Avoid Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⠦⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⣀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢀⣴⣿⠟⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⣿⡟⠀⠀⠀⠀⢈⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⢍⣵⣿⠟⠁⠀⠈⢻⣷⡄⠐⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⢟⡁⠀⢀⣄⠀⠀⢿⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣵⣿⠟⣥⣾⡇⠀⠸⡏⠀⠀⠘⣿⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢟⣵⣿⠟⣥⣿⣿⣿⡇⠀⠀⠃⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⣵⣿⠟⣡⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⣿⡆⠀⠀⠀⠀⠀⠸⢟⣵⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⡇⠀⢀⡄⠀⠀⢸⣿⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢻⣿⡀⠀⠀⠀⢀⣴⣿⠟⣑⣛⣛⣛⣛⣛⣛⣛⣛⣛⠃⠀⠀⠀⠀⢠⣿⡟⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢻⣿⣄⢀⣴⣿⠟⣡⣾⣿⣿⣿⡱⢎⣿⣿⣿⣿⣿⠇⠀⠀⠀⣠⣿⠟⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢿⣿⣟⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⣼⡿⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣶⣶⣶⣶⣶⣿⠿⠟⠋⠁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠰⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣶⡰⣄⣆⡖⢦⣤⣶⢢⡀⣶⢲⣰⢰⣂⢰⡖⣶⡦⢰⣆⣰⠒⠰⡖⣦⡴⢲⣰⣆⡆⣖
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠛⠈⠓⠋⠋⠛⠊⠀⠛⠊⠘⠐⠊⠈⠃⠋⠙⠋⠙⠘⠒⠈⠃⠋⠙⠚⠘⠈⠃⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣸⡏⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠶⠒⠲⢤⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⢀⣀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⣼⠁⢠⠖⠛⠋⠉⠉⠛⠓⢲⡄⠘⣆⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢘⡷⣄⠸⡇
⠀⠀⠀⢀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⣆⣀⣤⠶⠴⠶⠞⠁⠘⣦⡟⠀⠀⠀⠀⠀⣀⠀⠐
⠀⠀⠀⢸⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣀⠉⠀⠀⠀⠀⠀⠀⢀⡟⠀⠠⠀⠐⠃
⠀⠀⠀⠀⣇⠀⠀⢳⡀⠀⠀⠀⠀⢠⡟⠀⠀⣸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⠀⢀⡀⠀⣀⠀⢀
⠀⠀⠀⠀⢻⡀⠀⠀⣿⢦⣄⣠⠴⡏⠀⠀⢠⡏⠀⠀⠀⡀⠀⠀⠀⣀⣤⣤⣤⣼⠷⣄⡀⢀⣠⢾⣥⣀⣀
⠀⠀⠀⢀⡤⠷⠤⣶⠿⠀⠀⠀⠀⠷⣶⠤⠿⣤⡀⠀⠀⡇⠀⢠⡞⠁⠀⠀⠀⠹⡄⠈⠉⠉⣿⠀⠀⠀⠉⢳⡀⠀⠁⠀⢤
⠀⠀⠀⡏⠀⠀⠀⠹⣄⠀⠀⠀⠀⣠⠇⠀⠀⠀⢹⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠙⠦⣤⡴⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐
⠀⠀⠐⡇⠀⠀⠀⠀⠈⠓⠶⠶⠚⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠇
⠀⠀⠀⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢸⠀⠀⡇⠀⢻⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣼
⠀⠀⠀⡇⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⡇⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠃⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡞⠁⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣌⠻⣦⡀⠀⠀⠀⠀⢀⡴⢋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣌⠻⣦⡀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣌⠛⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠘⡇⢀⡴⢋⡴⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢋⡴⠋⠀⠈⠳⣌⠻⢦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡴⠻⣄⠀⠀⠀⠀⠈⠓⢌⠛⡄⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⣠⡤⢼⠀⢸⡀⠀⠀⠀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡤⢾⡇⠀⠸⠀⠘⢧⡀⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞⢹⠀⢸⡇⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢸⡂⠀⠀⠀⠀⠀⠀⠀⢄⣀⣄⣗⣵
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢲⣄⠀⠀⠀⠀⢀⠀⠀⢸⢳⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣭⣿⠇⠀⢐⠂⠀⠐⣦⡀⠀⠑⠒⠾⠏
⠀⠀⠀⠀⠀⠀⠲⣤⣴⣼⣾⠀⠀⠀⠀⠈⠻⢯⣯⣿⠆⠀⠀⠀⠀⢠⠐⠒⠐⢄⠀⠀⠀⠀⠸⡀⣀⣄⡇⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢱⢦⡀⠀⠀⢀⠀⠄⠀⠠⠄⢀⠀⠈⣄⡄⠤⣏⣗⠀⠀⠀⠀⠈⠲⠶⠽⠿⠀⠀⠀⠀⢀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⢼⣺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠑⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠑⠁⠀⠀⢀⡠⣤⣤⣄⡀⠀⠀⠀⠑⡄⠀⠀⠈⠀⠀⢸⠦⡄⠀⠀⠀⠀⣠⠤⣐⣶⠀⠀⡀⡄⢒⡩⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣴⡵⠛⡉⠁⢉⠙⠒⢄⠀⠀⠈⠢⡀⠰⢤⠴⡤⣔⡿⠁⠀⠀⠎⡵⠚⠉⠘⢦⣩⡵⠞⠋
⠀⠀⠀⠀⣀⣀⣀⠀⠀⡀⡀⠀⣀⣀⣠⣤⠷⠛⠀⠀⠀⠀⢀⠟⣆⠈⠢⡀⠀⠀⠈⠢⢀⣉⠹⠋⠀⠀⠠⣪⡞
⠀⠀⠀⠀⠉⠘⠛⠓⠛⠒⠚⠚⠛⠋⠋⠁⠀⠀⠀⠀⢶⡲⠖⢷⡳⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⢄⠵⠋⠒⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠉⢆⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠈⠉⠛⠃⠀⢀⠀⣀⡈⠒⢤⣀⣀⡤⣚⠴⠚⠀⠀⣀⣀⣀⣡⣷
⠀⠀⠀⠀⠀⠐⢦⠴⠰⣟⠇⠀⠀⠀⠀⠀⠀⢘⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⢦⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠻⠼⠿⠯
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠰⢤⠴⠶⣽⠆⠀⠀⠀⠀⠀⠀⠰⣤⠴⠼⣾⠅⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢐⢄⠀⠀⠀⠋⠉⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⢀⢈⠙⡄⠀⠁⠀⠀⠀⢽⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⠤⢼⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⡄⠀⠀⠠⠄⠀⠠⡄⠀⠀⠙⠭⣁⣈⣿⠇⠀⠐⢦⠠⠦⢶⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠘⠲⣒⣒⣧⡿⠀⠀⠀⠀⠀⡀⡏⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠙⢭⣝⣷⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠉⠀⠀⠀⠀⠉⠉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⠤⢄⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣴⢟⡡⠲⡀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⠎⠀⠀⡸⠒⢒⡪⠋⠁⠀⠀⠉⢑⣶⠟⠓⢇⠀⠀⠱⡄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⠃⠀⠀⡜⠁⠰⡉⣀⡤⠂⠤⣀⣴⠟⠁⠀⠀⠀⢣⠀⠀⠘⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢼⠀⠣⢄⣀⡜⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠢⢄⡀⠀⠀⠀⢳⣀⡤⠚⠀⡷
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠱⡠⠋⢳⠐⢢⣴⠟⠁⠀⠀⠀⢀⠈⠑⠤⡔⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠐⣅⣠⢃⣴⠟⠉⣆⡀⠀⠀⢄⠀⠑⢦⠴⠃⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢀⣼⠟⢇⣀⡎⠀⡜⠐⢤⣀⡹⠒⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣴⠟⠁⠀⠀⠀⠉⠈⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠉

⠀⠀⠀⠀⣀⣤⣶⣶⣿⡿⣿⣿⣷⣶⣤⣀
⠀⢀⣴⡾⠛⢩⡿⠋⠀⠀⠀⠈⠻⣷⡈⠛⢷⣄
⣰⡿⠋⠀⠀⣿⠃⠀⠀⣤⣄⠀⠀⢹⣇⠀⠀⠹⣷⡀
⢻⣧⠀⠀⠀⣿⡀⠀⠈⠿⠟⠀⠀⢸⡿⠀⠀⢠⣾⠃
⠀⠙⢷⣄⡀⠹⣧⡀⠀⠀⠀⠀⣠⣿⠃⣠⣴⠟⠁
⠀⠀⠀⠙⠻⢷⣿⣿⣶⣶⣶⣾⣿⣷⠟⠛⠁⣠⣾⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠁⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣤⣤⣤⣄⣠⣴⣶⣶⣤⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠁⠀⠀⠙⠏⠀⠀⠀⠙⠟⠁⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠛⠛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣰⡿⠁⢀⣀⠀⠀⣀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠻⣷⣾⠟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣤⣤⣴⣶⡶⣿⡄⠀⠠⣾⠟⠻⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⢀⣴⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣷⣶⣶⣾⣛⣋⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⢀⣀⣠⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢋⣭⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣌⣛⣴⣿⣿⣿⣿⣿⣿⣿⡟⣉⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⡋⠬⢹⣿⣿⣿⣷⣥⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢻⣿⣿⣿⢡⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣙⣛⣡⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠘⣛⣛⣛⣻⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠈⠉⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⡀⠀⠀⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡟⣋⠻⣿⣿⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⢹⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡃⢾⣿⣷⣌⠻⠇⠀⣠⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣷⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠎⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⡿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡅⢾⣿⡿⠋⠀⠀⠈⠹⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣀⡀⢤⠀⠀⣿⠰⠷⠀⠾⠆⠸⠷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠋⠉⣹⠊⠇⠀⠿⢤⣤⣤⡤⣤⡤⣤⢼⠇⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣠⠖⠃⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠉⠉⠉⣿⠀⠀⢰⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠽⠚⠉⢳⡀⠀⠀⠀⠀⠀⡟⠀⠀⣀⡀⠀⠀⣠⠀⡇⣀⠀⠀⠳⢤⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⢻⣀⣾⡥⠭⠿⢯⣹⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠴⣚⡷⠀⣀⡤⠖⠋⠉⠀⣧⣀⣀⣀⣨⡇⠀⠀⠀⠀⠀⠀⣠⠖⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⣡⠴⠋⠁⠀⠀⠀⣠⣾⡉⠉⠉⠉⣸⣃⠀⠀⠀⠀⠀⣼⠁⡼⢉⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣠⠴⠛⠹⡄⠙⠦⣤⠴⠃⡽⠉⠳⢄⠀⠀⢿⠸⡇⣾⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠴⠖⠋⠁⠀⠀⠀⠘⢦⡀⠀⢀⡼⠁⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⠀⠀⠀⠀⠀⠀⠙⠦⠎⠀⠀⠀⠀⢲⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠠⠄⡄⡤⠠⢠⢤⠤⡤⢠⠀⠤⡀⠆⣤⢤⠠⡤⠤⠄⠠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⣤⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠃⠀⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⢿⡿⠀⣴⣿⣿⣷⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣾⣿⣿⣾⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⢿⣿⣿⣿⣧⠘⣿⣿⣿⢿⣿⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⠈⠛⡟⠛⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠛⠀⠼⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⣀⣈⡻⣿⣿⣿⣿⣷⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣼⣿⣿⣧⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠤⠌⠛⠾⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢀⣴⣶⣶⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣿⣿⣶⣾⣿⣿⡟⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⢿⢿⣷⣿⣿⣽⡿⡗⠠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠄⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣤⣿⣯⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠁⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠘⢿⣿⣿⡿⠁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢫⣾⣿⣿⣿⣶⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⣠⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣹⣿⣿⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⢉⣉⣉⣉⣉⡉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⣠⠤⠖⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣀⡴⠚⠉⠀⢀⣀⡤⠴⠶⠒⠒⠒⠒⠶⠦⢤⣄⡀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⠞⠁⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢀⡼⠃⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⡞⠁⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠀⢀⡼⢳⡀⠀⠹⣄⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠏⠀⢠⠏⠀⠀⠀⠀⢸⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡽⠋⠀⣀⡴⠋⠀⠀⠹⡄⠀⠹⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡏⠀⢠⡏⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⡴⠲⣄⠀⢹⡄⠀⢻⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠀⠀⡾⠀⠀⠀⠀⠀⣅⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠈⡇⠀⣧⠀⡟⠀⠀⣧⠀⠈⣇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⡇⠀⢹⢸⡇⠀⠀⢸⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠸⡇⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⡇⠀⠘⣿⠀⠀⠀⢸⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠟⠀⠀⠀⣼⠀⠀⡿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡄⠀⢹⡀⠀⠀⠀⠀⠀⢸⣀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⠓⡆⠀⢀⡏⠀⢰⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢷⠀⠈⢧⠀⠀⠀⠀⢀⡼⠋⠀⢀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠈⠛⠁⠀⡾⠀⠀⡞⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⢧⠀⠈⢷⡀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡾⠁⠀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠈⢷⡀⠀⠙⠋⠀⢀⡴⣏⣀⣀⣀⣀⣀⣀⣙⣋⣀⣀⣀⣀⣀⣀⣴⠇⠀⠀⣰⠏⠀⢀⡞⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⣦⡀⠀⠰⢯⡀⠀⠀⠁⠉⠀⠁⠈⠉⠁⠉⠀⠀⠀⠀⠁⠀⢀⡴⠞⠁⢀⡴⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢦⡀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⣀⡴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⢀⣠⠴⠚⢉⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⠤⠤⠤⠤⠴⠶⠒⠋⢉⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣶⣶⣶⣶⣶⡷⢾⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣷⢾⠶⣶⡶⠶⠶⣶⡾⠷⡾⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⣄⠀⠀⣠⡖⢀⡁⢲⣄⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⠙⢷⣄⠙⠳⠌⠡⠞⠋⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣧⠀⠙⢷⣄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣼⣿⣿⣄⠀⠀⠙⢳⣄⠀⠺⣯⣉⣛⣭⡿⠂⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠺⠿⣿⡿⠿⠃⠀⢸⡏⢹⣿⣄⠈⠉⠉⠉⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣀⡀⠀⢠⣿⢠⡿⣡⡿⢷⣄⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢹⣧⢠⣿⣿⣿⣼⢏⣴⠃⠙⢷⣄⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣡⣤⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⢠⣶⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⡓⢦⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡏⠈⠳⣤⡞⠁⠈⣧⠀⠹⢿⣿⣿⠿⠃⠀⠀⠀⠙⢿⣿⣿⡿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠴⠋⠙⠆⢠⢇⣤⣶⣶⣶⣶⣶⣦⣤⣤⣠⣴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠖⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢻⣿⣿⡇⠀⢸⣿⣿⡏⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⢹⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⢸⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠃⠘⢿⠿⠃⠀⠘⠿⠿⠁⠘⠿⠿⠃

⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣴⣦⢀⣶⡆⢠⣶⡄⢰⣦
⠀⠀⣴⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣷⣶⣿⣷⣶⣿⣶⣿⣿⣷
⠀⠀⣿⣿⣿⣿⡇⠀⠶⠤⠤⠤⠄⠰⠤⠶⠴⠆⠶⠶⠶⠤⠤⠤⠄⠤⠤⠤⠴⠆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠈⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⣿⣿⣻⣿⣿⣿⣿⡇⢿⣿
⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠸⠿⠿⠈⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢀⡠⠄⠒⠒⠒⠢⢤⡀⠀⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠃⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⣨⠾⠛⠛⠋⠉⠉⠙⠓⠶⣤⡀⠀⠀⠀⡇⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢠⣇⢠⡄⣿⡴⢧⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣡⡤⢟⣽⠞⠳⠏⠀⠈⢧⢹⡀
⠀⠀⠀⠀⠀⠀⢨⡅⠀⢰⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢸⡄⡆
⠀⠀⠀⠀⠀⠀⢘⡇⠀⣸⠀⠀⠐⠋⢉⣁⣉⣛⣦⡀⠀⠀⢀⣤⣤⣴⣟⣁⡀⠀⠀⠀⠀⠘⡇⣿
⠀⠀⠀⠀⠀⢀⡤⣧⠞⠁⠀⠀⢠⠞⠁⠀⣀⡈⠙⣟⠀⠀⠀⢀⠔⠉⣀⡀⠉⠢⡀⠀⠀⠀⠓⣿⡀
⠀⠀⠀⠀⢀⣸⡀⣇⠀⠀⠀⠀⡜⠀⠀⢸⣿⣷⡆⢸⠀⠀⠀⢸⠀⣾⣿⣷⠄⠀⣣⠀⠀⠀⠀⡿⢹
⢀⣀⠤⣰⠋⠉⠙⡿⠋⠉⢢⠀⠈⠀⣀⣈⣙⠋⠀⠄⠀⠀⠀⠘⣄⠈⣉⣉⣀⡠⠃⠀⠀⠀⢀⡥⠊
⡏⠀⠀⢿⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡼
⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⣇⠀⠀⢸⠀⠀⠀⣷⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢴⡇⠀⠀⢀⡞⠁
⣿⠀⠀⠸⠀⠀⠀⠛⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠔⠉
⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠧⠀⢻⣓⣦⢤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⠶⢯⣁
⢿⠀⠀⠁⠀⠀⠀⢠⠋⠀⠀⠀⠀⢸⡇⠀⢠⠇⠉⠉⠻⠯⡭⠿⠋⠀⢻⡀⠀⠈⠳⣄
⢸⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⣇⡤⢾⠀⠀⠀⣠⣶⣶⣤⠀⠀⢸⠉⢧⡀⠀⠘⡆
⢸⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⠋⠀⣿⠀⠀⠀⠻⠻⠛⢿⠂⠀⠘⠆⢈⠇⠀⠀⡟
⠘⢧⣀⣀⣀⣀⣠⡽⠤⠤⠤⠤⠶⠃⠀⠀⡇⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡗⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠒⠲⠶⠶⠤⠤⢤⣤⣤⣄⣧⠰⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⠀⠀⡟⠛⠓⢲⡀⠀⠀⢈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠉⢙⡆⠀⢸⠛⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠒⢲⡖⠚⠀⠀⠘⠒⢲⡒⠒⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⣄⠀⠀⠀⠀⠀⢈⡷⠚⠛⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣫⣵⣦⣭⣭⣭⣭⣥⣝⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢛⠻⠿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⢀⣴⢟⣵⠿⡁⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢀⣴⢟⣵⡟⣩⡾⢋⣬⣥⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣡⡾⢋⣴⢆⣿⣿⣿⣧⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠟⠵⠿⠿⠿⠿⠿⠿⠏⣼⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣇⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣿⣿⣿⡄

⣿⣿⢻⣻⡇
⣿⡯⡸⣿⡃
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⢀⢀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏⢨⠿⠹⠹⠸⠺⡥⠀⢍⡶⢹⠸⡹⢺⡹⠀⠸⢾⡅⢎⠷⠏⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠿⠿⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⢀⣼⡿⣷⡄⠀⠀⠀⢀⣴⠟⠁⠀⠀⣄⣶⣀⠀⠙⢷⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢀⣴⡿⠋⠀⠘⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⡿⠋⠀⠀⠀⢀⣿⠃⠀⠀⢸⡇⠀⠀⠀⣷⣿⣿⣿⣴⠆⠀⣿⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣆⣴⡿⠋⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠘⣷⡀⠀⠀⢿⣿⣿⣿⠋⠀⣠⡿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠈⠹⣦⣀⠀⠉⠉⡀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⢀⡄⠀⢉⣷⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⢹⢸⢸⢸⣿⣿⣿⣧⠀⠀⠀⢠⡟⠀⠀⢰⣾⣿⡇⡇⠛⠁⠹⣆
⠀⠀⠀⠀⠀⢸⣿⣧⡒⢿⡆⣞⢰⡎⡇⣒⣽⣿⠀⠀⠀⢸⣿⣿⣿⣿⠘⠈⠘⠸⠟⣻⣿⣿⠄⠀⠀⢸⠀⠀⠀⢸⣿⣿⣧⣇⣤⠄⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⣷⣿⣶⣾⣷⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠸⣆⠀⢀⣸⣿⣿⣿⣿⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠹⣶⡟⠀⠙⠛⠛⠁⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢰⠱⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣎⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣀⣈⣀⣹⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⠿⠿⠿⠮⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣦⣀⣤⣤⣀⣤⣤⣤⣀⣴⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠛⠿⠛⡛⠿⠟⠛⠛⠿⠛⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⢈⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠩⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠍⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣖⠀⡠⠈⢐⡀⠀⠀⠀⢄⠀⣲⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠫⣿⠟⠻⣿⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⠀⣠⣄⠀⣦⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿






⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠫⡗
⠀⠀⠀⠀⠐⠥⡀
⠀⠀⠀⠀⠀⠀⠙⢔⠄⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡈⠡⢤⠤⠉⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠞⠉⢔⢆⡈⠄⢀⢜⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣒⠀⠀⠈⠆⠁⠋⡔⣁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢰⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⢉⣱⣢⢶⡐⠖⠀⠀⠩⢠
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠳⠛⠟⠋⠉⠀⢸⠑⢌⢄⢀⠎⠌
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡄⠠⢋⡡⠊
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⡤⢄⣀⣀⠠⠤⠀⠤⣀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠂⠁⠙⠍⠒⠀⠀⠒⠉⠀⠀⠀⠀⠀⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡇⠀⠀⠐⠀⠠⠐⠣⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢩⢡⠀⠀⢂⠀⠀⠀⠀⠁⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⠀⠠⠀⠀⠀⡀⠄⠁⠀⢠⠤⠤⠄⠑⠀⠀⠀⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⡄⠀⠀⠀⠁⠁⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⢀⠀⠈⠐⠠⣴⡀⠀⠀⠀⠐⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠠⠁⢀⠜⠂⠄⠀⡅⠀⠀⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡄⠀⠀⠰⠂⠈⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠣⢀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⢠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠊⠊⠀⠀⠀⠀⠀⠀⠀⡈⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠢⣉⠓⠲⠦⠤⠤⠤⠶⠒⣋⡡⠔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠉⠉⠐⠒⠒⠒⠈⠉⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠴⠒⠀⠲⠄⠀⠀⠀⢰⠄⠀⠀⠀⠂⠀⠾⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣰
⠀⠀⠀⠀⢀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢠⣀⠁⠀⣀⣀⣤⠄
⠀⠀⠀⠀⢸⠄⠄⠋⠈⢷⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⡟⠋
⠀⠀⠀⠀⢸⣶⠀⠀⠔⣻⡇⠀⠀⢠⣾⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⢶⣿⠀⠀⠀⠀⠀⠀⠀⠈⢛⡛⠁⠀⠀⠀⠐⢶⣀⢀⣠⡿⠂
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠚⠁⠀⠀⠘⣿⣿⣿⡾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⢠⣾⡄⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠁⠀⠀⠀⠀⡈⡀⠀⢀⣾⣖⣺⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠠⠀⠀⠀⠀⢡⠀⠀⠀⠉⠉⠀⣐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠴⠦
⠀⠀⠀⠀⠘⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠐⠲⠀⠀⠀⢐⣲⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠦⢶⠄
⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⢹⢿⣶⣶⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠆⠀⠁⠀⠀⠀⠀⠠⠄⣀⠤⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢟⣻⠿⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠃
⠀⠀⠀⢀⠀⠀⠾⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠸⣿⠇⢛⣔⠘⣿⣦
⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⠀⠐⠂⣼⣿⠷⣽⠉⠀⠀⠀⣀⢠⣿⣧⡀⡀⠀⠀⠀⠀⢀⣾⣿⡄
⠀⠀⠀⢲⣿⣿⣿⣷⠀⠀⠀⠀⠐⣾⣿⣿⣶⡆⠀⠀⠀⠀⢹⣾⣿⣿⣷⠃⠀⠀⠀⠐⣾⣿⣿⣿⡆
⠀⠀⠀⠘⠻⠛⠻⠛⠀⠀⠀⠀⠀⠛⠟⠛⠟⠃⠀⠀⠀⠀⠘⠻⠛⠻⠛⠀⠀⠀⠀⠀⠛⠟⠛⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠾⠿⡿⠿
⠀⠀⠀⠀⠾⣿⣿⡆⠀⠀⠀⠀⣴⣿⣦⠈⠓⢻⣿⡽⠀⠀⠀⠢⣴⣿⣧⡴⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⢺⣿⣿⣿⣷⠀⠀⠀⣶⣿⣿⣿⣶⢠⡞⢿⡁⠀⠀⠀⢲⣾⣿⣿⣷⠀⠀⠀⠰⣾⣿⣿⣷⡆
⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠸⠃⠈⠓⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⠃


⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠾⣤⣦
⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⣿⣿⣥⣤⠀⠀⠀⠀⠀⠀⠀⣿⣟⠢⡀
⠀⠀⠸⣿⣿⣿⣾⡆⠀⠀⣶⣶⣶⣶⡖⢀⡿⣿⣇⠀⠀⢀⣽⣿⣿⣿⣶⡆⠀⢸⣷⣷⣾⣿⠙⣧
⠀⠀⠈⠛⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠃⠞⠀⠛⠛⠀⠀⠸⠃⠛⠛⠛⠛⠃⠀⠘⠛⠛⠻⠛⠀⠰⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⣤⠀⣤⡀⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠙⠒⠚⢳⠱⠀⢸⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⠴⠃⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡇⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠉⠉⢹⠀⠀⢰⡃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠛⠉⠙⣆⠀⠀⠀⠀⢸⡇⠀⣠⣄⣀⣀⣤⢸⢠⠀⠀⠑⠦⠤⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⣆⠀⠀⠀⣀⡧⠾⣷⠒⠛⠓⢾⠃⠞⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣪⠵⢚⡠⠖⠋⠁⠀⢀⡼⠦⠤⠶⢺⠀⠀⠀⠀⠀⣠⠋⡥⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⣠⠶⡟⠢⣀⠀⣠⠛⡶⢤⡀⠀⠀⡇⢸⢠⠎⡥⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⠤⠖⠉⠀⠀⠙⢆⠈⠉⢁⡜⠀⠀⠈⢦⠀⠁⠈⠈⠁⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⠀⠀⠀⠀⠀⠑⢤⠎⠀⠀⠀⢰⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣿⡷⢸⠆⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠄⣠⡀⢈⡿⠛
⠀⠀⠀⠀⣶⢠⣶⡆⢰⠈⢻⣶⣶⠋⠀⠀⠀⠀⠀⠀⠀⢼⡅⠦⠔⠶⠰⠎⣷⠀⠀⠀⠀⠀⣼⣧⢤⣄⢻⡘⣿⠛⡾⢁⣤⣿⡦
⠀⠀⠀⠀⠙⢮⣭⣵⠋⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣶⣤⡀⢀⣰⣿⠀⠀⠀⠀⠀⠀⠉⠀⢸⠛⠓⠛⠚⠓⢻⠀⠉
⠀⠀⠀⠀⠀⢸⣿⣿⠠⠶⠟⢁⣽⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⣾⡁⢿⣤⣄⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⠀⣎
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣃⡀
⠀⠀⠀⠀⠀⠈⣁⣾⣗⢸⣿⣤⡉⠀⠀⠀⠀⠀⣇⢾⣿⢆⡇⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠠⡤⠄⣻⡿⣿⢰⣷⠻⡃⢤⡄⠀⠀⠀⠙⣷⣶⡟⠁⠘⠿⢿⡀⠀⠀⠻⠿⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠁
⠀⠀⠀⠀⠀⠈⠿⣷⣷⢸⣻⡻⠇⠀⠀⠀⠀⠀⠀⣿⣿⡁⢀⣐⠶⠚⣷⠀⣀⠲⠖⢻⡆⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠲⠁⠉⠉⡌⠉⠉⠳⠀⠀⠀⠀⣤⠶⠏⣸⡷⠀⠀⣿⠛⠛⠀⠀⢸⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠛⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠲⠒⠂⡇
⠀⠀⠀⣀⣀⣸⠿⠃⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⣐⡀⣀⣀⠢⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⡆⣴⣶⡄⣶⢠⣶⣦
⠀⠀⠀⠏⣼⣿⠿⡦⠆⠾⠇⣏⡽⠗⠺⢿⡹⠀⠀⠘⠀⠐⠀⠀⡇⡖⢀⢀⢀⠐⠀⠀⠀⠀⢠⣽⣯⡀⠻⣮⣯⣶⠃⢠⣽⣯
⠀⠲⢦⡾⠛⢿⡀⠀⠀⣤⣤⣤⣤⣤⣤⣶⡦⠀⠀⠈⠒⠒⠒⠲⠇⣇⠀⠀⠀⠠⠀⠀⠀⠀⢼⣛⣛⠁⠄⣟⣛⡋⠠⢘⣟⣛⠄
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇⠀⣿⠛⣷⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠁⠀⠛⠀⠋⠀⠘⠃⠙

⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⢀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣗⡀
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢠⠄⠀⠀⠙⢶⣄⠀⢀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠉⠙⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠸⠀⠀⠀⠀⢸⡞⠀⢸⣤⣄⠛⢻⡇⠀⠀⠀⠀⠀⠀⡏⠀⢠⣄⠀⢀⢠⡄
⠀⠀⠀⠸⠿⠿⠿⠿⠃⠀⠀⠀⢂⠀⠀⠀⡾⠁⠀⠈⣇⣀⣄⣨⠁⠀⠀⠀⠀⠀⢠⡇⠀⠸⣿⣀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠙⠛⠁⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠑⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢤⣀
⠀⠀⠀⠀⠀⢰⣶⣷⣀⣤⡿⠿⠃⠀⠀⠀⢀⡏⣉⣁⣠⣤⣶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠍⠀⠂⠀⢀
⠀⠀⢀⠀⡄⢺⣿⣿⣿⣿⡇⡀⢠⠀⠀⠀⠘⡇⠈⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠶⠀⣶⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠠⣾⣾⠇⣠⡄⠻⠇
⠀⠀⠉⠛⠋⠀⠈⠁⠀⠀⠈⠙⠛⠁⠀⠀⠀⠙⠛⠁⠉⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠁⠉
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢰⣦⠀⣾⡷⡆⠀⠀⠀⢀⢸⣧⡤⢠⣽⣄⢠⣿⡄⢀⣈⣯⢀⠀⠀⠀⢴⣶⡶⢧⣤⠀⠀⠀⢤⣴⣦⠀⠀⡀⠀⠀⠀⠰⢷⣄⠀⢠⣤
⠀⠀⠈⠿⣿⣿⣷⡆⡶⡇⠀⠈⠹⠿⣿⣾⣿⣿⣿⣿⣿⣿⡾⠿⠃⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣾⣷⡆⠀⠀⠀⣾⣿⣶⣿⣿⠆
⠀⠀⠀⠈⠀⠁⠈⠀⠁⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡀
⠀⠀⠀⠁⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣙⣿⠁
⠀⠀⠀⣴⣾⣿⣿⡷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⢖⣀⠀⢀⠘⣧⣆⢀⡾⠃⠠⣐⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠟⢹⣿⣿⣧⠀⢀⣿⣿⣧⣴⣿⠀⠀⠀⢿⡛⠛⠀⠒⣿⣿⣿⣿⣶⡆⠀⢸⣿⠃⠀⠀⢀⣿⣿⣧⣾⡖⠀⠀⠀⠀⠀⣸⣿⣿⣤⣶⡆
⠀⠀⠀⠀⠸⠿⠿⠿⠀⠘⠻⢿⡟⠛⠻⠀⠀⢀⡼⢹⡄⣤⡈⠿⠿⣿⠿⠿⠇⠀⢸⢹⠀⠀⠀⠸⠿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠿⢿⡿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⢀⡤⠾⠶⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠶⠖⢶⠈⠧⠀⠀⠀⠀⠤⣾⠀⠠⠀⢸⡦⠤⠀⠀⠀⠸⠁⠀⢀⡤⡿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠴⠋⠀⠀⢦⠀⠀⠀⠀⠈⠳⣤⡤⠞⠀⠀⠀⠀⣰⠀⠀⠀⠸⣄⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣺⡽⠞⢳⡀⠀⠀⠀⠀⠈⠃⡴⠚⠉⠉⠙⠓⠒⠶⢶⡖⠃⠀⢸⠋⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡟⢉⣉⣉⣉⣉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣀⣬⢷⡀⠀⠀⠀⣇⣼⢳⡞⠳⢤⣤⠼⣿⢻⡀⠀⠀⠈⠛⠛⡇⠨⠭⠍⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡭⠖⠋⠀⠀⢀⡀⠸⡇⢸⡇⠀⠀⠀⠀⡏⢸⡇⢀⣀⠀⠀⠀⠛⠛⠓⠚⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⠤⠀⠶⠒⠀⠛⠋⣁⣨⣥⠴⣇⣸⠇⠀⠀⠀⠀⣧⣸⠧⢤⣭⣀⡉⠙⠁⠘⠒⠂⠰⠶⠤⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⣠⣾⠿⣄⠀⠀⠀⣠⠿⣿⣦⡀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⠴⠋⠁⠘⢧⡈⠙⠲⠖⠋⣰⠃⠀⠙⠲⣄⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡓⠶⠚⠋⠁⠀⠀⠀⠀⠈⠳⣄⠀⢀⡴⠃⠀⠀⠀⠀⠀⠉⠓⠲⠖⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠤⣤⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⠀⠀⣠⠤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠖⠲⠂⠖⠲⠒⠓⠚⠲⠲⠒⠲⠂⠒⠢⠺⠒⠒⠒⠒⠒⠒⠂⠐⠺⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣚⠉⠉⠉⠉⠉⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠛⠀⢖⠀⢸⡁⠉⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠙⢦⠀⢈⣑⣸⠀⠀⠇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡞⠁⠀⣳⠀⡀⠸⣧⡠⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⣤⣼⣀⣀⣠⡃⢠⣷⠀⠙⡴⠟⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⠋⢱⠆⠉⠁⠀⠀⣾⣿⡇⠀⠀⠈⢹⠉⡗⠂⠤⣀
⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠁⠀⠀⢀⡴⠁⠀⠀⠀⠀⡼⠀⡠⡇⠀⠀⠀⡜⠀⡇⠀⠀⠀⠉⠒⠤⢢⡀
⠀⠀⠀⢠⡎⠁⠀⠀⠀⢀⠀⢀⡴⠋⠀⠀⠀⠀⢀⡜⠉⢹⡅⡁⠀⠀⠀⠁⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤
⠀⠀⠀⠀⠓⠦⢤⣥⠥⠔⠚⠳⢄⡀⠀⠀⠀⠠⠊⠀⠀⠀⠑⠒⠤⠤⢀⣀⣀⡠⠎⠀⠀⣄⠀⠀⠀⠀⠀⠀⡼⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡇⠀⠀⠀⠀⠑⢢⠀⠀⠀⡰⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢤⡄⣀⡏⠉⠁⠁⠠⡀⢀⣀⠠⢾⣄⣀⣰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡅⠈⡹⠀⠀⠀⠑⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡎⠉⠱⡀⣼⣆⠀⣠⣀⣇⠀⠀⠀⠠⣄⣀⣀⠀⢀⠎⠀⣀
⠀⡀⢀⠄⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡙⠻⢷⣄⠀⢀⡈⢣⠹⣧⡀⠑⠷⣄⠀⠀⢸⡱⢌⠲⣁⣶⣙⠈
⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠒⠦⡀⠀⠀⠀⠀⠀⠉⠢⣄⢉⠧⡀⢙⠉⠓⣌⢻⠢⣀⠈⢳⡀⢸⠀⠀⠑⠄⣀⡤⠔
⠀⠈⢄⠀⠀⠀⣠⠆⠀⠁⠀⠀⠀⠀⠀⠈⠂⢀⠔⠀⠀⠒⠀⠀⠉⠂⠼⢾⠀⠀⠈⢢⠑⢄⠙⠣⢽⣾
⠀⠀⠀⠉⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠱⠎⣰⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢠⡟⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠤⠤⡞
⢀⠔⠁⠁⠀⠉⠁⠀⠉⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡄⠀⠀⠀⠀⢀⠞⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡇
⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⡇
⠀⠀⡏⠉⠑⠀⠚⠒⠊⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠙⣆⠀⠀⠀⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⡓⡄⢣⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⣤⣤⢸⠀⠀⠀⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⢿⡇⠀⢧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣂⠇⡸⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⢸⠀⠀⠀⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⠀⠸⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣡⠔⠁⠀⠀⠀⠀⠀⠀⠘⡟⠻⠿⠿⠛⡟⠀⠀⠀⣿⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⢀⡼⣰⢃⠀⠀⠠⡹⡄
⠀⠀⣇⠀⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⡇⠀⠀⠀⢿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡞⢰⠇⠎⠀⠀⠀⢱⠘
⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣇⣾⢸⠀⠀⣀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣰⣶⣦⡀⠀⠀⡆⢀⣀⣀⣀⣀⣀⠀⡆⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠠⠤⠄⠠⠤⠄⠇⠀⠀⠈⢿⣟⣏⠁⠀⠉⠙⣿⡿⠀⠀⢃⢸⣯⣟⣿⣿⣿⢀⠇⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⣿⣦⣤⣤⠎⠙⠁⠀⠀⠈⠢⡙⢿⣿⠟⡡⠊⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⡖⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸
⡇⠐⠒⠀⢨⠀⠀⢸⠀⠀⡤⠚⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠐⣾⣿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⣿⠈⡇⠀⠀⠀⢸
⠀⠀⠀⣠⡇⠀⠀⠀⡀⠀⣠⣾⣿⣿⠀⠀⠀⠀⢀⣀⣤⣼⣤⣄⣀⠀⠀⠀⠀⠀⡠⠴⠶⡷⠶⠤⡀⠀⠀⠀⡧⠺⢶⣄⣀⡤⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⢹⡟
⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⣿⣿⡇⢠⣶⣄⢰⣿⣷⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⢠⣿⡌⢿⣿⣧
⣿⣿⡇⢸⣿⣿⢸⣿⠋⣤⣭⣭⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢸⣿⣿⠀⣾⡏⣡⣬⣭⣭⣤⣄⡀
⣿⣿⡇⢸⣿⣿⢸⣿⣆⠙⠿⠿⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⠀⣿⣷⡘⠿⠿⠿⢿⣿⣿
⣿⣿⣷⣈⠛⣃⣄⡛⢋⣰⣶⣶⠆⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⣃⣌⡛⣋⣰⣿⣿⣿⣷⣶⡶⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣴⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⠏
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⠿⠟⠋




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⢰⣾⣶⠀⣠⡶⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣉⣹⣿⡞⠛⠛⣾⣋⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢸⣿⣿⠙⠿⣭⣁⣀
⠀⠀⠀⢀⣀⡀⢀⣴⣦⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣿⠀⠀⠈⠙⠋
⣠⣤⡀⢸⣿⣿⢸⣿⡿⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⣶⡄⣾⣿⡇⢸⣿⣿⠀⣠⣶⡆
⣿⣿⡇⢸⣿⣿⢸⣿⡀⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⣿⣿⡇⢸⣿⣿⠀⣿⣿⡇
⢿⣿⠇⢸⣿⡿⠸⣿⡷⠈⣙⣛⡛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⣀⠻⠟⢃⠙⠿⠃⣼⣿⣿⠀⣿⣿⡇
⣦⣤⣶⣦⣤⣴⣷⣤⣴⡾⠟⢛⣃⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣥⣴⣶⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⣴⣿⣿⣿⣿⣿⣿⡇
⢻⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⠁
⠈⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁

⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⣤⠛⠛⢳⡀⠀⠀⠀⢀⣴⢶⠆⠀⢤⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⢠⣶⣯⠀⢀⠀⠀⠀⠀⠀⢠⣬⣯⣴⣦⠀⠀⠀⠀⢠⣿⠀⠀⢰⠄⠀⠀⠀⠺⠿⢾⠀⢤⣸⠀⠀⠀⠀⠀⣠⣬⣯⣤
⠀⠀⠸⠿⠻⠿⠶⠀⠀⠀⠀⠀⠻⠿⢿⠏⠉⠀⠀⠀⠀⠘⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠿⠤⠤⠾⠆

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⠒⠢⡀⠄⠀⠀⠀⠀⠀⠀⢸⣯⠃⠀⠀⠀⠀⠀⢠⡏⠀⢨⠂⠀⠀⠀⠀⠀⡶⠉⠉⢀⣄
⠀⠀⢠⣶⣷⣾⣷⠀⠀⠀⠀⠀⣼⣴⣰⣾⡆⠀⠀⠀⠀⢀⢈⣀⣾⣷⡄⠀⠀⠀⠀⠸⢷⣾⣼⡀⠀⠀⠀⠀⢸⣗⣀⣀⣿
⠀⠀⠘⠿⠿⠂⠀⠀⠀⠀⠀⠀⠻⠃⠀⠻⠇⠀⠀⠀⠀⠀⠾⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠈⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠰⣿⡷⠀⣛⠿⠀⠀⠀⠀⠐⠿⣿⠿⠀⠀⠀⠀⠀⢀⠒⠺⣿⠟⠀⠀⠀⠀⠀⠀⢘⣿⣋⠀⠀⠀⠀⠀⢰⣦⠘⢿⠟
⠀⠀⣴⣾⣶⡿⠋⠀⠀⠀⠀⠀⢰⡞⠛⢷⡄⠀⠀⠀⠀⠸⠦⣼⣿⣷⡄⠀⠀⠀⠀⢰⣿⠛⢿⣧⠀⠀⠀⠀⠙⠉⡻⠿⡴⡂
⠀⠀⠙⠙⠛⠃⠀⠀⠀⠀⠀⠀⠘⠓⠒⠘⠃⠀⠀⠀⠀⠀⠀⠛⠛⠙⠃⠀⠀⠀⠀⠈⠛⠀⠙⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛

⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⠠⡀⠀⣠⣄
⠀⠀⢀⣶⣹⣟⠀⠀⠀⠀⠀⠀⣠⠄⡀⣶⣶⠀⠀⠀⠀⢀⣽⣇⣹⣫⡄⠀⠀⠀⠀⢠⢄⡀⡀⣄⠀⠀⠀⠀⠘⠋⣁⢛⣟⡀
⠀⠀⠚⢻⣿⣿⣷⠀⠀⠀⠀⠀⣿⡾⠓⠿⠿⠀⠀⠀⠀⢈⣿⣧⣽⣿⡆⠀⠀⠀⠀⣿⡿⠓⢻⣿⠀⠀⠀⠀⠀⢈⣤⣿⣿⣧
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠰⢖⠶⣠⡀⠀⠀⠀⠀⠰⠀⠿⠀⠆⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠄⢰⣗⠀⠀⠀⠀⢀⣠⠀⣀⣀⢀⣀
⠀⠀⣴⠂⣺⢀⠾⠦⠀⠀⠀⠀⠀⡔⣒⠢⠀⠀⠀⠀⠀⢠⣼⣟⠁⣘⠀⠀⠀⠀⠀⠸⢦⣤⢿⣦⠀⠀⠀⠀⠹⣦⣽⣟⣹⣟
⠀⠀⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⢸⣧⣭⡼⠀⠀⠀⠀⠀⠸⠿⠿⠰⠿⠇⠀⠀⠀⠀⢠⣾⠻⢿⡟⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠠⢤⢠⢠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⣭⣍⠁
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢮⣯⣯⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠘⢿⣿⣿⠗⠀⣤⣦⡀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⠏
⠀⠀⠀⠀⠀⠀⣤⣿⣧⣴⣶⡍⠉⠁⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠙⠧⢟⠛⢻⢏⠃⠀⠀⠀⠀⠀⠀⠀⠲⡖⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⡇
⠀⠀⠀⠀⠀⢰⡐⠂⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣦⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢰⣤⣤⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴
⠀⠀⠀⠀⠀⠀⠙⣿⠏⣵⣾⣄⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⢀⣹⣀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠂
⠀⠀⠀⠀⠀⠀⣸⣿⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⢠⠄
⠀⠀⠀⠀⣤⣦⠈⢻⠋⣴⣿⣶⡀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⣿⣿⣭⡀
⠀⠀⠀⠀⠈⠙⠐⠛⠓⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠚⠁⠀⠀⠀⠀⠀⠀⠒⠿⠓⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠻⠛⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢛⡛⠀⠿⣿⣽⣷⠀⠀⠀⠄⠄⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠼⣿⣷⡞⠛⠋⠙⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⣀⣀⠠⠾⡏⢸⠎⠹⠛⢁⣠⠄⠖⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠊
⢀⠴⠒⢯⠁⡄⠀⠀⢠⠘⣄⠁⠀⠀⠀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠛⣿⣄⠀⠀⠀⠠
⠋⠀⡀⠀⢇⣧⡉⠂⢸⠀⢈⣦⢠⡀⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⢰⡶⠀⠀⢻⣼⡇
⠀⠀⠘⢄⠸⡏⡇⠀⢸⠳⡎⣿⠀⡂⢱⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡴⣠⠀⠈⠈⠀⣺⠏⢁⡇
⠀⠀⠀⠈⢳⠃⢻⠀⠈⣼⣱⡇⢠⠃⡈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠡⠆⠋
⠀⠀⠀⠀⢸⠀⠨⣇⢀⣷⠟⠀⠈⡠⠄⠘⠀⠉⠁⣀⢦⠖⠒⣶⠾⡷⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠐⠀⠰⠀⠀⠠
⠀⠀⠄⢂⣸⡀⠀⡽⢈⡀⣨⠔⠉⡠⠴⡂⢽⡽⠋⠠⣮⠣⣰⢯⡄⣷⠀⠀⢑⡖⠀⢸⣿⣶⢦⡄⠀⠀⠀⠀⡀⠂
⠀⠔⠸⠁⠂⠙⠚⠓⠋⠴⠧⠀⠊⢄⣌⢴⡟⠀⠀⢀⣺⣶⣿⣟⠐⢹⡣⡀⢸⡏⠀⢸⣯⣿⡇⣿⡄⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠀⡰⢡⣀⠀⠀⠈⠴⠈⡾⠁⠀⠀⠀⢹⣿⠃⣿⠀⢘⣇⠈⢻⠃⠀⢸⣿⣿⣿⣿⣷⡀
⠈⠑⠢⠤⢄⣁⢰⡇⣤⠉⣆⢣⠀⠀⠸⡁⠲⢴⡄⣆⢸⡇⢠⢿⣧⡨⣿⠀⢠⠀⠀⢸⣹⣿⣿⣿⡟⣷⠀⠘⠀⠀⠐
⠀⠀⠀⠀⡸⠀⢈⡵⡇⢠⠇⠐⡄⠀⠀⠻⢿⣶⣌⣻⣿⠀⠀⢸⣿⣏⣿⠐⢆⠀⠀⡏⣺⢏⣿⡿⢞⣿⡲⡄⠄
⠀⠀⠀⡴⢡⢖⠏⠀⡇⢸⡆⠀⣸⡄⣠⣶⣿⣷⣿⣿⣉⣀⢀⣊⣝⢟⡿⠀⠀⢳⡜⣘⡁⢺⣿⡟⢉⡑⠒⠽⣆⠀⠁
⠀⡠⠊⠀⠀⡎⠀⠀⢇⠈⡇⠀⣿⣿⠟⣫⡿⠗⠻⠋⢛⠛⢟⠟⢹⠟⠁⠀⠀⠀⠀⠀⠌⠋⡃⢋⣠⠀⡀⠀⢹⢿⡉
⠊⣧⠀⠀⢸⠃⠀⠀⠸⡄⢻⡀⠻⣟⡾⠉⠠⠂⠀⠈⠜⠇⠀⢰⡏⠀⡀⠀⠀⡀⠀⣀⡠⡎⠁⠡⠠⠞⣠⠆⢈⠏⠁
⣘⠈⢧⠀⡼⠁⠀⠀⠀⢳⢄⡉⠂⠹⣷⣶⣦⣤⣄⣄⣀⣨⣂⣸⡇⡠⠛⠉⠉⠉⠁⠁⠀⠈⠀⠀⠀⠚⠑⠋⠁
⠃⠁⠈⠳⡇⠀⠀⠀⠀⠘⣆⣯⣱⠦⠤⠛⠛⠋⠉⠁⢀⣤⣼⡇
⡰⠀⠀⠀⢱⠀⢠⣖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠤⢄⡀
⠀⠀⠀⠀⡇⠀⠘⢦⣙⠲⠤⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠠⡄⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⣷⠀⠀⠀⢻⡑⠢⢄⡉⠒⠤⣀⠀⠀⠀⠀⠀⠓⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀⡀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡿⠛⠻⠿⠿⠛⢫⡉⠈⠙⠛⠛⠃
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⢻⣷⣸⣦⣸⣷⣶⣤⣤⡤
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⠻⠁⠀⠀⠀⠀⢀⣴⣿⣷⣿⡿⠛⠛⠉
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⠀⠀⣴⣿⡿⣻⣿⣿⡧⠀⠀⢀⣀⡀
⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠳⠉⠀⠀⠁⢠⣾⣿⡟⣴⣿⣿⣿⠟⠉⡈⠁
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⠸⣿⣿⣿⣿⣇⣐⠃
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⣿⡟⢸⡄⣿⣿⣿⣿⠀⢠⠄
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣿⠇⠘⢿⡟⣿⣻⣿⣶⠐⠂
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣷⣤⣀⡀⠀⢀⣿⠀⠀⣾⠀⠏⣿⣿⣿⣧⣄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣶⣾⣷⣤⣤⣏⠘⠰⠆⠚⠋⠉⠉⠁

⠀⠀⠀⠀⠀⢀⠄⠈⠉⠐⠤⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠠⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⣀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠳⡀⠀⠀⢠⠞⠀⡘⠉⠉⠀⣾
⠠⠒⠊⠉⠉⠓⠢⣄⠀⠀⠰⡄⠀⢀⡤⠤⠤⠶⣌⡓⠲⠄⡀⠀⠀⠀⠀⠀⡏⠉⠉⠁⠒⠦⡀⠀⠁⠀⢀⠏⠀⠀⠁⠀⠀⢀⠇
⣧⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠸⡜⠁⠀⠀⠀⠀⠀⠙⢆⠀⠀⠑⠄⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠅⡠⠊⠀⠀⠀⠀⠀⠀⠃
⠈⠢⡈⠀⠀⠀⠀⠀⠀⠀⠘⠒⣼⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠢⠆⠀⠀⠀⠀⠈⠀⠀⠜⡜⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠘⠧⠄⡀⠀⠀⠀⠀⠀⠀⠹⣗⠀⠀⠀⢀⠈⠀⢀⡿⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠈⠀⢹⣦⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⢣⡀⠀⠁⠀⠀⠀⠀⠀⠈⠈⡟⠀⠈⠀⠀⢀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠒⠒⠀⠀⣰⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠴⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠐⣀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠤⠀⡀⠄⠀⣀⡀⠠⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠄⣀⡀

⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⢿⣶⣶⠀⠈⠁⠀⠀⠀⠀⣿⣿⣿⣇
⢶⣶⣶⣶⣶⣾⣵⣶⠆⠀⠀⠀⠀⠀⢀⣿⣽⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢧⣿⣿⡟
⠀⠙⢮⡻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢻⡟⠀⢀⣤
⠀⠀⠈⠻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⡀⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀
⠀⠀⠠⠶⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠖⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠠⠾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠀⠘⠿⠛⠿


⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⣴⣿⣧⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⡄
⠀⢰⣿⣿⣿⡇⡸⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⣻⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⡇⠹⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢿⡿⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣠⣤
⠀⢸⣿⣿⣿⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠈⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿
⠀⠸⠿⠿⠿⠇⠀⠼⠄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠠⡇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠸⠿⠿⠏⠀⠀⠀⠿⠿⠿⠿⠇⠿⠿


⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀
⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⢀⡀⣶⢀⡀⣶⢀⡀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣾⡇⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣶⠄
⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣫⣶⣾⣷⠟⢶
⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢟⣻⣟⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣶⣿⣿⣿⣿⠀⢸
⠀⢸⣿⣿⣿⠀⠀⢀⣤⡄⠀⠀⠀⣷⣄⣀⣀⢸⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⢹⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡗⠋
⠀⠘⠿⠿⠿⠈⠻⠿⠛⠀⠀⠀⠀⠈⠻⠿⠿⠷⠾⠷⠿⠿⠿⠇⠀⠀⠀⠀⠀⠘⠿⠿⠟⠀⠀⢠⡆⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⣤⠖⣿⢠⣀⡄⢻⠀⠀⠀⠀⠀⢸⠀⢿⢰⣷⡗⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠧⠾⣄⣿⠰⠛⠆⢸⠀⣠⡤⣄⡀⢸⠀⠒⠘⠙⠁⠐⠀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠈⠉⠀⠀⣠⠏⠀⡷⠴⠾⣿⠈⠛⠛⠛⠛⠛⠛⠛⢻⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠒⠋⠁⠀⠀⢳⣀⣀⡏⢀⣀⣀⡀⠀⠀⠀⡃⠸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⡟⠳⠞⢹⣯⣤⢦⡍⣧⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡟⠀⠀⢈⣧⣀⣼⡃⣿⠀⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠯⠿⠛⢲⡾⠉⠉⠛⠋⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠖⠚⠛⠛⠋⠙⠛⠛⠓⠲⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⢀⣀⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⣠⣶⣾⣿⠿⠟⠛⠛⠛⠋⠙⠛⠛⠛⠿⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⡿⠋⠙⢿⣿⣆
⠀⣰⠇⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢂⣴⣿⡿⠋⠀⠀⠀⠈⢻⣿⣆⠀⠸⡆
⢠⡏⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⠀⠀⢀⣀⠀⠀⠀⢿⣿⡆⠀⢹
⣸⠁⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⠀⠀⠸⣿⠀⠀⠀⠘⣿⣿⠀⠈
⣿⠀⢠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⠀⠀⠀⡿⠀⠀⠀⠀⣿⣿
⣯⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢹⡄⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣤⠀⠀⠀⢰⣿⡿⠀⢠
⠈⣧⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠁⠀⠀⢀⣿⣿⠃⠀⣼
⠀⠸⣆⠀⠸⣿⣷⡀⠀⠀⢀⣴⣿⡿⢋⣴⣷⣶⣶⣶⢓⡚⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⣰⠃
⠀⠀⠙⣆⠀⠙⣿⣿⣄⣴⣿⡿⠋⠰⠿⠿⠿⠿⠿⠿⢬⠥⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⣠⣿⡿⠃⠀⣰⠃
⠀⠀⠀⠘⢧⡀⠈⠻⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣿⣿⠿⠛⠉⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡶⡆⢶⢰⢢⡖⢲⡔⡆⡖⠲⡄⢰⡖⢲⣴⡆⣖⡂⢲⡖⢰⣒⡆⣰⣦⢠⡖⠒⢲⡖⢰⢠⡖⢲⣴⣦⣰⠰⣖
⣼⣭⣿⣬⣯⣬⣷⣾⣥⣧⣷⣾⣥⣼⣷⣾⣼⣧⣶⣯⣼⣧⣼⣬⣧⣯⣽⣮⣷⣦⣼⣧⣼⣬⣷⣾⣭⣥⣽⣴⣾
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⢤
⠀⢠⣿⡟⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⣀⠈⣇
⠀⠸⢿⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⢁⡇⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⣇⠀⢸⠃⠠⡃
⢠⡀⡸⠂⠾⡁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠹⠀⠀⠐⠒⠒⠒⠶⠶⠶⣿⠆⠀⠐⢷⠒⠒⠒⠒⠚⠓⠀⠙⣤⣘⣄⠜
⠈⢻⣅⠀⠀⠈⢢⣼⠋⠀⢀⣀⠤⠤⣤⣤⠋⠑⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡘⠁⠀⠈⡄
⠀⢸⠙⣦⠈⢳⣞⠉⠒⢊⡥⠒⠒⠚⠀⣿⡄⠀⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡄⠀⢠⢠
⠀⢸⠀⠈⣷⡏⡇⠉⠉⠁⠀⠀⠀⠀⠀⡏⡇⢰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⢸
⠀⠀⣀⡾⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⢸⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠀⠈⣼
⠀⣰⣿⠀⣀⣀⡟⢧⡀⠀⠀⠀⠀⠀⢸⠀⢳⢸⠆⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡄⡃
⠸⠋⡇⠀⠀⠀⢰⠈⠳⠀⠀⠀⠀⠀⢸⠓⣞⢉⡦⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠒⠏⢱
⠀⠀⢣⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⠏⢸
⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡌
⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠸⠤⣤⣤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠯⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠤⠤⢤⡇
⠀⠀⠀⠘⠧⠤⠤⠬⠃⠀⠀⠀⠀⠀⠀⠰⣉⣉⣉⣀⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢀⣴⣿⣿⣿⣿⣿⠟⠁⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠙⢿⣿⣿⣿⣿⣷⣄
⣾⣿⣿⣿⣿⠟⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣧⡀⠀⠙⢿⣿⣿⣿⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⢿⣿⣿⣿⠟⠁⠀⣠⣿⣿⣿⣿⣿
⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠹⠟⠁⠀⣠⣤⣄⡀⠈⢻⡿⠋⠉⠀⠀⠉⠛⢿⣿⣤⣼⣿⠟⠋⠁⠀⠈⠉⠻⣿⠋⠀⣀⣤⣤⡀⠀⠙⠿⠁⠀⣠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀⠀⠀⣠⣾⣿⣿⣿⣷⠀⠀⠀⢀⣴⣿⣷⣦⠀⠈⢿⣿⣿⠏⠀⢠⣶⣿⣶⣄⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠻⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠃⠀⣸⣿⣿⡀⠀⢻⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⣠⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⠰⢿⣿⣿⠷⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣧⡀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⢠⣀⣠⣴⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣀⣀⣀⠀⠉⠁⢀⣀⣀⡀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⡿⠃⠀⠘⠿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⢀⣠⣄⡀⠀⠹⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⢠⣾⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⡄⠀⠈⠻⣿⣿⣿⣿⣿⣦⠀⠀⠙⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢻⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣄⠀⠈⠻⣿⣿⣿⣿⠃⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠸⣿⣿⣿⣿⠟⠁⠀⣠⣾⡿⠀⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠊⠑⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠣⠄⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⣠⠤⡖⠒⠂⠒⠒⠂⠀⡀⠀⡼⠁⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢰⠀⠀⠀⡠⠂⠀⠈⠉⠀⠀⠀⡧⠖⠂⠀⠀⠀⠐⠲⢁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡋⠀⠌⠀⠀⡀⠀⠀⠀⣞⢈⣉⡂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⢀⡄⠀⢴⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠢⠈⠠⡈⠀⠁⣤⠐⠚⠒⠀⠀⠲⠀⠀⠂⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠐⠒⠀⠀⠀⠐⠒⠂⠉⢳⠀⠇⠀⡇⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠅⠀⢉⠉⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⣷⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠁⠈⠀⠀⠁⠀⠀⠁⠀⡟
⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠀⠀⠂⠀⣯
⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⢸⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⡇⠀⠸⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠂⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⡇⠀⠀⠑⠲⠤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⠀⠀⠀⠀⠀⠘⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠤⠂⠐⠂⣀⢴⠛⠀⣄⠠⠄⠉⠀⠀⠀⠀⣿⡄⠀⠀⡐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⣀⡠⠤⠒⠊⠀⠀⡰⠃⠀⠀⠡⠀⠀⠄⢸⠀⢰⣀⣀⠠⠤⠒⠀⠀⡟⠃⠀⠀⠂⠀⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡰⠃⠀⠀⡄⠘⠃⢰⠀⠀⠀⠀⠀⠀⠀⢁⠨⠒⠤⠄⡀⠤⠔⠂⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣂⣁⣀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢤⡤⠤⠤⠚⠁

⠀⠐⠀⠀⢤
⠀⠀⠀⠀⠸⡆⠀⠀⠀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⡀
⠀⠀⠀⠀⣠⣷⡿⠂⡖⠻⢿⣿⣟⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣶⡦⡄⠀⣼⣿⣷⣿⣾⡆
⠀⠀⠀⠀⠀⢿⡇⠀⠃⠀⢸⣿⠁⡼⠀⠀⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣴⣿⡟⣿⠟⠀⠘⣾⣿⣿⣟⠁⠀⠀⠀⠀⠉⠛⠉⠁
⠀⠀⠀⠀⢀⣻⡇⠀⣾⣷⣾⡟⠀⠀⠀⠀⠀⠐⢶⡀⠀⠀⠀⠀⣿⠗⠁⠀⠀⠀⠈⠻⣯⠁⠀⠀⢀⡿⢋⠔⠋⠀⠀⠀⠀⠒⠒⠚⠓⢦⣴⡆
⠀⠀⠀⠀⣿⠙⠀⠀⣿⣾⠙⠇⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⢹⡇⢧⡀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠘⣿⣷
⠀⠀⠀⠀⡏⠀⠀⠀⡁⠉⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣼⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⡏⠀⠀⠐⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡞⠀⠀⠈⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠆
⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠶⡆⠉
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣀⡀⠀⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣧⡀
⠀⠀⠀⠀⠻⠤⠤⠤⠤⠄⠴⠇⠀⠀⠀⠀⠀⠀⠀⠴⠆⠚⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠶⠀⠭⠁⠠⠦⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠿⠿⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡏⠙⡆⣀⡀⡲⢀⣀⢰⣿⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠓⠁⠐⠲⠖⠂⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡼⠣⠟⠳⣇⡿⢿⣼⢿⣧⣻⡤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠲⠀⠒⢻⡓⠂⣿⣶⠒
⠀⠀⠀⠠⡤⠀⠠⣤⣤⠀⢴⡄⠀⢠⡀⣀⡀⠀⢀⣀⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⡀⣠⣤⡄⢰⣿⠏
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡀⠀⠸⣇⢹⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⢠⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠙⠾⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⣿⣇⣸
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⡿⠟⠉⠹⣷⠀⠀⠀⠈⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⢀⢀⠀⠀⠸⠗⠒⠀⠘⠇⠀⠀⠀⠀⠀⠻⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢸⣟⠐⢺⡿⠛⢛⡟⢻⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⢿⡤
⠀⠀⠀⠀⠀⢠⣤⣤⣤⢀⡄⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⠻⣄⣀⣳⣶⢛⣁⡼⣿⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⡿⢿⠇
⠀⠀⠀⠀⢘⡟⢩⣭⡇⣭⠁⠀⠀⠀⢀⣀⣬⣛⣓⣠⣤⠆⠀⠀⢼⠠⠀⠁⠀⠀⠀⠀⣀⡿⠀⠀⠀⠀⣀⣀⡀⠇⠀⠀⠀⣤⣾⣿⣿⣫⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⡿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⣙⣿⠗⣠⠄⠀⠀⠠⠀⠀⢀⡤⠀⢀⠀⡀⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡎⣡⠞⡅⠀⠀⠀⠰⢏⡴⡋⠀⠀⠙⢲⠑⡤⠋⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡀⢀⡀⠀⠀⢤⡀⠀⠀⠀⢆⠀⠀⠀⢠⣄⠔⣿⣿⣏⣳
⠀⠀⠀⣠⠟⠁⠀⠃⠀⠀⢀⡴⠋⠀⠀⠁⠀⢀⡴⠋⠀⠁⠡⠀⠀⠀⠀⠀⠀⠈⠻⣦⡣⡴⢲⠨⠗⠀⠀⠙⢦⡁⢲⡤⠅⠀⠀⠀⠈⠳⣄⠛⢹⡤⡀
⠀⠀⠞⠁⠀⠀⠀⠐⠀⠰⠋⠄⠀⠀⠀⠄⠀⠋⠠⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠊⠌⠻⣦⡋⠆⠀⠀⢀⠊⡇⡙⢦⡘⠀⠀⠀⠀⢠⡇⢈⢳⣄⠆⢁
⠀⠀⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠁⠐⠂⠁⠀⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢈⠻⣦⡀⠀⠘⠀⠀⠀⠀⡙⢦⡀⠀⠀⠇⠐⠀⠀⠈⠳⣌
⠀⢠⢤⣀⠤⡀⢠⡄⡄⡠⢄⢠⠄⠀⣤⢠⠤⡀⠀⡄⠤⠄⡀⡠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡜⠈⠛⠀⠐⡄⠀⠀⠀⢀⠃⠙⠂⠀⠠⡀⠀⠀⠀⠀⡌⠓
⠀⠸⠴⠛⠤⠃⠘⠋⠇⠳⠜⠘⠀⠀⠛⠘⠴⠂⠦⠃⠢⠂⠉⠙⠀⠀⠀⠀⠀⠉⠒⠤⠖⠈⠀⠀⠀⠀⠀⠃⠤⠄⠔⠁⠀⠀⠀⠀⠀⠈⠣⠀⠤⠐⠃
⠀⠀⠀⠀⢢⠆⡖⠂⡆⠀⢰⣰⠀⢰⡀⢐⣆⠔⠂⢰
⠀⠀⠀⠀⠈⠀⠁⠁⠉⠉⠈⠈⠀⠈⠀⠉⠉⠈⠈⠈

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

⠀⠀⠀⠀⠀⠀⣤⣶⣾⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣤⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠉⠉⠛⠛⢿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠾⠿⠛⠉⠁⠈⠉⠓
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡠⢄
⠀⠀⠀⠀⠀⢠⠿⠉⠀⣀⣄⣉⠓⢬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠔⠒⠛⠛⠻⠦⣤⡀
⠀⠀⠀⠀⠀⠈⠁⠀⢾⣿⣿⣿⠀⣐⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣃⣿⠁⢰⣿⣿⣷⡀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠍⠉⠉⠠⢬⣛⣿⡟⠀⠠⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢘⠛⣍⡀⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠉⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⡴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⢀⣀⠴⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠐⠋⡈⠙⡍⠑⠒⠚⠉⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⠀⠀⠀⠀⠙⠀⠀⠀⠸⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠈⠛⢢⠔⠉⠀⠳⡄⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⡌⠀⠀⠀⠀⡁⠀⠀⠀⠸⡄⠀⠀⠀⡠⠤⠤⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠙⠦⠤⠤⠘⠀⠀⠀⠀⠀⠈⠓⠶⠊⣀⠀⡀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠈⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠠⡀⠈⠑⠢⠤⠤⠄⠐⠒⠉⣽⣿⡟⡱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠻⠿⣃⡀⠀⠀⠀⠀⠀⣀⠄⢰⣁⠴⢏⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠒⢦⠿⢙⠀⠀⢀⠔⢈⠀⠀⠊⠁⠀⠀⠓⠁⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣎⣀⣡⠞⠀⠈⠢⢤⣧⣠⠃⠀⠀⠀⢠⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡝⠉⢀⠔⠒⠢⠸⢿⠇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠒⣤⣤⠄⠂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⠏⠀⠀⠀⠀⠀⠀⠀⣀⠟⠛⡀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢹⠁⠀⢘⠒⠂⠤⠤⢀⡀⠀⠐⡅⠀⠀⢀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠦⢀⣠⣾⣦⣄⠮⠄⣀⠀⢤⠀⠤⠀⡁⠉⠲⡴⡦⡔⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⣿⣿⣿⠏⠀⢠⠉⠀⠉⣑⠂⠒⠒⠒⢷⠤⠧⠄⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡄⠉⠉⠉⡦⠤⡟⠒⠀⠐⡯⠤⠀⠀⠐⠫⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣼⣿⣤⣤⣬⣆⣠⡁⣡⡋⠁⡜⠲⠆⠐⠲⠋⠉⠑⣶⣶⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣮⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⠀⠉⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣷⣄⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠉⠉⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⢀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⢸⡟⠀⠀⠀⠀⢀⢀⣤⣴⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⠀⣢⠀⢸⠀⢸⣷⡄⠀⠀⠀⣿⣿⠁⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⠿⠃⢘⣻⡷⢸⠀⢸⣿⣧⣄⠀⠀⠉⠛⠄⠐⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣾⣦⠄⠌⡿⠁⢸⠀⢸⣿⣿⣿⡀⠀⣀⣠⠒⣀⡐⡤
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢁⣀⣆⣀⠀⠀⠘⠀⣿⣿⣿⣿⣧⣰⡿⠆⠈⠙⠉⠙⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣶⡶⢱⣾⣿⣿⣿⣿⡟⠉⠿⠆⢤⠀⢀⣀⠯⠉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⡇⢸⡿⠋⠩⠉⠩⠤⠀⠀⠄⠩⠤⠬⠍⠃⠸⠿⠷⠶⠶⠾⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠐⠘⢛⢛⣛⣛⣛⣋⢸⠀⢰⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⢠⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠙⡟⢻⣯⣞⠀⢸⡇⠀⠀⠀⠀⢠⢶⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⢸⠀⢸⣿⡟⣠⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠇⢸⣿⣟⠀⠘⠇⠀⠀⠀⠀⠸⣿⡟⠀⠇⠀⠀⠀⠀⣷⣿⣷⣀⣿⢸⠀⢸⠏⢸⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣛⣛⠃⠀⠀⠀⠀⠀⠀⣽⣷⣤⠀⠀⠀⠀⠀⠙⠻⣿⠟⠁⠈⠀⠘⠀⠀⣙⣷⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⣛⣋⣙⣙⣛⠆⠀⠀⡰⣶⣛⣛⠓⠻⠀⠐⠀⢀⣤⠴⡿⣿⡚⠀⢄⠀⠀⢴⠿⠯⠋⠁⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠉⠈⠀⠀⠀⠀⠀⠿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠻⠿⠛⠛⠋⠈⠀⠀⠈⠚⠿⠟⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠔⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⢠⡄⠀⠀⢠⠀⢠⠀⠀⠠⠀⠠⠀⠀⠤⠤⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠠⠠⠠⠀⠀⠀⠄⠠⠀⠄⠀⠀⠠⠄⠤⠠⠀⠤⠠⠀⠀⠀⠠⠀⠀⠄⠀⠠⠄⠀⠶⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠠⠆⠤⠄⠀⠶⠠⠄⠤⠤⠤⠀⠄⠀⠠⠤⠀⠀⠤⠤⠀⠐

⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣄⡄⣄⣀⠀⠀⠀⠀⠀⢰⠀⢡⢀⣰⡆⠀⠀⠀⠀⠁⠈⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡏⡟⢓⣛⣿⠀⠀⠀⠀⠀⢠⠀⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⣿⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡅⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⡒⢀⡀⢀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠋⠀⢉⢉⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠘⠂⠁⠟⠃⠀⠀⠀⠀⠀⠘⠂⠠⠴⠛⠃⠀⠀⠀⠀⠛⠀⠈⠛⠁⠀⠀⠀⠀⠀⠂⠡⠴⠟⠀⠀⠀⠀⠀⠀⠓⠂⠤⠤⠤⠚⠛

⠀⠀⠀⠀⢀⣀⠀⠀⢠⣠⣀⡀⠀⠀⠀⢀⠠⠀⠀⣀⠀⢤⠀⠀⢠⡔⠀⠒⠒⠀⢢⡄⠀⠀⢀⡤⠤⠠⠤⢤⣤⣀
⠀⠀⠀⠀⠘⢯⠀⠀⠘⢳⡾⠁⠀⠀⠀⠨⠀⠀⠀⠀⠀⠸⠀⠀⠀⠱⡀⠀⠀⢠⠞⠁⠀⠀⠸⠀⠉⠉⠉⠉⠉⠋⠀⠀⠀⠄⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠀⡀⣠⠄⠀⠀⠀⠀⠀⠑⡄⢀⠊⠀⠀⠀⠀⠀⠑⠀⡀⢀⡀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠻⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠁⣼
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡶⠃
⠀⠀⠀⠀⠀⢀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠄⠚⠃⢀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢂⠀⠀⠀⠀⢸⠄⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣀⠄⠄⠀⠀⢤⡄⠀⠀⠀⠀⢠⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠒⢢⡄⠀⠀⠀⠀⢀⠀⡀⢀⣀⣀
⠀⠀⠀⠀⠸⡁⢀⠀⢀⡉⢙⡇⠀⠀⠀⠀⠉⠙⠛⠛⢙⡿⠀⠀⠀⢘⠀⠀⠁⠀⡗⠀⠀⠀⠀⠸⠀⠀⠀⢹⡇⠀⠀⠀⠀⡿⠒⣲⣀⣶⠛⣿
⠀⠀⠀⠀⠀⢃⠀⠀⣀⢀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⣧⢚⠩⡿⢻⡀⣿
⠀⠀⠀⠀⠀⠸⡆⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠂⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠀⠀⠇⠁⠲⠄⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠷⠂⢀⡴⠇⠀⠀⠀⠀⠀⠘⠆⠀⠤⠾⠀⠀⠀⠀⠀⠇⠐⣠⡾⠁⠀⠀⠀⠀⠀⠣⠀⠀⠼⠀⠀⠀⠀⠀⠷⠤⠀⠀⠒⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢛⡁⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⡛⠀⠃⢸⡇⢸⣿⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⡆⣾⣿⡇⣿⣦⠙⢡⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⡃⣈⢁⣹⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣭⣥⡙⢿⡇⣿⣥⣾⣦⣽⡿⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⣿⣧⣾⣷⣿⣾⣿⣷⣾⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣤⣿⣿⣿⣿⡿⠃⠀⠀⠘⠿⠿⠟⢷⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠰⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⢶⣄⡒⢒⣰⡆⣰⣾⣿⠿⣷⣦⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡌⣿⣿⣿⣿⣧⢠⣤⣤⣶⠆⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢡⣿⣿⣿⣿⣿⠌⠻⠿⠋⢼⣿⡇⠀⠀⠀⠸⠇⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⢨⣭⠄⣉⣙⠻⠟⣡⣶⣷⣌⣉⣴⣶⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣾⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛







⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⢿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡃⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠼⠶⠆⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠀⠀⠀⠀⠠⢀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠛⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⢘⠁
⠀⠀⠀⠀⠀⠀⡼⢣⡀⠀⠀⠀⠀⡘⠀⠀⡻⣁⠀⠙⢧⡀
⠀⠀⠀⠀⠀⢠⠃⣁⠑⡀⠀⠀⠀⠀⠀⠀⡇⠈⢳⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⡞⠀⣿⠀⢘⡆⠀⠆⠀⠀⢸⠀⠀⠀⠙⣦⠀⠀⠻⢄⡀
⠀⠀⠀⠀⢰⣇⠤⠐⠊⠁⠀⢀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠳⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢬⡀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⣼⣷⡄
⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣉⣿⡟⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠽⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠛
⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠒⠀⠘⠓⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠹⠟⠀⡀⠀⠲⠆⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠪⣄⠀⠀⠢⠄⢳⠀⠈⢩⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⣆⡤⠒⠲⡞⠀⠀⠘⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡟⠀⠀⠀⡟⠀⣤⠀⢻⡶⢊⡥⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⠰⠄⢣⣤⠀⠤⠀⢀⣿⡆⠀⠁⠃⠂⢜⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠂⠁⢀⠞⠀⠀⠀⠀⠀⡾⣿⣧⠀⠀⠀⠀⠀⠀⡋⠉⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠉⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⡔⠀⣀⠼⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡜⠁⠐⣇⠘⠀⠀⠀⠐⠃⠀⠓⢤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠠⣄
⠀⠀⠀⠀⠘⢦⣄⣐⣀⣊⣀⠥⠲⢍⡀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠓⠤⢄⣀⣀⠀⠀⠀⢀⡠⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⢹⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⡰⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠠⠀⣀⠀⠀⠀⠀⠇⠀⠤⠼⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠠⠀⠂⠀⠀⠀⠀⠣⡀⡄⠀⠈⠉⠁⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡄⠀⢀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡌⠀⠀⠀⠀⢶⣧⡀⠀⠤⣔⢀⠀⠀⠀⠀⢦⣀⠀⡀⠀⡠⠋⣀⣠
⠀⡀⢀⠔⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠑⠿⣦⡀⠀⢠⣈⠣⡘⢿⣄⠀⠑⠷⣄⠀⠀⠈⡧⠷⢌⠻⢠⣞⣍⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⢙⢦⡀⠙⠉⠙⢆⠙⡗⢤⡀⠈⢳⡄⠀⡇⠀⠀⠳⢄⣀⣤⠴
⠀⠀⢃⠀⠀⠀⠀⢀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢⠀⢀⠄⠒⠒⠂⠄⠀⠉⠃⠀⢵⣬⠀⠀⠀⠳⡈⠢⠈⠛⠯⣙⣦⡇
⠀⠀⠀⠁⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠄⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⢤⣠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⣀⡽⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣶⣿⠀⣿⡇⢸⣷⢀⣀⠴⠋⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⢸⣿⢸⣿⣷⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣴⣦⢹⣿⠀⣿⡇⢸⣿⢸⣿⣿⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⢸⣿⣦⣿⣧⣾⣿⣼⣿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⣿⣿⣄⡛⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⣿⣿⣿⣿⣆⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⡴⠘⢿⣿⣿⣿⣼⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡋⠀⠀⠀⠉⠛⠻⠿⠿⠛⠋⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠛⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⡄⠸⠀⠀⢰⠙⣾⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⢀⠰⡉⢻⡄⢻⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢴⡏⠱⡅⢸⣇⠸⣇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣷⡈⢻⣄⠹⡄⣿⠀⣿⠀⢧⢀⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣧⠈⢷⡄⠻⡆⠘⡏⠀⡼⢢⠸⠞⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣷⡈⢿⡄⠻⡆⠘⣆⡇⢸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣷⠈⠋⠀⠀⠀⠈⠀⢸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠒⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠶⠾⠿⠛⠛⠛⠛⠿⠷⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠿⠟⠛⠛⠻⠿⠶⠶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣴⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢠⣾⠃⠀⠀⠀⠀⢠⣾⠋⠻⣷⡄⠀⠀⠀⠀⠈⠻⣶⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠘⣷⡀
⠀⠀⣾⠃⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠘⣷
⠀⣼⡏⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣼⡿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢹⣇
⢀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢀⣾⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿
⢸⡏⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸
⣾⡇⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣬⠿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⡴⠦⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠸⣇⠀⢈⡿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢿⡀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿
⣿⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠛⠉⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⣿
⢿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸
⢸⣇⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣸
⠈⣿⡀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠙⢿⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢀⣿
⠀⢹⣧⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⣼⡏
⠀⠀⢻⡄⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠂⠀⠀⠀⠀⠀⣠⣴⠶⠾⠟⠛⠿⣦⡀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⢰⡟
⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠈⢿⣦⣠⡿⠁⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠰⣿⣿⡶⠶⠾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣶⣶⣶⣶⠶⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠛⠛⠉⠁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣛⣩⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠋⣰⡾⢋⣥⣶⣾⣿⣿⣿⣦⣼⡟⠁⠉⢹⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⡟⢰⠋⣰⣿⣿⠟⠋⣉⣹⣿⣥⣶⣦⣤⣶⣿⠛⠁⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡀⠻⠀⠠⢏⡘⠁⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣴⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣿⣷⡀⢸⣆⢸⣷⡄⠀⠀⠀⠀⠉⠹⣿⣿⣛⣻⣿⣿⣿⣯⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣿⣇⠸⣿⣿⡄⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣿⣿⡿⣛⣋⣤⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⡇⢻⣿⣆⠙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡈⠻⣿⠄⠛⠀⠀⠐⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠀⣸⣿⣿⣿⠛⠳⣦⠈⠀⠀⠀⣀⣠⣀⣀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⣧
⠀⠀⠀⠀⠀⢀⣾⠇⣾⠟⣡⠀⠀⣿⡟⣿⣿⡀⢀⠴⠀⠀⠀⠀⠉⠋⣽⢿⡿⢷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣵⣿⠇
⠀⠀⠀⠀⣴⣿⣿⠸⣿⣼⡇⢸⡇⣿⡄⢻⣿⣇⠀⠀⣠⡄⠀⠀⠀⠀⠈⠂⠀⠺⢀⡿⠀⠀⡀⢈⠛⠛⠛⠑⢢⣿⣿⠿⠋
⠀⠀⠀⢼⣿⣇⣻⣿⠳⣽⣇⠸⣝⢆⠳⡄⢿⡟⠂⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡞⠹⣿⣶⠶⣦⣀⣾⡟⠁
⠀⠀⠀⣾⣿⣿⣿⡅⣀⠀⠙⣷⣜⢾⣧⣿⣿⣷⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠁⠘⣻⣿⣿⣸⣷⣦⡀
⠀⠀⠀⣹⣿⣿⣿⣼⣏⣀⣦⢸⣿⣷⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠲⠀⣄⠀⢸⠄⠀⠀⠀⢰⣿⣿⡝⡇⣿⣿⡷
⠀⠀⣼⡟⢻⣿⡋⢷⣌⣉⣡⡿⣿⣿⠟⢿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠐⠛⠀⠀⠀⣰⣿⣿⣿⢫⣿⣿⣿⣿
⠀⢠⣿⠃⢿⣿⣷⣼⣿⣿⡿⢟⣼⢟⠉⠈⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣬⡙⠷⣦⣄⡀⠀⠀⢀⣴⣿⣿⣿⡧⠞⣋⣡⣤⡜⣇
⠀⠘⣿⣷⣬⣽⣿⠿⠯⠿⢲⣿⠁⣯⣀⣴⣿⣿⡏⠹⣄⠀⠀⠀⠀⠘⣿⣿⣶⣤⡍⠉⣀⣴⣿⣿⣿⣿⣯⢞⣭⢥⣤⣤⡹⣾⡀
⠀⠀⣾⠟⡏⢹⣿⣶⣶⣶⡿⠻⣌⠻⠿⣿⣿⡿⠁⠀⠘⢦⡀⠀⠀⠀⠈⠙⠙⠉⢀⣤⡽⠿⠛⠋⢁⣠⣯⢞⡵⠛⢉⠉⠻⣾⣷
⠀⠀⢻⣦⡻⠶⣿⣿⣿⣿⣷⣾⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠉⣳⣤⡀⠀⠀⢀⣴⡏⢡⣤⣾⢟⣿⣿⢿⣿⡏⢠⣴⣿⣷⡤⠀⠙⢦⡀
⠀⠀⠀⠙⠻⠿⠟⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠏⣾⡿⣡⣿⣿⣿⣾⣿⢿⠟⠁⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⢸⣿⢡⣿⣿⣿⠿⣛⡵⠋⠀⠀⢦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣤⡙⢾⣿⣸⠿⣿⠶⠚⠁⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣶⠄⢰⡿⣄⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡛⠛⠛⠁⠀⠈⢷⣝⠳⣄⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠈⣷⠀⠈⠙⢿⡖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⡀⠰⣆⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠹⣆⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠉⢉⡿⠀⠙⠀⠀⠀⠀⠘⠛⢿⡀⠘⠃⢠⡟⠛⠁⠀⠀⠀⠈⠁⠀⠀⡾⠛⠋⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠋⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠛⢶⡞⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠻⣤⣤⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⢀⣀⣀⡘⠁⠀⠀⠀⠀⠀⡈⠁⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣻⡽⠞⠛⣆⠀⠀⠀⠀⠀⠀⠛⢠⡾⠋⠉⠉⠉⠛⠛⠲⠶⢶⠘⠃⠀⢸⡏⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⣼⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣹⣆⠀⠀⠀⠀⢸⢀⡶⣦⠶⢦⣀⣀⣠⣶⡶⣿⠀⠀⠀⠘⠳⠶⣿⠀⢛⣛⡛⠛⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⢞⣫⡥⠟⠀⠀⠀⠀⠘⣿⡇⢻⠀⠀⠈⠉⠁⢸⠇⢿⠀⠀⠀⠀⠀⠀⠻⠦⡤⣤⠤⠤⣤⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣋⡁⠀⣤⠤⠄⠐⠒⠀⣸⠀⢸⠆⠀⠀⠀⠀⢸⡀⢸⡀⠐⠒⠀⠰⠦⠤⠀⣤⣤⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠈⠛⠉⠀⠈⠁⠀⢀⣠⡤⠶⠞⠛⠋⢙⣧⣟⠀⠀⠀⠀⠀⣸⣧⣟⠉⠛⠛⠶⠦⣤⣀⡀⠀⠀⠀⠀⠉⠉⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠁⠀⠀⠀⣀⣴⠿⣿⠛⢷⣄⠀⠀⣰⠟⣻⠟⠷⣄⡀⠀⠀⠀⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣀⡴⠞⠉⠀⠀⠘⢧⡀⠉⠙⠋⠁⣠⡏⠀⠀⠈⠙⠶⣄⣀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⠞⠁⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⡀⢀⡀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠒⠛⠚⠛⠛⠓⠛⠚⠻⠛⠛⠓⠻⠒⠚⠓⠒⠓⠛⠛⠛⠛⠛⠓⠛⠒⠚⠿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠈⠈⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠂⠀⠠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⡄
⠀⢀⣉⠀⠀⠀⠀⠀⠀⢺⣷⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣉⠀⠀⠀⠀⡀⣋⣉⣩⣿⡇
⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢰⠀⠀⠀⠀⣴⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠠⠇⣤⠌⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠙⠶⣄⡄⠀⢀⣠⠞⠁⠀⠀⠀⠀⠳⡄⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣤⣆⡼⠃⠀⠀⠀⠙⠶⣄⠤⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⣾⡟⠀⠀⠀⠀⠀⢁⠤⡄⠀⠀⢠⣆⣁⣀⠀⠀⠀⠀⠀⠘⣏⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣤⠈⡄⢀⠃⡀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠿⣿⠉⠙⠁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⠈⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⢤⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣧⣀⣴⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⣀
⠀⠀⠀⠑⠒⠂⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠓⠒⠂⠀⠐⠒⠚

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠒⠒⠒⠠⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⡠⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⣀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠐⢧⡀⠀⠀⠀⠀⠀⡴⠒⠒⡄⢀⡀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢦⣀⠖⠉⠲⡇⠀⠀⡿⠁⠈⢳⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡰⠉⠲⣄⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⡇⠀⠀⢸⠚⠉⢢⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡰⠁⠀⠀⠈⠳⡀⠀⠀⠀⠀⠙⢦⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠙⢦⡀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⠀⠀⠀⠀⢠⠊⠙⢢⡈⢣⡀⠀⠀⠀⠀⠙⢧⡀⠀⠸⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⢳⣸⠈⠢⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⠀⠀⠀⢿⠀⠀⠈⠳⡀⠀⠀⠀⠀⠙⢦⡀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠙⢾⡁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⡜⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠙⢦⠜⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⣀⡤⠋⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠉⠁⠒⠒⠊⠉⠀⠀⠀⠀⣈⠆⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠉⠓⠦⠤⢀⣀⣀⡀⢀⣀⣀⡠⠤⠒⠋⠁⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠚⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧
⠘⣿⣿⣿⠃⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢹⣿⣿⣿⣿
⠀⢿⣿⡟⠀⠀⠰⠟⠋⠉⠉⠙⠛⠷⠆⠀⠀⠶⠟⠛⠉⠉⠉⠛⠷⠀⠀⢻⣿⣿⣿
⠀⣿⠃⠀⠀⠀⠀⣠⣀⡀⢀⣀⣄⠀⠀⠀⠀⣴⣶⣤⡀⢀⣀⣄⠀⠀⠀⠀⠙⢻⡏
⠀⣿⡀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠈⢿⣄⠙⢿⣟⠋⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢷⣄⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠻⣦⡀⣼⠟⠁
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢠⣤⣄⣠⣤⣠⣤⣤⣤⣹⣧⡀⠈⠻⣿⡀
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⣡⣿⣅⣀⠈⠙⠛⠿⠄⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠶⠶⢾⡏⠀⠉⠛⠻⠷⣶⡄⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⢻⣿⡶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⢿⡇⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣷⣿⣿⠟⠋⠀⠀⢀⡀⠀⠀⣾⣿⡆⠀⠀⠀⣠⡀⠀⠀⠀⠀⣹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⢿⣿⡆⠀⣿⣿⡇⠀⠀⣼⣿⣿⠀⣀⣢⣾⣿⣿⠟⠻⣿⣿⣆
⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⢀⣀⠀⠀⢸⣿⣿⠀⢻⣿⡇⠀⢠⣿⣿⣧⣴⣿⣿⣿⠟⠁⠀⠀⠹⣿⣿⣦
⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠸⣿⣷⡀⠈⣿⣿⡆⢸⣿⣿⠀⣼⡿⣫⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠹⣿⣷⡀⢻⣿⣷⣼⣿⣿⣴⣫⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣿⡿⣯⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡆⠀⠀⢀⣤⣶⣾⣶⣦⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⠟⠋⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠹⣿⣿⣷⡀⢠⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠟⠁⣸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣅⡀⠀⠙⠛⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⡄⢠⣿⠁⠀⠀⠀⠀⣿⠀⠿⠿⠛⢹⡉⣿⢻⣿⡆⠀⠀⢸⡋⠀⢿⡄⠀⣷⠀⠰⢦⣼⣿⣦⣁⠀⣿⣿⣿⣿⡿⢿⣇⠈⠉⢄⡙
⣠⣿⠃⠀⠀⠀⠀⠿⠿⠀⠀⣀⣾⣿⠻⡿⢻⣿⣿⣄⠀⣸⠀⠀⠈⢻⣦⣿⣴⡶⠛⣿⣧⠈⠙⠛⣿⣿⣿⣿⣍⣉⣿⣆⠄⠀⠑
⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣧⡌⠁⠁⢛⣭⣿⣿⣆⣾⠀⠀⣴⠋⠈⣉⡌⠛⢆⣿⣿⣿⠀⢨⣿⣿⣿⣿⠂⢺⡟⣻⣷⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⣦⡶⢠⡄⢶⣭⣭⠙⠻⣿⣦⣴⣷⠶⠛⠉⠀⠀⠈⣿⣿⢭⠁⠰⣿⣿⣿⡯⠀⣬⠙⣿⣿⣿⣷
⣿⣾⣷⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣷⢺⡷⣾⠛⠋⠀⠀⠀⠈⣿⡤⠶⠶⠶⠶⠶⠖⣿⣯⣶⣤⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿
⠇⣽⣍⢻⣿⣉⠁⣩⣿⡟⠸⣿⣿⠙⣷⠀⠀⠀⠀⠀⠄⠀⠀⠈⠛⠒⠒⣴⣶⡖⠋⠻⠛⠉⠻⠁⣽⣿⣿⠇⣟⡟⠀⢸⣿⡟⢥
⣶⣿⣷⣾⣿⣷⣇⣿⣿⣡⣷⣈⣤⣴⠟⠀⠀⠀⣆⠀⡖⠀⠀⠀⠀⣰⣾⣿⣿⡇⢠⣶⡄⠀⠠⠀⠈⣿⡇⠀⡿⠰⠄⢹⣏⠉
⣻⣿⢛⡿⡛⠟⠛⠛⠙⠛⠛⠋⠻⠧⠄⢴⣄⠀⠈⠀⢀⡄⡄⢀⣀⣽⣿⣿⣿⡇⠀⠈⠀⠀⠐⠀⠀⣿⡇⠀⠀⠀⠀⠈⢿⡄
⣻⢣⠟⢦⣹⠀⢸⠣⢀⠻⠠⠠⠆⠀⠈⢽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⢀⣀⣰⣄⣠⣿⣇⢀⣄⣴⣼⣤⣈⣿⣶
⣿⠿⠾⠛⢛⣽⣥⣼⣤⣦⣤⣤⣄⣴⡆⢹⣿⣧⣤⣤⣤⣤⣤⣤⣤⣶⣿⣽⣿⣧⡶⣠⡞⢉⠉⢹⣽⣿⣿⠋⣻⣿⣿⠉⠁⣭
⣷⣶⣤⣄⣨⠶⠉⡀⢹⠀⡉⣠⠀⡉⡁⠸⠿⠿⢿⣶⡦⢤⡀⠉⠉⠋⠉⣠⣴⣿⣾⣋⣾⣿⠂⣵⣿⣿⣿⣿⣩⣿⡿⠀⣤⣿⠷
⣾⣿⣿⠟⠛⠳⣤⡃⢠⠀⠃⠈⠜⠁⠀⠀⠀⠀⡠⠃⠀⢸⡇⠀⠀⢠⣶⢟⣿⣿⡿⠉⢛⢻⠄⣿⣿⣿⣇⣀⠈⡋⢰⣄⣤⣹⢷
⣾⣏⠀⠀⠀⠀⢀⣽⢷⣷⡟⠁⠀⠀⠤⠉⠚⠉⠁⠀⠀⠀⣇⠀⣠⣾⠛⠛⠛⣱⣇⣀⡀⠀⣠⣿⣿⣿⣿⣿⡿⢿⡿⠀⢰⣷⣦
⡇⠈⠓⢦⣠⠔⠋⠀⢸⣿⡇⠀⠀⠀⠠⠄⠠⠤⠴⠁⠀⠀⢻⣦⠀⠛⣤⣴⠞⣿⣧⣿⣅⠀⢛⣛⠛⣿⣿⠀⠀⠈⠁⠀⠛⢿⡌
⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣹⠿⠛⠁⠈⠻⣷⡼⣿⣯⣼⣯⣿⣼⣿⡟⠛⠛⠛⠛⠋⢉⣻⡝
⣷⣄⣀⠀⢹⠀⣀⣤⣾⣿⣇⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣷⣄⠀⢀⣤⠊⠈⠻⣿⢛⡛⢋⠉⠉⣽⣿⣤⣤⣤⠀⠠⢾⣿⡷
⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠋⢉⠟⢻⣿⡿⠻⡟⠻⡏⢩⣩⣽⠿⡇⠁⠀⠀⠀⣾⣦⣿⠀⠴⠴⣤⡶⠿⠿⠿⠿⣿⣶⣤⣬⣥⣿
⣿⣿⣿⣿⡟⠁⠄⠀⠀⠀⠀⠀⣰⢿⣿⣟⠀⠁⢀⣥⡾⠋⠁⠀⠀⠒⠀⠸⣦⣿⣟⠉⠀⠀⢀⢿⡗⠶⠦⠴⠶⠞⢻⣿⣿⢻⣿
⣿⠉⠉⠉⣷⡀⠀⠀⠀⠀⢠⣤⣄⣤⣭⣿⣤⡶⠛⠁⠀⠀⠀⠀⡀⠀⠀⠄⣿⣬⣁⣀⣀⣀⣹⣤⣽⣦⡀⠀⠀⣢⢞⣬⡿⢻⣶
⣿⡇⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠒⠀⠀⠀⠁⢠⡄⠀⢸⣷⢶⣄⢰⡀⣶⠺⠏⠹⠟⠃⢈⣡⣾⣷⣿⣿⣿
⣿⣛⣛⢋⣉⡀⠀⠀⠀⣀⣈⡄⠀⠀⠀⠀⡛⠁⢂⣴⣿⣿⣷⡄⠛⣾⣿⣷⣼⣿⠓⢿⣿⡄⣿⠉⠀⢀⣠⣾⣿⠿⠋⠉⢡⡾⠉
⣿⣿⠿⠿⢿⡿⢿⣿⠿⠿⠿⣛⣂⠀⣀⣼⣿⣿⣿⡿⠋⠉⢿⣿⣿⣿⡁⢸⣿⡿⣷⣶⣍⣻⣧⢀⣴⣿⠟⠋⠀⣀⡄⠀⢸⡇⢸
⠐⠀⠄⠐⠀⠘⠀⠀⠁⡀⠔⢙⡟⠛⠟⠉⠹⣯⡀⠈⠀⠀⣀⡼⠃⠀⣀⠸⢿⣿⡿⠛⠉⠛⢿⣿⣿⣇⣠⣠⣴⣿⣷⣄⣸⡆
⣶⣯⣤⣄⣀⣀⣡⡀⢠⠇⠀⠀⡇⠀⠀⠀⠃⣿⣟⣷⡞⠛⠋⢀⠀⠤⠀⠀⠀⠈⠁⠀⠀⠀⠈⣿⣯⣍⣽⣏⣻⣟⣟⣟⢻⣿⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣷⣶⣷⡄⠺⣿⣿⣿⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢈⣿⡛⣿⣿⣿⣿⣻⣽⡟⠻⣿⣿⣿⠃⠀⣿⣏⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣾⣿⣷⣿⣿⣀⣿⣿⣿⣿
⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⠋⠙⣿⠀⠀⣿⡿⠿⠇⠀⣧⣤⣤⣬⣭⡍⠙⠛⠛⠛⠓⠚⠻⣿⣿⣿⠿⣿⠾⣷⣿⣿⣿⣿⣿
⣿⡇⣿⡏⠀⠀⠈⠉⠉⠉⠛⢙⡝⠛⣿⠀⠀⣿⣦⣴⣆⣀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣾⣠⣿⠉⠁⠀⠈⠷⠃⠈⠶⠉⣿⣧
⢻⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠿⠀⣿⠀⢈⣿⣿⠟⢿⣭⣤⣦⣨⣶⠄⠀⠀⠀⠀⠀⠀⠈⢹⡏⢀⣴⡄⠀⣠⣦⠀⠀⢠⣼⣿
⢙⣿⠀⣴⠇⠀⠀⠀⠰⠿⣆⠀⠀⠀⣿⠀⠈⣿⣿⣷⣿⢷⣆⣸⠿⣧⣀⠀⠀⠀⡀⢀⠀⠰⣾⣷⣿⣿⣿⣼⣿⣿⣧⣴⣿⣿⣿
⣿⡟⡀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡟⣿⠟⢿⣧⣿⣿⣾⣿⣿⣆⣠⣷⣴⣆⣀⢀⣠⡆⠀⣠⣿⡏⠉⢉⠟⢻⠉⡟⢻
⡿⠁⠿⠂⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢸⣿⡟⡷⣿⠀⢺⣿⣿⣿⣿⣿⣿⠋⠁⢉⡯⠉⢉⣿⣿⣇⣼⣿⣿⣿⣠⠋⠀⠈⢦⣷⣿
⣷⣿⣾⣿⣷⣿⣿⣿⣷⣷⣶⣶⣷⣶⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿
⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣯⣿⣹⣿⣿⣿⣿⣿⣿⣻⠿⣿⣿⡿⢿⣿⢏⣿⣯⡅⣿⡿⢿⡿⠿⠿⢿⣿⣷⣿⣿⠇⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢰⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠂⡞⠐⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠈
⡄⠀⠀⠀⠈
⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⢀⡶⠒⠀⠀⠀⠰⠐⠀⠀⢐⠆⠀⠀⠀⠠⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣀⣀⣀⡀⠀⢀⡴⠟⠁⠀⠀⢀⣀⣤⠤⠤⠤⠈⠓⠀⠀⠈⣱⠤⠀⢀⠄⠀⠀⠀⠀⣏⣿⠄⠀⠀⠂⠠⠦⠤⠤⠠⠴⡴⠂⡀
⠀⠀⠀⠘⠁⣴⠏⢻⠉⠀⠀⣄⢀⡀⠘⡆⠤⠤⠄⢠⣴⣎⠀⠉⠁⠀⠊⢀⠀⠀⠀⠀⠘⠘⠊⠛⠃⣴⣀⠄⠀⣀⣀⡀⠀⠠⠇
⠄⠲⡆⡡⠊⠙⠉⢉⠁⠉⢀⠟⠀⠀⠀⠇⠀⢀⡄⣸⡿⠛⠀⠀⠠⠄⠀⠀⠦⠠⠄⠀⡄⢀⠀⢠⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠂⠔⣵⣤⣤⡤⠀⠤⠤⠀⠁⡀⠀⠈⠀⠀⠹⠊⠁⠀⠀⠀⠀⠀⠰⢁⡀⠀⠀⠀⠀⢠⠂⡰⢃⠎⠁⠀⠑⠤⠘⢦⡀⠀⠀⠀⠦⣀⠀⢠
⠁⠘⠶⠮⣍⡴⣶⡒⠲⣤⠄⠀⠑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢀⡠⠀⡤⢈⠲⣔⡴⠁⡼⠁⠃⠀⣤⣄⣀⠀⠀⡭⠇⠀⠀⠀⠀⠈⢂⠄
⠀⠀⠂⠀⠠⠄⠀⠈⠉⠛⠛⠲⠦⢤⣤⣤⣀⢀⠀⠀⠀⠀⠀⢠⣥⣀⡧⢿⣀⣃⠘⢀⡞⠁⢀⠀⠀⢿⣿⡟⠫⡂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠖⠀⠀⠀⡀⠁⠀⠀⠀⠌⠀⡞⢠⠎⢀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⠠⠤⠖⠒⠺⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⣡⠤⣾⣴⠄⠀⠉⠙⢆⠀⠈⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣦⡿⠛⠁⠀⣄⡎⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡶⠚⠈⠋⠁⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⢶⠿⡿⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣭⣵⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢴⠄
⠀⠀⠀⠀⠀⠻⠿⣿⠿⠟⠋⠉⠀⠀⠀⠙⠢⣉⣳⡄⢀⢀⣀⠤⠤⠔⠒⠊⠁

⠰⢆⠀⠐⣄⠀⢠⠊⠀⢠⠀⠀⠆⠀⠐⠁⠙⠀⠀⠸⠀⠠⠀⠀⠀⠸⠀⠀⢆⠀⠁⣅⠀⣠⠊⠁⢠⠀⠀⡄⠀⠘⠃⠙⠀⠀⠸⠉⠠⠃⠀⠀⢠
⠀⠠⠀⠀⠈⠉⠉⠀⠀⠀⠘⡄⠀⢀⣢⣬⠃⣀⠂⠀⠀⣀⣴⣇⠀⠢⠀⠀⠠⠂⠀⠈⠉⠁⠀⠀⠀⠘⡅⠰⢀⢁⣬⠃⣠⠂⠀⠀⣀⣰⣇⠀⠠
⠀⠀⠀⠀⠀⣰⠦⢶⡄⠈⢌⣁⠄⠀⠀⠜⠀⠃⠀⠀⠘⠉⢉⠽⡄⠀⢐⠐⡲⣒⡂⠀⣴⠶⢶⡄⠐⠈⠀⠀⠀⠰⠌⠄⠃⠶⠀⠸⠉⢉⠸
⣊⣷⣘⡄⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠢⣀⡠⠂⠀⢠⡀⠈⢐⡇⠀⢠⣈⣳⣘⣤⠀⠰⡀⠰⠀⠀⠀⡇⠀⠀⠀⠡⣀⡀⠀⠀⢘⡀⢈⢐
⠁⠀⠑⠃⠀⡄⡀⠀⠀⠀⠒⠑⠂⠀⠀⠀⠸⠅⠀⠀⠀⠉⠀⠈⠀⢀⠘⠀⠁⠋⠊⠀⠄⡀⠀⠀⠀⠒⠐⠂⠀⠀⠀⡸⠀⠀⠀⠀⠁⠀⠉⠀⡄
⣄⡄⠀⠠⠤⠬⠤⢄⣀⠠⡏⠉⠉⡵⠀⠈⠁⠉⠀⡒⠒⠐⠀⠀⠂⡞⠐⣄⡄⠀⠠⠤⠥⠤⢄⣀⠠⠏⠉⠉⠷⠀⠈⠁⠉⠀⠆⢲⢰⠀⠠⠀⠎
⠞⠁⠀⠐⢄⣀⣠⠞⠈⠀⠠⣞⠗⠀⠀⠆⠰⡆⠀⣇⣽⣬⠀⠀⢃⠤⢰⠟⠁⠀⠐⢄⣀⣀⠖⠀⠀⠣⣎⠇⠀⠰⠄⠒⠆⠀⣥⣽⣸⠀⠀⡄⢼
⠀⡀⠀⢀⠈⠉⠁⠀⠀⠀⠐⠀⠐⠀⠀⠦⠴⠃⠀⠠⠿⠝⠀⠀⠘⠈⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐⠀⠐⠀⠀⠦⠴⠂⠀⠢⠿⠏⠀⠀⠐⠉
⠀⠂⠀⠸⠀⢀⢁⣈⢡⠀⢰⣧⠀⣶⣷⠦⠀⡤⣀⠁⠀⠀⢐⠒⣒⣲⠀⠀⡆⠀⢒⠀⣐⢁⣀⢡⠀⠠⣧⠀⣾⣿⠦⠀⡤⣀⠁⠀⠀⢐⠒⢂⣒
⠀⠐⢶⠀⠀⠈⢠⢤⠊⠀⠀⠸⠀⠉⠋⠛⢏⠝⠁⠀⣀⡀⠨⠒⠤⠀⠀⠀⠀⢶⠂⠀⠐⠠⠤⠈⠀⠀⠀⠀⠉⠛⠙⢯⠝⠁⠀⣀⡀⠈⠒⠤
⠀⠠⠍⠤⠀⠀⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⡸⡀⠀⠀⡁⡇⠀⠌⠀⡄⠀⠀⠠⠌⠤⠀⠀⠉⠈⠁⠀⡂⠀⢱⠀⠀⠀⡘⠀⠀⠀⠁⡅⠀⠎⠀⡄
⠦⡀⠀⢠⠐⠂⠢⡀⠀⠀⡄⢸⠀⠀⢀⠀⣁⠉⠀⢀⠀⠠⠀⠀⠀⠀⠎⠦⡀⠀⢀⠐⠂⠢⡀⠀⠀⡁⠁⢘⠀⢀⠀⢁⠁⠀⢀⠀⠀⠀⠀⠀⠠
⠀⠁⠀⠄⡄⡤⢂⠾⠄⡀⠡⠤⠤⠀⠸⠀⢨⠀⠀⠐⠂⠄⠀⠀⠀⠏⠀⠀⡁⠈⠄⡄⡠⠂⠾⢢⠀⠯⠤⠼⠀⢈⠀⠘⠀⠀⡒⠂⠄⠆⠀⠀⠈
⢈⡀⠀⢀⠁⠉⠈⡸⠚⠀⠤⠤⡀⠀⠀⠄⢼⠀⠀⢠⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⢂⠃⠉⠀⠰⠚⢀⠤⠤⠀⠀⢸⠄⢤⠀⠀⢠⠄⠂⡇⠀⠀⠈
⠈⠑⠀⠀⠓⠒⠚⠀⠀⠘⢰⠂⠃⠀⠔⢀⡤⠀⠠⠀⠀⠀⢠⡄⠀⡌⠉⠁⠑⡄⠀⠑⠒⠚⠀⠀⠘⢰⠂⢃⠀⠴⠀⡤⠀⠠⠈⠀⠀⢠⡄⠀⠈
⠀⠈⠀⡀⠀⢀⣄⣀⠀⠠⠠⠅⠀⠘⠊⢳⣀⣖⡀⠀⠠⠧⠭⢵⡀⠀⢀⠀⠈⠀⠀⠀⢀⣀⣀⠀⢀⠠⠏⡐⠘⢈⢻⡀⣞⡀⠀⢠⠦⠭⢧⠀⠈
⡐⣮⠳⡂⠀⠘⠀⠐⠂⠀⠑⡖⠁⠀⠀⠅⠀⠀⡀⠀⠀⠀⠐⠈⡇⠀⢐⠐⣎⠷⣂⠀⢛⠀⢐⠃⠀⠑⡒⠁⠀⠈⠄⠀⠀⠀⠀⠸⠀⠐⠈
⠂⠛⠒⡧⠀⠀⠒⠀⠀⠀⣀⢃⡀⠀⠀⠈⢲⠎⠀⠀⠘⠲⠒⠖⠁⠀⢸⠒⠛⠒⠿⠀⠀⠒⠊⠀⠀⣀⢃⡀⠀⠀⠈⢲⠊⠀⠀⠘⠒⠒⠖
⠀⠀⠀⠀⠀⡡⡂⠀⠀⢀⣤⡐⡲⢄⠀⠠⠎⠲⠀⡀⣠⢐⠀⠀⡀⣹⠀⠀⠀⠀⠀⠀⡅⡀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠎⠲⠀⡀⣠⢐⠀⠀⡀⣁
⣿⠆⠀⢀⠀⠀⠀⣸⡰⠈⠃⡠⣄⠋⢀⣀⣀⡀⠀⡆⣖⢐⠀⠀⡄⢁⢀⣯⠗⠀⢀⠀⠀⠀⣱⡀⠈⡀⡠⡄⡀⢀⣀⣀⡀⠀⠂⣺⢰⠀⠈⠀⣀
⠁⠀⠀⠀⠱⠶⠞⠁⠀⠀⢈⣉⣠⠀⠀⣅⣈⡇⠀⠛⣿⢿⠀⠀⢰⢭⡼⠁⠀⠀⠀⠱⠶⠚⠁⠀⠀⢉⢉⣁⠁⠀⣅⣀⡀⠀⡟⣿⣾⠀⠀⢃⣭
⠀⣂⣀⢰⠀⠀⠐⠂⠀⠀⢀⡄⠀⣠⣤⡀⠀⢀⠠⠈⠉⠁⢀⠀⠀⢀⠀⠀⣀⣀⣐⠀⢀⠀⠒⠄⠀⠀⡄⠀⣠⣤⡁⠀⢀⠀⠈⠉⠁⢀⠀⠀⢀
⠀⠠⣀⠈⠀⠸⡀⢀⡸⠀⠘⢻⠀⠻⠿⢭⡰⢫⠂⠀⠀⠀⠰⣤⠀⠘⠀⠀⠣⣀⠔⠀⠲⡀⠀⡸⠀⠈⠛⠀⠻⠿⢍⡰⠫⠃⠀⠀⠀⠰⢤⠀⠒
⠀⠀⠽⡀⠀⠀⠼⠹⠄⠀⡐⠈⠄⠀⠀⠀⢱⠁⠀⠀⠎⡆⠈⠠⠉⠀⠀⠀⠀⠸⡀⠀⠀⠴⠱⠄⠀⡄⠀⢂⠀⠀⠀⢹⠀⠀⠀⡎⠆⠀⡠⠉⠈
⡀⠈⠀⠉⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠠⠎⠶⠀⠀⠂⠃⠀⠁⠀⠁⡀⡀⠈⠀⠈⡀⠀⡀⠀⠀⠀⠂⠀⠸⠀⠀⠀⠎⠶⠀⠀⠇⠃⠀⠁
⠠⠆⠀⡚⡀⠀⡈⣢⠀⠀⡇⠘⢀⠀⢸⠉⢹⠀⠀⢰⠀⠈⠀⠀⠀⡮⠠⠬⠆⠠⡒⠀⠀⡈⣢⠀⠀⡆⠀⢰⠀⠸⠉⠹⠀⠀⣈⠀⠈⡄⠀⠀⠴
⢠⣀⣀⣁⣇⣗⣀⣉⣈⣂⣀⣀⣀⣀⣈⣀⣘⣀⣀⣀⣀⣁⣀⣀⣀⣡⡀⣠⣂⣀⣁⣇⣳⣀⣉⣘⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣁⣀⣀⣃⣀⣀⣰
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠈⠀⢀⣰⢶⣶⢶⣄⠀⠀⠀⠀⠀⠘⠂⣀⡴⣶⡶⣦⡀⠛⠀⠀⠀⠀⠀⡉⣠⢲⣿⠢⡄⢁⠀⠀⠀⠀⠀⢈⠁⡤⣲⣖⠤⠈⡁
⠀⠀⠀⣭⣵⣿⡿⠈⠉⡀⠀⠀⠀⠀⠃⢠⣽⣶⣿⡇⠉⠁⠂⠃⠀⠀⠀⠀⢠⡿⠟⠋⠙⣿⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠁⠙⠇
⠀⠀⠀⣿⣏⠈⠁⠀⠀⡱⠀⠀⠀⠀⠀⠘⣿⡇⠙⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⢀⣿⡄
⠀⠀⠀⠘⠋⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⣙⣋⠁⠀⠉⣹⡃⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿

⣞⣏⣉⣉⣉⣉⣉⣉⢉⡉⢹⣶⣦
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣛⣋⣉⣽⣟⣛⡷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡋⠉⣿⠀⠉⣿⠛⢷⣝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⢸⡇⣼⠇⣴⠟⠉⢷⡝⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠒⠒⠒⠒⠛⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠚⠓⠛⠚⣳⡶⠞⠋⢿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠯⠶⠶⠶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡴⢿⣿⡳⠶⣤⣀⣼⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⠞⠁⠀⠻⡿⣦⡀⣹⢟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡔⠁⠀⠀⠀⠀⠙⣮⡿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⣿⣿⣿⡿⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⢈⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣼⠀⠀⣷⢠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠀⠀⠀⠀⢸⣀⣀⡏⠀⠀⠀⠈⠉⠉⠛⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣦⣤⣄⣀⣀⣀⣀⠈⠉⠉⠀⣀⣀⣀⣀⣠⣤⣴⣶⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⡄
⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠻⠻⢿⣿⣿⣿⣷⣆⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣰⣾⣿⣿⣿⡿⠟⠛⠿⢿⣶⣤⡄
⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⢿⣿⢝⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡽⡻⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢓⣢
⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢠⣀⣴⣶⣿⣿⣿⣽⣓⠶⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠐⣶⣿⡫⠁⠀⠀⠀⡉⠍⠛⢾⣍⢳⡄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⡀⠀⠀⣴⢋⣠⣤⠶⠾⢾⣷⣿⣿⣄⡀⡀
⠀⠀⠀⠈⠁⢸⣯⡎⠀⠀⣴⣿⣿⣿⣷⣄⠀⠈⠣⠙⠦⡁⡆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠠⠁⣠⣜⣤⢄⣼⡾⠋⠀⢀⣤⣤⣄⡀⠀⠺⢝⢿⣿⣄
⠀⠀⠀⠀⠨⠩⢬⣧⣀⠸⣗⣻⣿⣿⣿⡿⠀⣠⣿⣦⣀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢃⣼⣿⠁⠀⢰⣟⣿⣿⣿⣿⡀⠀⠈⠣⢻⣿⡧
⠀⠀⠀⠀⠀⠀⠨⠃⠬⢙⣋⢉⡘⠛⠛⠓⡛⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣿⡿⠿⠿⢛⣤⡀⠘⢿⡽⣿⣿⡿⠃⣠⢤⣿⣿⡟
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠙⠉⠉⠛⠝⡿⠺⢿⡟⢿⠮⡖⠏⠀⠠⠓
⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠁

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣄⣀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣄⠀⠀⠀⠀⠀⠈⣽⡂⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡠⢴⠀⣀⡀⠀⢸⠛⢷⣄⠀⠓⡀⣠⡙⢆⠈⠣⠀⢄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠲⠁⠁⠀⠈⠀⠰⢠⠄⠀⣧⡀⠀⣦⠀⠈⢀⡤⠂⠀⡄⠀⠱⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠼⠀⠀⠀⣿⠀⠀⠀⢛⠶⠀⠑⣿⣄⠈⠂⠀⠊⠠⡪⠖⠠⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠒⠀⠀⠀⡯⠀⠀⠀⣈⡱⠄⠀⠸⡝⢦⣐⣠⡴⢺⡄⠄⠨⠀⠅⠀⢡
⠀⠀⠀⠀⢀⣀⠀⠀⠀⡠⠃⠀⠀⠀⡀⠠⡄⢀⣣⠀⠀⠀⠀⢳⡙⠻⢃⠕⠉⡇⠀⡀⠀⠁⠀⠨
⠀⠀⠀⠀⠇⠙⠀⡀⣀⢀⡴⢂⠛⡑⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⢃⠀⡊⢐⠁⡀⠀⡀
⠀⠀⠀⠀⠜⣄⢀⠀⡌⢀⠀⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⡀⢀⠄⡨⠀⣇⠐⡄⡀
⠀⠀⠀⠈⠀⠈⢢⠃⣡⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡄⠈⠀⠡⡪⢊⡴⢹⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢸⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠠⡠⢂⢄⢄⡤⠞⠁⢀⢸⡀
⠀⠀⠀⠀⠀⠀⢨⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢊⠔⡐⠁⡔⠁⠇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠠⠘⣸⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢌⠤⠆⢧⠀⠸⡈⠎⢌⢄⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢪⠠⠄⠀⠄⠀⢠⠀⡠⡨⣢⡱⡄
⠀⠀⠀⠀⠀⠀⠁⠂⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠒⠁⠛⠈⢇⠈⡌⠨⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠄⠤⢤⣤⠀⣀⠀⠘⡆⠰⠂⢊⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⢛⢋⣩⣌⣤⣅⣤⣤⣤⣀⡠⣄⡀⢤⣌⣁⢰⠾⡶⠶⠒⠚⠕⠕⠙⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠸⣵⣾⡾⠿⡛⠛⠋⠉⠉⠉⠉⠌⠂⠠⠀⠀⢈⡩⠟⡡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠈⠉⢉⣋⣒⣓⣓⡓⠒⣚⡓⠓⢋⠉⠩⠀⠒⠈