Bait Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⢹⡀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡟⢉⣌⡙⢿⣿⠆⣟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣇⠙⠿⢃⡼⠃⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⣿⠛⠿⠚⠋⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠉⠀⠘⢹⣿⡿⠷⠦⢄⡀⠀⡀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣛⣁⡀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⢉⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠛⢻⠶⢿⣤⣀⣀⣀⣀⣠⣴⣾⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⣥⣤⣤⣤⣔⠁⠀⠀⠀⠀⠉⠉⢉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣛⣋⣁⣠⡾⠉⠉⠉⠑⠒⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠛⢋⣽⠟⠓⠒⠆⠀⠀⠀⠀⣠⠶⣟⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠾⠟⣷⡿⠧⠤⠄⠀⠀⢀⣠⠔⠋⠀⠀⠀⢙⡾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⢶⣦⡾⠯⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣁⡴⠟⠁⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣧⠾⠋⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⡀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣴⠿⣧⣈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡮⠗⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⡆⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⣿⠀⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠻⣷⠧⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⣴⣿⣿⣿⡀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⡿⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⣠⣴⠂
⠀⠀⠀⠀⠀⣀⢈⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣝⡿⢿⣿⡿⠟⠃⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⢹⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠻⠟⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡷⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇
⡀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⢀
⣿⣻⣻⣻⣛⣛⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣟⣻⣟⣻⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢀⣼⠟⠋⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠒⠛⠁⠀⠀⠄⠀⠀⠀⠀⢦⠀⠀⠀⠀⢁⠀⠺⢷⡀⠰⡇⠀⢀⢠⣿⠀⠀⠀⠀⢱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢀⡄⠀⠈⢠⣿⡇⠀⠀⠀⠀⢛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡅⠀⠀⢠⣼⡇⠀⠀⠀⡄⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⠀⠠⢼⡇⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⢀⠃⠀⠀⠊⠀⠰⢺⡇⠀⠀⠠⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣔⡶⣖⡦⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⢐⠻⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠹⣇⡿⣿⡧⣦⠀⠐⡆⠀⠘⠀⠀⠀⠀⠂⠀⠀⠁⠀⠈⠠⢶⠆⠀⠘⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⣺⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢇⠀⠀⠿⠀⢰⠀⠀⠀⠰⠆⠀⠀⠈⣽⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠈⠽⡏⠉⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⡓⢷⡰⠀⠀⠀⢺⣦⠀⠀⠀⠀⠀⠀⠌⡀⠀⢰⡇⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⣭⠈⣷⡆⠀⠀⠈⠈⠀⠀⠀⠀⢀⠀⠀⡂⠀⢨
⠀⠀⠀⠀⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠸⣿⣦⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢨⠀⠀⠐⠊⠇⠀⠀⠙⣝⢿⠆⠀⠀⠠⣀⣀⡴⠃⠀⠀⠀⠰⠄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠂⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠈⠂
⠀⠠⠀⠀⢸⡀⠀⠀⠀⠄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠈⠿⣿⣿⣿⣄
⠀⠐⠢⠐⠉⠳⠤⠁⠀⠐⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡾⣿⣺⣿⡁
⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣄⣀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠏⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣿⠈⠻⣿⢧
⠈⠀⠀⠠⠡⠘⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠛⠿⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠙⣷⡕⢄⠀⠀⠐⢀⠄⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠋⠀⠂⠀⠀⣞⢰⣿⣿⣿⡷⠂⠀⠙⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠘⠛⠉⠁⠀⠀⠀⠀⣀⣀⣤⡀⠘⠃⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⣿⣶⠀⠉⠛⢿⣿⣧⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢿⡿⠀⠀⣼⡄⠈⠉⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡠⢤⣾⠿⠁⠀⢠⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣢⠤⠒⠁⠀⠉⠁⠀⢀⣠⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁
⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢻⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢊⠉⡣⠋⠀⠀⡅
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⣻⠉⠁⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⣀⠔⠉⠀⠀⢠⣾⣿⣦⡀
⠀⠀⢋⡴⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠀⡀⣠⠃⠀⠀⠀⠀⠀⡠⠖⠉⠀⣀⠤⠐⢈⡏⠀⠀⠀⢀⣿⣿⣿⣿⣧
⠀⠀⠛⣿⣿⣿⣿⣿⣿⣶⢄⡀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠌⠠⠔⠒⠉⠀⠀⠀⡘⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠠⣶⠜⢿⣿⣿⣿⣿⣿⣿⣮⡢⡀⠑⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⣼⣭⣦⠀⠻⣿⣿⣿⣿⣿⣿⣿⣎⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⣷⣶⣼⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠤⢸⢹⣿⣿⣿⣇
⠀⣿⣭⣽⠀⠀⠀⠀⠀⠉⠛⠻⠛⡡⠜⠁⠀⡠⠄⠒⠀⡠⡴⠒⢤⣤⠀⠀⠀⠀⠀⠀⢀⡤⠖⢉⢥⡄⠨⠁⠀⠘⣏⣿⣿⡿⡼
⠀⢿⠗⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠑⠑⣀⣀⣄⡀⠀⠀⠀⣠⠴⠋⠀⠀⠈⠛⠁⡄⠀⠀⠀⠹⣮⣭⡾⠃
⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⡀⠀⣀⠀⢀⡩⠋⠓⠋⠀⢀⠔⠁⠀⠀⠀⣀⢠⠄⠐⠁⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠂⠉⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠠⠒⠉⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠠⠐⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠣⡰⠳⢄⠆
⡄⠀⣿⡀⢠⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠑
⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣄⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢦⣈⣏⣡⣤⣤⣬⣎⣟⣹⠉⣿⠲⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣩⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠉⠓⠉⠛⣿⡟⠻⢿⣿⡿⠁⠀⠀⠀⢀⡀⠉⠲⢤⠤⡄⠀⢀⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⢹⠀⠀⠀⠀⠀⠈⠛⠀⠀⣵⢿⠂⢠⠿⠀⠀⠐⢳⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢋⣀⣤⠤⠤⠤⠤⠤⣄⣀⠀⠀⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡟⠁⠀⢨⣶⠩⠗⠢⡈⡳⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠟⢶⠦⣤⣀⣀⠐⠀⠀⠀⠀⠀⠘⢿⣦⣄⣔⡯⠒⠤⠬⠤⠽⠗⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠉⠁⠈⠉⠛⠒⠒⠒⠒⠂⠘⠓⠚⠋
⠀⠀⠀⠀⠀⠀⣰⢻⣿⡿⠛⢄
⠀⠀⠀⠀⠀⢠⠃⠀⠉⠀⠀⠈⠢⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⡎⠀⣀⡤⠞⠙⠓⠶⠦⠤⢦⠵
⠀⠀⠀⠘⠛⠉⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣶⣶⣶⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⣤⡾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡻⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⠷⠶⢾⣿⠇⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠤⢙⣿⣿⣿⣿⣯⣿⣼⣻⡷⠟⠉⠀⠀⠀⠀⠙⠷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠤⠸⢿⣿⣿⣶⣿⣿⣿⣷⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢠⣶⡢⡜⢢⡒⢼⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠘⠿⠿⠊⠀⠀⣘⣼⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠀⠀⠤⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⢀⣠⣤⡶⣾⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⡔⠁⢀⣤⠄⠀⠀⣾⠺⣷⡉⠁⠀⠀⠂⠈⠀⠀⣨⣽⡿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⢰⠟⠃⠀⠀⠀⢹⣆⡈⠓⠀⠀⡠⢖⡡⠂⠠⣿⡿⠳⣍⠻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠗⠀⠘⠀⠀⠀⢁⠔⠁⠀⠉⠀⠀⠀⠁⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠰⢂⡀⠄⢤⠀⠀⠀⠀⠀⢰⣿⣿⡿⠃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⣝⠷⠚⠁⢤⡀⠀⠀⣩⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⡀⠀⠀⢨⡿⠂⣔⣿⣿⠋⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡗⣠⣶⣶⣶⣶⣦⡴⠋⢀⣱⣿⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠘⣿⣿⣿⣿⣿⣿⣶⣶⠛⠏⠀⠀⠀⠀⢈⠒⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠈⠁⠐⠈⠛⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⠂⢀⡨⠂⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠶⠂⠈⠻⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣀⣀⣀⣀⣤⣤⣤⣤⣤⡿⠥⠤⠾⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠂⠐⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢀⡠⠤⢤⠄⠒⣤⠀⠁⡤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠊⢠⠏⠀⠀⡔⠁⣞⠀⠀⠁⠀⠀⠘⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠘⠦⡄⠀⠑⣂⣀⠤⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡀⠀⠀⢀⠀⠁⠀⠀⠀⣀⣀⣀⠔⠁⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠔⠂⠀⠈⠳⢤⣈⣀⣀⡤⠞⠀⠀⢀⠞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠆⠀⠀⢠⡀⢰⠀⠀⠀⠁⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠁⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡴⠉⠀⠰⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠔⠁⠀⠀⢸⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⢶⣶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣴⣷⡆⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠙⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡏⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠷⢦⡀⠀⠀⣼⠋⣴⣶⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⡿⠛⠛⢶⡤⠾⢿⣿⣿⣿⣿⣿⡆⠀⣿⠀⣼⠃⠀⠙⠋⠀⠀⠀⠀⠀⢸⡇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⠟⠀⠀⠀⠀⠙⣦⣀⠀⠀⠀⢠⣿⡇⣠⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠹⡟⢷⢀⣠⣿⣿⠟⠉⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿⢠⣾⣦⠀⠀⠀⠀⠀⢻⡈⣿⡿⠋⠀⠀⢀⡿⠀⠀⠀⠠⢦⣤⣴⣶⣾⣿⠃⣀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠛⠁⠀⠀⠀⠀⠀⠸⣷⡟⠀⠀⠀⠀⣼⠁⠀⢀⣠⣄⡀⠙⠻⢿⣿⡟⢸⣿⡷⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣤⠀⠀⢰⡏⠀⣰⠟⠁⠈⢻⡀⠀⠀⣸⠃⠀⠈⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡆⠀⣸⠇⢠⡏⠀⠀⠀⢸⣦⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⡇⠀⠀⠀⠠⠶⣾⣿⣿⣿⣿⣿⠀⠙⣧⠀⡿⠀⣾⠁⠀⠀⠀⣼⣿⡿⠉⣿⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢠⡇⠀⠀⢀⣠⣄⡀⠈⠉⠉⢹⣿⠀⠀⡇⢸⠇⢰⠇⠀⠀⠀⢰⣯⠟⠁⢠⡇⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⡄⠈⣷
⢠⡾⠛⠛⠷⣤⡟⠀⠀⠀⢀⣇⠀⢀⣟⠁⠈⢻⣆⢀⣠⣿⣿⡀⢀⣧⣿⠀⣿⠀⠀⠀⣠⣿⡋⠀⠀⢈⣇⠀⠀⣠⡶⣦⣄⡀⠉⢀⣸⣿⣃⢀⣿⢀⣴⠞⠛⠛⢷
⠛⠀⠀⠀⣰⠛⣿⠉⠛⠋⠉⠉⠉⠛⠉⠋⠙⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠛⠉⠉⠉⠉⠋⠙⠋⠙⠋⠙⠉⠛⠉⠉⠉⠛⠉⠉⠉⠉⠉⢉⣿⠛⣧
⠀⠀⠀⠀⠙⠦⣼⣧⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣼⣧⠴⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡌⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠠⠍⢀⠀⠀⠈⠃⠀⠀⠀⠈⠛⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⢠⠂⠀⠀⣀⠀⠀⠀⠀⠀⢤⠀⠀⢀⠲⠿⣲⡯⠀⠀⠀⣼⣿⣿⣿⣾⣶⡆⠀⢀⢼⣤⡀
⠀⠀⠀⠀⢀⣠⣦⡄⠀⠀⠀⠀⢠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⠀⠃⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡼⣿⣿⣿⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣀⣀⠤⣀⣴⣦⣀⡶⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⠔⠂⠀⠀⠴⢿⣿⣗⣐⣾⡿⠻⣿⣿⠷⠄⠀⠀⠀⠀⡠⡀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⢀⣠⣴⠶⠛⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠤⢸⠀⢡⠉⡄⣠⡄⡌⠁⠀⠠⠀⠀⣀⣤⡤⠤⠠⠀⠀⢦⡞
⠀⠀⢀⡀⠀⣀⡤⢴⣤⠤⠀⣠⠤⠀⠘⠈⠠⣄⠀⠀⠀⠀⠞⠀⠈⠠⠈⡄⠈⠚⠀⠀⠈⠀⠠⠀⠉⠀⠁⠘⠁
⠀⠀⠀⠐⠂⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⢀⣰⣶⣄⡀⠀⢀⣰⣶⠀⠀⠀⠐⣆⠀⢀⠸⡇⠀⢠⡶⡄⠀⠄⠈⠁⠀⠀⣀⠀⠀⠀⢸⡽⢢
⠀⠰⠈⠉⠉⠉⠉⠉⠹⠛⠁⠀⠀⠀⢘⣾⠀⠀⠀⣧⣼⠀⠀⠀⡇⠀⢸⡷⡇⠀⠀⢐⣯⣷⣿⣿⡇⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠈⡇⡇⠀⠀⠀⠙⠛⠟⠉⠁⠀⢀⣀
⠀⠀⢀⣤⡤⠤⠀⠀⡉⠁⠀⠀⠉⠊⢀⠀⠀⠀⠠⠀⢠⡀⠀⣠⣤⡄⠀⢷⠁⠀⢀⡀⠀⠠⠀⠠⢄⣶⣾⣿⡷⠒⠆⢥
⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⡀⠀⠀⢀⣀⡄
⠀⠀⠀⣴⣦⣤⣄⣀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣤⠴⣒⡩⠥⠤⠄⠀⠭⠅⢀⡒⠂⠥⣤⠄
⠀⠀⠀⣿⠟⠛⠛⣿⢿⣿⣿⠋⢋⠉⣿⣿⣿⣿⣿⣷⡖⠀⠉⠨⠄⠔⠲⠂⠉⢚⡲⠂⣖⣤⠤⠀⠀⡀⠤⣢⡝⣂⡒⢡
⠀⠀⠀⣸⣹⣿⣿⣿⡾⠛⠃⠀⠀⡈⠁⣉⠉⢉⡉⠀⠰⠀⠀⠒⠶⢆⠉⠉⠭⠑⠪⠙⢀⡉⣜⡲⠬⠑⣉⠢⠤⠤⠰⠚
⠀⠀⠀⢹⣿⡟⠛⠋⠀⢈⢠⠈⠀⠡⡄⠉⠖⠀⡀⠊⠀⢀⡀⠀⠈⠑⠄⠢⠀⠈⠑⠚⠷⣕⢭⠥⠉⢥⠤⢤⠍⠐
⠀⠀⠀⠸⣿⠇⠀⣀⠠⠝⣁⣀⣛⡠⠤⠑⠚⠋⠻⠶⣦⢤⣈⠒⢢⡑⠢⡄⠢⠔⠀⠀⠀⠈⠳⢕⢦⣬⣕⣨⣅⣤⣀⡀
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠉⠉⠉⠁⠀⠀⠈⠁⠒⠰⠮⠭⢳⡟⣶⡷⠝⠛⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⠦⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠔⠓⢆⡀⢐⣦⣠⠼⠷⡖⠀⠀⢀⠠⡾⠀⣤⣤⣤⣭⣥⣀⣀⡀⠀⠀⠘⠿⠃
⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣮⣷⢶⡚⠙⢯⣀⣈⣶⣤⠀⠀⠀⠀⠀⢱⠀⢒⣒⣒⣿⣿⣿⠿⠇
⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣧⠟⢾⡉⠉⢳⣄⡈⠀⠀⠠⠄⠀⠀⠀⢴⠚⠛⠩⠉
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠺⠗⠀⠀⠀⠀⠀⣿⠶⠖⠛⠛⠩⡈⠀⠐⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣶⢆⣶⠁⢀⣀⠄
⠀⠀⠀⠀⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠰⠷⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⢸⣿⡆⢰⣿⢸⠇⣴⣿⣿⣿⣦⢠⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣾⡄⢿⣿⣿⣿⡿⢰⡿
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣾⢿⣁⠀⠀⠙⠿⢦⣽⡿⣿⡁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⢸⣧⠀⠀⣀⣠⣤⣤⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠒⠊⠉⠉⠀⢀⣀⠀⠀⠀⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠈⠁⠀⠀⣀⣀⣤⣤⠴⣾⣮⠻⣿⡿⣄⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢀⠔⠒⠉⠁⠀⢀⣀⣤⡀⣶⣔⢾⡟⢿⣇⠀⠻⠧⠘⠛⠁⠉⠀⠀⣀⡠⠤⠐⠂⢙
⠀⠀⠀⠀⠀⠈⡷⢄⠀⠀⠀⠈⠻⣿⣿⡾⠛⠛⠋⠀⠁⠀⠀⣀⡠⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡏⡆⠣⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⡇⠘⡄⠈⡢⡄⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⡇⣿⣿⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠠⠋⠁⠀⠃⠀⠀⠀⠀⢀⡀⢠⡄⣶⠺⣿⠋⢹⡟⠸⠿⠃⠋
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⢾⣷⠟⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⡆⠀⠀⣀⠠⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠐⠒⠒⠊⠍⠀⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠠⠤⠀⠒⠒⠉⠉⠑⠒⠃⠀⡆
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠰⡀⠀⠉⠒⠂⠸⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⠢⢤⡀
⠀⠀⠀⠀⢀⠨⣀⡀⠉⠒⠺⠦⣄⡒⠀⠀⠀⠀⠉⡶⢄⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠟⣀⡠⠗
⠀⠀⠀⠀⠑⢦⣈⠛⠷⣶⡤⣴⣀⡀⠑⠖⠦⠤⣼⣦⡀⠈⣳⢤⡔⡇⠀⠀⠀⣀⣀⠤⠴⢒⣊⡭⢴⠒⢛⠟
⠀⠀⠀⠀⠀⠀⠣⣉⠓⠦⣍⣛⠁⠉⣙⠲⢤⣀⣀⣈⢉⣶⣄⠠⠼⠗⣒⣊⢭⠤⠒⠒⠉⣉⡠⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣌⡉⠒⠦⠭⢁⣒⣭⣭⠭⠤⠖⠲⠋⠉⣁⣠⠤⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⢀⣀⣀⣠⠤⠔⠒⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣿⣟⣛⣟⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⢀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠢⢤⠀⢠⡠⠀⠀⣠⢀⡲⠶⠔⡉⢅⠉⠀⠀⢭⡉⠐⠦⡀⠀⠀⠀⠀⠨⠤⢉⡀⠄⠠
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠼⡋⠉⢹⠈⠑⠐⣀⠤⣶⣧⠴⠄⠊⢠⡦⢆⡷⠀⠽⠀⠀⡔⣩⣉⢀⡀⠴⣾⠇⣐⠀⠀⠀⠀⠤⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠙⠟⠛⣆⣉⡔⣀⡠⣔⣎⠰⠤⠚⠁⠀⠀⠀⡂⠉⣁⠉⣠⣤⣀⠅⠉⣠⠒⠷⠦⠊⡶⠀⠒⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⢆⠤⣄⡀⢌⡨⢵⠀⠌⢰⡀⢠⠰⠞⠁⠠⢁⣄⡘⠿⠈⡃⣠⠈⡛⢲⣋⡴⢣⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⠈⠔⠂⡀⠰⣄⣈⠐⠀⡤⠃⠤⣈⣧⡀⠧⠭⠀⣀⣤⠃⠀⠉⠉⢉⠔⠁⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠢⢄⡄⠀⠐⠃⠀⠎⠙⠛⠛⣛⠉⡑⣲⠐⢀⣣⠀⠀⠀⡀⡄⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠐⢠⡠⠖⠀⠴⠆⣤⣤⣠⣤⠅⡥⢨⠟⠀⠀⠀⡆⣆⡗⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠒⠒⠂⡠⢀⣐⡐⠲⠔⠒⠚⢖⠦⠚⠃⠀⠀⠀⠃⠓⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢀⠀⠀⠀⠀⠀⠈⠒⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠍⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡀⠀⠀⠀⠀⠈⣀⠤⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠐⢉⠁⠀⠀⠀⠈⡒⠰⢂⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠂⠄⢀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠶⠶⠟⠛⠛⠻⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠉⠙⠛⠶⣦⣤⣀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠾⠿⠟⠛⠛⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣴⡾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠰⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⡀⠀⠀⢿⡇⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠁⠈⣿⡄⢠⣤⡾⠋⠀⠀⠐⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⠈⠁⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⣴⣦⣤⡀⠀⠀⠈⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠈⠛⠉⣿⡇⠀⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠆⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠲⢦⣤⣤⣤⣤⣴⠾⠛⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠀⠀⢀⣠⣴⣾⠿⢻⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡈⠉⠀⠀⠀⠀⣀⣠⣴⡾⠟⠛⠉⠀⠀⠀⠈⠛⠿⢿⣶⣶⣶⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠀⢷⣄⠹⣷⡄⢀⣤⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠁⠀⠀⣠⣿⠄⠙⠃
⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⢀⣤⣶⠿⠛⠁
⠀⢀⣤⣾⠿⠋⠁⠀⠀⢀⣠⣴⡾⠟⠋
⣴⡿⠋⠀⠀⢀⣤⣴⡾⠟⠋⠁
⢿⣷⣤⣶⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡉⠐⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⣙⣤⠀⠀⠀⠀⠀⠀⠀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⣦⣫⡶⠋⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠿⣷⣋⡦⠀⠀⠈⢻⢲⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡲⠆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡾⠀⠑⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡔⠁⠀⢀⠄⣢⡆⠀⠀⠀⠀⠀⠀⠘⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⢀⡈⢧⠀⠀⠀⡇
⠀⠀⠀⢀⠊⢠⠃⠰⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣾⡇⠀⠀⠀⢠⣿⡟⠀⢳⠀⠀⡇
⠀⠀⠀⠊⠀⠁⠀⠀⠀⢠⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠘⠛⢁⠀⠀⡇⠀⡇
⠀⠀⠶⢀⡵⠀⠀⠀⠀⢨⣼⠁⠀⠀⠀⠀⠀⠀⢀⣠⣖⣮⣽⣦⣤⣴⣦⣤⣇⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⢄⡀⠸⡄⡇
⠀⢘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠛⠷⠟⠀⢻⡇
⠀⢨⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⣅⣀⠤⢊⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠁⠀⠀⠀⠰⠀⣠⡇⣸⡃
⠀⣈⡀⠠⠀⠀⠀⣸⡿⢳⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠒⠈⠙⠲⠤⠤⠤⡤⠤⠤⠔⠒⠚⡉⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⡷⠀⢀⡴⠏⢀⣼⣿⢠⠟⡇
⠀⡍⡄⠰⣴⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠐⠀⠀⠉⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡤⠖⠋⢀⣤⣿⣿⢿⣿⢘⡇
⠀⠳⡙⡄⠈⢿⣷⣤⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠤⣄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣠⣒⠋⠀⢀⣠⣴⣿⣿⣟⣵⣿⡇⢨⡇
⠀⠀⢹⣿⣷⣦⡙⢿⣿⣷⣤⣘⢮⣭⡹⠳⠂⠀⢀⢾⣷⠀⠀⠀⠀⠀⠀⠸⠶⠾⠿⠋⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠚⠀⢀⣴⣦⣴⣿⣿⣿⡅⣴⣿⣿⣿⠇⠰⡇
⠀⠀⠈⡄⠙⠿⣿⣜⠿⠿⠀⠙⢻⢞⡂⢀⡀⠀⠻⠞⣡⣶⡦⠀⠀⠀⠀⠀⠄⠀⣀⠠⠤⠤⠀⠀⠀⠀⠀⣀⣀⣠⣤⢰⣿⣿⣿⣿⡿⠟⠃⣼⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠱⡀⠀⠈⠿⢿⣶⣦⡀⠈⠛⢿⡿⣷⠿⠷⣾⣿⣯⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣯⣤⣴⣾⣿⣿⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠛⠿⣿⣦⣤⣤⣤⣤⣀⣀⡀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠟⠻⠛⢛⣿⣋⣽⣷⣶⣬⣟⣛⣉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠈⠉⠉⠛⠟⠿⢿⣿⣿⣦⣄⡀⠉⠛⠛⢉⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣴⠀⠀⠀⠀⠉⠈⠛⠿⣿⣷⣦⣤⣶⣿⠿⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⣤⣤⣀⠀⠀⠑⠛⠛⢿⣯⣶⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠠⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠀⠙⡁⠐⡖⠀⢠⠎⠉⠽⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠶⠖⠒⠒⠋⠉⠝⠁⠀⠀⠀⠈⠁⠖⠪⢼⣷⡒⣁⠀⠀⠀⠀⠙⠃⠉⠓⠲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠂⡄⠀⠀⠀⠀⠀⠉⠑⠂⠐⠦⣤⡤⠴⠊⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⢰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠦⣄⡀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠖⠒⠛⠋⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⡴⠒⠉⠙⠒⠦⡀⠀⠙⠿⣦⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⢤⣀⠀⠀⠀⠀⠀⡧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠙⠷⡄⡅⡆⠰⠀⠂⠔⠀⠀⡠⠀⠀⢀⡠⠤⠤⠬⠿⣷⡢⣄⠀⠀⡇⠠⢡⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠙⠣⢳⣤⢀⢠⠀⢠⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⣽⣶⣧⣀⠁⠀⢈⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠚⢼⣶⣠⠸⡔⢢⡤⡀⠉⠢⠄⠀⠀⠹⣿⡷⠈⢉⡗⠦⣄⣈⣠⠤⠴⠒⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠽⠏⠀⠁⠀⠀⠀⠀⠙⠲⠽⠗⠛⠛⢲⡄⠀⠉⠓⠦⣌⡀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠹⣳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⡀⠀⠉⠛⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⠀⠈⠙⠫⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢾⣧⣤⠟
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⡤⠤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠄⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠖⠉⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⢡⠖⠛⢛⢲⣞⠒⢇⠁⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠋⠉⠉⠓⢦⠤⠤⣤⠶⠤⣤⠤⠖⢉⣨⠽⠛⣿⣆⠀⠀⠇⢴⠊⢱⠛⠉⠁⠀⢓⠩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢀⣀⣈⣷⢖⣚⡷⢀⡔⠒⠛⠉⠀⢀⣴⡿⠋⠀⠀⠀⠘⠑⠎⡳⠖⢟⡠⠳⡄
⠀⠀⠀⠀⣠⣶⠿⠷⠈⠙⢷⣦⣀⡏⠉⠉⠁⠀⠀⠛⠉⠳⣄⣀⣤⡶⠛⢁⠀⣀⠤⠤⢤⡀⠀⠀⠁⢴⠈⢱⡆⣿⠐
⠀⠀⠀⠀⠛⣿⣄⠀⠀⠁⠀⡀⠉⠛⠷⠶⠶⢤⠶⠶⠶⠛⠋⢉⣀⣤⠖⠃⠰⢑⡄⠆⣤⢹⣄⡤⠦⠉⠙⠈⠡⠃
⠀⠀⠀⠀⠐⠈⠛⢷⣤⡀⣎⠉⠉⠀⠀⠀⠠⠀⠠⣄⢀⣠⣴⠿⠋⠁⠈⠀⠀⣈⠃⣄⠿⣸⠃⣠⠰⢤⡘⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⠴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⢿⢦⡀⠗⣤⡺⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣴⣻⡌⡇⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠚⢉⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⣡⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣽⣿⣿⠛⠛⠒⠤⣄⣀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣣⣿⣿⣿⡏⣠⣰⣿⣷⣶⣾⣽⣗⡦⣄⠙⠒⠦⣝⠂
⠀⠀⠀⠀⣠⡤⠾⣵⣿⡿⠻⢏⣴⢛⣭⡍⠻⣿⣿⣿⣿⣿⣦⡱⣦⣤⡈⠳
⠀⠀⢼⢸⡁⠰⠾⠟⠋⠀⠀⠚⢱⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⠘⣯⠛⣶⣤⡀
⠀⠀⠀⠀⢼⡿⠿⠵⠂⠀⠀⠀⠀⢻⣿⣿⡃⠺⣿⣿⣿⣿⣿⣿⡀⠁⠀⣹⢸⣿⣆⠀⣀
⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠺⣿⠿⠃⠀⠈⢻⣿⣿⣿⣿⣧⠀⢀⣏⣾⣿⣿⣷⣬⣙⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠘⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢃⠎⢸⣿⣿⣿⣿⣿⣨⣭⣽⣿⡟⠛⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣟⣂⣀⣀⣀⠀⠀⢠⣿⣿⣿⠿⠋⠁⢻⠀⣿⣿⣿⣿⣿⣿⣷⡽⣿⣿⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⢖⣥⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⠻⣯⢝⠛⠋⠀⠀⠒⠟⠿⠋⣴⣶⣶⡇⢸⠀⢻⣿⣿⣿⣿⣿⣿⣿⣞⢿⣯⣄⣘⣆⣠⠄⠀⢀⣮⠟⣵⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣷⣮⣅⣀⡀⢀⣀⡀⢀⣠⣾⣿⣿⠟⣠⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣭⡍⢹⠀⠀⠀⢿⠇⣼⣿⣿⣿⣿⡇⢀⣀
⠀⠀⠀⠀⠀⠀⢣⠸⣿⣿⣿⡿⢿⣿⣟⠛⠉⠀⠈⣉⣉⣩⣴⣚⣅⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⡀⠀⠀⠀⢰⣿⣿⣿⣿⡏⢳⠸⣿
⠀⠀⠀⠀⠀⠀⢞⢆⠻⣿⣿⣿⣷⣶⣤⣤⣶⡾⠂⣿⣿⣿⣝⢯⣻⠣⠀⠘⣿⡿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⠶⠂⠠⣿⣿⣿⣿⠻⠉⠈⣇⢯
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⣿⣿⣿⣿⣟⣛⡩⠔⣠⣿⣿⣿⣿⣦⣙⠢⣄⠀⠈⠻⣜⢦⠻⣟⢿⣿⣿⣿⣿⣿⣆⣀⣠⣴⣶⣾⣿⣿⣿⡇⠀⠀⣿⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡛⢷⡶⠚⢩⣿⣿⣿⣿⣿⣿⣷⣬⢱⠀⠀⠈⠊⠳⣌⠳⡿⣿⣿⣿⣿⣿⣧⡀⠈⠉⠛⠻⢿⣿⠁⠀⠀⣻⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡘⣟⣿⣿⣿⣇⢃⠈⢧⠀⠈⡇⣿⣿⣿⣿⣿⣿⠃⡸⠀⠀⠀⡄⠀⠈⠑⢦⡈⠻⢿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⢢⣀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿⠛⠛⣿⡿⡄⠀⠈⢆⠀⡇⡇⣿⣿⣿⠟⣡⠜⠁⠀⢤⠔⣣⠀⠀⠀⠀⠈⠓⠤⣉⠻⢿⣿⣿⣿⣷⣦⣤⣤⡴⢣⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⠉⠉⢷⡹⡄⠀⠀⠹⣄⣁⣩⡭⠖⠋⠁⠀⠀⠠⠞⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠭⠽⠿⢋⣴⣿⡟⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢷⡙⢄⠀⠀⠀⢠⢚⠄⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠏⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠢⡀⠉⠁⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠀⠀⠀⠀⢤⠴⢶⣻⣿⣿⡿⠟⠁⣼⠟⣠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⡠⠧⠞⠀⠀⠀⠀⠀⠀⠠⣼⢻⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠜⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠾⠂⠀⣀⣠⣴⠷⣪⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡢⣌⡙⠶⣄⡀⠀⠀⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠟⣛⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠲⠤⣉⠛⠲⠦⢤⣤⣄⣀⣈⣉⣉⣤⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⢿⣿⣭⣾⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⢿⣿⣿⣿⣿⣿⡏⣿⣿⡜⣿⣿⣿⣶⣤⣀⢔⣺⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠉⢿⣿⣿⣣⣿⣿⠁⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⡻⠿⠿⣛⠁⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣦⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⢹⣿⣿⣿⣶⣦⠤⢄⠈⠻⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⡀⠀⠙⠛⠛⠉⠀⠠⠐⠃⠀⠀⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢀⠎⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣄⠀⠀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⢹⣷⠤⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⢻⡀⠹⣠⠇⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡇⠀⠈⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣷⣦⢤⠤⡾⢃⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢠⣿⡿⠃⣠⣮⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫⣴⠿⠿⠗⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣴⣴⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣻⡻⠿⢿⣿⣿⠿⠿⠷⠖⠀⠐⠲⡃⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⣿⣿⣿⣿⠿⠿⠓⠢⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠁⠀⠉⢦⡀⢢⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⢹⠀⠘⠛⠒⠤⣜⠀⠀⠀⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠊⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣶⣶⣶⣤⣀⣀⣀⡀⡤⠰⣶⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠉⠉⠉⢉⠉⠙⠻⣟⠿⣿⣿⣯⡽⠃⠀⠀⠀⠀⣠⣞⢃⣀⣠⣴⢲⣠⣴⣶⣾⣿⣤⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠈⠙⠛⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠻⠞⠈⠉⠛⠿⠻⠻⡛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⠟⠻⠛⠿⠿⠿⠿⠷⢆⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⡀⣀⡀⠀⣀⣀⣤⣾⣿⣿⣿⣿⣛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠶⠶⠶⠶⠤⠶⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠿⠋⠀⠈⠙⠉⠙⠿⠿⣿⢿⣿⣭⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠃⠀⠀⢀⣤⣾⠏⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⣦⡅⠀⠀⠀⠀⠀⠠⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠟⠀⠀⠀⠀⠀⣾⠇⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠉
⠀⠀⠇⠀⢠⡀⠀⡇⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⠂⠀⠀⠀⠀⢺⣻⣿⠟⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣼⢟⠁⡆
⠀⠀⠈⠒⠋⠓⠂⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠋
⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠢⡆⠀⠀⠀⠀⣀⣴⠄⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠛⠀⠀⠀⢀⣠⣴⡶⠀⠀⠀⠀⠀⠀⣠⡴⠗⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠓⢀⣤⡄⠀⢰⡟⠉⠀⠀⠀⠀⢰
⠀⠸⠀⠀⣀⠐⢸⠀⠀⣠⣯⣥⣤⡴⠂⠀⠀⣴⠟⠻⠁⠀⠀⠀⠀⠀⣠⠞⡫⡡⠀⠀⠀⠀⠀⠀⡠⣾⡿⠓⠁⠀⡟⠀⠛⠿⠛⣁⠂⠀⠀⠀⠀⠘
⠀⠀⠑⠂⠙⠛⠀⢀⣞⣉⣭⠽⠂⠀⠀⢀⣾⡿⠀⠀⡌⠀⠀⠀⠀⠔⡡⠊⡔⠁⠀⠀⠀⠀⠀⠰⢏⠋⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣈⡉⠖⠀⠀⠀⠀⢠⡟⠋⠁⠀⠀⠈⠀⠀⢀⠪⠊⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣀⣠⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠄⠀⠀⠋⠉⠁⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠈⠀⠠⠃⠀⠀⠀⠀⠠⠆⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁
⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠄⢀⠖⠀⠀⠀⠀⠀⠁⠂⠀⠒⠒⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣲⣆⠄⠐⣊⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠸⠋⠽⠟⠉⠁⠀⠀⠐⠒⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⠁⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠂⠈⢺⡄⢤⣠⢀
⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣠⡈⡀⠀⠀⠀⠈⠀⠀⠔⡻⠿⢦⣄
⠀⠀⠀⠀⢀⣠⡵⠂⠀⠀⠀⠈⠀⠒⠦⣤⣄⡀⠀⠀⠀⠀⣥⠒⠀⠀⠀⠀⢀⣀⣆⢙⡃⠀⢠⣠⣤⡀⠀⢀⡶⢿⢷⢹
⡠⢷⡀⠀⡳⠘⠿⡎⡂⠀⠀⠠⠀⠀⠀⠀⠉⠑⠢⠤⣀⠀⣧⣤⠤⠄⠒⠂⠉⠀⠀⠁⠒⠒⠋⠉⠫⡙⠓⡊⢤⠾⠀⢺
⢥⡀⠑⠬⡻⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠬⠑⠢⠔⠁
⠀⠀⠀⠀⣀⣀⣁⣀⡶⠶⠒⠒⠒⠒⠒⠲⠤⠤⢄⣀⣀⠀⠄⠀⣀⣀⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣓⣛⣿⣭⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣲⣶⣖⣒⣂⡤⢤⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⢀⣀⣀⢀⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡁⢼⣿⣿⣿⣿⠏⠀⢀⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⣾⣭⣭⣽⣿⣿⡿⠟⠛⠿⢿⣿⣿⢏⣿⣿⣏⣿⣿⣼⣿⣿⠏⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⡿⠒⠀⢠⠀⠘⡾⡿⠘⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⡟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣽⣿⣿⣿⣿⣿⡿⠋⢠⣔⣤⡻⣷⣄⡁⢳⡀⠙⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⡜⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⡽⡋⠀⠀⣈⣈⣁⣱⣍⡻⢿⣆⠑⠄⠀⠹⠟⣿⡿⠃⣾⡟⢿⣿⣿⣴⣿⡿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⢏⠞⠸⠙⠿⣷⣾⣿⣿⣿⣯⠙⣮⣿⣦⡀⠀⠀⠀⠙⠇⣰⣿⠁⠀⢈⣩⡭⠤⠔⠊
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠈⣼⡠⡸⣿⣿⣿⣿⠿⣿⣿⠗⠈⢯⣓⠉⢀⠀⠀⡈⠉⢑⣁⣴⠿⠋
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢸⣿⣷⣮⣚⣉⠑⠒⠉⠉⣠⡴⢊⠀⠉⣻⠿⠛⠒⠿⠛⠛⠉
⠀⠀⠀⠀⠀⢠⣿⣿⢙⢿⣆⡀⠙⠟⠉⡁⠈⠂⠀⣠⣾⣿⣿⣯⢖⡭⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣾⡷⠉⠁⠀⠀⢢⢻⠂⠀⢰⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⢄⡀⠀⠀⢀⠴⢺⣻⣿⣿⣿⣏⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢲⡀⠃⠀⡴⠃⠀⠀⠓⠽⠓⠯⠞
⠀⠀⠀⠀⠀⠀⣿⠋⠀⢀⠀⠉⠑⢀⠞
⠀⠀⠀⠀⠀⠀⣿⣧⠀⣾⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣆⢹⣧⢠⠀⠘⣿⣷⣖⣲⣶⣾⣿⣿⣿⡿⢟⡽⠖
⠀⠀⠀⠀⠀⢠⢻⣿⡣⠻⣿⣧⢣⡐⣌⠻⣿⣿⣿⣿⡿⠿⢋⠔⠋
⠀⠀⠀⠀⠀⠨⠨⠋⠀⠀⠙⢿⣷⣵⣌⢳⣽⣿⣿⣶⠆⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣟⠛⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⡻⣟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⠻⣧⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠇

⠀⠀⠀⠀⢠⠔⣒⡲⢄⡀
⠀⠀⠀⠀⢻⡘⡀⠈⠐⠝⠦⡀
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠉⠒⠤⣀⡀⠀⠀⣀⡤⠖⠊⠉⠉⠉⠉⠉⠉⠁⠀⠀⡉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⡄⠀⠈⠑⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢎⣁⣳⣄⣀⣀⡀⣀⣉⡒⢤⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⢸⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠑⠢⢄⣱⠀⠀⠀⠀⡰⠃⠀⠉⠉⠉⠉⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠈⢲⡀⣠⠊⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⢀⣀⠤⠒⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠟⠒⠒⠒⣺⠞⠁⠀⠀⠀⢀⡴⠋
⠀⠲⠮⠭⢤⣄⣀⣀⣀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⣄⡠⠴⠚⠒⢒⣒⣋⣀⣀⣠⣤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠴⢥⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣦⣾⣿⣿⣿⣿⢟⣿⢿⣿⣿⡿⢟⣟⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢨⣿⣯⢻⡣⢁⣵⣵⣟⠸⠟⢉⡤⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⢻⢐⣤⣴⠿⠀⡹⢻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢸⣧⡛⣉⣔⠋⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣰⣏⡐⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡇⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⣛⡛⠻⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠉⠍⠒⠂⠁
⠀⠀⠀⠀⠀⢀⣀⡠⠤⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠤⣀⣀⡀
⠀⠀⠀⡖⠛⢩⡶⢾⣟⡈⣡⣀⣀⣀⠀⠈⠁⠀⠀⠀⠀⢀⣠⡀⠀⠉⠙⢲
⠀⠀⠀⠹⣷⣿⠿⠿⢿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠉⠛⠻⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣶⣶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠛⠻⠿⠿⠗⠒⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⠒⢦⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⢄⠐⠀⡀⠈⣆⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠖⠀⢀⡆⠀⠐⠒⠂⠰⠦⠂⠀⠀⢸⡶⣌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢺⣀⠐⣯⠂⠾⠋⠀⡀⠲⣀⣤⠀⠀⠀⠀⢀⣼⢡⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠈⢄⠉⢰⡞⠀⡾⢛⠛⠁⢀⣤⣤⣾⡿⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠰⠉⠀⠀⣷⣾⣶⣶⣿⣿⡿⠟⠁⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⡦⠒⡂⢶⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠊⠢⡀⠀⠀⠀⠀⠀⠡⠀⢓⢈⠈⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⣀⠔⠉⠁⠂⠀⠀⠑⣀⡀⠰⠋⠢⡘⠀⠣⠁⠀⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠖⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠅⠀⠢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠚⡁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⡶⠧⠶⠓⠒⠒⠛⠒⠒⠷⠴⠂⠀⠀⣀
⠀⠀⠀⠀⠀⡠⠃⠀⡇⠀⠀⠀⠜⡄⠀⠀⡲⣤⡀⠀⠺⡄⠀⠈⠀⠀⠀⠀⠁⠒⠻⢧⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠞⠃
⠀⠀⠀⠀⡰⠁⠀⠀⢡⠀⠀⡄⠀⢱⠀⠀⠙⣈⠻⣶⣄⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⡒⢲⠶⠒⠒⡒⠂⠢
⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⡐⠀⢰⠸⡄⠀⢄⠈⠓⠀⠙⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⠈⠢⠌⠀⣰⡇⠀⠀⢁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⠴⠁⠀⠞⠀⠈⡲⢈⠏⢀⠀⠀⠀⠈⢃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢢⡀⠈⡁⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⢶⡖⠒⣋⡵⢲⠀⠀⠙⢶⣄⠀⠈⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠐
⠀⢀⡜⣀⣀⣀⠤⣤⠒⠉⠀⠞⢁⡔⠁⠀⡎⠀⠸⡄⠀⠈⠻⢦⣀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠠⡄⠀⠀⠈⢠⡐⠂⠀⠀⠀⡀
⠀⠀⠀⠈⡁⠀⠀⠉⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊⠈⠑⠒
⠀⠀⠀⠀⠑⠲⠄⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀
⠀⠀⠀⠀⠀⠈⠰⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠄⢤⣀⠀⠀⠀⠈⠑⠢⢄⣠⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠨⢙⠳⢦⣄⣀⣠⢴⠞⠋⠀⠀⠀⢨⣦⠀⠀⠀⠀⠀⠀⠉⠒⢄⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠭⣙⠦⢤⣄⡀⣀⣠⡴⠋⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⢀⣤⠤⠖⠒⠒⠀⠐⠒⠂⠤⣄⠀⠀⠀⠀⠀⢷⡀⠀⣠⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⡐⣧⢻⡁⠀⠀⠀⠠⡤⠞⠂⠀⠤⣿⡗⣀⠀⠀⠀⠀⠹⣄⠒⢵⠲⠤⢾⠛⠁⠈⠢⡀
⠀⠀⠀⠀⠉⠛⣦⣌⠓⠒⠤⠤⠤⠤⠴⠞⢛⣁⡴⠛⢃⡄⠀⠀⠀⠙⣆⠀⠉⠐⠂⢀⣀⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠳⣌⠉⠁⠐⠒⠒⠒⠚⠋⢉⣱⠖⠚⠋⠀⠀⠀⠀⠀⠈⢷⡶⠾⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠤⠐⠒⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⢾⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠑⢄⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⣧⡴⠃⠀⠀⠀⠀⣀⣀⣤⣼⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣸⠏⠉⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⣷⣿⣿⣷⣿⠶⣄⠀⠀⠀⠀⠀⠀⠀⣐⢤⣬⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠙⠢⠖⠋⠀⠀⠀⠀⠀⠀⢠⣄⠠⣿⣿⣷⣶⣿⣿⠗⣶⠖⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢽⣿⣭⣿⠿⣿⡿⣿⣿⡥⠖⠊⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠭⠥⢽⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⡿⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⠁⠁⠎⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⢀⣀⣀⣀⣀⣖⣀⣀⣀⡀⠀⢀⣾⣿⣿⣶⣶⣧⣶⣶⣶⡤⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣛⣿⣿⣿⣿⣿⡟⠂⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠘⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠿⠿⣿⡟⠻⠃⠀⠀⠀⠘⠿⣿⠻⢻⡟⠻⠙⡁⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⣀⣀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⣇⠀⢈⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢻⡿⡿⠛⠛⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⡠⠆⠀⠀⠀⠀⠀⠀⠀⣏⠉⠁⠀⠀⢸⠄⠀⠀⠀⢠
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠸⣦⣀⣀⣠⠞
⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢢
⠀⠀⡆⠀⠉⠉⠀⠀⠈⢣⢀⠔⠒⠒⢄⠀⡏⠀⠈⠁⠀⠀⠀⠀⢇⠀⡰⠒⠒⠀⠒⡄⠀⡀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⡎⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢸⡜⠀⠀⠠⠀⣤⠃⠀⢹⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠫⡅⣸⠀⠀⠀⠁⠀⢳⠉⠁⡆⠀⠀⠀⠘⠋⠁⠱⣀⣀⣀⠀⠈⠣⠀⠸⠀⠀⡞
⠀⠀⡇⠀⠀⠀⠠⠴⢃⠇⠀⠰⠟⠀⠀⠈⡆⠀⢀⠈⠀⠀⢠⠀⠀⡠⠒⠒⠒⢵⡄⠀⣇⠀⠣⢤⠇
⠀⠀⡇⠀⠀⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⠀⡀⠀⢸⠀⢠⠇⠀⠠⠂⠐⠁⠀⢸⠀⡞⠉⣢
⠀⠀⠑⢲⢀⠤⠞⠰⣅⣀⡀⡰⢃⠀⠀⢀⠜⠀⢸⡀⠀⣀⡜⠀⠈⠧⡀⠀⠀⠀⠈⢀⠆⠀⠙⠚⠁
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⡘⠃⠈⣧⠒⠋⠀⠀⠀⢿⠉⠉⠀⠀⠀⢀⡙⠙⡄⡰⠂⠉⠀⠀⠰⠄
⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠛⠀⠀⣬⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠈⠁⢠⠈⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠔⠒⠒⠒⠲⠤⢄⣙⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄
⠀⠀⠀⣀⡠⠴⠒⡃⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⡔⠋⡍⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤
⠈⠢⢀⠐⠠⢀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠠⠀⠂⠙⡇⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⢀
⠀⠀⠀⠙⠢⢄⡈⠒⠤⣀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡢⣄⠀⣀⠴⡉⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠘⢍⠑⠠⢀⠁⠂⠄⡀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠆⠒⡒⠇⠈⠉⠀⠀⢀⠧⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠐⠤⣀⠁⠒⠄⠀⠀⠀⠉⠐⢒⠃⠁⠐⡊⠭⠐⠀⠁⠀⠀⠀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡉⠂⠀⡀⠉⠐⠂⠉⡀⠀⠂⠁⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣁⢲⡈⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣉⣲⣦⣤⣴⣎⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⡿⡿⠿⢿⣿⣿⣓⣄⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣻⣭⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⣳⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⡟⣭⣝⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡹⣽⣱⣧⣾⣿⠃⣿⣿⣿⡿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣽⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣽⣹⣻⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⡇⣿⣿⣿⣠⠉⠋⠉⢛⠟⢛⡫⢉⣤⣶⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣳⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠛⠸⡜⠀⠈⠻⣯⢆⠀⢠⠁⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣿⡿⡟⠯⠙⢿⣾⣿⣿⣿⣯⠳⣄⠀⠙⢦⡀⠀⠈⢮⠳⣾⡀⠙⣿⡙⠿⣿⣿⢿⣿⠋⠁
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠐⠢⢭⣿⣿⣿⣿⣇⣿⠳⣤⣀⠉⠒⠠⠀⠵⠮⡳⢄⡈⠉⠓⠦⣭⣟⣶⣯⣴⣤⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠑⠶⠤⣤⣾⣿⡿⠋⠙⢿⣷⣄⠀⠈⠉⠉⠂⠀⠀⠀⠀⠘⠳⠶⢦⣤⣬⣍⡛⢿⣿⣿⣿⠆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣴⣶⣶⣶⣶⣖⣒⣦⣤⣴⣾⠿⠛⠉⠙⠛⠚⠋⠁
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡤⠄⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⡟⠉⠋⠉⠉⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⠞⠙⣄⣤⣶⣛⣭⡥⠶⠶⠛⠛⠉⠹⠟⢁⢟⣿⢿⣿⡿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣀⣀⡼⠁⢒⣵⣜⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢢⠟⣿⢶⠇
⠀⠘⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣠⣴⣻⡿⣟⣀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⡟⡜
⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⢠⣞⣵⡿⠋⠀⢿⣿⣾⣹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⠞
⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⣮⡿⣿⡇⠀⠀⠘⣿⣿⠟⠋
⠀⠀⢀⣿⠻⣿⣿⣆⠀⠀⣿⠞⠋⠁
⠀⢠⣿⣿⣆⠹⣿⣿⣷⣤⣇⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣿⣿⣿⡿⠛
⢰⣿⣿⣿⣿⣆⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣚⣻⣿⣿⠏
⣿⣿⣿⣿⣿⣿⢷⣄⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣯⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣟⣿⣿⣿⣿⣿⡿⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣍⣻⣿
⠈⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡀
⢀⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣗⣿⣿⣇
⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣭⣭⣙⠻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣏⣼⡿⢿⣯⢿⣧⣈⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⡟⢸⢣⣿⣿⣿⣿⣿⣿⣮⣛⣿⣋⣿⣿⣿⡿⠿⠟⠉⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⡏⣧⠀⠸⣿⢿⣿⣿⣿⡿⠛⣛⣿⣿⣿⡿⠟⢋⣤⣶⡽⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣇⢻⠀⢆⠹⣦⣍⠛⠉⣰⣿⢟⡵⢋⠕⢀⣴⠿⣋⣥⣿⣶⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣬⣭⣟⣿⣿⣿⣿⣿⣯⡻⣧⠈⢦⣈⠛⢆⣸⣿⢏⣾⡇⡏⢰⣭⣶⣟⠫⠭⢥⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣸⣿⣿⣹⡝⢿⣿⣿⣮⡳⣦⣉⠑⢼⣿⣿⣿⣿⣧⣥⡼⠏⢊⣁⣤⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣇⢷⡀⠙⠻⣿⣿⣿⣿⣿⣷⣶⠿⠿⢿⡷⣶⣚⣻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣻⣿⣾⣿⣿⣿⣛⡿⠎⣷⡀⠀⠈⠙⠿⣷⣾⣶⣶⣤⣶⣿⣦⡉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡜⣷⡀⠀⠀⠀⠈⣹⣯⣿⡁⣼⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡡⣿⣿⣌⢿⣶⡊⠇⣴⣿⣿⡇⣿⡘⢹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠻⣿⣎⢻⣟⡿⠛⢟⢿⡟⡏⠉⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣧⠻⣿⣆⠀⠸⣷⡇⠀⣽⣿⣿⣿⣧⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣧⠹⣿⣷⡄⣿⣷⠀⣿⣿⣿⣿⣿⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢳⡹⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⡌⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡁⢹⣿⣿⡟⣼⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣷⡀⢿⣿⣽⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠁⠈⠉

⠀⠀⣠⣤⣼⣿⣿⣿⠶⠀⠀⠀⠐⠁⠐⡄⠀⠀⠀⡠⠤⠶⢶⣦⣀⡀⠀⠀⠀⠶⡄⠀⠀⠀⠀⠠⠞⠀⠠⠞⠁⠀⠴⠋⠀⠄
⠀⠀⠀⠉⠉⠋⡁⠰⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠄⠀⢾⣼⣿⡇⠀⠀⠀⠀⣿⠀⠀⢰⣦⣤⣴⣦⣀⣰⣤⣀⣠⣤⣄⣒⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠃⠀⠀⠰⠀⡿⠀⠀⠘⠛⠈⠡⠜⠆⠛⠰⠼⠧⠴⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠟⠛⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠀⠀⢀⡀⠀⣠⡆⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠀⠀⠀⠈⠁⠀⠀⢀⣴⡿⠛⠛⠻⠛⠢⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣽⣶⣿⣿⠻⠿⠇⠀⠉⠉⠀⠁
⠀⠀⢀⠐⠄⠀⠀⠀⠀⠂⢈⠄⠀⠀⠀⢀⣽⠏⠁⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⡄⠀⡀⠀⠘
⠀⠀⣄⠀⠁⠒⠠⠁⠀⠖⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠷⠀⠀⠀⠀⠇⠀⠀⢰⠘⡆
⠀⠀⠓⠺⢿⣈⡢⠞⠁⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⠀⠀⠔⠀⠀⠀⠀⠂⠀⠀⣠⠁⢀⠘⠞⢣⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡷⠈⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⡅⠀⢠⡇⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣻⣿⣿⣟⣩⠁⠀⠀⠈⢂⠀⠀⠸⣿⣏⠰⡇⠀⠀⠀⠀⠀⢁⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⡀⠀⠓⠂⠀⠀⠀⢀⡆⠀⠀⠀⠀⠿⠉⠉⠉⢙⠉⠹⠀⠀⠠⠀⠀⢻⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠇⡁⣹
⠀⠀⠀⠀⠈⢡⡆⢈⠜⢧⠠⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣄⣠⣄⠀⢀⠀⠀⣦⠀⠀⢆⠃⣿
⠀⠀⠀⠀⠀⠉⡤⠋⠀⠈⡆⠒⡀⠀⠀⠀⠀⢀⣠⠄⡐⢻⠷⠢⠀⠀⠀⠀⠀⠀⣸⠀⠀⡟⠛⠋⠉⠻⠀⠈⠙⠂⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢰⣮⣿⣷⣽⣭⣦⣴⣿⡤⠀⠆⠀⣠⠋⠀⠀⠷⠂⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣐⠶⢿⣯⣝⡳⣆⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠰⠃⠀⠚⠀⠀⠘⠆⠀⠀⠀⠄⠀⣹⠀⢀⡳⢶⣶⡼⢷
⠀⠀⠀⠀⣀⠀⠀⠛⢷⣦⠋⠟⠁⠛⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⠀⣴⠏⠀⠀⠉⠀⠋⠑⠚
⠀⠀⡆⠀⢹⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⢀⣤⢴⡶⠀⠀⠀⠀⠜⠀⣼⠇⠀⠀⠀⢀⡀
⠀⠀⡇⠀⠈⡱⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠜⠋⠀⠀⠀⠀⠀⠤⣴⠏⠀⠀⠀⠠⢸⢃⠐
⠀⢀⡇⠀⠀⡷⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠄⠀⠀⠀⢀⡴⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⡀⡀⠀⠀⠀⡘⠘⠀⣇⡁⡐
⠀⠸⡇⠀⢀⡘⠂⡁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠁⠁⠈⢹⡷⠁
⠀⠀⡇⠀⢠⠟⠀⠀⠀⠀⣄⡀⣔⡾⢤⠗⣤⢠⣆⠤⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⡄⠄⠆⢂⢐⡐⡀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣾⡾⠂⠀⠀⢃⣀⠐⣛⣂⢛⡚⣘⣋⢳⠞⠋⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸⡟⠒⠐⠂⠂⠃⠐⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢿⣿⠯⠋⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠒⠀⠀⣀⣰⣶⣷⠾⣿⣛⣦⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⡀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠭⣓⢲⣤⣶⣾⣿⣯⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡛⣿⣿⣿⡿⢋⢋⣴⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⠻⠋⠉⠹⣝⢎⣑⣵⢏⡿⢡⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⠀⠀⠀⠀⠈⠉⠉⢁⠞⢀⣿⣿⢿⣿⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢐⡭⠥⠤⠤⠀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⡀⠐⢄⠀⠀⠀⠀⣁⠔⠁⠠⠜⠋⠀⣴⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠳⣄⠀⢀⠰⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⠋⠀⠀⠀⢀⣠⣤⣤⣔⡲⠶⣶⣿⣷⣌⣙⠺⠷⠤⠍⠉⠀⢀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠺⣿⣿⣭⣶⡄⣀⠤⣻⡿⠀⠀⠉⠙⠛⣲⣶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣺⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⠿⠋⠀⢀⣀⣤⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠁⠹⣿⣛⠙⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠹⣿⣀⡄⣆⣿
⠀⠀⠀⢀⠀⠀⠀⣰⣿⡖⠲⠾⢤⠄⢻⣿⣷⡀⠀⠀⠀⠀⠀⣼⠏⠀⢄⣠⣤⣤⣶⣶⣶⣶⠉⠻⣿⣿⣿⠃⠶⠶⣀⠐
⠀⠀⢰⡇⠀⡀⣖⣌⠳⣦⠆⠀⠀⢈⠉⠙⠻⣿⣦⡀⠀⠀⠀⢻⣤⣶⣿⣿⣟⣛⣩⣭⣿⡃⣀⣀⣴⣯⣷⠖⠒⠋⢙⣠⢀⢇⢀⡷
⠀⠀⠈⢧⢈⣧⣿⡿⣷⣬⣿⣷⠦⡤⣝⡛⠒⠚⠻⣿⣷⣶⣄⣤⣿⣿⣿⣿⣥⣶⣾⡿⠿⣿⣭⣥⣶⢖⡂⣀⡤⠶⣱⢇⡞⡼⠃
⠀⠀⠀⠈⠳⣿⢮⣽⣷⣿⣿⣿⣷⣌⣀⣉⣻⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⣛⣓⠾⣛⣫⣥⣠⣤⣬⣥⡶⣟⣵⠯
⠀⠀⠀⠀⠀⠀⠉⠻⠟⠯⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⢿⣿⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠹⠿⠷⠶⠶⣶⣶⣾⣿⣿⣯⣽⠿⠛⠚⠛⠁⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣴⣧⠻⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⣿⣾⣿⠃⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠁⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠋⠁⠀⠀⠀⠀⢯⠀⣠⡔⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡆⠰⠒⠀⢸⣀⠀⠠⠄⢀⣀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠈⠈⠀⠈⠜⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠿⠿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣽⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡺⣿⣿⣿⣿⣿⣿⠿⠟⣂⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢟⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⣠⣾⣿⣿⡿⠟⣫⣵⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⣴⣿⣿⡿⣋⣴⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣀⠀⠀⠀⠀⠀⣠⠞
⢀⣾⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⠋⢡⣭⠻⣿⣿⣿⣿⣿⠟⣡⣾⣟⣹⣇⣠⣤⡶⠞⠃
⣸⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡰⣾⣿⢣⣿⣿⣿⠟⣡⡾⢋⣤⠏⠉⠉⠉
⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣇⠻⠿⠟⠁
⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡟⢋⣉⣤⣴⣾⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⠟⣡⣶⣄⡉⠙⠛⠛⠛⠛⠉⠁
⠀⠈⠳⠸⣿⣿⣿⣿⣿⣿⣧⣙⠛⠛⣁⡼⠟⠋⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢶⣶⣾⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠃⢀⣀⡀⡀⠀⠀⠀⠀⠈⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠉⠑⠦⡀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀⣀⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢰⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⠁⠲⣄⠀⠀⠈⣄⠀⠀⠈⠁⠸⠄⠀⠀⠀⠀⠀⡈⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⣤⠀⠀⣠⣤⣤⣤⡉⠢⢀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⣼⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠍⢠⡄⠀⢈⣤⠀⠀⠉⠂⠀⢷⡄⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠁⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠉⠂⠀⠀⠀⠀⢸⣧⠀⠀⠀⠸⠐⠢⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠁⠈⠄⠁⠤⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢄⠀⠀⠀⢀⣠⣤⣴⣆⣀⡀⠀⠀⠀⠀⢸⣤⠀⣈⠆⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠂⠠⠄⠀⡀⠂⠀⠈⠙⠻⠯⣭⠍⠿⠛⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⠴⠋⠀⢰⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠦⠤⠶⠒⠁⠀⠀⡰⠋⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⢲⣿⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠓⠦⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠒⠁⢠⠋⠀⠈⠸⣖⣤⣤⣤⠠⠄⡐⡅⠀⢀⣠⠂⣠⡊⠀⠀⠈⡉⣲⠶⢄⠀⢰⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠞⡵⠃⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⢉⠀⠐⡎⠃⢰⡟⠀⣦⠀⡾⢰⠃⠀⠀⠑⢬⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠔⡀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠀⢰⠀⣤⣼⠁⠀⠁⢸⠃⢀⡔⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠁⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⣿⡏⡜⠀⡠⡿⢸⣿⡧⠊⠀⣀⣈⢢⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⡟⢹⢹⡇⣼⢡⠇⡾⡏⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠶⠀⢸⠚⠀⠀⢸⣿⠃⠸⠈⢱⡇⠀⠁⡀⠀⡂⠀⠰⡀⠀⠀⡰⠀⠦
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠄⢰⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡏⠀⠆⠀⠀⢧⡀⠋⣀⣾⡗⠄⠀⢈⠉⠉⠀⢀⢰⠂
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⡀⣿⢣⠐⣄⠀⠻⣆⠀⠀⠈⠈
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⠇⠹⡈⡄⠈⢗⠀⠙⠂⠀⠀⡐
⠀⠠⠇⠀⠀⠀⠀⠀⠀⢀⢮⠀⢈⡇⠘⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡇⠀⠀⠈⠀⢰⠁⠀⠀⠀⠀⡞⠀⠀⠀⠛⠒⠠⠀⠀⠀⠐⠈
⠀⡧⠘⠀⠀⠀⠀⢀⡔⠁⠰⠰⠊⡅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏⠀⠀⠀⠀⣽
⠀⠸⡀⠃⠀⠀⡠⠋⠀⠠⠃⡦⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠃
⠀⠀⠑⣦⠤⠐⠁⢀⡴⠕⠊⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠈⠒⠶⠒⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⣒⣛⣯⡉⠭⣩⣽⡛⢒⣲⡦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣭⣶⣿⣿⣯⣝⡋⡽⠿⣯⣧⣛⣻⣽⣽⣿⣦⣍⣓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣩⣾⣿⠿⢟⣻⠿⢿⣿⣿⣮⢛⣷⡶⣽⣭⢾⢾⣽⡵⡿⣟⣷⣿⣋⣶⣄⡀
⠀⠀⠀⠀⠀⢠⡾⣡⣿⢿⣭⣄⠀⠉⣭⣁⠀⠉⢿⣿⣞⡷⠿⣭⣽⣛⣻⠶⢾⣟⡿⡯⣻⣿⣿⣶⣟⣲⣄
⠀⠀⠀⠀⣰⢏⣾⡿⠁⣿⠟⡭⡁⠀⢱⠙⠳⡄⠀⣿⣏⡿⠟⠶⠾⠽⠝⠛⠓⠓⠛⠛⠿⠟⠻⠿⣿⣿⣾⣳⣄
⠀⠀⠀⣰⢏⣾⡟⠀⣸⣿⢠⠁⠘⡄⢸⠀⠁⢻⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠈⡛⠻⠷⡄
⠀⠀⢠⡟⣾⡟⠀⠀⠙⠃⢾⢆⡠⠃⣘⢀⠁⣸⠀⠁⠀⠀⠀⢀⣀⣠⠤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⣽⣽⣛⡚⠁
⠀⠀⢸⢣⣿⠃⠀⠀⠀⠀⠉⠉⠀⡠⠁⠀⣰⣃⡐⢲⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠿⡉⠀⠀⠀⠂⢀⠀⣄
⠀⣤⡼⢯⢿⣆⠀⠀⠀⢀⣀⣠⠮⠴⠒⠋⠉⠀⠛⠙⢯⡉⢀⣀⠈⠁⠐⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡳⢤⠀⣀⡼
⠀⠘⠓⠉⠛⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣒⠦⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠦⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠛⢒⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢁⢠⠂⠀⠰⡁⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⢀⠀⡀⡐⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠃⠀⠃⠈⠈⠀⠀⠈⠘⠀⠁⠋⠀⠀⠁
⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⣶⡄⠀⠀⠠⣶⠀⠀⠀⢠⣶⠀⠀⠀⢠⣦⠀⠀⠀⣴⡆⠀⠀⠀⣶⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠃⠀⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⢠⣾⣿⣦⠀⢠⣾⣿⡆⠀⢰⡿⠷⡄⠀⣾⣿⣦⠀⢰⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣸⣧⡇⢰⣄⣀⣠⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠈⣿⣿⡇⠀⣿⣿⣾⡇⠀⢻⣷⡏⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠈⠛⠿⠟⠁⠀⠀⠒⠀⠀⠀⠙⠛⠋⠀⠀⠛⠛⠁⠀⠛⠛⠛⠛⠀⠈⠛⠁⠀⠘⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⣤⣶⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣻⣾⠿⠏⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣛⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠋⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡛⠿⣦⣌⣉⣉⣉⡉⣻⠉⠉⠉⠉⠉⠁⠀⠀⠀⣉⣈⣀⣀⣠⡤⠤⠖⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣀⡍⠉⠉⠉⠉⠉⠀⠀⠉⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⣀⡀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣿⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢷⣶⠿⠶⢷⢶⣶⠶⠾⣿⠿⠋⠉⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⣀⠀⠈⠉⠀⠀⠁⠠⠴⠀⠈⠀⠰⠒⠀⠀⠳⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣾⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⢻⡛⠭⠽⠋⠿⠿⣷⣦⣀⢀⡀⠀⣀⠀⢀⣀⣠⣤⡶⠖⠛⠛⠛⠉⠉⠁⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡶⣦⠄⣀⡀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣤⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⣶⣾⡿⠛⠛⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⡿⢿⡟⠅⠁⠀⠀⠀⠀⠀⠀⣠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣭⣟⠉⠉⠙⠙⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⠀⠀⣁⣤⣄⣤⣤⠶⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⡳⠦⢙⠻⢿⣿⣿⣿⣿⣾⣾⣶⣶⣶⣶⣦⣶⣶⣶⣾⣷⣾⣿⣿⠿⠛⠋⠡⠀⠐⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣈⡙⠛⠲⠶⠍⢉⡛⠛⠛⠛⠉⠉⠋⠙⠛⠛⠛⠛⠛⠋⠀⠐⠊⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢖⣒⢾⡉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣊⣭⣵⢆⠖⣈⡙⢟⠃⠻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⣵⣾⡿⢹⣿⠿⢢⠘⠿⠟⣨⠘⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⣾⣿⣿⢁⡟⠈⣷⠀⠀⠑⠲⠶⠋⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣱⣿⢿⣯⡝⣹⡋⠓⠘⢎⡣⣀⠀⠂⠤⠴⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⣼⡟⣹⢙⡋⢻⠛⣧⣦⢡⣄⠙⠦⢅⡀⠲⠄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣱⢣⣾⣋⣉⣹⣙⡏⠙⠉⠛⠁⠀⠁⠀⠀⠀⠈⠁⠒⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⢣⣿⣡⣈⣿⣨⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⢿⣿⣿⣿⢼⡀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣱⠏⢾⠛⡶⠷⠀⠀⠀⠀⠐⢧⣦⡀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣼⣏⣽⣹⣏⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠔⣠⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⡞⡧⠼⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡭⠗⠛⠧⣿⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣱⡿⢷⠐⠀⣶⠁⠀⢀⣀⣠⡤⠴⠖⠚⠋⠉⠀⠀⠀⠀⠀⠨⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣼⡿⠯⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢦⡈⣁⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡹⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢸⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣸⡄⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣴⠿⣷⣜⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣰⣧⠀⡇⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠞⠋⠻⠧⠗⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣲⣦⣄⡀⠀⠀⠙⢲⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣻⠿⢿⣛⠋⠁⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣬⣵⡆⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠻⣄⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⣼⠇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⢯⡁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⢀⣿⠂⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠸⣯⣕⣶⣤⣀⣀⣈⣳⣿⣅⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠻⡍⠉⠁⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⢻⡗⢦⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣶⡾⠷⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡀⢈⣏⢱⡄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡟⠛⠋⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⣠⠶⠶⣦⣝⣦⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠻⣦⣽⣶⣄⠘⠦⣄⠀⠈⠉⠁⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠷⠄⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣶⣤⣤⣤⣴⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⠻⠿⣿⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡀
⠀⠀⠀⢤⠂⠀⢨⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠇⠀⠀⠀⠀⣀⠴⠖⠒⠲⢤⡀
⠀⠀⠀⢹⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠸⠀⠀⠀⠀⢰⠃⠀⠀⠰⠂⠀⠙⡄
⠀⠀⠀⡆⠇⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠓⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⡇⠀⠉⢀⡀⠒⢀⢷
⠀⠀⠀⠡⣀⡀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢀⡌⠀⠀⠀⢀⡤⣀⢹⠀⡀⠀⠑⣄⠁⠈
⠀⠀⠀⠀⠀⠹⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡀⠈⠛⠀⠀⠰⠁⢀⠆⡆
⠀⠀⠀⠀⠀⠀⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢸⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⢧⠀⠠⠀⠁⣀⠜⠣⠀⡇
⠀⠀⠀⠀⠀⠀⢡⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡎⠁⠈⢧⠀⠀⠀⠀⠀⢠⠃⢨⠁⠀⠀⠀⠀⠀⠀⠈⢇⠀⠐⠊⠀⠒⠁⢀⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⠄⠒⠒⠊⢉⣇⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢱⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⠀⠀⠀⠀⣸⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢷⡻⡄
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠈⡇⠀⠀⠀⡇⠐⠞⢠⠀⠀⠀⠀⠀⢹⢀⠀⠀⢸⠀⠀⠀⢠⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠁⠀⠀⠀⠙⠦⠴⠋⠀⠀⠀⠀⠀⢸⠈⠀⠀⢸⠀⠀⠀⡜⠀⡌⡌⠀⠀⠀⠀⠀⡠⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⠀⠀⠀⠇⠀⢱⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠘⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⡴⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠡⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠉⠂
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡸⠀⠀⠀⠀⠀⢘⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢄⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⢀⡞⡇⢹⣗⠄⠀⠀⠀⠀⠀⢀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⣾⠁⣇⢰⠹⠈⠀⠀⠀⠀⠠⠁⠀⠀⠠⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠸⣿⡀⢘⣦⣀⡴⠀⠀⠀⡐⠁⠀⢀⠀⠆⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠯⠴⠋⠀⠀⠀⡐⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠈
⠀⠀⣠⠔⠉⠁⡰⠉⠉⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢱⠄⠤⠐⠁
⠀⠀⡇⠮⠵⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠⠀⠀⠀⠀⠀⡠⠊⢠⠃
⠀⠀⠈⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠻⣿⣦⣶⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠸⣿⣟⢶⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⢿⡛⢿⣷⣿⣹⣷⣹⣿⣿⣿⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⢸⣯⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣦⣝⡾⣽⣿⣿⣿⣇⣾⣿⣿⡄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣅⠈⠛⢿⣿⣛⣿⣿⣿⣿⠟⣹⣾⣦⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢲⣵⣮⣽⣿⣟⣋⣠⣼⠛⢿⣿⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⣿⣿⣿⣿⣿⣿⣧⠘⢛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⢀⣹⢃⠉⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣯⡛⣆⠀⠀⠀⠀⠀⢀⡠⠊⣰⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠃⠀⠉⠉⠛⠋⠉⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠟⠋⡡⠴⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠻⠛⠛⠛⠋⠉⠀⠀⠉⠙⠛⠻⠿⠿⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⠀⠀⠂⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⣀⣀⣐⣒⣒⣂⣀⣀⣠⣠⣤⣤⣤⣀⡄⡀⢠⣤⠤⠤⠤⠤⠐⠒⠉
⠀⠀⠀⠀⠠⠔⢶⠀⠀⠀⠈⠙⢲⣤⢀⣀⣀⣤⣤⠶⠛⠉⠉⠁⠀⣀⣴⡀⠀⠀⠀⠀⠐⠉⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⢍⡉⠁⠀⠉⠉⠀⠀⢀⣀⣤⣤⣖⣋⣁⡀⡁⠂⢐⠀⣋⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡎⢭⠀⠀⠒⠬⠍⠉⠓⠒⠒⠊⢉⡀⠤⠛⠉⠉⠩⠉⠉⠻⠄⠀⡠⠂⠀⠹⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⢀⣠⣴⠶⡋⠉⠉⠉⢀⠀⢀⠀⠀⠀⢀⣀⣀⠰⠀⢠⠀⠀⢀⡀⠄⠁⢀⣀⣀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⠴⢋⠉⠙⠂⠈⠛⠛⠒⠀⠀⠘⠁⠤⠄⠈⠀⠀⠂⠀⠐⠶⠚⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠛⠇⠀⠀⢠⠶⠶⣿⡆⠀⠀⠀⢀⠴⠖⠶⠄⠀⢀⠞⠳⠄⠀⠟⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢾⠀⠀⠀⠐⠤⠐⠁⠀⠀⠈⠃⠐⠂⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⢀⣀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣀⠄⠀⠀⠍⠄⠠⣝⠅⠀⠀⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠷⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠠⠀⢀⡐⡁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡏⣼⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠆⠀⠀⠋⡋⠉⣠⣤⣴⣿⠃⠸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⢿⣷⣾⣿⠿⠟⠃⠀⢸⢇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⡔⠁⠀⠢⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠋⠀⠀⢠⠄⠀⠐⡂⡘⠀⠘⠧⠤⠤⣀⣰⡀⢀⠄⣠⠀⠀⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠴⠃⠀⠀⣠⢆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡀⠀⠜⠁⠀⠐⠀⠀⠀⠀⠰⠀⠁⣰⠇⠘⢰⠁⠀⠀⠀⠀⠙⠂⠠⢶⣎⡁⠀⣤⡶⠿⠿⠉
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⣠⡏⡄⠀⠀⠀⢰⣿⣦⣡⣾⢿⡇⠀⣸⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⠀⠀⠀⠤⠀⠀⠀⠀⢠⠀⠰⠃⠉⣷⠁⠙⠁⠀⠀⠀⣿⢏⣿⣿⡿⠀⠙⠔⢁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡇⠀⠀⠸⢇⠀⠈⣿⣿⣿⣿⣧⢀⡄
⠀⠀⠀⠀⠀⢀⠔⠟⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠠⠁⠀⠹⣿⡷⠿⠿⠞
⠀⠑⠄⣀⣤⣶⣊⡤⠖⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠠

⠀⠀⠀⠀⠀⠐⠦⣀⢀⡀⠀⠀⠀⠂⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠈⠦⠈⠙⢤⣀⠀⠀⢀⠀⠀⢀⣠⠴⠞⠓⠀⠀⢀⠀⠠⠖⠀⠉⠀⠘⠋⠀⠀⣐⠣⠤⠛⣄⠂⢀
⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠙⢳⣦⣄⠈⡏⠁⠀⠀⠀⠀⠀⠐⢄⠠⣿⡀⠀⠀⠀⡂⠀⠀⠀⠀⣀⣀⡤⠿⣓⣤⠇
⠀⠀⠀⠀⠀⠒⠂⠳⠤⣤⣀⣉⢈⠻⣮⠑⠢⢤⣐⠂⠀⠀⠀⠀⠀⠈⠑⠂⢄⣼⠧⣀⣐⣚⡭⠅⠒⠀⠉⠁
⠀⠀⠀⠀⡂⠉⠒⠲⠤⢤⣏⣉⣭⡏⠀⠀⠀⠀⠈⠁⣢⣄⠀⠀⠀⠀⠀⠀⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠗⠃⠁⠀⠀⠀⠀⠀⠀⠀⢈⡥
⠀⠀⠀⠀⢠⠂⠉⠉⠉⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄⡀
⠀⠀⠀⠀⡆⠄⠀⢠⠞⠛⢿⡶⡅⠀⠀⠀⠀⡠⠄⢄⢀⢐⣀⡀⢀⡆⠀⡀⠒⠀⠀⡔⠉⠀⠀⠐⣩⠴⡔⠂⢲⡆
⠀⠀⠀⠀⠹⣦⣀⠸⠴⠿⠆⠻⢴⡄⠀⠀⡐⠀⠀⣠⣾⡇⢈⣧⢞⠀⠀⢠⠀⠀⠘⠀⠀⠀⠀⣠⠞⣷⡅⠀⢀⠇
⠀⠀⠀⠀⠀⠈⠙⠢⠑⠂⠀⠀⠀⠀⠀⠍⠹⣄⣰⠷⢊⣠⢞⠛⠉⠒⣈⣀⡷⠀⢠⡀⠀⠠⣴⣋⣀⠜⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣠⠍⠃⠭⠗⠂⠩⠭⣁⣀⣨⠇⠀⠈⠳⣴⡀⣿⠉⢁⣠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣀⣠⠼⠶⠒⠉⠉⠊⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡺⣿⣷⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⡈⣶⣌⣈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠊⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⢿⡿⢛⣅⠙⠂⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⡵⢋⣰⡾⠋⢠⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠟⠁⠁⣠⡼⠏⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⣀⡾⠏⠁⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢀⣀⣶⠟⠁⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢠⡼⠟⠀⠀⣀⠔⠉⠚⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢡⡼⠏⠁⢀⡠⠚⠁⠀⠀⠀⢹⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⣴⠏⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠞⢀⡴⠊⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⡽⠦⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢰⣇⠈⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠻⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⣸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠻⣧⠧⠎


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠓⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡟⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣾⣷⣾⣿⣶⣶⣄⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠩⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠣⠈⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠰⠆⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⣿⣿⣿⡿⣻⣿⣻⣿⣿⣧⠘⣞⠁⠉⢉⡙⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢃⣼⣿⣿⣿⣮⣿⣏⢛⣿⣿⣿⡿⣧⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣾⣿⣿⣻⣿⣟⣿⣿⢬⣿⣿⠿⠠⠋⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀⣾⣿⣿⣿⣯⣽⣿⣴⣿⣌⣿⣿⢰⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠞⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣿⣯⡙⠳⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣷⠼⣯⢳⡟⢹⣴⣿⡆⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⣿⣿⣿⣿⣷⣤⣀⡀⢀⣠⣶⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⣶⣾⣿⣧⣼⣿⣿⣿⣧⡀⠀⠆⠰⠄⠀⠀⠀⠀⠀⢠⣿⠷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢟⣿⣿⣿⣏⢻⣶⣄⡙⢿⣧⡤⢿⡹⢿⣿⣶⣤⣀⣠⠀⢀⣀⣠⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢫⣾⣿⠟⣛⣿⡍⣿⣿⣿⣯⡁⠢⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⠉⣳⣽⣍⠻⡇⠸⠏⣿⡿⠿⢿⣿⡅⢸⣿⣿⣿⣧⡰⠙⣿⠉⠾⣿⣿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣧⣿⣆⣄⣿⢃⠀⣠⡝⣿⣿⠻⠿⠿⣿⣧⡀⢡⢀⣰⣿⠏⣗⣸⣿⣿⣋⣩⣽⣟⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡉⣹⣿⣿⣿⣿⣾⢂⣙⣷⠏⠀⠀⠀⠀⠈⣿⣷⣮⡎⠁⠑⠞⣠⣿⣿⣿⠏⠀⠀⠉⢻⡎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⠏⠉⣿⡭⠿⣭⣟⣟⣥⣤⣾⣷⡦⠀⠀⣿⣿⣿⣷⡄⠀⡤⠻⣿⡿⠁⠀⠀⠀⠀⠈⣷⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢈⣅⡿⠈⠀⣴⣿⡁⠀⢈⡍⢉⡿⣿⠻⣿⠻⣿⠞⣿⣿⣿⣿⣷⣤⣶⡆⣿⠁⠀⠀⠀⠀⠀⢠⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣤⣿⣿⢿⡀⣀⠟⠀⣰⣶⣶⣤⣤⣴⣯⣴⠾⢶⡿⠶⣿⣿⣿⣿⣿⣿⣁⠈⣉⠉⠀⠀⠀⠀⡠⠋⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠿⣿⣿⣿⣿⣵⣄⢠⣾⣃⢉⣹⠯⠞⠥⢸⣾⠇⡄⠟⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⢀⡴⢶⣿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣩⢿⡏⣟⣸⠃⠀⠰⠀⡾⢁⡔⢽⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⣿⠋⠀⠚⢷⣤⣧⣈⣠⡶⠂⢠⡏⢀⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⣏⣀⣀⣤⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⣴⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠈⠉⠁⢀⡤⠟⠀⡸⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣉⣽⡛⢋⣱⡿⣧⡀⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⢫⣿⣿⣿⣿⣿⣿⣷⣾⠿⠻⠛⠶⢾⣿⠃⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠹⣿⣿⣛⠿⠁⠘⠛⣻⡄⠹⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢇⡻⣿⡿⣿⣾⡿⠛⠛⠁⠰⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⣿⣿⡷⣿⢿⣷⢀⡘⣿⢧⣴⠆⠀⢁⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⣾⣼⣿⠁⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⣽⡿⣿⣺⣿⣿⣿⣿⣸⠛⣥⢨⡁⢠⠄⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠘⠣⠾⠤⠤⠾⠛⠛⣫⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠹⣿⣿⠀⠀⠀⠀⠂⡀⣠⢟⡥⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⡗⣿⡆⣠⣴⣶⣿⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣠⣀⣀⢛⣷⡿⢻⡅⣼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⠿⠃⣨⣵⣿⣻⣿⣿⡿⣴⠺⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡷⠹⠿⠟⣿⡇⠘⢻⡄⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠋⣀⣼⣿⣿⣿⣿⣿⣽⣿⣿⠿⡥⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠁⠠⠤⡎⠀⠀⠈⡘⠀⠀⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⢺⣎⡁⡆⢹⣿⣿⣿⣿⠛⡫⣾⣃⠀⡞⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠛⢁⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⢀⠀⠹⣿⠹⣿⡿⠿⣻⣷⡟⣽⣿⢶⣬⣙⠷⡿⠛⠖⠃⠀⠀⣀⣥⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡈⠸⠋⠘⠁⠉⢱⠆⡀⢀⠀⠈⠁⡞⣤⣾⣿⣤⣦⣟⢿⣶⣿⣿⠼⠋⠙⠛⢜⣃⣰⣤⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡌⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⡟⢫⡿⠝⣿⣿⣿⡆⠻⣿⣿⠀⠘⢛⡛⣦⣿⡿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣠⣴⣶⣾⣶⣶⣦⡀⣄⣤⡦⠀⠘⢣⣄⣴⣶⣿⡷⠀⠉⠛⡶⠶⣾⡽⡿⠓⡄⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠊⠛⠻⠿⠊⠉⠋⠑⡟⠃⠀⠐⠶⠟⠋⠏⠉⠋⠀⠀⠀⠀⠈⠀⠁⠐⠟⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⡿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠉⠀⠘⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠁⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠋⠁⠀⠀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠘⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠈⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⣄⣀⠈⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⡀⢈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡎⢁⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⡘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⠿⢿⣷⣦⣀⠀⠀⠀⠀⣠⣿⣿⢿⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⠀⠀⠉⠻⢿⣿⣿⣶⣿⣿⢿⣽⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⠋⠀⠀⠠⢶⣿⣩⣿⣿⡟⣿⣟⣿⡿⠿⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⠻⣿⣿⠁⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣿⣿⣿⡏⣿⣷⣿⣿⣶⣦⣤⡸⣦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣷⣿⣿⢿⣿⡏⠙⠻⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⢸⣿⣷⣾⣿⣿⡇⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣄⡀⠉⠛⠋⠉⠀⢀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠀⠀⢰⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⡇⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠸⣿⣄
⠀⠀⢀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⢿⣿⡀
⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠘⣿⣧
⠀⢀⣾⡟⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⢸⣿⣆
⠀⢸⣿⠁⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠈⢿⣿⡄
⠀⣾⡿⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠘⣿⣧⡀
⢠⣿⠇⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠁⢸⣿⣇
⢸⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⣿⣿⡄
⣿⣿⠀⠀⠀⣰⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡀⠹⣿⣧
⣿⣿⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⣿⣿
⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⣸⢻⣿
⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢨⠸⣿
⣿⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡄⣿
⢿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣽⠛⣿
⠸⣿⡇⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣠⣿⢠⣿
⠀⢿⣿⡀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⣬⣿⣿⣿
⠀⠘⣿⣷⡀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢏⣿⣿⣦⣿⣿
⠀⠀⠘⣿⣷⡀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢏⣹⣿⣧⣾⣿⠇
⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⣣⣿⣯⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠙⠻⠿⣿⣷⣶⣤⣤⣤⣄⣀⣠⣤⣤⣴⣾⣿⣿⡿⢿⣭⣿⢿⣣⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠠⠈⠙⠛⠛⠿⢿⣿⣿⣿⠿⠿⠿⠛⠻⣥⣴⣾⣽⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣤⣉⡀⠐⠂⠀⠆⠀⡤⠈⣉⣵⣶⣶⣟⡻⠷⣎⣭⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣴⣦⣴⣶⣶⣿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡀⢰⣠⡆⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⠹⡷⠄⠛⠃⠈⠈⠁⠀⠐⠋⠁⡼⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠙⠛⣠⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡥⠐⠀⠀⠀⠀⠚⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣩⠶⢄⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡔⡖⡖⡦⡀⠀⠀⠀⠀⣰⢞⡁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠱⣷⣿⠎⠀⠀⠀⠀⢠⠏⠀⠀⢼⠽⠀⣇⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⢿⡀⠀⠀⠀⠀⡼⠓⢦⡀⠀⠀⡀⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣇⠉⠒⠤⠴⠚⠁⠀⡀⠙⣆⠘⣸⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⢮⡢⠀⠀⡀⡶⡸⣅⠔⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡦⠤⣄⡀⠀⠀⠀⠀⢀⠩⠒⣉⠜⡑⠡⠗⠊⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠦⣙⣂⣀⣀⣀⠀⠠⡊⣠⠊⠀⠂⠀⠀⠀⢀⣀⣀⣀⣀⡞⢩⠴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⣉⡤⠤⠤⣌⡙⢦⡀⢀⣈⣀⣀⠀⣠⠖⢉⣤⠤⠤⣬⡉⢳⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠒⠲⢄⢙⡶⠛⣉⡤⠤⣌⡙⠳⣴⢋⡤⠒⠒⢤⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣬⡟⣠⠏⣡⠴⠶⢤⡉⢦⠘⣿⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠳⠶⠶⠶⠷⠚⠛⠋⠁

⠀⠀⠀⠀⠀⢀⣤⣔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡀⡀⠛⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠦
⠀⠀⠀⢠⡶⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠐⠋⠀⠀⠀⠘⠋⠉⠀⡀⢀⠄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠁⠀⢀⣤⢀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠀⠀⢀⠀⠀⠘⡻⠋⠀⠀⡔⠁⠀⣾⠖⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⢸⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣀⠴⠂⠀⠀⠀⠀⡛⠁⠀⣤⣄⣀⠤⠚⠁⠀⠀⢰⡗⠃⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⡀⣠⣆⢒⣈⣀⣤⡤⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠰⡿
⠀⠀⠘⠻⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠁⠀⠀⠀⠀⠞⠀⠘⠟⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣤⣤⣀⡀⢀⣾⢃⣼⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠘⠲⣿⠆⠀⠀⠈⡁⠀⠐⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⣿⠃⠘⠟⠛⠙⠛⠛⠛⣿⡟⣛⡁⣘⠁⣤⢤⢿⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢸⣿⠇⢀⣶⠲⡄⣼⡏⢹⢠⣿⣏⣼⢣⡿⢸⡧⢊⣔⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⢂⣾⣯⣠⣧⣿⠣⡼⢻⡟⢿⣄⠘⢓⣉⣙⡉⢸⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⠞⠻⣧⠟⠁⠙⠚⢁⣀⣥⣼⠿⢿⣍⣁⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠈⠿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠞⠃⠀⠈⠉⠁⢀⡀
⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⠓⠚⠃⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⡾⠖⠋⠉⠁⠙⠒⠀⠀⠀⠰⠒
⠀⠀⠀⢠⠀⠘⠷⠀⠀⠀⠀⠀⣴⡶⠂⠀⠀⠉⠉⠀⠄⠀⠀⠀⡠⠂⠛⠁⠀⠐⠟⢃⡀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠘⢿⠁⣠⣤⠀⣀⣘⣀⣀⠠⣤⠀⠀⢤
⠀⠀⠀⣿⣧⠀⠀⣠⣄⢠⣴⣦⠀⠀⢠⣶⣤⠀⠔⢠⠆⠀⠀⣼⠃⠀⢛⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⢦⡤⠒⠉⠉⣉⠉⠉⠉⠓⢄⢀⡀⠁
⠀⠀⢰⣿⣿⣿⣧⣤⣤⣄⡀⢤⣄⣀⣀⣀⣀⣀⣴⡏⠀⠀⢀⡴⠀⠀⠙⠁⠀⠀⠀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⠀⠀⠋⠀⠀⢀⡀⠀⠈⠡
⠀⠀⠘⢿⣽⣋⡉⣉⡉⠹⠿⠟⠛⠉⠉⢉⣽⡷⣀⣶⠀⠀⠈⠳⠀⠀⠦⢀⠐⠀⠈⠀⠁⠚⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⣏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⢰⣿⣿⣿⣭⢤⣦⣦⣤⣤⣴⡷⠾⠿⠛⠙⠋⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠉⠀⣀⠀⠀⠤⠀⠀⣀⡀⠀⠀⢸⡇⣴⠄⠀⠀⠀⠀⢴⣆⠀⠀⠀⡀
⠀⠀⠀⢨⣥⣌⣉⠀⠀⣀⣀⣀⢠⠾⠆⠀⣀⣠⣤⣴⡛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⣿⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠛
⠀⠀⠀⢉⠾⠋⠉⢭⡉⠉⠋⢹⠟⢻⡏⠉⠉⠉⢈⣉⣤⠀⠀⠀⠀⠀⣀⣀⠀⠰⠖⠒⠒⠀⠉⠈⠁⠀⢠⡀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠶⠂
⠀⠀⢠⣿⡟⢛⡶⠶⣤⣤⣤⣤⡶⠶⠶⠞⠛⢋⡉⠡⢄⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠙⠁⢠⡆⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⢶⠆⣀⠤⠔⠁
⠀⠀⠀⠨⣷⣿⣿⣄⣀⠺⠷⠀⠀⠀⣀⡀⠀⣛⣣⠴⠊⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠉⠁⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠷⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⢛⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢟⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢏⣴⠟⠁⠀⠀⠀⠀⣠⣤⣿⣶⣦⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣱⡿⠁⠀⠀⠀⠀⢠⣾⣿⣳⣽⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢻⣿⣿⠏⣽⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣿⡟⣁⣀⣤⣤⣤⣴⣿⣿⡿⠟⠋⠉⠀⡀⠀⢠⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣾⢫⣿⢰⡏⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣼⣿⠿⠿⢭⣭⣛⣿⡿⠋⠁⠀⠀⢀⠀⡞⣷⠀⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⠁⣾⣿⢸⡇⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣛⣛⣓⣶⡟⠁⠀⠀⢀⡀⠀⣾⣧⣷⠸⣆⣿⠻⣿⣿⣿⡆⠀⠻⣿⣿⣿⡇⢀⣿⣿⢈⣧⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⣿⠋⠀⠀⡀⠀⢸⡇⢀⣿⣿⡿⠀⢹⣿⣷⣌⠻⢿⣿⣄⡀⠈⠙⠻⢶⣛⡛⠻⠿⣍⣙⠲⠦⣤⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣾⡿⠁⠀⣀⠀⣿⡄⢿⣿⣮⣿⣿⡿⠂⠀⠿⣿⣿⣿⡲⠬⣙⠛⠿⠿⠿⢶⣮⣽⣿⣿⣿⣋⡽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠟⣿⠁⢀⠀⣿⣆⣿⣿⣾⣿⡿⢿⣿⣿⣔⣠⠶⠛⠛⠛⢻⣷⣦⣤⣤⣀⣠⣶⣶⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⠇⢀⢸⣷⣿⣿⠿⢛⣩⡴⠖⠛⠉⠉⢉⣀⣤⣴⡶⢻⣿⣿⡿⠟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣿⣿⠀⣼⣿⣿⢟⣥⠞⢋⣁⣤⣤⠶⠾⠟⣿⢿⣟⡁⣠⡟⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⡏⠰⣿⡟⣱⢋⣤⣞⣫⣭⣤⣴⡶⠶⠞⠃⣿⣿⣷⢻⣷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢸⣿⡽⣡⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠈⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣼⣿⣳⣿⠏⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢻⣶⣿⣪⠽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣾⣿⣿⣷⡯⢽⣶⣦⣤⣀⣀⣀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠄⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣻⣯⣵⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣖⣻⡛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡻⢭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⡀⠁⠀⠀⢏⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⢻⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣧⣿⣿⣿⣿⠋⠴⣤⣤⡤⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⣿⣿⣿⣿⣥⡛⣦⡞⢡⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣟⣿⣷⣄⣠⣾⣿⣿⣿⣿⣻⣷⣿⣿⣿⣇⡻⣡⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣟⠫⠍⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣁⠽⠿⣿⣿⣏⡠⠖⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡿⢿⣿⣿⡏⣰⣿⣟⠙⠿⠋⠙⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⡿⢩⡶⠝⣫⡍⠃⠸⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⠿⣷⡨⣷⠀⠉⢠⡆⢸⠀⢸⡇⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣤⣶⣤⡂⠀⣤⣤⣦⣴⣄⣾⡇⠀⢰⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢝⡃⢁⣾⣿⣿⣿⣾⠟⡁⠀⣾⣿⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠺⠉⠛⠛⢉⠠⠈⠀⠀⠿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⢯⠄⡀⠀⠂⢠⡄⠂⠁⠀⠀⠀⠀⠀⠈⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⣿⣿⣿⣿⣿⠒⡫⠈⠁⢀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⡑⠀⣤⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⡌⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⢱⡇⢿⣿⡿⠟⠉⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣸⣷⡈⠁⠀⠀⣠⣾⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⡿⠓⢶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⡄⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡢⡀⠀⠀⠀⠄
⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⠔⠀⠀⠀⠀⢸⠷⣀⠀⠀⠀⠀⠀⢂⢘⡿⠂⠀⣀⡄⠀⠀⠀⠀⠀⠀⠸⣄⡀⡺⡟⡿⠄⠀⡀⠀⢀
⠀⠀⠀⠘⠺⠄⡀⠀⠀⢸⡂⣀⠤⣀⣥⠶⠛⠁⠀⠀⠀⠀⠀⠪⣑⣚⡴⠟⠉⠀⠀⠀⠀⠀⣄⠀⠀⢿⡿⣝⡋⢀⡠⠞⢀⡀⡾
⠀⠀⠀⠀⠉⠓⠲⠬⣌⣉⣩⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠓⠦⠅⢠⡤⠀⠠⠐⣈⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠤⠒⠚⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡉⣱⡗⠠⡀⣐⡆⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆⠀⠈⠀⠀⠉⠀⠀⠈⠉
⠀⠀⢀⡀⢀⣠⡴⠒⠋⠁⠀⠀⠀⠐⠄⡀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢢⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣄⠤⠒⣉⠁⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣿⣷⠖⠩⣦⣀⠤⢄⣀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠈⠁⠁⡢⡄⠀⠀⠀⠀⠀⠀⠤⠀⣠⠴⠊⠁⠈⠙⠿⣤⣬⠓⠀⠉⠀⠠
⠀⠀⠀⢄⣐⣿⣇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢰⣀⠀⢾⢿⠤⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠊
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣛⣁⢀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⣤⣼⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠐⠀⠈⠀⢸⠀⠀⠀⠀⣤⠀⠸⣿⣿⣿⣿⠋⢀⣾⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⣾⣿⣿⠿⢠⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠈⣄⠀⠟⢻⣿⡿⣿⣿⠁⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠙⠿⠿⠖⠛⠆⠀⠿⢿⠁
⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠛⠀⢻⡏⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠻⣿⣿⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣿⣷⡄
⠀⠂⠀⠀⠀⣠⣾⠁⠀⠐⠿⠀⠀⠀⠀⠀⠂⠘⠦⠁⠀⢀⣀⣀⠹⣇⠀⠀⠀⢰⢲⣀⠁⠀⠀⡀⠀⡠⠘⡇⠀⠀⠀⠀⠀⡠⣄⠐⠀⠀⠀⠉⢻⣷
⠀⢀⠀⠀⣸⠯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⡉⢡⣤⡈⠀⠀⠀⠀⠀⡙⠤⠄⠈⢀⣀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠈⠀⠤⠐⠛⢛⣀⠙
⠀⠀⠄⠀⣏⠀⢠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠟⣠⡎⠉⡰⠀⠀⠀⢠⣤⠔⠀⢀⣸⠏⠃⠀⣠⠖⠀⠀⠀⠀⠀⠀⣤⢀⣀⣿⣈⡙⢳⡔⠆
⠀⡀⠈⣆⠈⠳⠂⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠈⢁⡠⠾⠇⠀⠀⠀⠀⠙⢧⣠⢉⣥⠤⠒⠿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⡻⣿⣿⡟⠛⠁
⠀⠀⠀⠛⠙⠲⠤⣤⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠋⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⣀⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⡄⠘⠉⠉⠀⠀⠀⠀⠀⠀⠨⠛⢿⡉⠑⢄⠀⠀⠘⠙⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠐⠆⠀⠀⠀⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠛⢐⡶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠧⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢀⢠⠐⠊⠝⠁⠀⠀⠑⡀⠀⠀⠀⠈⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠰⢤⣤⣠⣄⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠉⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⡄⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⣀⡀⠀⠀⠀⠀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠋⠁⠀⠂⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠈⠉⠀⠀⠀⠛⠀⠀⠀⠠⢀⠀⢀⣀⠀⢠⡀
⠀⠀⠀⠀⠠⠤⣄⣀⠀⠀⠉⠀⠀⠀⡠⠾⠏⠃⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠒⠛⠁
⠀⠀⠀⠀⠈⠀⠀⠄⠈⠒⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠉⠰⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠁⠀⠺⠄⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠚⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠚⠋⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠃⠀⠀⠀⠉⣡⡀⠀⠀⢀
⠀⠀⠀⣴⠛⠝⠛⠛⠉⠉⠈⠁⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠖⠛⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠢⠄⠀⠰
⠀⠀⠀⠉⠳⠤⠦⡄⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠶⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠀⠀⠀⣠⠴⠖⠖⠲⠀⠀⠀⠀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⡼⠁⠀⠀⠀⠀⣰⡿⠀⠌⠀⠈⠇⠀⡀⠀⢀⡀⠁⠄⢀⠀⠀⢀⣆⡅
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⣰⣿⡟⢀⡀⠀⠀⢰⡿⠁⢰⠀⠀⡔⢀⣾⠏⠀⣿⣿⡿⣰⣿⠃⠀⠞⠻⠁
⠀⠀⠀⠀⠀⠀⢰⡿⠃⣰⣿⡟⣼⣿⣿⡗⢠⡿⠁⠀⣬⣠⠊⠀⣾⡟⠀⢀⡿⠋⣱⣿⠃⠀⠌⢀⠃⢀⡞⣷
⠀⠀⠀⠀⠀⢠⡿⠁⣰⣿⡟⢸⣿⣿⡟⢀⡾⠁⠀⢠⡿⠁⠀⣸⠏⠀⢀⡾⠀⢠⡿⠁⠀⠌⠀⠌⠀⣼⣿⠇
⠀⠀⠀⠀⢠⡟⠀⣰⣿⡟⠀⣸⠟⠉⣠⣾⠃⠀⢠⣿⠃⠀⣰⠏⠀⢀⡾⠁⢀⡿⠁⠀⡌⠀⠐⠀⣸⣿⡏
⠀⢀⣴⣶⣿⠀⠐⣿⠟⠀⣴⠏⣰⣾⣿⣿⠀⢀⣿⠇⠀⣰⠏⠀⢠⣿⠁⢀⡾⠁⠀⡘⠀⢠⠁⣠⣿⡿
⠀⣾⣿⣿⡿⠁⣼⠁⢀⣼⠏⣰⣿⣿⣿⣿⢀⣾⡏⣀⣴⡏⠀⣠⣿⢃⣴⣿⡇⠀⡐⠀⠀⠆⠀⡼⡟⠃
⠘⠿⣿⠟⠁⠀⠁⢀⣾⠏⠀⠻⢿⣿⣾⠃⠘⠋⣼⣿⣿⡇⠀⠿⠋⠺⢿⣿⠃⡐⠀⠀⡈⠀⢰⢱⠁
⠀⠀⠀⠀⠀⠀⢀⣾⠏⣀⠀⠀⠀⠈⠁⣀⡀⠀⠙⠿⠟⠁⠀⠀⠀⠑⡀⠀⠀⠀⠀⠐⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠁⣼⡟⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⢀⣤⣶⣤⠀⠀⠀⠀⢠⡶⠀⣴⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⣀⠀⣾⣿⣿⣿⠃⣴⠄⣰⣿⠁⢰⡟⠀⣰⣿⣷⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣠⡆⠀⣽⠛⠉⠉⣼⡿⠀⣿⣿⡿⠋⣼⠏⢰⣿⠇⢀⣿⠃⣰⣿⣿⣿⠀⠀⢀⠆
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣰⡟⠀⣼⠃⠀⠀⣰⡟⠁⢀⡟⠁⠀⣰⠏⠠⣿⡏⠀⣼⡏⢠⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡟⠀⠀⠀⣰⡟⠀⣰⠃⠀⠀⢰⠟⠀⢀⡞⠀⠀⢰⡏⠀⢀⡿⠀⢠⣿⢡⣿⣿⣿⣿⣿⠀⡄⠄
⠀⠀⣀⣴⣿⠀⠀⠀⢰⡟⠀⣰⠇⠀⠀⣴⡟⠀⢀⡾⠁⠀⢠⡟⠀⠀⣾⠁⠀⣾⡇⠈⠛⢿⡿⠿⠏⡘⠘
⠀⣼⠟⠙⣿⠀⠀⡠⠂⠀⣰⠏⠀⢀⣼⣿⠇⢀⣾⠃⠀⢀⡟⠀⠀⣼⠇⠀⣸⣿⠀⠀⠀⡼⠀⠀⢰⢡⠃
⠀⠿⠶⠿⠋⡰⠋⠀⠀⢰⠏⠀⠀⣼⣿⠏⢠⣾⠇⠀⣶⣿⡆⠀⣰⡟⠀⣼⣿⣿⠀⠀⢠⡇⠀⢠⠃⡌
⠀⠀⠀⠀⠐⣶⣶⣤⣀⠉⠀⠀⣼⣿⠏⠀⣾⠏⠀⡸⠻⡿⠁⢰⡿⠁⢸⣧⣇⡿⠀⠀⣾⠃⢀⡎⢰⠁
⠀⠀⠀⠀⠀⠉⠛⠿⠃⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠁⠀⠀⠉⠋⠁⠀⣸⣿⠀⠜⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠶⣶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⢿⠟⠋⠁⠿⠿⢇⢀⣴⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣾⣿⡿⣻⢵⡾⠃⠀⡜⠀⠀⠀⠧⠊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⢟⣵⠿⠛⠉⢧⡀⠸⡀⠀⣀⠄⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢟⡵⢋⣤⣶⠋⠀⠀⠈⠀⠈⢉⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡫⠋⢀⡋⠉⠁⠀⠀⠀⠀⣠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡿⠑⠁⢀⡾⡆⠀⠀⠀⣠⡴⠊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠁⠀⠰⠏⠀⣀⡤⠖⢉⣏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣀⣴⣶⠉⠀⢠⡾⠙⠁
⠀⠀⠀⠀⠀⠀⢀⡤⣤⠄⠀⢸⡟⢠⡞⠉⠉⠃⠀⣷⠀⠀⣤⣠⡀⠂⠠⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠶⠤⠼⠧⠿⠤⣤⣤⣤⣬⣥⣤⣤⠴⠿⠤⠤⠤⠶⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⢻⡟⣿⢿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡏⡾⠁⠇⠛⡙
⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⣿⢸⡇⣿⢰⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⢸⠀⢀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⣇⢹⠘⡇⢿⢈⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣨⣿⣅⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⠀⢸⢰
⢠⣦⡴⢛⣛⡳⠀⠀⢸⣧⣿⣾⣶⣷⣾⣾⡇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⢸⣾⠀⢸⠀⣼⡾⠀⠀⠰⣛⣛⡳⣤⣦
⡏⣿⠞⢣⣤⠙⡆⠀⠈⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⡇⢠⣿⠇⠀⠀⡞⢡⣤⠛⢾⡏
⢷⣿⠀⠘⣲⠞⠁⠀⠀⠀⠈⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠨⣿⠀⡇⢰⡏⠀⠀⠀⠙⢶⡚⠀⢸⣷
⠈⢿⣆⣾⢙⠃⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⢡⣿⠃⠀⠀⠀⠀⢛⣹⣆⣾⠏
⠀⠀⣙⣿⣿⣤⣄⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣿⣿⣾⣏⣀⣀⣀⣀⣤⣼⣿⣟⡁
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⢀⣴⠟⢩⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⢫⠙⢷⣄
⢀⣿⠃⠀⢾⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⢻⣇
⢰⡃⠀⠀⠘⣧⡀⠀⣈⡡⠠⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠠⠠⣉⡀⠀⣠⡟⠀⡀⢀⣿
⠀⢿⣶⡻⠆⡠⠉⠛⠋⠉⠀⠀⢿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠉⠛⠋⠩⣀⢾⣱⣿⠇
⠀⠘⠛⠭⠥⠄⠀⠀⠀⠀⠀⠀⢸⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠭⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃
⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣲⠟⣅⡿⡿⠋⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡆⠁⣷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡎⠀⠀⠀⠀⢀⡄⣻⣭⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢻⣿⣿⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣥⣾⣿⣿⣿⣿⢻⣇⠻⠿⠷⢻⠀⣀⠾⣛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠁⠀⠐⠀⣰⢻⣶⣿⣿⣿⣿⣿⡃⣿⣿⣷⣦⣤⣬⣝⣿⣷⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⣿⣿⣿⣿⣿⣿⠛⢛⡻⢯⣉⣈⣉⣀⣬⣾⡿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⣿⣿⣿⣿⣿⣿⠀⣾⡿⠀⠀⠀⠀⠀⡴⠋⠀⠀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⢸⠰⣿⣿⣿⣟⠁⣰⢯⡿⠛⠁⠀⠀⣠⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡾⣸⣿⣿⣿⣿⡁⣠⠞⣣⠀⢀⡴⠊⢹⣏⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡷⢽⣿⣿⣿⣿⣷⣧⢞⣥⡶⠋⠀⠀⠸⣽⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⣿⣿⣿⣿⠋⣴⣿⠟⠀⣠⠀⠀⠀⠀⠉⠁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⣿⣿⣿⡟⢸⣿⠃⠀⢸⠇⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢿⣿⣿⡇⣿⠇⠀⠀⣾⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣟⢻⣆⠀⠀⠈⠀⠐⠀⡀⠸⠿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡞⢿⣷⣦⣤⣤⣴⣦⣾⣦⠺⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠂⠢⡀
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣲⡘⣬⣦⣷⣷⣿⢰⡞⠛⢻⢻⣦⢰⣿⡷⣶⣤⣄⡀
⠀⢈⣈⣉⠛⠺⠶⣶⣦⣄⣀⣀⣀⣀⠀⠀⢀⣀⣀⣄⣾⠾⠿⢿⣿⣿⠣⠜⢆⢼⣿⣼⣩⣿⣭⣸⣿⣾⠇⠒⠜⡯⠛⣻⡿⣶⣦⣄⣀⡀⠀⠀⣀⣀
⠀⠈⠢⡉⣉⠙⠓⠲⠶⢭⣿⠛⢟⢻⣷⢫⣍⡍⣿⣿⣏⣲⣿⣘⣿⣿⣿⣻⠛⡿⣿⣿⣿⣼⣭⣏⣿⣿⡗⢰⡞⠀⣰⠇⠀⠀⠈⢻⡟⣿⡟⡶⣧⡼⠃
⠀⠀⠀⠙⢏⣉⣉⣩⣭⣽⣯⣽⣮⣿⣿⢳⡿⢾⣾⣿⣧⣯⣭⣯⣿⣿⢛⡛⣻⣿⣿⣿⣧⣤⣿⣿⣿⣿⣟⣾⠀⢠⡟⠀⠀⠀⠀⠀⠑⠬⠵⠃⢈⡇
⠀⠀⠀⢀⡜⢛⣉⡥⠶⣚⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣼⣶⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢻⣿⣿⣿⣦⡈⠁⠀⠀⠀⠀⠀⠀⢀⣄⣤⠞
⠀⠀⢀⠞⠋⣉⡤⢖⡫⠁⠈⠉⠛⠛⠉⠛⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⣸⡟⣏⣿⡈⠛⢷⣤⣀⠀⢰⢠⣴⠿⠋
⠀⢠⡯⢖⡫⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⡚⠖⠘⠁⡻⣦⢭⣙⣛⠻⡿⠿⣿⣿⣿⡏⣷⣶⣾⣿⣿⡇⣿⡿⠇⠀⢀⣈⡭⠿⠛⠉
⠀⠈⠛⠁⠀⠁⠀⠀⠀⡀⠀⠀⠀⢠⠊⢻⠉⠀⠀⠀⠈⠓⠕⠃⠀⠀⠙⠒⠒⠒⠒⠃⠣⠤⠄⠚⠛⠁⠛⠓⡿⠛⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⣠⡶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠋⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢠⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢻⣟⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣐⡪⢽⣿⣯⣷⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⡤⣺⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠠⢴⣶⣯⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⠃⣠⠞⢉⣾⣿⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⣽⣿⣿⣿⣿⣿⣿⣿⢸⠟⡛⠛⠋⠉⠀⢀⡜⠁⣠⠿⠟⣋⡝⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠈⢿⣻⢦⣀⡀⠀⠘⠳⠶⣃⡤⠞⠉⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣰⣿⣿⣿⣿⣿⣿⡿⠛⣁⣀⣀⣀⠀⠉⠓⠮⠭⠽⠟⠛⠉⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⣿⣿⣿⣿⣿⡿⣫⠖⠋⠙⠻⢛⣿⠃⠀⠀⠀⠀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⣿⣿⣿⣿⡟⠁⢹⠖⠲⠖⠒⠋⠀⠀⣀⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣿⣿⣿⣿⡿⠀⣴⠃⠀⠀⠀⠀⣀⡴⠋⢹⡙⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠃⣸⠇⠀⠀⠀⢀⡾⠋⠀⠀⠀⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠸⠀⡿⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⡇⠀⠀⠀⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⣷⠀⠀⠀⢻⣿⣟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣰⣄⠘⢿⣿⡗⠊⠀⠀⠀⠀⠀⠀⢀⡤⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⣿⣝⣿⣷⣮⡻⢿⣷⣤⣀⠀⣀⣠⣴⠏⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⢿⡆⠉⠛⠶⣬⣙⠻⠿⣿⣿⡿⠅⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣾⣿⣤⣤⣿⠯⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⡝⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠻⡌⠂⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣆⠀⠀⠀⢇⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣷⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡘⢳⠶⠒⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠷⡧⣸⣯⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣲⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⡶⣒⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠸⣿⣿⣿⣃⣠⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⠻⠟⠻⢋⡂⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣵⡟⣫⣭⣧⣿⠸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣿⣿⣿⣿⣿⣿⠆⠉⣼⣯⡼⢰⡟⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⢿⣿⣿⣇⠀⢾⣿⣿⡿⣼⡟⠀⢸⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⡓⠂⢰⡿⣿⣿⠟⠋⠁⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢛⡁⠀⡀⠀⣀⠀⠀⠀⠀⠀⠈⠻⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢡⠇⠐⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠀⢀⡀⣸⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡽⠁⢼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣹⡘⣿⠿⠟⠁⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣧⡀⠀⢀⣴⣿⣟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⡉⠙⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⡿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⠶⠶⣭⣭⣿⣿⣷⣦⡤⠀⡀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣿⣿⡿⠟⠛⢉⣁⣤⣤⣶⣶⣾⣿⢿⣿⣿⡿⣿⣿⣶⣶⣶⣯⣤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠋⣁⣤⣶⣿⣿⡿⣿⣾⣿⣿⣻⣿⣿⣯⣿⡷⣿⣿⣿⣿⢿⣿⣿⣿⣿⣯⣝⡳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⣿⡾⣿⣷⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⡿⢷⣮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣿⡟⣿⣿⣿⣷⢿⠿⠟⣋⣛⣭⣭⣽⣿⣿⣿⣿⣿⣿⣧⠸⢿⣿⣿⣿⣿⣯⣻⢷⣿⣿⣿⣷⣿⣦
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠛⣋⣡⣤⣴⣾⣿⣿⣿⣿⣯⣿⣿⡿⣿⣿⣿⣿⣿⣷⣶⣤⡝⣿⣿⣿⣿⣿⣟⡛⠛⠛⠿⠿⠇
⠀⠀⠀⢠⣾⢿⣿⣿⡿⠋⣠⣶⣿⣿⣿⢿⣿⣭⣿⢹⣿⣿⣿⣿⣿⡿⢟⣉⣭⣤⣶⣭⣿⣿⣿⣞⣿⣿⣿⠿⠿⠿⠿⠿⣷⠄
⠀⠀⢠⣿⠏⣿⣿⣿⠃⣾⣿⣿⡿⣫⡿⣫⢿⡿⢳⣿⣿⣿⣿⡿⢯⡾⣫⢟⣿⣿⡟⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠁
⠀⠀⣼⣿⠀⣿⣿⡟⠘⡿⠟⠉⠀⠙⢾⡵⠟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⠏
⠀⠀⢻⣿⡀⣿⣿⣿⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉
⠀⠀⠈⣿⣧⠘⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀
⠀⠀⠀⠘⢿⣷⡜⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⣿⡇
⠀⠀⠀⠀⠈⠻⣿⣦⣟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠖⣚⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⡿⠷⠶⠶⠟⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣛⣛⣛⣛⣛⣃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣶⣶⡶⠄⠀⠀⠀⢀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⣷⣄⠀⠀⠀⠈⠙⠛⠛⠛⠛⢁⡴⠟⠁⠀⣀⣠⣤⣤⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣠⣶⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⢶⣤⡀⢳⠀⠀⠀⠀⢀⠞⢁⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⠈⠳⡄⠃⠀⠀⠀⢁⠔⠋⠀⢀⣠⣤⡶⠶⠾⠿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠙⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣉⣀⣀⣀
⣿⣿⠋⣋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠉⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠫⣗⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⡂⠭⠐⠊⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⠉⠑⠢⢄⡀⠈⠓⠺⠕⡢⢄⣀⠤⢒⡪⠅⠒⠈⢁⡀⡄⢰⡆⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠄⠴⡁⠀⠈⠑⠢⢄⡀⠀⠁⠒⠤⠒⠚⠉⡇⡋⠀⣤⠀⢠⢸⠀⠃⠓⠈⢀⠠⢚
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠈⠑⠠⢀⠀⠀⠈⠑⡢⢄⡄⠀⢀⠃⡇⡇⠀⡯⠈⠈⢀⡀⠤⣒⠩⠔⠊⠁
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢈⠗⡄⡰⠁⡜⡃⠀⠀⡇⡇⡇⣀⠤⢔⡪⠅⠒⠉
⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡰⡳⠁⡜⠀⡇⠀⠀⣇⡧⠭⠒⠉⠁
⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⡠⠃⡰⡱⠁⡜⠤⣀⡧⠔⠊⠁
⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⡰⠁⣜⡴⢁⡜
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠾⠤⠚⠀⠉⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠘⠹⠤⠤⠀⠐⠚⠔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠓⠈
⠀⡄⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈
⠘⡀⢈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡀
⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠌⠀⠀⠀⡆
⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠐⢀⠀⠀⠂⠀⠁
⠀⠀⠀⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠐⠀⡄
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠀⠀⠀⠛
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡧⠀⠀⠀⠀⠔⡀⡀⡤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⠀⠀⠀⠐⢐⠛⠁⠀⠀⠀⠀⢋⣀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠂⠉
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠣⠈⠁⠁⠀⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠄⣀⠤⠶⠒⠀⠤⡀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠘⠁⠀⠀⠀⠀⠀⠢⢀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢶⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠈⠻⣿⣿⢦⡀⠀⠀⠀⠀⠀⠂⢠⠚⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠀⣿⠿⢿⠳⢄⠀⠀⣸⣿⣷⡄⠀⣠⠟⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠚⠁⣀⣠⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⢃⡄⣷⠚⢄⣀⣀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⠀⠔⠉⠀⢈⠀⠀⢲⣦⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠛⠔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠃⠀⠀⢀⠂⠀⠉⠁⠙⠿⠷⣤⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠆⢀⡀⠀⠀⠀⠀⣬⡗⠠⠤⠤⠄⠒⠉⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢠⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢰⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⢀⣡⣄⢠⡤⠎⠁⠀⢣⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢈⠀⠀⠀⠀⢠⡂⠀⠁⠀⠀⠀⠀⠀⠀⠙⢦⣸⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⢚⠀⠀⠢⠄⠀⢠⣿⠁⡆⠘⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⢩⡍⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱
⠀⠀⠀⠀⠀⠀⡐⠀⠸⠀⠀⠀⠀⠀⠈⠉⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠑⣄⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠀⠀⠀⡸⠀⠀⡄
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠉⠀⠈⠒⠂⣴⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣴⡾⠃⠀⠀⡇
⠀⠀⢀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰⠃⠀⢸⣿⠁⠀⠀⠀⢹
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⠁⠀⢀⡽⢰⠇⠀⠀⠀⠀⢸⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⢈⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠀⡇
⠀⠀⣆⠀⠀⠀⠀⠠⠔⠊⣡⢀⡄⣸⠔⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⣇⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠙⠢⠤⠴⠶⣶⠾⠷⠛⠉⠀⠀⠀⣿⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠁⢁⠀⠀⠀⠀⠀⢠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⠀⠀⠀⠀⠈⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⠀⠀⠐⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠑
⡿⢿⠿⠿⡿⣟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣻⣻⣿⣛⣿⣿⣟⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⣤⡐⠀⠠⢴
⠀⠀⢰⣿⣶⣶⣤⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⢀⠠⠠⠒⡠⠴⣒⡯⡔⠒⠒⠀⠀⠒⠀⠐⢉⣀⠲⠢
⠀⠀⠀⣿⣥⣤⣼⡿⣿⣿⠃⠡⠤⢿⣿⣿⠿⠿⠛⠀⠀⣀⣃⡉⢭⡥⢤⣫⢍⠚⠾⢯⣀⠤⢄⣍⠒⠀⢌⣁⡆
⠀⠀⠀⣿⣿⣿⣿⠋⠡⠀⠠⠰⠆⠰⢆⡀⠒⢀⠄⠁⠀⠀⠩⢖⢄⠈⠐⠬⣡⣴⡆⡳⠮⠭⠄⠒⠨⠝⡁⠁⠁
⠀⠀⠀⢿⣿⠀⠀⠀⠰⠎⠀⠦⠈⣀⣀⠤⠶⣤⣀⠀⠲⠤⡀⢄⣁⠀⡠⣀⠀⠈⠙⢌⣫⣈⠛⠍⡫⠄
⠀⠀⠀⠘⠫⠔⠂⠉⠁⠉⠉⠈⠉⠁⠀⠀⠀⠀⠉⠛⠷⡶⠾⠆⠜⠁⠀⠀⠀⠀⢀⣀⣑⢄⢬⣵⡻⠿⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠙⠁⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢦⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⠃⠀⠀⠀⠀⠀⢰⢇⡇⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣀⣤⡿⠒⠀⠀⠀⠀⠀⣿⠞⠀⠀⣀⡀⠀⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡿⠉⠁⢸⡇⣴⣦⣠⡶⣆⣼⣿⠆⠀⣾⠟⠁⣼⢋⡿⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣾⠐⣧⠏⠸⡵⠃⡾⢳⡀⡼⣇⣀⠜⣯⢼⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⡏⢀⣤⣤⣀⠀⣤⡁⠀⣭⠁⠀⠀⣀⣂⠸⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠈⢱⡏⠀⠀⢹⡆⡏⠳⣄⣿⠘⠃⠰⣦⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠘⢧⣄⣠⡼⠃⣇⠀⠘⣯⢰⣿⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣠⠖⠋⠐⣄⢀⣤⣤⣄⡀⠀⢀⣿⡏⠀⢠⣿⡏⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠞⠁⠀⠀⠀⣿⠀⠈⠁⠀⢀⣀⣼⡿⢀⣀⣼⡟⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣿⢴⡞⣿⢃⣾⢡⡿⠁⡾⢡⡿⢁⡶⣰⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡤⠚⣹⠃⠾⠷⣏⡼⣧⢾⡇⣸⣧⢾⠇⣸⠗⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⡴⠋⢀⣴⠃⠀⠀⠀⠋⠀⠀⠸⣧⠇⠀⠸⣤⢣⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡤⠞⠁⣀⣀⣠⣶⣿⡇⣶⣄⡀⣀⢀⡀⢠⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠃⠟⠁⠀⠀⠀⠀⢼⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡉⠡⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠫⡈⠄⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠈⠢⡁⢀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠊⠁⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⡴⠊⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⢼⠁⠀⠀⠀⠀⠠⡖⠁
⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠒⠁⠀⢸
⠀⠀⠸⣄⡆⢀⠀⡀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡇
⠀⠀⢠⠎⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⡀⠁⠠⠠⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡈⠀⢣
⡞⠀⠠⠀⠀⠈⠉⠉⠀⢀⣀⠀⠀⠀⠀⠀⠈⠭⠤⠅⠂⠁⠠⠤⠀⢀⡎
⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠊
⣶⣶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⠞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣥⡍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣽⡅⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⡐⢶⣾⣾⣿⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣉⣉⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢸⡆⠀⠀⠀⡆⠀⠀⠀⢰⠀⠀⠀⢀⠠⠄⡀⠀⠀⢰⡆⠀⠀⠀⢰⠀⠀⠀⠀⣶⠀⠀⠀⣶⠀⠀⢸⣷
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠿⠀⠀⢰⣿⠀⠀⢸⣿
⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠈⢲⣶⠁⠀⠀⠈⡇⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⣼⣦⠀⠀⣴⣷⡄⠀⢀⣼⣤⡀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⡆⠀⢸⣿⠀⠀⠸⡟
⠀⢷⠿⠀⠀⠙⠛⠁⠀⢻⣿⣿⠇⠀⠀⠀⠁⠀⠀⠘⠿⠿⠃⠀⠐⠂⠒⠀⠀⠿⠿⠿⠀⠘⣿⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⠀⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⣿⣧⠀⣿⡗⠒⢶⣶⠀⠀⠀⣿⠀⢸⣿⠀⢸⠀⢸⡇⠈⠃⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠙⡟⠀⢠⣤
⠀⢹⠋⠀⢿⣿⡿⠿⠿⠀⠀⠀⣿⠀⢸⠟⠀⣸⠀⢸⠃⢠⡇⠀⣭⠃⠀⢻⡟⠀⠀⠀⠠⠀⢀⡇⠀⠀⡏
⠀⢸⡀⠀⢀⠀⠀⠀⠀⠀⣀⠀⣿⠀⢸⡄⠀⣿⠀⣽⠀⢸⡇⠀⣻⠀⠀⢸⡇⠀⠈⠀⠉⠀⢸⡇⠀⠀⠁
⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠯⠀⠸⠇⠀⠿⠀⠾⠀⠸⠇⠀⠻⠀⠀⠸⠇⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⢀⡀⠀⣤⠀⣴⡆⢀⣄⡄⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢰⠆⠸⡟⠀⣿⠀⢹⠃⠀⢻⠁⢸⣿⡇⢀⣀⣀⣀⡀⠀⠀⣾⣿⣿⠀⢀⣼⣿⣿⣥⡀⠀⢸⣍⣉⣩⡇
⠀⢰⡄⢀⡇⠀⣿⠀⢸⠀⠀⢸⠀⢸⡟⠁⠀⠻⣿⡿⠁⠀⠻⣿⣿⣿⠀⢸⣿⣿⣿⣿⡷⠀⢸⣿⣿⣿⡇
⠀⢸⡇⢸⡇⠀⣿⠀⢸⠀⠀⢻⠀⢸⡇⠀⠀⠀⢻⠀⠀⠀⠀⠿⢿⣿⠀⠘⣿⣿⣿⣿⠃⠀⣸⣿⣿⣿⣇
⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉