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