⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠣⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⣷⣒⣚⣿⣿⣿⣿⣛⣓⣒⡾⠁⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣻⡭⣶⠀⠀⣹⠀⠉⠙⠛⠛⠛⠉⠁⠈⠧⣀⣀⡤⢟⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡤⡴⢤⡀⠀⢀⣠⣾⣿⣿⡿⠋⠀⠘⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣿⣷⣤⡀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣏⠃⠀⠹⢴⡯⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠠⣄⠀⠀⠀⢀⣀⠉⠻⢿⣿⢦⠏⠀⠸⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠦⡄⣀⠈⢙⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⢷⡀⢠⣾⠟⠀⠀⠀⡏⠀⠀⠀⢀⣞⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⢎⣼⠛⠒⠋⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣀⣀⣀⣈⣻⣿⣋⣀⣀⣀⡀⠈⠓⢺⣰⣿⣟⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⣸⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⣿⡿⠻⡏⠉⠉⠉⠁⠀⠀⢀⠙⣿⡜⣆⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣀⣼⡟⠀⠀⢻⡀⠀⣀⠀⠀⠀⠸⡄⠸⣷⠘⣿⢇⣴⢿⣷⠀⠀⠀⠀⠀
⠸⣉⠇⠈⠙⠿⢤⠀⠀⣾⠀⡀⠀⠀⢠⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⡆⠹⣏⠀⠀⠀⠈⢧⠀⣿⠀⠀⠸⣿⡿⢶⣽⣇⣿⡿⠋⢸⣿⠀⠀⠀⠀⠀
⠀⠳⠤⠤⣤⡀⣸⠀⢠⡇⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡇⠀⠈⢧⡀⠀⠀⠸⡆⣿⠀⠀⠀⣿⣧⠀⢈⣯⣿⣧⣄⢸⡿⠀⠀⠀⠀⠀
⠀⠀⠀⢰⡏⠉⠁⠀⣸⠁⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⡷⠾⠛⠛⠛⣆⠀⠀⢷⣿⠀⠀⠀⢻⣿⣴⣿⣿⣿⣟⢿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠃⠀⠀⠀⣿⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠈⠳⣄⠸⣿⠀⠀⠀⢸⡿⢋⣾⠇⣿⣿⡎⢻⣇⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⣤⠄⢀⠀⠀⠀⠈⠓⣿⠀⠀⣀⢸⡇⣸⡿⠀⣿⣿⣧⣸⡟⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣇⠀⠀⠀⢸⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣇⣤⣶⣿⣿⠿⣷⣶⡂⡟⠀⠀⡇⢸⡇⣿⣇⠀⣿⢸⣿⡿⠁⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡇⠀⠀⠈⡇⠀⠀⢸⡀⠀⠀⢻⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡿⢫⡟⠁⣨⡑⢿⡄⢳⡇⠀⢸⠇⣼⠕⡚⡟⠛⠟⢸⣿⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠈⡇⠀⠀⠀⢿⠀⠀⠘⡇⠀⠀⠘⣇⠀⠀⠀⢻⡸⡇⠀⠀⠀⠀⠀⠀⠀⡿⢸⠃⠘⡇⣀⢿⡃⢸⡇⢸⠇⠀⡿⢠⣟⡔⠁⢸⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣧⠀⠀⠀⠸⡆⠀⢲⣻⡀⠀⠀⢹⡀⠀⠀⠈⣧⢹⡀⠀⠀⠀⠀⠀⢰⡇⡿⠀⠀⠉⢽⣫⡽⠎⠀⡿⠀⢰⠇⣸⢻⠀⠀⢸⠀⠀⣸⡿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡀⠀⠀⠀⢷⡀⠈⢿⢷⠀⠀⠀⢷⡀⠀⠀⠸⣎⣧⠀⠀⠀⠀⠀⣼⢹⡇⠀⠀⠀⠀⠘⠘⠀⢰⡇⢠⡟⢠⣏⡞⢀⠆⣸⠀⢠⣿⡇⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⢳⡀⠈⢿⢧⡀⠀⠈⠇⠀⠀⠀⠸⣞⣧⣤⠴⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣞⣠⠏⡾⠴⢋⣴⣿⠀⡼⣿⡇⣿⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠻⣄⠈⢯⣳⡄⢀⣠⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⣠⡀⣠⣾⣿⣿⡼⢱⣿⠃⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣎⠳⡄⠀⠀⠈⠳⣴⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⢦⣄⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢟⣿⠟⠁⠀⣿⡇⢀⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⢻⣿⣶⣤⣜⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠡⠞⠁⠀⠀⢰⣿⠁⢸⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋⠛⠉⠀⠀⠀⠀⠀⣼⡟⠀⢸⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⡿⣿⣿⣿⣷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡟⠛⠛⢿⠉⠙⠒⠶⢤⣄⡤⠖⠋⠁⠀⣸⠿⢯⡹⡆⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡴⠊⠉⢳⡀⣀⣀⠀⠀⠀⠀⣀⠀⡼⠁⠀⠀⢇⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⡾⠉⠉⠙⠁⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠟⠁⠈⢳⡀⢠⠟⠉⠻⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣨⠷⠒⠛⠉⠻
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⡶⢤⣄⠀⠀⠀⠀⢀⣠⠿⣤⠀⠀⠀⠀⠀⠀⠀⣧⠏⠀⠀⠀⠀⠀⠀⡼⠓⠦⣤⣄⣀⠀⠀⢀⣀⣤⡞⠁⠀⠉⠁⠀⠀⠀⠀⢠⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢳⣶⠖⠛⠛⠀⠀⠀⠷⠄⠀⢤⠀⠀⠀⠛⠀⠀⠀⡴⡇⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⣻⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣧⣶⣶⣶⣦⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣏⠓⠀⠘⣿⡄⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⣻⣿⣿⣿⣿⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣿⠃⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀
⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢻⣇⠀⠀⠀⠀⣾⣿⠟⠁⢀⣾⣿⡿⢸⣿⣿⣄⠀⠉⠻⣿⣷⠀⠀⠀⠀⣞⡏⠀⠀⡟⠀⠀⠀⠀⠀⢀⣀⡤⠞⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢯⣹⣀⡀⠀⠀⠀⢸⡃⠀⠈⣟⡆⠀⠀⠀⠹⣿⣷⣶⣿⣿⣿⠇⠈⣿⣿⣿⣆⡀⢀⣨⣿⠇⠀⠀⣸⡏⠀⢀⡞⠁⠀⠀⠀⠀⠀⣘⣾⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⢷⣄⣀⣘⣿⡀⠀⠀⠀⠀⠉⠙⢩⣿⡟⠀⠀⣿⡧⠹⠿⠿⠿⠛⠁⠀⠀⣸⡝⠀⢀⣜⣁⣤⠤⠴⠖⠚⠋⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠶⠾⠿⠥⠤⠤⠬⠥⠤⠤⠴⠶⠶⠖⠒⠒⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀