Protozoa Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⣠⢤⡠⡦⢠⠤⡄⢤⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠟⠸⡇⠻⠼⠃⠓⠘⠒⠃⠛⠄⠻⡼⢧⡼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣤⣤⣴⣶⠶⠶⠶⠶⣶⣦⣤⣤⣀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⢀⣠⣴⡴⠶⠖⠛⠛⠋⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⣠⡾⠛⠉⣤⠀⠀⠀⠔⠄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣰⠞⠁⠀⠤⠀⡄⠀⣀⣀⣀⣀⣀⣴⠟⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣀⡴⠃⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⡾⠁⠀⠞⠈⠠⠐⠁⠈⠓⠒⢛⣽⠏⠀⣀⣤⠤⢴⡦⠖⢉⡤⠶⣿⠉⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣿⠀⠀⠒⠂⠘⠁⢀⣀⣠⡴⠞⠋⠁⢀⡞⠁⠘⣢⠞⠁⣰⠋⢚⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠉⣀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠷⠖⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠖⡾⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠋⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⠞⠁⠀⠀⠀⢸⠁⠀⠀⠀⣀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠁⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣠⠴⠖⠚⠁⠀⢀⠀⢀⠐⠢⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⠁⠀⠑⠒⠀⠀⠀⠘⠲⢄⡀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠉⢙⡆⠀⠐⠂⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡴⠋⠙⠒⠒⠋⠉⠙⠒⠦⢤⣤⣤⠇⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠳⠦⠶⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠶⠶⠶⠶⠿⠟⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠉⠀⠀⢀⣀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⠁⠀⠀⠀⢰⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣭⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠈⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⡏⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢀⡄⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⢀⢀⢀⢀⢀⠀⢠⡀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠳⠸⠸⠘⠘⠔⠙⠄⠑⠦⠊⠢⠞


⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⠒⠒⠒⠀⠤⣄
⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠒⠀⠀⠀⠀⠄⡀
⠀⠀⠻⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠙⠂⠄⣀⠀⠀⠀⠀⠈⠁⠀⠉⠢⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠒⠠⠄⠤⠒⠒⠈⠉⠉⠉⠉⠑⠒⠒⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠱⡄⢠⢠⠄⡄⠤⠰⠠⠀⡠⠠⠠⠠⠠⠀⢠⢠⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡆⡆⡔⢒⣿⢴⠢⢰⢰⣤⠒
⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠑⠋⠀⠈⠈⠈⠂⠈⠂⠈⠀⠘⠘⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠈⠝⠁⠈⠁⠈⠈⠀⠉
⠀⠀⠀⠀⠀⠀⡠⠄⡠⢠⢠⠀⣄⠤⡆⡄⢤⢰⠀⢠⢠⢤⢤
⣤⣤⣤⣤⣤⣤⣵⣦⣵⣾⣬⣶⣧⣶⣧⣷⣾⣼⣴⣾⣼⣼⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣾⣥⣭⣭⣬⣥⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠐⢦⡀
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠰⠃⣿⡲⡄⠀⠑
⠀⠀⠀⡀⠁⠀⣀⣘⠁⠀⠀⠀⢾⣿⡦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠦⠝⠓⠇⠀⠀⠆
⠀⠀⠀⠀⠘⠸⣟⣺⢢⠞⠋⠙⣦⣶⣦⠀⠶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠏⡇⡀⣤⡤⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠾⠀⠘⣦⣠⡴⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡔⡆⣴⠔⢣⣆⡇⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⢀⣴⠀⠀⠀⠘⢋⠈⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⡸⢠⡿⡀⣈⣨⣻⠀⠀⢃⠀⠀⠀⠃
⠀⠀⡀⠊⠀⢀⠀⠐⠈⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣅⣤⣋⠉⡟⡇⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣎⣹⣿⣿⣧⣇⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢇⢋⢛⢟⡿⡝⡘
⠀⠀⠀⠀⠀⠈⠁⠈⢢⡖⢻⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠾⣅⡸⢋⠞
⠀⠀⢀⠀⠀⠀⢀⣴⣿⣿⣟⣯⠟⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠖⠁
⠀⠀⠀⠀⠀⢀⣿⡿⠛⣿⠏⡧⣽⠥⡹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢀⡤⣒⡚⠴⠊⠓⢲⣤⡀
⠀⠀⠀⠀⠀⢸⣿⢹⠀⣿⠾⣛⣻⠵⡄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣅⠈⠙⠁⠀⠀⣾⣿⣟⣿
⠀⠀⠀⠀⠀⢸⣿⡬⣌⢿⡐⢖⣽⡁⠡⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣕⠦⢤⠀⢀⠈⡍⡀⡜
⠀⠀⠀⠀⠀⠀⣧⣷⣄⣢⣏⢹⣿⡟⡴⣿⡏⠀⠀⠀⢀⣠⠤⠤⠤⢶⣏⠙⠙⠛⢛⠏⠀⠀⠁⠠⣸⠛⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⢚⡿⡓⣸⣾⡿⠁⠀⠀⢰⡏⣤⣦⠀⢀⢈⠅⠀⠀⢰⡛⢉⠇⠸⠃⣈⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⠾⢷⣷⢿⡿⠃⠀⠀⠀⠀⢷⣈⡉⢐⢿⣏⣷⣯⣀⣀⣩⠥⠖⠒⠋⠁

⠀⠀⢿⡝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⣀⣀⣠⣀⣀⡀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣾⢽⣿⡄⠀⠀⢀⢞⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠈⠉⠉⠉⠉⠁⢸⣿⣿⡆⠀⣴⣿⣷⣾⢷⣈⠀⠀⠀⣁⢘⣀⡇⠀⠀⢸⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠴⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣟⣿⣿⠏⠀⠀⠀⠈⠀⠈⠁⠀⠀⢡⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠉⠑⠀⠘⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠸⠓⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⠤⠀⠀⠀⠀⠀⣿⣏⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⣿⡇⢠⣿⠃⠀⠀⠀⣀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⡄⠁⠀⢀⠆⠀⢸⡿⠋⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠐⢲⣤⣴⣶⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠙⠓⠒⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⠾⠑
⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠈⢢⣀⡆⠐⣏⢛⡗⠀⠀⠀⣀⣀⠀⠀⠀⣰⡦⡶⡦⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⢹⡶⢠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣄⠀⠀⠀⢻⠏⠀⠀⠀⣼⠱⣿⡆⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠰⠳⠆⠀⢀⣀⠀⠀⠀⠈⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⠄⠀⠀⢿⠀⠀⠀⠀⠛⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣽⣿⡿⠁⠀⠀⣀⠀⣾⣿⡀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⡄⢀⢈⣁⣀⠀⠀⣟⣿⠇⣿⡿⣁⠀⠀⠀⡇⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⡸⠿⠟⢨⣥⣭⡭⠁⠀⣉⣭⡤⠿⠟⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⢠⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠛⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣾⣿⣿⣮⠀⠀⠀⠀⢸⡇⠀⠀⣠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠤⢄⠀⠀⠀⡼⣿⣄⠀⠀⠀⣶⣿⣿⣿⡅⠂
⠀⠀⠘⣿⣷⣿⣿⠃⠀⠀⠀⢸⡇⠀⢸⣿⠇⠀⠀⢀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⡀⡱⠀⠆⠘⣇⣼⡟⠀⠀⠀⠚⢿⣿⣿⡏⠀⠂
⠀⠀⠀⠋⠉⠉⠃⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⣿⡷⠀⠘⢿⠇⣠⠀⠀⠘⣧⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⢀⣀
⠀⠀⠀⠀⢀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠒⠂⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠁⠐⠂⠠⠤⠤⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠈⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠄⣀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠀⠀⠈⠳⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠈⠉⠒⠤⢀⣄⣈⣦⣀⠀⣀⣍⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⢏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⣼⣠⠾⠋⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠊⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⣰⠞⣟⣻⡿⠡⠔⠒⠁
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏⠀⠀⢀⠀⢀⣴⣾⣧⣼⡏⠥⣿⢠⠞⠉⠙⢦
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠈⠀⣰⣿⠏⠋⢉⠨⠉⡻⣷⠟⠀⠀⠀⠈⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣿⣿⠄⠶⢠⣶⡄⢁⣿⣤⣀⠀⠀⢀⡇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠤⠃⠀⠀⠙⣿⡳⠄⠂⠙⠁⣠⠟⢈⡙⠳⢖⡟
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⡀⠀⠀⠀⠈⠢⢀⣿⡯⣳⠿⠒⠛⠁⠀⠠⠿⠔⢀⡇
⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠓⠁⠀⠀⠀⠀⣼⡣⠧⡟⠀⠀⠰⣁⠄⠀⣤⡆⢸⠁
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⡼⠓⠂⢸⠇⠀⠀⣀⠀⠀⣠⠀⠁⢺
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠣⠐⠀⠀⠁⠀⣼⠃⠀⠁⡾⠀⠀⠀⢀⠗⠀⠀⡀⠀⡝
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠠⠄⠀⣰⡇⠠⠂⢰⡇⠀⠀⠀⠀⠀⠀⠲⠃⠀⡇
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡾⠀⠀⠠⡿⢠⣄⡄⠀⢈⡗⠀⠀⠀⢰⠁
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣸⠃⠀⠀⢸⡇⠀⠉⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣿⠀⠀⠀⣽⠀⠀⠠⡄⠀⢪⠀⠀⣠⠎
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢠⡇⠀⠸⠂⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡇⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⡇⡀⣾⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⢀⡀⠀⠀⠀⣧⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠲⣽⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠰⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⠤⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⣿⠻⣻⣿⣋⡛⣿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣛⣛⣟⣛⣿⣟⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠐⠒⢳⣄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠒⡆⠀⣀⢠⡤⠀⠀⠁⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠓⢶⣞⣡⠨⠶⠟⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⠡⠀⠀⠔⢢⣄⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⢢⠜⣃⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠙⠛⠑⣲⣶⡮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⢞⣯⡉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠐⠦⣄⡀⠐⠏⠀⢀⠀⢁⠀⠂⡀⠀⠐⢩⣮⡙⠓⣄⠀⠱⡆⠀⠶⢲⠋⠁
⠀⠀⣀⠀⣄⠀⠀⣽⣷⣦⣿⣷⣆⠈⠫⡳⣄⠀⠀⠠⡌⣾⡿⠫⡄⠂⠀⠛⠳⣀⠀⠀⢀⠶⢸⣃⠙⡀⠠⠤⡀
⠀⠄⠃⠈⠠⠄⠸⠉⣧⣞⠛⠋⠄⠀⠀⠐⠘⢦⢀⠀⠁⡔⠰⠃⠀⠀⠐⠂⠀⠨⢧⣂⠈⠐⠀⡋⢀⡩⢀⠚⡁
⠀⠂⠸⠂⢁⠀⠀⠀⠓⠓⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠠⢁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡃⢀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠂⠀⠠⣶⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠤⡢
⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠻⢦⣀⣤⡀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢨⢝⢆⠠⠄⠈⠉⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠻⣦⣛⠘⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠞⣤⠁⣀⡀⠀⠀⢘⠆⠀⠀⠀⢸⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠱⢴⠺⢿⣽⣷⠺⠓⠀⠀⠀⠀⣸
⠀⠀⠀⣢⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⠀⠀⠀⠈⠒⠒⠏⠁⠀⠀⠀⠀⠀⢈⡏
⠀⠀⠆⠻⢇⠑⢀⠀⠀⠀⠀⢰⣦⡤⢿⣦⠂⠀⠀⠀⠀⠀⠀⠈⢦⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣦⠘⠁
⠀⠀⠈⠑⠀⠑⠈⠂⠀⠀⠀⠈⠈⣱⣾⣧⣀⣀⣤⣤⣀⠀⠀⠀⠀⠙⠦⠔⡄⠀⠀⠀⢀⣦⣄⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣤⡄⠀⠁⠀⠀⠀⠐⠹⠿⠋⠀⠀⠀⢀⠀⡍⠀⠀⠀⠀⠈⣊⠀⠀⠀⠀⠙⠚⢳⣴⣿
⠀⠀⠀⠀⠀⠻⠁⠀⢰⣦⣅⠀⢠⡀⠀⠀⠉⣦⢐⣺⡹⠁⠀⠀⠀⢴⡟⣋⡍⠉⣉⡽⣇⠈⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣒⡿⠞⠀⠀⢉⣉⠉⡘⠫⣦⡀⠀⠠⣶⠤⠀⠀⠠⣅⣀⣨⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠊⢀⡀⣀⡀⠀⢰⡶⢤⣙⠩⠁⠙⠓⠒⠐⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠽⠃⠘⠓⠒⠄⠐⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣶⣇⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⠋⣽⣿⣾⠇⠀⠀⠀⠀⠀⠀⢼⡝⠓⣟⢻⣿⡇
⠀⠀⠀⠀⠀⢂⠸⣏⢸⠀⠀⠸⠩⢻⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⢃⡏⠀⠀⠈⣮⣅⠀⠀⠀⠀⠀⠀⠀⣿⠀⢄⣈⠽⢻⡗
⠀⠀⠀⠀⠀⠀⠠⠑⠈⢃⣴⣗⣥⣷⣷⡀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⢹⢹⠧⣿⠆⢞⣛⡡⡁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢚⣤⣠⠋
⠀⠀⠀⠀⣠⠀⠀⢀⠀⣿⣽⢈⡘⣿⣯⠁⠀⠀⠀⢹⢿⡧⡀⠀⠀⠀⠀⢧⢞⡗⠀⢠⠁⢰⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣧⡧
⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⡿⢝⢳⣿⡇⠀⠀⠀⠀⠀⢒⢛⣏⢦⠀⠀⠀⠀⠙⢿⣚⣰⡼⠃⠀⠀⠀⠀⠀⠀⠀⢿⡄⣤⣶⣾⣿⡏
⠀⠀⠀⠀⢸⠀⢗⠨⣿⣿⣫⣶⠽⣿⠃⠀⠀⠀⠀⠀⠈⡎⣻⣢⣵⡀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣯⡿
⠀⠀⠀⠀⢸⣧⢉⡁⣮⡟⣱⣿⣤⠏⠀⠀⠀⠀⠀⠀⠀⠘⠜⢯⣿⣯⣂⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀⠀⠀⢰⣗⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡿⣾⡍⣾⣦⡁⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠈⢿⡛⠟⠩⠆⠀⠀⠀⠀⠀⣾⣷⣾⡷⠗⠀⠀⣼⠀⠂⣿⣿⡻⣿⣿⣇
⠀⠀⠀⠀⢸⡿⠑⠉⢿⣿⠀⡂⡔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠀⠀⠀⠀⠀⢠⡿⣭⣿⡀⠀⠀⢠⣽⣭⣄⣤⣵⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣧⢄⠈⠷⠄⢴⢫⣥⡗⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⣤⣤⣀⣀⡀⡴⡶⠴⠶⠆⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣶⣆⠐⢈⣦⡁⡲⠃⠀⠀⠀⠀⠀⢀⣤⣾⡿⣯⠲⠿⡾⡾⣷⡾⣿⠟⠛⠐⠺⡿⡆⠀⠀⠀⠻⡿⠟⣻⡟⣃⢀⣿⠃
⠀⠀⠀⠀⠀⠈⢿⡦⣹⢿⣿⣧⠏⠀⠀⠀⠀⠀⣰⡟⠉⡉⠀⠁⢘⣀⣰⠿⡈⣤⠀⠤⠀⠀⢐⣿⡏⠀⠀⠀⠀⠹⣞⣿⣶⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠈⡋⠻⠂⠉⠁⠀⠀⠀⠀⠀⠀⠿⣦⣀⣾⣿⣿⣽⣿⣧⣶⣿⣂⣿⣢⣼⣿⡶⠎⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠘⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠂⠀⠄⠀⠀⢀⡌⠄⡈⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠃
⠀⠀⢀⡀⠀⠀⠀⠀⢃⠀⠄⠀⠃⢀⡠⠂⢈⠠⠀⠩⠈⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠠⠀⡀
⠀⠀⠁⠈⠑⢄⠀⠀⠀⠀⠀⠖⠈⠤⣉⠁⠄⠄⠰⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡
⠀⠀⠢⡀⠈⠀⡈⠒⡐⠃⢉⣮⣶⠏⢉⣈⠈⠥⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⣰⠀⠄
⠀⠀⠀⠈⠁⠀⢄⠀⠐⢒⣾⣿⠟⣴⠈⠻⢦⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⣟⠀⠠
⠀⠀⠀⠀⠀⢀⡠⠃⣠⣀⠉⠻⡀⠒⠀⣿⠛⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠁⢿⡀⠀⠡
⠀⠀⠀⠀⠀⡂⠘⠃⠉⠀⠞⠁⣾⣿⣿⣷⢀⡐⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠳⠖⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠀⠙⠛⠟⠋⣌⠸⠗⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⢀⠀⠀⠀⡀⣁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠉⠁⠀⠈⠀⠑⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠂⠀⠈⠄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡄⠰⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⢆⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠁⠺⠀⠁⠀⠈⠐⠃

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠘⠑⠀⠁⠀⠀⠓⠘⠀⠃⠇

⠀⠀⠀⠀⣀⠀⠀⠐⠂⠒⠒⠀⠄⡀
⠀⢠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠂⠄⣀⠀⠀⠀⠒⠁⠂⢄⣀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⡄⡀⢀⢀⠀⡀⡀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠁⠀⠚⠀⠀⠊⠁⠈⠁⠈⠀⠀⠐⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡀⣀⡆⣀⢀⣲⣀⣆⢀⣀⣖⣢⠀⠀⠀⠀⠘⠤⠈⠐⠂⠅⠏⠱⣱⠁⠹⠂⠊⠉⠐⠀⠃⠒⠸
⣿⡏⣟⣛⣛⣛⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣠⣤⡶⠶⠿⠚⠓⠲⠦⠤⣼⣁⠀⠀⠀⠀⠀⣠⡾⠛⠃
⠀⠀⠀⠀⠘⠛⢷⣶⣆⡀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠦⣤⡞⠁⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⣀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⡠⠂⡀⠀⠀⠀⠀⠀⠁⠤⡀⠀⠀⠀⠀⠈⢷⡀⠀⣀⣠⣾⠿⠋
⠀⢰⣶⣦⣠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠀⠀⠀⢿⣼⠆⠀⠀⠀⠰⣺⣿⠣⡀⠀⠀⠀⠘⢛⡾⠛⠉
⠀⠀⠀⠉⠙⠛⠿⣷⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣼⣟⠉⠁⠀⠱⡀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠚⠛⣶⠀⠀⠙⣧⡀⠀⠀⣀⠀⠀⠀⠀⢿⠀⢀⣀⣤⣶⠿⠟⠁
⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⡶⣤⡀⢹⡇⠀⠀⢹⠀⠀⠀⠀⢸⡙⠛⠛⠉
⠘⠛⠛⠻⠟⠻⠿⠟⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣧⠀⠤⠾⠁⣾⠃⠀⠀⢨⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⠙⠁⠀⣀⣠⡾⠁⠀⠀⠀⢸⡄⠀⠀⠀⢸⡁⢀⣀⣀⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⡄⠀⠀⠉⠉⠉⠉⠁⠁⠀⠀⢞⣷⠘⡇⠀⠀⠀⠀⠛⠟⠛⠋⠉
⠶⠿⠶⠾⠿⢶⡶⠖⠲⠀⠀⠀⠀⠀⣿⠀⠀⠸⣏⢰⡀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠃⠀⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠿⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⢷⡀⢀⣀⣤⣴⡶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⣶⣦⠀⠀⠰⣉⡿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⢷⡙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣤⣦⣤⣶⣶⣶⣶⣾⡆⠀⠀⠀⠀⣿⠀⠀⠀⣤⡴⠞⠉⢀⣶⣰⡟⠀⠀⠀⠀⣀⡀⠀⠀⢻⣆⠀⠀⠀⠀⠙⢦⡀⠀⢀⣀⣠⣴⡶⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣐⣁⡾⠃⠀⠀⠀⢻⣦⡀⠀⠀⠀⠈⠻⢆⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠸⣇⠀⠀⠠⢴⣦⠀⢶⡠⡀⠀⠀⠀⠉⠉⠀⠀⢴⣤⣾⣿⠿⣳⣄⠀⠀⠀⠀⠈⠻⣦⡀
⠀⢀⣤⣄⣠⣀⣤⣤⣤⣦⠀⠀⠀⠀⠀⣏⠀⠙⠛⠛⠉⠀⠈⠳⣾⣧⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠙⣷⡀⠀⠀⢀⣀⣤⣴⣾⠷
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡴⠟⠋⣡⣄⡈⠉⠛⠦⣄⠀⠀⠈⢷⡄⠀⠀⠀⠸⢟⡾⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢀⣐⣪⣿⡇⠀⠀⠞⠀⠀⠀⢤⣼⠇⢠⣤⣄⠙⣷⠀⠀⠀⢻⡄⠀⠀⠀⠘⣷
⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠿⡇⠀⠀⠀⠀⠈⣧⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⠶⣄⠐⠛⣳⡦⠤⡿⠃⠸⣧⠀⠀⠀⣿⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢿⣤⣍⡛⠲⠛⣧⠀⢸⡟⠀⠀⠀⢸⡇⠀⠀⠀⠘⢧⣤⣤⣤⣦⣴⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠰⣻⡖⢀⠀⠀⠀⢀⣼⠃⠈⠛⠋⢠⡿⠀⣀⣤⡀⣸⡇⠀⠀⠀⢠⣗⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⢷⠀⠀⠀⠀⠀⠹⣤⡀⠘⠛⠋⠀⠀⠓⢄⣀⡀⠀⠀⢀⣠⣴⠟⠀⢠⡵⠟⢃⣿⠁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠛⠋⠁⠀⠀⠳⡀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠈⠉⠛⠛⣋⠉⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⠀⠀⠉⠣⢤⣄⡀⠀⠀⠀⠀⠀⠹⢶⡿⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⣸⠿⠿⢾⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠶⠶⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠙⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠈⠛⠶⠶⣤⣤⣀⣀⣀⣀⣀⣠⣄⣤⣤⣠⠶⡾⠟⠉⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣿⡉⠈⠉⠉⠉⠉⢻⣿⡃⠈⠉⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣝⠓⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣲⣶⣿⣿⣿⡟⠓⠀⠀⠀⢀⣀⠠⠠⢦⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠀⠀⠀⠀⠘⠁⠉⠟⠁⠀⠀⠀⠀⠀⠀⠻⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠶⠠⡄⠀⠀⠀⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⣀⠅⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠰⠶⠈⠀⠀⠀⢀⡀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⣨⢄⡀⠀⢨⣀⡌⣠⢠⠞⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢸⠖⠠⣄⠼⣀⠀⢳⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⠀⠀⢀⣬⢷⣼⡅
⠀⠀⠀⠂⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⢧⡀⣰⡃⠀⠈⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠶⠆⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⣤⡤⠤⠶⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡖⠛⡟⠋⠀⠀⢸⠘⠂⠀⠀⠀⠀⠛⢙⢛⠳⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⢻⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠐⡏⠛⠛⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡞⠱⡏⠙⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⢀⣾⠅⠀⡇⠀⠁⠀⢁⠀⠀⠐⣈⣀⢀⠀⠀⠀⡦⢄⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠼⠟⣷⠄
⠀⠀⠀⠀⠀⢰⠃⣼⠀⠐⠗⣤⠁⠀⣘⣎⢧⡔⠷⠈⣾⡌⢀⠸⠀⠈⠀⠀⡄⠐⠈⠁⠀⠀⠀⣤⠀⠀⠀⠊⢀⡜⠝⣧
⠀⠀⠀⠀⠀⠈⠰⣶⡞⢫⠠⠞⢿⣞⠹⣯⣞⢰⣆⠀⠃⢀⢀⠇⢀⠓⠁⠐⡞⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⡀⣋⣠⢚⣯
⠀⠀⠀⠀⠀⠀⠀⠸⢦⠸⠀⠀⡆⣯⠣⣥⣿⡍⠉⠶⣧⣼⢋⠀⢸⠀⡄⡀⠁⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠸⡁⢭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠄⠐⢞⣃⢤⣾⣟⣸⣾⣄⣰⣏⢿⣿⣴⣰⣶⠋⣀⣄⠨⠁⢀⠌⢀⡌⡀⡠⠋⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣍⣉⠈⠉⣻⡈⠁⢋⣀⣉⠿⣿⢷⠤⣧⣴⣤⣴⣿⠮⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢋⡿⢇⠀⠀⠘⢻⣦⢄⠉⠄⠱⠄⠄⣌⢽⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡼⢡⠞⠀⠀⠀⠀⠀⠀⠀⡌⣆⠰⠀⣦⠀⠸⠈⣧⡆⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢧⠳⠃⠀⠀⣇⢨⣿⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠸⡀⠀⠈⡂⠈⢆⠘⣦⠘⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⣆⠈⠈⣦⠈⢦⡛⢧⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⣆⠀⠈⢣⠀⠑⢞⢿⡌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣢⡀⠀⠑⠀⠀⠑⣿⡁⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠈⠳⡌⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣀⡆⠦⢰⡂⡀⠀⠀⢨⡠⠒⠘⡄⠐⠜⠄⠦⢁⠪⢪⠧⢀⠀⠀⢺⣩⡗⡀⢤⣀⡴⢩⠞⣀⠌⠀⠠⣂⠎⢐⠘⠐⠄⠀⠀⡠⣪
⠚⣜⢌⢦⠈⠑⢞⢀⡈⠢⡤⠐⢢⣄⠰⡰⡀⠀⠈⠉⠣⣀⠈⠐⢔⠈⠀⠀⣮⡅⢀⣍⠄⠛⡆⡐⢡⠃⠀⠀⢁⣴⠚⢶⠉⣿⠋
⠈⢠⠷⡄⢁⣬⡣⢫⡀⠀⢠⡔⠁⠘⠀⢀⡰⠀⠥⠕⠀⠈⠐⠤⠄⠇⠀⠴⠋⠀⢉⡇⠀⠀⠘⢀⠁⠀⢀⢶⡗⠀⢀⡚⣠⠌⠑
⠄⡀⡰⠛⣕⡭⠐⡰⠤⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⢸⠂⣀⠐⣎⠙⢄⠀⡠
⣞⠴⠑⠘⠁⠙⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠈⠀⠀⠀⠀⡁
⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠓
⡷⠶⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⣤⡀⣤⢠⢀⡤⠤⢠⠤⢤⡄⠀⡄⠀⣤⠀⣤⣤⠀⠀⣤⡀⢠⡤⣄⠀⠀⠀⠀⠀⠈⢉⡦⣀
⠀⠰⡞⠉⠀⠀⠀⠀⠀⠀⢧⣠⠇⡏⠳⣿⢸⠸⣄⣀⢸⣛⡸⣇⡀⣧⣀⢿⣀⡟⣿⣀⣰⠧⢷⢸⠗⣇⠀⠀⠀⠀⠀⠀⠀⠀⠛
⡲⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣷⢸⣹⠆⡞⢉⡁⣰⢻⡀⣿⣆⣿⢸⠰⣯⡁⡿⣄⣼⡇⢯⣁⠀⠀⠀⠀⠀⠀⠀⠐⠁⣦
⠞⠤⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠘⠈⠃⠙⠒⠃⠋⠉⠓⠛⠈⠛⠘⠐⠒⠃⠃⠉⠘⠃⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤
⡀⡀⠜⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⡦⡤⡤
⠀⠀⡀⢤⠄⠉⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣢⣠⢀⠈⠉⢉
⠠⢊⡤⠂⠀⢀⢴⠇⢀⡠⡪⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡒⠄⠀⠒⡎⣦⣠⠑⡂⠀⠙
⢤⢛⣁⣀⣴⢁⡎⠀⠘⡵⠯⠙⢀⡴⢞⠝⡆⠀⠠⠐⡠⠀⡀⠐⠀⠀⠀⠀⠀⠀⠊⡹⢆⡀⢘⠫⠅⣹⠀⢁⠕⢥⢀⠈⠊⠂⢆
⠁⠊⢛⠋⢸⠞⠀⠀⠀⠄⠀⠠⢾⡏⢁⡼⠁⢠⢡⠁⠀⠠⡰⠤⢔⡠⠒⠀⠠⡀⠙⢤⠐⠣⡀⠑⢶⠃⢠⢀⣄⠀⠠⠇⢸⣾⠆
⠀⢀⡆⠀⠉⠀⢀⠔⢱⡤⠖⠒⠛⠙⠙⠀⠀⡄⡀⠀⠠⢸⡘⠟⢀⡯⠄⠀⠢⠟⠀⢰⠂⠘⡅⠀⢸⡀⠈⡼⠥⡄⠐⠉⡖⡄⠺

⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠉⠁⠈⠈⠉⠉⠁⢀⡀
⠀⠀⠀⠀⠀⠁⠈⡓⠤⣀⠄⠊⠉⠀⠀⠀⡁⠀⠀⡀⠁⢐⠠⡀⠀⠈⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢁⠈⣦⠀⠀⢀⠂⡄⠠⠀⠄⠀⠀⠁⢀⣀⠂⠀⠁⠄⠂⠄⠀⠈⢣
⠀⠀⠀⠀⠀⡠⠊⠀⣾⠃⠰⠊⠀⠀⣀⠀⣀⣿⡿⢈⣼⠟⣄⣈⠂⠁⠀⡀⠏⠀⠀⢣
⠀⠀⠀⠀⡀⠀⠀⣄⠁⢥⣄⡆⡀⢱⡟⠲⠋⠀⠀⣿⡿⠀⠙⠙⠚⠂⠄⠐⠴⠤⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠂⢀⢠⡤⠘⠉⠀⠀⠀⢠⣿⠿⠈⠡⡻⠀⡿⠋⠈⠖⢀⠶⡆⠂⠶⣾⠎⢠⠄⢸⠁⡉⢀⡂⠀⠠⠀⠄⢀⠀⠀⣤⠠⢀⠄⡃⠀⠄
⢰⣁⡀⠀⠊⣌⡷⠈⡀⣀⠀⢰⠚⠃⡰⠰⣾⠃⢠⡇⢀⣈⣀⣢⣆⣥⠀⡀⠀⠀⠠⠅⠀⠧⠀⠂⠀⠀⢀⠀⠬⠁⠉⠀⠈⠈⠁⠀⠀⠀⠁
⠀⢀⠀⠃⠀⣈⠀⠀⠀⡀⢀⡀⠀⠤⢶⡟⢋⣨⡟⠋⣹⠳⠟⠹⠊⠂⠀⡀⢆⣠⠀⠀⠀⣠⡤⠀⠂⠈⠀⠀⢠⠆⠀⠀⠀⠀⢀⡀⢀⡄
⠎⠀⢀⠈⠐⠉⠀⠀⠈⠋⠂⠁⠀⠤⠂⢀⡬⡿⠛⣸⣷⢁⣴⠀⡲⠆⢀⣄⣼⠀⠂⠈⠁⢀⡘⠢⣀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠃⠀⢀⠐⠀⠀⢄⠀⠀⢠⡬⠤⠨⠉⢳⠾⠋⢩⣿⡩⣸⠿⠝⠊⠀⠀⠂⠶⠀⠀⢈⠉⠀⢤⡉⠐⠀⠦⡀
⠀⠀⠀⠄⠀⠌⠀⠀⠀⠀⠀⠄⠡⠤⣤⣀⡤⢀⢖⠾⣿⣯⠰⣯⡤⠀⠀⣀⢀⡀⠀⢠⠜⠀⢈⠥⠀⣁⠈⢍⠉⠁
⠀⠀⡠⠃⠀⠈⠅⠀⠀⠀⠀⠀⠁⠀⠈⠋⠀⠻⠒⠉⠉⢿⠞⢹⡉⡇⠘⠋⠀⠁⢀⠁⠀⡠⠁⠀⠀⠀⠉⠢⠇
⠀⢅⠀⠀⠀⠌⠁⠃⠈⢀⠀⠀⠃⠀⠀⠀⠀⡀⢀⠀⠀⠀⡈⠀⠀⠀⠐⠀⠀⣤⡎⠀⡰⠁
⠀⠀⠀⠀⠀⠈⠐⠀⠠⠢⠀⠀⠂⠀⠀⠀⢠⠀⠛⠀⢀⡶⠀⣠⣮⠂⠀⠀⠀⠽⠀⡰
⠀⠀⠈⠀⠀⠤⠄⢀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⢀⠈⠁⠀⠈⠊⠀⠀⢀⡜⠀⠐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠄⣀⠀⠀⠰⠴⠐⠈⠐⢄⡂⠘⠀⢀⠀⠀⠀⢀⠴⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⡀⢀⠀⣇⠂⢈⡀⠤⠀⠄⠀⠱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡠⠋⠀⠀⠀⣼⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡔⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣽⣿⢛⣧⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣷⣯⣿⣿⣯⣿⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣭⣿⣿⣿⣾⣏⢿⡿⣻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢘⣿⢹⣿⢿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⡝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣿⣿⣿⣿⣾⣿⣹⣿⢿⣿⣿⣿⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣻⣭⣻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣹⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣽⠿⣿⣯⣿⣿⣿⣞⣾⣽⣟⣯⣿⡿⣿⣿⣷⣿⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠳⣿⣿⣿⣿⣻⣿⣾⣿⣿⣿⣿⣿⢿⣾⣿⣿⣿⣿⣽⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⢿⣿⣻⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⢿⣽⣿⣿⣿⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⢯⣿⣿⣯⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣤⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣽⣿⣹⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⣿⣻⣻⣿⣿⣿⣿⣿⣿⡹⣿⣯⣿⣷⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣻⣿⣿⣟⣵⣿⣿⣿⣿⣿⣿⢻⣿⣿⢿⣿⣷⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣿⣿⠹⢮⣿⣿⣾⣿⣿⣿⣽⣿⣾⢿⣶⣾⣏⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡹⢿⣿⣿⣿⣿⢿⣿⣿⣿⡷⢿⣿⣶⣿⣦⣹⣯⣿⣿⣿⣿⣿⣿⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣟⣿⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣟⣿⣟⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠼⢛⠷⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⣿⣾⣿⣿⣻⢿⢿⣿⣯⣿⣿⢿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢉⠋⡛⠻⡿⣿⢿⡿⣿⠿⡿⢇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠚⠛⠛⠙⠉⠉⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⠂⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡀⠀⠁⠀⠀⠀⠀⠀⠿⠀⠀⠀⠾⠂⠳⠀⠈⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠙⣆⠀⠀⠢⡄⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⣴⣄⠀⠀⣠⣤⣌⠁⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠈⡀⠀⠀⠈⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢿⠄⠈⣿⣤⠿⠇⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠰⠛⣿⡆⠀⣾⠁⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⠆⠀⠀⠀⠀⠂⠀⠀⠀⢠⣤⠀⠀⢉⠀⠀⠈⠛⠁⠀⢶⠆⠀⢾⡧⢀⡀⠀⠀⠀⠀⠈⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡅⢀⠞⠀⠀⠀⡀⠀⠀⠀⠂⠀⣤⡴⠇⠀⢶⣤⠀⣀⣀⣠⣤⣤⣰⣦⣄⠀⠈⠹⠇⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣄⣴⡾⢿⠂⠾⠃⠀⠉⠀⠙⠃⢸⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⣆⢀⣄⣈⠉⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⢻⡾⣦⡀⠀⢣⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠀⠀⠁⠀⣿⣿⠓⠀⠀⠀⠀⣠⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠰⣏⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢴⣶⣀⠘⣿⡅⠀⠀⠸⠋⠀⠈⠁⠀⠀⠘⠟⠀⠀⠀⠀⠀⣶⣌⠐⠀⠀⠀⠀⣴⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠻⢿⡿⠉⣷⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⣿⠇⠀⠺⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⡄⠀⠀⠀⠈⠻⠃⠟⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⣤⡄⠘⠗⠀⠀⠿⠁⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⢀⡀⢠⡄⠰⣾⣷⡆⠛⠉⠁⠀⠀⠀⠈⠡⡟⠀⠀⠀⠀⣠⣶⡄⠀⠀⢠⠖⣶⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠛⠿⣀⣸⣿⣀⠁⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠏⠁⡀⠀⠀⠙⠛⠛⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣴⣅⠰⣿⠀⣤⡠⠿⢶⠶⠄⠀⣤⠀⣠⣤⠶⠟⢁⣤⠀⢾⣿⣶⠖⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠸⠟⠀⠙⠀⠀⠈⠋⠀⢀⣷⣿⣦⡀⠚⢁⡀⠀⢱⡶⠀⠀⢀⣿⡿⠋⢠⡜⠉⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⢀⣀⡠⣆⠀⡀⠀⠀⠈⠉⠈⣿⡿⣶⣿⡇⠀⡀⠀⣷⠒⠚⠉⠀⠀⠀⣀⡀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠄⡼⠟⠁⠈⠀⠁⠀⠛⠀⠀⠀⠉⠀⠈⠉⠹⠃⣀⣴⡄⠀⠀⠀⠀⠐⠛⠉⠁
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣼⡇⠄⠀⢠⡶⣈⠉⠁⠲⣦⠀⠀⢠⡀⠀⠐⣶⡀⠶⣼⣿⠇⠀⣠⠞
⠀⠀⢠⡟⠁⠀⠀⢰⣶⡄⢰⣿⠇⠀⣴⣶⣧⡟⣸⠃⡄⠀⣥⠀⠀⢹⡁⠐⠋⠀⠀⠀⠀⢀⡜⠁
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⣿⣿⣿⢱⣟⣰⠃⢸⠋⡦⠀⢸⣿⠀⠀⠀⠀⠀⣠⠎
⠀⣿⠁⠀⠀⠀⠀⠀⠀⣄⠉⠀⠦⠀⠘⣿⣇⣸⣇⡟⢠⣶⡤⠀⠀⣠⠏⠀⠀⠀⠀⡔⠁
⢠⡄⠀⠀⠀⠀⠀⢀⣠⡌⠻⠦⣄⢀⠀⠈⠙⠿⠟⠛⠚⠉⠀⢠⡶⠋⠀⠀⠀⢠⡜
⢸⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⢼⣦⡀⣀⣀⢀⣀⣠⣤⠖⠂⠀⠀⠀⠀⠀⠉
⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠶⣾⠏⠛⠋⠉⠛⠛⠉⠀⠀⣀⡀⠀⠀⠴
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠖
⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠈⠛⠛⠷⠶⣦⣄⣠⣤⣤⣤⣀⡴⠶⠚⠛⠉
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⣦⠀⠀⠀⠀⠠⣁⣹⠟⠀⠀⠀⡘⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⡿⠀⠀⠀⠀⠀⣨⡏⠁⠀⠀⠀⠹⡶⡟⠃⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡄⠀⠀⠀⠀⠐⣷⡀⠀⠀⠀⠀⣾⣇⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢦⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣽⡗⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⣀⣿⣿⣀⣀⣀⣿⣿⠀⠀⠀⢀⣴⡞⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣄⠀⠀⣀⣼⣿⠿⠟⠛⠉⠙⠉⠉⠛⠻⠿⣷⣤⣀⣴⣿⡟⠀⠀⠀⠀⠀⡄⠺⣷⡀
⠀⠀⠀⠀⠀⢠⣤⣶⡀⠀⠀⠀⠀⠀⠙⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣯⠀⠀⠀⣀⣤⡤⢿⠶⠿
⠀⠀⠀⠀⠀⠻⠃⣸⠯⣅⣀⠀⠀⠀⢠⡾⠁⠀⠀⢰⡶⠶⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠙⢷⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⠏⠀⠀⠀⠀⢠⡀⢀⣿⠇⠀⠀⠀⠀⠀⠛⠋⠛⣷⡀⠀⠀⠀⠻⣯⠀⠀⠀⠀⠀⠀⡤⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠀⠀⠳⢤⡴⡏⠀⠀⠀⠀⠀⢹⣧⣤⢶⣶⠶⠞⡷⣴⡿⠁
⠀⠀⠀⣴⡾⢷⣤⡀⠀⣀⣀⣀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠺⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠉
⠀⠀⠀⠻⠃⠼⠋⠛⠛⠿⠿⣿⠀⠀⠀⠀⠛⢻⣷⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⡄⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠚⠇⠀⠀⠀⠀⠀⠀⠰⠶⢷⣦⠀⠀⠀⠀⠐⠦⢾⡃⠀⠀⣻⣿⣶⣶⣤⣤⣠⡖⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠉⠁⠈⠝⢷⡾⠟
⠀⠀⠀⢀⣾⠿⢾⡴⠶⠿⠿⠛⠛⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣼⣏
⠀⠀⠀⠀⠻⠆⠚⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠋⠉⢙⣷⠀⠀⠀⠀⠀⠀⠰⠾⡋⠀⠀⢀⣼⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⢦⡀⠀⠀⠀⠀⠑⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠉⢙⣶⡏⢠⡦
⠀⠀⠀⠀⠀⠀⠀⣶⠶⣷⠚⠛⠉⠀⠀⠀⣼⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢾⣿⡿⣄⠀⠀⠀⠀⠀⠈⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡦⠘⠀⠀⠀⠀⠀⣼⣷⠏⠁⠉⢲⢶⣤⣤⣤⣄⣠⣠⣤⣤⣴⣿⡟⠃⠁⠀⠙⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡼⠟⠁⠀⠀⠀⣿⣿⠁⠀⠈⣿⣿⠓⠀⠀⠙⢿⣷⠀⠀⠀⠀⠀⢬⣿⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠳⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢹⡿⠀⠀⠀⠀⠈⢿⠄⠀⠀⠀⠀⠘⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣼⠾⣆⠀⠀⠀⢀⣸⡋⠀⠀⠀⠀⠐⣾⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡌⠀⠀⠀⣴⣏⢉⠂⠀⠀⠀⠀⠻⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠤⢦⡀⠀⠀⠐⢤⣀⠀⣀⡤⠖⠲⣄⠀⢰⡖⠒⠒⢂⡀⢀⠔⢲⠀⠀⣠⠄⠀⢠⠔⢒⠈⠒⣦⠀⣠⠴⠒⠒⢶⡄
⠀⢠⡟⠀⠀⠊⢙⡆⠈⠉⣁⣤⣤⡀⠈⠓⠄⠉⢠⡞⠉⠀⡏⠀⠞⢰⠏⠁⠀⠀⠀⡜⠁⢀⡾⠃⠘⠁⡠⠄⢀⠼⠁
⠀⣟⣀⠔⠀⡴⠋⠀⣠⠞⠁⠀⠈⣧⠀⡆⢰⠀⢸⠁⢀⠞⢁⡀⠀⠘⠲⠦⠤⠴⠚⢡⠀⠈⠷⣄⣀⣸⠆⠀⠘⠢
⢀⣤⢤⡀⢸⡁⠀⠒⠁⠀⠀⢀⠇⠘⢦⡼⢸⡇⠸⠀⠀⠀⢀⠇⣰⠂⠀⡠⠒⠒⢦⡈⠳⢤⣤⣀⠉⠁⠐⠉⠉⢳
⢸⡁⠀⠘⠀⠙⠒⠶⠶⠒⠚⣡⠴⣶⠀⠀⠀⠳⠤⠴⠚⣉⣁⠀⣇⢀⠎⠀⢀⣀⠀⢳⡀⠀⠄⠈⡇⠀⠀⡄⢀⡟
⠈⢷⡀⢀⠖⠛⢳⡄⠀⠒⠋⠁⡼⠃⠐⠊⢉⡷⢀⡴⠋⠀⣸⠇⠈⠁⠀⢠⡏⠉⢷⠀⠑⠶⠃⠀⠳⣤⠜⠀⢸⡀
⠀⠀⠙⣦⠀⢀⡾⢁⡴⢲⡄⠘⠧⠤⠀⣠⠟⢰⠋⠀⢠⠏⠁⠀⠒⠛⢶⠀⢧⠀⠈⢧⣀⠀⠀⢀⠀⠀⡄⠀⠀⢹
⠀⣠⣄⠈⠀⡿⠀⠎⠀⠀⢧⣀⡠⠀⢠⡏⠀⢈⠲⢄⠈⠓⠒⠒⣁⣠⠞⠀⢸⠀⢀⡄⠀⢀⡠⠊⢀⡴⠃⢰
⠀⢀⡟⠀⠸⣇⠀⠀⢸⠀⢀⣉⠀⠀⠈⣇⠈⢉⡇⠀⡄⢀⡴⠋⠉⡀⠀⠒⢁⡴⠋⠀⢴⠉⠀⠀⣟⠀⠀⠈⠉⢳⡄
⢀⡾⠁⠀⠀⠙⠲⠶⠊⠀⠇⢸⡇⠀⠢⠀⠀⡾⠀⠸⡀⠸⣄⣀⡀⠈⠳⡄⢸⡀⠀⠀⠀⡹⠀⠀⡈⠉⠑⡄⢀⡾⠁
⢸⡇⠀⡜⢠⠶⠒⠒⠶⡄⠀⠘⣇⣀⡼⢀⠀⢧⡀⣀⠇⡤⠒⢤⠈⠀⠀⡿⠀⠙⠛⠚⢋⣤⠖⠋⠁⢀⠜⢡⡟⠀⢀
⠀⠛⠊⠀⡟⠀⠀⡠⠞⠁⠀⠢⣤⡀⠀⢸⡄⠀⠉⢁⠎⠀⣀⣈⣀⣤⠞⠁⣠⠇⠀⢰⣏⣀⠀⠀⠀⠀⠀⢸⡀⣠⠃
⢀⡴⠒⠀⠃⠀⠺⠤⠤⠒⠀⢀⣨⠇⠀⣀⡙⠲⠴⠋⠀⢜⠁⠀⠀⣠⠖⠚⠁⢠⠴⢦⠈⢉⡷⠀⡴⠋⢱⠀⠉⡁
⠘⢧⣀⠀⠀⠀⣠⠴⠖⠛⠛⠉⠁⠀⣼⠁⠈⢆⡴⠋⡷⠀⠉⠀⢀⡇⠀⠀⣰⠁⢀⡾⠀⡏⠀⢰⠇⠀⡇⠀⠀⢹
⠀⠀⠙⢳⡄⠸⡁⠀⠤⠖⠛⡷⠀⢠⡏⢀⡤⠞⢁⡼⠁⢀⡀⠒⠋⢀⣀⡴⠃⢀⡾⠁⣠⣤⡤⠞⠀⠀⣁⣠⡴⠋
⠀⡞⠁⠀⣿⢀⡀⠀⠀⡌⢸⡃⢀⡼⠁⠈⠀⢠⠏⠀⢰⠋⢻⡀⠠⠋⠀⢀⠀⢸⠁⣸⠃⠀⢀⡴⠞⠛⠉⠁⠔⠊⣳
⠀⠑⠦⠖⠋⠈⠳⠤⠞⠁⠀⠉⠉⠁⠀⠀⠀⠸⠴⠂⠞⠀⠀⠙⠒⠂⠀⠈⠳⠤⠴⠋⠀⠀⠘⠀⠀⠖⠓⠶⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠴⠴⠖⠒⠒⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣠⠤⠒⠒⣫⣭⠉⣽⠿⣷⣔⠂⠀⠀⠄⠰⡿⠟⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠀⣠⠴⢆⡯⢥⠀⠀⠀⠈⠉⠉⠀⠏⣍⠉⠀⢀⣄⠀⡀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⡾⠯⠟⠓⠀⢀⠀⠀⠘⠤⠃⠦⠔⠀⠸⣅⣇⡰⠞⠁
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢰⣿⡷⢄⣩⣴⣤⣄⠑⢊⠀⠀⠀⡀⠠⠄⠀⣰⣾⣤⠋
⠀⠀⠀⠀⠀⠈⠳⠤⠖⠚⠉⠁⠀⠀⣹⠀⠀⠀⠀⡴⠋⠀⠀⡢⠀⠊⣃⣾⣿⣿⣷⣷⣷⡘⠀⠀⠀⠈⣡⠴⣄⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣠⣾⣆⣀⣠⡊⠁⢰⣄⠿⣽⣿⣿⣿⣿⣿⠷⠀⠀⢠⣿⠍⢛⡉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⢀⡠⠞⠁⡿⠥⠞⢁⠆⠀⢺⣼⠾⣷⣺⣿⣿⣿⠾⠁⠀⠓⠊⠉⠁⠁⡞
⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⢀⣤⠖⢋⠴⡦⡀⢸⠂⠀⠂⠀⠀⠰⡟⢃⡀⠩⠁⠉⠀⠀⠀⠦⠕⣰⡿⢀⠞
⠀⠀⣠⠖⠉⠀⠀⢀⡤⠴⢚⣩⣀⣀⡘⣚⠁⠈⠇⢀⠄⢈⣁⡀⡀⠘⠁⡁⠀⠀⠀⡘⠊⠀⠀⠀⠈⡰⠋
⠀⡰⠁⠀⠀⣠⢞⣁⠀⠰⣿⡿⠿⠛⢩⣴⠆⠀⠀⠀⠈⠟⠌⢙⠇⠘⠶⠃⢀⡐⣨⣤⣀⠄⠀⣠⠞⠁
⠀⠇⠀⠀⢠⡇⠺⡝⢀⢼⠏⠉⠄⢀⣬⡶⠾⢄⣀⡠⣨⣦⠀⠁⠀⠀⠀⠀⣿⣷⡇⡠⢋⡰⠞⠁
⠈⡆⠀⠀⠀⠛⠓⠳⣋⣄⣤⣴⡴⠿⢻⠀⠀⠀⣩⣬⣌⡉⢉⠁⢠⣞⡳⠀⢀⣡⠴⠛⠁
⠀⠘⢦⣀⣀⣠⠀⡜⠁⠀⣀⠄⢀⡀⠀⠑⠀⠀⠻⠿⠁⠛⢁⣀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠉⠉⠀⠀⠹⢧⣀⡙⡒⠋⠀⠀⢀⣀⣠⣤⠴⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠆

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⢀
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⠂⢀
⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡀⠀⠀⢒⣄⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠲⠀⠢⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡄⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡀⠀⣀⠄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠴⠰⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⣤⠀⠀⠤⠤⠠⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⢐⠃⠀⠍⠀⣀⠋⡀⠀⠀⠐⠀⠸⡀⠁⠀⢠⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⡀⠁⢰⠇⠀⠀⢀⠀⠀⡀⠒⡆⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠈⠂⠀⠐⠒⠁⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣰⣦⠀⠀⠀⠂⠀⠀⠀⢀⣠⠐⠁⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⢀⠀⠀⠁⠀⣠⠀⠀⠀⠀⢰⡀⠀⠀⠀⣠⠀⡘⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠠⠔⠀⠀⢠⡇⢰⠃⠀⠀⢮⠙⡇⠀⠀⢀⠀⠃⢠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⡀⠀⠀⠀⠈⠇⠀⠀⠫⠁⠀⢤⠇⠀⠀⠀⠀⠀⠀⠘⠂⠰⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢴⡄⠀⠀⠀⠀⠀⠰⢦⣄⣀⣠⠆⠀⠀⠐⢢⢀⣠⣀⠄⠀⠂⢀⠀⠙
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠶⠉⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠃⠠⠀⠀⠀⠘⠚⠠⠀⠄⠀⠁⠋⠑
⠀⠀⠀⠀⠠⢀
⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣐⣤⣦⣶⣤⣤⣇⣰⢤⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⢹⢫⠟⠉⠀⣀⠀⠀⠀⠀⠉⠀⠉⠙⠛⠾⢴⣁⡔⢀
⠀⠀⠀⠀⠀⠀⠀⢼⢠⠀⠀⢸⢱⡯⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠠⣍⠲⣄⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠹⠘⠀⠀⠀⠁⠈⠀⠀⠀⡆⢆⠀⠀⠀⢀⠀⠀⡠⠃⢈⠦⢙⢧⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠂⢇⢁⠀⠀⠀⠰⡂⠠⡀⠳⠞⠀⡔⢈⠏⠀⢈⠀⠀⠏⠀⠈⠣⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢂⠀⠀⠀⠈⣒⡁⡬⠋⢳⡍⠁⠀⠀⠀⠐⠚⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⢂⠀⠰⠥⠤⠃⢙⣶⠛⠱⠂⠢⡀⠀⠀⠀⠀⠀⠃⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢑⣥⡀⠀⠀⠀⠋⢀⠁⠀⠈⠂⠘⠃⠀⠀⣀⠀⠀⡠⠄⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠰⠊⠀⠀⠀⢀⠀⠀⠀⠈⠈⡇⠀⠑⠂⠀⠀⢻⣷⣤⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣄⠀⠀⠀⠀⢠⠁⡦⢱⠀⠀⡀⠈⠂⠠⢄⠀⠀⠘⣇⠀⠉⠻⣧⣤⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⢦⣀⠀⠀⠑⠒⠁⠀⠀⠘⠄⠀⣀⠜⠀⠀⠀⢹⣆⠀⠀⠘⣧⠈⠛⢧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⠀⠙⠲⠶⠶⠟⠀⠀⡄⠙⠪⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⢁⡘⡻⠀⠀⠀⠀⠀⠀⠖⡀⠀⠀⠀⠈⢣⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠨⢂⣀⠀⠈⠁⠀⠀⠀⠀⠀⠠⣄⠇⡔⠨⠁⠀⠀⢳⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⢦⡀⠀⠀⠈⠁⠈⠆⣔⣲⢉⠥⠄⡀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⢠⡌⠝⢠⢀⡈⠒⠂⠉⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠘⡜⠀⢐⢐⠄⠀⣠⡏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠁⠀⠀⢁⣠⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠻⠛⠋⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠚⠒⢶⣶⣶⠶⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠋⣿⠉⠉⠋⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠈⣁⣀⣀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠂⠀⠀⠀⠀⠀⢾⣿⣿⣿⡟⠙⠛⠁⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣦⡞
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢀⢰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠿⠧
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠃⢈⣵⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣟⢠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣿⣿⠏⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⠀⢻⣟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠟⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢯⠀⠀⠀⠘⣿⣼⣿⣷⠀⠀⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠄⠠⠤⠤⠤⠤⣿⣿⣿⣿⣿⡄⠀⠀⠀⢰⣿⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡣
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⣺⣿⢳⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣰⣥
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡅⢠⣄⣀⢹⣧⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⡄⠀⠀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣹⣿⢛⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢻⣷⡀⠀⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠂⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠸⠇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣁⠀⠃⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠦⠀⠀⠀⠀⠰⠃⠀⢹⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠛⢦⡀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⡤⠀⠰⠀⠀⡘
⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⡀⠀⠀⡰⠁
⠀⠀⠀⠀⢸⠇⠿⠀⣤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⡀⠄⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠲
⠀⠀⠀⠀⢸⡇⡄⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠁⡀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡆⠀⠄⣴⡄⠀⠀⢰⠀⠐⠀⠷⠈⠁⡀⠀⠀⡰⠁⠉⠉⠀⡀⠈⢱⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⢥⠀⠀⠉⠁⠀⠀⠀⢆⣨⡇⠀⠀⢀⡁⠀⠞⠁⠀⣀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⡘⢀⠀⠀⢀⠀⠂⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⠀⠳⣬⢀⠀⠀⠁⠀⡀⠀⠀⠘⠆⡀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠰⠗⠀⣡⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢠⡈⠀⠀⠀⠠⣶⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⣠⡄⠀⠀⠀⠀⠁
⠀⠀⠀⣠⠶⠀⠠⣠⠀⢀⠈⠁⠺⠂⠀⠀⠀⠀⠀⢀⠀⠢⠤⠜⢀⣠⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⢀⡴⠁⠀⠀⠀⠈⠀⢨⡄⠀⠀⢀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⢳⠀⠀⡀⠀⠁⠀⢠⠊⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⡔⠉
⠀⠸⠃⠀⠶⠀⠀⠀⠀⠀⠀⠁⠀⠸⠁⠀⠀⢚⣻⡷⠀⠀⠀⠈⠁⡄⠰⠄⠀⠀⠀⡀⠈⠀⠀⠴⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡗⠀⠠⠀⠓⠄⠀⠀⣈⠡⡀⠠⠴⠀⢠⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀
⢰⠄⠀⠀⠀⠐⡷⠀⠀⠀⠆⠀⠠⣤⣶⠶⢦⠀⠤⢠⠀⠐⠗⠀⠀⠀⠀⠀⠔⠁⠀⠐⠋⠁⠀⢀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⣰⠆
⡌⠀⠀⠉⠒⠂⠀⢐⡀⠀⠀⠀⢸⡿⣼⣿⣶⠆⠀⠈⠴⠀⠀⠀⠀⠈⢀⠈⠀⠀⠀⠰⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠒⢀⠀⠈⠀⠀⠓⠄⠈⠻⠿⠟⠿⠀⠈⣠⠶⡀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠡⠄⣚⠀⢀
⠀⠀⠀⠐⠀⠙⡃⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠶⠀⠀⠉⣉⠀⠘⠃⠀⠙⣠⠋
⠀⠀⠀⠀⠀⠐⣛⠈⠁⠀⠀⠖⠀⠤⠀⠀⠘⠀⢀⡀⠀⠂⣴⡆⠀⡀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⣀⠀⠠⠗⠀⠉⣀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠲⠶⢤⣄⠀⠀⠀⠀⠋⢀⠀⠀⠐⡄⠀⡀⢀⡤⠎⠀⠀⣀⠀⠈⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠟⡥⠀⠠⠀⠉⠌⠳⡄⠀⠀⠀⢀⡐⠂⠠⠀⠈⠉⢥⠀⠲⠄⠸⡽⢃⡀⠀⠀⠠⠀⣠⠆⠀⢀⠔⠒⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠃⢹⣇⠘⠂⣢⠄⠠⠘⢤⡙⢆⠁⠀⢉⡹⠀⠀⠀⠀⠈⠀⠀⠰⠆⢠⡆⠈⠁⠘⠂⠀⡘⠁⠀⠀⢨⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠳⣌⠁⠊⠀⠁⠀⠙⢺⡆⠀⢿⠇⠀⠴⠀⠈⠉⠙⡆⠙⠀⠐⠀⠒⠐⠀⠀⠀⠃⠀⠀⢀⠧⠀⠂⠂⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠘⢧⠀⠀⠀⠀⠀⢼⡇⠰⠀⠒⠀⠀⠣⣀⢀⠀⠃⠀⠛⠀⠊⠀⠒⠲⢤⠄⠐⠒⠒⠁⢀⠀⠠⠦⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣣⡀⠁⠀⢂⡼⠀⠀⠀⠀⠀⠀⠃⠀⢠⣤⠀⠀⠐⠂⠀⠰⠦⠘⠀⠠⠀⠈⠃⠈⣀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠋⢁⡀⠈⠀⣠⠖⠉⠀⠀⠀⠈⠙⠶⣆⢀⠆⠀⢰⡄⠐⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⣀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠀⠀⠠⡄⠀⠀⠻⠇⠐⢶⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⢀⡴⠂⠴⠀⠿⠀⡸⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣈⠁⠀⠀⠘⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠀⠀⠀⠀⠘⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠀⠀⠀⢀⣄⣻⠀⢀⡀⠠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢷⡾⢷⡆⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⢦⠉⢁⣀⠈⠁⠀⠘⢻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠃⠀⠀⠀⠁⠀⠀⣀⠀⠀⢹⣧⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠓⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⢹⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠀⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠛⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠚⠁⠀⠀⠀⠀⠀⢀⣌⣷⡄⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⡄⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢀⡀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠃⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⠀⠀⠀⠈⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡀⠀⠀⠀⠆⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡎⠁⣄⠘⢿⡄⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⣼⠋⠀⠎⠀⣸⡏⠀⠀⠀⠀⠀⠼⢿⡄
⠀⠀⠤⠀⠀⠀⠀⠀⣿⡉⠉⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⡏⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢀⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⡖⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠛⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⣧⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠃⠀⠀⠰⠄⢀⣄⠈⢻⡆⠀⠀⠀⠈⢠⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠋⠀⠀⠀⠘⠋⠀⢼⡄⠀⠀⡾⠀⢸⣧⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏
⠀⠀⠀⠀⠀⠠⣤⣘⣿⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⣿⠂
⠀⠐⠂⠀⠀⠀⠀⣿⡏⠁⠀⠘⠀⠀⠀⠀⢰⠀⠀⣤⠀⠀⠀⠰⣿⠀⠀⠀⠘⠿⠁⠀⠂⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠀⢸⠆⠀⡟⠀⠀⠀⢰⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⣀⠀⠐⠿⠶⢿⣿⡶⠂
⠀⠀⠀⠰⠖⠀⠀⣀⠀⠀⢉⣿⠇⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠚⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠶
⠀⠀⠀⠀⠀⣀⡈⠉⠀⢠⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠈⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠈⠁⢠⡄⠀⠾⠂⠀⠀⠀⠀⠀⠠⠀⠀⠒⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠦⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢾⣆⣀⠁⠀⠀⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠹⠆⠀⠀⠀⠛⠀⠀⠀⠘⠋⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠈⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠁⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⢠⡄⠈⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⣿⡇⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⣈⡋⠀⠐⢧⣀⠀⠀⠀⠰⠄⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠙⠛⠀⠀⠀⠘⠃⠀⠀⠸⠂⢀⡜⠛⠃⠀⠀⠘⠿⢆⠀⠀⠀⠀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠰⣇⣠⡿⠀⠀⢸⡷⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⢿⠁⠀⠠⠌⠁⠀⠀⣀⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠿⠀⢠⣾⠆⠀⠀⢀⡀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢀⡾⠃⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠘⣧⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠻⣦⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠚⠀⠀⠈⢁⣤⡀⠑⡶⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠰⠆⠀⡄⠀⣀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠴⢾⡿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠋⠉⠛⠛⠛⠛⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢀⣀⡀⢀⣀⣀⣀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⣤⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣿⡧⠤⠅
⠀⠀⠀⠀⠀⠀⠰⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠉⠙⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠿⠿⠷⠲⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢿⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢡⠀⣶⡿⠿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉


⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣾⣷⣌⣉⣻⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠸⢛⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⣿⠄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠙⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠿⠿⠿⠿

⠀⣀⠤⣤⡀⠀⠀⠀⠠⣄⡀⠀⠀⠀⢀⣠⡤⠶⣦⡀⠀⠀⣴⠒⠲⠶⠦⠀⠀⠀⣠⠴⢦⡀⠀⠀⢀⡀⠀⠀⢀⡤⠔⠒⠒⠲⢦⡀⠀⢀⣠⠴⠶⠶⢶⣄
⠀⠀⢠⡾⠁⠀⢠⠤⠤⣄⠙⠳⠶⠖⠋⠀⠀⠀⠈⠻⣄⡀⠙⠦⠀⣠⡴⠚⠁⣸⠃⠀⡼⠀⣠⠾⠋⠀⠀⠀⠉⢠⠖⠋⠀⣠⡾⠁⢰⠋⠀⠀⠀⠀⢀⣿
⠀⢠⡿⠁⠀⡀⠀⠀⣠⡾⠀⠀⢀⣤⠶⠛⠻⣆⠀⠀⠀⠉⠀⠀⢰⡏⠀⠀⢠⡟⠀⠀⠃⢸⡏⠀⠀⠀⠀⠀⣠⡟⠀⠀⢸⡏⠀⠀⠀⠀⡞⠋⠁⢰⠛⠁
⠀⢺⣇⣠⠞⠁⢠⡾⠋⠀⠀⡴⠋⠀⠀⠀⠀⢻⡆⠈⡇⠈⡆⠀⢸⡇⠀⣰⠟⠠⢄⠀⠀⠀⠙⠛⠶⠶⠶⠛⠉⠰⡄⠀⠈⠻⢦⣄⣀⣠⠇⠀⠀⠈⠓⠂
⠀⢀⣬⣄⡀⠀⣿⠀⠀⠠⠚⠀⠀⠀⠀⢀⡇⠈⢷⣄⡽⠀⣿⠀⠸⠇⠀⠀⠀⠀⣸⠃⣠⠆⠀⠀⣠⠴⠶⠶⣄⠀⠻⣦⣀⣀⣀⠈⠉⠁⠀⠞⠉⠉⢻⡆
⢰⡟⠁⠈⢳⠀⠹⢦⣄⣀⣀⣀⣠⣤⠴⠋⣀⣀⡀⠈⠀⠀⠹⣆⡀⢀⣠⠴⠶⠚⠁⢰⡇⠀⢀⠞⠁⠀⠀⠀⠈⣧⠀⠀⠉⠉⠉⢿⡄⠀⠀⠀⡀⠀⣸⡇
⠸⣧⠀⠀⠀⠀⢀⣀⣀⠉⠉⢁⣀⣀⡴⠞⢉⣼⠇⠀⣀⣠⣄⡈⠉⠉⣁⡤⠶⠶⣦⠈⢷⡤⠋⠀⠀⣠⠴⣦⡀⠘⢧⡀⠀⣱⠀⢸⡇⠀⠀⣰⠁⢰⡟
⠀⠹⣧⡀⠀⠞⠉⠉⢙⣧⠀⠈⠉⠁⠀⢠⡿⠁⠀⠋⠁⠀⣼⠇⣠⡾⠋⠀⢀⣤⠟⠀⠀⠀⠀⠀⢰⡏⠀⠈⢷⠀⠈⠙⠚⠁⠀⠈⠳⠤⠾⠁⠀⠸⣇
⠀⠀⠈⠻⣦⠀⠀⢀⣾⠃⢠⡴⢲⣆⠀⠸⣅⣠⠄⠀⢠⡾⠋⣴⠋⠀⠀⣴⠟⠁⠀⠐⠒⠛⠲⣄⠀⣷⠀⠀⠘⣧⡀⠀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠘⠀⢀⡿⠁⢰⠋⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠁⠀⠳⣀⡀⠀⠻⢤⣤⡤⠤⠆⠀⢀⡿⠀⢸⡆⠀⠀⠀⠉⠁⠀⠀⣰⠃⠀⣠⡾⠀⣠⠀⠀⠁
⠀⠖⠛⣷⠀⠀⣸⡇⠀⠀⠀⣠⠀⠘⠷⠤⠞⠀⠀⢸⠃⢀⡤⠤⡌⠙⠂⠀⠀⠀⣀⣤⠴⠶⠛⠁⢀⡼⠁⢀⣴⠃⠀⣠⡤⠚⠁⠀⣼⠋⠀⠀⢷⣤⣄⡀
⠀⠀⣼⠏⠀⠀⢹⣇⠀⠀⠀⣸⠀⢀⡤⣤⡀⠀⠀⠘⢧⠈⠀⣰⠇⠀⡠⠂⢠⡞⠋⠀⠀⣀⡀⠈⠉⠀⣴⠋⠁⠀⠸⣇⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠈⣿
⠀⣰⡏⠀⠀⠀⠀⠙⠶⠦⠶⠃⠀⡏⠀⢸⡇⠀⠠⢄⠀⠀⢰⡏⠀⠀⡇⠀⠸⣄⡀⠀⠀⠀⠙⢦⡀⢸⡇⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠉⠙⢷⠀⢀⣴⠏
⢀⣿⠀⠀⢰⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⢸⡀⠀⠀⢹⠀⢀⣈⣉⠉⠳⡄⠀⢸⣧⠀⠻⠶⣤⣤⠶⠞⢁⣠⡴⠞⠁⠀⢀⡾⢀⡾⠃
⠈⣿⣀⣠⠏⢠⡟⠁⠀⠀⠀⣹⡗⠀⠀⠈⠛⠶⠚⠁⢰⡀⠈⠳⠦⠴⠋⡴⠋⠀⠈⡇⠀⠀⠀⣸⠇⠀⢠⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠋⠀⣾⠃⠀⢀⠆
⠀⠈⠉⠁⠀⣾⠁⠀⠀⣤⠞⠋⠀⠀⠉⠓⢶⣄⠀⠀⠸⣧⠀⠀⠀⠀⡼⠁⠀⣀⣤⣤⣤⠴⠞⠁⠀⣠⡞⠀⠀⠐⣯⣄⣀⡀⠀⠀⠀⢀⡀⠀⢻⣄⣠⠎
⠀⣴⠶⠦⠀⠛⠀⠀⠸⣅⣤⠤⠖⠀⠀⠀⣀⡿⠀⠀⠀⠘⠻⠶⠴⠚⠀⠀⡾⠁⠀⠀⠀⢀⡴⠶⠛⠁⠀⣠⠴⢤⡀⠀⠉⣿⠀⢀⡾⠋⢙⡆⠀⠉⠁
⠀⣿⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠾⠛⠋⠀⠀⣴⠋⠉⢢⠀⣠⡶⢷⡄⠈⠒⠄⠀⠀⡟⠀⠀⠀⢀⡾⠁⠀⣼⡇⢠⠞⠁⠀⣼⠁⠀⣸⠁⠀⠀⠹⡆
⠀⠉⠻⣶⣄⠀⠀⣠⠞⠋⠉⠀⢀⣀⣄⡀⠀⠀⢸⡇⠀⠀⢈⣼⠏⠀⣼⠃⠀⠀⠀⣀⣴⠇⠀⠀⠀⡾⠁⠀⣼⠟⠀⠘⠀⠀⢠⡿⠀⠀⠳⠀⠀⠀⣰⠇
⠀⠀⢀⡀⠙⣷⡀⠹⣄⠀⠐⠒⠋⠉⣹⠇⠀⠀⣿⠃⢠⠴⠛⠁⢠⡾⠁⠀⣀⣀⠉⠉⠀⢠⡤⠶⠾⠃⠀⢸⠏⠀⣠⡴⠶⠶⠛⠁⠀⢀⣀⣤⣴⠞⠋
⠀⣸⠉⠀⠀⢸⡇⢀⠀⠀⠀⠀⡜⠀⣿⠀⠀⣸⡟⠀⠈⠀⠀⣰⠏⠀⠀⣸⠃⢹⡆⠀⢠⠏⠀⠀⠀⠀⠀⣿⠀⢰⡟⠀⠀⢀⣴⠶⠟⠛⠉⠁⠀⠴⠊⠙⡆
⠀⠻⣄⣀⣠⡼⠃⠘⣧⣀⣠⡼⠁⠀⠙⠳⠞⠋⠀⠀⠀⠀⠀⣿⣀⠄⢠⡇⠀⠀⠻⠦⣤⡄⠀⠀⠘⢦⣀⣈⣠⡟⠀⠀⠀⠸⡅⠀⠀⡤⠶⣤⣄⣀⣀⣴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠘⠓⠛⠁⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⢰⣆⠀⠀⠀⠀⠀⣾⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢿⣆⠀⠀⠀⠐⠿⠁⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠈⠿⠂⠀⠀⠀⣶⣄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣶⣶⣶⠄⣿⠙⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⡇⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡀⢨⡄⠀⠀⠀⢀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢷⣼⡇⠸⠿⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀⠙⠃⠀⠀⠀⢰⣧⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠻⠇⢈⣁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣦⣄⡀⠀⠀⠀⠀⠀⣀⠘⠃⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⣁⣤⡀⠈⠙⠻⣶⣤⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢠⣶⠿⠛⠉⠀⠀⠀⠀⣀⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣦⣄⡀⠀⠀⠀⠀⠈⠛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣈⠙⠻⣶⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣠⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢠⣦⠀⣿⢀⡀⠀⢙⠁⠀⣀⠀⠀⢸⣇⣀⡀⠀⠀⣀⡀⠀⢀⠀⣀⠀⠀⢀⣀⣸⡇⠀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣀⠀⠀⣿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⠁⣿⠉⢻⡆⣿⠀⣾⠋⠙⣿⢸⡟⠉⢻⡆⣾⡯⠽⠇⢸⡏⠙⣷⢰⡟⠉⢻⡇⣾⠋⠙⣿⠈⢉⡿⠋⢰⡟⠉⢻⡄⣿⠀⣿⠭⠿⠆
⠀⠀⠀⠀⠀⠘⠟⠀⠛⠀⠸⠃⠛⠃⠙⠷⠚⠛⠘⠛⠶⠛⠁⠙⠳⠞⠃⠘⠃⠘⠟⠈⠻⠖⠛⠃⠙⠳⠞⠛⠐⠟⠷⠖⠈⠻⠶⠛⠀⠛⠀⠛⠶⠞


⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣤⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣶⠶⣤⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣄⣠⡶⣿⡧
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣰⢇⡀⢛⣋⣿⣿⣟⣹⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣔⢿⣴⣾⢩⣽⣿⣿⣝⣿⣷⣄⣴⣄
⠀⠀⠀⠀⠀⠠⣀⠺⠟⣧⠍⢶⡧⠈⠿⠋⣙⣻⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠛⣿⠷⣿⣿⡭⠛⣟⣭⡆
⠀⠀⠀⠀⠀⠀⣿⢄⡼⠟⢦⠿⠠⠛⠚⢶⠈⢧⣼⣿⠿⢿⡄⠀⠀⠀⠀⢒⢾⡭⣿⣿⣿⣏⣷⣿⣾⣋⣱⣴⣿⣿⣧
⠀⠀⠀⠀⠀⢐⣿⣼⣿⢳⡗⢲⡀⢹⣤⠞⣳⣾⣿⣿⣦⣾⡇⠀⠀⠀⠀⠗⢾⣿⣿⣾⡥⣼⣿⣿⣿⣭⣻⣟⡳⠚⡏⢳⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣧⡿⣣⡴⢃⣶⠶⣾⡱⣏⣹⣿⣿⣿⡁⠀⠀⠀⠀⠀⣸⣯⣿⣗⣲⣼⠟⣿⠻⣿⡿⣿⣿⣿⣧⢮⡁
⠀⠀⠀⠀⠀⠀⠸⣿⣼⣷⠏⣒⠀⣿⣶⢟⣷⣿⣿⣟⣿⠋⠂⠀⠀⠀⠀⠘⢿⣙⣶⣿⡿⡙⢸⡗⠙⣿⣷⣯⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⢉⢿⣿⣶⣿⡟⣛⣯⣿⣥⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⢿⡕⢠⣿⣿⡌⢿⣿⠿⠷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠛⠻⠟⢿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠖⡞⠉⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣮⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠉⠒⢄⠒⢄⠱⢄⣠⡄⢀⡭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠢⡀⢡⣦⡽⣿⣜⣧⠾⠳⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃⢀⢜⣿⣵⣿⠀⢂⣤⠀⣄⢾⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⣹⡇⠀⠀⠀⠀⢠⣦⣤⡀⠀⢿⣭⣿⣿⣦⣾⣿⠲⡏⡲⡇⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠳⣍⣷⢠⣿⣿⠛⢙⣿⣿⠾⣿⣷⣤⡇⠀⠘⢨⢀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠋⠀⠉⠢⣼⣿⣆⣠⣾⠋⢨⣟⠈⢻⡷⡞⠁⠀⠀⢀⠠⢄⣀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣝⡷⣿⠿⠿⡿⠓⠀⠀⠀⠀⠀⢶⣾⣯⣴⠐⣮⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⣿⠟⠁⣠⠖⠀⠀⠀⠀⠀⠀⠀⢉⠞⢃⡾⠿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣛⣻⣭⣭⣭⣍⣛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣯⣽⠷⣶⣿⣯⣝⡲⣄
⠀⠀⠀⠀⠀⠀⢀⡞⢁⡴⢮⣉⠠⣾⠀⣨⡝⢷⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⡿⢫⢹⣶⣿⠛⣇⣭⣿⣮⢷
⠀⠀⠀⠀⠀⠀⡞⢀⡟⠀⢼⣿⣶⠿⢿⣿⣀⣀⣹⡯⢷⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⡿⢇⣞⣻⣟⣹⡔⣋⠛⡋⢻⡎⣇
⠀⠀⠀⠀⠀⢰⡇⢠⠙⢳⣾⣿⠁⣾⢆⣿⡿⠛⣻⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⡷⡿⣻⣿⣇⣿⢾⣿⣇⣰⣼⣇⣿
⠀⠀⠀⠀⠀⠸⡇⢘⡌⠛⣻⣿⣷⣶⣿⣿⡿⠿⣼⡿⣼⠇⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣧⣾⡿⣿⡿⣿⣗⣿⣯⣿⢃⡟
⠀⠀⠀⠀⠀⠀⢻⡀⠻⣼⠟⠁⣺⠛⢻⣿⣷⣼⠏⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠻⣧⣿⣛⡭⢷⣟⣻⣻⣿⢋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣈⠙⠶⢭⣗⣤⠟⠛⣡⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠿⣿⣶⢿⢛⣩⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠾⠿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁


⣿⣿⠟⢷⣾⣿⣷⣾⣷⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣏⣹⣧⣤⣤⣤⣤⣤⣤⣮⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣿⣾⣤⡾⠛⠉
⠀⠀⠀⠰⠿⣿⠿⠿⠛⠙⣿⠙⠛⠿⢿⣿⠷⠀⠀⠀⠀⠀⠀⠀⡄⣽⡿⣿⠿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣶⣿⣯⡤⠤
⠀⠀⠀⠀⠀⠀⠙⠻⠷⢾⣶⣶⣿⣤⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠿⠿⠄
⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠿⠟⢻⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠊⠙⢻⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠰⢶⣶⣶⠿⠿⠟⠿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣤⣼⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠻⢿⣿⠿⣿⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠤⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠂⠒⠒⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⢠⠠⡄⢀⠀⡀⠄⣄⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠘⠈⣘⠇⠈⠀⠀⠀⠁⠀⠃
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢠⡀⢀⣀⣀⢀⢀⢀⢀⢰⡄⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠃⠑⠐⠘⠈⠐⠔⠘⠄⠸⠤⠋⠠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⠤⠔⠂⠉⠉⠁⠀⠀⠉⠑⠒⠤⢄
⠀⠀⣤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠉⠉⠒⢄
⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⡀⠀⠠⠔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠈⠑⠂⠤⡀⠀⠀⠀⠀⢀⡀⣩⠷⠦⠤⠤⣀⣀⣀⣀⡀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠗⠈⠇⠽⠸⠐⠅⡇⠇⠅⠵⠇⠓⠇⠀⠇⠆⠇⠇⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⢠⢀⠀⡀⣀⣠⣀⢀⡀⣀⠀⡄⡀⡀⡀⡀⡀⠀⣀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢆⡀⢆⣸⢸⣸⠨⣈⡷⣀⡇⡇⣇⡇⡏⡉⢱⠀⠀⠀⠀⠀⠀⠳⠖⠀⠘⠊⠃⠒⠃⠀⠘⣸⠈⠂⠊⠂⠊⠂⠃⠛⠀⠛⠀⠛⠀⠃
⣿⠿⠿⠿⣿⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠄⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠩⠀⠀⠈
⠀⠀⠀⠀⢀⠔⠁⠀⠁⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠁⠀⠀⠀⠀⡁
⠀⠀⠀⢰⠁⠀⠀⠀⡇⠁⠐⠚⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠠⠁
⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⡠⡰⠀⠀⠀⢄⡀⠀⠀⠀⠀⢡⣠⠀⠀⠼⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠒⠤⣠⣼⡾⠟⠁⠀⠀⠀⠀⠉⠁⠈⠈⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⡀⢀⢠⡖⠀⣀⣲⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⢠⠠⠲⠊⠉⢁⣠⠀⠀⢠⡄⠀⣈⣭⣁⣐⣮⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠠⠞⠃⠐⠂⠐⠀⠠⠠⠬⠩⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⢀⠔⠒⠠⠀⠀⠀⠀⠀⠠⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢦⡤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠉⠀⠀⠀⠉⠒⠦⠄⡤⣀⣀⣺⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠲⠐⠀⠀⠀⠀⠉⠈⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠤⠞
⠀⠀⠰⠀⠀⠀⢀⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠣
⠀⠀⡀⠈⠀⠀⠀⠀⡀⠀⠀⢠⠰⠀⢀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢐⠊⠀⠀⠀⠐⠀⡈⠒⠂⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠪⡀⠀⠀⢀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠑⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠲⠀⠀⢱⡄⠀⠈⠀⠀⠀⢄⡔
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠠⠒⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢘⠀⠀⠀⠀⠀⠀⣈⠄⠀⡁⠦⣀
⠀⠀⠀⠂⠀⠀⠠⠆⠉⢰⠂⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠈⣻⠤⠄⠀⣀⠀⢳⡀
⠀⠀⠀⠀⠂⠂⠀⠂⠁⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⡂⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠂⠀⠺⠟⢤⣼⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⣿⣿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⢀⣀⣀⣀⠀⣴⡿⢛⣵⣾⠿⠿⠿⠿⣷⣮⣙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⣴⣿⢟⣛⣛⡛⣼⡟⣵⣿⢟⡵⣫⣶⣾⣶⣦⣿⣿⣦⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⢣⣿⣿⣿⢹⣿⢹⣿⡟⣼⢸⣿⣿⣯⣍⠙⢿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡸⣿⣿⣿⢸⣿⢸⣿⡇⣿⢸⣿⣿⣿⣿⣷⢢⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢭⣛⡊⣿⡎⣿⣿⡸⣮⡻⢿⡿⠿⣫⢟⣾⣿⢳⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⣿⣿⡜⣿⣬⡻⣿⣮⣝⣛⣛⣫⣵⣿⠟⣡⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣮⡻⢿⣮⣽⣛⣛⣛⣛⣯⣵⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⣀⣀⡴⢋⣿⣿⣿⣿⣿⡗⣭⠙⠛⠛⠛⠛⢛⣭⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢫⣶⣾⣿⣿⣿⣿⠿⣫⡾⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣮⣛⠷⣮⣭⣭⣭⡵⠶⠛⠉⠀⠀⠤⠠⠤⠤⠄⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠐⠂⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠃⠘⠛⠛⠀⠛⠛⠃⠀⠀⠀⠈⠿⡿⠇





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⡁⢀⠈⠂⠚⡠⢀⠆⡀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢇⢠⠀⢢⠸⠎⡀⡝⡅⢁⣴⠑⢂⠀⠁⠀⠀⡠⢔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠡⠑⠙⠈⢀⠀⢠⢁⣀⣃⣾⣡⡼⠧⠦⠲⠮⢔⡀⠭⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⡃⠰⢄⡀⠀⠀⣠⣷⣯⡟⡟⠻⠻⢿⣿⣿⣦⡒⠐⠊⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦⠙⢘⠆⠕⡀⠁⢀⢰⣯⠿⠟⡁⠀⠀⠂⠀⠄⠟⣿⣿⣿⣄⡛⠃
⠀⣀⠀⢀⠀⠄⠀⢆⡀⠀⡐⠵⡄⢻⢆⡃⢂⣤⣷⣯⣅⡀⠀⠁⠀⠁⠁⠀⠈⠽⣿⣿⣯⣍⣶⣤⣀
⠀⠀⠀⠀⠀⠀⡁⢀⡁⠀⡚⢍⠂⢤⡁⠵⣫⠚⠛⠩⠙⠻⣦⣆⡀⠠⠀⠀⠠⣒⣿⣽⣞⠯⠷⠕⠫⠑⠄
⠀⠀⠀⠀⠀⠀⠈⠣⣍⠁⠀⠰⢲⣄⣈⢱⡇⠄⠠⠀⠀⠁⠈⣿⣷⣷⣶⣤⣡⣽⡿⣿⣿⣷⡄⢵⣯⣲⡀
⠀⠀⠀⠀⠀⢀⣀⣈⣛⠙⠴⣥⢐⠘⢧⣜⢂⠀⠀⠠⢐⣀⣾⠛⢿⣍⣉⣿⡟⢿⣗⣖⣟⠛⣏⣫⢉⠈⠉
⠀⠀⠀⠨⠁⠀⠀⣉⣬⠉⠀⠀⢤⣨⡼⣿⣿⡾⡖⠒⢺⣽⣷⠦⣾⣿⡿⢿⣁⣠⣏⣵⣾⡶⢗⣒⡈⠂
⠀⠀⠀⢀⠤⠒⠀⠉⠀⠄⣀⡀⠀⢤⣤⠚⢻⠡⠀⠀⢀⣽⣧⣤⠍⢙⣧⣾⠏⠡⣿⣗⠊⠁⠬⣉⠁
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠔⢒⣢⡾⢵⡥⢞⡜⢷⣄⠀⣨⡿⠃⠈⣿⡿⠟⢷⣄⣠⡿⡛⠑⠦⠑⠘⡂
⠀⠀⠀⠀⠈⠡⠀⢂⡴⠔⠋⠁⠀⡩⢖⡩⡞⡪⣝⡿⢮⢿⣦⡀⣸⣷⣄⣦⣿⣿⢯⡻⡆⡐⣄
⠀⠀⠀⠀⠀⠠⠎⠈⡀⠀⣨⠕⠩⠈⢩⠞⢀⢰⡏⡙⢩⠉⢹⠛⠛⢁⠉⡍⠹⠈⠢⠙⠉⣌⡨⡰⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣔⣫⠔⡡⢀⣴⡦⠠⢂⠏⠔⠁⠀⡆⠀⡬⡄⠈⠇⠁⡀⠔⠐⠀⢬⠈⣂⠹⠸⡂
⠀⠈⠀⠀⠀⠀⢈⠑⠈⠔⠨⠒⣽⠟⠔⡁⠋⡜⢀⠄⢠⠁⠘⠀⠃⠈⢠⠀⢁⠀⡀⠀⠀⣊⠣⡀⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠐⣡⠂⠄⣠⡘⠇⠊⠀⢀⠴⠐⠮⠀⢰⠀⣳⠀⠀⠡⡀⠀⠘⠐⠅⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⢎⠠⠄⠐⣈⡀⠺⡀⠀⢀⡀⡄⠀⠃⠠⡸⡀⢠⠀⣀⠈⢛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠁⠀⠁⠘⠉⠀⠗⠰⠈⠘⠇⠀⠀⢀⠩⠀⠈⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠘⠀⠀⠈⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣄⣀⣀⡤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢼⠛⠛⠒⠲⡏⠛⠚⠉⠙⣏⠉⠈⢳⡀⠈⣧⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⣯⠀⠀⠀⠈⣇⠀⠀⠀⢹⡀⠀⠀⠀⢻⠀⠀⠀⣷⠀⠘⢆⠀⠀⠐⠟⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢠⡟⠁⠀⠀⠸⣆⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢹⡀⠀⠀⠙⣇⠀⠀⢙⣶⣤⡤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⠀⢘⡇⠀⠀⠀⣿⠀⠀⠀⠈⣻⣦⡀⣀⡼⡗⢶⠋⠏⢷⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⠇⠀⠀⠀⢿⠀⠀⠀⠀⢸⡆⠀⠀⠀⢧⡀⠀⠀⠸⡆⠀⢀⣴⠛⢏⠏⠉⠀⢸⢘⠀⠀⣸⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡼⡁⠀⠀⠈⢆⠀⠀⠸⣆⠀⠀⠀⠀⢳⡄⠀⠀⠀⠙⠶⣤⡀⢹⡟⠋⠀⠀⢸⢸⠀⠀⢸⡨⣆⣀⡏
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⢦⡀⠀⠈⢆⠀⠀⠘⠢⡀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠙⡿⠳⡀⠀⠀⡈⠈⡳⣤⣿⠊⡁⣩⡇
⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠹⣄⠀⠈⢢⡀⠀⠀⠉⠦⠀⠀⠀⠀⠉⠛⠳⢦⣤⡾⠁⡠⠙⢦⡄⠃⢀⡠⠜⠘⢍⣳⡄⠁
⠀⠀⠀⠀⠀⠀⠘⠓⢤⡀⠀⠀⣸⢦⡀⠀⠙⢆⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⣠⠃⠀⠁⠀⠈⢈⡿⡉⢫⡀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠓⢲⠄⠉⠶⢾⠁⠀⠙⠳⣄⠈⢳⣄⠀⠀⠀⠀⠈⢓⡦⠴⠃⢠⠋⠀⣀⠔⠃⠀⠘⢋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⠀⢻⠀⠀⠀⠀⠘⢧⡀⠈⠙⠢⣀⣀⡴⠛⠁⠀⡴⠛⠒⢉⠀⠀⠀⠲⡚⠈⠂⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠤⠼⣧⣀⠀⠀⠀⣀⣳⡤⠴⠖⠋⠉⠀⠀⠀⠀⡇⠀⣤⣀⡦⠀⠀⠀⠠⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘⣬⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠚⠉⢤⡆⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡄⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠄⠀⠀⠀⠀⠀⠀⣼⣷⠄⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣠⡄⠘⡄
⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣷
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠂⠀⣿⡆
⠀⠀⠉⠀⠀⠀⠀⠀⠈⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⣾⡆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣗⡚⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡃
⠀⠀⠀⠢⢬⠒⣈⡀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠑⠆⠀⠀⠄⠀⠔⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠃⠀⠀⠀⣿⡿
⠀⢀⣴⣦⡀⠀⠀⠀⢀⡀⠀⠐⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠈⠁⠠⠀⠀⠀⢐⡆⠀⣾⣿⡇
⠀⠸⣿⣿⣷⡀⠀⢈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠈⠁⠈⠸⠛⠁
⠀⠀⠈⠟⠉⠀⠀⠀⠐⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣀⣄⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠒⠑⠒⠏⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀⠄⠐⠁⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢰⣿⣿⠏⠀⢀⣀⡀⠀⣴⣾⣦⠈⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⠀⠀⢿⣿⣿⣷⣝⠛⠋⠀⠀⠀⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢻⣿⣿⠀⠀⠈⣛⠿⢿⣿⣷⣤⣀⣀⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⣿⣇⠀⢸⣿⣿⠄⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠈⢿⣿⣷⣌⠛⠋⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣶⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠙⠛⠿⠿⠿⠟⠁











⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡟⢿⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⡿⠿⠿⠇⠸⣿⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⣫⡵⠖⠒⠛⠙⠒⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣭⣭⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⣋⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⢿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⣻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⡿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣀⣀⣀⣤⣤⣀⣀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⢻⣿⣿⣯⣭⣭⣙⡛⣿⣿⣿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⣽⣿⣿
⠀⠹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣇⣀⡤⠖⠛⠉⠉⠉⠉⠓⠢⢤⣀⠀⠈⠛⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣙⣿⣿⣿
⠀⠀⠘⢿⣿⣿⣟⣴⡟⢨⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠈⠻⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡻⣿⣿⣿⣿⠏
⠀⠀⠀⠈⢿⣿⣿⣿⢃⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠈⠙⢿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣀⣠⣤⣤⣴⣶⣿⣍⣿⣿⣾⣿⡿⠋
⠀⠀⠀⠀⠈⣿⣿⢃⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣉⣙⠛⠿⠿⠿⠿⠿⠟⢛⣛⣩⠽⠚⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⠇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢆⣀⣤⠼⠄⣀⣾⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⢀⠔⠋⠀⠀⠀⠀⠀⠀⠑⢶⡋⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠋⠀⠀⠀⢠⡄⠀⠀⣠⡀⠀⠘⢦⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠀⠀⠀⠀⠀⠞⣋⡑⠺⡉⠀⠀⠀⢸⡀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣌⢓⡦⡿⠀⠁⠀⠀⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠤⠴⠁⣀⢸⠀⠀⢥⣤⠴⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠖⠂⠀⠀⢀⠀⢳⡄⠀⠀⠀⠐⠞⠸⡀⠀⢱⡀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⠀⠀⣰⠄⠀⡿⠀⠀⠀⢣⡀⠀⠳⡛⠉⠉⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢸⠀⠈⠁⠀⠛⠀⢀⡼⠃⢀⣳⡄⠀⠘⢲⡚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠈⡁⠒⠛⠈⢷⠆⠀⣀⡠⢭⣉⠘⠢⡀⠀⠙⣆⣀⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢡⠀⢀⣴⡦⠀⠚⠁⢸⡆⢤⡱⡄⠘⡄⠀⠸⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠉⠀⠀⠀⢧⠀⠀⠀⠀⠀⠽⣤⡟⢾⡁⡇⠀⢸⠀⠀⢷⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠳⣀⠰⠋⠀⠀⣘⣁⣨⠜⢠⠶⣸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⢰⣄⠀⠈⠓⠄⣀⠀⠀⠐⢦⠀⣀⡴⠁⠀⢠⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠑⢤⣀⠀⠀⠉⠉⠋⠉⠉⠁⠀⢀⡴⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠉⢲⡖⠒⠶⢶⡖⠒⠛⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣾⠁⠀⠀⠸⡇⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⣌⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣧⠻⣽⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡄⠳⣌⠓⣽⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠑⢦⡈⢷⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣀⠙⠳⣄⡙⢶⣾⣷⣿⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠀⡉⠻⢶⣬⣻⣿⣯⣽⣴⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠒⠶⣄⣉⠙⠳⣦⡝⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠄⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢩⡀⠀⠀⠀⠀⠀⣀⣀⣀⠉⠻⢶⣤⣙⣿⡶⣟⣛⣋⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠑⠂⠠⣮⣴⣶⣤⣌⣉⠛⠷⢦⣝⣿⣿⣿⣿⣿⣿⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣑⣾⠀⠀⠀⠀⠀⣰⣦⠄⠀⠀⠀⠀⠱⢶⣦⣭⣙⠻⣷⣭⣿⠿⢦⡤⠛⣋⣩⣴⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠐⡿⣁⣂⠀⠀⠀⠀⡈⢧⡀⠀⠀⠀⠀⣀⣠⣀⣙⡙⠻⣶⣿⣿⣿⣶⣞⣛⣩⣥⣴⠶⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⡧⡐⡹⢧⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠉⠉⠉⢛⠿⠿⣿⣳⢿⣟⣿⣷⣿⣿⣉⣭⣴⡶⠶⠖⠒⠦
⠀⠀⠀⠀⠀⠀⠠⠜⠤⠓⠌⠀⠀⠀⠀⠀⠙⠞⢃⣠⠀⠀⣀⡤⣤⣷⣤⣤⣬⣘⣻⣿⢿⣿⣿⣏⣧⡤⠶⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠠⠰⢴⠄⡐⠀⠀⠀⠀⠀⠀⠀⣾⣤⠀⠉⠒⠒⠺⠟⠃⠤⣦⣬⣋⡛⢯⣽⠿⠶⠖⠂⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠤⣽⠘⠷⠞⡷⠀⠀⠀⠀⠀⠀⠚⣧⡴⠃⣠⡤⠤⠴⠶⡶⣷⣾⣿⣷⣾⣶⡶⠴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⢡⢥⢠⠄⠄⠁⠀⠀⠀⠀⠀⠉⢈⣷⠚⢀⣶⣶⣶⣷⣾⡿⣿⣾⣿⣿⣛⠛⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⡡⡈⠀⡀⠦⡄⠀⠀⠀⠀⠀⠀⠈⠉⣴⡶⣭⣩⣭⣭⣭⣿⣟⣿⣿⣯⣙⣋⠒⠒⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠡⢂⡀⢀⡀⡀⠀⠀⠀⠀⠀⢠⠞⣁⣰⣧⣤⠶⠹⠶⢻⠿⠟⠻⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠦⠀⠃⠙⠀⠀⠀⠀⠀⠀⠀⠚⠉⢉⣹⣧⣤⠶⢋⣱⡶⠶⣤⠧⣄⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠒⠆⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣛⡁⠉⠸⠥⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣆⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⡷⠀⠠⣆⣠⣼⠉⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⢾⡁⣽⠀⣬⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡐⠐⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠁⢸⢀⣷⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⢁⡤⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⣄⡀⢸⣾⣧⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢛⠈⠎⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣈⣿⣇⣹⣿⣿⣯⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢶⡠⠤⡀⢤⡲⠦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠐⢤⣱⡀⠁⡈⢀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠻⠏⠃⠀⠀⠀⠀⠀⠊⢈⢿⣿⡿⢿⣿⡟⢿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠈⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⠀⠀⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⡋⣹⣯⣍⣽⣿⣷⣿⣏⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡟⠸⣙⣻⢳⣧⣿⣷⣿⣿⣿⣉⢙⣋⡯⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣃⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⣟⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⣤⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣧⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠈⠉⠀⠀⠀⢀⠀⠀⢀⡀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢶⡿⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣬⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣾⣿⡿⡆⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠈⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⠿⣹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣻⣗⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣻⣿⣷⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⢿⣷⡀⠀⠸⣿⣄⡈⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠹⣿⡄⠀⠈⠛⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣤⣄⠀⠀⠀⠀⣀⠀⠀⠙⣿⣦⡀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣆⠀⠀⠸⣦⠀⠀⠈⠻⢿⣶⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠈⣿⣧⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠉⠙⠻⢿⣶⣾⠟⠀⢠⣿⠇⠀⠀⢀⣠⣤⣤⣶⣶⡿⠟⠉⠁⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⡀⠀⠀⣀⣠⣴⣿⠏⠀⠀⢠⣿⠟⠋⠉⠉⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⣾⡿⠀⠀⠠⠶⠶⠶⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠟⠛⠁


⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⣰⣿⣿⣿⠿⠛⠻⠋⣉⢻⣷⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⢠⢰⣠⡀
⠀⠀⢰⣏⣧⣍⡑⢾⡛⢆⠑⠦⢙⣿⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⢌⠡⡝⠕⢕⢌⠄
⠀⠀⢸⡿⡎⢩⣙⣂⠅⣸⠟⠚⣋⡤⢉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣤⣓⣸⠸⢀⠥⢁⠦⡔⡂
⠀⠀⠘⣾⣽⢦⡤⠥⠾⠇⣸⣭⣥⣴⣾⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣚⡓⣁⢖⠑⠛⠒⠊
⠀⠀⠀⠙⢶⣯⢛⣿⣖⣾⣷⣯⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠑⠿⢿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸⣿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢻⣧⣤⣴⣾⣿⣦⣤⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣶⣿⠿⠟⠛⣿⣿⣿⣿⣿⣷⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⣴⣿⡿⠋⠁⠀⠀⠀⣿⣿⠟⠁⠈⢻⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣿⣿⠟⠁⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣯⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣾⣿⠋⠀⠰⠶⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡇⠀⣀
⠀⠀⠀⠀⠀⢀⡀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠘⠻⣷⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢿⣿⣄⣀⣤
⠀⠀⠀⢀⣀⡀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⣿⣿⡛⠉
⠀⠀⠀⠀⠛⢿⣿⣿⠁⠀⢠⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣷⣠⣴⡶
⠀⠀⣠⣄⡀⣼⣿⠃⠀⠀⠘⠛⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣍
⠀⠀⠈⠛⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠈⢻⣿⣶⣾⠟
⠀⣠⣀⣀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⡀
⠀⠉⠛⢿⣿⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠘⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠛
⢀⣀⣀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠈⠛⢻⣿⡿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠠⣿⣿⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠘⣿⣿⠿⠟
⣀⣀⢸⣿⡇⠀⠀⢀⡀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⢿⣿⡀
⠛⠛⢿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⣿⡿⠿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣄⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠘⠻⠿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠙⠛⠋⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣤⡄⠀⠀⣼⣿⡿⣶
⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠛⠃⠀⢀⣿⣿
⠀⠸⠿⠿⢿⣿⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣼⣿⠿⣶⡦
⠀⠀⠀⠀⢈⣿⣿⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠺⠟⠋⠻⣿⣷⣄⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢀⣴⣿⡿⠻⣷⡆
⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⢹⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⡏⠁⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠃⠀⠈⣽⡿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣿⡏⠁⠀⠹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⣿⠇⠀⠀⣿⡇⠀⠀⢸⡷⠀⠀⠸⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣂⣤⣤⣭⣷⣶⣶⣶⢦⣤⣤⣄⣐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⣵⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣭⣿⣿⣿⡗⡮⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢂⣥⣾⣵⣿⣿⣿⠟⠛⠉⠉⠉⠉⢉⡉⠉⠋⠉⠁⠀⠙⠻⢷⣦⣶⢽⢦⣔⢄
⠀⠀⠀⠀⠀⠠⢊⣴⢿⣿⣾⣿⣿⠋⠀⢀⠠⠐⢂⡉⠉⠉⠁⠀⠉⡑⠐⠀⠄⠀⠀⠙⠳⢷⣾⣿⣷⣕⢄
⠀⠀⠀⠀⠔⣴⣯⣷⣿⡿⠛⠉⢀⠄⠊⡠⢰⣄⠁⣌⡇⠀⠠⡤⣧⠁⣴⡄⠠⣄⣀⠐⠄⠈⠙⢿⣿⣼⣦⡡⡀
⠀⠀⢀⢊⣼⣟⢿⣿⡟⠀⢀⠔⠁⠀⣰⣁⢬⡀⠑⠈⠈⠲⠶⠰⢯⡆⢄⠁⡤⠄⠔⢉⡀⠀⠠⡀⠻⣿⣿⣿⡔⡀
⠀⢠⢃⣾⣻⣿⣿⡏⢀⡔⡡⣄⣤⡀⠁⢾⠇⠙⠀⣠⡀⢣⡷⠀⠀⠉⢟⠠⠛⠁⠀⢀⣕⠦⠈⠁⠢⡘⢿⣿⣿⡜⡄
⢀⠂⠾⣿⣿⠋⠁⢀⠎⠀⠀⠧⡅⠸⠀⣾⠐⣹⠸⣇⣠⠄⠀⡐⠶⢇⡈⠀⠰⠈⠱⢿⡎⡄⢢⠀⢀⠑⠀⢿⣿⣿⣰
⠀⢸⢠⠤⡟⠁⠀⠈⠀⠙⡆⡰⠃⠖⢂⠘⠯⠝⠀⢠⡄⠀⠌⢳⢄⠀⡀⣘⠳⠀⠀⠀⠑⠎⠳⡀⠀⠁⠁⠀⠋⠀⢧
⠀⡀⡀⢨⠀⠀⢠⠃⠘⠌⢁⠀⢴⡀⢁⠄⣀⡰⡵⡀⠰⡇⢠⠺⠟⠸⠀⢈⠿⠞⡲⢠⡎⠙⢈⠏⠡⡀⠀⢃⠀⠀⢸
⠀⡇⡇⡶⢰⢆⢸⠀⡀⢀⠘⠖⠡⠂⣮⣂⠀⢙⣋⠀⠀⠙⠀⠀⠀⠁⡔⠏⠒⠀⠁⢲⠉⠩⢀⡘⠅⠉⡔⣾⠀⠀⢸
⠂⡇⡇⡗⢸⢸⡀⣴⣿⠀⡇⢀⠤⠀⢈⡉⢢⠸⢿⠁⠘⠖⠀⣠⢭⡢⠑⠄⠁⠀⡉⠠⡄⠘⠂⢩⣜⠇⠒⢠⠀⠀⢸
⠀⢣⡰⢨⡘⠿⠁⡮⠚⠂⠁⣢⠀⠂⠀⠠⠂⠀⢸⣉⣀⡸⢛⠈⣿⣀⣦⣤⣐⡘⢳⡤⠦⣿⣈⠀⠀⢀⡄⠘⠀⠀⡸
⠈⠈⡄⠿⠀⠀⠀⢃⠘⠄⠈⡍⠰⣶⣷⢀⣀⣠⢚⣅⢈⢙⢄⣀⢻⠉⡅⢲⣦⠈⣯⣈⠀⠸⢇⠀⠀⠈⠉⠄⣤⠀⡇
⠀⠐⡸⣤⣶⣄⠀⠈⢏⠂⠄⢰⠐⣋⡤⠏⣭⡝⢮⠟⠉⠀⠹⣟⡃⠢⣼⡽⠒⠚⠋⠶⠭⠇⠀⠑⠤⢽⠃⣼⢿⡿
⠀⠀⠠⡘⣿⣿⣦⣀⠀⠹⡔⢧⣅⠄⣂⠶⠓⠛⠿⢴⢈⡱⠰⡁⠋⠀⣶⠀⠀⠠⡄⡘⣀⠀⠟⠥⠖⢀⣾⡷⣹⠃
⠀⠀⠀⠑⡜⢿⢿⣿⣧⣄⠈⠢⢵⠆⢿⠀⣀⡀⠰⢪⠂⠀⠂⠉⠶⠉⣿⠢⣀⡄⣢⠞⠛⣀⡰⢊⣰⣿⣗⡿⠁
⠀⠀⠀⠀⠈⢎⠻⣵⣿⣿⣶⣄⠀⠑⠪⡕⣊⡉⠀⡎⢿⣴⣯⣀⣴⣈⠀⠀⠳⣜⠡⠒⢉⣀⣴⣿⣾⡽⢋⠄
⠀⠀⠀⠀⠀⠀⠑⢌⠻⣿⣿⣿⣷⣦⡀⠀⠀⠈⠉⠉⠐⠒⠀⠀⠀⠋⠀⠉⢉⣡⣄⣶⠾⠟⣿⡿⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⡛⢿⣿⣻⣻⣿⣶⣶⣤⣀⣤⣠⡤⢤⣴⣴⣾⣏⣷⣿⣯⠕⡚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠉⣙⠛⠾⢯⣿⣿⣿⣿⣿⣾⣿⣿⠿⠛⣋⠩⠐⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣤⣭⣭⣉⣀⣀⣂⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⣠⣤⣿⣷⣿⣿⣿⣆
⠀⠀⠀⠀⣹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⣌⣷⣿⣿⣿⣯⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⠏⣠⡄⠀⠀⠀⠀⠀⠀⠘⡇⠸⠟⣡⣿⣿⣿⣿⣿⣏⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢿⡝⣟⣿⠇⠀⠀⠀⠀⠀⠀⠁⢇⠀⠐⢻⣷⡿⠯⢟⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⠁
⠀⠀⠀⠘⠷⣍⣻⣁⣴⠿⢁⠀⠀⠀⠀⠀⢸⡀⠀⠁⠀⠀⠒⠉⠀⠀⠀⠀⠀⢤⠀⠁⢠⠟⣯
⠀⠀⠀⠀⢙⣿⡜⣟⢁⣼⠏⣠⠀⠀⠀⠀⠈⣧⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡀⠴⠂⢀⠅
⠀⠀⠀⠀⣈⠻⢶⣽⡟⢁⣼⠃⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⢠⠈⠣⠄⠀⠀⠁⣠⠆
⠀⠀⠀⠀⡙⠿⣦⣜⣷⠿⢁⣠⣾⠗⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⢡⠎⡀
⠀⠀⠀⠀⠙⠷⣦⣍⢻⣾⠟⠋⣁⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠴⢊
⠀⠀⠀⠀⠀⠀⠳⣬⡉⢷⣶⠟⠋⣡⡶⠀⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⢈⠅⠀⠈
⠀⠀⠀⠀⠈⠛⢶⣌⡙⠻⣠⡴⠞⢉⣴⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠠⢐⡟⠀⠐⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⣇⣠⡶⠟⢉⣠⣄⠀⠀⠀⠀⡠⠴⠦⠄⠀⢾⡁⠀⠄⠊⡀
⠀⠀⠀⠀⠀⠙⠻⠟⠷⠾⡼⣃⣤⡶⠟⠛⠁⠀⠀⠀⢤⡤⠄⠀⠀⢠⡏⠁⡠⠾⠋
⠀⠀⠀⠀⠀⠰⠶⠶⢶⣶⡤⣏⣁⣠⣤⡶⠖⠀⠀⠀⠀⠀⠀⠀⠁⢾⠁⠀⠀⡤⣤
⠀⠀⠀⠀⠀⠴⠶⠶⠶⢤⣄⢙⣉⣁⣀⣠⡄⠀⠀⠀⠈⠘⠀⠁⢀⡞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⢸⠉⠉⠉⠁⠀⠀⠀⠀⠙⠁⠀⠀⢠⠂⠠⡉⡙⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠸⣿⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣎⡝⠊⣵⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣘⣿⠚⠋⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢠⡀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣾⣷⣶⣾⣾⣶⣶⣷⣤⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢈⣇⣠⠤⠤⠠⠒⠴⢤⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⡛⠉⡭⠀⠛⠓⢆⠀⠀⢁⠀⠀⡌⠹⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠐⠀⠀⠀⠁⡘⠠⠄⣛⢆⠁⢈⢀⠀⡀⠠⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⢀⠖⠁⠀⠤⠐⠂⡀⠀⠀⡁⠀⠠⠅⡈⡆⠀⠁⣀⠑⠀⠂⡄⠠⠀⠈⢧⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡤⠋⠰⠀⢰⠄⠀⣀⠃⠀⠀⡄⠉⠀⠂⠀⡇⠀⠸⣷⣤⠤⠂⠀⠔⠀⠄⠀⠑⡆⢁⠄⠂
⠀⠀⠀⠀⠀⠂⠄⡀⠀⡞⢁⠀⡄⠀⠙⠟⠋⣀⠀⡰⠈⡀⠡⠘⠀⢁⡷⠀⢠⠀⢰⡇⠈⡶⠀⠀⡸⠀⠀⠘⣧⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠀⢀⡝⠠⢈⠀⢳⡦⠂⡀⠀⠀⢰⣷⣠⣷⠀⠀⢀⣷⡇⢀⠌⠤⠀⡍⠉⠀⠳⠶⠇⠀⠃⠀⠘⣆
⠀⠀⠀⠀⠁⠂⠄⡜⠀⠀⠈⠁⠩⠀⠈⢁⠀⠁⠀⢈⠁⠀⠀⠀⠉⢙⠛⠁⠀⠆⠀⣰⡾⠉⠀⠀⡂⠰⠾⠇⠀⠈⡧⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠺⠀⢠⣄⣀⡔⠫⠀⠀⡜⠁⠀⡀⠀⢰⠄⠀⠀⠀⡸⠀⠛⠂⡰⠆⣙⠓⠂⠐⠀⠀⠀⡀⠀⠀⠀⠘⡊⠀⠀⠠
⠀⠀⠀⠀⢀⣀⠇⠀⠌⠉⠵⠄⠀⠀⠸⣷⡤⠔⠀⠠⠋⢀⡐⠀⠀⠃⢀⠀⠀⢀⡀⠓⠀⠀⣀⡰⠀⠀⡁⠰⠀⣀⠀⢋⡀⠄
⠀⠀⠀⠀⠀⡼⠈⢀⠀⠀⣨⡄⣠⠐⣀⠀⠀⣤⠞⠁⠀⢘⠃⠀⢈⠀⡀⠒⠀⠪⠇⢸⠃⠸⠛⢀⠀⢀⡁⠀⠰⠎⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠄⢈⡀⠰⠁⠈⠟⠀⣧⠀⠀⠃⣨⠀⠄⠘⠃⠀⡃⠀⠁⠀⠄⢀⡄⠀⣎⠀⣸⠆⠰⠋⠁⠀⢸⠀⠀⠀⢳⠂⠈
⠀⡀⠀⠀⠰⡇⠤⠀⠁⣀⠁⠀⠛⡀⠛⢀⡞⠀⠃⠀⠀⡐⠠⡐⢡⡤⡀⢀⠀⢈⠁⠀⠁⣴⣋⠒⠀⢀⠉⠀⠀⣶⠀⢀⢸⡁
⠀⠀⠈⠉⢸⠂⠠⣁⣠⠀⡉⢠⠀⠀⠀⠟⢀⠀⠈⠀⠀⠬⠀⠃⢺⣀⡇⠘⠀⢧⠀⠀⢈⠁⠹⣆⢀⣦⠀⠁⠀⠈⠀⠀⠈⡀
⠀⠀⠀⠀⢸⠈⠁⠀⠁⠓⠀⠸⡄⠀⢆⠀⠀⠀⠈⠀⠃⠀⡀⠀⠐⠉⣤⠔⠀⠀⠀⠀⠀⠈⠀⠀⡾⠀⠀⠂⠀⢀⡄⠈⠀⡄
⠀⠀⠀⠀⢸⠀⡄⠀⠀⡁⢄⡐⠠⠀⣾⠀⠀⢀⠀⠰⡆⠈⡅⠀⠐⣞⠚⠀⣠⠄⠀⢀⣀⣈⠀⠐⠀⢈⠆⠀⠄⠀⠀⠀⢀⡆⠀⠄
⠀⣀⠀⠀⠨⡄⠐⠀⠀⢢⠀⠘⣆⠠⢿⣄⢀⣤⠐⠀⠀⢠⠃⠀⠀⢋⠀⠆⠳⠀⠀⠺⣷⡀⠀⠀⡄⠐⣀⠀⠀⠀⠀⠀⠸⠂
⠀⢀⠀⠀⠄⢆⠀⢧⡀⡀⠷⠤⠇⠀⠀⠉⠙⠃⣀⠁⠀⣸⡀⠀⠀⢫⠀⠊⠀⠀⠁⠀⠈⢉⠀⠈⣆⣀⡿⠦⠒⠀⠠⠀⡗
⠀⠀⠀⠀⠀⠚⡀⠀⠀⠐⠀⡐⡀⠀⠀⠁⠂⠀⠀⠀⢠⠏⣇⡀⠁⡄⠀⠀⠁⠀⠀⠀⣤⢁⠀⡠⠅⠀⠀⠀⠀⠀⠂⣘⠁⠐⠂⠄
⠀⠀⠀⠠⠀⠀⢹⡄⠂⢠⡂⠀⠀⢸⣦⠀⠀⠰⣦⡴⠋⠀⢡⠘⢠⡄⠁⠀⠐⠀⠀⠀⠙⠛⠉⠀⠀⢰⠃⢀⡄⠈⣰⠅⠈⠀⠂
⠀⠀⠀⢀⠀⠀⠀⡰⣄⠁⠙⠒⠰⠞⠛⠄⠀⠀⠀⢀⠀⠀⠈⢦⣬⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠸⠾⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢀⠜⢢⡀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣨⡧⡀⠀⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠘⠣⢀⠀⠀⠀⠀⠄⠀⠈⠀⡠⠞⡇⠁⠉⢢⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠖⠒⠂⠂⠙⠈⠀⠀⠀⠉⠀⠀⠀⠉⠑⠂⠔⠠⠒⠖⠘⠁⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠡⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⢤⢄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠒⣼⢎⢴⠤⢄⡁⠑⢢⡀⠀⠀⠀⠀⠀⠀⣀⣴⠞⣩⡎⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⣨⡸⣦⠀⠄⠱⣀⢘⡀⠀⠀⠀⣠⡶⣋⣤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⢹⡏⠐⠐⠽⠮⠁⠀⠀⠸⠑⠊⠉⠀⠀⠀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠷⡼⢳⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢧⣠⢴⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢤
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⣎⠴⣗⠌⡃⠁⠀⠀⠀⠀⠀⠀⠀⢁⠐⡉
⠀⠀⠀⠐⠂⢄⡀⠐⡀⠀⠀⠀⠀⡔⠙⠋⠰⢋⡜⠢⣉⠁⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⣠⠠⡄⣀⠄⣴⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢤⠀⠑⢶⡚⠟⠚⠁⢄⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠈⣳⣄⣏⢡⣨⣠⠎⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣰⡌⠘⣷⡿⣷⡒⣃⣴⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⢤⡌⠁⠼⣿⣟⣻⣾⠉⠈⠑⠀⠀⠀⠀⠀⣀⣀⠤⠶⠌⠃⣀⠀⠀⢤⠀⣤⣀
⠀⠘⠉⠁⠀⠀⠁⣿⢹⡁⠀⠀⠀⠀⠀⠐⠀⠄⠀⠉⠁⠀⠀⢘⠈⠁⠀⠀⠀⠽⠿⠏
⠀⠀⠀⠀⠀⠀⠀⡗⢚⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⠀⣀⡀⣠⠂⠀⠀⠀⠀⡀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⡯⢘⡹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡷⠉⡀⠀⠀⠀⠀⢈⢺⣿⣆⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠑⣎⣎⣿⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⢸⢿⣷⣶⣾⣶⣶⣶⣿⣻⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⣼⡓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⢻⣇⣾⣷⣖⡛⢯⣿⣿⣧⣰⠉
⠀⠀⠀⠀⠀⠀⠀⣰⢡⠊⡌⢿⠈⣄⢀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠸⣿⣷⣿⣛⣻⣮⣽⣿⠟⠁
⠀⠀⠀⠀⠴⠾⠥⠤⠼⠶⠧⠤⠠⠤⠄⠬⢒⢤⣤⣤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠿⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠿⠿⠶⠶⠶⠶⠿⠿⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣽⣟⣛⣛⣻⣿⡿⠿⣿⣟⣛⣛⣛⣯⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣻⣿⣿⣿⣯⣿⣿⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣉⣿⣿⣻⣿⣿⣟⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣿⣷⣶⣶⣶⣶⡖⢶⣶⣶⣶⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣶⣶⣶⣾⣿⢯⣭⣭⣭⣭⣭⣭⣭⣽⣿⣶⣶⣶⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⣭⣟⣛⣛⣛⣛⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣿⣭⣭⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣭⣭⣽⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣯⣭⣭⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠿⠿⢿⣿⣿⣿⣿⣼⣿⣿⣿⡿⠾⠿⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣶⣖⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠶⠶⠶⠶⠶⠶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣶⣶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣀⣤⣄⢀⣿⡿⠏⣸⠄⣀⣤⣄⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢀⡴⠒⠲⢄
⠀⠀⢰⡯⣿⣾⡇⠀⠉⠉⠀⢹⡿⣿⣾⡟⠉⠛⠿⠿⡁⣀⠀⠀⠀⠀⢀⣀⢟⠿⠷⠋⣿⢀⡀⠀⢈⡇
⠀⠀⠀⠙⠷⠟⠁⠀⠀⠀⠀⠀⠙⢻⠛⠁⠀⠀⠀⢠⣟⣽⡽⣆⣀⣾⣿⣿⣧⠀⠀⠀⠘⠮⣥⡤⠞⠁
⠀⠀⢀⡴⣟⣷⡄⠀⠀⠀⠀⢀⡴⣚⣶⡄⠀⠀⠀⢨⢷⣭⠿⠃⠀⠻⢿⠷⡏⠀⠀⠀⠀⣴⣞⡲⣄
⠀⠀⠘⣿⣿⣳⢇⣀⣤⣤⣀⠼⣿⣿⣻⠗⢤⣶⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣖⡚⢿⣿⣿⡟
⠀⠀⠀⠀⠉⠁⠀⢿⣾⣿⣿⠁⠀⠉⠁⠀⠈⠿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠏⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⡇
⠀⠀⠀⠀⠀⣀⡴⠞⠳⢤⣀⠀⠀⠀⠀⠀⠀⢰⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⢠⡴⠋⠁⠀⠀⠙⠲⣌⡙⢢⡴⠖⠛⠁⠘⠘⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡄⠉⠙⠓⠰⢏⡀
⠀⠀⢸⡇⣷⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⡜⢡⠞⠀⠀⠀⠀⠀⣚⠁
⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡜⢠⠞⠉⠉⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⢧⣀⠀⠀⠀⢀⡤⠖⣃⡼⠧⣄⣀⠀⠀⠈⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠐⢚⣉⡤⠟
⠀⠀⠀⠀⠈⠓⠦⣄⣩⠴⠚⠁⠀⠀⠀⠈⠁⢸⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣭⣯⣭⣽⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⢉⣉⡉⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢀⣀⣀⣀⠀⡴⠋⣠⠖⠋⣉⣀⣈⡉⠒⢤⡈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠱⠖⠉⣠⠤⠤⡜⠀⡜⠁⣠⢪⠟⠉⠉⠙⠆⠀⠙⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡜⠀⠀⢰⠃⢸⠀⢠⠃⡇⠀⠀⠒⢶⣄⠀⠀⢹⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⢣⠀⠀⠸⡀⢸⠀⠸⡀⡇⠀⠀⠀⠀⡟⡇⠀⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⡝⠒⢣⠈⢇⠀⠱⣌⠢⠤⠤⢚⡵⠁⢠⠇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⠀⠈⠣⡀⠑⢄⣀⠉⠉⠉⢁⡠⠔⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⡀⣀⡜⣸⠀⠀⠀⠈⠲⡤⣈⣉⠉⢉⣁⡠⠔⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡨⠖⠂⠁⠀⠀⠀⣠⠞⡴⠀⠈⠁⠀⠈⣆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠒⠶⠶⠖⢒⣫⠵⠊⠀⠄⠠⠤⠤⠀⠘⡆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸⡉⠉⠉⠉⠉⠀⠀⠐⠒⠒⠒⠂⠐⠀⠀⠈⡄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠃⠀⠀⠀⠀⠀⠉⠈⠉⠉⠈⠉⠉⠀⠀⠀⠙⢄⣠⠏





⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⣛⣋⣛⠻⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣍⢻⣿⣿⣿⣿⣿⣿⣿⣌⢻⣿⣿⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣙⠻⣿⣿⡿⠛⣛⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣦⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⠻⣿⣿⣷⣦⣭⣉⡛⢿⡟⣛⣋⣉⣉⣉⣙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣷⣼⣿⣿⣿⣿⣿⣇⢸⡇⢻⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠇⠸⣿⣿⣿⣿⣿⡿⢡⣴⣬⢻⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⢸⡇⢸⣿⣿⣿⣿⣿⠿⢛⣡⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣩⣴⣶⣿⣷⣍⠻⠿⠿⢿⣷⣿⣿⣿⣾⣿⣇⣾⣿⣷⣸⣿⣿⢟⣥⠾⠷⣌⠻⠟⣋⣩⣴⠾⢛⣉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣬⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢼⠁⣾⣷⠘⡿⢛⣋⣭⣴⣾⣿⡏⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣛⡛⠻⣿⣿⣿⣿⡿⠖⠙⣿⣿⣿⣿⣿⣿⣿⣷⡘⢷⣭⣥⠾⢣⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣌⠛⣋⣥⣶⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⢠⡿⢠⣾⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠇⣾⠃⣾⣿⣿⣿⣿⣿⠟⣡⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⡟⣸⡏⣸⣿⣿⠿⠟⣋⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣭⣙⠋⣼⣿⣿⣧⣭⣍⣉⣰⣿⣠⣭⣥⣴⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⠟⠉⠚⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣀⣿⡀⠀⠀⣾⠋⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣤⣀⠀⠀⠀⠀⣨⠕⠚⠋⠉⠁⠉⠉⠓⠳⢤⡀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢠⣄⡀⠀⠀⢈⣠⡀⠀⠀⠘⣧⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢦⡤⠄⠁⠀⠀⠀⠀⠀⣠⡽⠷⠶⢤⡞⠋⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢻⡆⠀⠙⣆⠀⢠⡀⠀⠀⣷⣠⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢶⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠶⣄⢉⣽⢂⡟⠀⠀⡃⠀⠀⢽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣃⣀⣤⠞⠁⢀⠀⡇⠀⠀⠨⣀⣀⣤⣴⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀⠀⠀⠀⢠⠀⠀⣦⡀⠀⠀⠀⠀⠀⢀⣸⠇⣇⠀⠀⠠⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠿⠀⠀⠀⢀⡀⠀⠁⠀⠸⡀⠀⠀⢻⡄⢀⣠⣤⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢸⡇⠀⠀⣐⡷⠀⠀⣹⠏⠀⠀⠀⠀⢳⡀⠀⠀⠻⣍⠁⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠛⠛⠁⠀⠀⢸⡇⠀⠉⠉⠀⠈⠛⠃⠀⠀⣘⡿⠀⠀⠻⣄⠀⠀⠈⠳⣤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣀⣲⡆⢲⡠⡀⠀⠈⠁⠀⠰⠾⠿⠛⢦⡀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠿⠇⠀⠀⠀⢱⡀⠉⠀⠀⠀⠙⠛⠀⢀⡤⠖⡒⠒⠦⣄⠀⠉⢲⡀⠀⠀⢻⣤⡤⠶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢇⠀⠀⣠⣴⡶⠀⠐⠋⠀⠈⣽⡃⡤⣌⢳⡀⠀⢹⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠶⠿⠛⠋⡀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠐⣻⡆⢩⣿⣴⡏⠈⡇⠀⠀⣷⠀⠀⠈⣇⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠘⣦⠀⠀⠀⣶⡆⠀⠀⠉⣹⠆⠲⠏⣸⠃⣀⡀⣿⠀⠀⠀⣿⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⡄⠀⠀⠈⠳⣄⠙⠉⠀⠀⠂⠠⣄⣤⡤⠞⠃⠰⠟⢡⡏⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠈⢢⡀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠐⣦⡤⠀⢀⣴⠏⠀⠀⠀⣼⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠲⣄⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠖⠚⠋⠁⠀⠀⢀⡼⠃⠀⠈⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⢀⡿⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣹⡏⠉⠙⠛⠻⣿⠉⠉⠉⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠻⠄

⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⣤⣄⣀⡄⡀
⠀⠀⠀⠀⠀⠸⠻⠉⠀⡀⡀⠀⠀⠀⠈⠉⠛⠶⠤⢂
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠘⠃⠀⠀⢃⠀⠀⠀⠀⠀⡱⠨⡲⢶⡠
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠤⠀⡀⠤⠀⡀⠰⠀⠀⠀⠘⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢀⠁⡁⡎⢹⠍⠀⠀⠀⠐⠀⢀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠉⠀⠀⡞⠋⠘⠀⠠⡄⠀⠀⠈⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠒⡄⠘⠄⠀⠸⣧⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⡀⠀⠀⢌⠶⡱⠀⢀⠀⠀⢀⠀⠀⢧⠈⠙⢷⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⢧⣀⣀⡧⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⢀⡀⠀⠀⠀⢀⠶⠄⠀⠈⠁⠀⠄⠁⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢄⡀⠁⠀⠀⢀⠀⠠⠔⢄⠈⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⣄⠀⠀⠀⡅⢁⠲⠀⠠⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁⠀⢁⠃⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣄⣀⣈⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣾⣿⣧⣼⣿⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣿⣶⣿⣿⣿⣿⣿⣿⣶⣆⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣀⣶⣿⣿⣿⡿⣿⣿⡛⢟⡙⢿⣿⣿⣿⣧⡌⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣹⣶⣿⣿⣿⣯⣿⣦⠙⠻⠁⢸⣿⠾⠛⢻⣿⣿⣿⣆⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠰⣶⣿⣿⣿⣿⠏⠉⠉⠉⠁⣀⠀⠐⠊⠁⠰⠆⢀⣌⣿⣿⠟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⣠⣿⣿⣿⠛⠁⣠⡀⠀⣠⡞⢹⣷⠀⠐⠶⠆⠀⠘⠛⢡⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⢿⡿⢿⣉⠻⠆⠀⠉⠁⠀⠉⠀⠿⠋⠀⡄⠀⢀⣄⡀⠀⠀⢀⣿⣿⣶⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣶⡿⠃⠀⠀⢀⣄⠀⠀⠿⠀⠠⡄⠀⠀⠀⠀⠰⣾⢻⡇⢠⡿⢿⡟⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⢠⣦⠀⢠⣄⠘⠛⠁⠀⠀⠀⠙⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣰⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⡁⠀⢠⣿⠆⠀⠀⣤⣄⠀⠾⣿⣿⣿⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢀⣤⠋⠁⡀⠀⠀⠀⠀⠀⠻⠋⠀⠀⣼⡟⣿⣧⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⡿⠁⠀⠀⠛⠃⠀⠀⠀⠀⠈⣷⡀⠛⠛⠰⠿⠿⣷⣰⡞⠁⠀⠀⠀⠀⠀⠸⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠿⠿⣷⠀⣈⡀⠀⠀⠀⠀⣴⠆⠀⠈⣿⣆⠀⠀⠀⢀⣼⡏⠀⠀⠀⢀⣤⠄⠀⡠⠠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣻⣿⣷⣶⣿⡤⠛⠃⠀⠀⠀⠀⠀⠤⠀⣀⠛⢻⣷⣤⣴⣿⢿⣇⠀⠈⠁⠈⠃⠀⠰⡄⠠⣿⣿⣧⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⡿⢿⣦⠉⠰⠀⣾⡿⣷⡄⠀⠀⡀⠰⠿⠁⠘⣿⢿⣿⡷⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⣿⣷⣾⠟⢿⠆⠀⠻⢷⠿⠀⠀⠀⠁⠀⠀⢀⡀⠀⢼⡏⠀⠀⠀⠉⠻⠆⠀⠀⠞⠀⣠⡀⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⣿⣿⡿⢣⣶⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⣿⣿⢷⡌⠙⢳⣶⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢻⣿⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⠀⠀⠸⠿⠏⠀⠀⣠⣶⠶⡀⠀⣀⢰⣦⠈⣷⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠈⠀⠀⣾⣇⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢹⡟⢦⡀⠀⠀⠀⠀⠀⣿⣷⣾⠀⠀⠈⠘⠛⠿⠟⠀⠀⠀⣀⠀⠀⠀⠸⠟⠻⠆⠀⠀⠀⠀⢹⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⠿⠗⠀⠀⠀⠀⠀⢰⡾⠋⠉⠉⠐⠚⢳⣤⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢻⣿⣼⠿⠋⢠⡀⠀⢠⣤⡀⠀⢸⣿⠆⣤⡄⠀⠀⠀⣿⠿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⢿⡇⠀⠀⠀⣄⠀⠀⠀⢠⣼⠿⣦⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⡧⠀⠈⢷⠆⢀⣤⣄⠀⠀⢠⡄⠈⢿⣿⠀⠀⠀⠚⠃⣀⠀⢡⡴⢿⣤⡀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠉⠉⣻⣿⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⠀⠀⠀⠀⠸⣿⡞⠀⢠⠀⠀⠀⠀⠀⣶⣆⠀⠰⣶⠋⠀⠀⣠⡄⠀⢿⣦⠀⠀⢿⡋⢹⠆⠀⡀⠀⠀⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣴⣿⣿⣴⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⢀⣀⠄⠀⣴⡏⢀⣄⢸⣿⠀⠀⠈⠛⠃⠀⠈⠃⠀⠰⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡏⢰⣾⣛⡇⠀⣴⠄⠀⠙⠃⠀⠀⠀⠀⠛⠀⠀⠈⠙⢿⣤⣄⣀⠁⣈⠋⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠃⠈⠉⠛⠁⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠀⠀⠀⠀⠀⠰⠖⠀⠈⠁⠠⢾⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⠁⠀⠀⠈⠁⢀⡀⠀⠀⢿⣀⡸⠂⠀⠀⠀⠀⣞⠁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡶⠄
⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⡆⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠁⠀⣀⣀⡀⠁⠀⠀⠀⠈⠿⠿⠀⠀⢠⣾⡧⠀⠀⠀⣴⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣯⡁⠉
⠀⠀⠀⠀⠀⡀⠸⣿⣿⡏⠁⠰⠀⠀⢰⡟⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠁⠀⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣶⣶⣏⡁
⠀⠀⠀⠀⣀⠙⠻⣿⣿⠁⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⣀⠀⠀⠈⠉⠀⠀⣿⡄⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⣰⠾⢦⡀⠀⠀⠸⣿⣿⡏⠁
⠀⠀⠀⠀⣀⡀⢲⣿⣧⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠻⣧⠀⢀⠀⠀⣤⠉⣷⠀⠀⠀⣠⡈⠉⠁⢀⡀⠀⢡⣤⠈⠁⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⡠⠾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠃⠀⠐⠂⣀⡀⠀⠐⠶⠿⠏⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡰⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢀⣤⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⠇⢀⣟⣾⣿⣯⣀
⠀⠀⠀⠀⠘⠋⢸⣿⠀⠀⠀⠀⠸⣿⠁⠘⠆⠀⠀⠉⠁⢀⣤⣤⣄⠀⠉⠀⠀⠀⠀⠘⢻⣤⡀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠐⢶⣾⣿⡄⠀⠀⢀⡄⠈⠛⠋⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⢠⣤⣤⣤⠀⠠⣤⣠⡿⠟⠀⣀⠀⠀⠀⠀⢰⡄⠀⠂⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢤⣴⣿⣷⠀⠀⠈⡁⠀⠀⠀⢴⠄⠀⠘⠀⠀⠀⠀⠀⣀⠠⠭⠏⠀⣿⠀⠀⠀⠀⠀⠘⠛⠀⠰⠿⠃⠀⠀⠀⢙⣻⣿⡗⠰
⠀⠀⠀⠀⠀⣤⣿⣿⣇⠘⠁⠈⠛⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣾⠟⠁⣀⠀⠘⢷⡄⠀⢠⣶⡄⠀⠀⠀⣀⠀⠘⣧⣤⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣿⣿⡟⠀⣴⡆⠀⠀⠀⢀⡞⠛⢻⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠈⠓⠀⠀⣶⠗⠀⠀⠈⢙⠃⠀⠈⠀⢀⣽⣿⡿
⠀⠀⠀⠀⠀⢉⣹⣿⣧⠀⠉⠀⢰⡆⠀⠀⠑⠆⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠠⡦⠀⠀⠀⠉⠀⡀⠀⠀⢘⣿⣿⣇
⠀⠀⠀⠀⠴⠈⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠙⠃⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠇⠀⠀⣸⣿⡿⠛
⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡄⠀⢀⡀⠈⠀⡀⠀⠀⠀⠛⠛⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⢀⣄⠀⠀⠀⠀⠀⣠⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧⠈⠀⠈⢁⡀⠈⠁⠀⠀⠀⠀⠀⠐⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣿⣧⡐⠃
⠀⠀⠀⠀⠀⠀⠂⢰⣿⣿⣷⡀⠀⠈⠁⠀⠀⢀⣠⡀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⢸⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⢾⣉⣿⣿⡀⠀⠶⢀⠀⠀⢸⡋⠹⡆⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠶⡀⠀⠸⠷⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⡇⠀⠀⠀⠀⠀⠈⢱⠞⠁⣷⠂⠀⠀⠀⠀⠀⠀⡶⠀⠸⠟⠀⠃⠐⠟⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣧⠀⠰⠿⠂⠀⠘⢦⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣶⠀⢀⣼⣿⡏⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣶⠀⠀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠀⠀⢀⣤⣄⠀⠀⠛⠇⢀⡀⢠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⣿⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣿⣈⡿⠃⠀⠂⠀⠈⢀⣜⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⣿⣆⣀⠀⠀⠀⢠⡟⣣⠀⠀⠀⠀⣤⠀⠀⠘⠛⠃⠀⠀⠀⠀⢀⣸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⢿⣿⣶⡄⢀⠘⠿⠟⠀⠀⠀⣴⣦⡄⣤⣴⠚⠀⠀⠀⢀⣀⣼⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⣾⣿⣿⡄⠀⠀⣀⠀⠀⠙⠛⠀⡈⠁⢶⠄⣀⣠⡿⢿⡟⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠘⢋⣿⣿⣿⣆⣀⣿⡦⠘⠃⠀⠈⠛⠀⣀⣴⣿⣿⠶⠟⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠹⢿⠿⣿⣿⣿⣾⣿⣷⣾⣿⣿⡿⢻⡿⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⢸⠛⠛⠛⠙⣟⠉⠛⠋⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠚


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢁⣶⠴⣶⣿⠉⠀⠉⠉⠉⠉⣉⠉⢁⡈⠉⠉⣀⡉⠉⠉⠉⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣐⠃⢀⣾⣿⣷⣶⣦⡈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡘⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣷⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢣⣤⣦⣾⣿⣿⣿⡿⣧⣬⡻⣿⡟⣿⣿⣿⣿⣿⣷⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⢳⣾⣿⣿⣿⣿⣉⣹⣄⠻⠿⠁⢀⣶⣶⡽⠟⢿⣿⣿⣿⣆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠉⠻⠟⠋⠁⠀⠀⠐⠊⠙⠋⠀⠀⠉⠙⣿⣿⣿⣷⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠂⣨⣿⣿⡿⠛⠙⠁⠀⠀⠀⢀⣠⣶⣆⠀⢀⣄⢀⡄⠀⢺⣿⠉⣿⢙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣤⣼⡟⠛⣿⡷⠀⠀⠸⠿⠀⠐⠿⢧⣤⡿⠀⠀⠁⠀⠁⠀⠀⠀⣠⠜⢿⣿⣏⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣬⡟⠛⠟⠃⠀⠃⠀⠀⢀⡀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣴⣦⡀⠀⢀⣤⡼⠿⣿⣿⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣯⠁⠀⠀⠀⠲⠆⠀⠀⠈⠀⠀⠺⠇⠀⠀⠀⠀⠀⢸⣿⣭⡇⢠⣿⠛⢿⣾⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⠀⢸⡷⠀⠀⣤⡄⠈⠉⠁⠀⠁⠄⢀⣼⢿⣿⣿⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⣽⣿⠀⠀⠀⢠⣄⠀⠈⣿⣿⣿⣿⡁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠺⢷⣀⠀⠀⣀⣌⠃⠀⡇⠀⠉⠁⠀⠀⠀⠿⠟⠀⠀⢈⡟⠻⣿⣇⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⣿⣻⣿⠁⠀⠀⠸⠷⠀⠀⠀⠀⠀⠸⣯⡀⠐⠿⠟⢠⣾⣷⡶⢤⣤⠃⠀⠀⠀⠀⠀⠀⠈⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣠⣿⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠘⢷⣤⡀⠀⠀⠉⠀⣰⣿⠁⠀⠀⠀⠀⣀⠀⠀⠀⠈⢺⣿⣿⡧⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢻⣿⣷⣄⢉⣿⡀⢼⠷⠀⠀⠀⠀⠀⠉⡀⠀⠀⠸⣿⣧⣀⠀⣀⣼⣿⣇⠀⠐⠶⠀⠀⣿⠇⠀⣈⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⣾⣿⣿⡿⢉⠿⣀⠀⢀⣠⣤⡀⠀⠀⠀⠁⠀⣠⣧⠀⢿⣿⣼⣿⣿⣹⣿⣄⠀⠀⠀⠀⠉⠀⠈⠛⠀⠸⣿⣿⣿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⡟⣣⡀⣀⡈⠀⣾⣟⢙⣷⠀⠀⠰⠆⠈⠛⠉⠀⠘⠟⢉⣿⠟⠉⠙⠻⣶⣄⠀⠀⠀⢀⣄⠀⠀⠀⣿⣿⣿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⡿⠿⠟⠉⠛⠃⠀⠈⠻⠟⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢻⡁⠀⠀⠀⠀⠈⠙⠆⠀⠀⠈⠉⣠⣶⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢺⣿⣿⡿⢧⣴⣾⡆⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠸⠇⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠙⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢶⣿⣿⣿⣧⠀⠉⠉⢴⣶⣦⡀⠈⠀⠈⠉⠀⠀⠀⠀⠀⡜⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣷⠀⠈⣿⣷⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣉⠀⠀⠀⠺⣿⠟⠀⠀⠀⣠⣶⠦⠀⠀⢀⡄⢁⡈⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⢀⣿⣀⣍⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣿⣿⣿⣿⠙⠷⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠈⠻⠿⢧⣾⡿⠀⠀⠀⡀⠀⠀⠀⠀⣶⠿⢷⠄⠀⠀⠀⠀⠈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣷⢿⣷⠆⠈⠁⠀⠀⠀⢀⣤⡾⠿⠟⠳⠶⠀⣠⣄⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠁⠤⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠸⠿⣿⡛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⣀⣠⠄⠀⠈⠹⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣿⣿⡦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⠻⠇⠉⠠⠶⠄⠀⠸⠗⠀⠀⠀⣿⡇⠈⠛⠃⠠⠄⠀⠀⠠⠾⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣠⡈⢻⣿⡀
⠀⠀⠀⠀⠀⠙⠛⣿⣟⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⣠⠄⠀⢶⡾⠙⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠶⢶⣿⡿⠃⠀⠀⠛⠃⠀⣴⣶⠄⠀⠀⠰⡶⠀⠙⢿⣄⠀⠀⠀⠈⠣⡀⢠⡄⠠⠔⠳⢾⣇⡀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠈⠛⢹⣿⣿⠶⠦
⠀⠀⠀⠀⠀⠠⣤⣿⣷⡄⠀⠀⠀⠀⠀⠻⣿⡆⠀⣤⠀⠀⠀⠀⠀⠙⣷⣄⡀⠃⠰⠗⠀⠀⠀⣴⡆⠀⢿⣧⠀⠀⠸⣿⣁⣁⠀⠀⣤⠀⠀⢰⣿⣼⣿⡤⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⣠⡄⠀⢴⣾⠉⢠⣄⠈⢻⡇⠀⠀⠘⠛⠃⠀⠀⠋⠀⠀⢸⣿⣿⣿⠄
⠀⠀⠀⠀⠀⣈⣿⡿⠁⢀⣴⡟⣦⡀⣴⡄⠀⠀⠺⠆⠀⠀⠀⠀⠈⠃⠀⠀⠀⠿⢿⣤⣄⣀⠉⠁⠘⠋⢰⣿⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⣽⣿⣧⡀⠈
⠀⠀⠈⠛⠛⣿⣿⣧⠀⠘⠛⠁⠛⠀⠘⠁⠀⠀⣀⣤⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⠻⢿⣷⣶⣿⣶⠛⠁⠀⠀⠀⠀⢠⣦⠀⠀⠆⠀⢤⢘⣿⣿⠿⠈
⠀⠀⠀⠀⢘⣿⣿⡿⠀⠀⠀⠀⠰⠀⠀⠀⠀⢰⡟⠋⢛⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠁⠀⢀⡄⠀⢤⣼⣿⣿⡄⠰⠦
⠀⠀⠈⠛⢛⣿⣿⡇⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠻⣷⡟⠃⠀⠀⠀⠀⠘⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡷⠀⠀⠀⣶⣤⠀⠀⠀⠀⢻⣿⣿⠧⣤⡄
⠀⠰⠶⠾⠿⣿⣿⣿⡄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⡀⠀⠀⠀⢠⡄⠘⠛⠁⠀⠀⢰⣿⡄⠀⠀⠀⠀⣼⣿⣿⣤⣄
⠀⠀⢠⣤⣼⣿⣿⣿⠄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠰⣤⡀⣿⣿⣿⠀⠀⠀⠀⠀⢸⡈⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢀⡀⠀⠀⠀⠈⢻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⣿⣿⣷⡀
⠀⣤⡄⠛⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⣼⡇⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⢠⡾⠿⣦⠀⠀⠀⠀⢿⣻⣿⡁
⠀⣤⣄⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡀⠀⠀⠀⠀⢰⡏⢁⡀⠀⠀⠀⣠⡀⠉⠀⠀⠀⠀⠘⢷⣤⡄⠀⠀⠀⢃⣼⣿⣿⠋⠁
⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠘⢣⡄⠘⠳⠚⢀⡀⠈⠃⣤⡔⠿⠟⠀⠀⣤⠀⠘⠃⠀⡀⠀⠀⠀⠀⢀⠸⣿⣿⡿⠳⠶
⠀⠙⠀⢾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠘⠋⠙⠂⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣷⠀⠙⢰⣿⣿⡿
⠀⠀⠒⠈⣿⡇⠀⠀⠀⠀⢠⣾⠇⠻⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠂⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢶⣿⣿⣿⠃
⠀⢀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠻⣤⣤⠀⠀⠀⠀⠀⠀⠰⠿⠛⠻⢦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⠆
⠀⠙⠛⠿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢸⡇⢸⡿⠿⠗⢠⡖⠒⠀⠒⠋⠁⢠⣦⠀⠀⣀⣠⣄⠀⠀⠀⢀⣿⣿⣿⡀
⠀⠰⠾⢿⣿⡿⠁⣠⡀⠀⣀⠀⠀⠀⠐⠛⠀⠀⠈⠀⠀⠀⠀⠀⣠⣤⠘⠓⠠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠉⢩⡄⠀⠀⢈⣿⣹⣿⠋⠃
⠀⡀⠾⢿⣿⣿⠀⠉⠀⠀⠙⠀⠀⠀⠀⣠⣤⡀⠀⠀⢠⡀⠀⠸⠟⠁⠀⠀⠀⠀⢿⣧⠀⠀⡼⠷⡀⠀⠀⢀⣤⡄⠀⠀⢹⣶⠀⢸⣿⣿⠟
⠀⠀⠰⣿⣿⣿⠃⣠⣾⡆⠀⠀⠀⠀⣾⡟⢻⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠠⡾⠛⠀⠀⠘⣍⠀⠀⠀⠀⠀⢀⣶⣿⣿⠃⠒
⠀⠐⠂⣼⣿⣷⡀⠈⠁⠀⣴⠄⠀⠀⠙⠳⠞⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⣤⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⣹⣿⡿⠦
⠀⠰⠒⢩⣿⣿⡁⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠿⠇⠀⠸⠿⠀⠀⠀⠀⠈⠁⠀⢀⣀⠁⠰⡿⠀⠀⠀⠀⣹⣿
⠀⠀⠶⢿⣿⣿⣿⡆⠀⠀⠈⠀⠺⠂⠀⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠒⢈⣽⣿⣏⠁⢶⠀⠘⠓⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠻⠆⠀⠆⠀⠀⠀⣶⡟⠛⢁
⠀⠀⠐⠈⢉⣿⡿⣷⡀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⡄
⠀⠀⠀⠀⣀⠿⣿⣿⡷⠀⠀⢀⠴⠀⠀⠀⣰⡶⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣀⡀⠉⠁⢀⣄⠀⠀⠀⢹⣿⣿⣁
⠀⠀⠀⠀⠉⠲⣾⣿⣧⡀⠀⠙⢦⠀⠀⠘⢇⡀⢈⣁⡀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡿⠻⡦⠀⠘⠟⠀⢀⣤⣾⡿⠉
⠀⠀⠀⠀⠀⠰⣍⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⢻⠋⠉⣿⡆⠀⠀⠈⠀⠀⠀⠰⠶⠀⠰⠿⠃⠀⠁⠘⠃⠀⠀⠀⠀⢺⣿⣟⣡⠶⠂
⠀⠀⠀⠀⠀⠤⠈⣿⣿⣿⡆⠀⠀⢿⠆⠀⠀⠻⠦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⢀⣠⣿⣿⠟⠁⣀
⠀⠀⠀⠀⠀⠀⠉⢀⠾⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠘⠂⠀⠀⣀⣀⡀⠀⠀⠾⠂⢀⡀⠀⣼⣿⣿⠧⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠴⣿⣿⣟⢦⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡆⠀⠀⠀⢸⣏⢹⡟⠀⡴⠄⠀⠘⠃⣰⡿⢿⣋⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠛⢀⠸⢿⣿⡄⠀⠉⠁⠀⠀⢠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⢾⣿⠻⠏⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⣷⣾⡀⠀⠀⠀⢿⣧⣼⠃⠀⠀⠀⣀⡀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⣴⣿⣽⠦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠛⣿⡋⢱⣄⠀⠀⠙⠃⠀⠀⠀⢸⣿⣿⠞⠻⠶⢄⡀⠀⢠⣴⣶⣾⣿⡟⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⣿⣿⣿⡀⠰⠀⣠⡆⠀⡀⠙⠁⢠⣄⠀⠘⢋⣠⣶⣿⡉⠻⠏⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠛⠁⠹⣿⣿⣿⣦⡘⣿⣄⠘⢛⡀⠀⠈⠁⣤⣶⣾⣿⣿⡟⠻⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠛⢛⡹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡟⢻⡉⠰⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠁⢸⠋⠻⠛⠛⠟⠋⠛⠃⠈⠋⠀⡜⠁

⣶⣶⣶⣦⣤⣤⣶⣤⣶⣶⣶⣤⣶⣦⣴⣶⣶⣦⣴⣶⣦⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣴⣦⣤⣴⣶⣶⣶⣆⡀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢛⣒⠲⢤⣄
⠀⠀⡏⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣧⢿⣿⣿⣶⣌⠳⣦⡀⠀⢠⠂⠀⠀⡦⠆
⠀⠀⠀⠀⠘⠅⠂⠀⠀⠀⠀⠁⠂⠄⠀⠀⢹⡜⢿⣿⣿⣿⣷⡈⢳⡄⠀⡠⠂⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠈⠑⢄⠹⣄⠻⣿⣿⣿⣿⣦⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠳⢦⣉⡙⠛⠛⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠐
⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠃⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀⠀⣀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣦⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣯⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡽⢯⣤⡀⠀⠀⠀⠈⠠⣅⠀⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⡙⠺⣰⡴⣧⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⡐⣈⣀⠷⠉⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠀⠀⠀⠀⠈⠿⠟⠯⠙⠂
⠀⠀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢶⠀⠠
⠀⠀⠰⠀⠀⠁⡀⠡⠀⠀⢩⠁⠀⠀⠀⠀⠀⠀⠀⠈⢹⢶⠄
⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⡀⣀⡠⣸⣷⠈⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⣹⣿⣦⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠠⠁
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡀⣀⣀⢻⡜⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠉⠙⣿⠉⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⡟⢲⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠛⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣠⢠⠆⣴⢣⠎⣠⠖⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⣼⣸⡥⠷⠚⠋⠉⠉⠉⠉⠙⡆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡆⢇⡵⠊⠁⠀⠀⣾⣷⠄⠀⣶⠀⢠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣇⡷⠋⣠⠖⢒⢦⠀⠀⠁⠀⠀⠀⠀⡼⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢣⠎⢠⠎⣰⣾⡿⣻⠀⠀⣾⣿⠆⠀⣰⢃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣷⠋⠀⠹⣤⣨⠭⠚⠁⢀⡀⠀⠀⠀⣰⣃⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⢀⡷⠁⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⡇⠀⡴⢡⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠀⣃⠼⠋⣠⡄⠀⠀⢀⡠⠤⠤⠞⣠⠴⠚⣡⢞⣹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢇⡸⠟⣁⡀⠿⠋⠀⠀⠀⡏⠀⠀⣠⠞⢁⡠⠾⠥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣳⠎⠀⣼⣿⣿⠠⠶⠶⠂⠀⠙⠦⢔⡡⠖⠫⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠁⢠⡶⠈⡛⠁⣤⡀⠀⠀⢀⣠⣞⣙⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠛⠁⢸⡇⠀⠈⠛⢀⣴⣋⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢴⡷⠀⠘⠃⠀⠀⣠⢫⠤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠈⠀⣀⡤⠊⣭⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠭⢍⡙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡿⠿⠿⠿⠿⢿⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⢸⡟⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣾⡇⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⢀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠸⠿⠿⠿⠿⠿⠿⠟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡷⠿⢶⣦⣄⡀⠀⠀⢀⣼⡇⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠙⠻⠿⠿⠟⢿⡇⠀⠀⠀⠈⠻⠿⢶⣶⠶⠶⠶⠶⢶⣶⠾⠿⠋
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢿⣶⠶⠶⢶⣾⠟
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢰⣿
⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣶⣶⣶⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⣀⠠⠠⠄⠠⠄⠠⢀
⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⢀⠀⠈⠓⢀
⠀⠀⠈⠁⠠⠈⠀⣤⡄⠀⠀⠀⠀⠀⠀⢄⠙⢄⡀
⠀⠈⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠻⠃⠀⠀⠁⢀⠉⢄⡀
⠀⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠑⠄
⠀⢰⠀⠐⠀⡔⢠⣦⢱⠀⢀⠃⠀⠀⠀⠀⠠⣄⡆⠀⠠⡈⢁
⠀⠀⢆⠀⠀⠈⠲⠴⠚⠀⠀⢀⣀⣀⡀⠀⠈⠚⠃⠀⠀⠐⠀⠢
⠀⠀⠈⠤⠀⡄⠀⠀⠀⠀⠀⠙⠛⠿⣿⣷⣄⠀⠀⠀⣠⡀⠀⠄⠓⡀
⠀⠀⠀⠀⠡⡈⠄⣠⣄⠀⠀⠀⠀⠸⢿⡿⠋⠀⠀⠀⠉⠁⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠉⣆⠉⠁⠀⠀⢀⡶⡆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣇⠀⠀⢀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠐⠀⠀⠀⠀⠀⠤⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⡀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠛⠃⠀⠀⠀⠠⢀⡀⠀⢀⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢴⡆⣱⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡐⠀⠀⠀⢠⠶⡆⠀⠀⠀⠀⠀⠠⠑⠶⠖⠋⠀⠀⠀⠀⠀⠀⠄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠈⠀⠀⠑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠈⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⠾⢤⡼⠇⠀⠘⠬⠟⠀⠀⠀⠀⠀⠀⠀⠀⢄⡽⡆⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⡀⢶⠆⠀⠀⠯⣟⡷⢣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠡⠀⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⢸⢸⣣⣯⣿⠀⠀⠀⠀⠀⠀⢠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠠⡇⣷⢿⡵⣾⣡⠐⢦⡀⠀⠉⠁⠀⠀⠀⣄⣠⠀⢀⠀⠀⣴⡆⠠⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠀⠘⡆⣿⢮⣿⡿⠹⡀⣳⢻⡇⠐⠀⠀⠀⠀⡀⠺⠟⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠙⣌⢯⡙⢥⡼⣟⡿⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠛⠃⠀⠀⠀⠘⠀⠆⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⡄⠀⢠⣠⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄⠀⢀⡠⠀⠀⠀⠀⠤⢀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠈⠀⠘⠛⠃⠀⠀⠀⠀⡀⠤⢤⡀⠀⣠⠀⠐⠀⠈⠀⠁⠀⠀⠰⠶⠀⠈⡀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠁⠀⠀⠀⠀⠀⠐⡀⠿⢂⡇⠰⠏⠀⠀⠀⡀⠀⠈⠀⠀⠒⠀⠀⠀⢁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⢀⣠⡀⠀⠹⠀⠀⠀⠀⠀⢠⣤⣤⠈⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣄⠀⠁⠠⠀⠀⠀⣤⡄⠀⠀⠙⠁⠀⠂⠀⠀⣀⠀⠀⠘⠛⠋⠀⠁⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠄⡀⠀⠐⠀⠈⠀⠀⢀⢀⡀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢐⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠢⠀⠸⡛⠇⠀⠀⡀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢄⡀⠀⠠⠀⠀⠀⠁⠈⠀⠀⡌⢴⡦⢹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠖⠋⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡈⠐⠄⡀⠀⠀⠀⣠⡀⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄⡀⠁⠐⠀⠙⠑⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⢀⣀⡀⣀⡠⠎

⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⣴⣿⡿⠛⠉⠉⠉⣛⠻⢿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⡟⠀⣴⣦⣄⡸⣿⠇⠀⠙⣿⣿⣿⣦
⠀⠀⠀⢸⣿⡇⠀⠘⠿⣿⣿⣦⣀⠀⢀⣿⣿⣿⣿⣇
⠀⠀⠀⠘⣿⣷⡀⢰⣿⡆⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠿⣿⣦⣉⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⠟








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣽⣿⣽⣽⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠄⠏⠋⠰⡨⠲⠈⠸⠉⠪⠨⠅


⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠒⠒⠂⠤⢄⡀
⠀⠀⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠐⠀⠐⠤⢀
⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⠈⣑⣶⣄⣀⣀⠀⠀⠀⢀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⡀⡄⡀⢀⡀⡀⣀
⠀⠀⠀⠀⠃⠙⠨⠇⠅⠃⠭⠉⠉⠑⠇⠑⠅⠀⠈⠨⠸⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠣⠏⠠⡇⠃⠣⠁⠀⠛⠤
⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⣀⡀⣀⡀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣘⣤⣘⣬⣜⣤⣏⣤⣟⣤⣇⣇⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣿⣿⣽⣿

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣶⣿⣿⣶⣷⣾⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣭⡉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣤⣄⠀⢼⣿⣿⠀⣿⣿⡇⠀⣤⣄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⠆⠘⠿⢃⣀⡙⠟⠁⢸⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠻⠟⠀⢀⣠⣿⣿⣷⣀⠀⠈⠿⠟⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⠿⠿⠿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠛⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠉⠛⠿⣷⣄⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⠿⠁⠀⠀⠀⣼⣿⠟⠛⢿⣷⡄⠀⠐⠁⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⢿⣦⣤⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣰⠚⠑⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠁⠀⠀⢿⡀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠒⠀⠀⠀⠂⠐⠀⠀⠀⠒⠐⠀⠂⠂⠀⠀⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⢀⢀⢀⣀⡀⣀⣀⡀⠀⢀⠀⣀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⡠⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠶⡖⠢⢤⣀
⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⢀⡜⠒
⠀⠀⠀⠈⠒⠤⠄⣀⠀⠀⠀⠉⠉⠙⠙⠛⠛⠑⠋⠑⠂⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⡤⠤⠀⠀⠀⢈⡁⠀⢈⡙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠙⠰⠱⠞⠪⠱⠷⠀⣠⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⢦⠀⠀⠀⠀⠀⡰⠃⠀⠁⠀⠣⠄⠀⠀⡠⠤⠐⠂⠉⠑⠒⠠⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠓⢄⣀⡠⠜⠀⠀⠀⠉⠀⠀⠀⠤⠀⠀⠀⢰⣶⣶⠀⠀⢀⠇⢀⣠⠞⣋⡉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣀⡀⠀⠀⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠭⠁⠒⠒⠁⢠⡞⠁⠘⠿⠓⢀⡀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠰⡏⡌⣹⠇⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⢠⡟⠀⠊⠀⠀⢀⡀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠯⠃⠀⠀⠈⠀⣸⡁
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠉⠁⡠⠔⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠀⠀⠀⠀⠾⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⢀⡠⢖⡟⠀⠀⠀⢸⡇⠀⠀⣠⢤⡀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⡼⠀⠀⠀⠀⣼⠀⢀⡞⡴⣰⢛⡀⠄⡾⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⢀⡶⠖⠂⠢⠶⠀⣀⠤⠒⠉⠀⠐⠋⢁⡞⠁⠀⠀⢀⡾⠁⠀⢸⠋⠁⡇⠻⠃⢸⡃⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣴⣶⡎⠁⣠⠏⠀⠀⠀⠀⣿⠀⡀⠀⠘⢬⡿⠃⠀⢠⢋⡟⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠁⠀⠀⠀⠀⠀⢀⡤⠊⠡⠞⠃⠐⠀⠛⠻⣣⠞⠁⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⣧⠎⢄⡟
⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⣀⠤⠒⣡⠄⠄⢠⡄⠀⣠⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡄⢏⠇⣠⣤⠀⣾⣳⢄⡞⠁
⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⣀⠴⠊⠁⡀⠀⠉⠀⠀⠀⠀⣀⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⢿⠛⡾⠋⠁⠀⠉⣡⠞
⠀⡰⠋⠀⠀⠀⠀⢀⡠⡚⢱⣶⠶⠳⢦⠀⠐⠀⠚⡡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⠟⡿⡆⢀⡀⠀⣰⠏
⢰⠃⠀⠀⠀⠀⠀⠺⠤⢔⣻⣧⣤⠔⠋⠁⢈⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠙⠃⠿⢯⡾⠃
⠘⠦⣀⣀⣀⣠⠤⠖⠊⠁⠀⢏⣀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠐⠁⠒⠀⠒⠁⠐⠓⠀⠀⠀⠁⠀⠙⢧⣄⣠⡴⠋

⠀⢀⠄⡄⠀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⢐⡶⠄⠀⠀⠀⠀⠀⠀⠀⢠⠦⠄⡀
⠀⠺⢉⡆⠀⠙⡄⢣⠀⠀⠀⠀⠀⠀⠀⠰⣑⡀⠈⠃⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⡀⠖⠰⠀⡀⠀⠀⠀⠀⠀⠀⠘⠾⣄⠈⢱⡄⡀
⠀⠸⡄⠸⢄⠀⣇⠞⠀⠀⠀⠀⠀⠀⠀⠀⠡⣰⢄⠀⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⡙⠣⠀⡴⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠀⠑
⠀⠀⠙⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢡⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠒⣀⣆⠠⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢒⢦⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣐⣢⣼⠄⡀⣀
⠀⠀⠊⣼⣶⡬⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⣿⡄⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠤⢳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣷⣿⣧⣀
⠀⠙⡱⢈⢫⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⢿⣿⣏⠟⣏⠁
⠀⠋⠀⠀⠈⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋



⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⠀⠀⣜⠉⣳⠀⠀⠀⠀⠀⠀⢸⣇⠐⠛⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⡀⠸⡇
⠀⣀⣤⡍⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠲⠀⠱⡀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡘⠧⢄⡀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠹⣆⠹⢦⣄
⢉⢜⠝⠓⠫⡉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡐⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⢹⡄
⠀⠎⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠨⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⡰⠊⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⡿⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⣐⣾⣴⡤⠘⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣉⡿
⠀⠀⠠⠋⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢾⡟⣿⡷⡅⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄⠻⢿⣿⠃⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⢴⢻⣻⣿⡿⠁
⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢐⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⣤⣄⡀⠀⠀⠀⠀⣀⣤⡴⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⣡⣤⣶⣶⣶⣦⣌⠛⢦⣀⣠⠞⢋⣤⣴⣶⣶⣦⣤⡙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⣾⠟⠋⠉⠀⠉⠙⢿⣷⡄⠻⠃⣴⣿⣿⣿⡿⠛⠛⢿⣿⣦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⢀⣴⣶⣿⣶⡄⢻⣿⡄⠀⠹⠿⠿⠋⠀⠀⠀⠀⢹⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣷⣤⣾⠟⠁⠀⣽⡿⠀⢻⣿⣄⠀⣠⣴⣶⠿⢶⣦⣄⢰⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢉⣥⣤⣤⣼⣿⠁⠀⠀⠙⣿⣿⣟⣁⠀⠀⠀⢈⣿⣿⡿⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢰⣿⠉⠀⠈⣿⡟⠿⣶⣤⣴⡿⠛⠻⠿⠿⣿⣿⠿⣿⡿⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠻⢶⣤⣄⡙⢿⣶⣾⡿⠻⢷⣦⣴⣚⠛⢦⣀⣴⡿⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠉⢻⡆⠀⢠⡶⠶⠶⠟⠙⠛⠛⢻⡟⠉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⣷⣀⣸⡇⠀⢾⣉⠙⠓⠖⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡉⠩⣖⠒⠲⠼⠇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⣠⠤⣄⠀⠀⠀⠐⢤⣀⠀⠀⠀⢀⣤⠴⠶⣤⠀⠀⢰⡖⠒⠒⠒⠀⠀⠀⣠⠶⢦⠀⠀⢀⣀⠀⠀⢀⡤⠖⠒⠒⠲⣦⠀⠀⣠⡴⠶⠶⠶⣤
⠀⠀⣰⡟⠀⠀⠤⠶⢦⡈⠛⠶⠚⠉⠀⠀⠀⠈⠳⣄⡈⠳⠄⣠⡶⠋⠁⣼⠁⢀⡏⢠⡶⠋⠁⠀⠀⠉⢠⠞⠉⢀⣴⠟⠀⡞⠁⠀⡀⠀⢀⣿⠁
⠀⣰⡏⠀⢀⠀⠀⣠⡾⠁⠀⣠⡴⠛⠉⠻⣆⠀⢀⠀⡀⠀⢀⡟⠀⠀⣰⠏⠀⠀⠁⢿⣀⠀⠀⠀⢀⣠⠏⠀⠀⢸⡇⠀⠀⠀⢰⠉⠁⢰⡋⠁
⠀⠻⠤⠔⠁⢀⡾⠋⠀⢀⡴⠋⠀⠀⠀⡀⢻⡀⠸⡄⢹⠀⠈⡇⠀⠰⠋⠐⢦⠀⠀⡀⠉⠙⠛⠛⠋⠁⠐⣆⠀⠀⠙⠳⢤⣤⠞⠀⣀⣀⣉⠁
⢀⣴⠶⢦⡀⢸⡇⠀⠐⠋⠀⠀⠀⠀⣰⠃⠈⠳⠴⠃⢸⡆⠀⠃⠀⠀⠀⣀⡞⢠⡞⠁⠀⡠⠚⠉⠙⠲⡄⠙⠳⠦⣤⣤⡀⠀⠀⠈⠁⠀⠈⣧
⢸⡇⠀⠀⠇⠀⠙⠳⠶⠶⠶⠶⠚⠋⣡⡴⢶⡄⠀⠀⠀⠻⢦⡤⠖⠛⢉⣁⠀⢸⡁⢀⡞⠀⠀⠀⠀⠀⢹⡄⠀⠀⡄⠀⣿⠀⠀⠀⡄⠀⣸⠇
⠈⣷⡀⠀⢀⣤⠶⠶⣤⠀⠠⠴⠶⠛⠁⣴⠟⢀⠤⠖⠒⣦⠀⢀⡴⠛⠉⢉⣷⠀⠙⠋⠀⠀⣰⠋⠙⣦⠀⠹⢦⣠⠇⠀⢻⡄⢀⡴⠁⢰⡏
⠀⠈⠻⣦⡈⠁⠀⢀⡿⠀⢀⣤⡀⠀⢸⠃⠀⠀⠀⢀⡾⢋⡴⠋⠀⢀⡴⠟⠁⢀⣠⣤⣀⠀⢻⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⠀⠙⣆
⠀⠀⠀⠈⢧⠀⢠⡿⠁⡴⠋⠈⣷⠀⠈⠓⠉⠀⢠⡟⠁⢸⡁⠀⠀⢾⡀⠀⠀⣀⡀⠀⢹⡇⠘⡇⠀⠀⠙⠲⠂⠀⠀⢰⠀⠀⢨⡇⠀⡀⠀⠏
⠀⡤⢦⡄⠀⠀⣿⠁⠘⠁⠀⠀⠹⣆⣀⡴⠀⠀⣿⠀⠀⣀⡙⠲⡄⠀⠉⠉⠉⣁⣠⡴⠟⠁⢀⡇⠀⢠⡆⠀⠀⣀⠴⠃⢀⣴⠋⠀⢸⣁⡀
⠀⠀⣸⠇⠀⠀⣿⠀⠀⠀⢸⠀⠀⣀⣀⠀⠀⠀⢻⡀⠋⢀⡿⠀⢀⠄⢀⡴⠛⠉⠀⠀⠀⠒⠋⢠⡾⠋⠀⢰⡏⠁⠀⠀⢸⡃⠀⠀⠀⠉⠙⣷
⠀⣰⡏⠀⠀⠀⠙⠷⣤⣤⠞⠀⡞⠁⢹⡆⠀⢤⡀⠀⠀⣾⠁⠀⡏⠀⢸⡀⠀⠀⠈⠙⢲⡄⠀⣿⠀⠀⠀⠀⠙⣶⠀⠀⠈⠙⠓⢲⡄⠀⣠⡿
⢠⡿⠀⠀⡄⠀⣀⣤⣤⣤⣀⠀⠀⠀⢻⡄⠀⢀⡇⠀⠰⡇⠀⠀⢸⠀⢀⣉⡙⠓⢦⠀⠀⣿⠀⠙⠶⣤⣤⡤⠞⢁⣠⡴⠞⠀⠀⣰⠃⣴⠏
⠘⣧⣀⡰⠃⣼⠋⠀⠀⠀⢩⡷⠀⠀⠈⠳⠶⠋⠀⣦⠀⠙⠦⠤⠊⡰⠋⠀⠈⡆⠀⠀⢀⡿⠀⠀⡄⠀⠀⣠⡾⠛⠁⠀⠀⠀⠘⠁⢸⡏⠀⢀⡖
⠀⠈⠉⠀⠀⡏⠀⠀⣠⠞⠋⠀⠀⠉⠛⢶⡄⠀⠀⢻⣄⠀⠀⠀⡴⠁⠀⣠⣤⣤⠤⠶⠋⠀⣀⡼⠃⠀⠀⢿⣤⣀⡀⠀⠀⢀⣀⡀⠸⣇⣠⠞
⠀⣴⠒⠂⠀⠃⠀⠘⠧⠤⠴⠒⠀⠀⢀⣰⡟⠀⠀⣀⡙⠳⠶⠚⠁⠀⣞⠁⠀⠀⠀⣤⠖⠛⠉⠀⣠⠶⢶⡄⠀⢈⡟⠀⣰⠏⠉⡷⠀⠈⣁
⠀⣿⣀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠛⠋⠁⠀⢠⡟⠁⠙⡆⣠⡾⢻⡇⠈⠙⠂⠀⢸⡇⠀⠀⠀⡼⠁⠀⣼⠇⢰⠋⠀⠀⡟⠀⢠⠇⠀⠀⠈⣧
⠀⠈⠙⢷⣄⠀⢠⡞⠉⠀⠀⣠⡤⢦⡄⠀⠀⣾⠀⠀⢀⡴⠋⢀⡼⠃⠀⠀⢠⠤⠞⠀⠀⠀⣼⠁⠀⣼⠏⠀⠈⠀⢀⣼⠇⠀⠘⠃⠀⢀⣴⠏
⠀⢀⡤⠄⠙⣷⠀⠓⠀⠀⠉⢀⠀⣼⠃⠀⢰⡟⠀⠞⠋⠀⣠⠟⠁⢀⡴⣦⠀⠀⢀⡶⠛⠛⠁⠀⢸⠏⠀⣴⠟⠛⠋⠁⣀⣠⣤⡶⠾⠛⣁⣄⡀
⠀⣾⠀⠀⠀⣿⠀⣆⠀⠀⢀⡎⠀⢿⣀⣠⡾⠁⠀⠀⠀⢰⠏⠀⠀⡼⠀⠸⣧⠀⠚⠀⠀⠀⡄⠀⠸⠀⣸⠇⠀⠀⢰⠟⠉⠁⠀⠀⠀⠘⠁⢀⡗
⠀⠈⠓⠖⠚⠁⠀⠙⠷⠶⠛⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⠖⠋⠐⠃⠀⠀⠈⠙⠛⠂⠀⠀⠈⠲⠶⠖⠋⠀⠀⠀⠘⠂⠀⠐⠋⠙⠳⠶⠶⠟⠁

⠀⢀⠀⠀⠀⣀⢀⢀⠀⣀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠈⠐
⠀⠀⠀⠈⠈⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡄⢠⣤⠐⠂⠀⠤⢀
⠀⠀⠀⠀⠈⠀⢠⠖⠉⡍⠲⢄⡀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠇⠈⣜⢨⠄
⠀⠀⠀⠀⢀⠖⠁⠀⠀⣃⣄⢀⢙⣦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⢾⡼⠑⡀⢀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠉⢹⠀⠀⠇⠀⠀⠀⠀⠀⠀⢈⡈⠁⠀⠀⢈⠄⡀⣈⠜
⠀⠀⠀⠀⠡⡀⠀⠀⠀⢸⢀⣰⣶⠎⠀⠀⠈⢦⣀⠀⠀⠀⢀⣀⣳⣴⠂⠠⠼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠢⢤⣀⣼⣿⠿⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢄⣀⡀⠀⠠⡀⢀⣀⣶⡀⢀⣐⣶⡄⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡙⠂⠀⠁⠈⠀⠨⡁⡠⠴⠺⢉⠉⢀⡄⠀⠀⣀⡀⠈⠉⣉⠃⠀⡤⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠂⠆⠀⡠⠾⠡⠴⠀⠠⣀⣈⣉⣑⠒⠂⠉⠑⠊⠉⠉⠉⠉⠛⠉⠉
⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠀⢀⣠⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠏⠀⠀⠙⠣⡂⠀⠀⠀⠀⠉⠀⠈⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⣠⠤⠜⠑⠛⠣⠤⣉⡅⠀⠒⠃⠤⣀⣠⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⠀⠂⠀⠀⣀⠀⠀⠀⠤⢬⠄⠀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠣⠴⠴⠛⠀⠀⣀⠀⢀⡰⠋
⠀⠀⢀⡐⠂⠉⠉⠉⠀⠉⠉⠉⠉⠋⠥⡀⠀⠀⠀⠀⠢⠀⠀⠈⠀⡀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠁⠀⠂
⠀⠀⠘⠢⡀⠀⠀⠤⠤⠒⠋⠉⠈⠐⠁⠀⠁⠀⠀⠀⢀⠃⡤⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠈⠂⠀⢈⠄
⠀⠀⢀⠀⠁⠀⢀⠀⢀⡠⠐⠒⠐⡆⠞⠐⣀⠀⠁⠀⠀⠀⡌⢫⣀⡉⠁⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠐⡲⠋⠉⠀⠀⠒⠆⡀⠑⠦⠴⠃⠠⠀⣀⣀⠀⠀⡙⠀⠀⠀⠀⠀⠈⠒⢆⠀⠀⠀⠀⡜
⠀⠀⠀⠆⠉⠁⠀⠀⠀⠀⠀⠀⡌⠉⠢⡌⠢⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠆⠰⡄⠀⠘⣇⡀⠀⠀⡇⠀⠀⢀⡄⡰
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠃⡠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠲⠄⡿⠁⠀⠀⠀⠀⠀⠀⢺⠁⢰⠂⣀
⠀⠀⠀⢠⢀⠀⡈⢀⡰⠶⠌⣄⠀⠐⢄⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠡⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠠⠶⣿⠀⠀⢓⠉⠙⢢⡀
⠀⠀⠀⠄⠀⠀⡀⠘⣣⠄⠐⠃⡄⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⡀⠀⠀⠉⢣⠀⠀⠀⠔⠰⢛⡛⠃⢠⣶⠐⠀⣷
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠽⠠⠀⠀⠀⠀⠙⣏⠀⠀⠀⠀⠈⠀⠀⠉⠋⠲⠶⠟⠻⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠂⠀⠀⠂⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣
⠀⠀⠀⠀⠀⢠⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢣
⠀⠀⠀⠀⢠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⢠⣏⡠⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⢸⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⣈⣧
⠀⠀⠀⠀⠶⡈⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣼⢸⣤⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⡰⠆
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠈⢑⡄⣄⣀⣀⣀⣀⣀⣀⣀⠀⢀⠞⠉⠉⠉⡏⠉⠉⠉⠳⡄⠀⢀⣀⣀⣀⣀⣀⣀⣠⡀⠔⠉⠀⢀⡴⠊
⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡼⡸⠀⠀⠀⠀⠀⠀⢀⠏⣠⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⣆⠈⣇⠀⠀⠀⠀⠀⠀⠸⡸⣀⡴⠋⠀⠀⠀⠀⠀⠀⢀
⠘⣶⡀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠒⠂⠘⠛⠛⠛⠋⣰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣆⠈⠛⠛⠛⠛⠂⠒⠚⠃⠉⠀⠀⠀⠀⠀⠀⠀⣰⡞
⠀⠘⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⢁⡏⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢹⡀⢹⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡜
⠀⠀⠘⣆⠈⢢⢀⣀⣀⣀⣀⣀⣀⢠⠖⠉⢀⡎⢸⡇⠀⠀⠀⠀⡰⠋⠈⠢⣄⠀⠀⠀⠀⢸⡇⠈⣆⠈⠑⢦⢀⣀⣀⣀⣀⣀⡀⢠⠊⢀⠞
⠀⠀⠀⠈⢦⡄⡌⠀⠀⠀⠀⠀⢸⢸⣠⠔⠋⣤⠘⡇⠀⠀⣠⠊⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⠃⡷⣌⠓⢤⣸⢸⠀⠀⠀⠀⠀⠘⡄⣠⠏
⠀⠀⠀⠀⠈⠁⠓⠒⠒⠒⠒⠒⠋⠈⠁⠀⡼⠙⣆⠹⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠏⡼⠁⠹⣆⠀⠉⠈⠓⠒⠒⠒⠒⠒⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢛⠃⣼⠁⠀⠘⣦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢃⠾⡅⠀⠀⠹⡄⢛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠒⠉⠁⠀⢀⡼⢠⠇⠀⠀⢠⠇⠙⠦⣌⡑⠒⠢⠴⠶⠂⢋⣡⠖⠉⠀⢱⠀⠀⠀⢳⠸⡀⠀⠀⠉⠑⢢⣄
⠀⠀⠀⠀⢀⠔⠹⡼⣄⡤⠖⠊⠉⠀⡾⠀⠀⠀⡜⠀⠀⠀⢀⡏⠙⠒⢲⠒⠉⠁⠸⡄⠀⠀⠀⡇⠀⠀⠘⡇⠈⠉⠒⠦⣄⡼⡼⠑⢄
⠀⠀⢀⡴⠁⠀⡰⠃⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠇⠀⠀⠀⣼⠀⠀⠀⢸⡆⠀⠀⠀⢷⠀⠀⠀⢸⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠳⡀⠀⠳⣄
⠀⢠⣟⠤⣠⠞⠀⠀⠀⠀⠀⢀⡄⢸⠁⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠸⡄⠀⠀⠀⡇⠀⠀⢸⡀⣄⠀⠀⠀⠀⠀⠘⢦⡠⢼⣧
⠀⢸⠀⢑⡆⠀⠀⠀⠀⣠⠔⠋⡇⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡇⡏⠑⢄⡀⠀⠀⠀⠀⣖⠁⣸
⠀⠈⠀⡼⠀⠀⠀⡠⠞⠁⠀⢀⡇⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢈⡁⡇⠀⠀⠙⠦⡀⠀⠀⠸⡀⣿
⠀⠀⣤⠃⠀⢠⣜⠦⣀⢀⠴⠋⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠃⠈⠳⣄⢀⡠⣞⣤⠀⠀⢣⡆
⠀⠀⠏⠀⠀⢸⠀⠉⢪⠁⠀⠀⠀⠘⡇⠀⠀⠈⡆⠀⠀⠀⢻⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⠀⢹⠚⠁⢸⠀⠀⠈⠇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢳⠀⠀⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⡇⠀⠀⠀⢠⠃⠀⠀⢰⠁⠀⢠⠇⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠀⠘⡆⠀⠀⠀⢇⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⡞⠀⢠⠏⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⢀⣸⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢹⡀⠀⠀⠘⡄⠀⠀⡇⠀⠀⢀⡏⠀⠀⣰⠁⣰⠋⠀⠀⠀⠀⠀⠀⢸⣀⡀⢸
⠀⠀⠀⠀⠀⠨⢭⠠⠰⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣣⠀⠀⠀⢣⠀⠀⣧⠀⠀⢸⠁⠀⣠⡷⠚⠁⠀⠀⠀⠀⠀⠀⢀⡴⠀⢨⠭
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠼⠧⠤⠿⠤⠴⠟⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠚⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿