏⠀⠑⠘
⠀⠸⣿⣧⠀⡀⠀⠸⡀⠀⠀⠀⣳⣿⠏⠀⣠⣟⣻⣿⣿⢫⣬⣄⢩⣷⡟⣆⡧⠈⠀⠀⢀⠎⠀⠀⠀⠀⢸⣷⣴⣿⣿⠃⢸⡏
⠀⠀⠸⣿⡄⠀⠀⠉⢿⡄⠠⠀⣿⡟⠀⠘⣿⠐⣿⣿⣿⡈⣙⡋⠠⣭⠠⣿⡱⠀⠀⢠⠏⠀⠀⠀⠀⠰⠇⢀⡻⣿⠁⠀⣾⠁⠀⢡
⠀⠀⠁⣿⣧⠀⠀⠀⣸⣷⠀⠀⣿⡇⠀⠀⠙⣦⣽⣿⣿⣿⣿⣦⣠⠄⠂⢺⠄⠀⢠⠏⠀⠀⠀⠀⠀⣽⣾⣥⠔⠋⠀⢰⣿⠙⢶⢾⡇
⠀⠀⠀⢹⡏⣷⡀⠀⢸⣿⡅⢠⣿⡇⠀⣐⣶⣿⠿⠿⢿⣿⡟⠻⣿⣿⣿⡿⠀⣠⠋⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⢀⣿⣿⡆⠀⢠
⠀⠀⠀⢸⡇⣿⡇⠀⢰⣿⠃⠸⣿⠀⣸⣿⣿⣿⣶⣶⣿⣿⣿⣦⣤⣤⣿⣀⣰⠡⠴⣶⣾⣾⣿⣿⣿⣿⠏⠰⠀⢠⣾⡿⣷⠈⠀⠀⣻
⠀⠀⠀⣸⣷⣿⠂⠀⢸⣿⠀⢀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣉⣩⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⢏⠅⠀⠀⢀⣾⡿⠁⠈⠀⢀⣼⣿
⠀⠀⢀⡟⣿⣿⠀⢀⣾⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣼⣿⠿⢡⠟⠁⠀⠀⣾⣿⠃⠀⠀⢠⣿⡿⢹
⠀⠀⢸⠁⣿⡏⣠⣾⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠒⠶⣶⣾⡿⠛⢀⡏⠀⠀⠀⣼⣿⠏⠀⠀⢠⣿⡟⡰⠿
⠀⢀⡏⠀⣿⣷⣿⢧⠤⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢀⠀⡇⠀⠀⣴⣿⠏⠀⠀⣴⣿⠏⠀⠎
⠀⠸⠧⠄⣿⣿⡇⠘⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆⢀⠂⣴⣶⣦⠀⠀⠀⠈⣃⠀⢠⡇⠀⣰⣿⠏⠀⣠⣾⡿⠃⠀⠁
⠀⠈⠀⠀⣿⣿⣿⠀⠹⡄⠀⠀⠀⣿⣿⣿⣿⡏⠀⢸⣿⣿⣧⢤⣾⣿⣿⣿⣿⣄⠀⠀⠑⢄⢸⠀⣴⣿⠏⢀⣾⣿⠟⠁
⠀⠰⠤⢀⣿⣿⣿⡄⠀⢳⠀⠀⠀⢿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⢾⣿⢋⣴⣿⠟⠁⠀⠀⠀⠀⠀⡠⠚
⠀⢀⠀⢸⣿⣿⡿⠸⢄⠀⢇⠀⠀⠘⣿⣿⣿⣧⣠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠻⢷⣆⡀⠀⢱⣿⡿⠃⠀⠀⠀⠀⠀⢀⢞⠱⡀
⠀⠀⢀⣿⣿⣿⡇⠀⠀⢵⢼⡀⡈⠀⠙⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣦⣤⣤⠻⠿⢶⣶⣿⠛⠻⢶⣖⡂⠀⢀⡿⡘⡄⢳
⠀⠀⣿⣿⣿⣿⢇⠀⠀⠀⢧⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠻⢿⣿⣷⡀⠀⠀⠙⢧⡀⠀⠈⠉⠋⣿⠀⣡⠘⢉
⠀⢸⣿⣿⣿⡇⠘⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣟⠿⠿⣿⣿⢿⣷⣦⠀⠈⢻⣿⣿⣶⣄⣀⡀⠙⣷⣄⠀⣸⡿⣍⠀⠀⠈
⠀⣿⣿⣿⣿⠁⠀⠈⢧⡀⠀⠀⠀⢀⠀⠀⠀⡠⠞⠤⣈⡁⠀⠀⣰⣿⡇⠀⠀⣿⣷⡆⠈⣧⠙⢿⣿⣿⣿⣶⣌⣿⣿⣿⠃⠈⠳⢤
⠀⣿⣿⣿⡿⠀⠀⠀⠈⢐⢄⠀⠀⠀⠘⢦⡞⠁⠀⠂⠈⠃⢠⣾⠟⣼⠈⠃⠀⣿⣿⣷⡀⣷⠀⠀⢙⡿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⡝
⠀⢻⣿⣿⠀⠢⠀⠀⠀⠈⢢⠣⡀⢀⡴⠋⠀⠢⠀⠀⠄⣴⣿⡋⢸⠏⠒⠄⠀⠿⣿⣿⣧⢹⠀⠘⠁⠀⡈⣻⣿⣿⣿⣿⣿⣿⣷⣄⣱
⠀⣿⣿⣯⠀⢀⠀⡀⠀⠀⠀⢣⡸⠎⠀⠀⠀⠀⠀⢈⣼⠟⠁⣇⣾⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠂⣀⠞⠠⠃⠎⠳⣌⢉⠉⢻⠿⠿⢟
⠀⣾⣿⣿⠀⠈⠀⠛⠐⢂⠄⡟⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠠⢸⠇⠀⠀⠄⢀⡠⣾⣻⣿⣿⡁⠌⠡⠀⠀⠘⠀⠀⠟⣦⠹⣤⣀⢃⡘
⠀⢻⣿⣿⠀⠀⠀⢠⠜⠇⡄⠀⠀⠀⠀⠐⠀⢰⣿⠃⠀⠀⢀⡏⠀⢀⠤⠚⣡⠞⠁⣸⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠁⡷⡀⠈⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣶⣶⣾⣭⣭⣭⣭⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢩⠞⣿⣿⣿⣿⣿⣿⡿⠟⠛⢋⣯⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⣧⣿⣿⣿⣿⣿⡽⢁⣴⣶⣿⡿⠛⠉⠙⠉⡉⠙⢳⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣶⣶⣼⣾⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠸⣝⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⡆⠀⠀⠘⠉⠁⢀⠀⠀⠀⠀⢻⣝⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠻⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠉⠉⠈⢹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣦⣴⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣟⣛⣉⣭⡟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠘⠯⣭⠿⠋⠀⠀⠀⠀⢀⠐⠰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠛⠛⠛⠛⢿⠋⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣦⣴⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠲⡒⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡐⠄⠨⠁⠑⠒⠢
⠀⠀⠀⠀⠀⠀⢀⠎⠡⠀⠀⠼⠁⠀⠡⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⡠⠃⡀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠨⠑⠐⠂⠂⠉⠑⠤⠀⠀⢀⠀⢀⠠⠔⠉⠈⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⠈⠀⠀⠀⠂⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠤⢽⣀⡀⠒
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⢉⠀⢉⠳⠦⣄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐⣤⡀⠀⢀⣠⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢠⣴⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⣆⣄⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠙⠃⠀⠀⠉⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢦⡀⣈⠁⠀⠀⠉⠉⠃⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡐⠀⠀⠀⣀⢀⢸⡀⠇⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠁⠈⠈⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⣲⡂⡉
⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠂⠤⢀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢿⡷⠀⠐⠠⡀
⡆⠅⠑⣄⠀⠀⠀⠀⠀⠀⢸⣷⠰⠄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢄⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡀
⠅⠀⠀⠈⠁⠂⠀⠀⠀⠀⣾⣿⠀⠢⠤⠀⠀⠀⠀⢰⡆⠀⠛⠂⠀⠀⠀⠃⠀⠈⢲⠋⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠂⢩⣿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠐⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠈⠐⠠⠄⡀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠐⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠸⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⡆⢠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠃⠀⠀⠀⠀⠀⠀⠃⣀⡢
⣿⣿⡿⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⠲⠤⠤⠖⠒⠒⠛⠛⠛⠛⠋⠒⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⢸⣧⠴⠖⠛⠲⣄⠀⣄⣻⠉⠓⠲⠿⡍⠓⠲⠦⠿⠏⠛⠲⢦⣄⣀⠀⢦⣤⣀⠀⠑⠦⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠢⢾⡀⡮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⡞⠠⠌⠉⠓⠛⠀⠉⠓⠦⣌⣷
⠀⠀⠀⠈⠛⣟⣛⣒⣒⠒⠒⠠⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠶⢦⠀⠉⠁⠀⠀⠀⣀⣀⠀⢀⠴⠒⠓⣄⢠⠞⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⡠⠔⣊⣁⣀⣀⡤⠤⣤⠤⣤⣤⡬⠍⠛⠛⢛⣿⡿⠶⠶⠶⠦⠈⠧⣄⣀⣀⣠⡄⠀⢠⡴⢋⡽⢩⠅⢰⠋⠀⠀⠀⡜⠈⣦⠄⠀⠀⣸
⠚⠋⠉⠉⠉⢻⡅⢠⠖⢾⠉⠹⡇⠀⣵⣔⣚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢻⡉⣵⢻⠀⢾⠀⠀⠀⡰⠃⠀⠈⢟⠛⣛⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠳⠜⠢⡀⢣⠀⡇⠐⢋⣁⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣⣸⣿⠀⠀⠑⠒⠊⠁⢀⠀⠀⠨⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠉⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠳⣼⡄⠀⠀⠀⠐⢦⠉⠄⠀⢀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣇⢠⣤⣤⣀⣀⡀⣀⡰⢛⣶⡤⢾⣿⡆⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣙⡆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠛⠋⠉⠉⠹⣄⣈⣉⢙⡄⠀⠀⠉⠀⠁⠸⡅⢀⣼⠉⠹⣆⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⡟⠉⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣏⡃⠀⠀⠀⠀⠀⠀⠘⠲⢋⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢇⠀⠀⠀⠀⠀⠀⣎⠁⠀⠘⠛⠊⠁⠀⠀⠀⠀⠀⡀⠈⢧⣀⠀⠀⠙⠳⢤⣤⣤⣤⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⡾⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⢀⣨⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⢶⢲⠊⡿⠀⠀⠀⠈⠓⠦⣄⣀⣀⡀⠀⣰⠁⣧⠔⣹⢿⣿⣤⡀⠀⣴⢮⣉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣦⠁⠀⠀⠀⠀⠀⢰⠃⡏⠉⢩⠏⠁⠀⠉⠀⠀⠀⠙⠦⣙⢺⢁⣤⡨⢏⡁⣹⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣯⣄⡀⠀⠀⠀⠀⢿⠾⠗⠂⢺⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢰⡶⣄⣹⠁⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡜⠻⠏⡟⡿⡲⠶⡾⢶⣊⠻⠟⠻⠇⠉⢳⠀⣀⡠⠤⠒⠊⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⢀⣃⡤⢖⡽⠿⣧⡀⠈⢑⣲⣭⣛⣓⠶⢲⡏⣟⡓⠒⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣴⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⠥⣶⣛⡃⢂⣴⡧⠯⢬⢤⠖⠒⢺⠻⡉⠁⠀⠈⠉⡿⠿⣷⣶⣶⣤⣤⣍⣉⣉⠒⠒⠲⠦⠤⢴⠋⣠⣾⠟⣋⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡸⠀⢸⠉⡏⠻⣳⠀⢸⠾⡆⠀⠀⡇⢧⠀⠀⠀⠀⢹⣤⣤⣤⣌⣉⣉⡛⠛⠻⠿⠿⢿⣶⣶⣶⠾⣋⡥⠊⢹⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠇⠀⢸⠰⡇⠀⠉⠀⠈⠉⠁⠀⠀⢳⠼⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢤⣄⣼⡧⠚⠁⠀⠀⢸⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣂⣤⣴⣶⣶⣿⣿⣶⣶⣦⣄⡀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣠⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⡿⠟⠛⠛⣿⡿⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣧⠀⠀⠀⠈⠉⠛⠇⠙⠋⠁⠀⠀⠀⠙⠋⠁⠀⠀⡙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠛⠿⢿⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠋⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡿⠿⠃
⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣁⣤⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⡆⢠⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡏⠉⠙⢯⣠⢸⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⢸⡃⠈⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣠⡎⠓⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⠀⠀⠀⠚⠗⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠁⠀⠀⣾⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⢙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⣀⣠⡤⠶⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣿⣿⠟⠳⢤⣄⣀⣀⣀⣀⡠⠖⠉⠙⠲⢶⡿⠿⠋⠉⢉⡡⠤⠶⠒⠉⠁⢨⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⢨⠋⠈⠈⠀⠀⠀⠀⠀⠀⠙⢦⡀⠾⢯⣤⣄⣠⣤⣌⡀⠪⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣭⣿⡾⠛⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡇⠀⠇⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣄⠀⠀⢀⣴⡄⠀⠀⢀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡠⠍⠛⠯⠀⢿⠇⠨⠟⠋⠥⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠁⣤⡦⠀⠈⠀⠠⡦⡄⠉⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⠀⠮⠴⠀⠀⠀⠠⠬⠆⠀⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⡛⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠚⡛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠁⠀⢀⠀⠈⢳⣶⣦⣶⣶⠋⠀⢀⠀⠀⠡⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢀⠀⠀⠈⢹⠉⠀⠀⢀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⣀⠀⢑⠐⠂⢩⢐⠂⠒⠁⢀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⣄⠀⢈⠀⠘⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂⠀⢸⣿⣶⣅⣴⣾⣿⠀⠀⠚⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⣈⠀⠸⠿⠛⠁⠙⠻⠿⠀⢈⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⢿⢿⣴⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠟⠋⠁⠀⠀⠀⢸⡟⠀⠈⠉⢉⠛⡿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠈⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⣠⣶⣷⡄⠀⣤⣶⣶⠄⣠⣤⣤⠀⠀⠀⠈⠀⣠⢿⣷⢽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⢰⣿⠃⠾⠇⠀⣿⡇⠀⠀⣿⠉⠉⠀⠀⠈⠀⠀⠀⠘⣿⡓⠾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡀⠀⠘⠿⣶⣶⣄⠀⣿⣷⣶⠘⣿⡶⠶⠀⠀⠀⠀⠀⠀⠀⣿⣯⣳⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠠⣶⣄⠈⣿⣷⣿⡅⠀⠀⣿⣆⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣬⣛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠻⠿⠿⠃⠙⠿⠟⠃⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⣿⣯⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣾⡄⢀⣴⣾⣶⣦⡄⣠⡄⠀⣶⡄⠀⠈⠀⠀⠀⠀⣿⣟⠺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣾⡏⣿⠀⢹⡿⣿⡇⢀⣿⠇⡀⠀⠐⠀⠀⠀⣿⡯⣙⢺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠨⣿⠇⠻⢷⡿⠃⠹⣷⣾⠏⠀⠂⠀⠀⠂⠀⢱⣿⡷⢤⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣀⠠⠀⠶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣄⡀⠀⠀⠐⠠⠈⢠⠄⢈⣿⣏⡓⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣺⣴⣾⣾⣦⣬⣶⣶⣶⣶⣤⣤⣄⡀⠀⠀⠀⢀⢀⠀⠤⣂⣿⣿⣿⠮⢽⣿
⠀⠀⠀⠀⠀⠀⣄⠀⣶⡀⣿⣿⣿⣿⣿⣿⢿⣿⠟⠁⠀⠀⠉⠙⠻⣿⣶⣼⣎⣾⣿⣿⣾⣿⣿⣿⣿⣙⢺⣿
⠀⠀⠀⠀⠀⠀⠘⢷⣼⣿⣿⡟⡁⣿⠟⣿⣾⠃⣼⣷⡄⢰⣿⣷⠄⢹⣿⣿⡿⣿⣿⣿⣿⣿⠿⣿⣿⣮⣹⣿⣠⣶⡟
⠀⠀⠀⠀⠀⢶⣶⠾⠟⠋⠙⠁⡦⠘⡆⢈⣿⠀⢙⡋⢠⣬⡙⢋⡀⠈⣿⢻⠇⡈⣠⠉⢈⠁⠠⠘⢛⡿⠟⢛⡋⠸⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠈⠃⠑⠀⠣⡄⠁⠈⠙⠶⠾⣿⡘⢛⣡⣼⣿⠾⠃⠽⠠⠇⠀⠀⠉⠰⠀⠀⠟⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠹⣽⣿⣷⡿⠏⠀⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠙⠛⠻⠷⣶⣤⣤⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢰⡶⠲⣄⠀⠀⠀⠉⠉⠉⣿⡏⠀⣾⠛⡿⣸⠛⣷⣸⠿⣏⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣧⣠⣿⣆⠀⠀⠀⠀⣰⡟⠀⠀⣿⢠⣤⢻⡀⢹⣿⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⣦⣽⣿⡏⠀⣿⡄⠀⠀⢰⡿⠁⠀⠀⢿⡀⣿⡽⣇⣼⢿⡀⡿⢹⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠿⣿⣿⣿⣿⡇⠀⢹⣿⣷⣶⣿⠃⠀⠀⠀⠈⠛⠛⢇⢙⠋⠘⠿⢃⡘⠛⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣿⠏⠀⠀⢀⣈⡻⣿⡿⠁⠀⣼⣿⣿⣿⣯⣀⠀⠀⠀⠀⢠⣷⣿⢸⡇⣟⠻⢸⡇⢸⡏⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣋⣵⣿⣿⠀⣴⡶⠸⡟⢱⡟⠁⣠⣾⠟⠉⠏⠿⠻⠿⢷⣦⣀⠀⢸⡿⣿⢾⡇⣿⢶⡆⣿⢻⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣜⢿⣿⢿⣿⣤⡀⢠⣬⠴⢺⣧⡾⠛⠀⠀⠀⠀⠀⠉⠑⠁⠟⠿⣷⣮⣁⠙⠘⠇⠻⠿⠧⠿⠘⠇⠈⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⠙⣷⣝⢿⣿⣿⣿⢶⣦⣶⡿⠋⠀⠀⣄⡀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣧⡙⢿⣿⠀⣴⠟⠁⠀⠀⠀⠀⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⣿⣎⠻⣾⡏⠀⠀⠀⠀⠀⠠⣤⣴⠋⠉⠙⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⢘⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⢀⡤⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⢬⣿⣿⣿⢿⡿⡿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠁⠀⠁⠁⠀⠉⠉⠛⠻⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⣶⣦⣤⣄⣀⡀⠀⢀⣴⡿⠏⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣆⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠾⢶⣦⣤⣤⣯⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⡦⠐⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠈⢻⣿⢿⡿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⠀⠀⠀⡀⠐⡈⣋⠂⠀⠀⠀⠀⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⣿⣿⣶⣄⠀⠀⠀⠀⢨⣴⣿⣿⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠭⠿⡿⠟⠀⠀⠀⠀⠸⢿⡿⠯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⣀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣴⣶⠆⠀⠀⠀⠀⠀⠘⢧⣁⣙⣛⣉⣵⠋⠀⠀⠀⠀⠀⠐⣶⣦⣌⢲⣀
⠀⠀⠀⠀⢦⡴⣶⣶⡄⠈⠸⠛⠛⠁⠀⠀⡀⠀⠀⠀⠀⠈⢿⣻⢟⡿⠁⠀⠀⠀⠀⢢⠀⠀⠈⠛⠛⠃⠉⢠⣴⣶⢦⣴
⠀⠀⠀⠀⠈⠾⠿⡿⢷⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⡾⢿⠿⠷⠃
⠀⠀⠀⠀⠀⠈⡖⢄⢸⣧⣴⡶⠂⢀⠀⠀⠸⢷⣦⡀⠀⢀⣀⢀⡀⢀⡀⠀⢀⣴⡾⠏⡀⠀⡀⠐⢶⣦⣼⡇⣀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡬⣷⣟⡀⠀⠘⡦⡄⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠘⡏⢀⢴⡃⠀⠀⣙⣶⠥⣼⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠙⢿⣤⣌⣿⣁⡴⠊⠀⢻⠀⠀⢈⠌⠉⠻⡿⠋⠁⡀⠀⠀⡟⠀⠙⢦⣈⣿⣁⣤⡿⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠈⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠏⠀⠀⠀⠻⢿⣿⣿⡆⢰⣿⣿⣿⠟⠀⠀⠀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢈⡆⠀⠀⠀⠀⠀⡌⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠈⠑⠄⡀⠀⡘⠀⠀⠀⠀⠐⡀⠀⠀⡠⠔⠊⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠲⠁⠀⠀⠀⠀⠀⢡⠔⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠐⠱⠀⠀⠂⠀⢀⠀⠀⢠⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠯⣈⠙⠢⢄⡀⠸⠀⠈⠄⠄⠀⠀⠀⠠⡸⠀⠀⠀⢐⢁⠅⠀⣈⡤⠖⠉⣁⢽
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠉⠲⢄⠈⠓⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢁⡤⠚⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠑⣞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢠⠜⠉⠁⠉⠀⢶⠒⣲⣦⠀⡤⢤⠀⣤⠄⠀⠐⠐⢢⡄⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⡼⠁⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠉⣇⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠒⠒⠲⡁⠀⠀⠀⠀⢾⠋⣾⣿⡎⢱⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠀⠀⠀⠀⢸⠒⠒
⠀⠀⠀⠀⠀⠀⠉⠉⣙⡇⠀⠀⠀⠀⠈⠓⠼⠿⠕⠋⠀⠀⣀⡀⠀⠀⠈⠙⠖⠻⠈⠀⠀⠀⠀⣸⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠶⠤⠤⠤⠤⠤⠴⠒⠒⠉⠁

⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣟⣻⣿⣟⣻⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠿⠻⠿⠁⠀⠀⢀⣀⣤⠤⣄⡀
⠀⠀⢠⣾⣿⣾⣽⣷⣶⠄⠀⠉⠀⠀⠀⠀⠀⢰⣾⣿⣿⢿⡿⣿⣷⠄
⠀⠀⠀⠐⠠⠿⠿⠻⠝⠀⠀⠃⠀⠀⠀⠐⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣼⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣶⣶⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣻⣿⣆⣀⣠⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣷⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣧⣿⣿⣟⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣮⢿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⡼⢻⣧⣿⡏⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣽⣞⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⡇⣿⣾⠡⡇⣿⡟⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢿⣧⡿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡗⡇⣿⢀⢷⡏⢺⢻⣟⣿⣿⣿⣿⡦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢹⣿⣹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⢿⣾⣾⡇⢸⣸⣿⣿⣿⣿⢹⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠓⠿⠋⠀⣸⡿⢿⡿⠞⠃⠈⢿⡿⢿⣿⣿⣿⣿⣯⣿⣽⣷⣿⠛⠿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁

⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⢿⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⡟⠁⢠⡀⠀⠙⢍⣀⣠⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠈⠁⢶⣿⣷⣄⡀⠈⣿⣿⣷⣴⣶⣦⡄⠀⠀⢠⡖⠆
⠀⠀⠀⣿⣿⡿⣛⣾⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⠇⠀⣀⠈⣄⠀⠙⠿⣿⡇⠀⣿⣿⣿⣿⣿⢸⣿⣿⣶⢜⣗⠀⠁
⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⡏⣁⠀⠀⡆⠈⡷⠀⠀⣩⣷⣤⣿⠃⣰⠟⡋⠿⠉⠉⠙⠈⢻⣷⣄
⠀⠀⠀⣻⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⠋⠀⣤⡴⣿⡇⠀⠸⠀⢰⡇⠀⠻⢿⣿⡁⠀⠻⣾⣧⠆⠀⠀⠀⠀⣼⣿⡟⢳⡄
⠀⠀⠀⣙⣿⣿⣿⡆⠈⢻⡏⢻⡟⠀⣴⣿⣿⣿⣃⣀⣠⣤⣾⣿⡅⠀⢠⣼⣿⣷⡀⣼⣿⠆⠀⠀⣠⡄⠀⣹⣻⣿⣹⣏
⠀⠀⠀⣿⡌⢻⣿⣿⠀⠀⠇⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠙⠻⣿⣿⠰⣏⣡⣦⣤⠾⠋⠙⠀⢹⣿⣿⢹⡇
⠀⠀⠀⣿⣷⣴⣿⣿⣴⣶⣶⣿⣿⣿⣿⣿⣁⣿⣿⣿⣷⣿⣏⣿⣿⡇⠀⠀⠀⢻⠿⢿⣿⣿⡗⠖⢦⡀⠰⠀⣿⡿⢋⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣽⣇⡉⠛⣿⣿⣿⠇⠀⢀⣴⣿⠀⠙⣿⣿⠿⠒⠚⠙⠀⢠⣿⠁⠋⠟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣁⠀⠸⡿⣿⣿⠀⠀⠿⢿⣆⡀⠀⠀⠀⠈⠉⣦
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⣿⠀⠸⢿⣿⣇⠀⠀⠀⠀⠙⢻⣿⣿⠀⣠⠾⢃⣴⣶⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠿⣿⣿⣿⣿⣿⠁⢰⡏⠉⣮⡁⣀⣤⣤⣶⣿⣿⠃⡌⠁⢀⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠙⢿⣿⣿⣷⣿⡇⠀⠈⠁⠀⠀⠈⣹⢹⠏⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣟⠛⠛⠻⢿⣦⡀⠀⠙⢿⣿⣿⡷⠂⠀⠴⠶⠖⠾⠃⡟⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢤⣽⣷⠀⠀⠀⠉⠉⠶⠀⠠⠀⠀⠂⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣹⠃
⠀⠀⠀⢸⡏⠈⠁⢩⣿⣿⢻⣿⣿⣿⣿⣿⣿⠻⢷⣄⠈⢹⡆⠀⠒⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣼⠀⠀⣠⣿⡿⠁⠀⠹⣿⣿⣿⣿⢿⣦⣀⣹⣆⠈⠃⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠁⢀⣼⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⠿⠛⠉⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣡⡖⠂⠀⠀⠀⠀⠀⠀⠀⢀⢿⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⡻⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣤⣴⡴⠀⠀⠀⠀⠀⡀⠰⠒⠿⠿⣿⣿⣿⣿⣿⠈⢿⣿⡀
⠀⠀⠀⠛⢿⣿⣿⣿⣷⣤⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠃⢀⡠⠒⠊⠉⠉⠙⢷⣿⣤⡀⠀⠈⠛⠛⠿⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⡘⢿⡿⢿⣿⣿⣿⣿⣴⣿⡀⠀⠀⠀⠀⠀⠀⠻⡸⢻⣿⣶⣤⣄⠀⠀⠀⡀⠀⠀⣴
⠀⠀⠀⡤⠒⠛⠛⠉⠛⠛⠿⣿⣿⣿⣿⣿⣼⠿⢿⣿⣿⣿⣿⣷⣄⠀⣀⣤⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠹⠿⢺⣡⡄
⠀⠀⠀⣆⡀⠀⠀⠀⠀⣀⣠⠾⠟⠛⠉⠀⠀⠀⠀⠛⠀⣨⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠻⣿⣿⡄⠀⠀⢻⣇⡄
⠀⠀⠀⣿⣿⣦⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣤⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠘⠿⡇
⠀⠀⠀⣿⣿⣿⣷⡧⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⣿⣿⣿⣿⡵⠹⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣀⡀⢶⢽⣿⢷⠇
⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡭⢀⠆⠀⠀⠈⢁⠀⠀⠈⠁⠀⠈⠱⠏⠀⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⡀⣰⣤⣞⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣵⣦⣽⣷⡏⢿⣿⣿⣾⣃⣀⣀⠄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠛⠛⠁⠀⠙⠘⠛⠛⠂⠀⠀⠀⠀⠀⠈⠛⠊⠙⠛⠛⠿⠿⠿⠻⠻⠟⠻⠾⠛⠿⠿⠿⠿⠏⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠂⠀⠈⠐⠒⠤⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣘⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⠠⢤⡴⣂⠤⠐⠒⠒⠒⠂⠤⠀⠀⠘⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⢠⠞⠊⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⠈⡀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡄⠈⢦⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠉⠑⢄⠀⡄⡠⠒⠒⢄⢀⣀⡀⠀⠀⠂⠀⠀⠀⠃⢸⠀⠀⣀⣈⡆⡎⠀⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠷⡀⠘⡄⢿⣧⠴⠦⠀⢧⠀⠀⠀⠀⠒⠛⠛⠛⠿⠀⠣⡎⢝⢀⡁⢨⠊⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣹⠀⢰⠀⠀⠃⡜⠀⢀⡤⠔⠒⢠⠄⠀⠀⠀⢀⡿⠑⠂⠀⠀⣀⠼⢁⠎⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢂⡑⠋⠈⠳⣊⣀⠣⠀⢣⠀⠸⣄⠀⠘⠁⠀⠀⠠⠁⠀⠀⠀⠀⢀⡔⠁⡠⠊⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠑⠢⠤⣉⠋⠀⠀⠀⣀⠀⠀⡀⠤⠶⣭⠶⠊⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠊⠀⠀⠰⠁⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡑⠦⢄⣀⣀⡀⠤⠴⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⢏⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡇⠐⢢⡄⢸⡤⡔⠒⢰⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⢪⣀⠉⡆⣸⡅⠀⠉⣧⠠⠼⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠀⠀⠈⠳⡄⠉⠉⣽⠃⠐⣍⠗⡲⠶⠛⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣒⠲⠠⠤⠴⠗⣒⠬⠋⠀⠀⢟⢼⣀⠀⠀⠀⠀⡨⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠑⠢⠍⡉⣉⡉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠴⠶⠒⠶⠚⠰⠶⠖⠒⠂⠀⠀⠀⠘⠞⠧⠅⠂⠔⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢠⡀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⡀⠈⡉⠉⠛⠀⠀⠀⠀⠀⠘⠛⠁⠛⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠈⠓⠀⠀⠀⠀⠀⠀⠀⠰⠴⠶⢶⠄⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢿⡆
⠀⠀⠀⠀⣤⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣶⣿⡇
⠀⠀⠀⠀⠿⣿⣿⣿⣿⡿⠟⠛⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠙⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠿⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣃⣉⣋⣠⣉⣩⣏⣣⣿⣫⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⢀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠣⢀⠀⠀⢀⡀⠀⠀⠀⣸⣤⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣶⣿⣶⣼⣿⣷⣧⣾⣼⣿⣷⢂⣤⡀⠀⠀⠀⠀⡠⠔⠋⠀⢠⠖⠋
⠀⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⢤⡈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⡤⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣤⣞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠂⡈⠁
⠀⠀⠀⠠⠤⠤⠴⠒⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠐⠚⠙⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣱⣿⣿⣿⠻⠿⠋⠉⠿⢻⣿⠃⣼⣿⣿⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢈⣩⣽⡒⠀⠘⠋⠀⢘⣩⣤⣼⢿⣿⣿⠛⠁⠀⠀⠀⣀⠀⢈⡀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠈⠁⠛⠋⠀⠀⠀⠀⠉⠀⠛⠉⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⡤⠄⠀⠀⠠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⡀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣦⣄⡑⠀⠠⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⣈⣤⣾⣷⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣴⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠔⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠄⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡏⠀⢀⣴⣶⣄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⣿⣿⣇⡇⠀⠀⠀⠀⠐⠒⠋⠉⠙⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠻⠿⢫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢿⣿⣦⣤⣉⣉⠉⠛⠓⠒⠒⠒⢛⣋⣉⣩⣥⣼⣿⠏⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠈⠙⠻⠿⢿⣿⣿⠿⠿⠛⠉⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡛⢦⣀⣀⣀⣀⣠⣞⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠀⠉⠉⠉⠉⠉⠘⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠿⠽⠆⢠⡄⠘⠉⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠠⣤⠼⠧⡤⠄⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣴⠀⠀⠀⠀⣧⠖⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⠀⡴⠚⠛⠛⠋⠉⣿⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡼⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⣼⠈⢳⡀⠀⠀⠀⠀⠀⠀⣷⢰⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠐⣿⣀⡀⠈⠳⡄
⠀⠀⠀⠀⠐⠟⢃⣠⣴⣿⣴⣶⣿⠟⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠓⢿⣤⣶⡴⠿⠖⠲⠶⠶⠶⠖⠋⠁⠈⡿⠀⠀⡷
⠀⠀⠀⠀⠀⢰⣿⡿⣟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣾⢹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣌⡓⠿⢶⣤⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⢹⡀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣷⣦⡀⠈⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣦⣻⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠂⠀⠀⣿⡇⠀⠀⠀⠀⠀⣾⠁⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠈⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⢻⣿⣯⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⢶⣦⣄⣙⠷⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣉⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠉⠁⠀⠉⠉⠙⠃⠀⠉⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣀⣀⣠⣴⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⠦⠀⢀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢸⠠⡀⠠⠀⡜⢆⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠳⡄⢹⣧⠀⠈⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⣼⣷⡀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣣⡿⠏⠕⣘⠛⣄⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠀⠀⠄⠀⠀⣠⠥⣄⠀⢱⣺⠠⠾⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠀⠀⠀⠀⠒⠀⠉⠉⢦⡀
⠋⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠉⠀⠀⠀⢠⣤⠤⠴⠖⢆⢲⡄⢠⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠤⠠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢤⣄⡀⢀⣤⠈
⠀⠀⠀⣀⣀⣤⣴⣶⣤⢤⡄⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣀⠿⠠⢤⡀⠀⠀⠀⠀⠀⢈⠭⠟⠢⢤⡀
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⡇⠀⠀⠀⠀⠈⠁⠀⠆⠀⠁⠢⠤⠀⢀⠁⠂⢐⠋⠉⠠⠤⠒⢂⡀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢨⡧⠤⡄
⠀⠀⠀⠉⡁⠂⢸⣿⡇⠀⡇⠀⠀⠂⠀⠄⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⢀⠉⠀⠀⠀⠀⠀⡠⠁⠦⠄⠀⠀⠀⣴⡗⠈
⠀⠀⢠⣄⡀⠀⣸⣽⡇⠤⠇⠀⠀⠀⠀⠀⠀⠠⠄⠈⠁⠒⠀⠠⠤⠀⠠⡙⠢⡀⠀⠀⠐⡧⠰⠀⡀⠀⠀⣴⠹⠀⠀⡀
⠀⠰⣋⠒⠻⠁⣉⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢤⣠⠐⢤⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⡀⠀⠁⠀⢰⢣⠅⠀⠰⡇⠃
⠀⠀⠀⠈⠀⠀⠀⠈⠄⢀⠂⠀⠄⢀⠀⠀⠀⠀⠀⠙⠃⠁⠁⠀⠰⠞⠇⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦⡄⢻⡾⠀⠀⢰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠤⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡎⠀⠀⡰⠦⣄⣀⠀⠀⠀⠀⡀⢀⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠀⠀⠛⠓⠒⠚⠢⠤⣀⣀⡀⠈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠁⠈⠐⠐⠈⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠟⠛⢙⣙⡛⡛⡛⢻⡟⣛⢻⢛⡛⣿⠻⡿⢛⠛⠻⢻⣟⡟⣟⣛⢛⡻⢟⣟⡟⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⢻
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠘
⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠲⡄⠀⡀⡀⠀⠀⠀⣀⠀⠀⠐⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡔⠀⠀⠀⠀⢀⠂⠀⠀⠀⠘⠄⠀⠁⠀⠀⠀⠀⠠⠀⠁⠀⠀⣀⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⡀⡘⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⡜⠀⠀⢀⠔⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢀⠔⠈⠈⠀⠀⠈⠐⠠⡊⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠉⠀⢠⣄⡄⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠠⠁⣌⡥⠀⠀⠦⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⠀⠀⠂⠀⠀⡇⠀⠀⠠⠰⠁⠀⡹⠀⢘⠂⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠈⠠⡀⢈⡀⠔⠸⠁⢀⠞⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠈⠀⢀⢍⠥⠔⠀⠀⢨⠀⠐⣀⡪⠔⠣⡀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⡀⠀⠰⣤⠀⠁⠀⠀⠀⠀⠘⠀⢳⠎⠀⠀⠈⢀⣄⡠⠀⠀⠀⠈⠢⢄⠀⠀⠀⡐
⠀⠰⠂⠀⠄⡀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡉⠀⠈⠂⠑⠼⢡⠀⠀⢘⠁⢸⠁⠀⠀⠀⠀⡆⠀⠉⠐⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠢⠀⠀⠀⠜⠀⠀⡰⠀⠇⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠈⠁⠈⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠁⠈⠁⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣶⣿⣿⣷⣄⠀⠀⠀⢠⣶⣿⣿⣶⣄⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⣠⡶⣿⡿⢶⡄⠀⠀⠀⣠⢾⣿⣿⢦⡀
⠀⠀⠀⠀⣿⣷⣾⣿⣶⣿⡆⠀⠀⣿⣷⣿⣿⣶⣿⠄⠀⠀⣿⠷⠿⠿⠾⣿⠀⠀⠠⣿⣷⢿⢷⣾⣿⠀⠀⢰⡿⠾⠿⠷⠾⣿
⠀⠀⠀⠀⠙⢷⣽⣿⣵⠟⠀⠀⠀⠹⣯⣾⣷⣮⠟⠀⠀⠀⠻⣾⣛⣛⣷⠟⠀⠀⠀⠻⣷⣿⣿⣿⠏⠀⠀⠀⠻⣽⣛⣻⡵⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⣼⣏⣻⣿⣋⣿⡄⠀⠀⣼⣟⣿⣿⣟⣷⠀⠀⠀⣾⣟⣿⣿⣻⣷⠀⠀⠀⣾⣟⣿⣿⣻⣧⠀⠀⢠⣿⣉⣿⣟⣹⣧
⠀⠀⠀⠀⢻⣿⣫⣯⣿⣿⠃⠀⠀⢿⣟⠀⠀⢹⣿⠁⠀⠀⢿⣷⣿⣟⣿⡿⠀⠀⠈⣿⣿⢛⣛⣿⡿⠀⠀⠘⣿⣿⣻⣻⣿⡟
⠀⠀⠀⠀⠀⠙⠛⠟⠛⠁⠀⠀⠀⠀⠙⠳⠖⠛⠁⠀⠀⠀⠈⠙⠻⠟⠋⠁⠀⠀⠀⠈⠛⠻⠞⠋⠀⠀⠀⠀⠈⠛⠻⠟⠋

⠀⠀⠀⠀⢠⠾⠿⠿⠿⣦⠀⠀⠀⣠⢾⢿⣿⡷⣆⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⣰⣾⣿⣿⣷⣄⠀⠀⠀⡴⣻⢟⢿⡳⡄
⠀⠀⠀⠀⣿⢷⣶⣷⣶⢿⡇⠀⠀⣿⢮⣿⣷⣽⣿⠆⠀⠀⣿⣾⣿⣿⣷⣿⠀⠀⠰⣿⣿⣿⣿⣿⣿⠀⠀⢸⣷⣭⣿⣮⣵⣿
⠀⠀⠀⠀⠘⢿⣽⣿⣵⠟⠀⠀⠀⠙⢯⣿⣿⡵⠏⠀⠀⠀⠙⢮⣿⣿⡵⠋⠀⠀⠀⠻⢿⣧⣼⡿⠋⠀⠀⠀⠻⣿⣭⣽⡿⠃

⠀⠀⠀⠀⠀⣠⣤⣦⣤⡀⠀⠀⠀⠀⣠⣴⣦⣤⡀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⢀⣤⣴⣦⣄⠀⠀⠀⠀⢀⣤⣴⣤⣄
⠀⠀⠀⠀⣼⣏⣿⣿⣏⣿⡄⠀⠀⣾⣏⣿⣿⣹⣿⡀⠀⠀⣾⣇⣽⣿⣸⣷⠀⠀⢀⣿⣏⣿⣿⣹⣷⠀⠀⢠⣿⣹⣿⣿⣹⣧
⠀⠀⠀⠀⢻⣿⣛⣛⣻⣿⠁⠀⠀⢿⣿⣿⣯⣿⡿⠁⠀⠀⢿⣏⢭⡽⣹⡿⠀⠀⠈⢿⣿⠙⠉⣿⡟⠀⠀⠈⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣠⡾⢿⣿⠿⣦⠀⠀⠀⣰⡿⣿⣿⢿⣦⠀⠀⠀⣴⠛⣿⣿⠛⣦⠀⠀⠀⣴⠟⣿⡟⢻⣆⠀⠀⠀⣴⠿⣿⡿⢷⣄
⠀⠀⠀⠀⣿⣿⠟⠛⣿⣿⠇⠀⠀⣿⣷⣟⣛⣿⣿⠂⠀⠀⣿⣷⣿⣿⣿⣿⠀⠀⠐⣿⢻⣿⣿⡻⣿⠀⠀⠸⣿⡿⠿⠿⣿⣿
⠀⠀⠀⠀⠘⠿⣦⣤⡿⠋⠀⠀⠀⠘⠷⣿⣿⠶⠋⠀⠀⠀⠙⠿⣿⣷⠿⠋⠀⠀⠀⠙⢾⣯⣿⠾⠃⠀⠀⠀⠙⢿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠙⠿⣿⣶⣤⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠙⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⣿⣇
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⣼⣿⠟⠉⠁⠀⠀⠀⢻⣿
⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⣠⡆⠸⣿⡇
⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⡙⠻⣶⣤⣀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣾⣿⡇⠀⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣷⡀⠈⡙⠿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠊⠁⠀⠀⠀⣀⣴⡿⠟⠁⠈⣿⡀⠘⣿⣇
⠀⠀⣿⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠉⠛⠿⣦⠹⣧⠀⠀⠀⠀⣀⣴⡾⢿⣉⡀⠀⢠⠀⠸⣧⠀⢹⣿⡄
⠀⠀⠸⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⡀⠀⣿⡇⠈⠀⢠⣴⣾⠟⠋⢤⣬⢿⣿⣷⡏⠀⣀⣹⣧⠀⢻⣿
⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⣸⣿⡆⣿⣇⠀⠀⣾⣿⠀⠀⠀⠀⠉⠂⠋⠉⠓⠀⠈⢿⣧⣀⣾⡟
⠀⠀⠀⢸⣿⡆⠀⠀⠈⣿⣿⠀⠀⠀⠀⢸⣷⡀⠀⠀⣙⣛⠁⢸⣿⠀⢀⣿⡏⠐⢦⡀⠀⢀⣰⣶⠦⠄⠀⠀⠀⠻⣿⡟
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠹⣿⣄⠐⣿⠙⣧⢠⣿⠀⣸⣿⠇⠀⠈⢻⣶⡿⠻⣇⢠⣶⣄⠀⠀⠀⢻⣿⡀
⠀⣤⣶⠿⠟⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⠻⣷⣿⠀⣿⣿⠏⢰⣿⡟⠀⠀⠀⠸⠟⠁⠀⠈⣃⣈⡉⣷⣄⠀⢸⣿⡇
⢸⡿⢀⣤⣤⣤⣤⣤⣴⣿⣿⣧⣄⡀⠀⠀⠀⣿⡄⠘⠛⠀⣿⠇⣠⣿⡟⠀⠀⠀⠀⢀⣠⣴⣞⣛⣉⣍⣉⣻⣿⣷⣼⡿
⢸⢇⣾⠟⠛⠛⠉⠉⠉⢙⣻⣿⣿⣿⣷⣦⣄⠈⠿⣶⣶⡾⠋⡴⠟⣿⣷⠀⠀⠀⠀⠉⠉⠙⠓⠿⢬⣍⣋⣙⣹⢿⣧
⣼⡿⠃⠀⢀⣠⣴⣾⡿⠿⠿⠛⠛⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⣈⣉⣁⣼⣿
⠁⢀⣴⣾⣿⣿⣭⣄⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠉⠛⢿⣧⡀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣤⣀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⣄⡀⠀⠘⣦⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢿⣿⣷⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣻⣷⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣆
⣿⣷⠿⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠈⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡴⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠀⢀⣤⡀⠀⠀⠀⠀⠐⠋⠉⠙⠂⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⡏⠀⢹⡄⠀⠀⠀⣠⡤⠶⠂⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧⣀⡼⠀⠀⠀⠀⠈⠙⠳⠆⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⢻⡙⠲⠶⠶⠖⢋⡟⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⠲⠶⠶⠖⠋⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠉⠓⢲⡔⢉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡊⠳⣖⣉⠀⠀⠀⠀⠀⠀⠈⠙⠢⡴⠋⠀⠀⠀⠀⠀⢀⠔⠊⠁⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠚⠁⠀⠉⠓⣤⣀⣀⣀⠀⠀⠀⡇⠀⠀⣠⠴⠒⠚⠓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠉⢢⡤⠚⠦⣞⠀⠀⢀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢀⣠⠤⠤⠤⠞⠀⠀⠀⠘⠚⠉⠉⣩⣧⣤⣤⡤⢤⣨⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢮⡠⠔⢒⣒⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠒⠖⠙⢧⡀⢇⣰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣄⣽⣋⡩⠥⣴⣺⣇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣥⣤⠴⠶⢚⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⣿⠒⠚⠉⠍⢈⣉⣟⣲⠖⠊⠙⠻⣟⡿⠁⠀⠀⠙⢞⠋⠉⠉⠀⡇⠀⠀⠀⠀⡤⢤⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣌⠳⠷⠒⣛⣋⣭⣥⠃⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⣀⣸⠀⠀⡇⠀⢸⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⢹⡋⠉⠉⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⡤⠀⠀⢸⠀⠀⠀⠀⡇⠀⠸⡁⠀⠱⠤⡁⠒⠃⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠈⠂⢼⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠀⢠⢦⠀⠀⠀⣠⠏⠀⠀⠀⠀⡇⠀⠀⠈⢆⠀⠀⠈⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⢄⣀⣀⠴⠃⠀⠙⠒⠊⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡟⠀⢀⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⡇⠀⠀⠀⡴⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠞⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⣷⠴⠖⠋⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⠤⠄⠒⠊⠉⠀⠀⠀⠀⠀⠀⡷⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠾⠤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡴⠋⠁⣠⠤⠴⢶⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠏⠀⣰⠋⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡟⠀⢠⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣠⠞⠁⠀⠀⢹⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⠀⡇
⣰⠁⢀⣴⢤⡀⠘⢲⡄⠀⠀⢸⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⠄⠂⢰⠠⣹⠐⠸⠤⠃⢀⣀⡇
⣇⠀⡼⢀⡇⠙⠒⠊⠀⠀⠀⢸⡈⠢⢸⠀⠀⠀⠀⣐⢹⠰⣸⡠⠯⠸⢀⣐⣉⣤⡤⠶⠖⠚⠛⠉⠉
⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⢸⠅⠀⠀⣈⣀⣤⡭⠴⠶⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⡄⣤⢤⢠⣠⡄⢠⠄⠀⠀⠀⠀⠀⠀⢀⠀⣠⢦⡆⢠⣤⡦⡆⠀⡀
⠀⠀⠀⠀⡤⢀⠀⡀⠀⠀⢀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠇⠀⣧⠏⠿⣼⢻⢻⠀⠸⠄⠀⠀⠀⠀⠀⠀⢀⠀⠱⢴⡆⠤⠔⣰⢤⡦⠗
⠀⠀⠀⠀⢇⠔⠤⠀⠀⠀⠐⠆⠤⠃⠀⠀⠀⠀⠀⠀⠀⡶⣾⣣⣲⣰⣶⡄⢸⣷⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡷⠆⣦⠀⠀⠁
⠀⠀⢀⡀⢀⡘⠿⠟⠃⠀⠀⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⡏⡥⢡⠀⡥⢸⠇⡤⣤⣨⠇⠀⠀⠀⠀⠀⠀⢧⡾⣴⠧⡀⡗⠳⢐⢼⢻⣸
⠀⠀⢹⡏⣿⣿⠙⡟⠀⢸⠷⣿⣿⠭⠀⠀⠀⠀⠀⠀⠀⣠⣿⣼⣦⣇⣼⣤⣗⡽⣼⣧⠀⠀⠀⠀⠀⠀⢿⣽⠽⣿⣯⣯⡽⣽⡿⣿⣿⠇
⠀⠀⠈⠁⠉⠉⣤⣡⡄⣬⠀⣍⡍⡏⠀⠀⠀⠀⠀⠀⠀⠙⢯⠿⠿⠟⠿⠿⠟⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⣾⡄⢸⠀⢸⡴
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠶⠶⠢⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠠⠛⡇⠼⡤⠼⠿⠄
⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠁⠀⡼⠁⠀⢐⢟⠀⠀⠀⠈⠀⠀⠀⠀⠀⣎⠱⢀⣴⢚⠘⢛⢲⢰⣓⠶⠀⠀⠀⠀⢠⠶⠆⣶⡄⣦⣴⣰⠶⢴⣖⢰
⠀⠀⠀⠀⠠⠀⠀⠠⡄⢆⠈⠀⡀⠠⠂⠆⠀⠀⠀⠀⣀⢼⢳⡾⢧⠿⠽⢻⣿⢾⠼⣾⠀⠀⠀⠀⠘⠦⠼⠛⠷⢏⢿⡹⠶⢼⠯⠸⠦
⠀⠀⠀⢠⣤⡀⣠⣄⢠⡞⣀⣀⢀⣀⣤⡀⠀⠀⠀⠀⠈⠃⠚⠠⠶⠶⠤⠈⠰⠶⠬⠟⠀⠀⠀⠀⠀⠀⠀⢠⡀⡘⢛⠃⢠⢈⠀⢂⡀
⠀⠀⠀⢸⣾⣿⣏⣹⢿⢿⣿⣿⣸⡟⢳⡄⠀⠀⠀⠀⢸⢾⠞⡇⢾⣹⢷⠀⡧⡇⠆⠠⠀⠀⠀⠀⠀⠐⢈⣘⣃⡊⢀⡔⢙⣉⣉⣀⡀
⠀⠀⠀⢨⡭⣥⢩⢁⣌⢠⡍⡬⢩⢈⢩⠅⠀⠀⠀⢠⣖⢲⡲⢰⡀⣆⡖⣦⡶⣰⡀⣶⠄⠀⠀⠀⠀⠀⢸⠇⠟⠻⠸⡜⠣⠟⠻⠻⠅
⠀⠀⠀⢸⠀⡿⢸⠸⠈⠸⠿⣧⠸⡜⢸⠀⠀⠀⠀⠈⠛⠙⠙⠋⠛⠋⠃⠋⠙⠋⠛⠛⠀⠀⠀⠀⠘⠃⠀⠃⠀⠑⠓⠛⠀⠁⠓⠚⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠯⣄⡈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⢁⣠⢽⡀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠉⠓⠦⣀⠉⠳⢤⡀⢀⣀⡤⠤⡤⡶⠶⢶⠤⡤⢤⣀⡀⢀⡤⠚⠉⣀⠴⠊⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠳⣄⣀⡉⠋⢻⠀⠀⣷⡇⠀⢸⢸⡇⠀⢸⠟⣁⣀⣠⠞⠁⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⣰⠛⠃⠁⠀⢸⡀⠀⣿⡇⠀⢸⣼⠀⠀⣿⠀⠉⠘⠓⣆⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡖⠚⠀⠀⠀⠀⠀⠈⠳⠶⠋⠑⠶⠚⠈⠳⠖⠃⠀⠀⠀⠀⠀⠓⢢⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠒⠒⠒⡇⠀⠀⠀⠀⠀⢾⠋⢸⣿⣷⠈⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⠒⠒
⠀⠀⠀⠀⠐⠚⠛⠛⣧⠀⠀⠀⠀⠀⠈⠳⠬⣿⠯⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⢱⠖⠻⠀⠁⠀⠀⠀⠀⡞⠛⠓⠂
⠀⠀⠀⠀⠀⠒⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢻⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠚⠢⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⣴⠶⠄⢠⣶⡄⢠⣶⢶⡄⢀⣴⠶⠀⣰⣦⠀⣠⡶⠆⢠⣦⠀⣴⡆⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠿⣦⡀⣸⢻⡇⢸⣿⣼⠃⣾⡇⠀⠀⣿⢻⡄⠻⣶⡄⢸⣿⣤⣿⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⣤⣼⠇⣿⠛⣿⠸⣿⢸⣇⠸⣧⣤⢸⡟⢻⡇⣤⣼⠟⢸⠈⣿⢹⡇⠀⢠⣥






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⢰⣖⢰⣴⣶⢶⣴⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⢀⣴⡄⢠⣿⡇
⠀⠀⣶⣬⣯⣌⣩⣭⣬⣥⠞⣿⣤⣤⡤⣶⠆⠀⠀⠠⣆⠀⣤⣄⢠⡆⠀⠄⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣻⡿⡑⣿⢋
⠀⠀⠛⣛⠟⣘⣏⣿⡇⠟⣤⣟⠟⣿⠇⠻⠆⠀⠀⠀⢻⣎⣿⣿⡿⠁⠀⠀⢠⣴⣿⠀⠀⠀⠀⠀⠀⣤⣤⡄⣹⣿⣯⣨⣝⣁⡙⣿⣥
⠀⠀⠠⣿⡇⡿⡇⣹⢷⡿⢾⢿⢿⣿⣡⡄⠀⠀⠀⠀⡀⢛⡉⢛⡁⣀⢀⣸⣏⣛⠻⣢⣤⣄⠀⠀⠀⢿⠀⣿⢿⣯⡼⡿⢿⣿⠃⡏
⠀⠀⢰⣿⣶⣶⣾⣼⡿⣾⣿⠀⣷⣸⣸⣏⣾⡟⠀⠀⠿⣼⡿⢿⠟⢿⣼⠃⣿⣽⡗⢿⣿⣇⠘⢷⡄⠀⢺⣀⡀⣾⣿⢆⣿⡇⣼⣯⣤⠾⠁
⠀⠀⠀⢸⣷⠋⣮⡾⣽⣬⣏⣶⠉⠉⠀⠈⠈⠉⠀⢠⡶⢾⡟⢸⣿⣶⡇⣭⣬⣵⣄⣶⣾⢀⡾⠛⠁⠀⣬⣭⠁⣝⣛⡈⣟⣫⣿
⠀⠀⠀⠘⠃⢾⠿⠇⠙⠋⠉⠙⣋⣤⡾⠗⣤⡄⠄⠘⠳⠟⠀⢿⠿⠿⠷⡿⠿⠸⠿⢿⡽⠠⡅⠀⠀⠀⠀⠻⢾⡏⠿⠟⢿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡯⠬⠵⠖⠚⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣛⡿⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢠⣤⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⡤
⠀⠀⠀⢠⡄⣰⠿⣿⣰⡿⣧⣤⣰⣿⡇⠀⠀⠀⠸⣿⣧⣴⣧⣾⣿⣷⣦⣿⣾⣿⣿⠂⣶⣤⣶⣴⠀⠀⠀⠀⠀⠀⢿⣼⠇⠐⣶⢰⡖⠷⣰⡶⣷
⠀⠀⠈⠈⠛⠉⢶⣛⡌⢳⡛⣛⢈⠛⠃⠀⠀⠐⠶⢻⡍⣅⣍⣣⣄⣤⡿⢼⡇⣦⢖⣦⣯⣈⣈⣿⠀⠀⠀⠀⠀⢲⣤⠟⢿⡟⢿⣿⣇⠀⡿⢸⣿⡟
⠀⠀⡀⠀⠀⢀⡈⠿⠁⣿⠟⢿⡿⢶⠀⠀⠀⠀⣀⡸⠛⠿⠯⣻⣿⢋⣏⡰⠟⠟⠇⠙⠁⠙⠚⠁⠀⠀⠀⣿⢻⡗⢳⡛⣶⢋⣾⣿⣹⣿⣴⣶⣭⣭⢀
⡆⠀⡀⢰⠀⡌⢷⣾⢾⣿⢿⡿⣳⣿⡆⠀⠀⠀⣼⡇⢸⡿⣗⡯⣾⡟⡏⠀⢀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠸⠇⢿⠈⠻⠟⡙⠻⠚⠃⠋⠹⡌
⠀⠀⠐⡐⠀⠃⣬⣅⣬⣩⡍⣅⣨⢭⡟⣤⠀⠀⠙⠃⠚⠛⢉⣽⣾⣷⣤⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⢱⢢⡄⠡⣈⠔
⠀⣤⣤⣷⣶⡄⡏⢿⣹⡿⣿⣏⡛⢺⡿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⠫⠬⠀⠁⠀⠋
⠀⣿⣇⠀⣼⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠙⢲⡀⠀⣰⠞⠛⠛⠋⠙⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣔⣩⠤⣤⠀⠀⠀⠹⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠛⢉⡴⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⢀⢈⡗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠛⠁⠀⠀⠉⠓⠦⣄⣀⠐⢶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣄⡉⠳⣄
⠀⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣓⠮⣷⣄
⠀⣠⠶⠛⢉⢉⠙⠲⣼⠃⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠠⡀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠙⠂
⢰⡇⠀⠀⠉⠘⢳⠀⠹⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣦⢹⡀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⢰⣤⣀⠀⠙⢷⡀
⠈⠛⠲⠤⠤⠤⢤⡄⠀⠀⠀⣀⣤⣴⠲⡖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⣀⣀⠘⣆⠙⠳⢦⣽⣏⠓⢦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣾⣿⣿⣇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⢀⣀⣉⣉⣉⠀⠸⡆⠀⠀⠀⠉⠀⠀⠈⠛⢾
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⠀⠻⣇⣹⣾⠀⢹⣆⠀⠀⠀⢷⠀⠀⠀⠀⠀⣿⡴⠚⠉⠉⠀⠈⠉⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢧⣈⡷⡁⢸⢦⡀⠀⠘⣆⡀⠀⢠⢀⣿⠘⠀⠀⠀⠀⢀⣠⠞⠀⠀⢧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡄⠀⠀⠙⢦⣼⠧⣄⣹⢦⡀⠀⠉⠓⢶⡛⠙⠲⠴⠶⠖⠛⠉⠀⠀⢀⣠⠼⠛⠉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠻⣦⡀⠀⠀⠻⣦⣼⠳⣦⣽⡷⣤⣜⡂⢃⠀⠤⠆⣀⠀⢠⠢⠀⢀⢤⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠀⠐⣿⠻⢦⡐⠈⠀⠙⠓⢾⣟⠛⢮⣿⠛⠶⣦⣌⣀⣁⠀⠈⠈⠀⢈⠉⠉⠙⡏⠁
⠀⠀⠀⠀⣠⠞⠀⠀⢰⠃⠀⠀⠸⣆⠀⠙⠦⣀⠀⠀⠀⠈⠙⠺⢤⣉⣳⠟⠤⢤⣼⣉⣉⡟⠉⠈⠀⠀⠀⣿
⠀⠀⠀⡴⠃⠀⠠⠽⡟⠀⠀⠀⠀⠙⢆⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠉⠙⠒⠾⠧⢤⣯⡀⠀⠀⠀⠀⣰⠇
⠀⠀⡼⠁⠀⠀⠀⣸⠃⢀⣠⣤⣶⣚⡿⠃⠀⠀⠀⠀⡼⠻⡝⠛⠲⠦⢤⣤⣀⣀⣀⣀⣈⣀⣀⣠⡤⠞⠋
⠀⡼⠁⠀⠀⠀⠀⣯⠶⢋⣿⡿⠟⠛⢦⣀⣠⠖⣲⠴⠗⠒⠛⠒⠶⡞⣽⣶⠀⠀⠈⠈⠉⠉⠁
⡼⠁⠀⠀⠀⠀⠀⠛⠋⠉⠀⠙⠦⣄⡀⠉⠉⠈⡉⠓⠶⣄⠀⠀⠀⡗⢙⠿⠶⣦
⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠠⡈⠓⢦⡼⠀⠀⠀⠈⠋⠙⢺⠃
⠀⠈⢳⡈⠙⠓⠶⠦⣤⣄⣀⣀⣀⠀⣼⢈⠳⣄⣸⠦⡼⠃⠀⠀⠀⠀⡴⠒⠻⡆
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢉⣉⠉⣿⢸⡀⠀⠁⠀⠀⠀⠀⠀⢰⡋⠙⢶⡞⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠐⠀⠀⠀⠿⢿⡈⢧⠀⠀⠀⠀⠀⠀⠐⣺⡛⠛⠻⣄
⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠳⢄⠀⠀⠀⠀⠀⠹⡟⠲⠶⠋
⠀⠀⠀⠀⣼⢷⣶⢤⣤⣀⣀⣀⢀⣀⣠⣏⣀⣙⣦⣤⣀⣀⢀⣀⡴⢃
⠀⠀⠀⠀⠙⠶⢍⣛⠳⠤⣬⣭⣏⣁⣀⣀⣭⢿⣩⣃⣈⣉⡿⠁⢀⡾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠛⠛⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠶⠶⠾⣧⡲⣶⣤⠀⠀⠀⣼⣿⠃⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⢦⣤⣉⠋⠁⠀⠀⠀⠉⠁⠀⢀⡴⠞⠋⠉⠉⠉⠉⠛⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠃
⠀⠀⣿⡤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⠚⢿⠛⣿⣿⣿⠁
⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢏⣹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣟
⠀⠀⢨⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠁⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠉⠀⠀⠀⠀⠈⠉⠑⠢⢤⡀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡡⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠎⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠳⠔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡠⠔⠊⠉⣠⣫⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠴⠏⣠⡤⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠠⠶⣦⠀⠀⡼⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠦⣀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠁⡸⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⠇⠀⠀⢀⡼⢄⡀⠀⠀⠀⣠⢤⡈⢳⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢋⡥⠤⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⢸⠀⠀⠀⠸⡄⠀⠙⣄⣄⣠⡇⠀⢹⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⠙⠲⡄⠀⡇⢀⡴⠒⠲⣄⠀⢀⡀⠀⠀⠀⡔⠉⠀⠀⠈⢳⠸⡆⠀⠀⠀⠈⢢⢰⠁⠀⡝⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄⠀⠹⣀⣱⢺⣀⣀⣀⡈⢼⠁⠉⠆⠀⠀⠤⠶⠶⠶⢶⣾⠄⠳⣄⠊⡟⠉⠛⠘⠢⠞⠁⠀⣰⠃
⠀⠀⠀⡇⢀⠔⠒⢲⠃⠀⡝⠀⠀⡇⢀⠟⠋⠁⠀⢀⣈⠁⠀⠀⠀⠀⠀⠀⢠⣤⢄⡀⠀⠀⠏⠀⢳⡖⢲⠀⠳⠀⢠⠞⠁
⠀⠰⡀⣇⢸⡀⣠⠼⣄⢀⡇⢠⠀⠀⢸⠀⠀⡴⠋⠁⠀⡨⠝⠓⠲⠤⣤⠤⠖⠁⠀⠀⠀⠀⡰⠖⠋⣠⠃⠀⠀⠀⣾
⠀⠀⠳⡈⢦⠈⠀⠀⠈⠯⣤⡤⠑⠀⠘⣄⠀⠙⢤⣀⠈⠀⢀⣀⣠⠔⠁⠀⠀⠀⠀⠀⢀⡴⠁⢀⠜⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠙⠲⠤⣬⣉⣉⣉⣁⣀⣀⠠⠀⢠⡤⠖⠚⣧⡶⠖⠉⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡓⠤⣀⠀⠀⠀⢀⣀⡤⠔⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢢⡤⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣄⡀⠀⡾⢷⠀⠀⠀⣀⣀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢬⣇⠀⠉⣶⠇⢸⣖⣟⠉⠉⡽⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣧⣀⡈⣳⢀⣧⡀⠀⠀⢹⡤⠔⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⠀⠈⠙⢦⠀⠈⠉⢩⡟⠁⠰⡋⠿⢄⣶⠶⠿⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡓⠤⢀⣀⣀⠀⣠⣇⠠⢔⡻⠃⠀⠀⣿⣠⡋⠀⠀⠀⠀⠀⠀⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠢⠤⠤⠔⠒⠂⠉⠀⠀⠀⠀⠘⠪⣝⡒⠤⠤⠤⠴⠚⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂⠒⠀⠀⠀⠀⠰⠀⠶⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠒⠰⠂⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣸⣿⣧⠀⠀⠀⠀⣀⣈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡁
⠀⠀⠀⠀⢀⣾⠟⢉⡉⠙⢻⣿⠀⣤⣞⣋⠠⠤⠤⣤⣹⣿⣧⡀⠀⠀⠀⠀⣴⣿⠟⣉⣁⣉⣛⢶⣄⠀⠀⠀⢠⣾⠛⢉⣉⡛⣦
⠀⠀⠀⠀⣼⠃⠀⢀⣹⡆⠸⠟⢸⡏⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⣾⡿⠟⠛⠉⠉⠀⠀⠀⢛⡇⠀⠀⠀⠁⠀⣾⡟⠀⣼
⠀⠀⠀⠀⣿⡀⠀⢾⣯⣀⣠⡀⠈⢛⠲⠴⠤⠤⠤⠤⣼⣿⣿⠇⠀⠀⠀⣿⣦⣤⠀⢀⣀⣀⡠⠴⡟⠁⠀⢀⣀⣀⣀⣿⣷⢀⡿
⠀⠀⠀⠀⠸⠇⢀⣼⡿⠟⠋⢁⠀⠀⠘⠒⠂⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⣹⡿⠟⠛⠀⠩⠥⠄⠚⠀⠀⠀⠀⠉⠛⠿⣿⣧⡼⠃
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⣀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣾⠋⠀⠐⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠺⠶⠿⣦⣤⣠⣾⡿⠉⠛⠋⠀⠀⠀⠀⠀⣠⡶⠛⠉⠈⠉⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠉⢹⡇⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣼⣷⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠛⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠼⠛⠉⠙⠛⠻⠷⢶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠋⠉⠀⢠⡿⠁⠀⠈⠙⠻⣦⠀⠀⢻
⠀⠀⠀⣠⣴⠞⠋⠉⠉⠉⣀⣤⡶⠾⠶⣦⡀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣶⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢸⡅⠀⢀⣀⣤⣄⣼⡇⠀⠘⣷⣤⣄⡀
⠀⢀⡾⠋⠀⠀⠀⠀⠀⢾⣏⣁⡀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠞⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⣦
⢰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣷⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡄⢹
⣿⠃⢀⣴⠇⠀⣠⣤⣄⠀⠀⢀⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⣤⠖⢻⡇⠀⠀⠰⣆⠀⠈⣿⡿
⣿⠀⢸⡏⠀⠀⡏⠀⠙⠷⠾⠛⠋⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⠸⠀⠀⠀⠀⣿⠀⣰⡟
⠸⣧⣼⡇⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣾⣿⡆⠀⢺⡆⠀⠀⣠⡿⠞⠋
⠀⠀⠈⠙⠶⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣤⣿⡧⠀⠈⠻⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⠄⠐⠒⠒⠶⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠉⠉⢹⡿⠉⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠉⠙⠙⠛⠛⠛⠋⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀⣀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠉⠙⠛⠿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣤⣤⣤⣶⣶⡶⠟⠋
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾⣿⡛⢷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⢿⣶⡀⠀⠀⠀⠈⠛⠿⢿⣷⠶⣶⣦⠀⢀⣤⢤⣤⠀⠀⣠⣤⣤⣄⣤⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢀⣼⣿⠇⣀⣤⣤⣤⣶⡆⠘⣿⠀⣿⡿⢰⣿⠇⣰⡿⢀⣾⡿⠁⢈⣿⡟⠀⣿⡏⣽⠇⢰⣶⡄⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠈⠉⠁⢰⣿⡏⠀⢹⣿⡇⠀⡿⢰⣿⠃⣿⣿⠀⠀⠀⣼⣿⠁⠀⣼⣿⠁⢰⡿⣿⣄⣀⣼⣿⠟⢩⣿⣷⠶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⢻⣿⡇⠀⢸⣿⡆⢸⡇⣿⡇⢰⣿⣿⣄⣠⣾⢿⣿⣤⡾⣿⣯⣠⣟⡴⢿⣿⢩⣿⠏⢠⣿⡟⢁⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣶⠿⣿⣧⢀⣾⣿⣷⡿⠁⠻⠿⠟⠀⠉⠛⠋⠁⠀⠉⠉⠀⠈⠛⠋⠸⣧⣼⠟⢸⣿⢀⣿⠏⢠⣾⠏⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⢻⣿⡇⠙⠿⠿⠋⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⢀⣶⡆⠠⣄⠀⠀⣤⠀⠀⠀⠀⠈⠉⠀⣿⣃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⢀⣼⣿⡇⠀⢰⡆⠸⣿⣅⠙⠇⠀⠀⠀⠀⠀⢸⡏⢿⣄⡾⢹⡇⠀⠙⢷⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⢸⡇⢠⣆⠉⣻⡦⠀⣤⠶⣄⠀⠘⠃⠈⠛⠁⠘⠃⠀⠀⠸⠇⠀⠀⠀⢠⡶⣄⠀⢠⡶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠛⠋⠁⢀⣹⣷⢾⡷⣶⠀⠀⢰⣶⡴⢶⡆⢠⡖⢲⣄⣶⡆⣈⠳⣿⣀⣠⣿⡳⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⣷⡄⣶⡆⢀⣾⠂⣿⣷⠋⣿⣆⣿⣡⠟⣼⣸⡟⠀⠀⢸⡟⣀⣼⣿⣿⠷⢾⢿⡿⠀⣿⢃⣿⣿⣧⠾⣱⢧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡉⢀⣿⡇⢸⡿⠀⣿⣇⡴⣿⠻⣿⣀⣴⢣⣿⢁⡀⠀⣾⡿⣧⠟⠈⠿⠶⠋⠸⡷⠞⢿⡾⠹⣿⣀⡴⢋⣿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣏⢿⣧⡿⢷⡼⣿⡟⠞⠃⠀⠉⠉⠀⢸⣧⣼⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠃⠀⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢆⡠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠉⠉⠉⠉⠓⢦⡀⢱⠄⠈⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠁⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠱⡎⠀⠀⣠⠆⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠊⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⣇⠀⠀⡏⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐⠀⠰⠈⡄⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢠⠄⡀⢪⡄⠀⠀⢀⠀⠀⠀⡄⢰⣀⣀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡑⠄⡈⠁⡁⠀⠀⣀⣠⣎⣀⣀⠀⠹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠂⠉⠩⡭⠤⠀⠀⢈⡹⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠈⠑⠶⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣠⠴⠯⣄⡀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠛⠤⢤⣄⡀⠈⠉⠒⠢⠬⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡶⠤⣀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠙⢦⠀⠈⠙⠓⠦⠤⠀⠀⠤⠔⠂⡘⠉⡹⠁⠀⠈⠙⠲⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠐⠒⠠⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢤⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢈⡂⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠆⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠺⣳⠒⠊⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⢐⣀⡃⠀⠀⠀⠀⠀⣤⠔⠁⠀⠀⡇⠀⠀⢠⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⡇⠀⠀⠀⠈⢢⡘⢆⠀⢠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠹⣀⢠⣤⠴⠔⠃⠀⢰⠀⠀⠀⠑⢄⠢⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠊⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠎⢀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡌⢋⢒⠐⠖⠁⠀⠀⠀⠀⠀⢠⠃⡘⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠈⡆⢇⠸⡄⠀⠀⠀⠀⠀⠀⠉⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠰⠌⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⡂⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠐⠛⠦⣀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠂⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⢀⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠄⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⡀⠀⠀⠀⠀⠀⠿⡿⢿⡿⢻⠟⢛⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣹⠟⢻⣿⣿⠿⢿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣭⣿⣿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃⢀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠛⠿⣷⣄⠀⣀⣴⣾⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣏⣽⣷⣤⣹⡍⣼⠋⢉⣉⠷⣆⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠒⣀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠟⠀⣸⠃⣿⣺⣿⡆⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣾⠟⠛⣿⡟⠁⠀⠀⣀⠀⠀⠀⠀⠈⠷⠶⠶⠾⢷⣦⣿⣏⣉⣀⡤⠟⠀⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇⠀⠀⠀⠘⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⢘⣿⠀⢻⡆
⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⠁⠀⠀⠀⣠⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⢰⡿⢸⡇⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⢀⣿⡇⡠⢿⡄⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⢸⡇⣼⠁⠈⢻⣦⡀⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇⠀⠀⢸⡏⠈⢦⡔⠁⠈⠻⣦⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⢿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⢠⡄⠈⢿⣤⠊⠑⢦⣄⡀⠈⢙⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣧⡀⢷⡀⠈⠛⠷⠶⠶⠶⠿⠛⠋⠁⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠻⣤⡙⠶⣤⣀⡀⠀⠀⠀⠀⢀⡴⠛⠋⠛⢦⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠠⣿⣧⣤⣌⣛⠶⣤⣀⠀⠀⠀⢀⣴⠋⠀⠀⡀⠀⡈⣧⣤⣤⣤⣶⠶⠿⠛⠃⢠⠇
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠉⠙⠛⠿⣯⣝⠓⣶⠞⠁⠀⢀⠀⢻⡀⡂⢹⡀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠘⠿⠶⣶⣶⠶⠶⠶⢶⣤⣀⠙⣿⠋⠀⠀⢀⣼⡇⠸⡇⢷⣼⠛⠛⠛⠛⠛⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠳⣤⣀⣤⡶⠋⡘⢣⠀⣷⡞⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣘⢉⡽⠀⢠⠃⠈⡷⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣠⡏⠀⣼⣁⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⢿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⠈⠉⠉⠉⣹⠋⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣧⠀⠀⢰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣷⡤⠹⣧⢀⡿⠀⠠⣾⣧⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⡤⠶
⣤⣤⣀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⣿⠶⠟⠋⠁⢀⣼
⢳⡀⠉⠛⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣠⡿⠃
⠀⠻⢦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⠾⠛⠻⠿⠶⣶⡶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣤⣤⣴⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢹⢀⣀⠀⠀⣀⣠⡤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⠇⠀⠀⠀⠀⠈⠀⠀⠙⠊⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⣤⠴⠋⠉⠙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⣀⠀⣀⡴⠃⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⣀⣠⡖⠚⠁⠉⠉⣠⠞⠛⠁⠀⠀⠀⠀⠀⠘⠛⠓⠆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠚⠁⢀⡤⠶⠦⣄⡀⠀⢀⡴⠒⠒⠦⣄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠲⡏⠀⠀⠀⠀⠈⠳⡄⠀⠀⠹⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣹⣔⣦⠀⠀⠀⢀⡇⠀⠀⠀⠘⣶⠹⣋⡭⠭⠉⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠘⢻⣏⡁⠀⢀⣠⠞⢁⡤⠶⢤⡀⠘⣯⡽⠆⢾⣉⣌⡟
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⠀⠀⠀⠀⠀⠀⢀⣨⠦⠤⢤⣄⣀⠀⢠⠞⠀⠉⠉⠉⠀⢠⠏⠀⠀⢈⡇⠀⠘⣦⠔⠋⡡⠋
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠩⢍⣳⣌⣉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⠀⠀⠀⠀⠀⠀⠂⢠⢀⡠⠞⠀⠀⠀⢿⠀⡎
⠀⠀⠀⠀⠀⠸⡆⠀⢀⡈⢳⣦⣸⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠹⡄⣸⠶⠯⣄⣀⣠⡤⠴⠖⢦⡀⠀⠀⠠⠚⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡇⣗⠀⢀⣤⠤⠤⠤⠴⠦⠂⠛⠓⠤⣄⡀⠀⠀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⣆⡴⠋⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣟⣀⣸⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⢻⡉⠁⠙⢆⠈⠳⣞⠉
⠀⠀⠀⠀⠀⠀⢸⣀⣈⣉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⣶⠦⠤⠤⢤⣤⣶⣧⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⣯⣇⣀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢤⣀⠀⢀⡟⠛⠋⠉⠉⠁⠀⠀⠳⣤⣀⣀⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣦⡀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠙⠒⠲⠦⣤⣤⣤⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡜⠀⠀⠀⠀⠀⣀⣾⣰⠃⠀⢰⠃
⠀⠀⠀⠀⠀⠀⡞⠉⠀⡹⢍⢙⢓⣦⡤⣄⣉⣠⣏⣉⣁⣠⣾⢤⣤⠤⣀
⠀⠀⠀⠀⠀⠠⣷⣤⣀⣧⣠⣭⢭⣝⣶⠿⠛⠛⢶⣤⣮⡤⠤⠚⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠋⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠷⡖⠯⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣿⢀⣀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠼⠇⠀⠀⠀⠀⠉⠉⠉⠓⠚⠉⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠀⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⡤⠊⠀⠒⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⣠⠤⠞⠁⠀⠉⣠⠶⠛⠀⠀⠀⠀⠀⠀⠰⠶⠤⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⡟⠁⠀⠀⠀⠠⠞⠁⢀⣀⣀⣀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠉⠢⣠⠊⠀⠀⠀⠀⠙⢦⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢆⣠⣄⠀⠀⠀⠈⡇⠀⠀⠀⠹⣤⢔⡶⢒⣒⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠸⢿⠞⠋⠀⠀⠀⡰⠁⢀⣀⡀⠀⠹⣦⢻⣍⣠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡞⠑⠂⠤⠖⠋⢀⡴⠋⠀⠙⡆⠀⠹⡏⣀⠼⠂⡰⠂
⠀⠀⠀⠀⠀⢠⠒⠀⠀⠒⠒⠲⢶⡒⠒⠶⢇⠀⠀⠀⠀⠀⠐⠚⠉⠁⠈⠈⠐⠢⢤⡚⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⣰⠃⠀⠀⢿⠁⠀⠊
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡉⠛⣟⣲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⢸⡄⠡
⠀⠀⠀⠀⠀⠀⠁⠀⠠⣀⣀⠉⠒⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠁⠀⠀⠒⠒⠂⠉⠀⠀⠀⣄⠀⠀⠀⠀⠉⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⠀⣨⡆
⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢤⣀⠀⠀⠈⠉⠛⠲⠶⠤⢤⣄⣀⣀⡀⠀⡤⠤⠐⢶⠉⠂⣌⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠸⠤⠼⠧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠳⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠘⠇⠠⠤⠤⡤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⣏⣉⣉⣭⠭⠽⠛⠻⣄⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⠴⢲⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠙⠲⢤⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠄⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠉⠓⢂⡶⢶⠦⠤⢤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣤⣼⡀⠀⠀⠀⠀⢠⣼⣥⣏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡞⠁⠀⡨⠳⣀⣉⣳⡶⠦⣤⣤⡾⢤⣬⣤⠼⠷⣶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⣟⣦⣤⣷⣤⣤⣶⣒⡾⠿⠛⠉⠻⢶⣾⣗⡶⠖⠱⡄⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠿⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢶⡀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⠉⠉⠛⠛⠛⠋⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠉⠀⠀⣠⡏⠀⠀⠀⠀⢀⠐⢂⠀⢀⠀⠄⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⠋⠀⣰⠃⠀⠀⠀⠀⠹⠀⠘⢀⠀⠀⠘⣿⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⠚⣿⣧⠀⠀⠀⠠⠆⢂⣤⡀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠒⠐⠃⠀⠀⢰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⢨⣿⣷⠀⠀⠀⠈⠐⠖⠲⠂⠁⠰⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡟⠙⠲⢤⣀⢠⣿⡟⠒⠤⣀⣰⠀⠀⠑⠀⠂⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⡀⠄⠉⠛⢿⡇⠀⠀⠈⠙⠦⠄⠀⠀⠀⠀⠘⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠐⢀⠀⣸⠹⡄⠀⢀⣀⣀⡈⠄⠀⠀⠀⠀⢢⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⡁⠀⡌⠀⢀⡇⠀⠀⣀⢸⣿⠟⢁⠂⠀⠀⠈⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⡸⠀⠈⠸⠿⠿⠼⠃⠠⡃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⢀⠀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⡅⠀⠀⠀⠀⢨⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢲⣮⣉⡅⠀⠀⠀⡤⢪⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⠙⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⡴⠖⠛⠛⠛⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⡤⡤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠾⣿⡿⣿⣿⣾⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢿⡛⢻⡟⠈⠙⠛⠓
⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣦⣤⣴⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣀⣀⣐⣻⣿⣿⣿⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣍⠉⠉⠀⠀⠀⣴⡄
⠀⠀⠀⢀⣤⠤⣴⣤⣤⣤⣤⣀⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠙⡥
⠀⠀⠀⠀⢆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⢀⡀⠉⠉⠉⠙⠛⠿⣿⡀⠀⠀⠀⢞
⠀⠀⠀⠀⠀⠑⢄⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠉⠂⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣘⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⣻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⣠⣷⣶⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣸⡇⠫⢿⠌⢿⣿⣿⠋⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣻⠿⠯⣽⣿⣿⣿⣿⣿⡷⢂⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⡇⢠⣀⠀⠸⠟⠁⠀⠀⠀⠈⢿⣿⣿⠿⣿⣯⠏⠙⣛⣷⣶⠤⠼⠿⠿⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣿⣿⡆⠸⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠤⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⡶⠐⠒⠶⢤⡠⣰⣿⣿⣖⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠛⠓⠒⣹⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠙⠛⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠁⠈⠫⠹⣷⣦⣴⣶⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣤⣄⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⣿⠟⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣄⠀⠀⠹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣩⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡷⠀⢉⣠⣽⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⣀⣤⡴⢶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⣤⣔⠋⣁⣽⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣁⣒⣦⣄⣴⣿⣿⣶⡟⠩⣩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢸⡇⠴⠮⣿⣬⡹⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣤⣶⣶⠿⡟⠛⠉⠉⣯⣻⣿⣯⣽⣿⣻⡰⢻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃⣿⢓⡏⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢀⣴⣿⣿⡿⠓⠀⠽⠖⢁⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⠀⠉⠁⠀⠀⠙⢿⣿⡟⣹⣿⣿⣦⡄
⠀⠀⣾⣿⣿⡏⠦⢠⣠⡴⢶⠿⣿⣿⣿⡏⠀⠙⠛⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⣷⠀⠀⠀⠀⠐⣻⣶⣴⣿⡿⠋⠙⣁⣀
⠀⣼⣿⣿⣿⢠⠀⣾⣿⣆⣐⣾⡿⢿⡻⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⢀⣠⣴⣴⣷⠿⢹⣿⠏⠀⣠⣾⠿⠹
⠀⠸⣿⣿⣿⡌⠀⠿⢿⣿⠋⠉⠁⢏⣩⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠋⠀⠀⠈⠈⠀⣰⣿
⠀⠀⠙⣿⣿⣵⢦⣿⣼⣿⡄⠀⠀⠈⠛⠄⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⡀⠀⠀⢀⡧⠰⣽⡃⠶⠆⡰⢿⡏
⠀⠀⠀⣟⣿⣹⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠃⠄⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⣤⣾⠃⠀⠀⠀⠀⠩⠘⣦⣼⣿⣰⣿⡅⠈
⠀⠀⣠⣮⣿⡏⢿⣿⣿⣿⣿⣷⣶⣷⣶⣶⣤⣤⣈⣀⠲⣥⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⣄⣀⡀⠀⠀⠀⠂⠀⣄⣐⠇⠀⢠⣄⠀⠀⢴⠒⠛⠛⠀⠑⢟⢷⡄⢻⣦
⠀⢸⣿⣿⣿⣿⣜⠻⠿⠿⣿⣿⣿⣿⣿⣟⣭⣟⣽⡟⠛⠛⠷⠤⠯⣽⡿⠃⣀⣤⣶⣶⣿⣿⠁⠀⠀⠀⠈⠙⢿⡿⠀⢘⡏⠸⣳⡦⡞⠁⠀⠀⢀⢳⡄⢻⣿⣯⣿
⠀⠀⠐⠶⣴⠲⠿⠻⠶⣻⡿⠉⠛⠛⠛⠛⠿⠿⠿⣧⣤⣤⣄⡀⠀⠀⠀⠈⠁⠀⠀⠡⠙⣿⡀⠀⠀⠀⠀⠀⠀⢠⠿⠖⠀⣮⡿⠈⠅⠀⠀⠈⠤⠆⠙⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣤⡀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⡀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠠⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠂⠀⠀⠆⠄⠀⠆⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣚⣭⢽⣿⣄⠀⠀⢀⣴⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⣤⣀⠙⣿⠀⠀⠚⠋⠁⠀⠀⠀⠐⠒⠶⠶⣶⣶⣤⣄⣉⣿⠿⣿⣿⣦⡀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠮⣽⣇⠈⢤⡈⠙⠶⣤⣀⡀⠀⠀⠀⠁⠀⠀⠈⢛⣿⠟⠻⢦⣄⠙⢿⣿⣆⡠⠀⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⠀⠻⣶⣄⡀⠉⠙⠻⠷⣶⣦⣤⣤⡶⠟⠁⠀⠀⠀⠉⠳⣦⣹⠟⠁⣰⠏⢰⡏
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⡀⠀⡀⠀⠀⠀⠀⠈⣿⠻⢿⣶⣤⣄⣀⣀⣀⣠⡤⠀⠀⠀⢀⣠⡤⠖⠋⠉⠀⣀⣁⣠⡾⣱⠇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠠⣈⠑⠢⠄⠀⠀⠀⠀⣿⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣦⣄⣀⠀⠀⣀⣠⣤⣶⣿⣋⣉⣡⣾⣿⡞
⠀⠀⠀⠀⠀⠀⣿⣿⢶⣀⡈⠛⠶⢤⡤⠖⠒⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠛⠛⠉⠘⡛⠛⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⣀⠀⠀⣀⣠⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡑⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠙⡛⠟⠛⠋⠛⣉⢠⠞⠁⠀⠀⠀⠀⠀⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡌⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣍⡛⠶⠖⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⣤⠀⠀⠀⠀⠿⠋⠉⠙⠇⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣧⣬⣙⡛⠛⠋⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⢀⠄⢠⣤⣤⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣩⣭⣙⠃⠈⠁⠀⠀⠈⠓⠒⠈⠑⠶⠖⠛⠛⠦⡄⠄⠀⠀⢰⠇⢰⠏⡀⠈⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⢁⣄⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣉⠀⠀⢀⠀⠀⣿⠀⠈⢰⡿⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣯⠈⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⣀⠈⠀⠄⢀⡀⠀⠸⠃⠐⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡹⢄⡀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡿⠶⠖⠀⠀⠀⠀⢀⡀⠀⢈⣷⠀⠃⠰⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡉⠋⠀⠀⠀⠀⠀⠈⠁⠀⠈⠳⢦⣀⠀⠀⠀⠀⠈⠁⠠⡾⠟⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢶⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠬⠑⠲⠤⠒⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡜⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣹⣧⠀⠀⠉⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠑⢬⣿⡄⠀⠀⠀⠀⠈⠉⠛⠳⢶⠶⠶⠛⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠈⠙⠓⢶⠦⡀⠀⠀⠀⠀⢸⡷⠰⠶⠞⣩⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢀⡏⠀⠹⡄⠀⠀⠀⢸⣿⡀⠀⠀⣧⡀⠈⠓⠶⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⢦⡀⠀⠀⠀⠀⣼⠁⠀⠀⠙⢦⡀⠀⠀⢿⣷⡀⠀⡏⠹⣶⡴⠞⠁
⠀⠀⠀⠀⣀⣀⣠⣤⣤⠞⠋⠻⢧⣄⡀⠹⣄⠀⠀⣴⠃⠀⠀⠀⠀⠀⠙⢤⡀⠀⠻⣷⢀⡿⠦⣤⣄⣀⣀
⢀⣴⣾⣿⣿⠿⠛⠻⣿⣦⣄⣀⠀⠈⠙⠛⠛⣦⠞⠷⠤⢤⣤⣤⣤⣤⣀⣀⣙⣦⣄⣨⣿⣡⡴⠟⢁⣩⣿⣿⠿⠷⣶⣄⡀
⣿⠿⢿⣿⣿⣶⣤⣀⠀⠉⠛⠿⢿⣿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣤⣴⣶⡿⠿⠋⠁⣠⣴⣿⣿⠿⣦⡀
⠀⠀⠀⠙⠻⢿⣿⣿⣿⣶⣤⣄⡀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⢁⣀⣤⣶⣿⡿⠟⠋⠁⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⠿⠟⠋⠁

⠀⠀⣿⣿⣿⣿⣿⣿⣻⣙⣩⣍⣋⣋⣻⣿⣿⣿⣿⣛⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢸
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠀⠀⠀⠀⣠⡤⠤⠤⠤⢤
⠀⠀⡠⠖⠀⠀⠀⢀⠄⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠤⠤⠄⠀⣀⡤⠤⣤⣔⡀
⠀⠀⢴⣶⠒⠂⠀⢷⡖⢀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⣀⠶⠇⠸⡣⠤⠦⠿⠷⠟
⠀⠀⠀⠉⠉⠁⡠⠶⠓⠻⢤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢈⠁⠀⠈⠒⠒⠒⠁
⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⣷⠀⠈⠂⠄⣀⠀⠀⠀⠀⣀⠠⠒⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠈⡠⠉⠀⠀⠀⠀⠀⠀⠀⠸⢀⡼⠄
⠀⠀⠀⠀⠀⠀⠀⣍⠉⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣤⣀⣠⣼⠆
⠀⠀⠀⠀⡔⠉⣠⣭⡍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡸
⠀⠀⠀⠀⠱⡀⠈⠉⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⡏⢉⠉⠓
⠀⠀⠀⠀⠀⠈⠉⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⠉
⣀⣈⣀⣀⣀⣀⣀⣈⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣈⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣿⢻⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⠏⠀⠙⢮⣙⠛⠿⠿⠿⠿⠛⠋⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⢠⡤⠆⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⠟⠁⢠⡴⢦⡄⠀⠀⠀⠀⠉⠙⣓⡂⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣿⠀⠀⢺⠀⢸⡇⠀⠀⠀⠀⠀⣾⠉⣻⠀⠀⠀⠘⣿⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⣿⠀⠠⠘⠦⠞⢡⡟⠀⠀⠀⠀⠹⢶⣏⠀⠀⠀⠀⢻⡶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢻⡆⠀⣿⠀⠀⠀⠀⠀⠈⠳⣤⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡟⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⠙⠳⣇⠀⣾⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡟⣶⠶⡶⢶⢻⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠙⣷⡏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⡟⢰⡇⢸⣸⡇⠀⠀⠀⣿⢀⡶⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠳⣄⠀⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⢻⠟⢺⠛⢹⣿⠀⠀⠀⢸⡏⠘⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠘⢦⣄⣘⡷⢿⡇⠀⠀⡇⠀⠀⠀⠀⢀⣀⣸⢃⡟⠀⣼⠀⢸⡟⠀⠀⠀⢸⡇⠀⢹⡄⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡙⠳⢦⣬⠟⠁⢸⡇⠀⠀⣷⠀⠀⠀⣴⠋⠉⠉⠻⢧⣀⡏⠀⣸⠇⠀⠀⠀⣾⠁⢀⡼⠃⣤⠾⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣏⠀⣀⣤⡿⠁⠀⣠⣿⠀⠀⠀⠻⠴⢶⢦⡀⠀⠙⢷⣤⠟⠀⠀⠀⠀⣿⠀⡿⠀⢸⣁⡀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠁⢀⡴⠋⣿⡀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⠟⢠⣞⠁⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠷⠴⠟⠀⠀⢸⣷⣄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⣿⠀⢠⡟⠉⠁⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣀⡴⠛⠀⢿⡙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠃⠀⠟⠃⠀⠙⣳⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠖⠛⠉⠀⠀⠀⠀⢻⣄⠈⠙⠛⠶⣶⣶⡶⠶⠞⠛⢉⡼⠛⢷⡀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⠞⠁⠈⠻⣦⣀⣴⠟⠁⠀⠈⠛⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠛⠁

⠀⠀⢀⣀⡀⠀⠀⣀⣠⡄⠀⠀⠀⠀⢀⠀⠴⠺⠆⠀⠀⠀⠀⡤⠂⠀⠰⢤⠀⠀⠀⠀⠀⠠⠄⠀⠰⠲
⠀⠀⠈⠩⠥⠄⠌⣭⠁⡀⠀⠀⠈⠉⢭⠀⠀⠤⠄⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠐⠛⠻⠄
⠀⠀⠀⢀⡀⠀⠰⠶⠤⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠉⠁⢀⠀⠀⠠⠀⠀⠀⠀⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⢐⠲⠀⠠⢚⡀⠀⠀⠀⠀⠀⠤⠀⠠
⠀⠀⠘⢔⢂⡸⢄⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠉⠈⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⢾⡧⠀⠀⠀⠀⠀⣉⣀⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣏
⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⣛⡒⠐⣚⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣦⡄⠀⣤⣤
⠀⠀⠀⡀⠶⠀⠶⢀⡄⠀⠀⠀⠀⠀⠙⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣿⢰⣶⡆⣻⠁⠀⠀⠀⠀⠀⢆⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠈⠻⠈⠟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠘⠿⠿⠛

⠀⠀⠀⠀⠤⠀⠋⠁⠀⠀⠀⠀⢀⣤⣄⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡈⠁⠈⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒
⠀⠀⠀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠞
⠀⠀⠀⠀⠛⠓⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⢀⣀⣤⣴⡶⢿⡟⢻⣿⠉⣿⡉⢿⡋⢶⣬⣿⣿⣿⠟⢡⣶⣌⣿⠏⣹⡏⢩⣛⢻⣷⡶⣤⣄⣀
⠀⠠⣶⣿⣿⣿⣵⡆⢸⣿⣷⠘⠿⡂⢹⡇⢸⡷⣄⠙⢿⣿⣿⠀⢸⣿⠛⣿⠀⣿⠁⠾⠟⣠⣿⠁⣾⣿⣿⣿⣷⡦
⠀⠢⢤⣉⠛⢿⣿⣿⠀⢿⣿⡀⢿⣧⠘⡇⢸⣇⣻⣃⣼⣿⣿⣷⣜⣛⣀⣿⠀⡟⢠⣷⠈⣿⠇⢸⣿⣿⠟⢋⣡⠤
⠀⠰⠆⣽⠀⣾⣿⣿⣇⣸⣿⡷⠾⠿⠛⠛⠛⠋⠉⠉⣉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⣶⣿⣤⣛⣛⣿⡆⢸⡅⠶
⠀⠀⠀⠃⢸⡿⠿⠛⠋⠉⠀⢀⣤⡶⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⣠⣴⣤⠤⠀⠈⠉⠛⠻⠿⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠁⠀⠀⠀⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣈⡛⠛⠂⠀⠀⡀
⠀⢀⡀⠀⢀⣤⡀⣿⢰⡏⠈⣷⣤⠀⠀⢰⣄⡾⢹⡇⢼⡋⠻⠀⠀⠀⠀⣼⡏⠱⡄⣶⣰⠏⣹⠀⠀⣠⣀⠀⠀⣀
⠀⢸⣿⣦⣿⣿⠃⣿⣿⠀⣼⠏⣿⢀⣿⢸⣿⠁⣸⠇⣨⠳⣤⡤⠖⠀⠀⣿⠀⠀⡷⣿⡏⢀⣿⢀⡄⢻⣿⣧⣾⣿
⠀⠀⠙⢿⣿⠃⠀⢻⣿⣏⣁⠀⠛⠛⣿⠘⠟⠀⣿⣰⠇⣾⢙⣧⠀⠀⠀⢻⣆⣰⠃⠹⠃⢸⣇⡼⠁⠀⢻⣿⠟⠁
⠀⠀⣠⣤⣤⡄⠀⠈⢁⣤⣄⠀⠀⢠⣤⣤⣤⠀⣬⣥⣤⡍⢩⣥⣤⣤⠀⣤⣬⣤⠀⣤⡄⠀⠉⢠⡄⠀⢠⣤⣤⣤
⠀⣸⣿⠁⠀⢻⡀⠀⣾⡟⢿⡄⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢰⣿⡇⠀⣿⣷⠀⠀⢸⡇⠀⢸⣿
⠀⣿⡇⠀⠀⢼⠃⢸⣿⠇⢸⣧⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⣆⠀⢸⡇⠀⢸⣿
⠀⣿⣇⠀⠀⠀⠀⣾⣿⣀⣀⣿⠀⢸⣿⣀⣀⠀⣿⣇⣀⡀⢸⣿⣀⣀⠀⢸⣿⡇⠀⣿⣯⠹⣧⢸⣇⠀⢸⣿⣀⣀
⠀⠹⣿⣄⢀⣼⠀⣿⡏⠉⠉⣿⡇⢸⣿⠉⠉⠀⣿⡏⠉⠁⢸⣿⡉⠉⠀⢸⣿⡇⠀⣿⣏⠀⠹⣿⡇⠀⢸⣿⡉⠉
⠀⠀⠉⡛⠛⠁⠀⠛⠁⠀⠀⠘⣃⠘⠋⠀⢀⣀⠛⠁⠀⡀⠀⠙⠛⡛⠃⠛⡛⠛⠀⠙⡃⠀⢀⣈⠁⠀⠀⠙⠛⠛⠃
⠀⢠⡟⢹⡇⢹⣧⡟⣿⠀⢀⡀⣿⠀⠀⠀⣿⢩⡇⣴⠛⢻⡄⣿⡟⣿⢀⡾⢻⡄⣴⠟⢹⡆⣿⢹⣿⣧⡞⣿⡀
⠀⢈⡵⢾⡇⢸⣿⠁⣿⢠⡿⠙⣿⠀⠀⠀⢹⣇⢀⣩⠶⣾⡇⣿⢠⡿⢸⡇⠀⠀⣩⡶⣾⡇⢻⡄⣸⣿⠃⢸⡷⣷
⠀⣿⢡⣿⡇⢸⡏⠀⣿⢸⣇⣰⣿⠀⠀⠀⢠⣿⣿⡟⣰⣿⠇⣿⣟⠁⢸⣇⣼⢣⣿⣰⣿⡇⢰⣿⡉⣿⠀⢸⡇⣿
⠀⠈⠉⠘⠃⠘⠃⠀⠉⠀⠉⠙⠛⠀⠀⠀⠈⠉⠁⠉⠁⠛⠁⠛⠈⠁⠀⠉⠁⠀⠉⠁⠛⠁⠈⠋⠀⠋⠀⠈⠁⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠘⡏⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⢸⡇
⠀⠘⢿⠛⢹⠀⠉⢓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⢿⠢⣄⠰⡿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⢁⠔⠁⣼
⠀⠀⠈⢧⠈⣇⠀⠀⠈⠁⠒⠢⠬⢍⣛⠲⠤⣄⣠⣤⣈⣧⠈⠙⢧⠀⠉⠙⠛⠛⠛⠛⠛⠶⠶⢦⣤⡀⠀⠀⣠⡤⠶⠚⠋⠉⡀⠤⠐⠊⠀⠀⠀⣿
⠀⠀⠀⠈⣇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢬⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠀⠉⠐⢋⠁⠀⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠸⡆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠹⡄⠣⡀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⢆⠑⢄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢄⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠊⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡑⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢠⠎⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠰⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠀⠛⠃⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠀⠛⠃⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀
⠀⠀⠀⠀⠀⢤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣉⣳⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠴⢾⣭⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣙⣻⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠦⣄⡀⠀⠀⠐⠂⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠒⢒⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡠⠔⠊⢕⠒⠢⠤⣀⣀⠀⠀⣀⡀⠤⠴⠂⠀⠉⠓⠦⣀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣔⠉⠀⠀⠀⠀⠉⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣠⡤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⢲⡀⠀⠠⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⠶⢺⠋⠀⠀⣤⡤⠄⠀⣀⠄⠆⣐⣠⢂⠠⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡘⠀⠀⠀⢳⣦⠤⠄⢀⡠⠔⢫⠸⠒⠁⠀⠀⠈⣗⢈⡆
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⠾⠤⢤⢀⠞⠉⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠈⡟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠿⠶⠷⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡤⠴⠖⠒⠒⠒⠒⠒⠶⠦⢼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠔⠚⠛⡿⠻⠿⠿⢿⣿⣿⣶⣦⣤⣤⣀⣰⠀⢀⣠⣤⣶⣿⠿⠟⠛⠉⠁⣧⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠡⠿⠟⠉⠉⠉⠉⣿⠀⠈⠟⠿⠿⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠢⢀⡀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⡤⠀⠀⠀⠉⠒⠤⠤⠀⠠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢱⡀⢨⡗⠒⠶⠤⣤⣤⣤⠤⡤⣤⠤⠐⠒⠚⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠱⢦⡀⠀⣏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣧⠔⣆⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣄⣀⡀⠀⠀⣷⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠃⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣚⡋⠉⢉⣙⣓⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠋⣀⠤⠴⠿⠋⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡠⠴⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠊⠀⠀⠀⠀⠀⠀⠠⢎⠔⣀⡴⠋⠀⠐⠒⠲⢦⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⢿⣮⣥⠤⠄⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⢀⡴⢦⠀⠀⢸⡇⠀⠀⠀⠀⢹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢚⣷⡶⡄
⠀⠀⠀⠀⠀⢀⡎⠀⣼⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣹
⠀⠀⠀⠀⠀⣸⠀⠀⡟⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⡼⠁
⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠹⣦⡀⠀⣤⣤⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⢠⠔⠊⠉⠉⠉⢹⣾⡄⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⢧⡄⢀⣀⡀⠠⢾⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋
⠸⣄⠀⢠⠄⠀⢾⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉
⠀⢯⣀⠀⠀⠀⢚⡷⠃
⣀⣌⣷⣶⣶⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⣀⡠⠴⠖⠒⠒⠲⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠲⠦⢤⣀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⠞⢁⡤⠖⣒⠛⠛⣒⣲⢤⡈⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⢁⡤⣖⣒⠛⢛⣒⠲⢤⡈⠳⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡴⢩⠔⢉⣠⠤⠤⠤⣈⠙⠮⣳⡀⠳⣤⣤⣤⣤⣤⣤⣤⣤⠞⢁⣖⠵⠊⣁⡤⠤⠤⣄⡉⠢⡍⢦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞⢰⠃⡰⠋⣀⠴⠶⠤⣀⠙⢆⠙⢿⡄⢹⣿⣿⣿⣿⣿⣿⡏⢠⡿⠋⡠⠋⣀⠤⠶⠤⣀⠙⢆⠘⡆⢱⡀⢳
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⡇⢰⠃⢰⡃⢴⣛⠲⡌⢆⠈⡆⠀⡇⠈⡿⠿⠿⠿⠿⢿⠁⢸⠀⢰⠁⡰⢡⠖⣛⡦⢘⡆⠘⡆⢸⠈⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⢣⠈⢇⠈⠳⠼⠿⠀⡇⣸⠀⡇⠀⡇⢠⡇⠀⠀⠀⠀⢸⡆⢸⠀⢸⠀⣇⢸⠀⠿⠧⠞⠁⡸⠁⡜⢀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢷⠈⢧⠈⠳⣌⠑⠠⠤⠔⢊⡴⠃⡼⠁⡼⠁⡼⠀⠀⠀⠀⠀⠀⢧⠈⢧⠈⢧⠘⢦⡑⠦⠤⠄⠊⣡⠞⢁⡼⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠳⣄⡈⠉⠓⠒⠛⣉⠤⠊⣠⠞⢁⡼⠁⠀⠀⠀⠀⠀⠀⠈⢧⡈⠳⣄⠑⠤⣉⡛⠒⠚⠉⢁⣠⠞⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡙⠓⠿⠭⠥⠶⠒⠋⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⠙⠒⠶⠬⠭⠿⠚⢋⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠖⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠖⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣷⠀⢰⡆⠀⣼⠛⢦⡀⢠⡖⢲⠀⢰⡆⠀⣴⢳⡆⣾⣆⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢀⣾⢻⠀⣿⣤⡞⢁⡟⠀⠀⢀⣟⢿⠀⠈⢧⡀⣿⢿⡀⡟⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣹⢈⡏⢹⡇⣿⠈⣧⠘⣇⠀⣸⢩⡏⢹⡆⣏⢸⡇⣿⠘⣿⠃⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠁⠀⠃⠛⠀⠘⠂⠙⠛⠁⠘⠁⠈⠃⠙⠛⠁⠙⠀⠉⠀⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣴⠶⠾⠛⠛⠋⢩⠥⣴⡀⣶⡏⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⡴⡇⠠⡷⠀⠀⣿⠀⣿⢳⡿⣇⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣧⣸⡆⣯⠄⠀⢛⣂⣉⣤⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣚⣈⣋⣥⣭⣶⠶⠾⠛⠛⠉⠉⠁⠀⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠀⡦⡄⠀⣀⠀⠀⡀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢾⡩⢸⠀⠀⡇⠗⠪⡅⠀⠀⠙⢹⠎⢼⠘⢙⠻⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠀⣴⠶⡄⢰⡶⠆⢠⡶⠦⡄⢰⡄⠀⢰⢰⡄⣤⡶⡄⢴⡶⠂⣰⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣸⡇⠀⢸⡇⢠⡟⢸⣿⠀⡏⢸⡇⣿⡇⠀⣿⡇⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣧⢻⡏⠁⢸⡟⢯⠀⠀⣿⣦⡇⢸⡇⣿⡇⠀⣿⡏⠁⣀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠏⠸⠧⠄⠸⠇⠈⠧⠀⠹⠿⠀⠸⠇⠻⠧⠞⠹⠧⠄⠙⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⢸⣿⡿⠿⡛⡏⡿⢟⡿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢻⣿⣜⣳⣇⣇⣿⣻⣧⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠿⠛⠛⠛⠿⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣿⣷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠁⠀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⢀⣠⣤⣖⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⢀⡤⠚⠛⠚⠛⠉⠉⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡷⣀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢟⣿⣽⣿⣷⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⡕⣧⣾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣿⣿⣿⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⡎⠉⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠈⣿⡆⠀⠀⠀⠀⢀⣆⣀⣸⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣤⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⢀⣀⣀⠀⣀⡀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠙⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⢰⣯⣾⠛⠋⠈⠛⠛⠛⠛⠿⠿⠷⠶⣤⣄⣈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠙⠿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠋⠉⠉⠉⠉⠉⠙⠂⠈⠙⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦⣄⣀⠀⠀⠀⢀⣠⡴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣤⣤⣤⣴⣦⣄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⠉⠉⠉⠉⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠀⠀⠀⣀⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⡿⠟⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⠿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢠⣤⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣶⣶⣶⣶⣶⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣉⠛⠦⠀⠀⠀⠀⠀⠤⠚⠉⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣟⢻⣿⣟⢿⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢦⡀⠀⠀⠀⢠⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⢿⣿⣶⡄⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣈⣉⣙⠛⠦⠀⠀⠀⠀⠠⠶⠛⠋⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡛⢿⣻⡷⠀⠀⠀⠀⠀⠀⠀⠞⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣄⡀⠀⠀⠀⠀⢠⣶⠿⠛⠉⡀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣶⣿⣶⡾⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠟⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⣿⣿⣿⣿⣿⣿⣟⣏⣋⣩⣏⣋⣋⣻⣛⣻⣛⣻⣏⣿⣉⣋⣉⣿⣟⣟⣛⣟⣟⣿⣻⣿⣿⣿⣿⣿⡟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠑⠠⢀⡀⣀⠀⢀⢀⡀⠄⠂⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⠀⠀⢀⡹⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⣎⠢⢤⡀⠀⠚⠑⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣄⠀⠀⠀⠀⢀⡀
⠀⠰⠒⠒⠶⠒⢒⠭⠭⠭⡀⠀⣤⠀⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⢀⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠑⠒⡞⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠒⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠁⠀⣧⠀⠦⠀⡀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡐⠋⠳⠤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⠰⠀⠀⠀⢀⣀⣣⠞
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠐⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠔⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢄⣀⠀⠀⢀⡠⠐⠂⠉⠉⠐⢚⡖⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡍⡳⠥⣀⡀⠀⠀⠀⠀⠙⢶⠁⠀⠀⠀⣀⡔⠊⢸⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡠⠜⠒⠒⠒⢄⡀⣸⣀⡰⠊⠁⠀⠈⠉⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⣀⣀⣀⠟⠀⠈⠳⠔⠒⢲⣄⣀⣀⣀⣱⣀⠀⠀⣀⣀⣄⢀⡤⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢥⠤⣒⣒⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠭⠒⠻⡓⣇⣴⠋⠉⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣼⡭⣵⣖⡲⢧⣀⣀⣀⣀⣀⣠⣤⠾⠳⣖⣒⣛⡭⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡹⣧⣈⣤⡤⠴⢶⠟⠁⠀⠉⢷⠁⠀⠀⠈⡇⠀⠀⢸⡇⠀⠀⢰⠉⢱⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠙⢲⡖⠒⠚⠋⡙⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇⠀⠀⢸⡇⠀⢀⡸⠠⢤⠀⠸⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡇⠀⠀⠀⢰⠀⠀⠈⠀⡠⣀⠀⠀⡠⠃⠀⠀⢸⡇⠀⠸⣄⠐⠛⠒⢓⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠑⠤⠤⠖⠁⠈⠁⠉⠀⠀⠀⠀⢸⡇⠀⠀⢈⠇⠀⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢸⡇⣀⡤⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠈⢲⣤⣤⣀⣀⣀⣀⣤⣶⣿⡇⠀⠀⢸⣯⡤⠤⠒⠁
⠀⠀⠀⠀⠀⢀⣀⡤⠤⠧⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⠇
⠀⠀⠀⢠⠖⠉⡠⠴⠒⡶⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⢠⠃⢀⠎⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠈⢿⣿⡟⠁⠙⠁⠈⣿⠟⠀⠀⠀⢸⡇
⠀⣠⠞⠀⠘⡆⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠉⠓⠒⠐⠂⠊⠁⠀⠀⡀⠀⢸⡇
⡰⠁⣠⣤⡀⠓⣄⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠤⢐⠀⠧⠄⢀⣸⡇
⢧⣠⣃⠇⠉⠚⠁⠀⠀⢧⠀⢸⡇⠀⠀⠀⠸⠐⠧⠘⣈⣀⣬⠤⠶⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣇⣤⡤⠶⠖⠚⠋⠉⠉
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠛⠉⠀⠀⠀⠀⠀⠀⣀⢰⡞⣩⠗⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠠⣤⡐⠢⡀⠀⠀⢠⣿⠉⠁⠉⠀⠐⣸⡇
⠀⠀⠀⠀⠀⠀⡔⣨⣽⢭⣠⣤⣤⣤⣤⡤⠞⠃⠐⠉⠉⠛⠉⡌⠀⠂⠁⣀⡿⢋⣠
⠀⠀⠀⠀⠀⠀⠀⢡⣀⠊⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠈⠉⢄⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣝⠇⠀⠸⠟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠄⠘⣷⣤⣼⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⣠⣀⠀⠀⠀⠘⣿⠟⢠⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡉⠀⣀⠀⣀⡄⠀⠀⢸⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡤⣤⣀⣠⣬⣿⡿⠁⠐⠛⠐⠈⠀⠀⠀⠀⣷⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢹⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠟⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⡀⠀⠀⠀⡀⢀⡴⠟⢿⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡠⠄⠀⠺⠀⠀⠀⠀⡡⠊⠀⠀⢀⡷⠋⠀⠾⣿⣶⣢⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣟⣧⠘⣗⠀⣀⣀⣀⣠⡤⠊⠀⠀⠀⠔⠋⠀⡀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠦⠄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠒⠱⠞⢫⣴⣶⠂⠀⠀⡈⠂⡈⠉⠉⠉⢁⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠝⠛
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣿⣿⢟⡅⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⢼⢀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠂⠀⠀⠀⠀⠀⠀⢿⠁

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⣠⢊⡵⠋⠙⠉⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠦⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⢸
⣿⠀⢀⠎⠸⡇⣾⡵⢒⠛⠉⢽⡇⠀⠀⠀⠀⠀⠀⢠⠊⠈⠈⣠⢖⠈⣧⠇⠀⣠⣴⣶⣷⣦⣰⣊⠐⠦⢄⠑⢠⠀⢸
⣿⠀⢈⠆⢸⣇⣿⠃⠀⠓⡘⢻⡇⠀⠀⠀⠀⠀⠀⠸⠀⢰⡒⠁⠇⠐⢅⠀⠸⣟⠉⢉⡈⢿⡯⠘⡁⢲⢄⠀⢄⢹⣸
⣿⠀⠻⠦⠎⠈⠛⢆⠀⠀⠅⡜⠀⠀⠀⠀⠀⠀⢤⣃⢆⣄⣥⡀⣷⠾⣿⠀⠠⡗⢂⠀⢀⠘⡇⣜⢧⣠⡎⢳⠀⢢⣹
⣿⠀⠀⠀⠀⢀⠠⣤⡟⠑⢯⢀⠀⠀⠀⠀⠀⠀⠀⡜⠛⢯⣢⣸⠇⢀⠎⠀⢀⣧⣼⡟⢉⢼⡇⡏⣺⠕⢑⡹⢲⠴⢻
⣿⠀⠀⠀⢰⠋⡄⠸⡓⠚⢻⠀⣇⠀⠀⠀⠀⠀⣸⠀⠀⣠⠟⠁⣰⡇⠀⢀⠇⡆⢀⡗⠚⠀⣭⠉⡍⠐⢌⠑⠯⣄⣼
⣿⠀⠀⢠⠃⠀⢹⠀⠑⠴⠁⢰⢻⡀⠀⠀⠀⢠⠯⠷⠴⠧⢠⠚⣧⣧⠀⡿⠒⢁⠎⢀⡤⡀⣸⢴⣁⡇⠀⠑⠤⠴⢻
⣿⠀⠀⠈⠓⢲⠇⠀⠀⢀⠀⢸⢚⠀⠀⠀⠀⡞⠀⠰⠦⡁⠛⠋⢁⡗⠀⠓⠒⡇⠀⢰⣆⣈⣠⡴⡇⡿⡄⠠⣀⡀⢸
⣿⠀⠀⢰⠆⢸⠒⢲⠀⠀⠀⢸⢸⠀⠀⠀⢠⠇⠀⠀⠀⠉⠋⢹⠀⠀⠀⠀⣰⠳⠄⠀⠈⠀⠰⡇⡧⣇⣈⡀⢀⣀⣼
⣿⠀⠀⢸⢰⠸⣀⣸⠦⠞⡦⡾⣌⠆⠀⠀⢸⡴⠦⣄⠀⠀⡇⣸⠆⠀⠀⢀⡗⠒⠀⠤⠲⠶⢤⠷⡡⡇⠀⣉⠉⠁⢸
⣿⠀⠀⠈⠙⢦⠼⢛⠀⡀⡇⢧⢏⠀⠀⠀⠀⢧⠀⠉⠙⡷⠏⢹⠄⠀⠀⡜⠀⠀⡹⠛⡆⠀⠈⡆⠙⡇⠀⠈⡇⠀⢸
⣿⠀⠀⠀⠀⡇⡄⢸⠐⡇⡇⢸⢸⠀⠀⠀⠀⣇⠀⠀⢠⡇⠀⢨⠇⠀⢠⠇⠀⢠⠃⠀⡇⠀⠀⡇⢀⡇⠀⠀⡇⠀⢸
⣿⠀⠀⠀⠠⣇⡇⡾⠀⡇⢣⠈⣏⡇⠀⠀⠀⣾⠀⠀⢘⡧⠀⣸⠀⠀⢸⡄⠀⣿⠀⠀⣗⠀⢀⡇⠈⠓⡖⢶⢱⠖⢺
⠿⡤⠤⠤⠤⠬⢽⢿⠿⠷⡿⡿⠧⠤⠤⢤⠤⠧⠤⠤⠾⠤⠤⠽⠤⠤⠼⠤⢤⠼⠤⠴⠧⡤⠼⢧⠤⠤⠧⠼⠼⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣤⠀⠀⠀⠀⠀⣤⡄⣤⣤⠀⠀⠀⢠⣤⢠⣤⣤⣤⣤⣤⡄⠀⠀⣤⡄⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⠀⢸⣿⢸⣿⠛⠛⠛⠛⠃⠀⠀⣿⡇⣿⡟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⠀⢸⣿⢸⣿⣦⣶⡄⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠿⠇⠿⠿⠀⠀⠀⢸⣿⢸⣿⠋⠛⠁⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⢰⡷⠀⢠⣤⠀⠸⠿⠸⠿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⡷⠀⣿⠇⣠⢟⣿⡖⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢀⣠⣤⣤⣄⠀⠀⠀⠻⠚⠁⡺⠶⠏⣠⡄⠉⠁⣠⣤⠀⠀⠀⠀⠀⣠⣤⣤⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠉⣿⣿⠄⠀⣀⣀⡀⠸⣷⣤⣾⣿⠤⠶⣾⡿⣷⡆⣠⣤⣤⡀⢸⣿⣿⡿⠋⠉⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣀⣀⣀⣿⡟⢀⣾⡿⠋⣿⡆⠀⣿⣿⠀⠀⣼⣿⠁⢀⣾⡿⠁⣸⡗⠀⣿⣿⣃⡀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⢷⣾⠿⣿⡄⣼⣿⣥⣴⠟⠀⢸⣿⡇⠀⢠⣿⣿⠀⢸⣿⡷⠾⠟⠁⠀⣿⣿⣭⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⡆⠀⠀⢠⣿⡇⢻⣿⡀⠀⢰⡆⢸⣿⣇⠀⢸⣿⣿⡀⢸⣿⣇⠀⣸⡇⠀⣿⣿⠉⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣷⣈⠛⠉⠀⣠⣾⡿⠃⠘⠿⣷⣶⠿⠁⠈⠛⠛⠃⠀⠙⠛⠋⠀⠙⠿⠿⠟⠁⠀⢿⣿⡄⠀⢻⣿⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢿⡄⣰⣦⢠⡇⠐⣷⠘⢻⡟⠃⣾⠀⢸⡇⠀⠀⠀⠀⠈⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢰⡞⠳⣦⠀⠀⠀⠸⣷⠇⢻⣾⠁⠀⣿⠀⢸⡇⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠀⣹⣿⠀⠀⠘⠻⢶⣾⣷⢶⣶⡄⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⢀⣠⣄⠀⣠⣀⣤⣶⣦⣀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠈⠛⠋⢠⣴⡶⢶⣶⠂⣷⢀⣿⢇⣴⡟⢙⣷⣰⣿⠛⢻⣿⠁⣾⣇⠻⣃⣿⣿⠃⣸⣿⠏⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⢴⣿⡏⠀⣸⡟⢠⡇⣼⡟⢸⣿⠀⠛⢡⣿⡇⠀⣿⡇⢰⠏⣿⣿⠉⣿⡏⠀⣿⡏⢀⣾⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣿⡄⣿⣇⣠⣿⡇⣸⢳⣿⠁⣾⣿⣀⣠⠞⣿⣧⣼⢿⣧⡞⣾⠁⣿⡇⢿⡇⢸⣿⢀⣾⡟⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣰⣿⠇⠙⠛⠋⠙⠿⠃⣿⣧⣴⠏⠙⠛⠁⠀⠈⠉⠀⠈⠉⠀⠉⠛⠋⠀⠀⠀⠀⠁⢸⣿⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠂⠀⢸⣿⣧
⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣷⣄⠀⠀⠉⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠉⢋⡄
⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠉⢿⡻⠏⠆⠀⠀⣠⣶⣿⠁⠀⠐⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⡀⠑⠀⠁⠀⢺⣿⣿⣿⣿⣶⠤⠍⠀⠀⠻⣿⣶⣦⡙⣶⠀⢀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⢀⠰⢴⣶⣶⣶⣾⡄⠀⠁⠀⠀⠙⠋⠀⠀⠉⠀⠠⢀⠀⠀⠀⠉⠉⠙⠀⠀⣿⣿⣿⣝⡋
⠀⠀⠀⠀⠀⠀⢠⠀⠀⣽⣿⡿⠿⡇⠀⠀⣠⣀⠠⢰⠃⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢋⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠰⠻⠇⠀⠀⠤⣄⡂⣛⣯⣴⣿⠀⠂⠀⠀⠀⣾⠀⠀⠀⠀⠓⢶⣶⣾⣧⣰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠈⠙⢿⣿⡻⠇⠀⠐⠀⣆⠘⢉⣼⠀⠀⠉⠻⣾⣧⣤⣿⡿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢰⠀⠘⠀⡿⠙⢤⣀⠘⢿⣏⣀⣼⡟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣀⠀⡀⣀⣠⣴⣾⡴⠀⠀⠀⠇⠀⠀⠉⠙⢋⣩⣿⠋⢀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⡷⠀⠘⠀⠀⠀⠀⠀⢀⡀⠋⠁⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡿⠀⠀⠀⠈⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠟⠿⠿⠟⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣭⣭⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣧⡸⣿⡇⣶⣶⢆⣿⡟⣰⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣬⡛⢿⣿⣷⡙⣿⢨⣭⢸⡟⣼⣿⣿⢟⣩⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⢿⣿⣿⣷⡅⣼⣿⠈⣼⣿⣿⣿⢎⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿⡜⢏⣾⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⢸⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⣌⠿⠸⢋⣼⣛⣛⢛⢻⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠷⠰⠿⠿⠿⠿⠿⠿⠿⠸⠿⠿⠿⠿





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