Twig Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠒⠚⠋
⠀⠀⠀⠀⠛⠀⠒⠦⢄⡄⡄⣀⢀⠋⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⡇⠢⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⣼⣰⢃⡤⠀⠀⠀⣴⠿⠟⠲⠀⠀⠀⠀⠀⢀⣾⠛⠆⠈⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣾⢁⡾⡆⣰⣿⠀⠀⠀⠀⠀⠀⠀⢀⡘⢧⡑⢀⢀⡀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣾⣇⣿⣿⢀⣄⠀⠀⠀⠀⣠⣼⠏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠿⣿⣷⣶⣶⣾⣿⣛⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠛⠻⠇⣆⠀⠈⢻⡿⢿⣿⣿⡏⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⡀⠀⠀⢀⡀⣿⠀⣰⡞⠀⠀⣽⣏⠠⠛⠛⠠⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠲⢶⣶⣶⣦⣤⣄⡙⢧⡀⠀⠀⠘⠻⣍⢀⣿⣧⣤⣾⣿⣏⡀⠀⣀⣠⣈⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣠⣄⣀⠀⠈⡉⠙⠁⠉⠙⢷⣿⣿⣤⣿⠟⠋⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠙⠋⣉⣿⣿⣿⣿⣧⣠⡆⠀⠀⠀⠀⠀⠙⢷⡶⠶⠶⠶⠖⠚⠁⠙⣿⣵⡦
⠀⠀⠀⠀⠀⠀⢐⡶⠊⠀⠀⢰⣶⣿⣿⣮⣿⣿⣿⣿⠅⠠⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢷⡄
⠀⠀⠀⠀⠠⣼⡟⢀⣀⣤⡼⢟⡛⣿⣿⣿⣿⣿⣿⡋⢀⣀⣀⡀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠚⣿⡾⠟⠋⠁⠀⠀⠔⢿⣿⡿⣫⣿⣏⣿⢀⠉⠛⠷⣄⠀⠀⣰⣿⣿⠏⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠐⣻⣿⣿⠙⣼⣿⠋⠀⣀⣶⣤⣭⣾⡿⠿⠿⢶⣾⣤⠐⣀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡿⠀⢹⣷⢂⠜⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⢿⣷⣆⡀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣼⣧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣧⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠶⠶⠦⣄⡀⠀⠀⠈⠙⠓⠒⠀⠀⠠⠤⠀⠐⠂
⠀⠀⠐⠂⠦⣤⣀⣀⣀⣀⣀⣀⣀⠄⠊⠁⠒⠒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣠⡤⠴⠛⠛⠛⠃⠀⢤⣀⡀⠀⠀⣀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⡤⠄⠀⠤⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈
⠀⠠⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠴⢶⣶⠖⠒⠒⠛⠲⠆⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠤⠠⠤⠤⠤⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⡀⠀⠀⠀⠉⠒⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣠⣀⣠⡀⢀⡄⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠃⢉⡁⠒⠀⠐⠂⠀⠀⠂⠐⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠋⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠠⠤⠤⢦⡵⠤⢤⣤⣀⣀⠀⠀⠁⠈⢩⠅⠐⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠒⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡈⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠊⠁⠒⡒⠒⠓⠒⠒⠒⠲⠶⠤⠤⠄
⠀⠠⠤⠤⠤⠴⠶⠒⠈⢫⠉⠉⠙⠢⠈⠀⢄⠀⠈⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡠⠀⠀⢀⠇⠀⠀⠀⠀⡰⠁⠀⠀⠊⠀⠀⣸⡧⠶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠛⠛⠾⢷⣶⣖⡒⢒⣊⣁⣀⣀⣼⣠⠾⢛⣉⠀⠀⠀⠀⠀⠠⠤⠤⠀⠐⠤⠤⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡏⢳⠀⢠⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⢷⡄⠀⠀⢧⣿⣼⢀⣯⣾⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢧⢸⡄⡗⠀⠀⠈⣿⣏⣼⣿⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢦⡈⢧⠀⠀⠘⣿⣿⢃⣠⣤⣾⣟⡻⠿⣏⣛⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⢣⡀⢳⣼⡆⠀⢀⣿⠿⠿⢿⣖⠲⠬⣝⣆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡌⠻⣄⠀⠙⣄⢻⡇⣰⣟⣥⠴⠶⣶⣬⠙⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠸⡄⠙⣆⠹⡆⠀⠈⢿⣿⣿⡿⠒⠛⠉⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣇⠈⠙⣄⠀⠙⣆⠹⡄⢷⠀⢠⡿⠋⠀⠉⠉⠙⠛⠉
⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⢿⠀⠈⢧⠀⠘⣆⠀⠈⢷⣿⣼⣠⡿⢁⣤⠴⠶⠒⠲⣦⡀
⠀⠀⠀⠀⢠⡿⠿⣄⠀⠙⠳⣄⠀⠀⢣⡀⠈⣧⠀⢹⠀⠀⠈⣿⣿⣿⣶⣫⡤⠴⠒⠛⠉⣹⠇
⠀⠀⠀⠀⠀⢧⠀⠀⠑⣄⠀⠘⢦⠀⠀⠱⡄⠸⡄⢸⡇⠀⠀⣿⡿⠟⠛⠓⠒⠒⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⢧⡀⠘⣇⠀⠀⠘⣆⣧⢸⡇⠀⣼⡟⢀⣀⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⢳⡀⣿⠀⠀⠀⢸⣿⣼⢀⣾⡿⢚⣉⣠⣤⠤⠤⠤⣬⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣇⣿⠀⠀⠀⠀⣿⣿⣿⣿⣯⣍⣀⡀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⣀⣼⠟⠋⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣴⣾⣿⣗⣒⠛⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠛⠯⣍⠉⠙⠓⠲⠤⣄⡀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠙⣦⣼
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠚⠋⠁
⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠈⠛

⣟⣛⣛⣛⣋⣛⣛⣛⣻⣿⣛⣿⣿⣛⣟⣿⣻⣿⣿⣛⣿⣿⣟⣻⣿⣟⣿⣿⣟⣻⣿⣟⣻⣿⣟⣻⣿⣿⣻⣿⣿⣿⣟⣿⣻⣟⣻

⠀⠀⠀⣴⢦⡀
⠀⠀⠀⢿⠀⠙⢦⡀
⠀⠀⠀⠸⣇⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⡴⠋⢸⠃⠀⠀⣼⠋⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠋⣹⠇
⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠈⢷⠀⠀⢀⡞⠁⠀⣿⠀⠀⢠⡇⠀⠘⣇⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⢸⡇⠀⡾⠁⠀⢰⠇⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⢠⠟⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣆⢸⠇⠀⢠⡟⣀⡀⠀⢸⡇⠀⠀⢸⡇⠀⢠⡏⠀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⢠⠟⣰⠏⢻⡄⠀⣧⠀⠀⢸⠇⣠⠞⠛⠋⢉⡁
⠀⠀⠀⠀⠀⠀⠀⢠⠶⢤⣄⡀⠀⠈⣿⡴⠋⠀⡿⠀⠀⢻⠀⠘⣇⠀⣾⣾⠷⠚⠋⠉⠉⠉⠉⠛⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠙⢶⣄⢸⡆⠀⠀⡇⠀⠀⠸⡇⠀⠘⣿⠋⠙⠳⠦⣤⣄⣀⣀⣀⣠⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠙⢦⣷⠀⠀⢿⡀⠀⠀⣿⠀⡼⢃⣀⣠⣤⣤⣤⣄⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣈⣿⡄⠀⠈⣧⠀⠀⣿⣾⠟⠋⠉⠀⠀⠀⠀⠀⠉⢙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⢳⣤⡟⠉⠛⠲⠶⠤⠤⠤⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⣸⠁⣇⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣄⠀⠀⢸⣿⡽⣆⢹⣸⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⣷⢹⠳⣄⠈⢧⢻⣿⢈⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠳⣄⠀⠀⠀⠀⢿⠣⣧⣹⣆⠈⢷⣿⡾⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣇⢹⢦⡀⠀⠀⠈⠳⣝⢿⣹⠀⣠⠟⠿⣿⣯⢭⣙⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣿⢎⣆⢳⡀⠀⠀⠀⠈⢺⣿⡾⠋⠀⠀⠀⠉⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠹⣞⢯⣿⣸⡇⠀⠀⠀⣠⣼⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⢷⣄⠀⠀⠀⠀⠀⠘⠻⣿⣿⡇⢀⣠⠾⠿⣿⣿⢭⣝⣛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡳⡕⢝⢷⣄⠀⠀⠀⠀⠀⠈⣿⡷⠟⠁⠀⠀⠈⠙⠻⠷⢾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠹⣎⠪⣮⡻⣯⣳⡀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠈⠛⢮⣿⣿⣿⣧⠀⣴⣿⣶⣶⠖⠲⠶⢶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢸⣟⣷⣙⢶⣄⠀⠀⠀⠀⠈⠙⠻⣿⡾⠻⢿⣽⣛⠳⠶⠟⠫⣭⣭⣭⣿⠿⠋
⠀⠀⠀⠀⠀⠘⣿⢿⣝⢷⣌⠳⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠙⠻⠿⠿⣭⠴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢧⡹⡣⡙⢿⣾⣧⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣮⣷⣌⡻⢿⣆⢸⠇⣀⣤⣶⠾⠟⠛⠛⣛⣒⣲⢶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣽⣿⣿⣿⡷⠶⢊⣉⠭⠭⢥⣰⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠙⠶⣬⣍⣒⣂⣬⡵⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⣶⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⣾⣿⡦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡼⠋⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢠⢏⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣰⣶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠉⠉⠉⠲⡀⠀⢠⡿⢾⠒⠲⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠶⣼⣿⡿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠃⠈⢇⠈⠳⣄⠇⣰⠋⠀⠀⠙⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠠⢄⣾⣿⣧⣀⣀⡵⣄⡀⠀⠀⠀⠀⠀⡴⠊⠁⢸⠀⠀⠈⠓⠶⢻⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⢀⡤⠽⣷⣤⣀⠀⢸⠀⣰⠂⢸⠂⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡴⠋⢻⠉⠓⢮⡓⣿⠒⠋⠀⠀⠀⢀⡞⢀⡴⣚⠛⠓⠒⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣀⣀⡤⠔⠋⠀⠀⠀⠉⠻⣆⠀⠀⠀⢀⣾⠖⢿⡉⠉⠉⠁⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣤⠤⣤⣤⣀⣀⡀⠀⠀⠈⢧⡀⢀⡞⠁⠀⠀⠉⠙⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠉⠓⢤⡀⠀⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠉⠛⠦⣷⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠙⡾⠁⢀⡤⠖⠒⠒⠒⠒⠲⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⢋⣀⣤⣤⡄⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡼⢿⣉⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⣤⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⠀⢧⡦⣿⣟⣡⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢺⠀⣂⣴⣿⢟⣡⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⡷⣿⣿⣿⣾⣿⣫⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⡄⠒⢀⡆⠀⠀⠀⠀⠀⠀⠀⢀⢁⠸⠈⣿⣡⣿⣿⣛⠛⢩⣭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣿⡳⢂⡿⠁⣰⠀⠀⠀⠀⢠⢠⢸⣷⡆⣽⣿⣿⣾⣿⣭⣥⣅⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣉⢟⢿⣿⠷⣾⡧⠈⠀⠀⠀⠓⠆⣛⣿⣧⣷⣿⢿⣶⣛⡻⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡐⢌⣿⣯⡿⡣⢞⠀⠀⠀⠀⠑⠠⠀⠹⣿⣿⣿⣴⣶⠥⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣏⡞⣿⠋⠀⠀⠘⠀⢷⣔⣴⣤⣿⣿⢯⣍⣁⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣖⠨⣿⣿⣿⣷⠀⡀⠀⠀⠄⠀⢻⡟⣽⣿⣿⡯⢭⣭⠙⠒⠒
⠀⠐⠠⡈⠀⠀⢀⠀⠀⠀⠸⢦⡉⢑⣶⣿⣿⡷⠀⠀⢸⡇⠰⣸⣞⣾⡾⠿⣍⠙⠛⠥⠤
⠀⠀⠠⣈⢦⡄⡖⢠⡔⠀⠀⡀⠿⣿⣾⡷⣿⠚⢀⣠⠞⢧⢀⣿⠟⠉⠛⠛⠥⣀⣀⠀⠀⠀⠀⠀⢠⠀⠀⡠
⠀⣀⣢⣬⣿⣷⣧⡟⠀⠀⠠⣭⣷⣍⡺⣿⣏⣴⡿⣔⡠⣸⣾⠋⠀⠠⠤⠖⠂⠀⠀⠀⡄⣀⣤⡆⢸⡴⠟⢫⡾⢁⠖
⠀⠀⠙⢞⠿⣿⣿⣏⡟⢰⣠⠄⣙⠻⣿⣿⣿⡽⠟⡁⣸⡟⠉⠀⠀⠀⣰⢃⣄⡴⠂⠜⣇⣹⣿⣧⠿⣷⣿⣪⣼⣥⣴
⠀⠀⠀⠴⠿⢿⣿⣿⣧⣿⣷⠚⠦⣴⣮⣿⡌⠀⢈⣴⡿⠥⢀⠀⢀⣴⣡⡟⠁⠚⢂⣴⣷⣿⣾⣿⣷⣶⣿⣿⣟⡓
⠀⠀⠀⠀⠉⢐⠲⣿⣿⣿⡁⢸⣤⣟⡟⣿⣿⣠⡿⠯⢤⡄⣠⣷⣿⣿⣿⣿⣶⡾⣿⣛⣿⡻⠛⠓⠛⠛⢯⣻⠝⠿⠃
⠀⠀⠀⠀⠀⢤⣔⠲⠶⢿⣾⣿⢠⡴⠋⢙⡿⣿⣁⣄⣾⡿⣍⠉⢯⠙⠗⠉⢿⣻⣿⡂⠉⠙⢶⡲⠝⠦
⠀⠀⠀⠀⠀⠀⠀⠈⣟⡓⠻⣿⣿⣶⡴⣛⣽⡿⠋⠉⠙⠻⠦⣕⡢⢄⠀⠮⠀⠉⠉⠿⠄⠀⠑⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⣹⣿⠃⢀⡤⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⠿⠊⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⢻⠟⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣫⠟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠶⣄⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⢷⠀⠀⠀⡾⠁⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣼⠁⠀⠀⣧⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠳⡌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠈⠙⠛⠛⠛⢧⠀⠀⠀⠈⠛⠶⡴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⡄⠘⣧
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠹⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣸⠟⠲⠦⣤⣀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠉⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⢻⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠈⢳⡄⠸⡆⢸⠆⣠⠖⣶⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣴⠞⠛⢦⠀⠀⠈⠛⣦⡘⣧⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠈⠻⣝⠷⣄⠹⢦⣧⠟⢰⡏⠀⡏⢿
⠀⠀⠀⠀⣴⠋⠀⠉⠳⡄⠀⠀⠀⠻⣦⣠⡞⠓⠶⢤⣄⣈⠳⣿⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⢧⡙⢧⡀⣿⢀⡿⠀⢰⠃⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⡿⠤⠤⠤⣤⣤⣤⣤⣤⣀⣀⣀⣉⣙⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠹⣌⢷⣸⣼⠃⢀⡟⢠⡏
⠀⠀⠀⠀⠙⠳⠶⠶⠛⠁⠀⠀⢀⣤⣄⠀⠀⠀⠀⣀⣤⠾⣻⠏⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡜⢾⣿⡟⠀⡼⢁⡾
⠀⠀⠀⠀⢀⣤⠦⣤⡀⠀⠀⠀⢿⣀⣨⠟⠚⠛⠋⠉⣠⠞⠁⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣷⣦⠙⠛⢿⠇⣼⢃⡾⠁⣀⣠⣀
⠀⠀⠀⠀⣾⠁⠀⠈⣷⠀⠀⠀⠀⠉⠁⠀⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡜⣧⠀⣿⡾⣡⠞⣡⠞⠉⢠⠟⣷
⠀⠀⠀⠀⠙⠦⠤⠾⠋⠙⠒⠒⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢸⠁⠀⡇⠸⣇⣿⠛⢡⡞⠁⠀⡴⠋⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢸⡇⠀⢸⠀⣿⡇⣴⠋⠀⢠⠞⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⢿⠀⢸⡆⢻⡿⠃⢀⡴⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣀⣀⠀⠀⢰⡟⠻⣍⠛⠶⣄⡀⠀⠀⠙⣷⡄⠀⠀⠈⢧⠀⡇⣾⠁⣠⠞⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⢤⣀⠀⠈⠉⠙⠳⢦⣷⣄⠈⠙⢦⡈⠙⢦⣀⠀⠘⢿⣆⠀⠀⠈⠳⣿⣯⠞⢁⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⠈⠙⠳⠦⣄⡀⠀⠀⠙⢦⣄⠀⠙⢦⣄⠙⠳⣄⠈⢿⣆⠀⠀⣼⣿⡥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣙⠳⣦⣄⣀⠉⠛⠲⢦⣄⣉⢳⣦⣀⠈⠳⣄⠙⢧⡈⢿⡆⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢶⣶⣤⣤⣽⣿⣃⣉⡓⣦⣬⣹⣾⣧⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣩⡿⠿⠛⣉⡿⢿⠉⢉⣩⠟⠋⡽⣿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⣤⣤⣤⣤⣤⣤⣤⠤⠶⣲⡾⠛⠁⢀⡴⠚⠁⣠⢟⣴⠛⠁⣠⠞⢡⡏⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⣤⠞⠉⠀⣀⡴⣿⠟⠀⢀⡴⠃⣠⠟⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠞⠋⠀⣀⣤⠞⢋⡾⠁⣀⡴⠋⢀⣴⠋⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠛⠉⠀⠀⣾⣡⠞⠉⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠄⠀⠀⠀⠀⠀⠀⠀⢰⡤⣾⣿⣷⠆⣤⡞⢡⣾⣿⣯⣀⣀⣿⣿⣿⣿⠺⣿⣿⢿⡿⣿⣿⣧⣠⣿⣿⣿⣶⣽⣿⣧⠀⠀⠁⠀⠙⠻⣷⣤⣤⠟⣷
⠀⠑⠀⠀⠐⠐⠒⠳⠮⣶⣿⣿⣯⡞⠋⣉⣴⣾⣿⢷⣌⠈⢟⣿⣿⡇⠈⣿⡏⣿⡇⢹⡇⢻⣆⢸⣿⣿⠀⠉⢻⣿⣄⠀⠀⠐⠂⠀⠈⠋⠑⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣯⣤⣟⠛⢿⣿⡷⠎⢠⠄⢸⣿⣿⠋⡺⣿⡟⢻⡇⢸⡇⠀⠻⣯⢹⣿⠛⠒⠛⢻⣿⣆⡀⠀⠀⠀⠀⠨⢀⠀⣠
⠀⠀⠀⠀⣀⣤⡤⢾⣿⠃⣽⣴⡿⢼⡏⠀⢸⣿⠇⠀⠰⢂⢼⣿⣿⡞⠱⣿⡇⠸⣷⣸⡇⠀⠀⠙⣿⣿⠀⠃⠀⠀⢿⡇⠀⠘⠒⢤⣄⣀⠈⡀⠉
⣀⡤⠖⠚⠉⠀⢠⣾⢿⣶⠟⠛⡁⠈⢧⠀⢸⡟⠀⠀⠀⢀⣾⣿⡇⠐⠀⢿⡇⠀⣿⣿⡅⠀⠀⠀⠘⣿⣆⠀⠀⠀⠘⣷⡀⠀⠀⠀⠙⠿⣨⣿⣯
⠌⠢⠀⢀⡀⢠⣿⠉⢘⣿⣆⡀⣿⡄⣸⣠⣿⠇⣀⣀⠜⣳⣿⣿⡇⠀⠈⣿⡇⠆⣿⣿⠀⠀⠀⢀⡀⣿⣿⣏⠉⠀⠀⢻⣟⠙⠎⢛⣙⠛⠛⠉⠈
⠀⠀⠀⢀⡴⠎⢙⡃⣼⢣⠉⠀⣌⠀⠸⣿⡿⠖⢻⡍⠈⣹⣿⣿⠉⠢⠀⣿⣧⣀⣽⣿⡖⠀⠒⢥⠀⡿⣿⣿⣧⣀⠀⣸⣿⣷⢺⠊⠁
⣀⠠⠔⠀⠀⣀⣼⣵⡇⠀⠀⠀⠉⠀⢰⣿⡇⢸⣦⠀⠠⣿⣿⡏⣀⣤⣨⣿⠀⠉⢻⣿⠉⢤⠄⠀⠀⣿⣿⣿⠈⢿⣤⣿⣿⣿⡿⠓
⠈⠢⠚⠃⠀⣬⣿⣟⣁⠀⠓⠀⠀⠀⢿⣿⠀⣾⠁⠀⢸⣿⣿⠁⠏⠀⠘⣿⠄⠀⣬⢿⡄⠀⠳⠀⠰⣿⢹⣿⠤⢄⣹⣿⡋⢿⣧
⠀⠀⠀⠰⠃⣹⣿⠛⠁⠀⠀⠀⠀⠀⢸⣟⠐⢹⣤⣦⣼⣿⣿⠣⠀⠣⠸⣿⡀⠀⣿⣶⠇⠀⢀⣀⠀⡯⢸⣿⠀⠸⡟⠙⠿⣾⣿⣀⣤⡀⢀⡀
⠀⠀⢀⠖⢠⣿⡞⠀⠀⠀⠀⠀⡀⠀⣿⣏⣠⣼⠉⢁⣿⣿⣷⣦⠆⠀⠀⣿⡀⢰⡿⡿⢶⡆⠀⢀⡘⡿⢨⣿⠀⠘⠇⠀⠀⢻⣯⠻⡄
⠀⠀⠎⢐⠂⣹⣷⣠⡄⠀⠀⠀⢸⠷⣿⡿⣿⡿⠁⠲⣿⣿⠀⠈⡶⡾⠃⣿⡅⡾⢻⣿⠈⢳⣦⠀⠀⢠⣼⣿⡖⠀⠀⠀⠀⢻⣿⢷⣴⣄
⠀⠀⢣⠉⠀⠙⠃⠀⠀⠀⠀⢠⣰⣤⣿⡇⢀⡙⢦⣴⣿⣯⡐⠆⡀⠘⢶⣾⡇⣁⣰⡿⠇⠀⢻⣤⣤⣠⣿⢿⡇⢀⣶⣤⠀⠈⣿⣦⣻⣿⣄⠠⡄
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡷⠟⣨⠤⣾⣿⣿⠉⠼⢆⡀⠠⠈⣿⣿⠿⣿⣇⠀⣀⠤⡹⠿⠿⠁⣸⣧⠁⠘⠋⠀⠀⣿⡇⠙⡿⣿⣿⢿
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠞⠁⠄⢀⡿⠋⠀⠀⣠⠋⢀⣤⣿⢛⡀⣿⠻⢦⠀⠠⢄⠀⠄⠀⢸⠿⡀⠀⠀⠀⠀⣿⡁⠀⡄⣾⠉⠛
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⢸⡇⢠⠀⠸⠛⡿⠋⣸⠧⠻⣇⡧⠀⠀⠑⢦⡀⠀⡀⠂⢸⡘⡗⠀⠀⢀⡀⢻⡁⠀⡧⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢩⠿⡁⠨⢆⠀⠀⠀⣿⠓⣶⠀⠐⠋⠀⠀⣿⠄⣤⡟⣷⡀⠀⠀⠀⠙⢷⣀⠀⢸⠂⣽⠁⠀⠉⠙⣿⠤⣀⣇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⠁⠀⡇⠀⠘⢧⡀⢠⡏⢀⣿⣏⡀⠀⠀⢠⣿⡆⠁⠀⣟⠁⡀⠄⠀⠴⠟⠉⠀⢸⡇⠀⢻⡄⠀⠀⡿⠀⠠⠈⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠛⣾⡇⢾⢭⡿⢇⡀⡀⣸⣿⡀⠀⢠⣿⠆⠡⠔⠃⠀⡀⠀⠀⠸⠇⠀⢺⣇⡀⠈⠅⠀⠀⠀⠙⡇⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⡴⠀⠀⠀⠀⢠⡿⢣⡀⣾⠇⠺⣿⠁⣽⡿⣇⠀⡏⠁⠑⠀⠀⠀⠀⠹⠇⢠⡈⢰⣇⠁⣿⠁⠀⢠⠀⠀⣆⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⣼⠃⠀⠀⠀⢁⣽⣧⢸⡟⠃⠀⠀⢠⣶⣿⠒⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣴⢈⣦⠈⠁⠀⠀⣠⢀⠘⢢⣤⣣
⠀⠀⠀⠀⠂⠀⠀⠦⠀⠀⡆⠀⠀⢠⣶⠋⣾⣅⡈⠃⢰⠄⠀⠈⠙⣿⠁⢰⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠾⠈⠛⠒⠓⡴⠛⠀⠈⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠾⠋⠀⠀⠉⠀⠉⡄⠀⢀⡄⠀⢀⠿⡤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠐⠂⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠂⡌⠀⠀⠀⠰⣷⠀⢀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡘⢿⠂⠈⢶
⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠚⠓⣇⢨⠁⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⢰⡄⣠⠀⠀⠀⠀⠀⠀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠟⠂
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢁⠀⠀⠀⣠⠞⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⡇⢸⠀⢠⢞⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡏⢸⠀⠀⢠⠀⡄⢠⣿⡡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠀⠀⡄⠈⣇⠀⠐⣶⣧⡾⠛⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⡇⠀⠀⠀⣇⠀⢸⠀⠀⡿⠛⠓⠒⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⢣⠀⠀⠀⠸⣠⢸⠀⣜⡡⠔⠊⠁⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠳⡄⠀⠀⠀⠀⠀⠀⡇⠀⠈⢇⠀⠀⠀⢿⡟⡴⠋⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠢⣄⠀⠀⠀⠀⠱⡀⠀⢸⡀⠀⠀⢸⣿⣗⠡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠓⢄⡀⠀⠀⠱⡜⣼⡇⠀⣰⠟⠉⠀⢀⣀⣠⠤⠤⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠐⠄⠀⠙⢦⠀⠀⠘⢿⣇⡼⠁⣠⠔⠚⠉⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠢⣀⠑⢄⡀⢣⠀⠀⢈⣿⣵⣋⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠲⢤⣀⠈⠑⠢⢍⡲⣧⠀⣼⣿⣉⡠⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠑⠦⡀⠀⠉⠻⡿⠉⠀⠀⢀⣀⡤⠤⠶⠒⢒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⢜⣦⠀⢰⠃⣠⠴⠚⢉⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣾⠾⠷⠾⠤⠤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⡄⠀⢀⠀⢀⠀⠀⡄
⠀⠲⣄⢠⣹⡄⠈⣧⢸⠀⢰⡇
⠀⠀⠪⣽⣿⣿⣄⡘⣾⡆⣸⠃⡇⠀⢸
⠀⠀⢐⣲⣿⣿⣿⣿⣿⣇⣿⠀⣷⠀⢸⠀⠀⡀
⠀⠀⠀⠉⢉⣩⠖⠋⠉⣛⣟⠀⣿⠀⣿⠀⢰⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⡀
⠀⠀⠀⠚⠉⢀⣠⡾⠛⠛⠒⣿⣾⣻⣹⡄⠸⡈⡇⢀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⡆⣧⣣⠃
⠀⠀⠀⠐⠚⠉⠡⠒⠚⣋⣉⣥⠴⣞⣻⢧⡀⡇⢻⢸⣸⡇⠀⣧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡐⢜⣿⣿⠃⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣡⡴⢟⡫⠥⠖⠛⢳⣼⠘⣯⡇⠀⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣳⢿⣿⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠉⠊⠵⠖⠒⢛⣛⣫⣽⣷⢿⣿⡆⣿⢸⠃⡇⢰⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣹⣷⣼⡋⡴⢃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢛⣩⣿⣯⠥⠶⠿⣿⣿⣾⣰⠃⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣤⣈⠳⣾⣏⣴⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠚⠋⢉⡁⣠⡄⠒⣩⣥⣮⠭⠯⠵⣾⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣟⠻⣧⣾⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⣈⢧⡢⠈⠣⡙⢦⣟⠉⢀⠳⢤⣀⠀⠌⢈⡑⣦⡀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠘⢿⡟⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠘⢷⣦⣌⡲⣄⠱⣌⠻⣧⣙⢄⠙⢦⣉⠓⢄⡙⢂⣙⣻⣗⣺⣷⣻⣿⣄⡄⠀⠀⠀⠀⣀⡈⠙⢿⣽⣷⠟⡡
⠀⠀⠀⠀⠀⠠⢤⣤⣬⣷⣭⣿⣿⢶⣷⣾⣟⣿⢛⣿⡟⢉⠏⡽⢁⣯⣟⡟⠙⠁⠈⠉⢿⣿⣦⡀⠀⠀⠠⣍⣉⡓⣿⢧⣞⣁
⠀⠀⠀⠀⠠⣤⣶⣾⣿⣿⠟⡿⣡⢿⢿⡿⡿⢁⣾⠟⢠⢋⡼⢡⠞⢫⡟⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡀⢀⠠⣑⠮⣿⡾⢋⣶⠶
⠀⠀⠀⠀⠀⠤⠞⠋⡽⠁⢰⠇⠁⠞⠸⠘⡡⠋⠀⢀⡴⠋⠠⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⣀⡨⠿⣿⣦⣕⠦⡑⢾⡾⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡰⣶⣶⣶⣶⣶⣤⣴⢿⠖⠓⠋⠻⣿⣷⣌⠻⣷⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠹⢦⣑⣮⡉⠻⣟⣦⣾⡏⣿⣇⠆⠠⡀⠐⢄⠈⠻⡿⣷⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⢴⣮⣶⣤⣙⣢⣜⣦⡼⣿⣟⡾⡽⢾⡿⣍⡳⢦⣈⠳⣌⠳⢤⡙⢌⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣒⠦⣍⣓⣮⣿⣶⣭⣟⣿⠟⣼⢻⣿⣹⠛⠾⣷⣄⣹⠷⣿⢟⠾⠛⣺⠋⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣛⣛⣷⣿⢿⣿⣿⣯⢿⡿⣟⣿⣻⣷⣯⣿⡿⢋⡞⠅⠐⢟⡿⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢶⡿⡿⢡⣾⣡⡿⣿⡿⢛⠋⡿⢩⠔⢋⠜⠀⠚⠌⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠉⠀⠋⢸⠫⠏⠀⠉⠔⠁⠌⠀⠀⠐⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠄⠀⠀⠀⢀⡴⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠀⡇⠀⢀⡴⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠀⠀⠀⡜⠀⡇⢀⣎⢎⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠈⣇⠀⠀⢇⣠⢇⣾⣷⠯⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⡇⠀⠸⡄⠀⢸⣿⣾⣃⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢸⡀⠀⠀⠀⢹⠠⠀⡇⠀⡸⠃⢀⣀⠤⡔⢲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⠀⠀⠀⠈⣶⣦⠇⣰⡥⠚⠁⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢇⠀⠀⠀⠸⣟⡴⢋⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠑⢤⡀⠀⠀⠀⠀⢣⠀⠀⢸⠀⠀⠀⢠⣿⣟⠥⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⠢⣄⠀⠀⠀⢳⡸⣼⡇⠀⣰⠟⠁⠀⠀⣀⡠⠤⠤⠄⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠦⠀⠈⠳⡄⠀⠀⠙⣿⣇⡼⠃⣀⠴⠚⠉⠁⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠦⣀⠘⢦⡀⠙⣄⠀⠀⢈⣿⣡⣞⠁⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⠤⣀⠀⠈⠓⠦⣍⡳⣼⡄⠀⣼⣿⢋⣁⣀⣠⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠉⠲⢤⡀⠀⠉⠻⢿⣾⠛⠉⠉⠀⠀⣀⣀⣠⡤⠤⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⢄⠀⠀⢸⠃⠀⣀⡴⠒⠋⠉⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣷⣷⣄⣏⣴⣋⣥⠄⠒⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠛⠹⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣴⠗⠎⠉⣡⣀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⢸⣿⣰⣤⣶⡁⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⣸⠈⠁⠀⠀⠈⠀⠀⠙⣿⣿⣿⣦⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢠⣸⡇⢠⣼⣷⣶⡾⠗⠀⠸⣿⣿⣿⣿⣿⣷⡆⠀⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⠖⠀⠙⢿⢀⡌⣿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⣨⡟⠀⠀⡀⢀⡤⣴⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠿⠁⠀⣠⣿⡟⣸⣧⡉⢣⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣧⠀⠽⢿⠂⢰⠄⣘⣯⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣇⠀⠀⠀⣉⣹⣇⡸⢿⣵⣿⣆⠀⠀⢀⡀⠀⠀⠀⠀⠀⢹⣤⠀⣸⣶⣋⡄⢀⣿⣠⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⣿⣿⡅⣶⣿⣿⣿⣷⣶⣿⣄⣠⣴⣦⠀⢠⣾⣅⡀⠀⢹⡙⠛⢈⡉⢉⡁⣀⣠⠔⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⠻⢀⠀⠈⢙⣿⣿⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡷⣶⣿⡯⠀⠀⢀⡦⠀⠘⢿⣧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣄⣀⣌⢻⣓⣀⠀⠀⠿⠿⣿⣶⣆⠉⠈⠹⡍⠙⠉⠛⢷⣽⣅⠀⣠⡼⠏⠀⠀⠀⠈⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣠⠤⠀⣉⣻⡆⠛⣿⣧⠀⠀⠠⠛⠹⢻⡿⢷⣾⣤⣤⣤⡴⠀⠈⢻⣿⣿⣷⠀⠀⠀⠀⠒⠠⠌⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣋⡥⠶⠉⠁⠀⠉⠢⣹⠟⠀⢠⣴⠀⠀⠀⠀⢤⣤⣿⣿⡿⠀⠀⠀⣸⣯⣟⡋⠂⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠴⠶⢀⠉⡄⠀⢨⠃⠀⠀⠀⠀⠈⠛⡏⠋⠀⠀⠀⣠⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⣠⠆⠀⠀⠈⢄⢸⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣠⣤⣼⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⣴⠖⠀⠀⠀⠀⠀⠀⢸⡶⠊⠁⠀⠈⠑⢦⣄⣄⠀⠉⠛⠿⣯⣍⣥⡶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⡀⠙⢿⠳⣄⢀⡀⠀⠉⠙⠛⠓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠈⠁⠀⡀⠱⢼⣄⠉⠒⠲⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠻⠗⠦⠍⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠀⠀⠀⠠⠄⠄⠀⠐⠂⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡙⠁⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠠⠤⠶⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣀⣮⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⢀⠰⣄⢨⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣄⣠⡿⡄⢉⣿⣿⣿⣿⣆⢠⠀⠀⠀⠀⢄⣀⡁⠃⠠⡀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣗⣽⣿⣿⣿⣿⣿⣏⣳⣤⣦⣀⣀⠛⢿⣿⣷⣶⣴⣶⣿⣿⡃
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⡛⢿⣿⣿⣿⣿⣷⣿⢏⡤⣨⣿⣿⣿⣿⣿⣿⣛⠉
⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣥⢾⣟⣻⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣟⢻⣿⢟⣯⡰⠿⠿⠻⣿⣟⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣷⣾⣄⡉⠙⣿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠙⠻⠿⢿⡿⠟⣿⠋⠉⠛⠃⠈

⢀⠀⣤⣶⢰⡤⢈⠎⡆⠀⠓⣴⠶⢆⣰⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠏⠙⢻⣿⢹⣧⡾⠛⣿⡟⠉⣄⡗⣿⡷⣂⡔⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⠀⢀⠄
⣼⠀⢻⡏⣷⡟⣿⡴⣿⡟⣧⢿⣿⣿⡷⣾⣷⣮⠄⣟⠡⠒⣒⡒⣠⣶⡤⠤⠤⠴⠤⠬⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⢠⠀⢰⠿⠀⣼⠋
⣿⣧⢸⡗⡏⢳⣻⣷⣏⣷⣻⡏⣼⣿⠞⣱⠞⣩⢞⣡⠤⡤⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⣠⠎⠘⠃
⣿⣬⡼⣿⣷⠎⣿⣿⢟⡴⢿⡟⣱⢛⣼⡻⢾⡟⠁⠀⢠⣴⣿⣿⣿⣿⣽⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠂⠈⢷⣌⣇⠀⠀⠀⠀⠀⠠⠄
⢿⡎⣧⣿⣿⣀⣿⣿⠟⡵⣿⢿⣿⡿⣫⣾⣫⢠⢀⣼⣿⣿⣿⣯⣿⣿⣿⣟⣿⣿⣶⠀⠀⠾⠷⣂⡄⣠⣤⣄⠀⠀⠀⠀⠙⣦⡀⢀⠀⠀⠡⣾⣺
⢷⣿⣯⡷⣿⣿⣿⢃⣾⡼⣡⣿⣿⡚⠽⠟⣻⣿⣿⣾⣿⣿⢹⢻⣿⣿⡿⣿⣿⣿⣿⣷⣄⢠⣽⣿⣿⣿⣿⣿⣶⣆⠉⠀⠀⠘⣿⣎⠾⣭⣿⣿⣷
⣼⠹⣜⢧⣹⣿⣯⣾⣽⡿⣻⣟⣵⣠⢾⢟⣷⣢⣿⣿⡟⡇⣏⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣾⣥⣄⣼⢙⣿⣷⣜⣿⣿⣿
⡙⠶⣼⣆⣿⣿⣿⢋⣿⣾⢟⡭⠿⣵⣷⣿⣿⣿⣿⠟⠷⣷⡇⣿⣿⣿⣿⣿⣿⢸⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⢳⣄⡈⣿⣿⣻⣧⡞⣴⣿⠿⠃⠩⠴⠿⣿⠋⠀⢹⠀⢰⣷⠇⣧⣿⡞⣿⣿⡿⣿⢻⣿⣷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣾⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⣤⣼⢹⣸⣽⢻⠁⣿⢹⡇⣽⢸⢷⣿⢻⣾⡏⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⢃⠠⡔⢄⣠⣶⣶⣠⣼⣷⡾⠟⠛⠿⠿⠿⠟⠿⠷⠿⠿⢿⣿⣾⣿⣿⣼⣿⣷⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⡂⣿⣿⣿⣿⡎⢁⣿⢦⣿⡟⣛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠟⣿⣿⡇
⢴⢖⣿⣿⣿⣿⣿⣍⣉⢀⠀⠾⠷⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢔⡯⢕⣋⣿⣿⡇
⣴⡾⢿⣿⣿⣿⣿⣿⣿⣿⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⠀⠀⠀⣀⣀⣴⣚⣱⣞⡩⢔⡵⠒⣿⣿⡇
⠚⠖⢛⣽⠋⠉⠹⡏⠻⠟⠋⠑⠂⠀⠀⠀⣴⣿⡧⠀⠀⣶⡿⠆⠀⠀⠀⠀⢀⠀⠀⣄⣚⡥⣿⣿⣟⣷⣼⠿⠿⠿⠿⠿⠗⠚⢹⣾⣷⣿⣿⣷⣴
⠙⡿⠉⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⣦⠀⠀⠰⣾⣿⣿⣿⡿⠒⣿⣿⡏⠀⠠⡴⣶⢶⠒⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠉⢀⠠⠊⢀⣿⣿⣷⣦⠀⠙⠋⠈⠁⠀⠠⠶⠶⠷⢿⡿⡿⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠭⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣾⣿⣷⣶⣶⣄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠟⠿⣿⣿⠿⠃⠀⠀⠀⠀⣩⡀⠀⠀⠀⢀⡀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⣶⡛⠁⠀⠀⠀⠀⠘⠛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣠⡾⢹⣷⣤⠟⠁⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠐⢸⠟⠀⢸⡟⠑⠆⠀⣬⠀⠀⠀⠀⠀⠉⠀⠢⢀⣺⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣶⣂⠔⠊⠁⠀⠀⠀⠁⠀⠀⠀⠐⠻⠄⠀⠀⠀⡀⠀⠀⢀⣨⣿⣿⡦
⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⢿⣤⡀⠀⠀⠀⡀⢀⣠⡂⠀⠀⠀⠀⠀⠐⢠⡾⠁⠀⣀⣠⣿⣿⣿⢷⣤⡀⣤⠂
⠀⠀⠀⠀⢀⣿⣯⣿⣿⣿⣿⣿⣾⡑⣦⣦⣶⣷⠞⢁⣆⡶⠗⠢⠀⠐⠄⠀⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⣀
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣶⣽⣶⣿⣿⣿⣷⣶⣶⣶⡴⠶⣄⣼⠿⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠃⠀⠀⣀⣀⣠⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⢿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣾⣷⣴⣤⡄⠀⠀⠀⠀⠀⣤⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣛⣿⣿⣶⣿⠶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⡾⢋⣴⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠐⢾⣦⡘⢿⣿⣎⢿⣿⡰⣶⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠖⠥⠎⠑⢡⡴⠟⣛⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠙⠿⣷⣄⠙⢿⣆⢻⣿⡈⣿⡇⣿⣿⣾⣿⣇⣄⠀⠀⠀⢀⠔⠛⠁⠀⠀⠀⠀⠐⠉⣠⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡈⢿⣷⡘⣿⡆⣿⢃⣿⢫⡟⢡⡿⠟⣸⣿⠇⡀⢾⠁⠀⠀⠀⠀⠂⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣶⣦⣄⠹⣿⡎⡿⢃⣻⣤⣽⣬⣥⣊⡡⠟⢀⣖⡿⢃⣾⣿⠚⣷⠶⠋⠀⠀⢀⠠⣴⣾⣿⡿⠟⠛⠉⣩⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡙⢿⣷⠸⢣⣶⣿⣿⣿⣿⣟⣿⣿⣿⣬⠉⣩⣶⠿⢋⣵⣿⠰⡄⠀⠐⠈⠁⣴⣾⠟⠁⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢿⣿⣿⣦⠙⣾⣷⢿⣦⡹⡞⠟⢿⣿⣿⣿⣿⣷⠰⠶⠾⠿⣿⣿⡆⡇⠀⠀⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣬⣙⣷⣿⡷⠶⠦⢙⣾⢛⣥⣿⣿⣿⣿⣟⣐⠾⠿⢿⣿⣿⡇⡇⠀⢀⡼⠋⠠⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠟⠋⠉⠀⣈⣛⣉⣉⡛⢸⣿⣿⣿⣛⡘⠟⠘⣿⣿⣿⣿⣿⢥⣙⣛⣋⣉⣿⣿⢣⣡⡴⠎⣠⡌⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢿⣿⠿⠿⢺⣿⣿⡿⣩⡶⢂⡄⣙⢟⡛⣧⡰⣦⣍⣉⣉⣻⣿⢏⣾⣿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣴⣿⡟⠋⠀⢀⣤⣶⣿⣿⣶⡻⣿⣿⣿⣱⡟⣱⣿⣸⣷⣿⣿⣿⣟⣿⣿⣿⢿⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠹⠋⠀⢀⣴⣿⠟⠋⣩⣴⣶⡶⠄⢿⣿⣿⣧⣿⣿⠸⣿⣜⣿⣿⣿⣿⢟⣵⡿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⢠⣾⣿⠿⣫⣴⣾⡷⢂⣙⣻⠿⠿⡿⠿⠿⢿⣛⣭⣷⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠰⣿⡟⠀⢠⣿⡿⠁⣰⣿⣿⡟⣴⡿⠛⢉⠽⠻⠿⠛⠿⠻⡿⠛⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⠃⢀⣿⡿⣼⣿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣯⣽⣿⡄⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⣾⣿⣿⣿⣿⣿⣿⣦⡄⣷⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⢎⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣷⣶⣶⣦⣿⣶⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⣀⡠⢖⢀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠸⣯⠁⣼⣟⣛⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⡄⠀⢠⣦⡀⠀⣠⠖⡋⢸⠇⠀⠀⠈⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⣞⣹⠇⠀⢨⣟⡇⢀⡷⠭⠴⠋⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢸⠉⠁⠀⠀⠀⢻⣀⡟⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡾⠛⠶⠞⠛⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠸⣿⢸⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢻⣤⣀⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢀⣾⠞⠁⠈⠓⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡙⣆⠀⠀⢻⡏⠀⢠⠖⠖⢒⠒⣺⠏⡴⢲⠞⠉⠀⠈⠻⣦⠀⠀⣼⠁⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣻⡏⠀⠀⠈⡇⢀⣿⣾⣛⣋⣤⠟⠀⣿⡼⠁⠀⠀⠀⠀⢸⣇⣼⡏⠀⠀⠀⠀⢀⣾⣿⣃⡀
⠀⠀⠀⠀⠀⠀⣴⣶⠖⠛⠲⢆⢸⠀⠀⠀⣰⠖⠋⠀⠀⢉⣉⠀⠀⠘⠋⡴⠟⠀⡀⠀⠀⠘⣿⣿⠀⠀⠀⢀⣴⣿⠋⠀⠀⠈⡟⣷⡀
⠀⠀⠀⠀⠀⢸⠋⠥⠃⠀⠀⠀⠻⣄⠀⢰⡇⠀⠀⢀⡾⢿⡒⠽⣆⠀⣴⠥⢮⣉⡝⠀⠀⠀⢸⢿⠀⢀⣴⡿⠟⣥⠀⠀⠀⠀⠳⣈⣙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠀⠀⠀⠀⠀⠀⢹⡎⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⣿⠰⢊⡵⠋⢀⡴⢻⡆⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣾⠏⣿⠀⠀⠀⠀⣰⢇⡼⠋⢠⠀⣿⢁⣼⣥⣴⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⢀⣿⣋⡴⠃⠀⢀⣠⢔⣥⡾⣥⣾⣿⠀⣷⠞⢉⣩⡥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢧⠎⠉⣿⡄⠀⡴⢫⡶⠛⠉⢹⡏⡿⢸⣿⣯⣤⣟⡛
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣦⣛⣎⣀⠀⢻⣆⣀⣠⡤⣄⠀⠀⢸⠀⠀⠘⢿⡊⢠⡏⠀⠀⠀⠘⢧⣧⢾⠀⠀⣠⣼⣯⠽⠟⠛⣛⣡⣲⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠖⠲⠤⠊⠁⠉⠙⢿⣿⣦⣉⠙⠛⣷⠀⢸⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⣹⣎⠴⠚⠙⠦⣤⣤⣤⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣌⠛⠉⠉⠱⣼⣆⠀⠀⢸⡇⠈⣇⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠘⢿⣧⠀⠀⣿⠀⠘⣄⢴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⠘⡏⢧⡀⠸⡆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣭⣛⠀⠉⠙⠓⢤⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⠀⠹⡇⠀⢀⡄⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠘⣃⣼⣭⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⠀⢀⣟⣀⡤⠤⢤⣀⢛⣋⣛⣳⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣸⡶⣫⠟⠁⠀⠀⠀⠉⠱⢌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿⡼⠁⠀⠀⠀⠀⠀⠈⠒⠈⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣸⣿⣢⠀⢀⣄⠀⠀⢀⣴⡜⡉⣀⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡿⠿⠛⢉⣻⠘⣻⣇⣻⣾⣯⣼⣻⠷⣠⣼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⡟⠃⠈⡅⡀⢀⣼⣮⣹⠗⢈⣼⠦⢉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡤⣐⠠⠸⣿⡟⣾⢸⠠⠋⠚⠟⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡟⠉⠵⣬⣷⣍⢿⠀⠀⢷⡏⣴⠶⣹⢻⠿⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⠀⠈⠽⣿⣾⡇⠀⢻⣿⣯⣿⣿⣿⣁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠀⠀⠐⠋⠙⠣⣤⣾⢿⣿⣎⠛⠫⢶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠄⠀⠀⠀⠀⣭⡀⠀⠈⠉⣹⣿⣿⣯⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⡀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠊⠀⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⢿⣽⣿⡿⢻⣿⣷⣿⡟⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣀⡀⠀⠀⠀⠀⡠⣢⡀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠉⢰⡄⠀⡠⡪⢊⣀⣙⠲⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⠈⠀⡌⡡⣪⠊⢗⡾⣭⠻⣓⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⢫⠞⠀⠀⠘⠙⠚⠛⠋⠀⠀⠈⠙⢦⡀⣄⣀⣾⣍⣇⡀⠀⠀⠀⠐⠃⠀⢠⣴⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢴⠗⠒⠂⠀⠀⠒⠒⠒⠒⢒⣐⡒⠒⠒⣿⡶⣿⣭⣭⣭⣥⣥⣤⣤⣤⣿⠤⣍⣜⣟⠉⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠇⠀⢀⣈⣰⠂⠀⠀⠀⢚⣙⡇⠂⠀⣿⠃⠁⠀⠀⠀⠈⠀⠀⠉⠉⢳⠝⠣⣏⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡠⢰⡇⢀⠫⡂⠀⢠⣹⡥⠤⠀⠀⠠⠌⠃⡧⠀⠀⣿⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠠⠤⡭⣤⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠂⡐⣾⡶⣞⣥⣃⢘⡃⢾⢦⡧⠀⣠⣶⣴⡤⣤⣥⣤⣤⣤⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣰⣤⢐⡗⠇
⠀⠀⠀⠐⣶⡾⠶⠿⡟⣩⢫⡄⠀⢈⡀⠚⠔⠀⠏⠋⠉⠀⠀⠉⠉⠁⠀⠍⠁⠀⠁⠀⠀⠒⠶⠶⡶⠶⣶⣟⢿⡛⠮
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠠⠀⠄⠀⠐⠈⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁


⣿⣟⢻⡿⢿⣿⠿⢿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠄⢀⠀⠉⠁⠀⠀⠀⠀⠓⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠔⢀⠀⠀⠦⢂⣀⣠⣀⣀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠀⠀⠂⡀⠀⠀⠠⠄
⠀⠀⠀⠀⣸⢃⠤⡠⠀⠶⠜⠢⡀⠨⢉⣆⠲⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡃⠀⠀⠀⡔⠀⠠⢠⠀⠐⠂
⠀⠀⠀⠁⡈⠘⠃⠁⠀⠈⠈⠈⠀⠀⢀⢡⠂⠀⠈⢤⡀⠈⠆⡀⠀⠀⠀⣄⢀⡀⡆⠁⠸⠈
⠀⠀⠀⠀⠀⡀⠒⠁⠀⡀⠀⠆⠀⠀⠀⠀⠐⠂⠀⠀⠁⠀⠁⠀⠀⠀⡆⡁⢈⡤⠓⠉
⠀⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠤⠀⡇⠐⠙⠀⠁⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢂⠑⠀⠀⠀⠂⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠂⠀⠂⠀⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⠝⠊⠁⠹⠊⠀⠒⠂⠀⠀⠀⢀⡟⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠲⠔⠦⠶⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠗⠀⠀⠊⠙⠉⠁⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠀⠀⠀⠴⢊⡤⠄⠤⠴⠀⠀⠀⢀⣀⣀⡀⡰⡀⠈
⠈⠁⠀⠀⠈⠉⠋⠉⠀⠀⢀⠰⠒⠉⠉⠒⠒⠤⣀⡀⣀⡀⡀⠀⠴⠿⣷⡄⢀⣠⣴⡶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠉⠉⠀⠙⣅⡀⣠⠖⢗⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠁⢀⠛⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠚⢄⢠⠔⠤⠀⠊
⠀⠀⠀⠀⠀⣀⡴⢧⡶⠒⠲⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠑⠀⠀⠀⠁
⠀⠀⠴⠂⠉⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖
⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡀
⠀⠀⠀⢾⠓⠦⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢷⠀⠀⡠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⢆⠀⠈⠳⡀⠀⢀⡎⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡼⠀⡜⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳⠀⠀⢀⡴⠒⠉⣹⠃
⠀⠀⠀⠀⠈⠳⣄⡀⢳⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⢹⣸⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⡰⠋⠀⢀⡴⠁
⠀⠀⢠⡤⠖⠒⠲⠭⢟⣦⣿⠀⣸⠁⡠⢦⠀⠀⠀⠀⠀⢇⠀⠑⢆⢸⡇⣠⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢳⠀⢸⣰⠧⠴⠚⠁
⠀⠀⠀⠉⠓⠦⠤⠤⠤⠬⢿⡰⠃⡼⠁⢸⠀⠀⠀⠀⠀⠈⢦⡀⠈⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢱⡀⠘⣆⣾⠓⠚⠉⠉⠙⠒⣢⠄
⠀⠀⠀⠀⠀⢞⡉⠉⠙⠒⢬⣇⢸⠁⢀⡞⠀⠀⠀⠀⠀⠀⣀⡈⠓⠺⡇⣠⠖⢻⡆⠀⠀⠀⠀⢸⡀⠀⣇⢀⠞⣀⣉⣓⣒⡒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠶⠺⣟⡠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠙⢆⠀⡷⠁⠀⡼⠀⠀⠀⠀⠀⠀⢣⠀⢸⣾⣉⣀⣀⣀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣦⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢱⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢈⣴⣾⣿⣿⣿⣿⣿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⡋⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓
⠀⠀⠀⠀⠀⠉⢫⡄⣶⠀⢸⡇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⠐⢆
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⠇⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢸⣿⢸⣿⡇⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⢘⣔⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢯⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢋⢡⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⢸⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⢨⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉⣻⠟⢀⣩⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠛⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⣤⣦⣤⣄⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠻⡿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣽⣿⣿⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡿⠿⠿⠏⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢶⣤⣦⣴⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⠁⠈⠁⠀⠈⠉⠛⠻⣿⣿⣉⠉⠙⠿⠃

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠆⠀⠀⠀⠉⠛⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠄⠀⢀⡀⢀⠀⢀⠐⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣙⣿⣧⣼⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⣤⡴⠂⣠⡾⠟⠁⠴⢾⣯⡴⠓⠂⠀⣰⠾⣿⣿⣿⣿⣿⡷⠲⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾⣿⣏⣉⡀⠀⠀⠀⠰⣾⡋⠀⠀⠀⠀⠀⠠⢾⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠠⣬⣽⠿⠶⠖⠂⠤⣤⣼⡧⠀⢀⠀⠀⣀⠤⣾⣿⣿⡗⠊⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣐⣏⣙⢓⣶⠀⠒⠾⣿⣥⡤⣤⡟⠁⠀⠀⠀⠀⠋⠁⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠁⠀⣉⣵⡶⠤⠜⣇⠈⠻⣾⠁⢀⡀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠨⠟⢿⣯⣁⠀⢀⣴⡿⣦⡀⢹⣴⡏⢀⣠⠷⣀⠀⠀⠀⠠⠄⠀⠀⡤⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣒⢆⠀⠈⢷⠀⠀⢼⡀⠈⠻⣾⠯⠀⢸⣧⣀⣈⡄⠤⠀⠐⠂⡴⠞⠁⠀⠖⠀⠗⠀⠂⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⣿⢶⣾⡅⠀⢸⣷⡶⠤⠘⣣⡤⢻⠋⠙⠃⠀⠀⠀⢹⣿⣿⣿⣟⣐⡁⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠉⠣⣴⠟⠉⠀⣀⣼⠏⠀⣼⣄⣨⠄⠂⠘⠛⣿⣿⣿⣿⣿⣿⡥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⢤⣽⠅⠾⣛⣉⡩⠧⠀⣿⣝⡇⠀⠀⠀⠈⠉⠀⣾⠿⠛⠛⠿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠾⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢶⠂⢀⡀⣈⣤⠂⠀⣤⣴⡇⢠⢸⢷⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣦⣭⣩⡿⠛⠀⢀⡀⣐⣷⠄⠀⠋⠉⣼⣟⣀⢨⣿⡿⠡⠶⢵⡾⢥⡆⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⣿⣿⡋⠁⠂⠀⠀⠀⠀⠁⠄⠃⠒⠛⠿⠛⠛⠘⢿⡉⠃⠠⠆⡻⠀⠁⠘⢉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⣿⡄⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠛⠾⠶⠚⠁⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠺⣿⢿⠟⠉⠛⢻⠟⠏⠙⠫⡏⢸⣿⣧⢸⠇⢂⡈⠼⠶⣦⡄⠠⣀⣄⣀⠀⠀⢃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠁⢤⠀⠀⠀⠀⠶⢄⢡⡇⢸⣿⣿⣽⣶⢿⠀⠂⠀⠉⠁⠘⠚⠚⠃⣼⡀⠙⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠆⠀⠀⠀⠐⠀⠸⡏⠸⣿⡇⢸⣷⣾⣷⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠋⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠓⣦⢀⣾⡊⠀⢀⣀⠀⠀⠈⠡⠀⣿⣿⣸⣿⠛⠝⠶⠖⠒⢤⠀⠀⠀⠠⡍⢻⢋⣀
⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠄⢹⠾⡿⠟⠁⠀⠉⠀⠀⠀⠀⠀⢼⣿⡿⠛⠃⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠾⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣤⡝⠣⠀⠈⠁⠀⠴⣟⣀⣀⣤⡴⢶⣄⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠐⠀⠀⠀⠈⠛⠿⠚⠓⠶⠤⢴⣶⣾⣿⣿⣿⣿⡟⢛⠛⠛⣀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡎⠀⡀⠀⠀⠀⠀⠢⠀⠀⢰⣾⣿⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡅⢹⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡇⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢸⡿⠟⢻⣿⣿⣿⣿⣿⠻⡷⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡦⠔⠋⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠻⣿⠈⠃⠀⠈⠻⠿⣿⣿⡏⠢⠀⠈⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢀⣿⠀⡂⢲⡐⠒⢶⣿⣿⣗⠉⠀⡀⠈⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠃⠠⢶⣿⣿⣿⠀⣆⣥⠙⢰⡆⠀⢻⣿⡖⢰⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁⠀⠂⠀⠀⢨⠻⣿⣿⡆⣉⠋⠀⠁⠰⠓⣌⣿⢄⠀⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣍⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⢀⣤⣄⠀⠀⠀⠀⠀⠐⠄⠀⠀⠒⠺⠷⣼⢵⣶⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠦⠈⠀⢀⣶⣆⠀⠀⠀⣁⠸⢆⠀⠀⢀⡦⡽⣷⣤⣄⠀⠘⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⣗⠒⠖⠈⠛⠋⣤⣀⡀⢀⠀⠀⠑⢄⠀⠀⠀⠉⠖⠀⢉⠀⠶⠆⠀⠀⠠⡀⠀⠀⢼⡷⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣰⢛⣛⣁⣬⣈⣳⣶⠶⠿⠦⠤⠶⠲⢴⠿⣿⠦⠴⠿⠶⢤⣽⣤⢤⣀⡀⢀⣴⠆
⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⠉⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠒⠒⠂⠛⢷⣌⠉⠉⠛⣦
⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⠂⢠⣤⣄⠘⠀⠀⠈⠾⠀⠀⠔⡆⠀⠀⠀⢀⣀⠰⡆⠀⠀⠀⢀⣀⣹⣶⣦⣾⠗⠢⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢛⠛⠁⠀⠐⢀⡂⠂⠒⠚⠒⠶⢦⡄⣘⠟⠗⠐⢃⡬⠛⠉⠁⢀⣠⡤⠶⠋⠙⠒⢦⣌⡂⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡰⠇⠀⠀⠀⠈⠁⠀⠰⢷⠄⠀⠋⠐⠊⠡⠄⠚⢩⣄⡤⠒⠈⣉⠋⠀⠀⠀⠀⠀⣀⣼⠿⠶⣌⡑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⢩⣥⡄⠀⠚⠢⠀⠀⠀⠀⠴⢾⠉⠀⠀⠀⠀⠉⠲⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠐⠿⠋⠰⣦⡄⠀⠁⠀⣦⡀⠛⠋⠀⠀⠀⠀⢀⡴⣿⠀⢰⣿⡇⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⣆⡙⣃⠀⣼⠟⠠⠂⢰⡉⠛⠀⠀⠀⠤⠀⠀⠀⣴⡤⣔⠾⠋⣱⡌⠳⢀⣥⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠀⠀⠉⠀⠀⠈⢤⠄⠀⠀⠀⠀⢀⠤⠔⠊⣻⠿⠁⠀⠀⠀⠀⠛⠸⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⢠⣿⠀⠈⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣤⣆⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠅


⠀⡄
⠀⠄
⠀⠃

⢰⣶⣶⣶⣶⣶⣶⡖⠒⠒⠒⠒⠒⠲⠒⠖⠒⠶⠶⠶⢶⡆⢰⡖⠲⠖⠲⠶⠶⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡟⣿
⢸⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣰⣿⣿⡿⣿⡟⣿⣿⣿⢛⢿⢸⠡⠘⣽
⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⣾⣿⠳⣟⣿⡛⠆⣽⣿⢿⣬⡸⣾⠀⢠⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⣿⡧⣿⣿⣼⣽⣦⡸⢿⠈⣿⠇⡿⠀⡟
⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⡿⣾⣇⣙⡶⣜⢾⡀⡇⢀⡇⣸
⢸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠘⠿⣿⣟⡛⠙⢿⣿⣷⣿⣿⣷⣳⢸⢰⡉
⢸⣿⣿⣿⣿⣿⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠓⢮⠀⢸⣿⣿⣿⡇⣿⢸⣼⠃⢱
⢸⣿⣿⣿⣿⣿⣾⡈⢳⡴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⣾⣿⣷⣿⣾⣿⠴⠞
⢸⣿⣿⢻⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⡿⣿⣿⠛⠙⡱⢀
⢸⣿⣿⣿⣿⣿⡟⣾⢙⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣄⢘⡶⢫
⢸⣿⣿⣿⣹⣿⡟⣿⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣏⣸⣧⣿
⢸⣿⣿⣿⣹⣟⣷⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⣿⣿⠛⠟⠩
⢸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣋⣾⣿⠏⡀⢠⢂
⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠈⡴⠁⠀⡼
⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠂⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠶⠷⠷⠶⠷

⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⣤⡀
⠀⠀⠀⠀⠀⢸⢰⠞⠁⠁⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠈⣣⣼⡉⣠⠄⠄⠀⣀⠀⠠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢸⡤
⠀⠀⠀⠘⠦⠀⠿⠂⠘⠃⠀⠀⠀⢆⠰⠀⠀⢀⢰⡆⠀⠀⣩⣷⠀⠀⠀⠀⣤⡀⣄⢃⣨⡶⠗⠀⠉⠁⠀⢀⠈⠳⠄⢀⠀⣀⠀⠀⠠⢀⠀⠘⠟⠁
⠀⠀⠄⡀⠠⡿⠤⠤⠔⠂⠀⢶⠄⠈⢶⡀⠐⠁⡘⠀⠀⠀⠉⢹⠀⠀⠀⠀⠈⠛⣤⣏⡏⠒⣧⠔⠀⠀⠀⠀⠁⠤⢮⢠⣤⡄⠀⠈⠦⠐⢂⡴⠆
⣀⠀⠑⠼⣶⠆⠀⣀⡰⠀⠀⠀⠀⡀⠘⣧⡔⠀⠀⠀⠀⠀⠘⠝⡇⠴⠀⠀⠀⠀⠀⢸⣽⠻⠃⣤⠀⠀⠀⠙⢦⡀⠈⣥⡅⠀⠀⠐⠶⠂⠾⠆⠀⢀⡀
⠀⠀⢠⣤⠅⠀⢀⡴⠀⠀⠀⠀⠠⡈⣤⡼⠀⠐⠒⢰⠇⠀⢀⢠⣧⠤⡄⠀⠀⢰⡟⠂⢸⠀⣀⡰⠆⠀⠀⠀⠀⠐⠄⠘⡁⠀⠀⡀⢠⡀⢀⠀⠀⠘⠊
⠀⠀⢰⠉⢁⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣸⣀⣁⠠⠂⠀⠀⠘⠈⡁⠀⠀⠀⠀⠀⠘⠟⣛⣯⡌⠁⠀⠀⡶⠀⠀⠀⠈⣿⠀⠀⠀⠙⠛⣁⢨⠦⡤⢀⠬
⠀⠀⠀⠀⠻⠀⣀⠀⠀⠀⠁⠀⢢⠀⠀⠉⠀⠀⠤⢺⠃⠀⠀⠻⠟⠀⠃⠀⢀⠀⠀⣀⠈⢃⠀⡠⢪⡞⠁⠀⠀⠀⠀⠘⠀⠀⠠⣠⠀⠻⠂⣤⠀⠾⠆
⠀⠀⠃⣀⠀⠀⠀⠑⢄⠀⠈⣦⠀⠁⢀⡄⣃⠠⢀⠂⠀⠀⠸⠽⣷⡰⠀⠀⠈⠙⠦⠬⠽⢶⣔⡤⠊⠀⠀⢠⠎⡆⠀⠰⠀⠀⢠⠼⠦⠀⠀⠉⣀⡀
⠀⠀⠀⡌⠀⠂⠀⠀⠀⢢⠀⠈⠳⣇⣸⠁⠉⢀⠁⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠁⢃⡅⣀⠀⠀⠀⠈⠚⠁⡤⣄⢠⠉⠋
⠀⠀⠀⠆⠀⠩⡀⠀⠀⠀⡆⠀⠀⠈⠏⠁⠀⠁⠀⠀⠀⠀⠑⠊⠋⠀⠀⠀⠀⡤⠄⠀⠀⠘⠀⠀⠈⢇⠀⡆⢸⠃⡋⢀⠀⠀⠀⠐⣀⣿⠁⠀⡤⢌⡆
⠀⠀⠀⠘⡀⠀⢧⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢐⠀⠀⠐⠀⡀⠀⠀⢰⠀⠀⠀⡀⠀⠲⣆⢠⠀⠉⠂⠀⠀⠀⠠⠈⠀⢹⠁⡤⡑
⠀⠀⠀⠀⠀⠦⠘⠦⠀⢀⠃⠀⠀⠀⡆⠀⠀⠀⢈⣋⠀⠀⠀⠀⠀⠈⠄⢁⢰⠀⠀⠀⠀⠈⠀⠀⠈⠙⠁⠀⡹⠊⢀⠀⠁⢠⠞⡇⠀⠀⠀⠌⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠁⠀⢀⠀⠀⠐⠀⢀⡀⠉⠉⠰⢄⠀⠰⡄⣄⠈⢠⡌⠀⢸⠇⠀⠠⠀⠀⠀⠑⢵⣀⢻⠀⢂⠀⠠⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⠀⠀⠀⠈⠳⠀⠀⠑⢬⡃⠀⠀⠫⡁⡀⢣⡀⠂⠃⠀⠀⠀⠀⠰⣄⠀⠙⢢⡦⠂⠀⡡⠀⢠⠊⠇
⠀⠀⠀⠀⠀⡀⠰⣆⠀⢀⠀⠀⠘⢆⣴⡄⠀⠳⠇⠀⠀⠀⠀⠀⢀⡀⠀⠳⣘⢁⣅⡀⠐⡦⠀⠀⠰⡈⢆⢂⠀⠡⡀⣸⠃⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠐⡀⢈⣆⠈⠀⠀⠰⠻⢀⡀⢠⠶⢀⠀⠀⠀⠀⡀⠀⠆⠐⠦⢜⡆⣑⣀⠄⠀⠀⠀⠀⠈⠒⠶⣄⠀⠡⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠋⠀⠈⢆⠱⣅⠀⠀⡸⠁⠀⠀⠄⠠⠘⡗⠁⠀⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠢⣇⠀⠋⢀⡶
⠀⠀⠀⠀⠀⠀⠃⠈⡒⢷⠀⠀⠾⠀⠀⠀⠦⠀⠑⢾⠇⠈⢀⡀⠀⠀⢆⢀⠁⠀⠰⣠⠁⠠⠀⠉⠀⠀⠈⠫⣆⡃⠀⠘⡀⢀⣨⠌
⠀⠀⠀⠀⠀⠀⣤⠀⡀⠹⡇⣀⠆⠀⠀⠀⠈⢌⠄⢸⡆⠀⠘⠁⠀⠀⢀⠑⣤⡔⡐⢡⡎⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠀⡟⠉⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡟⠡⢂⠏⠀⠀⠀⠀⠙⢾⡇⡴⠰⡄⠀⠇⠈⠃⢸⢪⡴⠊⠁⢀⡄⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⡧
⠀⠀⠀⠀⠀⠀⠀⣤⠈⡘⢿⠔⣃⠀⠀⠀⢂⠀⠀⢸⡟⢁⡠⠁⠀⠛⠷⣷⠀⡼⠀⠀⡄⠘⠀⠀⠀⠀⢊⡁⠀⠀⠀⠀⠀⠀⠐⠠⠀⢱⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⡇⠀⠁⠀⠀⠀⠈⠢⡀⢄⣧⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢰⠀⡠⠁⠀⠠⠀⡤⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠈⠣⡀⠀⣀⡅⢐⠂⠀⠀⠀⠀⠄⢈⡀⡟⠠⠐⠹⠁⠀⠀⢰⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⢈⠁⠀⠀⠐⠃⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢈⠓⠻⠃⠍⣀⠀⠀⠀⠀⠀⠀⠈⢟⠐⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠃⠀⠀⠀⡤⡄⡠⠀⠨⠀⠀⠀⠤⠀⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢇⢰⣿⣠⠍⠀⠀⠠⢀⠤⠀⠀⡏⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⠼⠧⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⡠⢀⢍⠜⠀⠀⠀⠀⠀⠓⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⢶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠶⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠈⠀⠀⠘⠂⡀⠀⠀⠀⠀⠀⠀⠙⠂⢀⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⡀⢀⣤⡇⠀⢠⢀⡄⣠⠀⠀⡀⣆⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄
⢀⣿⣼⣿⣷⣿⣾⣿⡧⣿⣿⣼⣿⣿⣷⣿⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠐⠂⠾⠀⢠⠀⠀⠀⠀⠀⠀⠢⣼⠤
⠸⡿⠿⢿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠒⠤⢽⣖⡃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⣒⢻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⣤⣀⠤⢶⣆⠠
⠀⠀⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣾⡀⢤⡥⠽⣿⠷⢒
⠠⣸⣿⣧⣶⣷⣾⣷⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡂⣨⣿⠤⠀⠙⠶⢾⡟⣋
⠀⣿⣿⣿⣿⣿⣿⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⢀⠀⣑⣲⣿⣊⡁⠈⢓⣾⣟⣋
⠀⢿⣿⣿⣿⡿⠝⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣼⣥⡄⠠⠽⢿⠆⠆⠤⣻⡿⣿⣋

⠀⠀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡿
⠀⠀⢶⣴⣶⣿⣿⣿⣦⣦⣄⣀⣂⣴⣦⣶⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣆⣤⡄
⠀⠀⠀⠿⠿⠛⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠉⣿⣿⣿⠋⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠟⠋⠉⠁⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢠⣴⣿⣿⣿⣦⢾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣄⣤⣴⣤⣤⣤⣤⣾⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣾⣿⣿⡿⡧⠀⠀⠀⠀⠀⢤⣀⠉⠹⣾⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠈⠙⠻⠟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠁⠀⠁⠀⠀⠀⠈⠉⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢤⣿⣷⣦⣴⣶⣿⣶⣦⣠⣶⣧⣄⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣷⣤⣆⣀⢀
⠀⠀⠀⠀⢨⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠛⠛⠚
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⢿⡿⠛⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠀⠉⠉⠉⠉


⠀⠀⠀⠀⠀⢀⣀⡄⢀⣀⣀⣀⣀⣠⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⣾⣶⣶⣶⣄
⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖⠂⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠛⠛

⠀⠀⠀⠀⠀⢘⣳⣷⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⣶⡆⢀⠀⠀⠖
⠀⠀⠀⠀⢰⣿⠎⢹⡇⠀⠀⠀⠀⠀⠠⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢿⣯⡠⣿⡍⠉⢹⣿⣿⣷⣤⣾⣿⡀
⠀⠀⠀⠀⣾⣿⣷⣼⠀⠀⠀⠐⠽⠇⠀⠀⠉⠉⠹⣿⣿⣿⢿⣿⣿⣿⣿⣽⣿⣿⡟⠀⠀⠀⠘⣿⡇⠛⠿⠿⠾⢿⣾⠟⠉⠋⠉⠃
⠀⠀⠀⢠⣿⣛⣟⠵⡶⠃⢀⡀⠀⠈⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⢹⣿⣿⣿⣿⠃⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⢸⠃
⠀⠐⠘⠋⠈⠉⠁⠀⠀⠀⠰⢇⠀⠀⠀⠀⠀⠀⢀⣸⡟⢙⡿⠀⠀⣸⡟⠀⢙⣿⠆⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠛⠄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡤⠃⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠞⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⢞⡜⠃⠀⠀⠀⠀⠀⣄⠀⠑⠄⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⡩
⠀⠀⠀⠀⣀⣤⣴⡿⣿⣷⢉⡆⠀⠀⠀⠀⠀⠀⣿⠀⠤⠈⢶⡄⠀⢰⣿⣿⠶⠂⠀⠀⠀⠀⣤⠏⠐⢤⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⣾⣿⢿⢻⡞⢛⣷⡌⣥⠀⠀⠀⣴⣦⣀⡏⠀⠀⠦⠈⠀⠀⢸⣿⣩⡴⠖⠀⠀⠀⠀⠳⠶⠶⠟⣫⠷⠉⠉⠁⠀⠉⠚
⠀⠀⠀⢻⡇⠀⠸⡇⢀⢹⠃⠀⠀⠀⢠⣅⣿⣿⣇⠀⠀⠀⠀⠀⠠⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦
⠀⠀⢀⣄⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡝⡛⢫⡭⠿⠀⠀⠀⠀⣠⣶⣾⣶⣶⣶⣶⣦⣶⡗⠀⠀⠀⠀⠀⣤⡀⣠⣾⣿⣿⠏
⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠆⠀⠀⠈⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠿⠿⠿⣿⣿⠋⠀⠀⠀⠀⠁⠀⠈⣰⣿⣿⣿⣿⡟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠘⠹⡋⠁⠀⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠱⠄⠀⠀⠀⠚⠀⠰⠄⠀⠀⠀⢀⣿⣿⣻⣿⣿⣿⣿⠇
⠀⠀⠐⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⣿⣿⣿⠏
⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠰⠟⠛⠉⢷⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣿⣿⣶⣆⠀⠀⠀⠀⠀⠋⠈⢿⡿⠟⠒⢶⠛⡿⠶⣦⣀⡐⠿⠻⠿⠟⠟⠁
⠀⠀⢿⣿⣿⣿⡟⢿⣿⣿⣿⣸⣿⣿⣿⣿⠿⢿⠿⢷⣄⠀⠀⠀⢠⣴⣿⣦⣀⡀⣠⣧⣀⠀⣾⠁⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢸⣿⣿⡿⠇⠠⠿⠿⠿⣿⢿⡟⠉⠉⠑⠘⠦⠼⠟⠀⠀⠀⢠⡟⠁⠉⠉⠉⠉⠀⠁⠹⡏⠀⠿⠀⠀⢠⣤⣾⣟⣿⣿⣿⣷⡀
⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢃⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢰⢷⠀⠀⠀⠀⠀⠁⣿⠿⠿⠿⠿⢿⡋
⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠽⠿⠿⠿⡛⠿⣽⡿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠋⠋⠉⠐⠁⠂⠀⠀⠉⠀⠒⠀⠀⠈⠀⠈⠫⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⠟⠟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⡿⣇⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢻⣿⣧
⠀⠀⠀⢀⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆
⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⣠⣆⡐⣨⣥⣠⡀⠀⢒⣺⣗⡖⠃⠀⠀⣠⣴⣦⣤⣀⡀⠀⠀⢀⡞⢽⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⡟⣱⣾⣿⣿⣿⣿⣿⣿⠀⢠⣽⡿⠿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⡆⣿⣿⣿⡇
⠀⠀⠘⣿⣿⣿⣿⡿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⡇⠀⠐⣨⣿⣿⣿⣿⣿⣿⣿⣿⣧⠜⣧⣿⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⡏⡥⠀⠀⢿⡿⣽⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⡘⠉⢻⣿⣿⣿⣿⣿⠟⢡⡟⢀⡃⡀⠀⠈⠓⢼⡗⢹⣿⣉⣿⣿⡿⠀⣿⣿⣿⠇
⠀⠀⠀⢀⠸⣿⣿⣿⣧⣄⡺⢿⣿⣿⡟⠋⣴⣿⣷⣿⣏⣿⣆⠀⠀⠀⠛⣯⠼⣽⣿⠟⠁⠀⣿⢿⡿⢀
⠀⠀⠀⠈⠀⢹⣿⣿⣿⣦⣾⣿⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣿⣦⠀⢠⣾⣸⣧⣤⣄⣀⣀⣤⣎⣡⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡌⣾⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠁⠎
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠁⠀⠘⣿⣿⣿⣿⠿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⣾⣿⣿⣿⡟⣹⢻⠻⡇⠀⠀⠀⠀⠀⠀⠙⠟⣇⣸⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣏⢸⢹⡀⣿⡄⢀⠀⢠⠀⣆⢠⣴⠙⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣷⣧⣾⢸⡀⢹⠀⢸⠰⡇⢾⣽⠶⣼⣽⣿⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣩⢻⢙⡊⣇⡈⣱⠈⡁⢠⣹⣴⣰⣭⣿⠅⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣏⠿⣷⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣏⠛⣻⣿⢿⠷⠒⠋⢻⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣀⣠⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⡛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⡀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠃⠀⢠⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠐⡀⠀⠀⠀⢠⠘⠀⠀⢸⠀⠀⢸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢤⣇⠀⠹⡄⠀⠀⡏⢘⢂⠀⠸⡀⢣⣼⠀⠐⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⡀⠀⢹⡀⠀⡇⠀⠀⡂⠀⠑⣼⣃⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⢪⣇⠄⢸⡇⠀⠑⣄⠀⡅⢀⢜⡁⠤⠐⠒⠒⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠐⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠄⡟⠀⡸⠀⠀⠀⣸⣮⡞⢁⠋⠀⣀⣀⣀⣤⠔⠂⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⠀⠈⢧⠀⠀⠀⠀⠀⣠⠤⠤⠤⢬⣳⣄⢸⡧⣶⣵⣾⣿⡟⠛⠻⣍⠙⠍⣋⠁⠀⢀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⡇⠀⠀⢠⠎⠀⠀⠀⠀⣀⣬⣷⡧⠶⡯⣍⠉⠉⠀⠉⠛⢿⢽⣑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡀⠀⣿⠀⠀⡇⠀⣀⣿⣀⣰⢥⣿⡾⠿⠯⠿⣇⠀⢣⣨⠧⠤⣄⡀⢆⠀⠁⠨⠻⣤⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⢠⡟⣠⠞⠋⠉⠀⠉⠻⣯⡝⠁⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠉⢾⢢⡀⠀⠀⠈⠓⠄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⡟⠁⠀⠀⠀⣀⣤⡀⠈⢷⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠂⡇⠈⠒⠤⢄⣀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣇⠀⠀⠀⢸⠁⢂⡇⠀⢸⠠⣤⣤⡀⠀⢹⡀⡀⠀⢀⡴⠢⡄⣰⠇⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡣⠁⠹⣆⠀⠀⠀⠛⠉⠀⣰⠏⠓⠯⠭⠤⠶⠋⠑⢬⣦⣜⣧⣿⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡕⣀⣤⣔⢚⠛⢶⡦⠴⢖⠊⠙⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡜⠋⠁⠀⠀⠉⠙⠦⣤⣕⠲⡤⣀⣀⣀⣭⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⠧⠂⠀⠀⠀⠀⠀⠀⢠⠀⠊⣧⣽⡠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢠⣀⠀⠀⠀⠀⠆⢰⠈⣘⣾⢫⣁⡠⣄⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣯⣩⣷⣯⢿⡅⠀⠀⠀⠀⣸⢸⣾⣿⣿⣯⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣏⣉⣶⢻⠇⢠⡀⠘⣤⣾⣿⣷⣶⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⠛⠳⢤⣿⡭⡀⣼⠁⢸⣿⣷⣛⡛⠲⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡀⡆⢘⠘⢻⣟⣛⣩⠿⢳⢼⢹⣄⣼⠻⣽⠛⠶⠂⠚⠀⡠⠄⠀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⣷⣭⣿⣿⡿⠿⢿⣷⢶⣿⣵⡾⣿⡿⠿⡛⠟⠀⣀⣡⠾⠧⠌⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣾⣦⣀⡀⠀⠀⠀⠀⠀⠘⡀⣿⡏⣟⣿⣿⣶⣾⣿⣛⣿⣿⣷⣖⣽⡷⢾⣿⠿⠩⠽⠛⠀⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢧⣿⡿⣿⣾⣿⣷⣄⠀⠀⠀⠀⢄⠙⣯⡿⣧⣈⣿⣿⣧⣤⠿⣿⣿⣏⣿⣶⣿⣷⢻⠍⠶⢶⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢻⣶⣧⡝⡹⣿⣿⡀⠀⠀⠀⢈⢷⣸⣿⣿⣷⣿⣉⡿⢋⣼⣿⣿⣿⡿⢿⡯⢳⡌⢟⠻⠇
⠀⠀⠀⠀⠀⠀⠀⢈⣲⣾⣿⣿⣿⣥⣺⣿⡆⠀⠀⠘⠨⣽⣿⣿⢏⣨⣲⣶⣿⣿⣿⡥⡉⣬⢡⣿⣋⣋⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣖⣿⠟⢓⢿⣿⡄⣀⠀⠀⣡⡘⣿⣿⣿⣿⣿⡯⢵⢻⣶⣷⣿⣹⣎⡿⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠻⣿⣿⣧⣼⣋⢷⣽⢠⡼⢠⣿⣿⣿⣿⣧⣽⣿⣾⡿⢉⣼⡿⢗⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⣿⡭⠛⠛⢿⣿⣾⣟⣿⣧⡞⣿⣿⣉⣿⡏⣿⣷⣈⣿⣯⣙⣵⣶⠄
⠀⠀⠀⠀⠐⠀⢂⢀⠀⣟⠻⣿⣿⣛⢀⡿⢋⠑⣹⣿⢃⡿⣿⣿⣿⣿⡟⣳⣿⠻⠷⣿⠏
⠀⠀⠀⠠⣀⢠⡄⢀⣾⣿⣽⡿⠛⠻⣿⣯⣽⢹⣯⣿⣸⣿⣷⣄⣹⣿⣮⣙⣥⣵⣾⠆
⠀⠀⠀⢀⠈⠙⢿⣼⣏⣿⣿⣿⡖⠊⠘⣿⣿⣿⢹⣿⣿⣿⣟⡵⠾⣿⠗⣵⢟⠟⡁⠀⠀⡠⠎⡤⠔⣀⠀⡠⢊⣀⠤
⠀⠀⠠⠌⠲⠵⣬⣯⣿⣿⡗⣹⣿⣏⣁⡡⣾⣯⣼⣿⣿⡿⠟⠻⠖⠚⠿⢋⡮⢘⣠⡤⢊⣄⣥⣔⣯⣴⣾⣉⠡⠄
⠀⠀⠀⠀⠀⠑⣼⠛⣿⣿⣷⣬⣿⣿⣿⣏⣥⣼⡿⣩⡏⠀⠐⡀⢀⣡⣾⣻⠷⣻⡟⡔⣿⣿⠟⠛⠓⠶⠤
⠀⠀⠀⠀⠀⠠⠄⠉⠛⢻⣿⣿⣿⣎⣿⣿⣿⣿⣾⠟⢀⣤⠴⣶⠿⡿⠞⠛⠿⢿⣿⣷⣛⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠈⢻⣿⡎⠻⣧⣛⣿⠏⠐⢫⣄⡰⢥⣌⡛⢿⣿⣟⢾⣀⣚⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠓⢀⣼⠟⠁⠀⠀⠀⠀⠈⠙⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣶⣶⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣾⣭⣭⣉⠉⠙⠛⠿⢿⣟⣛⠉⠉⠉⠃⠀⠀⠀⠈⠁⠈⠻⣿⡿⠿⠿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡀⠀⠀⠀⠈⠛⠛⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠉⠀⠀⠈⠉⠛⠶⠶⠒⠒⠒⠲⢦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡖⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣾⣿⣿⣷⠂⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣗⡀⢀⣺⡆
⠀⠀⠀⠀⠀⢠⣷⠀⢀⣠⣿⣿⣿⣿⣄⡀⠀⢘⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⣦⣿⣿⣿⣿⠇⢨⣿⣿⡄
⠀⠀⠀⠀⠠⢿⣿⡄⢠⣽⣿⣿⣿⣿⣟⣁⡐⢻⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡄⢴⣿⣿⣿⣿⣿⣊⣼⣿⣿⣧⠄
⠀⠀⠀⠀⠼⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⡛⠂⠀⠀⠀⠀⠀⣀⣠⣿⠀⣶⣄⣀⠀⠀⠀⠀⠀⣰⣿⣿⣗⣲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⡿⠻⡿⠀⢸⡿⢿⣷⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣠⣤⣮⣦⣴⣷⣶⣾⣧⣤⣿⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠚⠛⠿⠛⠛⠛⣛⣛⣋⣽⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠚⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡞⢶⣣⠀⠀⠀⠀⠀⠠⣖⠤⠒⠒⠒⠒⠒⠒⢖⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢿⡀⠠⠌⠳⣄⠀⠀⠀⠀⠈⠃⠘⠀⣬⣥⠈⠃⠸⢤⡤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠈⢷⡀⠀⠀⢈⣷⡄⠀⠀⠀⢧⡀⠀⠜⠅⠀⠀⣀⡀⢠⡄⠉⠀⠰⠟⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⡉⠳⣄⠙⣦⣤⣿⣷⡇⠀⠀⠀⠀⠉⢲⡖⠒⣈⠁⠙⠁⣈⠀⠘⠛⠀⣤⠘⠓⠀⣷⡀
⠀⠀⠀⠀⢀⣠⠖⠹⡏⠁⢿⡀⠹⣇⠈⠳⣄⠈⠳⣇⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⡁⢀⠉⢠⣤⠀⠛⡀⠸⠇⢀⡀⠀⣶⠄⢴⢌⣑⡒⢠⣶⣶⠄
⠀⠀⠀⣰⡟⠹⣆⠀⢻⡀⠘⣧⠀⠙⣦⠀⠙⢦⡀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡏⢳⠀⠞⠉⠉⠉⠉⠑⠺⡏⢸⠉⡄⣇⠀⠈⠉⠉⠉
⠀⠀⢰⠅⣿⣄⡙⢦⡀⢳⡀⠈⣧⣀⠈⠷⣤⣤⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⢸⣴⠇⠀⠀⠀⠀⠀⠀⢹⣾⠀⣷⡇
⠀⠀⠀⠀⣇⢹⢹⡈⡟⠉⠉⠀⠀⠉⢻⡀⡏⢸⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⣀⢀⣀⣀⣀⣀⣠⣌⣡⣀⠈⣡⠴⢤⡀
⠀⠀⠀⠀⡟⣿⢸⣛⡇⠀⠀⠀⠀⠀⢨⣛⡇⠸⣦⡇⠀⠀⠀⠀⠀⣠⢤⣄⣶⢲⣮⠶⠭⠿⠾⠗⠚⠒⠛⠚⠓⠚⠓⠉⠉⠁⠈⠀⠉⠙⠓⠒⠒⠲⠦
⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠉⠁⠀⣠⠔⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⣙⣎⣱⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠓⢦⡉⠒⠒⠒⠒⠲⠦⠤⢤⣄⡀⠀⣀⠀⠀⣠⣤⠤⠤⠤⠤⠤⣤⡀⢠⣄⠀⣤⠴⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣀⠀⢀⣀⣀⣀⣀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡏⢧⡼⠀⠀⠀⠀⠀⠀⠀⠀⢧⠏⠸⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠳⠍⣤⠀⠄⡀⢠⠉⡿⠃⢸⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣞⠝⠶⠒⠒⠢⢤⣠⢤⡀
⠀⠀⠀⣠⠴⠒⠛⠋⠉⢹⡇⠀⠘⡄⠀⠘⢿⠃⠀⠀⡇⠀⢸⠀⠀⢀⡠⠟⠛⢧⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣣⣏⡀
⠀⣠⢾⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠑⠤⣈⣁⣁⡤⠞⠀⢠⠟⠀⣴⣋⣀⠀⠁⠘⡿⡆⠀⢀⡖⣋⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠉⠉⠓⠲⢄⠀⢀
⣰⡇⣪⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠙⣧⠶⠆⢻⣳⡄⢸⡄⠋⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡼
⠉⠁⣵⠀⡴⡄⢀⣤⣤⣤⣤⣤⣤⣄⠀⡬⡍⢹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣾⡿⣆⠀⠳⣄⣀⠈⠁⢀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⢰⠁⡇⣿⠀⠀⠀⠀⠀⠀⢹⠀⠇⢇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢟⣻⡀⠀⠀⠈⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠸⠞⠀⠣⠇⠀⠀⠀⠀⠀⠀⠘⠾⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢸⣭⣇⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⡤⠦⠤⠤⣤⠀⠀⣀⠀⠀⡇
⠀⠀⠀⢀⣠⣤⣄⢀⣀⣀⣀⣀⠀⢀⡠⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣄⡷⢼⡆⠀⠀⠀⣇⠀⣸⢻⠀⢀⠇⠀⠀⠀⢸⡀⣸⢿⠀⢀⠇
⠀⠀⢰⠋⠀⢀⡼⠉⠀⠀⠀⠈⢹⣏⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠁⢻⡒⣇⠀⠀⠀⠉⠉⠁⠈⠓⠋⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉
⠀⠀⡿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⠀⢹⣿⡿⠋⣭⣭⡉⠉⠉⠉⠋⢿⣿⡗⠲⣦⡀
⠀⠀⠹⣄⠀⢿⠙⠂⠀⠀⠉⠛⠀⠼⢤⡀⠀⣀⡼⠛⠓⠶⢤⡀⠀⠀⠀⠀⠀⠸⣿⠷⠀⠀⣤⡀⠀⠻⠟⠃⠀⣶⡆⠀⠀⢀⣶⣦⠀⢻⡀
⠀⠀⠀⣈⠉⠙⣆⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⠀⠀⠘⠿⠟⠀⠰⣾⡦⠀⠈⠀⢰⣶⡄⠉⠁⢀⠾⡇
⠀⠀⠸⣍⠳⠶⠋⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⢀⡶⡄⣾⠿⠤⠤⠬⠤⠶⠶⠖⠒⠻⡆⣴⡆⡜⠀⣧
⠀⠀⠀⠈⠑⠲⠤⠴⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠧⠀⠀⠀⣼⢸⠇⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡇⡇⡇⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡄⠀⠀⣀⣀⣀⣠⣀⣀⡀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣿⢸⠀⢻⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢳⠀⠀⡏⠀⠀⠀⠀⠀⢳⠀⢸⡇⠀⢸⠀⠀⠀⠀⢹⢸⠀⠸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣷⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣸⠀⠀⠀⠀⢸⣾⠀⠀⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠗⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠈⠓⠚⠁⠀⠀⠀⠀⠀⠘⠒⠋⠘⠒⠃⠀⠀⠀⠀⠀⠁⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠐⠀⠂⠒⠀⠃⠐⠀⠒⠐⠐⠂⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣴⣿⣦⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠀⢀⣆⠰⠖⠀⢻⣿⡌⠀⠀⠀⠀⠙⠛⠷⠂⢶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣉⠁⠀⢀⡴⣌⣭⡀⠀⣀⣠⣄⠀⡀⠀⠀⢸⣿⣿⠦⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⣼⣿⣿⣿⣾⣟⣡⣶⣿⣿⣿⣏⣰⣶⣿⣿⣿⣿⣿⣿⣿⣦⡈⠠
⠀⠀⠀⠀⣰⣤⣄⠀⠈⠀⡿⢋⠉⣿⣿⡱⣦⣼⣿⡏⣿⣿⡇⢯⠃⣶⣴⣿⢻⡁⣶⣝⣿⢻⡳⡄
⠀⠀⠀⠉⠹⢛⡂⠀⠀⢸⡇⠸⠧⢹⣿⣿⣶⣌⣿⡇⣩⣍⡥⣿⠀⣬⣿⣾⣷⣿⣶⡙⣟⠁⠀⠅
⠀⠀⠀⠀⠀⠸⠧⢀⣤⣾⣰⣿⣿⣆⣿⣝⠿⣋⣿⣇⣿⣿⣧⣸⣆⡟⠛⣻⣯⠛⠿⣣⣿⠀⣴⡀
⠀⠀⠀⠀⠀⠐⠲⡿⢻⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣻⣿⣿⣿⣿⠛⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠤⣵⡉⠁⠉⠹⣿⠟⠁⠊⣡⣿⣿⣿⣿⡟⠁⠀⠀⠀⣽⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠸⣯⣿⡍⠘⣃⢶⡄⠀⠀⠠⠀⠙⠿⠿⠋⠉⠁⣀⠠⠀⠀⠾⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⡐⢤⣶⣶⣾⣧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⠉⢐⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⣴⠋⢀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⣸⠃⠀⠉⢿⠞⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡷⠺⠏⠀⠀⠀⢀⠟⠀⠀⢀⢠⣧⠻⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡷⠀⠀⠀⠀⢀⣼⡇⠀⠀⢀⣾⣿⠀⠀⠘⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⣤⣶⠀⣼⡀⠀⡀⠸⠿⣟⠀⠀⠀⠘⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢁⣠⡶⣿⡿⠋⠀⠙⠁⠀⠈⠶⣄⠿⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠟⠀⠀⠀⠀⠀⠀⡀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠆⠀⠆⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⣤⣶⣵⠊⢿⣶⣄⣄⠀⣀⠀⢀⣠⣤⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠁⠀⠀⠀⠘⢷⡌⠉⠛⠛⠈⢿⡿⣿⡿⠙⠃⣩⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⡜⠀⠀⠀⠙⣦⡀⠀⠀⣀⣽⠟⢛⣤⣈⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣀⠀⢀⡉⠉⢉⠹⠠⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢻⡇⢼⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣿⠇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣿⣿⠃⠀⣠⡴⠁⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠟⠀⠚⠻⠃⢰⡟⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⡿⠋⠀⠀⠀⠀⠀⠐⣾⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠟⠙⠛⠛⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠈⣡⠆⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡶⠶⠿⠿⠿⣿⢿⣿⣯⡴⠞⠁⣤⠀⠈⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠔⣸⣿⠀⠄⢠⠇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⣼⠟⠁⠀⠐⠛⢋⣁⣤⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⣠⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺


⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⣠⠊⠉⠒⠉⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡁⢘⣆⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⢀⣠⡤⣤⡀
⠀⠀⠀⢀⡴⠊⠉⢠⡰⢏⠃⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣘⣁⠙⣉⣏⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣾⣿⢶⣤⠁
⠀⠀⠀⢺⢀⡷⠚⠃⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣍⢤⣩⣀⣯⠀⠀⠀⠀⠀⠀⢸⠟⠈⢲⡷⠊⠀⠘⠃
⠀⠀⠀⢸⠻⢥⣤⠤⠊⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⢲⠓⠂⠚⠀⠀⠀⠀⠀⠀⠼⠥⣌⣡⣨⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⠀⠀⡄⡀⠀⢰⣀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⢀⣠⡴
⠀⠀⢀⢤⠀⡊⠀⢈⡀⡏⣷⠀⠀⠀⠀⠀⠀⢠⡀⠀⡰⠟⢠⠖⠁⠀⠀⢈⠙⣶⠂⣿⡃⢀⣞⣥⠾⠤⠀⠀⠀⠀⢹⣧⠀⢀⢔⡿⠉
⠀⠀⢸⣸⣦⣰⠀⢸⣾⢟⠟⠀⠀⠀⠀⣀⣈⢳⣿⣠⢶⣟⣿⣀⠀⠀⠀⠋⠉⠙⠫⡹⡴⠟⠛⠻⢯⠀⠀⠀⠀⠀⠀⣯⣇⢎⡞
⠀⠀⠈⠳⠥⡟⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⢎⠳⡽⢡⢻⡟⣥⠴⠀⠀⠀⠀⠀⠀⠀⢧⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⣿⣼⡀
⠀⠀⠀⠀⠀⠱⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣇⣬⣬⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣇⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣸⣿⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⠤⡄
⠀⠀⠠⠶⢤⡄⡾⢁⣤⡆⠠⠀⠀⠀⠀⠀⣴⣸⣿⣷⢟⣾⣶⡁⠀⠀⠀⠀⠀⢀⠮⠭⠥⠬⠭⢷⠀⠀⠀⢀⡴⠅⣤⢤⠒⠶⠻⣄⣄
⠀⠀⠀⠑⠂⠉⣧⠟⠭⠔⠀⠀⠀⠀⠀⠀⣿⣿⣾⡏⣼⣿⣹⣃⠀⠀⠀⠠⣦⠞⠀⠀⠀⢀⠀⢈⢣⣄⠀⢨⠧⡶⠿⢀⡀⢐⠶⠀⠘⢶⡄
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡟⣾⡛⣿⢗⡿⡏⠁⠀⠀⠘⠓⠚⠙⢦⡜⠉⠑⠊⠀⠀⠀⠀⢱⠃⣠⠋⠉⠙⠋⠙⣦⠀⢣
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠟⠚⠃⠛⠂⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⡤⠢⡔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢟⣻⠒⢤⠀⡴⢫⠝⠋⠉⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀
⠀⠀⣀⣀⣃⢀⣀⡸⠊⠉⢿⠀⠀⠀⠀⠀⠠⡀⠀⡆⢀⡠⠔⠂⠀⠀⠀⠀⣀⠤⡤⡭⣿⣿⣧⡀⣲⢦⡀⠀⠀⠀⢀⠞⠁⠀⠀⠙⡄
⠀⠘⠢⡤⠈⣿⢉⡷⠠⢤⠚⠀⠀⠀⢀⣀⣀⡈⢦⣷⢞⣡⡤⠤⠀⠀⢠⣮⢴⠎⣰⠇⠈⠘⡆⢷⠈⠉⠁⠀⠀⠀⠸⠀⣄⠀⠀⢀⢱
⠀⠀⠘⠤⠚⢈⡉⡧⣸⠞⠀⠀⠀⠀⠉⠀⠀⠉⠻⣷⠋⠁⠀⠀⠀⠀⠈⠀⠈⡼⠀⠀⡇⡆⠙⣾⠀⠀⠀⠀⠀⠀⠘⠚⢻⠒⢲⠛⠋
⠀⠀⠀⠀⠑⠊⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠒⢒⠒⠻⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣾⣯⣭⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡿⠛⢻⠋⠉⠉⠉⠀⣿⠿⠛⠛⠛⠙⠛⠛⠛⠛⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⣿⠀⢸⡿⠿⠛⠛⠉⠁⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⣿⡇⠀⠀⠀⠀⢼⠇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⣀⣀⡠⠤⠀⣷⠆⢸⣿⣿⡇⠀⠀⠀⠀⢶⡆⣸⣬⣭⣉⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⣿⠀⢸⣧⣿⡇⠀⠀⠀⠀⢸⠁⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⣿⠀⢸⣗⣿⡀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⢀⣀⣋⣀⣈⣉⣭⣤⣤⡤⡤⣄⣈⣁⣝⣻⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡴⢶⣶⡋⠉⠉⠉⢀⣭⠏⢹⡿⣿⠇⣀⡀⠀⣀⣨⡽⢾⡿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠲⡄⠃⢸⠋⢉⣦⣴⡀⣿⠀⢸⡇⣯⣤⠼⠟⠛⢦⣝⡇⣸⡇⣿⣿⣿⣿⣿⣿⠀⢸⢧⣈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⣀⣸⡶⠉⠀⠀⠀⠛⠒⠘⣛⣉⣁⣀⣀⣀⣀⣙⣛⣻⣛⠿⠿⣿⣿⣿⣿⣄⣸⣠⡏⢸⣏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⣉⣩⣥⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣷⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢿⡿⠛⢿⣿⠿⠿⠿⣻⠋⣿⣟⣭⣼⣿⡿⠿⠿⣿⠿⢻⣿⡻⠛⠋⠙⠻⣿⣿⡟⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠦⣤⣮⡛⢦⡤⠘⢀⣿⣿⣿⠋⢀⣠⡴⢶⠿⠶⠚⠉⠀⠀⠀⠀⠀⠸⣿⣏⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⡟⠉⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣿⣿⣿⡌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣤⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣴⣾⠓⣣⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠄⠀⠀⠀⠀⠀⣷⡀⠀⠀⣿⣿⣿⣧⣤⣤⡔⠶⣛⡟⠁⣠⣴⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣠⡜⠁⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣿⣿⣿⡇⢙⣠⣤⣴⣶⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⠀⢄⠀⣿⣇⡴⠖⠀⠀⠀⠀⠀⣿⠀⠀⡆⣿⣿⣿⡟⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣧⡀⣶⣿⣿⡀⣀⣤⣀⡐⢲⣶⣿⠀⢠⠇⣻⣿⣿⣷⣀⣀⣠⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣩⣾⣿⣿⣿⣿⣿⣿⣋⣿⣿⣷⡿⠀⣾⡄⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⡿⡿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣾⣁⢀⣀⡍⠲⣿⠟⠁⢤⣤⡀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠿⣯⣭⣧⣾⣿⣿⢀⣤⣀⣼⣾⣿⣮⣿⣷⠹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣴⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⣽⣿⠻⣷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣼⣿⣞⢿⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣴⣄⠀⠀⠀⠀⠀⢀⣠⠀⡀⠻⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣯⣿⡟⣿⡿⣿⠟⠋⣽⣿⠈⢷⠄
⠀⠀⠀⢀⠀⢠⣤⣾⣿⣿⣿⣿⠀⠑⢠⡀⠀⣾⣿⡏⢈⡇⣠⣿⣿⣿⣿⠿⣿⣿⣿⠿⠟⠋⠙⠻⣿⣿⣶⣿⡦⣼⣿⠛⣠⣀
⠀⠀⠀⣾⣧⣼⣿⣟⣋⠉⠉⠉⠀⠀⠀⠉⠢⣿⣿⣇⣹⣿⢿⣿⣷⡻⢿⣿⣿⣿⣿⣶⣶⣿⣶⣿⣧⣼⣿⣿⣾⣿⣿⣤⣿⣿⣷
⠀⠀⠀⠹⣿⣿⣃⣨⡟⠐⣷⣦⣆⠀⠀⠀⠀⠺⣿⣿⣹⡇⠘⢿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⣛⣩⣿⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠘⠿⣿⡿⢇⣼⣿⣿⣇⣴⣦⠀⠀⠀⣻⣿⢹⡇⠀⠀⠈⠻⢿⣿⣿⡙⠻⠿⢿⣷⣶⣾⡶⣶⣿⣿⡿⠿⠛⠛⢻⠿⣯⡏⠉⢳
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⡼⠃⠀⠀⠀⠀⢨⣍⠙⠻⣷⣤⣤⣤⣤⣴⣶⣾⣿⣿⠿⠶⠶⠬⠅⠀⠀⢀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡵⣂⣼⣿⠿⠋⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣤⣤⡤⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⠿⡾⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣄⠘⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣻⣖⢤⣤⣤⣤⣄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠾⠿⣿⣿⣷⡝⢿⣿⣿⣿⠾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣷⣶⣤⡙⠿⠿⠷⡹⣿⣁⣀⣃⠰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣽⣷⣿⣿⡦⠂⣈⣽⣮⢿⣭⡙⡆⢼⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⠻⣿⣿⣿⢿⠻⣿⣿⣿⡿⠁⢰⣿⣿⣿⣧⡻⡿⢰⢠⡉⠳⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠐⠢⢬⣿⣿⣷⠤⠽⠋⠀⠀⢸⣿⣿⣿⠉⣷⡝⠈⢧⠀⠀⠀⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡤⠞⠙⠀⠈⣆⠀⠀⠀⠀⠈⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⢖⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢿⠸⣠⠇⠀⠀⠸⣆⠈⣩⡆⠀⠀⠀⠀⢀⠄⠀⠀⠀⣿⠀⢛⡛⠁⠀⠀⠀⠀⠀⠀⢀⡤⠘⢁⡀
⠀⠀⢈⠐⢋⡶⠀⠀⠀⠈⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡈⠚⡏⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠴⠛⠁
⠀⠀⠸⣦⡈⢠⢴⡆⠛⣩⠃⣞⣴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠇⠀⠀⠀⠀⠁⠀⠀⠸⣷⡄⣰⡦
⠀⠀⠰⣄⠙⣠⠶⠀⢤⣘⠧⢁⣴⠇⠀⠀⠐⠀⠀⠀⠀⣄⡻⠄⣀⡶⠀⠀⠁⠀⠀⠀⠀⣤⣽⣡⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠚⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡶⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠆⠀⠀⠀⡈⠗⠾⠇
⠀⠀⠰⡆⢚⡋⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⠰⠈⠛⠋⠀⠀⠀⠀⠀⠊⠸⣀⠁⢋⣠⠄⠀⠀⣻⢇⢴⣦
⠀⠀⢀⠐⢋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣠⡄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠋⢡⠆⠀⢀⡹⠀⢰⠇
⠀⠀⠘⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⡾⠍⡉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣗⡏⢠⡶⠘⣛⠁⢉⡙⠁
⠀⠀⠰⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣈⣹⣏⣤⠀⠙⠳⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣠⣤⣄⣄⣄⣀⣀⣀⣀⣄⣠⣤⣤⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠀⠿⠿⣿⣿⣿⠿⠿⠿⠿⠟⠻⢿⣿⡿⣿⣿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⠛⠛⠛⠻⠏⠉⠁



⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣷⣶⣶⣷⣶⣾⣿⣶⣾⣿⣿⣿⣷⣾⣿⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⠶⣶⠶⠶⠀⠂
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⠉⠀⠀⠀⠈
⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠟⠙⠋⠛⠏⠁⠙⠁
⠀⠀⠀⠘⢻⣿⣿⣿⢿⣿⣿⠛⣻⠻⠟⠛⠛⠻⠏⠛⠃
⠀⠀⠀⠀⠈⠀⠽⠴⠏⠊⠉





⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⡿⡿⠛⠟⡿⠛⠛⠉⠉⠉⠙⠛⠛⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⢠⡤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⡤⣀⡤⣠⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⣠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢆
⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣿⡿⣄
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⣀⣰⣿⢴⣦⣶⡿⢿⣿⠃⣏⠁⠘⢢⣀
⣿⣿⡿⠋⠀⣠⢤⣠⠄⡚⢻⡷⠾⠛⣩⣿⠱⠟⣼⠰⠸⠆⡀⠀⠉⠻⡒⠀⢄⡀⠀⡀
⠟⠋⠀⠀⣰⣿⣧⠚⢁⠴⠀⡠⢴⣿⣣⢀⣠⣾⢿⡆⠀⡀⠸⢆⠀⠹⠀⣀⠀⠉⡓⣧⡆
⠀⠀⠀⢸⣿⣿⡿⣻⠶⣛⣡⣼⡿⢣⢇⡞⡋⢡⠎⠹⡆⠀⠲⠎⠀⡄⣸⠀⣵⣇⣤⣧⣷⡀
⠀⠀⢠⣿⠟⠁⠀⣀⠛⣋⡙⠋⠀⣠⠎⢠⠉⢠⣆⠀⢀⠠⢠⡄⣀⣡⣬⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠙⠳⠚⠛⠛⠛⠛⠛⢿⡄⡄⠁⠁⣾⣇⣸⣻⣤⣸⣤⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⡅⣃⣣⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣧
⠀⠀⠀⠀⢰⡄⠀⣧⣎⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⢿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⢡⣀⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠘⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣸⣾⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⣿⣿⣿⡷
⠀⠀⠀⠘⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠟⠀⢀⣤⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣄⣠⢀⣾⡃⢀⣀⣤⣤⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⢾⡶⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠷⠀⠀⠀⠀⠀⠀⢲⠶⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠎⠁⢨⣿⣭⣿⣿⣿⣿⣿⣿⠟⠿⢿⣿⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⢼⡟⣽⣿⣿⣿⠿⠆⠀⠀⠁⠈⠃⠀⠀⣀⠀⠀⠀⢀⡄⠀⠘⠁⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⣿⠋⢸⣯⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⡼⢥⣾⡟⡽⠁⣧⣤⣾⢧⡆
⠀⠀⢠⣤⣤⣴⣶⣷⣟⠛⠁⠁⣀⠀⠀⠀⠀⠀⠀⠋⣀⢸⣿⠁⠀⠀⠀⠀⠀⡀⠀⢹⣿⣷⢶⣾⡇⠁⢠⠿⢋⣽⣿⡗
⠀⠠⡶⣿⣿⣿⠻⠻⡿⠀⣦⡀⠉⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠉⢿⡇⠀⠸⢶⣛⣻⣿⡅
⠀⣠⣧⣹⣿⣋⡄⢠⣤⣸⡍⠀⠀⠀⣄⠀⣐⠀⠀⠀⠰⠄⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⡇⠀⣴⣾⣧⡁⣿⣦⡀
⠀⣻⣿⣿⢟⢛⣶⣞⣀⢻⣿⡁⠀⠠⣿⣦⢿⡆⠠⠧⠀⠙⣿⡀⠴⢶⡄⠀⠀⠀⠀⠀⠈⠁⠰⠿⡁⠀⢸⣿⣿⣿⠟⢹⡇
⠀⠉⠿⣛⣻⡿⢏⣽⡍⠀⠙⣿⣿⡗⣿⢿⣾⣄⠸⠡⠂⡀⣿⡄⣅⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇⠀⠄⡉⢩⣯⠟⣹⠇
⠀⡀⠀⡜⣻⡀⠀⠿⣿⣶⣄⠐⣼⣇⣿⣷⣮⠽⠈⠉⠁⠁⣿⡇⠿⡍⢀⠀⡄⠀⠀⠀⢠⣴⣿⣿⠟⠰⣞⢡⡿⣟⠁⣿⡆
⠸⣅⢈⣀⢸⣦⣖⣤⣌⣸⠿⠃⠺⠁⠲⣿⣷⣶⡶⠀⠄⠀⣿⢃⣄⣿⠴⢾⠃⠀⠀⣀⣴⣿⣿⡅⢀⣀⠈⠐⢟⠿⣶⣿⠃
⢨⣾⣍⡈⣽⡏⠙⠛⢙⡟⣳⢸⡟⠀⠇⠘⣿⣿⡶⣂⡈⢠⣿⠈⠀⣿⣄⣞⡀⢧⣬⣿⣿⠿⣋⡁⠀⠟⠛⠳⣤⣀⣸⣿⡄
⠛⠀⠸⠓⢋⠁⢤⣳⢾⣷⢿⣟⢿⣤⠂⢽⣷⠳⣌⣻⣧⢸⣿⠆⠀⣼⣷⣦⣆⣉⠹⡟⠋⠀⠘⢁⠈⢰⠷⠔⣿⣿⣿⣿⠅
⠀⡐⠀⠀⣿⡇⠀⣿⣼⣆⠀⢻⣿⢉⢷⣴⢯⣯⠙⠋⢹⣼⣿⠀⠀⣻⣸⣿⣵⢭⣽⣶⣀⣣⡗⠨⠍⠉⠀⠀⣻⣿⣯⠉⠁
⢠⡀⣀⠀⣿⣷⡀⠙⣿⣿⣿⣿⣿⠏⣥⣿⣿⣿⣇⠀⠘⣿⣧⠀⣼⠿⣭⣿⣷⣌⢻⣿⡿⣿⣷⣀⣼⣤⣽⠦⠾⠶⠿⠛⠁
⠚⠛⠶⠥⠿⢿⣿⣾⣯⣽⣿⣿⡿⢠⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⢠⣿⣿⣿⢃⣼⣿⣶⣋⠀⢀⣨⠚⠁⢄⣤⣾⠋⢠⣄
⠀⠀⣖⢀⡀⠀⠀⠀⠀⠀⠉⠀⢼⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡿⠁⠀⠸⢿⣿⡿⠀⢂⠀⠈⠑⠒⠠⠌
⠤⢀⠸⣟⣐⣦⡄⠀⠀⠀⠀⢿⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠾⢠⡀⢈⠙⢷⣦⣆⠳⠐⣶⡤
⠀⠀⠀⠛⠫⠁⠄⠀⠌⠋⠒⣻⣿⣯⣿⣿⣿⣿⢿⣿⣿⣟⠛⢿⣿⣿⣿⣯⣰⡦⠞⠿⠱⠧⠤⠤⠤⠟⠛⠙⠓⠤⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣉⣛⣉⣈⣙⣛⣛⣉⣉⣃⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠁⠀⠀⢀⣀⣹⠀⠜⠀⠀⠀⠀⠠
⠀⠠⠀⠐⠦⢤⡀⠀⠃⣼⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆
⠀⠀⠀⠐⠓⠒⢿⣦⢰⡇⠀⠀⠀⠀⡀⠀⢐⣀⣤⠀⣠⡑
⠀⠀⠀⠀⠀⠀⠀⠈⡌⠃⠀⠀⠀⣀⣁⣀⣀⣀⣀⣀⠀⡌⠷⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠯⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣠⠞⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣀⣀⣄⣠⣤⣠⣤⣤⡌⠁⣠⠟⠁⠈⠐⠀⠀⣈⣁⣀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⠿⠗⠋⢩⠻⠀⠇⠸⠁⠿⠿⣿⠿⣿⡀⢀⣀⣀⡀⠐⠧⠀⣠⡀
⠀⠀⠀⠀⠀⠒⠚⠒⠀⠀⠐⠋⠛⠁⠀⠀⠀⠐⠃⠀⠘⠀⠘⠀⠰⠀⠐⠂⠐⠀⠊⠃⠠⠄⠉⠃⠘⠅⠀⠀⠀⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⢴⢀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡰⢺⣷⡆⠀⠸⢄⡀⠈⢤⣄⠀⠀⡄⠀⣤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⣰⡆⠀⣀⠀⣴⠾⠃⠀⠻⠲⠿⠛⠓⠓⠒⣾⣟⠋⠠⡘⠃⠇⠃⠀⠛⠀⠚
⠀⠀⠀⠀⠒⠊⠉⠉⠉⠋⠀⢀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠚⠉⡀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⣀⣀⠀⣰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠰⢂⡀⠀⠰⢿⡟⢶⢻⣙⡽⢖⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠁⠀⠀⣰⡟⠈⢿⣭⣲⣄⣾⣿⣿⣾⣷⣤⡋⠣⠀⣉⢀⡀⢨⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⡶⣇⠀⢸⣿⣗⠚⠉⠻⠟⣿⣿⣿⠛⠁⠀⢘⡻⠋
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣾⡷⠁⠀⢻⣿⣿⠇⠀⣶⣄⠉⠁⠄⠀⠀⠀⠉⣿⡟⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡁⠸⣾⠄⢀⣤⣶⢵⣶⡇⠀⣹⠓⠀⠀⠀⠙⠟⠀⠸⣽⣿⣷⠞⠀⠀⠀⠀⠈⣶⡛⢽⣯⢷⣿⠿⡄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣀⡠⣔⡍⣀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢿⣾⣿⣾⣷⣶⣾⣿⣿⣖⣦⣀⡄
⠀⠀⠀⠈⢙⣷⠾⣿⣿⡆⢤⣻⣽⢿⣿⣿⣷⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣘⣿⠻⢿⣿⣿⣿⣿⡿⣿⣾⠻⣷⢿⣧⠄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⢠⡶⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣴⣯⣯⣴⣜⠁⠈⠉⢛⣛⠧⠛⠤⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠛⠀⢶⡇⠹⢿⣿⣿⣿⡟⠻⡇⠀⠀⢾⡿⠁⠀⠀⠸⣧⣷⣀⠀⠀⠀⢰⣿⣿⣿⣿⠁⣤⣤⡄⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢁⣽⣿⣿⡾⢗⡀⠐⣿⠃⠀⠀⠀⠀⢹⣇⣾⡆⠀⠀⣿⣿⡿⣿⢿⣷⣾⣿⠿⣤⡈⠃⢰⣦⣀⣠
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣤⣾⣿⣿⣿⣤⣀⡇⣽⣿⠀⠀⠀⠀⠀⠀⣿⣿⣇⣡⠀⠘⣿⣇⣀⣾⣿⣏⠁⠾⠛⠃⠶⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⢿⡃⢨⡁⣿⣿⠀⠀⢠⡀⠀⢀⠸⠿⣿⣿⣼⡄⢻⣽⠄⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠁⠀⠀⣠⠉⣿⡆⢸⣦⠙⣦⡘⢦⣀⠛⠿⣿⣿⠆⠝⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠀⠘⠙⢦⡹⣷⣍⢿⣤⡹⣿⣿⣶⡯⠙⣂⢀⡀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠟⠿⢭⣿⣾⣝⣿⣦⡀⠄⠈⣵⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠂⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⠀⠀⠀⣿⣿⠋⣿⣆⡹⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⢸⡄⠀⠀⠟⠁⠀⣽⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢸⡇⢀⡆⠀⣠⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⠀⠀⢻⣿⣿⣟⢀⣬⣱⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣀⡀⠈⣿⣿⡿⠸⠏⠁⣤⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣇⡀⢿⣿⣿⣄⣠⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣾⣿⣿⣹⡟⣋⣠⣶⡿⠻⠋⣹⣿⣿⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣋⡻⠿⣹⡧⠀⠶⠖⠉⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣇⠙⣿⣷⣬⣇⠀⡀⢢⡀⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠈⣿⠟⠉⠉⠙⣄⠰⢶⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠿⠶⢷⣆⣨⣧⣼⣿⣿⡿⠿⠿⠛⠛
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⣸⠇⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠘⠋⠉⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⢸⡅⢰⣿⠄
⠀⠓⣤⡄⣿⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠸⣿⡟⠁⣀⡴⠀⣠⡞
⠀⠀⣿⠁⠙⣯⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣴⣾⣶⠚⠉⠠⢞⠛
⠀⠀⠀⠰⠃⠀⠀⠀⢀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡌⠻⣿⣿⣿⣁⠤⣠⣾⡵⠊
⠀⠀⠀⠀⠀⠀⢤⣴⣀⡷⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠙⣿⡄⠘⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡟⠁⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣧⡸⣇⠀⣾⠉⠁⠀⡀⢀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣶⠏⣠⡄⠟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠛⠀⠀⠀⢀⡀⠀⠘⣿⣿⡟⣿⣿⣟⠰⣿⣀⣠⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣴⡎⡹⣶⣄⢿⡟⠀⠈⢻⠋⢁⣴⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣷⣠⡆⠉⠘⣧⠘⣿⣿⠁⢠⠀⠀⣠⣟⣱⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣣⣄⠀⢠⣄⠸⣿⡄⠸⡆⠀⣹⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢿⣏⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⡋⠉⠙⠁⠈⢿⣧⢿⣷⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⢰⣦⣴⣄⠀⠉⠠⣿⣧⣿⠇⠀⠀⢀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⣾⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⡌⢻⣿⣿⣷⣤⣀⠙⣿⠇⢀⡄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣼⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡃⣼⠏⠻⣿⣿⠱⠀⢾⣠⡎⢡⡾⠃⠀⠀⠀⠀⠀⢦⣄⠀⢀⡁⠀⠙⡏⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠀⠀⠀⠙⠟⠀⠗⠀⢿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⠀⠀⢶⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣦⡀⠀⣾⠀⠀⢴⣧⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⣄⣸⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣴⡟⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠿⣿⣿⡅⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠙⠟⠀⠀⠀⡀⢻⡷⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡟⢻⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠐⡇⠀⣾⡇⢸⣿⡀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠘⣟⣧⡴⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣄⡀⠀⠀⠉⠻⢼⡆⠻⡇⠈⣿⠁⠀⠀⠀⣴⠀⠀⠈⠁⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣈⠳⣎⣁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠄⠃⢰⣧⠀⠆⣠⣴⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣮⣿⣦⡄⠀⣦⠀⠀⠀⢀⣴⣷⡆⠘⣿⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠙⢿⣿⣿⡇⠀⡷⢀⡄⢠⡾⠋⢹⣿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠙⢿⣧⣤⣥⢀⣴⠟⢀⣶⣾⣿⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠉⢿⡇⠈⢡⡄⢸⣿⠀⣈⢠⣶⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠈⡇⢀⣾⡇⢸⣿⣦⣈⡵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⣿⠈⢿⡇⠀⡻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣷⣄⠀⣿⡆⠘⠃⠀⢃⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⢿⣿⣿⠿⢇⣿⠀⢀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⡆⠈⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠛⣿⡄⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣰⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣷⣾⣷⣶⣶⣄⡢⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣾⣿⢿⡿⠿⠟⠛⠉⠉⠁⠂⠀⠉⢍⣉⣉⡉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⢀
⠀⠀⠀⠀⠀⠀⣚⣾⣿⣿⣿⣟⠁⠂⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⣀⠉⠡⢐⠩⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇
⠀⠀⠀⠀⠀⠈⢃⣿⠿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢄⣒⢠⣄⠒⠥⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣐⣺⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢩⣯⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⣸⣿⣿⣿⡿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢨⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠛⠁⠀⠀⠀⣀⣠⣤⣤⣤⣦⣤⣤⣠⣀⣀⠀⠁⢰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣃⡠⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡃⠁⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠘⣗⣡⣥⡀⢀⣄⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⠟⠉⢸⣿⣿⣿⣿⣿⣿⣯⣴⣿⡇⠙⠛⠿⣿⣿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠸⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣞⢻⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠉⣙⣛⢛⣻⣿⣿⣻⣷⠙⢻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣧⣀⡀⠀⠈⠻⣿⣿⣿⣿⡿⠿⣯⣈⣉⣩⣯⡍⢛⣋⣛⣛⣟⣋⣤⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣬⣙⡛⠛⠲⠶⠶⠾⠯⠷⠶⠞⢛⣛⣉⣥⣴⠶⠿⠋⠉⠉⠉⠉⠉⢹⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⡿⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⢰⠆⠐⠒⠢⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠈⠙⢃⣄⠀⠀⠀⠄⠀⠀⡐⠀⢹⡀⠀⠀⠀⠀⠀⠀⡀⠄⣈⣤⡊
⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⢤⡀⢰⠋⣓⠄⠀⢀⣾⡇⠀⠀⢂⡀⠈⠃⢀⡩⠀⠀⠀⠠⣦⠀⠀⢡⡏⠉⣟⢳⠲⡄⠂
⠀⠀⠀⠀⠀⠒⠁⠀⠤⣝⡲⢭⡨⣝⡊⠓⠀⠈⠉⠀⠀⠀⠀⠘⠚⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠹⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡒⠦⠝⠲⠍⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⢀⠀⡄⠀⠀⠀⠀⠀⠀⡧⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡁⠀⠀⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠋⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠞⠳⠃⠀⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠘⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠉⢈⠀⠀⠀⢀⠰⠔⠳⠄⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡇⠘⠋⢸⠀⠀⠀⠀⢁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⢀⡀⣠⠂⠀⠀⠈⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠃⠒⣁⠆⢀⡃⠐⢠⠀⠀⠀⠀⢸⠇⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡙⠁⠤⢠⠄⣀⡰⢀⡠⠀⠀⠀⠀⢀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡸⠁⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠘⠈⠓⣐⡃⠀⣃⡀⠀⢨⡆⠀⢤⠄⠀⠀⠀⢸⡆⠀⢶⣶⢆⡀⡀⠀⢀⠀⠀⠀⠀⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⠀⠁⠾⢃⡈⠀⠉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠾⠂⠈⠃⠀⠀⠀⠀⡀⠈⠁⣄
⠀⠀⠀⠀⠀⠀⣠⠠⢄⣀⣤⠖⠒⢄⡠⠞⠀⠀⠂⠀⠀⠀⠀⠀⣀⢣⡄⠀⠠⠄⠖⠀⠀⠀⠀⠄⠀⠀⠈⠈⣀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠜⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠲⠒⠶⠏⠋⠝⠉⠋⠙⠉⠀⠠⣭⣥⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠂⢂⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠙⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣆⠀⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⣐⣿⣿⣿⣿⣅⡀⠀⠀⠀⣴⣿⣿⣧⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⣰⣿⣿⣿⡄⠀⣀⣬⣿⣿⣿⣿⣿⣿⠯⠀⠀⣼⣿⣿⣿⣿⣯⡀⠀⠀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⣷⠆⠰⢿⣿⣿⣿⡶⠈⠉⣻⣿⣿⣿⣿⣿⣿⣿⠷⣦⣾⣿⣿⣿⣿⣶⣄⡀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⣺⣿⣿⣿⠃⣴⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⡡⢾⣿⣿⣿⣯⡀⢀⣾⡆
⠀⠀⠀⠀⠀⣼⣿⡇⣈⣽⣿⣿⣿⣷⣾⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⠦⢼⣿⣿⠄
⠀⠀⠀⠀⣼⣿⣿⡿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣺⣿⣿⡅
⠀⠀⠀⠀⣺⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡗
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠄
⠀⠁

⠀⠀⠀⠀⣀⠒⠴⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⡄⠀⠀⢄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⠀⠀⠀⢀⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠑⠀⠀⠀⠳⡀⠀⡀⢐⡒⢻⣄⠀⠀⢀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⣀⡀⣀⠀⠀⠀⡀⠀⡞⠻⠿⠵⣞⢻⡆⠀⠀⠀⡠⠐⠿⢣⣤⣄⡴⣖⣒⣂⣶⣬⣥⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠙⢧⣤⣈⣋⣴⣶⣷⡄⠸⡄⠀⠠⠋⢸⡇⠀⠀⢸⢅⣲⢛⢽⣿⡛⣯⣽⣿⣿⣿⡿⢛⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠂⠀⠀⢀⣠⣴⣀⠀⠐⠀⠘⣿⣿⣿⠿⣿⡿⠷⣤⡈⢠⡀⠐⣾⣀⣀⣀⠊⣸⣿⢿⣿⣿⣦⣿⣻⣿⠏⠀⠀⠈⠀⠠⠂⠀⡠
⠀⠀⠀⠀⠐⡀⠈⠊⠙⠛⠿⠿⠶⠀⡀⠀⠸⣿⣿⣏⣹⣿⣶⣶⣼⡇⢻⡏⠁⠨⣽⣷⣾⣭⣿⣾⣿⣿⣿⡟⠉⡁⠀⠀⠀⠀⠀⠀⠔⠚⠀⠀⢀⠄
⠀⠀⠀⢰⠄⠀⠀⠀⢈⡁⠠⡀⠀⠀⣧⠀⠀⠣⣻⣿⣾⣿⣼⣿⣿⣷⣶⣆⣀⡜⣿⣿⣿⣿⣿⣷⠟⡷⠏⡼⠁⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠈⢁⡈⢧⡀⠈⠀⠉⠒⠨⣹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡎⢠⠊⡥⠄⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢳⡄⠀⠀⠘⠂⠨⡈⠍⠙⠛⠋⢉⠡⡙⣿⣿⣯⠙⢿⣯⢹⠀⠀⠈⠘⡆⠀⠀⠆⠈⠀⠐⠊⠀⠀⠠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄⢙⠀⠐⠈⢢⡇⠑⠾⠀⢤⣀⢸⣳⣠⣿⣿⣿⣄⢸⠟⠘⢀⠇⢀⡼⠁⠀⡔⠔⠀⠀⡰⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⡀⠉⠲⡀⠈⢿⡀⠸⣦⣜⣧⣈⠓⠿⣿⣿⢿⡿⡇⠀⠠⠀⡾⠋⠀⠀⢈⠄⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠙⢤⡀⠑⠦⢈⡉⠙⢦⡍⢳⠈⠈⣿⠁⠁⡀⠄⣸⢁⢆⣄⠀⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡁⣁⣀⣰⡄⠳⡀⢹⣆⣁⡀⠙⢶⣶⣋⡄⣠⣦⣿⣿⣭⣭⣭⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⡤⢼⣿⣿⣿⠷⢚⣣⡷⢖⣶⣻⣭⣳⣾⣿⣻⠿⣿⣛⡻⣿⣿⣿⣙⣛⠓⠠⡀⡤⠐⠸⠓⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣉⠙⠉⠽⠻⣥⢞⣿⣭⣼⣿⠿⣛⣭⣽⡶⣝⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣶⣟⣶⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⡔⡐⠈⠂⠀⠈⠻⠟⠉⠛⠋⠁⠀⠉⠀⠀⠘⢽⠛⣈⣽⣿⣿⣿⣽⣿⣿⣿⡿⢿⢮⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠒⠒⠒⠒⠒⠒⠓⠀⠒⠒⠒⠆⠒⠒⠒⠒⠒⠚⠛⠓⠒⠚⠛⠛⠛⠛⠛⠒⠛⠛⠛⠃⠠⠄⠈

⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⢢⠁⢉⠁
⠀⠀⠬⡕⡀⠀⠔⣱
⠀⠀⠠⢅⡂⢸⠀⠀⢀⠰⡀⠠⠕⡀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢺⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠇⠘⠎⠀⠌⠣⣀⠔⠈⠁⠂⡄⠠⠗⠀⢀⠀⠀⠀⢀⠃⠁⢢⡤
⠀⡴⠀⠁⠥⠁⠀⠘⡇⢀⢀⡀⠈⠁⢀⣀⠏⢂⠐⠀⠄⠁⠀⠠⠡
⠀⠈⡁⡞⠃⣰⣧⣾⠿⡋⠛⠷⣴⠞⠉⠉⠱⠄⠄⡠⠒⠂⠠⠀⢠⠉⣉⠀⠀⠀⠀⣠⣀
⠀⠀⠉⢣⣼⠏⠁⠀⠀⠀⢐⡍⢹⠀⠀⠰⣤⠓⢲⠃⠀⠀⣤⠲⡆⠈⠀⠀⠀⢰⠒⢌⣴⠂⡄⠀⠀⡀
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢀⡈⠢⢛⠈⢴⢂⡴⠛⠛⠓⠶⡥⣤⢬⡀⢐⠰⡀⠈⠗⠈⠊⠉⢀⠈⠀⣘⠂
⠀⠀⠀⢀⣿⣷⣶⣶⣦⣴⣏⣀⡀⠈⢉⣠⣾⣥⡴⠶⠶⠶⠾⣤⣆⣈⣠⡴⠒⠋⣒⠂⠐⠀⣈⢈⣒
⣠⣤⣾⣿⣿⣋⣉⣉⡉⠉⠙⠛⠛⡻⠿⢯⣍⡀⠀⠀⠀⠀⠰⠂⠄⠀⠈⠉⢩⡫⠈⠑⠠⠍⡀⠀⠐⡀
⣿⠿⠟⠁⠀⠉⠉⠈⠉⠲⢤⠀⠈⢈⠀⡆⠀⠉⡓⡲⠤⠄⠀⠌⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠁
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠃⠄⠄⠊⠀⠀⠀⠈⠁⠀⠀⠒⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡈⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⡎⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡴⠋⠀⢀⣼⠃⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠿⠁⠀⢠⣾⡟⠀⢸⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⣇⠀⠀⢻⣿⠃⠀⣺⣿⠁⢸⡷⠶⠶⠶⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡟⠿⢿⣥⠊⣠⣴⣿⠏⢀⣾⣅⣀⣀⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⡶⢿⣿⣆⠙⢙⣃⣤⣾⣿⣧⣌⠉⠙⠛⠿⣿⣿⢻⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⢈⢿⣿⢿⣿⣿⣿⣿⣿⣧⣀⠀⠢⢤⣌⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣽⠘⣟⣿⣿⣿⣯⠛⠿⣿⣶⣾⣟⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠀⢀⡴⠁⢠⣿⠀⣿⣾⣿⣿⣿⣿⣶⣶⣏⣽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣴⠟⠀⠀⣾⣿⠀⣿⣿⡽⣿⠛⢻⣿⠟⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⣸⣿⡟⠀⣿⣿⣧⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣰⣿⣿⡇⢀⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣴⣿⣿⣿⣽⣾⣿⣿⣿⣿⡽⢿⡉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠈⠙⢻⣛⡛⣛⣛⣿⣿⣿⣇⠀⠉⣄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾⡟⣿⣿⣿⣿⣿⣟⢿⣆⠀⠸⣦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡙⠇⠀⠙⢿⢻⣿⣿⠘⢻⡃⣠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⢻⣇⠙⠚⠶⠘⠟⠛⣿⣿⣿⣿⣿⡆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠚⠁⠀⣰⣿⣿⣶⠶⠆⠲⣴⠶⢿⡿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢤⡾⠋⢹⣿⣾⣷⣷⣶⢾⣶⡍⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢇⡠⣄⠈⠀⢴⣿⣿⣿⣿⣿⡟⢠⣤⣦⡐⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣄⣤⠄⣰⣿⣿⠋⠀⢸⡁⠀⢻⣿⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠈⠉⠀⠀⠀⢺⣧⠀⢸⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⣙⢿⣷⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣯⣿⣿⣿⡟⢡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⢿⣿⣿⣿⣿⣿⠛⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢭⣭⡀⠀⠾⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣄⣀
⢀⣾⣷⣀⢠⣶⠖⡀⠀⠀⠀⠀⠀⣬⣣⣒⣻⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢀⠰⠞⠁⢀⠀⠀⠀⠀⠀⠀⣿⣭⣧⠼⠿⢟⠃
⢹⢿⡿⣿⣵⣥⡩⠒⠀⠀⠀⠀⢓⣑⣢⣶⠗⠆⠀⠀⠀⠀⠶⠀⣀⠈⡲⢤⡆⠒⡀⠀⡤⠆⠀⠀⢠⣶⣿⡿⣿⣶⣷⣿⣿⣄
⠠⠉⢲⣿⣿⢬⡟⠛⠃⠀⠀⠀⣷⣾⣶⣶⣾⠆⠀⠀⠀⠀⠀⠀⠖⣠⣄⣁⣴⠖⠒⢤⣀⠀⠀⠀⠸⠟⢠⣾⣿⣿⡿⠉⠛⠁
⠀⢂⡾⠜⣿⣷⡽⠲⠢⠀⠀⠀⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿
⠀⠸⠁⢈⡿⠹⡗⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟
⠀⠀⢀⣶⠅⢺⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⣀⣤⣴⣴⣦⣤
⠀⠀⠚⣯⠀⣿⠇⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⣾⠟⠘⠁⠀⠀⠀⠀⠀⠀⠀⣦⡀⣦⢀⣴⠟⠀⠀⠀⠀⠀⣠⢿⣿⣿⣷⣽⣖
⠀⠀⠀⢀⣸⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣇⣿⣿⣿⣥⣾⠃⠀⠀⠀⠀⠞⠋⣿⢻⡉⠙
⠀⠀⠀⠀⠱⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣼⣿⣿⡧⢾⣒⠀⠀⠀⠀⠀⠀⠀⣿⡄⠁
⠀⠀⠀⠀⠀⠃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⣽⣿⣿⣱⣆⠈⠁⠀⠀⠀⠀⠠⠶⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢮⣧⣼⢶⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡷⣼⣿⣦⣽⣿⡾⢿⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⣿⣦⣄⣙⣿⣿⢿⣵⣦⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠻⢿⣯⡉⠉⠻⣿⣏⠂⠌⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⣦⡹⣿⣧⣄⠘⣯⡶⠤⣽⣿⣄⡀⠻⣿⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠰⣹⡟⠻⢿⡛⠛⢿⣿⠲⠿⠿⢾⣿⣿⡟⠛⡡
⠀⠀⠀⠀⠀⠀⠢⢄⡠⡀⠀⢻⣏⣿⡻⢦⣬⣻⣶⢠⣿⡷⣦⣤⣀⣨⣿⣷⠒⣀
⠀⠀⠀⠀⠀⠀⠀⠲⢧⡈⣳⣄⣻⡘⢳⣤⣹⣿⡉⠘⢧⣄⣈⣿⣿⣋⣀⣩⣿⡛
⠀⠀⠀⠀⠀⠀⠤⡐⢄⠉⢪⡿⣗⠑⣮⣍⢉⠻⠛⠂⢼⣿⣅⣈⣝⣿⣿⣥⣴⠤⠒
⠀⠀⠀⠀⢄⡢⣄⡈⠢⣷⡀⠙⢎⠓⠹⣦⣜⣿⠛⠳⢂⡋⠩⠯⣙⣋⣧⣼⣛⣛⣉⣥⠤⠤
⠀⠀⠀⠀⠀⠉⠳⢽⡲⣌⠻⣄⠀⠻⡇⢻⣿⠻⢈⣷⣿⣿⣶⣚⣋⣭⣥⢭⣽⣧⠤⠄⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠄⠀⠉⠻⢷⣮⡃⠀⠙⢮⠻⢧⠾⠟⠛⢳⣾⣟⠛⠛⠛⠓⠲⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢽⣿⣦⡀⠤⣣⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⢉⡶⣥⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢦⣑⠨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⢕⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


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

⠉⠀⠀⠀⣀⣆⣿⣿⠃⡠⢀⣤⣾⡩⠴⢴⠊⠈⡱⠤⣈⣱⣄⢀⡾⢑⠤⠤⢦⣤⠤⠞⠁⠀⠓⠄
⣖⣛⣛⣭⣥⣼⣿⣿⣿⣿⡿⠿⣶⣶⣴⣇⣀⠀⠀⠀⣣⣠⣤⠟⠓⠶⣒⠒⠛⢄⠀⢀⡠⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣟⠉⠉⠉⠉⢩⡟⢉⣉⡹⡈⠉⢿⡿⢷⣾⡛⠹⣄⣠⠖⠛⠒⠛⠶⠖⠛⠛⠗⠒⠒⠛⠊⠓⠄
⠋⠙⢦⣏⠻⢷⣾⣶⣄⣾⣠⣴⠶⠚⣟⠛⠲⣤⠶⠒⠛⢿⣙⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢀⠠⠀⠀⠀⠂⠈
⠀⠔⠡⠈⠉⢉⣽⡿⠟⡟⠛⠻⣷⢶⣿⡀⠀⢀⣀⣄⣀⡠⢹⠓⠶⢦⡤⠦⢤⣴⣮⡥⠤⠬⠄⠄⠐⠒
⣤⣤⣴⠶⠟⠋⠁⢇⣸⠃⠀⡰⢩⠛⠚⣟⠛⢿⣅⣀⣤⣤⣤⣥⣀⡀⢹⠄⠀⠀⢀⠌⠑⢄⣀
⣿⣿⣧⠀⠜⠉⠀⠈⣟⠀⠀⠹⡌⠢⡀⠀⠱⡀⠙⢧⡄⠀⢳⠄⠙⠛⢾⠦⣼⡁⠀⠀⠀⢀⡀⢁⡠
⠈⠛⢿⣿⣤⣤⣀⡞⠸⠗⠦⠮⣧⡀⠀⠀⠀⠀⣀⣤⠇⠀⠯⣄⣀⠀⢸⡤⢄⡉⡑⠉⠀⠈⠁
⠀⡎⠀⢹⣿⣗⠪⠀⠀⠀⠀⠀⠈⡟⠐⠀⢠⠜⠱⣏⡀⠀⠀⠀⠈⠳⣄⡇
⠆⠀⢠⡞⠛⠻⠷⣤⣑⢀⡤⠴⢤⢤⡤⣶⡃⠀⠀⠀⠹⠉⠁⠀⠀⣠⠟⢱⠤⠤⠒
⣇⣀⠀⢳⠀⠀⠀⢈⡟⠛⠓⠶⠶⠶⣤⣀⡉⠀⠀⢀⠠⠠⠄⠀⢀⠓⢄⠈⠂⢀⠀⠀⠠⠄⠀⠀⠀⡀⡀
⠀⢹⡀⠺⠤⡀⠀⠈⣇⣀⠀⠀⠀⠀⠎⠻⣯⠝⠒⠦⠤⠔⣋⠉⠙⠀⠀⠀⠀⠈
⠀⠟⠀⠀⠀⠀⠂⠀⠀⠙⣦⡀⠠⠀⠀⠀⠘⢷⣤⣀⡘
⠀⠘⠀⠀⠀⠀⣠⠤⡤⠞⡃⠉⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣏⣽⣿⣿⣿⣿⡿⣿⣶⣽⣿⢿⣿⣷⣿⣛⣿⡿⢯⡟⣻⣿⣛⣿⣿⣧⣿⣿⣿⣻⣿⣿⣯⢿⣿⣏⣿⣿⣓
⢻⣽⡿⠟⠿⣾⣷⣿⠛⢟⣏⡸⣿⣽⢓⡿⣽⠭⡫⠯⠵⡿⢿⠽⣞⠻⡿⠿⢬⣟⢯⣟⣣⣽⡾⠟⠻⡼⣧
⠈⡧⠑⠂⠩⡹⡆⢋⡒⢘⠃⣀⡏⠘⠳⢉⠛⠈⡤⢓⢷⢅⠈⠉⠰⠙⠋⣧⣄⢇⠐⣊⠃⢾⣍⠀⠀⠬⠏
⠀⠁⢀⡀⠠⠄⠆⠀⠁⠀⠒⠀⠀⠀⠄⠚⠁⠐⠁⠈⠀⠙⡄⠉⠂⠜⣀⠀⠄⠒⠂⠁⠘⠆⣤⠀⣁⡀
⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠐⠀⠈⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⡠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⢠⡀⠀⠀⠀⠀⢠
⠀⡀⢸⡇⢠⠀⠀⠀⣽⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⣿⢰⣗⣾⡀⢸⡇⣿⡏⠀⣄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢿⣼⣿⣿⡆⢸⣷⢿⡷⠀⣿⣹⡅⢠⡄⠀⣄⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⢀⡄
⠀⠈⠙⠛⠿⢧⣸⣿⢿⡷⢸⣿⣻⡇⣸⡇⢠⣿⠸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠘⠰⠧⡿⠘⠃⠃⣤
⠀⠀⠀⠀⠀⠛⠆⠈⠙⠛⠺⠿⠿⠗⣿⣧⢰⣿⢸⣯⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⣯⣗⢰⡆
⠀⠀⠀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢴⠋⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢧⠢⠀⠀⠀⠶⡀⠀⢑⠀⠀⠀⡆
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠆⠆⠐⠚⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢦⡄⠀⠀⡘⣅⠀⠘⡂⠀⠀⠐
⠀⠈⠀⡀⠀⠀⣂⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠭⠠⢘⠀⠀⡗⢻⡂⢢⣤⡘⠇
⠀⠢⡸⡁⠰⣊⠁⣄⠵⠄⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠠⡀⠤⡀⠀⠀⠛⠏⠶⣠⢝⣌⢡⣶⡶⠾⠠⡼
⠀⣶⠌⡴⠞⠇⠒⠀⣄⠔⢃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢸⠃⠧⣉⣭⢀⠀⢳⣻⣼⣨⡕⢦⠙⣀
⢰⠛⠀⢾⠿⡓⢿⠷⠟⡘⡷⡲⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠠⡀⣠⣖⠑⢠⣠⣐⣗⣰⢾⣦⣾⣟⢾⣻⣃⣲⡽⠶⠄⠏⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⣀⣀⣤⣤⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡻⣭⠩⠁⠁⠄⠤⠆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⡿⣱⡟⣌⢧⡛⣀⣀⢀⣀⠁⠤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢻⣛⠛⠁⠛⠃⠻⠎⠻⣦⣀⣠⣴⣤⣤⣤⡖⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⠏⢻⣿⣥⢿⣄⢦⡸⣟⣼⢿⣏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⢹⡋⠈⠉⠂⡅⣿⠃⠀⠻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⠀⠀⠀⠘⡧⠶⣶⣦⠀⡇⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⣀⡀⢿⣠⢤⠀⢠⡏⠀⣽⡟⠀⣷⣀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢉⡕⠛⠶⣾⣿⣿⣿⠐⣻⣿⣇⣿⠿⠿⢛⣻⣧⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣦⠚⣻⣤⣶⡖⣫⣿⣷⣶⣐⣶⣾⣭⣿⣿⣿⣿⣏⡉⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢩⣿⣧⠀⠄⣧⠀⠀⠈⠉⣙⣇⢉⣿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣉⠀⠉⠻⠂⠀⠛⠀⠰⡶⠶⠛⢟⣻⣿⣿⣿⡟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣤⡄⠀⢀⡄⠀⢠⡆⠀⠀⡆⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⣿⣆⠀⠀⠀⣏⠀⠀⠀⠀⠀⣾⠇⠀⢸⣿⠀⠀⡇⠀⠀⠀⠠⠀⡊⢑⠄⠄⠀⠀⠹⣦⣄⣰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⠻⣿⠟⠂⠀⠀⠛⠀⠀⠀⠀⠀⠈⠃⠀⠈⠋⠀⠀⠣⠀⠀⠀⢐⠂⠈⡁⠐⡇⠀⠀⠀⠰⢿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⣐⠀⠀⢸⡇⠀⠀⠀⠀⠘⠙⢷
⠀⠀⠀⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⠀⡆⠂⢀⠂⢐⠂⠀⣰⠀⡄⡆⠂⠀⠀⠀⡇⢉⣿⣷⠈⢧⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠁⠀⡆⠀⠘⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠹⡇⠀⠀⠠⣿⣿⣿⣿⣿⣿⣦
⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠻⠏⠉⠟⠇
⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠸⠀⠀⠀⣠⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣤⣀⠀⠀⠀⢸⠁
⠀⢰⠋⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⣾⣟⡝⢾⡗⡀⠀⠀⠂⠀⣄⠀⠀⠀⠀⣀⠠⠂⠀⠀⠐⢿⡿⣜⣿⣦⠀⠀⠐⠄⠀⣰⣾⣿⣿⣿⣿⣆
⠀⢠⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣽⣿⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⣼⣿⡅⠀⠀⢈⣤⠀⠀⢨⠟⡏⢙⡟⣯
⠀⠘⢄⡀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠈⠛⣿⡀⠀⠀⠀⠂⠀⠀⠀⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⣸⡿⠋⠀⠀⠀⢸⠁
⠀⠀⠀⠈⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠒⡀⣠⠒⠀⠀⠀⣆⠀⠀⠀⠀⣠⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣦⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣽⡄⠀⠀⢀⣿⡄⠀⣿⠀⠀⠀⣤⡤⠈
⠀⠠⠴⠶⠶⠶⠿⠶⠤⠤⠀⠀⠀⠻⠃⠸⠿⠀⠀⠿⠁⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠦⠻⠀⠀⠘⠻⠀⠀⠁⠀⠀⠀⠛⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⣀⣰⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⣤⣬⡵⡭⣿⣆⣱⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠒⠶⣿⣻⡿⣏⠶⣾⠿⣚⣍⣹⠿⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡓⢢⣤⣀⣑⡢⢬⡓⣿⣽⣿⣱⣝⣛⣹⣷⣶⣿⣿⣦⡤⠶⣿⠿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣘⣶⣞⣿⡿⢛⣬⣬⡷⣿⢿⡿⡟⣯⠟⡟⢡⢏⡼⠁⠀⠁⢀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣙⣿⣿⣿⣿⠛⠉⠹⡴⢋⠄⣹⠀⢡⡼⢻⣁⢼⠏⠉⠑⠞⢒⣒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣶⡶⣿⠿⣩⣿⣿⣙⢣⡲⠃⢛⡰⠁⠀⠋⠃⠈⠁⡎⠀⠀⢐⣊⣭⣼⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢒⡶⢼⣿⣿⠟⣏⣧⣾⣿⣿⣿⠛⢻⣇⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⣐⣂⣐⣛⣵⡟⠰⣘
⠀⠀⠀⠀⠀⠀⡀⣀⣀⣴⢶⣿⣿⣟⡏⡴⠾⢻⣿⣷⣿⣟⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢼⣋⡿⣝⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⣈⣤⠔⣾⣾⣿⢆⣷⠢⣚⣿⣿⡛⣿⠗⠙⡰⠁⠀⠀⠀⠀⠀⠀⠀⠠⠬⠑⠲⢿⣋⢻⡿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⣪⣾⠋⠾⠨⣯⠉⣼⣿⣿⣗⣼⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⡲⠟⢣⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠂⣩⠖⣾⣿⠇⡆⠰⡀⠀⣾⢻⣿⣟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠅⠀⢀⣾⠀⣺⠙
⠀⠀⠀⠀⠀⠀⣠⣞⣿⡙⡀⢳⠀⠀⠀⢱⢟⡿⢹⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣷⣼⣧⠁
⠀⠀⠀⠀⠀⣜⣡⣿⣗⢧⠃⠘⠀⠀⠀⠋⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⣿⣿⣽⡜⡆
⠀⠀⠀⠐⡪⣼⢿⣿⣧⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣿⢸⡿⠀⡝⡘
⠀⠀⠀⠈⠊⠈⠸⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢳⢵⠀⠀⠁
⠀⠀⠀⠠⠀⠰⢀⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⣾⡟⠁⢻⡈
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣽⢻⣷⡜⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠁⠘⠉⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣈⡛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠿⠻⠟⠛⠿⠿⣿⣿⣶⣤⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠟⠉⠉⠉⢰⡦⠀⠀⠀⠀⠀⣉⠻⢿⣷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠋⠀⠀⠀⣀⠠⠀⠀⠄⠄⠀⢀⢀⣴⡏⠂⠀⠙⢿⣷⡌⢻⣦
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⡀⠂⠀⢹⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠀⠀⠀⠀⠹⡆⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠂⠙⣦⣄⠘⣇⠀⠀⠀⢴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⡄⠀⢹⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣟⠀⠀⠰⠧⣀⠀⠈⢿⠟⡤⣸⠃⠀⣰⣿⣿⣟⣁⣠⣤⠤⠀⠀⠀⠐⢁⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣼⣿⡿⠇⠀⠀⠀⠀⠈⢙⠶⠘⠿⠿⢾⣤⣴⣿⡿⣯⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⡆⣿⣿⣇⠀⠿⠏⠇⠀⠀⠀⠀⣀⣀⠀⢸⣇⣀⣼⣟⠀⠀⡀⠀⠀⠀⠂⠸⠅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⡇⢻⣿⡏⠀⠀⠀⢁⠀⢀⣀⣤⣌⡭⡝⣿⣿⡟⣿⠫⠿⡂⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣷⠸⣿⣧⠀⠀⠀⠀⠍⠉⠀⢠⣾⠷⣿⣿⣿⠀⠗⣽⡮⣿⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⢻⣇⢻⣿⣇⠀⠀⠀⠈⠂⡴⠋⠁⠉⠀⣿⡟⠀⠀⠘⡇⠘⠛⠖⠒⠀⠀⢠⣶⣶⣶⠀⣼⡿
⠀⠀⠀⠀⠀⢻⣆⠻⣿⣄⠀⠀⠀⠀⠀⠐⠀⠀⣠⡺⠃⠀⠀⠀⠻⠬⠈⠀⠀⠀⠀⠡⣬⢭⠅⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣧⡙⢿⣷⣂⠀⠀⠀⠀⠀⡌⠉⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡙⢿⣷⣤⡀⠆⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⢀⡀⠀⠁⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣬⡛⠿⣿⣶⣶⣤⣤⣬⣠⣤⣤⣤⣶⡿⠿⢋⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣬⣭⣍⣛⣛⣛⣋⣩⣭⣴⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣆⣢⣤⠈⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣄⠉⢟⣯⢿⣿⣿⣿⣿⣴⣄⠰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠯⣥⣦⣙⣿⢿⣷⣿⣿⣋⣿⣿⣿⣿⣦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢺⣟⣻⠛⡿⣷⣿⣿⣿⣽⢧⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣍⠍⣴⣯⣹⣦⣏⣉⣿⣼⠿⢿⣖⠛⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄⠀⢻⣮⢻⣿⣽⣯⣍⡷⢠⠉⣿⢟⣱⠛⢏⣴⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠀⢠⣦⡁⢠⣷⡬⡌⠁⠛⡾⡧⡇⡯⠉⠓⠈⠿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠀⠉⠁⢚⠎⠱⠀⠂⣀⣇⣀⣟⠻⠲⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡷⠇⠐⠁⣀⡀⠀⠗⣏⠛⢿⣛⠙⠃⢐⠧⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠸⣿⢠⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢿⣸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠤⠄⠄⠀⠀⡀⠀⠀⠀⠴⠖⠿⠙⠻⣦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠩⠄⠀⠀⠀⠀⠺⠋⠀⠀⠈⠀⠀⠀⠀⠽
⢠⣀⣀⠀⠀⢀⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⠤⡀⠀⠛⠃
⢨⡙⠉⠫⠽⣾⢋⣿⣯⡀⠀⠀⠀⠰⠛⠵⠂⠐⠿⢇⣠⢴⣶⠀⠀⠀⠐⢄⡀⢀⡀⠀⠀⠀⠀⣘⣿⠀⠛⠀⠀⠈⠀⠀⠀⠂
⠈⠁⠓⠂⠀⡿⠟⠋⠁⠁⠀⠀⣀⢠⠀⠀⡆⡰⠀⠀⠁⠀⠀⠀⢠⠆⠀⠀⠋⠉⠉⣠⣄⠀⠀⠀⠁⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠓⠀⠈⢹⡦⠄⡏⢣⠀⡆⠰⡄⠐⡄⠈⠤⡀⠴⠀⠀⠾⠿⠃⠸⠃⠀⠒⠚⠋⠀⢀⣿⣧⠀⣼⣧⡀
⠀⢰⣍⣿⣿⡾⢂⡄⠀⣀⣀⣀⣀⡄⠀⡀⠀⠀⠀⢠⠆⠀⠀⡀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠄⠀⠀⢸⣿⣿⢸⣿⣿⡇
⠀⠀⠙⠉⠡⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠢⢶⠘⠄⠀⠃⠙⠀⠳⠀⠃⠈⠂⠓⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠈⠀⢸⣿⣿⣧
⠀⠠⠠⠄⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠺⢿⡿⠻
⠀⠀⠀⠀⠀⠀⠈⠃⠀⡏⠀⠀⠀⢀⠴⠤⡀⠀⠀⠀⠀⢀⣠⠀⢀⣄⠀⠀⠀⠦⣄⠀⡀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡰⠀⠁⠀⠀⠀⠿⠏⣸⠳⠀⢠⠁⡀⠻⠃⢀⣄⣄
⢠⠀⠀⣀⠀⠀⠀⣼⠇⢀⣷⣿⣿⠀⠀⣢⣀⡀⣤⠀⠠⠀⠖⠀⠀⠦⠀⠄⠀⠀⠀⠀⠀⠀⠰⣤⣾⣦⣤⢀⢸⣿⣧⣿⡄
⠀⢀⠔⠁⠀⢰⡐⠋⠀⠀⣱⢛⡄⠘⡾⠉⠛⠛⠁⠀⠀⢢⠀⠀⠀⢀⠞⠀⠀⠐⠂⠛⠃⠀⠀⣼⣿⠻⠟⣾⠏⣿⣿⠀⠙⢷⡄
⠀⠸⡄⠀⠀⣸⠃⠀⢠⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠰⣰⡟⠀⠾⠟⠀⠀⠀⠃
⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠁⠀⠀⠀⠀⠈⠀⢠⡤⠄⠀⠀⠀⠂⠀⠀⠀⠉⠠⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⡀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⢹⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⢒⣵⣤⠀⠈⠘⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣶⣶⣴⣶⣏⡻⣉⣈⡿⠒⠂⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠛⢿⡙⠛⠻⣿⣧⡈⠻⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠉⠉⠉⠳⣦⡝⠆⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢳⣷⣔⣆⡀⠀⣿⠀⠀⠀⠀⠀⠸⠿⣦⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠶⠿⠘⠿⠦⣤⣀⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠓⣐⠞⣻⣷⠘⡁⠀⠀⠀⠀⠀⢀⢀⡀⠀⠘⣿⡀⠀⡀⣤⣌⠀⠀⠀⠀⠀⠀⣈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠍⣠⣾⣿⣾⣿⣿⡿⠃⠀⠀⠀⠀⠞⣋⣿⡝⣠⣾⣿⣡⣞⣸⣭⣉⣰⣴⡗⠀⣰⣏⣹⣸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠞⠟⠛⠛⠋⠀⠈⠐⣦⠀⠀⠀⠤⠙⠙⠻⣤⣿⣿⣄⣤⣼⠿⠿⠿⠿⢯⡽⠿⠚⢿⣯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⢰⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢀⣤⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⠶⢶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣏⣩⣽⣿⣷⡄⣤⣤⡶⢶⣀⠀⠀⢀⣠⣤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡄⢠⡀⣀⣤⣾⣾⣿⣿⡿⡷⠒⠻⠶⢄⡤⠆⢘⣢⣠⣤⣬⣍⣙⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠉⠻⠟⠉⣉⣀⠀⣠⣄⣠⣤⣴⣆⣰⣈⠀⣀⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠏⠀⠁⠉⠛⠟⠉⠁⠤⠤⣭⣥⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠇⠀⠀⠘⠀⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⣤⣤⣤⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢗⡐⣂⢁⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣿⣿⣿⣿⡿⣿⣏⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⡴⡰⣶⣄⣄⡀⠀⠀⠀⠀⠀⢺⣾⣾⣿⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⢀⢶⡽⡿⡗⠓⠺⢏⢿⣿⢇⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾
⠀⠀⠀⠀⠀⢀⣸⣿⠻⠀⠇⠀⢸⠀⠩⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠲⡀
⣿⡄⠀⠀⠀⢸⠭⠭⡆⠤⢈⠀⢸⠀⠀⡒⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢄⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣀⣀⢠
⣼⡆⠀⠀⠀⢩⡭⠭⡇⠀⠸⣶⣾⠀⠀⠳⢶⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣡⣀⡀⠀⢀
⣿⣷⠀⠀⠀⠐⣻⢭⣧⠀⠀⠈⠁⠀⠀⡸⣬⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⢶⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⡄⢾
⢿⡇⠀⠀⠀⠀⠠⡿⡷⣵⣶⣶⡖⣦⣥⣿⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠟⠀⠀⠈
⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢀⣀⣀⣀⠀⣠⣤⣀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢿⣿⣿⡟⠈⢿⣿⣿⠃
⠀⠀⢀⣀⡀⡀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢒⣦⡀⢼⠾⠗⣰⡨⠃
⠀⠀⠘⡉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⠉⠁⠈⢻⣱⣦⣄⡀
⠀⠀⠀⠀⠀⠄⣠⣄⠀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠾⡏⣯⣿⠂
⠀⠀⠀⠀⠀⢰⠉⠇⡃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣻⣇⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠤⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣒⣒⠂
⠀⠀⠀⢀⠀⢠⠀⠄⡀⠀⠠⠘⡇⠀⠀⠀⠀⠀⢀⡀⡀⠴⠒⠲⠲⠒⠤⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠙⠋
⠀⠀⠀⠀⢂⢀⠃⠈⢁⡠⠁⠀⠀⠀⠀⠀⠀⠀⣏⠾⠃⠀⠀⠉⡍⠀⠀⠈⣷⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠀⣴⣶⡄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠿⠿⠟⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⣾⣿⡇
⠀⠀⠀⢀⣤⡀⢀⣀⣀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠻⠿⠿⠟⣿⣿⣷⣶⡦
⠀⠀⠀⠀⠁⠈⣿⣿⣽⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⠀⢸⠀⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢻⠟⠐⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⠀⠸⠐⠀⢻⣿⣿⡿⠃
⠀⠀⠀⢀⡛⠃⣶⣶⡾⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠑⠂⠀⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠸⣾⠂⢸⣿⣿⣿⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢢⣳⠀⠀⠀⡼⠁⠀⠀⢀⣠⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⡝⠀⢠⣞⣠⣤⣤⣼⠕⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⣀⡴⠛⠉⠀⠀⠀⠀⠉⠓⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠟⠉⢀⠤⠴⠒⠒⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣕⣡⠤⠶⣗⡒⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⠡⠐⠊⠉⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣩⠖⣋⡁⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⡡⠖⠉⠉⠉⠉⠩⣉⠃⠒⠀⠑⠀⠀⠀⣰⠟⠚⠉⠈⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣁⣐⣉⣀⣀⣀⠀⠄⠠⣤⠄⠉⢉⣁⣀⣤⡤⠎⢥⣭⣘⣲⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⣨⠤⠖⠒⠚⠛⠛⢾⡛⠉⠁⠀⠈⠛⠻⢥⡀⠀⢄⡀⠀⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⡠⠞⠉⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦⠤⣄⣀⡀⠀⠀⢌⠓⠦⣈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⢀⡴⠋⠀⠀⠆⢀⢀⠀⣰⠞⣀⡀⠀⠀⠈⢷⠠⡀⠀⠀⠀⠀⠑⠤⣈⠑
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⢀⠔⠉⠀⠀⠀⠀⢀⣾⡴⠾⠟⠳⠟⠉⠙⠢⠀⠀⢱⠘⡄
⠐⠒⠒⠒⠋⠉⠉⠀⠀⠀⠔⠁⠀⢀⣀⣀⡤⠖⠋⠁⠀⠀⠀⠉⠑⠒⠂
⠀⠀⠀⢀⣀⡤⠤⠴⠆⠴⠶⠒⠶⠦⠭⠥⢤⣀⣠⣤⣄⣀⣀⡀⠀⣀⣀⣀⡀⣠⠖⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢀⠈⠀⠀⠀⠈⠉⠂⠀⠄⣒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢄⡀⠀⠀⠀⠠⡀⠈⠑⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠙⡆⠀⠀⠀⠉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠘⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⢀⠀⠀⠀⢲⣤⡄⠀⠀⢀⠀⠀⡸⣆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣿⣶⠘⣧⣿⡟⠀⠀⠈⠿⣿⣄⢐⣾⡷⢳⣽⡍⠀⠀⠀⠀⠀⢻⣖⠀⠀⣤⣶⠧⠀⠀⠀⢀⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⡤⣤⣠⡿⣌⢡⡄⢶⣷⣤⢨⣉⡱⣎⣠⡔⠊⠀⠀⠀⠀⠻⣤⣀⠘⣿⡦⠴⠛⣧⣴⠀⠀⠙⣾⣇⠈⣯⣵⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣴⡓⢿⡀⠘⡯⣞⠛⢿⠃⠠⣍⣡⣔⠛⣵⠋⠸⣿⡍⠀⠀⣰⡦⡿⠶⣏⢠⡟⢀⠀⠀⠸⠻⠄⢠⣄⡄⢉⣌⣰⡞⠟⠃⢰⣠⣆⡤
⠀⠀⠀⠀⠀⠀⡼⠟⠋⠀⢉⣭⡿⡿⢟⢧⣈⣯⣟⣯⠃⠀⠙⢻⡄⢀⠙⠠⣄⣤⠀⣀⣼⣄⡎⣻⠹⡿⠶⠂⠀⢀⡰⠀⠽⠻⡟⢣⠶⣀⢀⣤⣊⠃⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢠⣄⣰⠀⠀⠀⠀⠀⡿⡿⠙⣷⢼⡟⠀⠀⠀⠀⢻⣜⣿⣤⡽⢿⠆⡁⢀⠈⠁⢿⡘⣶⡂⣀⡴⣮⡛⢃⣀⣺⣡⡴⠞⣉⠁⠀⠨⠗⠢⣴⣧⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣽⠿⡃⢦⣾⣧⠄⠀⢀⠙⣲⣆⢺⣴⣧⢀⡄⠹⣿⢿⣟⢪⡴⣷⠿⣄⠀⡀⢹⡝⠁⣼⣟⣋⡉⢚⡿⠋⠀⢙⡘⠿⢾⡀⠀⢸⣷⠃⠉⠟⠷
⠀⠀⠀⠀⠀⠀⡼⠟⠋⢉⣿⢶⣿⡁⠀⠒⠻⠟⡿⠛⢿⡁⠀⢠⣿⣶⣿⠉⣽⢦⡶⣤⣤⠀⠀⠽⣾⢀⣔⠉⣀⠿⢳⢎⣴⠃⠀⢿⡿⠁⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠻⠃⠈⠙⣦⡷⣏⠀⢠⠄⠈⣷⠀⠀⠀⠀⠹⣦⡆⠈⠞⢳⢾⡇⠀⢀⡟⠋⠈⠽⣋⣠⢸⠋⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⣾⢷⣄⣀⣬⣷⡀⣹⣿⣶⠀⢿⡄⠀⠀⡽⡏⢻⡇⢀⠴⢶⡍⢉⡝⢺⡇⠀⢠⣴⣍⣇⢸⠀⠀⠀⠀⢀⢠⣤⠄
⠀⣀⣶⣼⣦⢤⣤⣄⡤⢿⣿⡿⠟⠋⠀⠈⠻⢶⡶⠿⠶⢾⣷⣦⣤⣅⣄⣼⡇⣸⠻⠎⢀⣻⡛⠀⣷⢢⣴⠚⠟⠙⣿⠀⠀⣳⢆⢸⠟⠁⢀⣦⡞⠀⢠⡀⣴⣀⡀
⠘⠉⠋⣸⠗⠋⢀⣹⠳⠈⠸⠆⠀⠀⠀⠀⢻⡻⠀⢀⢀⣤⣤⡋⠉⠉⠉⠛⢿⣷⡄⢤⡬⠉⣥⣠⣟⠁⠉⣉⣢⠀⡏⠰⣾⣈⣋⡇⠀⣀⣼⢷⠶⢖⣉⡛⠉⠈⠁
⢀⣀⡞⣋⣓⠀⠈⠏⠃⠰⣆⣀⡀⢲⡤⣶⣫⡝⢳⣿⠀⡴⣿⠏⠀⠀⣄⣲⠺⣉⣿⣎⣹⠏⣙⠈⣧⢼⡔⣊⢉⡾⢡⣸⡏⣙⣫⣷⡋⠀⠛⠻⠦⠈⢿⡦⠄
⣘⣽⠻⠁⣯⠀⠀⠀⢀⣏⣹⣻⣥⣿⠾⠧⣿⣛⣹⣏⣀⡀⠘⠂⠀⠀⠚⣫⡾⢹⠇⢻⣇⠰⣿⢷⡇⣰⣽⡛⢻⣇⣹⡿⠛⣉⣡⣸⣷⢀⣠⣦⣶⡀⠀⠛
⠉⡎⠃⠀⠀⠀⠀⠀⢶⣾⣣⡝⠁⣞⡆⠀⠀⣉⢹⡭⠛⠛⠻⠶⢦⣴⣿⣤⠈⣏⣻⣾⢻⣧⢀⣾⣥⡿⢟⡏⢹⡟⠻⢞⣰⡜⢻⣽⠖⢺⣿⢩⢧⣇⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁⠾⡿⠀⠙⠇⠀⠈⡽⠟⠃⠀⠀⢠⣠⡄⠈⢛⣿⣶⠋⠓⠀⠀⢻⣿⠋⣷⡤⠘⣿⣿⠘⢶⣸⣻⣶⠻⡄⢐⣤⠖⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣝⢫⣽⣍⠛⣦⡀⠀⠀⢰⣿⣆⠘⠿⠀⢠⡿⠤⠚⣹⡦⠈⢿⡉⠉⢩⢾⠃⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣯⠐⢻⡏⠀⠈⠻⣦⣀⡿⠀⣿⣄⣀⣀⣿⣁⣰⡧⠄⢹⢢⣼⣧⣗⢀⣼⣤⠤⠴⣚⣟⠻⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⣉⣿⣿⣽⣫⡴⠶⠂⡉⢸⣯⢀⡟⠀⠀⡀⠀⠋⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⣰⠄⠻⣷⣄⣼⣿⣿⡇⢡⡈⢯⣥⡄⠙⠛⠛⢻⣾⠖⣰⡮⠷⠍⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣼⣿⣱⣦⣿⣿⣿⣿⣿⡇⠺⣟⡄⡿⠀⠲⣿⠋⣸⣿⣉⠀⣻⣆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣤⣸⣿⡎⢸⣿⣿⣿⡇⠈⠉⢙⠇⠀⢸⣠⡾⣅⠀⠞⣗⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣹⣾⣿⣿⣿⣿⣶⣶⣾⠶⣤⣾⠻⣟⠟⠃⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡝⠀⢹⣿⣿⣿⣿⡏⢠⢷⡂⠸⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⡇⠀⢫⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢄⣰⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⢹⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⡉⣠⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡟⠉⠀⢻⣿⣿⣿⣧⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠈⠓⢾⣿⣿⣿⣿⣿⣿⡻⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⠈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠳⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠃⠒⠒⠢⢤⣾⣿⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠈⠓⢦⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⢦⣄⣀⠀⠉⠳⣕⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣦⣼⣩⣽⣶⣦⣄⡙⠻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠖⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⢀⣠⡼⠿⢿⣿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⣿⣿⣷⣦⣀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣤⣶⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢃⣼⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀⠀⠀⢀⠤⠤⠤⠤⠄⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠾⣿⣡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣤⣤⣤⣽⣿⣿⣯⣤⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⣟⠛⠻⠿⢿⣿⣿⡿⣆⣀⡀⠀⢸⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣮⣍⠛⠶⠶⠾⠯⠭⢭⠭⠽⠿⠷⠤⠴⠞⢛⣋⣙⣛⣛⣉⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⣿⡟⣿⠉⢻⠻⣿⣿⡴⢿⢻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⠀⣿⡇⣿⠀⢤⠀⠀⠃⠀⢸⡜⢻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣝⠿⣿⣿⣿⣿⣿⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⢸⣟⡉⠨⢹⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣹⣷⣮⡻⣿⣿⣿⣤⣿⣷⣿⣶⣶⣶⣶⣾⣿⣿⣿⣯⣿⡿⠿⠿⠿⠿⠛⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣛⣛⣛⣉⣉⣈⣉⣤⣤⣤⣤⣤⡶⠾⠷⠶⠶⠖⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢂⣥⣽⣿⣿⣿⣿⡿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣠⣬⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠶⠿⠋⢉⡄⠀⣶⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠁⣄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢸⣿⡟⣿⣿⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⡇⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡲⢎⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠈⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣧⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⣶⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣿⠁⣠⣴⣿⣿⣿⣿⣿⣿⣾⣿⣤⣤⣶⣤⣄⣀⣠⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣠⣶⣿⣶⣶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣻⣿⣿⣿⣛⡁
⠀⠀⠀⠀⠘⠻⠉⠻⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠃⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⢯⣷⣶⣶⣶⡦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⠀⠈⠉⠛⣿⡯⢥⠈⠉⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣦⡀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠀⠀⠀⢀⣀⣤⣤⣤⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠛⢿⣿⣶⣴⣿⣭⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣸⡿⢻⣿⣿⣯⡉⣩⣽⣿⠿⢛⣉⣭⣿⣿⠟⠛⣛⣉⣉⣉⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣿⡇⣸⡏⠛⢿⣿⣿⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣛⠿⠀⠿⠃⢠⣾⣿⣿⣿⣧⣾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣶⣦⡔⢛⣿⣇⡰⢿⣿⣎⠋⣋⡛⠙⢿⣿⣦⣴⣶⣿⣿⣿⣿⣿⣿⣷⠶⠶⠈⠉⠙⠛⠻⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣥⣼⣟⠋⠾⢿⣿⣿⠿⣠⣵⣶⣾⣽⠿⡄⣼⣿⣿⣿⣯⣿⣿⣷⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣰⣿⠿⣟⣛⣟⣣⣶⣶⣶⣦⣼⡛⢛⣛⣿⣿⡷⠐⣋⡛⠿⣿⣿⣧⣤⣽⣿⣿⡿⠿⢿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠷⠶⠾⣿⣏⣹⣛⡛⠛⣻⣿⣿⣥⣀⣙⡻⢯⣵⣿⣿⣿⣷⡸⡏⢻⣷⣄⣉⠛⠻⣿⣶⣮⣍⡛⠻⠿⣿⣿⣿⡳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣱⣿⣷⣶⣽⣟⣫⣍⣼⣿⡹⣿⣿⣾⣿⣿⣿⣎⢻⣿⣯⡛⠿⠃⠿⣼⡟⢿⣿⣿⣿⣿⣛⠛⠿⣿⣶⣤⣄⡀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣻⣿⣿⣿⠟⣫⣞⣯⣝⡿⣿⢹⣿⡛⢻⣿⣿⣽⣿⣿⣭⡬⣁⣶⡜⣧⠀⣹⣷⣿⣛⡻⣷⣦⣄⠉⠻⢿⣿⣗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡝⠛⣩⣾⣥⣿⣿⣿⣿⣿⠿⠜⣿⣿⣦⣍⡛⠛⠛⠉⢙⠾⢿⣿⢃⣿⡄⣿⣿⣿⣿⣿⠿⣿⣿⢿⣶⣤⡈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⢀⣠⣾⡿⢫⣾⣿⣿⣭⣶⣶⣿⣷⣹⣿⣯⣭⣶⣾⣿⣿⡆⣿⣶⣦⢸⣿⣷⣿⣿⣿⣷⡙⢷⣌⠻⣷⡌⠙⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⡿⢼⣿⣿⣫⣶⣷⡈⢹⡛⠿⣿⣿⣿⡇⣼⡇⢿⠀⢙⣛⠛⣻⡛⣿⣿⣌⠻⣷⣌⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠛⠛⢋⣡⣾⠟⢿⣵⣿⣿⣿⣿⣦⡙⣷⣮⣙⡛⠁⣛⠀⠀⠀⢸⣻⡋⣻⣏⢙⣇⠙⣇⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣴⣻⡿⢟⠅⠀⢻⣿⣿⣿⠿⢿⠿⣓⣌⢻⣿⣷⣾⠃⠀⠀⠀⠀⠹⠃⠈⢻⡄⠉⠀⠘⣧⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⣽⣯⣾⣫⣾⣿⣎⣿⢧⡐⣿⡹⣷⡙⢿⣷⣮⣩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⣛⡟⣻⡋⣿⡗⣨⣧⠻⣿⣧⠙⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣟⣽⣟⣷⣹⣾⣿⣿⣷⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣟⣽⠞⣿⣿⣷⡈⠫⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡝⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⢀⠜⣽⠋⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠠⠒⠉⠉⠁⠒⠤⠀⢀⣀⣤⡶⠏⠀⠀⠀⠒⡲⠖⠐⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢀⢀⡴⠅⠀⠀⠠⢤⣤⣶⣶⢿⢿⠭⠭⠓⠒⣚⠡⠄⠊⠁⠀⠀⠁⠀⠄⠀⠀⠀⢠⣤⣦⡾
⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⢀⠔⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢝⣿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⡴⠒⢒⣲⡖⠉⢉⣈⠉⠉⠋⠀⠀⡄⠐⠀⠆⠀⠀⡍
⠀⠀⠀⠀⠀⡸⣋⠄⠀⠀⠁⠀⠀⠀⠀⠀⠁⢐⠀⠀⠠⢀⡂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠄⠀⠀⠀⠘⠀⠀⢀⠄⡉⢆⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠁⠔⠁⠠⠁⠀⢀⡀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢾⣦⡀⣀⡜⠊⠅⢰⠂⠰⠀⠀⠀⠀⠚⠋⠁⠀⡀⠠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣦⣿⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⢀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠐⢂⡀⢬⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⠀⢠⢃⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡁⢠⠊⠁⠀⠀⢈⠡⠴⢒⠂⠒⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡾⠉⡿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⢛⣶⣧⡀⠂⢀⠀⢀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠓⣄⢸⣶⢸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢉⡿⠿⡇⠄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠞⠀⠐⢠⠄⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣯⡉⠛⠛⠿⠿⠿⣶⣄⠉⠻⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣤⣤⣄⣄⣀⣀⠀⠀⠀⠰⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠉⠉⠭⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⢀⡠⠄⢬⣭⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡶⣤⣩⣒⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣁⣀⣀⣀⣀⣀⣄⣽⣯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⢟⠅⡇⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠚⠂⠃⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣄⡒⢶⣷⣾⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠖⣀⠀⠀⠀⠀⢐⡤⢴⣾⣴⡞⢛⣯⣽⠀⠀⠀⠀⠀⣄⣶⣋⣴⠇
⠀⠀⠀⠀⣐⣬⣥⣽⣛⣯⣴⣧⣸⠂⠀⠀⠀⠀⢶⣌⡶⠛⠧⣨⣥⡴⠀⠀⠀⣛⣱⡿⠛⢽⣝⡯⣽⣿⣤⠀⠀⠀⠠⣼⣿⣿⡿⠒⡆
⠀⠀⠀⢀⢁⣈⣤⣎⣩⣉⣹⣿⣿⣷⠀⠀⠀⠲⣤⣄⣩⣛⣒⠛⢃⣀⡀⠀⢀⡘⠚⣻⠾⢿⡿⣷⣿⣿⡿⠀⠀⠀⢀⢠⢸⠋⣴⣾⡿
⠀⠀⠀⢨⣾⣿⢭⠍⢍⣙⣩⣟⣛⣭⡃⠀⠠⢦⣤⣁⣴⡾⣋⣙⢛⣉⡀⠀⠀⢛⡿⠻⣿⣿⣟⣙⣿⣿⡷⠀⠀⠘⢃⣸⣿⣷⣼⣿⣷
⠀⠀⠀⢢⣄⣀⣲⡼⠟⢉⣭⣉⣿⣿⡇⠀⠀⠈⠹⠿⡿⡟⠛⠿⠛⠋⠀⠀⠀⠈⠙⠻⠝⢿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠉⠛⠟⠋
⠀⠀⠀⠀⠀⠙⠋⠘⢷⠚⠋⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠤

⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⠠⣤⣤⠄⠀⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡰⠂⠉⠁⠐⢄⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢦⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠐⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠂
⠀⠀⠀⠀⠀⠐⡀

⠀⠀⠀⢴⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⢀⠀⠀⠀⠀⠣⠒⠉⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣠⡴⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⡠⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠃⠀⠀⠀⠀⣿⣰⣠⢸⡆⡏⢠⠂⠀⠀⠀⠈⠄⠀⠀⠐⠂⠐⠂
⠀⠀⢔⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣾⡇⠘⠈⢛⠀⠉⠓⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢁⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⡇⡄⣠⣼⣡⢠⣰⢰⢀⡤⡀⠀⠀⢹⠃⠀⠀⠄⠙⠃⠀⠀⠃
⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠸⠜⠀⠉⠓⠙⠿⠘⠈⠀⠀⠁⠁⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁
⠀⠀⢠⣆⠀⠈⠑⠀⢀⣂⣀⡀⠀⠀⠀⠠⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣀⢐⣀⠀⠀⠁⠀⠀⡄
⠀⠀⣄⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡀⡥
⠀⠀⣀⣄⡘⠃⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⠐⠒⠘⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠁⠀⠀⠠⠠⠀⠀⠃⠀⠀⠀⠀⡆⠨⠥
⠀⠀⠉⢁⡠⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠂⡠⠒⢠⠄⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠚⠐⠶
⠀⠀⠀⠈⠀⠒⠂⠀⠀⠓⠘⢂⡀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠫⡀⠰⠄⠈⠓
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⡓⠩⡬⡱⠂
⠀⠀⠀⠐⠰⠉⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠟⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢌⣙⣻⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⠀⠀⠀⠀⠲⡞⢟⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢈⣓⣲⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⢀⣻⡁⠀⢠⠀⠀⣭⢭⡿⣿⢗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢺⡁⠀⠀⠀⠀⠀⠀⠠⣯⠄⠀⣾⡄⠑⢛⣿⣿⣿⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣗⠀⠀⣀⠀⠀⠀⠀⠀⠰⢮⣻⡟⠀⠀⢸⡀⠀⠀⣬⣿⣯⠴⣿⣷⢿⣿⣿⣿⣿⣷⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡻⣿⡂⠀⣿⠂⠀⠀⠀⠀⣘⠢⠴⢞⣁⠀⣿⡿⠐⠀⠀⡀⠀⠀⣨⢛⠿⣿⣿⣿⡿⠿⠿⢦⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢙⣶⠇⢸⣾⡇⠀⠀⠀⠀⣉⡙⢿⢿⠋⢸⠟⠁⠀⠀⠀⠀⠀⠰⢡⠎⠎⣼⣯⡁⠀⠀⠀⠀⠈⢻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢠⣓⣷⡯⢸⣿⣗⠀⠀⢀⡄⣃⠸⣿⣿⣷⠋⠀⠀⠀⡤⠀⠀⠀⠀⠀⢠⣾⢿⣿⣇⠀⢠⠀⠀⠐⢄⡻⢿
⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣴⠬⢧⣦⣼⣟⣛⣛⠺⣿⣟⢿⣿⣿⣿⣷⢀⡀⣀⠀⠀⠀⠀⠀⡀⡰⠋⣡⡿⠴⠞⠢⡈⣷⡀⣷⡘⣧⠈
⠀⠂⠀⠀⠀⠀⠀⠤⠒⠋⠀⠈⠛⠛⠛⠛⠛⠛⠉⠉⠻⠷⣤⡀⠀⠀⠀⠀⢀⣉⠳⢦⣾⣶⢟⡴⡵⠋⠀⠀⢀⣀⠀⠹⠁⠈⠉⠉
⠀⡀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠟⠛⠀⠒⠆⢠⣴⣶⠀⠀⠀⠀⣀⣤⣄⠐⠺⣯⣍⣀⣀⣸⣿⣮⣞⣥⣀⠀⠰⠛⠉⠀⠐⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠈⠉⠛⠛⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠐⠀⠀⠀⠀⠒

⠀⠀⢀⠀⣀⣀⣀⣀⠤⠤⠀⠤⢤⢤⣤⣤⠤⣤⣄⣐⣲⣆⣀⣤⠀⠀⠀⢀⡀⠀⠀⢄⡀⣀⣀⣀⣀⡤⠤
⠀⠀⠀⠀⢀⠾⠗⠶⠿⠷⠶⠒⠒⠒⠲⠲⠶⠤⡄⠤⣤⣤⣭⣽⣟⣭⣭⣉⣀⣀⡀⣐⣒⣄⣀⣀⣐⡂⠒
⠀⠀⠀⠀⠴⠀⠀⠖⠒⠒⠲⠶⠞⠛⠛⠉⠉⠉⠉⠙⠀⠀⠀⠀⣀⣀⣀⣀⣠⢤⣉⠉⠈⠁⠀⠀⠘⠋⠛

⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⡔⠀⠀⠀⠀⠀⠀⠀⠀⢮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠸⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣖⣡⣠⠔⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⣠⣌⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⣠⣞⣥⣹⣃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣠⣌⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠢⡀
⠀⠀⠀⠀⠀⠀⠞⠃⠛⢛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡛⠋⠚⠲
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣀⣤⣒⣥⠆⠀⠀⣀⡀⢀⢀⡄⣀⠀⣰⣀⣠⣆⢀⡀⢠⢀⡀⣀⣄⠀⠀⠰⣬⣒⣄⣀
⠀⠀⠀⠈⠉⠁⠒⠓⠧⠴⠚⠋⠳⠻⠞⠓⠁⠙⡇⠈⠈⡍⠋⠈⠚⠳⠓⠎⠻⠓⠤⠞⠚⠆⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣖⣿⣿⡿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣿⣿⣿⣿⡿⣾⡟⣿⡏⢾⡿⣿⣿⣿⡏⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣉⠙⣹⠁⣿⢿⣫⢯⡟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡜⣿⢸⣿⡼⣿⡏⣿⠀⣿⠀⡟⡾⠀⡾⢀⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⢿⡄⢿⡇⢻⡇⢿⠀⣿⢠⡇⡇⢸⠇⣰
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢿⡿⣏⢷⡈⢧⠘⣧⠘⡇⢸⡆⣿⢰⣵⠀⣾⡴⠃⡆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⢧⡙⢌⠫⣷⠘⣇⢹⣆⢿⣸⣇⣿⣼⣾⣵⣿⡷⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⢦⠑⠈⢳⡈⢧⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡅
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠷⠦⣀⠀⠁⣽⣾⣿⣿⣿⣿⣿⠿⢻⠿⢿⡿⣿⣿⣦⣀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⢬⡀⠀⠀⠑⣶⣿⣿⣿⡿⢏⡁⠉⡄⠀⡂⢀⡀⠈⢋⠝⣿⣷⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣛⡀⠀⠀⠀⢀⣿⣿⣿⣿⢧⣄⣑⣢⣼⣿⣿⣿⣿⣷⣯⠈⠉⢽⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⠈⠭⠀⠀⣬⣽⣿⣿⣷⠦⠭⢽⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠐⠫⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣭⣥⠤⠤⠀⠸⣿⠿⠿⠟⠚⠻⠻⣿⡿⠿⠿⣿⠿⣿⣿⣿⢧⣴⣾⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣶⠒⠂⠀⠀⢿⢂⣷⣷⣵⢄⣴⢋⣿⣿⣍⣽⣾⣿⣿⣿⠯⢷⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶⡊⠁⠀⠐⠀⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡛⠓⠫⣛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣾⡅⢤⣀⡄⣊⡾⢻⣿⣿⣿⢿⢿⣿⣿⡏⠙⢄⠀⠸⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣫⣾⠘⠁⢠⠃⠃⡀⠀⠈⡅⠈⠁⡀⢀⡻⣮⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠆⡀⡾⢀⢀⡇⠀⡄⣸⡴⣤⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣽⣿⣿⣿⣿⣿⡧⣷⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉