⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣶⣶⣶⣯⣥⣷⣶⣶⠫⣾⣿⣷⢧⣿⠃⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠋⠉⠉⠭⠭⠭⠥⠭⠍⠉⠁⠈⠛⠁⠀⠀
⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀ ⠀ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⣈⡓⠀⠀⠀⠀⣀⡤⠖⠋⠉⠉⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀ ⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠊⠉⣛⠛⠛⣿⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀
⠀ ⠀ ⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣷⣄⠀⠀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⢤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⡠⠃⠀⠀⠀⠀⠀
⠀ ⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠋⠀⠀⠀⠀⡀⠐⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣤⠪⠀⠀⡤⠀⠀⠀⠀
⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠂⠀⠀ ⠀⠀⠀⣀⣿⣿⣿⢿⣿⣿⣯⠛⠛⠻⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢋⠡⣤⠀⠊⠀⠀⠀⠀⠀
⠀⠀ ⠀⣿⣿⣿⠘⣟⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠈⣿⣿⣿⣇⣀⣀⠴⣿⣿⣿⡇⠀⠀⠀⠀⢀⡔⠋⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀ ⠠⢿⢿⡟⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢆⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⣴⠋⠀⠀⢀⣔⡾⠿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀ ⠀⠀⠘⠁⠀⠀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣧⡀⢸⡿⠿⢿⣿⡿⣿⣿⣦⣬⢿⣹⣿⣧⣴⠾⠟⢁⡿⠃⠀⠀⠀⣸⠃⠀⠀⠀⣀⣵⡖⠀⠀⠀⠀⣀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠊⢉⠭⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⢀⣀⣰⣱⣈⠙⠛⡷⣿⡁⠀⠀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡿⢦⠔⡰⠐⢁⣠⠣⢠⠄⠀
⠀⠀⠐⠒⢶⣶⣤⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⢾⣿⡄⠀⠐⠻⠂⠀⠱⡘⣝⠂⢄⡀⠀⠀⢀⠄⠀⠀⠀⡀⡠⠐⢊⡨⢅⢠⠇⢞⠡⠖⠉⢸⠀⡏⠀⠀
⠀⠀⠀⠙⢷⣿⣿⣿⣶⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⢀⡀⢀⠀⠀⠀⠀⠈⡆⠀⠑⡄⠀⠸⣀⠀⠀⠀⠀⠀⣀⠄⠒⣳⠈⢈⣃⣶⡦⡖⠸⠀⠃⠀⠀
⣀⣀⣀⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠵⠋⠡⢁⢿⣿⣿⡏⠀⢀⣀⡀⠁⠒⠂⠠⣅⣀⣠⣧⠀⠀⠈⠁⠀⠀⠀⠹⠄⠠⢴⣇⣠⡇⠇⠉⠀⠀⠀⠀⠀⠀⠀
⠿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡇⢊⠘⣿⣿⣿⠀⠀⠀⠙⠢⣀⣠⠀⠈⠛⢻⣿⣷⡀⠀⠀⠀⠀⢀⢾⣷⢾⢼⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⡀⢣⠘⣿⣿⡄⠀⠀⠀⠀⢀⣽⠇⢠⣶⢸⣿⣿⣷⣄⠀⠀⠀⠘⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⠃⠻⢷⡔⠁⠀⢿⡎⠻⣷⣦⣤⣤⣴⣾⠟⠀⠘⣿⣸⢿⣿⡏⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⢿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠛⠂⠀⠃⠙⠂⠀⠀⠻⣆⠀⠀⠉⠀⠙⢿⣿⣿⣿⣥⣄⣀⠀⠀⠹⠈⢻⣿⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⠘⣿⠀⢿⣇⠙⠿⣿⣟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠻⡉⠛⠛⠛⢻⡆⠀⢀⠀⠀⠙⢧⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠐⠄⡀⠀⠀⠃⠀⠸⡀⠀⠀⠀⠑⢄⡀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠉⠐⠂⠤⢠⣅⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠈⠁⠒⠀⠀⠊⠐⠪⡳⣄⠀⠈⠢⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠉⢻⡟⠛⠻⠿⠿⠷⣶⠦⢤⣄⣀⠀⣸⠀⠀⢳⠿⣷⣄⠀⠀⣠⣼⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⢿⣿⣿⣶⣿⠀⣈⣙⠓⠶⣟⣽⣿⣷⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠐⠭⠿⢏⠹⣿⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣿⣿⣏⠻⣷⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⡇⠀⠀⠀⠀⠀⣼⠀⢰⣆⣠⣿⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢸⣿⣷⡀⠀⠀⠀⠀⠸⡄⣾⣿⣿⡟⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⢱⣿⣿⣿⡇⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢿⣿⣿⣦⣀⠀⠀⠀⢻⣿⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⠿⣿⣿⣷⣦⣄⡀⢻⣿⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⡄⠀⣾⣿⡿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⠀⠘⣿⡁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠘⣇⠀⠹⣷⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⠙⠛⠿⣿⣿⣿⣿⣷⣦⣼⡆⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⡄⠈⣿⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠙⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀