Parchment Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⡀
⢀⢀⡈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡿⡄
⡆⢀⣤⡄⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢤⣤⡀
⣇⠙⠛⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠘⢛⡋
⠘⢌⡠⠟⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⠸⠌⡠
⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⣴⡶⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢔⢢⠀⠀⡇
⠀⠈⡄⢀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢠⠁
⣀⣀⣈⣢⣤⣦⣤⣄⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣒⣂⣠⣤⣴⣤⣴⣁⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠔⠀⠒⠒⠚⠓⠒⠒⠘⠛⠓⠛⠓⠀⠛⠃⠒⠚⠒⠒⠒⠓⠒⠒⠒⠴⠄⠐⠖⠒⠒⠲⠦⠴⠀⠀⠒⠒⠒⠒⠒⠐⠒⢆⣴⣤⣤⣤⣄⡀
⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠈⠙⠻⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⢿
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣴⣤⣶⣄
⠀⠀⠈⢿⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⣠⠏
⠀⠀⠀⠀⠙⠻⣯⣭⣿⣿⣟⡻⢿⣷⣾⣽⡿⣿⠿⠟⢉⣩⣍⣓⡚⢉⡓⣒⡒⢀⣰⣦⣤⣤⣤⣄⣄⣀⣀⣀⡀⢀⡀⠀⠙⣃⡶⠏⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠁⠈⠁⠉⠉⠁⠈⠁⠀⠀⠀⠀⠈⠉⠈⠉⠉⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡿
⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⣾
⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠒⠒⠚⠛⠻⠶⠄
⢀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⢡
⢘⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⡀⣀⣀⣿⡿⠀⢸
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠋⠙⠉⠛⠛⠉⠁⠀⢸
⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠦⢀⡀⠀⠀⣀⣤⣾⠟
⠈⠳⢶⣔⣶⣶⣒⣖⣲⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣲⣿⣷⣾⡿⠿⠛⠁
⣀⣀⣀⣈⣈⣉⣉⣉⣉⣉⣉⣉⣙⣋⣙⣛⣛⣛⣧⣉⣉⣐⣛⣂⣠⣖⣈⣉⣉⣋⣛⣉⣉⣉⣙⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠿⠿⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻
⡰⠋⠀⡐⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⣠⠴⡫⠋⣸
⡇⠰⡋⢉⢳⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡤⠖⠋⡠⠊⢁⡴⠁
⠹⣄⠉⢉⣼⣧⠤⠂⠹⣄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠞⠉⠀⣠⠊⠀⡴⠋
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣠⠞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠀⠀⡴⠃⣀⣀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣄⠘⢀⡾⠁⠀⠈⣳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡢⠀⡼⢁⣀⣒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⠶⠺⢯⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠠⡈⡱⢈⡇⠀⠀⣀⣠⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣤⠾⠷⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠵⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⣶⣍⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⡇⠀⣀⣤⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣷⡇⣾⣿⣿⣿⣿⣿⣶⣄⡀⠀⢤⣶⣀⣀⠀⠀⠀⠀⠀⣿⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠃⠀⣾⣿⣿⣿⣿⣶⣄⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠟⠛⠋⢁⣀⡼⠟⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⠿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣻⣭⣭⣶⣾⣿⣿⣶⣭⣟⣛⣿⣿⣿⡁⠀⠀⠀⠀⠛⠛⠁⠐⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠉⠙⠛⠿⣿⣿⣷
⠀⠀⢠⣼⣷⣾⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⡆
⠀⠈⠛⠛⠻⠿⠿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣿⢳⠀⠀⠀⠀⠀⠀⣰⠁⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠳⡄⠀⠀⢀⡴⠃⠀⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⢿⡄⠀⠈⠻⢿⣿⡿⠋⠐⠏⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡯⠿⠀⠀⠙⢶⣴⠟⠀⠀⣼⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⢢⡀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⣦⠰⡀⡇⠀⡇⠢⡸⡄⢰⠀⢄⢢⣀⡠⢰⠀⠁⡧⠀⡇⡇⠰⡀⠀⢧⠀⡇⠀⡥⠃⠱⠀⠀⠀⢸⣶⠆
⠀⠀⠀⠀⠻⣦⠀⠈⢀⣘⡀⠈⠀⠘⠀⠁⠀⠁⠁⠀⠉⠀⠁⠈⠈⠀⠠⢛⣧⠀⠀⠁⠀⠁⠀⠁⠁⠀⠁⠀⠀⠁⠁⠀⠁⠈⠈⠀⢷⣷⣾⣿⣿⠄
⠀⠀⠀⠀⠀⣭⣆⠀⣞⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⣷⣼⠁
⠀⠀⠀⠀⡭⡿⣷⠀⣬⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⣶⣤⠘⠟⢀⠀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⢿⡷⠆
⠀⠀⠀⠀⠀⣇⢹⠐⡃⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣹⡿⠀⠄⠄⠘⠋⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠧⠾⠄⢦
⠀⠀⠀⠀⡻⣿⠘⡀⢀⣀⠠⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠈⡠⣼⣿
⠀⠀⠀⠀⠀⠬⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⣀⠣⡤⢺⡅⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠙⠩⠁⠀⠀⠀⠀⠀⢾⣿⠅⠀⣰⠀⠈⠀⠀⠐⡆⠀⢰⠄
⠀⠀⠀⠀⡀⠀⡀⠀⢠⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⡏⠀⠀⠁⠀⠀⠀⠀⠀⣧⣀⣈⢱
⠀⠀⠀⠘⣿⣷⣄⠀⡾⡇⢀⡅⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠈⠈⠺
⠀⠈⠙⠛⢿⣿⣿⣾⣿⣿⣾⣿⣶⡶⠶⠶⠀⠀⠐⠲⣀⣿⣷⣺⡗⠀⠀⠀⠀⠠⢴⣷⣾⣿⣌⠁⠀⠉⡀⠁⠠⠔⠀⢀⠀⠠⡊⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠚⠁⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠼⢷⣤⡀⣠⠄⠀⠀⠀⣠⣼⣿⠤⠤⣽⡯⠛⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⢠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠮⠽⢟⣤⢻⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠛
⠀⠀⠀⠘⢉⣧⣿⣿⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢬⡃
⠀⠀⠀⠲⡌⣿⣈⣿⡖⠀⠀⠀⠀⢰⡀⠀⠀⠀⠂⠀⠀⣠⢀⡀⠀⣄⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠁
⠀⠀⠀⠐⠀⠛⠋⠉⠀⠀⠀⠀⠀⡸⢇⠀⠸⠂⠀⠀⠾⠛⠏⠀⠿⠻⠇⠀⠀⠀⠸⠁⠆⠀⠀⠀⠀⠰⠒⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⢩⠀⣠⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⡄⣀⡄⠀⠀⠀⠀⠀⠀⢀⣴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣴⣿⣺⣿⣿⣿⣷⣦⡀⢠⡾⢾⣿⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣿⣿⣟⣿⣿⣿⣿⣆⣥⣼⣍⠻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⢻⣟⣿⣿⡟⡿⡿⣿⣿⣆⢻⣾⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⠀⠀⣼⣿⣿⣿⣷⣶⣿⢟⡿⣯⣼⣏⣉⡾⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣿⣿⢶⣿⡘⣿⣿⣿⣽⢿⣿⣮⣻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣽⣿⡿⣿⣿⣿⣿⣷⣾⣯⣿⣿⢦⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣿⣿⣿⣻⠿⢿⢿⡏⣸⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣻⣴⣿⣿⣿⣿⣿⡿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠠⣶⣿⡿⠿⠛⢿⣿⣿⣿⡁⣿⣿⣿⣿⢟⣥⣿⣿⣿⣿⣿⣟⣿⣿⡇⢲⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠙⣿⣆⠀⠀⠀⣼⠉⠉⠲⢾⣿⣿⣿⡘⣿⣯⣿⣿⣿⣿⡿⣿⠟⣠⡟⠀⠈⠻⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠸⣿⣦⡀⣰⠃⠀⠀⠀⠀⠀⠈⠉⠀⠹⣿⣿⣿⣿⣿⣿⣧⢰⣿⣇⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢻⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣯⣿⣾⣿⣿⣷⣶⣾⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣶⣤⣤⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⠏⠙⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣞⡻⠿⠷⣤⣤⢀⡿⠁⠉⠀⣠⣤⣀⣠⣶⣿⣶⣤⣄⣠⣶⡿⠋⠁⣼⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠁⠸⠁⣠⣴⣿⡿⣿⡿⣿⣧⠀⠀⣠⣿⡿⢟⣀⡀⢰⣿⡇⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣄⣀⣤⣤⣶⣿⡿⠋⢸⣿⣧⣿⡿⠂⢸⣿⠿⠃⠈⠙⠻⢾⣿⣧⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡿⠿⠛⠉⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠋⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠤⠄⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣀⠀⠄⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠏⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠉⠀⠀⠉⠽⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠂⠈⠉⠀⠀⡉⠀⠀⠀⠈⠈⠩⠶⠲⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡄⠠⠐⠀⠀⠀⠈⠁⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⢀⠃⠈⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⡶⣶⣿⣿⣧⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣿⠛⡗⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠨⢿⣿⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣾⢋⠀⠄⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣃⣴⠀⠄⠀⠀⠀⠀⣀⣠⣤⠤⠿⠿⣈⠗⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠄⠀⢠⡟⠁⠈⠓⠄⡀⢀⡴⠟⣉⠐⠖⠀⠀⣁⢉⣈⡀⠀⠐⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡅⠀⠀⢱⠃⠀⠀⠀⠀⠀⠈⠡⠒⠉⠀⠀⠈⠉⠀⡀⣀⡀⡃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠀⠠⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠈⠁⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢆⡰⠀⢀⡰⠃⠀⣀⠀⠉⠀⢸⡟⣻⣆⠀⠀⠀⠀⠀⠉⠁⣀⠴⠛⠛⠓⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⡶⠀⣠⠄⠀⠄⠀⠀⠀⣿⠛⠷⢻⣆⠀⠀⠀⠀⢙⠞⠁⠀⠀⠀⠀⠀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠞⢠⣿⠉⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⣦⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠐⡙⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣯⢰⠆⠀⡄⡀⡀⠚⡿⠀⠀⣴⠏⠀⠙⠂⣾⡇⠀⠀⠀⠀⠀⠁⢀⠇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡸⠓⡃⠀⠁⠞⣠⢂⡾⠁⠈⠅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠜⠂⢈⡼⠁⠀⠀⠀⠀⠀⢘⠶⠄⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⣠⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⠀⡠⠀⠀⢀⣤⠀⢀⡤⠤⠤⣤⣽⣦⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠋⢀⠀⠀⠀⠀⠄⠀⠘⣶⠄⠊⠉⠀⠀⠀⠸⡄⠀⠀⠀⠀⢈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⢠⡇⢰⡗⠀⠀⠁⡀⠀⠀⢠⣀⣿⣧⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢾⠀⠀⠀⠀⠈⢢⠀⠈⠀⢙⡾⠂⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⢈⣥⣼⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠳⢯⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠈⠈⣠⠀⠀⠀⠀⠀⢸⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⡆⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣹⢻⡄⠀⠀⣠⠀⢃⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⣧⠀⢠⠌⠡⢘⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⠃⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡇⠀⠀⢹⣇⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣀⠀⣪⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣘⣧⠀⠀⠀⢻⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣲⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⢀⡿⠀⠀⠀⠀⢻⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠶⠶⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⠶⠛⠁

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠠⠄⣀⣀⣀⡀⠠⠀⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡤⠂⠄⠀⠀⠀⠀⢀⠀⠀⠐⠋⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡂⠄⠀⠀⢀⠂⠊⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⠈⠐⠀⠀⢔⡀⣀⠄⠊⡀⠀⠀⠀⢇⠀⠀⠀⠀⠈⢧⠀⣠⡀⠀⠀⠀⡘⠁⠀⢀⣠⠀⢁⣀⢀⣀⢀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⢸⡅⠀⠀⡄⠀⠀⠁⢂⢀⠏⠀⢀⣀⠤⢀⠒⢉⠁⠀⢸⠂⠀⠁⠀⠀⠀⢻⡉⠟⠤⡀⠀⠃⠀⠰⠪⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈
⠀⠀⠀⠘⡅⢣⠀⣡⡀⠉⠀⢸⠀⠤⠐⠋⠀⠀⠳⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⢸⠁⠀⣥⠟⢁⠅⠲⠚⢁⠁⠂⠉⠀⠀⠀⠀⡺⠒⠀⠀⠀⡀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠁⠀⠐⠸⠯⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⢀⡂⢀⣧⠐⡠⠐⠀⠘⠱⠀⠂⠀⠴⣌⣐⢊⠀⠠⠀⠂⠉⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠈⡄⠂⠄⡀⠂⡀⠀⠀⠀⠄⠀⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠸⣧⠦⡰⢤⢷⠀⠐⢐⡀⠀⠠⠐⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢰⠂⡐⠓⢧⣠⠀⡀⠂⠀⠀⠀⠊⢌⠀⢸
⠀⠀⠀⠰⡄⢅⣴⠋⢉⣱⣰⣻⠉⠦⡁⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠁⣠⠐⠀⠀⠀⠀⠆⠒⠘⡇⠃⠀⣹⡢⠠⣐⠀⢄⡂⠀⠘⠒⠀⠀⠀⢸
⠀⠀⠀⠀⣗⣼⠑⡐⠂⠀⠍⠑⡀⠂⠡⡀⡠⠤⠒⠂⠒⠛⠚⠊⠉⠂⠀⠉⠀⠀⠈⢁⡀⠂⠀⠼⠁⠘⢹⣄⠁⠈⠀⠈⠂⠈⠐⠠⢐⠠⠀⠀⠈
⠀⠀⠀⠀⢛⣯⠊⠀⠀⢬⠀⡀⠹⡀⢂⠡⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠐⠈⠀⠀⠀⠀⢀⠀⠀⢿⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⣾⡹⠀⠀⢀⠂⡳⠒⡁⢀⠀⠀⠷⠗⠀⠀⠀⠀⠠⠈⣀⡀⠀⠑⠆⠀⠈⡁⠀⠀⠀⠀⠉⠁⠛⢤⣁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡯⣧⠀⠀⠉⠟⢁⠤⠠⠉⣃⢠⢭⠀⠀⡄⠀⣐⡠⠉⠁⠈⢂⠖⠻⠇⢀⡈⠀⠀⠀⠀⠐⠒⠂⠜⠁⠂⡌⠀⠀⠀⠀⠃⠐⠀⠀⠀⢨
⠀⠀⠀⢀⡹⣁⡈⠑⠶⢀⠀⠠⠄⠀⠂⠀⠈⠃⠀⠘⣛⣁⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠒⠈⢀⡠⠄⠀⠜⠩⠀⠁⠀⠀⠔⡩⠀⠀⢸
⠀⠀⠀⠰⠇⠀⢠⠃⠈⡅⠀⡂⠀⠘⡩⢠⠃⢻⢄⠄⠀⠈⠊⠀⠀⠀⠠⠤⠐⢄⠀⠀⠀⢰⠤⠀⠘⠉⠉⠀⠂⠁⠀⠀⣷⠖⠀⢀⠙⠁⠀⠀⢸
⠀⠀⠀⠰⠂⡜⠸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠆⠈⠀⠀⠀⠀⠀⡤⠀⠀⢀⠀⠀⠠⡉⠂⠀⠏⠀⠀⠀⠀⠀⢠⠀⠠⠄⠈⠏⠀⡠⠂⠀⠄⠀⠀⠰
⠀⠀⠀⢠⡋⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠠⠤⠴⢥⡈⡦⡎⠀⠀⢀⠀⠁⠘⠆⠀⠀⠀⠀⠀⠆⡀⠀⢁⡴⣀⠀⠀⠀⡌⠒⠤⠤⠐
⠀⠀⠀⠘⢁⠸⢀⠀⠀⠘⡀⠰⠀⠐⠀⢠⡀⠀⣀⡡⡄⠀⠈⣆⡙⠣⡐⢲⠖⠴⠆⡀⠠⠤⠤⠀⡄⠀⠀⠉⢈⠀⢰⠀⠀⠑⠦⠂⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⡄⠠⠄⠀⠐⠄⢰⠀⠀⠨⠰⠀⠁⠀⠰⠕⡄⠐⠀⢘⠑⡠⢬⠲⢄⠑⢄⡀⠀⠀⠀⠀⠈⠒⠒⠐⠠⠀⠀⠀⠀⠀⠒⠀⢀⠀⠁⠀⠀⡅
⠀⠀⠀⠀⠂⠀⠂⠀⠀⣀⣐⠀⠀⢀⣐⣀⠈⠀⠀⠈⠸⢦⡠⠰⠶⠀⠀⠑⢄⡙⢦⣈⠂⠀⣀⡀⠆⠰⠀⡈⠀⠀⠀⠆⠀⠀⠀⡀⣀⠁
⠀⠀⠀⠀⠗⣀⠂⠒⠁⠸⠟⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠄⣈⠒⠊⠁⢁⡀⠤⠄⠠⠄⠀⠀⠀⠠⠠⠄⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⢤⣀⠀⢸⡇⠀⣼⣿⣿⣳⡄
⠀⠀⠀⠀⠀⠀⢀⣺⣽⣿⣿⣿⣿⣿⣿⡷⣸⣧⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⡱⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠺⠇⢸⣿⣿⢿⣻⣵⡟⢿⣿⣿⣿⣿⣟⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⢠⣷⣾⣿⣿⣿⣾⣿⣹⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣻⣿⣿⣿⣿⣿⣿⣫⣷⣶⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣫⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣵⣾⣿⣭⣤⣻⡺⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡝⣿⣿⣿⢿⣾⣭⣿⣿⡟⣿⣿⣭⣿⡼⣿⣿⣧⣿⣿⣿⢿⣿⣿⣿⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⢹⣿⡿⠀⠈⠉⠛⠛⠃⠛⠛⠛⠉⠀⠙⠿⣿⣿⣿⣵⢿⣿⣿⣯⠀⠸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⠿⢷⣤⣀⣘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡻⢿⣿⣿⣦⣝⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠟⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⣽⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡷⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣟⣿⡟⠃⠀⣀⣴⣾⣿⡏⠀⠀⠻⣿⣶⢦⣄⠀⣿⡷⡄⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣾⣿⣿⡿⢋⣤⣤⣤⣤⣾⣛⣛⣿⣿⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠐⠛⠛⠛⢛⣻⣿⡿⢟⣯⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢿⣿⣿⣷⠀⠀⠀⢀⣠⡾⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⠀⠻⢿⣿⣿⣾⢏⣾⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⣠⣤⣧⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⣿⣿⠿⠛⢛⠿⣿⣿⣗⢯⣟⡿⢿⣿⣿⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⢻⣿⣿⣿⣿⣿⣿⣷⣄⣠⣿⣿⣿⣿⣿⣷⣭⣛⡷⢶⣥⣤⣄⣀⣻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⠉⠹⢿⣿⣿⠃
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⠻⣦⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⡆
⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠖⠙⠀⠈⢷⣀
⠸⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⣰⣶⣾⣷⣀
⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣄⣿⣿⠙⠿⣿⣷
⠀⠉⠛⠛⣻⣿⣿⢿⣿⣿⠏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀⠀⠁⠘⠁⢸⣿⣿⢽⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⡆
⠀⠀⠀⠈⠉⠉⠁⠘⠋⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣆⡀⠀⢀⡀⢸⣿⣿⠺⠟⠛⣻⣿⣏⣟⣿⣿⣿⣿⣿⣿⣿⠟⢸⣿⡇⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣷⣿⠃⠀⠶⠰⠿⠿⠿⠛⠿⠿⠾⠿⠛⠉⠀⠀⠈⣿⡇⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡿⠋⠉⠉⢩⣯⣿⠷⠟⠛⠉⠁⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆

⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⢶⣿⠛⠉⢸⡗⠓⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠄⠒⢲⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣶⡾⠿⠛⣿⡇⠀⢈⣿⢄⠀⢸⡧⣭⠀⠻
⠀⡇⢀⡄⠀⠀⠀⢀⠀⠀⣀⡀⢠⢴⡲⠊⠀⠀⠀⠀⢂⠁⠀⠀⠀⠓⢲⡖⠒⠒⠒⣞⠟⠉⠁⢹⣧⣤⢶⡾⠉⠀⠀⠀⣀⠱⠘⣿⠁⣠⣀
⣤⡇⣜⣁⠀⠄⠅⠀⢿⠠⡏⠄⢘⢸⣧⣶⣶⡆⠀⠀⠿⠷⠀⠀⠀⠐⠀⣷⠀⠀⠀⢻⠲⣄⣀⣸⡗⢿⠏⣀⠀⣛⠂⠀⠄⢷⡄⡇⢺⡏⡍
⠁⠙⣿⣷⠀⠀⠀⠀⠌⣆⣀⠁⠘⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡖⢀⢻⡴⠖⠚⠛⣯⠟⠏⠈⠁⡾⢀⣿⠀⠀⠀⠀⠐⢠⣄⢸⣿⠀⣄
⠃⠒⣿⣾⣄⢙⡆⡿⣳⢿⡀⠀⣾⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠳⣦⠈⡷⣆⠀⠀⠀⡈⠀⠀⠀⢻⣸⣿⠀⠃⠒⠦⠀⣸⣿⡾⢏⣀⠤
⠚⠃⠀⠜⢌⣉⣽⡽⠿⠻⣠⢾⠘⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣷⠀⠀⠁⢰⠁⠀⠁⠀⠁⣿⢿⠃⣄⣀⣀⢴⣿⡗⠉⢈⣁⢰
⠀⠀⠀⠀⢀⠀⢛⠀⠀⠀⡈⢾⢨⠶⣿⠉⢹⡇⠶⢤⣄⣀⠀⢀⣰⠀⠂⠀⠩⠐⠃⠒⢰⡇⢗⣠⠦⣅⠀⣠⡷⡔⣾⠕⠋⣡⣿⣿⠟⠀⠐
⠀⢀⣦⢠⠁⠊⠭⠀⠀⡀⡁⢸⣿⠀⣿⠀⠈⠁⠀⠈⡁⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⠉⣠⡏⠁⡴⣿⡟⠁⠀⠀⡏⡞⣼⡟⡿⠁⠀⠀⠈
⠀⢼⣿⢿⡇⢠⠀⠀⣘⡃⡿⠻⠛⠀⣿⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⣤⠾⡷⠶⣆⠜⢁⣐⣾⠷⢟⠠⠆⠀⡀⡇⣧⣿⠀⠀⢀⡀
⠉⠁⢻⠀⣱⢊⣤⡔⠋⠹⣷⠆⠀⠀⢿⠃⠂⠐⠂⠀⠁⠀⠙⠐⠖⠶⠾⣧⡀⢰⡞⠉⠙⣍⠃⠀⠀⠀⠡⣴⡞⡯⠦⣯⡿⣴⡿⢟⣻⠿
⡀⠠⢈⣺⣿⣽⣤⠤⠀⢀⡿⣿⣤⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⢻⠆⠀⠀⠁⠀⠀⠀⠀⠠⠼⢃⡅⣷⡸⡇⠀⢠⡤⠄⣄
⠤⢖⡗⣉⣿⣮⡉⠗⠀⠈⡏⢺⣿⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠹⢄⣀⡀⠀⣀⠤⠶⣏⠀⠀⢜⢧⡹⡇⠁⠀⠈⣀⣤⠀⠈
⠀⠀⡇⡴⠋⠀⠀⠀⠀⠀⠁⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡇⢀⣰⣇⠀⣤⣄⠓⠱⣤⠈⠀⠀⠀⠉⠀⡀
⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡗⠉⠀⠀⠀⡀⠈⠹⢦⡈⣎⠈⠀⠀⠀⠈⠉
⠀⠀⢇⣣⡖⢀⠤⠤⣤⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠘⠐⠵⠋⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣾⣆⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠩⣽⣛⣳⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠉⠉⠛⠆⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡧⢀⠀⢀⠀⠀⣠⣷⣿⡇⠀⠀⠀⠀⣹⣿⣿⣄⠀⡇⠀⣿⣿⣿⣿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⣠⡟⢋⣽⣿⡇⠠⡀⠀⣸⣿⣿⠿⢛⣡⣤⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣦⡀⡠⠀⠈⠳⠀⠈⠉⠁⠀⠙⠛⠉⢉⡠⢂⣭⣿⣋⡤⠛⠉⠉⠛⢻⣿⡏⠀⢀⣠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈⠁⢠⠟⠛⣥⣄⠀⠒⠦⣶⣄⠀⠀⠀⠀⠀⠀⢉⣴⣿⡿⢋⣡⢔⣪⣵⣶⡿⠋⢙⣣⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣶⠀⠁⠀⠀⠀⠉⡙⢦⣄⠀⠉⠓⢄⣀⠀⠀⡴⣿⣿⣿⣿⣿⡿⣟⣋⠉⠉⢀⣀⣛⣉⡀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠰⣆⡉⠳⣄⠀⠀⠙⠿⠶⠾⢻⣿⣿⣿⠟⠋⠉⠁⢀⣤⠚⠉⠉⠉⠉⢻⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⠀⠀⠀⠀⠄⢀⣠⣼⣷⣄⠈⠓⢤⣀⣀⣴⡶⠟⠋⠁⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣷⣄⣴⡿⣋⡥⠀⠀⠀⠀⠀⠘⢂⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣦⣷⢀⡄⠀⠀⠀⠀⠀⠈⣉⠛⠿⠛⢋⣠⣾⣿⣡⣤⡀⠀⠀⣈⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣥⣤⡔⣠⣴⡘⢣⣤⣤⣶⣿⣿⣿⡿⠋⢠⣽⣿⠠⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣠⣼⣿⣿⡇⡀⢀⣠⠀⡀⠀⣀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⣸⠏⢿⣿⣿⣿⣿⠟⠋⢀⣰⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣈⣻⣆⡄⣿⣦⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠄⢠⣿⠀⡈⠻⠿⠛⠁⣠⣴⣾⣭⣿⡿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠀⢸⣿⣷⠟⠀⢀⣤⣾⠿⠿⠛⢉⠁⠄⣐⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⠃⠀⠘⠋⠀⣠⣴⣿⣿⣥⠀⠀⠈⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡼⠀⠀⢀⣴⣾⣿⣿⣿⣿⠋⢀⣀⠀⠀⠠⠄⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⠀⢠⡄⠡⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣤⡴⠚⣿⣿⣿⣿⣿⣿⣿⣶⢿⣿⣿⣿⣷⣧⣤⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⡄⠀⠁⠀⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡟⠉⠀⠀⣿⣿⣿⣿⣿⣿⠛⠑⠾⠻⠿⣿⣿⡁⣿⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡄⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⢸⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠙⠃⠠⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡄⣿⣿⠟⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⠀⣸⣿⣿⣶⠀⠀⠀⣄⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⡀⣿⠹⠀⢿⢿⣿⣿⣧⡀⠀⣿⣿⢀⣦⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠁⡿⠀⡀⠘⠈⠋⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⡿⣿⣿⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢿⠿⡟⢸⣿⣿⣼⣿⢿⡿⢹⣿⣿⡇⣿⡏⠈⠻⡇⢹⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢸⣿⠋⠏⠀⠘⢁⣼⣿⣿⣷⣿⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣸⠹⠿⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠝⢍⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢐⢀⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡓⠼⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠛⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠿⢸⠸⠇⡇⠹⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⢠
⠀⠀⠀⠀⠀⡠⠔⠒⠸⠤⠤⣀⡀⠀⠀⠀⠀⢀⢀⠤⠤⠗⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠜⠀⠀⠀⠀⡠⠚⠁⠀⠉⠉⠉⠁⠈⠀⠑⢄⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠂⢀⠜⠁
⠀⠀⠀⠀⠑⡤⠤⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⢤⠊⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠜⠁⡴⠁
⠀⠀⠀⠀⢀⠃⢀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡄⠘⡄⠀⠀⠀⠀⠀⢀⠜⢁⡴⠃⣠⠊
⠀⠀⠀⢀⣜⣒⠚⣄⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡷⢒⣳⣄⠀⠀⠀⡴⠁⡠⠊⢀⠜⠁
⣿⠛⠛⠛⠛⢻⠋⠉⠉⠉⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢳⠨⠽⠋⠀⡠⠊⠀⠀⠀⡰⠃
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⡇⠈⠀⢰⢤⠜⠀⠀⠀⣠⠊
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢨⣿⡅⠀⠀⠀⠀⠀⡃⡗⠒⢺⢶⣕⢄⢀⠔⠁
⣿⣀⣀⣀⡀⠠⡧⠤⠤⠤⠄⣿⠤⠤⣿⠿⡿⠤⠤⠤⢤⠴⠁⢣⠔⣋⢶⣛⡟⠛⠂
⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⡇⠀⠃⠀⠀⠀⢸⠀⠀⠣⠬⠼⠒⠁
⠈⣿⡀⠀⠀⠀⠃⠀⠀⠀⢸⡿⠤⣀⡇⠀⢠⠠⠤⢄⡸
⠀⠘⣷⡀⠀⠀⢠⠀⠀⢠⣿⠛⠛⢂⠇⠀⢇⡚⠛⢓⡝
⠀⠀⠘⢿⡢⢀⣸⣀⢔⡿⢫⠉⠉⢸⠀⠀⢸⠈⠉⢸
⠀⠀⠀⠀⠉⠳⠯⠖⢋⡠⠚⠁⠈⠁⡧⠀⡇⠉⠀⠉⠢⣀
⠀⠀⠀⠀⠀⢠⡔⠊⠥⠤⠐⠒⠂⠉⠀⠀⠈⠁⠐⠒⠀⠤⠭⠲⡄

⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠩⡜⠻⠆⠒⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢌⠃⠳⠶⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⢀⣤⣶⣕⠲⡈⢳⡄⠀⠀⠀⢀⣠⠶⢾⡛⡻⣿⠟⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⣏⡬⣯⣇⠇⢻⡔⠹⡒⡇⠀⠀⣰⢋⡤⣸⣾⠷⣣⣰⠿⣇⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣐⣨⢟⣢⠽⢿⠿⢿⣳⢁⣟⣿⣿⣃⣘⡷⣿⢿⠀⢙⣯⣢⣯⡊⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⠦⠶⣾⡤⡄⣛⡿⢉⣿⡷⣁⣽⡇⠋⠨⠌⡟⡋⣿⣿⣿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠉⣹⣾⣭⣞⣯⠹⢟⠊⠞⢮⡏⣛⠛⣒⣒⣺⣷⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠋⠉⡍⣉⠉⣩⣿⣿⠿⣷⣉⢍⠢⡈⢆⢧⢨⠝⡛⠫⡋⣽⢻⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⠀⠀⢂⣠⣿⣿⠟⠁⠄⣿⣈⠣⡆⢁⠸⣎⠧⢤⠬⠒⢩⣭⣭⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠔⣀⠂⠔⣤⣼⣿⠟⢡⠄⡒⢠⠆⠨⠀⠟⢆⢤⠿⡍⢠⣼⠒⣁⣥⣸⣘⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠃⡴⠠⠘⢊⣽⡿⠋⠈⠊⠀⠀⠈⠁⢀⠀⢆⠸⠘⢩⡐⢻⡄⠀⠉⠀⣠⣤⣯⣛⣍⡉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡐⣄⠀⠛⠃⢀⣴⡿⠋⢀⡀⠈⡀⡠⠀⡠⠀⠀⠀⠸⡤⢳⣣⡩⠲⣧⠀⢀⣴⣿⡻⠿⠿⢉⠉⢸⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸
⢸⡾⠟⠲⠶⣄⠀⠔⢁⡿⢀⡔⢸⣿⣏⣳⣭⣭⣉⡛⠛⠚⠚⠳⠶⠷⠚⢓⣛⣊⣉⣳⣧⠘⠇⠉⣉⣠⣴⣶⣦⠀⡇⣎⣀⣤⣴⣶⡿⡟⠛⠳⣄⠀⢸
⠈⡟⠂⠀⠉⠀⡟⠀⣺⠁⡾⡇⢵⡟⠛⠓⠂⠁⠈⠩⠛⠿⣿⣿⣿⣿⠛⢻⠀⠛⠿⢭⡻⣆⣰⣿⣿⢯⢅⣼⠇⡜⣠⣿⡿⢿⢿⠟⢂⣠⣦⡰⠾⣆⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⢀⣤⣁⣀⠤⠖⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣿⣿⣿⣿⣶⣤⣴⣿⣾⣿⣥⣶⡶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⡿⣿⡛⠛⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣄⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣭⣿⣿⣿⣴⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷⡘⠻⣿⣿⣿⣿⣿⡿⠛⠧⢀⣰⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣟⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣨⣽⣿⣿⣏⠀⠀⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠛⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠧⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠹⢷⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠠⠔⢖⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⢄⠀⠀⠑⢁⡀⢢⠀⠀⠀⠀⠑⠆⡠⠠⣶⡦⠀⠀⠀⠠⠀⠠⢻⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡌⢢⡀⠘⠁⠀⠀⠆⠀⠀⠉⠈⠙⣄⡘⠁⠐⠃⢠⠐⠀⠀⠀⢹⣧⠀⠀⠀⠀⠠⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⠄⠑⡄⠀⠂⠀⠀⡄⠀⠠⠤⠓⣂⣕⠀⡀⠐⡡⠂⠀⠀⠀⠀⢩⢃⠀⠀⠀⠀⠑⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠀⠞⠂⠂⢀⡈⠳⡀⠂⠀⠀⠤⣼⣿⣥⣯⢤⣀⡀⣀⠀⠀⠀⢆⢣⠀⠀⢀⣠⠘⣐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢉⠄⠐⠂⣼⠀⢆⣨⡔⠿⠓⣀⣷⣁⣈⣀⣉⣈⣀⣈⣈⣆⣩⡉⠁⠁⠀⣽⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠠⠀⠂⠈⠈⡀⠡⡘⠓⠋⠉⠉⠉⠛⠃⡉⠉⠩⢭⠉⠉⠩⠟⣿⠆⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⠚⠙⡅⠀⠀⠀⠈⠀⢀⢤⠀⡀⠀⠉⠧⠴⠶⠶⠂⠀⠀⢰⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠀⠄⠂⣀⡇⠀⠀⠀⠀⠀⡆⠠⠰⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⡞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢸⡿⡝⠀⠀⠀⠀⠀⠀⢡⠀⡷⠀⢤⡖⢠⠀⠀⠀⠑⠀⡰⠁⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠋⡘⣞⡄⠁⠀⠋⡃⢇⣠⡀⢀⠠⡇⣀⢸⠀⢤⠀⠸⠣⡌⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⢡⡄⠀⡆⠁⢸⡿⠛⢻⡀⠄⡽⢙⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣖⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡀⠀⠃⠀⠃⠀⠀⠀⠀⣾⠉⠃⠀⠸⡀⠀⠆⠀⠀⠀⠀⠑⠄⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠀⠘⠄⠀⣰⣏⠀⢠⠀⠀⠘⢄⠰⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⡤⢠⢀⣿⣿⣄⢀⡀⢀⠀⠘⠀⠀⠐⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⢁⠀⠀⠀⠐⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⣿⣧⡈⣾⣿⣿⣿⡏⡅⡿⠀⢰⠗⠠⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠃⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⢿⠉⣱⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡟⠁⠀⢠⣿⣿⡟⠁⢻⣿⣿⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠀⡀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⢠⣿⡿⠋⠀⠀⠈⣿⣿⡀⠀⠀⠀⣾⣿⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⣰⣿⡿⠁⠀⠀⠀⠀⢘⣿⣇⠀⠀⠀⠛⠿⠿⣷⣄⠀⠀⠀⠂⠀⠔⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣀⣴⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡖⢶⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣤⣤⣀⠀⠀⠀⠀⢀⣴⠛⣳⣄⣤⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠉⠻⢿⣿⣿⣮⣿⠻⠯⢻⣆⠀⡰⠻⣿⡟⠋⠛⡝⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⣀⣈⣷⣀⣀⠸⣿⡜⠁⠈⣿⣶⠄⠉⠻⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠈⠻⢏⣹⡏⠉⢀⡿⠀⠀⠀⣿⠧⠖⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡁⠄⠀⠘⡏⠣⠀⣺⡇⠀⠀⢠⣿⡀⣀⣠⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠉⠁⠀⡠⠖⢚⠉⠀⠈⢿⡇⠀⠀⠸⣿⣿⡿⠛⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣾⡿⢿⣿⣿⡿⠇⣀⠙⢆⠀⠀⢻⣿⣿⠷⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣅⣿⡷⠊⣼⠋⣠⣿⡿⠠⠀⠑⠠⣴⣿⣿⣤⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣻⡿⠁⣠⢆⡼⣹⡧⠀⠀⠀⠒⣤⠚⢉⡿⣏⢾⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣴⣿⢣⣟⣽⡇⣰⢟⣿⣇⠝⠀⠠⠔⣲⣯⡀⠀⢰⣿⣿⢤⠹⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠰⢾⡇⣼⣏⣿⢀⣽⠋⣰⡏⠀⠀⣴⡖⠎⠀⠀⢤⡿⠃⠈⢿⡆⠑⢶⣜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⣡⠛⢰⣿⢻⣯⣜⠃⢀⡏⠇⢠⡾⠋⡀⠠⣠⡶⠛⠁⠀⠀⠀⠙⠳⣶⣤⣭⣛⣴⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⡏⠀⣼⣟⣿⣿⠋⠂⠈⣦⠄⣼⡶⠾⡷⣺⡟⠁⠀⠀⠀⠀⠀⢠⠖⠈⠩⡀⠀⣿⣿⣦⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣅⡈⠁⣚⣿⢻⣿⡿⠐⢄⡄⠱⢏⡿⢰⡿⠛⢁⠌⣠⡴⠆⠀⠀⠀⣠⠊⠔⢳⠍⢰⣿⡯⡿⠙⣴⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⠛⣿⣿⣻⡟⠀⠢⠀⡀⠀⣼⡷⢻⠁⠀⢁⠩⣠⣿⡆⠀⢠⡀⡁⠀⣰⡏⠀⣾⣿⣟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢈⢫⠀⡀⣿⡟⣿⡇⠀⠀⢠⣷⠔⠹⡀⢿⣎⢀⣠⡾⠋⢀⣠⣄⠀⢿⣿⣿⡿⠃⠀⣿⡹⡏⠀⣶⠿⢄
⠀⣀⡀⠀⣀⠤⠴⠿⠁⡀⡱⣿⣿⣿⡇⠀⠀⢸⣿⠃⢀⡇⠈⢻⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⡿⠀⠀⣰⠿⣧⠁⠀⠉⠀⠘
⠰⢿⡄⠋⠀⠀⠀⠀⢚⡽⣱⠿⣿⣿⡇⠀⢀⢻⣷⣶⣾⣿⣶⣿⢟⣹⣿⣿⣿⣿⣅⡀⠀⢤⠀⢀⣞⠿⠚⠛⠃⣀⠃⢀⣼
⠀⠀⢡⡀⢀⡄⠀⢲⣿⠿⠃⢠⣿⣿⡧⠀⠘⡚⠛⣩⣿⣿⡿⠿⣫⣿⣿⣿⣿⣿⠏⠁⠀⠀⠰⠿⠁⠀⠀⠀⠤⠿⠦⠜⠁
⠀⠀⠈⣿⣍⣤⣞⡃⢀⣀⡴⣿⣿⣿⠀⢀⡔⠀⣹⣿⠈⠿⠓⠈⣥⣤⠴⣿⢿⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠬⣾⣛⣿⣿⣿⣿⣷⣾⡅⠘⡆⠀⠀⠀⠀⠉⠀⠁⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⠷⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠈⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⠜⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣦⠀⢈⡉⠁⠀⡝⡇
⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀⠈⠃⢆⡀⣠⣝
⠀⠈⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠿⣉⣳⣊⢫⡍⢠⡄⠀⢀⣀⣰⣿⢃⡀⠀⠀⠀⠀⠻⣇
⠀⠈⠁⠀⠀⢀⣤⣄⣀⣀⡀⠀⠀⠀⢠⡟⠁⠀⣹⣯⣷⡈⠂⢠⢿⡄⢸⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣽⣯⠈⠁⢿⣿⣿⣿⠀⢸⣞⣛⣟⣡⡛⢛⣯⡆⠀⠀⠀⠀⠀⠸
⠀⠀⠘⣿⡿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⠀⢨⠟⢿⣿⣤⡘⠉⢿⡿⣀⠁⠄⠿⣀⣄⣀⡀
⠀⠀⢀⣿⣿⣽⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠞⠐⣿⣿⣿⣿⣦⡠⠇⢠⠰⠆⡀⠃⠀⠄⠹⠶⣾⣾
⠀⢰⠀⠘⠿⢏⡉⠷⣈⣿⣼⣿⣧⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⡘⣄⣠⠡⣠⡤⠈⠴⠀⠊⣿
⠀⢈⣀⡀⠀⠈⠃⠤⠟⢻⣿⣯⣿⣿⣷⣿⡇⡄⡀⠀⢸⣻⣿⣿⣿⣷⣰⣿⣄⢀⠠⠈⠁⠀⠀⢀⡀⢘
⠀⠈⠁⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⢻⣿⡿⢷⣾⣧⡅⢰⣷⣿⣿⣿⡇⠘⠻⠓⢺⠇⠀⠀⢀⠔⠀⠉⠘
⠀⢀⠄⠀⠀⠀⠀⣴⡿⠟⠁⠀⠀⠀⣿⡇⠀⠈⣿⣿⣿⣿⣿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠰⡀⠑⠄⠘
⠀⠈⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⢿⡇⠀⣶⣿⣿⣿⠟⠉⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⣹⢠⠀⠀⢸⣷⠀⢙⣿⠛⠁⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈
⠀⠀⡄⣀⠀⣀⡀⢀⣀⣀⣀⠼⡠⠀⠀⠸⠰⢿⡿⣀⠤⡀⠀⠤⠤⠙⡷⠀⢂⠂⠀⠄⢂⠰⠈⠁⡄⢸
⠀⠈⣗⠑⣖⠺⠖⠲⢟⠊⠛⠊⠒⠛⠛⣃⠛⠋⠓⠒⠒⠒⠒⠛⠓⠚⠛⠒⠛⠛⠛⠓⠚⡹⣓⣛⡟⡿
⠀⠀⣍⡑⣟⣓⠆⢀⣨⠓⣀⢀⣀⣀⠀⠈⢿⠀⢀⡀⠂⣀⣀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠈⠧⡛⡟⡆⠥
⠀⠀⣄⡂⠀⢩⣾⣿⡟⢐⣻⣿⣿⣿⠆⡺⠀⠈⢁⢠⣾⠿⢿⣿⣿⣶⣶⡖⠘⣢⠀⠀⠀⠀⠀⠀⠌⢒
⠀⠘⠇⠇⠀⠺⠛⢻⣧⡙⢳⣿⣿⣯⡄⠋⡎⡀⠒⠐⢀⣠⣤⣽⣿⡿⣃⣠⣾⣷⣶⣄⣀⠀⠀⠀⡆⢴
⠀⠀⣉⡇⠀⢀⣴⣿⣿⣿⠼⠛⠛⢩⡿⠀⡋⣄⠀⠀⢹⠋⠀⠀⠀⠀⠉⠉⠉⠛⢿⣿⡻⣶⣄⠀⡇⣹
⠀⠀⢡⢻⣾⡟⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠲⠈⢤⠐⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⡴⠇⡊
⠀⠀⠀⠲⠙⠞⣄⠈⡳⢷⠤⣄⢀⡀⡀⢠⡄⣀⠀⠀⠐⠄⠄⠠⠀⠀⠄⠤⠤⢤⠄⠤⠤⠴⠟⣁⠞⠁
⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠂⠚⠀⠒⠒⠒⠐⠀⠀⠀⠀⠂⠒⠒⠒⠂⠐⠀⠂⠂⠒⠐⠒⠒⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣴⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠋⠀⠀⡈⠠⠀⠀⠀⠀⢄⠀⠀⡀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⢤⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠿⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣠⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠐⠿⣻⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣁⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣉⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣹⠿⠿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠄
⠀⠀⠀⠀⠀⠀⢀⢰⢿⠁⠊⠀⠀⠀⠀⠈⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⢆
⠀⠀⠀⠀⠀⠐⢃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⢠⣠⡀⠀⠀⠠⠋⠀⠐⠀⠠⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⣁
⠀⠉⠛⠲⢦⠞⠀⠀⠀⠀⠀⠈⠁⠉⠛⠲⠤⣄⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡖⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⠿⠿⠿⠦⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⡀
⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠤⠄⠀⠂⠑⢄⠙⠀⠀⠋⠀⠐
⠀⠀⠀⢀⠴⠧⢄⡀⠀⠀⠀⣼⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⠀⠘⡄⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⣄
⠀⠀⠐⠁⠀⠀⢀⣠⠴⠋⠓⢦⣐⣀⠀⠻⡒⢄⠀⢀⡠⠊⠀⠀⢰
⠀⡶⢒⠒⠒⠋⠉⠀⠀⢀⠀⠀⠈⠉⠛⠒⢛⡲⡎⠁⠀⠀⠀⠀⣠⢔⡂⢄⠀⠀⠀⠀⠀⠀⢤⡀
⠀⡇⠉⠀⢠⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⠀⠈⠁⣇⠀⠀⠀⠁⠀⢬⠉⠉⡥⡀⠊⠀⠀⡀⠀⠀⠱⡀
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⡟⣆⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠈⢣⡀⠀⡐
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⢦⡀⠑⢄⠈⣀⡀⣀⣀⠄⠠⢀⣀⠀⣀⡰⠋
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠑⠄⠀⠑⠦⠀⡀⠔
⠀⢣⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡏⠉⢳⠀⠛⠂⠀⠘
⠀⠘⣆⠛⠀⠘⣄⠀⠀⠀⠀⠀⢀⡼⠁⠀⠛⡼⠒⠒⠚⠀⠀⠀⠀⠘⡀
⠀⠀⠘⢆⠀⠀⠈⢦⡀⠀⠀⢠⠞⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠈⠳⣄⠀⠀⠙⠦⠔⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃
⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⡀⠀⣠⡞⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⠴⠊⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃

⠀⠀⠀⠀⢠⡟⠿⢻
⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⢸⡇⣶⠿⠯⠭⠭⠽⠿⣆
⠀⠀⠀⠀⢸⡇⢻⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⣠⣿⣿⠟⠻⠛⠛⠻⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡴⠒⢺⢛⣡⢄⠀⠈⠀⠀⠀⠈⠁⠀⣠⠈⡛⣗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠙⠻⣿⣿⠟⠿⢿⣿⣿⡄
⢧⡀⢻⠀⠉⠉⠀⠀⠢⠄⠠⠔⠁⠀⠈⠉⠁⣽⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⠀⠰⠀⠀⠀⠰⠆⠀⠘⠷⢀⣀⣿⠿⣿⠃
⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⡁⠀⠐⠤⠔⠀⠀⠘⠑⠀⠀⠁⠀⠀⢸⠆
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣄
⠀⠀⠀⠀⠐⡿⢻⠛⡖⠒⠒⣲⠛⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣛⡖⠒⠒⠒⢛⡏⠛⣿⣿⡿⠟⠛
⠀⠀⠀⠀⢰⠃⡞⠀⠘⠒⠒⠁⠀⢻⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢳⡙⣟⠿⢿⠋⠙⠦⡀⢸⡿⠿⠄
⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠸⡆⢧
⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⡼⢁⠞⠀⠀⠀⠀⠀⠀⣇⠘⡄
⠀⠀⣠⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⠿⡖⠉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡀
⠀⠀⠙⠒⢺⡗⡶⢶⡦⠤⠴⢶⡶⢶⡞⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣷⡶⣶⠤⠤⠴⢶⡶⢶⠛⠒⠚⠁
⠀⠀⠀⠀⢸⡇⢧⠀⠈⡗⢺⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⢹⡒⢺⡅⠀⢸
⠀⠀⠀⠀⢸⡇⢸⡄⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢻⠀⢸⡇⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢈⣇⠀⣇⣸⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡆⠀⡇⠀⣧⠀⣸
⠀⠀⠀⣴⠚⠉⠉⣽⣀⡿⢹⣀⣺⠉⠉⠛⣢⠀⠀⠀⠀⠐⠒⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡗⠛⣇⣀⡏⠉⣿⣀⡸⠓⠒⠢⣄
⠀⠀⠀⡏⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡗⠒⠒⠲⠶⠶⠶⠒⠒⠒⠚⠉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠑⠀⠀⣀⡀⠀⠀⠈⢧⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠵⡒⠢⣄⠀⢺⡵⠋⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠰⣿⣮⠾⣷⠈⡄⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⡏⢰⡟⢀⢀⡾⠁⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢉⠹⠿⣦⣮⣼⣅⣱⡀⠀⠀⠀⠙⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⣀⠔⠩⠑⣄⠀⠙⢿⠻⠿⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡴⠞⠁⠀⠀⠀⠈⠑⠦⣤⣷⠀⠀⠀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⡴⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠠⣶⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⡦⠠⢿⠀⣤⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠙⠛⠛⠶⠶⠶⠚⠛⠛⠉⠀⣠⣾⣤⠼⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⡂⠄⡀⠈⠲⠂⡀⡀⠄⡂⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠂⡁⠆⣈⠀⠀⡁⠆⡁⢂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠃⠤⠀⠀⠄⠃⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠿⠶⢄⢀⢀⠤⠊⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⢿⡁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠝⠛⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣿⣻⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠋⠉⠉⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠁⠀⠁⠈⠁⠉⠑⠢⡀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⢆⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⢸⡑⢄⣧⡀⢰⠐⡖⠢⡘⣢⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡃⣤⡁⠙⠋⠳⠇⠀⡏⡇⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣠⡇⠀⠀⢀⠀⠇⠈⠻⠂⠂⠀⠀⢊⠀⠆⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡆⠀⣸⡞⠀⠀⠀⠀⠀⠀⣠⠚⣆⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣉⣢⡀⠀⢀⣠⣴⠎⠊⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⣚⣿⣭⢿⠙⢡⠇⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠝⠻⣄⡁⣀⡞⡊⠁⠀⣠⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⡥⠒⠁⠀⠀⢀⠜⠉⠀⠈⠉⠙⢧⡤⢄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠻⠒⠁⠀⠀⠀⢀⢼⠀⠀⠀⠀⢰⠀⠀⠀⣠⠃⠀⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡄⠀⠀⢀⡠⢧⠴⠊⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠘⡦⢚⠥⠖⠁⠀⠀⠀⠀⠀⠠⠀⠀⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡀⠀⠀⠀⢀⠞⠉⠅⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠮⢄⣀⣈⣑⢢⡴⠁⠀⠀⠤⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠦⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣚⣻⡍⠉⠁⠀⠲⠖⠒⠛⠓⠀⠀⠀⠀⠀⠲⠶⠦⣭⡉⠙⠓⠦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⢯⣍⠉⠉⠉⠛⠛⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠛⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣧⡆⠀⠰⠖⠛⠛⢻⡆⠀⠀⠀⠀⠀⢀⡏⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣻⣏⣀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⣸⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⢀⣄⣀⣤⣤⣼⡇⠀⠀⠀⠀⢀⡟⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣌⣽⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⠛⣻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⠁⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠀⠘⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⣿⠄⢠⣤⣤⣤⡤⣿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⣿⠒⠀⠀⠀⠀⠀⠾⠧⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣁⡀⠀⠀⠀⠀⠀⣼⡇⠀⠴⠶⠶⠚⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣤⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠖⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢿⣟⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠙⠳⠆⠀⠈⠀⠀⢉⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠷⠄⠀⠀⠀⣀⣴⠿⠛⣋⣉⣉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⢀⣼⠟⣡⡾⠟⢛⣙⠛⠿⣶⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⠀⠀⢀⣾⠃⣾⠏⣴⡾⠛⠛⢿⣦⠘⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠲⠀⠀⢸⡏⢸⡏⢸⡏⠀⠀⠀⠀⢻⡇⣹⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡶⠶⢶⣶⠶⠶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠙⢷⣦⡀⢸⡇⢸⣇⠸⣷⣄⠀⠀⢀⣾⢃⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⢿⡄⠀⠉⠳⠀⠀⠙⠃⠀⠈⠙⠂⠈⠻⠟⠛⢿⠶⣶⣤⣤⣭⣿⣿⣿⡄⠻⣷⣬⣉⣠⣴⠟⣡⡾⢃⡾⠃
⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⠉⠙⠷⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡙⠊⠉⠻⠍⠙⠿⣶⣬⣉⣉⣩⣥⡾⢋⣴⠟⠁
⠀⠀⠀⠀⠺⣿⡁⠘⠿⠆⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢷⣶⣤⣤⣄⣀⣉⣉⣉⣩⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢀⣠⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⣿⣷⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⢻⣿⣿⣿⣿⡿⠛⢉⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋⣠⠾⠋⣁⣠⣄⣀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⣿⣿⡟⢀⡾⠃⣰⠞⢋⣉⡉⠙⢻⣄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠁⣼⠃⣼⠃⣰⣿⣿⣿⣷⠀⢻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⣿⠀⣿⠀⢻⣿⣿⣿⣿⠃⣼⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⢶⣶⣦⡐⣶⣶⣄⠠⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⡄⠘⢷⣄⣸⣿⡿⠋⣰⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠲⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣷⣾⣿⣦⣶⣦⣀⢤⣀⠀⣀⣀⠈⠻⣦⣄⣉⣉⣁⣤⠞⢁⡴⠋
⠀⠀⠀⠀⢀⣠⡀⠘⢿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣾⣿⣷⣦⣀⣉⣉⣉⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠈⢻⣿⣶⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢿⠁⠀⢀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠶⠀⠀⡞⠠⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠐⢮⣁⣼⣧⣀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⢀⠀⢀⡀⠹⣯⡙⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣸⣿⣿⣿⡉⠀⠉⠀⠀⣾⣷⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡄⠀⢻⡆⠀⣴⣿⣿⢿⣿⣷⣀⠄⠀⠀⢸⣿⣿⣿⣿⡆⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡦⠞⠀⣸⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠈⢿⣿⣿⡿⣿⣄⠋⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠀⢠⣿⣿⣿⡿⠛⠉⠈⠀⠁⠀⠀⠀⠀⠉⠙⢿⣮⡛⠃⠀⢿⣤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣾⣿⣿⣿⣿⠀⠀⠲⣶⣠⣀⣀⣠⣤⣦⣁⠈⢿⣿⣦⠀⢿⣿⣾⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢿⣿⣿⠟⠁⠀⠀⠀⠀⠘⡃⣸⡇⠀⠀⢹⡟⠿⣿⣿⠂⢿⣿⣿⣹⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠘⣿⠁⠀⠀⣰⠀⠀⠀⢰⡇⢹⢰⠀⠀⠀⣧⠀⢻⡟⠀⢸⣿⣏⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⠀⠻⠀⠀⣴⣿⣴⠆⣀⣼⡇⠀⣿⣆⠀⠀⢹⠀⠘⠀⠀⣸⣿⣽⣿⣿⢟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⠀⠀⢀⣾⣿⣿⠏⣼⣿⡿⠃⠀⣿⣿⠀⣿⠘⣧⠀⠀⢀⣿⣿⣿⣿⡿⣈⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣿⣿⠃⣴⣿⣿⣿⡏⢰⣿⠟⠁⠀⢀⡟⣿⡇⢸⣇⠘⣷⣦⣾⣿⣿⣿⣿⣿⣼⣿⣷⡇⣤
⠀⠀⠀⠀⠀⠀⠠⣬⣿⣿⣿⡿⡋⣸⣽⣿⣿⡟⢠⠟⠁⠀⠀⢀⡾⠁⢹⣧⠘⣏⠛⢉⡴⠀⠉⠛⠛⠿⢿⣿⣿⣁⣾⣿⡇
⠀⡀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⣠⣿⣿⣿⡏⣴⠋⠀⠀⠀⣠⠈⠀⠀⠀⠿⡆⢻⣰⢣⠀⠀⠀⠀⢀⠀⠀⠉⠻⣿⣿⣿⡟⢠⣀⡀
⠀⣿⣦⣤⣼⣿⣷⣿⣿⣿⣴⣿⣿⡏⣽⢏⠜⠁⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠀⣀⣤⡴⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⡿⣯⡿⠻⠕⢸⡿⢰⡟⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⢭⠴⠞⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⠏⢽⣴⠏⠀⠀⢀⣿⠇⠘⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⣿⡏⠠⢄⡈⡻⣦⣾⣆⠾⠉⠀⠀⢿⡄⢸⣆⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿
⠀⢿⡤⣤⣴⣿⣛⣮⣿⣿⡄⠀⠀⠀⢸⣿⡌⢿⣿⣴⡀⢿⣄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿
⠀⣇⣻⣶⣤⣤⣭⣝⣿⡿⢿⣆⠀⠀⢸⣿⣿⡜⣿⣦⣦⡈⢿⣅⠀⠀⢾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣉⣿⣿⣿
⠀⠉⢱⣬⣿⠿⠛⠋⠁⠶⣾⣿⣶⠀⠀⣿⠉⣿⣸⣿⣿⣿⣾⡻⣷⣔⢤⠛⢷⣄⠀⠴⢦⣤⣀⣀⠀⢦⣄⠀⠀⠀⣈⠸⡅⠻⣿⣿⣿
⠀⢠⣿⠟⠷⠿⠶⣦⠤⡤⣠⣾⣿⣦⠀⠉⢠⠘⣿⠉⠟⢿⣯⠑⢘⣿⣿⣷⣠⣙⠻⣷⣶⡤⣤⣉⠉⡛⠻⠷⣦⣠⣇⠀⠁⢺⣮⡻⣿
⠀⣴⠟⠀⠀⠀⠊⣤⠀⠀⠈⠀⠙⣿⣧⡀⠈⢄⣸⡦⠌⠰⣤⡖⠉⠴⠛⠉⠁⠈⠑⠀⠉⠃⠈⠉⠀⠈⠀⠀⠀⣠⡴⠞⠻⣿⣿⣿⠎
⠀⣿⣶⣤⡀⠀⠀⣯⠀⠀⠀⠀⠀⢸⢿⣷⣤⣀⣹⢸⣆⣴⠇⠙⠶⠂⣄⣠⣤⠀⠠⢀⡀⠀⠀⣀⣀⠀⢤⡞⠙⠁⠀⠀⠀⠈⠢⠐⠛
⠀⠿⠻⣿⣷⣄⠀⢿⡄⠀⠀⠀⠀⠀⠾⣏⡉⣍⠝⢾⣿⣦⣄⣠⣴⣶⣿⠟⠁⠀⣴⠀⣅⣠⣿⣥⣤⣤⣽⣿⣂⡀⠀⠀⠰⣂⡴⣄⠐
⠀⠀⠀⢻⣿⣿⣅⠘⣿⣄⠀⠀⠀⠀⠀⠘⢿⡄⠦⡄⢛⢀⣿⠋⠉⢹⣿⠀⠀⢰⡟⠀⣇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠻⠿⣭⡿⠟
⠀⠀⠀⠈⢿⣿⣿⣆⢻⡙⣦⠀⠀⠀⠀⠀⢸⠇⢆⣿⢀⣿⣿⡀⠀⢼⣿⡆⠀⣿⠁⠈⠉⠛⠲⠶⣶⣤⣤⣤⣤⣤⣤⣤⡴⠂⠉⠄⣁
⠀⡀⠁⣄⠼⣿⣽⡿⢷⣤⣈⠳⣄⠀⠐⢤⣐⣶⣸⣿⡿⠋⠠⣿⡆⢨⡤⠷⡀⢞⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⡞⠁⠀⠀⣠⣿
⠀⣁⠀⠙⠳⣞⣿⣿⠈⠙⢿⣶⣌⡳⢄⡀⠀⣽⠇⢻⣱⣦⣶⠶⢁⠛⠀⠀⡇⢸⠀⠀⠙⢿⡷⠒⠠⢤⣤⡤⢤⣤⠬⣅⣤⣴⡟⠿⠿
⠀⠀⠣⠴⠀⢬⠙⣯⣦⡀⠀⠉⠉⠻⠦⠙⢲⡏⣠⠟⠁⠀⠀⢠⡀⣠⠀⠀⠁⣸⣄⣀⣀⣠⡹⣆⠀⠀⠀⠀⠸⠃⠀⠀⠙⠟⠁⢀⣴
⠀⠀⠁⠀⠀⠉⣝⣤⡹⠟⠷⠦⣤⣤⣤⢤⣼⠟⣯⠙⠷⠶⣦⠼⠃⠉⠀⠰⠚⠋⠩⠭⣉⡉⣁⣸⣷⣶⡄⠀⡀⠀⠀⢀⣀⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢧⣝⠦⠑⠀⠐⠋⠙⠇⠀⠃⠀⠙⠶⣷⣭⣤⣤⡠⢴⣤⣤⣀⡀⠀⠀⠀⠉⠉⠙⠋⠑⠊⣀⣤⣶⣿⣿⡿⢿⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠠⠄⠈⠀⠀⠀⠀⠄⠀⠀⠀⠈⠻⢷⠀⡁⠀⠀⠀⠉⠙⠛⠻⠿⠳⠞⠾⠛⠿⠿⠿⠛⠡⠈⠀⠁⠈⠉⠿
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠒⠂⠀⠀⠀⠚⠛⠒⠀⠀⠀⠀⠀⠀⠐⠑⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣄⠀⠀⠀⠀⠀⢀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠻⢆⣠⣾⣿⣿⣿⡿⣧⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣭⢩⣾⣿⣾⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⠟⠛⣯⣾⣿⣿⣭⣽⣿⣿⣇⠉⢹⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⢿⣽⣿⡿⣸⣾⣿⣿⣿⣿⣿⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣿⣿⣿⣧⣸⣿⣭⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡧⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠒⣤⠈⠛⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣟⡉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣠⡿⠃⣇⢻⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣇⡿⢿⡛⠋⠙⢻⣿⠃⢠⣿⡄
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⢹⣆⢹⣾⣿⣿⣿⣿⠇⠛⠛⠉⠁⠀⠀⢳⠀⢠⣾⠃⠀⢺⣿⠄
⠀⠀⠀⠀⠀⠸⣿⣾⣶⣶⣿⣿⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⠏⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⢿⡟⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⣽⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣉⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣠⣷⣯⠶⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣨⠟⣿⠀⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡎⠁⣀⣽⡆⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⣾⡇⣼⣿⣿⡆⠀⠀⠀⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢀⣿⣿⣷⣾⣿⡇⣤⣴⡶⢟⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⣏⠛⠻⢿⣿⠛⠓⠁⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⣿⣿⣿⣿⡿⢰⣾⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢉⠿⣿⣿⡇⢸⣿⠿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣬⣷⣦⣽⡇⢈⣠⣾⣯⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⢰⣿⣿⣿⣿⠋⠋⠉⢙⣿⣿⢿⠌⠀⠀⢀⡤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣿⡇⣤⠄⠀⠙⢿⣿⠟⠀⠀⠀⠈⢛⠕⠁⠀⢠⡾⠁⣀⡤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⣀⢀⣿⣿⡇⡄⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣍⡀⣠⣤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⠀⠐⠀⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣿⣿⣿⣿⡇⣿⡄⠀⢣⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡏⣿⡇⣿⡇⢀⣀⣳⣀⣀⣸⣿⣇⣀⣀⣀⣀⡀⠀⢠⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠠⠀⣼⣿⣿⣿⣿⡿⠁⣽⡇⠹⣗⠻⠿⠿⠿⠿⢿⣿⣿⠿⣿⠿⠿⠿⠂⣼⣿⣿⡿⣯⣿⡟
⠀⠀⠀⠀⠀⣠⣦⣀⣄⠀⠸⣧⣄⣿⣇⣀⣈⣤⡄⠀⠀⠀⢸⣿⡇⠀⠘⣦⠀⠀⢠⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣻⣿⣿⣿⢃⠀⠁⠈⠉⢉⣉⣉⣹⠀⠀⠀⠀⢸⣿⣿⣄⠀⠸⣧⠀⠛⠉⠻⣄⡀⠙⡿
⠀⠀⠀⠀⠀⠻⣿⣿⠃⣿⣿⡟⣤⠁⢈⡉⡡⡴⠀⠀⠀⠀⢸⡿⠏⡙⠷⠀⠀⠀⠀⢀⡀⣬⣍⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⡟⠃⣯⠀⠠⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣤⣰⣶⣿⣿⢸⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠰⠌⢿⠆⠐⠀⠒⠉⠀⠀⠀⢀⣠⣤⢶⣷⢻⡟⣛⣡⣤⠤⠴⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⢿⠄⠀⠀⠀⠰⡴⠿⠛⠋⠈⠁⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣀⡷⠒⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠛⠻⠿⢷⣮⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠀⠀⠀⠀⢤⡹⣷⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⡤⠤⣀⠠⡀⠹⣿⡅⠸⡄⠀⠀⠀⠀⠀⠀⠀⡴⡡⠋⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣠⠂⠀⠀⠀⠀⢀⡸⠟⠒⠿⢄⡀⠈⢢⢻⡎⣿⡿⠀⢇⠀⠀⠀⠀⠀⣠⣪⡞⠀⢀⣿⣿⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣿⣶⢿⠃⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠐⡝⡄⠈⡇⣇⢿⢀⡀⣞⠀⠀⠀⢀⢔⣵⠋⠀⢠⣾⣿⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠋⢡⡷⠁⠀⠀⠀⢸⠁⢀⠀⠀⠀⠀⠀⢸⣼⠀⢃⣿⢸⠘⡇⡏⠀⠀⡴⣿⠟⠁⢀⣴⣿⣿⡿⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣡⣠⡇⠀⠀⢘⣴⠃⠀⠀⠀⠀⠀⢸⢃⡇⢸⡏⠈⢰⡇⡇⣠⣾⡾⠋⠀⢀⣾⣿⣿⡟⠁
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⡇⢵⣼⡇⠀⢸⣷⢟⣿⠟⠁⠀⢠⣾⣿⣿⠏
⠀⠀⠘⣿⡄⠀⠀⠀⠀⢀⠿⠿⠋⠀⠀⠀⠀⠫⣗⡄⠀⠀⠀⠀⠀⠘⠀⠀⡼⣷⡄⣿⡏⡼⠃⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⣿⡇⠀⠀⡀⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⣀⣠⣦⣀⡼⣁⣿⣿⣿⠇⠀⢀⣴⣿⣿⠛⡁
⠀⠀⠀⣿⣿⣦⠀⣿⣇⡇⠀⠀⣰⡜⠀⠀⠀⠀⠀⠀⠀⢀⡼⣳⡵⠟⢈⣴⡿⠺⡹⣿⣤⣶⣿⠿⣿⣏⡼⠁
⠀⠀⠀⢸⣿⣿⡄⣿⡿⡇⠀⣦⣿⠁⠀⠀⣠⢤⠖⠒⠊⠁⠚⠁⢀⣴⠟⠋⠀⢀⣧⢇⡿⠛⠁⠀⢻⣿⣷⡆
⠀⠀⠀⠈⣿⣿⡇⠉⠀⡇⠀⣿⡏⠀⠀⡰⣱⡷⠀⠀⠀⠀⢀⡴⠋⠁⠀⣠⣴⡾⠟⢧⣧⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⢹⣿⡇⢰⡄⡇⠀⣿⠁⢀⡴⢛⣳⣶⡖⣲⣶⣴⠋⠀⠀⣀⣼⢟⣫⡖⠁⠈⢻⡄⠀⢠⡿⠃
⠀⠀⠀⠀⠀⢿⣷⠀⣿⣷⠀⣿⢠⢋⣾⣿⣻⠟⢳⡁⠹⡙⣠⣾⠿⢻⣗⢿⣿⡃⠀⠀⠀⠙⢀⡞⠁
⠀⠀⠀⠀⠀⠈⢿⡆⠘⢻⡆⠘⡎⡾⠚⡁⠠⡆⢸⡷⣄⠈⠛⢣⠀⡏⠈⠢⡙⠄⠀⠀⢀⣴⢟⡴⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡀⢹⡳⡄⠹⡆⢻⡀⣷⣹⣷⣭⠭⡾⠀⠀⠀⠈⠢⣖⡶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⡷⠊⠀⣨⣦⣹⠶⢛⣛⢩⣿⡇⡰⠁⠀⠀⠀⠀⠀⠀⣹⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠴⠋⢀⡤⡊⠑⠮⡑⢿⠙⢃⠏⣸⡰⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠂
⠀⠀⠀⠀⠀⠀⢰⠁⠣⣀⡔⠉⠀⢹⣦⡀⠈⢮⢳⣾⢠⡿⠁⠀⠀⠀⢀⣤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠚⠒⠒⠚⠁⠀⠀⠀⠙⠻⢶⣼⣿⣫⣏⣀⣀⣤⣶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣯⣀⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⡟⣤⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣮⢻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠛⣿⣿⣿⣿⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⠟⢾⣿⢶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⣇⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⣀⣰⣿⣿⣿⠷⠀⣠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⢻⣿⣏⣿⡄⣰⣿⣄⠀⠀⠀⢠⣿⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠁⢀⣼⠏⠉⠀⠀⠈⢿⣿⣷⡀⢀⣼⣿⣎⡽⠃⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢀⣀⣨⣤⠀⠀⠀⠀⠀⠈⣿⣿⡿⠁⣿⣿⣿⡀⣿⡷⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡟⢻⣿⣿⣦⡀⠀⣀⠀⣀⣿⡿⢡⣿⣿⣿⣿⣿⡗⠰⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢹⡇⢸⣿⣿⣾⣿⣶⣿⣿⣿⡿⢡⣾⡟⣿⣿⣿⣟⠀⠀⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡾⣿⣿⡿⢻⣿⣿⣿⠟⢰⣿⡿⠀⣿⣿⣿⣿⣦⠀⠈⠷⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠠⣿⡿⣀⣿⣿⣿⣿⣴⣿⣿⠇⠀⢿⣿⣿⣿⣿⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⡼⣶⡿⢣⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⣻⣿⠟⣷⡟⣱⣿⣿⣿⣽⣿⣿⣻⣿⠳⣆⣀⣠⣾⣿⣿⣿⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⣀⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠉⠉⠉⠙⣿⣿⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣙⡛⠻⢿⣮⣻⡶⢶⢶⣦⢤⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠛⠋⣿⢛⣿⣿⡟⠃⣽⣿⣿⣿⣿⣶⣦⣽⣿⣷⣿⣶⣷⣶⣯⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠄⠀⠀⠀⣦⢸⣿⣿⡧⢾⣿⣿⣿⣿⣿⡉⠀⠀⠈⠙⠙⠛⢻⣿⣿⡏⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⢸⣿⣿⣿⣧⡀⠀⠀⣀⡿⣾⣿⣿⣧⣿⣿⣿⣿⣷⣿⠟⣿⣄⠀⠀⣀⣴⡟⢹⡉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⣹⠉⠀⣸⣿⣿⡿⣿⣟⠛⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠛⠿⣶⣿⣿⣥⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⣿⡇⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠹⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⢿⣿⣿⣾⣿⡿⠛⠀⠙⢿⣿⣿⣿⣿⡿⠻⣿⣇⠀⠘⡋⠁⠈⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠹⠿⢿⣿⡇⠀⠀⠀⠀⣻⣿⣿⡟⠁⠀⠈⠻⣷⣄⠱⡄⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠙⠻⣷⣾⣦⡀⠀⠀⠀⠀⠀⠹⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⢸⣿⣿⡃⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠙⠀⠀⠀⠙⠻⣿⣧⡄⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠈⠙⢿⠖⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣼⣟⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠆⠀⠀⠀⠀⠻⣷⣤⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣿⡄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠻⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠂⠈⠉⠉⠁⠐⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠚⠓⢻⠀⠀⠀⡤⢤⠀⠀⠠⡟⠐⠓⠂⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⢄⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠁⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡁⢈⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⣀⠀⠀⢀⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣬⣀⣀⡀⠀⠀⠀⠐⠶⠶⠆⠀⠀⣠⡀⠀⠈⠉⡉⢉⠉⠉⠉⠉⢉⠉⠉⠉⡈⢉⠁⠉⠀⡀⢤⠄⠀⠐⠶⣤⠄⠀⠀⠀⣀⣀⣀⣡⡀
⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠆⠐⡆⠀⠀⡆⠀⠀⠀⠀⠀⡇⠈⠀⠀⠰⡇⠈⠀⠀⠐⡇⠈⠁⠀⠀⡇⠀⢀⠀⠀⠆⠀⢰⠀⠀⠀⠀⠀⠀⠈⢁⡀
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣄⠀⠸⠃⠀⠀⠃⢀⡄⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢀⠀⠀⠁⠀⢸⠀⠀⢠⡆⠘⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⣄⠀⠈⠁⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡄⠘⠀⠀⠈⠃⠀⢶⠀⠀⠀⠀⠀⠀⠀⠸
⠀⣄⠀⠀⠀⠀⠀⠤⠀⢴⡤⢴⣧⠤⠠⣿⠄⢸⡆⠀⠀⠶⠄⢺⠇⠀⢸⡇⠰⡇⠀⠰⡷⠀⢾⠄⠀⢸⠇⣴⡤⠤⢤⣴⠀⣴⠄⠤⠀⠀⠀⠀⠀⠀⣠
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⡀⠀⠀⡄⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⡄⠀⠸⠀⠀⠈⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠃⠀⠀⡇⠀⢠⠀⠀⠀⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⡀⠘⠀⠀⠀⠄⠀⢰⠀⢀⡇⢀⡀⠀⠀⡀⢸⡀⠀⡄⠀⢘⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⠀⠀⠀⢡⣴⣄⣀⣀⡀⢤⣀⣀⡀⣡⣌⣀⡀⢀⣡⢀⣀⣀⣠⣶⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠲⣀⠀⠤⠤⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠤⠤⠀⣀⠶
⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠐⠶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⠉⠀⠀⠙⠛⣿⠟⠛⠛⠀⠀⠀⠈⠁⠐⠀⠀⠛⠛⠻⣿⠛⠃⠀⠈⠙⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⢣⡀⠀⠀⣠⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣄⠀⠀⠀⠀⠈⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣄⠀⠀⠸⠆⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠓⠆⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡀⣠⡉⣀⡀⢄⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠘⠢⠬⣤⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢈⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣠⡉⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⢀⣤⠒⢫⡽⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢏⣁⠒⢢⣄⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡠⠚⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣌⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢸⡿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠱⠄⢤⣤⣤⣴⠒⢲⠶⠶⠶⠶⠖⠲⣦⠀⠀⠐⠛⠁⠀⠀⠀⢀⣌⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡀⠰⠆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢇⡀⣀⣰⣦⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣀⣽⡟⣈⠹⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠻⠧⠾⠿⠻⣿⠾⠗⠰⠟⠛⠛⠋⠀⠀⠀⠀⣿⣋⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠜⠉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣤⣤⣀⣀⣠⣠⠆⠀⠸⠂⠀⢒⠲⣶⣿⢴⣿⡛⣁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠙⠛⠫⠶⠶⠶⠆⠶⠾⠛⠉⠀⠀⠀⠘⢻⠉⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠰⠏⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⠊⣠⡇⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠳⢀⠀⠒⠀⠀⢠⠈⠀⢹⡇⢀⠠⠟⠓⢶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠲⠔⠶⡌⢱⠷⠀⠀⠈⠶⠀⠸⢷⡈⣠⠶⠶⠊⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠛⢤⣷⠀⠰⠀⣾⣧⡤⠋⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠠⠄⣴⣄⠙⠶⠁⠀⠀⠈⠶⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠛⠒⠓⠲⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠒⠛⠛⠛⠛⠓⠚

⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠰⣄⠀⠀⠀⡇⠀⡇⠀⡇⠀⠄⠐⡆⢦⠀⡇⠀⡇⢦⠸⡀⢰⠀⠄⢠⡀⠀⢠⠀⠈⢸⠀⢸⢸⠘⡄⠀⠰⡄⢸⠀⢸⠼⠊⢦⠀⠀⠀⣶⣴
⠀⠀⠀⡀⠘⣷⡀⠀⠃⢀⣃⠀⠁⠀⠐⠈⠁⠀⠃⠃⠀⠋⠀⠁⠈⠈⠀⠠⠿⣧⠀⠀⠑⠂⠘⠀⠘⠘⠀⠈⠀⠀⠑⠘⠂⠈⠀⠑⠊⠠⣶⣧⣾⣿⣯
⠀⠀⠀⠀⠀⢄⣣⡀⢰⡘⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡟⠻⣧⣻
⢠⠀⠀⠀⠀⣹⢻⡀⢨⡍⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠷⢀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⣶⢿⡽⠂
⠸⠀⠀⠀⠀⣸⠐⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⠐⠗⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⢿⣆⣿⠈⡄
⢀⠀⠀⠀⠸⣾⡇⣷⢸⡤⠔⣟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡖⠀⠐⠀⣰⣿⡀
⠈⠀⠀⠀⠀⠈⡣⠼⠃⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡅⠀⢄⣁⠭⡋⠀⠀⠀⠀⠀⠀⠀⣰⡠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⢀⠓⠼⠓⠀⠀⠀⠀⠀⠀⣶⡇⠁⠀⡀⠀⠀⠀⠀⢠⡀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠃⠀⠀⠀⠀⠸⡇⠀⠸⢄
⠀⠀⠀⠀⣾⣦⣄⠀⢀⣿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠛⠻⢘
⠀⠀⠰⠶⢾⣿⣿⣧⣼⣼⣤⣾⣦⣀⣤⣤⣄⠀⠀⣠⣄⠀⣿⣇⢤⡄⠀⠀⠀⠀⠀⣠⣤⣈⣿⡒⠂⠀⠀⠃⠀⠀⡀⠄⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠙⠋⠉⠹⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⣶⣟⢑⣮⡁⠀⠀⠀⠀⠀⠉⠉⠹⣿⡟⢀⠰⣤⠶⡄⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠉⠀⠘⠀⠀⠀⠀⢘⢻⣿⠏⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄
⠀⠀⠀⠀⡄⢦⢦⣄⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠛⠚⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡧⢠
⠀⠀⠀⠀⣇⢬⡛⢿⣏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠰
⠀⠀⠀⠀⠛⢸⣷⠾⠟⠀⠀⠀⠀⠀⣾⠀⠀⣠⠀⠀⠀⣠⣶⡴⠀⣠⣶⡶⠀⠀⠀⢸⡆⣀⠀⠀⠀⠀⠀⡀⣀⣦⣴⠄⠀⠀⠀⠀⠀⠀⠘⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠁⢀⠋⠀⠀⠀⠛⠀⠁⠀⠉⠉⡁⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠁⠙⠉⠙⠀⠀⠀⠀⠀⠀⠀⢀⣢⡴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⢉⣤⠶⠛⠛⠉⠉⠙⠦⣧⡀
⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠠⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣘⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢦⣤⣦⣄⢀⣾⣿⣷⣷⡄⣸⣿⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣷⡄⠂⠠⠄⠀⠀⠀⠀⠀⠲⣤⠄⠀⠲⣮⣻⣿⣿⣿⣍⠉⣥⣾⣿⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⡧⠛⠄⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤⣀⠻⣿⣿⣿⣿⣦⣿⣿⡧⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣧⢹⢿⣿⣿⡷⠀⠀⠀⠀⠀⢀⣴⣀⣀⠀⠀⢩⣿⠳⣿⣹⣿⣿⣿⣿⣿⣶⣿⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣆⡀⠉⠻⣣⠀⠀⠀⠀⣠⣿⣿⣿⣿⠇⠀⠸⠿⠂⣻⣟⠿⣿⣿⣿⣿⣿⣿⡏⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⣿⠷⡀⠀⠈⠂⢠⡔⠚⠉⢁⠀⠈⠁⠀⠀⠀⣤⣀⢸⣿⠦⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⢾⣿⣷⣤⣄⣀⣀⡁⠠⠁⠀⠀⣄⢷⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠘⠿⠁⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠙⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣦⣷⣠⡴⠋⢀⠀⠀⠀⠀⠛⢦⣦⣼⢃⣽⣿⣿⠟⠁⠀⠻⡀
⠀⠀⠀⠀⣤⣴⠀⢊⠁⠀⠀⢠⣼⣿⣹⡿⢻⣿⣿⡟⠁⣸⠇⠀⠀⠀⠀⠀⠛⢿⣿⣿⡋⠈⣡⣶⡇
⠀⠀⠀⣠⡿⠛⠋⠉⠀⠀⠀⠸⡏⠛⠁⠰⣿⣿⡉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⡃⠀⠀⢸⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠹⣿⣷⠆⠄⠀⠀⠀⠀⢠⡞⣻⣿⡟⠀⠀⢸⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠘⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⠀⠀⣄⣀⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣠⠀⡀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠟⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢾⣿⣷⣶⣶⡀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⠈⠀⠀⠀⠀⠈⠉⠀⠈⠉⣉⢉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⡠⣤⣤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣚⡉⠙⠲⡼⢽⢤⠒⡠⢄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⢨⠑⠒⣒⠀⢸⢘⣄⡳⠚⠉⡟⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⡇⠁⡇⡜⢩⢅⣱⣾⣈⣓⠮⣗⢄⠛⢓⣲⣷⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡏⡇⢳⠀⡇⡇⡰⠉⡴⠖⢂⣨⠵⡌⠧⢛⣁⣤⢼⠙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢱⠧⡇⡼⢠⠇⡇⡇⣠⠖⢫⣝⣒⣉⣧⢺⠭⠤⠭⣾⢧⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣾⠀⡇⠙⢒⡊⠆⣇⠓⠥⣀⠂⠄⠠⢸⢸⣉⠹⠆⢹⠂⢒⣚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡇⡇⡇⠾⣏⢫⠝⢈⡆⠠⣄⡉⠓⠢⡞⢸⣒⣒⣲⡟⣺⠉⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⣧⣷⡤⢯⢛⡴⠲⢭⡒⢤⡈⠑⡖⢳⠲⡦⠤⢴⣎⠙⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⢻⢹⢩⡟⠋⣂⠞⠓⢬⠳⡌⢦⡽⢨⠟⣁⡒⢦⢄⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠘⡅⢻⠁⢦⣧⡆⠱⡘⡆⢱⢃⢤⣀⡑⢂⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠘⢇⣈⣘⣉⣁⡷⢹⠘⢌⣹⠣⢍⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠴⠶⠒⢛⠠⡗⠎⠙⠉





⡟⠉⣿⣛⢻⢛⣻⣛⣟⣛⡿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣟⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⠀⠒⢢⣽⠃⠀⠉
⢀⡠⣀⣠⣦⠀⠀⢀⣴⣿⣆
⣷⣿⣿⡿⠿⠗⠒⠈⠁⢨⣿
⠈⠙⠉⠀⠀⠀⠀⠀⠀⡂⢸⠇
⠀⠀⠀⠀⠀⠀⢰⠶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠸⠒⠆⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠙⠋⠉⠛⠓⠠⠤⣤⣀⣀⠀⠉⠉⢻⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢰⡏⢄⠀⠀⠐⠀⠀⠀⡼⣿⠉⠉⠉⣻⠁⠐⠂⢄⠀⠀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⢤⡤⠀⠀⠀⠀⠀⢠⡇⠀⠘⣷⣌⣓⠦⠴⠦⠖⢋⡴⠏⠀⠀⢰⡇⠀⠀⠀⠈⡻⠞⢀⣤⠀⡒⠠⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⡀⠀⣼⠃⠀⠀⠸⡍⢛⡑⠋⠙⠋⠉⠀⠀⠀⠀⢸⣇⡇⢀⠐⠈⢀⡴⠋⠀⠀⠁⠐⠊⠀⢠⠝⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣠⡟⠀⠄⠀⠀⠹⣦⠈⠑⠢⠄⢀⠀⠀⠀⠀⣼⠉⣁⣠⡵⠚⢉⡀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠿⠀⡰⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⠉⠘⠢⠄⠀⡇⠀⠀⠀⠘⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣰⣃⠄⠀⠀⠀⠀⠀⢂⠙⠦⣄⠀⠀⠀⠀⣿⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢤⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⠀⡠⠀⠀⠀⠀⠀⠳⣄⠈⠉⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢧⠈⣷⠠⠀⠀⠀⠀⢠⠀⠀⠿⠀⠋⢠⡇⠀⣀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⢈⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢺⡄⢘⡆⢀⠄⠀⣀⣠⡀⠀⢠⡖⠒⠀⠹⠆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠭⣀⡙⠛⠂⠒⠒⠛⠛⢛⣛⠿⢳⡀⠹⡸⠔⠋⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣼⠶⠀⠀⣀⡉⠛⠛⠛⠛⠛⠋⠉⣀⣀⣼⣷⣀⢷⡀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠚⠁⠀⠀⠀⠀⠃⠈⣩⡟⠁⢰⠟⠉⠉⠁⠀⠀⠉⠒⢷⡀⠀⠀⠀⠁⠀⢹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⡴⠃⠘⢦⣀⡀⠰⠶⠤⠤⣄⣈⠀⢀⡀⠀⠀⠈⣧⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠀⠀⠀⠀⡀⢤⡐⢊⠁⣀⡤⠖⠁⠀⠀⠀⠀⠉⡀⠀⠉⠉⠉⠉⠉⠒⢦⣌⢲⣄⠀⠘⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⢀⡤⣔⡪⠽⠓⠋⢉⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠉⢦⡀⠀⠙⢷⡝⢦⠀⢱⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠖⠃⢚⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡀⠀⠀⠑⠄⠀⠈⢿⠈⠃⠠⠦⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣴⣫⠏⠀⠺⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⠁⠀⠀⠀⠀⢨⡆⢰⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡐⡦⠻⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠈⠀⠀⡿⢸⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠹⣠⡆⠁⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⠶⠆⠰⣇⢸⡆⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢰⡏⠑⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠙⣾⣷⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠞⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⠀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡄⠀⠀⢀⡴⠋⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⢙⡷⠀⠈⢣⡀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⠀⣰⠉⣴⠟⠛⠉⠉⠉⠙⠒⠒⠤⠤⠤⠤⠤⠔⠒⠋⠉⠀⠀⢀⣀⡤⠚⠁⠀⣀⣤⠟⣄⡠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⣈⠀⢀⣅⠰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⠀⠈⠋⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢩⡋⠈⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠁⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⠈⠃⠀⠳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣢⣄⣂
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⢋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⢀⠀⠀⣾⣾⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠃⠸⡄⠀⠈⢿⣦⡾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⢀⣀⠁⠀⠁⠀⠀⢀⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠏⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣽⣧⡈⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⣰⢀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠀⠘⠛⠛⠛⠛⠀⣊⡴⠿⣿⡳⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡿⠀⠉⢆⠀⠀⠀⠀⠀⣀⠀⡁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⣟⠏⠀⡙⡂⠤⣄⣀⣀⡀⠈⠀⠀⠀⠀⠃⠳⠀⠀⢼⠀⠀⠀⠀⢀⠟⠀⠛⠛⠚⢿⡠⡀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢹⡏⠀⠀⠻⠉⠗⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠶⠤⢴⡋⠀⠀⢀⠀⡚⠿⣿⣿⡀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⡀⠄⠀⠀⣅⠘⢧⠐⠒⣳⠀⠀⠇⡀⠐⠀⠀⠀⠀⠁⢀⣀⣴⣾⣿⣿⣿⢦⠀⠀⠀⠀⠀⠰⠶⠀⠀⡀⠉⠀⠀⠀⢸⡇⠀⠰
⠀⠀⠀⠀⢈⡓⢶⠀⢺⣃⣠⡴⠟⠈⠉⡉⠀⠀⢘⠇⢠⠀⠀⠠⢾⠏⠁⠀⠀⠀⢿⣿⠀⢣⣄⣀⣠⠀⠈⠀⠀⠀⣠⡀⠀⠀⠁⠾⡅⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠁⠠⠿⠭⠥⠤⠶⠿⠷⢶⣤⡴⠟⠀⠸⠴⠂⠀⠀⣀⣠⣤⣀⡀⢘⣿⡀⠙⠉⠉⣀⠀⠀⠀⡤⠶⠚⠛⠁⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣼⣋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠉⠉⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠉⢹⡇⠀⠒⠒⠀⢸⡏⠉⠀⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠘⠇⠀⠈⠉⠀⠸⠁⠀⠀⠀⠀⠀⠈⠢⢀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠐⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢀⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠄⠤⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢰⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠤⠤⠤⡤⠛⠀⠂⠐⠐⠣⢤⠤⠤⠤⣤⠀⠀⠀⣀⣀⢀⡀⠀⠀⡄
⠀⢀⠋⠀⠀⠀⠀⡄⠉⠄⠀⠘⢸⠀⠀⠐⠀⡂⠀⡆⠘⠀⠰⠀⢀⠀⠐⠀⡆⠀⢸⠉⠁⠀⠀⠀⠀⠙
⠀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠈⠀⠀⠀⠠⠇⠀⡇⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠈⡀⠘⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⣀⣀⣸⣀⣌⠀⢠⡀⣤⠀⠀⡇⢠⡄⠀⣧⢠⠄⠀⡄⢸⠀⢀⡇⢸⡄⢀⣇⢀⣄⣀
⠋⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠃⢠⠀⠀⠀⠈⠇⠀⡇⠀⠀⢀⡀⠸⠀⠀⠁⡄⠀⢠⠁⠨
⠈⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⢰⠀⡀⠀⢸⠀⠀⠀⡇⠀⠀⠘⠃⠀⠀⠺⠀⠁⠀⡄⠰⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠆⠀⠀⠀⠀⠀⢃⢸⠀⠀⢄⡇⠀⡈⣠⣀⣀⣡⣤⣀⣀⣤⡁⠀⠠⠘⠀⠀⠀⠆⠀⠀⠀⠀⠀⠰
⠀⠀⠘⣄⠀⠐⠒⠂⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠒⠀⠀⣠⠁
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⢠⡄⢠⣤⣄⣀⣀⠀⠀⢠⡄⠀⠀⣀⣀⣀⣤⣤⣤⡄⠐⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠐⠈⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠈⠒⢤⡀
⠀⠀⠀⠀⠀⠠⠛⠀⠢⡀⠀⣀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢀⠀⠀⡴⠊⠀⠦
⠀⠀⠀⠀⠀⡆⢀⠀⠀⡔⠋⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠻⡄⠀⠈⠀⡇
⠀⠀⠀⠀⠀⢷⠖⠐⠒⢃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠧⢀⣠⣀⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡄⠀⠀⠈⡛
⠀⠀⠀⠀⠀⠀⠐⣤⠒⠉⠓⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠈⠑⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢸⠂⠀⠀⣠⡆⡤⠤⠤⠤⢰⡦⠀⠀⢻⠀⠀⠀⣰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⠌⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠈⠆⢠⡞⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠀⠀⠀⠼⣇⣤⣤⣤⣄⣘⠷⠀⠀⠂⠂⠸⣿⠆⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠀⠀⢀⣠⡀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠢⠶⣖⣃⣀⣛⣉⡿⠛⠛⠘⣧⠧⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡀⠀⠀⠀⡇⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢘⡄⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠓⢄⡀⡇⠀⢀⠃⢸⠆⡠⠚⠢⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡙⠂⠣⣜⢃⠀⢈⢃⡘⢋⡜⠂⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠈⠻⠀⠈⠺⠟⠁⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⢀⡀⢄⡂⠀⠠⣠⠄⡀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⡀⠀⢠⡶⠉⠀⠀⡀⠀⠀⢾⡁⠤⠒⠀⠈⠀⠀⠀⢀⡄⠁⠀⠀⠀⠈⠀⠀⠀⡀⠀⡠⠖⠒⢠
⣴⠟⠛⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸⣵⣠⣦⢀⡀⠀⠀⠀⠀⠑⢄⠀⣀⠀⢀⢌⠅⢀⡀⠀⢰⡄⡀⠀⠀⠰⡀⠘⣁⣠⡴⣤⢣⡀
⣿⣆⡶⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠉⢿⣦⣄⡉⠀⠀⠀⠀⠀⠀⠀⢀⢫⣴⣛⢀⡶⣻⣦⣶⠞⠉⠀⠀⠀⠀⣰⣾⢿⣡⡼⢩⣿⣧
⠉⢷⣴⣶⠻⣀⡀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠔⢠⠀⢨⠏⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣹⢟⣿⣦⡬⡈⠁⠀⠀⠀⠀⠀⡜⢿⡿⠚⡻⠶⠁⠀⠈⠛⡄
⡾⣿⡿⢾⣷⣛⣳⡦⠀⠀⢿⡞⠠⠐⠚⠁⢸⣿⣷⣾⣤⣤⡖⡆⠀⠀⠀⠀⠀⠀⡰⠻⠉⠻⠉⣩⠼⠇⠀⠀⠀⠀⠀⣸⠇⢸⡷⢤⣤⠀⢸⣆⠀⡆⠐
⠇⡌⠈⠀⡄⣹⣏⡐⢨⡒⢄⠀⠀⠀⠀⠀⠈⢻⡽⠉⢿⠟⠁⠀⠀⠀⠀⠀⢀⡄⠂⠀⠀⠂⠒⠁⠀⠘⠀⠀⠀⠀⢠⡿⠀⡀⣿⣯⡈⢀⣰⣟⠀⢩
⢰⣷⣀⡴⡁⠘⣾⣷⡮⠥⠤⠀⠀⠀⠀⠐⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⠞⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣹⣷⡄⢉⠿⠀⣉⣴
⠞⢋⣁⣾⠃⠀⡆⣏⠀⠀⠀⠀⠀⠀⠀⠒⢸⠗⠲⠀⠀⠀⠀⠡⠀⠀⢻⣏⣻⠉⡋⡙⠡⠖⠀⠀⣯⠀⠐⠒⠀⠀⣿⠁⠀⡄⣿⣇⣈⣷⠸⠀⠘⢃⠜
⠈⢏⢱⣿⠰⡀⢠⠘⠈⢄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠸⣴⠆⠠⢉⣉⠓⡙⣉⣱⡀⢀⡠⠒⠀⠀⠀⠀⠀⢸⠀⠀⡇⠡⡿⠟⢀⡀⣈⡦⣬⠈
⣄⠘⡸⢿⡀⢰⣞⠀⣜⡀⣱⡄⠀⠀⠠⢶⢤⡟⠣⠄⠠⠄⠀⠀⠀⠀⠀⠀⠈⠉⠟⠾⠋⣲⠉⢀⠀⣀⡀⢀⢀⡀⠈⡆⢹⣧⢀⣅⡰⣏⠀⢎⢠⣄
⣷⡀⢣⠸⣟⢸⣁⣾⣿⠻⠉⠀⢰⠣⠊⢛⣻⢛⣛⠛⠒⠀⠀⠀⢀⣀⣤⠄⠐⠲⣷⣶⠞⠾⡇⠁⠀⠀⠀⠀⠀⠀⠀⣇⠾⠋⢨⣹⡿⡇⠀⢰⠞⠀⠜
⣿⣧⠀⡀⢸⠈⢸⡇⡿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠙⡇⠀⢤⣍⠙⠀⠀⠀⠘⢳⣆⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡠⠋⡀⢶⣿⠉⢡⠃⢀⠇
⣿⣄⠣⠙⠻⣠⠘⡃⠁⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⣈⠈⠻⠶⠀⠀⠀⠈⠟⠀⠀⠀⠈⠵⠁⠀⠀⠀⠐⡀⠣⠻⢾⣫⠃⡸⠀⡘⠀⠀⠁
⠘⣝⣳⣢⣴⡙⣾⢣⣄⡷⠐⠿⠇⠤⠀⠀⠀⣈⠭⠀⠀⢠⣦⡉⠉⠀⠂⠀⠰⠂⠀⠀⠀⡠⠁⠀⠀⠀⠀⠰⠀⠀⠙⡄⠀⠠⡀⢀⢃⠞
⠀⡼⣹⣿⢀⣨⠟⢂⡁⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⠀⠀⠸⠙⠁⠐⠀⠀⢀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠄⠀⠈⢇⠸⣦⣹⣆
⡼⠷⠞⠋⠀⠘⠶⢟⢮⡐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⡠⠂⠀⠁⠀⢀⡀⠈⠈⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⡀⢠⠀⠀⠀⠣⡀⢋⠘⣧
⢀⠀⢀⢴⠀⠀⣴⡀⠣⡀⣄⠀⠀⠀⠐⠂⢀⠀⠀⠔⠀⠀⠈⢨⠁⠀⠂⠀⠀⠳⡲⠄⠀⠀⠀⠀⣠⡿⢤⣠⡀⠠⡰⠛⠤⠸⢀⠀⠀⠀⡽⠊⢏⡂
⢨⠀⢁⡄⠸⠻⣷⠳⣶⠙⣟⢳⣄⠈⠓⠀⠀⠀⠀⠀⡰⠠⠆⠀⠀⠀⣰⡖⠀⠀⠈⠞⠀⠀⣠⣾⢻⣿⣷⡐⣿⣧⡶⠋⠀⠀⠀⠀⠀⠀⣠⠖⠀⢠⢰
⡄⠀⠉⠃⠂⠀⠈⠳⡝⠳⣄⡙⢌⠳⣬⡐⣤⡘⢓⣾⠿⢗⡄⢀⡴⢊⡼⠁⠀⠠⢤⣄⡤⠊⠁⢠⠎⣿⠋⠉⠍⠁⠆⢁⣀⠀⣀⣀⠀⢀⡈⢤⢁⣴⣷
⣄⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠙⠢⣌⠀⠉⠓⠮⣍⡉⢩⣿⢟⠁⣠⠋⠀⢠⡇⢸⢈⣿⣇⠀⢀⠄⣸⣿⡀⢀⣦⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠛
⣿⡗⠂⣠⡐⠀⣀⢤⣽⣿⣷⣦⡀⠈⠙⠶⣄⡀⢀⡉⢕⡚⣶⡽⠃⠀⠀⡀⣫⣿⡿⢿⣿⠏⠁⢠⣿⣿⡟⠋⣹⣿⠈⠼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⣸
⣿⣵⣶⢋⡵⣛⣿⣿⣿⣿⣿⣿⣿⣦⣀⣴⠛⠟⠷⠦⣼⣭⣽⣶⠶⢁⣌⡙⠳⢾⡃⠧⠍⠀⣠⡿⠿⡏⣰⣶⣿⡟⢳⣄⣿⣿⣿⣿⣿⣿⣿⢉⣁⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡟⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣶⣀⡠⡄⢠⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣷⣄⠹⠿⠋⣘⣿⣿⡄⠀⡀⠤⠂⠈⠛⠄⠆⣠⠹⠃⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠛⢋⣠⡐⢿⣿⣿⣿⣿⡴⢖⢆⠀⠀⠠⠀⠀⣉⠀⠀⠊⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣿⣿⣇⣠⣽⡿⠛⠁⠀⠈⢎⣷⡄⠀⢀⠄⠀⠀⠀⠀⠐⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠛⣁⣦⠀⠀⢀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡋⠁⠀⠀⠈⢻⣦⡴⠿⠒⠀⠜⣿⣦⣀⢀⢀⢀⣠⣴⠛⣆⡀⠀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠰⣶⠟⠙⣧⣀⠀⠀⢈⠌⠻⠏⠛⣉⡉⠉⠀⢿⣿⣟⡢⢴⠞⠂⣀⣀⠀⠀⢀⣀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠁⠀⠀⠸⠉⡀⠀⢽⡀⠀⠀⠀⠙⠀⠁⠀⢡⣝⢿⡟⠛⠓⠐⠏⠉⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠐⠁⠀⠀⢸⠇⠀⠀⠀⠀⠘⡄⠀⠀⣿⣿⡇⠀⠀⢠⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣿⡇⠀⠶⢺⡗⠲⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠃⠀⠀⣿⣿⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⢀⠸⣧⣀⣀⣀⣀⣀⣀⣿⣿⣶⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠘⠀⣿⠛⠛⠛⠛⠛⠛⢻⣿⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡴⠿⢻⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠶⢶⠂⠐⠀⠀⠞⣧⢱⠃⠘⡄⠀⠀⠀⠀⠀⣼⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡆⠀⠀⠀⡇⡘⣧⠀⠀⣇⠀⠀⠀⠀⠀⣿⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠆⢰⠁⠀⠀⠀⢣⠁⡘⠣⢀⠸⡀⠀⠀⠀⠀⣿⣿⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠂⡏⠀⠀⠀⠀⠘⠈⡇⠀⢡⠀⢷⡀⠀⠀⠀⣿⡿⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠈⣇⢡⡆⠀⠂⠀⠀⣘⡣⢰⡃⠠⢀⡁⠳⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡗⠀⠀⠀⠈⢹⣓⣃⣰⠢⢀⢑⠀⠘⠦⣀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⣬⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠁⠰⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡁⠀⠈⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠀⢀
⠀⠀⢀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢢
⠀⠀⠈⠁⣌⢆⠀⠈⢱⣿⣦⡀⠀⠀⣾⡁⣀⠀⠀⠀⠀⠀⢰⠀⢀⡀⣆⠶⠎⣠⠀⠀⠀⡇
⠀⠀⠀⡀⠹⠧⠀⢀⣈⡟⣾⣕⡀⠀⣮⢸⣿⢥⡀⠀⠀⠀⠈⠀⠛⣿⠁⠀⠀⣯⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⣤
⠀⠀⠀⠁⠀⠀⠀⠸⠿⢻⣹⣿⡇⣰⣬⡆⡽⢾⡇⠀⠄⠀⣦⣤⣿⣿⣤⡀⠠⡏⠀⠀⠄⣶⣠⡏⠀⠀⠐⣴⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣴⣿⣿⣇⣿⣇⠁⢡⣿⡇⠀⠃⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢴⡄⠠
⠀⠀⢠⠀⠈⠀⠀⢾⠃⢸⡃⠀⠸⣇⣿⢳⡌⠈⠁⠈⠆⠰⢿⣿⡿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣨⣀⢻⣦⠀⠀⣤
⠀⠀⠀⠀⠀⣁⠀⢰⠀⣿⡃⢀⢀⣾⣗⠱⠇⠀⠀⠀⠀⠀⠈⠻⠁⢚⣿⡿⠁⠀⠀⢠⠰⠄⢸⡿⠛⠈⠙⠃⠀⢸⡇
⠀⢸⣿⣿⣦⡏⢃⡴⠂⠻⠶⠿⠟⢹⡏⢁⡄⠀⠀⠀⠀⠀⡦⣾⣼⢷⡿⠃⠀⢠⠀⠈⠀⢤⡾⠌⣿⠀⣀⠠⣾⣿⡇
⠀⢸⣿⣿⠟⣷⡀⠀⠀⠀⠀⠀⠚⢿⡇⠀⡙⡾⢂⠀⠀⠀⠀⣭⡌⠘⠇⣀⠉⣿⡀⠀⠀⣸⠧⣆⣿⠀⣿⠀⢻⣿⡇
⠀⢸⣷⣿⣿⣿⠀⠀⠀⠀⣰⡇⣠⡞⣡⣴⠿⠀⠾⡀⠀⠀⢰⡿⠀⠀⢠⢿⠿⣿⣿⣦⡀⡁⠀⢿⣿⠀⢸⣤⣿⣿⡃
⠀⣾⣿⣿⡇⠁⡀⠀⢀⡼⠁⡀⣼⡇⠟⠀⠀⠀⠀⣿⣷⡄⠾⠃⢰⡆⠀⠀⠀⢿⣿⡿⢿⡖⠀⠀⢹⡄⢾⡟⢻⣿⠁
⠀⠿⣯⣿⡇⠀⠁⠀⢘⠀⢠⣵⣿⠁⠀⢠⣿⣀⣾⠛⡻⠃⠀⢴⣾⡇⠄⠰⡆⠸⠟⠡⣾⠃⠀⢀⣼⠧⢸⣿⣿⣿⠀⡀
⠀⠀⢾⣿⡃⠘⠀⠀⢸⣄⣀⢹⣿⣧⠀⣼⣤⣼⡽⣇⠀⣠⠖⠀⠈⣿⠶⡄⠀⢲⣾⢰⣿⣷⣦⣼⣧⣀⣸⣿⣿⣿⠀⣅
⠀⠐⡿⣿⣇⡀⠀⠀⣿⡟⣽⣧⣻⡟⡇⢹⣹⠋⢴⢿⡀⣹⠀⠀⢀⣽⠀⠀⠀⣾⣿⣘⣿⣿⣿⢿⡏⠈⣿⣿⣿⣿⠀⠉
⠀⢸⣷⠿⠿⠷⠶⠀⢹⢄⣿⣿⢿⣷⡇⣿⡟⠀⠐⣿⠇⡏⠀⠀⠀⣿⣇⣰⣿⠟⣿⣿⣿⣧⣿⣾⣀⣠⣿⠾⣿⣿⠀⢠⣀
⠀⠘⠃⣲⣶⣷⡆⠀⣾⣿⣿⣿⣹⣿⡏⢸⠃⢀⠈⡟⢰⡇⠀⠀⠀⣿⡏⢹⣿⠀⣿⠏⠉⠀⠈⠙⠉⠁⠀⠀⠀⢀⠀⡄⢳⡀
⠀⢠⠞⠻⠿⠛⠝⢫⣻⢿⣿⠏⣿⣿⡇⣨⣿⣾⣴⣷⡾⠀⠀⠀⢸⣿⡇⢸⡇⠀⣿⣄⣤⡆⠀⠀⠀⣀⡀⢰⣄⣹⣶⣿⣿⡇
⠀⠀⠀⣠⠀⠀⣤⢯⣯⣿⠟⢲⣿⣿⣿⡿⠛⠙⣿⡿⠁⠀⠀⠀⣾⣿⣷⡿⠋⠀⠉⠙⠋⢀⣤⣄⠀⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠿⠀⠀⠈⠉⠙⠛⠛⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⣶⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠦⠾⣽⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠠⠄⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⠁⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣺⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠋⢋⠉⠄⠰⠿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣉⢠⡤⠀⠆⣒⣂⣩⡤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⣶⢀⣃⡴⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣾⣟⣷⣶⣶⣶⣶⡶⡶⡖⣖⣒⣚⣛⣟⣋⣭⡩⡴⠦⠶⠒⡛⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠟⠛⢛⡛⢉⡉⢉⡉⢩⡅⢡⡅⠠⣄⠀⡆⠰⠇⠘⣃⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡄⠰⠆⠘⣁⣘⣃⣀⣥⣬⣥⣤⠥⠬⠥⠶⠶⠖⠒⠚⠋⠉⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣶⣾⣿⣿⣿⣯⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠖⡒⠋⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠿⠛⠛⢛⡋⠉⡭⠉⢩⠉⠁⠄⠀⢰⠀⠀⠇⠀⣓⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢩⠄⠀⡆⠀⠩⢀⢀⡃⣀⣁⣀⣨⣡⣤⣤⠤⠴⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣂⣠⠤⠶⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣷⣤⣤⣤⣤⣤⣤⡤⡤⢤⠤⠤⠤⠴⢶⢶⣶⣶⡶⢶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣷⡿⣶⣺⢹⣦⣞⣼⣣⣷⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠛⠛⠛⠚⠓⠓⠚⠛⠛⠛⣛⣛⣉⣩⡭⠿⠿⣛⠻⢴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⢖⠒⢋⢉⣩⠉⠤⡀⡶⠀⠿⢀⣚⡤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢐⠹⢂⣚⣡⣭⠴⠴⠴⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⡛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⡤⠔⠉
⠀⢰⠤⠀⠙⣆⡀⠀⠈⠉⢁⡄⠀⠀⠀⠀⢠⢀⣀⣀⠀⠀⣠⢶⣂⢤⡀⠉⢁⣠⡴⡄⠀⠀⢀⡌⢀⡌⠀⣠⠖⠚⠁
⠀⢈⠀⠈⠠⡄⠰⠤⡄⠀⠀⠀⠶⠀⡀⠐⠀⠋⠙⠿⠁⠀⠈⠙⢿⣰⡏⠐⣋⡿⠑⠉⢉⣠⡈⢀⠂⠑⠄⠀⠸⢎⡁
⠀⠘⠀⠀⠀⢀⠤⠀⠛⠦⠀⠀⠀⠀⢑⣲⡀⠃⠀⠀⠀⠀⠀⢰⠈⠁⠀⠀⠉⠀⠀⢙⡍⡼⠁⠀⠀⠀⠀⠀⠄⠘⠃
⠀⠠⠀⠀⠠⠋⠀⢠⡀⠈⠁⠀⠀⠈⠻⣿⠉⠠⣄⠀⠀⠀⡀⠈⠀⠀⠀⠁⠈⠉⠀⠋⢿⡤⠃⢀⡀⠄⡀⠀⢀⡀⠄
⠀⢀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⣈⠉⠳⠎⠀⠀⠈⠖⣺⡟⠙⣶⡀⠀⠀⠀⢀⣛⠉⢀⠄⠐⠀⢀⣁⠤⡅⠠⠃⡈⠃
⠀⢸⠀⠀⠀⣀⠈⠁⠀⠀⣈⡀⠉⠀⣤⡶⠦⣼⣿⣞⡛⢿⡾⣧⠘⣧⡠⠴⠛⢛⣻⣻⣦⡀⠀⢻⣫⡜⠻⠁⠔⢀⡄
⠀⠰⠀⠀⣤⡉⢉⠀⠀⣼⠻⠮⡙⢄⠉⠁⠀⠻⠏⠈⢉⠿⣤⣿⡆⣩⣧⠤⢾⡭⢥⣬⣉⣿⣶⠀⠉⠀⠋⠈⢐⢋⡁
⠀⠰⠰⠦⠀⠀⠀⠠⠶⣄⡱⡿⢡⡄⠀⠀⠀⠒⠒⣒⡟⣺⠏⠃⠉⣉⣿⣦⡜⡖⡚⣃⣬⠿⣦⡄⢀⠀⠀⠀⠀⣬⠂
⠀⠰⠀⠀⠀⠠⠄⠀⠀⡛⠭⣃⡘⠙⣼⣤⠠⠾⠁⠀⠀⡎⠀⠀⠴⠀⡺⣹⠶⠦⡄⠤⠤⠁⢾⠗⠀⣆⣀⠋⠐⠀⡇
⠀⠘⠀⠀⣐⠚⠃⢀⡘⠙⢲⠏⣙⠛⠅⠉⣗⣴⡶⢚⠽⡆⠠⠄⢀⣴⡷⣳⣖⣒⡀⠰⠊⡙⢢⠤⢤⢁⠲⣤⡈⠹⠇
⠀⢰⢀⣐⠋⢡⣄⡈⠉⠁⢺⣃⣤⡘⢉⣴⣿⣿⣿⣤⢊⡌⣰⣾⣇⠠⡘⣿⣯⣿⣏⣤⠞⠀⠂⢸⠀⢾⡂⣹⣆⠑⡄
⠀⢨⠍⠀⣀⠈⠉⠉⠉⠉⢹⠁⠀⠀⣼⣏⣿⣿⠀⡙⠉⡱⠟⢘⠟⠶⢻⣯⣿⠁⠈⢹⣇⠐⠀⠀⢈⠀⣻⡇⠈⠆
⠀⢈⠀⠘⠛⠓⠒⠀⠒⠐⢾⡀⠀⠀⡾⠉⢿⣿⠊⡀⠨⠐⣥⣸⠐⠀⢸⣿⡯⠚⠠⣿⣿⠀⢠⠲⢾⣁⣉⣤⠂⠁
⠀⠸⠄⠀⢠⡀⠀⡀⠀⠀⣤⣧⡀⡼⠛⢤⡺⣿⠀⠀⠀⠑⡀⡉⢀⡔⢢⡿⠁⠀⠈⠿⢿⣷⣿⡄⠀⠀⢃⡠
⠀⢠⣄⡐⡜⡓⣀⠀⠀⣀⡄⠈⠉⠉⠓⠉⠀⢹⡇⠀⠀⠁⠀⠐⠁⠀⣿⠙⣦⣄⡐⠒⠉⠀⠀⢘⣦⡀⠈⡔
⠀⢈⣿⠷⣤⢺⣧⢽⡿⢢⡀⠀⢤⡄⠀⠀⣀⢀⣿⣄⠀⣀⣵⠦⠀⠐⢿⣆⣀⣙⠛⠺⣭⠦⠚⠋⠀⠓⠋⠁
⠀⢰⣸⢋⡿⢿⣧⣴⠋⠻⠇⠀⠠⣠⡷⠯⣽⣥⡈⠉⠉⠹⠿⢾⣀⡴⠶⠏⠀⣈⣤⣬⡯⢿⠭⢀⠀⠀⠀⠀⠀⠀⠁
⠀⢸⢛⢿⣔⣙⢣⣠⣤⣤⣀⡀⢴⡏⠀⠀⠀⡉⠉⠲⢤⣜⡶⠀⠁⠀⣠⣦⡾⠋⣠⠀⠀⠀⣸⡟⢀⡄⡀⠀⠀⠀⡄
⠀⢘⡙⢻⢯⣭⢿⠏⠁⠜⢰⠉⢈⣿⣶⡇⠀⠀⢀⡀⢀⠹⣴⣤⢦⣤⣞⣻⠀⠀⠀⢦⣤⠖⠹⢇⠀⠄⠐⠖⣶⠂⠂
⠀⠘⠁⣬⣲⣓⡾⠒⠒⣶⢃⠴⠋⠀⠉⠙⠗⠲⣦⡀⠀⠠⣙⢷⣞⠋⠀⠀⣦⡤⠶⠟⠑⣆⠀⢰⡔⠀⠰⠀⠀⠀⡄
⠀⠈⣰⢏⠀⠸⠓⠀⢀⣆⠙⠶⣈⠀⠂⠀⠀⠀⠀⡰⢳⠾⣅⠀⠉⡓⢾⡽⢯⡀⠀⠀⠀⠈⣆⠈⠓⠀⠀⠀⠀⡏
⠀⠈⠹⡀⠀⠀⠀⠀⡘⠉⠀⠉⢻⠑⢦⢄⣀⣀⢶⠠⣟⡢⠊⠛⠶⢍⡀⠻⣄⠻⣄⣰⣄⠿⠝⠀⠀⢹⠀⡞⠈⢱⡆
⠀⢠⣀⣿⠀⢠⠀⢘⠀⠀⠀⢀⣬⠉⠉⢻⠉⠿⡌⢻⣙⣯⣐⢰⡀⠀⠉⣷⣟⠱⠈⢿⠈⠁⢷⠴⠀⠺⠀⡇⠀⠀⡇
⠀⠈⢹⣯⣀⣳⠀⢹⠓⠒⠚⠋⡙⢚⢛⠏⣹⠒⣽⣻⠟⠛⠫⣟⠝⠻⢾⣷⢾⡽⠳⣼⠀⠀⠀⡀⠀⠧⢤⡷⠂⠈⠁
⠀⠐⡼⠂⠛⢽⡇⢸⠙⠁⠀⠀⠀⠌⣤⡵⢢⡄⠀⠘⣆⠀⠀⣼⠆⠀⠚⣄⣀⠤⠒⢚⣒⣉⣉⣉⡉⠐⠂⢸
⠀⢸⠁⠀⠀⡰⢢⡟⠠⣌⠄⠛⠀⣎⠹⣏⣓⠋⠀⢀⡼⢣⠉⢻⠆⠀⠐⡋⠀⠀⣋⠩⣁⢍⠹⣀⣀⣻⠁⠘⣷⠄
⠀⢀⠀⢀⣊⠀⠀⠈⠙⠟⠚⠛⠛⠛⠲⠤⠥⠴⠖⠋⠙⠊⠃⢸⡆⠀⠀⣱⢦⡀⠀⠈⣉⣈⣉⣁⣈⣀⠲⠁⠙⡆
⠀⠈⠉⢹⠟⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣄⣶⣞⣡⠞⡟⠛⣿⠉⠙⡆⠀⠈⣿⣁⣠⣀⣈⡟⢳⣀⠀⠀⢁
⠀⠀⠀⣼⠦⠄⠀⢈⣹⠗⠲⠦⠤⣶⣶⣾⣿⠛⣽⠁⠀⠀⠓⣶⣶⡿⠀⠰⣋⣀⣤⠀⠽⡇⠈⠉⠑⣙⣶⣶⠆⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠋
⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣙⣿⣿⣽⡍⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠀⠀⠓⠈⠅⠀⠀⢰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠍⣾⠈⠙⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⡀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡄⠀⡥⠀⠀⠀⣰⠊⠁⠈⠻⣆
⠀⠀⠀⠀⠤⠤⠤⣤⡤⠴⣦⣤⡤⠤⢆⠄⡀⠀⠰⣄⣀⣀⡈⡟⠉⣍⠂⡺⢀⡀⠀⠀⢠⣴⠇⠀⠀⠀⠀⠈⠉⠳⢦
⠀⠀⠀⣀⣀⢀⣌⠁⠀⠘⢻⣿⡿⠆⠈⠃⣄⣀⡤⠻⣿⣿⡐⡟⠀⡈⣠⠇⠀⢉⡆⠀⠈⢿⣆⠀⠀⡀
⠀⠀⠀⠀⠀⢺⣤⡀⣶⣿⣿⠆⠠⢀⠀⠀⠸⠀⠀⠇⢸⢸⣽⣷⣰⡷⢺⠀⠀⢸⢁⠀⠀⠀⠉⠳⠮⠤⠖⢂⣢⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠟⠸⠉⠀⠀⡀⠀⠀⠘⠢⣂⡀⠀⢘⢻⣇⣻⡷⠞⠅⠀⠈⠈⠑⠠⡀⢀⣀⣣⣼⡿⠿⣯⣙⠓
⠀⠀⠀⠀⠀⠀⠘⢯⠀⠀⢀⡀⠀⡠⠀⠀⠀⠀⠀⠉⠓⠺⠞⠋⠘⣏⠔⠁⠀⢂⡀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠈⠛⠕
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠚⣮⡉⠉⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣧⠀⠀⠶⠀⢺⣦⠤⢤⠤⠶⠶⠛⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢧⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⡅⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣆⢡⠀⠀⠀⠀⠀⠠⣠⡴⠞
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡙⠳⣥⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⡙⢏⠛⠛⠯⣉⠁
⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣌⣳⣄⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢤⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠟⣯⠁⠠
⠀⠀⠀⠀⠀⠀⢀⡯⡁⠀⠈⠈⠉⠉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⡇⣿⡀⠁
⠀⠀⠀⠀⠀⠀⣟⠁⠚⡇⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢄⠨⠫⢸⠄
⠀⠀⠀⠀⠀⠰⣧⠀⠨⠇⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⡄⠈
⣶⣶⠶⠶⠶⣶⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⣄⣀⣤⣤⣼⠶⠞⠋⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡛⠉⠉⠁⣠⣤⡿⠋⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠛⠛⠛⠛⠋⠉⠀⢀⣤⠿⠿⠃⣴⠿⠿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠉⢣⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⣰⣿⡇⠘⠀⠀⠀⠀⠀⠀⣤⠾⠉⠀⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣀⣹⠇⠀⠀⠈⠛⠾⠷⠘⠋⠀⠀⠀⠀⣿⠟⠉⠙⠳⠄⠀⠀⠀⠉⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠈⣿⡟⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⢰⡇⠀⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⣿⡃⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣰⡿⠀⠀⢀⠀⠀⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠰⣿⠁⠀⣠⠞⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣟⣠⠀⠀⠀⠀⠠⢤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⢴⣿⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣉⣡⣼⠖⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠋⠉⠙⣿⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡦⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠶⣶⢿⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠈⣶⡀⠀⠀⠀⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣹⣷⣦⣶⣦⣤⣀⣤⣄⣤⣤⣠⣤⣤⣤⣄⣀⣠⣤⣼⠏⠀⠀⣿⡃⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡄⠀⣼⠏⠩⢴⡶⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⢷⡄⠀⠀⠘⢻⠀⠠⠞⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣼⡟⠉⡟⠁⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣎⣵⡄⠀⠀⠃⠀⣀⡀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡉⠐⠒⠙⣿⡿⠀⣿⠛⠒⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠉⠉⢉⣟⣧⡀⢰⠟⠀⠉⠻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢶⣆⠀⣿⠃⢰⡟⠀⠛⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠺⠋⢻⣧⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡀⢿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠖⠲⢿⣿⣿⡇⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠀⠀⢀⣤⡀⠀⠓⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠒⠒⠺⢇⣿⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠁⠀⢠⠉⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠈⠀⠻⠿⠛⠲⣶⣾⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡾⠉⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡇⠀⠀⠀⠈⠀⠀⢠⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢀⣸⣿⡇⠀⠀⠀⠀⢀⣾⡿⣍⠁⢹⡷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣶⠟⣦⣀⠰⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠻⢯⣽⠃⠀⠀⠀⣰⠀⣿⣧⢻⣷⠎⠀⠘⣠⠆⠀⠀⠀⢸⡄⠀⢀⡼⠋⠀⠘⠿⠷⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⣤⣼⣿⠀⠀⠀⡸⠃⢠⡟⠁⢸⡟⠀⠀⠀⠙⣀⠀⠀⠀⣼⣅⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦⠠⠼⠃⠀⣾⠇⢠⣿⠀⠀⠀⠀⠀⠻⠟⢻⡦⠘⠋⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⢀⡀⠀⢠⡆⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠶⠶⠰⠶⠿⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣤⣤⣤⣤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⠄⠀⡀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⡶⠎⢹⡇⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢠⣶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡷⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠋⠁⠠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⠄⠉⠡⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠃⠀⢰⣏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠘⠛⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣏⣁⣀⣀⠀⣤⠤⠀⠀⠰⠆⢿⡁⣹⡿⠿⠃⠀⠀⠀⠀⠀⠀⠰⠉⠀⠋⢁⣤⣴⠶⠂⠉
⠀⠀⠀⠀⠀⣠⠐⠾⡍⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠛⠉⠁
⠀⠀⢠⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶
⡠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠛⠉⡛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣀⣤⣀⣠⣤⣤⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠛⢿⡽⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠆⠀⠀⠰⡋⠙⣤⡤⠒⠀⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⢐⣷⠞⡫⠀⡀⠀⣠⣤⡼⡝⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠿⠰⢿⡧⠀⠰⠀⠀⠈⡹⡟⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⢤⡴⠖⣿⡁⠀⠜⠁⠠⠒⠚⡒⠁⠀⠰⠿⠇
⠀⠀⠀⠀⠀⠀⠞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⡰⠃⠀⠃⠀⠛⠤⣶⣖⣀⣀⣀⠰⣄⠀⣀⣱⢦⡀
⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠉⠁⠀⣠⠊⠀⠀⢹⠇⠀⠀⠀⠀⠀⠘⠀⡀⠙⢦⠉⢃⣁⡦⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡔⠁⠀⠈⠃⢀⣀⡀⢠⠶⠃⠀⠀⢀⣴⣾⠟⠁⡈⡳⠋⠐⠀⠀⠀⠀⠀⠀⠀⠛⣤⢁⠀⠆⠈⢳⠀⠀⢂⠰
⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠇⢄⡶⠈⡤⣂⣐⣀⢠⣤⣀⠀⡀⠁⡄⠘⠈⡇⠀⠀⠈⠀⠀⠀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠻⣧⠉⠁⠀⡛⡯⠭⠉⠠⠾⠁⠘⠀⠋⡅⠄⢹⡆⠀⢀⠀⠐⠀⠀⣼⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣀⢀⣀⣀⣴⠾⠯⡏⢹⣿⣒⠶⠶⠶⠦⠤⠤⠤⠤⠤⣤⣀⣄⣤⣤⣼⣋⣈⣹⣆⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠐⠄⠉⠀⠘⠃⠂⠀⠀⠧⠟⣴⣯⣻⡟⣛⣒⣂⣂⣉⠉⠉⠀⠄⠐⠒⠂⠀⣾⠂⡘⣽⢿⢻⡙⠉⠻⡿⠶⠶⠶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠹⠁⣀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠰⠿⠤⠷⢧⣼⣧⣴⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⣥⠞⠁⠀⠀⠀⠀⡐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⡀⢀⠞⠁⠀⠀⠀⠀⢀⢼⣄⡴⣤⣀⡆⠀⡀⠀⠀⠀⠀⠀⢰⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠸⡄⠀⠀⠄⠡⢴⡯⢴⠂⠀⠀⠀⠈⢛⡀⠀⠀⠀⢈⠀⣼⠛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠃⣦⠀⠞⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⠤⣞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢁⠀⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠤⢠⣸⡗⠀⠂⠀⠀⠁
⠀⣀⣀⣀⣠⡀⠀⠀⢸⣾⣇
⣾⣵⣿⣿⡿⠷⠤⠜⠋⠉⣿⡄
⠙⠛⠟⠁⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢨⡅⢤⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡤⢧⣀
⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠁⠈⠉⠑⣤⠰⠶⠦⢤⣀⢀⠀⠀⠀⠙⠿⠿⢿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⢰⣿⠀⠀⠀⢈⠀⠀⠀⢠⣤⡤⠤⠤⢤⡤⠤⢄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣇⠳⣤⣀⣈⣀⣠⡤⢊⡿⠀⠀⠀⣾⠁⠀⠀⠈⢷⣄⠜⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢥⣀⡀⠀⠀⠀⣼⠃⠀⠈⡿⠷⡮⢭⠭⠭⠥⠔⠋⠁⠀⠀⢸⣇⢠⠀⠀⡠⠔⠁⣠⠞⠑⠴⠂⢉⠔⠒⠒⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡆⢠⡟⠀⠀⠀⠘⣦⡑⠢⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠀⢠⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡾⠁⠐⠀⠀⠀⠈⢷⣄⠀⠈⠐⠂⠠⠄⠀⠀⣿⠃⣴⠖⠋⠁⠀⠢⡄⣀⠀⠠⢸⡇⠀⠀⠈⢁⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣊⠃⢠⠇⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⢻⡀⣹⠀⠀⠀⠀⠀⠀⠈⠙⠀⢸⡇⠀⠀⠀⠈⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣿⠔⠁⢀⠀⠀⠀⠀⠡⡀⠙⠲⠤⠀⠀⠀⢸⡇⠙⡆⠀⠀⠀⠀⠀⠀⠄⠀⢸⡇⠀⠀⠀⠀⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⠂⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⣷⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⠤⡾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⡄⠸⡗⠂⠀⠀⠀⠀⠘⠀⠈⠉⠁⠀⢾⣇⠈⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⢬⣷⣄⣀⣤⡀⠀⠀⠀⢀⣀⡠⢂⣨⣧⢀⢻⡀⡘⠀⣠⡤⠶⡀⠀⣾⠛⠋⠀⠙⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠑⢦⣄⣉⣉⣉⣉⣉⣉⣩⣭⠗⠙⣇⠈⣴⠃⠉⠀⠀⠀⠇⠀⢻⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⠟⠓⠂⠠⠄⣀⢉⣉⡉⠉⠉⠁⢀⣀⣠⡴⠿⣦⡘⢧⠀⠀⠀⠀⢸⠀⠘⣧⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠋⠀⠀⠀⠀⠀⠁⠀⣸⠟⠁⣈⢿⠁⠀⠀⠀⠀⠀⠉⠻⠦⡀⠀⠀⠘⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠁⠀⠀⠀⠀⠀⣀⡴⠛⠁⣀⠔⠁⠀⠳⣄⣀⣙⣛⣦⠤⠶⡤⣈⠠⢄⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠡⠐⠀⠁⢀⣠⡤⢒⣉⢁⣤⠔⠊⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠉⠉⠉⠲⢦⣝⢦⡀⠀⢹⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⣀⠤⣚⡽⠟⠛⠉⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠑⢄⠀⠀⠙⣧⡙⣆⠀⠳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡴⢊⠐⣋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠣⡀⠀⠘⣧⠘⡁⠐⠇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⢞⣾⠃⠀⠏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡋⠀⠀⢱⠀⠀⢻⡄⡇⠀⠀⣀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣈⠎⡿⠀⠘⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠃⠀⢸⡇⣿⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠈⣤⠀⠀⠀⠀⣲⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡲⠤⠄⢼⡇⢹⡄⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠀⣿⠉⠂⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢳⣼⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡸⠏⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⢳⡄⠀⠳⣄⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠔⢋⣤⠶⠟⠛⠛⠓⠒⠤⢤⣄⣀⣀⣀⣀⣠⠤⠖⠚⠉⠁⠀⠀⢀⡴⠟⠀⠀⢀⣸⢆⡀⣀⡀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⠛⠀⢘⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠲⠖⠛⠛⠁⠸⠋⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠉⠒⢲⠋⠳⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠂⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣦⣀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⣿⣮⡙⠿⠿⣿⣿⣿⣿⣦⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⡿⠇⣷⣿⣿⣿⣿⣿⣿⣦⠹⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠋⠉⠛⠋⠀⢹⣿⣿⣿⣿⣿
⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⢸⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠹⣝⣻⣾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣇⠀⠀⠀⢠⠞⠁⠀⠀⠀⠘⣿⣏⣽⣦⣽⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣦⢀⡴⠁⠀⠀⠀⠀⠀⢠⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣠⡀⠀⠀⣼⡿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣴⣄⣰⣾⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⡇⠀⠀⣿⠃⢸⣿⣿⣟⣾⣿⣿⣿⣿⣻⣿⣿⣿⣠⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣷⣤⣴⠍⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣏⣷⣾⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣱⣄⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠉⠉⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣳⠒⠚⠙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⢉⠖⠛⠿⣲⣶⠿⠋⠈⡝⠉⡹⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇⢎⢞⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢽⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⡡⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⢘⡆⠀⠀⠀⠀⠀⠀⠀⢸⣯⠊⢥⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣤⡴⡿⣄⡀⠀⠀⠀⢀⣀⡞⢿⢶⣯⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⢯⡼⢁⠷⣭⣛⠿⣻⡏⠭⢝⣤⡟⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡼⢹⡇⣿⢹⡶⣕⠲⣸⡽⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢤⣱⠃⡾⢠⡟⢸⡇⢻⡿⣾⣷⡷⠒⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡿⢽⡇⢸⠇⠸⡇⠸⣧⠩⢹⡃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣶⣯⣻⣿⣿⡰⣇⣿⣋⡽⠥⠾⣗⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⣻⠋⠈⢻⠀⠀⠀⢻⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⢀⡇⠀⠀⠘⣆⣠⣤⡟⣇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠽⣿⠆⠀⠀⠀⡿⠝⠿⡇⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣖⣒⣺⡆⠀⠀⠀⣿⣶⢲⣿⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠽⢻⠁⠀⠀⠀⢻⠫⠝⢹⠀⢸⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡟⠀⠀⠀⠀⠈⣆⠀⢸⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⡠⣷⠀⠀⠀⠀⠀⣰⣒⣮⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡐⢨⡾⠃⠀⠀⠀⠀⠽⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠋⠁⠀⠀⠀⠀⠀⠘⠦⣤⡶⠟

⠟⠒⣢⠤⡄⠴⢶⣶⢤⠶⢶⡷⠀⣸⠇⠀⠀⠀⠀⢄⠀⢸⠤⢤⡖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡋⢰⠀⡇
⠀⣀⠃⠀⠀⡀⠈⠁⣀⣠⣀⣳⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⣿⣇⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠁⠘⠀⠁
⠎⢙⣇⠀⢦⠿⡀⠀⠀⣮⣽⣷⠄⠀⢐⠀⠀⠈⠁⠈⠑⠀⠀⡙⢫⠀⠀⠀⠁⢐⣀⠀⢰⡀⢈⠀⠀⣀⠀⣀⠀⠀⠀⢰⣶
⠰⠋⠋⡁⠐⢤⣽⢀⠀⡏⢿⣿⣰⠀⠘⠃⢀⡀⡀⣀⢀⠀⡀⢫⡉⡃⠀⠀⣀⢘⣁⣄⡛⢀⠈⡀⠈⢳⣦⢾⡶⠀⠀⠀⠏
⠣⠀⢌⠦⣠⡜⢷⡎⠠⣇⢠⡀⢻⣼⢀⠠⣼⣧⣤⢻⣼⣼⢧⠾⢿⢹⣅⠀⢉⢉⣿⠁⠀⣤⠄⡶⠰⠘⠘⠊⡑⠄
⠀⠐⢿⠀⢂⠶⢈⣻⢂⡘⢇⣭⢉⠅⠈⠶⠿⣿⠙⠙⢿⢃⠘⠀⣾⡍⢻⣧⣌⠩⣟⣾⡖⢻⡇⠃⡇⣄⠐⣰⠁⣐⡚⠈⠰
⡆⠐⡀⢣⣀⠘⣀⠻⠛⠳⡘⠡⣘⣌⣴⠴⡶⣼⣇⠀⣺⢿⠀⠈⡿⠉⠉⡻⠟⣴⠓⢫⠇⠽⣤⢰⣧⣭⠶⣛⣥⣴⡀⢀⡀
⢱⣰⣛⣞⠧⠀⡤⠄⠀⢂⣿⡂⣤⣾⠴⠆⢹⣿⡏⠠⠈⢠⣴⣤⣠⡾⢶⡐⠠⡁⣴⡇⠖⣸⢁⡁⡔⠳⠄⠈⠯⠙⠷⣦⡽
⠛⠉⣆⠀⠀⢘⣧⡦⠀⠉⣭⣅⠃⡘⠂⠠⣬⢙⣿⠴⢴⣘⣦⣝⠨⠄⡰⠀⣈⢁⣭⡁⣬⡄⡀⠁⠀⢠
⣄⣴⡟⠄⠀⣁⡿⠈⢘⣧⡅⣠⣤⣆⣈⠷⠄⠭⠁⢠⣌⣁⣝⣇⠠⣦⢓⣆⣴⣰⣿⡀⣄⢠⣧⣀⣠⡴⣤⣴⣄⣠⡀⡆⠠
⣾⡿⠗⢀⡄⠈⣰⣞⣟⠙⠘⠩⢁⡑⠛⣀⠦⣤⣾⣿⣿⣿⣾⢹⣅⠾⣺⣵⠏⠟⠿⣷⡸⣿⡟⢹⣿⠅⣿⣿⣿⣿⣇⠇
⡩⠃⠀⢺⡤⠀⣸⣾⡻⢟⠙⣠⣬⡷⠞⣟⣂⣩⡛⡿⢡⣿⡏⡌⣿⡐⣄⣉⣴⡗⣀⣿⡾⣿⠀⣿⣿⢰⣿⣿⣿⣿⣿⣾⠻
⢕⣡⠀⢹⣿⡀⢼⡿⣿⠿⣷⣿⣿⢀⣿⣭⣭⣭⡟⠀⣏⣿⠇⡇⣿⡶⢩⡍⠠⠱⢈⡿⣷⣿⢰⣿⣻⢸⣿⣿⣿⣿⣿⣭⣤
⢿⣻⡆⢹⣟⡇⢸⠿⠿⣻⣿⣿⠗⣺⠙⣋⠛⠟⣟⢴⡿⣿⣤⣧⣷⣦⣿⣿⣵⣶⣶⣿⣯⣾⣿⣿⣯⣸⣿⣿⣿⣳⡿⠭⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣦⣄⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣍⣉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣦⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠈⠙⢿⣿⣿⣷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠛⠉⣀⣠⣤⣤⣤⣤⣄⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⠀⠀⣀⡀⠀⠠⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠛⠋⠉⠁⠀⠀⢀⣀⣤⣤⠀⠸⣿⣿⣿⡀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⣀⣿⣿⡇⠀⢠⣤⣶⣶⠀⠘⣿⣿⣿⡄⠀⢿⣿⣿⣇⠀⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠘⠿⣿⡟⠀⢀⣿⣿⣿⡇⠀⠘⣿⣿⣿⡄⠀⢸⣿⣿⣿
⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠈⠀⠀⣾⣿⣿⡟⠀⠀⣼⣿⣿⡿⠁⠀⣼⣿⣿⡿
⠀⠀⠀⠀⢿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⠙⢿⣿⣿⣿⣦⣄⠀⠈⠙⠛⠀⢀⣼⣿⣿⡿⠁⢀⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠈⠻⢿⣿⣿⣿⣶⣤⣀⠀⠈⠙⠛⠟⠁⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⡀⠀⠈⠉⠉⠛⠛⠻⠿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⣤⣤⡤⣤⣤⣤⢤⡤⠤⠤⢤⡤⠤⠤⢤⣤⠤⢄⠄⠠⠤⠀⠠⠀⠀⣠⡤⠄⠀⠀⡀⢀⠤⠀⠀⠀⠀⠀⣤⠠⢤⠠⠀⠤⠀⠀⠀⠠
⠀⡿⠟⠋⢻⣿⣄⠋⠄⠀⠀⠈⠑⠸⠃⠀⡱⠋⠉⠉⠀⠀⠀⣠⠀⣴⡷⠁⠘⢆⠀⠀⡞⠀⠢⠀⠀⠀⢰⢇⡀⠀⠱⡀
⠀⣿⢿⣷⣿⣽⣷⢀⢾⡦⠐⠂⠀⠀⠀⢱⡦⠄⠀⠀⠀⢤⣤⣀⣶⠯⠁⠀⢀⠈⢆⠸⡑⠲⡄⠀⠀⢀⠋⡈⠈⠀⠀⠙⢦⠹⣄⠀⠘
⠀⠏⠳⣹⣿⠉⢰⠟⠸⠁⠀⠀⠀⠀⠀⠸⢧⣐⣀⠈⠀⠰⠮⣿⡣⠁⠀⠐⡟⡀⣨⢦⠀⠀⣘⠠⢰⡎⠦⠀⠀⠄⠀⠀⠀⠑⣌⠢⣼
⠀⠀⠀⠘⣷⢄⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠿⠿⠷⠤⠴⠌⠋⠹⠷⠄⠳⠐⠿⠋⡁⠿⠔⠤⠀⠠⠀⠀⠀⠢⠸⠧⢸
⠀⣧⣤⣥⣤⢬⣿⠄⠀⢤⣤⢤⡤⣄⠠⠤⠐⠤⠂⠴⢆⠐⣶⠶⠂⠶⠶⠶⠶⢶⣶⠶⠢⡞⡚⢦⣤⣲⣆⠲⡖⢰⣶⣶⣶⡶⠈⢶⡆
⠀⡿⣿⠀⡿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠳⡈⠀⠀⠘⠒⣦⣔⢼⢸⡤⢸⡇⣸⡟⠹⠿⠋⠳⣄⠘⠚⠋⠈⠙⠛
⠠⣧⡾⠛⣡⡇⠀⣷⠀⡄⠙⠦⢀⠀⠀⢃⣽⠶⠒⠀⠂⠈⣽⢶⡂⠄⠀⠠⠀⠸⢚⠀⠀⡇⢈⠄⢀⣰⣞⡀⢁⠀⢀⠤⣤⠤⣤⣤⡼
⠐⣧⡿⠖⣿⠇⠙⠛⢿⠿⠚⠐⠋⠑⢦⡙⠢⠔⠒⠒⠀⠀⣤⠤⠀⠀⠀⠀⠀⢸⢸⠀⠁⠋⠃⣵⣉⠀⠀⠀⠀⢶⡜⠀⠁⡴⠇
⠐⣴⣳⠀⠉⠰⠶⠶⣶⠠⠶⠤⠤⠴⠤⠭⢃⢀⣀⠰⠶⠶⠿⠀⠀⠶⠶⠶⠦⠾⠶⠆⠛⣧⠀⠿⣿⣖⡒⢒⣻⡇⠴⠶⢶⢿⡄⠐
⠐⡿⠋⡜⣡⣰⡶⠶⠷⢤⠀⣒⠀⣀⣐⠒⣉⡩⠃⣀⡀⠀⣀⣀⢀⣀⢀⣀⣀⣀⣀⠀⡂⠑⠲⣠⣤⠀⣀⢠⣄⣉⣩⣥⣬⣿⣦⡸⣡
⠀⣷⡜⠐⠻⣛⠷⠚⠁⡖⠁⠒⠒⠀⡶⠴⠲⡄⢲⢿⣶⢦⡀⠲⠟⣟⢷⠮⡍⢯⡴⢋⣤⣼⡁⠀⠀⢀⣤⣤⣤⣤⣤⣤⡅⢠⣍⠀⣭
⠀⡏⢹⢉⣡⠄⠼⢦⣤⣦⣀⡖⡬⠄⠠⠤⣀⢱⣌⣡⠄⠉⠙⠲⠦⢤⣼⣶⣆⣲⣀⣻⣛⣒⣖⣀⣀⣠⣠⣽⣖⠂⠀⠤⠔⢾⡆⠷⠤
⣤⣭⣤⣭⣭⣭⣭⣥⣭⣭⣭⣭⣤⣤⣤⣤⣬⣭⣥⣬⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣬⣭⣥⣭⣭⣬⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⡶⠶⠒⠋⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠿⠛⠋⠉⠀⠀⢀⣀⣠⣤⣶⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠑⣄⣀⡤⠴⠒⠉⠉⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡼⡆⣏⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣤⣴⣶⣶⣦
⠀⠀⠀⢠⠖⠛⠙⣿⣿⣿⣷⣻⡜⢾⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠻⠿⠟⢿⣿⣿⣾⣿⣿⣿⠿⠿⠿⠇
⠀⠀⠀⠻⠖⠒⠉⠉⠩⣿⣿⣷⡟⠞⠀⣿⣿⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠈⣿⣏⣱⣶⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⣿⡏⠐⣶⣤⣤⣤⣤⣶⡀⠀⠀⢸⣶⣾⣿⣿⣋⠀⣰⣿⣿⡏⠿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⣾⣿⣄⡈⣩⣿⣛⣿⣿⠇⠀⠀⠘⢿⡿⠶⣻⣿⣿⣿⣿⣽⠃⠀⠀⠀⠉⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣻⣿⣿⣿⣿⣭⣭⣯⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⣶⡞⢶⢦⣤⣤⣤⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⡿⠿⠿⠛⠛⢳⠸⣯⣿⣿⣿⣇⣠⠤⠴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠬⠽⠛⠛⠋⠉⠉⠉⠀⠀⠀⢀⣀⣠⡤⣴⠇⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡊⠉⠉⠁⠀⠀⠀⠀⢀⣀⣠⣤⣤⡴⣶⣶⣿⣿⣯⠀⠀⠀⠘⠤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠒⠒⠛⠉⠉⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠦⣄⣀⣀⣀⣀⣀⣠⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣗⠲⠤⠤⠤⠤⠤⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣯⡟⠛⠛⠋⠛⠛⠛⠛⠳⣿⡿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⡿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⣯⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⢿⢟⣿⣶⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣥⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣵⣿⣿⣿⣿⣿⣬⡧

⠀⠀⢠⡆
⠀⠀⢸⡇
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠒⠐⠶⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⢷⣤⡉⠉⠢⠄
⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣀⣀⠀⠈⠛⠢⠄⠈⠦
⠀⠀⠂⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡁⠀⠀⠉⠉⠀⠐⠒⠂⠀⠀⠀⠀⠀⠰⢾
⠀⠀⢣⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠿⠟⠛⠛⠛⠓⠚⠓⠒⠀⠀⣀⠀⠀⣸⡇
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣚⣋⣛⠋⢻⣶⣴⡾⠿⠂⠛⠀⢀⣀⣿⠇
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠛⠛⠛⢻⡀⠀⠀⢀⣀⠠⠤⢛⣿⠇
⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣶⠶⠟⠉⣀⣤⣾⡿
⠀⠀⠘⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⢿⣯⣭⣴⣶⣾⣿⡿⠟⠛⠛⢦⡄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⢿⡟⠉⠛⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡉⢻⣧⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣄⠛⢂⡸⢳⣾⡇⢀⣠⡾⠃
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣼⡛⢧⣀⠉⠃⠀⢀⠿⠋⠁⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠘
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⡟⠃⠤⠈⠉⠀⣠⡞⠀⠐⠀⠘⢿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣶⣤⣤⣄⣀⣼⡇⠰⠆⠀⠀⠸⣷⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣼⡇⠀⠈⠉⠉⢉⡝⠁⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣏⡀⠀⠂⠀⢀⡼⠁⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⣤⡶⠞⠛⠻⠶⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠏⠑⠶⣿⠀⣸⠃⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣰⠾⠛⣿⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣇⠀⣀⣴⡇⠀⠀⠀⠀⢻⣿⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠈⢿⣆⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⣿⣿⢺⣿⣿⣿⡶⠿⠶⢶⣤⣿⣿⣶⣶⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠀⠀⠈⣿⣆⢠⣴⠁⠀⠀⠀⠐
⠀⠀⠀⠀⣿⣿⠈⢹⣿⠉⢠⡄⠀⢀⣿⣉⠉⠉⠉⠉⠛⠛⠋⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠋⠀⠀⠸⡏⢹⠟⠁
⠀⠀⠀⠀⢸⣿⣷⡞⠃⣀⠾⠿⠿⣿⠉⠙⠿⠿⠶⠶⢶⣶⠶⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠛⡀⠀⠃
⠀⠀⠀⠀⢸⣿⣯⣤⣀⣀⠀⢀⡼⠿⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⡀
⠀⠀⠀⠀⣸⣿⣿⣙⠛⠛⢶⣿⣇⡀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠈⣿⡿⠷⠶⠦⠄
⠀⠀⠀⢸⠛⣿⡇⠉⠈⠳⡞⠉⠙⠿⠷⢶⣤⣤⣄⣠⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⢰⡿⠀⣀⡀⠀⣿⣷⢤⣤⡀
⠀⠀⠀⠸⣗⣿⡇⠀⠀⢰⣶⣄⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⡿⠃⢠⠿⣷⣄⠈⠻⣿⠉⠻⢿⡄
⠀⠀⠀⠀⣸⣿⣏⣀⢰⡏⠈⠛⠿⢿⣶⣶⣦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠠⠞⠀⠈⠻⣷⣄⠈⠁⡀⠘⠁
⠀⠀⢀⣾⠋⣿⣿⣛⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⣄⡀
⠀⠀⢸⣇⠀⠉⠻⢿⡟⠙⠿⢷⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⣶⣤⣤⣀⡀⠀⠰⣄
⠀⠀⠀⠹⣧⣄⠀⠀⠑⡄⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠂⢀⣴⡾⠛⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⡄
⠀⠀⠀⠀⢈⣿⣿⣶⣾⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⢠⣾⣿⣿⣿⣌⣙⠛⣻⣿⡿⠿⢿⣶⣶⣶⣶⣶⣶⣾⣿⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⣾⣿⠀⡉⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠘⠛⠛⠉⠉⢹⡏⠀⡄⢀⣽⠇⠀⠀⠀⠀⠀⢠⣾⠏⠁⠘⣧⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⣀⣀⣴⣧⣤⣷⣾⡿⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠴⠶⠶⠶⠶⠆⠀⠈⣿⡀
⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢹⠇⠀⢀⣀⣀⠀⣤⡄⠀⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡧⠄⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢰⣷⠚⠋⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣄⣀⣀⠀⣸⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⣿⡿⠛⠀⠈⠉⣉⠰⣿⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠉⠉⢶⡿⠛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠈⣷⠀⣀⣀⣀⣀⣠⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠙⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢻⡧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⡼⢇⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⠀⠀⠀⢿⣅⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠸⣇⠀⠄⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⣇⠀⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⢊⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠗⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠺⠀⢀⡄⠀⢻⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢹⠄⠀⢀⡀⠀⠀⢀⣾⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⢰⡆⠀⠀⠀⠘⢷⡀⠀⠀⠀⠘⢄⣠⠟⠳⠄⠀⢸⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣾⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⡇⠀⠀⠀⠀⠈⠻⣆⠀⠀⣠⠟⠁⠀⠀⠀⠀⣬⠃⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠙⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⢿⡇⣶⡄⠀⠀⠀⠀⠙⢷⣄⠈⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡧⠀⢹⡆⠀⣿⣯⣽⠆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠧⣴⣄⣨⠟⣈⡅⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⣀⠀⠶⡶⠞⠉⠁⠀⠀⠘⠿⠞⠋⢉⣿⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠘⠉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢆⡀⠀⣸⡿
⠀⣶⠀⠀⠀⠀⠀⠀⢀⣄⣠⣀⣀⡀⢀⣠⣾⡇⠀⠀⠀⠀⠀⡐⠛⠉⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⡀
⠀⠻⢶⣶⣤⣶⣤⣴⣶⡷⠾⠿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣦⣶⣶⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦
⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠻⠉⠻⠟⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⢀⣀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣪⣵⡾⠷⢶⡭⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⡟⠁⠀⠀⠀⠙⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣏⣓⢦⡀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⡷⠿⣻⣿⣸⠛⣇⠀⢀⣟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣽⠞⠉⠀⠀⢈⠝⣡⣾⣿⡤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠖⠋⠀⠀⢀⡤⣚⡵⠞⣽⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣭⠞⠁⠀⢀⡠⢖⣽⠞⠉⢀⢾⠏⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠵⠋⠀⠀⣠⣖⣭⠞⣻⠏⠀⢀⣾⡟⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣠⣔⡿⠞⠉⠀⣸⠏⠀⢀⣮⡟⡇⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⣠⣴⡿⠞⠉⠀⠀⠀⣴⠏⠀⢀⢮⠟⠘⡇⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⠟⠁⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⣸⠏⠀⢠⢫⠏⠀⡆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⣀⣴⣾⡯⠵⠒⠲⣼⡿⣅⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⢣⠏⠀⠀⢱⡇⠀⣧⡇
⠀⠀⠀⠀⣠⣞⡵⠛⠁⠀⠀⠀⠀⡸⢠⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⣣⠏⠀⠀⠀⢸⡇⢠⢻⠃
⠀⠀⢀⢞⡵⠋⠀⠀⠀⠀⠀⠀⢰⠃⡞⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣰⣳⠏⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢀⣏⡞⠀⠀⠀⠀⠀⠀⠀⣰⢃⡜⢁⣿⡟⠀⠀⠀⠀⠀⠀⢀⣠⠷⢷⣴⣿⠏⠀⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢰⡿⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⡠⣿⡟⠀⠀⠀⠀⠀⢀⡴⣫⠴⠚⢯⢹⣿⣆⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠈⢿⣄⠀⠀⣀⡠⢖⣵⣪⠴⣫⡾⠋⠀⠀⠀⠀⠀⢀⣞⡼⠁⠀⠀⢸⢀⡇⢻⡆⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠙⠛⠿⠿⠿⠭⠥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⡼⣸⠁⠘⡇⠀⠀⠀⢠⣾⠽⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⣰⣷⠇⠀⣸⠃⠀⠀⢠⣿⠃⠀⠈⣿⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡆⠀⢀⡴⡿⠋⢀⣴⠃⠀⠀⠀⢸⣿⠀⠀⠀⢹⡇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣽⠮⢴⣺⠟⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⣸⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧⣣⣀⣴⣟⣴⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣉⣁⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡆⠈⣀⠀⣀⠀⣡⡔⢓⠀⠀⠀⠀⢀⠀⠟⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⢰⡄⠸⢿⡞⠋⠀⠉⡄⠀⣿⣾⠷⠾⠛⠲⠛⢠⡈⢻⣠⣾⣆⢀⣼⣁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣴⡄⠀⠀⠀⠀⠀⠀⠀⠿⢺⣿⡄⠀⠀⠀⠠⣀⣀⠿⣿⣿⣿⣷⣿⣶⣾⣆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣽⠀⠀⠀⢠⡷⣿⣿⡿⣿⡀⡂⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠀⠀⠀⢀⣿⣇⣰⡿⣿⣿⣾⣿⣿⣿⣟⣁⠘⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠚⣇⣸⡇⠀⣤⣾⡟⢯⣿⡿⡿⣷⣻⡄⠀⠀⠀⠀⠀⠀⠘⠛⠟⣀⠀⣀⣸⡯⣹⣿⣵⣟⣿⠉⠼⣿⣿⣧⣿⣦⡝
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣉⣡⣄⡁⠘⡁⠸⠯⣤⣾⣿⣿⣶⡤⠭⣤⣤⣤⡠⠀⣄⣮⣽⣿⣦⣿⣷⣾⣿⣤⣄⣨⣤⣀⣹⣿⡟⣿⢹⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢈⣥⠸⢿⣽⣿⣿⣷⠀⣐⣒⣁⣼⣿⣯⣷⡄⢰⣿⣿⣿⡧⠀⠀⠀⠿⡏⢵⡖⢫⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣾⢹⣿⣷⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀⣼⣿⣿⣏⡙⣿⣟⣿⣿⣿⣿⣿⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⡚⠀⠋⠙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣤⡿⢿⡻⢋⢅⣼⣛⣙⢻⣿⣿⣿⣷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣬⣝⢫⣆⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠐⠀⠁⠃⠞⠹⢉⠿⣦⣿⢣⣾⠋⠻⣏⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⡀⠀⢀⣀⣀⣀⣀⣉⣛⣛⣃⠀⡈⢂⡀⣀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠒⠋⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠰⠔⠊
⠀⠀⠀⠀⡀
⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⠖⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠴⠚⠤⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠙⠋⠲⠶⣦⣄⣀
⠀⡠⠠⣭⡀⠀⠀⠀⠀⠄⠠⣀⠀⠈⠀⠛⠁⠠⠠⣦⣤⡀⠀⠉⣡⣶⡀
⠀⢀⣤⣴⣶⣶⣶⣤⣤⣤⣀⣲⡤⢌⣉⡐⠒⠢⢄⠀⡌⣫⡔⠛⠋⠹⣿⡄
⠀⢸⣿⣿⠟⠛⠻⠿⠿⠿⢿⡿⠿⣿⣷⣾⣭⣓⢶⣴⡜⠇⠀⠀⠆⠀⠹⣿
⠂⣾⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⠿⣿⣧⡹⣿⣎⠀⠀⠀⢀⣀⠀⠠
⡀⡇⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⢹⡟⠀⠀⣴⢻⡿⠀⠀⠄
⡀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢿⣦⡀⢹⣆⣧⢀⢠⡄
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡿⠇⢀⡿⠈⠰⡌⡇
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁⢻⠀⢸⡗⡀⠀⢹⡟
⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠘⠛⡄⡜⡇⠇⠀⣼⠁
⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⠙⢱⡇⣆⠔⠁
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⡅⠈⣸⣿⣿⣆
⠀⠠⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⡶⠟⢝⡯
⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣦⠔⠁⠀⠈⠁⠒⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⢙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢄⣀⠀⠀⠸⣿⣿⠗⠲⠤⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠶⠀⠀⠀⢁⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⣱⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣷⡶⠀⠀⠀⢠⠖⠚⠩⠿⣧⡤⠾⠉⠙⠒⣄⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⠛⢸⡿⠋⠛⢆⢰⠟⣶⠀⠀⠀⠀⠀⠀⠀⢰⢶⠈⠔⣺⠉⠈⠉⢧⢨⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠸⠇⢀⢀⣘⡏⠀⢻⡂⢀⠄⢀⣀⠀⢀⢸⡀⢄⢧⠸⡄⠀⠀⠀⢣⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡌⠿⢇⠁⢹⡑⣿⠋⠀⠛⠿⣆⢈⡃⠈⠆⠑⣔⣦⡀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⡜⣷⠤⠊⠀⠀⠉⢳⠀⠀⡤⠀⡛⠁⠀⠀⠑⠬⣙⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⣰⡾⠄⠄⠀⣰⣧⣄⠀⠀⠀⠀⢸⠟⢷⠆⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⢠⣄⣀⣟⠛⠶⣶⣃⣶⣮⠿⠟⠃⣦⠀⠀⡸⢁⣿⠂⡞⣱⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣿⡿⠿⠳⣦⣬⣭⣁⡀⠀⠀⡰⣾⣷⣿⣠⡼⠃⠀⠬⢯⣽⠅
⠀⠀⠀⠀⠀⠀⢀⡀⣤⠀⠀⠐⢲⣾⡿⣷⣄⠀⠀⠀⠀⠙⡶⣦⡶⠀⠀⠀⠛⠙⢿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⣴⠉⡙⠁⣀⠄⢒⣶⣿⣇⠃⢸⡿⣄⡀⠠⢄⡉⠉⠤⠖⢀⣀⣤⣾⠏⠉⣿⡿⢷⡦⠀⠁⢤⡤⣀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢣⢶⣿⣿⣽⣿⣧⡈⠈⣙⡿⢷⣦⣬⣭⣻⣉⡬⣋⡵⢻⢆⡼⣿⡧⠚⡿⠆⣆⠈⠋⠻⣥
⠀⠀⠀⢀⡤⠃⠀⠀⠹⣎⠋⡀⢈⣿⣿⣿⣯⠲⣙⣿⣶⣿⣿⣿⣿⣿⠿⠫⢐⡿⠊⢱⣿⠁⢟⡳⢃⠿⠀⠀⠀⢹
⠀⠀⣰⠉⠀⠀⠀⠘⠢⣉⣁⡡⢞⣵⣿⣿⣯⠱⣜⣿⣶⣿⣿⣿⣭⣷⣦⡶⠏⡡⢠⣿⣿⣦⠀⣀⡄⠀⠁⠀⠀⠸⡄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣽⢿⣿⣿⣿⣿⣿⣿⣿⣷⣛⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⣛⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢻⣿⡿⢿⣿⣿⣿⣷⣭⣿⣿⣿⣽⣾⣿⣿⣿⡿⠋⠸⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⠛⠉⣸⡎⡿⠛⢿⣿⣏⡉⠛⠛⣻⣿⠛⠹⢿⡇⢠⡅⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠈⠛⠹⠀⠀⠈⠳⢌⣻⣿⣛⣫⠽⠃⠀⠀⢨⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢇⠱⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣠⢞⠡⠔⠂⠲⣤⣤⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠀⠀⣰⠋⢀⣀⠄⠀⠀⠀⠹⣧⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⡰⠀⣰⣿⠵⠆⠀⠤⣀⣼⣋⣙⣻⡤⠠⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠻⠦⠶⠁⡰⢋⣴⣾⠛⠉⣠⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣰⠗⠉⠉⠉⣩⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣬⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⣼⠟⠂⠀⠀⠀⠀⢀⣠⣶⣴⠾⣿⣿⢿⡿⠿⣿⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠰⡆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠀⢀⣴⣿⠷⠟⠏⠀⠀⠀⠀⠀⠀⣀⠀⣀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢀⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⣿⠟⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢤⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡅⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⢸⡇⠀⠀⠀⠀⠀⠀⢠⣎⡀⠚⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠀⠀⠀⠀⡇⠈⠀⠀⢀⣀⣀⣀⣠⠄⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⢘⠟⠛⠻⢷⡒⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⢿⠛⠻⠟⠀⠀⣠⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠊⠀⠀⠺⠟⡇⡿⠀⠀⣠⡤⠖⠂⢠⣴⠛⠐⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣸⣶⠤⠤⠄⠠⢴⡞⢣⡐⠲⠦⣄⣁⣤⠠⠿⠓⣄⠀⠸⢿⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠲⣇⠀⠉⠉⠐⠀⠀⠀⠀⠁⠀⠉⢀⠀⢨⡇⠀⢀⡄⢲⣿⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠰⢿⢯⣀⠀⠀⠀⠈⢶⣄⠀⠀⠀⢀⣄⠀⠀⠀⠠⣿⠀⠀⠇⠀⠀⠄⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠉⠈⠙⠀⠛⠓⠟⠷⢄⠀⡀⠾⠇⠀⢆⠀⠌⢛⡒⠂⢉⣠⢛⡑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠃⠀⠀⠀⠈⠂⠀⢀⢦⡀⢀⣿⡱⠀⠀⠀⠙⠦⠈⠛⡛⠤⠌⠙⠮⢹⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁⠀⠀⠀⠀⠀⠁⠀⠘⢸⡿⣿⣟⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠈⠀⢠⠓⣍⠉⠛⠷⣆⣁⣠⡴⠖⠀⠀⠐⠀⠀⡠⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⡐⠀⢀⡇⠀⠀⠀⠀⡽⠛⢿⡟⣀⡀⠀⣀⠀⠀⠞⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢰⡀⠀⠀⣤⠁⢀⡜⣤⠌⠀⠀⣰⡄⠤⣤⣧⠀⠀⠀⠀⠀⠀⠌⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⣤⠰⠃⠀⡬⠹⠏⠀⠀⣰⠋⠀⠀⠀⢻⣧⣀⡀⠀⢀⣼⠇⠹⣆⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⢹⡸⣿⡟⠁⠀⠺⠖⠁⠀⠀⣼⠇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⠇⠀⠀⢻⡀⠀⠀⠈⡄
⠀⠀⠀⡤⠀⠈⠃⠀⠀⠀⠈⣇⠉⠀⠠⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⣰⠿⠍⡂⣿⡿⣀⠀⠀⠘⡇⠀⠀⠀⠘⡄
⢀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠛⠚⠂⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣰⣷⠏⠐⠢⠄⠘⡏⠀⠀⣀⠴⢿⠀⠀⠀⠀⠘⡄
⠚⠉⠉⠉⠉⠙⠢⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠠⠁⠀⠘⣯⡠⠉⠁⠈⠀⣷⡀⠀⣻⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣿⢀⠃⠀⣠⡾⠁⠀⠀⠀⠀⢀⠉⠁⠀⠈⠳⢀⡿⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⢀⠆⠀⠀⠈⠀⠂⠈⢩⠃⠀⠀⠀⠀⠀⢸⡀⠈⠀⠀⠀⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡸⠤⣄⣀⡀⠀⠀⣿⠠⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡤⠄⣀⠀⠀⢠⣴⠁⠀⠀⠈⠉⠁⢰⣿⣿⡇⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⢀⣴⠾⠃⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠀⠤⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣀⣤⠎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣷⣿⣿⣿⣷⣯⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣽⣿⣿⣿⣿⠟⠋⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⠟⠀⠀⠀⠈⠉⠈⠓⢽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⢳⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⠀⠀⢀⣜⡾⠃⠀⠀⠐⠀⠀⠩⠭⠀⠀⠉⠈⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⣽⣢⣔⣯⡾⠁⠀⠀⠼⠿⢶⣶⣦⣤⣤⣤⣤⣄⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢽⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠋⠀⠀⣾⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣙⣲⣤⣤⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡳⠃⠀⠀⠈⠋⠀⠐⠈⢠⡤⠄⠒⠂⠉⠉⡈⡉⠈⠉⠉⠉⠉⠉⠙⣾⡆
⠀⠀⠀⠀⠀⢀⣀⣀⢴⣺⠽⠁⠀⠐⡀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻
⠀⠀⠀⢀⣔⡽⠛⠛⠉⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⣾⡞⠀⢠⠁⠀⠀⢀⢀⣀⣠⣤⣿⣦⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇
⠀⠀⠀⢻⢧⡠⠈⡄⠢⠀⣄⣻⡿⠉⢱⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠈⣿⣿⣧⢈⣠⣧⢻⣱⠃⠀⢸⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽
⠀⠀⢀⢔⡷⠃⢉⣟⠽⣞⣶⣫⠇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣯⡏⠀⠀⣏⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣹
⠀⠀⠻⣷⣦⣞⠝⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡳⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣿⢿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠫⢟⣿⢿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⣠⢖⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠿⠿⠿⣿⡿⡿⠿⠿⠗⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣴⣡⣉⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠄⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠈⣿⣿⣯⠈⡄⠀⠀⠀⠀⠀⠀⣀⠴⢻⠿⣟⢉⣴⢉⣤⡌⢱
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡾⠋⠀⠀⠀⠀⠀⣠⡞⠛⢛⣻⣇⣀⣀⠔⢆⠀⠈⢉⡶⢼⣆⡉⠀⠀⢀⣁⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣛⡿⣧⠀⠀⣀⡀⠤⠚⠉⢀⡔⡫⠀⡀⠛⠁⠀⠀⠳⡀⣯⡁⠀⢹⠝⣲⣖⣿⣟⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠲⣤⣍⡉⠻⢷⣆⣠⠀⣼⡏⢀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⡀⠿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⢿⣮⣿⣿⡿⢿⣯⡄⢳⡀⠀⠀⠀⠀⠀⠀⠹⠆⢈⣀⣄⡏⠉⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡍⠘⢉⣿⡟⢁⣿⣿⣿⣭⡷⣄⠀⠀⠀⠀⠀⠀⢻⣿⡏⠉⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠋⠀⢺⣿⢿⣯⠀⠇⠘⢷⣄⠀⠀⠀⠀⢠⠀⠈⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⠀⠀⣠⣶⡾⠑⣿⠃⣠⣾⡘⠒⠺⢷⣦⣤⣤⡜⣑⣰⡒⠐⢂⡖⢀⠙⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠳⠿⣿⠿⠖⠊⠉⢛⣿⣿⣷⣀⡰⠶⡏⠈⡇⠀⠉⠉⢀⠀⠉⠉⠉⠑⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡛⠬⢗⠈⣿⠇⢰⡗⠊⢡⠀⠀⠀⠌⠀⠀⠀⢀⡤⠉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠈⠻⠛⠛⢿⣿⠀⢸⣇⠀⡞⠂⠀⠀⠀⡀⢀⡔⠋⠀⠀⢹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⣷⢰⣆⢿⣄⠀⢿⡴⠇⢸⡆⠀⠀⣷⠏⠁⠀⠴⢄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠉⣻⣥⣸⡟⡄⠀⠃⠀⢠⡇⠠⠀⠀⠀⠀⠁⠐⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣥⣤⣶⣤⡔⢀⣾⣦⣴⣶⡖⠒⠶⠶⠞⢉⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠻⡿⠋⠀⠀⠀⠄⢹⣿⣿⣿⠁⠀⠀⠀⠀⠈⣿⠶⡏
⠀⠀⠀⠀⣰⠒⠰⣐⡯⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢀⣾⡿⡀⠀⠁⠀⠀⡇⠂⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠟⠰
⠀⠀⢠⡴⠓⣢⠂⠉⠁⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠃⠢⣼⠁⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⣀⣶⠀⢸
⠀⢠⡿⠃⡠⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠿⠇⠀⠀⠀⠀⠀⣌⠈⠓⡂
⠀⠈⣅⣠⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⢹⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠃
⠀⠀⡟⠁⢀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠁⣠⠊⠀⠀⠈⠉⠛⠛⠻⢿⣿⡶⢄⠀⠀⠸⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⢱⡠⠂⠀⠀⠀⠀⠀⢠⣖⠁⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠲⡀⠐⠦⡂⠈
⠀⠀⡜⠁⠀⠀⠀⠂⠀⢀⣿⣿⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠒⠀⠀⠱⠶⠀⠞⠃
⠀⠀⢳⡀⢀⠔⠁⠀⠀⠘⣻⣿⣏
⠀⠀⠀⠙⡏⠀⢠⣴⠂⠀⠛⣿⠛⠁⢀
⠀⠀⠀⠀⠘⠆⠛⠁⠀⠀⠀⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⢐⣄⣠⠀⢴⢄⣜⣧⡈⠱⠆⠐⢢⡠⠐⢴⠂⢻⡵⠋⠁⠀⠀⡛
⠀⠀⣘⣂⠠⠆⡼⣧⠙⣯⡰⠕⢧⠑⣆⡠⢱⡞⠀⢠⡄⠀⣰⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠋⠉⣀⠆⢀⠙⣰⠀⠊⢳⣏⡃⣑⢰⣿⠁⢸⠞⢛⡫⣔⠂⠘⠦⡠⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠒⠀⡆⢂⡀⢺⡅⠡⠀⡀⠀⠈⠳⣿⣿⡿⢀⠈⡄⠀⠀⣀⠀⠀⠠⡄⢸⠃⠀⠀⠀⢠⡖⠄⡀⠟⡆⢠
⠀⠀⠄⠀⡀⡄⢸⠅⣄⠀⠀⣰⠀⣉⢸⡏⢡⡾⣿⣵⠂⢀⠋⠀⠀⠀⠀⣻⡀⠀⠀⠀⣰⠤⣈⡃⣾⣿⣿
⠀⠈⠀⠋⠋⠁⢽⠀⣄⠈⠲⣥⡀⡉⢺⣿⡟⠙⠛⣇⣼⣦⣷⠤⢠⡀⣈⠼⣷⡀⣤⢾⡛⠿⣏⣷⡟⢿⣉
⢀⡀⣀⠐⡁⠁⢸⠀⢋⡀⠀⢊⠱⡅⢸⣿⠁⣸⡿⣆⠋⣿⠛⢛⡋⢀⡀⢘⣯⢫⡈⣿⣷⣞⣋⡙⠙⠛⢛
⠀⢀⡉⠈⠅⠣⠾⠖⠌⣤⡀⢎⡀⡀⢸⣷⣿⣿⣿⣾⠷⢷⣾⠋⠐⠶⢀⡨⣷⣏⣷⣿⡶⣶⣥⣍⣵⣾⣭
⡀⠌⠡⢦⣸⡁⢀⣴⠖⢺⣿⡆⣀⠷⢸⣦⠈⢿⡅⠀⠀⣈⡁⠐⡌⠈⠉⢘⣏⠉⣿⣿⣧⣾⣌⡛⠁⢈⣙
⣆⣰⡯⠶⢶⣧⡠⠖⠲⠊⣿⣀⡨⡆⢸⡏⣦⢺⡓⣽⣾⣯⣥⣀⣠⡄⠀⠘⣇⠀⠘⣿⣿⡿⡳⢬⡃⠙⢷
⠿⠉⠀⠀⡀⣿⣧⡀⠀⢺⡿⢩⠑⢌⢸⡧⠷⠾⢻⣏⠁⠉⠉⢹⣤⣄⣀⣻⡟⠐⠀⣿⣿⠖⠛⢡⣤⠀⣉
⠀⣤⡀⠀⢣⠘⣿⣿⢿⣿⣿⡈⡏⠀⢺⡧⡤⡀⠀⢈⣠⣴⣿⠿⠟⢮⣾⠿⣿⠀⠀⣿⣿⣻⣦⣞⣻⣅⣤
⡄⠬⣷⠀⠘⠆⢹⠏⡾⠛⠋⣽⡞⢄⢸⡇⣬⣽⡿⣻⣷⡿⡗⡄⠀⠀⠿⣼⣿⡧⠀⣧⣿⠘⢊⠉⠙⣓⠐
⠾⠷⢹⣧⡀⢀⡈⠸⠁⢀⣼⡿⠻⡴⢾⡃⣿⡙⠉⠘⡏⠐⢸⠘⢶⣶⣴⣿⣿⡿⣞⣇⡿⢧⣿⣿⣹⠛⡴
⡢⢶⡖⢿⣷⣾⠃⠒⢀⡜⠡⠀⠄⢷⢸⠇⣿⣷⣰⢰⣄⣀⠸⣧⣸⣭⠹⠛⠙⠻⣮⣃⣿⡀⣿⣧⠸⣀⣿
⢆⣀⣹⣿⣿⠗⣀⣠⣿⣄⡠⣠⣀⡸⣿⣿⣷⣾⣿⣾⣿⣿⣇⡘⠛⠻⢶⣦⣀⣀⣘⡿⡿⠍⡍⡿⠯⠏⠉
⠉⠀⠈⣟⠛⠛⠉⠉⠉⠉⣙⡅⢈⣳⡞⠋⠀⠠⣟⢭⠛⢿⣿⠑⠀⣤⣠⡛⢹⣬⢇⠙⣗⠆⡠⠜⠀⢀⠖
⠑⠀⠲⠋⠀⠀⡄⠀⠀⠀⠘⢥⢻⡛⠙⢢⣉⣉⣩⡷⠤⠟⢻⠀⠀⣿⠋⣿⠈⠉⠉⣿⣌⠳⡴⢦⠙⠃⢠
⠊⠈⠂⠀⠀⠀⠐⡀⠀⠀⢀⠀⢢⡐⠀⢠⣶⣁⣀⣄⠠⣀⠉⠀⣠⡟⢰⠿⡖⠂⠀⠈⢿⣯⡝⢆⡀⠀⢇
⡁⢀⢄⣀⢠⡄⠀⠡⠤⠤⠤⠤⠀⠱⠄⠄⠈⠉⠉⠉⠙⠒⠒⠺⠶⢶⣽⣇⣦⣀⠀⠀⠈⢻⡀⡄⢻⣶⣴
⠘⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣝⡻⠦⣤⡀⣾⣷⣅⣤⣿⣿
⣭⣁⡈⢉⣉⣉⣉⣁⠁⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⣉⣉⣉⣉⣉⣭⣤⣽⣽⣶⠀⣶⣟⠃⡸⠃⢻⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠘⢲⣹⠛⢋⣨⣽⣾⣏⡨⠍⢁⠔⠁⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⣿⣿⣿⠛⠉⠀⡠⢊⣴⠛⠑
⡤⡤⠤⣤⣤⠄⠀⠀⢤⡤⡤⠄⠄⠤⠠⠠⠤⠤⠄⠄⢤⠤⠤⠄⠤⠄⢼⡗⠟⠃⠀⢀⢠⠜⡡⠎
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣖⡂⢐⣶⣶⣶⣶⣶⠒⡛⠀⠀⠊⠁⣠⠋
⡿⡄⠀⠉⠉⠉⠉⠙⠻⠿⠿⠿⠿⠿⠷⠛⠿⠿⠯⠽⠿⠯⣿⣿⣏⣿⣦⣡⢴⣤⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⣿⠿⠶⣟⠱⣏⣉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠛⠦⢍⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠄⠀⠀⠀⠀⠀⢀⣜⠛⠋⠲⠢⠄⠄⠀⠈⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠞⠀⠀⠀⠀⠀⠀⠀⣟⢛⣛⣛⣛⠒⢂⣀⡀⠄⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢹⡷⠶⠶⠾⠛⠋⠁⡀⠰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠶⢞⣋⣡⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⡾⢉⣛⣛⠛⠛⠉⠁⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢀⡴⢿⡹⣆⢠⡅⠀⢀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⢿⡙⠢⠁⠨⡿⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣟⠂⠌⠀⢐⡎⠈⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡟⠛⠶⠶⢾⠏⠁⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡁⢐⠀⢰⠃⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⣠⣮⡤⠖⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⡎⠈⠙⢦⡟⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢺⣿⡷⠶⠶⣤⣿⣶⣶⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⡆⠀⢹⣆⣠⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢨⡏⢈⣖⣠⠿⢤⣌⣉⣉⣉⣉⣉⣉⣉⠁⠈⠥⠤⠥⠀⣝⠨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣀⡁⢀⠿⢦⣤⣄⣀⣉⣉⣀⣀⣠⣤⣤⠀⠀⠀⠀⣧⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡯⠍⣹⠿⣦⣄⡀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣿⠛⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⡇⠀⣼⣄⡀⠈⠉⠉⠉⠛⠛⠛⠉⠉⠉⠀⠀⠀⢠⡟⢀⣷⣌⠹⣟⠻⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡴⠃⠉⠛⠛⠶⠶⠶⠶⠦⠴⠶⠦⠀⠀⠀⣰⠏⠀⠈⠀⠙⠷⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⢷⡟⠳⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡀⠀⠀⠀⠀⠈⠙⠶⣦⣤⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣀⡴⠟⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣝⡿⢻⡷⢶⣶⣤⣤⣤⣤⣿⠉⠀⠂⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠿⢿⡿⠿⢻⡿⠁⠀⠀⠀⣨⠿⣅⠹⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣄⣤⣼⠁⠀⠀⠀⣴⠏⠀⠈⠀⢿⠀⠀⠠⣄⠀⠀⠀⠀⠉⠀⠀⣀⣀⣤⣠⠄⠠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠇⠀⠀⠀⣸⠏⠀⠀⠀⠀⢸⣇⠀⠀⢀⡛⣀⠤⠄⢰⡆⠀⠋⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡍⢀⠀⢰⣿⣤⠄⠀⠀⠀⠀⣿⡄⠀⠈⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠉⢁⣿⢰⡆⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⣿⣼⠇⠀⠀⠀⠀⠀⢸⠙⢧⡀⠀⠀⠀⠀⠀⠈⣷⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣸⣇⠀⠀⠀⠀⠀⠀⢻⡀⠘⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢨⣇⣤⣄⠀⠀⠀⠀⠀⠀⡯⢹⠊⣧⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣗⢸⠀⢸⣇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢀⠈⣧⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣼⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠉⢧⠀⠀⠀⢸⠀⢀⡀⠀⢰⡏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢟⠀⠀⠈⢳⡄⠀⠘⡤⠉⠉⠀⣾⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⣿⣇⠀⠀⠀⠀⠀⠀⢀⠞⠓⢎⣶⠀⠀⠀⠙⢦⡈⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠛⠀⢻⣼⠷⣿⠀⠀⠀⠀⠀⢰⣋⠐⢶⢎⣵⠁⠀⠀⠀⠀⠑⢄⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠟⠀⠀⠀⣠⠚⠉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠀⣰⡦⠀⠀⠀⠘⢇⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⠠⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠄⣸⣤⣿⣧⠶⠀⠀⠀⢸⣦⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠈⣾⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⢀⠀⠀⣠⢆⡄⢀⠀⠀⠀⠀⢀⣱⢬⠇⠀⡠⠀⠀⠀⠀⠀⠈⠁⠀⠉⢼⣿⡇⣠⠞⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠈⠁⢀⡈⠳⠌⠁⠀⠀⠀⠚⠀⡿⢀⠄⠀⠀⣠⣤⣄⠀⣤⠰⣶⢰⣯⡼⠿⠿⣋⠈⠛
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⣾⠀⠃⠀⠀⠀⠀⣄⣈⣦⣬⣿⡟⠋⢹⣾⠛⢠⣼⣾⣿⡤⠀⢀⣈⠀⠠⠄
⠀⠈⠁⠀⠀⠮⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠂⠀⠲⣆⣠⣄⠹⣿⣿⣿⡃⣘⣃⣼⣷⣾⣿⠏⢩⣷⠚⠋⠉⠻⢶⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣨⠁⠀⠀⠠⣿⣴⣎⣿⠁⠀⠀⢃⣬⣿⡏⠈⢀⡯⢸⣷⣶⣶⣶⣿
⠀⠈⠀⠀⣀⠀⢀⡀⠀⠀⢀⠀⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⢸⣽⣿⣦⣤⣟⣁⠎⢻⡟⢿⡥⣿⡇
⠀⠸⢿⣽⣿⠆⠈⣱⣬⡛⢛⣫⣶⢶⡜⠶⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠉⣿⡇⢀⠀⡀⢸⣿⣯⣿⣻⢏⠙⠀⠘⣿⣿⠟⣽⡇
⠀⠀⠀⣈⢛⣻⡶⣮⠉⢤⠘⣿⣿⠾⣿⢀⡄⠠⠜⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣀⣿⡇⣾⠁⣇⠸⢸⣿⡇⠋⣞⢣⡀⢀⣿⣿⡘⣿⡇
⠀⠀⠀⠉⠋⠉⠉⠒⢠⠤⠀⠼⠛⢀⠈⠀⢸⢧⡜⢧⠃⢴⣆⠠⠀⠀⠆⠀⠀⠀⠀⣀⣴⣾⠛⢿⣿⡇⣿⠆⡟⠰⣿⣿⣷⣄⠻⡀⠙⣾⠋⠈⠙⢻⡇
⠀⠰⠀⢉⣯⣵⠶⢴⡯⣶⠀⠠⣈⢀⠀⣷⢀⡀⣧⠀⢀⣠⠛⢒⢦⡀⠰⠀⡀⠀⣶⣟⡋⠀⠉⠙⢿⣇⣿⢀⡇⢀⣾⣿⣿⡏⠀⠛⡙⣿⡀⠀⠀⠈⠁
⠀⢸⣷⡿⠓⢿⠿⠶⠠⠜⠀⠀⠁⣼⣻⣿⣦⠰⠈⡷⠸⡿⠉⢙⡘⡇⢳⣾⠡⣾⣿⡟⠑⠀⢠⣴⣿⡟⣟⣼⠇⣸⣿⣿⡿⣟⢳⡄⠀⠫⠀⠉⣠⣀
⠀⠸⠟⠃⠀⠠⢀⣴⠶⠂⣠⠀⠀⣩⠈⠉⢠⣴⣂⠠⣶⡷⣾⡄⢁⣴⠏⠛⣠⠿⠿⣿⣷⠀⠈⣿⠿⠿⣿⡏⢠⣿⢿⣿⠃⣿⠀⢉⣀⣀⠀⠀⠈⠻⠇
⠀⢠⡄⣠⠀⠀⠘⠉⠉⠛⠛⠟⠀⢇⠀⢀⠘⠹⣇⠀⢹⣿⢿⡟⠛⣋⣤⣾⠁⡔⢲⣿⣼⡀⠀⠘⠀⠐⠸⢷⡀⠁⡀⠀⠀⠈⠀⠀⠈⠁
⠀⢸⡏⣿⠀⠞⠀⢸⣿⣿⡆⠰⠀⣤⡄⢀⠆⠀⡏⣠⡈⠻⣾⣧⣤⣿⡿⢀⣜⠀⢸⣯⢹⡇⠀⠐⡿⠀⢀⣀⠁⠀⠀⠀⠀⠛⢷⡄⠀⠀⠖
⠀⢸⣷⣿⠀⠤⠀⢸⣿⣻⠃⠀⠀⣉⣇⣀⣤⣤⣿⣿⡃⠀⢺⣿⣿⢛⠁⣸⡏⠀⢈⣛⣻⣇⣤⣤⣤⣿⣿⡏⢳⣆⠐⡂⢠⠀⠀⠘⢆⠀⠀⠀⠠⠄
⠀⢸⡏⢻⣤⠆⠸⡏⠉⣩⠋⣱⣄⣾⠟⠰⢦⣤⠀⣻⣷⣶⣤⡟⣥⠏⠀⣹⣛⠟⠛⠛⣿⡉⢿⣿⣿⡿⠙⣄⡹⣯⣡⠻⡎⠃⠀⠆⠈
⠀⢸⣧⣾⣿⣄⢀⣠⣻⣿⣴⢉⣬⠽⣶⡄⣦⣿⣇⣻⣿⣿⢟⡿⢻⢄⣴⣼⡏⠄⠀⢀⡿⢦⣄⢿⣿⣆⣦⣤⣿⣟⣽⣷⡟⠐⠀⠀⠐
⠀⠘⠋⠛⠛⠛⠛⠛⠿⠻⣿⣾⣧⠄⢸⡇⢨⣡⣄⣹⠟⣡⣿⣷⣯⣾⡿⠱⡇⠶⠀⢸⠅⠀⠙⠂⢻⣽⣿⣿⣿⣿⣿⣿⠃⠠⠀⢀⣼⣶⡀
⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠘⠛⠛⣤⠛⠛⢻⣿⣿⣧⣾⣿⣿⣿⣿⡿⠀⠈⣇⠀⠘⠀⠀⠀⠀⡀⢀⡑⣿⣿⣿⣿⣿⣧⣴⠀⡄⠀⠠⢶⣍⠒⣷⣆⡀
⠀⠀⣀⠀⠸⣥⠔⣤⠄⣠⠄⢰⡆⠀⠀⡴⠄⢻⠟⢃⣼⣿⣿⣿⣿⣷⡀⠀⢫⠄⢀⣤⠀⢞⣻⣤⠋⢰⣿⡟⣿⣿⣿⣿⣿⠀⡗⢀⣀⣀⠘⣷⣤⣿⠃
⠀⢠⣿⣄⣼⣿⠁⣤⠖⠁⠀⠀⣠⡼⡹⠇⢰⢀⣰⣿⣿⣿⣿⣿⠻⣿⣧⡀⣄⡷⠈⣳⠊⠛⣿⣅⠀⠸⣿⣢⣿⣿⣿⣿⣿⣾⠲⢾⣿⢋⣴⣿⣿⣿⡇
⠀⠸⣿⣿⣿⣥⣋⠨⠗⠀⠀⢰⡿⢀⠃⠀⣴⣾⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣷⣤⠄⡍⠳⡶⠙⠉⢳⣶⣿⣿⣿⣿⣿⣿⣿⡏⡷⣮⣽⡉⣿⣟⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣀⣰⠤⣦⣾⣷⣿⠀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣎⣿⠿⠛⢡⡿⣼⡷⢶⣷⣤⣶⣿⣿⡝⣿⠿⡿⠿⢿⣿⣷⣶⣌⣹⠷⣜⣿⣿⢸⡇
⠀⢸⣿⣿⣿⡾⢿⣿⡏⢶⡈⢻⣽⣧⡀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⡄⠚⣴⢿⡇⠈⠻⣅⣿⣻⣿⣿⣿⡷⠀⠂⠈⠉⠉⠀⠩⣽⢷⣿⣿⡇⣿⡇
⠀⢸⣿⠟⠻⢿⣾⡽⢻⡿⢿⣿⠛⠋⢹⣿⣿⣿⠛⠉⣿⣿⣿⣿⡿⢿⣿⠀⠈⠈⠤⣧⢠⣿⣿⣿⣿⣿⣿⣿⣥⠀⢠⣶⣿⠀⠀⣤⡘⣧⡼⣿⡿⣿⡇
⠀⠀⠀⠈⠛⠢⠬⠤⣄⠐⠊⠁⠀⠀⠘⣿⣿⢿⣿⠇⢘⣿⣿⣙⠓⣿⣿⡇⢀⡏⠐⡏⠀⢻⣿⣿⠽⠋⢩⡯⠁⣰⣾⣿⡿⠀⣤⣿⡆⣽⣷⣾⡷⠋⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠀⠀⠀⠀⡌⣷⣼⣿⢶⣿⣿⡿⠏⠐⠻⣿⣇⡾⢀⠇⠀⠀⠀⢻⠁⠀⣰⠏⢀⣤⣿⣿⣿⡁⢰⡏⢿⣁⣸⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠈⠉⠀⠈⠁⠙⠲⢤⣷⣈⣿⣏⣸⣿⠋⠀⠀⠀⠀⠉⠉⢀⡞⠀⠀⠀⠀⠈⠀⠀⢻⣤⠿⠛⠣⢻⣿⡿⠃⠄⠈⢹⣿⣿⠾⡦⣿⡀
⠀⢀⣠⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⢀⣈⡁⠉⣉⠿⠃⠀⠀⠀⠀⠀⠀⣠⣿⠃⢠⡦⡄⠀⠀⣷⡐⢶⠇⣰⡗⠀⣾⣿⡇⢠⣦⣴⡟⠁⢢⣶⣦⣨⡇
⠀⢸⡇⠀⠀⠀⠀⠰⣄⠀⣀⠀⠀⠀⣨⣿⠿⣿⠋⠀⢀⣀⣂⡐⠛⠋⠐⢹⣿⡃⢀⣿⣆⠀⠀⢻⣿⣷⡶⠁⢀⣤⣼⡟⠻⢿⣾⣿⣾⣿⡿⠁⣼⠟⠃
⠀⢨⣿⠀⠀⠀⠀⢀⠌⢲⠍⠀⠀⠀⠉⢿⡄⢹⡄⢿⣿⠉⠉⠉⠀⠀⠀⠐⠛⢛⣾⣿⣿⣷⣄⡐⠈⣻⣷⡾⠿⢿⣿⣧⡀⠈⢿⣿⣿⣿⣧⠘⠻⣿⡇
⠀⢨⣿⣤⠀⠀⠀⠲⡀⠀⠀⣀⠀⠀⠀⢸⣧⣘⣿⣮⠡⢶⣟⡀⠀⣯⠀⠀⠐⠛⠛⠿⣿⣧⣙⣛⠚⠁⢸⠃⠀⠀⠈⠙⠻⠄⠀⠿⠟⠛⢿⠐⢶⣾⡇
⠀⢸⡋⠁⠀⢠⣄⣀⡘⢻⡾⢿⣿⣁⠀⠀⠈⠉⠉⠿⣦⡀⠿⠏⠀⡄⠠⠄⠺⠷⠄⡀⠀⠀⠉⠉⠃⠀⢈⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⢸⣷⣤⡦⠾⠋⣭⡿⡟⠛⠙⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠰⠆⠀⠀⠀⠀⠀⠛⠀⠀⠚⠀⠀⠀⠀⣼⣿⡆⡀⠈⠁⢠⡀⠀⠀⢀⣠⡄
⠀⠘⠛⠃⠂⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠈⠉⠉⠁⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⠛⠂⠀⢀⠤⢤⠀⠀⠘⠃⠒⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⢸⠀⠘⠠⠜⠃⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠰⠄⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠠⠜⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⢃
⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠉⢱⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⠇⠀⠘
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠉⠈⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡟⠀⠀⡇
⠀⠀⢀⣇⡀⠀⠀⠤⠤⢤⡄⠀⠀⠀⡙⠉⠋⠉⠉⠓⢈⠉⠀⡀⢀⡈⠉⢁⠚⡉⠙⠋⡙⢃⠀⠀⠠⣤⡤⠤⠤⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃⠀⣠
⠀⢠⠈⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⡀⠀⠹⠀⠀⠀⠀⢸⠀⠀⠁⢸⡇⠀⢈⠀⠁⠀⠀⡇⠈⠀⠀⠒⠀⢰⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⠀⠀⣾⠀⠀⠁
⠀⡈⠀⠀⠀⠀⠀⠀⡄⠀⣶⠀⠀⡇⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠃⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠈⠀⠸
⡀⠃⠀⠀⠀⠀⠀⠀⠃⢀⠁⠀⠀⠀⢸⡀⠀⠀⣧⠀⣄⠀⢀⡄⢸⡄⠀⢸⡀⢀⡀⠀⢰⠀⡃⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⡆⠀⢸⠀⢀⡄
⡷⠀⠀⠀⠀⠉⠉⠙⡟⠙⡋⠁⠉⡏⠈⠁⠀⠀⠋⠀⢉⠀⠀⠁⢸⡇⠀⢸⠁⠈⠁⠀⡏⠉⡟⠀⠘⡟⠉⠛⠉⠉⠁⠀⠀⠀⠠⡷⠀⠀⠈⠁⠀⡄⠀⡀
⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠀⠁⠀⢺⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠈⠀⡇⠀⠀⠁⢀⠀⠀⣤⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡆⠀⠀⠃⠀⠃
⠀⠰⠀⠀⠀⠀⠀⠀⠈⢁⠀⠇⠀⠀⡀⠀⡀⠀⠀⡄⢘⠀⠀⠀⢸⡇⠀⡀⠀⡇⠀⡦⠀⠸⠀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⡆⠀⠼
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⣀⡐⠂⠀⢡⣤⠁⠀⠀⡱⠆⢀⣀⢴⣬⣀⣀⠠⠦⠀⠀⠐⠴⠀⠀⠘⢀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠃⢀
⠀⠀⠀⢣⣀⠀⠰⠶⠆⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠲⠶⠀⠀⣀⡔⠀⠀⠀⠀⣶⠀⢸⠀⡈
⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠉⢀⡀⠀⠇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢔⣁⠼⠧⢆
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠀⠀⠀⡄⠀⠀⣸⣉⡷⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠱⠖⠐⠶⣶⣶⣤⡤⠀⠀⠀⠐⠇⠀⠀⢠⣤⣤⣤⣴⠖⠲⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⢳⡞⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠋⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠠⣦⡿⢹⠇⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠚⠃⠘⢄⠀⠀⢀⡀⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢤⡀⠀⠀⢀⡠⠜⢣⠈⠢⡀⠀⣼⣇⣄⠠⠂
⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⣦⠐⠈⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⢠⠞⠑⠂⠒⢏⠀⠀⠀⠱⠄⠘⡿⠿⢮⠏
⠀⠀⠀⠀⠀⠀⣇⣠⠃⣀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡄⠃⠀⣀⠀⠀⢈⣤⡀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⢙⠋⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠶⢋⣀⣀⣀⡀⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⣀⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠔⢂⠤⠐⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠉⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣤⣀⡀⠀⠀⣠⠀⠀⠀⠀⠀⡀⢀⣀⣀⠀⠀⠒⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣀⠀⢀⠜⠀⠀⠀⢹⡇⡏⠉⠀⠉⠈⠃⢸⡇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⡀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⠀⠒⠚⠣⠤⠴⢦⣶⠤⠥⠼⠃⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⠀⠀⠀⠀⢀⣀⣀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠶⢶⣤⣴⠶⠋⠀⠙⠛⣣⣿⠿⠟⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠛⠙⠛⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠆⡀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠞⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠱⠂⠾⣁⠾⠄⠀⠀⠀⠀⠴⣎⠳⠞⢶⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠉⠲⣾⠀⠀⠀⢸⠷⠋⠁⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠃⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⠻⠀⠀⠀⠿⢂⣠⣤⠄⠀⠀⠀⠀⠀⠀⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⢒⡛⠭⠉⢂⣀⠒⠠⠤⠶⣤⡖⣂⡀⠉⠭⠝⣒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⠩⢔⡂⠭⣐⣂⣭⣑⣦⣶⣶⣗⣚⣛⣛⣓⣚⡾⢿⣷⣥⣶⢬⣕⣪⢙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⡪⢔⡨⣐⡬⣟⣿⠿⠛⡋⠋⠙⠉⠙⡏⠛⠻⠛⠿⣯⣭⣛⠲⢭⣛⠦⣍⠳⣝⠢⡙⢦
⠀⠀⠀⠀⠀⡴⢋⠚⡡⡪⢊⢵⠟⣡⠞⡏⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠋⡹⢶⠈⠳⣌⢧⡈⢷⡌⢎⢧
⠀⠀⠀⠀⣼⢡⠃⡔⡱⠱⢱⠁⣼⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣟⠀⠀⠸⣾⠡⠈⠘⠀⡊⡇
⠀⠀⠀⠀⣿⢀⠘⠀⡇⠀⢸⢠⡏⠀⠀⠀⠀⣀⣠⠤⠤⠴⠶⠶⠦⠤⣤⣀⡀⠀⠀⠇⣼⠉⠳⡄⣰⢃⠎⡜⢠⢀⣵⡇
⠀⠀⠀⠀⣿⡜⡀⢄⢇⠰⢰⠈⣧⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⠇⠀⣠⣿⠗⡡⢊⠔⡡⣫⠏⡇
⠀⠀⠀⠀⡇⣿⡐⡌⢎⢢⠑⡕⢌⠳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡖⢟⡫⢔⡨⢒⡥⣪⠞⠁⠀⡇
⠀⠀⠀⠀⣧⠉⠻⣌⠢⡱⢕⢬⡑⠬⢚⣽⣓⣶⢤⣤⣤⣤⣤⣤⣴⠶⠶⠞⢛⡋⠭⢐⠪⠅⡒⠥⣢⣵⣯⡥⠴⢶⠖⠟⠒⠶⠒⠖⠲⠶⠒⠲⢦⢤
⠀⠀⠀⠀⣿⠀⠸⠋⢻⢮⣑⠩⣙⠲⠦⣤⣉⣉⣀⣀⣈⣉⣉⠀⠀⠴⠆⠑⣂⠠⣉⣂⣬⣵⠞⢻⠉⠉⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠇⣼
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢹⠛⠶⢬⣍⣙⡛⠒⠖⠒⠒⠒⠀⢒⣈⣃⣽⣾⠿⠻⠋⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠇
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⠀⠀⠀⠁⢹⠉⠙⠛⠛⠛⡏⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡞
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠚⠋⠉
⠀⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣷⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡠⡤⣤⣄⡀⠀⠀⠀⠀⠀⠀⡐⠋⢿⣿⣷⠂⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣄⣠⣼⣷⣧⣼⣋⣿⠷⣄⠀⠀⠀⠀⠀⠀⠀⣨⡟⢸⣿⠏
⠀⠀⠀⠀⠀⠀⣿⡟⢛⣿⣿⣿⣿⣿⡷⣾⠿⢷⡀⠀⠀⣆⢀⣴⡟⠣⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣳⣿⣿⣿⣿⣿⣿⣿⣷⡖⢿⣇⠀⠀⢘⣿⣏⢁⣰⣿⣦⡀
⠀⠀⠀⠀⠄⢰⣿⣿⠋⠁⢸⣿⡧⠘⢻⣿⣤⣤⣿⡆⠀⢨⣤⣴⠿⡿⣍⣙⣷⡀
⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠈⢻⣷⠀⢸⣿⣿⣿⣿⣷⣾⣿⣻⠇⣸⠁⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⢲⡦⠤⠀⠀⠟⠒⢊⣩⣿⣿⣿⣿⣿⡿⢸⠆⠀⢠⣿⣿⣽⣤⡀⠀⠀⠀⠘⠛⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠰⢿⡏⣹⠟⠁⠁⠂⡹⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣀⣼⠘⡿⠀⠀⠀⠀⢠⣿⡿⠃⠀⢾⣿⣿⣿⣿⣿⠀⣀⣤⣤⣄⠀⠀⢉⣿⣿⣿⣿⣦⡄⣑
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣾⡇⠀⠀⠀⣠⡝⠟⠃⣠⡮⣿⣿⡿⠿⣿⣿⣿⣿⣿⠛⠙⢧⣾⠟⠻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⠥⠥⠥⢯⠤⠀⠀⠀⠀⣀⣸⠁⠀⣿⡇⠠⠐⢹⣿⣿⣿⠃⠀⠀⠈⠇⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠋⠠⠿⣄⠀⣀⣴⣄⣀⠀⠀⡀⠀⣸⢹⠀⢰⡿⠗⠤⠠⠤⣀⠉⠈⠀⠀⠀⠀⠂⠀⠀⣸⣿⣿⣿⣿⣻⡿
⠀⠀⠀⣠⣤⣶⣴⣦⣀⣀⣤⣿⣿⣿⣿⡗⠉⠈⠙⢲⡟⢸⠀⡿⠷⠶⠶⠶⠒⠛⠋⢹⣦⣄⠀⠀⠀⠀⢀⣼⡟⢩⣿⣿⣿⡇
⠀⠀⢰⣿⣿⠛⠛⠋⠉⠉⠁⠀⢿⣿⣿⡇⠀⠀⠀⢾⣧⢾⣴⡃⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⡘⠃⢠⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠉⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠈⢛⠛⣿⣇⠈⣿⣿⣿⣷
⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡧⠀⠀⢱⣿⣿⣆⢸⣿⣿⣿
⠀⠀⠀⠀⠈⢣⣿⣴⣦⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⢿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠃⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠈⠋⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⡍⠀⠀⠀⠀⡠⠦⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠄⠀⠀⠀⠀⣄⡀
⢸⡆⠀⠀⢠⠎⡀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣰⡟⣿⠉⠰⡆
⢸⡇⠀⢸⡟⣙⡅⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⢿⣇⣿⠀⠀⡅
⢸⠃⠀⢸⣿⣿⣿⡇⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢠⠄⠀⢸⡇⠀⠀⠀⠉⣿⡄⠀⡇
⢸⠀⢀⣿⣬⡛⠛⢁⣤⣤⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠨⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⡇
⢸⠐⢿⣿⣿⣗⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⡇
⢸⠀⠀⡏⠁⠈⢷⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⡇
⢸⠈⢠⡇⠀⠀⢘⠀⢠⠆⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠁
⢸⠀⠀⡇⠀⠀⢸⠀⠆⠀⠀⢀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠰⠀⠀⠀⠁⠀⠀⠀⠀⠘⠃⠀⢸
⢸⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢣⡀⠇⠀⢀⠠⡂⠀⠀⠀⠀⢀⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠄⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠋⠀⠀⢰⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸
⢸⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢢⡀⢠⠂⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⢸
⢸⠐⢄⡅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⡜⠁⢾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠈⡓⠒⠒⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠐⠒⠚⢲⡄⠐⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸
⢸⡄⠰⡇⠀⠀⢸⠃⠀⢀⠠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳⡀⢃⡄⠂⠀⠀⠄⠀⠀⠀⠀⠀⠸
⢸⠀⠠⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠸⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⢸⠀⢰⣇⠀⢠⠸⢆⠐⠀⠤⠀⢀⠀⠰⠀⠰⠀⠁⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠆
⣺⡀⠹⣿⣦⡈⠚⠛⠒⠒⣂⢂⡀⢂⣀⣀⠐⣂⢀⣄⢀⠒⠶⣀⡈⠉⠉⠁⠘⢾⠳⠀⠁⠅⠀⠀⠀⡀⠀⠶⣼
⢿⠑⢄⣿⣜⣷⡀⠀⠀⠀⠀⠀⠁⠘⠃⠈⠀⠉⠀⠁⠘⠂⠉⠀⠀⠀⠀⠀⠀⢘⣤⠀⠠⡇⠀⠀⠀⠀⠐⠂⢸
⢹⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣤⣤⣶⣦⣤⣤⣤⣄⡀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⢾⢀⢀⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⢮⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠐⠀⠳⠶⠒⠒⠄⡐⣾⣿
⣿⣼⣟⣿⣿⣿⣷⢶⣶⠶⠶⠶⠦⠾⠾⠿⠿⠿⠓⠛⠉⠉⠉⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠉⠈
⠈⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣝⣙⢳⣶⣠⣧⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⠌⢱⣯⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣋⠀⠀⣿⣿⣤⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣶⣿⡿⠟⠛⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣈⣳⡿⣷⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⣩⣬⣌⢟⢳⣏⣩⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⢊⡱⢋⠹⡽⣯⣾⣮⣽⣿⣿⣷⣿⣽⣿⡿⡗⢀⣠⣤⣰⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⢄⣪⡑⣼⣾⣵⣿⣿⣿⡏⠉⠹⣿⣧⡽⢭⡑⢠⠋⡄⠚⢉⣡⣶⣾⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡈⠁⣩⠻⢿⣿⣿⣩⣿⣧⣀⣐⠻⠭⣤⡀⠐⣄⣼⣇⠈⠓⢕⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣻⣧⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣷⣻⠖⣲⡆⢸⣿⣻⣷⣤⣿⣷⣦⣶⣾⣧⠄
⠀⠀⠀⣤⣤⣤⣤⣤⡄⡶⣄⠻⣟⣻⣿⣿⣿⠿⡿⣿⡿⢻⣿⣿⣿⣏⢻⡟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⢠⠞⠻⢿⡿⣿⣿⡿⠄⠙⣇⠀⡉⡽⢛⣉⠀⡘⣿⡇⣾⣿⣿⣿⣿⡂⠑⢴⢀⣼⣿⣿⣿⣭⣿⣽⣿⣿⣷⡤
⠀⣸⠀⠘⢯⡁⠸⣿⡇⢴⣾⣿⠛⢍⣱⠿⣿⢿⡿⣿⣿⣫⡿⠻⢿⣿⡿⠀⠘⣿⣿⣿⣽⢻⣿⣿⣿⣿⣿⣿⠇
⠀⢸⣄⢳⣬⣻⡆⢿⣷⣿⣿⣟⣧⠘⣾⡃⢄⣿⣧⣿⣿⣿⡇⠀⠀⢿⣧⣀⣀⣿⣿⣏⢹⣾⣉⣼⢿⣿⣿⣿⠐
⠀⠀⠻⣶⣽⣿⡇⠀⢸⣿⣿⣿⢾⢲⣇⢙⠷⠙⣿⣏⣿⣿⣿⣶⣄⠀⢿⣿⣿⣿⣯⡉⠙⠻⣿⣿⣶⣿⣿⣿⡀⠀⠀⣠⣶⣶⣶⡀
⠀⠀⠰⣴⣿⣘⣧⠀⢸⣾⣿⣷⡌⢃⣡⢬⣦⠵⠛⢻⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⡖⢯⣶⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⡘⠰⣿⣅⠀⠺⢷⣿⣧⠿⣭⡞⠉⠀⢀⠀⣸⣿⣿⣿⣿⡁⠀⢭⣿⣿⡏⠑⡀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣹⣿⣧⡀
⠀⠀⢨⠀⢘⣿⡟⠀⠀⠀⠳⢚⣴⣳⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠰⢿⣿⡏⣤⠙⣤⣶⣿⣦⣿⣿⡛⠻⠟⠛⠋⠀⠀⠀⣿⣻⣿⣇
⠀⠀⠀⠀⢹⣯⡂⠀⢀⠠⠂⢉⠵⣿⠷⠀⢙⣽⣿⣿⣿⣿⣿⣿⡛⢃⠈⣿⡿⢿⡀⣇⠘⢿⡄⠘⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠟
⠀⠀⢠⡁⣠⢻⠏⠰⢏⡠⠚⠁⠘⣡⠆⢱⣨⡾⡏⢻⣿⣿⣿⣿⡧⠈⠥⢿⣧⣘⣷⡎⠀⠈⣿⠄⣾⠇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠄
⠀⠀⠀⠻⣷⠆⣠⠶⣅⠀⢀⡰⢂⣡⠴⠿⡇⢸⠁⠈⠈⣹⣿⠟⠁⢤⡴⢼⣿⡟⣿⡷⡕⠀⢿⣯⣠⡞⠁⠀⠀⠀⠀⠀⠀⠘⣿⠟⠁
⠀⠀⠀⠀⢈⣠⠑⢾⡟⠐⣋⣼⣾⡅⢀⡾⠃⢈⠠⢠⢀⣿⣧⡀⠀⠀⠀⠰⠋⠁⠊⠁⠉⣄⠀⣿⣿⣿⣿⡟⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢠⠈⣿⠀⠀⠀⠀⠀⠙⢿⣿⢿⡄⠀⠸⠀⠀⠘⠻⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣟⠻⠟⠀⢰⠆⠘⢫⣿⣿⣄⣀⣀⣀
⠀⠀⠀⠈⠺⠁⠀⠀⡠⠀⠀⠀⢺⡗⠀⡟⢦⠀⢻⡆⠀⠀⠀⢹⡛⠯⣿⣷⡄⠀⢀⣀⣀⣀⣀⡀⠙⡿⣄⣀⡰⠿⣬⣶⠮⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⣐⣢⢾⣴⣠⣆⡀⠐⠳⡖⢿⣾⠀⣉⣀⣀⣀⣤⣤⢿⣆⣥⠘⢿⣾⣿⠽⠟⠃⠈⠹⠀⠀⡀⢹⣷⠀⠀⠙⠻⠞⠉⠙⠛⠛⠛
⠀⠀⢀⣸⣞⠉⠙⠿⣿⣿⣿⣧⣴⣾⣿⣦⣭⡿⠛⠋⠑⢉⠋⠉⠈⠉⡉⢻⣿⣿⣿⣷⣶⣶⣶⣶⡿⢄⣿⡇⠈⢓⣶⡀
⠀⠀⢈⡙⠚⠀⠙⠛⠃⠈⠀⣩⠔⣌⠹⠿⠋⠀⠀⠠⠶⠊⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠙⠛⠉⠉⠉⣿⣀⡉⣹⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣠⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢜⣻⣿⣿⣿⣏⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣦⣤⣶⣤⣤⣀⣀⠀⢤⣤⣶⣶⣿⣿⣿⣿⣿⣦⣦⣶⠂
⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⠉⠉⠉⠉⠛⢿⣏⡽⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣁⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⡯⠟⣿⣿⣿⣿⣿⡆⠉⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠛⠉⠀⣼⣿⣿⣿⣿⣿⣷⠀⢸⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡏⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⣿⣿⡿⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠴⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠱⠄⠀⠀⠀⠸⠄⠻⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣽⣽⣿⡟⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇

⣾⣾⠿⣶⠾⢷⣶⡿⢷⣷⢷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿⣷⣾⣶⣾⣾⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠜⠉⠀⠀⠈⠀⠀⠈⠀⠐⢠⡀⠑⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⣆⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⢠⢡⠀⠀⢻⣰⠀⡇
⣤⣄⡪⠎⠉⠑⢒⠒⠒⠠⠤⠄⢀⣀⡀⠀⠂⠀⠀⠀⠀⠀⠛⠁⠀⠁⡼⠀⠀⣽⢸⢀⡇
⠘⠋⠉⠋⠙⠓⠒⠶⠦⠤⢬⣄⣐⠀⠀⠉⠡⡀⡀⠀⠀⠙⠒⠒⠒⠊⢠⠔⠾⠗⠯⠴⠦⠀⠀⡀⠀⠀⢀⢄⣤⢄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠆⠈⡢⣄⣀⡀⠀⢀⣀⣬⡴⠒⣀⣞⣛⢳⡶⠴⡖⠀⠠⠮⠾⣫⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠓⠀⠀⠀⠰⠃⠘⠛⡏⣟⠐⠛⠖⣺⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢞⣡⠗⠖⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠉⠉⠛⠛⠻⠭⣛⡳⢶⣤⡀
⠀⠀⠀⠀⠀⣠⠾⠛⠓⠉⠉⠉⠉⠉⠒⠲⠤⣄⡀⠀⠀⠉⠳⣌⠛⢶⣄
⠀⠀⠀⢠⡞⠁⠀⠀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⣄⡀⠀⣸⠇⠀⠹⣧⡀
⠀⠀⢠⠏⠀⣠⡶⠛⠉⠉⠀⠈⠉⠉⠛⠷⣦⣄⠀⠀⠀⠉⠛⠁⠀⠀⢀⡼⣷⡀
⠀⢀⡟⢀⡼⠋⢀⣤⠶⠖⠒⠶⠦⣤⣀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣠⠞⣡⠴⠜⣷
⠀⢸⠇⢸⠁⢰⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠙⢷⡀⠠⡏⢠⠊⠀⠀⠘⣼⡇
⠀⢸⡀⢿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⢿⡀⠹⡌⡄⠀⠀⠀⣐⣷
⠀⠘⡇⠸⣀⡈⠉⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⣷⠀⢹⡐⣀⡴⠚⠉⢿
⠀⠀⡵⠦⢤⣉⠙⠲⢤⣀⠉⠓⢤⡀⠀⠀⠀⠀⠀⢱⠀⠀⢸⡀⠀⣇⡅⠀⠀⠀⠘⡄
⠀⢀⡇⠀⠀⠈⠙⠓⠤⣈⠛⢦⣀⠈⠲⣄⠀⠀⠀⢸⠀⠀⢸⡇⠀⣿⡀⠑⠲⠴⡾⠁
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠑⠢⡈⠳⢤⡀⠑⢄⣠⠏⠀⠀⡾⠀⢠⡟⣁⣀⢤⣾⡁
⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⡀⠁⠀⢀⠞⠁⠀⢼⣠⠤⠚⠉⠀⢹
⠀⠀⠀⠈⠛⠷⠶⠶⠶⣶⣤⣤⣀⡀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⣀⠴⠛⠻⢤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠲⠶⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠸⠿⠶⣲⡴⢤⣄⣀⣀⠀⠀⠀⢀⡖⠒⡶⠤⢤⣤⣈⣉⡭⠿⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⠶⠚⠁⠀⠀⠀⠀⠉⠉⠛⠓⠾⠿⣿⣶⣤⣤⣂⡀⠠⡠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣷⣴⣾⣷⠆⠀⠻⠛⣻⡗
⠀⠀⠀⠀⠀⠀⠘⠛⢉⠉⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣥⣁⣽⣿⣿⣆⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢻⣿⣿⡇⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣼⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⡶⣻⣿⣿⣿⡆⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣤⣽⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣽⣾⣿⣿⣾⣿⣿⠀⣼⠁
⠀⠀⠀⠀⢀⡏⣹⣿⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠟⠛⣿⠀⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣷⡶⢿⣆⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡀⠀⠉⠂
⠀⠀⠀⣠⣿⣿⠈⠉⠉⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠷⡀
⠀⠀⠈⠉⠁⠉⠀⠀⠀⠀⠰⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤⣤⡶⠂⠀⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣇⣼⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⣿⣿⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣶⣷⠄⠠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠹⣿⣿⣿⣁⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⢿⡟⠻⠿⠷⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣿⣿⣿⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣶⣶⣶⣿⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣿⣿⣿⣿⣿⣷⣌⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣿⣏⠿⣿⣆⠙⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠐⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠙⠛⠛⠛⠙⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣇⣠⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢩⣿⣁⣼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣧⣴⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠘⠟⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⡟⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣼⣇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠃⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡿⠀⠀⠰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠉⠉⠉⢉⣉⡉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡄⠀⢀⡤⠂⠉⠉⠉⠉⠓⠦⡈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⣇⣴⣿⠆⠀⣀⡤⠐⠒⠦⣄⠲⠦⣤⣭⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣿⣿⣄⠀⠀⠀⠈⠑⣄⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠿⠛⠛⠛⠿⢿⣿⣷⣦⡀⠀⠠⡈⢧⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣥⠶⠿⠿⠿⠷⢶⣤⣌⠙⢿⣿⣦⠀⠹⡄⠃⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣯⡀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠙⣿⠇⠀⢻⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠉⠛⠋⠓⢦⡀⠀⠀⠀⠀⠀⠀⠹⣇⠘⣧⠀⠈⡷⡀⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⣀⣠⣤⠈⢳⣷⣶⣶⡿⠗⣺⣿⡄⢿⡆⠀⡇⢹⣧⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣶⣶⣾⣿⣧⠷⠖⢿⣯⣽⣿⠆⠀⣿⣿⡇⢸⡏⢀⠃⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⣋⣠⠬⢿⣛⠯⠉⢠⡄⠀⠈⣿⣿⡇⠀⠀⣿⣿⠁⣻⡇⢸⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⢁⣀⡀⢩⣥⡀⠹⣧⡀⢻⡆⢠⣿⡿⠃⠀⠀⢸⡟⢠⡿⣗⠈⣧⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠻⣷⣦⠙⣿⣦⠘⠿⠄⢁⣼⡟⢹⡀⠀⢠⣿⠗⠋⠀⠈⠙⠃⠙⠲⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠘⢿⣷⠌⠛⢁⣠⣶⠟⣻⠁⣸⡧⠴⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣼⣿⣛⣥⠴⠛⠋⢧⢀⣠⣤⣾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡏⠁⢀⣀⣤⡶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⢯⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣯⣾⣿⣥⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠙⢯⣠⠟⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣟⣡⣿⡀⢰⣇⣀⣤⢶⠞⠛⠋⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣀⣀⣁⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠻⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⡿⣿⡂⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠆⢉⣉⣛⣫⣏⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⡇⣸⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣃⠀⣸⣦⣿⣿⣿⣿⡇⣴⣦⡀⢀⢳⣄⡀⠀⠀⠀⠀⠀⢀⡏⢧⡸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣯⠛⣦⡀⠀⠀⠀⡌⣤⣸⣷⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠳⢦⣄⡞⢱⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⣿⣿⠛⠉⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣶⣶⣾⣵⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⡀⢀⡠⢀⣠⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡰⣭⣛⠃⠻⠍⢠⣟⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠷⠿⣌⣱⣤⣆⠠⢸⠟⢿⣿⡀⠈⠉⠛⠛⠿⠛⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣠⣶⣾⠉⠉⠉⠐⢃⣴⣿⣿⣇
⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣶⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠟⠋⠿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠁⠈⠋⠀⠈⠉⠈⠉⠈⠉⠉⠈⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⠢⢍⣛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠑⠦⣌⣉⠒⠤⡉⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣑⠢⢭⣑⠢⢍⡙⠒⠤⣀⠀⠀⠈⠙⢷⣄⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠓⠮⣍⣒⠦⢍⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣭⣷⣦⣌⣁⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣯⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣄⣠⣤⣴⣟⣁⣤⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⣿⢿⣿⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡜⠘⣯⠉⣿⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠖⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠁⠀⠘⢧⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡟⣀⣠⣤⣴⣦⣬⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⣼⣿⠏⠉⠀⠀⠀⠀⠂⠠⠤⠤⠤⠤⢤⣭⣉⣉⣉⣉⣉⣉⣛⣛⣛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⢶⡾⠿
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠒⠒⠤⠬⠭⠭⠭⠭⠭⠍⠉⠁⠀⠀⠀⢠⡟⠳⢤⣀
⣼⠃⠀⠀⠀⠉⠑⠢⠄⠀⠀⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣒⡒⠒⠒⠢⠤⠤⠤⠤⢬⣭⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠄⢀⣀⣀⣀⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⠉⣛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⣀⣤⣽⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣭⣭⣭⣭⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⡦⠄⠤⠄⠀⠀⠀⠉⠁⣀⣀⠀⠀⠙⣟⠛⠋⠉⠉⡟
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣩⣭⡿⠿⠟⠛⠛⠛⠛⠛⠒⠶⠶⠾⠿⠛⠀⠀⠀⢸⠃
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠞⠛⠉⠁⢀⡀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⠶⠒⣛⣉⣉⣤⡤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣈⡳⣭⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡶⠶⠖⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣶⠖⠒⠶⠚⠛⠛⠛⠛⠙⣿⠟⠛⠻⠿⠿⢿⠖⠒⠲⠖⠒⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠂⠀⣰⣟⠀⠀⠀⠀⡽⠀⠀⠠⠠⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡆⠀⡀⠀⠙⣧⠀⠀⣸⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡷⡦⠀⠸⠀⠘⣇⣰⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⣀⡠⣤⡄⠀⣿⡔⠂⠀⠀⢸⡿⣶⠦⠀⠀⠀⢿⡏⣙⣂⣄⠀⠀⣁⣀⠀⠀⠀⠀⣿
⠀⢴⢶⡌⠀⠒⠂⠀⣤⣿⣿⣄⠂⠀⢸⡇⠈⠀⠸⡖⠁⢽⠬⣅⡀⠈⠣⡀⠀⠀⠠⠀⠀⣤⡁⡌
⠀⠀⠉⠀⠀⠋⣠⠛⣿⣿⣿⣿⣷⠆⠀⡇⢀⠀⠀⡇⠀⠘⠆⠀⠈⠑⠲⠭⠀⠀⠀⠀⠈⣹⡷⠙⣦⣀⢀⣠⡀
⠀⠀⢀⠀⠀⠀⢡⣼⣿⣿⣿⠿⠟⠀⠀⣇⡀⠀⡉⠛⠻⢦⣼⣤⣤⣤⣤⣤⡀⠀⠐⠀⣴⣿⡆⠀⢿⡏⠀⣀⠒
⠀⠀⠈⠁⠀⠉⠃⣤⠿⠋⢀⡥⠀⡀⠀⡰⣅⠀⠀⠀⠀⠀⠤⣉⣉⣉⠉⠉⠁⣀⣤⠾⠠⠞⠋⠉⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡁⢸⣷⡒⠁⠠⠄⠀⠀⠈⠙⠻⠟⠗⠀⠀⢀⠀⠴⠖⠀⠐⠛⠇
⠀⠀⠀⡤⠀⠚⠁⢛⡀⠀⠀⠒⠶⠶⠖⠒⠲⠛⠃⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠚⠃⠈
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠦⡀⠀⠀⠀⠀⠀⣠⠔⢲⢰⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⡆⡇⠀⠀⠀⠀⠀⣼⠕⠂⢀⠀⢠⠂⠀⠈⢆⠀⠀⠀⡜⠁⠀⢸⢸⠴⣚⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢸⠀⡇⡇⠀⠀⠀⠀⡸⠁⠀⢠⢇⡴⠃⠲⡀⠀⠘⡄⠀⠀⡇⠀⠀⢸⠔⠊⠁⢀⡇
⠀⠀⠀⠀⠀⠀⡞⢳⢸⠀⢸⠀⡇⡇⠀⠀⠀⢀⠇⢀⡴⣷⢍⠀⠀⡠⣈⢦⠀⠱⡄⠀⢹⠉⡣⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⢀⣧⣸⣼⣠⣼⣄⣧⣇⣤⡀⠀⢺⠊⣏⠀⠐⠆⠀⠀⠰⠀⠀⣳⢄⣹⢀⡼⠃⠯⡖⠒⠊⠁⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣷⡀⠀⣐⣂⠀⢀⠟⡏⢀⡞⠈⠉⠉⠉⠁⠀⡤⡄⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢹⢦⡀⠀⠀⠀⠹⣄⣇⠙⡦⢍⡡⡴⠋⢠⢃⡟⠀⠀⠀⠀⠀⢀⡴⠋⠙⢢⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠭⠬⠉⡇⠀⠀⠀⠀⠈⢁⣹⡇⠀⠀⣸⣾⡓⠋⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡴⠚⠉⠓⡇⠀⣛⣛⠀⡇⠀⢀⠔⢺⠉⠉⢣⡈⢲⡞⢁⠇⠈⠙⡶⠦⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠰⡁⠀⠀⡀⠳⠤⢤⠤⠄⠇⠀⠆⠀⣸⠀⠀⢘⠑⢻⡟⠋⠃⠀⠀⡇⠀⢹⠀⠀⢸⠖⠖⠖⠒⠲⠚⠳⠲⡇
⠀⠀⠀⢀⠔⠁⠀⢄⣇⡠⠀⠘⢦⠀⠀⢰⠀⠀⢻⠀⠀⠘⠖⠑⣞⠒⠂⠀⠀⡷⠀⠈⡄⠀⢸⠀⠀⠰⣉⣱⠀⠀⠀⡇
⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠀⠀⢠⠇⠀⢸⠀⠀⣸⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣧⠀⠀⡇⠀⢸⠀⠤⠤⠤⠬⠤⠄⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣇⣀⡀⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡸⠀⠀⢳⠀⢸⣀⣉⣉⣉⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠒⠬⣷⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣟⡩⠭⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡞⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣇⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢼⣄
⠀⠀⠀⠀⠀⠀⠤⢤⡴⣫⡿⠓⠒⢾⢶⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣳⣯⣻⣦⡤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠺⠛⡏⠀⠀⣀⠜⠓⠚⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃