⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠀⠀⠀⠀⠀⣤⣤⡤⠀⠀⠀⠀⠀⣴⣿⣦⠀⠰⣶⣶⠦⠀⢴⣶⣶⣶⣴⣶⣶⣶⡆⠀⠀⢴⣦⡄⠀⠀⠀⠀⣤⣶⢶⣦⣴⡦⠀⢴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⡏⠀⢹⣇⠀⢻⡇⠀⠀⠀⣿⠀⠈⣿⠉⠉⠹⡇⠀⠀⠀⢹⣷⠀⠀⠀⠀⣿⡇⠀⢻⣿⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⡏⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠘⣿⣤⣶⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢸⡟⠀⠀⠀⢹⡇⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⢸⣿⡆⠀⠀⢸⣿⡇⠀⣼⣿⣿⣿⢻⡏⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣴⣄⣀⣼⡇⠀⠀⠀⣸⣧⣀⣿⡄⠀⢸⡁⠀⠀⣿⠀⠀⠰⠀⠀⠀⠀⢸⢻⡇⠀⠀⡟⣿⡇⠀⡟⣹⣿⣟⣸⠇⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⢹⣿⣿⣫⣴⣿⣿⠟⠉⣿⢯⣩⣿⣿⣦⣀⣿⠀⠀⣠⠀⠀⠀⠀⢸⢸⣷⠀⠀⡇⣿⡇⢰⣧⣿⢿⣿⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⡟⠉⡼⢿⣦⣶⣦⣿⣿⣿⣿⣿⣿⢾⣿⣆⢀⡤⢹⣿⣿⣿⢻⣿⣷⣾⣿⠀⠀⠀⠀⢸⠘⣿⠀⢸⠃⣿⡇⣼⣿⣿⣿⣿⣿⡆⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⠇⢈⠀⠉⠙⣿⣿⠋⢡⣌⢻⣿⣿⡟⣧⣈⣭⢦⡖⣍⣻⣧⣸⣿⠀⠈⢻⠀⠀⠀⠀⢸⠀⣿⣆⣸⣠⡟⢃⡉⣉⣩⠙⣛⡿⠁⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⡔⠋⠁⠀⢀⣼⣿⣄⢀⣯⣾⣿⣿⣿⣿⡿⠦⢄⣻⣿⣍⢻⣿⣿⠀⠀⠸⠀⠀⠀⢴⣞⣽⣿⣿⣿⣿⣟⡛⠉⠉⠀⢸⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻⣓⠒⠚⠋⢉⡴⣿⡿⠻⣿⣿⣿⣛⢿⣿⣦⡀⠀⣹⡿⠟⢿⣿⢀⣀⡤⠴⠖⠒⠚⠻⣿⣿⣿⡿⠛⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⢸⡟⢻⠖⠉⠁⣠⣿⣷⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠷⠄⠀⠉⠉⣽⡳⣮⢇⣠⠔⢺⠁⠈⣿⠇⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣿⣷⣭⣦⣀⠀⠀⠀⣷⠘⡆⠀⣦⡘⠿⣹⢿⣿⠿⣟⣷⣟⢛⣉⣿⣿⣿⣷⣦⣴⡀⣀⣼⠿⣿⠋⠀⠀⢸⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡿⣋⣤⡤⢤⣭⡙⠲⠤⣟⡇⢱⠀⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣏⣀⣼⡏⠀⠀⠀⣼⡄⠀⢻⠀⢀⣿⣇⠀⠀⠀⣸⣿⡀⠀⠀
⠀⠀⠀⠀⢀⣿⣁⡟⢻⡷⣿⠗⠉⠛⣲⣿⠿⣬⡇⢸⣿⡇⠈⢿⣌⢻⣿⣿⣿⣿⣙⣻⡟⣿⠙⣿⣿⡿⣿⡟⠛⠃⠀⠀⠘⠛⠛⠀⠈⠀⠘⠛⠛⠀⠀⠘⠛⠛⠋⠀⠀
⠀⠀⠀⢨⣏⣽⣿⣧⡴⠺⠯⠤⠶⠉⣁⣸⣠⠿⣷⡀⢿⢿⣄⢸⣿⣷⣿⣿⢯⣤⢞⡯⠗⠋⠉⠛⢧⡙⠮⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣾⡿⢛⣛⣺⣿⣾⣓⣶⣴⣾⠟⣵⣿⣷⢻⣿⡜⣿⣿⣹⣿⣿⣿⣇⠀⠉⠉⢀⣠⣶⣿⣭⣙⠻⡄⠁⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⣿⡞⠉⠙⠋⣩⠷⢭⣽⠛⠋⣩⢿⡟⣹⠈⣿⡇⡟⠙⠟⢻⣿⣿⣿⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣧⡀⠀⣩⣤⣶⣿⡿⠿⠟⠡⠞⢠⣿⣧⣻⢛⡧⡀⠀⢸⣿⡏⣽⣿⣾⣾⣿⣿⣿⣿⣿⣿⠇⠀⠙⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣹⣿⠋⠉⠀⠀⠀⠀⡠⠀⣠⣿⣿⣿⡿⢿⣿⣶⣿⣾⣿⢋⣽⡧⢈⣿⣿⣿⣿⣿⣿⠛⠙⡄⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⡟⠟⠁⠀⣀⣤⣶⢊⡴⢋⣿⣹⣿⣿⠒⢻⣿⡿⢿⣿⣿⣼⡏⡇⣾⡇⣿⡇⢹⣿⠁⠳⠄⠸⡄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣾⢿⡏⠀⢀⣴⣿⠟⣹⣿⠏⣴⡟⣏⠀⠻⣿⣷⣅⠀⣻⣶⣿⣄⡿⣼⣧⣽⢠⣿⠛⠛⠿⠀⠀⠀⡰⠛⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠘⠀⣰⣿⡟⠁⣰⣿⡏⢾⡟⠀⡿⡄⠀⠉⠻⢿⣿⣿⣿⣿⣿⣡⣾⣭⣿⠟⠟⣦⣄⠆⠀⡼⠰⠃⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢇⠀⢀⣸⣿⣿⣷⣾⣿⣿⠁⡿⠳⣸⣷⣿⣦⣀⠀⠀⠈⠛⠛⠭⠽⠿⣿⣟⠋⠀⠈⢷⣤⣀⡜⠀⠁⢰⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢦⣼⣿⠿⠛⠉⠉⠀⠘⠀⢁⢀⣿⠛⠮⣍⠛⢿⢶⣶⣦⣶⡶⢾⡟⣿⣉⢳⡀⠀⣿⣿⣿⢀⠖⢲⣏⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡇⣼⣾⢹⡆⠀⠈⢳⣏⣾⣿⠏⢻⠤⠟⣿⡘⣿⣷⣷⣤⣿⣿⣿⣏⠀⢸⣿⢷⣦⣹⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣄⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⢻⣿⣿⡏⠁⠀⠀⣠⣾⣿⠋⠀⠚⣆⠀⡟⡇⡿⠀⢹⣿⣿⣿⣿⣿⣦⠤⢹⣷⣿⡏⢹⡿⠛⣿⠛⢻⡇⢸⣿⠉⠉⢻⡇⠈⣿⠋⠻⣷⠀⠀
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢿⣿⣷⣶⣾⣹⣿⠟⠁⠀⠈⢀⡞⠀⢰⣡⡇⠀⣸⢿⠻⢿⣿⣿⣿⡀⣼⡿⢻⠇⢸⠁⠀⣿⠀⠀⡇⢸⣿⠀⠀⠀⠇⠀⣿⠀⠀⢿⡇⠀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠳⣼⣥⡿⠋⠑⠲⣤⡇⣼⢠⠴⢿⣿⣇⡼⠋⠀⠀⣼⣿⡿⠉⠀⢻⠧⢸⠀⢸⠀⠀⣿⠀⠀⠷⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⠿⣿⣿⠟⠓⣄⠀⠠⢹⢱⣿⠀⠀⢸⢿⣿⠁⠀⠀⡀⢿⣿⠁⣠⠞⠋⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⢠⠀⠀⣿⠀⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⣿⣿⣿⡟⠀⢀⠎⠀⢰⣦⡿⣿⠀⠀⣌⢸⡿⡆⠀⠀⣇⣈⣹⡿⣿⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⣸⠀⠀⣿⠀⠀⣾⡇⠀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠿⠛⢹⡟⠀⠀⡞⠀⠀⣸⢹⠀⣿⡄⢠⣿⣿⣹⡁⠀⢠⠇⠈⡽⠁⣿⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣶⣶⣿⠀⠀⣿⣀⣴⡿⠀⠀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡸⠁⢀⡜⠀⠀⢰⠃⣸⠀⢻⡇⣼⣿⡙⣯⠳⣄⡾⠀⡼⠁⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠉⠉⢿⠀⠀⣿⢿⣿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⢃⡴⠋⠀⠀⢠⡏⢰⢻⠀⢸⣧⢹⣿⣿⣿⣤⠿⡀⢰⠁⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⢸⠀⠀⣿⠀⣿⡄⠀⠀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡎⡏⠀⠀⠀⣠⣿⣴⠇⢸⠀⠘⣿⣼⣽⣿⣿⠋⢀⡇⢸⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⢸⣇⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡜⠈⠀⣀⡴⠞⠋⢀⣿⠀⢸⠀⠀⣿⣿⣿⣿⣏⢀⣾⣷⣾⡆⠀⠀⣿⡀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⢠⠀⣿⠀⠘⣿⡀⠀
⠀⠀⠀⠀⠀⠀⠸⣷⠀⣸⣷⣿⠟⢻⡇⠀⠀⢸⡏⠀⢸⠀⠀⣿⣿⣿⣇⡘⢿⣿⠟⢻⠇⠀⠀⢿⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⡿⠀⣿⠀⠀⢿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⢿⣧⡉⣠⡏⠀⠘⣷⡀⢀⣿⠃⠀⢸⡇⠀⢸⣷⣿⣄⠀⠻⣇⠀⢸⡀⠀⠀⢸⣷⡀⢠⡏⠀⠀⠀⢀⣿⡄⠀⠀⢸⣿⣀⣀⣰⡇⢀⣿⡆⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠀⠀⠀⠘⢿⠿⠃⠀⠐⠿⠿⠂⠘⠉⢿⣿⣧⡀⢹⡍⠻⡇⠀⠀⠀⠻⡿⠿⠁⠀⠀⠀⠻⠿⠿⠀⠀⠟⠛⠛⠛⠛⠃⠺⠛⠻⠀⠀⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⡌⢿⣦⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣳⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀