⠀⠀⠀⠀⠀⠀⢐⠀⠀⠔⢂⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠈⠋⠀⠉⠡⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣴⣤⡼⣀⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⣈⣧⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣸⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢹⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠀⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣷⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⠥⡄⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣇⣀⠀⣨⣠⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⠤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣿⣝⣻⢿⣿⣶⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⢻⢸⡇⣿⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢮⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣮⠎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣹⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣠⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠁⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣠⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣟⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠁⡤⢵⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣅⣨⣄⠀⣀⣂⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢼⣷⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⢿⣿⣶⣿⡿⢟⣻⣷⣻⣿⡿⣿⠿⣿⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠷⣾⣻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢰⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⡬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢐⠀⠀⠔⢂⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠈⠋⠀⠉⠡⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣴⣤⡼⣀⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⣈⣧⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣸⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢹⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠀⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣷⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡮⢤⠈⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣸⣄⣀⠀⣁⣼⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣦⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣮⢿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠿⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢧⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢮⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣮⠎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡂⠺⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠁⠀⠙⠁⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⢦⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣁⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠉⢠⠬⡦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢘⣀⣄⠀⣀⣰⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⢿⣿⣶⣿⡿⣟⣻⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠸⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢰⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠒⠀⡤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠏⠙⠂⠀⠐⡄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣄⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣂⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣸⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢹⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠀⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣷⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡾⢤⠈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣈⣀⠀⣠⣇⣠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣾⡧⠄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠿⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠗⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣧⣽⢧⣴⣾⣿⣿⣷⣝⣏⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡂⠺⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠁⠀⠙⠁⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⢦⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣁⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢠⠬⡦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢘⣀⣅⠀⣀⣰⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⢿⣿⣶⣿⡿⣟⣫⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣵⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⢦⣄⡀⠀⠀⢰⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⡬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠒⠀⡤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠏⠙⠂⠀⠐⡄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⣆⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣂⣸⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣸⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢹⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠀⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣷⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡾⠤⠈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣈⣀⠀⣠⣇⣠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⡧⠄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣾⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣶⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠈⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣧⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⠿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣹⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡂⠺⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠁⠀⠙⠁⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⢦⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣁⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢠⠬⡦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢐⣀⣤⠀⣀⣸⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⣿⢫⣶⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣾⣿⡿⢟⣫⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢻⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⡬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠸⡦⠤⠤⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣣⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⠏⠉⠟⠒⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣶⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⣿⣷⠿⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢧⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⠿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣏⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣠⣀⠇⣀⣀⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠹⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣾⣿⡿⢟⣻⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡯⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⢦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⢫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⡏⠉⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣮⢿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠿⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣹⣄⣀⣀⣸⣏⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣀⣀⠇⣀⣀⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠹⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣶⣿⡿⢟⣫⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣵⠛⠉⠉⠉⣿⢸⢻⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠂⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⣵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⡏⠉⠏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠿⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣿⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡟⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣠⣀⠇⣀⣀⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠹⠉⠙⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣴⣿⡿⣟⣻⣷⣻⣿⡟⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⣇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⢹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⡬⣤⣤⣼⣦⣤⣤⠾⠯⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⡏⠉⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢻⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢿⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢿⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣮⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣎⣛⣛⣮⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣀⣀⠇⣀⣀⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠙⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣶⣿⡿⢿⣫⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠸⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⡿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⠏⠉⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢿⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢿⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣿⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣹⣄⣀⣀⣸⣏⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣠⣀⠇⣀⣀⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠙⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣴⣿⡿⢟⣻⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣄⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⡏⠉⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣶⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⣿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢧⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣿⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢻⣯⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣠⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣀⣀⠇⣀⣀⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠙⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣶⣿⡿⢟⣻⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢰⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣰⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⣝⣿⡏⠉⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢿⡿⣟⣾⣝⣻⢿⣿⣶⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⣿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢧⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢻⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠊⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣎⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣀⣀⠇⣀⣀⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠉⠙⣽⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣶⣿⡿⣿⣫⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⢫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⡬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⣇⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣹⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠳⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠁⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣧⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⡏⠉⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣿⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⡿⣷⠾⠉⠀⠀⢸⡏⡇⣿⠉⠉⠉⠛⢾⡻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢧⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣯⣽⢧⣴⣾⣿⣿⣷⣝⣯⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢯⢿⣯⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣯⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣻⣄⣀⣀⣸⣏⣛⣛⣪⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠒⠀⠐⠟⢱⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣐⢀⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣏⣐⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢼⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣿⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢡⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣻⣃⣿⣀⣀⣰⣇⣶⣶⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠙⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣇⣀⣀⠄⣀⣀⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠹⠉⠙⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣶⣿⡿⣟⣫⣷⣻⣿⡿⣿⠿⣽⣻⢿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⡇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢸⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⣫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣠⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢫⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⡬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⡤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡎⠘⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⡀⣅⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣒⣺⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠻⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣼⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠀⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣷⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⠖⠒⢢⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠇⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠀⣀⣀⣠⣀⣼⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⣝⣯⡏⠉⠋⠒⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣷⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⣿⣷⠾⠉⠀⠀⢸⡟⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⣇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣞⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣧⣽⢧⣴⣾⣿⣿⣷⣝⣏⢿⡼⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢮⢿⣯⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣮⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣹⣄⣀⣀⣸⣏⣛⣛⣮⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⠖⠒⢢⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢠⠒⠀⠐⠇⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣰⢀⣼⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣏⣒⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⣷⣶⢾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢼⢸⣿⡿⣻⡽⠛⠛⢻⡟⣿⡿⣿⡻⣿⣿⣇⣿⣿⣇⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡸⣫⠞⠥⠄⠀⠀⢸⡇⣾⣧⢤⠙⠷⣝⢿⣾⢿⡏⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⠀⢸⡇⡏⣇⠀⠀⠀⠈⣇⣫⣼⡇⣿⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢈⠿⠛⠻⠿⣀⠤⠤⣸⡿⡿⣿⠶⠦⠀⠀⠈⢱⠖⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡀⠀⠀⡼⠁⠀⠀⠈⣿⢹⢹⡄⠀⣀⣀⣀⣡⢴⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⢳⡶⣧⣀⣀⣠⠴⠛⠛⢾⣷⣿⣿⣿⣿⣿⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⣼⢁⡿⠿⣇⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡇⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡷⡣⠋⠀⠑⡌⣿⣶⣤⡔⣵⣿⠿⠿⠟⠛⠙⣧⢿⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡔⠉⠉⢱⣄⣠⠞⢻⣿⣿⠹⡅⠀⠐⠦⣀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠦⠤⠤⠞⣿⠃⠀⢸⣽⣟⣀⣱⡀⠀⠀⠈⠁⠀⢀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢳⣦⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡿⣾⡝⣿⠛⠻⠿⣿⣿⣿⣿⣿⠟⠛⠋⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣇⣛⣃⣿⣀⣀⣰⣇⣶⣴⣮⣼⣧⠤⠤⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⠴⠚⠁⠀⠀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⡤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡎⠘⠂⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣅⣀⣀⠇⣀⣀⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠉⠙⣿⢫⣶⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡟⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢷⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⣶⣿⡿⢟⣫⣷⣻⣿⡟⣿⠿⣿⣻⢿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⢸⡿⠟⣷⠛⠉⠉⠉⣿⢸⢳⡇⠀⠀⠉⠿⣾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢯⡞⢟⡛⠁⠀⠀⢠⣄⣿⡾⣼⠁⠀⠀⠀⣀⠈⢳⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⣇⠺⡅⠀⠀⠀⢸⡏⡏⣇⡟⠀⠀⠀⠀⣿⠃⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢳⡟⡶⠦⣄⡀⠀⠀⢰⡇⣹⣗⢲⡆⠀⠀⠉⠀⠈⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢯⡿⣹⢫⣾⣿⣿⣷⣦⡼⣧⣽⣯⠭⣄⣀⣀⣀⣀⣴⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⣭⡿⡵⣳⣿⣿⣿⣿⣿⢻⠀⠀⢸⠁⠀⠈⢻⣿⠀⠹⣼⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣵⠋⠛⠛⠻⠿⢿⣬⣤⣤⣼⣦⣤⣤⠾⠧⣄⠀⠙⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠃⠀⠀⢀⡠⠀⠀⢠⢃⣿⣿⡟⠀⡇⠀⠀⠈⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠀⠀⠚⠉⠀⠀⢠⣧⣘⣿⣿⠇⠀⣸⣶⠦⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡿⠿⠿⣿⣶⣶⣶⣿⣿⠿⣿⣿⣤⣤⡟⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⢤⣤⣠⣾⣵⣛⣛⣱⣏⣀⣀⣠⣟⡶⢇⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠉⠉⠉⠓⠒⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⡤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡎⠘⠂⠀⠒⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⣅⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣒⣺⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣶⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣾⣿⣿⣸⣿⣿⢟⣿⢿⡟⢿⡟⠛⠛⢯⣟⢿⣿⡇⡷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⢿⣿⣷⡿⣫⡿⠋⡤⢼⣷⢸⡇⠀⠀⠠⠬⠻⣝⠇⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⢸⣧⣽⣸⠃⠀⠀⠀⢸⢿⢸⡇⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣸⠲⣆⠁⠀⠀⠴⠶⣿⢿⢿⣇⠤⠤⣄⠾⠟⠛⠿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⢻⡶⣬⣁⣀⣀⠀⢀⡟⡟⣾⠁⠀⠀⠈⢧⠀⠀⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣼⡇⣿⣿⣿⣿⣿⣿⡷⠚⠛⠦⣄⣀⣀⣼⢶⡞⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⢸⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⣸⠿⢿⡈⢧⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣿⣸⠋⠛⠛⠿⠿⣿⣦⢢⣤⣴⣾⢃⠊⠀⠙⢌⢿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣀⠴⠂⠀⢈⡏⣿⣿⡟⠳⣄⣠⡎⠉⠉⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠁⠀⠀⢀⣞⣀⣻⣯⣷⠀⠘⣿⠳⠤⠤⠴⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣴⡞⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠛⠻⣿⣿⣿⣿⣿⠿⠟⠛⣿⢫⣿⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠤⠤⣴⣧⣵⣦⣶⣸⣆⣀⣀⣿⣘⣟⣸⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠁⠀⠀⠀⠀⠀⠓⠦⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⠖⠒⢢⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠐⠇⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣀⣀⣀⠠⣀⣀⣠⣀⣼⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⡝⣿⠏⠉⠋⠒⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢹⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣮⢿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢷⣿⣿⡿⣟⣯⠿⣿⢻⡿⣟⣾⣝⣻⢿⣿⣶⣿⡟⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢸⣿⣷⠿⠉⠀⠀⢸⡟⡇⣿⠉⠉⠉⠛⢾⠻⢸⡇⣿⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣯⡞⠁⣀⠀⠀⠀⠀⣷⢻⣻⣠⡄⠀⠀⠈⠛⣻⢳⡝⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⡇⠐⣿⠀⠀⠀⠀⢹⣼⢹⡿⡇⠀⠀⠀⢀⠟⣸⢧⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⠃⠀⠉⠀⠀⠰⡖⣺⡏⢸⡇⠀⠀⢀⣠⡴⢶⢻⣎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣦⣄⣀⣀⣀⣀⠭⣽⣧⣽⢧⣴⣾⣿⣿⣷⣝⣏⢿⡼⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⡏⠀⢻⡟⠁⠀⠀⡇⠀⠀⡏⣿⣿⣿⣿⣿⣎⢮⢿⣮⣞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⠋⠀⣠⡼⠷⣤⣤⣴⣧⣤⣤⢥⡽⠿⠟⠛⠛⠙⣮⠏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠁⠀⠀⢸⠂⢻⣿⣿⡘⣆⠀⠀⢤⡀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠳⠤⣶⣇⠀⠘⣿⣿⣇⣼⡄⠀⠀⠉⠓⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣹⢻⣤⣤⣿⣿⠿⣿⣿⣶⣶⣶⣿⡿⠿⢿⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡸⢶⣹⣄⣀⣀⣸⣎⣛⣛⣮⣷⣄⣤⡤⢿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠚⠉⠉⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀