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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠁⠀⠈⠈⠀⠁⠁⠀⠈⠀⠀⠁⠁⠀⠈⠀⠀⠁⠀⠈⠈⠀⠁⠁⠈⠈⠀⠀⠁⠀⠈⠈⠀⠁⠁⠀⠈⠀⠀⠁⠀⠈⠀⠀⠁⠀⠈⠀⠀⠁⠀⠈⠀⠀⠁⠈⠈⠀⠁⠀⠈⠀⠁⠀⠈⠀⠁⠁⠈⠈⠀⠁⠁⠈⠀⠀⠁⠀⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁⠁⠈⠈⠀⠀⠁⠁⠀⠈⠀⠀⠁⠁⠈⠈⠀⠁⠁⠀⠈⠀⠁⠁⠀⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠈⠻⣷⣶⣤⡀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⣴⣾⣷⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⣠⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣼⣿⣿⣿⣄⠀⠀⠀⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣾⣿⣿⣿⢻⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
┌───────┐ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣀⣠⣼⣿⣿⣿⢟⣫⣿⡻⣿⣿⣿⣿⣷⣄⣀⣠⣴⣿⣿⣿⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠷⠛⣻⣿⣿⣽⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
│ S E O │ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⣀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⠿⠁⠀⠀⠀⠀⣷⡄⠀⠈⢷⣤⠀⠀⠀⠀⠈⠉⠋⠁⠀⠾⣿⣿⠈⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⣼
│       │ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠀⣴⣾⣿⣿⡁⢀⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠞⢉
│  C K  │ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣤⡴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣏⠺⠿
└───────┘ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣶⠶⠄⠀⠀⣼⣇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣾⣿⣿⣿⣯⣭⡽⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣧⣻⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣿⣿⢠⣤
⠀⠀⠀⠀⠀⠀⠐⢺⡒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣸⠏⠀⠀⠀⠀⢿⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢸⣿⣿⣏⡻⣿⣿⣿⢩⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠃⠀⠀⠰⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉
⠀⠀⠀⠀⠁⠀⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡳⡏⠀⠀⠀⣄⠀⠈⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢻⣿⣿⣿⣿⡿⣳⣾⣿⣿⡏⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡮⡏⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠘⠟
⠀⠀⠀⠀⠂⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⢠⣟⢀⣤⣾⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣥⣥⣶⣿⣿⣿⣿⡿⢸⣿⣿⡯⣿⣿⣛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣦⡀⢀⣠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣠⢤
⠀⠀⠀⠀⡁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⠀⠀⣀⣿⣿⣿⣿⠃⣿⣿⣶⣛⠯⢞⣻⣽⣿⣿⣿⣿⢛⣥⣶⣮⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢸⣿⣿⣆⢻⣿⣿⣏⣿⣿⣿⣿⣿⣿⢫⣵⣾⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠈⠀
⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⣋⣙⣷⣿⣿⣿⡿⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⣿⣿⣹⣿⣿⢁⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⡽⠋⠀⠀⠀⢿⣿⣿⣿⣦⣴⣿⡿⢿⣿⣿⣿⣇⢺⣿⣿⣿⡄⣿⣿⣿⣿⡿⣿⣵⠿⠁⠀⠀⠀⣀⣠⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢹⣾⣿⣿⣿⣿⣿⣾⣿⣾⠝⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣷⣍⣛⣵⣾⣿⡿⣹⣿⣾⣿⣿⣿⡿⠛⠉⠀⢠⣶⣿⣦⡀⠙⠿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣷⣾⣿⠟⣴⡿⣟⣻⠟⠀⠉⠁⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣟⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢣⣾⣿⣿⢿⣿⠃⠁⠈⠋⠁⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠻⣿⣿⣿⣿⡿⢛⣴⣿⢸⣿⣿⣿⠟⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣀⠀⣭⠛⢿⣿⣇⢸⣿⣿⢿⢘⢻⣭⣿⣿⣧⣿⠃⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡄⠀⠀⢠⣤⡀⣀⣠⣤
⠀⠀⠀⠀⠄⠀⢠⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣝⣻⣿⣿⣿⣿⣷⣮⣭⡍⣡⣴⣿⣿⡟⠸⣿⣿⡇⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⠿⣧⡿⢏⣠⡈⣿⠇⠸⣿⣿⠸⢐⡘⢿⣿⡷⡟⣿⣤⣠⣴⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣿⡇⠀⠀⠈⢻⣿⡛⠋⠁
⠀⠀⠀⠀⡀⠀⢸⡀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⠏⠀⠀⠹⣿⣇⠠⣿⣿⣿⣿⢛⣿⣿⢟⣥⣾⠟⣀⠚⠛⣷⠊⠀⠀⢿⡏⣰⠟⠙⡆⢙⣷⣧⣻⣿⠿⠋⠀⠀⠀⢻⣿⣿⢛⣭⣿⣿⣿⣿⣥⣿⣻⣿⣿⠟⠀⠀⠀⠀⠈⣿⡆⣤⣦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⣿⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣟⢸⣿⣿⠇⢴⣤⣼⠷⠄⣙⠠⣿⣿⣿⡿⡿⣩⣴⡿⢋⣴⠶⠿⣿⣿⡌⢷⡇⠀⢸⢠⠇⠀⡆⣿⢸⡟⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋⠁⠀⠉⠙⠿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢿⡇⠈⠁
⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⡆⢿⣿⣛⠈⣿⣿⠀⣤⠌⠉⠛⠾⠶⢶⠎⣿⣿⣷⣑⢉⠁⠐⣾⣿⠀⠀⣸⣿⣿⣿⡇⠸⠀⣮⠀⠈⣷⡏⣘⣦⡴⠂⠀⠀⠲⣶⡾⠿⠟⠟⠋⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠺⣷⡶⠶
⠀⠀⠀⠀⡀⠀⢰⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠶⠶⣦⣄⠀⠀⠀⠈⠁⠀⠀⠉⢻⣿⣄⠙⠿⣇⠈⠙⠂⢃⣾⣿⠋⠉⠳⣦⠀⣠⡹⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣟⣱⢟⡟⢸⠀⠟⠀⣸⡟⠐⠛⠋⠑⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀
⠀⠀⠀⠀⠂⠀⠜⠀⡦⠂⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣮⡙⠿⣄⠀⠀⠀⠀⠳⣶⣿⣷⣿⡀⣇⠘⠢⠐⣶⣶⣿⣿⣄⣀⣴⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣷⡿⡗⣚⣯⠶⢿⡇⢸⢀⣶⠿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣷⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠤⠠⠠⡄⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠀⢀⣀⣀⡀⢾⣿⣿⣿⣿⡿⣿⣛⣛⢿⣿⣿⣿⣦⢻⡄⠀⠀⠀⠀⠀⠉⠁⠀⠑⠹⣆⡆⢀⡹⣿⣯⣛⣿⣿⣯⣵⣿⣿⡇⠺⣿⠿⣿⣿⣿⣿⠿⠟⣫⣥⠆⣾⡇⡎⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣏⣻⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠅⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠛⠛⢷⣽⣿⣿⣖⣧⠀⠀⠀⣤⠀⠀⠀⠀⠐⠃⠘⢷⣀⣁⢻⣿⣿⡿⢿⣿⣿⣿⣿⣷⣦⣾⠾⣛⣋⣭⣴⣾⡿⢻⡁⣼⡏⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣾⣿⢻⣿⣿⣻⣀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⡀⠀⠀⠉⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⡤⢀⠀⠀⠀⠈⠹⣿⣿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠹⣿⣧⡀⠶⣶⣴⣶⣶⣶⣿⣿⠿⢿⣻⣯⣷⣾⠟⣵⣯⠋⠀⠀⠄⠚⠉⠙⣶⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠈⢳⡄⠀⣴⣿⣧⣀⣠⣴⣿⣿⣿⠿⣼⣿⣿⢿⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⠀⠤⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⢰⠎⠙⠆⠀⠀⠀⣿⣿⠟⠁⠀⠀⢀⣀⣀⡀⠰⠾⠦⠀⠀⠀⠀⠀⣠⡆⠉⠻⣿⣦⣀⠧⣾⣶⣾⣿⣿⣿⣿⣿⠿⣟⣵⣾⠏⠁⠀⠐⠀⠀⠀⠀⡰⢋⣤⣾⠿⣟⣯⣿⣾⣭⣻⣿⣶⣄⠀⠱⠀⠙⢿⣿⣿⣿⣿⣿⡟⣩⣾⡿⣿⣿⡿⠋⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣿⣿⣿⠀⠀⠀⢸⡀⠈⠀⠀⠀⣴⡟⠉⠀⠀⣀⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠈⠙⠿⣿⣦⣽⣿⠿⠿⠿⠿⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⢀⡤⢋⣼⡿⣻⣶⣿⣿⣿⣿⣿⣛⣄⢹⣋⣹⣆⠀⠀⣀⣾⣿⢻⠹⢍⡁⣾⣿⣿⠀⠉⠁⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⣿⣿⣿⣆⠀⠀⡄⠈⠑⠒⠒⠉⣠⣾⣿⣿⣿⣿⣿⢛⢻⣿⣿⣿⣿⣧⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⢀⣠⠖⠉⣼⣿⡟⣽⣿⠟⣽⣾⣿⡿⣿⣿⣷⣿⣿⣏⣻⣿⡿⢿⠟⢿⠚⠀⠙⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⣾⣿⡿⣻
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⢿⣿⣿⣿⣷⣾⣷⣄⡀⢀⣶⣞⣽⣿⣿⡽⠟⠉⠉⠉⠓⢿⣿⣿⣿⢿⡀⣀⣐⡃⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⢆⣾⣿⡿⣹⣿⣫⣾⢟⣵⣶⢾⣿⡿⣿⣿⣿⣿⣥⣤⣈⣉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣳⢿
⠀⠀⠀⠀⠀⠀⠤⢼⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣽⣿⣿⣿⣿⣷⣎⣉⣤⣿⢿⣻⣿⠏⠀⠀⠀⠀⠀⠀⢈⣿⡟⠻⣴⣿⣿⣿⣿⣦⡀⠀⠀⣴⠉⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⣠⠀⣡⡼⠋⣤⣾⢃⣾⣿⣿⢠⣿⣿⣿⣗⡿⢟⣵⢾⣿⣿⣿⣿⡿⢟⣛⣻⣭⣉⣙⠿⢷⣤⣄⣀⠀⠀⠀⠀⣴⣿⣅⠀⣀⣼⣿⣣⣾
⠀⠀⠀⠀⡂⠀⠂⠘⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⢋⣏⢻⣿⣿⣷⣾⡉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣾⣿⡿⡿⡿⠟⢛⣋⣤⠶⡆⢧⠀⠀⠀⢀⠄⡀⣿⣿⣷⣀⡠⢒⣡⠞⠁⢠⣾⣿⢃⣾⣿⣿⣿⠚⣾⣿⠁⣾⣾⢟⣽⣿⠿⢛⣉⣴⣾⣿⠿⠿⠛⠙⠛⠻⢷⣮⣛⠿⢿⣶⣤⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠄⠀⠤⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠻⢿⣿⠿⠟⠋⠀⠀⠀⠤⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⣩⣴⣿⣿⣿⣇⣼⡃⠈⠳⣀⠸⢿⣿⡴⣿⡿⠟⣩⠵⠋⠁⠀⠀⠸⣿⡏⣾⣿⣿⡟⣿⢰⣿⢃⣾⡟⣡⣿⢟⣵⣾⠿⠿⠛⣩⣵⣶⢤⣄⠀⠀⠀⠀⠈⠻⣿⣾⣍⣙⠻⢷⣮⡙⢿⣿⣿⡿⠟⣰
⠀⠀⠀⠀⠅⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⣱⣿⢿⣿⣿⣳⣿⣸⡆⠠⣤⣈⡓⠪⢅⣴⠠⠞⠋⠀⠀⣠⣤⣶⣾⣇⣿⡇⣿⣿⣿⡟⢻⢸⡏⣼⠟⣰⣿⣿⣿⠋⢀⣠⣴⣿⣿⣿⠋⠀⠈⢻⣦⣀⠀⢀⣀⣬⣌⣉⠋⢤⡀⠙⢳⡌⠻⣍⡏⡉⠙
⠀⠀⠀⠀⡂⠀⠈⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡞⣰⣿⢧⡿⣻⣿⣿⠿⡟⢀⣼⣿⡿⠿⢊⠄⠀⢰⡀⢤⣶⣿⣿⣿⡿⢣⣾⡟⣇⣿⣿⢺⣿⠘⠀⠃⡿⢠⣿⣟⣿⠇⣴⣿⣿⡿⣿⣿⠋⠀⠀⠀⠀⢠⢉⡛⢛⣥⣶⡞⠋⢷⡄⠸⣧⡀⠙⠆⠈⠃⠂⠀
⠀⠀⠀⠀⠀⠀⠀⠓⢢⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢿⣿⣿⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⢀⣴⣿⣿⡷⠀⠀⠀⠀⠀⢾⣿⣿⣿⣟⠇⣿⣿⡿⢰⣿⣿⢣⣿⣻⡆⢹⣿⠢⣪⡎⠠⢀⡎⣇⡸⢻⣿⣿⣯⣴⡿⠿⣯⣿⡝⣿⣎⠻⡇⠀⠀⠁⣼⡿⢣⣿⠸⣿⣿⣿⡆⣿⡇⠀⠀⠀⠀⣀⣾⣿⣿⣾⣿⣿⠔⠂⠀⢻⣄⠙⠻⢷⣤⣤⣶⣦⣤
⠀⠀⠀⠀⡁⠀⠀⠒⠊⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠆⠀⠀⠀⠈⠻⢿⣿⢧⡆⣿⣧⢡⣿⡟⣱⣟⣿⡟⢁⣼⣿⣧⣌⡀⣤⣾⣧⣉⣤⣾⣿⣿⣟⢻⣧⡀⣿⣻⣷⡞⢿⣦⠘⠄⠀⠠⡟⣠⣿⢃⣴⣿⣿⣿⣿⣿⠁⡄⠀⠀⢠⠉⣿⣿⣏⣹⣿⠃⠀⠀⠀⠄⡉⠳⣤⣤⣤⣉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣯⣽⡅⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢸⣇⠁⠇⣿⢿⠟⢁⣿⠋⠀⢺⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⡿⢋⣴⠈⣿⣿⢿⣷⡹⣿⣆⠙⡙⠂⠀⣀⣴⣿⢃⣾⣿⣿⣿⣿⣿⡟⠙⠁⠀⡀⠈⢉⣿⣿⣿⣽⠃⠀⠀⠀⠀⠀⣽⠇⣰⣶⣽⠙⢧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⢄⡀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣶⣤⣀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣶⡇⠻⠀⣼⡿⢆⣴⡟⠁⠀⣾⣧⡉⢿⣿⠋⢸⣿⣿⣿⠟⢫⣍⣼⢿⣿⠀⠈⣿⣯⣿⣷⡙⣿⠖⣠⣴⣿⡿⠛⠃⣾⣿⣿⣿⣿⣿⣿⠆⠾⠀⢰⠇⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠶⡭⢿⣿⣿⠏⠀⠈⡇⠀
⠀⠀⠀⠀⠀⠀⢸⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣷⡚⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⡟⣠⣾⡿⣿⣿⠏⠀⢰⡄⣿⣿⣷⠦⠀⠀⠈⠛⢫⣴⣾⣽⣟⡷⡽⠃⣴⠀⠸⣿⣿⡟⢿⡏⣴⡟⠉⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣷⠄⠀⡜⢀⣿⣿⣿⣿⣟⠀⠀⠀⢸⠀⠁⣼⣗⣷⡾⠃⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⣀⠼⠏⠁⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠘⠿⠿⠇⠀⠋⣿⣿⣧⠀⠀⠀⠀⢠⣶⣶⣶⣄⣤⣾⣿⣿⣿⠿⡽⢏⣼⣿⢿⣹⡿⠃⠀⠀⠘⠃⠈⠀⠀⢀⣀⠀⣀⡀⠀⠀⠈⠉⠀⠀⠀⣲⠀⠀⠀⢹⣿⣿⣮⢰⣿⠀⠀⢀⡤⢀⣿⣿⣿⣯⣝⣛⠛⢰⡁⠀⣰⠁⣸⣿⣿⣿⣿⠋⠀⠀⠀⡆⢀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣶⣦⣴⡿⣟⣩⣿⣾⣿⣿⣿⣿⣿⠿⣡⠿⢟⣥⣾⠟⠁⠀⠀⣀⡄⢀⣤⣶⣿⢿⠃⢈⡹⠿⣦⣤⣀⡀⠀⠀⠈⠈⠀⠀⠀⠈⢿⣿⡿⠘⣛⣤⣴⠿⡁⣸⣿⠋⠙⠉⠉⠛⠿⠐⠁⢀⠏⢰⣿⣿⣿⣿⡏⠀⠀⠀⡼⠁⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣼⣿⣿⣿⣷⣼⣾⠯⢟⡥⣯⠿⣀⣼⡿⠋⠀⠀⠀⠸⠿⢡⠿⠟⠉⠁⠚⠀⠈⠋⠀⠙⠿⠿⣽⣷⣶⡆⢠⠤⠈⠁⠀⠘⣫⣴⣾⣿⣿⠿⠷⠄⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⣿⣿⣿⣾⡿⠀⠀⠀⣸⠇⣸⣿⣿⣿⠋⠀⠀⠠⠁⠀⠌⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠷⣶⣆⡹⣿⣿⣿⣦⡀⠀⠀⢀⣠⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⡷⡾⠟⢛⣉⣤⣶⣯⣠⣤⡾⠟⠉⠀⠀⠀⠀⡤⡆⠠⠀⣤⣶⣿⣿⣿⣤⣾⣿⣷⣦⡀⠀⠀⠀⠀⠈⠀⠄⠐⠁⠀⣰⣿⡟⣿⣿⣿⣷⣶⣦⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠙⠛⠻⣧⠀⠀⢠⡟⢠⣾⣿⣿⠉⠀⠀⢀⡎⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⣿⠟⣻⣿⣿⣭⡏⠁⢻⢽⣿⣿⣟⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢋⣨⣴⣾⣿⣿⣿⢻⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣼⣿⣿⣿⠿⠀⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡏⣽⡟⠉⠞⢿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢸⣿⠿⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣾⣿⠉⡖⣿⡿⠋⠉⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⠗⣩⣶⣿⣿⣿⣿⣿⣿⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠹⠿⠿⠛⠉⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢷⡏⠀⠀⠀⠀⠉⠹⡹⠌⠛⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢸⣿⣿⠠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⢿⢼⠼⢛⣩⣭⣤⣶⣾⣿⣿⢟⣯⡙⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⠇⣠⣴⣶⣾⡿⢂⣴⣶⣤⣤⡀⠀⠀⠀⠀⢼⠇⠀⠀⣀⣶⠇⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⡾⣿⣷⣮⣻⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⢸⣿⣿⣴⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⡿⢻⢎⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠿⠋⣁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣸⣿⠿⠟⠉⠀⠾⣿⣿⣿⣿⣿⡷⢷⡀⣠⣀⡀⠀⠀⠿⠁⠀⠀⠀⠀⠀⢀⣀⠐⠀⠀⠀⠸⠉⣷⣿⡿⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⢀⠼⠁⠀
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