Smart Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢉⣉⣇⣠
⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠠⠐⡄⣄⢤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⡀⠀⠄⠠⠀⢰⡇⠺⠠⠦⠀⠂⠀⠀⠀⠀⠆⠀⠄⠠⠘⠉⠉⠙⢻⠓⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⣄⠀⣠⠀⣀⣤⣶⣿⠗⠈⠃⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠤⢄⣼⣀⣀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠘⣆⠈⠻⣿⣿⣿⣿⣧⠠⠀⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡞⠉⠉⠰⣄⡀⢁⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢹⣿⣿⣿⣿⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⡁⠀⡆⠀⡣⠀⠀⠀⡍⠈⠉⠋⠠⠠⡔⠃⠈
⠀⠀⠀⠀⢃⠀⠀⠦⢿⡀⣿⣿⣿⣿⡟⠁⣶⠀⠀⠀⠀⠀⠀⢶⠀⠀⢀⢷⡎⠀⠈⠉⠀⠀⠀⠠⠀⣀⠀⠀⠀⠄
⠀⣠⣶⡄⠀⠀⠤⡄⠀⠃⢻⣿⣿⣿⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡏⠀⠠⠀⠄⠀⠀⠀⠐⠀⠄⢀⡀⠀⠘
⢸⣿⣿⣿⣆⡀⢀⠀⠀⢰⣿⣿⣿⣿⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⢙⠻⡶⡆
⢸⣿⣿⣿⣿⢁⡜⢤⣦⠾⠿⢿⣿⣿⣿⣁⣀⣄⣤⣤⣀⣀⡀⣀⠀⠀⡀⡄⠀⢠⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⣸⡌⡔
⠀⢀⠄⠀⢱⠶⠻⠇⠀⠀⠀⠀⠈⠘⠿⠿⣿⣿⡿⠿⣿⣿⣿⣿⠃⠁⠘⣧⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡜
⠀⠘⠒⣤⣤⣠⣦⠴⠛⠠⠶⠳⠿⣿⣶⡤⠚⢃⠀⠀⠀⠀⠀⠛⠁⠀⠰⠇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠷⠁
⠀⠀⠘⠙⡿⢿⡟⠃⠀⠠⠴⠄⠒⠂⠀⠉⣁⠀⠀⠀⠲⠀⠐⣒⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⠓⣒
⠀⡤⠘⣂⠐⠋⠈⡈⠙⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠻⠈⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠁⠉⣐⡀
⠀⠀⠀⠻⠒⠲⠤⠀⢰⣶⣤⣀⣀⣀⣤⡝⠙⠛⠚⠛⠃⠄⠄⠀⠀⠀⠉⠛⠺⣿⣦⣄⣀⣴⠾⠯⠤⠔⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡛⠋⠁⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡟⠧⣤⣀⣀⡀⠀⠐⠂⠉⠁
⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠐⡆⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢠⢀⡀⠘⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠊⢀⢒⡥⠒⠀⡆⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠌⠾⠠⠈⠀⠉⢶
⠀⠀⠀⠀⠀⠄⠢⠄⣐⣀⣘⡣⠄⠐⢬⣁⠀⢃⢻⡈⠠⣀⡤⠛⡀⠀⠀⢸⢳⠀⢰⣻⠀⠀⢀⢅⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠈⠹⣄⠀⠉⠀⡀⠾⣁⢒⡌⢀⡀⠐⠒⠒⠻⠧⡀⠀⠈⢻⠒⡟⠁⠀⠀⠞⠘⠃⠀⡟⠠⡄
⠀⠀⠀⠀⠀⢰⢣⡆⠀⣬⠄⣤⠀⢠⠎⠀⢸⡴⠉⠉⢣⠐⠑⠲⣠⡆⠀⠀⠰⠀⠃⠀⠀⢰⠒⠐⢦⣀⠀⠺⠄⣤
⠀⠀⠀⠀⠀⠘⡎⠓⢈⡤⠤⠤⠀⣏⠀⠀⠈⣁⡤⠤⢄⠔⠋⠉⠉⠀⡀⠀⠀⠀⠀⠀⢀⠀⢠⠤⠀⠘⠧⠀⠿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠀⠀⣀⣀⣠⡾⢁⡀⢾⠁⠀⠀⠀⠄⣀⡄⠨⠀⠀⠈⠉⠨⠁⠈⠃⠀⣠⡟⠀⣶⠈⠁
⠀⠀⠀⠀⠀⠉⣀⠀⡆⠀⠈⢦⠎⠁⠀⢸⠁⢸⠀⠈⢀⠀⠀⠀⢤⠈⠻⡌⣭⣤⣭⠁⠁⢀⡷⠰⠎⠉⠀⢴⡂⠛
⠀⠀⠀⠀⠐⠉⠉⣁⣇⢰⣃⣀⣀⡀⠀⠸⠀⢀⠔⢉⠀⠀⠀⠀⢈⡇⠀⢇⠈⠙⡅⠀⠀⡼⠁⢠⡔⠀⣯⠐⠋
⠀⠀⠀⠀⠀⠀⡞⠁⢠⠞⠉⡀⠀⠀⠀⠀⠀⡏⠀⢯⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡇⡠⠞⣁⠀⠊⠁
⠀⠀⠀⠀⠀⣈⠤⠀⡟⠀⣼⠁⠒⠉⢉⠃⠀⣇⠀⠘⣆⠈⠲⠤⠄⠀⠀⠀⢀⡼⠁⢠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠈⠒⢠⠊⠉⠉⠙⠚⠆⠀⠘⡆⠀⢰⡀⠀⠀⡂⠉⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠙⠄⠓⠶⠔⠂⠀⠘⡀⠀⢠⠀⢠⡄⠄⠄⠧⠀⠼⠷⠊⠉⠀⠀⠀⠴⠀⠄⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠃⠀⡀⠀⢼⡅⠀⠀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣀⡀⠀⢈⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠘⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡜⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠔⠁⠀⠒⠒⠒⠢⡀⢰⠁⠀⠀⠈⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⡒⢒⡂⢰⠀⠀⠀⡆⢣⡀⠈⠀⠑⣠⠆⠈⢆⣀⣀⠢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⠀⠘⠀⠈⠀⠀⠀⠀⠀⢈⣉⠉⠉⠀⠀⡀⠤⢄⡀⠀⠀⠀⣄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⡁⣠⣃⠀⠀⠂⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠉⠉⠂⠀⠀⠊⣉⢠⠸⡉⢉⢇⠀⡁⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠄⠁⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠩⠉⠉⠉⠉⡀⠀⢢⢠⠊⠀⠀⢁⣀⣉⣀⣀⣀⠀⠈⠉⠉⠉⠉⠉⡅⠐⠐⠓⠒⠗⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠁⠀⠘⠜⠀⠀⠀⠁⡀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠢⠀⠎⠔⡆⠀⡼⠄⢠⢀⡄⠠⢇⠈⠉⠉⠉⠁⠀⠁⠀⠀⢐⣁⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠁⠀⡀⠘⠀⠂⠀⠃⣠⠈⠉⠁⠭⠀⠉⠁⢀⠀⠐⠒⠒⠒⠒⠒⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠠⢵⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⡠⠠⡀⠘⠀⠰⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠛⠛⠛⠋⠳⠀⡠⠘⠢⡄⢹⠀⠀⠄⠂⠠⠀⠀⠀⠀⢀⠀⠢⡤⠂⢀⠀⠈⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⣀⠀⣬⠁⠁⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⢀⢄⠀⠀⠘⠀⡁⡠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠶⠦⠀⠈⠠⠀⡀⠀⢠⢾⠀⢴⠊⡀⡈⠢⠀⠀⠀⠀⡂⠈⡄⠐⠒⡄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⢤⢁⣸⠀⠸⠤⢤⠃⠤⠀⠄⠄⠀⢈⠯⠀⠈⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠈⠂⠀⠀⠀⠢⡈⠲⡉⠫⠙⢇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠄⠤⠀⠈⢂⠘⠀⠑⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⡄⠀⢠⠐⠀⠐⠁⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢉⠜⠉⠀⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠶⠿⠟⠛⠛⠛⠛⠛⠛⠷⢶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⣀⣀
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠿⠋⠉⠉⠉⠙⠛⠷⢶⣤⡀
⠀⠀⠚⠋⠁⠴⠦⢤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣴⣶⣶⠶⠾⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⣸⠷⢦⣤⣄⡀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠚⠻⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⣿⠀⠀⣸⠏⠙⠛⠛⠓⠶⠶⠶⠶⠦⠤⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣠⣤⣶⠿⠋
⠀⠀⣿⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢿⣿⣿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⣿⠟⠋⠉⠉
⢀⣴⡿⢦⣌⠁⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⣠⡤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣙⣛⠛⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿
⣾⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠉⠀⣀⣀⡈⠙⢷⡄⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠉⠉⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣶⣇
⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣰⣿⣿⡟⠛⢧⡀⢹⡆⠀⠀⠀⣰⠟⠀⢠⣶⣿⡿⠛⢦⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿
⠘⢿⣦⣤⣴⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣥⣤⣾⠇⢸⡇⠀⠀⠀⣿⠀⠀⣿⣿⣿⣇⣀⣴⡷⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣤⡿
⠀⠀⠈⠉⢿⡄⠀⠀⠀⠀⠀⠀⢻⣄⠀⠘⠿⡿⢿⡿⠛⢀⣼⠃⠀⠀⠀⢻⣄⠀⠹⣿⣿⣿⣿⡿⠃⢀⣿⠇⠀⠀⠀⠀⠀⠀⣴⡟⠛⠛⠋
⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣠⣴⠟⢁⡴⠀⠀⠀⠀⠻⣦⡀⠈⠉⠉⠉⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⢰⣟⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠈⠑⠾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠤⠤⠀⠀⢀⣀⣀⣤⡴⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠿⠿⠿⢶⡾⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠶⠆⠀⠉⠁⠀⠀⠈⠉⠑⠲⡶⠖⠒⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢀⣀⣴⠞⠃⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠖⠒⠒⠒⢲⠒⠒⠒⠒⢆
⠀⠀⠀⠀⠀⠀⢀⡤⠋⠙⢳⡄⢀⡴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢸⣷⠀⠀⠀⢸⠃⠀⠀⠀⠛⠃⠀⠀⠀⣸
⠀⠀⠀⠀⠀⡴⠋⠀⠀⣠⠞⠛⠋⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣤⠟⠉⠒⣤⡀⢸⠇⠀⠀⠀⣤⡄⠀⠀⠀⣿
⠀⠀⠀⠀⣤⠁⠀⠰⠶⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠁⠑⢦⡀⠀⠙⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠙⠲⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠷⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠛⠲⣄⡀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⢀⠉⠁⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣰⠂⣄⢈⡽⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⡈⠛⠂⠀⠀⠈⢰⠄
⠀⠀⠀⠀⠀⠀⢁⠀⠈⠏⠀⠀⣠⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣉⠶⠦⠀⠀⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠐⠷⢠⡤⠞⠉⢸⡟⠛⠛⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠖⠶⠲⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢢⠄⠂⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⡉⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⢛⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣶⠒⠒⠒⢶⠞⠃⠀⠀⠀⠀⢸⡆⢀⣠⣤⡀⠶⠇⣤⣄
⠀⠀⠀⢀⡀⠀⠀⣀⡀⠠⠤⠤⠈⠋⠀⠉⠉⠁⠀⢠⠗⠀⠀⠀⢀⡤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦
⠀⢀⠶⠋⠁⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠇⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⣀⣀⣀⣎⠀⠤⠤⢈⡙⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⢀⡼⠃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣁⣤⣤⡤⠶⠶⠋⠁⠀⠀⠧⣤⣄⠀⠀⠀⠀⠀⠀⠁⠈⠁⣨⡇
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠟⠛⠻⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣈⣀⣀⣠⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰
⠀⠀⠀⢀⠔⢛⠓⠲⢄⠠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠁⠘⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠐⠀⠀⠀⢀⠂⠒⠀⠀⢰⠐⡀⠂⠀⠠⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠐⠀⠐
⠀⡀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⢀⡐⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢈⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⢀⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⡔⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠤⠀⠀⠃⠀⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣬⣥⣤⣤⣤⣥⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⣠⣴⠟⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣼⠯⢤⣀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠛⠒⠒⠒⠛⠛⠛⠛⠛⢿⠓⠢⢄⡈⢳⡀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⣋⣀⣀⠀⠀⣠⡀⠀⢀⣤⣤⣄⠀⢸⡆⠀⠀⢹⡄⠹⣆⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⡟⠁⠀⠹⣄⡈⠛⠁⣀⡼⠀⠀⠘⡇⠈⣧⡤⠤⢤⡇⠀⢹⡤⠤⠤⣄⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡭⠭⢍⡉⠀⠀⠀⠀⡇⠀⣯⠀⠀⠀⣧⠀⢼⠀⢠⡄⠀⢻⣇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠏⠀⠀⠀⢹⡄⠀⠀⠀⡇⠀⡟⠦⠴⠚⢻⠀⢸⠀⠈⠁⠀⢸⣿
⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⡀⠀⠀⠀⢀⠇⠀⠀⠀⡇⠀⡇⠀⠀⠀⣿⠀⡸⠀⢠⡄⠀⢸⡿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠳⣄⣀⡠⠎⠀⠀⠀⢠⡇⢸⡷⠚⠓⢦⡇⢀⡇⠀⠈⠁⠀⣸⡇
⠀⠀⠀⠀⠀⢸⠀⡴⠒⠒⠒⢦⠀⡴⠒⠚⠓⢦⢸⡇⢸⡇⠀⠀⢸⠇⣼⠀⢀⣄⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢸⡜⣇⠀⠀⠀⣼⠈⣧⠀⠀⢀⡼⢸⠁⣾⠀⠀⠀⣾⢠⡇⠀⠈⠁⠀⣸⡟
⠀⠀⠀⠀⠀⠀⣇⡈⠙⠒⠋⠁⠀⠈⠙⠛⠉⢀⡾⢠⠟⠲⠤⣼⠇⡞⠀⣤⠄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠹⢭⣉⠉⠛⠛⠒⠒⠒⠛⠋⠉⠀⡼⠀⠀⢠⡏⡼⠁⠀⠉⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢷⠒⠒⠒⠒⠒⠚⠛⢿⡏⠉⢉⠞⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠻⣶⡏⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣀⣀⣀⣈⣹⣷⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⢬⠍⠁⠀⠀⠀⢀⣀⣐⠒⠀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠈⡁⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⢀⠄⠈⠀⠀⠀⡔⠁⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢠⠆
⠀⠀⠂⠀⠀⠀⠀⡠⠘⣤⣄⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣄⡤⠚
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⡈⢧⡀⠤⡀⠀⠀⠃⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠁
⢠⠂⠁⠀⠀⠀⠀⠘⣤⠞⠻⣼⡇⢀⠇⠀⠀⠁⠀⠒⠢⠆⠀⠀⠀⢠⣄⣒⣒⣂⣃⣅⠀⠀⠀⠊
⠻⠁⠀⠀⠀⡠⢂⡀⠀⠀⠀⠀⠀⠘⢄⠀⣯⠙⡍⠉⠑⠒⠀⠀⠀⠀⠀⠀⠥⠤⢨⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠁⠄⠀⠀⠀⠀⠀⠀⢈⡦⠇⡬⠀⠀⠀⠀⠀⠀⠐⠺⠶⠦⠤⠄⢠⠀⠀⠀⠀⠀⠁
⠠⡀⠀⠀⠀⠀⣠⡘⠢⡀⠀⠀⠀⡨⠁⣠⢄⣀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠛⠐⠂⠁⠀⢀⡄⢀⣔⢈⡸⡥⣼⢴⡟⢻⣹⠵⣄⣤⣐⢒⠠⠤⠤⣀⠀⢠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡠⠉⠈⠉⢹⣂⠀⠀⠭⠤⠿⣽⣶⡏⠓⠛⠉⠁⠀⣒⣮⣼⡇⠸⠤⠤⠴⠶⠦⠷⢀
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠐⠒⣞⡀⢤⣀⠂⠀⠄⢀⠉⠙⠢⢀⠀⢤⠈⠉⠀⣀⠶⠓⠒⢀⡀
⠀⣠⠤⠄⠤⡎⠄⢁⠐⠒⠋⠁⠀⠀⠀⠻⠛⣶⡦⣤⡁⡀⠀⠠⠀⠙⠉⠁⠀⠀⠀⠀⠁
⠀⡇⠄⠀⠀⠈⠢⠤⠤⠤⠤⢤⡀⠀⢠⠃⠠⠃⢀⣿⠓⠠⢄⣀⠀⠄⣀⠀⠐⠒⠂
⢰⠉⠢⡔⡀⢸⣭⣩⣉⣩⠅⠀⠸⣒⢁⠔⠁⢠⣾⠃⠀⠀⠀⠀⠉⠒⠦⢌⡙⠂⡏
⠈⢆⠀⠈⠢⣸⢙⣉⣁⣁⣤⣤⣤⠤⣭⠒⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢶⡶⠆
⠀⠀⠐⠄⠀⣸⠉⠉⠁⠀⠀⣀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠑⠿⠤⠦⠶⠂⠈⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⡠⠔⠒⠀⠀⠐⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠑⠆⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡠⢾⡄⠀⠀⡇⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠉⠱⡈⠆⠀⡸⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣤⠒⠒⠐⠒⠢⣄⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠒⢦⡠⠵⣊⣭⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⠈⠉⠒⠤⡈⢂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠋⠀⠈⠓⠿⠯⠖⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⠋⠱⣖⠢⠤⠄⣀⣈⣁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠰⠀⠢⠈⠀⠀⠀⠄⠤⢈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠆⠀⢀⠀⡀⠈⠑⡢⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡜⠀⠀⠀⠀⠀⠸⣄⠉⠛⢂⡠⡞⠁⠀⠀⠀⡴⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡰⠁⠀⠀⠀⠀⢀⡼⣀⠉⠉⣁⠔⡇⠀⠀⠀⡜⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠷⣦⣀⣀⣀⡠⠊⠀⠀⠉⠉⢀⡴⣶⠀⡠⠛⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠖⠁⡄⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢀⠴⠋⠱⡀⠀⠀⢟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠷⠂⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠠⠤⠤⠤⢀⣔⠁⠀⠀⠀⢱⠀⠀⠈⡎⣗⣀⣀⡀⠠⡔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠗⠐⠙⠈⠉⠁⢉⢾⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠤⠄⠄⣄⣤⣷⠶⣷⢒⡩⠄⠑⠒⠒⠶⡂⠈⠉⡩⠭⠅⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠻⣷⣷⣜⠁⠀⠒⢉⠉⠤⡀⠈⠢⡀⠀⠐⠉⠁⢀⣈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⢀⡠⠴⢂⣀⣶⠤⠌⣢⡔⠛⠛⢹⣋⡃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣽⠃⠸⢁⡒⠏⠩⠛⠤⠨⣿⠔⠉⠉⠀⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠄⠀⢠⠤⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠺⡀⠔⠛⠒⠒⠣⡀⠀⠀⠀⠤⡀⠈⠏⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠈⡀⠀⢀⡠⠁⢀⣀⡤⠴⢶⣶⡶⢶⣤⣤⣧⡤⢄⠀⠠⡈⠛⢢⣄
⠀⠀⠀⠀⠀⢠⣋⣖⠈⠉⠀⣨⣴⠟⠉⠀⠀⠀⢘⣏⣗⣚⡛⠛⠛⠙⠾⣗⡂⢆⠀⢎⠙⣦
⠀⠀⠀⠀⢀⡎⠀⠀⠀⣰⣾⡿⠁⠀⡀⠀⠀⡂⠭⠗⠂⠀⠀⠈⢐⠠⣴⣾⣷⡌⠒⠚⠀⠙⡆
⠀⠀⠀⠀⢘⠦⣀⡄⣰⡿⡡⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢮⡇⢹⡀⠸⠒⢶⢿
⠀⠀⠀⠀⢸⠀⠁⢠⡿⣾⠇⠀⠀⠀⠀⠀⠔⣡⠖⠛⠛⠛⢶⣄⠀⠀⠀⡈⣇⠀⣇⠀⠖⠺⢤⡇
⠀⠀⠀⠀⠸⡄⠀⢸⡗⢣⡀⠄⠤⠤⠀⠄⢠⡏⠀⠀⠀⠀⠀⢿⠀⣇⣀⡧⣿⠀⢿⠀⠢⡀⢈⡇
⠀⠀⠀⠀⠀⠈⠉⠚⠿⢭⡟⠉⠛⠉⡻⠛⠉⢁⠀⠀⠀⠀⢀⠼⠀⠈⠉⠃⡿⠿⢾⠀⡐⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠈⠤⢄⠀⣰⠗⡀⠀⢐⡗⠀⠀⠀⠀⣸⢻⣧⡿⣀⣈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠳⣌⠪⣽⠃⠄⢀⠉⡉⢹⣆⢀⠔⣱⣿⣿⡿⠤⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠑⢤⣉⡓⠚⠒⠛⢉⣉⡴⠊⣿⡿⢛⠚⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣌⠙⡿⠿⠉⡯⢋⣽⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢦⡤⣤⠔⡡⠻⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢛⣛⡛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⣠⣦⠀⣀⣀⣀⣀⣀⣴⣿⣧⣀⡀
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣠⣾⣿⣯⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣠⣤⣄⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣾⣿⡷⠀⠻⠿⠟⠉⠉⠉⠉⠉⠻⠿⠟⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠊⠁⢀⣀⡀⠈⠉⠙⠲⢄⣀⣀⣀⣀⣀⣀⣠⣶⣦⠀⠀⣾⣿⣦⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⢿⣿⡿⠒⠒⢤⡀⠀⠉⠉⠉⠉⠉⠉⠙⠿⠟⢀⡴⠛⠛⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣨⣇⡀⠀⠀⠀⠀⠀⣰⠒⠒⠋⠀⣾⣿⡧⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⣾⣿⡦⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠉⡟⠁⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠙⠛⠙⠲⣤⣤⣤⣤⣤⣤⣀⡀⠀⠈⠛⠁⠀⠀⠀⣠⠞⠀⠀⠈⠛⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢈⣶⣶⡄⠀⠀⠀⣠⣷⣄⠀⢀⣀⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣳⣶⣄⠈⠻⠟⠃⠀⠀⠀⠻⠿⠟⠰⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢟⣤⣄⠀⠀⠀⣤⣤⡀⠀⢀⡴⠋⠉⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠈⠿⠿⠋⠉⠉⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⠤⣄⡀⠀⠀⡴⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠽⠏⠀⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⢠⣄⢠⡩⠓⢲⠶⠆⠀⠀⠀⢀⠒⠈⠁⠒⠁⠀⠀⠀⠀⡶⠲⡶⣶⢶⡂
⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⡔⢢⣶⠄⠀⠀⠀⠀⠠⡤⠀⠠⡤


⠀⠀⠀⠀⢙⠋⠀⠀⠀⠀⠀⠀⠀⡒⡂⢀⣀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠁⢘⠆⠀⠀⠀⠀⠀⢀⠐⢀⠬⠈⠀⠀⠀⠀⠀⠀⠉⣠⣮⣃⠀⠀⠀⠀⠀⠄⡀⠀⡈⡄
⠀⠀⠀⠡⢀⡈⠆⠀⠀⠀⠀⠀⠸⠴⠄⠀⠐⠊⠀⠀⠀⠀⠀⠀⠏⠂⠇⠀⠀⠀⠀⠀⠀⠁⡠⠀⠁


⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⢲⡆
⠀⠀⠀⠰⣍⣹⠂⠀⠀⠀⠀⢀⡰⢃⠘⠿⠃⠸⠀⠀⠀⠰⣶⠶⣢⣶⠶⢾⠀⠀⠀⠀⣸⠀⢦⡆⢸⡇
⠀⠀⠴⠆⠾⠷⠰⠦⠀⠀⠀⠀⠙⠙⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⠤⢼⡇⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⡤⣀⢶⠆⣤⣤⡀⠀⠀⠻⣥⡦⠶⠶⢶⡄⠀⠀⠀⠀⠀⣴⣦⣶⡄
⠀⠀⣿⣿⣿⡑⠀⠀⠀⠀⠀⠀⡔⣿⡇⠀⠘⢽⡇⠀⠀⠀⢸⠟⣤⠛⢠⢇⡀⠀⠀⠀⣘⣼⣛⣻⣌⡂
⠀⠀⢿⠿⢿⠇⠀⠀⠀⠀⠀⠐⠙⠚⠃⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠿⠴⢷⡆⠀⠀⠀⠛⠛⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⡠⠂⠀⠀⢀⠄⣀⡀⢀⡀⠀⡀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⢀⣈⠀⠀⢀⡀⣀⢀⢀⠀⢁⣀⣀⠀⠀⡀⠈⠀⠀⠉⠀⠉⠀⢈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⣀⠀⣀⠄⢀⠔⢠⠔⡀
⠀⠀⠀⠀⠐⠊⠒⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠆⠐⠀⠄⠇⠀⠀⠀⠄⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠒⠒⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠐⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⡘⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢂⡤⠀⠀⠀⠅⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠈⡃
⠀⠀⠀⠀⠀⢀⠀⡀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠆⠀⠀⢐⠀⠈⠀⠀⠀⠀⠀⠀⠆⢸⠐⠠⠀⡅
⠀⠀⠀⠀⠀⠀⠄⠂⢰⠀⠀⠄⠰⠀⠀⠐⠀⠐⠐⠂⠉⠀⠀⠀⡠⠀⣤⠠⣄⣥⣸⣠⣥⣼⣅
⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠸⠀⠀⠀⡅⡀⢀⣄⠛⢛⢛⣛⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠂⠠⠐⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠉⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠐⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠐⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⣠⣶⣿⠟⠲⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⢀⢀⣀⡤⡄
⠀⣏⠠⢈⣌⣡⣿⡆⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⠀⠀⠀⠀⠀⠨⢥⡿⣸⢟⣊⣑⡫
⠀⠙⢷⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⡯⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣿⣿⠀⠀⠀⠀⠀⠈⠱⡸⢵⡰⣹⠹⠜
⠀⠀⢸⡿⠏⠻⠿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠈⢱⡇⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢠⡀⡰⣿⣀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣺⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣄
⠀⠘⡿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⡞⣿⡟⡆⠀⠀⠀⠀⠀⠀⢠⣴⡿⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣨⣿⣶⣿⣄
⠀⠀⠧⣼⢿⣅⠇⠀⠀⠀⠀⠀⠀⠀⠱⡷⡷⠁⠀⠀⠀⠀⠀⠀⢸⠛⣷⣶⠛⡇⠀⠀⠀⠀⠀⠀⠸⠋⣿⣿⣟⠹⠆
⠀⠀⠀⣿⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⢻⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿
⠀⠀⠀⠿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠿⠄

⠀⠀⣰⣾⡟⠓⠢⠀⠀⠀⠀⠀⠀⢠⡶⡿⡷⠦⠀⠀⠀⠀⠀⣾⣳⡴⣶⡶⠶⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⡛⣼⡶⠀⠀⠀⠀⠀⠈⢩⣶⣄⠀⠀⠀⠧⠀⠀⠀⠀⠀⢀⢠⢰⢈⡇⢀⡄
⠀⠀⣿⣿⣿⣻⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⡻⣆⠀⠀⠀⠀⠀⠀⠉⡷⣶⠀⠀⡬⠦⠀⠀⠀⠀⠘⡽⣖⣒⡃⠾
⣠⠸⠛⠋⢸⣏⣽⠇⠀⠀⠀⠀⠀⢸⣭⠛⢠⣿⠘⠈⠀⠀⠀⠀⢀⣴⣟⣋⣂⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇
⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⣀⣀
⠀⠐⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡞⢛⠛⡆⡞⣿⡟⡆⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤
⠀⢀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⡇⣿⡄⡇⡇⣛⡃⡇⠀⠀⠀⣿⣿⣿⢿⢹⢿⣿⣿⠀⠀⠀⠀⢸⠀⡀⠀⠀⡀⠂⢸
⠀⣰⣿⣸⣿⣸⣿⡄⠀⠀⠀⠀⡇⠈⠀⡇⡇⣒⡂⡇⠀⠀⠀⣿⣿⣇⠿⡿⢣⣿⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣸
⠀⠉⠽⠿⠿⠿⠭⠁⠀⠀⠀⠀⠳⠶⠶⠃⠳⠿⠗⠃⠀⠀⠀⠫⠿⠿⠿⠾⠿⠿⠽⠀⠀⠀⠀⠸⠿⠿⠯⠯⠽⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⣛⣋⣉⡭⣯⡭⣍⡽⣛⣻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣟⢽⡂⣿⢹⡿⠋⣿⡿⣨⡾⠟⡾⠿⣿⣇⣶⣭⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⡫⣪⣾⡿⣧⢿⡸⢹⣿⡟⢱⣿⣿⡶⡏⣾⣥⢘⣛⣯⣿⢧⠝⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣽⡶⣛⣴⣼⣿⣌⣳⠜⣿⣏⣷⣭⡷⣻⢧⣿⣭⣿⣧⢉⣶⣷⡾⣤⡻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣟⡚⢛⡿⢻⣥⡭⣷⣟⣻⢯⣽⣾⣥⣷⣶⣶⣶⣯⣽⣿⣯⣵⣶⢶⣦⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣝⣻⠾⢖⣻⡭⠶⢛⣭⣥⣿⣿⢹⣿⣿⢆⣭⣭⢛⣌⣻⡽⠿⣽⣃⣿⡃⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣝⢿⣿⣾⠟⡿⣏⣿⣿⢟⣭⡏⣭⣿⣭⣝⡻⣿⣿⣿⣿⢸⡟⣮⢿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⢰⣿⢏⣋⡿⣻⡽⣋⣻⠟⠿⢿⣬⣟⣋⣙⡾⢷⣝⢷⣮⢥⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠿⣾⣿⣷⢽⠷⠿⠿⢿⣿⣟⣎⣻⣿⣿⠿⠿⠟⡼⣫⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠰⡀⢠⡀⠀⢀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣤⡨⢉⣱⢀⠂⣀⠿⠶⠮⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣄⠀⠀⠘⡟⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣀⡇⠀⠀⠴⢇⡆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠁⣼⣿⣿⡏⠀⢸⣿⣿⣿⣿⣇⣀⣘⣇⠀⠀⠀⠘⡇⠐⠚⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣰⡀⠀⣿⣿⣿⡇⠀⢸⣏⣿⣿⢿⡏⠁⠀⠀⣀⡤⠶⠾⠧⣄⠀⢸⡌⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⡀⠉⣛⣛⠿⢿⡇⠀⠘⣿⣿⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡇⢾⣇
⠀⠀⠀⠀⠀⠀⠀⣼⢿⡿⠉⠐⠀⢭⠓⠺⡇⠀⢸⣿⣿⡇⢸⣇⣀⣀⣤⣠⠀⠀⠀⠀⢀⣀⣼⠃⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣟⠟⢀⠄⠀⡀⠀⠈⠂⠈⢆⣸⣿⣿⡇⢸⡏⠀⠀⠀⠈⠀⠀⠰⠶⠾⠉⠀⠠⡏⠀⣷
⠀⠀⠀⠀⠀⠀⢀⡿⡧⠦⡄⠀⠈⠄⠀⠀⠀⢾⡄⠀⢹⡇⢸⡇⠀⠀⠀⢀⡀⣠⠇⠀⠀⣀⣀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠎⠀⠀⡀⠀⠀⠈⣿⠀⠈⡇⢸⣇⠀⢀⣀⡰⠛⠁⠀⠀⠉⠁⠈⠙⡄⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠂⠉⠐⠀⠀⠀⠨⡄⠀⠀⢸⡟⠉⠉⠉⠻⢄⣀⠀⣀⣀⠆⠀⣸⠇⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⢰⢶⣺⡆⢐⣀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡄⠀⠈⠉⡉⠁⠀⢸⡏⠐⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢸⠿⣿⡇⠀⠁⠀⢀⠀⡀⢸⡇⠀⠀⣸⠋⠓⠚⠛⠙⠓⠒⠋⠙⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣘⠲⢼⠁⠀⠀⠀⣾⣾⣿⣾⡇⢀⣴⠋⠀⣠⠷⣤⠞⠉⠙⢦⡀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡖⣦⣉⠞⢀⡾⠀⢰⡅⣹⣿⣿⡷⠛⠓⠶⠚⠁⠀⣸⠇⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣹⣶⡞⣥⣷⣿⣿⣿⣿⣿⡇⠀⣀⣰⣦⡴⠟⠋⠀⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣿⣽⣿⣿⣿⣿⠃⣟⠚⠋⣉⠀⢀⣴⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠈⠒⠶⠟⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡴⠋⠁⢀⠀⠉⠳⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣼⠀⢠⡞⠉⠙⠂⣰⠛⠉⢀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⢦⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠟⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣄⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⠶⠶⡄⠀⠀⢀⣠⠛⣦⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠈⠉⠀⠀⠀⠀⢸⠀⠀⢠⡄⠀⠀⣤⠀⡇⠀⠀⠈⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠀⢸⠀⢀⣟⣹⡄⠀⣿⠀⡇⠀⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⣄⣀⡴⠛⠛⠀⢸⠀⠾⠁⠀⠳⠀⠿⠀⡇⠀⠛⠛⢦⣀⣠⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⣀⣀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠴⠇⠀⣀⣀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠉⠉⠀⢀⡄⠀⠀⢀⡀⠀⠀⢠⡀⠀⠉⠉⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠀⠀⠀⢀⣸⠇⠀⠀⢸⡇⠀⠀⠸⣇⡀⠀⠀⠀⢿⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣦⠀⠀⠈⠳⠞⠁⠀⠀⣴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⡠⡀⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡭⠟⢉⣁⠀⣠⠖⠦⣝⡎⠑⠢⡄⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⢢⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠐⠈⠲⣤⣯⣄⡀⠇⢀⣀⡘⠃⠀⠀⢹⡀⠒⠚⠒⠦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠘⠀⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠠⣄⠀⠀⠀⠀⠉⠛⢬⡉⠙⢳⣄⣀⣀⣀⠀⣸⠀⠀⡈⠀⠂⡀⠀⠀⡔⠐⠂⠄⢴⠀⠈⢠⡅⠀⣴⠟⠁
⠀⠀⠀⠀⠀⡠⠊⠢⢤⠀⠀⠘⣆⡤⠒⠀⠀⠀⠰⠋⠙⡆⢹⡄⠀⠈⠻⠇⢄⠀⣻⠔⠛⠃⠀⢠⠀⠀⠀⠀⠀⠂⠄⠒⠀⠈⠛⠠⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠤⣤⡀⠀⢀⡏⠀⡞⠀⠀⠀⠀⠒⢤⡀⠀⠇⠀⢀⠤⠤⠼⣇⠛⠓⢦⠀⠈⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢧⡀⠀⠀⡝⠀⠸⡅⠈⢧⡀⠀⠠⠚⠓⠄⠙⡄⠀⠀⣀⡀⠐⣄⠈⢳⡀⢈⣃⠀⠀⢀⠄⠈⠉⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢃⠀⠳⢄⡀⣧⠀⠀⠃⠀⠀⠀⠀⣰⠢⠀⠣⠀⡇⠀⣋⣀⡉⠁⢸⠇⢈⡇⠁⠘⠛⠈⠠⠄⠀⢀⠄⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢁⠐⢦⡀⠘⢦⡀⠀⠀⠀⠈⡉⠁⢠⠗⢀⠈⠑⠛⠁⠀⠙⠚⠋⢉⣀⣤⡴⢊⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⠿⠛⠿⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠹⠄⠀⠈⢉⣀⣀⡀⣇⡄⠺⢀⠀⠈⠐⠒⡖⠒⢀⣎⡀⢈⠀⠠⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢀⠀⠀⠀⠉⡹⠷⠐⠤⡈⠢⡀⠀⠙⢆⠀⠩⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠠⠀⡜⢂⠘⡂⠈⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠔⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⡠⠊⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣷⣶⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠈⣰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢄⠈⠀⢜⣦⡀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡂⢀⠀⠻⠓⢀⠀⠀⢀⡅⠀⠗⢶⣤⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⠂⠀⢠⠂⠀⢸⠈⠈⠀⠀⠉⠃⢀⠠⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠈⠀⠀⠉⠀⠀⠒⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢓⡦⢄⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣿⣧⡀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠻⣿⣿⣿⣿⣿⣷⠖⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢿⣿⡿⠋⠁⠀⠀⠀⠀⢻⡋⡻⠿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⠅⡩⠗⠀⠖⢬⠹⢄
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣾⣿⢦⡠⡬⠌⢒⣣⡀⠈⢻
⠀⠀⢠⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⢰⢠⣥⣄⠍⠉⢘⠣⣤⡀⠑⢄
⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠻⣿⣿⡿⠀⠀⠈⢓⡿⡟⠿⠔⠁⢰⣿⣿⡄⠀⠑⢄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⢅⠀⢐⣤⠜⢁⡷⢦⣄⣠⣞⠀⡇⠀⠀⠀⣼⢿⡟⢹⡀⠀⠀⢳
⣀⣶⣿⣷⣆⣀⣀⣀⣀⣀⣀⡀⡋⠀⣩⣟⣶⡤⠤⢮⣷⣀⢀⡾⢀⠉⣿⠙⢹⣉⡿⠀⠙⠟⢼⡀⠀⠀⡇
⠻⠿⣏⣉⣙⣛⡛⠛⠛⠛⠛⡱⠚⡾⠿⠿⠿⠿⣿⣷⣶⣦⣭⣝⣒⠦⡼⠦⣞⣓⣀⣡⣭⣭⣭⣭⣿⣶⣥⠤⠤⠤⠤
⣀⣀⣈⣉⣉⣉⣉⣉⣉⣛⣛⣒⣈⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣛⣻⣧⣟⣛⣛⣓⣒⣒⣒⣒⣛⣛⣛⣛⣃⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⡿⠁⢀⣤⠶⢤⡀⠀⣻⠶⠒⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣁⠀⠀⠀⠸⡇⠀⠸⠁⠀⠀⠀⠘⠁⠀⣤⠤⣤⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠈⠙⢦⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⣄⣀⡀⠀⠀⠀⠀⣶⠛⠛⠛⠛⠛⠛⠛⠛⣦⠀⠀⠀⣠⠾⠙⢧⡄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠉⠀⠀⠀⠀⠀⣿⠀⠀⣰⡆⠀⠀⣴⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠃⠀⣿⠀⢠⣏⣹⡄⠀⣿⠀⣿⠀⠘⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦⣀⣀⣠⠞⠛⠃⠀⣿⠀⠟⠉⠉⠳⠀⢿⠀⣿⠀⠘⠛⠳⣄⣀⣀⡴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠾⠉⠉⠁⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⡿⠀⠀⠀⠀⠈⠉⠉⠷⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢿⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠈⡿⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⢰⠞⠋⠀⠀⠀⢸⡇⠀⠀⠀⠙⠳⡆⠀⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⠀⢰⡏⢹⡆⠀⠀⠀⣨⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣼⠃⠀⠀⠀⠉⠉⠀⠀⠀⠘⣧⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠋⡀⢰⠂⠀⠀⣱⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣂⠘⠂⠀⢀⢞⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⡀⢥⣀⡃⢠⠁⠂⠀⠀⠀⠀⠀⠀⠀⣠⠇⠁⠀⠀⢀⠠⡀⡠⠀⠀⠌⠐⡊⢁⡐⠤⠂
⠀⠀⠀⠀⠈⢀⠀⠰⠂⠀⠀⠁⠂⠁⠀⠁⣤⠧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡀⠀⠀⠀⡄⠀⠆⠐⠏⠐⠀⠀⢄⡄⠀⠔
⠀⠀⠀⠀⠢⠀⡇⠀⠐⠀⠀⠀⠀⠀⠄⠄⡄⢢⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⢌⠘⠁⣀⠀⠐⠀⠀⠀⠂⠁⠐⠁⠄
⠀⠀⠀⠀⠀⠢⠑⠀⠀⢈⠀⠆⠈⠉⠀⡈⠁⠔⠂⠀⠀⠀⠀⠀⠰⠛⠓⢦⠏⠀⠀⠀⠀⠄⠀⠀⠀⠁⠰⠀⡂⠀⠠⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠒⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠦⣤⣄⣀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣾⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠈⠛⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣠⣤⣄⡀⠀⠀⠠⢟⠭⠤⢤⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠈⢉⡽⠛⠛⠳⣤⡴⢶⡏⠴⠶⢸⡇⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠙⣿⡇⠀⠀⢸⡇⠛⠁⢀⡟⠀⠀⠙⠦⠴⠋⠀⠀⡉⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠰⡎⢻⡄⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣟⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣠⣶⣮⣤⣄⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠏⢿⣧⡀⠀⠀⢀⣼⣿⣿⠿⠟⠻⢿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⢘⣿⠋⠀⠈⠁⠀⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠻⣿⣿⣿⣿⣆⠀⢀⣴⣦⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⣿⣿⣿⠏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠙⣿⣿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⣿⣿⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣽⣿⣿⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣇⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡄⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣧⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡧⠸⠿⠿⠿⠆⠀⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠄⠀⠀⠿⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠦⠀⠆⢠⡄⠉⠃⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠶⢈⠴⠂⠁⠀⠀⠀⠉⢠⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⢸⠀⠀⢀⠄⠂⠀⠀⣛⠰⠏⠄⠀⠀⠀⠙⢦⡀⠀⠤⠀⢷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠘⠄⠀⠀⠀⠀⡎⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⣠⠄⠀⠀⠔⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠚⠁⠀⠒⠀⣶⠀⠉
⠀⠀⠀⠀⠀⣀⠔⠊⠉⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠚⠀⣼⠃
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⠔⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠉⠁⠀⠈⣳
⠀⠀⠀⢀⠴⠲⠄⡴⠚⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠐⢦⡀⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣄⡤⠤⠤⠄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠱
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣀⣀⣤⡦⠺⣉⡀⠀⠀⠀⠀⣀⡤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠲⣶⡿⡑⢀⠈⡛⢷⣶⣖⣺⢍⠅⠀⠈⣉⣀⣀⣀⣠⣤⣴⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠕⠦⠭⠛⠛⠊⠁⠀⠀⠀⠻⠛⠛⠛⠛⣋⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠀⠀⠀⠀⣠⠔⠒⠋⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠞⢀⡠⠶⣄⠀⠀⡀⠀⠀⠀⠀⠈⢻⣿⡦⣄⠀⠛⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠬⠓⠀⠀⡞⢀⡞⠀⠀⠘⢦⠀⢻⣆⡀⠀⠀⠀⠀⢿⠃⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠙⠦⣳⡙⢦⡀⠀⠀⢸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡾⡀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠈⢳⠶⣼⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡇⠀⠛⠀⠀⢀⠀⠀⠛⠀⠀⢸⣴⠿⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠤⠄⠀⠀⠀⣠⠟⠁⠀⠈⢳⢀⡇
⠀⠀⠀⣀⡤⠤⠤⣀⣀⡤⢤⣀⡀⠀⠀⠀⢠⠎⠈⠓⠦⢤⣀⣠⡤⠔⠋⠀⠉⢢⡀⠀⠼⠋⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢾⡡⢖⣋⣑⣦⣥⣖⣲⣦⣍⡱⠀⠀⡏⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣯⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⢸⠁⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠤⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡞⠀⠭⣧⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡇⠀⢰⠋⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⢠⠼⠭⠭⠭⠭⠭⠭⠭⠭⢇⠀⠸⣷⢌⡽⢛⣏⣉⠑⠲⠒⠋⠙⣧⢂⡄⢈⡇⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠒⠓⠲⠶⠶⠶⠶⠶⠖⠲⠶⠚⠒⠒⠛⠻⣨⠴⠒⠒⢬⣒⣪⠥⠤⠬⢽⡶⠛⠒⠒⠺⠤⠤⠤⠤⠤⠤⠤⠤⠗⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⢤⣄⡧⣩⣴⣥⣍⣄⢁⠒⠀⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⢨⣐⡲⢦⢹⣿⡿⠧⠼⠶⠎⠾⠾⠿⠫⠶⠶⢂⣄⣤⣀⡀⠄
⠀⠀⠀⠀⠀⢀⣐⢶⡸⣜⠎⠓⣑⢴⠦⠀⠀⠀⢠⣤⣤⡄⠀⠠⢀⡈⠉⠉⠢⣦⣴⠆⣚⢁⣂⡀
⠀⠀⠀⠀⡈⢀⠡⠙⠃⠛⠛⠉⢀⠄⠀⠀⠀⠀⠚⠛⠛⠃⠀⠰⠊⡁⠾⠋⣗⠒⠒⣒⠮⠍⠁
⠀⠀⠀⢐⠨⠲⠥⠖⠴⠶⠶⠦⢁⣠⢖⢄⣤⣤⣤⣤⣤⣤⣤⣔⣦⣉⣀⡐⣙⣛⢉⣥⡓⢈⡃⢒⣒⡂
⠀⠀⠀⠀⢁⣴⠳⡎⣿⡡⡋⠛⠛⠪⠓⢉⠨⣻⣾⢛⠛⡹⡛⣟⣒⠒⣲⠭⢤⣔⢮⢲⡆⣷⡽⣎⡛⠟
⠀⠀⠀⠀⠈⠉⠑⠛⠣⢪⡪⣉⠋⠧⣒⡵⣪⠵⠟⣸⠻⡝⣦⣤⡦⣿⡦⢹⡏⣿⡜⣦⢹⣬⣁⠈⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢇⡟⣾⡞⣵⠊⣵⡇⣿⡿⢃⣧⢙⡏⡗⢿⣁⣿⣇⣿⡇⣹⣧⣉⡭⢳⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⡇⢻⣆⢹⣧⠘⠿⠸⠿⠿⣤⠃⣮⡙⠛⠉⠉⠁⠉⠉⠉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠺⢉⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣀⣀⣸⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠛⠻⣷⣶⣤⡘⠀⠀⠀⠀⠀⠘⠲⣿⡛⢻⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⣷⣜⠿⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⡄⠙⠟⣻⣿⣴⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣾⡟⠋⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⠩⣍⠛⠿⠃⠀⠄⢤⠆⠀⠀⠀⠀⠀⣤⠀⠀⠀⡿
⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠈⢎⡓⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⡀⠉⡉⠉⢏⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠻⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠒⠀⠀⠀⢁⣁⣠⡴⢚⣩⣼⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠯⣄⡀⠈⠳⡝⢦⡀⠀⢄⡀⠀⠀⠀⠌⠀⠀⣉⣀⢚⣋⣶⣶⣾⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⣙⣦⣀⡙⣆⠙⠢⠄⠁⠈⠡⠦⠅⠂⢀⡻⠷⠈⢘⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠤⠤⢄⣈⠑⢦⣹⠈⢧⣀⢻⢀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠸
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⠙⢦⡹⣆⠈⢿⡿⣀⣀⣀⣀⠤⣄⠀⠀⠀⢸
⠀⡀⠂⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣧⡈⢧⢯⠀⠀⠀⠀⢸⡆⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢤⣀
⠀⠀⠀⢠⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⢳⠘⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⢸⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠈⠓⠠⢤⡀
⠀⠀⢠⠟⠀⢀⠀⠡⡀⠀⠀⠀⠠⠤⢀⣀⡀⠲⢄⡿⠸⡄⣹⠀⠀⠀⠀⠀⣇⠀⠀⢀⢈⡀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠲⢀
⠀⢠⡟⠀⢀⠈⠲⢄⠙⣄⠀⠀⠀⠀⢀⣀⣉⣛⠒⠚⢳⣁⣿⣄⠀⠀⠀⠀⢹⢰⣊⣀⣵⣮⠆⡀⠁⠖⠰⢧⢀⠔⠁⠀⠀⠀⠀⠀⢀⡴⠋
⣰⠋⠀⠀⠈⠳⢤⣀⠙⠪⢿⣟⠛⠋⠉⠉⢉⣉⡡⠤⠼⢷⣿⣍⡃⠀⠀⠀⢸⠶⠶⠖⠋⠁⣀⠜⠀⠀⢀⠔⠁⠀⠀⠀⠀⢁⠀⣰⠟⠒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠽⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡧⠉⠀⠀⠀⡜⠀⠀⠄⠐⢮⠀⠀⠀⢀⠂⠀⠂⠀⠀⠀⡠⣣⣎⡙⠒⠒⠒⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⡟⠠⡤⠒⠚⠓⠒⠒⠒⠲⣦⣀⠀⠀⡞⠀⠀⠀⠀⡠⣪⡎⠙⢿⠷⠆⠀⠤
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣛⠴⣿⠻⠀⣠⠆⠀⠀⠀⠀⠘⡎⢳⠔⠙⢷⡄⠠⣪⢴⣫⡞⢢⠈⠀⠰⣶⣴⡆⠤
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⣦⡀⠀⠶⠛⠁⠸⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⣻⡄⡇⠀⠀⠙⠚⠁⠠⡝⠴⡆⡠⠊⡴⢠⣦⡌⢢
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠋⠙⢦⣄⠀⠠⠔⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⡝⠈⢁⡀⠤⠀⠐⠠⡀⠈⠳⣄⠑⠒⠠⠬⠍⠅⠼
⠀⣘⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠉⢻⡖⢶⡚⠛⠛⠉⠉⠉⢉⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⢄⠈⠁⠀⣀
⢰⠘⠧⢅⣒⠒⠂⠀⠀⠀⠀⢉⡉⣀⣄⣛⣚⠛⢲⣲⠤⣄⠙⢦⠙⢄⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠈⠳⣄
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠌⠃⠈⢳⡀⠣⡀⠀⠈
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⡄⢀⣀⠐⠒⠛⠛⠦⢄⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢸⡰⠋⠉⢳⠀⠙⠲⣄⡴⠊⠀⠐⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠠⢈⡤⠤⠄⠀⣏⠀⠀⠈⣁⡤⠤⢤⠴⠊⠉⠉⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀⠀⡼⣧
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⠀⠈⣀⣀⣀⡾⠃⡀⢼⠁⠀⠀⠀⢤⣀⢸⠁⠀⠀⠴⠒⡀⠀⠀⠘⢦⣇⣠⡷⠃
⠀⠀⠀⠀⠀⠈⡀⢀⡆⠀⢘⢦⠞⠉⠀⢸⠃⢸⠁⠈⠂⠀⠀⠀⢠⡈⠻⡄⠀⠀⣠⠖⠈⢀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠐⠉⠉⣚⣇⢰⡁⣀⣀⠀⠀⠸⠀⢀⡔⠊⡀⠀⠀⠀⠀⡇⠀⢹⠀⠈⡇⠀⠀⢠⠆⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡞⠁⢀⠞⠉⡁⠀⠁⠀⠀⠀⡞⠀⢮⠀⠀⠀⠀⠊⠀⠀⠘⠀⠀⣿⢀⠴⠃⠠⡀⠀⠀⠀⠀⢀⠀⠤
⠀⠀⠀⠀⠀⢈⣀⡀⡟⠀⣼⠁⠔⠋⠙⠂⠀⣇⠀⠘⣆⠈⠲⠤⠤⠀⠀⠀⠀⣠⠏⢀⡴⠋⠐⢫⣉⣉⣡⣤⣤⣶⣶
⠀⠀⠀⠀⠀⡇⠀⠀⢧⠀⠘⠦⢠⠞⠉⠉⠓⠺⠆⠀⠈⣆⠀⠐⡄⠀⠀⠐⠊⠁⠀⠚⠀⠐⠀⠨⠋⠉⠉⢉⣭⡅⣀
⠀⠀⠀⠀⠀⠑⠀⠢⠤⠵⠄⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⢠⡧⠖⠛⠩⠴⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡦⠄⠀⠀⢀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⡀⠺⠇⠀⣸⠃⠀⠀⢀⠈⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡬⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⠐⢒⡂⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠠⠈⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠆⠒⠒⢒⣤⠤⠴⠶⠦⣤⣤⣤⣄⣀⣀⣀⣉⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠊⠁

⠀⠀⠀⠐⠂⣐⣒⣒⣂⣐⣒⣂⣂⣒⡒⠒⢒⣒⣀⣀⣀⠀⣀⠀⠀⣀⣀⣀⣀⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⣀⣀⡀⢀
⠀⠀⠈⠁⠀⢻⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣧⢿⣿⠸⣿⠿⠿⠛⠿⠏⣉⠀⣿⣿⡾⣿⠽⠿⣿⣗⣒⣿⡏⠁⢸⣶⠠⡤⠀⠀⣀⡠⠤⠤⣤⡖⢻
⠀⠀⣦⣀⠀⢸⣿⣿⠿⠶⠀⠐⣶⣶⡇⠀⢸⠉⢹⣿⣷⢲⣿⣷⣶⠒⣶⣾⣶⠀⠟⢻⣿⣿⣦⣤⣤⣤⠤⣀⣴⣦⢸⣽⣿⣿⠖⠋⠁⠀⠉⠀⠐⢿⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⠀⣀⣿⢽⡇⠀⢸⠀⣸⣿⣿⣿⣿⠛⠛⠒⣛⣛⣻⠀⠀⢸⣿⣿⡾⠿⠿⠷⠤⠿⢿⣿⢸⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠋⠉⠉⠉⠛⠉⠛⠛⠛⠛⠻⠿⠇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠀⠿⠿⠿⠿⠲⠶⠶⠶⠰⠿⠿⠿⠘⠿⠿⠿⠿⠞⠃⠀⠀⠐⠒⠚⠛
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣤⡄⠀⠀⠀⣠⠀⠀⠀⠄⠀⣄⣤⣤⣤⣤⠀⣤⣤⡄⣤⣠⣤⠀⠀⠀⠠⢤⣤⣤
⠀⠀⣄⣾⡃⠀⠀⠀⠀⠀⢀⣤⣛⠋⠀⢉⡇⠀⠀⠈⣹⡿⠖⢓⡿⠻⢿⣿⣿⣿⣿⠀⣿⣿⠆⣿⣿⣯⣄⣀⠀⠀⢿⣿⣿⠷
⠀⠀⣾⣿⣷⡤⠶⣀⢀⣀⣾⣿⣿⣤⣀⣾⡇⠀⢸⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⣿⣿⡀⠛⢋⣿⣷⡯⠅⠐⢘⣉⣽⣶
⠀⠀⣿⣿⣿⡦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡟⠀⠚⣿⣿⣿⣷⣾⣿⣽⣿⣿
⠀⠀⠿⠿⠿⢷⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⣿⡟⠀⠀⢀⣿⣷⣷⣾⣿⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠶⠄⢰⡶⡖⠀⠀⠂⠐⠒⠒⠒⠲⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⢠
⠀⠀⡿⠦⠀⠀⣽⣿⣿⣿⡿⠿⠿⠿⠆⠀⠀⠀⠰⠿⡋⠁⣿⣿⣷⡆⢸⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⠃⢸
⠀⠀⣇⣠⣴⣶⣯⣭⣉⣉⣡⣤⣤⣀⡀⠐⡖⢻⣿⣿⣧⣴⡿⣿⣿⡇⢸⣿⣿⣿⣿⣧⠶⠄⢠⠔⢾⡇⠀⢸⣿⡇⠉⡻⢿⣿⣿⠿⠿⠿⠛⠀⢸
⠀⠀⣋⣹⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡇⠀⣤⣿⣿⣧⣽⣿⣿⣿⣿⡇⢸⡿⣿⣿⣿⣷⡠⣤⠰⣾⢋⡇⠀⢸⣿⡇⢠⠉⢶⣼⣿⣦⣴⣦⠶⠿⢿⡀
⠀⠀⠀⠠⠿⠛⠛⠛⠛⠛⠷⠖⠒⠒⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⠋⠙⠛⠛⠛⠛⠛⠒⠚⠛⠃⠀⠸⠿⠇⠸⠦⠼⠿⠿⠿⠿⠿⠄⠀⠸
⠀⢀⣀⡀⢀⡀⣀⣤⣤⣤⣤⣤⣤⠀⡤⠠⠄⣄⡀⢴⡶⠶⢂⣰⣶⠀⠀⣀⣀⣀⣠⣤⣄⡀⠀⠀⠀⠀⢤⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⢸⣿⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣾⣧⣬⣴⣶⣿⣿⣿⣿⣹⠀⣦⣍⣯⠉⠉⠉⢹⣿⡶⠒⣦⡤⣬⠀⢸⣿⣿⣿⣭⣽⣿⣿⣿⠟⠋⢀⣏⣈
⠀⠈⠉⣾⡄⢿⣿⣿⣿⣿⢿⣿⣿⠀⣿⣿⣿⣿⢫⣴⣿⣟⣛⢿⣿⠀⣿⣿⠉⣴⣶⣶⣾⣿⣤⣤⣿⣇⣿⠀⢸⡍⣏⠀⣿⣦⣄⣠⣤⣤⣴⣿⣿⣿
⠀⢀⠐⠒⣿⢿⡧⢴⠥⢶⣾⣋⣩⡄⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡄⣿⣿⣤⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠀⠘⣧⣿⣤⣿⣿⣿⣏⠉⠉⠉⠉⠉⣿
⠀⠈⠛⠙⠁⠐⠘⠛⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⣿⣛⣻⣿⣿⣻⣛⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣷⣿⣿⣉⡿⣷⣟⢹⣛⠿⣾⣿⣻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣬⢿⣯⣟⢻⡹⣿⢷⣻⡿⣽⣿⣻⣾⢛⣯⣯⣙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣹⡟⣻⠟⣭⢯⣷⡿⣿⣿⣩⣷⣿⣿⣿⣿⣿⣵⢿⣜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣼⣿⣛⣿⣟⣟⣿⣶⣿⢽⣿⢿⣾⣏⣥⣽⣿⣿⣭⣯⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣥⣭⣿⠿⣿⣿⡞⢿⢿⣿⣻⡿⢿⣽⣿⣿⣟⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣯⣿⣾⣿⣶⣿⣟⣿⣿⣶⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⣠⡴⠶⠶⠶⠶⣤⡀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢠⢀⠈⡀⠠⠂
⠀⠀⠀⠀⠀⠀⢠⣤⠀⣷⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠢⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠈⠠⠨⢠⠁⠠⠁⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠐⠠⠀⠐⠁⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡘⠀⠈⡀⢠⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⢼⣶⣶⣶⣾⡆⢠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡆⠈⠁⢀⡀⠈⠀⣼⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⢠⣿⡅⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⢀⠀⠀⢰⣿⣿⠟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡄⠀⢸⣿⡇⠀⢀⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠈⠁⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⢸⣿⡇⠀⣼⣿⣏
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡆⠘⠃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⢸⣿⡇⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡰⢶⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡸⣿⡇⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡧⢿⣿⣿⣿⣿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⠿⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⠶⠛⠛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣟⣻⣍⣷⡶⢶⡶⠶⠦⣤⡶⠶⠶⣄⡀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣧⡘⠻⣾⠛⠛⠟⠻⠷⣤⡀⠈⢙⣦⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⢼⠇⠈⠛⢶⣄⡙⢶⣄⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⡇⠀⠀⠀⠀⠉⠻⢦⣌⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣉⠳⣦⡀⠀⠀⠀⠀⣀⣀⠠⠤⠤⣘⣛⣻⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠶⢾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣌⡙⠲⠖⠶⢂⣉⣭⡶⠶⠾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢻⡏⠱⠀⢳⡜⠻⣯⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⢻⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⢸⣗⠀⢠⣈⠇⠀⠙⠷⣮⡙⠶⢦⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠹⢧⣤⣴⣧⠀⠀⠀⠈⠙⠷⣦⣭⣤⣔⣒⢒⣷⣦⣬⣽⠟⠉⠹⣧⡄
⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⡇
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⣿
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣿⡇⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠻⢦⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣿⠁
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⣤⣀⣀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠉⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⡛⠷⣶⣤⣤⡀⠀⠀⠀⠀⠀⢠⣤⡶⠛⠛⣉⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢈⡷⣦⡀⢀⡿⠃⠀⠀⠀⠀⠀⠘⣿⡀⣠⣾⡏⠉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⢸⠀⠈⠛⢿⣷⠾⣷⣴⣶⣤⣤⣶⣼⡿⠋⠀⣅⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡄⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⠀⠈⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⡶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡶⠛⠛⠉⠉⠀⠀⠉⠛⠛⠶⣤⣴⠶⠛⠛⠉⠀⠀⠀⠉⠛⠛⢶⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⢀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⢀⡽⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⢿⡀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠘⣷⡆⠀⢀⣀⣀⣤⣠⣤⠒⠂⣠⡀⠈⡁⢀⣄⡐⢲⣤⣤⣤⣄⣀⡀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠈⠛⠺⣶⡖⠿⠲⠷⠶⠾⠿⠶⠖⠒⠲⠿⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠿⠖⠒⠾⠶⠶⠶⠶⠶⠶⠶⢲⣾⠖⠛
⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡿⠛⠻⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠟⠛⣿⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠦⠛⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠣⠴⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣤⣴⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢂⣤⣤⣤⡾⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣤⣤⣤⣤⣶⣤⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠟⠿⠿⢿⣶⣄⠀⠀⠀⠀⠀⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⡅⠀⢨⠁⠀⠀⠂⡌⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣈⠀⠀⠀⠀⠀⠄⠠⠈⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣶⣦⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣀⣠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷


⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣿

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠠⠔⠂⠨⠉⠁⠉⠁⡐⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢔⣲⠄⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠒⠁⠀⠨⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣠⠀⠀⢀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡀⢰⠛⢇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠤⣀⡀⠀⠀⠀⠀⠫⡔⡊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢤⠄⠀⠀⠀⠀⠀⠰⠔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠼⠇⠉⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠴⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠸⣮⣤⠤⢤⣴⣤⣤⣶⡤⠤⣤⣤⡇⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣄⠀⢠⡀⠀⣄⠀⣠⡀⠀⣄⠀⣠⣀⣄⠀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢳⡄⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡈⠉⠳⣾⠛⠉⠀⢠⡤⠶⠤⢤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠙⠷⡀⠀⠈⠳⣄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⣸⠃⠀⠀⠀⠈⡠⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⢀⡟⠀⠀⠠⠀⡐⠀⠀⠀⠀⡀⣽⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠻⣔⣾⠐⠀⠀⠄⠀⠀⠀⠀⢰⣿⠘⠀⠀⠀⠀⠀⠀⣦⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢡⠻⣿⠟⢛⠛⠿⢶⡆⠀⠀⠀⡾⠉⠐⠶⡀⠀⢀⡀⠀⠉⠉⠉⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡾⠚⠉⢖⡉⠡⣠⡜⡇⠀⠀⣸⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢤⠈⡆⢈⡾⠁⠀⠰⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⣁⡅⢸⡇⠀⢰⣶⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠾⠾⡄⠀⠀⠴⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣤⣭⡵⠿⠅⢀⣼⣟⠀⠀⠀⢳⡄⡀⠀⠀⢀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠘⠀⡠⢤⡅⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠻⢯⡁⠀⠀⠘⠛⠁⠀⢸⠇⠀⢹⠟⠒⠚⠛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⣼⣇⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠐⣶⠀⢀⣴⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⣇⢧⢼⠶⠖⠋
⠀⠀⠀⠀⣰⡖⠉⠁⠉⠹⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⢸⠌⣧⣳⡾⡆⠀⠀⠀⠀⣰⡀⣀⡄⠀⠀⠀⠿⢿⡋⠉⡍
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⣿⣷⣇⠀⠀⠀⠐⠋⠉⠋⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⢸⠃
⠀⠀⠀⢀⣟⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣄⣼⣳⣴⠚⠉⠡⢼⣿⣿⣿⢽⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⢸⠀⠀⠿
⠀⠀⠀⠀⠉⠛⠛⠋⠉⠙⠒⠶⠶⠶⠖⠒⠒⠋⣽⣿⡓⠒⠒⣀⣈⣉⣉⣟⠾⠋⠀⠉⠓⠢⢀⢀⣤⣤⠴⠖⠛⢲⠦⢼⠒⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠦⣄⡀⠀⠀⠉⠉⢉⣠⠼⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠒⠒⠋⠉⠢⡀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⢦⣤⣄⡀⠀⠀⠀⠀⠀⣴⢶⣶⢶⡶⣶⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠉⠦⠀⠀⠰⠈⢻⣦⠀⠀⠀⠀⠿⠾⠷⠾⠷⠿⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠓⠒⠒⠒⠒⠒⠛⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡇⣠⠶⣆⢠⡄⢠⡶⣦⠀⣿⢻⡇⠀⠀⠀⠀⠘⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢾⡇⠘⠛⠋⠘⠃⠈⠛⠛⠀⣿⠶⠇⠀⣀⣀⠀⠀⠀⣘⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟⠛⠀⠃⠘⠃⠘⠻⣦⡟⠀⣴⠟⠉⠉⠻⣶⠟⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣿⠛⠉⢻⣇⣠⠟⠁⠐⣇⠀⠀⣰⠞⣦⠀⡆⠀⠀⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠛⠶⠶⠟⢻⡃⢀⠀⠀⠙⢦⡀⠿⠛⠿⠀⠇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠻⣦⣤⣤⣤⡟⣉⡉⠉⠛⢦⡀⠙⣦⡀⠀⣠⡾⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢠⣤⣤⣤⣤⡤⢾⣯⡿⠶⠶⢾⡷⠾⠋⠙⣾⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣦⡸⠷⠶⠶⠶⠶⠶⠏⣰⡟⠛⠛⠃⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠃⠈⠻⣶⠶⢶⡶⠶⢶⣾⠋⠛⠶⠤⠤⠤⠤⣾⣛⡧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣤⣤⡴⠿⠀⣼⢷⡄⠀⠉⠻⢦⣀⣀⣀⣀⣀⣈⣉⣁⣀⣴⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⡀⢀⡀⠀⢀⡀⠀⠀⢀⣀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣁⣉⣉⢀⣁⢉⠉⠈⣈⠁⢉⢉⣁⡈⡉⡈⡀⠁⢈⡁⣀⠈⢀⢁⣉⡁⠁⡀⡁⠉⣉⡀⡈⢈⢉⠁⢈⢉⣈⢀⣁⡀
⠀⠀⠀⠀⠀⠈⡁⡈⠀⠈⢉⠀⢀⢉⠀⡈⠈⡀⢁⠀⠁⢉⠀⢈⢀⠈⠀⠉⠈⠁⠀⠀⠁⡁⡀⡀⠈⢀⠈⡈⡀⢈⠈⠉⠀⠀⡁
⠀⠀⠀⠀⠀⡉⡁⡁⡀⡈⢀⡀⡈⢈⡈⠀⡀⠉⠀⠀⠀⠀⠀⠈⠈⠉⢁⢉⢀⡀⡉⠀⠀⠀⠉⡁⠀⠈⠉⡁⠁⡉⠉⣁⠀⡉⡀
⠀⠀⠀⠀⠀⣁⢀⣁⠀⠀⡁⠁⡁⡁⢀⡀⡈⡁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠁⠁
⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠀⠉⠁⠀⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⢀⡀⢀⡀⣤⡿⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠛⢀⣄⠈⢹⣼⠷⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣁⠈⠋⢀⣸⠿⠙⢲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣻⣿⣶⣷⣾⣿⡀⠛⢻⡯⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠒⠀⠀⠀⠀⠀⠀⠠⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠋⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠄⠈⣛⣉⡀⢰⠀⠀⠀⠀⠀⠀⠀⣉⠛⢤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⢥⡄⠊⠁⠀⠩⢿⣆⠀⠀⠐⠢⣤⠋⠉⠁⠀⠳
⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⢸⣦⠄⣀⣀⡴⠂⠈⠁⠀⠀⠀⠈⡇⠘⢆⠀⠀⠀⢀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠈⠋⠉⣉⠛⠉⠙⢦⢀⣀⣀⠀⠃⣠⠞⠉⢓⠿⢥⣀
⠀⠀⠀⠀⠀⠀⠢⣄⣀⣤⠞⠁⠠⠤⠤⣌⡷⠃⠀⠈⠋⠀⠈⠱⠀⣇⠀⠀⠀⣀⠀⠈⠛
⠀⠀⠀⠀⢢⣄⣀⠀⠈⠧⣀⡀⢀⡀⠀⠈⢣⠀⡇⠀⠀⡼⠀⢀⡤⠞⠓⢠⠎⠀⠀⢀⠖⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠀⠀⠳⣄⠀⠀⢸⡄⢳⡀⠀⢸⡀⡞⠀⠀⠀⢺⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠈⢳⣄⠀⠸⡆⠀⠉⠃⠈⣳⠀⠀⠡⠇⠀⠀⠐⠚⠧⠀⠀⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣄⣸⡀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠦⠤⢸⡀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠉⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠦⠴⢞⣿⣆⠀⠀⠀⠀⠀⠀⣰⣾⡳⠦⠴⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⠽⠊⠀⠀⠀⠀⠀⠀⠀⠀⠓⠯⣟⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣦⣤⣤⣀⣀⡀⠀⣠⣤⣼⣧⡾⣿⣦
⠀⠀⠀⣤⣾⣿⠿⠛⠛⠉⠛⠄⢹⡿⠛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⣿⣿⣿⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣾⣿⣟⠹⠄⠀⠀⠀⠄⠀⠈⡠⠀⡦⢰⢨⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⢛⠉⠋⠁⠈⠏⠈⠁⠀⠁⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣾⣿⣏⢭⡶⠀⠀⠀⠳⢀⠀⠀⠁⠆⠁⠛⠀⢋⣀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡈⠃⠻⠄⠡⠐⠀⢀⠀⡄⠀⠀⠈⠨⢿⣿⣿⣿⣿⣿⣿⡏
⢸⣿⣿⡷⣿⣥⣄⠀⠀⠁⠀⢀⠀⠀⠀⠛⠶⠿⠾⢟⣻⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣻⠷⡶⠦⠶⠀⠠⠀⠀⠀⠄⠀⠀⠨⣿⣦⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣯⢸⣿⣿⢛⠛⣡⠀⠁⠐⠈⠀⠀⢧⣢⣧⣴⣴⣿⣿⡷⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣼⣂⡶⠀⠀⠨⠁⠂⠀⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⢻⣿⣿⣿⡿⣷⣶⡿⣛⠀⠀⠀⡀⠀⣀⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣸⣷⣄⣽⣿⣧⣿⣿⣿⠏⠁
⠀⣿⣿⣿⣿⠛⡋⢁⢃⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡌⠛⣻⢿⣾⣿⣿⣿⠋
⠀⠈⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣾⣿⣿⡿⠋
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠋
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡋⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣁⠂⠉⠹⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢀⠀⢀⠀⠜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠈⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠆⠀⠒⡢⠋⠢⣀⣀⡀⠤⠄⣺⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣄⠀⠀⠑⠤⣀⢸⠁⠀⡤⢾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠸⢧⡀⠀⠀⠀⢸⠀⠀⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠒⠂⠈⢳⡄⠀⠀⢸⠀⠀⠈⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠔⢛⡢⣄⢨⠀⠀⠀⠸
⠀⠀⢀⠀⠠⣤⣤⠤⠤⠤⠤⢀⣇⡀⠀⠀⢠⠠⢐⣸⣯⠬⠆⣀⣀⡜⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⡌⠴⢉⣀⣀⡀⠠⠤⠤⠀⢈⣓⣌⢀⣀⡈⠉⠁⠈⠠⠤⠤⠍⢢⡄⠀⠀⠀⠀⣀⡀⠀⠈⠑⢄⡀
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣔⠦⠝⠂⠁⠀⠀⠀⠀⠈⠢⢀
⠀⣌⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠐⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣦⡀⠀⠘⢷⡀⠀⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⡿⢰⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣭⠧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⡍⠘⠀⣿⣿⣿⣿⢟⣥⣿⣿⣅⠈⠀⠀⠀⣧
⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠘⣿⣿⣟⣻⣁⠘⠃⠐⢺⡀⠉⠀⠀⠈⠿⣿⣿⡿⠓⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠹⢦⠄⠀⠀⠀⠈⢻⣿⣿⣷⣄⣀⣀⣼⡷⠀⠀⠀⠀⠀⢈⠻⡏⠀⢈⣧⡀⣼⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠻⣿⠂⠀⢘⣿⣏⠉⠀⠀⢦⣿⣿⣷⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢀⣄⣀⠂⠀⠀⠟⠀⢰⣿⠏⠀⠈⠁⠀⣸⣿⣿⣿⡿⡼⠀⠀⠀⠀⣀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡙⠀⠈⢻⢶⣤⣀⣠⢼⡏⠀⢀⡀⢀⣼⣿⣿⣿⡟⣴⣤⠇⠀⣀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠶⣄⠀⠀⠈⠟⡉⠛⠁⠘⠋⠀⠤⠠⣿⣿⢟⣼⣿⡏⢀⡀⠛⣕⠲⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣠⣸⣧⣄⣤⣆⣀⣠⣴⣿⣯⣾⣿⣿⣮⣥⣿⣶⣿⡿⠞⠛⠿⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣉⠙⢿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠠⠭⠷⠄⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⣿⠛⣛⣛⡛⠟⢻⠟⠉⠛⡿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⣩⣶⣿⠗⠋⠀⠀⠀⠀⠈⠋⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠐⠦⠀⡀⠀⢰⣷⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡴⠖⠲⠲⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣷⣾⣷⠖⠺⠿⢻⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠙⠛⠀⣀⣀⣤⣤⣤⣤⣀⡀⠀⠛⠃⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⡀⠀⠀⢀⣴⣾⡿⠟⠛⠛⠛⠛⠻⢿⣷⣦⡀⠀⠀⢀⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⣴⣿⠟⠁⠀⢀⣀⠀⠀⣀⡀⠀⠈⠻⣿⣦⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⣴⠋⠈⢹⡏⠁⠙⣆⡀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⢰⣿⡇⠀⢀⡯⠀⠁⠀⢸⡇⠀⠈⠀⢿⡀⠀⢹⣿⡄⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁⢸⣿⡇⠀⠸⣄⠀⠀⢶⣄⣠⡶⠀⠀⣨⠇⠀⢸⣿⡇⠈⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠫⣠⠆⠀⡏⢹⠀⠰⣄⠝⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠈⠢⠄⠇⠺⠠⠜⠁⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣶⣶⣶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠙⠒⠦⣀
⠀⠀⠀⡴⠋⢀⠴⠚⠉⢉⣩⠙⠢⡄⠈⢣
⠀⠀⡼⠀⣰⠃⠀⠀⢸⡉⠉⠂⠀⠘⡆⠀⣇
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣚⣋⣉⣉⣉⣛⡒⠲⢤⡀
⠀⠀⢣⠀⠹⣄⠀⢳⠖⠚⠁⠀⣠⠎⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⡞⠭⣴⣀⣠⠖⠛⢳⠾⢙⡆⠘
⠀⠀⠈⢷⣄⡈⠙⠒⠲⠖⠒⠋⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣈⡙⠒⠒⠒⠒⠒⠒⠚⣉⣤⣾
⠀⠀⠀⠀⠙⠿⣷⣶⣶⣶⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⠤⠤⠴⠶⠶⠤⠤⢤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⠔⣒⡏⠉⣩⠭⠿⣗⣲⢤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡈⠺⠭⢿⣟⣁⣀⣤⠥⠴⠚⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣀⣀⠉⠙⠿⣿⣿⣿
⠀⢀⣠⠤⠖⠒⣒⣒⡒⠒⠦⠤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡴⠋⠁⠀⠀⠉⠲⣄⠈⢿⣿
⡔⠁⡔⣊⣩⠉⢉⡽⠛⢯⡭⠖⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⣠⠄⠀⢀⣠⣀⠈⢆⠈⢿
⣷⣄⡙⠒⠮⠭⠭⠤⠤⠤⠖⠚⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠈⠹⣄⣠⠎⠀⣸⠶⢸⠀⢸
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠣⡀⠀⠀⠀⠀⠁⢀⡞⠀⡸
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠉⠛⠋⠉⠉⠉⠀⠘⢿⣷⣄⠉⠒⠤⠤⠤⠔⠋⠀⡴⠃
⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⣛⣛⠿⠿⠛⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⣾⡿⠀⣴⣶⣶⣶⣶⣾⣿⣿⣶⡄
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⢿⡉⠄⠑⡎⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣂⠈⠩⣶⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣤⠆⠙⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠙⠛⢿⣿⣯
⠀⠀⠈⠉⠉⠉⠙⠛⠛⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣘⠓⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣰⣿⡟⢀⣴⣦⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠸⠛⣿⣿⣿⠉⠁⠒⠚⠋⠉⠁⠀⠒⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠂⠀⠀⣀⠛⠙⠿⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⣀⠉⢻⣿⠃
⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠰⠶⢾⣦⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⣦⣤⣢⠿⠆⠀⠀⠀⠀⠀⠀⡀⠀⢀⠟
⠀⠀⠀⠀⠀⣀⡿⠀⠀⠈⠋⢀⡔⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠠⡂⠀⣀⠀⠀⡀⠀⠀⢴⠀⢶⡶⢏⣤⣴⠏
⠀⠀⠀⠀⠈⠋⠛⠀⠀⠰⣦⡈⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡜⣀⠺⠿⠀⢸⣿⡊⣀⠀⠶⠁⠀⠀⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⡄⠹⠀⠀⠛⠀⠀⠂⠀⠀⠄⠀⠈⠁⠀⠀⠀⠛⠀⠁⠈⠐⠂⠀⢙⣷⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⡀⢸⠿⠿⡏⠉⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣽⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢱⠌⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣾⣦⡀⠀⠀⠀⠀⣠⡍⢦⣄⣀⣠⣶⠉⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣻⠟⠋⠈⠙⠿⣷⣦⡀⡀⠀⠀⠀⠀⠀⠀⢶⣶⡾⠁⠈⢷⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣾⡙⡷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣯⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⣤⣠⣀⠀⠀⠠⠍⢀⡀⠀⠀⠙⣷⣄⣠⢴⡴⣄⢸⣿⣿⣍⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣷⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠋⡰⣶⣦⡀⢹⠀⠀⢠⠈⣩⢶⠶⠟⢻⣿⠓⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣒⣠⣶⣿⡿⠷⣶⣆⠀⠀⠀⢀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠁⣤⠀⠀⣀⣤⠄⠀⠠⣿⣏⣡⡿⠟⠛⠉⠀⠀⠈⢻⡀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢠⡾⠟⠁⠀⠀⠒⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠤⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠆⠀⠀⠉⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠉⠻⣦⠀⣸⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠘⣧⡿⠀⠀⠀⠀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠉⠙⠛⠿⣿⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⢀⣀⣠⣾⣿⣏⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠘⠛⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠘⠛⠻⣿⣿⡟⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⣀⣀⣀⣀⣤⣶⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁


⣿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠈⠉⠙⠛⠒⠦⣤⣤⣤⣤⣤⣤⡤⠔⠒⠋⠉⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠎⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢤⣤⣤⣤⣤⣤⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣀⡰⠶⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠶⢆⣀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠀⠀⠀⠀⢰⡏
⠀⠀⠀⢰⡆⠀⠀⠀⣴⠋⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⡀⠈⠳⣄⠀⠀⠀⡗
⠀⠀⠀⢸⡇⠀⠀⠘⠃⠀⣀⣠⣁⣀⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣁⣀⡀⠙⢃⠀⠀⢱
⠀⠀⢀⣼⡇⠀⠀⣼⣿⠟⠉⠉⠁⠘⠟⠉⠹⣏⠙⠻⣦⣤⣤⣤⡾⠋⢉⡟⠉⠙⣧⠀⠉⠉⠻⢷⣿⡆⠀⢸⣦⣀
⠀⡸⠋⠈⢁⠀⢠⣿⡇⠀⠀⠀⠀⡿⠀⣀⠀⢹⠀⠀⢹⣿⢿⣿⡅⠀⢨⠃⢀⡀⠸⠀⠀⠀⠀⠈⣿⡇⠀⠸⠃⠈⢿
⠀⣷⠀⠀⢹⠂⠀⢿⣇⠀⠀⠀⠀⣿⣼⣿⣷⣿⠀⢀⣼⠃⠀⣿⡇⠀⢀⣼⣿⣿⣦⡄⠀⠀⠀⢀⣿⠇⠀⣤
⠀⢇⠀⠀⠘⣦⠔⠾⣿⡄⠀⠀⠀⢻⣿⣿⣿⣿⠀⣼⠟⠀⠀⠘⣷⡀⢸⣿⣿⣿⣿⠃⠀⠀⢀⣾⣿⠀⢀⡏⠀⠀⢠
⠀⢸⡆⠀⠀⠀⠀⠀⡋⠙⢷⣤⡄⣀⣿⣿⣿⣷⠟⠋⠀⠀⠀⠀⠈⠛⢶⣿⣿⣿⡏⣀⣤⣴⠟⠁⢽⡀⡾⠀⠀⢀⡼
⠀⠀⠘⢦⣀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣾⡏⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠉⠑⠒⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡶⠠⠶⠤⠄⠐⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⢶⡆⠀⠀⠀⠀⠀⣿⠶⢦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠃⣸⠟⠋⠁⠀⠀⡄⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠻⣁⠀⠘⠛⠷⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣶⠃⠀⠀⠀⠀⠀⢷⣀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠻⣄
⠀⠀⠀⢀⣴⠉⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⢴⣷⠀⠀⠀⠀⠀⠙⢧⡄
⠀⠀⢠⣞⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡟⠛⠉⣁⣀⣠⣤⣤⣬⢋⡀⠀⠀⠀⠀⠀⢈⣵⡄
⠀⣴⠋⠉⠳⢆⡀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⡼⣧⣄⡀⠀⢀⡴⠏⠉⢷
⣦⠁⠀⠀⠀⠀⣹⣷⠔⠛⠉⣱⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠿⠿⠛⠛⠃⢸⣋⢰⡿⠏
⣿⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠙⠿⣿⣷⡀⠀⠀⠀⢀⣠⠤⠞⠛⠁⠀⠀⠀⠀⠀⢀
⠋⣄⠀⠀⠀⠈⢷⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠆⠀⠀⠀⠙⠳⠶⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠙⢦⡀⠀⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠞⠂⠉
⠀⠀⠈⢳⣄⠀⠀⠈⢳⡄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠛⠛⠋⠀⠀⠉⣤
⠀⠀⠀⠀⠙⠷⣄⠀⠀⢹⠆⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⣀⣤⣾⣿⡄⠀⠀⠀⠀⣀⡠⠶⠋
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠺⠟⠁⠀⠈⠛⠒⠛⣦⣀⠀⣀⣵⣶⣿⣿⣿⣿⣿⣿⣷⣤⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢠⡾⠛⠛⠛⠋⠉⠁⠀⠀⠀⠹⣿⣿⣿⡿⠟⠋⠈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠖⠶⠖⠶⠶⣾⠃⠀⠀⠀⠿⠷⠶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⡛⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⢛⠃⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠄⠶⠛⣷⠀⠀⠀⠀⠀⠀⣠⠿⠶⠶⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠋⠁⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠘⠛⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣋⣁⣀⣀⣀⣀⣀⡀⠀⠠⣴⣿⠀⠀⠀⠀⢺⣦⠄⠀⠀⣀⣀⣀⣀⣀⣀⣉⣣⣄
⠀⠀⠀⠀⠀⢀⣿⣍⣭⣍⣩⣭⣉⣉⣉⣻⠷⠖⠊⠁⠀⠀⠀⠀⠀⠉⠲⠿⢛⣋⣉⣩⣭⣭⣭⣭⣭⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⠒⠒⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⢀⢤⡔⠂⠤⣄⠀⠀⠀⠉⠒⢤⡀⠀⠀⠳⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡼⣁⣀⠼⠂⠀⡤⣱⡀⠀⠀⠀⠀⠈⠲⡄⠀⠀⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢀⡼⠤⠤⠥⠤⠤⠯⠤⠬⠧⡄⠀⠀⠀⠀⠀⠈⢦⣸⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠳⡙⠛⠉
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠉⡇⠀⡔⠉⠲⣤⠶⠶⢆⡔⠋⠲⡄⢸⠉⡇⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠒⡇⠀⠳⠤⠔⢹⠀⠀⠨⠳⠤⠴⠁⢸⠖⠃⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡹⢯⠀⠀⡨⠯⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣑⣒⣒⣒⣒⣊⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢠⠇⣀⣸⠀⠀⠀⢀⢀⡀⣀⡀⠀⠀⢸⣀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⡞⡸⠁⢹⠀⠀⠀⣮⢧⢷⡻⠀⠀⠀⡸⠀⢣⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⡏⠀⡇⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠒⠁⠀⠸⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢸⠀⠀⠀⠀⠀⢠⢏⡙⡄⠀⠀⠋⢹⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠳⡌⠙⠒⢄⣎⡎⣱⢱⡀⠀⣤⣼⠀⢀⡔⠋⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠟⠀⠀⡼⣨⣭⣥⡀⢧⠀⣿⣿⠀⠸⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠸⠷⠁⠀⠀⠘⠾⠇⠳⠟⠀⠀⠀⠀⠠⠚⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡄⠀⢀⡠⠀⣀⣴⠀⡀⠀⠀⠀⠀⠀⠀⡦⣤⣴⣆⣤⡤⣤⣄⣀⡄
⠀⠀⠀⠠⣐⢀⠆⠀⠀⢀⠀⠐⢀⣤⠶⠛⠋⠀⣀⣠⣷⣶⣿⣖⢾⡯⠽⠆⢳⠀⠀⠀⠀⠀⠘⢿⣾⣿⢿⣿⣿⣼⣿⣿⣷⣞⠃
⠀⠀⠀⠀⠁⠀⠄⢁⠔⠈⠠⣴⠟⢁⠨⣴⣶⣿⣿⣿⣿⣿⣿⡟⣹⠁⣨⠯⣵⣄⡀⡀⠀⠀⢀⣿⣿⡿⠚⠉⠉
⠀⠀⠀⠀⠐⠀⠀⠸⠀⠀⢂⣷⡀⣹⡞⢹⡿⠟⠋⠉⠉⢭⡧⣿⢾⣿⢿⡿⣟⣻⣷⡈⢢⡀⢨⡁⢸⡇⠀⠀⢀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣀⣀⣀⢤⡆⠈⣿⢻⣷⠏⠂⠀⠀⠀⠀⠈⣷⣧⣿⣿⠟⠀⠀⠱⣈⣿⣤⣱⡘⣧⣸⡇⠀⠀⠈⠻⢃⡀
⠀⠀⠀⠀⠛⡳⣯⡉⢿⣷⡀⠐⠉⢈⣿⣴⣷⠀⠀⠀⠀⠀⡉⣿⣿⣿⣷⡀⠀⢀⣿⣿⣿⣿⣿⣇⡉⢄⠀⠀⠀⠀⠈⠀⡀⠀⡠⠀⠀⠐⢉
⠀⠀⠀⠋⠃⠀⠋⠛⡄⠉⠀⠀⠀⠘⠵⡟⠀⠀⠀⠀⠀⠘⡗⠘⣿⣿⣿⣿⣀⠈⣹⣿⣿⣿⣿⣿⣿⡈⠂⠀⠀⠀⠀⠀⠘⢻⣤⣤⣔⣦⣀
⠀⠀⠀⠈⢢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡃⠀⠐⠂⠠⢠⡖⠙⣿⣟⣿⣿⣟⣲
⠀⠀⠀⠀⠀⢀⣀⣤⣵⣦⣶⣶⣶⢠⣿⠇⣠⣤⣴⣤⠀⠀⠀⠈⠀⠀⠈⠙⠿⡛⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠾⣥⡄⠀⡀⢘⣫⠽⠟⠛⠥⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣼⡁⣨⣟⡛⠉⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⠢⠀⠐⢛⢿⡟⣤⠉⠀⠔⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠭⠽⠿⢿⠁⣽⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣳⡇⠀⠈⢠⡷⠛
⠀⠀⠀⠀⠠⢁⢀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⠘⠲⢀⣡⣔⠒⠈⣽⣿⣇⠀⠀⠀⠀⢺⠉⣅
⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡹⣿⣶⢁⠰⡾⣿⣶⣷⡤⠀⢠⠀⢉⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠂⢀⡀⠀⠱⠀⠀⠹⣼⣃⡀⠒⠊⠉⠉⠉⠁⠃⠀⠈⡷⠾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢸⡗⠀⠀⠢⢀⡀⠺⠟⠿⠾⠿⣿⣶⣆⡢⠀⠀⠀⠆⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⠀⠒⢀⠎⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠔⢒⣒⣒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢖⣋⠥⠒⠂⠉⠉⠁⠀⠀⠀⠈⠡⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⣠⠏⣷⡏⠀⢠⣶⣦⣿⣿⠀⠀⠂⠀⠀⠀⠱⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠞⠫⠸⡆⡘⣿⡄⠈⠙⠻⠿⠁⠀⠀⠁⠀⠀⠀⣀⠱⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡜⠁⠀⠀⠆⢳⡱⠙⣞⡀⠀⢠⣤⣶⡖⠋⠀⠀⠀⠀⢹⣧⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠀⠀⠀⠀⢸⠳⡀⠹⡵⡀⠈⠉⠁⠀⠀⢀⡸⣷⡺⣧⢻⣧⠐⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠀⠀⠀⢸⠀⢳⡰⡹⡼⡄⠀⠀⠀⠀⠘⣿⡽⠷⠛⠃⠉⢁⡸⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣀⡤⠶⠒⠲⠾⡀⠀⢷⠁⠹⡔⡄⠀⠀⣀⡠⠤⠤⠐⠒⠈⠉⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢙⠀⠀⢏⡀⠀⠹⡀⠈⣧⠀⢻⡈⠉⠁⠀⠀⣠⠤⠶⠒⠚⠛⣗⠛⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡔⠸⡄⠀⠀⣀⠀⠀⠱⠀⠘⣆⢀⠳⣤⣤⠴⢾⡇⠀⠀⠀⠀⠀⡿⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠱⡀⠀⠈⠃⠀⠀⠱⡀⠘⢦⣑⣒⣖⣦⣿⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀⠀⠐⠲⡄⠀⠈⠢⡀⠀⠀⣸⢡⠇⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢆⠀⠀⠀⠀⠀⠀⠈⠑⢲⢣⠏⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣓⢦⣀⣀⣀⣒⣡⢖⣵⣋⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⠽⠶⠖⠛⠋⠉





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

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⢀⣄⢠⡀
⠀⠈⠻⢬⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢆⣀⠀⠀⠀⠀⣀⠀⢀⡀⠉⠃⠀⠀⠀⠀⠉⠀⠀⠀⢀⠄⠀⠀⣏⣩⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠢⣄⡄⢐⠀⠀⠀⠀⠀⠉⠀⠀⠀⡤⣰⡃⡿⠁⢠⣤⠆⠀⠀⠀⣠⣼⣲⣆⡀⠀⠁⠀⠈
⠀⠀⡠⠒⢰⠄⢨⣤⡾⠛⠃⠞⠃⢄⠀⠀⠀⠀⢀⠠⣀⠀⢽⠟⠁⠀⠀⠀⠀⡀⠔⠚⠉⠈⠉⠻⣷⢀⣦⠀⢰⡐⡄
⠀⢠⣠⠏⠈⠀⠘⠔⠀⠀⠀⢤⣄⡀⠈⠲⡄⠀⠀⠓⠂⣴⠿⢀⠀⠀⠀⡔⠀⢀⣠⣤⣀⠀⠀⠀⠀⠻⢦⠃⠈⠛⠃
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢠⢾⣿⠛⠀⠀⠈⠀⡀⠀⠈⠁⣨⠟⠁⠀⢈⣀⣾⣽⣋⣉⣁⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠻⢿⣿⠶⡀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⢻⣸⡓⠓⣚⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣌⣷⢿⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⣇⠀⠀⠛⠭⠭⠼⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠟⠀⠀⠀⠠⠃⢀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠈⠛⠲⠇⠀⠀⠀⠙⠢⠄⣀⣀⠤⠄⠀⠀⠤⠐⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠉⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛
⠀⠀⠀⣞⠱⠆⠀⠀⠻⣥⠀⠀⠀⡠⠀⠀⡞⠀⠘⠀⠀⠀⠀⠀⡁⠀⠀⢀⡀⠀⠀⢢⡄⠀⠀⣄⢄⣀⣀⡀
⠀⠀⠀⠙⢄⣀⣀⠀⠀⠀⠀⠀⠀⠁⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡇⠀⠀⠀⢿⠀⠀⢿⣆⠌⠿⠂⠋
⠀⠀⠀⠀⠀⠉⢹⠆⠀⠀⠀⠀⠀⡴⠋⠀⡀⠀⠀⠀⣠⡤⣰⡄⡇⠀⠀⠀⡇⠀⠀⠀⠘⠷⣄⣀⠘⠓⠂⠀⢰⣦
⠀⠀⠀⢸⣡⠄⠃⠀⢀⡤⠀⣠⠞⠀⠆⠀⠿⠀⡇⠀⠷⠖⠉⠀⡇⠀⠀⠀⡇⠀⠀⢠⡀⠀⠀⠉⠓⢦⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⣤⠀⠀⠀⡍⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠈⠃⠀⠀⠐⠀⠀⠉⠳⠶⡦
⠀⠀⠀⠀⠀⣤⠞⠁⠐⡆⠀⠀⣿⠀⠀⢠⡁⠀⡇⠀⠻⣙⡀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡆⠀⠂⠀⠀⢳⡄
⠀⠀⠀⢀⠔⠁⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⡇⠀⠆⢀⡆⠀⡄⢰⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠱
⠀⠀⠀⣉⠠⠤⠧⠤⠶⠧⠴⠷⠛⠒⠖⠾⠷⠖⠇⠀⠀⠀⠀⠀⠛⠛⠉⠋⠙⠛⠛⠛⠛⠒⠛⠒⠓⠂⠀⠄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⣤⠀⠀⠀⠀⠀⠚⠶⢤⡀⠀⠀⣀⠠⡄⠀⠘⣃⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢸⣷⠀⢸⣤⣴⣶⡆⠠⣤⢐⣏⠁⠀⠀⠋⣰⠃⠀⡰⠣⠁⠀⠀⠤⠤⠚⠃
⠀⠀⠀⠀⠀⣄⠳⠊⢁⡐⣛⠒⢕⣫⣾⢿⡏⠀⣼⠏⠉⠿⠀⠀⠀⠙⠉⠀⠀⠈⠉⠀⢀⡜⠁⠀⢀⠀⢦⠀⠀⠀⢀
⠀⢉⣈⣐⣒⡏⠀⠐⣟⣛⡂⠈⣿⠅⠀⠜⢉⣷⠶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⡄⠰⡏⠁⣸⠂⠀⠀⢠⡀
⠀⠘⢂⣀⠀⣿⣀⣀⡿⠖⢀⠀⠹⡄⠋⣀⠈⠛⠐⠉⠁⠀⣀⣈⠀⠀⠀⡀⠀⠀⡉⠁⠀⠀⠈⠂⠤⠴⠋⠀⣠⡎⠈⠁
⠀⠐⠊⠁⠀⡴⢿⣧⢿⡶⠀⠀⠑⢠⡎⠀⣀⠀⠐⢀⣠⡞⢹⢿⣐⢰⡻⠏⡙⠿⣆⠀⠀⠀⢌⠀⠀⢠⠴⢚⢧⠠⣄⠈
⠀⠀⠀⠲⡄⣀⡢⠥⠀⠀⢀⣚⠫⠓⠛⠉⠀⠀⠠⠄⠈⠫⠛⢲⣬⣿⣏⣈⣧⠴⠀⠀⠝⠓⠀⠀⠀⢐⠒⠀⠾⠀⠉
⠀⠀⠀⠀⠈⠉⠛⠒⠒⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⠶⢧⣿⠖⠖⡂⠢⠀⠀⠀⠀⠀⠀⠈
⣠⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⣛⣿⢉⡛⣛⡛⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠂⠈⠉⠛⠲⢤⡤⠖⠛⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠈⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⠛⠖⣃⣈⠳⠚⣶⠀⠀⠀⠐⠋⠀⠀⠀⠋⡷⠀⠀⠀⢸⠁⢈⡭⢤⡍⠙⢦⡀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠼⢣⡞⠉⠉⠹⡆⠻⣤⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⢀⡏⢀⣸⠀⠂⠽⢤⡀⢳⠀⠀⠙⡆
⠀⠀⠀⠀⢀⡔⠋⠀⠀⠘⢶⡜⣇⡀⠀⢠⠆⣴⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⡇⢾⠱⡄⢀⣀⣂⡇⣸⠀⠀⠰⣇
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠸⣥⠬⡍⢉⡤⢤⡽⠀⢀⣤⠴⠖⠒⠒⠒⠒⠦⢤⣀⠀⠹⣌⡛⠚⠋⠉⠉⠀⣧⠀⠀⠀⠘⡆
⠀⠀⠀⢸⠁⠀⠀⢀⣤⣄⠀⠀⠀⠙⠛⠁⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡄⠀⠉⠛⠛⠛⠙⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠸⡇⠀⠀⠀⣠⠟⠀⠀⢀⣀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠙⣆⠀⠀⠭⠀⠀⠀⠈⣸⠃⠀⣼⠀⠀⣠⠤⢤⣄⠀⠀⠀⠀⠀⠀⣠⣄⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢿⠀⡼⠁⠀⠀⠈⠘⠲⠦⠤⠴⠚⠁⠈⢧⣿⠀⠀⠀⠀⢈⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠘⠶⠤⣄⠀⠀⠀⠀⠀⠀⢠⠞⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣆⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣠⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣠⣀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠙⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢰⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢀⡴⢆⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⣠⠶⣄⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⢸⠉⠓⠲⠶⠖⠚⠉⢹⠀⠀⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠖⠒⢻⡙⣇⠀⠀⠀⠀⠀⠀⠀⢀⡟⣹⠓⠒⠶⠶⠄⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠳⢤⣀⣀⣀⣠⠴⢋⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢭⠥⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹



⣏⣹⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⢉

⠀⠀⠀⠀⠀⠀⠀⢸⡍⠀⡏⠉⡆⡆⠈⠁⢠⢳⠀⡯⢝⢠⠀⠀⡇⠀⠀⠪⣁⠈⡏⠀⣏⡕⢴⠉⢱⢐⣴⠁⣮
⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣧⣴⣥⣧⣤⣤⣬⣤⣥⣶⣼⣼⣦⣤⣧⣤⣤⣤⣬⣤⣧⣤⣥⣥⣬⣤⣮⣬⣤⣧⣷⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠠⠀⠀⠠⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⡀⠠⠋⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⠀⠀⠀⠈⠀⡀⠔⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠲⣆⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣰⣾⣿⡿⠀⠀⠐⠀⠂⠀⡿⠀⠀⠀⠀⣤⣈⣹⠂⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⢸
⠀⠀⢀⣾⡇⢹⠿⡇⠀⠀⠀⠀⠀⠐⠁⠀⠀⣠⠟⢟⣿⣿⢰⣾⣿⠟⠁⢀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⢨⡟⠙⠂⠀⠀⢸
⠀⠀⢸⣿⠃⠻⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣝⣽⣿⡟⢸⣟⣃⣤⣈⣀⣀⣀⣀⡤⢼⣿⣿⣿⢋⠁⠀⠛⠀⠀⠀⠀⢸
⠀⠀⠘⣿⣿⣟⣳⠢⠦⠠⡄⠀⠀⠀⠀⠀⠁⡆⣠⡭⠞⠁⠈⠀⠀⠀⠀⠀⠀⣹⣯⡇⠘⢿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠛⣼⠇⠀⠀⠉⠉⠁⠀⠀⠠⠀⠂⠀⠀⢸
⠀⠀⢠⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠔⠀⠀⠠⠶⠾⠤⠀⣐⠀⣼⠀⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣀⠴⠂⢠⠤⠄⣀⡤⠀⣠⠤⠤⠆⠠⠤⠤⠤⠄⠀⢀⡠⠀⢀⡤⠄⠤⢀⡄⢀⡠⠤⠤⠀⣀⡠⠄⠠⢀⣀⣀⣠⠆⠀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠀⠀⠳⠞⠃⠀⠉⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⣀⡤⣤⡀⠀⠀⢀⣤⢤⣀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⣼⠁⢀⡀⠙⡆⢰⠋⢀⡀⠈⣧⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⠖⠋⠉⠛⢿⡀⠀⡇⢸⠀⢀⡿⠛⠉⠙⠲⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢰⡇⢰⠋⠉⠀⠀⠁⠀⡇⢸⠀⠈⠀⠀⠉⠙⡆⢸⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣨⣧⡈⠁⢤⡀⠀⠀⠀⡇⢸⠀⠀⠀⢀⡤⠈⢁⣼⣅⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⣼⠁⣠⠉⠀⠀⠉⠉⠉⠙⡇⢸⠋⠉⠉⠉⠀⠀⠉⣄⠈⣧⠀⠈⠷⡼⠃
⠀⠀⠀⠀⢠⡞⢶⡀⠀⢷⠀⠳⣄⠀⠀⣀⣤⠀⠀⡇⢸⠀⠀⣤⣀⠀⠀⣠⠞⠀⡾⠀⠀⢀⡇
⠀⠀⠀⠀⠈⠓⠚⠁⠀⠈⢳⠦⠤⠤⠞⠳⡄⠀⢦⡇⢸⡴⠀⢠⠞⠳⠤⠤⠴⡞⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢻⡀⠀⢠⠴⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠦⡄⠀⢀⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠓⢦⡏⠀⠀⠀⡦⠀⡇⢸⠀⢴⠀⠀⠀⢹⡴⠚⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢷⡀⠛⠚⠁⣰⠃⠘⣆⠈⠓⠛⢀⡾⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠉⠛⠚⠋⠁⠀⠀⠈⠙⠓⠛⠉⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⡀⠀⢠⡴⢦⠀⠀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⠊⠉⠙⢢⡀⠀⠀⢠⠖⠈⠑⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡾⠘⠊⠉⠉⠀⠈⠉⠉⠒⠔⢢⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⠎⠀⣀⣀⣠⠖⠉⢷⠒⢒⣐⣐⡚⠳⣄⠀⠀⠀⠀⠀⢀⣘⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠈⠃⠰⡀⠀⠘
⠀⠀⠀⠀⢀⠇⠰⡏⠀⠉⡤⠦⣀⣸⠉⠀⠀⠁⠀⠙⢍⢆⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠈⡒⠠⢄⠱⡀
⠀⠀⠀⠀⢸⠀⠀⠙⢄⠜⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠆⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⢈⡡⠘⠛⠀⠀⡰⠁⡧⠔
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠑⠚⠒⢤⣀⠀⣀⠀⠀⠀⣔⠃⠀⠀⠀⠀⢸⠙⠉⠉⠉⠉⠁⠍⠀⠀⢸⠀⠑⠈⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡞⠀⠀⠒⠀⠀⠀⠉⣉⠈⡙⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠃⠀⠀⠂⠀⠀⠀⠠⠌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⣀⡠⠄⠀⠀⠀⠀⠉⡉⠜⠉
⠀⠀⠘⠣⡄⠀⠀⠀⣴⢧⣀⠀⠀⠐⠢⠠⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠌⣀⣀⡀⠀⢤⣔⡊
⠀⠀⠀⠀⠙⠚⢢⡼⠀⢀⠑⠭⠶⣦⣒⣂⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠳⠴⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⡄⠀⠘⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡆⠀⠀⠁⠰⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠘⠄⠀⢆⠀⠉⠂⠄⣗⠄⣀⠀⠑⠢⢄⣀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢐⠄⠀⠀⠀⠄⠈⠀⢀⠄
⠀⠀⢨⠀⠀⠀⠈⠁⠀⠑⠢⠤⢀⡀⠉⠙⣢⡰⢄⣠⠏⠑⠦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠀⠐⠉
⠀⠀⢸⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠈⠑⠲⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠲⠤⠄⢀⠠⠆
⠀⠀⠈⡄⠀⠀⢸⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠣⠀⠀⠜⠀⠀⠀⠐⠄⠀⣀⡠⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠋⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢂⠉⠄⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡀⠁⣀⡠⠭⠒⠂⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠐⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣩⠞⠋⢁⠀⢈⡴⠶⢤⡱⡌⠑⢦⡀⠐⠄⠤⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠶⣤⣞⣉⡁⠸⠀⠀⠀⠿⠂⠀⠀⣻⡀⠠⠒⠲⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠈⠉⠉⠙⠲⢎⡙⠙⠳⣦⡀⠀⠈⠀⠀⢰⡄⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⣀⠀⠀⠀⠹⡄⢀⣠⠄⠀⠀⠀⣿⠒⢦⠈⢿⠋⠉⠙⢲⡞⠁⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡷⠋⣠⠀⠀⠀⠀⠄⣀⠈⠁⠸⡇⠀⠀⠀⢁⣈⣦⠰⠧⣅⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠉⠳⡄⠀⢸⠃⠀⡇⠀⠀⠀⠀⣤⣌⠑⢦⠀⠁⠀⠐⠉⠉⡉⠙⢢⡀⠈⠳⡌⢂⠀⠀⠀⠡⠤⠄⡀⠀⠀⢀⠠⠓⠂
⠀⠀⠀⠀⠀⠀⠠⠈⢧⡀⠀⢠⡇⠀⠘⡄⠀⠙⠢⠀⠀⢁⡀⢰⠀⠈⡇⠀⡤⠶⠦⢄⠙⣆⠀⢳⠀⠴⢦⣔⠣⠰⠆⠀⠀⠀⠰⠀⠈⢀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠙⠢⠈⣧⠀⠀⠁⠀⠀⢀⣀⡴⠊⢀⠀⠐⢠⡇⣠⠤⠤⢤⣀⣠⣏⡀⠼⠒⠊⢁⡠⢴⠊⠃⢀⣠⣐⣥⣴⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⠲⡄⠈⠳⢄⡀⠀⠀⠀⢠⠀⠀⣾⠑⠄⠀⠈⠉⠀⠀⠀⠉⠀⠀⡤⠶⠶⠋⠐⠀⠈⢶⣿⣿⣿⡿⠿⠟⠋⢁⣠⣴⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⠙⠀⠀⠀⠠⠤⠤⣄⣹⣀⠘⠓⠄⡀⠀⠀⠀⣟⠀⠀⣿⡤⠀⡆⠄⠀⠋⠀⠀⠀⠀⠩⣁⠀⣤⣴⣶⣿⣿⣿⣿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠀⠀⠀⠀⠀⡨⠟⠪⠐⠀⡈⠢⡀⠀⠘⢢⠀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠘⢀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢀⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠣⠔⠈⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠄⢘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢫⢖⡀⠀⠀⠀⠀⠚⠐⣆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⠠⢀⠀⠈⠂⠀⠀⠘⠠⠤⠀⠓⠣⢏⠃⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⠃⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢐⠠⠂




⣶⣾⣿⣶⣶⡷⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣷⣿⣶⣿⢷⣶⣶⡿⠶⣿⣿⡿⠿⠷⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠁⠀⠀⢀
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠘⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⣀⣤⣾⣷⡄
⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⢹⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉
⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛
⠀⠀⠀⠀⠈⣿⡄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣄⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠘⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠛⠋⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣿⣻⣿⣿⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠛⡛⠉⠉⠛⠛⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⢡⣄⠀⢀⡓⠒⠄⡀⠸⠂⣀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠘⠄⠨⠔⠓⠓⢢⠄⠀⠅⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⣄⡐⠀⠠⡅⠀⠀⢸⠤⡄⠀⣀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠰⠆⠀⠸⢈⡁⡥⠀⢇⡀⠉⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣬⣥⣤⣄⡀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⠃⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷