Sensory system Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢓⣤⣴⣶⣶⣶⣮⣭⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⠉⣿⣴⣿⣿⡿⠉⠛⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠁⡄⠀⠀⠁⠀⣧⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠻⠿⠿⠵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠓⠲⠤⠀⠉⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠄⠇⠈⠨⠀⠀⠑⠄⠅⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⡴⣩⠶⠶⣬⠳⡄
⠀⠀⠀⠀⠀⠀⡀⠐⠀⣀⠀⠄⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⡇⠀⠀⠈⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣭⡖⢮⠶⣮⣕⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠈⢠⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⠋⠉⠛⠋⠉⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠗⠂⠀⠔⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⡇⠀⠀⠀⠀⠀⠀⡟⡜⠀⠀⠀⠀⠀⠀⠀⣖⢠⡀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⣠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢀⠠⠀⠀⠀⠤⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⠔⣀⢀⣤⣤⠀⣀⠂⠄⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠒⡨⠄⠀⠉⢸⣇⣼⠇⠁⠀⣦⠀⠦⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠉⠁⠀⠉⠐⠰⢦⡌⠥⠴⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠘⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠲⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠉⠑⠦⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⠤⠤⠤⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠴⠚⠉⠀⠀⠀⠀⠈⠙⠲⢄
⠀⣠⠊⠀⠀⠀⡤⠒⠒⠒⠠⣄⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠏⢡⠀⠀⠀⠀⠀⠀⠀⠑⣄
⢰⠃⠀⠀⠀⡎⢰⢋⠉⠉⠳⡌⡆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡂⢘⣀⡀⠀⠀⠀⠀⠀⠀⠘
⡆⠀⠀⠀⠀⠀⢹⠀⠑⠀⢀⠁⠃⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡄⠈⠀⠉⠉⠒⢲
⡇⠀⠀⠀⠀⠀⠀⠀⡸⢠⠃⡔⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠠⠰⡀⣇⢰⠃⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⢱⡀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⠇⢹⡆⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀
⠀⠳⡀⠀⠀⠀⠣⣀⣀⡠⠃⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠢⡀⠀⠀⠀⢀⡜⠀⠀⢀⠞
⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⢀⡤⢤⡀⡤⠤⢠⡄⠀⡄⣠⠤⡄⢠⠤⠀⡤⠤⡄⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠈⠉⠉⠉⠁⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⢤⡀⡷⠦⢸⠙⡄⡇⠻⠦⡄⢸⠴⠀⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠁⠧⠤⠸⠀⠹⠇⠷⠤⠇⠸⠤⠄⠦⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⡠⠒⢢⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣀⠤⠄⠒⠤⠴⠒⠤⠄⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⠁⠀⠀⡆⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡉⡏⠉⠉⢙⠋⠉⠉⠻⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡌⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢧⠀⠀⠘⠀⠀⠀⡖⠁⠀⠀⠀⡃⠀⠉⠉⠉⠉⠉⠁⠀⡇⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡐⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢎⠀⡀⠀⠀⠀⢀⠀⢸⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⠢⠆⠠⠊⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠉⠙⠢⣀⠴⠋⠉⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣁⣀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠟⠛⠻⠶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⢀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⢀⣤⠾⠋⣹⠟⠁⠀⣀⣤⣄⡀⠈⠻⣯⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠀⠀⠈⠙⣿⠋⠁⠀⠀⠉⠙⠻⢶⣄
⠀⠀⢀⣴⠟⠁⠀⢰⡏⠀⢠⡟⠉⠀⠉⢻⡆⠀⢹⡇⠀⠈⠙⢧⡄⠀⢀⣀⣾⠋⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⡙⣷⣀⡀
⠀⠀⠙⢧⣄⠀⠀⢸⡇⠀⠸⣇⠀⠀⠀⣸⡇⠀⢸⡇⠀⠀⢀⣼⠟⠀⠈⢹⡟⠙⠋⠉⣿⠋⠛⠙⠛⠷⣶⠾⠛⠙⠋⠙⣿⠋⠛⠋⢻⡏⠁
⠀⠀⠀⠀⠙⢷⣤⡀⢿⣄⠀⠙⠻⠶⠞⠋⠀⢠⡾⢁⣠⡶⠋⠁⠀⠀⠀⠀⠻⣆⡀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣷⣤⣄⣀⣠⣤⣶⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣀⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢦⣾⠋⠙⣷⡶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣴⠞⠋⠁⣀⣀⣀⡀⠉⠻⢦⡀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⢀⣾⠃⢀⣴⠟⠉⠉⠉⠙⢷⣄⠈⢻⡄⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⢸⡟⠛⢷⡄⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⢸⡇⠀⣾⡁⠀⠀⠀⠀⠀⠀⢻⡆⠈⣿⠀⠀⢀⣀⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠈⠁⠀⠉⠙⠳⣦⡀⠀⠀⠀⣸⠇⠀⣿⠀⢰⡟⠉⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣰⡏⠀⣸⠇⠀⢸⡇⠀⣿⠀⠀⣿⠀⠀⡿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢰⡟⠀⣰⠏⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣀⣠⡾⠁⢠⡟⠀⣰⡟⠀⠀⠀⢸⡇⠀⠙⠛⠶⢦⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠴⣤⣄
⠀⠀⠀⠀⠀⠛⠉⠁⠀⢠⡿⠀⢰⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣼⠃⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠁⢠⡟⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣠⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠁⠀⣀⣴⠟⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⣠⡾⠃
⠀⠀⠀⠀⠙⠻⠶⠶⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠾⠛⠋⠙⠛⠛⠛⠋

⠀⠀⣀⠀⢀⠀⡀
⣿⣏⢩⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠠⠀⠀⡄⠀⢀⡄⢠⠀⣄⡀⢠⠀⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠛⠀⠓⠀⠐⠃⠘⠂⠃⠃⠘⠃⠛⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠞⠋⠑⢶⣶⢶⣖⠋⠙⠲⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⣀⣀⣠⣠⣄⣀
⠀⠀⠀⠀⠀⡠⠞⠁⣀⡦⠴⢶⣿⣯⣿⣿⣯⣿⠴⠦⢤⡀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣳⣿⡿⠟⠛⠛⢿⣿⣿⣿⣶⣄⣀⣢⡄
⠀⠀⠀⠀⢰⢣⣶⣾⣿⡿⠿⠷⣶⣶⣶⣶⣶⣶⠾⠿⢿⣿⣷⣶⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⠟⠉⠀⣾⣀⣀⣴⣿⣿⣿⣿⡟⠿⣿⣿⣦
⠀⠀⠀⠀⢨⣾⣿⣿⡇⠀⠀⠀⠀⠃⢾⡞⠀⠀⠀⠀⠀⢿⣿⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⢠⣟⣿⠙⠃
⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⠼⠛⠻⠷⢦⣤⣀⡙⠿⣿⣿⡿⠟⢁⡴⣫⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠿⠧⢤⢤⢤⠤⠶⠚⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠀⢀⣠⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢀⣴⣿⣿⠋⣿⣿⡌⠀⠀⠀⠀⠀⠀⣀⢈⡄⢠⠆⠸⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢠⣾⠿⠿⣿⠂⣸⡃⠇⢰⠀⠀⠀⠀⠀⠁⠈⠁⠠⠀⠀⡄⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⡼⣀⣤⣤⣤⠿⠟⠶⢆⠙⠂⠀⠃⠁⠘⠀⠀⠀⠀⠀⠇⢺⡄⠰⠆⠚⠳⠤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠄⠈⢻⣟⠙⠶⢔⠛⠈⢠⠀⠀⡄⢀⠁⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣤⣶⠇⠀⠀⡘⠀⣸⠃⠂⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠙⠣⡄⠠⠀⣠⡴⠡⠂⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠈⠀⠠⠂⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⢀⠀⠀⠀⠀⣀⣿⠋⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠈⣆⣤⣴⣶⣄⠀⣤⣴⣿⣿⣶⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣠⡿⣤⣦⣤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠻⣛⣿⡿⢿⣤⣾⡿⠿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠁⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⡾⠋⠻⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⢀⣽⠶⠶⠶⢦⡀⠀⠀⠻⠶⠾⠃⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣤⠶⣤⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠛⠻⣇⣀⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠋⠻⣦⣀⣴⠛⠙⣷⢠⡟⠉⢻⡄⠀⠀⠀⣾⠋⠛⣆⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠻⠦⠾⠃⠀⠙⠶⠴⠏⠈⠳⣤⠞⠁⠀⠀⠀⠻⢦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠞⠳⣦⣀⢀⣀⣀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠻⣤⣤⠟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠀⢹⡷⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠂⠀⠄⠆⠰⠰⠐⠆⠆⠆⠂⠀⠀⠀⠀⢰⠄⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠢⣤⣴⢛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⣾⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠁⢈⠀⠀⠀⠀⠀⠀⢠⠉⢀⠀⠀⣠⣟⢲
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠃⠉⠐⣇⠀⠀⠀⠀⠀⠿⡿⠧⣼⠿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⡀⠀⣈⣉⣵⣿⣊⣉⣁⠂⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⣤⡴⣶⣤⣄⠀⢀
⠀⠀⠀⢰⣶⣿⠗⠳⠶⣶⠶⠶⠺⢿⣶⣆⠀⠀⠀⠱⠞⠷⠶⠆⠀⠀⠀⢰⡿⠋⢰⣤⣼⣿⣿⡟⢿⣦⡀
⠀⠀⠀⠙⢿⣧⠀⠀⠀⠉⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⢤⣈⠻⠿⡿⠟⢠⣿⠃
⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲⡀⠀⠀⡇⠀⠀⡇⠀⠀⠀⡠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢳⠀⠀⢷⠀⠀⢯⠀⠀⢰⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠉⠙⣿⣿⣿⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⡆⠀⢸⠀⠀⢸⠀⠀⡜⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢈⣿⠉⠀⠀⠀⠈⠻⢼⠿⠋⠀⠀⠘⡆
⠀⠀⢀⠤⢤⠀⠀⠀⠈⠀⠀⢹⡀⢸⠀⠀⢸⠀⢰⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⡴⠤⠂
⠀⠀⠈⣆⠀⠡⡀⠀⠀⠇⠀⠀⡇⢸⠀⠀⢸⠀⡟⠂⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠛⠋⠉⠉⠁
⠀⠀⠀⠘⢦⠀⠘⢀⠀⠘⡀⠀⠹⠼⠀⠀⠸⠜⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠈⢧⡀⠀⠑⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⠄⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣠⠤⠤⢄⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡜⠁⠀⠀⡰⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠃⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡀⠀⢀⠄⠀⠀⠀⠀⢧⡀⠀⢀⡤⠂⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠐⠀⠊⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⡀⠀⠔⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠈⠑⠢⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠀⠀⠀⠀⢀⠞⠀⣠⣖⠉⢉⣏⠉⢒⣄⡀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⣠⣼⠿⠟⠻⠿⠿⠿⠿⠿⣿⣦⡀⠀⠀⠀⠈⠳⣄⠀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠐⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉⠀⠀⠀⠀⠀⠈⠣⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠑⠒⠂⠀⠆⠀⠒⠊⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠤⠀⠸⠠⠤⠤⠀⠀⠴⠀⠢⠠⠄⠠


⠀⠀⠀⢀⠤⣄
⠀⢰⠁⢹⠀⢸⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⢉⣁⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠐⡀
⡤⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⢋⣀⣀⠀⠉⢆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢇
⡇⢸⠀⢸⠀⢸⠀⢸⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠉⡆⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸
⡇⠘⠀⠈⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡰⢃⠇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢇
⡇⠀⠀⠀⠀⠀⡠⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠁⠎⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠳
⢇⠀⠀⠀⠀⠀⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠈⠢⣀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠠⢄⡀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠉⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠂⠤⠔⠒⠤⢄⡀
⠀⠀⠀⠀⢀⡠⠒⢹⢁⠖⠉⠉⢢⠈⡆⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠠⣶⠮⢥⠤⠤⠤⠤⣀⠤⠤⠤⠤⡬⠭⣶⠆
⠀⠀⠀⠠⡋⠀⠀⢸⡘⢄⠀⢀⡸⢀⠇⠀⠀⠀⣀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠈⠑⠢⢄⣑⠢⠭⠭⠔⠋⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⢀⠉⠙⢦⡀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠉⠙⠦⡀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠈⠙⠲⣄
⠀⠀⠀⢠⠎⠀⠀⠀⠀⡅⠀⠑⡀⠀⠹⡄⠀⢀⡞⠀⠀⣀⠤⣀⡠⢄⡀⠀⠙⡆⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢻⠀⢼⠀⠀⢊⣡⣤⣀⣠⢄⡘⠀⠀⢹⠀⢸⠃⠀⢠⠂⠀⠀⠀⠀⠰⡀⠀⢸⡆
⠀⠀⠀⢻⠀⠀⠀⢀⠀⡀⠆⢀⠂⠀⠀⡾⠀⢸⡀⠈⠀⠀⠀⠉⠁⠀⠈⠉⠀⣼⠀⠸⡆⠀⠘⠀⠃⠀⠀⠉⠐⠁⠀⢸⠃
⠀⠀⠀⠈⢧⡀⠀⠀⠐⢀⠠⠃⠀⢀⡴⠃⠀⠀⢣⡀⠀⠈⠉⠉⠉⠉⠀⠀⡴⠃⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⠢⢤⣉⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣠⠴⠊⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣠⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠁⠀⠀⠈⠉⠓⢄⠀⠀⠀⠀⠀⣠⠖⠩⠍⠀⠀⠉⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⠀⣤⣀⣤⣤⣀⠀⠈⢳⡀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⣾⣿⠟⡿⢩⡍⢻⢧⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠹⢦⣘⣮⡥⠊⠀⠁⢀⡇⠀⢸⡄⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠳⡀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⡁⠄⢐⢰⡄⠄⠀⡂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⣿⣷⠀⢎⠠⢰⠄⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⡞⠛⢳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⣀⢰⡏⠉⡇⠀⢸⡋⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠉⠙⠲⣄
⣾⠉⢹⡇⠀⡇⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣴⠞⠛⠛⠛⠶⡄⠈⢷
⣿⠀⢸⡇⠀⡇⠀⢸⡇⠀⣇⡴⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⣇⡀⠀⠀⠀⠀⢹⠀⢸
⣟⠀⢸⠇⠀⡇⠀⢸⠇⠀⣿⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⠀⠀⠀⠈⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠒⠚⠃⠀⠀⠀⠀⣴⠃
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡃⢀⡤⠶⣤⡀⠀⠀⣀⡴⠦⣄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠈⢳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣤⠴⠈⠓⠶⠖⠋⠐⢦⣬⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠤⠶⠛⠁
⠀⠀⠉⠓⠶⠶⠶⠶⠞⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠦⣤⣀⢀⣤⠴⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡿⠛⠉⠀⠀⠀⠉⠻⣿⠶⣄
⠀⠀⠀⢀⣠⣴⠚⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣾⠁⠀⠀⣠⣤⠀⠀⠀⠈⣧⠈⠛⢦⡀
⠀⠀⠀⠈⠻⣍⡉⠉⠛⠛⠛⠛⠒⠒⠒⠚⠛⠛⠛⠋⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣇⠀⠀⠸⣿⣿⣷⠆⠀⠀⣽⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⢻⡄⠀⠀⠉⠉⠁⠀⠀⢠⠏⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣦⣄⣀⣀⣀⣠⣶⠿⠚⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠉⠛⠲⣄⡀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠙⠓⢦⡀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠉⠛⠲⣄⡀
⠀⠀⠀⠀⣠⠎⠀⠀⠀⠤⠠⢍⢢⠀⠀⠹⣄⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠄⠁⠀⠅⠃⠀⠀⢹⡄⠀⡾⠀⠀⡠⠒⠉⠑⠊⠉⠓⢄⠀⠈⣇⠀⢠⡏⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢹⡄
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡂⠀⡘⠀⠀⢀⡇⠀⡇⠀⠠⠔⠖⠯⠷⠾⠭⠶⠤⠀⠀⣿⠀⢸⡀⠀⠰⡇⣠⡀⠀⢀⣄⢘⠆⠀⢀⡇
⠀⠀⠀⠈⣇⠀⠀⠀⠂⠀⡐⠁⡠⠃⠀⠀⣸⠁⠀⢻⡀⠀⠀⠠⠤⡤⠤⠤⠤⠀⠀⢠⠇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠘⢦⡀⠀⠀⣀⠠⠚⠁⠀⢀⡴⠃⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣤⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⣤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣤⡤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⢀⡴⠚⠩⠅⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⣀⣀⣀⣀⠀⠀⠈⢳⡀⠀⠀⢠⠏⠐⠀⠀⠀⠀⠀⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣤⣿⣿⢿⡟⣛⠛⡿⡆⠀⠈⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠹⣄⠘⢄⣛⣡⠇⠈⠂⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣸⠃⠀⠀⢻⡀⠀⠐⠁⠀⠈⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⡀⢊⢸⡁⠧⠐⠈⢸⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⢈⣿⣿⠀⠸⠀⠀⠧⠨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⡠⡀⠀⢰⠀⠈⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⠀⠰⠀⠸⠀⠀⡇⠀⠰⠁⠰⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣤⡶⠞⠋⢻⣿⣿⠓⢄
⠀⠀⠀⠀⠀⠀⠁⠀⢇⠀⡁⠀⡇⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠸⣇⠀⠀⠀⠓⠟⠋⠀⠀⠇
⠀⠀⡏⠑⡀⠀⠈⠀⠘⡀⡇⠀⡇⢰⠀⢰⠁⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠉⠳⠶⠦⠤⠤⠔⠒⠋⠉
⠀⠀⠘⣄⠀⠀⠀⢀⠀⠧⠇⠀⠧⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢦⡀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⣤⡀
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠀⠙⡀
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠰⣀⢀⠠⠀⠐⠀⠀⠀⠀⠀⢀⠀⢀⡠⢀⡀⠀⠀⢃⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢀⠔⠁⡀⠄⠠⢀⠈⠂⠀⠈⢆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠂⣠⡾⠶⣶⡷⠶⣷⣦⡀⠀⠀⠑⢦⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠉⠀⠁⠈⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠀⠒⠊⠁

⠀⠀⠀⠀⠀⡠⠖⠛⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠊⢀⡴⠋⠉⠉⠓⢦⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣂
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠈⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠈⣿⣿⣿⣧⣽⣿⠛⠶⣤⡀
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠢⣸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠠⢀⣹⣷⣭⣭⣾⣿⣤⡶⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⢦⠀⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⢸⠂⢀⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⣀
⠀⠀⠀⠀⢀⣿⠁⢀⣠⠔⠁⠀⠘⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠘⢧⠜⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠏⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣠⠶⠶⠶⠀⠀⡠⠃⠀⢠⠋⠉⡆⠀⢸⠀⠀⢸⠀⠀⡰⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠐⠊⠉⠀⠀⠀⠸⠀⠀⢸⠀⢸⠀⠀⢸⠀⢀⡇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡇⠀⢸⠀⢸⠀⠀⢸⠀⢸⠁⠀⢸
⠀⠀⠀⠀⢀⡤⠴⠦⠤⠤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⡇⠀⠈⡆⢸⠀⠀⢸⠀⡎⠀⠀⡏
⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⡇⣇⠀⠀⡇⢸⠀⠀⢸⢰⠁⠀⢀⡇
⠀⠀⠐⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢡⢸⠀⠀⢣⢸⠀⠀⢸⡸⠀⠀⢸⠁
⠀⠀⠀⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣼⠀⠀⠘⠊⠀⠀⠘⠃⠀⠀⡸⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⣴⠞⠁⠀⣠
⠀⠀⠀⢀⠔⠊⠉⠑⠒⠒⠋⠉⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣠⡾⠁⠀⣠⡾⠋
⠀⠀⠐⠁⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⡸⠉
⠀⠀⠀⣠⣦⡤⠄⣀⠨⢄⡀⠸⢤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣤⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⠀⠛⠛⢦⡀⠀⠀⡠⠛⠛⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡎⠉⡆⠀⠀⡇⠀⢸⠀⠀⠀⡴⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣦⣶⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢁⠀⢹⠀⠀⡇⠀⠘⡆⠀⢠⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣴⡿⠟⠉⠀⢀⡸⢿⣿⠇⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢇⠀⡇⠀⠀⡇⠀⡜⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠾⣆⠀⠀⠀⠀⠙⠛⠉⠀⠀⢀⠇⡀
⠀⠀⢸⠉⠡⠀⠀⠀⠃⠀⠘⡄⡇⠀⠀⡇⢰⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠤⠤⠤⠤⠖⠚⠋⠉
⠀⠀⠈⢧⡀⠑⢄⠀⠰⠀⠀⢇⣽⠀⠀⣇⠇⠀⢰⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢳⡀⠀⠁⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣁⣀⣉⠐⠄
⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠛⠁⠀⢠⠛⢦
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡀⠀⡠⠂⠀⠀⠀⢣⡀⢀⣠⠔⠂⠀⠀⠢⠀⠀⠀⠀⢀⡀⠀⣀⣤⣄⡀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠠⠀⠀⠀⢀⠔⠁⢀⣀⡀⣀⠀⠈⠢⡀⠀⠘⢆⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠠⠋⢀⣴⣍⣀⣼⣤⣩⣦⣄⠀⠀⠀⠈⠳⣄⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠋⠉⠙⠛⠉⠉⠙⢻⣷⡄⠀⠀⠀⠈⠳⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⠀⢀⠀⣀⡀⠐⠁⡠⠀⠀⠀⠀⠀⠈⠢⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠈⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠤⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡕⠒⠪⣝⢆
⠀⠀⠀⠀⠀⡠⠒⣈⣩⡤⢤⣍⡑⠢⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⡑⠀⢸⢸
⠀⠀⠀⠀⠀⣴⠊⠀⡇⣤⡷⡁⠈⠑⡄⠀⠀⠀⠀⠀⠰⡞⠶⠤⢄⣀⣀⠤⠴⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠜⠀⡸⡆
⠀⠀⠀⠀⠀⠈⠑⠢⠌⣒⡊⠥⠔⠊⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣀⡀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣉⠔⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⢰⢹⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣒⣉⣉⣒⣉⣈⣑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠃⡗⢩⠃⠀⠀⠀⠀⠀⠀⠈⠻⡙⢻⠋⢻⠋⢻⢛⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⠘⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⡶⠶⠶⠶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢍⠓⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⢠⠊⠁⠑⠀⠀⠀⢀⠠⢀⣀⡀⠄⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⡁⠄⡀⠀⠀⠀⠈⢢⠄⠑⠂
⠀⠀⠀⠀⢸⠀⠀⠘⠡⠀⠀⠠⡈⢨⠴⡏⢁⠄⠀⠀⠠⠊⠀⠀⠀⠀⠀⠚⡒⠀⠀⠐⢚⠇⠀⠀⠀⠀⠃⣸⠁
⠀⠀⠀⠀⠀⠄⠀⠠⠁⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠢⠐⠤⠃⠀⠀⠀⠈⠒⠙⠂⠁⠀⠀⠀⠠⡐⡀⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠶⣶⣶⣶⡶⢶⣶⣶⣶
⠀⠀⠀⣾⣽⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠏⠹⡟⢀⡘⢿⣿⣿
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣆⣰⣷⣌⣡⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣴⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣏⣍⣉⣉⣍⣛⣅⣏⣩⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠞⠁⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣶⣤⣄⣀⣠⣤⣾⠿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⠶⣦⡄⠀⠀⠀⠀⠹⢾⣿⡿⠋
⠀⠀⠀⡠⠒⡶⠛⢽⣿⡿⠟⠋⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⠏
⠀⠀⡜⠠⠄⠀⠀⡆⢱⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢡⠀⢤⣭⣤⠀⣜⣿⣿⣷⣦⣤⣀⣤⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣼⣩⣿⣷⠀⠀⠀⠀⠀⢠⣎⣿⣷⡀
⠀⠀⢿⣽⣆⠿⠑⠊⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠈⠻⠿⠟
⠀⠀⠋⠐⠛⠋⠁⠛⠀⠃⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⣀⣠⣄⣀⠀⠈⢳⡀⠀⠀⠀⢀⡞⠁⢠⠴⠚⠓⢢⠀⠈⠳⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⡟⠀⡰⠋⣁⣰⣦⡌⠑⣄⠀⣷⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⠤⡄⠀⢻⠀⠀⠀⢰⡇⣀⡤⠴⠲⢤⠖⠢⢤⣀⠘⡇
⠀⠀⠀⠀⠀⣇⠀⢣⡀⠻⣿⣿⠇⢀⠞⠀⣿⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠈⣸⠁⠀⣸⠀⠀⠀⢸⡄⠙⢍⡓⠒⠒⠒⢚⣩⠏⢀⡇
⠀⠀⠀⠀⠀⠹⣄⠀⠉⠒⠦⠴⠚⠁⠀⣰⠃⠀⠀⠀⠸⣆⠀⢸⡀⠀⠀⣠⠇⠀⣠⠏⠀⠀⠀⠀⢷⡀⠀⠉⠙⠓⠋⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠈⠳⢤⣉⣉⣉⣁⣤⠞⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⣤⡤⣤⣤⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣠⣤⠤⣤⣀⡀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⣠⠤⠤⣄⠉⠳⣄⠀⠀⠀⠀⠀⣠⠞⠉⠀⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡎⣴⠋⠉⠋⢳⠀⠘⣧⠀⠀⠀⢰⠏⠀⠀⢠⠇⠀⢹⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠉⡇⢀⣴⡞⠀⠀⣹⠀⠀⠀⢸⠀⠀⢀⠞⠀⠀⠈⢧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢈⠙⠦⠞⡼⠀⠀⢀⡟⠀⠀⠀⠸⡆⠀⠸⣤⢤⢠⠤⡼⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠓⠶⠚⠁⢀⣠⠞⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⢰⣟⣛⢀⡇⣤⡀⣤⣠⣤⡄⠀⢀⣤⣤⢀⣤⣄⢠⣤⣄⢀⣤⣤⢠⣤⣄⢠⣤⡄⠀⢀⣏⢠⣤⣄⣠⣤⣄⢠⣤⣄
⠀⠀⠀⠀⠀⢸⡏⠉⠸⡧⠸⣿⠇⢿⣿⡇⠀⠸⣿⡿⠸⣯⡿⢸⡇⣿⠸⣿⡷⠺⣿⡿⢼⣿⡷⠀⠸⡿⠸⣧⡗⠻⣤⡿⢸⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠉⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠋⠉⣉⣀⣤⣍⣉⠙⠓⠦⣀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡴⠛⣩⠥⠖⠈⠉⠻⣦⡀⠈⠳⣄
⠀⠀⠀⡴⠁⢀⣴⠾⠿⣷⣶⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣰⠋⠀⡰⠃⠀⠰⣄⡀⠀⡈⢿⡀⠀⠈⢧⡀
⠀⠀⡼⠁⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⢿⣿⣦⡄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠇⠀⢊⡵⠖⢲⣤⠈⠁⠀⢱⢸⡇⠀⠀⠀⢧
⠀⢠⡇⠀⠀⢠⣀⣀⣦⣤⣶⣤⣷⣦⣤⣄⡀⠀⠈⠙⠛⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠙⣆⠀⠀⠸⢸⡇⠀⠀⠀⢸⡆
⠀⢸⡇⠀⠲⣶⣿⣿⠿⠛⢹⣿⣉⣽⣿⢿⡟⢶⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠀⣀⣍⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠸⡇⠀⠀⠀⠀⠹⣆⠀⠈⢷⣝⣛⣯⠞⠁⠀⠙⢦⡀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠋⣴⠄⡴⠀⣼⠁⠀⠀⠀⢸⠇
⠀⠀⢳⡀⠀⠀⠀⠀⠈⠳⠦⣤⣄⣉⣀⣤⠴⠆⠀⠘⠋⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⡾
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣠⡤⠶⠒⠒⠒⠒⠶⢤⣄⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠉⠀⠀⡄⠀⠀⠀⢠⡄⠀⠀⠉⠲⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀⣀⠀⠀⠀⠀⢀⣀⡤⠖⠋⢠⠞⠁⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠳⣄⠉⠳⠦⣄⣀⡈⠙⠋⠁⣀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠹⡄⠀⠀⠀⡞⠁⠀⠀⠀⠀⢀⣴⠀⠀⠈⢷⠀⠀⠀⢠⠏⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠓⠦⣄⠘⢦⡀⠀⠳⠄⠶⠶⠶⠀⠀⠶⠶⠶⠠⠋⠀⢀⡴⠋⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡴⠋⠁⠀⠀⠀⠄⠀⠀⠂⠀⠐⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⠙⠒⠲⠤⠤⠤⠤⠶⠚⠋⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⡾⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠈⠀⠀⡁⠀⠀⠀⠀⢷
⠀⢰⠇⠀⠀⣀⣤⠞⣫⡅⢀⠈⠉⠉⣡⠀⠈⣝⠳⢦⣀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢸⠀⠛⠛⡉⠦⠤⠤⠶⠾⢧⣴⣴⣽⠷⠶⠦⠴⠶⢊⠟⠛⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠸⡆⠀⠀⠈⢦⡀⠀⠀⠀⠄⠆⠐⠠⠀⠀⠀⠀⡴⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠇
⠀⠀⢳⡀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠤⠤⠤⠴⠒⠋⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡞
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⣀⣀⣠⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣄⣀⣀⠀⠀⣀⣀⣨⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠈⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⢁⣠⣤⣤⣤⣈⠉⠒⢤⡀
⠀⠀⠀⢀⡔⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠞⠋⡥⠖⠒⠂⠉⠻⣦⡀⠈⢢⡀
⠀⠀⢠⠏⢀⡴⠟⠛⠻⠿⠿⣿⣷⣶⣶⣤⣤⣀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢸⠃⠀⡜⠁⠀⠐⢤⠄⢠⠘⣧⠀⠀⠙⡄
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠙⠻⠿⣷⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠀⢸⠞⠉⠹⣧⠀⠀⠀⡇⣿⠀⠀⠀⠸⡀
⠀⢸⠀⠀⠀⢲⣤⣼⣶⣾⣷⡾⢿⣷⣶⣤⡀⠀⠀⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠘⠧⠀⠀⠃⡟⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠙⠻⠿⡏⠁⠀⣿⢶⣾⡿⣹⠏⠻⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢩⡄⢠⡶⢂⠀⡀⠀⡿⠀⠀⠀⠀⡇
⠀⠈⡆⠀⠀⠀⠀⠙⢦⣀⡈⠛⠒⠚⠁⣀⡀⠈⠷⠄⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢸⣠⠎⠀⠋⠘⠁⣼⠃⠀⠀⠀⢰⠃
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⢀⡞⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⠏
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠔⠊⠉⡀⠀⠀⠀⢀⠀⠉⠑⠢⣄⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊⣡⠞⠁⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠈⠳⡄⠙⠢⣄⡀⠀⠐⠶⠒⠋⠀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠚⠉⠉⠀⠀⡰⠁⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠉⠑⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢷⠀⠀⢀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⢄⣀⠀⠀⠱⡀⠀⠀⢼⠁⠀⠀⠀⠀⠠⠞⠀⠀⠈⡇⠀⠀⢠⠎⠀⠀⣀⣠⠤⡤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡙⢦⡀⠈⠓⠚⠛⠛⠀⠀⠛⠛⠃⠚⠁⢀⡴⢋⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠱⣄
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠉⠉⠛⠛⠉⠉⠁⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⢠⠇⠀⠀⠀⢀⣤⠶⠚⠲⠶⠶⠖⠒⠶⢦⣀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢸⠠⣤⡴⢞⣋⣑⣋⣠⣆⣀⢀⣾⣦⣘⣓⣉⣛⢦⣤⡤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠈⢢⡉⠉⠉⠠⢉⣛⢛⣋⢡⠨⠉⠉⡡⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠈⣆⠀⠀⠀⠑⠦⣤⣀⣀⣀⣀⣀⣀⣀⠤⠞⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣰⠁
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⢀⣡⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣴⠟⠋⠁⢠⣻⣿⠇⠙⢆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠻⣦⣀⠀⠀⠉⠁⠀⣀⡬⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣤⣤⡤⠴⠚⠉⢦⣀⣠⣴⠶⠛⠛⠋⠉⠉⠙⠛⠳⢶⣤⣀⣠⠎⠑⠢⠤⣤⣤⣤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠴⠀⠘⠀⠑⠂⠀⠀⠀⠀⠀⠀⠈⠠⠀⠄⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠂⠀⠀⠐⠀⣇⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣷
⠀⠀⠀⢀⣠⠤⠖⠒⠒⠢⢤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡄⠀⠀⠀⠀⢈⠄⠀⠀⠀⠉⠰⢀⠈⠀⣀⡨⠂⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠤⢤⣀
⠀⢀⠔⠉⠀⣀⡀⠐⠀⠀⠀⠈⠑⢆⠀⠀⠸⣇⠀⠀⠰⡔⠀⠀⠀⠤⠀⠀⠀⠁⠂⢤⠀⠰⠋⠉⠀⠀⠀⠀⠀⢀⡿⠀⠀⢀⠖⠉⠀⠀⢀⠀⠀⠀⠀⠈⠑⢄
⢠⠏⠀⠀⠈⠁⠀⠘⠓⡄⠀⠀⠀⠀⢳⡤⠔⢿⡀⠀⠀⠠⡀⠠⠀⢀⡠⣤⣤⠀⣀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠗⠤⣴⠃⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⢧
⡎⠀⠀⠀⠀⠀⠀⠁⠀⡸⠀⠀⠀⠀⠀⡇⠀⠈⢷⡀⠀⠀⠉⢰⣿⣷⣾⠿⠭⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈
⡇⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢳⣄⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠘⡇
⢱⡀⠀⠀⠱⣄⠀⠀⠔⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠳⡄⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣀⣀⣀⣀⣤⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡴⠃
⠀⠀⠈⠳⢤⣀⣀⠈⠒⣂⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⡀⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠛⠛⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠤⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠊⠀⠄⢀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁⡴⠿⠶⢾⠶⠾⢷⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⣀⠀⠛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⡀⠀⠁⠈⠀⠀⠀⡠⠃⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⠉⠒⠒⠂⠚⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⣀⡠⠤⠖⠉

⢠⣶⠶⠶⠶⠶⣶⡄⠀⢀⣤⣀⣤⡀⠀⠀⣶⡶⠶⣶⠶⣶⡆⠀⠀⢀⣄⣠⣤⡀⠀⢰⣶⡶⠶⣶⣶⣶⠀⠀⠀⢀⣄⡀⠀⠀⠀⣶⣶⡶⠲⣶⣶
⢸⣧⣤⣦⣴⣤⣼⡇⠀⠸⠋⠀⠙⡇⠀⠀⣯⣀⠠⡤⢀⣨⡇⠀⢾⣿⣿⣿⣿⠀⠀⢸⣿⠄⠀⠘⠋⢻⠀⠀⠀⣿⣿⣧⠀⠀⠀⣿⣿⠃⠀⢿⣿
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠘⣄⣀⣠⠃⠀⠀⣿⣿⣀⣀⣸⣿⡇⠀⠀⡄⢾⣿⡏⠀⠀⢸⣿⣄⣀⣀⣠⣿⠀⠀⢸⣿⣿⣿⡄⠀⠀⣿⣟⣀⢀⣘⣿
⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠚⠛⠛
⠀⠀⣠⣶⣦⣄⠀⠀⢸⣿⡿⢛⣿⣿⣷⠀⠀⢀⣠⣤⣄⠀⠀⢰⡟⠻⠿⢿⠟⢹⡆⠀⠀⠀⣀⡀⠀⠀⠀⣾⠟⠛⠛⠻⢿⡇⠀⢀⣠⣀⣀⡀
⠀⠀⢻⣄⣨⡟⠀⠀⢸⣿⠀⣾⣿⣿⣿⠀⠀⢼⣿⣿⣿⡇⠀⢸⡿⣷⠆⠀⠰⣿⡇⢀⣴⣥⣤⣬⣦⡄⠀⣿⣠⡄⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣄
⠀⠀⢬⣭⣭⠅⠀⠀⢸⣿⣦⣈⣻⣿⣿⠀⠀⠘⠻⠿⠟⠁⠀⢸⣷⣦⢰⣿⡄⣿⡇⠈⠁⠈⠉⠁⠀⠁⠀⣿⣿⣁⣻⣿⣠⡃⠀⠀⠙⠘⠘⠀⠁
⠀⢀⣀⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠁⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⣀⣀⣀⣀
⢸⣿⣿⠟⣿⣿⣿⡆⠀⣾⡛⣦⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⡇⠀⠀⢰⡆⢰⡆⠀⠀⢸⣿⣿⡛⢿⣿⣿⠀⠠⣶⣦⠀⠀⠀⠀⠀⡿⠿⠿⠿⠟⠻
⢸⡿⠁⢀⣈⢻⣿⡇⠀⢉⣿⣧⣴⣷⠄⠀⣿⣛⠉⠀⠀⠙⡇⠀⠀⠠⢽⣿⠄⠀⠀⢸⠟⠉⠀⠀⢈⣿⠀⠀⠸⣿⣷⣄⠀⠀⠀⡇⢀⣀⡀⣰⣿
⢸⣷⣾⣾⣯⣶⣽⠇⠀⠘⢿⣿⡟⠉⠀⠀⣿⣿⣿⣷⣾⣿⡇⠀⠀⠀⠸⠇⠀⠀⠀⢸⣧⣤⣤⣤⣼⣿⠀⠀⠀⠿⠿⡿⠧⠀⠀⣷⣿⣿⣷⣿⣿
⠀⠉⠉⠉⢉⡉⠉⠀⢀⣤⣤⣤⣤⣤⣄⠀⠈⠉⡉⠉⠉⠉⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠉⠉⠉⠉⠉⠁⠀⣠⣤⣤⣤⠤⣤⡀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢈⣿⡆⠀⢸⡋⠉⢻⡿⠃⣿⠀⠀⠈⢻⣆⠀⠀⠀⢸⣿⡿⠀⠉⢹⣿⡅⠀⠀⠰⣿⡷⠀⠀⠀⣿⣿⠻⠳⢠⣿⡇⠀⠀⠀⢻⣦
⠀⢸⣿⠿⣿⡟⠁⠀⢸⣷⣤⣄⠀⣠⣿⠀⠀⠀⠸⣿⣿⠆⠀⢸⣿⡀⠀⠀⢀⣿⡄⠀⠀⣼⣿⣿⡀⠀⠀⣿⣧⡀⠀⢠⣿⡇⠀⠀⢀⣿⣿⣧
⠀⠀⠐⠀⠀⠊⠀⠀⠸⣿⣿⣿⣯⣿⡿⠀⠀⠀⠀⠫⠉⠀⠀⠘⢿⣿⣦⣴⣾⡿⠂⠀⠀⠘⠟⠟⠀⠀⠀⢿⣿⣷⣤⣿⣿⠇⠀⠀⠀⠛⠟⠃
⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⣀⣀⣀
⢸⣿⣿⡏⠹⣿⣿⡅⠀⢀⣀⣼⣄⣀⠀⠀⣿⠟⠉⠉⠙⢿⡇⠀⠀⠀⢀⣰⣆⡁⠀⢸⡿⠛⠓⠂⠾⣿⠀⠀⣠⣾⣶⣄⠀⠀⠀⣿⣿⡇⠈⠹⣿
⢸⣿⣯⠃⠀⣼⣿⡇⠀⣿⣿⣿⣿⣿⡇⠀⣏⠀⠈⠀⠀⠀⡇⠀⢠⣶⣿⣿⠟⠁⠀⢸⡇⠀⠀⠀⢀⣿⠀⠀⣿⣿⣟⣿⡆⠀⠀⣿⣿⠏⠀⠀⣿
⠸⣿⣿⣦⣴⣿⣿⠃⠀⠈⠛⠛⠛⠋⠀⠀⢿⣦⣤⣤⣤⣾⠇⠀⠘⠻⠟⠁⠀⠀⠀⠸⣿⣤⣤⣤⣾⡿⠀⠀⠉⠛⠛⠋⠀⠀⠀⢿⣯⣔⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⢀⡀⢀⠀⠀⢀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡄
⠀⢠⣾⣿⣿⣷⡄⠀⢸⡇⠀⠀⠀⠈⣿⠀⢸⣾⡽⣿⢭⢸⠀⢸⣿⣿⠏⠹⣿⣿⡇⠀⠀⢀⡞⣄⠀⠀⠀⣿⡿⠿⠙⠿⢿⡇⠀⠀⠀⣈⣦⣠⠔
⠀⠘⢿⣿⣿⡿⠁⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣿⢭⣿⠼⢸⠀⢸⣿⠃⠀⠀⠘⣿⠅⠀⢀⢶⡼⡴⢆⠀⠀⣿⣷⠄⣀⢰⣿⡇⠀⠀⠉⢩⠟⢧⡀
⠀⠀⠀⠉⠁⠀⠀⠀⠸⠷⠶⠶⠶⠾⠟⠀⠈⠈⠉⠉⠉⠘⠀⠘⠷⠶⠶⠶⠶⠿⠃⠀⠈⠉⠉⠉⠉⠀⠀⠻⠿⠾⠿⠷⠿⠇⠀⠀⠀⠈
⢠⣶⣶⠖⠒⣶⣶⡀⠀⠀⢠⣤⡄⠀⠀⠀⡔⢲⣶⠒⣶⠖⠆⠀⣤⡄⢠⡄⠀⣤⠀⢰⣶⠶⠒⠶⢶⣦⠀⠀⢀⠤⠀⠄⡀
⢸⡟⠛⣤⣤⠛⢻⡇⠀⣤⣼⠛⢣⣤⡄⠀⡿⢏⣩⠿⣉⠿⠇⠀⣈⡽⢏⠸⠿⠀⠀⢸⠁⠀⠀⠀⠀⢿⠀⠐⢡⢊⣤⠑⠐
⢸⣧⣤⠛⠛⣤⣼⡇⠀⠛⢻⣤⡜⠛⠃⠀⣷⢎⣙⠶⣉⠶⡇⠀⢉⡱⢎⡱⢶⡉⠀⢸⡄⠀⠀⠀⢀⣾⠀⠐⠈⠈⠉⠔⡠
⠈⠛⠛⠒⠒⠛⠛⠁⠀⠀⠈⠉⠁⠀⠀⠀⠓⠘⠛⠐⠛⠂⠃⠀⠙⠁⠘⠂⠀⠋⠀⠘⠛⠓⠒⠒⠛⠛⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣄
⠀⠀⠀⠀⡴⠋⢀⡠⠤⣀⠀⠙⣆⠀⠀⠀⡴⠁⡤⠒⠒⡆⠈⠳⡄⠀⠀⠀⡴⠉⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⢰⡇⣰⠁⣤⣿⣆⠙⡄⢸⠀⠀⢸⠁⠀⡇⠀⠀⠷⡢⠀⢹⠀⠀⢸⠁⣴⣒⣑⢒⣑⡲⡄⢹
⠀⠀⠀⠈⣇⠈⠢⢍⣛⡡⠜⠁⡼⠀⠀⠘⡆⠀⡇⠀⠀⢠⠃⢀⡞⠀⠀⠘⣆⠈⠲⠤⠤⠴⠊⢀⡞
⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⡤⠞⠁⠀⠀⠀⠘⠦⣌⣒⣒⣁⡤⠊⠀⠀⠀⠀⠘⠢⣄⣀⣀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⠤⠶⠦⢤⡀⠀⠈⠉⠀⠀⢀⣠⠤⠶⠦⢤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⢖⠼⢖⢄⠉⢦⠀⠀⠀⣰⠋⠀⢠⠀⢀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠟⡄⣠⡼⠀⠘⡇⠀⠀⡇⠀⢀⠎⠀⠘⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡙⠤⣳⠃⠀⣸⠁⠀⠀⢧⠀⠸⠤⠄⠤⠜⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣉⣋⣁⣠⠞⠁⠀⠀⠀⠀⠳⢤⣀⣀⣀⡠⠞⠁

⠀⠀⠀⠀⣴⣖⢒⡆⡄⣠⣠⣄⠀⢀⣤⡀⣤⡄⣠⣤⢀⣤⡀⣤⡄⣠⣄⠀⢰⡆⣤⡄⣠⣄⢠⣤⡀
⠀⠀⠀⠀⠻⠉⠘⠇⠻⠇⠻⠿⠀⠘⠿⠛⠿⠏⠻⠸⠘⠿⠛⠿⠗⠻⠟⠀⠸⠛⠷⠇⠳⠞⠸⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡎⠀⢱⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡁⠈⡄⠀⠇⠀⢸⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⣴⡶⠒⠛⣿⣿⡗⠢⡀
⠀⠀⠀⠀⠀⠀⠠⠀⢱⠀⠀⠀⢸⠀⢀⠃⠀⠇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⣟⠁⠀⠀⠐⠭⠿⠃⠀⠘⠆
⠀⠀⡖⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⠀⣜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠳⠦⣤⢤⣤⡤⠤⠴⠚⠊
⠀⠀⠘⡄⠀⢄⠀⠁⠀⢸⡸⠀⢸⡠⠁⢀⠇
⠀⠀⠀⠘⣆⠀⠐⠜⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⡐
⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠉⢢
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠓⠀⠉⠀⠀⠀⠀⠀⠀⣠⠔⠀⠉⠉⠉⠑⠀⠀⠀⢣⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⡐⠁⣠⣎⣉⣭⣁⣦⣄⠀⠀⠀⠑⢄⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⣀⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⢴⡞⠉⠉⠉⠋⠉⠉⠻⡷⠄⠀⠀⠀⠱⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠄⠠⠄⠀⢀⠄⠀⠀⠀⠀⠈⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⢀⣀⡤⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣟⣛⣛⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠔⢊⣉⣉⡑⠢⡀
⠀⠀⠀⠀⡴⠁⠀⠀⠙⣆⠹⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⡶⢶⣤⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠠⣸⡀⡇⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠾⠄⠀⣿⣟⣛⣽⣇⣬⠟⠂
⠀⠀⠀⠀⠉⠉⠈⠑⡄⠘⡇⡇⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉
⠀⠀⠀⠸⣷⠀⠀⢀⠆⠀⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠈⠐⠢⠤
⠀⠀⠀⢸⣇⠔⠊⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡀⠀⢀⣠⠴⣦⠀⢀⠌⠀⠰⠒⢆⠀⡀⠀⢸⠀⢠⠒⢢
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠈⠀⠀⠀⢸⠀⢸⠀⡇⠀⢸⠀⡌⠀⢸
⠀⠀⠀⠀⠀⣈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢂⢸⠀⠘⠀⡇⠀⢸⢀⠃⠀⡌
⠀⠀⠠⠊⠉⠀⠉⠁⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠈⠀⠀⡇⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠒⠒⠛⠛⠓⠛⠛⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⡇⠀⣇⡇⠀⢠⡆⠀⢰
⠀⠀⠀⠂⠤⠤⠤⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⡇⠀⠈⠁⠀⠈⠀⠀⠘⠀⠀⠀⡠⠄⠒
⠀⠀⠀⡀⠤⠤⢄⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠎⠀⢀⠔
⠀⠀⠊⠀⢀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⡠⠁
⠀⠀⡠⠦⠐⠠⠜⠠⠄⠢⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣴⠆⠀⠀⠀⠀⠀⠲⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠀⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠂⠄⣀⡀⠠⠊⠀⠀⠀⠀⠀⠉⠑⠄⠀⠔⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡏⢳⡀⠀⢀⡴⡄
⠀⠀⠀⠀⠀⠀⢸⠈⣦⠀⠸⡇⠈⡇⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⢸⡄⠀⡇⠀⡇⢀⡇⠀⡏⢠⢤
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⢰⡇⠀⡇⡼⠀⡼⢡⠏⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⣚⣻⣯⣭⣟⣒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⠾⠀⠀⠉⠁⠀⡧⠏⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⡟⢁⣤⡀⠈⢣⠈⠙⠻⢦⡀
⠀⠀⢀⡤⢤⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⣀⠀⢧⡈⠻⠟⢀⡼⢀⣠⠴⠋
⠀⠀⠈⢧⡀⠙⢦⣀⡗⢦⡤⣄⣀⠙⠲⢤⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠚⠛⠋⠉
⠀⠀⠀⠀⠳⡄⠀⠉⠀⠀⠙⢦⠈⠙⢦⠀⠀⣸
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⡤⣤⣠⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⣉⣉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢠⡞⠉⠁⠈⠙⢲⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⢦⡀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⠀⠀⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⣀⡼⠀⣰⠃⣸⠁
⠀⠄⠀⠀⢀⡠⠖⠋⠉⠉⠓⠚⠉⠉⠓⠢⣄⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢀⣀⣠⣤⡀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢼⡴⠋⠁⠀⡴⠃⡴⠃
⠀⠀⠀⢠⣯⠤⠤⠴⠒⠒⠶⠖⠒⠒⠶⠤⠬⣵⠄⠀⠀⠀⠀⠀⠉⠓⠒⠂⠈⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠺⠟⢋⣉⣉⠙⠛⢿⣷⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀
⠀⢸⣿⣿⣿⡿⠿⠛⠉⠉⠉⠙⠛⠉⠉⠉⠙⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢀⡉⠑⠢⢀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢐⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡷
⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇
⠀⠀⠻⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⠟⠀⠀⣠⣤⣶⣾⣿⣿⣷⣶⣦⣄⡀⠈⠳⣄⣄⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⠿⠟⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣙⣛⡻⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⡿⠉⠀⢀⠞⠛⣭⣿⣿⣧⠈⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣂⣠⠀⠀⢸⣦⣤⣿⣿⣿⣿⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣤⣀⠀⠀⠹⣿⣿⣿⡿⠋⣀⣠⡾
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣷⣶⣤⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⠟⠁⣀⣤⣶⣾⣿⣿⣿⣶⡶⢤⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠳⣄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢦
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⡀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣦⣬⣍⠂⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡆⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣋⣡⣤⣤⣤⣤⣤⣀⠉⢻⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⣤⣤⣿⣿⣯⡶⢻⡍⢿⣿⣿⣿⢁⡇⠀⠉⠓⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠉⠉⠙⢦⡑⢄⡻⢦⣉⣉⣥⠞⠁⠀⠀⣀⣠⠽⠶⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⣓⡒⠲⠖⠒⣒⣋⡭⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡚⠷⢦⣄⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⠤⠴⠟⠁
⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠈⠛⢦
⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠑⠢⠄⠒⠋⠉⠑
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢰⣤⣦⣤⣤⣤⣄⡠⠤⠴⠶⢶⣶⣤⣤⣴⣶⣶⣤⣤⣤⣤⣀⣈⣲⣤⣤
⠀⠀⠀⠀⠀⠀⠱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢈⡩⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⠤⠤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢠⠖⢩⠏⢠⠓⡽⡍⠲⡄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡳⠼⣯⣢⣖⣽⠧⢚⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡆⠀⠐⠱⣆⡀⠈⠁⢀⣀⠈⠃⠀⣰⢁⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢤⣀⡀⠀⡠⠞⠉⠀⠘⢆⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⡰⠃⠀⠉⠳⢄⠀⢀⣀⡤⠤⠤⢄⣀
⠀⠀⠀⠀⢀⠔⠋⠁⠀⣀⡀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠑⠦⣄⣀⡀⠀⣀⣠⠴⠊⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⢰⠋⠀⠀⢠⠀⠚⠏⠦⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢠⠋⢳⠀⠀⠀⠀⠙⡄
⠀⠀⠀⡇⠀⠀⠀⠀⢠⢨⢰⠀⡀⠀⠀⠀⠈⡆⠀⠀⠀⢀⡤⠖⠊⠉⠉⠑⠒⢤⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡏⠀⢀⡃⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠐⡆⠸⠼⠼⢤⠇⠀⠀⠀⠀⡇⠀⢀⡔⠁⠀⠀⠀⡔⢢⠀⠀⠀⠈⢢⡀⠀⢸⠀⠀⠀⠀⡴⠁⠀⢸⡸⡄⠀⠀⠀⢸
⠀⠀⠀⢣⠀⠀⠀⠸⡀⢱⠊⢰⠀⠀⠀⠀⣰⠁⠀⡎⠀⠀⠀⠀⢀⡑⢊⡀⠀⠀⠀⠀⢳⠀⠈⣆⠀⠀⠀⣙⠢⣄⡭⠒⡁⠀⠀⠀⡜
⠀⠀⠀⠀⠳⣄⠀⠀⠙⠒⠒⠁⠀⠀⢀⡴⠁⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠈⢦⡀⠀⠉⠁⠀⠀⠉⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠈⠑⠢⠤⣄⣀⡤⠤⠒⠉⠀⠀⠀⠘⡄⠀⠀⠀⣀⡈⡇⢸⢁⣀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠒⠤⢤⡀⣠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠚⠤⠤⠍⠭⠄⠀⠛⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢀⣀⣀⣀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⣀⣀⣀⡀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠖⠋⠉⠀⠀⠈⠉⠓⢤⡀⠀⠉⠑⠒⠒⠊⠉⠀⢀⡤⠒⠉⠀⠀⠀⠉⠙⠲⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢀⣀⢀⡀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡤⢒⡒⢤⡀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣀⣔⣉⣀⣁⣀⣰⣄⡀⠀⠘⡆⠀⠀⠀⠀⢰⠃⠀⠀⡄⠀⠘⢸⠁⠈⠃⣧⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠛⠅⡏⠀⡇⠘⡇⠝⠋⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⢸⣖⠀⠈⠇⠀⢠⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠳⣀⢀⡰⠃⠀⠀⠀⢠⠃⠤⠤⠤⠤⠘⡄⠀⠈⠎⠀⠀⠀⠀⠀⡏⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠙⠦⠔⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣤⣤⣤⣤⣤⣇⠀⠀⣿⣿⣿⠿⠿⠿⣿⣿⣿⢿⣿⠀⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⣿⠟⠁⠈⠙⠉⠛⠋⠀⠙
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠁⠀⠀⠀⠀⠘⣿⡿⣸⣿⠀⠀⣼⡏⠀⠁⠀⢻⡇⠀⠀⣿⠀⠀⢠⣆
⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⣇⠀⠀⠀⠀⠀⣄⣉⣰⣿⣿⠀⠀⣿⡄⠀⠀⠀⢠⣿⠀⠀⣿⣷⣶⡈⠁⠤⠀⠀⣴⣶
⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⣿⣷⣦⣤⣤⣴⣿⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠘⠁⠀⠀⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⣶⣤⣤⣤⣤⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣶⣦⡀
⣿⡿⠋⠀⠀⠀⠈⠹⠃⣠⣿⠀⠀⣰⣿⣿⣿⣿⣿⣇⠀⠀⣿⠋⠛⠛⠛⠛⠉⠉⠉⢹⣿⠀⠘⣿⣿⣿⣄⣀⣀⣴⢀
⣿⣅⠀⠀⠀⠀⠀⠁⣼⣿⣿⠀⢠⡟⠻⣿⣟⣿⡿⢿⡄⠀⣿⣦⠤⢤⠤⠤⠤⠀⠀⢸⣿⠀⠀⠹⣿⣿⣿⣿⣿⠟⠈
⣿⣿⣴⣦⣾⣿⣾⣿⣿⣿⣿⠀⠘⢷⣾⣿⣿⣿⣷⣾⠃⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣶⣀⣠⣶⡄⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣻⣷⣶⣶⣶⣶⡟⠀⠀⣿⣷⢳⣻⣿⣿⣿⠋⠈⢻
⠀⢠⣾⣿⣿⣿⣿⣿⣷⡄⠀⡿⠁⠀⠀⠈⢿⣿⣯⡉⢉⣿⠀⢾⣿⣿⣿⣿⣿⣿⣿⡦⠀⣿⣿⣷⡧⢿⡿⠗⣻⣿⣾
⠀⠙⢿⣿⣿⣿⣿⡛⣿⠟⠀⣧⠀⠀⠀⠀⠈⠛⢛⣵⣿⣿⠀⠀⠙⢿⣿⣿⣿⡽⠋⠁⠀⣿⣿⣿⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠃⣼⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛
⡿⠻⢿⡿⠿⠟⠿⢿⡿⠟⢿⠀⣠⣤⣄⠀⠀⠀⣠⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣀⣤⣀⡀
⣿⣿⣶⡄⣀⠀⠀⠀⣿⣿⣿⠀⠻⣥⣼⣷⣶⣶⣧⣠⡿⠀⣿⡛⢿⣿⠟⠉⠉⠉⠙⠻⣿⠀⢀⣼⣿⣿⣿⣿⣿⣇⡀
⣿⣿⣿⠀⠟⠂⠀⡀⢻⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⡿⠋⣤⡀⠘⠀⠀⠀⣠⣿⠀⢸⣭⣿⣿⣿⣿⣿⣮⠃
⣿⣿⣿⡀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠻⢿⣿⡿⠟⠀⠀⠀⣿⣷⣾⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠛⠻⢿⣿⣿⠟⠋
⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿
⠀⠀⠀⣶⣦⢰⣿⠀⠀⠀⠀⣿⣟⡉⠛⠋⠉⠉⠋⢠⣽⣿⠀⣤⢤⣤⣴⣶⣦⣤⡤⣤⠀⣿⠟⠛⠁⠀⠀⠀⠀⠙⣿
⠀⠀⠀⢈⣹⣿⣇⣤⡀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠈⣿⣿⠸⣇⣸⣿⣿⣿⣿⣿⣧⣸⠇⣿⠆
⠀⠀⠠⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⠓⢰⣿⡆⠚⢠⣿⣿⠀⠀⠹⣯⣿⣿⣿⣽⠃⠀⠀⣿⣤⠀⠐⠂⠀⠐⠂⠀⢾
⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⣿⣿⣿⣶⣤⣤⣤⣶⣿⣿⣿⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⣿⣿⣤⣄⣀⣠⣀⣠⣴⣾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡿⢿⠿⠿⠿⡿⠿⠿⠿⠿⠿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⠿⠿⠿⠿⡿⣿⠿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣉⡉⣍⠙⠻⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⣿⣻⣼⣽⣿⣿⡿⣿⣽⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠃⡴⠛⠁⠀⠈⠹⢦⠀⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣽⣷⣿⣿⣿⣯⣹⣿⣧⣿⡿⢿⡿⢿⣿⡾⢿⣹⣿⣿⣿⣧⣼⡇⠀⠀⠀⠀⠀⠈⣧⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣶⣿⢺⢿⣿⣷⣾⣿⣿⣶⣿⡟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣛⣿⣿⣷⣿⣿⣿⣛⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠠⠤⢄⡑⠄
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⢸
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣉⡥⠬⢭⡅⣂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⡸⢸
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠈⠰⣶⠆⠁⠀⢉⡷⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠸⠀⡴⢁⠆
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠁⠒⠚⠢⠤⠴⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠈⡠⠊
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠑⠢⠄⠀⠉⠙⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⢠⠉⢂⠀⢀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡀⠸⠀⢸⠀⡘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡃⢠⠀⢸⢠⠃⠰⢠⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠼⠀⠘⠂⠀⣧⠂⡠⠁
⠀⠀⠀⠀⢀⠤⠒⠉⠑⠒⠂⠁⠒⠠⡀⠀⠀⠀⠀⠀⡠⠄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠰⡅⠀⠀⠒⠂⠀⠒⠂⠒⠀⢈⠆⠀⠀⠀⠀⠱⡀⠙⢄⡌⠒⢆⠠⢄⡀⠀⠀⡆
⠀⠀⠀⠀⠈⠑⠠⠄⢀⣀⣀⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠂⠀⠙⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⣀⡀⣀⠠⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣻⣟⣙⣫⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⡆
⠀⠀⠀⠀⢠⠒⢢⠀⠀⢸⠀⠀⢧⠀⠀⠀⡤⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠘⠀⠈⡆⠀⢸⠀⠀⢸⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣤⣶⠿⠛⠉⠈⣹⣿⣿⠈⠱⢆
⠀⠀⠀⠀⠀⢃⠀⠸⡀⢸⡀⠀⢸⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠘⠳⠿⠋⠀⠀⠈⠇
⡖⠐⡄⠀⠀⠘⠀⠀⢣⠀⡇⠀⢸⠀⣸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣤⣀⣀⣀⣠⡤⠴⠚⠒⠁
⠱⡀⠘⢄⠀⠀⠆⠀⠸⡀⡇⠀⢸⢠⠋⠀⣸⠁⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠹⣄⠀⠑⢄⠰⠀⠀⠉⠁⠀⠈⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠠⡀
⠀⠀⠙⢦⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠾⠉⠉⢙⠶⣌
⠀⠀⠀⠈⢷⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠎⠀⠈⡆
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⡄⠀⠀⢀⡠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠄⠘⠀⠀⠀⠀⠀⠙⠓⠈⠉⠀⠀⠀⣀⠔⠀⠀⢀⡴⠊⠁⠈⠉⠁⠉⠒⢄⠀⠀⠱⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⢠⠊⢀⣴⡊⠉⣂⡉⢂⣄⠀⠁⠀⠀⠙⢦⡀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢁⣤⠾⠟⠛⠻⠿⠛⠛⠿⣷⣄⡀⠀⠀⠀⠙⢆⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⢹⠒⠀⠀⠀⠀⠀⠀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠁⠀⠀⠀⠀⠀⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠉⠉⠀⠃⠀⠉⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⣠⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡉⣿⣿⣿⣿⡏⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣡⡶⠞⠳⠶⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⣿⣿⣿⣿⡇⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠶⢿⣿⣷⡌⣷⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⣸⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⣿⡆⣿⣿⢃⡿⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢋⣴⠋⣴⠞⣡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠸⣯⣉⠻⣿⣿⠟⢋⣽⡗⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣼⣿⣾⢃⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣴⣶⣬⣩⣴⣶⣴⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⣿⡟⣸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠟⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠻⢿⣶⣤⣤⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡿⠿⠿⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡏⣴⠛⡛⣶⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⠻⢦⡴⠏⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣿⣿⡿⢛⣛⡛⠿⣿⣶⣶⣿⣿⣿⣿⠟⠙⣿⣇⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠘⣿⡏⣴⣿⣿⣿⣷⣦⣉⡛⠛⠛⣋⣡⣾⣿⡌⠿⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⣀⡛⢡⣿⣿⠿⠛⣛⠛⠿⣿⣛⣛⠛⠿⣿⣿⣿⣄⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣿⣿⣿⡟⣡⣾⣿⣿⣿⣷⣎⠻⣿⣿⣷⣌⢻⣿⣿⡏⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠈⠻⡿⢠⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡄⠿⠋⠀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣂⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣋⣙⠻⣿⣿⡟⢸⠟⣿⣉⡛⢿⡆⠀⠀⠀⠀⠀⠀⢤⣄⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠉⠀⠀⣀⣤⣶⣶⣿⢃⣿⡇⣾⠩⠌⣷⢸⣿⢁⡏⢸⡋⠌⣻⠈⣿⠸⣷⣶⣦⣄⡀⠀⠈⠳⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⠃⠀⠀⣠⣾⣿⣿⣿⣿⡟⣸⣿⣷⣌⠓⣾⣇⣾⡏⢸⣿⣌⡛⠞⣋⣼⣿⡇⢻⣿⣿⣿⣿⣦⡀⠀⠈⢷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⡿⠡⡿⠿⠿⠿⠿⠿⢿⠿⠡⠄⠿⡿⠿⠿⠿⠿⠿⡿⠘⠿⠿⠿⠿⠿⠿⠄⠀⠈⠁



⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣠⠔⠒⠉⠒⠒⠊⠛⠒⠤⡀⠀⠀⢷⣀⣤⣤⣀⡾⠀⠀⣠⠔⠋⠋⠁⠀⠀⠉⠉⡓⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⢿⡤⠤⡤⠤⠤⠤⠤⠤⡤⠤⢬⡷⠂⢸⡇⠀⠀⢸⡇⢰⠊⠀⠀⠀⠀⢰⣿⣤⡀⠀⢁⠀⠈⢳⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⡇⠀⠀⠀⠀⠀⡶⠊⠁⠀⠀⢸⠇⠀⠀⠸⡇⠈⠑⢤⡀⠀⠀⠈⠛⠋⠀⠀⠄⣠⠔⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡾⠀⠀⠀⠀⢷⠀⠀⠀⠈⠓⠲⠦⠤⠤⠶⠒⠋⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠢⠄⠀⠤⠊⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠉⠙⠓⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠛⠉⠁⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠚⠒⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠛⠉⢀⣀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⡠⠐⠒⠒⠒⢄⠈⠙⢦⡀
⠀⠀⠀⡴⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢀⠎⣠⠖⠀⠑⠢⡀⠱⡀⠀⠹⡄
⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡏⢠⠠⡀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⠀⢣⡀⠀⠀⠀⠁⠀⡇⠀⠀⢻
⠀⢠⡇⠀⠀⠀⡠⣀⡀⠘⠃⠘⠀⠛⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⠀⠈⠑⣦⠀⠀⡰⠁⠀⠀⢸⡇
⠀⠸⡇⠀⠀⠀⢡⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡼⠀⡰⠚⠁⠀⡰⠁⠀⠀⠀⣸⠁
⠀⠀⢷⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢣⡀⠀⠀⠀⡸⠁⠀⠀⠀⢠⠏
⠀⠀⠈⢳⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⡸⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢣⠀⠀⣠⠃⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⠦⣄⡀⠀⠑⠒⠂⠒⠈⣀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⣀⢀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠈⠛⠢⣤⣀⣑⣘⣁⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠚⠛⢧⡀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢰⠁⠀⢣⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⣸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠸⢖⣲⣄⠀⠀⢠⡶⣲⠞⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠁⠊⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣄
⠀⠀⠀⠀⠀⠀⡞⠛⡇⠀⣿⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡟⠙⡇⠀⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡴⠚⠛⠓⢦⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⣿⠀⣧⡞⠓⡆⠀⠀⠀⠀⠀⠸⠁⠺⣤⡀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⡇⠀⠃⠀⠃⠀⠛⠀⣿⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡼⠁⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⣾⠀⣀⣤⡀⠀⠀⠀⣠⣄⡀⠘⡇
⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣥⠤⠉⠲⠴⠚⠁⢤⣥⠴⠃
⠀⠀⠀⠀⠀⠀⠙⠒⠲⠖⠒⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠛⠉⠉⠉⠛⢶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣚⡉⠀⠀⠀⠈⠉⠁⠀⠀⠀⢉⣓⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡾⠁⠀⣠⣤⠀⠀⠀⢳⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢮⣉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⣉⡴⠋⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣇⠀⠀⢿⣿⡿⠀⠀⢸⠃⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣄⣀⣠⣤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣽⣄⠀⠀⠀⠀⠀⣠⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠛⠋⠁

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

⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠐⠁⢀⡴⠚⠋⠛⠳⢤⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣒⣒⡒⠒⠂
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⢧⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣿⡿⣽⣿⣉⣿⣿⠶⣤⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⠘⡆⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⢻⣷⣛⠿⣋⣼⡟⢀⣨⡿⠓⠆
⠀⠀⠀⠀⠀⠘⠗⠢⠤⠤⠤⡀⠀⠱⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⡄⠀⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⡰⠃⠀⡟⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤
⠀⠀⠀⠀⠀⣼⡏⢀⡤⠴⠊⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠻⠴⠋⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡴⠶⠿⡶⠀⠀⣀⠎⠀⠀⣸⠉⠉⡆⠀⢀⡇⠀⠀⡇⠀⠀⡔⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠴⠊⠉⠀⠀⠀⠀⢹⠀⠀⢹⠀⢸⡇⠀⠀⡇⠀⢰⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⠀⠀⢸⠀⢸⠀⠀⠀⡇⠀⡸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡆⢸⡀⠀⠘⡆⢸⠀⠀⠀⡇⢀⡇⠀⠀⡾
⠀⠀⠀⢀⠔⠋⠁⠀⠀⠈⠁⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⠈⡇⠀⠀⡇⢸⠀⠀⠀⡇⢸⠀⠀⠀⡇
⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡄⡇⠀⠀⣇⢸⠀⠀⠀⡇⡎⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢇⢳⠀⠀⢸⣸⠀⠀⠀⣧⠃⠀⠀⢸
⠀⠀⠀⠈⠑⠢⠤⠤⠤⠤⠤⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡠⠴⠒⠢
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡰⠋⠀⠀⠀⡠
⠀⠀⠀⢀⠔⠊⠉⠉⠉⠓⠚⠉⠉⠉⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⠞⠁⠀⠀⡠⠊
⠀⠀⠠⠁⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⠎
⠀⠀⠁⢀⣴⣄⣸⠀⣀⣸⣁⣀⢠⢇⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⣀⣴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⣠⡴⠆⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢃⠀⣠⣄⡀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⣀⣀⡠⠊⠉⠉⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⣼⣇⣛⣡⣿⡀⠀⠀⠀⢸⠸⠿⡉⢊⡇⠾⢭⡑⢣⡇⠆⢸⣿⣿⣿⣿⣿⣿⡀⣤⢀⣾⡇⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⢰⡟⣻⡿⣟⢻⣧⠀⠀⠀⠐⣮⡇⢛⣇⡳⡞⡘⢳⣆⠳⡄⢸⣟⠙⠻⠿⠛⣹⣿⣿⣿⣿⡇⢸⣿⣿⢀⣄⣄⠀⠀⠀⣸⣿⡇
⠀⠀⠀⣾⢢⣿⠀⢹⣆⢻⣆⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢸⣿⡇⠀⡀⢠⣿⣿⣿⣿⣿⡇⢸⣿⣿⠈⣷⡏⣿⣿⣿⣿⣿⡇
⠀⠀⣸⡇⣾⣿⣿⣿⣿⣆⢻⡄⠀⢸⡟⠁⠀⣠⣄⡉⠙⢿⣿⣿⡇⠘⠿⠿⠾⠿⠿⠿⠿⠿⠿⠿⠇⢸⣿⣿⠀⣺⣢⡼⠉⣿⣿⣿⡇
⠀⠀⠿⠾⠏⠀⠀⠀⠀⠿⠿⠗⠀⢸⡇⠀⣾⡱⠎⣿⠆⢸⣿⣿⡇⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣿⠀⣮⡇⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⡛⠁⠀⢾⣿⣿⡇⠀⠀⣠⡾⣛⣽⣭⣙⢻⣦⠀⠀⢸⣿⣿⡆⢽⠅⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠴⠶⠶⡷⠶⠶⠂⠈⢻⡇⠀⢰⣯⣴⠏⠁⠈⠹⣧⣽⡇⠀⢸⣿⣿⣇⣀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠐⣄⠈⢆⠀⢆⠀⠀⠀⡀⠀⢸⡇⠴⠶⠶⡷⠾⠿⠖⠀⣼⡇⠀⢸⡛⣿⠀⠀⠀⢀⣏⣩⣿⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁
⠀⠘⢦⡈⢦⡈⣆⠸⡄⢦⠀⡇⠀⢸⣇⣀⣀⣀⣃⣀⣀⣤⣾⣿⡇⠀⢸⣿⣿⠀⠀⣴⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠰⣤⡙⣦⡳⣜⣆⣷⢸⢠⡇⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠘⣿⣿⣷⣤⣼⣿⣿⣿⣿⡆⢸⣿⣿⠟⢩⡍⢈⣭⡛⢿⣿⡇
⠀⠘⢦⣝⣾⣿⣿⣿⣿⣿⣿⣷⡇⢸⡀⢀⣾⣿⣧⣿⣿⣿⣿⠿⡇⠀⠀⠈⠛⠻⠿⠿⠿⠛⠉⠋⠀⢸⣿⠃⣶⡄⠀⠈⠛⢃⡈⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⠋⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⣀⠾⠗⣀⣈⣵⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣉⣿⣿⣿⣷⣶⣿⣿⣇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⡇
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢸⣿⡿⠛⠛⣿⣿⠉⣿⣿⣷⡇⠀⠀⠀⢀⣀⣴⣶⣦⣀⡀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⡇
⠀⢸⣿⠉⠉⠉⠉⠉⠛⢿⣿⣿⡇⢸⠏⡠⠀⣠⣿⣿⣿⣿⡟⣻⡇⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⢸⣿⠀⣰⠂⣹⠘⢷⡄⠙⣿⡇⢸⣿⣶⣿⣿⣿⠻⡛⠛⢿⣿⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⢸⣿⠀⣾⠀⢼⠀⠘⡇⠀⣻⡇⢸⠟⠹⢻⣿⣿⡄⠈⠀⠀⢹⡇⠀⠀⣤⠐⠇⣤⠰⠃⣤⠀⠀⠀⠀⢸⡏⣹⡆⠀⠀⠀⣟⢉⡇
⠀⢸⣿⠀⠓⠀⢸⠀⠐⡋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁⠾⠂⠀⠀⠀⠀⠀⢸⡟⢻⣇⠀⠀⢰⡯⣹⡇
⠀⢸⣿⠐⠶⠦⠈⠰⠛⢁⣼⣿⡇⠀⠀⣾⠻⣷⠀⠀⣾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣜⢿⣦⣴⡟⣥⡿
⠀⢸⣿⣤⣤⣤⣤⣴⣶⣿⣿⣿⡇⠀⠀⣿⠠⡟⢀⣼⠃⣼⠏⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⠻⣿⠶⣿⠋⠁
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⣿⣿⣿⣿⣡⣿⠋⠀⠀⠀⢸⣿⡋⠉⠩⣉⡉⠙⠛⢿⣿⡇⠀⠀⠀⠀⠀⣿⣭⣿
⠀⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⠄⣿⠻⣿⡹⣧⡀⠀⠀⢸⣿⠁⣸⣿⣛⡻⣿⣄⠈⢿⡇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⢸⣯⠀⠀⢠⣿⡿⢟⣝⢿⣿⡇⠀⠀⣿⡁⣿⠀⢸⣇⣺⣇⠀⠀⢸⣿⢀⠻⣷⣭⣼⣿⠩⠀⣸⡇⠀⣄⣀⣀⡀⣤⡀⣠⣤⣀⣤⡄
⠀⢸⣿⣴⣶⣾⣿⣯⣿⣿⣿⣾⡇⠀⠀⠙⠓⠛⠀⠈⠛⠛⠋⠀⠀⢸⣿⠀⣤⠀⠈⠁⢂⣀⣴⣿⡇⢠⣧⣿⡏⣹⣯⣙⡏⣿⣭⡞⡁
⠀⢸⣿⡿⡹⣿⣿⣿⣾⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢈⣷⠺⣿⣿⡴⢻⣿⣿⢯⣻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠻⣿⣿⣿⣿⡇
⠀⠀⢰⣤⣤⣤⣤⣤⣧⠀⠀⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⢸⣿⡟⠁⠀⠀⢀⣿⣟⣻⣿⡇⠀⠀⢰⣶⡄⣾⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡑⠃⠀⠀⠀⠀⠋⣽⣿⠀⠀⢸⡟⠀⠉⠀⠹⣧⠀⠀⢸⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⣙⡿⣯⣠⣀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⣤⣀⣀⣴⣿⣿⣿⠀⠀⢼⡇⠀⠀⠀⢀⣿⠄⠀⢸⣿⣇⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⣾⣿⣷⡏⣿⣿⡇
⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠃⠀⠀⠀⠐⠋⠀⠀⢸⣿⣿⣦⣄⡀⠀⠀⣠⣿⣿⡇⠀⠀⠈⠉⠋⠛⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡜⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⡿⠿⢿⣿⠿⠿⠿⢿⣿⠟⢿⠀⠀⠀⣶⣤⣤⣤⣠⣤⠀⠀⣿⡟⢿⣿⡈⠻⣿⣿⣇⠙⣿⡇⢠⣴⣦⡄⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡿⠿⣿⠿⢿⣿
⣿⣶⣶⡄⣀⠀⠀⠀⣶⣶⣿⠀⠀⢠⣿⣿⣿⣿⣿⣯⠀⠀⣿⣷⣤⣿⣿⣾⣿⠙⢻⣿⣿⡇⠈⣿⣿⣟⣄⡀⢀⣠⡄⠀⢸⣿⡥⠀⠀⠀⠀⠀⠀⢶⣿
⣿⣿⣿⢁⠿⠂⠀⠀⢻⣿⣿⠀⢀⡿⠿⣿⡿⣿⡿⠿⡆⠀⣿⡟⢿⣿⣧⠈⣻⣷⣴⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⡇⢀⡀⣀⣀⢀⡀⢸⣿
⣿⣿⣿⡀⠀⠀⠀⠀⣼⣿⣿⠀⠈⢷⣶⣿⣿⣿⣷⣴⠇⠀⣿⣧⣠⣿⡿⢿⣿⣿⠙⢿⣿⡇⠀⠀⠀⢙⠛⠛⠋⠀⠀⠀⢸⣿⣧⡀⠀⠻⠟⠀⣠⣾⣿
⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⣿⣿⣿⣿⣧⣀⣿⣿⣷⣼⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⣩⣭⡉⠉⠉⠉⠉⠉
⠀⠀⢀⡤⠖⣒⠒⠢⣄⠀⠀⣿⣟⣿⢛⣻⣿⣿⣿⣿⣿⣿⠀⠀⢺⣷⣤⣤⣤⣤⣾⠂⠀⢸⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⡇⠀⠀⣿⣿⣿⠀⢸⣿
⠀⡴⢁⠖⣉⣭⣉⠳⣌⣧⠀⣿⠋⠁⠈⠉⢻⣿⣿⠛⠛⣻⠀⢴⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣶⣿⡿
⣴⣿⣿⣿⣧⠀⣸⣿⣿⣿⡶⣿⠀⠀⠀⠀⠈⠻⠟⣣⣾⣿⠀⠈⠙⢿⣿⣿⣿⣛⠟⠋⠀⢸⣿⡛⠛⠓⢓⢚⡂⠀⢀⣿⡇⠀⢰⣿⣿⣿⡏⠁
⠈⠙⠛⠛⠋⠀⠈⠉⠙⠁⠀⣿⣶⣤⣤⣤⣤⣴⣾⣿⣿⣿⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢇⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⡿⡻⣿⠀⢀⣀⣀⠀⠀⠀⢀⣀⡀⠀⣿⣇⣉⢿⣿⣿⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠛⠛⠿⠿⣿⣿
⣿⣿⣿⠛⣿⣿⢰⡹⡇⣿⣜⠀⢿⣏⣹⣦⣶⣤⣿⣉⣿⠀⣿⣿⣜⣞⣿⣿⣿⣣⣀⠘⣿⡇⠀⠀⠘⢀⣤⣄⡠⠂⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠙⢿
⡿⠻⣿⢸⣎⢿⢸⣷⡀⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⡏⠁⠀⣿⣿⣿⡞⠘⠛⠓⣱⣿⣿⣿⡇⠀⠀⠄⢿⣿⣿⡇⡀⠠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡇⣷⡍⢸⣿⣆⣸⣿⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⣿⣿⣿⢁⠀⠀⢀⠘⣿⣿⣿⡇⠀⠠⠒⠀⠍⠩⠰⡄⠀⠀⢸⣷⡆⠀⠁⠈⠀⠉⠀⢘⣿
⣧⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣿⣿⣿⣦⣀⣁⣈⣼⣿⣿⣿⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣿⣷⣶⣤⣴⣦⣤⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣷⣦⣴⣾⣿⣤⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠟⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⠟⠁⠀⠙⠛⠛⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⣠⣢⢏⠜⣿⣿⠉⠋⠀⠀⠀⠀⠋⣁⣀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠘⠷⣵⠷⣚⠿⠿⠦⠀⠀⠀⠀⠀⢿⣾⠇⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⣄⣼⡠⡠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠟⠛⣉⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠀⠀⠋⢠⣎⠈⠁⠈⠦⡀⠀⠀⠀⠀⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠜⠁⡜⠀⠀⠀⠊⠹⣎⢧⡀⠀⠀⠈⡟⠒⠢⠤⢤⡤⠃
⠀⠀⠀⠀⠀⣠⣋⣀⠜⠀⠀⠀⢸⠀⠀⠙⢦⣙⠶⣤⠼⣳⢦⡀
⠀⠀⠀⠀⣼⣽⢾⠋⠀⠀⠀⠀⣆⠀⠀⠀⠀⠉⠛⠓⠛⠋⠀⢳⡀
⠀⠀⠀⠀⠘⡷⠁⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⢧
⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡇
⣤⣤⣤⣤⣤⣼⣦⣤⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣯⣵⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⣀⠠⡴⠒⠀⠉⠀⠒⢦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⣀⡀
⠀⠀⠀⢀⡠⠐⠉⠀⠈⠀⡐⠉⡐⠹⡆⠀⢡⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠀⠈⠒⠁⠀⠀⠁⠒⠠⢀
⠀⠀⢐⠁⠀⠀⠀⠀⠃⠀⢪⣍⠀⡠⡧⠀⠨⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⢠⣂⡥⠀⠠⠄⢀⠠⠤⢤⣄⡤⠤⢄⡀⠠⠄⠀⢈⣀⡄
⠀⠀⠀⠉⠃⠄⡀⠀⠈⠢⡀⠉⠉⠉⢀⠐⠁⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠉⠉⡃⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢀⠁⠀⠈⡁⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢧⠀⠀⠀⠀⠀⠀⠀⠤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⡀⠄⠊
⠀⠀⠀⠀⡀⠤⠤⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄
⠀⠀⡠⠈⢀⠤⠀⠀⠤⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠀⢹⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⢄
⠀⠠⠀⣔⠀⡀⠀⠀⠀⠀⠢⠀⠡⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢂
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆
⠀⢀⠀⢂⠀⠀⠀⠀⠀⢀⠔⠀⠌⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠈⠀⠀⠀⣧⠌⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠀⠀⢂⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠊⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⢀⠌⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢠⠀⡁⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⡄⠂⠀⢠⠁
⠀⠀⠢⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠐⠠⠤⠤⠐⠈⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢹
⠀⠀⠀⠀⠀⠀⠀⢠⠊⢢⠀⠀⠀⡇⠀⢸⡄⠀⠀⢀⠔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢇⠀⠀⡇⠀⠈⡇⠀⠀⡞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣰⣶⡿⠛⠉⠈⢹⣿⣿⡇⠉⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⡀⠀⡇⠀⠀⡇⠀⢠⠃⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠈⠳⠼⠟⠁⠀⠀⢹⠄
⠀⠀⢠⠖⠢⠀⠀⠀⠰⠀⠀⢱⠀⡇⠀⠀⡇⠀⡎⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣄⣀⣀⣀⣀⣀⣀⣠⠴⠶⠚
⠀⠀⠈⣆⠀⠡⡀⠀⠀⠀⠀⠸⡆⡇⠀⠀⡇⡸⠉⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠈⢧⠀⠈⠢⡀⢠⠄⠀⠓⠛⠀⠀⠓⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⠄⡀
⠀⠀⠀⠀⠈⢳⡀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠆⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⡶⠒⠒⢦⣌⠂
⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠃⠀⠀⢀⡏⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠐⡀⠀⡔⠀⠀⠀⠀⠘⣦⣀⠤⠴⠊⠀⠀⠀⠑⠀⠀⠀⠀⢀⠤⠄⠤⠤⠒⠤⣀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠀⠀⢀⠜⠁⢀⣀⢄⡀⢄⡀⠀⠑⠄⠀⠀⢣⡀⠀⠀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⢀⠊⢀⣴⣯⣤⣤⣧⣤⣬⣶⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠉⠉⠙⠋⠉⠉⠙⠻⣷⠄⠀⠀⠀⠀⠙⢦⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⢀⣀⡀⠀⡀⠀⣀⡀⠄⠁⢠⠀⠀⠀⠀⠀⠀⠓⠤⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣤⡤⠤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣪⠭⠭⣕⢢⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⢒⣋⣓⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠻⡒⢄⠈⡇⡇
⠀⠀⠀⠀⠀⠊⡠⠖⢩⢏⢩⣍⡏⠓⢤⡙⠀⠀⠀⠀⠀⠀⣠⣔⣋⡀⠈⠉⠁⠀⣉⣒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⡇⡇
⠀⠀⠀⠀⠀⠘⠢⣄⠈⠮⣛⡥⠃⢀⡤⠋⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠉⠒⠊⠉⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⡸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢌⣒⣊⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠤⠀⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢾⢹⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⡇⢸⢸⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣉⣤⣤⣭⣥⣤⣬⣱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠃⠘⠘⠀⡟⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠫⡉⢻⠉⢹⠉⠙⡟⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠘⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢱⣶⣶⣶⣶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢌⠑⠤⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠄⣒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⡤⡴⠒⡒⢦⢤⡀⠀⠀⠀⠈⢨⡀⠅⢸⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⢄⠂⣈⠁⠀⠀⠈⢂⠀⠀⠈⠻⣿⠛⠛⠛⣿⠟⠁
⠀⠀⠀⠔⢇⡇⡄⢶⡿⢀⢸⡨⠂⠀⠀⠀⠰⢄⠀⡄⠀⠀⠀⡄⠀⠀⠀⢡⠀⠀⠀⠀⢀⠜⡡⢂⠀⠀⠀⡸⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋⠉⠀⠀⠀⠀⠀⠈⢁⠜⠀⠀⠀⠌⣀⠀⠀⢀⡀⠁⠀⠈⠐⠥⠊⠉⠣⠴⡠⠊⠀⠀⠀⠀⠀⠈⠢⠄⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠈⠉⠙⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡍⣉⣁⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⡶⠶⢶⢶⡲⠶⠶⣶⢶⣶⡶⠶⠶⢪⡄⠿⢆⣭⠟⢷⣶⡶⢖⣶⠖⣖⣶⡶⣶⠶⣤⣤⣤⣤⣴⣤⣭⣦⣭⣤⣥⣤⣤⣤⣭⣤⣭⣤⣭⣤⣭
⢿⣤⣼⠿⢇⡰⣬⢘⣏⣴⢄⣄⠐⣷⡞⢱⣟⠍⡙⢦⠀⡉⢻⢰⡎⠛⢰⡎⣿⠀⣿⡇⡿⠀⣉⡀⠌⠠⣌⡉⠁⣉⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⠛⠳⠀⣤⣾⡟⠋⣍⠻⣷⣦⡉⠇⢁⣏⣳⠌⠃⢸⠀⡛⡏⣼⠀⠀⠀⡇⢰⡁⣿⡇⢿⣬⡥⠀⠀⠀⠀⢻⡿⠿⠿⠿⣿⠉⠛⠉⢿⢿⡿⠿⠿
⠘⣿⣴⡛⢷⣿⣇⠛⠟⣠⣹⡾⠛⣊⡿⠆⣝⡠⢀⠆⣰⡕⣡⠟⠀⠀⠀⠱⡜⠇⣈⣑⡴⢋⣴⣣⠄⠀⢀⡼⢣⣮⣇⢂⢿⠀⠀⠀⡸⠸⠪⣿⠛
⣶⣿⠯⠧⡵⣌⣍⠛⠻⣭⣡⣶⣞⣿⠇⢸⣇⣠⠎⢠⣇⠀⣏⣤⡀⠀⢠⢄⡘⢨⢩⣼⢲⡋⢬⡓⠺⡴⠋⣴⣇⣀⣿⠙⢨⡗⣒⢪⡄⣕⡁⣿⣾
⠻⣿⣶⣦⣤⣼⣿⠶⠶⢿⣯⣤⣽⣿⣶⣷⣤⡤⣀⣾⣉⠉⡦⠵⠏⣍⠱⠄⣀⡀⣯⠢⡌⣁⣬⠟⣋⡸⠃⣀⣿⡟⢋⡇⢸⢦⠛⢖⣌⢷⣳⣟⡿
⣽⠅⠁⠀⠚⠉⠏⢳⣤⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⠴⠒⣚⡓⠒⠤⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢱⠀⠀⠀⣀⡀
⠀⠀⢀⡜⡡⠚⣉⠁⠈⠉⠲⡌⡆⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⣴⠛⣆⠀⠀⠀⡇⠀⠘⡇⠀⢸⠁⠘⡆
⠀⠀⡞⠘⢁⠞⠁⡀⠀⠀⢠⠇⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⡇⠀⡞⡴⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢧⠀⠀⣷⠀⠀⣧⠀⡟⠀⠀⡇⠀⠀⡤⢤
⠀⠀⢹⠀⢹⡳⡀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⡆⠀⢹⠀⠀⢸⠀⡇⠀⠀⡇⠀⢸⠁⠸⠂
⠀⠀⠘⡆⠀⠑⠈⠙⠢⠽⠂⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢧⠀⠀⢳⠀⢸⠀⠀⢸⠀⡇⠀⠀⡇⠀⣸⠀⢸
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢸⡀⠀⠸⡆⠸⡄⠀⢸⠀⡇⠀⠀⡇⢀⡇⠀⣸
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⡠⠀⠀⠸⡌⢦⡀⠀⠠⠖⠒⠲⠤⢀⡇⠀⠀⣇⠀⠀⢣⠀⡇⠀⠸⣄⡇⠀⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢹⠀⠀⠈⠛⠁⠀⠀⠉⠀⠀⠀⢧⠞⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⠤⣀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠴⠚⠉⠉⠑⠒⠉⠉⠉⠲⢤⡀⠀⠘⢆⠀⠀⠈⠑⢄⠀⠀⠀⠈⡇⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠠⠵⠒⠒⠛⠒⠲⠤⠖⠚⠛⠒⠒⠥⠀⠀⠀⠑⢦⡀⠀⠈⢧⠀⠀⢸⠃⠀⠒⢄⡀⠉⠉⠒⠢⢤⣀⠀⠀⠀⢸
⠀⠀⠀⠙⠒⠦⠤⠤⠤⠤⠤⠤⠤⠔⠚⠁⠀⠀⠀⠀⠀⠙⡄⠀⠀⠑⠒⠋⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠊⠁⣀⡤⠴⠒⠚⠋⠉⠉⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡸
⠀⠀⠐⠚⠉⠁⢀⡤⠖⣒⣒⡒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⢞⡥⢚⠍⣩⠍⠉⡍⠒⢬⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠡⣄⣈⢧⣈⣋⣰⣃⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣿⣿⡏⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠛⣿⣿⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⢹⣿⣀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣼⣿⣿⣀⣿⣿⣷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣈⣿⣿⡟⢻⣿⣿⣿⢙⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢋⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⠉⣿⣤⣿⣿⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⣹⣿⣿⡋⢻⣿⣧⣸⣿⣷⡀⠀⣿⣿⣤⣿⣿⣁⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⠛⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣇⣹⣿⣿⣿⡄⠘⣿⣿⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣷⡀⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⠋⠁⢀⣿⣿⣿⣿⣿⣿⠁⠀⢰⣾⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿
⢀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣷⣤⣿⣿⣿
⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠸⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠀⠀⠘⠿⠿⠿⣿⠀⠀⢹⡿⠿⠿⠿⠿⠛⠋⠉




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

⠀⠀⠀⢀⠠⠀⠐⠒⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠈⠱⣄
⢀⠎⠀⠀⡠⠔⡒⠛⠒⠢⢤⡀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠂⠀⠀⠢⠀⠀⠀⠀⠘⣷
⢸⠀⢠⠎⠀⠈⠀⣴⡄⠀⠀⠙⠕⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠘⡀⠐⠤⣀⠐⠀⠈⠀⠀⢀⠠⠂⠀⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠱⡀⠀⠀⠉⠂⠒⠐⠈⠈⠀⠀⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⣰⣻
⠀⠀⠈⠲⢄⣀⡀⠀⢀⣀⣠⡴⢿⡿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠈⠒⢄⡀⠀⠀⠀⣀⣀⣤⢾⡿⠋
⠀⠀⠀⠀⠀⠙⠛⠯⠿⠿⠟⠛⠁⠠⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠉⠛⠯⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡎⠀⠀⠀⠢⡀⠀⠀⠀⢠⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠘⢤⡀⠄⠀⠄⠃⠀⠀⢀⣜⡿⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠒⡄⠌⠐⢤⡀⠀⠑⢤⣀⠀⠀⠀⠀⢀⣀⣴⣯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄
⠀⢀⠄⠀⠤⠀⠀⡔⢰⠃⠂⠀⠀⠙⣦⠀⠀⠈⠛⠻⠿⠯⠿⠿⠛⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠈⠀⠀⠰⠑⠆⠁⠈⠀⢀⢈⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⠐⠐⠊⠉⠀⠠⡀⠀⢿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠎⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡂⠀⠀⠀⠀⠀⠒⠒⢂⠏⠀⢸
⠀⠐⡀⠀⠀⠘⠀⠀⠀⡠⠂⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠂⠄⠀⠀⠀⠄⠂⠁⠀⢀⣿
⠀⠀⠀⢄⠀⠀⠀⠀⠐⠀⠀⠀⢀⣴⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡾
⠀⠀⠀⠀⠉⠲⢤⣤⢤⡴⣶⣾⡯⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡤⣤⣶⣶⣿⠝⠋