Parasitism Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⠏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠇⢿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢛⠵⠒⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣥⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣤⣴⣶⡶⠶⣤⣤⣾⠿⠿⠛⠛⠛⠛⠻⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⣿
⠈⢿⣿⣿⣿⢾⣿⣷⣀⠤⠔⠒⠒⠒⠀⢄⡀⠙⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣿
⠀⠈⢻⣿⣯⣿⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠈⠙⢿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⢀⣠⢶⣿⣿⣿⡟
⠀⠀⠀⢻⣿⣣⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⢭⣛⣛⣛⣛⣛⡯⠝⠛⠁
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⡀⠠⠚⠉⠁⠀⠀⠀⠀⠁⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠐⠢⡀
⠀⠀⠀⠀⢀⠤⠖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠀⠀⠂⠀⣀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠤⠤⠶⠠⠤⠄⠔⠤⠅⣀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠊⠀⠁⠀⠒⠲⠤⣀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠜⠁⠀⠀⠀⣠⡀⠀⠀⢀⠀⠀⠉⠓⢄⠀⣠⠞⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣀⠎⠀⠉⠉⠉⠉⠉⠓⠤⣄⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⢸⡔⠁⠀⠀⠢⡄⢀⠀⠀⠀⠠⢄⠀⠀⠙⣆⠀⠈⢆⠀⠀⡀⢅⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⡀⠀⠀⣀⠄⠚⠉⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠱⣄⣀⠀⠂⠀⠀⠀⠀⣻⢀⠀⠈⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⡄⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠈⠁⠒⠓⠐⠖⠚⠡⡀⠈⠀⣨⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠘⡎⠀⠀⠀⠀⢀⠀⠀⠁⢂⠐⠀⠀⠈⠉⢒⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡼⠃⡆⢂⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠄⡴⠋⠀⠀⠈⠉⠑⠒⠦⠤⠤⠤⠄⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢀⠀⠀⠀⠀⠂⠰⠀⠀⠀⠂⠠⠆⠐⠀⠀⠀⠀⠀⠈⠀⠠⠀⠈⢇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢃⠀⡠⠠⠄⠀⠁⠀⠠⠈⠀⠐⠐⠤⠉⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⡆⠀⠀⢀⠂
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢥⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠤⠒⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠢⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠤⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠠⠄⠠⠤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠋⠉⠲⠄⠀⡇⠀⢀⣀⠀⠀⠀⠁⡀⠀⠀⢀⣂⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⢀⠤⠤⢌⠠⡊⠤⢼⡄⠀⠀⠀⠈⠑⠰⣼⠻⣤⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥
⠀⢀⠄⠂⠦⢄⣈⠳⢤⣤⡛⡀⠀⠀⠁⠀⠀⡀⠤⠒⡂⣿⣀⣿⣗⠒⠠⠀⡀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣌⡀
⠀⡅⠀⠀⠒⠁⠚⠉⢹⠀⠁⠀⠃⠀⠀⠀⠀⣀⠔⠊⡅⠈⠉⢁⠧⠑⠢⣀⠀⠀⠀⠀⠒⠉⣞⡀⢀⣴⢠⠣⣣⠈⠂
⠀⠀⠀⠋⠀⠀⠀⠀⢧⠉⠢⠤⠀⠀⠀⠀⠈⠀⠀⢐⠀⠐⠒⠋⠀⡇⠀⠀⠁⠀⠀⠀⠀⠋⠀⠠⠋⢠⠜⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣧⡀⠱⣷⠰⠆⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡙⠛⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣴⡄⢰⡄⠀⠀⠀⠀⠀⠀⢸⣿⣽⣾⣿⣷⣶⣶⣿⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⠋⣡⣈⠁⣤⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠋⠁⣸⣿⣿⣄⠉⠀⠀⠀⠀⠀⠀⢀⡀⣾⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠻⠿⠃⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣟⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⣀⣠⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣽⣿⣤⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠶⣾⣶⣶⠷⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣶⠀⣀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⡉⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠘⠿⣿⡿⠟⠸⣿⡾⣿⡦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡼⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠋⠉⠙⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠠⣄⣀⢀⣀⡀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢰⣶⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣤⠙⣿⣧⣄⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠘⣿⢇⣶⣿⡆
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣷⡌⠀⠈⠉⢻⣷⠀⠀⠀⠀⠀⣀⡈⠀⣴⣶⣾⣷⣯⣛⠋⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡷⠀⣧⠀⠈⠻⠀⠀⠀⠀⠈⠛⢁⡴⠋⢫⠟⢟⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠇⠀⢈⠇⠈⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋



⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⢠⣾⣿⣧
⠀⠀⠀⠀⠀⢰⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⡿⠀⠀⠛⠿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢰⣶⠀⣡⣶⣌⠁⣿⡇⢀⡰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠯⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⠛⣠⣿⣿⣿⡄⣼⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠋⠛⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠈⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⡀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⡌⠀⠀⠈⢄⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠎⠀⠀⠰⠁⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⣀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠠⠁⠄⠀⠀⠈⢄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⡐⢰⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡇⠡⠆⣀⡀⠀⠀⠈⡄⠀⠀⠀⢰⠀⠀⠀⠀⢀⡅⠀⠈⠠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⠠⡀⢨⠀⠀⠀⠆⠀⡠⠀⠐⠃⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠇⠀⢠⠞⠁⠈⠀⠉⠉⠉⠉⠉⠑⠄⠀⢳⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠎⠀⠀⠀⠀⠘⡄⠀⠀⠢⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢄⡀⠀⠀⠀⠑⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠃⠀⠀⠀⠀⠱⡀⠀⠀⠈⠢⢀⡀⣦⡆⢠⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠔⣰⠏⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⢄⣇⡀⠀⠀⠋⠁⠀⠦⢤⡂⠄⠀⠀⠀⠀⠀⠀⠠⡀⠈⢫⠀⢀
⠀⠀⠀⠀⢀⡞⠁⠀⡀⠂⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⠂⠠⠤⠤⠐⡇⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡘⢆
⠀⠀⠀⠔⣣⠀⡒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠄⠀⠀⠀⠀⠀⠘⡒⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠊⢡
⠀⣠⠊⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠂
⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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





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

⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠒
⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠉⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠠⢄
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠁⠒⠠⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠈⠳⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠈⠑⠒⠤⠄⣀⡀⣉⣢⣀⣀⣀⣀⡃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠹⡀⠀⠈⡟⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣼⡖⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⣠⡤⠶⠾⠛⠛⠛⢛⣿⣶⣤⣼⡶⠛⠁⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠁⠀⠈⠀⠀⣠⠟⢋⣧⡼⢿⠀⠒⣉⣥⣄
⠀⠀⠀⠀⠀⢀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠇⠁⠀⠀⡄⠀⣠⣴⣿⣿⣶⣾⣇⠴⣾⢀⡾⠃⠀⠉⢳⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠉⠀⣰⣿⠏⠛⠁⡁⠬⠈⡻⣯⡾⠃⠀⠀⠀⠀⢣
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡆⢫⡔⢥⣦⡀⠉⣻⣅⡀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢀⡱⠀⠀⠀⠻⣿⠶⣈⠀⠘⠿⠁⢈⡿⠉⠛⠳⣤⣤⣴⠃
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠀⠀⠀⠈⠀⠀⠤⡀⠀⣽⣿⢥⣠⣤⣄⡤⠞⠁⠀⢳⡖⡈⠐⣿
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⢂⡀⠀⠀⠀⠄⠃⢰⡟⢻⢱⡟⠉⠀⢀⡀⠀⠀⠋⠉⠁⢠⡇
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡿⠔⠓⣾⠁⠀⠀⠘⠣⠕⠀⠠⢮⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠄⠀⠀⢀⡞⢙⡀⢠⡏⠀⠀⠀⣀⡀⠐⢀⡤⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⡚⠤⠃⠀⠀⠉⠀⢀⡾⠡⠈⠁⣼⠁⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢠⡀⠀⠀⣾⠁⣀⠄⢀⡟⠀⠀⠀⠀⠀⠀⠀⠠⣼⠇⠀⡟
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣸⡓⠣⠀⢀⣾⠃⣀⠀⠀⠀⠐⣦⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢀⡿⠀⠀⠂⢠⡟⠀⠣⠞⠀⠀⠙⠋⠀⠀⠀⢀⡟
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡇⠀⠈⠢⣹⠇⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⠀⠠⠀⣾⠀⠀⠐⠳⡄⠀⠵⠂⠀⣠⠞⠁
⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣿⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠂⠀⠀⢸⠃⠀⢈⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⠃⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣻⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢸⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣁⡀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠲⣧⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠘⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣿⣛⣛⣛⣛⣃⣙⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣭⣭⣯⣭⣭⣿⣯⣭⣭⣭⣭⣩⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⣠⣤⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢛⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡳⣆⣿⠀⠀⠀⣀⣌⡀⠄⠸⢿⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠴⠊⠉⠁⠀⠀⠀⢠⣿⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⢠⠤⢶⠶⣆⠠⠄⠠⢤⣴⡁⠄⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠠⠂⢰⡞⡿⠟⠻⡀⠀⢠⠞⠀⠀⠀⠀⢳⣶⣍⣷
⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠈⠶⡏⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠘⣆⠻
⠀⠀⠀⠀⠀⢰⠞⣷⣴⠏⠀⠀⠠⠿
⠀⠀⠀⠀⠠⡷⢰⣮⡛⠀⠊⠁⠂
⠀⠀⠀⠀⠀⠁⢈⡟⠁⠀⠀⠿
⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⢀⠐
⠀⠀⠀⠀⠸⣷⣮⣿⣻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⣆⠀⠀⠀⠀⠀⠀⠘⢀⡔⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠲⢼⣦⣤⣀⣀⣠⣴⣷⡜⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⢧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠾⢫⠉⢉⡉⡉⠡⠄⠀⠂⠀⠀⠀⠠⠊⣀⠠⠤⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢧⡀⠀⠀⠀⠀⠀⠀⢠⢘⣷⡀⠀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⣃⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⢤⠀⠀⠀⠀⠀⠀⠈⢛⡿⠿⠟⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠳⡀⠀⠀⠀⣀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢹⡄⠀⢺⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢧⡀⠀⠀⠀⢅⡀⠻⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠆⠹⢾⣧⣶⣿⡁⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠿⣿⠃⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⢻⡀⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢸⠀⠐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢰⣽⣯⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⡼⠛⠀⢲⣦⠀⢸⣄⠉⣿⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠈⠉⠁⠀⠀⠈⠉⠙⣿⠁⠀⠀⠀⠙⣿⣦⠻⣆⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⣿⣼⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠃
⠤⠤⠄⠀⣀⠴⠾⠟⠄⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⣢⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡄
⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣴⣶⣦⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⣶⡀
⠀⠀⠀⠀⢀⠤⢲⠉⠽⠍⢵⡒⡄⠀⠀⠀⠐⣼⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢦⣵⣶⣾⡴⢛⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⡿⠁
⠀⠀⠀⡰⢩⠊⠁⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠘⣆⠘⢆⡀⠀⠀⠀⠀⠀⠀⣤⠭⣿⣿⡿⣟⣿⠭⠴⠀⠀⠀⠀⠀⠀⡠⣿⣟⡿⠁
⠀⠀⠐⠥⠃⠀⠀⢀⠔⡥⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠉⣷⣀⠀⠀⠀⠀⠀⢒⡹⢿⣷⣿⢿⣑⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉
⠀⠀⠀⠀⠀⠀⣆⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠕⠆⠀⠀⠀⠀⠀⠙⠁⠃⣼⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢠⠔⣉⠉⢉⡙⠢⡀⠀⠀⠀⠀⠀⠀⠀⣱⠟⣻⡇⠉⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢄⡀
⠀⠀⢠⠃⣠⠘⠂⠋⠀⢦⢹⠀⠀⠀⠀⠀⠀⣴⢋⡶⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢊⣥⡶⢂⠇
⠀⠀⠸⡄⢁⣠⠈⠉⢁⡀⢸⠀⠀⠀⠀⠀⠀⣇⠎⢰⢀⣘⠂⡄⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠁⠀⠀⠀⠀⠀⡰⢊⡵⣞⠽⠒⠉⠁
⠀⠀⠀⠑⢤⡀⠰⠃⣚⡴⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡗⠠⠀⠀⠀⠀⠀⠀⠈⠒⠿⠁⠀⠀⠀⠀⠀⠀⠳⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠹⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠰⣤⠞⠋⠉⠙⠺⡅⠀⠀⠀⠀⢠⣿⣿⣧⠀⡠⠔⠊⠙⠆⠀⠀⠀⠀⣾⡿⠘⢛⡛⣦⡀⠀⠀⠀⠀⣠⡀⣰⣶⣾⣶⣦⠄⠒
⠀⠀⠀⢸⠃⠀⠀⢀⡀⠆⣿⠉⠀⠀⠀⠀⣿⣿⣿⠎⠀⢀⡀⢤⠇⠀⠀⠀⢰⣿⠋⣾⣿⣿⢈⣏⠀⠀⠀⠀⠉⢹⣿⣿⢛⣿⢿⡆⣀⡀
⠀⠀⠀⠘⣆⠀⠀⠀⠰⢢⡏⠀⠀⠀⠀⠀⠙⠛⡇⠀⠀⠀⡠⠋⠀⠀⠀⠀⠈⠛⣾⣮⣭⣽⣿⠿⠀⠀⠀⠀⣤⠐⠲⣿⣿⣿⠞⠁⠈⠁
⠀⠀⠀⠀⠋⠙⢲⠖⠺⠁⠉⠀⠀⠀⠀⠀⠀⠀⠣⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⢩⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⠦⢄⡀
⠀⠀⠀⠀⢀⣵⣦⣀⣤⣤⣠⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⢼⢋⣡⢤⠞⠀⡤⠷⣦⠀⠀⠀⠀⢠⢎⣀⠃⠨⡁⠠⠝⣆
⠀⠀⠀⢀⣿⣶⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⠾⣿⣿⣧⣤⠀⠀⠀⠀⠈⠉⠀⠀⣠⠎⠀⡠⠏⠀⠀⠀⠀⢸⠀⠠⡀⠖⠀⠀⠄⢸
⠀⠀⠀⠐⠻⣽⣿⡿⣿⣿⠿⡀⠀⠀⠀⠀⠀⠼⣷⠼⠿⣛⡿⠂⠀⠀⠀⠀⠀⠀⢴⠁⢀⠾⠁⠀⠀⠀⠀⠀⠘⣄⠃⠀⠃⣀⠰⢀⠏
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠴⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢱⣿⣿⣿⣿⣿⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢶⡀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠋⠛⢶⠂⡀⠀⠙⠿⣿⣿⠁⣀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⢤⣾⣿⣿⣼⣿⠁⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⡷⠀⠰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⡿⠋⠈⣿⣃⠀⣸⣿⣆
⠀⠀⠀⠀⠀⢐⣶⣄⠀⠙⠟⠋⠀⣠⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⢀⣾⣿⢲⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⢹⣿⣷⣄⠀⢀⡴⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠗⠛⠻⡿⣷⣄⢀
⠀⠀⠀⠀⠀⣀⣿⠋⠀⠉⠉⣼⡄⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡼⠟⠉⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⢰⣿⢋⣿⠀⠀⠀⣸⡟⠀⠀⢀⣸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠄⠐⠒⠶⣛⡷
⠀⠀⠀⠀⢻⣿⣿⣀⣀⣴⣟⣀⣀⣀⣀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣲⣤⣤⣄⠀⠀⠀⣺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠛⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠴⠶⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢠⠞⠁⠠⣾⠇⣀⡈⢦⠀⠀⠀⠀⠀⠀⠀⣠⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⢝⡧⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣿⣿⠁
⠀⠀⢸⠘⠛⡓⠀⠀⠙⠗⢸⠄⠀⠀⠀⠀⠀⠲⣿⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠈⠉⠉⣿⣿
⠀⠀⠘⣆⠸⠿⠀⠺⠿⢆⡞⠀⠀⠀⠀⠀⠒⠷⣶⣦⣼⠇⠀⠀⠀⠀⠀⠀⢀⠜⠛⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⠶⠂⠀⠿⠿⠁
⠀⠀⠀⠈⠑⠶⠤⠴⠖⠉⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⢀⣼⣷⣏⢹⣶⣧⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠙⠋
⠀⠀⠀⠻⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠁⠀⢇⢡⡌⣱⠇⠈⠀⠀⠀⠀⠀⢸⣿⢿⡿
⠀⠀⠀⠀⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⢀⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣶⣣⡤⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⢀
⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣯⢞⡁⠀⠀⠀⠀⣄⣼⣿⣿⡤⣜⣿⣿⣇⡄
⠀⠀⠘⢿⣿⣿⣯⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⢹⣿⣿⡟⠁⠈⠉⠙⣿⠃
⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡟⣻⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⡀⠀⠀⠀⠀⠀⣴⡶⠶⡶⠶⠶⢶⣦
⠀⠀⠀⠀⠈⠙⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠟⠁⠀⠀⠀⠀⠀⣿⣀⣂⣐⡒⠂⠸⠛
⠀⠀⠀⠿⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠍⠁⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⡶⠂⠀⠀⠀⠀⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠝⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢰⣷⣾⣿⣿⠇⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⡀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠋⠽⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣶⠊⠀⡀⢠⣤⣤⣤⣤⣤⣤⣤⡤⠿⠆⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠙⠟⠁⠾⠃⠚⠿⢿⣿⣿⣿⣿⣿⣥⠠⠀⠀⠀⠀⠀⡀⠘⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣦⣤⣤⣤⣴⡌⣉⡋⠻⢷⣿⣿⠀⠀⠀⢀⡀⠰⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣾⣿⣶⡾⠁⢙⣿⣿⣿⣿⣻⡶⠞⢻⠋⠀⣠⣿⠀⡟⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠠⠿⢻⡤⠞⣹⣿⠿⠿⠃⠠⠁⠀⠀⢀⣸⠽⠁⢠⠀⡄⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣷⣦⣀⠀⠀⠜⠁⠰⠋⢹⠀⠀⠀⠀⠀⠀⡿⠟⠁⣠⣴⡁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣚⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⠀⢀⠀⢋⡟⠀⣧⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠀⣴⠋⠀⠀⠸⡄⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢴⡆⣠⣷⠀⠀⠀⠀⢧⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠋⠉⠀⠀⠀⠀⠸⡄⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠁⠀⠀⠁⠀⠀⠀⠀⢣⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡿⠛⠓⠀⠀⠀⠀⠈⠉⠙⠛⠛⠂⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠿⠛⢩⠤⠆⠀⠉⠁⣤⣶⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⢄⠀⠀⢷
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠠⠰⠆⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⣿⣷⣶⠀⠄⢀⡀⠀⠀⠀⠀⠀⡂⠀⠘⣇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣴⣾⣿⣿⣿⡄⠀⠀⡐⠁⠀⠀⠀⠔⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⡠⠊⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⣿⣧⡀⠀⠰⠿⠟⠃⠀⢀⣾⣿⣿⣿⣿⣿⢯⡄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡆⠀⠀⠀⠀⡶⠿⠿⠛⠉⢙⠟⠋⠀⢳⡄⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠠⢠⡔⠈⠁⠀⠀⠀⠈⢷⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢶⡀⠈⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠙⠦⣄⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⢐⡄⠁⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠰⠉⠛⠿⠷⠂⠀⠀⠀⠰⠂⠀⠠⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠴⢤⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣠⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠀⡖⠒⠂⢲⡄⢰⡆⣶⠒⠒⢠⡖⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⣟⣉⡁⠈⣧⡾⠀⣿⣉⣉⢸⡟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠁⠀⠈⠁⠀⠈⠉⠉⠀⠁⠀⠉

⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠶⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲
⢸
⢸
⢸
⢸
⢸
⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢸⠀⠸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡰⠻⣥⣤⡤⠶⠃⡂⠀⠀⠀⠀⠀⠀⠈⢳⣞⡋⠁
⢸⠀⠀⢻⠿⠀⠀⢀⡀⠀⠀⠁⠀⢠⣠⣴⠿⠿⠿⠇⢀⢔⠒⠂⠀⡐⠚⠉⠀⠀⠀⠑⡀⠀⠘⠻⢦⣤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⢈⣸
⢸⠀⠀⠀⣶⣶⣾⣿⣿⣦⠀⠀⠀⠀⠀⢿⢷⣄⣀⣴⠟⠘⠁⠀⠀⠙⠂⢠⡀⠀⠀⠀⢣⡀⢀⣴⠾⠛⠂⣷⣶⣄⠀⠀⠀⠀⠀⢀⣿⠉⠈⣼⣿
⢸⣤⣴⡞⠁⠀⢻⣿⣿⡛⠁⡀⠀⠀⠀⢀⣀⡈⠉⣀⣤⡄⠀⠀⠀⢀⣴⣿⣿⣿⡛⠳⠤⣤⣾⣿⡂⢀⣸⣿⣷⡿⢦⠀⢷⣶⡶⠿⢹⣶⣴⡿⢿
⢸⡿⠋⢣⣀⣤⣾⣿⣎⠛⠀⡇⠀⠀⠀⠈⠉⠉⢛⣛⣛⠒⠀⠀⣠⣾⣿⡽⣿⣿⣿⡀⠺⢿⣿⣶⢿⡟⣏⡿⢻⡆⠀⠀⠈⠙⣿⣾⣿⣿⣿⣗⢸
⢸⡇⠀⣼⣿⣿⣿⣿⣿⣆⠐⣧⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⢰⡸⢿⣿⡿⠀⣿⣿⣿⣿⠀⠈⢩⡿⠻⣿⣿⣿⣼⠃⠀⠀⠀⢸⡟⢿⣟⣿⣿⣿⣻
⢸⢀⣼⣿⣿⣿⣿⣿⣀⣿⣸⣿⠀⠀⠀⢠⣄⠘⢿⣿⡏⠉⠹⣧⣿⣿⣷⣶⣿⣿⣿⣯⡄⠀⢾⣿⠀⣿⣻⣿⣿⠀⠀⢠⣤⣿⣅⣠⣿⣿⣿⡿⢿
⢸⡾⢿⢹⣿⣿⣿⡟⠙⠛⠋⣿⡆⠀⠀⣤⣿⣶⣾⣧⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣸⣿⡗⢿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⣯⢸⡇⢸
⢸⡇⠀⢸⣿⣿⣿⠃⠀⠀⠀⠸⣧⣴⡧⣿⣿⣿⣿⡟⠉⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⢸⣾⣿⠛⣀⠀⣀⣈⠉⠛⣻⣿⢿⣿⠇⢸
⢸⣷⠀⣿⣿⣿⣿⡄⠀⢀⣴⠛⠿⣷⣦⣿⣿⣿⡿⠀⢀⣾⣿⣿⣿⣿⡟⠋⠉⡿⠃⠈⣿⡿⠿⣿⣿⣸⣿⣿⣶⣿⣿⢻⣯⢀⣴⡿⢋⣼⣿
⢸⠁⠀⣿⣿⠛⢿⣿⣦⡿⠃⠀⠀⢨⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣹⡇⠀⠀⠀⠀⠀⣾⣷⣄⣀⣭⣦⣿⠋⠻⣿⣿⣷⡟⢸⣿⠀⠚⣻⡟
⢸⠀⢠⣿⡏⠀⠀⠙⢿⣿⣦⠀⠀⢸⣿⢹⣿⡏⠀⠀⣼⣿⠁⠙⠿⣿⣷⣄⡀⠀⠀⠀⠁⠀⠉⢹⣿⡿⣿⠀⠀⠹⣿⣿⣿⣶⣿⡆⠀⠘⣿⣄
⢸⠀⢸⣿⡇⠀⠀⠀⣰⣿⣿⠀⠀⠸⣿⣜⣿⠀⠀⢀⣿⣿⠀⠀⢀⣸⣿⣿⡿⠀⠀⠀⠀⠀⢴⣿⡿⣿⣏⣠⡀⠀⢹⣿⢿⣿⣿⣷⣀⡀⠈⠿⣶
⢸⠀⣼⡿⠁⠀⠀⠞⢿⣿⣧⡀⠀⣴⣮⣿⣿⣦⣄⣸⣿⡇⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⠈⡋⢿⣿⣿⣿⣷⡀⢸⣿⠘⣿⣿⣿⣏⢻⣷⢀⣿
⢸⣾⡟⠀⠀⠀⠀⠀⣿⣟⢿⣷⣼⣿⡟⠛⠛⠛⣿⡿⠛⠃⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣷⣼⣋⣤⡿⢿⡿⣿⡼⣏⠈⠉
⢸⡿⠁⠀⠀⠀⠀⣼⣿⣏⠀⠙⠛⢹⣇⠀⠀⣼⡿⠁⠀⠀⠀⢠⣾⣿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣬⡾⣏⠁⢀⣾⣷⣿⣿⣷⡇
⢸⠁⠀⠀⠀⠀⠀⠈⠛⢿⣄⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠘⣿⣿⠖⣦⠋⣠⣿⡿⢋⣉⠁⣿⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣮⢷⣴⣻⣿⣿⣿⡿⢷⣿⣿
⠸⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠼⠿⠿⠷⠦⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠷⠦⠤⠤⠤⠶⠿⠿⠿⠿⠿⠷⠶⠾⠿⠷⠶⠿⠤⠤⠿⠿⠴

⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣻⡽⠉⠁⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣻⣿⣿⡿⣿⡵⢖⠛⢛⡉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣋⣴⣾⣿⣟⣽⣞⡁⠀⣩⠾⠛⠹⣤⢖⡀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣉⣴⢾⠿⠗⢗⣩⣿⣿⣿⣾⣷⣴⣿⡴⣃⣀⣸⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⣠⣾⣿⣿⣋⣠⣼⣿⠋⠚⣽⣷⣿⣿⠿⣫⣷⠆⣀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⡿⣿⣿⠚⠟⠛⢉⣴⣩⣿⣿⣧⣾⣿⣿⣷⡿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣧⣀⣤⢏⣯⣶⣿⣿⡿⣿⣿⣿⡟⠁⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣵⠟⠻⠛⣉⡷⠞⣹⣿⣯⣄⣀⣤⠀⠃⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⡠⠞⠁⢰⣿⠟⠁⠀⠈⣻⡍⠤⠤⠔⠀⠚⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢦⣀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣴⠒⠋⠉⢹⠄⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠁⠒⣾⠉⠉⠉⠠⡀⠈⢣⡀⠀⢸⣤⣼⡿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣿⢳⡀⠀⠀⠀⠀⠀⢿⡀⠈⢏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣼⠇⠀⠻⣦⠀⠀⠀⠠⠈⢻⣄⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠉⠉⠉⠈⠛⠁⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⢰⡿⠀⠀⠀⠙⣷⣀⠀⠀⠐⠀⠹⣆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⡟⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⢈⣧⡴⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠋⠁⠀⠀⠀⠀⣠⣶⣶⣤⣄⠀⣠⡿⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⣰⠟⠋⠳⣌⠁
⠀⠀⠀⠀⠄⠀⠤⠀⠤⠀⣀⠀⠀⠀⠀⠀⡠⠊⣿⣿⣿⠟⠉⣷⡟⠁⠀⢀⣴⣿⣿⣷⣤⠀⠠⠤⠒⠒⠛⠛⠒⠛⠿⣿⠀⠀⠀⠀⠁⠢⠀⢠⢒⡆⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⢶⡄⠀⠀⡀⢧⠀⢀⣨⣿⣟⡛⠊⠙⡿⠛⠛⠋⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣀⣀⣄⣀⠀⢀⣹⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣏⠀⠀⠸⡆⠀⠀⢀⣴⠟⠋⠁⠉⠉⠙⢧⣴⠋⠀⠀⣰⠏⠀⠀⠀⠀⠤⠤⠤⠤⠤⠬⠭⠍⢉⣉⣙⣛⣛⡛⠛⠻⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠛⢦⡴⢸⡇⠀⠀⠀⠀⠁⠀⢀⣿⡶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣤⠶⣲⣷⣿⡇⠀⠀⠀⠀⠀⣠⢞⡿⢀⣴⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⠟⠁⣼⡿⠇⠀⢸⡾⠷⣄⠀⠤⠖⠊⣡⣾⠷⠛⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⢹⣿⣿⠁⠀⠀⠉⠀⢰⣤⣍⠁⠀⢻⣄⣠⣴⣾⣏⡀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠙⠚⠿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣿⠁⠈⠙⠋⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⣿⢻⣿⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⢶⣤⡀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⢺⡟⣠⣾⠟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢻⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢠⣦⣭⣼⣷⡬⠄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⡶⢸⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠾⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⡿⠁⠀⠀⠀⣠⡶⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠿⠛⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⠀⠀⠀⠸⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⡾⢛⣃⣀⣴⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠠⣾⣿⠷⢬⣝⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣰⡶⢸⣯⡸⣷⡌⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣿⡇⠈⠉⠁⠘⣿⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠙⠻⢶⠦⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠿⠿⠿⠛⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠉⠁⠉⠹⣿⣿⣿⣿⣦⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⣠⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠒⠂⠀⠀⠀⠀⢀⣀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠈⠂⠀⠀⠤⠊⠀⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢊⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠒⠙⡒⠠⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡇⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣀⡀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠂⣧⣤⡇⣿⣭⠐⢯⣃⠉⡟⠁⠀⣾⣹⢆⡾⡄⢸⢸⢧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠃⠀⠃⠓⠒⠈⠒⠋⠀⠃⠀⠀⠛⠀⠘⠉⠙⠘⠘⠀⠛

⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⢿⣿⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⢱⠁⣸⠤⡀⠀⠀⠀⠀⠀⢠⣶⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣠⣤⡀
⠀⠀⠀⠀⠈⠸⣯⣽⣦⠀⠀⠀⠀⠀⠀⡀⠀⢈⠁⠁⠂⢁⡀⠁⠀⠀⠀⠀⠀⠀⢍⠻⣟⣎⠂⠀⠀⠀⠀⠀⠀⣙⣿⠟⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠈⠈⠻⠑⠄⠀⠀⠀⠀⠀⠚⠏⢠⡌⡄⢄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠁⢹⣻⠀⠀⠀⠀⠀⠀⠀⠻⣿⢾⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠒⡘⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠄⡄⠀⢄⣀⠨⠀⠀⠀⠀⠀⠀⢀⣴⣴⣤⡀
⠀⠀⠀⠀⡘⡈⢁⠐⢀⠐⠘⡀⠀⠀⠈⠷⢦⡄⠻⠀⠘⢋⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠤⠐⠀⡐⠀⠄⠠⠀⠀⠀⠀⠀⣀⣴⠂⢸⡆⠈⠻⣤⠀⠀⠀⠨⠙⡏⠀⢀⠀⡠⢒⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠆⠡⠰⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠊⠃⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠘⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣭⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⡾⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠰⡤⢄⡀
⠀⠀⠀⠀⢸⣻⣿⠉⢹⡦⠀⠀⠀⠀⠀⠐⢄⠀⠘⠀⠐⠃⠀⠀⠀⠀⠀⢠⡴⣿⡟⠁⢰⡿⠀⣰⠆⠀⠀⠀⢐⡌⠀⡨⢄⠀⠳
⠀⠀⠀⠀⠀⠻⣿⣶⣿⡻⣆⠀⠀⠀⢀⠄⠀⡁⠀⠀⠀⠠⣊⡥⠀⠀⢰⠟⠁⠈⠁⢀⣬⣿⡶⠇⠀⠀⠀⠀⢨⣄⢁⠪⠐⠄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠼⣺⡯⠌⠀⠀⠀⠀⠋⠩⠙⢂⢠⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢦⠅⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠸⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢠⣀⣠⠀⠀⠀⠀⠀⠀⢲⡶⣿⣻⡷⣴⠀⠀⠀⠀⠀⠀⠐⠉⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢚⠀⢀⠀⠀⡔
⠀⠀⠀⠀⢸⣶⣿⠿⣿⣦⡄⠀⠀⠀⠒⣿⣻⡽⡾⣿⣻⡷⠀⠀⠀⠀⠀⠀⢄⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢲⠸⠢⠬⡄⢄⠈⠁
⠀⠀⠀⠀⠘⢿⣿⣀⣼⣿⠇⠀⠀⠀⠀⢹⣯⣿⣻⣯⣿⡁⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠱⣉⣠⠂⡆
⠀⠀⠀⠈⠂⠘⠛⠻⠇⡀⠀⠀⠀⠀⠀⠈⠀⠉⠯⠀⠈⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠨⠕⠀⠀⢤⢙⣮⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠈⠒⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠳⣄⠀⠀⡴⠋⠀⠀⠀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣰⠖⠓⢦⡀⢰⡇⣶⠛⡇⢸⠀⢸⠃⢸⠙⠳⠟⢹⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢋⣉⠳⡄⢰⠃⡟⢙⡆⣧⢸⡇⣨⣉⠁⢸⠀⢸⠀⢸⡀⠀⡴⠾⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣦⢸⡇⠯⠼⠃⡇⢸⠀⠉⠉⠀⣿⢸⡇⢧⡘⡆⢸⠀⢸⡄⠀⣙⠒⠛⢦⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⡞⣳⢸⠘⡇⠀⠀⢀⡇⢸⡆⠀⠀⢠⡇⠀⡇⠀⠉⠀⢼⠀⢈⡇⠸⣍⡹⠂⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⠁⢸⠀⣧⠀⠀⠈⣄⣠⠇⠀⠀⠀⠙⠚⠁⠀⠀⠀⠈⠓⠋⠀⠀⠉⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸



⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⡖⠒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠒
⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢠⠤⢄⠀⡀⢤⠤⠀⣠⠀⢠⠤⢄⢠⠀⠀⡠⠄⠀⢀⠠⠄⠠⡤⠄⡤⢄⡀⡠⠤⡀⢠⠀⡀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢸⠀⢈⠀⡇⠀⡂⢰⠄⡆⢸⠒⢣⢸⠀⠀⡗⠀⠀⠈⠑⠄⠀⡇⠀⡇⢺⠠⡇⠀⡸⢸⠻⡀⢸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣶⣾⣷⣷⣾⣾⣶⣿⣶⣶⣶⣷⣶⣶⣷⣶⣷⣶⣶⣾⣶⣷⣾⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


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


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

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


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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠉⠉⠉⠛⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠙⠻⣿⣦⡀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⡿⠃⠀⠀⠀⣠⣶⣾⣿⣶⣤⡀⠀⠀⢠⠖⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠿⠋⠀⠀⠀⠀⢰⣿⡟⠁⠀⠉⠻⣿⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⠿⣷⣶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣰⠞⠙⠂⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠀⠀⠀⢿⡄⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠻⢷⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁






⣀⣀⣀⢀⡀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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


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

⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠈⠹⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠂⠀⠀⠴⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠘⠲⠐
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⡜⢲⠊⢱⠊
⠀⠀⠀⠀⠀⠰⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠘⣿⡿⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣼⠉⢹⠋⠙⠓
⠀⠀⠀⠀⠀⠀⠀⣷⣤⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠂⠘⠋⠀⠀⠀⠀⠀⣶⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠘⠃⠀⠀⠀⠀⠐⣿⠗⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⠉⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢠⠀⠀⠀⠀⠻⠂⠀⠠⣿⡷⠀⠀⠙⠛⠃⠀⠀⢼⣿⠀⠀⠀⢀⠀⣿⣿⡿⠿⠿⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠛⠀⠀⣾⣿⣿⠀⠐⠉⠀⠁⠀⢹⡋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣆⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠀⠈⠉⠃⠅⠀⠤⠤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠒⢸⣿⡇⢸⣿⡇⠐⣿⡗⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣄⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⣿⡇⢸⠀⠀⠐⠀⠀⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣬⣭⣤⣤⣭⣥⣤⣭⣥⣼⠀⠀⠀⢠⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣤⣤⣬⣭⣭⣭⣥⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣲⡆
⠀⠀⠀⠀⠀⡤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢏⡿⣿⣿⣿⣁
⠀⠀⠀⠀⠐⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⠀⠀⠀⠀⢙⠈⢿⡿⣣⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣦⡀⠀⠀⠈⡟⢻⣿⣿⢽⣿⢿⣿⣿⣿
⠀⠀⠀⠀⡰⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠻⠟⣿⠙⣾⣿⠘⢿⡟⠛
⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⢸⣦⣀⠀⠀⠀⠀⢿⡆⠀⠀⣦⢡⣤⣿⣿⣉⠺⣿⢴
⠀⠀⢰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣠⡀⣿⠟⠂⠀⠀⠀⠈⣿⡄⠀⢻⣿⢿⣿⣿⣿⣿⢿⣼
⠀⠀⣘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣾⣿⣿⣿⣦⡀⠀⠘⣿⡀⠈⠏⠈⠻⣿⣿⣿⠿⠋
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠸⣷⠀⠀⠀⠀⠈⠿⠟
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠏⣿⣿⣿⣿⣿⣿⡄⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⢻⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢿⡇⢿⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⢿⠄⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠘⣤⠀⢼⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⢾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠲⣦⣄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠐⣧⣸⡆⠀⠀⠀⠀⠀⠀⠀⠁⠃⠘⠛⠛⠿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣄⣀⣠⡿
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉

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

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

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

⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀⣀
⠀⠀⡞⠉⡆⠀⠠⣄⠠⠈⡁⠄⠀⣀⠀⢰⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠈⣻⡉⢸⣄⠀⠀⠀⢄⣃⠤⡀⠀⠀⢴⣘⣷⣾⣿⣤⣿⣣⣤
⠀⠀⡇⠀⡇⠀⢀⢈⣶⣿⣽⣳⡎⠅⠀⠈⠠⡀⠀⠀⠀⣰⠶⣿⣿⣿⣷⣤⠀⠀⠐⣅⣸⡿⠛⢻⣇⣈⠇⠀⠉⣾⣿⡇⠇⠰⠶⣾⣿⣷⡿⣷⣾⡯⣿⣤⣤
⠀⠀⠘⠶⠁⠰⠖⠀⢿⣿⣿⡿⡓⠦⠄⣀⡴⠃⠀⠀⠀⠻⠷⠿⠿⠿⠿⠃⠀⠀⠐⣃⣸⢿⣶⡾⢏⣈⠆⠐⠒⣿⣿⡿⢩⠀⣶⠿⣷⡟⣿⣽⣿⣷⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠎⠈⠍⣆⠈⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠⠾⢄⠸⠁⠀⠈⠁⠌⠉⠄⠃⠀⠀⠐⠿⠋⣿⡙⣿⠹⠇⠉⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢴⣆⢤⣻⣱⣰⣶⠀⠀⢀⡶⠋⠉⠏⠙⠲⣄⠀⠀⣸⣿⡗⠀⠀⡴⣺⣵⣓⣫⡭⠭⠭⢽⡲⡄⠀⠀⣀⣿⣄⡀⠀⣴⠟⠉⠉⠙⣦⠀⠀⢠⣶⣷⡀
⠀⣀⡐⣾⣿⣿⣿⣷⣒⣀⢀⡏⠔⠀⢀⡀⠁⠀⢸⡆⠀⣾⣿⡇⠀⠸⠰⡇⠀⠀⡰⣪⠭⣝⢦⠹⡘⡄⠈⣹⣿⣿⠉⢸⡇⠀⠀⠀⢠⡿⠀⠀⢸⣿⣿⠂
⠀⠛⠩⢿⣿⡿⣦⣻⠭⠋⠀⣇⠀⠀⠀⠀⠀⠉⣸⠇⠀⢿⣿⡇⠀⠀⢰⠨⠀⠀⡡⡧⣀⢊⡜⢠⢇⡇⠀⣿⣿⣿⠂⠘⣧⠀⠀⠀⠈⣇⠀⠀⢸⣿⣿⡄
⠀⠀⠾⠏⠞⣽⡙⠈⠿⠀⠀⠘⠢⣄⣘⣀⣠⡴⠋⠀⠀⠸⣿⡗⠀⠀⠘⠊⠀⠀⠑⠌⣛⣓⣒⣫⠞⠀⠀⢹⣿⡿⠀⠀⠘⢧⣌⣀⣂⣹⡇⠀⠘⠿⠿⠁
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠁⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⡀⠁
⠀⠀⠀⢠⣦⣆⡆⠀⠀⠀⠒⠚⠚⣒⠒⠋⠉⠂⠀⠀⠀⠀⠀⢀⢠⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠐⠐⣆⣈⡓⠀⢀⠀⠀⠀⣠⡞⠋⣉⠛⢷⡄
⠀⠀⠘⣾⣿⣿⡴⠁⠀⠀⢰⣾⣿⣿⣦⠁⡜⠀⠀⠀⠠⣚⣭⠿⢿⣿⣿⣿⣿⡿⢯⣝⡒⠀⠀⠙⣿⠁⠀⠨⠀⣴⣿⣿⣿⣶⠚⠀⠀⠀⣿⢠⡟⢙⡷⢠⡇
⠀⠀⢀⣿⣿⣿⣷⡀⠀⠀⢺⣿⣿⣿⣿⠀⢈⡧⠀⠠⠚⠁⢠⠖⣩⠿⣿⡿⢿⡑⢦⡀⠙⠢⠀⠀⡼⠂⠀⢠⡤⠿⣿⣿⣽⠟⠲⠆⠀⢰⠇⠈⠻⠦⠴⠛
⠀⠀⠀⠹⣿⣿⠛⠈⠀⠀⠈⠻⠿⠿⠁⠀⠒⡄⠀⠀⠀⠐⠁⠘⠁⠀⠋⠃⠀⠙⠀⠑⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠙⣅⠀⠐⠀⠀⢻⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠺⣿⡿⠃⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢿⣷⡄⠀⠀⠀⠘⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⢻⣿⣧⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⢀⣀⣤⣿⣶⣶⣶⣾⣿⣤⣀⠀⣰⡟⠁⠀⠀⠀⠀⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣤⣾⡿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⢶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡶⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠾⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠷⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⡿⠛⠁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠈⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⠀⠀⠀⠀⢀⣼⠏⠀⠉⠛⣿⣿⠿⠿⠿⣿⠛⠉⠀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⢻⣿⣧⠀⠀⠀⢰⣿⡇⠀⠀⠀⢿⡄⠀⠀⠀⠙⣿⣷⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⣿⠁⠀⠀⢠⣾⣿⡆⠀⠀⠀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⣤⣴⣦⠄⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⣿⣿⢧⣶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⣠⣦⠀⣼⣿⠰⢸⢻⣿⡎⡯⡼
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣄⠉⠁⠀⣸⣿⠿⣷⡇⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠸⢿⣿⣿⣧⠀⠀⣿⡏⠀⠀⠀⠉⠉⠉⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⣠⢤⡄⠀⠀⠀⠞⠀⠀⣿⣿⣿⡆⢠⣿⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⡾⠉⢡⣼⠃⠈⢧⡀⢠⠁⠀⠀⠀⠉⠙⠝⠃⢸⡿⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⢠⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⡟⠀⠀⠀⠙⠋⠀⠀⠀⣀⡰⠒⡆⠀⣾⡇⠀⠀⣾⡄⢀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡀⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣇⢲⣬⡁⠀⢰⠿⡇⠈⠉⠛⠛⠒⠲⠶⠶
⠀⠀⠀⠀⠀⣠⣀⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡎⣿⣤⣤⣿⠀⡇
⠀⠀⠀⠀⠀⠙⠻⠾⠿⠒⠒⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠈⠛⠃⠈⠻⣿⠀⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠿⠿⠿⠟⠛⠚
⣿⡿⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⡠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠦⡀⠀⣠⠴⠶⠶⠶⢶⣶⣶⡶⠶⠶⠶⠶⠶⠦⡄
⠀⡇⠀⢻⣽⠓⠒⠒⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣀⡀⠤⢤⣖⡲⠭⠭⡽⠅⡇⠀⡇⠀⠀⡠⠖⡩⠖⠒⠚⠀⠀⠀⠀⠀⠀⡇
⠀⡇⣀⠯⠼⢄⡀⠀⠀⠀⠁⠄⡀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣯⠒⠉⠉⣿⠀⠀⠀⠈⡜⠀⡇⠀⣇⠔⢋⡴⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇
⠐⣿⣁⣀⣀⣀⡧⠀⠀⠀⣀⣀⣀⣀⣀⣀⡇⠀⡇⠀⠀⠀⣫⠔⠚⠃⣿⡀⠀⠀⠀⣿⡆⡇⠀⣧⠔⠉⠀⠀⠀⠁⠈⠀⣁⣀⣀⣀⡀⣀⡇
⠀⣿⠀⠀⠀⠀⣷⠀⠀⢸⣒⠒⠒⠂⠤⠒⡇⠀⡇⠀⠒⠉⣾⠀⠀⠀⣿⠇⠀⠀⢀⣿⠆⡇⠀⡇⠀⢀⡀⠀⢀⠠⠔⠩⣗⠒⠂⠤⠤⠖⡇
⠀⣿⠀⠀⠀⠀⡟⢠⣒⠥⡚⠁⠀⠀⠀⢀⡇⠀⡇⠀⠀⢀⡥⠒⠁⠀⣿⠀⠀⠀⠀⣿⡆⡇⠀⡇⠀⠈⠀⠀⡨⢠⣖⡭⠖⠋⠀⠀⠀⠀⡇
⠀⣿⠒⠒⠒⠒⢃⡾⣒⡪⠕⡀⠀⣀⠔⠋⡇⠀⡇⠛⠉⠁⣞⠤⠤⠤⣿⣀⣀⣀⣀⡿⠁⡇⠀⡇⠀⣆⢰⠀⠣⡸⢕⣊⠥⣂⠀⢀⡴⠊⡇
⠀⣏⠓⠒⠒⠒⠋⠉⠓⠾⠭⠒⠋⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠂⠤⠼⢁⡇⠀⣇⠀⠀⠀⠀⠀⠉⠓⢶⡭⠕⠊⠁⠀⢀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡞⠉⠉⢉⡩⠭⠭⠟⠛⠻⢭⡉⠉⠉⠙⡆⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆
⠀⡇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡇⠀⡇⢀⡴⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡜⠉⠙⣆⠀⠀⠀⠀⠀⠀⡞⠉⠙⡆⡇
⠀⡇⢀⡇⠀⡴⢄⣼⠢⣼⠳⣄⠀⠈⡆⠀⡇⠀⡇⡞⠒⠒⠚⠤⣀⡀⠀⠀⠀⢀⣠⣄⡀⡇⠀⡇⠑⠤⠴⠃⠐⣲⢦⢤⢤⠀⠳⠤⠴⠃⡇
⠀⡇⠈⣇⡞⠀⣤⠀⠀⠀⢠⠈⠙⣆⡇⠀⡇⠀⣷⠁⠂⠀⠀⠀⠀⠈⠉⠉⠛⡇⠀⠀⢱⡇⠀⡇⡴⠋⠙⣆⢈⡉⠉⠈⢈⡀⡔⠋⠑⡄⡇
⠀⡇⠸⡉⠙⠚⠋⠉⠉⠉⠉⠛⡎⢉⡕⠀⡇⠀⡏⡆⠀⠄⠀⠀⠀⠀⠀⠀⢐⡇⠀⢠⠇⡇⠀⡇⡇⠀⠀⡏⠙⠉⠉⠉⠙⠁⡇⠀⠀⡇⡇
⠀⡇⠀⠈⢲⡀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⡇⠀⡇⢱⡀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠔⠁⠀⡇⠀⡇⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇
⠀⡇⠀⠀⠀⠈⢳⠶⠤⠔⡖⠉⠀⠀⠀⠀⡇⠀⡇⠀⠉⠒⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⡇⠀⡇⢸⣤⣤⠇⠀⠀⠀⠀⠀⠀⢿⣤⣤⠇⡇
⠀⠧⠀⠠⣶⣭⣽⣤⣤⣤⣼⣭⣵⣦⣤⡴⠃⠀⠣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠃⠀⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠇

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

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


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

⠀⠀⠀⣀⠤⠤⠤⢄⠀⠀⠀⠀⢀⡤⠤⠤⢄⡀⠀⠀⠀⠀⣀⠤⠤⠤⣄⠀⠀⠀⠀⢀⠤⠤⠤⢄⡀
⠀⠀⠜⣁⠀⣀⠀⢀⠑⡄⠀⡰⠁⣴⡄⠄⠀⠈⢦⠀⢠⠎⠀⠀⠀⠀⠀⠱⡀⠀⡔⠁⠀⢠⣴⡆⠙⠆
⠀⢸⠀⡇⠀⠀⠀⣸⡂⠱⠀⡇⢘⡀⣃⢀⣶⣄⢘⠀⢸⢎⠉⠀⠀⠀⠁⡑⡇⠰⠀⠀⠀⠈⢙⠉⢁⢸
⠀⠈⢆⠑⠒⠒⠒⠚⣠⠃⠀⠱⡘⠀⠓⠀⠁⢀⡜⠀⠘⣄⠉⠀⠀⠀⠉⡰⠁⠀⢣⡀⠀⠀⠀⠃⢊⠇
⠀⠀⠀⠉⠒⠖⠒⠊⠁⠀⠀⠀⠈⠐⠂⠖⠒⠉⠀⠀⠀⠈⠑⠒⠀⠒⠊⠀⠀⠀⠀⠉⠒⠂⠖⠚⠁
⠀⠀⠀⡠⠤⠒⠢⢤⡀⠀⠀⠀⢀⠤⠐⠒⠤⣀⠀⠀⠀⢀⡠⠔⠂⠢⢄⠀⠀⠀⠀⣠⠤⠒⠢⠤⡀
⠀⢀⠎⠀⠀⢄⠢⡀⠘⡄⠀⡴⢁⣤⠀⠀⣠⡌⢣⠀⢠⠋⠀⠀⠀⣀⢤⡱⡄⠀⡞⣀⣦⣆⡀⠀⠈⢆
⠀⢸⠀⠈⠆⣠⡴⠢⠀⡸⠀⡇⢰⣷⠀⠀⣾⡆⢸⠀⢨⠀⠀⡐⠀⠑⠉⠀⡇⠰⡀⠺⠿⡟⢤⣤⡀⢸
⠀⠀⢣⡁⠁⠙⠃⠀⡢⠃⠀⠱⣜⠛⠁⠀⠛⢋⠎⠀⠈⢆⠋⠀⠀⠀⢀⡴⠁⠀⠣⡀⠀⠀⠈⠛⣡⠃
⠀⠀⠀⠈⠒⠒⠒⠋⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠈⠑⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣀⣀⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡶⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⢋⣹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠈⠙⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠉⣳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣩⡄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⢰⡖⢛⣛⣓⠶⠋⠁⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⢤⡇
⠀⠀⠀⠀⠀⠻⣌⣭⢍⡅⢠⠶⠛⠋⠉⠛⠲⢤⣀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⣀⣼⡛⠲⣼⠃
⠀⠀⠀⠀⠀⠀⠹⡆⠾⣡⠟⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠉⠙⠛⠒⠒⠚⠛⢹⠄⣃⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⣤⣤⣤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠋⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢈⣂⠀⢸⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⠀⠀⠑⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠢⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠖⢃⠀⠀⠀⣄⣀⣀⠤⠀⢸⠀⠀⠀⠀⠀⠀⡥⠊⠀⠙⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈
⠀⠀⠀⡄⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠣⠀⠀⠀⣠⠃⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠌⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⢬⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣀⠀⣠⠃⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⢆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⢡⡘⠁⠉⠃⠀⠀⠀⠀⣀⢀⡴⠉⢀⠠⠁⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢠⣷⢸⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠶⠂⠀⢒⡈⢄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠁⠀⠀⠀⠀⢀⠆⠀⠃⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠐⢂⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋⠄⠀⠀⠀⠈⠔⠢⡦⠤⠄⢀⠀⢀⡀⠀⠄⣀⣠⠤⠆⢈⣳⠔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢓⣦⣀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁

⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⣀⣛⡻⠿⠿⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠘⢫⠋⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⣿⡇⢿⣯⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⣿⡇⠸⣯⣾⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⡄⢃⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣿⣿⣻⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢹⣮⣦⣄⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠠⠖⠉⢙⣫⣷⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⣄⣿⡇⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢸
⣿⠀⣠⣥⣴⣾⣿⠿⠛⠋⠛⠿⣿⣶⣄⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸
⣿⠼⠛⢻⡉⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣾⡿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠂⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠖⢺
⣿⢷⢶⣼⠑⡒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⣉⣁⣴⣮⡿⡿⢾
⣿⣲⡩⠶⢡⢸⢹⡶⣌⠑⡲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢀⠴⠚⢁⣠⣼⣟⡽⢿⡿⣻⣷⣿⢾
⣿⢁⡥⣙⡖⠤⠍⡷⢎⣏⢽⠑⡂⠑⠢⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⣃⣤⣾⣿⣻⣿⡟⢿⣿⣾⡿⣿⢿⣿
⣿⠘⢋⡼⢸⣀⠐⢇⣬⠠⠋⢀⣄⠀⠈⠀⠁⠢⣀⠀⠀⢸⡇⠀⠀⢀⡴⢯⢿⠚⠒⠙⡛⣟⣛⣡⣿⣍⣙⡀⢞⣪⣾

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⡠
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠠⡱⡄⠀⢸⣄⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⢠⠀⡙⢻⣦⡀⠀⣄⢳⡹⣆⣨⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠛
⠀⠀⠀⠀⠀⠈⠀⠁⠐⠚⠂⠈⠁⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠀⣤⡗
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⠚⠅
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⣾⣿⡀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣿⣷⡄⠀⢠⣄
⠀⠀⠀⠀⠀⢾⣿⣶⠀⠀⢰⣾⣿⠇⠀⠀⠀⠰⠃⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠁⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⣿⡏⠀⠀⠀⠀⢰⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡸⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡟⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⣆⠀⠀⠀⠃⠀⠠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠰⡏⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠇⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⢻⠀⠀⠀⠀⠀⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠶⠶⠶⠾⠄⠀⠀⠀⠶⠶⠶⠿⠷⢦⣄⣀⣀⡀

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣿⡿⠟⢛⣋⣀⣀⣀⠄⠁⡰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠙⠏⠉⠉⠉⠉⢁⣀⠤⠤⠤⠰⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢄⠚⢸⠀⠀⠀⡀⠀⠀⠀⠀⣸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⣄⠀⠀⠈⠂⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢠⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠁⠀⠀⠀⠲⢶⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⣆⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣄⢻⣿⣶⣦⣴⣦⣭⣉⠛⠛⠁⠀⠀⠀⠀⣼⣿⣿⣷⣾⡿⠿⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣶⣿⣤⡀⢰⡄⠈⠳⣄⠀⠀⡆⠀⠇⠀⠀⡰⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⣸⢻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢠⣾⡷⣻⣿⣿⣿⣿⣿⣿⠠⡇⠀⢀⠈⠀⢠⡇⠀⠀⡆⢲⣷⣽⡇⣙⣇⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⡄⢀⣴⡿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡜⠀⠀⠀⠙⢿⡶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⢰⣿⣿⣿⡡⢸⣿⣇
⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣤⡶⠶⣆⣌⠂⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠟⠿⠿⠏⡙⠰⢶⠻⢿⣿⣿⡥⣸⣿⣃⠄⠄
⠀⠀⠀⠸⠟⠙⠛⠿⣿⣿⣿⣿⡿⢤⣀⠚⠉⠉⠁⠀⠀⠀⠈⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣶⣬⣭⣤⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠙⠛⠋⠋⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣹⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠂⠀⠀⠀⠣⡤⣄⠀⢀⡄⠘⡏⠀⠀⠀⣀⡀⠀⠀⠔⢰⢀⠔⠂⠀⠀⠀⠳⣤⣄⠀⢀⡄⠘⡏⠀⠀⠀⣀⡀⠀⠀⠔⢰⢀
⠀⠀⣤⡄⠀⠀⠈⠀⠊⠀⠐⠁⠀⠀⠉⠁⠈⠒⠂⠀⠀⠉⠀⠀⠀⣤⡄⠀⠀⠈⠀⠊⠀⠐⠁⠀⠀⠉⠁⠘⠒⠂⠀⠀⠉
⠀⠀⠉⠉⠳⡀⠀⠠⠤⣀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠋⠱⡀⠀⠀⢤⣀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢰⠁
⠤⠆⠀⠀⠀⣀⠀⠀⠀⣀⠠⠀⠀⠀⠰⠂⠀⠔⠛⠁⠀⠀⠠⠤⠆⠀⠀⠀⣀⠀⠀⠀⡀⠠⠀⠀⠀⠰⠃⠀⠖⠚⠁
⡠⠄⠀⢀⠀⠁⠡⡀⠀⠉⠈⠃⠀⠠⡀⠀⠀⠠⢦⠀⠀⠀⠀⡠⠄⠀⢀⠀⠁⠣⡀⠀⠈⠈⠂⠀⠠⡀⠀⠀⠠⢦
⠀⠀⠀⠨⣱⠀⠀⠀⠀⠀⠀⡀⠀⠠⣃⠤⡄⠀⠘⠀⠌⠙⠤⠀⠀⠀⠨⣱⠄⠀⠀⠀⠀⠀⡀⠀⠠⣇⠴⡄⠀⢘⠀⠌⠙
⠀⠠⢄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀
⠀⠀⠈⠛⠁⠐⠀⠀⠢⡀⣀⠀⠙⢦⠀⠀⡠⠄⠀⠀⠘⠒⠆⠀⠀⠈⠛⠉⠙⠀⠀⠢⡀⢀⠀⠙⢦⠀⠀⡠⠄⠀⠀⠙⠒
⠀⣸⠀⠀⢀⡀⢀⣀⡀⠈⠁⠀⠀⠀⠀⠀⢑⣲⠀⠀⢀⠀⠀⠀⣸⠀⠀⢀⡀⣀⣀⡀⠈⠁⠀⠀⠀⠀⠀⢑⣲⠀⠀⢀⠄
⡿⠁⠀⠀⠘⠿⠃⠀⠀⠀⠀⠸⠤⢤⠀⠀⠃⠀⠀⠀⠋⠀⢸⡿⠁⠀⠀⠘⠿⠃⠀⠀⠀⠀⠸⠤⢤⠀⠀⠃⠀⠀⠀⠋
⠀⣠⠎⠀⠀⠀⠀⢲⢰⠀⠀⠀⠀⠈⠧⠀⠀⣀⠰⠓⢄⡠⠀⠀⣠⠎⠀⠀⠀⠀⠒⢰⠀⠀⠀⠀⠈⠇⠀⠀⣀⠰⡒⢄⡠
⡀⠀⠀⠈⣆⠀⠀⠀⠉⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⡀⠀⠀⠈⣆⠀⠀⠀⠉⠀⢸⡿⠀⠀⠀⠀⠀⠀⠉⢀⢀
⠀⠀⠀⠀⠈⠛⠳⠀⡰⠃⢀⠇⠀⠀⠤⠖⢢⣀⡀⠁⠘⠴⠁⠈⠀⠀⠀⠈⠋⠳⠀⡰⠁⢀⠇⠀⠀⠤⠖⢢⣀⡀⠁⠘⠴
⠀⠀⢿⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠿⡧⣄⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤
⠀⡀⠀⠀⠈⠂⠀⠈⠙⠒⠀⠀⠀⠀⢑⡀⠀⢀⣤⠜⠀⠀⠀⠀⡀⠀⠀⠈⠂⠀⠈⠉⠒⠀⠀⠀⠀⢑⡄⠀⢀⣤⠜
⠉⠁⠀⠀⠀⠆⠄⠀⠀⠶⠚⡀⠀⠀⠈⠀⠀⠁⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠖⠄⠀⠀⠲⠸⡀⠀⠀⠈⠀⠀⠁
⠊⠁⠀⠰⢄⠀⠈⠂⠀⠀⠀⠀⠀⠈⡆⢀⠀⠈⠹⠀⠠⢠⢠⠊⠀⠀⠰⢆⠀⠈⠂⠀⠀⠀⠀⠀⠈⡆⢀⠀⠀⢱⠀⢠⢤
⠀⠀⠀⠈⠘⠀⠀⠀⠀⠤⠴⠂⠀⠀⠒⠉⡇⠀⠀⠀⠁⠀⠉⠀⠀⠀⠈⠘⠁⠀⠀⠀⠤⠴⠂⠀⠈⠓⠉⡇⠀⠐⠀⠁
⠀⠈⠱⣤⠴⢦⠀⠀⡀⠀⠀⠀⢤⡀⠀⠀⠀⡀⠀⠀⢤⣒⡀⠀⠈⠱⣤⠴⢦⠀⠀⡀⠀⠀⠀⢤⡀⠀⠀⠀⡀⠀⠀⢴⣒
⠀⢠⠀⠀⠀⠀⠀⠀⠈⠢⠖⠀⠀⠙⠂⠀⢎⣁⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⠙⠀⠀⢎⣁
⣤⠊⠀⠀⢰⣦⡔⠒⠂⠀⠀⢀⠀⠀⠀⠀⡀⠊⠀⠀⡴⠁⢠⣤⠚⠀⠀⢰⣆⡔⠒⠂⠀⠀⢠⠀⠀⠀⠀⡔⠚⠀⠀⡴
⠋⠀⡠⠀⠀⠉⠀⣀⢀⠀⠀⠀⠉⠹⡄⠀⠀⠀⠀⣄⠀⠀⠘⠋⠀⡠⠀⠈⠉⠀⣀⢀⠀⠀⠈⠉⠹⡄⠀⠀⠀⢀⣄
⠀⠘⠁⠠⡀⠀⠀⠘⠼⠀⢠⣤⠀⠀⠉⠀⠀⠲⠬⠀⠑⠊⠀⠀⠘⠁⠠⡀⠀⠀⠐⠼⠀⢠⣤⠀⠀⠁⠀⠀⠒⠬⠂⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠃⠀⠀⠀⣀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⢠⠇⠀⢀⣠⠖⠀⠀⠀⠀⠐⠁⠀⠀⢑⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢣⣀⡭⢶⢾⠚⡶⢤⣞⠀⠀⠀⠀⠀⠀⠘⠦⠠⠰
⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⣈⡽⠋⣀⠀⠀⢠⠀⣠⠌⠉⢣⡤⠤⠒⠀⡠⠀⠀⠀⠀⠀⡅⠀⠀⡄⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣸⡁⠀⣀⡀⠀⡈⠀⠀⢀⠀⠉⡇⠀⠀⠀⠁⠀⠀⠈⢢⣠⠵⠔⢶⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢼⡂⠘⠁⡀⠠⠖⠀⠂⠉⠁⢰⠗⠂⠀⠀⠀⠐⠢⢤⠟⢀⠀⠆⠀⠈⢫⣀⡀
⠀⠀⠀⠀⠐⠄⠀⠈⠁⢀⡼⠳⣀⡀⠠⠂⢠⠀⠑⠀⢲⠿⢄⠀⠀⠒⢂⠀⢒⡎⠀⡀⠁⠀⠁⠄⢈⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠉⢳⡤⣤⣈⣀⡤⠿⡏⠀⠘⠂⢀⠀⠀⠀⠨⢧⡀⠀⡄⠴⠈⠀⡰⢇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠠⠇⠈⡇⠀⠇⠀⠱⠀⡀⠀⠁⠀⠀⠀⠐⠈⡹⢦⠤⡦⠴⡋⠣
⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⠢⠀⠁⠈⠀⠁⠀⠈
⠀⠀⠀⠀⠊⠀⠀⠀⠀⣀⡴⠚⠒⠆⡀⠀⠈⠁⠀⠀⠀⠀⢃⠀⡄⠀⡇⠀⠀⡰⠀⠀⠀⠀⠀⠀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢥⠀⠀⠀⠀⠀⡀⠲⡤⠓⠛⠀⠛⠲⢷⣅⢀⠤⠴⠂⠀⡜⠁⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⡀⠏⠀⠀⠀⢀⣀⠈⡞⠀⠐⠂⠀⠌⠀⠤⠈⢻⠀⠀⠀⠀⠐⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢀⡩⡇⠀⠠⠂⠐⠀⠀⠀⣀⢸⠃⠒⠂
⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣆⠀⠀⢠⠀⠑⠀⢠⠯⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠟⢲⢤⣀⣀⣤⠝⢣⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠊⠀⠈⠀⡆⠀⠘⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⠋⠉⠙⡿
⠀⠀⠀⠀⠀⢀⣤⡀⢀⣅⠐⠀⠲⠀⡷⠀⠑
⠀⠀⠀⠀⢂⡽⢿⣿⡷⠊⠃⠠⠀⠸⠉⠳⡀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⡖⢀⣠⣤⣿⡟⢷⠀⠀⢻⡾⠃⠀⠞⠙⣿⣆⠀⢧⠀⠀⠀⠀⣀⣀⢠⠦⠀⠀⠀⣸⡇⠠⠀⠀⠀⡀
⠀⠀⠣⢄⠐⠁⢽⡇⠀⠀⠀⣿⡏⠀⠀⠀⣼⠹⣿⣶⡆⠀⠀⠀⣰⡿⠟⠀⣀⠄⠀⠀⠻⣧⠀⡎⠀⠘⠁⠉⠱⠤⣆
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢠⠸⡡⠀⠀⠀⠇⠀⠊⠁⠀⠀⠀⠀⡿⢃⠀⢀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠃⠀⢈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠋⠀⠀⢤⢠⡄⢘⠀⠃⠀⠀⠀⠀⠀⠐⠁⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠂⠀⠀⠀⡼⠇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⠋⠁⡿⠀⠀⠀⠀⠀⣠⡤⠀⠀⠘⠀⣿⠃
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⣿⡀⠇⠠⠡⠿⠴⠀⣄⡘⠁⠀⠈
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⡇⠀⠀⠸⣿⣿⠀⠀⠀⠀⠁⣾⣛⡓⠀⣀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣸⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⡀⡀⠀⠀⢰⣿⣿⣧⣤⣴⡆⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠣⠀⠀⢀⠴⢏⡉⣹⡦⠄⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⡿⣿⣿⣿⣿⣽⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣼⠿⠸⠟⢻⠿⠘⠟⠛⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢀⣿⢀⣤⣤⡀⠀⠀⠀⢤⣦⡀⢸⢻⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡄⢸⣿⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⢸⡇⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣧⢸⣿⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠈⣇⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠜⠉⠀⠒⠒⠃⢒⣀⣩⢽⣆⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠀⠀⠒⠒⠉⢁⣀⡠⢼⡏⠀⢠⡏⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠊⢉⣀⠤⠐⠞⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⡤⠤⠤⠬⠭⠔⠊⠀⠀⠀⠀⢸⡇⣇⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠈

⣿⡯⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⠿⠿⠿⠿⢿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠋⠉⠀⠀⣀⣀⣀⡀⠀⢀⣀⠀⠈⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⢀⣤⣶⠀⠀⣿⣿⣿⡟⠁⢀⣿⣿⣿⣷⣦⡄⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⣶⣿⣿⣿⡀⠀⢿⣿⣿⡇⠀⢸⣿⣿⣿⡟⠁⢀⣠⣄⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣀⠀⠹⣿⣿⣿⣷⡀⠈⠻⠿⠃⠀⠘⠻⠿⠏⠀⣠⣿⣿⣿⡆⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣸⣿⣦⡀⠈⠛⠿⡿⠋⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⢻⡿⠿⠟⠛⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⢻⣿⣿⣿⣶⣤⡀⠀⢀⣾⡿⠛⠋⠉⠉⠙⢻⣷⡀⠀⠀⣀⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⡀⠀⠛⠿⣿⣿⠿⠁⢀⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⣿⣿⣿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣿⣷⣤⣀⣀⣀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠉⠉⠛⠃⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⣿⣿⣿⣿⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⣼⣿⣶⠄⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠟⠛⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣷⣶⣶⣶⣿⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⢿⣿⣿⣿⠿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⠀⠀⠀⢀⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠛⠛⠉⠀⠀⢸⣿⡇
⠀⠀⠀⣠⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⣶⣶⣶⣶⡷⠀⢸⣿⠃
⠀⢠⣾⠟⠉⠉⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠘⣿⣿⣿⣿⣿⡇⠀⢸⣿
⢀⣿⠇⠀⣰⣷⡀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠈⠛⠿⠿⠿⠃⠀⣾⡿
⣼⣿⠀⢠⣿⠿⠗⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢰⣿⣶⣤⣀⣀⡀⠀⢠⣿⡇
⣿⡇⠀⠈⠀⣀⣠⣤⡀⠈⠻⣷⣄⡀⠀⠀⠀⠀⣀⣠⣴⡿⠏⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⣼⡿
⣿⣇⠀⢀⣾⣿⡿⠛⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠉⠀⣀⣴⣄⠀⠙⢿⣿⣿⠏⠀⣰⣿⠁
⠸⣿⡄⠀⠻⠋⠀⣠⣶⣿⠟⠀⢀⣀⡀⠀⢀⣀⣤⡀⠀⣾⣿⣿⣿⣷⣄⣀⠀⠀⠀⣴⣿⠃
⠀⠹⣿⣄⠀⠀⢼⣿⣿⠏⠀⣰⣿⣿⡇⠀⢿⣿⣿⣧⠀⠘⢿⣿⣿⣿⠿⠋⠀⣠⣾⡿⠁
⠀⠀⠘⢿⣷⣄⡀⠈⠙⠀⠀⣿⣿⣿⣷⠀⠘⣿⣿⣿⣷⠄⠀⠈⠉⠁⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠈⠛⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



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



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

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


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



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


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

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

⠀⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⡠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠦⡀⠀⣠⠶⠶⠶⠶⢶⣶⣶⡶⠶⠶⠶⠶⠶⠦⡄
⠀⡇⠀⢻⣽⠛⠛⠛⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣀⣠⠤⢤⣒⡲⡭⠭⣭⡅⡇⠀⡇⠀⠀⣠⠖⡩⠖⢒⠚⠀⠀⠀⠀⠀⠀⡇
⠠⡇⣀⠯⠼⢄⡀⠀⠀⠀⠁⠄⡀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣿⠒⠉⠉⣿⠀⠀⠀⠈⣿⡅⡇⠀⣇⠔⢋⡤⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇
⠈⣷⣁⣀⣀⣀⡇⠀⠀⠀⣀⣀⣀⣀⣀⣠⡇⠀⡇⠀⠀⠀⣫⠖⠚⠃⣿⡀⠀⠀⠀⣿⠇⡇⠀⣧⠔⠉⠀⠀⠀⠁⠈⠀⣁⣀⣀⣀⡀⣀⡇
⠀⣿⠀⠀⠀⠀⡷⠀⠀⢰⣒⠒⠒⠂⠴⠒⡇⠀⡇⠀⠒⠉⣾⠀⠀⠀⣿⠇⠀⠀⢀⣿⠆⡇⠀⡇⠀⢀⡀⠀⠀⡠⠔⠩⣗⠒⠂⠤⠤⠒⡇
⠀⣿⠀⠀⠀⠀⡏⢠⣒⠭⡚⠁⠀⠀⠀⠀⡇⠀⡇⠀⠀⢀⡥⠒⠁⠀⣿⠀⠀⠀⠀⣿⡇⡇⠀⡇⠀⠈⠁⠀⡨⢠⣖⠭⠖⠋⠀⠀⠀⠀⡇
⠀⣿⠓⠒⠒⠒⣇⡾⣒⡫⢕⣂⠀⣀⠔⠋⡇⠀⡇⠛⠉⠁⣞⠤⠤⠤⣿⣀⣀⣀⣀⣿⡃⡇⠀⡇⠀⣆⣰⠀⠧⡸⢕⣊⠥⣂⠀⢀⡴⠊⡇
⠐⣏⠓⠒⠒⠒⠋⠉⠓⢾⠭⠒⠋⠀⠀⢀⡇⠀⣇⠀⠀⠀⠈⠉⠉⠙⠒⠒⠒⠒⠬⠼⢃⡇⠀⣇⠀⠀⠀⠀⠀⠉⠓⢶⡭⠔⠊⠁⠀⢀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡞⠉⠉⢉⡭⠭⠽⠟⠛⠻⢭⡉⠉⠉⠙⡆⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆
⠀⡇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡇⠀⡇⢀⡴⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡜⠉⠙⣆⠀⠀⠀⠀⠀⠀⡞⠉⠙⡆⡇
⠀⡇⠀⡇⠀⡴⢄⣼⠣⣼⠳⣄⠀⠈⡆⠀⡇⠀⡇⡞⠒⠒⠚⠤⣄⡀⠀⠀⠀⢀⣠⣄⡀⡇⠀⡇⠑⠤⠴⠃⠐⣲⢦⢤⣤⠀⠳⠤⠴⠃⡇
⠀⡇⠈⣇⡞⠀⣤⠀⠀⠀⢠⠈⠙⣆⡇⠀⡇⠀⣷⠃⠂⠀⠀⢀⠀⠈⠉⠉⠛⡇⠀⠀⢱⡇⠀⡇⡴⠋⠙⣆⢈⡉⠉⠈⢈⡀⡔⠋⠑⡆⡇
⢀⡇⠸⡉⠙⠞⠋⠉⠉⠉⠙⠛⡞⢉⡕⠀⡇⠀⡏⡆⠠⠄⠀⠀⠀⠀⠀⠀⢐⡇⠀⢠⠇⡇⠀⡇⡇⠀⠀⡏⠙⠉⠉⠉⠙⠁⡇⠀⠀⡇⡇
⠀⡇⠀⠈⢲⡀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⡇⠀⡇⢱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠖⠁⠀⡇⠀⡇⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇
⠘⡇⠀⠀⠀⠉⢳⠶⠤⠔⡖⠉⠀⠀⠀⠀⡇⠀⡇⠀⠉⠒⠦⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⡇⠀⡇⢹⣤⣤⠇⠀⠀⠀⠀⠀⠀⢿⣤⣤⠇⡇
⠀⠣⠀⠠⠖⢫⣽⣤⣤⣤⣼⣭⣵⣦⣤⡴⠃⠀⠣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠃⠀⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠇

⠀⠀⠀⣤⣶⣦⡀⠀⠀⠀⢰⣶⠟⣷⡆⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⣴⣖⣶⣢⡀
⠀⠀⠘⣇⠛⢉⠃⠀⠀⠀⠰⣾⡟⣻⡧⠀⠀⠀⠀⢿⣳⡟⠀⠀⠀⠀⠀⠸⣝⠅⠀⠀⠀⠀⣟⣙⣋⣹⡇⠀⠀⠀⣥⣭⣭⣭
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣼⣿⠇⠀⠀⠀⢴⣿⣿⣿⡆⠀⠀⠐⣿⣿⣟⢛⣛⠀⠀⢀⣴⣾⡿⠟⢢⠀⠀⠈⢻⢟⡻⡟⠃⠀⠀⠠⣞⣲⣋⣇⠂
⠀⠀⠰⣿⣿⠛⠀⠀⠀⠀⠹⣌⣉⣽⠃⠀⠀⠀⢾⣿⠟⠉⠁⠀⠀⠈⠦⣶⡤⠾⠋⠀⠀⠀⣼⠮⠷⣧⠀⠀⠀⢰⡿⠛⠩⠛⠃
⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠛⠀⠀⠀⠀⠁
⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠠⡦⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⣶⡄⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⣰⢢⡄
⠀⠀⢰⡿⠓⠻⡇⠀⠀⠀⢸⣻⠏⠁⠀⠀⠀⠀⣸⡇⠀⢀⡀⣹⣿⣏⣀⠀⠀⢀⣼⠀⠀⠀⠀⣘⣲⣿⣷⡞⠀⠀⠘⠹⡏⠧
⠀⠀⠈⠳⠤⠴⠃⠀⢠⣶⡏⠀⠀⠀⠐⣿⣿⣿⠟⠀⠀⠐⠚⢿⣿⡿⠧⠀⠀⢸⣇⡀⠀⠀⠀⠨⠿⠿⢯⠁⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁

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

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

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