Kingship Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⡤⠟⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠈⡌⢷⣄⣰⡆⢹⡄⡀⠀⣤⠀⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢆⠳⠄⢀⣼⣿⣦⣈⣉⣠⣿⠀⢀⣠⠞⣰⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣚⣽⡷⠾⠿⠷⠶⢟⣽⣷⣦⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠟⠄⠁⠀⠀⠀⠀⠀⢬⣙⠻⢾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡾⢲⣶⡆⠒⠶⣟⡛⠂⢿⣷⣦⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠋⠁⠀⠈⠉⠉⠀⠘⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣠⣄⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣿⠟⠻⢿⣦⡀⠀⠀⣼⣿⣷⣝⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠙⠛⠂⠒⠉⠿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢿⣿⣤⣤⠀⠀⠀⠀⢿⣿⣿⣷⠂⣤⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⠛⠋⠀⠀⠀⣀⣤⣾⠛⠁⠀⠤⡴⣦⠀⠀⢀⣴⣿⡿⠛⢉⣽⣿⣿⣿⣶⣿⣷⠶⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣷⠐⣾⣿⣿⣿⣿⡇⢠⣠⡄⢘⠃⠈⠻⣶⣿⣿⠋⠀⣰⡾⠛⠛⠉⠉⠀⠋⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⡇⣿⠟⢠⡟⠀⠀⢀⣿⣿⠏⢀⣾⣿⠃⠀⠀⠀⠀⠂⠀⠀⡾⠿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠛⠉⠀⠀⠀⢸⣿⣿⣿⠇⣿⠀⣸⣥⠀⠀⢼⡻⠏⠀⣾⣿⠟⡁⣠⡀⣠⠃⠀⠀⣰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⡄⠀⠀⢀⣼⣿⣿⣦⢾⣷⠈⣉⠤⠀⢠⣶⣷⠞⣰⣿⡇⢠⠇⠌⢠⣯⠀⠀⢠⡧⠀⠀⢸⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣦⣿⣦⣿⡀⠀⣿⣿⣤⣾⣿⣿⡇⠀⠄⠀⠸⠁⠀⢀⣿⠃⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⠉⣿⣿⣿⣿⣿⡟⠁⢈⣫⠛⡛⠻⠚⣉⠽⠉⢙⣿⡿⠇⠀⠁⢀⣦⢠⡞⢸⣿⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣷⠈⣿⣶⡇⠀⢀⣿⣿⣿⣿⡿⠋⣄⠀⢠⣿⣋⣿⠃⢀⣇⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⡿⠻⠟⡇⠠⣟⣿⣿⣿⣿⣿⣿⣰⣥⡙⣧⣀⣸⣿⣿⣿⢿⡷⠀⠘⠃⡹⢡⣿⠋⣸⣿⣿⡆⠀⠀⠀⠀⢸⣏⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⠀⠀⠀⠀⠉⣻⣿⣿⣿⡟⢣⢾⡇⢸⡁⠀⣀⡑⣨⣿⡟⢛⡀⠀⠾⢤⣰⣏⡠⢀⡋⢿⣷⣶⠶⠂⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣆⢨⡇⠸⠧⠄⢾⣿⣿⣿⣷⣿⡆⣾⣤⣼⡧⣿⣷⣿⣁⣾⣟⣡⣤⠴⠤⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡃⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠙⢿⣿⠙⠛⠙⠛⠐⠚⠈⠉⠛⠫⣿⣥⣤⣤⣤⡄⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣀⣀⡀⢀⣼⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣸⣩⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠀⣻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇⣀⡀⠀⠀⠀⣀⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣿⣷⣿⢿⣿⡿⣿⣿⣧⡄⠀⣤⣭⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⣽⢿⣽⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡇⠐⠛⠿⠋⠀⠀⣿⣶⣿⢣⣿⠀⠀⢸⣿⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠋⠀⠂⠚⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣿⠀⠀⢰⠏⠏⢹⣿⡿⠀⠀⠀⠀⠀⠀⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢙⡿⢻⠀⠀⢸⡇⠀⠈⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⠍⡀⢸⠀⠀⠸⠀⠀⠀⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠆⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠠⡀⠀⠄⠀⠈⢀⠐⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠈⠘⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢠⠀⠀⠠⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠠⠀⠀⠂⠀⠛⠘⠋⠂⠀⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⠀⡀⢂⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠊⠀⠈⠁⠁⠉⠂⠄⡀⠀⠈⠀⠀⠀⠈⠉⡀⠄⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠐⠂⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠑⠢⠄⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢤⡀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠉⠡⠦⠂⠑⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠈⠀⠈⠓⣄⠀⠀⠀⠑⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠒⢭⣔⣒⣒⠂⠝
⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡀⠀⠀⠀⠁⠒⠠⠄⡀⠀⡠⠃⠀⠀⣠⠎⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⣀⠴⠼⣉⠉⡉
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠈⠲⠦⣤⡀⠀⠀⠀⣈⠛⠠⠔⠆⠊⠁⠀⣠⠞⠀⠠⢀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠳⣇⠠⡄
⠀⠀⠸⡄⠀⠀⠀⠀⠀⣄⣰⠠⡄⠀⠀⠖⠀⠀⠀⠈⠛⠶⠤⣤⣤⠤⠖⠋⠉⠛⠒⠒⠶⠒⠚⠉⠈⠃⠄⣠⠆⠀⠉⠳⠦⠄⠀⠀⣤⣠⠴⠃⢀⡇
⠀⠀⠀⠙⢦⣄⡘⠑⠎⠀⠉⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠚⣷⣤⣄⣀⣤⠀⠀⠀⠐⢄⠀⢣⡄⠀⠀⠀⠀⠀⠀⠒⡏⠑⠲⠒⣉⡄
⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢳⡦⣀⠀⠀⣰⠆⠙⢄⣀⠀⠀⣄⠀⠄⠁
⠀⠀⠀⢀⠄⠠⣗⠢⡦⠀⠀⠈⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣤⣤⡿⠿⠟⠛⡛⠀⠀⠀⠀⠀⠀⣷⣋⠤⠊⠀⠀⠀⠀⠈⠀⠀⡆
⠀⠀⠀⠘⠈⠁⠀⠃⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠔⠀⠀⠀⠀⠀⠀⢠⠇⢀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁
⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⣀⠜⠁⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡁⠔⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⢰⠀⠀⠀⠀⠀⠀⢨⠀⢸
⠀⠀⠀⠀⠀⠀⡏⠈⠁⠀⠉⠀⡀⠀⠀⠀⠈⠁⠒⠠⠤⠀⠀⣀⣀⣀⣀⡀⠀⠤⠄⠐⠂⠉⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠏⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⡶⠾⠿⢿⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⡏⠉⠁⠀⠀⠈⡹⠁⠀⠀⢸⠋⠀⠀⠀⠀⡌⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢸
⠀⠀⠀⢠⠎⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠁⠀⢤⠀⣨⡆⠀⠀⠀⠀⡇⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣠⠏⠀⠀⠀⠀⠀⡞⠀⠀⠄⠀⢠⡄⣄⣘⠀⠀⠀⠀⢸⡄⢠⡼⢧⠀⠙⢦⠀⣀⠀⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠆
⢀⡴⠁⠀⠀⠀⠀⠀⣸⠁⠀⠀⠆⠀⠀⠳⡄⠉⠢⣄⢀⠞⠁⢀⠎⠀⠈⡆⠀⠈⠋⠀⠀⣠⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠻⢤⣀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⠋⠀⠀⣾⠠⠄⠐⡷⠀⠀⠀⠀⢠⠃⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠈⠉⠁⠒⠒⠒⠂⠀⠀⠁⠉⠉⠉⠉⠉⢻⡄⠀⠀⠀⠀⡇⠀⢀⠔⠁⠀⠀⠀⠀⠞⠀⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠀⠉⠓⠀⠀⠀⠀⠙⢦⠳⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠼⠁⠀⠀⠀⠉⠒⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠐⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠰⣦⡄⣠⣄⠀⠀⠀⣿⣿⠇⠀⠀⠀⠙⠛⣿⣿⡛⠛⠁⠀⠀⠀⣿⣿⠇⠀⠀⢠⣤⣀⣴⡶
⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⣰⣿⣿⣧⡀⠀⠀⠀⢀⣾⠉⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣅⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡄⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⢻⡟⠛⠛⠛⢶⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡏⠀⠀⣠⡴⠟⠁⠀⠀⠀⢸⡇⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⡇⠀⠘⠃⠀⠀⠀⢠⡶⠛⠛⠃⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣴⠞⠋⠀⠀⠀⠀⠀⠀⢠⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣤⡙⢳⡄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃⢸⡧⠤⠼⠿⠦⠁⠀⠀⣾⠛⠷⠶⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⣧⠀⠀⠀⠀⠀⣼⠃⠀⣠⡾⢷⣄⠀⠀⠀⠠⣟⠛⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣩⡴⠟⠁⠀⠀⠈⠻⢦⣀⠀⠙⢳⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠳⣄⠰⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠆⠁⢀⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠀⠰⢶⡶⣶⠶⢶⡶⢶⡶⠶⣶⢶⡶⠆⠀⠀⣸⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢰⡏⠀⠀⢸⡇⣿⠀⢸⡇⢸⡇⠀⣿⢸⡇⠀⠀⠘⠉⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠀⠀⢸⡇⣿⠀⢸⡇⢸⡇⠀⣿⢸⡇⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⢀⣸⣇⣿⣀⣸⣇⣸⣇⣀⣿⣸⣧⡀⢰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡄




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣴⣿⡇⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⠻⠿⠿⢿⣿⣿⣷⣦⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣬⣛⣛⠛⢛⠿⠷⠶⣈⣭⣥⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣌⣿⣿⣛⢛⡻⢿⣿⣷⣎⢻⣷⣾⣿⣿⡏⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣿⣿⣿⣿⣿⣿⣿⣿⣌⣩⣬⣿⣿⣿⣇⢿⣿⡝⢿⣵⡟⣱⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠛⠁⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡌⣥⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡄⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⢋⣿⡿⣸⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⣄⣀⣤⣾⣿⣿⡿⣋⣴⣿⡿⣡⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠿⠿⠟⣋⣵⣾⣿⡿⣫⣴⣿⣿⣿⢣⣿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⣴⣿⣿⡿⣋⣵⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠡⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⡿⣰⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠙




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣰⣿⣿⣷⣤⣤⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣛⣿⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⡘⢻⡿⢂⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡴⠋⠈⠉⣻⣟⠋⠉⠈⣧⠀⠀⠀⠀⠀⢠
⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠁⠀⠀⢠⣤⣽⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⣀⣆⠀⠄⠀⠀⠀⢠⣷⣿⡶⢲⡆⠀⠀⠀⣴⠙⣛⣦⠤⠀⠀⢰⠖⢶⣿⣿⡄⠀⠀⠀⠠⠀⣢⣀⡄
⠀⠀⢠⡿⡿⡄⠀⠀⠀⣴⠛⠛⠿⠄⠀⠃⠀⣀⢂⣭⣸⣏⣭⡀⣀⠀⠘⠀⡰⠿⠛⠛⣦⠀⠀⠀⣠⢿⢯⡄
⠀⠀⠈⠀⠀⣾⡂⡠⠀⠈⢀⣤⠴⢖⠓⠋⢉⣉⣉⣉⣉⣉⣉⣉⣿⡿⠿⠛⠲⠦⣄⡈⠁⠀⣅⢰⡿
⠀⠀⠀⠀⠀⠈⠅⣻⡷⠋⣁⡤⢶⠞⠛⠛⣉⠀⢠⣾⣀⣀⣷⡄⠀⣉⠛⠛⠲⡦⢤⣈⣱⣾⣟⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⡁⠀⢠⣤⣴⣾⣷⣶⣶⣭⣷⣾⣭⣶⣶⣾⣷⣦⣤⡄⠀⣈⠻⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢞⣹⢾⠋⠋⠉⣀⣀⣠⣤⣤⣤⣤⣴⣶⣏⣁⣉⠉⠙⠙⠷⣏⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠈⠉⠧⠄⠀⠀⠀⠀⠀⠈⠙⠛⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣷⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢷⣦⣀⡀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⣀⣤⡾⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⠟⠉⢻⣶⣿⠋⢸⡟⠀⠀⠀⠀⢺⡇⠙⣿⣶⡟⠉⠻⣦⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣴⣿⣿⣿⣆⠀⢀⣀⣾⣧⣀⡀⠀⣰⣿⣿⣿⣦⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣧⠈⣻⣿⣿⡟⢁⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣀⠈⠙⢿⣿⣿⠻⡆⢉⣹⣏⡁⠸⠻⣿⣿⡿⠋⠁⣀⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠈⠛⠛⠋⠀⣾⣿⣿⣷⠈⠛⠛⠛⠁⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⣿⡾⠛⠛⢻⡟⢻⡟⠛⠛⢷⣿⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠶⢠⣿⠀⠚⠂⢸⡇⢸⡇⠀⠛⠀⣿⡄⠶⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠶⠘⣿⠀⢛⠀⢸⡇⣼⡇⠀⡛⠀⣿⠃⠶⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠰⠆⢻⣆⠘⣁⢸⡇⣿⡇⣈⠋⣰⡟⠐⠆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠙⣷⣜⣃⣿⣦⣉⣸⣷⣿⣄⣉⣴⣿⣈⣣⣾⠋⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠈⠂⠀⡀⢀⣺⡇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢺⡅⠠⣠⠋⢸⡎⣣⣴⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠐⣸⠧⢘⡀⣤⣀⡿⠵⠎⠠⠂⠀⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⢃⡰⢛⠐⠉⣀⠠⣿⣛⣥⣀⠯⣀⣠⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⠀⠹⡄⣛⢛⣽⣿⡓⢀⣋⣸⣿⣿⣿⡿⠚⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⠘⠶⣄⠾⣋⢿⢃⡟⣜⡛⠉⠁⠈⠌⠀⠀⠀⠀⢀⣤⢴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢹⣦⢡⣸⢯⠼⣿⠀⢢⡄⢀⡤⢀⠀⠀⢀⡋⢻⣿⣿⣿⣿⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⡏⢍⣧⡰⠁⠀⣪⣷⠘⠫⠀⠀⠀⡘⠁⠀⢹⣟⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⠈⡼⠟⠢⠛⠉⠀⠀⠀⠀⠀⠀⢨⣐⣶⡞⠉⢹⣿⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⢤⣴⣤⣦⡞⢈⣙⣽⣤⠶⠾⠂⡂⠀⠓⢂⡲⣦⡵⠔⡧⠀⠀⠁⠛⢶⣤⣴⣤⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢧⡟⢻⣿⣿⣎⠙⠂⠉⠻⣿⣏⣴⣷⣤⣾⣷⣮⣯⡿⠁⢨⢀⡴⠊⠀⠀⠀⠈⢿⡟⠛⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣙⡳⢦⣾⣿⣿⡿⠤⣤⣤⣤⣀⠈⠉⠉⠛⠛⠛⠾⢛⡑⠛⠋⠉⠀⠀⠀⠀⠀⠀⠘⣷⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡿⠋⠉⠙⠃⠀⠀⢀⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣁⣶⣶⣤⣿⠀⣠⡼⣿⡿⠛⠛⠛⢛⡿⣿⡿⠋⢻⣿⣤⠤⠤⠴⠤⠤⠄⠀⠀⣼⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡿⣿⣼⣿⣷⡾⣥⣄⡈⢣⡀⠀⢁⡞⣠⢋⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡿⣻⣿⣿⣟⣠⣅⡸⠙⡆⠉⠛⢋⡴⠃⡾⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣴⣿⣿⠿⠛⠟⠉⠀⣴⠇⢀⠞⠁⠀⣠⢳⣄⠀⠙⠛⠙⠛⠣⢔⠚⠒⠢⣀⠘⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠻⢅⡀⠈⢀⠀⢠⠼⣀⡹⠁⣀⡼⠱⠘⢅⠀⠀⣀⢀⣀⡠⠞⠀⠀⠀⠈⢢⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠻⣿⣷⠋⡔⠱⠦⣤⠁⠀⡀⠀⣼⡗⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⣠⣔⡩⠂⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠀⠀⢀⡸⠖⠛⠉⠉⢀⡴⠃⣠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⣴⣏⠀⠀⢀⣠⠖⠉⠳⣤⠃⠓⠤⢀⡀⢀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠙⠊⠁⠈⠳⢤⡴⠃⠀⠀⠀⠀⠈⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣆⠀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠄⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠐⠋⠃⠛⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⣶⡄⢀⣠⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⣐⡟⢫⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⠄⠀⠀⠀⢹⣽⣭⣽⣿⡿⠀⠀⢀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣄⣈⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⢈⣿⣿⣿⣏⣠⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇⠀⠀⠉⠁⠈⠉⢁⢹⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⡧⠀⠉⠋⠻⠿⠟⠛⠓⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⢀⠠⠈⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣿⣃⠀⠀⠀⢀⡐⢐⣿⡟⢙⣿⣿⣿⣿⣿⣿⡿⠁⡆⠀⠀⢀⣼⣿⡂⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣀⣩⣽⣿⣿⣿⣧⠚⣆⡀⡰⣼⢼⣿⠇⣾⣿⣿⣿⣿⣿⣿⠇⠰⠃⢀⡴⠋⠀⠀⠙⡐⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠯⠾⣿⡅⣿⣿⣿⡟⠶⣧⣄⣶⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣈⣁⡀⢠⣿⣶⣧⢠⣀⠀⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠰⡀⠈⠇⢻⣿⣿⣿⣿⣗⢞⣿⣿⣣⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡏⠛⣛⠿⠿⢿⡈⣿⣼⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠱⣾⣾⣤⣭⣾⣯⣿⢿⣍⡿⢡⡈⠈⢻⣾⠛⠈⠛⠻⢈⡫⠼⠟⠿⠉⣰⣯⣶⡶⠿⠃⠉⠃⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⠿⠿⠿⢻⠟⢿⡿⣧⡦⠦⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠈⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠔⠚⠁⠀⠞⠢⠀⠊⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢌⠁⠀⠆⠀⠀⠀⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠃⠁⠄⠀⡀⠀⠀⠀⣂⠅⠀⠐⢂⠀⠀⠀⠘⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠀⡅⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⡄⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡴⠠⠃⢀⠀⡀⠦⠀⢀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠁⡼
⠀⠀⠀⠀⠀⠀⠸⠆⠀⡞⠉⠀⠀⢈⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠈⠀⠘⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠐⠀⠀⢸⠅⣧⠀⠄⠊⣈⡠⠞⠀⠀⢀⠘⠀⠀⠀⠀⠀⢠⠀⠁⠂⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠃⡰⠊⠀⠀⡆⢸⠀⠀⡀⠀⠂⠀⠀⠄⠀⠁⣿⠀⡆⠀⠀⠀⠀⠸⠀⠀⠚
⠀⠀⠀⠄⠐⠀⢠⡇⠰⠀⠐⠃⢻⡆⠄⠁⠀⠀⠀⠀⠀⠉⠀⠛⢀⡇⠀⠀⠀⠀⠃⡆
⠀⠀⠀⠲⡄⠀⣼⠀⡄⣀⡼⠐⢸⠷⢃⠀⠀⠀⠀⠀⠀⠰⠠⢄⢸⡧⠀⠀⠀⠀⠀⢱⠲⠂
⠀⠀⠀⠀⠈⠀⣇⠀⡄⠀⠀⣎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⢠⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⡏⠀⣤⠀⠁⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠘⠃⠘
⠀⠀⠀⠀⠀⠀⡇⠀⣯⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⢩⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠂⠀⠀⡇⠀⠂
⠀⠀⠀⠀⠀⠀⣘⢀⡈⠀⡇⠰⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠉⠐⠇⡶⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠐⠒⠒⠁⠐⠂⢀⠀⠀⠀⠠⠄⢤⢀⡐⢀⠀⠰⣠⠀⠀⠟⠐⡤⠤⠰⣴⠶⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠃⠀⠀⠀⠀⢀⣤⣤⡤⠤⠤⣤⣄⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣴⣾⡧⣴⡤⡀⠀⠀⢈⡉⠳⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣴⡿⠛⠀⠀⢺⡀⠈⠂⢀⠀⠙⣷⣄⡙⣿⡉⢿⣶⣶⣄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⣿⣷⡄⠀⠈⢳⣿⣿⣿⡿⢿⡇⠈⠁⠈⣿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠸⣿⣷⣶⣟⠐⣿⠄⠹⣦⣀⣀⣩⡿⠋⡠⣿⡗⢀⡀⣠⣿⡿⢸⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⡈⠛⠛⠉⠠⣿⠂⠀⠬⣉⣉⣡⠔⠊⠁⣾⡇⠈⠙⠛⠋⡀⢸⣷⣾⡷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠛⡏⢋⢐⣿⣀⣤⣤⣧⣬⣴⣤⣦⣧⣼⣗⢘⠛⡟⢣⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠴⠿⠿⢛⣛⣛⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣛⣛⣛⠿⠶⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⢟⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢻⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣻⢙⣿⣟⣁⣤⣤⣤⣄⣀⠀⠀⠀⢀⣤⣶⣶⣶⡤⠄⠀⢰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⣿⣿⠿⠟⣿⣿⠿⢿⣷⣄⠀⠀⠉⠹⣿⠏⠙⠂⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣏⣠⣴⠀⠉⠉⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣿⣿⣿⣇⠀⠀⠀⠀⢠⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡏⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⢿⣿⡇⠀⠀⣠⠆⠛⠛⢦⠀⠐⠂⢠⣤⣤⡀⠀⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣧⠀⠀⣿⣿⡿⠿⠃⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣶⣾⣿⣏⣠⣶⣶⡓⠠⣄⣀⣄⣀⠙⠉⠙⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠫⢭⡹⠿⠛⠋⠉⠉⠀⠀⠈⠻⠟⠛⢃⣠⣤⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢄⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⢀⣤⠄⠀⠀⠀⠀⡀⢸⣆⠀⣈⠛⡏⠀⠘⣿⠋⢹⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⣿⣿⣿⣿⣿⣌⠁⠀⡀⠀⠀⢀⣷⣌⣿⣯⠁⠀⠀⠐⣦⠀⣠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣆⢡⣴⣶⣿⣿⣿⣿⣿⣶⣶⡶⠟⢀⣾⣿⣿⣿⣿⣷⠀⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠛⢿⣿⣄⠙⠛⠿⠿⠿⠿⠿⢿⠟⠛⠃⠀⣠⡿⠋⠀⠀⠉⠛⢿⡇⠀⠀⠈⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠁⠀⠀⠀⠈⠙⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣄⢹⡟⠃⠈⢻⣇⢠⠶⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣉⣂⣘⠃⣤⣈⢋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡥⣤⡄⠀⣤⠀⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡀⠃⢠⢆⢠⠀⠐⠀⣹⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡁⠸⠆⠘⠂⡤⠀⢠⠟⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣤⠤⡄⠠⡞⠁⠀⠀⠀⠀⣰⠋⠻⢻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡴⢋⠋⠀⠀⠤⠀⠙⡙⠓⠲⢦⣀⢻⣤⡆⣜⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⣠⠋⢠⣾⣷⣶⢀⡄⢶⣦⡿⢆⠀⠀⠈⢢⡿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⡰⠃⡰⠃⠀⠉⠃⡈⣁⠊⠁⠀⠈⣆⠀⡄⡰⠰⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢠⡰⠁⡴⠁⠀⠀⠀⢨⠗⣻⠀⠀⠀⠀⠸⡀⢠⢡⠃⠈⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠗⠒⠤⣤⣤⣷⣾⡇⠀⠀⠀⠀⠘⠟⠿⠀⠀⠀⠀⢀⣷⢂⠎⣀⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⢹⠀⠘⣿⣷⠀⠀⠀⠀⠘⠷⠿⠀⠀⠀⠀⣨⡏⣈⡌⠀⠙⣆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢾⢧⣀⠔⠺⡄⢀⡄⠻⢧⡀⠀⠀⠀⣶⢿⠀⠀⢀⣀⡿⢰⣿⠀⠀⢀⣾⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡀⠠⢷⡋⢘⠓⡉⠙⣿⠾⠿⡄⠴⠻⠿⠛⢻⢃⠋⡟⠀⢀⣼⠇⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠐⠈⠛⢢⠀⠀⠀⣸⣇⡴⠀⠀⢰⣿⠂⠀⠀⣠⠚⠉⠁⢀⣜⡾⠁⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⠀⠀⢀⡴⠛⠢⠶⣜⠛⠀⠀⠀⢸⡷⠦⠀⠴⢿⣶⠤⠐⠁⠀⠙⢲⡀⠀⢰⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⡶⢿⣷⠞⠋⠀⣿⠀⠀⠀⠀⠈⠣⡤⠀⢄⡀⠀⡠⠄⠂⠈⠙⠀⠀⠀⠀⠀⢀⢡⣰⣟⣁⣤⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣷⣤⡏⠀⠀⢠⣿⠀⢀⠀⠀⠀⠀⠰⡀⡀⠙⢄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⠟⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⢿⡇⢰⡀⢸⣿⠀⢸⠀⠀⠀⠀⠀⠃⠑⡀⠀⠙⠒⢼⡇⢀⡇⠀⠀⠀⠀⠀⣆⣿⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⡇⢸⡧⢼⡇⠀⠀⠀⠀⠀⢠⠀⢸⠄⢱⠀⠀⠀⢸⡇⡘⠀⠀⠀⠀⠀⠀⢹⢹⣇⢠⡖⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡶⠋⠙⡃⠸⡇⢸⡇⠀⠐⠇⠀⠀⠈⡆⠀⡆⠀⠀⠀⢠⣀⢳⠁⠀⠀⠀⠀⠀⠀⢸⡈⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⡇⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠹⡄⢹⠀⠀⢀⢺⣼⠇⠀⠀⠀⠀⢀⣠⢤⣄⡇⢁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⡀⠀⠈⣾⠟⠀⠀⠀⠀⠀⡾⠇⢀⣼⡷⠈⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠟⣿⠶⢶⡄⠀⠀⠀⠀⠀⠀⠙⣟⡇⢠⡞⠀⠀⠀⣀⣠⢼⢡⠟⡏⠀⠀⠀⢰⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⢿⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⢣⣞⣤⡴⠖⠋⠛⠀⠈⠈⣠⡇⠀⠀⠀⠈⢧⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡶⠈⢣⠀⣤⡟⢷⣄⡀⠀⣠⣶⠾⢛⣯⠀⠀⠀⠀⠄⠀⠂⢀⠋⠀⠀⠀⠀⠀⠈⢣⡏⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢈⡾⠁⠛⠘⠦⠜⢾⣧⣴⡟⢼⣿⠦⣤⢠⣄⣁⢀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡉⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢁⣀⠴⠤⠄⠚⠉⠀⠀⠀⠀⠀⢀⡸⠃⠀⣀⠞⣿⡀⠞⠋⢣⡉⠉⠙⠉⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠀⢳⡆⠀⠀⠀⠙⢧⠀⠤⠻⢀⡀⠀⠀⠀⠙⠓⠦⣤⣀⣀⠀⠀⢀⣀⣀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠠⣄⠞⠀⠀⠀⠀⠀⠈⠳⣆⠐⡛⣹⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠙⠓⢛⣩⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⣤⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠉⠀⢧⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠾⠄⠀⢸⣄⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠛⠩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡞⠙⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⡲⠒⠦⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢀⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡾⠘⢦⣈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠉⠘⠃⣼⠀⠀⠀⠀⢀⣀⡀⢀⡇⣤⠠⠅⠀⡆⡇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⢦⣤⢤⡏⠀⠀⠀⠀⣇⠈⠁⢠⣧⣀⢀⡄⡸⠁⡇⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦⠔⢻⠀⠀⠀⠀⠀⠈⠓⠀⠁⠈⠿⣭⠶⡁⠀⠙⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠘⡧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠁⠀⠀⠀⢷⣿⡄⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣰⠄⠀⢤⣸⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠁⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠀⢠⠛⣤⠆⠀⠀⢄⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠃⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡇⢸⡇⠉⠉⠀⠀⠀⠀⢨⣀⣀⣀⣀⣀⣀⡀⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⢹⠋⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⡏⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠘⠶⠦⠄⠀⠀⠀⠀⠀⠀⢀⣤⡼⠀⠀⠘⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢧⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⢤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣏⣼⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢸⣶⣴⣿⣾⠣⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣄⣽⣿⠿⠿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣽⣿⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣰⣿⠋⠉⠙⣿⣧⠀⠀⠀⠹⣿⣿⣷⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⣠⡾⣿⠋⣀⣀⣴⣿⣿⡆⠀⠀⠀⢻⣿⣿⡇⠀⣠⣾⣿⣿⣉⢉⣿⣿⡿
⠀⠀⠀⠀⠻⡶⣟⣿⡿⠋⠀⣹⣿⡇⠀⠀⠀⢸⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠺⠟⠁⠀⢰⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣾⣧⣼⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⢀⣾⣿⣿⣿⣧⣀⣸⣿⠿⠛⠛⠻⢿⣿⣿⣿⢧⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢿⣿⣤⣼⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⣠⣽⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣯⣿⣿⣿⣟⣿⣿⣿⠁⠀⠀⢀⣴⠆⣾⣿⠟⣿⣿⣿⠿⠿⢿⡿⠋⠉⠉⠉⠛⢿⣿⣿⡀⠉⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣶⣄⣨⣥⣾⣿⠿⠢⣿⣿⣿⣾⣿⠗⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠈⡝
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡿⠟⠉⠀⠀⢹⣿⣿⣿⣤⣸⣿⠋⠀⠀⠁⠀⣠⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡅
⠀⠀⠀⠀⣴⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣾⡿⠁⠀⣴⠰⡀
⠀⠀⠀⠸⣿⣿⡟⠛⠛⠛⠿⢿⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⣤⣶⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣿⣿⠁⠀⠀⢸⣧⡙⢦⣀
⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠈⠙⢻⣿⣿⡿⢏⣿⡇⣿⣿⠿⠟⠛⠛⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠈⢿⣎⠲⣍⣓
⠀⠀⠀⠀⠀⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⣿⣿⣷⠄⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣯⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠈⠙⠪⠛⠇
⠀⠀⠀⠀⠀⠈⠙⢿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡄⠀⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠿⠛⣦⠀⠀⠹⣏⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⢹⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⡁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⠁
⠀⠀⠀⠀⣶⣶⣶⣶⡶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⢶⣶⡗
⠀⠀⠀⠀⣿⡆⣠⣤⠤⣤⣤⣤⣤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣽⣿⣿⣟⣛⣛⡃⣿⠁
⠀⠀⠀⠀⣿⣏⣛⣥⣾⠟⣩⣤⣤⢤⣤⡉⠻⣷⣦⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠉⣀⢠⡄⣠⡉⠻⣷⣾⣟⣃⣿
⠀⠀⠀⠀⠉⠉⢹⣿⠃⣄⢿⠰⣦⣀⡙⣷⣳⡌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠃⢠⠃⠟⣨⣥⣴⠀⡇⡈⣿⡟⢛⠋
⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⡄⢰⡏⠙⣷⢸⣧⡃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣼⠀⢸⠃⠀⣹⢰⣇⡧⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠈⢿⡌⢿⣵⡈⢛⢿⣏⢤⠻⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⡏⠘⣼⡷⢞⡃⣾⡏⢁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣐⣿⣮⣁⣛⣘⣃⣙⣈⣴⣿⣷⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣼⣿⣦⣁⣀⡂⠛⠃⢉⣠⣾⣟⠁⠁
⠲⠿⠿⠿⠿⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣤⣤⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣐⣯⣀⠀⠀⣼⢷⣾⠟⠲⠶⣤⡀⠀⠀⠀⠀⢀⡤⠄⠀⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣦⠤⣄⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⣠⠟⠋⠉⠀⠀⠀⠀⠉⠛⢷⡀⠀⠀⠙⢷⣄⣀⡀⠜⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠓⣠⠶⠲⢦⡴⠋⠀⠀⠀⠀⢿⠀⣼⠟⢷⠀⠀⡾⢻⡆⢸⠇⣀⠀⠀⡾⠉⠈⠙⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠉⢀⠤⠌⠁⢀⣀⡔⠂⣤⡀⠑⠊⠀⠀⠶⠶⠒⠀⠑⢉⡼⠤⠐⡦⠀⠈⠙⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⢀⣀⠀⠀⠀⠸⠇⠉⢁⣘⣃⣀⣤⣶⡿⠀⠀⠀⠳⣦⣤⡥⢀⣀⡀⠀⢀⣀⡛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠉⠐⠶⠶⠶⠖⠋⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣀⣀⣀⠒⠒⠠⠤⢤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣀⣀⣤⡤⣤⡶⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠂⣤⠉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠖⠛⢉⣵⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣿⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠸⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠈⠁⢹⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡇⠀⠠⠞⠋⠉⣀⣉⡙⠻⣷⣦⠄⠀⠀⢀⡾⣛⣛⡋⠀⠈⠻⣦⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠀⠀⠀⠀⢀⡿⠀⠀⣀⡴⣫⣭⣭⡍⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⠙⠆⢠⠆⠈⢻⡄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠠⠞⠀⠀⠀⠴⠋⠁⠀⠀⠉⠀⠀⠽⢯⣤⡶⠀⠀⠀⠀⠀⢸⣧⣭⡥⣶⠀⠈⠀⠀⠀⠁⣰⠟
⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠁⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠀⣿⠀⠀⠀⣄
⠀⠀⠀⠀⠀⣼⠛⠉⠉⠡⣄⡀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⠀⠀⠀⢿⠀⠀⠀⢰⠏⣦⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⢠⡶⠒⠈⠻⣀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⢠⣤⡄⠀⣸⠇⠀⠀⠋⠀⠈⠳⢦⣄⠀⣿⠂
⠀⠀⠀⠀⠀⠹⣄⢿⠁⠀⠀⢀⣈⡓⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⣀⠀⡘⢧⡀⠀⠀⡇⠀⠀⠀⠀⠙⢀⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⢰⡟⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⡴⠒⠛⠁⠀⠋⢒⢿⠛⢳⣄⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢠⠷⣦⣀⣀⠀⠠⠟⠀⣤⠞⠩⣍⣁⠒⠛⠈⣉⣽⡇⠘⣿⡀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠁⠀⠈⠉⠻⣏⠀⠀⢀⡟⠀⠀⠈⠛⠿⣿⠿⠛⠙⣧⠀⠹⣷⣖⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠳⢶⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⡄⠀⠀⠀⠀⠉⠀⠀⣠⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⢰⡟⠿⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⠛⠿⠿⢻⡇⠀⢷⣄⣀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠀⠈⢿⠗⠀⠀⠈⠻⣶⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢠⡘⠃⠀⠀⠉⢿⣦⡀⠀⠀⠈⣧⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠦⣄
⠀⢀⣴⠾⠋⠁⠀⢀⣠⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠙⠿⣶⣆⢸⡃⠀⠀⠀⠀⠀⢸⡀⢀⣠⣾⡏⠀⠀⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠐⠋⠁⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠘⣿⣮⠳⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠀⠈⠻⢿⣷⣤⡀⠀⠈⢻⣇⠀⠀⠀⢰⣿⠁⠀⠀⠀⢲⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⣆⣠⠌⠉⠛⠳⠄⠀⠙⠳⣆⠀⠘⣿⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢶⣼⣿⣽⣧⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣾⠆⠀⠐⣞⣿⡏⢹⣿⣣⠆⠀⠰⣿⣿⠂⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢀⡀⠀⠀⢲⣶⣿⣶⠇⠀⣨⣿⣿⣿⣿⣇⠀⠘⣷⣿⣦⡖⠀⠀⢀⣀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣇⠀⠀⠀⣼⣿⣿⣶⣾⣡⣿⣧⣼⣷⣌⣿⣶⣿⣿⣿⡀⠀⠀⣸⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣖⣺⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣒⣮⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣾⣿⣿⡿⣿⣿⣋⣿⣿⢿⡽⢿⣿⣯⣻⣿⡟⣿⣿⣷⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⣽⣿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢻⢿⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⣙⣵⣶⣾⣿⣷⣷⡔⣰⣾⣶⣾⣿⣶⣮⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⢿⡿⣿⣿⣿⡿⣳⣼⠻⣿⣿⣿⠿⣿⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢡⡊⢓⡉⣉⡁⢰⣿⣿⣧⢀⣉⠩⡓⢃⣦⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣶⣤⣶⣶⣹⠇⢿⡿⢿⠿⠹⢟⣴⣶⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⣇⣀⣀⡀⡀⣀⣀⣬⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣶⡦⢤⠥⠤⢶⣿⣯⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠁⠉⠛⠛⠛⠛⠛⠛⠋⠀⢈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⢀⣈⣀⣀⣠⣀⢠⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢆⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⢿⣦⡿
⠀⠀⠀⠀⠀⠐⠂⠀⠀⣴⡏⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠⣷⡄⢠⣿⣯⣤⡀⠉
⠀⠀⠀⠲⠔⠋⠉⠉⣾⣿⣧⣠⣴⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠸⣾⡀⠀⠀⠀⠀⠀⣠⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢦⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡈⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠸⣷⣤⠖⣱⡿⠋⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢐⠀⠈⢂⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣿⣇⣾⣯⢀⣠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣯⠈⢳⣄⢺⣧⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣿⣥⡈⠙⠯⠙⠹⢿⠛⣿⠟⣻⣿⣿⠋⠥⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛⠀⢀⣀⠱⣾⣾⣿⣿⣩⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⢹⣿⣿⣿⣟⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣜⠼⢿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⣴⣿⣿⣶⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠈⠂⠹⠿⣿⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠚⠳⠟⣩⣆⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⢿⠛⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡀⠀⠘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⠿⣟⢫⠟⠋⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠱⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢠⠀⠀⠈⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁⠀⠀⣂⠀⠀⠁⢘⡻⠿⠟⢿⣿⣟⠍⠉⢿⠿⠋⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠵⠒⣩⣿⣯⣱⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠻⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⣴⢫⠻⣄⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⢀⣯⣝⣒⣋⣵⡾⢷⣬⣓⣚⣉⢧⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⠟⠉⣉⣰⠤⠤⠤⠤⣀⣉⠙⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣷⣯⣥⣶⣶⣶⣶⣶⣶⣬⣙⣾⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠉⠛⠿⣿⣝⠃⠨⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠉⠛⢷⣿⣿⣿⣿⣗⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⢴⣿⣿⣇⣉⣙⡒⠀⠀⠀⠐⣚⣩⣹⣿⣿⣿⡆⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⡇⠑⠈⠁⠀⠀⠀⠀⠈⠒⠁⢸⢡⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡘⣿⣿⣇⠀⠀⣀⣀⣀⡀⠀⠀⢠⣿⣿⣧⣻⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⣺⠇⣿⢻⣿⣄⠀⠙⠲⠖⠋⠀⣠⣾⣿⣿⣏⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⢞⣭⡆⠀⣼⣿⣿⣿⣦⣤⣤⣴⡾⠟⣿⣿⣿⣯⢿⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡿⣡⣾⣿⣿⣿⣿⡟⠿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣻⣿⡿⠇⠀⠀⠀⠀⠀⢹⣿⡿⠻⢿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⣈⡿⠯⠄⠀⠀⠀⠀⢀⣠⣤⣿⣿⣦⠀⣵⢻⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⢿⡳⣄⡀⠀⠀⠁⠀⢀⣠⠴⣫⣿⣿⣿⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠂⠀⠙⠲⠭⣟⣓⣒⣻⡭⠗⠋⠀⣸⠟⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣽⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡸⣹⡿⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡠⣜⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠻⣾⣷⠏⠀⠀⠀⣿⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⣴⣿⡿⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣬⣿⣷⡄⠀⠀⢸⣧⣠⣾⣤⣤⣤⣴⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣭⡙⢻⢿⣶⣤⡾⣿⣿⠛⣫⣭⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣥⣞⣿⡿⠁⠘⠿⡿⢧⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠿⠟⠛⠋⠜⠛⣛⣦⣾⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢷⠔⣛⠉⣁⠀⠈⠉⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠦⠬⣿⣿⣾⣿⣾⣿⣴⣤⣿⣧⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣇⠀⠉⠲⡂⢤⡤⠄⠀⢀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠀⢠⠀⠀⢠⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠐⠄⢰⡀⠀⠀⢠⣷⠈⣤⣤⣾⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⠀⠀⢰⡄⠄⠀⠈⠻⠤⣿⣿⣿⣿⡻⠂
⠀⠀⠀⠀⡄⠣⣴⢀⣀⢠⡀⢠⣷⡄⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠂⠸⡇⠀⠀⠀⠀⠀⠀⣻⣿⣿⡅
⠀⠀⠀⣌⣶⣿⣿⣿⣏⠉⡤⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⢀⡄⢰⠈⢀⠀⠀⠀⠀⠁⠈⣿⣿⡷⠂
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣖⣿⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢀⡤⠀⣿⠀⠀⠀⠀⢸⣆⣿⡿⠇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⣿⣿⣷⣶⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣥⢀⣧⡴⢂⢀⣤⣸⣿⣟
⠀⠀⠀⠀⠀⣈⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠛⣻⡟⢙⣿⣿⣿⣿⣿⣿⡷⢠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠰⠂⠈⠛⠻⡿⠯⠁⠚⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠁⠈⠁⠀⠙⠛⠛⠛⠛⠁⠁⠀⠀⠀⠈⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣾⣿⠟⢈⣿⣿⣾⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⠀⠹⣿⣶⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⣶⣿⠀⠀⣠⣿⣿⣿⣴⡿⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣰⣿⣿⣿⡆⢸⣿⣻⣿⣯⣿⠁⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⣿⡿⢋⣠⣾⣿⣿⣿⣿⣿⣷⢀⣹⣿⣿⡿⣣⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⡛⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⡿⢋⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⠘⣫⣿⣿⣿⡿⠿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣇⠸⣿⣿⣏⡻⣿⣷⣶⣄⡉⢉⣉⣽⣼⡟⣿⣿⣏⠳⣀⠟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⡌⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢸⣿⡄⢻⡆⠀⠀⠀⠀⢰⣿⠏⣸⣿⠃⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠘⣿⣷⠈⣿⠀⠀⠀⠀⣿⡿⢠⣿⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⢹⣿⡆⣿⡇⠀⠀⢰⣿⠇⣼⣿⠇⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⣿⣇⢸⣷⡀⢀⣼⣿⢠⣿⣿⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣟⠛⠻⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⢻⣿⣀⠀⣠⣀⣀⡀⢀⣿⡋⠁⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⡿⠛⠀⣸⣿⣿⡿⠿⠛⠛⠻⢿⣿⣷⣄⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⠥⠤⣴⠏⠉⠙⣧⠶⠲⠶⢤⠀⠀⠌⠻⣆⣒⣻⣧⣀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠂⢀⠉⠄⠐⣿⣿⣿⣿⠃⠀⠀⠀⣽⣷⣦⣶⡏⠀⠀⠀⠉⢹⣬⣭⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠙⣿⣿⣿⠀⠀⠀⠀⢾⣿⣿⣿⣆⠀⠀⠀⠀⣸⣿⡿⠧⢈⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⡦⢴⠶⠾⣿⣿⣿⠿⠶⢶⢤⣺⣿⣿⣷⣤⣀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣰⡤⠀⠀⣤⣖⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠀⢈⠛⣯⠃⣝⣶⣽⣈⣡⡶⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣽⣵⣠⣳⣶⣶⣶⣷⣿⣿⣿⣿⣿⣷⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠈⢻⣯⣿⠟⠟⣽⣿⣿⣿⣿⡿⢿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⢊⣥⡾⣷⣿⣿⣿⣿⣶⣶⣿⣾⣿⣿⣿⣷⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢉⡎⠉⣠⣯⣻⣿⣿⣿⣿⡿⣿⣿⣿⣏⠙⠉⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⠡⣽⡿⣯⣽⡏⢻⣻⠞⠉⠉⠀⠀⠀⢀⣆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠙⠀⠀⠘⢈⠀⣷⠊⠀⠀⠀⠐⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠁⣀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠈⠉⠻⠛⠿⠷⠴⠇⠾⠴⣾⣷⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢰⡟⣳⠀⠸⣯⣽⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣽⣇⠀⣽⢿⣄⢠⡿⢿⡄⢀⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠷⡟⠀⠙⠟⠁⠈⠻⠟⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣀⣀⠀⠀⠀⠀⣀⣀⣴⡟⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⠀⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⣶⣶⣤⣤⣶⣶⠶⠶⢾⡆⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢸⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⡤⠀⠀⠀⠀⠀⠀⠉⠁⠸⣿⠘⡀⠀⡆⠈⣇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⣤⣄⠀⣿⠀⢣⠀⠘⡄⠹⣄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠸⣷⠀⠿⠟⢰⠋⠀⢀⣠⣝⡋⠀⢻⡇⠀⠀⠀⠈⠀⠈⣮⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⣀⠹⣧⣠⠞⠻⣤⡴⠟⠁⠈⠛⣦⣬⣿⡀⠀⣴⢟⠛⢿⡯⣿⡟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣧⣤⣟⣿⣶⣿⠴⠶⠤⠞⠒⠚⠉⠁⣸⣷⠶⠟⠿⣧⣼⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡞⢦⡈⠈⣹⠋⠉⠛⠿⢶⣦⣤⣤⣶⡶⠟⠋⠁⠀⠀⠀⣸⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣍⣹⡗⠘⠟⠛⢻⡀⠀⢀⡄⠚⢿⣙⣻⠟⠃⠀⠀⠀⠀⠀⢰⡏⠈⠛⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⡌⠁⠀⢀⣀⣸⣇⠀⣸⡇⠀⣼⢻⠇⠀⠀⠀⠀⠠⣄⠀⣸⣷⣄⠀⠀⠀⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣤⣴⠟⠉⠉⣽⢟⣿⠁⠀⡏⢸⠀⠀⠀⠀⠀⠀⠉⢿⡙⣷⣿⢠⣦⣘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣾⠷⣦⣬⣧⣸⣇⣀⣀⣠⣤⡴⠶⠟⢷⡘⣿⡿⠁⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢸⡇⠀⠀⠀⠀⣉⣉⡉⠁⠀⠀⠀⠀⠀⢸⡇⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠋⠀⢸⣇⣀⣼⠷⠛⠉⠻⣄⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⢹⡆⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠈⠛⢷⡾⠋⠀⠀⠀⢀⣴⠛⠛⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠻⣧⡶⢿⣅⠀⠀⢀⣤⡾⣧⠀⠀⠹⢦⡀⠀⠀⢀⣴⡾⠛⠛⠻⠶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⠀⠀⠙⢿⣶⠛⠉⠛⠛⢷⡀⠀⠈⢿⡶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀⠀⣠⡶⠛⠁⠀⠀⠀⠀⣼⣷⠶⠶⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⠋⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠋⠉⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣠⣤⣤⣤⡶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⢸⣿⠀⣼⣁⣴⠞⢁⡆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢹⡀⣿⠀⠀⣷⣿⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠁⠀⡟⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡎⣷⣿⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⣼⠃⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣷⢸⣿⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣸⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣤⣀⣽⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣟⣿⣿⣿⣦⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣟⣫⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣄⣸⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢟⣿⠿⣷⢀⣤⣤⣤⠘⠿⢿⣯⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣾⣿⡀⠀⠘⠛⠛⠋⠁⢀⣹⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠼⢻⣿⣿⣆⠀⠀⠀⠀⣠⣾⣿⠏
⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢈⣷⣼⣿⣷⠚⠛⢾⣿⣯⣿⠀⠀⠀⠀⠶
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿⡿⣿⠀⠀⢸⢿⣿⣿
⢸⡇⠀⠀⠀⣠⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⢡⣤⣤⣤⢾⣿⣿⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣷⣿⣿⣿⣿⣼⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⡈⠁⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡖⣾⣹⠃⠀⠀⠀⠀⢀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⢀⡴⠋⡒⡁⣉⡒⠂⠄⡒⠚⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠓⠒⣾⣥⠜⠣⠚⣓⣘⣛⡛⠛⠏⠴⢋⣬⣧⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣯⡾⠒⡺⠕⠒⣉⣁⣤⣤⣤⣤⠍⣉⣐⠂⠭⡙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡫⣐⣬⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⡪⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⡿⠛⢁⣶⣿⡿⢋⣾⣿⣝⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠟⡋⠀⠀⠁⣡⢟⣴⣿⡏⠹⣽⣦⡙⠿⣿⣟⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣤⡾⢀⣴⣶⣿⣷⣿⣿⡟⠀⠀⠈⠻⣿⣶⣜⣮⣷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣿⣷⣿⣿⣿⣿⢟⡵⠋⣀⣠⣤⠶⠦⠌⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠻⠿⡄⠀⠘⠋⡡⢴⣶⣶⡶⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡝⠿⠗⠁⠀⠀⠀⠁⠈⠋⠉⠀⠀⢸⣿⡛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⣿⣿⣿⣿⣧⡀⠀⠈⠛⠢⠖⠘⠀⠀⠀⠀⠀⠉⣽⢀⣿⠟⣿⢸⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣴⣯⣿⣿⣿⣿⣿⣷⡀⢠⣴⣶⠶⠶⣤⡤⠀⠀⠀⣠⣿⣽⠏⣶⠇⣼⣿⣎⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡙⠷⠶⠖⠁⠀⢀⣤⣾⣿⣿⠃⣸⣯⣶⣿⣿⣿⢳⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣿⣿⣧⢠⣿⣿⣿⣿⣿⠇⣸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠹⣿⣿⡝⣿⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢹⣿⣿⣿⣿⣿⡙⠷⠃⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⠀⣿⣿⣧⡡⢁⡿⢱⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠸⣿⣿⣷⡞⡕⣰⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡻⣿⣼⣇⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⠴⠚⢋⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠃⠸⠟⠁⠀⣠⣴⣿⣿⣿⣿⠋⢩⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡀⠀⢀⣤⣾⣿⠿⠟⠋⠁⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡔⠂⣄⠀⢠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠸⢇⠀⠏⠀⠘⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡅⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠆⡼⠂⠀⠃⡴⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⠀⠀⡈⠀⡀⠀⠀⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣤⣀⣀⣉⣀⣉⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠎⠙⠋⠉⠉⠉⠉⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⡆⠀⠀⠀⠠⣄⠀⠀⠀⠘⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡈⠁⠀⠀⢨⡅⠐⠀⠀⠀⢀⡀⢀⡀⣀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠀⠀⠀⠀⡸⠃⠀⠘⢻⡿⠃⠀⠘⢿⣟⠀⢼⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣤⣤⣤⠾⠁⠀⠀⠀⠀⠀⣀⡼⠒⠀⡙⠧⣼⡇⢠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠷⠤⠀⠈⠉⠙⠛⠷⠤⠤⢄⣀⠈⠃⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⡘⠻⢀⠀⠀⠀⠀⢀⣀⡀⠒⠶⠒⢶⡄⠉⠻⣦⣄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⣁⡀⠀⠀⠀⠀⠛⠃⠀⠀⠉⠛⠋⠀⠈⠀⠀⠘⠓⠈⠻⣦⣀⠀⠉⠛⣷⣶⡄⠀⠀⠀⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢴⡆⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠐⠿⠏⠀⠈⢻⡅⠀⡄⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠠⠾⠟⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢀⡀⠛⠃⠀⠀⠀⢀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠛⠁⠀⣠⢤⣤⣄⠀⠀⠀⣤⡈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⢠⡶⠋⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⣁⡴⠏⠀⠙⠓⠤⢀⠀⠀⠈⠛⠳⣶⡦⢤⡤⠤⠤⢤⣤⡀⠀⠈⠀⠀⠏⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠁⠂⡀⠀⠀⠀⠀⠀⠁⢠⣤⠚⠀⠉⠐⣶⡆⡸⢷⣄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠠⠆⣀⠋⠁⠀⠀⠀⠀⠹⠃⢰⣦⣈⠉⠀⠄⠠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⡏⠐⠆⠀⠀⠀⠀⠰⣄⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠁⠒⠒⠒⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⣠⠀⠀⠰⠄⠀⠀⠀⠷⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⣰⠏⠀⠀⣠⠀⠿⠂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡘⠃⠀⡴⠋⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⡄⠃⡄⠿⠀⡀⣀⠀⡀⠀⢀⡀⠀⠸⠇⠀⠀⢸⡇⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠸⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⣴⡧⢸⡃⢰⡆⠀⢿⠀⢠⡸⠃⡴⠀⢠⡄⠀⠀⠈⠃⠀⠀⡈⠀⠀⡇⠀⠀⠀⠀⣀⣀⣤⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠈⠓⢤⣅⠀⢸⠀⢸⠇⢀⣤⠒⠋⠙⠀⠀⠀⡀⠀⠠⡇⠀⣴⡟⠀⠀⠀⠸⣯⣀⣠⣄⣁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣤⡴⠤⠶⠤⠿⠦⠼⠆⣤⣀⣘⣁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠁⣀⣀⠠⠴⢯⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢰⣷⠶⠐⠒⠚⠉⠁⠀⠀⢨⣥⣄⡀⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠁⠒⠒⠒⠶⢤⣤⣤⣄⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠒⠒⠒⠐⠒⠚⠛⠓⠉⠉⠉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢦⣤⣤⣤⣤⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⣶⡶⠾⠷⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣟⠛⠻⠿⠶⠞⠻⠟⢻⡟⠉⠉⠛⠛⠻⣿⣿⣷⣿⠛⠛⠋⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⢦⣴⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠷⠿⠷⣤⣤⣤⣤⣤⡴⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣀⠀⠀⢠⡄⠀⢀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡿⣿⠛⢇⡰⠿⡟⠇⣌⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡀⢀⣽⣿⠇⣶⣅⣹⣧⣾⣿⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣲⣧⣿⣿⣿⣿⣿⣿⣿⣽⣿⣾⠿⠷⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⡇⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢿⣾⠟⠐⠛⣽⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣿⣧⣀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣟⡀⠀⣈⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣙⠛⠻⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⡆⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠈⠉⠛⠻⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡮⠴⠚⢉⡐⢂⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⠦⣤⣞⡵⠛⣟⣵⣯⣾⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣆⠀⠀⠀⠀⠀⠀⣠⠞⡁⠀⠀⠈⢷⣀⣀⣀⣠⣤⣤⣾⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣿⣿⣛⠛⠷⠤⣤⣄⣀⡀⠈⠙⠻⠶⣭⣉⠙⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢹⠀⢸⡇⠈⠙⠻⠶⣤⣄⣀⠉⠙⠓⠶⠢⣤⣀⡉⠳⣦⡘⡇⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⡇⠀⣼⠇⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣤⣀⣀⡈⠉⠛⠦⣽⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡄⠀⢸⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢷⠀⠀⠀⠒⠿⡁⠀⠀⠀⠶⠄⠶⠂⠀⠀⢠⠇⠀⠀⢠⣟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠘⣄⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢳⣄⡀⠀⠀⠈⠛⠶⠤⠤⠴⠶⠛⠁⢀⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡿⠉⣷⡶⣶⢶⣶⠶⠶⣲⠶⢺⣿⢿⡛⢿⡙⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃⣾⠑⢤⡿⠱⣛⣆⠙⠷⡻⠉⣀⣾⡟⠀⢟⠀⢱⡌⢧⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠃⣸⠇⠀⢸⡇⠀⠉⠿⢷⣤⠶⠶⣯⠟⠀⠁⠘⣇⠀⢻⡄⠳⡄⠙⣆
⠀⠀⠀⠀⠀⠀⣰⠏⢰⡏⢠⡟⠀⠀⣿⠁⠀⠀⠀⢻⡇⠀⠀⢸⡇⠀⠀⠀⢻⡀⠈⢿⠀⠻⡄⠘⢷⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠏⢀⡾⠀⠀⢀⣿⠀⠀⠀⠀⠘⢿⣤⣤⡾⠃⠀⠀⠀⠸⣇⠀⠘⣧⡀⠛⡄⠈⢷⡀
⠀⠀⠀⠀⣰⠇⠀⠸⠀⣼⠃⠀⠀⢸⡿⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣿⡀⠀⠘⣧⠀⢱⡄⠀⢳⡀
⠀⠀⠀⢠⡟⠀⠰⠆⢠⡏⠀⠀⠀⣼⡷⢦⣀⣠⡀⠀⠀⠀⠀⢀⣀⣀⡤⠶⠋⢹⣧⠀⠀⢹⡆⠀⢻⡀⠈⣷
⠀⠀⠀⣸⠃⠀⠖⠀⡾⠀⠀⠀⠀⣿⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⡴⠟⣿⡄⠀⠀⢷⡀⠈⢣⠀⠘⣇
⠀⠀⠀⡿⠀⢰⠀⢰⠇⠀⠀⠀⢠⣿⠛⠳⢶⠶⠤⠤⠤⢴⠖⠲⠞⠉⠉⠀⠀⠀⠘⣷⠀⠀⠈⢧⡀⠘⡄⠀⠸⡄
⠀⠀⢸⠃⠀⡇⢠⡿⠆⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠄⠀⠸⣇⠀⢻⡀⠀⣷
⠀⠀⡜⠀⠀⠁⣿⣿⣄⣀⣀⣀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣡⡴⣿⡀⢸⡇⠀⢸⡀
⠀⢀⡇⠀⢰⠀⣿⢰⠈⠉⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠀⣸⣷⠀⣷⠀⠈⠃
⠀⢸⠃⠀⠈⠀⣿⣸⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣴⠏⢹⠀⢻⡀⠀⣰
⠀⣿⠀⠀⠀⠀⢸⡇⠙⠳⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡅⠀⢸⠀⠀⡇⠀⢸⡄
⠀⣿⠀⢰⡂⠀⠘⣿⡔⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣰⠏⠀⠀⠃⠀⠀⡇
⠀⡇⠀⢸⡇⠀⠀⠘⢿⣄⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⢸⠀⠀⢻
⢸⠁⠀⠘⠁⠀⠀⠀⠀⠉⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⠀⠀⢸⡂
⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⡆⠀⠘⡇
⢸⡇⠀⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣇⠀⠀⣿
⢸⠇⠀⡇⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⠀⠀⢿
⡌⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸
⡇⠀⠀⡏⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢹⠀⠀⠘
⡇⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸
⣿⠀⠀⢥⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢸
⣿⠀⠀⠱⣤⣀⣠⣄⣿⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣿⡴⠞
⠸⣆⠀⠀⠀⠈⠉⢻⡟⠷⠲⠶⠚⠉⠁⠐⠙⠳⠶⢤⣤⡤⠶⠋⠑⠒⠙⠳⠶⠤⠴⠖⠋⠁⠈⠛⠉⢉⣉⣿⣇⠀⠀⠀⢀
⠀⠉⠳⢦⣤⣄⣠⣾⣷⣤⣄⣀⣠⣤⢴⡶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣴⣶⠶⠶⠖⠲⠦⠤⣤⣤⣴⡾⠛⠉⠉⠙⠑⠛⢈⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⢟⠁⠀⠀⣻⠀⢸⡄⠀⠀⡾⠂⢨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠀⢿⡀⠀⠀⢸⣆⠀⠀⣿⣆⣾⣿⡄⣾⠃⣠⣟
⠀⠀⠀⠀⠀⢀⣤⡀⢠⣄⠀⠀⣿⣧⡸⣷⣄⢀⢸⡿⣷⣄⣹⣿⡿⣽⣿⣾⣾⣿⣏
⠀⠀⠀⠀⠀⠈⠛⠣⣄⠉⢳⣦⣽⣿⢿⣿⣽⣵⣿⣿⣯⣿⣿⣿⣷⡿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⡽⠿⠿⠿⣿⣿⣿⣿⣟⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⢿⣻⣭⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣛⣽⣷⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠟⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣼⠁⠀⠀⠀⠀⠀⡠⢰⣿⣿⠋⠙⢅⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⠀⠀⠀⠀⢀⣀⣠⣿⡿⠁⠀⠀⠈⢶⣤⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⢫⣽⣷⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣷⣀⣀⣶⣀⣴⣿⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡛⠛⠛⢿⣦⣿⣿⣿⣻⣿⢿⡻⢿⡿⠿⡏⢻⠛⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡛⣿⣿⣥⣤⣸⣿⣿⣏⡝⣿⠿⠋⠿⠜⠿⠿⢾⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠁⣹⠛⠃⠀⠙⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣷⠀⠀⠀⠀⣿⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠀⠀⠀⠀⠸⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⢻⠀⠀⠀⠀⠀⣿⠀⠀⣠⣀⣄⣠⢀⡀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⣾⣆⣀⣀⣴⣤⣿⣴⣶⣿⣿⣿⣿⣾⣧⣆⠀⢀⡀⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠻⡝⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣷⣶⣿⣾⣿⣿⣿⣷⣽⣛⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣾⣶⣶⣾⣿⣶⣤⣀⣀
⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⠀⣀⣠⣤⣤⣤⠄⢹⠿⢻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⠃⣔⡒⡷⣆⣹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⡟⣿⣿⠄⠙⣁⣱⣿⣇⣿⢿⣷⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡴⠀⠀⣼⣿⣯⣿⣿⣿⣷⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⣿⣿⣿⣿⡗⠖⠂⢸⣟⣿⣿⣷⣯⣽⣯⡻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⣿⢹⣿⣿⣿⣿⢰⣷⠸⣿⡏⠴⡌⣿⣻⡿⣇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠸⣿⠘⣿⣻⣿⡏⣾⣿⣧⡙⢿⣿⣵⡿⣿⣿⣮⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢿⢠⡘⢿⣿⣻⣿⣾⣿⣿⣧⠀⠀⠛⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣾⢆⠀⣯⣬⡻⣿⠶⠿⣿⣷⡄⠀⠀⠀⠈⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣌⡉⠙⢦⡘⢿⣷⡹⣸⡇⣬⣙⣿⣆⠀⠀⠀⠘⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣿⣦⣙⢿⣶⡁⠻⣌⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠻⢶⣶⣮⣭⣷⣯⡙⠻⣷⣌⠑⢝⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣶⣄⠙⢷⡀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢳⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⢻⢆⣸⡀⠀⡜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡾⠀⠁⢡⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣲⠬⠡⠤⣤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠟⠀⠀⡠⠲⠈⢧⡠⠆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣶⣶⢤⡄⠀⠀⠀⠀⢾⠄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣝⣿⣟⣿⣆⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⣿⡀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣤⣀⣸⠀⠀⠀⠀⢀⡞⠉⠉⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⡠⢣
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠟⠛⠉⠁⠀⡄⠀⣽⡙⠓⠆⠀⣾⣀⠀⠀⠀⢀⣈⣵⡄⠀⠀⠀⠀⠀⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⠀⢻⣇⡀⠧⡜⣿⣈⠉⠉⠉⠉⠉⢀⣹⣄⠀⠀⠀⢠⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠿⣷⣾⡇⠘⠗⠫⣅⣒⣚⣛⣛⣉⠽⣍⠓⠒⠦⢤⣀⡇⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣀⠈⢲⣶⣄⠀⠲⢄⡀⠀⠀⠀⠰⠘⢍⠣⠀⠀⠀⠈⠳⢝⡺⢭⣉⣉⠙⣆⠀⠀⢠⡞⢁⣀⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠙⢻⡛⠛⠷⢤⣬⣝⡄⢀⠀⠀⠀⢠⠑⢄⠀⠀⠀⠀⠀⠈⠑⢾⣯⣽⣿⣷⡄⠛⠉⠁⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠠⠤⠤⠤⠿⠏⣛⢁⡀⠤⣀⣍⠙⠌⣧⡀⠀⡀⠀⠀⠑⠦⣀⡀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⢋⣑⡦⠤⠁⢍⣂⡙⠦⠿⣅⠀⢀⣈⣷⠟⠋⠉⢃⠀⢀⣔⣹⠦⡀⠀⠱⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠒⠒⢤⣽⣁⢀⣵⡀⠀⠀⡝⡗⠃⣿⠟⢤⡘⣶⣄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠷⣖⣨⡟⠉⠹⡀⠀⠘⠀⠀⠙⣿⡄⠙⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠙⣝⣟⡈⠧⣀⣀⠀⢰⡀⠈⠃⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⢌⠉⠉⢣⡀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠈⠑⠢⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⢱⣾⢍⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⠄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡇⢀⠀⠀⠀⠀⠈⣷⠴⡞⠀⠀⢠⡎⠑⠀⣧⣀⣠⠇
⠀⠀⠐⠀⠈⡆⣰⠒⢆⠀⠀⢻⠀⢳⠈⠃⢁⠀⠀⠀⣿⠃⡇⠀⠀⢀⠉⠋⢠⣿⢹⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠘⠒⢖⢳⡘⠲⡊⠄⠀⠈⠀⠈⢲⠀⠈⢆⠀⣰⠏⠀⠱⡄⢀⡎⠀⡴⠻⠃⠈⡀⣀⠔⡱⠒⣠⡾⣡
⠀⠀⠀⠀⠈⢧⠙⠦⣠⣀⡤⡈⠀⣀⠀⠑⣂⣘⣫⠟⢀⢤⡀⠘⣟⣒⣂⠐⠀⢀⠀⢽⠁⣸⣡⠴⠋⣼⠃
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠒⠊⠁⠰⠉⢧⠀⠀⠈⠁⠀⡎⠀⢳⠀⠀⠁⠀⢀⡞⠁⢂⠈⠙⠚⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠑⡀⠸⠀⠀⠀⠀⠀⠱⣄⠊⠀⠀⠀⠀⠀⢇⠀⠈⠀⠀⠀⠀⡸⢹
⠀⠀⠀⠀⠀⠀⠣⡘⠀⠀⠀⠀⠀⠈⠁⣀⣀⣀⣠⣤⣶⣤⣤⣄⣀⣀⣀⣀⠈⠀⠀⠀⠀⠀⠀⢧⠎
⠀⠀⠀⠀⠀⠀⠀⣿⠴⢖⠒⠛⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠓⠤⢲⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⢄⡠⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⣷⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⣤⡤⠤⠔⢿⣿⣿⡄⠀⣿⣿⣦⡾⠁⣀⣀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣧⠀⠰⡔⠢⡄⠀⠀⠀⣠⣿⣧⣀⣀⣠⣼⣿⣿⠇⠀⠀⢉⣉⣀⡼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠹⠜⣶⣿⣄⢠⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠟⠁⢹⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢰⡊⠰⡿⣿⣟⣿⣥⠀⢋⠙⢯⣿⠿⠿⣿⡟⠀⠀⢸⠀⢴⣷⣠⣤⣴⣶⣒⣂⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⢸⣿⣿⣷⠀⠀⠀⠉⠻⠿⠋⠀⠀⠀⣾⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢤⡀⢠⣿⣿⣿⡆⠀⡠⢀⣠⣤⣀⣰⣤⣶⣿⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠿⢿⣿⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠰⡎⠉⠻⠿⢿⠟⠀⣴⠶⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡆⠀⠀⠀⠈⡃⢩⠙⠄⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢬⡷⢤⡄⠁⢸⠃⠀⠀⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⢀⢔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠖⠁⠉⠑⠕⠚⠕⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⡛⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⡄⠀⠸⠀⢇⠀⢠⢏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠈⠲⠇⠀⠸⠞⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢀⣀⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⢁⠀⠈⠙⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠁⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡔⠡⡀⠀⠀⠀⠀⣼⡀⠀⠀⢀⢿⠉⢦⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠀⡔⠑⠦⢤⡴⠎⠤⠉⢒⠒⠋⠀⢣⠀⠉⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠆⠀⡰⠁⠀⠀⠀⠀⠀⠈⢆⠀⠸⡀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠀⠀⠞⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠳⠀⢠⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣀⣠⠔⠃⠀⠀⠀⠀⠸⠀⠀⠀⠐⢯⢄⡀⠀⠀⠀⠀⠀⠀⡘⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠰⠀⠀⠀⠀⠀⠑⠂⣤⣶⠀⠀⣀⣆⠀⠀⠀⣤⡀⠀⢀⣶⣤⠐⠁⠀⠀⠀⠀⠈⡆⠈⢢⠀⠀⠀⠀⠴⠧⢤⠈⢘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣀⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⡆⠀⠀⠈⢄⠀⠉⠐⢃⡤
⠀⠀⠀⠀⠀⠀⢀⡠⣤⠔⣀⠺⡀⠀⠈⠢⢤⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡤⠴⠁⠀⠀⡗⡐⢦⡤⠊⠀⢠⠒⠁
⠀⠀⠀⠀⣠⠞⠁⠀⡇⠒⠀⢸⠳⡀⠘⠀⠀⠆⠀⠀⠀⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠈⠿⠿⠛⠁⠀⠀⠀⠀⠀⢠⠁⠐⠃⢀⠜⣧⠐⠂⢱⠀⡠⠃
⡠⠔⠂⠉⠁⠀⡔⠈⢣⡉⠀⠇⠐⠈⠢⣄⠐⠊⠒⠄⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠤⠒⠑⠂⣠⠔⠁⠂⠘⡊⢉⠎⠉
⠳⣔⡛⠗⠀⢰⠃⠀⠀⠈⡙⠀⠀⠀⠀⠀⠉⠲⠤⣀⠛⠈⠓⠠⣄⠒⡛⢤⡀⠀⠀⡠⠟⠒⣠⠴⠊⠁⠛⣀⠤⠒⠉⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠈⠢⠜⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⠀⠈⠀⡇⠀⠈⠛⠉⠀⠀⠀⠉⠀⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠀⡃⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠴⠀⡇⠠⠀⠀⠀⠀⠀⠀⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠤⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠚⠀⣧⡀⣀⣉⣀⢀⣤⠀⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⣬⣽⣿⢸⣿⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⣿⣷⣶⣶⣶⣾⣿⠀⠉⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡨⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⣠⠀⣿⣿⣿⣿⣿⣿⣿⠀⣀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⣀⠀⠉⠉⠉⠀⠒⠒⠒⠒⠂⠘⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠓⠐⠒⠒⠒⠒⠂⠈⡉⠉⠁⣀⠀⠀⠤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠦⠤⠄⢠⣄⡈⢀⣀⣘⣃⡀⠀⠂⠀⠀⠲⠀⡏⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠐⠀⠀⣘⢃⣀⣀⠁⣠⡄⠠⠤⠴⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣼⣿⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣿⣯⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡆
⣶⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣾⣶⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣭⣽⣏⣥⣤⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢨⣿⣏⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡆
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⣤⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⡀⢠⡒⠂⣀⠀⠀⠀⠀⣄⣧⣿⣶⠶⣶⡄⠀⠀⠄⢠⣜⢛⣓⣤⣠⠄⠀⠀⣴⠶⢶⣿⣼⣧⠄⠀⠀⠀⢀⠀⠒⣦⠀⡀
⠀⠀⠀⡹⣿⣿⡀⠀⠀⠀⠀⣴⠿⠿⣿⡇⠀⠸⠃⠀⠀⡠⢛⠛⣼⡟⣏⠢⡀⠀⠀⠻⠀⢈⣿⠿⠿⢶⡀⠀⠀⠀⠀⣸⣿⢟⡁
⠀⠀⠀⠛⠉⠙⣷⡄⠀⡀⠈⠿⠀⢀⢈⣌⣦⡤⠤⠴⠾⠖⠛⠛⠛⠛⠻⠷⢾⣶⣤⣤⣤⣊⡍⢀⠀⠼⠃⠀⠆⢀⣼⡋⠉⠹⠃
⠀⠀⠀⠀⠀⠈⠻⠦⣾⣅⣀⡠⠔⠛⠋⣙⣦⣤⡤⠤⠖⠒⢒⡖⠒⠲⡖⠒⠛⠿⢭⣭⣀⣀⠉⠙⠲⢄⣀⣀⣵⠰⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠑⢼⣿⡋⣠⡴⠚⠻⠋⠁⠀⣠⡤⠀⠐⣽⣷⠤⢴⣿⠇⠀⠠⣤⡀⠀⠙⢿⠓⠶⣤⣾⣿⣿⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⢩⡄⠀⢠⡶⣬⣾⣿⣿⡷⠾⠴⠿⠿⠿⠧⠯⣾⣿⣿⣿⣧⡴⣆⠀⢀⡬⠹⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣎⣷⢾⠏⠋⠉⠁⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣟⣁⣀⡀⠉⠉⠙⠿⣼⣉⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⣸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠻⠿⢿⠛⠋⠉⠛⠻⢿⣛⡛⠛⠛⠛⠉⠉⠉⠛⢻⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⣤⣤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⡿⠿⢿⣿⡿⣷⣿⣿⣿⣿⣇⡀⡗⡲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣷⣿⣷⣿⠿⢆⢸⡾⣧⣀⣿⣿⣿⣿⣿⠁⣮⡹⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠉⣾⠛⣿⣿⣿⣾⣄⣨⡟⠂⢘⠿⠿⣋⡍⠻⣿⣟⣡⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣵⣤⣖⣼⣿⣿⣟⣛⣛⠵⠒⠿⠉⠀⢘⢋⡁⠲⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠾⠟⣁⠿⠄⠐⠒⢀⣄⠈⢰⣆⣀⣘⣂⣀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⠈⡰⣠⣨⣥⣶⡶⠶⠗⠘⠋⠉⠉⠉⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣻⣽⣷⣿⣿⡿⢣⡷⡀⠆⠀⠀⣠⡤⣦⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣉⠋⠀⠁⡀⠀⢸⣱⣼⡿⠀⠻⡀
⠀⠀⠀⠀⣠⠞⠛⣶⡄⠀⠀⠀⠀⢸⡿⠛⢿⣼⣿⣿⣧⢾⣡⠄⠀⠈⠀⠉⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢿⣶⣼⣿⣷⠀⠀⠀⢀⣼⣿⣿⣾⣿⢻⣷⡇⣠⣯⡀⠀⠀⠀⠀⠀⢀⢀⡀⠱⡀
⠀⠀⠀⠀⠈⠹⠿⠯⣿⣠⢄⠠⣿⣿⣿⣿⣿⣼⣿⡿⠘⠞⡎⣑⠀⠀⠀⠀⢀⣼⣟⣻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣏⠀⠙⣿⣿⣿⣿⣿⠟⠁⠀⢰⣶⡎⠀⠀⠠⣤⣿⣼⢟⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢿⣦⡀⠙⣿⣯⣍⡈⠀⠀⠀⠈⢿⣻⣧⡆⢆⡐⠋⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⣝⢄⣿⣿⣿⣿⣷⢦⣄⡀⢸⣿⣿⣿⣶⣐⠆⠀⣌⡀⢄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⣶⣾⣦⡙⢿⠟⠿⠛⠛⠓⠛⠿⣿⣿⣿⣿⣳⣄⠂⠀⠀⡀⠁
⠀⠀⡤⠀⣠⠀⠀⠀⣴⣿⣿⣿⣿⣯⠿⣦⠁⠄⠀⠀⠀⠀⠀⠀⠹⣿⡿⢿⣿⣣⠄⠁⠬⠀⢀⡄⢀⣀⣀⡀
⠀⠀⢉⣩⡟⢅⣫⣾⣿⣿⡿⠻⠛⢀⢄⡸⣷⣄⠢⡀⠀⠀⠀⠀⠤⠌⠻⣦⣿⣯⣾⣻⣷⣤⣲⣿⣿⣿⣿⣛⠧⢀
⠀⠀⠠⠾⢒⡠⣾⣿⣿⣿⣷⣶⣿⣿⣾⣽⣿⠿⣇⢌⠢⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⣶⣺⣟⣻⣯⣛⣛⣲⡥⠆
⠀⠀⠀⠢⣑⢼⡿⣿⣻⣿⣿⣿⣶⣿⡭⡨⠏⠛⠚⠷⣤⣈⠢⡀⠀⢔⠛⠁⣀⠉⠻⢿⣿⣿⣧⣻⠿⣷⣍⡙⠀⠁
⠀⠀⠀⠀⠑⢋⠭⠏⠕⢛⡯⠛⣋⠘⠃⢀⠄⠀⠀⠀⠙⢯⣆⡘⢆⠈⠤⠦⡄⠀⠀⠀⠈⠻⢿⣿⣧⣮⣄⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠙⢿⣄⣃⠁⠶⣄⡀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠓⠘⠂⢀⣀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⣼⠙⣐⠠⣶⣿⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣯⣕⠆⠉⢛⠑⣁⡭⣫⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣧⣷⣮⣬⣥⣽⠾⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⡤⣄⠀⠐⢪⣥⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⠟⠋⡃⠀⠀⠉⠀⠀⠈⣁⣽⣿⣿⣿⣷⣾⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⢿⣿⣿⡀⠐⠅⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣥⣾⣿⣿⣿⣤⣾⣛⠟⠀⠀⣠⠞⣿⢟⣿⣿⣿⡻⡟⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣅⣻⣿⣿⣿⣯⣄⣠⡴⠚⠁⠀⢿⣿⣿⣿⣏⢿⡈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⡿⠟⠛⠛⣿⠿⠟⣋⣁⣁⠀⠀⢀⣀⠀⠀⠸⣿⡿⠛⠻⠿⢿⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣏⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⡿⠛⠀⠀⠐⢿⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⢛⣿⣧⡀⡹⠓⢀⠇⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⠀⠀⠀⠀⠀⠛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⡯⠓⣄⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣇⠀⠀⣶⠂⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣽⣻⣿⡇⠀⢻⡶⠛⠳⣄⡀⠀⠀⣀⢀⡀⠀⠀⣀⡴⠋⠉⠳⣶⣿⠀⠀⢰⣿⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢠⠏⠀⠀⠀⠀⠙⠶⣤⣿⣼⡧⡴⠞⠉⠀⠀⠀⠀⠸⣿⡀⠀⠁⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄⠀⠀⠀⠀⠀⠀⠈⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⣰⢿⡄⠄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣤⡤⠤⠄⠂⠁⠈⠐⠢⠤⢤⣤⣤⠤⠞⠁⠘⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⣀⣀⣳⣼⣁⣠⣤⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣼⣿⣿⣿⡇⠉⣩⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣿⢿⢿⣷⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣴⣶⣄⠀⠀⠀⠀⣀⣴⣾⣿⡿⣫⣿⣿⣝⢿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣶⣤⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⣴⣶⣿⣿⠿⢛⣡⣾⣿⣿⣿⣿⣷⣬⡛⠿⢿⣿⣷⣦⣼⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠈⠻⣿⣿⢿⣿⣿⣯⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣭⣿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⣠⣿⡇⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠀⢸⣿⣅
⠀⠀⠀⠀⣷⣾⡇⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣾
⠀⠀⠀⠀⢿⣿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣿⣿⣿
⠀⠀⠀⠀⣾⣯⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣽⣿
⠀⠀⠀⠀⠙⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡟⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⡏⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁⠀⢸⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠒⠒⠋⠉

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

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡒⠶⠎⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⠡⠆⡆⡴⣤⡙⢁⡴⡾⣭⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣏⡠⢔⣒⣒⣦⢼⡟⢠⢏⣒⠂⡤⢿⡄⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠐⢻⣣⡞⣉⠨⠦⣮⣝⢠⢿⣦⡷⣬⠽⣟⠷⡜⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣳⡘⠺⠗⢸⣱⡋⠙⣙⢷⡱⠆⢒⣽⣛⠟⠟⣓⣒⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣇⣼⡗⠢⡤⠤⣐⣾⢹⣦⣣⡈⠘⠕⢋⠙⠉⠁⠾⣹⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡜⢿⣷⣄⠈⠲⢬⢝⣛⣳⢍⢿⠏⢏⢸⣭⠑⠺⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣑⠠⢠⡱⣯⣿⡄⡿⣆⣲⢣⢋⡴⠂⠁⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⣙⢦⣓⠛⣛⣴⠷⢾⠏⣹⣿⣷⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣯⣭⣏⠂⢡⡟⠼⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣚⠞⠠⠷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣍⣲⢋⡆⠸⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡋⡠⠺⢁⠒⡒⣸⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣏⠈⠞⣯⠘⣷⡑⢦⠃⠂⠆⠌⡜⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢻⠐⡈⡜⣶⠊⠻⣷⣧⣉⡹⢍⣉⣒⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⢇⢡⠄⢍⠷⣝⢠⣍⢛⣛⠛⠿⣛⣻⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠈⢦⠲⢈⠰⣈⠳⠦⣬⣙⣛⣠⣭⠵⠞⡩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠑⢞⡀⡄⢉⡈⠢⠤⠼⠤⠔⠚⣍⠀⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠓⠮⣥⣘⣂⡸⠀⠻⠀⠙⢁⣘⡯⠜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠲⡀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⡴⠉⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢈⡟⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢈⠟⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈
⢳⣀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠜⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⡴
⠀⠈⠉⡟⢄⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢀⡔⠁⠑⡄⠀⠀⠀⠀⠀⢀⢎⠄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⠞⡏
⠀⠀⠀⢰⠈⠳⡀⠀⠀⠀⡰⡽⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢠⣮⠎⠀⠀⠈⢦⠀⠀⠀⢀⢮⡟⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠔⠁⢸
⠀⠀⠀⠀⡇⠀⠘⢆⠀⢰⠻⠁⠀⠀⠀⠀⠀⠀⠘⡄⠀⣴⡿⠃⠀⠀⠀⠀⠀⠳⡀⢀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠱⡀⡠⠋⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠱⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⢰⣓⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣩⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣆⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⡀⠀⠀⢿⠧⡀⡴⡄⢸⡟⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢿⣄⠀⡏⣑⠄⡄⠀⢮⡳⢹⠀⣡⡚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢶⡀⢤⢰⢋⠘⠯⠉⠼⢁⣉⣠⣭⣥⢩⣙⢫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⠈⠛⣿⠈⣁⠴⣺⠟⣩⣾⡛⢅⣧⡾⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣔⠀⣠⠞⣱⠎⢠⠞⠁⠀⠀⠀⠁⠀⣯⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⢾⣯⣈⣋⡡⡒⠁⠀⠐⠒⢤⡄⠀⣠⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⢶⣬⣭⣽⠆⠀⠀⠻⠿⠏⠀⢰⠷⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷⣠⢫⢯⡓⠀⠀⠀⠀⠀⠀⠀⣀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣝⣿⣌⣿⠑⡀⠀⠀⠀⠀⠤⠾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣌⣥⣤⠿⠀⠈⠲⣤⣀⡀⠈⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣥⢾⠀⠀⠀⠀⠙⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⢰⠀⠀⠀⣼⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠲⠶⠊⠁⠀⠀⠈⣇⠀⢠⠋⣀⣑⣌⠿⠋⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠈⠀⠀⠀⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣳⠀⠀⡄⡀⠀⠀⡼⢁⠘⣆⠀⠀⢔⡄⠀⢸⣉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢦⡀⡈⢢⢀⡜⢑⠁⠀⠌⢧⣀⠊⢇⢠⠎⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⣃⠠⠭⠀⠀⠀⠀⠀⠀⠁⠦⠬⢋⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠤⠖⠒⠚⠋⠉⠉⠉⠉⠉⠉⡓⠒⢲⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⢦⣄⡁⠀⠀⠐⠁⠈⠀⠀⢀⣈⡴⠋⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠁⠀⢈⡉⠛⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠈⢦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣿⠃⠀⢀⣭⣍⠁⠀⠀⠀⠀⠀⠀⠀⣫⣭⣝⠀⠈⣿⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⡏⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⢸⣁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⠀⡉⠉⠩⠍⠉⠉⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⣄⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠉⠛⠶⣶⠤⠤⠤⠤⠤⠤⣶⠶⠛⠉⠀⢀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⠉⣷⠒⠤⢤⣬⡑⠦⣀⣠⠔⣩⣤⠤⠤⠒⢾⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⢿⡇⠀⢀⠎⠀⠙⢆⠈⢀⠔⠁⠈⢢⡀⠀⠘⡿⢥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣰⠇⠀⡄⠀⠀⢳⠋⠀⠀⡀⠀⢱⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⣴⣤⣤⣇⣀⣀⣸⣀⣀⣀⣤⣠⣼⣬⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠹⢤⣤⣤⠼⠇⠀⠀⠸⠧⣤⣤⡤⠿⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣷⣖⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣯⣵⣮⣉⣿⣶⣦⡀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⣼⣿⣿⣿⣿⣿⡀⠀⠛⠛⢚⣽⣿⣿⣿⣿⣿⣟⠛⠳⠤⣄⣠⠟⣽
⠀⠀⠀⠀⠀⠀⠀⢀⢯⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣆⡴⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠉⢸⣼⣿⣿⣿⣿⣿⡿⣿⣿⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣬⢿⣾⣻⣿⣿⣿⣿⣿⣮⢹⣤⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇⠲⣿⣟⠛⠙⠻⢷⣄⡀
⠀⠀⠀⠀⠀⢨⠀⠀⠈⢷⢻⡉⠻⢮⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡒⠿⢿⡦⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠈⣏⠀⣄⢸⠃⠛⢦⣄⡙⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣥⣿⣿⡿⢿⡳⣄⢼⡇⠈⡆⠘⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣷⣨⢨⣰⡄⠀⠉⢿⡿⣸⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣛⣄⠘⣧⠙⢻⡇⠀⢸⠀⢹⣿
⠀⠀⠀⠀⢠⡀⠀⢻⣷⣿⣿⣿⣧⡄⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣸⣿⣗⠀⣿⢀⡾⡁⢸⡌⢧⢸⡏
⠀⠀⠀⠀⠀⠳⣄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡟⠀⢹⣧⣿⣿⠰⣿⠚⠀⠀⢰⡆⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠘⡷⢀⣿⣿⠏⠻⣿⣿⣿⣿⣿⣿⠃⣿⣿⠃⠀⠀⠀⢻⣿⣿⢡⣿⠀⠀⠀⢸⠇⠀⢸⢧
⠀⠀⠀⠀⠀⠀⠀⢷⢸⣿⡿⠀⠀⢸⣿⣿⣿⣿⠋⢸⡿⢁⠆⠀⠀⠀⠸⣿⣿⡾⡏⠀⠀⠀⡞⠀⠀⡾⠈
⠀⠀⠀⠀⠀⠀⠀⡼⣸⡟⠀⠀⢀⣾⣿⣿⡛⠏⢠⡟⠀⡜⠀⠀⠀⠀⠀⣿⣿⡅⣷⠀⠀⠜⠁⠀⢰⠃⠘
⠀⠀⠀⠀⠀⠀⡴⢱⣿⣧⠀⢀⣾⣿⣽⣿⡃⠀⣸⠇⠀⢳⠀⠀⠀⠀⠀⣿⡟⢠⡇⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⣴⣃⣾⠿⣟⠀⢸⡟⣿⠟⠋⡁⠐⠹⠀⠀⠘⠀⠀⠀⠀⠀⢻⠀⣸⡇
⠀⠀⠀⢀⠜⠁⣹⠃⢀⣣⠜⠉⠑⠋⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇
⠀⠀⠀⠂⠠⠴⠓⠐⠊⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄
⠀⠀⠀⠀⠀⠀⡄⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⠄
⠀⠀⠀⠀⠄⢠⠀⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗⡁⠂⠄
⠀⠀⠀⠀⠲⢿⠇⠀⢢⡑⡄⠀⠖⣠⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⣠⡀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⠐⠢⠀⣰⣿⠇⠀⠿⠶⠂
⠀⠀⠀⠀⠀⠈⢂⢂⠘⣇⠈⢂⠲⣿⡀⠀⠀⣰⣿⣿⣿⣿⣧⡀⠀⠸⡦⡈⠀⠀⠐⠿⢿⣿⣿⣧⠀⠀⠀⣸⣟⡢⠃⢘⣿⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢳⣷⣿⠀⢹⣿⣿⣷⢄⣴⢟⣿⣿⣿⣿⣿⣽⣶⡤⠛⠓⠀⠀⠀⠀⠀⢹⣿⡞⣧⡀⠴⢟⣋⠀⢠⡜⢷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠀⢸⣿⣾⣟⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣮⣝⣛⣽⠃⢀⣿⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⢀⣿⡿⠏⠀⠈⠻⣛⣛⣛⠛⠛⠛⡛⠓⠀⠀⠀⠀⠀⠰⠾⠿⠿⣿⣿⣿⣿⡇⠀⣸⡿⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠾⣿⣛⠏⠉⠒⢻⣿⣿⣿⣿⣿⣿⠿⠟⠉⠉⠉⠀⠀⠐⠂⠐⣦⣖⣒⣛⣶⡥⠬⣙⣷⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡔⠘⣿⣿⡀⠀⠀⢨⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣄⣀⣼⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⢼⣛⣻⣿⣶⣶⣿⣿⣷⣤⡤⠤⠤⠤⠤⠀⠀⠀⣀⣀⠀⠀⠀⣀⣩⠿⠿⠽⠿⠻⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⣿⣾⣿⣿⣿⠿⠶⠶⠶⠶⠦⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢶⢶⣀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠶⢤⣬⣭⣭⣤⣤⣤⣤⣄⣀⣀⣨⣭⣭⣭⣭⣽⡿⠿⠷⠿⠿⠻⠟⠛








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢩⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⣞⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠾⢰⡋⢐⡄⠀⠀⣿⠂⡞⠀⠀⢀⠀⠀⠀⢸⠋⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⡀⠹⠈⣄⠀⢀⠃⠀⠃⠘⠃⡎⠀⠀⠀⣸⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⢦⡀⢀⠄⠊⠀⠀⢦⠀⠀⡇⠀⠀⢰⡿⢠⡇⠀⠀⠀⡴⠙⡄⢠⠏⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠋⠑⠂⠀⣤⠀⠸⡆⢀⣧⣀⣴⠟⠀⠸⠄⠀⢀⡄⠙⠖⢁⡼⣧⣤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡃⠀⠀⠀⠀⠠⠀⠸⡇⠀⠐⠎⠹⠞⠁⣠⠀⠀⢇⡴⠊⠀⣨⡴⡿⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠁⠀⠀⠀⠀⠘⣀⠴⠁⠀⠀⠀⠀⠐⠋⠀⡇⠀⠻⣕⣶⠬⠀⠈⠀⠀⣇⣀⠤⠲⣀⠁⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⠤⠶⠦⢤⣀⣀⡁⠀⠀⠀⠀⠀⠀⢰⢀⡴⠃⠀⠀⠈⠀⢀⡤⠶⠀⢰⡍⢁⠔⠃⣀⣴⠞⣣⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠐⠀⠋⠉⠐⠚⠣⠴⣶⣬⣁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠓⠟⠉⠋⠉⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣷⣶⣤⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠦⣤⣀⠀⠀⠷⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣽⣿⣿⣿⣿⡏⠘⠀⠀⠀⡴⢶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⢠⣌⠙⠖⣠⠀⠀⠀⠀⠀⢠⠉⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢻⣭⣴⣄⢴⠀⠀⠈⣡⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⣷⡀⠀⠀⠈⠔⣢⢄⡀⡸⠂⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢟⣽⣿⣿⣿⣿⠘⢠⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣇⠀⠀⠀⠀⠀⠉⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣾⣿⣿⣿⣿⣿⣿⠃⠈⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠠⠀⠀⠀⠀⠈⣿⣿⣿⠿⣿⣿⣿⡿⠁⠀⣠⣤⣤⡀⠀⠈⠉⠙⠛⠛⠛⠁⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠠⠀⢀⠄⠀⠀⠈⠛⠁⠃⠸⠿⠋⠀⠀⠊⠉⠩⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⡿⢶⠦⣠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠃
⠀⠀⠀⠾⣀⣄⡀⠉⠀⠘⠋⣱⣧⠴⣦⠀⢄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠰⡶⠡⣘⢙⡷⣾⣦⣤⣾⠃⢠⠟⢳⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣦⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠐⢲⡄⣁⠀⠋⣹⣿⣿⣶⣮⣀⡤⠀⡙⠚⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡇⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⡇⢹⣷⣶⣿⣿⣿⣿⣿⣿⣷⣄⡁⠰⠙⣷⣰⣿⣿⣿⣿⣿⣿⣿⢛⣻⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⠁⣀⠂⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⠛⢁⡼⠁⠀⠀⠀⡐
⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⣀⣠⣜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣛⣀⡀⣀⠔⠁
⠀⠀⠀⠀⢼⠷⠐⠆⠾⢿⡿⢿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠛⣿⡿⠟⠉⠀⠀⠀⠀⠀⣸⠛⠋⠉
⠀⠀⠀⠀⢿⣴⡀⡀⢠⢸⡇⢰⠀⡌⠈⡅⠈⠁⠀⠉⠀⠀⢹⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠛⠷⢾⠎⢁⠸⠧⠟⣠⡧⣌⠣⠴⣶⠟⠛⠉
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠑⠒⠀⠠⣤⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣼⣀⡀⠀⠀⠀⡠⠞⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠞⠁⠀⣿⣿⣿⣿⣭⣍⣀⣀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣅⢀⠆⠀⠀⣿⣿⣿⣿⣯⡉⠉⣉⣥⠀⢉⡑⠠⣄⡠⠔⠊⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠙⠉⠀⠀⠀⠹⣏⠙⠛⡻⠖⠋⠁⡆⠀⠀⠈⠛⢶⣤⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⠇⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢸⠀⢤⣀⠀⠀⠀⢽⣿⣶⡜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⡴⢦⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⣆⠀⠀⠨⣹⣶⣶⣿⣿⡿⢿⣦⡠⠀⠐⢂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠂⠀⠀⠀⠀⣠⠄⠀⠑⣄⠀⠀⢠⠿⣄⠀⡇⠀⠀⠈⠓⢤⡀⠉⠛⡿⠟⠋⠀⣸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⡾⠞⠁⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⡀⠀⠙⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠂⠀⠀⠀⠁⠀⣠⠀⠀⠀⠀⠀⢀⡠⣴⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠏⠓⠀⠄⠀⠀⠀⠀⣴⡂⡎
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⣰⠇⠀⠀⠀⠀⠀⣸⠥⠂⠀⠀⠀⠀⠡⠚⠁⠀⠀⠀⠀⢀⡤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠁⠁
⠀⠀⠀⠀⠀⠀⢀⡮⠀⠸⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠉⠐⠒⠒⠢⢦⡤⠤⠐⠃
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠞⢁⡀⠄⠒⠢⢄⠀⠀⡀⣤⠀⢀⣀⡇⡇
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣡⠞⠛⣉⢽⣓⡢⢄⠐⣺⠇⠁⢄⣡⡚⣇
⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡘⠁⠀⢀⠼⡛⢹⡁⣿⠎⠀⢄⠀⠀⠺⣡⠟⠛
⠀⠀⠀⠚⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠞⠁⠀⠈⠁⠀⠈⠉⠉⠙⡆⠀⠀⠀⠀⠀⠙⣄⡀⠈⡀
⣀⣄⣀⠤⠴⠾⠋⢀⠀⢀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⢀⡿⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⣠⡀⠀⠀⠀⠘⡇⡶⠴
⠀⠀⠀⠀⠶⠐⠺⢧⡖⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢡⡼⠀⠀⠀⠀⠀⠀⢀⣀⣬⢀⣠⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⣀⠀⠀⠈⣁⡔⡁⠀⠀⡀⠀⠀⠀⠈⠀⠆⠀⠀⣀⠴⠚⡱⠚⠁⠀⠈⠉⠁⠐⠫⡀⠀⣰⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠉⠁⠀⠚⠉⠒⠈⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⡼⢋⡥⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠟⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠘⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠄⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⢀⠀⣠⣠⣶⣿⣿⡷⠤⠔⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠁⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠏⠻⣋⠛⠟⣟⡹⠷⣄⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⣀⣠⠴⠞⠉⠀⠀⠁⠀⢈⠉⡉⠉⠀⠀⠈⠉⠀⠚⠋⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⠣⠀⠈⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⢂⠠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠚⠆⠀⠀⠤⢀⣀⣀⡀⠀⠀⠠⣤⣤⠤⠶⠒⠛⠉⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠆⠲⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡐⣁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠦⠀⡙⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⡴⠛⠛⢻⡇⡠⢎⡀⠀⠀⠀⠀⠀⠀⠈⣹⡷⠀⠈⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⡀⢾⡟⠃⢀⡶⠾⠋⣠⢮⣿⡀⠀⠀⠐⢄⠀⠀⢈⣷⢘⡇⠐⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⡷⠀⣰⣿⡤⠛⣡⠷⣤⣿⣿⣏⣵⢟⣆⡀⠀⠀⠉⠘⠋⣿⠸⢇⡤⢠⠇⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⠏⠀⣸⡿⢃⡰⣮⢀⣼⣿⣿⣍⣩⡾⢶⣙⠛⠛⢠⡶⣶⠟⠁⠀⠉⠀⠁⠀⠀⠒⠈⠀⠀⠀⠤⣀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣟⠃⠀⠀⣿⠦⠻⠇⣠⣶⣤⠟⣁⣾⠷⣶⢎⣭⣙⣿⣤⣾⣧⢀⠀⠀⠈⠀⠀⢠⣀⡴⠂⢡⡀⣠⡆⣬⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣮⠀⠀⠐⢋⠀⠈⢶⣿⣿⣶⣾⣿⣷⣯⡷⣿⣥⣮⠑⢋⣼⠶⠋⠀⠀⠀⠀⠀⠟⣹⢰⡶⢿⡏⠠⡄⠻⢃⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢿⠏⠀⢠⣿⠋⢠⣴⣿⡿⢻⡟⢻⣯⡟⠉⣽⣿⡟⣡⠛⣱⡄⠀⠀⠀⠀⠀⣠⣾⠷⡛⠿⠃⢸⡀⣦⢹⡀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣴⠂⢠⣿⢃⡔⣧⣹⣯⣴⣿⣿⣿⣿⡷⠰⣧⣼⣿⠁⣾⠋⡀⠀⠀⠀⣠⠖⢩⡟⢠⠁⠆⢸⣼⣷⡏⢸⣿⡎⢿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠇⠀⢸⡟⠏⢰⣿⠟⣼⠟⠛⠉⠁⠈⠳⠀⣥⠘⣧⠒⣡⠖⢀⣤⠰⢾⡷⢾⣏⣰⣿⡜⠆⠁⢁⡟⢡⡜⠘⠁⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣼⠇⠈⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣠⡶⢪⠀⠀⣿⠀⢡⣶⣿⣵⠔⠋⠀⠲⣾⣧⢀⣤⠘⢣⣾⣠⣿⣿⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠠⠀⠰⠀⢠⠁⠀⠀⠀⠀⣰⣿⠋⣼⠃⠀⣴⠃⣀⣷⡍⠀⠀⠀⠀⠀⠀⠸⠏⢹⣇⡶⢻⢻⠏⢹⣿⡄⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣱⠄⠀⢰⠀⢸⠀⠀⠀⠀⣠⣿⣏⣴⢟⣤⡽⠞⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢇⡀⠀⠀⣼⠉⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⠀⢀⣿⣦⠈⠀⠀⠀⠀⣿⣿⢏⠲⣟⣥⣴⢖⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣾⢷⣤⠀⡟⠀⠉⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⡆⢸⣿⣿⣆⡆⠀⠀⠀⣿⣿⣼⣷⣿⣇⣱⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠀⣤⢸⣿⢰⣇⢀⡄⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠈⣿⠁⢸⣿⣿⠿⣧⠀⠀⢀⣿⣿⣫⣿⡟⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⣿⣿⣿⠀⡇⢸⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⡀⢸⡟⣻⠄⠘⠂⠀⢚⣽⡾⢛⣿⣷⡸⣿⣿⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⢿⣿⣿⣾⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⢸⣏⡷⠀⠀⠀⣱⡟⣥⠛⡿⠉⣽⣷⡌⢻⣯⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠈⢋⠹⡇⣟⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢳⠀⠀⢀⢿⣿⠁⠀⠠⣼⠉⠛⡷⣾⡍⠉⠉⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠹⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⢸⠀⠀⢸⠈⠙⠷⠀⡷⢈⣦⡀⠲⣿⣧⠛⠿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠂⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠈⠀⠀⠈⠂⠀⠠⠆⠀⠀⠀⣠⠞⢩⣿⢰⣿⢻⣿⠷⣄⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠶⡖⠀⠒⢺⠇⣰⡏⣹⣿⣏⣽⠿⠀⠻⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⡤⠙⠶⣤⡄⣾⣟⣤⡋⣴⣻⣿⢇⡀⠀⠀⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣇⠀⠠⡄⠀⠉⠻⡿⢹⡏⠻⢿⣼⣷⣄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⢻⡄⠀⢸⢶⡆⠀⠀⠹⡗⠀⠘⠙⡏⠻⣿⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⢻⣦⡀⠸⣷⠱⣄⡂⠈⠠⠔⠚⠀⠰⠞⣷⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡻⠆⠈⠢⡙⢷⣌⡀⠀⠀⣀⣴⣶⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣿⡿⢣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣥⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢀⣿⠿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠐⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢸⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠟⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⡤⠀⣀⠀⠀⠀⠀⠡⠀⠀⠀⠉⠢⡀⠀⠀⡌⠸⠊⣯⠔⡏⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠉⠑⠢⢄⠡⠀⠀⠀⠀⠈⠢⡴⠀⠀⠀⠀⠐⠁⠖⢸⢁⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⢡⡅⢤⠞⠃
⠀⠀⠀⠀⠀⠈⠑⢄⠉⠀⠁⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠐⠣⡀
⠀⠀⠀⠀⠀⠴⢐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠶⠬⡄
⠀⠀⠀⠀⠀⠶⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⣣⡀⡖⠒⠲
⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠉⠉⠠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠋⣟⠀⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⢈⠧⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⢄⠀⢠⡀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠙⠀⠘⠁⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠠⣤⢤⠔⠈⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠐⠈⠉⠉⠁⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠂⠀⣀⠀⠀⠀⡀⡀⠀⠀⢀⡀⢀⠶⢀⠀⣆⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⡀⠀⡄⢻⡀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠆⠀⠀⠀⡆⠀⠀⠀⢨⠀⠀⠀⠀⢀⠀⣀⠀⢷⠸⡇⠀⠀⠀⠀⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠀⡇
⠀⠀⣀⡀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⢀⠀⡀
⠰⠁⠀⢀⠽⠆⠀⠎⠀⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠁⠀⡆⠀⠚⢅⠀⠈⠱
⡇⠀⠀⠀⠤⠤⠊⠀⠀⠴⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠠⠄⠀⠘⠢⠤⠴
⢣⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠈⠐⠦⣄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠙⠠⠀⠴⠖⠀⠋⠁⠀⠀⠀⠮⠄⠠⠤⠄⠒⠀⠀⠀⠒⠠⠤⠄⠀⠽⠀⠀⠀⠀⠉⠂⠒⠶⠢⠀⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢀⠀⣀⠀⠸⣷⠠⡀⢿⣷⡡⡛⢔⣶⡷⢀⠄⣴⠃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⢢⡳⣿⡟⢮⣤⡌⢋⡀⡀⣀⣃⡈⠁⠀⠓⢂⣉⡎⣶⣿⣰⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡿⠰⢦⣬⠹⣷⠬⡻⢃⡈⡰⣒⣛⣞⣛⠲⠍⢁⠻⠿⢀⣿⠛⣊⣂⣀⢄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⣬⡽⢻⣔⣮⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣥⣜⣅⢸⣿⠄⠠⢋⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢛⣧⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠈⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣿⣿⠀⢀⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠀⢀⣤⣤⣌⡙⠛⠀⠀⠀⠰⠶⠛⢋⣉⠉⠂⠀⣿⣿⡿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠀⠙⠿⠿⠿⠋⠂⠀⠀⠀⠀⠠⠺⣿⣿⣿⡶⠀⣿⣿⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠠⠄⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠂⠀⠀⠀⠀⠀⠂⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⣃⣠⣀⢛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣰⣬⣵⡦⠀⠈⠉⠻⣿⡿⠟⠛⠀⠀⢴⣬⣁⡄⠤
⠀⠀⠀⠀⠀⠀⢠⢀⡀⠸⠛⠿⣿⣫⠄⠀⠀⠀⢷⣿⣷⡏⠀⠀⠠⢤⣽⣿⠿⠛⠃⢀⡀⡄⠴⠃
⠀⠀⠀⠀⠀⢨⣷⣿⣡⣄⠋⠹⣿⣿⣄⠘⣆⢀⣾⣿⣿⣇⣀⣶⠁⣠⣿⣿⠏⠁⠀⣈⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⣷⣴⣿⡿⢿⣻⣛⣟⣻⣏⣻⣛⣟⣻⣻⠿⢿⣿⣦⡾⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣏⣿⠶⠛⠛⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠛⠻⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣇⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣸⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠙⠛⠿⠿⠿⠿⠿⠻⠻⠟⠛⠛⠉⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⢀⣀⡀⠀⠀⢀⣀⡀⠤⠤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡄⠀⡀⣤⡄⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⣾⡇⢸⠝⠁⠀⣁⠀⣀⡀⣀⡀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠽⣏⡗⠉⡶⡄⠈⣿⠀⢻⡏⢻⡇⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⣷⣦⠖⣧⡷⠀⣿⠄⢼⡇⣼⡧⢸⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⣄⡼⣧⣠⠀⠀⠀⠀⠠⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠶⣤⣬⢙⣃⠀⠀⣀⡽⣰⣎⣯⣄⠀⢀⣰⣋⣇⣠⠴⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠠⠊⣻⣗⠭⢛⡋⢭⠞⣫⣏⠛⠿⠿⠿⠿⣿⣿⣧⣤⣤⣴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠐⠉⣿⣷⣤⣀⣤⠤⢿⢿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣷⠚⠯⠉⢉⣹⠿⠀⣈⣥⡀⢾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣿⣷⡶⠞⠛⠛⠉⠉⠻⠟⠉⠙⠿⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢢⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⡿⠞⠁⠁⠠⡀⠀⡀⢀⠀⠀⠠⣈⡂⠛⢿⣾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢳⣿⣿⣿⣷⣾⣿⡿⢿⣾⣷⣿⣿⣿⣿⡖⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢿⣽⣿⡿⢿⠟⣴⣦⡻⠿⢿⣟⣿⡹⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⡠⢀⣀⣰⣶⢾⣿⣻⣷⣿⣶⣄⣀⠀⣤⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡽⠉⠙⠋⡙⠋⠘⢿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣿⢮⣭⣱⣆⣧⣽⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠻⣿⠷⠶⠷⠶⣿⣿⣟⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⡐⠈⠀⠀⠀⠁⠞⣿⠷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠉⠋⠉⠙⠋

⠀⠀⠀⡆
⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣶⣿⢃⣤
⠀⠀⠀⡇⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⡇⢀⣴⣿⣿⣿⣿⣿⣿⡁⠘⣿⣿⣿⠿⣿⣿⣿⣿⣿⣏⠂
⠀⠀⠀⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⢆⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡾⣿⣿⣿⠿⢿⣿⣿⡅⡀
⠀⠀⠀⡟⢻⣮⣿⡟⠁⠈⠙⣿⣿⡇⠙⠿⣿⣿⣿⣮⣭⣵⣿⡈⢿⣿⣄
⠀⠀⠀⡇⠀⠻⣿⣧⢀⣤⣾⣶⣿⣿⣆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣦⡳⠙⠃
⠀⠀⠀⡇⠀⠀⢫⣿⣿⡿⣿⣿⣿⡿⣿⢿⢞⢻⡿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⡇⢰⣶⣾⣿⣿⣥⣻⣿⣿⠁⠙⠘⠄⠁⠃⠀⠙⣿⣿⣿⣿⣿⠁
⠀⠀⠀⡇⣾⡟⠋⢻⣿⣿⡿⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⣷⡟⠁⠀⠈⢿⣿⡟⢻⣿⣿⣿⣧⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠤⠖⠁⠀⠈
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠉⠃⣸⡿⣿⣿⣮⢻⣷⠀⠀⠀⠀⠀⠀⠘⣁⣤⡔⠁⠀⠀⠀⢀⠀⢀⠔⠀⠠⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡟⣷⠀⠹⣿⣧⠹⣄⡀⣀⣤⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣿⣞⠥⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣤⠊⢁⢂⣠⡔⠀⣂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣈⣟⡇⠘⠬⣿⠏⠠⣹⣿⠃⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⠠⣲⢀⡀⣀⡀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠀⠀⣁⠂⠑⠈⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠇⠤⠤⠭⢤⠤⣼⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣔⠂⠂⠀⣘⣤⡈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆⠓⠃⠀⠀⠓⠃⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⠀⠤⡀⢀⠀⠘⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⢠⠀⡤⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⢀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡚⠀⠀⢡⠈⠁⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡏⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠨⢇⡀⠀⠀⠁⠀⣀⠀⣰⡀⢠⢶⡀⢀⠀⠀⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣄⠀⠈⠀⢄⡠⠾⢷⣿⠊⣹⣻⢮⡾⠚⢘⡇⠸⡌⢀⣍⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⢄⠀⠈⠑⠠⢤⣄⡀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠈⣠⠵⠋⠁⠠⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠢⣁⡀⠀⠀⠀⠈⠡⡄⠀⠀⠀⠀⠀⠀⢀⢀⠔⠈⠀⠀⠀⢀⢀⠔⠉⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⢀⠁⠀⠀⠀⠀⠀⢳⠀⣐⣤⣤⡀⠀⢀⢸⠀⠐⠀⠀⠀⠈⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢩⠁⣉⠁⠘⠠⢐⠈⠀⠀⠀⡇⠅⠸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠁⠀⠀⠂⠀⡈⢀⡄⠳⠙⠻⠃⠴⠃⡏⠀⠀⠀⢐⠀⠀⡧⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠀⠀⠀⠨⠀⠁⡀⢐⠂⡠⣀⣀⣀⠤⡀⣂⠥⠐⠈⢀⠀⠀⡅⡀⠅⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⠚⠃⠐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣷⣄⣼⣧⣠⣞⣡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⡟⠉⠁⠉⠉⠻⣯⡉⢛⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⢈⠀⠀⠀⠀⠀⢹⣿⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⢠⡿⢻⡅⠀⠀⠀⠺⣤⣀⣤⡴⢂⣾⡿⠀⠉⠛⠻⣿⡆⠀⠀⠀⡀⠀⢀⡄⠀⢀⡄⢀
⠀⠀⠀⠀⠀⢸⠃⠈⠛⠃⠀⢉⣒⣺⣿⣿⠾⠟⠋⠀⠀⠀⠀⠀⡿⠁⠀⠉⠲⣼⣶⣼⣷⣴⣿⡶⠋
⠀⠀⠀⠀⣠⠜⠀⠀⣠⡶⡟⠛⠉⠉⠀⠀⠐⣒⡦⠄⠀⠀⢤⡂⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣥⣤⣤⣄⡀
⠀⠀⠀⣼⡏⠀⢠⣾⢫⡾⠁⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠐⠋⠈⠀⠀⢀⣴⠟⠋⢡⣾⠟⠉⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⢸⣿⠀⠀⣿⡇⡟⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⠄⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⢸⣿⣦⡟⣿⣧⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠁⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠠⣼⠏⠹⣇
⠀⠀⠘⣿⣿⡇⢹⣿⣷⣤⣀⣴⡇⠀⠀⠀⠀⠀⠰⠾⠿⠃⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣦⣤⣄⣀⠀⠀⠀⠿
⠀⠀⠀⠘⠿⣿⣄⠙⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡀⠀⠐⣲⣭⣁⣀⠀⠀⠀⣌⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠉⠉⠁⠀⠀⠀⠈⣿⣆⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢹⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠀⠀⠀⠀⠀⠸⣷⣤⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⣰⣬⣫⣀⣴⣀⡜⣣⣤⣠⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣧⣈⡏⣠⣤⠀⠀⠘⠛⠙⢿⣿⣿⣿⣿⠿⠛⠯⠀⠀⢀⣤⣄⠻⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣦⣍⣱⣿⣿⣿⡿⠉⠀⠀⠠⠂⠒⣦⡹⣿⡟⣡⣆⢀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣎⣭⣴⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡆⣤⡄⠀⠘⠛⠙⠻⣿⡟⣴⠅⠀⢀⠀⠀⠀⢹⣷⣿⣷⣿⠁⠀⠀⠀⢠⠖⢺⣦⣻⣿⠟⠋⠙⠃⠀⣠⣤⢲⠄⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣿⣿⢁⣀⠀⡎⠙⢷⣾⣿⣿⡀⠀⢸⣼⠀⠀⣼⣿⣿⣿⣿⡀⠀⢐⣶⠃⠀⢠⣿⣿⣷⡾⠛⠅⠀⠀⠉⣿⣶⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⢿⣷⣾⠉⠳⣆⠀⠘⣿⣿⣿⣿⣦⣴⣿⣦⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣶⣿⣿⣿⣿⢁⠀⣠⠀⠀⣇⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣤⣻⣶⣿⣿⣿⠿⢛⣟⣫⢏⣹⢯⣠⣿⣅⣼⣭⣼⢻⣩⢯⣛⡟⠻⢿⣿⣿⣿⣼⡏⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⢻⣽⣤⣶⠿⠷⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣶⣼⣿⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣯⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣄⣀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣤⣀⣀⣠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠁⠉⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠇⠃⠁⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢻⣷⡀⠂⢠⡄⣴⣿⡀⡰⠀⠠⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⠓⣵⣿⡇⠘⣏⡿⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠿⠿⣿⢻⡃⣼⣏⡀⠀⠀⠰⣿⡖⢠⣾⣳⡤⣾⣳⠆⢠⣴⡞⢿⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⡖⠻⣧⣼⠃⠀⢸⡿⡆⠀⠀⣿⡇⠀⢸⣿⠁⢸⣿⠀⢠⣿⡇⢸⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⣤⣭⣤⣄⣤⢾⣄⢷⡄⢀⣿⣇⡀⣸⣿⡆⣸⣿⣀⢸⣿⣇⣸⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠉⠉⠻⠿⠁⠌⠻⠟⠀⠈⠙⠋⠀⠙⠟⠁⠉⠿⠁⠊⠙⠋⢹⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢘⡃⠀⢰⡆⠀⣨⠁⠀⣠⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣇⡀⣿⣧⢸⣿⣀⣸⣿⣶⣿⣦⣰⡿
⠀⠀⠀⠀⠀⠀⠀⢦⣦⣘⣿⣼⣿⣷⠾⢟⡋⣝⡿⣆⣠⡠⡬⣽⠉⡉⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣧⣴⣄⣿⡷⠌⢓⣛⣛⣓⣂⣀⣀⡴⢾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⢷⠾⣛⣭⣶⣾⡏⠉⠉⠀⠀⠀⠀⠀⠈⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠃⠰⠚⠛⣷⣶⠤⠀⣰⣶⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠁⠀⠰⠶⠖⠚⠟⠁⠀⢿⠟⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣌⣉⣿⣿⣧⣀⠀⠀⠀⢀⣴⣂⡀⢀⣴⡄⣺⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣾⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢀⣶⢿⣋⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣰⣇⣸⢡⣿⣿⣿⣿⣿⣿⡿⠃⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⠛⠘⣿⡏⣿⣏⠟⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⢸⣿⢻⠘⣿⣷⣷⣿⡿⠀⠀⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠘⢼⣿⣱⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢿⡿⡿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠛⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⠟⢻⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠻⢷⣶⣶⠿⠏⣼⣧⠹⢷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⢴⣾⣿⣿⠸⣗⡤⠔⣒⡛⠻⠿⣟⣒⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⠷⣢⠴⣋⣠⣥⣶⣶⣶⣿⣿⣷⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⣿⣿⣿⣿⣟⡥⣢⣾⣿⣿⣿⣿⣿⠏⠀⠈⢿⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣿⣿⣾⣿⡿⠿⣟⣛⣉⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡾⠀⠀⢀⣀⣈⣙⠛⠃⠀⠴⠞⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⣧⠀⠀⠈⠛⠘⠟⠁⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠁⠈⠈⠓⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠟⣿⣿⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇⣤⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠐⢶⣟⣻⣇⣾⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⢸⣿⣿⣿⣿⣿⠉⠳⣦⣄⣀⠀⠀⠀⠀⢁⣾⡿⠿⣯⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧⡻⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣷⣶⣶⣾⣿⣷⣄⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢹⣿⣿⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠻⣿⡇⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⡛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⣿⣍⡛⠻⠷⣌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣾⣷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣠⣾⠟⠹⣷⣄⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣸⣿⠉⠉⠁⠀⠀⠈⠉⠉⣿⣇⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠹⣷⣤⣤⣤⣤⣤⣤⣤⣤⣾⠏⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣀⣀⡀
⠀⠀⠀⠀⣴⡾⠿⠛⠛⠛⠛⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠛⠛⠛⠛⠻⠷⣦⡄
⠀⠀⢀⣴⣿⣶⣤⣄⣀⣀⣀⣤⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣀⣀⣀⣀⣤⣴⡿⢷⣤⡀
⠀⢠⡾⠋⠀⠀⢿⣏⠉⠉⠙⣿⠁⠙⠿⠶⢿⣦⠀⠀⠀⠀⣼⡶⠾⠛⠉⠉⣿⠋⠉⠉⣹⡟⠀⠀⠙⣷⡄
⠀⣿⠃⠀⠀⠀⠘⣿⡄⠀⣠⣿⣄⡀⠀⠀⠀⠙⠿⢶⡶⠿⠋⠀⠀⠀⢀⣸⣿⣄⠀⢰⣿⠁⠀⠀⠀⠘⣿
⢸⣿⣀⠀⠀⠀⠀⢹⣷⣸⡟⠉⠛⠻⠿⢶⣶⣦⣤⣤⣤⣤⣴⣶⠶⠿⠛⠛⠉⢻⣇⣿⠇⠀⠀⠀⠀⣀⣿
⣼⡟⠛⣿⡄⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⢠⣿⠛⢻
⣿⡇⠀⣿⡇⠀⢾⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣷⠀⢸⣿⠀⢸
⣿⡇⠀⣿⡇⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣾⠟⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⢸⣿⠀⢸
⣿⣇⣀⣿⣇⣀⣺⣇⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣸⣟⣀⣸⣿⣀⣸
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠭⠭⠍⠥⠅⠤⠬⠭⠌⠤⠤⠥⠤⠭⠬⠭⠭⠬⠭⠌⠍⠭⠩⠩⠩⠥⠥⠤⠤⠤⠤⠤⠠⠥⠨⠥⠡⠁⢀⡉⠉⠉⠭⠉
⡇⣠⠶⠞⠁⢈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠓⠢
⡇⡇⢀⠈⠁⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡧⠶⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⡐⡂⠀⡀⢰
⠀⢇⣙⣣⡼⣚⣽⣷⣦⣤⡤⠤⠤⠰⠴⠒⠂⠈⠀⠐⠋⢰⣁⠀⠀⠀⠁⠁⠀⠀⠐⠂⠀⠀⣀⣠⡿⣿⣦⣙⠦⣈⣃⠟
⠀⠀⠙⣷⢾⢿⡋⠛⠯⣝⡙⠛⠛⠂⣠⢰⣤⢊⣇⣶⣰⠷⣶⠟⣦⡖⠶⠖⣆⣤⡉⠁⠀⠀⣉⠥⠞⢋⣿⡿⣿⡏⠁
⡀⠀⠀⡟⢸⢿⠛⠢⣄⣤⡈⠀⠀⠠⡷⣅⣹⡰⢦⣤⠶⣄⡷⣠⠄⣤⣜⣈⣧⡨⢐⠀⠀⠉⠀⣀⠴⠊⠀⣍⠻⣧
⠀⠀⠀⡗⢸⣿⠀⠀⠀⢉⡑⠒⠒⠒⢳⡽⢿⣶⢬⣹⠶⠧⢾⠼⣓⣿⠤⢰⣿⢁⠚⠒⠒⠉⢁⡄⠀⠀⢠⡋⡄⣿
⠀⠀⢸⡏⢸⡿⡄⠀⠀⠘⠇⠀⠀⣦⠀⠉⢻⣥⣭⠒⣂⣻⣽⣏⣱⡖⡵⢚⡏⠀⠀⡆⠀⠀⠘⠀⠀⠀⢨⡅⡇⣟⡆
⠀⠀⠸⣇⠀⣿⡇⣠⠀⠀⠀⣤⢤⡤⡤⠤⣤⠿⠶⠿⠯⠿⢽⡽⢿⣿⣶⣿⣤⣤⣤⣤⣤⣄⠀⠀⠀⣴⠸⠃⡇⣏
⠀⠀⡴⣿⢠⢻⡇⠛⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⣸⠀⡇⣿⠀⡀
⠀⢀⡇⠂⠈⢸⣷⠀⠀⢰⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⡧⢸⠇⢿⠀⡇
⠀⢸⡇⡀⠘⢸⣿⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⡇⠘⠀⠀⢰⡇
⠀⢸⠁⠇⠀⠀⣶⡀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⢠⣇⠀⠀⠀⠀⠇
⠀⢸⣀⣤⡤⡾⢟⡆⣻⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⢨⣍⠛⠲⣄⠃⡄
⠀⢸⠋⠃⠀⢐⡹⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢬⣙⠲⢼⣈⡃⢸
⠀⣎⣤⠶⠛⠛⡇⠀⠀⢀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⢸⡛⠛⡳⣮⣳⣸
⠀⣿⣒⣒⣀⣀⡇⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠇⠀⠀⢸⣥⣤⣄⣈⠝⢹
⠸⢈⠻⠉⡍⠉⡃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢸⡇⠄⢈⣀⡀⣸
⠀⠈⡀⠀⣡⡾⠁⠀⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣴⠀⠀⠙⢶⣬⡛⣇⣾
⠀⠘⣣⠞⠋⣀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠹⠄⠀⢠⠀⠙⠿⡿⢿
⠀⢰⡏⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀
⠀⠈⠻⣦⣀⠁⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡴⠀⠀⠀⠀⢀⣠⡞⠃
⠀⠀⠀⠈⠙⠻⠶⣦⣤⣀⠁⠘⢧⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡴⠋⠀⢁⣠⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡀⠀⠈⠉⠉⠉⡍⠉⠙⢦⣀⡴⠋⠉⠉⠉⡅⠀⠀⠚⠁⣠⡴⠟⠋⠉
⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⢶⠀⠙⠷⢦⣤⣤⣤⣤⣤⣄⣀⣙⣀⣀⣠⣤⣤⣥⣤⣤⠶⠞⠃⢰⠃⠈⠛⠚⠛⠛⠛⠓⠓⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢤⠸⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⠂⠀⠀⠄⠂⠀⠀⠐⠀⠂⠂⠐⠐⠀⠀⠀⠂⠀⠀⠀⠀⠒⠘⠒⠒⠒⠒⠒⢒⣒⣒⣒⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⣶⣦⣄⡀⣤⣸⣧⣠⠀⣠⣤⣶⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣷⣤⢤⣤⣈⠙⢿⣿⣿⣿⣿⡾⠛⢉⣠⣤⣴⣾⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣞⠉⠉⠉⠉⠙⠳⠈⠻⣿⡄⢿⣿⣿⣿⢁⣾⠟⠉⠐⠋⠉⠉⠉⢙⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⢻⣿⣿⣿⣿⣶⣶⣤⡀⠀⠈⢻⣼⣿⣿⡇⠞⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⡟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣖⠒⢛⣉⣤⡦⠍⠍⡉⢿⣿⣿⣄⢿⣌⣦⣿⣿⣠⢠⡾⣠⣾⣿⡯⠉⡉⠉⢬⣥⣉⡓⢒⢲⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⢿⣿⣿⣯⣴⣦⣴⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⣶⣶⣶⣮⣽⣿⡿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡲⠄⠔⠒⣚⣿⣿⣿⠋⠙⣻⣿⣿⡿⠿⠿⠻⠿⢻⣿⣿⡟⠿⠿⠿⠿⢿⣿⣿⡟⠋⠹⣿⣿⣿⣟⠒⠦⠤⢖⣲⣶⣦⡀
⠀⠀⠀⠀⠀⢀⣿⣯⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣦⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣼⣿⣿⣿⣿⣟⠿⠿⠿⣿⣿⣽⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⢀⣀⣤⣄⣶⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣠⣤⣀⣀⡀⠉⠛⣿⡇
⠀⠀⠀⠀⠀⣾⣇⢀⣴⣿⣿⠿⢟⡛⠝⠛⣩⣿⣿⣿⣿⣿⣶⣿⣿⣦⣀⠀⠀⠀⠀⢀⠀⣀⣴⣾⣿⣶⣿⣿⣿⣿⣿⣯⡻⢯⣟⡿⢿⣿⣿⣷⡄⣼⣷
⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⢁⠉⠁⠄⣾⡿⢟⣽⣿⣿⡇⠙⠿⣿⣾⢽⣿⣿⣿⠠⠤⢿⣿⣿⣿⣧⣿⠿⠃⢸⣿⣿⣮⡙⢿⣷⠠⠀⠁⠈⠛⢿⣿⣿⣿⡆
⠀⠀⠀⣰⣿⣿⡟⠁⠐⠈⠁⣀⣴⡿⠋⠀⢉⣵⣿⣿⣷⣄⠀⠀⠉⠻⣿⣿⠃⠀⠀⠈⢿⣿⡟⠋⠁⠀⣀⣼⣿⣿⣮⡉⠀⠙⢿⣦⣀⠀⠁⠂⠙⢿⣿⣿⣄
⠀⠠⠾⠛⣹⣯⡁⠀⠀⢠⣶⣿⠋⠀⠀⣤⣿⣿⣿⣿⣿⣿⣷⣷⡆⠀⣹⠃⠀⠀⠀⠀⠈⣯⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠹⣿⣷⣄⠀⠀⢈⣿⡏⠛⠦
⠀⠀⠀⢠⣿⣿⣷⣴⣶⣿⡿⠁⢀⣔⣴⡿⢟⣽⠟⣉⣽⣿⣿⣿⡇⠀⢿⣂⠀⠀⠀⠀⣀⣿⠂⢨⣿⣿⣿⣿⣍⡛⢿⡙⢿⣦⣲⡀⠈⢿⣿⣷⣴⣾⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⢀⢮⡿⢋⣴⣞⣥⣾⣿⣿⣿⣿⡿⠁⠀⠀⢫⣿⣶⣶⣿⡿⠃⠀⠀⢻⣿⣿⣿⣿⣿⣦⣹⣦⡙⢿⣷⡀⠈⢿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡋⣎⣿⣿⠟⣵⣿⣿⢟⣿⣿⣿⣿⣧⠤⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠤⣼⣿⣿⣿⣿⠻⣿⣿⣯⠻⣾⣷⡇⡹⡌⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⣑⣸⣿⠃⣸⣿⡟⠁⣾⣿⣿⣿⣯⠉⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠈⣟⢿⣿⣿⣷⡈⠻⣿⣧⠸⣿⣿⠉⣻⣼⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠏⠀⠈⣿⣿⣿⡿⣿⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣤⣀⠀⠀⣸⣿⣿⣿⣿⠁⠀⢻⣿⠃⢹⣷⣿⣿⣿⡟⣿⣿⡿
⠀⠀⠀⠀⢹⣿⡟⣾⣿⣿⡟⣿⢳⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣤⣾⣿⡿⡟⠻⠿⠿⠛⡿⣿⣿⣷⣦⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⡎⣿⢻⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣧⢿⣿⣿⣇⣿⢸⣿⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢹⣿⣷⣿⣿⣿⣿⡇⣿⡿
⠀⠀⠀⠀⠀⠹⣇⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣄⠀⠀⢠⣀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠹⡄⢿⣿⣿⣿⣿⣿⡇⠀⠀⡿⢸⣿⡟⢻⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡇⣽⠆⠀⢈⣿⣟⣿⣿⣿⡟⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣷⠀⠀⠁⢸⣿⡗⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⠛⢀⣿⠇⠸⠀⠀⢸⣿⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⣿⣿⣿⣯⣀⣀⠈⣿⣷⡀⠹⣿⣷⢠⣿⣿⣯⢻⡟⢿⣿⣿⠠⣈⣿⡟⠀⣼⡿⠀⣀⡄⣴⣿⣿⣿⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠸⣿⣧⠀⣿⣿⣿⣿⣿⣿⠀⢀⢏⣿⣿⣿⣼⣿⡑⣰⣿⣿⣼⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣷⣽⣿⣦⢙⣿⣿⣿⣿⣗⠀⠀⣻⣿⣿⣿⣿⡥⣰⣿⣯⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣷⣾⣿⣿⣿⠿⢋⣾⣿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡷⠾⠛⣩⠴⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡾⢷⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣼⣧⣼⣧⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢤⣶⣿⡿⣧⣼⢿⣿⣶⡴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⣀⢀⡤⠄⣓⢷⣿⣏⠋⠙⣹⣿⡞⣚⠠⢄⡀⣀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢦⢀⡀⣏⣹⡞⢉⡏⢠⢪⠶⠺⣿⠙⣄⣰⢉⡿⠗⠶⡕⡄⢹⣉⢢⣏⣹⢀⡀⡴⡒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⣾⡟⣩⡭⣭⣽⣿⣷⣿⡏⠀⠀⢺⣻⣿⣿⣟⡇⠀⠀⢹⣿⣾⣿⣯⣭⣭⣍⣿⣷⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣷⠟⢛⢻⣿⣿⣿⡇⠀⢞⣭⣛⣿⣿⣋⣭⡧⠀⢸⣿⢿⣿⡛⡛⠻⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⢿⣿⣿⠀⢸⣛⢨⣇⣿⡷⢦⠈⢉⣿⣿⣿⢯⡉⠀⡴⢾⣿⣸⠅⣙⡇⠀⣿⣿⡿⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⣬⡙⢧⡻⢮⣼⣿⣿⣧⡾⣀⢄⢙⣿⣿⡋⡠⣀⢷⣾⣿⣿⣦⠥⢏⡴⢋⣥⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⠟⠺⣷⡀⢶⣦⣿⣿⣦⣘⣋⣼⣷⣿⣧⣑⣃⣶⣿⣿⣴⡶⢠⣾⠞⠹⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣼⣯⣾⣟⡿⢿⣿⠿⣿⣿⣟⢻⣿⣿⠿⣿⣿⣿⣿⣷⣽⣧⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⣿⣍⣿⡷⠾⠿⠛⣛⣛⣛⣛⣛⣛⡛⠿⠷⢾⣿⣟⣿⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣷⣲⣭⣿⣿⣷⣿⣶⣶⣿⣿⣿⣯⣭⣖⣾⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⡎⢱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢲⡀⡀⠀⢨⡀⠀⠀⠀⠀⣌⠀⠀⡀⡰⢢⠄
⠀⠀⠐⠇⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣄⣼⠀⣆⠙⠀⠄⣿⣄⣼⣿⣷
⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⡄⣷⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⠷⠿⣶⡶⠿⠿⣽⣿⣿⡄
⠀⠀⠀⣶⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡞⠁⠀⠀⠀⠀⠀⠀⠙⣶⣿⣿⣧⡀
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⣭⣿⣿⣤⣀⠀⠀⢀⠀⠀⢀⣠⣼⣿⣯⣻⣗⠄
⠀⠀⣠⢠⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣧⣿⠁⠈⣽⠟⠁⠻⠁⠙⢿⡍⠀⢿⣯⣷⣯⡃
⠀⠀⠻⠼⠁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⣼⠇⠀⢄⡀⣠⠄⠀⣿⠀⣸⣿⣿⣿⡂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠻⣯⣽⣷⣸⣧⢀⡈⠙⢁⡀⣰⣟⣼⣿⣹⣟⣿⢷
⠀⠀⡿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣻⠏⢻⡟⣸⠛⠻⠛⢿⠘⡿⠉⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠖⣿⣗⢟⣼⣷⣼⡇⠀⠀⣿⣴⣷⣝⢷⣿⡷⢦⠁
⠀⠀⣤⢤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣻⣿⣍⣰⣿⡋⠛⢶⠞⠉⣿⣧⣉⣽⣟⠓⠄
⠀⠀⠚⠛⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠿⡿⣸⣿⣄⠀⠀⠀⢀⣼⣧⡻⡿⢮⠑
⠀⠀⢀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣷⣿⠿⠿⢷⡿⠿⣿⣧⡟⠷
⠀⠀⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠜⠄⢀⣞⠀⠆⠼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⡾⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢩⡦⠀⠀⠀⠀⢀⡀⠀⡇⢸⠉⣻⠹⣇⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⢤⠞⠋⠁⠀⣠⠀⠀⣰⠇⠀⠀⠟⠁⢀⡇⢸⡴⢡⢀⣯⠟⣹⡁
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⣰⠇⠀⢀⡄⠀⠘⠀⠀⠀⠀⡀⣄⣤⠠⣼⡔⠉⠱⠙⢘⠅⡼⣉⢳⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⠇⠀⠀⣦⠀⡤⠂⢹⣾⢻⠀⡿⣌⡶⠤⣤⣬⡞⠁⠉⠀⣷
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠷⣦⣄⡇⠀⠀⡀⠀⠀⠀⠁⣼⢧⠀⠘⠛⠀⠀⠻⣯⡝⠻⣚⡟⠀⢰⠇⠀⢸
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠈⣷⠀⠀⣤⣤⡿⢶⠀⠀⠀⠀⠀⠀⠀⠉⢩⣥⠖⡇⠀⢀⠀⡼⠓⢆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⠶⠶⠌⣇⠀⠀⠀⠐⡇⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠋⢠⡇⠀⠀⢳
⠀⠀⠀⠈⣇⠀⠀⠀⣠⠟⠉⠀⠆⠀⠀⠘⢦⡀⠓⠆⠙⢮⡆⠀⠀⠙⢶⣆⠀⠀⢀⡴⠟⠂⢸⡇⠀⢀⡼⠀⢀⣀⣸⡄
⠀⠀⠀⠀⢹⣆⠀⣾⠉⠀⠀⣤⡀⠀⠓⠀⣀⣽⣦⣀⣤⠘⡇⠀⠐⢤⣀⡉⠀⠀⠠⣄⣤⠆⣸⢇⣠⠛⣦⠞⠛⠉⢉⡉⠓⢦⡀
⠀⠀⠀⢠⡿⠉⣿⠇⠀⣆⠀⠘⠃⠀⠀⡾⠁⠀⠀⠋⢼⣁⡙⢤⣀⣀⠉⠀⣀⠀⢠⣀⣠⠾⠛⠉⠀⠀⣿⠀⠺⠀⠘⠃⠀⠆⣧
⠀⠀⠀⣸⡇⠀⣯⠀⠀⠈⠁⠀⣠⣤⠀⣧⠀⠀⠀⠀⡆⠀⠉⠙⠺⠋⠉⢻⡙⠦⡼⠋⠀⠀⠀⠀⠀⢰⡏⠴⠆⠀⠰⠂⢠⣄⡿
⠀⠀⠀⢹⡇⠀⠻⣄⠰⠶⠂⠀⠀⠀⠀⠈⠙⢶⣤⣄⠠⣤⠤⠖⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠘⠉⠉⠙⢢⣠⠞⠁⠀⡀
⠀⠀⠀⠘⣧⠀⠀⠙⠶⣄⣀⠐⠾⠃⠀⢀⣠⠎⠀⠉⠙⠮⡖⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⣄⡀⠀⠀⠈⠙⠛⡿⠁
⠀⠀⠀⣼⠟⠣⡀⠀⠀⠀⠉⠛⠲⠶⠚⠋⠀⠀⠀⠀⠀⣠⠶⣦⡀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣼⡴⠏⠙⠳⠶⠶⠖⠋
⠀⠀⣸⡏⢀⡲⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣷⠟⠁⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠸⣧⡈⠓⠀⢲⡌⠙⠲⢦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣴⠿⠋⠁⠀⠀⢀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠘⠻⣦⣀⠀⠠⢤⠄⠘⠆⠀⠉⠉⠩⠭⠽⠿⢯⣥⣤⠤⠶⠶⠛⢋⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠲⣦⣤⣤⣀⣀⣙⣛⠀⠀⠶⠖⠀⠀⠟⠃⣀⣨⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣋⡉⠛⣯⣿⠙⠉⠙⣿⡿⠿⡿⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⢛⣿⣿⣿⣦⣄
⠀⠀⠀⠀⣿⣿⡯⣿⡙⣿⣿⣿⡷⣶⣆⣼⣿⣿⣿⣿⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⣿⣧⢻⣿⣿⣿⣿⡟⠻⢘⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠁⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠷⠶⣤⣄⠀⠀⠀⠀⠀⢸⣦⣷⣿⣡⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠳⠙⣽⣿⣸⣿⣿⡀⢀⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣄⠀⠀⠀⢸⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⢹⣿⣿⡇⢻⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡆⠀⠀⣿⣿⣿⣿⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣶⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁⣼⣿⣿⣿⣿⣇⠀⣐⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣮⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠁⠀⠀⣹⣿⣿⠏⠉⠛⢿⣿⠹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⢸⣿⣧⣿⠀⠀⠀⣿⣿⡿⠏⠙⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢰⣿⡟⠀⠀⠀⠀⠀⠈⢷⣿⣿⡀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠄⠹⣿⡁⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⢀⣾⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢀⣿⣿⡆⠀⠀⠀⠀⢀⣾⡟⠁⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⠆⠀⠀⣠⣾⡿⠁⢠⣾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠛⡏⠉⠁⠞⠁⠓⢆⠔⠚⠃⠀⠓⠒⠀⠉⠑⠛⠈⠁⠦⠀⠈⡱⡀⠀⠙⠒⡶⠉⠀⠀⢀⠖⠉⡠⠒⠀⠚⠑⠎⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⣠⢾⠎⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠃⠀⠀⡟⡵⡀⠀⠀⠇⡀⠀⠰⠁⠀⡜⠀⠀⠀⠀⢸⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠐⣏⠁⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡵⡀⠀⠀⡆⠀⠇⠀⠀⠃⠀⠀⡀⠠⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠇⠀⢘⡇⠱⡄⠀⠡⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠘⣵⢀⡆⠀⠀⠃⠀⠀⡇⠀⠈⠉⠁⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠰⡀⠀⠳⣀⣀⠀⠀⠀⠀⡇⠀⠀⠀⡃⠀⠀⠀⠀⠀⠐⢋⢧⠀⠀⠈⠢⡀⠰⣄⠀⠀⠀⣴⡝
⠀⠀⠀⠀⠀⠛⠓⠒⠒⠙⠃⠀⠑⠒⠒⠀⠈⠋⠓⠒⠒⠋⠃⠀⠋⠑⠒⠚⠙⠃⠀⠀⠀⠛⠀⠀⠀⠀⠈⠒⠈⠤⠌⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣂⠀⡈⠀⠀⣍⠇⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⡐⣇⢰⠆⢹⠈⣥⠐⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠛⠀⠀⠓⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡇⠀⠀⠀⠀⣸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠨⢀⣀⣀⡠⠃⠀⠀⠀⠀⠉⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⡠⠀⠂⠲⣄⠀⠀⠀⠈⠉⠀⠀⠀⡠⠀⠀⠠⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡅⠄⠉⡱⠄⠃⠈⠁⠀⠀⠈⠁⠚⠠⣎⠈⡁⠰⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⠃⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡋⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⡀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠉⠐⠂⠀⠠⠤⠤⠄⠀⠒⠀⠈⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠀⠈⠐⠠⢀⣀⠀⠀⣀⡀⠤⠐⠁⢻⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠃⠀⠀⠀⠀⠀⠋⠉⠉⢩⠀⠀⠀⠀⠀⡚⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⢠⠜⠀⠀⠀⠀⢦⡤⣤⠄⠖⠁


⣶⠖⠲⢶⣶⢶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⡀⢂⣸⣧⣠⣬⣨⣅⣥⣤⣭⣧⣼⣥⣤⣥⣼⣡⣄⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣦⣤⣦⣤⣿⣿⣤⣤⣤⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡀⠀⠀⢠⠏⢻⠀⠀⠀⢠⠖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⢣⡀⠀⡾⠀⠘⣇⠀⣠⠟⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⡨⠶⢇⠀⡔⠛⠚⢅⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⡆⣅⣀⣀⣈⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡈⠉⢉⣀⡀⠂⠂⢄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠀⠀⢹⠀⠀⠀⢨⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⣈⣦⠤⠤⠦⠷⠤⢴⣀⡀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣯⡤⠖⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠦⣬⣏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠑⢄⡀⠀⠀⠈⠓⠲⠤⠤⠤⠤⠤⠴⠒⠋⠁⠀⠀⣀⠔⠉⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣠⢤⡀⠀⠀⠀⠀⣀⢤⣀⡠⠴⠚⠁⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⢀⣴⠋⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⢳⣄
⠀⠀⣀⣠⣴⠋⠸⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣸⠈⢳⣤⣀⡀
⡔⠋⠁⠀⠀⠀⠀⢙⡶⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠰⣾⠁⠀⠀⠀⠀⠉⠓
⠱⡖⢒⣦⠀⠰⡞⢹⠇⠀⠀⠀⠀⠀⠉⠓⠶⠤⠞⠳⢦⣀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠦⠴⠖⠋⠁⠀⠀⠀⠀⠀⢻⠙⡶⠀⢠⣖⠒⣾
⠀⠉⠁⠘⠦⠴⠃⡞⠀⠀⠀⠀⠀⢀⣀⡤⢤⣄⡀⠀⠀⡏⢹⡶⢦⠶⣾⠉⡇⠀⠀⣀⡤⠤⣄⣀⠀⠀⠀⠀⠀⠘⡆⠳⠤⠞⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠙⢷⡀⡇⢀⡀⢸⠀⣀⠀⡇⢰⠞⠁⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣧⡇⠘⠃⢸⠀⠙⠀⣇⡟⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠰⠆⢸⠀⠶⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⣽⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⢸⠁⣉⠉⠁⣨⣇⠀⠀⡀⠀⠠⠤⠀⠐⢺⡀⢀⡀⢸⠀⣀⠀⢸⠂⠀⠠⠤⠀⠀⣀⠀⢀⣯⡀⠈⢉⡁⢹
⠀⠀⠀⠀⠀⣿⠀⠉⠁⠀⠈⣸⡄⠀⠀⠀⠀⠀⠀⠀⠈⡧⣈⠁⣸⡀⢉⡠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⠀⠀⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠹⣄⠀⠀⠀⠀⠀⠀⣸⠃⠈⠉⠁⠉⠉⠀⠹⡀⠀⠀⠀⠀⠀⢀⡴⠃⠈⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣤⣤⣤⣰⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣤⣤⣤⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣠⡄⣠⣶⡮⠙⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠐⣶⣿⣿⣾⣿⡷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠉⠀⠀⢿⣿⠙⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⡀⢀⣤⡴⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣇⡀⠀⠉⠂⠀⠀⡀⠀⠀⠀⠐⢡⣴⣿⣷⣽⣿⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢰⣧⣬⣿⣿⡇⡀⠀⡰⠀⠀⢋⠀⠀⠀⠀⠀⠫⠉⠉⣿⣿⠃⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⡛⠋⠹⣿⣇⠑⢴⠃⠀⠀⠀⠈⠀⠠⠤⠀⠀⠀⠀⠸⣿⢀⣲⠊
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿⡟⢀⠐
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠐⠒⠀⠀⠀⠀⢲⡞⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠇⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⠁⡜⡀⠀⠀⠀⠀⡠⠔⢀⣀⠀⢀⣄⣠⢄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⡉⠀⠀⠀⠀⠐⡷⣿⣿⣦⣿⣿⡅⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠉⠈⠁⣿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⢃⢸⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠃⠀⠀⢀⣄⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠋⣡⠾⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠛⢻⣿⠛⢀⣀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢄⣾⠟⠛⠛⠹⣿⣿⡿⠟⠛⠋⠻⣤⠀⠀⠀⠀⠀⠀⠀⢀
⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠐⠛⢿⡆⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⢠⡖⠀⢀⣄⠀⠀⠀⠀⢰⣼⣷⣿⣿⠿⢾⣇⠀⠀⠠⣄⢢⣶⠙⣛⣲⣴⣦⠄⠀⠀⢰⡶⠿⣿⣿⣾⣧⡄⠀⠀⠀⠀⢠⡄⠀⢳⡄⢀⡄
⠀⠀⠀⠹⣿⣿⡃⠀⠀⠁⠀⠀⣠⡾⠿⢿⣿⡇⠀⠸⠏⠀⠀⠀⡠⢚⡟⢤⣿⠛⡋⠤⠀⠀⠀⠹⠇⠀⢸⣿⡿⠿⢷⣄⠀⠀⠈⠀⠀⢘⣿⣿⢏
⠀⠀⠀⠿⠛⠋⢿⣤⠀⠀⠀⠘⢿⡄⠀⠀⢙⡙⣤⣀⣀⣤⡴⠶⠴⠾⠷⠾⠿⠾⠿⡦⢶⣦⣄⣀⣀⣤⠋⡍⠀⠀⢠⡿⠃⠀⠄⠀⣰⡿⠙⠛⠿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣁⣴⠂⠀⠀⢁⣴⠶⠾⠿⣉⠉⢀⣀⣀⣀⡤⢤⣤⣤⣤⣤⡤⡤⢾⣿⣛⡛⠋⠉⠙⠓⠶⣦⡈⠀⠀⠐⣦⣀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⢿⡁⢻⣿⡾⠛⠁⣀⣠⣴⣶⠟⠛⠛⠉⠁⠀⠀⣴⣆⠀⠀⣰⣦⠀⠀⠀⠉⠛⠛⠳⢶⣦⣄⣀⡈⢙⣶⣿⡿⢁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⣦⣴⠟⠁⠈⠉⠀⠀⣀⣴⣿⣁⣀⣀⠺⠿⣍⣹⠿⠗⣀⣀⣨⣿⣶⣠⠄⠀⠉⠀⠙⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⣶⠀⣀⣼⣻⣿⡿⠿⠿⠿⠮⠙⠛⠛⠛⠛⠛⠛⢫⣭⡿⠿⢿⣿⣿⣟⣏⣀⠀⣶⡀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣸⠿⣟⠋⠉⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣿⣭⣭⣭⣤⣀⣀⡀⠀⠉⠈⠛⠿⣯⡹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣽⡶⠾⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⢶⣤⡴⣮⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡤⠀⠀⢀⡀⠀⠀⠀⠀⢼⣤⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⣀⣤⣤⣴⣶⣄⡠⢌⡁⠀⢀⣀⡠⣠⣴⣿⣿⣶⣶⣤⣀⠀⢻⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠃⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⠁⠀⠀⢹⠄⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠈⢷⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⣿⣿⣿⡛⢫⡅⠛⢿⣾⣿⣿⣿⢮⠙⠻⢬⣽⣿⣿⣿⡟⠀⣼⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠛⣋⣁⣌⡄⠀⠀⢠⣿⣿⢾⣿⣿⣶⢀⣄⣀⠉⠙⠁⠀⠘⠛⢛⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢞⠿⢟⡛⢶⣸⣧⣼⠙⣿⣸⡟⣘⣿⣿⣷⡾⣿⣶⣴⣧⣴⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠻⣾⣅⠸⣡⠇⠻⣷⡟⢹⠿⣏⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠿⠀⠇⠀⠀⡔⠻⠿⠶⠿⠙⣧⣼⣿⠏⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣹⣿⢤⣏⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣷⣌⠀⠂⢠⠀⢲⠒⢀⠰⡆⢹⣽⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣶⣷⠀⠀⠶⠀⠸⠀⠃⢙⣿⣿⣿⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⣼⡧⡆⠀⠀⠀⠀⠀⢀⣉⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⣧⠀⠀⠀⠀⢀⠦⣬⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⣸⠣⠀⠀⡴⡉⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠑⠾⣴⣤⣧⣿⣾⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢶⡿⠿⠿⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠘⣄⣀⣤⣄⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠟⠻⠿⢿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠩⠁⠓⠉⠉⠚⠉⠀⠉⣽⣿⣶⣴⣶⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠛⠛⠛⠀⢲⣶⣿⣶⣭⣶⣾⣼⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠁⠀⠀⠀⢻⡇⠈⣻⡇⠀⠀⠀⠀⣿⣿⣿⣿⠃⢰⣿⡇⢰⣿⣿⣿⣷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠏⢻⠃⠀⠀⠀⠀⣸⡷⠚⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣭⡁⢹⣿⣿⣿⣿⡇⣰⣄⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡆⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣶⣿⣿⣿⣿⠇⣿⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠹⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣟⣸⣿⣏⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⠀⠀⢀⠨⠍⠹⠄⡀⠀⠀⡀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢢⡄⢾⠀⢈⠄⠀⠨⠤⡀⢀⠤⠇⠀⠰⡁⠀⡱⢤⡔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠎⡓⢂⠹⡁⠀⠀⢀⡴⠀⠀⢦⡀⠀⠀⢉⠏⡰⢚⠱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⣇⠚⠀⠉⠚⢒⠉⠀⠠⠄⠀⠉⡓⠚⠉⠀⠑⣰⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⡈⠤⠤⠤⠤⠤⠤⠅⣀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠉⣀⣠⣤⡤⠤⠤⠤⣤⣤⣀⡈⠙⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠳⠗⢚⠉⠨⠤⢤⡁⠀⠀⢀⡤⠦⠅⠈⡟⠲⠗⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣀⡸⠁⠀⠀⠰⣿⢠⠄⠀⣄⣿⠆⠀⠀⡁⢧⣀⣍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠫⢄⡤⢚⠂⠀⠀⣆⠀⠀⠀⠀⢰⡄⠀⠀⡟⠦⣄⠼⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡣⢀⠞⠉⠀⣀⡤⠔⠊⠉⠉⢑⡋⠉⠈⠑⠢⢤⣅⡀⠈⠓⡄⢀⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢺⠀⠘⣿⡀⠀⠀⠑⠤⣤⣀⠠⢴⣊⣙⣦⠤⢀⠠⠤⠚⠀⠀⠀⡸⠉⠀⠉⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⡀⢸⠀⠀⢹⠳⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣶⠞⠁⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠟⠒⠚⢄⡀⡄⠀⠀⡆⡄⠳⡍⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⠔⠉⠀⠀⡠⠔⠊⠉⠁⠈⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⣇⡤⠄⠠⠀⣙⡇⠀⠀⢻⠗⢑⡁⠢⣝⠲⠦⣤⡀⢀⣤⠶⢒⡫⠚⡉⠀⠀⣰⠊⠀⠂⠀⠀⠀⠈⠙⢶⠃
⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠴⡁⢱⠀⠀⠁⢢⠀⠈⠆⠀⠉⠑⠲⠛⠛⠓⠋⠉⠀⠀⢀⡆⠠⠙⣆⣠⠊⠁⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢧⠈⠀⢠⠉⠹⠀⢀⡇⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣼⣷⠀⠀⠀⣀⠠⠤⠤⣤⠃
⠀⠀⠀⠀⠐⡆⠀⡐⠀⠀⠐⠀⠚⠓⠊⢉⣷⠛⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣠⠞⠉⠻⣇⡤⠈⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠸⣤⣳⠂⠀⠀⠀⠀⣤⡤⠎⠉⡳⢄⡀⠉⠒⠤⢀⠀⠀⠀⠠⡀⠐⠈⠀⡠⠖⢹⠀⠀⠀⢹⡇⠀⣴⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⡇⠀⠀⡇⠀⠉⠒⣤⣄⣀⣐⢄⡀⢀⠄⠤⠂⠛⠀⠀⠀⠀⠀⠀⢀⣀⣼⣍⠄⢀⣆⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠤⣄⠀⣀⣴⠾⠣⠀⠠⡇⠀⠀⠀⠁⠀⠁⠉⠐⠂⠁⠀⠀⠀⠀⠀⠀⢠⠦⠤⠤⠘⡟⠀⣯⣀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠀⠀⢣⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⢰⠁⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣈⡦⠀⠀⠸⢦⡀⠀⠀⠢⡀⠀⠀⠀⠀⢀⡠⠊⠀⠀⣠⠇⠀⠀⢴⡃⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢠⡞⠉⠀⠈⠐⠠⠀⠙⣦⣄⡀⠈⠑⠢⠤⠒⠉⠀⣀⣴⠏⠀⠔⠀⠉⠉⠉⢲⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠘⠤⠠⠤⠀⠼⢄⣀⣀⣀⣀⣀⡀⠤⠧⠤⠽⠶⠤⣀⢀⡤⠶⠯⠥⠼⠥⠀⣀⣀⣀⣀⣀⡸⠤⠀⠤⠤⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⣴⠃⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠹⡄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡐⠀⠀⠀⢠⡟⢀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢧⡀⠀⠀⢠⡄⠀⢻⡇
⠀⠀⠀⠣⠀⠀⠀⢻⣤⠜⠁⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠁⢴⣾⡇⠀⠀⢀⡂
⠀⠀⠀⠹⣖⡀⠀⠘⣿⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡿⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠈⠛⠲⠶⣿⣿⣦⣤⡆⠘⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡸⠀⢦⣠⣼⣿⡷⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡻⢿⣿⣶⣾⣶⢿⡀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣧⣦⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠛⢿⣿⢳⣆⠀⠀⠀⠀⣴⠰⣿⠟⠋⠁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢄⣻⣿⡿⠒⢰⠄⠲⣿⣿⣆⡴⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢧⠀⠏⠀⠀⡔⠁⠀⠀⠀⠈⠉⠠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢢⣠⣄⠀⠘⠒⠒⠂⠀⣠⣀⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠀⠀⠀⠀⠀⠘⠛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠠⢋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠑⠄⣰⣶⣶⡀⣶⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠘⠪⣛⢛⣶⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⠟⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠀⢀⡴⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣾⣦⡀⠙⢦⣀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣟⣻⣄⣴⣉⡳⠦⣍⠉⠁⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠟⠛⠛⠛⠻⠦⣀⢀⣼⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣷⣶⣶⣶⣶⣶⣶⣤⣌⡙⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣻⣿⠟⠑⢬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⣿⡏⠀⠀⠀⠙⠺⢽⣻⡿⢿⣿⣿⡿⢿⣟⡛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡓⠶⣄⠀⠀⠠⠶⣛⡉⠉⠛⠋⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡙⢿⡇⠀⠀⠘⠛⣿⠿⠂⠀⠀⠈⢿⣿⡟⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣴⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣤⣠⣄⣀⡀⠀⠀⠀⠀⠀⣸⠘⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡹⠿⠿⠋⠀⠀⠀⢀⣠⣾⣿⢇⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣾⣿⢷⣀⠀⢀⣀⣤⣶⡿⠁⠀⢹⣿⣿⣿⡼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⣿⣿⣿⣷⣝⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣤⣾⣿⡟⠀⠀⠀⠀⢂⣠⠖⠋⠁⠀⠀⠉⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠛⠉⠁⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⠀⠀⠀⡟⢇⠀⠀⠀⠀⠀⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠈⡗⢄⠀⠀⠀⢰⠀⠈⢆⠀⠀⠀⠀⡠⢺⠃⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡧⣄⠀⠀⢣⠀⠑⢄⠀⡇⠀⠀⠘⡆⠀⣠⠊⠀⡘⠀⠀⣠⢴⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠑⢤⡘⡄⠀⠀⠛⠀⠀⠀⠀⠘⠞⠁⠀⢠⢃⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠤⠽⠗⠒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⠺⠯⠄⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣸⡇⠤⠤⠤⠀⠒⢒⣒⣒⣒⡒⠂⠀⠤⠤⠤⢼⡀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣀⡠⠔⠊⠀⠀⠀⠈⠢⡀⠀⢲⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣣⣊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠓⠼⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡤⣄⠔⠁⠙⠋⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠓⠀⠀⢇⡤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢸⠀⣾⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⢸⡇⡁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠢⡟⢆⠀⠀⠀⠉⠁⠀⠀⠀⠤⠦⠄⠀⠀⠈⠉⠀⠀⠀⠀⡜⡷⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠸⡘⢆⠀⠀⠀⢀⠔⣲⣈⣉⣁⣉⣁⣶⠢⡄⠀⠀⢀⠜⣸⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⡀⠑⠦⠤⠊⠀⡆⠒⠊⠁⠓⠂⡰⠁⠐⠒⠊⠁⢠⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠈⠢⢄⣁⠠⠞⠁⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⣶⠺⡍⠀⠉⣩⢉⢰⡌⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⠖⠃⠀⠀⠙⠢⠖⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠆⠀⠋⠰⠀⠰⠀⠠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⣒⣒⡒⠒⠒⠒⠒⠤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠸⣿⠲⠀⠀⡀⣾⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⠀⠀⠀⠀⠀⣤⡼⡠⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠒⠀⠀⠀⠀⣼⠷⣧⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⢇⡀⢀⣼⠟⠀⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⡟⠁⢃⣴⠟⠁⠀⣼⣌⠆⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⠶⠛⠁⠀⠀⡜⠤⡞⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⢠⡅⢰⠁⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠈⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡀⠀⠀⠀⠀⠀⠃⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⡄⢨⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠛⡋⡭⣿⣿⣿⣿⣟⠘⢆⣀⣀⠴⢿⡋⡀⠀⠀⡀⠀⣼⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣀⣴⣛⡓⣂⠔⡋⠉⡥⣫⠜⠅⠈⢀⠀⠀⠱⠀⠀⠄⠐⢠⣿⡇⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⣿⠿⠿⠿⠟⠻⢶⣯⣰⡞⢤⠇⢸⣁⣽⢀⣠⣤⢠⣾⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢱⢏⢡⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣶⣾⣿⣻⣼⣿⣷⣾⣿⡿⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⢀⣿⡆⠀⠀⡀⠀⠀⠠⡲⠛⠀⡠⠛⢿⣿⠿⠟⣋⣼⠟⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⢸⣿⠁⠀⣸⠀⢀⡾⠀⠐⠀⡼⠁⡴⠊⣠⠴⠞⠛⠁⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣿⡿⡀⢀⡇⠀⡎⡇⠈⠛⣼⠁⢰⢁⣾⠁⠀⡄⠤⢄⠐⡒⢣⡍
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢰⣿⡇⠁⣸⠀⢀⠃⠁⠘⠃⣿⢰⠈⣼⣿⠀⠀⠀⠃⢗⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣻⡏⢸⣿⢃⠀⡟⠀⢸⠀⠠⠰⠀⣿⠀⠀⣻⡏⠀⢠⡋⠓⢠⠅⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⣼⣿⢸⢸⡇⠀⢸⢀⠀⠀⢰⡟⠀⠀⣿⡇⠀⢸⠀⠀⡎⠀⠘⠄⢐
⠀⠀⠀⠀⠀⠀⢀⡤⢾⡿⣾⣿⠖⣾⡇⠀⠛⠈⠀⠀⣸⡇⣷⠀⣿⡇⠀⠀⠂⠈⠀⠉⢳⠀⠈
⠀⠀⠀⠀⢀⣤⣤⣶⣿⣷⣯⣝⣿⣿⡧⢀⡇⡄⠀⠀⣿⡇⡿⠀⣿⡇⠀⢀⠐⠂⠐
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠉⠉⠻⠿⢦⣷⣷⣌⣸⣿⣇⣠⣄⣿⡇⠀⠈⠀⢀⣀⣀⣤⠤⠔⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⣶⣦⠀⣀⣴⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⣆⠀⣲⣤⣯⣉⣺⣿⣿⣿⣿⣿⣷⣀⣰⡇
⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣟⣦⣾⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⡧⡄
⠀⠀⠠⡀⠀⠐⣉⡀⠀⡃⡀⠀⠾⢷⣄⠀⠀⣆⣀⣠⣦⣶⣾⣿⡃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈
⠀⠈⡺⣟⢒⣼⢇⣉⣧⣿⣷⣯⣮⣰⣌⣳⣶⢾⠟⠻⠿⣾⣿⡿⣾⠏⢡⠶⣿⣿⣿⡷⠿⣿⣽⣿⣿⣟⢻⢿⠁
⣤⣬⣷⣾⣿⣿⣻⣿⣿⡿⣿⣿⣏⣽⣿⣿⣤⣈⡀⠀⣸⣇⣭⣷⣤⣿⡿⠛⣿⣿⠃⠀⠀⠀⢻⣷⣿⣿⡟⢿⣇
⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣘⣤⣿⣿⡿⣿⣾⣟⣻⣶⣿⠿⣿⣿⣿⣋⢰⣿⣿⠃⠀⠀⠀⠀⢠⣿⣿⣿⣏⡉⢻⣦
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⣝⣿⣿⣸⣧⣶⣿⣿⣿⣿⣼⢫⣽⣹⣆⢩⣿⣿⣇⠀⠀⡀⠀⠸⢿⣿⠟⠥⢳⣾⣿⡧⠀⠀⠠
⣾⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣽⣿⣿⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⣍⢻⣧⡑⠄⠀⠀⠀⠁
⠷⣏⠁⢌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠀⠈
⠀⡿⢄⠀⠁⢻⣿⣿⡿⢿⣿⣿⣧⡈⠉⠙⠛⠛⠛⠛⠛⠉⢻⣿⣿⣿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣀⠒⠆
⢰⡇⠀⠛⢀⣿⣿⠏⠀⠀⠈⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆⠈⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⡃⠀⠠⡀
⠀⡧⠀⠀⢿⣿⣿⠀⠀⠀⣠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣸⡧⣸⡇⠀⠀⠈⠂
⠀⡇⠀⠀⠀⠙⠿⣷⣀⠀⠻⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢻⣿⣧⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⠄⠐⠁
⢐⣧⠀⠀⠀⠀⠀⠈⠻⣷⣤⡈⠻⢿⡀⠀⠀⠀⠀⠀⠀⠀⢾⣾⣠⣿⡿⠟⠀⠀⠀⠀⠀⠈⢻⣷⡆⠀⠀⠀⠀⢠⡿
⣼⣿⣴⣄⠀⠀⠀⠀⠀⠈⣿⣷⣦⠀⠙⠳⢦⣤⠀⠀⠀⢠⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣳⠀⠀⠀⠀⣾⠄⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠐⢻⡧⠀⠀⠀⣷⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠻⣇⠀⠀⣸⢿⣦⣤⢴⣿⣧⣄⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡀⢽⣿⠞⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⡖⠉⠿⢊⠿⠛⢋⣹⣿⢷⡭⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⢑⣠⡾⢿⣖⠉⠉⠁⡴⠚⠛⢆⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣵⣶⣦⣈⣰⠀⣶⣴⣿⣿⡆⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡯⠻⠿⠿⡿⡿⠈⢻⣿⣿⣿⡟⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⣶⣶⣶⡆⢰⣿⡜⢧⣶⣶⡞⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⠛⠋⢩⣿⣿⣻⣷⣾⠙⠚⠛⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣷⣶⢾⡻⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⣿⣿⣷⣶⣥⣾⡿⣿⣏⡀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⡿⡉⢻⣿⣧⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⡎⠀⣠⣸⡇⢫⣿⣦⢼⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⣿⣿⡇⢄⣿⢻⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠿⠫⣿⣿⣿⡟⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠬⣦⠀⠀⣜⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⢒⢈⡇⠐⠣⠒⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣹⣤⠼⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠠⠔⠷⠂⠀⠘⠀⠀⠀⡄⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣢⠈⢄⣋⠴⢂⠐⣒⠀⠀⠀⠀⠀⡄⠓⢢⢅⠃⠤⣡
⠀⠀⠀⠀⠀⠀⠀⠀⢎⠹⠂⡉⢤⢞⡉⠣⢓⠔⠊⠀⠀⠀⠀⠀⠀⠈⠁⢀⠘⡮⠲⡈⠙⢤⠇
⠀⠀⠀⠀⠀⠀⠐⢰⠌⣰⠋⠀⠁⠁⠸⢠⠃⠀⠀⠀⠀⠈⠁⠀⡔⠀⠀⠀⢀⠀⡄⢱⠀⢄⠳⣈⡢
⠀⠀⠀⠀⠀⠀⠈⠉⢰⡇⠀⠀⠀⠀⡃⡆⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⠉⡀⠈⠀⠀⠂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠅⢸⡇⠀⠀⠰⡐⠛⢇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢈⣂⠈⠸⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠱⠃⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⡀⡀⠀⣀⣁⠀⠴⠀⠖⡰⠧⢰⠁⠔⢀⡤⢴⠀⣈⡀⠀⡀⡀⢠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠎⠉⠀⡁⠰⡍⢽⠀⢀⠴⡀⡨⠉⠫⡀⢀⡀⠀⠢⠩⡝⠀⠋⠐⠂⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡙⠀⠀⡡⣁⣀⡴⠦⠤⠤⠴⠤⠤⠛⠦⠤⠴⠌⠈⡤⢄⠑⢀⡃⠂⡀⠔⡠
⠀⠀⠀⠀⠀⠀⠀⠐⣛⡀⠭⠅⠀⠀⠀⡂⠀⠀⢀⠀⠋⠁⠀⠉⠀⠐⠒⠒⠀⠠⠭⠀⢐⣒⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢓⢀⠀⠀⠀⠀⢀⡋⠂⠐⢃⡈⠺⠟⣀⠔⡈⠀⠠⣶⡄⠀⠰⣆⢀⣉⠙
⠀⠀⠀⠀⠀⠀⠀⠑⣲⠿⠒⠐⠀⠀⠈⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣀⠉⠁⠀⠐⠂⠦⢵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢨⡔⢢⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣅⠢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⣙⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⢛⡉⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣴⠶⠒⢲⡦⠄⠤⢀⡠⢄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣩⡜⡃⠘⣇⠀⠀⠀⣾⠠⠈⢷⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠛⠛⠁⠀⣀⠀⠈⠑⠚⠈⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣱⠆⠀⠹⠀⠶⠀⠰⠀⠈⡰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠥⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣳⣟⣃⣀⠀⠀⢀⢡⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠨⠿⠌⠁⠀⠀⠊⢸⡤⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠆⠀⠀⠀⠀⠀⠀⣶⡾⢁⠆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠒⠓⠊⠑⠀⠀⠀⣼⠷⣿⣴⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⡞⠀⣽⠀⠀⢀⣼⡿⠀⠙⣿⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⢛⡼⠂⠟⠉⣤⡿⠋⠀⡠⡂⠺⡉⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣀⣈⣤⡾⠋⠀⠀⢠⡟⠛⡗⠀⠀⠀⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⡆⠉⢳⠀⠀⠀⠀⠈⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡝⠃⠀⠀⠀⠀⠀⠀⢠⠄⠸⡆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⣤⡇⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⡃⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⠀⠰⠀⡤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇⣿⡄⠀⠀⠀⠀⠀⢀⠃⠐⣼⠆⠀⠀⠀⠀⠀⡆⠐⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⡿⣿⣿⣿⣿⣿⣿⡇⣦⠀⠀⠀⠀⢸⣀⣼⠋⠀⠀⠀⠀⠀⠀⡇⡆⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠉⠉⢉⡉⠤⠬⢿⣿⣿⣟⢿⡀⡀⢀⠤⣤⠼⠛⠿⡀⠄⠀⠀⠡⠀⠀⢸⢷⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡄⢃⣞⣛⠛⠛⣤⠒⠩⠈⣠⢚⣡⠘⡕⠌⠀⠀⠀⠀⠱⠈⡄⠐⠀⢰⠀⣿⣿⠘⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠟⠛⣟⣿⣿⣿⠿⠶⢿⣤⣼⠃⣸⢸⢈⢼⠀⣾⠉⣻⡀⡇⢠⡆⢠⣼⣿⡏⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣌⠲⠏⠈⠁⠀⠀⠀⠀⠈⠉⠛⠻⠶⣾⣼⣀⣿⣷⣿⢰⣿⣿⣇⣼⣿⣿⠑⠀⡁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣠⣏⠂⣼⠀⠀⠀⠀⢀⠂⠀⠀⠀⡀⣰⠝⢙⢿⣿⣿⣾⣿⣿⣿⡿⢟⣿⡯⠁⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣸⡏⢰⣿⡇⠀⠀⢀⠂⠀⠀⡠⠊⠊⠙⠀⡠⠊⠉⢻⡿⠟⠛⣁⣴⣿⠋⠁⢠⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠀⣿⣿⠀⠀⢀⡟⠀⢀⣼⠃⠀⠃⠆⡾⠃⢠⠞⢁⣠⠶⠞⠛⠉⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⡇⢰⣿⣿⠀⠀⣾⠁⠀⡎⡟⠀⠚⠣⣼⠃⠀⡝⢠⣿⠁⠀⢀⠠⠤⠤⠔⠒⠒⣊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⣼⣿⡆⠇⢀⡏⠀⢸⠃⠇⠠⠼⠀⣿⢠⡄⢃⣿⣿⠀⠀⡇⢰⠀⡏⠄⠙⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⣿⣿⡇⠀⢸⠃⠀⢸⠀⠀⠀⡀⢀⣿⠈⠁⠸⣿⡇⠀⠀⣧⠤⡀⠐⠦⠀⠠⠈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣿⠀⣿⡈⢤⠀⣿⠀⠀⣿⠐⠀⠈⠁⢸⡿⠀⠀⢀⣿⡇⠀⠀⡇⠈⠀⢦⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢰⣿⡇⢸⢀⣿⠀⠀⡏⢰⠀⠁⠀⢸⡇⠀⠀⢸⣿⡇⠀⠀⡇⠀⠘⠎⢠⠀⠁⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⣿⣷⡿⢷⠆⣾⣯⠀⠀⡇⠈⠀⠦⢄⣾⡇⣶⡄⢸⣿⡇⠀⠀⠐⠀⠉⢀⠋⠳⣆⠀⡅
⠀⠀⠀⠀⠀⠀⢀⣀⣡⣴⣿⣧⣽⠛⢻⣶⣿⣿⡀⠀⡇⡀⠀⠀⢀⣿⡇⣽⠀⢸⣿⡇⠀⠀⠀⡤⠀⠠⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠰⠿⠿⣿⣿⣿⡿⠿⠛⠛⠿⣿⣷⣌⡛⡇⣇⢰⠀⣾⣿⡇⠃⠀⢸⣿⠃⠀⠀⠤⠀⠒⠂⠀⠉⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠒⠿⠿⠷⠞⠛⢾⣿⣇⣀⣀⣠⣤⣤⠴⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣀⣤⠶⠚⠋⠉⠉⠉⢹⠉⠉⠉⠉⠛⠲⢦⣄⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠻⣆
⠀⠀⠀⢠⡟⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣫⡟⠀⠘⣏⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⣦
⠀⠀⢠⡟⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⣤⣾⣩⡴⠞⠋⠀⠀⠀⠈⠛⠶⢬⣿⣦⡄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⣇
⠀⢀⡾⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢹⡆
⠀⣸⠃⠀⠀⠀⠀⣾⠀⠀⠀⣠⠞⢻⠀⢀⣤⣴⣶⠶⠦⣤⣀⠀⠀⣀⡤⠶⢶⣶⣤⣄⡀⢻⠛⢦⡀⠀⠀⢹⠀⠀⠀⠀⠀⢿⡀
⢠⡟⠀⠀⠀⠀⠀⣿⠀⣠⠞⠁⠀⣾⣴⡿⠋⠉⠉⠉⠛⢦⡉⠉⠉⢡⡶⠛⠉⠉⠉⠛⢿⣾⡆⠀⠙⢦⡀⢸⡇⠀⠀⠀⠀⠸⡇
⢸⡇⠀⠀⠀⠀⠀⢿⡾⠁⠀⠀⣰⢯⡏⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⠏⠀⠀⠀⠀⠀⠀⠀⢿⢧⡀⠀⠀⠻⣾⠃⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣼⡏⢸⡇⠀⠸⣇⡀⣠⠟⠀⢀⡇⣿⠀⠀⠻⣄⣀⣼⠃⠀⢸⢈⣷⡄⠀⠀⠹⡆⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⢸⡇⠀⢠⡞⠁⢿⡈⣧⠀⠀⠈⠉⠁⠀⠀⣼⠃⠸⣆⠀⠀⠈⠉⠀⠀⢀⡟⣸⠃⠙⣆⠀⠀⣿⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠈⢷⣠⡟⠀⠀⠈⢷⠈⠳⢦⣄⣀⣠⡴⠞⠁⠀⠀⡘⠳⣤⣀⣀⣤⠴⠋⣰⠏⠀⠀⠘⣇⣰⠇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠃⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡿⢿⡀⠀⠀⠀⠀⠀⢸
⢻⡀⠀⠀⠀⠀⢰⡇⠀⠹⣆⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠐⠶⠶⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢠⡞⠁⠈⣧⠀⠀⠀⠀⠀⣾
⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠰⠶⠶⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡴⠋⠀⠀⢀⡟⠀⠀⠀⠀⢀⡿
⠀⢿⡀⠀⠀⠀⠀⣽⣄⠀⠀⠀⠈⠳⣄⠀⠀⠈⡿⢦⣀⠀⠀⠀⠀⠀⣀⣴⢾⡇⠀⠀⣠⡴⠋⠀⠀⠀⢠⣾⡁⠀⠀⠀⠀⣸⠇
⠀⠸⣧⠀⠀⠀⣼⣿⠈⠳⣄⠀⠀⠀⠈⢳⡄⢠⡇⠀⠈⠙⠛⠛⠛⠛⠉⠀⢸⡇⢀⡼⠋⠀⠀⠀⢀⡴⠋⢸⢷⡀⠀⠀⢠⡟
⠀⠀⢹⣆⠀⢸⡇⢿⠀⠀⠈⠳⣄⠀⠀⠀⠹⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡞⠀⠀⠀⢀⡴⠋⠀⠀⣼⠀⣧⠀⢀⡿⠁
⠀⠀⠀⢻⣆⠸⡇⠘⢧⡀⠀⠀⠈⠳⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⡴⠋⠀⠀⠀⣰⠏⢀⡟⢀⡾⠃
⠀⠀⠀⠀⠹⣆⠹⣦⡈⠻⢦⣄⠀⠀⠘⢧⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡴⠋⠀⠀⢀⣤⠞⢁⣠⠞⢡⡾⠁
⠀⠀⠀⠀⠀⠙⣧⡀⠉⠛⢻⠟⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠻⣷⠚⠋⠁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⡟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⢀⣴⠏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣏⡉⠀⠀⠈⠙⠶⣤⣀⣀⣀⣀⣀⣤⠶⠃⠀⠀⠀⠉⣉⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⡀⠈⠉⠉⠉⠉⠀⣀⣀⣤⡴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⣶⡀⠀⠀⣴⣾⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠘⢿⣦⣾⡿⠁⠀⠀⠻⣷⣴⣿⠏⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣨⣿⠃⠀⢠⣿⢿⣷⡄⠀⢀⣼⡿⣿⡇⠀⠀⣿⣯⣨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⣄⣼⡿⠀⠹⣿⣤⣾⡟⠁⢹⣷⣀⣴⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠛⠹⠇⠀⠀⠙⣿⠏⠀⠀⠘⣿⠟⢩⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣤⣾⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣴⠟⠋⠉⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠁⠀⠀⠀⠀⠉⠻⣿⠛⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢠⣖⢀⠀⢹⡇⠀⠀⠀⢸⣿⣿⣤⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣾⣷⠀⠀⠀⠈⠹⠿⠟⠁⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢿⣶⣤⣄⣀⣠⣤⣾⠟⠁⠀⠈⢿⣿⣦⣄⡀⠀⠀⠀⣀⣠⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠙⢿⣿⣆
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠀⣀⠀⠀⠀⠀⢶⣿⣿⡆⠀⢰⣿⣿⡏⡆⠉⠉⠉⠉⠉⣀⣀⠀⣠⣤⡙⣿⣧⡀
⠀⠀⢰⣿⡟⣀⠙⠻⠿⠟⠁⠀⠈⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⡴⠲⠷⠜⠁⢈⣙⡿⠟⠁⠘⣿⣷
⠀⠀⢹⣿⡜⠿⠇⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡤⠴⠒⠋⠉⢁⣴⣦⡀⢀⣿⡿
⠀⠀⠀⠻⣿⣦⡀⠀⠈⠛⠋⠀⢠⣶⣿⣿⣿⣦⡀⢀⣄⠀⠀⠀⠀⠀⠀⢿⣅⣀⣀⣤⣴⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠙⠿⣷⣤⣄⡀⢠⣿⣿⣿⣿⣿⣿⣿⠙⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢿⣿⣿⣿⣿⠟⣩⣤⣴⣶⣤⣄
⠀⣠⣾⣿⣿⣿⣶⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠟⠛⣿⣿⣿⡿⢻⣍⣹⣿⣿
⢸⣿⣿⣿⡿⢠⣿⣿⣿⣿⣷⠀⣉⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⣀⠀⣈⣻⠛⢿⣷⡈⢹⣿⣿⣿
⠘⣿⣏⣩⢠⣿⣿⣿⣿⣿⣿⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⡇⠈⣿⡇⠈⣿⣿⣿
⠀⠹⣿⣆⠸⣿⣿⣿⣿⠿⠃⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⠀⢸⣿⡇⠀⢨⣿⡟
⠀⠀⠹⣿⣿⣿⣷⡀⠀⣴⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣠⣿⣿⣄⣰⣿⠟
⠀⣠⣾⡿⣿⡿⠛⢿⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⢫⣿⣿⣿⣿⣄
⣰⣿⠏⣠⣴⣶⡘⠃⣉⡟⠿⣷⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣾⣿⡟⠋⣁⡀⠀⢙⣿⣁⣹⣿⡇
⣿⠿⠿⠿⠋⢻⣿⣾⣿⣿⣦⣿⡟⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠸⣿⣧⣾⡿⣿⣦⣼⣿⠿⠿⠿⠇