Cute kitten Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⢶⣶⣶⣤
⠀⠀⠀⠀⠀⢠⣿⠁⠱⣄⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⢀⡴⠋⢻⣧
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠈⢧⡀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢈⣻⣤⡴⠶⠶⠶⠟⠛⠛⠛⠻⠶⠶⠶⢶⣥⣄⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢻⡀⠀⠀⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢦⣄⢠⡞⠁⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠃
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠉⢳⡄⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⢠⡾⠛⠉⠙⠻⢦⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠙
⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⢀
⠀⠈⠉⠉⠛⠛⠶⣿⣄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⢦⣀⣀⣀⡼⠻⣄⡀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⣠⣼⡷⠶⠚⠛⣿⠋⠉⠉⠉⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢹⣯⠉⠓⠀⠠⠒⠀⠀⠀⠂⢀⠀⠀⠀⡏⠉⠁⠀⠀⠀⠉⠉⡯⠀⠀⢀⠀⠒⠐⠂⠠⢀⠀⠀⠀⠁⠀⣿⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣇⣠⠔⠊⠉⠉⠑⢲⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠋⠀⠁⠐⠒⠀⠈⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⠐⠀⠤⠄⠐⠉⠲⢦⣼⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠙⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠙⠻⣦⣄⠀⣿⠀⠀⠀⠀⡿
⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠞⠛⠁⠀⠀⠀⠀⠀⠙⠳⣿⡆⠀⠀⢸⡇
⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⣠⡞⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⢦⡎⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⠞⠉⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡎⠀⡜⠀⠀⠀⣿⠀⡠⠀⢠⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠷⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠷⠖⠒⠊⠙⢶⣧⣀⣇⣀⣤⡶⠿⢥⣄⣀⣀⣀⣀⣉⣉⣉⣩⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠰⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⣄⠀⠆⠰⠒⠐⠂⠀⠀⠀⠠⠄⠀⠰⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠾⠀⠀⠀⠰⠦⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠟⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠓⠓⠒⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠈⠙⠂⠀⠋⠀⠀⠀⢀⣀⠀⠀⠀⠁⠒⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢰⡿⠁⣀⡆⢄⠀⠀⠂⠤⠐⠂⣦⠦⡀
⠀⢠⠄⠀⠈⣤⠀⠀⠀⠀⣸⡏⢀⣀⣸⠴⡂⠘⠷⠖⠙⠓⠘⠁⠀⠀⠀⠀⠈⠀⢀⣿⡆⠀⣤⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠓⠋⠛⢉⡁⣀⠤⠀⠀⠀⠀⣀⡀⢲⣦⠀⠀⠀⠛⢛⣥⣄
⠀⠀⢀⡀⡀⠀⠀⠀⢀⠀⠀⠀⣰⡄⠛⠁⠀⠘⠗⠇⠀⠀⠈⠷⠼⠟⠀⠠⠀⠂⢀⢀⡟⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠶⠋⠀⠀⠶⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠂⠀⠀⠀⡀⠈⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄
⠀⠀⠀⠀⠀⠉⠀⠀⠘⠆⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⠀⠀⠰⠀⠀⠰⠀⠀⠀⠠⠸⠆⠀⠀⠛⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠷⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠐⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠀⢠⠀⣆⡄⢘⡆⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⡇⠠⡇⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠤⠀⠀⠀⠀⠰⡀⠹⠃⠘⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⡹⠀⠆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⣠⣌⠃⠀⠀⠀⡡⣤⢤⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠉⠀⠁⣖⠀⠀⢸⠃⠉⠀⠀⠀⠙⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠉⠁⠀⣠⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠤⠤⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠈⣀⠀⠀⠤⠒⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠏⢀⣀⣀⡀⢀⠀⢠⡀⠀⠀⠀⠀⠀⡀⠈⠙⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠅⠀⠀⠁⢬⠿⠿⠟⠉⠁⠀⠈⠹⡟⠗⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠚⢁⠀⡀⠀⣠⣀⣀⡀⠤⢦⣀⡆⠉⣸⣯⣾⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⣷⣦⠉⡏⠀⠀⠀⠀⠉⢹⣾⠟⠛⠊⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡈⠁⠘⠆⠐⠀⠀⠀⠠⠏⠀⣤⢤⡸⢿⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠴⠆⠀⠜⢀⠤⢤⣀⠀⠀⣀⡆⠀⡀⠀⠀⠶⠦⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢳⠋⠉⠛⠀⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈


⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣶⣶⣶⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠏⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡟⢦⣸⣦⡘⠀⠀⠀⢀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣿⡸⡏⠛⡇⠀⣴⠺⣆⠀⠈⢀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠙⣯⡻⣿⣶⡙⠄⠀⠈⣟⡻⠃⠀⠀⣛⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣀⣠⡴⠾⠛⠚⠙⠛⠐⠺⢟⣛⠛⠀⠀⠀⠀⠈⠠⠔⠊⠁⣡⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢀⠔⠁⠀⠀⠀⠈⠉⠓⠒⠠⢤⡐⢂⣤⠄⠀⠀⠉⠓⣤⢤⡀⠀⠀⠀⢀⡄⠀⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣿⠀⠀⠀⢸⣘⡷⠋⡆⠀⢠⣾⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣰⠁⠀⠀⠀⣾⡅⠀⠀⠀⠀⣧⠁⢀⡄⠀⠐⠀⣀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠸⠀⠀⢾⡁⠸⠄⠀⣼⠆⠀⠁⠄⠸⢁⠛⠂⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠘⠀⠀⠀⠠⣸⡇⠐⡆⠀⢿⣦⠈⠳⡆⢠⣆⣀⣀
⠀⠀⠀⠀⠀⠘⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢰⡀⢧⢡⠀⢱⠀⢻⠊⠃⠀⢀⠀⡉⠉⠛⠇⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡀⠀⠀⠀⢀⠌⠀⠀⠀⠐⡀⠀⠀⡀⣷⢸⡄⠀⢤⡆⠀⠤⡀⠀⠀⠉⠑⠀⠢⠶⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢷⠸⣄⢱⡙⣆⢻⡄⠘⣷⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠢⡘⢶⢇⠙⡎⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠀⠓⠈⠈⢇⠃⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⢠⣤⣶⠶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡭⠴⠾⠟⠀⠀⡛⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣦⣴⣿⡷⠶⠆⢰⡶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⠼⣭⣏⣁⣁⣤⠠⠀⣤⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⢄⡀⠀⠀⠀⠀⠹⡉⠙⠉⠛⢠⢰⣧⣀⣴⠃⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠴⢦⣀⣷⣄⠀⣤⣼⠦⡙⠯⣁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠊⠁

⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⡀⠀⢀⠀⡀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠊⠠⠇⠀⠐⠀⠀⠀⠀⠠⠈⠀⠘⠒⠀⠔⡀⠀⠀⢰⢀⠒⠉⠀⠀⠀⠀⠈⠉⠱⠀⡇
⠀⠸⠂⠀⢀⠀⠀⠀⠀⢀⡀⠈⠶⠀⠀⣄⠀⠠⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⡇⠀⣀⠀⠁⠀⠀⠀⡀⠀⠘
⠀⠀⠀⣿⣯⣿⠀⠀⢸⣯⣿⡗⠀⠀⠀⠈⢠⣶⣲⡄⠀⢠⣒⣶⡄⠃⠀⠀⡷⢿⣯⣵⠀⠀⢠⣿⣽⢳
⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⢆⠀⠙⠿⠃⠀⠘⠛⠛⠀⡀⠀⠀⡅⠈⠛⠛⠀⠀⠈⠛⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⡀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠒⠠⢄⢀⣀⡀⡤⢠⠮
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠘⠁⠀⠀⠀⠡⠄⢀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠁⠀⡆⢀⠄
⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠊⢡⡀⣁⣀⠃⡘⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠿⠿⠻⢿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡴⡄⠀⠀⣴⢳⠀⠀⢰⣦⡄⠀⣴⡶⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠀⠀⢀⣶⣄
⠀⠀⠘⠓⠑⠂⠀⠈⠉⠀⠀⣿⠿⠃⠐⠻⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠀⡌⠃⠀⠀⠀⢛⣻⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢓⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⣽⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⣼⢿⣿⡿⣷⡄⠀⠘⠋⠙⠃
⠀⠀⠠⣆⠀⠀⠀⡶⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡿⠚⠉⠛⢿⡏⠀⠀⢀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣫⠽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⢠⠀⠀⠀⠦⠀⠀⠀⠠⠀⠀⠀⠄⠀⠈⠳⣇⣠⠼
⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⢀⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡐⠟⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠻⠇⠀⠛⠁
⠀⠀⠀⠀⣝⣻⣿⡀⠀⠀⠀⢸⣫⣿⣇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣤⠀⠀⣠⠂⢀⡤⡀⠀⠀⢰⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠴⠶⠷⠿⠀⠣⠾⠿⠛⠗⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠠⠀⠑⠀⠒⠂⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡖⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡿⡾⠻⠀⠀⠀⠀⠀⣀⠹⠄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡺⣤⣽⠇⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠈⠠⡀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠁⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠒⠐⠀⠒⠒⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠏⠀⠀⡅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⢀⠏⠀⠀⠀⠁⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢀⠎⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢸
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠤⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⢠⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠠⠊⡔⠀⠘⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡐⠠⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠀⠀⠀⢷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡇⠀⣠⠞⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠞
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡗⠋⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣟⠈⠉⠉⠉⠉⠀⠀⠀⠀⣰
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢹⠀⢀⣀⡤⠚⠉⠀⢺⠀⠀⠀⠀⠀⢿⠀⠀⢀⣀⣀⣀⣠⠴⠚⠁
⠀⢏⡑⠢⠤⠤⠤⢚⠉⡋⠉⢿⠀⣀⠠⠤⠼⢏⡉⠀⠳⡰⣄⣠⣯⠐⠊⠉⠁⠉⠙⠻⢍⠉⠉⠉
⠀⠀⠉⠓⠚⠓⠛⠉⠉⠀⢀⡎⠁⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣀⠀⡄⠀⣠⣤⡤⠞⠀⠀⠀⠳⠤⠶⣤⣤⣧⣤⣤⠗⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣀⠎⠀⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠁⠀⢀⠜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠂⠀⠄⡀⠁⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⠆⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⡀⠎⠠⠄⠀⣠⠤⠀⠠⠠⡀⠐⢄⢀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⠄⠀⠀⠐⣍⠀⠀⠀⡄⣤⠈⠢⡈⢣⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠁⠀⣀⠦⠀⢘⡀⠳⠀⠁⠁⠐⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⡨⠂⠀⣈⠤⠤⠄⠄⡀⠂⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠔⠈⠀⠀⠀⠀⠀⠀⠀⠁⠂⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡔⠐⠀⠀⠀⠀⠀⠀⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⡆⠀⡀⠠⠄⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⢀⣀⣀⣀⠄⠚⢇⠀⠦⠀⣀⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠠⠀⡀⠐⠀⠀⢀⣀⠀⠉⠁⠂⠀⠃⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠒⠀⠐⢀⠀⠂⢺⠀⠀⠍⢆⠀⠈⠢⣀⠀⠀⢄⠀⠈⠆⠐⠲⢦⣄
⠀⠀⠀⠀⠀⣠⣄⠒⠂⢠⣀⡀⡀⠄⠀⠀⠀⠀⠀⠀⠁⠀⢈⡆⠀⠠⡈⠢⠀⠀⡦⠀⠀⠈⢆⠀⡏⡁
⠀⠀⠀⣰⣟⠕⠁⡠⠂⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⠑⠚⠉⠉⢦⡀⠀⠀⡤⠊⠁⠀⠀⢀⠀⠸⠀⢁⣈⣀
⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠁⠁⣈⢀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠤⠰⠁⠀⠃⠀⡸⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⣀⣤⠴⠶⠚⠛⠛⠒⠒⠶⠶⢤⣄⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣩⣭⡳⡄⠀⠀⠀⠀⠀⠀⢠⣞⣭⣍⢳⡄⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢳⡸⣿⣧⣾⠇⠀⠀⠀⠀⡀⠀⠸⣿⣧⣼⣣⠇⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠆⠖⠒⠻⣆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⢰⡀⠀⠈⠛⠛⠁⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠉⠉⠉⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⠀⠀⠀⢉⣀⣀⣤⠴⠞⠋⣧
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠁⠀⠀⠀⠀⠸⡄
⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠘⠳⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡇⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⢤⣤⣤⣀⡀⠀⠀⠀⠀⣿
⣧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢦⣄⢠⡏⠢
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⢱⡀
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣀⣠⣀⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⡀⠻⡄⢳
⣿⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣴⠶⠛⠉⠁⡏⠀⡀⠀⡉⠛⣦⠀⠈⢢⡀⠘⣆⡿⢸⡇
⢹⣆⠀⠀⠈⠙⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⣀⣀⣈⣧⣼⣄⣼⣥⣤⣿⣄⣠⣼⣁⣴⠏⢀⣼⠃
⠀⠹⣧⣀⠀⠀⠀⠀⠈⠉⠙⠙⠛⠛⠛⠒⠶⠒⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠉⠋⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠻⢶⣟⠁
⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⢦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⢠⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣛⣻⣶⣾⣛⣁⣀⣀
⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣶⣿⠛⠛⠛⠿⣿⣿⣿⡷⣶⡖⠶⢦⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⣠⣴⣻⡿⢻⣿⠶⣦⣤⣤⡀⠀⠈⠛⠀⠁⣄⠀⢿⣿⣟⠉⠙⠛⣛⣫⣽⠿⠃
⠀⠀⣽⠁⣿⠠⣾⣿⣿⣶⣌⡙⠃⠀⣄⠀⢤⣲⡈⢧⣀⠙⠫⠞⢻⣿⣿⠟⠁
⠀⠘⣿⠂⣿⢀⣿⠟⠛⠋⠙⠁⠀⠀⠈⢳⡌⢻⣷⡘⣿⣄⠀⠀⠀⢻⡏
⠀⢠⣷⠆⢿⣌⠁⠀⠀⠀⠀⠀⣄⠀⠀⢈⣷⡀⣿⣷⢸⣿⢀⠀⠀⣾⠇
⠀⢾⣿⠟⠀⠀⠀⠀⠀⠀⠨⢽⣾⣧⠀⣾⡿⣷⣿⣿⣾⣯⣾⣄⠀⢻
⠀⢸⣇⣀⣀⣠⣤⠴⠤⠀⠈⠛⠛⠛⠂⠻⡇⢻⣿⣿⡿⠋⠟⠉⢀⣾
⠀⣸⣿⣿⣿⡿⠿⠟⠛⠓⠖⠀⣶⣶⣤⣄⠀⢸⡿⢿⡇⣶⣖⠀⠛⡟
⠀⢹⣿⡿⠿⠛⠒⠀⠀⠀⠤⣄⡘⠿⠷⠛⣀⠈⠀⠀⠘⠛⣃⡤⢠⠇
⠀⠈⢿⣶⣤⣐⡲⠶⢶⣦⣄⠉⠛⠓⠁⠀⠻⠆⠀⠀⣧⠻⡟⢁⡏
⠀⠀⠸⣿⣿⠉⠁⣀⣠⣽⣿⣷⡀⠐⡢⣐⠀⠐⢟⡽⠃⢸⣿⣾⡁⠈
⠀⠀⠀⣿⡇⠀⠀⣀⣠⣄⠀⠘⠻⢦⣤⣀⣤⣴⣛⠦⠴⣿⣟⠙⠿⣦⣀
⠀⠀⠀⣿⣇⣴⣿⠟⢁⣠⣤⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⢬⣿⡆⠀⠈⠛⠿⣶⣄⡀
⠀⠀⢰⣿⣿⣿⣷⡟⣋⡄⠀⠀⠉⠛⢾⣦⣄⡀⠀⠀⠀⢠⣽⣿⣄⠀⠀⠀⠀⠉⠛⠳⣶⣤⡀
⠀⠀⣾⡿⣿⣿⣿⣾⠋⠀⠀⠀⠀⠀⠀⣈⣩⡿⠷⠶⠶⢦⡈⠛⣿⡆⡀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⣸⡿⠁⠘⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢻⡄⠘⣿⣿⣄⣄⠀⠀⢀⡀⠀⢀⣉⣷⣤⣤⣄
⠠⣿⣷⡶⠀⠈⠙⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣻⣿⡄⠈⠛⠿⣿⣿⣶⣿⣿⣶⠛⠏⣩⣋⣤⣿⣧
⠀⣿⣿⡷⠀⠀⠀⠀⠀⣉⡻⣿⣿⣶⣄⠀⠀⣄⠀⠳⣄⣤⣹⣿⣿⣦⠀⠀⠈⠉⠻⣿⣿⠃⣶⡦⠷⢿⣿⣿⡿
⢸⣿⡿⠀⠀⠀⣠⣶⠟⠛⠋⠙⠛⠛⠛⢧⣰⣬⣧⣀⣾⡟⢻⠛⢿⣿⣧⠀⠀⡀⠀⣽⣿⡀⣭⣴⣶⣾⣿⣿⠁
⣼⣿⡇⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⡈⠃⠀⠀⠀⣿⣿⡄⠀⡇⣼⣿⠏⠀⠈⠙⠛⣻⣿⠇
⢹⣿⡅⠠⢿⣿⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠈⢿⣷⠀⣿⣿⡇⠀⠀⠀⠀⢠⣿⡟
⠈⣿⣷⣠⣿⡿⢠⣿⠀⠀⠀⠀⣤⠀⠀⢶⠀⠀⠀⠀⢻⣷⡷⠀⠀⠀⠈⢿⣷⣿⣿⡇⠀⠀⠀⠀⣾⡿⠁
⠀⢻⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⢿⣆⠀⠘⣧⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⢨⣿⣿⣿⣄⡀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢳
⠀⠘⣿⣿⣿⣧⣄⢻⣿⣧⠀⠀⠘⣿⣄⠀⠙⢷⡼⣦⡀⠀⢻⣿⡄⠀⠀⠀⢻⣿⣿⣿⣷⣦⣾⣿⠏⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣸
⠀⠀⠘⢿⣿⣿⣿⣷⣽⣿⣷⣄⡀⠹⣿⣷⣄⡘⣿⣿⣷⣄⣘⣿⣷⣤⣀⣠⡿⠛⠉⠉⠉⠙⠻⠧⣤⣀⣀⣀⣀⣤⠶⠋⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠙⠙⠿⡟⠿⠿⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠙⠻⢿⣿⣦⡀⢠⣀⠀⠀⣄⡀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢷⣾⣿⣷⣶⣶⡶⠿⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣀⣀⣀⣀⡀⢀⣴⡾⠏⠀⠀⠀⠀⠀⡠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣀⠀⠰⠁⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣠⣌⡛⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣧⣇⠀⢠⠂⣼⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⢀⡠⠱⣦⣷⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣵⣿⣿⣿⣝⢻⣿⡇⠈⣨⣥⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣟⣦⣿⣿⣿⣅⣽⣏⠻⣿⣿⣿⣿⣿⡏⢹⣿⢷⣛⣋⣉⣿⣿⣫⣼⣥⣈⣿⣌⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⢻⡿⢿⣉⣿⡿⠻⠻⡟⢫⣿⣿⣿⡟⢻⡨⣃⣭⣭⣯⣗⠛⢛⣛⠟⢿⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠟⢛⣿⣾⣿⣿⣿⣿⣶⡀⠙⢿⢿⠟⢡⣿⣿⣿⣿⣿⡿⡟⠛⢛⣤⡀⠙⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣵⡆⠘⣿⣿⣿⣿⡇⠀⠈⠀⠀⣨⣿⣯⣿⠿⠿⠝⠀⣤⣶⣎⠛⠻⢶⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣽⢷⣨⣁⣸⣿⣧⠀⠀⠀⣠⣿⣿⣿⡇⢤⠤⢤⢞⣫⣴⣁⢶⣿⣿⡿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢿⣿⣶⡯⢹⣿⣿⣿⢀⠀⣀⣿⣿⡿⠉⡄⠄⠠⢴⡾⠲⢾⣿⣯⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠹⠶⠓⠈⠉⠈⠙⢭⡉⢻⣥⠞⠋⠉⡀⠀⠀⠚⠉⣲⡻⠿⠯⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⡈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⡩⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆⠀⢈⠷⠖⠒⠒⠒⠒⠒⠺⠤⣀⠀⡔⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠋⠁⠙⠆⠀⠰⡆⠀⠠⠊⠉⠳⡄⠀⠀⢸⠀⠀⣠⠞⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⡧⠄⠀⠀⠀⠀⠐⡤⠔⠓⠤⡄⠀⠀⠀⠁⠀⠀⢺⠓⢹⠋⠉⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⢥⠀⠀⢀⠃⠀⠀⠀⠀⢀⣠⠏⠀⣼⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠙⠲⠤⣄⡀⠀⠀⠑⠒⠀⠀⠀⠀⣀⡠⠔⠉⠙⠢⣼⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢦⡀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢄⡀⠀⠀⠀⣸⠀⠀⠀⠀⢳⣀⠔⠁⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠦⠀⠧⣄⣄⣠⠞⣀⣀⣀⣠⣺⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⣀⡀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣭⣝⢿⣿⠃⠀⠀⢻⣿⢟⣭⣝⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣿⣿⣿⣾⣿⠀⠀⠀⢸⣷⣿⣿⣿⡗⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣛⠿⣟⣿⡇⠀⠀⠀⠈⣿⣟⠿⢟⣵⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠟⠋⠉⠀⠀⠄⠀⠀⠈⠛⠻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⡿⠟⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⡠⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠒⠀⠈⣿⣶⣿⡏⠉⠉⠉⣿⣿⣿⣄⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣑⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⢙⣿⣿⣿⣷⣄⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣃⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣆⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⡻⣷⣼⣿⡄⠀⠀⠀⣾⣷⣼⡿⣹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠒⠣⣄⢠⡄⠓⠛⠛⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠉⠀⠀⠀⡴⠚⠉⠉⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠉⠓⠒⠂⠀⠀⠀⣀⣤⣶⣾⣿⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠐⠲⠁⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠙⠲⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⡀⢀⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠉⠻⣿⣿⣿⣿⠆⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⣿⡿⠋⢪⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠁⠀⢹⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠚⠉⠉⠀⠀⠈⠙⠛⠒⠒⠶⠶⠶⠶⠒⠒⠒⠶⠶⠖⠋⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⡟⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢲⡄
⠀⠀⠀⢸⡇⠈⠢⡈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡠⠊⠠⣿
⠀⠀⠀⣾⡇⠀⠈⠭⢦⡀⠉⠳⣦⡀⢀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⠀⣠⡴⠋⠁⣠⢚⣒⡀⠀⣿
⠀⠀⠀⣿⡇⠀⠀⠙⣉⣛⣄⠀⠀⠙⠋⠉⢹⣿⠀⠀⣿⡿⢉⡟⠉⠛⠉⠀⢀⣞⣯⡭⠄⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠴⠶⠈⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⠁⡼⠁⠀⠀⠀⠀⠉⠈⠡⠶⠂⠀⠀⣿
⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢰⡏⠀⠀⠀⣠⣶⣶⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠓⣶⡦⣄⠀⠀⠀⢿⡀
⠀⠀⠀⣿⠀⣀⠀⡼⢱⣿⣿⣷⣤⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣴⣿⣷⠘⡆⠀⡀⠘⣧
⠀⠀⢰⣿⡿⠋⠀⣇⠸⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠙⣿⣿
⠀⠀⠸⣿⣤⣀⡀⠘⢦⡙⠿⢿⡿⠟⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⠿⢿⠿⠟⣁⡜⠁⢀⣠⣬⣿
⠀⠀⠀⢿⡟⠛⠁⠀⠀⠉⠓⠒⠒⠋⠁⠀⠀⠀⢤⣶⣦⠄⠀⠀⠀⠉⠛⠒⠒⠋⠁⠀⠀⠈⠛⢻⡟
⠀⢀⢀⣈⣷⣤⠔⠒⠲⠶⠶⠒⠤⠄⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠒⠒⣴⡿⣀⡀⢀
⠁⠈⠁⠀⢀⡹⣷⣶⡂⠈⠉⢁⠠⠀⠀⠀⠘⠦⠖⠋⠓⠶⠖⠀⠀⠀⠈⢍⠉⠁⠀⣒⣴⡾⢏⡀⠀⠀⠉
⠀⠀⠀⠊⠁⠀⢀⡽⠛⣷⣶⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣶⡞⠻⣁⠀⠀⠈⠑⠀⣠⡶⠶⣤
⠀⠀⠀⠀⢀⠔⠁⠀⣼⠃⠀⠈⠙⠛⠻⠶⠶⣶⣤⣤⣤⣤⡶⠶⠶⠟⠛⠉⠁⠈⢻⡄⠈⠑⠄⠀⠀⠀⢿⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⡄⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⢀⡿⣶⣤⣼
⠀⠀⠀⠀⠀⠀⣸⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠸⣇⠀⠀⠀⢀⣾⡁⠀⢩⡟
⠀⠀⠀⠀⠀⠀⡿⠀⠘⣷⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡿⠀⠀⢿⠀⢀⣴⠟⠹⣷⣤⠟
⠀⠀⠀⠀⠀⠠⡇⠀⠀⢿⠀⠀⠀⠐⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠀⠀⠀⢸⡇⠀⠀⢸⡟⢻⣿⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣾⠀⠀⠀⣸⣁⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣷⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢠⡏⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣻⡆⡆⢠⡄⠈⣇⣀⣀⣀⣀⣀⣀⡏⠀⢠⠀⡆⣼⢓⣦⠾⠋

⠀⢀⢀⣤⡀⠀⢀⣀⠀⣀⡀⢀⣠⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠄
⠀⠨⠡⠂⢁⠰⣄⠃⢀⢹⠀⠀⠈⡤⠄⠘⠀⠀⠸⠄⠠⠄⠜⡄⠀⠀⠈⠀⠄⠀⠤⠀⡎⠀⠀⢈⠁⠀⢙
⠀⠈⢀⠀⡺⠀⠋⠐⠀⢄⡀⠀⠘⡇⠀⠀⠀⢀⠀⠀⠀⠐⠀⢸⠀⠀⠀⠢⠀⠉⢀⡴⠁⠀⠀⠀⠃⠂⢃⠄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡀⠀⠀⠈⠳⠄⣀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⣸
⠀⠀⢀⣀⡀⠀⠀⣠⢤⡀⣼⡀⠀⠈⡥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣤
⠀⠀⢸⠄⢨⠀⠀⠈⠀⠄⢈⠁⣀⡁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠡⡀⢀⠴⠀⢀⠞⠀⠀⠀⠶⠤⠀⠠⡴
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣤⡦⡀⠀⠀⠀⠀⠀⠀⠓⠉⣤⡤⣬⠀⠀⠀⠐⠠⠄⠠⣀⠇
⠀⠈⢇⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠣⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢼⠀⢸⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠆⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⡄
⠀⠪⠀⠠⠋⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠓⠀⠓⠁⠀⠀⠃⢁⠤⡄⠀⠐⡆⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣨⣀⠀⢀⡉⠲⠀⠀⠀⡀⡀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⢑⢲⠃
⠀⠀⢀⡴⠲⠶⠦⢤⠀⠀⠀⠙⠋⠄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄⠀⡉⠀⠀⠀⠐⣄⡈⠀⡆
⠀⠀⠀⠈⡯⣿⡏⠁⠀⠀⠀⠀⠑⠒⢲⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠂
⠀⠀⢀⠀⣇⣀⡗⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠰⠂⠐⠺⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠢⠤⠂
⠀⠀⠘⠒⠄⠀⠆⠋⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠂⠐⠃⢢⠀⠀⠀⠀⢂⡄⠀⠀⢀⠄⠀⡤⣀
⠀⠀⠀⠀⠀⠀⣠⣤⠶⠂⠁⡀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠁⠀⣠⠇⠀⢀
⠀⠰⠅⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⣀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⢫⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⣎⡠⠔⣁⡀⣼⠄⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⢠⣤⢫⡁⢨⢣
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⢨⡆⠀⠀⠀⠀⢌⠀⠀⣪⡗⢃⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⢀⠀⠀⢀⡂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠒⠒⠋⠉⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⢸⣿⣿⣧⡀⠀⠈⠄⠀⠀⠀⠀⣹⣿⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⢸⣿
⠀⠀⠀⠀⠘⠀⠀⠀⡌⠀⠀⠀⠀⠀⣠⣾⣧⢠⣶⣶⣶⣷⣄⠀⠐⡀⠀⠀⠀⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⢰⣄⡀⠂⢠⣾⡛⠙⠻⣿⣿⣶⡄⠈⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡔⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⠏⠠⠠⣿⠋⣡⡀⠀⠙⢿⣿⡄⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢼⣩⡅⣩⣭⣭⣭⣭⢈⣭⣭⣭⣭⣭⡈⠀⢀⠀⠀⠀⠄⠀⠀⢆⢠⠋⠀⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠸⢀⣠⢠⠤⣬⣿⣇⡿⠿⠟⠛⠛⠚⢛⡟⠛⢿⡿⣁⣀⠄⠀⠐⠀⢀⠀⢀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⠀⠀⠀⠐⢀⠈⢰⡏⠁⠐⡄⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡡⠄⠀⠀⠀⠙⠄⠀⠀⢉⡀⠈⡷⢖⣶⣞⣻⣄⣁⠔⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠉⠭⣽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠀⠀⠀⠀⠀⢀⠓⢒⡄⠀⠒⠦⡤⠤⠤⠴⠒⠋⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠒⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⢃⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠴⠀⠀⠀⠀⠀⠀⣿⣿⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠉⠉⠐⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠁⢁⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠘⣄⣀⡠⠤⠄⠤⠠⠤⡤⢀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⠔⠃⠁⠀⠐⠀⠀⠀⠀⠀⠀⠆⠈⠑⠳⢄⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠄⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡴⠁⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⡀⠄⢀⡀⠀⠀⠀⠀⠣⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢠⠃⠀⢠⣶⣿⡄⠀⠀⠎⠀⢀⣴⣾⣆⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⡄⠀⢻⣿⣿⠇⠀⠀⢆⠀⢸⣿⣿⡟⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡷⠰⠀⠀⠐⠂⠈⠑⠀⠛⠁⢪⠉⠉⡄⠑⠂⠛⣋⡀⠀⠀⠀⠐⠐⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣀⠠⠔⠊⠉⣀⠀⠀⠠⡀⣱⣎⢀⠄⠀⢀⡀⠀⠉⠐⠠⣄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⡰⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣨⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠃⠀⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠤⠄⠀⠐⠂⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢠⡄⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠹⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠠⠃⢂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠀⠀⡾⢸
⠀⠀⠀⠀⡇⠀⠈⢢⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢦⡀⠀⢀⡼⠁⠸
⠀⠀⠀⠀⢡⠀⠀⠀⠑⠤⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢃⠀⠀⠀⠰⡈⠈⡡⠃⠀⠀⢄
⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠀⢠⣤⣬⡅⠐⠦⢼⣤⠤⠀⣀⠑⢞⢅⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠯⠁⠀⠈⠳⡸⠀⠁⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢺⡟⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣼⠟⠳⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠛⣧
⠀⠀⠀⠀⠀⢠⡟⠀⣠⣤⣀⠈⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣠⣄⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠁⣀⣤⡄⠀⢻⡄
⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠉⠛⠲⣤⡀⠈⠻⣦⣀⣤⠴⠶⣿⣿⡏⠉⠁⢹⣿⣿⡏⠈⠉⢹⣿⣿⠶⠦⣤⣀⣴⠟⠁⢀⣤⠖⠛⠉⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⣾⠁⢰⡿⠀⠀⠀⠀⠀⠙⠦⡀⠈⠉⠀⠀⠀⠈⢿⡇⠀⠀⠈⣿⣿⠁⠀⠀⢸⡿⠁⠀⠀⠀⠉⠁⢀⠴⠉⠀⠀⠀⠀⠀⢿⡆⠈⣷
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢹⡏⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⢹⡆⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢰⡏
⠀⠀⠀⠀⠀⠸⣇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⣀⠈⣻⣟⠁⣀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣽⣄⣀⣀⡀
⠀⠀⠈⠉⠉⠹⣏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣴⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣦⣄⡀⠀⣿
⠀⠀⠀⠀⠀⢀⠼⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠙⠳⠶⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠶⠞⠋⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠘⡇⢸⠃⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠛⠲⠶⢤⡀⠀⠀⠀⠀⣀⣀⣘⣷⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣃⣀⣀⠀⠀⠀⠀⢀⡤⠶⠖⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⠛⠂⠀⠀⠀⢻⡟⠀⠀⠀⠐⠛⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⡠⠴⢦⣄⠀⠀⠀⣼⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣈⣓⣒⣒⣒⣒⣂⣀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠁⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡄⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠙⢿⣦⣷⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣿⣿⣿⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⣿⠋⢰⠿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⣴⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠶⢷⣶⣶⢾⠿⠿⠾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠤⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣉⡯⠀⢀⣿⣿⣿⣿⠟⠁⠙⡿⢩⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠛⠙⢿⠿⠇⠰⢾⠿⠿⠟⠁⠀⢀⣀⠁⠀⢙⣻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣷⣶⣶⣿⣄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣴⣏
⠀⠀⠀⠀⠘⢿⣶⣭⣀⣁⣀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⠙⠛⠋⠙⣿⣿⣿⠟⠛⠇⠹⠿⢾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁⣉⣉⣹⣭⣿⣿⣯⠀⠀⢀⡀⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣟⣛⣿⣿⠋⣹⣿⣿⣿⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⡀⠟⣩⣿⣿⣷⡄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⣹⣿⡟⠙⠋⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣥⣤⣶⣶⣶⣾⣿⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣤⣴⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠉⠉⠉⠀⠀⠀⠀⠁⠄⠠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠀⠐⠆⠀⠀⢣⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⡆⠀⠀⠀⠐⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠄⠈
⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠐⠖⠀⠀⣀⡀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠠⠀⠀⡔⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⡠⠤⣀⠂
⠀⠀⡐⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢐⡰⠒⠀⠁⠀⠄⠀⠀⠀⠢⢄⡈⠀⢀⣀⡠⠊⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠐⠒⡒⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⡠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠊⠀⠠⠄⠐⠂⢐⠆⢀⠠⠗⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠄⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠙⠓⠂⠀⠀⠀⢀⡄⠀⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠉⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠤⠤⠤⠌⠀⠀⠤⠄⠀⠤⠤⠀⠂⠂⠩
⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⠶⠒⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠙⢷⣄⣀⣀⣄⣀⣀⣠⡾⠋⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡽⣿⠉⢹⣯⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣮⠁⠁⠀⠈⠈⣷⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣦⣭⠈⠀⠀⠀⠁⢭⣴⣿⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠶⣀⡰⠀⠀⠀⢀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠃⠠⢀⠀⠈⠀⡀⠀⠈⠙⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⢀⣴⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢹⣿⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡼⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢸⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡼⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⠀⡀⢸⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡎⣿⣿⣿⡟⢸⣧⠸⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡈⠛⠉⠡⣼⣿⡀⠉⠛⠋⣸⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⠿⣿⣷⠀⠀⠀⢻⣿⠁⠀⠀⣰⣿⡿⠟⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡀⠛⠀⠀⠀⢸⣇⠀⠀⠀⠛⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠐⢇⡀⠀⠀⢀⡸⢧⡀⠀⠀⢀⡰⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⣿⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⡿
⠀⠀⠀⠀⠀⠸⡆⠈⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠙⢷⣤⣠⣤⣴⣶⣶⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣤⡀⠀⠀⠙⠟⠋⠀⠀⣴⣻⣿⡆⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠛⢛⡛⠳⢦⡀
⠀⠀⠀⠀⢀⣸⣇⣤⡤⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⢻⣿⡆⠀⠀⠀⠀⠉⠉⠙⠛⢻⡟⠒⠂⠀⢸⡇⢸⡏⠙⢷⡀⣿
⠀⠀⠘⠛⠋⠉⢧⠀⣀⣤⡤⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠶⠤⣤⣤⣾⣀⠀⠀⠀⠀⠻⠶⠃⢀⣼⠁⣿
⠀⠀⠀⠀⠀⣀⣼⢿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⢿⣀⢉⡥⠶⠦⣤⣤⣤⡴⣟⡁⣰⡟
⠀⠀⠀⠀⠾⠋⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⢶⣤⣤⣴⠟⠀⠀⢀⣠⡴⠚⠁⣼⣿⣿⣄⣠⣤⣤⣀⣀⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣤⣤⣄⣀⣀⣠⣤⡤⠶⠚⠋⠁⠀⠀⠀⢻⣿⣿⣿⣯⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡤⠶⠶⢤⣤⣤⣄⡀⠀⠀⠀⢀⣴⠞⠳⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢠⡞⠁⠀⠀⠀⠀⢸⡅⠈⣷⡀⠀⣠⡟⠁⠀⠀⣷⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠻⠾⠋⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⢠⣤⣴⣶⣶⣶⣦⣄⣀⣀⣀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡴⠊⣇⠑⢄
⠀⠀⢀⠤⠒⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢆⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢣⢂⣀⣁
⠀⠎⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡄⣀⢈⣄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠀⠀⠀⠠⠤⠤⠀⠀⢀
⢰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⠀⠩⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠖⡀⠀⠠
⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠁⠀⢀⠔⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢈⡴⢨
⠀⡄⠀⠀⢹⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⣶⠟⠃⠂
⠀⠘⡄⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢊⣴⣾⠟⠁
⠀⠀⠈⠢⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠈⠀⠀⠀⠀⡠⠃⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣾⡿⠊
⠀⠀⠀⠀⠸⠒⠠⠀⠀⠉⢩⠍⠋⠖⡀⣶⡒⠒⡘⠁⠐⠀⠶⣓⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⠀⠀⢰⠀⢠⠄⡄⠀⠀⠀⡜⠀⠀⠩⠄⠀⢀⠀⠀⠀⠀⠉⠉⢀⠒⠄⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣴⡾⡄⠀⠀⠹⣶⣦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠆⢁⡠⡆⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠻⡀⠀⠔⠛⢿⣿⣿⣿⣿⣿⣿⣿⡖⠒⢶⣻⣿⠿⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣈⠩⠀⠉⠉⠛⠛⠿⠿⡇⠀⡟⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢣⡙⢄⠀⠈⢻⡛⠛⠛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠈⡀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⡃⣇⠀⠀⠘⣧⠀⣀⠄⠀⠀⠀⠀⠀⠀⠁⠑⠀⠤⠤⣤⣤⣴⣶⡿⠉⠀⣹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢌⣿⡅⡇⣠⣔⠒⠛⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⣠⣾⡿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡆⠀⢩⣞⣅⡀⡀⢤⠠⠀⠀⡀⢀⣴⣶⡿⠟⠁⠀⣠⣾⣿⣿⢣
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⠹⣿⠯⠐⠚⠛⠛⠟⠛⡷⠿⢷⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⣿⣿⡟⣾
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⠀⠀⣀⡀⣔⠀⠀⠀⠀⠀⠁⢀⢀⣀⢀⣴⣶⣿⣿⣿⣿⣿⠃⠨
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣟⣒⠀⢀⠀⠀⢠⣯⣷⣿⣾⣿⣿⣿⣿⣿⣿⢏⢄⡇
⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢛⣉⣙⠛⠛⠛⣛⠋⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⢿⡼⠁
⠀⠀⠀⠼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣷⡶⠦⠾⠿⠶⠷⠳⠤⠴⠤⢤⣤⣴⠷⠋⢵⡟
⠀⠀⠀⠘⠻⠻⠿⣿⣿⡟⣿⣿⣿⣿⣷⣄⡀⠹⣶⣤⣴⣤⣤⣄⣄⣠⣤⣤⣤⣤⠤⢤⣀⡀⣼⠛⠓⠚⠓⠒⠛⠛⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣦⣌⠛⠛⣟⠿⠿⠛⠿⠻⠿⢿⣶⣶⣤⣬⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠤⢠⠄⢠⣄⣀⣀⠀⢀⡁⣁⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡧⣉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⡆⠈⠓⠀⠀⠀⠀⢀⠀⢀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠫⠄⠆⠀⠀⢀⣾⠛⠟⠛⠋⢉⡤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⣀⣀⡀⠀⠀⠀⠸⠁⠀⠀⠠⡄⣇⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⠈⡏⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠑⠑⠭⠭⠞⠀⠀⠀⠀⠀⠐⣨⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠁⠀⠀⠀⠐⠠⠀⠀⠀⠈⣛⣟⡿⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⡀⠈⠀⠀⣀⣀⠀⠀⠀⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡿⠉⠀⠑⠄⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣾⡤⠀⠀⡠⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠈⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣄⡥⠀⠀⠉⠀⣀⣀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠄⢄⣀⡀⣾⠋⠁⠀⣠⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠑⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠙⠒⠲⠶⠒⠒⠓⠒⠢⠴⣆⣠⣀⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠢⠠⡁⠀⠀⠀⡠⠀⠒⠲⠶⢦⡄⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡈⠀⠀⠀⠀⠀⠈⠙⠱⡄⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠁⢈⠀⠄⣄⠠⠀⡐⠀⠀⡀⠈⡀⠃⠐⠠⡀⠀⢐⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠁⠴⢡⡄⣀⢠⡄⠲⠀⠀⠀⠀⠀⠄⡀⣀⣴⠇⢀⠀⠀⠉⠁⠐⠓⠄⡀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠃⠀⠘⢺⠟⠋⠙⠙⠿⡗⠃⠀⠀⠀⠀⠀⠀⠉⠁⢆⡭⠌⠀⠂⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⣎⠀⠀⠴⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢎⡀⣀⠀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠀⠈⠀⠁⢹⡏⢀⠁⠀⠠⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠍⠀⠀⠀⠀⠀⠹⡌⠇⡞⠁⠀⠀⠀⠘⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣤⣦⡄⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡀⠀⠘⣷⣿⣿⠀⠀⠀⠀⢸⣷⣿⡇⠀⢀⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠂⠀⠉⠁⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠂⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠄⣂⠄⡀⠀⠀⠀⠀⠀⠐⠂⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠐⡂
⠀⠀⠀⠑⠽⠬⠠⠀⠀⠀⠀⠀⠙⡀⠀⠀⡲⠠⡤⡄⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣴⢠
⠀⠀⠀⠀⠐⢴⡄⠀⠀⠀⠀⡀⠀⠈⠠⠀⠐⠀⠓⠂⠀⠁⠀⠐⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡎⡀
⠀⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢀⠀⠐⣋⠀⡀
⠀⠀⠀⠀⠀⠘⢤⡄⠄⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢿⠀⠀⣱⣄⠀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠃⠰⠉⠁⠊
⠀⠀⠀⠈⠀⠀⡱⢔⡹⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠄⠀⠀⠰⠒⠢⡄⠁
⠀⠀⠀⢀⡐⡊⠠⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢠⠧⠀⠀⢤⠀⠀⠄
⠀⠀⠀⠐⠐⢣⠬⡀⣀⡼⠃⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠁⢀⡀⠔⠐⠒⠀⠐⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⢀⡀⠀⠀⠀⠆⠰⠀⠀⠀⢠⡀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠐⠂⠁⠀⠀⠈⠒⠒⠁⠀⠀⠁⠈

⠀⠀⠀⠀⠀⢀⡴⠂⠁⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡀
⠀⠀⠀⠀⠠⠈⠁⢀⠀⠀⠀⠀⠀⠄⠠⡤⠄⢤⡤⢠⡄⠶⠀⠛⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠡⣤⠀⠈⡄⠀⠀⠀⡧⠔⠒⠗⠒⠒⠳⠼⠧⠤⢀⣥⣿⣦⠀⢴⠆⠀⠀⢠⣶⣄⠀⠀⠿⠀⠀⠀⠀⠐⠠⠄⡀
⠀⠀⠀⠀⠀⡩⠋⣑⣷⡀⠀⠀⠁⠤⡴⠃⠂⠂⠁⠀⠀⠀⠀⠀⠉⠫⡀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⢠⣿⣷⠀⣶⠀⠈⠐⢄
⠀⠀⠀⠀⠠⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢶⠄⠀⠀⢰⡦⠀⠀⠈⠛⠋⠀⠀⠀⣠⣤⠀⠐⢄
⠀⠀⠀⠀⠸⠟⠐⣶⣾⠁⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⠃⠀⡀⠁
⠀⠀⠀⠀⣀⣀⡠⠛⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⢀⣠⡜⠀⣿⣿⠂⢀⣄⠀⢠⣾⣦⠀⠀⠛⠁⠀⠀⢀⣄⠀⠀⠘⠋
⠀⢀⣒⠁⣶⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⠈⣥⢀⡈⠁⠀⢈⡉⢀⠈⠿⠏⠀⣀⠀⠀⢀⣤⠀⠉⠐⠀⣤⠈⡀⠀⡀⢀
⠀⢿⣿⡇⠀⠀⠄⠀⠀⠀⠛⡤⠇⣀⣀⣀⠀⣀⣀⣀⡀⢀⠀⠀⢀⠀⠰⡎⠉⠁⠉⠀⠀⡀⠝⠒⠄⠀⠛⠀⠀⢿⣿⠇⣴⠁⠀⢈⡤⠚⢈⣵⣦⠀⣨⡄⢀
⡆⠈⠛⠡⣦⠀⠐⡀⠲⠀⠀⠈⠐⡆⠠⠄⢀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠒⠲⣤⠦⠀⢠⣾⠀⠀⢀⢀⡠⠀⠀⠠⢤⣥⣴⠁⠀⢀⠋⢴⠆⠘⢿⠟⠀⠈
⠇⠿⠂⢰⣿⣧⢠⡀⠀⠁⠀⠀⠀⠈⢠⠀⢠⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⢠⠟⠁⠰⢲⠟⠉⠁⣿⡄⠀⠀⢠⣤⡀⠀⣠⡄⠀⣰⣶⡄
⠀⢄⠀⠈⠛⠃⠈⢁⡀⢀⠀⠀⠀⠀⣨⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣬⠏⠀⠀⠀⣿⣧⢀⠀⢻⣿⠇⠈⠀⡀⠀⠹⠿⠃
⠀⠀⠈⠐⠀⡄⠠⣿⣿⡆⠀⢀⡈⢁⣸⠠⢥⢀⠑⠢⣀⠀⠀⠀⠀⢠⠠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢩⣿⣾⢰⡆⠀⠀⠀⣿⣿⡄⠀⣦⡠
⠀⠀⠀⠀⠀⢘⠀⣉⠉⠙⣦⡙⠁⠀⠊⠀⠉⢚⣴⣶⡄⠱⠆⠀⠀⢲⡗⢦⠅⣀⠀⠀⠀⠀⠀⣀⣀⡠⡧⠴⠐⠂⠈⣿⣿⣍⠀⣠⣤⠀⠀⠙⠋⡶⠈⠉
⠀⠀⠀⠀⠀⢸⠀⠛⠀⢸⣿⡇⠀⣶⢼⣶⠂⠀⠹⠿⠇⢠⣾⣷⣀⣀⡀⣰⣶⣦⠉⠿⠀⠉⠈⢿⡿⣀⠀⢰⣿⣷⠀⠛⠋⣶⠀⢿⣿⠇⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠉⣁⠚⠛⠃⠠⠤⠊⠈⠂⢺⡇⠀⠀⣴⣦⡀⠀⠟⠀⠘⠿⠋⠀⠀⠰⡷⠀⠀⠀⠛⠁⠈⠻⠋⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠄⠒⣁⠀⠀⢸⡏⣿⡀⡄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠆⠜⠛⠛⠿⠥⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠑⡤⠀⡠⠈⠀⠀⠒⠄⠉⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢀⠀⠀⠀⠀⠠
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠠⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠄⠰⠁⠀⠀⠂⠀⠐⠀⠠⠀⠀⡄⠀⠀⠀⠀⠱⡀⠀⠀⢀⠄⠀⠀⣀⡀⠀⠀⠄
⠄⠀⠀⠀⡀⠀⠈⠁⠀⠄⠀⠀⠄⠀⢀⠀⠀⠀⠠⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠴⠁⠂⠁⠈⠁⠀⠀⠀⠀⠀⠓⠀⠚⠁⠀⡠⠁⠀⠈⢂⠀⠀⠄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠐⠀⠈⠀⠀⠀⠀⢀⠔⠀⠀⠀⠄⠀⠘⠉⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠱⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠱
⠀⠀⠀⠀⠀⠀⢀⠔⢄⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⡔⠘
⠀⠄⠀⠀⠁⠉⠀⢀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠈⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠐⠂⢀⡀⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠑⠄⡀⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠒⠢⣌⠀⠀⠀⠀⠀⠃⠄⠈⠂⣀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠈⠀⠄⠀⠀⡀⠀⢂⠀⠀⠀⠀⠁⡀⠀⢈⠋⠀⠈⠁⡀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠂⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠒⠂⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠠⠀⣀⠀⠄⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⣠⠀⢤⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠈⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣭⠒⡋⢱⠀⠀⠀⠀⠘⠁⠀⡙⣥⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠀⢀⠼⠄⠀⠀⠀⠈⢦⡀⠀⠀⣀⣤⠤⠤⠤⣄⣤⠀⠀⠀⢠⠃⠀⢀⣘⠃⡌⢳⠀⠀⠀⡞⠁⠘⠟⠀⢒⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠯⠶⠼⠅⣅⣠⠋⠀⠈⠉⠃⠸⠀⠀⠈⠛⠀⠔⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣛⢙⠉⠀⢻⡰⠂⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠸⠤⠄⠀⣠⠶⠋
⠀⢀⣀⣤⡼⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⡠⠔⡏⠨⠇⠀⠀⡖⢨⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⢡⠁⡄⠁⠀⠀⠀⠀⠀⠀⡀
⢀⣼⠀⠀⠀⠀⠀⢀⣀⣀⠀⠺⢤⡀⠀⣌⣀⣀⡂⠈⠅⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⢣⠀⢨⠆⠀⠀⠀⠀⠀⡼⠁
⠛⠢⠤⠖⠉⠙⠛⠃⠀⡠⠤⢤⠴⠛⠋⠉⠁⠈⠹⡄⢙⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠃⠀⠀⠀⢰⠋
⠀⠀⠀⠀⡀⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠆⠀⠀⠀⡜
⠀⠀⠀⢮⡁⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣠⠤⠤⣶⠫⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢐⣊⠤⡀
⠀⠀⠀⠀⠻⠤⠶⠚⠛⠚⠋⠀⠉⠁⠀⠀⠀⠀⠈⠙⠣⢿⣄⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠙⠚⢢⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠲⠖⠓⠲⠶⠤⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣲⣞⣛⣶⣴⣙⣓⣒⣋⣉⣉⣈⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⣸⡷⣄⠀⠀⡀⠀⠀⠀⣠⣬⡁⠀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢩⣿⡿⠟⢻⣏⠹⣾⡋⢿⣼⣆⣸⠿⣯⣄⣘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⠶⠀⢛⠉⠢⠉⣹⣝⢻⡯⠈⣋⣇⠙⢻⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡀⠨⠄⡇⢾⣿⣿⣽⢰⠀⡿⢠⢶⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠾⣋⢹⡼⠓⣆⠈⢉⣇⡟⣘⣤⢤⣥⣴⣖⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠀⠀⠉⢛⣿⣯⣹⡀⠀⢽⣧⣿⣦⣾⣶⠷⣙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠉⠀⠀⠈⣹⡿⢿⡇⡀⣸⣿⣿⡟⢿⣿⣓⣻⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⡄⠀⠀⠀⠀⠈⠿⠛⠙⠻⠿⢦⣬⣭⣿⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢀⠀⠀⠀⠐⠓⠢⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢶⣤⣄⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣏⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⢨⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣾⢿⣼⣿⡟⣻⣿⣷⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁

⣿⣿⡿⠿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿
⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠈⠙⠋⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠁⠀⠀⣤⣦⠰⣦⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢺⣷⡿⠛⠛⢦⣌⣛⣾⣿⣿⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠉⠁⠀⠀⠀⠀⣛⣛⠛⠛⠒⢺⡷⠶⠶⠦⠤⠤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠉⠉⠉⠉⠛⢻⡟⠓⠒⠲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⠟⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣾⣿⣿⣷⣶⣦⣾⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⠆
⣿⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠉⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣾⣿⣶⣦⡀⣠⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡴⠛⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠳⣄⠀⢀⣠⠴⠖⠚⠛⠉⠉⠉⠉⠒⠢⢄⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⡹⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⢦⡀⣸⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠱⡏
⠀⠀⠀⠀⠀⠀⢹⡄⢀⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠒⠠⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢳⣸⠀⠰⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣶⠆⠱⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣴⣿⣆⣠⠱⠀⠀⠀⠀⠀⠀⠣⠍⠡⠔⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢠⠻⠿⠋⡠⠀⠀⣠⣐⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⠀⠈⠀⠀⠀⠉⠙⣏⠀⢀⠄⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠠⠁⠀⠓⠄⠔⠁⠉⠁⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠤⠔⠚⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣾⠁⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣧⠀⠀⠀⣸⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⢠⡟⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⡄⠀⠀⠘⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣀⣴⠏⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡀⣇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠉⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⡩⠛⠂⠀⢀⠇⠀⠀⠀⢠⠇⠀⠀⠀⢠⡷⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠾⣧⣤⣤⣤⣼⡀⠀⠀⡀⣸⣅⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⢰⣖⣴⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⡼⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡴⢎⠾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠷⣿⡿⣯⣙⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠘⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡟⣻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠉⠁
⠀⠀⣼⠷⢦⣄⡀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡀⠀⠀⠀⠀⠀⢀⢤⡠⠄⠤⢄⠀⠀⢠⢊⠋
⠀⠀⠀⠀⠀⠈⠛⢗⢄⠀⠀⡇⠀⢱⠀⡀⠤⣀⠀⠀⠀⠀⠀⣏⣉⠉⡇⠀⠀⠀⠀⠰⡅⠀⠑⣴⡄⠀⠑⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡕⢤⡃⡠⢼⠉⠀⠀⠀⢑⠖⠢⢄⣸⡀⣩⡧⠧⠒⠊⠉⠒⡎⠀⠀⢀⣩⠉⣶⠎⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⢌⡐⠪⢅⣢⡎⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠺⠵⣴⣝⣢⣁⠔⠁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠑⠤⣈⡏⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠉⠐⠂⠀⠤⠦⠤⠤⠤⠤⠒⠂⠉⢛⠉⠉⢹⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠤⠄⠐⠂⠉⡠⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣀⣸
⠀⠒⠒⠂⠒⠂⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣤⡅⣦⡍⠆⠀⠀⠀⢂
⣠⠔⠀⠉⠈⠁⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠉⠒⠀⠀⠀⠀⠈
⠣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⣴⡆⠄⠀⠀⡜⠀⠄
⠀⠑⡀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠙⠁⠀⠀⠈⠃
⠀⠀⠘⡄⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢀⠔⠒⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠉⠓⡀⠀⠀⠸⠀⡀⡇⢄⢀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠄⠁⠈⢑⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠈⠀⠐
⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠤⠀⠀⠠⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⢠⢈⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⣀⣀⡤⣊⡁
⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠈⠉⠉⠁⠈⠐⠂⠤⠤⠼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣶⣦⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⣶⣟⠄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⣿⡿⠿⢤⡀⠀⣀⠀⠠⠤⠤⠔⠒⠶⠚⠁⠀⢾⣧⠂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣧⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣏⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣺⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠈⢡⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠑⠛⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠘⠃⣀⣀⣄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⢠⣴⠖⢄⠀⠀⠀⠀⠀⣾⠹⣿⢢⠄⠠⠕⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢤⡛⢟⡼⠃⢠⠈⣁⣀⠈⠋⠈⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⠀⠀⠀⠉⠉⠀⠀⠰⠌⣁⡬⠃⠀⠀⠀⠰⠆⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⣹⣅⠀⠂⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠙⠣⠤⠄⠠⠚⠉⠉⠑⠂⠀⠀⠋⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢳⡴⠦⠠⠀⣀⠀⠀⠀⠉⠉⠀⠀⣠⡤⠏⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣾⡁⠀⠀⠈⠉⠙⠀⠛⠲⠶⠟⠛⠃⠀⠀⠀⣷⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠅⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠸⣶⣋⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡟⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤⠀⠀⠀⠉⠁⠾⣶⣤⣀⣄⠀⢺⡀⠐⣄⣀⣲⣼⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠟⠀⠉⠋⠙⠋⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⣃⢽⡄⠀⠀⠀⢀⡤⢰⠆⠀⠀⠀⠀⢈⣆⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠃⠈⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠁⠤⢄⣄⣀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢀⣤⡼⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡄⠀⠀⠘⢁⠀⠀⠀⡀⠀⠀⠀⢻⣆⡄⠀⠀⠀⢹⡴⠶⠛⠋⠋⠛⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡀⠀⠘⢻⠛⠾⠛⠁⠀⠀⠀⣨⠛⢹⣇⠀⢼⠁⢡⠀⠀⠀⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⡆⠀⢸⠆⠀⠀⢠⣀⣄⣀⣿⠀⠈⢿⡀⣿⠀⢿⡀⢀⣠⣠⡶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠁⠘⢆⠘⠃⠀⠉⠈⠹⣧⠀⠀⠐⢻⡏⠀⣿⠟⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠿⣆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⡟⠁⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢸⡀⠀⡀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣀⠀⣸⡇
⠀⠀⠀⠀⠘⠦⠄⢀⣀⣠⣤⣀⣀⣽⣇⣀⣀⣰⡟⠒⢂⡠⠄⠛⠠⣄⡇⠀⠙⠀⠀⠀⢰⡇⠀⢀⣠⣦⣄⣼⠀⠐⠋⠉⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⠊⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢹⡄⠉⠀⠈⠃⢹⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢸⡆⢀⣵⡾⠀⠀⢀⡤⣄⡀⠸⡟⠢⠤⠴⠷⠾⡄⠀⠀⠀⠀⠀⠘⠧⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⢸⠀⠀⢀⠀⠀⣤⠀⠀⢹⡀⠀⠀⠀⠀⢹⡆⠀⠈⠉⠀⠀⢠⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣸⣆⣀⣼⣄⠴⠎⠀⠀⠀⠀⠀⠀⠛⢦⣀⣠⣧⣀⣰⡶⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣀⡤⠤⢤⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⡅⠙⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡾⣏⡷⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠒⠦⠀⠀⠂⠰⠀⠀⠈⠠⠆⠀⣼⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠉⠤⠀⠠⠔⠒⠒⠀⠀⠒⣠⣾⣿⣿⡿⣿⣷⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⢿⣿⣿⣿⠿⠿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣾⣷⣄⣀⣤⣴⣶⣶⣶⣀⡀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⠛⠿⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢸⣞⣿⣷⣄⣀⣀⣀⣀⠀⠀⠀⣼⠿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣾⣿⣿⣿⣿⠛⠛⣿⣿⣿⣾⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣵⡏⣿⠇⠀⢀⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠋⠛⠒⠊⠁⠒⠂⠘⢸⠃⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⢍⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡦⠉⠓⠒⠖⠚⠉⠀⣴⣾⣿⣿⣿⣿⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠻⠿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡔⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄⠀⢀⠧⠊⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠚⠀⢒⣠⣤⣄⣐⣒⠒⠂⠈⠉⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣓⡦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡴⠖⠚⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠁⣠⡶⠉⠀⠙⠳⣝⡆⠀⠀⣠⡴⠋⠉⠙⠋
⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡟⠀⠀⣦⠀⠀⠀⠉⠲⣴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠈⠁⠀⠚⠁⠀⠀⠀⠀⠀⠘⣾⡄⠀⢸⠁⠀⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣟⠛⠃⢀⣴⠀⠲⠦⣀⠀⠀⠀⠀⣸⠀⠀⢹⣷⡀⠘⣇⠀⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⠃⣀⡼⠀⠈⠓⠀⠀⣠⠏⠞⠀⢀⡇⠱⡄⠈⠳⢦⡙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡀⢀⣀⣀⡀⠔⠁⠀⠀⢀⡼⣧⠀⠙⣆⠀⠀⠙⠲⣝⣦⡙⠦⣄⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣝⣒⣾⠋⠉⠛⠲⢾⣉⡁⠈⠳⢄⡈⠳⢤⡄⠀⢀⣹⣿⣷⣄⣙⣦⣤⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣒⣒⣒⣒⣀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠷⠶⠶⢄⠁⠉⢙⡛⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⠀⠀⢀⣀⣀⣀⠹⣎⢻⡄⠈⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠉⠉⠉⠙⠇⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢁⣀⣈⡉⣀⣶⢤⡐⠦⡤⠤⠤⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠀⣨⣿⢭⠳⣤⡼⠀⢙⣷⠀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠠⠒⠎⠀⠤⠤⠠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠁⠈⠐⠳⣌⣑⣦⠪⡾⠀⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠂⠀⠀⠀⢠⠀⠄⢸⠀⠫⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠀⠀⠈⡿⠀⡐⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⣁⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⠤⠀⠀⠀⡠⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⢠⣀⣀⣀⣀⣀⣈⢸⠀⠰⢸⡇⠀⠀⠀⠀⢠⡤⣤⣤⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⢀⢀⡣⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣄⣶⡂⠐⠀⠀⠀⠀⠀⠘⠀⢀⣺⡇⠀⣠⠔⢈⢉⣁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣠⠿⠵⠶⠶⠇⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠒⠓⠛⠓⠒⣶⡴⢶⣶⠒⠒⠓⠛⠒⠛⠊⠀⠀⠀⠑⡍⠀⠀⠀⠀⠐⠀⠀⠀⢀⠴⠊⠁⠄⡀⠀⠀⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⢼⢇⣰⣂⣀⣀⣘⣉⣉⣉⣉⣉⢓⣒⣲⣶⣶⣖⡆⠔⠲⡀⠀⡀⠀⠀⠀⠐⠂⠌⢀⡖⠁⠀⣀⡀⠀⠀⠆⠀⠀⡞⢈⡅
⠀⠀⠀⠀⢀⠀⡈⣆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⡇⠙⢞⠁⠠⢣⡀⠀⠀⠀⣀⣂⣈⡁⠀⣊⡀⣱⠀⠀⠰⢀⠈⠀⠸⠁
⠀⠀⠀⠀⠹⡭⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠁⣦⡀⠀⠀⣇⠀⠉⠁⠈⠀⠀⠀⠈⠣⡀⠀⠉⠁⡀⠠⠚⠁⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⡇⠘⠛⠉⠉⠑⠒⠴⠤⠤⠤⠤⠤⠤⠾⠢⠤⠤⡶⠥⠟⠒⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⢊⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡃⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣶⣦⣴⣶⠶⠲⡶⢲⣀⣸⣨⡗⣷⣷⣷⠤⢦⣤⠤⠤⢤⡬⢬⣤⣤⣤⣤⣀⣀⣥⣥⣼⠶⠶⢶⠾⠃⠈⠑⠂⠉⠀⠄⠒⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⠀⠉⠉⠉⠉⠙⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⣠⣶⣿⣿⣷⣈⣿⣝⢿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣀⣀⠀⠀⣠⣾⣿⣿⣟⠷⠞⠉⠉⠉⠀⠟⠿⢿⣿⣷⡄⠒⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢸⣿⣯⣹⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠐⢶⣷⣦⣿⣿⠀⠀⠉⠉⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡜⠁⢀⢀⣿⣿⣿⣿⣿⣏⣻⠀⠀⠀⣀⠈⠑⡄⠀⠀⠀⠀⢙⣿⣻⣧⡤⠖⠚⠉⠁
⠀⠀⠀⣠⠴⠚⠛⢛⣷⣶⣾⣅⡀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⡿⢿⠛⠉⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⣠⠟⠋⠁
⠀⠀⠀⢀⣐⠔⣀⠄⠉⠛⣿⣟⡿⣶⣄⠀⠀⠀⢸⠙⢿⡿⠻⠷⢋⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠉⠣⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠉⢻⣷⣶⣤⣠⠄⠘⠿⣾⣿⣷⣄⢀⡼⡆⠠⢐⠀⠀⠸⡄⠀⠀⢸⣶⣄⠀⠆⠀⠙⢢⠖⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣁⠄⠈⠿⣭⡻⣿⣼⠃⠄⠨⠀⠀⠀⠙⣆⠀⠀⠙⠛⣀⠤⣢⠖⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠘⢶⣤⡤⢀⡀⠙⠿⣿⣦⡀⢀⠈⠹⣯⣿⣶⣂⠎⠀⠀⣀⣤⣾⡷⠒⠒⠉⠁⣘⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣔⢀⡀⠀⠘⠿⣷⡴⠂⠀⠈⠉⠛⠦⣀⠹⠿⠛⠁⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡤⠀⢀⠀⠀⠉⠑⠲⢶⣄⣀⠄⠈⠙⠲⢤⣤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠥⣖⠁⠀⠀⠀⠀⠉⢻⣿⣦⣴⠄⣀⠈⢙⣦⣀⠈⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⣁⠤⠀⠀⠹⢿⣿⣯⡁⠀⣀⠘⢿⣷⣄⠀⠀⠉⠓⢦⡀⠀⠀⢴⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠙⠻⠿⣌⠄⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓

⠀⠀⠀⢀⣠⠴⢲⡄
⠀⠀⠀⢻⡀⠀⠀⢳
⠀⠀⠀⠀⢧⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠘⡆⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⡴⠖⠲⢦⠈⢣⡀
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⣀⣀⣤⠤⠞⠀⠀⠀⢸⡇⢠⣶⣸⢀⡏⠙⠦⣀
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠹⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠖⠃⡾⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⣀⣈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⢠⠀⣆⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⣄⠈⢧⡈⠓⠂⠀⠀⠀⠀⢀⡼⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠓⠤⣉⣁⠀⠀⣀⣠⠔⢋⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣌⣉⣭⡥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⢾⠹⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⣤⡶⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠙⠻⣷⣦⣀⣀⣠⣾⡋⠁⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡁⣷⢎⡇⠀⢘⠸⡶⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⠀⠀⠸⣿⣿⣿⣶⡿⠃⠀⠹⣷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣤⣤⣴⣿⣿⣿⣏⠀⠛⠞⠃⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠗⠀⠀⠐⠾⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣦⡀⢀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⢿⡇⠀⠀⠈⣿⣿⣿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀⣠⡀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣤⡶⠿⠿⠿⢿⣿⣯⣽⣶⣿⣭⠔⠻⡟⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠻⠯⣼⣗⣦⣶⢆⣠⡼⠓
⠀⢀⡤⠶⠚⠓⠒⢒⣾⣾⡿⢟⣍⣻⢯⡞⠁⠀⡘⢀⡠⠔⠒⠈⠛⠁⠒⠢⢤⡀⠀⠀⠀⢰⣆⠀⠀⠈⡻⠉⣧⠙⣒⡦⠄
⠰⠋⠀⠀⠀⣠⡾⠛⠋⠁⢠⣬⣳⠁⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⡄⠀⠎⣿⢗⠄⠚⠒⠉⠈⠉⠁⠀⠘⠛⠃
⠀⠀⠈⠀⠈⠁⠀⠀⣠⠀⠛⠃⠁⠀⠀⠀⣰⠃⠀⠀⠰⠆⢀⠁⡄⠈⡄⠐⠂⠀⠀⠘⢆⠀⠀⠘⣷⣄
⠀⠀⠀⠀⢰⣷⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢩⡄⠡⣀⣣⣧⣼⣀⠌⣤⠋⠀⠀⠘⡄⠐⠰⢻⡙⠃
⠀⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⢠⣤⠀⢸⠀⠀⣈⠀⠈⣐⣾⡷⣾⢎⣷⡊⡠⠄⡉⠀⠀⡇⢀⠆⠀⣿⣄⠁
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠈⡄⠀⠀⢀⠛⡐⠛⡿⢲⢫⠉⢂⠘⣃⠀⠀⢠⠃⢎⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⠄⠀⠱⡀⠀⠀⠀⠄⠐⡁⠐⢀⠁⠠⠄⠀⠀⠀⠌⠀⠈⠀⢠⢿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣌⠓⠀⠀⠘⠄⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠿⡠⡄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⠸⣿⡟⠷⢦⣄⡀⠀⠀⡶⠀⠀⢀⣀⠀⠠⡤⠐⠀⠀⠀⠀⢀⡀⡀⠃⠀⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⢻⠀⢨⡦⡈⠙⢆⠀⠃⠀⠀⠀⡋⠀⠀⡅⠀⠀⠀⠀⠰⡸⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣆⠈⠁⣨⡅⠘⠀⢰⠄⢲⡀⠀⢨⡇⠀⠀⠀⠃⠀⠀⠀⠀⡘⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠓⢀⠀⣰⠀⠈⠁⠀⠸⡻⣦⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡃⠀⠀⠀⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡀⠀⠸⠀⠁⡑⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢃⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡨⠒⢾⡿⠿⠶⠪⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢉⠋⠀⠀⠀⠀⠀⠀⠀⠈⢫⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⠃⣈⣥⣄⠀⠀⠀⢀⣤⣌⡀⢳⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣷⣾⡇⠀⠀⣾⣶⣿⡧⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡠⠘⠉⢁⡀⠀⠀⢀⠉⠙⠠⡀⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠘⠁⠀⠀⠉⠁⠀⠀⢰⠂⠀⠨⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠤⠙⠀⢀⠔⠁⠀⠀⠘⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠙⠳⠶⠒⠒⠂⠉⠰⣆⠀⠀⠀⡔⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⡀⠀⠀⠀⠀⣤⣤⠀⠀⠛⡄⣤⢊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣿⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⡟⠁⡀⡀⠀⢱⠃⠀⠀⡀⢠⡆⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⣓⠁⠀⠀⠈⢹⠉⠀⠀⠀⢙⡃⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠒⠁⠑⠂⠀⠒⠁


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

⠰⣿⣿⣿⠀⠒⠀⠀⠀⠀⠰⠀⠀⠗⠄⠀⢰⣰⣤⣄⣀⣀⣰⡴⡜⣶⣶⣿⣿⣿⣿⣆⣆⣤⣴⣦⠴⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠳⠀⠀⣀⠠⡛⠦⠐⠀⠂⠈⢆⢺⣿⣿⣿⡿⢻⣿⣿⡍⠙⢿⣿⣿⣿⣿⣿⠟⣾⣽⣯⡆⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⡠⣙⣆⠲⣌⣧⣷⡄⠀⠀⠀⠐⣜⣷⣿⣿⣿⣇⡙⠻⢿⣧⠀⠀⣿⣿⣿⣿⣿⣇⡿⣽⣿⣿⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣜⣿⡿⢬⡃⠉⠀⠀⠀⠀⢤⣬⡻⣿⣿⣿⠟⠃⠀⢸⣿⠀⠀⣨⣿⣿⣿⣿⣾⢣⣿⣿⣿⠀⠀⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⢠⣶⣾⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠠⣜⢿⣿⣿⣿⠀⠀⠀⢸⣿⡆⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣾⣿⣿⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠋⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⠀⠀⠀⢸⣿⢷⡶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡀⢀⡸⣷⢹⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠋⠁⠀⣿⡿⠉⠉⠙⠋⠉⠉⢹⣿⣿⣿⣿
⡀⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠀⠀⠉⠀⢀⣼⣿⠃⠀⠀⣀⠀⠀⠀⢘⣿⣿⣿⣿
⣹⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣮⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⢶⣶⣾⡿⣿⣧⣴⠿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣟⣉⣫⣦⣿⠿⠿⢻⣷⣄⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣴⣿⣯⣸⣻⣷⣭⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠋⠛⠹⢿⣿⡉⠀⠀⣸⣿⣿⣀⣠⣰⣿⣿⣿⣣⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠟⣛⣛⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣶⣾⠟⠉⠁⠈⠹⢿⣧⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠈⠀⠁⢨⣽⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠉⣛⣻⡷⠞⠀⣀⣰⣖⣒⣺⡟⠉⣉⣿
⠀⠀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣀⣀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⠏⠀⠀⠈⠛⠛⠁⠀⠀⠉⠉⠉⠀⠉⠉⠉⠛⠂
⣾⣿⣆⠀⠀⠀⠀⠀⠀⣀⠠⠄⢻⡞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡿⠛⠉⠿⣿⣿⣿⣿⣿⣷⣿⣾⣿⠏⠀⠀⠀⠀⠀⣀⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⠁⠀⠀⢀⣤⣤⣾⡥⠀⣀⠀⠾⠏⠛⣚⣿⣩⣬⣿⣿⣿⣿⣿⣿⡝⠁⠀⠀⠀⠀⠉⠉⠉⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠁
⣿⠃⠀⠀⠀⠀⠛⠙⠉⠴⠪⠾⣷⣯⣤⡮⢭⣿⡿⢿⣿⣿⠿⠛⠙⠉⢁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠠⠚⠁⠀⡀⣤⡞⢾⡾⠃⠀⠉⠉⠀⠀⠀⠀⠀⢀⡤⣾⣽⣿⣿⣶⣤⢢⣦⢶⠀⠀⠠⢀⣀⢤⡈
⢀⣠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⡴⠶⠋⢠⠀⣴⢴⣏⣴⣾⣿⣿⣿⣿⣿⣿⡿⠈⢰⣦⣤⣼⣿⣧⡀
⣿⣿⣿⣯⣴⣾⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⡿⠻⠟⠟⠁⠀⣰⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣤⣴⣾⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣿⡿⠟⠋⠁⠁⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⡏⠉⠁⠂⠀⠀⠀⠀⠀⠰⣄
⢻⣿⣿⡿⢿⠟⢉⣷⡾⢿⣿⣿⣶⢟⡔⢁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⡇⢫⠃⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣏⡙⢏⣁⡀⢀⡨⣶⣄⠀⠀⠀⢻⣿
⠰⠟⡋⠀⠀⢠⡿⠿⠃⠁⡨⢊⠁⢡⣾⣣⣶⣶⡏⣴⣾⣆⣤⣶⣠⣿⣷⣇⢀⣿⣿⣿⣿⣶⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠚⠻⠛⠛⠻⠙⣏⣤⣸⣷⣌⣻
⠀⠀⠀⢀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠞⣼⠿⣿⡿⢸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣼⣏⣿⣿⣿⣯⢻⣿⣧⡁⠹⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠨⠻⣿⣿⣿⡟
⠀⠀⠀⠼⠶⠟⠁⠀⠀⠠⢄⠀⠀⠀⢁⠾⠃⠀⠋⢿⣿⣿⠟⠁⠙⣿⣿⣿⡏⢙⣻⣿⣿⣿⣿⡿⢿⠽⠂⠹⣿⣿⣿⣿⣿⡿⣿⣷⣄⠀⠀⠀⠀⠀⠘⢿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠂⠀⠈⠀⠠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠠⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⢹⠄⠤⢀⠄⢳⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠿⠄⠤⠀⠀⠓⢃⠀⣸⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠈⢆⠐⠂⠘⠚⢤⣤⠠⠤⠔⠁⠘⣧⠯⠀⢹⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢺⣧⠘⣄⡀⠀⠀⣠⡿⠀⠀⡘⣿⢠
⠀⠀⠀⠀⠀⢀⣴⠻⣷⣄⠈⠻⢷⡘⣿⡼⢋⣭⡆⠀⢰⡇⣿⡄
⠀⠀⠀⠀⠀⣿⡹⣧⡙⣿⡟⠀⠀⠉⠀⠑⠿⣿⡇⠀⢼⣇⢸⡇⠆
⠀⠀⠀⠀⠀⢿⣱⡜⣷⡌⢿⣷⣄⠀⠀⠀⠀⠀⠁⠀⠘⢿⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠈⢿⣽⣮⢿⣆⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡈⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⣾⠷⣻⣧⡙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠆⠀⠀⠀⠈⠀⠀⠈⠀⠀⠒⠀⡖⠠⠠⢴⠐⠂⠈⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡟⣝⢷⣌⢻⡄⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉⠑⠤⣤⣤⣤⠼⢦⡀⠈⣤⣶⣆⣠⣞⣁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣻⣿⣦⠻⡀⠀⢀⡾⠁⠀⠀⢀⣤⣴⣮⣥⣤⣤⣤⣄⣀⡀⠀⠀⠀⠘⠛⠿⠀⣷⣦⣤⡜⣻⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣷⣷⣠⣾⡿⢷⡶⢿⣿⣿⣿⣯⣭⣽⣿⣛⣛⡿⠿⠷⣶⣤⣬⣽⡛⢾⣿⠠⠛⠉⠉⠙⠒⠤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠾⠷⡾⣧⡼⠿⢼⣿⣸⣿⢸⣷⠉⣾⡏⣻⣟⣛⠷⣶⣶⠙⡿⠃⠈⠀⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢰⡶⠂⠀⢰⡿⡵⢞⡪⠓⠋⠁⠀⠀⠀⠀⠀⡀⢉⠉⠓⠛⠣⠿⣇⣽⡏⡸⠟⠁⠀⠀⠀⠠⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⣸⡥⠊⡡⢤⠇⣢⡀⢀⣰⠀⠠⠙⠙⠉⠑⠀⠀⠀⠀⣸⡟⠉⠀⠀⢀⡠⢀⡥⠚⠁⠀⠀⡄⠀⠆
⠀⠀⠀⠀⢀⡸⠀⢀⡀⢶⠸⠟⠏⡾⢠⢗⡀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢌⠟⠀⠀⡀⢔⡡⠞⠁⠀⠀⠀⠀⠀⠊⠀⠁⠶⡄
⠀⠀⠀⢆⠈⠀⠙⠨⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⡶⣀⠐⣞⢀⡠⢊⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠛⠳⡌⠛⠛⠹⢸⡿⣫⣞⢡⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣩⠞⠋⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⣦⣤⣤⣦⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⣀⣤⣤⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠑⣀⡁⠶⡉⣣⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣟⡄⡸⣾⠀⠀⠀⠐⠠⠀⡐⠂⡂⢢⡀⠀⠀⢀⠰⣄⠩⢤⠘⢣⣄⠹⣆⡙⣥⠚⡾⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⠛⠘⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡈⢤⡈⢤⡀⠻⡤⠑⢢⡈⢵⠆⠓⠀⠈⠈⠀⠸⢡
⠀⠀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠦⠉⠠⠚⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢠⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⡤⠧⠀⢀⡤⠀⠀⠀⠀⠀⠀⡤⢀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⡀⠛⠀⣠⡶⠀⠀⠀⠠⠇⠉⠉⠉⠉⠛⠉⠉⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⢀⡤⠖⠀⠀⣀⣠⢠⣤⣤⠶⠶⠴⠁⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀⠰⠁⠀⡀
⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠀⠀⢲⣶⣾⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠘⠲⠆⠒⠊⠀⠀⠀⠀⠈⠉⠠⠈⠀⠁⠉⡗⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠁⣀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠄⢤⠀⠀⠧⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠹⠁⠠⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⡦⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠘⠀⠀⠃⠀⠅⠀⠐⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣐⢁⠺⠁
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⠤⠊
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⢤⠄⠈⠀⠀⠀⠈⡀
⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⢆⠴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⠁⠉⠀⠀⠀⠂⠀⠠⠤⠄⠤⠔⠈
⠀⠀⠀⢀⡀⡧⠀⠴⠆⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠓⡼⠆⠀⣀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠐⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠐⠒⢤⡄⠀⠈⠁⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈
⠀⠀⠀⢀⡀⠀⠀⠟⠀⠀⠴⠀⢁⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠁⠀⠷⠀⢰⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠐⠆⠀⠀⠌⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⡶⢠⡀⢒⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠠⠈⠃⠀⠀⠀⣀⠀⠀⠀⠈⠇⡈⠕⠉⠀⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢁⠀⠃⠠⠄⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠒⠠⠖⠒⣷⠒⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⡕⠀⢀⡀⠁⠀⣀⢱⠀⠀⠀⠀⣀⡤⠶⣖⠂⠀⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠉⠁⠀⠀⠉⠒⢄
⠀⢧⠀⠈⠁⠀⠀⠁⢸⠓⠒⢞⠣⠈⠁⠀⠀⠀⠀⠀⠀⠈⡏⠐⣄⠀⠀⠀⠀⠑⠄⠈⡄⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠲⢻⡗⠂
⠀⡜⠓⠢⣀⣀⡠⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⢆⠘⣃⠀⠀⠀⠀⢐⠀⢡⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⣈⣢⡀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠆
⠀⠘⠒⡞⠁⠀⣠⣇⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠀⠀⠀⠐⡄⢸⠀⢹⠀⠀⠀⠀⠈⢄⠈⢦⡀⢀⡠⢆⠀⠀⠀⠀⠈⠐⠖⡇⠀⠈⢇⠀⠰⠆⠠⠈⠃⡄
⠀⢠⠎⠀⢀⠔⠉⠉⠉⠒⠒⠒⠒⠉⠁⠀⠀⠀⠙⡆⠀⠀⠃⢸⠀⢸⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣠⠜⠢⣀⠀⠀⠀⣀⠀⠑⠒⢤⡀⠀⠉⠦⠠⡔⠈⢒
⢰⠁⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠢⠤⠞⠉⠉⠘⠤⠤⠞⠒⠋
⠈⠑⠊⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢡⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢹⡟⠋⠐⠈⢹⠀⠀⠀⠀⠘⠤⠴⠊⠘⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠑⠲⠒⠂⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢦⠐⠂⠠⠀⠐⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠠⡦⠀⡀⢰⠆⠈⢲
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣀⠼⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣘⡆⢠⡀⠀⠀⠀⠀⡠⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠀⠀⠀⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡔⠈⠀⠀⠀⠀⠀⠋⠀⠲⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⡀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠣⡉⠵⡄⠀⠀⠀⠀⠏⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⡀⣠⠕⠈⠀⡠⠒⡒⠚⢂⠀⠀⢸⠀⢸⠀⡥⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠒⢄⠈⠑⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⠎⠀⠀⠉⢀⣀⡅⠀⢸⠐⠁⡠⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡀⠂⠈⢄⠀⢈⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⣠⠀⠀⠸⠖⠊⠀⠀⠀⠀⠀⠀⣇⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣰⠁⠑⠢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢻⡏⠋⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⠴⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⢲
⠀⠀⡰⠇⠀⠐⡉⠉⠃⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠀⡄⠰⠄⠀⣼
⠀⡜⠀⠘⡀⠀⠘⠆⠀⢀⡀⠀⠀⠶⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢀⡠⠋
⡸⠀⠀⠀⠱⠀⢀⡃⠀⠈⠁⠀⠀⠀⡰⠈⠀⠀⠀⠀⠀⠤⢴⡶⡤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠁⠉⠉⠑⠤⣄⡀
⡇⠀⠀⠀⢀⠀⠀⠑⠒⡦⠤⠤⠔⢏⠀⠀⠀⠀⠀⢠⠀⠀⠈⠃⠀⠀⠠⡀⠀⠀⢀⣞⡀⠀⠀⠀⠀⠀⢫⠑⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⢃⠀⠀⠀⣼⠔⠈⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠤⠍⠁⠀⠀⠀⠀⢨⡄⠈⡄⠀⠀⠀⠀⡸⠀⠀⠀⡀⠀⠀⠀⠀⢸⠐
⠈⠢⡀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣄⠭⣀⣂⡈⣁⡰⠁⢀⠤⠜⠂⠀⠀⠀⠀⠀⣠⡾⠃⢀⠃⠀⠀⠀⢮⣀⣀⡄⠀⢙⣀⣠⡀⠀⡸
⠀⠀⠀⠁⢻⠀⠀⡀⠀⡀⠀⢈⠀⠈⣲⠀⠀⠀⠀⠀⢧⣀⡨⣇⡀⢨⠒⠲⢇⠀⣀⢴⠿⠓⠒⠚⠛⠉⢀⡤⠊⠀⠀⠀⠀⠀⠀⢀⣑⣓⣉⣀⣀⡈⠼⠃⡠
⠀⠀⠀⠀⠈⠂⠒⠋⠒⠓⠤⠴⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⢇⣀⠄⠐⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⢈⣀⠠⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠶⢲⡆⠀⠘⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣀⣤⠴⠦⣄⡀⠀⠀⠀⠀⠀⢁⣴⠋⡀⠀⣴⣏⣤⡄⠀⣴⣋⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠙⠏⠁⠚⠉⠀⠀⠀⢸⡇⣀⡀⠀⠀⠀⠛⠛⠋⠁⠀⠉⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⣀⡀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠙⢳⡄⠀⠀⣠⡤⠴⠶⠶⢦
⠀⠀⠀⠀⠀⣽⠆⠀⠀⠈⢮⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⢿⣤⣤⣤⡄⠀⠀⠀⠘⢿⣦⣼⠃⠀⠀⠀⠀⢘⡷⠶⠤⣤⣀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⢸⡷⠶⠶⠞⠋⠁⢀⠀⠤⣄⡀⠈⠙⢦⡤⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠈⢷⣶⠒⠒⠦⢤⣀
⠀⢀⣠⣾⣅⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⣾⠁⢠⠶⢦⣄⠀⢀⣀⣠⡾⠁⠀⠀⠀⢻⡄⢀⣴⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⡏⠳⡄⠀⠀⠈⠉⠓⠦⣄
⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡇⠀⣿⠀⠀⣹⡦⠈⠉⠁⠀⠀⢢⡶⠃⠘⣷⡾⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⡄⠀⠀⠀⠀⠀⠈
⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠛⠶⠚⢿⠃⠀⠀⣰⠀⠀⢸⡇⠀⢰⡿⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷
⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠘⠦⠤⠟⠁⠀⠀⣸⠇⢀⡾⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠸⡆
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣷⠀⠀⠀⢀⣀⡠⠋⠀⣾⠃⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⠂⠛⠋⠉⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⡳⣶⣶⣤⣤⣴⡿⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣴⠾⠛⣃⠀⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣠⡶⠟⣉⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⣋⣴⠿⣫⣾⡿⠛⢾⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⣤⣄⣀⣀⣀⣈⣿⣿⡿⠿⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠖⠀⠤⢄⣀⣠⣤⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢁⡀⠒⠀⠀⠀⠀⠀⠀⠬⠽⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠐⠁⢀⠀⡀⣄⠀⠀⠀⠀⣀⣀⣀⡈⢻⠇⢀
⠀⠀⠀⠀⠀⠀⣸⣷⡀⡀⢀⣼⠿⢋⣡⡴⢚⣀⣠⡴⠞⠛⠋⢀⣠⣴⣾⠿⠋⠉⠉⠻⢿⣶⡾
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣾⣿⡴⢛⣭⠶⠛⠉⠀⣀⣤⣶⣿⠿⠿⠓⠒⠲⠶⢦⣤⣀⣀⠙⠁
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣾⠟⠀⢠⣴⣿⣿⡿⠟⢉⣁⣠⣴⠞⠛⠒⠒⠲⠶⢶⣽⣦
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠋⣄⣴⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡃⠀⠀⢀⢠⣤⡤⢾⣿⣦
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣍⠛⠋⢉⣻⣿⣿⣿⣿⣿⣵⣶⣤⣶⣿⢿⣾⣯⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⠟⡋⠁⠀⣴⣿⣿⣿⣿⣿⣁⣘⣤⢾⣿⣿⣷⣴⣦⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⡿⠿⡿⡿⠟⠋⢻⠿⠷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠁⣝⣛⣽⠏⠀⠀⠀⠘⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡮⠉⠀⣴⣀⣀⣠⣤⣬⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡻⢟⣿⣿⣿⣿⡿⠿⡃⣀⣠⣾⠙⠿⡿⢿⣿⣿⣷⡄⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣦⣀⣠⣤⣭⡟⠁⢀⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡉⠻⣿⣿⣯⣽⠏⢀⣴⠏⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢾⣧⡄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡸⣿⣿⠋⠀⣸⠏⢠⡾⠋⣸⠟⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣁⣀⠀⠀⠀⠀⠀⢸⡟⢠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠖⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣤⡈⠁⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⢿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠻⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⡶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⡡⠐⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠊⠍⡛⠶⣤⣀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠶⠟⠁⢀⠊⠀⣠⠄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠠⡄⠀⠈⠑⠄⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢩⣆⠤⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠶⣆⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⠀⠀⠀⠀⠀⠀⣨⣶⠌⡆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢘⠟⣀⠆⠀⠀⢀⠀⠃⠉⢊⣀⣀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠠⠀⠀⠛⠀⠀⠄⢀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣇⣀⡀⢀⠌⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢄⣉⡨⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠁⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣴⡏⠉⠻⡆⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡿⠀⠀⠀⢻⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⢸⡁⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⢸⠃⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⣇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢦⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠈⢳⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠊⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠚⠷⣄⠀⠀⠀⠀⠀⠈⠑⢴⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡄⠀⠀⠈⡆⢸⡛⣧⠀⣀⠀⠀⠀⠀⣠⠀⡇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠴⠥⠴⠥⠴⠖⠋⠀⠈⠻⠧⣤⣴⣥⢤⣿⠊⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⣀⣠⣄⣀⣸⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣳⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⠿⣷⣄⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣥⣌⣿⣿⣧⠄⠀⡀⡀⠒⠻⢀⠀⠉⠙⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣾⣿⡿⢩⠡⡃⠈⠁⣧⠀⠁⠀⠀⠚⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣟⣭⣼⡂⠠⠀⠀⢨⡾⣿⡿⣘⠛⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⣏⠿⢙⣿⡰⠂⠀⡬⠓⠂⠁⠁⢀⣀⣽⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⣥⡤⠤⠂⡏⢁⢀⣰⠀⠀⠀⠁⣠⣬⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷⣯⠀⠈⢻⡍⠀⠀⣈⣵⣿⡅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣶⣿⣿⣿⣷⣶⠛⠉⠀⣹⣿⣿⣝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣴⣿⠟⡾⣯⢨⡾⠟⠟⢿⣿⢟⣯⢛⠁⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣒⣲⠿⠋⠘⠻⠇⣀⠈⣷⠄⠁⠉⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣩⣴⣆⠀⠀⠀⠀⠀⠁⠙⠋⠀⠀⠀⠀⠀⢦⠁⠀⠀⢀⣈⢉⠁
⠀⠀⠀⠀⠀⢈⣽⣿⠇⠀⠀⠀⣈⡓⠦⠈⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣀⣄⣀⡀⢀⢀⠎⠁⠢⡄⠠⡄⠀⠀⠘⡀⢠⠸⡀
⠀⠀⠀⠀⢀⡼⠛⠉⠋⠉⠉⠉⠙⣷⣷⣤⠄⢠⢹⣄⠀⢀⠀⠀⣧⣦⣼⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣄⣧⣼⢨⡀⢸⡄⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠺⢇⠀⠀⠀⠄⠀⠁⢀⠀⠙⠛⠿⠟⠘⠳⡼⢾⠛⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⢻⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡽⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⣰⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⡄⠀⢀⠀⢴⠀⠀⠀⠈⠛⢿⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢧⠤⣾⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠒⠒⠂⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢂⢀⣀⡈⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠘⠁⠶⠾⣿⣿⣿⡆⠀⢠⣿⣿⣿⡿⠿⠓⠈⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣀⠈⠛⠛⠁⠒⠀⠙⠛⠛⠀⣠⣴⣾⣿⣿⣷⡄⠀⠀⣠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⡀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡇⣠⣞⣩⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⠻⠿⠿⠿⠿⠿⠿⠛⠟⠛⠛⠛⠛⠀⠐⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀⣀⣀⣀⡀⠠⡆⢀⣀⣀⣀⣀⡀⠀⠀⣿⠀⣀⣀⣀⡀⠀⠀⢠⢀⠀⣀⣄⣀⣀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡍⠛⠉⢿⣟⣛⣋⠉⠉⠙⠛⠛⠛⠉⠉⠉⣿⡍⠙⠛⠚⠛⠯⡽⠟⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⠛⠻⠶⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⢷⡐⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡏⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠀⠀⣠⣤⣄⡀⢠⠔⠆⠀⣀⣀⠰⣇⠆⢰⠀⣦⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⣛⣯⣍⢹⣿⠷⠤⠞⠿⠷⠥⠤⠇⠀⠘⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣰⠛⠂⢀⣄⣜⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠎⠛⠛⠛⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⡀⡀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⣧⣉⣭⣿⣟⣿⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⠉⣉⡉⣙⣻⠀⠀⡁⠀⠀⠀⠀⣁⠀⠀⠀⢰⣶⣶⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⠶⠓⠀⢰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡖⣴⠾⣾⠏⣽⠐⣮⡀⣞⠚⠛⣧⣟⣶⡄⠀⠎⠉⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⣴⣖⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠚⣿⣇⣼⣄⠁⠈⠅⢈⡀⣁⠀⢄⡀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣿⠷⣶⠀⠀⠀⣤⣀⣼⣯⣈⣉⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⡘⣿⣿⣿⣿⣿⣾⣿⣿⣤⣼⣿⣿⣿⣽⣽⣿⣿⣷⣤⣤⣄⣀⣠⣾⣥⣤⣿⣿⣭⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠀⢈⠉⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⠀⠑⠀⠂⠀⠰⠂⠀⡀⣉⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠈⠀⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠠⠤⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⣀⣀⠀⣠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢿⣿⡟⠈⣿⣿⢿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣿⠿⢃⣀⠻⢿⣷⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⢄⣀⢧⣀⠄⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠳⠤⠄⠉⠠⠴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⢀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠉⠋⠉⠻⠿⠿⠿⠿⠿⠋


⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⡀⠀⠤⡔⢷⣦⣠⣀⡤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢠⡍⠉⣁⣬⣆⣻⣿⣿⣷⣯⣏⡛⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣠⣶⡿⣿⡏⢉⣉⣁⡸⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⡖⢠⣾⠟⠁⠈⠋⣠⣿⠋⠁⠀⠙⠻⠮⣝⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⡞⢠⡿⢁⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠈⠳⠙⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣾⢡⡟⢀⠀⠀⢀⣼⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⡟⠀⠄⢀⠀⣾⠃⠈⠀⠀⢀⣤⣤⣤⣄⠀⠀⢰⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⡿⢰⣾⢡⠇⣸⣿⡇⠀⠀⠰⢃⡀⢈⣿⡞⢿⡀⠸⣯⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⠗⠊⣿⠏⣰⣿⣿⠇⠀⠀⠐⠿⠏⠛⠐⠁⠈⢷⡄⠱⣦⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⢸⣦⣿⣟⠗⢶⡄⠿⢿⠿⠋⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⢧⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣠⠿⢿⣿⣯⣽⣿⣿⣆⠸⣷⡀⠀⠀⠀⠀⢀⠳⣄⠀⠀⠀⠀⠀⠀⢄⣠⣾⣷⣦⠴⢶⣆⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⣴⣟⣭⣅⣠⣤⣿⣿⣄⡈⠑⣶⡀⠀⠀⠈⢤⠘⢷⡄⠀⠀⠀⠀⠀⢀⣼⡇⣠⣶⡆⢻⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⢀⢺⣿⣋⣩⣽⣿⣿⣷⣿⢻⣿⣶⡟⠳⣄⠀⠀⠀⠀⢿⠁⠀⠀⠀⢴⣶⣋⣙⣧⣼⡏⢧⠸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⠿⠖⣉⣛⠉⠹⠏⢻⡛⠛⠛⠋⣿⠃⠀⠉⢓⠤⣀⠀⠘⠀⠀⠀⠀⠀⠀⢛⣡⠟⠋⠀⠀⣠⡿⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣶⣴⣿⡂⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠤⢄⣤⣤⣴⡶⠛⠁⠀⠀⢀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠿⠿⢾⡭⠉⢉⣛⠿⡦⠀⠀⠀⠈⢦⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣛⣾⠇⣤⣄⣴⡀⠀⡀
⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢄⠀⠈⠳⣤⣀⠂⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣸⠿⣿⡇⠘⣿⣿⡇⢻⠇
⠀⠀⠀⠀⢸⣿⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⣀⠂⠀⢹⣀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⡿⠀⣿⡇⠀⠘⣿⣧⠀⠆
⠀⠀⠀⠀⣸⣿⠀⠀⢰⡁⠀⠀⠀⠂⠀⠀⡀⠀⠘⣿⣷⡀⠉⢻⣷⣄⠀⠀⠀⢰⠋⢸⡿⠀⡇⢺⣿⡇⠀⠀⢹⣧⠀⢠
⠀⠀⠀⢠⡏⣿⡀⠀⠈⠀⠀⠀⠀⠀⣀⣢⠈⠀⠀⠸⣿⣧⠀⡌⣆⠉⠀⢤⡄⠀⠀⢸⡇⢰⠇⠀⣿⡇⠀⠀⢰⣿⡄⢸
⠀⠀⠀⣾⠀⠻⠇⠀⠀⠤⠤⠀⠀⠂⠉⠻⢿⡳⠆⠀⣿⣿⡄⢁⢹⡄⠀⠀⡤⠀⠀⣿⠀⣾⠀⠀⣿⡇⠀⠀⠈⣿⡇⠀⡆
⠀⠀⢰⡏⠀⣠⣴⣶⣶⡶⠶⠶⠶⠦⢤⣤⣀⣁⠀⡀⢹⣿⡇⢸⠈⣷⠀⢸⡇⠀⠀⠘⠀⠀⠀⢸⣿⡇⠀⠀⠀⢿⣷
⠀⠀⢸⠁⠀⠈⠓⠾⢿⡇⠉⠁⠀⠀⠀⠀⠈⠉⠝⠛⢼⣿⡇⢸⡆⢻⡄⡺⠁⠀⠀⠀⠀⠀⠀⣼⠏⠃⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠘⠇⠁⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠈⠻⣦⣄⣠⣤⣤⣴⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣻⣿⣿⣿⣿⣿⣇⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠐⠟⠋⠁⠉⠙⠙⠛⣛⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⢾⡿⠆⠀⠀⣾⠿⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣐⣀⡀⠏⣀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⠿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⢿⢿⣟⣿⡿⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡿⠟
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠛
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈
⠀⠀⢾⣿⣿⣿⣿⣿⠟⠛⠛⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣰⣿⡟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⣿⣿⡇⠀⠀⢰⡀
⠀⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠁⠃
⢰⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⣀⣀⣴⣧⣀⡀⣀⣠⡴⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠒⠒⠒⠂⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠐⠂⢄⡴⠁⠀⠀⣀⠔⠂⠀⠉⠀⠈⠉⠀⠒⠠⡀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⠁⠀⠀⠀⠀⠑⡄⡠⠊⠀⠀⢀⠤⠤⠀⠠⠤⢄⡀⠀⠈⢧⡀⠀⠀⢣⡀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⡠⠊⢀⣠⠤⠤⠤⢄⡀⠈⠢⡀⠀⠻⣤⠊⠅⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠲⡏⠁⠀⠀⠀⠀⠈⢢⠀⣰⠤⠤⡅⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠠⢄
⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⢈⠀⠀⠀⢈⠀⠈⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⠤⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠡⡐⡀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠋⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠓⠔⠁⠀⠀⠀⠓⠔⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠙⠢⢀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡴⠂⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠤⠐⠊⠑⠂⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠄⣀⠀⠀⢀⡠⠦⢄⡀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣄⡀⠀⠀⣠⠔⠒⠂⠄⠀⠀⠀⡀⠠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠹⠿⠟⠻⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠠⣶⣦⣄⡀⠀⠀⠀⠀⢫⢻⡀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⣿⠿⣦⠀⣄⣀⠈⠇⡷⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠹⠀⢟⡇⢻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⣀⣤⡀⢰⡀
⠀⠀⠀⠀⣀⠄⢀⣨⣿⡐⢄⢠⣠⡖⠀⠀⠀⣀⣀⠀⠀⠀⠀⠁⠀⠀⠈⢳
⠀⠀⠀⡰⠋⠀⣻⣿⣿⣷⡘⠿⢿⡇⠀⠀⠰⠋⠨⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⢐⣿⣿⡇⢻⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠻⣿⣄⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠁⡜
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠈⠛⠿⢷⡤⡀⠀⠀⠀⠀⠀⠀⠶⠂⢸⠁
⠀⠀⠀⢠⡀⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⠈⠐⠢⠤⢄⣀⣀⣀⡰⠇
⠀⠀⠀⠀⠹⣦⡀⠀⠀⠸⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠟⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⣿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣻⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣶⣦⣤⣤⣤⣶⡟⢻⣧⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡟⢻⣿⣯⣨⣿⣿⣿⣷⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠛⠿⠿⠛⠉⠻⠭⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣄⣠⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿