⣿⣻⢹⢹⣿⣿⣿⣿⣿⣿⣫⣿⣾⣿⣿⣿⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⡏⡿ ⡏⠋⠘⣼⣿⣿⣿⣿⢏⣿⡿⠋⠁⢀⣸⣝⣯⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠃⢳ ⡇⠀⠀⢘⣱⣿⣿⡿⣿⣿⠁⠀⠾⣻⣿⠏⠿⣿⣿⢿⣿⣿⣿⣿⣿⣯⡛⠀⠀⢸ ⣿⠀⠰⣿⣿⣿⣿⣿⣿⣿⠁⠀⠊⠉⢡⣠⣶⣶⣿⡏⢿⣿⣿⣿⣿⣿⣿⡶⠀⣾ ⣿⠁⠘⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣸⢁⡾⠿⢿⣷⡘⣿⣿⣿⣿⣿⣿⠇⠀⣿ ⣿⠀⢠⣿⣿⣿⣿⣿⣟⠟⢻⠀⠀⠀⢹⡿⠶⢿⣿⣿⡇⣿⣿⣿⣿⣿⡿⠄⢸⣿ ⣿⡂⠈⢸⣿⣿⣿⣿⣿⠚⠀⠀⠀⠀⠈⠀⠀⢺⣿⣿⡇⣿⣿⣿⣿⣿⠃⠀⣸⣿ ⣿⡇⠀⠀⢿⣿⣿⣿⣿⣷⡌⠁⠀⠀⠀⠀⢀⣼⣿⣽⣿⣿⣿⣿⣿⠏⠀⠀⣿⣿ ⣿⣷⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣐⣤⣤⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠿⣿⣯⣿⣼⡍⢂⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣬⣽⣼⣿⣾⣿⣣⣾⣾⣶⣾⣦⣄⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠋⠉⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣈⣿⡿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⠁ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣟⠋⠉⠀⠀⢹⣿⣿⣿⣿⣿⡿⠋⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠋⠉⠋⠉⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣤⣄⣀⣀⣾⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⡆⠙⠿⣿⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣹⣿⣿⡿⠀⢠⣿⠃⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⡉⠁⢠⣶⣿⠇⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠯⡶⢀⣿⡿⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢶⣶⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢿⣿⣦⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣦⣍⠐⢼⣿⣿⣧⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⣿⣿⣿⣦⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣒⣚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡽⣿⠟⢃⣿⠟⡉⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢯⣭⣿⢹⡟⢶⣗⠀⢸⠇⢁⣠⠫⠟⣹⣿⣯⡭⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣄⠣⠁⠈⠓⠊⠐⠃⠀⠀⣰⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠀⠀⠀⠀⠀⠀⠀⣿⣯⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⡀⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣀⣀⣿⡿⠃⢢⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡿⣷⡿⠿⣿⢿⡿⠩⢹⣿⡟⠉⠉⠉⢹⣿⢻⠛⠛⠻⢿⣿⣷⣿⣿⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢶⣿⣦⣿⣷⡇⢁⠚⡇⠐⢀⣿⣷⣤⣤⣤⣿⣧⣀⣠⣇⡰⢀⡿⣿⣿⣷⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣵⡎⠹⣿⣿⣿⣁⣾⢻⣿⢻⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⡿⠛⠛⢛⡟⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣞⣰⠞⣩⣿⣻⡇⢠⠃⠸⠉⠙⣿⣿⣿⣿⢈⡁⠤⠁⠌⢉⢙⣿⣐⣧⣤⣾⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⢋⡍⢇⠛⣿⣧⣼⣦⣴⣔⣠⣿⣿⣿⣯⣦⣤⣦⣤⣦⣄⣸⡻⢛⠋⠉⠻⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣆⣷⣴⣶⠿⣿⡿⣿⢿⣿⣿⣿⣿⣿⢿⡿⢿⣿⢉⠉⣿⣿⣷⣾⣤⣭⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⣿⣿⣿⠐⣉⣌⠇⠐⠁⡄⣿⣿⣿⠈⡠⢾⡇⠠⠠⣼⣽⡟⢣⠄⣹⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⠁⣿⣷⣿⡟⢿⡵⣾⣶⣦⣿⣿⣿⣴⣷⣾⣷⠖⡴⠿⣿⣿⠷⠚⢛⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣿⣿⣿⣧⠐⡅⢘⠃⣿⣿⣿⣟⠿⡿⣿⣧⣈⠴⢃⠘⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⠃⢸⣿⣶⣿⡿⡟⢉⣿⢿⣻⣼⣧⠂⠛⢿⣷⣟⣀⣥⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⡙⣿⣾⣿⣿⠿⢄⠉⣸⡿⠁⠉⠉⠟⢬⢛⢚⠻⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣅⣙⣿⣧⣮⣬⣽⡟⠀⠀⠀⠀⢶⣶⣮⢧⢠⠙⣰⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣈⣽⣶⣿⣿⣷⣼⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣏⢿⢯⣿⡟⠀⣿⣿⣿⣶⣄⣠⢈⠲⠟⣛⠏⣍⣏⠬⣄⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣃⣧⣿⣿⣿⣿⣿⣧⣒⣴⣦⣾⣶⣾⣶⣿⡇⠀⠀⠀

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⣭⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⢔⡾⣹⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢳⣿⣿⣿⠋⣻⣿⣿⣿⣿⣿⣿⣾⣿⠟⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⣼⣷⣻⣧⣴⣿⣏⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠤ ⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣏⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀ ⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣝⣿⣯⣾⠋⣇⠀⠀⠀⠀⠀⠀ ⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⣽⣝⠋⢡⣯⣀⠘⢦⡀⠀⠀⠀⠀ ⠀⠀⡷⡁⠀⡄⠀⢠⠻⠀⠀⠀⢸⠙⠀⠀⠀⠙⡇⢹⣧⠛⠂⠀⢤⣉⠢⡀⠀⠀ ⡠⢊⠚⢇⣰⢣⠀⡞⠒⠣⠀⠀⠘⡄⠘⠓⠲⢆⣳⠀⠀⣠⣄⣀⣀⠙⢯⣾⡄⠀ ⣇⣇⡌⠈⡜⡌⢳⣧⣤⣄⡑⠄⣀⣳⢀⣠⣤⣴⣾⡆⠀⣿⠖⣠⣮⠀⠀⣿⠇⠀ ⠈⠛⢇⠀⠿⠷⡘⣿⢙⠿⡏⠀⠈⠉⢻⣻⣿⡏⢹⡟⣢⣿⣟⡻⠋⢀⡴⠁⠀⠀ ⠀⠀⠈⠢⢤⣀⣋⡿⢮⡉⠁⠀⠀⠀⠈⢉⣙⠷⠛⠺⣿⣙⣛⠭⠝⠋⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠒⠶⠲⠞⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣵⣕⣉⣫⣿⣦⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡾⡿⡟⣻⣿⡏⠱⣮⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣰⢿⡛⣿⣾⣜⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣾⡏⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣵⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀