⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠉⠁⠀⠀⠀⠉⠉⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠶⠶⠶⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⢿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠙⠘⡀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⠀⣧⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢏⢀⢿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣣⢎⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠏⠹⠉⠉⣿⣏⣩⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⡄⠤⠀⠠⡀⢼⣿⣿⣿⣿⡿⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⡒⠉⠁⢀⢰⢱⡆⠀⢀⢣⣸⣿⣿⣟⣥⠾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⣢⣠⠎⢸⢸⠃⠀⢸⣸⡇⠇⠀⠙⠓⠒⠻⠋⠙⢟⣿⡶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣉⣽⢇⣤⣾⡎⠀⢀⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢝⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⠉⠑⠒⠒⠛⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠁⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣏⠁⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢄⣠⣠⣬⣽⣻⠇⠀⠀⠀⠀⠀⠀⢇⠘⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣰⢏⡀⠀⠀⠀⠀⠀⡐⣵⠋⠀⠀⠀⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣤⡏⠓⠈⠑⠢⣄⡀⢰⢃⣧⡀⠀⠀⠀⠈⠛⠿⣌⡉⠻⣦⣤⣤⡀⠀⠀⠀⠀⠀⡇⠱⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢩⠀⠀⠀⠀⠀⣠⡟⢩⠉⠹⡏⢦⡀⠀⠀⠀⠀⠀⠉⠓⢆⡙⠛⣿⣿⣿⣾⣿⣿⣧⠀⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢈⣟⠢⣄⠀⡔⢻⠿⣹⡀⣠⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⣿⣿⡟⢋⣿⡓⠺⣻⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣑⠐⠦⠿⠛⠻⠵⢦⣭⣀⡏⢠⣣⠿⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⣠⣿⣿⣿⡷⠛⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠚⠁⠀⠀⠀⠀⢸⡆⣶⣶⠈⣧⣼⣧⠶⣆⡄⣀⡀⢻⢦⡀⠀⠀⠀⠀⢀⠞⣹⠟⣡⣾⢿⣿⣿⡟⠁⠀⡏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡞⢛⣃⠉⢰⡀⠀⠉⠉⠛⠛⠛⠙⢄⠀⠀⣀⡟⣿⣿⠾⠃⠀⠀⣿⡽⠀⠀⠈⡗⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⠸⠿⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣈⡙⠉⠛⠓⠓⢦⡀⠈⢁⠀⠀⣠⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⡆⣀⡀⠀⠀⠛⢦⡀⠀⠀⠀⣼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⡀⠀⠀⠈⢦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⠤⠤⢄⡀⢱⣴⠏⠉⠑⣄⠀⣀⣸⣷⣖⣛⣉⠽⠢⡄⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠙⠷⢄⡀⠀⠹⡄⢰⡇⠀⠀⠠⢄⠄⠀⠀⠀⣆⢸⡆⠀⠀⡇⠀⡷⣄⣄⣘⡛⣃⣀⣀⠧⢘⠁⠀⠀⠀⠈⢷⠁⡟⢧⠆⠀⠀⠀⡼⠁⠀
⠀⠀⠀⠀⠀⠀⡴⠋⠟⠿⣷⣀⣀⡙⠀⠀⢱⡓⡗⠲⠶⠖⢶⡶⠒⠒⠒⡎⢸⡇⡤⡀⠁⠀⣷⠀⠉⠉⡟⡏⠀⠀⢸⢸⠀⠀⠀⣀⡔⣸⠈⢛⣳⠖⠶⠶⠖⣇⠀⠀
⠀⠀⠀⠀⠀⠘⣇⠉⠑⠂⠀⠀⠀⠀⠀⠀⡞⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⡇⠈⡟⠷⠓⣀⠀⡏⠀⠀⠀⡇⡇⠀⠀⡀⢸⠀⢀⡴⣣⢏⠀⠀⠀⠀⠀⠀⢀⠀⠙⠷⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⡇⠀⠀⡹⢠⡇⠀⠀⠀⡇⡇⠀⠀⡇⢸⠀⣾⣱⠃⠈⠁⠒⠒⠒⠒⠈⠁⢀⠔⣀⡇
⢀⡶⣶⣶⣶⣾⡷⠒⠂⠀⠀⠀⠀⠀⠀⠀⡿⢸⡃⠀⠀⠀⠇⡇⠀⠀⣰⠀⢸⡇⠀⠀⠅⠰⣅⣀⠀⠀⡣⠃⢀⣀⡇⢸⠈⠁⢹⣕⠲⡦⠀⢀⠀⠀⠀⣴⠁⠺⢻⡇