Lucky Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⢰⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⣧⠀⠀⠀⢹⠀⠀⠀⢸⢀⠀⢸⠀⠀⠀⡏⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠘⡆⠀⢆⠈⠆⠀⡇⠈⢸⠀⡈⢠⠃⠸⠀⢀⠀⣰⠋⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡀⠳⡄⠘⢆⠘⠀⠘⡀⠀⠀⠇⠀⢸⠀⠀⡘⢠⣶⢦⡉⡰⠃⢀⠄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢦⡀⠘⢦⠘⠀⠈⢆⠀⠀⠁⠀⠀⢠⠀⠘⠀⠀⠁⣾⠃⣸⡇⠁⣠⠋⡠⠋⠀⡀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠑⢤⡙⠀⠀⠓⡀⠀⠈⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢸⣿⡴⠛⠀⠐⠁⠀⠀⠰⠊⣠⢖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠄⡀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⢸⣿⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠈⠛⠲⢤⣀⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⣰⠏⠠⠒⠉⢸⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣧⠀⠀⢀⣼⠏⠀⠀⠀⡀⢸⣿⠐⢁⣀⠤⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⣏⢸⣿⣧⡿⠈⠳⠶⠋⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⢀⣤⣴⠆⠀⠀⣰⣿⡶⡟⢰⣿⡇⠀⣿⠀⠙⠛⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣉⣁⣀⣤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣼⣧⠀⢀⣿⣿⠀⣿⠈⠻⡷⠼⠃⠀⠀⠀⠀⢹⡇⠀⣤⡀⠀⠀⠀⠀⢀⢸⡿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⢰⣿⡇⢀⣿⣿⣆⡾⢿⣿⢀⡿⠀⠀⠀⠀⠀⠀⣀⡀⠀⣸⣧⣼⣻⠇⠀⠀⠀⠰⡄⢠⡄⠀⢀⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⠃⣿⣿⡃⠈⣙⠛⠁⠀⠀⠀⠀⠀⣸⣿⢻⡄⣿⣿⣿⠟⠀⠀⠀⠀⢸⡇⠘⠷⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠙⠛⠁⠀⣼⣿⣇⠀⣿⣧⠀⢀⣀⠀⣤⠀⣿⡯⠀⢰⡿⣿⡟⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⠀⣸⠏⣿⣿⠀⢸⣿⡀⢸⣿⡀⣾⠀⢹⣿⣠⡞⠁⢿⣧⠘⣿⡄⠀⣸⠇⠀⠉⠒⠦⠄⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⠄⢀⡠⢠⡟⠀⣿⣿⠀⢸⣿⡇⠸⣿⡇⣿⣇⡾⠁⠀⠀⠀⠀⠀⠀⠈⠻⠶⠏⠠⢄⡀⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣡⠖⠉⠀⣾⢀⠄⢸⣿⠀⠸⣿⣇⣼⡿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⢄⠙⠂⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣀⠔⢰⡏⠀⠀⣼⡏⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠑⢆⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⡔⢠⡟⠁⡀⢠⠈⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⢀⠘⣆⠀⠱⣄⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣈⠳⠴⠋⠀⡜⢀⠏⠀⡄⢸⠀⢀⠀⡄⠀⠀⢃⠀⢀⠈⣆⠀⢳⡈⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⡼⠁⠈⠀⢰⠀⠟⠀⢸⠀⡇⠀⣆⠘⡀⠀⢆⠈⠀⠀⢳⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡏⠀⠀⠀⣾⠀⠀⠀⢹⠀⠀⠀⠸⡄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠃⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣽⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⢰⣶⡶⠿⢿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⣆⠘⠁⠀⠀⠀⠈⠳⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣽⣏⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⠀⡄⢀⡤⣄⠀⣄⣤⠠⣄⣠⠄⠀⣠⢤⡄⢀⣤⡀⠠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀⢿⣤⠇⠸⣧⡤⠀⡟⢧⠀⢹⡏⠀⠀⢿⣤⡄⣼⠿⣧⠀⣿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠓⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠑⠦⣀⡀⣸⡇⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣰⠋⠀⠈⢻⡆⠀⠀⠀⠉⣿⠓⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⡠⠃⢀⠔⠒⢦⡀⠀⣿⠀⠀⠀⠀⣷⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠉⠀⠈⠁⠀⠀⢺⣴⠀⠘⡇⡠⢻⣿⡀⠀⠀⡟⠀⣼⠃⢀⣿⡆⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣴⡆⠀⠀⠀⠀⣸⣿⣒⡺⠋⠀⢸⣏⠙⠓⣾⠓⠋⢿⣧⠞⠙⣿⣄⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⠔⠁⢉⣉⡀⠀⠀⠈⠻⠶⠞⠁⠀⠀⠀⠀⢀⠀⠈⠛⠷⠦⠤⠔⠁
⠀⠀⠀⢀⠴⠒⠛⠛⠛⠛⠛⠛⠓⢻⡟⠋⠉⠀⢠⡞⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⡏⢀⣤⡀
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢠⡿⠀⠀⠀⢀⠏⠀⠀⠀⠀⣰⠏⠉⢹⠀⣾⡗⠁⣾⠁⠀⠀⠀⠀⣀⣀⡀
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠑⣾⡧⠤⠄⠒⣡⠀⠀⣾⠀⠀⣿⠀⠀⠀⢠⣿⣷⣞⣁⡀⠀⢀⠔⠋⠁⠈⢻⡆
⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⠀⢰⣿⠀⢰⢿⣦⣤⠴⠃⠸⠇⠀⠈⢙⠿⣧⣤⣀⣀⣠⠞⠁
⠀⠀⠀⠙⠷⣶⣤⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⡸⣿⣤⠞⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠔⠒⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣠⣤⣤⣈⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢊⣡⣤⣤⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠑⢄⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣿⣿⠏⠐⠈⠉⠁⠀⠀⠉⠉⠐⠈⢿⣿⣿⣿⠇⡘⠀⠀⠀⠠⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡫⠋⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠌⠫⡛⢀⠃⠀⠀⡆⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠑⡋⣿⢿⡅⠄⠀⠀⠀⠀⡊⡿⣿⡍⡗⠁⠐⡆⠀⠀⡠⠃⢄⠆⠄⠄⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠈⠉⠈⢀⣤⣤⣤⣄⠈⠈⠉⠈⠀⠀⠀⢠⠀⡐⠀⠀⠀⠉⠘⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢁⣀⠈⠥⢰⠁⠙⢿⠟⠁⢳⠨⠍⢁⡈⠉⠀⠸⠠⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⢁⠠⠒⠉⠹⠀⢀⠜⢄⠀⡸⠉⠐⠢⡀⠉⢀⠃⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡁⠀⠀⠀⠀⠙⠢⠤⠤⠚⠀⠀⠀⠀⠀⣰⡇⠘⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡟⡷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⠃⠃⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠑⠻⠾⠭⠿⢿⣿⣿⠿⠿⠿⠟⠛⠉⠢⡀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢄⠀⠀⠀⠀⠀⠈⢦⣀⣠⣮⣐⢄⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠂⠀⠑⠄⣠⡀⣀⣠⣿⣿⣿⣿⣿⣷⡑⡀⠀⠀⠠⠂⠉⠑⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⢂⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⠊⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⡠⠄⠂⠀⠐⠠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⠠⡀⠠
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣨⠻⣿⣿⣿⣿⣿⣿⣿⣿⢫⡁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡂⣂⣂⡣⠗⠓⠱⢍⠻⠿⣿⡿⠿⢛⠵⠁⠒⠦⣆⡆⡄⣄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢛⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠤⠶⠄⠀⠀⡠⢊⣵⣿⣿⠷⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⣱⣶⣯⡉⠳⢦⣀⣠⣤⠴⠶⠖⠚⠛⠓⠻⠷⠿⣿⣿⣋⢸⢸⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⢙⠹⣿⣿⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡇⠀⣠⠄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡰⠂⠘⠛⠓⠶⣍⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢤⡀⠀⠀⠀⠈⢷⡶⡟⠛⠲⣤⡈⠻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⡴⠞⠛⠓⢦⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠉⠃⠀⠀⠀⠀⣷⢰⠀⠀⢀⢹⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠇⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡷⠷⢸⣶⡇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸⡿⡷⠿⠆⢺⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣾⠀⠀⠀⠀⢸⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠓⣶⠆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣴⠟⣹⣿⡄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢮⣛⢶⣬⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣋⣥⣪⢕⣿⠇⠀⠀⠀⠀⢰⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣼⢫⡑⠮⣷⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣾⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⠦⣭⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⡿⢴⣶⣿⡟⠛⢫⡍⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⡷⣻⣿⢧⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⠖⠀⠀⠀⠀⠀⠀⠀⢸⡀⠰⢛⠿⣿⣻⣿⠿⠁⣩⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠿⠃⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡏⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⣾⣷⣀⠀⠀⠀⠀⠀⠐⠏⢀⡾⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢋⢿⡆⠀⠀⠀⠀⠀⠈⢿⡄⣼⠟⠉⠉⠉⠉⠀⡀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠈⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⣦⣰⠀⠀⡀⠐⣀⣼⠻⣆⡀⠀⠀⣒⣢⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠈⠉⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⠄⡀⣄⠍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠒⠉⣂⡬⣁⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡴⢊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠀⠀⡴⡫⠊⢁⠄⠠⢄⠀⠀⢀⠔⠉⠲⠀⠀⠀⠀⠀⠀⣎⡉⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡜⡁⢠⡞⡔⠀⢀⡤⠀⠀⠀⢣⠀⡎⠀⠀⠐⠥⢀⠀⠀⠀⠀⠰⠀⢭⠤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡾⡸⢠⡓⡜⠀⠐⠀⠀⠈⠢⣀⠀⣣⠅⢠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⡐⠀⡼⠳⠁⡞⠜⠀⠀⠐⢄⡀⠀⡀⠘⢣⣽⣷⢃⣀⣢⠔⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠁⢰⡷⠃⢰⢍⠂⠀⠀⠀⠀⠀⡉⢭⠓⢿⢿⡿⢟⡉⡉⠉⠑⢆⠀⠀⠀⣘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⣼⣠⠀⢸⠪⠀⠀⠀⠀⢠⠊⠀⠈⠶⠉⢸⡗⠆⠙⠑⠀⢀⡼⠀⠀⣰⠃⢠⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠆⠀⢯⣼⠀⢰⢽⠀⠀⠀⠀⠈⠢⠤⠄⠀⠀⡘⢃⠀⠀⠀⠀⡅⠀⠀⣰⠓⢀⠿⡔⠀⡜⠁
⠀⠀⠀⠀⠀⠰⡆⠀⢸⠙⡆⠈⣟⡆⠀⠀⠀⠀⠀⠀⠘⠂⠒⠁⠈⢳⣤⠔⠊⠀⢀⡜⠃⣠⠟⡩⠀⣼⡙
⠀⠀⠀⠀⠀⠀⣧⠀⠘⣷⠚⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡑⢄⡠⠋⢀⣴⢋⠟⢀⡜⡬⠁
⠀⠀⠀⠀⠀⠀⢈⡀⠀⣈⠀⠈⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠊⠀⣴⣿⡷⠃⢠⣾⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠗⠾⡀⠐⢤⡀⠀⠀⠀⠈⠲⠄⡀⠀⣀⣀⡠⠔⠀⢁⣠⣶⠚⡲⠋⢀⡴⣿⣾⣷⡶⠶⣲
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠢⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠾⠛⣉⠕⠋⠀⡠⡛⡥⠋⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠒⢄⡈⢙⠲⠦⣀⡀⠀⠀⠀⠀⠀⠓⠉⠀⣀⠰⢫⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣠⠈⠑⠢⠤⢀⣨⣇⡄⠀⢀⣀⡤⠔⣋⣥⣖⠉⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠓⠮⢦⣠⣠⡻⠲⠗⣚⣙⡡⠜⠚⠁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠂⠡⠀⠨⠐⡠⢀⠄⡉⢍⠡⡀⠄⡀⢔⠠⣐⠄⡂⠄⡂⠅⡂⢅⡪⠑⠌⠂⠅⠪⠂⠐

⠀⠀⠀⠀⠀⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣆
⠀⠀⠀⠀⣼⣿⣿⡿⢿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢤⣶⣾⣿⠿⣿⣿⣿⡄
⠀⠀⠀⠰⣿⣿⡗⡀⢨⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢾⡁⠛⢙⠆⠀⠀⠀⠘⣿⣿⣯⠀⡀⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣁⣀⣷⡖⠋⣉⡆⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢠⡋⡀⠘⣿⢤⡬⠂⢀⣠⡄⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⣠⣶⡶⢶⣼⡟⠀⣀⣀⡀⠀⠀⠀⠉⠑⠚⢹⡄⠀⠀⣼⣿⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠀⣼⣿⠏⠀⢀⣿⢃⣾⣿⣏⣻⣄⣠⣾⡿⢷⡀⣸⣇⣤⣤⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢠⣿⣿⡏⠙⣿⣿⡇⠀⣿⣿⣤⣴⣿⠇⣾⣿⠃⠀⢹⣿⣿⡏⠙⢺⡟⣿⣿⠏⢹⣿⠀⠀⠀⢠⣿⣿⣿⠉⢿⣿⣷
⠀⠀⣸⣿⣿⣀⣸⣿⣿⠁⠀⢈⣉⣭⣴⡿⠚⣿⣿⠀⣠⡟⣿⣿⠀⠀⣾⢹⣿⡏⠀⣾⡏⠀⠀⣠⠟⢹⣿⣿⣄⣸⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⣾⡿⠋⢁⡿⠁⣀⣙⡛⠛⠋⠀⠻⣿⣦⠞⠁⠘⢿⣷⡴⠛⢷⣤⠾⠋⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⡾⢁⣾⠛⣿⣿⢀⣤⡆⢀⣤⠀⠀⣠⣤⣄⠀⢠⣷⠀⠀⣰⡿⠃⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⠉⢻⣿⣿⠛⠿⠟⠋⠀⣾⠃⢠⣿⠏⢸⣿⠃⢸⣿⠀⣼⡟⢰⣿⠀⣼⡏⢀⡴⠏⠀⠀⢀⣿⣿⡏⠹⣿⣿⣿
⠀⠀⣿⣿⣿⣀⣼⣿⣿⠀⠀⠀⠀⢰⣏⣴⡿⠋⠀⣿⡿⠀⣸⡟⢸⣿⠁⠀⠁⣠⣿⠿⣯⠀⠀⠀⠀⢸⣿⣿⣇⣰⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣾⡿⠋⠀⠀⢠⣿⣧⣤⣿⣇⣼⣿⠀⢀⣴⣿⡏⠀⣿⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣄⣠⣴⠟⠛⠛⠋⠈⠛⠻⣿⣿⠟⠁⠿⠃⠀⢹⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⣿⣿⣿⡏⠹⣿⣿⣷⡀⠀⠀⠙⠛⠉⠑⠶⢤⠀⠀⢀⣶⡄⠀⠠⠶⠖⠂⠀⠀⠀⠉⠁⣠⣿⣿⣿⠋⢹⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣧⣀⣿⣿⣿⣿⣄⠀⠀⠠⠤⢤⠤⠄⠀⠻⣏⠁⢉⡿⠃⠠⠶⠶⠶⠆⠀⢀⣴⣿⣿⣿⣇⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣠⠴⠂⠀⡷⠖⠾⡇⠐⠲⠦⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⣤⣀⠀⠀⢀⣀⣀⠀⠀⢀⣠⣤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣄⡈⠙⠛⠛⠉⠉⠉⠙⠛⠋⠁⣠⣤⣄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠉⠙⠁⢀⣀⣀⡀⠘⠉⠉⠙⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠐⠒⣃⠀⠀⣀⠀⠘⣻⡟⠁⢀⡄⠀⢀⣘⠲⢰⡿⠞⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠁⠀⠀⠈⠛⠛⠋⠙⠛⠋⠀⠀⠀⠉⣡⠟⠀⢀⡀⢀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠁⠀⠲⣼⣧⣼⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⠃⢀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆⠀⠘⠷⣽⡵⠞⠀⠀⠀⢠⣀⣀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠳⣤⡴⠶⠒⠲⠶⢦⡀⠀⠀⠉⠙⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣙⣦⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⢦⣄⡀⠀⠰⣬⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠛⡟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⡤⠶⠶⣤⡇⠀⠀⠀⠀⠀⠀⢸⣤⠴⠶⠦⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡄⠀⡄⠀⢹⡄⠀⠀⠀⠀⢠⡟⠁⢠⡀⢠⡌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠷⠶⠷⠶⠟⠛⠳⠶⠶⠚⠛⠻⠶⠾⠷⠾⠷⠟⠁


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

⠀⠀⠀⠀⠀⢀⣶⠞⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⣿⠃⣠⣤⡀⠙⣷⠀⠀⠀⠀⠀⣴⠟⠉⠀⠈⠙⣦
⠀⠀⠀⠀⠀⣿⡀⣿⠀⠹⡄⢸⡇⠀⠀⠀⢰⡏⠀⢀⣄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢹⣇⠸⡆⠀⢻⠀⣧⠀⠀⠀⢸⡇⢰⠏⠉⣻⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡄⢻⡄⢸⡆⣿⠀⠀⠀⢸⡇⣸⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠙⢷⡀⠀⠀⠀⠀⠀⠀⣰⢟⣉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠻⡈⣷⠀⡇⢿⠀⠀⠀⢸⡇⡟⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠞⢦⠻⣄⠀⠀⠀⠀⡼⢣⠞⠉⣧⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠃⠀⠘⠒⠋⠙⠛⠁⠀⠸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⠀⠐⠃⠉⣾⡋⠛⠐⣶⠈⠈⠓⠋⠈⠃⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠀⠄⠀⠀⠀⠀⠀⠀⢀⢠⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⡸⣱⢛⡶⢠
⠀⢀⠰⣖⢲⡻⡄⠀⠀⠐⠋⠉⠓⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠀⠀⢳⣿⣼⡅⢰
⠀⢱⠀⣽⣬⣷⠇⠠⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠐⠀⠀⠀⢸⠃⠀⠀⢸
⠀⢸⠀⠀⠀⢿⡀⠀⠒⠀⠀⠀⠀⠦⠚⠳⠴⠂⠀⠀⠀⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⠉⠈⠋⠀⠀⠀⠀⠀⠀⢠⠏⡠⠖⠒⣼
⠀⢸⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣻⠀⣧⣀⣠⡟
⠀⠘⣇⠀⠀⠀⠐⣏⠛⠳⢶⡶⢦⡤⢤⠤⣴⠶⠖⠚⢛⣡⠞⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢦⣍⣉⠛⠛⣳⠖⣒⣾⡛⠛⡉⢉⣠⠶⠃⠀⠈⢀⡟
⠀⠀⠘⢦⡀⠀⠀⠈⠛⠲⠶⠤⡜⠐⠒⠒⣻⠧⠴⠖⠋⠁⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠟⠉⠙⠛⢻⣈⣥⡠⠿⠏⠉⠻⣷⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠉⠀⠀⠀⣤⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠐⠒⢄⠀⣧⠊⠁⠀⢀⡌⠉⠀⢈⣉⠀⣠⠇⠀⠠⠏
⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⣠⠔⠀⠀⠒⠒⣊⣷⡶⣮⡴⠂⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣠⠘⣮⠱⠶⢤⣤⣮⣤⣤⡬⠷⠚⠁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⡰⠁⠀⠀⠀⢠⣾⠟⢃⡀⠛⣾⠩⡤⢶⡟⢣⠀⠀⠀⠀⠀⠀⠀⠻⣧⠚⢁⢠⡼⠂⢀⣤⡈⢷⣧⡀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣆
⠀⠀⠀⠀⠀⡿⠀⠀⠀⡄⠀⠀⠀⢠⢿⡏⢡⡗⡿⢠⣶⣛⡆⠠⣷⠘⡄⠀⠀⠀⠀⠀⠀⡴⠈⠛⡞⠋⣤⠄⢹⡌⡃⠀⢻⣧⠀⠀⠀⠀⡆⢆⠀⠀⣷⢸⡄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⣎⣿⣀⡾⠀⣾⣾⠉⢹⡇⠰⡏⠀⡇⠀⠀⠀⠀⠀⢠⡇⠐⢢⣧⠈⣹⡞⠀⢯⡿⠂⠉⢿⡇⠀⠀⠀⡙⢦⣀⡤⠟⠀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⢰⣿⠇⠤⠄⠀⠀⠁⠀⠟⣀⣹⠃⢠⡇⠀⠀⠀⠀⠀⢸⡇⠈⡆⣿⡔⠀⠉⠀⠀⢒⢦⠄⢘⣿⠀⠀⢰⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠹⢤⣸⣿⣉⠀⢴⠄⠐⣽⡟⠆⢠⡟⠀⣼⠁⠀⠀⠀⠀⠀⠈⣿⣮⡴⠋⢷⡂⠀⢷⣆⢙⣿⣷⠈⣿⠀⠀⠀⣧⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠈⣿⠀⠤⣼⡂⠼⣿⠋⣈⡿⠁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠘⢷⣀⠛⠏⢀⡉⠅⣺⠏⠀⠀⠀⡿⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⢠⡎⣠⢀⡄⠀⡆⠀⠛⢦⣄⠀⠍⢙⡋⣰⠞⢡⠰⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⡶⢠⠄⠛⣶⣌⣀⣤⣾⣥⣤⣤⣄⣀⡇⠀⡆⢰⣸
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠚⠁⠀⠀⠀⠈⠉⠛⠉⠉⠙⠓⠛⠚⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠩⠓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠯⠍⠛⠻⢶⣄⣿⢀⢀⠄⠀⠈⠠⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠭⠁⠀⠠⢄⡢⢝⡿⢧⢊⠞⡠⠀⠀⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⡂⠀⠭⣓⠬⣓⢽⡣⠋⠊⠔⢁⠔⣩⠇⣠⡴⠶⠖⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠄⢐⡂⢭⠒⠅⠹⠀⠀⠀⠀⣾⢋⡾⠁⠁⠀⠀⠀⠈⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠟⠛⡗⢦⣝⠳⣶⡭⠁⠀⠀⢼⡧⢎⢀⡞⣡⠏⠀⠈⡔⣠⢃⠄⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠄⠀⠀⠀⡈⠢⠉⠛⢦⣉⠳⢦⡐⢼⡧⢫⢏⡴⡁⡠⠀⠀⠐⡱⣣⣎⢔⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⡐⢤⡑⢌⡂⠀⠀⢄⡈⢕⠦⣝⣶⣷⣷⣊⣯⣼⠖⠊⠉⢙⡪⣓⢭⡻⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⠨⣕⡮⣑⢤⠄⠈⠲⠶⠶⡶⣲⠾⣿⢿⠧⣍⡂⠀⠀⠠⡑⢍⠪⡑⢍⠢⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⡟⡵⡱⡳⣢⠂⢀⠀⢈⡴⢋⡼⣹⣎⣗⢦⡙⠦⣠⡕⢌⠢⡑⠌⠂⠁⡀⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠜⡼⡱⢱⢣⢣⢇⣦⡞⣠⠞⠀⠁⣿⡤⡅⠙⠲⣌⡻⣦⣠⡀⢄⢠⣰⣌⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠐⠡⠁⠃⠃⠈⣸⢏⣼⠕⡡⠀⢀⠈⡗⠀⠀⠀⣉⠛⢦⡉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⠃⡄⡀⣰⣰⠶⢳⢋⠁⢊⠔⡡⣢⢠⡇⣔⠤⡑⠠⠈⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡟⠉⠀⠀⣿⠥⢀⠵⣪⢞⡵⡳⣫⡪⡑⠬⡑⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣿⢂⠀⠊⠵⠫⠞⣼⣿⡟⠈⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⠃⠀⠀⠀⠀⠘⢧⣒⡡⢂⣠⡴⠋⠈⠻⢦⣤⣤⠼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⣠⠴⠤⣄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⡀⠀⢀⣀⣀⡀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⢀⠇⠀⠘⠊⠉⠉⠉⠶⣅⠸⠄⠀⠀⠀⠀⠀⡤⠤⣄⣰⠇⠀⠈⢣⢤⠴⠚⠉⣇⠀⠀⠀⠀⠀⠸⡈⢢⡀⠉⠉⠉⠙⠉⠉⠉⠁⢠⠊⢰⠃
⠀⠀⠀⠀⢸⠃⠀⠀⡀⠀⠀⠀⠀⠀⢹⠤⢤⡀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣱⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⢻⠖⠲⢦⠀⠀⣠⠖⠳⡄⠁⠀⠀⠀⠀⠀⠀⠀⡰⠓⠦⡀
⠀⠀⠀⠀⡼⠢⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⠀⠀⡏⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢻⠀⠐⢇⡠⠖⢧⡀⡀⠀⠀⠀⠀⠀⢠⠟⠤⣀⠟
⠀⠀⡠⠞⠄⠀⠀⠐⠀⢠⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢣⠀⠀⠀⠈⠐⠠⠠⠤⠤⠤⠔⠺⠃⠀⠀⡸⠀⠀⠀⢧⡀⠈⢣⠀⠀⠈⠉⠀⢀⠏⠀⢠⡟
⠀⢸⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢀⡴⠃⠀⠀⠀⠈⢳⠄⠀⠉⠂⠈⠤⠚⠁⠀⢤⠋
⠀⢸⠁⠍⠳⢉⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠈⣆⠀⡹⣹⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠘⡇⠀⣤⡀⠀⠀⠀⠀⣠⡄⠀⡼
⠀⠀⠸⡄⢹⠟⠛⢀⣀⣀⣀⣠⠂⠀⠐⢦⠃⠀⠀⠀⠀⠀⢳⡀⠀⢣⡀⠀⠀⢀⡞⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠑⣄⣀⣀⡴⠉⠀⢰⠁
⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠉⠒⠶⠔⠋⠀⠀⠀⠀⠀⠀⠀⠓⠶⠒⠛⠒⠒⠚⠦⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠞⠃⠀⠀⠙⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⢩⡓⣄⠀⣠⣞⡍⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⠀⠀⣀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠤⢤⣤⡤⠤⠔⠉⡇
⠀⠀⢀⡠⢄⡴⠛⠉⠠⠠⠄⠉⠛⢇⠀⠀⠀⠀⠀⠀⢠⠡⠀⠀⠈⠉⠉⠀⠀⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠉⠀⠀⠀⠐⢇
⠀⠀⠸⣤⣴⠁⠂⠐⠀⠀⠀⠀⠐⠈⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣀⠀⠀⠀⠀⢀⢀⡀⠀⡴⡏⠀⠐⠒⠄⠀⠀⠔⠒⢂⢨⡄
⠀⠀⢀⠇⠇⠀⠂⠐⠀⠀⠄⠂⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠐⠆⠀⠀⠀⠰⠂⠀⢹⡤⡈⢣⠀⠀⠰⡃⠀⠈⠻⠀⢠⡀⠀⠀⠀⠸⠀⠀⢐⢋⢉⢱
⠀⠀⡜⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠀⠀⠀⠀⡸⠀⠀⢸⠀⣀⣀⡏⠈⢠⣶⣾⣿⣼⣶⡄⠄⠀⠀⠄⢼⠉⠉⢹
⠀⠀⢣⡀⠀⠀⠈⠀⠠⣈⣣⠆⠥⠄⠈⢆⠀⠀⠀⣠⠋⠓⠤⠀⠀⡀⠀⠀⠠⠞⠁⠀⣠⠃⢸⡀⠈⣁⣴⠻⡯⡙⠛⢋⢭⠻⣦⣜⣀⠔⠃⠰⣿⡼
⠀⠀⠀⡜⠀⠀⠀⣠⠊⣩⣹⡦⡤⣀⣀⡼⠀⠀⢰⠁⠀⠀⠀⠀⠘⠄⠅⠀⠀⠀⠀⡔⠁⠀⠀⡹⠃⢸⣏⠘⠷⠇⠀⠸⠾⢂⢻⡦⠹⡄⡴⠋⡈⡇
⠀⠀⠀⡇⠀⠀⠘⡇⢸⠶⢯⠀⠇⠀⠀⢸⠀⠀⠘⡶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠛⠄⣾⢧⡙⠒⠀⠀⠄⠒⣃⡼⣿⡄⡿⠀⠸⠟⡰
⠀⠀⠀⢣⠀⠀⠨⡷⢿⣗⡇⣸⠀⠀⠀⡜⠀⠀⠀⢃⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡸⠀⠀⠀⠀⠀⠸⣮⣿⣻⡟⠛⣿⣿⣧⣷⣋⣼⡷⠴⢦⠔⠁
⠀⠀⠀⠀⠙⠳⠄⠞⠂⠖⠚⠣⠰⠛⠋⠀⠀⠀⠀⠀⠓⠾⠿⠒⠒⠒⠒⠒⠻⠿⠒⠁⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠤⠿⠿⠛⠁⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠀⠀⠀⣠⣴⣶⣤⡄⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠀⠀⣀⣤⣤⣀⠙⠻⠿⠟⠋⠀⠀⠀⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠁⠀⠀⣀⣀⠀⠻⢿⣿⡿⠃⠀⠀⠀⠀⠠⣾⣿⣿⣷⡄⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣷⣤⣀⠀⠰⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⠈⠉⠉⠉⠀⠀⣀⣤⣾⢿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡉⠉⠀⠀⢀⣤⣤⣤⡈⠙⠛⠛⠋⠀⠀⢀⣠⣴⡾⠟⠋⠁⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠋⠙⠻⣿⣿⣿⣶⣤⣀⠙⠿⠿⠿⠋⠀⠀⠀⢀⣤⣶⠿⠛⠉⣠⣶⡆⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⢹⣿⣿⣿⣿⣿⣷⣦⣄⡀⢀⣠⣴⣾⠟⠋⠁⠀⠀⣾⣿⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡿⢷⣦⣾⣿⣿⡋⠈⠙⢿⣿⣿⣿⣿⠟⠉⢠⣶⣿⡆⠀⠀⠙⠋⣁⡀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠙⢿⣿⣿⣧⡀⠀⠘⣿⣿⣿⡏⠀⢠⣿⣿⡟⠁⠀⠀⣠⣾⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⡀⣸⣿⣿⡿⠿⢷⣾⣿⣿⣿⡇⠀⠀⠉⣁⣤⡀⠀⠀⢿⡿⠟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠋⠙⢿⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⡇⠀⠀⣼⣿⣿⠇⠀⠀⠀⣠⣶⡆⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⢹⣿⣿⣿⣦⣄⣠⣿⣿⣿⡇⠀⠘⠿⠟⠁⠀⠀⠀⣾⣿⣿⠃⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣾⣿⣿⡋⠉⠙⢿⣿⣿⣿⡇⠀⠀⢠⣶⣿⡆⠀⠀⠙⠋⣁⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣧⡀⠀⠘⣿⣿⣿⡇⠀⢠⣿⣿⡟⠁⢀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣾⣿⣿⣿⡇⠀⠀⠉⣁⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣇⣠⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⠲⣄⠀⠀⠀⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⠀⢃⡨⠟⠐⢝⠤⣠⡥⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⢀⠉⠁⠀⠀⠀⠀⠉⡒⠺⢥⠞⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠁⠀⠸⡁⠃⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⠀⠀⠀⢀⡀⠀⠉⠉⠀⠀⠀⢹⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠒⠂⠠⠄⠀⠀⠘⠠⠀⡸⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⠒⠳⣾⣟⢳⡀⠀⠀⠀⣀⣀⠀⠀⠔⠁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣢⠇⠀⠀⢸⣟⣲⣽⣯⠷⣭⠭⢽⠍⠀⠀⠀⢱⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡅⠀⠢⠠⠀⠈⠱⣄⠀⠀⠀⠀⢸⠀⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠓⠚⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⣰⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠐⠒⠒⠶⠖⠛⠀⠣⠒⠒⠋⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡔⠒⠠⢍⡒⠤⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠔⢊⡩⠔⠒⢢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⣀⠬⠒⠚⠉⣿⣿⣿⣿⣿⣿⠙⠑⠒⠥⡀⠀⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠊⠀⠀⠀⠀⢀⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠑⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠊⠀⠀⠀⠀⠀⠀⠀⠢⢙⠛⢛⠡⠈⠀⠀⠀⠀⠀⠀⠀⠱⡀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡷⠁⠀⠀⢀⠠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠘⢎
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠣⡰⠁⠀⠀⠔⣵⣾⣿⣶⣌⢂⠀⠀⠀⠀⠀⠀⡞⣡⣾⣿⣶⣌⢆⠀⠀⠘⡄
⠀⠀⠀⠀⢸⠀⡀⠀⡀⠀⡄⡇⠀⠀⢘⢸⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠨⠄⣿⣿⣿⣿⣿⠌⠀⠀⠀⢱
⠀⠀⠀⠀⠈⠣⠷⠀⠗⠒⠁⡇⠀⡀⠈⢎⡻⠿⠿⠟⡣⠂⠀⠀⠀⠀⠀⠀⠱⣙⠿⠿⠿⢋⠜⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⢳⠀⠄⠁⠀⠈⠁⠀⠉⠀⠀⠀⡀⢻⡋⡀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠤⢀⠆
⠀⠀⠀⠀⠀⠀⠘⣶⣶⣬⠢⠀⠱⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠁⠀⠀⠱⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠊⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢀⢔⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡝⡉⠿⢛⠽⠛⠉⠁⠀⠀⠀⠀⢢⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡻⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠠⡨⡠⠊⢁⠀⠀⠀⠀⢀⡸⠟⢙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⢀⡀⠀⠀⠀⡀⠈⠀⠀⠢⡀⠐⢞⠲⡁⣠⠄⢐⠊⠁⠀⡌⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⠟⠀⠀⠀⠀⠱⡀⠀⡎⠀⠀⡀⠳⡀⠈⢢⡀⠑⠀⡐⢠⠃⠀⠀⠀⢡⠻⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢱⡀⠅⠐⠄⠐⢄⠈⠢⡀⠁⡠⠂⢀⠇⠀⠀⠀⠀⠀⠁⠂⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⣀⣇⠡⡀⠈⠂⠀⠑⠀⠠⠂⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠊⠁⠀⠀⠀⠀⠈⢳⣈⡑⡒⠐⠂⠁⢀⣀⣀⡎⠀⠀⠀⠀⠀⠈⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠒⠂⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠤⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⢀⣀⡀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠛⠛⠛⠛⠛⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠉⠉⠓⠒⣦⡀
⠀⠀⠀⠀⠀⢰⡿⢭⣁⣀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡤⠶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠉⠉⠓⠲⣯⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠉⣸⠋⠉⠉⢻⡈⠙⠒⠦⣤⣀⣀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡷⢤⣄⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠖⠚⠫⣤⠞⠀⠀⢠⠏⠀⠀⠀⠀⢱⡄⠀⠀⠙⣏⠉⠉⠓⠒⠚⠁⠀⠀⠀⠀⣀⣸⠇⠀⢀⣈⡭⢿
⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢀⡞⠲⠦⣤⠴⠚⠋⢻⡀⠀⠀⠈⢧⡀⠀⠀⠀⣀⣠⠴⠖⢛⣩⡤⠖⠛⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⡶⠤⣤⣀⣀⡼⠁⠀⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠳⡴⠚⣉⣡⡴⠖⠛⠉⢸⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣟⣀⣤⠀⠀⣼⣁⡀⠀⠀⡇⠀⠀⣀⣠⣼⣧⠀⠀⣤⣀⠹⡏⠉⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠁⢸⢀⡼⠁⠈⠉⣿⠒⠷⠚⠋⠉⠀⠉⢹⢧⣸⠁⠈⠛⠃⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⣀⣤⠼⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣿⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⠴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⣀⣸⢀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢘⣿⡄⡀
⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⢀⠀⠀⠀⠀⣀⣤⠀⢠⣀⠀⡄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⢿⣿⠟⠇⠟⣦⣀
⠀⠀⠀⠀⠀⠀⢘⠛⠛⢰⣿⣿⣷⣦⡀⠉⠁⢠⣾⡛⠊⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⠟⠃⠀⠀⢀⣸⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠁⣤⢼⣿⠟⠀⠉⠻⢿⣷⡤⠸⣿⣧⣾⠏⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢀⣹⠶⡗⢫⡉⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣦⡀⠤⠐⠊⠉⠀⠠⢠⣥⣤⣤⠰⣞⡂⠀⠀⣀⣀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠰⠃⠻⠟⢠⣾⠃⠈⣇⠈⠋⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣭⡻⠟⠋⠸⣷⣌⠳⣄⠀⠀⢀⠎⠀⠈⢛⣶⡘⢿⣟⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠸⡀⠀⠈⠛⢿⣧
⠀⠀⠀⢆⣛⣴⠟⠀⠀⠀⠀⠘⢿⣧⠹⣆⢀⠸⠤⠖⠊⠁⠀⠀⢀⣿⣷⡄⠈⣯⣍⡙⠲⣄⠀⠀⠀⢀⠀⠀⠀⢀⣤⢤⣶⡟⡁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⣤⣶⡇⠀⢀⣾⣀⣀⣀⣀⣘⣩⣭⣤⣶⣿⣿⣿⣷⡈⠃⠀⠀⠂⠀⠀⠔⢋⣤⡾⢋⣮⣤⣤⠶⢁⣼⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⡄⠀⠚⠩⢯⡘⢿⠿⠖⠒⣸⣿⣿⡷⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⢛⠁⠀⠛⠢⠰⣇⠀⠀⢀⡼⠹⣿⢟⣾⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠴⣾⠏⠀⠀⠈⠳⣤⠏⠀⠀⢵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠿⠙⠁⣴⣿⣿⢿⠿⠇⠙⠁⢀⣀⣤⡄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣅⣀⣀⠐⠿⠂⠀⠀⣤⣌⠲⠾⠿⠟⠁⣰⢺⡆
⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⠀⠠⠤⢠⡶⠊⠉⠉⠁⢀⡠⠀⡠⣶⡌⢣⡀⠸⣿⣷⡿⠛⠻⠦⣀⠠
⢠⣤⣤⣶⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠈⢠⣮⣾⣿⣿⣿⣿⣐⡼⠋⠀⠀⠀⠀⠈⠉⠁
⠈⠿⠛⠛⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣳⠋⣹⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠉⡁⣀⢀⠠⢤⣄
⠀⠀⠀⢀⣦⠀⣠⣤⣤⡶⠛⠉⠙⠓⠶⣶⢦⣷⠀⠰⠞⠳⡄⠙⣻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣀⠀⠠⣤⣤⠆⠀⠤⠼⠟
⢀⣀⣴⠋⣠⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⢹⡌⡏⠐⠀⠀⠀⠀⣰⠌⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠀⠘⣤⣴⣤⣴⡄⣀⠷⠆⢀⡄⠀⠀⠀⠀⡀
⠈⠽⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⢰⡆⠀⢀⣸⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠘⠿⠯⠁⣴⠿⠛⠇⠀⠘⢡⣾⣌⡙⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠠⠴⠾⠧⠦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣶⠖⠀⠀⠀⠀⠀⠚⠂⠀⠀⣿⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠛⠓⠲⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⣰⣼⣿⡿⠂⢰⡾⠟⣀⣶⡽⠋⠀⠀⠀⠀⠑⣆⠀⠐⠆⠀⢻⣿⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⣿⡿⠿⠛⠉⠀⠀⢋⠀⣸⡟⠁⢀⠀⠀⠀⠀⢰⢠⣿⣿⣷⡆⢻⣶⡿⠛⢧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠎⠀⢠⡏⠀⠀⣿⣄⡟⢛⣷⢸⣿⣿⣾⣏⠀⠀⠀⠳⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡶⠚⠒⠦⣄⠀⠙⢦⠀⠀⢀⡴⠋⢀⡤⠖⠛⠉⠙⢦⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠈⠳⡀⠀⢳⣀⡞⠀⣰⠋⠀⠀⠀⠀⠀⢸⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⢇⠀⠀⠀⠀⠀⠹⡄⠀⠙⠀⡼⠁⠀⠀⠀⠀⢀⡴⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⢤⣀⠀⠀⠀⢹⡀⠀⢰⠇⠀⠀⢀⣠⠖⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠉⠙⠓⠲⠶⠇⠀⠼⠴⠒⠋⠉⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣲⠆⠀⠀⣠⣄⠀⠀⢴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⣾⠀⠈⣧⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢰⠏⣿⠀⠀⡏⢧⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⠏⠀⡟⠀⠀⡇⠈⣇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⢶⣠⠏⠀⠀⡇⠀⠀⡇⠀⠘⣆⡾⢦⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡇⠀⠀⡇⠀⠀⠹⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⡇⠀⠀⡟⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⣀⣴⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⠚⠇⠀⠀⢠⣾⣿⠿⠛⠉⢠⠷⡄⠴
⠀⠀⠀⠀⠀⠤⢒⣲⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⢤⣤⣤⣴⣟⢣⠀⣀⢀⣤⠟⠉⠢⢤⣾⣿⣤⣈⣢⣄⠀⢂
⠀⠀⠀⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡿⢛⡿⠀⠉⠁⠈⠉⡷⠖⠒⡾⣆⣀⠀⣀⣀⣉⠳⠄⠈⠀⠀⠀⢀⣾⣿⠋⠀⠉⠉⠙⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣻⣿⣷⠄⠀⠺⠦⠀⠀⠀⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢠⣴⣤⣶⣶⡲⠄⠀⠾⠿⣥⠀⠀⢀⡀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠀⠀⠀⣠⠾⠛⠛⠛⠛⠻⢷⣤⣄⣀⠀⠀⠀⢠⣷⡀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣲⣾⣿⠟⠉⢀⣠⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣾⣿⣷⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢉⣦⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡗⠒⠶⣦⣌⠳
⠀⠀⠀⠀⣀⣤⣐⣢⠞⠁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠉⠉⠁⠀⠀⡰⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡓⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⣄⡀
⠀⠀⠀⠀⣀⣄⡀⠀⡀⣿⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠚⠛⠛⠟⠿⠛⠿⠛⠿⠟⠛⠛⠋⠝⠛⠛⠋⠉⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠠⠤⣶⠄⢠⢄⠽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⢸⣿⠀⣾⢋⣤⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⢹⡇⠀⠰⠂⠀⠀⡿⠀⠃⠈⠉⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢀⠀⠟⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⡀⠀⣠⡶⢶⣿⠁⠀⠠⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠⡾⢉⠇⣰⡿⢠⠇⣴⡟⢠⣿⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣧⡼⣾⡏⢻⡾⢾⡟⣠⢿⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣤⣴⣶⡄⣼⡏⢀⡞⢠⣿⣠⠏⠀⠸⡷⠋⣾⡇⠀⢀⡾⠁⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉⢁⡼⠃⠹⠷⢋⣠⣬⡿⠷⠦⣄⠀⠀⠀⣿⡇⢀⡾⠁⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡶⠛⠉⠀⣴⠂⠀⡼⠀⠀⠀⠘⠳⠛⡁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠸⣿⣄⠀⢀⣠⡾⠋⡀⠀⣼⠏⠀⠀⠀⢰⡏⣠⠞⣱⠆⢀⣾⠀⣰⠟⣹⠇⠀⠀⠛⣥⣴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠛⠛⠛⠁⠀⠀⠁⢸⡏⠀⠀⠀⢀⣿⢿⡅⢠⣿⡴⢻⡏⣸⣿⠞⠁⣠⠄⣀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⣷⣄⠀⠀⣼⡇⠀⡟⠀⠉⢀⣾⠷⣿⠁⠀⣰⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠗⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⡼⠁⢀⡴⢻⡟⠀⠻⣶⠾⠋⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠻⠟⠁⣠⠏⠀⣾⠁⢠⣶⡆⠀⠀⠀⢰⡿⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⢠⡏⢀⣼⠃⢠⡞⣯⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⢁⡀⠀⠉⠀⢁⣀⣠⣿⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⣠⡞⢹⠄⠀⣴⠋⡷⠀⢀⣾⠟⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⣿⢃⣾⣿⠀⣜⣠⣾⢿⣴⣃⣠⣿⠁⣠⢿⡟⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣼⠇⣼⡇⠘⡿⠉⢸⡇⣴⠋⠉⢸⣇⡴⠃⢸⡇⠀⣠⡟⠀⠀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⣀⣀⣤⣿⣿⠀⣿⣠⡞⠁⠀⠸⠟⠁⠀⠀⠀⠉⠀⠀⣼⠛⠞⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⢁⣾⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡆⣸⠇⢀⣤⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢠⣦⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣼⠀⣰⡇⢀⣾⠃⠀⣴⣿⣾⠿⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⢇⣼⣿⠀⣼⠇⣠⢞⣿⣿⡁⠀⠀⠀⢀⣾⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣾⡿⠀⠀⠀⢀⣀⡀⣿⠟⢸⣧⡾⠻⠶⠃⣸⠇⠘⢷⣤⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣿⠷⠶⠿⠿⠟⠛⠁⠀⠀⠀⢉⡀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⢀⡾⢹⡇⠙⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠆⢾⣋⣉⣛⣋⣩⣉⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠊⠉⡦⠞⠉⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠢⢼⡀⠀⠳⠦⠀⣗⡶⠚⠁⠀⢈⣦⠚⠉⠙⢦
⠀⠀⠀⠀⠀⣴⡒⠒⠋⢹⢀⠀⠀⠀⣀⣙⣧⣦⡀⣰⣿⢽⠇⠀⠀⣼⣧⣖⣲⣦⡜
⠀⠀⠀⠀⠀⢻⣿⣦⠽⠯⣿⣷⢠⢸⣿⣿⣿⣿⢳⡿⣀⣾⡼⠗⠛⠛⠛⢯⣯⡿⠗⠊⠲⡄
⠀⠀⠀⠀⠀⢠⣟⠁⡠⣠⣾⢿⣿⣮⣯⣿⣿⣻⣾⣮⣿⣿⣿⡆⠀⠀⡀⠈⣟⣭⣁⠀⢠⠃
⠀⠀⠀⠀⠀⢸⣯⣳⣮⠇⠈⣛⣓⣛⣛⣉⣿⣿⣿⣿⣿⣿⣵⣡⣼⣖⣥⣾⣿⣿⣿⣞⣁
⢀⡠⠔⠒⠩⠶⢿⣭⣽⠀⣾⢽⠋⠉⠀⠀⢀⣽⣿⢿⣿⣿⡄⠀⠀⢹⣿⠛⠟⠀⢀⣠⢴⢟⡇
⡎⠉⠑⠒⠤⣄⣀⠭⠛⠷⡶⢾⣇⡆⡄⡀⢸⣭⢠⣿⣿⣿⣵⣵⣴⣼⠤⠉⠓⢿⣍⠴⠣⢺⡇
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣵⡛⠻⠛⠳⠾⠿⠛⠃⠁⣀⣤⡼⣻⡀⠀⠠⠆⠘⡗⠱⢠⣸⡇
⢧⣀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠈⠉⠒⠢⠤⣠⡴⠒⢻⡿⢋⢬⣺⡇⠀⠣⠢⠀⣿⣵⣿⡏
⠀⠹⠛⢶⣦⣤⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢺⡺⣲⢟⢕⠅⣮⣾⡇⢀⡤⠔⢀⡯⡈⢺⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠓⠂⠀⣧⣀⡀⠀⠀⠀⢸⡾⣀⢀⣶⣾⢞⣿⠇⠈⠀⠀⢸⣟⠶⣹
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠉⠙⠛⠒⢦⣼⢶⣾⣿⢟⣕⣕⣿⠀⠀⠀⠀⠈⣿⢈⣾⡀
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣗⡹⡾⣫⢺⢝⢕⣿⠀⠐⠒⡁⠀⣷⢋⣾⣿
⠀⣛⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢅⣾⡫⡣⣢⡗⣿⠀⠀⢇⠄⠀⣯⣿⣿⠒⠂⠀⠀⠈⢁
⠀⣏⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿⣕⢿⣪⡪⡫⣪⣿⠀⠀⠁⢁⣠⣿⡋⢠⠄⠉⠁⠀⡠⠄
⠀⠀⠉⠛⠲⢤⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢕⢕⢟⢝⢋⣾⣿⢀⣤⡶⠛⠉⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠚⠷⣦⣤⢸⡃⠀⠀⠀⠀⣿⣱⢷⢕⣵⠿⠿⠭⠋⠱⠖⠤⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠶⢤⣀⣿⡥⠟⠉⠩⠥⠄⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⢂⡀⠀⠁⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢢⡡⠺⣍⠄⠢⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙
⠀⡀⠐⢤⠁⠐⠁⠀⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠑⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆

⠀⠀⠀⠠⡠⠀⠠⠀⠀⠀⠀⠀⠴⠰⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠸⡃⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⡄⢀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑
⠠⣀⡠⠃⠐⠋⠀⠀⠀⠀⠐⠀⠠⠀⠠⢐⠀⠀⠀⢀⠀⠐⣀⠖⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠊⠤⠖⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡁⣲⠜⡑⡤⠃⠀⠀⠀⠀⠀⡠⠲⠢⠀⠀⠀⠀⠘⠄⠸⠻⠖⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣰⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣄
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢁⣤⣤⣶⣶⠦⡄⣼⣿⠋⢱⡆⣠⣾⡿⣀⠀⣰⣿⠏⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡟⢼⣿⠟⣻⣿⠃⠀⢸⣿⠃⢀⣿⢰⣿⠏⢠⣿⢰⣿⠏⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣴⣿⠃⠀⠀⢿⡟⢀⣼⠃⣿⡿⠀⣾⠇⢸⣿⣠⡟⢹⣿⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣇⣠⣾⠟⠁⠀⠀⠀⠈⠛⠚⠁⠀⢿⣧⡼⠋⠀⠀⠉⢡⣾⡆⠙⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⢀⣾⣿⠀⢀⣀⠀⠀⣤⣾⣿⣧⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⣾⣿⡏⣠⣿⡟⢀⣾⡿⠫⣿⡟⣠⣾⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢸⣿⣟⣴⣿⣿⠁⣼⣿⠁⠀⠈⠀⣼⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢈⠻⠛⠁⣿⡇⠀⢿⣇⣠⡴⠂⢠⡿⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⠏⠀⠀⠀⠘⠓⠚⠁⠈⠉⠀⠀⠘⠃⠀⠹⣿⣧⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣟⠟⠉⣳⠀⡄⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠛⠓⢤⡿⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠙⢷⣄⣻⡟⠷⠀⢢⠄⠀⠀⢉⠉⠙⠡⠄⠶⣄⠀⠀⠀⢷⣄⡀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⠤⠉⢿⣤⠄⠈⠀⢰⣧⠾⠃⢠⡇⢸⡦⠋⠐⠾⣭⡦⣿⣍⣧⣀⡀
⠀⠀⠀⠀⠀⠀⡷⠶⠾⠧⣤⡀⠀⠠⠈⠁⠀⠤⠀⠄⠒⠀⢸⡟⠀⢀⡾⣩⣿⡽⡦⠾⠿⣷⣶⣚⠣⡀⠁⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠿⡄⡀⠀⢀⣠⡔⠁⠀⠀⠀⢀⡞⠀⠀⣸⣧⣿⢽⡄⡼⣷⢼⣿⣏⠉⠛⣤⣬⣾⣟⢻⡛
⠀⠀⠀⠀⠀⣼⠋⠀⠀⡀⣰⡇⢤⣴⣟⡉⠰⣾⡗⢀⣀⣺⡔⠃⣤⠙⣿⣻⡾⣷⡍⠘⠀⠙⠻⣶⣦⣼⣿⣿⣿⣧⠱⣄⠀⣄
⠀⠀⠀⠀⠀⢁⠂⢠⣶⣹⡏⢷⢠⡉⢻⣿⣿⣿⣿⣶⣯⡙⠓⢊⠁⡀⣿⣿⣧⣿⣷⣶⡦⠀⠀⠈⠻⢤⡈⣿⣿⣿⡟⠻⠂⠻
⠀⠀⠀⠀⢠⡆⡦⢸⠁⣶⣿⡜⣧⢳⣀⡝⠉⢻⣿⢿⣧⣷⠂⠈⢳⡉⣿⣿⣿⢿⣿⣿⠃⠀⠀⠀⠀⠀⢿⣻⡼⠃⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠓⠀⠀⠀⢹⣷⣿⡈⠂⠻⡇⠀⠘⢻⣷⣙⣯⠅⢠⣤⣿⣿⣿⣯⡘⣿⣿⠀⠀⠀⠀⠀⠠⡆⢻⣃⠀⠀⣰⡷⠂⠀⣰⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠃⠀⢀⠀⠀⠀⠀⠹⣿⣥⣶⢰⢿⣿⣿⣿⢻⡆⢿⡟⠀⠀⠀⠀⠀⠀⠃⡼⠁⠀⠰⣿⣄⡀⠀⠹⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⡖⠁⠀⠀⠀⠀⠀⣿⣿⡟⣼⢿⣿⣿⡟⠻⣿⡈⣿⣄⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣿⠇⠀⢰⠋⣧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⡇⢲⣦⣀⠀⠀⠀⢻⣿⠞⠁⠈⡛⠘⣷⣤⣬⣁⡉⣻⣆⠀⠀⠘⣶⠃⠀⠀⠀⠀⠀⣸⠇⠀⠈⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣀⢣⡀⠙⣿⣷⠀⠀⠘⠁⢲⣄⣾⣷⠠⢤⣿⢿⠛⠿⢿⠿⣿⠀⠀⠁⠀⠀⠀⠀⠀⢀⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠦⢼⡋⠀⠈⠙⠢⣼⣿⣿⠁⠀⠀⣿⣈⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣼⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⣀⡄⠀⠀⢺⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⣴⡀⠀⠀⠈⣿⠇⠀⠀⠸⡇⠀⠀⠀⣼⠛⣿⣹⠛⠀⠀⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣿⣷⡀⠀⠀⢸⣷⣇⠀⢸⣿⠀⠀⣤⣿⠷⠛⠋⠀⠀⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠿⡷⠤⢶⣀⠙⢿⣷⠜⣿⣶⣜⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⣂⡄⡀⡔⠒⡀⠀⢀⠀⡤⠔⠒⠠⡀⠀⠀⢀⠀⡤⠔⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠀⡨⣜⢰⢅⠃⡰⢁⠊⠅⡄⠂⠄⠑⠀⡰⢡⠊⠁⡄⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠡⡀⡁⡘⢹⠁⢀⠃⡌⠀⠃⡈⣒⠽⠀⢠⠁⡀⠀⢆⡈⣒⢽⠀⠇⠀⠀⢀⠠⠤⢤⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠊⠀⠀⠆⢘⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠇⠀⠀⠀⠀⢸⠀⡀⢀⢊⠔⠁⡑⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠇⠀⠀⠀⠸⠀⠠⠀⠀⠀⠀⡘⠀⠿⠀⢠⠀⠀⠀⠀⡌⠀⠃⠁⢌⠀⠀⠇⠁⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠄⠄⠀⢣⠀⠑⠤⠤⠊⠀⡔⠀⢣⠀⠑⠤⠄⠊⠀⡌⠀⡀⠐⡀⠀⠈⢻⠧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠤⠊⠀⠀⠀⠑⠠⠀⠠⠄⠊⠀⠀⠀⠑⠠⠀⠠⠔⠊⠀⠀⠑⠢⠄⠁⠩⠊⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠀⢈⠀⢡⠀⠠⠪⢪⢢⠀⠀⠀⡠⡊⠥⢈⠑⢄⠀⠈⡀⡅⢀⠆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢤⠀⠀⠈⡄⢸⠀⢀⢠⠌⡘⠀⠀⠌⠔⠀⠄⡂⠂⢘⠀⠀⢠⢠⡌⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠈⠎⢀⠁⠀⡘⠘⠀⠀⠈⠢⠤⠊⠀⠀⢠⠌⡞⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠆⠁⠀⠌⠀⡌⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⡆⠃⠈⢢⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠧⠊⢀⠌⠈⠀⠆⠁⠰⠀⠀⠀⠀⠀⠠⠀⢰⠀⡇⠀⠀⠀⢃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡇⣗⠂⠠⠤⠤⢲⡆⠀⠀⢀⠌⠀⠀⠀⢰⠐⠄⠑⠄⣀⠀⢠⠖⠀⠠⣀⡣⠁⠀⠈⢌⠒⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠜⠀⠈⠁⠒⠂⠁⠑⠤⠐⠁⠀⠀⠀⠐⠊⠀⠀⠑⠒⠒⠂⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣤⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠹⣯⣩⠟⡀⠀⠀⠀⠀⡀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⡀⠀⣅⡰⢀⡠⡷⠐⠚⠛⠋⢉⡀⠙⠟⣦⣄
⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⣀⠄⠀⢀⡴⠤⠀⠠⣠⠊⠀⢀⣷⡄⠘⢾⣌⢙⣒⡬⠟⠛⠿⠾⠷⠔⠴⠾⣟
⠀⠀⠀⠀⢿⡉⠻⠏⢛⡻⠟⣵⡿⠟⠁⠀⠀⠀⠀⣸⣦⠀⠈⢳⠛⠁⡤⣽⡯⠁⣄⡀⠀⠀⠀⣴⠀⣠⡄⠈⡧
⠀⠀⠀⠀⠻⣭⡛⠛⢉⣤⠞⡩⡜⠀⠀⠀⠀⠀⠐⠃⠉⢣⡀⠀⠂⠀⠙⠠⠝⠋⠀⠀⠐⠤⠘⣏⢣⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⠁⡇⡇⠀⠀⠀⠀⠀⢁⡶⠖⠉⢁⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠢⠰⣶⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠘⠃⠀⠀⠀⠀⢠⣙⣰⠆⠸⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⠀⠀⠀⠈⢂⣉⡠
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⢸⣦⡀⠀⠀⡠⠊⠑⠷⣌⡄⠀⢣⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠡⠼⠃⠉⠂⠈⠀⠀⠀⠀⠀⠁⠐⠒⠒⠷⢄⣀⢢⡀⠀⠀⠀⠲⣯⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠢⣀⠀⠀⠑⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠐⡧⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠈⠀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢷⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⢉⣢⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈


⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣬⣿⣶⣥⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣀⣤⣤⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣝⡻⢿⣿⣿⣿⣿⠟⢋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣵⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⡿⠃
⠈⢿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⣄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡮⠁⣴⠠⢹⡅⠀⠀⢠⣤⣤⣴⣿⣾⣷⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠂⠒⠨⣛⣇⣀⠀⣿⡿⠿⢛⣥⣾⣿⣿⣿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡗⢿⡿⠿⠛⠛⣷⣴⣵⣾⠿⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣟⠀⠸⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠉⠑⠛
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⢀⣀⠀⢀⠀⢰⠂⠀⠀⠀⠈⠶⣿⠖⢠⡆⠠⣤⣀⣸⡿⠁
⠀⠀⠀⠀⢀⣾⣿⣿⢿⣆⠰⣦⣿⣷⣮⣻⣿⣿⡀⠀⠀⠀⠀⣿⠖⠚⠀⠑⢮⣿⣿⠇
⠀⠀⠀⢀⣿⣿⡿⠛⠚⠉⠀⢸⣽⣜⠯⡁⣿⡟⠿⣷⣦⣤⣤⣽⣇⠰⠇⠀⣠⣿⠿
⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠻⡓⠀⣿⠇⠀⠀⠈⠉⠉⠉⠛⠛⠛⢼⣷⣟⣶⣤⣀⡀
⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣏⠉⠻⣿
⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠐⠂⡀⠀⠀⠴⠶⠾⠟⠁⠀⠀⠀⠀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢸⣿⠀⠉⢁⠀⠛⡹⢾⠍⠀⠀⠀⢀⣀⣀⡠⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢶⣿⡆⠀⢹⣤⠈⠁⠀⢀⣀⣴⣾⡿⠟⠋⠻⡌⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⠂⢀⣿⠁⠀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠘⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⠤⡤⠀⠠⣄⣀⡀⠀⠐⠉⣠⡖⢀⣀⡉⠛⠻⠟⠉⠀⠀⣠⠆⢀⣤⣀⡀⠹⣿⣄⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠶⠀⠀⠀⡏⠀⠀⠀⠈⣿⡟⣀⣾⣿⠇⣠⣬⣭⣷⡶⠋⠑⣰⠞⠓⢒⣾⠋⠙⠶⠞⣷⡟⣿
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⣶⣿⡿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢲⠧⠄⠀⠀⠀⢈⠘⠻⣿⠋⢠⠀⠀⠀⠀⠀⠠⡾⠃⠀⢠⣾⠃⠀⠀⠀⠀⠈⠻⣿⠄
⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠠⣿⣃⠀⠀⠀⠀⠀⠀⠀⢮⠳⠒⠲⠤⣄⡀⠉⡖⠼⠂⠀⠀⠀⠀⠀⣤⡀⠁⢀⣴⠟⠁
⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⣽⠏⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⠄⠀⢸⢀⡀⠀⠀⠀⠀⠈⢁⠘⠂⣀⣀⠀⠀⠀⠸⣿⣶⣿⡇
⠀⠀⠈⠒⠛⠋⠁⠐⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⠻⠿⠅⡀⠀⢸⠘⢿⣶⣄⡀⠀⠀⣿⠀⠸⡇⠀⠀⠀⣀⠔⢹⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢀⠀⠀⠈⢸⣷⠚⠽⣿⣿⣶⣿⣀⠀⡧⣠⠏⢀⣠⣀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡄⠀⠀⠉⠙⠿⠿⠿⣇⣿⠂⠀⠙⣿⣏⣀⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡀⠈⠋⠉⠉⠉⠉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢀⣀⣉⢀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⣸⠆⠀⠈⠻⢿⣤⣤⣤⣤⡘⠷⠀⠀⠀⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⣦⣼⣿⣿⣿⣿⣿⣿⣿⠋⣰
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣌⢡⣾⠟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡀⠀⢠⡄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⡟⠻⣏⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠛⠻⠿⠋⠈⠻⣷⣿⡆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠙⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣿⣿⡆⠀⣀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠛⠋⠉⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⠿⠟⠛⠁⠀⠀⣰⣿⣿⡿⠟⢁⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣟⣋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⡏⢸⣿⣿⡇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡟⠀⢸⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣟⠛⠋⠉⠉⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⡿⠀⠀⠀⢿⣿⡧⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣸⡿⠁⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⠃
⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶
⢜⣿⣿⠇⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿
⠘⣿⣿⠀⠀⠀⠀⠀⠀⡿⢿⣿⡆⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡄⣤⣴⣶⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠁⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠒⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡆⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⢯⡛⠉⠻⣤⣴⠞⠉⢀⢀⣀⡁⢔⣺⣿⠿⣯⣄⡀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⢄⠘⠛⣿⣤⠞⣽⡿⣇⣠
⠀⠀⠀⠀⠠⢶⣶⡿⠟⠉⠉⠀⠀⠀⠀⣶⣟⣤⣌⠁⠀⠀⢺⡇⣸⣿⣯⠻⠛⠁⠀⠀⠈⠁⠲⠾⠿⠋⣾⠀⠸⣋⣠⣿⠜⣦⣄⠀⠀⣰⠋⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⢿⣪⣿⡿⠀⠀⣴⣿⣿⣿⣝⣏⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣋⡴⠞⠉⠙⠻⠿⢟⣛⡶⢼⠇
⠀⠀⠀⠀⠀⠀⠀⣤⠴⠀⠀⠀⢀⡤⢿⣗⠟⠓⠒⢦⣤⠁⠉⠉⠉⠙⣯⠓⠦⣄⠀⠀⠀⠀⠀⠸⠈⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⠐⢄
⠀⠀⠀⠀⠀⣠⢎⠐⠋⠁⠐⠚⡡⠊⠉⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⠂⣜⣆⠀⠀⠈⠃⠀⠻⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠈⠉⢡⠁⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠂⢀⣀⢀⣀⣀⢈⡀
⡀⠀⠀⠤⢤⣤⣤⣤⠤⠬⠭⠭⠭⠯⠽⠽⠿⠿⠿⠿⠧⠛⠓⠉⠉⠁⠀⠁⠁⠀⠈⠀⠉⠉⠀⠈⠙⠋⠍⠸⠿⠭⠿⠉⠉⠉⠙⠁⠉⠉⠁⠉⠁⠈


⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣤⠀⠀⠈⢢⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣘⠷⣄⡀⠀⠳⡌⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⡠⠒⠉⠀⢈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠉⠙⠻⢿⣦⣀⠙⠻⣀⣀⣀⠀⠀⡠⠋⡠⠊⠀⠀⠀⣀⠞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣀⠀⠀⠙⠲⠶⢤⠤⠭⠝⢺⠉⠈⠋⣟⠩⠴⠛⠒⠊⠉⠉⠉⠉⠙⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠉⠀⣀⠈⠀⠐⢢⡤⠤⣤⣤⣤⣼⣿⡀⢸⡇⠉⠒⣌⢛⠛⠛⠛⠛⠃⠐⠀⣸⣀⣩⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠆⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠙⠛⠛⠙⢤⣤⡘⣦⠈⠙⠛⠲⠆⠀⣼⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⢄⣀⡀⠀⠀⠀⠀⢀⣨⡇⠸⡇⠉⠉⠓⠒⠢⣼
⠀⠀⠀⠀⠀⠀⠐⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠐⡞⠉⢠⠇⠀⣟⠳⢦⣤⣤⡴⠟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠜⣀⡤⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡞⠀⠀⡿⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡞⠀⠀⠀⡇⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢰⠧⣀⠀⡄⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣀⣠⡀⠀⣼⠀⠈⠳⡇⠀⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠹⣠⠇⠀⠀⠀⡇⠀⠀⠁⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡇⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⡀⠀⠈⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⢰⠀⠀⠀⡇⠀⠀⠘⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠁⢸⠀⠀⠀⡇⠀⠀⠀⠘⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠜⠀⠀⢸⠀⠀⠀⠇⠀⣀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣤⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢀⠤⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⢤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⡀⠀⡇⣀⠤⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢹⣟⡛⣻⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢄⠀⢀⠎⠀⠀⠈⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⣄⠀⢀⣾⣽⣶⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠱⡘⠀⠀⠐⠄⠀⠀⠎⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡰⠋⠉⠐⠀⠀⠀⠹⡄⠀⠀⠀⠀⣤⡤⠃⠀⠀⠀⠀⢠⣾⣻⡿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢧⠠⡀⠀⠀⠀⢈⣑⠴⡠⠔⠺⠿⠅⠤⢄⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣿⣿⣯⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠛⠋⠉⣸⠓⠄⠀⠀⠀⠀⠄⠀⠘⡄⠀⠀⠀⠙⠛⠛⠛⠋⠉⣰⠟⣿⡻⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠠⡀⠀⠀⠀⢻⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⣿⣷⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢆⣂⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠂⠔⠉⠉⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣶⣦⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⢤⠅⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⣿⣿⣿⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⢣⡀⠀⠁⠀⢰⡿⠖⠉⠀⠀⢱⠀⠀⠀⢀⣠⣤⣤⣘⣿⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⠀⠙⢒⢄⣐⠝⠀⠀⠤⠱⠶⣟⠀⠀⠀⣾⣵⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⢣⠔⠒⠀⠀⠉⢠⠴⠶⣤⡀⠀⠀⠀⠀⣽⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣠⣴⠟⠀⠀⠈⢿⣆⠄⢰⠾⠟⠀⠀⠀⠰⣷⣾⣿⣿⣿⣿⣿⣿⡻⣿⣿⡿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⢋⠃⠀⠀⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢹⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢡⡀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠉⢵⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠕⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⠄⠀⠀⠁⠀⠀⠐⠋⠀⠀⠀⡆⠀⠀⠠⣷⠄
⠀⠀⠘⠀⠀⠸⢥⡾⠿⠀⡀⠀⠀⠈⡀⠀⠀⢘⠀⠀⠀⠤⠀⠀⢀⡇⠀⠀⢀⡀
⠀⠀⢀⢀⠀⠀⢮⣀⡀⠀⠱⠀⠀⠀⠁⠀⠀⠸⡠⡀⠀⠀⠀⢀⠎
⠀⠀⠀⢫⢢⠀⠀⠙⠀⠀⠀⢆⠢⢀⡀⢀⡴⠁⠑⢌⣒⣀⠶⠋⠠⠴⠶⢢⣤⣤⠴⠊
⠀⠀⠀⠀⠳⣑⠠⢀⡀⢠⠎⢀⠁⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠈⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠐⠂⠠⡴⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠀⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡠⡀⠀⠀⢀⡔⠣⡐⢄⡀⣀⣠⠴⠇⠀⠀⡸⠀⠀⠀⠀⠀⠒⠼⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠃⠄⠤⠄⠚⠀⠀⠈⠑⠒⠈⠁⠘⠒⠯⠒⠁⣀⠤⣀⡀⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠠⠕⠓⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠡⣑⡦⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠓⢍⠀⠀⠑⢶⠒⠒⠤⣤⠎⠁⠀⠀⡨⠝⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣴⠞⠓⠤⣑⠢⢀⠀⠑⢤⡤⠁⠀⢀⠤⣊⠠⠐⠓⣄⡀
⠀⠀⠀⣀⡠⠤⠒⠊⠉⠁⠀⢸⠁⠀⠈⠑⠂⠭⣙⣛⣿⣈⣑⣾⣛⣉⠡⠐⠂⠁⠀⠀⡩⠉⠒⠒⠤⢄⣀
⠀⠀⠀⡏⠉⠒⠢⢤⣀⠀⢀⣀⠬⢒⣂⡤⠝⠛⠭⠑⠋⡹⢌⠙⠊⠉⠛⠫⠍⣛⠪⢍⣀⠀⣀⠤⠒⠊⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢙⡻⠖⠊⠉⠀⠀⠀⢀⡠⠔⠊⠀⠀⠉⠒⠤⡀⠀⠀⠀⠉⠲⣎⡏⠀⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⠖⠯⢄⣀⠀⠀⠀⠀⢀⣀⠼⠢⡀⠀⠀⠀⠈⠣⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠙⠒⢢⠞⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠈⠉⠒⡖⠊⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⣼⠤⠒⡋
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣠⣋⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⣢⡀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⠜⠁⠈⡏⠒⠤⢄⡀⠀⠀⠀⡇⠀⠀⢀⡠⠔⠚⡇⠀⠐⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠃⠀⠀⠀⡇⠀⠀⠀⠈⠉⠒⠤⡧⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠢⠤⣀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠢⢄⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀⣇⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣿⡫⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠸⠂⠀⠀⠛⢀⡄⠖⠀⢀⣀⢀⣠⡀⠀⠀⠀⠀⠀⣨⠀⠗⣈⠀⠀⠠⡦⠀⠀⠀⠀⠘⠋⠀⠃
⠀⠀⢀⣄⠄⠖⠀⠀⠛⠀⠀⠀⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⢈⠀⢀⠀⣀⠀⠀⠀⠀⠀⣴⠄⢈
⠁⠀⢀⣠⠀⠀⠀⠘⠓⢀⠰⠆⠀⢠⡄⠀⠀⠀⠛⠀⠀⠀⠀⢉⡀⠀⠀⠀⢠⠀⠹⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⢰⠄⠀⠸⠂⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠴⠗⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⡄⢀⡈⠉
⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⣙⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠰
⠀⠀⣄⠄⠀⠀⠈⠁⠀⠀⠀⣛⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠛⠀⠀⠀⠋
⠀⠀⠉⢀⠀⠠⡤⠀⠐⠇⠀⣛⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠿⠀⠀⡀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠶
⠘⠃⣤⠘⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⣀⠀⠈⠃⠀⠈⠁⠀⠤⠾⠄⠀⢴
⠘⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠒⠶⠀⠀⡀⠀⠀⠀⠐⢀⣌⣿⣤⣤⠀⠀⠀⠀⡤⠀⠀⠖⠀⠀⣀⠘⠗⢀
⠐⠇⣠⠘⡁⠨⠃⠀⠀⠀⢠⡄⠰⠆⠘⠃⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠀⠀⠛
⠀⠗⠈⠀⠛⠀⠀⠀⠀⢀⠈⣤⠀⠶⠀⠀⠶⠀⠀⠀⠀⠙⢿⣿⠁⢀⠀⠀⠀⠚⠂⠀⠙⠈⢀⡶⠀⠀⣦
⠀⠀⠀⠀⠁⠐⠗⠀⠀⠛⠀⢉⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⡀⣀⡀⠐⠀⠈⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⣰⠆⠀⠖⠀⠶⠀⢀⠈⠁⢠⡄⠀⠰⠆⠀⠀⠀⠀⠀⠐⠁⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠰⠆⠀⠀⠀⠶⠀⠀⠀⠘⡃⠀⠀⣄⠀⠀⠀⠀⠀⠀⢠⠄⠠⠰⠆⠀⠰⠆⠀⠀⠰⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠴⠀⠀⢛⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠠⠄⠶⠈⠀⠀⠐⠀⠘⡁⠀⠈⠀⠀⠀⠀⠴⠠⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⢢⡤⠀⠄
⠲⠀⠀⠀⠗⠀⠐⠀⠀⠛⠂⠈⠃⠀⠂⠛⠀⠀⠂⠀⠶⠀⠀⠀⠰⠆⠀⠀⠀⣤⡄⠀⠀⢀⣄⠀⠀⠸⠂
⠈⢃⠘⠃⠀⠀⣠⡀⠀⠀⠀⠂⠻⠀⢠⡄⠀⢀⣄⠀⠀⠀⣤⠀⠀⠀⠀⣤⠀⠀⠀⣠⡄⠈⠁⠀⠀⣀
⠀⠉⠀⠰⠆⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠉⠀⠀⠀⠀⠀⠁
⠐⠁⠀⡀⠀⣛⠀⠀⢀⠀⠐⠇⢀⠀⠘⠃⠀⠰⢒⡂⠀⠀⠀⠀⠛⠀⢠⡉⢀⠁⣁⡘⠃⡀⠀⠀⢰⡆
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣽

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⡀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⠀⠸⣯⣷⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠉⢻⣿⣦⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⠻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡉⠛⠛⣿⣿⣿⣿⡿⠿⠇
⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⢐⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠈⠻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣀⣠⣤⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠔⣻⣿⠏⠀⠀⣼⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣿
⠀⠀⠀⠀⢿⡏⠀⠀⠀⠿⢻⣿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠁⠸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣵⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣶⣋⣿⣿⣯⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⣸⣺⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⡿⠟⠉⠙⣿⠉⡅⡈⠋⠻⢭⣽⡆
⠀⢠⣿⣿⣿⣿⣿⣿⣥⣿⡿⡝⢿⣾⣯⢻⣯⠝⣦⣷⡽⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⠾⡴⠂⠂⠉⠊⠀⡡⢎⠡⠀⠈⠋⠄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣸⣇⣻⡎⢟⣷⠾⣿⠿⣿⣿⣿⢋⣾⣿⣛⣿⣁⣳⣋⡈⣀⢀⡠⠁⢁⠀⠀⠀⠄⠀⠈⠉⢠
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣿⡿⣶⣷⣶⡶⠄⣄⠀⡈⣗
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣆⢺⣧⣽⣿⣿⣷⣤⠄⠀⢘⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣛⡛⢻⡿⠛⠃⠀⢀⣾⠃
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣯⡟⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡉⠉⠉⠻⠋⠁⢁⣈⣈⣦⣤⠶⠛⠁
⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠋⠉⠁⢀⣴⣿⠋⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⠁⠈⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠟⠁⢀⣯⣿⡟⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⠞⠉⠀⠀⠀⣿⣿⣷⡿⠝⣿⣿⣿⡟⢭⣽⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣻⣟⣦⣿⣟⣿⣿⣿⡿⠿⢿⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢻⢟⢻⣿⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣻⣿⣻⣿⣾⣿⣿⣿⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⢩⣪⣿⣿⣿⣿⣽⣿⣭⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⣿⣿⣿⣿⣽⣿⡿⠿⢿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢟⣻⣙⣯⣍⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠋⠃⠀⠘⠛⣛⠛⠋⠀⠀⠀⣠⣄⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣷⠄⣀⣴⣾⠿⠿⠿⠿⠿⣷⣦⣈⠻⣿⣆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣯⣿⣿⢁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣊⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠚⣿⣿⣿⣻⣿⢯⡍⢉⣉⡉⠛⠒⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⡦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡄⠀⠀⣿⣿⣷⣺⣇⣻⡆⣏⠀⣹⢰⠟⠳⡄⣠⣌⡉⠓⠶⢤⣀⡀⠀⣿⣯⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⡿⠛⣿⣿⣷⣤⡤⣌⣉⣁⡘⠦⠾⢡⡏⣀⡗⠀⢠⠆⠈⠙⠳⢿⣟⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣋⡀⠙⣿⠁⢠⣿⣟⡺⠿⢠⣿⡏⠉⠉⠛⠓⠦⣌⣉⡀⠠⢯⡀⢠⡟⢀⡆⢈⡙⠧⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡴⠋⠀⢸⣿⡒⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⣘⠲⠟⣴⠋⠙⠆⣈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣾⣿⢭⣀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣌⡓⠃⡾⣷⠒⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠧⣿⡟⢳⣤⣙⠀⠀⣰⠏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢾⣿⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠋⠁⠀⢠⣿⠃⠙⢳⡞⠁⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⢣⣅⠀⣠⣿⠃⠀⢀⣾⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣲⣿⠏⣴⣿⠁⠀⠀⢸⡟⣫⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⢀⣴⣿⡿⠍⠈⠻⢃⣾⡿⠁⠀⠀⠀⠀⣿⠋⠉⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣯⡀⠀⠰⣿⣿⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠛⠛⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠘⣿⣿⡀⠀⠈⠉⠳⣦⡀⠀⠀⠀⠀⠀⢀⣼⠛⣿⣿⠃⠀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣄⣀⡀⣹⣿⠷⢤⣀⡀⢀⣼⣿⣦⠀⠀⠀⢠⡟⠁⢀⣿⠿⣄⣀⣤⣶⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠉⠉⢛⣻⣧⣀⠀⠀⠙⠻⣿⡿⠉⢷⡀⢠⣿⣷⣴⠋⠀⠀⢸⠿⠛⠋⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢿⣿⣿⣿⡿⠛⠲⢤⣄⣈⢧⡀⠈⢷⡞⢹⡟⣁⡠⢤⣶⣇⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠤⢤⣄⣉⠙⢻⡦⢸⡇⢿⣿⣅⣠⡴⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠿⣿⣿⠉⠋⠉⠉⠉⢻⠋⠉⠉⠉⠛⠛⠓⠒⠒⠲⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⡇⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠦⣤⣤⣤⣤⡼⠷⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⡾⠃

⠀⠀⠀⠀⠀⠀⣤⡴⠆⢀⠀⠀⠀⢠⣴⡶⢂⡀⠀⠀⣀⣤⠖⣠⣤⡴⢀⡀⠀⠀⠀⠀⠀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⣀⣠⣤⣤⣽⣷⣾⡟⣷⣤⣤⣼⣿⣷⣿⣿⣤⣤⣹⣿⣴⣿⣿⣶⡿⢿⣄⣀⣀⣀⣀⣹⣽⣶⣿⢿⡄⠀⠀⣀⣤⣤⣿⣿⣶⣿⣆
⠐⠛⠉⠙⣿⠿⠿⢿⣿⡿⠛⣿⠿⠿⣿⣿⢋⣽⠿⠿⠿⣿⣿⣿⣿⣿⣇⠉⠉⣩⢿⡿⠿⠿⣿⣿⣇⡀⠐⠛⠛⠻⣿⡿⣿⣿⣟
⠀⠀⠀⠀⠙⣆⠀⠀⠃⠁⠀⠈⠦⠤⠼⠬⠞⠁⠀⠀⠠⠇⠘⠧⠄⠈⠳⠤⠊⠁⠏⠀⠀⠀⠀⠘⠄⠑⠀⠀⠀⠀⠹⣦⠀⠘⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠟⢙⣿⣿⣿⣿⡿⠛⠋⣹⣿⣿⣿⠿⠟⢙⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣿⣿
⣿⣿⣿⡿⠿⠷⠄⠘⠉⠀⠈⠟⠛⠃⠀⠈⠁⠈⠻⠯⠀⠀⠋⠀⠀⠈⠀⠈⠻⣿⣿⠿⡿⠄⠀⠋⠉⠈⢻⣿⣿⡿⠿⠯⠀⠀⠊⠉⢻
⣝⣁⣠⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢰⣷⣎⣉⣁⡀⠀⠀⠀⠀⢀⣶⣤
⣿⣿⣿⣿⣄⠻⣿⢓⣠⢸⣿⣤⠚⠿⠆⠂⠁⣰⣿⡿⠿⢰⣄⠐⢖⠂⠀⠿⣣⣔⢡⣾⣿⣷⣶⣆⢲⡌⢿⣿⣿⣿⡀⠚⢖⠆⡠⣸⣿
⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣦⣼⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣦⣾⣿⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠛⠋⠉⠁⠀⡸⠉⠙⢮⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢣⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⣆⠀⠀⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠢⣄⠈⠳⣄⠈⢣⡀⠀⠈⣿⠀⠀⢀⢀⡠⠔⠋⠁⠀⠀⠀⠈⠉⢉⣹⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣢⡤⠤⠤⣀⣀⡈⠑⠦⣌⡑⠀⠙⢦⡀⢸⡷⡆⠉⡙⢄⣀⣀⣀⣤⠤⠤⠒⠊⠉⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠯⣭⣉⣀⣀⣀⠀⠀⠀⠉⠉⠓⠒⠩⣵⣶⡤⡟⠒⢇⢸⡄⡈⠌⢧⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠓⠛⠛⠩⠭⠥⠤⢄⣀⠤⠭⠙⠛⠛⣛⣋⠭⠟⠒⠺⠧⠖⠻⢄⡇⢠⠀⢨⡭⣙⣒⠒⠒⠒⠶⣾⠭⠭⠭⠭⠚⠛⠒⢲
⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⡎⠉⠉⠉⠙⠒⠒⠒⠒⡶⠒⢺⠁⠘⠀⢸⠉⠉⢹⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡆⠀⠸⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠁⠀⠀⡇⠀⢸⡇⠀⠀⠨⡇⠀⠀⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠉⠉⠉⠉⠁⠒⠚⡇⠀⠀⠀⡧⠤⠤⢄⣀⣀⣀⣀⣀⣇⠀⣿⠀⠀⠀⠀⡧⠤⢼⡇⠀⠀⠀⡗⠒⠈⠉⠉⠉⠉⡏
⠀⠀⠀⠀⠀⠀⢀⠜⠁⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⣿⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠛⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⢡⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⢀⡀⠀⢸⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⠊⠁⠈⠑⠚⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠻⠤⣄⣀⠼⠤⠄⣤⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣄⠠⠤⠴⠧⠀⠠⠤⠛⠚⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣶⣷⣶⣿⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣷⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠟⠛⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡍⢀⠠⠀⠠⠀⠀⣴⡆⠸⢿⣦⠀⠀⣀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣷⣾⣿⣿⣿⣿⡿⢶⣿⣿⣿⣿⣷⡖⢀⡈⠀⠀⠀⠘⠋⠀⠀⠀⠀⣠⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠉⢿⣏⣿⣿⢿⣷⠂⠑⠊⣿⣿⣿⣷⣦⣄⠀⠀⣴⣶⡆⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⡉⠙⣿
⢕⣾⠘⡹⢉⠿⠗⠶⠈⠘⠛⢿⣿⡿⣿⡿⡟⣾⣿⣏⠀⠄⠀⠀⢀⡸⢿⣿⢿⠿⠏⣹⢿⠛⠛⠲⣦⠈⠫⣗⣾
⢄⣹⣷⠏⠀⠀⠲⠄⠃⠀⠀⠀⠁⠘⠚⢧⠀⠈⢻⣿⣀⣀⠀⠰⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣿
⠈⢻⡏⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠉⢿⣅⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⢸⣇⠐⠲⠂⠀⠀⢠⣸⣿⠇⠀⠀⠀⠀⠀⢰⣿⡧⠀⠀⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢺⡏⣉⣻⣆⢀⡔⡵⠛⢉⡙⣆⠀⠀⠀⠀⢀⣼⡇⣀⣶⣶⣿⣿⣿⢀⡀⠀⠀⢀⡀⣀⠀⠀⠀⠀⣀⣀⡀⣸
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣽⣷⣿⣴⣷⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣾⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠩⠙⠀⡸⢉⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡄⢀⠃⢠⠰⠀⠀⢠⠈⠀⠀⠀⠂⠀⠀⠀⢰⡄⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠞⠈⡆⠀⠒⡾⠤⠒⣁⡀⠀⠀⠀⠀⠀⠀⠘⢻⡏⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠃⠈⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠉⠢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣾⢐⣸⢄
⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠤⠐⢄⠀⠀⠀⠀⠀⠀⠀⠙⡿⠈
⠀⠀⠀⠀⠀⠀⠈⠒⢠⠔⢠⠀⠀⠀⠀⠀⠈⠀⢀⣁⠐⠁⠂⠀⠂⠀⠈⠀⠀⠀⠰⣮⡸⢣⢊⡷⡦⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⠃⠀⠀⠈⠢⠄⠃⡴⠀⢧⠧⢵⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢢⡁⠅⡠⣺⠇⡆⡄⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠄⠘⢂⠚⢽⡠⠤⠴⠀⠀⠀⠀⠘⠠⠆⠹⣪⣤⡃⠀⠀⠬⡄⠀⠀⠀⠺⠃⠀⠁⠰⠆
⠀⠀⠀⠠⠀⠀⠀⠄⠅⡀⠀⠀⠄⡀⠀⠀⠈⠡⠖⠋⣈⠟⣿⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⡀⠀⠶
⠀⠀⠀⠀⠠⠀⠀⠘⠀⡉⠀⠀⠄⠁⠀⠀⠀⠀⠀⡒⠂⠀⡇⠈⡀⢍⠁⠀⠀⠀⠀⠀⠀⡤⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠎


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⢤⣀⠈⠐⠤⣀⣼⠁
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠠⠃⢀⣀⠀⠀⡞⠀⠀⠹⡆⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠢⠤⠀⠀⠀⢰⡁⠈⣧⠀⢷⡄⠀⠀⡇⢀⡔⠛⣿⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⡄⠀⠀⠀⢸⣷⢠⡟⠀⣾⡛⠶⣴⠗⠺⣧⡼⠻⣆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢘⣿⣀⣀⠔⠁⣉⣉⡀⠀⠘⠳⠒⠁⠀⠀⠀⣠⠀⠉⠓⠦⠄⠊
⠀⠀⡠⠒⠉⠉⠉⠉⠉⠉⢹⡏⠁⠀⢠⠏⠀⠀⢹⠀⠀⠀⠀⣠⠤⡀⢰⡇⠰⣦
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢄⣿⣀⢀⠀⠊⠀⣄⠀⣼⠃⠀⠀⣸⣇⡴⠃⠀⠀⡠⠔⠒⢦⡀
⠀⠸⣆⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢸⣏⠀⠀⢸⠇⢰⣏⠀⣿⣆⣀⠔⠹⠏⠉⠓⡶⣮⣀⣀⣀⡼⠁
⠀⠀⠙⠳⠦⠤⠤⠖⠋⠀⠀⠀⠀⠀⢿⣄⣀⠾⠧⠜⠛⠊⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⡉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⠀⠈⠛⢿⣿⣦⣄⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠙⣿⣷⣦⡀⣠⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄
⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠉⠉⠛⣿⣿⣿⣿⠿⠛⠉⠉⠀⢿⣿⣿⣿⡆⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡏⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣰⣿⠟⠙⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣠⣴⣾⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣀⣤⣴⣶⣿⣿
⠐⢹⣿⡟⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠺⣿⡁⠀⠀⠀⠀⡏⢻⣿⣄⡀⠀⠀⢀⣀⣤⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⣿
⣤⣦⣼⣥⣤⣤⣤⣤⣴⣤⣽⣿⣥⣤⣤⣼⣿⣿⣿⣯⣼⣿⣷⣤⣤⣤⣤⣼⣿⣭⣤⣤⣤⣦⣤⣤⣤⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠡⢴⣿⣴⣤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢴⡆⠹⠆⠋⠉⠈⠘⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢀⣤⣾⣤⡂⠉⠀⠀⠀⠀⠰⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡀⣼⡿⠿⠛⠻⢿⠟⠿⠿⣦⡀⠀⢨⡹⣯⠱⣿⡇⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣤⠿⣇⣸⠂⠀⠀⡠⠃⠀⠀⠀⠀⠉⣳⣮⣠⠚⠋⠉⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠙⠃⢭⣷⠱⣀⣄⡴⡁⠀⠀⠀⠆⠀⠀⠘⣿⣿⣿⣙⡿⢈⠁⠈⠛⢏⠀⠈⠉⠉⠉⠙⠉⠙⠛⠉
⠀⠠⠬⡥⢦⣼⡋⣹⣴⠟⠋⢸⣿⠃⢰⣿⠂⠒⠀⠀⠧⣽⣿⣥⡽⠋⢁⠀⠀⠀⠡
⠀⠀⠐⠀⠀⠈⠉⠁⠀⠀⠀⠀⠹⡤⢈⠛⠢⡀⠀⣀⡴⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣧⣄⠈⠙⣿⣄⣀⣀⣀⡤⠔⠒⢠⡀⠀⠀⠀⡔⠂⠄⠛⢿⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠈⠀⠈⠀⠈⠒⠀⠊⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡋⠉⠉⠙⢢⡄⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢺⠉⢦⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡄⠀⠙⣄⠀⠀⢳⠔⠒⠒⠛⠛⠛⠋⠉⠑⠢⣤⠏⠀⠀⠀⣀⣠⡬⣦⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢷⠚⠉⠉⢦⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⣴⠚⠉⠀⠀⠘⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠉⠙⣦⡈⢧⡀⠀⠈⢷⡀⠸⡀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠈⢓⣦⡀⠀⠀⢹⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢦⡀⠈⢷⠀⢷⣄⠀⠀⠀⣰⠋⣠⠶⢋⣠⠴⠞⠉⠉⠙⢷⣄⠀⣧⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣝⢦⡘⣇⣨⡿⠷⢤⣜⣡⣾⠿⠚⠉⠀⠀⣀⣠⣤⠶⠚⠛⠣⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⣎⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠏⠀⠀⣾⡋⠉⢀⣀⣤⡶⢞⣛⣉⣡⣤⣤⣤⡴⠶⢋⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠩⣯⡀⠀⠀⣏⠉⠙⠛⠛⠛⠋⠉⢉⣉⣀⣠⡤⠤⠶⠚⠋⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠑⢦⡀⢀⣠⣤⠴⠖⠛⠉⠀⠀⠀⠀⣈⣽⠿⣟⠉⠉⠉⠉⠉⠻⣍⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⢙⡏⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠠⣀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢀⣈⣿⣦⠤⠒⠉⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠑⢤⡀⢸⡇⠀⠀⢀⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⣦⡴⠞⠋⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⣷⣄⠀⢸⡇⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠈⢷⣼⠇⠀⠀⠀⠀⠈⠢⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣠⣿⠴⠖⠚⠋⠉⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠷⣄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠞⠛⢹⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣸⠀⠀⢀⡿⠀⠀⠑⠢⢄⡀⠀⣿⠀⠀⠀⣀⡠⠴⠚⠉⢹⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠙⢦⣷⠖⠚⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣼⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡀⠀⣀⣀⣤⡤⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣀⣠⣼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⡿⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡇⢀⡴⠋⠁
⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⢉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⣥⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠶⠶⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠉⢁⣈⠙⠻⢿⣿⣿⣿
⣿⣿⣿⣿⠏⢠⡄⠀⠀⠀⠀⠀⠠⠶⠌⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⣴⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢴⡆⠀⠀⠀⠉⣤⡄⠀⠉⣿⣿
⣿⣿⣿⠋⠀⠈⠁⠀⣤⡀⠀⠀⢀⣀⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠿⠀⠘⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠴⡦⠀⠀⠀⣤⡄⠀⡀⠰⣾⣿⣿
⣿⣿⣿⣌⠿⠀⠀⠀⠙⠁⠠⠖⢉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⠷⠀⠀⠀⣶⡄⠀⠨⣇⣼⣿⣿⣿
⠿⠿⠿⡿⣷⡀⠸⡆⠀⣠⠤⢖⡩⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢌⣋⠀⠀⠀⠀⠰⠗⠀⠀⣁⣀⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣄⡀⠀⠾⠆⠀⠴⢀⣽⣿⠿⠿⠿
⠀⠀⠀⠀⠙⠿⣶⣄⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡶⢦⣄⣤⣴⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠷⣤⣀⣤⡾⠷⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠎⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣿⣿⠟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠎⠁⠙⢻⣷⣤⣀⣂⠀⡄⠘⠀⣶⡇⢸⠀⠀⠀⠀⠀⣠⢀⣦⢞⣽⠟⠉⠎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⠁⠀⠀⠀⠨⠿⠿⣿⣾⣷⣠⣧⣿⡇⢸⠀⢠⠀⣤⣠⣷⣾⣵⡿⠁⠀⠀⠂⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠐⡄⠀⠀⠀⣌⣉⡭⠾⢿⣿⣿⣇⣸⣿⣼⣿⣿⠿⠿⠃⡄⠀⢀⢀⢰⣥⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢭⣦⣤⣴⣾⣿⠷⠷⠛⠛⣻⣿⣿⣿⣿⣯⣭⣭⣬⣮⣤⣴⣿⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⢒⣿⣿⣿⡏⠻⣆⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⣋⠩⢴⣿⣿⢿⣧⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣾⠹⣰⣿⡆⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⢿⠔⠏⢻⡇⠀⠀⠘⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡏⠘⠀⠀⣿⡇⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⣠⣾⡿⠉⠁⢀⣴⣿⣿⠃⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣾⣷⣿⢟⠙⢁⠖⢺⣫⣿⡿⠃⠀⠀⠀⠀⠀⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣴⣯⣾⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⢯⣽⣭⣷⡌⠉⠙⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⠁⠉⠀⢈⣉⣭⣍⣁⣀⠀⠀⣶⣦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣿⡿⢃⣤⣾⠿⠛⠋⠙⠛⠛⠿⢶⣤⡙⣿⠆⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣾⠿⠿⣾⣿⣥⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⢹⣿⣶⣻⡿⢝⡗⣠⠶⢦⡀⣉⣙⠛⠲⠦⢄⣀⠀⠀⠀⠀⢻⣿⣆⣠⡀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⣸⣿⣿⣿⣿⠞⠃⠻⣤⠼⢹⡃⢨⡇⡾⠲⡄⠈⠙⠓⠦⣤⣸⣿⣖⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣷⡿⠁⢹⣿⣛⣿⡟⣳⡶⠶⢦⣭⣉⡘⠧⠴⠃⢀⡾⠀⢀⡄⠉⠛⠶⣿⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠐⠛⠳⢦⡄⣨⠏⠀⣿⣟⠊⠙⠁⣿⡇⠀⠀⠀⠈⠉⠓⠶⠦⣬⣓⠆⢾⣁⡾⣡⠶⣌⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⢠⣿⡷⠄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣍⡀⢧⣤⢈⣴⣂⡌⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣿⠯⣷⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣷⣤⣘⠹⡇⠀⠀⣸⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣙⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡳⠉⠛⢠⣿⠛⢦⣀⡞⠁⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠉⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣓⠀⠀⣠⣿⠃⠀⢠⡏⢀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣵⣿⠇⣴⡿⠃⠀⠀⣿⢿⠻⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣠⣼⣿⡛⠹⣿⢋⣼⡿⠃⠀⠀⠀⢻⡾⠛⠲⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣓⠀⠀⢴⣿⣏⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣆⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⣴⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⢠⡄⠀⠀⠀⠀⠀⠀⢠⣾⠏⡜⣦⠛⢽⣉⣷⣤⠀⠄
⠀⠀⣠⣴⡾⠿⣶⣶⣡⠞⢭⡧⠀⠃⠀⠀⠀⠀⠀⣴⢯⣵⡄⣽⠘⢷⣄⣘⡁⠀⢡⣐
⠀⣴⣿⣿⣿⣇⡀⢘⣻⢷⣶⣦⣠⣦⡀⠀⢀⣀⠀⠙⠊⠁⠘⣿⡆⠀⠙⡿⣏⡙⢿⢾⣦⡒⠀⠀⠀⡀
⣰⢿⣿⠯⠟⠻⣿⣶⣬⣛⠿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣆⣤⣄⣹⣿⣄⠀⠁⠙⠿⣼⣾⣿⢃⣷⠀⠀⠘⠿⣷⣦
⠙⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣉⣿⣿⣿⣿⣿⣿⡟⠉⠋⢹⣿⣿⡀⠀⠀⢂⠀⣠⣾⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⡟⣷⣿⠿⠛⢿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣷⣤⠀⠛⠻⣏⣉⡲⣿⢀
⠀⠀⠀⠀⠀⢀⠘⣡⠞⠛⠙⠋⠉⠉⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠻⠜⠃
⠀⠀⠀⠀⡀⢈⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣧⠀⠀⠀⠀⢸⡵⠧⡘⠇⠀⠰⡆
⠀⠀⢀⡴⠙⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⢧⣤⡄⠀⠀⠁⠈⠳⡀⠀⠛⠼
⠀⠀⠀⠀⣰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠒⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢠⡾⠛⣷⠀⠁⠀⠀⢠⣿⠇⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⡀⠀⠀⠀⣾⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⣿⠃⠀⡟⠀⣀⠀⢀⣾⣿⣴⣾⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣿⠁⢀⣠⡀⠀⠀⣿⣇⣼⠁⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠤⠀⠀⠀⠀⢀⣿⡏⠀⣠⡿⠃⢠⡟⢁⡟⠀⠀⢻⣿⡁⠀⢀⡴⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣥⡾⠛⠁⢀⣿⣡⠞⠁⠀⢠⡾⣿⣧⣴⠏⠀⣿⣿⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⠙⢳⣄⣀⣼⡿⠁⠀⢀⣴⠋⢀⣿⣿⠁⠀⢸⣿⡟⠀⠰⠂⠀⠀⢰⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠙⠿⣷⣦⣤⠆⣸⣿⠇⠀⠀⣿⠁⢸⣧⣀⣴⠟⠁⣠⡿⢻⣿⠀⠀⣼⣿⡇⠀⠀⠀⠀⢠⡿⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣿⠀⠀⠙⢋⣁⠀⣴⠏⠀⣼⣿⠀⠀⢿⣿⡇⠀⠀⠀⣴⡿⠁⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⣿⠀⠀⣼⠇⠀⠀⢀⣾⠏⣼⠃⠀⢀⣿⡏⠀⠀⠘⣿⣧⣤⣴⣾⠟⠀⠀⠉⠴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⢻⣿⣀⣼⠟⠀⡤⣄⡼⠁⢰⡏⠀⢀⣾⠟⠀⠀⡀⠀⠈⠙⢛⣿⠃⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠛⠛⠉⠀⣼⣳⢯⣧⠀⠈⠷⠶⠟⠉⠀⠀⠀⠁⠀⠀⠀⣼⡏⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠗⠀⠀⠀⢰⡏⠀⠛⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢰⣿⠁⣠⠏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⡄⠀⠀⢀⣿⠇⠀⠀⠀⢿⠃⠀⠀⠀⠉⠀⠀⠀⢠⣾⠟⠙⢃⣿⡏⣰⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣿⠀⢠⣿⠋⠀⠀⣹⡛⢳⡻⠟⠁⠀⢰⡆⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠾⠇⠀⢀⡾⠀⢠⣿⡇⢀⣾⡏⠀⣠⣾⣻⣻⡛⠁⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣾⠇⢠⢿⣿⢀⣾⣿⠁⣴⠏⢸⣿⡟⢿⣦⣀⠀⠀⢀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⣰⣿⣠⡟⢸⣿⡾⠁⠻⠟⠁⠀⢸⣿⠁⠀⠙⠻⠿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢀⣤⣶⠿⠛⣿⠟⠀⠈⠛⠁⠀⠐⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣠⣾⠿⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠸⠂⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡙⢿⣿⣿⣿⣿⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⣿⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠉⠉⠙⠛⠏⠋⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⢠⡆⠀⣦⠰⣦⠀⢧⠈⢠⡀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢈⢠⣾⣇⣸⣿⣆⣿⣷⣾⣦⣼⠉⢦⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢚⠀⢸⡾⢋⣽⡟⠛⠋⠛⣩⣮⡝⡟⠀⠘⣾⡟⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣇⠘⢿⡗⡠⠀⠀⠹⠿⠃⡇⠀⢰⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡀⡀⠄⠀⠀⠀⠀⠀⣰⣥⠀⢸⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣇⠀⣻⡶⠛⠀⠈⣁⣼⡀⢺⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣿⣿⢻⣦⣴⣾⣿⣿⡛⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣾⣟⠲⣾⣿⣿⣿⠿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣼⣻⢰⡿⣿⠋⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣯⣿⣿⢿⣼⣿⣧⣽⣦⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢸⣿⣿⣿⣤⡯⢿⣿⣿⣷⢃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣀⣿⣿⡿⠋⣿⠏⠂⠻⣿⣿⣮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⣿⣿⠀⠀⣨⣄⠀⠀⣿⣧⠀⠀⠀⠀⠀⡆
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣱⣶⣿⣿⠀⠀⠀⠀⠠⠇⠀⢀⣤⣤⣤⡀
⠀⢰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣄⡤⢤⠴⠁⠀⣤⣿⣿⣏⣭⣵⠄
⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡃⢻⣿⣿⣿⣯⠀⠀⠀⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⠧⢞⣿⣿⣿⣿⣆⠀⠀⠈⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣩⣼⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⢿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢡⣾⠟⠉⠀⠀⠻⣜⢧⠀⠀⢀⣤⠄⠀⠀⢀⣴⡿⠋⠀⠀⢀⣴⡿⡟⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⠟⠁⠀⠀⠀⠀⠀⠹⡌⣆⠀⠘⠿⠃⠀⣠⡾⠋⠀⢀⣠⡶⠟⠁⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⠏⣀⣠⣤⣀⡀⠀⠀⠀⢻⡜⡄⠀⠀⣀⣼⣯⣄⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣴⡿⠟⠛⠛⠛⠿⣷⣄⠀⠀⠹⣿⣴⠿⢛⣛⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠸⡿
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⢷⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⡼⠱⣿⠿⠿⢿⡏⠀⠀⢠⣤⣤⣄⣀⣠⣤⣤⣤⣤⣴⣿⣤⣀
⠀⠀⠀⢸⠿⢿⣿⣿⣯⣤⣤⣬⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣿⣧⣷⣶⣶⢶⣾⠟⠃⢰⣶⣿⣿⠿⢿⣿⣿⡟⢛⣛⣛⣋⣁⣬⣿⣿⣷⣶⣶⡄
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⢛⣿⡿⠛⣫⡿⢻⡿⠟⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠛⢻⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣼⠏⢠⡟⠁⠀⠀⠀⠀⣌⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⡾⠁⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣿⣿⣶⣶⣶⣦⣤⣤⣤⣄⣰⠟⠀⠀⠀⡼⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣤⣤⣤⣶⣶⣿
⠀⠀⠀⣤⠀⠀⣿⠉⠉⠙⠛⠛⠛⣿⠋⠀⢀⠀⠀⠁⠀⣼⣥⣀⣀⣀⠀⠀⠀⠀⢹⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢻⠋⠉
⠀⠀⠀⠿⠀⠀⡏⠀⠀⠀⠀⠀⣼⢁⣠⣾⠿⠀⠀⠀⢰⠏⠙⠛⠿⠿⣿⣿⣶⣦⣼⣤⣶⣿⣿⠿⢧⠀⠀⠀⣤⡀⠀⠀⢠⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⡾⠋⠀⠀⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠋⠉⠁⠀⠀⠸⡀⠀⣼⠋⠻⡄⠀⡾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢳⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇⢰⠇⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣶⣶⣤⣤⣀⣀⠀⣿⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⣿⣿⡿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢠⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠒⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡤⠀⡠⢀⠤⠀⣀⠄⣢⠄⢀⡠⠂⡾⠁⠀⠀⣼⠁⣠⠄⠀⣠⠄⢀⡤⡰⠄⣰⣏⠤⣶⠄
⠀⠀⠀⠀⠀⠻⢷⣤⣀⠠⢴⡞⠁⢾⡤⠜⠓⠸⠧⠤⠛⠂⠻⠖⠈⠃⠀⠀⣰⠇⠐⠿⠖⠈⠻⠄⠺⠧⠄⠸⠋⠐⠮⠅
⠀⠀⠀⠀⠤⠤⠤⠤⠤⣤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢀⠤⠒⠚⠛⠓⠒⠶⢤⣄⡀⠀⢀⣾⠋
⠀⠀⠀⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠿⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠈⠉⠛⠓⠲⠶⠒⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢀⣠⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⣉⣉⠉⠻⡿⠋⣀⣀⣈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠉⠉⠻⠃⣼⣷⡄⠛⠉⣀⡀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣄⠙⠋⠠⡄⠹⣿⠁⠤⠈⠋⠀⠶⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠠⢴⣶⣦⡤⠀⠿⠠⠴⣶⠿⠖⠂⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢤⡄⠁⢠⠀⣶⠀⠖⠀⢠⠆⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠈⢿⣦⡀⠁⠻⠆⠠⣿⡆⠐⠛⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣤⡾⠿⠛⠉⡉⠛⠟⠿⣷⣄⠙⢿⣧⡀⢲⡄⢻⠀⡿⠀⣼⡿⠋⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣾⠏⢠⠄⢰⡄⠉⣰⡆⢀⡀⠙⢷⣄⣸⣷⣄⡀⠘⠀⣠⣾⠏⠀⣠⣴⣾⠿⠛⠛⠛⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⢿⣆⠘⢉⡀⢻⡷⠈⢉⣀⠈⠛⢿⣿⣿⣿⣶⣾⣿⣷⣴⣿⠿⠋⠀⣴⠂⣰⣆⠘⢁⡄⠀⠙⣿⡆
⠀⠀⠀⠀⠀⢿⣦⣈⣠⣶⣄⡁⠸⠁⠜⠀⢉⠀⠷⠄⠙⢿⣿⣿⣿⣿⠟⢉⣠⡄⠰⠖⢀⠰⣿⠃⡀⠛⠁⣸⡇⢸⡿
⠀⠀⠀⠀⢠⡿⠋⣉⠛⠿⠛⠋⣛⠀⠛⠻⠿⠿⠒⠒⠀⠈⣿⣿⣿⠁⠤⠤⣤⣤⣦⣄⣉⠀⣋⢀⣉⣤⡈⠛⢀⣼⠃
⠀⠀⠀⠀⢸⡇⢰⡿⠁⣀⠙⢀⣿⡄⠋⣠⡄⠰⠟⢁⣴⠾⣿⣿⡿⣷⡀⠰⣤⠈⠉⢉⡍⢀⡋⠙⠻⣿⠿⠓⢿⣇
⠀⠀⠀⠀⠸⣷⡈⠃⠰⠟⠀⠺⡟⠀⠶⠀⢀⣴⣾⠟⠁⠀⣿⣿⠀⠘⢿⣦⣀⠐⠛⢀⠀⣾⡇⠘⢂⠀⢶⡆⠈⣿
⠀⠀⠀⠀⠀⠙⣿⣶⣄⣈⡉⣀⣁⣀⣤⣶⣿⠟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠹⣿⣧⡀⠙⠀⢹⠓⠀⢻⡆⠸⠃⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠈⠙⢿⣶⣤⣄⡀⠓⢀⣠⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢿⠉⠀⣻⠏⠛⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣣⣠⣇⣤⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣢⠔⢊⣩⠟⡇⢸⠙⢦⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⣠⡎⠀⠀⡇⢸⠀⠀⠳⣄⡽⠆
⠀⠀⠀⠀⡰⠶⢍⡔⠛⠛⠋⣷⢺⠛⢿⣇⣀⣀⡇⢸⣀⣀⣀⣹⠁
⠀⠀⠀⡞⠁⠘⣤⣇⠀⠀⠀⣿⡞⠀⢀⣿⠁⠈⡇⢸⠈⠉⢹⡇⠀⣀⣤⣤⠴⢦⡀
⠀⠀⠀⠘⢦⡠⠮⠿⢄⣤⠼⠟⢦⣢⣞⣿⠐⠀⡇⢸⠀⠀⢸⣧⣼⠃⠀⠀⠀⠀⣧⠴⠲⢦⣀
⠀⢠⠴⠖⡾⠁⠀⠀⡼⠳⡄⠀⠈⢯⠉⠉⢻⠂⡇⢸⣤⠞⠋⠰⡂⡆⠀⠀⠉⠀⡿⢠⠃⠀⠉⢷
⠀⢸⡀⢰⡷⠺⡆⣼⠁⠀⢷⠀⠀⠘⣆⡀⣼⡉⡇⢸⠾⡄⠀⠀⠱⣽⠀⠀⠀⠀⣿⠃⠀⠀⢀⡟⠜
⠀⢆⢻⠛⠛⠛⠛⣿⠀⠀⠸⡇⢠⣄⣿⢹⡇⠈⡇⢸⠀⢹⣄⣠⠤⠼⢧⣀⡤⠼⠓⢦⣀⣴⣏⣀⣀⣀⡀
⠀⠀⢸⡆⠀⠀⠀⣿⠀⠀⠀⣷⡏⠈⠛⢸⡉⠀⣧⡟⠉⠉⡽⠁⠀⢠⠞⠁⠑⣄⠀⠀⠹⣆⣀⣀⣀⣀⡇
⠀⠀⠸⡇⠀⡀⠀⣿⠀⠀⠀⡏⠀⠀⠀⢸⠀⠀⡏⢳⡶⢶⡷⠲⣄⡞⠛⠛⠛⠛⣆⢠⢤⣸⡏⠉⢹⠋
⠀⠀⠀⣇⠀⠀⠀⢻⠀⠀⠀⡇⡄⠀⠂⣸⠐⠀⡇⣸⠸⡄⠀⠀⠈⠁⠀⠁⠀⠀⠹⠋⢀⢈⣋⣀⡟
⠀⠀⠀⠹⠤⠤⠤⢿⣤⣤⣤⠷⠶⠶⠖⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠁⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⣿⣿⡟⠉⣼⣿⠿⠛⠛⠛⠿⠿⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡒⠻⣿⣷⣄⠈⠛⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠶⢂⠞⢻⣀⡤⠀⠙⢯⣽⣶⢤⡟⠀⠀⠀⠀⢒⡫⠽⠟⠂⠀⠀⢠⡄
⠀⠀⠺⣷⣀⡴⣫⠴⢋⠁⠀⠀⠀⠀⠻⣿⣷⣤⡀⠀⡆⠀⠀⠀⠀⠀⣀⣠⣤⡾⠁
⣾⠁⠀⢿⡟⠋⣁⣤⠿⣧⠀⠀⠀⠀⠀⠈⠛⠿⠿⠗⠘⣶⣶⣶⡿⠿⠛⠛⠉⠀⠀⠀⠒⠒⠒⠚⠛⠛⠓⠢⣄
⠻⣦⡴⠈⠳⠆⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠐⢶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢿⣿⣷⣄⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠿⠛⢉⣀⣴⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠸⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠿⠛⠻⣿⣿⣿⣿⣿⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣠⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠿⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⠴⠚⠉⠀⣼⡄⠀⠀⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡁⠀⣀⠀⠁⠀⠀⠀⠀⣼⠟⠛⠛⠉⠀⠀⠀⠀⢰⣿⣷⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⠋⠉⠉⠀⡴⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢀⣼⡇⠀⠔⠁⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠿⠛⠁⢀⣤⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢀⣀⣸⣿⡿⠃⠀⢀⡴⠂⠀⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠿⢿⠟⠁⠀⠀⢀⡞⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠕⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢤⣶⢰⣿⠆⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠶⠋⠁⢤⣾⣿⢼⣿⣧⣤⡖⠀⠀⢤⠀⠀⠀⠀⣠⡆⣀⣧⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⡀⠀⣤⣠⣴⣫⣿⣿⣏⢻⡍⣠⣀⣤⣤⠈⠀⠀⠀⠘⢿⣿⣿⣿⣷⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⢻⣷⡀⣼⣼⣿⣿⣿⣿⣿⣆⠀⠙⣉⣹⠁⠀⠀⣠⣾⣦⡀⠉⠀⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠷⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣘⣿⣿⡆⢀⣠⣿⡿⠛⢡⡎⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠿⠿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣾⣿⡟⠁⣰⣸⣿⣠⡾⠀⠀⠃⠀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠰⡄⢀⠔⠀⠂⠊⠉⠀⠀⠀⠉⢻⣿⣿⠟⣉⡘⣷⣽⣿⡟⠀⢰⣟⡘⣿⡛⠃⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠂⠄⠀⢀⠀⠀⠸⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣽⣿⠏⠀⠀⡴⠺⣷⡼⣿⡿⡄⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠘⠀⠉⢄⡀⣴⣿⣿⣦⣾⣿⡿⢋⣩⣵⣿⡿⠉⠀⠀⠀⢀⢁⣿⣿⣹⡷⠄⢀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡀⠚⣿⣿⣿⣿⡋⡩⠁⣠⡿⣻⠿⡻⠁⠀⠀⠈⢠⡟⠼⢿⣽⢷⡑⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⡟⢡⠞⠁⠀⠀⡰⠀⠀⠀⠀⠀⣾⠁⣸⣿⣾⣿⣿⣋⡗⢶⣦⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠛⠋⠀⣠⡎⠀⠀⠀⠀⠀⡸⠀⣸⣿⣿⢏⣼⣏⡐⠁⠀⠀⠀⠀⢇⠀⠀⠀⠀⢰⡏⠀⣿⣿⣿⣿⡏⠉⠑⠠⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⣿⠇⠀⠀⠀⠀⢠⣧⣴⣿⣯⢱⣿⠋⠁⠙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣾⡇⢸⣿⣿⣿⣿⠃⠀⠀⠴⠿⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠙⠀⢀⢀⠀⣾⣿⣿⠀⠀⠈⡼⠃⠀⠀⠀⠀⢀⠀⠀⢀⠈⡀⠀⠀⣸⣿⠃⣾⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣄⡀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣇⠸⢿⣿⣿⣿⢶⣤⡃⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠁⠀⣰⣿⡏⣴⣿⣿⣿⣿⣿⡆
⠀⢀⣠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡈⣿⣿⣦⣌⢻⣏⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣦⣤⣈⣉⣉⣩⣩⣿⣿⣿⡇
⠀⢏⣩⣿⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣷⡄⠀⣉⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⡀⠀⠀⢠⠘⣿⣿⣿⣿⣿⣿⣄⣿⣿⣸⠀⢸⡆⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⡿⠋⣸⣿⣿⣿⣿
⠀⠀⠀⠴⣿⣿⣿⣿⠿⠿⠿⠿⠛⠁⠀⠀⠘⣧⠀⠀⠘⣷⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣦⣀⣄⠀⠁⠀⢀⣿⣿⣿⠏⢀⣾⠿⢿⣿⣿⡇
⠀⠀⠀⢬⠽⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⠇⠀⠙⢿⣿⣷⣿⣿⠀⠀⠻⠿⢻⡏⢸⡆⣄⣼⣿⣿⣿⠀⠞⠁⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠛⢿⣿⢠⣶⠀⠀⠋⢠⣿⣿⣿⣿⡿⣀⠀⠀⢀⣴⣿⣿⣿⠿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣔⣾⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⡄⣿⠃⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣶⡀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢻⣷⡆⠀⠀⠀⠰⡄⢻⣿⡿⢿⣿⠿⠛⢿⣿⣿⣃⠀⣄⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠹⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⣽⣶⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠙⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⢀⠀⠀⠀⠀⠘⢿⣿⡼⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⣶⠋⠀⠀⣠⡄⠀⠠⠻⣧⣷⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠀⠐⠾⣿⣿⣠⢀⣄⠉⣉⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢠⠈⠉⠙⢾⡿⠿⠿⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⠠⠂⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢀⣀⣤⣀⠀⠀⢀⣠⣤⣀⠀⢰⡄⠀⢀⡴⠛⣉⠉⠁⠈⠉⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠉⠳⡶⠋⠁⠀⠙⣧⠀⠟⣠⣿⢁⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢰⠏⢹⣿⠀⠀⠀⠛⢦⣄⠀⠀⠀⢰⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⣿⠀⠀⣿⣄⠀⠀⢠⣤⠍⠛⠶⢦⣼⠇⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⠶⠚⢛⣿⣃⣤⡄⠀⠈⠳⠝⠳⣤⣠⡾⠋⠀⠀⠀⠀⣿⠀⠀⢹⡅⠀⠀⠀⠀⣀⠀⢠⡴⠋⠀⠀⠀⠈⣻⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣡⡶⠋⠁⠀⠀⠈⠉⠉⣸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⢷⣄⠀⠛⠛⠋⣠⣾⣁⣠⣤⣤⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢈⡀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠞⠋⢉⣛⡆⠀⢸⣏⣀⣨⡟⢸⠇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢈⣿⠛⠲⠖⠚⠛⠉⢘⡇⠀⠀⠀⢀⡴⢶⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⠞⠻⢿⣍⠷⢤⡾⢋⣽⠟⢠⣾⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡞⠁⣠⡴⠀⠀⠈⠛⠻⡇⠀⠀⢠⡟⠀⠘⣧⠀⠀⠀⠀⠛⠀⠀⠀⠸⣇⠀⠀⠀⣹⣇⠀⣴⠟⠁⣴⠏⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠉⠀⠀⠀⠀⢀⡀⣿⠀⠀⢸⡇⠀⠀⢻⡐⠂⠀⣠⠶⣄⠀⢠⡄⠙⢦⣀⣴⡏⠙⠟⠁⣠⡞⠁⣴⡾⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣧⠀⠀⠙⠶⠶⠛⣠⡟⠀⠀⠈⠻⣄⣀⢸⣇⠀⣴⠛⠾⠻⡆⠀⠀⠀⠀⣿⠋⠻⢦⣤⡴⠋⠀⢸⡏⢻⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡤⠤⣴⠞⠋⠀⠀⠀⠀⠀⠈⠉⣛⣿⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⡟⠻⢦⣼⣷⣤⣄⡀⠀⢀⣠⣶⡟⢉⣙⠳⠶⠶⣤⣀⣴⠟⠀⠀⠀⣿⡛⡛⣻⣟⣻⣟⡛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠛⠷⢦⣿⣿⣧⡼⠿⠞⠋⠁⠀⣹⡟⠉⠛⠶⠶⠛⠉⠁⠀⠀⠀⣴⠏⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠛⢳⡶⠖⢶⠟⠻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⢄⡾⠛⠻⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠻⡇⢸⡇⠀⠈⣷⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⠁⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧⣤⣤⣿⡀⣸⡏⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⣸⡃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⡟⠀⠀⠀⠈⠻⠛⠻⠶⣼⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⣿⠋⠉⢹⡆⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⣶⣀⡀⠀⠀⠀⠀⠘⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠀⠀⠀⢸⠏⠉⠉⠛⠛⣶⠀⠀⢹⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣤⠿
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿
⠀⠀⠀⢸⠏⠉⢹⡟⠛⠛⠉⠉⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣆⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻
⠀⠀⠀⢸⠀⠀⣸⣧⣤⣤⣤⣤⠴⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⢻⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣼⡇
⠀⠀⠀⢸⣦⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣤⣤⣬⣽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⢀⣴⣿⣿⣷⡄⠀⠀⠀⢀⢸⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣿⢸⡿⠿⠿⠷⠙⠿⣿⣿⣿⣿⡟⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣿⣿⣧⢰⣿⣿⣿⡿⠷⣮⣝⣛⠿⠇⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⢰⣾⣿⣿⣿⣿⣿⣿⡿⢟⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⡾⣫⣵⣿⣿⣿⣧⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣎⢿⣿⣯⣛⣫⣾⣿⢣⣿⡿⢿⡿⠿⠛
⠀⠀⠀⠀⠀⣴⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠀⠙⠛⣿⣿⡏⠀⠀⠀⣰⡇⣼⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣍⢛⣛⡿⢟⣛⣫⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣄⠿⣿⢇⣤⡄⣠⣿⠇⣿⣷
⠀⠀⠀⠀⠀⠈⢿⡟⣱⣿⣿⣿⣿⣿⣟⣛⡇⠀⠀⠀⠀⣠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡝⣿⣶⡿⣫⣾⡿⣣⣶⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣟⣴⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⣤⣙⢋⣼⣿⢏⣼⣿⠟⣼⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣟⠻⠿⠿⣩⣿⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠾⢂⠀⠀⠀⠀⠀⠈⢋⣜⠿⣿⣿⡿⣡⣿⡿⠀⢻⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠿⠿⠿⠂⠀⠀⠀⠀⠀⢻⣿⣷⣮⣭⣾⣿⣿⡇⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡘⢿⣷⣍⢱⡄⣤⣄⠀⠀⢀⣤⣤⡻⠟⠛⠷⣶⣶⡮⠛⠋⠀⠀⠀⠀⠀⣘⡛⢿⣃⣿⣓⣟⣻⣃
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣥⣤⣤⣭⣵⣶⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣝⡻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣌⢿⣿⡿⠿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡆⣶⣶⡰⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⡟⣼⣿⣷⣹⡟
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⡇⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣆⣿⢗⣿⣿⣷⢻⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡅⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢘⣟⣛⣿⠸⣿⠇⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⢃⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠛⣛⣛⣛⣛⣋⣾⣿⣿⣿⣷⣴⣶⣬⣝⡋⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣿⡦⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣷⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣿⣿⣿⠃⠀⠀⠀⠈⠉⢻⣿⣿⡇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢱⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣴⣶⣤⢀⣤⣤⣤⣶⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⣿⣿⣿⠸⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⣿⣿⣿⠿⣿⣿⠊⣿⡁⠀⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣸⣿⣿⣿⣿⣇⣀⣸⣿⡆⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⣸⣿⣿⣿⣿⣇⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣄⣸⡒⠃⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣻⡽⣿⠁⠀⠀⣠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠱⣿⣄⡻⠦⣀⡴⣻⣿⣿⡿⠿⢿⣿⡄
⠀⠀⠀⣀⣠⣴⣶⣦⣄⠀⠀⠀⠀⡠⠴⢶⣶⣄⣀⣀⣀⣀⣀⣠⢌⠋⢁⣤⣶⣷⣿⣿⣿⠇
⠀⠀⠘⢿⣿⡿⠿⠿⢿⣷⣤⣤⡾⠃⠀⠈⠛⠿⡿⣩⢙⣵⡿⠛⣤⣴⡿⡿⠁⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⣦⣶⣾⣉⡆⠘⠿⠗⢺⣿⣿⣿⣿⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⣿⡿⣿⣿⣿⡗⣦⣤⣼⣿⡿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠃⣿⡿⠋⠃⠽⢿⣿⡿⣽⣦⣨⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠋⢀⣾⡟⠁⠀⠀⠀⠀⠀⣾⡿⠁⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠙⠲⢤⣀⠀⠀⣸⠏⠀⠀⠀⣈⣿⠇
⠀⠀⠀⠀⠉⠁⠲⠀⠄⠀⠀⠀⠀⠀⠘⣷⠀⢀⣀⣀⣉⣶⠞⠉⠘⠻⠶⠋⠁
⠀⠀⠀⠀⠀⠀⢂⢢⡄⢠⠀⠀⠑⠀⢈⠀⠸⢭⠉⠉⠉⠀⠀⠀⢀⣰⣄⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠙⠻⠿⠛⠛⠻⠿⠟⠛⠒⠛⠛⠘⠒⠂⠋⠑⠛⠻⠉⠋⠛⠙⠃⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⣀⠀⣀⣀
⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠉⠉⠉⠈⠉
⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠁⢀⠀⠁
⠀⠂⠐⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠂⠰
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠒⠀⠄⠰⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠁⠈⠁⡂⠀⠁⠒⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢀⡁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠈⡁⠉⢁⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣄⣀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠒⠒⡾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣰⣤⠀⡟⠆⠀⠆⠀⠀⠀⠀⢀⡶⠔⠤⢼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢽⠟⠏⠀⡀⠀⢀⠀⠀⠀⠀⠐⠤⢄⣀⠀⡀⠀⠐⠄
⠀⠀⠀⠀⠀⣀⠀⢤⠄⠂⠄⠀⣀⠀⠉⢒⣤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡘⠛⠈⠀⠀⠀⠈⡄
⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠂⢠⠀⠀⢀⠀⠀⠀⢄⠀⠀⣤⡄⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣒⠶⠒⠀⠀⠀⡀⠀⠀⡀⠀⠀⣀⣦⣤⠖⠉⠉⠀⠀⠀⠀⠀⢹⠄⠀⡀⠠⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠈⠄⡄⠚⣇⡀⠀⠈⠉⠙⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⣕⣈⣄⣀⠤⠤⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢒⠀⠀⠈⠀⠀⠀⠀⠀⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠤⠼⠶⠖
⠀⠀⠠⠄⢀
⠀⠀⠈⠀⠀⠀⠠⠀⢀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐
⠀⠐⠀⢀⠀⠀⡀⠤⠀⠀⡀⠀⠀⢀⠀⣠⠀⠀⠀⡀⠀⠀⡀⠀⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠁⠀⠁⠋⠀⠁⠀⠁⠀⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⢀⡄⠠⡀⠀⢀⠀⠀⠀⠀⠒⡀⠀⠀⡀⢠⠀⠀⠀⢀⠀⠀⣤⢀⡄⡀⡠⠀⠀⠀⠐⠀⠀⣴⠠⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⢶⣆⠀⠀⠀⣀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢘⠁⠀⠈⠏⠀⠀⠀⠉⠁⠉⠁⠀⣸⣷⣤⣷⣾⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠘⡛⣿⣿⠀⠰⣴⣷⣶⣾⡇⣠⠠⣶⣿⣿⡿⠟⠛⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠊⣩⣿⣄⣰⣛⠋⠀⣹⢧⣭⣇⠉⠉⠉⠀⠀⣠⡀⠘⠻⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠁⠋⠈⢉⣿⡿⠿⠃⢸⠃⠈⠃⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡦⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⠿⠲⠶⣶⣶⣾⡿⠃⣤⣠⡿⠋⠀⠀⣀⠀⠀⡼⠀⠀⠀⢀⠐⢤⣀⠀⣈⣷⣄⡀⠀⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠁⢀⠤⢺⡿⢁⣠⣿⠟⣡⣴⣾⣿⣿⣏⣴⠇⠀⠀⠀⣾⣠⣾⡿⠋⠉⠉⠛⢇⠘⠛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠠⣦⠀⠀⢠⠀⠀⠀⣠⠟⣀⣴⠟⠁⣼⣿⣿⡿⠃⠸⣿⡏⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠰⡿⠀⢰⣧⣖⡀⢀⢡⣴⣟⠁⠀⠸⢻⡇⠈⠁⠀⠀⢿⣧⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⠀⠀⠀⣽⡄⠋⠉⠻⣄⡀⠙⠿⠛⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠙⠷⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢋⡿⢷
⠠⣄⣀⣀⠀⢀⣠⣴⣿⣿⣧⠈⠆⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠙⡿⠚⠁⡿
⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡀⠀⠀⢸⣿⣷⣄⡀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠸⠿⠟⠛⡿⠟⠁⠀⠀⠀⠀⠹⣦⡄⢸⡏⠻⣿⣿⣦⣄⡈⢿⣿⣧⡔⠀⣀⠀⡀⠀⠀⠐⠢⣿⣾⣁⣠⠆⠀⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠌⡇⠀⠈⠻⣿⣿⣿⣿⡟⢿⣷⠾⡁⠀⠙⢶⡲⠤⠶⢿⣿⣿⠁⠀⠀⠲⣤⣌⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢇⠀⠀⠀⠘⣿⣿⣿⠀⠘⠿⢦⣀⠀⠀⢄⠙⢄⠀⠀⠙⠿⠿⠖⠚⠋⠉⠉⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣠⠘⣦⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠈⠉⠓⠾⣷⣌⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣄⡜⢷⣄⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣍⠙⠮⠻⣿⣿⠀⠀⢠⣤⣴⣦⣀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡆⠀⠙⢿⣇⠀⠀⠙⠿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣎⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣦⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣟⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣷⡀
⠀⠀⢠⠟⠋⠉⠉⠛⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢋⣩⠤⠶⢦⡈⢳⡀
⠀⠀⡟⠀⢰⡏⠉⠉⠛⠲⠦⢤⣍⡓⠦⣄⡀⠀⠀⠀⠀⣀⡤⠶⠿⠷⠶⠦⣄⠀⠀⠀⢀⣠⠴⠚⣉⣤⠴⠚⠉⠀⠀⠀⣠⠇⠀⡇
⠀⠀⢳⣄⠀⠙⠲⠤⣄⣀⣀⣀⣀⣉⣙⣛⡻⣶⣤⣀⢰⡏⠀⠀⠀⠀⠀⠀⢸⣇⣠⣶⣿⢗⡚⣉⣁⣀⣀⣀⣀⣤⠴⠚⣁⣠⣞⣁⣀⣀⣀
⡟⠛⠛⠉⠛⠲⠦⠤⢤⣤⣭⣭⠭⠭⢭⡿⠛⠛⠋⠉⢉⣻⣶⠦⠤⠤⠶⠖⠛⢿⣉⠛⠛⠛⠛⠛⠛⠿⠿⠟⠗⣶⠛⠋⠉⣻⠉⢻⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠋⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠋⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣿⠀⠀⢀⣿⡀⣸⣀⣀⣀
⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠉⣿⠉⠉⠉⣿⠉⢻⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⣄⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠓⢤⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣸⣀⣠⠤
⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣇⣀⢀⣀⣀⡀⣸⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣿⣀⣤⠴⠟⠛⠉⠁

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠂⠀⠀⠀⠰⠷⠁⠀⠀⠀⡀⠀⠀⠀⢰⠲⡀
⠰⠀⠀⠀⣴⢠⡤⣲⡂⡀⠀⠀⠀⠘⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠃⠁
⠀⠀⠀⠀⠀⠟⠉⠉⠀⠐⡀⠀⠀⠀⠠⠀⠀⠀⢀⡂⠀⠀⠀⠉⠀⠀⠀⡘⠁⠀⠀⠀⠀⠂
⢠⠰⠀⠀⠈⢣⡤⠄⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠘⢧⠣⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢣⠱⡀⠀⠀⠉⠁⠀⠀⠘⢎⠀⠄⢀⣀⣠⡴⠃⠈⠳⢌⣒⣐⠰⠛⠁⠰⠬⠝⣒⣒⣒⣲⠶⠊
⠀⠀⠳⣈⠢⢄⠀⢀⣀⢠⠃⠀⠉⠐⠒⠒⡏⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠈⠑⠂⡠⠄⠒⠁⠀⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⡲⢄⣠⠎⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⡠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠆⠴⠀⠀⢠⡷⠃⠀⠀⣠⠖⠀⡖⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠊⠀⠀⠸⢡⠃⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠑⠜⠐⠀⠀⢀⠞⠂⠀⠛⠀⠀⢀⠃⠋⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⠀⠀⠀⢀⣠⠊⢆⡑⠄⣀⡀⣀⣀⠔⡏⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠒⢪⠥⢶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠧⢀⣉⣁⡉⠴⠃⠀⠀⠈⠒⠀⠂⠚⠁⠰⠦⠏⡂⠒⠃⠀⣀⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⣀⣒⣂⠭⠕⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠢⠄⠉⠁⢧⠑⢄⢀⣀