Athlete Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀
⠀⠀⠀⠀⠀⣄⢠⣼⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠃⠘⢻⡆
⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣤⣤⠛⠓⣶⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣦⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢺⡿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣰⠀⠀⣠⣾⣿⡄⣴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠈⠉⠉⠙⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⣀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠙⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣻⠗
⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣀⡠⠧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠛⠛⠉⠉⠛⣷⣆
⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⡄⠄⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⠻⢷⡄⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢺⣿⣿⣿⡄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠙⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣶⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠙⠛⠛⠋⠉⠉⠉⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⢀⡀⡀⡀⠀⡆⠀⢀⠄⠀⠀⠀⢀⠀⣀⣈⢁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣠⣼⣿⡟
⣿⣿⣷⣟⣞⣵⣶⠇⠀⣼⣦⣻⣿⣿⣿⣿⣿⣷⣶⣶⣼⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠛⢛⡋⢛⣿⣯⡉⠀⢾⣿⣿⣿⣿⡯⢉⠹⣉⢛⠻⢛⡋⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠁
⣾⠶⠿⢷⣿⣿⣿⣮⣿⣿⣿⣿⣿⣷⣷⣾⣾⣷⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠉⠉⠐⠢⣤⡤⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣀⣤⣤⣴⣶⣶⣤⣤⣀⠀⠻⣷⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠟⠛⢉⣉⡀⠀⠀⠈⠉⠙⢿⡄⡹⢧⡀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣡⠖⠉⠀⠀⠀⠉⠁⠐⠒⡄⢰⠉⠳⡘⡌⠓⠢⡄⡀⣀⡉⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠖⠲⠀⠀⢘⡛⠓⠀⠀⢰⡎⠀⢀⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠿⡄⠀⠘⠋⠀⠀⠀⠀⠙⠲⠏⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⣅⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⡤⠤⠐⢉⠀⠀⠀⠀⠀⣈⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢬⡒⠒⠋⠀⠀⠀⣠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠸⡤⣤⠄⠒⠈⠉⡕⠒⠒⠂⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠙⠤⠀⠠⠔⠛⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠢⡀
⢀⡀⣀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡇⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⢀⡎⠁⠀⠀⠈⠑⠢⡀⠀⠙⣄
⡏⠀⠈⢠⣿⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⢆
⠀⠀⠠⣾⣏⠀⠈⠈⠒⠦⢀⡀⠀⠀⡠⠐⠁⠀⠈⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⢆
⠀⠀⠀⠀⠀⠉⠂⠤⣀⠀⠀⠀⠈⠁⠀⠀⣀⠔⠋⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⠉⠉⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢆⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠙⠦⡀⠀⠀⣀⠤⠊⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠸⡓⠈⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢀⠞⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣼⣿⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣿⡇⠀⢀⡿⡃⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⣼⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢀⣼⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣋⠀⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢋⣤⢞⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⡶⡫⠖⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣭⣈⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠘⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠈⠣⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⢀⣨⠽⠓⣶⣦⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡖⠛⠉⠁⠀⠀⠸⣿⣿⠃⠀⠀⠀⢸⣇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⢀⣿⡈⠀⠀⠀⠀⠈⠙⢦⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣸⠿⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⢀⠟⠂⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣿⣿⣷⣴⣶⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠉⠉⠉⠉⠉⠛⠻⠿⠿⠶⠶⠶⣦⣦⡀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰
⠀⠀⠀⠀⠀⢰⣶⡶⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠙⣿⢳⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠨⣄
⠀⠀⠀⠀⠀⣼⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⣶⣿⢿⣶⡆⠄
⠀⠀⠀⠀⠈⠉

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⣿⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠺⣦⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡿⣿⡇⠀⠉⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠸⠿⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣠⣾⣦⣄⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣤⣤⣠⣼⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⢀⣴⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⣼⠿⠋⣩⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠿⠿⠟⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⣾⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠈⠻⣿⠏⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣆⠻⣿⣿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠙⣆⡀⠀⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠋⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣤⣶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣷⣄⠘⠿⣿⣿⣶⣦⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⠀⠀⠈⠻⣿⣄⡀⠀⠉⠙⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠈⣻⣿⡿⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢛⡛⠃⠀⢀⣿⣿⡆⠀⠀⣼⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠉⠀⢀⣤⣾⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⠋⠁⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⣤⣾⣿⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠀⠀⢀⣀⠀⠀⠀⡔⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⣄⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣿⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡀⠀⠀⠀⣀⡀⠈⠣⢄⡀⠀⣠⡂⢸
⡠⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣤⠀⠀⢠⣤⠈⠀⠀⠀⠅⣠⠁⠙⡄
⢇⠀⠀⠈⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡔⠀⠀⠉⠀⠀⠀⠨⠶⣁⣀⠰⡏
⠀⢣⡀⢀⣤⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⢋⠌
⠀⠀⠻⡿⠋⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⣀⠠⠐⠆⠀⠀⠀⠀⢠⠏⠁
⠀⠀⠀⠘⡄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠩⡑⠤⠠⠔⠁⠀⠀⣀⠔⢃
⠀⠀⠀⠀⠘⢄⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⢀⡶⠈⠁⠀⢻⣀
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⣀⢴⠖⠉⠳⣄⣀⣠⠔⠀⠙⢶⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠉⠀⠂⠀⠉⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⣀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠢⡀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠟⠀⠀⠀⠀⠐⠒⠒⠰⠇⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡾⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠐⠒⠂⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢀⡀⠀⠀⠀⠀⠀⠉⠒⠤⢄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢀⡀⠀⠀⠀⢨⠁⡏⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡎⣸⡠⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⠤⠐⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣟⠆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣒⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⢣
⠀⠀⠀⠀⠀⢀⠤⠒⠋⢝⣉⣀⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠌
⠀⠀⠀⠀⠀⢸⣒⣛⠫⠭⠉⠒⠂⠈
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⠀⠀⠀⠁⠀⠄
⢀⠀⠀⠀⠀⠐⢈⡀⠀⡤⢠⢤⣀
⠀⢶⠚⠛⠀⠀⠂⠀⠊⢰⠁⠀⠈⠙⢆
⠀⠈⢳⡆⠀⡰⠃⠀⠀⠈⢀⠀⠀⠀⠘⠧⣄
⣀⣀⠀⢙⠈⠀⢀⠀⠄⠀⠀⠂⢄⠠⠐⠀⠈⠣⡀
⡠⠀⠐⣀⠉⡀⠂⠄⢀⣀⠀⠀⠠⠑⠒⠄⠂⠀⠙⢦⡀
⠀⢀⡔⠂⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠠⢀⠀⠀⠒⠀⠀⠀⠀⢇⠄⠀⠀⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠀⠉⠀⠁⠀⠀⠀⠀⠀⠰⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠐⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠂⠀⡸⠤⠐⠈⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⢀⠀⠈⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⡀⠀⢳⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠎⠉⠐⠀⠀⠔⠄⠑⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢄⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠁⠀⠠⠀⠀⠀⢦⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⡀⠨⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⣤⣔
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣤⣹⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⢦⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⡿⠋⣹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⢀⣶⡆⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠁⢠⣿⣿⣿⡿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣴⣿⡟⠀⠀⠀⣀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⢿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⣀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣽⡽⠀⢀⣿⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠖⠀⢠⣶⡂⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠂
⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠘⢃
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣤⣤⣴⣶⠾⠛⠉⠁⠀⠀⠈⠉⠻⢷⣄⠀⠀⠀⠀⠀⣠⠤⠒⠋⢠⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣐⠄⠀⠀⠀⠀⠀⠀⠙⣓⣈⡁⢀⣠⣄⠀⠀⢀⡷
⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⣿⡿⠿⠿⢿⣭⣭⣈⣻⡿⠿⠯⠭⢿⡿⠁
⠀⠀⠀⠀⠀⠙⠷⣶⣤⣤⣤⣤⣤⣤⣤⣴⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⣩⣿⡧⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣤⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣍⣉⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⢻⡟⠁⠉⠓⠦⢤⣦⣄⡀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣤⡤⠤⠒⠯⠀⠉⠁⠈⠉⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡅⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠖⠚⠛⠉⠉⠉⠉⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⣁⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡌⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢶⣄⣰⡀⠀⠀⠀⢠⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠭⠍⢻⣧⣤⣀⣀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⣿⡿⣿⠛⢶⣄⣴⣶⣤⣤⣤⠀⠙⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠉⠀⢺⣿⣿⣿⣿⣿⣿⣠⣄⠀⠿⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣷⣦⣀⠀⠀⠀⠈⣿⣿⡿⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣤⣀⡟⠛⢃⣠⣤⣀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⡉⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠓

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢣⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠙⢄⠀⠀⣰⠉⠉⠀⠘⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⢙⡴⠁⠀⠀⣀⠀⣀⣠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢓⣞⣋⠤⠤⠤⠤⠼⠿⣏⡀⣻⠁⠀⣀⣀⣀⣀⢀⣀⣀⣀⡀⠤⠤⢀⣀⣀⣀⣠⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣇⣀⡠⠤⠒⠢⢄⣀⣴⣿⠛⢶⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡶⠒⠒⠂⠈⠉⠁⠒⠀⠉⠁⠀⠀⠀⣀⠤⠖⠁⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠇⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠸⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠑⣶⠒⠂⠉⠉⠉⠉⠉⠉⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⠴⠮⠅⠈⠉⠲⢄⣸⠀⣀⠤⠔⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣨⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠁⠀⠀⣔⣠⠖⣧⠠⡀⢤⢠⢠⢠⢠⢀⠀⢠⢠⠐⠀⡠⡄⠀⣠⡄⠔⠂⢠⠀⠠⢀⡄⠄⡆⠀⡉⠢⠉⠤⢡⠘⢱
⠀⠀⠀⢀⡠⠖⠊⠉⠀⢀⡠⠔⠒⠈⠉⠀⠀⠁⠈⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠈⠀⠀⠋⠀⠈⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠉⠀⠁⠀⠁⠀⠁
⠀⡠⠚⠁⠀⠀⠀⢠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣾⣿⣿⣿⠃⠀⠀⢈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠉⠉⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⠉⠉⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣤⣄⣠⣄⠀⠀⠈⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣧⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣿⡄⠀⠀⠀⠀⠙⠁
⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⡠⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠲⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠻⠿⠿⠿⠛⠋⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⣀⠙⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⣰⡀⠀⢸⣦⡀⠀⠈⠈⠀⠀⠀⠿⠋⠀⠁⠀⠹⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⡿⠻⣆⠈⢿⣿⣷⣤⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣤⣦⣀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣷⣤⣼⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⣼⣿⣿⣿⣷⣦⣄⠀⣠⣿⣆⠀⠙⢿⣿⣿⣿⣿⡿⠟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣠⣤⡴⠾⠿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣄⠀⠀⠀⣀⡤⠚⠋⠉⠉⠉⠉⠉⠁⠀⢰⡿⠛⠻⠟⠓⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⠆⠀⠜⠻⣷⠄
⠈⠓⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠁



⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⠀⠀⠀⠀⠀⠀⣸⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⡳⢦⡀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⢀⣤⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣦⡀⠀⠀⢀⣠⣤⠤⣤⣤⣀⠘⠁⠀⠀⠀⢀⣤⠾⣋⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣠⡾⢋⡥⠶⠶⠶⢦⣍⡛⢦⡀⢀⣴⣿⡷⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡴⢋⡴⢖⣚⣒⠶⣬⡻⣆⠻⣆⠉
⠀⠀⠀⠀⠀⠀⠿⣭⣉⣉⣉⡛⠛⠶⠶⢤⣀⠀⠀⡟⣸⢡⡏⡼⣫⠥⣭⡙⣮⢳⡜⣇⢻⡄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠋⣠⡀⣇⢻⠸⡞⣟⢧⢾⡆⣷⢸⡆⡇⣿⢸⡇⠶⢿⣿⣿⣯⣭⣭⣭⣭⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⡽⣎⢷⣙⢮⣽⣭⠾⢃⡾⣱⢇⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠻⣌⠳⣭⣓⠲⠖⢚⣋⡴⢋⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣦⠀⠈⠳⣌⡛⠦⣭⣍⣉⣩⡥⠞⣋⡴⠋⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢋⣥⠞⠁⠀⠀⠀⠀⠉⠓⠶⠦⠤⠶⠖⠛⠉⠀⠀⠀⠈⠻⣍⡳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠠⡶⡄⠀⠀⠀⠀⠈⠙⢦⣉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡟⠀⠀⠀⠀⠀⠀⢷⠹⣄⠀⠀⠀⠀⠀⠀⠉⠳⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣸⠇⠀⠀⠀⠀⠀⠀⠘⣇⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠃

⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣤⡀
⠀⠀⠀⢰⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⠤
⠀⠀⠀⢰⣿⠀⣿⢀⣴⣦⠀⠀⠀⢠⣤⣴⣶⣶⣾⠿⣿⣿⣿⣿⡿⠿⣿⣟⠋⠙⠻⠿⣿⣿⣿⡄⠀⠀⠀⠻⣿⣿⣏⠁⠀⠀⠉⠁
⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠨⠽⣷⣶⠾⠿⠿⠛⠉⠀⠘⣿⣿⣿⣦⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠙⣿⣿⣷⣄
⠀⠀⠀⢸⣿⡄⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡆
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⢸⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⢿⣿⣿⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣄⣽⣿⣿⣇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⢙⣿⡿⢿⣿⣿⣿⡏⣿⣿⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⣭⣭⠿⠇⠀⠀⠀⠀⠀⣸⡿⠁⢸⣿⠟⠋⠀⠻⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣭⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⡇⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠶⠾⠟⣻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣿⠁⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⠀⠘⠻⢿⠟⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠿⢤⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⢀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠸⠿⠷⣄⠀⢀⣾⣶⣄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠈⠙⠛⠻⠿⠿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣰⣿⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡾⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣿⣿⣿⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣽⣿⣿⣿⡿⠉⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⣴⣿⣿⣿⠿⠿⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠋⠀⠀⠀⣹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣿⣿⣿⣏⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠻⢿⡿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠒⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉

⠄⠀⠤⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣠⢤⠄⠤⠀⠀⠀⠀⠀⠀⢠⡤⢠⣄
⠀⢏⠐⠁⢴⠊⠀⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠌⠁⢎⡱⠂⠠⠆⠀⠀⠞⢠⠋⡸⠉⡝
⠀⠀⣆⠰⡀⠜⠸⣇⠀⠀⠀⠀⠈⡆⠀⠀⡤⠊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢈⠀⡀⠠⠃⠀⠀⠀⠀⡄⠠⢁⠂⠰⣶⠁
⠀⠀⠈⠂⠀⢀⠞⠹⡄⠀⠀⡀⠀⠘⢆⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠣⠘⡌⠐⠉⢂⠀⣸⠄⢀⢀⠂⣠⠃
⠀⠀⠀⢈⡴⠊⠐⠌⠱⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠀⡀⠀⠂⠀⠈⠀⠁⣤⠋⡰⠁⠎⢱⠏
⠀⠀⠀⠀⠸⡀⠈⠒⠆⠑⡀⠁⠈⠂⠀⠰⡄⠠⠂⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠡⠀⠀⢀⡞⠀⠀⠀⠀⠠⢀⠚⠠⠂⣐⠇
⠀⠀⠀⠀⠀⢹⡂⢈⠢⠜⢣⠀⠀⠀⠀⠀⠫⡀⢀⠀⡠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⠀⠀⠀⡀⠀⠀⠀⠀⢀⡰⠀⡰⢠⢠⡜
⠀⠀⠀⠀⠀⠈⠷⠠⡀⠀⡉⠆⠀⠀⠀⠀⠀⢴⡀⠀⠔⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢈⠁⢄⠐⢁⠐⣇⠀⠀⡾⠄⢀⣠⠈⡩
⠀⠀⠀⠀⠀⠀⠈⢃⠀⡁⠀⠘⣆⠀⠀⠀⠀⠀⢧⠀⠀⠁⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠊⠁⠈⠰⠁⠈⠢⠀⢀⢾⣁⡄⠀⣥⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠂⠎⠑⡄⠀⠀⠀⠀⠈⠶⠀⠀⠀⠱⠃⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡰⠆⠀⠀⠀⠀⣼⢃⠐⠁⠄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠁⠀⠀⣱⡀⠀⠀⠀⠀⠐⠇⠀⠀⠂⠘⠆⠀⠀⠀⠀⠀⣘⠄⢀⠀⢠⠜⠀⠀⠀⠀⢰⠁⠀⠋⣝⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⡔⢁⠲⡀⠀⠜⠀⠀⠘⡀⠀⠀⠀⠈⢀⠀⠀⠀⢴⡁⠀⠀⠑⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠂⠀⢷⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⠈⡄⠀⢠⠘⢦⠀⠄⠐⠀⡄⡈⠀⢠⠀⢈⡐⠂⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠘⡀⠀⠀⠐⢩⣀⡧⠀⠀⠘⡶⠁⠀⢀⠀⢀⣎⠠⠀⠛⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠠⠀⠘⡆⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠙⢷⡀⠐⠐⠁⠀⠀⠀⢀⠞⠠⠛⠶⢣⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠁⠸⡄⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⡇⠀⠀⢀⠀⠀⠀⣼⡀⠁⢤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⢹⣶⡶⣦⠂⠀⡠⠀⠒⠖⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡐⠀⢳⡀⠀⠀⠀⠀⠀⠆⠐⠀⠀⡀⢿⣯⠿⡀⣨⠁⠠⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠠⠈⠇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⢯⣼⣷⡉⢆⠡⠀⢄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⡄⠀⠀⢁⠀⠀⠀⠀⠀⠐⡄⠀⠀⡄⠘⣿⡟⢿⡄⠂⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂⠀⣠⠋⠀⠀⡀⠠⣤⡤⠼⠀⠒⠐⠀⠈⣷⣤⢮⡁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠙⠋⠀⢠⡀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠘⠿⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠀⣈⣰⡶⣾⢷⠴⣿⠿⠿⠾⣷⣿⣷⣶⣂⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠽⠀⠋⠉⠉⠀⢀⣀⣀⣀⣀⣀⣀⣈⠉⠙⠛⠻⠿⣶⣟⡢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣀⡠⠴⠓⠚⠉⠉⠉⠉⠉⠉⠙⠛⠻⠦⣀⡀⠉⠙⠻⣾⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⢤⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣢⡄⠀⠀⠀⠈⠙⠢⣤⡀⠈⠩⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣱⡟⠁⠀⡠⡶⠋⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢿⢼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣸⠎⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⠛⠛⠻⠿⠿⠿⠷⣦⠈⠳⣄⠰⣦⠈⢿⣷⡌⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⡻⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣦⡄⠈⢳⡌⠀⠀⢻⣿⡄⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⢼⠒⣾⠁⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠹⣾⣿⠀⢹⣿⡀⢿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠊⣿⠃⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠈⢹⠇⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣏⢹⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠠⡆⠸⡅⣸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡆⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠂⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⡀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢙⣆⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣌⠃⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢻⡆⠀⠀⠐⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢠⡛⠀⢈⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⢿⡄⠀⠀⠀⠁⠀⠀⢰⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡔⠀⢜⡎⢠⠒⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡀⠀⠈⠻⣧⡀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠋⠠⣼⠎⠀⠁⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⣤⡄⠈⠿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⣰⠞⠁⢰⡄⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢿⣄⡀⠈⠙⠷⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡕⠊⠁⢀⡠⡿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣴⡆⠀⠈⠙⠛⠳⠶⠶⠒⠀⡀⠀⠒⠦⠤⠙⠒⠈⠁⠀⣰⠎⠲⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠝⢓⠶⠶⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣦⣌⠓⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⣉⣉⣋⣉⡙⠋⠙⠿⠟⠉⠉

⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢺⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⣴⡷⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⢻⣷⡌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢋⣼⡿⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠹⣿⡌⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠙⣿⣆⢻⣷⡀⠀⠀⠀⠀⠀⣼⡿⢡⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠹⣿⡄⠀⠀⢀⣾⡟⢡⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠹⣿⣆⢠⣾⡟⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡘⣿⣿⠏⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⠏⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠁⣰⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣤⣶⣶⣿⣶⣶⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠱⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢥⠆⡈⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⢓⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡺⣹⣷⡄
⠀⠀⠀⠀⠀⠀⢮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⡴⠀⠀⠀⠀⠀⠀⢠⣿⡓⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⢞⣿⡄
⠀⠀⠀⠀⠀⢌⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡥⢰⠀⠀⠀⠀⣿⡷⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⢾⣿
⠀⠀⠀⠀⠀⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠖⢨⠀⠀⠀⠀⣿⣟⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⣻⣿
⠀⠀⠀⠀⠀⠩⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠯⠍⠘⠀⠀⠀⠀⣿⣶⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡲⣶⣿
⠀⠀⠀⠀⠀⠀⢚⡵⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⡒⢠⠃⠀⠀⠀⠀⠘⣿⡭⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣭⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠫⢇⡄⡀⠀⠀⠀⠀⠀⢀⢠⡲⣝⠂⡰⠃⠀⠀⠀⠀⠀⠀⠘⢿⣯⡼⣹⢿⣿⣿⣿⣿⣿⣿⡿⡟⣏⢮⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠽⠃⠀⠀⠀⠸⠝⠂⢁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣯⣞⣼⣿⣿⣿⣇⣣⣵⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⣻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠉⠙⠛⠻⠷⣦⣤⣀⣠⣴⠾⠛⠁⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⣀⣴⣤⣀⠀⠀⠀⢠⡿⣿⡏⠁⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⣀⣤⡾⠟⠁⠈⠙⠛⠷⣶⣿⠃⠹⣷⣄⣀⣴⠿⠋⠁
⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⣹⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⡿⢰⡿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⠛⢷⣿⠿⠷⣦⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⣿⡆⠀⠘⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣿⣇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣯⣴⠿⠁

⢀⣀⡀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢹⣿⣿⣿⡇⠙⣿⢸⠃⠇⠈⢸⡏⣻⡏⣿⢿⣿⣿⢛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⢟⢿⣿⣷⠀⢹⣿⠻⡅⠀⢹⠘⠀⠀⠀⠸⠀⢃⠀⡿⠣⡾⠁⠠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣎⠳⡀⠂⠫⠃⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠜⡰⠱⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢿⡉⠻⣯⡂⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣹⠿⣿⢿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠻⣤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⡕⠋⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣍⠻⢦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢋⢴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣪⠛⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢉⣿⣿⣿⣿⣿
⣿⣿⣾⣍⡋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣟⣿⣿⣿⣿
⡷⠷⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣞⣿⣿⣿
⣿⣯⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠏⣴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⢛⣻⣿⣿
⣾⣯⢤⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠋⠀⠈⠹⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣠⣿⣿
⣿⡿⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⢠⡀⠀⢹⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣿⣭
⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠘⠀⠈⢳⡄⠀⢁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢬⣭
⡿⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⣿
⣿⣖⣛⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⡶⢶⣮
⣿⣿⡻⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡞⣷⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣙⠿⣿
⣿⣮⣡⣀⣄⣤⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣶⣷⣿⣿⣿
⣿⣿⣿⣿⡟⠍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣽⣻⣿⣿⣿
⣿⣿⣿⣿⣾⣛⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣉⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢴⣽⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢟⣡⣾⡣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠈⢢⣀⢵⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⢀⣠⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⣰⠇⠀⢀⠀⠀⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠐⡀⣄⣆⢰⣄⠰⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⡿⡕⠀⣼⠀⣴⣧⢎⡄⢸⢠⡄⣇⠀⡀⡈⠀⠀⣿⡄⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣹⣾⣿⣿⡄⣾⣼⣧⣿⡆⣷⣾⣇⡰⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣾⣶⡤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣉⣤⣶⣾⠿⠿⠿⠿⣷⣶⣤⣉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡘⠿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣌⡙⠻⠿⢿⣿⣿⡿⠿⠟⢋⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠦⠴⠶⠶⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⣨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠟⠀⠈⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆⠀⠀⠀⠲⠀⣇⣀⣤⠤⠶⢶⡶⠚⠉⠉⣇⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠏⢡⠞⠃⢠⡀⠀⢤⡶⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⠉⠣⠀⠀⠘⢦⠀⠀⠙⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⠇⠀⠀⠀⠀⢠⠂⠀⡄⠀⢀⣄⣀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⢠⡏⠀⢸⣃⡴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡾⠁⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⡙⠉⠉⠙⠛⠓⠶⢒⡶⢄⡀⠀⠠⠏⠀⠀⠀⢀⡾⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⣤⣤⣤⡄⠀⠀⠢⢍⡂⣤⠀⠀⠀⣠⡾⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠔⠁⠐⡁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣮⣅⡀⠀⠀⠀⠀⢀⣴⠊⣠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠀⠈⠛⠿⣿⣶⣶⣶⣿⣿⣿⣿⣯⣉⡙⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⠻⣷⠀⠀⠀⠀⠀⢸⣿⠛⠿⠛⠛⠛⠛⠿⢿⣿⣿⣮⣙⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠁⠀⠀⠘⣆⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠙⠛⠳⠎⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⣾⣧⠀⠀⠀⠀⣠⣤⣤⣴⣧⣤⣤⣀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠙⢻⣿⣤⣴⠶⠛⠉⠉⢿⣛⠒⠶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢠⡾⠛⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣸⣇⡔⠀⢰⡆⢸⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣿⡟⠀⠀⢸⣇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⠀⠀⠀⡆⠀⡇
⠀⠀⣼⠏⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⡇⢀⡇
⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣄⣸⡇⢸⡧⢄
⠀⣼⣁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠒⠚⠃
⢼⣋⣉⣉⣓⣦⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⡾⠋⠛⣷⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣠⣄⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡄⠻⠧⠛⢁⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣛⣛⣉⣉⣁⣀⣀⣀⣉⣉⣉⣙⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⢻⡏⠛⠷⢦⣤⣀⡀⠀⠈⣿⢋⡭⠥⢄⠀⡀⠈⠉⠉⣉⣉⡉⣿⣁⣀⣤⣤⠴⠶⠞⠛⠋⢉⡿⠀⠀⠀⢀⣤⣦
⠀⠀⠀⢀⣀⣤⣤⣿⠀⠀⠀⠀⠉⠛⠻⢾⡏⡆⠀⠀⠘⠈⡇⠉⡇⠈⠀⠀⠈⡿⡏⠉⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠰⣿⣷⣿
⣠⣾⣆⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠇⠀⠀⠀⠁⡇⠀⡇⠀⠀⠀⠀⢇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣦⣄⡀
⠉⠛⠛⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⠀⠀⠀⠈⠀⡇⠀⠃⡆⠀⠀⠀⠘⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠃
⠀⠀⢀⣤⡴⠾⠛⠀⠀⠀⠀⠀⠀⠀⢰⠇⡂⠀⠀⠀⠀⠀⡇⠀⡆⡇⠀⠀⠀⠀⠇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡾⠠⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠸⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⢀⣹⡧⠀⠀⠀⠀⠀⠀⢰⠇⡆⠀⠀⠀⠀⢠⠇⡄⠀⡇⡇⠀⠀⠀⠀⠀⡇⢻⡄⠀⠀⠀⠀⠀⠀⠰⣿⣀
⠀⠀⢠⣶⡋⠁⠀⠀⠀⠀⠀⠀⠀⡞⡸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠰⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡦
⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⣸⢣⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⢿⡆⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⣹⡷⠀⠀⠀⢠⠏⡎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⠀⠀⠀⢀⣴⡾⠋
⠀⠀⠀⣠⣶⠛⠁⠀⠀⠀⠀⣾⢠⣁⠀⠀⠀⠀⠀⠀⡂⡸⠀⠀⡇⢢⠀⠀⠀⠀⠀⠀⢸⠄⣿⠄⠀⠀⠈⠻⣦⡄
⠀⠀⠀⠈⠙⠻⣦⣀⠀⠀⠀⠹⡄⢀⣉⠉⠑⠒⠀⠒⠀⠓⠠⠤⠃⠘⠦⠤⠴⠒⠚⣉⣁⠀⡿⠀⠀⠀⣀⡴⠟⠁
⣀⣠⣤⣤⡄⠀⠀⠙⣷⠄⠀⠀⠹⡼⡄⠀⠉⠉⠉⡆⠀⡖⠒⠒⡆⡰⠒⠉⠉⠀⠀⢀⡌⡼⠁⠀⠀⠺⣯⡀⠀⠀⢾⣿⣿⡆
⢻⣏⣼⠏⠀⢀⣤⠞⠁⠀⠀⠀⠀⢻⡱⡀⠀⠀⠀⢰⠀⡇⠀⠀⠃⡇⠀⠀⠀⠀⢀⠎⡼⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠛⠁
⠈⠋⠁⠀⠀⠛⠛⠛⠛⢿⡆⠀⠀⠀⣷⠱⣀⣀⠀⢸⠀⣇⠀⡘⢰⣀⣀⠀⠀⡠⢂⡼⠁⠀⠀⠀⢠⣶⠶⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣴⠶⠛⠻⣷⣤⠬⠭⠍⠀⢈⣉⣀⣀⣀⣀⣀⣀⣀⣾⣁⠀⠀⠀⣠⣄⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣅⣀⣀⡀⠀⠀⢀⣀⣀⣀⣬⣿⠏⠁⠈⠛⠶⣤⡏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣈⣉⣉⣉⣉⣉⣥⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣎⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⢙⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠘⠿⢿⠿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢰⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠙⠛⠛⠋⠙⠻⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣠⣤⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠆⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠴⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢿⣿⣿⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⢠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢿⣿⣄⣄⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠩⠍⠉⠉⢻⣿⣦⣤⣀⣀⡀⠀⢀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃⢀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⣦⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠀⢹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣴⣶⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⢀⠤⠖⠒⠛⠲⠲⢤⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣀⡤⠶⠒⠛⠋⠉⠛⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟
⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡆⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁
⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⢛⡶⠯⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠸⣄⠀⢰⠋⠀⠀⠀⣠⡴⠟⣁⣀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠈⠳⢦⡤⠤⠶⠚⢉⣴⠟⢉⡴⠂⠀⠋⠁⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠳⣄⠀⠀⣾⣁⡴⠛⠀⣠⠞⢹⠉⠉⡗⢦⡀⠀⠀⣿⡿⠂
⠀⠀⠀⠀⠀⠈⠉⠻⡏⠁⠀⠀⠀⠁⣰⠃⠀⠀⡇⠀⣧⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠤⠴⡋⠁⠀⢀⡞⠁⣠⡏⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣾⣭⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣴⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⣿⡲⠶⣄
⠀⠀⠀⠀⠀⢽⣇⠀⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠑⢦⡀⠀⠙⢦⣀
⠀⠀⠀⢀⣾⡿⢿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠉⠓⠀⠀⠀⢀⡶⠋⠉⠁⠀⠙⢦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠘⣦⣸⣷⣶⣣⣄⣀⣀⣘⡇
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢻⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⡟⠛⠿⠿⠻⠿⣯⣍⣹⠇
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠹⣆⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣤⡀⠀⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠈⠑⠦⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⣰⠁⠀⣠⡿⠁
⠀⠀⠀⠀⠀⣠⠃⠀⣰⡟
⠀⠀⠀⢀⡴⠁⠀⢰⣿⣷⣤⣀
⠀⢀⣴⣿⣄⣀⣀⣀⡈⠛⠿⣿⣇
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠒⠤⣀⠀⠀⠀⠀⠀⣠⠖⠉⠛⠲⢄⠀⠀⠀⠀⠀⣠⠔⠒⠢⢤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠑⠒⠤⠒⠊⠁⠀⠀⠀⠀⠀⠑⠦⠤⠤⠞⠁⠀⠀⠀⠀⠈⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠤⢄⡀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠦⣄⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠒⠤⠤⠤⠒⠉⠀⠀⠀⠀⠈⠳⢄⣀⣀⠴⠃⠀⠀⠀⠀⠉⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⡶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⠀⠀⠀⠀⠀⣀⡴⠛⢷⡄⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⠉⠉⠉⠉⠉⠡⠀⠀⠈⢡⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠶⠷⠶⠶⠷⠶⢶⣶⠟⠀⢀⡀⠀⠀⠀⣀⣠⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⣀⡀⢀⣀⣠⡾⠋⠠⠀⠀⠈⡙⠉⠙⢉⣰⠟⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⡟⠛⢿⣿⣿⣿⣶⣶⣷⢶⢾⠶⣶⣶⡿⠿⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣥⠶⠛⠋⠉⠉⠁⠀⠀⠀⠈⠉⠉⠙⠛⠶⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠛⢦⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⣀⡼⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠒⠒⠶⠶⢬⣝⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠶⠶⢶⠶⠄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⠁⠀⠀⢀⡏⠀⠀⢀⡄⠀⠳⣤⣤⣤⣤⣤⡀⠀⠀⠰⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⢠⠏⠙⢦⣀⣀⣬⣤⠤⠾⠃⠀⠀⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⢠⡶⣄⡀⠀⠀⠀⣠⡼⠇⠀⠈⢻⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠄⠀⠀⠈⠳⣄⠙⠷⠶⠟⢋⣀⣼⢷⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⠢⡀⠀⠀⠈⠛⠶⠶⠞⠛⠉⠀⠀⠙⣆⠀⢱⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⡇⠀⠀⠀⢀⡠⠊⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⠈⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠘⢦⡇⣀⡠⠖⠉⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣴⣀⠀⠈⠉⡙⠛⠓⠒⠒⠖⠒⠛⡉⠉⠠⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣡⣁⣂⡂⡈⡀⡈⣠⣁⣤⣴⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⣀⠀⠀⡠⠴⠂⠀⠀⠀⠘⢳⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣠⠀⠠⠊⢃⣠⠀⣀⣀⢤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡄⠉⠁⠀⠀⡄
⠘⣭⠁⠀⠀⢀⠤⠀⠀⠀⠀⡠⠂⠀⣀⠵⠛⢿⡁⢀⠄⠁⠸⠋⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠠⠁⢈⠁⠀⠤⡇
⠘⠫⠀⠀⠀⠀⠀⠀⠤⠤⢀⢰⠗⠂⠀⠀⠀⣀⠄⠁⠀⢀⠄⠐⢡⠀⠀⠀⢀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⢌⠀⠀⠠⠈⠀⠀⠄⠄⠀⠀⠀⠴⠖⠋⠀⠀⠀⣀⠉⡔⠃⠹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⡐⠀⠄⢀⠀⠠⠉
⠀⠀⠈⠐⠒⠒⠓⠒⣚⠉⡇⠀⠐⢠⡈⠁⠀⠀⠀⠀⡤⠖⢀⡀⢠⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠠⢀⣉⡥⣦⠠⠔⢛⠀⡆⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣠⣾⣿⣽⣽⣯⣽⣿⣿⣿⡿⠋⠉⠉⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⢉⣍⠀⠀⡞⡇⣠⠎⢀⡶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡄⠀⠄⠠⢀⡤⠀⡁⠀⠓⠈⢀⡜⢸⠀⠄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⢀⢲⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠤
⠀⠙⠀⣠⠀⠈⠀⠀⠛⠂⠀⡄⠊⠀⡏⠀⠀⠐⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⢋⠘⡀⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⡤⠀⠂
⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⡆⠀⢸⠁⡆⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⢋⠠⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠂⣠⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣤⣮⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣩⣭⣹⣍⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡶⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⣄⣼⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣾⣗⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢠⡄⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⡀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢨⣿⣾⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠆⣶⣶⣾⣷⣄⠀⠓⢯⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠞⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⠀⠀⠀⠀⢸⡇⢿⣿⣽⣿⣿⣗⣦⡀⠙⠂⠀⠲⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⡄
⠀⠀⠀⠀⠀⡀⢤⣿⡿⡇⢠⡄⠀⠀⠈⣿⣅⣽⣯⡟⢉⣯⣿⣿⣢⣀⠈⠲⣿⣫⡴⠀⠀⠩⠀⠀⠀⠀⠀⠀⠠⣿⠀⡇⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⣿⣦⣾⣵⣿⣿⣗⠋⠘⣛⡽⣿⣿⣱⣋⣵⣄⡈⠙⢿⡷⠁⣀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠰⠇⢠⣤
⠀⠀⣠⣾⠿⢃⠀⠀⣾⠸⠲⠿⣿⣿⢿⣿⣿⣿⡟⠿⠛⣴⣿⣿⣿⡿⣻⡏⣹⡶⣞⣷⣾⣿⠃⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⢾⠟⠁⠀⠀⢷⠞⣹⡗⠀⢀⣈⡿⢨⣱⣷⣽⡿⢟⣚⣵⣿⣿⣿⣷⣿⣵⣿⣿⠾⠿⣿⡏⠀⠐⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣯⣟⣳⣢⡟⠻⣶⣿⣿⣯⣟⣿⢺⣯⡟⢿⣁⣿⣿⣿⣿⣿⣿⣦⡀⢈⠃⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⠿⣤⣿⡿⠿⠟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢢⡀⠀⠀⠋⡀⠀⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣀⠀⠘⠿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣯⣿⣯⣽⣿⠇⠀⠐⠂⠀⠊⠛⢿⣿⣿⣷⣄⡀
⠀⠀⠀⢶⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⠻⣿⣿⣿⣿⣿⡷⣦⡀⠀⠀⠀⠀⠀⠈⠛⢿⣏⣽⡆
⠀⠀⠐⣲⣆⡚⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏⢿⠛⣻⡗⢾⣟⠳⣄⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠉⠀⠁⠀⠀⠀⣠⠞⠋⠟⡿⠉⠉⠻⠛⠛⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⡳⠷⡘⢌⣽⣦⡙⠇
⠀⠀⠀⢠⣄⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣚⣿⣿⣿⣿⣿⣿⣿⣿⣾⠉⠿⣮⣻⣽⣷⣄⣀
⠀⠀⠀⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⠀⠁
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢀⡀⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣿⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡼⣻⢿⣤⣾⣿⠀⢹⣿⣿⣿⣿⣿⡻⢱⣾⡿⣿⣿⣟⢻⣿⣿⣿⣷⣤
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⠾⣻⣿⣟⢀⣶⣿⠿⣿⣿⣯⣿⣟⠾⠛⢌⡻⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠐⠟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡿⣿⣦⣼⠿⣻⣿⠈⠀⣀⠀⠑⢤⣙⣿⣻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠀⠈⠀⠉⠒⠿⠃⠀⠴⠿⠶⠀⠈⠻⣿⡿⣏⠙⠿⣿⣿⣇⡀
⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢗⠄⠙⠏⠙⢆⢀
⠀⠀⠀⠙⠛⠉⠁⠀⣾⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠋⠉⠉⠀⠀⠀⠈⠉⠙⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⡠⣀⣵⡾⠟⠋⠉⠉⠉⠉⠉⠙⠛⠷⣮⣔⢄⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⢀⢈⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣔⠄⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠄⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠊⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠁⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠊⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡡⠁⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠑⢝⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⡋⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠁⠊⠙⠻⠷⣶⣶⣶⣶⣶⠶⠟⠛⠅⠊⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⣤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⣶⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣥⡀⠀⢠⠀⠀⠀⡠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡟⣃⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⡄⠀⣿⠛⡿⣦⡀⠠⠚⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣷⡀⠀⢀⠀⠈⣩⣾⣻⢌⡵⠶⡆⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣐⣴⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠢⣐⠳⣶⣟⢁⣱⣿⣳⡐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢿⠁⠈⠹⣦⣤⡤⠶⠶⠶⠶⠶⠒⠖⠀⠐⢴⣄⠀⠉⢩⠙⣤⣼⠇⢣⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠘⢧⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣵⡄⠑⠄⡇⢘⣻⡗⠷⢄⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠃⠀⠀⠀⣸⡿⠀⠀⢀⢿⢀⠉⠉⡱⠞⠻⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⣦⣤⡀⠈⡿⠶⠒⠙⢦⣀⠀⢰⣿⣟⡀⢀⠂⠀⢸⡦⡀⠈⠢⡀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠒⠚⠂⠀⠀⠐⠀⢾⣿⠇⢳⡀⠇⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⡟⢰⠂⠀⢷⠀⠀⠀⠈⠑⠆⠀⠈⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢳⢠⠀⠀⠀⠀⠀⠀⠙⡆⣻⣿⣶⣿⠀⠀⢠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠈⣀⠄
⣠⢤⠶⠀⢐⡂⠀⠐⠒⠓⠲⠎⠀⠀⠈⠘⠃⣠⠖⠐⠛⠃⣴⠦⡤⠿⠔⠠⢖⠒⠲⠶⣴⣷⣿⣿⣿⣿⡤⣀⡀⣄⣀⠀⠀⠀⣠⣤⣷⣤⣀⣈⠂
⠀⠀⠀⠀⠀⠉⠠⠐⠒⠈⠀⠀⠈⠈⠁⠀⠀⠣⠐⠰⠄⠀⢀⠉⠀⢀⠀⣀⠀⠀⠀⠀⠤⠤⡀⠀⠘⠁⠀⠈⠛⠁⠀⢀⠢⡤⣁⠉⠉⠀⠀⢉⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⡶⠟⠳⢦⡀⠀⠋
⠀⠀⠀⠀⣠⡞⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣴⣾⡿⣭⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠆⠀⠈⢛⣿⣯⠾⠃⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢺⣦⡀⠀⢸⣿⣧⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠿⠟⠀⣤⡀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣼⣿⣅⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡾⠁⠀⠀⠻⠿⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠈⠁⠀⠀⠀⠘⣿⡯⣿⣿⡿⠂⠀⣀⡴
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠀⠀⠀⠑⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣸
⣿⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣶⣿
⠻⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠰⣾⣿⣿⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢈⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢽⡄⣦⣶⡞⣰⢷⢰⡞⣴⢛⡇⠀⡇⣴⠃⣾⢻⡔⡟⣷⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠈⠁⠈⠉⠈⠁⠈⠉⠁⠀⠁⠈⠉⠈⠉⠀⠁⠉⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⡦⠀⠀⠀⠀⠀⠀⢰⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⢾⡜⢳⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣼⣟⠄⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠿⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠁⡴⠋⠀⠀⠀⠀⢰⣮⠉⠈⢯⡆⠀⠀⠀⠀⠀⠠⣶⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡄⠀⠀⠀⠀⠀⠀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⠀⠀⠀⠀⢶⣿⣿⡶⠂⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⡜⡵⠟⠙⣮⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠰⢟⣟⠇⠀⠀⠀⠀⠀⠀⣬⡍⠋⣥⠀⠀⠀⠀⠀⠳⡻⢶⣶⢟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⡆⠀⠀⠀⠀⠀⠀⣌⣭⣭⡗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢶⣤⠀⠀⠀⠀⠀⠀⢙⠻⠿⣇⠀⠀⠀⠀⠀⢰⣶⣿⣿⣷⡶⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠘⡜⣿⡿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⣶⠷⠁⠀⠀⠀⠀⠀⠻⠇⠈⠟⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⣠⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⡾⠇⠀⠀⠀⠀⠀⠀⠘⠻⡿⠛⠀⠀⠀⠀⠀⠀⠈⡿⢿⡅⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⣘⣑⡄⠀⠀⠀⠀⠀⠀⠈⣨⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣆⡀⠀⠀⠀⠀⠐⠲⣷⣶⡗⠂⠀⠀⠀⠀⠀⠹⣷⣸⡟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⡘⠿⠿⡋⣀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣩⣥⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠟⠀⠀⠘⠟⠂⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠚⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣽⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡿⣛⣿⡿⠟⠋⠁⢿⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠀⠀⠀⠀⠈⢿⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠻⠋⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣄⠀⣠⣶⡿⣿⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠈⢿⠁⠀⠉⠀⠁⠀⢰⡶⠀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⠀⠀⢂⠀⠀⠀⠁⠀⠀⢰⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠈⣹⣏⣁⠀⡀⠀⠀⠀⠀⡸⠁⠙⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢩⣭⡅⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⣀⣀⣤⣾⣿⡄⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠴⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡈⠉⠁⣤⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⢣⢃⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋⠁⠀⠑⠿⢶⣿⣶⣭⣝⡲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣮⣿⣿⣿⣿⣿⣿⢋⡿⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⠋⠿⠿⣿⣿⣖⣙⠢⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡾⠁⠀⠀⠀⠀⣼⣿⣿⣶⣴⣆⣀⡀⠀⠉⠿⣿⡿⡗⠑⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⡵⠛⠀⠀⠀⠀⠀⠠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⣙⣾⣿⣤⠡⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠸⣿⣿⠟⣾⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⠀⠂⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⣀⣾⠟⢹⣿⣿⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⣝⣛⢛⣛⣋⣴⣿⣿⣿⣿⣿⠀⠈⠄⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⣼⣿⣿⡇⢲⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣦⠀⠈⠒⢄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢚⢿⡿⠁⣿⣿⣿⣿⠟⢿⣿⣦⠀⠀⠀⠀⠀⠀⢤⡹⣿⡿⢛⣻⣿⣿⣿⣞⣿⣿⣿⡿⠿⢋⣴⣖⣤⠈⢶⣤⣀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠈⠀⠉⠁⠈⠉⠉⠀⠀⢸⣏⠟⣿⣦⣤⣄⣀⠐⠘⠃⠀⠑⢤⡀⠘⢿⣿⣿⣿⡖⢲⣶⣾⣿⣿⣿⠿⠀⠀⠈⠛⠷⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⡾⢻⡏⢿⡿⠂⠀⠀⠀⢠⠀⠀⠉⠙⠂⠙⠻⢿⣿⣾⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⡇
⠀⢀⡏⠀⡤⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢳⡄⢡⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⣇⠰⠃⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠹⣮⠃⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠈⠻⣷⡀⠀⠀⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠘⠻⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣦⣀⣀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣾⣿⣿⣿⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⣶⡆⣤⢤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⣟⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣙⣃⡻⣧⣼⣼⣿⣿⣿⣿⣿⢿⣿⣿⣋⣿⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣷⣦⣍⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢿⠻⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⡟⢿⣿⣿⠙⡽⣯⠇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠀⠀⠛⠿⠁

⠀⠀⠀⠀⠀⠀⢀⠀⡄⢀⠇⢰⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡀⠀⠂⠠
⠀⠀⠀⠀⢀⠑⠀⡣⠸⡸⠀⠎⠈⠈⣠⠄⠀⠀⠀⠀⠀⠊⠀⣀⠑⠄⠄⠀⢈⠀⠀⠂⠀⠀⠀⢠⡀⠀⠛⠁⣀⡀⠀⡠⠊⢁
⠀⠀⠀⠀⠀⠉⠒⢬⡤⠋⠉⠑⢶⠊⠠⠔⠚⠉⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠧⠀⠁⡔⠁⠀⠀⠑⡈⠁
⠀⠀⠀⠈⠀⠩⠄⠲⣇⠀⠀⠀⢀⠒⠒⠒⠒⠀⠀⢀⠏⠀⠐⠀⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⢠⠀⢢⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠔⠚⢉⡌⡲⠄⠐⢯⡉⠒⠄⠀⠀⠀⠀⠀⠈⣰⠂⠀⠐⠀⠈⠠⣀⢀⠀⠀⠀⠈⢄⠈⠃⠠⠉⠀⡈⠄⠀⡀
⠀⠀⠀⠀⠐⠉⠐⡋⢀⡏⠀⠰⠀⠱⣄⠰⠄⠀⠀⠀⠀⠀⢇⠀⠈⠈⢦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡈⠀⠗⠳⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄⢀⣠⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠍⠙⠛⠛⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠻⣿⡿⢿⡟⠛⠋⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⢸⣿⠿⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⠁⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⣀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣄⠀⠀⠀⠰⣦⡄⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣧⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠛⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢿⣿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠒⣀⡀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⣷⡀⠀⠀⠈⢶⡄⠤⠽⠋⠀⠐⠓⢶⠂⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣙⡛⢀⠀⠀⠘⣧⣶⣤⣤⣶⣾⢟⣿⣷⣿⠋⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣰⠖⢙⣻⠿⣿⣟⡟⠂⠒⠀⠀⢹⣿⣿⣤⠈⣀⣼⣿⣿⣿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⣍⣹⣿⣦⣻⣿⣿⣿⡟⠃⠀⠀⢠⣾⣿⣿⣯⡠⢾⣿⣿⣿⡿⠉⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠙⠉⠁⣸⣿⣿⡷⠀⢠⡀⣀⣬⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠈⢻⣯⣸⣿⣛⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⢴⣿⣏⢸⠃⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⢠⣼⢿⣿⣿⣦⡀⠀⠙⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠁⠈⣿⣿⣿⣷⣄⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣆⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠙⠿⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⠄⠀⠀⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⠀⠀⢀⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣾⠁⢙⠴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣀⢋⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⣠⣤⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣀⡀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢦⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣻⣿⣿⣿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠙⠉⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡏⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⢠⣿⣟⢀⣀⣠⣤⣶⠶⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣤⣴⣾⣿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠾⣿⠿⠋⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣾⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⢀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⢿⣿⣿⣇⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣾⣿⣿⠆⠀⠀⣰⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢸⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⡿⠁
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⢿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⠿⠟⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⢤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠠⠴⠛⡻⠛⠉⠙⠻⢿⣿⣷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⡿⠟⠋⠉⠛⢿⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⣿⣿⣦⣴⣤⣄⣀⡀⠀⣾⣿⣿⣿⣿⣷⠀⠀⣀⣠⣤⣴⣤⣾⣿⣿⡿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⢺⡶⢦⣝⣿⣿⣿⣿⣯⣿⣿⣯⣯⣿⡛⠛⣋⡤⢶⣗⢓⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣒⡒⠒⠿⠯⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣽⣿⣶⣿⣿⣿⡿⠭⠿⠓⢒⣒⣒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⢛⡉⠉⠉⠉⠽⠧⠿⠿⠛⠛⠛⠛⠻⠿⠼⠿⠍⠉⠉⢉⡛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⠀⠀⢠⠄⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢂⠀⠀⠀⢡⠀⠀⠀⠈⠄⠠⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠘⡇⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠐⡀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠆⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠌⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢡⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⡌⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢡⠀⠀⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠤⠤⠤⠤⠦⠤⠤⠤⠦⠤⠤⠤⠼⠶⠶⠶⠿⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⣀⣀⣀⣀⣀⣠⠐⢢⣀⣀⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⡉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣧⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⣀⣤⣤⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠈

⢀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠒⠉⢡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣶⢖⣿⣷⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⢾⡿⠿⠋⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣌⠳⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣿⣭⣁⠀⠀⠀⠀⠀⡀⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⣟⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣷⣝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⠈⠢⣄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣰⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⡇⡟⣰⠃⠀⠀⠀⠀⢀⡠⠖⠚⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⢠⣈⠳⡄⠀⢿⡷⠁⠀⠀⠀⢀⡴⢋⡤⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣙⣻⣄⢸⠁⠀⠀⠀⢠⣮⢖⣉⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⢻⠀⠀⠀⢠⠟⢋⡩⠴⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠙⢦⠀⠀⢸⡇⠀⣰⣋⣴⣟⣚⠋⠉⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠛⠉⠛⡒⠦⢤⣀⠘⣆⠀⡄⢧⠀⠈⡇⢰⠟⠁⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠉⠓⠢⢬⡓⢬⡳⣜⣾⡀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⣬⣙⣿⣆⠙⢧⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢾⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡴⠊⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠛⠛⠓⠲⢤⡀⠀⠀⠀⠀⠙⣄⣷⠀⠀⠀⠀⠀⢠⣞⣤⢞⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠉⠓⠢⣌⣣⡀⠀⠀⠀⠸⣿⠀⠀⠀⣠⠴⠛⠋⠉⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠤⢽⣷⠀⠀⠀⠀⡇⠀⣤⡾⠵⠶⠶⢿⡿⠯⠤⠈⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢰⣧⠞⠁⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢉⣒⡦⡀⢳⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣤⣤⣭⡿⣄⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣓⣒⣒⣒⣒⣒⣶⠾⠿⠯⠭⠭⢭⡿⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠘⢄⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢁⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⢄⠀⠀⠈⢂⠹⣷⣒⣒⣂⣈⣭⣭⢭⣭⠟⠀⠂⠀⠀⠀⢠⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠑⠄⠀⠀⠁⠘⢿⡉⠁⠀⠀⠀⢠⠊⡠⠀⠀⠀⡠⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠈⠢⠀⠀⠀⠈⢷⠀⠀⠀⣰⠃⡴⠁⠀⠀⣔⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠑⠀⠀⠀⡌⢧⠀⣰⠃⡜⠀⠀⢀⣾⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢆⠀⠘⡌⣷⠃⠎⠀⠀⡠⣣⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣠⣤⠴⠶⠷⠶⠶⢦⣴⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣀⡤⠴⠲⠖⠲⠒⠦⣌⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡠⢊⡥⠀⠀⠀⠀⠀⠀⠀⠠⡉⠳⣌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠞⣁⢹⠶⠃⡀⠀⠀⢀⡐⠶⣆⣐⡳⠘⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢠⡏⢠⣛⠛⠁⠀⢇⡩⠉⡆⣳⠀⠙⣮⣑⢝⣘⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⣆⣩⠀⠀⠀⠀⠀⠀⡀⢣⡀⠀⠶⠧⣽⣡⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡌⣯⠰⠂⠀⢢⡀⠀⠀⠀⣰⠇⢀⡻⢶⣎⣸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠱⡈⢀⠿⣠⠀⠙⠲⠶⠚⠉⠀⣬⢻⡎⣱⠃⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣳⡀⠘⢮⡻⠿⣾⣳⣆⠀⢰⣠⣷⠽⢷⡴⠁⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢝⣦⣀⠙⠲⠭⣽⣟⣓⣺⣏⡥⠖⠋⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢟⡶⣦⣤⣤⣬⣭⣤⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠛⠛⠋⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣶⣄⡀⠀⢠⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⣌⣻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⣻⡿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡻⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠙⠛⠛

⣿⠿⠿⠿⡿⢿⠿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣛⣛⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠻⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡄⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡻⣿⣿⣿⣿⣿⣷⣦⡻⣿⣿⣿⣿⡿⠿⠛⠋⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⣿⣿⣿⡄⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠛⠿⠟⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⡀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠈⠛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠈⠛⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⢺⣿⣿⣿⣿⣿⣿⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠉⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⠻⣧⢀⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡀⣼⠟⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⠀⠙⢿⣿⢻⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡟⣿⡿⠋⠀⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠈⢻⠈⣿⣜⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣣⣿⠁⡟⠁⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠫⢄⠀⠀⠣⠘⣿⣷⣮⡙⠛⠛⠿⠿⠿⠟⠛⠛⢋⣵⣾⣿⠃⠜⠀⠀⣠⠝⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣷⡄⠀⠀⠀⠁⠈⢿⣿⣝⡳⢦⡀⠀⠀⢀⡤⢞⣫⣿⡿⠁⠀⠀⠀⠀⢠⣾⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⠛⠿⣿⣌⢦⠀⠀⢤⣽⣮⣭⡀⠙⠀⠐⠋⢀⣩⣵⣯⡄⠀⠀⡴⣡⣿⠿⠛⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣶⣬⠌⢑⣷⡄⠀⠉⠻⢿⣿⡄⠀⠀⢨⣿⡿⠟⠉⠀⢠⣾⠊⢀⣥⣶⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣉⢛⣷⠄⠈⣽⠻⠦⠀⠀⠈⠋⢻⡄⢠⡟⠙⠁⠀⠀⠴⠟⣯⠁⠠⣾⡛⣋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⠋⠉⠴⢿⡂⠈⣿⡘⣇⣀⣠⠀⠀⠀⢹⡏⠀⠀⠀⣄⣀⣸⢃⣿⠁⢐⡿⠦⠉⠙⠛⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⠀⠀⣻⣥⠄⠛⢿⣮⣭⣵⣿⣦⣴⠟⠻⣦⣴⣿⣮⣭⣵⡾⠟⠠⣼⣟⠀⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣀⣩⣿⣷⣀⣈⡀⣰⠏⢠⢿⡟⠀⠀⢻⡿⡄⢻⣄⢀⣁⣀⣿⣿⣍⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠁⡞⠁⠀⠉⠙⢻⠏⠀⠀⠀⠀⢻⠀⠀⡟⠀⠀⠀⠀⠹⡟⠋⠉⠀⠀⢣⠈⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡇⢀⠀⡀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠀⡀⢸⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣿⣶⡇⠀⢰⡷⢾⣇⣀⣄⠀⠐⠂⠀⣠⣀⣸⡷⢾⡆⠀⢸⣦⡷⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠟⣷⠀⢸⣇⠈⠛⠙⣿⣦⠀⠀⣴⣿⠏⠛⠁⣸⡇⠀⣿⠿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⣧⠸⣿⣦⣧⠀⢻⣿⣧⣼⣿⡟⠀⣼⣴⣿⠇⣼⠏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠏⠻⣧⡈⢿⡌⢠⡿⢁⣼⠟⠹⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠇⠸⠿⠟⠁

⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢀⠀⠣⠆⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠈⣿⡇⠀⠘⣷⠗⠀⠀⠀⠀⠀⠀⠀⠈⢛⡿⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⠀⣰⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⡴⢋⡤⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠚⣟⠁⣰⠟⣩⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⢸⠇⠀⠀⠀⠀⢠⣞⣤⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⡄⠀⠀⠀⢠⠖⣲⢏⠢⠹⡄⡥⠲⡲⢄⠀⠀⠀⠀⠀⠀⢠⣿⠋⠈⠀⠀⠀⠀⣰⠟⠉⣠⠞⠉⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢿⢧⡀⠀⠀⠀⠸⣖⢋⡹⠀⠚⠁⢯⣝⣋⡽⠀⠀⠀⣴⣶⠀⡾⠈⢷⡀⠀⠀⠀⠀⠃⣀⣼⡧⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡍⡶⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢄⠴⠶⠶⢶⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢧⣴⣶⡄
⠀⠀⠀⠀⠀⠢⠴⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠢⣄⠀⠀⠀⠀⠿⢻⣿⣧⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣤⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠐⠛⠶⣦⣀⣤⣴⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⠋⠻⣇⠀⠀⠀⣠⣿⠸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣄⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢰⠃⠀⠀⠀⠈⠁⠀⢰⣿⠁⠉⠀⡿⠹
⠀⠀⠀⠀⠀⠀⠤⠿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠐⠚⠑⠁
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣄⠀⢀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣼⣯⣄⣀⣀⣤⡀⠀⠀⠀⠈⠘⢦⣀⣼⣷⡄⠀⠀⠀⠀⠀⠀⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠸⡇⢀⠀⢠⠇⢰⣿⠇
⠀⠀⠀⢠⢟⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠻⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢻⣿⣷⠟
⠀⠀⠀⠋⢰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟
⠀⠀⠀⣠⡿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⡇⠀⠀⠀⠀⠀⢀⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢻⡄
⠀⠀⢼⡋⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣇⠀⠀⠀⠀⠀⣸⠃⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⣴⡆⠀⠀⠀⠀⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠆⠋⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⣤⡄⢠⠆⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢷⡀⠀⢀⡾⠀⠀⠀⡄⣾⣿⠟⠛⠛⠀⠀⠀⠀⠀⠘⠉⠀⣰⣿⡿⢷⡼⠁
⠀⠀⠀⠀⠀⣴⣿⣶⣯⣤⣤⠀⠀⠀⠀⠀⢻⡄⣿⣿⣷⡒⠙⠀⠀⠀⠀⠀⠀⠀⣽⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣤
⠀⠀⠀⠀⠀⢿⣿⣀⡀⠀⠀⢲⡄⠀⠀⠀⠀⠀⠘⢿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠻⣆⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢿⡀⠀⠀⠀⠀⢾⠉⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣴⣿⣿⣿⣿⡿⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣀⣀⠀⣼⣿⣿⡯⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⣿⣿⠟⠋⢩⣿⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⠈⠑⠀⠀⠀⠀⢸⢿⣿⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠂⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠃⠀⣻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⠏⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⢿⣿⣷⣤⡀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⣀⣀⠀⠈⠙⠿⣆⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣴⣿⣿⣿⣿⣦⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠟⠁⠀⠀⠀⠈⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⣀⠀⠀⠀⠀⠠⠾⠛⣛⣩⣥⣶⣶⠶⠷⠶⠶⢦⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣟⣁⣠⡤⠶⠚⠋⠉⣉⣉⣉⣉⣙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣬⣉⣉⣉⣁⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⢰⣿⣿⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣸⣿⣟⣴⣿⡟⠀⠀⣴⡆⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⠃⠀⠀⠀⠀⢹⣿⣿⠃
⠀⣼⡇⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⢿⣧⣤⣤⣤⣤⣼⣿⣿⣀⡤⢿⢦
⢠⡟⠁⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣿⣿⣿⣿⣧⣼⣧
⢿⣧⣤⣤⣀⣤⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠉⠙⠛
⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⠸⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⢻⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⢸⠇⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢹⣿⡟
⠀⠀⠀⠀⢹⣿⡟⢻⣿⡟⠈⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠘⣿⡇
⠀⠀⠀⠀⢸⣿⠇⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣷
⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⡟
⠀⠀⠀⠀⢸⣿⠀⠀⢹⣿⠀⠀⠀⢠⣶⣶⣿⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⠀⠀⠀⣿⡁⠀⠀⠀⣸⡃
⠀⠀⠀⠀⢸⡟⠀⠀⢸⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣼⣿⣇⣀⣀⣀⣿⣷⣀⣀⡀
⠀⠀⠀⢠⣾⣿⠀⠀⢸⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣶⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶

⠀⠀⠀⢠⢢⢢⣠⢄⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⢦⡤⣀⣠⡄⣦
⠀⠀⠀⠈⠁⠉⠛⠚⠀⠀⠀⠀⠀⠀⠀⡶⢴⢴⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⠋⠛⠃
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠸⠐⠂⠇⠀⠀⠀⠀⠀⠀⠀⢠⣄⡄⢤⡤
⠀⠀⠀⢰⢢⣰⢶⣴⡆⠀⠀⠀⠀⠀⠀⠈⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠱⠐⠒⠘⠑⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠘⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡆⠀⢀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠘⢇⠀⠇⠀⠀⠀⠀⠀⠓⠋⠐⠊⠋⠀⠀⠀⠀⠀⣶⠀⡆⠀⢶⠀⢰⠚⠀⠀⠀⠀⠀⠀⡄⢠⠀⣰
⠀⠀⢀⠀⡀⢀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢰⢰⢸⠀⠀⠀⠀⠀⠀⠿⠄⠇⠀⠸⠀⠸⠤⠀⠀⠀⠀⠀⠀⢃⠈⠈⠘
⠀⠀⢸⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠆⠰⠀⠄⠀⠀⠀⠀⠀⠀⠈⠮⠆⠇⠥
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⠄⠀⠆⠆⠆⠦⠀⠀⠀⠀⠀⠀⢰⠖⠰⣀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀
⠀⠀⠀⣀⢀⡘⢁⢀⠀⡀⠀⠀⠀⠀⠀⡶⡄⡆⠀⠀⠀⠀⠀⠀⠀⡐⠂⣀⣀⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠈⡇
⠀⠀⢸⡬⢫⡼⣜⡞⣾⠁⠀⠀⠀⠀⢄⢁⠁⡁⢀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠈⡁⣈⠁
⠀⠀⠈⠀⠀⠁⠉⠁⠁⠀⠀⠀⠀⠀⣈⢉⡉⡀⢈⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⢀⠄
⠀⠀⠀⠤⠀⠠⠀⠄⣤⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠁⠀⠀⠀⠀⠀⠘⠀⠊⠉⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣟⠛⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⡿⣦⣾⡿⢃⣤⣶⡿⠛⠉⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣯⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⢀⣀⣀⣠⣤⣤⣾⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠐⠶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠋⠉⠉
⠀⠀⠀⠀⠈⠛⠋


⣶⣶⠶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⡿⠉⠈⠙⠶⣄⡀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⢼⣿⣷⣄⡀⠐⣿⣿⣷⣤⣾⣿⣿⣤⠀⠀⠀⠀⠀⢀⣤⣾⣿⡯⢄
⣿⣿⣿⣿⣿⡟⠀⠀⢀⠀⠀⡤⣿⣿⣿⣿⣦⣻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣴⣿⣿⣿⣿⢤⡀⠀⡀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠀⣠⣿⣿⣿⣿⣿⣿⡋
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠹⣿⣿⣿⡜⠃⣰⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠈⠛⣛⣿⣾⣿⣻⣿⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣙⠋⠉
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠛⠁
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⢿⠎⢻⣿⣿⣿⣿⠏⢿⣯⢉⡀
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣤⣀⣴⣿⣿⣿⣿⠁⠀⠸⣿⣦
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⡄
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠋⠈⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢻⣧
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠁⠀⢿⣦⣄⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖⠚⠉⠛⣷⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡒⠀⠈
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢈⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠉⠙⢷⣦⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠙⠉⣼⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⢸⡇⠁
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡍⢷⣤⡀⠀⠀⠉
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⣽⡋⢰⡿⠁⠈⣿⡇⢦⡄
⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣾⠀⠈⠀⠀⠀⠃
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⡅
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠁
⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡄⠀⠸⣇⠀⠀⣧⠇⠀⠀⠀⢀⡞⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠹⡹⡀⠀⣿⡄⠀⠛⢀⡆⠀⢠⣿⠁⠀⣠⣾⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠘⠿⡀⠙⠇⠀⣧⢧⠀⠀⣼⡇⢠⢯⠇⠀⣜⡵⠁⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠰⣆⠀⢻⢸⡄⠀⠿⢠⠏⡾⠀⠘⢋⡤⠀⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠈⠻⣟⡲⢦⣄⡀⠀⠀⠈⠻⣗⠦⣀⠈⠀⣸⣤⠷⠤⣤⣏⣼⠁⠀⠴⠟⠁⣀⣠⣴⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢭⣷⣤⡀⠀⠈⠑⢮⣹⠞⠁⠀⠀⠀⠀⠈⠙⢦⣀⣤⠶⣛⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡴⠋⠁⠀⠀⢀⣀⡠⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠿⠿⠿⢾⣿⣟⣛⣛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⠤⠤⣤⣌⡙⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠦⣍⡉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⡭⠋⣹⣦⡀⠀⠀⠀⠀⢀⣠⠞⠛⠲⠦⣍⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠒⠋⠁⠀⣴⢋⡴⠋⠛⠒⢲⡟⢻⡀⠘⣗⢄⡀⠀⠈⠉⠛⠻⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⠏⠀⢀⡄⠀⠀⢳⡈⣇⠀⠘⢦⡑⣄⠀⠀⢤⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⣠⣿⠞⠁⠀⢠⢿⠃⠀⡄⠈⢧⢻⡀⠀⠀⠙⠾⣆⠀⠀⠈⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣴⠟⠁⠀⠀⠀⣿⠋⠀⠀⣿⡀⠈⣿⡇⠀⠀⢲⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠘⣿⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣷⣦⣄⠀⢀⣤⣄⡀
⠙⠿⠿⣿⣷⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢈⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠛⠙⠛⠋⠉⠉⠀⠀⠀⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡿⠿⠟⠛⠛⠛⠿⢿⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣡⣴⣾⣿⣿⣿⣿⣷⣦⣌⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣠⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣦⣉⠛⠛⠿⠿⠟⢛⣉⣴⣾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡉⠻⠟⢿⡿⢿⡿⠻⠟⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣿⣄⣴⣦⠀⣴⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣿⡏⠀⠸⣿⣿⡿⠁⠀⠉⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢹⡟⠁



⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⣶⣶⡄⠀⢠⣶⣆⠀⠀⢠⣶⡄⠀⠀⣴⣶⡀⠀⢠⣶⡄⠀⠀⢠⣶⡄⠀⢀⣴⣦⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⠀⣽⢿⠀⠀⠀⠿⠇⠀⠀⠚⠛⠗⠀⠀⠟⠻⠂⠀⠚⠛⠛⠀⠀⢸⣿⠁⠀⠀⣿⣟⠀⠀⢨⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠈⡀⣶⠠⠟⠀⢻⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣇⠀⠀⢸⣿⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠻⢌⠃⠀⠀⣷⡼⣷⣄⡀⠀⡦⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈⠱⢦⠀⠈⣷⠈⠀⠻⡀⠰⢀⡆⢰⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠸⣦⡀⠄⠀⠀⠀⠃⠀⠀⢃⡐⣻⣿⣤⣝⢁⡄⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⡀⠀⢀⠈⣷⣯⣿⡿⡍⠧⣉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡀⠀⠀⠀⠀⠀⠛⢦⣙⢿⣦⢫⡘⣿⡄⠙⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⡀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⠀⠀⠀⠰⠾⠯⣙⣶⠿⠋⠙⠿⠀⠀⠀⠀⠀⠀⠀⡰⢻⠟⢀⣀⠀⠀⢀⡀⢀⣾⡟⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣛⡛⠛⠉⠒⠂⢠⠆⣡⡴⠋⣠⠀⢰⡉⠂⠠⠤⠤⠤⠀⠠⠚⠁⣡⡴⠛⣉⡼⢓⣀⠄⢺⡿⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⠦⠄⠀⣸⣿⣿⠆⠛⠁⡤⠀⠻⢦⣄⣤⣄⡠⠶⠞⠛⠛⢉⣴⠟⣩⠞⠈⠁⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠶⠾⠟⠹⣿⠁⠀⣰⣿⠣⠀⡀⢚⣛⣋⣡⣤⣤⣄⠀⠀⠋⠀⢀⠰⠾⠿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠀⠀⠙⠀⠀⢻⣿⣄⠀⣘⠷⢦⡄⠀⠈⠉⣉⣉⣳⢶⣤⣀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠙⡸⠇⠀⠙⡲⣤⠠⣴⠒⠾⡋⠙⠻⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣈⣹⢷⠻⣿⢢⣾⣷⣙⠳⠀⠻⢆⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣻⠀⢻⣿⢿⡴⡟⣸⡌⢻⣻⣄⡀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠶⠞⠀⠻⣆⢻⣮⠻⠃⡿⢠⢄⠻⢮⣭⡁⠘⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠖⠀⠘⠀⠀⠘⢆⡙⠳⡀⢳⣿⢸⣦⡀⠙⠛⠷⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣤⠈⢳⡄⢹⣾⣿⠈⢻⣿⡄⢠⠒⠝⠣⠀⠈⠁⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣀⣴⣿⡿⠀⠀⠁⠀⠻⣿⣷⡜⢦⡙⠂⠉⠂⢴⡀⠀⠀⠀⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡁⠀⣠⢀⡆⠀⠀⠀⠙⡿⣿⡄⠙⣦⡀⢠⡄⠱⣄⠀⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣴⢶⣾⣷⠟⣡⣾⠃⣰⠀⢠⡀⠻⣿⣷⠀⠹⣿⡌⠳⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡍⠀⢸⣿⡏⢰⣿⠃⡼⠛⣇⠈⢷⣀⠙⣿⣷⡀⠙⢿⣦⡀⠀⠐⠒⢉⣡⣤⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢼⡇⢨⠁⠀⣿⣷⡀⢻⣷⡈⠻⣷⣄⡀⠙⠿⠶⠶⠾⠛⠛⠉⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⠀⠀⠀⠘⢿⣇⠈⠻⣿⣄⡈⠙⢿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⢳⡄⢀⠹⣷⣄⠈⠙⠛⠳⢶⣍⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣧⠈⠻⠿⠿⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣌⠒⢶⣤⣄⠉⠻⠿⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠂⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠈⠃





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

⠀⠀⢠⣴⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣶⡀⠀⠀⢠⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⢠⡏⢸⢀⣤⡀⠀⠀⣤⣤⣤⡴⣶⣿⣿⣿⡿⣿⡟⠛⠿⢿⣿⣷⡀⠀⠈⢿⣿⣟⠉⠉⠉⠉⠁
⠀⠀⢸⣇⢸⣿⣿⡷⠀⠀⠀⠐⠻⠶⠛⠛⠋⠁⠀⢻⣿⣿⣶⣿⣿⣿⠃⠀⠀⠀⢻⣿⣷⡄
⠀⠀⢸⣿⢸⣿⣿⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⢻⣿⠛⠁⠀⠀⢠⣴⣿⣿⣿⡇
⠀⠀⣼⣿⣸⣿⣿⡿⠋⠉⠉⠉⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣄
⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢿⣿⣿⣶⣼⣿⣿⡆
⠀⠀⠈⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠀⠀⠀⠀⢸⡿⢹⣿⠿⠃⢿⣷
⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣀⣀⣀⣀⠀⣩⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢸⡇⠀⠀⠀⠻⣧
⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠛⠉⣯⡉⠉⠻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣼⠃⠀⢸⠁⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⠈⠉⠁⢹⣿⣿⠀⠀⠀⠀⠀⠛⠂⠀⠈⠉⠀⠀⠀⠀⡈⠛
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢀⣰⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⡇
⠀⠀⠀⠀⣿⣄⠀⠀⣀⡀⠀⠀⠀⠀⠘⢿⣿⠿⠿⢿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⣿
⠀⠀⠀⠀⠈⠉⠁⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⠀⠀⠀⠀⠀⢠⣾⣥⡴⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣰⣿⣿⡿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⠿⠁⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⠀⣾⣿⣿⠉⢡⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⢶⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⣿⣶⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⠁
⠀⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇⠀⠀⠈⠉⠛⠛⠻⠿⢿⠿⠿⠟⠛⠛⠋⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⣀⠁⠑⡌⠦⠀⣤⣀⠉⠀⢀⡷⣤⡀⠀⢕⣠⣀⣛⣀⣄⡊⠀⠀⠰⣿⡍⢿⡀⠀⠀⠈
⠀⠀⠁⠀⠐⠀⠀⠀⠈⠁⠀⠀⠛⠛⠋⠁⠀⠀⠀⢀⣄⠀⠀⠀⣀⣠⣄⠘⢷⠈⣿⠀⠠⠴⠤⠴⠄
⠀⠀⠀⠀⣠⡟⠛⢦⠀⠈⠉⢉⣳⣦⣶⠀⠀⠀⢀⣸⣇⣈⡂⠈⠁⠶⢿⣦⣀⣀⠈⠀⢀⡀⠀⠀⡠
⠀⠀⠰⣾⣿⣿⠃⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠉⠉⢉⠉⠁⠀⠀⡀⠀⠈⠉⢉⠀⠊⠛⠶⠖⠚⠁⠁
⠀⠀⠀⠀⢈⡄⣠⠄⠀⠀⠀⠰⠤⣄⣠⣄⣤⠤⠆⠀⠀⠉⢟⣿⣾⣿⣿⣗⠏⠁⠀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⢀⣼⡿⢋⣵⠕⠀⠀⠀⠈⠉⠉⠉⠉⠙⢉⣁⡀⠀⡤⠤⢤⣀⣶⣀⡤⠤⡤⠀⠀⢀⣄⠈⠙⠦
⠀⠀⠉⠉⠚⠋⠁⠀⠀⢠⣾⣿⣦⠀⢀⠀⣤⣟⣵⠍⠃⠀⠐⠘⠛⠛⠛⠃⠀⠀⢠⣶⣮⣩⣶⣦
⠀⠀⠑⢦⣀⣀⡤⠊⠀⣼⣁⣦⣹⡄⣀⣀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⣤⢿⡿⣤⠀⠀⠀⠀⠀⠀⠀⠙⣿⢻⠟⠀⠀⠀⠀⠀⣠⣀⡀⠈⠀⢠⣤⠀⢀⣼⣿⣥⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠔⠀⠀⠦⠀⣠⣛⣌⡂⠀⠀⣠⣆⣼⣷⠤⠤⠀⠀⢾⣿⣆⠀⠀⠉⠙⠛⠛
⠀⠀⠶⠀⠀⠛⠀⠀⠁⠀⠋⠀⠀⢀⣉⡉⠉⠀⠠⢄⡉⠉⠉⢀⡠⠄⠀⠀⠀⠉⠛⠿⠶⠶⠶⠦⠤⠄
⠀⠀⠀⠀⢀⣤⣾⠅⠀⠀⠀⢀⣴⣟⣉⠉⠓⠀⠀⢀⣙⣿⣿⡉⠀⠀⠙⠑⠶⣤⣀⡤⣄⣠⡴⠖⠙⠁
⠀⠀⠀⠈⠋⠙⠃⠀⠀⠀⠀⠋⠙⠛⠋⠀⠀⠀⠀⠘⠋⠛⠋⠁⠀⠀⠀⠀⠈⠚⠛⠛⠛⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⢀⣶⣾⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⢠⣿⣿⣿⣿⡟⠈⢨⡿⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠊⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡤⠐⠐⢠⣿⣶⢞⣷⡀⠀⠀⠀⣰⠀⢤⢀⠀⠙⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⠻⢿⣾⡿⡇⠀⠀⠀⡟⢀⢀⠀⢠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠉⢠⡀⠀⠋⠔⠁⠀⠀⢰⠀⠀⠸⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠀⡘⠈⠀⠀⢱⣶⢶⠀⠀⠀⠀⠸⠀⠀⠀⠩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢚⡄⠀⠀⡠⠛⢁⣘⡄⠀⠀⠀⣀⡀⠀⠈⠓⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠉⡉⠔⠋⠀⣠⢷⣌⠁⠀⠀⠀⣹⡁⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣤⣤⣀⣠⡇⠘⠁⠀⠀⠀⠀⣼⣿⣶⣄⠀⠲⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⣄⠐⠛⠋⠉⠟⠛⣛⡻⢾⡷⡂⠀⠀⠈⢲⣅⠈⠉⢻⣧⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣷⡿⠷⢦⣤⣀⣀⡨⠄⠡⠀⠀⢡⠀⠀⠀⠈⠘⠀⠀⠀⢹⡄⢋⠇
⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠉⠻⣧⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢸⠿⠁
⠀⠀⠀⠀⠀⠀⠢⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢦⡀⠂⡴⠿⠶⣄⠀⠀⠀⠀⠠⠶⠖⠋
⠀⠀⠠⡠⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠤⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡗⠀⢸⣿⣿⣿⠀⢹⢈⡇⠀⠀⠀⠀⡆⠀⠀⢸⣿⣿⣿⣿⣿⣯⣉⣿⡿⣿⣿⣿⣟⡋⠁⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢰⣿⣿⣿⣿⣿⣿⣷⢸⡇
⣿⣿⣧⠀⠈⣿⣿⣿⠀⣾⡾⢣⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣏⣹⡏⣿⠉⠁⠀⠙⠓⡂⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣾⣿⣿⣿⢸⣿⣿⣿⠀⡇
⣿⣿⡏⠀⡀⣿⣿⣿⠀⣼⡇⠛⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣬⠹⠿⣿⠇⢹⡦⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⡟⣼⣿⣿⣿⢀⡏
⠹⣿⡇⠀⢹⣿⣿⣿⡀⣿⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⣟⣥⣤⣄⢿⠀⣶⡀⣀⠀⠀⠀⠀⠀⠀⢠⣴⡄⢀⣀⣀⣤⣤⣺⣿⣿⡇⠈⣿⢿⣿⣷⡇
⠃⣿⡁⠀⢸⣿⣿⣿⡃⣿⡷⠀⠀⠀⠀⠀⠃⠀⠀⠺⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣀⣀⣴⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣹⠇⡄⣿⢸⣿⢹⡇
⠀⣿⡇⠀⢸⣿⣿⣿⣧⣿⡇⠀⠀⠀⠀⠀⡇⠀⠀⡘⢼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⣿⣿⣿⣯⣥⣾⣿⣿⣿⣯⢸⣦⣡⡏⢸⣿⢸⡇
⠀⡿⠃⠀⠘⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⢰⣿⡅⠹⣿⣿⣿⣿⣯⣉⣀⣰⣿⣿⣿⣿⣿⣿⡗⠘⣿⡿⠿⢯⣉⣉⣽⣿⣿⣿⣿⢰⣿⣿⡇⢸⡏⠈⡇
⠀⡇⠀⠀⠀⣿⣿⣿⣼⣿⡅⠀⠀⠀⠀⠀⢰⠀⢠⠀⣿⣿⣷⡶⠛⠉⠁⠷⠾⢷⡷⠶⠿⢟⡋⣹⣿⣿⡿⠀⢿⣿⠛⠿⣤⣬⣭⡟⣻⣿⣿⢸⣿⢿⡇⠈⡇⠀⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢰⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣽⣿⣿⡟⠀⠘⢿⣆⠀⠀⠉⠁⢺⣿⣿⡟⢘⣿⣼⡇⢠⡆
⢸⡆⠀⠀⠀⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⡿⣿⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⢞⡛⢿⣿⡋⠀⠀⢸⡿⣿⣦⣀⠀⢸⣿⣿⡇⣸⣻⣿⡏⠘⡇⢸⠂
⠸⣅⠀⠀⠀⣿⣿⣿⡇⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡟⠀⢀⣐⣶⣿⣿⢛⣩⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⢿⣿⣆⣽⡿⣿⡇⣼⣿⣿⣧⠀⠇⣼
⢠⣿⠄⠀⠀⣿⡏⢿⡇⣿⣿⠀⠀⠀⠀⠀⢸⠀⢻⡇⢹⣿⣿⠧⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⡇⣿⣿⣿⢿⠀⡆⣿
⢸⡿⠀⠀⠀⣿⡇⣨⡇⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸⡇⢸⣿⣿⡆⠷⢣⣿⣿⣿⡿⠿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣻⣿⢸⠆⠁⣿
⢰⣏⠀⠀⠀⢺⣧⢹⡇⢸⡇⠀⠀⠀⠀⠀⢘⠀⢸⡄⣸⣿⣿⣿⣾⡘⣟⢻⣿⣿⣿⣿⣏⣹⠋⢛⠉⡏⠙⡏⢿⣹⣿⣿⣿⡿⣻⣿⣖⣯⢀⣿⣿⣿⢀⣧⠀⣿
⢸⣿⡄⠀⠀⢸⣿⢾⡇⢸⡇⠀⠀⠀⠀⠀⢸⠀⣿⣴⣿⣿⣿⣿⣿⣷⡙⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢃⣯⡨⣿⣻⣿⣾⣿⠀⣿
⢸⣿⡇⠀⠀⢸⣿⢸⡇⢸⣿⠀⠀⠀⠀⠀⢸⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣌⠸⡿⣿⣎⣩⣙⣋⣯⣯⣽⣽⣾⣿⣾⣿⣿⣿⣿⣿⣿⣼⠇⠿⣿⢻⣿⣿⣿⠀⡟
⠸⣿⡇⠀⠀⢸⡇⢸⡇⢸⡟⠀⠀⠀⠀⠀⠘⡂⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣠⣌⠻⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢠⠄⣿⢸⣿⣟⣟⠀⡇
⠀⣿⡇⠀⠀⢸⡇⢸⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣜⣿⣿⡿⣿⣿⢿⣿⣏⣿⣿⣿⣿⣿⣿⡟⣿⣸⠘⣿⢸⣿⣿⢿⣳⡇
⢸⣿⡇⠀⠀⢨⡇⢸⡇⢸⢿⠀⠀⠀⠀⠀⢠⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣋⠁⣀⣼⣧⣽⣿⣿⣿⣿⣿⣿⡷⠠⣿⣿⠛⣿⢸⡟⢸⢸⢻⡇
⢸⣿⡇⠀⠀⢸⡇⢸⡇⠘⠸⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣿⣿⢸⣿⣼⡇⠘⢸⢺⡇
⣼⣿⡇⠀⠀⢸⡇⠸⣇⠀⣱⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣟⣿⣷⣸⣶⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡄⠀⠀⠀⠀⠀⢀⠁⣠⡔⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⢿⡀⠀⠀⠀⣀⣴⠾⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠈⢳⣤⣴⠾⠛⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⣿⣶⣿⣿⠟⣫⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋
⠀⠀⠈⠻⢆⣿⡄⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿⣿⠉⠙⠻⣿⡿⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠉⠙⠙⠲⠦⠴⠁⡆⠀⠀⠀⣿⡏⠀⠀⢠⡿⠀⠀⠙⢷⣤⣀⢀⣠⣤⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⢰⡿⠁⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡉⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠗⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠄⠀⠀⠀⣶⠟⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠟⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠋⠀⠀⠀⢀⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣤⣤⣤⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢀⣀⡀⣀⠤⠴⠞⠋⢉⣧⣄⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢳⣄⠘⠄
⠀⠀⠀⠀⠀⠀⠐⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠶⢶⣖⣒⣚⣒⣲⣦⠼⠧⣀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢢⣻⡤⠴⠒⠒⢒⣒⣛⣛⣛⣯⣭⡭⠭⠭⠶⠶⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄⠙⠢⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣹⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠊

⠼⣅⠃⠐⢄
⢫⡀⠀⠤⠎⠦⠀⢢
⠀⠙⠦⣄⣀⡀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠠
⠀⠀⠀⠀⠉⠹⣆⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⣇⡤⠒⠀⠀⠀⠀⢀⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠈⠓⢄⡀⠀⣹⣦⣤⣀⢀⣤⣶⠟⢈⡄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠶⣄⠀⠀⠀⠀⠑⢦⡘⣿⡉⠉⠉⠉⠀⠀⠘⣷⡀⢀⡴⠃⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠘⢷⣄⠀⠀⠀⠀⠉⠸⣿⣦⠐⠀⠀⠀⢀⣿⣷⡈⠷⣄⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⢄⡀⠀⠀⠀⣿⣿⣦⣀⣀⣠⣾⣿⣿⣿⣇⠈⠓⢄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢯⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠋⢻⡀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠈⢧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⢸⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠲⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⡄⠀⣀⡄⠸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣏⠀⣠⠀⠀⠀⠀⢠⠎⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣿⣿⣿⣿⡀⠁⠂⠀⠀⡐⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⢿⣿⡏⠀⠀⠀⣄⠀⠸⠆⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢀⡤⠀⠀⠀⠀⣶⣾⠟⣠⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠋⠐⠂⠀⢄⣠⡿⠷⠚⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠟⠀⣠⠖⠒⠚⠧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠸⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡤⠊⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠏⢠⡴⠋⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣉⣽⣦⣀⡀⠀⠘⢀⠤⠒⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⣹⣿⣿⣿⣶⠆⠁⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠋⠉⢈⣿⣿⢀⡼⠁⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⣈⣭⠷⠞⠛⣿⠟⠀⣸⣿⣿⣿⣴⣾⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢛⣭⡶⢾⣿⣿⠏⠀⣼⣿⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⣾⣿⣿⠏⠀⢸⣿⣿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⣿⣿⣿⠿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣹⣿⢛⣇⣶⣤⣼⣿⣿⠟⣦⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⢻⣿⣿⣿⣿⣿⣿⢋⡠⠏⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠹⣮⠍⠛⢻⣿⣿⢿⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡇⠀⠀⠀⠀⣼⣿⡇⠘⢿⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢩⣿⣿⠁⠀⠈⣷⣄⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇⠈⠀⠀⢻⣹⡟⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⢸⡇⠀⡆⣿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠸⠿⢘⠃⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠘⢁⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣾⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠸⡇⠸⢿⣧
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⠀⠄⠛⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣶⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠘⢷⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⣀⢻⡄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⢻⡇⠀⠀⠀⠀⣼⠁⠙⠲⠦⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢸⡇⠀⠀⠀⢰⠇⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠦⣄⠀⠀⠀⠀⠳⡄⠀⠘⡇⠀⠀⠀⡾⣷⣶⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⡗⣀⠀⠀⠀⠹⣄⠀⡇⠀⠀⢸⠃⠀⣩⠋⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢥⣄⡀⢸⡇⡿⠳⣄⠀⣀⠼⠶⠓⠶⠦⢯⣀⡜⠁⠀⠀⠀⢀⣠⠴⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠉⠙⠓⠧⣄⣠⠟⠁⠀⠀⠀⠀⠀⠀⠉⢳⡄⢀⣠⠖⠋⠁⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⠒⠦⢤⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⣀⣀⣀⣤⠤⠤⠶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠚⠋⠉⠉⠉⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣄⠀⠉⢉⣽⡷⠒⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⣠⠞⠙⢲⣤⣄⣀⣀⣤⡴⢾⠁⠀⠈⠓⢦⣾⡟⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⡔⠁⠀⠀⡾⠀⢠⡇⠀⠈⡇⠀⠳⡄⠀⠀⠀⠘⠦⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡴⠋⠀⠀⠀⣼⠃⠀⢸⠃⠀⠀⢻⠀⠀⠙⣆⠀⠀⠀⠀⠈⠳⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠋⠀⠀⠀⠀⣸⣧⠤⠴⠾⠀⠀⠀⠸⡆⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣰⠚⠛⠉⠉⣹⠀⠀⠀⠀⣷⠀⠀⢀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠲⢦⣿⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⡴⠒⠒⠀⠄⠀⠀⠀⠀⠀⠀⣠⡀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠐⠤⣠⢼⣷⡞⠁⠀⠀⠀⠠⠂⠉⠀⠀⠁⠐⢤⡀⠀⠀⢸⠁⣀⡀⠀⡰⢀⡔⠁
⠀⢠⢸⣧⣤⣽⣆⣀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠑⡄⠀⠘⣷⣿⠷⢊⣱⠏
⢀⠀⢸⡟⠉⠻⣍⠉⠀⠀⠀⠀⠀⢰⠟⢧⠀⠀⠀⠀⠀⠀⠀⢳⡀⡀⠀⠀⠀⠀⠰⠀⠀⠹⣷⣶⠟⠁⠀⠀⠀⢀⡀
⠀⠳⣄⣧⠀⠀⠈⡣⠂⠀⠀⠀⠀⡟⠀⢸⠀⠀⠄⠀⠀⠀⠀⠀⢿⣟⣀⣠⡄⠄⠀⠀⠀⠀⢿⠃⠀⠀⢀⡤⣤⣿⡃
⠀⣆⠀⢹⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⢀⣀⣀⣀⣀⣀⣼⡟⠀⢀⣀⣠⠀⠀⠀⠀⢸⠂⠀⠀⠜⠀⣿⣏⢳
⠀⣧⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢩⠉⠉⠉⠉⠛⠛⠋⠉⠁⣀⠂⠀⠀⠀⡿⠀⠀⠀⢠⡾⠿⣿⠨
⠀⠻⠦⣤⣾⢟⣥⠈⠳⠤⠀⠀⠀⠀⠀⡶⠂⠐⠂⠀⠙⠢⠄⣀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠘⢧
⠀⠀⠀⠘⢿⣶⡏⠁⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠨
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠃⠱⣄⠀⠀⠀⠀⣤⡀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⠶⠟⠂⠀⠀⡼⢻⣧⣴⠟⠁⠀⠀⢀⣴⣆⣀⠀⠀⠀⢸⣯⠉⠛⠒
⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣼⣇⠈⢻⡇⠀⠀⠀⢲⡿⠶⠶⣿⡀⠀⢠⡾⠿⢦⣀
⠀⢸⣰⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⢷⣿⠁⠀⠀⠀⠈⠙⢶⣶⡿⠃⠀⠘⣿⣶⠇⢹⡆
⠀⢰⣿⠷⢤⠴⠁⠀⠀⠙⢦⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⡄⣀⢸
⠀⠈⣿⣶⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠛⠂⠛⠁
⠀⠀⡿⣿⠀⠀⠀⢀⠀⠀⠀⠀⣼⣟⢛⣷⡤⠦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠤⠤
⠀⢰⡇⢹⠀⠀⠀⢸⠀⠀⠀⠀⠙⣿⣭⠀⠹⣆⠘⠂⠀⠀⠀⠀⣀⠀⢸⢃⡀⠀⠀⣀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⡠
⠀⠸⠁⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠑⣾⣽⣇⡴⠊⠀⠀⠀⠊⠀⣇⣤⡤⢀⣴⡞⠁
⠀⠀⣀⠀⠀⠀⠀⢸⡇⠀⠐⠀⠐⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣚⣩⡞
⠀⠀⠈⠀⢿⣇⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣄⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠡⢄⡙⣻⣏⣀⡀
⠈⠀⠀⠠⢼⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠏⢀⡘⢿⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠢⢠⢼⣏⠐
⠀⠀⠀⠀⠉⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠻⣿⠟⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣯⠉⠉⠛⠣⢖⣒⣒⡒⠒⠒⠒⠒⠒⣒⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣡⡟⣠⠊⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠈⠄⠙⣶⣒⣒⣒⡒⠂⠠⠤⠤⢤⣾⠟⡰⠁⠀⠀⢀⠌⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⡀⠀⠀⠈⠀⠘⢯⠉⠉⠉⠉⠛⠛⢳⡿⠃⠜⠀⠀⠀⠐⠁⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⢄⠀⠀⠀⢢⡈⢳⡀⠀⠀⠀⢀⡿⠁⠀⠀⠀⣠⠊⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠱⡀⢷⡀⠀⢀⡾⠡⠀⠀⠀⡴⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡹⡷⡀⠀⠀⠘⣄⢳⣀⡞⣡⠃⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣎⢂⠀⠀⠘⢂⣻⡀⠃⠀⠐⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠿⠟⠛⠉⠉⠉⠛⠓⠲⠦⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣤⠴⠚⠉⠉⠉⠉⠉⠒⠶⣄⠀⠉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢡⡾⡋⢰⠒⠀⠀⠀⠀⠀⠀⠀⠐⢦⡙⢦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢰⣏⡐⣅⣀⣿⠛⠂⠀⠀⠀⠀⠀⠙⢾⢇⡄⠳⣄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡞⡘⢍⣩⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣋⡆⢸⡆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣹⣭⢭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣰⠀⡅⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⠙⣧⡴⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⣽⢣⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⢧⠛⣴⡿⣃⠀⠀⠀⠀⠀⠀⠀⢀⡠⣷⠀⠋⡜⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠈⢳⡉⣽⢿⣶⡀⣄⠀⠀⣤⣶⣼⣿⠞⣡⠎⠀⣰⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠙⢦⣌⡛⢛⣷⡠⠾⣿⣛⣃⡵⠞⠁⣠⡞⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣈⠉⠛⠓⠒⠓⠉⣉⣁⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢯⣿⣿⠿⠿⠟⠋⠑⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣭⣥⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡤⣤⣤⣄⣀⣀⠀⣠⣤⣶⣾⣵⣰⠄⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⣾⣯⣻⣿⢿⣦⡷⠈⠻⣿⣭⣙⠻⠿⣿⣿⣿⣿⣇⠈⠈⠁⠐⢦⠙⠒⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣷⣍⠓⠄⠀⣠⡆⣼⣹⣿⣿⣶⣤⣙⠻⣿⣧⠠⠞⠓⠀⠘⠀⠒⢢⠳⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠉⢜⣽⣿⣿⠿⣿⣿⣿⣿⣬⣛⣷⣄⣰⠋⠛⠢⢤⠤⠖⠛⠻⣄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣮⡿⣿⣷⣦⡘⢲⠰⢀⠍⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣶⣮⣭⣡⡀⠀⠀⠉⠛⢿⣿⣿⣞⣿⣿⠀⣦⣤⣀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠙⢻⣿⣿⣿⡆⠲⡄⢹⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠀⠀⢀⠻⢿⡟⢻⠀⠘⣿⣿⣿⣿⣿⡢⠙⣿⣿⣷⠀⣁⣬⣔⣊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⣴⠛⠲⣏⠙⡆⠀⠀⡇⠀⣈⠓⡤⠙⠊⢳⡀⠘⣿⡟⢊⡡⠴⠒⠆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡭⣝⠲⣄⢈⣧⡘⢄⠓⠒⠊⣉⢹⠖⣉⡁⢸⡇⣾⡿⠡⡋⠠⣒⣚⡛⠛
⠀⠀⠀⠀⠀⠀⢀⠤⡘⢁⢤⡤⣙⢮⡣⠀⠉⠲⣍⣒⣠⠞⢀⣈⣁⣬⣾⣿⠟⠁⣀⠴⠋⣉⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⠥⣙⡲⠬⠕⠂⣠⣴⣾⣿⣿⣿⣿⣿⡅⠀⠤⠤⠥⣄⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡾⠛⠉⠻⢿⣿⣿⣿⣿⡇⠈⠁⢀⡢⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⢙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⡆⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠘⠁⠀⢀⣾⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⣠⣾⡇⠀⠀⠁⠀⠀⠀⠀⠀⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠀⠀⠹⣦⡀⠀⠀⣠⡿⣧⣀⣠⡾⠋⠸⣧⣀⣀⣠⠄⠀⢀⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠁⠀⠀⢻⡟⠶⠾⠛⠀⠈⠉⠁⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⣀⣼⠇⠀⣀⣤⠶⠖⠛⠛⠲⠶⣤⡀⠀⠈⠻⠶⠶⠶⠶⣲⡿⠋⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀⠀⣸⡇⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⢰⡟⠀⠀⠀⠚⣉⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠻⣟⠛⠋⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠘⢧⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⢀⡿⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢠⡟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡅⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣷⡀⠀⠈⠙⠓⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣾⡃⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢈⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⢠⡾⠋⠉⠛⠛⠛⠲
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⢀⣼⡀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⢀⣀⡴⠟⠁⠀⠀⢠⡾⠻⢧⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢹⠄⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠀⠀⣠⣟⣤⡴⠶⠲⢦⡀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣤⣀⠀⢸⡆⠀⠀⠄⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠸⠷⠶⣦⠀⣴⠟⠉⠙⠿⠁⠀⠉⠻⢦⣷⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢸⡇⠀⠀⢠⡆⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⠀⠀⠀⠺⠇



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠋⠙⢿⣿⣿⣷⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⣀⡠⢻⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠋⠉⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠉⠉

⠀⠀⢀⣤⣤⣤⣴⣶⣆
⠀⠀⠛⠛⠙⠛⠿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⢫⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⡀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣠⣤⣤⣿⣿⡆⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠙⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢦⡀⠀⠀⠀⠀⢰⠙⠻⢦⢀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢣⡀⠀⠀⠀⡸⠀⠀⢘⠀⠀⣀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠱⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠎⠀⠀⠀⠀⠑⠤⠀⣼⠃⠀⢀⣿⣆⡀⠄⠀⠀⠀⣰⠃⠀⠀⠀⠀⠑⢄⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠎⢠⠂⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡙⠅⣒⡀⠔⠐⠚⠁⠀⠀⠀⠀⠘⢦⡀⠙⠦⣈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⠔⠁⡰⠃⢠⠇⢀⠀⠀⢀⡴⢻⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⢄⠀⠙⠢⣀⠈⠑⠦⣄⡉⠙⠒⠦⠤⠤⠤⠤
⠀⢀⣀⣀⡤⠖⠋⣡⠔⠁⣠⠞⢀⡴⢃⡠⠋⠐⠂⠁⠀⠀⠹⣆⡀⠀⠀⠀⠀⠲⣦⣄⣀⡀⠀⡙⢤⡀⠑⢤⡀⠈⠑⠦⢄⣀⠈⠉⠓⠒⠶⠒⠂
⠀⠀⠀⠀⠐⠒⢉⡠⠔⢊⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠈⠙⠿⣤⣀⠀⠉⠒⠀⠉⠓⠦⢤⣀⣈⠉⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢔⠐⠖⠒⠈⢀⢨⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⠀⠀⠈⣎⢾⣿⣦⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠼⠃⠀⠀⣹⢸⣿⣿⣧⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠁⠉⠀⠀⠀⠀⢠⡟⣸⢿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣰⡟⣼⢳⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣼⠋⣰⢏⡿⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢠⡿⠃⣰⡏⣼⢣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣿⠃⢠⡿⢰⡟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠀⣸⡇⣼⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⣿⡇⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣷⠀⢹⡇⢸⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠘⣧⠈⢿⡀⢿⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⢷⡈⢳⡌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢦⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣦⣿⣿⣿⣿⣟⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣫⣷⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⠀⠀⠀⠀⠈⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣫⣿⣿⣿⢫⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢟⣵⣿⡿⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠉⢻⣿⣿⣿⣿⣷⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠙⢿⣿⣿⠿⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢄⡀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢀⣈⣳⡀
⠀⠀⢀⣀⠀⠀⢰⠃⠀⡠⠔⡺⠍⠉⠈⢯⣳⠀⢀⠀⠀⠠⠾⠃⠀⢀⣤⡄
⠀⠀⠈⠁⠉⠀⡇⡠⢊⡀⠀⠁⡞⠷⠀⠈⣿⠰⠟⣁⡠⠴⠒⠒⢦⠈⠁⠀⠀⢀⣤
⠀⠀⠀⣠⣀⠀⢹⡕⢁⢤⡄⠀⠀⢠⡶⠀⣿⠔⠊⠁⠀⢀⣤⠀⠀⢣⡀⠀⠀⠛⠁
⠀⠀⠀⠈⠉⠁⠀⠻⡄⠀⠘⠂⣀⠀⠀⢀⠇⠀⠀⠀⡠⠊⢀⣱⢤⡤⡧⣀⣐⣦⡀⠐⠊⠉⠉⠉⠑⠂
⠀⠀⠀⢀⡤⣄⣀⣀⠼⣦⣤⣀⣙⣅⡴⠛⠓⠚⠣⠴⢽⠦⣜⣡⣀⡫⢧⣄⣁⣼⡁⠒⠓⠒⠒⠊⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⣄⢄⡠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠉⠀⠀⠴⠤⠤⠤⠤⠤⠤⠦⠄
⠀⠀⠀⠀⠀⠑⠒⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣀⡤⠤⠤⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠒⠒⠓⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉