⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣀⣤⣶⣶⣶⣵⣒⠤⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡴⠿⠥⡒⠛⠢⡀⠙⢯⣿⢀⢻⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⠊⠃⡀⡀⡀⢢⣀⡸⠻⣧⢹⠘⢾⠸⡄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡟⣴⣀⢐⠤⠶⠚⣛⡁⡇⠈⣼⡀⣼⡇⠹⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⣩⣤⣶⣿⡟⣻⢹⣿⢳⡀⠟⠔⢿⣷⡀⡇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢻⣿⢧⢸⣚⣇⣧⣼⣿⠨⡀⡏⣷⣤⣂⣇⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡎⣿⣿⠟⠫⣉⡀⡀⢐⢃⣶⡀⢸⢸⡀⢹⣸⡷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡇⢰⡿⠘⣀⡀⡀⡀⠷⢹⡀⡇⣼⠙⡄⣿⣿⡟⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠅⠸⣿⡀⡩⠄⣠⣤⡀⡟⣼⣠⠑⣰⡾⡼⢽⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡟⣽⣿⣿⣿⣿⣿⣿⣿⣯⣿⠹⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠈⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡠⡀⣿⠟⢸⣿⣿⣿⣿⣿⣿⡄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⣴⠋⢼⣽⡀⡺⡀⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡎⡠⣕⠢⠃⡀⡀⡀⢹⣿⣿⣿⣿⣿⣿⣧⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠻⣿⣿⣿⣿⣿⣿⣿⣧⣁⠐⠵⠁⣾⢀⢌⡠⢀⣾⣿⣿⣿⣿⣿⣿⡟⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠇⢓⣈⣴⠁⠘⣿⣿⣿⣿⣿⣿⣧⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣾⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⠁⡀⡀⠈⠙⣿⣿⣿⣿⣿⣧⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢷⣤⡹⣦⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡵⢜⡲⡌⢿⣿⣿⠿⠃⡀⡀⡀⡀⡀⢀⣿⣿⣿⣿⣿⣷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣓⢝⣫⡼⠖⡏⡀⠐⡀⡀⡀⡀⡀⢀⣿⣿⣿⣿⣿⣿⡏⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠛⡄⡀⡀⠘⡀⡀⢷⡀⡀⣀⡀⠸⣿⣿⣿⣿⣿⣿⡇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⣀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠓⠤⡀⡀⡀⡀⡀⡀⡀⡀⠉⣀⣬⣿⣿⣿⣿⣿⣿⣿⡿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⡀⡀⠙⢤⡀⡀⡀⡀⡀⡀⡀⡻⣿⣿⣿⣿⣿⣿⡏⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠉⠉⡀⣻⣿⣿⣿⣿⣿⣿⣿⢻⡀⡀⡀⡀⡀⡀⠈⠦⡀⡀⡀⡀⡀⠰⡀⠈⠻⣿⣿⣿⣿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠺⣟⣯⡿⣻⣿⠽⠷⠿⡿⣄⡀⡀⡀⡀⡀⡀⡀⡀⠳⣀⡀⡀⡀⡄⡀⡀⠠⠙⣿⣿⢷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⣯⢟⠩⢆⠊⠁⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠳⡀⡀⠨⡀⡀⡀⠈⢄⠙⠆⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣠⣾⣿⡇⠊⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⡄⡀⡀⡀⡀⡀⡀⠡⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⢀⣶⣿⣿⣿⣿⠠⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠘⡄⡀⡀⡀⡀⡀⡀⢣⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢀⣾⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢰⢻⡀⡀⡀⡀⡀⡀⢀⡐⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⢀⣾⣿⡿⢋⡀⣿⣿⣿⣿⡆⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢘⡮⡀⡀⡀⡀⡀⡀⡈⡃⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⢀⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠐⠰⢾⡀⡀⡀⡀⡀⡀⡫⠃⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⠩⠤⡠⣀⡀⢀⣈⣥⣴⣃⠲⢄⡀⡀⡀⡀⠈⡀⡔⡠⣸⣱⡀⡀⡀⡀⢀⡜⠇⣾⣿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⢿⣿⣿⣿⣿⣿⣷⣤⣴⣭⣿⣿⣿⣿⣦⣘⣛⣛⣾⢣⣿⣿⣿⣷⢫⡀⡀⡀⠂⠘⣑⣍⣕⢁⢣⣽⣖⡆⢎⡡⠉⣾⣿⣿⣿⣷⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⡇⢬⣼⠷⣋⠴⡇⠫⠽⠣⢑⣨⣶⣽⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠓⠒⠒⠿⠿⠿⠿⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⣾⣿⣿⡛⢿⠿⣿⣿⣿⣄⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠈⠉⠛⠿⠿⠿⠿⠛⠛⠿⣿⣿⣿⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠻⣿⣿⣿⣿⣿⣿⣷⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢻⣟⣿⣿⣿⣿⣿⣿⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢻⣿⣿⣿⣿⣿⣿⣿⡄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠘⣿⣿⣿⣿⣿⢻⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠈⠛⠛⠛⠛⠋⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀