⠀⠀⣀⣠⣤⡤⢤⠤⡴⢦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠛⠛⠛⠛⠛⠛⠛⠳⢦⣔⠨⣉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡿⢢⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠾⠟⠿⠶⠶⢤⣤⣼⠏⡛⢦⣤⣤⣾⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣼⣿⣤⣼⣿⣤⡿⠤⠧⢇⢄⠠⣼⢠⠘⡄⡘⢄⠠⢄⠛⠻⠿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠛⠉⠁⣠⡴⠛⣍⣒⠌⡡⢃⠌⢢⠡⡟⡄⢃⠆⡱⢈⠒⡌⠬⣕⣨⠐⡌⢛⢆⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡴⡟⣡⣵⠟⣝⢠⠊⡔⢡⠊⡅⠚⢇⠸⡀⢎⠰⡁⢎⠰⡁⢆⣉⠛⠦⣇⠢⡙⢦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⢿⣫⣾⠟⠁⠀⠀⠈⡍⣉⡉⣉⡴⢹⡎⣧⠘⡄⢃⣾⠶⢓⣾⠒⠦⠙⠒⠌⠑⢧⡊⢷⡀⠀⠀⠀⠀
⠀⠀⢠⡞⣑⣿⣾⠋⣀⡤⠔⣢⡿⡁⢆⠢⣽⠡⣹⢎⣿⡔⡈⠆⣿⢠⠃⢼⡿⣳⢖⠤⢄⡀⠀⠙⢮⣷⠀⠀⠀⠀
⠀⢠⣟⣰⣾⡟⡓⢌⡞⡄⢳⣿⠣⡑⢌⢲⡏⢴⠏⡭⣷⣷⠈⢆⠽⢆⡍⠢⣧⡙⢯⣷⡀⢎⡑⠢⣄⠹⣂⠀⠀⠀
⢠⣿⠞⢁⣿⠑⡌⣼⢃⢬⣿⡟⢦⡵⣬⣏⡗⡞⠀⣗⡟⡾⡇⢌⢺⠈⢦⢃⢳⠙⣆⣽⣷⡠⢌⠡⣷⣕⠘⣦⠀⠀
⠿⠁⠀⢸⡇⢎⡐⣿⢀⣾⣽⡇⢢⣰⣏⣀⣿⣏⣒⣿⡇⢹⣿⡀⢾⢠⣜⣻⣻⡉⠈⢲⡌⢷⡌⢒⣸⣾⠷⣽⣇⠀
⠀⠀⠀⢾⡃⢆⣴⣧⠊⣿⢾⡐⣡⢿⢿⣿⣿⣿⣿⠻⡇⠀⢳⣇⢺⡜⢻⣿⣿⣿⣿⣷⠟⣄⢻⡄⠸⣿⡆⠈⢿⡆
⠀⠀⠀⢸⣅⡾⠋⣿⢐⣯⣿⢰⣿⠸⣘⠉⢻⡟⢿⠀⠁⠀⠀⠻⣆⡇⠘⠛⢻⡿⣿⢸⢸⣿⣆⠍⡒⣿⠀⠀⠀⠁
⠀⠀⠀⢸⡾⠁⠀⢻⡆⣿⡸⣿⣻⡀⠉⣯⡒⢔⣴⠀⠀⠀⠀⠀⠹⣇⢸⣧⠨⣅⡛⠂⡿⣞⡟⣦⠡⣿⠀⠀⠀⠀
⠀⠀⠀⠘⠃⠀⠀⠀⢷⣿⡿⣿⢽⡇⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠛⠴⢰⡿⣽⡴⢋⣷⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡗⡿⣞⣧⡀⠀⠀⠀⠀⠀⠀⣄⣠⣀⡰⠀⠀⠀⠀⠀⠀⣸⡿⣹⡗⢂⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠇⡹⣿⣟⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣠⣤⣶⢿⠛⡄⣿⡏⡐⣿⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡇⠰⣸⢿⣼⡿⣜⣯⣿⣿⢏⣍⠹⣿⣿⣯⣳⣿⣯⣿⠘⣰⡿⣷⢘⡟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠱⣏⡑⢼⣟⣾⣿⣿⡹⣖⢿⠩⠉⠓⣿⢣⢯⣽⣿⡷⣿⢈⢼⡿⣽⢨⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣿⡐⢺⣿⠊⠙⢿⣿⣯⣞⡇⠀⣸⣧⣿⣿⣿⠟⠙⣇⢊⣾⡟⣽⣇⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⣿⣧⢺⠁⠀⢀⠈⡟⢿⣿⣷⢠⣿⣿⠿⢻⢁⠀⢀⡇⣺⣯⣝⡶⣯⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢣⡷⠋⢫⡆⠀⡌⠀⠱⢤⣙⣻⡿⢛⣁⡢⠏⠘⡀⢸⠜⠁⠹⡾⣵⣹⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠊⠀⠀⠀⠁⢰⠀⢀⠖⡉⡽⢟⣄⠉⡙⠢⣄⠀⡇⠀⠀⠀⠀⠘⢷⡇⣧⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⡆⠀⡞⠠⠐⣀⡞⠘⢆⠐⠠⠙⡄⢠⠀⠀⠀⠀⠀⠈⢿⢹⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⣿⣟⢦⣀⠀⠀⠀⠀⣣⣤⣧⣴⣷⣮⣠⣠⣀⣙⣲⣥⣧⣸⠀⠀⠀⠀⠀⢀⡼⣿⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡿⠻⣯⣿⢯⣷⣤⡴⣾⣿⢻⣟⣻⢻⣟⠿⣿⡻⢿⣟⡿⣟⣿⣷⣄⠀⠀⣴⣿⣿⣿⢿⡆⠀⠀⠀
⠀⠀⠀⠀⠰⣏⣠⣤⣈⢽⣷⠿⣯⣝⣾⢫⡿⢮⡵⣿⢺⡝⣭⣛⡭⣷⣹⢳⣻⣷⢯⢷⣾⡽⣻⣟⠁⣀⣙⣦⠀⠀
⠀⠀⠀⠀⠀⠀⠙⣿⠀⣾⣭⡟⣶⢻⣧⢻⡏⣷⠚⣯⢳⡝⣾⢱⡏⣽⡖⣯⣷⢺⣯⣷⢹⡟⣿⢹⡍⢹⣯⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⡌⣿⢧⣻⣵⣿⣾⣾⣽⣲⣻⣝⣮⢳⣽⢧⣛⣽⡷⣙⣾⣭⣿⣞⣳⡽⣣⢿⡠⢙⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠣⡐⣿⢯⡳⣟⡿⣶⣿⣭⣽⣾⣶⣾⣿⣿⣿⣶⣶⣿⣾⣾⣿⡷⣿⣳⣟⣭⣟⡧⢉⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢡⠘⣿⣏⡷⣏⡾⢧⢯⣽⠀⠈⠁⢻⢯⣟⡽⣇⠙⠁⠒⢯⣗⢯⣳⣿⡾⣼⣺⣿⢸⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⡆⢱⣿⣞⣿⣽⣚⡯⣟⣾⠀⠀⠀⢸⣟⣿⡞⣽⡀⠀⠀⢸⣻⢞⡿⣿⣏⣷⡟⣻⣸⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣧⠹⡟⢿⣻⡷⣯⣳⢿⣞⡇⠀⠀⢸⢯⣳⣽⣿⡇⠀⠀⢸⣯⡿⠁⣿⣾⠋⠀⣿⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⠀⠻⣧⠙⠷⣿⣿⣧⣀⣀⣸⣿⣽⣾⣿⣿⡀⠀⠈⣿⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠙⣿⠀⠄⣿⠂⠙⠯⠛⠹⣏⠉⡉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⢻⡔⢈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣷⣿⡀⠀⠀⠀⠀⢸⣷⣾⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣟⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀