Biochemical Emojis & Symbols

⠀⠀⠀⠀⠀⣀⠤⠒⠀⠈⠉⠉⠁⠐⢄
⠀⠀⠀⢠⠊⡠⠀⠈⠁⢀⡀⠀⠈⠑⢄⠑⡄
⠀⠀⠀⠁⠎⠀⡀⠈⠠⠀⠀⢄⠐⡀⠀⢀⠘⡀
⠀⠀⠈⠀⠀⢰⠀⠈⠀⠀⠀⠀⠆⢰
⠀⠀⠸⠀⠂⠀⢆⠐⠄⠀⡀⠄⢀⠂
⠀⠀⠀⠑⠀⠀⠀⢢⠈⢋⠠⠒⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠤⠴⠉⠉⠉⠧⠀⠄⠤⠏⠉⠉⢱⠤⠤⠠⠜⠉⠉⠹⠆
⠀⠀⡀⠀⠀⠀⠀⠀⠇⠤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠢⡤⠤⢀⠦⠀⠀⠠⢴⠤⠄⡼⠄⠀⠀⠴⡤⡄⢰⠄
⠀⠀⠀⡄⠀⠀⢀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄
⠀⢀⠀⡀⠀⠀⠈⠢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢱
⠠⠀⠀⢐⠀⢀⢰⠒⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠃⠀⠀⠀⠀⢡
⠈⠁⠐⠜⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡠⠊⠁⠈⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡜⠀⢀⠀⠀⢀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⣄⠀⠀⠂⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡂⠀⠀⠀⠈⠄⠀⢨⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠂⠒⠒⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠒⠒⠶⠖⠶⠶⢾⠶⠶⠶⠶⣶⠶⠶⡴⠶⢦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢐⡊⠀⢘⠀⠀⠀⠀⣿⠀⢸⡁⠀⢸⣎
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⡄⠀⠀⠀⠀⣆⢀⣰⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⢀⢤⢔⡁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⠀⠀⠀⡃⠀⣼⠀⠀⠀⠀⢸⠀⢠
⠀⠀⠀⠀⣴⣩⠊⠁⢑⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉
⡄⠠⠀⠤⠿⠧⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢

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

⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⡀⠀⠠⣀⠀⠀⠀⠴⠄⠀⠀⠀⠰⠂⡐⢀⡀⠂⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⢀⢀⢀⠀⠀⣹⠀⠀⠀⠀⠠⣀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠐⠠⠐⡂⠀⠆⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡖⠉⠈
⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⢻⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢡⠀⠀⠀⠂⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠉⢀⡀⣉⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⣀
⠀⠁⠀⠈⠐⠠⠀⠀⡀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⠇⠱⡄⠀⠀⠀⣀⡤⣄⠀⠠⠀⣠⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠨⠀⠀⠔⠁⠈⠀⠁
⠀⠀⠄⡀⢆⡀⠀⠱⠀⠀⠀⠐⠀⠸⠀⠠⠁⠀⠀⠀⠁⠟⢅⡰⡞⠉⠁⢸⠁⡇⣛⡅⠀⡀⠆⡇⢰⠋⠉⢘⣟⠁⠀⠃⡇⠀⠐⢠⡄⠀⢲⠇⠂
⠀⠀⣆⠅⠬⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠥⠬⠞⠁⠑⠴⠒⠯⠄⠣⠡⠆⠨⠗⠠⠃⠪⠠⠖⠨⠥⠆⠀⠠⡇⠀⠄⠀⠈⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⢠⣤⣤⡤⠄⠤⠤⣤⠤⠤⣤⡤⠤⡤⣤⢤⡤⠤⡤⡤⠤⠤⣤⣤⣤⣠⣴⡅⢀⢥⠀⠀⣀⠀⢈⠀⢤⡀⠀⠐⠀⡀
⠀⠀⠀⡀⠐⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠁⠀⠡⠀⠀⠓⠀⠩⠅⠐⠃⠀⠈⠐⠁
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠠⠒⠀⠀⠀⠘⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⡃⠀⠀⠴⢿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠂⠀⢀⠀⢀⡄⠀⠰⢀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠁⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠦⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠦⠤⠔⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⡀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠄

⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠁⠀⠀⠀⠀⠤⠤⠥⢤⠤⠤⠄⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡷⠚⣱⠤⠀⢸⡄⠀⠀⠀⢠⠚⠣⠘⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠶
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠒⢾⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠸⠃⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⡆⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠑⠦⠤⠞⠁⠀⠀⠀⠀⢀⡴⠛⠋⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠈⠀⠀⠁⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⡶⠄⠀⠀⠀⠀⠀⠀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠦⠊
⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠂⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠠⠄⢱⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠄⡀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⢢⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠂⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⢘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⣦⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠛⠲⡄⠀⠀⠀⠤⢀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠀⠀⠀⠉⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠄⠠⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⣤⡄⠀⠀⠀⠀⡼⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⢸⠀⠀⠀⠀⢸⢇⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢸⠀⠀⠀⠀⢸⠀⠑⢦⡀⠀⠑⢄⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⠸⡆⠻⣍⠉⠙⢧⣄⣀⡙⣦⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠀⠹⡄⠈⠳⣄⠀⣯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠿⣿⣿⡆⢳⡀⠀⠈⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣿⠚⣿⣿⣿⡄⢳⠰⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⣏⣉⣿⣿⣿⣿⠈⡇⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠭⠭⠭⠬⠭⠤⠴⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⣤⠶⣄⠴⢠⣀⡄⡖⢠⣠⡄⠤⣆⠰⠆⣔⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠈⠁⠁⠉⠀⠉⠁⠈⠉⠁⠁⠈⠀⠀⠁⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⡷⣀⠀⠀⠀⣀⢾⣿⣀⡀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⢾⣿⠈⠀⠀⠀⠈⢹⣿⠎⠀⠀⠀⠀⠱⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⢾⣿⢀⠀⠀⠀⢀⢰⣿⢆⠀⠀⠀⠀⣀⠀⠺⠿⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣿⡷⠉⠀⠀⠈⠉⢾⣿⠈⠁⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠉⢸⣿⠆⠐⢛⣛⣛⣛⡛⣪⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⢛⣛⣛⣛⣛⡻⠿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣷⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠹⢹⣿⣿⣿⢟⣛⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃⠛⠿⠿⣿⣿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦
⠀⠀⠀⠀⢀⣠⣤⠹⣿⣿⡿⢃⣦⣀⠀⠀⠀⠀⠀⣀⣴⡄⢿⣿⣿⠟⣠⣄⡀⠀⠀⠀⠀⠀⣀⣴⡶
⣰⣿⣿⣧⠙⠋⠀⠀⠀⠀⠀⠀⠈⠙⢡⣾⣿⣿⡎⠛⠁⠀⠀⠀⠀⠀⠀⠉⠋⣴⣿⣿⣶⠘⠋⠁
⠻⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿
⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⣰⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⢿⣿⣿⡿⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡸⣿⣿⣿⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠈⠁⠀⠙⠻⠷⢠⣾⣿⣶⡀⠾⠛⠁⠀⠉⠁⠈⠛⠿⠂⣴⣿⣷⣆⠰⠟⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⣀⢹⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⢠⣶⠿⠋⠀⠈⠉⠁⠘⠻⢷⢀⣶⣶⣶⡄⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⡉⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠿⢿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣤⣍⣉⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢙⣉⣩⣤⣤⣤⣤⣤⣉⣉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣭⣭⣤⣤⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⠛⠛⢿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⠺⠿⢀⣿⣿⣿⣿⣿⣿⣿⠟⣉⣍⡙⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⠇⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⡀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣋⣁⣈⣻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠛⠛⠛⠛⢻⣇⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⠀⠀⢠⣾⠟⠉⠉⠉⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⢸⣇⣤⣴⣶⣶⣶⠶⠿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⣾⠟⠁⠀⠀⠀⢸⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠉⣿⡟⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⢸⡏⠙⠛⠿⠿⠿⠿⠶⣦⣄⠀⠀⠀⠀⠀⠈⠛⢿⡿⢿⣿⠶⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡅⠀⠀⠀⠀⣶⣄⡀⠀⠘⢿⣦⣄⣀⣀⣀⣀⠀⠈⢻⣧⣀⠀⠀⠀⢠⣦⣼⣿⠈⣿⡆⠀⢹⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣀⡀⠀⠿⠿⠿⠿⠛⠻⢷⣦⣀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠟⠛⠿⣦⣄⡀⠈⢻⣿⣤⣽⣧⣀⣸⡇⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠁⠀⣶⣶⣶⣶⣤⣴⡾⠟⠉⢀⣀⣤⣤⣤⣴⣶⣶⣶⣦⣤⣶⠟⠋⠁⠀⣼⡏⠈⣹⡟⠛⣿⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡃⠀⠀⠀⠀⠿⠋⠁⠀⢠⣾⠟⠋⠉⠉⠉⠉⠀⢀⣼⡟⠉⠀⠀⢰⣦⣠⣿⠁⢀⣿⠃⢸⡿⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣇⣠⣤⣴⣶⣶⣶⠾⠟⠋⠀⠀⠀⠀⠀⢉⣿⠿⢷⣾⣿⣶⣿⠿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠙⢿⣦⡀⠀⠀⠀⢸⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⡾⠋⠀⠈⢿⣦⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠿⠿⠿⠿⠿⣿⡄⠀⠀⢸⡏⠛⠻⠿⠿⠿⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠘⢿⣦⣀⣀⣀⡀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣼⡏⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣍⡉⢉⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠉⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠈⠙⢿⡦⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠞⣡⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠘⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣾⣿⣿⣿⣿⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⢶⣶⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣸⡏⠉⠉⠉⢹⡏⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⣠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠇
⠀⠀⢠⣿⣾⣿⣿⣿⣛⡉⠈⢿⣦⣤⣴⡿⠁⠀⠀⠀⠀⠻⣿⡟⢁⣶⣦⣼⣯⣤⣤⣄⣀⣀⣤⣴⣶⣣⣤⣄⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠉⠻⣿⣿⣿⡿⠛⠻⠶⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃
⠀⠀⢠⣤⠀⣤⡤⠀⣤⡄⢠⣤⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠋⠁
⠀⠀⢘⣛⠀⠛⠓⠐⠛⠃⠘⠛⠀⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠲⢦⡀
⠀⠈⢹⣯⠉⣿⡏⠉⣯⡍⢩⣿⠉⣩⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡓⠬⠤⢎⣱⡆
⠀⠀⢸⣿⠀⣿⣇⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣷⣞⣿⣿⡇
⠀⠐⠺⠿⠒⠻⠟⠚⠿⠓⠺⠿⠒⠿⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠿⠿⢿⠏

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⢸⣗⣒⣚⣒⣲⣿⣿⣿⣗⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⣿⡟⡟⠛⣿⣇
⠀⠀⠀⢸⣇⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠐⠀⠇⢘⣿⣻⣿⡯⠔⢂⣤⣶⣶⣶⡆⠀⠀⠀⠙⠃⠀⠀⣿⡇⡇⠀⠘⠿⠂
⠀⠀⠀⢸⣯⣭⣽⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⡇⣴⢠⡾⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⣘⣟⣀⣀⣀⣀
⠀⠀⠀⢸⣿⠿⠿⢿⣿⡿⠿⢿⣿⣿⣿⠀⠀⠀⢀⣿⣽⣯⣿⣽⣿⣿⡿⡹⣳⢳⢷⢟⣿⡿⠁⠀⠀⠀⢸⡟⣁⠈⠿⡿⠇⠄⣿⣿⡆
⠀⠀⠀⢸⣏⠀⠀⣿⣿⣏⣰⣸⣿⣿⣿⠀⠀⠀⣮⣭⣽⣿⣿⣿⣿⣿⣾⣿⣷⣶⣶⣿⡿⠁⠀⠀⠀⠀⣿⣷⣭⣲⣶⣷⣶⣦⣿⣿⣧
⠀⠀⠀⠸⠿⠀⠀⠼⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣽⡿⢿⣿⣿⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡿⢿⣷⡄⢹⣿⣧⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣧⡈⢿⣿⣦⢻⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⣸⣿⣧⣈⢿⣿⣦⡻⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣾⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⠹⣿⣦⡹⣿⣿⣿⠏⠉⠙⠛⠛⠻⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡘⣿⣷⡝⣿⣷⣹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢻⣿⡌⢿⣿⡌⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣖⣤⢤⣤⣿⣿⣿⣇⢻⣿⣄⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⢻⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣎⢿⣷⡝⢿⣷⣨⣿⡿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣎⢻⣿⣄⣿⣿⡿⠁
⠀⣰⣿⣿⣿⣶⣦⣤⣤⣤⣾⣿⣿⡇⣻⣿⣤⣿⣿⣿⠟⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⣿⣿⣿⡻⣿⣎⠙⢿⣿⣿⠋⠉
⣾⣿⡿⣿⣇⠙⣿⣷⣿⡿⠃
⣿⣿⣧⣿⣿⣿⣿⠿⠋
⡿⠿⠿⠟⠛⠉

⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀
⠀⢠⠅⠄⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢆⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠂⠌⢠⡝⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠿⡿⠿⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠸⠀⠀⠀⢀⡀⠈
⠀⠀⠀⡀⠀⠸⠟⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠂⠀⠀⠀⠀⠂⠒⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠐⢢⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⢄⠀⠀⣸⠀⠀⢀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠈⠀⢢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠐⠀⠀⡀⠀⠾⠶⠶⠒⠛⠚⠳
⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣆⣀⣶
⠀⠀⠀⠀⠀⠐⠲⣤⠃⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⡀⠁⠀⠁⠀⠃⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠾⠋⠋⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⢀⡼⠀⢰⢐⠉⠀⡆⠀⠏⢘⣀⠎⠓⠈⠄⡇⠀⠀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⢀⡇⡸⠘⣀⠈⣇⡀⠄⡘⠙⣈⡧⢀⣀⠇⠀⠀⣿⣿⠛⣿⣿⡅⠀⠀⠀⠀⠀⠀⠲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⢈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠑
⠀⠠⠂⠀⠀⠐⠂⠘⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠰
⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠄⠀⠀⡄⠀⠀⠀⠀⠀⢠⣄⡄⠠⠄⢀⠀⠀⠀⣾⠋⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠈⠄⠀⠀⠀⠈⠑⠀⢀⣷⡀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠁⠈⡳⠒⠀⠠⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃⠠⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠠⠎⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡤⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢐⠀⠤⡂
⠀⠀⠀⠀⢀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⢨⠿⠂⠀⠀⠀⠀⡝⠐⠀⠒⣠⠁
⠀⠀⠀⠀⠀⢂⡤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠈⠂⠀⠈⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠈⢁⡠⠄⠀⠀⠀⡀⠀⠀⠠⠨⠈⠀⠀⠀⠀⠀⡀⠵⠮⠤⠄⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⢀⠠
⠀⠀⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠧⠈⠀⠐⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠐⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢸⢀⠊⠀⠃⢰⠀⠀⠄⠠⠀⠐⠁⠀⢡
⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⢈⠷⠨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⠀⠠⠈⠈⠀⠀⠀⠁⠀⠈⠂⠐⡆
⠀⠰⠰⠀⠀⠴⡆⣠⣄⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⡯⠭⠭⠭⠭⠭⠀⠀⠀⠀⠚⠉⡈⠑⠂⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⢦⣘⣠⠄⠀⠀⠐⠈⠐⠀⠀⠀⠈⠀⠀⠘⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠂⠈⠄⠅⠔⠀⢀
⠀⡰⠒⢢⠀⠀⠀⠀⠀⠀⢀⡦⡄⠠⣶⠄⢠⡴⢦⡀⢰⠀⠀⢀⡴⢶⡄⠀⡴⠶⡄⠀⣦⠀⣴⠀⠀⠀⠀⠰⠁⠀⠈⡆⠒
⠀⠀⢀⠀⠇⠆⠸⠇⠃⠀⢸⡷⣇⠀⣿⠀⣿⠀⢸⡇⢸⠀⠀⢸⡇⠀⣿⢸⡇⠀⣦⠄⠘⣿⠃⠀⠀⢀⠀⠀⠀⠀⡀⠁⠈
⠀⠁⠀⠊⠀⠀⢸⠀⠀⠀⠘⠷⠛⠐⠛⠂⠘⠷⠚⠁⠘⠳⠒⠈⠳⠞⠃⠈⠳⠶⠋⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⣤⠄⠀⠀⠀⠀⠀⣼⣃⣘⠻⠃⡋⠀⠀⠀⢀⠃⠀⠒
⠀⠸⠇⠀⠁⠀⠸⠆⠀⠀⠀⠘⠇⠀⠀⠐⢄⠁⠀⠀⠀⠉⠀⠀⠀⠠⡀⠀⠀⡀⡄⠀⠀⢀⡠⠔⢁⣠⠀⠀⠀⠁⠀⠀⠁
⠀⠀⡄⢀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠒⠇⠀⠀⠉⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠃⠂⠇⠁⠹⠀⠀⠐⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡐⠨⠀⠀⠀⠀⢀⢀⠂⠀⢢⠀⠀⠀⠀⠀⠀⠀⠂⠊
⠀⠀⠄⡀⠁⠀⠀⠀⠀⠁⠘⠀⠀⠤⠀⠀⠀⣀⢀⠀⢀⢀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⡐⠀⠃⡀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠀⠀⢈⠔⠁⠀⠸⣀⡅⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⡀⠀⠂⠀⠄⡐⠀⢂
⠀⠀⠀⠀⠀⡴⠁⠀⠠⠄⠀⠀⠀⠈⠸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠐⠀⠁⠀⠀⠀⢈⠁⡀
⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠻⠃⠀⡐⠄⠈⠁⠀⠄⠀⠆⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡠⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠲⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡂⠀⣢⣀
⢰⠉⠉⠀⣠⣄⠀⠸⡀⠀⠀⠀⠀⠀⢀⠖⠉⠹⡎⠉⠑⡄⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⣆⠄⠁⠀⠀⠀⠓⡀
⢀⠕⡀⠘⢿⡿⠃⠀⣱⠀⠀⠀⠀⠀⠸⠀⢠⡌⣧⣦⠀⢰⠀⠀⠀⠀⠀⢸⡀⠀⠀⢿⠗⠀⣿⣿⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⠀⠞
⠘⠄⢤⠀⠀⣀⠀⢸⠁⠀⠀⠀⠀⠀⠸⠀⢈⢰⡧⡁⠀⡈⠀⠀⠀⠀⠀⠈⢿⣶⣤⣤⣤⣴⣿⡟⠀⠀⠀⠀⠀⠲⡦⠀⠀⠀⢀⣰⠤
⠀⠀⠘⠤⠎⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠣⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠗⠫⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⡃
⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢄⠵⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⣴⣿⣿⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡞
⠀⠠⣔⠀⣲⠅⠠⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣿⣿⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇
⠹⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠓⠛⠀⠀⠀⠀⠓⠚⠇⠀⠀⠀⠀⠀⠀⠸⠇⠸⠇

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠑⢆
⠀⠀⠀⠏⢋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⢀⣄⠀⠀⠀⢱⠬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠉⠉⠉⠀⠀⠀⣀⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠲⣄
⠈⠻⣷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠤⠴⠦⣤⣶⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⠴⣿⣦⣄
⠀⠀⠈⠀⠀⠀⠀⣀⣁⠀⠀⠀⠀⠀⠙⢿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⢰⣿⣷⣷⡆⠀⠀⠀⠙⠋⠁⠒⠒⠒⠒⠈⢳

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠄⢹⡆
⠀⠀⣰⠏⠹⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⡠⠄⠀⠀⠀⠀⣼⣿⣿⣦⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⠴⣾⠂
⠀⠀⢻⣧⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣶⢄⠠⠔⠉⠀⠀⠀⠀⠀⠀⠉⠩⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣍⠀⡁⣉⣧
⠀⠀⠘⠛⠻⠉⠁⠀⠀⠀⠀⠀⠀⠰⢭⡻⢟⡯⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣂⣐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣽⡆
⠀⠀⠉⠉⠉⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣿⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⣠⠀⣴⠂⠄⣰⠏⢢⠰⡶⣴⠀⠀⠀⠉⠛⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⢻⠀⣿⣠⠆⢻⣄⡜⠀⣰⠇⣶⠄⠀⠀⠀⠀⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⢀⡀⠀⢀⣀⣀⣈⠀⠈⠁⠀⣀⣀⠀⠀⠁⠀⠀⢀⡀⣀⣀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⢸⣿⡏⠁⠀⠀⠈⢹⣦⠀⠀⢸⣿⡇⠀⠀⢠⣾⠏⠀⠻⢿⣷⡀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠀⠀⠀⣿⣇⣀⣠⣤⣴⡿⠋⠀⠀⢸⣿⣿⠀⢠⣿⡏⠀⠀⠀⠀⠘⣧⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⣿⣿⠿⠿⠛⠛⠓⢦⡀⠀⢸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿⣿⠀⠸⣿⣇⠀⠀⠀⠀⠀⡟⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⢻⣿⢾⣿⣶⣶⣾⠟⠁⠀⠈⢿⣿⠀⠀⠹⣿⣦⡀⠀⣀⠔⠀⠀⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠉⠁⠀⣸⠁⣀⠀⠀⠀⠀⢀⡔⡠⠉⠉⠉⠁⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⠀⠀⠀⠀⠀⠠⠀⢀⡏⢨⡤⣤⡶⢰⡴⢸⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢸⠋⠈⠀⠁⠀⠀⠀⢸⠀⠔⠀⠀⢀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠼⠀⠀⠀⠀⠀⠀⠀⢁⣀⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠟⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⢠⣾⢛⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠐⠂⢲⣿⠏⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣸⣿⠜⣴⠟⠁⡹⠀⠰⣬⣀⠀⠀⢠⡞⠉⡿⠀⠀⣼⠋⠁⠀⣿⠀⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣿⡏⣼⡏⠀⢸⡇⠀⢀⠀⣹⠀⢠⡟⠀⢀⡇⠀⢸⠇⠀⠀⢠⡿⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⣾⠃⢠⠇⠀⣿⠀⣸⠃⠀⣾⠀⢠⣿⠊⠀⠠⢸⡇⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠂⠁⠀⢸⡿⠀⠻⠦⠞⠙⠛⠁⠀⠀⠻⠆⠋⠒⠊⠉⠐⠃⠙⠂⠀⠁⠀⠀⠀⠀⠀⣀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠛⢻⡟⠓⠒⠒⠲⠲⠶⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⡤⠤⠶⠶⠖⠚⠋⠉⠁⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡾⠀⠀⡝⠂⠀⠀⢀⠠⠄⠀⠀⣠⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⣀⡠⠊⠀⠀⠀⠇⠀⢆⡠⠊⠀⠀⠀⠠⠁⠀⠀⠈⠁⠀⠀⠀⠀⢰⡆⠀⠀⠠⠐⠆⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠼⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠙⡀⠀⠃⠀⠘⠀⠀⢈⠙⢤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠁⠈⠁⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣕⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠱⠦⣄⡀⠀⠀⣀⡤⡔⠎⠂⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡒⢄⣀⡀⢄⠺⠞⠟⡢⢀⣀⠀⡰⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠨⠑⢈⡀⠄⠋⠀⠀⠀⠀⠀⠙⠦⡈⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡡⢐⡩⠄⠂⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠢⠀⠐



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

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢀⠤⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡤⠀⠀⠀⢀⡦⢤⡎⡴⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⣆⠃⠀⠀⠀⠀⠀⠀⣤⠆⣚⠑⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠰⠗⠐⠀⣹⠃⠀⠄⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢰⡟⢳⡄⠠⠀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠘⠤⠃⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣳⢸⣴⠛⣾⣧⡟⣧⡼⣻⣶⢸⡆⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠦⠤⠘⠓⠚⠉⠘⠘⠚⠙⠋⠓⠋⢙⣿⠘⠛⡇⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣵⠴⠳⠀⠀⠀⠐⠺⣄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠠⠞⠁⠀⠀⠀⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠀⠀⠀⠀⠀⠳⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠿⢆⡀⠐⠇⠀⠀⠀⠀⠰⠛⣖⠻⣿⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠛⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠹⠧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢤⠤⢷⢶⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⡸⡆⠀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⡇⠀⠀⠀⢤⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⢾⢾⠶⠶⠶⠶⠿⢶⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠛⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⡅⡆⠀⠀⠲⣗⢸⣿⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠿⠶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢃⢰⠀⢀⣀⢒⢸⣿⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠶⠆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⡸⡄⠀⠤⣤⢸⣿⠀⠀⠀⠀⠀⢸⣠⠤⠤⢤⣤⠤⢤⠤⢤⣼⠀⠀⠀⠀⠀⠀⠀⣿⡇⠒⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠇⣇⠀⠀⠭⢸⣿⠀⠀⠀⠀⠀⠈⢿⢦⡀⠈⠚⣤⡘⢃⢞⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⣲⣤⣼⣧⣬⣤⣴⣫⠞⠀⠀⠀⠀⠀⠀⠈⠓⢿⣶⣦⣿⣾⠵⠋⠀⠀⠀⠀⠀⠀⢠⣾⣻⣧⣤⣴⣿⣳⣄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⠶⠖⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣉⣉⣉⣉⣉⣉⣉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠒⠒⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠉⠉⠉⠉⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠤⠤⠤⠞⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠤⠤⢤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣎⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠈⣉⣿⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢀⣐⣳⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢀⡚⣧⢿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠤⠼⡜⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⣏⡷⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠠⠬⢿⣹⣧⠀⠀⠀⠀⠀⠀⠈⣯⢧⠀⠀⠀⢶⡦⢰⣦⠀⠀⠀⠀⠀⣠⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣹⠀⠀⠀⠀⠀⠀⠀⠈⢯⡳⠤⣀⣀⣀⣀⣀⣀⣀⣀⡠⠔⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡟⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣴⢿⡦⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸
⠀⠀⢀⣇⣿⠀⠀⠠⡿⠀⣶⡄⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡏⣸⣿⠀⣾⠇⠀⠀⣿⡏⢷⠀⠀⢸
⠀⠀⢸⣿⡿⠀⠀⣰⡆⠰⣿⡇⢠⣾⣦⠸⣿⠃⠀⣠⢾⣷⠀⠀⠀⠀⠀⠀⣾⣿⡇⣿⡇⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⣿⣸⢁⣿⡿⢠⣶⠀⢠⣶⡿⣧⢸⡆⠀⢸
⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⣿⠇⡼⢹⣿⠀⣿⠀⣼⣵⠟⠁⠀⠀⠀⠀⠀⢀⣿⣿⠃⣿⡇⣰⠻⣿⡇⠀⠀⠀⠀⠀⠀⣿⡟⣼⣿⣥⣼⣿⠀⣿⣿⠃⠈⢹⡷⠴⢺
⠀⢀⣸⣿⣥⣤⣤⣿⡇⣰⣿⣾⠃⢸⣿⢰⣿⣾⠹⣷⠀⢀⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣧⡏⠀⣿⣇⡇⠀⠀⠀⠀⢀⣿⡗⠋⢁⣼⣿⣿⢀⣿⣿⡄⠀⣸⠃⠀⢸
⠿⠿⠛⠋⠉⠉⠉⠙⠿⠏⠛⠋⠀⠘⣿⡿⠻⠃⠀⢻⣇⣼⠃⠀⠀⠀⠀⢸⣿⣿⠀⠙⠛⠀⠀⢿⣿⠃⠀⠀⠀⠀⠘⣿⠃⡰⣣⠟⠉⠻⠟⠁⠿⣷⡶⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠶⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠢⠄⣀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⡀⠄⠠⠀⡀⠀⡀⠑⠢⢄⡀⠀⠄⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢇⠀⠀⠀⠈⠀⢈⣀⠁⠀⠀⠀⠈⠑⠢⠄⡀⠀⢀⠀⠀⠀⠒
⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰
⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢘⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠤⠤⠄⠀⠤⠤⠤
⠀⠀⠀⠀⠚⠀⠀⠀⢀⠀⠀⡤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⢸⠂⠃⠀⠅⠀⠅⠀⢩⠀⡅⠀⠨⠰⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠘⠘⠀⠀⢀⠀⠐⠀⠀⠈⠀⠀⠀⡀⠀⢀⠀⢸⠀⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠂⠄⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⢁⠐⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠠⠀⠈⠈⠀⠀⠈⠈⠁⠀⠀⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣲⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⠭⠍⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣤⡌⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣆⠀⠀⠀⡇⡖⠳⡤⠤⢤⡀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣽⣿⣿⡇⡿⣡⡝⣛⣛⡻⣿⣿⣿⣿
⠀⠀⠠⢶⣚⠞⠓⠀⠀⡇⠙⡞⠉⠉⢳⢳⠀⠀⠀⢸⣿⣿⣟⡉⡭⣥⣽⣿⡇⣿⡎⣵⣶⣶⢹⢸⣿⣿⣿
⠀⠀⠀⠀⠘⠀⠀⠀⠀⣷⣾⠁⠀⠀⢸⢸⠀⠀⠀⢸⣿⣿⣿⣿⣵⣿⣿⣿⣷⠉⠡⣿⣿⣿⢸⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣤⣤⢔⡣⠏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣛⣛⡛⠛⠛⢛⡯⢞⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠆⢻⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣯⣽⣨⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⠤⠤⠤⠴⠷⠤⠤⠼⠦⠤⠄⠀⠀⠀⢸⣿⣿⣿⣿⣟⣛⣛⣛⣋⣘⣛⣛⣃⣛⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⢀⣴⣾⣿⣷⣤⣤⣤⣤⣤⣴⣤⣦⣶⣶⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠂⠠⠀⠀⡂⠀⠀⠤⠀⡂⠀⡇⠀⠂⠀⡆⠀⠂⢠⠀⠀⠀⠀⠀⠀⢰⠀⠰
⠀⠻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣯⡀⠀⠐⠃⠐⠐⠀⠂⠂⠂⠐⠀⠒⠀⠐⠂⠂⠒⠃⠐⠃⠘⠐⠂⠂⠒⠀⠂⠐⠂⠚
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢂⡀⠀⢀⡐⢄⠀⠀⠠⢀⡁⠀⠀⣁⠢⠀⠀⢀⠔⠈⢠⣄⠈⠂⡀⠀⠀⡄⠊⠀⠀⢈⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠠⡀
⠀⠈⢀⣼⣶⣒⣾⣧⡀⠄⠠⢡⣿⣷⣀⣰⣿⣧⠀⠀⠀⠀⣠⣼⣿⣆⡀⠈⠀⡐⠀⠀⣴⠟⣦⡀⠀⠂⢀⣀⡤⠀⠤⠄⠀⢄⣡
⠀⢀⠈⠈⠧⣶⠼⠀⠁⠀⠐⠀⠀⠀⣽⣅⠀⠀⠀⠀⡀⢸⣟⠁⠋⢿⡇⠀⠀⢡⠀⢠⠿⠾⠋⠁⠀⡄⠈⠹⣀⣀⠞⢆⣀⠼⠘
⠀⠀⠁⢐⠚⠛⠓⡂⠐⠀⠀⠐⠄⡘⠛⠛⠀⠔⠁⠀⠈⠨⡉⠉⠉⠉⠡⠂⠀⠀⠲⠼⠀⠀⠀⡀⠈⠀⠀⠑⠄⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢀⠀⠁⠀⠀⠀⠈⢠⠀⠀⡐⠁⢰⣆⢠⠎⠐⠀⠀⠠⠊⣼⣷⣠⣶⣶⡄⠀⢀⠀⠀⠀⣦⠀⠀⠢⠀⠀⠔⠁⠐⣺⡀⠀⠀⡀
⠀⠀⢠⣶⣤⣶⣤⣤⣄⠇⠰⢰⣧⠀⡁⠀⣤⡄⠐⠀⣤⣶⣿⣿⣿⣿⡿⠃⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠠⠀⠀⠐⠀⠁
⠀⠠⡴⣿⣿⡿⢿⣿⠟⠀⠈⠀⠉⠤⠀⣠⣀⠀⡐⠀⠟⢿⠿⠟⠛⠉⠀⠀⠀⠃⠀⠀⠀⣇⠀⠀⢀⠂⠈⡀⠀⣶⣷⣦⡄⠀⡀
⠀⠀⠐⠠⠀⠀⠀⠄⠀⠀⠀⠈⠢⠇⠈⠉⠩⠀⠀⠀⠀⠀⠄⠀⠀⠠⠐⠁⠀⠀⠑⠠⠴⠿⠦⠄⠁⠀⠀⠈⠠⠿⣿⡿⠇⠊

⠀⠀⡠⢢⣶⣿⣷⡄⠢⠀⠀⢠⣾⣿⠀⠀⠈⠂⡀⠀⢀⢀⡵⠒⠒⠢⣔⠄⠀⠀⣤⠀⠀⠀⠀⠑⢀⠀⠀⡠⠂⠀⠀⠤⠀⠢⡀
⠀⠀⢀⣼⣏⣩⣹⣣⡄⠁⢀⠁⠀⠠⠀⣀⠀⠀⠀⠀⢠⡏⢰⣿⣿⡟⠛⡆⠀⡈⠹⣿⣦⣄⠀⠀⠀⠀⠀⠂⢸⣷⣦⠀⠀⠀⢠
⠀⠠⢀⣠⠽⣿⡿⣄⡀⡀⠈⡀⢠⣶⣶⣿⣿⣶⡶⠀⠘⣧⡴⣿⣿⡇⢠⢇⠀⢁⠀⠈⠙⠛⠳⠀⠀⠀⠀⠀⣼⣿⣿⣶⠀⠉
⠀⠀⠙⠅⢀⠀⢀⠨⠋⠀⠀⠐⠀⢙⠛⢛⠛⠉⠁⠀⠈⠪⠓⠦⠤⠒⠅⠀⠀⠀⠀⠄⡀⠀⢀⠲⠊⠀⠀⠑⢾⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠐⠂⢩⣷⣶⣄⠀⠀⠀⠐⠊⣇⠀⠗⠢⡀⠀⠀⠄⠂⠈⣩⠁⠢⡀⠀⠀⠀⠊⢀⠀⠈⠐⢄⠀⡀⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠾⠿⠿⠋⠃⠀⠛⠲⣀⡼⠦⣄⣀⣀⠀⠈⣰⣆⠀⣿⠀⣿⠀⡀⠀⠀⠀⣤⠷⣏⠀⠀⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⣼⣁⡃⣸⡋⢉⠀⢸⣿⣿⣿⣿⣧⣿⠀⠃⢀⠀⢸⡄⠒⠨⠅⠀⠀⠀⠀⠀⠄
⠀⠀⠻⠛⠛⠒⠛⢃⠐⠁⠀⠘⠣⡎⠁⠉⢻⠄⠃⠀⠈⠻⣿⣿⣿⣿⡿⠊⠀⠀⠢⢀⢷⣤⡶⠡⠊⠄⠀⠀⠀⠀⠄⠠⠄⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠠⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⢀⠀⠀⠂⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠠⠀⠀⠀⠰⡄⠀⠀⠐⠠⠠⠀⠠⠤
⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐⠀⠀⠀⠘⠛⠇⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠐⠐⠀⠂⠀⠆⠂⠖⠀⠐⠀⠒⠂⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠁⠁⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠐⠁⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠐⠴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢴⠆⠃⠀⡀⠀⢴⠄⠀⠐⠀⠀⠁⣠⣀⠓⠀⡕⠀⠀⠄⣀⠂⠀⠀⠐⠂⠀⠀⠀⡀⠀⠒⠧⢀⡠⢀⣀⠄⠀⠐
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⣰⢉⠩⢠⠀⡀⡀⠀⠀⠙⠇⠠⡦⠀⠀⠀⠀⢰⠞⠀⠀⠀⠈⠀⠠⠟⠀⣀
⠀⠀⠀⠀⠗⢀⡴⣖⠀⠐⠀⠀⠀⠀⡀⠁⢈⣦⣀⢀⡇⠁⠀⢀⠢⠀⠀⠀⠀⠀⡄⠀⢠⣰⠀⠈⠟⠁⠘⠃⢴⣶⣿⡀⠇
⠀⠀⠀⠄⡀⡀⢣⣠⣤⢨⣀⣤⡆⣤⣷⣦⢨⣿⣼⣾⡇⣤⣷⣈⣽⣤⣤⣆⣶⣴⣿⣂⣨⣿⣿⣴⣴⢶⣼⠀⢀⣌⣿⣇⣥⢦
⠀⠀⠀⠘⠁⠛⠛⠃⠉⠛⠛⠛⠓⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠚⠛⠛⠛⠛⠙⠙⠛⠛⠋⠛⠛⠚⠛⠁⠙⠛⠛⠙⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣦⣤⣴⣴⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣴⣣⡞⢁⡴⠀⠀⠀⠀⠀⢠⡞⠁⠀⣴⣷⠀⠈⠻⡄⠀⠀⠀⠀⠀⢠⢎⠁⠰⡇⡦⠀⢁⢆
⠀⠀⠀⠀⢠⣾⡿⠷⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡜⡽⢋⡾⠋⠀⠀⠀⠀⠀⠀⣾⠰⠾⢶⠉⠁⠘⢿⡆⢹⠀⠀⠀⠀⠀⢣⠈⢁⡂⡇⠆⣂⠉⢠
⠀⠀⠀⣠⣤⣬⣥⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣴⢋⣠⠔⠂⠀⠀⠀⠀⠀⢹⡀⢠⣶⡄⢠⣶⣤⠀⡾⠀⠀⠀⠀⠀⢃⡸⠀⣁⡇⣂⡁⠰⡠
⠀⠀⢀⣤⣬⣿⣿⣿⣯⣤⣤⡀⠀⠀⠀⠀⠀⠛⣾⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣉⠁⠀⣉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⣈⠇⣁⡤
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⣄⣠⣿⣿⣄⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⠄⢀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⢲⡄⣸⡇⠀⣧⡟
⠀⠀⠀⡀⢸⣿⣿⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠁⠙⣄⠀⠀⣡⢄⡰⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠈⢻⣿⠁⢠⣿⡄⢠⢀⡀
⠀⠀⠀⠀⢙⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⢋⠀⠀⣩⣉⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⠈⠈⠁⢸⡾⠁
⠀⠀⠀⠀⠎⠘⠿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠈⡋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠙⠇⠀⠀⡾⢿


⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⢤⡄⠛⠃⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⢶⡄
⠀⠀⠀⣠⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⠀⠀⠀⠙⢻⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⢨⠉⡅⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠻⠃⠀⠘⠟⠀⠀⠀⠀⠀⠀⢻⠙⠲⠠⣶⣶⣦⡄⠀⣸⠀⠀⠀⠀⠀⢀⣀⠀⢐⡛⡿⢿⠯⠀⠀⠀⠀⠀⠀⠠⣿⠴⣷⣶⣾⠄⢸⠂⡇
⠀⠀⠀⠀⠀⠾⠆⣠⡐⠿⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠙⠛⢿⣇⡴⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣈⣉⣁⣀⣾⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⣠⡼⠻⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠚⠃⣀
⠀⠀⢀⡞⣿⠻⣆⢻⠄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⡁⠀⢴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⢈⣄⢸⣦⣾⢉⡔⡀
⠀⠀⠘⢧⣙⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠀⠈⠛⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⣠⢿⣿⣿⢣⡀⠼⠆
⠀⠀⠀⢀⣹⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠘⡋⠻⠀⣐
⠀⠀⠀⠈⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⣠⣴⢤⡤⣄⡴⡄⣠⡀⣤⣠⢤⡄⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠹⠻⠍⠻⠳⠿⠇⠋⠛⠈⠃⠛⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣴⠟⣷⣴⠟⣷⠛⡼⠃⠀⠀⠀⠀⠀⠀⠘⡧⠀⡇⡾⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠙⠃⠛⠘⠐⠋⠰⠑⠗⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣴⣠⣄⣞⣦⣞⣆⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠿⡈⠉⢩⣚⠟⡄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠁⡏⣹⠀⣏⡁⢰⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⠀⡄⠄⣤⠀⢠⡄⠀⠀⠀⠀⠀⠀⠇⠀⠏⠧⠇⠯⠤⠸⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠃⠀⠓⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⢄⢀⡄⢀⣀⡀⣀⣤⠄⠀⠀⢀⠀⢀⡤⣠⡀⠀⢠⡄⣴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠙⠋⠘⠛⠃⠛⠉⠇⠀⠈⠷⠁⠀⠤⠁⠸⡧⢻⢢⣿⡪⢿⣷⢳⠀⠘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠈⠀⠀⠘⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠁⠀⠐⠀⠈⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⣿⣹⢸⡂⠸⡇⢿⣻⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣁⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⢿⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣍⠛⢦⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠛⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣉⡛⢶⣌⡛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠷⣤⠾⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⠉⣻⡆⠈⢛⣶⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠙⠛⠋⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣴⠏⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⡆⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣠⣤⣄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠃⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠻⢦⠟⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠶⢤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⡀⠀⠀⢻⣄⠀⠀⠀⠀⢀⣀⣠⣤⠶⠞⠛⠉⠁⠀⢰⡟⢷⡄⠀⠀⣿
⠀⠀⠀⠀⠀⣠⡿⠶⠶⠚⠛⠛⠓⠶⠶⠶⢦⣤⣤⣤⠶⠶⠛⠋⠉⢷⡀⠀⢸⣏⠛⠛⣻⣿⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⢸⡇
⠀⠀⠀⠀⣰⡟⠀⠀⣴⠶⣦⠀⠀⣶⠛⣶⠀⠀⣠⣤⣤⡀⠀⠀⠀⠈⢿⡄⠀⢻⡄⠀⢿⣼⠟⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣰⠏⠀⠀⠀⠙⠷⠋⠀⠀⠘⠛⠃⠀⢰⣏⠀⠀⣿⠀⠀⠀⠀⠈⢻⡄⠀⠻⣦⠀⠀⠀⠀⠀⠸⣇⣀⣠⡟⠀⠀⠀⠀⢀⣰⠟
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⣀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠈⠿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠞⠛⠉


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⢀⡀⠤
⠆⠐⠦⢀⠒⠈⠁⢀⠤⠒⠂⠚⢳⠞⠒⠒⠒⠒⠃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠐⠁⠠
⠐⠀⠀⣀⣀⣀⣠⣎⠀⡀⢀⡀⠌⠀⠀⣤⣀⣀⣀⡼⢋⣠⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠌⠀⡀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠊⠁⠀⢠⠇⠀⠀⠈⠁⠀⠀⢠⣤⣾⡉⢉⣿⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠘⠉⠐⢢
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠙⠿⣿⡆⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠐⠚⠉⠻⠆⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⣄⣴⡾⠂⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠲⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡐
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢈⡀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⣿⡟⠀⢀⣠⡀⠀⢀⡀⠀⠰⣴⣇⡀⠀⠀⠀⠀⢀⣹⣎⠁
⠀⠀⠂⠀⠀⠀⣀⠄⠘⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠘⠿⠃⠀⠙⠁⠀⣀⣏⠀⠀⠀⠠⠀⠀⡎⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⣾⠉⠉⣳⠀⠀⠀⠀⠀⠑⢄⣀⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢈⣿⠀⠀⠰⠆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⠀⠀⠐⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡷⠿⠿⠿⣶⡄⢰⡆⢠⣶⢶⣄⢸⣿⢀⣶⢶⣆⠀⣰⡾⣷⡄⣷⠀⢰⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⠀⠀⣸⡇⢸⡇⣾⡇⠈⣿⣼⣿⣾⡇⠀⣿⣶⣿⢀⣿⡇⣿⡀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠠⣿⣧⣶⡶⠿⠋⠀⢸⡇⠹⣷⣾⠟⢸⣿⠹⢷⣶⠿⠈⢿⡾⣿⡇⠻⣷⠟⣿⡃
⠀⠀⠀⡀⠀⢀⡴⢢⣤⣀⠀⠋⠙⣎⠙⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⠇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢄⠄⠀⣀⡤
⠀⠀⠀⠁⣰⠋⣤⣤⣤⣾⣷⡄⠀⠈⠒⠁⠀⠐⢦⣀⣣⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡾⠏⠀⢠⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣶⠟⠋
⠀⠀⣠⠞⢁⣼⡿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⢸⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠞⠛⠉⠠
⠀⠀⠀⣴⣿⠋⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢳
⠀⠐⣾⡟⠁⠀⠀⠀⠠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⢹⡶⢤⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣼⣤⡀⢀⣠⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣻⡄⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⣹⡛⠻⢿⣿⠇⠘⠛⠓⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠻⡇⠀⠀⠙⢀⣄⣀⣄⡀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⠙⠙⠱⣦
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⡦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⡾⠃⠀⠄⠠⠀⠉⢷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡿
⠀⢄⣤⡶⠇⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠌⠀⠀⠈⠼⡽⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠘⠂⠁⠀⠐⠀⠠⠰⠗⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣷⣶⣶⡶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠈⠉⠉⠉⠀⠂⢄
⠀⠀⠀⠀⡠⠊⠁⠄⠒⠂⠉⠀⠀⠀⠀⠢⡀⠑⡄
⠀⠀⠀⡐⢁⠎⠀⢀⠔⢀⡀⢁⡁⠐⢄⠀⠈⠄⠘⡄
⠀⠀⢠⠀⠀⠀⢰⠁⠰⠁⠀⠀⠀⢡⠀⠃⠀⠀⡀⠰
⠀⠀⠨⠀⠀⠀⢨⠀⢠⠀⠀⠀⠀⡈⠀⡄⠀⠀⠁⠀⡀
⠀⠀⠀⠣⠀⠂⡀⠣⡀⠱⣄⠤⠊⢀⠔⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠠⠀⣉⢱⡀⠰⠔⠊⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠀⣄⣀⣀⡀⠀⠀⠀⠀⣠⣀⣀⣠⠀⠀⠀⠀⢀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⠀⠚⠀⠀⠀⠛⠆⠀⠀⠖⡇⠀⠀⠘⠒⠀⠀⠐⠋⠀⠀⠀⠗
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⡧⠤⠄⠦⠂⠀⠀⠀⢦⠤⠤⢰⠇⠀⠀⠀⠴⡤⠤⢤⠧
⠀⠀⠀⢰⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠
⠀⠀⠀⠸⠀⠀⠀⠈⢆⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸
⢀⠀⠁⠈⠤⠀⠀⢠⠄⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠀⠀⠘⡀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠘
⢠⠄⠀⢀⣦⠀⠈⠐⡒⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠁⠈⠀⠀⠀⣠⠄⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⠀⠀⠀⠤⠀⠂⠀⠄⠀⠀⡁⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠨⠀⠑⢆⡀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢀⡆⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠒⠐⠐⠂⠒⠒⠒⠒⠐⠂⡖⠐⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠂⠀⠂⠀⠀⠀⠀⠀⡇⠀⢢⠀⠀⠀⠀⠀⢰⠀⠐⠂⠀⠀⠀⠁
⠀⠀⠀⠠⠀⢀⣴⢺⡧⠅⠠⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⣇⣀⡆⠀⠀⠀⠀⠀⢡⣀⣸⠀⠀⠀⠀⠀⠸⣀⣀⠆
⠀⠀⠀⠠⢠⡇⡔⠁⠀⢈⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡄⠀⠀⠐⠘⠛⠧⠀⠔⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣀⢀⡀⠀⣀⢀⡀
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠰⠄⠀⠀⠴⠂⠀⠀⢀⠀⢀⠀⠀⠀⠘⠓⠬⣶⠄⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠉⠀⠃⠀⡏⠈⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⡧⠀⢤⠀⠀⠀⠀⡏⠀⠀⡇⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⡄⢉⣀⠉⠀⡉⠀⡉⠀⡁⠀⡁⣰
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠔⠁⠀⠀⠑⠄⠀⠀⠀⣈⠤⠤⠤⢄⠂⠀⠀⡇⢸⠀⠀⠀⢂⠀⡄⠀⡇⢰⡇⡇
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⠰⡔⠒⠒⠒⢒⡄⠀⠀⠀⡏⠀⠀⠀⠘⡇⠀⠀⡇⢸⠀⡄⠀⢀⠀⡇⠀⣇⢸⠇⡇
⠀⠀⣠⠣⠤⠼⢄⠀⠀⠀⠳⠤⠴⠃⠀⠀⠙⠦⠤⠤⠞⠁⠀⠀⠀⠧⠤⠤⠤⠤⠕⠀⠠⡃⠀⠛⠀⠀⠈⠋⠀⠀⠈⠋⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠆⠰⠒⢲⠀⠀⠀⠀⠀⡟⠒⠚⠀⠀⠀⠀⠀⢎⠀⠡⡀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠠⡂⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠆⠀⠨⠀⠀⠀⠀⣞⠀⠡⡀⠈⢄⠀⠀⠀⠀⡀⡔⡀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣀⣂⣀⣂⣐⣀⣘⣀⡄⠀⠀⢰⠋⠀⠉⣇⠀⠀⠀⠀⠀⠀⠈⢫⡈⢣⡀⠀⠀⠠⠐⡃⠀⠀⢸
⠀⠀⠠⠊⠀⠀⠀⠁⢀⠀⢸⠀⡇⢰⠁⢈⡤⣸⠈⠀⠀⢠⠏⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠙⣆⣹⡄⠀⢰⠚⣼⠀⠀⢪
⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⢸⠀⡇⢸⠀⢸⠀⣿⠠⠀⠀⡼⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠄⠀⢸
⠀⠀⠹⣄⣀⣀⣀⣀⡞⠀⠸⠤⠱⠽⠠⠈⠭⠉⠰⡄⠸⣁⣀⡀⣀⣀⢀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣐⢇⣀⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢷⣤⣉⠀⠈⠦⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⣙⠟⠡⢈⣱⣄⠐⠀⣄⡀⣼⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡤⠛⢁⠄⠀⠀⠀⠀⠹⣏⠸⣳⣾⣥⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣠⣿⠄⡰⠎⠐⠀⠉⠂⠀⠤⠠⠄⠀⠘⠓⡟⠦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⡔⠃⠀⠀⣀⠀⡀⠀⣀⠀⠶⠀⠀⠀⠹⢦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠋⡿⢻⡁⠀⠀⠀⠉⠢⠀⠄⠀⠀⠀⡀⠀⠀⠀⣠⡆⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠘⡷⠐⠒⠂⠀⢠⡴⠄⠀⠒⠈⠁⠒⠀⠠⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⣤⠀⠠⠄⠀⠀⢆⡀⢀⡀⢀⣀⡄⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠻⠆⠀⠀⣤⣔⠛⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠰⠦⣀⠉⠁⠀⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣦⡀⢨⡀⣠⠀⢀⡀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⣴⡀⣠⠆⣐⣘⣃⠌⢫⡍⠉⠐⠿⣧⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡎⠀⠀⠀⢨⣽⢏⡀⠋⠻⠟⠁⠀⡈⠓⠦⣀⠘⠛⡧⣌⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⢀⣼⠛⣿⠗⠀⠀⣀⡀⠰⠶⢖⣀⠂⠐⢮⣅⡈⢻⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⢨⣄⠀⢀⣀⡀⠀⠀⠀⡨⣿⣄⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿⠆⠘⠂⠋⠘⠃⠓⠀⠉⠙⠋⠉⠀⣠⠚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⢠⣤⣴⡷⠀⠀⠤⠀⠠⠄⠁⠀⡴⠌⣡⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣈⣽⠋⠀⠀⠀⠀⢀⡀⣄⠀⣠⣤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡀⠐⡶⠂⣤⠒⢘⣳⣎⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣶⠁⡀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣞⠁⠉⢤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⣻⣷⣀⠀⠲⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠇⠀⠀⣾⢤⣴⠀⠀⠿⠶⡄⢀⠄⡘⠷⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡬⠛⠁⠁⠀⠀⠀⠈⠓⣽⣦⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢼⡇⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠠⡀
⠀⠀⠀⠀⠀⢀⡄⠁⠉⠀⠐⢋⠸⠁⢘⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠢⠄
⠀⠀⠀⡀⠆⠁⠁⠀⢴⢾⠀⠀⠀⠙⢯⠸⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⣤⢼⡇
⠀⠀⠀⠃⠀⠀⠀⠘⠆⠀⠀⠀⠘⣿⡎⠘⠆⡀⠀⠐⠀⠀⠀⢀⠙⠐⠛⠻⡟⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰⠋⢃⡒⠀⠈⠅⢀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠀⠀⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⣠⠴⠿⠀⠀⠈⠳⢤⣄⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠞⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡟⠛⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⡇
⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣦⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⢘⠿⣿⣟⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⢀⡠⠃⠀⠆⠹⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠉⠋⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢧
⠀⢠⠊⠀⠀⠿⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⢀⣴⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⢧
⠀⣶⣶⣶⣦⣄⡀⠀⢀⣰⡆⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠲⣖⠛⠒⠦⠞⢧
⠀⠸⣿⣿⣤⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⡶⠀⠀⠀⣾⣿
⠀⠀⠈⠻⢿⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣤⣤⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⢿⡿⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣀⣸
⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡒⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡟⢻⣿
⠀⠀⠉⣸⣷⣔⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣿⣟⣻
⠀⠠⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣂⣀⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣟⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣠⠊⠱⣄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⢢⡀
⠳⣄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣀⣬⡵⢮⣧⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⣄⠀⣶⠃
⠀⠈⠑⡄⡴⠋⢶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠠⡎⠉⡳⡏⠰⠆⢹⢞⠈⣱⠆⠀⠀⠀⠀⠀⠀⣀⣧⣀⣺⠋⠀⣙⡍⠁
⠀⠀⠀⡇⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠑⢟⣲⣖⡛⠊⢏⠁⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠈⡧⠤⠟⠁
⠀⠀⠀⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⡄⢠⠛⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠞⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⡇⠀⠀⠀⣴⡖⢀⡀⡾⠛⠿⢿⣶⠛⠛⠻⠿⠿⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⠋⢈⠃
⡇⠀⠀⣠⠋⣈⠀⠀⠀⠀⠲⣹⠟⠲⠤⠤⢀⢀⠀⠸⠧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠼⠢⡀⠞
⡇⠀⣸⣧⣶⣿⣿⣷⣶⣌⣭⣀⠛⠲⠒⠒⠒⣂⣀⣶⣶⣿⣿⠀⠀⠀⠀⢀⡞
⡇⠀⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⣙⣛⣿⣿⡏⠀⣿⠀⠀⠀⢀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⣿⣧⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠿⠿⢿⣿⠿⣷⠀⣿⡆⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⠉⡇
⡇⠀⣿⣿⣿⡿⠛⠛⠒⠒⠂⠀⠀⠀⠀⠀⠒⠚⠛⠶⢿⢸⣿⠓⠀⢀⣿⣃⣀⡀⠀⢀
⡇⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⠿⠟⠀⠀⠈⠛⠻⢿⡇⣶⣤⠘⢦⣻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣀⠙⠿⡄⡀⢀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠴⠀⣀⠀⠀⠔⠀⠀⠀⠐⠀⠁⢀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣤⣿⣿⣿⠇
⡇⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠄⡀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⢿⡿⠿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠂⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠠⠈⠀⠀⠑⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠐⠂⢨⠿⠂⠀⠀⠀⠀⡝⠐⠀⠒⣠⠁
⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⡀⠀⠀⠀⠀⠍⠁⠈⠂⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⡀⠀⠀⠠⠨⠈⠀⠁⠀⠀⠀⡀⠵⠮⠤⠀⡀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⣄⡀⢀⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠧⠈⠀⠐⠀⠀⠀⠆
⠀⠀⠀⠁⠀⠀⠀⠀⢷⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢸⡀⠊⠀⠃⢸⠀⠀⢄⠠⠀⠐⠁⠀⢡
⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⢈⠶⠨⠀⠀⠈⠀⠈⡇⠀⠀⠀⠀⠐⠀⢈⠠⠠⠈⠈⠀⠀⠀⠀⠀⠈⠂⠐⠆
⠀⠰⠰⠀⠀⠴⡆⣠⣄⠀⠀⡀⠀⠀⢀⠀⠄⠀⠀⠀⠯⠭⠭⠭⠭⠭⠀⠀⠀⠀⠚⠁⡈⠑⠂⠀⠀⠐⠀⠁⠄
⠀⠀⠀⠀⠀⢆⣘⡠⠄⠀⠀⠐⠉⠐⠀⠀⠀⠈⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠤⠇⠑⠀⢀
⠀⡰⠒⢢⠀⠀⠀⠀⠀⠀⢀⡦⡄⠠⣶⠄⢠⡴⢦⡀⢰⠀⠀⢀⡴⢶⡄⠀⡴⠶⡄⠀⣦⠀⣴⠀⠀⠀⠀⠠⠁⠀⠈⡆⠒
⠀⠀⢀⠀⠆⠆⠸⠇⠃⠀⢸⡷⣇⠀⣿⠀⣿⠀⢸⡇⢸⠀⠀⢸⡇⠀⣿⢸⡇⠀⣦⠄⠘⣿⠃⠀⠀⢀⠀⡀⠀⠀⠀⠁⠈
⠀⠁⠀⠈⠀⠀⢰⠀⠀⠀⠘⠷⠛⠐⠛⠂⠘⠷⠚⠁⠘⠗⠒⠈⠳⠞⠃⠈⠳⠶⠋⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡠⠀⠀⠀⠀⢀⠀⣼⣃⣘⠿⢃⡋⠀⠀⠀⢀⠃⠀⠒
⠀⠘⠇⠀⠁⠀⠸⠆⠀⠀⠀⠘⠇⠀⠀⠐⢄⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⡀⡄⠀⠀⢀⡠⠔⠀⣠⠀⠀⠀⠁⠀⠀⠁
⠀⠀⡄⢀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠒⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠃⠂⠇⠁⠙⠀⠀⠐⠂⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠨⠀⠀⠀⠀⢀⢀⠂⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠄⡐⠁⠀⠀⠀⠀⠁⠈⠀⠀⠤⠄⠀⠀⢀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⡐⠀⠃⡀⡀
⠀⠀⠀⠀⠈⠀⠠⠀⢀⠠⠀⠀⠀⢈⠔⠁⠀⠸⣀⡅⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⡀⠀⠂⠀⠄⠐⠀⢂
⠀⠀⠀⠀⠀⡴⠁⠀⠠⠄⠀⠀⠀⠈⠸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠐⠀⠁⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠹⠃⠀⡐⠄⠈⠁⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠤⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⠀⠶⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⢠⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠈⠁⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⠀⠀⠀⢰⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣢⠀⠀⠀⠀⢀⣠⠖⢁⣤⠖⠃⠀⠀⠀⠀⠀⠈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⠈⠉⠉⠒⠤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡟⡩⠳⠀⣀⣴⣯⠵⠚⠉⡗⠒⠒⠒⠒⢲⠒⢲⠐⠏⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠉⠀⠒⠤⠀⣐⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡵⢀⣴⡾⠟⠋⠁⠀⠀⠐⢟⠓⠒⠒⠒⢲⡓⠒⢚⠟⢠
⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠷⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⢀⡀⠤⠂⣠⡞
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠳⡀⠀⠀⠀⢀⠼⠞⠒⢿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢴⠶⠞⠉
⠀⠀⠀⣀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠆⠘⡄⠀⠀⡎⠀⠀⡀⣼⠱⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠣⣸⢄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠘⠏⠁⠀⠀⠈⠦⣐⠃⠀⢰⠋⡟⢀⡾⠁⣀⣀⣿⣿⣿⠀⠀⠀⠀⠉⠙⠉⢹⠈⠁
⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⢀⢤⠙⠀⡤⠃⠉⠉⣼⣅⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣸⣿⡇⣿⣇⡀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣆⠠⠋⠀⠀⠇⠀⡀⠀⠀⢰⣿⢿⣾⣿⣻⣿⣿⣿⠀⠀⠀⢀⣼⡇⠀⢸⡇
⢠⣿⣟⣷⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠣⡀⠀⣀⠠⠚⠁⠀⠀⠀⠀⠈⣈⠖⠦⠀⣿⣿⡀⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⠟⡀⠀⢸⡇
⢸⣿⣯⣿⣿⣿⣿⡜⠆⠀⠀⠀⠀⠀⢺⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣾⣏⡼⡇⠀⣸⣧
⣘⣛⣛⣛⣽⣿⣿⣇⠀⠳⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠴⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⢲⣿⠟⠒⠒⠾⠒⢻⣿⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⠳⣄⠀⠀⠀⠀⢀⣀⠠⢰⡋⠀⡄⠀⠀⢀⠀⠀⣠⣤⡤⠤⠤⠠⠀⠀⠄⠀⠾⠃⠀⠀⣰⠰⠄⢾⣿
⠀⠠⠤⠤⠴⡌⡃⠀⢻⠀⠀⠀⠀⠑⠀⠄⠒⠉⠀⠀⠈⠁⠀⠇⠀⠐⠘⣀⡀⣡⣎⢰⣶⠌⢻⣏⣠⣤⠀⣠⣀⣠⠆⠈⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⡇⡃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣐⣈⣻⣡⣭⡷⠶⠻⠋⠋⣿⣽⠃⠈⣩⡹⣇⠀⣹⡇⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠇⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣯⡿⢩⠀⣄⡄⠀⠀⡀⠀⣿⣿⡆⠒⡇⢠⠙⠛⣻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⣇⣼⡄⣽⡷⠘⣤⡃⡀⢸⣛⣇⡞⠁⠈⢢⣴⣿⣿⣟⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⢚⣿⡏⣿⡇⣼⡷⢶⣿⣿⣿⠿⠿⢿⡗⢲⣶⣼⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⣀⣠⠤⠄⠒⠊⠉⠀⠀⠀⠀⣠⣼⣿⣿⣿⣷⡿⠟⠛⠉⠁⠀⠀⠀⠘⣟⠀⣿⠃⠠⠦⠄⢳⠈⠋
⠀⠀⠀⠀⣀⣀⠾⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣻⣤⣤⣤⣤⡇
⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟

⣿⣿⠛⣿⣿⣷⣿⡿⣴⣿⣿⢿⣼⣿⣿⣿⣿⣿⡾⣿⡗⠴⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿
⣿⠿⣿⣿⣿⣿⣿⡧⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠟⣛⣰⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢃⣽⣻⣿⠃⡾
⣿⣦⣾⣿⣿⣿⠋⢀⣀⠀⢹⣿⣿⣿⣧⣾⣿⣿⣧⠰⣍⢩⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠻⣿⣟⢻⣿⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣮⡿⢿⢾⡳⣾
⣿⣿⣿⣿⣿⣿⡀⠘⠋⢀⣼⣿⣿⡟⢿⣿⣿⣿⠿⣠⠋⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⡻⣿⡏⢿⡏⣷⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣆⠹⣬⣁⣾
⣿⣿⣿⣀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣾⣿⣿⡏⢰⢭⡄⢸⣿⣿⣿⣟⣿⣿⣿⣿⡿⠿⠷⣬⠛⡉⢅⠛⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡦⣼⣻⣷⣿⣾
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠶⣆⢨⡸⣇⠏⣼⣭⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⢋⣴⢥⣄⢿⣷⣮⣍⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣧⢸⣿⣮⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⣩⣀⢾⡷⢂⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣻⠝⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⣿⣿⠘⣿⠟⣤⠜⠛⠢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠷⠾⡿⢻⣬⣯⣱⣽⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⢉⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⡉⠞⠳⠖⠣⣳⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣻⠿⣿⡿⢛⣿⣭⣽⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃⠄⣛⢃⡄⠊⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠚⣁⣰⣗⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣴⡟⣱⣿⣿⣬⣿⣿⣿⣿⣻⣿⣿⣿⣿⣟⠀⠀⢉⡀⠄⢁⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⣄⠀⠈⠁⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⣿⣿⣿⣭⣯⣞⣲⡶⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣽⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣹⣿⣿⣿⣿⣿⣿⠿⢉⠛⢿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⢳⣷⣶⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠘⣥⣋⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡯⣔⡅⢋⣾⣿⣿⣿⣿⣿⣟⣿⣿⣾⣿⣿⣿⣿⣿⣏⣶⣄⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣦⡀⠀⢹⣿⣿⣿
⣿⣿⣿⣿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠚⣽⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡄⠀⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡾⠛⢷⡌⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⡋⠻⢦⡈⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡍⠐⣀⠨⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠻⣶⡿⢃⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠙⢿⡷⠟⣠⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠁⢁⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢫⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⡟⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠀⠆⠀⠆⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠂⢈⢡⠁⠀⣂⢉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⣿⢋⣙⢿
⣿⣿⣿⣿⣿⣯⣹⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠇⠉⠘⣄⣉⠩⣈⢠⡸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠴⢀⣠⣌⠙⢦
⣿⣿⣿⣿⣿⢫⣿⣽⠟⣷⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣈⢱⠁⢰⢄⡀⢠⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⠃⣓⡈⠡⠦⠌⢻
⣿⣿⣿⣿⣿⡛⣿⣩⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⢉⠰⠀⠣⡄⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢐⣘⣫⣾⣿⣿⣶⣾
⣿⡿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠘⠀⡤⣭⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣦⠙⠏⠱⠂⣤⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⣄
⠀⠀⠀⠀⠈⠓⠒⢲⠄⢀⣈⣀⣀⣔⣀⣀⣈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠂
⠀⠀⠀⠀⠀⠀⢠⣜⣤⣤⣤⣤⣽⣿⣯⣤⣤⣤⣤⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣷⣿⡷⣿⣟⣿⣿⡾⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠁⠁⠈⠀⠉⠁
⠀⠀⠀⠀⠒⠦⢤⣴⠞⠋⠁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠸⠀⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢰⠿⣶⣶⡾⠟⠉⠀⠖⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠒⠒⠠⠀⠐⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢠⠀⠀⠳⠖⠒⠲⠖⠂⠀⠄⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⣽⣦
⠀⠀⠀⠀⠀⠀⠀⣂⠐⡀⠀⠀⠀⠀⠀⠀⠐⠀⣐⡂⢀⣀⢀⣈
⠀⠀⠀⠀⠀⠀⢠⠿⠁⡸⠖⠒⠐⠠⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢡⣧⡄⠀⠀⡇⠀⠀⠀⠀⢠⣆⣀⣀⣀⣀⣐⡒⣒⡒⠒⠂⢐⠒⣶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⣿⠆⠓⠀⢸⠀⠈⠀⣿⢸
⠀⠀⠀⠀⢤⣴⣄⣠⠶⠀⠀⠀⠀⠀⠀⢀⡀⠀⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⠉⠀⠀⣾⣾⠀⠀⠰⡧⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣶⣿⣧⠀⠀⠀⠨⠛⢤⣤⣤⣤⣿⣶⣿⣿⣿⣷⣶⣿⠧⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠉⠀⢀⣜⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡟⠛⣿⣟⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠐⡭⠉⡕⠀⠀⠀⢸⠉⢹⠀⠀⠀⠀⢫⠉⠹⠀⠀⠀⠀⠀⠯⠉⠝⠀⠀⠸⠍⢩⠇⢺⠍⢩⠂
⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣇⣸⠀⢰⠈⢿⠀⠀⠀⢠⠿⡄
⠀⠠⠂⠁⠀⠑⠢⠀⠀⢸⢀⣿⠀⠀⠀⢀⠆⠀⠘⡀⠀⠀⢀⠖⣡⡨⠌⠢⡀⠀⣿⣿⠀⢸⠀⣿⠀⠀⠀⡖⠻⢲
⠀⣇⣀⣠⣀⣠⣀⡁⠀⢸⠀⣯⠀⠀⠀⡌⠀⠀⠀⠱⠀⠀⣦⣤⣮⣭⣤⣤⡄⠀⣿⣿⠀⢨⠀⣿⠀⠀⡜⠉⠉⠈⢡
⠀⠹⣿⣿⣿⣿⡿⠁⠀⣸⠐⢿⡀⠀⢸⣿⣿⣿⣿⣷⡇⠀⠘⢿⣼⣿⣿⡿⠃⠀⣿⣿⠀⢸⠐⢻⠀⢸⣶⣶⣿⣶⣶
⠀⠀⢀⠉⠉⠁⠀⠀⠀⠉⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠈⠉⠉⠉⠉⠉
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣖⠒⡲⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⣀⠤⡄⠀⠀⠀⠀⢰⣏⣹⡆
⣀⡀⣸⣇⣀⣀⠀⢀⣀⣇⣀⣇⡀⠀⠀⠀⠀⠀⠀⠈⡄⠈⢣⡜⠁⢠⠃⠀⠀⠀⠀⠈⠛⠛⢧⡀⠀⠀⠀⠀⣠⣀⡄
⠉⠉⢹⡏⠉⠉⠉⠉⡩⢋⡉⠛⢍⠀⠀⠀⠀⣀⠤⠶⠺⣶⢛⡛⣖⠞⠢⠤⣄⠀⠀⠀⠀⠀⠀⣿⡉⣹⡖⠉⠉⠋⠁
⠀⠀⢸⡇⠀⠀⠀⡌⠠⢀⠰⠆⠀⠇⠀⠀⠀⠑⠦⠤⢤⢿⣜⣋⡞⣤⠤⠤⠚⠀⠀⠀⢠⡒⣶⠛⠿⠿⠛⠲⣆⣶
⠀⠀⢸⡇⠀⠀⠀⢸⣿⣻⣷⣿⣿⠇⠀⠀⠀⠀⠀⢀⠆⠀⡸⢇⠀⠸⡆⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠘⠤⠊⠀⠀⠑⠤⠇⠀⠀⠀⢮⠍⠹⠀⢺⠍⢩⠂⠐⡭⠉⡗
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⣀⠀⠀⠀⠀⢀⣀⣸⣀⡅⣀⢸⣀⣸⢀⡀⣇⣀⣇⣀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⡄⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣤⠔⣧⠀⠀⠀⢸⡏⢸⣿⡇⠈⢸⣿⣿⠈⠁⣿⣿⡇⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠎⠉⠋⠉⢂⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠘⠿⠂⠀⠀⢸⡇⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡀⣹
⠀⠀⢸⡇⠀⠀⠀⣸⣶⣶⣷⣶⣾⡆⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⣿
⠀⢀⣼⣧⣤⣤⣤⣬⣿⣿⣿⣿⣿⣥⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣼⣧⣬⣭⣥⣤⣤⣭⣥⣤⣤⣭⣭⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣆
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣦⣤⣄⠀⠙⠃⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠈⢿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠘⢿⣦⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣿⣿⣷⡆
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⣀⣠⡼⠿⣟⠋⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⣿⡿⠋⠉⠀⠀⠈⠉⠻⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣾⣿⠿⠋⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣿⣤⣀⣀⣀⣀⣠⡴⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡿⠾⠿⠿⠿⠿⠷⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠴⠀⠀⠉⠒⠂⠀⠀⠛⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⠀⠀⠀⣸⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠀⣤⣴⣶⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠿⠂⠀⠀⠀⠀⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠃⠀⠸⡿⠿⠿⠿⠛⠁

⠀⣁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠶⠀⠀⠈⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣠⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⣄
⣴⡗⠀⠀⠈⠃⠀⡠⡖⠀⠀⡏⠀⠀⠰⢦⠎⢢⠀⠀⠈⣁⠀⠀⠀⡑⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠠⠂⠁⢶⠀⠀⠀⣲⣿⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠉⠀⠀⠀⠀⠀⢀⢴⡜⠃⢰⡂⠀⠀⠀⠀⠈⠀⠀⢰⣿⠃⠐⠇⠀⡠⠠⠀⠀⠀⠀⡀⠀⠄
⣆⠀⠀⠀⠀⠈⠛⠶⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠠⠄⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡌⠀⣧⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⢀⠠⢀⡀⠀⠀⠀⠊⠀⠀⠀⠀⢀
⣘⣣⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠠⠀⢐
⠸⣾⠍⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠐⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠁⠀⠀⠨⠀⠈
⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⣠⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠈⢠⡀⠀⠨⣀⠀⠒⠀⠀⠀⠀⠀⠀⢀⠀⡀
⡄⠃⠚⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠉⠀⡇⠀⢱⣄⡀⠈⠐⢸⢀⠰⠀⠠⠀⠀⠿⠟⠠⠠
⠐⠘⠀⢀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠇⠀⠈⣡⡠⢀⡀⠀⠈⢄⠸⠠⠙⠢⡀⡼⠀⠀⠀⠀⠄⠰⣅⠀⠀⠈⠃
⠀⠀⠢⣿⣀⣀⡁⠉⠀⢀⠀⠀⠀⠀⠀⠂⠜⠉⠈⠀⠀⠠⠀⠁⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃
⠀⠀⠀⡇⠀⠀⠉⡆⠀⠈⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠠⠄⣀⠀⠀⠀⠓⡀⠀⠐
⠀⡎⠀⠈⠲⠤⣀⣇⣀⣀⠀⠀⠀⢢⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠄⠀⠀⠀⢠⠠⠴⡖
⠘⠀⠀⠀⠀⠀⢸⡁⠀⠈⡄⠀⠀⠀⢤⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⠊⠁
⢀⣀⣄⣨⣤⡄⠀⠙⠒⡾⠦⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠜⠃⠀⠈⠛⠧⠂⠀⠘⢇⡀⢸⠀⠀⠀⢀⠐⠀⠀⢀⡄⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢺⡇
⡠⡀⠀⣀⡝⠀⠀⢄⠀⠀⠈⢹⡒⡄⠀⠀⢀⣤⣄⣀⡀⠀⠀⠀⠠⠄⠀⠄⢀⣀⠀⠀⠀⡀⠀⠂⠀⠀⠈
⠀⠀⠂⠈⠀⠀⠀⠘⠉⠀⠀⢀⠙⠧⠄⠀⠀⠉⠉⠙⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⢐⡂⠀⠀⠃⠀⠀⠈

⠀⠀⠀⠀⢰⡿⠛⠶⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣄⡀⠀⠀⠀⢀⣀⣀⠀⠤⠤⣤⡀⠀⢠⠖⠁⣀⡉⠒⠀⠀⠀⢠⣴⣶⣄
⠀⠀⠀⠀⢈⣤⠀⣴⠀⢠⠄⢙⠀⠒⠐⠄⠀⠂⠀⠀⠀⣠⣾⣯⣾⣿⣿⣿⣦⣀⡀⠀⣿⣿⠀⢤⣤⠿⡟⠀⠄⢈⣡⣍⣡⣄⠀⠡⡀⢸⣿⡈⠹⣷
⠀⠀⠀⠀⠈⢁⣀⡝⠀⣤⡠⢇⢤⡈⠂⠐⠇⠈⠀⠀⠀⣿⣿⣾⡿⣿⣿⣿⣴⣿⠿⣶⣄⠈⠠⢤⣀⣷⡀⠈⢰⠙⣣⡄⢿⠟⣀⠀⠐⠀⢹⣷⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⣸⣌⡷⣚⠦⡆⠰⠀⠀⠀⠙⢿⣿⣿⢛⣿⠏⣡⣬⡛⢿⣿⣿⣦⡀⠉⠉⠁⠀⠈⣿⡀⠀⠀⣠⡛⣶⡇⠀⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⣾⡤⣖⣂⣝⡛⠒⠀⠀⢠⣆⠀⠀⠙⠻⠿⠿⣿⣷⣶⣿⡷⣿⣿⣿⣿⠀⠀⠀⠀⣤⣼⣶⠆⣀⣟⠈⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣋⣉⠓⣱⠧⢿⡛⠽⡇⢰⠆⠀⢿⡿⢿⡏⠐⢒⡤⣤⡈⠙⠿⣿⣿⣿⠿⠃⠀⠀⠀⠃⣿⠉⠀⢀⡅⢉⡉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⣿⣤⣤⣿⡕⠃⠉⠀⠀⠈⠳⠦⣤⣐⣂⣰⠖⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⣾⠁⠈⠉⣿⡇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢦⡄⠀⠐⢦⠈⠛⠙⢈⠻⠊⠀⠀⠀⠀⠀⣤⣤⣍⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣯⡄⣴⣿⣧⠙⣷⠀⠀⠀⢰⣿⢿⡇
⠀⠀⠀⠀⠳⠤⣀⠑⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣴⡾⢧⠿⣿⢻⣿⡄⠀⠀⠀⠀⠀⢀⠀⢰⡄⠀⠀⠀⠀⠀⢡⡘⣄⣿⣿⣿⣴⣿⡇⠀⠀⣾⣯⣿⠇
⠀⠀⠀⠀⣑⡂⠬⡷⢬⠳⡄⠰⣷⣦⣀⠀⠀⠀⢸⣏⢀⣬⣤⢰⣄⣽⡇⠀⠀⠀⠀⠂⢸⡀⢼⠁⠀⠀⠀⠀⠀⠀⠐⡌⣼⣿⣿⡟⣿⣧⠀⠀⢿⣿⠏
⠀⠀⠀⠀⠈⠑⠿⣾⡻⣧⣌⠒⣀⡉⠁⠀⠀⠀⠀⢿⣯⣵⠿⠈⢛⣉⠀⠀⠀⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢿⣿⣿⣿⣿⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣈⣙⡻⣍⣻⠷⢶⣦⡽⣖⡲⠄⠀⢈⣥⣾⣷⢸⣟⣿⣷⡄⠀⠀⠀⠀⡸⢦⡅⢀⠈⡀⠀⠀⠀⠀⠀⠈⠢⡙⠿⠿⢃
⠀⠀⠀⠀⢰⣿⠉⡸⠓⣻⣷⢞⣩⡍⢿⣿⣽⣯⡶⠀⠘⣿⡕⢋⣄⣹⣽⣿⡄⠀⠀⠄⠑⠂⠀⢷⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠁⠸⢃⣾⠉⣉⡛⠳⣜⣧⢹⣽⣧⣽⡇⠀⠙⠿⠶⠟⠈⠛⠉⠀⠀⠀⠀⠀⠈⠀⣼⢀⣿⣶⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡀⠈⢁⡞⢀⠙⣤⣿⣽⡸⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠙⢀⠀⠀⠀⠀⠀⡀⣠⣾⢿⢾⣽⡇⠀⠄
⠀⠀⠀⠀⣼⣿⢿⡆⠘⣷⠛⡿⡟⠋⣸⠛⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢀⣇⠇⠀⠆⠀⠀⠀⠀⠀⠰⣽⣷⢟⣻⡿⣿⠃⠈
⠀⠀⠀⠀⢸⣧⢸⡏⠀⣴⢾⠆⡷⠀⠉⠀⢀⠈⢡⣀⡀⠔⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠠⠐⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢻⣏⣼⣿⣿⡇⠠
⠀⠀⠀⠀⠀⣿⣼⡇⠀⣿⠆⠀⠀⠀⢠⡦⠉⠙⢀⡄⢸⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠉⡽⢿⡗⣄
⠀⠀⠀⠀⠀⠿⠛⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⠚⠁⢠⠙⢨⠃⡅⠀⠀⠀⠀⠀⠀⢀⡙⣤⠂⣌⡀⠁
⠀⠀⠀⠀⠀⠀⣠⠶⠟⠓⢦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠰⠷⡈⠁⠀⡄⠀⠀⣤⠔⠁⠀⠀⠀⠸⠀⠀⣷⠄⠀⠀⠀⠀⢀⠈⠠⠐⢊⣈⡄⠁
⠀⠀⠀⠀⠀⢰⠏⠀⠠⠤⠀⠉⢢⡀⠈⠃⠀⠀⠀⠀⠀⠰⡖⠓⡆⠀⢱⠀⠀⠀⠀⠁⠀⠀⠋⡀⠀⠈⠉⡄⠀⢀⣾⡿⢛⡷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡟⠐⠀⠀⢀⡀⠀⠀⡇⠀⣀⢤⣤⣤⣤⢦⣤⡈⠆⣀⣤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⡿⢂⠀⠀⠋⠀⠑⢲⠇⢸⣉⠩⠉⠛⠀⠀⠛⣵⠀⠀⠈⠉⠳⡄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⡼⢃⡄⢠⣤
⠀⠀⠀⠀⠀⡆⢸⠀⡀⠰⡦⢀⠎⠀⠸⣭⣆⠿⣿⣿⢶⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠶⠊⠛⢶⣄⠀⠙⠛⠶⣾⡿⠵⠦⣖⡛⢛⣋⠉
⠀⠀⠀⠀⠀⣷⠘⠀⡇⠠⠄⣸⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠀⠀⣀⣀⢀⣤⣄⢀⣤⣶⡆⢀⣀⠀⠀⠙⠛⠒⠛⠛⢉⣠⡶⠚⠡⣿⣻⣽⣦
⠀⠀⠀⠀⠀⣿⠀⢠⠃⠐⢠⡇⠀⣾⢻⣧⣞⢻⣆⣤⡀⠀⢀⣤⣿⡿⢣⡏⣽⠏⢉⣿⠰⣿⠛⢿⡄⠀⠀⠀⣠⠴⠋⠁⣠⣄⠀⣤⡛⠁⠉⣰⣾⡆
⠀⠀⠀⠀⠀⣾⠀⢘⠀⢀⠞⠀⠀⣽⣿⣍⣙⠋⢿⣉⣧⠀⢻⡏⠻⣶⡟⣤⣿⣿⣿⠿⣇⣿⣤⣾⠃⠀⢀⡴⠁⢀⡴⠞⠉⢡⡄⠉⠁⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠳⠾⠖⠋⠀⠀⠀⠛⣽⣟⠙⢷⣶⠙⣏⠀⠨⣷⣴⠟⠹⠷⣾⣏⡹⢦⣿⠎⠉⠁⠀⢠⠞⠁⠀⢻⡀⠀⠀⠀⢡⣤⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡷⣦⠁⠀⠀⠀⠠⠾⠷⠚⠛⠒⠚⠛⠀⠀⠀⠀⠀⠘⢧⠘⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡶⡄⠀⠀⠀⠀⠀⠰⢋⣉⢉⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⢃⡐⠶⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣾⣿⠎⠙⢿⣧⣴⡀⠀⢰⣶⡤⢻⣟⢼⣇⣸⣯⣤⠀⠀⠙⢢⣄⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠸⣿⣙⣤⢢⣄⠛⠋⠀⠀⠀⠉⢰⡟⠻⣯⣹⣿⣿⡟⠃⠀⠉⠀⠈⣙⠶⠦⣬⣦⣼⣤⡜⠀⠀⠀⣦⠀⠀⢀⠜⠀⣠⠄
⠀⠀⠀⠀⠀⠉⢭⣤⣯⡿⠀⣴⣿⡦⠀⠀⠈⠿⣶⡿⣿⣷⡾⠏⠀⠀⠀⠀⡎⢀⣤⡤⢴⢾⡿⠟⠓⠒⢐⡾⠧⠴⣮⡁⠐⢶⣥⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣾⠙⣯⣵⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠤⠊⠀⠀⠀⠁⠀⠀⢀⡠⠒⠂⠀⠀⠀⢀⠈⠐⠤⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠠⠠⠆⠀⠀⠈⠀⢠⠀⠰⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠂⠀⠀⠐⠂⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠂⠀⠀⢀⠀⢀⡀
⢀⣶⠀⠀⠀⠸⠂⠂⢀⣄⠀⠀⢸⡇⠀⠀⢀⣴⣩⢣⠀⠀⠀⠐⠂⠀⠀⠀⠁⢀⠀⠫⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⣿⡟⠀⠀⠀⠀⠀⠀⠈⠶⠀⠀⠘⠁⠀⢀⣈⠅⢡⠈⠂⠀⠀⣠⣾⠀⢀⠀⠐
⢯⠁⠀⠀⠀⠀⠀⠂⠀⠀⣇⠀⣀⠀⠀⠈⠀⠀⠘⠄⠀⠀⢠⣾⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⣀⣤⠺⠗⠁⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠁⠀⠉⠀⠠⠊⠑⠀⢀⡀⠀⠠⠀⠈
⣣⠀⡀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠒⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡎⡀⢹⣦⡀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣄⠀⠀⠀⢠⡢⢂⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠄
⣐⣃⡜⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠄⠀⠘
⡘⠾⡟⠢⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠋⠀⠄⠀⠂⠀⠉
⠁⠓⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⠀⠀⠐⠀⠀⠀⠘⢀⠀⠈⠀⠒⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⠀⠀⠀⠀⠀⠘⠂⠀⠆⠀⢳⡄⠀⠉⠃⢀⠐⡄⠀⠀⠀⠀⠀⡀⣶⡦⢄⣀⠐
⢃⢠⡄⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⢀⡇⠀⢀⢤⠀⠀⠀⠀⠀⠳⠀⢀⡟⠳⢄⡀⠀⢰⠀⠊⠁⠀⠁⠀⠀⡍⠁⠀⠀⠵
⠈⠀⠀⢠⡄⠀⢀⣀⡤⠔⠀⠀⠀⠄⠀⠁⠀⢈⠤⣷⡍⠼⠀⠀⠀⠑⢤⠁⠁⠀⠑⢤⠏⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⡀
⠀⠀⠈⢺⠇⠤⠤⣄⠀⠀⠠⣤⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠸⡀⠀⠀⠈⡇⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡁⠶⠀⢄⠀⠀⠀⠈⢖⠀⡔⠋
⠀⡞⠁⠀⠙⠲⢤⣀⣇⣀⣀⠀⠀⠀⠀⢦⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠢⡀⣶⢴⠆
⠸⠇⠀⠀⠀⠀⠀⢸⡁⠀⠈⢃⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⡷⠊
⠀⠀⠀⠠⣂⣀⠀⠈⠓⠦⢤⣧⣄⡀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢃
⢰⠏⠉⠃⠿⣿⠀⠀⠀⠀⣟⠀⠈⣷⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠐⡆⠁⠀⠀⠀⠈⠒⠤⣿⣀⠀⠀⠀⠂⠀⠀⠀⠒⠁⠀⠀⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛
⠚⠐⠀⠐⠔⠁⠀⠀⢱⠠⠀⠀⠀⠻⣌⡇⠀⠀⠐⠲⠿⣶⣶⣄⣀⡀⠀⠀⠉⠁⠁⠈⠒⠲⠢⠄⠀⠐⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⠂⠀⢤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣄⣨⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⢰⠆⠚⢀⡄⠀⠀⢀⣀⣤⣴⡾⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠍⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠞⠣⠜⠂⠂⠀⠀⠸⠏⠉⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠴⢠⡤⠉⠀⠀⠰⠋⠀⠀⠀⣤⢀⡀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⢰⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠀⠙⠁⢠⡄⣤⠤⠊⠀⠈⠁⠀⠀⠀⠀⢸⡇⠀⠐⠀⠀⠀⠀⠀⢀⣤⡠⣀⣀⣀⠖⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣠⣴⠆⠀⠀⡰⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⣐⣤⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠛⠋⠀⠀⠶⠁⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠀⠀⡀⠀⠐⠀⠀⠉⠩⠫⠀⠀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⢀⠀⠀⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶⠐⠁⠀⠀⠰⡤⠀⠀⠀⠀⠀⠰⠇⠀⠂⠀⢎⠀⡀⠀⠀⠀⠀⠈⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠒⠄⠈⠀⠀⠀⠀⠰⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠖⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠛⠄⠀⠀⠋⠁⠀⠀⠀⢴⢦⠆⠀⢀⡴⠋⠁⠀⠀⠦⠀⠀⠀⠐⡁⠐⢌⡓⠋⠹⡀⢀⡄⠀⠀⠀⠀⣀⡴⠁⢟⣶⡇
⠀⠀⠀⣼⠗⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠈⢧⠀⠀⠀⠀⠐⠻⣂⠿⠦⠔⢏⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠾⢿⡇⢠⡘⣧⣤
⠀⠀⢀⠇⠀⠀⠀⠀⣋⠀⠆⠠⠈⠐⠀⠀⠈⠛⠁⠀⠂⠀⢀⠔⠋⠉⣀⠄⠐⣀⡜⠀⠐⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡀⠀⠘⠒⢉⣹⡬⣥⡯⣘
⠀⠀⣡⠂⠀⠀⣴⡆⠀⠀⠀⠀⠀⡁⣟⠀⠀⠀⠀⠐⠀⠀⠸⠀⢠⠎⠁⢠⠄⠀⠁⠐⠁⠀⠀⣠⣞⡙⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠒⠟⣿⣷⣿⠀⠌
⠀⢀⡷⠀⠠⡀⠉⡁⠀⠀⢀⣤⠀⠁⠀⠀⠲⠀⠀⠀⠀⠀⣠⠀⠋⠀⢠⡎⢲⡀⠀⠾⠃⠀⠀⠡⠀⠀⠀⠀⠀⠐⣿⠀⡀⠀⠁⠀⡐⠀⠀⠀⠀⠀⠸⢻⡄⠃
⠀⢀⢯⠀⠀⠈⠀⠤⣤⠄⠈⠁⠀⠀⠀⠀⠄⠀⠀⣤⠒⠥⣠⠞⣁⠂⣿⠃⠘⢻⡄⠀⠀⠀⠐⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠘⡟⠀⠠⣶⣄⠀⠀⠀⠀⠀⠐⠖⠁⠸⢧⡄⠀⢣⠃⣄⠋⢃⠀⣽⣿⡁⢎⡹⠀⠀⠀⠀⠀⠀⠀⢀⠼⣷⠀⠀⠀⠀⠀⢀⡄⠠⠈⠀⠁⠀⠐⠘⠃⠈⠁⡦
⠀⠸⠇⠀⢸⠛⠏⠀⠀⠀⠀⠀⣈⠐⠀⠆⠀⠀⠀⠘⣦⠀⠀⢦⡾⠣⠎⠻⣸⠃⠀⡠⠤⣄⣠⠛⡇⠈⠈⣿⠃⠀⠀⠀⠠⠄⠀⠀⠋⠀⠀⠀⠀⠀⠘⡇⢶⠐
⠀⡀⣤⡄⠾⣧⣄⡈⠀⢀⠀⠀⠬⣆⣀⠂⠐⠃⠀⠀⠘⢧⣀⡾⠔⠁⢀⣠⠃⠀⠀⠘⢣⠾⣿⣏⣢⠀⠀⠁⠀⠘⠄⠂⠀⠈⠓⠷⠴⠤⠊⠀⣿⢀⡄
⠀⠐⠛⠓⢀⠉⣥⣿⠞⠁⠀⠀⡀⠠⢁⣀⢀⡠⡐⠉⠀⠀⠙⠇⡀⢏⡠⠀⠀⠀⠀⠀⠠⠘⠀⠋⠀⠀⠀⠀⠀⢠⢖⣶⣶⠀⠀⢀⣤⠶⠖⠒⢻⡋
⠀⠀⠀⠀⠀⢀⢁⡄⠀⠀⠀⣀⣌⡐⠶⡈⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠓⠃⠀⠀⣾⠁⠀⠀⠀⢸⠃⢀
⢲⣄⠀⢀⣠⠞⠈⠀⠀⠂⠁⠉⣤⠶⠋⠉⠈⠓⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⣰⠇⠤⠀⠀⣠⡖⠀⠀⠀⠀⠀⢀⡀⣿⣀⣀⣤⠶⠋⠀⣜⣷
⠈⣂⢷⡰⠟⠀⡀⠀⠀⠀⡀⠜⠁⠀⠀⠀⡄⠀⠈⠀⠀⠀⠢⢀⠀⠁⠀⠠⡜⠀⠀⠀⠀⠛⠐⠠⠄⣈⡹⠁⠀⠀⠄⢠⡟⠛⠙⢻⡏⠉⠀⠀⠀⠀⠀⢸⣇
⢸⣿⠿⡇⣲⠏⡁⠀⠀⠈⠠⠢⠘⠈⡃⠐⡜⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢸⡄⠀⣀⣼⠇⠀⠀⢤⡀⠀⠀⠀⠁
⠀⠉⠻⡟⠀⠀⠀⠀⠀⠄⡁⠀⡁⠄⠀⣄⠀⠢⠀⠀⠀⠀⠀⠀⠉⠲⠖⠋⠀⠈⠂⠀⠀⡀⠐⠀⣄⠢⠀⠀⠀⣠⡶⠞⢿⡏⠉⠁⠀⠀⣴⣦⣿⣠⡤⣤⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠁⠆⡀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠂⠔⡆⠀⠀⢠⠉⠁⠀⠃⠰⠀⠁⠄⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠰⣄⡿⠟⠋⠉⠀⠺⢿⡄
⠀⠀⢠⣧⠀⠐⡄⠀⠉⢠⡀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠊⠀⢠⠀⠐⠦⠄⠀⠈⠢⣘⠡⠀⠀⢀⣀⣿⠶⠚⠉⠀⣤⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⠁
⠀⠀⢸⡿⠁⠀⢹⡄⠀⠀⠀⠀⡴⠄⠀⢠⡤⠶⠖⠚⠛⠚⢉⠋⠉⠀⠀⠀⠀⠀⣤⣤⣴⣦⣤⡄⠀⠀⠀⣿⣡⡿⠀⠀⠀⣤⢸⡏⠀⠀⠀⠳⣼⡆⠰⠖⠻⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠙⠆⠀⠀⠀⠁⠁⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠋⠉⠁⠀⠀⠠⠴⠟⠉⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠉⠑⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⢠⠄⠾⢀⡄⠀⠀⠀⣀⣀⣴⣶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⠉
⠀⠜⢂⠳⠈⠐⠀⠀⠺⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⡄⠓⠀⠀⠰⠛⠉⠀⠀⣀⢀⡀⠀⠀⠀⠀⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡠⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠀⢠⡀⣄⠠⠔⠉⠈⠁⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⡀⠀⢀⡴⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⢤⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢸⡇⠀⠐⠀⠀⠀⠀⠀⢌⣠⣠⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠚⠿⠋⠀⠀⠬⠁⠀⠀⠀⠀⠈⠃⣀⠀⠀⠀⠀⠀⠀⠉⠒⠚⡟⠀⠐⡀⠀⠀⠀⠀⠘⡉⠛⠀⠀⠀⠐⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⢀⠀⠀⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠴⠐⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢰⠆⠀⠄⡀⠄⠀⠀⠀⠀⠀⠀⠘⣿⣷⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠀⠠⠄⠈⠀⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠔⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣄⠈⠛⡁⠀⠀⠛⠁⠀⠀⠀⣠⣤⠆⠀⠀⡠⠛⠃⠀⠀⠄⠀⠀⠈⠰⠉⠐⠘⡲⠊⢱⠀⢁⡀⠀⠀⠀⠀⢀⣴⠙⡿⣤⣇
⠀⠀⠀⢄⡆⠀⠀⠀⠀⠀⣀⣀⠀⠉⠀⠀⡀⠘⣧⠀⢀⠀⠠⠔⢦⡌⣿⡄⠤⠋⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⣶⡿⡇⣠⠘⣏⣡
⠀⠀⢀⡎⠀⠀⠀⠀⠊⠩⡄⠀⠁⠀⠀⠀⠀⠖⠉⠀⠂⠀⠀⠤⠊⠙⡀⡀⡀⢈⡶⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠰⠖⠋⢱⣝⣃⡍⢘
⠀⠀⢦⠄⠀⠐⣥⡄⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⢸⡀⢠⠞⠁⣠⠀⠘⠃⠠⠂⠀⠀⢀⡖⠳⠦⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠃⠀⠠⢲⠿⣽⣧⣿⠀⡌
⠀⢀⣧⠀⢀⠀⠙⠁⠠⠀⠀⣀⠈⠃⠁⠠⠂⠀⠀⠁⠀⠀⣀⠁⠏⠀⢀⡞⢆⠀⠁⡴⠂⠀⠀⢇⠉⡁⢀⠃⠀⢰⣶⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠹⢿⡘⠃
⠀⠀⣏⠀⠀⠁⠂⢀⣀⡀⠈⠉⠁⠀⠀⠁⡀⠀⠀⡤⠔⡂⣩⡴⢂⠀⣾⠇⠘⢷⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠸⡞⠀⠀⣤⡀⠀⠀⠀⠀⠀⠠⠶⠂⠰⣇⡀⠀⣧⠃⡰⠫⠆⠉⣷⣿⠀⣰⠺⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠀⠀⠀⠀⠀⢀⠀⠠⠈⠉⠀⠀⠠⠹⠃⠈⢁⡆
⠀⠰⠇⠀⢰⠿⠋⠀⠀⠀⠀⠀⢉⠀⠀⠄⠈⠡⠀⠹⣄⠁⠠⢎⡼⢋⡜⢷⢪⠃⠀⢀⢀⡀⢠⠖⡆⠈⠸⣿⠂⠀⠀⠀⢀⠌⠀⠀⠂⠔⠀⠀⠀⠀⠸⠂⣤⠠
⠀⠀⣀⡀⣼⢆⡀⠒⠀⢀⠀⠀⢉⡄⠀⠄⠐⠀⠀⠀⠙⣆⢀⣜⠀⠀⢀⡠⠋⠀⠀⠳⢆⣽⣿⡖⢄⠀⠀⠁⠀⠺⠄⠀⠀⠐⠲⠤⣤⡥⠔⠀⣶⠀⡀⠀⠈
⠀⠫⠿⠏⠀⠙⣫⣷⠤⠀⠀⠀⠀⣉⠁⠀⠀⢀⠀⠒⠀⠈⠻⣕⠀⠇⢈⠁⠀⠀⠀⠀⡌⠐⠁⠏⠉⠀⠀⠀⠀⢀⡤⣠⡤⠀⠀⠀⣠⣤⠤⠦⢿⠊⠁
⠀⠀⠀⠀⠈⠀⢏⠁⠀⠀⠀⢀⣐⠠⡬⠙⢻⠁⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⠭⠃⠀⠀⣼⠁⠀⠀⠀⢸⠅
⢤⡀⠀⠀⣀⡼⠈⠀⠀⠄⠒⠉⣩⡵⠒⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰⠦⢀⡀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡴⠋⠀⡘⣆⠂
⠘⢋⣆⠨⠟⠀⠀⠀⠀⠀⠠⡼⠁⠀⢀⠀⡀⠀⠁⠀⠀⠀⢄⠀⠠⠂⠀⢀⡔⠀⠀⠀⠀⠛⠠⢠⡀⠈⢻⠁⠀⠀⠀⢠⡶⠛⠛⢻⡟⠉⠁⠀⠀⢠⠁⢹⡆⠁
⢸⢿⢾⡇⣤⡖⠁⠀⠀⠈⠡⠤⠀⠐⠀⠠⠰⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⢸⡀⠀⠀⣸⡇⠀⠀⣀⠀⠀⠀⠈⠃
⠀⠛⢷⡿⠉⠀⠀⠀⠀⡀⠉⠀⠀⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠒⠤⡤⠚⠉⠑⠀⠀⠀⠀⠀⠀⣂⠌⠀⠀⠀⣠⣤⠾⣷⠒⠛⠉⠀⠀⣤⣬⣵⢀⣀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠦⠀⠁⠄⠁⡀⢀⠉⠘⠀⠀⠀⠀⠀⠀⢀⢄⡐⠄⠀⠀⢀⠐⠂⠀⠇⢠⠈⠉⡀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠠⡀⣿⡿⠏⠙⠃⢽⣷⡀
⠀⠀⢠⣇⠀⠠⡀⠀⠒⠀⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠒⠀⢠⢀⠨⣄⡀⠀⠐⢄⠐⣂⠀⠀⠀⢀⣧⣤⠴⠋⠀⣀⠀⠀⠉⢁⡇⠀⠀⠀⠀⢈⢋
⠀⠀⢸⡟⠂⠀⢣⡀⠀⠈⠁⠀⡠⡀⠀⢀⣀⣤⡴⠴⠶⠖⠐⠛⠂⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⡈⠀⠀⠀⡾⢉⡿⠀⠀⠀⣀⣹⠗⠀⠀⠈⢧⣴⡀⠠⡴⢯
⠀⠀⠈⠃⠀⠀⠈⠳⡄⠀⠀⠀⠓⠃⠀⠉⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠤⠴⠶⠿⠿⠛⠛⠉⠁⠀⢀⣤⡿⠋⠁⠀⠀⠀⠈⠙⢇⠀⠀⠀⠀⠉⠓⠂⠤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠉⠉⠉⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⡀⠀⠁⠂⠀⠀⢄⠀⠀⠀⠀⠀⠈⢀⡠⠀⠺⠠⠤⠀⠀⠀⠀⠂⠂⠉⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠘⠆⠔⠀⠂⠀⠀⠀⠠⠂⠀⠲⠀⠘⣲⠎⢥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠲⣆⠀⣶⡄⠀⣀⡀⠀⣶⡄⢀⠠⠊⠀⠀⡘⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠈⠐⢨⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠈⠉⠁⠀⠉⠁⠈⠉⠉⡃⠀⠀⠀⡄⠈⠢⡀⠀⠈⠀⠀⠀⠀⢀⢔⠕⠄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡆⠀⠀⠄⠂⠁⢢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⠁⡀⢁⠀⣁⠈⢀⠀⡀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢋⡃⠀⠀⠀⡠⠁⠈⠀⠀⠀⠀⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢈⠁⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣉⡀⠀⠀⠀⠐⡀⠀⠀⠈⠈⠢⣀⡌⠀⠀⠁⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠍⣉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⠀⠈⡀⠀⡐⠀⠀⠀⢀⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢰⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠡⠀⠡⡀⠀⠀⠀⢀⠒⠂⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠛⢶⠇⡇⠀⠀⠀⡖⠈⠀⠉⠐⠀⠀⠀⠸⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⢀⠘⠀⠄
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣀⣀⣀⣀⠀⢢⠀⠀⠢⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠂
⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠧⠄⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢳⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡞⠀⣾⡇
⠀⠀⠀⠀⠀⢰⣿⡇⠀⣶⠀⠀⠀⠀⢀⡀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡞⣠⣾⡿⠁⢰⡆
⠀⠀⠀⠀⠀⣾⣿⠁⢀⡀⠀⣴⡄⣰⢿⣿⠀⢸⣿⣟⣀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣦⡀⡴⢻⣷⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠛⠁⠀⣀⠀⢠⣤⣤⣤⡤
⠀⠀⠀⠀⢠⣿⡟⠠⣿⡇⣸⣿⣿⠃⢼⣿⢀⣾⣿⡿⠋⢀⡤⠀⠀⠀⢸⣿⢃⣼⣿⡿⠁⣿⡟⢀⡄⠀⠀⠀⠀⣸⣿⠏⠛⠛⠷⣄⢸⣿⢀⣿⣿⣸⠇
⠀⠀⠀⠀⣸⣿⠇⠀⠙⠛⠉⠻⠃⠀⠈⠻⠛⣿⡟⠧⠴⠋⠀⠀⠀⠀⠀⠙⠋⠙⠿⠃⠀⠘⠻⠋⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣿⡏⠛⠋⠙⠛⠋
⠀⠀⠀⠀⣿⣿⣤⣤⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣥⣤⣴⣾⣿⡿⠃
⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠶⠶⣶⣾⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠔⠀⣋⣉⡉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⡛⠉⠀⢀⣶⠷⠛⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⢰⣿⣿⢶⣯⣭⣭⣖⣛⡒⠦⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢖⣻⠿⠛⠛
⠀⠀⠀⠀⠀⠈⣴⣇⠀⠀⠀⠀⢸⣿⣿⡿⠋⠁⠀⠀⠀⠉⠉⠉⠑⠒⠒⠶⠮⣭⣥⣶⢶⣾⢿⣿⣇⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢳⡄
⠀⠀⠀⠀⠀⠈⣿⡃⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⠞⠁
⠀⠀⠀⠀⢠⣶⣿⣷⣄⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⣼⠋⠹⣷⡄⠀⠀⠀⠀⠀⠀⣾⣇⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⢀⣼⣿⣯⣽⣿⣄⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠹⣿⣄⡀⠀⠀⣀⣼⠿⠋⠀⠀⠀⣠⠴⠶⣄⠀⠀⣠⠎⠉⢻⡅
⠀⣠⣾⠟⡵⠁⢳⡇⠀⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢿⣀⢀⡿⠉⠉⠙⢤⣤⣼⢧⣀⡠⠤⣄
⢸⣿⢳⡾⠦⣤⣤⡧⠠⠤⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣠⣤⣾⠁⠀⠀⢸⣂⠀⣼
⢿⣟⣿⣿⠲⣿⡟⢓⠒⠒⠒⣿⣽⠇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣿⡆⠀⠀⠀⠙⠛⠋
⠈⠻⠿⣿⣿⣟⣛⣛⣁⣭⡽⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣅⡐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀⠀⢀⣀⣀⡀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠀⠄⡀⠤⡀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠿⢿⠃
⠀⠀⠀⣿⡿⢿⡿⠀⢻⠭⠽⣿⠀⢻⠿⠿⣽⠃⢻⠯⠭⢿⠃⢸⢼⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⣀⡀⠀⠀⢸⠅⠀⠀⢸
⠀⠐⡆⣽⣄⣤⡇⠀⢸⠀⠀⣿⠀⢸⣶⡟⢻⠀⢸⢀⠀⢸⠀⢸⡟⠛⠛⢠⣄⠀⠀⠀⠀⠀⠀⠻⣯⣭⠽⣼⠇⠀⠀⢸⠃⠀⠀⢸
⠰⣺⣧⣼⣿⣿⡷⠀⢾⠀⠀⣿⠀⢾⣿⡇⢰⠀⢸⠛⠑⣻⠀⢸⡇⠀⢀⣖⣿⠛⠀⠀⠀⠀⠀⠀⡇⠁⠀⢸⠀⠀⠀⢸⠃⢀⠀⠘
⠸⢿⡽⠭⡿⢭⠥⡤⠿⡿⠿⢯⣤⠿⠿⠛⡏⢁⠙⠻⡛⢫⡁⠘⢻⡿⣟⣥⣽⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠀⢰
⢰⢾⡏⣻⣿⢿⣿⠉⣿⣟⡿⣿⠉⢹⣿⡟⢹⠉⢹⡿⠛⢻⠉⢹⡟⠛⢻⠉⣿⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡞⠀⠀⠀⠀⠀⠀⢇
⠰⢺⡇⢹⠉⠘⣿⠀⣿⠀⠀⢸⠀⢸⣿⡇⢸⠀⢸⡇⠀⢸⠀⢸⠇⠀⢸⠀⣿⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄⢰⣧⣶⠶⠶⠒⠂⢤⣼⡄
⠀⢸⡇⢸⠀⠀⡿⠀⣿⠀⠀⢸⠀⢸⣿⡇⢸⠀⢸⠀⠀⢸⠀⢸⡀⠀⢸⠀⣿⠀⠀⠀⠀⠀⢀⡴⠇⠀⠀⠘⠧⣿⣯⠉⠉⠉⠋⠉⠉⠁⣷
⠀⢈⡆⢸⠀⠀⡇⠀⣿⠀⠀⢸⠀⢸⣿⠀⢸⠀⢸⡀⠀⢸⠀⢸⠃⠀⢸⠀⣿⠀⠀⠀⢀⡔⠁⢀⣀⠀⠀⠀⠀⠈⠛⡀⠀⠀⠀⠀⠀⠀⣼
⠸⢠⡇⢸⠀⢀⡏⠀⣿⠀⠀⣼⠀⢸⣿⠀⢸⠀⢸⡇⠀⢸⠀⢸⡄⠀⢸⠀⣿⡃⠀⠀⣿⣯⣭⣍⣈⣁⣀⣀⣀⣨⣽⣿⠀⠀⠀⠀⢀⣾⣿
⠘⢹⡧⣾⣤⣼⡇⡀⢾⣠⣴⡿⡤⢼⣬⣂⡼⠤⢼⣇⢠⣾⠤⢼⣇⣀⣾⢤⣿⠃⠀⠀⣿⣿⣗⠀⡄⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣥⠿⠿
⠰⢺⣁⣙⡛⣻⢷⣤⣤⡍⣉⣙⣉⡀⣙⢿⣥⣄⠀⡙⢛⢋⣍⣉⣹⢿⣿⣧⣽⡧⠀⠀⠘⣿⣿⡀⣤⠀⠀⠀⣼⣵⣿⠃
⢠⣼⠋⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⢻⡟⠀⠀⠀⠈⠛⠿⢿⣿⣶⠾⠿⠛⠁
⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠂⠠⠈⠈⡄⠀⢤⠀⠀⢀⠀⡀⠀⠀⠐⠀⠀⡈⠀⠀⠀⠀⢀⠨⠶⠥⠄⠈⠂⠑⠀⡀⢀⠄⢈⠀⠴
⠀⠀⢈⠁⠃⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠈⠀⠁⠢⠀⠁⠁⠁⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠐⣀⢀⠂⠂⠰⢑⡁⢐⡴
⠀⠀⠈⠀⠁⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡒⠀⡀⠀⠀⠐⠐⠐⠀⠸⠀⣀⠀⠀⠀⠀⠀⠂⠐⢂⢀⠒⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⠂⠀⠂⠢⠀⠲⢦⡀⠐⡥⢄
⠀⠈⠀⡄⢀⠀⡀⠀⠀⠘⢃⢂⠀⠈⢀⣄⠀⠀⣀⠀⠀⠀⢀⣁⠓⠈⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢥⡤⡄⠀⢰⠄⢀⠀⠀⠛⠀⠑⠊
⠀⠀⡀⠓⠀⠄⠠⠼⠀⠀⠈⠀⠀⢀⠐⠀⠁⢨⠙⡞⡛⠀⠈⠈⠀⡂⢠⠁⠂⠀⠊⠁⠀⠛⡏⠏⠭⠉⠁⠀⠁⠀⠀⠐⠐⠀⢀
⠀⠀⠀⠠⠖⠐⠦⠀⠀⠀⠀⠀⠀⢐⠴⠀⢠⠀⠑⠊⠁⠤⠁⢊⡀⠀⡸⠀⠀⠐⠀⠀⠠⠀⠀⠀⢀⣀⢠⡀⠀⠀⠀⠒⠀⠠⠄⡀
⠀⡎⠉⢣⠀⠇⠀⠀⠀⠥⠀⠀⡀⠈⠶⠀⠪⡨⠀⠂⠤⠐⠁⠃⢀⣈⠀⠀⠀⠀⠁⠈⠀⠀⢼⡥⣞⠛⡘⠁⠂⠈⠂⠴⡆⠠⠀⠂
⠀⢄⠂⡴⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠆⠀⠀⠀⠀⣤⠀⠀⠀⠠⠌⠀⠸⠂⠀⣤⠀⠂⢴⡀⡠⠔⠊⠤⠄⠀⠐⠍⠀⠐⠄⠀⡀⢀
⠀⠀⣀⠀⠀⠀⠐⠆⠀⠀⠀⠉⠁⠀⠀⠀⠤⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⠀⠂⢀⠈⢀⠈⠀⡀⡀⠀⠀⠀⢀⡐⢿⠂⣀⠀⠀⠉⢙
⠀⠀⣏⡂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠄⠀⠀⠈⠀⠀⠠⠔⠀⠘⠀⠀⢃⠀⠠⠀⠀⢀⠐⠐⠀⠀⠀⠀⢹⠀⠀⠀⠁⠀⠑⠁
⠀⢀⠀⠀⠐⠀⠀⠠⠀⠀⡄⣤⢠⠀⡀⠂⠀⠀⠀⡀⠄⢶⣲⣤⠂⠀⠀⠀⠀⠄⠀⠙⡀⠀⠀⠈⠈⠀⡰⠁⠀⠈⠀⠀⠀⠠
⠐⠪⢬⠐⠲⠀⠀⠀⠀⠀⠀⡆⠀⠐⠠⠂⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠴⣂⠀⠀⡀⢠⣒⡆⢀⠀⠀⠁⠀⠀⠠⠂⠄
⠀⠤⡀⢃⠂⠀⠀⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠄⠀⢰⣖⠀⠀⠀⠀⠀⠀⠸⠁⠐⠈⠁⠀⠀⡁⡀⠁⡒⢀⠁⠂⠘⠄⠀⠃⠀⠰⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐⠈⠀⠀⠐

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢐⠀⠒⢛⠒⠒⠒⠂⠰⠆⢘⡋⢀⠍⠈⠭⠁
⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢠⡄⠠⠬⠥⠄
⠠⠀⠄⠀⠀⠐⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠈⠉⠀⢤⠀⢠⠐⠀⠀⢐⣃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠰
⠀⢀⣠⣀⣴⣒⣶⡒⠲⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡤⠀⠀⠰⠘⠈⠀⠈⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡌⠙⠁⡒⣋⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠧⠈⡓⠀⢀⣶⣿⣿⣧⠀⠀⠡⠖⡀⠀⣀⠀⢀⣁⡁⠀⢠⣠⣤⣤
⠚⣧⠀⠀⠽⠷⠤⠀⣀⡉⠉⠀⠃⠰⣦⡃⢀⠀⠀⢠⡄⢁⠀⠒⠂⢀⠀⢙⣿⣿⣸⡿⠂⠀⣤⡁⢀⠈⠀⠄⠉⠅⢀⣈⣉⠛⣿
⠙⠁⠀⠀⠂⠒⢀⣊⣍⠥⢭⠀⠋⠙⠑⠂⠀⠁⠈⠩⠀⠨⠝⠠⠖⠈⠀⣚⡟⠁⣟⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣯⣍
⠁⢸⠁⠐⢃⣿⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠁⢠⣶⣦⠘⢁⡄⠉⠁⠀⠈⡀⠄⠀⠀⠀⢰⢁⣌⣈⠀⠀⠲⠮⣿⣿⣟
⢉⣉⣁⣠⠈⡰⡗⠒⠌⠀⠀⠀⣁⡘⠁⠀⢠⠀⠀⠐⡄⠀⠙⠋⠀⠈⠀⢀⠘⠈⠀⠓⠀⠈⠉⡁⠐⢀⡀⢐⠒⠀⠀⠀⡈⠛⠛
⠠⠀⠀⠀⠀⣦⣄⡼⡴⡆⠀⠈⠀⠀⡂⠀⠀⠈⠁⠀⡀⠀⠀⠦⠄⠾⢆⡀⠀⠒⠀⠐⠈⠐⠒⠒⠀⣀⣀⡀⣀⠀⣀⠀⠀⠀⠤
⢀⠐⠚⠁⠀⠉⢉⠓⣾⠇⠀⠀⢁⡐⠁⠀⠉⠀⠄⠀⠁⠀⣿⣃⢀⣉⣉⣅⡉⢀⠈⣀⣉⣁⠈⠀⠀⠐⠛⢉⣉⠠⠥⠀⠐⠀⠐
⠉⠩⡤⠄⠀⡀⢛⣛⡒⠐⠀⠀⠀⠀⠁⠀⠠⠀⠈⠀⠆⠀⣢⠀⠐⠢⣤⠤⠂⠰⠶⠖⠀⠀⠀⣤⣤⣶⣖⣁⠠⠤⠚⠁⠀⠄⢯
⠛⠘⠁⠀⢈⡉⠉⠉⠉⠀⡀⣀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣛⡀⠤⠐⠉⡋⠁⠁⢉⢀⠀⠀⠒⠛⢲⢟⣁⣀⣀⣠⠀⠀⡀⠀⠈
⠐⠤⢰⣶⣏⣻⠂⠀⠀⠈⠉⠉⠉⠛⠀⠀⠀⠊⠕⠀⣽⣿⣿⣧⣿⠆⠿⡿⠆⠀⠀⠀⠈⡆⠀⢀⣈⡀⡈⠉⠉⡡⠄⠈⠁
⠀⠀⠀⡿⢋⣽⠄⡀⠐⠶⠒⠒⠂⡀⠀⠀⠀⠈⠀⢀⠈⠙⠁⠀⣀⡀⠕⠠⠠⠐⠂⠤⠊⠀⠀⠶⠀⠀⠒⠀⠀⠀⠀⡔⢛⠓⡄
⢄⠀⠈⠷⠺⠉⠉⠀⠀⠉⠖⠀⠉⠀⠀⠀⢚⡇⢠⠀⠀⡀⡾⠀⠄⠜⣠⡄⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡀⠀⠉⠂⠌
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣶⡇⠀⠀⠈⠀⠆⠇⠀⠁⠐⠶⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⢖
⣹⠀⠀⠢⠰⠄⠐⠀⠀⠀⠀⠘⠿⠿⠃⢀⠀⠀⢀⣀⣛⡋⠉⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⡇⢐⢒⠄⣷⡂⢀⡶⠂⢀⠀⠀⢐⣶⣦⣎⡀⠀⡠⠀⠠⠖⠤⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢑
⠇⢀⣁⠀⣈⡁⠈⠀⠤⠬⠁⡠⠝⣿⣿⣿⠛⠀⠁⠒⠀⠙⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡈⣅⠛⣎⠠⣵⣾⡵⡄⠀⠀⠄⢀⠋⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠒⡀⠀⠀⣠⣶⣶⣦⡀⠒
⠉⠀⠈⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣥⣯⠀⠀⢿⣿⣿⣿⠃⠀⠀⡴⠒⠐⢦⠀⠀⠀⠰
⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠄⠀⠀⠀⠀⠀⢠⡂⠀⠀⢠⣤⡤⠄⠈⠋⠠⠀⠀⠉⠉⠁⠀⠀⠀⢧⡀⠀⣨
⠀⠀⠀⠀⠀⠠⠀⠂⢀⠠⢆⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠃⠒⠒⠀⠂⠀⠀⠀⠐⢀⠆⠀⠒⠒⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⡆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⢰⠖⠆⠴⠲⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲
⠀⠀⠀⡏⠀⡙⢙⠀⠀⣀⡀⠀⠀⠀⠀⢠⡦⡧⢼⢼⠤⠦⠤⠴⢴⡆⠀⠀⠀⠀⠀⠀⡏⠀⡙⢙⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⠀⡸⠀⢨⡀
⠀⠀⠀⡧⠰⣧⣤⠇⠀⠀⠂⠀⠀⠀⠀⢸⡇⡇⢸⢸⠀⡂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡧⠰⣧⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⣀⡀⢀⣈⡆
⠀⠀⢀⣇⣀⣈⣁⣀⣰⣶⣾⡀⠀⠀⠀⠸⡇⠓⠚⠘⠒⠃⠒⠚⢸⡇⠀⠀⠀⠀⠀⢀⣇⣀⣈⣁⣀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⢀⡴⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⡆⠸⠛⣷⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣋⡓⠢⣄
⠀⠀⠀⠀⠀⠸⣇⠀⠀⣸⠇⠀⠀⠀⠀⢠⡞⠉⠉⢉⣉⠉⠉⠑⣦⠀⠀⠀⠀⠀⠀⢧⣤⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⡞⣤⣄⠛⠁⣶⡜⡆
⠀⠀⠀⢀⣠⡾⠋⠛⠛⠁⠀⠀⠀⠀⠀⠘⠧⣀⡠⠋⠈⠆⣀⡠⠛⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⣦⠀⣴⣬⢁⠇
⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠇⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⣄⡠⠴⠋



⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢃⡀
⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢴⣷⣮⡤⠀⡀
⠀⠀⠀⢐⣯⡫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣬⣥⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⡿⣿⠑⠈⠁
⠀⠀⠐⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠿⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠳⠀⠀⠁


⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣬⡀⠛⠛⢩⣥⠀⠀⠀⠀⠀⠀⢀⢀⡠⠦⠀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⣀⠰⠆⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢠⣤⣀⣈⡄⢀⣦⣤⣆⠀⠀⠀⠀⠀⠀⠀⠃⠀⢨⡄⣀⠰⠦⡄⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⡟⠁
⠀⠀⠈⠛⠿⠛⠁⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦⢄⠈⠠⠟⣹⠯⠫⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠃⠠⠿⠀⠉⠠⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠈⠃⢈⠀⠆⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠈
⠀⠀⠀⠀⠀⠀⡆⠡⠐⠀⢴⠆⠀⠁⠀⢦⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⡀⠊⢦⡍⠆⠀⠐
⠀⠀⠀⠀⠀⠀⢻⠐⡀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠘⢆⠀⠀⠨⡠⠈⣳⣙⠆⠀⢊⡀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⠆⠈⠿⡆⡀⠱⣊⣴⢛⣦⠁⠒⠀⡁
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠠⠀⠀⠁⠀⢄⡀⠀⠀⠀⠀⠂⠑⠌⠰⠄⠀⢌⠉⣣⣈⠈⣀⡀⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣁⠀⠀⠀⠀⠀⠙⣁⡔⠀⡀⠀⠐⡀⠃⣈⠁⢀⡛⠀⠂⠲⠖⠐⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠉⠄⠀⠀⠈⠒⠲⠀⡰⠡⠳⠆⠰⠄⢤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠃⠀⠂⠠⠀⠉⠀⠀⠐⠄⠈⠀⠁⠀⠀⠀⠄⢠⠂⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠂⠀⠅⠀⢴⠀⠄⠀⠀⠀⠀⠀⢀⠀⠴⠀⠀⠀⠀⠈⠑⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠰⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠀⢈⠠⠀⠀⠘⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠤⠄⠀⠂⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠁⠀⠆⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠁⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠠⠈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠠⠠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠩⡈⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢡⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣆

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠐⠀⢨⠅⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠈⢢⣔⣁⠀⠔⠠⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠠⡄⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠤⠤⠦⠤⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠃⢐⠄⠈⠩⠩⠉⠉⠉⠀⠐⠀⠀⠈⠀⠰⠆⠀⠀⠀⠁⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠱⢚⡃
⠀⠀⠀⠂⠀⠀⠀⠀⠉⢸⣳⠀⢿⠁⣼⠉⢧⢰⡇⠀⢰⠏⢳⡄⡞⠉⡃⠈⢧⣰⠃⠀⠀⠑⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠂⣼⡀⢿⣤⡞⠸⣧⣤⠸⣄⡼⠃⢧⣀⡟⠁⢨⠇⠀⠀⠈⠐⠀⠐⠂
⠀⠀⠀⠐⢠⠀⠠⠐⠀⡀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⡐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠄⠀⠀⠈⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⢠⠀⢂⠀⡀⠈⠂⠠⡄⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠉⠂⠀⠐⠈⠀⠀⠐⠄⠂⠐⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⠀⠀⠐⠋⠀⠙⠂⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢡⠃⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠂⡀⠀⠀⢈⡙⠒⠢⠄⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⣀⠤⠤⠀⢀⠀⠈⠈⠙⠲⢤⣀⠀⠀⠠⢀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⢰⠊⠉⠀⠀⠀⠸⠁⠘⠏⠀⠋⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠑⢦⡀⠀⠐⠁⠀⠰⠂⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠂⠀⠭⠇
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⢠⣰⠄⠀⠀⠂⠀⠀⠀⠀⠤⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⢿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢺⠒⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠆⡀⠄⠀⠀⠀⠀⠀⠀⠖⠐⠐⠁⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠰⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠨⠅⢀⠀⠠⠀⠀⡀⠀⠀⠄⠐⠀⠀⠀⠀⡄⠀⠀⠁⠀⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡐⠀⠀⣀⣀⣀⠀⠀⢀⣁⡂⠂
⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠂
⠀⡀⠀⠀⠀⠉⠀⠀⠀⢈⢀⠶⠀⠀⡖⢢⠀⠨⡁⠀⢈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⡄⠀⠠⡄⠀⢠⠀⠀⣤⠀⢠⡄⠀⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⡇⠀⠈⡅⠀⠭⠀⠀⢡⠀⢸⠅⠀⢠⠀⠿
⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠗⠀⠀⢀⠀⠀⠴⠀⠀⠀⢸⠀⡇⠨⠀⠇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠂⠀⠀⠒⠀⢸⠀⡇⢀⠀⡀⠀⢠⠀⠀⢸⠀⠸⠀⠠⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡔⠀⠀⠒⠂⠐⠂⠀⠀⠉⠀⣀⠀⠄⠄⠀⢸⠀⢣⣀⡀⠁⠀⠘⢀⣀⠜⠀⠀⢇⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡀⠀⠀⠀⠀⡠⠀⠀⠉⠀⠀⠀⠀⠼⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣀⡀⠰⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠱⣆⠴⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣰⣾⡿⠀⠀⠀⠀⢨⡆⣄⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣶⣦⣠⡤⠾⣧⣿⣟⠉⠻⢧⠀⠀⠀⣶⢸⣷⣿⠁⣸⣿⣅⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠙⠿⠈⠫⣄⠀⠙⣿⣿⡀⢀⡀⠀⢴⡆⣻⣿⠛⣽⢢⡿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠘⣿⣷⣸⡏⠀⢸⠃⢀⣏⠀⡢⠿⣷⢶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⢰⡄⠈⠁⠀⢀⠀⠀⣤⠼⣿⣿⡇⠀⢹⡗⠈⠉⠈⠀⢸⣿⡿⠉⠉⡉⢉⣷⡷⣦
⠀⠀⠀⡠⠖⠛⠛⠛⠉⠀⠓⠀⠀⠘⣿⣼⣿⣇⠀⢸⠇⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⡾⠃⠀⠈⠓
⠀⠀⠀⠶⣤⡤⠄⠄⡀⠀⠀⢀⠈⠰⣼⣿⣿⣿⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠈⠉⠶⣝⣻⣿⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡌⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠳⠶⣥⣌⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡇⠄⠀⠀⠀⠀⠠⠚⡀
⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣿⣶⣶⡄⠀⠰⢷⣿⣿⣩⡖⠛⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣷⣶⠶⠚⠋⠉⠁⠀⠈⠻⢿⣿⣶⣤⣀⠈⢹⠉⠙⢷
⠀⠀⠀⠀⠀⠸⣇⠠⠈⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣞⣻⡿⢿⠆⢺⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠙⠷⠶⣦⣤⣀⣀⣀⡀⠀⠀⢐⠊⣿⡶⠉⣘⣿⣤⡷⠾⠈⣿⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⣿⠛⠃⠀⢴⠛⠋⠁⣠⠞⠀⡟⢧⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠁⠀⠀⢠⠋⠀⢀⡌⠁⠀⢸⠃⠈⡆⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⣾⠀⠀⠀⠘⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠃


⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡠⠤⢀⠠⠐⠤⠄⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⠄⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡤⣂⡒⠐⠒⠂⠀⢠⡀⡄⠀⠀⠀⠀⠠⠄⣖⠀⠀⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠔⠋⠶⠙⠛⠀⠂⠀⠀⠀⠠⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⠀⠀⢾⠀⠀⠀⡀⠀⠀⠀⠠⣤⡀⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠤⠀⠁⠀⠰⠀⠀⠀⠀⡅⠀⠊⢀⡞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠃⠙⢎
⠀⠀⠀⠄⠀⠄⠙⠀⠀⠀⠄⠀⠈⢀⠀⠀⠀⠀⠀⠀⢠⠀⡴⠋⠑⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⢰⠀⠀⠀⠐⡆⠀⠀⠀⡀⠴⠀⠀⠀⠀⠠⠄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣀⠀⠀⠀⠀⠀⠘⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠈⠐⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠒⠰⡄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠤
⠀⠀⠀⠀⠈⠀⡤⠙⠀⠀⣀⠄⠃⡑⠀⠀⠀⠀⠀⢃⡀⠀⡄⠀⠀⠐⠂⠀⠀⠀⠀⡀⢀⠄⠀⠀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⢀⡀⠄⠀⠚⠓⠁⠀⢩⣤⠀⠀⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣶⡔⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠄⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡰⠎⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠠⠀⠀⢠⡄⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠻⣿⢆⡛⢧⠀⠀⠀⠀⠈⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⡴⢲⡴⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⡐⠀⢀⠀⠀⠀⠀⠁⠀⠵⠆⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⡀⢀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠢⡈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠤⠂⠀⢄⢅⡀⠀⠂⠁⠂⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠄⡀⢀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠒⠒⠀⠨⣀⠼⠀⠀⠀⠈⠈⢀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠔⠀⠂⠀⠀⠀⠀⡆⠀⠒⠀⠢⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠂⢀⣀⠒⠁⠀⠀⠀⠠⠤⠤⠾⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⢀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠐⢂⠴⠀⠠⡀⠀⠚⡛⣛⠛⠉⡉⠉⠀⠀⠄⢀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠘⣤⠀⠈⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⢇⢠⣦⡄
⠀⠀⠀⠀⢄⢠⠀⠀⠀⠀⢲⡄⣀⣁⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⡀⠀⣀⠀⢀⡀⠀⠀⢦⡘⢁⡤
⠀⠀⠀⠈⠙⢚⠀⠐⠀⠀⠀⠀⣿⣹⠇⠈⣿⠀⣸⠏⠈⣷⠀⣿⠀⠀⢠⡏⠀⢻⡄⣼⠁⢈⡁⠀⠘⣆⣼⠃⠀⠀⡀⠉⠁⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⡇⢀⣿⠀⢻⣄⣀⡿⠀⣿⣀⣀⠸⣇⣀⣼⠃⢿⣀⣈⡿⠁⠀⣸⠇⠀⠀⠠⠁⠠⡁⠀⠤⠆
⠀⠀⠀⠀⢀⠀⠠⠀⠄⠀⠄⠀⠉⠉⠀⠈⠉⠁⠀⠉⠉⠀⠀⠉⠉⠉⠀⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠉⢀⠀⡐⠀⠐⢄⢀⡂⠂
⠀⠀⠀⠀⠀⠀⠓⠀⠈⠀⠀⠀⡁⢈⠆⠀⡰⠢⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠆
⠀⠀⠀⠀⠀⠠⠒⠤⠒⠀⠀⠈⠂⠁⠀⠈⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠁⠒⠈⢀⠀⡘⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⡀⠀⣖⠁⡐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⠄⠀⠀⠀⠈⠁⠀⠀⠠⠰⠀⠀⡐⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠰⠂⠛⠂⠀⠀⠘⠃⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣗⠀⠃⠀⠃⠀⠆⠐⡵⠀⡔⠀⠠⠀⠀⢲⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠀⠀⠄⠀⠁⠐⢀⠐⠈⠁⠀⠀⠀⠆⠑⠁⠐⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⡄⠀⠀⠠⠟⠂⠐⠢⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢴⠂⢮⡉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⢠⠀⣾⠁⢹⠂⠀⠀⠀⠈⢠⣿⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢤⡀
⠀⠀⠠⢼⠋⠀⠙⣗⠉⠉⠉⠀⠀⠀⠀⠐⠀⠛⢛⣧⣴⠀⠀⠀⠀⠀⠀⠀⠐⢀⢰⣦⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠉⣤⣒⡤⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢿⢁⡀⠀⠀⠀⠀⠐⠁⡀⠀⠙⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⣥⣤⡐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢈⣶⠸⠟⣥⡷⠀⠀⠀⠀⠀⠆⣣⣔⣒⠆
⠀⠀⠀⠀⠈⠻⢛⣧⣦⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠮⠀⡀⠨⢶⠆⠄⠀⠀⠀⠀⣿⠻⠿⠆⠀⢱⡆
⠀⠀⠀⠀⠁⠀⠨⠙⡛⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣳⣄⠻⣦⠀⠀⠀⠀⠀⠁⡁⠁⠀⠈⡀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⢀⠄⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠐⠂⢊⡀⠀⠁⠀⠀⠀⠀⠀⠀⢀⢠⡀⠄⠁⠀⠀⠀⠀⠀⣄⡷⣿⣹⣨⣿⢾⡀⠀⠀⠀⠀⠈⣁⡀⠥⠰
⠀⠀⠀⠀⠀⠳⡡⠊⠐⢀⡀⠀⠀⠀⠀⠀⢾⡘⠇⠋⠀⠀⠀⠀⠀⠀⠀⣹⣿⣻⣿⣼⣺⣿⣵⠀⠀⠀⠀⢀⡄⢤⡤⠂⠀⢰
⠀⠀⠠⠪⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣷⣟⣧⣠⣭⣬⡉⠀⠀⠀⠠⢯⣽⠑⠛⢤⣠⠞⠷⠦
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢠⢤⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣾⣿⠋⢩⡿⢷⡀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠀⠀⠀⡀⢾⡿⠿⠿⣿⣿⡇⠀⠀⠀⠀⢸⣤⣌⣿⣿⣧⣈⡇⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂
⠀⠀⠀⠹⣿⣧⣀⠀⢀⡿⠁⠀⠀⠀⠀⠘⢿⣀⣹⣇⣀⡹⠃⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠈⠙⠛⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⢟⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠁⠁⠀⠈⠈⠩⢿⣿⣿⣿⣿⠀⣾⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣻⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣿⣿⠀⣿⣿⣿⠿⣿⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⣿⡟⣷⣿⣿⠿⠿⠯⠀⠘⢷⣦⣭⣙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠘⠛⠀⣥⣤⠄⢴⣿⣷⣿⣿⢿⣿⣿⣿
⣿⣿⣿⡿⣡⠆⢀⣿⣿⣿⣿⠷⠁⢦⠀⠰⠀⢀⣿⣿⣿⡛⠛⣻⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⡄⢰⣿⢀⣿⣿⣐⠱⣿⣿⣿⣿⡿⣹⣿⣿
⣿⣿⢣⣾⡟⠀⣾⣿⣿⣿⣿⡇⠀⡳⣶⢓⠶⣟⣻⣿⣿⡇⠠⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠃⢀⣉⣾⣿⣯⣧⡴⣿⣿⣿⣿⣇⣿⣿⣿
⣿⣡⣿⣿⠇⢰⣿⣿⣿⣿⣿⡇⠀⣿⠿⠞⠓⠛⢋⣁⣤⡄⠐⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌⠿⢿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠏⡾⣾⣿⣿⡿⠛⣛⡁⠀⣤⢶⣶⣿⣿⢹⡟⢷⡧⠠⢽⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⣾⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣟⣿⣿
⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⢆⠀⣆⣄⢌⡍⡛⢻⢾⡆⡇⠀⠈⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠰⣈⣃⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⣹⡎⠀⢻⣿⢿⢾⣆⠄⠀⠈⠀⠀⠀⠀⢠⣮⣧⡤⠯⠏⠙⠛⠀⢈⣄⠀⠈⢛⢻⠏⠫⠿⠉⠙⣿⣿⣿⣿⣿⣿⣧⣻
⣿⣟⣵⣿⣿⣿⣿⣿⣿⣹⣿⡆⠀⣿⣿⣿⣿⣿⣧⠒⠄⠀⠀⠀⠀⡟⠛⠀⣄⢤⡄⢀⠀⠀⠀⡿⢦⠀⠀⢰⣦⢰⡶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠇⠀⣿⣿⡏⠙⣿⣏⡅⠀⠀⠀⡐⣼⣅⣴⣀⠁⠉⡁⠀⢤⣆⡀⠹⣿⣦⡀⢰⡄⠸⠕⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⡃⠀⣿⣿⡷⠀⢛⠛⠂⠀⢀⢌⡜⣡⣌⠃⡩⣷⡶⠂⠀⡽⣿⣁⡀⠛⡼⢷⡄⠀⠐⠈⠌⠩⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣥⣠⠏⠟⠃⢀⡉⠁⠀⢠⣿⣿⠁⣿⡿⢸⣆⣿⣿⣦⡀⣿⣋⣙⣷⠠⣞⣾⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡗⠀⠀⢸⡉⠙⣧⣉⣡⡾⣽⣿⣿⣿⣿⡿⢿⣿⡶⡦⠠⢾⣿⡿⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣏⣯⣿⣿⣶⡛⣷⣶⣶⣾⠿⠃⠀⠀⣀⣘⣿⣿⣿⣿⣿⡟⠛⠛
⠀⠀⠀⠉⠹⠟⠟⣻⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⠴⣏⣽⣿⣿⣟⣿⣿⣾⣾⣿⠏⠀⠀⠀⠀⠃⠘⠿⠍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣝⣻⣿⣽⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠉⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣶⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⢸⡇⣴⠟⢦⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⡇⢸⢟⣥⡙⢛⣛⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣀⣛⣹⣦⠆⠀⢸⡇⠳⣤⠾⠓⠒⢲⡝⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠴⠏⠛⣻⣿⡇⢸⣌⠛⣣⣬⣭⣍⢷⠹⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠃⣶⠀⠀⠀⠀⢸⣇⣀⣸⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣼⢋⣿⣿⣿⣿⡇⠸⠿⠇⣿⣿⣿⣿⢸⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠽⠿⠧⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣂⣀⣘⣿⣿⣿⣿⢸⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣤⣤⣤⡴⢚⣵⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡛⠛⠛⠛⠛⢛⡥⢞⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣤⠉⣧⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⠛⣷⠘⣩⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢸⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⠔⢒⣚⠒⢄⠀⠀⠀⠀⠀⠠⣄⢰⠀⢰⡴
⠀⠀⠈⠈⣄⢀⡩⠄⣐⡀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡧⠶⡜⠃⠲⡎⡇⠀⠀⠀⠀⠀⠹⣾⠀⡼⡇⢀⣀
⠀⠀⠀⠈⠀⢈⠤⣀⠴⠤⠀⠀⠀⠀⠀⠻⣟⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⡰⠇⠰⠷⣠⠃⠀⠀⠀⠀⠀⢠⡟⣧⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠁⠁


⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠠⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⢴⣶⣤⠀⢾⣶⡄⠀⠀⠀⠀⠀⣶⠖⠲⠲⠰⠺⠐⢶⡆⠀⠀⠀⠀⠀⢰⢿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⠁
⠀⠀⠈⠻⠿⣀⡀⠉⠑⠂⠀⠀⠀⠀⣿⣀⣀⢈⣀⢘⣀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡃⠐⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠈⠉⠈⠉⠘⠁⠀⠀⠀⠀⠀⠸⣶⣾⡾⠀⠀⠀⠀⠀⠀⠀⠙⠇


⠀⠀⠀⠀⣀⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶
⠀⠀⢠⣾⣿⣿⢰⣿⣷⡀⠀⠀⠀⠀⠐⡇⢀⠛⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠙
⠀⠀⣾⣿⣿⠇⢸⣿⣿⡇⠀⠀⠀⠀⠀⡇⠲⣶⠇⢀⠀⡀⠀⠀⠀⠀⠀⠀⣨⡙⠀⡹⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣈⣴⣶⡆
⠀⠀⠘⠛⠁⠀⠘⠿⠿⠃⠀⠀⠀⠀⠰⠷⠶⠶⠦⠸⠿⠗⠀⠀⠀⠀⠀⠰⠶⢿⠿⣭⠴⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿


⠀⠀⢀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠀⣿⠿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣺⡠⠤⣠⡇⠀⠀⠀⠀⠀⠀⠀⠘⣀⣦⣔⣋⠀⠀⠀⠀⠀⠀⠰⡏⠀⣹⠀⠈⡏
⠀⠀⣼⢏⣾⡇⣷⡀⠀⠀⠀⠀⠀⠀⠋⢹⡛⣈⢻⠍⠃⠀⠀⠀⠀⠀⠘⠀⠜⣿⡿⠤⠘⠂⠀⠀⠀⠀⠀⠉⣟⠁⢠⠏⠙⣦
⠀⠀⠈⠺⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠽⠻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠈⠓⠖⠃

⠸⠿⠟⠀⠀⢰⣶⠀⣰⠚⣿⣭⡁⢳⣿⠋⠀⠘⠇⠀⠀⠈⠙⠛⠉⠀⠙⠛⠋⠀⠙⠓⠋⠁⠈⠛⠚⠉⠈⠑⠒⠀⡀
⠀⣴⣤⣴⣶⡦⢤⠀⣿⣤⣿⣤⣥⣬⣭⣤⠀⠀⠀⠀⠀⠎⠈⢳⡞⠁⢱⡞⠁⠹⣶⠁⠈⣶⠁⠈⣷⠏⠀⢳⠋⠀⢹⡀⠀⠀⠀⠀⠀⠀⠺
⢀⡈⢵⠌⡋⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸⢧⠀⢠⢿⡀⢀⢿⡄⠀⡾⣇⠀⡸⣇⠀⡸⢷⡀⣠⠀⣶⣦⣀⣶
⠛⠟⠀⠺⠿⠀⠀⠻⡼⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠈⢁⡁⠀⠀⠱⠄⠉⡌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣁⡀⠀⢻⣿⣿⣿
⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠐⣦⣷⣷⣾⣷⠲⣞⠀⠀⠀⠀⢀⣀⡀⠀⠁⠠⠀⠀⠋⡙⠛⠉
⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣧⠀⠀⠀⠀⠚⠢⠀⢀⠀⠈⢑⣾⣿⣟⠋⣋⣿⣿⣿⣿⡥⠖⠀⣾⣿⣷⠀⢸⠀⠶⢐⣇⣇⣐⠈
⣛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠽⠿⠽⠀⠀⣀⣠⣤⠖⠀⢠⣄⠰⢼⣿⣿⣿⡴⣿⣿⣇⣨⣿⣧⣀⡀⠈⠉⠁⠀⠈⠀⠀⢉⡙⠉⢛⣀
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢐⡀⡊⠉⠁⠀⢙⠀⠐⠊⢀⠽⢷⣿⣿⣤⣽⣿⣾⣿⢻⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⣇⠀⠈⠩
⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⣸⠃⢈⣶⡶⢾⣉⠀⠀⠄⠂⠐⠊⢻⣧⣼⣿⣿⣾⠿⡃⠈⠀⠀⠄⢀⣀⣤⣤⣤⠀⠐⠦⠄⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠙⢧⠀⣉⠀⠀⠀⣈⠀⠀⠉⠄⠀⠐⠃⠀⠈⠏⠃⠘⠄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣸⠁⡨⠷⠚⠛⢥⠀⠀⠀⣠⡄⠈⠀⠠⠀⣤⣦⡄⠀⠀⠀⣴⡶⣾⣿⠿⠿⠛⠋⠁⠸⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠲⣧⠀⠲⣀⣀⣠⡴⠀⠀⠀⠨⠀⠘⠀⠀⠀⠻⠿⠃⠀⠀⠊⠉⣈⣉⣁⡀⠀⢀⡀⠘⠳⣶⠤⣿⣿
⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠇⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣴⡄⠀⠀⠀⠀⠉⠙⠋⠁⠉⠁⠀⠰⠶⠀⠶⠈⣁
⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⣀⡉⠀⠰⠤⠀⠀⠀⣘⣃⢀⠀⠤⠀⠠⣶⡶⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⣀⡄⣄⠠⠀⠀⠂⠂⠘⢿
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠈⠛⠘⠿⠿⠺⠗⢠⣴⣄⠈⣿⣿⡿⣧⣶⠆⠀⡉⠉⠀⣿⣿⣿⣿⣷⠀⠀⠀⠰⡀⠂⠀⡃
⠀⠀⠒⠒⠀⡍⢳⣧⣻⢽⠀⠀⠀⠀⣀⣄⠻⠃⠈⢹⣷⡄⠛⠛⠋⠉⠿⠀⠘⠃⢠⠀⠻⣿⣷⡿⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠦⡈
⠀⣾⠀⠀⠀⠀⡾⠃⡵⡞⠀⠠⠀⣀⡿⡿⠃⣠⡰⣾⣿⣵⣶⣦⠄⠀⠠⢴⣲⡀⠀⠠⡦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠃⢀
⠂⠹⠂⠀⠀⢰⡱⣿⣻⠋⡄⠠⢀⣛⠁⣾⠆⣽⣧⡋⠘⢿⠓⢮⠑⠀⠀⠋⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢠
⠄⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠈⠁⠈⠛⠀⢸⣿⣧⠀⢀⡀⣀⣠⡤⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⡀⠀⠾⢶⡄⠀⠀⠀⠀⢠⡴⣶⣿⣽⣦⡄⡀⢻⣿⣷⣽⡯⠽⠋⠀⠀⣀⠀⠀⠀⠠⠀⠄⠀⠁⠀⢠⣶⡄⠀⠀⠄⣁⣨⣀⡀⠄⠈⠁
⠠⡇⠰⠶⠀⣄⠰⠆⠀⢠⣿⣿⠏⢻⣿⣿⣧⡀⠈⢻⣿⢿⣶⡾⣶⠀⠈⠙⠀⠀⠀⣀⠀⢠⣦⡄⠀⠈⠙⠁⠀⣰⡿⣿⣿⣿⢻⣶⡀
⢠⡇⢴⡤⠀⢿⠀⣦⠀⠀⠻⣿⣿⣶⣾⣽⡇⠀⢀⣿⣿⣾⣿⣷⠿⣷⣦⠀⠈⠀⠀⠈⠁⠈⡟⠁⠀⠿⠂⠈⢠⣿⣷⣿⣿⣿⣿⣿⣇⠈
⠈⠃⣀⣀⡀⢀⣀⣘⠀⠀⠀⠈⠉⠻⢋⠁⠀⣀⠀⢉⡛⠻⠿⣿⣶⡆⢰⣾⣶⣄⠛⣶⡦⢾⣿⡧⠰⣿⡆⠐⠂⢿⣿⣻⣿⣿⣯⣿⠇⠒
⣴⣿⢶⣆⣙⣻⢶⡨⡕⡲⡦⠀⠀⠄⠀⢠⡀⠈⠃⠈⡉⠛⠀⠀⠈⠀⠀⠙⠋⠀⠘⠃⠀⣠⣯⠀⠰⠆⠰⡶⠀⠈⡙⠿⢿⠿⠟⡉⠀⢀⣤
⣪⣿⣼⣟⡡⠪⠿⠄⢅⣧⣤⡽⠀⣀⣀⡿⠁⢰⠄⣴⠏⠐⠞⠛⢲⣤⠖⠛⠓⢦⡶⠛⠓⣎⡵⠚⠓⠲⣤⠖⠓⠂⠀⠁⠘⠀⠁⠀⠀⣿⢿
⢠⣧⣄⠀⠠⠴⠶⠀⠀⠀⢀⣄⣼⣿⣿⣿⠀⢠⡄⢠⡴⠀⠀⠀⢠⣟⠀⠀⢀⣿⡀⠀⠀⣸⡃⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⡒⠶⢶⠤⢄⠀⠀⠑⠊⠀⠀⠀⠀⠀⢀⠔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠠
⠠⢀⠀⡇⠀⠀⠀⡠⢤⣶⣆⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⢀⠆⠀⠀⠀⠀⠀⠉⢄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠸⠀⠀⠠⠂⠀⠀⠀⠀⠑⠇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠄⠉⠉⠉⢠⠁⠀⠀⠀⠈⠳⣤⣀⣀⢴⢣⠀⠀⠀⠀⠀⠠⠎⠉⠑⡄⠀⠀⠀⠀⠸
⢀⠇⠀⠀⠐⠄⣀⡀⠀⠀⢀⡀⠈⡀⠤⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡼⠶⣄⠀⠰⠋⠙⠳⡄⠀⠀⠈⠧⣀⡴⠁⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢺⡀⢀⡸⠀⠘⢤⣀⠴⠃⠀⠀⠀⠀⠀⡶⠖⠢⡀
⠀⡠⠶⢤⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡞
⠀⣇⠀⣠⡦⣀⣀⢄⡀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠚⠁
⢀⣈⠉⠁⡸⠋⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠁⠈⢳⠄⣇⠀⠀⠀⢠⠆⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠀⠈⠀⠀⠀⠘⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣀⠠
⢤⠴⠋⠀⠈⠲⠤⠖⠃⠀⠀⠀⠀⠹⢄⣀⠇⠀⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⠀⠀⠆⠻⠟⠀⠚⠂⣤⠄⠀⠀⠀⠀⠠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁
⢀⣤⠆⢱⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠⠉⠐
⠉⠀⠀⠀⢢⠀⠀⠘⢅⣀⡼⠓⠢⢼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠄⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠾⢁⡤⠦⠠⣄⠀⠀⠀⡠⠄⢄⠀⠀⠀⡠⠲⠶⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⢄⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⡧⠀⠘⢅⣀⡼⠀⠀⠀⠈⡂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⠢⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠢⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠁⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠁
⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠉⢣⠀⠀⠀⠀⢠⠀⠲⡎⠉
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠘⢄⡀⠀⠀⣠⡎⠀⠀⠀⠀⠸⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠃⠀⠀⠀⣸⣙⠉⠉⢱⠼⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢴⠘⠣⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢱⠀⢣⡀⠀⣨⠃
⠀⠀⠀⠀⠀⢠⠒⠒⠄⠀⠀⠀⠀⠀⣇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠡⠀⢀⠌⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄

⠀⠀⠀⠀⠀⢀⡠⡖⢛⡄
⠀⠀⠀⠀⣞⢫⡱⠞⠉⠳⡄
⠀⠀⠀⠀⠈⠉⢧⡀⠀⠀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠳⡄⢀⣠⠔⠉⢨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠑⢌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⠳⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠀⠀⠀⠈⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢠⡟⣲⣐⢦⡀⠀⠀⠀⠙⢞⢦
⠀⠀⠀⠀⠀⢀⠔⢁⠀⠀⢺⡠⢯⡼⢧⡇⠀⠀⠀⠀⠈⠳⡱⡄
⠀⠀⠀⢀⡴⠁⡴⠁⠀⠀⠀⠑⠛⠚⢻⡀⠀⠀⠀⠀⠀⠀⠘⢮⢦⡀
⠀⠀⢀⠎⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣄⠀⠀⠀⠀⠀⠀⠀⠑⢝⢦
⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠈⠳⡵⣄
⠀⡜⢠⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⢀⠁⡎⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡹⢦⡀
⡘⢠⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣀⠔⡫⠔⢉⡤⠊⣱⠄
⡇⢸⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠴⣪⠔⠈⡠⠖⢁⠴⢎
⡇⢸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡁⠊⣠⠔⢉⡠⠚⠁⠀⠈⠳⣄
⠇⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⣡⠴⣏⠀⠀⠀⠀⣠⠔⣩⠗
⢰⠘⡄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠔⢋⡤⠚⠁
⠘⡄⢣⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢣⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉
⠀⠈⣆⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢁⡠⠔⠉
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⡠⠖⠉
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢉⡠⠄
⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⢀⠰⠛⠟⠉⠒⣦⣄⠔⠊⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡟⠦⣤⡜⠦⡞⠀⠈⡶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡠⠼⢦⢤⡟⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⣄⣀⠇⠀⢱⣀⠜
⠀⠀⠀⢀⣠⣤⣤⣤⡤⠴⢧⠤⠤⠬⠍⠛⠛⠋⠿⠄⠤⠤⠤⠤⠤⠤⠶⠤⠤⠤⠤⠤⠤⠤⠴⠦⠤⠤⣤⡀
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣
⠀⠀⠀⡇⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠙⠛⢶⡀⠀⠀⠀⣸
⣤⣤⣤⣿⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤
⣿⣿⣿⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠛⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⡀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠿⠃⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠛⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠀⠀⠀⠀⠀⢠⣴⡞⠋⠀⢰⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣦⠀⣠⡴⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⠂⠀⠀⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡶⠀⠀⠀⢀⡀⢀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠟⢸⡏⣤⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⢀⠀⠘⠓⢀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠋⠛⠋⠁⠀⢀⡴⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠀⠤⣤⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠈⠳⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡆⠀⠀⠾⠶⠿⠂⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣽⢿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⢛⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣿⣿⡏⠀⠀⠀⣂⣠⣼⣆⠀⠰⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡄⠀⠀⠀⠙⡁⠤⣿⡆⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⣤⡄⠀⢰⣦⠀⠀⠆⠀⠀⠀⠀⠆⠻⠁⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡤⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⡀⠛⠙⢃⡀⠀⠀⢠⡄⠀⠀⠀⠀⠐⠿⠂⡀⠀⠀⠀⠀⡀⠲⠀⢠⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢃⠰⠻⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠀⠀⠀⠘⠣⡀⠀⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢠⡄⠀⠀⣿⢋⠤⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆
⠀⠀⠀⠀⠀⠀⠼⠁⠀⢀⡀⠀⠀⢼⠁⠀⠀⠀⠈⢱⣄⠰⠄⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠸⢠⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠹⠶⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠃⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠈⡻⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠋⠛⠓⠀⠀⠀⠀⢾⡗⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⢦⠀⠋⠀⣴⠶⡄⣤⠀⠀⠲⣦⣿⠀⠀⢠⠹⡇⣼⡏⠀⠙⣄⡾
⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⡆⠀⠀⠀⠀⣶⡟⣴⠂⢹⡄⢸⡇⣻⣿⠀⢀⣿⢀⡇⣿⢠⣄⢸⡿⠁⠀⢠⣆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠰⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡿⠒⠾⢃⠸⣇⠏⠻⠤⠤⠻⠎⠁⠻⠾⣧⡾⠁⢀⣤⣨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⡀⠀⠀⢠⠏⠀⠀⠀⠹⣧⡾⠀⠛⠀⠀⠀⠴⠠⡤⠄⠀⠀⠀⠀⠁⠀⠀⠀⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣽⣶⣶⠀⠀⠀⠀⠐⠂⠀⣀⣀⣀⡴⠟⠁⠀⠀⠀⢀⣤⡶⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢾⣽⠛⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢿⠶⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢆⣴⠁⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠐⢧⠀⠀⠀⠰⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣠⡤⠂⠀⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠘⠘⠋⠀⠀⠀⠀⣀⣀⢤⡤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⡾⠂⢠⣾⣧⡄⠀⠀⠄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣻⣿⡦⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤
⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠁⢠⡟⢁⠀⠀⠀⠐⠁⣁⢀⡠⠃
⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠂⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡄⢠⠈⠈⠁⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⡀⢠⡄⠀⠀⠀⠒⠒⡄⠀⠀⠀⠀⡀⠑⠀⠀⠠⠘⠀⠀⠑⠁⠀⠀⠀⡀
⠀⢨⠀⠀⢀⣤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⣒⣓⣒⣢⠀⠀⠀⠠⠒⠀⠢⠄
⠀⠀⠀⠀⠰⡀⠊⡉⠀⠀⠡⠀⠠⠈⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⡀⠒⠄⠀⠈⠁⠀⠀⠀⡤⡄⠠⡤⠀⣠⢤⡄⢠⡄⠀⢀⡤⢤⡀⢠⡤⣤⠀⢤⠀⣤⠀⠀⠀⠀⠄⠐⠀⡀
⠀⠀⠀⠈⠠⠀⡆⠐⠀⠀⡷⣇⠀⡇⢰⡇⠀⣿⢸⡇⠀⢸⡃⢘⡇⣿⠀⢴⡄⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠁⠐⠁⠀⠀⠆⠀⠀⠀⠓⠋⠐⠓⠀⠛⠚⠁⠘⠓⠒⠈⠓⠚⠁⠘⠒⠛⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⡀⢀⣤⡤⡄⠀⠀⠀⢸
⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠘⠐⠀⠀⠀⠀⠀⠼⠥⠬⢛⠡⠁⠀⠀⠠⠂⠀⡁
⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠒⠉⠠⠒
⠀⠄⢂⠀⠤⡀⠀⢀⡐⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠈⠀⡀⠀⡀⠀⠀⠀⠀⠈⠁
⠀⠀⡀⠀⢀⠀⠀⠀⠔⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠌
⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⡰⠀⠈⡁⢀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠂⠀⢀⠀⡀⠠
⠀⠀⠀⠀⢀⡔⠀⠀⡀⠀⠀⠀⠈⡦⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠄⠀⠈⠀⡀⠀⠀⠀⠀⠶⠀⢀⠄⠀⠂⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠠⠉⠉⠄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠠⠂⠤⠤⠐⠕⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠉⠉⠅⠀⠀⠀⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠆⠀⠀⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠉⣀⠀⠁⠂⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠊⠀⡐⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢑⠂⠉⠀⠀⠀⠤⠉⠒⠉⢁⡀⡇⠀⠀⠀⠀⠀⢀⡅⢀⡰⢘⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠰⠀⠀⠀⢠⠒⠀⠀⠀⡇⠀⠀⠀⠀⡠⠁⠈⠐⠖⠁⠈⠀⠂⠄⠈⠂⡀
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠑⢕⠄⡀⠀⢀⡠⠀⡀⢐⠅⠀⠀⡇⠀⠀⠀⣔⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⠄⠈⠄
⠀⠀⠀⠀⠡⠀⠀⠠⠃⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠗⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠶⠚⠠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⢒⠓⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡐
⠀⠀⠀⢀⠀⠀⠘⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠠⣤⠙⠠⡀⠀⡠⠊⠁⠀⠈⠀⡤⠊⢀⠔
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠄⡀⠙⠀⠀⠀⠀⡄⠀⠈⡔⠁
⠀⠐⠂⣀⠀⠀⠈⠕⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠀⠀⠠⠀⠀⠉⠀⠀⠈⠀⠁⠀⠀⡇
⡀⠄⡀⠈⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠇
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠜⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠐⠛⠛⠛⠛⠛⠛⠂⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣸⠇⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠂⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠛⠛⠛⠛⠛⠓⣂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⢳⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢹⠃⠀⠀⠈⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡤⠴⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⢂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡡⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠚⣛⣛⣛⣻⣟⣛⣛⣋⠛⢶⡀⠀⠀⠀⠀⠀⢸⠁⠈⢛⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠉⠉⠉⠉⠉⣀⣀⣠⡾⠁⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠈⠉⠉⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⡆⣿
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣀⠀⢸⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⣿
⠀⠀⢰⡇⠀⠀⠓⠛⠀⠂⠠⠤⣤⡤⠤⠖⠚⠉⠁⠀⠀⣀⠼⠆⠀⣇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⢀⡇⣿
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠒⠈⠉⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣅⠀⠀⢀⡇⣿
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣸⠇⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠓⠒⠊⢀⡟
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣰⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠚⠋
⠀⠀⠀⠀⠀⠻⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⣠⡾⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠐⠒⠒⠒⠀⠈⠀⣀⡴⠞⠉⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠛⠉⠁⠀⠀⠀⠀⢀⡾⠃⠀⠀⠈⠙⠳⢤⣤⡤⠤⠶⠚⠋⠉⠙⠲⣄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠲⠤⠤⣤⠤⠤⠚⠉⠳⣄⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠑⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠉⠉⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠷⠀⠸⠗⠀⢶⣶⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣀⣀⣀⣀⣀⣿⡏⠀⠉⠉⠛⠻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣶⣦⣤⣀⣠⣿⡇⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣶⡿⠿⠆⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⢀⣴⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣶⡿⠋⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡄⠀⣠⣤⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⡟⠁⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡾⣿⡆⠀⠀⢿⣧⣴⡿⠋⠀⢀⣴⣶⠿⠿⠛⠋⠉⠻⣿⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⣼⡟⠋⠉⠀⠀⢿⣧⠀⠀⢸⣿⠋⠀⠀⠀⣿⡏⠀⠀⠀⠀⣀⣠⣼⡿⠉⠉
⠀⠀⠀⠀⠀⠀⢻⣷⣄⣤⣤⣶⡾⣿⣤⣤⣼⣿⣀⠀⠀⠀⠙⢿⣶⣶⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠈⣿⡟⠉⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠻⣷⡄⠀⠀⣠⣾⠿⠿⣷⣄⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⣾⡟⠛⢿⣦⡀⠀⠀⠀⠙⣿⣆⠀⠀⠙⣿⣆⠀⢿⣧⠀⠀⠘⢿⣦⠀⣠⣴⣾⠿⠛⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠙⢿⣦⠀⠀⠀⠈⢿⣧⣀⣀⣼⡟⠀⠈⢻⣷⡀⠀⣈⣻⣿⠟⠉⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⡿⣷⣄⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠹⣿⣾⠟⠛⣿⣦⣴⣾⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣿⠏⠉⠈⠙⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣠⣾⠟⠿⣷⣦⣄⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠘⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡾⠟⠁⠀⠀⠀⠉⠛⠃⠠⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠯⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣿⡃⠀⠀⢸⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠘⣿⣷⣄⠀⢸⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢿⣿⣿⣏⠀⢸⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⣾⡟⢻⣿⣯⢹⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣾⣷⣾⣿⠛⠻⣇⣀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠘⡟⣿⡿⢿⣥⣶⣶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⢾⣷⣶⣾⣯⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠀⠀⠀⢻⢿⣿⠿⡚⣿⡿⢶⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢀⣺⣤⣿⠆⣧⣸⣿⠌⠳⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣧⣤⣶⣾⣿⣿⣿⣦⣠⣥⣤⣄⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣭⣯⣭⣍⣋⣛⣛⣛⣛⣛⣋⣩⣥⣼⣶⡶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣿⣿⡙⢻⡿⣿⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⠰⢾⣷⣾⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⢹⣿⣠⣾⣛⣒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡝⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣯⣽⣿⣿⣿⣿⣿⣯⣧⣿⣿⣿⣷⣿⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣭⣭⣉⣛⣛⡛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⠉⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠤⠤⢤⡀⠀⠀⢀⣀⡤⣴⣂⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠤⢴⣆⡈⠉⠉⠁⠀⢤⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⠀⠀⠀⠙⣶⢤⡴⠖⠲⠤⢤⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠀⠀⢉⡴⠃⠸⡇⠀⠀⣀⠀⢹⠟⡷⠄⠀⠀⣠⣴⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉⢄⠀⢀⣳⣤⣀⣁⡴⠋⠀⢇⣠⠴⠋⣨⢼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄⠀⠀⠀⠀⠙⠢⣌⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢊⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡻⣄⡀⠀⠀⠀⠀⠀⡴⠛⢳⡶⢶⠶⢶⠖⢻⠯⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣯⠹⡄⢙⠒⠦⣀⣠⠔⢣⠀⢰⠈⠁⢈⠋⣴⠋⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠗⠠⣄⡜⠀⠀⠀⠁⠀⢸⠀⡌⠀⠀⠈⢰⠹⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⣣⡀⠀⠀⠀⠀⠏⣤⠃⠀⠀⢠⡇⠀⡆⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⡏⠈⠁⠒⠒⢸⢠⠃⠀⠀⠀⠀⠇⠀⣥⣀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠀⠆⠎⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⢰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⢠⠴⠚⠒⢄⠀⠀⠀⢠⠎⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡹⠢⠄⣨⠖⠀⠀⠀⡱⠤⠚⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠃⠀⠀⠈⡿⠤⠶⢊⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠏⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⡠⠀⠀⠀⡀⠀⠀⠀⢔⠒⠒⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠁⠂⠄⢀⢀⠃⠀⠀⠈⠉⠉⠀⠉⠉⢫⠀⠀⠀⢘⠉⠁⡛⠀⠀⠀⠀⢨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠑⠲⠤⠤⣀⡸⠂⠠⠀⡀⠀⠀⠀⠀⠀⢘⡄⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠈⠁⠀⠂⠀⠀⡇⠀⠀⡸⠀⠀⠠⡀⠀⠀⠀⠰⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠙⠦⠤⣄⣀⣀⢸⠂⠈⠉⠑⠒⠠⠤⢤⣀⣀⣀⡇⢠⠊⠀⠀⠀⠀⠈⢢⠀⢠⠃⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠇⠀⡆⠁⠀⠈⠉⠀⡆⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠀⢰⠂⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⡃⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠣⣀⠀⠀⢀⡠⠃⠠⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⡀⠓⠀⠁⠀⠀⢠⡀⠉⠀⠀⠀⠀⠀⠀⠐⠈⠐
⠀⠀⢀⣠⠄⡀⢀⠀⠀⡰⠠⠈⠀⠀⠀⠒⠀⠐⠀⠀⢰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠃⠊⠀⠀⢬⡰⠢⠤⠖⠂⠀⠠⠀⠀⠀⠤⠀⠑
⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠂⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠘⠂⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢄⠙⠂⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢽⠛
⠀⠠⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⢤⠀⠀⠀⠀⠘⡤
⠀⡀⠀⠀⠀⠀⡤⠁⠈⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠄⢠⠂⡲⠊⠀⠀⠐⠀⠀⠀⠀⠀⠊⠘⠀⠀⠀⠀⠀⡎⠈
⠀⠈⠀⠀⢀⣰⡇⢄⢀⡀⠀⠀⠀⠐⠂⣁⣀⣀⣀⣀⣬⣈⣋⣀⣀⣀⣀⣈⣁⣁⣀⣠⣀⣀⣀⣈⣀⣀⣀⣁⡀⠀⢠⠂⠃⠀⠀⠔⠀⠁⠀⠀⠀⢱⡔
⠀⠀⣰⢮⡙⠅⡇⠉⢉⡁⠐⠄⠀⠀⢸⣿⣿⣿⣿⣿⡿⡿⡿⢿⣿⢿⡿⠿⢿⡿⢿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⢰⠉⠈
⣀⣀⣧⣲⣝⣶⣃⣀⣀⣀⣁⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣶⣿⣷⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⡘⣆⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⡉⡉⢋⣸⣏⣨⢃⢳⢰⡆⡇⢽⣁⣸⣈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢫⢾⡻⡍⡗⠛⠋⠁⠉⠉⠉⠀⢸⣿⣿⣿⢿⠿⡿⠿⠿⣿⣿⠿⠿⣿⠿⠿⣿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⠋⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⡏⢉
⢀⣀⣏⢢⡙⣦⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢀⠼⣏
⠙⠧⢸⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠰⠀⠀⠀⠀⠀⣀⠀⠀⠒⠠⠁⠓⠀⠀⢀⠀⠢⠀⠀⠉⠀⠀⠀⠈⠤⣸
⠀⢀⢞⠀⠀⠀⢤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠰⠄⠈⡀⢀⢄⢀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠘⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⠁⠀⡀⢀⠀⠀⠀⠀⠉⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠁⢀
⠀⠀⠄⡈⠃⠐⠀⠠⡠⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⠂⠁⢀⠀⠰⠀⠀⠀⠀⡀⠐⠆⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⣰⠄⠀⠀⢠⠃⠀⠀⠀⠀⠆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡜⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠀⠀⠀⡜⠀⠀⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠈⠰⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⣀⢤⡀⠀⠀⠀⠒⠙⠲⢤⣀⠉⠀⠀⠀⠈⠙⠰⢤⣀⠀⠈⠢⠑⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣂⢁
⠀⠀⢠⠔⠚⠉⠉⠀⢨⠆⠀⠖⠀⠘⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠈⢳⠀⠀⠀⢰⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠉⠆
⠀⠀⠈⠳⢄⠀⠀⠀⠎⠀⢀⡄⣴⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢄⣀⠜⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠒⠒⠲⠂⣲⢒
⠀⠀⠀⠀⠀⠑⠄⠈⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⠀⠒⠂⡀⠀⠀⠀⢸⠍⢹
⠀⠀⠀⠀⠀⠀⢐⣐⡄⠀⢀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠄⠀⠐⡀⠀⠀⢠⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⠀⢴⠀⡁⠀⠀⠀⠈⠚⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⣀⠀⠐⢤⡀⠀⡀⠀⠀⠸⡀⠀⠀⣭⣀⣀⠇⠀⠀⠀⠰
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠁⠀⠀⠀⠀⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⠁⠀⠐⠂⠂⠒⠂⠀⠀⠀⠈⠀⠀⡐⠀⠀⠀⠁⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⡀⠈⠉⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠂⠀⠁⠀⠀⠠⠥⠀⠁⠄⠀⠀⠀⠀⠀⣀⠀⠐⠒⠒⠋⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠴⢃⡀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠐⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠆
⠀⠀⠀⠀⠈⠑⠆⣄⡄⠀⠀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠠⠀⠤⠤⠠⠀⠄⠠⠤⠀⠀⠀⠀⠤⠤⢠
⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣁⡀⠀⠀⢀⢀⣀⣀⣁⣀⣀⡇⠀⠀⠀⡊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠜⠀⠀⠸⠰⠂⠀⠢⠀⠀⠄⠀⠀⠀⠀⠅⠀⠀⠀⠀⣤⠠⡇⠀⠰⡇⠀⢼⠀⠀⢽⢀⢠⠇⡀⢨⣇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠀⠀⠀⠀⠀⡏⢹⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⢘⡇⢀⠀⡇⠀⢺⠀⠀⢸⠀⢘⡇⢀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢋⠈⠀⠄⠂⠀⢹⠀⡄⠀⠀⠀⠃⠘⠃⠀⠀⢤⠠⠐⠣⡀⠀⠀⢸⠀⡇⠐⡀⡇⠀⠘⣀⠀⢸⠀⠀⡇⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⢀⠀⠌⠀⠃⠀⠀⠀⠈⠉⠀⠀⢀⣀⠈⠁⡠⠈⠁⠀⢸⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠐⡇⠀⠂⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣶⢁⠄⠈⠀⠁⠀⠀⢀⠀⠀⢠⡤⠐⠣⠀⠀⢸⠀⠳⠤⠴⠁⠀⠈⠦⠤⠞⠀⠀⠣⠤⠴⠃⢸
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠐⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠈⡹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⡌⠛⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠄
⠀⠀⢀⠃⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⡸⠀⠀⠀⡎⠘⠛⠉⠐⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡇⠀⠀⠀⠡⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢠
⠀⢸⠀⠀⠀⡀⠀⠑⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜
⠀⡇⠀⠀⠊⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠈⠉⠁⡄⠀⠀⠀⠀⠱⢤⠤⠴⡟⡀⠀⠀⠀⠀⡜⠁⠈⡆⠀⠀⠀⢠⠃
⡜⠀⠀⠀⠒⠠⠤⠀⠀⡤⠠⠌⠒⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠚⠓⣄⠀⡏⠈⠙⡆⠀⠀⠘⠢⠾⠁⠀⠀⠀⠀⠢
⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠦⠤⠃⠀⠙⠒⠊⠀⠀⠀⠀⠀⡜⠉⠑⡄
⢰⠋⠈⣷⠀⠀⠀⠀⠈⠙⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠴⠃
⠀⠑⠒⢉⡞⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠉⠙⣦⢸⠀⠀⠀⢠⠂⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⠈⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠤⠔⠉⠈⠳⠤⠖⠁⠀⠀⠀⠘⢄⣠⠃⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠘⠛⠃⠓⠀⡤⠀⠀⠀⠀⠠⣶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⣤
⡤⠞⠁⢆⠀⠀⢠⡔⠒⣄⢀⣀⡴⠛⠲⡄⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠈⢂⠀⠈⠓⠒⠉⠀⠘⣄⠀⠀⣽⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⡀
⠀⠢⠀⠀⠀⠱⣀⡀⡀⠀⠀⠀⠈⠉⠉⠀⡞⠁⠀⠱⣄⡀⣞⠁⢘⠄⠀⠀⢢⠈⠀⢸
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠙⠦⣄⠴⠁⠀⠀⠁⠁⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⠈⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⡀
⠀⠀⡇⠀⡀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡞⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢰⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠰⣄⡠⠇
⠀⠀⠰⡀⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠐⠒⠚⡄
⠀⠀⠀⠈⠂⠤⢀⣀⡀⠄⠊⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⠀⠑⢤⣤⠤⠞⣀⠀⠀⠀⠈⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⣧⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⢀⠔⠛⢄⠀⡎⠉⠉⡆
⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠞⠀⠘⠒⠚⠁
⠀⠀⠀⠀⠘⡀⢀⡅⠀⠀⠀⠀⠉⠒⠂