Airspace Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⠿⠿⠿⠿⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⢀⣤⣶⠿⢿⣿⠛⠉⠙⠛⠛⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⣠⣾⠟⠉⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣼⠟⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⣿⡉⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠹⣷⣄⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡄⠈⠻⣷⣄⠀⠀⠙⢷⣄⡀⠀⢀⣀⣠⣤⣤⣶⣶⠾⠿⠟⠛⠋⠻⣷
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠙⢿⣿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⣀⣤⣶⠿
⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠙⢷⣦⡀⠀⠀⠀⣀⣤⣶⣿⠛⠉
⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⢷⣄⠀⠀⠙⢿⣦⣴⠿⠛⢿⣮⣿⡇
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠗⠀⣼⡏⠀⠀⠻⣷⣄⠀⠀⠻⣿⣄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠉⠻⢷⣶⣤⣄⣀⣀⣀⣠⣤⣴⡾⠟⠋⠀⠀⢀⣿⠃⠀⠀⠀⢨⣿⣷⣄⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢠⣿⠃⠈⠻⣷⣄⠀⠈⠻⣷⣤⣤⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣿⡻⣷⡀⠀⠈⠻⣷⣄⠀⢈⣩⣭⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣰⡿⠛⠿⠟⠀⠀⠀⠀⠈⣿⠃⣾⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠃

⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⢫⣿⣿⣷⠀⢀⣾⣿⠿⠿⠿⠿⠛⠛⠛⣿⣿⣿⡄⠀⣾⣿⣿⣄⠙⢿⣿⣿⢿⣿⣿⣿⡆
⠀⣿⣿⣿⡍⠉⠉⠀⠀⢰⣿⣿⣿⣿⡇⢸⣿⣿⣶⣦⡀⠀⠀⠀⢸⣿⣿⣿⣇⢸⣿⣿⣿⣿⣷⡄⠉⢠⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣧⠄⠀⠀⠀⣾⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣤⣀⠀⢸⣿⣿⣿⡇⠸⣿⣿⣟⠻⠏⣠⣶⣄⠙⢿⣿⣿⡿
⠀⠘⢿⣿⣿⣶⣧⣤⣄⣿⣿⣿⣿⠟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⡿⠁⠀⠻⣿⣿⣷⣌⣿⣿⣿⣷⣤⣿⡿⠃
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢀⣉⣉⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⢋⣽⣿⣧⠀⢀⣾⣿⣿⡛⠿⡿⣿⣿⢿⣿⣿⣷⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⣩⣿⣿⡄
⠀⣿⣿⣿⡛⠛⠛⠛⠉⣰⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣶⣄⠈⠡⣾⣿⣿⣿⡇⢰⣿⣿⡟⠛⠛⠛⠛⢡⣾⣿⣿⣿⣿
⠀⣿⣿⣿⣛⠃⠀⠀⢸⣿⣿⣿⣿⣿⠇⢸⣿⣿⣟⡛⠟⣵⣷⡄⢸⣿⣿⣿⡇⠸⣿⣿⡛⠓⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣷⣀⣷⡈⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣦⣿⣿⣿⣦⣿⣿⡿⠁⠀⢻⣿⣿⣧⡄⣿⡄⢹⣿⣿⣿⣿⠃
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⣉⣙⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⣉⠉
⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢠⣾⣿⣟⠻⣿⣿⣿⡿⠛⣿⣿⣦⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⣩⣿⣷⡀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⣾⣿⣿⣿⣷⣄⠙⠋⣠⣾⣿⣿⣿⡆⢸⣿⣿⡟⠛⠿⠍⠉⢁⣾⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣷
⠀⣿⣿⣿⣟⠻⠋⣠⣄⠙⣿⣿⣿⣿⡇⢸⣿⣿⡿⠆⠀⠀⠀⣤⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⡛⣁⣀⣀⣀⣘⢻⣿⣿
⠀⠸⣿⣿⣿⣷⣌⣿⣿⣷⣌⣿⣿⡿⠀⠈⢿⣿⣿⣷⣀⣦⡀⣻⣿⣿⣿⣿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣤⣄⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣤⣄⣠⡴⠀⠀⠀⢸⣿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠙⣏⠀⠀⠀⠀⠀⣯⣤⣤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠊⠿⠿⣶⣶⣶⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠏⣙⡛⣁⣬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣤⣀⣀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣯⣄⡀⠀⠀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠙⠛⠿⣿⡿⠋⠀⠀⠀⠀⠀⣿⢹⢸⡇⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⢸⣾⣾⣷⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣠⣤⣤⠴⣂⣀⣀⣀⣤⣴⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠈⠛⠛⢻⣿⣿⣿⡟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⠏⣀⣉⠉⣤⣤⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢻⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣏⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣟⣿⣿⣿⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠠⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠻⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡃⢀⠛⣈⠳⡟⢋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠰⡀⠛⢀⡇⣹⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣰⠒⠀⠀⡼⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠶⢃⠀⠀⠂⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠁⠀⢀⡼⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢬⣁⣛⡥⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠉

⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿
⠀⠈⠉⠙⠿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣴⣿⣿⣿⠁⣠⣴⡖⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⡻⣿⣿⣷⣦⣄⡀⢀⣤⡄⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡤⠄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⢿⣿⠂⠀⠀⢀⣠⣤⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠹⣧⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠙⠶⣤⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠴⠿⠟⠛⢻⣿⡏⠉⠉
⠀⠀⠀⠙⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⢝⣿⣿⣿⣿⣷⣶⡶⠖⠀⠀⠀⢠⣷⣾⣿⣷⣾⣿⣷⣄⠀⠀⠀⠀⠀⢳⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠈⣿⣿⠛⣿⡯⠉⠀⠀⠀⠶⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠖⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⢿⣿⡙⠛⠂⠐⠚⠋⠉⠻⢿⣿⣷⣶⣤⣤⣾⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠐⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡤⠤⠶⠚⠛⠉⠹⢿⠆⠀⠉⠉⠛⠛
⠀⠀⠀⠘⢿⣷⣤⣀⣀⣠⣤⣤⣤⣼⣿⣷⣤⣤⣴⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠜⠛⠻⠿⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⢰⣄⡉⣲⣶⣦⣤⣤⣤⣴⣶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⣭⣭⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣭⣭⣭⣭⣭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣤⣤⣬⢻⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣛⣛⣿⣿⣿⣿⣿⣿⢟⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⢟⡱⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⡿⢛⣭⣽⣿⠟⣵⣿⣿⡜⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣭⣭⣭⣭⣭⡍⣭⣾⣿⢱⣿⣿⢟⣵⣷⢸⣿⣿⡷⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⡱⠶⣹⣿⣿⣿⡿⣸⣿⣿⡸⢗⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠹⣿⣿⣿⣮⣭⣫⣭⣾⣿⣿⡿⣱⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣬⡻⢿⣿⣿⡿⠿⣿⠿⣋⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣾⣭⣜⢟⣴⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠠⣶⣶⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣧⡀⠀⠀⠀⠈⠙⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠋⠉⠀⢿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠀⠀⠀⠀⠙⢄⠘⡃⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡾⢿⠃⠀⠀⠀⢀⠠⠐⢉⣦⡈⠒⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠈⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⠿⠿⠿⠇⠿⠿⠿⣿⡿⠿⠿⠇⠿⠿⠿⢿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠾⠿⠶⠦⠤⠤⠤⠤⠶⠿⠷⠦⠤⠤⠤⠤⠴⠾⠿⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠛⠛⢿⡿⠛⠛⠛⠀⠀⠀⠀⠚⠛⠛⢿⡿⠛⠛⠃⢀⣀
⠀⠀⠀⠐⠻⠿⠿⠿⣿⠿⠿⠿⠓⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠚⠿⠿⠿⣿⠿⠿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⣇⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣸⠇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣠⣤⠼⣿⣿⣿⣿⣿⣿⣿⣿⠧⣤⣄⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠁⠀⠀⠀⠀⣠⠴⣾⣷⠦⣄⡀⠀⠀⠀⠈⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣧⣼⣃⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⣀⣀⣄⢸⢀⣄⣀⡀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢠⡞⠛⢶⠚⠋⠉⠀⠀⢸⠀⠀⠈⠉⠛⠲⣤⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡼⠷⠶⠟⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠀⠀⠙⠳⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢀⣤⠶⠚⠃⢸⠀⠛⣟⠉⢹⡆⠀⠀⠀⠘⢷⡀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠀⠀⠙⠓⠛⠳⣄⠀⢀⣠⠾⢿⡉⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠀⠀⢰⠏⠙⣦⢸⠀⠶⣄⠀⢀⣠⡼⣿⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⡾⠀⠀⠀⠘⠳⠞⠃⢸⠀⣀⡼⢿⠋⠁⠀⢸⡆⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠒⠶⠶⠶⠖⠲⠶⠶⠶⠖⠲⠶⠶⢾⠟⠻⠶⠖⠶⠶⠶⠶⠲⠶⠶⠶⠶⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠘⣧⡀⠀⢸⠀⠀⣠⡟⠀⠀⠀⣸⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢷⡀⠀⠀⠈⠙⠒⢸⠐⠚⠉⠀⠀⠀⣰⠏⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠹⣆⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⡾⠁⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠉⠓⠶⠦⢸⠠⠶⠖⠋⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠛⠶⣤⣄⣀⡀⢸⠀⣀⣀⣤⠴⠞⠋⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠉⠉⢸⠀⠉⠁⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠈⠻⣧⡀⢸⣗⠀⣸⡇⢠⣿⠙⠋⣽⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠒⠚⠛⠓⠘⠛⠛⠂⠙⠛⠛⠐⠛⠛⠀⠐⠛⠛⠛⠛



⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⣼
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠿⠤⡄⠀⠀⠀⠀⠀⣀⡀⠠⠀⢀⣠⠴⠞⠋⠉⠁⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠰⠒⠂
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⢠⡇⠠⠤⠤⠤⠬⠤⠬⢰⠛⠛⠀⠀⠐⢂⣀⣀⣀⣀⣀⣈⣉⣁⡒⠒⠂⠒⠈⠀⠁⠀⣀⣀⠀⠀⠀⣐⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣗⡤⠙⠒⠒⠒⠒⠒⠒⠒⠒⢿⡟⣉⣙⣿⠟⢉⡙⢛⢹⠋⡁⢻⠘⠃⠉⠀⠀⠀⠀⠀⠀⠉⠁⢀⣀⠀⠈
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣯⣙⢏⠀⠀⣾⡇⠀⣾⣿⠀⢰⣶⣾⠀⠋⠁⡃⢸⠏⠀⢨⠨⠀⠁⢻⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠇⢸⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠋⠀⠀⠙⠃⠀⠙⠋⠀⠈⠉⢹⠀⠀⠀⠇⢰⠀⠀⠸⢸⠀⠀⢸⠀⠀⠉⠀⠀⠈⠀⠀⢈⡉⡀⢈⣉⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⣦⠀⠀⣤⡄⠀⣦⡄⠀⢹⣟⠈⠀⠀⣾⡇⠈⠀⣀⡀⠸⢲⣶⠈⠀⠂⣿⡇⠈⢸⣏⠀⢹⣿⠁⠀⣿⡇⢁⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣿⣶⣶⣿⣷⣶⣿⣿⣻⣿⣟⠁⠀⠀⠛⢙⢓⣶⠛⣧⠾⢟⡛⣛⣿⣿⣿⣗⢲⣛⣻⣓⣟⢛⡶⠒⣿⣧⡄
⠀⠀⠀⠀⠈⠉⠠⣋⣉⣉⣉⡁⣉⠉⠉⠉⢉⠈⠁⣀⠀⠉⠁⠀⠀⢈⣉⣈⣉⣉⣉⣋⣉⣉⣉⣀⣈⣀⣉⣉⣁⣡⣍⣨⣍⣉⢥⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠚⠛⠉⠉⠉⠉⠉⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠟⠻⠮⠄⠓⠰⢄⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⠶⡉⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢧⡤⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠈⠀⠈⣤⣾⢻⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈⠀⠀⠀⣿⣤⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠌⠉⠀⠀⠛⣇⣸⠏⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠐⠆⠀⠀⠐⢃⣨⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠀⠀⠀⢠⣼⣏⣴⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⠀⠀⠀⠀⣡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣄⣀⠀⠀⠹⠍⣛⣿⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢀⣀⠀⠔⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠈⠛⠶⠧⣀⠮⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣀⣤⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠶⠾⠿⢷⣴⠃⢈⣄⣠⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢡⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡀⡀⠀⣀⣤⠴⠟⣷⠸⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠆⠈⡁⣻⡾⢋⠀⠀⢀⠈⢿⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⠀⠀⠘⣧⢀⡄⠀⢿⡀⢻⡀⢀⣿⣧⣌⢧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢛⣿⣦⠄⠀⠀⠀⣿⣿⠃⠀⠈⢛⡙⢷⡐⢟⣾⣧⡈⠂⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠻⠿⠆⠀⠀⠴⠤⠎⠹⠦⠽⠗⠋⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠉⠈⠀⠉⠈⠁⠉⠉⠈⠀⠉⠉⠁⠀⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠈⠁⠈⠉⠁⠉⠉⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠀⠆⠀⠆⠐⠀⠒⠂⠀⠚⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⠄⠀⠴⡮⢀⡀⠀⠀⠀⠀⠈⠁
⠠⢴⣆⠀⠀⠀⠐⠀⠀⠀⠀⢰⡄⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠂⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⣸⠀⠀⠀⠀⠸⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎
⠀⠀⠠⠀⠁⠀⠉⠋⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠐⠀⠁
⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⡀⠀⠀⠐⠀⠂
⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣥⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢦⡞⣏⠰⣶⣶⣮⣴⢞⣤⣤⠰⣦⣀⠐⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⣾⣿⠖⢤⢀⣀⡨⣀⣀⣦⣗⢰
⢐⡻⡄⠋⢙⠿⣟⣡⠈⢨⢸⡗⣫⠩⡟⠊⡩⡼⢀⣀⣀⢀⣄⠀⣀⣠⣠⠟⣤⢝⡎⢼⠨⣏⣮⠹⣿⠫⡐⢓⢻⠤⢳⠾⣇
⠈⠇⢑⠦⡋⠏⢹⡼⢎⣽⢨⠶⣨⡷⠚⣶⢑⣯⣘⣿⠹⣏⠋⣿⣿⢿⣟⣿⢍⣜⣄⡼⡔⠿⢆⠓⢿⡧⠙⢠⠙⡏⣸⢳⡟
⠐⠀⠾⡖⣩⠓⢞⠋⠂⠶⢊⢌⡽⠋⢔⣠⠋⡐⣬⡱⢏⢀⣀⠈⠗⠞⠣⠊⢄⣿⡸⢆⡲⢌⣆⠛⠶⢦⢘⠀⢅⠱⢗⠊⢍
⠀⠀⣰⠀⡌⡙⠂⣀⠀⠀⠀⠀⠀⠋⠀⠓⠀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠀⣆⠁⢲⠌⢺⡷⠄⠈⠀⠁⠙⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠄⠀⠀⠀⢀⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡀⠀⣻⣤⡲⢀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠠⡼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⡀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣌⡆⠀⠀⢠⠟⠞⢃⡾⣹⢧⡹⡏⡜⠛⠁⠀⠀⠀⠀⣀⡤⠒⠋⠘⠁⠠⣶⡀⠀⢸⡞⢻⣀⡀⠀⠀⢠⣷⡄⠀⠀⠀⡩⡀⣥⣾⡀⣠⣿⣄⣾
⢀⣄⢠⣿⠀⣰⡰⡚⠛⣸⣦⣴⣃⣶⣤⣴⣏⠀⠙⠿⠀⠘⠁⢀⡠⢔⣾⡏⠀⠀⠀⠀⠀⢴⣿⣆⠀⠼⣇⣾⡍⠁⠀⠀⣦⢷⣰⢀⠀⢰⡟⢻⣠⡏⢳⣿⣸⣇⢿
⢸⣿⣫⣭⣷⣿⢉⣿⢉⣻⡿⢩⠇⢸⣧⠁⣾⡆⣸⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⣎⣻⢸⡤⣀⣽⣏⠹⠶⠀⣠⣤⣿⡉⢼⠁⠼⣷⣚⣡⣇⣸⣧⣹⠙⡛
⢸⣿⣽⡉⣿⠟⠁⣈⢈⡉⢹⣿⣠⣾⣧⡖⢁⣿⣏⠈⡝⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣷⢿⣌⣥⠈⠾⡟⠳⠇⠀⢈⣡⣯⡛⠆⣉⠰⣾⡿⢋⣍⠁⢹⡻⡦⣿
⢸⣯⣴⣿⡝⠛⠋⠈⣿⡛⠛⣿⣿⣿⡿⣥⣿⢿⣿⣆⣻⣿⣾⣧⣦⣷⣶⣴⣄⣤⣄⡤⠚⣇⣼⣮⣙⣥⡀⡇⠀⣀⣴⡌⢀⣧⢥⠔⢣⡴⢤⣿⣯⣿⣷⣼⣿⣁⣿
⢸⣿⣿⣿⠧⠾⣀⣼⠞⢷⣶⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣯⣛⣧⣿⣿⠖⣉⣿⠈⣧⡙⢯⢻⣿⣟⡣⠾⣙⣿⠅⠞⡔⠢⠗⢤⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡄⣠⠏⠈⣄⠀⢳⣀⣻⡇⠿⠛⢿⠈⠟⠻⡿⣿⠟⠙⠛⠙⠿⠿⣿⣿⡿⢛⣡⣻⠆⠁⣹⡚⠻⡿⢓⠦⠓⠆⣹⢇⡠⠃⡤⠧⡘⠾⢿⡿⡿⡻⠻⠻⣿
⠈⠋⠻⢯⣀⡴⠦⠤⠊⠓⠛⢿⣄⣀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⣡⠀⣥⠀⢈⣍⠠⡬⠃⠈⠋⠸⡀⠤⣹⠇⡴⢳⠀⡉⠙⡈⢃⠁⠀⠀⠀⠁
⠀⠀⠦⠴⠁⠀⠦⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠐⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠾⠉⠋⠁⠈⠻⠇
⠀⠀⠀⠀⢤⠀⠐⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠤
⠀⠀⠠⠄⠀⠐⠀⠀⠂⠁⠀⠀⠂
⠀⠀⠀⠀⠸⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠂⠀⣰⡀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⠏
⠀⠀⠠⠀⠀⠙⠓⠚⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⢀⡠⠀⠀⠀⠀⠂⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠤⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠐⠋⠀⠐⣇⠁
⠀⠀⠀⠀⠀⢀⣮⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠲⠆⠀⠀⠀⠀⢀⠀⡃⢸⡿⢧⡆
⠀⠀⠄⠂⠀⠉⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣹
⠀⠀⢀⠀⠀⠀⠀⠀⠀⢐⡀⠀⣱⠄⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣈⣁⣠⠴⠖⠒⠒⠤⠠⠤⠖⠒⠒⠐⢤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⣾⣶⣦⣶⡖⠚⠛⠛⠛⢻⣿⣿⣶⣦⣶⣤⣴⣶⣦⣴⣶⣶⣶⣾⡿⠿⠿⠿⠿⠿⢿⣾⣧⣴⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⡀⢠⠄⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠈⡀⠀⠀⢀⡀⢹⢦⣘⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠄⠄⠀⠀⡁⠤⠘⣁⠠⠐⠂⢢⣁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠰⠤⠐⠞⢀⣀⡀⠀⠀⠠⠉⠱⠒
⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠶⠵⢆⡠⠀⠠⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⡠⠐⠀⠀⠀⠀⠒⢄⠀⠀⠀⠐⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠴⠀⠔⢄⡀⠀⡠⢁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡀⠘⠀⢰⡆⠀⠰⡦⢠⠁⢘⠅⠕⣨⣀⠀⢀⡄⠀⡇
⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⡀⠀⢨⠀⠂⠀⣤⠀⠀⣀⠁⠁⠀⢀⡔⠁⠈⠑⢀
⠀⠀⠀⣠⠀⠀⠪⡜⠒⠒⡄⠀⠀⠀⠀⠈⠆⠀⠀⠀⠤⠈⠃⠀⡄⡠⢚⠀⠀⠀⡀⣶⢓
⠀⠀⠀⠁⠀⠀⠁⠉⣉⣩⢇⡀⠀⠀⠀⠀⠀⡀⠆⠤⠌⢠⠄⠊⢰⠀⢐⠦⢈⠀⠈⠃
⠀⠀⠀⠀⠀⠄⡀⠤⢤⣤⠺⠂⠀⠀⠀⠀⡀⡘⠉⠁⠐⠉⢄⣧⢀⠓⠂⠉⠁⠒⢘⠀⠀⡄⡀⡌
⠀⠀⠠⡏⠈⠁⢀⠀⠮⠘⠈⢀⠀⠀⠀⠀⠳⡃⠠⠬⠉⠐⠀⢙⠃⠀⠨⠀⣍⡍⢩⠑⠉⠉⠁
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⣴⠊⠁⠀⠀⠀⠀⠑⡒⢒⠒⠊⠀⢠⠀⠀⠀⠐⠈⣣⣼
⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⣃⢇⠀⠀⠢⢀⠀⠀⠀⠉⡩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠹⣵⠋⠳⢴⣖⢦⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⠦⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠁⣟⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣠⢤⠀⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⢆⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠳⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠤⡄⠀⠀⠀⠝⠃
⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⡇⠉⡵⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠼⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⡋⠁⣀⢀⢀⠀⠀⢠⠇⠀⡇⠀⢀⣀⡠⠔⠐⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⢧⡀⠀⠠
⠀⠐⠀⠀⠀⠀⠉⠛⢹⣸⣠⠃⢀⠞⠀⠀⠉⠉⣉⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⡞⡳⠋⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⢠⠒⢢⣀⡄⢠⠤⠼⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡙⠁⠀⠀⢰⢇⡠⣢⠖⢠⠤⠤⣄⠀⠀⣇⠀⠀⠈⠁⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣣⠞⣵⡵⢦⡀⣳⠀⠈⠓⠂⠀⠀⠤⠦⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡡⠚⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⣱⣄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⠒⠂⠒⠒⡖⠒⠲⠶⣶⠶⠶⠤⠀⠀⣀⣤⡴⣶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋⠁⠀⠀⠀⠀⠀⢦⣄⠀⠀⢀⡀⠉⠓⠋⠀⠀⠀⠛⠟⠀⠀⠴⣦⠀⠁
⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⣠⠞⠀⢀⠠⣤⣄⣀⣠⣄⡏⣉⣶⠀⠀⠀⠀⣀⣁⣀⠀⠀⠀⠨⠁⠀⣀⠁⢠⡀
⠀⠀⢀⡴⠋⠀⡀⠀⠀⣀⡤⣄⠀⢀⡾⠗⣠⣤⢭⢥⣼⡟⢫⣤⣤⠀⣉⢸⣤⠽⢯⣽⣷⠆⢀⣤⣞⢉⣬⣜⣤
⠀⣠⠟⠁⠀⠠⢝⠀⢸⣿⠺⡥⣴⡈⢉⢁⣱⠾⠛⠢⠚⠛⣒⠶⠶⠟⠀⠀⠈⠉⢑⠂⣤⣶⣿⢛⣵⣿⠟⢘⡿⢷⡄
⡠⠇⠀⠀⠀⣴⠋⠀⠀⣀⣽⣿⠣⠈⠀⣽⣷⠾⠓⠀⠀⠀⡉⢀⣰⠀⠄⠀⢀⣠⣀⠀⠩⠭⡡⣾⣾⡁⠀⠻⠷⣬⣷
⠷⣄⣀⠀⢸⢿⢳⣄⢀⣦⣹⠟⠀⠀⢀⣠⣤⠤⣤⡤⠤⢤⣤⣈⡙⠿⣷⣀⣜⠛⠋⠁⣀⣜⣻⣭⣟⣁⠀⢤⣥⣼⠻⠗⡀
⡇⢠⣿⣵⣾⣿⠿⣷⡿⠋⠁⠀⢠⡾⠋⠁⠀⠀⠀⠀⢀⣀⠀⠉⠙⠳⣄⠈⠙⢧⣦⡘⠊⢻⣟⠛⠓⠈⠀⠈⠉⠉⣀⡾⡇
⡇⠨⣹⡄⢠⣱⠊⠉⠁⠀⠀⠀⠈⢷⣄⠀⠀⢠⡴⠛⠛⠛⢛⠓⣦⠄⠙⢷⡀⠀⠻⠋⠐⣯⣩⡿⠇⠀⢀⣴⣾⡛⣿⠿⠃
⢙⠀⠀⡞⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣝⠂⠐⢣⡞⡅⠀⠀⢀⠸⢻⣷⣆⣸⠇⠀⠀⣿⢦⢾⣿⠀⠀⢠⣼⢟⣟⠟⣁
⠘⡆⠉⠂⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡆⠀⠞⠀⠙⣓⢒⣉⣠⣄⣹⣿⣿⡆⠀⢈⣤⣍⠂⠁⠀⠠⣀⣵⣾⢁⠎⠘
⠀⠀⠀⣀⣀⡂⡀⠺⠖⠠⠄⠀⠈⠀⣩⣽⣦⠀⠰⣾⡿⣯⡉⠁⠈⠙⢷⣟⠛⠿⠛⢋⡹⣶⣠⣾⠞⣁⠌⣽⣿⢷⣶⡆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠐⠾⣿⡿⠋⠀⣶⠻⣷⠸⠇⠀⠀⠀⣸⠏⣀⡶⣶⣿⣿⣿⣇⠀⠈⠛⠁⠀⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠳⡟⣦⠀⠀⠀⠀⠀⠀⣿⣼⢫⡇⠈⢁⠹⣿⣿⣷⡜⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⢻⣇⠀⠐⠁⢸⣿⣌⣹⣧⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠆⠀⠀⠀⠀⠀⣿⠀⣩⠿⠀⠀⠈⠛⢻⣿⣻⣦⡀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⢱⣶⡄⠀⠀⣿⠉⡄⠀⠀⠀⠀⠀⠘⠻⠿⠄⠿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⡇⠀⠀⣿⢸⡃⠀⠀⠐⠃⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡇⠀⠀⢻⡈⠁⠀⣠⠄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⣼⠇⠀⠀⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⡿⠞⢂⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢸⣷⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠈⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡀⠀⢁⠆⠀⠀⠁⠆⠀⠀⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠤⠀⠀⠀⠀⠀⠀⠀⠀⢑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⡠
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⡆⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠌⠢⡀⠀⠈⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⡏⠀⠀⡠⠂⠀⠀⠀⠈⢆⠀⢀⡀⡨⠓⠀⠀⢀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆⠀⡀⠀⠀⢠⣮⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣰⠀⠀⠀⠀⠈⠸⡁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠈⠉⠀⠸⠀⠠⠀⠀⡀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠠⠀⣀⠀⠀⠰⠀⡠⠀⣀⡠⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⢱⡁⠉⠀⢀⣀⡀⠄⠁⠀⣠⠂⠀⠀⡀⢠⠀⠀⡈⢁⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠠⠆⠀⠀⠀⡀⠀⢀⠀⢅⠊⠓⠤⣄⣀⣀⡠⠤⢚⠀⠠⠒⠉⠁⣈⢀⠀⠘⢇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠁⠈⠐⠐⠀⠉⢄⡰⠁⠀⠀⠑⠂⠤⠀⠤⠤⠵⠒⣉⠒⠤⠤⠤⠚⢉⠀⠄⠀⠀⠉⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡣⣀⠐⠁⠎⠐⢀⠂⠈⠀⠀⢠⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠊⡑⠒⠒⠒⠚⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠧⢄⡠⠤⠤⢀⣰⡦⢄⣰⡖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠔⠁⠀⠀⠠⠤⠤⠀⠢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢠⡄⠈⡀⠀⠀⠀⢀⠄⠒⠔⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢀⢶⣶⣶⣶⡖⢁⣾⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠉⠁⠀⠀⠀⠑⠒⢦⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠱⡤⠒⠄⠴⣱⠞⠁⠀⠀⠀⠀⠀⠀⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡞⠁⠀⠀⠀⢠⡀⠀⠂⢠⠀⡀⠈⠑⠒⠢⢄⠠⠤⢤⠔⠚⠃
⠀⠀⠀⢀⣤⣴⣎⠉⢳⡞⣦⡀⠀⠀⠀⠈⠀⣜⠀⠀⠀⠜⠁⠀⠀⠐⠀⠀⠠⠄
⠀⠀⠀⠈⢶⣾⣟⡛⢚⣿⣩⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠙⠚⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢖⡐⢤⢀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠁⠀⠀⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⠊⠐⣤⢏⣀⣞⡂⠀⠐⢿⣿⠿⠷⠂⠀⠀⠾⠟⠁⠀⠀⠐⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠈⢜⠷⠞⠚⠒⠓⠺⢭⣨⡅⠀⠀⠸⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠕⠐⠀⠆⡌⠀⠀⠀⠀⠀⠿⢿⣶⣿⣶⣶
⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⢦⠇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡂⠠⢀⡀⠠⠄⠀⠠⣔⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⡂⠠⠶⠂⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠂⠚⠉⠉⠀⠦⠀⠀⠠⠌⠃
⠀⠀⠀⠈⠂⢀⠈⠁⠀⠀⠠⣁⠖⡀⡡⢀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠄⠙⢁⡦⠀⠀⠀⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⣤⣤⠷⢶⡶⣄⣨⡵
⠀⠀⠀⠀⢠⠄⢄⠀⠀⠀⠀⠀⠀⣴⣦⡴⣶⡶⡆⠀⠀⠀⠀⣠⠄⠒⠂⠉⠉⠻⣶⡀⠀⣼⡟⠀⠀⠀⠀⢻⡏
⠀⠠⠢⢀⡈⣆⠈⠖⠐⠀⠀⠀⣴⣾⣿⣿⣾⣿⣿⣤⠀⠀⠸⣯⡀⠀⠀⠀⢀⢸⡟⠀⠀⠙⣧⠀⠀⠀⣠⣾⠇
⠀⠀⠉⠄⠘⠀⢆⠓⠀⠀⠀⠀⠈⠁⠈⠓⠋⠋⠙⠛⠧⠀⠀⠈⠻⠶⠚⠑⠺⠛⠁⠀⠀⠀⠈⠛⠛⢷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠂⠄⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠠⡭⡄⠐⠈⢡⡀⢠⠀⠥⢈⣱⡄⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣄⠀⣠⣾⣿⣿⣆⠀⠀⠀⠁⠈⠋⠀⠀⠰⠦⣉⣙⣣⣿⣧⠉⠺⣄⠀⢤⣼⣷⡏⠳⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆⠠⠀⠈⠰⠀⠀⠀⠈⠀⠀⠠⢭⡿⠖⠠⢉⣀⠀⠨⣿⡦⠀⠻⣄⣴⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⡄⠈⠓⢾⣿⡀⣆⠀⢹⣇⠀⠀⡂
⠀⠀⠀⠀⠀⢀⣴⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠚⠆⠀⠀⠀⢐⠤⠄⢨⠙⠒⠶⢼⣿⣇⠸⣧⠀⣿⡜⢀⡏
⠀⠀⠀⠀⣼⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢔⠒⠦⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡖⢿⣄⢸⣧⣸⣇⡠
⠀⠀⠀⢠⣿⣿⣿⠟⠋⣿⣟⣛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⢳⣤⣤⣠⣠⣶⣾⡟⢻⣿⡿⠀⢹⣿⣿⣿⡏
⠀⠀⠀⠈⣿⠏⠁⠀⠀⠈⠨⢻⣷⡀⠈⡿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣻⠀⠴⢊⣠⣾⠋⠉⠙⠛⠛⠻⣿⣷⡀⠞⣿⣿⡿
⠀⠀⠀⣼⣃⣀⣔⣦⢀⣤⣴⣿⣿⣿⠀⣿⣿⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣼⣩⣷⡀⢐⢀⣲⡄⢠⣄⣿⣿⡧⢠⠿⣟
⠀⠀⠀⠈⠉⠀⠚⢻⣛⣛⠛⠟⠋⠉⠈⠀⠙⠁⠁⠃⠛⣡⣽⣏⣩⣿⣭⣴⣤⣬⠤⠄⠉⠀⠀⠀⠘⣷⣤⣼⣿⡈⢳⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⠞⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣠⣶⣿⣿⠟⠛⠁⠀⠛⠉
⠀⠀⠀⠀⠀⠴⢶⣾⣧⣤⣤⣤⣤⣤⣄⣤⣤⣤⣄⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣾⣿⣿⠟⠛⣋⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⢀⠀⢠⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡆⠀⠀⠀⢀⠀⠀⠀⠈⠀⢀⠂
⠀⠈⠀⠀⣤⣤⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠛⠃⠀⠘⠂⠀⠀⠀⠈⠁⠀⠀⢒⣀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⢡⡍⠀⠀⣾⡍⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠘⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠀⠠⣼⣧⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠁⠐⠀⢀⣀⣤⠀⠀⣤⣤⡤⠶⠖⠓⠀⠀⠛⠉⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣤⡄⠶⠶⠀⠈⠁⠀⠹⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠄
⠀⠷⠿⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⢠⠍⠀⠀⠉⠁⠀⣆⣤⣥⣤⠀⠀⠀⢻⡄⠀⠈⠁⠈⠄⠒⠀⠀⠃
⠀⠀⠀⠀⠀⢰⣆⠀⠀⠰⣄⡀⠀⢸⠆⠀⠰⠀⠀⠀⠉⢀⡤⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠘⠁⠀⠀⠐⢀⡀⠒⠀⠀⠀⠉⠁⠀⢰⡆
⠀⠄⠤⠀⠀⠀⠀⢰⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠈⠀⠀⠠⠀⠀⢸⡇⢽⠀⠀⠀⠀⠰⠂⠟⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠉⠓
⠀⠀⠂⡄⠀⠃⢀⠸⠇⢽⡄⠀⠀⡀⠀⠀⠷⠷⠶⢶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠟⠀⠀⠙⠲⠆
⠀⠀⠂⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠂⠀⠂⠀⠁

⠀⠀⣸
⡇⣠⠿⠀⡀⣤⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⣿⡉⠀⠺⣳⠀⢸⡟⠀⡄⣸⠀⠀⠀⢸⡄⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⣾⣥⣌⠉⣠⡼⠂⠸⢁⣘⢻⣯⢀⣼⣴⡯⠤⣾⣠⠀⠀⡀⠁⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡿⢿⡶⠂⠠⠤⠀⢀⣬⡍⠼⣥⠀⣿⣿⠛⠂⢹⣿⣄⣼⢁⡀⢀⠁⡀⠀⣰⠀⡀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢀⡀⣀⠀⢠
⡷⠨⣙⣟⡳⠒⠀⢸⡟⣀⣘⠟⡯⣿⠛⡘⣋⡀⣽⣿⠟⣛⣦⣿⣼⡇⢰⡿⢻⣧⣶⡾⠇⣴⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⣶⣿⣡⡏⣾⣿⣵
⠀⠀⠀⠀⠈⠈⠀⠘⠛⣷⠍⠂⣼⡻⠂⠠⠤⠀⢐⣿⣧⠆⠰⣿⡗⠭⢿⡥⠄⢿⣿⡷⠼⣿⡿⢲⣾⣆⡀⢀⣄⣤⣿⣿⣿⣿⣿⣿⡿⠥⠽⠿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢯⡃⢦⣤⠄⠀⠉⢻⣇⠐⠈⢻⡿⢠⢽⣏⣀⣋⣿⡥⠀⢹⣇⣈⣿⡿⣿⣿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠶⠾⠿⣯⣘⡛⣷⣀⠊⢿⡧⢴⢺⣄⣐⠈⣷⡆⢸⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠚⠛⠶⠿⠿⠿⠒⠛⠉⠉⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⡄⠀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⣖⣚⠑⣒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⣠⠤⠐⠒⠒⠤⣄⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠓⢄⡀⣀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠑⢄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⠒⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡞⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠑⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⡄⢀⡀
⠀⠀⠀⠀⠀⣰⠖⢆⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠊⠠⣐⡄⠠⠄⠀⠀⠀⠀⠀⠈⡀⠬⠖⠦⢄⡀⠀⠀⠀⠀⠰⠥⢤
⠀⡤⠒⠲⠎⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠑⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠒⠢⡄
⠘⢦⢀⡤⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱
⠀⠀⠈⠀⠀⠉⠉⠋⠹⣁⣀⠉⠈⠉⠁⢀⣀⠉⠛⠀⠛⠉⢁⠘⠒⠐⠂⠒⠁⠉⠘⠛⠋⡛⠒⠻⠭⠭⠤⠔⠒⠙⠒⠂⠐⠒⠛⢯⣲⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠉⠀⢽⣉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠃⠀⠀⢄⢀⡀⣀⠀⠀⠀⢀⡠⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⠚⡩⠽⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠚⠻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢼⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗
⠀⠒⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⠊⠁⠀⠀⠈⠢⠤⠀⠀⡀⠉⠉⠋⠉⠉⠁⠈⠉⠁⠈⠉⠒⠒⠒⠉⠙⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂⣀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡤⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⠠⡔⠒⢄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⡤⢄⣀⣀⢄⣀⣀⣀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣒⣢⣤⣔⣒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠀⠀⠘⠒⢲⣀⣀⠀⠀⠀⠀⢰⠉⠉⠑⠂⠐⡄⠀⠀⠀⠀⣀⣀⣰⠋⠀⠀⠈⢀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡀⠀⠀⢫⠀⠀⠀⠀⠀⠀⣰⠀⠀⡸⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠒⠐⢄⡀
⠀⠀⠀⠀⠀⠑⠤⠤⠤⠶⣤⣤⣤⣴⠴⠦⠤⠴⠞⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠀⠀⠉⠛⠒⠒⠲⠶⠶⠶⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠚⠉⠁⠉⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⡀⠀⠀⠀⠀⠀⢀⡄⠒⠁⠀⠀⠀⠀⠀⠀⠉⣠⠵
⠀⠀⠀⠀⠀⠀⡀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢄⠀⠀⠀⠀⠙⠒⠒⠠⠔⠒⠢⠤⠚⠉⠉
⠀⠀⠀⠀⠀⠈⠒⠀⠠⠤⠤⢤⣀⣀⣀⣀⣀⡤⠤⠤⠤⠟⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢤⠋⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣦⣀
⠀⠀⠀⠀⠀⠀⢠⠴⠚⠀⠀⠀⠀⠹⠒⢦⠖⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⢀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠘⠦⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠐⠁⠀⠀⠀⠈⠒⠂⠀⠤⠤⠤⢄⠀⠀⠀⢀⣤⡤⣄⠀⠀⠀⣠⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠔⠒⠚⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣲⣶⣿⣿⣿⡿⢿⣿⣿⣷⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⠿⠋⢹⣿⠟⣛⣩⣶⣶⣾⡿⣿⠓⢁⣼⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡟⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠃⢶⣶⣿⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠙⠁⠀⢀⣄⠜⠈⠈⢷⣼⡿⠍⣁⣴⣦⣴⣿⣿⣿⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⠃⠀⠀⣠⣦⠎⠊⠢⢤⣄⡀⠹⣿⡳⣿⣿⣿⣿⣿⣟⠠⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⠀⡠⣞⣢⣵⣶⣶⣾⣧⣄⣝⢷⡈⢻⣿⣿⠿⣿⣿⡉⢺⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⣸⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⣙⣦⠈⢟⠿⣿⣇⣼⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⢀⣾⣻⠀⢀⢶⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢿⣷⣦⡀⡠⠏⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢿⡏⢿⣸⣼⣇⣸⣌⣯⣗⣿⠛⡭⢦⣌⠋⠗⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣿⣿⠟⠡⣷⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣳⣤⣿⣿⣿⣿⣿⣿⣿⡏⣾⠇⠀⠠⢯⠛⠀⠀⢄⡹⣿⣿⣿⣿⣿⣿⣿⡿⢇⣼⡿⢋⡰⡗⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡛⣷⣷⣿⣿⣿⣿⣿⣟⢻⣛⣿⣾⠉⠀⣐⣳⡃⠀⠀⠀⠈⠻⢮⣹⡛⡟⢻⡛⣅⣽⣿⢝⡯⠾⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣫⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣶⡿⣿⡭⣭⣹⡦⠀⢢⣄⢱⣿⣿⣿⣿⣿⣿⣿⠕⠻⠂⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣼⡿⠏⠟⣛⣉⣭⣭⣶⣿⣿⡿⢿⢛⣏⣧⣿⣿⣿⣿⡿⠣⣄⡳⣿⣿⣿⣿⣿⣿⡿⠻⠥⢇⣶⣴⡶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡥⠒⠉⠀⠀⠀⣠⣾⢿⢻⣤⣾⠿⣿⡛⢋⣛⣿⣟⣾⣷⣬⡻⢿⣿⣽⡻⠉⠉⠀⢀⡤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⣠⢿⣿⣷⣿⣛⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⠀⠁⢀⣀⡰⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⣾⣷⣿⣿⣿⠿⣭⡭⣯⣿⣯⣿⣿⡟⠛⠉⠁⠀⢀⣠⣴⣶⣿⣿⣍⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣿⠿⢿⠁⠀⠀⠀⠛⣾⣟⠿⣯⡊⠀⢀⣠⣴⣾⣿⣿⣽⣿⣿⣏⠂⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠋⣥⣌⠚⠳⡀⠀⠀⢤⡈⠻⣧⣈⠛⣮⣯⣿⡿⣿⣿⣿⣿⣿⣿⣏⡒⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡴⠛⣡⣿⣇⡀⣈⣷⣶⡔⠁⠀⠈⢙⠋⣽⣿⣿⣷⢿⣿⣿⣿⣿⣿⠮⠭⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠆⣠⣿⣿⣯⣷⣾⣿⣿⣿⣦⡀⠀⡚⢰⣿⣿⣿⣿⠴⣿⣿⣿⣿⠛⢁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⢁⡠⣾⣿⣿⣿⣿⣿⣿⠿⠏⣃⠀⢁⡶⢣⣿⣿⠟⢩⡏⢰⣿⣿⣿⢟⣊⣵⠟
⠀⠀⠀⠀⠀⠀⠀⢋⠟⢀⠴⠶⢋⠼⣻⣿⣿⣿⣿⡿⠃⣀⣀⠟⠈⠉⣰⣿⠟⠁⢀⡾⣠⣿⣿⠿⢟⣯⠿⠋
⠀⠀⠀⠀⠀⠀⢠⡎⠀⢀⡀⠐⠁⠈⠙⠫⠛⠉⠁⣠⣾⡿⠫⠀⠀⠀⠉⠁⢀⣰⣿⣾⡿⣋⣤⠾⠛⠁
⠀⠀⠀⠀⠀⣠⣿⠀⠀⡡⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠋⠀⠁⠀⠀⠀⠀⢰⣿⣿⠿⠾⠛⠉
⠀⠀⠀⠀⠀⠉⠋⢀⣾⢃⠔⠀⠀⣀⡤⠊⡴⢋⣀⡔
⠀⠀⠀⢠⠂⠘⠀⢜⣁⣤⠞⠓⠋⠉⠀⠀⠴⠟⠉
⠀⠀⢀⣤⣔⠀⠰⠋⠁⠀⣠⠄⠀⠞⠃
⠀⠀⠈⡻⠁⣀⣤⣶⡾⠛⠁
⢀⠀⡰⠷⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠞⠛⠛⠛⠛⠶⣦⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡶⠶⠶⠶⢶⣤⡀
⠀⠀⠀⢀⣀⣴⠟⠋⠉⠻⠋⠀⠀⠈⢻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⡀
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠙⠛⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠟⠉⠉⠉⠉⠻⡶⠛⠛⠛⠷⣦⣰⡟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠷⣦⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣽⣿⢭⡇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢤⡆⠀⠀⠀⠀⠀⠀⠀⠈⠾⠽⠶⠁⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⣺⣗⢺⠂⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣻⣿⣻⡟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠃⠀⠀⠀⠀⠀⢾⡷⣽⠆⠀⠀⠀⠀⣴⡖⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠹⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠸⣗⡿⣗⡟⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⢲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠙⠀⠀⠀⠈⠩⠣⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⣺⢿⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣸⠁⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠦⣄⡀⠀⠀⣤⣀⡌
⢀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠠⠀⢠⠃⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢳⡾⡴⣻⠉⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⡀⢠⠞⠒⠂⠀⠀⠐⠊⣒⣤⣤⣤⠤⠤⠤⠤⠤⠤⠴⠦⠴⠶⢾⠶⠖⠒⠒⠒⠿⣅⡷⠃⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀
⠀⠤⠤⢀⡤⢎⠀⠐⠈⠁⡰⠃⠀⢀⣠⡤⠖⠚⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⠀⠀⠀⠀⠀⢀⡞⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠑⠁⠀⠀⠀⢸⢁⡴⠊⠉⠀⠀⢀⣠⠤⠤⢶⡶⠖⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⡟⠉⠉⠉⠉⠉⠛⢶⢤⣀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠎⠀⠀⢀⡴⠚⠉⠀⠀⠀⠆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠓⠦⣄⡀⠈⠉⠒⢤⣀⡀
⠀⠀⠀⠀⠀⡠⠀⢐⠞⠙⢦⣀⡴⠋⠈⠢⡀⠀⠀⢰⠀⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠉⠑⠲⢤⣀⠀⠉⢙⠒⠦
⠀⠤⠖⠂⠈⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠈⠢⡀⠆⠀⠀⠲⠀⠀⠀⠀⠀⠀⢀⠀⡄⠼⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠈⠙⠲⢮⣀
⠀⠀⠀⠀⣠⡞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⢂⠀⠀⠀⠀⠀⣸⠊⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⡾⠻⣄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠲⡀⠐⡀⠀⠀⡠⠊⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀
⢀⣴⠟⠁⠀⠹⣦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣡⡠⠊⠀⠀⡆⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⠤⣄⣀
⠟⠁⠀⠀⢀⡴⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠂⠐⠂⠓⠂⠀⠠⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠙⢲⠦⢄
⠀⠀⢀⡴⠋⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⣠⠁
⠀⣰⠟⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠀⡐
⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⡀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢱⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠘⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⠄⠙⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠄⠀⠤⠂⠀⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢦⣀⣀⠀⠀⠀⠀⠀⣀⠤⠘⠳⣄⠀⠀⠀⠀⠀⣀⡤⠐⡚⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠴⠂⠉⠀⠀⠀⠀⠀⢙⣦⣄⡂⠈⠀⢀⠜⠀⠀⠐⡄
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣾⣿⣶⣶⣶⣶⣶⣶⣿⣷⣾⣾⣿⣷⣶⣾⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠒⠒⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⣠⠾⠋⠈⠳⣄
⠀⠀⠀⢤⡒⢊⣉⡡⠬⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠋⠁⣠⠞⠁⠀⠀⠀⠀⠈⣣⡀
⠀⠀⠀⠀⠙⠣⣤⠤⠒⢶⡉⢑⣦⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠟⠋⠁⢀⡴⠊⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠈⠳⢖⠉⣙⣧⣤⠼⠢⣄⠀⠀⢀⣎⣀⣠⣤⣴⣶⣿⣿⣿⠿⠟⠋⠁⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢋⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡤⠜⠳⣎⡉⣑⣄⣼⣿⣿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢋⣴⡿⠋⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣲⣊⠉⠙⣦⣠⡽⠂⠀⠀⢀⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⢁⣴⠟⠋⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠈⠻⡯⣥⠖⠻⡗⣒⣈⣭⣭⡳⡀⠘⡏⢳⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠟⠉⣠⠴⠋⠁⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡜⣧⠀⢱⢸⣟⢻⡍⠁⢡⠀⡗⠊⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⣀⠴⠊⠁⠀⠀⠀⠀⠀⣠⡾⠟⢁⡇
⠀⠀⠀⠀⠀⠀⠸⡁⣀⡠⠤⠖⡄⠀⠀⢱⢹⠀⢸⠊⣉⣀⣹⣶⣧⣴⠿⣿⣿⡿⠿⠛⠋⠁⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣴⣿⠃
⠀⠀⡠⢤⣤⣖⠮⡭⠒⠒⠒⠉⠁⠀⠀⢸⡿⠒⠺⣿⣿⠿⢍⣉⠬⠷⣖⠋⣣⣄⡤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⢀⣴⡿⠟⡏
⠀⠀⠉⠊⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠱⣤⣤⣤⣷⠶⡒⠊⣙⣧⣠⠵⢤⡀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⣠⣶⠟⠋⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠔⠉⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠈⠫⣄⡠⠽⠶⣖⠙⣢⣀⡤⠔⠊⠁⠀⢀⣠⣴⠿⠋⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣀⣠⣵⣶⡒⠉⣻⣄⠬⠷⢄⣠⣴⡾⠟⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⡉⠉⠀⠀⣙⣣⣀⣀⣤⠴⠒⠋⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣟⡛⠉⠁⠀⠀⠀⢀⣀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠛⠛⠛⠋⠉⠈⠻⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠛⠛⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣁⢠⢢⢤⡾⠧⡒⠂⡀⠀⠀⢀⡜⠁⠀⠀⠀⣁⠀⠀⠀⢀⠂⣤⡤⠍⠀⠀⠀⠀⠀⠤⠤
⠀⠀⢀⡐⢋⡀⠀⠀⠀⠀⠀⠉⢀⠙⠚⠼⣧⠉⠨⠀⠔⠂⠀⠐⠀⠀⠀⠊⠀⠀⠐⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠉⠁⠠⡇⠀⠀⠀⠀⠀⠈⠰⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠘⡃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠠⡄
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⡄⠛⠉⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠤⠀⠁⣂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠷⠾⠁
⠀⠀⠀⠀⠀⠀⠿⠀⠰⠛⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠒⠢⠀⣀⠀⠀⢀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢁⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠔⠁⠀⠨⡀⣴⡁⡙⠀⠘⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠃⠀⠀⢀⡈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠒⠀⡀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠰⠀⠰⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣤⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⢀⠜⠈⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡮⠀⠀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⠔⠁⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠌⠀⠀⢄⡀⠀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢩⠀⡆⢨⠙⣆⠀⠀⠀⠀⠀⠀⠰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠤⢭⣉⡃⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣘⠀⠤⢘⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡀⠀⠀⡀⠀⠀⠀⣀⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠞⢁⠤⡈⡆
⠀⠀⠀⠀⠀⠈⠀⠀⢈⢀⠎⠀⢸⠀⡀⠂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠅⢸⠀⠈⡄⠀⠀⠀⠀⠀⠀⢆⡸⠔⡁⠀⢀⠥⠄⢠
⠀⠀⠂⠐⠀⠀⠀⠀⢀⠠⠊⠀⠂⠀⠀⡀⠁⡆⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠰⠈⠀⠀⠈⠑⠒⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⡀⠤⠊⠀⠀⠀⡩⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⢂⠐⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠊⠀⠀⠀⠀⣀⠠⠔⠁⠀⠀⠀⠀⠒⠒⠠⣄⠀⠀⠤⠯⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣀⠈⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢨⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠁⠁⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠬⠄⡂⠨⠄⠀⢅⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠂⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢎⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡡⠝⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁⠉⠉⠉⠀⠀⠀⠈⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣞⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⡟⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠾⠿⢛⣛⣛⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣟⣟⣟⣿⢯⡇⣿⡧⠽⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⣟⣽⣭⣭⢿⣾⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣽⡗⢒⣹⢸⡏⠉⠉⢹⠁⣿⣿⣶⣶⠇⠀⠀⠀⠀⠀⠀⣠⣮⡭⡥⠶⠶⠶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿⣚⡺⣺⠠⢬⣸⣸⠇⠀⠀⢸⠀⣿⣿⢸⣿⠀⠀⠀⠀⠀⢀⡶⠷⠒⢒⣚⣛⣻⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⢍⢭⡭⡭⡏⠀⠀⡉⠁⠀⠀⠀⣾⠘⠉⢛⢻⣿⡌⢠⡄⠀⣰⠛⠛⠛⣛⢋⣩⡏⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢖⢓⢒⢖⠓⠂⠁⣀⣴⣽⣿⣿⢿⣿⣿⣷⣶⣶⡟⠁⠀⠀⢠⠾⢟⠛⢛⣟⣟⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⣩⣭⣉⣭⠉⠁⠀⢛⣿⣿⣿⣿⠆⢰⣿⢻⡹⣿⠟⠀⠀⠀⣴⣛⣛⣋⣹⣯⣭⣍⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢤⠭⠭⠦⠤⣦⠖⠀⠀⢀⠔⢩⣏⣟⣹⣯⣴⣾⣥⣮⣿⠏⠀⠀⢠⣾⣭⣭⣩⣭⣬⣬⣭⠋
⠀⠀⠀⠀⠀⠴⠷⠴⠶⠶⠴⠶⠊⠀⢠⠔⠁⢠⣶⣿⡿⣿⢸⡟⢻⠟⣏⠋⠀⢀⣴⣯⣤⣴⡤⢤⠶⢶⠔
⠀⠀⠀⡴⠶⠒⠒⣒⢓⢗⠒⠀⡠⠂⠀⠀⠀⠀⣿⡿⢹⣷⣚⡧⢾⣦⣏⠀⣠⠾⠶⠶⠶⠶⣶⠚⡷⠋
⠀⣠⠶⢛⠛⣋⣋⣛⠙⣁⠔⠉⠀⠀⠀⠀⢀⣬⣶⡏⣿⡟⣽⠛⢻⠟⣿⠒⣿⠻⣿⣿⢿⠛⠛⠋⠁
⠚⠊⠙⠉⠙⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣳⣽⣧⣿⠅⢸⠀⢾⢰⣿⠀⢹⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡹⠋⠟⢳⡿⠤⣾⣾⢾⠦⠺⡟⠉⠇⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡹⢁⣀⣀⣸⣧⣤⣴⣿⣤⣤⣀⣀⣀⡀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠛⠋⢹⠉⢸⣿⡏⠀⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠀⠀⠀⢆⣘⣂⣚⣿⣷⣶⣿⣿⣿⣹⣾⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢃⣠⣴⣾⣿⠭⢿⣿⣿⣻⣻⣮⣫⣝⣯⣀⣩⣭⣽⣯⣻⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣯⡿⡿⠷⠿⢿⡿⠩⣿⡄⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡀⣿⣿⣿⠿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣄⣾⣿⣦⣣⣾⡏⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠽⠿⠮⢿⣿⣝⣿⣿⣿⠿⠿⠿⢿⡯⢽⣿⣏⣀⠽⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢳⠀⠀⠀⢀⣀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⡳⠒⠊⠉⢀⠎⠀⢡⠀⠀⠀⢀⣀⠤⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠕⠒⢦⡀⠀⠀⡜⢁⠤⠒⣤⣏⠀⠀⠀⡵⠒⠉⠁⠀⡰⠁⠘⡄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⡠⠤⡙⢲⡎⠀⢸⠀⡰⠁⡜⠀⠀⡜⢀⢤⡀⠀⡴⠁⠀⠀⠘⣄⣀⠤⠔⠚⠉⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢧⠀⡧⠃⠘⠒⢋⡜⠀⢀⠑⢦⣎⠴⠋⢀⠇⡼⠁⡎⠉⡇⡰⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡠⠔⢁⡜⠁⣀⣀⣀⠞⠯⢇⡿⣁⠎⡤⡀⣠⢋⡜⠀⠰⣇⠀⡙⠉⢱⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠘⠤⢤⠞⠀⡞⠁⢠⢯⠀⡔⠉⢠⠿⡐⠦⠺⢴⡟⠹⡄⠀⠈⡛⠲⠒⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢠⠒⠲⡄⠀⣠⠋⠀⡜⠁⡰⠃⢸⠀⠹⣤⣋⠜⠁⠀⢀⠞⡠⠴⠁⠀⡼⠁⠀⠀⠀⡰⢫⠃
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣀⠞⠀⠘⣅⡴⠳⡀⠀⠳⡴⠁⠀⠻⠤⣶⢧⡀⠀⠀⢀⠎⠀⢳⠀⠀⡼⢁⠤⠒⠤⠚⢁⠏
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡇⠀⠀⢀⠞⠁⠀⠑⣄⣼⠇⠀⠀⠀⡰⠁⠰⢅⠀⢀⠞⠀⡴⠋⢀⡼⠁⠘⠒⠲⠚⢉⠏
⠀⠀⠀⢠⠞⢣⡀⠀⣠⠤⣹⠀⣰⠋⠀⠀⠀⢀⠟⠉⠀⠀⠀⡰⠁⠀⠀⠈⣇⠎⠀⠀⠧⠚⡽⠹⡀⠀⢀⠀⢀⠎
⠀⠀⠀⠹⡄⠀⠙⡆⠈⠲⢋⡶⠛⠉⠒⠤⣤⣏⡀⠀⠀⠀⣼⠕⠒⠒⢤⣾⡟⠀⠀⠀⠀⡰⠁⠀⠉⠉⠉⢱⡎
⠀⠀⠀⠀⠹⡄⠀⡇⠀⢠⢞⣀⣀⠀⠀⣠⠃⠀⠉⠳⠤⡼⢃⠔⠦⠼⢁⠞⠙⠶⠋⡗⡰⢁⣠⣀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠘⢆⠱⣤⠟⠉⢀⣈⠵⠒⡇⠀⠀⠀⢀⡞⣜⠁⢀⡤⢴⣮⠀⠀⠀⣰⣱⢡⠎⠀⠀⣓⠀⡜
⠀⠀⠀⠀⠀⠀⠈⠶⠧⠒⠋⠁⠀⠀⠀⠸⡀⠀⢀⡞⠓⣮⣉⡁⣀⠞⠈⡆⠀⢠⣿⢃⡈⢦⡀⠠⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⠏⣸⡠⠴⠒⠉⠀⢆⠀⠹⡤⢣⠃⣇⠙⠒⠃⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠈⡆⠀⢠⠃⠀⣈⣙⠢⣄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠏⠀⠀⠓⠒⠃⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠦⣄⣀⣠⠴⠒⠶⠶⠒⠋⠳⠤⣀⣀⣀⣠⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⡤⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣤⣤⣦⠀⢀⠋⠙⠉⠐⠲⠽⡟⠋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠹⣿⣿⣰⣃⡀⢀⣀⣠⡞⠸⣍⡉⠿⡀⢰⡶⠆⢀⣤⣄⠀⠀⢀⡀⠀⠀⢠⠄⠀⣀⠀⠀⣀⡠⠖⠂⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣄⣲⣦⡽⠶⠛⠋⠉⠀⠈⠈⠙⠻⡀⣿⣤⠀⢀⡀⡄⠀⠀⠙⠒⠋⠰⢋⣽⠟⠀⠀⠀⢀⣾⠤⠀⣤⠘⢶⣤⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣀⣼⣿⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠞⠈⠁⠀⠰⣿⠿⡟⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣟⠲⣶⣤⣄⡀⡀⠀⠉⠐⠠⠎⠀⠀⠀⢀⣶⠋⣠⡇⠀⢀⡀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠤⠒⠊⠙⠻⡏⠃⠠⢄⣐⠀⠀⠀⢠⣾⠁⣾⣏⣷⣠⠉⣁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠓⡤⣤⠖⠋⠈⢣⣄⣠⠴⠒⠉⢻⣷⣶⣦⣿⣄⣀⣀⠀⣿⢐⡏⡿⠗⠸⢁⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⢀⠀⠼⠂⢞⡉⠀⣈⡱⠶⠒⠉⠁⠀⠀⠀⠀⠀⠘⠉⠛⢿⡿⣶⣬⣽⣟⠈⠁⠘⠒⢳⣿⢁⠆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠘⠲⢎⠀⠀⣀⣤⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⢿⡀⠟⠏⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠁⠀⢀⣀⡤⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠴⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⡴⠚⠛⠛⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢀⠀⠀⠈⠉⠙⢧⠀⣀⡀⣠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠙⠁⠀⠈⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠸⠅⠘⠀⠀⠁⠀⢀⠀⠀⠉⠆⠀⠀⠀⢤⠀⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⣤⣃⡀⠀⠤⠊⠀⠀⠀⢠
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠉⠉⠙⠛⠿⠿⢯⣭⣍⣉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠲⠶⠶⠦⠤⣤⣥⣤⣦⣤⣀⣀⣀⣀⠀⣀⠀⠀⠀⠐⢶⢺⡇⠀⠀⠐⠠⣄⠀⠀⠀⠁⠈⣂⢀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣏⡙⠂⠂⠀⣸⠘⣧⡄⢀⠀⠀⠀⡄⠀⠀⠀⠀⢻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⠶⠃⠀⢸⡀⠈⠀⠉⠈⠀⠀⠁⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⣤⣄⣤⡼⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⣉⣀⣠⣄⣈⡉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⣠⡶⠟⠋⢉⣉⣉⡉⠛⠷⣦⡈⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠋⢀⣴⠟⠛⣉⣉⠛⠷⣦⡈⠛⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⠈⢡⡶⢟⣋⣙⡻⢶⡌⠁⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⢰⡟⢀⣀⣠⣿⠉⠉⣿⣀⣀⡀⣿⡄⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣠⣤⢶⣿⠛⠛⠉⠉⠻⢶⡶⠟⠉⠉⠛⠻⣷⢶⣤⣀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡶⠟⠉⠀⢸⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡄⠈⠙⠻⣦⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⣿⡷⠶⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠷⣦⣀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠃⠀⣠⣴⠟⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⠙⠻⠾⣿⣤⣤⣀⣀⣀⣸⣇⣀⣀⣠⣤⣴⡿⠾⠛⠉⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠸⣧⠈⠉⠉⠉⢻⡏⠉⠉⠉⢁⣿⠃⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠹⣧⡀⠀⠀⢸⡇⠀⠀⢀⣾⠃⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣼⣷⣤⣀⣸⡇⣀⣴⣿⣥⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠲⢦⣄⠀⠀⠀⢀⣠⡤⠶⠶⠒⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠛⠛⠧⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⣀⣀⣀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠈⣻⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⣀⡴⢾⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠿⢫⡟⠛⠛⠻⣤⣞⡃
⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠙⢦⣤⣀⣀⣠⣤⡞⠉⢳⣦⣤⣀⣀⡀⠀⢀⣀⣤⣏⡿⣠⣭⡗⠀⠀⠀⣿⠉⢿⠄⢀⣄
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⠹⣤⣤⣤⣽⡜⢯⣩⣟⢻⣍⣧⠙⣴⣧⣴⠟⣦⣤⡾⠡⣿⡭⠤⠟⠼⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣶⢂⡶⢶⣲⡶⣠⠶⠶⣦⡴⢶⡶⣀⠰⡖⢶⣔⡶⠒⣆⢲⡒⠒⠓⢶⣘⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢾⡉⢰⡇⠀⠀⢸⣗⠛⢰⠋⣷⠙⣻⡏⠀⠀⠘⣧⠛⢶⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⢧⣤⣤⣤⠟⡴⠞⠛⠳⢷⡙⣦⣀⣀⡴⣷⣦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡴⠒⢲⢆⡘⣇⠀⠀⠀⣰⢃⡄⢶⠶⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡞⣷⠙⢲⣶⠚⢡⡏⠹⣎⢧⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⢁⢺⡄⠙⠶⠛⠹⠷⠖⠋⠀⣸⠂⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠟⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠡⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠀⠀⣈⡖⢂⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠀⠈⡆⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⢀⣤⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠁⠀⠀⡁⠐⠐⢦⡀⠀⠀⠁⢢
⠀⠀⠀⠀⠀⢀⢠⠖⠃⠀⠤⠀⢅⣄⠀⠀⠀⠐⠂⠀⠚⠲⣄⠀⢀⣤⠀⣀⣀⣄⣀⣀⣀⣰⠤⡀⠓⢀⡀
⠀⠀⠐⠒⠓⠒⠒⠲⠢⠴⠚⠒⠒⠒⠤⢤⣀⣤⣉⡤⠤⠤⠤⠭⠡⠑⣟⡠⠠⠀⠀⠀⠀⠀⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢳⡔⠁⡹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⢉⠶⡁⡨⠻⣀⠔⠙⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⡠⠚⢢⡔⠉⡲⡊⢠⠟⢦⡀⠀⢀⣴⣿⣷⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠁⡸⠪⣠⠞⢳⡴⠻⣄⣴⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠖⢉⡶⠋⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠛⢦⣴⣷⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⢿⠟⠳⡤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣦⣤⣀⠸⣿⣿⣿⣿⣿⣿⠗⠀⡠⠾⣀⠔⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠛⠿⣿⠟⠙⣶⠚⢣⡔⠉⡲⡎⢀⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣧⠀⠀⠀⠈⠳⣔⠁⡨⠮⣀⠜⠣⣤⠊⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠈⠻⣄⠔⠉⡦⡊⢈⠲⣅⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⣀⠜⢣⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⣠⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⣿⣿⠿⠿⠿⢷⣦⡀
⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⡆⠀⠀⠀⢹⣇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡆⢰⣿⠿⢿⡇⠀⣿⡿⢿⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡇⢸⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡇⢸⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣷⣶⣿⣷⣾⣿⣶⣾⣷⣶⣿⣷⣾⣿⣶⣶⡆
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⣀⣸⣿⣄⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣧⣀⣀
⠀⠀⢸⣿⣄⢀⣴⣶⣶⡿⠋⠙⠻⣷⡄⠀⣿⡏⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⢸⣿⠛⢿⣯⣀⣀⡀⠀⠀⠀⣽⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠺⠃⠀⠿⠀⠘⠗⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠉⠉⠙⠿⣶⣶⡾⠟⠀⠀⣿⡇⠀⣠⣶⣦⡀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣿⡇⠘⣿⣅⣽⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⣄⢀⣴⣶⣶⡿⠋⠉⠻⣷⡄⠀⢀⠀⠀⠈⠉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠛⢿⣯⣀⣀⡀⠀⠀⠀⣽⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠉⠉⠙⠿⣶⣶⡾⠟⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠘⣿⣄⢀⣴⣶⣶⡿⠛⠙⠻⣷⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠛⢿⣯⣀⣀⡀⠀⠀⠀⣽⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⣶⣶⡾⠟⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣴⣶⣶⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠓⢓⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠈⠀⠠⠀⣶⡶⠀⠀⠀⠠⠀⠀⠀⠀⠐⠖⠓⠶⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠚⠓⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠑⠛⠙⠚⠓⠓⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠀⢠⠖⠀⠉⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠚⠐⠳⠖⠤⠂
⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠌⣠⠖⠈⠈⣧⣽⠋⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡐⠧⠢⠔⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠀⠈⠐⠉⠈⠒⠚⠀⠙⢆⢀⡠⢀⢀⠀⢠⡄⠤⠒⢲⣴⣧⣶⡜⢚⣒⠘⢀⠀⡀
⠀⠀⠀⠀⠀⡠⠐⡐⠒⠊⠉⠁⢡⣹⣷⢰⠀⣰⣶⡆⠀⡄⣯⡴⣄⢅⠠⡌⡛⣿⣏⣴⠀⠌⠁⢉⡁⡀⣀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣌⠄⠈⠉⢅⣐⣋⠏⡁⢹⠀⢀⡇⠀⣠⠤⠼⠂⠂⡁⠀⠈⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⢀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠋⠉⠁⠉⠐⠄⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠟⠁⠀⠈⠚⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡠⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠥⠒⠥⠀⠀⠀⠁⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⢀⠁⠄⡀⠔⢹
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠐⠁⠀⠀⠒⠦⠛⡊⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡈⡈⡀⠊⢀⠄⠀⢰⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢀⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠊


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠾⠁⠀⠀⢳⡤⣤⠀⠀⢠⡞⠁⠀⠀⢛⡆⠀⠀⢀⣠⣤⣤⡾⠋⠀⠀⠀⠉⢳⣀⣀⡀
⠀⠀⠀⠀⠀⣰⠞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠉⠉⠛⠉⠉⠁⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢧
⠀⠀⠀⠀⠀⠹⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⠃⠀⠀⠀⠀⠀⣠⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⠞⢦⣤⡤⠶⠶⠞⠉⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠛⢦⡄⠀⢹⣆⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣀⡀⢀⠀⠀⢛⡆⠀⠙⠳⠶⠴⠞⠦⣄⣀⣀⣹⣄⠀⠀⠀⠀⢀⡰⠦⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣤⣀⡤⠤⣄⡈⠉⠓⠚⠛⠋⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠛⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⣀⣰⠁⠀⠀⠀⠛⠀⠀⠀⢷⣀⠀⠀⠀⠀⣴⠒⠏⠉⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣄
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢳⡄⠀⠰⣏⣠⠀⠀⠀⣁⣹⠇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⠀⠸⠦⣄
⠀⠀⠀⠀⠀⠘⢦⣄⣤⠀⠀⠀⢠⡄⠀⢀⣄⣠⡼⠁⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⣠⣤⣀⠀⠀⠀⠀⠹⢦⣄⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠞⠛⠒⠋⠀⢀⣀⣠⣤⣄⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠿⠀⠀⠀⠈⠛⣧⠀⠀⠀⠘⢧⣄⠀⣄⠀⣲⡼⠇⠀⠀⢀⣤⠶⢦⣤⠤⣤
⠀⠀⠀⠀⠀⣠⠟⠉⠁⠉⢳⠀⠀⠀⠀⠈⠷⣤⣤⣤⣤⣤⡤⠼⠃⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⡞⠉⠁⠀⠀⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠙⠚⠦⠶⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠳⢴⡀⠀⠀⣀⠀⣠⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠀⠀⠇⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢴⡞⠉⠀⠈⠻⣤⠤⣤⡀⠀⢷⣀⡆⠀⠀⡀⠀⠀⠀⢀⣸⠇⠀⢀⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠇⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢻⡆⠀⠈⠳⣤⣤⠷⣤⣤⡴⠋⢀⣀⣴⠋⠀⠀⠀⠀⠙⣧⠴⠶⣤⡀
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠻⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠹⣧⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⣴⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣄⠀⠀⣀⣀⠀⣀⡼⠛⠛⠉⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⠟⠛⠛⠿⢛⣿⡿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⠛⠛⠛⠟⣫⣿⡿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⡛⢩⣦⠈⢹⠀⢸⠛⢱⣦⠈⣹⠛⢱⡆⠈⣵⡄⠉⣿⠛⢱⣦⠈⣻⡇⠀⣾⡅⢨⣄⠘⣿⠀⣼⡏⠀⣼⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⡇⢘⣛⣀⣼⠀⢸⠀⢘⣛⣀⣿⠀⢹⣷⠀⣿⡇⠀⣿⠀⢘⣛⣀⣿⡇⠀⣿⡇⠈⠻⣠⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⡃⠸⣿⠛⣿⠀⢸⠀⠸⣿⠛⣿⠀⢹⡏⠀⢿⡇⠀⣿⠀⠸⣿⠛⣿⡇⠀⣿⡇⠸⣆⠘⢿⡀⣿⠇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣴⣾⣿⣦⣴⣿⡟⠀⠸⣿⣦⣶⣿⣿⣦⣶⣿⣦⣾⣿⣦⣾⣿⣦⣴⣿⣿⣧⡶⠛⠓⠰⣿⣦⣤⣽⣷⠄⢀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡛⠿⠿⠋⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⡿⢿⣿⣿⣿⣿⣿⡿⢛⣭⡗⡶⡰⡶⣴⡪⢝⠻⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣠⡶⣤⣴⣄⣹⣿⣿⣿⢯⣦⣻⡼⠾⢯⣿⣿⣿⣥⣋⣾⣎⢿⣿⣿⣿⣥⡖⣰⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⡃⠹⣿⣿⢿⣿⣿⣿⢸⣿⣿⣏⣐⣀⣨⡀⠸⢻⣿⣿⣿⢸⣿⣿⣿⠟⣃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠸⣿⡄⠽⣫⣿⣿⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣦⣾⡿⣣⣿⣿⣿⣿⣸⢿⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣬⣝⣛⣛⣛⣛⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣄⣠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⡏⣤⡖⠛⣷⡄⣿⡇⣤⡖⠻⣷⡄⣤⡖⢻⣶⠜⢻⣦⠀⣤⡖⠻⣷⡄⢹⣿⠉⣾⡷⢻⣧⠀⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡧⠤⠿⠇⣿⡇⣿⡧⠤⠿⠂⣿⡇⢸⣿⠀⢸⣿⠀⣿⡧⠤⠿⠇⢸⣿⠀⣽⣧⣀⡿⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣇⠀⣤⠀⣿⡇⣿⣇⠀⣤⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⣇⠀⣤⠀⢸⣿⠀⣿⣇⢻⣦⠠⣿⡀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⠙⠋⠁⣀⣿⣇⠈⠛⠋⠁⠀⠙⠋⠈⠛⠋⠈⠛⠋⠈⠛⠋⠁⠀⠘⢛⣤⣽⡟⠀⠛⠷⠂⠘⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠷⣄⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡚⠁⠉⠈⢱⠤⡀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠌⠀⠀⠀⠠⢫⠐⢌⠐⢄⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠂⠀⠀⠀⡠⠁⣀⣄⠀⠈⠢⣈⠢⡀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡡⠁⠀⠀⠀⡔⠀⠀⠙⠛⡘⠁⢩⠋⠁⠀⠑⠄⣀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠑⢤⡴⠊⠁⠐⠂⠠⠌⡀⠀⠀⠀⡐⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢅⡀⢀⠄⠔⠓⠒⠛⠭⢥⣒⠠⠈⠂⢄⡜⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⡊⠀⠀⠀⠈⢃⠀⠈⠆⠀⠀⠀⠀⠈⠙⠦⣧⡀⠁⢠⠃
⠀⠀⠀⠀⠀⡔⠁⠀⠈⠐⠤⡀⠀⠀⠢⡀⠈⠆⠄⠀⠀⢀⡠⠒⠈⢳⡜⠃
⠀⠀⠀⠀⠀⠀⠑⠠⣀⠀⠀⠈⠐⠄⡀⠈⠂⢄⡖⣤⠒⠁⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⢈⡰⠔⠈⠐⠉⠛⠁⠒⠤⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⣤⣀⡀⠀⢀⣠⠴⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠙⣿⣷⠋⠀⠀⠀⠀⠀⠀⠉⣧⣀⣤⠤⣤⣀
⠀⠀⠀⢀⣤⣶⣿⠀⠀⠀⠀⠀⠀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢙⣦⡤⠖⠒⠲⢤⣀
⠀⠀⡴⠁⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠈⢧
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡴⠟⠋⠉⠉⠙⠲⢤⡀
⠀⠸⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠙⠂⠀⢠⠴⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀
⢀⣴⠿⠦⠀⠀⠀⠀⠈⠛⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠰⠟⠀⢠⣤⠀⠁⠀⠈⠙⠲⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠉⠳⣄
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠬⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠈⢳⠀⣰⠂⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠹⣄⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⢀⠀⠀⠀⠀⠈⡇⠙⠶⢶⣄⣀⣀⣀⡠⠇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⡤⠖⠲⢾⡁
⠀⠈⣳⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣠⣼⠃⠀⠀⠀⠘⣯⠉⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠘⠧⠀⠀⠀⠙⡆
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⡽⠀⢀⣀⣀⡀⠀⠀⠉⠳⠶⠤⠶⠀⢀⣀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⣄
⠀⢻⡀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠸⠋⠀⠀⠀⠈⠙⠻⣟⡛⢶⣶⣤⣤⣤⣄⣀⣉⣛⠛⠛⠁⠀⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⣸
⠀⠀⠓⢄⣀⣀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠉⠲⣍⡛⠿⣿⣿⣿⣾⣭⣭⣭⣉⣛⠛⠛⠛⠿⢿⠧⠤⣤⣄⠤⠴⠋
⠀⠀⠀⠀⡾⠛⠉⠙⢿⡀⠀⠀⠀⠰⣄⣀⣀⣠⠋⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠙⡄⠀⠀⠙⢦⡈⠙⠲⣍⡻⢿⣶⣤⣍⣙⣿⡶⠶⠯⠄⣀⣀
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠙⢦⡀⢠⡝⠳⢮⣙⠳⢯⣿⠲⠤⣄⡀⠀⠉⠉
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠙⣾⠷⠤⠤⠽⠻⢯⡀⠉⠒⠦⣍⡃
⠀⠀⠀⠀⠀⢹⣦⣄⣀⡀⠀⠀⠀⠘⢶⣤⣤⣀⣠⣤⠶⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⢶⣦⠤⠤⠖⠋⠉⠳⡄⠀⠀⠀⠀⠉⠲⣄⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠻⣇⠈⠁⠀⠀⠀⠀⣠⠟⠉⠉⠁⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⣿⠦⢄⣀⣀⣀⣠⡤⠞⠁⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠆⠂⠀⢀⣀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⣶⣿⡆⠚⠊⠀⢀⣰⠴⠏⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣤⣾⣿⣿⣿⣷⡄⢠⡶⠟⠛⠁⠀⡰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⣿⡿⣫⠵⠿⠏⢿⡿⣦⡃⠀⠀⠀⣰⡇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⣿⡟⡝⠀⠀⠀⠀⠀⠈⢿⠻⣦⣁⠰⠎⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⣤⢀⢠⠏⣠⢿⣟⠘⡃⠀⠀⠀⠀⠀⠀⢸⠇⠈⡿⢷⣿⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠈⠀⠀⠀⠀⡘⣰⣿⣣⣼⣿⡷⡀⠀⠀⠀⠀⢀⠞⢀⣾⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠂⣀⠼⠿⢓⣸⠿⣿⣷⡾⠓⠒⠒⠛⣁⣴⣶⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣤⣤⡤⠄⠠⠖⣉⠀⠄⠂⠀⠁⣠⡿⠇⠁⠀⠀⠀⣀⣻⠟⣟⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⡠⠊⠀⢂⣁⠄⠰⠖⢢⠞⠁⠁⠢⣄⣰⣶⡿⢟⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣀⡤⣦⢀⣀⠔⣥⣤⣶⣶⡿⠟⠋⢩⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢕⣠⣶⢿⣿⣿⣎⠐⣄⡻⢿⡟⠋⠁⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡴⠟⣹⣽⣿⡝⠿⣿⣷⣮⣹⠊⢁⢀⠀⠀⠀⢀⣭⠅
⠀⠀⠀⠀⠀⢀⡄⠀⠀⢂⣴⣿⣿⣿⣄⡀⣌⣿⠿⠇⠀⢰⢸⡄⠀⠀⢼⠏⡀
⠀⠀⠀⠀⠀⡻⠁⠤⢪⣿⣿⣿⣿⠟⢁⡹⠛⠉⠊⠀⠀⢀⠟⠀⣀⠖⠋
⠀⠀⠀⠀⣰⠀⡠⠂⠀⠈⠙⠁⣠⢶⠟⠃⠀⠀⠀⠀⠀⠀⠄⠋⠁
⠀⠀⠀⠘⠃⣠⠃⠀⢀⡠⠄⠞⣁⠄
⠀⠀⠐⠀⢀⠥⠖⠋⠁⣀⠀⠚⠁
⠀⠀⠿⢀⣠⣤⠖⠁
⠀⠐⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠁⢀⡸⠀⠀⠀⠀⣠⠚⠉⠉⠈⠲⡀⠀⠀⣀⡀⣀
⠀⠀⢀⡀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠠⠄⠐⠉⠀⠀⢀⡸⠀⢠⠊⠉⣁⠠⢤⡀⠀⠈⠁⠉⣷⠀⠀⠀⠗⠠⡀
⠀⠈⠉⠀⠀⠀⠒⠠⠍⡑⠂⠄⣀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠂⢮⣀⣀⠜⠁⠀⢸⠀⠀⠓⠀⠀⡱⠀⠀⠀⠴⠋⢀⣀⠀⠀⠤⠘⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⢀⠉⠒⠤⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢀⠀⢀⡀⠀⢣⡀⡀⠠⠐⠁⠀⠤⠄⠀⠀⠉⡸⠀⠀⠐⠁⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡐⠀⠀⢀⢀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠈⠉⠁⠠⢀⣤⡤⠴⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠢⠤⢠⣀⣁⡀⠒⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣒⡒⠢⠭⡂⢄⡀⠀⠀⢠⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠢⡀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠂⠤⠤⠤⠤⣀⠀⢁⣹⡀⡀⠈⡆⠈⠉⠉⠁⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠲⢄⡀⢀⠤⠤⠤⣀⡀⠀⠀⢠⡄⠀⠀⠀⡹⠉⠈⠦⠤⠔⠁⢠⣀⠀⠀⠈⡧⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⢬⠉⠒⠀⠀⠠⢿⠀⠀⠈⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⡦⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣆⠀⢀⡤⠤⠚⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠲⠄⠠⠴⠆⢀⠠⠔⠁⠈⠁







⣿⣿⠿⢿⡿⢿⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡐⢤⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢁⣀⣱⣮⡀⢸⡀⠀⠀⠀⠠⠝⠿⣄⠁⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡁⠀⠀⠈⢁⡨⠿⠯⠀⠀⠀⠀⠀⠀⠀⠛⣇⣀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠟⠃⠀⠀⠀⠻⠀⠀⠀⠈⢈⡧⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠐⣀⣠⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⣤⣦⡀⠀⢀⣄⠀⠀⣸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣶⣶⣤⣶⣾⣿⣶⣶⣦⣀⣠⣴⣿⣿⣷⣶⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⢿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠿⠛⠋⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⡀⠄⠀⠀⠘⠓⠲⣾⣿⣏⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠒⠇⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠀⢼⣿⣿⠟⠉⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢈⣿⣇⣀⣀⣠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⢶⣿⣶⣾⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢸⣎⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠸⠿⠯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢰⣿⣿⣯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢀⠀⠸⢿⣿⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢈⡴⠒⢒⣶⣌⠻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠄⡞⠀⣰⣿⣿⣿⡇⢻⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠻⣄⢿⣿⣏⡹⢃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠐⠈⠁⢛⣩⣴⣿⣿⣿⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢜⣹⣆⠀⠀⢀⡞⠱⠀⠀⠀⠀⢀⢼⣿⣿⣿⣿⡇⣝⢦⠀⠀⢀⡾⠡⢝⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⣿⣿⣤⠴⣋⡀⠀⢁⠀⠀⢠⠁⠸⢿⣿⣿⡟⣼⣿⢏⡑⢦⣾⠁⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⣿⣿⣷⡆⣤⠅⣀⡤⣆⠀⠁⠀⢸⣷⣹⣿⡶⣽⡃⢥⡬⠀⠉⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⢟⡷⠊⠁⠀⢸⡇⠀⠀⢸⣿⣷⣿⠀⠀⠉⠳⣄⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣷⠏⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠈⢻⡀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⡀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠸⣯⠏⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⢧⡀⠸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣙⣿⡆⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢸⣿⢷⠏⠀⠀⠀⠀⠀⠀⠀⣾⣋⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠸⣯⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠓⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⡟⠄
⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⢴⠋⠀⠀⠀⠀⠠⠠⠎⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣼⠀⢴⢎⡠⢤⡀
⢠⣆⣼⡗⣠⠀⠀⠀⠴⠂⠠⠀⠀⠈⢀⡀⠀⠀⠠⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⢀⣼⡆⠛⠛⠿⣤⣠⡄⣀
⣿⣿⡟⣷⣿⣠⣠⠔⢐⠁⡄⣀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⣆⠀⠀⣯⠃⠀⡜⢛⠉⣻⡁⣀⠹⣀⠠⣿⣀
⣸⡿⢀⢻⣿⣿⣿⢞⠁⠈⡁⠀⠡⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢀⢘⣤⠓⣄⡔⠀⠶⢌⣯⠈⠣⠈⢳⣮⣿
⢨⡿⢛⢰⣿⠟⠃⠌⠁⡀⠀⠀⢂⢘⣿⣷⣿⣶⣼⣤⣄⣀⡀⣠⣤⣤⣷⣤⣼⠴⠅⣘⣋⠀⣀⡥⠋⣁⠀⠈⢿⠀⠀⠠⡀⠈⣿
⢠⡀⠊⢈⣥⠖⠋⠀⠚⠀⠀⣄⡀⠑⢿⣿⣟⣹⣽⣽⣿⡿⠟⠟⠿⠿⣿⣿⠟⠀⡀⣈⡑⡆⢈⡶⢋⡠⢦⠜⢦⡂⠀⠀⠉⠙⢿
⠘⠋⠀⢯⡥⠀⠁⠚⠁⢤⡜⢳⡀⠱⠦⢷⣀⢼⠷⠛⣡⠤⣄⠀⠀⠀⠈⠻⠀⠀⣀⢉⠻⣄⣾⢉⢁⠀⠀⡁⠂⢄⡀⠀⠢⢀⠠
⠀⠀⢠⡎⠀⠂⣏⡀⠔⠀⠈⠤⠷⠀⣀⢀⠀⠈⠰⣭⡼⠏⢿⣽⠶⠾⠟⠁⣨⠈⢁⣁⣉⣻⣿⡷⢛⣧⣀⣿⣷⢄⣻⣇⣧⠈⢿
⠀⠠⠋⠀⣀⡤⠶⣅⠀⠀⠈⢳⡠⠲⡍⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⢩⠉⠛⠙⢦⠠⠉⠻⣏⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠠⠀⠊⠁⠐⠄⠀⠠⠳⠄⠸⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠁⠀⠀⠑⣄⠀⠀⠀⠀⢀⠔⠒⠢⠤⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⠒⠠⡄
⠀⠀⢀⡴⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠼⠀⠀⠀⠀⠀⠀⠑⠒⠦⠒⠢⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠎⠁⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠆⠀⠀⠘⠛⠛⠦⠰⢄⠀⠀⠀⠀⠀⣀⢀⡠⠬⠭⠭⠕
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⣁⠤⠤⡀
⠀⠀⢠⠊⠉⠉⢢⣀⣀⠤⠒⠤⡀⠀⠀⢀⠤⠤⠊⠀⠈⢳⠀⠀⠀⠀⣀⠤⠤⠤⠔⠉⠉⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠁⠀⠀⠈⠦⢄⡀
⠀⢀⠎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠳⡄⠀⡇⠀⠀⠀⠀⠀⠸⡁⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⡀⠀⠀⠀⢀⡴⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢
⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠁⠀⠈⢇⠀⠀⡀⠀⠀⡇⠀⠘⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⠀⠀⠱⢄⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠜
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠢⡀⠀⠀⠀⠀⠀⠈⠒⠊⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠓⠠⠔⠊
⠀⠀⠀⡰⠉⠉⠋⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⢀⠔⠊⠉⠒⢄⠠⢄⡀⠀⠀⠀⢀⣀⡔⠁⠀⠈⢢⢀⡀⠀⠀⠀⠀⠀⣀⢀⠔⠉⠉⠑⢄
⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⡰⠉⠀⠉⠀⠀⠀⠀⢸⠒⠒⢯⣉⡉⢡
⠀⠈⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⠴⠃⠀⣏⣁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠇⠸⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠱⣀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠁⠈⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠒⠀⠒⠉⠒⠊⠁⠀⠀⠀⠀⠑⠒⢆⡀⢀⡠⢀⣀⣀⠜⠀⠀⠈⡀⠀⠀⣀⠀⠀⢀⠆⠀⠀⠀⡤⠃
⠀⠀⠀⡠⠒⠠⣀⣀⡰⠊⣉⠵⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠉⠘⠣⠤⠒⠤⠴⠃
⠀⢀⠜⠀⠀⠀⢈⠔⠈⠒⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⢁⠔⠋⠑⠒⠉⠉⠐⢄⠀⠀⠀⠀⢠⠂⠉⠑⠴⠊⠉⠲⡄
⠀⠈⠢⠤⠤⡔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢴⠊⠀⠀⠀⢀⡔⠊⠋⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⡤⠶⠤⠴⠉⠉⠑⣄⣀⠱⠒⠢⠔⠒⢢
⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠉⠉⠉⢳⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⡀⣀⡀⡉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠂⠀⠀⠈⠉⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⡠⠴⠁⠀⠀⠙⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠂⠁⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠒⠒⠢⡀
⠀⠀⢠⠔⠊⠀⠀⠀⠀⡠⠤⠎⠁⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡤⠒⠓⢤⡀⢀⡠⢄⡀⠀⠀⠀⠀⠀⣀⠤⠀⠤⡎⠀⠈⠀⠀⠀⠀⠥⠤⡀
⠀⠀⡇⠀⠀⠀⢀⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡀⠀⠀⠀⠈⢲⠀⡠⠜⠀⠀⠀⠀⠁⠉⡆⠀⠉⠙⡆⠀⠀⢀⠃⠀⠀⠀⠘⠒⠢⡔⠋⠉⠉⢢⠀⢈
⠀⠀⠘⠦⢀⠰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢠⡤⣤⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠾⣁⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡎
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠓⠔⠢⢄⣀⠤⠊⠈⠁⠀⠀⠀⠀⠉⠉⠑⠈⠢⠄⠀⠤⣀⡠⠆⠔⠊⠉⠉⠉⠉⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠜⠁

⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠈⢲⠂⠀⠀⠀⠀⠀⠁⢽⠀⠀⣀⠄⡰⠃⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⢀⡼⠋⠀⠀⠀⢀⠿⣦⡀
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⣀⣑⣄⣠⠤⠴⠶⠖⠛⣏⠀⠀⠀⠀⠐⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⡟⠀⠀⠀⠀⣠⡾⠁⠀⠀⢀⠔⠣⠀⠈⢻
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡀⠀⠀⠀⠌⢀⠞⠁⠠⠀⠀⢀⡾⠁⠀⠀⠀⣰⠏⠀⠀⠀⡠⠋⠀⠀⢀⠔⠁
⠀⠀⠀⢀⡽⠆⠀⠀⠀⣀⣤⠶⠛⡁⠀⠈⢳⡀⠀⠀⠀⢀⣀⣤⠴⠶⠛⠋⠉⢯⠉⠉⠛⠓⠦⣾⣄⣀⠀⠀⢠⠞⠀⠀⠀⢀⣼⢃⣀⡠⠠⠮⠀⠚⠒⠈⠉
⠀⠀⠀⢸⠀⢀⡤⠖⠛⠁⠀⠀⠀⠈⠂⠄⡀⠹⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠗⠂⠀⠀⠈⠸⡇⠉⡙⣶⣏⠀⠀⠀⢠⡞⠁⠓⢄
⠀⣀⡤⢾⠋⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠒⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⡸⠠⣀⠀⠀⢀⡿⢈⡼⠃⠈⠙⢦⣴⠏⠂⠀⠀⠂⡑⠦⡀
⠊⠁⠀⠘⣆⣸⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⣿⠀⠀⠀⡈⠁⠒⠤⣤⡠⢀⣠⡁⠀⠀⠉⠒⠾⣤⣾⡀⠀⠀⣠⡾⠉⠓⠦⣄⣀⣀⣀⣐⣁⣀⡀⣀⣀⣀⣠⠴
⠛⠓⠶⠦⣞⠉⠀⠀⠀⣀⡠⠄⠒⠂⠈⠉⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠙⡆⠀⠈⡄⠀⠀⣰⡾⠋⠀⠙⢳⣴⣏⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⠉⠙⢍⠉
⠀⠀⠀⠀⠈⠛⢱⠤⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠻⣄⣠⡾⠋⠀⠀⠀⣠⣾⡁⠉⠛⠶⣤⣀⠀⠀⠀⠀⡿⠀⠀⡠⠊⠑⢆
⠀⠀⠂⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣴⠟⢧⡀⠀⠀⢀⣼⠋⠀⠹⡄⠀⠀⠈⠙⠳⢦⣄⣸⠃⠀⠀⠇⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣯⣀⣀⣀⣀⣠⣤⡴⠶⠟⠋⠀⠀⠀⠙⢆⣰⠟⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⢻⠷⠶⢶⣤⣤⣬
⣀⣀⣤⣤⣤⣾⠶⠶⠶⠶⢶⣄⠀⠀⠀⠀⢉⣟⠉⢹⠉⠉⠉⠁⠀⢀⣀⠠⠢⠐⢄⠀⠀⣠⡾⠛⠄⠤⠤⠤⠤⣸⡀⠀⠀⠀⠀⣰⠇⢀⠀⠀⡘⠀⠀⠀⠀⠀⢨
⠉⠁⠀⠀⠀⣧⠀⠀⠀⠀⢀⡿⠒⠛⣏⠉⠉⠁⠀⢸⡂⠁⠈⠉⠁⠀⠀⠀⠀⠀⠈⢱⣴⠋⠀⠀⠀⠀⠀⠀⠀⣸⠉⠲⢄⡀⢸⠇⠀⠀⠀⠠⠁⠀⠐⠀⠀⠀⡎
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⣼⡇⠀⠀⠸⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠒⠀⠀⠀⠀⣠⠟⠉⠣⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣹⠳⣄⡀⠀⠀⠸⠀⠀⠀⠀⠀⡬⠙
⠀⠀⠀⢀⣀⡨⡧⠶⠒⠋⣿⠀⠀⠠⠀⢷⠀⢈⣀⣈⣇⣀⣀⣀⣀⣤⣤⣤⣤⠾⠋⠀⠀⠀⠘⢆⠀⠀⠀⠀⣼⡀⠀⠀⢠⠁⠀⠀⠙⠦⣀⡇⠠⠀⠀⠀⡜⠁
⠒⠚⠉⢩⡇⢀⣻⣤⣤⡶⢿⠶⠶⠛⠛⠛⡟⠋⠙⠉⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⡼⠁⠀⠁⠲⠇⡀⠀⠀⠀⠀⡌⠛⢦⣀⢀⡜⠁
⠀⠀⠀⠀⢹⡟⠙⡀⠀⠀⠈⠳⡄⠀⠀⠀⢸⡀⠀⠀⢻⠀⠅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣁⡀⠀⠀⠀⠀⠀⠑⠢⢄⡜⠁⠀⠀⠈⠻⣄
⣀⣀⣀⣀⣾⡁⠀⠇⠀⠀⠀⠀⠙⢆⠀⠀⠀⢷⠴⠶⠺⠒⠒⠚⠚⢛⡟⠲⢤⣄⡀⠀⠀⠀⠀⠐⠀⡼⠃⠀⠉⠒⠤⢄⠀⠀⠀⢠⠎⠉⠢⢄⠀⠀⠀⠈⢣⡀
⠉⠀⠀⣼⠋⠛⢶⣄⠀⠀⠀⠀⠀⠈⣳⡤⠖⠚⣆⠀⠀⠀⢀⠀⢀⡞⠀⠠⠀⠀⠉⠓⠦⣄⡀⠀⡼⠁⠀⡄⢀⡀⠀⠀⡘⠉⠉⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠉⠲
⠀⠀⣼⠃⠀⠀⠀⠙⢷⣄⠀⢀⡴⠊⠁⠻⣄⠀⢹⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡿⠳⢤⣄⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⢄⡀
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠹⣶⠋⠀⠀⠀⠀⠈⢧⡀⣇⠀⠀⠀⢸⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠈⠙⠳⠦⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠱⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠐⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠾⠛
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠲⠶⢦⣤⣼⣀⣀⣀⣀⣹⣤⣤⣤⣤⣤⣤⡤⣴⠷⠶⠶⠶⠞⠾⠛⠛⠻⣟⠛⠛⠋⠉⠉⠙⠻⣭⣍
⠉⠉⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⢉⣿⠉⠀⡀⠀⠀⠀⣠⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠛⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠉⠉⠁⠉⠉
⠀⠀⠀⠀⠀⣔⣌⢣⣀⣎⣻⣆⠀⠛⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢐⡀⢃
⠀⠀⠀⠀⠀⠉⠋⠛⠙⠀⠉⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠉⡀⠀⠀⢀⣴⢟⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⠀⠀⢀⠰⠒⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⡐⠀⠀⠀⡀⠉⢠⣶⣛⠉
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠁⠉⠀⠃⠀⠀⠀⠀⢀⣤⡿⠃⠀⠉⠐⠐⠒⢲⠒⠒⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣂⠀⠀⠀⠀⣠⣤⣤⣄⣤⡴⠶⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠷⠾⠿⠛⠻⠶⠾⠿⠛⠁⠜⠤⠖⠂⠀⠁⠀⣶⠀⠀⠀⠀⠀⠀⠀⢀⣁⠴⠒⠚⠛⠉
⠾⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣤⣤⣶⣶⣶⣶⣶⠾⠟⠁⠀⡀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣆⡀⠀⠂⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤
⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠈⠀⠀⠀⠀⢀⣀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠂⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣀⣤⠀⠀⠐⠄⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⠛⠛⠻⠿⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⠦⢤⣼⡿⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠛⠛⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⣠⣴⠞⠋⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣿
⠀⠀⣿⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⡟
⠀⠀⠹⣷⣁⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠄⠀⢈⣴⡟
⠀⠀⠀⠙⢿⣦⣁⡂⠀⠀⡀⠄⠀⢀⣀⣠⡾⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⡀⠂⠥⣑⢂⣡⣶⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠻⠿⢶⣶⣶⠶⠿⠛⠯⣅⣂⢐⠠⠀⠄⠀⠀⠀⡀⡀⠰⠐⢜⣿⣯⣐⡀⡀⠀⠀⠀⠀⠀⢀⢀⢠⣼⠟⠁⠉⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠾⠶⣷⣷⣷⠶⠶⠟⠛⠉⠀⠉⠛⠷⢾⣦⣵⣤⣬⣬⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠂⢀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⢢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⡀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠄⢠⠠⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠐⢀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⡀⡀⠂⠀⣀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢐⠀⠂⠁⢁⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢀⠐⠁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠃⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⡀⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⢁⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡠⠁⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠅⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠠⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠛⠓⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠏⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠈
⠀⢀⡟⠁⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠒⠈⠉⠉⠉⠙⠒⠢⣄
⢠⠟⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠈⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠈⢳⡆⠀⠀⠐⠒⠉⠉⠉⠀⠀⠀⠀⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠄⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏
⠀⠀⠉⠒⢲⡶⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢄⡀⠀⠀⠉⠉⠒⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠐⠻⣍⠉
⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠄⠀⠀⢀⡴⠒⠲⣌⠢⠀⠀⠑⠲⢄⡀⠀⠀⠀⠉⠁⠀⠀⠀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⣀⡼⠀⠀⠀⠀⠀⢀⣈⠑⠄⠀⠀⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢯⠈⣹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠰⠀⠀⠐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡶⠀⠀⠀⠀⠀⠉⡙⠐⠋⠀⠀⠀⠀⠠⠈⢄⡜⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠐⢛⠘⠟⣾⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠘⠻⡉⠀⠀⢀⠔⡀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠋⢣⠀⠠⠎⠔⠂⡠⠆⠁⠄⠀⠀⠀⠀⠀⣁⣩⡁
⠀⠀⠀⠀⣠⣀⣠⠀⣤⠤⣤⠄⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠛⠚⠒⢥⣤⣠⣀⡀
⠀⠀⠀⠀⡮⠤⢬⡦⠶⠢⠤⠤⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⢣⣤⠠⣤⡀
⠀⠀⠀⠠⠴⠤⠴⡖⠰⠴⠄⠲⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⣈⣉⢁⡉⠑⢢⡐⠀⠈
⠀⠀⠀⠐⠒⠒⠶⠦⣄⢤⡠⢴⡔⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡀⠁⠉⢐⡇
⠀⠀⠀⠈⢐⡇⠂⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠈⠃⠀⠀⢐⠂
⠀⠀⠀⠀⢈⡁⠀⠀⠠⠜⠆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣴⣿⣿⣶⢿⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⣰⣟⢩⣭⢿⣷⠋⣶⣾⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠞⣁⣴⣿⡿⠛⡯⢚⣿⣿⣿⣟⣻⣻⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣰⣿⣿⣿⣿⣿⣶⣿⡿⣬⡟⠛⠛⠟⠛⠦⠐
⠀⠀⠀⡀⠀⠻⢱⠜⣿⣿⣿⣿⣿⡿⢿⠋⣽⠛⠓⠃
⠀⠀⠀⠸⠷⢸⣿⣼⣿⣿⣿⣿⡿⢶⠖⢛⡇
⠀⠀⠀⠀⠀⠹⠿⣿⡿⢿⡛⢿⡷⣬⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠈⠁⠀⠀⠈⢻⣷⡠⢀⣀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠤⠄⠀⠀⠚⠛⠚⠻⠿⠶⠿⠿⠿⠷⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣉⣉⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣙⣋⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡋⢛⣛⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⡭⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣹⣏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⣩⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⡄⣤⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠻⢦⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠶⢈⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠉⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡾⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣏⠝⣟⣿⡟⠛⠙⠿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣬⣭⣍⣻⠿⣿⣿⣿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣝⠙⣿⡿⠁
⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⠏⠾⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠛⠙⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹⠖⠛⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⣝⣴⠶⠀⠐⠻⢷⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠺⠀⠀⠀⢈⡔⢢⠀⡴⠋⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡤⠤⠼⣦⢸⠀⣧⡀⣨⠷⠒⠒⠀⠐⠂⠄⠀⠀⠠⠒⠋⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠊⠉⢹⠀⠀⠀⠈⠧⠤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⡳⢤⣀⡐⢄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠐⠒⠀⠐⠀⠀⠸⠤⠄⠀⠠⢏⠉⠉⠉⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⢶⣚⣠⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠉⠀⠀⠀⠂⠈⠁⢈⢧⡀⠀⠀⠉⠀⠈⠙⠒⠦⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡶⠃⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠓⡤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⣠⠞⠳⢄⣀⠀⣀⣠⠴⠋⠳⣄⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⣁⣤⣴⠶⠶⣦⣤⣉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣴⠟⠋⣠⣤⣤⣦⣤⣈⠙⢷⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠀⠁⠰⠟⢉⣤⡴⢦⣤⡙⠷⠄⠀⠀⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⣰⠆⠀⠛⣵⠞⠷⣮⠙⠀⢠⣄⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣀⣴⣿⠶⠶⠾⢿⣄⣠⡿⠷⠶⠶⣿⣤⣀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣤⠾⠛⠉⣿⠃⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠸⣏⠉⠛⢶⣄⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⣄⡀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡿⠀⢀⣤⡾⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠉⠛⠳⢿⣧⣤⣄⣀⣀⣸⣇⣀⣀⣠⣤⣼⡷⠟⠋⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢿⡄⠉⠉⠉⢹⡏⠉⠉⠁⢠⡟⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠈⢿⣄⠀⠀⢸⡇⠀⠀⣰⡟⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣿⣷⣤⣸⣇⣤⣾⣯⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁





⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠀⠀⠀⠀⠀⠀⠘⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢧
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠠⠴⠐⠐⠚⠁⠉⠉⠉⠁⠉⠉⠁⠀⠐⠒⠀⠀⠤⠤⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠠⠠
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠁⠈⠐⠀⠒⠒⠐⠂⠒⠒⠋⠉⢉⠅⠤
⠀⠀⠀⠀⠀⠀⢀⣀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠠⣀⠂⠄⠈⢹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠐⠂⠀⠠
⠀⠀⠀⠀⠠⣈⠒⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⢠⡀⢀⣀⠀⡆⡆⠁⢠⡀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄⣖⣿⢸⡀⢟⣚⠀⡇⡇⡏⢸⡀⣗⡚⠈⣳⠄


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀
⠠⠀⠀⠀⠘⠃⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠖⠀⠀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠴⠆⠀⠀⢼⣇⠀⠐⠒⢄⡀⠀⠀⠀⠀⠶⠀⠀⠀⣀⠀⠀⠀⠀⠀⡐⡁⣒⠪⢂⠀⠊⠉⢂⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣌⠌⠡⣅⠈⠒⠠⠬⢁⡒⡄⠀⠀⠀⠀⡰⡿⠀⠀⠛⠀⠀⠇⡅⠛⢁⢀⠀⠑⠒⠠⡁⠢⠀⢰⢷⠀⠣⠉⢐⢄
⠀⠀⠀⠀⢸⠀⠘⠃⢸⠀⢀⣀⢀⣤⣁⣶⠀⠀⡤⠴⢸⠤⠷⢤⡀⠐⡍⠥⠃⠀⠘⢜⣈⠆⢈⠀⣲⡅⢠⠏⠈⠧⢀⠠⡠⣿
⠀⢠⡆⠀⠘⢘⡃⣘⢘⣴⡿⣿⣿⡿⠈⢿⣀⣀⡇⢠⣆⢀⣦⠀⣦⣠⡇⡼⣵⢠⢿⡆⢠⣊⠕⢉⠙⠗⣴⣶⢳⣶⡕⠈⢎⠪⡆⠀⢀
⢐⢕⣿⠄⠀⢠⡄⡤⠸⠁⡀⠀⡀⠀⢘⠀⠁⣈⡀⠀⠀⠀⠀⠀⣠⡀⡇⠉⢉⡤⡍⠁⠈⠀⡀⠻⠃⡀⠹⣈⣉⣸⠰⠄⠀⠤⢠⡐⢛⣳⡄
⠀⡺⣷⡁⢴⡠⠄⠷⠀⠀⠛⠐⠋⠀⣈⠀⡆⠛⠃⡔⠂⠃⠒⡄⠉⠁⡇⠀⢸⣁⡇⠀⡇⢸⣿⠀⢸⣿⢀⠿⠿⢿⠀⠠⣶⠀⣈⠀⣶⡄⡄
⠨⢚⠣⡂⢽⡇⣶⣺⠀⡆⠷⠰⠇⢸⣉⡇⠁⠀⢘⠠⠂⡅⠄⢠⠀⢰⡧⢀⣞⣒⣣⡀⡇⠼⠻⠆⠼⡻⠄⣿⣭⢿⠀⠀⠛⠀⣿⠀⡭⡄⡇

⠑⢄⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣱⡀
⠀⠀⠑⣄⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠙⡄
⡠⠔⠊⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⠢⣄⢀⡀⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⡇
⠀⠒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⠤⠔⠎⠀⠀⠀⠙⠒⠢⢄⡀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠈⢫⠉⠁
⠈⠁⠀⠀⠀⣀⣀⠠⠄⠚⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠒⠂⠉⠉⠁⠁⠈⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠰⠀⠠⠤⠀⠀⠁
⢄⠀⠀⠀⠀⢀⠀⠀⠐⠒⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⠉
⣲⠄⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠶⠀⠁
⠀⠀⠀⠁⠂⠀⡀⠀⠀⠀⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠀⠀⢀⠘⡤⢤⣀⣀⡀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠉⠻⣷⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣷⡄
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠹⢷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⣉⠉⠀⠂⠀⠉⡗⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡨⠥⠒⠛⢏⡰⢲⠤⠤⠄⣀⡀
⠀⠀⠀⠀⡤⠒⠚⡍⢆⠂⠈⠠⠭⠆⠂⡐⢀⠚⠚⡜⠀⢬⢳⠢⢄⡀⠀⣀⠤⠔⢊⠧⠀⣎⠁⠀⠀⠀⢠⢾⠁⣏⠔⠂⠉⠑⢎⢹
⠀⠀⠀⠀⡇⠀⠀⠘⡌⢣⣁⠁⢀⡈⡴⠵⣆⣀⣀⢐⡀⠸⠘⠀⠀⠈⡉⠀⠠⢄⣿⠀⢧⠢⡙⢄⠀⠀⡇⡘⢸⡃⢀⡤⠒⠒⠁⢸
⠀⠀⠀⠀⣧⠂⠐⠀⠈⢦⡙⢦⡀⢨⣁⣀⣁⠸⠛⡄⠀⡇⣇⠞⠉⠳⠠⢀⡇⠘⣅⣀⣽⠀⢷⠈⠢⠜⠀⢠⢸⠑⠋⠆⠁⠀⠰⠎
⠀⠀⠀⠀⠃⢱⡎⠀⠂⢢⠙⠢⣈⠒⢄⡀⢠⣁⣀⣠⡼⠰⡙⠀⠀⢠⠃⢺⢆⣀⢾⠉⠿⣤⣼⡥⠔⠒⠀⠉⠀⠉⠉⠀⠒⠒⠤⡇
⠀⠀⠀⠀⠀⡠⠀⠄⢲⠊⢀⠤⡈⠑⠦⣈⠑⠤⣀⡰⢡⢣⠁⢀⡠⣊⠤⢐⡒⠢⢄⠀⡠⠒⠁⠀⠀⣀⠠⠤⠤⡀⣤⠄⣀⡀⠀⡇
⠀⠀⠀⠀⢸⠐⣴⠂⠀⣀⠎⠀⠈⠒⢄⠀⠙⠢⡜⡱⠧⡘⠒⡩⡪⠒⠉⡊⠙⡕⢆⠉⠀⠀⢀⠔⡩⠒⠈⠑⡄⠃⢐⠠⠀⢀⡉⡇
⠀⠀⠀⠀⢸⢄⡁⠀⠪⣀⡀⠀⠀⠀⢈⠃⢀⠎⡔⠉⠢⡈⠻⡋⢈⠁⡡⠈⣎⡴⠁⠀⠀⢰⠁⠊⠀⠀⠀⠀⡄⡆⡌⣄⠃⠶⠾⠆
⠀⠀⠀⠀⢸⠢⢌⡑⠤⣀⠀⠑⠤⠤⠋⠀⡎⡜⠐⢀⠁⠌⢦⠘⢆⠻⢀⡠⠊⠀⠀⡠⠛⢆⠑⢕⠂⢄⠀⢀⣵⡱⠰⠒⢎⠀⢠
⠀⠀⠀⠀⢸⡎⠑⠪⠵⣦⡑⢄⡠⠄⠒⣾⠀⠑⠓⠒⠒⠒⠧⠵⠀⠑⠉⠀⠀⣠⠞⡔⠒⠢⡑⢄⡑⠤⣉⣪⣧⠃⠅⠀⠴⠀⢸
⠀⠀⠀⠀⠸⢸⠀⣠⣴⠗⠛⠄⢧⠀⣺⠀⢷⣀⣀⠦⠀⠀⠀⠀⠀⠀⣀⠀⣎⠁⡜⠀⠀⠀⠈⠲⡉⡒⠤⢀⣉⠒⠒⠠⠤⠤⢼
⠀⠀⠀⠀⠀⠈⡺⠋⠁⠀⢀⣼⢸⠫⣭⠀⠨⣦⣟⠀⣫⠉⢰⢒⢉⠕⢬⡳⡈⠢⡱⡀⠀⠀⠀⠀⡧⠁⡇⡆⡔⠛⠛⠭⡀⠀⢺
⠀⠀⠀⠀⠐⡋⠀⢀⡠⠊⠁⡸⢸⢀⣥⣽⡁⢹⡇⢀⣰⣂⡆⡌⠠⡀⠀⠑⢌⢦⡘⢮⣄⠀⠀⠀⠱⡀⡇⢿⠀⠀⣀⢤⣊⠤⢼
⠀⠀⠀⠀⠀⣧⠔⡁⠘⠀⢠⢃⡏⠛⣹⡉⠫⡫⡀⠉⠁⡜⡰⢠⠤⡘⠦⠤⠤⠃⠑⢄⠙⢷⡂⢄⡀⡸⠱⡘⢅⣩⢖⢅⡲⠭⠯
⠀⠀⠀⠀⠀⡕⠀⠀⠂⡠⢃⢮⣿⡰⠉⠑⣀⠈⠪⡒⠞⡰⡱⠁⠀⠈⠒⢦⢀⠘⠰⠀⡑⠦⣈⠒⠬⣁⠂⡝⢌⠱⣅⡏⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠰⢀⠔⡡⠃⠂⠀⢧⠤⠤⠄⡀⣡⢊⢜⠜⠀⠀⠀⠀⢰⠡⠈⠀⠀⠄⢀⡰⠜⠓⠢⠬⣑⣢⣀⠑⣌⢪⢢⠀⡇
⠀⠀⠀⠀⠀⡗⡠⣪⠎⠀⠘⠒⣒⠸⠦⠠⠤⣕⣡⠊⢸⡀⠀⠀⠀⠀⠘⠀⠠⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠘⠦⢇⣉⡇
⠀⠀⠀⠀⠀⢯⡼⠥⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠬⢉⣐⠢⠄⠔⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠉⠁⠀⠀⠉⠉⠉⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢕⡠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣔⠩⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠒⠤⠌⢉⣉⠉⠛⠛⠛⠛⠛⠛⢉⣉⡉⠡⠄⠊⠉⠐⠠⣀
⠀⠀⠀⢠⠐⣂⠈⠵⠦⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡆
⠀⠀⠀⠀⠉⠢⣒⢘⣯⡬⠿⣥⠒⣂⡠⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⣀⡒⠢⠬⠑⡞
⠀⠀⠀⠀⠀⠀⠈⠑⢄⢚⣳⣭⠵⢄⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣦⡚⠛⣯⣉⠦⠤⢄⣀⠀⣀⣀⠀⣀⣀⣀⣀⠀⠤⠤⠐⢂⠝
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠈⠻⣷⡒⠻⢣⣭⣥⣯⢆⠸⡉⣆⣀⣀⣀⣀⡀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡀⠀⢡⢳⠀⠋⠯⠙⣄⣸⣠⠁
⠀⠀⢀⢄⣠⡤⣓⠬⠁⠐⠃⠀⢀⡿⠤⠓⠀⠩⣊⡙⢧⡝⢢⡀
⠀⠀⠈⠈⠀⠀⠈⢄⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠈⠲⢂⣉⠷⣬⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢖⣈⠹⢤⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣂⣉⡁⠬⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣎⣣⡾⡾⡔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠈⠋⠀⣀⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡥⣴⡏⣀⡁⠋⠘⣙⢀⡀⠀⢠⣀⠀⠂⣈⡀⠈⠀⠚⠆⠤⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢮⡁⢛⠈⢉⠁⣠⠸⠬⢈⡁⠀⠀⠀⠀⠀⠈⠁⢐⠀⠀⢡⡄⣠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠉⠚⠃⠉⠙⠘⠃⠐⢣⠘⠊⠀⠀⠀⠀⡻⠟⠀⠀⠰⠦⠀⠛⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡩⠀⠞⠀⠀⠀⠀⠀⠠⠀⠒⠂⢈⣤⡀⣁⠀⠀⠀⠀⠀⠀⠈⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠉⠁⠓⠓⠈⠁⠀⠐⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠰⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁