⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠄⠀⠀⠀⠀⣀⣀⣀⡀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⣃⣠⣶⡶⠋⢸⣿⣿⠇⢸⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡟⢡⡞⠀⠿⡋⢸⠣⢁⠁⣶⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣿⣿⠿⡟⢠⡱⣼⡀⢔⡃⢸⡞⠼⡂⢿⠀⢿⣜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠞⣱⣿⣿⣿⡿⢡⣥⢉⣴⣶⣿⣿⣷⣅⡀⠩⡟⠀⠸⣿⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⢻⣿⡿⠿⢣⢟⣵⣿⣿⣿⡿⠿⠻⣿⣿⣄⠀⠀⣴⠹⣷⡙⠢⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡾⡡⣿⣿⢿⡏⠀⣾⣿⣍⡁⠴⣲⠷⢶⡼⡿⢿⡢⢐⣿⣷⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⣧⢿⣿⠃⠠⠀⢸⣿⣿⡃⢈⢈⢨⠁⢀⡰⣿⠇⡈⠾⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡎⣿⡞⢸⠈⢖⣠⣾⣿⣿⣷⣬⣬⣥⣶⣿⣿⡥⣨⡿⠟⢂⣭⣷⣝⠻⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣥⠥⠭⣉⣜⠀⡉⠉⣩⣿⣿⠿⣿⣿⣿⣿⠿⠩⠜⠤⠰⣮⡻⣿⣿⡿⣷⡄⠈⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢌⠱⢦⣓⠜⢿⡼⣬⡨⣽⣻⡵⠿⠟⠛⠉⢁⣶⣿⣿⣿⣿⣇⡱⡹⣿⣷⡘⣿⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⣺⡬⡒⢵⣫⡶⠑⣬⣦⣓⢅⣻⣏⡀⢀⡀⠒⠛⠛⠀⠀⡛⣻⣿⣿⣤⡹⣼⣔⢹⣿⡜⡌⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣾⣹⢫⢮⣶⡜⣡⠴⡂⢐⣒⣣⡌⠩⢍⣉⣀⣀⣈⡁⣀⡀⡰⣿⣿⣿⣿⣿⣿⡄⣿⡼⣿⣷⠸⡁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢣⡟⡾⣱⠸⠐⠏⠁⣢⠈⠡⣶⡯⡲⡍⢷⡿⠟⣵⣿⣇⢷⣮⡻⢿⣿⣿⣿⣷⢻⡆⢻⣿⡏⢡⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⡿⣼⠃⣼⠏⠁⢀⠀⠘⠡⡾⢣⡮⡛⢶⣾⣷⣾⣿⣵⣶⣷⢌⢭⣭⣤⣍⠙⠻⣿⣼⡇⣧⠻⣿⠘⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⢀⣿⣇⡿⣼⡵⠁⢀⡔⣰⡾⣣⡄⠛⢼⢏⡶⣨⡛⢿⣿⣿⣿⣿⢧⢺⣿⣿⣿⣇⠀⢐⣿⡇⣿⣇⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⢏⡎⣿⢹⣇⠟⠄⢀⣏⣾⢏⣾⡟⣵⡎⢂⣈⠑⠟⣼⡇⣌⡝⡻⢿⡇⣧⢻⣿⣿⡿⢠⣾⣿⢷⣿⠇⡿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣾⢹⡿⣾⣏⣤⠋⢸⣸⡏⣾⡟⣼⢯⣾⡿⣱⣿⣗⣬⣜⠻⣼⢏⣾⢆⡩⠑⠙⠟⠁⢸⣿⣿⠸⢏⡾⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠏⣿⡇⣿⠟⢃⠔⠈⢿⢱⣿⢃⣋⢹⢻⢳⣿⡟⣼⣿⢣⣷⢆⡈⡋⢾⢧⡟⡴⢦⢆⡼⢿⣿⠐⠋⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣙⠟⣫⣆⠁⠀⡄⡼⣾⠙⣾⡾⡜⣏⢰⣝⠹⠿⢧⣿⡟⣾⡇⠇⡄⢎⡁⠲⠎⣾⣹⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠋⠉⠁⠀⡇⢣⢻⡇⣿⡇⣷⠏⣿⡟⣿⡟⣾⡿⢜⣵⠣⢂⣰⡟⡵⣵⢰⣿⠇⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡌⢈⢻⣽⢳⡝⠖⣿⢧⣿⢳⣿⢳⡟⠁⡀⢃⣡⢸⡇⣫⠞⣡⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⠁⣏⣘⡿⣼⣿⣸⡽⣼⣏⣿⡏⠏⠀⠈⠑⣼⡏⠋⡄⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⢠⡟⣄⢏⣿⢧⡿⣇⣿⢧⣿⢜⣫⠘⢠⢦⣵⠹⣋⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢫⣾⢣⡟⣼⢏⣾⡟⣾⡷⣼⡿⣨⡅⢛⡵⠨⠘⠘⡡⠘⠁⠂⠀⠀⠀⠀⠀⠀⡴⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣼⡐⡟⣤⢿⣸⣿⣱⣜⢣⣿⣇⡵⢆⡭⢊⡀⢡⢪⢂⡵⣸⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣿⠃⢀⡞⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢨⠈⠸⢈⢻⡡⣛⢧⢿⠏⣾⠟⢨⣺⢠⣷⡆⣦⠊⣆⣿⢳⡿⠀⠀⠀⠀⠀⣰⠸⡟⣿⣿⡇⣤⠘⢱⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⡱⣠⡁⠻⠆⣛⠏⣾⣿⣿⡞⣿⡿⣿⠇⡫⡘⣸⢏⣿⠇⠀⠀⠀⠀⠀⣿⣴⠓⠝⠻⣿⣿⣶⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⡀⠘⠹⢧⢦⣮⢡⡄⣉⣉⢙⡉⣡⣌⠅⢰⢀⠏⡾⠁⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⣶⣿⣿⣿⡘⢇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⢰⣍⠻⣄⠻⣄⢤⢀⠈⠉⠘⠓⠃⠛⠘⠃⠋⠁⠉⠈⠀⠈⣼⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⣦⡠⠀⠀
⠀⠀⠀⠀⠀⢸⣟⠹⢿⣦⠄⣁⠈⠁⠈⠀⣿⡿⠿⠒⠤⠔⠒⢲⣄⡙⠂⡆⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⣿⣿⣿⣄⡀