Haughty Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠛⠛⠛⢧⣤⡀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢁⡀⢤⣤⡀⠀⠀⠉⠻⣿⠏⠁⠀⠀⠉⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠛⠁⠀⢻⠇⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠘⠉⠑⠿⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠾⢿⠿⠆⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⣤⣤⣾⡟⠃
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠖⢠⣦⣀⣿⠁
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠶⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⢰⣞⠉⠀⠀⡀⢙⣿⡇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⡆⠀⠀⠀⣏⣀⣀⣀⣀⡀⢀⣴⡄⠀⠀⠀⠀⢸⣿⠀⠸⣿⠿⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠙⣧⣀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠛⠛⠛⠁⠈⡻⠃⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⣀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣄⣀⠀⠀⠀⠀⠙⠤⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢻⣷⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢶⠶⠉⠋⠁⠘⣿⡆⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣴⡿⠁⠀⠈⠙⠛⠳⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣠⡶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠶⠶⣄⣀⣠⣠⣴⡶⠿⣷⣶⣦⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠀⣰⠆⠘⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠛⢷⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠈⠛⠛⠿⣷⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⡄⠀⢀⣀⠀⠀⠀⠀⣠⣤⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⠛⠛⠓⠀⠠⣤⣤⣤⣤⣤⣴⣤⣼⠃⠀⢸⡿⠛⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠃⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⡿⠿⠿⠶⠶⠒⠀⠐⠒⠒⠲⠖⠐⠺⠿⢿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠛⠓⠂⠀⣀⣤⣤⣄⠀⠀⠒⠊⠙⠿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⣠⣤⣀⣤⣤⣀⣀⣀⣀⣤⣶⣦⡀⠀⠀⠀⠐⠲⠶⠶⠾⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⠿⠟⢻⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣶⣶⣶⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢻⣵⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⢮⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⡖⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡬⠀⠉⢃⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠠⠤⠤⠀⠀⠀⠘⠄⠃⠀⠀⠘⠇⠻
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠒⠒⠒⠢⠭⠤⠤⠀⠀⠀⠀⠀⠀⠐⠀⢀⣤⣤⣶⣶⡶⠾⠿⠿⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠂⠚⠉⠉⠁
⠀⠀⠀⠀⠐⠒⠀⠒⠀⠒⠀⠐⠒⠒⠒⠒⠀⠀⠐⠒⠒⠐⠂⠒⠂⠐⠐⠒⠀⠐⠂⠀⠂⠒⠒⠒⠒⠐⠀⠒⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⢿⣷⣦⣄⠀⣴⡞⣛⣻⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣡⣶⣶⣤⡹⣿⡟⠛⠛⠉⠁⠖⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢻⡘⠿⢾⣿⡻⠁⡿⠀⠈⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⣡⡾⢧⡳⢤⣉⣁⡤⠞⠃⠀⠀⢀⣠⣤⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⡋⢀⣼⠏⢸⡆⠙⠛⠓⢀⣀⣤⠴⣖⣿⣿⣿⣿⡿⠟⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⢂⣴⠟⠹⣤⣼⣃⣠⠴⠞⣋⣥⣶⣿⣿⣿⣿⠟⠉⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢷⣾⣶⡿⠿⠞⢛⣫⣭⣷⠶⢿⣿⡿⠟⢋⣵⡿⠟⠁⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠐⠀⠉⢿⣦⣤⣶⠟⠋⢠⠤⠖⠛⣉⣠⣴⣿⠿⠛⠁⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⣀⡤⠊⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠄⠀⠳⡀⠀⠀⠀⠀⠈⢄⡀⣀⣠⡤⠴⠒⠋⠁⠀⣀⣴⡿⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠴⠀⠉⠙⠛⠛⠓⠛⠛⡉⠁⠀⠀⢀⣠⣤⣶⣿⣿⢏⡴⢋⣭⡷⠚⣉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠐⠀⠀⠈⠋⠉⠛⠲⢤⣀⣤⣴⠶⠛⠛⠋⠉⠙⠛⠛⠿⠶⣿⣫⣴⡯⠭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡀⠀⠀⠀⠀⠆⠀⠀⠀⠃⠈⠙⠳⢤⣄⣤⡤⢤⣀⠀⠀⠀⠀⠀⠙⣷⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠄⠀⢘⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠳⡄⠀⠀⠈⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠉⣉⣽⣶⣶⣴⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣼⣆⠀⠓⠦⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣥⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣭⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⣴⠒⡖⣦⡖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠁⢹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⢠⡿⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⢿⣄⠀⠀⠀⢸⡇⠀⠙⢧⡀⠀⠀⠀⠀⢀⣾⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠙⢷⣄⠀⢸⡇⠀⠀⠀⠙⢶⣄⡀⠀⣼⠃⠀⠘⢧⡀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠙⠳⣾⣧⠀⠀⠀⠀⠀⠈⠻⢾⡏⠀⠀⠀⠈⢳⣄⠀⢀⣴⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡾⠁⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠻⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⣿⡛⠶⢦⣄⡀⠀⠀⠀⠀⠈⠉⢹⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠈⠛⠳⠶⣮⣿⡆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣻
⠀⠀⠀⣰⡿⠿⠿⣶⣦⣾⠋⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢿⡉⠉⠉⠛
⠀⠀⠀⠸⣷⣄⠀⠀⠉⠛⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣉⣻⣦⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣴⡶⠆⠀⠀⠀⠀⠈⠛⢯⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣴⡿⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢀⣿⡇⠀⠈⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠛⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠟⠷⣦⣼
⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉
⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⠟⠀⠀⠀⠀⠀⠀⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠃⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣴⣶⣄⡀⠀⠀⣿
⢻⣧⣀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣸⡟⠀⠙⠿⣦⣼⡏
⠀⠙⠿⠿⡿⠿⠿⠟⠛⠉⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⢠⡿⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⢷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣄⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢿⣯⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡟⠛⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣾⡇⠀⠀⠀⠈⠉⠛⠷⣶⣤⣀⣀⣤⡾⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠷⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠇

⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⠋⠉⠷⢤
⠀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠘⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠗⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⣀⡀⠌⠉⣀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠍⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠤⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣹⠏⢳⠀⠀⠀⠀⠀⢸⡿⠙⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠱⠀⠀⠀⠀⠀⠲⢾⠗⠈⠜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠄⠀⠀⠀⠀⠰⢬⡼⠻⣄⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡡⣠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⢀⠀⠀⠀⠀⢀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡒⠂⠀⠈⠀⠀⠀⠀⢀⣀⠁⠀⠈
⠀⠀⠀⠐⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠞⠅⠀⠃⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠙⠛⠿⠿⠿⠿⠛⢧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣍
⠀⠀⠀⠀⢀⣀⣀⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠶⠶⣶⣶⡷
⠀⠀⠀⠀⠈⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⣿⠿⠿⠿⠿⠿⠇
⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠋⠁⠰⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⣿⡿⠟⠛⠉⠙⣿⡇
⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇
⣿⡇⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢀⣤⣄⣼⣿⣿⠇
⢿⣧⡀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠰⠿⢿⣏⠀⠀⠾⢿⣷⣶⣶⣦⣰⣿⡿⠀⠀⠀⠀⠀⠒⠀⣉⡁⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣴⣄
⠀⢿⣧⡀⠘⠃⠀⠀⠀⠀⠀⣴⠆⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡚⠛⠻⣷⣶⣦
⠀⠀⠛⢿⣦⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡆⠀⣿⡟⢹⣷⣀
⠀⠀⠀⠈⣿⣦⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⡇⠀⢹⠇⢸⣿⣿
⠀⠀⠀⠀⠈⠛⠿⠶⠶⢶⣶⡄⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠸⠀⢸⡏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⢿⠀⠀⠀⠸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⡾⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣤⣤⣄⣀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠈⠻⣦⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠐⢾⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠈⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠶⣾⣿⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠛⠛⢣⡄⠀⣿⠀⠀⠀⠀⠈⠛⠻⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠤⡀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⠀⠀⠀⠘⢣⡀⠛⢀⣤⡜⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠸⠟⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠈⠳⢀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣄⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⠀⢀⣀⣴⡿⠋⠀⠀⠀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠻⠛⠋⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠛⢿⣶⣦⣄⣀⣀⣀⣀⣴⣶⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠋
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡀⠀⠀⠀⢀⣸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠛⠿⠿⠿⣷⣦⣾⠿⠻⣿⣆⠀⢿⣿⠟⠁
⠀⠀⠀⠀⠘⢿⡿⠖⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣿⠇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⢸⡇
⠀⠀⠀⢸⣿⠁⠐⣶⣤⡀⢷⣤⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇
⠀⠀⠀⣾⣿⣆⠀⢺⣿⡁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠘⢻⣿⣷⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡟⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣀⠀⠠⠴⠶⠶⠶⠆⠙⠋⠉⠀⠀⠀⠀⠀⠀⠉⠁⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⢤⣤⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢓⣶⣶⣦⣤⣤⣤⣼⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠒⠷⢶⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⣀⣀⣼⡿⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⢤⣶⡿⠷⣶⣄⣠⣄⣤⣤⣤⣠⣤⣦⡀⠀⠀⣀⣀⣭⣿⠿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠉⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣄⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⢸⡟⢻⣄⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢳⣄⠀⠀⢸⡇⠀⠙⢦⣀⠀⠀⠀⣸⠏⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠷⣤⣸⡇⠀⠀⠀⠙⠳⣤⣰⡏⠀⠀⠙⣦⠀⠀⠀⣠⡶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠙⠗⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⣤⡾⠋⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⣠⣤⣀⡀⠀⠀⠉⠙⠛⠷⠶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠹⠷⣭⣝⣳⣦⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣴⣶⣶⣤⣄⣾⠏⠀⠀⢀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⢿⡟⠛⠻⠿
⠀⠀⠀⢿⣄⠀⠉⠛⠿⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⣄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⣤⡶⠆⠀⠀⠀⠀⠙⢿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢰⣟⠁⢠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣠⣾⠟⠀⠀⠀⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠈⢳⡄⠀⠀⠀⠀⢰⣦⣄⡀⣿
⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠀⠀⢸⡇⠀⠀⠀⠀⣸⡏⠉⠛⠿
⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣀⡾⠃⠀⠀⠀⠀⢸⠃
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣻⡛⠛⠉⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠉⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣴⣶⣄⠀⠀⣾
⢻⣧⣄⣀⣀⣀⣀⣤⣴⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⡏⠀⠙⢷⣶⡿
⠀⠙⠛⠻⠛⠛⠛⠉⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣄⣠⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢻⡿⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣼⡇⠀⠀⠉⠛⠳⢶⣤⣀⡀⢀⣴⠟⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢒⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣏⣉⣭⣉⣉⣉⣿⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠀⠈⠛⠛⠋⠉⣛⣛⣛⣁⡀⢤⣌⣻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⢀⢈⣥⠄⠀⠀⠀⠀⢀⡉⢉⣉⡉⠁⠀⠙⢿⣈⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢳⣮⠿⡏⠀⠂⢀⡀⠀⠙⠛⠛⠉⠿⠓⠀⠠⣬⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠳⣌⣷⣯⣤⣴⡾⠆⠛⠀⠠⣴⣶⣶⣶⣬⣛⠀⢤⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⣾⠟⠉⠉⠙⢿⣿⣾⣅⣽⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⠟⠁⠀⠈⢹⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⢿⣿⣧⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢷⡀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⡟⠀⠀⠀⠈⣣⡀⠐⠀⣿⣿⣿⣿⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⣠⡄⠀⠀⢰⡇⠀⠀⢰⡿⠿⣧⠘⠀⠁⢤⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠂⠀⠀⡀⠛⠿⠆⠀⠀⢃⠀⠀⡈⠀⠀⠸⣤⣄⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦⠀⣘⣁⣀⣀⣴⣿⡇⠈⡧⠾⣿⣶⣶⣤⣾⣿⡿⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠾⣿⠿⠿⠿⣿⠗⠀⠀⠡⠀⠁⠀⠈⢉⡽⠛⢳⣶⣿⣿⣏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣄⠀⠀⡴⠶⠂⠀⠀⣀⠀⠻⣦⣄⠸⠟⠿⢶⣄⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⡍⠀⠀⠄⣀⣰⣄⣠⣯⠀⠀⢻⣿⣷⡄⠀⠈⠡⡞⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣉⢻⣿⣿⠉⠀⠀⠀⢀⠟⣿⣿⣿⣿⣦⣤⣼⣿⣯⣴⣆⠀⠀⢹⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⠇⠈⠻⣿⣿⣿⡟⠋⠛⠋⠛⢿⣤⣤⡞⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄⠀⢰⠶⠞⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⡉⠉⣡⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠁⠹⣷⣼⣆⠸⣿⣷⡄⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣇⣰⡿⠻⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡦⠻⣿⣏⡀⠈⣿⣿⣱⣶⣤⣀⠈⠀⢀⣴⣿⠛⣻⣿⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⢀⠘⣿⡟⢶⣭⣭⣿⢿⣯⣿⣷⠘⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⣾⡟⢿⣷⣦⣼⣿⡿⣿⡿⢁⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⢿⣮⡉⠻⠻⠷⠶⢀⣠⡎⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣿⣷⣠⠀⠙⠿⣷⣶⣶⡾⠛⢃⡼⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠙⠛⠶⠀⠐⠛⡯⠿⣄⡀⢀⡀⠶⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⣾⠀⠀⠀⠀⣿⠁⠉⠁⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⢠⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⢄⣿⣤⣤⣄⣴⣄⣀⣴⠟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⣉⡈⠉⠉⠙⠛⠋⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠠⠄⠀⠀⠀⠀⢰⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠰⢦⡽⠋⠁⣦⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⣅⠀⣶⣿⠟⣰⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠻⣿⣿⣿⠿⠟⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣣⡐⠆⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠣⡘⠛⢼⢛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠠⡉⠱⣎⡹⣿⣿⣿⣿⡿⠀⠰⢦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣤⡁⢐⣉⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⠂⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣄⡀⠄⠉⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣘⠉⣦⣡⣄⡈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠙⢿⣶⣿⠃⠈⠻⣷⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⠻⣿⣿⣿⣇⠀⠀⠀⣀⡈⠃⠀⢀⣹⡿⠆⠈⣦⣄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⣈⠓⠤⣀⠿⣹⣷⣤⣦⡈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⡆⠀⣀⠈⠳⣄⠈⣿⠛⣷⣄⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣇⠀⠙⠷⠂⢀⠙⣿⣹⣯⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⡟⠓⠤⣦⣄⠘⠿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠿⢷⣀⣀⣴⣶⣶⣾⣿⣿⣿⣿⠀⠈⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠉⠉⠉⠀⠈⠛⠻⢿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀
⠀⠀⣠⣴⣾⣿⠿⣿⣭⣥⣍⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⣧
⢀⣼⣿⣿⣠⣶⢿⡿⣟⣋⠿⠿⡶⣦⣌⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠹⡄
⣼⣿⣿⣿⣿⠛⣾⣾⣿⣿⣿⣷⣮⣛⣷⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢃⣿⡇⠀⡃
⣿⣿⣿⣿⠇⠀⠈⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣞⣛⠢⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⠁⢈⡑⣭⢓⠛⡅
⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣯⣿⠚⢛⢉⣩⡓⣢⢤⡀⠀⠀⢠⠊⠀⠋⠀⠸⢿⡟⢻⡿⠿
⠀⠹⣿⣿⡄⠀⠀⢿⣿⣿⣿⣿⣯⣭⣿⢿⣿⡿⡯⢐⢴⣖⣶⠊⠀⠀⣠⡤⠬⠰⠋⠀⠀⠀⠀⠠⣿⣷⠈⢣⢰
⠀⠀⠈⢿⣷⡀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣹⣫⡞⣩⣴⣋⣶⣋⡄⠀⡘⠋⡰⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣶⣿⣿
⠀⠀⠀⠀⠹⣧⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⢂⣿⢋⣿⣿⣫⣾⣿⢣⠖⡃⠀⠀⠀⠀⢠⡖⣰⡏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠳⢄⠀⠘⡿⠿⢿⣿⣿⣿⣿⣬⡇⣨⣿⣿⣿⣻⣿⣧⡿⡀⠀⠀⠀⢠⣼⣷⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠛⠉⠻⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣡⠀⢠⢀⡨⣯⣼⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣴⠺⢿⣿⣿⣩⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⢚⢏⣯⣿⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠃⠐⡟⣤⣿⣿⣿⣿⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡛⣮⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⢯⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⢧⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⢩⠷⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠽⠋⠁⢳⢱⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠀⠀⠈⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠙⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⡃⢀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡛⠿⣦⣴⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣳⣦⡀⠙⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⣻⣿⣿⣿⠟⠋⠀⠀⠀⠐⠒⠲⣾⡿⢧⣍⣀⠉⠛⠿⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠦⢝⡳⣌⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣠⣿⣿⣿⣿⣶⣦⣙⣦⡀⠀⠀⠀⠀⠀⣿⣿⡿⣶⣍⢢⣄⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠀⠀⠀⠀⢀⣿⣿⡃⠪⣟⣿⣮⡳⣄⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣷⣬⣷⡜⠻⣾⡳⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣮⣻⣿⣿⣶⣙⣿⣿⣿⣿
⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⡛⠛⠛⠛⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠏
⣠⡿⠋⠀⢀⣀⣀⣀⣀⠀⠒⠒⠚⠿⠿⠿⢿⣿⡦⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⢀⣁⠈⠀⠈⣿⢃⣿⣿⣿⣿⠏
⢿⡿⠾⠿⠿⠛⠛⠛⠛⠿⠿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣾⣧⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣟⣫⣴⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣾⡟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣿⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣤⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠁⠈⠙⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣅⣺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠶⢦⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠢⣆⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠨⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣾⣮⣷⡜⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣌⣲⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡻⢿⣂⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣕⡢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡵⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠙⣉⡙⣏⢁⣿⠉⣹⡝⢉⣿⠟⣭⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠶⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⣀⣠⡤⠴⠶⠚⠿⠛⠋⠙⠉⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣭⣒⣠⣤⣤⣴⣶⠶⠶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠂⠀⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢐⣂⣥⡴⠶⠛⠋⠉⠉⡏⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⣩⡴⠾⠿⠭⣄⡀⠀⠀⠀⠀⠀⠁⡿⠛⠉
⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⢀⣠⠔⣫⡵⠚⠉⠀⠐⠲⠿⠟⠢⠁⠀⢀⡬⢤
⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⢖⣩⣶⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⢀⣀⣠⠴⣊⣴⣿⣡⣶⣿⠻⣌⣉
⠀⠀⠀⠀⠈⠛⢿⣶⣦⣭⣭⣭⣭⣶⣾⣿⣿⡿⠛⣿⣿⣦⣈⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉⠁⢸⣧⣀⠘⢿⣌⢉⡙⠉⢠⠀⠀⠀⠀⠀⠀⠙⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⠈⠛⢷⣽⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⠷⠒⠀⠉⢛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⢶⠶⠀⠀⠹⣮⡝⣦⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⠀⠀⠀⢻⢻⣾⡗⣦⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠛⢈⡿⠁⠘⣿⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡏⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣥⠆⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣻⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣭⣽⣿⣽⣯⣿⣿⣽⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⣀⣴⡦⢴⣯⡉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⢴⣿⣿⢗⣺⣿⣿⡄⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠻⣿⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⣿⡟⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⡿⠁⡄⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡄⠀⣾⣿⣿⣿⠀⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⣴⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣦⣄⡀⠀⠁⠀⠀⣁⣴⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠀⢹⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠰⣆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⢶⣷⡦
⣰⠄⠀⠀⠀⠠⣿⣆⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠙⢇⠘⠁⠀⠀⠀⠀⠀⠀⠀⣤⣥⠀⠀⠀⣤
⡈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠁⠀⠀⠠⣤⠈⠾⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠟⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸⣼⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠈⡛⠋⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣊⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⠀⠘⠁
⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡮⠀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⡏⠀⠀⠀⠠⡠⠀⠄⠐
⠈⠇⠀⠀⠀⠀⠀⠈⢰⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠐⠆⠘⠁⠀⠀⠀⣸⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⣠⡘⡀⠈⠒⠒⡈⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠃⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢀⢠⠶⣦⠴⣦⣶⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠈⠀⠐⣄⠠⢀⣬⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢢⣤⣤⣶⡖⣺⣟⣷⣿⣤⡿⠀⡶⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠛⠛⠛⠛⠛⠋⠏⠽⡮⢶⢀⣾⣧⣷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠉⢹⠀⠶⠦⠠⣤⣤⣥⣤⣴⣤⡤⠖⠒⠊⠉⠺⠷⢦⡾⢡⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠀⠀⣀⢠⣤⣤⣤⣤⡀⠀⠀⠀⠀⣴⣤⣄⣠⣦⡀⠓⣀⡼⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠳⣌⠛⠻⠿⡿⠿⠟⣷⠶⡶⣤⡛⠿⠿⢿⠿⠟⠉⣿⡿⠁⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠁⠰⠶⠶⠷⠶⡄⠡⡄⠀⣼⣷⣄⡠⠶⠤⠴⢶⠛⠀⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢘⡃⣻⣇⡀⠀⠀⠀⣀⡀⢀⠱⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡧⢠⡀⠀⠀⠀⠀⠀⠈⠠⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⢹⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠢⢤⡀⠘⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠀⠀⠽⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂⣄⢀⣠⡄⠀⠀⠀⠀⣀⠀⢠⡜⢦⡜⠋⠉⠀⠂⠀⠀⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⢀⠄⠀⠶⠀⠀⠀⠀⣴⠖⠀⠉⠉⠊⠷⣤⣭⣉⣥⡶⡉⠋⠀⠁⡀⠈⠀⠀⢳⡀⠰⣷⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠳⢈⡹⠀⠐⠀⠀⠀⠀⣸⡍⠀⠀⠀⠀⠀⠈⠀⢉⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⡘⢧⠀⠙⣿⠘⠳⣄⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢨⠄⠀⡄⠀⠀⠀⢸⡇⠀⠀⠀⣠⠶⣻⠋⡛⣻⡟⢻⣥⣤⣭⣤⣹⣏⣁⡤⢈⠑⢮⡄⠀⢿⡉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠠⡆⠀⠀⡇⠀⠀⠀⣿⠀⠀⡠⢞⡈⣴⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠳⣮⡱⢄⠻⡀⢸⡟⠲⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⡆⡈⠠⡇⠀⠀⡃⠀⠀⢐⠃⠀⢶⡡⣼⡟⠉⠀⣠⣀⣀⣠⡴⣤⢴⡶⠶⠶⡰⢦⡴⣤⣻⣎⡰⠃⠠⣟⠀⠤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠧⣀⠂⠀⠀⠀⣇⠀⠀⠘⠀⡞⢢⡜⢁⣴⠞⠟⠁⢉⠍⡁⢐⠤⡀⢀⠀⠂⣀⠦⢔⣛⡧⣼⠡⠆⢈⣛⠓⠦⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠚⠂⠀⠀⢹⠀⠀⠀⠘⠃⣼⢛⠋⢁⠼⢃⠔⠣⠐⠔⠢⠀⠐⠈⠃⠐⠤⣐⢻⡍⠟⠁⢀⠠⢈⣉⡑⠲⠤⢨⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠒⠀⠙⡇⠀⠀⠈⣧⠀⠀⠣⣀⠉⠈⠛⠣⠶⣥⣤⣠⣄⡄⢠⣀⡄⣠⣴⣼⡶⠛⠋⠁⠀⠐⢀⣶⠈⡯⠄⠐⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣏⠑⠀⠀⡇⠀⠀⠀⢨⣇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠊⠉⠉⠀⠀⠀⠠⠄⠁⠀⢀⠀⢀⠓⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⣯⠙⠠⢈⡇⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠙⢶⣤⡂⢀⠀⠀⠀⠀⢀⡀⣀⣠⣤⡆⠀⠀⠀⠀⡈⠀⢠⡬⡁⠐⠠⣄⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣧⡈⠢⣈⠓⠀⠀⠀⠀⠂⠀⠓⣤⠀⠀⠀⠀⠀⠙⠉⠓⠲⠒⠚⠒⠋⠉⠁⠀⠀⠀⠀⠀⠐⠋⣐⠳⢤⡈⠓⠒⢄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠧⡈⠒⢀⢹⡆⠃⠀⠀⠀⠐⢉⡐⢯⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣉⡑⠦⣄⣉⠒⠦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡶⠈⠁⠠⠈⢳⠀⠀⠀⠀⠈⠰⢟⣯⠿⣿⡿⣷⠠⣶⡆⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣿⣟⣻⣶⣾⣿⠿⢿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⡗⠀⢈⠀⠀⠈⣷⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠄⠄⢀⠀⠄⢀⣦⢛⣿⣻⡷⣾⣝⣻⣶⣿⣙⣿⢷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⢀⡀⠀⠿⣷⠀⠀⠀⠀⢉⡤⠶⣾⡿⠿⠷⣶⡾⢿⣿⣟⠾⡷⠂⠞⠋⣷⣿⣯⣿⠿⠾⢛⣿⡿⠿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠄⣁⠀⠤⠀⣘⣧⢿⣀⢠⣼⣿⣤⡃⠀⣠⡀⣀⢸⣃⢤⣤⣤⣄⠀⠀⣀⢄⣠⡄⢀⣀⣀⡠⠤⣽⣛⣻⡷⣿⣛⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡂⠠⠄⠐⠀⠄⢠⡄⠉⠁⠈⠣⡝⢻⣾⡿⢿⣿⣿⣿⠋⣩⡿⠻⢷⣾⠟⠛⣿⠛⠛⠿⠉⡛⠻⠿⠍⠛⠛⠫⠭⡿
⠀⠀⠀⠀⠀⠀⠀⠸⠅⠐⠂⠠⠀⠀⠈⣥⠀⠀⠀⠀⠀⣰⠟⢱⠾⠛⢋⡁⣿⡯⠁⠀⠀⠤⠀⢠⠋⠤⠀⠒⠀⠤⠀⠀⠂⠀⠄⢀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⠀⠀⠀⣤⠁⠀⠀⢠⣴⠿⠟⠛⣦⠀⠀⠀⠀⢠⡟⠒⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠂⠁⠀⠐⠘⣷⠀⠀⢸⠁⡤⠒⠈⠚⠉⠀⣠⣼⣿⡀⠀⠀⠀⡜⠀⠀⠀⠒⠀⠀⠁⠐⠀⠤⠀⠀⠂⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⢈⡀⠒⢈⡁⢺⡇⠀⢸⠀⠔⠁⠀⢠⢶⡿⠟⠉⠹⣏⠀⠀⣤⡉⠐⠢⠈⠉⠒⠂⢀⣉⠐⠲⢤⣈⠑⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡀⠂⠉⠀⠀⠈⣷⡀⠈⣁⢄⠒⢈⠀⠁⠀⠀⣀⣶⡋⠀⢀⠒⠀⡁⠀⠀⢈⣉⠀⠤⠄⠉⠓⠀⣈⡁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠄⠀⠀⠀⣉⠀⢺⣦⠀⢟⡠⠀⠀⠀⣤⣶⡿⠛⣿⡈⠀⣜⠃⠠⠄⠘⠓⠀⣉⠁⠠⠄⠈⠑⠂⢀⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈⡁⠠⢀⠀⠀⢈⡀⢿⡀⢠⡇⢀⠀⠀⠌⠁⢀⣀⠷⠀⣴⠁⠀⠀⢀⡀⠒⠀⠀⡁⠀⠀⢀⡁⠀⠄⣨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠄⣀⠀⠀⡁⠠⠄⡸⢏⠶⣿⡄⠀⠀⠀⣀⣴⣿⠇⣤⠃⠄⣉⠀⠠⠀⣁⠀⠠⢀⡀⠀⠄⢈⡀⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠤⠘⠀⠄⣙⠢⠄⣻⡃⢼⠀⠀⠠⢾⠿⢳⣟⠰⠏⡁⠀⠤⣀⠐⠠⢄⡉⠂⠤⣀⠐⠲⢌⡉⠢⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠒⠀⠀⠀⠀⠐⠂⣬⡿⠀⠖⠀⠄⣀⣤⣼⣍⡿⠂⠀⠐⠀⢄⡀⠒⠤⠈⠑⠂⣄⠉⠢⢀⠉⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠚⠋⠘⠛⠛⠛⠛⠀⠀⠉⠉⠿⠿⠵⠗⠂⠈⠓⠂⠀⠐⠒⠀⠘⠓⠀⠘⠲⠠⠾⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣦⣄⡀⠈⠛⠃⠀⠀⠀⠀⠛⠋⠀⣀⣤⣶⣾⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣍⡙⠛⠃⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠛⠛⣉⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢻⣿⡄⠀⢠⣿⡟⠋⠉⠉⠛⣿⣧⠀⠀⣼⣿⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣾⣿⠁⠀⢾⣿⠀⠀⠀⠀⠀⢸⣿⠆⠀⢹⣿⡆⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⢸⣿⡇⠀⠀⠘⣿⣧⣀⣀⣀⣤⣿⡟⠀⠀⠀⣿⣿⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠘⢿⣷⣦⣀⠀⠈⠛⠿⠿⠿⠟⠋⠀⢀⣠⣶⣿⠟⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠉⠻⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠋⠁⢀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣠⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣿⣿⣤⣤⣤⣿⣧⣤⣤⣼⣿⣧⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠐⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⡀⡈⣉⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⡘⢓⠀⣀⠀⠀⠀⡼⡷⣿⣿⣿⢣⣟⣈⠀⠀⠀⠀⠂⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣘⡻⠋⢀⠁⢹⠏⠀⠀⠀⠀⠀⠁⠁⠀⠈⠉⠀⠀⠀⠠⡀⡈⢠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢤⠿⠉⣀⡀⠶⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡈⡎⣾⡉⠛⠷⠀⢀⣠⢴⣗⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⢴⣄⣷⠈⡖⠀⣠⣀⡀⠄⠀⠈⠹⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠄⠰⢿⠿⠇
⠀⠀⠀⠀⢀⠐⢻⡐⣷⣞⣼⡃⢹⣤⣅⢛⣫⡤⣷⡐⠟⠂⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣯⠱⢷⡌⠀⣀⠀⣧⠀⣤
⠀⠀⠀⠀⢯⣇⣵⣜⠢⣹⠿⠛⢻⣧⣀⣾⠿⣇⡿⠓⢠⡄⠰⠅⡅⠁⡤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⣀⡆⠉⣬⡁⠁
⠀⠀⠀⠀⢠⡄⡤⢸⢷⣶⣶⡶⢿⠟⣟⠵⡆⢈⣷⡀⢌⠀⠈⠦⢈⡀⣰⡆⠀⠀⠀⠀⠠⠀⠀⠘⠆⠀⠀⡏⠐⠒⠈⠸⠂
⠀⠀⠀⠀⠀⠀⠐⠞⠳⣍⠀⠰⡎⠀⣀⣀⡴⠋⠀⠁⠎⠀⠀⠘⠛⠃⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠘⠾⠇⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡳⣾⡎⣿⡿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣻⣿⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣘⣛⡻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣻⠿⠿⠿⣛⣿⣿⢸⣿⡿⣫⡶⢞⣛⡷⢾⣝⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⣼⢏⣾⣿⣿⣿⣷⡹⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⣿⣿⣿⣿⢇⣿⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣵⣶⣍⣿⣿⣿⣿⣿⢸⣿⡝⣷⣝⠿⠿⢿⣫⡾⣣⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣛⣛⡻⣿⣿⣿⣿⡟⣬⣭⣭⣭⣭⡝⣿⣛⣭⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⢿⣿⠇⣿⢱⢖⣒⢶⣶⡶⠶⠶⣿⣶⣶⣶⣶⣶⡝⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⣿⢸⣮⣭⣾⣫⢞⣭⣟⢮⣻⣿⣿⢸⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣧⢿⢻⣿⣿⣻⢧⣿⣿⢸⣿⡇⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⢸⣿⣿⣿⣮⣓⣲⣞⣫⣾⣿⣿⣸⣿⡇⣿⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠛⠛⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⠠⠤⠄⣀⡠⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠁⠀⠀⠀⠀⠀⠉⠀⠂⠀⠈⠀⠀⠀⠈⠈⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠁⠁⢈⢀⠀⣁⡉⠈⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⢠⠄⠈⠀⠁⠀⠉⠁⠀⠀⠊⠁⠂⠐⠋⠂⠀⠐⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠈⡑⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⠚⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢯⡀⡀⠈⡁⠀⡹⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠷⢺⠀⢻⠟⠀⢠⣧⣿⡞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⡀⣸⢹⡆⡟⢀⣾⡀⠀⣇⡾⣿⠑⡂⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠞⠙⡜⣧⡇⣼⣿⡇⣸⡾⠁⣹⠲⡱⢡⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⢹⣷⣿⣿⢃⡿⡡⣀⣀⠀⡛⢮⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠿⠛⠁⣼⡁⠚⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃


⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠸⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠓⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⢿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⣿⢸⣿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡟⣿⢸⡙⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⠁⡟⢸⡇⢻⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠲⠄⠀⠀⠀⠀⠀⠀⠜⠁⠀⡰⠁⢰⡇⠀⡇⠀⢳⡀⠀⠈⠛⠿⣿⣿⣿⢟⠥⠒⠁⠀⠀⢉⢻⣿⣿⣿⣿⣿
⠀⠀⣀⠠⠤⠤⢀⡈⠢⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢡⠀⠀⠑⠀⠀⠀⠀⠀⣈⡵⠃⠀⠀⠀⣀⣤⠾⢠⣿⣿⣿⣿⣿
⣴⠟⠁⠀⠀⠀⠀⠈⢰⡞⠈⠉⠀⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠂⠤⡀⠀⠀⠴⠋⠥⡐⢾⣿⣿⣿⣿⣿⣿
⠉⠋⠒⠒⠀⠄⠀⠀⠘⠧⠀⠀⢀⠐⠖⠶⢬⠑⡄⠔⠒⠢⠀⠀⢠⡤⠀⠀⠠⠀⢀⠞⠉⠁⠀⠭⠂⡣⠤⢤⠀⠀⠀⠀⠘⠟⡟⠉⠉⠉⠛
⠀⠀⠀⠀⢀⠠⠤⠀⠀⣤⠀⠤⡤⠇⠀⠀⣠⠋⢰⠀⡀⠠⠑⢄⣀⢠⡀⡆⠀⠈⣱⢆⡄⠑⡄⠀⠀⠈⠀⠀⢀⠖⠀⠀⠀⠀⠄
⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⠣⠤⡀⠀⢊⠀⠑⠐⠚⠀⠈⠄⠙⢄⠉⠉⣠⠜⠀⠘⢧⡌⠴⠚⠀⠀⠀⠑⠀⠃⠀⠠⠂⠀⠀⠀⢾⣦⣤⣤⣴
⠀⠀⠀⠱⣄⠀⠀⠀⣀⠀⠀⠀⠈⠂⠤⡀⠀⠉⠀⠀⠐⠁⠀⠀⠁⠈⣤⠀⠀⢢⡄⠀⠀⠀⠀⡄⠀⠐⠀⡆⠀⠀⠑⠀⢀⣰⠿⣭⢿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠁⠙⠶⠤⠤⢤⣤⣄⣈⠂⠤⠀⠐⢦⡀⠀⢀⣀⠴⠋⠀⠀⠀⠀⠑⢄⠀⠀⢈⡵⠀⠀⢈⣉⠁⢉⣉⣭⡤⠾⢏⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠄⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⡃⠀⠀⠀⢉⣉⠉⡤⠤⢐⣥⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⠉⠁⢽⣿⣿⣧⡐⠦⠴⢚⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡱⢒⠤⣿⠶⢄⣨⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣰⢁⣀⠀⠀⠠⠔⠀⢣⠸⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⡏⠀⠀⠀⡀⠴⠀⠀⢸⣄⡇⠈⢏⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢳⡀⠀⠀⠀⠀⠀⢀⡼⠽⠉⣲⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣉⣓⠚⠒⠢⡀⠀⢰⢁⡠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠛⠻⢽⠁⠀⠈⡧⠒⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢳⠢⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠫⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⢠⠊⠑⢄⠀⠀⠀⠀⠀⠀⣰⡅⠈⣆⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡪⣶⡇⠀⠀⠀⠀⠀⢿⣫⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠯⠭⠤⠤⠤⠤⠤⠤⠤⠜⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣛⣟⢻⣛⣿⣷⣿⣻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⡿⣻⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⢡⣟⢻⢿⠟⡇⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣸⣷⣾⣾⣼⣧⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⡿⡟⣿⢛⢟⡿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣷⣵⣿⣾⣿⣧⣧⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⣤⠀⣴⡶⠶⠀⣴⣶⠦⠀⣶⡆⣦⠀⣠⣤⣄⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠙⠀⠀⣀⣸⣿⠀⣿⣿⣥⠀⢼⣿⡷⠀⢿⣿⣿⠀⣙⣿⣯⠀⠀⠀⠃⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠁⠀⣀⡀⠀⠀⢀⠀⠐⢀⢀⣀⡀⠀⠈⠉⢍⡀
⠀⠀⠀⠀⠀⡀⠄⢠⠄⠀⠀⠀⠀⠸⢯⣍⠀⣰⣿⣧⠀⣿⢸⡏⢙⣷⠀⠀⠀⠀⠀⠀⡄⠠⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠶⠘⠛⠛⠐⠋⠀⠘⠃⠛⠘⠛⠋⠁⠀⠶⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠄⢾⣋⡁⣿⣿⠀⣿⢻⡇⣿⣏⠀⠠⠀⠀⠀⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⡤⠀⠉⠀⠈⠁⢀⣡⡌⠀⠈⠉⠠⡀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⢀⡀⠀⢈⡁⣀⢈⣀⡀⠀⠀⠁⠀⠀⠢
⠀⠀⠀⠀⡀⠀⠀⢠⣴⡶⣶⣾⡎⢿⡉⣿⣻⡆⢸⡟⣿⣿⡯⠁⠀⣾⣧⢸⣷⣶⣴⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠙⠖⠀⠘⢿⠟⠻⠛⠃⢘⡁⠉⣩⠀⢨⣅⡍⡍⠉⢡⡈⠋⠛⠛⠻⠟⠿⠿⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡴⣼⣿⡇⣇⣀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⡤⠀⡄⣄⢠⣄⡤⠤⠀⠠⣌⣩⢀⣭⣥⡀⡄⢠⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⡇⠹⡟⠸⣯⣍⠀⠀⢹⠅⠘⢧⡴⠇⢷⡼⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠄⡀⠀⢠⣶⣤⠀⣶⠶⠄⣤⣶⡄⠰⣶⡶⠀⠀⣀⠀⠀⢰⡀⡀⠀⠀⡀
⠀⠀⠀⠀⢀⡀⠸⠀⡷⠀⣤⡇⠀⠸⠿⠿⠀⠿⠥⠄⠼⠿⠆⠀⠿⠇⠀⠀⢸⣠⠀⢾⠁⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢸⣾⣴⣶⣲⡾⣦⣦⣶⡂⢸⢹⠁⣘⢫⣯⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠞⡀⢠⡖⠀⠀⡀⠈⠁⠀⡼⠂⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠉⠠⠐⠄⠋⠒⠤⢽⠠⠐⡂⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⡰⠒⠀⢀⡀⠀⣴⡶⠄⠠⠃⠆⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠒⠀⠀⠀⠩⠝⡁⠐⠂⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠿⠗⠀⠠⠀⠀⠀⠐⠀⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⡸⠠⢤⣠⡀
⠀⠀⠀⠀⠀⠀⠩⠗⠜⠋⡌⠡⠠⠶⠂
⠀⠀⠄⣠⡤⠈⠁⢴⠘⠐⠂⡰⠖⠂
⠀⠀⠹⠙⠀⠀⠀⠐⡄⣠⡤
⠀⠀⠀⠰⡄⠀⠀⠀⢹⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⢴
⠀⠀⠀⠀⠑⣤⡄⠀⠌⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠢⡄⠿
⠀⠀⠀⠀⢒⡖⠰⣗⣯⠘⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠛⠃⠁⢀
⠀⠀⠀⠀⠀⡋⠽⡢⢇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠁⣶⠊⠀⣠⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣤⠤⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠼⢃⣄⣀⣤⣤⣀⣠⠶⢾⠀⢀⠃⠀⠀⡀⠐⠀⠶⢦⠄⠀⡀⢀⠔⠋⠲⢰
⠀⠀⠀⠀⠀⢀⡞⠁⣮⠙⠁⣯⡠⢬⢇⢉⠃⠘⡖⢤⣠⠁⠀⠀⠀⢀⠀⠄⠀⠀⠈⠋⠂⠀⠌
⠀⠀⠀⠀⠀⢸⡃⠀⠈⠀⢄⡼⠰⠂⣰⣯⡐⠀⠛⠀⡆⢿⣠⠀⠀⠀⠲⠲⢢⡅⠀⢶⡦⠊
⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠠⡾⠀⠠⣀⣂⣰⡤⠤⠠⢧⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠈⠀⠉⠒⠀⠈⢆
⠀⠀⠀⠀⢀⡤⠤⠤⢀⣀⣄⡈⠁⠀⡁⠸⡀⡴⣊⠷⠄⠈⠷
⠀⠀⠀⠀⠀⡆⢠⡤⣤⡀⠀⠈⢣⠀⠀⣠⠤⣀⣠⠄⠤⣼
⠀⠀⠀⠀⠀⠘⠲⠴⢾⣍⣤⣤⢰⡆⠀⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠤⡄⢀⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠰⠶⠀⣧⠀⡍⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠄⠘⢆⠳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡜⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢫⡁⢀⡒⢈⡕⢰⠁⠀⠶⠖⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠘⢦⡉⣡⣴⠯⠃⢸⠀⠀⠀⠢⡀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠦⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡄⠀⠀⠀⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠰⠄⢱⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⡐⠂⡁⠄⡀⠄⡠⠀⡀⠀⠀⠘⡄⢀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠢⠀⢣⠀⠀⢀⠔⠋⡀⠆⠚⣀⣀⠀⠠⠀⡡⠐⡠⠐⡠⠐⣀⠀⢀⠀⠀⢣⡎⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⠳⠞⢁⠰⠆⢁⠄⠊⠁⢸⠀⠀⠉⢐⡡⠒⡡⠔⣉⠄⠀⢁⢄⠀⠈⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⢐⣠⠴⠊⠀⠀⠀⠀⠈⠄⠀⠀⡠⠔⣉⠔⢊⡡⠒⠀⡟⡀⠄⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⢈⠴⠊⡠⠐⡩⠔⠀⠠⠀⠱⡀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡀⠔⡉⠔⡈⠄⢊⡀⢠⡀⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠠⠔⡀⠐⡈⠤⠊⡡⠄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠔⢉⠄⢊⠤⠒⢉⡠⠒⡉⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⡀⠈⠠⠊⣀⡤⠂⣡⠔⢁⡄⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡇⠰⠚⣭⠖⢋⡴⢞⣭⠖⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⣠⢄⣁⡐⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠉⠐⡏⢙⠋⠚⠁⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠷⠀⠻⠞⡿⠚⣫⠖⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣉⠔⢉⠔⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠔⡊⠄⢊⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡨⠔⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠂⠁⢒⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢤⡂⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡢⠄⣒⠢⠐⣂⠠⠄⢒⠈⠁⣐⠢⠄⢂⡀⠄⢠⡇⠀⠀⠘⣦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠉⠐⠂⠁⠄⠒⡨⠔⣂⠭⣒⠿⣄⠀⠀⣿⠪⢅⡲⢍⡲⢝⡢⣔⠮⢐⡠⢐⠤⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⣴⠏⠈⠁⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⡿⠿⠛⠋⢉⣁⣹⣥⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡿⠏⣉⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠋⠉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣸⣿⣿⡿⢋⣰⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⢰⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⡶⠀⠀⠀⠹⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠀⠀⠀⠀⠀⠹⣧
⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⢦⡀⠀⢻⣦⣤⣦⣤⡀
⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⢶⣶⠶⠾⡇⠀⠀⢿⣇⠠⠈⣿
⠀⠀⠙⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠈⠉⠀⣰⠆⠀⠀⢸⣿⠀⢀⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢸⣇⠀⣀⣤⣤⣴⣾⠛⠋⠀⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⢻⣿⠛⠃
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⡟⠉⠁⠀⠀⠙⠋⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⠀⠀⠀⠀⠙⠻⣦⣤⣤⣤⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣠⡶⠿⠿⠿⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⡀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⣼⡟⡀⠒⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣀⢀⡀⠀⢀⣀⣼⠟⠀⠀⣸⠟⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠻⣷⠁⠦⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⠁⠀⢀⡼⠃⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠟⠋⠈⢿⣄⠀⠀⠀⠀⠀⠀⢸⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⣤⣄⠀⠀⠀⣤⣤⠜⠛⠁⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠛⠛⠛⠻⣶⣄⣀⠀⠀⠀⠀⠀⣀⣠⣶⡟⠋⠉⠛⠛⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠋⠁⢀⣠⣤⡶⣾⡆⠈⡙⢿⣿⠿⣿⠿⠛⠛⢻⣿⢿⠀⢿⣶⣶⣤⣄⡈⠙⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⢀⣤⡾⠿⠛⠉⠉⠀⢸⡧⠀⠁⣄⠙⢷⣄⠰⠒⠓⣼⠏⠀⠆⠀⣿⠀⠉⠉⠻⠿⣶⣤⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⢻⣧⠀⠀⠀⠀⠀⣸⡿⠀⠀⠈⠑⡄⠙⣧⡀⣰⡟⢀⡀⠀⠀⢘⣷⠀⠀⠀⠀⣿⠋⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⣹⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠒⢀⡈⠿⠋⡀⠞⠁⠀⠀⠈⢿⡆⠀⠀⣾⡏⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⡾⠿⢧⣤⣤⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣽⠿⢷⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⡿⠿⠿⠿⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⣾⠇⠸⠿⣿⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⠀⠀⠀⠀⢀⣸⣿⡿⠛⠒⠶⠶⠲⠶⠶⠟⠋⠉⠛⠶⠶⠶⠶⠖⢼⣿⠀⠀⠀⠀⠈⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠛⠛⠋⠙⢷⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠛⠛⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣷⠀⢻⠏⠸⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣼⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢻⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢾⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢸⣿⣤⣤⣼⣤⣤⣤⣿⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡾⠿⠿⠟⠿⣿⣿⣿⡏⠉⠉⣽⠉⠉⠉⢻⣿⣿⣿⠟⠛⠛⠻⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠙⣄⠀⠀⠀⠀⠙⠉⠛⠁⠀⠀⣿⠀⠀⠀⠈⠋⠙⠁⠀⠀⠀⠀⡰⠋⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠟⠛⠿⠿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⡀⠀⣀⣀⣀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠶⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⡇⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣾⣿⣾⣼⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⣿⣿⣽⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⢹⣾⣿⣿⣾⣿⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣿⣿⡿⣷⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣿⣿⣿⣿⣿⣾⣇⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣟⣿⣽⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⢄⠀⡇⠈⠢⣀⠀⡼⠻⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠳⠀⠀⠈⠙⠃⠀⠈⢦⡴⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢶⣲⣄⠀⠉⠉⢟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⠶⣾⠁⠀⣄⠈⠉⠁⠀⠀⠈⠳⢄⡀⠀⠀⠂⠀⠀⠀⢹⡉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠀⠀⠀⠈⠓⠦⠴⠆⠀⠀⠙⢏⡁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢤⣀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠈⡆⠀⠀⢰⠶⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣈⣡⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠐⠂⠀⠀⠀⠀⢸⠀⠀⣠⣄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠶⠚⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⢰⠇⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢻⠦⢤⣀⠀⠀⠀⠀⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢺⠀⠀⠈⠙⠓⠶⠞⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢤⡀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⡤⣤⢤⡀⠘⣆⠀⠀⡠⠊⡁⠀⠀⠀⢀⡉⠳⣄⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⢆⠀⣰⠃⠀⡿⠀⣸⠁⢸⡁⠀⠀⠀⠀⠙⠳⡜⣆⠀⠀⢰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢰⢒⡒⠓⠒⠒⠚⣇⣀⠀⠉⠁⣀⡴⠁⢠⡇⢀⡞⠙⠳⢦⣀⣀⠀⠀⠙⣿⠄⠀⢸⠀⠀⠀⠀⠠⡏⢓⡦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣈⣋⣀⣉⣉⣀⣀⣈⣉⣟⠉⠁⠀⠀⠘⣇⣸⠁⠶⠆⠀⠀⠹⠿⠓⣆⣻⠇⠀⢸⡄⠀⠀⠀⠀⠛⠁⠀⠀⣦⡀⢠⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣀⠀⠐⠂⠀⠀⠐⠂⠀⣀⡟⠀⠀⠀⠓⠦⠤⠤⠤⠤⢴⠛⠛⠁⣉⠛⠙⢻⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⣐⣂⠀⠀⣸⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠞⠙⢲⠆⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠸⣗⣦⣍⣉⣤⠞⠁⠃⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠓⠒⠛⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⠀⠀⢸⢦⡀⠀⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⡤⠤⠤⠤⠤⠤⠤⠤⠤⣿⠀⠀⠀⣀⡤⠶⢺⠏⢀⣇⣀⣀⣸⠀⠹⡖⠶⠿⣇⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠖⠒⠲⠀⣿⠀⠀⢰⠁⠀⠀⢹⣿⢾⠐⠀⠀⢸⣷⣿⡏⠀⠀⠈⢇⠀⠀⠀⡇⢰⣒⣒⣒⠒⠂⢨⠗
⠀⠀⠀⠀⠀⢸⣀⣉⣉⣉⣁⣀⣀⣀⣀⡿⠀⠀⡇⠀⠀⣄⣼⣽⢿⣄⠤⣖⣾⠏⢿⣧⢠⠄⠀⢸⡄⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠇⠀⠀⢸⡇⣿⠈⠁⢺⠭⣿⠂⠸⣿⣿⠀⠀⠀⡇⠀⠀⠀⢀⡞⢛⣛⣛⣛⣛⣛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⢹⣀⠀⠀⢸⠀⠀⠀⢸⣧⠇⠀⠀⣀⡿⢿⣄⠀⢿⣿⠀⠀⠀⣧⠀⠀⠀⠈⢧⣬⣭⣭⣭⣭⣭⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠋⠹⡇⠀⠈⣿⠂⠀⠸⣶⠲⠤⣼⠏⠀⢰⡞⢩⠀⡡⢻⡄⠈⢿⣀⣀⣠⣽⠀⢀⡖⠒⠒⠒⠒⠒⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡾⡟⠛⠢⠿⠀⠀⠀⢸⠀⠐⠃⠀⢠⡟⠀⠀⠈⢠⠈⣧⠀⠀⠉⠁⠘⠋⠀⢸⣇⣉⣉⣉⣉⣉⣋⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⣾⢁⣀⠶⠘⠤⢒⡸⣇⠀⠀⠀⣉⠀⠀⠀⠈⠈⠈⠉⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠿⣷⣬⣥⣭⣭⣭⣥⣶⣿⣷⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣌⠝⣿⢿⡿⢾⣿⣿⡛⠋⡉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠁⣾⣿⣿⠇⡂⣿⣘⣷⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠃⡀⠌⣩⡛⠰⠃⡛⠛⡏⠀⣶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠢⠉⠀⠁⠈⠀⠀⠁⠁⠈⠈⠨⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣀⠀⠀⠀⢀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣡⣤⠶⠿⠿⠿⣿⣶⣄⣀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠋⠉⠉
⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢲⡶⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠎⠀⠘⠉⠀⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠃⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣷
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣿⣿⣇
⠀⠀⠀⠀⢀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡇⣿⣿⣿⡀
⠀⠀⠀⠀⠘⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣽⣷⠾⠿⠛⠛⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠱⣤⣤⣶⢶⣿⣿⣿⣽⠿⠿⠛⠛⠉⠉
⣀⣠⣴⣶⣤⣼⣟⣀⣴⣾⣷⣿⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣏⣿⣿⣧⡇⣤⣰⣾⣿⣶⣤⣤⣀⠀⣀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⡷⢿⠳⢻⣿⣿⣯⣭⣽⣷⣕⢬⠑⢤⡀⢀⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣷⠶⣷⡊⠈⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣰⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣼⣆⣀⣼⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣼⠏⢠⣿⡇⢀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠶⠤⡘⣿⣿⣿⣽⡷⢊⣯⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡏⠀⠈⣿⡟⠉⢽⣿⡗⠀⠀⣾⡿⣿⣿⣿⡿⠋⢹⣿⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⠀⠀⠸⡇⠺⠿⠟⠁⠀⢸⠏⠀⠀⠉⠉⠀⠀⣠⣿⣿⣿⡿⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠒⠇⠀⠀⠀⠀⢀⣨⣧⡀⠀⠀⠀⠀⣺⣿⣿⣻⠿⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⣉⠁⠀⠀⠀⠈⣃⣿⣿⡷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠰⢞⣛⣛⣿⠖⠀⠀⠀⢡⠟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⢀⡀⣀⣡⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣄⣠⣴⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠉⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠙⣷⢦⠀⠀⠀⠉⠛⠏⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠿⠋⠀⠀⠘⢷⠀⠢⣿⡀⠀⠀⣷⣿⣧⡀⠀⠠⡤⠴⠒⠋⠉⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⡀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⣿⡆⢸⡟⣿⣿⣝⠶⣤⣤⣤⣤⣤⣄⣠⠶⠊⠁
⠀⠀⠀⠀⠀⠀⣰⠛⡡⠔⠪⠭⠿⠿⣿⣿⣭⣿⣿⣯⡭⠹⠟⠻⠷⠶⢾⣿⢸⣿⣿⠿⠿⠛⣳⣛⠿⢿⣻⢏⠟⣆
⠀⠀⠀⠀⠀⢰⠁⠈⠀⠀⠀⠀⣠⢼⡵⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠑⠈⠉⢀⠡⢖⣫⠟⠁⠘⢀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⢀⡀⠔⠋⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠀⠀⠐⠈⠀⠀⠀⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⠃⠀⢠⠀⠀⠀⠀⠀⣀⠈⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡌⠀⠅⢸⠀⠃⠆⠀⠀⠀⠀⡄⠣⠀⠡⡘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠸⠀⠐⡀⠑⡈⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠰⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢇⠀⢰⠀⢁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠄⠘⠄⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠈⠀⠃⠀⣴⣂⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣦⠎⢸⣄⠈⠳⢄⡀⢣⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢀⢠⠐⠃⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠀⠀⣀
⠀⠀⢀⠔⠁⠈⢦⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⠀⢀⠎⠀⠀⠀⢠⠞⠁
⠐⠉⣇⠀⠀⠀⠀⠑⢄⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢠⠃
⠀⠀⠈⢦⡀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡟
⠀⠀⠀⠀⠉⠓⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠞⠛⢿⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⠀⢤⣤⣤⣤⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⠀⠀⠀⠉⠛⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⣀⣴⣶⣾⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣷⣤⡀⣾⣿⣧⣐⠿⠀⠀⡀⠀⠸⢃⣼⣿⣿⡇⣠⣾⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄⠁⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠐⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠃⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⢆⠀⠀⠀⣸⠠⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢒⡖⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣀⠀⢇⡿
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⠖⣌⣻⣫⡕⣦⣄⡂⡀⠲⠖⠒⠖⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠈⠻⠋⠀⠘⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⢡⡄⠀⠀⠀⠀⠀⠀⢐⢙⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⢃⣤⡞⠁⠀⠀⠀⠀⠀⠀⠘⣧⡛⣠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢀⠀⠀⡀⣀⡤⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠘⣿⠉⠉⠀⢠⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠄⠀⠀⠀⣿⠀⠀⠀⠨⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢻⡆⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣼⡇⠀⠀⣾⣿⣇⠀⠀⠀⡀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢸⣿⡇⠀⠀⣿⣿⣿⡀⠀⠀⠃⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⣀⣹⣷⠠⢶⣿⣿⣿⣧⠀⠀⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣷⣿⣾⣿⣇⡀⠀⠀⠀⢠⣦⠀⢤⣣⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⠀⠀⠀⠀⠀⢻⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⡿⣿⡇⠈⠂⣈⠁⠐⠸⡀⡼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢸⣷⡿⠀⠀⠀⠐⠂⠀⠀⢣⠀⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⣼⣿⡇⠈⠁⠀⠘⠉⠀⠀⠀⡇⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⠀⠀⠀⠀⠰⠀⠀⠀⠀⢹⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠙⣿⢇⣠⠤⠤⠄⠀⠀⠀⠀⠀⠈⣧⠞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠄⠀⡔⠂⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠸⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⣁⠂⠱⠄⠀⢵⠉⣡⡰⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⣤⣤⠀⠠
⠀⠀⠀⠀⠀⠙⠛⠢⠀⢀⠠⠀⠸⠌⠿⢁⠀⠀⡔⠐⠀⠑⠠⠀⠉⠐⠀⠀⢀⠀⢀⠀⢆⣂⠀⢠⣤⣤⡀⠈⢡⠈
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢸⣾⡷⡄⠈⠁⠀⠀⡀⠀⠀⢀⢀⠀⠀⠀⠘⠀⠉⠉⠁⠀⠈⢿⣄⡀⢘⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠃⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣏⠁⢺⠄
⠀⠀⠀⠀⠀⡑⠈⢆⠰⡇⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠤⠀⠀⠀⠈⢀⣉⣽⣧⣄
⠀⠀⠠⠐⠀⡈⠁⠀⠀⠙⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠄⠀⠀⣒⣦⡀⡤⡀⢂⠀⠙⠒⠂⠼⣿⣿⢿⡟⠂⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠓⠁⠀⠀⠙⠎⠀⠠⠄⠀⠀⠀⠀⣬⣤⡇⠀⠀⠀⠀⠁⠀⠈⠀⠀⡀⠀⠀⠀⠀⠆⠆⠀⣿⣿⠛⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠂⠀⢠⠀⠀⣠⣿⡇
⠀⠀⠀⠐⠐⠄⠀⠠⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⠀⠀⠐⣿⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡌⠀⣄⣠⣤⠐
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢠⡟⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⠟⠛⠀⢸⣏⡽⢞⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠉⠛⠀⢨⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⣤⣀⠀⣀⣀⣀⣄⡀⣀⣀⣀⢠⡀⣀⢀⡀⡀⡄⡀⣀⣀⣀⡀⢀⣀⣀⡄⠀⣀⣠⡄⠀⣀⣀⣠⢄⣠⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⣈⣉⠉⠻⢧⡀⠀⠀⠀⠀⣿⣿⣿⡷⠒⠉⠉⠉⠉⠱⢶⣦⣀
⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡋⢾⣵⣿⣿⣽⣿⢟⣶⢦⣹⣆⣀⣀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣯⣍⣉⣀⣙⣛⣽⣾⣿⣋⡿⢾⣧⣾⠛⣿⣿⣶⣭⣉⠋⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠿⠟⡉⠉⣷⣧⣤⡿⠈⠛⠿⣿⠉⠻⠳⠒⠛⢷⣦⣤⣤⡀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠈⠉⠉⠁⠀⡀⢁⣴⣿⣿⣿⠇⠀⠀⢠⠃⠘⠋⠉⠁⠀⠘⠉⢁⡀⣱⣄⣀⡸
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⠀⠀⠀⠀⠀⠁⣸⣿⣿⣿⡟⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⣿⡿⠋⠀⠀⠀⠀⠀⢩⣄⣀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡋⠉⠉⢁⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠹⠿⠋⠁
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣴⣻⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⢨⠉⣭⢉⣀⣤⢤⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣣⣿⣿⣯⣽⣼⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⢫⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣿⣿⣿⣿⣽⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢣⡆⠄⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⠈⣿⠿⣯⣿⡿⢿⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⢀⠸⣿⠌⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠷⠀⠁⠀⠀⣿⣳⣦⣼⠆⠀⠀⠀⠀⢸⡿⣿⣿⡇⣿⠁⠈⠀⠹⣷⣌⣙⣻⠟⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡧⠶⠶⠂⠀⠀⠀⡿⣿⢇⣿⡇⠀⠀⠀⠀⣿⠠⠀⠋⣴⡿⠀⠆⠐⠀⣸⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣶⡶⣶⢶⣷⣶⣟⣿⡇⠀⠀⠀⢸⡏⡁⠈⢃⣿⡇⠐⠀⠀⠀⢼⣇⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣾⣿⣟⣻⣾⠃⠀⠀⠀⣾⢡⠁⠊⢸⡏⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⠏⡀⠐⠀⣿⠁⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⡟⣿⢟⣿⣿⣿⣿⣿⣇⠀⠀⢀⡿⢀⠤⠄⢰⡿⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠋⠛⠙⠋⠉⢻⣏⣻⣵⠇⠀⠀⢸⣧⣀⣀⠀⣼⡇⠠⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠈⠋⠉⡅⠀⠀⠀⠀⠀⡏⠉⠛⣿⠉⠠⠀⠐⡆⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⢸⢀⠞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣻⠶⠶⠶⢦⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⡏⠺⣄⣤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⣾⡌⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣿⣧⢿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠻⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣄⣼⣿⣿⣿⣷⣾⣽⣄⠀⣠⣾⢠⡾⠇⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⠙⣳⡆⠀⡆⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠙⠛⢻⣿⠛⠛⠛⠛⢹⣿⣿⣿⣿⣿⣏⠈⠉⠀⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣤⠀⣧⣤⣄⣀⣀⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣬⠸⣿⣽⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢭⣿⣿⣿⣟⣋⣭⣅⠶⠇⠀⣏⠈⢿⣿⣿⣿⢟⣉⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣾⣿⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣩⠂⣸⠇⢀⣤⡞⠛⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⡴⠀⠀⣿⡶⠛⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣠⡄⠀⣠⡾⠃⠀⢀⡮⠀⠀⣸⠏⠀⢀⣤⠶⠛⠛⠛⠿⣦⣄
⠀⠀⠀⠀⠀⣴⠏⠙⢷⠏⠀⠀⣀⠟⠀⠀⣼⣃⣤⣴⠏⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣥⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⢠⡾⠀⠀⡀⠀⠀⣠⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⢸⡇⠀⣀⠁⣠⡾⠋⠀⠀⢀⣴⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⢀⣀⣿⠀⠀⣿⣼⠏⠀⠀⠀⠀⠈⠛⠛⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡆⠀⠀⠻⣧
⠀⣘⣿⠉⠳⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⢻⣦
⠀⣻⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠁⢶⡀⠀⢻⣦⡶⢶⣄
⢀⡷⢀⠀⣼⠁⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠒⠀⠀⠀⣿⠦⣶⡆⠘⠷⠀⠘⣿⡀⠀⢹
⠈⣿⣮⠛⢿⡄⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⠀⣤⣼⢿⡀⠀⠀⠀⣶⠀⠀⠀⠙⣤⡈⢀⡴⠃⠀⠀⢹⣇⣠⡾
⠀⢸⣇⡴⠾⣧⠀⠀⠀⠀⢸⣧⠠⠶⠖⢺⣿⠇⠀⠀⢸⡇⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠈⣿⡁⠀⢙⣧⠀⠀⠀⠈⢧⡄⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⢿⣴⠎⠹⢷⡀⠀⠀⠈⠳⣤⡀⠀⠀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠈⢻⡆⠀⠘⢷⡀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⣠⣤⡿⢷⣦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠄⠀⠀⠀⣾⠃
⠀⠀⢰⣿⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⠀⡀⠀⠀⣠⡾⠃⠀⢠⠿⠀⠀⢠⡿
⠀⠀⠘⣷⡀⠀⠂⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠰⠃⠀⠀⢀⣾⠇
⠀⠀⠀⠈⠛⠻⠟⠉⠁⠸⣧⠀⠀⠀⠀⠀⠀⠏⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠶⠦⠀⠰⠶⠖⠉⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠛⠻⣷⣄⣀⠀⠀⠀⠀⣀⣠⣶⡟⢉⠙⠛⠷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠛⠉⣁⣠⣤⡴⢿⣷⠀⠛⠻⣿⠿⠟⠛⠉⢻⡿⢇⠈⢿⡷⣦⣤⣈⡉⠙⠷⣤⣀
⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⣶⡟⠋⠉⠀⠀⢸⣿⠀⠀⣄⠘⢷⣌⠋⠉⣾⢇⠘⠀⠸⣧⠀⠈⠉⠛⣷⡆⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⢷⡄⠀⠀⠀⢸⣿⠀⠀⠀⢸⡀⠹⢦⣴⠋⢠⠀⠀⠀⢿⡆⠀⠀⢰⡿⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣠⣼⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠂⢨⣥⠂⠀⠀⠀⠀⠈⣿⠀⠀⣿⣦⣄⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠙⠋⠳⣦⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠋⠁⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣽⡷⢶⣤⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢸⡇⢠⣴⣿⡷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⠀⠀⠀⠀⠀⣾⣷⠶⠶⠶⠶⠶⠶⠶⠟⠋⠻⠶⠶⠶⠶⠶⣾⠁⠀⠀⠀⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⢻⣦⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⣀⣀⡀⣠⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⡇⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣧⣤⣤⣧⣤⣤⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠟⠛⠛⠛⣿⣿⣿⠃⠀⢨⡅⠀⠀⢹⣿⣿⡟⠛⠛⠛⢳⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠙⡆⠀⠀⠀⠈⠀⠁⠀⠀⢸⡇⠀⠀⠈⠁⠉⠀⠀⠀⠀⡸⠀⠀⠉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣷⣶⣾⣿⣶⣶⣶⣶⣶⣶⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⠻⠆⠀⠀⠀⠀⣀⠀⠶⠀⠙⠁⠈⠁⠈⠉⠀⠛⠀⠶⠀⣀⠀⠀⠀⠀⢰⣟⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡛⣻⡆⠀⠀⢴⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣤⠀⠀⢠⣝⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠰⠇⠀⠀⠀⢀⣴⠞⠛⣷⣤⣴⣦⣤⣴⠟⠷⣦⡀⠀⠀⠀⠰⠆⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⣴⠟⠁⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⣦⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⣴⠟⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⣦⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣴⠟⠁⠀⢀⣴⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣦⡀⠀⠈⠻⣦⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠸⣧⠀⠀⠀⠸⣧⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⣼⠃⠀⠀⢀⣼⠇⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⠷⠀⠀⠀⠻⣦⣤⣤⡄⠉⠙⠋⠉⣠⣤⣤⣼⠏⠀⠀⠀⠾⠃⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⢲⣴⣶⣴⡟⠀⠀⠀⠈⠻⣦⠀⢀⣴⠏⠀⠀⠀⠀⢻⣦⣶⣦⡖⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⢠⣿⣤⣿⠀⠀⠀⠀⠀⠀⠙⢷⡾⠃⠀⠀⠀⠀⠀⢠⣿⣤⣿⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⢿⣿⣿⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣿⣿⡿⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠶⢤⣿⣿⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⢠⣿⣿⣿⡀⠶⢤⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡉⠓⢀⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡉⠒⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣿⣿⣿⡿⠁⣤⠀⣀⡈⢁⡀⢀⣄⠈⢿⣿⣿⣿⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⡆⠀⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣢⣄⠀⠀⡠⠠⠿⢿⣷⣿⣽⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠷⠄⠐⢶⡙⢿⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⢿⡆⠀⠀⠀⠀⣴⣞⣸⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡌⡓⠀⢀⣠⡀⠊⡽⢷⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣠⠇⠈⡿⠍⠃⠾⠾⠷⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠉⠀⣀⢸⠇⢀⠘⣦⣴⣶⣶⣿⣿⣷⣤⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠛⠉⢄⣜⣀⣙⣿⣿⣿⡿⢲⡞⢛⣿⣿⣶⣶⣦⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡆⣀⣀⣤⣨⣼⣿⣽⣾⣿⣾⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠉⡟⠈⣧⠞⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⠀⠤⢤⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⣠⡴⠏⠸⠐⡟⠋⢁⣁⣾⣿⣿⣿⠟⠻⠷⠦⠶⣤⡉⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢟⡟⠓⠲⣤⣶⣿⣿⣿⣧⣽⣿⣿⣷⣶⡶⠶⠦⠤⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣤⠶⠞⣛⢳⣟⣿⠟⣯⠙⠛⠶⣿⣿⢿⣿⣷⣶⡤⠶⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠙⢷⣎⠓⠢⠀⣿⣿⠿⠛⠁⠀⠀⠀⠀⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⣷⣁⠸⢻⣏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡦⠀⠈⣿⣰⣿⣽⡧⠀⢠⣄⢠⣀⠤⡐⣦⠱⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⣾⢁⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⣸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⣹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠑⠖⣀⠀⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡦⠀⠀⠀⠀⠀⢀⡀⠀⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⡀⠀⠀⠀⠀⠈⠅⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠘⢤⣤⠶⠖⠤⠾⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠀⠀⠀⠀⢨⡿⠀⠀⠀⠂⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠤⠀⠀⡀⠀⢸⡇⠀⠀⠀⠀⡴⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣙⣻⣀⣠⠀⣧⠀⣸⣧⣴⣂⣀⣴⣷⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⣩⠉⠀⠏⠜⠀⠚⣹⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠊⢠⣴⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡇⣤⡴⠾⣻⡿⠛⠙⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠛⠛⠀⡘⠀⠀⣴⣿⡿⠒⠖⠒⠂⠙⢿⣿⣿⣿⣦⡛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠌⣠⢴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣷⣄⣰⣾⠏⠤⠠⠀⠂⠐⠂⠀⠀⠀⠀⣠⣿⡿⠛⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣈⣁⣀⣠⣤⣴⣶⣶⣶⣾⡷⠦⠼⠏⠡⠴⠶⠷⠶⠖⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣶⣤
⠀⠀⠀⠀⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣴⡄⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠋⣵⣶
⠀⠀⠀⣴⣿⣿⣶⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⢠⣶⣿⣧⠀⠀⠐⠶⢾⣿⣿⣷⢰⣿⣿⣯⣤⠀⠀⠀⠀⣴⣿⣿⡇⣼⣿⣿⢾⡄
⠀⠀⠀⠙⣿⣿⡋⠀⠀⠀⣴⣿⣿⡿⣿⠆⠄⣸⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⠉⣼⣿⣿⠛⠋⠀⠀⠀⠀⢿⣿⣿⠁⣽⣿⡟
⠀⠀⠀⢺⣿⢻⡇⠀⠀⠀⠈⢻⣿⣧⣤⡄⠀⠿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠘⣿⡿⣦⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣀⣼⣿⡧⢸⡇⠀⠀⣿⢸⡇⠀⠀⠀⠀⢸⣿⣿⠀⣰⡟⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠉⠈⠁⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⠀⠀⠀⢳⡀⠀⣀⠀⢀⡾⠀⠀⠀⠀⢿⣛⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠰⣿⣿⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠹⣾⣭⣷⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⢰⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢠⣶⣞⣻⠃⠀⠀⠀⠀⠀⢀⣴⠿⠟⠿⣦⡄⠀⠀⠀⢀⣀⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠇⠀⠀⠀⢠⠎⣿⣿⠁
⠀⠀⠀⣾⣿⠿⠛⢷⣤⠀⠀⠀⠀⣸⡏⠀⠀⠀⠸⣇⠀⠀⣀⣿⣿⣿⣿⣿⣿⣷⣂⠀⠀⠀⠀⢸⣿⡯⠀⠀⠀⠐⠳⠆⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⠄⠀⠀⠀⣀⡀⠠⠂⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿⡹⣷⠀⠀⢠⡀⠀⢀⣀
⠀⠀⠀⠻⠟⠀⠀⢶⠞⠉⠀⠀⠀⠘⢿⢇⠀⠀⠀⣰⣿⡿⣶⣟⣇⣀⣶⠀⠀⠀⠀⠀⣴⣦⠈⠛⠛⠋⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣾⣿⣾⣿⣿⣿⣿⣷⠀⠀⠰⣿⣿⣿⠇⠀⠀⠹⣧⣾⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⣾⣾⣷⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⢠⠀⠘⣿⣿⡇
⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢤⠀⠀⠀⠀⠀⢸⡿⣿⠛⠀⠀⠀⠀⠀⠀⢸⣿⢹⡇
⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⣿⠃⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠁⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠲⠦⠤⠤⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⠄⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠶⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠐⠙⠀⠁⠒⠈⠉⢮⡂⣂⣰
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡌⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠰⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣒⣶⣶⣶⣒⣖⣲⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠋⠋⢉⡉⢉⣹⣧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠛⠛⠛⠛⠻⠟⠛⠻⡷⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⠘⠛⠒⢲⣶⣾⣿⣷⣿⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠙⡇⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣽⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢿⠘⢹
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⡟⡆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⡇⡅⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣆⣀⣀⣀⢠⡀⣂⡀⠀⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡇⠫⠄⠀⠀⡄⠀⣤⠄⢞⠘⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⣿⡟⣿⣿⣿⣿⣿⣷⣄⣀⣀⡠⡼⠀⠀⠀⠀⠠⠃⠈⣯⡆⠈⡄⠡
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠘⠆⠱⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡤⠠⣄⠤⢤⣤⣠⣤⣤⣤⣴⡆⣴⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠛⠛⠛⠛⠓⠛⠟⢻⣿⣿⣿⡄⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣴⠀⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠿⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⢠⣿⣿⣿⣿⡄⢤⡤⠤⠤⠄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⠃⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣀⣀⣀⣀⠀⣠⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠄⠀⠀⠙⠻⠿⠿⠿⣿⣠⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢀⢆⡒⠒⠄⠠⡀⠀⠀⠀⠠⠀⠠⡄⠀⠄⠀⠀⠄⠀⠤⡤⠤⠠⠤⠄⣀⡠⣀⡀⣳⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠃⣰⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⠟⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠛⠛⢛⣦⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠛⠓⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⢀⣴⠟⠉⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠟⠁⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣦⠀⠀⠘⣿⣆⣤⣤⣤⣀
⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠛⠻⢷⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡅⢀⡀⣀⣀⣼⣷⠀⠀⢹⣿⡏⠉⠙⢿⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⠋⠙⢿⠋⠀⠈⣿⠀⠀⠀⢿⣦⠄⠀⢘⣿
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢼⣿⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢳⣤⡀⠀⣤⡾⠀⠀⠀⠀⠘⣿⣄⣠⣾⡟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣤⣤⡤⢾⣿⣿⠃⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠉
⠀⠀⠀⠀⠠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠺⣦⣼⣿⣦⣤⡄⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⣾⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⢀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠓⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⣀⡀⠀⠀⠀⢰⣿
⠀⠀⠀⢠⣿⠋⠠⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⠀⣠⠟⠃⠀⠀⠀⣾⡟
⠀⠀⠀⠸⣿⡀⢤⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣤⣾⠿⠋⠀⠀⠀⣤⠟⠀⣾⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠻⣷⣄⣀⣀⣠⣴⡿⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠋⠀⠀⠛⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠛⢧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⠛⠣⣤⣤⣤⣄⣀⣤⣤⣤⡴⠛⠋⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠘⠛⠋⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡾⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡾⠿⠛⠋⠁⠀⢀⣀⣌⠛⢿⣶⣦⣤⣤⣄⣤⣤⣤⣴⣾⠿⣿⡇⠀⣤⣄⠀⠈⠙⠛⠿⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉⠀⠀⣀⣤⣴⣶⡿⠻⣿⡇⠀⠈⡉⠻⣿⣟⠛⢿⠉⠉⡀⠙⣿⠛⢇⠀⠹⣿⠟⢿⣶⣦⣄⣀⠈⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⢀⣤⣾⠿⠛⠋⠁⠀⠀⠀⣿⡇⠀⠀⠛⠀⠙⢿⣦⣀⠛⠛⠛⣴⡿⠀⣼⠀⠀⢿⣧⠀⠈⠉⠙⠻⢿⣷⡄⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠃⠀⠙⣿⣄⠀⣠⣿⠁⠛⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⣠⣿⠃⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠲⣤⠀⠈⠻⣶⠟⠁⣠⠟⠀⠀⠀⠀⢹⣷⠀⠀⠀⢠⣿⠏⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⣠⣴⣿⡋⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⣭⡤⠚⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢹⣿⣦⡄⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⠉⠙⠛⠛⠳⣦⣄⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡞⠛⠛⠛⠉⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⢻⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⠀⢸⣿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢐⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣶⣶⡄⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⣶⣶⣿⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠀⠀⠀⠀⠀⠀⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠻⣶⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠉⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣀⣠⣤⣄⣤⣴⠿⣿⡉⠉⠁⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢈⣿⣧⣤⣀⣀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠘⢿⣦⠛⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠉⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣷⠀⠘⣿⠟⠛⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣤⣠⣿⣄⣤⣤⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⡟⠛⠛⠛⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣿⡍⠁⠀⠀⠀⠀⠻⠿⠿⠿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠿⠿⠿⠟⠀⠀⠀⠀⠀⢈⡿⠛⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣿⣶⣶⣾⣿⣷⣶⣾⣶⣶⣿⣶⣶⣶⣿⣶⣶⣾⣿⣷⣾⣶⣶⣶⣶⣿⣶⣶⣶⣷⣶⣾⣿⣷⣶⣶⣶⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣦⣤⣤⣴⣶⣶⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⠈⠢⠁⣵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢲⣤⣄⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠂⡕⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡠⠀⠙⢿⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣧⠀⢀⠄⠁⠀⠀⡇⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⢀⣙⣬⢿⡾⠂⠀⠀⠀⡻⠀⠂⠐⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⣩⢁⣄⠈⠁⢤⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⡄⠀⠉⠓⠲⡞⠁⠂⢀⠈⠲⠀⠤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⡀⠀⠘⠢⣄⠀⠀⠉⢢⡀⠈⠐⠄⠀⠀⠈⡏⠀⠀⠀⠀⠞⠀⠀⠀⠆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠃⠐⠀⠀⠉⢀⠀⠀⠙⢧⠀⢀⠰⢐⡀⠀⢠⠀⠀⠀⣘⢼⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠥⠃⠀⠀⠀⠐⢖⠒⠀⠈⢂⠀⠀⠀⠀⠀⠱⡋⢴⡀⢠⠀⡀⠐⠚⠂⢀⠀⡀⠈⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⠐⠂⠀⠀⠀⠀⠡⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⠈⠙⢿⠀⠁⠀⠀⠄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠔⣚⠈⣠⣘⠖⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⣈⠂⣀⣀⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢉⡩⣬⠅⠚⡂⠀⠁⠀⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡿⠧
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠺⠈⠉⠉⠁⣀⣤⠴⠖⠒⠒⠚⢱⣄⠀⠀⠀⠀⡶⢲⣤⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠊⠀⠀⠀⣠⠄⠀⠀⠉⠙⠂⠤⠀⠀⠇⠺⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠊⠀⠀⠀⠀⠐⢱⠃⠘⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡴⠐⢢⡆⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠚⡁⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠉⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠟⠶⢯⠁⠒⠀⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠠⡀⠑⠊⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢬⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠡⠄⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠠⠀⠦⠠⠴⠤⣁⠀⠀⠀⠀⠀⢀⠤⡆⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⡶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⢀⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⣤⣦⠈⣷⠀⠀⢀⣀⣀⣄⣐⣶⣖⢂⡀⠀⠀⠀⠀⢀⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⡟
⠀⠀⠀⠀⠀⣰⣿⠃⢶⣶⠀⠀⠙⠛⠀⢸⣇⠀⠿⠿⢿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠉⣻⣿⣿⣧⣀⣤⣶⠀⠀⠀⢀⣼⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠉⠙⢷⡈⠉⠀⢠⣴⣶⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠴⠾⠛⠿⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢿⣿⣧⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠉⠉⠀⠀⠀⠀⠀⠐⠛⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⣀⣀⣤⣴⣶
⠰⣶⣹⣧⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢰⣿⣿⡝⣿⣌⣁⡀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠘⠿⠟⢻⣏⢻⣯⣿⠆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⣀⣀⣽⣷⡿⢿⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠿⢿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠈⠉⠉⠉⠻⠊⠁⠀⠀⠀⢠⣿⣿⣿⡟⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠰⠶⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠾⣭⣿⠟⠀⠀⠘⢧⣬⡿⠟⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠉⠉⠀⠀⠰⠀⠀⠉⠁⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⢀⡀⣠⠶⠤⣤⣀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠀⠀⠈⠉⠻⢿⣿⡽⠋⠀⠀⢰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠲⢤⣄⣀⣀⣀⣀⣠⡤⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣀⣀⣀⣸⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠯⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠂⠉⠉⠉⠉⠉⠉⠐⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢨⠀⠸⡀⡉⠘⠈⠃⢰⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠄⢤⢸⠀⢠⡀⠄⠄⠠⠀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠂⠚⠘⠈⠃⠘⠀⠂⠘⠀⡙⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠤⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢀⠔⠒⠃⠀⢡⠐⡜⢦⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⡏⠀⢌⠀⡃⠈⠈⠀⠁⠀⠸⠀⠀⠀⠀⢀⠜⠁
⠀⢀⡎⠁⣀⠤⠒⠊⠉⠉⠑⠲⢜⣆⡀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠈⠀⢀⡠⠖⠁
⢰⠁⠀⢸⠁⢀⣀⣀⣀⡀⠀⠀⠾⠛⣏⠀⠀⠀⠀⠀⠀⠀⠈⢑⠆⠀⠀⢀⣀⣀⣀⣀⣀⠠⠤⠒⠊⠁
⠏⠀⠀⡜⠀⠈⠉⠀⠄⠁⠀⠀⠐⠀⠈⡆⠀⠀⠀⠀⠀⠀⠴⠡⠤⠔⠚⠉
⢣⡠⣤⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡯⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠖⠒
⠈⠂⣔⢇⠀⠀⠀⠀⢠⠞⣀⣤⡤⠤⠔⡲⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠄
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⣀⣤⠚⣱⣺⡅
⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⢀⠜⠁⣀⡠⠤⠒⣈⠥⠐⠛⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢈⡱⢦⠀⣠⠟⠊⢉⡠⠔⠊⠉
⠀⠀⠀⠀⠀⢀⡴⠁⠙⠋⢿⠛⠻⡎⠁
⠀⠀⠀⠀⢠⠊⢀⡴⠀⠀⢸⡗⠉⢱
⠀⠀⠀⡰⢁⠔⠁⡇⠀⠀⢸⡗⠂⠀⢇
⠀⠀⡘⢠⠋⠀⠀⢰⠀⠀⢸⣿⠤⠀⢸
⠀⠀⢱⢸⡀⠀⠀⡼⠄⠀⠀⠁⢀⠀⣸
⠀⠀⠨⣉⠌⣱⠀⣯⣀⣀⣀⠠⠤⠊⡙
⠀⠀⠀⠙⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⢿⣿⡆⢴⣤⣤⣤⣤⢠⣤⡄⠀⣤⣤⣤⢠⣤⣤⣤⠀⠀⠀⠀⣼⣿⣿⣧⣿⣿⣿⡄⢀⣤⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⢀⣀⠈⠙⣿⣟⠉⠘⢿⣷⠀⣿⡟⠉⢸⣿⣯⠁⢀⣤⡀⢠⣿⡏⢹⣿⡿⠻⣿⣧⣿⣡⠟⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣄⠀⣻⣿⢇⣀⣿⣿⣀⡀⠘⣿⣤⣿⡇⠀⢸⣿⣤⡄⠀⠉⠀⢸⣿⡇⠀⣿⠃⠀⣿⡿⣿⣧⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠋⠘⠛⠛⠛⠛⠃⠀⠙⠛⠋⠀⠀⠘⠛⠛⠃⠀⠀⠀⠘⠿⠃⠀⠈⠀⠐⠿⠃⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠗⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢠⣤⠀⣤⠀⠀⠀⢠⣤⣤⡀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠻⣿⣆⠀⠀⠀⠀⢸⣿⣆⣀⣀⣼⣿⠀⣿⠀⠀⠀⢸⣿⣿⠁⣿⡿⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⣤⣤⣿⣿⡀⠀⠀⠀⢸⣿⣿⠿⠿⣿⣿⠀⣿⡇⠀⠀⢸⣿⣿⠈⣿⣇⠀⢈⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⣾⣿⡇⠀⠀⣻⣿⠀⠻⣿⣶⣶⣿⡿⠃⠀⠻⣿⣦⣤⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠉⠛⠒⠲⠶⠦⠤⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣼⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⡹⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣷⠞⠉⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⡟⠋⠁⠀⠀⠀⠀⣠⣤⡀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣠⣤⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠈⠛⡿⠋⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡷⠿⠿⠿⡿⣿⡿⢿⣿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⠶⠶⠶⢶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⢀⣤⣤
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢠⣶⡄⢹⡆⠀⢀⣀⣀⣤⣐⣶⣖⣢⡀⠀⠀⠀⠀⢰⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⢠⣿⡇⠘⣿⡿
⠀⠀⠀⠀⢀⣼⣟⠀⣾⣇⠀⠈⠉⠁⠀⣷⡀⠺⠛⠻⡿⠿⠿⣿⣿⣿⣿⠀⠀⠀⢈⣽⣿⣿⣄⣤⣴⡆⠀⠀⠀⣼⣿⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠁⠙⢧⡈⠀⠀⣶⣾⡆⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⠞⠋⠻⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⣿⡿⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠉⠀⠀⠀⠀⠀⠈⠉⠁⠈
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⣠⣤⣴⣶⣿⣆
⠘⣿⣽⣷⡶⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢸⣿⣿⣎⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠈⠉⠉⠹⣧⣻⣷⡟⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠶⣶⠾⢿⣏⠷⠀⠀⠀⠀⢰⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⠉⠁⠀⠀⠀⠉⠉⠛⠻⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢠⡴⣶⣶⠀⠀⢠⡖⢲⣶⡆⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠈⠻⠿⠁⠀⠀⠄⠙⠛⠋⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠶⠞⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⠀⠘⠓⢾⣷⣶⣿⠟⠀⠀⢰⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠠⣄⣀⠀⠈⠉⠁⢀⣀⠄⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠛⠛⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣀⠀⠀⣸⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠙⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡴⠣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⡀⠀⠀⠀⠀⠀⢤⣮⠁⡤⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⠀⠀⢀⡤⠒⠃⠀⠸⡅⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢸⣿⠙⣿⣿⣧⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣶⡗⠖⢊⡀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠻⠿⠻⡇⠈⢿⡏⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠡⢀⡌⠁⢰⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣿⣿⡆⠀⢠⠃⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⠀⠀⠀⢈⡶⣫⣿⡀⠈⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠈⠢⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⠟⡡⠊⠀⠂⠀⡰⢋⡼⠅⠉⣗⠀⢿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⠁⠀⠀⠠⠤⣀⣲⠆⠀⠀⢀⣸⣿⣿⣿⡿⢃⠔⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⠑⢌⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣿⣠⣤⣶⣾⣿⣿⣿⣿⠟⠠⢀⠔⠁⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠡⡼⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡀⠀⢀⢔⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠡⣀⢆⠀⠀⠸⢿⢿⣿⣿⣿⣿⣿⠋⡠⠊⠀⡴⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠢⣤⡿⠃⠀⠀⣦⠶⠻⣿⣿⣿⠟⢁⠀⠀⡠⢪⣾⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⢠⡄⠀⠙⠉⢲⡎⠀⠀⠀⠀⡏⡷⠦⣙⣿⢃⡄⠀⢀⢜⡵⢫⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠛⠃⠉⡈⠶⢶⠶⠞⠃⠀⠀⠠⠰⣹⡇⠀⠀⠀⠀⠀⡰⡣⠋⠀⣾⣻⠃⠀⠀⢸⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠰⣈⠲⠤⢾⣺⠛⠛⠳⠶⠶⡿⠇⢀⠔⠁⢠⢊⠜⠁⠀⡠⣿⡏⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢰⣿⣿⡿⠁⠀⠀⢀⠀⠑⣄⣈⡹⠃⠈⢿⡀⠀⠀⠀⠀⢐⡳⠊⢀⠔⡡⠃⠀⢀⡬⠢⠹⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⢼⣿⣿⣷⠀⠰⡀⠈⠢⢄⡠⢚⠀⠀⠀⠈⢷⡀⠀⠀⢀⠎⠀⣠⠊⠴⠀⣤⠔⠂⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⡀⢸⣿⣿⣿⣄⠀⠙⠲⣦⣾⠀⠀⡄⠀⠀⠀⠈⢧⠀⡠⠁⢀⠞⣡⠁⠀⠀⠀⠀⢀⣞⠚⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⠟⢘⠆⠀⣧⠀⠀⠀⠀⢀⠋⠀⡰⢋⡞⠻⣤⣀⡀⡀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠂
⠂⢀⣤⣿⣿⣿⣿⣿⣟⡀⠰⠸⠀⠀⢸⡄⠀⠀⡔⠁⢀⠞⡴⠁⡇⠀⠩⢿⡺⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣜⠂⠀⠀⠇⡠⠊⠀⡴⢣⠎⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠜⠁⣠⢎⡴⠃⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡠⠃⢀⡔⣡⠎⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠁⠄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡠⢊⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⡀⠔⠈⠀⠀⠀⠀⠂⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠦⠤⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⠀⢹⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⡻⠇⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣃⣁⣠⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠋⠀⠀⠀⠙⠦⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠂⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢿⣿⣦⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⣿⢿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢿⡌⣿⡋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⢸⣧⢸⣷⢃⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⢸⣿⢸⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣸⡟⣸⣯⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣸⣿⣿⢇⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⣿⡇⢿⣸⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⢿⡈⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⣿⣿⣈⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⣿⣏⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⣿⣿⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣿⣿⣿⣿⣹⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢹⣿⢿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⢯⣿⠿⣿⣿⣿⢿⡏⣿⣜⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⣱⣿⠏⣾⣿⢸⣿⣿⣿⢸⣷⠘⣿⣏⢻⣿⣿⣿⡿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⣤⣤⢀⣤⣤⣴⣦⣶⣾⣿⣿⡿⢛⣥⣾⣿⠏⣼⣿⡏⣾⣿⣿⣿⡄⢿⣧⠸⣿⣷⡹⢿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⣴⣶⣿⣶⢾⣿⣿⣿⠿⠿⠟⢋⣭⣶⣿⣿⣿⠋⣼⣿⣿⠇⣿⣿⢻⣿⣇⠸⣿⣆⠹⣿⣿⣦⡻⣿⣿⣿⣾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣦
⠀⠉⠛⠛⢿⣿⡿⢿⣿⣤⣶⣾⣿⣿⣿⣿⡿⢃⣾⣿⣿⡿⢸⣿⣿⢸⣿⣿⡀⢿⣿⣆⠹⣿⣿⣷⣌⡻⣿⣿⣿⣿⣮⣻⢿⣿⣿⣿⣿⢿⣷⣟⢧⣄⡀
⠀⠀⠀⠀⣾⡿⢷⣤⡉⠛⠟⠋⠉⠑⠈⠟⣡⣿⣿⣿⣿⠇⣾⣿⣿⢸⣿⣿⡇⠘⣿⣿⡄⠸⣿⣿⣿⣷⣬⡻⣿⡟⣩⣿⣦⣬⢿⣿⣥⣿⣿⡟⠀⣹⠃
⠀⠀⠀⠀⢻⣧⡄⠙⠿⣷⣶⢿⣶⠀⢠⣾⣿⣿⣿⣿⣿⢰⣿⣿⣿⠘⣿⣿⣿⠀⢹⣿⣿⡄⠘⣿⣿⣿⣿⣿⣯⠘⣿⣟⢠⣶⣆⠀⠀⠘⣿⡆⠺⠟
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⢻⣦⣄⢀⣼⢿⣿⣿⣿⣿⡇⣾⣿⣿⠿⡀⣿⣿⣿⣇⠀⢿⣿⣿⡄⠘⢿⣿⣿⣿⣿⡇⣿⣿⠀⢉⣿⡟⠛⠛⢛⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢇⣈⣿⣦⣭⣛⣻⡟⢘⣿⡿⠝⢢⣤⡄⢠⣶⣏⠀⠈⣿⣿⣿⡄⠈⢿⣿⠿⠟⢠⣿⡿⠀⠿⣯⣽⣆⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣯⣝⠻⠿⠿⠿⠛⢉⣤⣤⡾⠿⢷⣎⠉⣛⣿⣿⣿⠉⠛⢷⣄⠠⠿⠐⠾⢷⣤⣀⣤⣾⡟⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠻⢿⣶⣾⣿⠿⣿⡿⠿⠄⠀⠻⠿⠿⢛⣻⣏⡉⣀⣠⣤⣠⣄⣀⣠⣴⣾⡧⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠋⠁⠙⠛⠻⠟⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠀⢴⠄⠠⣦⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⣈⠛⠀⠀⠀⠀⠀⣴⠄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣤⠀⠀⠀⠀⠀⢻⣧⣈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⣹⡷⠀⠀⠀⠾⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢶⣌⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠛⠃⠀⠀⠀⢀⣴⡾⠛⢻⣷⣴⠶⠶⣦⣴⡟⠛⢷⣦⡀⠀⠀⠀⠀⠟⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⢀⣴⠟⠁⠀⣰⡿⠋⠀⠀⠀⠀⠙⢿⣆⠀⠈⠻⣦⡀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⣴⠟⠁⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⠻⣦⡀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⣴⠟⠁⠀⠀⢀⣴⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⡀⠀⠀⠈⠻⣦⡀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣿⡃⠀⠀⠀⢰⣿⠁⠀⠹⣧⡀⠀⠀⠀⠀⢀⣼⠏⠀⠈⣿⠆⠀⠀⠀⢘⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⣷⡄⠀⠀⠀⢻⣆⠀⠀⠈⠛⠷⠶⠶⠾⠛⠁⠀⠀⣸⡟⠀⠀⠀⣠⡾⠃⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⡈⢉⡀⢀⣤⠈⠛⠛⠛⢿⣆⠀⠀⠀⢀⣴⠟⠛⠛⠛⠀⣤⡀⢀⡉⢁⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⢿⡟⠻⣿⠇⠀⠀⠀⠀⠀⠹⣷⡀⢠⣾⠋⠀⠀⠀⠀⠀⠸⣿⠛⢻⡿⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⢸⣿⠶⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⢰⣿⠶⣿⡇⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣴⡟⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⢻⣦⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣦⡀⠀⠻⣦⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣴⠟⠀⢀⣴⡀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⣿⣙⠻⠂⠀⢻⣇⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⣸⡏⠀⠠⣿⣋⠻⠇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⢰⣏⣛⣿⠄⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠁⠀⠠⣮⣙⣿⡆⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠙⠟⠁⠀⢀⣼⡟⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢻⣧⡀⠀⠈⠻⠋⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⣶⣶⡶⠟⠋⠀⠘⠛⠀⠻⠂⠰⠗⠀⠛⠁⠀⠙⠻⠶⣶⣶⣶⡶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣼⠛⣿⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢿⡀⠀⠀⠀⠀⣠⡾⠁⠀⣿⠀⠀⠀⢀⡾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⢀⡾⠃⠀⠘⣷⡀⣀⣴⠞⠁⠀⠀⠀⣿⠀⣀⣴⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠙⢿⣄⠀⣠⠟⠁⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⢠⣿⠟⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⢠⣾⡟⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⣈⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣿⣅
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠋⠀⠀⠘⢿⣆
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡶⠾⠟⠛⠋⠉⠀⠀⠀⣀⣠⣤⡄⠈⢿⣧
⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⣶⣛⣩⣥⠾⠃⠀⠈⣿⡀
⠾⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠻⠛⠋⠁⠀⠀⠀⠀⠀⠹⣷⣀⣠⣤⣴⣤⡄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢀⣴⠞⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⢼⠿⠛⠉⠉⣹⡿
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⡿⠋⠀⠀⠀⠀⢠⣤⣀⣀⣤⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣧
⢠⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠀⢰⣿⡀
⣼⣧⣤⡶⢿⡇⠀⠀⠀⠀⠀⣾⠁⠀⠰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠁⠀⠀⠀⠀⠀⠙⣿⣤⡀
⠈⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠙⠳⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣠⣴⣷⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠉⠛⠳⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢿⣦⡾⠟⠁⠈⣿⡄⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⣦⣤⣀⣀⣀⠀⣀⣠⣾
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⣷⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠉⠛⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣦⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⢷⣄⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠋⠉⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⣿⠿⠟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢫⠜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡠⣴⣒⢖⡈⡀⠀⠄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢬⡭⠃⡉⢶⡇⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⡊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿
⣶⣾⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠛⠉⠉⠉⠛⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⠀⠀⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⣠⣴⠿⠛⠋⠉⠉⠛⠻⣷⣄⡀
⠀⠀⢀⡌⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠿⠶⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣸⠇⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⣿⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⢸⣿⠀⠀⣿⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⢸⡏⠀⠀⣿⠁⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⡇⠀⠀⣿⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣶⣶⣤⣤⣶⣶⣶⡶⠶⠶⢿
⠀⢸⣿⠀⠈⣿⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⠀⠀⣿⡇⠙⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⣿⡆⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿
⠀⠀⣿⣧⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣶⠶⠶⠶⠾⠿⠟⠛⠛⠋⣿⡿
⠀⠀⠸⣿⡄⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⢹⣧⠀⠀⠹⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠹⣇⠀⠀⠹⣿⣟⠛⠷⠶⢶⣦⣴⣶⡶⠶⠶⠾⠟⠛⠉⢁⣴⡿⠁⠙⢿⣷⣦⣀⣀⠀⢀⣀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀⣀⣀⣀⣠⣴⣾⣿⠟⠁
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⡿⠿⠿⠛⠉
⢀⣾⠋
⣼⣿⣷⣄
⣿⠙⢿⣿⣷⣄
⡟⠀⠀⠙⠿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣿⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⣁⡀⠀⢀⡀⠀⠰⠤⠀⠁⠀⡤⠀⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣠⠆⣸⡟⠁⠄⠀⣨⡀⠉⠂⢄⣀⠶⠛⠁⠁⠀⢀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠐⠋⢀⠀⠼⡀⠛⠁⠶⢢⣄⣻⠠⡤⠶⠂⠋⣒⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡡⣠⢈⠀⠀⠀⠄⠀⠀⠈⣆⣀⠐⠃⢠⡆⣀⠉⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⡖⠁⠉⠀⠀⣤⠌⣁⣰⢿⣏⢛⢂⡐⢉⠈⠉⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡠⠚⠛⠉⠂⠶⠀⠁⢨⡺⡏⠀⠘⠆⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣾⣿⣤⠈⠁⢠⠃⠀⠀⠀⠀⢘⠀⠀⠀⣢⠟⡤⠳⠄⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠛⠂⠀⠀⠘⠀⠀⠀⣀⢀⡉⠀⠀⠀⠑⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠷⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣦⠄⣸⠿⣍⠛⢢⡀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⠉⡟⡄⡆⢸⠀⠀⡛⡉⢬⠁
⠀⠀⠠⣖⣀⡾⡵⠊⠀⠀⢰⣇⠑⠈⠲⠀⣠⠿⠛⠁
⠀⠀⠀⢺⡞⢱⠁⠀⠀⠀⠈⣧⠀⠀⢀⡐⠁
⠀⠀⠀⠈⠁⢸⠀⠀⠀⠀⠀⢸⣆⡼⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢀⠃⢳⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠀⡀⠠⠄⢈⠓⣾⠄
⠀⠀⠀⠀⠀⠀⠹⡄⢀⡀⠀⢀⠌⢷⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤⢤⣀⣀⣸⣯
⠀⠀⠀⠀⠀⢀⣀⣉⠙⠖⣤⢸⡤⢻⣷⠂⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠴⠂⠖⣭⣤⠉⣀
⠀⠀⠀⠀⠀⠈⠉⣿⠁⡙⣿⣝⡧⠊⢹⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠙⠉⠀⠀⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠠⠞⣛⣿⠧⣥⡅⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⣄
⠀⠀⠀⠀⠀⠠⠖⢘⣉⣠⠖⠀⠙⣼⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠉⠽⠥⡰⠄⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣀⢀⠀⠀⠘⠓⡆⠀⠀⠀⠀⢿⣟⠚⠇⠄⢀⠤⠢⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡟⠁⣀⣀⣀⣠⣠⣄⡀⠀⣠⢤⢾⡗⠤⠒⠘⠃⠀⠀⠀⡀⠁⠃⠠⠴⠧⣦⡄⠈⣀⠀⠀⢀⡤⠚⣉⡳⠄⡦⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠕⠂⡟⠓⣹⠌⣶⢫⡉⠈⣟⠛⠲⢤⠜⢠⣊⡡⠐⠃⢘⢆⠏⠀⠀⠠⠀⠀⠀⠁⠈⠉⠁⠑⠻⡦⢤⠤⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠈⢷⠘⣡⠀⢻⡆⠉⡻⠟⢦⣠⠖⠙⠀⣫⡁⠳⡾⢆⡉⠄⠀⠀⠀⠀⣞⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢡⣀⡼⣅⡹⠏⢁⠽⢷⣌⠒⠈⠓⠉⠀⡃⠺⣯⣀⡔⢀⠀⠀⠈⠳⠚⠟⢧⡄⢀⢀⡵⣾⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠋⠉⠉⠁⠀⠀⠄⢘⣤⠇⠠⡊⠡⢄⠀⢡⡀⢀⠂⢀⣠⡇⠐⠤⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠉⠲⠤⠬⢵⡟⠣⠋⡈⠉⠹⡄⣀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡃⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣻⡏⢻⡾⠿⣒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⣿⣹⣿⣰⡿⢆⢰⡆⢀⣀⠀⢀⣄⣷⢟⣼⣡⣿⠀⠀⣤⣀⢻⣇⣙⣗⣶⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡟⢏⣿⡇⣿⣿⣿⣵⣇⢹⠃⠸⡿⢿⡿⢿⠃⣡⣞⣿⢿⣰⣷⡝⣿⣾⣿⣿⢦⣸⠎⣿⢿⣶⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢿⡇⡟⣿⣇⣼⣿⡿⠙⠇⣿⣿⢻⡞⣧⣄⠀⠸⡿⢸⡧⠱⢟⡉⣸⣿⡿⣿⣿⣿⣟⠻⣦⡙⠿⣽⣼⣿⢻⠿⠟⠓
⠀⠀⠀⠰⠶⠶⠛⠡⠾⠦⠽⠿⣿⣿⠿⠷⠦⠀⠋⠿⠾⠿⠿⠿⠾⠷⠿⠦⠶⠿⠍⠁⠀⠉⠛⠉⠉⠁⠘⠓⠶⠿⠧⠬⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠭⠭⠍⠉⡏⠉⢉⠭⢭⡉⢉⣉⣉⣉⣉⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠸⣄⡠⠃⠠⠤⠤⠤⠤⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠒⠒⠓⠒⠲⡄⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠋⣠⠖⠒⠲⢤⡙⢦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠃⡼⠁⠀⠀⠀⠀⢹⠀⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡄⢳⠀⠀⠀⠀⠀⣸⠀⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⠄⠑⠦⠤⠤⠞⢁⠜⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢦⠀⠀⠀⠀⢀⣀⣀⣰⣋⣉⣉⣉⠉⣹⣆⣒⣊⣁⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⡼⠀⠀⠀⠀⡏⡤⠤⡄⠀⣀⡤⢤⣀⠀⠀⠀⣀⠀⢹⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡇⠉⠉⠁⡞⡵⠚⠓⢎⢣⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⢇⡀⠀⠸⣸⠀⠀⣿⠀⢸⡧⠤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣀⣀⣀⣠⡇⠀⠀⠀⠘⠲⠭⠥⠖⠁⠀⠀⠛⠀⢸⡇⠐⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠲⠀⠑⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠤⠀⡄⠤⠀⠀⠠⠀⢠⠀⡄⢠⠀⠀⢄⣀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⠀⠀⠀⠁⠁⠈⠁⠀⠈⠉⠘



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣒⣒⠀⠀⠀⠀⢁⣴⣾⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣾⠯⠛⠒⠑⠋⠛⠀⢰⡿⣩⣴⠶⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠛⠀⠁⠀⠀⠀⠀⠀⠀⣠⠎⣔⣿⣭⣁⣼⣷⣄⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠈⠉⠀⢀⣠⠾⠵⠁⠂⡎⠉⠀⠀⠀⠈⠉⣷⣶⣯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⡈⠉⡉⠉⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⢸⡐⠈⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣾⠗⢋⡔⠑⢂⣀⢀⣀⠨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢙⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⢀⡴⠛⣠⣶⡿⢫⠀⢀⠀⠀⠀⠀⠢⠄⠀⠠⣖⣀⢁⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠔⠫⠂⢜⡵⠃⠤⠂⠶⠻⠇⠀⠀⠀⠀⠘⠛⠃⠛⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠊⡀⠨⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡔⠳⠀⠀⠀⠀⠀⠠⠀⠤⠤⠂⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⢀⠰⠌⢠⠀⠐⠤⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⣦⢠⠐⠂⠀⠀⠂⠁⠛⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣤⡈⠈⠿⠀⠉⠋⠁⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⢠⠆⠄⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠄⣠⠾⠃
⠀⠀⠀⠀⠀⠠⠀⠃⡄⠀⠀⣧⠀⠀⠀⠀⠀⢈⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠘⠀⠀⠀⡀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠛⠛⠛⠛⠛⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢉⡽⠒⠚⠻⣿⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⡩⠉⠉⠉⠀⠈⠂⠀⠀⠒⠂⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠚⠉⠉⠙⠻⡻⣿⣿⣿⣿⣿⣿⠟⡿⠛⠒⠦⢭⣻
⣿⣿⣿⣿⣿⡘⢰⠧⠀⠞⢌⠀⠀⠖⠚⠫⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠉⠉⠉⢛⣽⡵⠒⠉⠁⠀⠀⠀⠀⡈
⣿⣿⣿⣿⣿⡇⠼⠥⢲⠦⠖⣠⢼⣄⢠⣤⠄⠀⠠⣀⠄⠀⣿⣿⣿⣿⣿⣶⡴⠖⣀⡠⢀⠀⣀⡠⠞⣿⣵⣤⣤⣤⠤⠦⣀⣀⠏
⣿⣿⠿⡫⠥⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢁⡀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠾⠷⣶⠿⣿⠛⣛⠿⢿⠿⣦⣤⣴⣾
⢏⡕⠀⠁⢀⡴⡾⠀⠀⢀⠏⠈⠈⠁⠳⠀⠀⠉⠙⠓⠀⣶⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠎⠙⠀⠀⠀⠈⠁⠀⠀⠁⠴⠒⠂⠙
⠀⠀⠀⠀⢨⠜⠁⢀⠀⠀⣀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⣼⢿⡟⢻⣿⣿⣿⣿⣿⠿⠿⠃⠂
⠀⠀⠀⠀⣄⠀⠀⡀⠀⠢⠔⠘⡠⠃⡟⣸⠁⡇⠀⠀⠀⣿⠀⠇⠀⣿⣿⣿⣿⣿⠀⢀⣠⣶⡀⠀⠤⡴⢴⣤⣴⡄
⠀⠀⠀⠀⣿⡆⠀⠀⣀⣀⣀⣀⣀⣀⠀⠉⠉⠀⠀⠀⠀⣻⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣦⠀⠀⠀⠻⣧⣠⣿⣷⣶⣶⣶⣾⣯⣭⢹⣭⣽⣿⣿⣟⡛⠀⠀⠀⠈⣿⣭⣭⣙⠲⠠⠀⠀⢀⠈⠀⠀⠀⢤⡄⠀⠀⢠⣶⣶⣿
⣿⡆⠀⠀⠀⠘⣿⣿⣿⣿⢟⡋⠛⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⠋⠁⠀⠉⢱⠜⠟⠷⠁⠘⠂⠐⠃⠘⣠⣤⣤⠜⠻⢿⣿
⣾⣿⠀⠀⠀⠀⠘⢿⣿⣿⢿⣷⣶⣶⣌⡙⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⢀⠀⠀⠸⣴⠀⠀⠆⠀⡂⢠⠇⢀⠀⠐⠾⠟⠆⠀⠀⣿
⣿⣿⣧⠀⠀⣰⣶⡈⢿⡇⠁⠀⠀⠙⠛⣷⠹⣿⡿⢛⣥⡀⠀⢰⣿⣟⠀⠑⠀⠀⣿⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⡿⠈⢀⣿
⣿⣿⣿⣦⣴⣿⣿⣿⣌⣧⠠⠀⠀⠀⣴⣿⣀⣴⣾⣿⣿⣿⡶⠀⣿⣿⣷⡦⠀⢴⣭⣿⡿⠛⣇⣶⣄⣒⣒⣠⣤⣴⡾⠃⠀⠀⣿
⣿⣿⣿⡟⠋⢿⣿⣿⣿⣆⠀⠁⠂⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁⢠⣿⡿⠟⠁⢀⣼⣿⣿⣧⠀⠈⠻⣝⢿⣿⠿⠋⠁⠀⠀⠀⣰⣿
⣷⠞⠫⠌⠀⠀⢻⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⡸⠋⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣠⣾⣿⣿
⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⢳⣿⠀⠈⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⣀⢤⣲⣾⣿⣿⣿⣿
⣿⣷⣦⡄⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣠⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣒⣩⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢤⣄⡤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⡤⢂⣌⡫⠥⠄⠠⠶⠖⠲⠶⠄⠀⠒⠰⠚⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠶⡀⠀⢁⢐⣒⣠⣶⣠⣴⡾⠶⠟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡄⠀⡄
⠀⠀⠀⠀⠀⣰⣧⣾⣄⣴⣵⣿⣤⣼⣿⡿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠦⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⢀⡂⠀⠀⠀⢀⠄⠀⠀⢀⣀⣼⣷⣶⣶⣶⣿⣶⣶⣶⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡠⠀⠈⠐⠀⠀⣠⣾⣋⣽⠟⠛⢛⣩⠋⣩⣉⡙⢻⣿⣻⣿⡾⣿⣶⣆⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⡠⣀⣄⣴⣿⢿⣛⣏⣵⣿⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⢻⣿⣷⣞⢪⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠤⢠⣀⣜⣻⣯⣟⣋⣵⣾⣿⣿⣿⣿⡿⠟⢟⣿⢟⣹⣿⣻⣛⠛⢿⣿⣿⣯⣾⣿⣯⣛⣿⣿⣯⣝⡠⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⣨⢫⡜⠛⣼⣿⢿⢿⣾⣿⢿⠇⡐⢷⣼⣷⣿⣿⣿⡛⠉⠉⠒⡿⠉⠻⣿⣿⣿⣿⣻⡯⣩⣿⣿⣍⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣹⡼⢈⣼⠟⠋⠁⢸⣿⣿⣾⣤⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⢠⣷⡶⠀⠹⡟⠿⣿⣿⣿⣿⣧⣭⣹⣷⠉⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠸⠃⢒⣴⡟⣠⣾⡃⠀⠀⠀⣼⣿⣿⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠶⣿⣿⣗⠂⢤⡇⠀⠙⠻⣿⣿⣿⣿⣿⠞⠁⢀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠔⢛⣯⣼⣿⣿⡇⠀⢀⠀⢸⣿⣿⣿⣠⣽⣿⡿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠗⠀⣸⠇⠀⢉⣫⣿⣿⣿⣿⣟⠛⢋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣤⣼⣿⣿⣿⣼⣯⣆⡨⠆⠈⣿⣿⣟⡋⢽⣿⣷⣭⣙⣻⣿⣿⣿⣿⣿⡿⠂⣬⡟⠀⠀⣠⣿⡿⣫⠿⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠻⠿⣿⣿⣿⣿⣓⠅⢀⠈⠿⣿⣿⣾⣿⣹⣿⣿⣿⣿⣿⣿⢿⢫⣴⣾⢟⣁⣴⣾⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠻⣿⣾⣧⣄⣀⡈⠻⢿⣿⣿⣷⣄⣯⣿⣾⣾⣶⣿⣿⣷⣞⡯⢿⠛⠛⠀⠉⠀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⢶⠶⠶⢷⣿⣿⣿⣿⠿⣿⡿⣿⣿⡛⢯⠄⠒⠉⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠠



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⠋⠁⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⢻⡆⠀⠀⠀⠀⠀⢀⡴⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡗⠀⠈⠳⡄⠀⠀⠀⠀⠈⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣷⣔⣤⣼⠇⠀⠀⠀⠀⢀⣼⣷⣦⣄⠈⠣⠀⠀⠀⠀⠀⣴⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣷⣶⣿⠟⠉⠀⠀⠰⠶⢶⣿⣯⣙⡙⠛⢷⡄⠀⠀⠀⠀⠀⢿⣿⡭⣓⢮⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠟⠋⠁⠀⠀⠀⠀⠀⣠⣟⣥⣿⣿⣿⣷⣦⣹⣦⡀⠀⠀⠀⢸⣿⡿⣾⣕⣯⣌⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣼⣿⣧⡙⢿⢿⣿⣦⣹⣿⣿
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣽⡻⣿⣷⣍⢻⣿⣿⣿
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⢒⣒⠛⠛⠛⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⣶⣿⡿⣿⣿⢿⣿⣿⣿⡟
⣴⣟⣁⣀⣤⣤⣄⣀⣀⣛⣛⠛⠛⠿⠿⠆⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣠⠀⠀⣸⢏⣾⣿⣿⠟
⠙⠛⠋⠉⠉⠉⠉⠉⠙⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⢂⣴⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠋⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣼⠆⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣁⣀⣉⡙⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠣⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣼⣾⣷⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣞⣯⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⣓⠄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠰⠂⠠⠄⠀⠀⡆⠀⡄⢠⠄⠀⠀⠀⡀⠄⠀⠀⠀⠂⠀⠂⠠⠀⠀⠀⠐⠀⠀⠤⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⣶⠖⠶⡖⠶⣤⡀⠀⠀⣠⡶⠛⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⡃⠀⡜⠁⢒⠄⠙⡄⠈⢷⣴⠟⢉⡠⠄⠀⠸⣧
⠀⠀⠀⣰⣦⣀⡀⠀⠀⠀⠀⢰⡏⠄⠀⠀⡇⠀⠉⠲⣎⣠⣤⣴⣦⣴⣷⣴⣧⣤⣀⠀⠀⣿⡀
⠀⠀⠀⢹⡏⠉⠙⠛⠳⠶⢦⣿⠀⠀⠀⠀⣣⣴⠞⠛⠉⠀⣀⣤⣤⣤⣤⣬⣷⣿⣿⡿⢦⣽⡇
⠀⠀⠀⢸⡇⠀⠈⠁⠒⢲⣤⠾⢧⣆⣠⣾⠋⠀⠀⢀⣴⡞⠉⠀⠘⠋⠀⠀⠉⠛⢋⡀⠀⢙⣧
⠀⠀⠀⠸⣿⠀⠀⢠⣼⣿⠁⠀⠀⠈⠉⠛⣛⣳⠶⠞⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣾⣿⣧
⠀⠀⠀⠀⢻⣆⣼⠋⢛⣛⠀⠀⠀⠀⠀⡾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣶⡄⠁⣹⡄
⠀⠀⠀⠀⠀⣿⣷⣤⡘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⣿⠛⠁⠀⠀⠉⢡⣾⣿⡇
⠀⠀⠀⠀⢸⡟⠉⠉⠁⣤⣤⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠲⡄⠀⠀⣤⣸⡇
⠀⠀⠀⠀⢸⣿⣿⡶⠆⠙⠋⠀⠀⣰⠏⠁⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⢿⠐⠶⠛⠻⠀⠀⣰⣿⣤⣄⣀⡀
⠀⠀⠀⠀⠸⣟⠁⠀⠀⠛⠀⠀⢠⡃⠀⣀⣠⣴⠶⠈⢻⠀⠀⠀⠀⠀⠀⠈⠳⠶⠖⢁⡠⠚⠁⠀⠀⠀⠉⠛⣦
⠀⠀⠀⠀⠀⢹⣦⣶⠶⠋⠀⠀⠘⡏⠉⠀⠈⠉⠀⠀⡸⠀⣀⣤⡴⠶⠶⠶⢦⣄⡠⠊⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠻⣥⣤⣤⠤⠔⠒⠛⢦⣀⡀⢀⣠⣞⣡⣾⡏⢠⠀⠀⢀⣠⣼⣿⣷⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠛⠳⡆
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠿⠟⠁⠀⣠⠄⠀⠀⠀⠀⠀⣰⠇
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠙⠳⠦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠓⣤⣀⠀⠀⠀⠀⠀⠃⢀⣠⠾⣍⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⣨⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡧⠛⠛⠂⠐⠛⠛⠋⠁⠀⠈⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣄⣀⣀⡀⣠⣀⣀⠀⠀⣠⡤⣤⣠⣶⣾⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡿⠛⢛⣯⣉⠋⢻⡇⠀⠈⣻⡞⢣⡈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠶⣼⠿⠛⠷⠞⣻⡋⠉⠙⠛⣷⢤⣴⠿⣄⠈⢻⠟⠙⠻⢿⡴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⢧⡀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇⢸⣿⡆⠹⡆⠈⣷⠀⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠳⣄⠀⠀⠀⣿⠀⠀⠀⢸⣇⡞⣧⠙⣦⣿⠀⢹⡆⠀⢰⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠛⠛⣶⡟⠀⠀⠀⠸⠟⠀⡗⠀⠈⠉⠀⠸⣟⠛⠋⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣀⡀⠀⢸⣇⠀⠀⠀⠀⢀⣤⣧⣤⡀⠀⠀⠀⣿⠀⣀⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠉⠀⣼⡟⠛⠛⠛⠚⠻⣤⣀⣠⡽⠛⠛⢻⡟⠈⠛⠳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠚⠛⢷⡤⢀⠀⠀⠀⠀⠉⠀⠀⠀⣠⡾⠓⠒⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣷⡄⠀⠀⣀⠄⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⢁⡾⣿⠁⠀⠀⡅⠀⠀⢸⡇
⠀⠀⠀⣀⣤⡴⠶⡶⠛⠛⠻⠟⠉⣉⣋⣤⡶⠋⠀⢿⣤⣤⣤⣧⣤⣤⣤⡿
⠀⣠⡞⠛⠛⠀⣠⡴⠶⠚⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿⠰⠷⠀⣿
⣰⠿⠇⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡇⠀⣿
⢿⡀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣂⠀⣿⡄
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣉⣉⣋⣉⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠛⢿⣿⣿⠉⠉⡋⠉⠙⣿⣿⡟⠛⠓⠲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⠀⣘⠀⠈⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣯⣽⣿⣬⣿⡼⢧⢽⡤⣽⠤⡿⢿⣯⣥⣾⢭⡭⢿⣥⠤⡮⠿⡼⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣴⣶⣦⣀
⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⢭⢹⣿⣿
⠀⠀⠀⠀⢸⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⢀⠀⠈⠛⡟
⠀⠀⠀⠀⠀⢳⣄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢾⡗⠂⡠⠄⣀
⠀⠀⠀⠀⠀⣘⠻⣧⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⣴⣄⣀⣸⣷⠠
⠀⠀⠀⠀⠀⠈⠁⢻⡆⢄⢳⠄⠀⠀⠀⢀⠔⠁⠀⡠⠀⢀⣿⣷⡊⠙⠃⣾⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠟⠀⠀⠁⠐⠁⢀⠄⠊⢡⣤⣾⣿⣿⣿⡀⡀⢻⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠂⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⠀⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠢⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠟⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠃⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣸⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⡏⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢰⣇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠸⢨⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⣿⠀⡄⠘⢷⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡇⠀⠘⠻⣦⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠙⠃⠈⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⠄⠂⡀⠄⠼⢉⣀⠀⢂⠀⠀⠀⠐⠛⠛⠀⠐⣅⠉⡔⠀⠆⠘⠛⠛⠂⠀⠀⠀⡠⠀⠀⠐⣄⣀⢀⠤⡀
⠀⠀⠀⠀⡠⠂⡤⢠⠇⢀⡏⠶⢭⡄⢞⡀⠐⢦⣄⠀⠻⠇⠀⠈⠉⠙⠀⠀⠻⠃⢀⣤⠶⢀⣀⢇⣀⣋⡅⣗⠀⢉⢀⡈⠄⡀
⠀⠀⠀⢰⠀⢚⣣⡄⢰⡆⠈⠚⠃⡱⢠⣽⢿⡶⢈⣳⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⡑⢾⠿⣥⠀⡎⠱⠤⠋⢢⡌⣇⡿⡆⠠
⠀⠀⠀⠈⠁⣀⣩⡀⡻⠃⠤⠤⠊⣱⠟⢡⡞⠱⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⢷⡈⠻⡟⢦⣀⡀⠿⠗⠬⡛⠁⠲
⠀⠀⠀⠀⠉⣭⣒⡏⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣻⢝⣛⠺
⠀⠀⠀⠐⠤⠈⠴⠒⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠽⢘⣋⣀
⠀⠀⠀⠀⠀⢰⣄⡾⢋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢷⣆⣤
⠀⠀⠀⠀⢰⣄⢻⠗⣫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠶⣿⢃⡴
⠀⢠⣾⠀⠀⠙⢆⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠞⠁⠀⢰⣤
⠀⠈⠛⠀⠿⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠴⡆⠘⠋
⡂⠈⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠗⢂⡄⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢔⡋⠰
⠁⠂⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄
⠀⢠⣶⠀⠿⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠺⠇⢠⣄
⠀⠈⠻⠀⠀⣴⢏⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⢷⡄⠀⠸⠛
⠀⠀⠀⠀⠘⠁⣾⣟⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠶⣿⣄⠛
⠀⠀⠀⠀⣀⣘⠁⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡾⠯⠛
⠀⠀⠀⠀⢀⣴⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠑⣶⣠⡈⠐
⠀⠀⠀⢠⠲⠶⣩⠧⠎⠀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠘⣀⣗⢪⣭⣄
⠀⠀⠀⠸⠆⠴⣶⡐⣿⡆⠀⠈⠳⣿⣦⡙⣦⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣴⢟⣤⠞⡠⠞⠉⠁⣼⠤⢨⣍⢀⣼
⠀⠀⠀⠈⡀⢻⣆⢥⠉⢢⡔⣛⣦⡇⢚⣽⡿⢓⣨⠵⠆⠀⠀⠀⠀⠀⠀⠀⠐⠺⣥⡐⠻⣿⣝⠰⣄⠴⠦⣀⠟⢀⠋⣵⠂⠘
⠀⠀⠀⠀⠀⢁⡀⢠⢄⢀⡃⠡⠆⠀⡉⠀⠰⠛⠁⠀⢶⡆⠀⠠⡤⣄⠀⠀⡶⠄⠈⠛⠓⠀⢠⠂⢉⡝⢃⠏⠀⡋⠘⢁⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠑⠀⠤⠐⠁⠀⠀⠀⢲⣶⡶⠀⠒⠠⢓⡄⡓⠀⢰⣶⡦⠂⠀⠀⠀⠃⢀⡀⠤⠃⠈⠁⠒⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣴⣄⣈⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠠⡄⠀⠀⠀⠀⢹⣧⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⣷⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣤⣔⣒⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣠⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⡆⠀⠀⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⠏⠀⡀⠀⢠⣾⣿⠇⢀⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⡞⠀⣰⣿⣿⡿⠀⣼⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣢⡞⢀⣼⣿⣿⣿⠃⢸⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣱⣾⣿⣿⣿⣇⢀⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⡈⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢈⣀⣀⡀⠀⠀⠀⠈⠷⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠁⠀⠀⡠⠤⣄⠀⠾⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣠⠀⠀⠀⢀⠎⠀⠀⠀⠑⣄⠒⠺⣄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⢊⠏⠁⠀⠠⠊⠁⠀⠀⠀⠀⠑⠈⢢⠀⢈⣧⠀⠀⠀⢀⠜⢁⡠⡙⢆⡠⢔⠂⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡠⠀⡇⠀⠀⣀⣀⡀⣠⠒⠐⠀⢣⢀⠊⠀⠀⡠⠃⠴⠁⠁⣸⡤⠒⠁⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⠁⠀⢃⢰⠋⠀⠀⠈⠉⠁⠀⠀⠃⢺⣆⠀⡔⢡⠊⠀⣢⠴⡇⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⢄⣈⠉⠑⢄⡄⠄⠀⠀⠠⡶⠄⣀⠠⠃⠀⠓⠊⡔⡁⠖⠊⠁⠀⠈⢳⣈⣁⠀⠀⢀⡄⠈⠄
⠀⠀⠀⠀⠀⣀⣀⡀⡇⠀⠠⠒⠲⠤⣭⣐⡚⠷⠀⠀⠉⡉⠀⠀⠀⠈⠐⢄⣠⠊⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁⢠⠊⠁⢁⠀⠠
⠀⠀⠀⠀⠘⠁⠀⠐⠛⠒⠲⢤⣤⣀⣆⣀⣸⣑⡒⢤⣔⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⢱⠔⠋⠀⠀⠈⡆⠀⠇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣀⠕⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⣀⣠⣴⡿⠿⢿⡃⠀⠀⠀⠀⠀⠀⡘⢈⡠⠒⠁⠈⢣⠜⠀⡰
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣹⠉⢆⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠛⠛⠛⠁⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠃⠉⠀⠊⠉⢁⠆⢀⠜⠁
⠀⠀⠀⠀⠀⠈⡆⠠⡤⠭⡀⠉⢳⡳⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢁⣀⡀⠀⣀⡔⢁⣀⣀⡀⣤⠎⣠⠣⢄⡠⠔⢋⡔⠁
⠀⠀⠀⠀⠀⠀⢡⠀⢇⠀⠘⠤⠜⠣⡈⠢⡀⠀⠀⠀⠑⣄⣀⣤⠚⠁⠀⠉⠙⠛⠡⠭⢀⡠⠖⠁⡰⣁⠔⠉⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠆⠈⡆⢰⠚⢉⢑⣛⠂⠈⠲⢆⣛⣉⡱⠚⢀⡔⠀⠀⠀⠀⣀⠝⠓⠲⡤⠴⠚⢏⢀⡠⠚⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢄⠘⠂⠀⠀⡝⠒⠤⠤⠥⠤⠒⠁⠈⠉⠉⢏⠁⠱⡤⠦⠴⠁⠀⠀⠀⢣⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠅⢂⡀⠚⠢⠤⠤⣀⣀⣠⠒⠒⠢⠤⠖⠋⣄⠠⠋⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⡐⠁⠈⢢⣀⣀⡤⠒⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⡰⠁⠀⠀⢠⡇⣇⢘⠒⢤⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢲⠃⠀⠀⠀⢸⡇⡏⠁⠀⡸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⢸⢽⠁⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⢀⠤⡀⣸⣿⡿⠀⢠⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡤⠤⠔⠒⠊⢫⠀⢆⠀⠀⢀⠔⠁⠀⢳⡜⢿⠃⡠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢣⣈⡦⠒⠁⠀⠀⠀⠈⠧⣏⠔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠑⢄⠀⠀⠀⠀⠀⠐⠚⠉⠱⡀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⢔⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠻⢿⣟⣛⠿⢿⡿⣿⣭⣭⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢘⡏⠒⠀⠈⡉⣉⣉⠛⠺⠿⠿⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠀⠠⠤⠤⠄⠒⠒⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠢⣄⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⡫⠷⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⣵⡄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⣩⠅⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡟⠩⣟⢦⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡻⢟⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣼⠃⠀⠈⠳⢝⣦⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⣩⠖⢋⣴⡟⠃⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡏⠡⢠⠚⡍⠁⠀⣠⠒⠛⠻⠟⠉⠸⠛⠛⠻⣆⠀⠀⠀⢀⣴⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡿⠁⠀⣸⡇⠀⠐⣟⢻⢄⡀⠀⠀⠋⠁⠀⠛⢾⡟⠠⢄⡀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⠁⠀⡞⢹⣯⠀⢰⡿⣥⡈⠁⣀⢄⡄⢚⣀⣤⠾⠀⢀⢎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⠀⣠⠟⠻⣿⡄⠘⣄⢈⡻⠧⠂⠈⢣⠊⠰⠁⢀⣠⣞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢸⠿⡌⠁⠀⠀⢸⣧⠀⢸⡁⠀⢀⣀⣠⣏⠀⠁⠐⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠸⣀⡀⠀⠀⠀⢈⡇⠀⣸⡿⠛⠋⠉⣹⣿⣃⣇⣠⡿⠿⠷⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣤⣴⣿⣧⡀⢀⣠⣾⣷⡞⠁⢀⣀⣤⣾⣿⣿⣿⣧⣀⣀⣤⣤⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⣛⣻⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣟⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡯⠤⠼⣧⣿⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣿⣿⣼⠧⠤⢼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡾⠛⠛⠯⣵⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣿⠵⠛⠛⢷⣶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠦⠤⠤⠬⢦⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⡴⠥⠤⠤⠶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⣀⣙⣦⣀⠀⠙⢷⣄⠀⠄⢠⠀⣠⡾⠋⠀⣀⡴⣋⣀⣀⣀⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠙⢧⣾⣿⣼⠋⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⣼⡟⢹⣷⣥⣄⣤⣤⣶⣶⣶⣶⣶⡶⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠃⠉⢋⠭⠉⠉⠉⡈⢿⣿⣼⣿⢁⠈⠉⠉⠤⣚⠉⠉⠛⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠒⠀⠀⡠⠎⠀⢀⡤⠊⡀⠋⣿⣿⠉⢀⠁⠀⡀⠀⠘⢤⡀⠀⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠔⠂⢀⣄⣠⠤⠲⠋⠀⠐⢀⠂⣿⣿⡀⠀⠣⠀⠈⢓⠠⢄⡠⠀⠀⠂⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⠊⠀⡌⠀⣴⠃⢀⠎⢀⢂⠎⢸⣿⠐⠔⠀⠱⡀⠈⢦⠀⠱⡀⠀⢦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣞⠀⣠⠁⢀⠎⠀⡠⠂⠀⠘⠏⠀⠈⢦⠀⠐⡄⠀⣃⠀⣱⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣾⣤⡾⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢢⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢀⣤⡤
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⣿⣿⣿⠳⠟⠁
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠉
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⠶⠿⠿⠿⣿⣿⣶⣶⣶⣦⡄⠀⠀⢠⡿⠋⠉
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⢙⠛⠻⠟⠛⠁⠀⢀⡤
⠀⠀⠀⠀⠹⣿⡿⣋⠛⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⡤⢂⡭⠻⡿⢰⡖⠀⠀⠐⣾⡋⠉⠁
⠀⠀⠀⠀⠀⠘⠻⠉⢿⡄⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢷⠚⡉⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣁⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠙⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢀⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠙⣻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠙⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡈⠙⠿⡏⠀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⣿⡆⢹⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣾⣿⣿⡇⠈⣿⠃⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠀⠀⠀⠰⣾⣿⠀⠀⠟⠀⠀⠀⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⢛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⢻⠟⠀⠉⠋⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⢻⣿⣿⣿
⣿⣿⣿⢿⡿⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠛⠁⠀⠀⠀⠀⠀⠀⠹⠿⣿
⣿⣿⢥⠟⠓⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣏⠛⠓⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢳⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠹⠟⠛⠻⠛⠉
⠘⠂
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