⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠛⠛⣷⡀⠀⣀⣼⣿⡟⠚⣿⣿⠻⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠙⣿⣧⣠⢼⣿⡟⠋⠁⢸⣵⣴⣿⣿⢟⠛⠻⣿⣶⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢉⣿⣴⠿⣿⣯⠁⢸⣥⣴⣾⡿⠿⣻⠁⣹⣿⣶⠾⢻⣏⠻⣷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠙⡷⣀⣸⣿⣿⠿⠿⣯⠁⣀⣤⣿⣿⠋⢻⣿⣤⡀⢻⣆⣨⣿⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣀⣤⣶⣿⣿⡟⠹⣷⣧⣴⢿⣿⠋⠁⠀⠹⣧⠀⡿⣹⠶⢿⢿⣏⠈⣿⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢹⣇⠀⣀⣤⣻⣿⠱⢯⣣⣬⡉⣿⣿⠿⣷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠛⠛⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣷⣿⣿⠊⠁⢀⣻⣷⣤⠙⢺⣷⡄⠿⣤⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣾⡿⠟⢿⣏⢳⡄⣻⡵⠚⢿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⡀⢀⣴⢿⡯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣤⡶⢿⣟⠉⣿⣷⡀⠘⣿⠃
⠀⠀⠀⠀⢀⣶⣟⠉⠁⠀⠀⢨⣟⣟⣟⡵⠋⠀⠀⠀⠀⢀⡀⠀⡠⠀⠀⣶⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⠈⢟⡄⣈⣿⣷⣾⡏⠀
⠀⠀⠀⣰⣿⠗⠁⠀⠀⠀⠀⣼⣿⡿⠋⠀⢀⡠⠊⢀⠔⢉⡠⠊⠀⠀⠀⣿⡀⠙⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣞⣿⡗⣩⡿⠋⠀⠀
⠀⠀⣰⢿⠃⠀⠀⠀⠀⢀⣼⡿⣿⢻⠀⣠⠊⡠⠆⢀⠔⠉⠀⠀⠀⠀⠀⠘⣗⣄⠙⢦⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣼⡿⠋⠀⠀⠀⠀
⠀⢠⡟⡎⠀⠀⠀⠀⢠⣿⣿⡇⣿⣼⠑⠧⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣦⣌⠳⣄⡀⠀⠀⠀⠀⠈⢻⣤⡀⠀⠀⠀⠀⠀⠠⡀⡎⣿⣇⡿⣇⠀⠀⠀⠀⠀
⠀⣹⠃⠀⠀⠀⠀⣠⣿⣿⣿⣇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠽⢟⠦⠽⠦⢄⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⢿⠃⢸⣿⣷⡿⠀⠀⠀⠀⠀
⠀⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⡄⢸⣿⣿⣧⠀⠀⠀⠀⠀
⠀⣿⠀⠀⠀⣼⣿⢫⣿⣿⡽⠋⠀⠀⠀⠀⠀⠀⢠⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⠀⠀⠀⠀⠀
⠀⣻⡌⡀⠀⣿⠏⣾⢹⣿⠇⠀⠀⠀⠀⢀⡆⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣆⠀⠀⢀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀
⠀⢹⡇⡇⢠⣿⣶⡏⣿⣿⠀⠀⠀⠀⠀⣘⡇⢀⣿⣌⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡄⡟⣿⣆⠀⠘⡆⠀⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀
⠀⠸⣷⢸⡘⢻⡿⢿⣿⣿⠀⠀⠀⠀⠀⣿⡇⢸⡟⠈⠺⣷⡀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⢳⡟⠙⢿⣧⠀⢷⡄⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀
⠀⠀⣿⢠⠁⠀⢧⠀⣗⢺⠀⠀⠀⠀⠀⢹⢧⢸⠇⠀⠀⠹⣷⢄⠈⢿⣄⠀⠀⠀⠀⠘⣦⡀⠀⠘⣾⠃⢀⣀⡬⠓⠾⠿⢾⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⡄⠀⠀⠀⠀
⠀⣠⣿⠇⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⢸⡌⢿⡗⢳⣶⣦⣬⣻⢷⣌⣻⣷⠦⣤⣀⣀⣈⣻⡶⢤⣹⣿⣿⣿⣇⠀⠀⠀⡘⠀⠀⠀⠀⠀⣸⣦⣿⣿⣿⣷⠀⠀⠀⠀
⢀⣿⡝⠀⠀⠀⡇⠀⣿⢿⡄⠀⢀⠀⠀⢿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⢻⡿⠀⠀⠀⡇⠀⠀⠀⠀⢀⡟⣿⣿⣿⣿⣿⡇⠀⠀⠀
⢸⢿⠁⠀⠀⢀⠃⣴⣿⣾⣷⠀⢸⠀⠀⠸⡇⠀⠀⢸⣧⡀⠈⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣾⡷⠆⠀⣸⠇⠀⠀⠀⠀⣾⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀
⣿⡘⠀⠀⠀⡼⣴⠟⠛⠻⢾⣧⠘⣇⠀⠀⢳⠀⠘⢁⡝⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢉⣀⣼⣻⠀⠀⠀⠀⣸⡟⢋⣿⣿⣿⡿⠃⠀⠀⠀⠀
⣿⣇⠀⠀⠀⣷⠟⠀⠀⠀⠀⠛⣷⣿⣷⣄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡇⠀⠀⠀⠀⣿⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀
⠸⣿⡄⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⡠⡉⢿⣷⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢻⡇⠀⠀⠀⠀⣿⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⢹⣿⡄⢸⠻⣦⣀⠀⠀⠀⠀⠙⢿⣎⢻⡧⠀⠉⠙⠻⠶⢦⣤⣤⣄⣀⢀⣀⣀⡤⣤⣤⣶⣶⡚⣿⣟⡉⠁⠀⠀⠀⢿⡀⠀⠀⠀⣿⡀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⢿⣌⠇⡏⣿⡀⠀⠀⠀⠀⢀⣿⣾⠏⠀⠀⠀⠀⠀⣠⣴⠿⣿⣷⣄⠉⢫⡿⢭⡟⠁⣠⣾⣿⣿⣿⢦⣄⣀⠀⠘⣧⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⡆⠇⣸⡇⠀⠀⠀⠀⠾⠟⠁⠀⠀⠀⣠⡶⢻⡙⣵⣀⣹⣿⣿⣟⣀⠷⣾⣆⣤⣾⣿⠿⢻⣿⣾⠋⢹⣷⡄⠘⣧⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣸⣧⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣧⠀⠱⣾⣿⣩⡟⣿⣿⡿⠀⢹⣨⣯⠁⣻⣷⣶⢿⠏⠀⡼⣽⣷⠀⣿⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⢷⣄⢈⣿⣿⠾⢻⣿⣿⣿⣾⣿⣿⣿⣯⠿⣿⢧⣤⣾⣿⡿⢻⣤⣿⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠾⢻⣿⡟⢿⣻⡿⢿⣖⣻⡟⣝⣀⣠⡤⣿⣤⡄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⢾⡿⡿⣦⣿⣿⣏⡉⣯⣧⣾⣿⣿⢿⢏⣻⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⣯⠉⠛⠀⠿⣾⣩⣿⣯⡿⠛⢿⣏⣮⣵⡾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣻⣦⣠⣶⠾⢿⣿⢭⣿⣿⣶⣾⣽⣯⣵⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠯⢭⣿⢿⣿⣏⡀⣀⣤⣿⣿⡿⢿⣿⡁⣹⣿⣿⣿⣿⠿⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⡌⢣⠙⠻⣿⣿⣯⣥⣿⣦⣼⣿⣿⣿⡿⣿⡏⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣼⣴⠞⠛⠙⠛⠛⠛⠻⠛⠛⠻⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀