⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⡀⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠁⣤⣄⡀⢿⣧⣤⣁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣬⡛⠛⡏⠘⠛⡋⣸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⣿⣿⡘⣧⣼⠦⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣽⣻⣿⣿⣿⡿⢯⣷⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠒⢖⠒⣤⣭⢷⣻⣿⣿⣿⣿⣏⠁⠀⣿⣿⠏⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢽⣾⣿⣛⢟⡍⠩⠄⠳⣻⣿⣿⣿⣿⣿⣶⣾⠏⠀⠐⠂⢀⠛⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢊⣤⣿⢈⣜⡓⣻⣶⣔⣂⡀⢶⢹⣿⣿⣻⣿⠟⠁⠀⡰⠂⢐⠋⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣯⣿⣯⣿⣵⠽⠿⣾⣛⣯⣴⡏⠟⠳⣽⣏⡀⠀⢰⠃⠒⢸⠀⠀⠀⠀⠀⡅⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣀⣂⡀⠀⠈⠁⢉⠙⠺⢶⣤⣌⣀⠀⠀⡀⠀⣷⣸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣍⣛⣛⣷⣤⣀⣙⣞⣫⡓⣯⣟⣿⣶⣿⣶⣿⣟⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⣛⣉⣽⢿⣿⣿⣿⣿⣿⣾⣏⣻⡏⠀⠈⠙⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠉⠙⢿⣿⣿⣿⣿⠛⠛⣻⣿⡟⠛⠋⠉⠐⠻⠓⠙⠻⣿⣿⣿⣿⢿⡆⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⡄⠀⡼⡳⢻⢸⣷⠀⢿⡙⢄⠉⠓⠒⠒⠂⠀⠀⠀⠒⠿⢿⣿⣿⣷⣿⣼⣀⣤⠀⠈⢦⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⠀⠀⠨⠿⣟⡟⠁⠘⢾⣿⠳⣮⡙⠶⢶⣤⣤⣄⣀⠀⢀⣠⣤⡤⢿⣿⣟⣿⣿⣿⢿⣷⡶⢶⣶⣷⠀⠀⠀
⣴⣾⡿⣲⡄⠀⠀⢸⣿⣛⣳⣦⣄⣐⣿⠁⠀⠀⣸⣿⣿⣷⣿⣯⣧⣀⡈⠉⠀⢀⣀⢀⣠⣴⣾⣿⣷⣿⣿⣿⡿⢭⡐⢺⣿⡾⡆⠀⠀
⣏⣿⣿⣦⡌⠑⢄⡸⣿⡟⣡⠌⣹⣭⡧⡤⡤⣤⣿⡻⢿⣦⣿⣿⣿⣿⣿⣷⠂⢉⣿⠿⣻⣿⣿⣿⣿⣟⣿⣟⣿⡿⣦⣦⢼⣇⣇⠀⠀
⠈⠓⢿⣿⣿⣾⣿⣽⠿⠯⠭⣁⠛⡞⠀⣷⣿⣧⣿⣯⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣛⠽⣿⣯⣿⡇⠈⣟⣿⣏⣉⠩⣦⢹⠀⠀
⠀⠀⠀⠈⠫⣻⣿⡿⠷⠛⠛⠿⣽⠁⠀⣷⢿⣿⣷⣿⣿⣿⣷⣤⣉⣛⡻⠛⠙⠟⣛⣋⣩⣴⣾⣿⣿⣿⠧⡀⠈⣿⣶⡾⠛⢻⣤⡇⠀
⠀⠀⠀⠀⠀⠘⣿⣷⣶⠯⠟⣿⠃⠀⠀⠈⢾⣿⣿⣿⣿⣿⣷⣶⣽⣿⢟⣶⣛⣻⣯⡽⢿⣛⡿⠏⣽⣿⣆⠜⡆⠘⡿⣿⠖⠚⠻⢧⠀
⠀⠀⠀⠀⠀⠀⢸⣷⣦⡦⠴⠟⢢⠀⠀⠀⢸⣿⣿⡛⣿⡟⢿⣿⡿⣿⣿⣯⢽⡥⠶⠚⢩⣬⢹⣶⣿⣿⣿⣶⣿⣠⣯⡉⢰⣮⣹⣾⠀
⠀⠀⠀⠀⠀⠀⢮⣿⣴⠖⠒⣺⡈⢣⠀⠀⠸⣿⣿⣷⡸⣿⣄⢹⣿⣿⠛⠉⠳⡀⠀⣠⣿⠃⣿⣷⡃⠸⡟⠻⡿⣿⣟⣠⣿⡏⢿⡏⣇
⠀⠀⠀⠀⠀⠀⠘⣽⣿⣀⣿⡿⣿⡿⠀⠀⢸⣿⣿⣿⣧⣹⣿⣿⣿⣿⠀⠀⠀⠹⣴⡿⠃⣼⣿⣿⣇⠀⣻⣿⢷⠿⣤⣥⢿⡧⡼⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠫⡿⣧⡙⣯⢦⠀⣿⣿⣿⣿⣿⢴⣿⣻⣿⣿⣶⣀⢀⣴⣿⣏⢐⣿⣿⣿⣿⡆⣻⣽⠋⠀⢀⡟⢛⣿⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⢿⣌⠳⡷⣗⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣽⠀⠀⠘⠪⠿⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡙⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿