Antler Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢱⠂⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣾⠀⠀⠀⢰⣿⢠⢀⠀⠀⠀⠀⠀⠀⠀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢧⣰⡀⠀⠀⠀⠀⠀⠀⣆⡼⠁⠀⠀⠀⢸⣿⡟⠁⠀⡆⠀⠀⢠⠀⠈⣷⣿
⠀⠀⠀⠀⠑⢄⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡆⠀⠘⣦⡴⠋⠀⠀⠀⠀⠀⠀⠙⠛⠶⢴⣇⠀⠀⣸⡤⠶⠟⠛
⠀⠀⠀⠀⠀⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉



⢦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠔⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡄
⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡎⠀⠀⠀⠀⣦⢇⠆⠀⠀⠀⠀⠀⠀⡘⣴
⢀⣸⣼⠀⠀⠀⡀⡀⠀⠀⢀⣧⣇⠀⠀⠀⠀⢹⣸⠆⠀⠀⠀⠀⠀⠐⣇⡟⡀⠀⠀⠀⠀⣂⢀⡇⠀⠀⠀⠀⠀⠀⣰⠀⣀
⠀⠉⠙⠳⣤⣠⠇⢹⣀⡤⠞⠉⠁⠀⠀⠀⠀⠙⠷⣄⣆⠀⠀⠀⣰⣠⡼⠛⠀⠀⠀⠀⠀⠿⣼⣇⡀⠀⠀⠀⠀⢠⣸⣤⠟
⠀⠀⠀⠀⠀⢹⡆⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⠀⠀⡾⠛⠉

⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠙⡇⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⣔⣀⡇⠀⠀⠀⠀⠀⢷⣈⡆⠀⠀⠀⠀⢰⠃⠄⠀⠀⠀⠀⠀⠀⠠⢘⡄
⠀⡀⢸⡆⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⢿⣀⠀⢠⡀⠀⠀⢀⠀⠀⣠⡟⠀⠀⠀⠀⠘⠿⠧⣤⡄⠀⢦⣤⠞⠾⠁⠀⠀⠀⠀⢼⣴⡆⠀⠀⡀⢀⠀⠀⢰⣬⡇
⠀⠈⠛⠳⣄⠀⢀⣤⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠚⠷⠀⠠⠟⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⠀⠀⠷⠛⠉⠁
⠀⠀⠀⠒⠉⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢰⡌⡶⠀⠀⠀⠀⠀⠐⣾⢰⠀⠀⠀⠀⠀⣧⣠⡤⠃⠀⠀⠀⠀⠰⢤⣀⣧⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⠟
⠘⠳⣧⣀⠀⠀⠀⢀⣀⡿⠟⠀⠀⠀⠀⠀⠙⠻⠤⠤⢤⠀⠀⡤⠤⠤⠟⠁⠀⠀⠀⠀⠀⠷⠦⠠⣄⠀⣦⠄⠶⠾⠀⠀⠀⠀⠀⠀⠀⠈⡶⢄⠀⠀⡤⠞⠁
⠀⠀⠀⠉⠓⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⣰⠆⡀⠀⠀⠀⠀⡀⢴⡀⠀⠀⠀⠀⠀⢠⠸⠂⠀⠀⠀⠀⠀⠀⠐⢆⡆⠀⠀⠀⣦⠄⢠⠗⠀⠀⠀⠀⠀⠺⡄⠠⣶⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢢⠄
⢀⠙⣖⠁⠀⠀⠀⠀⠘⡾⠁⠄⠀⠀⠀⠑⠺⠠⠤⣄⠀⠀⠀⠀⣠⠤⠄⠓⠁⠀⠀⠈⠳⢿⣀⡀⠀⠀⠀⣄⣠⠿⠚⠁⠀⠀⠀⠐⠒⢣⠀⡀⠀⠀⢠⢀⠞⠒⠂
⠈⣗⠛⠲⣇⠀⢀⡷⠚⢓⡏⠀⠀⠀⠀⠀⠀⠀⠠⢌⣣⠀⢀⣞⡅⠄⠀⠀⠀⠀⠀⠀⢐⠲⠮⣿⡄⠀⢠⡿⠵⠶⠀⠀⠀⠀⠀⠀⠀⠐⠛⠳⠀⠠⠞⠛⠂
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠘⣇
⠰⣧⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠰⣤⣠⡄⢸⠏⠀⠀⠀⠀⢻⡆⢠⣄⣤⠂⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⢠⠚⠉⠀⠀⠀⠀⠙⢳⡔⠀⠀⠀⠀⠀⠀⣦⠟⠁
⠀⢎⣷⡀⠀⠀⠀⠀⣠⣎⠄⠀⠀⠀⠀⠈⠙⠓⠻⢷⣆⠀⠀⣴⠾⠗⠚⠋⠀⠀⠀⠀⠠⠐⠛⢦⡀⠀⢀⡴⠛⠂⠄⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⢀⡴⣇
⠀⠀⠀⠝⠳⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⠀⠀⠟⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠷⣄⢀⠀⠀⡀⣤⠞⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⣧⠌⠀⠀⠀⠀⠐⣤⠇
⠀⠀⠀⠀⠈⠙⠆⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡄⣠⡤⠞⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⢀⣴⠃⠀⠀⠀⠀⠀⠘⢦⣆⠀⠀⢠⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠋
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣏⢄⠀⠀⠀⠀⠀⢀⢤⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠱⢆⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠙⠶⢦⣄⠀⠀⣰⠼⠾⠃⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠙⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠀⠛⠋⠁
⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠈⠚⢆⠀⠀⠀⠀⠀⠀⡴⠒⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⢱⠆⠀⠀⠀⠀⠰⡜⠁⠀⠀⠀⠱⡴
⠀⠀⠀⠚⠓⢄⠀⠀⡤⠚⠒⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⠢⣁⠀⠀⠀⠀⠀⠀⣁⠞⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠄⡷⠁⠀⠀⠈⠳⡇⡄
⠀⠀⠀⣏⠄⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣜⠁⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⢳⡇⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠙⢧⣄⠀⠀⣡⠜⠉⠀⠀⠀⠀⠀⠀⠈⠢⠄⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⣠⠼⠁
⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠔⠀⠀⠀⠀⠀⠐⢾⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠘⣄⠁⠀⠀⠀⠀⢁⡜⠀⠀⠀⠀⠀⢰⡎⠀⠀⠉⡆
⠀⠀⠈⠓⠶⣄⡀⢀⣠⠶⠚⠁⠀⠀⠀⠀⠳⢤⣄⠀⠀⣠⡴⠗⠀⠀⠀⠀⠀⠘⠦⣄⡀⢀⡤⠞⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣠⠋

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡆⣴⠀⠀⠀⠀⠀⠀⠀⠰⡄⣷⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠂⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢤⣸⡀⠀⠀⠀⠀⠀⠀⢀⣇⡠
⠀⠀⢸⡸⡄⡀⠀⠀⠀⠀⠀⢠⢰⢇⠆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠙⠛⠷⣄⣁⠀⣄⣠⠶⠛⠋⠀⠀⠀⠀⠘⠳⣤⣔⠀⢀⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠽⣆⠀⠀⢰⡾
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⣸⢠⠂⠀⠀⠀⠀⠀⡄⢧⡀
⠀⠀⠀⡇⣇⡄⠀⠀⠀⠀⠀⢀⣸⢸⠀⠀⠰⣇⡜⠁⠀⠀⠀⠀⠀⠀⠙⣄⡿⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢰⡿⡇
⠀⠀⠀⣧⢸⠁⠀⠀⠀⠀⠀⠈⡇⣼⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢠⠀⣼⠁⠀⠀⢴⡸⡇⠀⠀⠀⠀⠀⠀⢸⢇⡧
⠀⠀⠀⠙⠾⣄⣀⠀⠀⠀⣀⣠⠷⠋⠀⠀⠀⠀⠹⢦⣇⣆⠀⠀⣆⣧⠾⠁⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠙⠆⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠰⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢀⡀⠀⠆⠀⠀⠀⠀⠀⠀⢆⠀⣄⠀⠀⡞⠀⣠⠖⠀⠀⠀⠀⠐⣄⠀⢱
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠘⣇⡼⠀⠀⠀⠀⠀⠀⠀⠘⣄⡿⠀⢀⢳⣼⠥⠖⠋⠀⠀⠙⠒⠬⣧⡾⡄
⠀⠀⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⢮⣿⠀⡀⠀⠀⠀⠀⠀⠀⣾⡵⠁
⠀⠀⠘⠺⢷⣀⠈⡀⠀⢀⠆⣠⡾⠗⠃⠀⠀⠀⠹⣦⣐⡀⠀⠀⣐⣠⡾⠁⠀⠀⠀⠀⠙⢷⣧⡀⠀⠀⢀⣼⡾⠋
⠀⠀⠀⠀⠀⠈⠛⠇⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠟⠁

⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⣾⡇⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠈⣿⡄
⠀⠀⠀⠀⢰⣿⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢹⣷
⠀⠀⠀⠀⢸⣿⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣸⣿
⣆⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠎⠀⠱⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⢰
⠸⣆⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠐⢠⢊⡤⠆⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣠⡟
⠀⠹⣷⣤⣀⡈⢿⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀⡈⢦⣞⠁⠒⠊⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢠⣿⣃⣠⣤⣾⠟
⠀⠀⠀⠉⠛⠿⢿⣿⣷⡄⠀⠀⠀⢸⣇⠰⠖⠦⡒⠟⠟⢦⣢⣄⣠⠃⠀⠀⠀⠀⠀⠀⠰⠈⠈⡇⠀⠘⠆⠀⠉⢱⠆⠀⠀⢸⡇⠀⠀⢀⣴⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⣾⡧⡒⠒⠦⠬⢍⡶⠂⠀⠀⡇⠀⠀⢀⣴⣓⠀⠢⣌⠁⢄⠈⢢⠀⠈⠳⡀⠀⠀⣄⠀⢸⡇⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣾⣷⡓⠤⣤⣵⠮⠞⠀⠀⡜⠁⢠⠄⠧⢓⠟⡡⠔⣣⠀⠠⢀⠀⢣⠀⡐⠲⠜⡏⠈⢠⣿⣷⠟⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠿⠿⢿⣶⣤⣠⠀⠀⠀⠀⠇⠀⠈⠉⠋⠁⠸⡠⠁⠈⠦⠀⠟⠀⠁⢠⠁⠀⠀⠈⣶⠿⠿⠷⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠭⠄⠴⠀⠀⢀⠔⢷⠒⠤⠦⠃⡀⣾⠀⠀⠀⢀⠗⡰⠃⡠⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢁⠄⡰⠊⢀⠀⡇⠀⠀⠀⠀⠀⠀⠏⠳⡀⠀⢀⡖⠊⡁⠈⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡐⢁⢂⠀⠀⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠀⢄⢀⣤⢀⣠⡩⠷⠀⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠂⢸⢀⡠⢃⡞⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⢀⡴⡉⠃⠜⠦⠌⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠈⠉⠁⠉⠓⠐⠉⢢⠀⠀⠀⠀⢀⡉⣉⠍⠣⠴⠄⢀⡬⠄⡐⣤⣤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣶⣶⣭⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣥⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢸⣿⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠸⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣿⠇
⠀⠀⢀⡆⢻⣧⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣼⡟⢠⡀
⠀⢰⡟⠀⠈⣿⣧⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣼⣿⠁⠀⢻⡆
⠀⣿⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⣿
⠀⣿⡇⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⣸⣿
⠀⠘⣿⣆⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⣰⣿⠃
⠀⠀⣸⣿⣦⣀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⣀⣴⣿⣇
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⢿⣿⣿⣷⣄⡀⠀⢰⠆⠀⠀⠀⠀⠰⡆⠀⠀⣠⣶⣿⣿⡿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣦⣼⠀⠀⠀⠀⠀⠀⣧⣴⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡅⠀⠀⠀⠀⢠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠛



⣿⣿⠿⢿⠻⠿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠐⡘⡄⠀⠀⠰⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡕⠀⠀⠀⡜⡐
⠀⠀⠀⠀⠐⠈⠓⠂⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠊⠔
⠀⠀⠀⠀⠀⠈⢢⠂⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢙⢊⢷⡄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣴⢏⢚⠙
⠀⠀⠀⠀⠀⠈⠲⠀⠄⠈⠤⠬⠴⠏⣷⣽⢝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢽⣴⡏⠵⠬⠤⠄⠀⡔⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣬⡳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣗⢌⣶⡶⡄⠀⠀⠀⠀⡴⣶⣎⢽⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠾⠀⠚⠟⠐⠀⠀⠀⠀⠀⠐⠘⠟⠐⠰⠖⠒⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⠇⢀⡜⠀⠀⠀⠀⠀⠈⢆⠀⢱⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢰⠀⠀⢀⠄⠀⠀⠀⠀⠀⢄⠀⠀⢠
⠀⢸⡀⣼⠀⠀⠀⠀⠀⠀⠀⡘⡆⢸⠀⠀⠀⢧⡆⣿⠄⠀⠀⠀⠀⠠⣿⢰⡜⠀⠀⠀⠈⢷⠔⠁⠀⡀⠀⠀⠀⡄⠀⠑⣴⠏
⠀⠈⠻⣷⡇⠀⡀⠀⠀⡀⠀⣷⣿⠏⠀⠀⠀⠈⠳⣇⠀⡄⠀⠀⢠⠀⣽⠞⠁⠀⠀⠀⠀⠈⠳⠦⣤⣧⠀⠀⢠⣧⡤⠶⠋
⠀⠀⠀⠈⠙⠦⣶⡀⢠⡧⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠀⣾⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⢸⢰⠀⣠⠀⠀⠀⠀⠀⠀⡄⢸⡄⡆⠀⠀⣰⠁⢠⡠⠊⠀⠀⠱⣄⡆⠈⣄⠀⠀⠀⢸⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹
⠀⠸⣼⠖⠁⢀⡄⠀⠀⢄⠀⠙⢾⡿⠁⠀⢆⢹⡀⡼⢠⠄⠀⠀⠠⡈⣧⢀⡏⡴⠀⠀⠸⣜⡇⢺⡀⠀⠀⠀⠀⠀⣸⠂⣞⡼
⠀⠀⠈⠳⢶⡏⠀⠀⠀⠀⣷⠶⠊⠁⠀⠀⠈⠙⠓⠧⣼⣄⠀⠀⣰⣧⠾⠚⠋⠀⠀⠀⠀⠙⠷⣼⣅⡄⠀⠀⠀⣄⣽⡴⠟⠁
⠀⠀⠀⠀⠀⠙⢦⡀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⡀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣅⠀⠀⠸⠙⢆⠈⠙⢦⣈⠓⢦⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠙⠦⠁⢀⠀⠀⠀⠀⡀⢀⡉⠙⠦⣍⠇
⠀⠀⠀⠀⠀⢾⠋⠀⠀⢀⣠⠎⠀⠀⣀⠀⠙⣿⣿⣦⠀⠈⢇⠢⡀
⠀⠀⠀⠀⠀⢸⠃⠰⣳⠋⠁⠀⠀⠀⠛⠀⠀⣿⢿⣿⡆⠀⠈⢀⡇
⠀⠀⠀⠀⢀⡟⠀⠸⡏⠀⠈⠄⠀⠀⢠⠀⣠⣿⠸⣿⡇⠀⠀⠀⠙⡄
⠀⠀⠀⠀⢸⠀⣾⠀⠱⡀⠀⡄⠀⠀⠀⣸⠏⣿⠀⡿⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠘⢦⣧⢢⠀⠅⠀⡅⢰⡄⠀⣿⠀⢹⠀⠀⠀⠀⣀⡠⠞⠉⠢⢄⢀
⠀⠀⠀⠀⠀⠀⠈⠓⡇⠁⠀⠘⠿⣀⣴⣿⡆⣸⠀⢀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⠒⠒⠦⣄⡀
⠀⠀⠀⠀⣠⠴⠶⠾⠁⠀⠀⠀⠀⣿⣿⣿⠁⢁⠴⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄
⠀⠀⠀⢸⡡⢒⡶⠒⠀⠀⠀⠀⢀⣿⡿⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⢠⣾⡷⣋⢀⢔⡤⠀⠄⠐⢁⢿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣦⠀⠀⠀⠀⠀⠉⢿⡇
⠀⠀⢠⣿⢞⡏⡾⢁⠤⠤⠐⠒⠁⣠⣿⣸⣆⣀⡘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠂⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠿⣾⠀⠀⢇⡀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⠟⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢸⡿⠦⣤⣀⡀
⠀⠀⠀⠀⠹⡆⢰⠟⠀⠀⠄⠒⠁⠘⠛⠛⠛⠁⡸⠋⠀⠀⠀⢀⣴⣦⣄⣀⣤⣤⣤⣤⣄⣀⣤⠴⣿⣷⣾⣦⡀⠀⠀⠀⠀⠾⡇⠀⠀⠈⠙⢶⡄
⠀⠀⠀⠀⠀⣹⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⡀⠄⠂⠘⠛⠛⠛⠛⠛⠛⠛⠉⠻⣿⣷⣶⣿⣿⣿⡿⢀⡍⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠘⠶⠾⠁⠀⡆⠠⠀⠀⠀⠀⠀⠀⣸⡅⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠏⠀⢸⡇⠀⠳⣤⡴⠀⡄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⡛⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠻⣿⡄⠀⠀⢧⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⢠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠙⢿⠃⣶⡆⠀⠀⢠⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⣷⠀⠀⠈⢧⡀⠀⣿⡀⠀⠂⠀⠀⠀⠸⡧⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡇⠀⠀⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⣷⠀⠀⠈⢳⡀⢻⡇⠀⠀⠀⠀⠀⠀⢻⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠿⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢾⡟⠀⠀⠀⠀⢻⡄⢿⠃⡀⠀⠀⣼⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡞⠀⠀⠀⠀⠀⠀⣷⢸⠀⠇⠀⠀⣽⢷⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⣿⢸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢷⣴⡇⠀⠀⠀⠀⠀⢀⡴⠃⠘⣦⣼⠃⣰⢃⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣁⣀⡇⠀⠀⣀⡤⠤⠴⠚⠉⠀⠀⢰⣯⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⢁⡴⠻⠒⠒⠒⠒⠲⠶⠖⠚⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡙⢦⠐⠀⠰⡄⠀⠀⠀⠀⠀⢀⡆⡠⠔⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⢒⣦⣤⠀⣀⣀⡴⣂⣈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢁⣁⠁⠈⣋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⢺⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⣾⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⡿⠂⠀⠀⣿⣤⣾⣶⣿⣿⣿⣷⠀⠀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⢸⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠆⢀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣴⣿⡟⠻⠛⠛⠛⠛⢻⣿⣷⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡴⢚⣿⡿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠸⣆⠀⠀⠀⠀⣀⣀⠀⢀⣛⣀⢸⣇⣀⣀⣀⣀⣀
⠀⠀⠠⠶⠒⠒⠉⠐⠒⠻⠷⠾⠶⣶⠶⠶⠟⠿⠾⠟⠛⠛⠛⠋⠛⠻⠛⠛⠛⠛⠛⠉⠉⠋⠉⠉⠉⠉⠋⠑⠒⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⢀⠀⠀⠀⠀⠀⠀⠀⣀⣭⢤⣄⣀⠀⠈⢇⠐
⠀⠀⠀⠀⠀⠀⣾⣿⣿⢀⠄⠠⡀⠀⠀⣴⠻⣿⣿⢿⢾⣽⣳⣲⢄⣖⡀
⠀⠀⠀⠀⠀⢷⣿⠙⢸⠁⠀⠀⠀⠀⡜⠾⢩⢿⠷⠦⢴⡯⣛⠿⣿⡿⠛⢷⣿⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢠⣷⡴⢩⣵⠛⡷⢾⢦⡏⣹⣏⢫⣶⣾⣿⣿⣷⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠲⠧⠀⠀⠀⠀⠘⡟⠞⠋⢠⠒⡇⢺⠉⢠⣶⡼⢹⣟⣿⣿⣷⡿⣿⢿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠁⡞⠐⡇⣯⠀⢶⢀⣜⢻⡦⣽⣟⣿⣿⡿⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠬⢀⡿⡗⠀⣤⠈⠁⡻⠿⢷⣬⠛⣿⣿⣿⣿⣇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡂⢸⡇⢰⠄⢍⠁⢨⣉⣟⢠⡜⠓⢻⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣂⣸⣁⣼⣿⣮⣤⣹⡗⢿⠈⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣋⢪⡿⣿⣏⠉⠃⢸⡇⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠁⢢⠀⠀⠀⠀⠀⠺⣿⠋⠙⢻⣿⣿⣾⣾⣷⣤⣼⣧⣼⢿⡿⣿⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⡰⠃⠈⢢⣀⣘⣀⡀⣤⡀⠀⠀⣿⡀⠀⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣆⢷⡹⣧⡘⣿⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠡⠄⠀⠀⠛⢉⣿⡗⠈⠁⠈⠉⠉⠁⠀⠀⠀⡟⠈⠻⣿⣿⣿⣿⣿⣾⣎⢳⣜⢿⣿⣇
⠀⠀⠀⠀⡔⠉⠀⠀⠀⡀⠂⣀⡴⠛⠛⠃⠀⠀⠀⠀⠀⠐⠀⠠⣸⠃⠀⣼⣿⠻⣝⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠡⠄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠠⢤⣄⣀⠀⣀⠘⢦⣴⠁⠀⠀⣘⣇⠀⠀⠈⣿⠋⠻⣿⢿⣿⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⣠⣶⣶⣾⣾⣿⣿⣿⣤⣾⣿⣿⡇⡀⡤⢿⣤⣶⣤⣿⡀⠀⠘⢷⡈⠘⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣙⡛⠉⠙⠛⡛⢛⠿⢿⣿⣿⣷⣷⣧⢸⣿⣿⣿⣿⡇⠀⠀⠈⢹⡆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠏⠳⠦⠤⠀⠀⠴⣿⣿⣿⣿⣿⠘⢿⣿⢁⣿⠁⠀⠀⠀⠀⠗⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠀⢘⣿⢀⡿⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠀⠌⠈⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡿⠟⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠛⠁⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⣶⣦⡀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠓⠶⣤⣄⣀⠀⠀⢰⣄⠀⠀⠀⠀⠘⣿⣿⣿⠢⣀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⢠⣼⣿⡄⠀⠀⠀⣀⢻⣿⣿⠀⠈⠻⠒⣿⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⣹⡇⠀⠀⠀⠈⠀⢹⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢃⣀⡤⠀⠠⣦⣌⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⡇⠀⣄⡰⢇⡀
⠀⠀⠀⠀⠀⢀⠐⣀⢀⠀⠀⠀⠀⠀⣤⡤⠈⠀⠁⠀⠀⠐⢿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠱⠄⠰⠃⠀⠈⣇⣼⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⡼⠋⠀⢰⣤⣤⣔⣤⡄⠈⢿⡀⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠝
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⢀⣼⡿⠛⠙⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⡛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢀⣀⣀⠀⠀⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠺⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡈⣿⡧⣶⣶⣾⠿⠛⠋⠚⠒⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣶⠿⠿⠿⠟⠛⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣥⣴⡶⠿⠿⠛⠋⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⡃⣴⡄⠀⢀⣭⣾⡶⠿⠈⠀⡈
⠀⠀⠀⠀⠳⠤⠤⠔⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣴⣶⠟⠛⠉⡀⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠏⠉⠂⠄⠀⠁
⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣤⡀⣷⠀⣼⢀⡴⠎⠀⠀⠀⠀⢐⢺⡏⠄⠀⢀⠉⢲⠀⠀⠀⠀⠀⠙⠛⢶⣄⠀⢰⠟⠀⠘⣆⢣⣈⣴⢀⡔⠀⠀⠐⣿⣄⠀⠀⠀⢀⣹⣧⣤⣤⣤⣾⠟
⠀⠀⠀⠈⢷⣼⣼⣷⢟⣀⣀⠀⠠⢤⠔⠀⣹⣇⠄⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠻⣦⠓⠀⠐⡖⣲⠀⢟⢃⣿⡵⠂⠀⠀⢸⣿⣤⣶⣿⠿⠟⢛⠉⠉⠉⠁
⠀⠀⠀⣀⠀⠙⠿⠟⠛⠉⠁⠀⠈⢻⣏⡀⠨⣧⠄⠀⢆⢀⡀⠀⠊⠵⠀⠀⠀⠲⠀⠀⣟⠀⠀⠀⠐⠈⡖⣆⢨⠄⠁⠀⠀⢀⣼⣿⡿⠋⠀⣀⣤⣬⣷⣷⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⣿⡠⠐⣷⠄⠘⣾⣫⠄⠀⠉⠐⣀⠀⠀⠀⡆⠀⠀⠈⠡⠀⠀⠘⠿⠿⢃⠀⠀⠀⢴⣿⣿⠏⠀⠀⣀⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⢁⠄⠀⢀⠀⠀⠁⠀⠀⠀⠠⠿⣬⣿⠊⢱⡿⠄⢀⠀⠀⠀⠉⠀⠤⠀⠈⣀⠠⠲⠀⠀⠀⠰⠖⠚⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⣸⠿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣿⣹⠛⠂⢠⡉⠦⡒⠤⠄⠀⠐⡸⠐⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⣀⢀⡀⡀⠀⠀⢱⡄⠀⠇⠀⠀⠈⠿⠉⠁⠀⠰⢐⠣⠼⠀⠆⠠⠈⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠛⣿⡇⠀⣴
⠀⢀⣀⡀⠀⢀⡀⠁⠀⠀⢨⣵⢀⣟⠀⣰⠀⠀⣀⣤⣄⡀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣀⠀⠀⠀⠐⠂⠀⢰⠀⠀⠀⠀⠀⣿⡇⠀⡿⠀⠀⠀⠀⠈
⠀⣿⣿⣿⡀⠸⣿⣾⣷⠀⢨⣶⢾⣭⢴⡍⠀⢀⣛⣓⣉⣵⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠠⡄⠀⠐⠁⠀⠀⠀⠀⠀⣿⣧⡾⠃⠀⠀⠀⠠⠀⠄
⠀⢹⣿⣿⠇⢠⣿⡿⠃⠀⠀⢰⣿⡿⡦⠀⠀⠘⣿⠿⠷⠾⣟⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠻⠿⢀⡤⠁⠀⠀⠀⠀⠀⣭⣻⡰⡞⠁⠀⢰⠶⢖⣻⣯⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣿⡇⠀⠀⠀⠀⠀⠎
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡾⠏⠋⠀⠀⠀⠘⢿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠃⠀⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠀⡀⠀⠀⠀⠀⣶⣾⣷⣀⠀⢀⡀⠴⢃⡄⢀⣠⡄⠀⠦⠀⠀⠀⠐⣄⠀⠄⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣴⡇⠀⡆⠀⠀⣿⣿⣿⣿⡀⢲⡙⡘⣆⣃⠀⡁⡦⢀⡃⠈⠀⠀⠠⡀⢂⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢶⠘⡁⠈⠁⠀⠘⡛⡴⠁⣠⣶⣤⠀⠈⠆⣡⠀⠠⢀
⠀⢠⣭⣏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⣿⣿⡏⠀⣀⠈⠁⠀⠈⠀⠀⠁⠀⢸⣿⣿⣿⡇⠀⠀⠈⠛⠟⠋
⠀⠈⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⢹⣿⣿⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠄
⠀⠀⠍⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⠀⠙⠀⠀⠠⠄⠀⢸⣿⣿⣿⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣠⣼⣦⠀⠀⠀⠐⠀⠛⡛⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣤⢀⡀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⢠⠰⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⢀⡄⣸⣿⡼⠇⠀⠀⢸⣇⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠔⡆⠉⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣀⡏⣦⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⠀⠸⣿⣿⡯⠀⠀⢀⠀⢨⣷⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠖⠀⠀⠀⠸⣄⣧⢰⡄⣿⠃⡏⣤⠀⠀⠀⡄⢀⡠⠂⠀⠀⢀⣤⢄⡀
⠀⠀⠀⠙⣿⡇⠀⠀⠼⢷⣘⣏⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢻⡄⣇⣿⢠⡇⣿⠀⠀⠀⢃⠀⢀⠀⠀⢀⠿⠇⡋⢿
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣶⢿⠃⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⣇⠰⠄⠀⠀⠀⢣⢹⣿⣸⣇⠇⠀⠀⠀⡎⡀⣻⠀⠀⠈⡆⠃⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠅⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⡁⠀⢫⡔⡮⠁⠀⠀⠀⢻⣿⣿⠞⠀⠀⠀⠀⠰⣃⣿⠀⠀⠀⠑⡀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠠⠀⢀⡿⡿⢗⢐⡱⠿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⣻⣿⠀⠀⠀⠀⠱⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⢰⠆⠐⢆⠀⠀⠀⠠⠀⠰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢃⠘⣾⡄⠀⠈⢱⣦⣀⠀⣧⠀⢳⠶⠄⠀⡇⢰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠓⠛⠿⠿⣷⣀⠉⠛⢯⠉⠙⠛⢄⠚⡇⠀⣣⣇⣼⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣷⣤⣀⣠⣾⣷⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⡏⣹⣿⠶⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⣶⣤⣛⣛⣯⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣶⢿⣻⣿⡭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣵⣾⣴⣖⡀⢀⡦⠄⣿⠁⡀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣹⣿⣦⣐⣧⠦⠧⣀⠀⣽⣿⣿⣿⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣶⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣤⣶⣦⣿⣿⡋⣉⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠟⢻⣿⣿⡟⠉⠙⠛⠿⣿⣿⣿⡿⢿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠋⣠⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠋⠀⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⢻⡇⠀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⢸⣇⠀⣀⠀⢸⣷⢀⣁⣀⣀⣀⢀⠀⣿⡀⠀⠀⢀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣽⣿⣿⣿⣿⣷⣶⢾⡖⠀⠒⡤
⠀⠀⠀⠀⠀⠈⠋⠩⠉⠝⠛⠋⠏⠟⠛⠛⠛⠉⠙⠙⠛⠛⠃⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⢰⡄⢸⣆⣀⡀
⠀⠀⠀⠀⠀⢀⣿⠛⢷⣿⡿⣾⣿⣷⣶⣖⣂⡀
⠀⠀⠀⠀⢰⠟⠛⠂⠀⠀⠀⠈⠙⢿⣿⣿⣵⡤⠄
⠀⠀⢀⡔⠁⠀⢠⠀⠀⡀⠀⠀⠀⠀⠙⢿⣿⡿⠃
⠀⠀⣯⡖⢀⡤⠐⠓⠐⣿⡀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠈⠛⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⢀⣀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣥⡀⠀⠀⠀⠀⠀⠀⠇⠀⠻⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣷⣦⣴⣇⡀⢀⣀⣀⣤⣤⣶⡿⠿⠿⠛⠛⠛⢿⣿⣦⡀⠀⠀⠀⢠⡆⠀⠀⠀⠈⠛⠿⢿⠷⡤
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠈⠉⠉⠙⠋⠁⠉⠻⣿⣿⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⢦⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠤⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢧⣀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠹⣿⡄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⡛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⣿⣦⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⣿⢿⢿⢿⠿⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣷⢶⣶⣶⣶⡶⢶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠒⠲⢤⣤⣤⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣿⣿⣶⣤⡉⠙⠛⠁⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣚⣿⣿⡏⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣄⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⡤⠒⠉⠁⠀⠀⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠊⣿⣿⣦⣤⣤⣼⠉⠉⠉⠉⠉⠉⠉⠉⣉⣶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⢂⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠻⢿⣯⣭⣽⣿⣿⣭⣽⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⢻⣿⣧⣤⣀⡀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠏⠘⢹⣿⣻⠿⠟⠃⠁⣹⣿⣿⢫⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⡀⢻⡇⠀⠀⠀⠐⠛⠛⢁⣉⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠉⢸⡿⡇⠀⠀⠀⠀⠀⠀⢹⣷⡶⠶⠏⠀⠙⠿⠞⠁⠀⠀⠀⣸⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡶⡽⠆⠀⠀⠀⠀⠀⠀⠀⢸⣟⡓⠂⢶⡄⠀⠀⠀⠀⠀⣀⣰⠟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣿⢸⡇⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠦⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⣠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⣾⣧⣠⣆⡀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣄⠀⠀⣀⣤⣴⠇⠀⠀⢀⠔⡌⠀⠀⢠⠄
⠀⠀⠀⠀⠀⡀⠀⡄⠀⢨⣤⣶⣿⣿⣿⣿⡿⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣷⣏⠴⠃⡸⠀⠀⠀⠘⡀
⠀⠀⠀⡆⠀⡇⠠⡗⢧⣈⡉⠈⢹⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⡰⠁⠀⠀⠀⠀⠈⠢
⠀⠀⣸⣿⣦⣷⣴⣿⣿⣿⠇⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⢋⣠⡶⠀⠞⠀⠀⠀⠀⣠⠁
⠀⠈⠋⣿⣿⣿⣿⠟⠁⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠶⢟⣁⡀⠀⠀⠀⠀⠀⡴⠃
⠀⠈⡄⠈⠿⡟⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡁⣰⣶⠂⢸⠉⢆⠀⠀⣠⠊
⠀⠀⢱⡀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠿⢋⣿⠟⠹⠁⠀⠘⠀⠀⢣⡼⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⠬⠿⡿⠁⠀⣴⢿⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢛⣿⠃⠀⡼⢣⣿⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢈⡴⠉⣿⡆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣁⠤⠀⠒⠉⠀⠀⠸⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣇⡶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠻⡄⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠒⣦⣵⣸⡟⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢆⡁⢂⠘⠈⡌⠘⢀⠠⠆⠀⠀⠀⠀⠔⢁⠄⠊⡁⠄⠂⢘⠻⣜⣻⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢤⡀⠙⢦⡸⣿⡤⠋⠀⠴⡳⣖⠠⣄⡎⠠⢂⡤⠊⠀⡠⠊⠁⢠⢹⣮⣳⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⣢⣌⡙⣿⣿⣿⣾⠧⠒⢖⠾⠎⢣⠀⠱⣧⠊⠀⢀⠜⠀⠀⢀⠃⠈⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢦⡤⠀⣡⢚⡟⢇⡉⣗⠢⡎⡄⠀⢡⠀⠘⣤⠐⠁⠀⢀⠀⠁⡰⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠃⡼⡵⠛⠁⡎⢁⠈⠁⣾⠀⠀⠰⠀⠀⢣⠀⠰⠄⠄⠀⠁⡠⠞⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠭⠭⢉⡚⠀⠀⢡⠀⠲⠀⠈⠟⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⣶⣤⣢⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠤⠤⠀⠀⠀⡅⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠂⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢫⡁⠒⠠⠤⠄⢚⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣁⡂⠐⢒⡩⡆⠇⠀⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠐⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠈⠀⠀⢸⠀⠀⠸⠀⠀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⠰⡀⠠⠀⠀⠀⠇⠀⢸⠀⠀⡄⠀⢠⠁⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣄⣥⣀⣆⠀⡀⢰⡀⢨⠀⢀⣇⣀⣼⣤⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠷⣧⣭⣩⣍⣿⣍⣏⣉⣋⣍⣩⣥⣤⠴⠟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠔⠒⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⢾⣽⣿⣿⣿⣿⣯⡗⠀⠀⠀⠀⢀⣤⠴⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠇⣀⠁⣀⣀
⠀⠀⢿⣿⡿⠁⠎⢻⣿⡿⠀⠀⠀⠀⣾⣿⣷⠰⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠉⠘⠃⠨⠄⠉⠁
⠀⠀⢫⢿⣿⠀⠀⢸⣿⣍⠁⠀⠀⢸⣏⠉⠀⠁⠈⠉⣽⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠘⣿⣤⣤⣼⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡖⣴⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣀⣈⣉⣩⡅⠀⠀⠠⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⣿⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⠀⡀⢀⡀⢀⡀⣀⢀⢀⠀⢀⢀⢀⠀⣀⢀⠀⢀⠀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀
⠀⠀⠀⠀⠀⣔⣆⠆⠀⠀⢶⣌⠉⠀⠀⠀⣱⢸⠀⠺⡇⣿⢾⢸⠇⢿⢸⢹⠀⢿⠈⡇⢸⠀⢻⡇⡟⡇⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠘⢹⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣤⠄
⠀⠀⠀⣀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣿⡎
⠀⠀⠀⠙⠁⠙⠛⠈⠋⠀⠀⢠⣤⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢶⣦⠿⠚⠿⣧⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠱⠶⣿⡇
⠀⠀⠀⢠⣤⡀⢀⣀⡄⠀⠀⠈⣿⡆⠀⠀⠀⣸⡟⠀⠀⠀⣷⣶⠇⠀⠐⠀⠘⣷⣶⠀⠀⠀⠀⠀⢿⣿⡤⡀⠀⠁⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⢐⣂⣀⣉⣭⡅⠀⠀⠀⠘⠇⠀⠀⠀⠿⠁⠀⠀⠀⣶⣾⡀⠀⠀⠀⢀⣿⣶⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⢿⠀⠀⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⢰⣿⢲⣶⣾⢿⡧⠉⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⣷⣶⡇
⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣉⡛⠈⠀⠀⠀⠀⣴⣶⡖⠀⣨⣭⣅⠀⠐⣶⣶⠆⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣿⢟⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣤⡏⣿⡉⠠⠀⠀⡄⠀⠀⢀⣴⢆⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢌⠑⡿⠀⠀⠀⠀⢿⣦⣶⣾⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠏⠀⠀⠀⠁⠀⠀⠀⠤⣾⣿⣿⡗⠂⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠒⠶⠆⠖⠀⠀⠠⢤⣤⣴⣿⣿⣿⣿⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣄⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠙⠢⣄⡀⠠⠀⣿⣿⣿⣿⣿⣿⡀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⢰⣆⣉⠳⢤⣿⣿⣿⣿⣿⡏⢹⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣉⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠀⠀⠈⣋⣝⠓⡿⣽⣿⣿⣿⣿⣿⣿⣿⣯⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢹⣿⣀⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠞⠁⢳⡀⠀⣸⣿⣿⣿⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⡀⠀⠀⠞⢀⠀⠸⡇⣀⣿⣿⣿⣾⣎⡿⣯⣽⣟⣿⣻⣿⣿⣿⡿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⢀⣴⣾⣥⡟⠀⠁⠀⣾⡁⠀⠐⠚⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠐⣯⠱⠄⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⢐⣿⣿⣿⣿⠛⣿⣾⠀⢸⡄⠀⢱⢌⣹⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣷
⠀⠀⠀⠐⠀⠚⠹⢿⠁⠀⠹⣿⡷⠺⢻⣕⠁⢸⣿⣎⣾⣿⣉⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠙⣿⡀⠀⠿⢿⣿⣿⣿⣧⣿⣉⢩⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠟⠻⣿⣷⠊⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢿⣀⣴⣸⣿⣿⣿⣿⠁⢹⡏⡉⠻⢿⠫⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣷⣟⠁⠀⠀⠈⠓⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⡆⠀⠀⡀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⣀⡀⣀⡄⠀⣀⣠⠴⡿⢁⠐⢯⡥⠆⠐⣧⡶⣋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠐⠒⠆⠐⠐⠠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⠀⠀⣤⣤⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣾⣿⣿⠿⠟⠛⢻⡏⠋⢘⣼⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⣠⣶⣾⣿⣿⠿⠛⠛⠛⠉⠁⠀⠀⠀⢰⣿⣶⣾⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⣀⠀⠀⠹⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠉⠽⠿⠭⠁
⠀⠀⠀⠀⠀⠀⣀⣄⡠⠖⢋⣁⣤⣶⣶⣶⣤⣤⣀⣀⢀⡀⠀⠀⠀⣠⣾⢻⡏⣷⡄⠀⢐⣳⣦⡀
⠀⠀⢀⣠⣶⣾⡿⢋⣰⣾⣿⠿⣿⣿⡲⣿⣿⠉⠀⠀⠀⠀⣴⣾⣿⣿⡇⠈⣸⡿⣿⣄⠈⠙⠋⠁
⢀⣾⣿⣿⣿⡏⣴⣯⣿⣿⣿⡞⡋⠁⠀⠀⠸⠅⠀⠀⠀⠀⠀⠻⣿⢿⣇⣰⣿⠃⠈⢿⣦⡀
⠘⢱⣿⣿⣿⠇⣾⡿⠟⣹⡏⠉⢻⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⣿⡟⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⣀⣀⡀
⢠⣽⡿⠟⠁⠀⠀⠀⣨⣿⣧⡀⠈⣿⢛⠓⠒⠁⠀⠀⠀⠀⣠⠌⠣⠁⠀⠻⣿⡠⣄⠀⠀⠀⠙⢿⣶⣄⣠⡾⠟⠉⠁
⡿⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣈⣸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠓⠂⣶⡆⡖⠀⢈⣛⠈⠁⢀
⠃⢀⣴⣿⣷⣤⣴⣾⠁⠀⠈⠉⠛⠛⢿⣿⣶⣦⡀⠀⠤⡄⠀⠀⠀⢠⣴⣢⡄⠀⠀⠀⠈⠳⣄⠠⣿⢟⡀⠀⣿⡆
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣯⣻⣿⡄⡼⠀⠀⠀⠀⢺⣯⠔⠌⢀⠀⠀⠀⠀⠈⠛⣙⠘⢷⡀⠙⠿
⠀⢀⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⡀⠘⢿⣿⢹⣿⠁⠀⠀⠀⠔⠈⣿⣟⠁⣾⠀⠀⠀⠀⠀⠀⠖⠁⢄⠳⣤⣄
⠀⠸⠱⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣧⠀⠀⢻⣷⠙⠀⠀⠀⠀⠀⢀⣿⡟⠘⣿⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠛⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢹⠄⠀⠀⠹⠈⣇⠀⠀⠀⠸⣷⣿⣿⠀⢻⡇⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡿⢀⣠⠄⠀⢸⣏⣿⠀⠀⠻⣦⣤⣤⠟
⠀⠀⠀⠀⠈⠛⢿⣿⣇⣷⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⢿⠉⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡌⣇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⠀⠀⠀⠀⢸⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠳⡄⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢖⣺⠟⠉⠈⠀⠉⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣔⣻⡿⠋⠀⡀⠈⠀⢰⡇⠀⠀⠀⠠⢤⡽⠛⠋⠀⠀⠀⠉⠙⠀⠀⢀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⠶⠤⠬⠽⠟⠋⠀⠀⠀⢻⣄⠀⠘⣇⠤⠶⠖⠂⠀⠀⠀⠀⠀⣄⠀⠠⠀⠁⠈⠉⠉⠉⠀⠀⠀⠘⠷⠄⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠈⢧⠀⡹⠀⠀⠀⠀⡀⠀⠀⠀⣼⠉⢣⡀⠀⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙
⠀⠀⠀⣀⣾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡆⢸⠆⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢰⠃⠀⠀⠹⡄⡙⠀⠀⣠⢆⡶⠋⠉⠧⣄⠙⠀⡆
⢀⡤⢞⣭⢞⣿⡄⠈⢷⣄⠀⠀⠀⠳⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⠀⠀⠀⠀⡞⠁⡉⠀⠀⠀⠀⠈⢃⢀⢹
⡞⡟⠉⠐⠉⠀⠉⢀⠀⢹⡟⠲⠤⠠⡄⠀⠀⢠⡆⠀⠐⢦⣠⣤⣀⠀⡄⠀⠀⣦⣬⠞⣙⠒⠺⡇⠀⠀⢀⡄⠠⠾⢧⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣷⡼⠀⡄⠻⣄⠀⣿⠈⠛⠲⠦⡉⠃⠀⠀⠳⣄⠀⣿⠙⢦⡘⢦⠀⢷⠀⢰⠋⠀⠈⠓⠲⡌⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡇⢸⠁⠀⠙⢦⠘⣆⠀⠀⢰⠁⡇⠀⠀⠀⠈⠃⢸⡆⠀⠈⢈⠁⠘⡆⢸⠀⠀⠠⢤⠜⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⣈⢣⠀⠀⠈⡇⣼⠀⣀⣨⢀⡇⠀⠀⣀⡔⣠⠏⢀⣀⡄⣸⠀⢰⢃⡞⠀⠀⠈⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠸⡇⠀⢀⠁⡿⠘⡧⠖⠋⠀⠀⠘⠵⠖⠁⠀⠨⠖⠋⢁⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠣⠁⠀⣇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⢿⣄⠄⠀⢤⡤⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⡶⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠸⣽⣷⣦⣀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢂⣶⣿⣷⣾⣿⣷⣦⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠁⢀⢤⣽⣿⣿⣷⡄⠀⠀⠀⡇⠀⢸⠀⠀⢹⣿⣿⣿⣿⣿⡟⠻⣷⣒⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡿⣏⠋⢹⡇⠀⠀⠀⡇⠀⢸⠀⠀⢸⠋⣿⣪⡟⢸⡇⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⠀⣃⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠉⠁⠉⠉⠐⠟⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡃⠀⢘⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣴⣆⣴⣤⣄⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠄⢀⡀⠴⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⢿⡏⢛⡻⣿⣗⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⣠⣯⣥⣬⠆⠀⠀⠀⠀⡇
⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⣿⣍⣃⡈⢠⣿⣿⣦⠄⠀⡇⠀⢸⠀⠀⢀⣀⢂⣩⣟⣻⣝⡁⠁⠀⠀⠀⡇
⠀⢀⣻⠁⠀⠛⢿⡇⠀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣶⣾⣿⣿⡟⠀⠀⠀⡇⠀⢸⠀⠀⠀⠈⠁⢿⢹⡯⡟⠐⠓⠁⠀⠀⡇
⣾⡿⠋⠤⠠⠀⢤⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢿⢺⡿⠿⠋⠁⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢀⣷⣶⠄⠀⠀⠀⠀⠀⡇
⠉⠃⠈⣶⣿⣤⡷⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⡇
⣧⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣶⠤⠄⠀⠠⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⠿⢿⠿⠿⡿⠿⢿⡿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⡶⣷⣶⣾⣶⣴⣤⣄⣀⠀⢀⣤⣶⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡟⠀⠀⠠⠴⣶⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⢸⢹⣧⠀⠀⠀⠈⠙⣿⣿⡟⠀⣿⣿⠿⢿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢧⠀⠹⣿⣦⡀⠀⠀⠀⠈⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⡟⠛⣿⡄⠀⠀⠴⠣⢤⣀⣨⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣷⠀⣴⣿⣿⣾⣿⣿⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠘⢦⣀⣿⣾⡯⠀⢰⣤⣤⡟⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢹⣿⡀⠈⢿⣿⣿⣷⣤⣈⠉⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠸⣿⡇⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡟⠃⠀⢿⣷⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⠃⠀⣠⣾⣿⣆⠀⠀⢷⣾⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀
⠀⠀⢀⣴⣿⡿⢿⣛⣫⣽⣿⣿⣿⠀⢰⣿⣿⣿⣿⣦⠀⠈⠋⢠⣾⣿⡿⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⣠⡿⠟⠉⢀⠈⢿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣷⡀⠀⣩⣿⠟⢠⢟⣾⣿⣿⣿⣿⣿⣿⠏⣠⣿⡿⢷⣄
⢰⣿⣷⡄⠀⣼⣆⠈⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠏⢐⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡄
⣾⡿⣿⢻⠀⣿⣿⣷⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣿⣧
⣿⢂⢿⡌⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣧
⡏⠈⣸⡇⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡘⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠁⣷⠀⣿⡿⠿⠿⣿⣿⣿⡿⠿⠿⣿⠿⣿⡇⢻⣷⣿⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠀⢻⠗⠤⢍⣛⡛⠿⠟⠛⣋⣩⣴⣿⡇⢸⣿⣿⢸⣿⣿⣿⣬⣛⠻⢿⣿⣿⣿⣿⣿⠭⣿⣿⣿⣿⢿⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢮⣛⢿⣿⡟⣿⡇⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣶⣮⣤⡬⢷⣶⣾⣿⣿⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠃⠿⣧⢸⣿⡇⣉⢹⣿⠿⠛⠿⠿⠛⠫⠀⠀⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⠀⢰⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿⣿⣧⢸⣿⣇⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣶⡄⢀⢰⡇⢠
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡄⢰⢀⡇⡇⠀⠀⠀⢿⣿⣙⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⡇⣼⠈⣷⣼⠀⣶⣦
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⣧⢸⣸⣿⠃⡇⠀⠀⠈⠛⣿⣿⡿⠿⣿⡿⠋⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣾⣿⡏
⠀⠀⢸⠀⠀⢰⣿⠁⠀⠀⢸⣿⣿⣿⠏⠸⠇⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣶⣾⣾⣦⣼⣿⡇⠀⠰⣸⣿⣿⣿⠀⢤⣼⠀⠀⠀⣸⣿⡿⡇⢸⣿⣇⢀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡓⣿⣿⡖⣿⣿⣿⣿⣿⣿⣾⣿⣷⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⢿⣿⢟⣷⣿⣿⣿⣿⡿⢯⣬⢄⡹⣿⣿⣿⣿⣿⣿⡿⢹⣿⡿⣭⠞⡁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⡬⣯⣡⣽⠂⡽⠏⠟⣟⣷⣾⣇⡀⠀⢹⣿⠛⠻⣿⡿⢋⣹⣿⡁⠁⠐⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣿⢻⠅⣈⣥⠹⡇⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⢾⣻⣿⣿⣿⣷⣾⡶⢶⡿⠿⠿⢿⠿⡿⠿⠿⠿⣿⣿
⢸⣿⠷⠀⠈⠉⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿
⠰⠆⠒⠤⠤⠠⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠶⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⢦⣤⣤⠶⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣾⣿⣋⣉⠉⠉⠉⠉⠒⢒⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣍⡲⠤⣄⠀⠀⢾⣿⠟⠛⠛⠉⠉⠀⠀⢀⣠⠴⣚⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⣒⠋⠉⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠻⢿⣿⣯⣯⠁⢡⣶⣾⣿⣍⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠄⠉⠈⠁⠽⠃⠀⠙⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⠠⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡆⢰⠿⠆⠻⠻⠶⠶⠋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣠⣶⣧⠸⣶⣄⠐⠂⣀⠈⠉⠈⠀⢀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠦⣄⡤⠐⣧⡀⠀⠀⠀⠀⠀⢀⣷⣿⡱⠉⠀⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣴⠇⠘⠓⢶⣤⡀⠀⢠⠞⣿⣄⣀⣠⣄⡠⠜⠁⣿⣇⠠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⣾⡄⢀⢀⣘⣇⠀⢀⣠⣴⣿⣿⣿⡋⠙⠀⠀⢰⣿⢻⣶⣎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢄⠀⣿⡇⠀⠀⠉⣿⠁⠀⢀⣸⣿⣿⣿⢿⡶⠤⣠⣾⠇⣸⣿⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠙⢻⡄⠈⠙⡟⠓⠉⣩⢿⣿⣿⢿⣦⡀⠀⠀⠀⣴⣿⠋⣿⠁⠀⠀⠀⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⢸⣷⣤⡤⠾⠒⠊⠁⠈⢻⣿⣮⣻⣿⣆⣼⣼⣿⣧⠈⠁⠀⠀⠈⡇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡙⠇⣿⢿⣿⡟⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣤⣿⣿⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣽⡄⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣮⡿⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠰⣴⡄⠀⠸⡿⣿⢿⡀⣰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⡴⣫⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢀⣤⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⣿⣿⣿⣿⣿⢿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠉⠁⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣰⡀⠀⠀⠀⠀⠀⢠⣦⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠢
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢿⡄⡄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⢿⣦⣴⣶⣴⡇⣿⣿⣿⣿⣿⣦⡙⠢⠄
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣴⣿⣿⣿⠿⠿⢧⡙⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠻⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣀⣀⣠⡤⠐⠀⣀⣠⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣤⣶⣶⣿⣿⡿⠋⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣊⣁⣀⣀⡀
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠘⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⣾⣿⣿⣿⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣀⣀⠀⣾⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠙⢌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡘⠉⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⠙⠿⡟⠛⢿⣯⡛⠛⠿⡿⢿⣯⡝
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠋⠀⢹⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⢠⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠻⣿⣿⣼⣿⡀⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣤⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠋⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡄
⣿⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣦⡀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋
⣿⣾⣷⣄⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄

⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠁⠀⠀⠠⠈⢁⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⢀⠳
⠀⠀⠀⠀⠀⠈⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠈⢁
⠀⠀⠀⠀⠀⠐⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠄⠀⠐⠈
⠀⠀⠀⠀⠀⠌⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠂⠀⣤⢀
⠀⠀⠀⠀⠀⠀⣦⠠⢽⣀⢀⣤⠠⠀⠀⠀⠀⠀⠀⢀⣨⡄⠐⠃⠐⠶⠦⠰⠎⠁⠀⠀⠀⠀⠀⠀⠠⡰⢻⣹⠖⠒⠇⢱⣿⠋
⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠈⠙⡟⠄⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⢿⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⠂⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡄⠀⠀⠊⠀⠀⠀⠀⠀⠠⡄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡔⠸⡡⡄
⠀⢠⡀⢇⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢰⠐⣇⢂⠑⡇
⠀⠈⣰⡬⠁⠀⠲⠀⢠⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣴⠀⠀⣴⡆⠸⠀⠀⠀⠀⠀⠀⠈⡘⣄⠀⣰⣤⠍⠳⠏
⠀⠄⠉⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠉⠉⠀⠈
⠀⠁⠀⠘⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢰⠆⠄
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢼⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣧⠄⣰⠷⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⡦⣤⡄⠀⢀⡠⣶⢾⠻⣇⡑⠶⠒⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⡷⣿⢇⢴⣆⠀⠀⠛⢧⣹⣷⣷⣾⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣍⡍⠉⢿⣻⣧⡈⡻⡇⣩⣳⡈⢀⠰⢩⡿⢿⣿⡿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣀⣀⣯⣍⣷⡘⢿⣟⣿⡗⢸⣷⠎⠀⠈⣿⠃⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⠀⠀⠀⢰⡏⠀⡜⠻⠟⠙⠛⣿⡻⣿⠃⠀⢹⣦⣾⣷⡏⠀⠀⠀⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣎⠳⣤⣿⣆⠢⠀⢠⣿⣶⣾⣿⣷⣿⠃⠀⢸⣿⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⣿⣿⡻⠆⢠⣴⠟⠉⠉⠛⠛⠿⢟⡂⠀⣸⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⣿⢟⣿⠃⣴⠟⠁⠀⠀⠀⠀⠀⢀⣆⣡⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢅⣿⣯⡾⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠸⣿⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠰⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠈⢻⡀⠀⠀⠀⠀⠀⠀⣄⢇⡀⢀⠂⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠟⠋⠁⠀⢈⣧⡜⠃⡦⠑⠷⠦⣿⠶⠽⠟⠳⠋⠻⠷⠶⠦⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⢷⣴⣷⣿⣿⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⣀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠂⢄⠀⠘⠒⠶⠀⠸⠇⠀⠀⠀⠈⠁⠀⢸⡀⠀⠐⢀⡀⠀⠙⠃⣤⡄⠁⠌⠁⠠⠀⠀⠀⠙⠂⠀⠒⠊⠀⠈⠠⠀⢀⠐⠂⣀⡀⠀⠉
⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠰⠆⠀⠀⠄⢀⣀⠀⠉⠁⠉⠀⣀⡄⠘⠦⡷⢛⣀⠀⠀⠀⠙⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠋⠁⠀⠀⠠
⠀⠀⠀⠀⠀⣶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠯⠀⠘⠗⠀⠺⠿⠀⠀⠀⠀⠀⠀⣠⠈⠈⠀⠀⠛⠀⡐⠀⠀⠀⢀⡀⠀⠠⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠄⠬⡙⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢐⠣⠀⡆⠀⣀⡰⢶⣶⡀⠀⠀⠑⠀⢀⠠⠀⠦⡀⠀⣠⡦⠀⠀⠀⠀⠀⠀⣠⠄⠀⡉
⠀⠀⠀⠀⠉⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢛⣁⠼⠏⠀⠀⠤⠄⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⠄⠀⠈⠉⠀⠈⠉⠁⢀⣂⠒⣤⠀⠀⠀⠠⠀⠀⠀⠤⠾⠀⠸⠀⠀⠀⠀⣀⠀⠐⠂⠴⠆⠀⠠
⠀⠀⠀⠂⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⡺⣿⡿⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠒⠀⠈⠀⠀⠠⠆⠀⠀⠀⠀⠀⠦⠀⠀⠛⣾⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠂
⠀⠀⠀⢀⠄⡀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣀⠀⠀⡆
⠀⠀⠀⠀⣀⡀⠀⠀⠤⠤⠦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣠⣿⣿⡿⢿⠿⠿⠿⢿⢿⣿⣧⡀⠀⠀⠀⢰⡲⡀
⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠙⠋
⠀⠀⠀⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠀⠂⠻⠀⣀⠈⠀⠀⢀
⠀⠀⠀⠀⠤⠀⢁⡀⠀⠀⠀⠀⠀⠀⠰⠀⣤⣴⡇⠀⠀⠀⠻⣿⣤⣬⣧⣬⣥⣤⣧⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⣠⡈⡀
⠀⠀⠀⠘⡀⠀⠀⠠⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠆⠀⠘⠓⠓⠀⠀⠀⠀⠁⣈⡁
⠀⠀⠀⠀⠴⡀⠀⠀⢘⣻⣿⠩⣬⠀⠀⠉⠀⠀⠀⠉⣸⠏⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠰⠆⠀⠛⠈⠶⠆⢐⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇⠀⠠⠠⠀⠀⠀⠰⠒⠚⠀⠐⠁⠀⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⠀⠄⢠⡄⠀⠀⠶⠐⠀⡴⡆⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠠⠤⠴⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⢸⠀⢨⡀⠛⠀⠀⢰⠛⣆
⠀⠀⠀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠔⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠐⠀⠘⠐⠂⠀⠀⠑⠉⠀⠀⠉⠉⠂⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠲⢷⣤⣶⠟⠀⣀⢀⠀⠈⠠⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⡀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⣀⠈⠉⠀⠈⠀⠸⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠄⠀⠀⢤⢰⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⡀⡀⠀⠈⠁
⠀⠀⠀⠠⠀⠐⠂⠀⠀⠀⠀⠈⠈⠉⠀⣠⠈⠉⠀⢤⣰⠔⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡠⠀⠀⠀⠀⠉⠁⠀⠉⠈⣀⠈⠉⠁⠀⠈
⠀⠀⠀⠀⠀⢀⣀⣀⠆⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠥⠠⠀⠀⠀⠄⢈⠀⠀⠀⠐⠀⠀⢀⠌⡉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠢⠀⠁⠉⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠄⠲⣄⠀⣨⠀⠀⡠⠀⠀⠀⠈⠂⠁⠂⠀⠀⠀⠀⠁⠀⢀⠀⠀⠠⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠐⠀⠀⠈⠃⠃⠀⠴⠆⠀⠶⠀⠀⠀⠂⠉⠀⠰⠆⠈⠛⠀⠀⠀⠘⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠿⢛⣋⣉⠉⠉⠉⠙⠻⢷⣶⣿⠀⠀⣠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⣋⣥⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢟⣥⣾⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⣀⠀⠀⠁⢀⣤⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢛⣩⣿⣿⠟⢡⣾⣿⡿⠿⠛⠛⠛⠛⠂⠀⠀⢀⣠⡞⢁⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠿⠋⢠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⣰⡏⠀⠀⠀⠺⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⣰⣿⡇⠀⠀⠀⠸⣿⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⣰⣿⣿⣷⡀⠀⣀⠀⢿⠀⠈⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⢰⣿⣿⣿⣿⣷⠀⢿⣆⠈⠆⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡟⠀⠀⠙⢧⡀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠉⠉⠛⠿⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⢳⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⣿⣿⣿⠟⠁⣠⣄⠀⠈⠃⠸⣧⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣼⣧⣀⣀⣤⣤⣶⣶⣦⢻⣿⣿⡿⢸⣿⣿⣿⣧⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⣿⣿⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⡟⠛⣋⣵⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⣿⡏⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢈⣿⣷⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠋⣿⣿⠃⠀⣿⠇⠀⣿⣇⢠⣿⣿⣷⣾⣷⣦⣤⣤⣤⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⢸⣿⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⢠⡿⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣴⣦⠀⢠⣤⣿⡿⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣆⣠⠿⠛⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⢸⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⠏⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣴⣿⣿⣿⣿⡿⠃⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠁
⠀⢀⡼⢿⣿⣿⡿⠋⡠⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⠏
⠀⢉⣼⣿⡿⠋⠀⢸⠇⠀⣠⡿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⢿⣿⡿⠋
⢠⣿⠟⠁⠀⠀⠀⠘⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠨⠋⢀⣠⣾⠟⠁
⡿⠋⠀⠀⠀⠀⠀⢸⠃⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁
⠃⠀⠀⠀⠀⠀⠀⠈⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⣿⣆⣰⣿⠃⠀⠀⠀⠀⢀⣴⠀⢠⣾⡿⠁
⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣧⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠿⢷⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠀⠀⢸⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⣿⠀⠀⠸⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⠋⢿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡇⣿⠀⣷⠀⣿⠀⣴⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠈⠃⢸⡷⠘⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡂⡼⠀⠛⠀⣻⡄⠀⠀⣠⣾⣮⣴⣆⠀⠀⢰⠀⣿⡀⢰⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⡇⢹⣾⠆⣼⠛⡇⠀⢀⣛⣹⣿⣿⣿⣤⠀⠈⢠⠏⠃⠈⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡇⢹⣇⣼⣡⣤⣀⣶⡄⠀⢹⣿⣿⣿⣿⣠⣿⢠⡀⢸⠀⠆⢸⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿⢻⣿⡿⢿⠞⣷⠀⣆⣿⢹⣿⢻⡉⣿⠿⠁⠸⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⣼⡏⣧⣸⠁⠈⠀⠆⢾⣿⣿⢸⣿⢹⣷⣿⡀⠀⡾⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⣿⠃⠟⢹⠀⠀⠀⢰⠸⣧⡏⣸⣿⣸⣧⠿⠧⢘⡇⢠⢠⡿⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⠀⡀⢙⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣶⣿⡿⣦⣸⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⠀⠁⢸⠀⡄⢰⡆⠀⢸⣇⣿⣿⡧⡻⢇⣿⡛⡧⠙⣿⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⠀⠀⢸⡀⠀⠈⠇⠇⢸⣿⣿⡟⠁⡇⢈⢸⠁⡇⠸⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠃⠸⡇⢠⠀⡆⠀⢸⢻⣿⡇⢀⠃⠘⢌⠀⠁⠘⢸⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠁⠀⡇⠀⠀⢿⠀⢸⣿⣿⡇⠸⠀⠀⠈⠀⠀⡆⢸⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠐⡄⢿⠀⠀⣼⠀⠸⢿⣿⡟⠀⠂⠀⡜⠀⠀⢸⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢱⠸⡇⠀⢸⠀⢸⣿⣿⡇⠈⠀⢸⡇⠀⣀⢰⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡆⢰⠀⢸⠀⠸⣿⣿⣧⠈⡀⠐⡆⠀⢉⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠁⠘⠀⢮⠀⣸⣿⣿⣿⠀⡇⢰⣇⠀⡄⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⡄⢀⠟⣸⠀⢿⣿⣿⣿⡆⡗⣸⡏⣰⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⣿⣿⣿⣿⠈⣐⢻⣥⣾⣿⣿⣿⣿⣴⣿⣷⣿⣿⣿⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⡾⠟⠟⠯⠬⠭⠍⠿⢭⠭⠍⠉⠉⠙⠛⠛⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡙⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠦
⣤⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣽⣿⡿⠿⠧⣽⣿⣿⣿⣿⣿⢿⣿⣷⣶⣶⣦⣤⣤⣄⣽⣿⣷
⠛⢻⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠶⠶⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣒⣦⣤⣤⣶⣶⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠟⠛⠛⠛⠛⣽⣿⣿⣿⣿⣧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣇⣀⡀
⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⠿⢿⣿⣶⣦⣄
⠀⡴⠞⣉⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣤⣄⡉⠙⠿⣷⣄
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠉⠉⠓⠄⠈⢿
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⠃⣹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣾⠿⠗
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣼⡸⣯⣿⣿⠀⢱⣾⣿⣿⣿⣿⣿⣿⠸⣿⡟⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣕⣻⡌⢿⢰⡇⠙⢿⣿⣿⣿⣿⣿⡇⠹⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⠉⢱⠘⠇⣿⠀⠀⢹⣿⣿⣿⣿⣯⣄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣿⣿⣿⠀⢸⣆⠀⠉⠀⠀⢞⣽⣿⣿⣿⠻⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⢼⣿⣿⢷⣆⢸⣿⣷⣤⡀⠈⠻⣿⣿⣿⣿⣷⠀⡟⠁
⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⣙⠻⣷⣿⡻⣿⣿⣿⣿⣷⣦⣘⠛⠛⣛⣛⡸⠁
⠀⠀⠀⠀⢀⣠⠿⠀⣰⣿⣶⣌⠳⣌⠻⣿⣾⣝⢿⣿⣿⣿⣿⣿⢏⠉⠉
⠀⠀⣠⡴⢛⣡⣴⣿⣿⣿⣿⣿⣷⣽⣷⣮⡻⢿⣷⣝⢿⣿⣿⡟⣼⣆⣀⢤⡀
⠀⠚⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⣷⣝⢿⣧⣿⣿⣿⣿⣻⣷⡄
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣷⡌⠻⣧⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠘⠉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠙⢿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣷⡑⠀⠀⠹⣿⢹⠁

⠰⠀⣐⣰⣆⢀⠂⢲⠀⠀⠀⠖⠀⢀⠀⠀⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠰⠀⢬⣿⢮⡙⠀⣸⠁⡆⡄⣹⠀⣯⠀⠀⠀⠀⠁⠉⠉⠉⠀⠈⠉⠉⠋⠉⠋⠋⠉⠘⠉⠛⠊⠹⠏⠿⣻⣿⣿⣿⣿
⠸⡆⢸⠛⡾⣴⡀⢹⡆⣣⡄⣟⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣾⣿
⢶⡀⢸⣷⡇⢻⡟⠿⡇⣹⡂⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣿
⣸⡁⢸⣿⡏⢸⡓⠀⣇⡿⠇⠉⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⢀⡀⣦⢴⢂⠀⠀⠼⠷
⢸⡇⢸⣿⡧⣘⡇⠠⢾⡇⡀⣵⠰⠀⡄⢚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢤⣼⣦⡿⣥⣿⣿⣾⡆⠀⠀⢲
⢸⠀⢠⡿⠓⢿⣿⠀⢺⡇⡇⢿⣀⠰⡇⣬⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠿⠏⠙⠟⠀⠀⣉
⢸⡀⠲⣿⠀⠈⠈⠂⢼⡏⠓⠾⣿⢴⣿⣿⠀⣟⢦⣶⣿⠟⠁⠈⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡘⢏
⢸⡄⠀⡟⠀⢀⠀⠀⢸⡇⠀⠘⠀⠾⡇⠀⠈⠙⠾⠗⠐⠆⢀⣠⣶⡄⠀⠀⢀⡠⣄⣢⣄⡠⠾⠸⠂⠀⠁⢸⡿⠿⠿
⠸⠇⠾⣇⠀⠀⠀⠀⢨⡇⢀⠀⠀⠲⣷⠀⠠⠀⠀⠀⠀⠈⠻⣟⠁⠀⠀⠀⠈⠹⠿⣋⣉⢀⡀⢤⡀⢶⣤⡞⣷⡢⡷
⠀⠀⠠⢿⠀⠀⠀⠦⡀⡇⠀⠀⠀⠀⠷⠀⠀⠀⢤⡀⢠⠀⠀⠸⣿⣮⠀⠀⢀⠀⠀⠀⠉⠙⠛⠛⠓⠛⠿⠿⠶⢤⠯
⠀⠀⠀⢺⡄⠀⠀⠀⢰⣷⠀⠀⠀⠀⠻⠀⠀⡠⠀⢄⠈⠁⠀⠀⠈⠙⠓⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣀⣀⠀⡀⠀⠀⠄⠀⠀⡀⠐⢾⣦⡁⢁⠀⡀⢠⡶⠂⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉
⢀⢠⠀⠀⠀⠀⠀⠛⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠻⠰⡄⠀⠀⠈⡁⢸⣦⡴⡈
⢈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⣠⣿⠄⠀⠀⠠
⢰⡄⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⢠⣾⣻⠀⢷⡇⠀⠀⣿⣿⣿⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾
⢸⡇⠀⠀⠀⠀⠀⠄⠀⢸⣧⠀⠀⠀⠀⢹⣛⣿⠀⠈⡧⣆⢀⣾⣿⠃⠁⠀⠘⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠈⢋⣿⣿
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⣾⡟⠁⠀⠀⣷⣿⣿⣿⣿⣤⣆⣤⡄⠹⠟
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢩⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣀
⠀⠙⢿⠀⠀⠀⡄⠀⣠⣄⠐⠂⠀⣂⢰⠀⠀⠀⠂⠈⠠⠄⢹⣿⣿⣿⣿⣿⣿⠋⠿⣿⣿⡆
⠀⠀⢿⡅⠞⠃⠀⠀⠀⠀⠁⠀⠴⡈⣿⠁⠀⢦⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⡗⢿⣿⣿⣆
⠀⠀⣼⡇⠀⠀⠀⠚⠁⠀⠀⠐⠤⣬⡽⠀⠀⣸⠶⢷⣤⠴⠛⢉⣿⣿⠟⠻⣿⣿⣴⣶⣿⣿⡽
⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡇⠈⠀⡾⠉⠀⠒⠀⣿⣿⣿⢾⣿⣿⡇
⠀⠐⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡼⣿⣿⠘⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⢰⣿⡷⠀⠀⠉⠀⠘⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⡰⣹⠇⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠠⠄⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀
⠀⠀⠀⠈⠚⠉⠳⠀⠀⠀⠀⢀⢪⠉⠋⠄⠀⠀⠀⢠⣵⠌⠦⢡⠸⠀⠀⠀⠀⠊⢠⠀⠀⠀⢠⡾⣿⢆
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠗⠁⠀⠀⠀⣸⡑⡆⢰⠀⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⢠⠁⡍⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⠀⠂⡄⢚⣧⣥⣬⡴⡏⠀⠠⠀⠁⠀⠀⠀⠀⠀⠂⡅⡇⣷
⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⢰⠀⡠⡀⠀⡃⠀⢠⠃⠀⠀⠉⠉⠀⠀⠀⠠⠀⡀⠀⠀⡀⠀⡀⠐⣡⣇⡇
⠀⠀⠀⠀⢱⠀⠇⠀⢠⠃⠀⠀⠑⠤⠒⠒⡌⠀⠀⠀⢀⣠⣶⣀⠀⠀⠀⠐⠄⠀⠀⠏⠀⠀⠀⠹⠿⠁
⠀⠀⠀⡀⠙⠀⡤⠠⠃⠀⠀⠀⠀⢂⠀⠀⠄⠀⠀⡜⢡⠊⠉⠑⠙⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⢀⠞⠁⠉⠢⠀⠀⠑⠌⣄⣀⡠⠖⠁⠀⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠈⣦⠃⠀⢀⡠⢅⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠙⠀⠀⠐⡤⠊⠉⠈⠑⡄⠤⠚⠀⠀⠀⢠⠂⠢⠀⠀⠀⠀⠉⢀⠀⢀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠒⠆⠀⠬⠇⠀⠀⠀⠀⢆⠀⠀⣰
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⢄⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣩⣴⡀⠀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠟⡀⠀⠀⠀⣀⠴⣈⡷⠖⡢⡄⠀⠀⠲⠃⠀⠈⡋⠀⠀⠀⠀⠐⠂⠀⢠⡀⠜⠄⡀
⠀⠀⠀⢠⡌⣷⡄⠃⠀⠀⡜⠁⠂⠀⠁⠀⠈⠸⠀⠀⠀⡠⠄⠀⠁⠀⠀⠀⠀⠀⠀⠠⡀⠺⡇⡼⣂⠂
⠀⠀⠀⠘⠹⠇⡅⠀⠀⠀⡇⠀⠀⠀⠈⠀⢀⢀⠃⢀⡠⣷⢣⠀⠄⠀⠀⠀⠀⠀⠀⠀⠋⠁⠛⠋⠉⠁
⠀⠀⠀⠀⢰⠀⠇⡀⠀⠀⠰⡀⠂⠀⠀⠀⡌⡜⠀⠃⠀⠈⢣⢠⠀⠀⠀⣀⠀⣀⠀⣀⠴⡶⡄⢀⢠⠒
⠀⠀⠀⠰⣎⠀⢸⡷⠀⠀⠀⠑⢄⢆⢐⣰⠜⠀⠀⢃⠀⠀⢸⠸⠀⠀⠸⣇⢀⠾⡖⣇⡄⡇⢇⣸⢨⣱⠄
⠀⠀⠀⠀⠚⣧⣸⠃⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠠⡀⢙⡆⠀⠀⠀⢀⠀⣄⣦⣀⠀⠄⢀⠀⡠⠄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⣠⣴⣶⣷⣿⡟⠛⠛⠛⠓⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣄⣤⣴⡿⠋⠁⣿⢻⣿⡇⠀⠀⢀⣢⣶⣦⣰⣬⣙⢷⣦⡀
⠀⠀⠀⠀⣠⢞⡿⠛⣯⣻⣿⣿⡟⠀⢼⠀⠙⣿⣿⡇⠀⠀⠀⠉⠻⣿⣿⢿⣿⢷⣬⣿⣦⡀
⠀⠀⠀⠈⠁⠃⣠⠞⠻⠛⣫⣿⣇⠀⠈⢻⣀⢀⠙⢷⣄⡀⠀⠀⠀⠙⣿⣆⠻⣇⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⢰⣿⣠⡴⠛⢡⣿⡘⣄⣄⠈⠻⣾⣶⡄⢈⠛⢦⣣⡰⡄⠘⣿⡄⠹⡄⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⢸⣧⠳⣌⠻⢦⣀⠀⠙⠷⣌⢷⣄⠻⣧⢻⡄⢸⣷⠀⠀⠀⡀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠁⠙⠚⠷⣤⣿⣽⣒⠤⡈⠙⠿⣷⣻⡞⣧⠀⣿⡀⣧⠀⢷⠸⣿⣿⢿⡀
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⠛⠮⣑⣄⠘⢿⣷⣿⠀⣿⡇⣿⡇⢸⡇⣿⣿⣇⢃
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⠈⢧⠀⠈⢿⡆⠸⣿⣿⠀⣽⣇⣿⡇⢸⣷⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠈⡇⠐⡀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢸⡆⠀⠘⣿⠀⣿⣿⠀⣿⢿⣿⣧⡾⢻⡏⣿⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢸⣿⠇⢰⡏⡼⠁⣿⠃⠸⢹⣿⣿⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⠀⡀⢸⣿⣿⡟⢀⡿⠀⠀⠀⠋⡆⢠⣿⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⠀⢸⡇⠀⢠⠀⢠⡶⢫⡇⣰⠀⢞⣿⡿⢁⣾⠃⡀⢠⠀⣸⣣⢟⣿⣿⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⡇⠀⣾⠀⡞⠀⣾⠀⡇⠀⣼⣿⢁⣾⠃⡼⢀⣾⣴⡿⢫⣾⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠈⣷⠀⢸⡜⠀⢸⡇⠀⡇⢸⣿⢣⣾⢏⡾⣡⣿⡿⢋⣴⠟⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣇⠀⣾⡇⠀⣇⣿⣇⣿⣫⣾⣿⠟⢋⡴⠟⢁⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⢀⡿⠷⠏⠀⢿⡇⢁⣿⣿⣿⣿⠿⠋⠁⠀⠉⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠀⠀⠘⠀⣠⠆⠀⢸⡇⡘⣿⣿⠋⠀⠀⠀⠀⡠⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠴⠟⠉⠀⠀⢠⣷⣰⣻⣇⣴⢎⢀⡴⢋⣴⢋⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⣧⣿⣿⣷⡷⢋⣴⠟⣱⠋⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠜⣿⣿⣯⡴⠛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣆⣌⡿⣿⣤⡞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣰⣿⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠉⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⣿⡿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣶⡿⢿⡿⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠿⠿⠟⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣾⣿⣤⣈⣁⣪⡍⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠿⠿⠿⠿⠿⠷⢦⣤⣭⣉⣙⡛⠶⣦⣄⡀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢶⣭⡻⢶⣄⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢷⣦⡀⣿⠆
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡉⢿⣿⡀
⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠈⠻⣦⡙⢷⣄
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠈⢿⣆⠙⢷⡀
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠞⠛⠛⠀⠹⣦⡀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⠓⠶⠶⢦⣄⣙⠛⠛⠛⠛⠛⠛⠛⢦⣙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣋
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠈⠙⠻⢷⣶⡀⠒⠀⠒⠚⠛⠳⣮⣙⠳⠶⣦⣤⣤⡶⠾⠋⣠⡾⠋
⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⢶⣦⣤⣄⣀⣀⣀⣉⣻⣶⣶⣶⠶⠶⠚⢛⣻⡏
⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡄⠀⠈⠉⢻⣧⣀⣀⣀⣀⣀⠀⠀⠀⠈⣉⣉⣉⣩⣉⠁⢠⣿
⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣠⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠷⣼⣧⣤⣭⣭⣭⣭⣭⣭⣍⣉⣉⣉⡒⠶⠖⢀⣿⠃
⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣠⢠⣟⠛⠲⠶⠖⠶⠶⠶⠶⠖⠒⠒⠉⠉⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣬⣤⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢸⡟⣼⡋⠉⠒⠒⠒⠲⠶⠶⠶⠶⠒⠒⠉⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⠀⠀⣀⣠⣞⠉⠀⣤⣀⣀⣠⣤⡴⠋⠁⠀⠀⣀⠐⠻⣦⠀⢹⣏⠙⠛⠓⠒⠒⠒⠒⠒⠞⠒⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢤⣀⡉⠉⠙⣛⣛⣉⣉⣻⣄⠀⠀⢹⠉⢁⡸⢧⣄⠀⠀⠰⣿⣷⣴⠟⠀⢸⡟⢛⠛⠒⢒⣠⣶⣶⣶⠾⠛⠉
⠀⠀⢀⣠⠴⠛⠉⠛⠃⢠⣤⠼⠟⣻⡆⢹⣿⣩⣿⠉⠙⠻⣆⠘⣿⣉⠀⣀⠽⠛⢦⣤⣤⣤⣴⠆⠀⣼⣷⣿⣿⠿⠿⠛⠋⠉
⠰⣿⣿⡇⠰⠟⠀⠴⠶⢀⣁⣤⣾⡟⠛⢾⣿⣿⠿⣷⠙⠃⢿⡄⠈⠛⠻⠧⠴⠴⠛⠋⣁⣈⣥⠴⠛⠉
⠀⠈⠉⠻⣿⣾⣿⡿⠿⠛⢋⣥⣼⠀⣠⣿⠿⣿⡆⢸⡧⣶⡌⢻⣦⣤⣀⣀⣀⣀⡴⠟⠋⠉
⠀⠀⠀⠀⡿⢛⣥⡀⠾⠆⠀⢉⣽⣶⠿⣿⠶⣿⢇⠈⣷⠀⠀⣠⡍⠻⣿⡟⠙⢿⡆
⠀⠀⠀⠰⣧⣠⡠⠶⠒⠈⢙⠟⢻⡷⣾⣿⠾⣿⡆⠀⢹⣶⣿⡀⠁⠀⠠⠿⢦⡾⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣾⣟⢛⣿⣿⣿⡴⠟⠀⠀⣸⢿⣿⡇⠘⠿⠃⢠⣌⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⢙⣟⢻⠟⠁⠀⠀⠀⠀⠀⣿⢶⣿⣿⡿⠇⣤⣦⠀⣠⡿
⠀⠀⠀⠀⠀⢘⡿⢉⣹⡿⠿⠟⣿⡆⠀⡀⠀⠀⠀⣸⣿⣼⣧⣼⣷⠀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⢈⡿⠋⢀⡀⢠⠞⠻⣷⣌⠛⠀⠀⣰⣿⣷⣾⣷⣾⣽⣿⣋⣀⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣆⠉⠀⢀⣠⣤⣿⠙⢷⣤⡾⠿⣿⣛⠛⠛⠛⣿⣿⣿⠋⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣤⡴⠟⠋⠉⠙⢆⠀⠈⠀⠀⠀⠙⢿⡛⠛⣹⣏⢩⣥⣤⡈⢙⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠙⣶⣏⠁⠀⠀⢠⡞⠋⠁⢈⣻⡄
⠀⠀⠀⠀⠀⠀⠘⢃⣀⣀⣀⣀⡤⢤⡾⠀⠀⢀⣀⣤⣤⣤⣀⡈⢻⣅⠙⠛⢁⠙⠲⠶⠿⠛⠻⣦
⠀⠀⠀⠀⣀⣄⣐⣀⡙⠁⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠛⣦⠹⣇⡉⣉⡁⠀⣤⠀⠀⠀⠘⡇
⠀⠀⠀⢰⡏⢉⣿⣧⣹⠂⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⡰⠀⠠⠤⢿⡆⢻⡅⠤⠄⠀⠈⠳⣄⣀⣀⡜
⠀⠀⠀⠘⠿⠿⢄⣈⡁⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⢿⣼⡄⠀⠸⠇⠈⣷⠆⠀⠀⠀⠀⠀⠉⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢀⡛⠿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢻⣤⣴⣿⣿⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠦⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢀⣤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⣞⠉⣹⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣼⣧⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⢹⣖⣉⣉⡤⠲⠆⠀⣠⣤⣤⣤⣤⣤⣤⣴⣦⡀⠀⣀⣴⠟⢻⣿⣿⣿⣷⣶⣤⣤⣴⠊⠁⠀⠀⠀⣀
⠀⠀⠈⠉⠉⠁⠀⠀⠄⠀⠹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠿⠿⠏⠹⣿⣿⣿⣿⣷⣄⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠘⢿⣿⣿⣿⣿⠄⠀⠀⢀⣀⣐⡖⠀⠘⠿⠛⠛⠿⠏⠀⠀⠀⠠⠤

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢟⣭⣭⡉⢙⣛⣉⠉⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣟⠻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠛⠛⣿⣿⣷⣿⣾⡄⠈⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⡀⣀⢼⣿⣿⣭⣴⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⣄⢈⣿⣿⣿⣿⣷⠈⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⠍⠡⣨⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣇⢸⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⠃⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣯⣶⣾⣶⣾⡀⠙⣿⣼⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣾⣧⡀⢪⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⠦⢹⣏⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⢾⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⢹⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠏⠀⠉⠠⠈
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣟⡆⠀⠀⠀⣿⣿⢿⡄⠤⠖⠂⠀⠀⠐⠲⠤⠀⡀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⠀⠀⠹⣿⡏⠃⠀⠀⢻⣿⡞⣧⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢹⡟⠄⠀⠀⠀⠙⠷⠏⠀⠀⠉⠳⠦⠤⠤⠛⠓⠒⢤⡀⠀⠀⠀⠀⠠⠀⠡⣀⡤⠖
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⡀⠀⠈⠓⣆⡀⠀⠐⢂⣹⠷⠊⠁
⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠀⠉⠻⣄⠀⠛⠙⠿⠷⠶⠟⠿⣿⣿⣿⣷⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⡳⣄⡀⠀⠐⠤⡀⠀⢄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡆⢹⠳⣄⡀
⠀⠀⠀⠀⣿⣦⠀⠛⠋⠛⠁⡛⠛⠣⢄⠀⠀⠉⠢⣿⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠸⠃⢸⠓⡄⠀⠈⢄⠀⠀⠐⠂⢤⡀
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣶⣶⣾⡗⠀⢲⣼⣾⣿⣿⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢷⡜⠀⠀⠀⠄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢸⡇⠀⠘⠃⠀⠀⠀⠻⢿⣿⠇⣦⢠⣼⣿⣿⣿⣷⡀⠈⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢸⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠑⡄⠀⢰
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⡀⢁⢀⠀⢀⠈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⣿⣿⠻⢿⣷⣡⣄⣴⡈⣰⡏⠀⠀⡠⠀⠀⠀⡠⠂⡴⠁⢠⠀⠃⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠂⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⣿⡛⢳⣴⣿⠀⠀⠊⠀⠀⢀⠔⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣆⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠄⠀⠀⠀⠠⠊⠀⠀⠐⠋⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣾⠟⣩⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⡔⠁⠀⠐⠁⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⢤⡄⠀⠀⠀⠞⢋⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠈⠀⠀⠀⠀⣠⣾⡿⠋⣀⠄⠀⠀⢀⠔⠁
⡰⠃⠀⠀⠀⠙⠰⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠎⠀⠀⠀⠀⠀⠀⠀⡤⠚⠋⣁⠴⠊⠁⠀⢀⡴⠃
⠑⡀⠀⡼⣷⣶⡀⡆⣄⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢂⡤⠊⠀⠀⠀⠀⠀⠋
⠀⢷⣿⣆⣿⠿⠿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠙⢿⣿⣷⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣃⣉⣋⣀⣏⣉⣯⣋⣭⣟⣽⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠋⡝⠁⢹⠉⢻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡘⢠⠀⡎⠀⡈⡻⣧⡀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠻⠛⢷⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠇⠀⣇⠈⢰⡇⠀⠁⠱⢈⣷⠀⠀⠀⠀⠀⣤⡿⡼⠀⠀⡆⢸⠛⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡇⠃⢸⡇⠀⢠⠰⡏⠸⡷⠁⠀⠀⠀⣾⠁⡇⢰⠀⢁⠀⡇⢹⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠀⣄⡇⠀⢸⡇⠀⠎⢠⡆⠀⣿⠀⠀⠀⠙⣿⠐⣧⠀⠀⢸⠰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⢸⡀⠉⡇⠀⢸⡇⠀⡆⢐⠇⠀⣿⠂⠀⠀⠀⣿⠀⢸⠀⠈⠸⠀⠘⠀⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⢶⢷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠃⠀⡇⠃⠀⠃⠀⠀⢀⡇⠀⢹⠀⠀⠀⠀⣿⠀⠻⡀⠀⣀⠇⠘⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⠃⢸⡆⠹⢳⡔⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⡇⠀⠀⣴⠇⠀⢸⠇⡔⣸⡇⠀⠀⠀⢹⡄⠀⡅⠠⢫⠀⣼⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢾⡁⢰⡄⣿⣄⡤⠀⠀⠠⣼⡇⠈⠀⢰⡇⠰⠀⡇⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⣸⡗⠀⡆⠀⢸⠀⣯⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⡘⠁⢸⠇⡏⣿⠀⠀⠀⠀⢸⡇⡀⠀⠀⡇⠀⠀⡇⢀⠆⠀⠀⠀⣾⠀⠀⠀⠀⢸⣇⠀⡝⠀⢸⢀⡗⢠⡗
⠀⠀⠀⠀⠀⠀⢠⣯⠀⢹⠀⠇⡠⢟⠀⡇⣿⡀⠀⠀⠀⢸⡇⡇⠲⠀⡇⢀⠀⡀⠀⠀⠐⠀⢀⣿⠃⠀⠀⠦⣸⠏⢶⠀⠀⡜⢻⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⡃⠀⠸⠀⡷⣿⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠈⠀⡅⠀⠀⢰⡀⠀⣿⠀⠀⠀⣨⡟⠀⡜⢀⣴⠃⡬⢠⡿
⠀⠀⠀⠀⠀⠀⠰⣧⠀⢸⢢⠙⡆⢀⣄⡆⣿⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠓⠀⠃⠀⠀⣸⠇⠐⣿⣀⣤⠾⠋⣠⡎⠀⣠⠃⡰⢁⣾⠁
⠀⠀⠀⠀⠀⠀⠰⣿⠀⢰⠀⢰⡀⠘⡇⠆⣿⠀⠀⠀⠈⢺⡇⡿⢆⠀⡇⠀⢠⡄⠇⠀⣧⠀⢠⡿⠭⠼⠒⢉⡳⠤⠚⣃⠟⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣆⠘⣱⠀⠇⢰⢿⡄⠀⠀⠀⢸⡇⡇⠀⠀⡇⠂⢸⡇⠀⠀⣿⠀⢸⡟⠒⢻⠋⣉⡹⠶⢺⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢣⠀⢻⠀⢸⠘⡜⢷⡇⢠⣠⣸⡇⡇⠀⢠⣇⠀⢸⡇⡄⠀⣻⡄⢸⣏⣹⣯⣥⣴⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣇⠘⢦⠈⢧⡈⠫⠿⠿⠿⢹⡇⡇⠀⠀⠏⠀⢘⠃⠈⠀⣿⠁⢸⡭⠍⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠘⡗⠈⠳⢄⣉⠚⠳⣆⡉⠨⣷⡇⠐⣄⠀⠀⠀⠀⠀⠀⣿⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠘⠷⣤⡤⢉⣉⣉⡩⠴⣲⣿⡇⠀⢹⠀⠀⠈⣦⠂⠀⡹⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⣄⣀⣀⣀⣴⠶⠛⠁⢸⣿⠀⢸⠀⡄⠸⠏⠀⠀⢷⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢿⢸⠀⢀⠀⠁⠰⠀⢠⠀⠼⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⠀⠸⢦⠀⡀⠙⠀⢸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠐⠈⠀⣷⠀⢰⠸⡆⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⢸⠀⠀⠿⠀⠀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢘⠀⠀⢰⡄⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣀⠀⢸⠁⠀⢸⠀⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠰⠈⠆⠘⠀⠘⢸⠀⣿
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⡀⠀⠿⠀⠀⣈⠀⣿⠆
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⢾⣿⠀⢸⠃⠀⣟⠀⡔⡏⠀⡿⢶⣤⢤⣤⣀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠚⠋⣩⡀⠀⠁⢸⣿⠀⠈⠀⡄⣿⠀⠀⡇⢸⡇⠀⠰⠄⠠⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⠄⠀⠒⠂⠻⢿⡷⠂⡆⠀⡿⢀⣀⣷⠟⠀⠠⠒⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⠙⢻⡷⠶⠿⠟⠋⠀⡀⠀⠙⠀⠰
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⢀⡀⠀⠈⠥⠤⠀⠀⠀⠤⠄
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠜⣙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡹
⠀⠀⠀⠀⠀⠀⠀⢻⡉⣙⣦⣽⢦⣄⠀⠀⢀⣀⢀⣤⣶⢴⣦⣄⣀⣀⠀⠀⢀⡤⢾⣤⣟⡉⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢟⢉⠴⢟⣿⣦⣷⣞⣛⣿⣯⣿⣟⣙⣿⣯⣿⣟⣛⣾⣧⣾⣟⠶⢌⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⡋⣹⡮⢏⣿⣿⡾⢧⣿⢾⣝⢾⣯⠿⣾⣿⣙⠯⣾⡙⣳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢈⣷⡀⣨⣾⣽⣿⡿⢿⣸⡿⣸⠿⣷⣿⣽⣯⡀⣸⣏⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡝⡢⣍⣗⡍⣈⠉⢹⠖⠺⣷⢼⡷⠒⠾⠁⢉⠈⣹⣋⡵⡚⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⢿⡏⣌⢿⣦⡶⢊⣼⡼⢿⣼⡍⢲⣦⣾⢇⡎⣿⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠙⢸⣇⣿⠢⠌⣙⣲⢏⠇⡈⢏⢷⣚⡉⠤⢺⣇⣿⠘⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡏⡞⣛⡻⣷⣄⣄⣽⣿⡺⡿⣺⣿⣅⣄⣵⡿⣛⡛⡌⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣵⣿⡿⣾⠿⣻⣿⡿⣿⣿⠿⣿⣿⡻⢿⣾⣿⣧⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⡛⠙⣲⣟⠛⠳⣿⣿⡶⠛⢛⣷⡚⠙⣋⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣭⣭⣍⣡⣹⣻⣥⣉⣭⣭⣽⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣨⡒⠍⣸⣡⣧⣹⡈⠕⢫⣈⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣝⣿⢽⡂⠀⡇⠀⣿⢽⣟⣴⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢹⣿⢷⣀⣃⣸⣿⣿⢹⡿⠁
⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡻⣷⣾⣽⣶⡿⡻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣿⣿⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⡀⠀⠀⠀⢰⣿⡿⢋⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣥⣂⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣦⡄⢙⡿⣷⠟⠻⣷⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠙⢻⣿⢟⡭⢻⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⣻⡟⠁⠉⠟⠛⠛⠁⠀⠘⠁⠺⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆⣙⣒⠊⠁⠢⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢿⠙⠋⠙⠛⡦⠀⠀⠀⣠⠏⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣻⣿⣿⡁⠀⠀⠀⠈⠁⢀⣤⡾⢃⣼⢏⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡀⢦⢠⣤⣴⣼⣋⣴⣿⣷⢟⣠⣴⣾⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣬⣽⣿⣿⣮⣶⣿⣿⣿⣿⣿⣿⣿⡏⢉⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣿⣿⣿⣿⣿⠀⢀⣹⣏⠤⣸⡟⠛⠋⣹⣿⣿⣾⣿⡿⣻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣏⣴⣿⣿⡟⢋⠀⠙⢦⣰⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⡟⠉⣵⣿⡟⠁⢸⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠊⡼⠋⠁⠀⠈⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠚⡁⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢀⣬⣼⠏⠀⠀⠀⠀⠀⣿⣅⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣥⣾⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⢸⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⠛⠀⠀⠈⠀⡞⠙⠛⠟⠙⠳⠛⠻⢿⣿⣿⡿⠿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠿⠒⣴⠞⣴⠛⣱⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⣿⡏⣀⠁⡐⢃⡴⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠔⠚⠘⣿⠿⢆⣿⣴⢳⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣈⣴⣿⣦⠀⣘⠙⠿⣟⠻⠈⣾⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣤⣤⢾⣴⣫⣍⡻⡎⣶⡕⢸⡿⢼⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠋⣾⣏⠙⢓⣣⣿⣷⢈⣽⠀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⢿⣦⣭⣭⣿⣧⣤⡰⣾⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⣮⣽⣿⡟⠁⠘⣷⠘⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠛⣋⡾⣮⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠉⠙⣿⣿⣿⣿⠟⠳⣬⢳⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡀⠀⠸⣿⣿⡇⠀⡄⠘⣷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣇⠀⣠⣿⣿⣿⣶⣿⣆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢻⡟⠛⠉⣿⠏⠉⠙⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢻⣦⡀⣿⡆⠀⠀⡿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢿⣾⣷⡀⠐⠁⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣟⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡟⠋⣴⣤⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣖⣴⠛⠁⠀⢠⠊⢹⣀⡀⠀⢠⡄⠐⢧⣀⡏⠁⠀⠀⠈⢹⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣆⣈⠓⠃⠀⠈⣧⣄⠀⠉⠀⢠⣄⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠉⠋⠀⠀⡤⣼⠀⢱⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⣤⠀⠉⢡⡤⡞⠀⠰⣵⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠚⠀⠀⢀⣉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣚⡭⠍⠉⠀⠀⠀⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡞⠉⠀⠀⠀⠀⠀⠀⠀⡴⠚⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⡤⠤⠤⠤⣶⣚⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠶⢤⣤⣀⡹⡍⠉⠉⠉⠉⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠽⢖⡋⠉⠙⡗⢢⣍⠻⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡞⠁⠀⠀⠉⡗⣸⠁⡞⢈⡷⣿⡦⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⣀⣴⣚⡛⡾⢷⣷⣋⣇⡏⡼⢹⡗⢹⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢀⢏⣮⡼⣹⢻⢿⠯⣾⣟⣻⠷⢿⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⢸⡘⡿⢬⣛⠹⠽⣿⣾⣼⣼⣹⢿⡓⣾⣠⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⡘⣆⢨⢟⢶⣬⣅⣳⠯⠥⢞⡽⠉⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⡙⠾⣾⣞⣀⣁⡤⠖⠋⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢤⠤⠤⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⣙⣿⡚⢉⣽⣷⣾⠿⠶⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⣮⣽⣿⠟⠋⠁⢀⣀⡤⢚⠉⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣿⣿⣿⣟⣻⠟⠁⠈⣠⠾⣫⣿⠀⠀⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⠿⠛⠁⠀⢠⡾⢁⣾⣿⠋⠳⢴⣗⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡠⢔⢒⡒⠀⠒⠒⠒⠚⠁⠀⠀⢤⣴⠟⢠⡿⣼⠇⠀⠀⠀⠙⢄⢠⣤
⠀⠀⠀⢀⣲⣌⣳⣤⣴⣊⣿⣿⠋⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢼⣧⣿⣆⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⢀⡈⠻⢿⡿⢁⣿⣿⠇⠀⡎⡇⣇⠀⠀⠀⠀⠀⢀⢀⠁⠰⢦⡀⣇⢸⣿⣿⣿⡆
⠀⠀⠀⠈⢽⣿⣑⣺⣿⣿⣿⡆⠀⠣⢻⣜⣷⣤⣀⡀⣀⣸⣿⣷⡄⠀⣠⣿⣼⣯⣿⣿⣤⣀
⠀⠀⠀⠀⠠⣬⣭⣛⣙⣾⡿⠛⢷⣤⣄⣙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡴⡃⠀⢌⠛⢿⣿⣿⣶⣷⣶⢤
⠀⠀⠀⠀⠀⠐⠛⠟⠁⠀⠀⠀⠀⠈⢙⣤⣀⢹⡿⣃⠀⠀⠀⠀⠀⠈⠋⠑⠒⠛⠓⠯⣟⣦⠀⠉⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣌⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢍⣳⣄⠀⠀⠀⠀⠀⠀⠀⢰⣶⠇⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠬⣷⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡏⠀⠀⠀⠠⠂⠀⠀⠀⠀⡎⠀⡸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣇⣠⠄⠀⠀⠀⣱⠛⠁⠐⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠞⠉⠀⠀⠀⠀⠀⠃⣀⣴⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⢶⣄⠀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢻⠸⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠈⠁⢠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⡆⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠿⣛⣵⡿⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣼⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠈⢱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⣿⡿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⣶⣺⣿⣿⣧⠀⠀⢸⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⡏⠙⠟⠟⣿⣿⣂⠀⣰⣀⣠⣿⣿⣿⣿⣿⡆⠀⠀⢸⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣄⠀⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣗⡀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⣼⣿⣿⡿⠟⠋⠁⠻⣿⣿⣿⡟⢿⣷⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡈⢻⣧⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠙⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠈⠀⠀⠀⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠟⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠁⠀⠀⠀⠀⠀⠀⠰⠛⠃

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣴⠀⠝⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣯⡤⠀⠴⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⣠⠀⠀⠈⠃⣴⠃⣴⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⡿⣿⠏⠀⠀⠀⣄
⠀⠀⠀⠀⠈⣷⠀⠀⣤⠃⣰⡇⠀⠀⢠⠀⠀⠀⣀⣀⠀⠀⠀⠈⠁⣿⣄⡀⢀⡴⠊⠀⠀⠀⠀⠀⠀⣀⠀⣠⠆⠃
⠀⠀⠀⠀⠀⠸⣆⠀⢁⣠⢿⠁⠀⠀⠀⠀⠀⠀⠙⠙⠗⠀⠀⠠⠆⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢻
⠀⠀⠀⠀⡄⣰⠹⣶⡈⣧⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠂
⠀⠀⠀⠀⢱⡏⠐⠋⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠈⠁⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣆⣠⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠄⠀⠀⠀⢀⠤⢴⠲⠶⠶⠛⠉⠃⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢿⡟⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⡿⢦⠄⠤⠼⠛⢻⠀⠇⣠⠀⠀⠀⡈⣠⢆⠀⠈⠇⠀⠤⠆⢠⡖⠀⠀⠀⢸⡇⣠⠛⠢⠀⠐⠁⢀
⠀⠀⢠⠃⣰⡇⠀⠀⠀⠀⠀⠀⣇⣼⠇⠀⢀⣴⠿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢰⠃⠀⠀⠀⠀⠀⣸
⠀⠀⢸⣾⡁⠧⠶⠄⠀⠀⠀⠀⠸⠿⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠀⠀⠀⢀⠀⢠⠀⠀⡀⠀⡀⠀⠀⠀⠐⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢸⣧⡎⠀⠀⢻⣼⡇⢄⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⣀⣴⣷⣿⣿⣴⣄⣀⣤⣿⣿⣾⣷⣠⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣰⣮⣾⣝⡛⠿⣿⣿⣿⣿⣿⣿⠿⢛⣫⣷⣶⣎⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠈⣿⣿⣿⣿⠁⣰⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠿⣿⣿⣿⣿⡋⠀⠘⠟⠻⠃⠀⡙⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣵⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣶⣶⣦⣀⣀⣀⣀⣴⣶⣶⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢿⣏⢙⠻⠿⢿⣇⣸⡿⠿⠛⡋⣹⡏⠁⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡂⠈⠹⣷⣤⣴⠤⣿⣿⡤⣦⣤⣾⠏⠁⢖⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢠⣿⢻⣇⡀⠈⠁⢀⣼⡿⣿⡀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠯⣭⣩⠽⢿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⡿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⡄⡄⣄⣠⢠⡤⣠⡀⣄⡄⢤⠄⢠⣠⢀⣄⢠⣠⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠁⠉⠉⠈⠀⠁⠁⠁⠁⠈⠀⠈⠈⠈⠈⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡶⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡙⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢿⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣼⣇⠀⢧⡀⠀⠀⠀⡼⠀⢰⣇⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣾⡷⣀⣀⣠⣷⣶⡿⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⣻⠂⠿⣯⠟⠸⣏⣴⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠶⠙⠓⣾⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢤⣴⣶⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣭⣭⣿⣣⣤⣴⣬⣭⣭⣑⡲⣶⡀⢀⠀⠀⠀⠀⢠⣰⡆⢘⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⣿⡟⣿⣿⢻⡿⠛⠁⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠘⢃⣼⡇⠀⣤⠀⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠹⢳⣿⣿⣿⣿⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⡁⢠⡟⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⡀⢠⣴⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣭⣀⣾⠃⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠘⢠⡿⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⢉⡁⢀⡀⠀⢠⣼⡇⠀⠀⠀⠀⠀⢐⣀⣼⣿⣿⠻⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠋⠐⣯⣿⣿⠷⣾⣿⣿⣷⣴⣤⣤⣶⣶⣾⡿⠛⢿⣿⣦⠾⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣽⣿⠀⢀⣼⠿⠿⠿⠿⠟⢿⣿⣿⣿⣇⠀⠀⠻⣿⣇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠙⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠃⠀⠀⠘⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⠀⢀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣴⡿⢣⣀⣴⣿⣶⣶⣴⠆⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡿⢃⣀⣀⡀⣿⡇⠴⠶⠾⠖⠛⣿⠟⠛⠙⠉⠉⢁⡌⢃⣴⣿⣷⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⣻⣿⣿⠉⣤⣿⣷⠖⢚⡓⠐⢋⠁⠈⠛⠋⠁⢾⣽⡯⠉⠈⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠑⠒⠂⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠛⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⢀⣀⣠⣤⣤⠤⠄⣀⣀⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠶⠖⠲⣤⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣶⣿⣦⡀⠀⠈⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⢠⣴⣶⡄⠈⠻⣿⣿⣽⡿⡣⠄⢄⢠⠜⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠘⠛⠉⠁⠀⠀⠚⢛⡭⠻⠦⣤⣴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢚⣓⠒⠄⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣼⠟⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⢿⠿⠿⠛⠋⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠾⠷⣶⣻⣿⢿⣿⣿⠀⠀⠀⠀⠀⢀⣼⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡄⠘⣿⣿⣸⣿⣿⡇⢁⣠⣠⣴⣟⠁⠀⡈⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠙⠉⠈⣿⣿⣷⣿⣿⣿⣿⠟⢣⡞⠁⣀⣀⣁⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠉⠀⠀⠀⠀⣴⠞⠉⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢻⣴⠂⠤⢀⡀⢀⣀⣴⣾⣿⣿⡁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣬⡗⠂⠀⣀⠀⢠⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⣿⢆⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⠀⠀⣀⠀⠀⣀⣿⠇⠀⠀⠉⢀⣿⣿⡿⠀⣿⣿⠛⠐⠦⠤⠤⠔⣿⡎⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢤⣤⣡⡴⢿⣧⠀⠀⠀⢀⣾⣿⠿⠃⣾⡿⠋⠀⠀⠀⠀⠀⠐⢻⣧⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⡏⠀⢸⣷⡐⠒⠀⠸⠋⠀⠀⠀⠟⡁⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⠀⠀⣾⠋⠑⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠈⢰⠾⣿⠆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠈⠀⠠⣾⡏⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⣿⡒⢦⡀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡶⠋⠉⠀⠀⠈⠀⠀⣂⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡧⠂⠀⠀⠉⠃⠙⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠘⢿⣶⡄⠛⠜⡀⠀⣠⢤⡀⣼⣶⣤⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⠣⠀⣘⣁⠂⠈⣸⠷⠋⢁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⣀⣀⣠⣴⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⢳⣶⣄⣤⣤⣤⣤⣤⠈⠀⠀⠀⠀⠘⢀⣀⣠⣤⣶⣶⡶⣿⣛⣉⣭⣿⣷⣿⣶⣿⣿⠀⣱⣶⣶⠄
⠀⠀⠀⢀⣺⣿⣶⣶⣶⣶⣴⣿⣿⣇⠠⠘⢦⡙⣿⣿⣿⣿⣿⠋⠀⠈⠀⠄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣙⣧⡼⣿⣿⣿⠃⠐⣟⠛⠓⠂
⠀⠘⠀⠀⠻⠿⣿⡿⠿⠿⠿⠿⠿⠿⣷⠒⠀⠟⠛⠛⠛⠛⠃⠠⠀⠀⢀⠀⢼⡉⠉⠉⠉⠉⣿⠉⠉⠁⢠⡏⢸⡇⠀⠀⠀⡆⠀⠀⡃
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣤⣠⣆⠀⠀⡀⠀⠀⠀⠀⠀⠉⠓⠤⠤⢤⣿⠀⠀⠀⣸⠁⣾⣧⠀⠀⠀⢰⠀⠀⢥
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡿⢨⣿⡿⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢸⣿⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠆⠸⣯⢠⠀⠀⠀⠀⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠣⠀⣿⡈⠀⠀⠀⠀⡇⠀⠀⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢽⡅⠀⠀⠀⠀⡀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣇⡐⠀⠀⠀⠇⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆
⠀⠀⠀⠲⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆⡼⢠⠆
⠀⢀⠀⠢⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠠⠇⠀⠉⠁
⠀⠀⠀⢆⠆⠀⠀⣸⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⡅⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⢠⠀⠰⡄⠀⢀⢀⡎
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡆⠀⡇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢘⢣⡄⢠⡀⠀⠘⠆
⠀⠀⠀⢤⣠⡄⠧⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣰⠀⠀⠀⠀⡿⠛⡷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠈⠻⡄⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⢀⠀⠀⠀⠀⢦⠀⠀⠈⠁⠀⠀⢀⣠⠁⠀⢸⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡇⢠⡆⠀⣰⠧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⣆⠀⠀⠀⡈⣤⣀⣰⠇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠱⣌⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠙⠓⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠰⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠼⠁⠀⠀⢷⣤⠀⠀⠔⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠋⠀⠀⢘⠒⠒⠐⠦
⠀⠀⠀⢀⡞⠋⠀⠀⠀⠈⠉⣇⢀⣀⠀⠀⠀⠀⢰⣟⡆⢸⣻⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠤⣤⢼⣿⣤⣥⣤⣤⢤⣤⣯⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠧⠀⠐⠸⠀⠂⠒⠠⠄⠘⠰⠸
⠀⠈⡞⠀⠗⠃⠘⢺⠖⠀⠘⢿⠋⠙⠍⠀⠐⡖⠖⠀⣖⣒⣒⠒⠂⠀⢰⠀⠀⠀⡰⠀⠐⠀⢹⠈⠈⢸⡏⠉⠉⠸⡇⠁⠈⠹
⠀⠀⡇⣺⢓⣷⢰⡟⠛⡇⢰⣟⣰⠀⠀⠀⠀⡅⠀⢸⡏⠉⢽⠤⠀⠀⠘⠀⢀⠜⠄⠀⠀⠀⣿⣍⣉⣿⢺⠁⠈⠀⡇⣍⣭⣘
⠀⠀⡇⠈⢹⠉⠸⡇⠀⡇⠀⢹⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⢸⠀⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⢺⠀⠀⢸⡐⠀⠀⠀⡇⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠓⠛⠛⠛⠛⠓⠒⢒⠒⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⠐⡖⠀⠀⠎⠁⠀⠐⠋⠛⠒⢲⠒⠒⠒
⠀⠀⠘⠛⠋⠙⢻⠛⠛⠛⢻⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠛⠋⠉⠀⠐⠃⠁⠀⠘⠛⠚⠙⠃
⠀⠀⠰⡶⠶⠶⢶⡶⠶⠶⡾⢦⡴⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⢁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⢀
⠀⠀⠀⢠⡤⣤⠸⠁⣀⣀⡈⢈⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣤⣤⣤⣤⠀⠀⠀⠐⠀⠀⣤⢤⣤⡄⠓⡏
⠀⠀⠀⠨⣇⣸⠨⠨⡇⠈⡇⢰⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠿⠤⠤⠇⠀⡇
⠀⠀⠀⠀⠀⠉⠨⠠⡇⠀⡇⠀⠉⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠂⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⣠⣤⣤⣤⣤
⠀⠀⠀⠋⠃⠇⠟⠛⠂⠀⠀⣀⣀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⠀⠀⠀⠀⢠⠀⠀⠆⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠘⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⡰⢠⠀⠴⡀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⣀⣀⣀⣀⡄⠀⠀⣤⣤⡂⡆⢴⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀⠀⠁⡠⠊⠀⠀⠴⠀⠀⠀⠀⠀⡇⠀⠀⠤
⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠉⠈⠁⠉⠈⠀⠀⠀⠐⠂⠀⠀⠐⠀⠁⠀⠀⠈⠳⠬⠀⠐⠀⠀⠀⠈⠀⠀⠃⠀⠀⠒⠀⠐

⠀⠀⠀⠀⢢⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠨⠳⣼⡀⣶⠋⠀⠀⠀⠁⠀⠀⠀⢠⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⢱⡄⢘⠀⢰⡄
⠀⠀⠀⠀⠀⢣⣘⠇⠘⢦⡀⠀⠸⣆⠀⠀⠀⢸⡇⠀⠀⢠⠇⠀⠀⠀⢀⣠⠞⣷⡧⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠿⢶⣦⣽⣆⠀⠀⢠⣷⣀⣴⣏⣀⣠⣤⣶⣿⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣮⣿⣷⣾⣿⠛⡛⠛⣻⡟⠉⢉⠀⠀⢀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡞⠣⠟⣿⣿⣿⣿⣧⣼⠿⠃⠀⠀⠱⣄⠈⠳⡄⠀⠁⠀⠀⠀⠀⠀⠀⣼⠀⣠⢀⠀⠂⢀⡀⢀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⣠⣠⣦⠴⠰⠀⠀⠠⠆⢀⣯⠀⣀⣨⣀⠂⢀⠤⠀⢠⠟⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣤⡔⠛⠀⠩⠴⠆⠀⢀⠛⢺⣽⣿⢼⣿⠿⠖⠀⠈⢀⠀⠀⢰⣏⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠀⠘⠁⠣⣀⢀⠶⠖⠁⠈⠀⠀⣩⣿⡟⣻⣿⣶⠀⠀⠊⡼⠀⢠⢍⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣄⠀⠿⠖⠀⠀⠀⢠⢿⣿⣿⡿⣿⣷⡀⠀⡟⣰⠋⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⢩⠳⠈⣤⣄⣤⣲⣙⣿⣿⣿⣿⣪⣿⡅⣀⡴⠃⡠⠄⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⡀⠀⠀⠀⢡⡈⠻⣿⣿⣿⠿⢿⣿⣿⣻⢿⢿⡿⣁⡀⠊⠁⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠉⠀⡈⠷⣿⡗⢿⣿⣿⡙⠪⣠⢣⣽⣟⣀⠘⠀⠤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠘⠓⣷⣾⣿⣿⣧⠜⢃⣿⣿⣿⣿⣧⠄⣤⡿
⠀⠀⠀⠀⠀⠁⣀⠀⠀⣀⠄⠀⠀⢿⣩⣍⣈⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣮⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⠷⠉⣽⢆⣎⣸⣿⣿⣿⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⢡⠄⡉⠁⠀⠀⡀⠈⢷⣿⣧⠆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⠶⠂⣀⣻⡏⣼⢶⣿⣿⣿⠿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠐⠂⠂⠠⡀⠐⠀⠀⠀⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠠⠤⣬⢛⣠⣿⠉⡃⣿⢺⢿⣟⣸⠛⢿⡗⠃⠀⠀⢀
⠀⠀⠀⠀⢄⠀⠀⠐⠐⢂⡄⡶⠀⠀⠀⠘⣿⣿⡏⣿⣿⣿⠛⠿⠿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣏⠀⡔⢁⡙⢋⣤⣒⣻⢻⣼⠞⣳⣽⣞⣏⠁⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠞⠐⡕⠲⡀⠀⢹⣿⣇⣽⣿⡇⠀⠀⠀⠈⢿⣿⣧⠀⠀⠈⠙⣿⣿⣿⣆⠒⠈⢋⢼⠑⣿⣿⣯⣿⣿⡷⣛⡿⠀⠲⡒⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠐⢀⣠⣌⣣⣴⣠⡄⠈⣿⣿⢺⣿⠀⠀⢀⣀⡀⠀⠹⣿⣷⡄⠀⠄⡄⠉⠛⢿⣷⣶⡄⠈⣰⣿⣿⣿⣿⣻⣿⣁⣴⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢄⣀⠀⠀⠈⠠⠀⠈⠗⠀⠸⣿⣾⡇⡀⢰⣬⣟⢃⣄⢠⣿⠏⠀⢀⣤⣷⣶⣯⣠⡌⢿⣿⠶⣿⣿⣿⣿⣿⣿⣿⣋⣉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠠⠄⠀⠘⠥⠀⠀⢀⡀⢠⡿⢿⣦⣯⢻⣇⠀⠀⠱⡏⢻⠛⣿⣷⣦⣶⣾⣿⣿⣿⣿⣿⢿⣴⢻⣶⣿⣿⣿⣿⣿⠻⡛⠻⣿⣋⠣⠤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡈⢛⣶⣄⡦⡀⠂⠀⣿⢈⣿⡆⠠⡤⢷⣇⡔⠈⠷⢿⠿⢿⣿⠿⣿⡿⣿⠸⠁⢈⣿⣾⠿⠿⠿⢷⡾⠯⡀⠀⠀⠂⠀⡀⠨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠧⣤⠌⠀⠀⣀⣸⠟⠁⢸⣷⣾⠕⡼⢿⡀⣀⣠⢰⣠⡄⣠⠀⠬⠩⢠⡶⠞⠛⠛⣓⠂⠰⠦⠀⠂⠀⠒⠀⢀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠐⣠⣿⠂⠉⠠⣠⠠⠐⢀⠀⢀⣬⣿⣿⣿⣿⣾⣷⣴⣿⣷⡴⠊⠉⣠⣶⡆⠓⠁⡀⠀⠀⠀⠸⡁⠲
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠁⠤⠁⣤⠂⠈⢹⣶⣤⢂⠀⡉⢉⡉⠛⠛⢿⣿⣿⣿⣋⡤⠎⣼⡿⠋⠀⠀⠘⠇⠐⡄⠀⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠗⠊⣏⠀⠾⡆⠋⠀⠒⠘⠉⠉⠉⣸⠧⠀⠀⠀⠀⠘⡀⠂⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⡄⠀⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣭⡙⠿⢿⣦⣿⣯⣻⣿⣶⣤⣤⣄⣤⣤⣶⣦⣤⣤⣀⣴⣿⣿⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⠘⠛⠿⣧⣿⣶⣤⣬⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣤⣤⠶⠄⠀⠈⠉⠿⣿⣷⣦⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⠖⠈⠉⠀⠘⠒⠲⢶⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠠⢀⣴⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠖⠛⣛⣩⣭⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷
⠀⠀⣠⣾⣿⣿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⠇
⠀⠀⠹⠛⢉⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡏⠀⢾⣿⣿⡆
⠀⠀⠀⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⣄⡄⠀⢿⣿⠀⢇
⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⠀⠈⢻⣆⠈⡀
⢀⣤⡾⠛⢙⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢹⣧⣄⠈⡋⠀⢱
⠀⠁⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣷⣇⡀⢀⢷⡄
⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⠻⢿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣏⢻⣷⣄⠾⣿⡀
⠀⢀⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡷⠈⠙⣿⣦⣿⣿⡂⠹⣷
⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣼⣿⠂
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣼⣿⣿⣻⣿⡿⠃
⠀⠀⠀⠀⡟⠻⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠁⠀⠛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠘⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣱⡿
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠛⠙⣿⣿⣿⣿⣿⢯⡿⢿⣿⣿⣿⣿⣿⡏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠱⠜⢀⡑⠝⠫⠝⠻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠋⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡺⠋⠋⠛⢂⠀⠀⠀⠀⠀⢠⢡⢋⠉⠇⢡⠣
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠐⢸⠀⡇⠁⡄⠀⠀⠀⠀⠀⠀⠐⢀⠁⡆⠀⢰⠘⠀⠀⠀⠀⠀⠆⡆⠘⢰⠸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠈⢠⢸⠀⡇⢀⢠⠀⠀⠀⠀⠀⠀⡇⠸⠀⠁⠀⠈⠀⡆⠀⠀⠀⠘⢠⠀⠀⠘⠀⡀⠇⠆
⠀⠀⠀⠀⠀⠀⢠⠀⡇⠈⢸⠀⡇⠸⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⢸⠀⠀⢀⠀⡇⢠⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠄⡄⢀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⡇⠘⠀⡇⢸⢸⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠘⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⠃⠸⠀⠇⠀⠀⠀⠀⢸⠀⠇⡁⢠⠀⢸⢸⠀⠀⠀⠀⠰⠹⠀⢁⢩⠣
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⡇⢸⢠⠀⠀⠀⠀⠀⠀⠀⠂⠘⢠⠀⠃⠸⠀⡄⠘⠀⠀⠀⠀⡇⠀⢨⢸⠈⠀⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⢰⢸⠀⡇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢀⠀⡄⠀⠀⠀⡄⡆⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⢸⠀⠀⠇⠀⠀⠀⠀⠀⡀⢁⠈⡀⠆⠸⠀⢡⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠃⡄⠈⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⡇⠈⠀⠀⢠⠀⠀⠀⠀⠀⠆⠸⠀⢁⢸⠀⠇⠠⠀⠀⢸⢰⠀⡇⠀⠀⢸⠀⢠⠀⡄⠀⠀⢠⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠃⠀⠀⠃⢸⠀⠀⠀⠀⠀⠰⡀⢣⠸⡀⠆⠸⡀⠣⡀⠀⠀⠀⠃⠀⡇⠀⡄⠈⠀⡇⠀⠀⠸⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⢠⠀⢰⠀⡄⢸⠀⠀⠀⠀⠀⠀⠱⡀⢆⠱⡈⢆⠑⢄⡈⢣⠀⠀⢠⠀⠁⠀⡇⢠⠀⡆⠀⠀⡄⢀⠀⠸⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠂⢀⠀⡇⢸⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠱⡈⢆⠱⣄⠑⠠⠌⣹⠀⠀⢸⠀⠀⠀⡇⢸⠀⡇⠀⠀⠇⢸⠀⡆⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⢌⡑⠢⠄⣸⠀⡆⠀⠀⡇⠀⡇⢈⠀⠀⠀⢠⠀⡆⢠⠀⡆⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠘⠀⢸⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢈⡐⠂⠰⠀⡇⠀⠀⡇⠀⠀⢸⢀⠀⢠⠃⡘⠀⡆⠀⠁⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡆⡇⢠⠀⠃⠀⠂⢸⢀⡦⠁⡰⠃⡘⠀⡐⠀⠇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⠀⠀⡸⠊⢀⠎⢀⠜⠀⡰⠁⡜⢠⠃
⠀⠀⠀⠀⠀⠀⡀⢠⠀⢠⠀⡇⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⢸⠀⠀⢀⠖⢁⠔⠁⡠⠋⠀⡔⠁⡸⠀⠎
⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢁⠊⡔⠁⡠⠊⠀⢠⠊⠀⡰⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠾⡘⢀⠎⠀⢀⠔⠁⡠⠊⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⡀⣧⢂⡠⠒⢁⠠⠊⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⠀⡝⠻⡄⣉⠠⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⡇⢰⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⢸⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡄⢸⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠃⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⡈⠙⠲⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣴⣶⣶⣶⣒⣴⣧⡀⠀⠙⣿⣷⡄⢹⡎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⣮⣿⣿⣿⢷⣄⠀⠈⣿⣧⡀⠁⠂⠹⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣋⣾⠁⣿⣷⣶⣦⣾⣿⠍⠀⠠⠄⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣤⡛⠿⣿⣿⡟⠛⠀⠀⠀⢀⣴⠀⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡧⢀⠀⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⢻⣿⠛⠻⣤⠉⠛⢠⠈⠁⠈⠉⠁⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢠⣿⣿⣿⣷⣾⣦⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⣿⡿⠋⢉⣿⡿⣷⣀⣀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄⣾⡿⠃⠸⠛⠛⠋⢻⣿⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⠁⠉⠛⠛⠛⠉⠉⠀⡀⠀⠀⠀⠘⢿⠀⠀⠘⣦⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⡤⠄⠀⠀⢀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣿⡔⠃⠀⠀⠈⠉⣉⣱⡦⠶⠆⠀⢀⣴⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢀⡀⢁⡠⢔⣰⣿⣿⣏⣴⣶⣶⡾⠟⠛⠁⠀⠀⠀⠀⠃⠀⠀⠀⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⠿⣬⣶⣼⣿⣿⣿⣿⠟⠛⠁⠀⠰⣤⣄⠂⠀⠀⠀⠄⠀⠀⠀⠀⢀⣷⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠘⠋⠀⢠⣰⠂⠀⠀⠀⢂⣤⣿⣿⣿⣧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣶⣾⣯⣧⣿⣿⢻⠛⣿⠁⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣀⣤⡶⡚⠋⠉⠁⠀⠀⠈⠉⠙⢓⢶⣤⣀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠦⠠⣾⣿⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣿⣿⣷⠄⠴⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣆⠘⠿⢿⣟⠻⠿⠆⠀⠀⠀⠀⠠⠿⠟⣻⡿⠿⠃⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣡⣤⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⣤⣌⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣷⣿⣾⢢⠇⢠⠀⠀⠀⠀⡄⠸⡔⣷⣿⣾⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡙⠿⠿⠿⡞⠁⢀⡼⠀⠀⠀⠀⢧⡈⠫⠷⠿⠿⠿⢋⣿⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢈⠈⠛⠦⠤⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠴⠛⠁⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⠸⣦⣄⠐⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⣠⣴⠇⢋⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣿⣃⣀⠀⠀⠀⠠⠤⣤⣄⠀⠀⣠⣤⠤⠄⠀⠀⠀⣀⣘⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⣁⣤⣤⣀⡀⢤⣌⣿⡇⢸⣿⣡⡤⢀⣀⣤⣤⣈⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠹⢧⣭⣽⠿⣿⡿⠀⠀⢿⣿⠿⣯⣭⡼⠏⠀⢀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠉⠁⠺⡿⣷⣿⣿⣾⢿⠗⠈⠉⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁⠀⠀⢀⣿⡟⢻⣿⡀⠀⠀⠈⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⡇⠀⠀⠀⢸⣿⡇⢸⣿⣷⠀⠀⠀⢸⢸⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⣠⢀⠀⠀⠉⠀⠀⠉⠀⠀⡀⣄⣿⢸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⣌⡟⣾⣤⡄⢀⠀⠀⡀⢠⣤⣷⢻⣩⣼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠹⣮⡑⣿⢈⡟⢻⠳⠟⡟⢻⡁⣿⢊⣵⠏⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠳⠾⣎⣯⢹⡚⢓⡏⣽⣡⠷⠟⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠁⠘⠋⠛⠛⠙⠃⠈⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠠⢀⣀⣀⣀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠂⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣿⠆⠁⠀⠀⠀⠀⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⠀⠀⠘⠰⣷⡈
⠀⠀⠀⠈⣴⠋⡌⠀⠀⠀⠀⠀⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠄⠀⠀⠀⠀⢣⠹⣧
⠀⠀⠐⠀⠋⢸⠁⠀⠀⠀⠠⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠄⠀⠀⠀⠘⡇⠙⠠⠂
⠀⠀⣧⠀⢀⠀⠀⠀⠀⠀⠃⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠐⠀⠀⠀⠀⠁⡀⠀⣴
⠀⢠⠃⢠⡟⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⡀⠀⠀⠀⢻⡄⠙⡀
⠀⠈⠀⢸⠁⠀⠀⠀⡄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢁⠀⠀⠀⠘⡇⠀⢁
⠈⠁⠀⣿⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿⠀⠘
⣆⠀⠀⣿⠀⠀⢰⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⡆⠀⠀⣿
⣿⠂⠀⣿⠀⠀⠀⠀⢨⡏⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⠃⠀⠀⠀⠀⣿⠀⠸
⣿⠀⣦⠘⡆⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⢼⡤⠀⠀⠀⠀⠀⢰⠀⢸⠃⣴
⢿⣿⣿⡄⠱⠀⡇⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣦⠀⠈⠀⠏⢠⣿⣿
⠀⢿⣿⣷⡀⠈⢱⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣷⡄⠀⠀⣾⣿⡿
⠀⠘⢿⣆⠁⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠈⣴⡿⠃
⠀⠀⠀⠙⢷⣄⠀⠀⠈⠡⣄⣿⣶⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠀⡄⠀⠀⠀⠀⠀⠀⠀⣶⣿⣠⠌⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⢿⣶⣦⣀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠸⢾⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡷⠇⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⡈⠛⠄⣀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⣀⠠⠛⢁⣴⣾⣿⠟⠁
⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣿⣿⣿⣿⣦⣀⠀⠉⠀⣉⡄⣸⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⣇⣠⡁⠀⠉⠁⣀⣴⣿⣿⣿⣿⣶⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢌⡉⠻⢿⣿⣷⣶⣿⣿⣿⣬⣿⣶⠷⠶⠶⢾⠿⠿⡷⠶⠶⠾⣶⣿⣡⣿⣿⣾⣶⣾⣿⡿⠛⢉⡡⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠰⣦⡀⠈⢻⣏⣻⣿⣿⣡⣶⣶⣶⣷⣶⣶⣄⣠⣶⣶⣾⣶⣶⣶⣬⣿⣿⣟⣹⡟⠁⢀⣴⠌⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢰⣿⣄⡀⠘⠛⣿⣿⡟⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⠛⠁⢀⣬⣿⡄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠘⠇⠘⠿⠏⠀⢰⣿⠋⣿⣿⢶⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⡶⣶⣿⠙⣿⡆⠀⠹⠿⠃⠼⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⢀⣀⣾⣿⣇⡿⠿⠚⠋⢸⣿⣿⡿⠛⠛⢿⣿⣿⡇⠘⠓⠿⢿⣴⣿⣷⡀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣧⣤⠀⠀⣼⠟⠁⢀⣤⣤⡀⠈⠻⣧⠀⠀⣤⣼⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣶⣾⣏⣄⣴⡟⠛⠛⢻⣦⣠⣿⣷⣶⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠈⣁⠀⣿⡝⣯⣿⣿⣿⣿⣽⢻⣿⠀⣈⠁⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣾⣿⡄⢿⣇⣽⣿⣿⣿⣿⣧⣸⡿⢠⣿⣷⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⣌⣿⡉⠀⠀⠀⠀⢉⣿⣡⣿⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⣿⣿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⣿⠁⠀⠈⣿⣿⠿⠛⠛⠿⣿⣿⠁⠀⠈⣿⣿⣿⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣖⣒⣒⢲⣶⢶⣶⡶⠶⠶⠶⠶⠶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣾⡷⠀⠀⣶⣶
⠀⠀⣀⡀⠀⠤⠤⠀⠐⠒⠒⠂⠄⠠⠐⠀⡀⣿⣿⣿⣿⣿⡿⠿⠷⡀⠀⠀⠠⠀⠤⠤⠤⠤⠄⠐⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⡠⠘⠦⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠔⢈⣆⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠘⣿⠀⠀⠀⡀⠠⠐⠊⠉⠂⠜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣤⡤⡀⠀⠀⠀⠂⢀⣠⣾⣿⡄⠀⠠⣤⡄⠀⠀⠀⠀⢸
⠀⠀⠐⠟⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡇⢹⠀⠔⠀⠀⠀⣿⡄⠀⠀⠀⠀⢿⡀⢀⣘⠿⠷⠾⠿⠿⠿⠿⢓⣀⠀⣼⠃⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⣀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⢱⠀⠀⠀⢿⡸⣄⠀⠀⠀⠈⠳⢜⡿⣿⣶⣤⣤⣤⣴⣾⠿⣫⠞⠁⠀⠀⠀⢀⡞⣾
⠀⠀⠀⢰⡟⢸⡟⢸⡏⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⡿⢠⠀⠀⠀⠘⣷⣌⠳⣄⣀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⣼⠏
⠀⠀⠀⠠⡧⠸⡧⢸⡆⠀⠀⠀⣴⠀⠀⠐⠊⠉⠉⢷⣾⣀⣀⡄⠀⠘⢿⣷⣦⣭⣍⣉⢻⢿⠖⠢⠴⣿⠢⠄⠒⠞⡟⣉⣉⣩⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⡀⠀⠀⢰⠛⢷⠀⠀⡀⣹⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠠⣄⣠⠀⣂⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⡄⠀⠀⠀⠀⠈⣾⡷⠀⠀⠈⡠⡈⠀⠀⢞⣷⠅⠀⠀⠀⠠⣾⢟
⠀⠀⠀⠀⠀⠀⠀⢸⠿⡃⠠⠤⢼⣿⡷⣺⣷⣷⣶⣿⣷⠶⠶⠶⠞⠉⠁⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⠀⠁⠀⠀⢈⣿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣀⠴⠚⢻⣿⡿⠿⡿⠟⠋⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠘⣿⡇⠀⠈⢹⡖⠀⠀⠀⡰⣿⠁⠀⠀⠀⠀⣤⡄⠀⢰⣶⢷
⠀⠀⠀⣰⠁⢠⠞⠉⠣⡑⡠⠃⠀⠀⠀⠐⢎⠀⠐⢎⠀⠒⢎⠀⠀⠸⣝⠀⠀⠀⢸⢹⠀⠀⠀⠱⣧⣀⣸⡇⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠁
⠀⠀⣴⠃⡀⢸⠀⠀⠀⢀⠤⢤⡀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⢡⡸⢄⣀⣠⠞⡸⠀⠀⠀⢰⣹⣿⣿⡰⠀⢀⡀⠾⠀⢀⠀⠀⣀⣀
⠀⠈⠛⠉⠉⠉⠁⠀⠀⡏⠀⠀⠉⠩⠍⠁⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠙⠺⠛⠻⠊⠁⠀⠀⠀⠀⠛⠛⠛⠃⠀⠈⢛⣈⣈⠋⠀⣹⣿⣿
⠀⠀⣠⣤⣀⠀⠀⠀⡸⠀⠀⠤⠂⠀⠀⢲⣾⣿⣿⡄⠀⠀⠀⣠⣤⡀⠀⣠⣴⣤⣀⣀⣀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠉⠀⠈⠀⠀⣽⠿⣿
⠀⠀⣿⡏⠉⠑⢤⣀⣧⣤⣀⣀⣤⣴⣾⣿⣿⣿⣿⣷⣀⠴⠋⠉⢻⣿⢰⡿⣿⣿⣿⣿⢧⣴⡄⠀⠀⠀⡸⣿⣿⣇⡀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⢻⣧⠀⢀⣤⣈⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⢋⣡⣤⠀⢀⣾⠇⠀⠿⢻⣻⠟⣁⡈⠓⠁⠀⠀⠈⢉⣿⣿⡿⠟⠀⠀⠀⠐⠃⠀⠀⠃
⠀⠀⠀⠙⢷⡘⢿⣿⠉⠲⠤⣤⣄⣀⣀⣤⣠⠤⠒⢹⣿⠟⣠⡿⠃⠀⠀⠀⠘⠋⠀⠙⠃⢰⠀⠀⠀⢰⣌⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠉⠳⠬⣦⣅⡤⠂⢄⠀⠀⠤⠀⣄⣼⣾⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⣿⡏⠁⠀⠀⢹⣿⣿⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⣟⣶⣤⠀⢀⣴⣺⣷⡎⡇⠀⠀⠀⠀⠀⠀⠀⠋⠰⣀⠀⠀⠀⠀⠀⠀⠀⢐⠓⣿⣿⣿⡷⠀⠀⢸⠿⠿⠿⠯⠿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠙⣤⣉⠛⠛⡅⡈⠛⠋⣉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣠⠰⠀⠀⠠⣘⣿⣿⣧⡄⠀⠀⢹⣶⣦⣤⣤⣶⣿
⠀⠀⣀⠴⣯⣬⡷⢄⠀⢸⡉⠳⠐⠀⠐⠀⠛⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⠂⠐⠀⡀⠀⠀⢡⣿⣿⡟⠋⠀⠀⢸⣿⣯⣿⣿⣿⣿
⠀⠀⠘⡾⣭⣭⠷⡚⠀⠈⠳⣤⠁⢤⠄⢡⡤⠞⠀⠀⠀⢰⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣾⠃⠀⠠⠮⣿⣿⣿⡆⠀⢠⠟⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠐⠚⠓⠒⠀⠀⠀⠀⠈⢧⣀⣀⠞⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣄⣀⣀⣠⣴⠞⠛⠋⠁⠀⠀⠈⠰⠿⠿⠏⠀⠀⠀⢀⣤⣤⣴⣴⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⣿⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣿
⠀⠀⢸⡏⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⢹⡇
⠀⠀⢸⡇⠀⡈⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢁⠀⢸⡇
⠀⠀⢸⣧⢰⠀⢹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠀⡆⣼⡇
⠀⠀⠀⣿⡀⡆⢆⢻⣿⣿⣿⣿⣦⣤⣀⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⣤⣴⣿⣿⣿⣿⡿⡰⢰⢀⣿
⠀⠀⠀⢹⣧⡜⡌⢇⢻⣎⢿⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣿⣿⣿⣿⣿⡿⣿⡟⡸⢡⢣⣼⡏
⠀⠀⠀⠀⢿⣷⡼⢎⠳⡙⢷⣝⢿⣿⣿⠏⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠹⣿⣿⡿⣿⡿⢫⠞⡱⢧⣾⡿
⠀⠀⠀⠀⠈⢻⣿⡜⢆⢌⢢⣹⣿⠛⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠛⣿⣏⡔⡡⡰⢣⣿⡟⠁
⠀⠀⠀⠀⠀⣸⠙⠿⠛⠛⠟⠋⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠙⠻⠛⠛⠿⠋⣇
⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⣀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⡻⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠶⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⢇⡄⠀⣰⠆⠀⠀⣺⠁⠀⡀⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠹⠿⠦⢤⣴⡞⠀⠙⠉⠁⠀⠀⢸⠃⡆⠘⣷⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⢸⣇⣧⠐⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢢⠀⠀⠀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠹⣿⣿⣶⣧⡟
⠐⠀⠀⠀⠀⠀⠀⣿⣁⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠛⠛⠋⠀⠀⠀⠘⠗
⣸⠇⠠⠄⠀⠁⢀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⢰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡦
⠁⠀⣧⠀⠋⠀⡀⢠⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣀⠀⠦⠄⠀⠀⠀⠁⠀⠀⠀⢀⠀⡀⠀⠀⠈⠉
⠀⠀⢸⡖⠂⠈⠀⡼⠀⠀⠀⠀⠀⠀⠰⢼⣦⠀⢸⡇⠀⠀⢀⠇⠀⠀⠀⢨⠁⢢⢠⠀⠀⠀⢸⣀⠀⠀⠸⠶⠃
⠀⠀⠀⠀⠰⢪⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⢾⣧⣼⠷⠃⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⡤⠚⠋⠀⠀⡆⢠
⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⠘⣧⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⢛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡅⠀⠀⢸⠀⠀⠀⠈⠋⢀⠠⠀⠀⠰⣄⡴⠀⠓⠂⢀⣀⣀⠰⠁
⠸⡤⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⡀⠀⣆⣸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢰⠀⡀⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢠⡄
⡀⠀⣀⡀⠀⠀⠀⢠⠂⠀⠀⠀⣠⣀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠈⢢
⠧⢀⠈⣇⠀⠘⠠⡇⠀⠀⠀⠀⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⡔⠀⠄⠀⠀⠀⠀⢦
⢷⣸⠀⠹⠃⠀⠀⠁⠀⠀⢠⢏⡆⢰⡞⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⢀⠠⡁⠀⠀⠀⠀⠀⡀⠀⢸⠇
⢀⠹⠀⢠⡄⠀⠀⠀⠀⠀⣹⣼⡆⣾⢰⠇⠀⠀⠀⠀⠀⠀⠀⠘⣇⣧⢡⠀⣇⡿⠀⠀⠀⠀⠀⠀⣘⠟⠀⢀⠠⠀⠀⠙⣿
⠾⠃⠀⡾⠛⠀⠀⠀⠀⠀⠻⣿⣷⣷⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠳⠷⠋⠁⠀⠀⠀⠀⠀⢀⡀⠈⠀⣀⠀⠂⠀⠀⢻
⡅⠀⢸⠃⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠰⠀⠀⢠⡜⠀⠀⠈
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣀⣥⠞
⠐⠀⠚⠛⠁⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣼⠀⠀⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣆⡄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⠴⠀⠀⠀⠀⣴⣿⡶⠆⠀⠀⠀⠀⠀⢈⣽⠿⠋⠀⠀⠀⠀⠀⠺⣿⣟⠷⠃
⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⡀⠀⠀⠀⣠⡒⠉⡀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⡀⠀⠀⣰⣯⡉⢛⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢠⡀⢠⣿⣧⢨⠀⠀⣀⠀⠀⠀⣿⣿⡄⠍⠃⠀⣹⡀⢰⣿⣧⠀⡾⠀⠌⣶⡀
⠀⠀⠀⠐⠒⠒⠛⣿⣷⣞⠓⠚⠳⡞⠛⢳⣶⣶⡄⠛⠋⠒⠒⠚⠻⣿⣾⣷⠖⠻⠗⠚⠛⢻⣾⣇⣠⡐⠛⠁
⠀⠀⠀⠀⢀⣀⣀⣿⣿⣏⣤⣀⣀⣇⣀⣿⣿⣻⣹⡄⢤⣀⠀⣀⣰⣿⣿⣧⣇⢀⡁⠀⠀⣾⣷⣿⢻⠄
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢹⣿⡏⠉⡏⢹⣿⣿⠉⠉⠛⣺⣿⡅⠀⢹⣿⡟⢻⡇⠈⠁⠀⣸⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠤⢤⣿⣿⠧⠬⣿⡇⠴⡧⢼⣿⣿⠤⠤⣴⣿⣿⣷⠤⢼⣿⡧⢼⣏⣴⡦⣴⣿⣿⠇⠠⢹⣇
⠀⠀⠀⠀⠀⣸⣿⣿⡄⢀⣿⣗⠀⠁⠸⣿⣿⡄⣸⣿⡇⠻⠋⠀⢸⠿⠀⠈⣿⣿⡀⣿⣿⣿⠀⠀⢸⡏
⠀⠀⠀⠉⢉⣿⣿⣿⠉⢹⣿⣿⠈⡏⢹⣿⣿⡋⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⣿⣿⡋⣿⣿⡏⠉⠉⠉⡇
⠀⠀⠀⠠⠼⠿⡿⠛⠤⠼⠿⡿⠶⡧⠼⠯⢿⠻⠤⠤⠤⠠⠤⠤⢿⡇⠤⠤⠿⠿⠷⢿⡿⠅⠀⠤⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠁⣀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⣀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠉⠁⠉⠀⠈⠁⠈⠁⠀⠈⠀⠉⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⢠⢊⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⢄
⠀⠀⠀⠀⢠⠃⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⠆
⠀⠀⠀⠀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⡇⢸⣏⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠐⠢⣤⠔⠒⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⢘
⠀⠀⠀⠀⢃⠸⣿⡎⠓⠀⠀⠐⠂⠈⠉⣹⠀⠉⠀⠀⠀⠈⢀⠀⠀⠊⠁⢋⡉⠉⠐⠒⠀⠀⠒⢡⣿⡇⡸
⠀⠀⠀⠀⠀⠓⢿⣿⣦⣤⣤⣴⣶⣶⣿⣿⠆⠀⠀⠀⠘⢿⠃⠀⠀⠀⠠⣿⣿⣷⣶⣶⣤⣤⣾⣿⡿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣓⠛⠻⠿⢿⣿⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⡿⠿⠿⠛⣛⠋⠁
⠀⠀⠀⠀⠀⠀⠰⠖⠋⠉⠻⢷⣶⣾⡗⢾⠳⢦⣀⡄⢀⣿⡀⢠⣀⣠⠞⡿⠊⣷⣶⡾⠟⠋⠉⢳⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢨⣧⠀⠙⠻⡿⠀⠀⠀⠀⠀⢻⠟⠋⠀⣸⣍⠐⠀⠤⠄⠘⣁⣤⡄⢐⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⡆⢠⡟⠄⠀⠀⠀⠀⠿⡄⢠⣿⣿⣿⡄⠀⠀⠀⠈⣟⠫⡈⣉⠈⠓⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢿⠸⣿⣷⣈⣷⣄⠀⠀⠀⢠⣾⣃⣾⣿⠇⣿⠹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⣿⣿⠋⣉⡹⢧⣤⡴⠏⣉⠉⢿⣿⠀⠁⠀⠳⠤⠭⢌⢢⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢰⣿⣿⣀⡸⣷⣶⣿⣶⣾⡏⣀⣼⣿⡇⡀⢀⠀⠀⠀⠀⢮⡓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⣾⣎⠟⣿⣠⣿⣿⡿⠿⢿⣿⣿⣥⣿⠫⢣⣷⡞⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠹⣿⣦⢹⣿⣿⣏⣀⣀⣀⣹⣿⣿⡿⣴⣿⠟⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠮⣿⣇⠉⢿⣿⣿⣿⣿⣿⣿⠋⢰⣿⠥⠊⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠈⠻⣦⣸⣿⣿⣿⣿⣿⡏⣴⠟⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠳⢦⡈⢳⣿⣿⣿⣿⣿⡞⢁⡤⠞⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠹⣄⡀⠀⢀⣿⣄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⣨⣉⣡⡬⠭⠯⢍⣓⡒⠀⠀⠸⣤⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠤⠤⠠⠤⠶⣶⡶⣖⣀⡀⠀⢻⣿⣷⠄⠀⢀⣠⠈⠉⠉⠒⠒⠂
⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠘⠉⠛⠻⢆⣸⠏⢀⣀⣴⣾⣋
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⡄⠀⠀⢸⡟⠀⣿⣿⣿⠫
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠸⣧⠀⠀⠘⡟⠀⠘⣿⠯
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⠀⠀⠙⣦⣦⣄⠡⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠠⣤⡈⠻⣷⢿⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⢙⠿⣦⣷⠀⠀⠀⠀⠀⠀⠀⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣧⣦⣠⡀⠂⢺⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃⠃⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣠⡄⠂⠀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣰⣿⢀⡆⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣼⠋⠀⠘⠄⠙⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⠀⠀⠀⡼⠛⠿⠟⠋⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⠀⠀⣀⣠⣤⣾⣿⣯⣷⣞⡿⣿⠀⠀⠐⣡⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠀⣴⣏⠀⣴⣾⣷⣾⡿⣿⠛⠻⣿⡿⣁⣿⣆⠀⣎⠀⠀⠀⠀⠘⠀⠀⣼⣻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣷⣿⢿⣼⣿⣿⣿⣿⠋⠋⠒⠶⠿⠟⠛⠋⢹⠀⠹⡄⠐⣄⢳⢁⣼⠏⠙⢿⣜⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⠟⣾⣿⣿⣿⡟⣿⠘⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠸⡆⢸⡎⢈⠃⠀⠀⠀⣿⠁
⠀⠀⠀⠀⣴⠟⠻⣿⣉⣿⣿⣿⣿⣷⣿⡿⢻⡟⢻⣿⣭⠿⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠗⠉⠀⡀⢀⡀⣿
⠀⠀⠀⢸⣿⠀⠀⢹⣿⣿⡿⠿⠟⡛⣋⣀⠟⢁⣸⣿⣿⠀⠘⣦⢻⡿⠓⠿⢳⡶⠆⠀⠀⡆⠀⠘⢆⡀⢠⠁⠀⠀⣿
⠀⠀⠀⠸⣿⣦⣶⣿⣿⣿⣷⣤⣤⡿⠯⡉⣀⣾⠈⠻⣿⣔⣛⡁⠈⠐⢦⣆⣈⣧⠀⠀⢤⣷⠐⠂⢨⣿⣿⠂⠆⢰⣿
⠀⠀⠀⠀⣸⣹⣹⣿⠿⠥⠾⠷⠦⠶⣾⣿⣌⡙⠛⠚⠚⠛⠛⠓⠒⠂⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⢠⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⢆⡞⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣽⣿⣿⣧⢼⣿⠗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣴⣶⣶⣤⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣴⢊⣵⣟⣵⣾⡿⣋⢠⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⣿⡿⢉⢹⡿⣠⣆⡀⡀
⠀⠀⠀⠀⠀⠀⢀⣼⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⣿⣿⣁⡛⡍⢀⣃⡤
⠀⠀⠀⠀⣴⣶⣿⠃⢸⡇⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⡄⠀⠀⠀⠀⢰⣃⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣀⣋⣱⠦
⠀⠀⣤⣾⣿⡟⠁⠀⣼⣷⣿⣿⣿⣿⣿⣿⣏⡀⢀⣷⣿⡂⠀⠀⠀⢸⣿⣿⣿⡄⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡽⡵⣊
⣤⣿⣿⡿⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣤⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠋⠉⣵⣴⣯⠞⢹⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣰⣿⠟⢹⠏⠉⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷
⣩⣶⣿⣿⣿⠟⠀⢻⣿⣿⣿⣽⡿⣿⣿⡿⢿⣿⡿⢻⣿⠀⠀⣴⣿⠛⣠⣿⡇⢰⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⡿⠀⠀⢸⣻⣿⣿⣿⢷⣼⠇⡹⡢⠘⣇⠀⡃⠀⡐⣈⣿⣴⣟⣿⣀⢺⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢈⣵⡃⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⣱⣆⠛⠰⣷⣴⣿⣿⡿⣩⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢷⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⢸⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⢿⢿⠀⠀⡀⠀⠸⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⢠⣿⣿⣷⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠈⠇⠐⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⠏⠀⡿
⠀⢻⣿⣿⡆⠀⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠰⣽⣿⣿⣿⡿⠁⠙⠛⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠌⠈⠀⢀
⠀⠠⠈⢿⣧⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢻⣿⣿⡟⣸⡟⠂⠛⡛
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣿⣿⡿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⠑⢀⠔⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠘⠇⠀⠈⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠑⠀⠈⠀⠀⠀⠀⠀⠙⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠠⠀⠘⠆⠀⠀⠈⠇⠀⠀⠿⢸⡇⠀⠀⣄⠀⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠞⠀⡇⠀⠀⡁⢠⡏⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⡄⠑⣤⠀⢀⠀⢰⠀⠀⠀⠀⡇⢸⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠛⠀⠈⠀⢠⠀⠀⠀⠀⠁⢸⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀⠀⠀⠀⠤
⠀⠀⠀⠀⠂⠀⣿⠀⠀⢸⠇⢠⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⢤⡀⠀⠀⠀⠀⡄⠀⡇⠀⠠⠀⡇⠈⢺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⣈⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⣠⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠸⣇⠀⠀⢿⠀⠄⠀⠀⠀⠀⣆⠀⠀⣠⡟⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠳⣄⡈⠃⠤⣈⡲⢀⡀⠀⠀⠀⠁⠀⠾⢋⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⢀⣌⣀⣀⣀⡀
⠀⠀⠀⠐⠃⠘⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⣾⠿⠿⠿⠿⠟⠿⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⡿⠾⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣄⣲⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠋⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⢀⡀⠊⠀⠈⠙⢦⡀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠰⠀⠀⠀⠀⣾⣿⡏⣴⡆⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⡧⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡶
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⡿⢀⠀⠀⠀⠀⣿⡟⢡⣿⡇⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⢸⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⢿⣿⠟⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⢀⠀⣿⡄⠀⢹⣿⡇⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⣿⣿⠀⡄⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠠
⠆⠀⠀⣿⣧⣿⣇⠀⠈⣿⡇⠀⠘⠀⠀⠀⣿⡀⠸⣿⠀⠈⣿⠇⢰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠀⠀⠀⠀⠀⡃
⡄⠀⠀⢻⣿⢿⣿⠀⢁⢻⣧⠀⢸⠀⠀⢰⡟⠃⢸⡟⠀⢰⡟⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⡀⠰⠀⠀⠀⠀⢰⡇
⠰⡆⠁⣸⣿⣆⠙⣧⠘⠀⣿⡀⢸⠄⠀⢸⡇⠀⣿⠃⠀⢾⠁⣴⣿⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠸⣄⠛⢿⣿⣧⡈⢣⠀⠸⢧⠈⠀⠆⢸⡇⢰⡟⠀⠀⠀⣴⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠈⠂⡀⠙⢿⣦⠠⠖⠀⠈⠀⠠⠀⠀⠁⡞⠃⢀⣠⡾⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡀⠀⠈⢦⣀⠙⢷⣄⡀⠈⢤⠀⠀⠰⠂⠀⠀⠛⠉⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠁⠐⠒⠈⠉⠱⠦⣬⠻⠷⡄⠡⠀⠰⠀⢀⣤⠶⠚⠁⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣶⣿⣶⣤⣤⣤⣼⣿⣿⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣾⣿⣀⢀⣀⣨⣿⣿⣷⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⠻⠿⠿⠿⠿⠿⠿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⠓⠘⠛⠈⠉⠃⠐⠀⠈⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣄⣱⠖⢀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⣄⡠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡖⢠⠚⠑⠶⠚⣯⣹⡖⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⢢⠜⠉⠀⠈⠀⠀⠠⠀⠀⠈⠁⠐⠠
⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⡐⠈⠀⠁⠠⠎⣿⠙⣏⡉⠳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⠡⡀
⠀⠀⠀⠀⠰⠃⠀⠈⠀⠰⠁⠈⠁⠀⣉⠰⡏⠘⠃⠈⠱⠞⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠆
⠀⠀⠀⠰⠂⢰⡁⠀⠸⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⢠⡆⠀⡉⠛⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠃⠀⢧⡤⠐⠇⠐⠁⠀⠀⠀⡄⠀⢀⠀⠀⠀⠠⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀
⠀⠀⢠⡄⠀⠀⠀⠀⢠⠀⠆⠀⢢⠀⠁⠀⠈⡀⠐⠂⠀⠁⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⢁⠀⠰⠏⠀⠈⢃⠀⠀⠀⢀⠀⠀⠀⠱⠀⠀⠀⠀⠉⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠦
⠀⠀⠀⠘⠃⠀⠰⡶⠀⠈⠡⣄⠀⠀⠃⠀⠀⠰⠀⠶⠘⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⢀⠚⠃⠀⠀⡇⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡄⠈⠂⠀⠀⠘⢄⡀⠀⡄⠀⠰⢀⡀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠰⡷⠀⠀⠀⠀⠀⠀⡇⠀⡀⠈⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⠀⠈⠻⠀⠀⢀⣁⣈⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣴⣾⣿⣷⣶⣷⣶⣿⣿⠿⡿⠟⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣦⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠀⠀⠉⠉⠁

⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⢤⢀⣿⣿⣿⣿⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⣤⣄⣀⣀⡀⠐⣶⣞⣹⣀⣀⣤⣤⣤⣤⣾⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⢛⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⠿⣿⣃⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣱⣷⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⣼⣦⠶⠶⠶⠶⢤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣦⡀⠀⠀⠀⢀⣠⣶⡿⢿⡿⣿⣯⣿⠀⠀⠀⣀⠀⠀⣀⠉⢉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣼⣿⣿⣦⣶⣶⣿⠟⠁⠌⠘⣿⢻⣿⣿⠁⠀⠀⠐⢬⣿⣾⣿⣶⣿⣷⣦⣙⠿⣷⣄
⠀⠀⠀⠀⠀⢠⡶⣟⠿⠟⢿⣍⣿⣿⣿⣿⡏⠀⢸⠀⡄⠹⣾⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣿⣿⡻⣷⣌⢻⣷⣄
⠀⠀⠀⠀⠐⠏⡴⠁⢀⣴⠿⡿⠟⠿⣻⣿⡇⠀⠘⢆⡇⡀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠘⢿⣿⡌⢻⣷⡈⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⡁⠀⣠⣴⠟⣿⡏⢧⢠⡀⠈⠿⣦⣠⡀⡄⠉⠻⢦⣄⠠⡀⢄⠀⠈⢿⣿⡀⠹⣧⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠞⠉⠀⢰⡟⣿⡘⢧⣷⡀⠀⠈⠻⣿⣷⡀⢶⣄⠙⢷⣽⡌⢷⡀⠈⣿⣧⠀⠙⠆⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠘⣷⣌⠷⣄⠙⠿⣦⣀⠀⠀⠙⠿⣦⡻⣷⡄⢻⣿⡘⣷⠀⢸⣿⡀⠀⠀⠀⢀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠙⠈⠓⠼⣳⣦⣠⣯⣛⡶⢄⣀⠈⠙⠿⣿⣧⢻⣇⢿⡇⠀⣿⡇⢸⣇⠀⢸⡆⢹⡿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠷⢮⣕⠢⡀⠈⢻⣾⣿⣼⣿⠀⢻⣿⠀⣿⡄⠘⣿⢸⣷⢿⣷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡟⣧⡀⠉⠳⣌⢦⠀⢹⣿⣿⣿⠀⢸⣿⢠⣿⣧⠀⣿⣇⣿⢺⣿⣇⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠃⠘⣧⠀⠀⠹⣯⡇⠀⣿⣿⣿⠀⢸⣿⣼⣿⣿⢠⣿⣿⣿⣸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⢣⠀⠀⠀⠀⠃⠀⠀⠀⠀⣶⠏⠀⠀⢻⡀⠀⠀⣿⣿⠀⣿⣿⡏⠀⣾⢿⣿⣿⣿⣼⠋⣿⡏⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠇⠀⠀⢸⣿⢠⣿⣿⠃⢠⣿⢠⡿⠈⣿⡏⠀⣿⢱⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣾⣏⣾⣿⡏⠀⣾⠇⠘⠁⠀⡿⢀⠀⢁⣾⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⠀⠀⣾⠀⠀⠀⡄⠀⢀⠀⣠⣾⡏⢀⠀⢀⣿⣿⣿⡟⠀⣼⡟⠀⠀⠀⠀⠁⣾⢀⣾⣿⣿⣿⠃⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⠀⠀⣿⠀⠀⢠⡇⠀⣸⡿⠋⣾⠁⣼⠀⠸⢣⣿⡿⠁⣼⡿⠀⡄⠀⣶⠀⣼⣇⡾⢻⣿⣿⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⠀⠀⢸⠀⢠⠟⠀⣼⡇⢠⡇⠀⢠⣿⣿⠃⣼⣿⢁⣼⠁⣼⣏⣼⣿⠟⣱⣿⣿⣿⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⡇⠀⢸⣇⡞⠀⢠⣿⠁⢸⡇⢀⣿⣿⠃⣼⡿⢃⣾⢃⣼⣿⣿⠟⢁⣾⠟⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠛⠀⠀⠙⠋⠀⠀⣿⠀⠀⣸⡏⠀⢸⡇⢸⣿⢇⣼⡿⢡⣾⣣⣾⣿⠟⢁⣴⠟⢁⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⡀⢹⡇⠀⣿⡇⠀⢸⣧⣿⣿⣼⣿⣵⣿⣿⠿⠋⣠⡴⠛⠁⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡀⠀⠀⠀⣸⠟⠿⢸⠂⠀⣿⡇⠀⣮⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠁⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠛⠀⠀⠀⠛⠀⠀⡿⠀⠀⢹⣇⠰⢹⣿⣿⡟⠋⠁⠀⠀⠀⠀⢀⠔⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣄⣀⣀⣠⣶⠾⠃⠀⠀⠈⣿⡀⡞⣿⡟⠃⣀⡴⢀⠀⣠⡶⢁⣼⢟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣸⠿⣧⢸⣹⣿⣾⣯⡖⣡⠞⢁⣴⠟⣡⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣧⣿⣿⣿⣿⠞⢁⣴⠟⢁⡾⠋⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⠻⣿⣿⣿⣥⣴⠟⢁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⡄⡘⢿⣿⣿⠀⣴⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣷⠋⣹⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⣿⣿⣿⣰⣿⣿⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⣄⠀⠀⢃⠀⠀⠀⠈⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠲⣶⣾⣿⣿⣿⣿⣧⠀⠀⠀⠉⠀⠀⠐⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣴⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠐⠋⠀⢸⡖⡄⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⣀⣴⣦⡀⠀⠀⠀⠘⣿⣷⣦⡄
⠴⠖⠀⠀⠀⠀⠘⠛⢠⠀⢀⣠⣤⣄⡀⣱⣿⣯⣶⣾⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣶⣔⡄⠀⠈⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣁⡉⠉⠙⠛⠛⠻⢿⣿⠿⠿⠿⠷⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣿⣿⡛⠂⠀⠀⠀⠀⠀⢀⣶⡀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣷⣶⣦⣤⣶⣾⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠲⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠏⠉⠉⠉⠉⠁⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠛⢿⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠉⠻⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠽⠞⠋⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⣄⣾⡇⠀⠀⠀⠀⠠⣄⢺⢀⣼⡀⢰⡀
⠀⠀⠀⠀⡀⠼⠛⠀⠉⠳⣄⠀⢠⠄⠀⣹⠌⠘⠉⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡀⠀⢻⠀⢠⡇⢠⣶⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣤⠿⣴⣿⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢩⣌⣫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡄⠀⠀⠨⢃⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣰⡾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⠀⠈⢸⡇⠙⢳⣶⠠⠄⠒⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⢠⠃⠀⠈⠁⠐⡄⠀⠀⠀⣰⠆⠻⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⡰⢃⡜⠀⠀⠀⢀⣠⡀⠄⠀⠋⢂⠀⠙⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠣⣿⠁⠀⠀⡀⣿⠉⠒⠀⠀⠀⢸⣤⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣇⢠⡆⠘⣻⣁⣀⠀⠀⢀⣠⣿⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡟⢻⢀⡞⠛⠛⠛⠛⠻⣿⠛⡟⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡄⠘⣸⡇⠀⠀⠀⠀⠀⠙⢧⣷⡌⢿⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⣿⡆⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠛⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀
⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⡀⠀⠀⠼⠁⠀⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⢐
⠀⠀⠀⠁⠀⠀⡀⣤⣀⡀⠀⣡⣄⣀⠐⠀⠀⠀⠀⠈⠀⠠⠘⠛⠈⠋⠀⠀⠀⠀⠀⠂⠐⢀⠂⠐⠶⠢⠶⠀⠄⡠⠀⠤⡀⠀⠀⠘
⠀⠀⠀⡆⠀⠁⢈⠀⢈⠀⠀⡀⠀⣸⠐⠂⠐⡆⠀⠲⠖⠶⠤⢠⢀⡤⠴⣀⡀⠀⠀⠀⢀⡀⢋⣉⠛⠋⠛⣛⡛⠀⠀⠀⠈⠂⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠦⢼⠀⠀⡇⠴⢸⣿⡷⢸⠉⡇⠸⠑⢀⠀⣼⠈⢹⠈⠯⠝⠀⠀⠀⠛⠛⠀⠾⠀⠉⠁⠺⠄⠀⠀⠐⠛⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠃⠀⠘⠀⠀⠈⠀⠁⠀⠀⠘⠀⠙⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡈⣿⣦⡀⠀⣀⣤⣤⣤⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⣀⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣶⣞⡫⠽⢛⣛⣶⣀
⠀⠀⠀⠀⠀⠀⠀⣼⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣛⣉⠁⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠘⢉⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠋⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠮⢭⡭⠭⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⠭⠲⣬⣓⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷⢶⣶⣤⡉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡿⠄⢈⠋⠻⣷⣶⠤⠔
⠀⠀⠀⠀⠀⠀⠀⣨⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⠒⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣿⠿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠬⠍⠉⠉
⠀⠀⠀⠀⠀⢰⣿⣏⣄⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢋⡴⣻⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠛⠡⠊⠰⠃⣸⣿⣿⣿⣿⣿⢟⣿⣿⣿⡿⣿⢿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣿⣿⣟⣁⣀⣀⣠⣤⣾⣿⣿⣿⡿⣫⣯⣾⣿⡿⢋⡼⢃⠞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⠿⠟⣛⣩⣭⣭⣶⣿⣿⣿⣿⣿⣿⣯⣥⣾⣿⠿⠛⢉⡴⠟⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣥⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣏⣉⣁⣀⣤⠴⠚⠉⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⣿⠿⠛⣉⣡⠤⠤⠶⠒⠒⠒⠲⠖⠶⠖⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢞⠽⠿⠛⠁⠈⠉

⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷
⠀⠀⠘⣿⣿⣗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣿⣿⡏
⠀⠀⠀⠈⢿⣿⣷⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣵⣿⣿⠟
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠻⢿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⣸⠃⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢳
⣿⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣦
⠹⣷⣤⣾⢏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢿⣠
⠀⢻⣿⡇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣏
⠀⠸⣿⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⢻
⠀⠀⠻⣇⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢸⡇⠀⢸⣿⣿⢀
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣾⠿⣣⡼
⠀⠀⠀⠀⠈⠻⣿⣶⣄⡀⢸⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣟⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⣀⡙⠻⢿⣷⣿⣷⣤⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⡿⠟⣋⣥⡄
⠀⠀⠀⠀⠀⢸⡿⠟⠳⢄⠈⠙⠻⣿⣯⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢉⡴⠋⠁⠀⠃
⠀⠀⠀⠀⠀⠸⣷⠀⠀⣀⠉⠂⢄⠀⠙⣿⣏⣀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢿⡏⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⠿⠀⠈⠇⠀⠛⠉⠀⠴⠀⠲⠶⠀⢱⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢉⣁⡀⠀⣠⡖⢠⣤⠄⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠄⠀⠀⢠⠀⠀⠀⠉⠀⢹⣿⣿⠋⠙⠿⣿⣿⠀⣠⡅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡻⣿⡿⠄⠀⢸⣿⠇⠀⠀⠀⠙⢿⡄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⠁⠀⢀⣀⣀⣀⣀⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⣤⡄⢀⠀⠀⠈⣻⣿⣿⣿⡋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠛⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠘⡧⠀⠰⡞⠀⠀⠁⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⣴⣍⠀⠀⠀⣠⣴⣦⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠿⢫⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣛⣩⣽⣿⣿⣦⡝⠯⢻⣶⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⣠⣤⣤⣤⣴⣾⠿⢋⣸⡟⡤⠛⠁⠀⠀⠙⣯⠳⢀⣿⣮⣻⡉⠛⠛⠿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢉⣁⡯⢥⡼⠖⣾⣿⣿⣾⠁⠀⠀⠀⠀⠀⣿⠀⢸⠋⠛⢿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠛⠋⠁⠀⠈⠙⠿⣿⣤⡄⠀⣀⣀⣠⡟⠀⣼⠀⠀⠀⠈⢶⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⢰⡟⢰⡿⠷⢤⣀⠀⠀⠻⣄⠁⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡄⠀⡜⠀⢻⡇⠀⠀⠉⠓⠀⠀⠘⣶⣴⠗⠺⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⢻⣶⡇⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢹⢰⣟⠉⠛⣄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢷⠀⢸⡼⠛⠡⣄⡈⢷⡀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄⠲⣤⡀⠀⡎⡇⢸⡏⠀⠀⠀⠁⠘⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠙⢄⠈⠹⣿⢳⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⡹⡆⠀⠀⠀⠀⠀⠀⠀⠨⠳⣾⠿⣸⣧⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢘⣿⣄⠀⠀⠀⠀⠀⠀⠀⢤⡋⠀⡻⠉⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢍⡁⠙⣦⠀⠈⠓⢦⣄⣀⣼⠇⣠⠇⠀⠀⢨⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠨⣷⡀⠀⠀⠈⠙⡏⠐⣿⠦⠤⠄⠀⢿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣕⣢⣀⠀⢰⡇⢀⡇⠀⠀⠀⠀⠀⠸⣿⣿⣶⡄⣀⣶⣚⡻⠿⡛⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⢿⢻⣿⣾⡇⠈⣇⠀⠀⠀⠀⠀⡴⣻⡟⠛⣩⣽⣦⣶⣶⣶⡶⠛⠷⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢾⣷⣎⣷⠀⡗⢿⣦⣤⣤⣶⠯⢴⣿⣤⡿⠟⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢾⣶⣽⣿⣿⣷⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢆⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣄⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⣷⣄⣿⠄
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣾⠃⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⣿⣟⠋⠙⠛⠋⠁
⠀⠀⠀⢄⣀⠴⠖⠲⢼⣿⣷⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢼⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⢠⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡈⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣸⡇⠀⣠⢔⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠒⢄⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡟⣡⡾⠋⠀⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠢⢮⣠⠄⢦⡀⠀⣿⠀⢀⣀⣀⣀⠀⢀⣿⣿⣿⣿⡟⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⣿⡦⠀⣿⣤⣻⡷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⣿⣷⡀⠈⠃⢀⠉⣟⣠⣏⣡⠼⠻⣿⠀⠉⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠹⣿⣿⣿⣯⣾⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⡀⢻⣿⣿⣿⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠇⢋⣿⣟⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢰⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⠘⢻⡿⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠉⠻⢋⣼⣆⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣀⣈⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠁⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⢿⣿⣿⣿⣿⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⠿⢟⢫⡟⠋⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠱⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⡌⠁⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠋⠀⠀⢂⡀⠀⠈⠀⠛⣿⠿⠿⣿⣿⣟⠉⠙⣿⡿⠋⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢙⠤⠐⣙⣿⡏⢡⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⠷⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣬⡽⣭⣭⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣤⠀⢠⡉⡇⠈⣹⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠠⠘⠛⠣⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⡜⠛⠓⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣹⣿⣿⣶⣶⣾⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣿⣿⣿⣿⣿⡟⡿⣿⣽⢿⣿⠟⣡⢿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣥⢢⡜⣿⣿⣿⣿⣿⣿⣷⣾⣿⣬⣤⣾⣿⣿⣻⣿⣿⣿⡄⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⣿⣾⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣾⣿⣿⣿⣿⣿⣧⣸⣆⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠊⢉⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡠⡠⢠⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⡏⢻⣿⣿⣿⣴⣬⣿⣿⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠐⡜⠁⡄⠀⣠⣿⣧⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⡿⠹⣿⣿⣿⣿⡿⠘⢧⡄
⠀⠀⠀⠀⠈⣴⡁⠀⠆⠀⠉⢹⣿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣺⣿⣿⣿⣿⣿⣿⣿⠇⠀⢹⣿⣿⣷⣶⣦⣤⡄
⠀⠀⠀⠀⠸⣷⠻⣬⠐⠂⣀⠸⣷⠀⠀⢸⣿⠿⠿⠿⢿⣿⣿⣿⡆⡏⢻⡿⣻⣿⣿⣿⣿⡟⠀⠀⢀⣿⣿⡿⠶⠖⣲⡇
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣛⣪⣑⣨⣀⣈⣾⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣯⣭⣵⢂⣼⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⢿⣍⠲⣝⣻⡿⠿⠿⣿⣿⣯⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⢸⠟⠁⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣮⣵⣾⣿⣿⣿⠁⠀⠀⠛⢿⣿⣿⣐⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⡷⠶⢚⣫⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣴⡿⣻⣿⣿⣿⣿⣿⣿⣷⣿⡀⠤⣀⡳⣾⣿⣭⣽⡛⢿⣿⣿⣿⣿⣿⡀⠀⡼⠡⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣥⣿⣿⣷⣿⣿⣿⣉⢠⡅⠀⠘⡿⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣈⢹⣿⡄⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠆⡐⠁
⠀⠀⠀⠀⠀⠀⣖⣦⡀⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣧⣼⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⢏⣁⣴⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⢸⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⣾⡋⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⢰⡟⠀⠀⠀⣴⡟⠛⢧⡤⠖⠒⣉⣡⡤⠤⠶⠶⠶⠤⠤⣬⣉⡓⠢⢤⣀
⠀⠀⢀⣿⠁⠀⢀⣾⡟⠀⠐⢉⡤⠒⠉⠉⣿⠀⠀⠀⢀⣤⣶⣶⣶⣬⡉⠓⠦⣈⠳⢤⡀
⠀⠀⣸⡏⠀⢀⣾⡿⠀⢀⢴⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡏⠀⢹⣿⣇⠀⠀⠈⠳⣄⠙⢦
⠀⢠⣿⠁⢀⣾⠟⢁⠔⠁⣸⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⠈⠳⣄⠳⡄
⠀⣼⡟⠀⣼⠏⢠⣾⠿⠿⠿⠿⣿⡿⠿⠿⣿⡿⠿⠿⢿⣿⡿⠿⢿⣿⣿⠿⡿⠿⠿⣶⣄⠙⣆⠹⡄
⢠⣿⡇⢰⡏⢠⠏⣿⡀⠀⠀⠀⢸⡇⠀⠀⣿⠇⠀⠀⢸⣿⠃⠀⢸⣿⣿⠀⡇⠀⠀⠈⢻⡆⠘⡆⠹⡀
⣸⣿⠀⢸⠀⣾⠀⢹⡇⠀⠀⠲⣿⣿⣇⣿⣿⣃⡤⣄⣸⣿⣀⡄⢸⣿⣿⠀⡇⠀⠀⠀⢸⡇⠀⢹⡀⢇
⣿⣿⠀⡄⢰⣿⠀⢸⡇⠀⣀⣀⣼⣿⡿⣿⣼⣿⣧⡿⣿⣿⢿⣧⣼⣿⣿⣤⣧⣤⣤⣤⣼⡇⠀⠈⡇⢸
⣿⣿⠀⡇⢸⠹⣷⣼⣷⡾⢛⠛⢻⣿⡇⣿⢹⠁⣿⡇⢹⣿⢸⣿⣾⣿⣟⠈⡏⠉⣉⣽⡿⠁⠀⠀⡇⢸
⢸⣿⡀⠀⢸⣦⣬⣿⡿⠿⠿⠿⣿⣿⡇⢻⣿⠀⣿⡇⢸⣿⠀⣿⣿⣿⣿⣿⣟⣛⠛⠋⠀⠀⠀⢀⡇⢸
⠘⣿⣇⢸⠀⣿⠿⣿⡧⠤⠤⠤⣾⣿⡇⣸⣾⠀⣿⡇⢸⣿⠀⣏⣉⣉⣉⣉⣻⣿⡇⠀⠀⠀⠀⣸⠀⡆
⠀⠸⣿⡄⢇⠸⣤⣽⣧⣀⣀⣤⡿⣿⣧⡿⢸⣶⡗⣷⣾⢿⣴⡏⠉⠉⠉⠉⢹⡿⠀⠀⠀⠀⢠⠃⡸
⠀⠀⠙⣿⣜⣆⠹⣍⠙⠻⢿⣥⣤⣤⣽⣥⣤⣽⣤⣬⣥⣬⣯⣤⣤⣤⣤⣴⡿⠃⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠻⣾⣧⡘⢦⡀⠀⠀⠉⣹⣭⣯⣽⣭⣯⣯⣿⣿⣽⡍⠉⠉⠉⠀⠀⠀⠀⢀⠔⢁⠔⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠐⠒⠲
⠀⠀⠀⠀⠀⠈⠻⣿⣆⡙⢦⡀⠀⢸⣿⣯⣽⣿⣯⣿⣿⣿⣿⣇⣀⡤⠴⠒⠚⠂⠐⠀⠀⣈⡀⠠⠄⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣌⠓⠾⣭⣏⣹⣩⣏⠯⠟⠛⠋⠉⢀⣀⡀⠀⠀⡀⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣌⣉⡁⠀⠀⠐⠒⠒⠛⡉⠁⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢻⣇⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠊⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⣿⠿⠷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣷⣮⣝⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢟⣿⣿⣗⢦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠀⣿⣿⠉⠁⢃⣸⠟⢿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠦⣾⣧⡀⠀⠀⠃⠀⠀⠀⢸⡇⡠⠌⣿⡟⠀⣦⣠⡿⣜⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣈⢽⣦⢤⣿⠀⠀⠀⣸⡿⣁⣴⢿⣧⣴⢿⣯⡾⢿⣏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢹⣷⣝⢷⣼⣦⠀⣰⣿⢟⣥⢟⣾⡷⣿⢻⣷⣿⠰⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠻⣿⠇⣿⡿⠿⠿⡻⢿⣯⣿⣹⡿⣱⡿⣿⡟⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢡⣭⣦⢿⣷⣀⣤⠀⠌⢻⡿⠻⣕⣻⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣿⣿⡏⠂⣴⡿⣿⣶⣿⣿⡏⠙⣿⠀⠀⠀⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠻⢦⡀⢀⣽⠋⢹⣿⣷⡄⢻⡄⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⢀⣾⣿⡏⠀⠀⢸⡇⠘⠀⣸⣿⡅⠀⣿⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣾⣿⠟⠁⠀⠀⠘⠁⠀⠀⠁⢸⠇⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⢿⣿⣇⠀⠀⠀⢤⣄⣴⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣿⣿⣧⠀⠀⢸⣿⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢿⣧⡙⠄⠀⢸⡟⠁⡐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠐⢠⡿⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⣿⠀⠀⡴⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⣿⠀⢰⡿⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠘⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣟⠳⣦⣀⠀⠀⠀⡠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⢶⠀⠀⠀⠀⠐⣼⠀⠀⠀⡠⠔⠙⢛⣿⣄⠀⠀⢈⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠁⠀⢰⠉⢃⠀⠰⠎⣸⠏⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣀⠄⢹⣿⣷⣖⢾⢺⣿⣛⡆⠀⠀⠀⠀⡀⠀⢀⠈⡆⠈⢤⠄⡀⣇⡃⠾⠹⠂
⠀⠀⠀⠀⠀⠁⣠⣤⣿⣿⣯⣮⣿⠿⢟⡿⢠⡿⡄⠀⢸⣧⠀⣌⣀⠱⣀⣘⡀⣻⢻⣇⠀⣠⡤⠖
⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⠜⠟⠉⠠⠏⠁⣼⠁⠣⠀⠎⢿⠀⠙⠛⣧⣤⣭⣴⢏⡼⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠁⠀⢹⣤⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤