⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠖⠉⠠⠀⠀⠀⠁⠀⠀⠀⠁⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡋⣴⠀⢇⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠼⠢⠿⠀⣱⠃⠀⠀⠀⠀.⠀⠀⠀⠀⠀.⠀⢀⠀..⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⠛⣸⠗⡒⢴⠃⠒⠀⢲⠀⠀⠰⡀⠀⢣⠀⠀⠀⢇⠘⡌⠀⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠐⣱⡯⢪⣚⡏⠀⠀⠀⣼⠠⠀⡆⡋⡄⠈⡆⠀⠀⠸⣆⢳⢈⡔⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀meow⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⣄⣿⢩⡉⣿⣸⢡⠀⢼⣹⢠⢸⡹⠀⠸⡀⢳⡆⠀⠀⣿⡈⡏⠐⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣸⡿⣴⢺⡿⣿⢎⠀⣾⠇⢠⡿⠁⠀⠀⢳⠀⣼⠀⠀⢸⡇⡇⢸⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣺⡇⠷⣿⡟⡇⡠⣰⡿⢀⡾⠁⠀⢀⠀⠜⣆⢻⣸⠀⢸⢻⣏⢸⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⡷⣶⣿⡁⡗⣨⣯⣫⢞⣠⡴⢾⣷⣶⠉⣇⢸⣿⠀⢬⢸⡏⢼⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⣩⣿⣭⡿⣿⣧⣿⡳⠃⠀⠉⠀⡈⠛⠛⠜⢹⣹⡏⠀⢘⡞⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⡖⢿⣿⠧⢸⠿⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⣿⡇⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢪⢺⣿⡀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢀⣿⣿⡇⢸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣉⣿⣿⣆⡀⠀⠀⠄⣀⡀.⠤⠀⠀⣠⠞⡿⠂⠀⢸⣿⢿⣷⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⢹⣿⡿⣿⣶⣤⣀⠈⠁⡠⡔⠷⡁⣼⠋⠀⠀⡇⢿⡿⣿⢿⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⠿⣿⢿⣿⣿⣿⣿⣿⣿⣵⣿⡜⢹⠏⠀⠀⠰⡀⠜⠛⢻⡿⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⣭⡵⣟⡄⣿⣧⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⠀⠀⢠⣯⡀⡢⢔⠁⠀⣀⠽⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⣉⡉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡾⠋⠀⣠⡾⠛⠛⠻⣷⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠖⠿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⢷⣶⣿⣤⣤⣀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠙⢷⣄⠀⠀⣠⣾⣇⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡋⠀⠈⠻⣦⡀⠀⠀⠀
⠀⢀⣴⠾⠋⠉⠁⠀⢀⣀⣀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠈⢿⡶⣤⡀
⢠⡾⠁⠀⠀⠀⠀⠀⠈⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡷⠦⢤⡄⠀⠀⠀⠀⠙⢷⡄⠈⣧⠈⣿
⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣋⣰⠟
⢸⣆⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⠁⠀
⠀⠙⠷⣤⣄⣀⣀⣀⣀⣀⣻⣮⡉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⣤⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠛⠒⠒⠷⠴⠶⠤⠶⠤⠤⣤⣦⣤⣤⣤⣿⣦⣀⣀⣀⠀⢀⣀⣀⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠞⠈⡑⢀⡷⢲⣿⣿⣿⣿⣿⣿⢾⡟⣿⣿⡿⠁⠀⠈⠎⠠⠛⢳⢦⣞⠑⢄⢀⡀⠌⠒⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣴⠉⠀⢀⠌⢤⣾⢏⣿⢟⠛⠿⣿⣿⣿⣿⣻⣟⣻⠃⠀⠀⡜⠔⠁⢀⠌⠀⢻⡷⣎⠑⢄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⡇⠁⠀⠈⢨⣴⢿⡼⢒⣵⠿⠓⣤⣈⣉⠉⠉⠉⣩⠏⠀⠀⢠⠋⠀⠔⠁⣠⠖⠁⠈⠪⢷⡀⠱⡀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⡗⡰⠀⠀⣔⡷⠛⢁⣴⠗⠁⡠⠊⠀⠀⠀⠀⠐⣇⡿⠀⠀⢠⠇⡠⠀⢀⠞⠁⠀⠀⠀⠀⠙⡻⡄⠐⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⠑⠁⢀⠜⠁⣠⡺⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⡪⢚⡇⢰⠀⣌⠂⠈⡂⠁⠀⠀⠀⠀⠀⠀⠀⠨⣳⡀⠰⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣾⠃⠉⠀⢅⣴⠞⠋⢀⠔⠁⠀⠀⠀⠀⠀⢀⢔⠕⠕⡿⢀⡇⢠⡁⠠⠈⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⡇⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢤⡎⡑⠒⢴⠀⣿⠁⠢⠀⡊⠋⣶⠶⣦⣄⡠⡂⠠⠊⠀⡇⡸⢱⢸⣿⡄⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⣿⡀⢰⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣾⣷⡄⠀⠀⠀⠾⣵⠒⠒⢺⠀⢿⠀⠀⠐⠀⠀⠥⢀⡉⠭⠎⣀⡣⠤⠀⢇⣇⠤⢽⣿⣿⡄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡿⠇⢀⣀⠀⠠⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢧⣧⣕⢝⢷⣄⠀⠀⣙⠒⢲⣦⡆⢸⡇⢸⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠁⠉⠘⢷⠒⢚⠻⣿⣷⠠⠤⠤⠤⠬⡀⠤⠒⠒⠈⠁⣽⡏⢡⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢪⣭⡭⣧⣪⠏⠀⢠⢡⣷⣼⣟⣧⢸⡇⢸⠀⠀⠀⠀⠀⢰⢠⢠⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢉⡄⠠⠠⠤⠤⣤⠤⠔⠒⠒⠒⠰⣶⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣼⢧⣔⣾⣣⣯⣫⡉⢺⡿⣿⣷⣿⣻⣧⡘⠀⠀⠀⠀⠀⠨⠸⢸⠀⠀⠀⠀⠀.⠀⠰⠀⠀⢀⡮⢡⠤⠤⠠⠤⠤⡀⠤⠔⠒⠒⠒⣿⠐⡖⠊⠉⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢆⣾⣠⣼⢝⠀⠙⢷⣼⣆⠊⢛⢟⣿⣼⠈⡀⠉⠉⠉⠉⠹⠐⢺⠤⢀⡀⠀⠀⠀⡆⠀⠀⣸⠁⠀⠂⠀⡠⠀⠤⢤⣤⣤⣤⣤⣤⣿⣤⣥⣴⠒⠉⠉⢉⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠷⢿⣹⣵⣟⠆⠀⠀⢿⣿⣆⠠⢊⣿⣿⠀⡇⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠐⢤⡀⠀⠀⢠⡯⠠⠤⢄⠎⠀⠀⠀⠘⣿⣿⣿⢷⣿⢿⣟⣿⣿⡠⣔⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠭⣾⣷⠋⠀⠀⠀⠊⢿⣿⡄⡄⢸⠛⡷⣇⡀⠠⠠⠤⠤⣇⡅⠀⠀⠀⠀⠀⢸⠁⠀⡼⠀⠀⠀⡌⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠟⠋⠁⠀⢸⡇⠀⣀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⢫⠀⠇⠀⠀⠀⠀⠇⠈⠻⣥⠓⠆⣤⡇⡆⠀⠀⠀⠀⠀⡇⡇⠉⠓⠢⣄⠀⢆⡀⠠⡷⠀⠀⠂⠀⠀⠀⠀⠀⠀⣄⠉⠉⠁⢳⡄⢀⣠⠤⠖⢟⡇⠀⠀⠈⢇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠖⠉⠠⠀⠀⠀⠁⠀⠀⠀⠁⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡋⣴⠀⢇⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠼⠢⠿⠀⣱⠃⠀⠀⠀⠀.⠀⠀⠀⠀⠀.⠀⢀⠀..⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⠛⣸⠗⡒⢴⠃⠒⠀⢲⠀⠀⠰⡀⠀⢣⠀⠀⠀⢇⠘⡌⠀⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠐⣱⡯⢪⣚⡏⠀⠀⠀⣼⠠⠀⡆⡋⡄⠈⡆⠀⠀⠸⣆⢳⢈⡔⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀meow⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⣄⣿⢩⡉⣿⣸⢡⠀⢼⣹⢠⢸⡹⠀⠸⡀⢳⡆⠀⠀⣿⡈⡏⠐⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣸⡿⣴⢺⡿⣿⢎⠀⣾⠇⢠⡿⠁⠀⠀⢳⠀⣼⠀⠀⢸⡇⡇⢸⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣺⡇⠷⣿⡟⡇⡠⣰⡿⢀⡾⠁⠀⢀⠀⠜⣆⢻⣸⠀⢸⢻⣏⢸⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⡷⣶⣿⡁⡗⣨⣯⣫⢞⣠⡴⢾⣷⣶⠉⣇⢸⣿⠀⢬⢸⡏⢼⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⣩⣿⣭⡿⣿⣧⣿⡳⠃⠀⠉⠀⡈⠛⠛⠜⢹⣹⡏⠀⢘⡞⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⡖⢿⣿⠧⢸⠿⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⣿⡇⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢪⢺⣿⡀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢀⣿⣿⡇⢸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣉⣿⣿⣆⡀⠀⠀⠄⣀⡀.⠤⠀⠀⣠⠞⡿⠂⠀⢸⣿⢿⣷⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⢹⣿⡿⣿⣶⣤⣀⠈⠁⡠⡔⠷⡁⣼⠋⠀⠀⡇⢿⡿⣿⢿⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⠿⣿⢿⣿⣿⣿⣿⣿⣿⣵⣿⡜⢹⠏⠀⠀⠰⡀⠜⠛⢻⡿⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⣭⡵⣟⡄⣿⣧⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⠀⠀⢠⣯⡀⡢⢔⠁⠀⣀⠽⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⣉⡉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡾⠋⠀⣠⡾⠛⠛⠻⣷⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠖⠿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⢷⣶⣿⣤⣤⣀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠙⢷⣄⠀⠀⣠⣾⣇⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡋⠀⠈⠻⣦⡀⠀⠀⠀
⠀⢀⣴⠾⠋⠉⠁⠀⢀⣀⣀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠈⢿⡶⣤⡀
⢠⡾⠁⠀⠀⠀⠀⠀⠈⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡷⠦⢤⡄⠀⠀⠀⠀⠙⢷⡄⠈⣧⠈⣿
⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣋⣰⠟
⢸⣆⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⠁⠀
⠀⠙⠷⣤⣄⣀⣀⣀⣀⣀⣻⣮⡉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⣤⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠛⠒⠒⠷⠴⠶⠤⠶⠤⠤⣤⣦⣤⣤⣤⣿⣦⣀⣀⣀⠀⢀⣀⣀⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠞⠈⡑⢀⡷⢲⣿⣿⣿⣿⣿⣿⢾⡟⣿⣿⡿⠁⠀⠈⠎⠠⠛⢳⢦⣞⠑⢄⢀⡀⠌⠒⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣴⠉⠀⢀⠌⢤⣾⢏⣿⢟⠛⠿⣿⣿⣿⣿⣻⣟⣻⠃⠀⠀⡜⠔⠁⢀⠌⠀⢻⡷⣎⠑⢄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⡇⠁⠀⠈⢨⣴⢿⡼⢒⣵⠿⠓⣤⣈⣉⠉⠉⠉⣩⠏⠀⠀⢠⠋⠀⠔⠁⣠⠖⠁⠈⠪⢷⡀⠱⡀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⡗⡰⠀⠀⣔⡷⠛⢁⣴⠗⠁⡠⠊⠀⠀⠀⠀⠐⣇⡿⠀⠀⢠⠇⡠⠀⢀⠞⠁⠀⠀⠀⠀⠙⡻⡄⠐⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⠑⠁⢀⠜⠁⣠⡺⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⡪⢚⡇⢰⠀⣌⠂⠈⡂⠁⠀⠀⠀⠀⠀⠀⠀⠨⣳⡀⠰⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣾⠃⠉⠀⢅⣴⠞⠋⢀⠔⠁⠀⠀⠀⠀⠀⢀⢔⠕⠕⡿⢀⡇⢠⡁⠠⠈⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⡇⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢤⡎⡑⠒⢴⠀⣿⠁⠢⠀⡊⠋⣶⠶⣦⣄⡠⡂⠠⠊⠀⡇⡸⢱⢸⣿⡄⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⣿⡀⢰⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣾⣷⡄⠀⠀⠀⠾⣵⠒⠒⢺⠀⢿⠀⠀⠐⠀⠀⠥⢀⡉⠭⠎⣀⡣⠤⠀⢇⣇⠤⢽⣿⣿⡄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡿⠇⢀⣀⠀⠠⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢧⣧⣕⢝⢷⣄⠀⠀⣙⠒⢲⣦⡆⢸⡇⢸⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠁⠉⠘⢷⠒⢚⠻⣿⣷⠠⠤⠤⠤⠬⡀⠤⠒⠒⠈⠁⣽⡏⢡⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢪⣭⡭⣧⣪⠏⠀⢠⢡⣷⣼⣟⣧⢸⡇⢸⠀⠀⠀⠀⠀⢰⢠⢠⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢉⡄⠠⠠⠤⠤⣤⠤⠔⠒⠒⠒⠰⣶⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣼⢧⣔⣾⣣⣯⣫⡉⢺⡿⣿⣷⣿⣻⣧⡘⠀⠀⠀⠀⠀⠨⠸⢸⠀⠀⠀⠀⠀.⠀⠰⠀⠀⢀⡮⢡⠤⠤⠠⠤⠤⡀⠤⠔⠒⠒⠒⣿⠐⡖⠊⠉⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢆⣾⣠⣼⢝⠀⠙⢷⣼⣆⠊⢛⢟⣿⣼⠈⡀⠉⠉⠉⠉⠹⠐⢺⠤⢀⡀⠀⠀⠀⡆⠀⠀⣸⠁⠀⠂⠀⡠⠀⠤⢤⣤⣤⣤⣤⣤⣿⣤⣥⣴⠒⠉⠉⢉⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠷⢿⣹⣵⣟⠆⠀⠀⢿⣿⣆⠠⢊⣿⣿⠀⡇⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠐⢤⡀⠀⠀⢠⡯⠠⠤⢄⠎⠀⠀⠀⠘⣿⣿⣿⢷⣿⢿⣟⣿⣿⡠⣔⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠭⣾⣷⠋⠀⠀⠀⠊⢿⣿⡄⡄⢸⠛⡷⣇⡀⠠⠠⠤⠤⣇⡅⠀⠀⠀⠀⠀⢸⠁⠀⡼⠀⠀⠀⡌⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠟⠋⠁⠀⢸⡇⠀⣀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⢫⠀⠇⠀⠀⠀⠀⠇⠈⠻⣥⠓⠆⣤⡇⡆⠀⠀⠀⠀⠀⡇⡇⠉⠓⠢⣄⠀⢆⡀⠠⡷⠀⠀⠂⠀⠀⠀⠀⠀⠀⣄⠉⠉⠁⢳⡄⢀⣠⠤⠖⢟⡇⠀⠀⠈⢇⠀⠀⠀⠀⠀
