⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡐⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣉⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠚⠋⠉⡇⡿⣾⢯⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⠿⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡯⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⢼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⣻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠙⠛⣁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣿⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡧⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠐⡦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⡀⠀⠈⠙⡗⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠶⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡏⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠧⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣀⡇⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠤⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢻⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣻⡽⠟⢻⡿⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣾⣿⡏⣿⠗⠛⠉⠉⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣻⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠊⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⣟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⣸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⡭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠖⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣁⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢿⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠇⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⢿⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⣻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠙⠛⣁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣿⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡧⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠒⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣆⣀⡀⠀⠊⠙⡗⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠾⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣸⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠁⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠁⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠉⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡇⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠶⠂⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢹⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡿⣿⣿⣿⡿⣿⣻⡿⠟⢻⡽⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣾⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠊⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⠀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣃⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢿⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡿⣾⢧⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡜⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⢛⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠺⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡉⠙⠋⣉⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠈⡁⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢯⣟⣿⡿⣿⣿⣿⣿⣹⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢽⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢾⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣃⣿⣿⣿⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡧⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⣭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠐⡶⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⡀⠀⠊⠙⡗⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢹⢸⡏⠙⠛⠶⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠒⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣸⠁⢠⣀⣤⣤⠄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠁⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠁⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢎⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠶⠒⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠻⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡿⣿⣿⣿⢿⣿⣿⡿⠟⢻⡽⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠉⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢿⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠙⢹⣿⣇⣷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⣚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢹⣁⣉⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣻⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠶⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡼⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠼⢥⣿⡜⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⢿⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢹⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠊⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⣻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣹⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠉⠛⣁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢯⡟⠻⢯⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣘⡻⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢽⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣛⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⡶⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⡀⠀⠊⠙⡗⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢹⢸⡏⠙⠛⠶⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠒⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣸⠁⢠⣀⣤⣤⠄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠁⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠁⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢎⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠤⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⠉⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⡿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣮⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡇⣿⠗⠛⠉⠙⠀⠀⢸⡇⢿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣋⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠊⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⣿⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢢⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⢹⣁⣁⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⠟⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡾⡇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠄⠀⠀⢀⣼⡏⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⢽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⣱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠉⠊⣉⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣽⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣼⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡞⣿⣹⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⣿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣿⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠐⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⡀⠀⠊⠙⠗⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡾⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⠸⣾⣧⣿⣿⣿⣟⡼⣷⢹⢸⡏⠙⠛⠾⢭⣝⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⢟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⡤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⢸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡝⠉⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢎⣷⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠤⠒⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠉⣅⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡿⣿⣿⣿⠿⣿⣿⡽⠟⢻⡿⣯⣟⢿⣮⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣾⣿⡇⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣋⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡭⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⡀⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢹⣁⣀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠶⠚⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠃⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⠟⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⣻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠉⠋⣉⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣽⢯⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣘⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣃⣿⣿⣿⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣛⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠶⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠊⠘⠇⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢹⢸⡏⠙⠛⠶⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠃⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡏⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⢽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⠧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⣆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡗⢸⣿⣿⣿⠀⠀⠈⢹⡉⠙⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣀⡇⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⠽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠤⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣾⣿⡏⣿⠗⠛⠉⠉⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⣟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠨⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣛⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢹⣁⣁⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠶⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⠿⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⡇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣶⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⣱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡉⠉⠋⣁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⣿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣛⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠐⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠈⠙⡗⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢹⢸⡏⠙⠛⠶⢭⣝⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣾⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡏⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⢽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⡆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠧⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡇⣶⡶⢎⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠴⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠸⢉⣅⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⡿⣿⣟⢿⣏⢷⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡇⣿⠗⠛⠉⠙⠀⠀⢸⡇⢿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⣿⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠉⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡯⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡀⠒⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⢙⣁⣈⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠚⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠃⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⠛⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡯⠷⣾⢽⡇⣿⠀⠀⠉⠁⣶⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠈⠉⠋⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⢿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡞⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣕⣻⢸⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣛⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠽⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠒⡦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠊⠙⠗⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⠸⣾⣧⣿⣿⡿⣟⡼⣷⢻⢸⡏⠙⠛⠶⢭⣟⣿⢻⣟⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⢽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⠧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⣆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡇⣶⡶⢮⣷⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⢉⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣎⢷⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣋⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠊⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢿⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢡⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢿⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡼⣷⠇⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⠟⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⡇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢺⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡏⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣶⣤⣤⣤⣀⣼⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⣱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡈⠙⠛⣁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢯⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡞⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⢤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣛⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠶⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠊⠙⠗⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠶⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣷⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣸⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡏⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⢸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⡆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠉⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣀⡗⣶⡶⢎⣷⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠴⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠹⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⡿⣿⣿⡿⠟⢻⡿⣿⣟⢿⣏⢷⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣾⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢷⢿⠀⣠⣤⣀⡀⡀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⣿⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⣷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠨⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣁⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢿⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡿⣾⢧⣻⣿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⢛⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢹⡇⣿⠀⠀⠉⠁⣶⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡅⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠄⠀⠀⢀⣼⡏⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠉⠋⣁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⣿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣿⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠒⠶⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⡀⠀⠊⠙⡇⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢹⢸⡏⠙⠛⠾⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣸⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⣇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣽⣿⣿⣿⡄⣆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠤⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠹⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⣿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣏⢷⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⣿⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠉⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢹⣁⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡿⣾⢧⣻⢿⣿⣿⡾⣷⠇⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⠟⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⡇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⣻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡏⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣶⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⣱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡁⠉⠋⣁⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡅⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢯⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣃⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣿⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠽⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠐⠦⠀⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠊⠙⡇⠐⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⠸⣾⣧⣿⣿⡿⣟⡼⣷⢻⢸⡏⠙⠛⠾⢭⣟⣿⢻⣟⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣸⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⢽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⢸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠉⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢮⣷⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⠽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠤⠒⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠙⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣻⡽⠟⢻⡿⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡇⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⢿⣛⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠊⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠙⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣛⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠲⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣉⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠇⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⢿⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢾⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠊⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡉⠙⠛⣀⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠈⡁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢯⣿⣿⡿⣿⣿⣿⢿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢸⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢻⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⡀⠀⠈⠙⡇⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠾⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⠄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠁⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⣆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠧⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠤⠒⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠙⢉⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣻⡽⠟⢻⡿⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⢿⣛⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠊⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠙⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣛⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠲⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣉⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠇⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⢿⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢾⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠊⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡉⠙⠛⣀⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠈⡁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⢿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢸⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢻⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠌⠙⡇⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠾⢭⣟⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⠄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠁⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠁⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⣆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠧⠞⠁⠀⠀⠀⠀⡿⠘⣿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢮⣷⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠴⠒⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠹⢉⣁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣾⣿⣏⣿⠗⠛⠉⠉⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠉⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⢛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡂⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣁⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢿⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠷⠛⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠃⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⢟⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣾⢹⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣶⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⠉⠙⠋⣁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣷⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡞⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡧⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣛⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠒⠦⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⡀⠀⠊⠙⡗⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠾⢭⣟⣿⢿⣟⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣸⣼⠁⢠⣀⣤⣤⡄⣿⢸⣸⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢈⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⢸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠉⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠙⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡇⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠤⠒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠻⠉⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⣿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣎⢷⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⣷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣻⢻⡟⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢡⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠶⠚⠋⠉⡇⡟⣾⢧⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⢟⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢹⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⣇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡏⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⢽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣶⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⡱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣇⠈⠑⠚⣁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⡁⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢯⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡞⣿⣹⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⣿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣿⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠴⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠊⠙⠇⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⡿⣟⡼⣷⢻⢸⡏⠙⠛⠾⢭⣟⣿⢿⣟⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⡬⠧⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠃⠀⠀⢹⣼⣸⠁⢠⣀⣤⣤⡄⣿⢸⣼⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣿⣿⣿⡄⡆⣀⣎⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢮⣷⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⠽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠤⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠛⢉⣅⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣿⡽⠟⢻⡿⣿⣟⢿⣎⢧⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠙⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢿⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣷⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡭⡏⠙⠛⠛⢻⣛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢡⣟⠀⠀⢠⣟⣘⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣁⣉⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢿⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠶⠛⠋⠉⡇⡿⣾⢯⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡜⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣧⢛⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⡇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⣻⢹⣿⣿⣿⣿⠒⢲⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣶⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⣱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡉⠉⠛⣁⣠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇⠈⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢯⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣽⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡟⢸⡇⠀⢀⣀⣤⣄⠀⣿⡞⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⢤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠶⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠊⠙⡇⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⠸⣾⣧⣿⣿⣿⣟⡼⣷⢻⢸⡏⠙⠛⠶⢭⣟⣿⢿⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠚⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣸⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡏⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⢽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⠤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢷⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣿⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠣⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡇⣶⡶⢎⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠤⠒⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠹⢉⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⠿⣿⣿⡽⠟⢻⡽⣿⣟⢿⣎⢷⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⣿⡏⣿⠗⠛⠉⠉⠀⠀⢸⡇⣿⠼⣷⣻⡞⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣛⣿⢷⣿⠀⣠⣤⣀⡀⠀⢸⡇⢸⡆⠈⣷⢻⣿⡿⠒⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡟⠁⣿⣸⣿⠀⠀⠉⠉⠉⣁⢸⡇⢿⡷⠶⢾⣧⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣧⢠⡇⣿⣿⠀⠀⠀⠀⠀⣏⣽⡿⣸⠃⣠⠞⠁⠘⠦⢤⡤⠚⠀
⠀⠀⠀⠀⠀⣀⡀⠀⡿⢟⣪⣿⡇⡇⣿⣤⣀⣀⣠⡤⠤⣶⡇⣿⡏⠇⠀⠀⠀⢠⠎⠀⠀⠀
⠀⠀⠀⢰⠋⠁⠈⠹⡟⠹⢹⣿⣇⢷⣿⣿⣿⠲⣶⣿⣾⣿⣿⣿⣿⣌⠢⡀⢠⣿⠀⠀⠀⠀
⠀⠀⠀⠸⣄⠀⠀⣠⠇⠀⢸⣿⣿⢸⡟⣿⣿⠀⣸⣿⣧⣿⣿⣿⣿⣿⣿⣾⡏⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠙⣟⡚⡏⠀⠸⣾⡇⠈⢿⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠈⣧⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡇⠀⠴⠚⠁⠀⢸⠀⣿⣿⡿⡇⠀⠀⠀⠀⣇⠀⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⣀⠀⠀⠀⠀⢸⡀⠈⠛⠀⡇⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣿⣿⣶⣶⣦⣼⣿⣿⣿⣿⣇⣀⣀⣠⡤⠔⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡯⡏⠙⠛⠛⢻⣛⣛⣻⢻⡟⠿⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢠⣟⠀⠀⢠⣟⣸⣟⣛⣸⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣭⠿⠋⠉⠉⠉⠉⠻⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡀⠲⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣷⣝⢷⡟⡓⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣻⡿⣿⣻⣭⠶⠚⠋⠉⡇⡿⣾⢧⣻⢿⣿⣿⡾⣷⠇⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢥⣿⡝⣾⠁⠀⠀⠉⠁⠀⡇⡇⣿⠀⣙⣷⡻⣿⣇⣿⢰⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣇⡇⣿⠠⣤⣤⣀⡄⠀⣷⣇⡟⠀⠀⠈⣷⢛⣼⠟⠋⠓⣄⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠷⣾⢽⡇⣿⠀⠀⠉⠁⣴⠶⣿⣼⡟⠛⠛⠋⢻⣼⡇⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⣀⡼⢿⠻⣿⡿⡇⣿⣶⡶⠤⠤⣼⣶⣿⢸⣇⣀⣤⡴⠚⠁⠓⠦⣤⠖⠁⠀
⠀⠀⢀⡤⠒⠻⡄⠀⠀⠀⣿⣷⢻⢹⣿⣿⣿⣿⠒⢺⣿⣼⣿⣿⣇⠆⠀⠀⢀⣼⡟⠀⠀⠀
⠀⠀⢸⠀⠀⠀⢹⣀⢠⢠⣿⣿⣿⣮⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣦⣵⡶⣻⢹⠃⠀⠀⠀
⠀⠀⠈⠳⠶⠒⠋⣽⢻⠋⢉⡏⠁⠀⠀⣿⣿⣿⣇⢸⠉⠛⠛⠛⢟⠋⢻⠙⢮⣸⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠊⠀⠀⠀⠀⣿⢿⣿⠃⢸⠀⠀⠀⠀⠈⠳⠜⡆⠀⠻⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡸⣶⣦⣤⣤⣤⣀⣸⣿⣷⣶⣾⡀⠀⣀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⣿⠀⠀⠀⣾⣱⢶⣶⢸⣆⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠿⠿⠶⠞⠓⠛⠛⠉⠙⢯⣭⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣏⡉⠉⠋⣁⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇⠈⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣼⣱⡿⣻⣿⢏⡟⠻⢿⣿⣿⡿⣿⣿⣿⣿⣻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⢳⣟⣼⠧⣿⢸⡇⠀⠀⠋⠉⠛⠛⣿⢹⣿⣿⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠔⠚⠒⢿⣿⡞⣾⠁⠀⡏⢸⡇⠀⢀⣀⣤⣄⠀⣿⡟⣿⣙⡿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⠀⡿⣼⡷⠶⢾⡿⢸⡇⣈⠉⠉⠉⠀⠀⣿⣧⣿⠈⢻⡍⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠑⢤⡤⠴⠃⠈⠑⣄⠀⣧⢿⣯⣹⠀⠀⠀⠀⠀⣿⢿⢸⡇⣼⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠑⡄⠀⠀⠀⠨⢹⣿⢸⣶⠤⠤⣄⣀⣀⣤⣿⢸⢸⣿⣗⣻⢼⠀⢀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣄⢀⠄⣁⣿⣿⣾⣿⣷⣾⣶⠖⣿⣿⣿⡿⢸⣿⡏⡏⢻⠟⠉⠈⠙⡆⠀⠀⠀
⠀⠀⠀⠀⢸⢹⣷⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⣿⣿⢻⡇⣾⣿⣇⡅⠸⣄⠀⠀⣠⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣾⡇⠈⠉⠉⠉⠉⢹⣿⣿⣿⠀⡿⠁⢸⣷⠏⠀⢹⣗⣻⠋⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⠁⣸⠁⠀⠀⠀⢸⢿⣿⣿⠂⡇⠀⠈⠑⠦⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⢸⠀⠛⠃⠀⡷⠀⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠓⠲⢤⣤⣀⣀⣸⣿⣿⣿⣿⣧⣴⣶⣶⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠻⡿⣟⣛⡛⣟⠛⠛⠋⢹⣯⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⣛⣻⣇⣻⣄⠀⠀⣸⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠉⠉⠉⠉⠙⠿⢭⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠐⠶⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⡀⠀⠈⠘⠇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⢞⢛⡿⣫⣾⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣧⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⠸⣾⣧⣿⣿⡿⣟⡼⣷⢹⢸⡏⠙⠛⠶⢭⣝⣿⢻⣏⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⡇⣿⣸⣿⣟⣿⣋⠀⣿⢸⢸⡇⠈⠉⠀⠀⠈⣿⢣⢿⣬⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠒⠙⠛⣧⡻⣼⠁⠀⠀⢹⣼⣼⠁⢠⣀⣤⣤⡄⣿⢸⣸⡇⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⠀⠀⢘⣧⡟⠙⠛⠛⢻⡇⣿⠶⣶⠈⠉⠀⠀⣼⢸⡇⣷⠾⢽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠲⣤⡤⠚⠈⠑⠢⣤⣀⣸⡇⣿⣷⣯⠤⠤⠶⣶⣿⢸⢧⣿⠟⡷⢧⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣷⡀⠀⠀⢠⣸⣿⣿⣧⣿⡗⠒⣿⣿⣿⣿⡟⡟⣾⣿⠀⠀⠀⢀⠟⠒⢦⡀⠀⠀
⠀⠀⠀⠈⡇⣟⢶⣦⣵⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣵⣾⣿⣿⡄⡆⣀⣞⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⣟⡵⠏⡜⠙⣻⠛⠛⠛⠉⡷⢸⣿⣿⣿⠀⠀⠈⢹⡉⠉⡟⣯⠉⠒⠶⠞⠁⠀⠀
⠀⠀⠀⠰⠟⠀⢠⠧⠞⠁⠀⠀⠀⠀⡿⠘⢿⡿⣿⠀⠀⠀⠀⠑⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀⣀⠀⢀⣷⣶⣾⣿⣿⣀⣤⣤⣤⣴⣶⡏⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⣠⡗⣶⡶⢮⣷⠀⠀⠀⣿⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⡽⠋⠉⠛⠛⠚⠳⠶⠿⠿⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀