Tomato Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⣿⡟⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠿⢿⣿⣿⣶⣦⣄⣈⡇⣿⢹⣿⠃⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣟⢿⣿⣿⣧⢿⣸⣿⣿⡿⣟⣻⣭⣭⣭⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⠿⢿⣶⣝⡿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣟⣾⣿⣟⣽⣾⣷⢻⣿⣿⣿⣶⣽⡻⣮⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣼⢟⣽⣿⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣮⡋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡇⡿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠓⢶⣤⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣡⡀⠀⠀⠘⠁⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠄⠀⢻⡌⠦⡰⠀⢼⣟⣀⣀⣀⣀⣀⡀⣀⣀⣠⣤⣤⡤⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠻⣧⣀⠀⢿⣴⡇⢨⣾⣷⣡⡤⠴⠚⢛⣩⣵⣖⣻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣀⣠⣤⡤⠤⣤⣌⣻⣷⢽⣿⡀⣾⣿⣴⣦⣶⣿⣿⣟⡛⠋⠉⠉⠉⠛⠂⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉⠉⠉⠉⠙⣻⠷⠶⣾⣯⣿⣯⢽⣿⣿⣿⣾⣷⣿⣿⣷⣾⣿⣷⣶⣶⣷⣴⣠⣌⢠
⠀⠀⠀⠀⠀⠠⠊⠜⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠛⠉⠁⠀⠀⠀⠙⣿⣷⡄⠀⠉⠙⠈⠀⠈⠉⠉⠙⠛⠻⣿⣿⣿⣷⡶⡤⠄
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠹⢷⣷⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢧
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣷⣃
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣿⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣯⣿⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣿⠃
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢠⢀⣷⣿⡄
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⢳⠁
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣿⡟⡞
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣩⡿⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⣘⣿⣿⣟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣾⣽⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⣠⣔⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢬⣀⡀⠁⠀⠀⠀⡀⠀⡀⠀⠐⠀⡀⠁⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠶⠴⠶⠶⠶⠷⠶⠦⠷⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣴⡠⢀⡀
⠀⠀⠀⠀⠀⣠⠶⢛⣽⣳⢄⣾⢿⡛⢶⡀⠀⠀⠀⣴⣾⢿⣟⣣⣀⣿⣿⣿⣦⡀⠀⠀⠀⣴⠶⣶⣶⣿⣯⣭⣽⣶⣦⡄
⠀⠀⠀⠀⢠⡏⣴⠈⠁⠀⠀⠛⠀⠀⢢⣿⠀⠀⣸⡏⠀⠊⠁⠀⠘⣿⣿⣿⣿⢷⠀⠀⢰⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⢻⣧⠀⠀⠀⠀⣠⣿⣿⣿⡿⠘⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣜⠆⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⢿⣿⣶⣶⣾⣿⣿⣿⠟⢁⠎⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣯⠥⠒⠁⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣜⣂⣿⣚⣠⡴⠁⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⣤⠄
⠀⠀⠀⠀⠀⢠⡴⠿⢹⡿⣿⣿⣭⠶⣄⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠿⢿⣷⣦⡀⠀⠀⠀⢀⠀⢠⣞⠧⢴⡿⠦⠀⢀
⠀⠀⠀⠀⢰⡏⠀⠀⠘⠀⠈⠃⠈⠃⠘⣧⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⢠⡏⠀⠈⠀⠀⠀⠃⠀⠀⠈⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠙⠳⠦⢤⡤⠴⠞⠋⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣤⠤⡴⢲⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣄⡇⡄⢀
⠀⠀⠀⠀⢠⣶⢿⡿⠗⣀⠀⣙⢲⣶⣦⡀⠀⠀⠀⠀⡀⠀⢰⠟⢿⠻⣄⠐⣦⣄⠀⠀⠀⠀⣀⡤⠙⠷⣾⣿⠶⠋⢤⣄
⠀⠀⠀⢠⣿⢤⣼⣷⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣶⡹⡆
⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢀⣾⣿⡗⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢹⣿⣿⣦⣤⣤⣤⣶⣿⣿⣿⠇⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⡾⣱⠇
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣀⣬⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢶⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢋⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⡀⡏⢸⣉⡉⠛⠛⢳⡶⠖⢛⣛⣿⣿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣯⢭⡤⣄⣸⡇⢡⠞⢑⠏⣠⠶⢃⡤⠚⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢉⡥⠤⠤⠴⠶⢍⠲⢍⣠⠃⠀⡾⢞⡅⣠⣏⡀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⣑⡄⠀⠀⢰⡏⠀⠉⠀⠠⢍⡓⢤⡀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠉⠀⠀⠀⠀⠀⡠⠎⢁⣀⣤⡀⠡⠐⠂⠋⠓⠒⠶⣌⡂⢹⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡞⠄⡼⠉⠀⠸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡜⠀⠀⣀⣤⠶⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣷⡜⠁⠀⠀⠀⢱⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣀⢀⠞⠁⠀⡀⠀⠠⠤⢤⡀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣀⡠⠤⠤⣿⣊⠜⠀⠀⠈⠀⠐⠀⠀⠀⠙⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠞⠁⢀⠀⠄⢀⠀⠀⠀⠔⠀⡔⢅⠶⡀⠒⠄⡘⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡜⠁⢠⠖⠁⢀⠐⡂⠑⢄⠀⣄⠀⡄⠀⢙⣃⠔⢊⡀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡼⠀⢠⠗⢉⠁⢠⠜⠉⠱⢬⠇⠈⠈⠀⠀⠈⢄⡔⠉⠁⢤⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣝⠋⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇⠀⢸⠄⣂⠐⠓⢠⡄⠀⠀⡠⢄⠀⡖⠐⠄⢀⣼⠢⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣂⣈⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢣⠀⠀⢧⣀⡈⠵⢊⣠⡄⠀⣠⢼⡆⡻⡤⡈⠟⠈⠠⢠⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠩⣤⢭⣓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣹⡆⠀⠈⣆⠀⠀⠀⡴⠈⠈⠊⠱⠒⠣⠀⡣⠁⠘⠦⠴⠒⠓⣉⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢶⡌⠉⠛⢛⠃⠈⣙⠛⠃⠈⢉⠩⠝⠛⠉⠉⠉⠀⢹⣆⠀⠘⣆⠀⠀⠳⡜⠠⠠⠀⡅⣠⠖⠁⢀⣠⣤⣴⣶⣟⣉⣙⣛⠛⡷
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠞⠋⠁⠀⠀⠀⣀⠀⠀⠙⠒⠋⠀⠀⢀⠄⣀⡤⠶⠒⣾⠈⠳⣤⡈⠣⡀⠀⠉⠒⠢⠴⠖⠁⢀⣴⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⣠⠞⢉⡀⠤⠒⠒⠈⠉⠉⠙⠁⠀⠀⠀⠀⠌⣠⠄⡰⣡⠞⠁⠀⠀⣠⠏⡀⠔⢹⠙⢳⡾⣷⣤⣄⣀⣤⡤⠶⠋⠁
⠀⠀⠀⣴⣧⣬⡥⠤⠤⢤⣤⣤⣀⣀⣀⣀⣤⠖⣃⣠⠞⣯⣼⠞⠋⠀⢀⡴⣛⣥⣤⠀⠀⡦⠀⡀⠳⣄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢼⠷⠛⠉⠀⠀⠉⠀⠀⠀⠀⠘⢋⡵⡚⣁⠀⠩⠁⡀⣿⠳⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣹⠏⣠⢋⣼⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡷⠛⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠹⡄⢸⡆⠀⠀⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡆⠀⡇⠘⡇⠀⠀⡇⡇⡠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣏⡛⠒⠲⠶⣻⠄⡟⠉⡇⠹⠛⠓⠶⢁⡏⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠁⠉⠢⣁⢿⡰⡡⢠⠀⡦⠀⠈⣡⠾⡛⠍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡼⡁⠀⠀⢠⠜⠊⠉⠉⠈⠉⠛⠻⠚⠉⠉⠉⠉⠛⢣⢤⠀⠘⢛⢦
⠀⠀⠀⠀⠀⠀⣼⠓⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⡀⠪⣮⣧
⠀⠀⠀⠀⠀⢀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠀⠂⡀⡔⢔⢅⠀⣹⣿⡇
⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⠀⠀⠀⠀⠺⠱⠰⢈⣎⠃⢿⣿⣷⠀⠀⢀⣀⣠⠤⢦
⠀⠀⠀⠀⠀⢸⡏⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣠⣀⡤⠼⠶⠶⠴⠶⠗⠒⠚⠛⠛⠋⠉⠉⠐⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠈⣇⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⣼⣿⠋⠁⠀⠀⠀⠒⠒⠀⠄⠀⠀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠹⣤⠀⠘⠀⠀⢀⣤⠀⠸⠀⢰⠏⢀⡀⠠⠀⠐⠒⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣶⡷⠄⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠐⢄⠃⡜⣿⡀⡶⠄⣿⠀⠠⠰⡀⠀⢀⡀⣟⢛⡬⠇⣴⢦⡄⣤⢄⠠⡰⣶⣦⠏⢰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡉⠂⢷⢟⢽⡇⠂⣿⠀⠀⠣⠳⣄⠀⠙⢮⡚⠲⠊⠊⢸⡁⣿⣾⠘⣷⢘⡷⢠⠊⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⣘⠒⠁⢈⡾⣇⠀⢑⠵⡜⢮⡓⢀⠙⠲⢦⣜⣁⠙⣉⣠⣛⣉⠥⠚⠁⠰⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣬⣧⣀⢈⠚⣷⣿⣦⣀⠰⢦⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢭⣛⣿⣿⣿⣽⡶⣤⠶⠀⠴⠦⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠛⠛⠚⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐⣤⣤⠀⡎⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⣹⣷⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⡀⢽⡄⢀⠆⠀⠀⡤⠄⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢀⣂⣫⡾⣶⡋⠀⠀⠁⢠⠀⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⢹⡿⠟⠿⣟⠉⠀⠀⣾⣷⣎⠁⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠈⠂⢠⠀⢈⢹⠧⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⡀⢸⡇⢀⣽⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⣀⣴⠏⠀⢧⡾⠉⡍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⣉⣰⣼⣾⣡⡦⢈⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⣨⠠⠵⢔⣢⡀⠀⠀⠀⠉⠛⠻⢿⣷⡼⠇⠀⠰⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢂⡠⠠⠔⠀⣿⢿⡃⠀⠀⠀⠀⠀⠀⠀⠉⡷⠹⣶⡀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠙⠛⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣶⣶⡆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⠒⢲⢀⡆
⠀⠀⠀⠀⠀⣀⣤⣾⣿⣶⣿⣿⣷⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢯⢳⣺⠀⢸⢊⠧⣀⣀⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⠶⣖⣚⡀⢸⢀⡇⠮⢔⠽⠛⢦⡀
⠀⠀⣰⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠎⡀⠀⠀⠀⣀⢎⠤⡉⣄⠐⠥⣀⣀⡀⢳⡀
⠀⢀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡸⢰⠁⠀⠐⠛⠛⠁⠀⣷⠀⠉⠑⠒⠋⠁⠀⣇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢳⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢱⠃⠀⠀⠀⢀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡙⠘⠦⠶⢮⣽⠟⠋⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠙⠳⡦⠤⣴⣾⠉⣠⢰⡦⣼⡧⠤⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠒⣶⣒⣒⡚⠛⠉⠉⠛⠋⠛⢶⣄⡴⣻⡥⠼⠟⡠⠀⠈⠀⠀⠀⣞⢠⢿⠀⡇⠀⠈⠳⢤⣶⠆⠀⠘⣦
⠀⠀⢀⠔⠁⠀⠀⠈⠉⠛⠓⠶⣶⣶⡷⢢⠋⠸⣤⡤⢤⣴⡃⠀⠀⠀⠀⠀⢯⠊⠈⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⢀⡤⠋⢳⠀⠈⠹⠷⣌⡻⡄⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠰⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡤⠚⠉⠀⠀⠘⡆⢰⠀⠀⠘⢧⣻⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⠑⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠈⠀⠀⠁
⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠈⠳⡒⢄⠰⡎⣢⠄⡲⠋⠀⠉⠙⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⢤⣀⣸⠈⠢⠃⡇⢰⡥⠤⠐⢒⡩⠃⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠰⠯⠀⠠⠤⣀⡀⠠⠤⠔⠀⢠⠔⠊⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⡠⠔⠢⠄⣉⡐⢄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⠔⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⡀⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⡆⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠒⠛⣛⣯⡿⢱⠁⢰⣤⣄⠤⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⡶⢶⣤⠤⠶⢿⣇⠈⢆⡾⣻⢳⢆⣼⡼⠙⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣵⣾⣋⣲⣾⡿⣷⡶⢶⣦⣈⡘⣾⡿⢡⣯⣾⣯⣑⡀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠞⢚⣻⣿⣷⣿⠣⠀⠈⠀⠸⠛⠻⣏⠽⠒⣡⣁⣠⡴⠶⠦⣍⡲⢤⣀⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⢈⣩⣾⣿⡖⢀⣤⣤⣤⣄⡈⠙⠦⡄⠀⠀⠀⠀⠀⠈⢷⡜⠉⠙⢿⡃⠀⠀⠀⠈⠛⠋⢰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣼⣿⠟⡩⠋⣴⣿⡏⠋⢠⡄⠩⣷⣄⠘⢆⠀⠀⠀⠀⠀⠀⣿⢂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⣧
⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⠖⣛⣉⣀⠀⠀⣿⡏⣁⠀⠀⠀⠀⠀⢻⣄⠈⡆⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢠⠃⠠⠈⣴⡿⠟⠉⠉⠻⡔⠿⠏⢹⠄⠀⠀⠀⠐⠂⢹⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⢢⡇
⠀⠀⠀⠀⠀⡌⠀⠀⢸⣿⠁⠀⠠⣎⣁⡀⠀⠀⠉⠀⣰⠀⠀⠀⢀⣸⡇⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠁⠀⠀⢸⡯⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠌⣸⠁⡾⢀⠖⠘⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣧⠏
⠀⠀⠀⠀⠀⢰⡈⡀⠘⣧⡀⠀⠀⠀⠉⠉⢀⡠⠀⢀⠉⠀⠀⠀⣤⠋⢰⢷⡦⠲⢀⡀⠒⠀⠀⠀⢀⠀⠀⡠⠀⢀⣼⡞⠁
⠀⠀⠀⠀⠀⠀⢦⢳⠀⠘⣇⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠛⠋⠀⣠⣿⣾⣲⣆⣞⢚⢈⣀⠤⠀⠃⣠⣠⣧⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠈⠃⠀⢀⡀⠀⠀⢀⣠⠟⠀⠀⠀⣠⠾⣿⣵⣿⣿⣿⣿⣷⣶⣧⣶⣟⣻⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣢⣄⣈⠉⠙⠛⠉⠁⣀⠠⠐⠊⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠲⣄⠀⠀⠀⣿⠹⣄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠙⠶⣄⣹⡀⠈⢷⡀⡼⢻⠏⠀⠀⠀⢀⣠⠴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⡷⠦⠀⠉⠃⠀⢀⡿⠁⡾⠶⠖⠚⠋⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⠖⠛⠋⠉⠉⠙⠳⣤⠀⠀⠈⠛⠚⠁⠀⠺⣿⣉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣤⠶⠚⠛⠋⠙⠲⠦⢤⣬⣙⣶⣦⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⢀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠑⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⡈⠳⣌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠹⣦⢻⠀⣀⣀⣀⣄⣠⣤⣄⠀⢀⣀⠸⠄
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠈⠃⠈⣿⢧⠉⠃⠁⠈⣧⠴⢶⡄
⢀⡤⠞⠋⠀⠀⠐⠒⠛⠁⠀⠀⠀⣶⠛⠒⠀⡴⠚⣾⡗⠀⠁⡄⢶⡄⢀⣤⠾⠗⠋⢻⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃⢀⢠⡄⠚⠉⠛⠛⠀⢙⡛⢡⠀⢀⣄⠐⠚⠋⢀
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⠶⠆⡈⠋⡀⠀⣰⠀⠐⠤⠞⠷⡼⠃⠘⠾⠇⢠⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣀⣷⣀⣿⠋⠹⣄⡀⠳⠾⢤⣀⣀⡦⣈⣷⣄⣹⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡏⢀⣾⢦⣄⣽⠋⠀⠉⠁⠉⠁⠀⠀⠀⠀⢣
⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠈⠉⠛⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠳⢤⣤⣀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⠀⣹⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⢦⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠞⠋⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠀⠀⣼⠉
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠑⣶⣤⡀⢰⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣛⣛⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢁⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣄⣀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⡇⢀⢸⣿⠃⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠈⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣦⣴⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠠⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⡀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠸⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⡸⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⢸⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡏⠀⢸⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋
⣿⣿⣿⣿⣧⡀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣭⣤⣤⣄
⣿⣿⣿⣿⡿⣾⣿⣿⣿⣏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⠀⠀⢸⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠆⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡼⠃⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣌⡙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠠⠈⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣷⣦⣤⣄⣀⡀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⠿⢿⣿⣿⣿⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣤⣀⣀⠀⠠⢀⣄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣐⣒⡂⠐⠢⣐⣢⣵⣦⣶⣀⣬⡙⣷⣤⡢⢄⡑⠢⣥⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠖⠀⠂⠀⠈⠓⠢⡀⠙⠿⣯⣻⢿⣿⣾⣿⣿⣷⣭⡲⣴⡙⢳⣟⢿⣷⣆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠛⠛⠿⠿⣿⣯⡻⣟⢾⣝⢿⠻⣿⣧⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠢⠙⢎⢷⢩⠻⣏⣿⣧⢌⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠈⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠠⡀⠀⣀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠰⢀⣡⢰⣿⣼⣼⣆⣿⣷⣿⣿⣿⣷⣶⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣌⢇⣳⣿⢷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢳⣷⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⣿⣿⣿⣿⣻⠿⣿⣿⠛⠻⠿⢿⣿⣿⣿⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⡋⠹⠤⠚⣿⣽⣿⣿⣾⠀⠰⣿⠀⠀⠀⠀⠈⠉⠛⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣭⣿⣿⣿⣿⠿⠟⠁⠀⠘⣆⠀⠀⠀⠀⠉⠙⠓⠁⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠒⠂⣤⠄⠀⠐⠘⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠂⠔⠂⡀⠀⠈⠉⠛⠓⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⢠⠃⣆⣸⣄⠀⣀⣴⠾⠋⠀⠀⠀⣴⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣼⠁⣄⠋⣠⢤⠤⢌⣉⠑⠢⣄⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢿⣄⣘⠾⣿⠿⠐⠳⢤⡈⠒⢄⢣⡜⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⡇⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣠⠞⠁⠠⡙⣤⣧⠀⣀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡴⠋⢀⡤⠈⣧⠈⣿⠀⠀⠘⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡢⠄⣠⠀⢄⣀⣁⠒⠤⠀⡀⠠⠤⢴⠋⡜⠁⠰⠃⠀⠈⣼⡀⢻⡤⠊⢀⠓⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡊⠑⢄⠀⠙⢦⡀⣤⠛⣤⠗⠀⠀⠴⠁⡸⠀⠐⠀⡠⠔⠒⠊⠛⢦⣵⠴⠉⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡉⠐⢌⣷⠰⠋⢀⣤⠞⠛⠛⠳⡄⠀⣸⠁⠀⢾⠒⠁⠀⠀⠘⢲⠋⢉⠄⣈⠀⠘⢀⣤⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠉⠢⢀⡬⠤⠚⠁⢅⠀⠀⠀⠀⣿⠀⡇⠀⠀⠘⡀⠀⠀⠀⠀⠂⠀⡸⠀⠀⠀⠀⠀⢠⠋⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠌⠀⢰⠃⠀⠄⠁⢀⣀⠀⣎⣄⢣⡀⠀⠀⠘⢄⠀⠀⠀⠀⡄⠃⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⢡⡀⢀⠀⠀⠒⠀⠉⠙⡛⣸⠋⢦⡱⡀⠀⠀⠀⠙⠤⣀⣠⡇⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠦⡥⠦⠄⠀⠀⡔⣰⠛⠒⠒⠻⣧⣠⡶⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⣠⠎⡰⠁⡸⢇⠀⠀⠿⣿⠛⢢⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠐⠒⠒⠒⠂⠨⡁⢐⣒⠶⢞⡛⠁⠰⠁⠀⢇⠈⠢⠤⣿⣥⡦⠔⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠄⠂⠀⠀⠀⢀⡴⢺⡿⠾⢙⠏⡈⠀⢃⠀⠀⠀⠁⣦⠴⣿⠟⣀⡔⢁⣴⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠑⡀⠈⢲⣯⠄⡁⠀⠈⣿⡆⠁⠼⣧⣷⡆⠀⢰⠃⢀⠫⢀⣉⣟⡡⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠄⠒⠈⠀⠛⠃⢀⡀⠷⠌⢻⡟⣰⠃⠈⡿⣏⠰⠓⠈⠁⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠃⠀⠀⠀⠾⠀⠶⡄⠀⠀⢀⡾⡽⣇⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣟⠁⠀⠀⠀⠀⢹⠀⡃⠘⣧⣰⣺⠞⠀⠈⠣⣀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠈⣧⠃⢀⡜⣽⠃⠀⠀⠀⠀⢈⣙⣿⣦⣴⡦⣤⠴⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠚⠓⠚⡟⡸⠃⠀⠀⠀⠘⢓⠠⣿⣿⣇⣤⠗⠗⠛⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡀⢰⠀⣧⠴⢀⡄⠀⡔⠉⠢⣈⠻⠯⠴⠂⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣷⡆⣾⣽⣤⠾⠁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡀⠊⠉⣨⡿⣛⡛⢃⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠚⠁⠈⠉⠁⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠙⠲⠤⠄⣀⡀⠀⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠀⠤⠐⠶⠚⠁

⠀⠀⠀⠀⠀⠠⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠙⠛⠀⠀⠀⠀⠀⠀⣰⠋⠓⢈⣛
⠀⠀⠀⡀⠀⠁⠀⠀⣤⠀⠠⠰⠧⠆⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣀⡼⠟⣰⠂
⠀⠀⠀⠙⠠⠔⠇⠀⠀⠀⠀⠘⠃⢺⠋⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠖⠀⠀⠀⠀⠢⣭⣭⡤⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢆⡀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠛⢃⠰⡄⠨⠁⠀⠂
⠀⠀⠀⠀⠀⠠⠄⠐⡈⠐⢆⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠆
⠀⠀⠀⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠙⠒⠸⠀⠴⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⡐⢤⣤
⠀⠀⠀⠃⠀⠀⠀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠠⢶⡄
⠀⠀⠀⠀⠀⠈⢀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠻⠦⢀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣯⣽⣛⢟⢿⣾⣿⣿⣿⣿⣟⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢉⣴⣿⣿⣿⣿⣿⢷⣌⣿⣿⣿⣿⣿⣿⣿⣾⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡏⣁⠀⣘⣿⣿⣿⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠉⣱⣶⡿⠛⠉⠉⠙⠛⠛⠛⠿⣆⡈⠛⠛⢋⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣷⣿⣿⣿⡿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠋⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢷⡙⢿⠀⢻⡟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⠀⠀⢳⣛⣶⠦⠄⠀⠀⠤⢶⣞⣛⣃⠀⠈⢻⣦⣐⠀⠉⠀⣤⡠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⡇⠀⠹⠯⠻⠓⠀⠀⠀⠀⠛⣟⡿⠟⣷⣄⠀⠉⠛⢷⣤⡀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⠈⢿⠁⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⢾⣳⣿⢷⣾⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡟⢻⣿⣦⣠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠏⠋⠀⠈⠀⠙⠘⣯
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⠙⠧⡀⠀⢀⣤⣤⣤⣄⠀⠀⣠⠀⠀⣀⣾⢮⣴⠆⢀⡴⠹⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡏⢻⢿⡿⠃⢀⣴⣿⣦⡀⠙⠒⠚⠁⠀⣬⣈⠀⣼⣿⡇⢘⠻⢾⡟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠈⡏⢿⣤⣀⣀⣠⣾⢏⣿⣿⣯⠀⠀⠀⠀⣰⣿⣿⣀⠃⠑⠁⠠⠭⢿⡟⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣻⣭⣛⣻⣷⣿⣁⢩⣿⠀⢀⡠⠼⣽⠟⠋⠻⣷⣤⣤⣴⢾⣿⠀⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⣿⣿⠍⠙⣿⢿⣿⣿⠃⢮⡑⣫⣣⢍⣁⣤⡬⠀⠀⠙⢳⣝⢿⣧⢞
⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⣣⢆⣤⣿⣾⡿⠃⠀⢨⣿⢏⡔⣽⠋⠀⠀⢀⣤⣾⣿⣿⣿⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣥⣾⣿⡿⠋⠀⠀⠀⠸⣿⣼⢰⣿⠆⠀⢠⣾⣿⡿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⢼⣿⡿⠿⠋⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣿⣿⣿⣭⡭⢝⣛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⢷⡮⡳⢔⠭⣊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣕⡭⣊⠕⠊⠔⠪⣦⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠙⠀⠀⠀⢸⣿⣿⡿⣿⣝⡻⣮⢉⠓⠦⣤⡐⢿⣿⣾⣯⣉⣉⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡷⣝⢪⡲⣕⢦⡀⡳⣝⣦⣽⣿⡯⠓⣤⣄⡑⣷
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣮⡳⡱⡌⢳⡑⢈⢻⣿⣿⡟⡠⡺⠛⢽⣿⣾
⠀⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠙⠋⠉⠻⣗⠔⡙⣄⠙⢦⡹⣿⣿⣿⠈⢀⠀⣿⣿⡃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠘⣎⢪⣳⣽⣿⣿⣿⢸⢨⢦⢫⢸⠃
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⡳⣌⠿⣾⣿⣿⣿⣿⣿⣿⣊⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡝⣝⢮⣓⣚⣫⠵⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡻⣝⠮⣕⣒⣲⡴⠞⡽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣕⠮⣟⣒⣒⣦⠴⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡫⠖⢒⣒⣬⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋

⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣰⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠴⢿⣿⡟⠋⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣽⣶⡶⠆⠀⠀⠀⠀⠐⠞⣿⣿⠿⠂
⠀⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣄⡀
⠀⣼⣿⣿⣿⣷⡆⠀⠀⠀⢠⣶⣾⣿⣷⣶⡄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⠀⠀⢰⣿⣿⣿⣿
⢠⡟⢿⣿⣿⣿⡇⠀⠀⠀⣾⠛⣿⣿⣿⠙⣧⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠙⢷⡄⠀⠈⢿⣿⣿⣿
⢸⡇⠘⣿⣿⣿⡇⠀⠀⠀⣿⠀⣻⣿⢛⡀⣿⠀⠀⠀⣸⡇⢹⣟⣛⠁⣠⡴⠛⠁⠀⢠⣾⣿⡿⢿⡄
⠀⢻⡀⢼⣿⣿⣧⠀⠀⠀⠘⠷⢀⠉⠐⠄⠇⠀⠀⢰⡟⠀⠀⠈⠉⠘⠉⠁⠀⠀⠀⣸⣿⣿⣷⠈⣷⡀
⠀⢸⡇⢨⣿⣿⣿⠀⠀⠀⠠⠂⠀⢄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣷⡈⢠⣶⠀⢀⣿⣿⣿⣿⡄⠙⣇
⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠑⡆⠀⠀⠁⠀⠀⠘⠇⠀⠀⠀⠀⠠⣿⣷⣸⠃⠀⣼⣿⣿⣿⣿⣿⠀⠙⠁
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⢻⣿⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⣸⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣴⠇⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄
⢸⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠈⠀⢄⠀⠄⠀⠠⠤⡄⠀⠀⠀⡀⠀⠀⠀⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠂⠀⠀⠀⠒⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣀⣀⣐⣀⣀⣀⣀⣁⡉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠀⠀⠠⠠⢄⠀⠀⠀⠀⠀⡇
⠀⠀⡇⢀⠤⠀⠀⢄⠀⢠⠔⠢⡀⡔⠒⢄⠁⠀⠀⠀⠀⠀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡸⠂⠀⠀⠀⠀⠁
⠀⠀⡇⡎⠀⠀⠀⠀⢡⠘⢄⠤⠃⠣⣠⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠶⠲⡄
⠀⠀⡇⠇⠀⠀⠀⠀⠸⢰⠁⠉⡄⡋⠈⠁⠀⠀⠀⠀⠀⠀⡉⡈⣁⠀⠀⣀⣀⡀⠀⢀⠀⢠⠀⠀⠀⠘⢄⡀⡀
⠀⠀⡇⠈⠢⢄⡠⠔⠁⠈⠒⠊⠀⠁⠒⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠉⠈⠉⠉⡉⡉⠁⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⡀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢩⡋⠽⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠇⡄
⠀⠀⢰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⢀⠀⠀⣇⠆⠀⢹⢠⠎⡗⠀⡔⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠘⢤⣠⣤⣎⡫⠃⠀⢇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠢⠈⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀⠀⢀⡀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⢡⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣤⠖⠛⢂⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣀⣠⡯⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠶⢶⡶⠆⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣿⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣄⣺⠗⠁⠀⠀⠠⠀⠉⠁⠉⠉⠀⠷⡿⠀⠁⠀⠁⠀⠀⠉
⢰⠿⠉⠁⠀⣀⠀⠀⠂⠠⠀⠀⠐⠋⠀⠋⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠺⠇⠀⣤
⠀⠀⠀⠀⢈⠻⠈⠀⠀⢀⡄⢰⡄⠐⠂⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁⠰⡇⢰⡾
⠀⠀⠀⠀⡎⠀⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂⠀⠀⠐⣃⠠⡆
⠀⠀⠀⠀⠩⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡨⠖⠺⠛⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⣀⡄⠐⠋⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⠿⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠃⢀⣠
⡀⠀⡄⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⢿⣯⡀⢠⣟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠄⠐⠋⢀
⠀⠀⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠃⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡙⡁⠀⠀⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠒⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡷⠰⠂⣇⣸⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠢⠀⠀⠀⠀⠤⠆⠐⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⢘⣿⣿⣿⣿⡇⠀⣳⠿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠀⢠⣶⣶⠀⠀⠀⢀⣖⣂⠸⠯⠌⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢜⣭⣽⣙⠃⠠⠘⠀⡿⠻⠄⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠈⠀⠈⣸⣿⣠⠴⠉⠀⣿⣇⠐⢃⡘⠉⣀⣀
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠘⠙⣬⠂⡶⠀⣿⡻⠃⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠀⠀⠀⠀⢸⣿⣴⡖⢻⡟⣯⡍⢨⣯⢨⣿⡇⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠛⣧⡿⠺⣧⡤⢾⠳⠄⢀⡆⠀⠀⠐⡆⠀⠀⢤⡾⢶⡿⠿⡿⡆⠇⢹⠇⠘⡇⠸⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠊⠣⠀⠻⠇⠀⣸⡗⣾⡧⠒⠀⠐⠀⠀⠐⠐⠒⠀⠈⠀⢈⣡⣤⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠫⡹⣆⠔⡀⠳⡄⠀⠀⠊⠸⠃⠱⠠⡤⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣃⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⡔⢱⡷⠀⠙⢜⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠁⣽⣇⠴⠆⠊⡹⠅⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣃⠞⣹⠾⠆⠀⠀⢡⣡⣜
⠀⠀⠀⠀⠀⠀⠀⢰⠇⣦⢻⢃⠘⠃⠀⠀⠀⠀⠀⠣⣿⣣⣎⡀
⠀⠀⠀⠀⠀⠀⢠⡿⢚⣀⢃⣾⠄⠀⠀⠀⠀⠀⠀⢰⡀⣳⣵⠰⡄
⠀⠀⠀⠀⠀⠀⠀⡼⢡⡟⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⣱⢎⡁⢤⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡙⢼⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⣴⠶⣿⣿⣿⣅⡀⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠋⠀⣠⣿⢉⣧⡙⡙⢷⡀⢀⠢⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣀⡴⢡⣿⠘⣿⣧⢹⡌⢻⣄⢣⠘⢦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⣤⣤⠶⠛⣁⣴⠟⢹⡆⠙⠛⠦⣷⡀⢻⣎⢧⠀⠙⠷⣤⣌⣙⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣾⡿⠛⢉⣠⣶⠿⠛⠁⠀⠀⢿⡀⠀⠀⠸⣷⠄⠹⣷⡀⠀⠀⠀⠀⠉⣉⠬⠙⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⣧⡀⢉⠻⢦⣤⡤⠖⡋⠁⠀⠀⢀⣤⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⣴⣿⣿⣿⠒⠶⣄⠀⠀⠀⣠⡴⠞⢿⣄⠀⠀⠈⠻⢦⣿⣶⡶⢖⡉⠄⠀⢖⣤⣿⣗⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣾⠁⣸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⡴⢾⣿⣿⣮⢳⣦⣤⣤⣤⣴⣶⠿⣷⣶⣾⣿⣎⢿⣏⠳⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣿⣿⡏⠉⠉⠀⣰⠀⠀⠀⠈⠈⠉⠉⠘⢿⣽⣿⣿⣿⣿⣿⣷⡘⢆⠹⣷⡹⣟⢿⣷⣶⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⡿⢹⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡜⣿⡿⢹⡟⡞⡄⢹⣷⢹⣶⣽⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⢀⣿⣿⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸⢧⡿⠁⣸⠧⠀⣿⢸⣿⡇⣿⡆⢿⣆⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣷⡄⢠⣶⣶⢶⣤⡄⠀⠀⠀⠀⠀⡸⢁⣴⠟⠁⣰⠏⡀⡰⡏⢸⣿⣷⠘⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⢫⣶⣿⣿⣿⣷⡈⠛⠷⠖⠉⠀⠀⠀⢀⣤⣎⣴⠟⠁⣠⡞⠁⣰⠟⣸⡇⣼⣿⣿⠀⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠁⢠⣿⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣴⣿⡿⠋⢀⣠⠾⢋⣠⠞⠁⢰⣿⠁⢿⣿⣿⡾⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡏⢸⣌⢻⣿⣿⣿⣷⣶⡾⣿⣿⡿⠋⣀⣴⠟⣡⠶⢋⣤⠶⠖⠛⣸⣢⠸⣿⣿⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣷⠀⠻⣷⣝⣿⣿⣿⣿⠇⣿⠟⢠⣶⢟⡵⠞⠁⣠⡯⠔⣲⣯⡴⠛⠻⣦⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⢳⡀⠘⣦⠑⠸⡏⢻⣿⣿⠏⣰⡏⣰⣿⢣⡞⠁⣤⠞⠁⣠⠞⣽⡟⠀⠀⢀⣼⣦⣿⣿⣯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⠇⢸⣷⡀⠘⣧⠀⢰⣼⠟⠁⣴⣿⣀⣿⢣⡟⢀⣾⠁⠀⣴⠏⠘⣿⠀⠀⢠⠟⠁⠀⢿⡄⢧⠻⣮⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⢠⣿⣿⣷⠀⠘⣇⠈⠁⣠⣾⢻⣿⣿⣿⠸⡇⣼⡇⠀⢸⡿⠀⢸⡟⠀⠀⠰⠄⠀⠀⠈⢧⠈⢧⡈⠻⣦⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢸⡇⢹⣿⠀⡀⢿⣠⡾⢿⡿⢸⣿⣿⣿⣇⢸⣿⠁⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢻⣦⡈⠛⢦⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⣿⣠⣷⠿⠋⣠⢪⠃⣼⣿⣿⣿⣿⡌⣿⠀⠀⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣿⢦⣄⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣷⡴⠟⠋⠀⢀⡴⢡⠎⣰⣿⣿⡟⢸⣿⣷⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢳⡍⠳⢦⡻⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠟⠁⣀⣀⡤⠔⢁⣴⢋⣴⣿⣿⣿⡇⡄⣿⣿⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣄⠀⠙⢶⣄⡉⠻⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠗⠛⣉⣉⣡⣤⡶⢟⣵⣿⣿⠏⣼⣿⢰⡇⣿⣿⣿⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠳⣦⣀⠉⠻⢶⣤⡿
⠀⠀⠀⠀⠀⠀⢠⠞⣁⣤⣶⣿⣿⡿⡿⠿⠿⣿⣿⣿⠋⣼⣿⠇⣼⢡⣿⠏⣿⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⣿⣷⣌⠙⠿⣿⠛⢹⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣭⣥⣴⠾⡷⢶⣶⣞⣋⣡⣾⣿⠟⠠⢣⣾⡟⢰⣿⠀⠀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣻⡿⣿⠈⣷⣀⡿⢀⠆
⠀⠀⠀⠀⢠⡾⢋⠏⢰⣿⣿⡿⣡⢫⡆⢾⣿⠉⣿⣿⡿⢋⣀⣴⠿⡿⢁⣿⡇⠀⢀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣀⣴⣾⠟⠀⠿⣳⡿⢋⡴⠃
⠀⠀⠀⠀⢻⣅⠈⢧⠘⣿⣿⡇⣿⢸⣧⡈⠉⢶⣮⣛⣻⣛⣋⣥⡞⢁⡾⣿⠁⠀⣸⣷⠟⢧⣀⣀⢀⣀⣠⠴⠒⠋⠉⢉⠾⠋⠁⢀⡰⣶⣯⠖⠋
⠀⠀⠀⠀⠀⠙⠳⣤⣙⠺⣿⣿⣼⣆⠻⠦⣭⣜⠛⢿⣟⣿⡿⠋⢀⠞⣽⠃⠀⣰⣿⡇⠘⣷⣬⣉⣉⣁⣠⣤⣤⣀⣤⣤⡤⠶⢚⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⡙⠛⠿⠶⠦⠽⠷⠾⠛⠉⢀⠴⣋⡾⠁⢀⣴⡿⣡⣯⡾⠛⠛⠛⠋⠉⠀⠀⣽⣿⣿⠙⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣭⣙⠲⠤⣀⡀⠀⠀⠀⣈⣤⠞⢋⣠⡴⠻⣿⠀⣿⠁⠀⣀⠔⠒⠒⢒⣒⣚⡋⠭⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡋⠛⠻⢿⡿⣷⣶⠞⠋⠹⡇⠰⠙⠦⠬⢉⣉⠴⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠛⠛⠉⠀⠀⠀⠀⠈⠑⠒⠻⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢴⡖⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠶⠿⣦⣿⠑⢢⣤⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⡆⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠊⠹⡄⢸⠇⠱⡄⠀⡋⠐⢦⠀⡀⠙⠲⣗⣠⣴⣿⡒⠀⢀⣙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡺⣧⣀⣞⡄⠰⠤⣴⠛⣼⣶⠞⣿⠥⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠙⢤⡀⢒⡿⢿⡄⠀⠀⣨⣿⡞⢯⡆⢇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⡗⢦⡰⡆⠀⡄⢙⠀⠀⠀⠈⠉⠀⠈⢷⠀⡼⡟⠐⠹⠉⡆⠈⠶⠁⠀⡠⠚⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⡏⠓⢴⢶⡀⠻⣦⣡⣎⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡿⢙⠀⢸⢀⠜⠁⠀⠀⠀⣸⠁⠀⡇⠀⡴⠒⠒⠒⡘⠋⣩
⠀⠀⠀⠀⠀⢯⠉⠓⢄⣣⠀⡈⢧⠀⠙⣅⠯⣹⠙⠷⣼⣧⡀⠀⠀⠀⠀⠀⠐⣇⠏⠀⠀⠋⠀⠀⠀⠀⠀⠡⣼⣶⡴⠶⠻⡗⣂⣉⠤⠚⠁
⠀⠀⠀⠀⠀⠘⢆⠀⡀⠃⠑⠼⣦⣀⣀⣋⣀⣈⣀⢔⡯⠯⢛⠦⡀⠀⠀⠀⠀⡇⡀⠀⢀⠛⢦⠀⣶⣀⣴⠞⡻⠳⢎⠐⠢⡄
⠀⠀⠀⠀⢀⣀⠀⢑⣿⣷⣶⡟⢻⡟⢯⠉⠉⢩⡿⠛⠩⠭⣜⠳⢬⣆⠀⠀⠀⣤⡇⠀⢸⢀⢀⣧⣿⡿⣄⠀⠱⣄⠀⠑⠀⠈⡄
⠀⠀⡠⠚⣩⠤⠞⡟⠁⡎⠃⡼⠺⢀⡼⢠⡤⠾⢅⣀⡀⠀⠀⠉⠢⡈⠢⡀⢰⢿⡇⠀⠈⢳⢟⣿⡤⠄⠛⠠⡀⠀⠉⠓⠦⣀⡇
⢀⠜⠀⠀⢀⣠⠜⠁⠀⡤⠞⢁⠾⠋⠒⠋⢃⡘⠚⠇⠉⢢⠀⠀⠀⠈⢆⠱⣀⢸⡇⠀⣠⢋⠎⠀⣈⠲⠤⣀⠈⠆⠀⠀⠀⠈⠁
⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⡇⠀⠀⠀⠘⡀⢿⢸⠃⡴⢣⠃⠀⢀⠋⣆⣠⡄⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢣⡆⢴⣞⣴⣅⣹⣗⠺⣴⢥⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢸⢀⡶⢄⡺⡉⠁⢹⢀⣤⣌⣻⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠤⠄⠒⠊⠁⠀⠀⠀⠀⠀⠀⡇⠸⠁⠀⠱⣧⠀⠘⠋⠀⠀⠀⣠⠞⠉⡙⢒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⠀⠀⠀⣿⠀⠀⠀⣠⠊⠩⠗⠉⢶⠉⠙⠛⠋⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠸⠀⠀⣤⣀⣿⣠⡄⠠⢧⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣇⠞⠑⢶⢧⡀⢰⡤⠆⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⡏⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢸⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠈⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆⠀⠑⠤⢤⣀⣀⡠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣴⠤⠤⢤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠤⣀⠶⢤⡔⢒⡵⠋⠉⢫⣵⠂⡰⠶⠶⠈⢿⣶⣒⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡿⠋⢙⣣⠊⠉⠠⡿⠋⠁⠀⠀⢀⣮⠃⠙⡍⠉⠑⠺⣓⡙⠹⢗⢄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠰⠟⠀⠀⠀⠈⠪⢗⣤⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣿⣿⡿⠿⢿⡟⢻⠿⢿⢿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠉⢳⡀⠀⢀⣴⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⢠⣾⣟⣿⣿⠟⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠒⢤⣀⠀⣀⣠⠤⠚⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠶⠦⢤⣀⣤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⣤⡄⢴⢶⣄⣶⣤
⠀⠀⠀⠀⠀⠀⢀⡾⣟⡿⠖⠙⢿⢿⠀⠀⠀⠀⠀⠀⣠⠊⡴⢿⢽⣧⡀⠀⠀⠀⠀⠠⣴⠿⢯⡝⠵⠿⡿⣖⠄⠀⠀⠀⠀⢘⣿⠋⠻⠏⢿⣾⣻⡃
⠀⠀⠀⠀⠀⢀⡤⣞⠁⢀⣤⠀⠀⣿⠇⠀⠀⠀⠀⢰⣟⠈⠀⡇⠀⠐⣵⡄⠀⠀⠀⣸⢋⣆⠀⠀⢀⠖⢄⡈⣾⣆⠀⠀⠀⣼⣃⠀⠀⠀⠉⠹⣧⣵⡀
⠀⠀⠀⠀⠀⣾⣥⡏⠀⠀⠀⠐⣼⢻⡇⠀⠀⠀⠀⣿⢟⣷⠂⠀⠫⠄⠹⣷⠀⠀⠀⠈⣎⢿⢄⡆⠀⠀⠀⠁⢻⠘⠀⠀⠀⣿⣿⡆⠀⠉⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⢿⣿⣧⡀⠠⣄⣼⠏⣾⡇⠀⠀⠀⠀⣯⣾⠀⠀⠀⠀⠀⣤⣏⠀⠀⠀⠀⢫⣸⢹⣷⢀⢘⠀⣀⣾⡃⠀⠀⠀⠻⡇⣷⡄⠀⠀⣠⣿⢾⣿
⠀⠀⠀⠀⠀⠀⠛⡏⠻⠶⢿⢻⡏⣿⠀⠀⠀⠀⠀⣿⢛⣄⠘⠑⠀⢠⣧⠉⠀⠀⠀⠀⠸⣝⠿⠿⣿⣿⠉⡹⣧⠁⠀⠀⠀⠀⣱⠺⡷⣴⣾⣿⡏⠓⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣽⡷⣿⠀⠀⠀⠀⠀⠈⣾⣿⠁⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠈⠁⢰⢸⠿⣸⠇⡿⠀⠀⠀⠀⠀⣿⡆⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠰⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠛⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⠠⣤⣀⣀⡀⢀⣀⣠⣤⠶⠟⠋⢡⣾⠏⢸⡇⠀⠘⢿⣿⣯⡛⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⡠⠴⣶⠛⠉⠉⠉⠀⠀⠀⢀⣴⣿⠃⠀⢸⡇⠀⠀⠈⢻⣯⠙⠳⢦⣤⡭⠟⠋⠉⠛⠓⠒⢦
⠀⠀⠀⣀⡠⣤⠔⠋⠀⠑⠦⣄⣀⣠⡴⠾⣻⡿⣿⠀⠀⢸⡇⠀⠀⠀⠀⢿⣷⡀⠀⠈⠙⠳⣦⣤⡀⠀⠔⠀⠆⠑⢶⣀⠂
⠀⠀⠀⠀⠞⠀⠀⣈⣽⠟⠋⠉⠁⠀⠀⣴⠟⠀⢿⠀⠀⠘⣷⡀⠀⠀⠀⢸⡏⠻⣄⠀⠀⠀⠈⠣⠈⠳⣦⡀⠈⢢⡀⠀⠑
⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⣀⢤⣾⠋⠀⠀⠸⡄⠀⢠⣿⠙⠦⡀⠀⢸⡇⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⢏⠢⣄⠁
⠀⠀⠀⣠⠖⣫⠀⠀⠀⠀⠤⠊⠁⡾⠃⠀⠀⠀⠀⠑⠀⢸⠸⡇⠀⠀⠀⠘⣧⠀⠀⠀⠹⡦⡀⠀⠀⠀⠀⠀⠀⠈⡧⡐⠢
⠠⠒⠉⠀⠀⠸⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⢻⠀⠀⠀⠀⠘⣆⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⢧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⠃⣄⠀⠀⠀⠀⢀⡤⠊⡅⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠈⢸⠉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠂⢄⠀⠀⠀⠀⠀⠀⠀⠌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⣀⣀⣤⡤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠖⠉⡐⠒⢲⣿⣿⣮⣍⠉⠛⠶⣬⣣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⣫⣠⠞⠛⠉⠭⢭⣿⣿⣶⣬⣿⣿⣄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢿⡀⢰⣏⢄⣢⡴⠶⠛⠉⠉⢹⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣪⢏⣼⡇⢸⣛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⢿⡻⡀⠑⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⣷⢿⣿⣇⡌⣉⠁⠀⠐⠚⠒⠲⢤⠀⠹⣿⢿⡀⠹⣌⠳⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡞⣵⣿⣿⠟⠁⠙⠁⠀⠀⠀⠾⠳⠄⠀⠀⢻⣷⣝⢦⡈⢗⢌⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⡜⣟⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⡧⡠⡙⠢⢭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢸⢸⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡋⢇⣹⣌⡛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠈⢀⢂⣿⠀⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡆⡆⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⣰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢥⣙⡻⣿⣿⣦⣤⣤⡤⠠⠀⠂⠀⠀⠀⠀⢸⠀⢧⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣺⠟⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣜⠌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣡⢀⠜⠋⠻⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠁⠀⠠⣺⢿⠈⣿⣹⢿⡆⠀⡐⠁⠀⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⡞⡱⣾⠀⣼⢋⡞⣠⢣⢟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣧⡁⢳⡀⣯⠞⡰⡱⣣⣣⣿⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣻⣿⡇⠁⡸⠐⣱⠿⡫⢾⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣇⡇⡇⡇⡏⠞⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⢻⣌⣄⠡⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣪⠿⣿⡴⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣀⣁⣀⣀⣌⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠤⠄⣄⠀⠐⠒⠢⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣤⠀⠀⢦⡁⠀⠀⠀⠀⠀⠀⡀⠒⢂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢣⠀⠠⠀⠀⢀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣇⠇⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠇⠘⡠⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣒⣈⣉⣀⣒⡀⠀⣠⡖⡁⢰⡏⠀⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠛⢉⣀⡰⢤⣩⠶⣤⡟⢿⣇⣐⣋⡒⢇⠀⣠⠜⠋⠰⣶⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠂⠳⠀⢀⣠⣴⠾⠧⢼⣷⣨⣿⣶⣾⠻⣆⠁⠣⢹⣿⣎⣿⣿⣤⠄⠀⠀⠉⠻⣌⣩⠜
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠁⠀⠀⠀⣀⡤⠋⠁⡀⠀⠠⠤⣀⢹⣿⡿⣮⣧⢾⠿⠃⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠨⢿⣈⡐
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠠⡔⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣟⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣆⣧⣈⣠⡑
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠞⡼⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠀⠛⠙⢦⡰⠠⠀⠀⠀⠀⠰⡁⣿⠇⠀⠀⠙⠚⠓
⠀⠀⠀⠀⢀⠄⠀⠀⣀⣠⢀⣡⢧⠁⠀⠀⠀⠀⠀⠠⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⣟⡟
⠀⠀⠀⠀⠖⠊⠉⠉⠁⠈⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⢺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⢀⣀⣗⣤⠴⣎⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣥⢀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⢧⣴⣿⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡉⠿⣧⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⣾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣿⡿⠘⣄⠀⠀⠀⠀⢀⣰⡹⠈⠤⣂⡴⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠤⢤⡬⢤⠖⠒⡿⠉⠘⣆⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠊⠐⣶⠀⡁⠰⠀⠉⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠂⣏⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣾⣿⣿⣶⠤⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⣛⣉⣤⡀
⠀⠀⠀⠀⠋⢻⣿⡿⠀⠀⠀⠀⠀⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠋⠁
⠀⠀⠀⢀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣀⣀⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⣴⣾⣿⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡆⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⡿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠛⠿⠿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠙⢿⣦
⠀⢀⣿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡜⢿⡇⠀⠀⠀⠀⠀⣠⣿⣃⣈⣿⣿⣿⡿⠁⣿⡀⠀⣸⡏⢹⣿⣿⣿⠁⠀⢀⣴⡿
⠀⠈⢿⡄⠈⢛⣿⣯⣿⠄⠀⠀⢀⡴⢀⠘⠄⣸⡇⠀⠀⠀⠀⠐⠛⠛⠉⠙⣿⣿⡿⠁⢀⣼⠇⢠⣿⠁⠸⠛⠷⣖⣤⡶⠛⠁
⠀⠀⠈⣷⠀⢾⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠐⠤⣀⠀⠀⠀⣠⣿⣿⣿⣧⡾⠛⠁⢀⡿⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣿⡄⢀⣿⣿⣿⣿⠀⠀⠘⠘⠀⠀⠀⣿⡀⠄⠀⠀⠈⠙⣷⢰⣿⣿⣿⣿⣿⠇⠀⢀⣾⠁
⠀⠀⠀⠈⠁⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⠀⠈⠙⠀⠈⠢⣄⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⡀⢠⣤⣿⡀⠀⣀⡤⠂⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⢠⣿⡟⠀⢤⣴⣤⠄
⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⢰⣿⣏⠙⠀⠉⠁⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠻⠅⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣸⠏⣿⠇⠀⠀⠀⠀⢸⠏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡿
⠀⠀⠀⣠⡏⣠⣿⠀⠀⠀⠀⢀⣿⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿
⠀⠀⣰⡿⠠⢻⡿⠀⠀⠀⠀⢸⡏⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡄
⠀⠘⠟⠁⠀⠸⣇⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠆⠀⠀⠀⢺⣅⠀⠀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠄⠀⠀⠠⠀⠟⠁⢐⡡⠖⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠘⠻⢾⣶⡁⠀⣀⢠⠖⣋⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⢠⣈⠹⣟⠉⠀⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⣌⣧⡀⠀⠁⠻⠄⠠⠂⢂⡤⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⠉⠻⣆⠛⢛⠋⠀⡀⠋⡀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣀⠛⠛⢉⣛⣉⣉⣿⡁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⢛⣿⣿⣉⣽⠛⢿⣷⣌⣷⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠏⣹⣏⡄⠈⣿⠄⠚⢧⢈⡋⣁⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡏⣸⣛⣿⡃⢀⣠⣆⠀⠴⠀⢹⣁⠀⠐⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣬⠁⠀⠓⠉⣤⠈⢳⡀⡀⠊⠻⡁⠃⠘⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠁⢸⣣⠖⡉⠱⡄⠀⠀⣸⠿⡤⠴⠆⠘⡎⠰⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣇⣦⠼⣿⠬⠷⠤⠽⡴⡊⣠⡀⠹⣦⠔⠊⢣⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡼⡇⠀⠈⢦⣠⠖⠋⠙⢎⠉⣉⣀⡌⠁⠷⠄⠳⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣧⣿⣀⡤⠘⢓⠶⠿⠄⠸⡏⢡⡄⠈⣷⠀⢀⡄⡆⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢉⢻⣧⠆⠀⢠⣠⠴⠐⢧⡔⠋⣩⠜⢷⡊⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠙⠄⣹⣆⣈⣼⡅⢠⣦⠀⠳⡜⠁⣠⡄⢡⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣥⡁⢹⡁⢨⠤⠐⠛⢄⡁⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠯⠾⢩⠃⠀⣤⠀⠬⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⣿⣿⣶⣶⣾⣭⣝⡲⣬⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣝⣛⡻⠿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⡴⡍⡊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢻⡜⣆⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⡘⣿⣿⢸⣿⢸⣿⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣿⡏⣾⣿⢸⣿⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣾⣿⣯⣿⢋⣾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣵⠟⠡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢫⣾⣻⠇⣼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡏⣿⣼⣯⠁⠐⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡸⣯⡘⢿⣿⣄⢠⣞⣚⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣮⡳⠌⠻⣶⣝⢻⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⣾⣿⣦⡈⣷⡝⠧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢉⣵⣿⠿⠃⠐⢵⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠯⠖⠾⠶⠿⠼⠶⠮⠽⡶⠀⠰⠿⠴⠾⠾⠯⠀⢰⠵⠾⠧⠗⠾⠷⠦⠾⠶⠾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡰⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⣤⠠⠄⡁⠀⠀⠀⠒⠀⠰
⠀⠀⠀⠀⠀⢡⠀⢀⠀⡀⠳⡄⠁⠀⠀⠀⠀⠀⠀⠈⢈⡇⠠⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠰⣄⠀⠀⠀⠀⠀⡀⠀⢀⣠⣬⡁⠟⠻⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠈⠓⠉⣿⣶⡆⠨⠀⠀⠈⠺⡁⠂⠀⠁⠌⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠨⡆⠰⠿⠟⠁⠀⠈⠲⠀⠒⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣨⠤⠄⣀⠄⠉⠀⠀⢄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠂⣁⣀⠖⢙⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢸⠏⢠⠈⡆⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⡆⢑⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠧⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠤⣤⣷⣠⠀⠀⠀⠐⢀⠀⠊
⠀⠀⠀⠀⠀⢈⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠚⠀⠀⢀⠾⢾⡿⠓⠂⣄⠀⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢤⠀⠀⠀⠀⡀⠀⠌⡔⠁⠀⢠⡾⠃⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠃⠙⠈⡄⠙⢄⣀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢢⣀⣄⠐⠁⢀⡀⠁⠀⠚⠻⣞⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠈⠀⠀⠀⠑⣾⡋⢹⡧⠂⠉⠂⠠⡆⣀⠀⠉⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⢁⠬⠩⢿⠒⠂⠈⠛⡿⠟⠙⣄⠀⡀⠐⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠘⢈⣢⡶⢀⠐⡀⠠⡄⡈⠂⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⢈⡞⠈⠁⠀⠃⠐⠀⠀⠀⠈⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠴⠶⠚⠿⢉⡼⢸⠃⠠⣤⣀⣀⡀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣮⣅⣤⡀⣀⣩⣵⣾⡇⠘⡄⣸⠿⠿⣤⠀⡘⣹⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⡶⢤⣩⣦⣼⣯⣭⣄⣉⠉⠣⢠⣸⡯⠚⡠⢊⣾⠻⠇⠀⠀⠁⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠫⣾⣾⡷⣲⣾⡛⠋⢻⠋⠉⡿⡷⢶⡆⢿⣛⡐⠿⠟⠟⠓⠒⠂⠤⣀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠤⢴⠶⠖⡋⠉⠉⠉⠙⠿⣭⣇⠀⠀⠀⠀⠀⠀⠈⠘⢧⡘⢀⢴⣧⣴⣾⠛⠛⠛⠓⠦⣝⣶⡴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⣠⣤⣾⣼⡶⢋⣠⣶⣶⠶⠦⣄⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣎⠁⠀⠙⢇⡀⠀⠀⠀⠀⠀⠉⠀⢨⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠤⣶⣿⠛⡩⠎⢠⣿⣯⡅⠁⠀⣤⠀⢽⢶⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⢼⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡰⠖⢛⣋⣁⠀⠀⠸⣗⠁⠀⠀⠀⠈⠀⠀⠀⢿⡀⠀⣧⠀⠀⠀⠀⠀⠀⡾⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠐⢉⣴⣿⠟⠛⠛⢷⡄⠴⣿⠜⢻⡀⠀⠀⠀⠀⠠⠈⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⡸
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⣾⡟⠀⠀⠀⡄⠀⠁⠠⠀⠀⠂⠀⢠⡀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠇⠀⠀⠀⠙⠒⠒⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠻⢸⡇⢸⡄⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡌⡾
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⣿⠁⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⡼⠀⣞⠀⢊⢀⠁⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠞⠁
⠀⠀⠀⠀⠀⠀⢱⠀⡄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⠀⠀⣀⣀⣀⠞⠁⣰⢛⡿⠃⠃⢠⢀⠉⠀⠀⠀⢀⠄⠀⠀⠐⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢢⡹⠄⠀⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠉⠉⠁⢀⣴⣿⣿⢏⣧⡠⡁⡋⢀⣀⠐⠀⡈⡀⣠⣰⣦⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⡠⠾⣿⣵⣾⣿⣿⣿⣳⣥⣶⣦⣶⠞⣙⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡐⢄⡀⠈⠙⠛⠒⠒⠊⠁⠀⢀⠠⠐⠉⠀⠀⠀⠈⠉⠉⠛⠛⠻⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠂⠐⠒⠂⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠂⠀⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⣠⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂⠀⠀⠀⠹⠉⠛⠋⢁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡞⠻⣤⣶⣿⣁⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⣤⠀⠀⡄⣄⡝⢃⡼⠷⠃⠈⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣧⠄⠠⠀⠹⣶⠁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠡⣀⠀⢠⣿⣟⡀⠠⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⢸⣏⢷⠈⢢⣜⣳⢆⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡀⠀⢰⣿⡇⠀⠀⠈⣿⢸⡇⡄⠙⢣⣴⠋⠑⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣾⠿⠃⠀⠀⠀⠈⢸⠇⡇⠀⢈⠁⢀⣉⡋⠓⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠙⢷⣼⣥⠀⠀⠀⠀⠀⠀⣸⠀⡇⢀⣼⠟⢩⣤⣿⠓⠀⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠌⢻⡀⠀⠀⠀⠀⢀⡇⠀⠁⢹⠿⠉⠉⠛⠋⢦⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡘⠀⡀⢠⠌⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⣤⠠⠉⢹⣰⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⣀⣀⣀⠴⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠆⠀⠄⠀⡤⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠸⢇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠆⠹⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠦⣄⠀⠀⠀⡇⠳⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⠙⠦⣄⣻⠀⠘⢦⢀⡞⣹⠃⠀⠀⢀⣀⣤⠶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣷⠆⠀⠙⠁⠀⢨⡟⢠⠷⠒⠒⠛⢉⣡⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠛⠉⠉⠉⠙⠳⣤⠀⠀⠈⠓⠛⠀⠐⠿⣏⣉⠉⠙⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡖⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⣁⡴⠖⠛⠋⠉⠙⠲⠦⢤⣭⣽⣶⡦⠀⠀⠈⠓⣆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠳⡀⡀⢡⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣙⣦⣿⣆⣟⣀⣬⡿⠾⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢉⡀⠉⠐⠀⠈⠉⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠐⠊⠉⠉⠈⠛⠛⠛⠙⣿⡙⠻⠛⠛⠿⢶⣶⣤⡀⠀⠀⠀⢀⠞⠀⣴⠿⣯⡹⠇⠀⠀⠀⠀⠀⠲⣛⣷⣦⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⢙⡂⠀⠀⠀⡞⠀⣾⣯⠽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⠭⠶⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠁⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣛⣽⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⡀⢸⣿⢟⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢇⠀⣿⣯⢵⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠈⡄⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡿⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠘⢦⡈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⠤⠴⠧⠀⠠⠄⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠚⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣐⡀⠉⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡴⠟⠃⠉⠉⠉⢁⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡟⠁⠀⠀⠀⠀⢀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⢀⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⢐⡟⠀⠀⠀⢀⣠⢤⡆⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠿⣷⣶⣄⡖⣸⡇⣀⣶⡶⠉⠀⠸⢰⣦⡻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣀⣤⣤⣤⣤⣽⣿⣽⣿⣿⣿⣯⣄⣠⣴⡆⢸⡇⠑⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣰⢟⠛⠋⠉⠁⠀⠀⠀⠉⠙⣿⣉⠛⠛⠛⠛⡇⣿⣤⠄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠑⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⣼⠃⣿⢿⣶⣄⠀⠐⠹⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⢤⣀⡀⠀⠀⠀⠀⣀⠻⡀⠠⣊⣼⠇⠘⣷⡹⣷⣄⠰⢿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⠤⠤⢤⣌⣻⣦⡈⢻⣧⣀⣒⣾⣟⣡⡾⠟⢻⡋⠀⣬⣿⡇⠈⣷⣦⢾⡿⠁⠀⣠⣴⠶⠶⢤⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠁⠀⠀⠠⠴⠶⠛⠛⠛⠒⠻⠿⣏⣿⡟⣫⡄⠀⣤⣹⢿⣷⣿⣧⣦⣼⣿⣾⣧⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠗⠛⠉⠉⠉⠉⠉⠉⢉⡩⠥⠶⠶⣾⣯⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣦⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⣾⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣎⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣿⣿⡟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡙⡇⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⣧
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⢻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡌
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠉
⠀⠀⠀⠀⠀⠀⠠⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢲⣟
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠃⡂⢠⠀⠀⠀⡀⣄⠄⣀⣀⢤⣄⠀⣤⣠⣴⣿⠿⠯⠉⠁⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢻⣿⣶⣾⣿⣷⣾⣿⣿⣿⣿⣾⣧⣿⣿⡿⠛⠁⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢾⣷⣞⣿⣿⣽⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⢢⣴⣼⣾⣷⣶⠟⣐⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⢶⣶⡤⠙⠁⠀⠀⠈⠀⣾⣿⣋⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣩⣀⠀⠀⢀⣤⣀⠠⣽⣿⠛⠛⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡇⠋⠋⠀⠀⠀⠛⠟⠓⣤⡋⠈⠁⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⢿⡁⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠈⢫⠛⠉⠀⠀⡾⠋⠈⠀⠀⡷⠹⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⢼⢤⣀⠀⠀⣴⠋⠀⢀⡞⠋⢰⣄⣻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣻⢷⣄⠀⣈⣵⣾⣧⠀⠀⠈⠃⣠⠻⠻⣿⣟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡖⠋⢺⣿⣿⣿⡿⠷⠍⠃⠀⠀⣠⡏⢲⣁⠹⣯⢷⣤⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⢄⠀⠉⠉⡀⠀⠀⢀⣴⣶⣤⣿⣿⣮⣋⢁⣼⣿⣿⠡⣀⣀⠀⠉⢲⣖⣀⡒⠀⠀⠀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⣸⣽⣿⠿⠿⠀⡄⠀⣾⡿⣿⣟⢿⣻⣙⣟⣿⣿⡐⣿⣿⣿⡀⣿⡿⠃⠀⠀⠉⠉⠉⣛⠓⠺⠿⠟⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢿⣿⣿⠀⣰⢄⣦⠀⢈⠇⠟⢏⣸⡏⠿⣿⣿⠽⣿⣿⣿⣿⣦⡆⠀⠀⠀⡀⢤⡶⢿⡾⡿⣿⣶⣾⣷⣿⡾⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⣿⡿⣿⣿⣵⣴⣯⣾⣶⠾⠟⠁⠀⡌⢩⣂⣿⣿⣿⣿⣿⣿⣎⠀⠀⠘⡿⠛⢛⣿⣿⣿⣿⣿⣿⣷⣿⡇⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠋⠁⠀⠀⠉⠙⠛⠋⠁⣀⠀⠀⠀⢘⣦⣾⣿⣿⢟⠛⠋⠉⠩⠭⠇⠤⠄⠹⣦⣿⣿⣿⣿⣿⣿⣿⡛⣿⣷⣟⠈⣧
⠀⠀⠀⠀⠀⢀⡀⣄⢀⣄⠀⣀⡙⠒⢢⣴⣿⣶⡶⠾⠊⢁⣤⣴⣶⣤⣼⠋⠋⡀⠈⢀⣤⣀⣠⣤⣤⣤⣤⣤⣡⣄⣯⣙⣿⠿⠿⠟⠛⠛⠋⣁⣠⠟
⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠘⠓⠒⠛⠛⠛⠛⠛⠛⠂⠉⠀⠀⠀⠀⠀⠙⠒⠛⠉⠉⠉⠉⠉⠉⢫⣖⡤⠤⠤⠶⠶⠶⠶⠒⠒⠛⠛⠉⠉⠉

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢠⣀⠀⠀⡠⠄⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠰⣮⣉⣷⡿⢿⠛⠣⠄⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣿
⠀⠀⢸⠘⠀⠀⠀⠀⠈⠠⠤⠀⠀⠀⠀⠀⡇⠀⡇⠀⠰⠿⢋⡜⣧⣤⠁⠀⠠⠄⠀⢸⡇⢸⠀⠀⠀⢀⣀⣀⠀⠀⣾⡿⢦⠀⠀⣿
⠀⠀⢸⠀⠀⣳⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣻⡇⢙⣿⣇⠀⠰⠶⠀⢸⡇⢸⠀⠠⠞⢋⡭⠍⠁⠀⠈⠁⠁⠀⠀⣿
⠀⠀⢸⠀⠀⣿⡇⠀⠀⣤⠄⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⢸⡇⢸⠀⠀⠰⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠈⠛⢶⣿⣧⡀⣀⡔⣠⣾⣿⡇⠀⡇⠀⠀⠀⠀⢙⠁⣿⣿⣧⣝⣄⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣀⣀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⢀⠀⢾⣿⣝⣩⠍⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢠⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠸⠄⠀⠉⠉⠉⠀⠀⢸⡇⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠛⠋⣿⠿⢿⣿⠟⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣷⣦⣤⣤⣤⣤⣄⣾⣷⣾⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢿⣿⣿⣿⡿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠈⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠾⣿⣿⣾⣿⣶⣦⡀⠀⠀⠀⠀⠀⡀⢸
⠆⢀⠀⠒⢶⣾⡇⠀⠀⠀⠀⠀⣠⠊⠁⢸⣿⡿⠿⠟⠛⠉⠀⢀⣀⣀⡀⠀⠛⢾
⠐⢻⣇⠀⡀⢸⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠉⠙⠻⠀⢺⡀⠀⠀⠀⠀⠀⢰
⠰⠋⠁⣆⣿⣬⠆⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⡀⢀⣤⡤⣤⣠⡀
⡂⠒⠂⠿⠿⠛⠂⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣾⣀⠀⠰⠶⠶⠲⣤⣤⡄⣀⣼⣭⣍⠐⠇⠀⣿⠀⢸⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⣄⣤⡤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠁⠀⠙⢅⢢⠐⡇⡧⠐⡞⠁⠀⠈⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡠⠤⠤⠸⠀⠑⠃⡇⠸⠄⠒⢂⡩⠞⠀⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠐⠓⠂⠐⠒⠤⢀⠐⠂⠊⠀⠰⣊⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠔⠈⠑⠂⠤⠍⣢⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣸⣱⣹⣏⣏⣃⣹⣸⣱⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉


⠀⠀⠀⠀⠀⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣸⣿⣄⣰⠥⠀⠀⡀⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢐⣠⠾⠐⠂⠀⠀⠒⠠⢀
⠀⠀⠀⠠⢤⠿⢟⡩⠒⠁⠀⠀⠀⠀⠉⠢⢕⢦⡀⠀⠀⠀⠀⠲⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⣰⠐⠉⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠑⢵⡄⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢰⠁⠀⠀⠀⠀⢢⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠪⠢⠀⠀⠐⠀⠀⠀⠀⠁⠁⠁⠀⠀⢀⠬⠁⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠑⠩⡢⡀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠕⠁⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤⠤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠀⠀⠀⠐⠊⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣀⣬⡤⢀⢀⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣲⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⢿⡿⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡅⠀⢠⣾⠏⠙⢿⣿⣿⣿⣿⣿⣿⡓⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣿⣃⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣤⠴⣤⣤⠴⠦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣒⣷⣤⡴⠆⠠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣤⣄⣉⣛⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠿⠿⠿⠿⠟⠻⠿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢾⣭⣭⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⣈⢉⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⢋⣵⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠉⢁⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣀⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠛⠻⠿⡿⠿⠟⠋⠁⠀⢀⣠⠄⠀⣄⣀⣀⣤⡄⢀⡀⠀⠀⢀⡄⡄⢀⡀⠀⠀⠀⠀⣸⣿⢀⣴⣶⣴⣿⣧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣄⢀⣶⣾⣿⣿⣿⣶⣏⣴⣶⣴⣿⣿⣾⣾⣿⣿⣿⣿⣶⣭⣿⣶⣶⣿⣿⣿⡿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠇⣀
⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠨⠿⢦⣀⣉⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⡿⢟⠛⠋⠙⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠄⠀⢀⣲⣿⣿⡿⢿⠿⣟⣻⣭⣷⡾⢛⣩⣿⣿⣿⣿⠿⠻⠟⣩⡿⢗⠟⠁⡀⠈⠀⠀⢉⣙⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠉⣠⡤⠒⢊⣩⣽⠿⢛⣫⡵⠞⠋⠉⢿⠟⠋⠀⠰⢶⠿⠁⠀⢀⡤⠀⠀⠀⠀⠄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢀⣀⣤⣾⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢾⣿⣠⣾⣿⢟⣯⣥⣾⣏⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣿⣿⣶⡾⠿⠿⠿⠿⠿⢿⣿⣏⡩⠟⣛⡿⠿⣿⣷⣦⣄
⠀⠀⠀⢀⣤⡾⠟⣫⣽⣿⣿⠟⠋⠁⠴⢒⣛⣿⣿⣿⣷⣦⣝⠻⣦⡀⠚⣫⠽⢿⣿⣿⣿⣦⡀
⠀⠀⣠⡿⠋⠀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠩⠶⣾⣿⣿⣿⣿⣿⣿⣮⣷⡀⠐⣫⣭⣿⣿⣿⡿⣿⣆
⠀⣼⡟⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣫⢽⣿⣿⣿⣿⣿⣿⣿⡟⣷⠀⣚⣿⣿⣿⣿⣿⢽⣿⣧
⢸⡿⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⢸⣿⣿⣿⣷⣯⡾⡟⣿⡆
⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⠛⠛⠛⠛⢛⡛⠿⠯⢹⣿⠛⢻⣿⠿⣿⣿⣿⣿⣿⣷
⣿⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣠⣾⣯⣥⠤⠤⢀⠀⠀⠀⠀⠙⣶⡄⢸⣿⢀⡟⡟⠀⣼⣿⣿⣿⣿⣿⣶⣤⣀
⢻⡇⠀⢄⠈⢒⠤⢤⣄⢀⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣾⣿⣿⣿⣷⣶⣿⣿⣿⣥⣠⣄⣤⣤⣽⣷⣄
⠘⣷⠀⠠⣝⠶⣭⣓⣶⠟⠁⠀⠀⠀⠀⣠⣠⣤⣤⣤⣤⣬⣽⣿⣿⡽⣯⣿⣿⣿⣿⣿⣿⣷⣬⣙⠷⣮⣼⣝⣿⣿⣿⣷⣦⡀
⠀⠹⣧⠀⠈⢝⣲⣿⠃⠀⠀⠀⣴⣾⣿⡿⢟⠋⠁⠀⠀⢀⣤⣶⣾⡿⢟⣿⢿⣿⣿⣿⣯⡙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠙⣷⡀⠑⢾⡏⠀⠀⢠⣿⣿⣿⠏⣴⡊⠀⠀⢀⡶⠛⠑⠋⠁⠀⠉⠐⠉⠔⡿⢿⣿⣿⣷⡄⠀⠊⠵⣚⡵⣫⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⢦⣿⡇⠀⢰⣿⣿⣿⢃⣾⣻⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⠷⣻⣿⣿⣦⠀⠀⠡⣚⡵⢞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⡇⠰⣿⣿⣿⡇⡺⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠵⣚⣽⣿⣧⠀⠀⠥⢞⣫⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣷⢠⣿⣿⣿⠀⣿⡮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⣭⣿⣿⣿⡄⠀⢘⣉⡵⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀⣷⣝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⣿⣿⣿⡇⠀⠨⠴⢞⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⢸⡻⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣛⣽⣿⣿⣿⠁⢀⣒⣯⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⠙⢾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⣷⣿⣿⣿⠇⠀⠠⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣄⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣒⣻⣿⣿⡿⠃⠀⣤⣭⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⣀⠀⠀⣀⣀⣀⡀⠤⣴⣶⣾⣿⣿⣿⡿⢋⣀⣒⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⢾⣭⣭⣭⣿⣿⡿⠿⠿⠿⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⠁⠲⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣋⣱⣾⣿⡿⠏⠙⠛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⡿⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣟⣿⡟⠙⣻⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢠⣿⣿⣿⣟⡏
⠀⠀⠀⠀⢠⣾⣿⣾⣿⢱⣿⣿⣿⣿⣿⣿⠟⠉⣁⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠏⣿⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣿⡜⣿⣿⣿⣿⣿⣴⡶⣾⡿⣡⣿⣿⡿⠿⢿⡿⠸⣿⣹⣿⢍⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠿⣿⢷⣼⣿⡟⠻⢷⣿⣄⣿⣿⠿⠛⣁⣀⣀⣉⣀⣤⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠻⠄⠀⢠⣿⣿⡿⠶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⠙⢉⣿⠛⢿⠿⢿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⢿⣿⣦⠴⠟⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡦⠴⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠇⠀⠈⢻⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢀⠤⠶⠶⢦⣤⠀⠀⠀⠀⣠⣶⡶⠶⠦⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣖⣾⣶⣄⠀⠀⠀⢰⣿⣿⣿⣗⡆⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠈⢙⡧⠿⠋⠉⠀⠀⠀⠘⠁⠙⠻⠥⡟⠁⠀⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢾⣟⣙⣋⣛⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠉⠉⠉⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⢀⣠⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⢸⣿⣿⣹⣿⣧⠀⣀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⢻⣿⣿⢻⣿⣿⢸⣿⡿⣻⣿⣿⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢿⣿⣧⢿⣿⡟⣿⣿⣻⣿⠇⣿⣿⢣⣿⣿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣠⡿⣿⠮⠟⠁⡘⠉⠈⠋⠀⠻⢣⣿⠿⣁⢺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⣿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⠛⠙⠛⠿⣶⣦⠀⠀⠀⠀⣴⣶⠿⠛⠋⠛⠦⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⡀⠀⣀⣤⣶⣶⣄⡄⠁⠀⠀⠀⠀⠈⢠⣠⣶⣶⣤⣀⠀⢠⡹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢺⡇⠀⠙⠣⠴⣿⠭⠿⠆⠀⡁⠈⠀⠰⠿⣭⣿⡦⠜⠋⠀⢸⡗⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣂⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡾⢐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢂⠀⠀⠀⠀⠀⠠⡵⠦⢄⣠⠴⢮⠄⠀⠀⠀⠀⠀⠰⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⣨⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠰⠶⠫⠤⠤⠤⠤⠽⠲⠆⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠦⣄⣀⣀⣠⠴⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠦⣄⣀⣀⣀⣀⣠⡴⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣤⣤⣶⡾⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡀
⢶⡿⠿⠟⢛⡿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣼⣿⢿⡙⠻⠿⢿
⠘⣷⡘⢃⡞⠺⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡟⠻⢳⡤⢠⡶⢶⡄⢤⡾⠛⢻⡍⣿⢾⣿⣿⣿⣿⣿⣿⣿⠛⠗⢳⡘⢃⣾
⠀⠈⢿⣌⢡⡍⢉⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠳⣤⣼⢧⣽⡦⢠⣿⡼⣧⣤⠞⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⡉⢩⡌⣡⡿⠁
⠀⠀⠀⠻⣦⣀⡮⠔⠚⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⡗⠂⢵⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣄⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣦⣿⣿⣿⣧⣿⡇⠀⢸⣟⡙⢋⢻⣿⡻⢟⣿⡿⡙⠫⣹⣇⡈⢘⣿⣿⣿⣿⣿⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣧⣿⡏⠠⠤⠃⣤⠈⣌⡕⢁⡈⢪⣡⠁⣤⠈⠲⠅⠙⣿⣯⣽⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠷⠶⣤⣤⣄⣙⣀⣈⣁⣀⣉⣠⣤⣤⠶⠾⠛⠋⠁⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣋⣉⣿⣀⣬⣉⣭⣉⣸⣟⣩⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠭⠉⠉⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠏⠉⠋⠹⠿⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠿⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⠀⠙
⠀⠀⠀⠀⠀⠈⠹⣶⣄⣀⣀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⣴⣷⣄⣀⣀⣤⣶⠟⠀⠀⠀⠀⠀⠀⢾⣿⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣧⡤⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠃
⡿⢿⢿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠛⢛⣩⣿⣷⣶⣤⣠⣤⣤⣄⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⢀⣶⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⢀⣴⣿⣿⠟⠁⠀⢰⣿⡏⢿⣷⡉⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⣡⣾⣿⠟⢁⣤⣀⣀⢸⣿⣇⠈⢿⣷⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣿⣿⠟⠁⠀⢠⣤⣤⣌⠉⢻⣿⡄⠘⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⢻⣿⠋⠁⢀⠀⡀⠀⠀⠀⠈⠿⠷⣧⠀⢻⡇⠀⣿⡇⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣼⡿⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠈⢦⣠⡴⢚⣉⣠⣾⣿⠃⠐⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣯⣾⣿⣿⢻⡀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠉⠛⣻⣿⣿⡿⠋⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⡏⢸⠁⠸⣿⣿⣿⡍⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢋⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣿⣿⠟⠀⡼⠀⣰⡎⢻⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⡿⢫⣶⣿⣿⠁⠀⢠⡆⠀⠀⠀⠀⠀⠀⠐⠚⠁⣿⣿⣿⠏⠀⡴⢁⣴⣿⠇⠀⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣧⢯⡏⣿⡏⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⡼⢡⣾⣿⡿⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⡏⢸⠁⢻⣇⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢰⠃⣾⣿⣿⡇⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠘⣇⠘⣿⣄⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠓⠘⠀⢿⠸⣿⡇⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡘⣿⣆⢀⢿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠈⣷⣌⢧⠈⠓⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣿⣷⣿⣿⡇⠇⠀⠀⠀⠀⠐⠉⠁⢻⣿⡀⣸⣿⣿⡦⣀⠈⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⣿⣿⣧⡟⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢫⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠚⠉
⠀⢀⡀⢀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣿⠿⢿⣿⡟⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⠀⠙⢿⣷⣦⢭⣑⣒⡾⠤⣬⣙⠮⡻⡿⣿⣿⣿⣿⣿⣿⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠃⠀⠀⠀⠀⠀⠀⠈⠛⠳⣶⣤⣍⣙⡓⢮⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⡛⠛⢻⣿⣿⣿⣿⣟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠀⠀⣀⠤⠐⠛⣿⣿⣿⣷⡀⡀⣿⣿⣿⣿⣿⡣⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣤⠶⠆⠀⡤⠂⠀⠒⢿⡿⠛⠚⠁⠀⠀⠀⠀⠸⡏⠹⣿⢻⣶⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⢤⣀⢤⣠⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⡧⠔⠏⢺⠉⢢⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠠⠘⠁⠀⠉⠒⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡱⠢⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠘⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠢⠀⢀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢟⢆⠀⠀⠀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⢻⣿⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡈⢧⡀⠀⠀⠀⠠⡀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⢘⠻⣿⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢥⠈⠳⣄⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⢈⠴⠋⢦⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠘⡏⠳⣄⠀⠀⠀⢳⠀⠘⣧⠀⠀⢀⡴⠋⡿⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⠶⣄⣀⠀⠀⠙⣄⠈⠛⠦⠄⢸⠇⠀⣿⡥⠞⠋⢀⡞⠁⠀⢀⣀⡴⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠉⠉⠓⠒⠚⠆⠀⠀⠠⣟⠀⠀⣸⠀⠠⠤⠚⠒⠒⠋⠉⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠤⠤⠤⠤⠙⠷⠶⠫⠤⠤⠤⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡟
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠂⡼⠁
⠀⠀⠀⠀⠀⠹⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣀⣀⣀⣠⣴⣶⣦⣄⠀⠀⠀⠀⢀⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠿⣶⣦⣄⣴⡿⠋⠀⠀⠉⠙⠛⠛⠉⠁⠀⠈⠻⣷⣤⣴⡾⠟⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠈⣹⡟⠀⡰⠊⠉⠓⠢⠤⠴⠒⠚⠑⠒⣄⠈⣿⡇⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣄⣀⣀⣿⡇⠀⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠀⢸⣿⣀⣀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠉⠙⢻⣿⡄⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⢀⣼⡿⠛⠋⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⠒⠉⠁⠀⠀⠙⣿⣆⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠉⠳⡀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⣀⣠⣧⣧⣀⡀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡏⠀⣸⠀⢀⣀⠀⠀⠀⠀⠀⢀⣽⡿⡿⢿⡟⢟⡻⡿⣿⣏⠀⠀⠀⠀⠀⠀⢀⣀⢹⠀⠈⢿⣄
⠀⠀⠀⠀⠀⣼⡟⠀⡰⠁⠀⠀⠉⠛⠻⢶⡒⢶⣿⣿⣇⡕⠚⠁⠉⠲⣏⣻⣻⣷⡤⢶⣶⠾⠛⠉⠁⠈⢦⠀⠘⣿⡄
⠀⠀⠀⠀⢰⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⠧⡆⠀⠀⠀⠀⠀⠈⡧⢟⣿⡏⠉⠀⠀⠀⠀⠀⠀⢈⡇⠀⣿⡇
⠀⠀⠀⠀⠀⢻⣦⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⣿⣏⠖⡇⠀⠀⠀⠀⠀⢀⠗⣎⣿⣧⣀⠀⠀⠀⠀⠀⠀⠘⣠⣾⣟
⠀⠀⠀⠀⠀⣠⣿⢷⣤⣀⣀⣀⣠⣤⣴⡶⠿⠻⣿⣕⣏⡦⢄⣀⣀⠤⣎⣛⣽⡿⠉⠛⠻⢷⣶⣤⣤⣴⡾⠟⠉⢿⣦⡀
⠀⠀⢀⣠⡾⠟⠁⠀⠉⣿⡿⠛⠋⠉⠀⠀⠀⠀⢸⢿⣷⣷⣞⣧⣎⣶⣷⣿⢿⠀⠀⠀⠀⠀⠀⠉⠛⢿⣄⠀⠀⠀⠙⢿⣦⡀
⠀⠴⠿⠿⠷⣶⣶⣤⣼⡟⠀⡤⠀⠀⠀⠀⠀⣰⣣⠊⠈⠉⠛⢻⡿⠋⠉⠈⢪⣧⡀⠀⠀⠀⠀⠀⢠⠈⣿⣶⡶⠾⠿⠿⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⢣⡀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⡜⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠙⠲⢤⡈⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⣠⠤⠒⠊⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣤⣀⠀⠈⠲⣄⠀⠀⠀⠀⠀⣼⣧⠀⢀⠀⠀⠀⣠⠎⠀⢀⣠⣴⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⣄⠀⠈⠙⠚⠉⣠⣾⠟⠻⣷⣄⠉⠛⠉⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣴⡶⢿⣿⠁⠀⠀⢨⣿⠿⣶⣦⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⢶⡋⠙⣶⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡋⠉⣿⠀⠀⢻⡄⠹⣆⠀⢘⣧⢤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦⣾⡁⠈⠻⣆⠸⡆⠀⠀⣿⠀⢹⡀⣼⠁⠀⢻⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠈⣇⠀⠀⠘⢧⣹⠀⠀⢸⡇⢸⣿⠇⠀⠀⢸⡇⠀⠀⡿⠳⣄⣤⡴⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣧⡀⠀⠈⣷⡀⠀⠈⢻⡇⠀⢸⡇⢸⣿⠀⡀⠀⣸⠁⠀⣼⢃⡴⠋⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠙⢦⣤⠼⣷⡄⠀⠈⣇⠀⠸⣇⣿⢿⡀⡇⣠⣿⠀⣼⣷⠟⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠸⣇⠀⠈⠻⡄⠀⢸⡀⠀⣿⣿⣾⡿⣿⣿⣷⣾⡟⠁⣠⠆⠀⢀⣠⡴⠟⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⣤⣄⡀⢌⣳⣄⠀⢹⡄⠸⣧⣠⣿⣛⣿⣿⣿⣿⣿⣾⣿⡞⠁⣀⡴⠛⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠈⠉⠓⢦⣝⣷⠋⠙⢦⣿⣏⣹⣛⣿⢿⣿⣿⣿⣿⣿⣷⠞⠉⠀⠀⠀⠀⣀⣀⣤⣴⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠄⠈⠻⣦⣀⠙⢻⣯⣨⡟⠻⡾⣿⡿⢯⣽⣏⠀⠀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠈⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⣀⣀⠀⠀⢀⣀⣄⣉⣉⣿⡉⣿⣇⣶⣧⣨⣇⢀⣿⡴⠶⠋⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠟⣩⡿⠋⠁⢀⡠⠜⠛⢉⣿⡿⠉⢿⣿⣿⣿⣷⣖⣶⣶⣾⡟⠋⠉⠉⠉⠙⠛⠿⣟⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⡾⠁⠀⠀⠀⠁⠀⣠⡴⠛⣷⠦⢶⣤⢿⣦⣤⣤⣽⣿⣿⣭⡛⠳⠦⣤⣀⣀⣀⣠⣿⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣸⠛⢷⣌⠛⠶⠮⣝⡛⠛⢿⣿⠶⣤⡀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠘⢶⣄⢀⣠⠞⣻⡄⠀⠀⢀⣀⣠⠟⢧⠀⠈⠳⣄⠀⠈⠙⢶⣄⣬⡷⠄⠙⠲⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠋⠉⠉⠁⢠⡏⢹⠗⠚⠉⠉⠉⠀⠈⢧⡀⠀⠘⢧⠀⠀⠀⢻⡀⠙⢦⡀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠾⢧⠀⠀⣼⠃⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠃
⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣸⠃
⠈⢯⡙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡏
⠀⠘⣧⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃
⠀⠀⠘⣆⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿
⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡇
⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⠶⣀⠀⠀⠙⠦⣄⣸⡇⢸⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠓⢬⣄⠀⠀⢹⠁⣼⠟⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⠿⣦⣾⣤⣾⣶⣤⣤⡤⠤⠄⠐⢛⣋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⣻⣤⣀⣀⣀⣀⣀⣤⣴⠶⠿⠿⠟⠛⠛⠻⠿⠿⠶⠦⠤⣤⣀⣀⣀⡀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⢸⠉⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿
⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠙⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠙⠉⠷⡶⠋⠋⠉⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿
⡄⢹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⢠
⠀⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⡆⣲⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠈
⠐⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠁⠀⠶
⠀⠀⠀⠀⠈⠿⣿⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⡦⢀⠀⡄⠈⢿⣿⣿⣿⣿⣉⠛⠿⠿⠛⠛⣩⣿⣿⣿⡟⠀⠀⡀⠀⠉⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠯⡅⡏⢱⢘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢰⠉⠀⡷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠙⠿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣷⣿⣿⣶⣶⣶⣤⣲⣞⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡾⡻⣿⡁⢿⣿⣿⣿⠿⡌⠙⠉⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⣤⣠⣏⣳⣾⣇⠈⠁⠱⢷⣶⠃⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⡇⠻⢿⣿⣿⣷⣵⣤⣀⣀⠀⣴⣿⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠢⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⣄⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⠛⡿⣿⣿⣷⡄⠀⠀⠈⠛⠻⢿⠿⠿⣿⠘⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠘⠀⠀⢸⣿⣿⣿⣶⣷⣶⣿⡆⣠⣶⣾⣿⠰⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠁⠀⠛⠈⠉⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣳⣷⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⠟⠷⣿⣿⣯⣽⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢸⡟⡿⠿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⠀⢘⡀⢠⣴⠟⠋⠀⠑⠆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣿⣷⣄⣈⣛⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠒⠿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠐⠂⡀⠰⠀⠀⠀⠀⢰⠁⣴⣿⣿⣯⡙⠿⠿⣻⣤⣙⣲⣦⣄⡀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠂⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⢿⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⡿⢛⣦⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢷⢽⣿⢇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠷⠋⠀⠀⠹⠠⠄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢘⢉⡉⣻⣟⠉⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⢿⡷⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣐⠛⠿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠂⠀⢀⣈⠁⠀⠀⠀⠂⠀⠀⠀⠄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠶⠿⠟⠁⠳⠶⢶⣀⡤⠀⠀⠛⢷⣦⣭⣼⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠴⢧⠠⣤⠀⢸⣷⣽⣿⣿⣿⣷⡀⢃⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⠷⠢⠀⠀⠠⠿⠶⠼⠿⠿⢿⣿⣿⣿⣷⡀⢷⡄⠀⠀⠀⠀⠀⠈⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣉⡀⠀⠀⠀⢠⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⣴⡶⣠⣶⣶⡆⠀⣴⠶⠂⠀⣀⣀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢋⣶⣿⣿⠁⢿⣿⣿⣧⢸⢃⣤⣾⡿⠿⠻⢿⣿⣿⡿⠟⠋⠀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⣿⣿⣿⣿⣄⠈⠻⢿⣿⡿⢻⠟⠁⠀⠀⢴⣿⡿⠧⠀⠀⠀⠀⠐⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡹⣿⣿⣿⣿⣶⣄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡼⠃⠀⠀⠈⠑⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⢀⣐⣦⣀⠀⠀⠀⠙⠳
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣝⡻⢿⣿⡌⣿⠿⣿⠃⠀⠉⠛⠿⣿⣿⡶⠄⠀⠀⠀⠀⠀⠰⠶⠶⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠙⠉⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣭⣿⣮⣅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠲⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⠇⢰⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢸⡀⠀⠀⠀⠀⠀⠈⣿⣿⠻⠭⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⠈⠻⢿⡶⠴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣌⠳⡄⠀⢸⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⣀⡤⠀⠠⠷⡘⡄⢸⢠⠖⢉⡤⠭⠶⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠭⠒⠂⠉⠑⢄⡇⠈⠎⠐⠉⠉⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠁⠀⠀⠀⠀⠊⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⣙⠢⡀⠀⠀⡜⠀⠀⣀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⡤⠀⠀⠀⠀⢀⠄
⠀⠀⠀⣀⠴⠊⠀⣀⣤⠤⣌⣦⢱⠀⢰⠃⡠⠊⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⢦⠀⢠⠀⡠⢖⡶⠖⠋
⠀⢀⠞⠁⠀⣀⡭⠿⠯⢍⠲⢝⢺⡇⣸⡼⠛⣉⣉⣲⣄⡊⡀⢀⡆⢸⠀⠀⡤⠀⠁⠀⠃⠘⢦⡈⢧⡘⣾⢡⡿⠚⠉⠩⠭⠉⠙⠲⢤⡀
⢠⠏⠀⡠⠚⠁⠀⠀⠀⠀⠤⠞⠛⠀⠞⣁⠋⠉⠓⢦⡙⢋⡁⠈⢀⢸⠀⠈⣠⠤⣄⣀⣀⡀⠐⢽⡆⠙⢿⠈⡄⠀⠀⠔⠒⠒⢦⣈⠀⠙⢄
⣿⠆⠀⠠⠊⠀⠀⠀⢀⡠⠀⠀⠀⠀⠚⠁⠀⠀⠀⠢⠙⡄⠙⠦⣿⠘⣶⢏⣴⠾⢭⣙⠋⠀⠀⠀⣹⣤⡤⠜⠁⠀⢠⡀⠀⡀⠀⠈⠳⡀⠈
⢿⠀⠀⠀⠀⠀⢀⡴⠫⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆⠀⠸⡀⣀⣮⠀⠛⣠⠀⠀⠀⠨⠑⢦⡀⠰⠋⠀⠀⠀⡀⠀⠀⠳⡄⠈⠳⡄
⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⠀⠃⡟⠉⠉⠉⠈⠁⠀⠀⠐⠲⣄⠀⠱⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⠀⢀⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⡆⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡀⠀⠹⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⢦⣀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡰⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⠔⣪⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⠄⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠜⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⢦⣀⠀⠀⣀⡠
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣤⠶⠶⠿⠛⠿⠿⠷⣶⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⡶⠿⠿⠛⠿⠷⠦⣤⣀
⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠉⠛⢻⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡾⠛⠉
⠀⠀⠈⠉⠩⠭⢟⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡻⠭⠍⠉⠁
⠀⠀⠀⠀⠀⠘⢩⠵⢛⡵⢋⡾⢹⠏⡟⡿⠻⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠈⠟⢿⢻⡹⡏⢷⡙⢮⡛⠮⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠃⠙⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠻⠷⠦⠄⠀⠀⠀⠴⠾⠿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠉⠉⠙⠛⠛⣻⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣤⣝⣴⣬⣩⣷⣥⣿⣯⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠐⠒⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠚⠓⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠈⠉⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣈⠂⠀⣀⠂⠠⣀⣀⠠⠤⠏⠠⠴⠷⣄⣀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠙⠀⢨⡀⠙⠀⠀⠉⠀⠀⠀⢀⣀⣠⡤⠔⠀⢀⣉⡟⠂⠤⠀⢀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠰⠀⠀⠈⠉⠉⢩⡇⠀⠀⠀⠀⢀⣂⡰⢀⡁⢈⡀⠀⣀⠨⠍⠀⢉⡒⢂⠤⠂⣀⣀⢷
⠀⠀⠀⠀⠍⠀⠀⠈⡅⡅⠀⡀⠀⠀⠀⠉⠉⡔⠲⠄⢌⣳⢸⠑⡚⠵⠬⠍⠚⠿⠖⠤⠆⠤⢀⣤⠶⠴⠯⠀⠀⢈
⠀⠀⠀⠠⢆⣀⣀⣤⠴⢒⡰⡁⠐⠐⠒⠒⠁⣉⠀⠀⢀⣻⣠⢀⡀⡀⠀⣀⡈⣁⠈⠀⠀⠀⢀⣀⣦⠤⢂⣁⡤⠼⠀⠚
⠀⠀⠀⠀⠀⠀⠨⢄⢀⣄⣈⣄⠀⠀⠀⢐⣠⠷⠇⠤⠬⣚⡀⠥⠌⠉⠉⠉⠙⠛⠿⠷⢂⣈⠡⠖⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠘⡅⠂⠀⠐⠒⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠉⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣙⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣀⠠⡌⣿⡿⠛⣻⠻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣦⣤⣀⣤⣀⣀⣀⣤⣴⣤⡄⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢸⣿⣉⣛⣾⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠀⢿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣙⡻⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⢻⣦⠀⠀⠀⠀⠀⠀⠀⣠⡙⠦⣬⣿⣿⣿⡇⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⢧⡀⠀⠀⠀⣠⣾⠋⠙⠳⣾⣿⣿⣿⣿⣆⢸⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⢻⣦⡀⣰⣿⣿⣧⣀⠀⠈⠁⠎⣙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣯⣿⣿⣶⣶⣤⣀⡛⠲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡟⣿⠟⣿⡋⢿⣿⣿⣿⣿⣷⣦⣙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣿⢸⡟⠀⠻⠇⣆⠀⠈⠉⣿⣯⡙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣼⢿⡿⠀⢸⠀⠀⠏⠀⠆⢰⠁⠈⠹⣦⡙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠁⣾⠃⠀⣀⣀⣠⣤⣀⣀⣵⠀⠀⠂⠛⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣼⠇⣸⠇⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⡐⠁⠀⠇⢀⣾⣿⠉⣿⣿⣿⠉⠉⢻⣿⣿⣧⣳⡄⢻⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢀⡤⠚⠁⢀⡼⠀⢸⣿⣿⡀⠹⣿⣿⠀⠀⣾⣿⣿⣿⣷⠹⡈⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡴⠋⠀⠀⠀⡈⠁⠀⣾⣿⣿⣇⠀⢻⣿⡇⢠⣿⡟⣿⣿⣿⡆⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⠁⠀⠀⠀⢀⠃⠀⢀⣿⣿⣿⣿⡄⠀⢿⡇⢸⣿⣇⡿⢸⣿⣿⡆⠈⡿⣿⣿⣧⠹⣷⡈⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢉⠀⠀⠀⢀⡾⠀⠀⣸⣿⣿⣿⣿⣷⠀⢸⡇⣸⣿⢻⠇⡜⣿⣿⡇⠀⠀⢛⣿⣿⣧⣹⣷⡄⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⠃⠀⢀⡞⠁⠀⣴⢿⣿⣿⣿⣿⡿⠗⣾⣿⣿⣿⢸⡀⠇⣿⠹⣷⡄⠀⠀⣿⣿⠿⡿⢿⣿⣆⠀⠸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠿⠀⠀⠎⠀⢠⡾⢁⣿⣿⣿⣿⣿⠁⠀⣿⣷⣿⣿⢸⡇⠐⢻⡇⣿⣧⡑⠀⣻⣟⠀⠹⣄⠙⣿⣷⡀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠁⠀⠀⣼⣇⣾⣿⣿⣿⣿⡿⠀⠀⣿⣧⣿⣿⢸⡇⡆⢸⡇⣿⡟⣿⣄⠹⣿⡀⠀⠈⢧⡘⣿⣿⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠋⣦⣿⠀⢸⡇⣿⡇⠉⠻⣦⣿⣿⡒⠢⡈⠳⡉⢻⣿⡟⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⠁⠀⠀⠀⢠⣾⢯⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⡟⠀⠁⣿⠀⢸⡇⢿⡇⢃⠀⢿⠉⠻⣿⣦⡈⢢⠳⣼⣿⣇⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⠀⠀⠀⣿⠀⣼⡟⠘⣷⠀⠡⠸⠀⠀⠈⠻⣿⣄⠀⠘⣿⣿⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⢿⣆⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⠀⠀⢠⡟⠀⣿⡿⠀⠹⣧⠀⠀⠸⠀⠀⠀⠈⢿⣆⠀⢹⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⢀⣶⠟⢋⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⢀⣿⠀⠀⣿⡇⠇⠀⣿⡄⠇⠀⡇⠀⠀⠀⠸⣿⡆⠘⣿⢃⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣷⣤⣿⣿⣿⣿⠇⣹⣿⣿⣿⣿⣿⡟⣷⣼⡏⠀⠈⢻⠀⠀⢸⠇⠸⠆⣿⣧⠀⠀⢰⣦⣠⣀⣀⣽⠃⣠⣾⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡟⢰⡟⠘⣷⡆⠀⠄⠀⠀⣼⣷⣶⣾⣿⣿⣦⠀⠘⣿⠟⠛⠋⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠙⢿⣿⣿⡿⢿⡇⢸⣧⠀⢹⣷⣦⣦⣴⣦⣾⡿⠛⠛⠻⠿⣿⡿⠶⠿⠀⠀⠀⠀⠠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⣿⡿⠁⢀⣨⣏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠋⡛⠻⢋⡉⡛⠛⢻⡛⠋⣛⡿⢛⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣀⣀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠢⡀⢀⣠⠤⠒⠊⠇⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠘⡉⠀⠀⠀⢸⠀⢠⠀⡀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠉⠓⢄⠒⢤⡖⠁⠐⡀⣼⣋⣉⡵⡆⢠⢈⡜⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⡠⠓⢄⠈⠢⡀⠈⢦⠙⣆⠀⡧⢊⠎⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⢀⠞⠀⠀⠀⠑⢄⠈⠢⣀⢱⡏⢸⠁⢘⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣑⡤⠤⣉⣁⠂⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠖⠲⣘⠀⠄⠀⠠⠔⠺⠍⠒⠒⠶⢤⣽⣤⣔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢰⠁⠑⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡎
⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⠶⠾⠛⠋⠁
⣿⡿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⢷⣾⣯⡳⣄⠰⣤⡐⠒⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⡿⢉⡴⣆⣾⣯⠛⠉⠂⠁⠛⣿⢷⡜⢿⡶⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⢏⣀⣾⣷⣿⣿⢯⣳⡄⠀⠀⠀⠸⡄⠳⠘⣇⠙⠁⠢
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⣡⣼⣿⣿⣿⣿⣿⢿⡻⣗⢄⠐⠀⠀⠑⢄⠀⠹⣆⡆
⠀⠀⠀⠀⣠⢖⣿⣿⣿⣿⠏⠈⠉⢽⣿⣿⣿⣿⢍⠣⠙⢮⡀⠀⢀⠀⠀⡀⢀⠀⠹⣾⣦⣷
⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⠙⢿⣷⣷⣄⢦⣷⣄⢠⣵⣄⢴⡌⢳⠀⠈⣿⣿⠂
⠀⠀⠀⠸⣿⢫⣿⣿⣿⣧⠀⣴⠋⠁⠀⠀⠀⠀⢻⣿⣻⣿⣿⣿⣿⣿⣿⢸⣦⡌⢣⣷⣙⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡸⠇⢄⣠⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡏⣿⡇⢨⣿⣿⣟⠗
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠈⠉⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡟⣋⣠⣿⣃⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣼⢟⣾⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣴⣖⠀⠀⠀⠀⠀⠀⠀⠋⠉⠈⡽⠁⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⠶⠆⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⢿⢻⠹⣿⣿⡿⣤⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠉⠛⠈⠄⣀⠤⠔⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠨⠁⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠀⠰⡆⠀⠀⠀⠀⠀⢀⣀⠌⠀⡀⠈⠒⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢄⠀⠀⠈⠋⢉⢹⠁⠀⠀⠀⠀⡠⠀⠁⠀⠀⠈⠀⢵⡤⣄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠔⢊⡔⠳⣌⠳⣀⢠⠀⠀⢾⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠂⣀⡸⢄⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢾⡳⠊⣡⠔⠉⠀⠀⠈⠢⣈⠰⢄⠀⠨⠀⠀⠀⠀⢐⠀⡀⠀⡀⠀⠊⠤⠀⠊⠃⢤⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⡄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⡑⠴⣆⠀⠀⠀⠠⡁⡘⠤⠜⠃⠀⢬⠕⠂⢀⡈⡥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢉⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣉⠂⢄⣀⠀⠑⢆⠀⠂⠔⢐⡠⢄⠤⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣁⣤⠖⠉⠀⠀⣀⣀⠀⡤⡤⢄⠀⠀⠀⠀⣠⣀⣀⢀⣀⣀⢌⠉⠒⠤⣍⣒⣦⠑⢶⢤⢄⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡁⣹⠡⠈⢸⠀⠀⠀⠀⡿⢠⠈⠀⢠⠀⠐⠀⠀⠀⠈⠈⠉⠀⠈⢞⠶⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡍⣏⣠⣸⠀⠀⠀⠀⢱⣅⣸⠠⡁⡤⡇⠀⠀⠀⡇⠀⠀⠀⠀⠈⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠉⠁⠈⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠠⢄⣒⣂⣀⣉⣀⠲⠀⠀⠰⡆⠢⠀⠐⢖⡆⠀⠀⠀⠁⠴⠀⠀⡠⣀⢿⠰⠀⠀⠠⡀
⠀⠀⠀⠀⢠⠁⠃⠰⠉⠄⠂⠱⠀⠀⠀⠀⢠⢘⡃⠈⠈⠀⠈⢸⠀⠀⢰⢀⠀⢄⡐⢆⡇⠀⠀⢀⡎⠀⣦⢼⠀⢻⢾⠀⢱⣴⠀⠁
⠀⠀⠀⠀⢸⠀⢸⠇⠀⠿⡇⢀⠿⠀⠀⠀⠈⢸⡅⠀⠀⠶⠀⢸⠀⠀⢸⣅⡰⢻⣀⢼⡇⠀⠀⠈⠁⠀⡃⠘⠀⡟⢻⠀⠈⠉
⠀⠀⠀⠀⢸⠀⢰⡤⠀⢰⡇⢰⣤⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠸⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⢰⡆⠀⡤⠔⠀⡧⠾⠀⢥⠤⠀⠇
⠀⠀⠀⠀⠀⠠⢼⡹⡀⢊⣡⣘⣼⣚⣒⠒⣒⣸⣀⠀⠀⠀⠀⠸⠀⠠⠠⠠⠤⠄⠤⠄⠀⠠⠬⣎⡁⢤⡉⠁⡀⡟⢃⠀⣛⢁⡀⡃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠋⠋⠉⠙⠙⠉⠋⠘⠉⠉⠋⠋⠁⠛⠙⠙⠛⠙⠋⠙⠙⠉⠋⠙⠛⠃⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄⣀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡀⠀⠙⠲⣦⣼⣿⣿⣷⣿⡁⠀⠀⣀⡤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣧⣤⡤⠀⠀⠙⠉⠉⠁⣹⣿⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⣿⣤⣿⣶⣤⣀⣤⣴⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡿⠛⠉⢽⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣠⣴⣦⡤⣤⡤⣶⣿⣾⣿⢡⠀⠀⠀⠙⠛⠿⠓⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠛⢷⣾⣷⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠁⠀⠙⠟⠋⠉⠛⣷⣶⣴⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⢀⡀⠈⣼⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣇⠀⠀⠀⠀⠀⠌⠓⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠟⠋⠁⠘⢶⣤⣒⣀⠀⢀⣰⣷⠏⠀⠀⠀⠀⠀⠀⢀⢦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠘⣤⣼⣿⣠⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣯⣿⣹⣟⣟⣿⣟⢻⣿⣿⣬⣽⣿⠀⠀⠀⠀⠀⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣾⠿⡿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠞⠻⠂⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠡⢤⣤⡤⢤⡀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠠⢤⠀⠀⠉⠲⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠀⠀⠲⡅⠀⠀⠀⠀⠀⠀⠈⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢱⡀⡁⠀⠀⡀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢰⡇⢃⠀⠠⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢙⡾⠁⣠⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡤⠤⠦⠬⣤⡀⠰⣯⣤⡸⢀⣠⢤⡒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠂⠀⠁⢈⡤⠔⡜⠳⣶⡛⣿⣏⠻⠾⠶⢦⣌⢣⣤⣾⠁⠀⠛⠳⣄⡄⡠⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⠀⠈⠉⠁⢠⠴⠟⠛⠛⠛⠻⢶⣿⣿⡻⡈⣧⣤⡛⠚⠻⢿⡿⠿⠒⠀⠀⠀⠈⠱⣶⡊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⣀⣰⠖⠁⠀⠀⠀⠀⠉⠁⠜⠿⣯⣷⣿⣻⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣷⠢⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠂⡍⡵⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠳⡖⠲⣶⣄⡀⡀⠀⠀⠀⠀⠀⡁⣸⡟⠒⠔⠢⣁⣄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠊⢸⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣧⠃⠀⠀⠀⠀⣷⣼⡇
⠀⠀⠀⢀⣌⠤⠤⠴⠒⠳⠒⣯⡇⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢀⣠⠖⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⡆⠀⠰⡇⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣇⣸⣾⣛⡯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⢧⣧⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⡄⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⣿⣧⠰⡀⠀⠀⠀⠀⠀⣠⢲⠐⡁⢄⣼⠿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠭⢁⣀⣀⣀⣚⣉⡥⠴⡖⠋⠳⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠌⠠⡔⠀⡂⣤⠉⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢀⠄⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣟⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⢀⡤⠒⠚⠋⠉⠉⠙⠲⣄⡀⠀⢀⢀⠀⠀⠀⠀⠀⢀⡠⠐⡄
⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⠑⠍⠲⢄⡀⣰⡋⠀⠀⢸
⢇⡶⠊⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⢻⡏⠀⢰⠀⢸⡤⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣠⡤
⠈⠁⠀⠀⠀⠈⢣⡱⢄⡀⠀⢠⠤⣤⣤⣤⣤⡀⠀⢳⠀⠀⢀⡟⢠⠃⢱⡀⠀⠀⠀⠀⠀⣀⡤⡆⡠⠊⠁⡠⠊⢰⡇⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⢬⣾⣿⣤⣶⣟⢻⣵⠦⡀⣨⣧⡀⢸⠁⠈⠀⢈⡇⠀⡀⠀⢀⡞⠁⠀⡟⠁⠀⠀⠀⠴⠃⠀⠀⠀⠉⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣯⡥⣿⣿⣿⣹⣿⡏⠀⢸⣿⠀⠀⡀⠃⠀⢀⣸⣗⡉⢧⠀⡞⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣦⣿⣯⢹⣿⣿⣷⣲⠟⠃⠀⠀⡀⣀⠀⠉⠳⢿⣷⢌⣷⡇⠀⠀⡠⠊⠀⠀⢀⠠⣄⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠟⠛⠛⠐⠚⡛⠶⠺⠬⣿⣿⣿⣿⡿⣎⠓⢹⡷⡄⠀⡠⠀⠀⠀⠀⠐⠒⢚⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⣄⣀⠀⣀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢮⡱⣌⢻⠘⣆⠁⠃⢰⢁⠀⢀⡤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡏⡀⠉⠙⣟⡗⠀⠀⢰⢀⣤⡤⣀⣠⡴⠖⠂⠀⢸⡿⣿⣻⣯⣿⣧⡄⡇⠀⢰⠟⡲⢒⠂
⠀⠀⠀⠀⠀⡠⠊⣿⡷⠛⠻⣯⣄⠰⠁⠁⣼⣹⣤⡿⣿⡅⠀⠀⠀⠀⣷⡯⠜⢿⣿⡟⢰⡇⠀⠇⠘⠀⢸
⠀⠀⠀⠀⠀⡁⠀⠈⠓⠀⠤⠈⣹⠓⠀⠀⠘⢿⡶⢶⡟⠃⡄⠀⠀⠀⣿⣶⡶⣿⠿⡁⠘⠃⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠱⠄⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⣠⡇⠀⠀⠈⠉⠐⣃
⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠾⠀⠄⠀⢀⣠⣺⣷⠶⠟⠉⢀⡀⠀⠀⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠀⠀⠀⠹⠿⠿⡀⠀⡀⢸⣧⣤⡀⠀⠀⠀⠀⠈⠑⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠟⠉⠉⢀⡠⠀⠀⠐⠂⠀⠀⠐⠒⢂⡀⠙⢾⣶⣬⣾⣿⡻⠒⠊⡙⢷⣖⣶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠠⢴⠋⠀⢀⠤⠖⠁⠀⢀⡠⠔⠒⢀⠦⠠⠀⠀⠼⠋⠙⠛⠒⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⡠⠛⠉⠁⠀⠀⢀⡴⡀⠀⡴⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠀⠀⠀⠀⠀⢼⠱⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠋⠀⠀⠀⢠⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣰⠁

⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋
⣼⡿⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⢀
⠋⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⣀⣴⣿⣿⣶⣶⣄⡀⠀⠀⢀⣭⣿⣶⣶⣤⣀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡟⠁
⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠋⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣠⣾⠟
⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⢀⣀⣆⣀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣾⠟⠁
⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⠟⠁⠀⠀⠀⠀⠀⣠
⠁⠀⠀⠀⠀⠀⣠⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⢁⠀⠙⠙⠋⠉⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣴⡿⠋
⢀⣴⡿⠁⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠈⠉⠙⠛⣻⣿⠿⠿⠿⠿⠿⢿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠟⣿⣿⠟⠛⠋⠁⠀⠀⠀⣴⡿⠋
⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⢀⣴⡿⠁
⣠⣿⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠿⠃⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⠟

⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠙⢆⡀⣰⡂⢀⡀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⣤⣷⣼⣧⣿⣤⣶⣟⣛⣓⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⣀⣤⣀⠈⠉⠈⠀⠀⠀⣀⡀⠉⠒⢄
⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠈⢻⡌⠉⠉⠉⠛⠻⢷⡖⠀⠀⠀⢠⠏⢠⡾⣛⡷⠙⠁⠀⠀⠀⠀⠈⡭⣿⠷⣆⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⢣⠀⠀⠀⡏⢀⣿⣳⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣻⣯⣿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠁⢸⣿⣭⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠇⠘⣿⡶⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠸⡀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠑⣄⠙⠿⢿⠆⠀⠀⠀⠀⠀⠀⠾⠟⠁⡤⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠑⠤⢤⡀⠠⠴⠄⠄⠀⢀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⡛⢶⣿⣶⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡄⠈⢯⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠈⠙⠻⣿⡦⠀⢿⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡀⢀⡀⡀⠀⠘⣿⣿⣶⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠙⠛⠁⠀⠙⣷⡿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⢰⣿⢡⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣸⡀⠣⠀⣼⠇⠀⢀⠀⠀⠸⣿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡀⠀⠀⠶⠘⠏⠀⠀⠀⢻⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⠀⢰⠆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢹⡆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⢸⡧⣰⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠈⠉⠙⠻⢿⡿⢿⡿⠿⠿⠿⠛⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣾⠀⠀⠀⠀⠀⡰⠀⠀⢠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣾⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄