⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⡹⣦⠀⠘⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢷⣤⡀⠀⠀⠀⢷⡛⢳⡀⢳⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⢻⣿⣦⣄⠀⠀⠳⣆⠍⣶⣌⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠳⣤⡈⠻⢦⣽⣦⡀⠙⣦⠉⢿⣎⢳⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢻⣷⣤⡙⢿⡿⣦⣌⣿⣌⢿⣧⡙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠒⠤⣀⡘⠻⣿⡶⣷⣎⣻⣯⣿⣿⣿⣷⣌⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠈⠋⠀⠋⢻⣿⣿⣿⣾⡍⣿⣿⣿⣿⣿⡍⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯
⠀⠀⠀⠀⠠⠤⡀⠀⠉⠻⢿⣿⣿⣾⣿⣿⣯⣻⣷⣄⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣋⣴⠋
⠀⠀⠀⠀⠀⠀⠉⠛⢶⣦⣦⣬⣿⣾⣿⣿⣿⣿⡿⣮⣿⣾⣿⣷⢶⡀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⢟⣩⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⡽⠿⣿⣷⣤⣉⠉⠙⢻⣻⡛⢿⣮⡻⣷⣄⠀⢀⠞⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣻⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠩⢻⣿⣿⣿⣷⣾⣿⣿⣛⣿⣽⡟⢦⣠⡾⣿⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⢟⣭⣴⣾⣟⣻⣿⣿⣛⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣟⣡⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⣀⠀⠀⠀⣀⣴⡿⢟⣹⣿⣏⡽⠿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣴⡿⡿⣿⣿⢿⣿⣿⣿⣿⣼⣿⣿⠛⠿⠿⠁⠀⠀⠀⢀⣠⢶⠿⢿⣷⣦⡿⠻⣟⠂⠀⠐⢹⣥⣾⡿⢭⣭⡯⢩⡍⣉⣉⣙⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡟⢽⣾⡿⠁⣾⣿⣿⣿⣿⣷⣯⣿⣷⣀⠀⢲⠀⠀⠀⢸⡷⣞⠛⢶⣿⣷⠀⠀⠈⢦⠀⣵⣾⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠈⢉⡟⠀⢈⣽⣿⣿⣿⣿⣾⣻⢟⠛⡿⣇⣸⡇⠀⠀⣸⣍⣬⣀⣸⣿⣿⠀⠀⠀⠈⣿⣅⣿⡆⣿⣿⣟⣿⣿⣿⣿⣿⠟⢶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡀⢀⣼⡿⣋⣾⣿⣿⣻⣿⣚⢷⣿⣾⣿⠁⡀⠀⢻⡿⣻⣿⢿⣿⠏⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣿⣷⣄⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠎⠀⢡⡟⣰⠟⠻⣿⣿⣿⣿⣻⣦⣻⣿⣷⠀⠙⠷⣾⢁⠀⣳⣿⣇⣠⣴⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⡆⢹⣷⣷⢻⣝⠍⠐⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿⠟⠁⠀⠰⠛⣽⣿⣿⣿⡏⣿⠛⣿⡄⡀⠰⣼⣯⡷⠉⠉⠻⣟⣿⣹⣿⣿⣿⣏⣿⣾⣿⡻⣿⣷⠀⠀⢻⠓⠊⠻⠳⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⣼⣿⣷⣿⣶⣿⣿⠉⢦⣏⠈⠁⠀⠀⢀⣿⠿⣿⣿⣟⣿⣿⣿⡟⡟⣿⡘⣙⢦⢠⡈⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⢷⠬⣿⣖⣿⡁⠀⠀⣤⠴⣿⣿⣿⣟⣿⣿⢿⡟⡼⢻⠀⠸⢣⡸⡄⠀⣱⣡⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣟⢉⠉⠁⠙⢂⣤⣤⣖⣾⣿⣿⣿⡿⡿⠀⠛⠀⠀⠀⠀⠀⢳⡇⢠⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⢠⣟⠀⠉⠓⣶⣿⣿⣧⡿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠻⣿⠟⣛⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡞⢿⣿⠏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣷⣼⣴⣿⡿⢷⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢻⣿⣿⡅⣿⣇⡤⠛⢷⣶⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠀⠀⠀⠀⡞⠿⣷⡦⣽⠿⣿⣿⣿⣤⡿⠿⠛⠛⠒⠒⠻⢷⣍⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⣤⡀⡉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠐⠠⠤⠤⠛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀