Strike Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣀⣠⠀⠀⠀⠀⣤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣭⣭⡄⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠝⠦⠀⣠⡶⡀⢠⠆⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⢠⣥⣀⣠⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣡⡶⠛⠘⣇⠀⢀⣀⣰⢸⠀⠀⣄⠀⠄⣀⠰⠶⠶⣪⠖⠐⠀⠀⠀⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠈⠴⣫⠵⠒⠛⢀⢱⠃⠀⠀⢘⡄⠀⢀⠏⠀⠀⢀⡀⠚⠯⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⠈⡁⠀⠀⠀⠀⠸⠀⠀⠀⣼⠁⠰⠷⠀⣀⡀⠉⠀⠀⠀⠈⠃⠀⠀⢀⣤⠶⠒⠢
⠀⠀⠀⠀⠀⠊⠀⠀⠁⠂⡀⠀⠘⣆⠀⠀⢳⣄⠀⠀⠀⢀⡛⠷⠶⢶⡧⣀⠛⢳⢣⡍⠁⠀⠀⠀⠀⠀⠀⣀⠴⣭⡀
⠀⠀⠀⠀⠸⣄⠀⠀⢠⡞⣱⠖⠈⠈⠓⠂⠀⠹⣦⡀⢀⢘⠛⠛⠛⢿⠞⠁⢠⡟⠀⠀⢀⣠⠶⠖⠒⠉⠁⢈⣻⢔⢠⡄⠀⠠⠂
⠀⠀⠀⠀⠀⠈⠙⠂⢿⣸⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠃⠀⡀⠀⠈⡀⢠⣿⠁⡠⠰⠋⠀⠀⢀⣠⣴⠞⠋⠀⠹⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⣷⣄⣧⢀⣇⢀⣿⣯⣾⠁⢀⣠⣴⣾⠟⠋⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⡠⣄⡀⠀⠀⠀⠟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⠟⠋⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡛⢿⣶⣤⣈⣿⢞⡯⠑⠚⠓⠶⢯⣛⢿⣯⣶⣶⠾⠛⠁⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣀⣀⣀⣙⣻⣿⢯⣿⣀⠀⠀⠀⠀⠀⠙⣎⢿⣿⣶⣶⣶⣶⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡜⠸⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡕⣌⡻⣿⣿⣿⣿⡿⠛⠐⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢸⣧⣴⠏⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠈⣿⣏⣀⡀⠀⠀⢀⣴⣿⣷⡶⠀⠀⠀⢰⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠸⠷⣦⣄⡀⠀⠀⠀⠈⠁⣿⠁⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⣶⣤⣠⡇⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠸⣿⠟⠁⠀⠀⠀⠀⣴⡾⠁⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠿⣷⠆
⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⡿⢃⡀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⣠⠆⠀⠀⢿⣿⣟⣡⡄
⠀⠀⠀⠀⠀⠸⠿⣿⡟⠀⠀⠀⣼⣿⣷⡾⠀⠀⠀⣼⣿⣿⣶⠆⠀⠀⠀⣠⣾⣃⡀⠀⠀⠘⢉⣿⠟
⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠈⠁⣿⣿⠃⠀⠀⠀⠚⠛⣿⡿⠁⠀⠀⢀⣾⣧⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⢈⣼⡟
⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠂
⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⣠⣾⣤⠀⠀⠀⠿⢻⣿⠟⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠾⢿⣿⠇
⠀⠀⠀⠀⠰⠿⣿⣥⡤⠀⠀⠀⠚⢹⣿⠃⠀⠀⠀⠀⠾⣿⡶⠀⠀⠀⠀⠈⢁⡟⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢸⡏⠛⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢼⡆⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⡿⠁⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠞


⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣥⠂⠀⠀⠀⠀⠀⠀⢚⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣀
⠀⠀⠀⠀⢠⣿⣏⠀⠀⠀⠀⠀⠈⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⡖⠀⠀⠀⠀⠀⠀⠋⣿⠃
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠂⠀⢠⡄⠀⠀⠀⠠⣤⠀⢸⠄⢀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⢸⡇⠀⠀⣤⣤⠀⠀⠀⣾⠁⢠⣤⡄⠀⠀⣼⠃⢀⣤⣄⠀⢻⡀⣶⡀⠀⠀⢀⣤⣄⠀⠀⠀⢰⡆⠀⠀⢠⣤⡄⠀⣿
⠀⠀⠀⠈⠻⣦⣀⣛⣛⡀⠀⠀⠙⣦⣘⣛⣃⠀⠀⠘⢷⣈⣛⣋⣰⠟⠀⠈⢷⣤⣀⣘⣛⣋⣀⣀⣤⡟⠀⠀⠀⣘⣛⣃⣴⠏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣷⡀⠀⠈⣿⣿⣿⡏⠀⠀⠀⠀⠈⠹⣿⣿⣿⡟⠉⠁⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣏⡿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢺⡉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⠿⣿⡿⠇⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠘⠷⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⡆⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠸⠇⠸⠇⠀⠀⠀⠀⠿⠀⠿⠀⠀⠀⠀⠿⠃⠻⠇⠀⠀⠀⠀⠀⠀⠿⠃⠻⠇⠀⠀⠀⠀⠀⠀⠿⠀⠿


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣽⣯⣬⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠀⠀⠀⠀⠀⢿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡂⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢀⡅⠐⣼⠀⡀⠤⠀⠀⠘⣷⠇
⠀⠀⠀⠀⢀⠀⢠⣇⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣻⠀⠀⠀⠀⡀⣄⢸⢺⠀⠀⠀⠀⠀⠀⢸⡇⠤⢿⠀⣧⠈⡆⡀⡰⡞
⠀⠀⠀⡼⣤⡣⡼⣿⢷⠀⡀⣄⠀⠀⠀⠀⢠⣿⡎⣻⠀⠀⠀⣼⣶⢪⠊⡛⠀⠀⠀⠀⢀⠀⢊⠁⣿⠁⠀⣿⣃⣀⠗⡡
⠀⠀⠘⣿⠀⡙⢿⡻⡦⢤⢿⠈⡁⠀⠀⠀⠀⡻⣯⡩⠇⠀⠀⣿⠢⢤⡔⠁⠀⠀⠀⠀⣼⠤⣾⣄⣸⠀⡀⣷⡼⣯⠂⢻⡄
⠀⠀⠀⢸⣇⢣⠈⣴⣿⡉⢿⠀⡇⠀⠀⠀⠄⣗⣿⣷⠀⠀⠀⡿⠃⠢⡅⠀⠀⠀⠀⠀⠈⣷⠦⢮⠀⠈⢷⣈⣄⣸⣏⣉⡃
⠀⠀⠀⠘⣸⠴⠣⠃⡀⠃⠈⠙⠃⠀⠀⠀⡁⡷⠀⠋⠀⠀⠀⢧⡤⢤⡇⠀⠀⠀⠀⠀⢁⣭⡤⣈⠃⠀⢹⣏⣻⣈⡏⠈⡏
⠀⠀⠀⠀⠘⠄⠈⣤⡳⠀⠀⠀⠀⠀⠀⠀⠉⠃⢸⡀⢂⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⢦⢰⠀⠀⠀⠀⣏⡀⠸⡇⠀⡇
⠀⠀⠀⠀⡀⢰⣴⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⢘⡇⡰⠀⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⡄⡎⠀⠀⠀⢠⢸⢡⢰⡆⠘
⠀⠀⠀⠀⡕⣹⣻⣻⡿⠂⠀⠀⠀⠀⠀⠀⠀⢇⢸⣴⡁⠀⠀⠈⠀⣇⠀⠀⠀⠀⠀⠀⠀⡃⢧⡀⠀⠀⢸⠸⡼⠀⠇⠠⠄
⠀⠀⠀⠀⠙⠃⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣄⠀⠀⠀⠀⢰⡀⠀⠀⢈⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⣤⣀⣠⣿⣧⣄⣀⣠⣤⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡶⠶⠶⠶⠀⢠⠂⠀⠀⠀⠃
⠀⠀⠴⣶⣶⣿⡻⣿⠟⠋⠛⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠙⣿⡟⠀⢠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠻⣿⠛⠁⠀⠀⠀⠀⣠⠀⢠⠟⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠸⣿⠀⢠⣿⣿⣿⣵⣿⣿⣿⣻⣿⣿⣿⣶⣶⠶⠿⠷⣄⠀⢀⠀⠀⢧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⢹⣦⣿⣿⣿⣯⣾⣿⡷⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠃⢸⣦⠀⣴⡏⢰⣇⠀⠃⠀⣰⠇⠀⠀⣸⡆
⠀⠀⠀⠀⢠⣿⡿⠛⠟⠉⠹⠟⠁⠀⠀⢹⣿⡿⢿⣿⠀⠀⠀⠀⠀⢰⣿⣧⣠⣿⣇⣠⣿⣷⣀⣴⣯⠀⠀⢸⡇
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠀⠀⠈⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢻⣿⠏⠉⢹⣿⣿⣿⠋⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢫⣴⣾⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⣛⡉⠀⠀⣸⣿⣿⡇⠀⠀⠀⢠⣿⠟⠛⠻⢿⣿⣿⣿⣿⣷⠈⠛⠿⠊⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠏⠀⠐⣿⣿⣿⣿⣤⣤⣤⣾⣿⡀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠀⠀⠀⠀⠀⠀⣹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠿⠟⠿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢴⡆⣤⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣦⣤⣤⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠟⠉⠻⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⠛⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠋⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠁⠀⠀⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠀⠀⠀⠀⢀⣤⣾⣿⡟⠀⠀⠈⣿⣿
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠋⠀⠀⣿⣿
⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿
⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣾⣿⣿⠏⢿⣿⣿⣶⣤⣄⣤⡤⠄⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⠇
⣿⣿⣿⠀⠀⠻⢿⣿⣿⣯⣀⠀⠀⣀⣿⣿⣯⠀⢤⣄⣀⣀⣀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⡀⠀⠀⢠⣿⡟
⣿⣿⣿⡄⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣦⣄⣈⣉⣙⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⠀⣸⣿⣿⣿⠿⢉⣠⣴⣿⠟
⢸⣿⣿⡇⠀⣠⡖⠀⠀⠀⠀⣤⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⣭⣉⣉⣤⣾⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠸⣿⣿⣧⠀⣿⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈⠉⠉⢹⣿⡏⠀⠀⠀⠉⠛⠛⠿⠿⠛⠋⠁
⠀⢻⣿⣿⣷⣿⣄⣀⢀⣀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠲⠒⠀⠐⠀⠀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠂⠀⠀⠁⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡉⠄⠀⠀⠀⡀⢀⣢⡀⠀⢤⣄⣀⠂⠀⠲⠤⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠋⠀⠀⡠⠖⠋⡟⠉⠉⠻⣆⠘⡗⡩⣽⢶⠉⡷⡖⠓⠲⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⡠⠪⠀⠀⠀⠘⠂⠀⠀⠘⡆⠃⢳⡈⠢⡉⠈⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⣼⢆⠊⢳⠀⠁⢄⢣⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠉⠡⢈⠳⡀⠅⡠⣬⢡⡱⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⡿⠄⠀⠀⠀⠀⠀⠀⡜⢁⡆⠀⠀⠊⠀⡈⠐⠨⢈⠈⠙⢦⡀
⠀⠀⠀⠀⠀⢠⠁⠈⠀⢸⠀⠀⠠⠐⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠖⠉⠡⣤⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠲⡆⠀⠀⠀⢸⠶⠁⠀⠀⠀⠀⠀⠀⠐⠀⠴⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢇⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⢀⢠⡶⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⢸⣆⠀⠀⠑⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡼⢉⣀
⠀⠀⠀⠀⡆⠘⡦⠀⠀⠀⡈⠘⢆⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⠂⠀⠀⠀⠘⢢⡀
⠀⡀⠁⠀⢀⠀⣀⣐⠚⠠⢨⡤⠌⢣⡁⢸⡄⠀⠀⠀⠀⠀⠀⣼⠈⠁⠐⠀⠀⠀⠀⡚⢀⡄
⡀⡠⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⣉⡠⠀⠨⠀⠐⠁⡤⠆
⣄⠈⠀⠀⠀⢀⣠⠤⠒⠓⠉⠃⠁⠀⠀⠸⠷⣗⢄⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠂
⠈⠀⠀⢀⠊⠁⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠉⠲⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⢀⡄⠠⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢹⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⠠⢀⠀⠀⣸⠀⠂⠐⠆⠀⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠲⣆⠀⠉⠑⠢⠤⠤⠴⠀⠀⠀⠓⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠘⡀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠁
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⣤⠄⠀⢀⠤⠒
⠀⠀⠀⠀⠀⠀⠁⠐⠤⣄⣀⠀⠀⠀⠀⣀⢀⡐⠤⠖⠂⠈
⣿⣿⡿⢿⠾⠿⣿⡿⣷⡶⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⡆⣿⣿⡇⢸⣿⣷⢀⣤⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠻⡗⠒⠶⠦⢤⣤⡾⠁⠀⣿⣿⡇⣿⣿⡇⢸⣿⣿⢸⣿⣿⠀⠈⢷⣤⡤⠴⠶⠒⢺⠟
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣉⣭⣥⣬⣉⣃⡘⢿⡗⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⠿⠿⢠⣾⣶⣤⣦⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡄⠀⠀⣾⣿⣿⣿⡃⠿⣿⣷⣾⣿⠟⣫⣽⣿⠀⠀⠀⠀⢠⣞⣀
⠀⠀⠀⠀⠀⠲⢿⡉⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡌⣿⣿⣏⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢉⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠐⠀⠀⡀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡀⠈⠂⡀⠀⠀⠄⠘⠁⠀⠰⠀⠒⠀⠀⠄⠠⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⣤⠚
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⡂⠤⡆⠀⢀⡀⠀⠀⢀⠈⡀⠀⠀⠆⡀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠉⢂⠠⠈⠀⠂⠀⠀⠙⡆
⠀⠀⠀⠀⢀⡅⠀⠉⠐⠀⠠⢀⠀⠐⠐⠃⠠⠀⠈⠄⢥⠀⠀⠠⠄⠀⡴⠀⠦⠤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭
⠀⠀⠀⠀⣂⡃⠄⠀⠀⠀⠀⠀⠀⠀⢍⠐⠒⢶⡀⡀⠀⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⡰⡦⢴⢧⠄⠤⠄⠀⠀⠀⠀⠀⠀⠳⡆
⠀⠀⠀⠀⠲⠠⡀⠀⠀⠀⠐⢡⠀⡠⠀⠈⠈⠀⢀⠀⢁⡄⠀⠀⠀⠀⠀⠠⣖⡌⠧⠆⠤⠰⠈⠉⠀⣔⠁⠂⠀⡀⠀⠀⠀⢧
⠀⠀⠀⠀⡀⠀⠀⠁⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠺⡌⠢⠀⠀⠀⠀⠲⢌⠠⠀⠠⠌⠂⢬⡨⠂⠈⠉⠓⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀⢠⣶⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠀⠀⠈⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⣀⣀⡀⠄⠠⠐⠂⠀⠀⠀⣄⡠⠀⠀⠠⢀⡶⡆⠀⠓⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠬⠀⣀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣘⡨⢆⣀⣠⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠰⢤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣤⠄⢤⣤⠄⣤⣤⡀⣤⢠⣄⣤⢀⣤⡤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⣻⡆⠀⡇⠀⣿⢿⡁⣿⢸⡿⣄⠸⣿⣃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠘⢧⡀⢠⡾⠛⠛⢷⡀⢀⡼⠃⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢷⢸⡇⠀⠀⢸⠇⡾⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠻⣤⣴⠟⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠙⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠳⠾⠛⠶⣤⠀⠀⢸⡆⢸⡇⢸⡇⠀⠀⣤⠶⠛⠷⠟⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⢸⡇⣸⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠈⠁⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠚⠛⢧
⢰⡿⣟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀
⣾⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⢷⡀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠙⢦⡀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠙⢦⡀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠻⣄⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢀⡴⠞⢯⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⠳⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣦⡀
⠀⠀⠀⠀⠈⠙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠢⣄⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⠀⠀⠀⠙⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠓⢄⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⡇
⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠑⢦⡀⠀⠀⠀⠙⢦⠀⠀⠀⢀⣴⠋⠀⠀⠸⣧
⠀⢀⡏⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠈⠢⣄⠀⠀⠈⢷⡀⣠⠞⠁⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⢸⡗⠂⢒⣸⠇⠀⠀⠀⠐⢿⠶⠤⢤⣹⣦⡀⠀⠈⠑⢦⡀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠟⣇⣀⡤⠤⠒⠚⠋⣩⢾⡇
⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⢀⠔⠁⡾⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣸⠃
⠀⠀⠀⣿⠛⠭⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢰⠇
⠀⠀⠀⢸⣄⡀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⣀⣤⠶⠋
⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠈⠓⠦⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠞⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⡠⠤⢦⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡠⠔⠊⠉⠀⠀⠀⢸⡀⠀⠀⠀⣠⠎⡏⠳⣄⠀⠀⠀⠀⣩⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⣀⣤⡤⠶⠖⠚⠛⢻⡇⠀⢀⠜⠁⠀⡇⠀⠈⠢⡀⠀⠀⣿⠛⠛⠛⠛⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⢸⡇⣠⠏⠀⠀⣠⢧⣄⠀⠀⠈⢦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⣿⠁⠀⣠⠞⠁⠀⠈⠓⢦⡀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⡞⠋⣩⡿⠀⠀⠀⠀⠀⠀⢸⣤⠞⠁⠀⠀⠀⠀⠀⠀⠉⠳⣼⠇⠀⠀⠀⠀⠀⠀⢿⠢⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡔⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⣸⣿⣶⣄⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣷⣦⡀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠉⠻⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⢿⡿⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠈⠉⠉⠉⠉⠁⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣤⣶⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣴⣶⣶⡇⠀⠀⠀⣴⣶⣾⠇⠀⠀⠀⢀⣀⣤⣶⣾⠇
⠀⠀⠀⠀⠀⢰⣾⣿⡏⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⢰⣿⣿⠃⣀⡄⠀⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⢀⣿⣇⣤⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢠⡿⠀⢀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⡏
⠀⠀⠀⠀⢰⣿⣿⣧⡶⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⢸⣿⣿⣃⣤⠀⠀⣠⣿⣷⣿⠏⠀⠀⢠⣿⣿⣿⠃⡀⠀⠀⠘⠋⢁⣿⡟⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠸⢻⣿⡿⠁⠀⠀⠀⢠⣿⣿⣤⠂⠀⠀⣾⣿⣿⡿⠁⠀⣰⠿⢻⣿⠏⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣿⣿⣿⡶⠃
⠀⠀⠀⠀⢀⣿⣿⡷⠂⠀⠀⠀⠞⣻⣿⠃⠀⠀⠀⠋⣾⡿⠁⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠾⠟⢻⡿⠁⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠈⢁⣾⣿⣥
⠀⠀⠀⠀⠘⣽⠟⠁⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⢻⡿⠁
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⣠⡄⠀⢀⡀
⠀⠀⠀⢠⣶⡀⠀⠀⠈⢿⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡟⠁⡴⠋⢀⣴⣿⣿⠆
⠀⠀⣀⠈⠻⣿⡄⠀⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠊⢀⣴⣿⠟⠋
⠀⠀⠈⠁⠀⠈⠻⣄⠈⠣⠈⠃⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⣀⣀⡀⣀⣀⣤⣤⣀⣀⡀⠀⠀⠀⣴⠟⠋⠀⠀⣀⣤
⠀⠀⢠⣄⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⣠⠞⠱⣒⠓⠀⢻⡍⠉⠁⠀⠀⠋⠔⠚⠛⠲⣄⠀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⣠⠔⠒⢿⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⣤⡴⠶⠶
⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⣠⠎⠈⠁⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢢⣼⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⣞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⡄⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠈⠇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢰⣾⡆
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠚⠉⣉⡁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠘⣿⡄⠀⠠⠤⠖⠂⠀⢸⣿⣦⣈⣉⣉⣡⣶⠟⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣀⡤⠹⣆⠀⠀⠤⠤⠀⠘⣿⣄⣐⠒⣚⣉⣤⣾⣿⣿⣿⠟⠛⠉⠁⣰⡇
⠀⠀⣠⠀⠀⠀⠀⠀⠀⠙⠳⣤⣐⣢⣴⡿⢧⣄⣐⣒⣁⣴⡿⣿⣿⣟⣿⡯⢭⣍⠉⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠙⠛⠉⠁⠀⡏⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣴⠟⠁
⠀⠀⠐⠿⠛⠉⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠶⠒⠒⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⠀⠙⠂
⠀⠀⠀⠀⠀⢀⣤⣾⠏⠀⣰⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠻⣦⡀⠈⠳⣦⡄
⠀⠀⠀⣠⣴⣿⡟⠁⠀⣰⡏⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣧⡀⠀⠙⢿⣦⡀⠈
⠀⠀⠈⠻⡿⠋⠀⠀⠀⠉⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣤⡀⣾⣿⣿⣿⣇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢶⣶⡜⣿⣿⣠⣿⡿⠀⣿⣿⣧⣤⡀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣴⣿⣿⣿⣿⡄⢸⣿⡇⢸⣿⣿⣿⣿⣤⣿⣿⠛⠛⠁⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⠋⢾⣿⣿⣄⣼⣿⣅⡀⣿⣿⠀⢿⣿⡄⠙⢿⢿⣿⣷⣾⠃⢰⣶⡆
⠀⠀⠀⠀⠀⢰⣿⠗⠈⠛⠙⢿⣿⡄⠈⠁⠻⣿⣟⠻⢿⣿⠿⠿⠂⠈⢩⠀⠈⢦⠀⠀⡄⠀⠀⠈⠉
⠀⠀⠀⠀⢠⣿⣯⣀⣀⣠⣄⡈⢿⣿⣄⢣⠀⠻⡿⠂⠀⡁⠀⠀⠀⢀⠀⢣⡀⢸⡀⢀⠃
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⢿⣿⡌⠿⠛⠀⢇⠀⢤⢀⡤⠖⡲⢦⡀⢸⢀⠀⣹⠚⠛⢦⡀⢠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠈⢢⠞⠉⠀⠀⠻⢿⠇⢸⠘⠀⡏⠀⠀⠀⠙⣆⢠⠏⠀⢀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠆⣰⠋⠀⠀⠀⠀⣠⠏⠀⢸⠀⠀⡵⢤⡀⠀⠀⠘⡏⠀⡠⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣠⠳⢦⣀⣠⡴⠚⢡⠈⠀⠸⠀⡎⠀⢀⡇⠀⠀⣀⣻⡰⣡⠃⡐⠁⡠⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠳⣄⠀⠞⠉⢠⠖⢉⠉⠁⠰⡀⠘⡄⣠⠚⢴⣦⢠⠋⢤⠀⠿⢿⡿⡰⢁⠜⢀⡜⠁⠀⢀⠔⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠂⡙⠲⢤⣬⠑⠒⠲⠶⢦⣀⢣⠈⠄⢣⠀⡽⠁⠀⠀⠉⠀⢳⠀⡑⠒⡉⠐⠀⠊⡰⠋⠀⢀⠔⠁⣠⠖⠁⡠⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⢹⡀⠀⠀⠀⠀⠈⠳⢄⠈⡌⢴⠃⠀⠀⠀⠀⢀⠆⣐⠀⡔⢰⠀⠙⡆⠀⣠⠞⢁⠴⢊⡠⠂⣼⠀⠀⣠⠃⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣠⣄⣨⠀⠀⢘⣀⣤⣤⣤⣤⣾⣄⡁⠈⠀⢸⠀⢰⠃⠞⠁⠖⣁⠔⠉⢀⣴⡃⢠⣞⡡⠖⠉
⠀⠀⠀⠀⣤⣄⣀⣉⠙⠲⠦⣄⡀⠙⠦⣀⣸⣷⣋⢜⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡼⢴⢸⡀⠀⠴⠞⠀⢠⠴⠋⠙⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠉⢉⡑⠒⠒⠤⠌⢀⣀⠁⠉⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢣⠀⠀⣀⣴⠋⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣩⡍⠀⠐⣛⠀⣼⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⠉⢰⠇⠀⠀⠀⠀⢀⡏⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⣀⣀⣀⡈⣷⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⠀⢸⣄⠀⠀⠀⢠⠞⠉⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠈⠉⠀⠀⠀⠀⢸⣿⡶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣻⢛⣷⠃⠀⠘⢿⣛⣀⠔⢡⣤⡤⠠⠤⠤⠤⠤
⢀⡴⠖⠒⠚⠉⠟⡧⠀⠀⠀⠀⠀⢀⣠⠔⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠁⠐⢶⠤⢀⡤⠒⠈⠁⠀⡄⡐⡆
⠋⠀⠀⣀⡤⠚⠋⠉⠉⠛⠛⠛⠛⢉⡠⠄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣀⣉⣠⠜⠉⠀⠀⠀⠀⠀⢷⣧⡞⠠⠤⢤⣀
⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⠠⠐⠈⡀⠔⠊⢁⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠘⡿⢻⡁⢀⣠⠾⠷⠤⢤⣀⡠⠤⠤⣞⠁⠈⠑⠒⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⠐⠉⢀⡠⣀⠅⠀⣀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣀⣱⣄⡉⠉⣄⠀⠀⠈⠒⢤⣐⠠⠤⣀⠉⠑⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⡠⢀⡡⠊⠁⢀⠔⠁⠀⠀⠀⡀⠀⠉⠉⠉⠉⠉⡀⠀⠀⠙⠁⠀⠉⠓⢦⡑⢤⡀⠀⠀⠈⠙⠢⠄⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠔⠉⠀⠀⡰⠃⠀⠀⡐⠀⡰⢀⠃⠀⠓⢸⠀⠀⢱⠀⠄⢰⠀⠀⠀⠀⠀⠹⡔⢌⠂⠀⢀⡤⠖⠋⠁⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⡠⠊⠀⢀⠌⡜⠀⠀⢀⠆⠀⠀⠀⠈⠀⠀⠀⡆⠀⡘⣆⠀⠀⠀⠀⠀⠙⠀⢠⠖⠋⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠃⠀⠀⠀⠀⡜⠀⠀⠸⢰⠀⠀⠀⠀⠰⡀⠁⠈⠧⣀⣴⣾⡷⢘⡷⠃⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠙⢛⣲⣿⢡⡀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠞⠋⠉⢀⣸⠿⠷⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣤⡴⠋
⣿⣿⡟⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡅⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣄⣀⣀⣤⣶⣶⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣯⣷⣦⣬⡽⠿⠟⠛⠉⠁⠉⠉⠛⠛⠳⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠫⠫⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡔⣴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢿⣿⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡛⣿⡇⠀⠀⣀⣀⣈⡙⠀⠀⠐⠶⣋⣥⣄⣀⠀⠈⣿⣵⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡟⠘⠁⠔⠊⠉⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⡧⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⢀⣤⣤⠀⡀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⡔⢀⣀⠄⢈⣀⠈⠢⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢰⣿⣷⣶⣶⢿⣷⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠈⠋⠙⠫⠉⠉⠉⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢉⡋⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠘⣆⠀⠀⠀⠀⠀⡴⠃⠀⣤⣀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣶⣿⣿⣷⠀⠀⠈⠳⠄⠀⣠⠞⠁⠀⢠⣿⣿⣿⣶⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣿⣿⣿⣿⣧⠀⠀⣠⣶⣾⣷⡀⠀⣠⣿⣿⣿⣿⣿⡇⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢻⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠛⠁⠙⠃⠉⠃⠁⠈⠁⠁⠉⠉⠉⠋⠋⠛⠃⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡏⠀⠀⠀⣿⠶⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠈⢻⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⢷⡀⣿⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠲⡄⠀⠀⠀⠀⠀⢻⡆⠀⠀⢹⣿⠀⢠⡶⠛⠛⠛⢶⣴⠞⠛⠛⠳⣦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠳⣆⣀⡀
⠀⠀⠀⠀⣠⡴⣴⡇⠀⠀⣿⠛⢻⡆⠀⠀⠘⣷⠀⠀⠈⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⠀⠀⠀⠀⠀⢠⡶⢶⣿⠀⠀⣿⠋⠙⣧
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⣇⣀⡀⠀⢻⡆⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⣀⣀
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡏⠉⢻⡆⠘⣷⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣠⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠉⢹
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠟⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠸
⢀⣤⡀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣼⠃⠀⢀⣤⣄⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿
⣿⠁⠹⣦⣿⠀⠈⠃⠀⠀⠁⠀⠘⠇⠀⠈⣧⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢿⡅⠙⣧⣿⠀⠀⠉⠀⠀⠀⠀⠀⠛
⢸⡇⠀⢻⣿⠀⢀⣠⣤⣀⣀⣤⣤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣇⠀⠸⣿⠀⢀⣠⣤⣄⣀⣤⣤⣄
⠈⣷⠀⠈⠿⢀⡿⠁⠀⠙⠋⠁⠈⢹⡆⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠀⠀⠻⠀⣿⠁⠀⠈⠋⠀⠀⢹⡇
⠀⢸⡄⠀⠀⠈⢻⣄⠀⠀⠀⠀⢀⡾⠃⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠻⣦⡀⠀⠀⠀⢠⡾⠃
⠀⠈⣷⡀⠀⠀⠀⠙⠷⣄⣠⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣦⣠⡾⠋⠀⠀⢰
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣷⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠿⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠛⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢡⣤⡀⢠⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠄⠀⠈⢿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠑⡾⠀⡇⠀⠉⣼⠀⠁⢳⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⡏⠁⠀⠀⠀⠀⠀⠈⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠹⣴⠃⠀⠙⠢⠔⠃⠀⠀⠘⡄⢀⣀⠤⠒⠒⢉⣉⠼⠒⣪⠇⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⡀⠀⢀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠀⣀⡔⢲⣍⠀⠀⢀⡇⣀⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⢀⡞⠁⡴⠃⠀⡘⠀⠀⠀⠀⠀⠀⠀⢠⠖⢤⣸⡤⠊⠁⠈⠙⠮⡑⠦⣈⠉⠁⢹⢤⠖⡖⢲⢦⠀⠀⠈⡀
⢀⡞⠀⣼⠁⠀⢰⠃⠀⠒⠮⣍⣉⠉⠉⡻⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠓⠤⣉⠓⠾⣾⡧⠷⠾⠚⠀⣀⣤⠗⠀⠀⠀⠀⢄⠈⢆
⢸⠁⢠⡇⠀⠀⢘⠷⣤⣀⡀⠀⠈⠙⢲⡇⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣄⣀⠀⠉⠒⢻⠀⣀⠤⠖⠋⢁⠏⢧⠀⠀⠀⠀⠈⠆⠘
⠈⠀⠈⠁⠀⠀⡞⠀⠀⢙⠿⣓⢦⣄⣸⠀⠀⠀⠀⠀⠀⠀⡠⢠⡀⠀⠉⠙⢿⣶⣄⣸⠋⠀⠀⠀⢠⠎⠀⠈⣇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⠉⠚⠳⠤⠥⢄⣀⣀⠤⠤⠖⢋⠵⠊⠀⠀⠀⠀⠀⠻⣿⠃⠀⠀⢀⡴⠃⢀⠜⠀⢸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠉⠉⠒⠒⡶⠤⠤⠤⠤⠴⠒⠊⠁⠀⠀⣀⡀⠀⠀⠀⢠⡇⢀⣠⠖⠉⣠⠔⠋⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⣉⣉⣉⣉⣁⡤⠀⠀⣀⡤⠟⣉⣤⡖⠋⠁⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠯⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠽⠗⠚⠉⠀⠀⠈⠑⠤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠃⠀⠀⠉⠉⠑⠒⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⡜
⠀⠀⠀⠀⠺⣤⡤⢄⣀⡸⢆⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣴⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⢤⣤⣤⠴⣄⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⢰⡃⠈⠁⠀⠉⠀⠀⠀⢹⠖⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠴⠶⠶⠤⢤⣤⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⣧⠀⢀⡤⢤⡇⠀⠀⡤⢢⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠁⢀⡞⢀⡟⠀⡜⠉⣧⡤⠀⠀⠀⠀⣤⡞⠙⡏⠙⡖⢢⠹⠴⠁⠀⢻⠀⡼⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢳⡤⢤⠔⢉⠔⠉⢀⠜⠀⡰⠁⠀⠀⠀⠀⠀⠀⢷⠀⢣⠀⢃⠀⢣⣀⡤⣀⡞⠀⡇
⠀⠀⠀⠀⠀⠀⣾⠀⡴⢣⡎⠀⠀⠀⠐⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠃⠀⠂⠀⠻⣀⠎⢻⢀⣇
⠀⠀⠀⢀⣠⠤⢷⠘⣄⠀⡁⠀⠀⠀⠀⢠⠟⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⡇⠀⠀⠀⠀⠀⣿⠘⣯⠜⠉⠛⢦
⠀⢀⡾⠉⠀⠀⠈⢷⣋⡴⠃⠀⠀⠀⠀⡿⠀⣈⠁⠀⠣⣀⣀⠴⠃⠀⠙⢂⡽⠀⠀⠀⠀⠀⠽⠀⣿⠀⠀⠀⠈⢷
⠀⢸⠀⠀⠀⠀⠐⡾⢋⡇⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣿⠲⢿⡉⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠠⠴⠃⠸⡆⠀⢠⠞⠉⣿⠀⡈⡀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⡿⢦⡀⠀⠀⢨⠇⠀⠀⠑⠀⠀⠀⣸
⠀⠘⢦⠀⠀⠀⠀⠀⠀⢧⡴⠋⠀⢰⣧⣀⣈⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣆⠘⢦⣠⠏⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢧⠀⢀⠏⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠉⠁⠀⠀⢈⣧⣔⠁⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠙⢦⣀⠀⠀⢀⣀⡠⠗⢻⡀⠀⣀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠒⠤⠤⣀⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢠⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠁⠘⡆⢹⡛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠙⠒⠒⠶⠒⠒⠚⠋⠉⠁⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡟⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⣄⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⢰⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⡇⠀⠀⠸⣧⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⣰⡇⠀⣰⡿⠀⠀⠀⠀⢻⣆⠀⠸⣧⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠞⣰⡟⠀⣼⣿⠃⠀⠀⠀⠀⠈⣿⣧⡀⠹⣧⠙⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣼⣿⢁⣰⡟⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠹⣧⡈⢿⣷⡤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⡇⣸⡟⠀⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⠀⢻⣧⢸⣿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣿⣿⣟⣷⡿⡁⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⢀⢿⣿⡟⣿⣿⣿⣧⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠹⣿⣿⣿⠁⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠈⣿⣷⣿⡟⣿⣿⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢸⣿⡆⢿⣿⡏⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⢸⣿⣿⠁⣿⡏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⣿⡆⢿⣔⣾⡿⣿⣿⣿⣿⣿⣿⣇⡄⠀⠀⠀⠀⠀⠀⢴⣽⣿⣿⣿⣿⣿⣿⢿⣯⣸⡿⢡⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⢻⣿⣎⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣡⣿⡟⣸⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡻⣧⡻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣴⢟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡽⣟⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡶⣻⣧⣤⣤⣼⡟⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡿⣫⢟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣮⣷⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠟⠁⢲⠂⢊⠉⠙⠏⠉⡙⢄⡞⠈⠻⣿⣿⣿⣿⣿⣿⣿⡯⣹⣿⣞⣵⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣽⣮⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡆⠀⠈⠀⠆⠈⢃⡘⠁⠰⠀⠁⠀⢰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡵⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢷⣿⣿⣯⣙⣿⣿⣿⣿⣿⡇⠞⣀⢠⠀⠀⠀⡀⠈⠁⢀⠀⠀⠀⠀⣀⠇⢸⣿⣿⣿⣿⣿⣫⣼⣿⣿⣾⣯⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣗⣾⣿⣿⣭⣿⣿⣿⣿⣿⡇⢒⣽⣾⡆⠀⢠⣍⣦⣴⣫⡄⠀⢰⣗⣿⡒⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣚⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⠯⢿⣻⣿⣿⣽⣿⣿⣿⣿⣷⣿⡟⢁⣠⣤⣤⡌⢻⡟⢡⣴⣤⣄⡌⢻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⡿⠽⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣂⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⡞⢷⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣷⣿⣿⣿⣆⣐⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣾⣿⣿⣿⣾⣛⡿⣿⣿⠸⣿⣿⣿⣿⡿⣤⣤⢿⣿⣿⣿⣿⠇⣿⣿⢿⣟⣵⣿⣿⣿⣷⣿⣟⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣭⣿⣿⣿⣿⣿⣿⣿⡿⠦⠛⢋⣁⣛⣱⣿⣿⣌⣋⣊⡙⠛⠴⢿⣿⣿⣿⣿⣿⣿⣿⣭⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣩⣽⣿⣿⣿⣿⡷⣶⣿⣿⣿⡟⢿⣿⣿⡟⢛⣿⣿⢿⣶⣾⣿⣿⣿⣿⣿⣍⡛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠻⣿⡿⠃⣿⠇⣿⣾⣿⣴⣂⣐⣣⣿⣷⣿⠘⣿⠃⠻⣿⠻⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⣿⠿⢾⡭⡧⡿⢿⣿⠷⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣤⣴⣦⣴⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠉⠳⢄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠞⠁⠀⠀⠀⠀⠀⢱⡄⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠔⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⡠⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠻⠤⢶⠆
⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⠀⠀⠀⠀⡠⢄⡀⣀⠔⠁⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⢋⣀⠏⠀⠀⣰⢋
⠀⠀⠀⠀⠀⠀⠀⣬⠽⠋⠀⠀⠀⢸⡀⠐⢢⠀⠀⠀⠀⠈⡗⠒⠲⡊⠁⠀⢀⢀⡀⠀⠹⡉⠉⣇⠀⠻⡅⠀⠀⢈⣉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢒⣶⠞⣀⠀⠙⠲⠤⠲⡀⠀⠀⠀⡌⠀⠀⡕⠀⡰⣓⠦⣹⡄⢀⠇⠀⢏⠀⠀⢸⣀⣀⣀⠰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣽⣋⣠⡤⢦⡤⠞⠦⣄⠀⡄⠀⢀⠄⠀⠈⠁⠀⠈⠀⠀⡣⠀⢸⠀⢀⡞⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠏⠉⠓⠦⣤⡆⠀⠀⠀⠀⠀⠀⠀⣇⣠⡼⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣈⣉⠑⠒⣶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⢰⠞⠉⠉⠉⡆⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⣇⠀⠀⠀⢧⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⠀⠳⣤⣄⠀⠀⣀⡀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⠤⠼⢦⡀⠚⠓⣶⠒⠚⠲⣾⢻⡟⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⠖⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠃⢸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠰⠒⠂⡄⠀⠀⠀⠀⠀⡠⠔⠒⢂⠄⠀⠀⠀⠀⠀⠀⠀⣄⠐⠁⣀⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣄⠐⠈⠀⠁⠀⠀⠀⠀⣰⢠⡥⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠉⠃⠶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠊⠀⠂⠀⠀⢀⣲⣾⣿⡖⣁⠋⠊⣀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠲⡖⢲⡀⠘
⠀⠀⠀⠀⠀⠀⢰⠀⣤⣀⠈⢃⡒⢊⠉⣭⠐⠬⣀⣉⠍⠉⠀⢀⣤⢀⠒⠔⠁⠀⠀⠀⠀⠠⠉⢹⣾⠁⠠⡇
⠀⠀⠀⠀⠀⢀⣈⡠⢸⣿⠦⣠⡉⠠⡅⠚⣖⠶⠃⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⣠⠤⠄⠚⠛⠛⠀⠀⠙⡆
⠀⠀⠀⠀⠀⡎⡝⠉⢘⠛⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⡀⠀⠀⠤⠂⠀⠀⠤⠤⠤⢤⣤⣤⡀⠀⣈⣶⣄
⠀⠀⠀⠀⠀⡷⣾⠀⠸⠀⣀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣻⡥⠆⠈⠣
⠀⠀⠀⠀⠀⡇⣾⠈⢂⠀⠀⠀⠘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⢀
⠀⠀⠀⠀⢠⡇⠁⠃⢀⠉⠲⠠⣤⡤⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣹⡀⠉⣉⣒⣺⡥
⠀⠀⠀⠀⠈⠓⣠⣤⢤⣤⡤⠀⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣹⢿⣿⠿⠏⠴⠧⣄⡀
⠀⠀⠀⠀⠀⡸⠁⠀⠁⠈⠀⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡴⣆⣁⠄⠀⠉⠒⠶⠯⢤⣉⣓⣄
⠀⠀⠀⠀⡔⠡⠀⠀⠀⠀⠀⠀⠀⠱⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡀⠁
⠀⠀⠙⠓⠮⢦⣀⠀⠀⠤⢤⣀⣀⣀⣓⣐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣮⣭⣤⣤⡶⠶⠶⠶⠶⠶⠖⠶⠶⠶⠤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠇⢀⡰⠖⠛⠋⡉⠉⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢸⣷⡴⣺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⢀⠄⠀⠀⠀⠸⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⠕⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⠀⠀⠉⠑⠦⣄⠀⠀⠀⠀⠀⣿⡿⠿⠌⠁⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⢷⣤⡀⠀⠀⠈⠳⣤⣄⡀⣠⠟⢣⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠙⠻⢦⣤⣷⠀⠀⢻⣿⡁⠀⠘⣧⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡏⠛⠀⠀⠸⣿⡇⠀⠀⠙⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⢀⡿⡇⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠹⣼⢱⣇⣀⣄⣴⣿⣦⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢄⡰⢃⣾⣿⣿⣿⣿⣿⢿⡗⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢸⣧⡴⢊⣴⣿⣿⣿⣯⣾⡿⡿⠈⠛⢦⣀⠀⠉⢢⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⢰⣾⣿⣿⢟⡵⠋⣰⠃⠀⠀⠀⠙⠳⢤⣧⣯⡞⡩⠖⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢠⣿⣧⣼⡿⠿⠿⠶⣿⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠉⢩⡽⠷⣶⣶⣦⣰⢶⣶⣤⣤⣤⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠛⣤⣴⣾⣿⣿⣿⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣌⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣛⠳⠍⠓⠉⠂⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣳⠀⠀⠀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⣿⣿⣧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣤⠺⣭⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠀⢀⡀⢀⠄⠠⠀⠀⠀⠈⢿⣾⢷⠀⠀⠀⢠⣴⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣴⠀⠈⡷⣬⡅⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⠙⠒⠂⢀⣠⠤⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣦⣴⣦⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠐⠁⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠹⠆⠀⢸⣶⠎⠁⠙⣦⠀⡀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡤⠖⠋⠘⠀⠀⠙⠋⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠶⠈⠳⠿⠚⠋⠀⣠⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡤⠴⠖⠛⠁⢀⡤⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠀⢀⣀⣤⡴⠀⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⡀⠀⠀⠀⠀⠀⠠⠀⢀⢀⡀⠖⠲⠟⠗⠀⠀⠀⣀⠟⠁⢡⠀⠀⠀⠀⠀⢀⣿⣿⣷⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢯⢹⡄⠳⣁⢠⣄⠰⠂⠁⠀⠣⠀⠀⠈⠒⢤⣀⠀⠀⠰⠛⠁⠀⠀⠈⣆⠀⠀⠀⠀⣼⣟⢉⣥⣀⡍⢿⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣎⢣⣙⢶⣛⠿⠃⠀⠀⠠⠄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠀⣀⣼⣿⣿⣿⣿⣿⣿⣦⢧⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠵⡘⢢⠘⠿⡹⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⡙⢣⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡾⢷⠀⠀⠉⠛⠿⢿⣿⣿⣿⢶⡝⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣤⡄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣶⡖⣿⣇⣿⡏⢸⣿⣭⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⣴⣿⠿⣿⡆⢻⣧⢹⣿⡿⣷⣿⣿⣉⡁⣘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠿⢿⣿⡈⠛⢻⣿⠿⣷⣾⣿⠀⠛⠃⠪⡉⢉⠉⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣤⣤⣻⣷⡰⡀⠻⠇⠈⠁⠀⠀⡀⢣⣀⣇⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢉⣿⡇⠉⠁⠱⢀⡤⠊⢹⣷⠄⡧⠀⡏⠉⠳⣔⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⠀⢀⡜⠀⡧⢀⣇⡀⠀⠘⡎⠀⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢉⡤⢛⠦⡤⠖⠩⠄⠀⠃⡎⢀⡆⢀⣠⣵⢈⢊⠌⡠⠂⠀⡠⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⢄⣙⡦⠭⣴⣇⠠⠀⠈⡄⢠⠋⠘⠛⠂⠘⠮⢝⡫⠀⢊⠜⠁⡠⠊⡠⠔⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⡇⠀⠀⠈⠑⢧⠐⣶⠁⠀⠀⠀⠰⢀⢀⢢⠀⢢⠀⣠⢎⠤⢊⠄⡺⠁⡸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢤⣀⠀⠙⢄⠀⢀⣴⠬⣁⣠⣶⣶⣶⣶⣷⣤⣁⠸⢀⡎⠘⢁⡡⠊⣀⢴⣧⡾⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠲⠈⢭⢵⣆⣙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹⡀⠉⢁⡞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⠽⠷⢾⣁⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢵⠈⡿⠀⠀⠀⣰⣃⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠠⠤⠀⠀⢻⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣤⣤⡼⠀⣷⣤⢀⡴⠋⠈⠁⢀⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⣷⠀⠀⠀⠀⢀⡠⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠡⢤⣄⡩⠛⠈⢉⡉⣄⠀⠁
⠀⠀⠀⠀⠀⠈⢅⣠⠴⠊⠉⠉⠉⠉⣉⠡⢔⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠥⠬⣴⡋⠀⠀⠀⠘⣿⠟⡐⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⠄⠊⢀⡴⠀⠙⠿⣿⣿⣿⣿⣿⣿⣧⡀⢹⡙⠦⠒⠉⠙⢂⠒⠊⠉⠓⠤⣄⡉⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⢀⠐⠋⢀⠞⠀⠄⠀⠉⠙⠛⠛⢉⡁⠈⠋⠉⠛⢿⡆⠀⠀⠈⠓⠀⣉⣲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠔⠁⢀⡔⠁⠐⡔⠀⢀⠀⢠⡏⠀⠘⡆⠐⡄⠀⠀⠀⠙⠂⠀⣠⠔⠋⠁⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠋⠀⠔⠈⠀⠀⡎⠀⠘⡇⠆⠀⢳⠀⠙⢦⣀⣤⡶⣦⠞⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠀⠀⠈⠇⠀⠀⢉⡻⠾⣯⡄⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣀⡴⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣉⠉⠉⠉⠛⠛⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⡈⠉⠉⠙⠒⠀⠀⠀⠀⠘⢧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡈⠉⠓⠲⠤⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣍⡉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⡟⢹⡷⠶⣄
⠀⠀⠀⠀⢦⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢀⣰⣟⠀⠀⣀⡠⠤⠀⠤⣤⡀⠀⠀⠀⣴⠋⢀⡞⠀⡼⠁⣸⠏⣷⡀
⠀⠀⠀⢠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠈⠉⠉⠀⠀⠀⠀⢠⡟⠛⢦⣀⡜⠁⠀⡜⠀⡸⠁⢀⠏⢀⡏⠙
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣸⠃
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠶⣴⠇⠀⡆⠀⣸⠁⠀⠀⠀⣠⠤⣄⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⢸⠀⣸⠀⢠⠃⢀⠆⠀⡐⠁⠀⠈⠻⣧⣀⣤⡤⠶⠶⠒⢦⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠳
⠀⠘⢴⡇⢀⡎⢀⡾⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠙⠛⠷⠾⠷⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢀⡀⠀⠒⠦⣤⣍⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡆⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣄⣀⠀⠈⠒⠒⠤⢤⣀⣈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡶⠦⠤⠤⠌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⠟⠀⠀⠀⠀⠀⣠⣀⠀⢠⣄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡞⠁⠈⠉⠛⠛⠋⠓⢮⣷⠦⣀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡾⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢳⡀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⠛⢦⣼⠀⠉⠀⠀⢀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⢹⣾⣧⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡇⠀⢺⣿⠀⠀⠀⢰⠋⠀⢠⣾⡇⠀⠀⠀⣠⠖⢦⠀⡟⣸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢧⠀⠀⠉⠀⠀⠀⢯⣄⣀⡈⢻⠇⠀⠀⢰⠃⣼⣹⠇⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠖⠒⠚⠛⢮⡛⡇⠀⠀⠀⠀⠀⢀⣿⡉⠉⠀⠀⠀⢸⣄⣈⡽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⠟⠁⠀⠀⠀⠀⣀⣀⣀⠙⢧⠀⠀⠀⠀⣾⡆⠉⠀⠀⠀⠀⠀⠀⣿⢎⡅⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡏⠀⠀⠀⠀⢀⡞⠁⠀⠈⠳⡌⢧⡀⠀⠀⠀⢰⣆⣤⣶⣒⣲⣤⡀⣀⠸⢟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢸⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢹⠘⡟⢦⣦⠀⠈⠉⠀⠀⠀⠈⠉⠙⢿⡠⢞
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠙⣌⡻⢍⣳⠶⢦⡤⠤⢤⣶⠚⠉⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⡞⢣⡀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠘⡆⠀⠸⠿⠦⠤⠴⠋⠀⢠⠏⠸⡄⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣾⠀⠀⠙⢦⡀⠈⢻⡖⠒⠒⠒⢒⡟⠁⠀⠀⠀⠀⠀⠀⢀⡴⠟⢶⠞⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡀⣷⠀⠀⠀⠀⠉⠳⣜⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠙⣿⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀⣏⢧⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢷⠒⠛⠉⠁⠸⡞⢆⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣸⣳⡀⠀⠀⠀⠓⠒⠦⣄⣀⡇⠈⢧⡀⠀⠀⠈⠓⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠷⠷⠿⠷⣄⡀⠀⠀⠀⠀⣀⠴⠃⠉⠛⢦⣀⣀⡀⠀⠀⢠⡽⠀⠀⠀⠙⢄⡀⣤⣄⠀⠀⢨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠛⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢵⢟⣩⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠏⣿⣳⣿⣿⣯⣞⣯⣿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠜⠁⠙⠙⠉⣾⠋⡩⣻⣟⢕⠿⣻⢻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠺⠿⣷⠿⣻⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢊⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣠⡔⠁⠀⠀⠀⠀⠀⠀⣠⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣻⣿⣾⣿⣟⠕⠀⠀⠀⠀⢀⠀⡺⣫⣽⢟⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁⠀⠀⠀⠀⠁⠀⠈⢫⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢇⣬⢠⡔⠀⠀⠀⠀⠀⠀⢀⣤⠚⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣾⣿⣿⣻⣷⠛⠔⠀⠀⠀⠠⢛⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⠺⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣰⣶⣟⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⡙⠛⠲⠿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡻⢷⣤⠰⡭⢤⣫⡅⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣰⠀⠀⠀⡰⡐⢀⠼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠒⠴⠛⣂⠾⣻⠛⠀⢸⡷⠶⠶⢤⡀⠀⠀⠀⠀⠈⠟⠉⣇⠀⢀⠁⠀⣨⠷⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⣸⡀⠀⠠⠝⠂⠀⣿⠀⠀⠀⠀⠻⢆⣀⡀⠀⠀⠻⡄⣿⡤⣌⣫⠶⣿⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢫⠉⢙⣦⡘⠖⠀⢸⡧⠀⠀⢰⣿⠀⠀⣤⡷⠶⠶⢶⣿⣍⠛⠉⠉⠀⠀⠘⢯⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠋⠉⠈⡤⣷⣤⡴⢾⡅⠂⢒⣺⣿⡶⢤⣿⣅⣀⡀⠀⠀⠽⠄⠀⢠⠀⠀⠀⢀⡂⠀⠛⣿⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠙⠒⢛⢺⣧⣿⣇⣀⠈⠙⠿⣯⠁⠀⠙⢦⠀⠈⠋⠙⠋⠶⠶⣶⡖⠁⠀⠀⢀⢄⣼⠀⠀⠘⣧⠀⠘⢂
⠀⠀⠀⠀⠀⠀⢀⡿⡀⠀⢠⡞⠉⠉⠈⠉⠉⠀⠙⠶⡶⠼⠷⠄⣤⡀⢳⡄⠀⠀⠀⠀⠀⣬⠑⠀⠀⣀⡤⠞⠁⠀⠀⠐⠈⢻⣄⠺⢧⡀
⠀⠀⠀⠀⠀⠀⢠⣷⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠸⡇⠚⠛⠋⠉⠀⠀⠀⠀⠀⠐⠂⠀⢸⡇⢢⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢈⡷⢠⢙⡆⠀⢀⢀⡀⠤⠤⣼⣭⣀⢤⣤⣤⣤⣤⠤⠴⠛⠀⠀⠀⠀⠀⢿⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠑⠀⢡⡀
⠀⠀⠀⠀⠀⠀⢸⢿⣸⠈⠳⣤⡐⠉⠀⢀⣰⢿⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢤⠀⡀⠀⠀⠠⣤⣤⣽⠃⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠟⣄⣿⡄⠀⠈⠻⠖⠒⠛⠃⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠠⡀⣁⣤⠴⠾⠿⠛⠋⠙⢷⣤⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠳⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠔⠋⠉⠈⠁⠀⠀⠀⢀⣤⣼⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠆⠀⢀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠀⢀⣀⣤⠖⠋⠉⠀⠈⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠃⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠔⠋⠁⠁⢆⠀⠀⠀⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢍⠙⠳⣤⣤⡤⠤⠶⣻⣽⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣞⡿⣬⣃⣀⣀⣀⣀⣂⣀⡿⣏⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣥⡈⠉⢉⠙⠛⠉⠉⠙⠀⢠⠐⣕⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡞⠁⠀⠀⠙⠀⠀⠀⠀⠀⠈⠀⠐⣹⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠟⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣀⠿⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆

⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⡶
⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠈⠉⠉⣿⡿⠃⠀⠀⠀⠀⠘⠛⣿⣿⠟⠀⠀⠀⠀⠀⠸⢿⣿⠋
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣀⡀⠀⠀⠀⢰⣿⣿⡗⠀⠀⠀⠀⢀⣾⣿⣇⣀⣀
⠀⠀⠀⠀⠛⢿⣿⣷⣦⠀⠀⠀⠀⠘⣻⣿⣿⣟⡀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠘⢻⣿⣿⠟⠁⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⠉⠁⣸⣿⡟⠁
⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠉⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⢶⠄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢋⠀⠀⠀⠀⠠⡀⠀⠑⣄⠈⠀⠣⡀⠀⠙⣄⠀⠀⠆⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⡀⠙⢦⠀⣈⠃⠀⠀⠈⠀⠀⠈⢣⢐⣦⠀⠢⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢿⠖⣀⠹⣦⠀⠀⢦⡀⠤⣳⣤⣾⠿⠁⠀⠀⠈⠢⡀⢷⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠈⢹⡈⠇⠀⠙⠛⠛⠿⠟⠁⠈⡀⠀⠀⠀⠀⠙⢆⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢳⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢦⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⢀⣀⣀⣀⠀⠀⠀⠀⠀⠰⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⢱⡄⠘⠋⣁⣍⡉⠀⠢⡀⠀⠀⠶⠸⢹⣿⡋⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⢀⠠⠘⠁⣀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠃⠀⢹⠋⢿⡍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠁⢀⡔⠊⠀⠁⠀⢠⡓
⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⠻⡄⠀⠀⣠⠎⠀⠀⠀⠀⣀⠙⢅⠀⣄
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⡆⡆⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠑⢾
⠀⠀⠀⠀⠀⠈⡟⠃⠀⠀⠀⠱⡸⣄⠀⠀⠀⠀⣀⣀⡠⢤⣤⣄⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⡼⠁⠀⠰⠊⠁⠀⢀⠕⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⣆⠀⠀⠐⢿⣟⣿⡿⣽⡿⠃⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠑⣄⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡀⠀⠀⠶⠌⢦⡀⠀⠈⠛⠒⠚⠋⠀⠀⠀⠀⢀⣼⣿⣿⡇⠀⠀⠀⠈⠀⢀⠄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠳⣿⣯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣨⢿⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⠀⠀⠄⠀⠀⠈⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠼⠃⠀⠀⠀⠀⢰⠇⠈⠓⠶⢤⡴⠶⡿⡿⠿⠿⠛⠛⠁⠀⡄⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠚⣂⠴⠋⠀⡀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢷⡀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢐⡂⠀⠀⣀⠀⠀⢴⣋⣵⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣯⡀⠀⣀⣭⠵⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⢀⠋⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡬⠗⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⣀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡽⣦⡀
⠀⠀⠀⢀⡾⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣠⣶⣯⣾⣿⣷⣄⡀
⠀⠀⢀⣼⣿⣶⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⣾⣫⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣀⣠⣴⢞⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣆⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⢿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣿⡇⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⡤⠞⠁⠀⠀⢀⣴⡿⠟⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⠁⢀⠜⠀⢀⣤⣤⣾⡟⠉⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀
⠀⠀⠀⣠⣤⣤⣤⣴⣶⡾⠟⠋⠁⠀⢸⡟⠀⠁⠀⠚⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣇⣀⣀⡀
⠀⠀⢰⠟⠀⠀⠀⣠⡟⠁⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⢠⣾⣿⡟⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠈⠁⠈⢳
⠀⠀⡻⠀⠀⠀⣾⠁⠀⠀⠀⠀⣠⡤⡼⣷⣴⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⢻⣷⣴⣶⣶⠶⠶⠶⣦⣀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠃
⡠⠞⠁⠀⠀⠀⢻⠀⠀⠀⢀⠞⠁⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠘⠀⠀⠎⠀⠀⢀⣀⣠⣤⡶⠶⠟⣿⡟⠋⠉⠙⣿⣿⣆⠀⠀⠀⠀⠙⠷⠶⣤⡀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣤⠤⠤⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣇⠈⠉⠛⠿⢿⣧⣄⠀⠀⠓⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠈⠿⢶⣶⣤⣤⣤⣄⣀⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⢠⡞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠀⢻⣆⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⣫⣿⣿⠟⠀⠀⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⠁⣠⣾⣿⣿⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣾⣿⠟⠁⢻⣇⠀⠀⠀⠈⠻⣆⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠁⠀⠀⣴⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣸⣿⠇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣆⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠙⠦⣀⠀⠀⠀⠀⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠞⢻⣿⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠷⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢠⣤⣤⣤⣴⡀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡐
⡀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠢⠀⣶⣫⠭⠾⡧⠀⠠⠋⠁⠒⠠⣀⠀⠔
⠀⠁⠒⡀⡀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠁⠻⠀⠀⠀⠐⡀⢈⡒⠒⠄⡀⠀⡋
⠀⠀⠀⠇⠀⢈⠐⡤⣀⠀⠀⠑⢄⠀⠀⠀⠐⣆⢀⣄⡀⣱⠋⠀⠀⢀⠞⠊
⠀⠀⢀⠂⠈⠀⠀⠟⠾⢉⠖⠤⠄⠈⠢⢀⣄⠭⢟⡾⠞⠉⠀⠀⢠⡃
⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣨⠡⡴⠋⠀⠀⠀⠀⢠⣗⣅
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡿⢋⣤⠂⢀⠀⠀⢀⢷⡥⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣶⡿⢃⣴⠏⣰⢃⠎⣜⢽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠱⢵⣯⣋⣼⣇⠎⡜⠉⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠎⠀⠙⣆⠀⠘⡿⠯⣧⠉⠀⣠⣾⠟⠙⢆
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⢄⠀⠀⠸⠒⠈⠀⢠⣽⣶⢶⢿⣧⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢈⡀⠙⠉⠀⠀⠀⠈⠘⠉⠁⠸⠢⡳⣦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢌⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠁⠀⠀⠑⢍⠣⣠⡀
⠀⠀⠀⠀⢀⠄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡀⠀⠀⠀⠑⠈⠻⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡶⠒⠛⠛⠙⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⠏
⠀⠀⠀⢠⡇⡰⣊⠩⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡟
⠀⠀⠀⢸⡇⠁⠁⠀⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠒⠒⠲⢦⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⠛⠛⠉⠉⣹⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣹⠇⠀⠀⠀⡜⠀⣀⣀⡀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠛⢉⡟⠀⠀⠀⠀⣼⠃⠀⠀⢀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣯⣤⣄⠀⠀⠀⢲⡏⠀⠀⠀⠀⠉⠉⢉⡽⠃⢀⢀⡼⠁
⠀⠀⠀⣿⣀⣠⣤⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⣏⣴⡶⠟⢋⡏⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣢⡟
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⣁⣀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠟⢛⡿
⠀⠀⠀⠀⠀⠰⡇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢠⠏⠀⠀⠀⠀⠿⠛⠒⡖⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠟
⠀⠀⠀⠀⠀⢸⡇⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋
⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠠⡇⢷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡴⠶⠖⠒⡒⣲⠆
⠀⠀⠀⠀⠀⢰⣶⠛⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢀⡇⠈⢣⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡶⠛⠁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⢠⡶⠞⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡞⠉⠀⠀⢀⢂⡴⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣰⢏⠀⠀⠀⠀⢠⣿⣃⣀⣀⡀
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣠⣭⠇⠀⠀⠀⠘⣇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠐⠉⠉⠉⣩⠞⠁
⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧⣤⡄⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠠⣞⠃⠀⠀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠓⢲⠀⠀⠀⠀⠀⣸⠀⢀⡀⠀⢠⠞⠀⠀⠀⢰⡯⠶⠖⠚⣿⠀⠀⠁⣴⠋
⠀⠀⠀⠀⠹⣆⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡜⡀⢸⠀⠀⠀⠀⠀⠙⠋⢹⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡄⠀⠀⠀⠀⠀⢀⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠹⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡇⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⠀⠀⠀⣰⡿⠃⠀⢀⡄⠀⠀⢀⡴⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⠏⢠⣾⣿⡿⣿⣿⣿⡿⢻⣿⣿
⠀⠀⢀⣾⣧⡤⠀⠀⣿⠟⠀⢀⣾⣤⠀⠰⢿⡧⠀⠀⢸⣿⣿⡟⠁⠘⢻⣿⣿⠀⣼⣿⡟⠐⢻⣿⣋⠀⢻⣿⣿
⠀⠀⠉⢹⠟⠀⠀⠀⡿⠁⠀⠈⡽⠁⠀⠀⢹⡟⠁⠀⢸⣿⣿⣶⠆⣰⣿⣿⣷⣠⣿⣿⣷⢠⣾⣿⣷⠆⣠⣿⣿
⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⢀⠀⠀⢰⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣟⡀⠀⠀⣴⠀⠀⠀⣼⣿⡃⠀⠀⣠⡟⠀⠀⢸⣿⣿⠇⠰⢿⣿⣿⢫⣿⣿⣿⠁⠀⣾⣿⣿⠏⣸⣿⣿
⠀⠀⢛⡟⠁⠀⠨⣿⠁⠀⠀⣾⣟⠀⠀⠴⣿⣿⠃⠀⢸⣿⣿⡤⢠⣿⣿⣇⠀⣿⣿⡿⠀⢸⣿⣿⣃⠀⢈⣿⣿
⠀⠀⠸⠁⠀⠀⢠⠋⠀⠀⠀⡞⠁⠀⠀⠀⡿⠁⠀⠀⢸⣿⣿⣷⣿⣿⣿⣟⣼⣿⣿⡿⣰⣿⣿⣿⡿⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠤⢄⣿⣿⣿⡇⣀⡿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⠿⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠛⠃⠀⠀⠒⠚⠛⠋⠙⠉⠁⠀⠀⠙⠁⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠰⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠋⠀⠀⠀⠀⠉⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⠁⠈⠑⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠓⠚⠓⠲⣶⣶⣶⣶⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⣿⣦⣤⣿⣿⠓⠢⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣶⣶⣶⣶⡟⠣⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣿⣿⣿⣿⢿⣷⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠉⠉⠙⠛⠲⠶⠤
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠹⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⠟⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠘⣿⣿⣇
⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡿⠁⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠘⣿⣿
⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⠸⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢷⣾⣿⣿⣷⠾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢡⣤⣾⣿⣿⣿⣿⣷⣤⡌⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣹⣿⡟⠀⠙⠛⠿⣿⣿⣿⣿⠿⠛⠋⠈⢿⣿⣏⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⡟⢀⡀⠀⠀⣀⣐⣿⣿⣂⣀⠀⠀⢀⡀⢻⠙⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠸⠁⣿⣿⡷⣦⣄⣈⣼⣧⣁⣠⣴⢾⣿⣿⠈⠏⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣆⠀⣨⣿⣶⡿⡿⢿⡿⢿⠿⣶⣿⣅⠀⣰⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢂⣲⣸⣏⣗⡑⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠏⢹⠪⣹⣿⣿⢏⡑⠏⠘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⣿⣧⠎⠀⣸⣿⣿⣿⣿⣷⠀⠸⣜⣿⣿⠇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣧⠀⣿⣿⡏⢠⡴⢿⣿⣿⣿⣿⣿⢧⣄⢹⣿⣿⠀⣼⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡀⠙⣿⣷⡀⢺⣿⣿⣿⣿⣿⣿⡗⢀⣾⣿⠏⠀⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢿⣿⣿⡧⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⡿⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠼⢻⠿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠐⠫⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠆⠀⠀⠀⠀⣰⣿⣿⡶⠛⠀⠀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠼⢿⣿⡿⠋⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢈⡽⠟⠁⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⠀⠀⠀⠀⠐⠁⢠⢌⠁⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⢃⠔⠃
⠀⠀⠀⠀⢀⢀⡘⠂⡰⠀⠀⠀⠀⠈⠀⡪⠐⠁⠀⠀⠀⠀⠈⣱⠃⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢴⡩⠇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠠⣸⣁⣀⡀⠀⠀⠀⡠⠋⢠⣎⠁⠀⠀⠀⣠⠊⢘⡂⠖⠀⠀⠀⠀⢠⠎⣠⠄
⠀⠀⠀⠀⠰⢓⢖⡴⠋⠀⠀⠀⠈⣩⡿⠒⠁⠀⠀⠀⠀⣩⠏⠋⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠰⠎
⠀⠀⠀⠈

⠀⠀⠀⣠⡤⠤⠶⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⡕⠀⠀⠀⠀⠀⠀⢀⡔⡆
⠀⠀⢰⡇⢔⠂⢀⣾⠀⠀⠀⠀⠀⠀⢀⡴⢣⡇⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⠃⠀⠀⠀⠀⢀⡴⠋⡜
⠀⠀⢸⠃⠀⠀⠈⡇⠀⠀⠀⠀⠀⢠⠞⠀⢸⣡⡤⠤⣤⠀⠀⠀⣸⠀⠀⠀⠿⢤⡀⠀⠀⡰⠋⠀⠀⠁⠀⣲⠆
⠀⠀⢸⠀⠀⠀⠀⠧⢶⡄⠀⠀⢠⡏⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣟⣀⠀⠀⠀⡾⠀⠀⠐⠳⠒⣲⠂⠀⡴⠃
⠀⠀⢸⡤⣤⠀⠀⢀⡞⠀⠀⠀⣾⡴⠖⠛⡇⠀⣼⠃⠀⠀⠀⠀⠀⢸⠀⠀⣸⠁⠀⠀⠀⢀⡜⠁⢀⣞⡁
⠀⠀⠀⠀⣼⠀⢀⡞⠀⠀⠀⠀⠉⠀⠀⢰⡇⡸⠃⠀⠀⠀⠀⠀⠀⡜⠀⠐⠗⡦⠀⠀⠀⣾⣀⡀⠀⣹⠋
⠀⠀⠀⠀⣿⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢧⣤⠀⣰⠁⠀⠀⠀⠀⢠⡇⡴⠃
⠀⠀⠀⠀⡟⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣰⠃⠀⠀⠀⠀⠀⣸⡾⠁
⠀⠀⠀⠀⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢰⡖⠚⠛⣻⠀⠀⠀⠀⠀⡷⡀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠄⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⣤⠼⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠹⡄⠀⠀⠀⠀⠸⠀⠀⠀⡼⠀⠀⠀⠀⠀⢀⡼⠋⠉⠉⢀⡶⠋
⠀⠀⠀⠀⣿⠀⠀⠀⡖⠋⠀⠀⠀⣠⠞⠁⠀⠘⡄⠀⠀⠀⡇⠀⠀⣸⠁⠀⠀⠀⠀⢠⠞⠀⠀⠀⣰⠟
⠀⠀⠐⣏⠁⠀⠀⢸⡇⠀⠀⢠⠞⠁⠀⢀⡴⠋⠀⠀⠀⢀⠀⠀⢠⡏⠀⠀⠀⠀⣠⠟⠀⠀⠀⠸⠯⠴⡶
⠀⠀⠀⢻⠀⠀⢸⠉⠁⠀⠀⠈⢇⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⠀⢸⣀⣀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠐⣎⠀⢀⡼⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⡆⠀⠀⠀⢸⣀⣄⠀⡔⠁⠀⠀⠿⠒⠛⡿⠀⢀⡼⠁
⠀⠀⠀⠸⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⢁⠀⠀⠀⠀⢀⢏⠞⠀⠀⠀⠀⠀⠀⢰⡇⢠⠎
⠀⠀⠀⠀⠘⣆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣻⠀⠀⠀⠀⣸⡞⠀⠀⠀⠀⠀⠀⠀⣼⣴⠋
⠀⠀⠀⠀⠀⠈⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣛⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣠⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠞⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣭⣙⣛⣿⣝⡳⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠙⠻⢽⣻⣷⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣟⡷⣦⣤⡙⢷⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠻⣷⣿⣎⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣐⣷⣿⣿⣿⣍⡉⠳⡀⠀⠀⠀⠀⠙⢿⣿⣿⡿⣿⣿⡟⢿⡀⢮⢻⣿⣞⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠈⠋⠛⣷⣿⣿⣍⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⠊⣿⣌⣷⣿⣿⣿⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠀⢀⠤⣴⣿⣿⣟⣯⠟⠋⠣⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣽⣿⣿⣷⢬⡻⣿⣿⡇⢻⣇⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠁⠀⠀⠉⢀⣸⢋⡉⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣧⣸⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⡀⢦⠀⣀⠑⠈⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠷⣤⠴⢿⡗⠈⠀⠩⣶⣄⠀⠀⠶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⡁⢿⠟⠹⣿⡞⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣄⠀⠀⠀⠀⠣⡀⠀⠘⠯⠛⠲⣦⡄⠀⠀⠙⠯⠿⠻⠿⠁⠈⣿⣰⣶⣄⠀⢹⡧⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⣷⣴⡄⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠅⡹⢿⠇⡜⣱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣥⣭⠟⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢄⣀⡉⡀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠟⠻⠿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⢷⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⡴⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⠛⡿⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⡆⣠⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⢿⣿⣿⠿⢿⣿⣿⣆⠀⠀⢣⠇⠀⠀⠀⠘⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣾⠏⠀⠀⠘⠋⠙⣿⣆⠀⢠⡁⠀⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣆⠀⠀⢀⡾⠿⣿⡆⢸⠃⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⠆⠀⡈⣿⠆⠙⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⡾⣧⣾⣇⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢻⣿⣷⣃⣿⣿⡇⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣠⣾⣿⣿⣿⡍⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣤⣄⡀⠀⠀⠐⠚⠵⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣤⣾⣿⣿⣿⣿⣿⣿⡼⣿⣏⣿⣿⣿⢦⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⣄⣙⣿⣿⣿⣷⣦⣀⠀⣠⢄⡠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⠁⢿⣿⡿⣧⣾⣿⣿⠏⣿⠚⣷⡶⡾⢿⠷⠈⠙⠻⢿⢿⣿⣿⡿⣋⣽⣿⠿⢿⢿⣿⣦⡰⣫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢾⡿⠁⠀⣿⣿⣿⠀⠈⠙⠟⠀⢀⠀⡄⠓⠀⠀⣠⣾⡿⣡⣾⣿⣿⣱⣈⣳⣾⣶⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣿⡀⠈⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠈⠁⠀⢀⣠⣶⣿⣿⣿⣽⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⢰⡟⠁⣤⡶⠆⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣎⠁
⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣶⡶⠾⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣀⣀⣀⡄⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠉⠉⢀⡇⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿
⠀⡀⠀⣠⡶⢻⡄⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠙⣦⠄⠀⠀⠀⠀⠀⠘⣿⠙⠛⠳⠿
⠀⠱⠞⠁⠀⠈⢷⣀⣴⠟⠉⣷⠀⠀⣠⠔⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⣤⣄⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⣧⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⠀⠀⠀⠀⠀⢀⡔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢦⠀⠻⣌⣆⠀⠀⠀⠀⢀⠣⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠑⣾⠦⢄⡟⠉⠑⢄⠀⠸⡁⣼⠟⠓⢄⠀⢀⡤⢀⠊⡱⠠
⠀⠀⠀⠀⠀⠀⠀⡊⠑⢆⠀⠀⢸⠀⠀⠁⠀⠀⠀⢧⡰⠛⡇⠀⠀⢘⣧⠥⠒⠻⠏⡀⠒⠐⢒⡦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⣆⠀⠈⠚⠀⠡⣣⡀⠀⢸⠃⠀⢱⡀⢀⣸⡏⠀⢠⢰⠀⠻⠤⠀⠊
⠀⠀⠀⠀⠀⢠⣀⠀⠈⠛⢿⠓⠢⣄⡈⠢⠒⠥⠴⢾⣀⣀⡌⠈⠻⠿⠧⡴⠗⠁⠀⣂⣤⠖⠋⣱
⠀⠀⠀⠀⠠⢄⠀⠀⠤⠀⢗⠀⠀⠀⠈⢢⠀⣠⣾⣟⡏⣀⠀⠙⢦⢀⠔⠡⠂⠀⣀⣹⡟⠉⢉⣀⠔⢂⣉⡆
⠀⠀⢠⡖⠓⠲⠭⣒⣄⠠⢼⣷⡀⠀⠀⠀⢷⣿⣿⢫⡌⠛⠀⠀⠀⢇⠀⢠⡴⠋⠁⠀⠀⠀⣸⡟⢉⠉⠁
⠀⠀⠀⠉⠛⡛⠒⠛⠻⡀⠀⠀⠑⠦⣀⢀⣨⣿⣿⣿⡀⠀⠀⠀⠀⡸⠒⣻⣀⡀⣀⣒⠤⠊⠽⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠓⠛⠲⠤⠤⠤⠼⠯⠽⢟⢮⣛⠿⣶⣶⣶⣿⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠑⠒⠒⠂⠀⠒⠛⠚⠻⠿⠿⠿⠀⠁⢈⣉⠿⠯⣭⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠳⣼⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣦⡇⠀⠀⠀⠀⠀⠠⢴⡶⠶⡿⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠠⡄⠀⢀⢻⠀⠀⠀⠀⠀⠀⣸⢧⣀⡠⡶⠀⠀⠰⠞⠙⠲⣤⡀⠀⠀⣰⠋⠀⠠⣤⠀⠀⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣏⠀⢀⠀⠀⠁⠀⠈⢸⠇⠀⠀⠀⠀⣉⣁⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡇⠀⠀⠀⠀⠀⢀⠁⢸
⠀⢠⣤⣄⣠⠶⠚⠋⠉⠉⠉⢹⠟⢆⡈⠃⠀⠀⠀⣀⠞⠀⠀⠀⣠⠎⡄⢈⠟⢦⠀⠀⣸⠋⠁⠀⠀⠀⠀⠀⠀⢹⠿⣄⠀⠀⠜⠠⡽⢀⠞
⠀⢻⠏⠹⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠉⣗⣶⣶⣾⠷⠶⠶⠶⢾⢸⠀⢽⣉⡀⡸⠀⠀⡇⠀⢠⠤⠴⠲⢄⡀⣰⠏⠀⢾⡹⣖⣲⢖⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡿⠀⡟⠀⠀⠀⠀⠀⠉⠑⠺⠗⠚⠁⢀⠤⠛⠂⠀⢳⡀⠀⠀⠉⠛⢒⣲⠶⠋⠉⢡⣾⠇
⠀⠀⢀⣀⣀⡀⠀⠀⠀⡟⠀⠀⢠⠴⠊⢱⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⡏⠀⠀⢰⢷⠀⠙⠒⢒⡏⠉⢉⠀⢀⣠⠶⠋⡾
⠀⠰⢥⠤⠋⢳⣄⣀⣠⣧⠀⠀⣮⣀⡤⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡴⠶⣶⣾⡏⣽⠟⠛⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⢤⣀⣀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⢮⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣀⡀
⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢂⠉⠉⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠁⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡠⡀⡤⢌⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣢
⠀⠀⠀⠀⠀⢌⠠⠊⠀⠀⠀⠀⠀⠀⣀⠤⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⢀⠀⠀⠀⠀⠀⣀⠀⢀⡁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠔⠀⠉⠀⠂⠠⠐⠂⠉⠀⢀⣀⣀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠈⢱⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠁⢠⠊⠀⠀⡠⠀⠂⠀⠄⣀⠀⣀⠔⠁⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠇⠀⠀⠜⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠤⢀⡰⠂⠈⠀⠐⠄⠀⠀⢸⠀⠀⠀⠀⠀⠆⡠⠐⠀⠢⡀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠘⠀⠠⠐⠀⡁
⠀⠀⣀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠄⠀⠀⠀⢀⠂⠀⠀⠠⠃
⡐⠁⠀⠈⠲⠀⠀⢰⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡢⢀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⢘⢀⠠⠐⠁
⠇⠒⠐⠦⠀⠆⠀⠀⠑⠠⢀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⠋
⠐⢄⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠉⠣⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡃⠠⠤⠀⠐⠂⠉⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠑⠠⠄⠤⠓⠄⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⢀⡀⠀⠀⢀⡀⠀⠤⠀⠀⠒⠈⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠠⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠄⠀⠀⠀⠱⠀⠆⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡀⠂⡉⠐⠠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠢⢀⣀⡘⣄⡀⠀⠀⢙⠀⠀⠀⠰⢀⠤⠀⠊⠬⡄⠀⠀⠀⠀⠐⡀⠊⢀⢀⠀⠡⡑⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢀⠒⠉⠀⠀⠀⠀⠁⢢⣀⢄⠀⡔⢻⢅⠀⠀⠀⠀⠈⠂⠀⠀⢠⠂⠉⠉⠀⢀⠇⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠔⡇⠀⠈⡁⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⡌⡀⢴⡆⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⠀⠀⢠⡀⠀⡠⠒⠰⠁⠀⣰⠁⠀⠠⠐⠁⠀⠀⠀⠀⠈⠒⠒⠈⠑⠱⠄⠂⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠔⠒⠒⠠⠤⠤⢄⣀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠈⠉⠁
⢈⠉⢆⠀⠀⡤⠂⠉⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠒⢤⠀⠀⠀⡠⠢
⠀⠈⠈⠀⢰⣀⠀⠠⠄⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠤⠀⣀⡀⠀⡃⠀⠰⠄⠊
⡠⠀⠢⠀⡸⢆⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢠⡇⠀⢀⠀⢀
⠑⠊⠀⢠⠃⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣀⣀⣀⡀⠀⠀⠀⠈⠀⠂⠈⠀⢸⡀⠈⠢⠤
⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠉⠑⣦⡀⠀⠀⠀⠀⠀⠀⠉⠳⡀
⠀⠈⠢⢴⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡤⠜
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠙⠉⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠹⠿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠟⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⡨⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣁
⠀⠀⠀⠀⠀⣜⠉⢃⠀⠑⢍⣲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢊⠕⠋⠨⠗⢦
⠀⠀⠀⠀⠰⣁⠀⠁⠀⠀⠀⠀⢀⡀⠉⠙⠒⠶⠤⠤⠤⠤⠤⠴⠒⠚⣩⣀⠀⠀⠀⠀⠐⠀⠚⡄
⠀⠀⠀⠀⠀⠘⠤⠔⠒⠊⠉⠉⣹⣁⡤⠒⢤⠀⠀⠀⠀⠀⢀⠤⠢⢄⣿⠈⠉⠙⠒⠤⣄⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠉⠈⠄⠀⢸⠒⠢⠤⠬⠕⠺⡄⠀⠈⢀⠋⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⢸⡀⠀⠊⠀⠀⠐⡁⠀⠀⠀⠈⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⢃⠧⢀⣠⣄⣀⢼⣱⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢀⡸⠁⠀⠀⠀⠀⠀⠀⠙⢄⠀⣀⠜
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠓⠶⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠠⠀⠀⠡⠄⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣦⡀⣾⡖⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣷⠙⠃⣉⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠒⢒⣒⣻⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡀⠊⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⡿⠟⠁⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢦⣀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣰⠖⢄⢀⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⣴⡦⣄⣨⠟⡲⢴⣿⣿⣿⣿⣿⣿⣷⡄⢨⠏⠀⠀⠀⠐⠄⠀⢤⣤
⠀⠀⠀⠀⠀⠈⣉⠀⠠⠤⠬⢶⣹⡂⡀⡉⢙⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠈⠉⠀⠂⠠⠄⢲⡿⠒⠲⣅⡩⢽⣿⣿⣿⣿⣿⣿⣿⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠛⠙⠻⠿⠟⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠈⢙⣛⠛⠷⢦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢁⡀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣠⡄⢰⡤⠐⠓⠀⠀⠀⠀⠀⠀⡾⢠⠂⠀⠀⠀⢀⣼⠅⠀⣴⠌⣇
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣄⠀⡄⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⣀⡀⢠⣾⡿⠃⢨⠃⠀⠀⠑⠀⢤⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠾⣿⣿⣿⣿⣾⡯⠀⠀⣠
⠀⠀⠀⢶⣤⣤⡼⣿⣿⣿⣿⠟⠋⠀⠙⣥⣤⣴⠆⠀⣀⣠⣤⣶⣶⣽⣦⡀⠉⠙⠿⠿⠗⣠⣴⣥⣴⣶⣦⣤⣄
⠀⠀⠈⠛⠻⣿⣷⣿⣿⣿⣶⣶⣶⣶⣶⣿⡿⠛⠓⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡀
⠀⠀⠀⡰⠻⢿⣿⣿⣿⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⢀⡆
⠀⠀⢀⡇⠀⣼⣿⡿⠃⠀⠀⢤⠠⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠈⠇⠀⠈⠈⠓⠂⠀⠀⠒⠀⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢐⡁⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠈⢢⠐⠄⠀⠀⠀⢠⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⢀⣾⠋
⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⣈⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠰⠋⠡
⠀⠀⠀⠀⠀⢠⣶⣤⣲⣄⠈⠉⠁⠀⠀⢸⡙⣒⠶⠶⢗⠀⢲⠀⠀⠀⠀⠀⠠⠄⡰⣄⣀⣀⣀⣀⣀⣀⡀⠀⠓⠶⠶⡄⢀⡆⢀⠄⠀⠀⠒
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣼⣏⣿⣿⣿⣿⣿⡇⠇⠀⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠛⠒⠐⠒⠒⠂⠒⢢⢽⠀⠀⣀⠲⠀⠐⠁⠘⠓⠦⠒
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠛⠉⠉⠉⣷⠱⠀⣴⣓⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢤⡀⢰⠸⣤⠊⠁⢀⠀⠠⠀⣀⡴
⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⢿⠸⠀⠨⠉⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⠄⠈⠆⠟⢆⠀⠂⢀⠖⠉⠉
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣏⠉⠛⢿⣿⣿⣶⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⢠⡴⠀⠀⡀
⠈⠿⣿⣿⣿⣿⣿⣿⡏⠈⠀⠀⠀⠀⠉⠁⠀⢤⠤⠀⠁⠠⠤⠤⡤⣀⣒⣒⣒⣂⣀⡀⠀⠀⡀⠀⠀⠀⢀⡠⠔⠉⠀⢀
⠀⠀⠈⢏⡛⠿⠿⠋⠀⠀⠘⠓⢀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⡤⠖⠉⣩⠞⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠊⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢰⠢⠤⠥⠆⣴⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠢⣄⢀⡠⠀⡌⠳⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠜⠀⠑⣦⡀⠀⢱⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣠⠀⢀⡟⠉⠀⠸⠇⠐
⠀⠀⠀⠀⠀⠈⠰⡶⠴⡶⠀⠀⠀⠑⠐⣷⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠚⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⣠⠰⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠆⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠇⠂⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡇⠀⢸⡗⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⡇⠀⢸⡇⠀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⡇⠀⢸⡇⠀⣿⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⡀⢻⠀⠀⡇⠀⠸⠇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢳⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠲⣄⣀⠀⠀⢻⠀⠀⣿⢀⡶⠋⠙⠲⠞⠋⠛⢦⠀⣿⠀⠀⠀⠀⡀⡶⠛⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⡇⠀⡇⢸⣇⠀⠸⡇⠀⠈⠸⡄⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⣾⠉⡇⠀⣯⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⢸⡏⢹⡄⢷⠀⠀⠀⠙⢦⣀⠀⠀⣠⠞⠁⠀⡏⠀⠀⠀⣿⠀⡇⠀⣿⠀⡏⠙⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⡇⠀⡇⢸⡇⢸⡇⠈⢧⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⢰⠇⠀⣀⠀⣿⠀⡇⠀⣿⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢺⠉⢷⡇⠈⠁⠀⠁⠘⠃⠀⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡍⢳⣿⠀⠁⠀⠀⠀⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⣴⠚⠳⠖⠛⢦⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠿⢰⠚⠛⠞⠛⢳⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⣄⠀⠀⣠⠟⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠘⢦⡀⠀⣠⠞⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠓⠞⠁⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢳⡀⠀⠀⠙⠚⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⢰⣿⣶⡆
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⣼⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣼⣏⣀⡀⠀⠀⠀⢰⣿⣯⣤⡄
⠀⠀⠀⠀⠰⢫⣿⣿⢏⣠⠄⠀⠀⠀⠀⢰⡿⢀⣠⡄⠀⠀⠀⢰⣿⣿⣶⠀⠀⠀⠐⠛⣻⡟⠀⠀⠀⠀⠞⢻⣿⡿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢛⣿⣿⠃⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠚⢁⣿⣿⠃⠀⠀⠀⠀⠀⠟⠋⣽⡟⠀⠀⠀⠀⠀⣸⣿⣿⡶⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣸⣿⣃⡀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠟⣿⡿⠁⠀⠀⠀⠼⠿⣿⠁⠀⠀⠀⢀⣿⣿⡟⠁
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣹⡟
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣶⣶⣶⡶
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣼⣿⣿⣿⣶⡦
⠀⠀⠀⠀⣰⣿⡟⠀⠀⣀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣀⠀⠀⠘⠛⢻⣿⣿⣿⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣤⡄⠀⠀⢀⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠛⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠾⢿⣿⡿⠁⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⣼⣿⣿⡶⠀⠀⠀⠀⠉⢹⣿⣿⠃
⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢠⣿⡿⢁⣠⣴⠂⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠟⠋⠉⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣁
⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠋
⠀⠀⠀⠀⠀⢰⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⢀⡀⠀⢠⡀⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⢸⣿⣄⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣤⣤⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⣼⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣿⡏⠈⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⣄⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⣤⡀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠸⣿⣷⣬⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⣿⣿⠆
⠀⠀⠀⠀⠀⢻⣿⡛⠻⢿⣧⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠹⣦
⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠹⣧⠹⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠀⠈⢁⠌⠁⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠰⠘⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣇⠸⠋⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣤⡁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠘⣋⡆
⠀⠀⠀⠀⠀⠰⠗⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁⢸⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⡀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⢠⠆⠀⢌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡴⢿⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠃⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠢⠀⡤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴
⠀⠀⠀⠀⠀⠀⠀⠘⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣉⣛⣛⣛⣛⣛⣃⣈⣋⣻⣋⣙⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣿⣭⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣤⣀⣀⣀⣀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠠⣄⣀⡀⢠⣤⣤⣤⣴⣶⣶⣾⡏⡿⢁⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⠹⡇⣿⡿⠟⠿⠿⠿⠿⠷⠙⠉⢀⣀⣤⣀⣀⣀⠈⠉⠁⠀⠋⠉⠉⠉⠉⣿⣿⣸⡇⣾⣿⠉⠉⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠻⠿⠿⣿⣿⣿⣶⣦⣤⣤⣄⣀⢀⣿⣿⣿⢠⣿⣿⣶⡞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⢇⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢳⣼⣿⢻⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣶⣤⣀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠛⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⣿⣿⡿⠃
⠀⠀⠀⠀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠉⠙⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢀⣤⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⢴⠻⠻⠃⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠀⠀⠀⠈⠻⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢀⠀⠙⠆⠀⠐⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢳⡀⠀⠀⠀⣀⣤⡴⠆⠒⠒⠻⡆⠈⡆⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠔⠛⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠁⣠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠱⣄⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠒⠂⠀⢀⡠⠄⠀⠉⠀⠀⠐⢮⡀⠀⠀⠀⠀⠀⠙⣤⡀⠀⠀⣀⠀⠀⠒⠁⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⡐⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠃⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠱⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣀⣤⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣆⠀⢷⣠⠤⠤⠤⠞⠛⠉⠀⠀⠀⠈⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢦⠀⠀⢀⠀⠘⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣣⡀⠀⠂⠀⢈⡷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⢠⣠⠦⠗⠋⠁⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⠿⠿⠿⢿⡛⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣻⣛⣿⣟⣻⣛⣿⣛⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⡀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⠀⠀⠠⡄⠈⢢⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠀⠀⠀⠸⠀⠀⠚⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠐⠉⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠈⠀⠁⠀⠀⠀⢠⠄⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠢⢅⣀⠤⠀⠀⠀⠐⠈⠀⢄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠁⠀⠐⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠉⠃⠂⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⠄⣀⣀⣀⠠⠦⠄⠀⠔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⠀⠀⠀⢡⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢂⠀⣅⠣⢠⡝⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⢀⠜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣒⣒⣒⣂⣁⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠫⣎⠝⢆⠀⠀⠀⠀⢳⠰⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠆⠀⠀⢀⠂⠂⠋⠀⠀⠀⢠⡿⡳⣻⡿⠋⣠
⢄⠀⠀⠀⠈⠣⠀⠁⠀⠀⠀⠀⠡⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⢜⣿⠏⠀⠔⢀
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠣⣳⠟⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⢀⢔⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢑⠕⡪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⠶⢦⡄⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣸⠇⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠉⠛⠶⣤⡾⠋⠉⠛⡷⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠛⠶⣤⣀⠀⡜⠁⠉⠀⠀⠀⢀⠀⢀⠿⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠈⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠒⢶⡁⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⣀⠀⡠⠎⠀⠉⠛⠶⣄⡀
⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠘⡟⠉⠀⠀⠁⢀⣤⡀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠓⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠁⠀⠀⠀⠀⠉⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⡆⡀⠀⠀⢰⡟
⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢰⠁⠀⠀⣼⠇
⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢝⢦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠲⠤⣄⠀⣴⣦⡼⠛⣋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣴⣎⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⢠⣹⣇⣰⣦⠀⠀⢀⣾⠋⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡜⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣍⢹⣿⣀⠀⣸⡏⠆⡉⠥⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢸⣷⣦⣶⣦⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣜⠿⣿⣥⣽⢀⡬⢂⣠⣤⣤⠀⠀⠀⢾⣿⣿⣿⣿⡆⠀⣿⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣌⢱⡼⣿⣿⣾⣣⣿⣯⡾⣿⣷⡀⠀⠸⣯⡶⠛⠋⠑⠛⠰⣶⣦⣤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⣿⣿⣿⡿⠿⣞⡥⣶⣻⣿⣿⣄⣇⠐⢿⡇⠀⠀⠀⠀⠹⡏⠸⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣻⣷⡻⠋⠉⠛⣿⣿⣿⣿⣿⣷⣾⣿⠶⣄⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡩⢴⣿⣿⣿⠂⠀⠀⢠⣿⣿⣿⣿⣿⣟⣿⡏⠀⠘⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⢿⣿⣿⠢⠀⠀⣿⠈⣿⠟⠋⢉⣉⣀⣀⡙⠂⠀⠀⠀⡗⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⣟⠺⣆⠀⣙⣾⠃⣴⣾⣿⡟⣿⣿⣗⣦⡄⢰⢠⠃⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠿⢿⣷⣧⠹⣶⣿⠟⠛⢛⣁⣐⣾⣿⠃⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠭⣙⣛⣿⣿⣷⡘⣿⠷⠟⠋⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠒⠒⠒⣒⣒⣶⣿⣿⣿⡌⠃⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⢈⣉⣩⡭⠭⠶⣒⣿⣿⣧⣤⣤⣴⣾⣿⡾⣟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⢾⠀⠀⠀⠀⢀⣠⠴⠖⠋⣩⣶⢟⡿⢫⡟⢹⠻⡇⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⠀⠈⠉⠀⢀⣴⡿⠋⣠⠏⢠⠟⠀⠼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠚⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⠄⠒⠀⠛⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⡿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢁⣰⣿⠟⠉⠀⠀⠈⠳⢾⣿⣿⣿⢯⠁⠀⠁⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣾⣿⣿⡁⠀⠀⣀⣀⣀⡀⠀⠈⣷⠦⣀⠀⡀⢀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⡀⠀⢋⣥⣤⣄⠀⠀⠳⣶⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⢹⣿⡟⠻⠁⠀⠈⠙⠁⠀⠀⠀⠀⠻⠱⡍⢋⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣠⣌⣠⠄⠀⣸⣿⡇⢠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⣡⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⢀⠀⢸⣿⣙⣿⣶⣾⣿⣿⣷⢠⡤⠤⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡄⢍⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⣼⠀⠀⠉⣻⠿⠿⣿⣿⣿⣿⣇⠓⠛⠀⠀⠀⠀⠀⢀⠀⠉⣿⣿⣿⠟⢸⣿⡿⡷⠀⠀⠀⠀⠀⡧⢸⢸
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣷⣿⢸⢠⣧⣾⣿⣶⣦⣤⠈⣿⣿⣿⣦⣤⣤⣤⣴⡶⠞⠁⠀⠀⣿⡏⢀⢰⡿⠁⢀⢧⠱⡄⠀⢀⡄⡇⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢳⣎⠹⣿⠘⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠸⣷⣿⣿⣷⣠⣺⣿⣆⣁⠀⢸⢻⠃⣿⠏⣰⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣻⡀⣿⣿⠛⠿⣿⣿⣟⣭⣀⡀⠙⢋⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣇⠀⣏⡿⢀⣯⣤⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣎⣿⣽⣿⣿⠋⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣴⢞⣉⠴⠊⠉⠉⣛⣿⣿⣿⣄⣿⡇⣿⣯⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢿⣘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠀⠀⣀⣾⣿⣿⡿⠛⠁⠀⠀⠀⠉⠉⠉⠉⠻⣿⣿⣴⢃⡿⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣎⢻⣯⣠⣿⣇⣈⣉⠙⢛⣿⠃⠈⢿⣿⣦⡶⢟⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠟⣛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠻⡿⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢏⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣬⣻⣿⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢦⣠⣾⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠿⠟⠋⠁⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡝⠀⠙⠻⠿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣙⣦⠤⣺⣍⠛⣦⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠂⠀⢀⣤⠞⣩⣷⠤⣴⣋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣿⡛⢡⣬⣛⠶⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⠶⣛⣥⣌⢛⣿⡷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠽⡷⣯⣽⣿⣿⣿⡿⣿⣷⣾⣿⠿⣿⣾⣿⣯⣽⢾⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠍⢻⣿⣆⢠⣠⣆⡄⢠⣿⡟⠉⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡛⢛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠉⠛⠋⠋⠙⠉⠛⠻⣻⠿⣿⡛⠁⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠄
⠀⠀⠀⠀⣾⢟⣿⣿⣿⠿⣻⡿⠋⠁
⠀⠀⠀⣰⠋⠞⡹⠋
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣦⣤⣤⣤⣄⣞⣶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣿⢿⣿⣿⣼⣷⣦⣿⣷⣯⣿⣿⣟⠕⣡⡶
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣯⣽⣾⣿⡿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣮⠷⣆⠄
⠀⠀⠀⠀⠀⠐⡁⡀⠀⠀⠀⠀⠀⣠⡿⣫⣽⣿⠟⠋⠁⠀⠀⠀⠘⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣬⣧⣂⡀
⠀⠀⠀⠀⠀⠈⢊⣄⣠⣤⣦⡶⠿⢉⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⢶⣿⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡴⡾⠿⠿⠏⠛⠀⠈⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣾⡿⣧⣾⣶⣤⡉⠏⠻⣿⣿⢶⣯⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠉⠐⠂⠀⠈⠀⠰⢒⣤⠤⣄⣀⠀⠀⠀⡈⠻⣿⣿⣷⣿⣿⡿⠟⠀⣀⣥⣾⣿⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⢀⡼⠄⠘⠈⠁⠡⠛⢓⣶⡶⣶⣤⣷⣶⢶⣶⣾⡿⠿⠿⠏⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠉⠙⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣿⠿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣦⣤⣄⣀⡀⠀⠀⠀⣀⡀⢀⠀⠀⠀⣼⡟⠁⡸⠀⢸⠈⠹⠛⣼⣿⣿⣿⣿⣿⣿⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃⠀⠌⠀⡤⠀⢨⣿⣿⣷⣤⣤⠈⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⣿⣿⣿⡄⢰⣆⠀⠀⠀⠃⢠⣾⣿⣿⣿⣿⣿⠄⠀⠙⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡿⣴⣇⡀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⢿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⡻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠰⠾⠛⠋⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠋⠉⠀⠀⠁⠈⢻⣿⣿⣿⡿⠿⠶⠤⠴⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣄⣀⣤⠀⠀⠀⣀⡀⠀⠀⢸⣿⢹⡴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡇⠀⠀⠀⠀⠉⢻⣦⣼⣏⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣽⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣻⣆⠀⢀⠀⠠⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣳⠀⠀⠀⠀⠀⣼⣿⡿⢸⣿⢹⣿⠀⣾⠂⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠳⣶⣶⣤⣾⣿⣿⣷⣾⡿⢸⣿⢰⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⣿⣿⡇⠀⠀⠐⠿⠿⢿⡿⠿⠿⠿⠫⠛⠛⠧⠺⠷⠀⠉⠉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠉⠉⠁⠀⠀⠀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣯⠴⠖⠒⠒⠂⠀⠄⠉⠉⠉⠙⠒⠦⣤⣹⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⠀⣰⠃⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠻⠿⢿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⢠⡀⠀⠀⣴⠲⣦⡀⠀⢢⠀⠀⠀⠀⠀⠈⠓⠻⣿⣷⣯⡷⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⠈⢿⣦⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣍⣳⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣷⣾⣧⠀⠈⢿⣦⠀⠀⠹⣄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣏⣣⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⡿⣿⡄⠀⠀⠻⣷⡀⠀⠙⣧⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣹⢷⣝⢦
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣿⣧⠹⡗⢄⠀⠀⠙⣷⡄⠀⠈⣧⠀⠹⡄⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠹⣿⣿⣟⣧⣷⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⡸⢹⣿⡀⢹⡇⢀⠀⠀⠹⣿⡄⠀⠈⣧⠀⢹⡀⠀⠀⢰⡟⠉⢻⣄⠀⠀⠀⠘⣬⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡗⠈⣿⣧⠀⢻⡄⠀⠀⠀⢹⣿⡀⠀⢹⡄⠀⢳⠀⠀⣻⡇⠀⠘⣿⣆⠀⠀⠀⢻⣏⠻⣿⡞⢷⡀
⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢰⠃⠀⢻⣿⡀⠀⢻⣆⠀⠀⠈⣿⣧⠀⠀⣧⠀⠘⣇⠀⢹⣇⠀⠀⢹⣿⢆⠀⠀⠀⠿⡆⠹⣿⣼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡌⠀⠀⠸⣿⣧⠀⠀⢻⡄⠀⠀⢹⣿⡀⠀⢻⠀⠀⢻⠀⣸⡏⠀⢠⡾⣿⣷⡄⠀⠀⠀⢽⡀⠹⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢿⣿⣆⠀⠀⢻⡀⠀⢸⣿⠇⠀⢸⠀⠀⢸⠀⢸⡇⢀⡿⠁⣿⠟⣿⡄⠀⠀⠘⣿⡀⢿⢋⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠈⣿⣿⠀⠀⠘⡇⠀⢸⣿⠀⠀⡿⠀⠀⢸⠀⢸⡇⣾⠃⢀⣿⠀⢸⣧⠀⠀⠀⢸⣧⢸⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢀⣿⡏⠀⢀⠇⠀⠀⢸⠀⣼⣿⡏⠀⢸⡏⠀⣸⡿⠀⠀⠀⢸⢹⡆⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢻⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣤⣿⡟⠀⠀⡜⠀⠀⠀⡌⢀⣿⣿⠁⠀⣿⠇⠀⣿⠇⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣇⣠⡞⠉⠙⠋⠉⠳⢄⠈⠉⠙⠉⠀⢀⡜⠀⠀⠀⢰⠁⣸⡿⠃⠀⢸⡿⠀⣸⣿⠀⠀⠀⠀⠀⣿⠛⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⡠⠋⠀⠀⠀⠠⢃⣼⣿⠃⠀⣠⣿⠃⢀⣿⠇⠀⠀⠀⠀⠀⠸⡇⢁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠄⠀⠐⣶⠟⢡⡏⠀⢀⣿⠃⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⣇⠸⣿⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⣾⠁⠀⣼⡟⠀⣼⡿⠀⠀⠀⠀⠀⢀⡆⢀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢹⡆⠈⠙⠶⢶⣤⣀⣀⣀⣤⣠⣴⠖⠛⠉⠀⠀⠀⣸⠃⠀⠀⣿⣁⣼⠟⠀⠀⠀⠀⠀⡀⡾⠀⢸⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣿⡀⠀⠀⠀⠈⠉⠉⠉⠉⢳⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣴⡶⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣨⠿⠋⠀⠀⠀⣿⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⡉⠛⠦⢤⣀⣀⡀⠀⠀⠀⠈⠙⠲⠾⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠊⠁⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢿⣷⢤⠀⠀⠀⠈⠉⠛⠓⠶⠤⢤⣤⣀⣀⣀⣀⣀⣤⠤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⡟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡺⠿⠷⣬⣙⠛⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠐⠂⠉⠓⠦⣄⣀⣉⣙⠳⠦⢄⣀⣀⠀⢀⣀⣀⣠⡤⠤⢒⣋⠶⠶⠶⠾⠿⠛⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣘⣛⣠⣭⣥⣤⣤⣴⣚⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠻⠿⠛⡿⠿⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠁⠄⠉⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⡀⠀⣤⣠⣶⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣆⣿⠏⠿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠒⠻⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⣿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⡵⡀⠀⡼⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣀⡶⢦⣬⣿⣿⣿⣿⣿⣿⣯⠀⢈⠆⠀⠀⠀⢠⣤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠒⠂⣁⣂⡀⢬⢻⠏⠉⠒⠺⠾⣿⣿⣿⣿⣿⣿⣿⣧⣼⠀⠀⠀⠁⠀⠀⠀⠹⠛
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠱⣿⠥⠤⣍⣚⣻⣿⣿⣿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠛⠁⠰⠟⠛⠿⠿⠿⠟⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⡀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⣠⡄⠠⣤⠀⠚⠀⠀⠀⠀⠀⠀⣾⢠⠂⠀⠀⠀⣠⣽⡁⠀⣦⠸⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⠀⡄⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⢠⣾⡟⠁⢨⠃⠀⠀⠃⠠⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠺⣿⣻⣿⣿⣿⡏⠀⠀⣤
⠀⠀⠀⠲⣦⣤⡼⢿⣿⣿⣿⠟⠋⠀⠘⣥⣤⣶⠆⠀⣠⣤⣤⣶⣶⣿⣦⣀⠉⠙⠻⠿⢃⣤⣶⣥⣶⣶⣦⣤⣤
⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡟⠛⠋⠀⢨⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠙⠛⠛⠉⠉⠁⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣯⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⠖⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣿⠋⠀⠀⠀⠀⠀⠰⢿⣿⡿⠃⠀⠀⠀⠀⠴⠟⠛⣿⣿⡿⠋⠀⠀⠀⠀⣾⣿⣯⡴
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣥⠀⠀⠀⠀⠀⠀⢀⣼⣏⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠆⠀⠀⠀⠀⠀⠸⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⢩⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣵⠀⠀⠀⠀⠀⠀⣰⠿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⢟⡻⢻⣻⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠠⠃⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠲⣀⠀⠀⠈⣄⠈⠀⠀⠀⠰⠆⠀⠀⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢤⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠈⠁⠔⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐⠀⠀⡐⠀⠀⢰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⢰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠑⠀⠀⠀⠀⠰⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠄⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢌⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠁⠊⠓⠀⠀⠀⠀⠀⠀⠠⣤⠤⠶⢦⣤⡤⢤⣀
⠀⠀⠀⠀⠩⠉⠉⠉⠙⣟⠀⠁⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠓⢦⣌⡑⠢⢄⡀
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠘⢧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠈⠉⠲⢄⣈⠒⠤⡄⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⢧⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠁⠉⠢⡀⠈⠒⠤⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢣⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠈⠔⠠⠀⢸⠀⡧⡀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠱⡐⠄⠀⠀⠈⢀⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠰⠃⠀⢁⠀⠀⣇⢀⡄⠀⢀⠐⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠹⡌⢀⠀⠀⠀⠑⠈⠂⠠⢍⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠸⠅⡶⠋⡎⠀⠵⢃⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠂⠀⠀⠀⠈⠂⠀⢀⠁⣀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠠⠄⠣⠀⢀⣀⠈⠀⠀⠊⠁⠀⡘⡀⠦⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠣⠁⡀⠀⠀⠀⠀⢦⡀⠈⠦⣐⠀⠠⠺⠀⠀⠀⠄⠠⠊⠀⠀⠀⠁⢁⠀⠙⡀⠐⠀⠀⠀⡰⢻⠁⣰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠡⡘⠄⠀⠀⠀⠀⠠⡉⠲⠀⠀⠁⠐⢇⠰⡀⢀⠀⡇⠀⠀⠀⢠⠰⠀⠀⢰⣆⣀⣠⢎⠔⠁⠘⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢂⠀⠀⠀⠀⠘⢄⠐⢄⠀⠀⡀⠀⠐⣈⠂⠁⡀⠀⠀⠎⠀⠱⡀⠌⠁⠀⠜⠁⠀⢀⡤⠔⠈⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠈⢀⠢⡀⠀⠀⠀⠀⠁⡀⠈⠐⠇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠔⠘⠀⠀⢀⣌⠐⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠈⠂⠑⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠁⠀⠀⠀⠃⡀⠱⠤⠴⠋⠁⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠈⠠⠀⠠⠀⠀⠀⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡌⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⡏⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠈⠀⠄
⠀⠀⠀⠔⡈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠐⠄⡀
⠀⠀⠀⢆⣿⣿⣟⣤⡀⠀⠀⠀⢠⣀⣠⠀⠀⠀⠀⠀⠀⡀⠀⣠⣦⢂⣄⢹⣿⣿⠉⢠⠄⠀⠀⠀⠀⠀⣳⣀⣄⡀⢀⣤⣤⣵
⠀⠀⠀⠈⠻⠿⠿⠟⠉⡲⢬⣿⣿⣿⣷⢦⢀⣀⣀⣦⣤⣶⣿⣿⣿⣿⣿⢸⡿⣯⣧⣴⣦⣀⠀⠀⢀⣼⣿⣽⣿⡽⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⣉⣉⣠⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠛⠛⣿⣶⣽⣻⣿⣿⣿⣿⠉⠀⠀⢯⡉⠛⠉⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠓⢄⠀⠈⢳⡀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠈⠀⠀⠢⣿⣦⣦⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⣿⡛⠛⠂⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⣷⣷⠇⠀⠀⠀⠀⢀⣾⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⣾⣿⣿⡗
⠀⠀⠀⠀⢠⣿⣿⠏⠀⣀⡀⠀⢀⣾⣿⣁⣠⣤⣤⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⣾⣿⣿⣾⣿⠏⠀⠀⡼⠿⠟⠛⣻⡿⠁⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣷⡟⠀⠀⠀⣸⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠸⠛⠋⣹⣿⠏⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⠖⠀⠀⠘⢉⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣾⣿⠏
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⣾⣋⣤⡤⣤⠀⠀⠛⠛⢹⣿⣿⠋⠀⠀⠀⠀⣾⣿⣿⡤⠀⠀⠀⢸⡿⠟⢩⡿⠃
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡴⠁⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠘⠋⣿⠟⠁⠀⠀⠀⠈⠀⢠⡟⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠉⠉⠓⠦⡀⠀⠀⢀⡤⠖⠲⠦⣄⣠⠴⠚⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀⣀⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠛⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠲⣄
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣠⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣤⡤⠶⠛⠁
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠲⠦⡴⣶⠋⠈⠛⠶⢤⣤⣤⣤⣤⣤⣴⠾⠦⣤⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠉⠓⠶⣤⣤⣬⣥⠶⠟⠉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⡀⠀⠀⠀⠀⠀⠠⠀⠛⢛⣿⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡠⠂⠀⠾⠓⢫⠟⠀⠀⠀⠀⠀⠀⠀⠠⣰⠟⠁⠀⢀⠄⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⢠⠎⠀⢀⠞⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⡰⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢀⣼⣯⡴⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣀⣠⡄⠀⠀⠀⠀⠀⢈⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢁⡞⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⢠⠞⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⡴⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡴⠋⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋

⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⣰⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣶⣶⣶⠆⠀⠀⠀⢀⣶⣶⣶⡆
⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⡀⠀⠀⢠⣿⡟⣀⣀⣠⣤⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⣸⣿⣿⣷⣾⣿⠃⠀⢠⣿⡿⠿⠟⣿⡟⠁⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⣼⣿⣿⣯⣶⠆⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠠⠿⠟⢻⣿⡿⠁⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⣴⣿⣿⣧⣤⡤⠀⠀⠐⠛⣻⣿⣿⠏⠀⠀⠀⢰⣿⣿⣧⣶⣾⠟
⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⢀⡾⠃⣀⣀⣀⠀⠀⠾⠿⢻⣿⣿⠟⠁⠀⠀⠀⢠⣿⣿⣏⡀⠀⠀⠀⣼⣿⠿⢻⣿⠋
⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠟⠛⠉⠉⡱⠃⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠠⠿⢻⣿⠟⠀⠀⠀⠀⠋⠁⢠⡿⠁
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⡀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠼⠁
⠀⠀⠀⠲⠤⣐⢆⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⠉⠉⠉⠉⠙⠃⠘⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢊⡷
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠂⠁⠈⠳⣄⠀⠀⠐⠶⠛⠛⠒⠛⠉
⠀⠀⠤⠤⠭⠭⠶⠦⠄⢸⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⠀⠠⠖⠛⠛⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⠀⠰⢦⣚⡛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⠃⠀⠀⠀⠀⠈⠉⠓⠲⠦⢤⣄
⢀⣀⣀⣤⠴⠀⠀⠀⠀⣼⠋⣝⠂⠀⠀⠀⢀⡠⠞⠁⠘⠛⠟⠓⠀⠀⢀⠀⣀⡋⠛⠒⢶⡄
⠈⠉⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠉⣀⠴⠈⠉⠀⠀⠀⠐⠀⠐⡾⠃⠰⠏⡜⢁⡤⠄⠀⠀⠙⠢⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⠁⠀⠀⠀⣠⠌⢻⣛⡷⠀⣠⣤⣤⣀⠀⢏⣰⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠈⢀⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⢀⡾⢱⡏⠉⠒⠶⠤⢤⡄⣠⡾⠃
⠀⠀⠀⢀⠤⣲⠟⠀⣾⠀⠀⠀⠀⠀⢠⠏⣼⠿⢶⣤⣀⡀⠣⠴⠋⣀⡠⠴⠞⠋⠀⠈⠃⠀⣀⣠⣴⠶⠛⠉
⠀⠀⠀⠈⠛⠁⠀⠀⢿⡀⠈⠀⠀⣠⠟⢠⡿⠀⠀⠀⠈⠉⠁⠀⠉⠀⢀⣴⠶⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣢⣴⣾⣥⡶⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⣰⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠻⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⣦
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀
⠀⠀⠀⠀⠀⠀⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡤⠈⢹⣦
⠀⠀⠀⠀⠀⠀⣿⠻⡷⠦⣤⣤⣄⠀⠀⠀⣀⣀⣀⣀⣠⡴⠟⠀⠂⢸⠌⢧⡀
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠙⢦⡈⢉⡉⠉⠉⠉⠁⠀⠀⠀⠀⡿⠀⠀⢹⡆
⠀⠀⢀⡾⠛⠳⠒⡙⠆⠀⠀⠀⠀⠀⠙⢦⡀⠉⠒⠦⣤⣀⠀⠀⣼⠃⣠⡴⠛⣄
⠀⠀⠸⡇⠀⣴⠚⠉⢠⡤⠤⠤⠤⠤⠤⠤⠽⠶⣄⡀⠀⠉⠃⣰⠟⠛⠁⠀⠀⠘⣦
⠀⠀⠀⠹⣆⢻⣆⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠼⢃⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⠳⣍⡻⣦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⣧⠀⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠈⠻⣮⣙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠳⠀⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠛⠶⢤⣄⣀⠀⠀⠀⢀⣼⠏⠙⠛⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣻⣋⣙⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣽⣿⣯⣭⣿⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⣠⡤⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣠⠞⠛⠘⠁⠀⠉⢙⣮⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠁⠀⠀⠀⠀⠈⡀⠀⣨⣧⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⡄⢠⡿⠟⠀⠀⠀⠀⠀⠀⠐⠀⢀⣸⠇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⣼⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⢰⡾⣃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⢤⡼⣫⣴⠟⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⢄⡽⣫⡾⠛⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠀⢰⣪⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠲⢄⡰⠅⠀⠙⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣲⠿⣛⡽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⡼⠃⠀⣘⣽⢯⣀⣀⣀⣀⡤⠤⠔⣒⠯⢑⣊⣥⡶⠟⠋
⠀⠀⠀⠀⠀⠀⣴⠞⠋⠛⠛⠛⣿⠓⠀⣴⣿⣯⣄⣀⣀⣀⣠⣤⣤⡴⠶⠿⠛⠛⢛⡛⠉⠛⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣀⠀⢀⡟⠐⠰⣿⡿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠉⢈⣉⣁⣀⣀⣉⢙⡆⠀⠘⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⡄⠘⣿⡇⠈⠻⣷⣖⡒⠒⠚⠛⠛⠉⠉⠉⣉⣉⣩⡿⠛⠀⢀⢜⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣿⣿⣄⠀⠀⠩⠝⢛⣛⣛⡻⠿⠿⣿⠟⠉⠉⠈⣁⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣦⣤⣄⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢰⣿⣶⡆
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⣼⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣾⣏⣀⠀⠀⠀⠀⢰⣿⣯⣤⡆
⠀⠀⠀⠀⠰⢻⣿⣿⠏⣠⠄⠀⠀⠀⠀⢠⡿⢀⣠⡄⠀⠀⠀⢠⣿⣿⣶⠀⠀⠀⠐⠛⣻⡟⠀⠀⠀⠀⠞⢻⣿⡿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢛⣿⣿⠃⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠚⢉⣿⣿⠃⠀⠀⠀⠀⠀⠟⠋⣽⡟⠀⠀⠀⠀⠀⣸⣿⣿⣶⠂⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣸⣿⣃⡀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠟⣿⡿⠁⠀⠀⠀⠼⠿⣿⠁⠀⠀⠀⢠⣿⣿⡟⠁
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣹⡟
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣶⣶⣶⡶
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣼⣿⣿⣿⣷⡦
⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⢀⠀⠀⠸⠟⢿⣿⣿⣿⠁
⠀⠀⠀⢀⣿⣿⣷⣿⣿⠏⠀⠀⠀⢠⣿⣿⣤⡄⠀⠀⢀⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠛⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣷⣶
⠀⠀⠀⠈⠀⢠⣿⡿⠃⠀⠀⠀⠀⠾⢿⣿⣿⠁⠀⠀⠀⠀⣽⣿⡿⠁⠀⠀⠀⠀⠀⣼⣿⣿⡶⠀⠀⠀⠀⠉⢹⣿⣿⠃
⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠒⠒⠲⠤⣀
⠀⠀⠀⢠⠞⠁⢀⣀⣤⣤⣀⣀⡀⠀⣀⡩⠷⠶⠶⠖⠲⠶⠦⠤⣄⡀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠖⠲⢶⡤⠞⠉⠀⠀⢀⣀⣀⣤⣤⣀⡀⠈⠳⡄
⠀⠀⢠⠏⠀⢰⠁⠀⠀⠀⣀⠴⠛⠉⠀⣀⣀⠤⠤⢤⡤⠤⣄⣀⠀⠉⠳⡄⣠⠞⠁⠀⣀⣠⠤⢤⡴⠋⠀⢀⣠⠴⠚⠛⠦⣀⠀⠀⠀⠈⡆⠀⠸⡄
⠀⠀⢸⠀⠀⣿⠀⠀⣠⠞⠁⢀⡤⠒⠉⠁⠙⢦⡀⠀⢳⡀⠀⠀⢙⡆⠀⣸⣇⠀⢰⡋⠀⠀⢀⡞⠀⢀⡴⠋⠀⠉⠒⢦⡀⠈⠳⣄⠀⠀⣿⠀⠀⡇
⠀⠀⢸⠀⠀⢸⣀⡜⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⢇⠀⡰⠋⠀⡰⠃⠘⢦⠀⠙⢄⠀⡼⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢣⣀⡇⠀⠀⡇
⠀⠀⠘⣇⠀⠈⢿⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⠎⠀⣠⠊⠀⠀⠀⠀⠑⢄⠀⠳⣇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢻⠁⠀⣸⠃
⠀⠀⠀⠹⡄⠀⠘⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣇⢠⠏
⠀⠀⠀⠀⠙⣆⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠘⡀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⠀⢸⡏
⠀⠀⠀⠀⢸⠏⠳⣄⠀⠈⠓⠦⣄⣀⠀⢀⡠⠞⠁⢀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠣⡀⠈⠳⢄⠀⠀⣀⣠⠴⠚⠁⠀⣸⠀⠸⡇
⠀⠀⠀⠀⢸⡆⠀⡏⠙⠲⠤⣄⣀⣠⠝⠋⢀⡠⠶⠉⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡈⠶⠐⠒⠉⠉⢀⣀⣠⠤⠖⠋⢹⠀⢰⡇
⠀⠀⠀⠀⠈⡇⠀⢳⠀⠀⢀⡠⠊⠁⣠⠔⠉⠒⠀⠀⠀⠁⠊⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠈⠀⠀⠀⠒⠘⠫⣍⠈⠑⢄⠀⠀⠀⡞⠀⢸⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠳⠴⠋⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠙⠦⠜⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠲⣄
⠀⠀⠀⠀⠀⣰⠃⠀⡴⠲⣄⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣠⠖⢦⠀⠘⣆
⠀⠀⠀⠀⢀⡇⠀⡸⠀⠀⠀⠑⢄⡀⣙⣦⡀⠤⠀⠀⠀⡀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⡠⢀⠀⠀⠀⠤⣀⠔⠋⢀⡤⠊⠀⠀⠀⢧⠀⢸⡀
⠀⠀⠀⠀⢸⠇⠀⣇⣠⠴⠒⠋⠉⠁⣀⣀⠤⠐⠶⡈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣁⠶⠊⠁⣠⣔⠋⠉⠙⠒⠦⣄⣸⠀⠸⡇
⠀⠀⠀⠀⢸⡆⠀⡏⠀⢀⡤⠖⠋⠉⠀⠀⠑⢦⡀⠈⢣⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡜⠁⢀⡴⠊⠁⠀⠉⠙⠲⢤⡀⠀⠙⢦⣠⡇
⠀⠀⠀⠀⣸⡇⠀⢣⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠱⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⡄⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠹⣄
⠀⠀⠀⣰⠃⢱⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡄⠀⠘⣆
⠀⠀⢠⡏⠀⠀⣧⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢹⢆⠀⠙⢄⠀⠀⠀⠀⡠⠋⠀⡴⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣷⡀⠀⢹⡄
⠀⠀⢸⠀⠀⢸⠉⢣⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡞⠀⠑⣄⠀⠱⡄⢠⠞⠀⣠⠊⠀⢳⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⡜⠁⡇⠀⠀⡇
⠀⠀⢸⠀⠀⣿⠀⠀⠙⢦⡀⠈⠓⠤⣀⠀⣠⠞⠁⠀⡼⠁⠀⠀⣈⠇⠀⢹⡇⠀⠸⣅⠀⠀⠈⢧⠀⠈⠳⣄⢀⣀⠤⠚⠁⢀⡴⠋⠀⠀⣿⠀⠀⡇
⠀⠀⠘⣆⠀⠸⡀⠀⠀⠀⠉⠲⣤⡤⠖⠋⠁⠀⣠⠞⠓⠒⠚⠉⠀⣀⡴⠃⠙⢦⣀⠀⠉⠛⠒⠚⠓⠒⠒⠉⠉⠀⣀⣤⡖⠉⠀⠀⠀⢀⠇⠀⣰⠃
⠀⠄⠀⠘⢦⡀⠈⠉⠛⠛⠉⠉⠁⠀⠀⣀⡴⠞⠷⠤⠤⠶⠖⠒⠋⠁⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠴⠶⠶⢶⣛⠉⠀⠈⠉⠉⠛⠛⠉⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠉⠓⠦⠤⠤⠤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠤⠤⠴⠚⠉