Behaviour Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢿⣾⣿⢾⣶⢿⣾⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⠛⢿⣿
⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣧⣿⣧⣿⣿⣾⣯⣶⣯⣧⣽⣽⣯⣽⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠹
⣿⡟⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠈⢻⣿⠟⠉⠉⠉⢹⣿⠿⠛⠛⠛⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣏⡀
⣿⡧⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠘⢟⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⢰⣶⣿
⣿⣧⢀⡁⠀⢠⣭⣭⡁⠀⠀⠀⠀⢾⣿⣦⡄⠀⠠⠶⢀⡀⠀⠀⣀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⣿
⣿⣿⣿⡁⠀⠀⣻⣿⣷⠀⠀⢠⣤⣿⣿⣿⡆⠀⠀⣴⣿⡤⠀⠀⢹⣿⣿⣿⠀⣿⠉⣤⣤⣤⣤⣤⣤⣿⣁⣧⣨⣿⣿
⣿⣿⣿⣯⣘⣄⣿⣿⣿⣇⣴⣠⣿⣿⣿⣿⡧⠀⠀⠈⣁⣤⣄⠀⣾⣿⣿⣿⠀⠀⢰⣿⣿⠉⠙⠁⢰⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣶⣿⣿⣿⣴⣿⣿⣿⣿⣠⣇⣸⣿⣙⣦⣤⣤⣾⣿⡿⠟⠿⣿⣿
⣿⠁⠀⠀⠀⠈⣿⣿⠿⠛⠛⠉⠛⢿⡿⠉⠀⠀⠈⠛⠋⠁⠉⠉⢻⠟⠛⠛⢿⣿⣿⡟⠛⠉⠙⢻⣿⠋⠀⠀⠀⠀⢻
⣇⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢨
⣏⣛⠁⠀⢙⣋⣇⠐⠀⠀⠀⢀⣠⣾⣿⠆⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⢠⣿⣿⠦⠀⠀⠀⢀⣿⣿⠇⠀⠀⠻⣿
⣿⡍⠀⠀⢨⣽⡿⠛⠂⠀⠈⠛⢿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣶⣶⡆⠀⠀⢀⣹⣿⣿⡀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠘
⣿⣷⣶⣶⣾⣿⣃⣀⣀⣀⣀⣀⣈⣿⣿⣿⠛⠃⠘⠀⠰⠀⠛⠛⠃⠰⠀⠛⠻⣽⠏⠁⢀⣤⣠⣿⣿⣤⣤⣤⣦⣤⣤

⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⢀⡀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠇⠚⠀⣀⡤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⡇⠀⠄⠀⠀⠀⠠⠀⢀⠀⠂⠀⠀⢀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠃⠀⠠⠀⠀⠀⠠⠀⠀⠀⠂
⡇⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⠆⠀⠀⠈⠉⢉⡍⢈⡃⠐⠠⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⣀⣤⣀⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠁
⡇⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇
⡇⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⣠⠆⠈⠉⠁⠘⠓⢤⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠠⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⡇⠀⠀⠀⠃⠀⠀⠀⡄⠀⠄⠀⠀⠆⠀⠀⠐⠂⠀⢠⠀⠀⣀⠀⠀⡀⠉⠀⡄⠀⠀⠀⠀⠀⠀⢡⠀⠐⠀⠒⠀⠂⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠐⠀⠀⠀⢢⡀
⣷⣶⣶⣶⡀⠁⠀⠀⠀⠀⠐⠀⠀⣀⡴⢶⣶⣆⠀⠀⠀⠀⠀⠀⢠⣵⣶⡆⠠⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇⠀⠀⠀⡀⠀⠀⢀⣾⣿
⣿⣿⣿⣿⣷⣤⣤⣤⠆⠒⠒⠞⠉⠉⠀⠛⠿⣿⣿⣶⠄⢤⣤⣶⣿⣿⣿⣿⣤⣀⡀⣀⡀⠀⣀⣀⣿⣿⡏⢈⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⣦⣤⣀⣀⣠⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣶⣄⠀⣥⣴⣚⣛⣋⠀⠀⢻⣿⣿⣿⣿⣿⣿⡯⠀⠈⠁⠀⣽⣿⣿⣿⣇⠀⠀⠀⠀⠀⠉⠀⠀⣾⣶⣶⣶⠀⠀⢰⣌⡻⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⡀⢀⡀⣀⣸⣿⣶⣿⣿⣿⣿⣷⠄⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⢰⣿⣷⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⣹⡏⠻⣿⣿⣿⣿⣿⣿⣿⠿⠢⡄⢼⣿⣿⣿⣿⣿⣿⣿⡷⢰⡤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⡦⢼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣦⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣧⣴⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣴⢿⣿⣿⡆⠘⠞⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠁⠀⠀⠀⠈⠋⠀⠀⠀⠀⣸⡏⠀⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⢠⡶⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⣠⡤⠾⠛⠛⠀⠘⠻⠋⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠈⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⢸⣿⡏⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠘⣿⣇⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣰⣾⣟⣿⣦⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⠤⢀⣄⡀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠶⠀⣀
⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⣇⠀⠀⠀⠀⠀⡈⢂⡀⠄⢀⠂⡑⠀⠀⠀⠀⢀⡈⠀⠀⠀⢉⣀
⠀⠀⠈⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠂⠡⠤⠄⠂⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⣤
⠀⠀⠀⠾⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠸⠿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⠶⢎⠱⣮⡇⠀⠀⠀⠀⠀⠊⠁⠺⠑⠀⠔⡀⠀⠀⠀⢠⣾⠁⠀⠄⠀⠹⡄
⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠹⣿⣯⣿⣿⡿⠀⠀⠀⠀⠀⠁⠐⢩⠌⠀⡀⠃⠀⠀⠀⢠⡄⠀⠀⡄⠀⢠⣄
⠀⠀⠀⣹⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⠃⠀⠀⠀⠀⠀⠁⠠⠉⢉⠀⠈⠀⠀⠀⠀⠀⠱⢤⣀⣀⣠⠞⠁
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⡴⠿⡹⡟⣷⡄⠀⠀⠀⠀⠀⢰⣿⣭⣿⠿⣆⠀⠀⠀⠀⠀⠀⣴⡴⠿⠧⣶⡄⠀⠀⠀⠀⠀⣴⣿⣿⣽⣷⡄
⠀⠀⢿⣷⣿⣼⣾⣧⠀⠀⠀⠀⠀⢹⣍⣿⣿⣿⣿⡀⠀⠀⠀⠀⣴⡏⠀⢈⠀⠘⣶⠀⠀⠀⠀⠀⢿⣔⣷⣷⣿⣧
⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣷⣄⣈⣠⣼⡅⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛

⠀⠀⢠⣴⢶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣶⢶⣄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤
⠀⠀⣿⢏⣈⣹⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠢⠈⣙⠉⠠⢃⠀⠀⠀⠀⠀⣿⢿⣯⣾⣿⡇
⠀⠀⠐⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣀⣘⣋⣀⡀⠀⠀⠀⠀⠀⢈⠠⢱⣾⡟⠡⡈⠀⠀⠀⠀⠀⠹⣷⣶⣶⣿⡟
⠀⠀⠀⣸⣿⣿⠛⠃⠀⠀⠀⠀⠀⢰⣶⠰⣶⠠⣶⠀⠀⠀⠀⠀⠀⠀⠂⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣶⡄⢰⣶⣶⣶⣶⣶⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣶⣶⣶⣶⣶⡆⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⠙⣿⠇⢸⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⡇⢸⡿⠉⠉⠉⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⣠⣴⣶⣾⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣷⣶⣦⣄
⠀⠀⠀⢠⣾⡟⢉⡉⢻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣶⣤⣁⣈⡛⠛⠻⠿⠿⠟⠿⠿⠿⠟⠛⢛⣀⣉⣤⣦⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⠿⠿⣿⣿⣷⡀
⠀⠀⠀⣿⣿⡇⢸⡇⢸⣿⢠⣄⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⠏⣠⡶⢃⡘⢶⣄⠹⣿⣷
⠀⠀⠀⣿⣿⡇⢸⡇⢸⣿⢸⣿⢸⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣄⠙⢧⣈⣁⡼⠛⣠⣿⡿
⠀⠀⠀⠘⢿⣇⣈⣁⣸⣧⣈⣉⣸⡿⠁⠀⠀⠀⢀⣄⣽⣿⣯⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣦⣹⣿⣧⣤⡀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣤⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⢿⣿⣿⣿⣿⣿⣿⣄⣉⣤⣿⣿⣿⣿⣿⡟⢁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣦⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣉⣴⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠛⣿⣿⣿⣷⣶⣶⣌⡛⠛⠛⢋⣠⣶⣶⣿⣿⣿⣿⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⠿⢋⣉⣤⣤⣉⠛⢿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣀⠘⠿⠛⠛⠛⠃⠰⣿⡿⠟⠻⣿⡿⠂⠙⠛⠛⠛⠿⠃⣠⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⡆⣾⣿⣿⣿⡇⢸⡇⠒⠂⢼⡇⢸⣿⣿⣿⡇⢰⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠃⠺⠿⠿⠟⢁⣾⣅⠛⠃⣸⣧⡈⠻⠿⠿⠇⠘⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣦⡙⠿⣿⣷⣾⣿⠟⢁⣴⣶⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⢀⣠⣄⢀⣠⣄⡀⢀⡀⢀⡄⢀⣤⡄⢠⠀⢀⡀⣠⡀⢀⣄⠀⣠⣤⠀⣠⣤⡀⠀⢀⣤⣀⠀⣤⣤⢀⡄⠀⣀⠀⢀⣄⠀⣄⠀⣀⢀⡀⢀⣠⣄⠀⣀⠀⢀⡀⣠⣤
⣿⠁⠀⣾⡁⠈⣷⢸⡿⣼⡇⠹⢦⡄⢺⠀⢸⡇⣿⢿⣼⢿⡀⣿⠶⠀⣿⢼⡃⠀⢸⡶⣿⠀⡷⠶⢈⣷⠶⣿⢀⣿⣹⡄⢻⣠⡏⢸⡇⣾⠁⢸⡇⣿⠀⢸⡇⣿⢴
⣽⣷⣶⣬⣷⣾⣥⣼⣥⣽⣧⣶⣾⣧⣼⣷⣿⣥⣿⣼⣧⣼⣥⣿⣶⣦⣿⣼⣧⣤⣼⣷⣿⣤⣿⣶⣬⣧⣤⣿⣼⣯⣬⣷⣬⣿⣤⣼⣧⣬⣷⣮⣥⣬⣷⣿⣤⣿⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠐⠰
⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠




⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀
⣿⣏⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠈⢤⠈
⠀⠀⠀⠀⢀⣤⡒⠚⠉⠉⠉⡙⣿⡄⢀⠈⠁⠰⣶⠤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠴⠠
⠀⠀⢀⡴⠿⠿⠿⣿⣷⣴⢫⡀⠈⠘⠻⠣⡀⢰⡿⠀⠈⢢⡰⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡎⠀⠀⠀⠀⢈⣷⣺⠺⠅⠀⠀⠂⠃⠝⠠⠄⠀⢀⠜⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⢸⣡⣤⣴⣦⣤⣼⢏⠀⣐⠀⠀⠀⢆⢀⠉⢲⣤⣠⠎⠀⠀⠀⠀⡠⠖⠁
⠀⠈⢿⠋⠉⠉⠉⠛⣬⣧⣤⣷⣦⣄⠠⣄⠷⣾⡿⠁⠀⠀⢀⠔⠊⢉⣽⡿⠛⢩⣿⣿⡟⠶⡒⢤⡀
⠀⠀⠈⣆⠀⢢⢄⣀⣿⣻⣿⣿⣿⣿⣿⣄⠔⠚⣓⠲⠀⠀⠈⡇⢀⣾⡿⠁⡠⢟⡯⠙⠒⢦⡇⠀⢻⡀
⠀⠀⠀⠸⡄⣴⡟⢁⣾⡿⠉⢀⣾⣿⠟⢓⠦⢬⡈⡵⢀⠰⠦⠤⣼⠿⡓⠕⠊⠁⠀⣠⠀⡾⡇⠀⢸⠇
⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⣼⣿⠇⠊⠀⠀⠀⠙⢦⣿⠀⠀⠀⠊⠉⠀⢀⣀⠤⡞⠉⠀⡇⣇⠴⠋
⠀⠀⠀⠀⡇⠻⢿⣌⣿⣷⡀⠸⣿⡄⠀⠀⠀⠀⠀⠈⢟⠀⠀⠀⢀⣀⡴⣋⠀⢰⠃⠀⠀⣧⠇
⠀⠀⠀⠀⡇⠀⠈⣿⣿⠉⠻⣿⣟⠻⣦⣀⠀⠀⠀⠀⠘⣤⡤⠴⢫⠀⠀⢻⡄⡇⠀⠀⢰⢻
⠀⠀⠀⠀⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠘⣿⣧⠀⠀⠀⠀⣿⣌⡥⠄⠐⠂⠉⡸⠀⠀⠀⢸⣎⣴⠢⣿⡦⢄
⠀⠀⠀⢰⣗⣠⠤⢿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⠔⣸⡤⠤⠤⣲⣿⠛⠻⣏⡏⠉
⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠝⠛⡠⠤⣿⣿⡇⡀⠠⠄⠴⠖⠉⢻⣏⡀⠓⠛⠛⠛⠛⠒⡒⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠾⠿⠿⠗⠒⠒⠀⠀⠘⣿⣷⣤⣀⠀⠈⠁⠀⠁⠀⣀⣤⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⢿⣷⣀⣠⡔⠚⠉

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢀⡠⠀⠀⠀⠀⢀⡀⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣼⣿⣤⡀⠀⠀⠀⠀⠀⠀⢀⣬⣭⣥⡿⠀⠀⠀⠀⢀⣿⣁⣈⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⣿⣿⡧
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⡇⠀⠀⠀⠀⠀⢠⣶⣾⡉⢁⣴⠀⢠⣶⠾⠷⣶⡄⠀⣦⡈⢉⣷⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡿⠷⠿⠋⣀⣿⣿⣷⣾⣿⣿⣀⠙⠿⠿⢿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⠛⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣶⣶⣶⡄⡿⢉⣿⣿⣿⣯⠉⣿⢠⣶⣶⣶⣿⣿⠇
⠀⠀⠀⠀⢸⣿⡇⢿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣰⣶⣶⣿⡍⢹⣿⠃⡇⣼⣿⣿⣿⣿⣧⢹⠸⣿⡏⢩⣿⣷⣶⣆
⠀⠀⠀⠀⢸⣿⠇⢸⣿⠄⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⢸⡇⣿⣿⣿⣿⣿⣿⢸⡇⣿⡇⣿⣿⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢄⠐⠿⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠹⣿⣿⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠇⠀⠈⠀⠀⢰⣄⠀⠻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⠁⠀⠛⠋⠉⠁⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⢿⣿⠟⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⡎⠁⢀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣦⡀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⢃⣶⣶⣦
⠀⠀⠀⠀⢠⣾⣿⣦⣼⡿⠛⠀⣶⡄⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⢠⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⢀⣶⣦⢸⣿⣿⡟
⠀⠀⠀⠀⣿⣿⡏⠉⠩⠤⢤⣤⣿⠦⠤⠀⠀⠀⠀⠺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⢿⣿⡄⠀⠉⠀⠀⠀⠀⣀⣠⣼⣿⣿⣾⣿⣿⠇
⠀⠀⠀⠈⠿⣿⣿⣿⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣿⣿⣿⣧⢸⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢿⣿⣿⣿⣿⠼⡿⠀⣀⣼⣄⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠿⠇⠻⣿⠿⠿⠿⠿⢿⡿⠃

⠀⠀⠀⠀⠀⢀⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦
⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠛
⠀⠀⠀⠀⠀⠀⣼⣷⣶⣄⠀⠈⠂⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣄⠿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⠄⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣿⣷⡄⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣠⣤⣄⣀⣤⣿⣿
⠀⠀⠀⠀⠀⢿⣿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠐⡒⠀⠛⠛⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⠀⠙⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠐⠒⠀⠀⠛⠛⠛⠛⠛

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠊
⡇⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠸⠓⠴⠠⠖⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂
⡇⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⢀⠀⢀⡀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⡇⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠠⠀⢥⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠉⠀⠀⠀⠠⠀⠠
⡇⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒
⡇
⡇⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢶⣷⣶⣷⣤⡀
⣷⣶⡄⠀⠀⠀⢀⣼⢀⠀⠀⠀⢀⣼⣿⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠴⢶⣶⣶⣶⣦⡆⠀⠀⠀⠀⠩⡈⣿⣿⣿
⣿⣿⣿⡆⠀⠀⣽⠟⣛⠒⠠⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣶⣴⣿⣿⣿⣿⠧⠀⠀⠀⢀⣴⣷⣿⣿⣿
⣿⣿⣿⣇⠀⠀⢘⣏⢁⡀⠀⣢⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠆⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠶⣶⣞⢛⣿⣷⠄⠿⣿⠤⠬⠥⠯⡭⠽⠤⠬⠬⢽⣿⡇⣄⢸⣿⣿⣿⣿⣿⣿⣿⣄⣀⣨⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣷⣾⣿⣾⣷⣶⣶⣿⣿⣥⣷⣼⣿⣿⣿⣿⣿⣿⣿⣥⣿⣨⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢄⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⢀⠀⡸⠃⠀⠀⠀⢀⡷⠶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣤⠰⠿⠆⣹⠀⠸⣷⠀⠀⠀⠀⠀⢰⡷⠈⠀⢾⡗⠀⠀⠀⢸⡰⡷⣸⠂⠀⠀⣖⠰⡶⠄⣰⠀⢰⣶⠶⣤
⠀⠀⠈⠙⣿⣿⡅⠀⠀⢸⣷⠀⠀⠀⢰⣿⠀⠀⣾⣿⣿⣇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⢻⣿⣾⠋⠀⣾⣿⣿⠁
⠀⠀⠀⠀⣼⣿⡇⠀⠠⢾⡟⠀⠀⠀⠸⣿⠙⠃⠿⣿⣿⠿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⢿⡆⠘⢿⣿⣿
⠀⠀⠀⢠⡿⢸⡇⠀⠀⢼⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⣰⡏⢸⡇⠀⢸⣿⣿
⠀⠀⠀⠛⠁⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠀⠀⠀⠛⠛⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠘⠋⠀⠘⠁⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢀⣸⣿⡼⢀⣾⣿⡀⠀⠀⠀⠀⢀⣻⣧⣤⡀⠙⢫⡄⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣿⣿⣿⠀⠸⣿⣿⠇⠀⠀⠀⠀⢺⣹⣿⣧⠇⠀⣸⣿⠀⠀⠀⠀⢸⡿⢾⣿⣿⣄⣿⣇⡀
⠀⠀⠀⠀⠈⢿⣿⣿⠆⠀⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠈⣿⠃⠀⠀⠀⠀⠈⣿⠀⠈⠁⢉⣉⣹⡧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⡟⢸⡇⠀⠀⢻⡆⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠁⠁⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⣇⢠⣶⣦⢀⣠⠄⠀⢠⣦⢀⠀⠐⠀⠀⢀⣤⡔⢂⡀⢀⣤⡀⠀⠀⠀⠀⢠⣤⡄⠀⢴⣦⠀⠀⣴⡄
⠀⠀⠈⢻⣿⡟⠋⠀⠀⣾⣿⡿⠾⠀⠀⠀⣠⣶⣿⡴⠞⠀⣤⣿⣤⡀⠀⠀⠀⣴⣯⠀⢠⣶⣷⣦⠀⣹⣧⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⣿⣿⣿⠀⠀⣸⣿⣿⣿⣇⠀⠀⢰⣿⣿⡟⣿⣿⣿⣿⢛⣿⣿⡇
⠀⠀⠀⣼⠃⣿⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⢸⡇⣿⠀⠀⠀⢻⣿⡟⠀⠀⠤⣼⣿⡟⠁⠀⣿⣿⠀⠈⢻⣿⣧⠤
⠀⠀⠀⠿⠀⠹⠇⠀⠀⠀⠀⠿⠀⠀⠀⠀⠿⠀⠹⠇⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠿⠀⠀⠿⠿⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣏⡀⠛⡟⢠⣿⡿⢹⣿⣿⠃
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠁⠘⠻⢣⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠔⠒⠒⠄⠀⠀⠠⠖⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⡉⣧⠀⠠⠾⠯⠀⠀⠀⠀⠚⠓⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣿⣿⣿⣿⣶⡷⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣇
⠀⠁⠈⠀⠀⠀⢠⣤⣼⡁⣻⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⢛⣧⣀⡀⠀⠀⠀⢠⣿⣶⣶⣾⠁⢰⡋⠁⠀⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠛⠧⢤⡀⠀⠘⣏⠀⠈⡟⠠⠛⠀⠀⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠁⠐⣄⡙⠦⠴⠧⠀⠂⠀⠀⠀⠀⢐⡡
⠀⠀⠀⢀⣤⠾⠁⠀⢠⡟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢥⣀⣀⣤⠃⠀⠀⠀⢀⣴⠋
⠀⠀⣴⠋⠀⠀⠀⠀⠈⢷⣿⣿⡿⣿⡿⢿⣦⣤⡀⠀⠀⡠⠀⠙⠃⠀⠀⢀⡄⠉⠙⢧⡀
⣠⠞⠉⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠘⢧⡞⠀⠀⠀⠀⠜⠀⠀⠀⢠⠀⠀⠀⢡⠀⠀⠀⢳⡀
⠻⡄⠀⡾⠉⠀⠀⠀⠀⠉⢆⠀⠀⠀⡸⠁⠀⠀⠀⢠⡀⠀⠀⠀⠈⡇⠀⠀⠈⢇⠀⠀⢀⡗
⠀⠙⣆⢻⡀⠀⠀⠀⠀⠀⠘⡀⠀⠠⠁⠀⠀⠀⢠⠃⠱⡀⠀⠀⠀⣿⠀⠀⠀⠈⢆⠀⡼⣧
⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⢁⡠⠁⠀⠀⠀⢀⡆⠀⠀⠘⣄⠀⢀⡟⡄⠀⠀⠀⠀⢳⡃⢹⡀
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡜⠀⠀⠀⣀⠘⣆⡜⢀⡇⠀⠀⠀⠀⠀⠳⡄⠃
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠐⠒⠚⢻⠀⠸⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠉⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⡲⠬⠶⠎⣿⡇⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⢰⠉⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠛⠫⢄⠠⣢⣵
⠀⠀⠀⠀⢠⠚⡃⠁⠁⠀⠀⠰⣥⣠⠄⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠃⠀⠀⠀⠀⠀⠈⠙⠻⣧
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢾⣧⡄⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⢥⣿⣖⠁⠀⣿⣧
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⠼⢂⠀⠀⠀⠬⢀⠁⠰
⠀⠀⠀⠀⢸⠘⡆⠀⠀⠀⢀⠤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠼⣏⠄⠀⠀⣄⠒⠤⠿⠑⢢⡀
⠀⠀⠀⠠⠘⡆⠇⠀⠀⠀⣍⠀⠐⡀⠀⢀⢠⠆⢰⠄⠀⠀⠀⠆⠀⠀⠀⠀⠃⠀⠇⣿⢅⠀⠄⢠⡀⠀⠀⠔⡋⢳⡀
⠀⠀⠀⠈⡆⣷⢀⠀⠃⣀⣿⡀⠸⡏⣭⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢩⡎⣂⠀⠀⠳⠀⠃⠈⠁⠀⠓
⠀⠀⠀⢀⠥⢸⡄⢆⠊⢩⠔⡃⠀⠾⢿⣷⠀⠙⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⠅⢈⠀⣇⣰⢼⡑⠂⠀⠀⠀⢀⠀⠁
⠀⠀⠀⡇⠖⠈⣷⢤⣀⡀⠀⠁⠠⣤⣄⣉⣴⣠⡠⡽⠆⠀⠀⠀⢀⣴⠀⠁⠀⠀⠀⠀⠀⢻⠉⢲⠉⠁⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠁⡀⠀⢨⡿⣜⢲⡦⢲⡆⢳⠀⠀⠙⠃⠰⠄⠘⠶⠒⣞⠭⡁⡀⠀⠀⠀⠐⠀⠀⠿⠀⠘⠂⠀⠀⠀⠀⠀⠀⠉⠒⠤
⠀⠀⠀⠀⠄⠀⣼⠽⠟⠋⠈⢸⡇⢨⡆⠀⠀⠂⠀⠀⠀⠀⠈⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡇⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠻⢡⠀⡀⠀⠘⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⢾⢁⠄⢸
⠀⠀⠀⠀⠀⠀⠸⠈⣴⠃⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢓⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⠘⠀⢸⠃
⠀⢀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠈⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠀⠀⡜
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⡤⠀⣲⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢾⡭⡀⠀⠂⢠⣤⠿⣾⠇⠀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⣅⡀⠀⠀⢀⣠⣤⣤⠤⣾⣿⢿⣬⣿⠷⠤⢤⣄⠀⠀⠀⠀⢸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠈⠉⠁⠀⠉⠉⠉⡙⠻⠶⣼⠟⠉⠀⠀⠠⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠀⠀⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡟⠀⠀⠠⣤⡄⢿
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⣾⡟⠀⣴⠶⣦⡄⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⠉⢹⡟⠙⠻⣷⡀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⡆⠀⠘⣷⠀⠀⣿⣧⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠹⠇⠀⠈⠉⠀⢸⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⢿⣶⣤⣤⣄⣀⣀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠈⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⢻⣿⠀⢀⣠⣴⡿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⢿⣧⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⣿⣦⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠹⢷⡄⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡇
⢀⣴⣶⣶⣶⣶⣶⣶⣾⠿⠟⠃⠀⠀⠀⠀⠀⣠⣾⠟⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⡀⣀⣠⣾⠋
⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⡟⠉
⠀⠀⠹⣿⣦⣠⣤⣶⠿⠛⢿⣧⠀⠀⠀⣸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⠿⠟⠁⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠋⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠚⢿⡏⠙⣿⡾⣿⣷⣾⣶⣿⣿⠀⢸⣿⣷⣶⣶⣾⣷⣾⠟⠉⡿⠗⢲⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⣽⡷⣠⣿⣦⡾⠁⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠹⣦⣼⣧⣴⣾⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣨⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣁⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⣿⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠛⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠐⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⢦⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣄⠈⠙⠳⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠁⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢳⣄
⠀⠀⠀⣠⡟⠁⢀⡴⠋⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⢻⡑⣆⠀⠀⢻⡆
⠀⠀⣰⡟⠀⢠⡟⠁⣴⠃⠀⠀⠀⢀⠶⠶⠶⠶⠶⣶⣶⣶⠶⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠘⡇⠘⣧⠀⠀⢿⡀
⠀⢠⡿⠀⢀⡟⢀⡞⣿⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣷⠀⠘⣇⠀⠸⡇
⠀⣼⡇⠀⢸⠀⡾⠀⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠋⠙⠃⠀⠀⠀⠀⠀⠀⠚⠛⠆⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⣿⠀⠀⢻⠀⠀⣷
⠀⣿⠁⠀⠀⢰⡇⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⢀⡤⠶⠒⠒⠶⣄⠀⠀⠀⢀⡤⠖⠒⠲⢦⡀⠀⠀⠀⠀⢻⠀⠀⠀⡧⠀⠀⢸⡆⠀⣿
⠀⣿⠀⠀⠀⢸⠀⠀⢿⠀⠀⣸⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠘⡆⠀⡴⠃⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡄⠀⢠⡇⠀⠀⢸⠁⠀⣿
⠀⣿⡂⠀⠀⠈⠀⠀⠘⡆⠀⡟⠀⠀⠀⠘⠀⠀⣠⣤⣤⠄⠀⠀⠃⠘⠁⢀⣴⣶⡶⢄⠀⠈⠀⠀⠀⠀⠘⡇⠀⣸⠁⠀⠀⢸⠀⠀⣿
⠀⢻⣇⠀⠀⠀⠀⠀⢀⣹⣄⡇⠀⠀⠀⠀⠀⢸⣿⠟⠳⣴⡆⠀⠀⠀⠀⢸⡿⠋⢳⣾⠀⠀⠀⠀⠀⠀⠀⣇⣠⣇⠀⠀⠀⠈⠀⠀⡿
⠀⠘⣿⡄⠀⠀⠀⢠⠏⠁⢸⡇⠀⠀⠀⢂⠀⠀⠻⠦⠴⠟⢀⣀⣀⠀⠄⠈⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⢷⠀⠀⠀⠀⢠⡇
⠀⠀⠹⣿⡀⠀⠀⣿⠀⠉⢿⠁⠀⠀⠀⠀⠁⢄⣀⣀⣀⣴⡾⠁⠀⠉⠂⠐⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠓⡼⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢹⣷⠀⠀⠸⣦⡀⢸⠀⠀⠀⠀⠀⠀⡔⡉⡁⠀⠸⡇⠀⠀⠀⠀⠀⠀⠔⠐⠐⠀⠀⠀⠀⠀⠀⠀⣿⢀⠔⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢻⣧⠀⠀⠈⠉⢹⡄⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣿⡄⠀⠀⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠀⠀⠀⣿
⢀⣴⠂⠀⠀⣾⠁⠀⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⡇⠀⠸⣇⠀⠀⢿⣆
⢸⣧⣀⣠⡼⠃⣠⠀⠀⣿⠃⠀⢸⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠀⢸⣧⠀⠀⠻⣷⣄⣈⠻⣷⣶⠞
⠘⢿⣿⣥⣴⣾⠏⠀⣼⠏⠀⢀⣞⡴⠁⠈⠳⣄⠀⠀⠀⠀⠀⣀⠳⠞⠁⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⢻⣧⡀⢿⣦⣀⠀⠈⠙⠛⢛⡟⠁
⢤⣤⣶⣾⣟⣁⣤⠞⢃⣀⣴⠟⠋⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠈⠓⠂⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠙⠻⠶⠽⠿⠿⠿⠶⠛⠉
⣀⣈⣁⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣶⣶⣶⣶⣟⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣒⣦⣤⣀⣠⣤⡤⢤⡀⠀⠀⠀⢀⡤⠖⠚⠃⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⢀⣴⣪⢏⢔⡀⠲⣾⣿⣿⣟⣋⠉⢢⠉⢢⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⣠⣿⠿⢗⠕⢡⠞⠋⠉⠉⠉⠛⠻⢷⡄⢣⡀⠙⣆⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⡴⠋⠀⠔⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣷⡄⡜⣦⣈⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⢸⡇⣧⣤⣾⢃⣀⣀⡀⠀⠀⠀⠀⢀⣀⡠⠤⡈⣿⣿⣿⡿⡆⠀⠀⠉⠒⠒⠒⠁⠉⠛⠛⠋⠁
⠀⠈⢧⡹⣿⣏⠈⠀⣈⣉⠁⠀⠀⠀⠉⣁⣀⣀⣀⢸⣿⡟⠁⡇
⠀⠀⠈⣿⣿⣿⠘⠿⣩⣯⠀⠀⠀⠀⠀⢹⡯⠝⠁⣿⣟⣜⡰⠁
⠀⠀⣰⡿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⢸⣿⡵⣣⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⢸⣯⡘⣿⣿⣧⠄⠀⠀⠀⠀⣀⣐⠀⠀⠀⠀⢰⣿⡟⣿⡇
⠀⠀⠛⢿⣮⣿⡇⠀⡐⠀⠀⠈⠛⠋⠁⠀⠀⣠⣿⡿⠁⡹⠁
⠀⠀⠀⠀⠙⣿⡇⠀⠃⢠⡄⠀⠀⠀⠀⣀⢴⣿⣿⠇⣼⠁
⠀⠀⠀⠀⠀⣻⡇⠀⠀⠈⢠⠗⠒⠒⠊⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⣴⡿⣫⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⡛⡀
⠀⠀⢠⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡇
⠤⠶⢌⢿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⢹⣿⡿⣉⠴⠦⠤⣄
⢢⠀⠀⠑⢍⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⠖⠁⠀⠀⡜⠁
⠸⡄⠀⠀⠰⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠉⠉⠉⠀⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠆⠙⠦⣄⡴⢛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⣴⣶⣌⠻⣏⠙⠲⣽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠁⣠⣬⡳⣄⠈⠻⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣾⣿⡿⠃⠈⠳⣄⢹⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⠀⣠⣦⡈⠻⡵⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠡⠀⠀⠀⠀⠀⠀⠀⠈⠻⣴⣟⣿⡿⠋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡀⠀⣠⣤⣘⣧⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠅⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⣤⣤⣤⣤⣀⡙⠾⣿⣿⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠛⠛⠋⠉⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠉⠙⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢋⠭⢄⣀⣠⣤⣶⠶⠾⠿⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⢛⡿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⣿⠉⢉⣀⡀⠀⠀⠀⣿⠛⢻⣿⣿⠂⢀⣶⠶⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢻⡟⣷⠀⠀⠀⡟⠀⠈⣽⣿⠀⢸⡇⢸⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢟⣿⡄⠀⠰⡇⠀⠀⣿⡟⠀⢸⠀⡨⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢸⡇⠀⠀⡇⠀⢸⣿⡇⠀⣾⠀⠘⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢼⡇⠀⠀⡇⠀⢛⣿⡇⠀⡿⠀⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⡇⠀⢸⣿⡇⠀⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⠀⠀⡇⠀⣼⢻⡇⠀⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡇⠀⠀⣇⠀⣿⢸⡇⢀⡇⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣷⠀⠀⣿⠀⠐⢿⡇⢸⣇⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣃⡀⠀⠿⣤⣤⡼⠇⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠻⢿⡿⠟⠁⠈⠻⣿⡿⠟⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣦⡀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⠟⠁⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⢿⣿⣿⡿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⠏
⠀⠀⠀⠸⣿⣿⠂⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣭⣥⣤⣤⠀⠀⠀⢠⣤⣤⣭⣥⣾⣿⣿
⠀⠀⠀⠀⣿⣿⡀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠛⠛⣿⣿⣿
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃



⠀⠀⠀⠀⠀⣴⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣈⣝⡋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢙⣫⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣋⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡟⠻⢿⣿⣶⣿⡿⠟⢻⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣆⣿⣿⣿⣿⡇⠀⠀⠀⢾⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⠀⠀⠉⠀⠀⣴⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠃⠈⠻⠿⠛⠁⠀⠀⢀⣸⣿⣿⣿⣟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠛⣿⣿⡟⠛⠋⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡂⠲⣶⣶⡖⢶⣆⠀⢀⠀⠀⢄⣤⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⣀⠀⠈⢳⡈⠀⠀⠀⠹⢷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣗⡈⠀⣀⡴⠉⠀⠿⣷⣤⣄⠉⠳⣤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡋⣉⠀⠀⠠⢈⣿⣿⡿⢿⣿⣿⣿⣿⣶⣽⠇⠀⣠⣾⠛⠻⢿⣷⡄⠀⠀⠄⠀⠀⢴⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⡻⣥⣶⣶⣾⣯⣾⣿⣧⣤⣿⣿⣿⡏⠸⣿⣆⣼⣿⣿⣟⣢⣼⣿⣿⣷⠀⢠⣤⣼⣽⢽⡿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⡀⢀⠈⠙⠀⢹⣿⣿⡟⣿⣿⣿⣿⢶⣀⡀⡐⣸⢦⣴⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⢠⣉⣿⣿⣤⣀⡀⠘⠻⣷⣼⢀⠀⠀⢸⣿⣿⣇⣿⣿⣿⣿⣿⣏⣿⣤⢊⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡿⣻⢛⣿⣿⣷⣿⣤⣏⠀⠀⠟⢧⠀⠁⠁⣀⠀⠀⢀⣼⠟⠃⠉⠙⠿⣿⣿⣿⣉⠇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣭⢡⣿⣿⣿⣿⡟⠛⠛⡛⠀⣸⣿⣿⣿⣻⡷⣄⣾⡿⠀⠀⠀⠀⠀⠀⠹⢿⣿⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣸⣿⣿⣿⣿⣶⣷⣾⣿⣶⣿⣿⣿⣿⡋⠻⣻⣿⣿⡃⣱⠆⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠻⢿⣿⣻⣿⠿⠟⠹⣿⠧⠻⢝⣶⣄⣰⡋⣿⣿⣿⣧⣄⣴⣲⣦⠀⠀⠀⣠⣹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣤⣠⢚⢫⣯⣤⠀⠀⠈⠛⣦⢈⣿⣿⣯⣻⣎⢻⣿⣿⣿⣿⣿⣿⣆⢠⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣾⣾⣿⡿⠠⠀⣤⣤⣼⡆⢻⣿⣿⣿⣿⣿⣿⣽⡇⣿⣿⣿⣿⣿⣿⠿⣿⣿⡻⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠿⠿⠿⣿⣿⣘⢤⣩⠿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣋⣤⣶⣶⣾⣿⡷⡿⢿⡿⢹⡿⣿⣿⡇⣿⡿⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⠴⠚⠛⠉⠉⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠳⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⣦⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠙⠒⠒⠖⠚⠉⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠤⠶⠚⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠒⠆⠀⠠⠤⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠶⠀⠇⠀⣤⠀⠀⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠞⠀⠀⢀⠀⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⡎⠁⠀⠀⣰⢄⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠢⠤⠚⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡀⠁⠈⠁⠀⢀⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢻⠃⡄⠉⣶⡖⠂⠈⠉⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⠀⡇⠀⡃⠈⣿⢁⠀⠄⠀⡴⠃⠀⠀⠉⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢣⠀⠀⠀⠀⣰⢻⠀⠀⠀⠀⠀⣠⠇⢀⡗⢸⣿⢈⠆⡀⠀⠇⠀⠀⠀⠀⢀⠔⠂⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⣰⠃⢸⠀⠀⠀⠀⠀⡇⠀⢸⠀⣿⣿⠎⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⡄⠀⢠⠃⢠⠈⠀⠀⠀⠀⠀⢰⠀⢸⢠⣿⠏⠀⠀⣦⠄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡇⢠⠏⠀⡀⠆⠁⠀⠀⠀⠀⠀⡀⢸⢸⠏⣠⠎⡜⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⡄
⠀⠀⢀⣤⠤⠤⣀⠀⠀⠀⠀⠀⡗⢣⠖⢆⠀⢉⠸⢷⠀⠀⠀⠀⠀⠇⢸⡬⠊⢁⢀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⣇
⠀⠀⠘⠓⠒⠒⢺⠀⠀⢰⠉⣉⣉⠉⠀⡶⠁⠀⢀⡾⡆⠀⠀⠀⠀⢰⡼⠁⡠⠓⠘⠃⠲⢄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻
⠀⠀⢀⠀⠀⠀⢸⠀⠀⢰⠁⠠⠭⠉⣉⡤⠀⢀⡞⠀⢧⠀⠀⠀⠀⠈⠁⠀⣏⠀⠂⠀⢰⠛⢦⣀⣀⣀⠀⠸⢠⠤⢀⡀⠀⢸⡄
⠀⠀⢸⠀⠀⠀⣼⠀⠀⠰⡇⠀⣀⣐⣒⡇⢠⠎⠀⠀⢸⡄⠀⠀⠀⠀⡃⠀⢯⡀⠀⢀⡉⠀⠀⠙⡌⡆⠉⠉⠉⢳⣄⠘⢄⠈⡇
⠀⠀⢸⠀⠀⠀⢟⠀⢀⠤⠜⠓⠀⠒⠖⠒⠛⠲⡶⠦⠤⢧⠤⠤⢄⣀⣀⠀⠘⠦⠤⣀⣀⠀⠀⡸⠃⡇⠀⠀⠀⠀⠻⠄⠀⠷⡇
⠀⠀⢸⣀⣀⡄⣼⠀⠈⠳⠤⣄⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣉⡱⠶⠤⠤⠤⣍⣉⣀⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠙⠛⠂⠋⠀⠀⠀⠀⠀⠀⠉⠐⠂⠐⠂⠤⠴⠶⠦⠴⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠐⠒⠒⠓⠒⠶⠶⠦⠤

⠀⠀⠀⠀⠐⠶⠒⠒⠒⠒⠲⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣌⣈⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⢀
⠀⠐⠀⠀⠀⠀⠙⠃⠛⠛⠓⠐⠒⠷⠀⠀⠆⠀⠀⠉⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⡀⡀⢀⡀⣁⢂⢄⠀⡐⠹
⠀⠀⠐⠀⠀⠀⠀⠀⠋⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⢸⣀⠀⠀⠨⡀
⠀⠀⡀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠀⡄⠀⠘⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡋⠀⠀⠀⠸⠀⠀⠛⠛⠃⠀⠀⠀⠁⠃⠀⢐⢢⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠄⠀⠀⠂⠈
⠀⠀⡄⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠨⠅⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠄
⢀⠀⢠⡄⠀⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢀⠀⠀⢸⠀⢨⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢥⠳⠸⠘⠏⠏⠏
⠀⠠⠀⠀⠀⠀⠀⣀⡒⢃⢁⣰⢀⠛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⡈⠉⠙⠛⠉⠉⣘⢛⡂
⠀⠀⠀⠀⠀⠀⠤⣄⠈⢽⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡀⡀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⣀⠀⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡀⠀⠲⢈⡙⠗⡦⣐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣜⢦⣀⠑⠠⠉⠒⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠛⢿⣷⡤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⣠⣤⣀⣴⠀⠀⠀⠀⠀⠀⠠⣤⣶⡖⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠙⠉⠉⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⡶⠄⠀⠀⠀⠾⠀⠀⠀⠀⠺⠛⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⣠⡴⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠶⣤⣀⡀⠁⠀⠀⣀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠦⣀⡀⣀⠀⠤⢶⠶⣖⣶⣶⣷⣶⡶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠖⣿⡿⣫⣿⠟⠋⣠⢐⣨⣵⣾⣿⣿⣿⣟⣭⣶⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⢛⡟⢡⣾⣯⣾⡟⣡⡨⣶⣵⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡿⢿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣾⣿⠀⣾⢁⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⠟⢋⣴⣿⠿⣿⣿⣿⣿⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣉⣤⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠿⠻⠟⠻⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣄⣠⣤⣄⣀⣠⣄⡀⠀⠠⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡯⠀⠇⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⢠⣾⣿⠟⠋⠉⠉⠉⢉⣭⣍⡉⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠉⠉⠙⠻⣿⣿⡆⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣴⣾⣿⣎⡸⢏⠀⢀⣤⣴⣶⣷⣿⣿⣿⣿⣶⣦⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⠀⠙⣧⠀⣿⣿⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡾⣍⣿⡏⠁⠿⣉⣽⢻⡟⢩⣽⣭⡙⢿⣿⡿⠿⠃⠀⠠⣿⡟⣹⣿⡿⠛⣩⣭⡉⠻⣿⣿⣧⠀⠀⠀⢿⣷⠟⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣽⠿⡇⢸⠀⠉⠳⣾⠇⣼⣽⣯⣥⣤⣿⠆⠀⠀⠀⠀⣿⠘⠙⠿⢠⣜⣛⢻⠃⠀⣿⡿⠛⠃⣤⠀⡼⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢇⠀⠀⠈⠀⠀⠀⠙⠾⢷⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣉⣡⣤⠾⠏⠀⠀⠀⠘⡶⢈⡆⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⢲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣘⣇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡅⠘⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⡀⠀⠀⠀⢠⡀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠔⢸⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢻⡟⢿⢿⣦⣤⣼⠿⣿⣼⣿⣿⠀⠀⠀⠀⠀⡀⠃⡜⠀⣽⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣹⡿⣿⡿⣿⣿⣟⢿⣏⠀⠀⠀⠀⠠⠎⠀⠀⡼⠀⣤⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⡆⠀⠀⠀⠀⢠⣾⣷⢞⣇⢼⣴⠆⣜⡷⢾⣦⡀⠀⠀⠀⠀⢿⡶⠁⣔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⢷⡄⠀⠀⣰⣿⣾⢷⣾⣿⠷⠿⢾⣿⣿⡾⣿⣷⣆⠀⠀⢊⠼⣿⢘⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠸⢧⡿⣶⣾⣷⣾⣿⣿⣿⣷⣤⣶⣶⣾⣿⣿⣿⣧⣺⣷⡶⣾⣾⣟⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠛⠿⣿⣿⣿⡟⠁⠹⡿⠿⢿⣿⣿⡿⠉⢻⡿⣿⣿⠿⣻⡏⠂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢠⣽⣿⣿⣿⡄⠀⠈⠁⠉⠈⠁⠀⢀⣾⠁⠰⠂⣰⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣾⣿⣿⣿⡏⠛⠦⣄⣤⣤⣤⡤⠖⠟⣁⠔⠀⠌⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⣀⣂⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡖⠲⣯⣍⣉⣩⣭⣩⣉⣨⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⠄⠀⠀⠀⠀⢠⡴⠶⠶⣶⠶⣶⠶⢶⡶⠶⡶⠶⠶⣦⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⠤⠤⣿⠤⣿⠤⢼⡧⠴⡷⠤⢤⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀⠀⠀⡀⠀⡃⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣿⣤⣿⣤⣼⣧⣤⣧⣤⣼⢃⣾⠃
⠀⠀⠀⠀⠁⠀⠀⠉⠀⠉⠀⠈⠁⠀⠍⠀⠀⠁⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣿⣀⣿⣀⣸⣇⣠⣧⣴⠏⣼⠃
⠀⠀⠀⠀⠘⠀⠀⡇⠀⠉⠀⠈⠁⠀⠇⠀⠈⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠏
⠀⠀⠀⠀⢀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣃⣀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠇⠀⠀⠀⠀⠀⠀⠀⠐⢍⠇
⠀⠀⠀⠀⠉⡟⣏⠉⠉⠉⠉⠉⠉⠉⠉⡟⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠚
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⣤⣤⠄⠀⠀⠀⠀⢀⡤⠤⠤⡤⠤⢤⠀⠠⠤⠤⡤⠤⠤⣤⠀⢠⡏
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠸⣧⠀⠠⡧⠤⢼⠀⢸⡄⠠⡇⠀⢠⡟⢀⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣇⣀⣸⣀⣸⣀⣀⣇⣀⣼⠃⣾
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣇⣀⣸⠀⠸⡀⣀⣇⣠⠏⣼⠇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⣤⣴⠏
⠀⠀⠀⠀⢀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⢉⠇
⠀⠀⠀⠀⠙⡟⢟⠛⠛⠛⠛⠛⠛⠛⠛⡿⢟⠁
⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠋⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡀⣠⣄⢀⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠚⠉⠁⠀⠉⠈⠙⠙⠢⠤⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣻⠠⠄⠀⡤⠀⠀⠰⠀⠀⠀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠋⠉⠛⢦⠘⢆⡀⠐⠒⠒⠃⠀⣀⠤⠚⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⠃⠀⡀⠠⣌⡀⠀⣇⡠⠜⠋⢚⣊⠭⠟⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠁⢀⡰⣇⣀⡤⠒⠉⠀⠀⠀⢀⣀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢤⡔⠒⠉⢰⡍⠀⢣⣀⣠⣤⠒⠋⠉⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢆⢠⠖⢲⠀⠀⡜⠁⣰⠃⠀⠀⠀⠷⢤⠊⠁⠀⠘⡄⠀⠀⠀⣀⣀⣠⠤⢼⠀⠀⠀⠀⠀⠀⢀⣤⡖⢒⡶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡼⠀⣏⣠⣀⣧⣀⠟⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⢠⠏⠀⡏⠉⠙⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠠⠀⠀⠄⠈⢦⣠⣿⡛⠋⣳⢄⣀⣤⠔⣯⠽⡇⠀⢠⠇⠀⠀⡠⢄⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⢸⠀⡸
⠀⠀⠀⠀⠀⠀⣴⣯⣇⣀⣀⣙⣊⣀⣀⣸⣀⣀⣈⣶⣁⣀⣮⠼⠊⣀⠀⡇⠀⢸⠀⠀⢠⠃⠈⣆⠀⢀⣀⡼⠀⠀⠀⠀⠀⠑⢤⣧⣀⣀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢸⠀⡇⠀⣸⣉⣙⠏⠀⠀⠈⣯⠥⠴⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠔⠋⡠⠣⣖⣁⣀⡼⠀⠀⠀⢴⣃⣠⠴⠃⠀⢀⠴⣒⣒⠛⣒⠢⡀
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠐⠃⠚⠘⠒⠃⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠯⠗⠒⠒⠫⠝⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣤⣄⣉⣉⡒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⣿⠟⠛⠏⢹⣿⡿⣿⣷⣦⣅⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢫⠟⣹⡿⠟⠁⠐⠁⠚⠀⠀⠈⠁⠀⠀⠈⠉⣉⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠁⣰⣠⢀⣠⡄⡀⠀⢠⠀⠀⠀⢰⣦⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣷⡆⣸⣾⠇⢀⣯⢿⣿⠛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣆⠀⡼⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣴⢸⡏⢸⣿⢀⠀⠹⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⡞⣸⠃⣸⠏⣿⣿⣿⣿⣿⣿⣿⣧⡾⢱⠇⡾⢁⡷⢿⡿⠶⠄⢿⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢱⢃⡟⢀⣿⢸⣿⣻⣿⠿⠏⠙⠁⠋⠀⠋⠘⠁⠈⠀⠈⠀⣀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⣿⣸⠃⣸⡇⣾⠑⠋⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠿⠁⠂⠀⢨⣿⡇⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⡏⠀⠐⠛⠱⠃⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠈⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡏⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢣⡇⡀⣼⡦⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡎⣼⠀⠁⣿⢁⣏⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠇⠀⠀⠌⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣿⣼⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⡟⢀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⢀⣿⠁⡾⣼⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠢⡘⠿
⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⠿⠛⡿⠁⠀⠀⠀⠀⠀⢠⠏⡀⠁⠒⠠⢄⣀⣀⠀⣀⣠⡔⠉⠇⠀⠀⠀⠀⠀⠀⠈⢶⡀
⠀⠀⠀⠀⠀⢸⢠⣿⣿⡿⠟⠋⠁⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠲⠤⢀⡀⠈⠉⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠳⡤⣀
⠀⠀⠀⠀⠀⡿⣼⠛⠁⢄⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠳⠶⠌⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⣱
⠀⠀⠀⠀⠀⣾⠃⠑⢄⠀⠐⢄⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡛⣄⠀⠀⠀⠀⠁⠀⠀⠀⢀⡈⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠢⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠷⢶⠾⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠀⠀⢀⣾⣿⠛⠁⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⣼⣿⠁⠀⣴⣦⠀⠀⢀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⣿⣿⠀⠀⠘⢿⣷⣾⣿⠟⠁⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠹⣿⣆⠀⠀⠈⠛⠋⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⣷⣦⣤⣤⣤⣴⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⣿⣿⢿⣿⣶⣶⣶⣾⣿⢿⣿⡟⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⢿⣿⡟⠀⢻⣿⡀⠀⠉⠉⠉⠁⠀⣸⣿⠃⠘⣿⣿⢿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⠟⠛⣿⡏⠀⢸⣿⡇⠀⠘⣿⣧⡀⠀⠀⠀⠀⣠⣿⡟⠀⠀⣿⣿⠀⠈⣿⡟⠛⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⣿⡇⠀⢀⣿⡇⠀⠈⣿⣧⠀⠀⠈⢻⣿⣦⣤⣤⣾⣿⠋⠀⠀⢠⣿⡏⠀⠀⣿⣧⠀⠀⣿⡏⠙⠻⢿⣷⣄
⠀⠀⠀⢠⣿⡟⠁⠀⠀⢀⣿⡇⠀⢸⣿⡇⠀⠀⠹⣿⡆⠀⣠⣾⣿⡟⠛⠛⣿⣿⣦⡀⠀⣾⣿⠁⠀⢀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠙⣿⣧
⠀⠀⠀⣸⣿⠁⠀⣀⣄⠀⣿⡇⠀⠘⢿⣿⣦⣄⡀⢻⣿⣾⡿⠛⣿⣧⣤⣤⣿⣟⠻⣿⣾⣿⠃⢀⣴⣿⠟⠋⠀⠀⣿⡏⢀⣄⡀⠀⢹⣿⡀
⠀⠀⠀⣿⣿⠀⠀⢹⣿⣦⣿⡇⠀⠀⣠⣾⡿⠋⠀⠈⣿⣿⠀⢸⣿⡟⠛⠛⣿⣷⠀⢸⣿⡇⠀⠈⠻⣿⣦⡀⠀⠀⣿⣧⣾⣿⠃⠀⢸⣿⡇
⠀⠀⢠⣿⡇⠀⠀⠀⢻⣿⣿⡇⠀⠀⠸⣿⣧⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⢀⣿⡿⠁⠀⠀⣿⣿⣿⠃⠀⠀⠀⣿⣧
⠀⠀⢸⣿⠇⠀⢀⣀⠀⢻⣿⡇⠀⠀⠀⠹⣿⣦⠀⠀⠀⢿⣷⣸⣿⡇⠀⠀⣿⣿⣸⣿⠇⠀⠀⢠⣾⡿⠁⠀⠀⠀⣿⣿⠃⢀⣀⠀⠀⢿⣿
⠀⠀⣿⣿⠀⠀⠻⣿⣷⣤⣿⡇⠀⠀⠀⠀⠹⣿⣆⠀⠀⠸⣿⣿⣿⠃⠀⠀⢻⣿⣿⡿⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⣿⣯⣴⣿⠿⠃⠀⢸⣿⡇
⠀⠀⣿⡏⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⢻⣿⣿⠀⠀⠀⢸⣿⣿⠃⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠈⣿⣧
⠀⢸⣿⠇⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠈⣿⣿⠀⠀⠀⢸⣿⡏⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⢿⣿
⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠸⣿⡇⠀⠀⣾⣿⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⢿⣿⡀⣸⣿⠇⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⣷⣿⣿⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⣰⣦⣤⣄⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣤⣴⣶⡀⠀⠀⢸⣿
⢹⣿⡄⠀⠀⠈⠉⠛⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠋⠉⠀⠀⠀⣼⣿
⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟
⣤⣿⣯⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣬⣿⣧
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠑⠠⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⡀⢀⠀⠀⣀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠃⠁⠄⢀⠌⠀⠄⠀⠀⠀⠑⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠈⡀⠈⢢⡀⠀⠀⠀⣤⣨⣥⣤⣬⣿⣇⠑⠾⢁⣷⣤⣄⡀⠀⠒⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠇⠀⠁⢁⠀⠸⢃⡁⠀⣠⠟⠛⢿⣿⠏⣍⣿⣿⣾⣿⡈⠉⣿⣷⠀⢀⠀⠆⠀⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠐⠛⠃⢃⡘⠀⠏⠼⢫⣿⠿⠿⠒⠌⣙⠋⠀⠀⠸⠁⢀⠊⠀⠀⠀⠀⠀⠠⢂⡀⠀⠠⠎⠀⠃
⠀⠀⠀⠀⠀⠤⠀⡀⠑⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⢃⡴⠋⠁⠐⠀⠀⠀⠈⢢⠄⠀⣾⣄⠠⡁⠢⡀⢂⢀⠌⡁⠀⢀⡀⡤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣶⡾⠁⠀⠀⠀⠒⠀⠠⠀⣀⠄⢸⠋⡼⠁⠐⢀⠀⢀⠀⠀⠀⠀⣱⢴⣇⢹⣷⡤⢀⣀⣁⠉⠀⠀⡀⠈⡧⠉⠀⠀⠁
⠀⠀⠀⠈⠀⢈⡴⠶⣄⠀⡹⠁⠀⡄⡈⠁⠀⠈⢦⠦⡉⢲⠂⡰⠁⠀⠀⠀⠃⠐⠀⣶⠄⠀⢰⠆⠈⠀⠋⠀⢀⢡⠝⠄⠀⠒⠉⠉⠁⠀⠀⡀⠈⠉
⠀⠀⠀⠀⠈⠘⠷⠆⣼⡷⠁⠀⣀⡀⣉⠇⠀⠤⡄⢰⠏⠋⠐⢛⡀⠀⠀⠀⠀⠀⣼⠃⡐⠀⣼⣷⡀⠀⠀⣄⣈⣙⠁⠀⠀⠀⠀⠀⠀⠀⢈⣟
⠀⠀⠀⢀⣠⡿⠟⠉⠙⢁⠀⠀⠈⠛⠉⠀⠤⡀⠁⢨⡳⠀⠀⠱⢄⠀⠀⠀⠀⣸⣯⣴⠁⣤⠸⠛⠇⠀⣀⢹⣿⣽⣿⣤⠀⠀⠀⠀⠀⠀⣻⣿⣇⡀
⠀⠀⣸⣿⠃⠠⠆⡀⠀⠀⠠⣀⣀⠀⠀⠀⠈⠃⠀⡾⡃⠀⠀⠀⠀⡇⠀⠀⠀⢿⣿⠃⡸⠃⡀⠀⠔⡡⢺⡄⢿⡀⢹⣿⡏⠀⠂⠐⠀⠀⠉⢿⣿⣿
⠀⠀⠈⠙⢣⠀⠀⠄⠀⠀⠀⠀⡉⠁⠀⣤⣀⠀⡎⡈⡑⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡼⠀⣰⢋⢉⠐⠆⡜⠁⠀⠀⠈⠭⢄⠀⠀⢀⣈⡇⠀⢾⡿⠟
⠀⠀⠀⠐⠄⠑⠤⠀⠀⠀⢠⡜⢡⠴⣿⣿⣿⠛⢴⣦⠃⠀⠀⠀⠙⠀⠁⣤⢠⢤⠀⠠⠤⠄⡘⠀⠀⢠⡦⠀⠀⢀⡀⠰⠋⠀⠤⠜⠉⠢⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⣀⡀⠀⢩⠃⠀⠰⠛⠦⡄⣉⠓⠠⢾⡏⠀⠀⠀⠀⠀⢀⠀⢙⣀⠀⣃⠰⠒⠂⠀⠀⢠⠌⠓⡁⠞⢟⢻⡀⢷⠿⡀⢈⠀⢀⣂⣀⠀⠀⠀⡀
⠀⠀⠀⠀⢿⠏⢠⣾⣿⠀⠁⠀⠀⠀⠀⠉⠉⠒⠠⢹⡀⢣⠀⠀⠀⠘⠀⠁⡠⠌⠁⠌⠀⠀⠒⢄⠠⠦⠐⢁⠊⠛⣽⡧⠈⡄⣀⠀⠡⡈⠛⢿⡷⠀⡆
⠀⠀⠀⠀⠀⠀⢾⣟⡛⠄⠉⡙⠠⠄⡀⠀⠀⠀⠀⠀⠓⣤⠠⠀⠀⠀⠘⢢⠍⢀⡀⠠⠄⠀⠃⠺⡅⠀⣠⠂⠀⣼⣿⡇⢣⡅⠉⢢⣀⣼⣶⡾⠁
⠀⠀⠀⠠⠀⠀⠈⠙⠁⠈⠂⠈⠂⢤⣨⠜⠒⠂⢤⣄⠀⣿⣦⣤⣠⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠃⢀⡚⠉⠀⡆⠀⠀⡀⠀⠹⣿⡿⡁
⠀⠀⠀⠀⠉⠀⠀⠀⢀⠤⠐⠚⡟⠉⣀⠧⠄⠤⠤⠀⣀⠈⠉⠙⢹⠆⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢚⠃⢀⠎⠀⠄⠀⠱⠀⠀⠀⠀⠀⡋⢀⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⡏⠈⠀⠀⠀⢠⡒⠒⠠⠈⠦⠀⠒⠀⣎⠀⠀⠀⠀⠀⠉⠀⠀⣀⠎⡰⠏⢂⠠⠦⢀⠀⠒⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢿⡇⠀⠀⠀⠀⠠⡁⣀⠀⠈⢂⠠⠀⠊⠉⡕⠒⠄⣀⢀⢀⡠⠿⡏⡴⢠⠞⠣⠀⢡⠰⠤⠴⠓⣀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠻⠀⠀⠿⠀⡔⠒⠀⠀⠀⢈⠡⠄⢠⠆⣅⠀⠤⠸⣄⡀⠀⠙⠳⢌⡲⠌⢹⠀⢀⠀⢰⣾⡷⠆⠒⠫⢵⠄⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠢⡈⠁⠀⠀⢉⣴⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⣠⣤⣼⣶⣤⡀⠠⢈⠐⠸⢀⡈⢱⣾⡏⠀⠀⠀⠀⠀⠑⢄⠀⠠⠀⡂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠙⢿⡀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠫⠞⠛⠛⠉⢉⣉⢈⠁⡶⢄⡀⠀⡁⠘⣿⡇⠀⠀⠀⠀⠀⠀⢂⢂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠡⡈⠄⠀⠀⠀⠀⢀⣀⣼⠐⠀⠋⠁⠀⠉⠀⠀⠀⠀⢀⣀⣿⣷⡈⠋⠙⠇⠀⢠⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⡙⠒⠒⠀⠀⠀⠙⠛⠻⠦⢤⣤⣦⣴⡶⠲⠶⠶⠉⠉⡉⢉⢉⢉⢋⠀⠘⠨⠃⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠊⠀⠀⠀⠈⢻⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠠⠁⠀⠀⠀⠀⠀⠈⡇⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠘⠷⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠗⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠀⠀⠀⡀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢡⡄⠀⠀⠋⢠⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⣿⣿⠛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡀⠀⢸⠀⠀⢠⡀⠀⠀⢠⡏⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⢠⡀⢠⣿⡏⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠃⠀⢸⠀⠀⠰⠃⠀⠀⡾⠀⣶⠀⠀⢸⠀⢠⠀⢿⠀⢸⡇⠀⠀⣿⢸⣿⠇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡦⠀⢼⠀⠀⣄⠀⠀⢰⠃⠀⠘⠀⠀⢸⠀⠘⢧⠈⠀⢸⡇⠀⢀⡿⢸⣿⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢸⠀⠀⠸⡆⠀⡟⠀⠀⠀⢷⠀⢸⠀⠀⠈⠧⠀⠸⡇⠀⠈⠀⠘⠋⠀⠀⠀⠙



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⣿⣿⡏⣠⣝⠿⠋⠀⠀⠀⠀⠀⠈⢷⡄⠉⠒⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠘⣿⣷⣿⣾⡇⠀⠀⠀⢀⡀⠀⢀⣲⣆⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⡀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠑⠂⣼⣿⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣇⡀⢠⣿⠟⢧⡑⣭⣛⣋⣀⣤⣤⣌⠙⢻⡁⠀⢀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠘⢿⣷⣼⡇⠠⣀⠀⠉⠛⠛⠛⢛⠿⠿⢷⣄⢃⢠⣾⣿⣿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠹⣿⣡⡶⢖⠮⠀⠀⠀⠀⠴⢶⠶⣤⣹⣿⡿⠟⠉⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠀⠀⣸⠋⠒⠈⠁⠀⠀⠀⠀⠀⠘⠒⠉⡟⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢷⣶⣦⣷⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣦⣤⣶⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠙⠻⠟⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⠛⠛⠛⠉⢉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢠⠀⢰⣼⣧⠀⠀⣠⣄⣤⡀⠀⠀⠀⢸⣧⣤⣧⣶⣶⣰⣿⢋⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⡀⠀⠈⠛⢃⡀⠈⠻⣛⠋⠋⠁⠀⠀⣸⠏⠛⠉⠛⢻⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡓⣀⣤⣿⣷⡀⠀⠠⠄⠀⠀⢀⣼⣿⠷⠦⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠤⠠⠤⠞⠋⠁⢹⡶⡀⡀⠲⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡰⠀⠀⠀⠀⠻⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⡹⢉⠝⠀⠉⠁⠀⠐⠒⢤⡀
⠀⠀⠀⢀⠚⠉⠀⠀⠉⠂⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢧
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠤⠤⠄⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠷⠁⢠⢾⠂⣫⣿⡷⣤⢠⣤⣀⠐⠀⠄⠿⣃⣤⣬⡙⡭⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠐⠒⠿⣵⣿⠿⣷⡗⣒⣘⣋⣥⣤⠀⣸⡻⠀⣼⢣⣟⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⢳⡻⡭⠀⡺⢷⣿⡿⢓⠁⠁⠀⠨⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡐⠒
⠀⠀⠀⠀⠀⠀⢺⣿⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠛⡶⠀⢸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣤⡌⢀⣠⣆⠙⠓⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⡏⠉⢢⠀⠀⠼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣶⠟⠉⠀⣆⢸⣷⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠛⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠴⠒⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠲⢤⣀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡠⠊⢁⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄⡈⠓⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠠⠊⣠⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣶⣄⠑⠄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠠⠚⠉⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠛⠆⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣠⣷⠶⠛⠓⠾⢷⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡾⠷⠛⠻⠶⣾⣤⡄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢈⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⡦⡄⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⣿⠀⠀⠀⢀⣴⡟⠁⣀⠀⠀⢱⠘⠀⠀⠀⠀⠋⡎⠀⠀⣠⣶⡖⠂⡀⠀⠀⠀⣿⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢿⠀⠀⠀⠘⣿⣻⣿⡿⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⣛⣷⣾⠇⠀⠀⠀⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠒⡇⠀⠀⠀⠘⢆⠀⠀⠀⠈⠉⠉⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠉⠉⠁⠀⠀⠀⡰⠃⠀⠀⠀⡗⠛⢦
⠀⠀⠀⠀⢀⢃⢤⡀⡇⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⡠⠖⠁⢀⡤⠂⠀⠀⠠⢄⡈⢮⡢⢄⣀⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⡇⣠⠄⡇
⠀⠀⠀⠀⠘⡇⢰⡛⢳⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⠀⠀⠀⠀⣏⠀⡠⠀⠀⢀⠀⣳⠀⠉⠓⠶⢤⣤⡤⠶⠊⠁⠀⠀⠀⢀⠟⣳⠀⡇
⠀⠀⠀⠀⠀⠹⣄⠁⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠒⠚⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣤⣈⣷⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠙⠁⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣤⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡑⢄⡀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠶⣦⣤⡤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣤⡄⢸⣿⡅⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣤
⠀⢰⣿⣦⣼⣿⣧⠀⢠⣼⣧⡄⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⣿⣿⡇
⠀⢸⣿⣿⠿⠿⠛⠀⣏⣿⣿⣿⠀⠀⠀⠀⠀⠀⠏⣰⣿⣿⣿⠀⠀⠀⠀⠀⣠⣼⣥⣾⣿⣿⣯⠀⠀⠀⠀⠀⠀⣴⣮⣴⣿⣿⣿⠷
⠀⢠⣿⣷⠀⠀⠀⠀⠛⣿⣿⠛⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⠉⠁⠀⠀⠀⠀⠛⠻⠿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠸⣟⣿⡿⢸⣿⡇
⠀⠀⡟⣿⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⠘⠟⠛⠛⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠉⠇
⠀⠀⠃⠉⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⡄
⠀⠀⠀⢀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣃⡀⠀⠀⠀⠀⠀⠀⠀⢛⠞⠃⣠⡀
⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⣿⣏⠳⡀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠹⠛⠲⠤⣭⣶⡄
⠀⠀⠀⣿⡏⠈⠓⣄⠀⠀⠀⡀⠀⠀⠀⠠⣶⣄⠈⠠⣿⣿⣿⡧⠁⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⣿⡇⠀⠀⣿⠥⠭⠽⠃⠀⠀⠀⠀⠈⠛⠁⠠⠤⠟⢨⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡿⡇⠀⠀⢻⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠿⣧⣼⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣄⣠⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⣄⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣐⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⡀
⠀⠀⣾⣿⠛⣶⣦⣴⣾⠛⣿⣧⠀⠀⠀⠀⠀⠁⣿⣶⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⠙⣶⣶⡄⠀⠀⠀⠀⠀⢠⡿⣿⣿⡇
⠀⢀⣿⡇⠀⢸⣿⣿⡇⠀⣸⣿⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⢻⣇⠀⠀⠀⠀⢀⡘⣠⣿⣿⡁
⠀⠸⣿⣿⡄⠀⠀⠀⠀⢘⡿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⠈⠁⠈⠉⠀⠀⠀⠀⠻⢿⠿⣿⣿⠁
⠀⠀⠏⠈⠃⠀⠀⠀⠀⠘⠀⠙⠀⠀⠀⠀⠀⠀⠀⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡿

⣶⢶⠶⢾⡿⠷⣿⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠶⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠊⠀⢀⠀⠀⠠⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡌⡜⠰⢡⠂⠄⠄⢀
⠀⠀⠀⠀⠀⠀⠋⠀⠠⢀⠄⠀⠀⠀⠀⠀⠀⠠⡄⣀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⣠⠔⠀⠀⠀⢀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢡⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠁⠀⠀⠀⢠⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⡈⠂⠀⠀⠀⡀⠀⠀⠀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠊⠀⠀⢀⡴⠁⠀⠀⠀⠘⠤⠼⠀⠀⠐⡄⠀⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠑⠤⢀⣀⣀⠀⠀⠀⠀⠈⠉⠉⢉⡝⢉⠁⠀⠀⠀⠀⠐⠂⠁⠀⡀⠠⡀⠀⠀⠀⠀⠠⠄⢂⠈⠢⢅⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡈⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠈⠑⠒⠒⠀⠐⢒⡂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠠⡀⢀⣀⣀⢀⣠⡏⠀⠀⠀⡇⠀⢀⡰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⡤⠀⠀⠀⡄⠀⠠⠐⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⣀⣀⣠⠇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠐⠒⡀⠀⠀⠀⢺⡑⠀⠀⠀⣀⠀⠲⡁⠀⠀⣀⣈⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠲⠄⠀⠀⢔⣂⡲⠀⠀⠀⠀⠀⠀⠤⠤⠇⠂⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠐⠚⠉⠀⠄⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠤⠤⠄⠀⠀⠒⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠓⠀⠀⠐⠒⡏⠉⠀⡀⠒⠒⠒⡎⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⡀⡀⠀⠀⠈⢀⣀⠀⠀⠀⠀⠁⠤⠄⠀⠀⠀⠀⠃⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇

⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢘⡠⠸⠲⡶⡄⠀⠀⠀⠀⠀⢀⠀⡀⠄⠀⠈⠁⠄⠀⠤⠀⠀⠀⡤
⠀⠀⠀⠡⡈⠺⡃⢾⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠲⢶⠖⢃⣤⠤⢀⣖⠶⠶⣶⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠙⢏⣀⣶⠆⣭⡕⠂⠽⣤⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠲⢿⡁⣰⡷⠀⢑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠃
⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣠⠤⠤⠤⠬⣍⣓⠂⠿⠏
⠀⠀⠀⠀⢀⠖⢚⣉⡡⠤⠴⠖⠒⠒⠒⠶⡤⠇⠀⠀⠘⠛⠦⠰⢤
⠀⠀⠀⠀⠀⠉⢫⠴⢦⠀⢸⠉⣿⠀⠏⢭⠇⠀⠀⠐⠒⢠⠀⢐⡩⠐
⠀⠀⠀⠀⠀⠀⠸⡀⢸⠀⢸⠀⣿⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⢸⠀⡇⢸⢀⡇⢠⠒⠒⠶⣤⢤⡀⠀⠀⠀⠈⢔⠀⠒⠼⠁⠈⡁⠡⢊⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠘⢴⣁⣸⣤⣗⣨⠞⠀⠀⡽⣆⠀⡤⠬⠽⠆⠀⠀⠀⢀⠐⠀⡰⠋⡊⢸⣞⣩⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⣶⣶⡶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣫⣵⣶⣿⣿⣿⣿⣿⣿⣿⣶⣯⣝⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣨⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣵⡜⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡾⣫⣾⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣼⣿⣷⡉⣬⣭⡍⣩⣭⣍⢩⣿⡍⣩⡍⣩⡾⣫⣬⢹⣿⣿⣎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣧⠻⠿⠿⠸⠿⠿⠸⠿⠇⣩⡾⠫⠾⠿⢇⣿⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣆⢿⣿⠸⣿⣿⠸⣣⡾⣫⣾⢹⣿⡟⣼⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⡔⣶⡆⠶⣢⡾⣫⣄⣶⣶⣰⡶⣸⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣷⠉⣡⡾⣫⣬⣭⣭⣭⣥⣭⣥⣿⣿⣿⣿⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⣿⣿⣿⡿⣫⡾⣫⠞⣛⢿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡻⡿⣫⡾⣫⣶⣷⣜⣛⣤⣶⣦⣛⣃⣴⣶⣶⣿⣿⣿⢏⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣫⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠿⠿⠿⠶⠶⠛⠋⠁

⡀
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣫⣵⣶⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣻⢦⣄⠀⠀⣠⠊
⠀⠀⠈⠱⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢓⣽⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣗⢻⣷⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⠇⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣼⣿⣿⣿⡿⠉⢁⣀⡚⢥⣀⣀⣠⠶⠶⢶⣤⡀⠀⣿⣿⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢱⠀⠀⠀⢼⣿⣿⣿⣿⠃⠀⠛⠉⠛⡸⠃⢻⠁⢴⡟⢶⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⡀⢊⣾⣿⣿⢹⡟⠓⠦⠤⠤⠚⠁⠀⠘⢆⡀⠀⠀⣀⡴⠁⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡨⠂⢿⣿⣿⢰⡇⠀⠀⠀⠀⠀⠂⠘⠀⠀⠈⠉⠉⠁⠀⠀⣿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠁⣹⣿⣧⣾⢀⡀⠀⠀⠀⣴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠈⠁⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠼⠀⠋⣿⣿⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⠀⠀⢰⠀⠀⢸⣿⣿⣿⡟⠀⠀⢰⡀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢿⣇⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣮⣿⡜⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⣼⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣨⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣫⡖⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⣻⣿⣿⣷⠃⣿⠇⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⡿⣿⡿⣿⣿⢟⣾⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⡿⣿⣿⠷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣶⡀⣶⡀⣰⡆⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣀⣿⠛⢛⣛⣛⣛⣛⠛⢻⣆⡀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⠙⣿⠀⣿⠛⠛⠋⢻⡇⢸⡏⠃⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠛⣿⠀⣿⠀⠀⠀⢸⡇⢸⡟⠃⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠻⣿⠀⠛⠻⠟⠻⠟⠃⢸⡿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠙⠛⣿⠛⣿⠟⢻⡟⠛⠁⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠉⠀⡉⠀⠈⠁⠀⠀⠀⠀⢠⡶⢶⣄⣀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡹⣧⡀⠀⠀⠀⠀⠀⣿⣻⡧⠀⠀⠀⠀⠀⠀⠈⠻⠞⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠙⠉⣀⣈⡻⢦⣀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠏⢀⣾⠋⠉⠙⣯⠹⣿⡟⠛⠟⠋⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠈⢷⣄⣀⣠⡿⠀⠘⣷⡀⠀⠀⠀⠈⢳⡾⠃⠀⢸⡟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣀⠉⠛⠉⣀⣘⣿⢸⡇⠀⢰⡿⠿⠟⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⢿⣴⡟⠉⠙⣫⣴⣿⣤⣼⣷⣤⣤⣤⣤⣤⣾⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⡄



⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⢀⡀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⣟⣺⠰⡇⠃⡏⢹⢠⢸⡎⢦⠃⡇⣏⣹⢸⠃⣤⣿⢸⠀⠀⡇⡏⢹⢸⡁⡇⠛⢸⢸⢸⢰⣏⠃⡇⠓⢸⢹⡆⣏⡁⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⢀⣀⣤⣤⣶⣶⣶⣯⣤⣤⣀⡀
⠀⠀⠀⠀⠸⢷⣦⡀⠀⠀⠀⠀⠀⠁⠀⣰⢋⣴⣿⣿⣥⣤⣈⣉⡙⠛⠛⢻⣿⣿⣵⡢⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⠀⠀⠀⣠⣤⣽⣿⣿⡿⠟⠟⠻⢻⣿⣯⣶⡶⠇⠈⠉⠽⣿⣷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣿⣥⣴⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣒⠀⠙⠻⣿⣷⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠋⣉⣠⣤⣤⠤⠤⣴⣤⣀⣟⣻⠻⣿⣿⣿⣿⣿⡦⠍⠺⢿⣿⣿⣧
⠀⠀⠀⠾⠿⠛⠛⠋⠉⠁⢠⢾⠃⠈⣁⣀⣠⣤⣤⣤⣤⣤⣏⣉⡗⢀⢸⡏⠉⣻⣿⣿⣷⡉⠳⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡟⠀⠉⢉⣁⣤⣤⣤⣭⣴⡿⢷⡄⠀⠛⠛⠁⢠⣾⣿⠟⠿⣶⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢡⡶⠶⠛⠉⢩⡿⢯⠻⢿⣿⣦⣉⠀⠀⠀⢠⣿⣿⠟⠃⡀⠠⣽⣿⣿⡽
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣼⣧⡰⣤⣸⣆⠀⠈⠃⠀⠁⠈⠋⠁⠀⠀⠘⣿⣷⡀⠀⣙⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⡼⠀⠀⢀⣴⣿⡿⠥⡈⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣸⢯⣤⣽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⢧⡀⠳⣿⣿⣿⣋⣉⣓⠢⡄⠀⣿⠀⠀⠀⠀⣀⣴⡿⠇⠘⠿⠿⢫⣵⣶⡌⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⣿⣭⣍⣉⡉⠙⢿⠃⢋⠀⠀⢀⡼⠋⠁⠀⠀⢀⣠⣶⢾⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠠⠴⠖⠋⠀⠀⠊⠀⠀⠀⢀⣼⡿⢧⣌⡥⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣦⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠛⠢⡀⠀⠀⣀⣀⠀⢀⡤⠄⠀⠀⢐⡖⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡄⡎⠁⠈⢳⢸⡀⠀⠀⠀⢸⡇⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢺⡄⠀⠀⠀⡸⠁⠑⢦⡴⠋⠀⠉⢻⡛⠓⠒⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠷⠤⠴⠚⣁⠴⠋⠉⠉⠙⠦⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⣁⣤⠀⠀⠀⠀⣤⣈⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣽⠀⠀⠀⠀⡏⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⣿⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⡿⠀⣼⡇⠀⣿⠰⣏⠓⣋⡙⠞⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⡇⠀⡇⠀⡏⣿⠀⢻⢶⡋⢼⠉⢙⡆⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣧⣠⣧⠀⣇⢀⡇⢸⡀⣸⢰⢋⣨⡓⣋⡀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠉⠀⠛⠃⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠈⠀⠁⠀⠃⠂⠂⠀⠀⠀⠃⠀⠐



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

⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢰⡏⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢹⠀⢠⡇⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣦⠀⠀⠀⠀⠀⣴⠖⠒⠒⠒⣶⢰
⠀⠀⣿⣶⣯⣾⠋⠉⠉⠉⠻⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⣾⣿⢲⣦⡾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸
⠀⠀⣿⣿⣿⠛⠒⠒⠒⠒⠚⠃⠀⠀⠀⢸⣦⣤⣤⣤⣤⣾⠜⠁⠀⠀⠀⠀⠀⢸⣷⠽⣯⣿⠇⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣿⢸
⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⠤⢤⣤⡤⠤⠄⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⢠⡤⠤⠤⠤⢭⣶⡶⣄⠀⠀⠸⣿⣷⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠻⡏⡭⣱
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣦⣤⣤⣤⣼⣷⣭⣿⠀⠀⠀⠀⠉⠉⠻⢯⣶⣄⠀⠀⠀⠀⢠⡾⣿⡁⠀⠀⠀⢹⣷⠿
⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⣼⣽⣿⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠘⣧⣀⣷⣤⡴⠚⠉
⠀⠶⠷⠶⠦⠀⠂⠀⠰⠶⠦⠶⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⢠⠴⠦⠤⣤⣶⣶⣶⢦⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠂⠀⠀⢀⣴⣤⣤⣴⣶⣼⣾⣤⣦⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣷
⠀⢸⣿⣿⣿⡟⠉⠙⢿⣿⡇⠀⠀⠀⠀⢠⣼⣰⡈⠉⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⣸⣧⣤⣤⣤⣤⣤⣿⡀
⠀⢸⣿⣿⣿⣧⣀⣀⣼⣿⡇⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⠿⢿⠿⣿⣿⡟⠉⠛⢻⡆
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣤⣾⣿⣿⡇⠀⠀⠀⠈⠛⠉⠉⠉⠉⠉⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢀⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀
⠀⠀⢸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⣦⣤⡄⠀⠀⠀⠀⠀⠀⣟⠀⢠⢉⡉⣳
⠀⢠⣼⡿⢿⣧⡤⠤⢤⣤⠀⠀⠀⠀⠀⢀⠇⢰⣿⣿⢾⣷⠀⠀⠀⠀⠀⢀⡾⣱⣾⣿⣿⣶⡹⣧⠀⠀⠀⠀⠀⣸⣻⣯⣷⣲⣵⡇
⠀⢸⣿⣿⣿⣿⣦⣤⣼⣾⠀⠀⠀⠀⠶⠊⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⢀⡟⣾⣹⢧⡗⣾
⠀⢸⣿⣾⣿⣿⣿⣦⣾⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣻⣮⡻⠿⠿⣋⣼⡇⠀⠀⠀⠀⠸⢵⣇⣟⡽⢰⠇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣋⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠠⠴⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿⣿⣿⣿⡿⢋⣿⣿⡿⠿⠋⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠼⠛⠉⠀⠀⣼⣿⡿⠟⠋⠉⠀⣼⣿⣿⡿⠟⠋⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⣠⣼⣍⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⣨⠀⠀⡀⠀⠀⠀⡾⠛⠻⠆⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⠀⠑⠲⡄⠀⠀⠀⠀⠀⠈⠛⠁⠀⡼⠃⠀⠀⠀⠱⢤⠾⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⠄⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠚
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣠⠇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣤⣠⠟⠉⠁⢰⡄
⠀⠀⠀⠀⠀⠘⣧⡀⢀⣀⣿⣆⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢹⠈⠏⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠸⡟⠋⠉⠀⢹⡆⠀⠀⠀⠀⠀⠙⠷⣤⣄⣈⠻⣿⣿⠟⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⢠⡇⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⠀⢉⡟⠛⠛⠷⠾⣿⣿⣿⣉⡁⠀⠀⠀⠀⠀⣰⠏⠉⠙⠒⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡇⠀⢀⣠⢴⡞⠁⠀⠀⠑⠲⠖⠒⠚⠋⠁⠀⠉⣿⢄⠀⠀⢀⡾⠃⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⣴⠟⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⣤⡴⠶⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠻⢶⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⠉⣿⣿⡝⠻⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠈⢿⣿⡆⢘⢏⣠⣤⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⡦⠜⠛⠁⠀⠈⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣽⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⢦⡀⠀⠀⠀⠴⠖⠒⢄⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡖⠦⠈⠀⠀⠀⢐⣶⣦⠀⠀⢠⠟⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡄⠀⠀⠀⠀⠀⠉⠈⠀⠀⢸⣬⣷⣭⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠐⠛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠤⠤⣾⡇⠀⢸⡿⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣡⣠⣿⣟⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠂⠀⠀⢿⡀⠈⣿⣿⡿⣿⡇⠀⠀⠀⠀⢠⣤⣾⣏⣉⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠠⠀⠈⠱⠀⣿⠀⠀⣸⡇⠀⠀⣠⠔⠾⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡐⠀⠀⠀⠀⠀⠀⢆⢠⠻⣦⡴⠟⣀⡄⠀⢹⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢳⣀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡅⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⢄⠀⠀⠀⠀⢹⠟⠉⢏⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⠃⠀⠀⠀⡄⠀⠀⠈⢢⠀⠂⠈⠉⠉⢧⢈⣿⠃⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡏⠀⠀⠀⣰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡾⠁⠀⠀⠀⢀⡤⠶⠚⠛⠛⠳⢦⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢀⡞⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢠⡟⡆⢀⠞⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠎⠀⢱⡀⠀⠀⠀⠀⠱⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⡏⡴⠋
⠀⠀⠀⠀⠀⠀⡼⠀⢹⠏⠀⠀⠀⠀⠀⡟⢧⠀⠀⣠⠏⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⣀⠆⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠃⡰⠃⠀⠀⠀⠀⠀⢀⡇⠈⢆⣰⠃⣀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠋⠁⠈⡇⠀⣿⠓⠶⠤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢁⠀⢻⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣴⣦⣤⡄⢀⣤⣤⡄⠀⣴⡦⣤⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁⠀⠀⢸⣿⣿⣿⣿⣷⣤⣿⣿⡇⠀⣿⡷⣿
⠀⠀⠀⠀⠀⠛⠛⣛⣿⠿⠀⠀⠀⠀⠙⢿⣿⣿⣟⠻⣿⣿⣿⣿⣿⡏⣿⣤⣠⣷⣤⣄⣀
⠀⠀⠀⠀⠀⣶⣿⣭⣤⣤⠀⠀⠀⠀⣀⣠⣼⠿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣧⣽⠿⠇
⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⣠⣴⣿⣿⣿⣿⡉⠁⠉⠛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿⣟⠥⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⣿⣷⡿⢿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠻⠛⢿⣿⣿⣿⣿⣿⣿⣫⠤⠄⠀⠀⠀⠀⠀⠀⢿⣿⡇⣿⣿⣷⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣭⡅⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⢠⣷⡄⢹⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣧⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⢿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠻⠛⣿⠀⠹⣿⡟⢷⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠸⡷⠀⠀⠀⠿⠀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠟⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣏⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠀⠀⠀⠀⣀⣴⣿⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⣇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢰⣿⡆
⠀⠀⠀⠀⠸⣿⣿⡄⠀⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣶⣶⠀⠀⠀⠀⣀⠠⣶⢸⣿⠻⣿⠇⡀
⠀⠀⠀⠀⠀⠉⠛⡄⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠿⣆⢸⡇⢰⠿⠿⠃⠀⠀⠀⠀⠿⠀⡇⠈⡟⠀⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⡇⠠⡇⠀⡇⠀⡋
⠀⠀⠀⠀⠀⠀⣠⠏⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡏⠳⣄⠀⠀⠀⠀⠀⠀⣿⠀⣧⢠⡇⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠰⠏⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠚⠃⠀⢸⡇⠀⠈⠛⠀⠀⠀⠀⠀⣿⠀⣿⢸⣷⠀⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣉⣀⣀⣀⡀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠠⠆
⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠉⠁
⠀⠀⠀⢀⠄⠉⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠢⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠻⣄
⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑

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

⠀⠀⠀⠀⠀⠀⢼⠢⣄⠀⠰⡖⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠉⠒⠼⠆⠀⠀⠉⠒⠢⠤⠤⠤⠤⠶⠒⠒⠚⠛⠋⠉⠉⢳⡀⣀⣀
⠀⠀⠀⢠⣄⣀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢅⠠⠼
⠀⠀⠀⠈⢧⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠈⣛⠶⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡴⠂⠀⠀⢀⣀⠤⡴⠂⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⡳⢛⡫⠿⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠸⣇⢶⠀⠀⢀⡤⠞⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠖⠛⠓⢦⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠈⠈⠙⠚⢁⣤⣒⣲⠆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠏⠀⠀⠀⠀⣳⣀⠤⠤⣌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠛⠋⠉⠙⠿⡽⠉⠒⣤⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠣⡀⠀⠀⣀⢧⡀⠀⠉⢀⠇⠀⠀⠀⠀⠀⠀⠀⡠⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣨⠍⡉⠁⠀⠙⠣⠉⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠣⠤⠆⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢠⠀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣂⠔⠋⠀⠀⠀⠀⢸⣉⡹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣠⠤⠖⠋⡙⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣀⣀⡀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⡏⡟⠉⠈⠉⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⡟⠗⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⡿⠗⢾⣦⣀⣀⡀⠀⠀⡇

⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⡄⢀
⠀⠀⠐⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡀⠈⠂⠃
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢻⣿⡏⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⢯⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠄⢐⣶⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠈⣼⠇⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠓⠒⠂⠒⠒⠒⠚⠓⠂⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢁⣀⣠⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠙⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠁⠀⠀⢀⠐⣦⡀⠀⠀⢠⠛⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢳⣂⠀⠀⠀⢀⡲⠅⠰⣏⠑⠂⠀⠀⠀⠀⠂⠀⠀⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠛⠀⠄⠀⢀⠀⠀⠀⡙⠲⠢⠀⠀⢠⣤⣤⣤⣤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠀⠀⠀⠠⠤⠀⠀⠀⠀⠤⠀⠤⠆⠀⠀⠀⠬⠀⠀⡀⠀⠀⣿⠀⣿⣿⣼
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡟⣿
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠄⠀⣄⢀⣀⡀⢀⣟⠀⣿⡇⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠰⣢⣀⣀⣈⣉⣭⣼⡇⢀⠿⠁⡟
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⡄⠀⠀⠀⠐⢒⢂⠀⣿⡛⠛⠛⡟⠛⢻⡇⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣿⣿⣀⣀⣇⣀⣸⣧⣤⣤⣤⡇
⠀⠀⠀⡴⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢰⠟⡿⠉⠉⠉⡏⠉⢹⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠿⠀⠀⠀⠃⠀⠸⠇⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠀⠀⢿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡟⠀⠀⠠⣤⡄⢿
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⣾⡟⠀⣴⠶⣦⡄⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⠉⢹⡟⠙⢻⣷⡀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⡆⠀⠘⣷⠀⠀⢿⣧⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠹⠇⠀⠈⠁⠀⢸⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⢿⣶⣤⣤⣄⣀⣀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠈⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⢻⣿⠀⢀⣠⣴⡿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⢿⣧⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⣿⣦⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠹⢷⡄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡆
⢀⣴⣶⣶⣶⣶⣶⣶⣶⠿⠿⠋⠀⠀⠀⠀⠀⣠⣾⠟⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⠀⣀⣠⣾⠋
⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⡟⠉
⠀⠀⠹⣿⣦⣠⣤⣶⠿⠛⢿⣧⠀⠀⠀⣸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⠿⠟⠃⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⠇⣿⣿⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⢙⣿⡆⢸⣿⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠚⢿⡏⠙⣿⣶⣿⣷⣾⣷⣿⣿⠀⢘⣿⣷⣿⣶⣾⣷⣾⠟⢉⣾⠗⢲⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⣽⡷⣠⣿⣦⡾⠁⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠹⣦⣼⣷⣤⣾⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣨⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣀⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⣿⣯⣉⣉⣉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣶⠟⠛⠛⠦⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠊⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⢴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠤⠀⠈⢀⡞⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡠⢦⡀⠀⢸⣤⣴⠟⠓⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠿⠞⠀⠀⠀⠈⠁⢀⡄⠀⠒⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡆⠀⠀⠀⠀⠀⠀⡟⢛⢫⣥⣀⠀⠀⠀⠀⠀⠀⢀⠞⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣠⡴⣦⠀⢸⠀⠀⢀⣽⣤⣶⣤⣀⣠⣤⢋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡕⠂⠘⠓⠒⠁⠀⣻⣿⣿⣿⡿⣟⣝⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣤⣶⣿⣇⣠⣴⣿⣿⣿⣿⡷⣌⢀⣾⡀⠀⠀⠀⠀⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠟⡵⠫⣄⠀⠉⠲⣤⠶⠋⠉⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢋⠞⢁⣠⡽⣦⡀⠀⠈⣦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⡙⢿⠿⠿⠿⢿⠟⡡⠛⣾⣋⣁⣀⣈⡙⠲⠾⠋⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡼⠃⠀⠀⣰⢋⠞⢵⠋⠙⠿⣶⣄⠀⠀⠈⣹⡟⠑⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣾⠀⠀⢀⡜⡵⠃⠀⠀⡇⠀⠀⠀⣻⣷⣄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⡤⠤⠀⠀⠀⠀⠈⢷⡄⠘⡶⢊⢫⠞⠁⠀⠀⣰⠃⠀⣴⣿⣿⣿⢯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢠⢂⡾⠆⠀⠀⡴⠃⠀⣀⣸⣿⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣮⠞⠁⠀⢀⡞⠁⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣡⠋⠀⠀⠀⢠⠎⠀⠀⣀⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠜⠁⠀⠀⠀⣰⣧⣤⣀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠋⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡇⠀⠀⠀⠀⠀⣠⠶⠚⠒⠲⣿⡞⠁⠀⠀⠀⠀⠘⠋⠉⠉⣇⠀⠀⣿⢆⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠹⠀⠀⠀⢀⠜⢡⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠘⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⡀⠈⠢⡀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⢻⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⡠⠊⡀⠀⠀⠈⢢⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡟⠀⠘⣇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠰⠞⢋⣁⡤⠂⣠⠔⠊⢀⡠⢂⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⡇⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠩⢭⣴⠲⠭⢐⣠⠔⣋⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠬⢭⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠈⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢸⣿⣿⠁⠀⠈⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⢸⣿⣿⣷⣤⣾⣿⣿⡇⢠⣤⣶⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⣤⣤⡎
⠀⠀⠀⠀⠀⠀⣠⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⡤⣶⣦⣀
⠀⠀⠀⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣯⣷⣿⣿⣿⣾⣽⡲⢤⡀
⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⢿⢿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⣤⣤⣤⣤⣄⠠
⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⢹⣿⣿⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀
⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠈
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢠⠀⠸⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠎⢱⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⡂⠈⢦⣤⣶⣶⡄⠀⢸⣿⣿⣿⢿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣷⣦⠀⠀⠀⠇⠀⢸⣿⣿⣿⡇⠀⠀⠉⠉⠁⠘⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢀⣤⣧⡄⣼⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠲⠤⠴⠒⠋⢩⠟⠒⢦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢶⡋⠀⠀⢠⠶⠦⡄⠀⠀⡰⠃⠀⠀⠚⠀⠀⠀⠈⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠙⣆⠀⠀⢧⠀⠀⠸⢄⣀⢀⣀⠞⠁⠀⡠⠖⠒⠲⢤⣀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠘⣆⠀⠀⢀⣀⣀⠀⠈⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢈⡧⣀
⠀⠀⠀⠀⠀⣀⠴⠚⢧⡀⠀⠀⠀⠘⡆⠀⢯⠀⠈⢆⠀⠀⠀⡜⠁⠀⠀⣄⠀⠀⢀⡜⠀⠀⠀⠀⡰⠋⠀⠈⠓⢦⡀
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠳⢄⠀⠀⠘⡄⠈⠁⠀⠈⡆⠀⠘⣄⠀⠀⠀⠈⠙⠚⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⡏⠀⠀⣀⠤⣀⠀⠀⢈⡆⠀⠀⠘⢄⣀⣀⡠⠃⠀⠀⡼⠓⢤⣀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠂⠉⠓⠦⣀
⠀⠀⢠⡇⠀⡼⠁⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⠞⠀⠀⠀⠈⠉⠓⠒⠚⠉⠀⠀⢀⡞⠁⢀⠀⠀⠀⠈⢹
⠀⢀⡼⣄⡀⠙⢄⠀⢀⡴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠑⣄⠀⢠⠋
⢠⠋⠀⣀⡁⠀⠀⠉⠉⠀⣠⠔⠉⠉⠉⠲⣄⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠶⠶⠶⠦⠤⢄⠀⠀⠀⠙⠢⠤⠎⠀⢸
⡷⠒⠉⠉⠉⢣⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⡞
⢧⠀⠀⣰⠂⠈⣇⠀⣇⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠹⡄⠀⠑⢤⣀⠀⠀⣀⣀⣀⣀⠴⠋⠀⠀⠀⣀⠤⠴⠒⠉
⠘⡄⠀⠹⣄⣠⠞⠀⠘⣆⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠘⠦⣄⣀⣀⣀⠀⠀⠈⠑⠒⠋⠉⠀⠀⢀⣀⣀⣀⣇⣀⣀⠀⠀⠀⠀⢠⠎⠉⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⡼⠙⠒⠒⠲⠶⠶⠒⠒⠋⠉⠀⠀⠀⠀⠀⡨⠋⠙⠒⠒⠾⠧⠤⠤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠋⠠⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⣿⣻⠄⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⡀⠀⠀⠷⠂⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣶⣶⣿⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⠘⠗⠉⠀⠀⣶⣶⣦⠀⠐⠋⠋⠛⠋
⠀⠀⠀⠀⠀⠠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⡀⠀⠀⠀⠀⠀⢀⡀⠶⠘⣃⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠈⣁⠰⠆⢋⠀⣴⡿⢻⣿⠀⠀⢠⣶⣠⣦⣴⣆
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠟⠻⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⢉⣤⡌⠀⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠻⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣶⣄⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⠋⠙⠛⠛⠛⠛⠛⠉⠛⠛⠋⠛⠃
⠀⠀⠀⠀⠀⠀⢰⣸⣿⣿⡏⠉⠙⣿⣿⣯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡀
⠀⠀⠀⠀⠀⠀⣏⠈⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠷⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⠀⠒⠀⠀⠀⠀⣂⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠈⠙⠃⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣴⡿⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣤⣤⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠆⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠛⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢛⡙⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⢿⡿⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠶⠶⠾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⣶⣶⣶⣶⡄⣿⣿⡿⢀⡀⠀⠀⠀⣠⣾⠋⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣷⣿⠟⣡⣿⠇⠀⢠⡾⠟⠁⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢉⣤⣾⠟⣡⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠻⣿⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡶⠟⠋⠀⠀⠀⠀⢈⠻⠿⠟⣠⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠋⠉⠁⠀⣀⣀⣀⣀⡀⠀⠀⣿⣷⣶⣾⣿⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠙⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢆⠀⠀⠀⠀⠏⠑⡄⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡂⠀⠇⠀⠀⡘⠀⢠⠁⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠂⠐⠂⠠⢎⢀⡈⠀⠀⣰⠷⠀⠊⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡴⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠒⠀⠉⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⡄⠀⢌⠢⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠡⠌⠦⢕⠚⠂⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠈⠢⠐⠀⠀⠀⣀⡼⠀⠀⠀⣀⠠⠄⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⣨⢃⡹⠊⠁⠀⠀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢘⡠⢶⠟⢹⠀⠀⠄⠒⠈⠁⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡖⠀⠀⠀⠀⠀⣸⠂⠁⠄⢜⠁⠀⠀⠀⣀⣀⣀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢡⡶⢀⠀⠤⠞⢙⣈⣤⣠⢭⠉⠁⠒⠤⢠⢡⠀⠀⠤⠠⡔⠠⠀⡠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡼
⠈⢆⠀⠀⠀⠀⠀⣐⡏⡀⠘⠀⠀⠀⠀⣈⡀⠀⠀⠀⢀⠟⠄⢊⣀⠐⡈⠑⣀⠌⠀⢠⠂⢄⠀⢀⠂⠀⠀⠀⠀⠀⡇
⠀⠘⣆⠀⠀⠀⡰⢹⡠⠉⠙⠂⠨⣉⢡⠀⠈⠦⠒⡖⠁⠀⠀⠀⢠⠀⠀⠉⠓⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢄⠀⠀⠄⠃⢇⠀⠀⠀⠀⠀⡄⢰⡄⠳⠈⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇
⠀⠀⠀⠀⠒⠰⠀⠐⣚⣧⣐⣀⣀⣀⣈⠁⠈⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣤⣤⣴⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠈⠙⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⣿⣿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢻⣿⣄⢀⣀
⠀⠀⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⠀⠸⢿⣿⣿⣿⣷⡀
⢠⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠁⢹⣿⣿⣿⣿
⠘⢿⣿⣿⣿⡟⠁⠶⠦⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⣴⣾⣿⣿⣿⣷⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠈⠛⠋⢻⣿⣿⣿⠿⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⢻⡿⢶⡄⠀⠀⠀⢠⣿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠈⣇⠀⣷⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠸⣧⠀⠑⠀⠉⠀⢹⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠛⠛⠶⣶⣶⣶⣿⠀⠀⠀⠀⣠⠟⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣷⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀⠀⠘⢯⣠⡴⠟⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢹⣧⡴⠛⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠙⢦⠀⠀⠈⣷
⠀⠀⢀⣤⣤⣤⣤⣤⠾⠋⠀⠀⢀⣴⠟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣾⠁
⠀⠀⠙⣷⠶⠶⢶⣶⡄⠀⠈⢿⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⡶⠛⠁
⠀⠀⠀⠘⠳⠶⠟⠉⠻⣦⣤⡾⠁⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣶⣶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⡇⣿⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠒⡞⢳⡶⠶⠶⣾⡇⢹⡶⠶⠶⢶⠛⡖⢲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣀⣙⣑⣉⣉⣀⣀⣀⣸⡇⣿⣀⣀⣀⣈⣊⡑⣟⣁⣈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣰⣼⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣇⣀⣴⣾⡷⠳⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠻⣿⡿⠋⠀⠀⠀⠈⠻⣾⣿⣿⣿⣧⠿⠓⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠁⢀⣀⣤⠶⠛⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠙⠷⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠉⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⢰⣾⣶⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡇⠀⠀⠀⠀⢻⣿⣷⣶⠖⠀⠀⠀⠀⠀⣴⣤⠀⠀⣀⡀⠈⠛⠃⡀
⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⣾⣿⡇⠀⠀⠈⠉⠁⠘⠛⠁⠀⠀⢸⣿⡷
⠀⠀⠀⠘⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠀⠈⠉⠀⠶⠛⢀⣤⡀⠀⠠⣾⡿⠀⢀
⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⣼⣿⠀⢠⣶⣄⠀⣴⣿⠀⠀⠀⠀⠸⢿⠿⢀⣤⣄⡀⠼⠿⠃⠀⣾⣿⡆
⠀⠀⠀⢀⣀⠀⠀⠀⣶⣷⡆⠀⠻⠟⠀⠘⠿⠋⠀⠉⣁⣀⠀⠀⢀⣀⠀⠀⢿⣿⣿⡇⠀⣠⣤⡀⠀⠉
⠀⠀⠀⢾⣿⣷⡄⠀⠈⠛⠁⢠⣶⣶⠀⣴⣦⡀⠀⠀⠿⠿⢁⣶⣿⠛⢠⣤⣤⠉⠉⠀⠀⠹⠿⠃
⠀⠀⠀⠀⠙⠛⠁⠰⠾⠿⠂⠀⠙⠋⠀⠈⠙⠁⠀⢀⣤⠄⠈⠻⠟⢠⣶⣶⣆
⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⢾⣦⡀⠀⠉⠉⠀⠀⠀⠀⠘⣿⣿⡯
⠀⠀⠀⠀⠀⠈⠛⠋⠀⣶⣶⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠇⠀⠀⢹⠀⠀⠀⠀⠀⢀⠔⠚⠓⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⢸⣿⠀⠀⠀⠀⢠⣇⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⣆⠀⢀⠞⠀⠀⠀⠀⢸⠿⠃⠀⠀⡸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠁⠊⠉⠑⠀⠀⠘⢤⣀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠛⢿⠛⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢰⠉⠣⠤⢤⣀⣈⣙⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢄⡀⢀⡠⠤⠤⠤⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⠈⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠉⠉⠓⠲⢤⣤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠹⡄
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⡰⠒⠊⠁⠀⠀⠀⠉⠢⢤⣀⡽
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢃
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠘
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡀
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⠀⣀⡀⠐⢤
⠀⠀⠙⠦⣀⠀⣠⠖⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⣠⠯⠤⡌⢁⡴⢐⡦⠽⡀⢈⡀⢈⡀⢀⡁⠀⢩
⠀⠀⠀⠀⠈⠷⢧⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⡠⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠉⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡩⠙⠛⠋⠉⠋⠓⠀⠀⠐⠊⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠥⠄⠀⠒⠒⠐⠒⠀⠒⠀⠀⠀⠤⠭⠵⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠘⡆⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⢒⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⡠⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⣠⣴⣶⣶⣄⡀⠀⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠁⢀⠠⠐⠀⢡⣿⣿⠀⣽⣿⡆⠁⠒⠄⡀⠈⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠒⠂⠀⠀⠀⠉⠁⠀⠒⠃⠀⠀⠤⠈⠁⢀⠀⠀⠈⢿⣿⣁⣸⡿⠃⠀⠀⠀⠀⠁⠂⠀⡀⠈⠑⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠐⢈⡀⠢⡈⠉⠀⠒⠂⢄⠀⠀⠈⡒⠐⠙⠛⠛⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠃⠀⠀⠀⠀⠁⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⢿⡛⠀⠀⠀⠀⠀⠀⠀⠒⠄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠈⠒⠀⠁⠀⠀⠀⠀⠘⠿⠃⢀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠄⠀⢀⡠⠎⠠⠌⢁⣀⠀⠀⠀⢀⠄⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠈⡏⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⢒⠀⠈⠭⠘⠐⠆⠐⠀⠤⠅⢀⣁⡀⠀⢰⡇⠀⠤⠤⠤⠤⠄⠄⠀⠁⠀⠀⡏⠀⠀⢹⠁⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠄⠤⠤⠤⠄⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡀⠀⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⠉⠁⠢⡀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠄⠤⡀⠀⡤⠤⠤⠤⠧⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠦⠀⠀⡼⠤⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⠀⠃⠀⠓⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠔⠒⠀⠀⠀⠐⠂⠀⠃⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⣔⠀⠐⠒⢒⠖⠶⠴⠦⠦⠤⠄⠀⠤⠴⡶⠖⠒⡒⠒⠒⠒⠒⠀⠤⠤⠌⡒⠄⣀⡀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣑⠠⢀⠀⠀⠀⠀⠀⠈⠠⠄⢉⡛⠶⡤⠈⢑⠄⠀⠀⠀⠀⠀⠀⡠⠆⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⣌⣁⣀⣀⣄⣒⠒⠒⠒⢶⣿⣭⣭⣭⣤⣤⣭⡽⠿⠿⠿⢶⣶⣶⣶⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣭⣭⣵⠿⠷⠿⠿⢿⣿⣿⣭⣭⣭⣭⣿⠿⠿⠿⠿⣿⣿⣿⣯⣭⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣯⣿⣿⣷⣾⣿⣿⣿⣯⣭⠭⠭⠽⠿⠷⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⣀⠀⣀⢀⡀⢀⣀⠀⠀⢀⣀⡉⠙⠋⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣶⣶⣶⣶⡶⠂⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠈⣯⠉⠉⠈⡏⠉⠉⠉⡏⠉⠉⢩⡏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⠶⣿⠶⠶⠶⡷⠶⠶⢶⡷⠶⠶⢾⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠛⠛⢻⠛⠛⠛⡟⠛⠛⢻⠟⠛⠛⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣸⣆⣀⣀⣇⣀⣀⣸⣀⣀⣸⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠉⢹⡏⠉⠉⡏⠉⠉⣿⠉⠉⣹⠁⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠈⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣼⣧⣤⣤⣧⣤⣤⣿⣤⣤⣿⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇⠀⠀⡇⠀⠀⡟⠀⢠⡏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⠶⣿⠶⠶⡷⠶⢶⡷⠶⢾⡇⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡇⠀⢸⡇⠀⣾⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠒⠚⠛⠒⠚⠓⠒⠛⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠑⠀⠐⡂⢀⠀⠠⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡄⠀⠀⠀⠆⠀⣆⠀⠀⢸⠀⠁⠀⢸⠉
⠀⠀⠀⠑⢦⠠⠀⢥⠘⡀⠄⢸⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠒⠈⠆⠀⠀⣀⣸⡀⡏⢆⠀⢸⠀⠀⠀⢸
⠀⠈⠳⠤⠘⠁⠀⠘⠀⠹⠀⠸⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠤⠤⠁⠀⠳⠇⠈⠿⠠⠤⠤⠀⠸
⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠢⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠐⠀⠀⠀⠀⠂⠀⠂⠄⠀⢀⢉⡙⠂⠀⠢
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠃⠒⠀⠀⠀⠈⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠐⠰⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠤⡀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠄
⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠢⣀⠄⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⣤⣤⣤⣤⣴⣴⣶⣶⣶⣦⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠊⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡔⠉⠉⠙⢦⠀⠀⠀⠀⠀⢀⠖⠉⠉⠓⣄⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⢀⣴⣦⡄⣧⠀⠀⠀⠀⡏⣠⣤⣄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣼⠀⠀⠀⢸⣷⣿⣿⣿⡇⠀⡧⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⢿⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⣿⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⣋⡝⠁⠀⠀⠀⠀⠙⣝⠛⠋⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢻⡿⠛⠿⠿⣿⣿⡿⠿⠿⣿⡟⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣶⡿⠟⠙⢷⡿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠿⠟⠉⠀⠀⠀⠀⠙⣾⣿⣿⣿⠞⠋⠀⠀⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠈⠻⣯⡀⣀⣤⠶⠛⠉⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⠻⠿⣿⣷⡀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠙⣿⣿⣄⢀⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⣿⡿⠀⠸⠟⠁
⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠀⠀⠀⠀⠘⣿⣿⠟⠁⣤⣤⠀⠀⠘⢿⡆⢠⣴⠆⠀⠀⢠⣤⡄
⠀⠀⠀⢰⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣥⡀⠈⠻⠟⢀⣠⡄⠀⠀⠀⠀⣠⣤⠀⠘⠛⠃
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢠⣤⠀⠀⣀⡈⠁⣠⣤⠀⠈⠀⠀⣾⣷⡄⠀⠙⠋⢠⣴⠄⠀⣠⣤
⠀⠀⠀⠀⠀⠁⠀⢀⣤⣄⠀⢸⣿⠁⠸⣿⡿⠀⠛⠋⠀⠀⠀⠀⠈⠉⢰⣿⣿⣦⠈⠁⡀⠀⠛⠛
⠀⠀⠀⣶⣶⣄⠀⠘⠿⠟⠀⣠⣄⠀⢀⣀⠀⠀⠰⣿⣷⢀⣤⣾⠇⣀⣈⠛⠛⠁⠀⢾⣿⠇
⠀⠀⠀⠙⠿⠿⠀⣠⣤⡦⠀⠻⠿⠀⠘⠿⠃⠀⠀⣀⠀⠸⣿⠇⣠⣭⣍⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠛⠋⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠹⠿⠃⢠⣤⣄⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⢸⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠰⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⡆⠀⠀⢀⠀⢀⠀⠠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢤⣤⣀⣀⡀
⢸⠀⢰⠒⢶⣶⣶⣶⠖⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⢋⡿⣿⡿⣿⣷⣶⣶⡤⢤⣀
⢸⠀⢸⠀⣞⣴⣾⣵⡀⣹⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⠷⣶⣾⢐⡛⠎⠺⢽⡿⡿⠈⢀⢀⡏
⢸⠀⢸⠉⢛⠛⠛⠛⠛⣿⠀⠀⠀⢰⡆⠀⠙⢩⠍⡩⠁⠄⡆⢸⠄⠀⢿⣿⣘⣙⠻⠵⣷⢦⢶⠀⠀⣹⠃
⢸⠀⢸⠀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠘⡷⠀⠀⠘⠓⠈⠐⢀⠁⠀⠉⣷⠛⠛⠛⠿⠿⠿⣶⣶⣦⣄⢀⡟
⢸⠀⢸⠩⠭⠭⠭⠍⠀⣹⠀⠀⠀⠀⣮⡄⠀⠀⠐⠊⠀⠃⠈⣁⣐⡿⠸⡭⡿⣷⣖⡴⣦⢤⠌⠀⠈⢱
⢸⠀⢸⢀⣀⣀⣀⣀⣤⣤⠀⠀⠀⠀⠋⠉⠙⢿⡿⢿⡿⢿⡿⣿⣿⡇⠀⠀⠐⠂⢁⠌⠉⠉⠀⠊⣶⡏
⢸⠀⢰⠀⠘⢊⢟⡝⠁⣿⠀⠀⠀⠀⡆⠀⠀⢯⡶⢇⡽⢏⡹⣏⣿⣧⣤⣤⣦⣴⣾⣮⣦⣤⣦⢴⣾⠃
⢸⠀⢸⣤⣶⣿⣾⣷⣶⣿⠀⠀⠀⠀⡇⠀⠀⠀⢀⣶⠛⣶⣷⡄⣿⣿⣽⢾⣡⡞⣻⣾⢑⠖⢲⣿⡿
⢸⠀⢸⠟⠻⠚⠻⠑⠙⣿⠀⠀⠀⠀⣧⠤⠤⣤⣭⣤⣭⣤⣭⣥⣿⠉⠈⠈⠁⠍⢉⠉⠁⠀⠉⢸⡇
⢸⠀⢸⢀⣤⣤⣴⣤⢤⣿⠀⠀⠀⠀⣙⠀⠀⣸⢟⡛⣻⡿⠿⢿⣿⣶⡄⣤⢄⣠⣭⡥⣄⣤⣠⢸⡇
⢸⠀⢘⠉⢻⡟⠿⣟⠟⣿⠀⠀⠀⠀⡯⠀⠐⣐⠛⠖⠧⠟⠾⡎⢸⣿⣧⣿⠾⠦⠛⠃⠈⢹⡿⢸⡇
⢸⠀⢸⠀⣰⣖⣲⣞⣲⣿⠀⠀⠀⢰⡗⡀⣀⠧⠛⠭⠿⠶⢶⣯⠉⠋⠀⠀⠂⠂⠁⠀⠀⠸⡟⠛⠃
⢸⠀⢸⠛⠛⠛⠛⠛⠉⣫⠀⠀⠀⠈⠙⠓⣿⠿⢿⣿⣷⣶⣶⣿⡀⠀⠀⢢⣤⡄⠀⠀⢀⣀⣿
⢸⠀⢸⠀⠀⣤⡄⠀⠀⣧⠀⠀⠀⠀⠀⠀⣯⡀⠀⢾⣶⢶⣶⣾⣇⣀⣀⢤⣿⠗⠒⠂⠉⠉
⢸⠀⢸⡁⢹⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⣉⠉⠉⡉⠉⠁⢸⣿
⢸⠀⠸⠤⢤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⠹⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣣⠀⠀⠀⢸⣂⣀⣀
⠀⣀⠤⠤⠀⢶⡧⡈⣻⣵⠞⠙⠉⠀⠀⣀⣭⣵⣦⣀
⢸⡱⠉⠀⠀⢹⡿⡞⡏⠀⡀⣄⠀⢤⣀⡈⠉⠉⠉⠛⠷⣦
⠀⠙⣆⠀⢀⡞⣁⣻⡷⠶⠿⠿⠾⢦⣬⣭⣥⣤⣄⡀⠀⠈⢳⡀
⠀⠀⠈⢻⡟⢤⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⢦⠀⠀⡇
⠀⠀⠀⢼⡌⢾⡏⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⡟
⠀⠀⠀⠘⢞⣻⢀⣀⣀⣀⠉⠃⠀⠀⠐⠚⠋⠉⠙⢺⣿⣧⡞⠁
⠀⠀⠀⠀⠘⢹⠈⠛⢍⡿⠁⠀⠀⠀⠀⢐⣟⢻⡶⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠉⠀⠀⠀⠀⣴⠟⠋⠉⠉⠓⠦
⠀⠀⠀⠀⠐⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢆⠀⢠⡴⠒⠚⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠲⣠⠉⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠠⠙⠛⢋⠀⠈⢢⠀⠐⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢀⣼⣟⣿⡇⢄⠀⠀⠀⠀⠠⡄⠘⠀⠀⡀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⣾⠸⠘⡿⣇⠀⠁⠒⠀⠒⠃⠉⠀⠀⠀⠃⠀⠀⠉⠈⠉⠉⠢⢤⣀⣀⠤⠒⠤⠤⠴⠶⠛⠁
⠀⠀⠀⠀⣿⡀⣀⠈⠚⢧⡀
⠀⠀⠀⠀⡈⢿⠂⠙⠳⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠠⠐⠂⠁⡀⠈⣷⡄⠀⣲⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠌⠂⠂⠄
⢀⠀⠀⠂⠂⢹⠟⣡⡾⠯⣀⣀⡀⠀⠀⠄⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠄
⠨⠌⠀⠐⠀⠘⠦⡽⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠡⠁⠠⠀⠀⠈⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣾⣿⠿⡿⠿⢿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠟⠋⠁⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠤⠚⡁⠀⠘⠲⠄⢀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠋⢊⠉⣶⢇⡔⠉⠉⠒⠢⡄⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢿⣄⣠⢤⠞⠁⠐⢆⡀⢂⣀⡴⠃⠀⠉⠣⡒⡎⣷⢆⣠⡄⠀⠀⠀⣀⣀⣼⣇⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠍⠉⠁⠀⠀⠀⠀⣀⠣⢃⣿⣿⣋⣤⣴⡶⠛⠉⠀⠀⠀⠀⠀⢾⣿⠧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⡇⢠⠔⢤⣤⣤⠔⠓⠢⢄⡀⠀⠀⠀⠸⢻⠉⠙⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡴⠋⡽⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠆⠀⠀⠀⠈⠻⣦⡑⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⡞⠹⠀⠑⣶⠤⣀⣀⠤⣤⡴⠖⠊⠁⢀⠎⠀⠀⠀⠀⠀⠀⠙⣆⠐⠀⠀⠀⠀⠀⣀⡼⢷⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣺⠃⠀⣸⠁⠀⠆⠒⠺⡌⣀⠠⠞⠓⠠⡀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣧⣠⣤⣶⡟⠁⠀⠀⠉⠻⠟
⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡇⠀⠀⢀⡖⢰⣷⣍⣁⠀⠀⠀⠈⠖⠁⠀⠸⡤⠀⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⠉⠙⢻⠁⠀⢀⣧⣜⡛⢿⣊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠀⠀⢀⣀⣠⡤⠤⣼⣿⡏
⠀⠀⠀⠀⠀⠀⡎⢀⣠⢾⡾⡛⠳⣤⡉⢷⣤⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⣶⠋⠀⠀⠀⠀⢸⣿⣷⡖
⠀⠀⠀⢀⣀⡔⠿⣍⠻⣦⢈⣬⠗⠚⠿⢭⡅⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⡇⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠠⢾⣦⣽⠃⠉⠉⣋⣀⣀⣀⣀⣀⣰⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣴⠀⠀⠀⠂⠀⢹⣲⣄
⠀⠀⠀⠀⣾⣭⢀⡄⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢁⣠⠀⠀⠀⠀⢀⣽⣿
⠀⠀⠀⠀⠀⠻⠞⠛⠛⢻⣿⣿⡟⠛⡟⡛⢿⠻⢹⠟⠻⢻⡛⡟⢻⠻⡟⣛⡟⡛⢿⣿⣏⣠⣦⢀⡶⣤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠟⡁⠇⣼⢠⢸⡢⢹⡘⢁⡇⣀⢘⡇⠼⡇⡀⣿⣿⣿⣿⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠿⢿⣿⣿⣿⣿⣿⡟⠉⠻⠿⣿⣿⣿⣿⠿⠿⠿⢿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠳⡄⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⢀⣼⣷⢮⡄⣽⡤⠀⠦⠤⢤⣄⣀⣀⡥⠼⣻⠅⠤⠤⠤⢤⣀⣀⠀⣀⣀⡠⣳⡄
⠀⠀⠀⠐⠈⠛⠿⢮⣝⣒⠳⠤⢤⣚⣿⣶⣾⣿⣿⡦⠾⣡⣎⣠⣆⡴⢆⣼⣻⣿⣷⣾⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢠⡄⢤⡀⠀⡄⠀⠀⠀⢀⡁⠀⠀⠀⠀⠲⡄⠀⠀⠘⠁⠀⠀⠀⣠⡴⠦⠄⡇⠂⠀⠀⠀⠀⣀⠀⠀⠀⠹⠟⢀
⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡇⠐⠀⢀⡀⢀⣀⡈⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⣍⠀⠀⠐
⠀⠀⠀⠁⠀⢏⠈⠁⠀⣠⣾⠐⠂⣶⡼⠃⣠⣶⣦⣄⠈⠀⠀⠀⠀⠋⠀⢸⣿⣿⡗⢿⡿⠀⣰⣤⡀⠘⠃⠀⢀⡀⢠⣿⣿⣿⣶⠀⢠⡧
⠀⠀⠀⠀⠠⠞⠂⠀⠰⠙⠉⠂⠀⠿⠿⠿⣿⣿⣿⣿⠄⠀⢠⡆⠠⠂⠀⠀⠉⣸⠇⠸⡇⠀⠻⠛⠀⠀⢠⣾⣿⣷⠈⠉⠁⣿⢿⣿⡇⠐⠀⣀⢀
⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⢸⠏⠈⠉⠀⠀⠀⣻⡾
⠀⠀⢙⠛⠛⢿⣷⠀⠀⠀⠀⠀⠐⡄⣠⢠⡿⠿⣿⡇⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡦⠀⠾⠿⠏⠿⠀⠀⠀⠀⠀⠁⠀⠀⢈⣳⣶
⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠴⠃⠀⠈⢟⢸⡇⠀⠿⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⣴⠆⣰⣦⠀⠠⣦⠃⣀⠀⠀⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣟⣿⠂⠀⣠⣴⠶⣦⣄⠀⠐⣿⣻⡗⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⣼⠗⣠⠶⣄⠺⣧⠘⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⠆⠹⣗⣘⠛⣃⣺⠋⠰⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⣆⠿⠶⠟⠀⣰⠋⠙⠚⠋⠹⣆⠀⠚⠶⠿⢀⡶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠳⢦⡀⣿⢀⡴⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⡀⠉⠀⡀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⠀⠀⡄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠁⣀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⢠⣾⡿⠇⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠘⡀⠈⠒⢀⠀⢀⠉⠁⠀⠀⢰⡆⠢⣌⠙⠃⢀⠀⠀⡄⠀⠰⡏⠀⠀⠀⠀⠀⠀⣤⣀⠠⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠠⠀⠘⣇⡐⣶⠇⠰⠘⠁⠀⠀⠀⢰⠂⠀⠁⠀⠀⠠⢀⢹⡟⣡⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠁⠀⢠⠀⢠⣼⠿⠄⠀⠀⠀⢑⠂⡾⠇⠈⠁⠈⠀⠀⠀⠠
⠀⠀⠀⢠⠊⠀⠀⡐⠇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⠀⠐⠀⢠⠀⠆⠀⢐⡀⠀⠘⡤⡁⠐⠦⠀⠀⠀⠄
⠀⠀⠀⠈⠃⠀⡀⠀⡂⠀⠀⠀⠀⠀⠀⢠⡿⠁⢨⢂⠂⠦⡤⢆⠤⡋⠁⢦⣤⣀⡜⠀⠁⠀⠷⠀⠂⣴⣶
⠀⠀⠀⣀⡤⠐⠠⣁⣩⠦⢠⡀⠤⠚⠉⡀⠀⢂⣸⠂⡔⠤⡐⡆⠜⠀⠀⠀⠉⠈⠉⠁⠒⡶⣌⠐⠢⠈⢻⡧⠄
⠀⠀⠀⣿⣇⠀⠄⠯⠀⠄⠀⠀⠀⠀⠀⢄⠠⠄⡑⣀⠱⠟⢋⠀⡀⠀⠀⢀⡁⠀⠀⠀⡀⠑⠘⣧⡀⠑⡀⠓
⠀⠀⠀⡟⠉⡉⢃⠄⡀⠐⠄⠀⠀⠀⠀⠈⡉⢀⣁⠘⠁⠈⠀⠀⠐⢀⢠⢸⣇⠀⢒⡔⠀⠤⢊⡻⠗⣊⠽⠀⠦
⠀⠀⠀⠃⠀⢀⡌⠁⠹⠄⢈⡄⠀⠮⢰⢀⡀⠀⠂⠱⡀⠀⠀⠀⠄⠀⠁⢄⡛⠾⡿⠠⡔⣋⠤⠒⠊⠀⣀⠤⠒
⠀⠀⠀⠀⢀⣌⠀⠀⠀⢀⠆⡔⠂⠀⠥⠭⡀⠀⠐⠲⢈⠠⠂⠀⠀⠀⠀⣄⣱⡷⠉⠉⠉⠉⠡⠂⠎⠁
⠀⠀⠀⠁⣿⣿⣤⠄⢖⣡⣭⡧⠀⡀⠀⠀⡀⠴⠤⡀⣻⣶⡆⠀⠁⠀⠀⠁⠥⣾⡖⠀⢰⣦⣛⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡉⠉⡥⢸⡟⢉⡹⢧⠁⠁⠀⠀⠱⠀⢀⢁⡿⠻⠁⠂⠆⠀⠆⡂⠀⣽⠇⠀⠈⢥⡀⢁⠀⠀⡀⠄⠀⠂⡀
⠀⠀⠀⠀⠱⠀⠠⠘⠧⠀⠀⣦⣥⠀⠈⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠠⠄⢀⣸⡇⠀⠀⠀⠀⠈⠞⠡⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⣙⠃⠀⠛⠂⠤⣾⠃⠀⠀⠀⠀⠐⣬⠦⠚⠙⠃⠀⠀⠒⠀⠐⡇⠀⠤⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠁⠺⠃⠸⠀⠀⠀⢙⣤⠄⠐⠂⠁⠀⠀⠀⠀⢀⠄⡀⠂⠀⠀⡇⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⢀⠐⠋⠀⠁⠲⣄⠀⢀⠠⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠀⠅⠀⠀⠀⢀⣠⡔⠁⠀⠀⠀⠀⢀⣼⣶⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠈⢹⠛⣿⣿⣶⠒⠒⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣆⡤⠔⠒⢈⣽⣋⡤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠡⠞⠁⠀⠀⠈⠉⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⢛⣉⡠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠧⣀⢀⣴⠒⠒⠲⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠀⠀⢰⠃⠀⠀⠠⠂⡜⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣰⡁⠀⣇⠀⠀⡠⠔⠓⠒⢤⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⠀⡎⠀⠀⠀⢀⣾⣄⣀⡠⠓⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⣉⠛⠟⠀⠑⠦⠘⠻⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠈⣀⡄⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⢀⠈⡧⡾⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣥⠀⠀⠀⡏⠉⠁⠛⠓⠒⠒⠒⠀⠀⠒⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠸⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⡠⠤⡀⠰⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⢇⠀⠑⠸⡀⢰⠀⡰⢁⣍⡓⠦⣀⣤⣀⠀⡵⠠⣙⠇⠀⢀⠜
⠀⠀⠀⢰⠋⠛⢶⣄⡐⠧⢸⡰⢁⠂⠀⠉⠓⡞⠀⠈⠉⠳⡀⠙⢦⣀⣈⣣⣄⣀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠢⠤⠬⠥⠤⢄⡀⠚⡤⠀⠀⠘⠳⣤⡀⠀⢀⠳⠔⢣⡙⢹⢻⠁⠀⣷⡨⠭⠄⡁⠀⠀⠀⡴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⣅⠀⠀⣠⠛⠉⢧⣠⠾⠠⠄⠀⣙⠣⠤⡆⡰⠁⠀⠀⠀⠀⠀⠀⡰⠁⡰⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢄⡨⢶⡊⠫⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡠⠄⠲⠊⣠⠴⠓⠉⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣽⠖⠦⣬⣉⣐⣀⠀⠀⠤⠄⠀⠐⢂⣹⠀⣀⡼⢃⠴⢤⡐⠶⠶⠦⠖⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⠀⠶⠀⢈⡁⢽⡏⢉⡉⠻⠟⢉⣉⡟⠸⣀⡠⠖⠀⠀⠈⠓⠤⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⠀⠛⠁⣤⡄⠘⠃⣠⡄⠘⠋⢠⣤⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⢆⣉⡁⣼⡟⠛⣟⣻⣿⣉⠩⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⠒⢒⣒⡀⣴⠛⡿⠭⠭⠭⢥⣏⣿⠠⠤⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⢹⣂⠊⠉⠉⠩⢤⠖⠃
⠀⠀⣀⣀⣀⣀⣠⠄⠴⡒⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⡒⠶⠒⣲⣄⣀⣀⡀
⠴⡭⠥⠀⠀⠈⠭⠭⠙⠴⠤⣤⠤⠤⣤⣤⣀⣀⠀⡀⠀⠀⢀⣠⣤⠶⡛⣩⣩⣁⣀⣀⣀⣀⡽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠛⠉⠉⠁⠀⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣫⠅⢀⣀⣠⣤⣤⠤⠤⠤⣤⣤⣤⣄⡀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⠋⢶⣿⠭⠷⠶⠒⠒⠒⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⡉⠙⠀⠈⠉⠐⠚⢛⣛⡛⠲⠶⠶⠶⠦⠤⠤⠤⠒⠃⠀⠀⠀⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⡾⢡⢃⢀⣠⢰⣦⣙⠳⣶⣭⣭⣛⠒⠲⠦⠤⠄⠀⠀⠀⠀⢀⡴⠁⣰⠀⡼⢻⣆
⠀⠀⠀⠀⠀⠀⣾⠇⣠⡟⡱⠃⡞⡼⡟⣾⠉⠻⠷⣦⣍⣙⠻⠿⣿⣒⣶⠤⠤⠤⠤⠒⠉⣀⠴⢁⡞⠁⠈⣿⣧⡀
⠀⠀⠀⠀⠀⢰⡏⣴⢏⡞⠁⡼⣼⣿⡽⠃⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣍⣉⣉⣛⣛⣋⣩⡴⠞⠁⠀⠀⠀⣿⠸⣷
⠀⠀⠀⠀⠀⣼⣿⠏⡞⠀⡼⣱⣿⣿⠁⠀⠀⠀⠀⠴⠚⠳⣦⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠞⠳⣄⠀⠀⣿⡆⣿⡇
⠀⠀⠀⠀⠀⣿⡟⢸⠁⣸⢡⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣇⢻⣿
⠀⠀⠀⠀⠀⣿⡇⣿⠀⡏⣾⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠹⣅⢻⠀⠁⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⣿⢻⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⢸⣿⢹⣿⡄⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠁⠀⠀⢸⣿⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠋⢧⢻⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⣿⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣿⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⣾⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠦⣄⡀⠀⠀⠀⠀⠸⣇⣠⠇⠀⠀⠀⠀⢀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡇⠀⠀⠉⠓⠲⢤⣄⣓⣂⣀⣀⣠⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢷⡀⠀⠀⠀⠀⠀⣿⠯⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠙⠷⣤⣄⣀⣼⠏⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣿⠟⢿⣤⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣾⠏⠀⠀⠈⠻⣦⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⡏⠀⠀⠀⠀⠀⠀⠙⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡄⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⣩⣭⣍⠃⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡘⠿⠿⠿⠋⣿⣿⣿⣿⠟⣉⣭⣙⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡇⣿⣿⣿⡇⣾⣿⣿⣿⡇⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠃⠻⠿⠿⠇⠙⣿⣿⡿⠃⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⣿⡄⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⢻⣿⣿⡿⢃⣿⣿⣿⡇⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣧⣴⣾⣷⣦⠹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⣭⣛⣛⣥⣾⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡇⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⣀⡀⣶⣶⣶⣶⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣆⠻⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⣥⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⣀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠘⠿⠿⢿⠿⠿⠛⠉⠀⣾⣿⣿⣿⣿⣶⣵⣦⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢠⣤⢤⡄⠀⢀⣠⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠉⠉⠁
⠀⠀⠀⢸⣷⣶⣾⣷⣿⣿⣿⠟⠈⣿⣿⣿⣿⠋⠀⢿⣿⣿⣿⣿⢻⣿⣿⣿⣶⡄
⠀⠀⠀⠈⠛⠻⠿⠿⠛⠏⠙⠀⢠⣿⣿⣿⠃⠀⠀⠀⠙⠻⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⣤⣶⣶
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⣿⣷⠀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⢹⣭⣵⣦⣤⣶⣿⠀⢠⣜⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣷⣿⡿⢿⡦⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⢻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠘⢧⣽⠛⠛⠻⠷⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⡿⠃⠀⠀⠈⢡⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣀⣶⢻⡟⠛⠀⠈⣿⣶⣦⣤⣤⡶⠶⠶⠠⠤⠄⠐⠉⠀⠀⠀⠀⣠⣶⣶⣾⣦⡀⢠⣾⣿⣿⣿⣿
⠀⠀⠀⢸⡟⠴⠟⠟⠈⠁⠀⠀⠀⠸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⣀⡤⠆⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣷⣶⣦⣤⣤⣤⣤⣾⣿⣿⣦⣄⣄⣀⣤⣖⠛⠉⠀⠀⠀⠀⣀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⣿⣿⣿⣿⣿⣶⡄⠀⠀⠸⣿⣷⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣽⣿⣷⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⡿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢌⡉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠀⠤⠄⠤⠤⠤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢪⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⡴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⢀⠎⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠎⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡯⠯⠛⣛⣒⣒⣒⣋⣉⣟⡛⠒⠒⣒⣓⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠁⠐⠒⠒⠀⠀⠤⠤⣷⠤⠭⠤⠤⠤⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠬⠍⢁⣀⡐⠒⠒⠒⠘⠛⢲⡶⢾⣋⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡑⠒⠤⠤⠤⠤⠄⠀⣀⣀⣀⣈⡇⢠⣄⠤⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠒⢲⢲⠒⠒⠒⠒⠒⠒⡇⠀⠙⣏⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠸⡀⠀⠀⠀⠀⢰⠃⠀⠀⢥⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⡇⡇⠀⠀⠀⠀⠈⠓⠒⠋⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣉⠓⠠⠍⠘⠛⣟⠒⠟⠁⠀⠀⢀⣠⠴⠚⠉⣆⡀
⠀⠀⠀⠀⠀⠀⢀⡔⡫⠛⠉⣩⠭⠿⣖⡦⠤⢀⣀⠈⢹⠒⠒⠒⠉⠉⠁⠀⣀⣠⢤⡶⠯⠭⢬⡑⢄
⠀⠀⠀⠀⠀⠀⢸⡚⠦⠾⠥⠧⠤⠤⠬⠅⠂⠨⠶⠾⢭⡯⠽⠯⠽⠿⠭⠭⠤⣈⣉⣸⣓⣀⣀⣙⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠑⠂⠉⠉⠉⠁⠐⠒⠒⠤⠤⠤⠭⠥⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠏⠿⢿⡿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠟⠛




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⣼⢰⣿⡄⠀⠀⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠟⠈⠋⢠⣿⣾⣿⣿⣿⠿⣿⣿⣿⣷⠀⠿⠈⠛⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡆⠀⡇⢸⣿⣿⣿⣿⣻⣛⣿⣿⣿⣿⠀⡆⠀⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠈⠻⠿⠿⠿⠿⠛⠁⠀⠀⠗⠀⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠛⣿⣿⣛⡛⠒⠒⠒
⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠉
⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢸
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠆⠀⠀⠴⠿⠿⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠎⠺⠸⠀⡤⠤⠾
⠀⠀⢸⣿⣿⣿⣆⣠⣀⣀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡊⠸⠆⣶⣤⣤⠤⠤⠀⠀⣿⣆⠀⠀⠀⠀⣿⡀
⠀⠀⢸⣿⣿⡏⡙⠉⡁⣀⠀⢀⡀⢀⡉⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠀⢠⣄⠩⢇⡐⣶⡀⣀⠀⣀⡀⣀⣈⠛⠓⠒⠀⠀⢿⣷⣤⣄
⠀⠀⠀⢿⣿⡿⣠⠀⠂⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⡷⠉⠱⠭⠕⠚⠟⠶⠶⠶⠶⠖⣂⣀⣀⣀
⠀⠀⠀⣸⣿⠇⢰⠀⠀⠀⠠⠠⠀⠠⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠆⢶⠀⠖⡖⠴⠰⠀⣶⡀⣶⠆⢰⠆⢰
⠀⠀⣰⢿⡿⠀⠀⠀⠀⠀⠐⠔⠀⠒⠀⠲⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⠘⢿⡟⣿⠚⡞⠂⣿⠃⣿⡆⢺⡇⠘
⠀⢼⣿⣸⣧⣀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⣠⣀⠀⠘⣇⠻⠭⠛⠀⠛⠀⠛⠂⠘⠃⢀⡀
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡄
⠀⠀⡀⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣡⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠣⠀⠀⡹⠇
⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⡖⠒⠂⠀⠀⠀⢘⠄⠔⢀⠎⡕⡆⠀⠀⠀⠀⠀⣠⠂⡀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⢵⠀⠀⠀⡤⣏⢁⠀⠀⠀⠀⠐⢁⠎⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠆⠀⠀⠀⡹⠇⠀⠀⠔⡫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⢀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⡇⠀⣰⣇⣀⣀⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠔⠖⠀⠊⠠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠒⠁⣰⢟⡁⠀⠀⠀⠀⡘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⡐⠡⠀⠀⠀⠀⠀⠀⡏⠺⠀⠁⠤⠂⠁⢾⣏⣱⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⣠⠈⠉⠉
⠀⠀⡰⠋⠙⢊⠀⠀⠀⠀⠀⠀⠀⢙⠀⠋⠄⠀⠀⠀⢉⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢀⡜⠁⠀⢀⠞⠀⣀⠤⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠈⢋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠊⠀⠀⣠⠋⠀⣼⢡⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡏⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⡿⠁⠸⣿⡆
⠀⠀⠐⠁⠀⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡸⠁⡯⠉⠉⠉⠉⡴⠆⠀⠀⠀⠀⠖⣶⠀⠀⢋⡽⡀
⠀⠀⡠⠄⠒⢒⠀⠂⠀⠉⠉⠀⡁⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⡇⠃⠀⡀⡰⠚⠙⠂⠋⠉⠉⠃⠉⠁
⠀⢸⠀⠊⠀⠈⠀⠃⠀⠀⠀⠉⠠⠀⠸⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⢸⠀⠀⠁⢡
⠀⠘⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡏⠀⠀⠀⠀⡞⡄⠀⢸⠀⢰
⠀⡇⢀⠆⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⠤⡆⠀⠀⠀⢸⠃⠀⠀⠀⠠⡇⠁⠀⠘⠀⢸⠀⠀⠀⠀⢀⠀⠀⡀⢀⣂
⠀⠀⠘⠀⠐⠒⠀⣉⠁⠤⠄⠤⠀⠀⠐⠃⠀⠀⠀⡞⠁⠸⠧⠀⢸⢱⠀⠀⠀⠀⠂⠀⠄⠃⠀⠈⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣚⣭⡙⢦⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠿⠃⠀⠈⠳⣤⡞⣩⣉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢀⣴⣿⣶⡄⠙⢧⡈⠙⢦⣟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠋⠀⣠⣤⡙⢦⡀⠈⢿⡯⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣠⣾⣿⣿⠇⠀⠙⢦⡀⢳⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⣠⡀⠙⣞⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⣠⠾⢿⣿⠆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡼⠋⠁⠀⠀⠀⢻⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⢼⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⢀⡴⣷⣦⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠶⠶⠒⠒⠛⠛⠛⠷⠶⠶⠶⣦⣤⣙⠻⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⢹⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢁⠤⢀⣀⣠⣤⣤⠶⠶⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⣿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠶⢿⠋⠉⠁⠀⠀⠀⠀⠀⢰⡶⠖⢷⢿⢿⡆⠀⢠⣤⠶⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⠾⠿⣆⠀⠀⠀⢸⡇⠀⠀⣛⣽⠇⠀⣼⠃⠰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠂⢻⡀⠀⠀⢸⡇⠀⠀⡾⣿⠀⠀⡿⠀⠘⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢠⠁⣿⠀⠀⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⣸⢻⡟⠀⢀⡇⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡇⠀⠀⢸⡇⠀⢁⡼⡇⠀⢸⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇⡇⠀⢸⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸⡇⠀⢸⠀⡇⠀⢸⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣷⠀⠀⢸⡇⠀⡌⢀⡇⠀⣸⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⠀⠀⢸⡇⠀⠁⢸⡇⠀⣿⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣸⡟⠀⠀⢸⡇⠀⠀⢐⡇⠀⠛⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣉⣛⣿⣋⣡⣤⡾⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣶⡆⣶⠀⠀⠀⠀⠀⢠⣿⡿⣫⣭⣭⡻⣿⣧⠀⠀⠀⠀⠀⠀⣴⣶⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⠘⠓⢦⡀⠀⢸⣿⢹⣿⣿⣿⣿⢹⣿⠆⠀⠀⠛⠛⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⠀⠀⠉⠀⠘⣿⣧⣝⣻⣛⣥⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣛⢿⣿⣿⢟⣋⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿⣿⣿⣿⣟⢘⣛⣛⢘⣿⣿⣿⣿⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⣿⣿⣿⣷⠰⠿⠷⢰⣿⣿⣿⣿⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⢿⡛⣿⣿⣿⣿⣿⡿⣭⣭⣭⣭⣭⣭⣭⣭⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⠾⣿⡇⠶⠶⠶⠶⣶⠶⣿⡷⠶⠶⠶⠶⢿⣿⣇⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣯⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⡀⡀⣀⢀⢀⢀⢀⢀⡀⣀⡀⢀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠣⠟⠟⠇⠵⠺⠜⠘⠟⠿⠇⠟⠿⠫⠗⠛⠻


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣯⣿⣯⣿⣿⣿⣿

⢲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⢸
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠈⠀⡇⠀⠁⠢⡀⠀⠀⠀⢀⠴⠚⠈⠁⠈⠁⠈⠉⠙⠲⣄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡄⠀⠀⠀⠀⡄⠁⠀⠀⠀⠁⡀⠀⢠⠋⠀⡴⠂⠀⠀⠀⠀⠀⠢⡄⠈⢷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠲⣄⡀⠀⠇⠀⠀⠀⠀⠀⡀⢀⠇⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠙⢤⡁⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⡐⠁⠘⡆⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡜
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢀⠶⠾⢄⣠⠞⠀⠀⠀⢹⡄⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠺
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠆⣾⠀⠀⢘⠁⠀⠀⠀⠀⠘⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠨⠓⠶⢻⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠄⡀⣀⠠⠴⢌
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠆⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⢰⠀⠀⢀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠑⡄
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠀⣸⢠⡄⠈⡇⠀⠀⢀⠓⠀⡼⠀⠀⠄⠄⠀⠤⠄⠀⢸⡤⠞⠇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠀⠀⠀⠀⠩⠀⠀⠀⡷⠀⠀⠸⡀⠀⠄⠀⠀⢈⠀⠄⠀⠀⠀⠀⠀⠀⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠀⠀⠉⠁⠀⠀⠀⢤⠀⡀⠀⠀⠀⠀⠀⢣⠀⢠⠀⠀⠈⠀⠀⠈⠀⠀⠰⠀⠰⠁
⢹⠀⠀⠀⠀⣀⡀⠀⠀⠀⡏⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⡤⠾⠀⠀⠀⠀⠀⠣⡀⠂⡀⠀⠀⠀⠀⢀⡠⠀⡠⠁
⢺⠀⡾⠁⠈⠛⠷⠒⠀⠀⠀⠀⠀⠀⠘⠑⠂⠫⠤⠤⠤⠥⠤⠔⠃⠀⠀⠀⠀⠀⡟⠉⠀⠀⠉⠉⠀⠀⠈⠙⡆
⢼⠀⠈⠢⣄⠀⠀⠂⣀⠄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄
⢸⠀⠀⠀⠈⠳⠶⠖⢡⣦⠾⠄⠤⠐⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⢀⡊⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⢸⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢠⣤⣀⠀⠋⠀⣠⣤⡁⢛
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡴⠀⠀⠀⠈⠃
⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣐⣒⣒⡒⠐⡗⠐⣒⣀⣒⣊⡆
⢸
⢸
⢸
⢼⠀⠀⠰⢿⣿⠗⠀⣀⣄⣀⡀⣿⡆⣀⣄⠀⢀⣠⣀⠀⠀⠀⣰⣷⣀⣀⠀⢀⣄⣀⣀⣀⢀⣀⣀⣀⠀⣠⣤⡀
⣸⠀⠀⠀⢸⣿⠀⣸⡟⠉⢹⡇⣿⣷⡟⠁⢰⣿⣯⣽⡷⠀⠀⢹⣿⠉⣿⠀⢸⡟⣿⠟⠋⢹⡿⠉⣿⡎⠿⣯⡁
⢸⠀⠀⠀⠸⠿⠀⠘⠿⠶⠿⠇⠿⠏⠻⠧⠈⠻⢶⡶⠆⠀⠀⠸⠿⠀⠻⠷⠿⠃⠿⠆⠀⠸⠇⠀⠿⠇⠶⠾⠏⢴⠆
⢹

⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣽⣷⣄⠀⠀⠀⠀⠀⠀⠿⠿⠰⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡙⠃
⠀⠀⠀⡴⢲⡆⣄⣶⠢⡀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢰⣿⣿⠿⢟⣫⡆⠀⠀⠀⠀⠀⠀⢼⣿⣙⣷
⠀⠀⠀⠃⠀⠻⠸⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠸⠿⠇⠀⠀⠀⠀⠀⠘⢴⣿⣿⠟


⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣄⣀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠛⠛⣻⡛⠀⠀⠀⠀⠀⠀⠀⢸⠠⠠⠤⠈⡇⠀⠀⠀⠀⠀⠰⠂⣼⠂⣤⡄⠖⠀⠀⠀⠀⠀⠴⠟⠋⠉⠉⠉
⠀⠀⠀⠀⣠⣜⠟⣥⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⣷⣯⠁⠀⠀⠀⠀⠀⠀⣀⠘⠷⡶⡶⠂
⠀⠀⠀⠸⣿⠟⠿⠇⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠶⠃⠀⠀⠀⠀⠀⠀⠰⠿⠟⠻⠀⠀⠀⠀⠀⠀⠀⠙⠃


⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⡀
⠀⠀⠀⡏⠩⠭⠭⠍⢹⠀⠀⠀⠀⠀⠀⢈⣻⣃⣺⣟⡀⠀⠀⠀⠀⠀⠀⠀⣯⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣙⡯⠁
⠀⠀⠀⡇⠀⠉⠉⠴⢸⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣵⡀⠀⠀⠀⠀⢀⡔⢻⡀⣼⠲⡄⠀⠀⠀⠀⠀⠀⣮⣾⡟⠋
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠀⠈⠙⠃⠂⠃⠀⠀⠀⠀⠀⠘⠁⠁


⠀⠀⠀⠀⠀⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⣀
⠀⠀⠀⠀⢰⣷⠯⢭⡇⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣠⣷⠟⢿⣶⡁⠀⠀⠀⠀⠀⠠⣴⢿⣧⠀⣿⠃
⠀⠀⢀⣀⣼⢯⣭⣽⢃⡀⠀⠀⠀⠀⠀⢀⢇⢸⣿⣷⠀⠀⠀⠀⠀⠀⡬⠽⠷⣶⣦⣽⡒⠀⠀⠀⠀⠀⠀⣶⢰⣶⣶⣶
⠀⠀⠈⠉⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠛⠛⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠃⠘⠋⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠻⣷⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠹⡇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⣾⣯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣶⠇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢉⠟⡀⠀⠸⡄⠀⠀⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠲⢦⣄⣀⣤⠞⠋⠉⠉⠁⠘⠦⣤⡤⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢀⡀⠀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢆⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠞⠁⠋⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⣀⣀⣀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢠⣤⡤⠴⠶⠲⡆
⠀⠀⢸⣇⣀⣷⠀⠀⠲⣦⡴⠲⣦⡴⢦⡀⠀⠘⢻⡇⠀⠀⠀⣠⢤⣀⣀⣄⣀⣀⣀⣠⣀⣀⣀⠀⢠⡤⣄⠀⢀⣠⣤⣠⠄⣀⣤⣄⣤⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⢻⡏⠉⠿⢀⡄⠀⣿⠀⠀⣿⠀⠀⢻⡄⠀⢸⡷⠶⢦⡀⣁⣤⣿⠈⣿⠁⠈⠋⢸⡟⠁⠙⠁⢈⣠⣼⡇⣿⢤⣤⣿⠘⡧⣤⣼⡇⣴⠋⠛⣦⣴⠋⠙⣿
⠀⠠⠼⠷⠤⢤⣼⠇⣠⣿⣄⢀⣿⣄⠀⣈⣧⣀⢸⣁⠀⣸⢿⣁⢀⣿⣀⣿⣀⠀⠀⣈⣇⡀⠀⠀⣏⡀⣸⡃⡿⠦⠤⠟⢸⡷⠤⠴⠋⢿⡋⠙⡛⣧⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⠉⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡄⠀⢀⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡴⣿⢁⡴⢋⣴⡫⠖⣺⠞⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢱⣷⠏⢠⠞⢋⣴⠿⢛⣫⠭⠟⠋⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢰⠁⡼⠁⠀⠈⠀⣩⣴⣾⣭⣤⣤⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⢈⡡⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⠀⠀⠀⠀⠰⠁⠀⠀⠔⠀⠀⠀⠁⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣖⠨⠷⠀⠀⠀⣰⢁⠀⠱⡄⣰⠚⠓⠢⢀⠀⢠⠐⠈⠑⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠛⠳⢯⣥⣀⠀⠀⠠⠀⠀⠉⣦⡌⢻⡀⠀⠀⠀⠱⠁⣠⡀⠀⣿⠀⠠⠀⠀⠠⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠾⢭⣥⠤⠀⠀⠀⠀⠀⠀⠈⠳⡄⠣⡀⠀⠄⠀⠃⠈⠁⢀⡏⠀⠀⣀⣀⣤⠤⠶⠒⣶⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠴⠚⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠂⠤⠐⠏⠒⠐⠛⠒⠛⠉⠉⠀⠀⢀⣠⠞⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⢲⢟⣥⡾⠃⣀⣤⠆⠀⠀⡴⢊⣉⡯⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠶⣶⠚⠋⢁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢋⣾⡴⠛⢡⠏⡀⠀⠀⠣⣬⡤⠖⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡘⠋⠀⠀⣾⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢨⢉⡉⢸⠋⠛⢀⡜⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠌⠀⢷⡿⡇⡿⣆⠀⠓⡀⠀⠀⠀⢀⢁⣇⡈⢠⠃⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠇⠈⠢⠄⠀⡁⠀⠄⠀⠀⠀⠀⢉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢰⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡔⠀⠠⠤⣀⣰⠀⠙⣤⣶⣉⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡇⣷⣀⣀⣀⣀⣈⠓⠀⢣⣈⡀⢀⣞⣓⣒⣒⣒⣒⣒⣒⡒⠂⣵⣢⣭⣷⡆⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠰⣧⠞⣛⠃⠧⣈⠉⠁⠀⢠⠃⠀⠀⣸⠀⠉⠳⠞⠁⠀⢀⠀⠀⠀⠀⢙⠋⠲⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⢱⠏⠙⠞⠀⠀⠀⡏⠀⠀⠈⠇⠀⠀⠀⠐⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣶⠂⠈⠁⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢹⡄⢹⡅⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⣹⠀⣹⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⡟⠀⠰⡎⠠⣞⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠤⢤⠾⠗⢚⣿⣥⡭⠗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠀⠀⠘