Santa Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣴⣲⣶⣦⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠤⠐⠛⠛⠛⠛⠠⠶⠤⠶⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡧⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠿⡘⠦⢤⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⡶⣄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡂⢈⠉⠿⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⡳⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠋⠉⠐⠛⠚⠀⣹⡟⠀⠀⢘⠷⣶⢶⣟⡛⠓⠶⣄⡀⠀⠀⢸⣿⣄⠀⠀⢸⣿⣿⣿⣯⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠈⠋⠙⠷⠆⠙⠂⠁⠀⠀⠀⢿⣵⣉⠁⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⡁⠀⠀⠀⠀⠀⡐⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣊⣏⢠⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠁⠄⠀⢀⣤⠾⠛⠛⠻⠷⠾⠿⣷⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⣧⡿⠃⠸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣄⠁⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⡏⠇⠠⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⠦⣤⣤⣤⣴⣶⣶⡴⠤⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠠⣤⣠⠞⠋⠀⠀⢹⣤⠤⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠋⠛⠉⠛⠿⠛⠛⠟⢳⢦⣤⣀⠀⠀⠀⠑⣠⣴⠇⠀⢸⡏⡄⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠀⠉⠁⠀⠠⡞⢀⡇⠀⣠⠼⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠨⠀⣞⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀
⠀⠀⠀⠀⠀⠀⢈⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⡄⠀⠀⠈⠱⢶⣴⡤
⠀⠀⠀⠀⠀⠀⠈⠀⠹⣆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠞⠋⠀⠀⠀⠀⠀⡠⣧⠃⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠏⠀⠀⠀⠀⡰⢧⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⣄⠋⠀⠀⠀⡠⠈⠇⠸⣄⠀⠓⠂⡜
⠀⠀⠀⠀⠀⠀⠀⢈⠁⠷⠘⠣⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣿⠁⠘⠀⡀⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⠆⣰⡇⠀⢀⣼⠇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠀⠁⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠤⢤⠴⠊⢹⡗⠀⢊⠏⠀⠀⠀⠀⠀⠀⢻⡄⠀⣰⡆⠀⠀⠀⣰⠏⠙⠖⣦⠟⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⡎⠀⠀⢀⠀⠀⠀⠀⡀⠘⠃⣿⠀⠀⠀⣰⠋⢀⣠⡞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠠⣿⠀⣀⡄⢿⡀⠀⠀⠀⠙⠠⠀⠁⠀⣠⠞⠁⠉⠋⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠁⠀⣴⢻⡞⠉⠀⢸⣧⣀⠀⢀⣠⠆⢀⡴⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⡇⠀⠀⠀⢾⡏⢸⠟⢻⡇⢰⡏⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠚⠀⠁⠀⠀⠀⣼⠁⠀⠀⡸⠀⢸⡾⠀⠸⣿⠀⠙⠗⠀⣷⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠃⠠⠤⠊⠀⠀⠈⠇⠀⡄⠋⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⠀⣀⠤⠒⠉⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⢀⡳⠤⢤
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠋⠀⠈⠁⠀⠀⠀⠙⢄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⠜⠈⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⢠⣠⣀⣷
⠀⠀⠀⢀⣇⠀⢀⡴⠤⠤⢄⣀⣁⠂⣠⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠁⠙⡂⠦⣵
⠀⠀⢀⠃⠘⡔⢣⣀⡀⢀⡦⠚⢍⡏⠀⢀⡠⣰⠟⠂⠒⠀⢀⠀⠀⠈⠀⠀⠰⡇
⠀⠀⡎⠀⠀⠈⠀⠁⠀⠁⠲⠤⠊⡠⢲⠁⠀⠁⠙⠁⠒⠐⠋⠀⠀⡤⢀⠀⢰⣉⠉⠉⠓⢦⣠⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢰⡔⠉⠀⢠⠀⠀⠠⠀⠀⠀⣢⠤⢄⣰⠼⠿⠿⡏⠁⢑⠄⠀⠀⢹⠗⡖⠦⣄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠴⠋⠁⠀⠀⠀⠀⢸⡃⠀⢀⠇⠀⠀⠀⢹⠖⠛⠂⠤⠤⠊⠀⢸⠀⠀⢵⡄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞⠀⠀⠀⠀⣠⠔⠂⠀⠛⠿⠅⢀⡀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠱
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⠠⠊
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢰⠁⠣⡁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡳⠤⠠⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢣⠀⠀⠀⠀⠀⡠⠃⠀⠀⠈⠢⢄⣀⣀⣀⣀⠀⠤⠴⠤⡖⠅⠤⠙⠈
⠀⠀⠀⠀⢣⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢀⠄⠒⠚⠤⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⣠⠛⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠳⡄⠀⠀⠀⠀⠀⡴⠏⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠈⠐⠒⠒⠊⠋⠀⠀⠀⠈⠑⠒⠼⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠞⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠓⠦⠤⣀⣀⣀⡀⢸⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠄⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠘⠁⡈⠛⠆⠁⡠⣤⣶⡿⠛⠓⠎⠻⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⢂⣀⠰⡁⠉⣠⣾⡅⣈⠉⠀⣀⣀⠀⠀⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢀⠉⠀⠀⠀⢁⡀⠙⠛⠃⡀⢈⠀⠀⠘⠳⣦⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⣿⣌⡀⠐⠀⢰⣾⣇⠀⠀⢀⡜⠈⠑⠦⢤⠀⠉⢳⡀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠆⣀⣀⠀⠀⡄⠹⠻⣿⡦⠛⢙⡏⢿⣿⣿⡿⣁⠀⢀⠀⠀⠀⠀⠀⢹⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣆⠀⠘⢸⡀⣸⠃⠈⠀⠉⡇⠀⠀⣧⠀⠈⠙⢿⣿⣾⢷⣦⣄⣀⣴⠶⠟⢰⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⡟⠆⠀⠀⡇⡿⠃⢀⣦⠀⢳⠀⠀⢿⠀⠀⢐⣿⣿⣿⣼⠟⢿⡇⠀⠀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠃⡀⠀⣀⠟⣠⣴⠞⠛⠄⠀⡄⠀⠸⠆⠀⠼⣛⣽⣿⣼⣿⣷⣦⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠛⠦⠀⢭⣼⠃⠐⠋⠀⠀⠀⡖⣤⣴⣶⣶⣶⣶⣿⣿⣿⠿⠧⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⡸⢻⣿⣿⢿⠿⡿⠛⠋⠉⠀⠀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠈⠁⠀⠈⠁⠀⠀⣀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠙⠲⢦⣤⣴⠏⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⣿⢻⡘⣇⠀⠀⠂⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⢀⡔⠁⠀⠡⡀⠑⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣼⡏⠀⠀⠀⠀⣡⣤⣤⣶⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢹⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢓⡛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠛⠛⢿⣦⡈⢻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾⠟⠚⠓⠐⠒⠲⠿⢷⡄⠙⠛⠙⢿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣿⠀⠀⠀⠀⠈⠹⣍⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠘⠏⢙⣷⣼⣇⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⢀⡀⠠⠀⠀⠰⠟⠅⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣶⣶⡤⠶⡶⣻⣷⣶⣶⣶⣄⣀⣸⣷⡀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⢯⣦⡀⠀⢹⡿⠶⣶⣭⡼⠉⢿⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠷⣿⠛⢿⠀⠀⣈⠉⠁⠀⠀⣀⣼⣽⡇⠀⠀⠀⠀⠀⣼⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀⢠⣎⡀⠁⠨⠳⢶⣦⠰⠛⣷⣿⡇⠀⣀⣰⠁⣼⠏⢩⡄⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠿⠷⣿⠟⣩⣤⣄⣀⣀⣤⣤⡾⠋⣹⠀⢀⠾⣡⡼⠋⠀⠈⢃⢸⣿⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣷⣴⣷⣿⣯⣙⣉⣤⡿⠛⣋⠶⠒⠛⠦⢺⣾⠳⣤⠾⣏⠝⠉⠁⠲⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⢿⠋⣿⡏⢹⠻⢹⠛⠁⣴⠏⠀⠀⠀⠀⠈⢻⣷⡷⢋⡤⠄⠒⠒⠐⠀⠈⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⠀⠞⠙⣧⢀⠄⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠸⢫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠿⣿⠀⡀⠀⠉⢁⡞⠁⠀⠸⣿⣦⣤⣀⣀⣠⣴⣿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡖⢹⣧⡇⡀⠈⠉⠀⠀⠀⠀⠈⣻⣿⣿⣿⠏⢻⣿⠁⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⣨⢿⠇⣧⡀⠀⠀⠀⠀⠀⠈⠈⠁⠃⠁⠀⣿⡏⠀⣰⣿⠏⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣼⡿⣿⣦⣤⣼⡟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⣿⠋⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢀⡔⠒⠀⠐⠛⠛⠒⠚⠻⠟⣧⣤⣹⡀⠀⠀⠀⠀⠀⢢⣀⣠⡄⢸⣿⡇⣿⢃⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠸⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡄⣐⣃⡀⠀⠐⠉⠉⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸
⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠛⠉⠉⠙⠉⠓⠓⠒⠾⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠘⣹⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣉⣼⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⢷⣿⢃⡿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⢿⣾⣇⠸⣇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡌⠂⣿⡄⣧⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡜⢿⣄⡘⢷⣽⡿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣤⡒⠀⠀⠀⠀⠘⣿⣿⣷⣄⡙⢿⣎⠻⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢷⣶⣶⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⣶⣤⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⢤⠤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⠗⠀⠀⠀⠀⠀⠀⠀⠘⠤⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠰⢧
⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠯⠔⠛⠒⠃⠀⠀⠀⢎
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣽⠁⠀⢀⠀⠀⠀⠀⢀⠿⡞⠁⠀⠀⠀⠀⠀⠐⠲⠄⠀⣤⢃⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⡀⠈⠁⡄⠀⠀⡖⠋⢀⣄⠛⢀⠀⠀⠀⠀⢊⠉⡉⢃⡀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⢳⠀⢸⣠⡄⠈⠁⢠⡞⠐⠤⠟⠑⠂⠀⠙⠈⠉⠂⣀⠀⢀⠌⢠⢞
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠛⡄⠀⢠⡼⠀⠀⠀⠃⠤⡀⠀⠀⠀⠀⢀⡬⠧⣄⡀⢺⢢⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⠳⢾⠀⢨⠀⡄⡖⠀⡤⠂⠀⠈⢙⠅⡀⢀⡀⠁⣋⡼⠀⠁⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡜⢀⡜⠀⠈⠂⢇⣑⡊⠀⣠⠤⠖⠋⣐⠈⠅⠈⠉⠁⠀⠀⠀⠀⠿⢀
⠀⠀⠀⠀⠀⠀⢴⣟⠁⠀⠀⠃⠀⡝⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣆⠀⠀⠀⠀⠀⢀⠞⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢶⣆⣀⡳⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⢦⡀⣠⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⡀⠀⠀⠀⢈⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣇⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢤⣤⡬⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠂⢲⣄⢆⠀⠀⢈⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢳⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡠⠖⠈⢀⠀⠰⠦⣄
⠀⠀⢸⣦⣾⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠺⡟⠛⠛⠛⠛⠲⠉⣛⡶⣄
⠀⠀⣸⣿⢿⣿⣷⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠞⡓⢀⣠⣀⣀⣈⠻⣿⣷⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠔⢾⣿⣷⠾⠟⠉⣁⣤⡶⠞⠂⠀⠀⠀⠀⠀⢠⡟⠡⢊⣾⠋⠁⠀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⣾⣧⣶⠀⡆
⠀⠀⠱⠾⠿⣛⣵⡾⢻⣿⣟⢁⣤⡤⠀⠀⠀⠀⠀⢺⣿⠇⢱⡿⠅⢠⣀⡀⢀⣠⣤⢿⣿⣿⡟⠀⠀⢰⡆⢿⣿⣿⣾⠃
⠀⠀⠀⠀⠾⣿⣿⣷⣾⣿⡟⢻⡉⣁⣴⣶⡶⠖⠀⢸⡟⢠⣿⡁⠙⠛⠛⠁⢸⡟⠛⢨⣿⣿⠁⠀⠀⣾⠇⢸⡿⢋⣉
⠀⠀⠀⠀⠀⣬⣱⣿⣿⣿⣷⣿⣿⠛⠉⢁⣀⡀⠰⣿⣅⢸⣿⣳⣄⣠⠠⠤⠸⢿⣄⣸⣿⡏⠀⠀⢠⣉⠀⠀⠀⡈⠉⠛⡀
⠀⣀⣠⣤⣼⣿⣿⡿⣧⣼⣿⣿⣷⣶⡿⠟⠉⠀⠀⢿⣿⣟⣿⣯⡏⢉⣤⡶⣷⣄⠀⠙⡿⠃⠀⠀⢸⡿⢀⣷⠄⠀⠀⣾⡿
⠀⠘⠻⠟⠉⠁⠀⣼⣿⣿⣿⠿⣿⡉⢽⣶⣤⣤⣀⡘⠻⠿⠛⣿⠅⠚⠋⠰⠀⠈⠳⢶⣴⣶⣶⣦⣿⠁⢘⠿⠶⠄⠤⢉⣀⣤
⠀⠀⠀⠀⠀⢀⣴⠟⢻⣿⣿⣤⣿⣿⣾⣿⣿⣿⣏⣤⣰⣿⡬⠁⠒⠣⠰⠆⠀⢀⠀⢈⢻⣿⠋⠉⢠⠒⠻⡀⠀⠀⠀⡟⠉⢁⡆
⠀⠀⣀⣠⡿⠋⠉⢐⣾⣿⣿⠿⠻⣿⣿⣿⣿⣿⣏⣿⣿⣟⠀⠀⠐⠄⢀⠀⢦⠀⠠⠈⢹⣿⣄⣄⣤⣭⣾⣿⡄⠀⠀⠛⢀⣽⢡
⠀⠈⠀⠉⠀⠀⢐⣿⣿⣿⣇⠈⠓⣿⣿⣿⡿⢃⡼⠛⢿⣿⡇⣤⠀⡀⠀⠀⠀⣀⠀⢀⣼⣿⡏⠝⠛⠛⠉⠁⠔⠀⠀⠐⣮⠙⠇⠀⡁
⠀⠀⠀⠈⠿⠶⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣹⣿⣆⠀⣸⣿⣧⣈⠀⠀⣴⡞⢠⠀⠀⠀⢸⣿⡧⠀⣉⡠⠀⢢⡄⠀⠀⠀⠀⠛⠀⢻⣇
⠀⠀⠀⠀⠀⣤⣶⡈⠛⢶⣿⣀⣿⣿⣿⣿⣿⡟⠁⠉⠉⡹⠿⠿⠶⢦⡘⠃⣤⣈⡀⠂⣼⡏⣐⣆⠀⠂⠀⣠⢿⣿⡆⢠⣄⣁⡴⣿⡿⣧⡀
⠀⠀⠂⠀⢻⡽⢿⣿⣾⣿⣿⠋⠛⢟⡛⠛⠿⠁⠀⠀⠀⣠⡼⠃⠆⠀⢹⣿⡾⠛⣫⠞⢻⣷⣿⣻⣷⣦⣀⣾⣿⡿⢁⣼⣿⡇⠸⣧⠛⣻⡇
⠀⠀⠀⠀⢠⡄⠀⢺⣿⡿⢇⣤⠈⠊⡱⠉⢀⣶⢾⡶⢦⣾⣯⣤⢈⠁⣜⣿⣤⣿⣷⢀⡀⢸⣿⢿⣿⢻⣌⣉⣿⣤⣼⣾⣿⣿⠀⢹⣾⣻⣿
⠀⠀⠀⢠⡏⠀⢀⣾⠋⠀⠀⡟⠻⣧⣴⢀⣾⢃⣾⣷⣿⣿⣷⣾⡄⢀⣽⣫⣾⣿⡿⠻⠀⣾⠃⣿⣇⣀⢻⣿⣿⣿⣿⣿⣿⣵⡆⢸⣧⣿⣿
⠀⠀⠀⢼⠃⠀⣾⣿⠟⠋⠀⠄⠒⣈⠛⢿⣯⣾⣯⡼⠧⣈⠋⢠⣿⣿⣿⣿⡿⣿⣟⢓⡞⠙⠿⠟⡭⠁⣄⠬⣽⣛⣿⣿⣿⣿⣷⢘⣿⣿⠏
⠀⠀⠀⠸⡆⠀⣼⣿⣿⣿⣶⣶⣶⡌⢠⣤⣤⠙⠿⢿⣦⣴⣿⢿⣿⡿⠟⣣⣹⣿⣟⠸⡀⠀⠐⠂⣤⠀⠀⠉⠁⠘⢿⢹⣿⢻⡿⣻⠿⠃
⠀⠀⠀⠀⢿⡀⣿⣿⣿⢿⣿⣿⠟⣠⠀⡤⠀⢀⡛⠛⣻⣿⣭⡆⢡⣴⡿⢇⣀⣀⣈⠂⢟⠀⠀⠀⠻⠀⠀⠀⠀⠠⠀⠈⣿⢈⡏⠁⢀
⠀⠀⠀⠀⠀⢳⣿⣿⣿⣾⣿⡏⠀⣹⣿⠀⠀⠀⠬⢺⣿⣆⠙⢿⣏⠙⠁⠻⠛⡉⢿⡀⡿⠀⠀⣤⠀⠀⠀⢀⡀⡖⠀⠶⠀⣈⠃
⠀⠀⠀⠀⠀⢸⣯⢹⣮⣿⣿⣶⣄⣽⣿⣷⣤⣄⣶⡟⣿⣿⣷⣄⠹⣧⡀⠁⣶⣵⣤⡀⣿⣦⠀⢉⡀⠀⣠⣯⠀⣞⢉⣠⡄⠙⠁⠀⢀
⠀⠀⠀⠀⠀⣸⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⣹⣿⠀⠹⣿⣿⣿⣧⡘⢿⡌⠛⣉⣻⡇⢸⣿⠃⣼⣧⡄⢈⣠⡄⡿⠉⠁⣀⠀⠇⢠⣼⣧
⠀⠀⠀⠀⠀⠁⢰⣿⣿⣿⣇⢿⡿⣹⠿⠟⠀⢿⣿⡇⠀⢹⣿⣿⠻⣿⣦⡹⢦⣉⠿⠀⢺⡇⠀⠸⠀⣰⠟⠁⣴⠀⠀⠀⠛⢖⠰⢶⣿⡿⣷
⠀⠀⠀⠀⠘⡄⠀⣻⣓⠽⠿⠈⡿⠋⠀⠀⠀⢸⣿⣇⠀⢹⣿⣿⣦⡈⢿⣧⠈⣿⣷⢶⣶⣦⣤⣴⣤⡿⢄⣾⣗⡀⣽⠆⢠⡌⢠⡞⢫⣴⠏
⠀⠀⠀⠀⠀⣷⣾⣿⣿⣶⣦⢼⡟⠀⠀⠀⠀⢸⣿⡇⠀⠈⣿⣿⣿⣷⣄⠹⣆⢸⣿⣷⣶⣾⣤⢸⣇⣤⡀⢠⠉⢱⣮⠀⠀⣨⣏⣽⠋⠁
⠀⠀⠀⠀⣠⣿⠟⣦⠀⠘⣯⢸⡏⢶⡄⠀⣼⡟⣿⣿⠀⠀⢿⣿⣿⣿⣿⣦⡀⠁⣿⣿⣿⡟⣹⣿⣷⣿⠃⣠⣶⣩⣿⠖⢸⡇⠘⡇
⠀⠀⠀⠀⣿⣷⣾⣼⣷⣠⣿⣧⣾⣿⢿⣦⣸⣿⢿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣦⣈⣳⣶⣾⣿⣿⠟⢡⣾⣿⣿⢿⣷⡄⢸⠇⠄⣧
⠀⠀⠀⠀⣿⣟⣿⠛⣿⡟⣯⡉⣩⣿⡿⣧⣽⣿⣶⣮⣥⣤⣘⣿⣿⣿⣿⡙⣷⣿⢿⣟⠻⠟⢀⣰⣿⢦⣭⡹⠿⠿⢃⡞⣰⠆⢀⡿
⠀⠀⠀⠀⣿⣿⣧⡿⣯⣿⣟⣷⣟⣳⣾⡿⣿⣿⣿⣿⠀⠀⢉⣿⡿⣿⠻⣿⣟⢿⣾⣿⣧⢸⣾⡏⢻⣼⠟⣻⣷⠀⣿⠀⠀⠈⣵⠿⠂
⠀⠀⠀⠀⠹⣿⢿⣽⣿⣻⣟⣿⢿⡛⣬⡿⣟⣿⣿⣿⠀⣤⣾⣿⣿⠈⠉⠘⢿⣦⣿⣧⣥⣿⡇⢹⣿⣧⣸⣿⡟⠋⠻⣿⣄⡀
⠀⠀⠀⠀⠀⢻⣿⣏⣿⣏⣿⢿⡃⠐⣯⣴⠿⣣⣼⣿⡘⣿⣿⣿⣿⡆⠀⠙⢪⣿⣿⣿⣿⡿⣿⣿⣿⡟⣿⣼⡇⢠⣿⢻⡏⢻⣦⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣟⣿⣧⣾⠷⠟⢻⡵⢟⣿⣿⣿⣿⣿⣿⡿⠛⠶⠀⠀⠀⠹⣿⣿⣿⢷⣿⣿⡏⣸⣿⣿⣷⣾⠋⠋⠀⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡷⣴⡶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⡀⠈⢿⣟⢸⠿⡟⣰⡿⣿⣿⣿⣿⡷⣶⡾⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠋⣸⠟⣸⢋⣀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢸⣿⣿⣿⣿⣿⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⡘⠛⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠸⣷⣿⣿⣿⣿⠀⠂⠹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⢸⠇⢸⣿⣿⡇⠀⠛⠀⠘⣻⣿⡟⠛⣇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⡇⠀⣴⢾⣆⢻⣿⡇⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⠀⠀⠀⣙⣿⣿⣧⠀⠴⣿⣿⣿⣿⣿⣿⣯⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣶⣶⣿⣷⣿⣿⣴⣄⡞⢻⣿⠿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⢻⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣿⣿⣿⣿⡿⠿⠋⠀⣸⡿⢋⣛⣿⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⠀⠀⠛⢿⣿⣇⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣶⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⢀⡀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣒⣋⣛⣲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣤⣉⡛⠿⣿⣿⣷⣄⣉⠙⠓⠒⠶⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠴⠿⡿⠿⠛⠻⠿⢶⣾⣝⣿⣿⣷⡄⠰⣄⠀⠈⠹⣷⡈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢾⣵⣿⣶⡾⠿⢧⣴⣾⣿⣷⣦⣭⣝⡻⠿⢿⣿⣦⡙⣷⣶⣶⣸⣷⡀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠙⠿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣍⠈⣿⣷⡀⠰⡄⢉⡛⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣄⢷⣄⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠙⠛⣿⣿⣿⣿⣿⣿⣷⠀⢹⣾⣿⣆⠀⣶⣤⡌⣷
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⡄⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣆⢿⣿⣿⣧⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⣀⣤⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣇⣿⠏
⠀⠀⠀⠀⠀⠀⢼⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠉⠻⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⢻⣿⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣾⡁⢀⡟⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⡴⠖⠂⣩⡿⠈⠙⢷⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⠸⡆⣰⡿⠷⣶⡚⠉⠋⠻⡟⠛⠀⡴⢀⣤⣶⣶⣆⠀⢹⣿⣿⣿⠹⣿⣿⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⡤⠻⣾⠃⣲⠶⣶⣤⢱⣴⣶⣶⣤⣴⣿⠹⠿⠟⣻⣾⣿⣿⣛⠿⢀⣄⢿⣿⠛⢿⣿⣿⠏⢉⣹⣿⣿⣿⣿⠿⠋⠉⠉⠙⠳⢶⣀
⠀⠀⠀⠀⠀⠀⣴⠋⢠⣶⣩⡞⠿⣷⣛⣭⣿⣹⣿⣿⣿⣧⣈⠓⣀⣨⣥⣤⣌⣿⣿⡇⢸⣿⠀⠏⠀⠀⠋⡏⣰⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⢸⠃⠀⠸⣿⢻⣿⣶⣿⡿⠛⡏⣿⣿⣿⣉⣟⡛⠉⠉⠉⠁⠙⢿⣿⣿⣷⠀⠁⠀⠀⠀⠀⠀⢘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⡿⠀⠀⠀⢿⡆⣿⠋⠁⠀⣸⠃⠈⠙⠛⡁⠀⣿⣦⡀⠀⠀⠀⠀⢹⣿⣿⣷⡄⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠰⢶⣿⣤⡀⠀⠀⠘⣧⠛⠀⢀⣾⣿⣷⣦⣶⣿⣷⣾⣿⣿⡻⢷⣤⣀⣀⣨⠟⣹⢁⡇⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⠜⠝⠁⠀⠀⠀⠀⠀⢠⣾⡿⢿⠟⠋⢈⠛⠉⠋⠀⠙⠛⠂⠈⠀⠀⠀⠀⠁⣼⠃⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⢀⣴⣿⣷⣶⣶⣶⢶⣤⣀⠀⢤⣀⣀⣲⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣿⠏
⠀⣽⣿⡴⠋⠀⠀⠻⡉⠒⠊⠉⠀⢀⣀⢶⣿⡟⠛⠛⠛⣉⡥⠞⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⠀⣀⣀⣀⣤⠞⠃
⠀⠈⣿⡀⠀⠀⠀⠀⠉⠓⠲⠤⠚⠛⠁⠀⠙⣿⣿⣷⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣿⣿⣿⣿⠉⠉
⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⡀⣄⠈⠻⢿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠇⠛⢼⣿
⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡈⠲⣦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢤⣤⣤⠞⠁
⠺⣿⣿⣟⣧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠈⠛⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿
⠀⠀⠀⢻⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⣄⣸⣿⣿⠃
⠀⠀⠀⠀⠻⣿⣷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡞⣿⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠉⢻⣾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⣶⢀⢠⣀⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣄⣏⣼⢠⣿⢟⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⣧⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣴⣏⠔⣱⣿⣆⠀⠀⠀⠀⠀⠀⣦⣀⢀⢀⠀⡄⡄⢰⡄⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣮⣦⣦⣦⣹⣦⣬⣙⣿⣾⣞⣢⢿⣼⣮⣷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠴⠶⠶⠖⠒⠒⠒⠲⠶⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠈⠘⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠏⠀⠀⠀⠀⠀⡠⠖⠒⡶⠒⠚⠓⠚⣏⠉⠳⡄⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣆⣀⣀⡤⠖⠛⢄⡤⠞⠳⡄⠀⠀⣼⡉⠻⡟⠉⠀⣸⠃⢀⡴⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣰⠋⢴⣋⣀⠀⠀⢸⡀⣴⣿⡧⠤⠄⣿⣷⣆⣼⢀⡀⠙⣄⣀⡽⠂⢹
⠀⠀⠀⠀⠀⣀⡤⠤⣴⠒⠲⢤⣿⠀⣹⠀⠀⠀⠈⠙⠒⠋⠉⣿⠁⠀⠀⠀⠀⡍⠀⠈⠉⠉⠉⠉⠁⠀⢀⡟
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⣿⠀⠘⢲⡀⠀⠀⠀⠀⠀⠀⠈⢓⣤⢤⣶⣎⡁⠀⠀⠀⠀⠀⠀⣀⡴⣯⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⡋⠳⣤⣀⣀⣀⣤⠤⣖⢯⣭⣭⣥⣾⢟⠗⠲⠶⠖⠚⠋⠁⢰⣃⣀⣀⡤⢺⢌⠁⣸
⠀⠀⣾⠀⠀⠀⠀⠀⠀⢐⠒⠛⠁⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠈⢦⣙⢿⣛⡡⠊⠀⠀⠀⠀⠀⠀⠀⢾⠉⠁⠀⢀⣼⡧⠋⠁
⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠑⠦⣤⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣰⠏
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⢀⠜⡽⠁⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⠶⠚⢋⡞⠁⠀⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡠⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⡞⠀⣀⡜⠁⠀⠀⠀⠀⠀⠉⠙⠋⠳⢤⣀⣀⡠⣄⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⢣⣌⣒⡹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠚⠉⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠓⡒⠦⣄⠀⠀⠀⢀⣀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣀⣀⣀⣀⡞⢱⣉⣉⣩⣥⣽⠈⣷⣾⣿⣿⣿⠿⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠿⠿⠿⠿⠿⢷⡈⠿⠛⠛⠛⢛⡴⠛⠋⠉⠁⠀⢀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⡶⠤⣄⣀⣀⣀⡉⠛⣲⠚⣏⣉⣀⣀⣤⠤⠖⠚⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⡖⠋⠉⠉⠉⠁⠀⠈⠉⠉⠉⠓⢦⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣯

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⢶⡄⢀⡙⢛⣠⣾⣿⣦⣤⣾⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢉⣉⣀⠀⠉⠙⠛⢷⣦⣀
⠀⠀⠀⠈⠛⠁⠀⠛⢿⣿⡿⠿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⢛⣻⣿⣿⣿⣿⣷⣶⣤⣈⠙⠻⣶⣄⡀
⠀⠀⠀⠀⢀⡇⢰⣿⡦⠙⡿⠛⠛⠻⠁⣠⡀⠀⠀⠀⠀⣀⣀⡰⠂⢴⡿⠶⢾⠿⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣷⣦⡀⠛⢿⣷⣦⣄⣀
⠀⠀⠀⠀⢸⣇⠀⠉⠀⠚⠀⠀⠰⢶⣶⣍⢻⣿⣁⣴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣧⡌⠻⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢀⣿⡻⣷⣶⣾⣦⣄⡄⢁⢸⠿⠿⣷⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢲⣦⣬⡝⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠈⠉⠻⣎⠻⣿⣿⣿⣿⣬⣄⣀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣶⠂⠀⠾⣿⣿⣷⡦⢹⣿⣿⣇⠈⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⣽⠟⠋⠉⢀⡀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠃⣴⣦⣛⡛⣿⡷⢌⣿⣿⣿⡆⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⢨⣻⠟⠻⢋⡀⠹⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡀⠀⠀⡤⠔⠀⠈⠛⠛⠻⢿⣿⣿⣷⣶⣶⣶⣦⣀⡀⠀⠀⣶⠄⣷⣿⣿⣶⣾⠿⠿⡿⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢷⠀⠀⠀⠀⠀⠒⢶⣄⡉⠉⠁⠀⠀⠈⠉⠉⠻⠿⣿⣿⣆⠀⠀⠀⠛⠛⠿⢏⣵⣶⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠴⠿⠿⠟⠚⠒⢶⣭⡻⢧⠀⠀⠀⠀⠀⠀⠠⠄⢸⣿⣿⡄⠀⠀⠀⢠⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣠⡶⠶⢶⣿⣿⣿⣿⠈⠳⢶⣶⣂⣤⣴⣶⡶⣧⡸⣷⠀⠀⢀⣿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣤⠀⠙⠁⠀⣰⣶⠟⢡⣿⠃⢴⡎⠉⠁⡙⠛⣿⡿⢿⣡⡜⡄⠀⢸⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣾⣿⠃⠀⠀⠀⢉⡁⠂⠀⠿⢍⣁⠀⠉⠒⠸⠿⡿⢿⣯⣼⣯⡀⠀⠘⢸⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⢠⠀⠀⣾⣿⡕⢠⣧⣴⣶⣿⣅⢰⣧⡤⠊⠉⢙⣦⡀⠀⠀⢈⣿⣿⣝⠿⣷⣶⠂⠘⣿⡟⠻⣿⡆⠀⢹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣀⣘⠁⠠⣥⡦⠀⠀⣋⣽⣄⡀⠀⠈⠉⠈⠻⠶⠿⠶⠟⠿⠟⠃⠛⠶⣶⣿⣿⠿⠛⢆⣾⠻⢿⡆⢻⣇⠀⠈⠁⢀⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢀⠔⠛⠉⠀⣀⣠⣤⣤⠤⠬⠍⣻⠟⠛⠲⢮⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⡐⢲⣤⡙⢷⣶⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⣀⣀⡠⠞⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡛⠲⣄⣀⠀⠠⣤⣤⣤⡾⠿⠿⣋⣀⡒⣿⣿⣮⣿⠫⠭⠝⠋⠉⠙⠻⣶⣄
⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠒⠞⠛⠿⣷⣤⣄⠀⠉⠁⠈⠙⠋⠀⠀⢀⠉⢻⣿⠿⠛⠁⠈⠃⠀⠀⠀⠈⠀⢦⣻⣿⡓
⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣿⣧⠀⠀⡀⠀⠀⠀⠀⠀⠱⣌⣿⠏⠀⠀⠀⠀⠀⠀⠀⠒⣠⣎⣿⣿⣿⣷
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⢀⠆⠠⠀⡀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⣹⠀⣆⠀⠀⢠⠀⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⡟⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡀⠀⢧⡄⢁⡀⢠⠰⠁⠀⠀⠀⠀⠀⠀⠋⠀⣯⠀⢀⡞⠀⣽⡏⣿⡀⢦⡀⠀⢀⡀⠀⠀⢨⣿⣿⣿⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠴⠃⠠⠈⣢⣈⣤⢿⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠙⠂⠸⠋⠀⠻⢠⣿⣿⣦⣽⣶⣬⣛⣶⣶⣶⡟⣿⡟⢡⣌⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣧⠠⡍⠀⢸⣆⠀⠘⠀⠀⢸⠇⢀⣾⠂⠀⡄⠀⣿⠀⣠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡔⢸⣿⣆⠻⣮⣷
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠇⠈⢀⡼⠋⢠⢸⠳⠊⠀⠀⠀⣾⠀⠘⢿⣃⣾⣷⡄⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⢟⣀⠙
⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡈⢋⠀⡶⢄⠀⡨⠀⢀⣼⡇⢸⣧⣌⢻⣿⣿⠁⣾⠁⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣿⠃⣼⣿⣿⡾⠿⠃
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⣴⠂⡈⠀⠸⠋⣀⣶⠋⠈⢙⣸⣿⣿⣮⣿⣿⡽⠃⠈⣼⣿⣿⣯⣻⣿⣿⣿⣦⣌⠙⣷⣰⣿⢟⡁⣴⠿⠆
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⣶⠏⠂⠁⠀⡰⠊⠈⠁⠀⣄⣾⠟⣿⣿⣿⣿⣿⠇⢀⣾⣿⣿⣿⣽⣿⣿⣿⣿⡿⠃⢀⡾⠟⢁⣙⣻⣿
⠀⣄⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠚⠉⠃⢀⡔⠀⠀⠀⠀⠀⠀⣼⡿⠋⣸⣿⣿⡿⣿⡿⢂⣽⣿⣿⣿⣯⣾⣿⡿⠋⣁⠀⡌⠉⠀⢁⣼⣿⡿⣫⡦
⠀⠣⡤⠛⠀⠀⠀⠀⢀⣀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⡤⢀⣾⠿⠃⣰⣿⣿⣿⢋⢷⣼⣿⣿⣿⢻⣿⡖⣿⣿⣷⣾⣧⣶⣶⣧⠀⢻⣿⢟⣵⠿
⠀⠀⡄⠀⠀⠀⡴⣪⠛⠉⠁⠀⠠⢀⣀⣀⣀⠀⠀⣀⣤⣶⡾⠛⣩⣾⡿⢛⣠⣾⣿⣿⡟⣧⡼⠿⣯⣿⠽⠟⣛⠉⠥⠙⠒⠛⠛⠒⠒⠾⠧⠴⠿⠓⠚⠃
⠀⠀⢧⣴⠀⠈⠔⠁⣠⡶⣿⣿⣿⣿⣿⠋⠁⣠⣾⣿⠟⡉⢠⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣭⣿⡿⠕⠀⠂⠉⠀⠀⠰⠚⠉⠉⠉⠛⠛⠲⠒⢤⣀
⠀⠀⠀⠁⠀⠀⠀⠘⠩⣷⣶⣿⣿⣿⡁⠀⠈⢠⣿⣇⠊⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠙⠻⠿⢿⣯⠄⠀⣾⠿⠇⠘⢛⣹⣽⣿⣿⠿⠻⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠟⠛⠛⣛⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠋⠀⢤⣀⣀⣴⣿⣿⣟⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣥⢦⣿⣴⣿⠿⢿⣿⣿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣼⣿⣿⣻⠞⣿⡍⠉⠃⣰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣼⣿⠀⠈⠁⢀⣀⡉⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⣩⢿⣤⠀⢀⠈⡙⡅⢀⣒⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣟⠋⠑⠋⠿⣿⣿⡿⠉⠉⠛⣿⣿⢏⡀⢺⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠛⠻⣿⣿⣿⣏⣠⡆⠈⠉⠰⠛⠛⠛⠈⠩⠐⢽⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⣠⣙⣿⣿⣿⣿⣤⣀⣠⢠⡀⠀⠈⢓⣔⣶⣿⣿⣿⣷⣤⣤⣤⠤⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣧⣄⣀⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⢰⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣈⠃⠉⠈⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣿⣿⣿⣿⣿⣿⠀⡀⠀⠸⣦⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠐⢀⢠⡍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢖⠂⠠⢾⣿⣎⠻⣿⣿⣷⣷⣯⠐⢹⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⢠⠄⢴⢲⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠆⠀⣿⣿⡿⠀⠀⠀⠀⠙⠛⠟⠋⠉
⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠻⠿⠥⠶⠶⠶⠛⠁⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⣾⡿⠟⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣄⡀⠀⠀⠀⡉⠉⡛⠛⠛⠛⠿⡾⠿⠿⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣤⣥⣄⣁⣀⣉⡀⠈⠁⠠⢆⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢉⢳⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡀⢸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠠⣹⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⢰⣿⣻⣿⣿⣿⣿⣿⣿⠟⠀⠸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠈⠀⠉⠙⠛⠿⠿⠿⠿⠿⡿⠀⠀⢀⣿⣇⠁⠹⣿⣿⠿⠛⠁⠠⠀⢂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⢀⠄⢀⣿⠿⣷⢀⡀⠂⠀⠀⢀⣤⣦⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠿⣶⣶⣦⣦⣤⣤⣬⢤⠴⠶⡾⠃⢠⡈⠻⣷⣤⣶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣶⣄⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣶⣶⣿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠐⠂⠒⠒⣖⠒⢲⡶⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠂⠉⠁⢀⠀⠀⠈⠛⠃⠐⠾⠟⡐⠉⠀⠀⡀⠀⠀⠀⢈⡒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋⠀⠀⠂⠀⠀⠀⠄⠀⠀⢀⡴⠂⠀⠀⣡⠀⠀⠈⠙⠦⡀⠀⠀⠀⡯⠋⠙⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⡢⠀⠐⠀⠴⠷⢋⣀⣀⣀⣛⣃⣀⡐⠢⠄⢀⠈⠧⠀⠘⠃⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⣠⡮⢐⣨⣴⠶⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⠷⢶⣤⣅⡂⢄⡀⠀⠀⠀⡌⠀⠒⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡶⠄⡰⢛⣥⣾⠟⠉⠉⠀⠀⠀⠀⠀⢴⣤⠀⠀⠀⠁⠀⠀⠀⠉⣿⢿⣶⣬⡠⠄⠀⠠⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⢀⠌⠐⠢⠐⠖⠚⠉⠠⢪⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⣦⡑⢄⣶⡆⠀⣄⠀⠀⡧
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠙⡄⢀⢈⣴⣿⣿⡿⠃⣤⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣦⡫⣤⡄⠀⠐⠸⠆⠳⡀
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠸⢡⣾⣿⣿⣿⢓⣿⠿⠋⠁⠀⢥⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢿⣿⣿⣿⣷⡜⡉⠀⠄⠀⠀⠀⢑
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠠⢡⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⠀⠈⠀⡎⠄⠀⠀⠀⠀⢀⣠⣤⣤⣶⣤⣄⡀⠸⡇⠀⠸⣿⣿⣿⣿⣿⡌⣤⣶⡄⠠⠀⠈⢧
⠀⠀⢠⠀⠀⠀⠀⠀⢠⢃⣿⣿⣿⣿⠏⣿⣿⣿⣿⣵⣿⣆⣠⣶⣿⠀⠀⣠⠴⠛⠉⠁⠀⠀⠘⠉⠉⠙⣷⣷⣤⣤⣿⣿⣿⣿⣿⣿⠘⣭⠅⠀⠈⠑⠈⡀
⠀⠀⠆⠀⠀⢀⠒⠀⡀⣼⣿⣿⣿⣿⣖⡚⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣿⠿⠛⠀⠀⠀⠀⣴⡔⠛⠛⠁⠈⣿⣉⢿⣿⣿⣿⣿⣿⣿⣇⢨⡁⠀⣄⡀⠀⠃
⠀⠀⡃⠀⠀⠀⠀⢰⢀⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⠑⢙⣟⣀⣴⠖⢶⣿⡀⠈⠳⠀⠀⠀⢠⠤⠄⠘⠙⠛⣿⣿⣿⣿⣿⠀⠈⠀⣄⠘⢢⢴
⠀⠐⠃⠀⠸⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⡟⠷⢿⣿⣿⣿⣿⣿⣦⡀⢉⠉⠀⠀⠀⢙⡿⠄⠀⠀⣗⠀⠈⠀⠁⠒⠀⠀⠀⠙⢿⣿⣿⠀⡇⠀⠸⡄⠀⢘
⠀⠀⡀⠀⠀⠀⣰⠀⢸⣿⣿⣿⡟⠛⠟⢩⠋⠐⠀⣸⣿⣿⡏⠛⢻⣿⣿⠃⠠⢀⣤⠾⣤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⡅⠀⠀⡀⠀⢘
⠀⠀⠄⡆⠀⠀⠃⠀⠘⣿⣿⣿⡯⢆⡐⠀⠀⣠⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⠆⠀⠀⠀⠀⢻⣿⢀⠀⠀⠀⢧⡀⠸
⠀⠀⠀⠸⡀⠀⠀⠀⠁⢻⣿⣿⣇⡞⠁⠀⠴⢿⡿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⣤⡤⠔⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠈⡟⣰⡆⠀⠀⠈⢻⡆
⠀⠀⢠⠀⢳⡀⠀⠀⢈⣼⣿⡟⠊⠄⠈⠀⢀⡀⠀⢘⣿⣿⣿⣆⠀⠀⠀⡀⣰⣾⣿⣿⠛⠻⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠇⠀⠀⠺⣤
⠀⠀⠀⢆⠀⠙⣢⣴⣿⣿⣿⠃⠀⠀⠀⠋⠀⠀⠉⠀⢹⣏⣈⠻⣿⡆⠀⡁⠀⠀⠅⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠰⡟
⠀⠀⠀⢨⣴⣾⣿⠏⠉⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⡌⠀⠘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣴
⠀⠀⢠⣾⣿⠏⣴⠋⠀⠨⠆⠀⠀⠀⠀⠠⠈⠀⠒⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⣰⣿⣿⠇⠺⠁⠀⠀⠀⠀⠀⠀⠀⠄⡍⢀⣼⣿⣿⣿⣏⢀⣤⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⢠⣿⣣⠀⡄⠰⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⢺⣿⣿⣻⣿⣿⣻⣟⡀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⣿⡁
⢸⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⡌
⠘⠃⠸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⠁⠀⠈⠙⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠻⠛⠿⢿⣿⣿⣧⠀⢐⢿⢀⡄⡄⠀⠀⠀⠌⢀⣷⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⡄⢀⢺⣧⠁⣶⢆⣠⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠶⣾⡾⠟⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⠦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣼⣿⣿⣦⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠼⣇⢠⣿⣿⣿⡿⣟⣿⣿⣯⢿⣿⣿⣠⣷⣬⣿⣿⣿⣿⣿⠿⢿⣿⣷⡶⣾⣻⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡴⠷⢶⣅⣾⣿⡟⠽⠟⠋⠉⠉⠽⣤⠹⡽⠉⠉⠉⠉⠉⠛⠛⠻⠿⠶⣌⠻⣿⣿⣏⡟⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⣷⣀⣼⣿⡋⠁⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠌⠳⣜⢿⣿⣯⣱⡿⠂⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡉⡿⠉⡿⣿⢿⡱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣷⡿⣿⣟⣿⡛⡩⢛⣶⣿
⠀⠀⠀⠀⠀⠠⣤⡴⠛⡻⢿⣶⠃⠉⢹⡇⡧⠀⠀⠀⣀⣀⣤⡴⠶⠶⠚⠛⠻⠷⠶⠶⣶⣤⣤⣀⢀⣀⣽⣿⣯⢰⣦⣄⢹⠈⠉⠳⡿⢯⡈⠳⢦⣤
⠀⠀⠀⠀⠀⠀⣿⠑⢄⡘⢦⣍⠟⣿⣾⣇⣹⣶⣶⡞⠉⣁⣀⣀⣀⣀⣀⡀⢰⠀⠀⠀⠀⠀⢀⠉⠛⠿⣿⡟⠿⣸⠃⢈⢻⡤⠤⢤⣹⡄⠉⢻⡥⣿⡀
⠀⠀⠀⠀⠀⢼⡃⢠⣀⠉⣘⣿⣷⢹⣿⡉⣽⡿⢿⡇⢈⣩⣾⣭⣦⣭⣿⡟⠊⣿⣿⢿⣛⣛⣿⣷⠶⣤⡌⣝⢿⣷⣴⡟⢦⣤⣤⠄⢈⡓⠛⠳⡶⠛⠛⠂
⠀⠀⠀⠀⡿⢦⣽⠦⣬⣿⡻⣧⣄⡤⣾⠃⠟⣠⣿⡗⠀⠉⠀⠙⠛⠿⠟⣦⠀⢸⡟⠛⠭⢾⣿⡯⢷⡟⢈⡛⢠⣟⣝⣠⣶⡀⣿⡤⣾⡇⠀⠀⠀⣠⣤⣄⡀
⠀⢀⣠⠼⢃⣤⣢⡣⣶⣻⣻⣆⠛⣿⣿⠀⠀⠉⠸⣧⠀⠀⠀⠀⠀⠀⣀⠹⠀⢸⣧⠈⠙⠋⠉⠀⠈⠁⢸⣿⢈⣿⠛⣿⡟⡟⢋⡶⢁⣙⣦⣄⣸⠏⠿⣽⣷
⠀⠀⣷⢰⠎⢉⣥⠞⠉⢉⡭⢭⣹⣿⣿⢀⠄⠀⠀⣸⡇⠀⠀⠀⢀⣴⡀⠀⠀⢸⢯⡀⠀⠀⠀⠀⠀⠀⢀⢻⡺⣿⣼⣿⣥⠖⡛⠉⢉⣾⢟⠉⢻⣦⣤⣾⠟
⠀⠐⢿⡈⡾⣿⢃⡖⢻⣿⣦⣄⣾⣇⢷⠃⢀⡤⢠⡟⠛⢁⡄⠐⠁⠈⠙⣷⣶⠟⠞⠻⠦⣤⡄⠀⢤⣀⣾⣿⠁⣿⣟⣛⡉⢉⡽⡿⢾⣿⣽⣲⣾⣷⣿⡟
⠀⠀⢈⣿⣥⠿⠾⡇⢰⣟⠛⢻⣿⣾⣧⡏⣼⣷⣄⣠⣾⣃⣀⣠⡤⣰⢀⣹⣿⣤⣄⠀⠀⠙⢟⠛⢛⡻⣟⣫⣾⣼⡃⠸⢳⣾⢔⠕⡟⣽⡟⡹⢻⣿⣭⡶⠖
⠀⠀⡼⠋⠀⢠⣞⡋⠙⢿⣷⠟⢻⣭⡿⠇⠀⢩⡿⠿⡟⠹⠿⣿⣻⢿⣟⣛⣛⣾⣯⠻⢦⢴⣿⡷⢬⠛⢛⣿⡟⣹⣵⠾⢿⡷⠿⠾⣾⡛⠋⠻⣝⢦⡾⠁
⠀⣀⣄⣀⠀⢘⣏⣷⣖⡾⢿⣴⣺⣿⣳⡀⠐⠋⠀⠀⣰⠀⣴⡻⢿⡿⠛⢻⣿⠀⢿⣷⡶⣄⢀⡀⣼⣧⣙⢿⣱⣿⣵⣶⣾⣍⢂⡬⠶⢾⣄⡶⠛⠛⠃
⣾⠛⠳⡟⣿⠅⠚⠉⠉⠀⢀⣿⡫⣧⣾⢁⡦⠀⣤⢸⠁⠘⣱⠁⠀⠀⠀⠀⠛⠂⠘⢟⣄⠈⠻⣵⣌⠸⣟⠃⢻⣴⠛⠛⣿⣯⣿⣤⣤⣶⣿⣦⡀
⠻⣤⣸⣷⠏⢀⡼⢛⠳⢶⣻⣗⣺⢻⣿⠟⢁⢞⡷⠉⠀⠀⠀⠀⠀⢀⠆⠀⠀⢠⣄⠈⢿⡀⠀⢹⡷⡇⠈⠀⡾⣿⠳⠶⢿⡯⠒⢹⡿⢏⠉⢈⡇
⠀⠀⠀⠘⣏⠩⢤⣕⡿⠛⣁⠜⠏⢀⣿⡆⢨⡟⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠘⣿⡀⠀⠁⠀⠀⣿⡆⠀⢀⣿⡯⢿⡿⢿⣷⣤⣖⡻⠿⠾⠛⠁
⠀⠀⠀⠀⡿⢀⣶⠶⠾⣿⠋⢸⣿⢏⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⠀⣾⡇⠀⠀⠀⢰⣿⠀⠀⣼⣿⢿⣟⢁⢏⣜⡭⠛⢷⣀⠀⣠
⠀⠀⠀⠸⠿⢭⣀⠀⢰⠃⠀⠸⣴⢯⠿⣷⣄⠉⣙⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣰⠉⢹⡀⣉⡻⣾⢡⣄⣄⣄⣹⡻⣿⡄
⠀⠀⠀⠀⠀⠀⠉⢷⡈⣠⡞⠓⠛⡐⣶⣹⣿⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⠀⠀⢿⣸⡿⣽⣻⣿⣿⡄⠈⣇⠈⠻⡍⢻⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣿⣠⡾⢣⢹⣌⠛⣿⣇⠀⠀⣠⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠃⢸⡀⢠⢠⣿⠛⣉⡄⠉⢿⢇⠀⠸⣀⡤⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡟⠁⢸⣬⣾⠉⣛⡋⢳⣀⢘⡬⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⢠⣀⡞⠁⣸⣿⠃⣷⣈⣓⣠⡾⠛⠛⠳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⡀⠀⠇⢸⠦⣸⣿⣟⡿⠻⣿⣁⠀⠀⠚⣧⠀⠀⠀⠀⠀⠀⢺⣏⠤⣶⡽⠋⣠⡷⣾⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⢤⣌⣠⠏⠹⣯⣶⣤⠏⠙⣧⠀⠀⠻⣷⡀⢀⡀⢠⣷⢸⡆⠀⣷⢹⠚⢿⣟⠻⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠉⠀⠀⠀⠈⠳⣤⣀⠹⡇⠸⣿⣶⣿⠀⢀⡴⢷⣨⣤⣌⣁⡷⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⠀⣿⠛⣽⠛⠋⠀⠈⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡞⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⣀⣀⢀⣀⡀⣹⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣰⠟⠀⠀⠀⣄⠉⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⣷⣚⠰⡥⠘⠗⠊⣡⢹⣿⣿⣿⠉⠓⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢈⣢⣥⣤⡤⠾⠿⠟⠋⠹⠟⡿⠿⠋⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢴⣶⣤⣶⣶⡾⠁⠀⣤⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣟⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠦⣤⣼⠿⠛⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠈⠙⠛⢿⠀⣉⣉⠙⡿⠿⠟⠛⠋⠁⠀⣸⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠸⣿⣿⣷⣶⣤⣤⣼⠀⡟⢹⠃⣇⣀⣀⣤⣴⣶⣿⣿⠃⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠻⢿⣿⣿⣿⣿⣦⣤⣤⣤⣿⣿⣿⣿⣿⣿⠿⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠉⠛⠛⠻⢿⡿⠿⠿⠟⠛⠛⠉⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣤⣀⣀⣀⣴⣿⣷⣄⡀⠀⠀⣀⣀⣤⣴⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣴⠋⠀⠈⡽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢦⢄⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠀⠹⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⡁⠀⠈⠉⠙⠛⠿⣷⣯⣿⣿⣮⡻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡙⢷⣮⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⣉⣤⣶⣟⣷⡬⠅⣶⣂⡄⠀⡀⠈⢢⡀⠈⢿⡇⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢾⠉⣿⣿⡭⠁⠒⠀⠊⠙⠻⠿⣷⡄⠀⠻⡄⠞⠃⣿⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠱⠀⠺⢏⡬⣙⣛⣿⣿⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⣶⣦⣶⣾⣦⠀⠨⠿⢂⣠⠀⠸⡧⠀⢀⠆⠉⠈⠉⠉⣠⡿⠋⠉⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⡀⠉⠑⣪⠙⡉⠈⡑⢨⣀⠉⠀⠀⠀⠹⢄⠀⣀⠀⠀⠀⠚⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢃⠘⢾⣷⡶⠋⢁⠀⣀⣀⡀⠀⠠⠑⢶⣶⢻⡀⠀⡀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠘⠛⠀⣾⠰⠰⣿⣿⡿⠢⠑⠤⢀⣀⠀⠀⠀⣔⠀⠛⠃
⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠨⡃⠀⢉⣀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠙⠲⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢘⣤⠀⠀⠀⠀⠀⠙⠀⠀⠀⠌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣥⣮⣍⡅⣦⣰⠄
⠀⠀⠀⠀⣀⣾⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢻⣤⡆
⠀⠀⠀⠀⢿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⣾⡛⠁
⠀⠀⠀⠀⢬⣿⣿⠟⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣈⠉⢩⣿⣿⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠁⠃⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠙⢦⣬⣴⣾⣿⣿⣿⣷⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠀⠀⡤⠀⢀⣀⠀⣀⡀⣿⣦⣀⣀⠀⣀⡄⢀⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⠃⢠⠀⠀⠚⣠⣿⡐⠿⢿⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠛⠛⠛⠻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡼⠉⣠⡴⠒⠛⠛⠒⢦⣄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠋⠁⠀⢀⣀⣀⣀⠀⠈⠙⠻⢿⣦⠀⠀⠀⠀⠀⡀⢁⣾⡇⠈⠡⠀⠀⠏⠁⢹⡖⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢀⣤⡶⠛⠉⠛⠛⠛⠙⠳⣦⣄⠀⠙⣇⠀⠀⠀⢀⣄⠰⠁⠲⠤⠐⠀⠀⠢⢤⠖⠈⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⢄⣾⣏⡋⠐⠛⡉⠀⠀⢈⡛⠓⠘⡉⣷⡤⠟⡀⠀⠰⣿⣿⠓⠦⠤⠠⠄⠶⠦⠠⠠⠠⠖⠁⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢾⡿⣇⠀⠈⠈⠃⠀⠀⠘⠋⠈⠀⢀⠿⣇⠀⢸⠀⠀⠈⢫⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠱⢏⠀⠹⢦⡤⠴⠒⠦⠐⠒⠲⢤⣴⠞⠀⠸⠃⣼⣀⠀⠀⠘⢧⠘⣆⠀⠀⠀⠀⠀⠀⣈⠀⡼⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢦⣄⡈⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠈⠉⣀⡠⠼⢙⣿⣷⡀⠀⠘⢶⠘⠦⣦⣄⣀⡰⠒⢀⠖⠁
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠈⠉⠓⠓⠒⠉⠉⠓⠚⠉⠙⠒⠒⠊⠉⠁⠀⠸⣿⣿⣿⣦⡀⠀⠙⢄⣀⠈⢀⣀⠞⠉
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣶⣄⠀⠀⠙⠁
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⠆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣼⣿⣿⡿⠋⠉⠉⠉⠉⠛⢿⣿⣶⠀⢾⡤⣦⢀⡀⠀⡀⢰⠦⠾⠀⢶⣾⣿⣿⠟⠁⣀⠀⠀⠀⠙⢿⣿⣷
⢿⣿⡿⠀⠀⣠⣴⣿⣿⠶⠶⠿⠛⢢⣤⠀⠙⠋⠻⠞⠙⠋⢀⣤⣴⣿⠋⠉⠛⠛⠻⢿⣿⣷⡄⠀⠈⣿⣿
⠸⣿⡇⠀⢰⣿⣿⠋⠁⠀⠀⠀⠀⣼⣿⣦⣤⣄⠀⢀⣤⣤⣼⣿⣿⣿⡷⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣿⡟
⠀⠹⣧⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠙⠛⢛⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⠏
⠀⠀⠈⢳⣌⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣸⣧⠞⠁
⠀⠀⠀⠀⠉⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⢃⣬⣭⣭⠹⣿⣿⣿⣿⣿⣷⣤⣀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⢸⡟⢿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣘⠿⠾⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣷
⠀⠀⠀⠀⠸⣇⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠉⠀⣿⠀⠁⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣰⡟
⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠉⠉⠉⠉⠁⠀⢀⣿⣄⠀⠀⠉⠉⠉⠉⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣶⣶⣴⣶⣾⣿⡿⢻⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠤⢄⡈⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠤⠄⠠⠤⠄⣀⠀⠀⠀⠀⠡⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⠤⠒⠉⠁⠀⠀⠀⠀⠌⣂⠄⠑⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡂⠀⠀⠀⢌⠀⠀⠀⠀⠶⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠤⡀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠐⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⣀⢀⠔⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠈⠳
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡊⠁⠑⠄⡀⠀⠀⠀⠈⢆⠀⠀⠀⠠
⠀⠀⠠⡔⠀⠀⠀⠀⠀⠀⠐⠤⡠⠁⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠦⣀
⠀⠀⠀⢠⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⢀⠀⠠⢀⠈⠣⡀
⠀⠀⢠⠸⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢢⠀⠀⠑⢄⠘⠄
⠀⠀⢸⠀⠳⣀⠤⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⠈⢇
⠀⠀⠀⢣⡀⠈⠃⠩⡉⣁⠤⠒⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠰⠃⠀⠀⢰⠇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠑⢄⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠑⠲⠤⠖⠁⠀⠀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠤⠤⠴⠒⠋⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠊⠀⢀⣆⠀⠀⠀⢉⠐⠒⠒⠒⠒⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆
⠀⠀⠀⠀⠀⢠⠀⠀⣸⠟⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣸⠀⠀⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⣰⡟⠀⢠⡀⠀⠀⠁⡰⠀⠀⠀⠀⠀⣀⣤⡴⠦⠤⠄⣀⠀⠀⠹⡇
⠀⠀⠀⢰⣿⡇⠀⢸⣇⠀⠀⠀⠃⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠈⠣⠀⠀⠏
⠀⠀⠀⡟⢿⡇⠀⢸⣿⠆⠀⠀⠀⣰⣛⠅⠀⠒⠒⠀⠀⢀⡠⠀⠀⠀⡎⡀
⠀⠀⠀⣯⣨⣿⡀⠈⣿⡄⠀⠀⣰⣿⡁⠀⣉⡛⢲⣶⡄⠈⠘⠋⠉⠀⠘⠁
⠀⠀⠀⣻⣿⣷⡁⠀⣿⣿⢀⣼⣿⡿⠀⠾⠭⠀⢼⣿⣿⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢸⣿⣿⣷⠀⣿⣿⡿⠋⠻⢃⣀⡀⠀⠀⣰⣿⡿⠶⠂⠚⠛⠀⠐⠀⠘⠃
⠀⠀⠀⠀⠙⠻⣿⠀⢿⣿⣗⡳⠀⠈⠙⠛⣣⡽⢋⣁⣀⣠⣶⣤⠖⠀⠒⠃⠀⠀⠠⠄⢰⡄
⠀⠀⠀⠀⠀⢠⠜⠄⠐⠚⠿⣷⣷⣢⡄⠙⠓⢶⠛⠉⣍⣉⣁⣡⠄⠀⠀⠀⠀⠀⠀⠠⠿⠻⠦⡤
⠀⠀⠀⢀⢠⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀
⠀⠀⠀⣾⣼⡕⣤⣤⡀⠀⠀⠀⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⣦⣈⣿⣿⣾⣿⣿⣻⣴⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⣌⣻⣿⣿⣿⢿⣿⣿⣿⣿⠟⣹⠟⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣉⢠⡀
⠀⠀⠀⢹⣿⣿⠟⡼⢫⣿⡿⡁⠀⡔⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟
⠀⠠⡾⠿⠿⢿⣿⣿⣿⣿⣿⣧⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡰
⠀⠊⠈⠀⠀⠀⠞⢉⣭⣛⣛⣻⣿⡀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⢁⣼⣶⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⠙⠻⣿⣿⢿⣷⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠖⠳⣖⠒⠂⠀⠀⢀⡠⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠐⠀⠀⠂⠂⠐⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⠤⠤⠤⠤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠨⠭⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣠⡤⣤⣀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠀⠀⠀⠀⠀⣀⡴⠚⠉⢹⡟⠉⠉⠉⠉⣹⣄⡀⢈⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⣷⣀⣄⣀⡴⠚⠉⠳⣤⠴⠋⡒⡄⠀⠀⢰⣏⠈⠛⣏⠁⠁⢀⡟⠀⢠⡖⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⠟⠡⡞⣋⡀⠀⠀⠀⡇⠀⣤⣿⣧⣤⣄⣸⣿⣷⣆⣸⠀⠀⠈⢣⡀⣈⡽⠃⢸⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⢤⣀⣿⠀⢸⠀⠀⠉⠉⠉⠳⠶⠖⠛⠛⡟⠁⠀⠀⠀⠀⠈⡋⠉⠉⠛⠉⠑⠋⠉⠉⠀⠀⣸⠁
⠀⠀⠀⠀⢀⣴⠋⠁⠀⠉⠀⠀⠀⠈⣿⠀⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣴⢻⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠛⠒⠚⣳⣤⣄⣀⣀⣀⣀⣤⠴⠞⢉⡿⠀⠀⠀⠀⣞⠢⠒⠲⡄
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡇⠀⠀⣧⡀⠙⠳⠶⠶⠶⠖⠛⠉⢧⡻⣿⣿⣿⣿⠟⡼⠁⠉⠉⠉⠀⠀⠀⠀⢸⡧⠴⠶⠶⠋⢬⠱⣄⠴⠃
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢬⡉⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠟⢭⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣠⠾⠖⠁
⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⣤⣤⣤⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⠏⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡾⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠲⠤⠴⠶⠋⣴⠃⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠏⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣼⠁⠀⢀⡼⠁⠀⠀⠀⠀⠈⠓⠶⠤⠴⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢾⠀⢧⣀⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡖⠛⠓⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⣨⣗⢦⣭⠼⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠉⠆⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠋⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠤⠋⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠓⠒⠦⣄⠀⠀⠀⠀⢀⣠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣀⣀⡀⠀⠀⢀⡞⢱⡊⢉⣉⣉⣉⣷⠈⢳⣶⣾⣿⣿⣿⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣇⠈⢿⡿⠿⠿⠿⠟⢀⡾⠿⠛⠛⠉⠁⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣄⣀⡀⠀⠀⠈⠀⠉⠳⠤⣤⣤⣤⠴⠚⠉⠀⠀⠀⣀⣠⡤⠶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡉⠛⠓⣲⣶⠶⠤⠤⠴⠃⠈⠷⠴⠶⠶⣟⡋⠉⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧

⠀⠀⠀⠀⠀⠀⣤⡄⢀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⢤⣄⣀
⠀⠀⠀⠘⠟⠐⢶⣾⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⣤⣤⣤⣄⣉⠛⠦⣄⡀
⠀⠀⠀⠀⣈⢰⡶⠙⠿⠛⠻⠋⣄⠀⠀⠀⣀⣀⠀⠰⡾⠵⡶⠶⠟⠿⣿⣿⣿⣿⣦⣌⠛⢷⣤⣀
⠀⠀⠀⢀⣷⣄⣀⡈⠀⢠⠺⣷⣍⣿⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣄⠻⣿⣿⣷⣄
⠀⠀⠀⠚⢷⡝⢿⣿⣿⣆⣈⠉⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⣿⣶⣎⠻⣿⣧⠘⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠿⠾⢿⡽⠋⠉⢡⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡗⣤⡼⣻⣿⡅⣻⣿⡇⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡼⠃⠀⠀⠀⠀⢠⣄⣤⣤⣀⡀⠀⢀⠀⠉⠂⠸⠞⣙⣡⡈⢻⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⠀⠀⠀⠀⢤⣀⠘⠛⠛⠛⠛⠻⢷⣦⣄⠘⠃⠛⣿⢽⡯⢛⡙⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠿⠒⠒⢂⣭⡳⠀⠀⠀⠀⠀⠠⠨⣿⣇⠀⠀⠀⣴⣾⡿⠁⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⣛⣿⢟⡇⢐⢲⠶⢶⣶⣾⣧⠸⠄⠀⣰⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣿⠃⠀⠀⢒⠉⠁⠾⡃⠘⠂⠠⣦⣼⣿⣱⡎⠀⠀⣿⣿⣿⡆⠸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⢠⣿⠇⠴⢤⣶⣿⡀⢢⠄⣁⣑⣦⡀⠀⣸⣿⣝⠿⡶⠀⢹⡟⠿⣧⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⢀⠤⠐⠀⠀⢊⡀⢐⠾⣷⢄⣀⣀⡈⠛⠛⠋⠉⠉⠀⠙⠲⠟⠛⠉⢰⠈⠛⡈⢧⡀⢀⣠⣾⣿⣿⣿⡟
⠀⠀⢀⣀⠠⠒⠋⠉⠉⠀⠀⠈⠁⠀⠈⠉⠙⢷⣶⣶⡶⢤⡀⠀⠀⣀⣀⣴⡿⢖⠀⣹⣾⣿⡿⣛⠿⠿⠿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠚⠛⢶⣤⡌⠀⠀⠉⠋⠀⠀⡙⢿⡾⠛⠉⠃⠀⠀⠀⠀⢜⣿⡂
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠀⡀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⡀⠀⠀⠀⠘⣾⣏⡀⠀⠀⠀⠀⢀⡠⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⠁⠄⠄⠀⢀⠆⠀⠀⠀⠀⠀⠇⣀⠀⢠⠀⣿⢿⡀⡀⠀⠀⠀⠘⠛⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⡀⠀⡀⠘⠀⡇⠀⢋⣼⣷⣬⣶⣌⣲⣤⣬⢻⣟⢤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡌⠀⡸⠀⠀⠀⢰⠇⢸⡇⣠⠀⢡⠀⣾⣿⣿⣿⣿⣿⣿⣿⣥⠀⣿⣮⡻
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⡤⡀⠀⠀⢀⣞⢴⣄⢳⣿⡦⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣶
⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⣠⠒⠄⠀⠁⠠⠞⠁⢉⣼⣿⣧⣿⡷⠃⢠⣿⣿⡿⣿⣿⣿⣤⠛⣇⣾⢟⢉⡶
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠏⢀⠀⠀⠀⠀⠀⣴⠜⢹⣿⣿⣿⡏⣰⣿⣿⣟⣿⣿⡿⠟⢁⠴⠋⢀⣬⣿⡃
⠀⠦⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⢄⡼⠟⣠⣿⣿⠃⣯⣶⣿⠿⢿⡍⡿⢏⣤⣈⡄⣄⠰⣾⣟⣱⠋
⠀⢀⠀⠀⠐⠉⢀⣠⣄⣠⣥⡦⠀⣀⣤⡶⠟⣩⣴⣟⣭⣾⡟⣯⠳⢘⢋⠱⠀⡋⠅⠁⠀⠂⠉⠒⠒⠥⠞⠓⠚⠁
⠀⠈⠁⠀⠀⠐⢻⣭⣿⣿⡋⠀⠈⣽⠏⢀⣼⣿⣿⣿⣿⣿⣨⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠀⠰⠿⠃⣚⣩⣽⡿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠒⠒⠒⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠓⠈⠘⠊⠉⠀⠀⠙⠦⠔⢦⠤⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⡼⠃⠀⠀⠀⠀⣀⣀⠠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠁⠀⠀⢀⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸⠇⠀⢀⡠⠌⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⠿⠖⠛⠓⠉⠉⠁⠀⠰⠟⢢⢠⣄⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣧⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠠⡎⠛⠁⠀⠀⠀⢉⡆⠀⠀⠀⠀⢈⣀⡀⣀⠈⠙⣇⡀⠀⠘
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⡥⠖⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⢁⣀⣀⠈⠒⣿⠈⣧⠻⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣼⠃⢹⠋⠀⡠⢞⣉⠉⠲⣄⠀⠀⠀⠀⢠⡶⢣⣤⡀⠑⣆⢀⢰⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢿⣀⠀⠀⠀⠀⠠⠁⢀⠎⠀⢸⠁⣹⣿⣷⠀⠈⣆⠀⠀⠀⣾⡓⠾⡿⠃⢀⣨⣰⡄⠳⢄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⢽⡀⠀⢰⡿⢆⠀⠈⢧⡀⠥⢤⣙⣿⡥⢄⡠⠊⠀⠀⠀⠻⡉⠒⠒⠊⠁⠈⠉⠉⠐⠎
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡴⢇⡏⢹⡦⣉⣠⡾⠁⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠙⣆⠀⠚⠀⠢⠀⠀⠀⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠐⠒⠚⠁⢎⡤⠊⠁⠷⠀⠀⠀⢈⠉⠀⠀⢀⠠⠧⠤⢤⣀⣀⠤⠛⠒⠤⠤⠖⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠐⢄⣀⡴⠋⠀⠀⠠⡆⠀⠀⠀⠀⠐⠄⠐⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣤⣴⠲⠿⢿⣿⣿⣾⡿⠒⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠃⠁⠀⠉⠳⢤⣀⣈⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⠀⠀⣠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠓⠚⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⡀⠠⠎⠀⠑⠉⠀⠐⠈⢻⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡌
⢱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠈⢢⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠈⠲⠃⣄⢀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⠺⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢧⡈
⠀⠀⠀⠀⠀⠘⠣⠐⠢⠄⠣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡟⠦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣴⡄⠀⣯⠗⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣘⡆⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠲⢄⣀⠐⠂⢤⣤⡤⠄⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠛⠉⠉⠉⠉⠛⠛⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⢀⡴⠋⠁⣀⣀⣤⡤⠴⠶⠶⠶⠶⠤⣤⣭⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣴⣿⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠶⠶⣶⣭⣭⣙⡲⠦⣄⣀⡇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆⠀⠀⢀⣠⠶⠛⢋⣡⣤⣠⣄⡀⠀⠀⠀⣠⣤⣠⣍⡙⠻⢦⣭⣁⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡇⣷⡶⡋⠁⠀⠀⠤⣤⣤⣌⣀⡿⠀⠀⠻⣁⣁⣄⣤⠤⠀⠀⠈⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣼⢁⠛⡼⠇⠀⣀⣤⡚⣿⣿⠀⠀⡆⠀⠀⢰⠀⠀⢿⣿⠛⣦⣀⠀⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣰⢃⣾⠀⠀⠀⡀⠀⠀⠤⠤⠥⠉⠀⠁⠀⠀⠀⠀⠈⠤⠔⠤⠄⠀⢀⠀⠀⢼⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣦⡀⠐⢁⡼⣫⡆⠀⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⢷⣄⡾⠀⡽⠃⠀⠻⡆⠀⠀⠀⠀⠀⣀⡦⠶⠄⠀⠴⢤⣀⡀⠀⠀⠀⠀⢰⠿⠁⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⣸⣏⠀⡼⠁⠀⠀⠀⢻⣆⡀⠀⣰⠏⠁⠀⣀⣤⣦⣀⠀⠀⠙⢦⠀⢀⣰⡞⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣴⣄⣠⣤⣴⣿⠏⢸⣷⣂⡀⠀⠀⠈⠙⢃⣼⣟⣀⡤⠊⡙⠛⠛⠛⠙⢦⣀⣸⣧⡈⠋⠁⠀⠀⠀⣀⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠟⠛⠋⠀⠈⢻⣿⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⠀⠉⠛⠛⠋⠁⠀⠉⠉⠉⠛⠂⠀⠀⠀⠀⢻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠞⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣇⢰⣷⣾⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡞⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⡶⠶⠶⡶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠒⠚⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠋⠉⠁⣀⣤⣶⠿⠛⠶⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⢠⠂⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠉⡢⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⣴⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢤⣿⣿⣿⣆⣁⣀⣀⡀⠀⠀⠀⢀⣀⡀⠁⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣾⣿⣿⣿⡭⠉⠉⠛⠛⠛⠛⠋⠉⠛⠛⠒⠒⠲⢶⣘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⢀⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⠌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⢸⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣾⣃⣦⡴⠞⡋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢦⡀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢈⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠿⣿⣄⣼⣾⣿⡿⢿⢟⠻⠳⣦⣤⣴⣶⣶⣤⣶⣤⣄⠳⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⣿⣿⠿⡫⣛⣉⣉⣁⡘⠋⣷⢰⡆⠀⡾⢤⣍⣉⣛⠻⣟⠓⢀⣆⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⠁⣼⣿⠏⣜⡼⠛⠉⠉⠉⢿⣿⡅⠸⠿⠈⢶⣤⡤⠶⠿⣿⣾⣦⣸⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⣿⣾⣟⠄⠄⣠⠒⠒⠺⠻⡟⠀⠀⠀⣾⠀⠄⠀⢤⢬⣏⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⠋⣠⣤⣾⣿⣿⣶⡤⣿⣇⠀⠀⢸⡿⣷⢾⣻⡿⠾⣿⣟⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢰⡟⣿⡿⠛⠛⠉⠁⠠⠛⢧⣀⢀⡟⢄⠀⠈⠳⣦⡜⠁⠀⠀⠀⢹⣷⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⡀⠆⢠⡿⢛⡏⣾⠀⠀⠀⠀⠀⠀⠀⡈⣻⡷⠉⠑⡀⠀⠘⢷⣦⡀⣀⣀⣼⣿⡍⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢂⠘⣠⣿⡶⢸⣷⣿⣷⣤⣴⣶⡶⣶⣿⣿⣏⣀⣀⡠⡆⠀⠀⠀⣿⣉⣭⡏⠉⣿⠇⢰⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⠰⠈⣰⠏⢸⡇⠚⢿⣿⣿⡋⠈⣿⡾⠟⠉⠉⠿⠿⣿⣿⣀⣶⣂⣴⣿⠟⠉⠿⡾⠁⠀⢰⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢄⠂⢀⣼⡏⣴⣽⣷⡀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⣦⠉⠋⣌⠁⠀⠀⠀⣠⣾⡷⠃⡀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣐⠄⢠⡾⣿⣴⡟⠁⡈⢻⣿⣷⣾⣦⡀⢀⠠⢤⣄⡛⠓⠤⣂⣠⣿⣶⠶⣿⣿⣷⣾⣿⡛⠿⣮⡻⡻⣿⡇
⠀⠀⠀⠀⢰⣖⠒⠶⢤⣀⡟⣀⣤⠤⣴⡏⠀⣿⡟⠀⠚⢰⡿⠛⣿⣿⡿⠿⣿⣾⣿⣿⣯⣿⣛⣛⣋⣁⣤⣤⣿⣿⠿⣻⣄⢩⠀⠘⣷⠈⠘⣷
⠀⠘⢻⠿⠏⠉⠀⠀⠀⠈⠉⠁⠀⠀⠙⢦⡀⣹⠃⠀⠀⣸⣷⠟⣱⠏⢠⠾⠋⣸⠟⣿⣏⠻⠿⣿⣿⣿⡿⢿⣿⡿⢧⡁⢻⡀⠃⠀⢸⠤⣢⢸⡇
⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡱⣿⠀⣆⣤⡟⠉⡴⠏⠀⢀⡶⠸⡁⢼⢿⣿⣷⣶⣤⣦⣤⣾⡿⠛⠀⠀⢱⢸⡇⢂⠘⣯⢇⠻⡆⢿⡀
⣸⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣪⠞⠝⠁⡄⠃⠇⠀⣼⠁⠀⡇⢸⠀⠀⠋⠙⠉⠉⠉⠀⠀⠀⠀⠐⡀⡆⣇⢾⠀⠱⡘⠃⠘⠈⢷⡀
⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣿⡟⠀⠀⠠⠁⡠⠀⠀⣿⡆⠀⠁⠘⣰⠀⠀⠀⠀⠀⠀⠀⠀⡄⢱⢀⣷⡇⣿⠈⠀⠀⠑⠄⠀⠀⠀⠑
⢸⡇⣶⣶⣦⡑⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠁⠀⣀⣴⠞⡐⠀⠀⣿⠇⠀⡀⣄⠻⡆⢄⠳⠀⠐⡀⡄⡜⢡⡻⣾⠷⠀⡼⡆⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢧⡈⠹⣿⠻⣦⣍⡓⠀⠀⠀⠀⠀⢠⣎⡽⢇⣀⣸⣿⠏⢘⡀⠀⢠⠏⠀⠀⡳⣜⠓⢹⣦⡹⣦⣰⡜⣷⡰⠟⣱⠏⠀⠀⠘⣱⠄⠀⠀⠀⠀⠀⢀⡞
⠀⠈⠳⣄⠈⠁⠈⠉⢻⣳⣤⠤⢊⡴⠛⠋⠀⠀⠀⢠⣏⣠⠟⠀⠐⡫⢂⢀⡀⠘⣟⢦⡀⠙⠻⣎⢻⢿⡝⠸⢰⡏⢠⠂⠀⠀⠈⠳⠄⠀⠀⠙⢶⡞
⠀⠀⠀⠈⠓⠤⡷⢦⣤⣰⣖⠛⠉⠀⠀⠀⠀⠀⠀⠈⠸⡁⠀⠀⠀⠐⣡⠸⢁⠀⠌⢧⡙⢶⣆⠘⣿⠀⠀⠀⢿⠁⠆⠀⠀⠀⢢⠀⣀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢡⡾⠀⠈⡄⠙⢄⠻⣆⢸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⣨⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⡟⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠁⠇⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⢤⣤⣤⣀⣦⣄⡀⠣⠀⠀⡄⣰⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠂⠉⠉⠉⠉⠉⠁⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠀⠠⡀
⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠠⠠⠀⠁⢀⡅⡖⠀⠀⠀⠀⠈⠂
⠀⡜⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠁
⠘⠀⠀⠠⠁⠀⠀⠀⠀⢠⣀⣀⣠⠤⢴⡁⢀⡱⢤⣈⡀⠀⠈⠂⠄⠐⠀⠀⠊⢱
⡆⠀⠀⡈⠀⠀⠀⠀⠀⢀⡧⣀⠀⠀⠀⢈⡁⠀⠀⠀⠉⣠⠋⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠤⠄⠀⠀⠀⡀
⡇⠂⠂⠀⠀⠀⠀⠀⠀⢸⠁⢱⠉⠒⡒⠊⠉⠒⠶⠒⢛⠋⢣⠀⠀⠀⠀⢀⠜⠀⠀⠀⠈⡃⠀⠀⠰⠋
⢠⠀⠀⠀⠀⠀⠀⢀⡴⢸⠀⠀⢳⡀⡄⠀⠀⠀⠀⢠⠃⠀⢸⠀⣀⣠⠐⠁⠀⠀⠀⠀⠐⠀⠒⢆
⠀⠣⡀⠀⠀⠀⢠⡈⢀⡀⡆⠀⠀⠑⣆⠀⠀⢀⡴⠃⠀⠀⠎⠀⠀⠲⡂⠀⠀⠁⠀⠼⢀⠀⠀⠀⠀⢀⡠
⠀⠀⠈⠐⢐⠆⠌⠜⠁⠐⡜⡄⠀⠀⠈⣙⢋⡁⠀⠀⢀⠎⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠀⠈⠉⠀⠉⠁
⠀⠀⠀⠀⠁⢀⢀⠀⠀⢀⠃⠘⡄⠀⠀⠈⠉⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠃⠀⡠⠈⠢⠤⠞⠄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢠⠁⠀⠒⠒⠉⠊⠀⠀⠀⠐⠇⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠃
⠀⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠔⠊⠀⠂
⠀⠢⣈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡠⠔⠈⠁
⠀⠀⠀⠑⢄⠁⠀⠄⠄⢀⡀⠀⠀⠀⠀⣀⡄⠧⠔⠒⠉
⠀⠀⠀⠀⠀⣈⡁⠐⠢⢤⡨⠤⠀⠀⠠⠀⠀⠀⣀⡀⢀⠠⢄⡀
⠀⠀⠀⠀⢎⠀⠈⠁⠐⢂⠠⠀⠀⠀⠀⠀⠠⣁⢉⣉⡀⠀⠀⡑
⣤⣤⣤⣤⣤⣤⣶⣮⣥⣴⣤⣤⣤⣤⣤⣤⣤⣶⣾⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣷⣶⣶⣦⣄⠀⣀⣀⣠⣤⣀⣤⠀⣤⣤⡀⠀⢰⣶⣶⠾⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣌⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⠟⠛⠿⠷⠲⠿⡿⢿⡀⢀⡀⠀⢸⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠁⢿⣿⣯
⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣯⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢷⡤⣭⢽⠟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣋⣉⡉
⠀⠀⠀⢀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡐⠐⡆⢺⣯⣯⣿⡿⡿⡿⢿⣿⠟⠋⠸⠇⠸⡷
⠀⢠⡴⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠃⠀⠀⢿⣿⣿⠏⠁⠀⢡⣼⣿⡓⢂⣀
⠀⣀⠀⣉⣿⣟⣿⣿⣿⣿⣿⣿⣷⣟⣿⣿⣿⣿⣿⣿⡟⠛⣟⠻⢿⣦⣄⣴⠞⣿⡟⢶⡀⠀⠈⢼⠏⠉⠙⠳⢦⡈⠂
⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣧⡄⠙⠈⠋⠂⢁⣡⡟⣁⡀⠀⠀⠀⠀⠈⢿⣷⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣴⣿⣿⠉⠀⠉⠀⣦⣐⣶⣯⠷⠈⠁
⣿⣿⣿⣿⣿⣯⣩⣿⣿⣿⣿⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⢀⡶⣿⡽⠏⠙⢀⠐
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠋⠀⢀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⣀⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠿⣿⣿⣿⣿⣿⣷⣬⣿⣏⡁⠄⢀⣆⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⣻⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡟⢛⣟⣋⡟⠛⠋⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣶⣦⡉⠉⢀⣁⢀⡀⠀⠀⠐⠠⠀⠀⠚⢾⣿⡦
⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣽⣶⣿⣩⣷⠀⠰⠆⠀⠀⡄⢠⣿⡿⠋⠻⠟⠛⠿⣿⣿⣿⣷⣿⣿⣿⣿⡶⣶⣄⣀⢉⣀⣀⣾⣿⠙⠲⠆
⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣨⣭⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣶⡾⠛⠁⠀⠀⢠⡈⣦⡈⢻⣟⠛⠛⠁⠀⠀⠀⠀⠀⠠⠿⠿⡟⠋
⠀⠀⠀⠀⠀⠀⠈⢉⡿⠙⣛⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠐⢸⡁⠈⠉⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠉⢡⣿⣿⣿⣿⣿⣿⡍⠀⢨⣾⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣸⣿⣿⣿⣿⢿⣟⣻⣶⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣿⣿⣿⣿⡟⢃⢹⡿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠛⠛⠛⠉⠀⣼⣿⡷⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠆⠀⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣶⣤⡿⠱⠀⠀⠀⠀⠀⠀⠀⢀⠀⣹⠀⠀⢀⠀⢄⡀⠀⠀⠀⢀⣾⣁⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣶⣤⣀⣤⣐⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⢶⣿⣿⡿⠛⠓⠖⠀⢀⠼⣃⠉⠀⠰⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡯⠉⣉⠿⠀⠀⠆⣠⠞⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠈⠛⢻⣿⣿⡿⡿⠇⠲⠀⠀⠀⠀⠀⠀⠀⠈⢩⡴⠂⠀⠙⠂⠒⠛⠁⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢺⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠄⠀⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⣄⣘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⡴⠒⠃⢁⢁⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⣆⣘⢳⣔⡔⡱⣶⡏⢹⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣄⡘⣿⢻⣦⡈⠛⠯⠞⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⢻⢻⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢟⠿⠿⠷⠶⠛⠙⠛⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣴⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡉⠀⢍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣽⡇⠀⣾⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣿⣿⡿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⠐⠒⠲⠀⠀⠀⠉⠈⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠿⠾⢁⣦⣤⣼⣤⠷⠤⣤⡀⠀⠀⠀⠀⢿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣴⡿⢻⣿⠴⠟⣧⡀⠀⢀⣉⢻⣦⣄⣲⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣶⣶⣤⣄⠈⠛⢹⣿⠇⢀⡾⠉⠛⠛⠻⠃⣰⣿⣍⣁⢸⣿⣿⣀⣁⣤⣶⣿⣿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⠃⠀⠸⣇⣤⣠⡴⢦⣤⣨⠁⠉⠀⢨⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢉⣿⠁⠙⠟⢻⣿⣿⣿⣿⣿⠁⢾⣶⠶⣾⣿⣷⣦⣾⣧⣥⣽⠲⠶⣿⠛⣿⣿⣿⠋⠀⡐⢻⣿⣿⣿⣿⣿⡁
⠀⠀⢀⣀⠀⠀⢿⣿⠆⠀⠐⠄⠈⣽⣿⣿⣿⣦⣬⣥⣴⣾⢭⡶⣷⣭⣭⡽⢿⠳⢈⡙⠺⣿⣿⣯⠶⠂⠈⠉⠻⣿⣿⣿⣿⡿⠀⠀⡀
⠀⠀⡿⣩⣷⣄⠀⠉⠀⠒⠀⠀⠀⣠⡼⠛⢉⣽⣿⢻⠻⣇⡘⠋⠁⠈⣷⠀⠸⢦⡿⢇⣴⣿⣿⣷⡀⠴⠵⠆⢀⣿⣿⣿⣿⣿⣶⠚⢟⠂
⠀⠀⡀⣘⣷⣿⠿⠖⠻⣤⣤⣤⣾⠋⠀⣴⠟⠉⣿⣟⠀⠞⠁⢶⠄⢀⡿⠇⠘⢈⠁⣄⣿⣿⡛⠻⠿⣦⣤⡶⠟⠉⠀⣀⣡⣼⣿⣣⠘⢻⡇
⠀⠀⠈⠉⠉⠙⢀⠀⠀⠀⠈⠙⠛⢳⡾⠋⠀⣸⣿⣯⠄⣤⣠⡜⢂⠘⠳⡄⠀⣼⣺⣿⣿⣿⣁⢢⣠⣿⣏⢀⣤⣤⠞⠛⣿⣇⣈⣉⣷⡎
⠀⠀⠀⠀⠀⠰⠞⣻⣇⠠⢤⢠⣀⣀⡀⣀⣼⣿⢻⣿⣦⠙⡋⣀⢸⡄⠛⣷⢄⣸⡛⠿⠿⠛⠋⠛⠉⠁⣤⢉⣉⡇⢸⣶⠀⣉⣿⡿⣿⣷
⠀⠀⠰⠀⠀⠀⠀⢉⣽⠿⠿⠷⠟⢻⣿⠿⣿⣿⠆⢿⣿⣿⣷⣟⣛⣿⣤⣧⡿⠋⣡⡶⠶⠶⣦⣠⡶⣿⡿⣦⡙⠃⢸⣿⢿⣿⣾⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠂⢀⡿⠀⠀⠀⠚⢛⠛⠁⠒⡉⠿⠓⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡀⠀⢉⠿⠉⠇⠈⣿⡌⡛⣶⣿⣿⣀⡿⣷⡿⢩⡷⡇
⠀⠀⠀⢀⠀⣄⠸⣧⠀⠀⠀⠀⢙⣛⣻⣿⣄⣀⣠⣸⣿⣿⣿⣤⠉⠿⠟⠻⢿⣿⠿⣷⠶⠈⢀⡶⢃⠀⣿⣿⣴⡘⣿⣻⣿⠷⠈⠛⢿⣁⠃
⠀⠀⡀⠀⡠⠾⡇⢹⡷⢶⣦⣤⣼⣿⠋⠀⠈⠛⠉⣹⣿⣿⣿⡄⣶⣾⣯⡿⠿⠧⠈⠀⠀⠀⠁⢀⡏⢠⡿⢿⣏⠣⠘⠛⢻⡏⣤⠀⢸⣿⢀
⠀⢠⣬⣯⣥⣤⣴⣾⠁⠀⠉⠉⣹⣿⠀⠀⠀⠀⣺⡿⠛⢛⣿⣿⣧⣾⣏⣷⣶⣤⣔⠺⡟⢲⡖⠉⣿⣧⣴⡶⠛⢛⣾⠟⠈⠿⣿⣿⣾⣿⣿
⠀⠈⢿⣿⣿⣿⣿⣷⣶⣦⣴⣹⣿⣿⠃⠀⠀⠰⣿⠃⣿⣿⢫⣿⠟⢁⣼⣿⣿⣿⣟⣿⣧⣉⣿⠛⠿⠟⠋⣠⣾⣋⣥⣶⡷⢦⣼⣿⣿⣿⣿
⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⡯⢿⣿⣿⣷⣷⠲⣿⣿⣤⣿⣿⢸⣿⣄⡹⠟⢋⣿⣿⣿⣿⣿⣿⣿⣧⣤⣷⣾⠻⣿⣿⣿⣤⣴⢶⣿⡿⢿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣽⡇⣿⠆⣼⣿⣿⣿⣿⠁⣼⣿⣿⣾⣷⣿⣿⣿⠷⣄⠈⠉⠛⠻⣿⣿⣿⠿⢿⣯⣽⣸⣿⡟⣿⣷⣮⡹⣷⣜⠇
⠀⠀⠀⠀⠀⢀⣀⣿⠀⡟⢰⣿⣿⣿⡿⢿⣧⢰⣿⣿⣿⣧⣌⡉⠶⠙⠿⣿⣾⣷⣶⣤⣌⠁⡐⠻⣿⣟⡴⠟⠀⣿⠿⣟⣛⣶⠟
⠀⠀⠀⠀⢸⣿⣿⣦⣀⠉⢾⣿⣿⣿⣧⡄⢹⡾⠻⢿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡶⣤⣀⠰⠆⠉⠛⠋
⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣾⣿⣿⣿⣿⣶⣾⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣭⣴⣶⣿⣿⣿⡧⠘⠀⢹⣧
⠀⠀⠀⠀⠀⣠⡶⠿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣠⣾⣿⣿⣿⣿⣿⣿⣏⣿⣟⣯⣿⣿⣿⣿⣿⠿⠂⢘⣠⣾⠿⢿⣧⣶
⠀⠀⠀⠀⡸⠃⠐⠀⠐⠈⠉⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣟⣿⣿⡿⠿⠛⠋⠉⠛⠀⠀⠀⠈⠙⣷⡄
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⡇
⠀⠀⠀⠀⢸⡖⠁⠀⠀⠀⠀⡀⢷⠀⠀⠀⠀⢼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⣿⠟
⠀⠀⠀⠀⠈⢿⣤⢀⡀⠀⠀⠈⢀⠁⢀⡀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠲⣶⣦⣤⣤⣤⣀⠈⣷⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⢀⣀⣠⡔⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⠟⠳⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠙⠛⠿⠿⠿⠋⠉⣿⣿⡷⠀⠘⣿⣏⠙⠿⠿⠿⠿⠛⠁⠀⢈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⡀⠀⠀⢀⣤⠔⠀⣉⣿⠇⠀⠀⠻⣿⣴⣀⣀⡀⢀⣀⣠⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣶⣶⣦⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣷⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠦⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⢀⣴⣿⣭⣹⣯⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠀⠐⠂⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢠⠾⠐⠢⣀⣀⠀⠀⠀⠀⠀⠀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣀⣀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣤⠓⠒⢑⠞⠀⠘⠂⠤⡄⢄⡀⠀⠆⠀⢦⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣹⠸⠒⠒⠓⠀⠙⢦⢀⠀⠀⠀⠀⠀⣰⠀⠀⠈⠮⡀⣲⠦⢄⠀⣀⠈⠢⡜⠀⠡⡀⠀⢠⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢤⡉⠀⠀⠠⠚⠉⠀⠉⠢⡀⢠⠎⠈⢦⡀⠀⠀⠀⠙⠂⠄⠬⣛⠀⢠⣙⣀⣀⣅⢼⠡⡉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠴⠁⠀⢀⡔⠉⠑⢣⡇⠀⠀⠀⠈⠉⠀⠀⠛⠳⡀⠀⠀⠈⠀⠀⢀⡴⠁⠘⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡞⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢥⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣀⡀⠀⠀⠀⠀⠉⠐⠤⠀⠤⠤⠤⠤⠚⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡆⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⠀⠀⠀⠈⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢍⡉⠑⢤⠚⠐⢇⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡳⠤⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⣉⠀⠸⡀⠀⠀⠙⠢⢀⡠⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢇⠋⠉⠁⠐⠄⢹⠀⠀⠀⠉⠉⠉⠁⠀⠘⠥⠤⠤⡀⠘⢄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢇⢄⣀⠀⠀⡠⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠁⠈⠙⠓⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡀⠀⠨⠀⠀⠉⠘⠋⠏⠁⠛⠒⠒⠒⠒⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡄⠱⢀⠞⣄⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⢇⠀⠀⢠⠈⢆⠈⠓⠄⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠊⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢝⡄⢨⣄⣎⠀⠀⠑⢄⣀⠀⠀⢘⡏⠒⠒⠂⠀⠀⠐⠒⠒⠈⠉⠀⠀⠀⣀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠓⢬⣻⢚⡛⠁⣛⣁⣈⣙⠯⠈⠉⠹⠈⠂⠤⠄⠀⠀⢠⡤⠤⠄⠒⠒⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⢁⡜⡋⠑⢢⠸⠀⠀⡄⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡠⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡅⢸⠀⡇⠀⠸⠀⡇⢀⠄⠀⠀⠀⠀⠰⠒⠤⢀⣠⠠⠴⡀⠁⠒⠒⠀⠉⠁⣀⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢆⣸⡇⠄⠀⣠⠶⠋⠁⢀⠔⠊⠻⠉⠑⠢⢄⣀⣀⡀⠤⠘⡋⠀⠀⠀⠉⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢫⠀⠀⠀⠀⡏⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠀⠐⠠⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣬⢦⡀⠀⡗⠄⣀⣀⣀⣁⣀⣀⠀⠤⠠⠔⢚⢇⡌⠀⠀⠀⠀⠀⠀⣠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠑⠠⢀⣀⣀⣀⣀⡀⠤⠤⠒⠂⠙⢄⠣⣀⣀⣀⣀⠤⠚⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⣀⠀⠤⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢖⠒⠒⠂⢒⠐⢒⡶⠆⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⢀⠀⠀⠈⠉⠀⠒⠻⣑⠭⠀⢀⣀⠀⡀⠀⠉⡒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⠂⠀⠀⣰⠆⠀⠀⠉⠲⣄⠀⠀⢸⡗⠁⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠆⢀⠁⠀⣛⣛⣬⣬⣭⣭⣭⣥⣼⣒⡀⠤⢈⡃⠀⠜⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠀⠀⢠⡴⢓⣨⣴⠾⠋⠉⠉⠀⠀⠀⠀⠀⠉⡉⠛⠛⠻⣶⣮⣔⡠⡀⠀⠀⠰⠀⠘⠉⠙⢦
⠀⠀⠀⠀⠀⠀⡠⡁⠀⠀⠀⣠⠞⢃⢜⣥⣾⡟⠉⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣦⣕⠄⡀⢀⡀⠄
⠀⠀⠀⠀⢀⠌⠀⠉⠩⡍⠀⢀⢜⣵⣿⣿⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣔⢾⠟⠀⠱⢆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠘⡤⣢⣾⣿⣿⡿⣤⣾⡿⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⣿⣿⣿⣧⡹⠿⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡰⣱⣿⣿⣿⣿⣵⣿⡍⠀⠀⠀⠻⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢳⡀⠀⢿⣿⣿⣿⣷⡐⡀⢡
⠀⠀⢐⠀⠀⠀⠀⠀⠠⢱⣿⣿⣿⡿⣿⣿⣿⣷⣦⣶⡀⢀⣹⡆⠀⠀⢀⣤⠶⠚⠛⠉⢹⠿⠷⢶⣮⣇⡀⢸⣿⣿⣿⣿⣿⡘⠻
⠀⠀⡎⠀⠀⠠⡆⠀⢣⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣷⣿⣿⣧⣴⣾⡥⠄⠀⠀⠀⢠⣠⡤⠤⠈⢻⡟⢿⣿⣿⣿⣿⣿⣿⣧⢹
⠀⢠⠀⠀⠀⠉⠀⣤⢸⣿⣿⣿⣿⣷⣗⢹⣿⣿⣿⣿⣿⣿⡟⠛⢿⠃⢀⡤⢤⣷⡀⠙⠇⠀⠀⠀⢈⣉⡉⠻⠿⢿⣿⣿⣿⣿⡈
⠀⠸⠃⠀⣦⠀⢀⠇⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣿⣿⣿⣿⣷⡀⠘⠛⠉⠀⠀⠙⣿⡄⠀⠀⡆⠀⠊⠐⠀⠄⠀⠈⠙⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡌⠁⣿⣿⣿⣿⠿⠿⠫⠟⠐⠈⢺⣿⣿⡏⠻⢿⣿⡇⠢⠄⣀⡴⣁⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠸⠀⡀⠀⠘⠁⢠⢻⣿⣿⣿⣢⡐⠀⠀⣠⣶⣿⣿⣿⠃⠀⠀⠈⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⠀⠀⠀⠀⢘⣿⠃
⠀⠀⡆⢇⠀⠀⠀⠀⠸⣿⣿⡟⡼⠁⠀⠰⠿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⣤⠖⠛⠛⠉⠙⠁⠀⠀⠀⠀⠀⣿⣰
⠀⠀⢐⠀⢷⡀⠀⢀⣠⣿⣿⠋⠄⠂⠀⠀⠀⣀⢸⣿⣿⣷⡄⠀⠀⢀⢰⣿⣿⣿⡋⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠈⠆⠀⢙⣴⣿⣿⣿⠟⠀⠀⠀⠉⠀⠀⠀⡀⢻⣇⢍⠟⣷⠀⡈⡀⠐⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣾⡿⢟⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⣴⣿⡟⣡⠏⠐⠀⠳⠀⠀⠀⠀⢀⠀⢀⣋⣠⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣾⡟⠟⠀⡉⠀⠁⠀⠀⠀⠀⠀⠰⠀⣠⣾⣿⣿⣿⣧⣴⣿⠃⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣦⠀⡆⠙⠀⠀⠀⠀⢠⡴⠂⠀⠀⠙⣿⣿⣽⣿⣿⣿⣶⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⢺⠀⠀⠀⠀⠀⠀⠀⠌⠐⠊⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠲⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠚⠓⠦⠤⣤⣈⡓⠦⣄⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⣟⣀⣀⡈⣳⢤⣤⠤⠤⠤⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢭⡭⢭⡷⣼⡲⢤⡀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠃⠀⢸⢹⠶⢬⣷⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⡿⠀⢀⡞⡼⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠁⠀⢸⡤⢷⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠊⠉⠉⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠒⠶⠤⠤⠶⣶⠾⠛⣓⣲⣤⠀⠀⠀⠀⠀⠀⣤⣖⣛⠛⠶⣶⠦⠤⠤⠶⠒⣆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢳⢀⡇⠀⠀⠀⠠⠾⠖⠋⢩⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⠙⠲⠷⠄⠀⠀⠀⢸⢀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⢱⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⡄⠀⠀⠀⠀⠀⠀⠀⠈⡿⠒⠤⣀⣀⣀⣀⣀⣀⠤⠒⢿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⣿⡁⠀⠀⢈⣿⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠉⠒⠈⠀⠉⠉⠉⠉⠀⠉⠒⠉⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢤⡀⠀⠀⠀⠀⠀⠀⠻⣤⣤⠟⠀⠀⠀⠀⠀⠀⣀⠴⠊⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠲⠤⢄⣀⣀⣀⠀⠀⢀⣀⣀⡠⠤⠒⠋⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⣋⡉⠉⠉⠙⠛⠿⢿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡀⠀⠁⠀⠐⠖⠂⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣁⠤⠚⠛⠛⠋⠀⠤⢤⣦⣈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄⠈⠉⢉⡙⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⢀⡐⡂⣬⠟⠀⡟⠁⠾⢉⡇⠀⠢⡄⡙⢦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠀⠀⠀⠘⠁⣠⡤⠄⢾⡷⠦⢴⣾⢧⡆⠀⠈⠹⣶⣯⣝⡲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⠉⠈⠀⠀⠀⠀⢨⠙⠉⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠙⠻⠿⠂⠀⠀⠘⠦⠀⠀⠀⢰⣶⣶⣶⣤⣄⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⢿⣿⣿⣿⠟⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣴⡟⠛⣿⣿⣿⣶⣼⣿⣿⣏⣀⣉⣤⢄⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣮⣿⡏⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠚⠛⠛⠛⠛⠉⠋⠙⠉⠉⠉⠁⠉⢿⣿⣿⣿⣄
⠀⠀⠀⡠⠖⠒⢲⣟⠛⢻⠾⠋⠀⠀⠈⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣿⣿⣿⣿⡇
⠀⠀⡜⠁⠀⢠⠊⢼⡤⠃⠀⠀⠀⠀⣒⠢⠟⢛⠁⠀⠴⢬⡥⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠒⠒⢀⡂⠀⢻⣿⡿⡛⣿⣷⡀
⠀⡰⠁⠰⠛⠁⣎⠟⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡤⠀⠀⠀⠀⠈⠛⠽⡇
⠠⡇⠘⠄⠀⠀⠈⠱⡆⠀⠀⠀⠀⠀⠀⠀⠃⠤⠤⠤⢤⣤⣄⠀⠤⣤⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡂⢀⠌⠁⠀⢀⡀⠀⠀⠀⠀⢱
⠘⢇⣀⢄⠁⠀⠀⠀⣿⠀⠠⡄⠀⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡉⠀⠙⠓⠆⢈⠣⠀⠀⠀⢸
⠀⠈⢻⠦⠁⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣭⡽⠁⠀⠀⠀⠀⢳⡀⠀⠀⣸
⠀⠀⠀⠑⠤⢤⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠴⠐⠀⡀⠀⢀⡇⠘⠀⠏
⠀⠀⠀⠀⠀⠀⠉⠁⢐⡶⠚⠓⠒⠒⠒⠒⠲⢶⡶⠤⠤⠤⠤⣤⡤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣦⠀⠀⠀⠀⠐⠊⠡⠂⣠⠃⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣀⣤⣴⣶⣾⣿⢿⡷⣶⣶⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀⣀⣤⠞⠓⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠛⠉⠑⠉⠙⠀⠈⠠⠒⢋⠥⠖⠛⠿⠒⢫⣭⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠐⠉⠈⠉⠼⠟⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠶⠟⠛⠉⠤⠛⠋⠛⠋⠩⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠀⠒⠀⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠴⠏⠒⠢⠤⢀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⢀⠤⢀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⡁⠱⡄⠀⠀⠀⠀⠀⠀⠀⡎⠲⢉⣉⠬⢋⠀⠉⠐⠲⡐⢄⠀⢰⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠈⢉⣉⢁⠀⣈⠦⠤⣀⠀⠀⠀⠴⠀⠀⠀⠑⠠⣝⠓⠢⢠⡄⠈⢺⠀⠈⠆⢀⡰⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⢀⠌⠁⠀⡀⢌⣱⢠⠊⠀⠑⠢⠄⠀⠀⣠⡅⠀⠒⠀⠄⠓⠒⠀⡶⠚⠈⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⡔⠁⠀⠈⢻⠀⠀⠀⠀⠀⠀⠠⠈⠙⠦⠤⠀⠀⠀⠤⠊⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠈⠒⠢⠄⠀⠀⠤⠤⠤⠒⢍⡁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠊⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢦⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠇⠀⠀⠀⠑⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠑⠢⡔⠁⠣⡀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⠶⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣖⢉⡁⠀⢆⠀⠀⠈⠢⠄⠠⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠘⠉⠉⠁⢒⠈⡇⠀⠈⠉⠉⠉⠀⠀⠘⠀⠀⠄⡈⠃⣄⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢔⣤⣀⣀⣠⢀⠃⠀⠀⠀⠀⠀⣀⣀⠠⠤⠖⠊⠉⠉⠀⡍⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠲⡄⠀⢨⠀⠀⠀⠉⢉⠁⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡃⢘⠃⠘⢰⠉⢄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⡙⠨⠀⡘⠀⠣⡀⠉⠂⠀⠚⣧⢀⣀⠀⠀⠀⠀⣀⣀⠠⠤⠔⠒⠈⠁⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡑⢷⣈⣷⠧⣤⠀⠈⢲⠄⠤⠤⡛⢄⠀⠈⠉⠉⠁⠀⠀⠀⣀⡀⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠒⢁⠜⢚⣶⠦⡀⢲⠒⢲⠊⠁⠀⠀⠡⠀⠉⠉⠀⠀⠘⡉⠉⠁⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⡄⠀⢸⠈⡀⢸⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⣀⠙⠠⠤⠤⠄⠄⠒⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣸⠀⡇⢁⠀⢀⣠⠷⠊⢀⡠⠤⡄⠰⢄⡀⠉⠉⠁⣈⢱⠄⠀⠀⠀⠀⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡌⢖⠉⠀⠀⢠⠃⠀⠀⠈⠂⠀⠀⠉⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⠄⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣑⣄⠀⢘⠄⡀⠀⠀⢀⠀⠀⢀⣀⠀⠤⢰⢆⠀⠀⠀⠀⠀⠀⠀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠀⠈⠢⠄⣉⣉⣉⣉⣁⠠⠤⠤⠒⠚⡈⠄⣀⣀⣀⡠⠔⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⡠⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣷⣮⣙⠿⣿⣷⣌⠻⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢀⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣮⠻⣿⣷⣈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣠⡶⠋⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠳⠈⢻⣿⣧⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠞⠁⢀⣤⠾⢿⣒⣶⠶⠤⠴⣿⣓⣂⣀⡀⠀⠀⠀⠀⠳⡀⠀⠀⢿⣿⡆⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⣴⠫⢫⣞⣿⠉⠉⠭⠭⣄⠀⠤⠭⠿⢶⣶⣶⣿⡀⠀⠀⠹⣆⠀⢆⢸⠇⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠿⠀⣲⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠻⣿⣄⠀⠀⠙⣇⠈⠀⢀⣿⡿⢹⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⠆⠀⢺⠟⣡⡤⠉⣙⣛⣿⣿⣿⣿⣶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⣠⣤⠀⣠⣤⣶⡄⠀⠀⠐⣾⡶⠄⣠⠀⠀⠈⣿⡄⠀⠀⠀⡠⠘⠋⠴⠛⠛⠉⠉⢙⣿⠟⠁⠉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⠀⢠⣀⡿⠛⣺⡿⢿⡿⠿⣆⡀⠈⣉⣉⣤⠴⠏⠀⢀⢻⡅⠀⠀⠚⠁⠀⠀⠀⠀⠀⢀⣴⡿⠋⠢⣀⡀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡝⠀⡄⠀⠀⠈⢉⣥⡉⣄⠀⠀⢈⠁⢤⣄⡀⠀⠀⠀⠀⢀⠀⠻⠦⡀⠀⣀⠀⠉⠀⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⢷⢸⡇⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠤⠤⢸⡆⠀⠠⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠈⠢⡈⣼⣿⡿⠁⢀⡄⢀⠀⣰⣷⣿⣷⣀⠀⢄⠀⠀⠈⠛⠳⠦⠒⢀⠀⠐⠀⠀⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠐⢄⠀⠉⠁⠀⠀⣾⠀⠺⠈⢿⣿⣿⣿⠏⠓⠀⠙⠦⢤⣤⡤⠄⠀⠀⠀⠰⣄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⠀⢢⠀⠀⠀⠀⠀⠀⡽⠀⠀⠐⠋⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠠⢁⠀⠀⠠⠀⠈⠀⣦⠄⠀⠐⠓⠁⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠒⠶⠦⠄⠀⠄⡀
⠀⠀⠀⠀⠀⢀⣨⡶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣴⣿⣷⣦⣠⡆⣿⡆⣾⠑⠄
⠀⠀⠀⠀⣾⣿⠋⠁⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣸⣿⣿⣿⣿⣿⣿⡿⢟⣀⣦
⠀⠀⢸⣧⣿⣿⡄⠲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢀⣾⣿⡟
⠀⠀⠀⢿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣯⡄
⠀⠀⠀⣼⣿⣿⡿⠟⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠨⠉⠉⣼⣿⣿⣿⣿⣿⣿⢿⣿⡿
⠀⠀⠀⠙⠿⠹⠂⠀⠀⢰⣿⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠙⢓⣂⣠⣿⣿⣿⣿⡿⢟⣡⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣷⡀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠠⠀⢀⠄⠀⠀⣀⣀⠀⠀⣀⡀⢿⣷⣤⣀⠀⠀⠀⠀⢀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇⠀⠀⠀⠅⠀⡾⠀⣿⣯⠀⣾⣿⣷⣬⣻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⠀⠠⠃⢄⣀⣀⣠⢴⣿⠿⠆⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠘⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠦⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢢⡀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠢⢀⡀⠀⠀⠀⠀⠀⠈⠑⠂⠘⣆⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠁⠀⠢⢄⣀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠲⢤⠀⠀⠀⠀⠀⠈⠙⠢⠀⢀⡤⠶⠲⢤⡀⠀⠀⢀⠔⠀⠐⠀⠙⠦⣄⡀⠈⠢
⠀⠀⠀⠀⠄⠊⡠⠀⠄⠄⠀⠀⠀⣠⠃⠀⢸⠀⠈⠐⠤⠇⠐⠒⢈⡡⠀⠀⠀⡀⠀⠀⠉⠁⠀⡔⠒⠒⡄
⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⢠⠀⠒⠁⠀⠀⠀⠀⢰⣶⢢⠀⠀⠀⠸⠟⠀⠀⠀⠐⠒⠄⠀⠀⠀⠑⢀
⠀⠀⠀⠘⢲⠆⠠⠄⠀⠀⠈⠦⡀⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⡀⠀⠀⠀⠘⡀⠀⠚
⠀⠀⠀⠐⢣⠟⡀⠀⠀⠀⢀⡠⠀⡈⠂⠠⡄⢠⠀⠀⡀⠔⠉⠉⠉⠉⠈⠀⣤⠐⠁
⠀⠀⠀⠀⠉⢩⠃⠀⠀⠈⠉⠀⠈⠀⠈⠓⠴⣌⠋⢀⡠⢄⠪⢅⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠢⡀⠀⠠
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣾⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠸⡄⢠⡄⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠘⠶⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡎⠀⡀⠀⠀⠂⢀⠀⢣⠢⡀⠀⠃⠀⠀⠠⠟⠤⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣈⢦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠄⠀⠈⠂⠃⠀⠁⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠓⠄⠰⠄⡀⠀⠀⠀⣀⣀⠀⠀⠀⢠⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⣶⣿⣿⣿⣿⣿⣿⢿⣿⣶⣶⡶⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣷⣮⣝⢿⣿⣦⡙⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣌⠻⣿⣦⡜⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣠⠞⠃⠀⣠⠤⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠃⠘⢿⣿⡌⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⢁⣤⡶⣛⣛⣭⣽⡷⠒⠚⠿⣭⠤⠤⠀⢀⡀⠀⠈⢣⡀⠀⠸⢿⡇⠰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢾⠋⢩⣯⣿⣄⣉⠐⠒⠆⠀⠒⢚⠛⢻⣿⣿⣧⡀⠀⠀⢻⣄⠰⡘⠇⢸⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢘⠀⠱⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣆⠀⠀⠻⡄⠐⢀⣾⡟⠘⠿⣿⣿⣿⣶⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠘⢷⡀⠀⠉⠀⠘⢋⡴⢞⣠⣤⣴⣿⣿⣿⣿⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢺⣿⣤⣶⣾⣿⣧⠀⠀⠨⠿⠏⠀⣦⠄⠀⢨⣷⠀⠀⢀⠖⠀⠀⠉⠁⠀⠀⢀⣼⡿⠁⠀⠀⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠄⠀⠐⠀⠿⠥⠞⠫⠙⠋⠙⠣⠄⠐⠠⠖⠋⠉⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠟⠋⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⣸⡀⣿⠀⢀⣠⡴⠟⠃⠒⠀⠀⠐⠀⠙⠒⢦⡀⠀⠀⠀⡆⠀⠉⠃⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⢄⠙⢃⣿⣼⡟⠁⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠙⠳⣭⣀⣙⡀⠄⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠤⠀⠀⠁⠘⠛⠋⠀⢰⠇⢰⠠⣿⣿⣿⣿⡷⢄⠳⢤⡀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⢲⡄⠂⠀⠴⠙⠃⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⡢⢀⠱⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢱⡤⠐⠂⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣀⣶⣦⣄⢠⠆⣶⣈⡖⢄
⠀⠀⠀⠀⣴⡾⠟⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣶⣿⢿⡏⠀⡅
⠀⠀⢸⣆⣿⣧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣧⣿⣾⣿⣿⣿⣿⣿⡿⢀⣾⣿⡏
⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣯⡀
⠀⠀⠀⣼⣿⣿⡿⠟⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠛⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠙⠟⡻⠀⠀⠀⣼⡗⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠛⠒⠒⣠⣾⣿⣿⣿⡿⢟⣡⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣾⣿⣿⣿⣿⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢸⠀⢀⠄⠀⢀⣀⣀⠀⢀⣀⠈⣿⣷⣄⣀⠀⠀⠀⢀⣠⡀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⡟⠀⠀⠀⠈⠀⡶⠀⣿⣯⠀⣾⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠀⠀⠊⠤⣀⣤⡠⠾⢿⠟⠀⠉⠉⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠈⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⢿⠛⡟⢻⡿⢻⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠁⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠒⠒⠀⠀⠀⠀⠠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠉⠉⠉⠒⠂⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⠞⠁⠀⠀⢀⠀⠀⠀⣠⠒⠉⠁⠈⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⣁⠄⠀⠀⠀⠀⠀⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠁⠀⠉⠀⠀⣀⠀⠀⠀⠀⠀⣭⡬⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣀⣀⠀⠡⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣠⠖⠋⠉⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠉⠂⢾⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠿⣌⠀⠑⠀⠀⠀⠀⠀⢀⡠⠔⠀⣤⢤⡤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣤⠈⠃⠀⠀⠀⠄⠀⠀⠀⠀⣠⠾⣁⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢸⠹⡇⠘⠂⠀⠀⠀⠀⠀⠤⠴⠒⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠧⣼⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢺⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡄⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⠇⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⢀⡀⠀⠀⠀⢈⣀⡤⣀⠀⠁⠐⠒⠀⠀⠴⠚⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠈⠳⢤⣀⣀⠀⠀⠀⢀⣀⣤⠖⠋
⣀⣀⣀⢀⡀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⣀⣂⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⠋⠡⠤⠤⢀⣄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠚⠛⠟⠛⠛⣿⣽⣭⣹⣯⣥⣤⠄⠀⠀⠀⠀⠀⢈⡙⠻⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡤⢀⣴⣾⣷⣀⡀⠀⠀⠀⠀⠀⠀⠈⠡⡀⡀⢿⡜⣿⠋⠀⣀⣀⣤⣤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣇⣸⣧⣿⣷⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢾⢿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣧⢳⡄⠀⠀⠀⠠⠤
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣠⣦⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡸⣿⣿⣯⣭⣟⣉⣻⣿⢾⡇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣾⡟⠀⡀⢠⣶⣿⣏⠉⠀⠀⠘⣿⣿⣝⣉⡟⠛⠛⠛⠁⠀⣽⣿⣿⡏⠉⢹⣟⠉⠉⠁⠀⠈⣻⡿⡁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢹⣿⣴⣿⣿⣿⠿⣿⣷⣾⣦⡤⣿⠁⠈⠛⢿⡀⠀⠀⠀⢘⣿⣿⣿⣧⠀⠘⣿⢷⣤⣴⡶⠿⠋⢹⣇⡀⠀⠀⠀⠈⠫
⠀⠀⠀⠀⠸⣿⣿⣿⣿⠯⠀⠙⠿⣿⣿⣶⣿⡀⠀⠀⠈⠙⠛⠒⢛⣿⣷⣶⣿⣟⠀⣠⠇⣴⣿⠀⠀⠀⠀⠈⠁⠐⠂
⠀⠀⠀⠀⠀⢹⣿⣿⡿⠂⠀⠀⠀⢀⣾⣿⣿⣷⣄⣀⠀⣀⣤⣶⣿⣿⠛⠛⠿⣿⣿⣿⡿⠋⠀⠐⠲⣶⢄⣴⠀⠀⣀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠙⢿⣠⣤⣴⢶⣾⣿⣿⢿⣫⣿⣿⣿⣿⣿⠟⠛⠁⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠘⠀⠉⠚⠛⣿⡧⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⢿⠟⠀⢘⡷⠻⢛⠍⠉⣿⠃⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣶⠟⢉⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠊⢀⣠⣤⡤⠀⠋⢠⣤⣦⣴⣶⣿⣿⣿⣿⡟⠻⠿⡿⠟⠛⠅⠉⠉⠻⣦⠀⠀⠙⡀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⢷⣴⣠⡶⣿⡿⠛⣩⢫⡏⠉⠙⠀⠐⠀⠀⠙⠀⠀⢤⠁⠀⠐⠘⠁⠴⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠏⠀⡆⠀⠰⡀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠈⡃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠠⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⠋⢉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠑⠒⡄
⠀⠀⠀⠰⣜⢁⡀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⣀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠠⡞⢷⠃
⠀⠀⠀⠀⠀⠀⡨⠔⠒⠒⠂⠤⣤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠔⠒⠒⠒⠤⣤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠄⠒⠒⠒⠤⣤⠤⠤⡀
⠀⠀⠀⠀⢀⡎⠤⠐⠒⠒⠩⠶⣾⠀⠀⡱⠀⠀⠀⠀⠀⠀⢀⡬⠤⠐⠒⠒⠫⠵⢶⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⣼⠤⠐⠒⠒⠋⠷⢶⡃⠀⢸
⠀⠀⠀⠀⣎⡤⠮⠽⠉⠹⠭⠶⢤⡵⠒⠁⠀⠀⠀⠀⠀⠀⣎⡤⠮⠽⠋⠙⠭⠴⢤⡵⠒⠁⠀⠀⠀⠀⠀⠀⢮⡤⠶⢚⡍⢩⡓⠶⢤⣵⠖⠁
⠀⠀⠀⠰⡟⡄⠀⣁⠤⠤⡁⠀⡈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠟⡄⠀⢃⠤⠤⡘⠀⡸⠻⠀⠀⠀⠀⠀⠀⠀⠀⠿⢅⠀⢂⠤⠤⣘⠀⢸⢹
⠀⠀⠀⠀⢣⠈⠒⠁⠅⠌⠘⠊⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠒⠁⠙⠋⠘⠒⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠓⠃⠆⠡⠈⠒⠁⡌
⠀⠀⠀⠀⠀⠑⠂⠤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠤⠐⠋

⠀⠀⠀⠀⠀⠀⢰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠸⡀
⠀⠀⠀⠀⠀⠀⣀⠄⠂⠒⠠⢄⣠⠤⢄⠀⠀⠀⠀⠀⠀⠸⠔⠙⣀⠤⠒⠒⠢⠄⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠀⠤⣀⠤⠤⡀
⠀⠀⠀⠀⠀⣜⠤⠄⠒⠒⠢⢶⣬⠂⠀⠡⠀⠀⠀⠀⠀⠀⠀⣜⠡⠄⠒⠒⠺⠶⣬⠂⠀⠱⠀⠀⠀⠀⠀⠀⠀⣴⡡⠄⠒⠒⠺⠶⣦⠆⠀⢰
⠀⠀⠀⠀⣎⡠⢔⢖⠋⠓⠲⡦⣈⡣⠔⠁⠀⠀⠀⠀⠀⠀⣎⡡⠄⠴⠊⠙⠦⡲⢌⡱⠔⠁⠀⠀⠀⠀⠀⠀⢌⣁⢔⠥⠎⠙⠮⡠⢄⣱⠄⠃
⠀⠀⠀⠰⡟⡀⠀⢃⠤⢄⡚⠀⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⢒⠤⠤⡒⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢐⡤⠤⡐⠀⢸⢹
⠀⠀⠀⠀⢃⠘⠲⠁⠔⠆⠙⠒⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠲⠃⠢⠔⠘⠖⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠲⠃⠐⠂⠘⠖⠃⡔
⠀⠀⠀⠀⠀⠑⠠⠤⠄⠠⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠠⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠠⠤⠄⠊

⠀⠀⠀⡠⠤⠤⠤⡄
⠀⠀⠀⠣⠤⠤⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡩⠔⠒⠒⠒⠤⣤⠤⠤⡀⠀⠀⠀⠀⠀⠘⠽⠀⡡⠔⠒⠒⠒⠤⣤⠄⠤⡀⠀⠀⠀⠀⠀⢀⠑⠀⡠⠔⠒⠒⠒⠠⢤⠄⠤⡀
⠀⠀⠀⠀⢀⡮⠔⠒⠒⠒⠋⠲⣾⠀⠀⡱⠀⠀⠀⠀⠀⠀⢀⡮⠴⠐⠒⠒⠉⠳⢶⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⡬⠤⠀⠒⠒⠋⠳⢶⡃⠀⢸
⠀⠀⠀⠀⣯⡴⠂⠘⠉⠙⠁⠒⢤⡵⠊⠁⠀⠀⠀⠀⠀⠀⣮⡤⠮⠮⠋⠙⠗⠵⢤⣵⠒⠁⠀⠀⠀⠀⠀⠀⢮⡤⠦⠭⠋⠉⠯⠵⢤⣵⠒⠁
⠀⠀⠀⠐⡟⡄⠀⡣⠤⠤⡈⠀⡈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠟⡄⠀⣃⠤⠤⡘⠀⡸⠻⠀⠀⠀⠀⠀⠀⠀⠀⠿⢇⠀⢉⠤⠤⢘⠀⢰⢹
⠀⠀⠀⠀⢣⠈⠒⠁⠈⠀⠘⠊⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠑⠁⠉⠉⠘⠊⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠑⠃⠀⠜⠈⠒⢁⡄
⠀⠀⠀⠀⠀⠙⠂⠤⠀⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠄⠠⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⠠⠤⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢲⠠⠀⠒⠂⣤⡀⣀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠉⠙⠛⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢠⡞⠀⠀⠀⠀⡀⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⡿⠀⠀⠀⡘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸⠃⠀⡀⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣦⠤⠴⠦⠖⠒⠂⠀⣴⠤⡄⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⢚⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠋⠁⠈⠈⠲⡀⠀⠀⠀⠀⡀⠈⠀⠁⠛⢷⡀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢹⡆⠁⠀⠀⠀⠀⠀⠀⢀⠀⠈⢀⡤⠖⠒⠒⠲⠒⠁⠀⠀⠀⠀⠉⠈⠉⠁⠲⢸⡏⢣⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⡏⢻⠞⠁⢀⠴⠒⠒⢦⡀⠀⠀⠀⠀⢠⡰⣫⣉⠙⢢⡈⡁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⢛⡀⠀⠀⠀⠀⠀⠟⠀⡼⠀⢰⠃⣻⣿⣦⠀⠹⡄⠀⠀⠀⣾⠣⣾⡿⠂⠀⣧⢡⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⢻⡁⠀⢠⣶⡄⠀⠀⣇⠀⣠⣀⣛⣿⣃⣀⡀⠇⠀⠀⠀⣿⠒⠀⠠⠒⢉⠛⠋⠒⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣡⠶⡜⠻⣄⡑⢂⣾⠃⠀⠈⠉⠀⠀⠀⠀⠀⡠⠀⠀⠚⠓⢲⡀⠀⠖⠒⠄⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠤⠞⠃⠘⣠⠜⠉⢰⠀⠀⠀⠐⠙⠀⠀⠀⣀⠦⠀⣀⣀⣀⡴⠗⠤⢀⡠⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠘⢆⡀⣠⠞⠁⠀⠀⡄⠀⠀⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡽⠃⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠿⣿⣿⣷⣾⡖⠂⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠒⠒⠂⠁⠀⠙⠲⣄⣀⡉⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠉⠀⡄⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡼⠇⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾
⠀⠀⠀⠀⠀⠀⠀⠴⠁⠈⠊⠀⠀⠉⠹⠞⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⡸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡔⠁
⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠋⠀⠀⠀⠑⢄⢰⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠁
⠀⠀⠀⠀⠈⠋⠐⠄⣀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠙⠀⠻⢦⡀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠉⠒⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠏⠓⢤⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⢿⣄⡸⠗⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢬⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣄⣁⡶⠶⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣵⣍⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠤⠤⠤⢽⣷⡌⠻⠟⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⣀⡀⢻⡀⠀⠀⠉⠻⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠈⢀⡽⠺⣇⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣠⣀⣤⣤⣤⣤⣌⡈⠀⢻⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⣧⡀⠘⣿⢭⣍⣫⠛⢿⣿⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡻⠋⠇⠀⢠⡁⠀⠀⣠⡾⣿⠀⠀⢀⠀⣼⣧⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣧⣰⣧⠄⠐⠉⠛⢂⣡⠿⡿⠀⣠⠏⡼⠃⢰⠄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣥⣿⣷⣟⢟⣛⡿⢟⡯⠤⠧⣰⣵⢏⣠⣖⠒⠒⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⡟⣿⠋⠟⠛⠛⢠⡞⠀⠀⠀⠈⢻⣾⢗⡥⠀⠒⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⠈⠈⣧⢐⡀⠀⢸⡀⠀⠀⠀⠀⣘⡕⠁⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⡭⢣⡰⠀⠠⠚⠀⠀⠘⢿⣿⣶⣶⢿⣿⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⡸⡷⣠⠀⠀⠀⠀⠀⠊⠋⠉⠁⣸⡏⢀⣴⡟⠀⢀⡐⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢢⣴⡟⢿⣶⣴⠋⣤⠀⠀⠀⠀⢀⠀⠀⠀⣿⡇⣾⠏⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠉⠉⠙⠲⢾⡀⠀⠄⠀⠘⠶⠶⣤⣿⣧⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠲⣆⠀⠀⠀⠀⠀⠀⠀⣀⡹⠚⠒⠒⠢⠦⠤⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⢪⡟⠃⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣦⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣫⡾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣞⣿⣽⠁⡏⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠹⣯⠃⢿⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠸⣿⣧⡙⢧⣜⢧⣿⢿⠄⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣯⣥⣀⣀⠀⢿⣿⣿⣦⣽⣧⣿⣷⡄⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠙⠛⠿⠿⠿⣿⣿⣦⣄⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣝⢿⡏⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⢾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣧⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣠⣴⣶⠶⠿⠿⠿⠿⠷⠶⣶⣤⣀⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣠⣾⠛⣉⡤⠀⠠⡀⠀⠀⠀⢀⣀⢠⠬⢿⡿⢦⣄⠀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢻⡇⠸⠶⠶⢤⣤⣤⡀⠀⠀⣼⣦⡴⠖⠲⠽⢸⣿⢷⣸⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⣿⠀⢠⠞⢿⣿⣿⢇⠀⠘⣿⣿⡿⠷⡄⠀⢻⡏⠸⣿⣱⠏⠁⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⡏⣞⠉⠀⠈⣉⠴⠃⠀⠈⠦⣉⠁⠀⠀⢹⣎⣿⣦⣈⣿⣶⣀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠏⢸⣉⣀⣴⠿⠿⠰⢤⣤⣴⠶⠿⠿⣦⣀⣸⣿⠎⢿⣛⡟⠻⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠀⣄⠉⠉⠀⠀⠀⠀⠀⣠⡀⠀⢀⠀⠀⠈⠉⢁⡾⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡷⠂⠀⠘⢷⣶⣶⣊⣤⣶⠞⠛⢛⣶⣮⣳⣶⣶⣶⠿⠁⠐⢶⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠁⠀⠙⠉⠉⠁⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠰⡀⣰⠊⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣄⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⣿⡇⠀⠀⠀⠙⠀⠀⠀⠀⠲⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠊⠀⢸⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⣷⣸⡿⢻⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢦⠀⠙⠛⠋⠀⠀⠀⠀⣿⣿⣷⣿⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣿⣿⣇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡀⠀⣼⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣆⢿⡄⠀⠀⣤⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣼⢃⣼⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⡀⢸⣧⡀⠀⠀⠞⣡⡇⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣦⣻⣿⡄⠀⣰⣿⣠⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣧⠄⠉⠙⠻⢿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⣀⣁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣾⣿⡿⠛⠙⠛⠙⠋⢉⣙⠒⠤⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⢿⣿⣿⣿⣿⣯⣤⠤⠀⠀⠀⠈⠉⢠⠴⠂⢀⣡⠤⢠⡀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡷⠀⠀⠐⣿⣿⡝⠉⠙⡷⠶⠦⠀⠀⠀⠀⠠⠞⡋⠀⢖⡸⡇⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠿⣿⡄⢤⣤⡶⢣⣿⢟⣠⠶⣂⠀⢠⣾⣀⣠⠀⠀⠋⠷⠶⠈⣁⣼⠀⠀⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣤⢿⡿⡼⠱⠏⣉⣀⣿⣿⣿⣅⣤⠀⠲⠾⠾⠾⢿⣿⡇⢠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠸⣤⠀⠀⣤⠟⠃⠸⣿⣿⣷⣿⠿⠿⠿⣿⣧⢷⡀⠀⠐⢰⣦⣀⣴⡿⠷⠉⠚⣻⡿⠋
⠀⠀⠀⠀⢀⡤⠿⠆⠘⢦⠀⢀⡀⢹⣿⣯⣁⣀⣠⠴⠞⠃⠃⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠋⠶⠆⠀⠀⠀⠀⠈⠓⠆⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⢀⣀⠀⠚⠁⠀⠠⣤⣤⠴⡀
⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠐⠒⠒⠀⣀⣀⡀⠐⣈⡀⣤⣴⣾⠟⠛⡡⠉⠉⠉⠀⠀⠀⠀⠀⠀⢄⣴⠏
⠀⠀⠀⣞⣋⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠙⠛⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣶⡀
⠀⠀⣾⣿⣿⣿⣿⣿⣶⡃⡌⠀⠀⠀⠰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⣀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⣷
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣤⡶⠂⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡅
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣀⣠⣧⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⡼⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠉⠛⠻⣿⠿⠟⠛⠙⠻⠶⠤⡤⠤⠀⠒⠛⠛⠛⠀⢦⣀⣷⢀⣴⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠶⠒⠒⠒⠛⠛⠛⠒⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⡠⠄⠐⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡴⠊⠁⠀⠀⠀⢀⣀⢀⣀⣀⣀⣠⢤⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢀⣤⡞⠉⣸⠋⠉⠀⢀⡽⣤⣨⠧⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣸⠖⣲⠞⠁⠀⢱⠋⣁⣼⡀⠀⢸⣦⡀⢳⠀⠀⢞⠀⠸⡍⢱⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡇⠀⡇⠈⠓⠒⠢⡤⠼⠶⡿⠋⠉⠉⠉⢻⠛⠛⠲⠒⠼⠓⠊⠁⢸⠃
⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠈⡇⠀⠣⡄⠀⠀⠀⠀⠀⠀⠱⢤⣀⣠⣤⠊⠀⠀⠀⠀⠀⠀⢀⣴⡋
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠻⣄⣀⡀⣀⣀⣠⡴⣎⣉⣉⣩⡷⡦⠤⠤⠤⠴⠚⠉⣼⠁⠀⢀⡼⣕⠈⢱
⠀⠀⡏⠀⠀⠀⠀⠀⠐⠒⠺⠁⠀⠘⡆⠀⠉⠉⠉⠁⠀⠙⣌⢻⣟⢛⠕⠁⠀⠀⠀⠀⠀⠠⡟⠉⠉⠁⣨⡮⠂⠁
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠞⠁
⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⣠⢯⠞⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠚⠋⣡⠋⠀⠀⠀⠣⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⢠⠂⠀⠀⠈⠳⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡠⠤⡇⢧⣠⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⡖⠒⠶⠚⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠣⠤⢼⣲⠶⠚⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⢑⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠈⠐⢉⣀⣷⣀⣀⡀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⡔⡏⠒⠒⢒⣸⠙⣦⣤⣴⣶⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⣇⠹⠿⠿⠿⠿⣠⠿⠛⠛⠉⠉⠀⣀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡤⣄⣀⣀⡀⠈⠓⠒⡶⢶⠋⢁⣀⣀⡤⠴⠒⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⡜⠋⠉⠉⠉⠁⠈⠉⠉⠉⠲⢤⣤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡍
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⠉⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠙⣆⣠⡤⠖⠚⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠰⣯⣭⣤⣤⣤⣤⣀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡼⠁⠀⠀⢀⡬⠗⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⢀⡴⠛⠒⠒⠒⠋⠉⠉⠉⠉⠙⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⢿⠹⡏⡶⡆⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⡔⢱⣇⠀⢀⡞⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⣤⣸⠞⣇⡴⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣧⣤⡄⣺⣾⣦⠟⣠⠶⡟⠉⠁⣽⠷⢛⡛⣏⠉⠁⠙⠳⢦⣄⠀⠀⠀⠀⠀⠈⠳⣼⠀⠿⠸⠏⢠⢇
⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⢰⣽⠞⢁⣾⣧⡀⠀⢀⣀⣸⣛⣋⣀⡀⠀⢀⡀⣌⢻⢲⣄⠀⠀⢀⣤⣽⠀⢰⡆⠀⢟⣩
⠀⠀⠀⠀⠀⠀⣨⠷⢤⠤⠴⢚⡟⠁⠀⢸⡙⠟⠛⠚⠋⠁⣼⠋⠁⠀⠉⠳⣜⡛⠃⠀⢿⠉⠳⣄⡎⢸⡑⠦⠈⠁⠐⠋⣹
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⠎⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣰⠏⠙⢳⡄⠈⠀⠀⡿⡇⠀⠉⠳⠤⣀⡴⢻⠁
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⠲⣄⣀⣀⣀⣀⣬⣿⡖⠚⠁⠀⠀⠀⢿⣀⣀⡰⠁⡟⢦⣀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠦⡤⢬⣷⣀⠀⠀⠀⠀⠀⣼⡿⣵⠀⢹⠀⠈⠉⠓⠒⣞⠁
⠀⠀⠀⠀⢸⡇⠀⢀⡴⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡗⣾⠁⠈⠳⠦⣤⣄⣀⣤⠶⠋⠀⢸⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⣧⡴⠋⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠳⣄⠀⡿
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠙⣾⠁
⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠸⡆
⠀⢰⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡄⣀⣠⡤⢶⠖⠦⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢠⡞⢧⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡜⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡆
⡏⢡⠈⠳⣄⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡇
⢇⠈⢳⡀⠈⠳⣄⠈⠙⠲⠦⢤⣤⣀⠀⠀⠀⠀⢸⠇⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⣀⡴⢻⡄
⢸⢧⠀⠙⢦⡀⠈⠙⠒⠤⢄⣀⡀⠈⠉⠉⠉⠉⢻⠟⠓⠛⠓⠛⠳⡖⠒⠢⠴⠤⠤⠤⠒⠒⠋⠁⠀⣀⠤⠚⠁⠀⣸⢷
⠈⡎⠳⣄⠀⠙⠲⢤⣀⠀⠀⠀⠉⠉⠒⠢⠤⣤⠼⠧⣄⣀⣀⣀⣠⣃⣀⣀⣀⣀⣀⡠⠤⠴⠖⠒⠉⠁⠀⠀⠀⣠⠏⢸
⠀⠘⢦⡈⠳⣄⡀⠀⠈⠙⠲⠤⣄⣀⣀⣀⣀⡿⠀⠀⣽⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢀⡇
⠀⠀⠀⠙⠦⣄⡙⠳⠦⣄⣀⠀⠀⠈⠁⠈⠉⠀⠀⢀⣜⠀⠀⠓⠶⠤⠤⢤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠚⠋⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢸⠙⠳⢤⣀⡉⠉⠙⠛⠃⠐⠶⠒⠛⠉⠉⠓⠲⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢖⡿⠃
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠉⠙⡳⠶⠤⢤⣤⣀⣀⣀⣀⣀⡀⢀⣀⣀⣉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣥⠤⠖⠋
⢀⡴⠒⠒⠒⠚⢧⡄⣀⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢿⠉⠉⠉⠉⠉⠁⢀⡇
⡞⠀⠀⠀⠀⠀⠀⠃⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⣀⣀
⠘⢦⣀⡀⠀⠀⣀⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠙⠒⣶⠊⠉⠀⠈⠉⠳⣄
⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⣤⠷⣄⣀⣀⣀⣠⠴⠋
⣿⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣠⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⡀⠄⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⡂⢄⠀⠀⠀⠆⠐⠁⠁⠈⠈⠁⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠃⠰⠀⠀⡀⠔⢂⡈⠁⠀⡀⠐⠠⠀⠀⠂⠀⠄⠀⠀⠀⠀⢀⠚⠁⠀⠐⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⣀⠔⠀⠀⠂⡈⠥⠘⠢⡄⠀⠈⠀⠄⡍⠔⠙⢢⠀⠀⠀⠀⠀⠂⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⠑⠄⠠⠔⠈⠒⠠⠀⢂⠁⠂⠀⠀⡇⠀⢰⣀⣰⣦⣮⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⡀⠐⠀⠀⠀⠈⠡⠆⠘⠂⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠰⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡠⠠⡄⠀⠈⠀⠒⠁⠁⠀⠀⠀⠁⢀⣔⠀⠈⠁⠢⣀⠀⠘⠀⠡⠈⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⡝⠂⠠⠤⠤⠄⠀⠐⠈⠀⢛⡹⠀⠀⠀⠀⠀⠈⠀⠀⡇⢈⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⢁⡌⡀⠀⠀⣰⣦⡴⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠠⠤⠤⠄⠓⠈⠀⠁⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠁⠐⠁⠈⠻⡻⣿⡿⢸⣶⣶⣶⣶⣶⣿⣿⠿⠖⠡⠤⠠⢄⣀⡠⠀⣀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠄⠚⠁⠈⡂⠀⠀⠀⢠⠄⡀⠀⡑⠂⠉⠛⠛⠛⠋⠉⠉⠀⡀⠌⠀⠀⠀⠈⠀⠀⠈⢻⠙⠢⠅
⠀⠀⠔⠁⠀⠀⢛⣀⠀⣠⡀⠀⠀⠀⠐⢀⠀⠔⠀⠀⠀⠀⡀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠄
⠀⠀⠀⠀⠀⢀⢘⠛⠃⠉⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⣾⠀⠁⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠊⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢩⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠐⠄⡀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⣶⠆⠀⠈⠓⠀⠙⢉⠁⠀⢠⠄⣤⠄⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢰⣮⢁⣆⠀⠀⠀⠀⢀⠀⠤⠊⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⢀⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠀⠐⠖⠷⠀⠀⠀⠉⠉⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⣿⠶⠀⣄⣀⣸⣄⣷⣄⣀⡀⠀⠀⠀⠀⢿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⡶⢿⣯⣭⡷⣏⠀⠀⠈⡑⢲⣦⡀⢠⣶⣿⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠹⠎⣹⣿⠋⢠⡾⠓⡶⠾⢿⠇⢠⣾⡟⠛⠹⣿⣿⠋⠉⢀⣠⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢿⠉⠀⢿⠃⣀⢀⣀⣄⣀⢙⠗⠛⠋⠀⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⣿⠿⣿⣿⣿⣿⣿⡿⢃⣤⣄⣤⣿⣟⣻⣁⣸⣟⢛⣳⣤⣤⣼⡿⣿⣿⣿⣿⠛⠋⣹⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⠀⠄⠙⠿⣿⣿⣿⣧⡈⠾⣏⣈⣹⢛⣭⣛⣛⠛⣻⣿⣦⡀⠿⢧⣿⣿⣿⣣⣄⠘⠶⠻⣿⣿⣿⣿⣿⣦
⠀⠀⣰⠿⣦⡀⠈⠛⠋⢀⡀⠀⠀⠘⢛⣿⠿⠿⣿⡿⣿⢿⡏⢹⠷⠛⠛⣟⠃⢹⡄⢰⡇⢀⣿⣿⣿⡉⠀⣴⣤⡀⠹⣿⣿⣿⣿⣧⣀⣴⣦
⠀⠀⠍⢸⣿⣿⣤⣄⣠⡆⠀⠀⣀⣷⠟⠁⣠⡾⢿⣧⡾⠈⡳⠂⣄⠀⠀⣿⡆⠸⠸⠏⠉⣿⣿⣿⣿⣷⣄⢉⣉⣤⡾⠟⠋⠛⠛⣻⣧⡄⢷⣶⡆
⠀⠀⠀⠶⠟⢿⠉⠁⠀⠙⠛⠻⣿⣧⣀⣼⠟⠀⢸⣿⠃⠈⠁⠀⢹⡁⢰⣏⠀⠀⢠⡄⣿⣿⣿⠖⢀⠉⢻⣿⠟⠉⠀⢀⣤⢶⣿⡛⠻⠶⣄⠾⠃
⠀⠀⠀⠀⠀⠀⢱⣤⡆⠀⠀⠀⠀⠀⠛⠃⠀⣤⣿⣿⣯⠀⢷⡿⠃⢴⠀⣭⣕⠀⠹⣾⣿⣿⣿⣷⢮⡶⠿⠛⠢⠾⢿⠏⡁⢸⠙⠷⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠸⢿⣮⣵⣆⣲⣤⣶⣤⣾⣿⣯⠹⣿⣷⣴⣢⣧⣼⣥⡄⡟⣱⣾⠇⢉⣉⣀⠀⠀⠀⢠⣼⣅⠲⣾⠁⣸⣧⣤⣶⢿⣯⣿⣿⡄
⠀⠀⡈⠀⠀⢀⠀⣴⠾⠋⠉⢉⣩⣭⣿⠏⡙⢿⣿⡀⢻⣿⣿⣿⣶⣾⣷⣾⣷⣟⣡⣴⡟⠉⢩⣿⡶⣿⠛⣿⡿⢦⣌⣀⣿⡏⠻⣿⠛⣿⢿⣏⡇
⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⢀⣤⣄⣠⣥⠊⢉⠀⣸⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣇⡀⠠⠙⠃⣨⠀⢸⣿⡀⠙⣿⣿⣿⣼⡛⠛⣋⣸⠟⡇
⠀⠀⠀⠀⠀⢲⡌⢿⣄⡀⠀⠀⠠⣤⣼⡿⠷⢶⣤⣶⣼⣿⣿⣿⠒⠈⢉⣭⣝⣻⡷⢋⠟⠛⠁⢠⠞⢃⡄⢸⣿⣷⣧⠘⣿⣽⣿⡁⠀⠈⢻⡆
⠀⠀⢄⣀⡄⠚⠛⢀⡿⠉⠿⢿⡶⢿⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⡿⢿⣿⠟⢛⠃⠀⢀⣀⡀⠀⢠⣮⣠⠞⠋⢿⣤⡄⢉⡉⢿⣅⣶⣀⣸⣿⣠
⠀⠰⣾⣷⣶⣶⣾⣿⠃⠀⠀⢀⣠⣿⣿⠀⠀⠀⢀⣿⠏⣉⣩⣿⣿⣷⡿⢧⣿⣿⡷⣶⣉⡿⢩⣿⡀⣼⣿⣾⡿⠋⣉⣽⠟⢃⡘⠛⢿⣿⣿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⠀⠀⠀⣸⣿⠀⣿⡏⢸⣿⠋⣠⣾⣿⣿⣿⣿⣿⣷⣦⣽⣏⠛⢋⠉⣠⣾⣫⣴⡾⠿⠟⢳⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⠿⣿⣿⡿⣿⡿⠹⣿⣿⣿⣿⣟⠛⣿⣿⣶⢿⣧⣼⣿⣦⡸⠟⢡⣿⣿⣿⢿⣿⣾⣿⣷⣴⣽⡿⡏⢻⣿⣿⠶⣶⡶⠾⢿⡟⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡗⢰⣿⣿⣿⣿⣧⢀⢿⣿⣿⣿⣷⣿⣿⣿⠻⣆⡀⠀⡊⠉⠾⠿⣿⠿⢟⣿⣼⡧⢿⣿⢹⣿⣷⣎⣛⢷⣸⠃
⠀⠀⠀⠀⠀⢀⣀⣿⠃⠘⠁⣿⣿⣿⣿⠿⣿⣆⢼⣿⣿⣿⣶⣤⡙⠊⠉⠻⣿⣿⣿⣶⣦⣤⡈⣀⣙⣿⣿⡵⠞⠉⠸⡿⢿⣏⣽⣾⠃
⠀⠀⠀⠀⠸⣿⣿⣷⣤⡈⠹⣿⣿⣿⣿⣦⠀⣻⡟⠛⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢿⣶⢦⣤⡄⠖⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣿⠃⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣭⣴⣶⣿⣿⣿⣿⠔⠃⠈⣿⣆
⠀⠀⠀⠀⠀⢀⡴⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⣠⣾⣿⣿⣿⣿⣿⣿⣏⣿⣟⣿⣶⣾⣿⣿⣿⡿⠓⠀⣁⣴⡿⠿⣿⣶⡆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠊⠉⠹⠿⠿⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣟⣻⣿⣿⠿⠿⠋⠋⠙⠋⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡏⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⣿⠇
⠀⠀⠀⠀⢸⣦⠂⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⠀⣸⣿⠟
⠀⠀⠀⠀⠀⢿⣤⠀⡠⠀⠀⠀⠁⠀⠀⣀⠀⠀⣿⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⣧⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠻⠷⢤⠤⣤⣤⣤⣀⣈⣀⠀⠹⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣶⣶⡿⢦⣤⣴⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⠿⣿⣿⣿⣿⠿⠿⣿⣿⣆⠀⢰⣿⡟⠿⣿⣿⣿⣿⣿⠟⠋⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠻⣿⡇⠀⠀⢿⣿⣀⡀⠀⠀⠁⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⣀⣀⣘⣉⣤⣤⣾⣿⠀⠀⠀⠈⠙⠻⣷⣶⣶⣶⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⢿⣿⣿⣷⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⣿⣟⣛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡈⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠛⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣶⣷⣿⣿⣿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⠶⠶⠶⣶⣶⠶⠶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠚⠛⠉⠉⠉⠉⠉⠛⠒⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠋⠀⠀⠀⢀⡤⠤⡤⠤⠤⠤⡖⠲⢄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢧⣀⣀⡤⠖⢫⣠⠜⠡⠀⠀⢰⡙⠲⡚⠂⢰⠇⢠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠰⣋⣀⡀⢐⣇⣰⡿⠇⠀⠿⣷⡦⣇⣀⣈⣄⡨⠇⢸
⠀⠀⠀⢀⡴⠒⠒⠊⠉⠒⡇⠀⣇⠀⠀⠀⠉⠉⠀⢻⡀⠀⠀⣀⠄⠀⠀⠀⠈⠀⠀⢀⠞
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡇⠀⡸⢧⡀⠀⠀⠀⢀⣠⡼⠋⠉⣓⣤⣄⣀⣀⣠⠤⠖⣹⠀⠀⢀⣐⠄⠠⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠤⠴⠃⠀⢳⠄⠉⠛⠛⠉⠉⠱⣝⢿⡿⢟⠝⠀⠀⠀⠀⠀⠀⡟⠒⠂⠉⣰⡱⠚
⠀⠀⢧⠀⠀⠀⠀⠀⠑⠤⣄⣀⣨⣦⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⣠⠞⠁
⠀⠀⠀⠑⢦⡀⠀⡀⠀⠀⣠⣫⠋⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠠⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⣰⠃⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠃⢀⡴⠀⠀⠀⠈⠳⠤⠤⣄⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⠂⢧⡑⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠉⠋⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⠓⠙⠟⠉⠈⢧⠀⠀⠀⠀⠀⠀⠠⣀⠠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠔⠒⣛⣒⡲⣄⠀⣀⣀⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣤⣤⣤⡇⢧⣤⣶⣶⡧⢸⣿⡿⠿⠟⠋⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣉⣉⠉⠉⠉⠙⠮⢭⣤⡤⠖⠁⠀⢀⣀⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣉⡩⠗⠒⠒⠚⠁⠙⠒⠛⠫⢍⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏

⠀⢀⠄