⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⠸⠿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⠿⠃⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⣀⣀⣀⡀⡀⡀⡀⡀⡀⡀⡀⢀⡀⡀⡀⢀⣀⣀⡀⡀⡀⢸⣿⡀⡀⡀⣀⡀⡀⡀⣀⡀⣀⣀⡀⡀⡀⠘⠿⡀⡀⡀⣀⣀⣀⡀
⡀⡀⢸⣿⡀⡀⠘⠛⠉⠉⢿⣧⡀⠘⣿⡀⡀⡀⣿⠃⡀⣾⡟⠉⠙⣿⡀⡀⢸⣿⡀⡀⡀⣿⡇⡀⡀⣿⡿⠉⠙⣿⡆⡀⡀⡀⡀⡀⣿⡟⠉⠉⠋
⡀⡀⢸⣿⡀⡀⡀⣤⣶⣶⣾⣿⡀⡀⢿⣧⡀⢸⡿⡀⡀⣿⣤⣤⣤⣿⡇⡀⢸⣿⡀⡀⡀⣿⡇⡀⡀⣿⡀⡀⡀⣿⡇⡀⡀⡀⡀⡀⠻⣿⣦⣄⡀
⡀⡀⢸⣿⡀⡀⣿⡏⡀⡀⣼⣿⡀⡀⠈⣿⡀⣿⠁⡀⡀⣿⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⣿⡇⡀⡀⣿⡀⡀⡀⣿⡇⡀⡀⡀⡀⡀⡀⡀⡀⠹⣿
⡀⡀⢸⣿⡀⡀⠙⣿⣶⡾⠻⡿⡀⡀⡀⠻⣷⡟⡀⡀⡀⠙⠿⣶⣶⣶⡀⡀⠸⣿⡀⡀⡀⣿⠇⡀⡀⣿⡀⡀⡀⢿⡇⡀⡀⡀⡀⡀⠿⣶⣶⣿⠟
⡀⡀⣼⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⠘⠿⠛⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀

⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣤⣶⣶⡀⡀⣴⣦⡀⡀⢠⣶⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢰⣷⣶⠿⣿⣦⡀⡀⣾⡆⣴⣿⠇⡀⢀⣶⡿⠿⣷⣄⡀⡀⣶⣿⣿⣶⣶⡀⡀⣾⣷⡀⡀⢸⣿⡀⡀⢀⣶⡿⠿⣷⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⣿⣇⡀⣿⣿⡀⡀⡀⡀⣿⡏⡀⡀⡀⣿⡆⡀⡀⣿⣿⡀⡀⡀⡀⣿⣿⡀⡀⢸⣿⡀⡀⣿⡏⡀⡀⢸⣿⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⣿⡿⡀⣿⡇⡀⡀⡀⡀⣿⡇⡀⡀⡀⣿⡇⡀⡀⣿⣿⡀⡀⡀⡀⣿⣿⡀⡀⢸⣿⡀⡀⣿⡟⠛⠛⠛⠛⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢸⣿⣦⣀⣤⣿⠁⡀⣿⡇⡀⡀⡀⡀⠹⣿⣄⣀⣴⡿⡀⡀⡀⣿⣿⡀⡀⡀⡀⣿⣿⡀⡀⢸⣿⡀⡀⠹⣿⣄⣀⣀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢸⣿⡀⠉⠉⡀⡀⡀⠈⡀⡀⡀⡀⡀⡀⡀⠉⠉⠁⡀⡀⡀⡀⠈⠁⡀⡀⡀⡀⠈⠁⡀⡀⡀⠉⡀⡀⡀⡀⠉⠉⠉⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⢸⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⢀⣴⣶⣿⣶⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⣿⣿⣷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋⣤⣴⣤⣄⠻⣿⣿⣿⣿⣿⣶⣤⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⠈⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⠹⣿⠻⣿⣿⣿⣿⣟⣯⢿⣿⣿⡿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⠙⢷⣄⠉⠻⠿⣿⣿⣿⣷⣶⣿⣿⡿⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⠈⠻⣿⣦⣀⠈⠛⠿⣿⡿⢽⣷⡢⢔⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⠘⣷⡀⠉⠳⣄⡀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⠹⣷⡀⡀⡀⠈⠙⠒⠤⠤⡀⡀⠉⠛⠻⠿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⠘⣿⣆⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⣿⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣷⣄⡀⡀⡀⡀⡀⡀⢀⣾⣿⣿⣦⡀⡀⡀⡀⡀⡀⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣶⣿⣿⣿⣿⣿⣦⣀⡀⡀⡀⣿⣿⣿⣿⣿⣿⣄⡀⡀⡀⡀⡀⡀⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⡀⡀⡀⡀⡀⡀⡀⡀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⣄⡀⡀⠹⣿⡿⣿⣿⣿⣿⣦⡀⡀⡀⡀⡀⡀⣠⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡀⢀⣤⣦⣄⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡛⣿⣶⣝⣿⣷⡻⣿⣿⣿⣿⣄⡀⡀⡀⢠⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡀⢿⣟⣿⣿⠛⠛⠛⣭⣿⣿⣿⠛⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣷⣻⣿⣿⣿⣿⡀⡀⡀⣿⣿⣿⣿⣿⣿⡿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⡀⡀⠉⠉⢁⣤⣾⣿⣿⣿⠟⡀⡀⠿⣿⣿⣤⡀⡀⠉⢿⣿⠟⣾⣷⣿⣿⣝⣿⣿⣿⣿⣶⣶⣿⡯⣝⣿⣿⣿⣿⣿⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⡀⡀⡀⢰⡏⠈⣘⡟⠁⡀⡀⡀⡀⡀⣾⢛⢛⡿⣀⣀⡀⡀⢰⡿⣿⣿⣿⢵⣿⣿⣿⠉⠉⠁⢸⣿⡽⢽⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⠃⡀
⡀⡀⡀⡀⠙⠛⠉⡀⡀⡀⡀⣤⣤⡀⠈⠛⠛⢸⣿⣿⣷⣄⣿⣼⣿⣿⣫⣿⣿⣿⣿⡀⡀⡀⢸⣿⢵⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠃⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣏⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣿⣟⣽⣿⣿⣿⣿⡿⡀⡀⡀⡀⢻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠃⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⢛⣉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀⡀⡀⣠⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⡀⡀⡀⢸⣿⣿⣿⣟⣭⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⢶⣾⠟⠛⠛⠛⠋⠁⡀⠈⠉⠙⠛⠉⡀⡀⡀⡀⡀⡀⡀⣩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣾⣿⣿⣿⣿⣿⣿⠿⢋⣿⣿⣿⡿⠛⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⣤⣤⣴⣿⣿⣿⣿⡿⣿⣿⠿⣿⡿⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⠿⢿⣿⣿⣿⠟⠉⡀⢿⣶⣶⠟⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⠛⠛⠋⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀