Relieved Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣼⠿⠟⠿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⣅⠀⠀⠈⠻⣿⡆⠀⠀⠀⣰⢟⣷⣶⠆⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⣿⣧⠄⠀⠀⠀⢻⡇⠀⠀⣼⠋⢸⣿⣿⡖⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠻⠟⠋⠀⠀⠀⠀⢸⡇⠀⡸⠃⠀⠈⠙⠉⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠘⣧⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⣶⠾⠛⠛⠓⠦⣄⠀⠈⢷⣄⣠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢀⠄⠒⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠂
⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⡤⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⣿⡄
⠘⢳⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡿⠃
⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⠉
⠀⣀⠀⠁⠀⠀⠀⠀⠀⠹⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⣀
⠀⢿⡀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠈⠻⣦⣤⣤⣤⣴⠾⠃
⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠘⢻⡏⠉⠛⣷⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣸⣧⢤⣶⠋⠀⠀⢀⣀⡀⠀⠀⣿⠀⠴⢶⡆⠀⠀⣀⣀⠐⢶⡶⠀⠰⣿⠃⠀⣀⣀⠀⠀⢀⣀⣩⣷
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢰⣏⣉⣙⣷⠀⣿⠀⠀⠘⣧⢠⣟⣉⣙⣷⠈⣷⡀⢸⠇⠀⣾⣉⣉⣻⣤⡟⠉⠉⣿
⠀⠀⠀⠀⢀⣼⣷⠄⠀⢷⣠⣾⣏⠉⢉⡉⠀⣻⠀⠀⢈⣿⠘⣿⠉⠉⣉⠀⠘⣷⡿⠀⠀⢿⡉⠉⣉⢻⣄⠀⢀⣿⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣭⣿⣭⣥⣾⣿⣷⣤⣿⣿⣯⣬⣿⣭⣥⣤⣤⣭⣥⣤⣤⣤⣽⣯⣭⣤⣭⣭⣭⣽⣿⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⢿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⡿⠿⠿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⡀⠤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⢄⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠢⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠴⠃⢀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣦⣄⣀⣸⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⣇⣀⣠⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡒⠒⢂⡸⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠿⣿⣿⣿⡿⠟⠁⠉⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⢹⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠤⠀⠀⠀⠀⠠⠤⠐⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣷⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢟⠄⠀⠀⠈⡜⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠓⠁⠀⠀⠀⠀⠈⠪⠛⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋⠀⣠⡄⠀⠀⡠⢠⣭⠀⠀⠀⢸⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢡⠀⠀⠀⠙⠃⠀⠀⡇⠀⠉⠀⠀⠀⢰⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⣿⢿⠄⠀⠀⠀⠉⠉⠀⠀⠀⢠⣿⡿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢇⠄⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⣾⣿⡇⠀⠀⠀⠀⠉⠢⠤⠔⠀⠀⠀⠀⢻⣿⡇⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣶⣶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⢵⣾⣥⠋⣧⡋⠐⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠋⠈⠀⠀⠃⠉⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠹⠁⠁⢀⡀⠀⠀⠰⡀⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠐⠊⢀⣤⠄⠀⠭⠶⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠈⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠀⠀⠀⠠⢤⠄⠀⢸⠷⢄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⣠⡴⡀⠀⠀⡀⠀⠀⠉⢣⢤⠀⠀⠀⠈⠁⠈⠉⠀⢀⢠⡀⡰⠖⠉⢆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢩⣑⠈⠲⣄⡀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⢀⡀⠀⣀⠀⠀⠀⣀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠹⢲⡄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠄⠀⢀⣀⡴⠿⣿⢿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠢⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢏⡞⢴⡵⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠃⠨⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣷⣄⠀⠀⠀⠀⠀⣠⡾⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣥⠀⠀⠙⢿⡆⠀⠀⢀⡾⣿⣶⠆⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢼⣿⣷⠀⠀⠀⠘⣧⠀⢠⡞⠰⣿⣿⠇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠉⠁⠀⠀⠀⠀⡟⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠸⡄⠀⠀⠀⠀⠀⢀⣀⣤⣴⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⣀⣴⣿⣁⠀⠀⢷⡀⠀⢀⣴⡾⠟⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣻⠟⠉⠀⠀⠉⠓⠀⠀⠙⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣤⣀⣤⣤⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⡈⠓⠀⠀⠀⠀⠐⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⠴⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡤⠤⢤⡀⠀⠀⠀⠀⢠⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡼⠃⠀⠀⠀⠀⠀⢸⠀⢤⣭⡀⠀⠀⠀⠠⣤⡄⠀⣴⠆⠀⠀⠀⠀⠀⢀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⠀⢠⣟⣉⣻⡄⢸⡇⠀⢸⡇⣰⣋⣙⣧⠘⣇⢠⡟⠀⣼⣋⣙⣦⡼⠋⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠸⣦⡾⣯⣉⣉⠁⣸⡇⠀⣸⣇⠹⣍⣉⡍⠀⠹⣾⠁⠀⢻⣍⣉⡹⣧⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠀⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠟⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣛⣻⣛⣻⣿⣿⠻⣿⣟⠿⣿⡛⣿⡿⠻⢿⣟⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⣁⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⣴⣿⣿⣶⣶⠄⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⣿⡛⡀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢀⣼⠋⣾⣿⣿⣧⡄⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠘⣿⡆⠀⢀⣾⠃⠀⠻⣿⡿⠟⠁⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⢀⣀⣀⣤⣴⡾⠿⠿⣦⣄⡀⠀⠀⠻⣦⡀⠀⢠⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⡿⠋⠁⠀⠀⠀⠀⠈⠛⠆⠀⠀⠉⠻⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠷⠶⠶⠶⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅
⠀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀
⠀⣠⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣿⣧
⠻⣷⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿
⠀⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠔⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⣠⡾⠋
⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⠉⠁
⠀⢀⠆⠈⠁⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠶
⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠈⠙⠿⣶⣶⣶⡶⠿⠋⠁



⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡆
⠀⠀⠀⠀⠀⠀⢠⣿⡶⢶⣶⡏⠀⠀⠀⠀⣀⣀⡀⠀⠀⢸⡇⠀⠶⠿⣿⠀⠀⠀⣀⣀⡀⠘⢻⣿⠂⠀⠙⣿⠛⠁⠀⣀⣀⣀⠀⠀⢀⣀⣄⣼⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⡀⠀⢠⡿⠉⠉⠻⣧⠀⢸⣧⠀⠀⠀⣿⡇⢠⣾⠋⠉⠙⣷⠀⢻⣆⠀⣸⡟⠀⠀⣾⠋⠉⠙⣷⣠⡾⠋⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⢹⡇⠀⣼⣿⠿⠿⠿⠿⠃⢸⣿⠀⠀⠀⢿⡇⠸⣿⠿⠿⠿⠿⠃⠈⣿⣄⣿⠃⠀⠸⣿⠿⠿⠿⠿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠘⠿⠾⠋⠿⣦⣤⣴⠆⢠⣼⣿⣄⠀⣤⣾⣧⡄⠻⣦⣤⣴⠶⠀⠀⠘⣿⠏⠀⠀⠀⠹⣧⣤⣴⠶⠘⢷⣤⣤⠶⣿⣦⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣶⣾⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⡤⠊⠁⠀⠠⠾⣖⣋⠉⠀⠀⠠⠂⠄⠸⡄
⠀⠀⠀⠀⠀⢄⡀⢠⣄⠲⢶⣶⡀⠁⠰⣶⠁⠔⢖⡋⠀⢶⣀⡀⠒⠒⠂⠒⠻⠫⣙⡃⠋⠠⡀⠒⠲⣄⠰⠒⠒⠒⠒⢲⡆⡀⡒⡆
⠀⠀⠀⠀⠀⠀⠑⠤⠀⠢⢀⣙⢣⠀⠀⠉⢧⠀⠈⠱⡄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠏⡆⠀⠀⠀⠀⠉⠀⠉⠁⢠⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⣀⡀⠈⠈⠂⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⢀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠀⢀⠱⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠑⠄⠘⠡⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⡂⠀⠙⠒⣠⡀⠈⠢⢈⡆⠀⠀⢀⣀⠀⠀⠀⠄⠀⢀⣀⡀⡀⠀⢠⡀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠙⠒⠒⠒⠒⠚⠛⠚⠛⠛⠀⠀⣿⣿⣧⠀⠀⠐⠚⠃⠤⠉⠛⣠⠝⡳⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⠠⠁⠀⠀⠀⠐⠊⠀⠔⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠶⠶⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⣀⣤⠴⠖⠛⠛⠋⠉⠛⠛⠲⠶⣤⣀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣤⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢠⡞⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠉⠉⠙⠳⢦⣤⠞⠁⢀⡴⠋⠀⢳⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡟⠀⢀⣴⠞⠋⠉⠁⠀⠠⠶⠛⠉⠉⠛⣶⠞⠁⢀⡴⠛⠳⣦⡀⠀⢻⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣼⠁⠀⡾⠁⠀⠀⠀⠀⠀⣠⡴⠖⠛⣻⠞⠁⢀⡴⠋⠀⠀⠀⠈⢷⠀⠈⣧⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠟⢛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡟⡛⠻⢦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢾⣉⠙⣧⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣉⡷⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠻⢦⣉⣙⢻⡀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣋⣉⡴⠞⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣯⠉⠙⠳⣄⣠⠞⠁⢀⣴⣏⣀⣀⣀⣀⣀⣤⣤⠄⠀⠀⣠⠞⠛⠉⣽⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠘⢧⠀⣠⠞⠁⢀⣴⣯⣀⣀⣀⣈⣁⣀⣀⣀⣤⡤⠶⠚⠁⠀⠀⡼⠃⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠛⠁⢀⡴⠋⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠻⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠉⠛⠲⠦⣤⣤⣄⣀⣤⣤⠴⠖⠛⠉⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠶⠶⠶⠶⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣠⣴⣶⣶⣄⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣠⣴⣶⣦⣄⡀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣸⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⡄⠀⢸⡏
⠀⠀⠀⠀⠀⣿⡇⠀⣾⠀⢸⡆⠀⢰⡏⢀⡀⠀⣤⠀⠀⣿⠀⠀⣿⠀⣸⠆⠀⣿⠄⠀⣷⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠄⣼⡇⠀⢿⡀⢸⡇⠀⢿⡇⢸⡏⠀⠰⣿⢻⡇⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⢾⡇
⠀⠀⠀⠀⠀⠘⠇⠀⠈⠀⠻⡀⠀⠀⠁⠘⠧⠀⠀⠀⠸⡇⠀⠀⠁⠈⠣⠀⠀⢻⠀⠀⠈⠀⠘⠇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠾⣆⠐⠙⢷⠀⠀⠴⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⢠⡎⢁⡀⠉⡆⠀⠀⢸⡯⠑⢂⠈⠀⠀⡀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⢨⣁⣀⢀⠀⠀⠀⢘⢷⣺⣁⡰⠃⠀⠀⢸⣿⠴⢤⠠⠘⠇⠃⠀⠀⠀⠀⠀⠈⠻⣿⠆⢀⣤⠈⢈⠟
⠀⠀⠀⠀⠀⢸⣥⣌⠘⢃⠀⠀⠈⢰⡟⠁⠀⠀⠀⠀⠐⣿⠙⣄⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠌⡐⡲⠃
⠀⠀⠀⠀⠀⠈⣿⣻⡓⠓⠛⠳⠊⡈⡇⠀⠀⠀⠀⠀⢸⣷⣘⣿⢄⣶⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⢻⣯⠀⠀⢸⡆⠘⠘⠀⠀⠀⠀⠀⠀⢈⣿⣷⣄⢀⠩⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⢸⢻⠇⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢅⡈⠱⣂⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠰⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⠒⠲⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠰⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⣠⠐⠘⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠂⠂⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡁⢀⠀⢠⡈⢆⣂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣻⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣼⣇⠹⠾⠿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠖⠚⠸⠇⠀⠀⠀⠀⢀⢰⡀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠻⣿⣶⠖⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠖⣠⡀⠄⠀⠀⠀⠀⠀⠀⢸⣅⡰⠇⠁⠘⠅⡅⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣇⣀⠀⠂⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠈⠈⠛⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣀⠀⢰⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⣈⣀⡀⠰⠀⠀⠀⠀⠀⠸⠏⡍⠉⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡴⢶⡽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⢀⡀⠀⠀⠀⢀⣠⣠⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⢇⡄
⠀⠀⠀⠀⠨⠵⢞⣳⡟⠻⠂⠀⠀⠀⣠⣀⣤⡀⠀⠀⠀⠈⢢⣄⣠⡄⠃⠀⠀⠀⣿⣿⡟⠁⠀⣀⣾⣿⢋⠈⠀⠀⠀⠀⠀⠀⠐⣯⠁⡄
⠀⠀⠀⠀⠀⠀⠁⢻⠀⠀⠀⠀⠀⠰⣿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠠⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠃⠀⠀⠀⠈⢾⣿⣽⡞⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠉⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢘⣦⠾⠁⠀⠠⠐⣦⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢠⣤⢞⠀⠀⢀⠈⡄⢠⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠔⠋⠀⠊⠀⠀⠀⠀⠀⠀⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡟⠋⠉⠁⠀⠀⠈⠀⢑⡺⡍⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⠈⠰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣨⠁⣉⠀⠀⠀⣶⡶⠀⠀⠀⡇⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠀⠀⠆
⠀⠄⠀⠀⠀⠀⣀⠀⡦⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡃⡿⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠀⠀⠀⠾⡷⠄
⠀⠂⠀⠀⠀⠀⠈⢰⡖⠚⠀⠀⠸⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣾⠆⠄
⠀⠂⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⡂
⠀⠁
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⢀⢋⠀⠰⠄
⠀⠀⠀⠀⠀⠀⢀⡀⢤⠐⢒⠊⠁⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠒⠴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⢁⡈⡒⠀⠠
⠀⠀⠀⢠⠊⢹⠀⠀⣈⠠⠼⠐⠲⠄⠤⠆⠀⠀⠀⠀⠀⠑⠀⢀⠀⠂⠀⠀⡀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠛⠉⡟⠀⢀⠈
⠀⠀⠀⠈⠒⠊⠉⠁⠀⠀⠀⠀⠀⠧⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢤⠅⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣩⠀⠙⡤⠤⡤⠤⠤⡤⠤⠤⠄⢀⡀⣀⠀⠀⠠⠠⠤⠄⠐⠒⢌⡀⠀⠀⡀⠄⠀⠀⠠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠨⠏⠌⠙⠒⠏⠉⢳⠉⠁⠈⠁⠉⠁⠐⠒⠒⠘⠋⠥⠹⠗⡠⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢝⡐⠒⠒⠂⢀⠼⠦⠴⠦⡤⠤⠤⡤⠤⠤⠤⠶⡂⠐⠒⠒⣂⠯⡼⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠐⠢⣔⠁⠀⠀⠀⠀⣈⠗⢏⡀⠀⠀⠀⠀⠈⣢⠒⠉⠀⠀⠘⡇⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠄⡀⠀⠠⠊⠀⠉⠢⢄⡠⠊⠁⠀⠀⠉⠢⡀⡠⠔⠉⠀⠑⢄⠀⠀⣀⡆⠀⠀⠀⠀⢄⠀⠀⠀⡄⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡑⠣⣀⠀⠀⢀⠊⠑⠦⡀⠀⠀⢀⠴⠋⠢⡀⠀⠀⢀⡨⢶⠉⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠑⣔⠁⠀⠀⠀⠈⢢⣔⠁⠀⠀⠀⢙⣤⡔⠁⠀⠀⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡶⢄⠀⠀⢀⠊⠀⠑⢄⡀⠀⡔⠁⠀⠱⡀⢀⠔⠁⠀⠑⡄⠀⠀⡠⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠉⢢⡃⠀⠀⠀⠀⡝⠪⡀⠀⠀⢀⠜⢣⠀⠀⠀⠀⢘⡔⠈⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠃⠈⠒⢄⢀⠌⠀⠀⠈⢢⡔⠁⠀⠀⠱⡀⣠⠞⠁⢡⢀⠜⢃⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⡳⡀⠀⠀⠀⠜⠢⠀⠀⠀⢀⣽⠀⠀⣀⠔⠁⠀⠈⠅⠂⠅⠀⠀⠀⠨⠅⠂⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠬⠦⢀⣎⣀⣀⣱⡀⠠⠎⠄⠓⠉⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣠⠄⠚⠂⠀⢺⡆⠐⠃⠀⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⠀⠀⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠙⠊⠙⠃⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡴⠛⠉⢻⠓⡶⠤⠄⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠐⠇⠙⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠱⠃⠀⠀⠀⠀⠳⠤⢀⡏⣸⠟⠀⠀⠀⠀⠀⠀⢀⣿⠃⣰⡿⠁⠀⢀⣾⠟⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡂⠀⠀⠀⠀⠀⠀⣸⢡⡿⡶⣠⣾⡖⣰⣷⡗⣼⡟⢡⣟⣴⣧⡖⡼⠃⠀⠀⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢘⣘⣛⣉⣉⣛⣉⣘⣋⣉⠝⢻⡇⢈⢉⣉⣙⡁⠀⠀⠀⠀⠀⠀⢀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣤⣄⠀⣠⡉⢭⣭⠅⣤⠄⣨⡥⢠⠄⣠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⣇⣸⠇⣿⣧⢠⡇⠀⣇⠀⣿⡀⢸⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⡇⣿⣇⢰⡏⣿⠈⡇⠀⡟⠁⣿⠁⢸⡀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠙⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣋⠘⠚⠃⡘⠀⠃⠀⠛⠒⠙⠀⠈⠛⢃⣘⠓⠀⠀⠀⠀⠀⠀⠀⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡸⠀⢠⠎⡹⢀⠤⠤⠤⣄⡀⠀⠀⣠⠋⡰⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡀⠀⠀⠀⠀⠀⠀⡼⠛⠳⠀⣏⠔⡁⠈⣀⡀⣀⢀⡉⠉⣛⠟⠉⠀⠀⠀⠀⠀⠀⢀⠀⡳⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠁⠀⠀⠀⠀⠀⢠⠗⠒⡲⢻⠁⣰⣏⢜⣧⢴⣧⠾⣣⠾⢿⠀⣠⠀⠀⠀⠀⠀⠀⠈⢁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠘⠒⠚⠁⠈⠛⠁⠀⠈⠁⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠀⠞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢦⠠⣄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠢⣞⠀⠀⠀⠀⢀⡀⠀⠶⠀⠀⠀⡠⠻⠆⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠛⠀⢉⠉⠒⠂⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣄⠀⠀⣿⣿⠿⡿⠿⠿⠿⠿⠿⣿⡿⠿⠿⣿⣿⠿⠿⠿⠿⠿⡿⢿
⠀⣏⢁⣦⣽⣏⣁⣐⣶⡞⢹⡏⣱⣶⣂⣈⣽⣯⣄⡈⣽⠀⠀⣿⣿⢲⡿⠻⢂⡶⠿⢯⡉⢳⡆⣶⡏⡩⠿⠿⢆⠺⠻⣷⢺
⠀⢹⡶⡟⣵⠚⢒⣦⡞⣤⣼⣧⣤⢷⣴⡚⠳⣌⢻⢶⡇⠀⠀⣿⣿⡎⣛⢰⢋⣽⣯⠻⣹⢛⠃⢛⢻⡼⢫⣿⣍⢳⣌⡋⢸
⠀⡾⠀⣷⢧⣠⡌⠛⣤⠳⢸⡇⠞⡬⠛⣠⣄⠞⡟⠀⣧⠀⠀⣿⣿⢱⣿⢸⣹⠟⢷⣴⢻⣼⡆⣾⣽⢳⣴⠟⠿⣹⢸⣿⢸
⠀⣇⣻⡷⢚⣭⣑⠲⠾⣆⣾⣷⣴⠷⠒⣪⣽⡒⢿⢟⣹⠀⠀⣿⣿⢾⣌⣩⡽⢶⢯⣭⣁⠻⠁⠹⢋⣩⣭⢷⡾⣭⣉⣸⣾
⠀⣿⡷⠶⢾⣷⡶⠶⣷⣿⣿⣿⣿⣿⠶⢶⣾⡶⠶⢾⣿⠀⠀⣿⣿⠀⣛⣛⡃⢉⣛⣛⠉⠀⠀⠀⠈⢙⣛⣋⠁⣛⣛⡃⢸
⠀⡏⣵⣦⣜⡓⢒⣤⣵⡟⣿⣿⢳⡮⣄⡒⢚⣢⣶⣎⢹⠀⠀⣿⣿⢶⠋⣙⠧⣭⡽⢟⡋⣠⠀⢠⡌⣛⠿⣭⡥⢟⡉⢳⣾
⠀⣧⠉⣷⡴⠻⠃⣀⠾⡠⢸⡇⣄⠧⣀⠺⠟⢦⣧⠁⡿⠀⠀⣿⣿⢸⣾⢹⢫⣄⣽⢿⣱⢿⡇⣿⢿⣸⢿⣅⣬⢻⢹⣾⢸
⠀⢸⡦⣏⠳⣄⣈⠟⣆⠷⢿⡷⠾⣼⢻⣁⣠⠞⣹⢴⡇⠀⠀⣿⣿⡇⣛⢸⣜⠿⠷⣠⢻⣘⡀⣈⣹⢣⡴⠿⢟⣽⢌⡋⢸
⠀⡏⠐⠿⢳⡖⠒⢒⣾⢆⢸⡇⡲⣷⡒⠒⢲⡞⠟⠂⣷⠀⠀⣿⣿⣸⣿⣠⡍⣭⣭⡽⢃⣽⡇⣿⣍⠸⣭⣭⡍⣥⣰⣿⣸
⠀⠛⠓⠚⠛⠛⠛⠒⠒⠚⠛⠛⠓⠒⠒⠛⠛⠛⠒⠚⠛⠀⠀⣿⣿⣤⣭⣭⣶⣶⣭⣭⣭⣥⣤⣤⣭⣭⣭⣵⣶⣾⣭⣥⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣊⡥⠐⢲⠄⠠⡐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠎⠀⢊⢂⡈⠀⡠⠊⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⠄⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⣸⠀⢀⡀⠂⠁⠡⠎⠄⠀⣁⢰
⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣁⣀⣠⣼⠵⢳⡜⢇⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠁⠀⠀⢸⠈⠢⣢⠔⡡⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⣀⠥⠊
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠒⠒⠒⠈⢁⠩⠳⡦⢄⡀⠀⣀⡠⠔⠂⠉⠉⠉⠲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡘⡀⠀⠀⠠⠂⠁⣠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⡄⠀⠁⠀⠡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢁⢁⠀⠀⠠⠀⠀⠀⠀⢀⠸⠁⢀⠀⠀⠀⠀⠀⠀⠀⢘⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠀⠀⠁⠁⠄⠂⠁⢠⠀⢀⠁⠀⠁⠀⠀⠀⠀⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠠⢈⣀⠀⠀⠀⠀⢠⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡰⠀⢀⠆⠀⠈⠓⠀⠀⠀⠀⠂⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⠀⠀⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠰⡀⠀⠀⢀⢀⠂⠀⠀⠀⠀⠀⢀⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢬⡑⠒⠶⠤⠤⠄⠴⠂⢒⣩⠔⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠉⠁⠒⠒⠒⠊⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠂⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢀⠒⢄⠀⠂⣤⠔⠢⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡴⠋⡤⠯⠀⠀⣻⡄⠰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠋⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠷
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⡰⠀⠀⢀⡔⢀⠀⠰⠁⠔⠀⢠⠇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢇⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠐⠀⠈⠀⠃⠀⡆⠐⠀⢠⠇⠀⠘⢰⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈
⠀⠀⠐⢦⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠐⣄⣣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⡉⠉⠀⠠⠠
⠀⠀⠈⠁⣴⣦⡄⠀⢀⡤⢀⣠
⠀⠀⠀⡀⠈⠉⢁⡔⠻⠇⡉⠣⡄⣁⠉⠁
⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠉⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣙⢿⣾⣿⣿⣿⣶
⣿⣟⣴⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⡷⣿⡿⢶⣿⣷⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠁⡀⠰⣿⣍⢛⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡿⢿⣿⣿⣿⣿⣿⣿
⣿⣯⠀⠀⠀⠈⠆⠙⠿⠿⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡠⠿⠛⠛⢻⡿⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⢠⣄⠀⡴⢠⡄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⠀⠒⠂⠚⠉⠁⠂⢀⣿⣿⡆⠀⡏⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠤⠶⠒⢂⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣠⠤⠤⠤⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⢀⠀⡠⠤⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⠈⣭⣿⣥⣤⠀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠛⠂⠙⠉⠉⠀⠠⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢤⣤⣰⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢊⣿⠖⠳⡀⡴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣡⠿⡃⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⢹⡆⠀⠀⢷⣏⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⣠⣄⣤⣤⡾⢻⣦⠌⢀⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣟⣻⣿⡿⣿⣧⣾⡿⠋⣷⣶⡜⠁⠐⠁⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣉⣩⣭⡛⠿⣍⣁⠀⣿⠟⣿⣷⣴⠀⠀⠀⠀⠀⠆⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣹⣿⣿⣟⣥⣶⣶⣤⣙⠛⠿⠷⠾⢿⣿⢿⣷⠂⠀⢀⣄⣀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⣿⣿⣇⣾⣿⣿⣿⣶⣦⣤⡐⠚⢋⣈⣿⣶⣾⣿⡿⠋⠀⠀⡆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣦⣽⣿⠃⣿⣿⣿⣿⣿⣿⠾⠿⣿⣷⣆⣉⣹⣿⡿⠟⠋⣡⣤⣶⠔⠀⠀⠀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⣀⣼⣫⣾⣿⣋⣼⣿⣿⣿⣿⣿⡳⣄⠀⠀⢉⣉⣻⣿⣿⣷⣾⣿⣿⣿⣷⣦⣶⡗⠛⠽⠛⠁
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣷⣜⣦⣤⣄⠉⣻⣿⡿⢻⣿⣿⣭⣍⡙⠛⠭⠀⠀⠒
⠀⠀⠀⠀⠛⣿⠀⣸⣿⣿⣿⡿⠋⠀⢉⣀⣿⣿⣿⣿⣿⣯⣿⣿⣛⢃⡞⠉⠉⠁⠈⠁
⠀⠀⠀⢼⣿⣏⣴⣿⡿⠿⢿⡧⣄⣀⣹⣿⣿⣿⣿⣿⣿⠏⠿⢿⠛⢿⣯⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠻⠿⡿⠟⠀⠀⠈⠉⢈⣽⣿⣅⣠⣿⠿⡿⣿⣜⡇⠀⠀⠀⠈⠛⠿⢷⣶⣶⣶⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠚⢻⠟⠓⠧⣽⠿⡟⠤⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠃⠀⠀⠰⣿⠆⢰⠸⡄
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⢸⠁⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡾⠒⠠⢤⣀⠀⢰⠆⢀⣀⠀⠀⢻⣷⠀⢀⡔⣱⠋⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡧⠀⠀⠀⠀⣀⠉⠀⠴⠀⠀⣠⣿⡿⠒⣾⠀⣿⠀⠀⠀⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣇⠀⠀⠀⠀⢸⠀⡤⢀⠄⠀⠈⠉⢹⠀⢻⡀⢿⡀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⡀⠀⠀⠀⠸⡄⠉⠁⠀⠀⠠⠂⠀⠀⢸⡇⢸⡇⠀⢸⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠧⠤⠄⠒⣏⣈⣆⠀⠀⡠⢦⣀⠀⠀⠀⣧⠸⡇⠀⣿⣿⣿
⠀⠀⠀⠰⢸⡆⣤⣀⠀⠀⠀⠀⠀⡀⣀⣀⡀⢰⡀⠀⣀⠀⠀⠀⢀⠀⠀⢀⣼⠋⠒⠋⡀⠀⠈⠙⠓⢦⣿⠀⡏⢸⣿⣿⡇
⠀⠀⠀⣧⣿⡇⡇⢻⡆⠀⠀⠀⠸⡇⠈⢷⠀⠀⢀⠇⠈⠣⠀⢠⠎⢀⣴⠏⠁⠐⡀⠀⢹⣀⡀⠀⠀⠀⢸⢰⡇⣿⣿⣿
⠀⠀⠀⢹⣿⡟⢳⣸⣇⢶⣖⠲⠦⠥⠤⢈⣃⣀⣈⣃⣀⣁⣀⣈⠐⠛⠁⠀⢀⡞⠀⠀⢀⡟⠉⠛⠒⣠⡾⠛⢸⣿⣿⡏
⠀⠀⠀⠻⠽⠇⢰⣺⣿⡦⠙⣧⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⡀⠈⢁⣠⣄⣤⣴⡶⠿⠟⠛⠁⠀⣼⣿⣿⠃
⡰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣉⣛⣂⠉⠙⣒⠒⠼⠷⠦⣄⣀⡀⣿⢠⡇⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⠀⠀⠀⠀⠀⢸⡄⢀⣀⣀⣀⡀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠈⠀⡄⢠⠀⢀⠀⡀⠀⠇⣀⣀⡀⡤⠤⠤⠀⠙⠀⠀⠀⠀⠀⠀⠀⢿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⡇⡞⠀⠘⠀⠁⡄⢰⠀⠀⠀⠀⢧⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣇⣀⣀⣀⣀
⠀⠐⠄⠃⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠸⡄⠇⠐⠶⠤⠤⣤⣀⣀⠀⠸⠀⣿⠀⠀⢀⣀⣤⡤⠤⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠆⠀⠀⠰⠀⠀⡌⠉⠉⠉⠉⠉⠉⠉⠉⣱⠀⠁⠀⠀⠀⠀⢗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠚⠴⠿
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢨⠁⠂⠀⢠⡇⠀⠀⠀⠀⠡⠀⠀⠲⢦⠄⠀⠀⠀⠀⠶⣾
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠀⠀⠘⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠂⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠌⠀⠀⠀⠑⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠄⠄⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠰⠀⣀⣄⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡝⠃⠀⠀⡀⠰⠂⢀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠃⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⠁⠆⠀⠀⠠⠄⠟⠀⠀⠀⠀⠀⠀⠀⢩⠄⠀⠠⡝⠓⡐⠀⠀⠀⠀⠀⠀⠖⠄⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⡦⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉
⣿⣿⣿⠿⣿⠿⢿⣿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣮⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⠀⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣇⠀⢸⣿⡇⢸⣿⡇⠀⢀⣾⡿⠁⢸⣿⣿⣿⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⢀⣿⡿⣿⣇⠀⠀⢸⣿⣿⡀⢸⣿⡇⢸⣿⡇⢀⣾⡿⠁⠀⢸⣿⡇⠀⠀⣺⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⢸⣿⡇⣿⣿⠀⠀⢸⣿⣿⡇⢸⣿⡇⢸⣿⣧⣾⡿⠁⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣄⣿⣿⠀⠀⢸⣿⠃⢿⣿⠀⠀⢸⣿⣿⣿⢸⣿⡇⢸⣿⣿⡿⠁⠀⠀⠀⢸⣿⣧⣀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠿⣿⣿⠀⠀⣼⣿⠀⢸⣿⡄⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⡅⠀⠀⠀⠀⢸⣿⡿⠟⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⣿⣿⣿⡇⢸⣿⣿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣷⠀⢸⣿⡇⢹⣿⣿⡇⢸⣿⡏⢻⣿⡄⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿⠀⢀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣿⡇⢸⣿⡇⠀⢻⣿⡄⠀⢸⣿⡇⠀⠀⢻⣿⣆⣠⣿⡿⠀⣿⣿⣴⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠿⠿⠀⠿⠿⠀⠸⠿⠃⠀⠀⠿⠿⣤⠼⠿⢷⣿⣿⠿⠇⠸⠿⠇⠀⠀⠻⠿⠀⠸⣿⣷⠄⠀⠀⠛⠿⠿⠛⠁⠀⠿⢿⣿⡿⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⡟⠛⠛⠻⠂⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢠⡞
⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣰⣿⠇⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣼⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠏
⢠⣾⡿⠉⠉⢹⣿⣷⣄⠀⢻⣿⠶⣿⣿⡟⠀⠀⣠⣾⡿⠉⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⠋⢻⡄⠀⣿⣿⠁⠀⢠⣿⣿⠀⢰⣿⣷⠀⠀⣿⣿⡇
⣿⣿⠇⠀⠀⣾⣿⣿⠀⠀⣾⠀⢀⣿⣿⠁⠀⣰⣿⡟⠀⢰⣿⣿⡇⠀⢀⣿⣿⠀⠀⠀⠀⣼⣿⡷⠶⠾⠃⣸⣿⡿⠀⢀⣼⣿⡇⠀⣾⣿⣿⠀⠀⢿⣿⡇⠀⠀⢀
⠹⣿⣆⣀⣼⢿⣿⡿⠀⣸⠇⠀⢸⣿⣿⠀⣴⣿⣿⣇⣠⠟⢻⣿⣇⣠⡾⢿⣿⡄⠀⢀⣴⢿⣿⡇⠀⣀⡼⣿⣿⡟⠛⠋⢿⣿⣇⣼⠛⣿⣿⠀⣠⡿⣿⣧⠀⢠⡾
⠀⠈⠙⠋⠁⣸⣿⣧⠾⠃⠀⠀⠀⠻⠿⠛⠁⠈⠛⠛⠁⠀⠈⠻⠿⠋⠀⠈⠻⠿⠿⠋⠁⠈⠻⠿⠞⠋⠀⣿⣿⣿⡄⠀⠘⠿⠟⠁⠀⠹⠿⠿⠋⠀⠙⠻⠿⠋
⠀⠀⣀⡴⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢿⡀
⠀⣴⠋⠀⢀⣿⣿⣤⣤⣤⣤⣀⠀⠀⠀⣤⣤⠀⠀⠀⣠⣤⣤⣄⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⢀⣀⣤⣄⡀⣿⣿⣤⣼⣧⣤⠀⢠⣤⣤⣤⣤⣀
⢸⡇⠀⠀⣸⣿⠃⣿⣿⠿⠿⣿⣷⠀⠀⣿⣿⠆⠀⠀⣿⣿⡿⠿⠀⢠⣿⣿⠿⢿⣿⡆⠀⢠⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⡿⠿⢿⣿⣷
⠘⢧⣀⣰⡿⠃⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⡄⠀⠀⣿⣿⡇⠀⠀⣾⣿⡇⠀⠸⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⠻⣿⣿⡟⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⢹⣿⣧⠀⠰⠿⠿⠀⢸⣿⣧⠀⠸⠿⠿⠀⣼⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡿⠀⠀⣿⣿⡇⠀⠀⣿⣿⣇⣀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⣿⣿⣇⣀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⣦⠀⠀⣿⣿⡇⠀⠀⣿⣿⡿⠿⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⣿⣿⡿⠿⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⠙⣿⣿⡄⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣴⣶⡆⠀⢸⣿⣧⠀⢴⣶⡆⠀⢸⣿⣧⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⢿⣿⡇⠀⢸⣿⣿⠀⢸⣿⡇⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⠃⠀⣿⣿⣷⣦⠀⣿⣿⣷⣦⠀⠘⣿⣿⣶⣾⣿⠇⠀⠘⣿⣿⣶⣾⣿⠇⠀⣿⣿⣷⣶⠀⢸⣿⣷⣶⣾⣿⡿
⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣤⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣥⣤⣤⣤⣬⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢺⠟⢟⠀⠤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠆⠀⠀⠀⢀⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⡉⠉⠂⢰⠤⠐⠓
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠃⠀⠠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⢀⠀⠀⠀⠠⡀
⠀⠉⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡠⠀⠤⠶⠈⠊⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⡆⢀⡀⠑⠠⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢦⠧⢄⠉⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⢠⠀⡀⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠠⠃⠈⠑⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡶⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡻⠻⠆⠀⠀⠀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⠀⠀⢀⣠⣆⣠⠀⠀⠃⠀⢀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠁⠀⢠⡋⠉⠹⢃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠆⠂⠀⠺⢿⢇⡀⡀⠀⡸⠿⠛⢀⡿⢶⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢻⡃⢠⠇⠀⢠⠂⠀⠠⠀⠀⠀⣴⣿⠟⠲⣄⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠏⠀⠀⡇⠀⢠⣿⠟⠻⠀⠀⢹⠆⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⣄⠀⠘⢀⡄⠀⠃⠀⠉⠉⠀⡴⠞⠀⠀⠀⠀⢠⡆⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠞⢷⡛⠱⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐⠿⠃⣀⣸⡟⠦
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡄⢲⡆⠀⢠⣷⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡎⠑⠇⠀⢘⠉⠘⡇⢢⣿⡶⠊⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⡘⢾⡄⠀⣏⡼⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⡄⡟⢀⣶⣇⠀⠀⠀⠲⢀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⡀⢠⡄⢱⠙⡄⢸⣿⢋⡀⣘⣀⠀⠘⠛⠛⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣾⡇⢸⡄⠁⠀⡇⢈⠷⠛⡟⢃⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⣤⣾⣶⡷⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠈⡇⣸⣷⣷⡿⠟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⢳⡏⣿⠥⣶⣶⣽⣿⡴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠠⣄⡀⠀⠀⠀⣸⠇⣿⣠⣨⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠙⢦⡀⠀⣿⡔⡿⠃⠛⠛⢿⣼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢿⡆⢿⡿⡇⠀⢀⣀⡀⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣿⠾⠃⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠦⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠉⢠⣄⣀⠀⠀⠀⠀⢱⣼⣷⣾⠄⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡀⢤⣚⣿⢿⡟⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠤⣄⡀⠀⠀⠀⠀⠀⢈⡄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣿⣿⣻⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠂⠄⠐⠰⢄⣀⢀⡠⠀⠠⠉⠀⠐⠀⠀⢹⢿⣿⡿⠄
⠀⠀⠀⠀⠀⠸⠅⠀⠀⢀⠄⠉⠀⠀⢰⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠃⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠀⠀⠸⡺⣄⠀⡀⠀⠀⢃⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠤⠄⣀⠈⠻⢿⣾⣷⡤⠂⠁⠀⠀⠸⠁⠀⣤⣀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⡠⢮⡿⠁⡀⠀⠠⠬⠿⢻⣿⣿⢷⡦⠤⠖⠚⠁⠙⢷⡏⠿⠛⠀⠀⡁⠀⠐
⠀⠀⠀⠾⠁⠉⠀⠀⠈⠁⠀⠀⡄⢀⣨⠶⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠜⠃⢨⠀⣠⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠌⠀⢀⡏⢓⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠠⣿⡃⠀⠀⠑⠒⠀⠒⠂⠁⠀⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠠⠄⠀⠀⠈⠀⠛⠐⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣶⣶⣶⣦⣤⠀⠀⠈⠙⠲⠤⠰⠘⢀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⣾⠇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠛⠁⠀⢿⡖⣦⢀⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⢀⣀⣶⣦⡄⠀⠀⠀⠀⠈⢿⣿⣷⣦⡄
⢠⣤⠄⠀⠀⠀⠀⠀⠻⠟⢀⡀⠀⠀⣤⣄⡀⠀⢈⣾⣿⣯⣵⣶⣿⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣴⡀⠀⠈⠉⠛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠓⠙⠛⠛⠷⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠻⠟⠓⢲⣦⣤⣤⣤⣄⠈⠙⠛⠁⠈⠉⠉⠉⢀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠷⠶⠿⣿⣿⣿⣧⣄⢀⡀⠁⠀⠀⠀⠀⢀⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠛⠛⠛⠋⠉⠉⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⢿⣿⣯⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢫⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠘⠃⠀⠤⠂⠀⢀⠀⢀⡀⠚⡀⠀⠁⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⠠⡀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠒⠀⠸⡀⠄⠁⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠀⠠⡆⠀⠀⡀⠊
⠀⠀⠀⠠⠄⠂⠀⠀⣂⠂⠀⠁⠁⢀⠤⠄⠀⠂⣀⠄⠀⠢⢄⠀⡴⡄⠀⠀⠀⠉⡉⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠨⠃⠀⠀⠂⠀⠒⠁⠀⠀⣂⠀⠀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢠⡀⠀⠀⢀⡀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⠄⣀⡧⢐⠠⠁⠀⠊⠀⠁⠀⠀⠈⠀⠑⠂⠈⠄⡂⢼⣀⠠⠠⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⢚⣡⠖⡀⠂⠀⠀⠀⣠⡴⣤⡀⠀⠀⠀⠀⠀⠐⢀⠲⣌⡓⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣫⠚⠛⠉⠀⠀⣤⡄⣤⡀⣼⠇⠀⢹⣥⡀⣠⡄⠀⢠⡄⠀⠀⠀⠉⠛⠓⣝⣆
⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠀⠀⠀⠀⣿⡇⣿⣷⣿⠀⠀⣿⣻⡇⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⢀⢨⢿
⠀⠀⡄⠤⢀⠀⡜⣰⠂⠄⠀⠀⠀⠀⠿⠇⠿⠙⠿⠀⠰⠿⠙⣷⠿⠷⠾⠺⠿⠾⠀⠀⠀⠀⠠⠐⣆⢳⠀⠀⠤⢠
⠀⠂⡖⡕⢀⣸⢱⢣⠡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠙⠷⠿⠀⠀⢀⣀⠀⠀⠀⠀⠉⠌⡜⡎⣇⡠⢪⢲⠐
⠀⠀⠀⣧⡆⡇⣼⠏⣁⣀⣀⢀⡀⣠⢀⣄⣀⡀⣰⠏⣠⣟⢀⣤⣄⠀⢀⣤⡙⣩⣷⡀⣀⣠⣠⣌⠹⣧⢸⢰⣼
⠀⠀⠀⢿⣷⡷⠃⠀⠈⣻⠉⢸⣧⣿⢸⡏⢸⣿⣿⢸⣟⢋⣈⣿⣏⠀⣿⡏⣃⢾⡿⣿⡟⢸⡷⠆⠀⠘⢾⣾⡿
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠛⠂⠘⠃⢿⡘⣛⠛⠋⠛⠈⠛⠛⠋⠛⠋⠀⠙⠛⠛⠛⠃⠛⠃⠘⠛⠃⠀⠀⣸⣿
⠀⠀⠀⠀⠹⡇⢓⠢⢔⠀⠀⠀⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⡢⠔⡂⢸⠏
⠀⠀⠀⠀⠤⢿⣸⣸⡞⠀⠀⠀⣶⣶⡆⣶⠄⣶⠀⣶⣦⢰⣦⢸⡏⣶⣴⡆⣶⣶⠀⠀⠀⠀⠀⢳⣇⢇⡿⠤
⠀⠀⠀⠀⠂⠣⣧⢻⡁⠀⠀⠀⠀⣯⠀⣿⠶⣿⢸⣯⣿⢸⣿⣿⡇⣿⣿⠀⣽⣿⡀⠀⠀⠀⠀⢈⡟⣼⠜⠉
⠀⠀⠀⠀⠀⡄⣌⣷⣳⡀⠰⡀⠌⡛⠀⠛⠁⣿⢚⡃⢹⡞⢣⡙⠃⠛⠘⣧⣙⡋⢀⠠⢀⠆⢀⣞⣼⣡⢠
⠀⠀⠀⠀⠀⠓⠊⠣⡙⢏⠙⢎⠐⢞⠀⠀⠀⠙⠿⠏⠀⠙⠛⠁⠀⠀⠀⠈⠛⠁⡳⠂⡱⠋⡹⢋⠔⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⢤⡙⠾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠷⢋⡤⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠓⠤⣉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣉⠤⠞⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⣈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣧⠀⡜⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⢻⣷⡇⢰⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⡤⠎⢻⣿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢇⠀⠀⢿⣧⠀⢷⣄⠘⣇⠀⠀⣀⡀⠀⡠⠛⡧⢀⠘⢇⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣼⡟⠇⠀⠀⠀⠀⢩⠀⠀⠉⢷⡼⣄⠀⢹⣤⡞⢁⠀⠉⠸⡆⠀⠩⣿⣷⢲⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢫⣙⡿⢫⢡⠐⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⠂⠀⠉⠉⠻⣄⠀⠀⠹⠀⠀⠘⢧⡙⠻⢿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠏⠀⠀⣻⣧⣝⡀⠀⣀⣀⠀⢀⣥⠀⢀⡼⠿⣦⣤⠀⠀⢈⠓⢄⠀⠀⠀⠀⠀⠛⠲⠀⠙⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠁⠀⠀⠀⠉⠉⠉⠉⠲⠛⠉⠋⠉⠀⠁⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠒⠒⠂⠀⠀⢀⡉⠀⠇⠀⠘⠀⠀⠀⠈⠉⠙⠁⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⣀⡤⠖⠛⠋⠁⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠙⠒⠤⣄⠐⠒⠀⠀⠉⠻⡆
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣀⣀⣀⣀⣀⣙⣉⣀⣉⣁⣀⣀⣀⣀⣤⣤⡤⠞⠁⠀⠀⢀⣀⣴⠇
⠀⣘⠳⣦⣤⣄⣀⣀⣀⣀⡀⠀⠀⢀⡉⠉⠉⠉⠉⠉⠉⠉⠉⣀⣀⣀⣀⣠⣤⡤⠴⠶⢶⠿⠛⠉⠀⠈
⠀⠈⢩⠘⠓⠶⠬⠭⠍⣉⣭⣉⣉⣉⠉⠀⣈⣍⣛⠉⠛⠋⠉⣩⡍⠉⠀⠀⠔⠀⠀⠀⠀⢀⣀⣠⡄⠂
⠀⠀⠸⠂⠠⢤⣄⣀⣀⣀⡀⠀⠉⠀⠀⠐⠁⠀⠀⡀⠀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠖⠛⠋⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠂⠀⠃⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⠀⣹
⠀⠀⠀⠀⣷⠀⠈⠀⠀⠀⡠⠤⠤⠄⠀⠀⠀⢀⣠⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿
⠀⠀⠀⠀⢿⡑⠢⠄⢀⣐⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠈⠁⠂⠀⠆⠀⠀⠀⢀⡀⠀⠀⢈⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣨⠓⠚⠛⠶⢤⠠⠤⢤⣤⡤⠶⠦⢶⣶⣖⣉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⠀⠀⣠⠊⠀⠀⠀⠐⠶⠀⠀⠉⠙⠢⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢻⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣯⠀⡀⠀⢀⡀⠹⣆⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠁⠀⠀⠇⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠄⠲⠀⢈⡓⣤⡤⣠⡤⠶⠀⠀⠀⠈⠀⠈⠁⠂⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠰⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⢠⡾⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠋
⠀⠀⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⠶⠦⠤⠴⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠶⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠒⠂⠒⠉⠉⢡⠀⠐⣾⣿⣿⣿⣮⣵⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢄⣀⠂⢿⠉⢰⡇⢻⣿⣿⣿⣿⡄⠀⠙⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠠⢀⠉⠒⠀⢸⣷⠸⣿⣿⣿⣿⠇⠀⣀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢲⡦⢤⣄⣀⢠⡙⣆⠀⠸⣟⣧⡙⠿⣿⠏⠀⠀⣽⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢿⣳⣌⠳⣄⢳⡄⠙⢻⡿⠁⠀⢀⠔⣠⡾
⠀⠀⠀⠀⠀⠠⢄⣀⣀⠀⠀⠈⢙⢿⣷⣜⢦⡻⣄⠉⠠⠠⠚⠀⢘⡽⠀⠀⠀⠀⠀⠐⠄⢀
⠀⠀⠀⡜⠀⠀⠀⠉⠙⠻⢓⡦⢤⣀⠉⠻⣧⡐⣿⣶⡀⠀⠀⠛⠉⠀⠀⠀⠲⢖⣂⣀⣀⡘⣆⠰
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡈⠻⠮⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣻⣽⠟⠓⠲⠤
⠀⠀⠀⠀⢲⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈
⠀⠀⠀⠀⢈⢈⡉⠀⢀⠀⠀⡀⠠⢀⡀⠀⠓⠲⢬⢙⠿⣦⡀⠁
⠀⠀⠀⠀⠀⠱⡻⠷⢮⣭⣆⣀⣱⣦⣌⠓⠦⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠛⠻⠿⢻⡿⢾⣿⣷⣶⣄⡉⠀⠻⣄⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠴⠶⠶⢦⣤⣄⣠⡄⠀⠈⠙⠛⠉⣈⣧⢀⠀⠘⠁⠄⠀⠀⠀⠉⠉⣁⣒⠦⡌⢐⣒⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠁⣀⠀⢠⡤⣴⣤⣶⣶⣾⣧⣴⣿⣶⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣤⠾⣟⠖⢠⡻⡴⢦⣾⡟⠉⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢠⣦⣤⠤⢄⣠⣠⣦⡴⠖⠿⠛⠣⠾⠛⠋⣀⠀⡾⢰⣿⡷⢙⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⢄⡴⠙⣾⣷⣿⣻⠏⠁⠋⠉⢟⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢺⣿⣰⣟⠛⢿⣏⠀⠀⠀⠁⢌⡣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣰⣿⣵⣾⡏⠀⠈⣿⣆⠀⠀⠀⢀⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡼⣴⣿⣿⣿⡟⠀⠀⠀⠘⣿⣤⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣥⣶⣿⣾⣿⣾⣿⡟⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⢀⣤⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠔⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢀⣴⠛⠉⡷⠀⣠⠶⠒⠢⢀⣿⠀⠀⠀⢀⡌⠀⠀⣀⠀⠀⠠⢤⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢀⣾⠃⠀⠈⠀⣼⠃⠀⠀⠀⣼⠇⠀⠀⠀⣾⠁⠀⢰⡟⠠⠀⢠⡟⠀⣴⠋⠀⠠⣿
⠀⠀⠀⠀⠀⠰⣉⣿⢦⠀⠀⠻⣄⣀⡠⠂⢻⣀⠀⡠⠀⣿⠀⠀⠀⠸⣇⡠⠂⢸⡷⠁⠀⣾⠁⠠⣏⢀⡔⠁⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⡀⢀⠔⠀⠉⠀⠀⠀⣀⡀⠸⣇⡴⠁⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣴⠞⠁⠹⠀⠀⢀⡴⠲⡄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡤⢤⣀⡾⠁⢀⠔⠁⠀⠀⡞⠀⠀⠃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣴⠞⠋⢙⢦⠀⢎⠀⠀⠑⣄⣇⣀⠈⢢⡀⣴⠏⠀⢀⣿⠡⠒⠁⠀⠀⠀⠀⢇⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⢸⡇⠀⡰⠃⢸⡆⠀⣿⠋⠀⡟⢼⠁⠀⣸⠀⣿⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉
⠀⠀⠀⠀⠀⠈⠳⠋⠀⠄⢺⠃⠀⠹⠶⠞⠁⠈⠛⠛⠁⠀⠈⠛⠁⢻⣄⣠⠄
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠈⠡⠀⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣤⣦⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢏⡜⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠈⠙⡆⠈⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡼⣄⣀⡂⠀⠀⠀⣇⠀⠀⠠⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣡⠤⠶⠚⣟⠁⠀⠀⢠⠝⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠾⠵⠒⠶⣅⣀⢸⡆⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠈⢣⡀⠀⢰⠀⠈⠉⠑⢢⡤⠂⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡤⠀⠀⢀⠀⡀⠈⠳⡀⠀⠀⢸⡆⠀⠀⢁⢀⠀⠀⠇⠀⠀⠀⡔⡠⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⠀⠀⠀⠀⣀⣀⡀⠐⠄⠀⠀⠙⠀⠐⢌⡴⢂⠀⠀⠀⠀⣐⣎⠋⠀⠉⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠢⠀⠀⠸⡄⠀⠐⠀⠀⠈⢢⠹⢦⣾⠠⠂⠀⠀⠠⠀⠐⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠁⢀⡊⠀⠀⡄⠤⠆⠀⣤⡀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⢠⡀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠⠶⠂⠒⡻⢆⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣷⡋⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠐⠚⠞⠁⠀⣈⣀⠤⠤⠬⢶⡀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠆⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⣀⡤⠂⠉⠁⠀⡀⠠⠀⠐⠳⠆⣀⠀⠀⡇⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢀⠄⡜⠱⠶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⡄⠤⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠖⠉⢶⠞⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⠦⣴⡟⠠⠄
⠀⠀⠀⠀⢀⠀⠂⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠆⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡈⣤⠀⣤⣄⡂⠀⠀⠀⠀⠀⢀⢤⠔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠙⠉⠉⠛⠲⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠎⠁⠈⣇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣤⠶⠶⠚⠉⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣀⣰⠇⣴⡶⣿⣶⣄⠀⠀⠀⠐⢾⡵⣶⣬⠇⠀⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾⠉⣯⠀⠀⢰⡦⠀⠘⣧⠀⠀⠀⠠⣶⠀⠠⣷⠏⠉⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡄⢿⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢰⡏⠀⣸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠛⢻⡆⠀⠀⠀⠀⠘⠳⠖⠀⠀⠀⠀⠀⠀⣿⠞⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⡴⠛⠋⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠛⠲⣤⣀⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠿⣍⡁⠀⠀⠀⠀⠀⠉⠛⠦⠤⣤⠤⠶⠛⠁⠀⠀⠀⠀⠀⢉⡹⠟⠛⠦⣤⡀
⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠈⠛⠶⢤⣀⣀⠀⢀⣤⣄⣀⣀⣠⣤⡀⠀⢀⣀⣠⠤⠞⠋⠁⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣧⠙⣯⠻⣏⣼⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡌⢷⡈⢧⡙⣧⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢾⡟⣆⠳⣄⠻⠜⠷⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡼⠁⠀⠀⠹⣜⠳⡝⠃⠀⠀⠀⠈⢧⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣸⠇⠀⠀⠀⠀⢈⣷⣄⡀⠀⠀⠀⠀⢸⣇⡀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡾⠋⠛⠶⣤⣀⣠⠞⠋⠀⠀⠙⠳⣆⣀⣤⠶⠛⠙⢶⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠀⠀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⠀⠀⠀⠙⠷⣼⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢹
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢇⠀⠀⠀⠀⠀⠀⣠⡴⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣄⠀⠀⠀⠀⠀⢀
⠈⠳⣤⣤⣤⠴⠞⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠳⢦⣤⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠲⠶⢶⡶⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⣤⠴⠶⠶⠶⡶⠶⠶⢶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣼⣤⣤⣤⣤⣧⣄⣸⣿⣀⣀⣠⣿⣤⣤⣤⣤⣷⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡹⠁⠀⠀⠈⣟⠀⢹⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠇⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣉⡠⢼⣖⢤⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣐⠄⢻⣀⣒⣘⠢⢔⠘⡁
⠀⠀⠀⠀⠀⢀⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠛⠀⠠⢭⢿⠅⠀⠘⢃⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢰
⠀⠀⠀⠀⠀⠀⠋⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣄⡀⠈⠻⠃⠀⠰⣿⣷⣴⣿⣿⠀⠀⠻⠋⠀⢀⣴⣷⢄
⠀⠀⠀⠀⠀⢸⣷⡦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠻⠿⠀⠀⠀⠀⠀⠹⣿⣿⡿⠃⠀⠀⠀⠀⠸⠿⠛⠁⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠷⠀⠀⠀⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠘⠀⢰⣿⠀⠀⠀⠀⢀⠀⠀⠙⠛⠃⠰⣶⠶⣄⠀⣶⠖⠂⣶⠀⠀⠀⣼⡇⠀⣷⠀⣾⠈⠛⠉⠀⢀⣅⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣷⣤⣀⠀⠀⢠⣿⢀⣿⠀⣿⠀⠀⣿⠀⠀⠀⣿⣇⠀⢸⣦⡏⠀⠀⣠⣶⣿⠟⠀⠀⠀⠀⢀⣤⣄⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⢂⣀⡄⠀⠉⠛⠛⠀⠀⠀⣿⣟⠁⠀⣿⠓⠀⣿⠀⠀⢰⡏⣿⠀⠀⣿⡇⠀⠀⠉⠉⠀⠀⠀⣄⣠⠀⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⡿⠀⠀⠰⣷⣦⠄⠀⣿⢹⡆⠀⣿⠀⠀⣿⠀⠀⣸⣧⣿⡀⢀⡿⣷⠀⣄⣤⣶⠆⠀⠀⢿⠟⠀⣤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣾⡿⠃⠀⢀⣤⡀⠀⠀⠉⠀⠀⣿⠀⣿⠀⣿⣀⡀⣿⣀⡀⣿⠀⢸⡇⢸⡇⠘⣧⠈⢟⠁⠀⠀⢀⣀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢸⣿⡇⠐⠁⣶⡤⢀⠁⠀⠈⠀⠉⠉⠁⠉⠉⠁⠁⠀⠀⠁⠀⠀⠀⠀⢶⣦⠓⠀⣠⣿⡟⠀⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣉⡀⠀⠀⠸⠿⠁⠀⠈⢐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣎⠀⠀⠸⠿⠀⠀⠀⠉⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠃⠀⣠⣄⠀⠀⠀⢀⣼⡟⠀⠀⠀⣾⡇⠀⠀⠠⣶⡀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⢰⣿⠏⠀⠀⠀⠀⠉⠁⣰⣦⠄⠙⠁⠀⠀⠀⡿⠧⠀⠀⠐⣿⣧⠀⠀⠀⠀⠀⠘⢿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠁⠀⠀⢠⣶⠄⠀⠀⠻⠋⠀⠀⠀⠀⢸⣦⡤⢀⡀⠀⠀⠈⠛⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠿⢿⠀⣀⠀⠀⠀⣰⣶⠄⠀⠈⠉⠀⠀⡀⠁⠂⠤⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠁⠉⠛⠲⠆⠀⠠⢠⣾⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣞⣆⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⢟⠶⠶⠦⠶⡪⢕⢆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠾⠟⡸⠀⠀⠀⠀⠀⠀⠀⠇⣮⡾⠋⠈⠒⠒⠀⠙⢷⣵⣡⠀⠀⠀⠀⠀⠀⠀⢇⠻⠷⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⠒⡇⠀⠀⠀⠀⠀⠀⢸⣼⣿⠠⠶⠆⠀⠀⠰⠶⠄⣿⣧⡆⠀⠀⠀⠀⠀⠀⢨⠒⠂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢃⠀⠀⠀⠀⠀⠀⣼⠀⠟⠐⠉⠂⠀⠀⠐⠉⠂⠻⠀⣷⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡀⠀⠀⠀⠀⠀⡇⠉⢇⠀⠀⠀⠀⠆⠀⠀⠀⡸⠉⢸⠀⠀⠀⠀⠀⢀⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢧⡀⠀⠀⠀⠀⢃⠀⠈⢆⠈⢿⡉⢉⡿⠁⡰⠁⠀⡌⠀⠀⠀⠀⢀⡼⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⠉⠑⠒⠚⠠⠤⠤⠵⢦⠤⠤⡴⠮⠤⠤⠔⠓⠒⠊⠉⠉⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠒⠖⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⣀⠤⠤⠤⠤⢄⣀⡤⠖⠓⠒⠒⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠷⠤⠄⣀⡀⣼⠀⠀⠀⠀⢸⠦⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⡖⠒⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠒⡇⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⣧⣴⡏⠉⠉⠉⢹⠀⠀⠀⠀⢸⣴⠃⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠃⢸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢤⡜⠀⠘⠤⠤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⢤⡃⠀⠈⢦⡤⣜⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠼⠎⠑⠒⠤⠤⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⣿⢧⠀⠀⠀⠀⠀⠀⡔⡩⣚⡟⠙⠋⣻⣔⢕⡄⠀⠀⠀⠀⠀⠀⡼⣿⢓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣉⣉⡜⠀⠀⠀⠀⠀⢠⣱⡟⢁⡀⠀⠀⢀⣈⢻⣿⡀⠀⠀⠀⠀⠀⢠⣉⣉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡆⠀⠀⠀⠀⠀⣸⠹⡇⠭⠍⠀⠀⠩⠭⢸⡏⣇⠀⠀⠀⠀⠀⢸⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠀⠀⠀⡏⠒⡀⠀⠀⠀⡄⠀⠀⢀⠒⢹⠀⠀⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢃⡀⠀⠀⠀⠇⠀⠱⡀⠻⣉⣉⠟⢀⠎⠀⡘⠀⠀⠀⢀⡸⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠉⠉⠁⠘⠒⠒⠚⠷⡂⢒⠾⠓⠒⠒⠃⠈⠉⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠒⠒⠒⠒⠒⠲⠯⡉⠈⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠤⣀⢀⡀⣸⠀⠀⠀⠀⣄⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠒⢂⠀⠀⠀⢸⠀⠀⠀⢸⠒⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡶⡾⠁⠀⠁⢹⠀⠀⠀⢸⢶⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⢸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣀⣀⣀⣀⣀⠏⠉⢏⣀⡰⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠑⠍⠂⠤⠌⠱⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠰⢧⡤⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣋⠝⢃⠂⡀⠀⣀⣀⢴⡠⠀⠠
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠤⠤⠀⠁⠀⠈⣴⢤⡀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠱⡛⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣉⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠁
⠀⠀⠀⠀⢀⠈⠉⠀⣬⠧⠀⠀⢤⡄⠀⠀⣠⡄⠀⢀⡀⠀⡀⠀⢤⡄⡄
⠀⠀⠀⠀⠀⠀⠔⠂⠉⢷⡀⠢⠾⠷⠫⠞⠿⠅⣤⢛⠉⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⣠⠴⠈⡈⠀⠄⠲⠄

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠊⠈⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠐⡖⢠⠀⡆⡴⠀⠀⠀⠀⠸⡀⢀⠀⠀⠉⠀⠀⢀⣀⣠⣔⡒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⡇⠈⣆⠁⡗⠀⠀⠀⠀⠀⣧⠃⠀⠀⠀⣠⠞⠉⠀⠣⣀⣨⠆⠀⠐⡀⠁⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⠀⡧⠀⠿⠀⣇⡀⠀⣴⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⢄⠀⠀⠊⠄
⠀⠀⠀⢀⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄⠁⠒⠒⠒⠊⠁
⠀⠀⠀⢸⣷⣀⣸⡇⠀⠀⣠⡤⠀⢰⣢⢶⠆⣾⣧⠋⢠⠛⣶⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⣰⢂⡄
⠀⠀⠠⠚⣿⠁⢸⡇⣼⡇⣿⣼⣧⠾⠃⠺⠴⡿⠻⠤⢠⢲⡿⠄⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠆⠀⠚
⠀⠀⠀⠀⢻⡀⢸⣷⠻⡷⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇⠀⠀⠀⠠⠈⢀⠀⡀
⠀⠀⠀⠀⠈⢧⠎⠋⠀⠀⣴⣷⣷⣦⡄⠀⠀⠀⠀⠀⡀⡆⡇⣿⠀⠏⢸⢸⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⣷⢻⠃⣿⠀⢰⢸⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⠒⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠘⠀⠛⠂⠈⠘⠈⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⢹⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⡕⠀⣀⢠⡹⣶⡀⢂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⣀⢻⡄⢻⡌⡿⠾⠛⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠐⠉⠀⠈⠀⠀⠀⠀⣉⡀⠀⠀⠀⢀⡀⠀⣤⢀⡄⠀⠈⢹⡇⢸⡿⠀⣿⡌⠻⠋
⠀⠀⠀⠀⠀⠃⠀⡄⠘⠄⠀⠀⠀⣼⣫⣶⡀⠀⠀⡿⣸⡇⣿⠞⢃⡟⣵⣆⢧⡼⠧⠞⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂
⠀⠀⠀⠀⠀⠡⠀⠀⠠⠀⠲⡄⠀⠈⠁⠉⠀⠀⠀⠛⣹⠇⠛⠀⠈⠛⠉⠉⠀⠀⡀⣀⢀⡀⠀⡀⢀⡀⣀⡀⢀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠬⠁⠀⠀⠀⠀⢸⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣸⢸⡀⠀⢇⢸⡞⢸
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠁⠘⠀⡖⠠⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⢸⠀⡸⢼⠈⢇⢸⡄
⠀⠀⠀⠀⠀⠰⠃⠘⠁⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢾⣷⡄⠉⠈⠈⠃⠁⠈⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠿⠛⠛⠛⠉⠁⠙⠋⠘⠛⠛⠋⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢀⠐⣦⣶⠶⠿⢿⣿⣶⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⢳⣾⡟⢀⡴⠶⢶⣤⣿⣿⣇⣶⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡄⠀⠀⠀⠀⠀⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢴⡙⣏⠋⠀⠀⠀⡿⡎⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠻⡀⠠⠴⢠⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢵⡈⢿⣦⠶⢆⠄⠁⢉⣤⢿⣿⠃⠀⣀⣀⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠐⠀⠛⢦⡀⠀⠈⠉⠀⠂⣄⣵⡶⠶⣦⣄⡀⢀⣀⠀⡀⠈⠳⡄⢣⣄⡀⠀⠐⠨⠕⠋⢻⡴⡉⠉⣿⠿⡬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠈⠛⠦⣤⡈⠃⠘⠛⠛⠳⣦⡤⣬⣽⣿⣴⠶⡴⠃⢁⣿⡐⢏⠙⠆⠀⠀⠀⠀⠀⠁⠀⣠⡛⣖⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⢚⣉⣉⡛⠂⠈⠀⣤⡀⠉⣹⣣⠤⠶⣌⡙⣷⠈⢿⠈⣶⣄⣀⢀⣠⠤⣾⡟⠀⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠛⢻⡙⣻⠛⢯⠊⢀⣀⣩⣭⣷⣿⣷⣦⠀⣸⣧⣿⢋⣩⣤⠶⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣤⣾⣿⣷⠀⣿⣿⣾⣷⣴⣿⣁⣴⣀⠸⣿⡿⣿⣿⢻⣿⣿⣿⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣷⣿⣿⣿⠿⣶⠿⣟⣽⣿⡏⢹⣿⠛⠻⣛⠻⣷⢦⡉⣛⠎⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⠛⠍⣽⡟⢸⣿⣿⠛⢣⠘⣿⡝⠳⠾⠷⣌⠀⢹⣏⣾⣃⠀⠀⠀⣠⠴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢼⣟⣿⣤⣈⣢⡈⠳⣍⡶⠾⠿⠿⠋⠁⣱⣿⡄⠀⢈⣍⢠⠉⣶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣸⡛⢮⣥⣌⣙⠛⠷⠶⠿⠿⣿⣿⣤⡀⠀⠉⢻⣶⣿⡿⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⣼⣿⣿⣦⣉⣉⣻⣤⠶⠻⣶⣶⡄⠈⠉⢙⡀⠀⠀⠙⢯⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠠⣴⢿⡟⢻⣄⠀⠀⠀⢈⠀⠀⠀⠀⣀⣹⣿⣿⣿⣧⣍⣭⣤⣶⣿⠿⣿⣿⣾⣆⠀⠹⣦⠠⡀⠸⣿⣻⡄
⠀⠀⠀⠀⠀⡀⠀⣠⣾⡿⡏⢻⡆⠀⠀⠘⡄⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣟⡁⠉⠳⠿⠿⣷⣿⣿⡄⠰⣘⡇⠿⣦⡙⠷⡁
⠀⠀⠀⠀⢰⠃⣰⣿⠋⠀⠸⠀⣷⠀⠀⠀⢻⣆⠀⠛⢻⣿⡿⠛⠀⣿⠀⡙⢿⡄⠀⠢⡀⠀⠹⡎⢷⠈⢯⠱⠆⢹⡇⠀⠄⠠⠴⠆
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣰⠃⠀⠀⠀⠀⠉⠒⠚⠋⠉⠀⠀⢀⡿⢀⠰⠀⢻⠀⠀⢻⠀⠀⡀⢸⡀⢸⡇⠳⠈⣧⣤⣀⣿⡶⠆
⠀⠀⠀⠀⠀⢠⡀⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⡶⠊⠀⣼⣇⡃⢨⡧⠀⠸⠂⢰⢁⣼⡇⠘⠁⡀⠈⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢳⣌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⣤⡿⠛⠁⣀⣜⡿⣟⣡⡞⠃⠀⠀⢠⣿⣾⣿⠳⠀⣸⠃
⠀⠀⠀⠀⠀⠀⡀⠙⢦⡀⠀⠀⠀⠀⠀⢀⢠⢾⠟⢉⠀⠀⢐⣺⣭⣿⣿⣿⠋⠀⠀⠀⢀⣾⠙⢻⠋⣶⠻⡁⠤⣀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠁⠀⠀⠀⠛⣾⣿⣦⣿⡿⢉⡤⢚⣋⠡⢀⣠⣤⡴⠔⠊⢀⠔⠛⠋⠀⢹⣄⠘⣤⡽⠒⠠⠬⠉⠀⢠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⡀⠀⠉⢉⣀⠉⣰⣉⠞⣩⣶⣾⡿⠛⠁⣀⢠⡾⣡⢞⠁⠀⠀⠀⢻⣦⡉⠒⠄⠀⠀⠀⠀⠁⠀⣹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⡏⠀⣿⣃⡼⢋⡟⠉⢠⣄⣼⢁⣿⢰⡏⠠⠀⠀⠀⠀⠀⠙⠧⠙⠶⡤⣴⣦⠈⠠⠂⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣿⠟⢠⡿⡁⠀⠀⠉⣻⣿⣝⢦⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠤⠴⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠁⠀⠀⠀⢾⣿⢛⡀⠛⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢠⡀⠀⠀⢸⠋⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡈⢆⢿⣦⣀⣇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠄⢐⣻⣦⣭⡦⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿
⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠋⠑⣤⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠀⠀⠉⠖⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠧⠤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠚⠋⠉⠀⢰⢯⡼⠛⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⣠⣶⣟⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠔⠋⠀⠀⣀⠀⠤⣄⠀⠀⣀⠠⠄⣾⢇⠉⢿⡿⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠙⠂⠺⠁⠀⡏⠀⠀⢸⠒⡉⠜⢆⣄⠙⠞⡾⡌⣵⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠶⣶⣿⠛⣇⠀⢁⠀⠀⣭⢁⠁⡸⣤⠯⡙⢶⠘⠀⣹⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣟⢳⡐⡿⡷⣄⡀⢰⠂⠃⠀⠱⠘⢆⡙⢦⢡⠇⢸⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢽⡢⠙⢦⠈⠈⠅⢫⠀⠠⠀⡧⣲⡌⢃⠀⠈⡞⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢔⠿⣿⠀⣧⠀⢲⡀⠣⠀⡀⠀⢌⢷⠈⢰⡇⣣⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣷⣝⠃⢤⡀⠲⣄⣈⢦⠺⠦⠁⡆⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣶⣭⣄⣴⣦⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

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



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣽⣾⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⠛⠍⢙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⣰⢀⠁⠁⠘⢧⠴⢦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⠟⣄⠠⠁⢠⠀⣠⠃⠀⠙⣴⣯⣾⣷⣡⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣽⠷⠾⠓⠠⠼⣋⣁⣁⠠⠔⠉⠹⢿⠛⠭⡛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢀⠄⣀⡀⠀⣻⣗⣀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠱⡐⠋⠉⠠⠆⠉⠿⢿⣃⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢿⣀⠀⠈⣁⣠⡶⣶⣶⣏⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡽⠿⠋⠻⣽⣟⣽⣠⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⣤⣶⠘⠟⣧⣷⣚⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⣭⣀⡀⠀⠀⠈⢻⡄⠆⢹⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⣈⡁⠒⢦⣆⣻⠈⠀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠛⣷⣰⣿⣷⠶⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⣽⣿⡽⠐⡏⠉⠃⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡻⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢆⣼⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣧⡘⠀⠄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⠉⣴⣿⣿⡿⠗⠉⢋⠉⠙⠁⠀⠉⣡⣤⣄⢀⡀⠈⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢇⣬⣥⣭⡹⣄⠹⠿⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣿⡿⠃⣹⡷⠀⣿⢾⣿⣿⣿⣟⣷⣶⡄⠀⠀⡈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⣸⠁⡀⠀⣩⣷⡏⣰⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣴⣶⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢸⡟⢾⣿⣏⣠⣥⣼⣿⣛⣟⣛⣿⡟⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇⡀⢹⣿⠋⠁⠀⠂⠙⠀⠉⡛⢿⣷⣦⡄⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⠐⣥⡾⠃⠄⠈⠛⠀⠀⠀⠀⠐⠰⡦⡉⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣙⢤⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⣿⠀⢰⣦⡀⠀⠈⠀⣀⡀⠁⠀⠀⠀⣀⠀⠆⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⢸⣿⣃⠀⠀⠀⠈⠙⠲⠖⠂⢹⣿⢷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠁⢸⣿⣋⡓⣶⡶⠦⣀⡀⢀⣸⡿⣿⣆⠀⢊⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣿⡿⣿⣿⣿⣿⣗⣀⣛⢋⡿⣿⣿⣿⢿⠀⣘⠧⢒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠛⠻⠷⠛⠋⣿⡻⣿⡋⠉⢉⣿⣿⣿⣿⢿⡀⠑⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢻⣷⡷⢠⣇⢨⡦⠂⣠⣄⡀⠀⠉⠉⠘⢣⡱⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣀⡄⠀⠁⠀⠀⠀⠀⠀⠀⣟⡀⢀⣀⣀⡴⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⡿⠾⠿⠶⠀⠀⠀⠸⢦⣿⠿⢿⢟⠁

⠀⠀⠀⠀⣀⣀⣬⣿⣿⢿⣷⡴⣤⣴⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⢷⣿⣿⣿⣿⣧⣰⠆⠀⣀⡀
⠀⠀⣴⣾⣿⢿⢿⣿⣿⣿⣿⣿⡫⢭⣿⣵⣾⣧⣤⡄⢈⠀⢀⣤⣴⣞⣽⣿⣼⠿⣿⣿⣏⡽⠿⡿⢿⣿⣯⣭⡐
⠄⠀⢈⣿⣿⣾⣿⣿⣏⣹⣯⢹⣷⣿⡿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⣿⣿⣦⣿⣿⣿⣿⡿⡿⢿⣿⣿⡿⠃
⢠⣾⣿⣩⠟⢻⣿⣿⣍⣙⣛⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣿
⠀⣿⣿⣯⢀⣺⣿⠿⢿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⡼⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣤
⠀⠼⠮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣏⢹⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⢺⣿⣿⣷⣿⣿⣿⣿⣗⣿⣿⢿⠿⣿⡿⠐⠻⢻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠟⠒
⠀⠀⠀⠠⠈⠁⠛⠿⠛⠋⠙⠉⠁⠀⠀⢀⣿⣿⡀⠀⢼⣿⣿⣿⣿⣿⠀⠈⠈⠙⠙⠿⠻⠟⠿⢿⣿⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡐⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣠⣿⣿⣿⡛⣉⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⣤⠄

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠤⡆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠢⢍⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠩⠀⠁⠈⠩⢖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠉⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡜⠀⢰⡐⢢⠀⡝⢱⠀⢣⢀⡀⠲⠶⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠄⡠⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠁⠀⠁⠓⡔⢹⡀⠠⢈⣀⠀⢸⠈⠛⠀⠶⡦⠀⠀⠀⢀⠔⠁⠀⠰⠀⡰⢁⠀⠂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⢚⠃⠃⠀⠈⢿⠊⣁⡀⢙⢸⠀⠀⠀⢀⠀⠄⢀⠔⠁⠀⡐⠀⠠⠎⠐⡡⠀⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠃⠀⠀⢀⢊⠜⠁⠈⠋⠇⢀⣀⢒⡒⠢⠔⠁⡀⡄⠅⡡⠊⠀⠀⠀⠀⠜⠠
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⣀⡠⠀⢀⡠⠆⡀⢀⠄⡀⢀⠀⢀⠀⠀⠀⠀⡠⡆⠇⣡⠎⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡐⠊⠀⢀⢠⠤⠴⡴⡢⠫⢁⢊⠔⡰⢁⠜⡠⢂⠔⣡⠊⡠⠀⠁⣃⡥⠊
⠀⠀⠀⠀⠀⢈⠲⢤⣘⠡⠪⠊⠈⠔⠱⡡⢊⢎⡔⢁⠞⡰⢋⠔⢁⠔⡁⡔⠈⠁
⠀⠀⠀⠀⠀⠢⠍⠀⠀⠉⠉⠐⠒⠒⠀⠀⡡⢊⠔⡡⢊⡴⢁⠔⡡⠊⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠔⡡⢊⡤⡫⠚⡡⡪⠊⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡠⢊⠔⡩⢊⢔⡩⢊⠄⡴⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠖⡡⢊⠔⠕⡫⢊⠡⠂⢧⠀⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⣩⢔⠅⡄⠃⠐⡡⢂⡡⠘⡄⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡊⠔⡡⠀⠃⠃⠀⣊⠥⣊⠄⢱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⢰⠀⠀⠀⠔⡊⠔⠀⡷⠀⠈⣒⠭⢒⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡅⠀⡆⠀⠀⢋⠔⣪⠄⢱⠀⠨⠐⣊⠥⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠱⡀⠀⢐⡪⢒⠅⠈⡇⠈⠥⠐⠪⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠱⡀⠀⠈⠁⠉⠀⠀⣤⡤⣤⠤⡊⠙⡡⠔⣀⠄⢀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠩⠄⣂⠥⣚⢠⣀⣀⡀⠀⣀⡀⠽⡑⠌⢂⠡⢐⡩⠔⣊⠅⠂⠅⠊⠡⠊⠙⠊⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠋⠩⠔⠉⠐⠊⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠖⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿⠟⠁⠀⠀⠀⠀⠀⢉⡠⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⢤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⣾⣄⣀⣈⠁⠀⠀⠀⠁⠉⢹⡿⠃⠾⠛⠀⠀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣅⣠⣤⣶⣾⡿⠃⠈⠀⠀⣀⣠⠀⠀⣀⣤⠀⠱⠀⠀⠀⠁⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣾⣿⣿⣿⣿⠟⠁⢀⡤⠖⠋⠁⠠⠒⠋⠉⠀⠐⠀⠠⠟⠀⠀⠀⠴⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⡉⠀⠀⢁⣀⣠⣤⠀⠀⠀⠐⠚⠁⠐⠦⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠛⠁⠀⣰⣿⣶⣾⣿⣿⣿⠟⠀⠀⣠⣤⠀⠀⠀⠀⠚⠁⠄⠀⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣀⣤⣾⡿⠟⠛⠛⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢿⣿⠟⠁⠀⡀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⣠⠀⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠟⢁⡀⠐⠉⣀⣴⢟⣋⣁⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⠄⢠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⣠⣾⣿⣿⣟⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⡠⠔⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠋⠀⠀⢀⡀⠀⠀⠀⣠⡮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⢋⡠⠖⠀⠀⠀⣹⣿⣿⠿⠛⠁⠀⠀⣠⣿⡿⠋⠀⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣠⠔⠋⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⢀⣠⣾⣿⣯⣤⠄
⠀⠀⠀⠀⠀⠔⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠿⢿⡭⠁⠀⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⡀⢤⢴⣶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣶⣒⠤⠖⠛⠀⠀⠀⠰⡟⠁⠀⠀⠀⠀⠂⠀⣀⠤⠒⠉⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠿⠿⠿⠿⢿⣿⣿⣿⣥⠀⠀⠤⠴⠖⠀⠙⢨⠅⠀⢀⡠⠔⠈⠀⠀⠀⢀⣠⡄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠁⠀⠀⠀⢀⣀⣠⠴⠒⠀⣠⠞⢁⣀⣠⣤⠴⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣟⣁⣤⣤⣤⣶⣶⣿⣿⣝⣁⡄⠀⠀⠀⠈⠉⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠂⠀⢀⡤⠀⠀⠀⠀⣿⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⠄⠀⠀⠈⢸⡇⢸⡟⢠⡄⢀⣤⡄⠀⡀⣠⣦⢸⡧⣡⠀⣼⡇⢀⡀⣀⠀⣿⡄⠀⠀⠀⠀⢴⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠠⠤⠄⣾⠀⡸⡿⣾⠡⣿⣽⣇⣼⡟⢸⡇⡼⣿⣏⠀⣿⡇⢼⣃⣿⢀⢿⠀⡐⠤⠄⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠔⠁⠀⠙⠃⠀⠈⠋⠀⠀⠘⠋⠀⠉⠙⠞⠺⣧⠈⠁⠳⠂⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢠⠀⠠⠤⠄⡀⠀⠀⢀⡄⠀⠤⠤⡤⠄⠤⠤⠠⠄⠬⠄⠤⠠⢤⠀⠀⠠⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡘⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇⠀⢠⢠⠏⠀⢰⠀⢀⠂⠀⠀⡇⠀⠀⠉⠁⠀⡇⠉⠀⠀⡸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠻⠄⠰⠤⠀⠥⠠⠅⠀⠈⠴⠀⠀⠀⠀⠀⠤⠠⠀⠧⠀⠀⠀⠐⠦⠤⠀⠀⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡄⢠⢠⢐⠂
⠀⠀⠀⠀⠀⠀⠑⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠀⠐⠂⠹⡄⡇⢾⢾⠸⠇⠂⠀⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠊
⠀⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⠦⣰⠂⠀⠀⢀⠀⠀⠀⠐⣆⠤⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠓⢂⣤⠀⠈⠀⡍⠨⢭⡉⢦⡀⠀⠈⠐⠚⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡆⠐
⠀⠀⠀⠀⠀⠈⠙⠓⠂⠁⠀⠀⠀⠀⠀⢰⣿⡌⠀⠀⣿⠁⠃⠀⠀⠘⠆⠀⢙⣳⡇⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⡀⠀⠈⠐⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠃⣠⠄⢻⣿⡁⣄⢀⣿⡜⢀⣶⠂⢰⣇⢠⣖⠀⠃⣴⠢⢀⣴⢾⣿⠀⠐⠠⣄⠘⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠻⢠⣿⠏⢿⡄⠸⢿⡴⠋⣿⣯⢻⣖⠉⢿⡤⠊⠻⠺⣿⡰⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠘⠷⠃⠀⠀⠀⠛⠋⠘⠁⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⡿⢓⠂⢠⢠⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠆⠰⣊⡽⠢⠄⠈⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠄⠠⡀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠒⠠
⢺⡀⠀⠀⠀⠒⠀⠈⠉⠀⠀⠀⠀⠀⢒⡀⠔⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡁⠂⠀⠀⠠⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣬⡍⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⠖⠂⠨⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢌⠋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⠀⠓⠒⢃⡬⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠽⣶⣶⠂⠀⠀⠙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⠤⣤⣤⣤⡤⠤⣤⣤⠤⣤⣤⣤⢤⣤⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣻⠀⣼⢃⡘⣿⢰⡇⣿⠰⣇⠿⢸⠀⢸⠀⢿⠀⣐⣠⢿⡶⠄⠀⠄
⠀⠀⠀⠀⣷⣶⠀⠀⡈⠀⠀⠘⠛⠛⠛⠛⠋⠛⠛⠛⠛⢛⠛⠛⡛⠛⠛⠛⠛⠋⠀⠀⠌⠉⠁
⠀⠀⠀⠈⠩⠋⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠃⠀⠀⠀⢀⡐⠀⢀⣵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡇⢀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢲⣭⣷⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⡘⢉⡡⢤⣐⠘⢰⢄⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⡁⠈⠊⠏⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⡀
⣱⠀⠀⠀⠀⡀⠀⠀⠀⠐⠛⠃⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠿⠿⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⢼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠂⠁⠀⠀⠀⠀⠂⠀⠀⢀⠀⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠰⠻⡟⠂⠠⣀⠀⢈⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡁⡜⣻
⠀⠀⠀⠀⢀⠤⠤⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⢧⡄
⠀⠀⠀⠀⠸⠄⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡄⠈⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣂⣮⣿⣿⣏⣱⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⢷⡄⠤⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢨⣿⢗⠁⠻⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣀⣾⣁⠀⠀⢠⣷⡇⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠸⣿⣷⣭⣤⣾⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⣧⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⣠⠏
⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠁⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡤⣄⣀⣸⡶⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡟
⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⣀⣀⡀⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠂⢲⣶⣶⣶⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡀⠀⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠒⠒⠀⠀⠉⠆
⠀⠀⠀⠀⠀⢸⠀⠀⠛⠿⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠂⠶⢀⡻⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀⣀⣀⣀⣀⢀⣀⣘⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⡃⠀⠀⠀⠀⠀⢸⣷⣀⣀⣀⣄⣀⣉⣀⣭⣭⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣻⣿⠔⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠨⢹⡏⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⡀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣨⡤⠈⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣼⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠦⣀
⠀⠀⠀⠀⠀⠀⣿⣻⣗⣋⡈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠭⠍⠛⠀⠠⠴⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉