White violet Emojis & Symbols

⠀⠖⠒⠂⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣰⡟⡿⠿⣧
⠀⠀⠀⠀⠀⢨⡛⢿⣧⣾⣏⠀⠀⢠⣾⣿⣿⣿⣦
⠀⠀⠀⠈⠋⠀⠈⠀⠙⣿⣿⠀⣀⣀⣸⠿⠿⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠶⠀⢶⣤⣄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠚⢻⢻⣿⣷⣄
⠀⠀⠉⠓⠀⠀⣀⣶⠤⠛⠶⢫⣦⣀⠀⠀⣀⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣤⣾⠿⣿⣷⣦⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣤⡔⣿⣀⣼⣾⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⢿⡀⠀⠀⠀⠀⣺⡗⣏⢿⣩⣯⣥⣤⣀⣀⣀⡀
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢚⣭⠁⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⢹⣿⠤⣿⣿⡌⡿⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠈⠲⣦⣤⣀⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣂⣴⣿⣿⣾⣿⣿⠧⢰⠄⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠑⡀⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠙⢦⡀⠀⠀⠀⠀⡏⢻⡝⠛⠿⣿⣿⣿⡄⠀⠀⠆⠀⠀⠀⢈⠿⠀⠀⠀⠀⠀⠀⠘⠛⠃⣦⡀⠙⠦⠙⠛⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢇⠀⠙⠶⣄⡘⣿⣟⠶⠟⠛⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣄⠀⠀⠀⠀⢀⠀⠀⠀⢢⡀⠀⠈⠳⣌⡙⠃⠀⠀⠀⠀⣠⠊
⢀⠀⠀⠀⠀⠀⣴⣆⠀⠀⡰⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠂⡀⠀⠈⠁⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⡀
⠸⠀⠀⠀⠀⢾⣿⣉⣿⡿⠉⠠⠂⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⣭⣶⡀
⢠⠀⠀⠀⠀⢈⣿⣿⠟⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡆⠀⠀⡴⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⣿⣿⣿⣇⣤⣀⡀
⢸⠀⠀⠀⠀⢪⣿⣄⣤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡀⠈⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢟⣯⣿⣿⣿⣿⣷⣦
⠈⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠲⢤⡀⢰⠀⢠⣯⣾⣁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⢻⣿⣿⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⢦⣿⠋⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⡴⣱⣿⣟⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⣼⠁⣿⣿⣿⣿⠟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⠇⠀⠘⠿⠟⠋⠀⠉⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡨⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⡄⠁⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠆⠐⠃⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⣨⣏⠯⠁⢀⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⡦⠄⠛⠚⠀⠀⠀⠾⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⢀⠐⡀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠾⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⣦⠀⣴⣶⣶⣷⣶⡔⣠⣼⡇⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣧⣤⣾⠏⣿⣃⣠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢃⣴⡟⣻⢿⠛⠙⠃⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣍⢃⣉⠙⣏⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣴⡤⠀⠀⠐⣄
⣻⣿⣿⣿⣿⡟⣿⢿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢼⣤⣿⢻⣿⣿⣿⣿⡟⢿⣿⢿⡆⠀⠀⠀⠁⠼
⣿⣻⡿⢋⠞⠛⣳⡜⢁⠀⣼⣿⠁⠤⢺⣿⣿⣿⣿⣿⣿⢸⠈⢡⣼⣽⠙⠋⠋⢱⣿⣿⠽⣇⣀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠠⠖⠛⠙⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠂⠀⠀⠀⠚⣿⡖⢤⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⡆⠘⠈⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠴⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⣠⠞⠉⠀⠈⢻⡄⠀⠀⠀⠐⢀⡾⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠀⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣠⡞⠁⠀⠀⠀⠀⠈⢧⠀⢀⠀⠀⠸⠀⢂⡀⠀⠀⡀⣀⠘⠁⠀⠀⠀⠠⠀⡚⠁⠀⠀⡀⠀⠐⠀⠀⢠⠀⠋
⠀⠀⠀⠊⠘⠂⠀⠘⠃⠮⣀⣀⡠⠒⠛⠀⠀⠀⠂⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡁⠀⠄⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠞⠋⠂⠛⣯⡃⠀⠀⠀⠀⠀⠀⠈⠆⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠟⠀⡎⠀⠀⢀⠁⠀⠀⡀⠀⠄⠀⠀⠄⢀⠎
⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠙⠉⠳⣦⣀⣀⠀⠀⠀⠰⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠠⠇⠀⠀⠈⠀⠀⠈⠃⠀⠂⠀⣌⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡇⠈⠃⠉⠉⠐⠂⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡘⠀⡼⠤⠐⠂⠈⠁⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⡶⠲⢆⣀⣘⡇⠀⠀⠀⠀⣤⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣴⠈⢻⡄⠀⠉⠻⣀⣤⠀⠀⠀⠈⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠀⠀⠉⠢⡱⢶⣬⠟⠓⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⡄⠀⠀⠀⡜⠻⠷⢤⡀⢀⠑⣀⡔⠃⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠀⠀⢰⠃⠀⠀⠀⠀⢉⡘⠁⠀⠀⠀⠀⠤⠤⠄⠀⠁⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠘⡆⢀⡛⠀⠀⠀⠀⡴⠋⠀⢀⡴⠃⠀⠖⠀⠀⠀⢀⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢳⣜⠃⠀⠀⢀⡰⠀⠀⠠⠾⠃⠒⠘⠣⢄⠀⠀⠉⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡟⠀⠀⠀⢮⠃⠈⠀⠁⠐⢂⡀⠀⠀⠈⢰⣄⠀⠀⠀⠄⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣾⠁⠀⠀⡼⠃⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⢩⣤⠀⠀⠀⠀⠂⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣼⠿⠀⢀⡜⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢄⠠⠄⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⡇⣸⠀⣬⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢰⠀⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⢋⣩⣻⠟⠻⠿⠿⠟⠛⠿⠛⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⡿⢿⡆⠛⠏⠸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠀⠀⠀⠃⣀⢠⡟⠉⢉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡀⠀⢀⣴⢠⣿⠈⠁⠀⠞⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣤⣽⡸⣆⠻⠰⢟⡀⠀⠘⠠⠁⣾⣿⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠙⠂⠉⢻⢻⣿⣦⡈⠀⣀⠀⡃⢀⣠⡷⠞⠋⢀⡼⠛⠋⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⠀⠀⠀⣠⣄⣾⠀⠉⢿⣦⣽⣷⣿⣿⠃⠀⣠⠄⢩⠀⠀⠐⠋⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⠓⢂⠀⠈⠛⢿⣿⣶⣰⣆⠹⣿⣿⣿⣿⣶⣶⡃⠐⠁⣀⣠⣤⣶⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣄⣠⣤⣽⣶⣾⣷⣬⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣢⡔⢛⡉⠀⠀⠉⢉⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⡿⣿⡟⢻⣿⣿⣿⠿⠷⠦⣀⠐⠛⠷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⣀⣤⡿⠛⠋⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣀⠀⠉⠁⢤⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡂⠀⢈⡡⠈⢀⡴⢿⡿⢻⣿⣷⣟⢻⣋⣻⣿⣿⣿⣯⠛⢿⣶⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠿⠿⠿⣾⡟⠀⠋⢀⡰⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡌⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⢀⠴⠋⠀⠀⠋⠈⠁⡸⠟⢫⠛⢿⠀⠀⠀⠀⡄⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡏⢀⡄⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠸⡄⠘⡆⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣾⣃⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⣦⠈⠀⠀⢠⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡧⡇⡀⡆⣤⡄⠀⣀⣶⣤⣶⣾⣿⣄⣠⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣷⣿⣷⣾⠟⠋⠉⠛⠛⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢿⠇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡅⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣿⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠀⠀⠀⠀⠀⢸⠀⢀⠄⣤⠘⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⡄⠁⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠃⠀⠀⠀⢿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⠈⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⢀⡖⠁⣀⡷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠰⡄⠀⠀⠀⠘⣿⣿⣿⡗⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠀⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠁⠀⠀⠀⠀⢻⣿⠻⣿⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠸⡷
⠀⠀⠙⠀⠒⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⢶⣤⠟⢣⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣿⣦⠀⢰⡄⠀⠀⠀⠖⠀⠀⠀⠀⣼⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⣿⠂⠀⠻⠀⠀⠀⠀⠀⠀⠀⡘⠋⠁⠀⠀⠀⠀⠀⠀⢠⠀⣾⠟
⠀⠀⠀⢀⣄⠀⣐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠐⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠐⠈⢻⡇⠀⠀⢦⡀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⡆
⠀⠀⠀⠈⠋⠀⠋⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠏
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⣾⡟⠀⠞
⠀⠀⠀⠀⢀⣤⡶⡦⠀⠀⣀⣠⣀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡄⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠛⢀⡘
⠀⣶⡶⠋⠈⢉⣄⠀⣶⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⡏
⠀⠻⠿⣦⣶⣿⠿⣿⡿⠿⠿⣿⣿⣿⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡑⢃⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⣸⠇⢀⡤
⠀⠀⠐⠂⢹⣿⣶⣏⢀⡀⠀⠀⣉⠙⢧⣿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠘⠿⠣⡿⢳⠿⠀⠀⢀⣴⣾⡿⠀⠀⣿⠀⣿⠂⣴⣤⠀⠔⠀⣴⡏⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⣇⡀⠈⢰⣾⣿⠀⣿⡇⢴⣶⣄⣸⣿⠀⣾⣇⣰⠦⠀⠀⣾⣿⣿⣿⡇⠀⠀⠱⠈⠿⠀⢉⣹⣿⣄⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢉⣤⠀⣴⣿⣿⢰⣿⠀⣯⣉⣽⣿⡇⣬⢣⣼⡟⠀⢠⣀⡹⢿⣿⡹⠀⠀⠀⠀⢰⡗⠀⠸⣿⠰⣿⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⣿⡐⣿⣿⡏⢸⣿⢠⡿⠿⣉⡙⢁⣽⠟⣿⡖⠀⠀⠘⠻⣿⣿⣴⠀⠀⡄⠀⣹⣧⡀⢀⡘⢳⣿⡇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠉⠙⠿⠁⠸⢿⠀⡥⠾⠟⠁⠛⠋⠉⠉⡇⠀⠀⠀⠀⠈⢻⠴⠂⠀⠀⠘⡛⠛⢁⠈⠛⠁⠀⠄⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣄⣤⣷⣤⣴⣦⣄⣴⠀⠀⠀⠀⠸⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⠀⢰⡇⠀⠀⠀⠀⢰⣾⣿⣿⣿⠂⠉⠉⠁⠸⣧⡁⢰⠂⠀⠀⣤⡀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠃⢤⠀⢀⡀⠉⢿⣿⡇⠀⠀⠹⣿⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⣀⣿⣶⣦⣴⣿⡇⠀⠀⠀⠂⠀⡀⠀⣈⠀⠀⠀⠀⠀⠀⠘⠛⠉⠂⠀⠛⠻⠋⠛⠃⠀⠈⠛⢿⠀⠀⠀⢿⠀⠀⢀
⠀⠈⠀⠘⠟⠛⣉⣽⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⣷⠀⠸⠁⠀⠇
⠀⠺⠋⠀⢀⠛⠡⢿⣿⠛⠙⠀⢀⠘⠀⠀⠀⣰⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⡟⢀
⠀⡆⠈⢠⡟⠀⣠⠜⠋⠀⠀⠀⠀⢀⠀⡄⢠⠍⠀⠀⠀⡴⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠇⣤⠀⠈⢢⡙
⠀⢡⡀⣾⣏⣾⠇⠀⠤⠉⠀⠋⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠁
⠀⠈⢻⣦⡜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡞⠁⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠰⡇
⠀⠀⠀⠙⢷⡟⠷⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⡞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⢀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣾⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠐⠂⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⣀⡠⠤⠄⠤⠴⢶⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠎⠉⠑⠒⠒⠒⠒⠀⠬⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠏⠀⠁⠀⠀⠀⠀⠀⠋⠸⠃⠀⠀⠀⠀⠀⣀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⡆⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣆⠀⣧⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢙⢒⣿⡿⢷⣾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁⠀⠒⠤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣤⡀⣀⡠⠖⠛⠉⠉⠀⢠⣾⣿⠏⠀⠙⢷⣶⣦⣤⣤⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠅⠀⠀⠀⠀⠙⢿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⡄⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢀⣿⠀⠀⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣸⡟⡤⠛⠉⠠⠄⠐⠒⢓⣒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⣤⢸⣦⣀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢙⢸⣿⡁⠀⣶⣶⣶⣦⣤⣤⣤⣴⠞⠿⣿⣳⠮⣉⣉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣀⣠⣴⣿⠟⠉⡀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠏⢼⣿⡁⢸⣿⠇⢹⠇⠿⠛⠋⠀⠀⡇⠈⠁⠀⠀⠀⠈⠑⠢⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠑⠄⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠉⠃⣼⣿⠀⢰⠂⠀⠀⠀⠀⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣌⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡁⠸⠷⠤⣾⠂⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢠⢘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⡍⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠤⢙⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠿⢾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⣼⠿⠒⠋⠉⠠⣄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡌⠀⠀⠀⠀⠀⠸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⠀⠀⠀⣴⡿⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⣛⠛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⠀⣃⠀⣯⡟⣀⠖⠾⠉⠁⠀⠀⢠⣤⣄⡀⡀⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⣶⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠙⣢⡀⠀⢀⣬⣷⢲⣿⠀⣉⠚⠋⠁⠁⠈⠓⠒⠚⠻⠿⢿⢧⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡿⠋⢀⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡈⣿⠟⢻⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⢀⣾⡟⠀⣷⣤⣤⣀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣸⣿⣿⣷⠉⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠺⢿⡧⢈⢛⡛⠻⡿⠁⠈⣿⣿⢹⣯⠹⣿⣿⣿⣿⠹⡋⠉⠛⠛⠒⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⡀⠀⣿⣿⠈⣿⡄⠘⣿⣿⡿⠀⢿⢦⣀⠀⠀⠀⠀⠀⠈⠁⠛⠂⠀⠀⠀⠀⢀⣤⡿⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⡄⢿⣇⢠⠚⠁⠀⠀⣾⠆⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡶⠿⢛⣹⣿⣻⡟⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⡔⣼⣿⣷⣾⣿⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠈⡱⢦⣉⠈⠨⠁⣈⣘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⣶⣦⣙⣿⣿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣷⣮⢩⢻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣡⣾⡿⣿⣿⣿⣿⣿⣿⡼⣿⣿⡳⣤⣀⣀⣀⢀⣠⣴⠘⣿⣿⣿⠏⢀⡀⠀⠀⠀⠀⢀⢃⡀⠀⠀⢠⣄⣀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣧⡘⢿⣧⠙⢷⡀⠀⢠⣿⡿⠀⢸⣿⣿⠀⢸⣿⣦⣤⣀⣠⡆⡈⠠⣀⣼⣿⣿⣿⣧⢸⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠿⠿⠿⣿⣿⡿⠟⢻⡿⢿⣿⡟⠿⠟⠁⠀⠸⣿⡆⠈⠻⣆⠸⣿⡇⠀⢸⣿⣿⣇⠀⢿⣿⣿⣿⠿⠃⠻⣾⣹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢻⣿⡀⠐⢿⣿⣿⡇⠀⢸⣿⠯⣿⣆⠀⠉⠉⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⢹⣿⡇⠀⢸⣿⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠙⠀⠀⠀⠈⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠊⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⠀⠀⠀⡀⢾⣹⡇⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⣐⡀⠀⠀⠀⠐⠀⠠⠈⠀⠀⠈⡄⠀⠀⡠⠀⠀⠓⢎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣸⡅⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠄⠀⡰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⡄⠀⠀⠐⠀⠂⠀⠀⠠⣶⡟⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠸⠛⠚⠂⠀⢀⠀⡀⠀⣠⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢈⣯⡄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢢⡥⠊⠁⠀⠀⠀⠂⢀⣰⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⢻⣿⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠄⠉⠟⠁⠀⠀⢠⣀⣠⢼⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠧⠀⡜⠀⠈⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⠀⠀⠀⠠⠀⠀⠰⡶⠖⠂⠊⠀⠂⠀⠀⠀⠀⢀⠄⠀⠀⠐⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠐⠉⠀⠀⠀⡇⡀⠀⠀⠂⠀⠀⠀⢀⣠⢯⣤⠄⠀⠀⠸⡳⠛⠁⠀⡔⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢿⣍⣷⠀⠀⠀⢘⢷⣄⣀⠀⠐⠀⣀⢀⠸⣿⡼⠓⠠⡀⠰⣺⠁⠀⠀⢠⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠋⠉⠀⡀⠀⣼⣷⢈⠁⠀⠐⠠⣠⠃⠀⠀⠀⠀⠀⢀⢴⢈⠔⠈⠁⠘⣦⣀⠠⠂⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⣀⢤⠤⣿⢧⠀⠀⠄⠀⢀⡼⠂⠀⠀⣠⠀⠠⠚⣰⠁⠀⠀⠀⢿⡿⣿⣖⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠁⠀⠝⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠀⢠⢄⣹⠂⡤⠔⢲⠟⠻⠿⠁⢠⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠚⣷⡏⠣⠀⠀⣀⢀⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠓⢂⠀⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠆⠀⠀⢀⠀⠀⠑⠀⠀⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠐⣂⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡈⠉⠈⠠⠄⠄⠂⠈⠉⠃⠀⠀⡀
⠀⠀⠀⠀⠠⠔⠀⠀⣀⣉⣀⠀⠰⠆⢁⠀⠀⠀⠠⡇⠐⡍⠀⡀⠀⠀⠀⢀⠠⡄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢮⣡⠀⡜⠘⠁⠈⣄⣳⠃⣀⠀⠀⠃⠀⣠⣳⡎⠀⠁⣀⠀⠀⠀⠠⣤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠠⣼⠈⢫⡁⠀⠠⡼⡛⠉⠀⠀⠀⠉⠙⠓⠂⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠛⠀⠀⠀⠀⠀⠲⠶⠖⠊⠹⣾⠰⢠⠃⡠⡀⢀⠀⠔⠳⢢⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠧⢀⣀⠀⢀⡀⠀⠀⢀⣠⠀⠈⠀⠙⠇⠛⠻⠭⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣭⠍⠓⠛⠻⠁⠀⠀⠀⢠⠂⠁⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣇⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣤⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⣀⣄⣄⢀⡀⠀⠀⣲⡀⠀⠀⠀⠲⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠉⠀⠀⢠⣄⠀⠙⠂⢀⠈⠀⠁⣉⠋⣷⡆⠀⠀⣬⠖⠠⣇⠀⠈⡲⠄⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣄⠀⠀⣀⠀⠀⠄⣆⢬⡼⠓⠀⢠⣰⡧⣀⣀⣻⣇⡀⠀⠀⠀⢀⠉⢀⠙⣄⠈⠉⡒⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠀⣧⡄⡈⣥⢾⣜⡓⢉⣁⡀⠀⡀⠈⠁⠀⠀⠀⠈⢛⣁⠐⠤⠤⢀⡉⠁⠳⢽⡄⠀⡇⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡈⢳⣶⣿⣦⣿⠿⢋⠁⡠⢈⣨⠍⣁⠐⠦⠀⠀⠀⠀⠀⠈⢥⡆⢀⠀⠨⡇⠢⣀⠀⣿⠢⣅⠀⠀⠹⣿⠀⢀⡄
⠀⠀⠀⢀⠀⠀⠀⠎⠀⣠⣿⡏⢱⢋⣴⠥⠐⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢀⡆⣷⢀⠈⠀⢹⠆⢺⣷⡀⢣⢹⣦⠞⢹⡆
⠀⠀⠀⢸⡄⠀⡄⠀⣸⠿⣸⠷⣿⠁⣀⢂⡂⣔⣬⣰⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢿⠓⠾⢬⣤⡈⢧⡑⣼⢠⣿⣦⣿⠁
⠀⠀⠈⡀⢧⣠⠓⣤⣿⢟⡟⣸⡿⠬⠋⠣⠑⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠌⢳⣄⠸⡋⠙⢫⣷⣼⠁⢸⣿⠁⢰
⠀⢢⠀⢧⠸⣧⠑⣾⢧⡾⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠙⣿⡗⢾⣿⣊⣋⠄
⠀⠈⣧⢨⠃⠸⣿⣣⡾⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⣄⣿⣧⠸⣿⠈⣯⠆
⠀⠀⠘⣷⣀⠀⢻⣿⡡⣺⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⢻⣿⠀⣿⣇⡳⠂
⢦⡀⢰⡛⣿⣦⣾⣯⣠⣿⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠹⣿⠳⣿⣧⣟⠁
⠈⢷⡌⡾⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣱⣿⣧⣿⣿⠃
⠀⠈⢷⡸⡇⢸⣿⣿⠁⣀⡀⠀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⣿⠇
⠀⠀⠘⣿⠷⣸⣿⣿⠘⡿⠛⡸⡯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⠀⠈⠣⣼⣿⣿⣿⣆
⠠⣼⡇⣹⣯⢸⣿⣿⡏⠀⠐⣮⣿⢅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢷⣫⣴⣶⣼⣿⣿⡇⢸
⠀⢨⣟⡵⢻⣿⣿⣿⣇⠀⢐⡫⢷⢝⡂⠀⢄⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⣷⡋⠡⣼⣿⣿⡇⢸
⠀⠀⢻⣫⠌⢻⣿⣿⣿⡆⠀⢩⠽⣝⣃⣨⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣗⡂⢀⣿⣿⣿⡃⣾⠀⢸⠁
⠀⠀⠈⠧⣞⢄⢹⣿⣿⣿⣄⠈⠬⣯⢲⣵⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠬⢽⣿⠒⣾⣿⣿⣿⣶⠃⠀⠅
⠀⠀⠀⠀⠻⢎⢔⣿⣿⣿⣿⣦⠨⣧⣾⣿⢓⣂⣤⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣯⣄⠀⡄⠀⢀⣦⣄⢺⣿⣃⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢏⣿⣿⣿⣿⣿⣷⣾⣿⠻⠛⢻⣟⣿⣿⣤⣤⠀⠀⠀⢀⠀⠀⠐⡩⣻⣽⠇⡇⣇⡼⣻⠏⡽⢷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠿⣿⣿⣷⣬⣧⣼⣽⣽⣿⣿⣀⠀⣈⣄⣄⣀⣈⡴⠖⠓⣚⡛⠛⣿⣾⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠷⣤⣀⡀⢹⣿⣟⠁⠀⠀⠀⡀⢀⠌⠋⠙⠻⣧⡊⠁⢀⣅⠚⢻⣛⣠⣿⣿⣿⣿⣿⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠿⢿⣯⣭⣭⠾⢿⣤⣀⢀⠠⠊⢀⣱⡤⢶⣿⠖⡛⠛⠋⠛⠛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣧⣼⣾⣾⠾⠷⣦⠾⡉⠐⢣⡏⠀⠀⠀⣈⡥⠖⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣙⣇⣀⣜⣁⣀⣈⣠⣎⣈⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠆⠘⠃⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣇⠀⠀⣿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠁⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣀⣠⣦⡀⠀⢀⠀⠀⠀⠈⠉⠀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠙⢉⡟⠁⠀⠀⠀⠀⢤⠀⠀⠀⣷⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⢰⣤⣠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿⡄⠀⢰⣿⡇⠀⠀⢠⠀⠀⠀⣀⣂
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢰⠛⡚⣅⠀⠀⠀⠀⢸⣿⡯⠀⠀⡞⠃⠀⠀⣿⣧⠀⠀⠈⠉
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠁⢶⣀⠀⠀⠀⠀⠹⡿⡛⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠄⠈⢆⡀⠀⠀⠐⢍⣣⡀⠀⠻⠿⠿⢷⣿⣽⠄⠀⠘⠀⠶⠀⠀⠀⠄⡠⠀⠀⠢⣄⣠⡆⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠉⠁⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠖⠛⠇⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠁⠀⢈⣴⠛⣹⣤⠀⠀⠀⣠⠖⢀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡩⠘⠁
⠀⠀⢠⠀⠀⠀⠀⠀⢼⣦⡄⠀⠀⠀⠀⠈⢹⡿⣿⡇⠀⠀⢀⣒⣀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠀⠀⢰⠶⠶⠆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠶⠦⠄⠀⠀⠀⣅
⠀⠀⠀⠀⠰⣛⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠛⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⣰⡾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⣆
⠀⠀⠀⠀⢰⡿⢀⣤⡀⠹⣧⠀⠀⠀⣠⣶⡀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣶⡄⢀⣤⣀⠀⠀⠀⠀⠀⢀⣠⡶⣻⡇⢀⣼⠏⢠⣤⡀⢿⡄
⠀⠀⠀⠀⢸⡇⠈⡏⠉⠀⠙⣷⡀⣰⠏⠸⣇⠀⢠⡶⠋⢹⡇⢀⡼⠋⢺⡅⢸⡏⠙⢷⣄⠀⢀⣴⠟⡁⠀⣿⢀⣾⠃⠀⠋⢹⠁⢸⡇
⠀⠀⠀⠀⢸⡇⢀⡏⠀⠀⠀⠸⣿⣟⢠⠀⠻⣶⡟⠀⠀⠀⢿⡾⠃⠄⢸⣧⡾⠑⠀⠂⠙⣧⡾⠃⢀⠀⠘⣿⣿⠃⠀⠀⠀⢻⡀⢸⡇
⠀⠀⠀⠀⢸⣧⡟⠀⠆⠀⠀⠐⡌⠫⠻⣆⠀⢙⠃⠀⠆⠠⠈⡃⠀⡀⢀⢹⠇⡈⠀⠂⠀⠻⠁⠀⠀⢨⠚⠝⢡⠀⠀⠀⠸⠈⢿⢼⡇
⠀⠀⠀⠀⢸⡇⠻⡀⠀⠀⠀⠀⠉⠀⠀⠘⣆⠈⣠⠒⢠⡄⠄⠀⠀⠀⠀⠀⠑⣄⠀⠒⠂⠇⠀⡄⡠⠁⠀⠀⠉⠀⠀⠀⠀⠀⠋⢸⡇
⠀⠀⠀⠀⢸⣷⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠘⠁⠀⢲⢄⢳⣀⠀⠀⠀⣠⠉⢈⠗⢀⡁⠘⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⣿⠇
⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⠀⠀⠀⠀⠐⢜⣦⡃⡘⣈⠑⡀⠃⣨⣄⣇⣆⣄⡐⢀⠠⠁⠈⠀⢠⡆⣴⡃⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿
⠀⢀⣄⠀⠀⣿⢁⡞⠀⢰⠀⢰⡀⠀⠀⣺⡟⠈⢳⢄⡠⠐⠋⢀⠍⠁⠉⠈⠡⡀⠒⠀⠠⢴⡞⠀⢿⣗⠀⠀⢀⠆⠀⡀⠐⢷⠘⣿⢠⣤⡀
⠀⢼⠉⠻⠾⣿⠀⠃⠀⠉⠀⠀⠁⢠⡵⡽⠃⠀⠸⡠⣀⠀⠰⢀⠀⠀⠀⠀⠀⡀⠦⠀⡄⢄⠇⡀⠘⠿⢮⡆⠈⠀⠀⠁⠀⠚⠀⣿⡿⠹⣇
⠀⢸⡄⠄⠀⠘⡆⢠⡀⠀⠀⠀⣠⠾⠃⣴⣿⠑⠒⠁⠁⡀⠀⠋⠀⠀⠀⠀⠀⠑⠀⢀⠈⠘⠒⠈⣿⣦⠘⠿⣄⠀⠀⠀⢀⡄⢰⠋⠀⠀⣿
⠀⠘⣇⠀⠀⠀⠙⣄⠛⠲⢶⣴⣧⠄⣰⣿⠏⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠹⣿⡆⠠⣽⣦⡶⠖⠋⣠⠇⠀⠈⢀⡿
⠀⠀⠹⣧⠀⠁⡦⣌⣳⣦⣀⠈⠀⢰⡿⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠈⢻⡆⢀⠁⣠⠖⠚⣁⠠⡤⠀⣼⠃
⢀⣤⡶⠿⠓⣄⡌⠨⢤⠂⡐⠤⡙⢷⡃⠀⣾⢿⠯⠭⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠭⠽⡿⣦⠀⡸⠾⢧⠀⡍⠒⢃⣹⠾⠄⠺⠷⠶⣤⣀
⠻⠶⣦⣄⡈⡏⠀⠀⠉⠁⠈⡦⠀⠀⢱⠀⠹⣌⣔⡊⢨⣿⡔⢦⡄⠀⠀⠀⢠⠴⣢⡿⠅⢑⣶⣥⠋⠀⣿⠀⢙⠰⡃⠀⢬⠁⠉⠈⢣⠂⠀⣠⠿
⠀⢀⡴⠟⣉⡆⠰⠌⠀⠁⠚⠀⠐⠇⣾⡀⠀⠈⠉⠛⠚⠉⠉⠻⢃⡄⠀⢠⡘⠟⠉⠉⠓⠛⠉⠀⠀⢀⡆⠐⢜⠀⢅⠁⠴⠞⠁⠸⡍⣡⡾⠋
⣰⠟⠀⠐⠚⡇⠒⠄⠈⠁⠃⠀⠀⠀⠹⠳⡄⠀⠀⠀⠀⠒⠂⠀⣼⣇⢀⣼⣇⠀⠐⠒⠀⠀⠀⠀⢠⠿⠃⠀⢌⢁⡀⣉⠀⠶⢂⡄⠻⣯
⠻⠚⢶⣦⡬⠄⠉⡐⠱⠶⠄⠀⠀⠄⠀⢂⠉⠀⡄⡀⠀⠀⠀⠀⢟⡻⠻⢟⡻⠀⠀⠀⠀⠀⣰⠀⠁⠀⠀⠀⠸⠐⠤⣤⠷⠒⠤⠛⢦⠙⣧
⢠⣶⣛⡁⠀⠀⠐⠀⠤⠘⡆⠀⠀⣖⡤⠻⣦⡀⣘⡇⡀⠁⠀⠀⠙⠷⡤⠾⠋⠀⠀⠀⢀⢸⠃⢀⣴⣰⣆⡀⠀⡴⠀⠄⠙⠤⣐⣶⣤⡀⢹⡄
⠈⠉⠉⠛⠻⠶⣤⡀⠀⠀⠘⢣⠴⡿⡈⠛⢠⠀⡟⠓⣄⠀⠀⠁⢀⠤⠒⠤⡀⠈⠀⠀⣠⡈⠖⠁⠂⡛⡀⣤⡶⠀⠀⠀⠀⠀⠀⠙⢿⣝⠿⠃
⠀⠀⠀⠀⠀⠀⠈⢛⣷⠦⠢⣍⠀⣤⠐⠂⠬⠀⡅⠸⡈⠛⣿⣿⡿⠿⣿⠿⢿⣿⡿⠛⢁⡆⠃⠘⢠⡄⠐⣶⠀⢢⡀⠠⠀⢀⣠⡶⠟⠋
⠀⠀⠀⠀⠀⢀⣴⠋⠀⡀⢰⡟⠀⣈⠉⠠⠖⠠⠋⣀⣻⣦⠘⣋⡤⠄⠀⠠⢤⣛⠁⣴⣷⠟⠈⠣⡀⢁⡨⠥⢀⡿⢤⣴⠞⠛⠁
⠀⠀⠀⠀⠀⣾⠁⠀⠀⡠⠀⡜⠑⢨⣥⡒⠀⢠⠚⠿⡍⠛⣧⠀⠀⠀⠤⠀⠀⠀⡼⢻⠉⠀⠓⢶⠄⠒⢤⠄⠋⠂⠀⠹⣆
⠀⠀⠀⠀⠸⣧⠶⠶⠶⠴⠦⠶⠶⣿⠁⠛⠀⣠⣆⠀⠂⠀⠈⠳⣄⣀⠀⣀⣤⠞⠀⠈⠀⡐⠀⠈⠀⠂⣤⣀⡀⠀⠈⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣴⠏⠙⢷⣀⠀⠀⠀⣯⠙⠏⢩⣿⠀⠈⣠⡾⢿⠀⠀⣠⡾⠋⠉⠙⠳⢶⣄⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡟⠀⠀⠀⠙⠳⠶⣤⣽⣆⠀⣿⣏⣴⠞⠋⠀⠈⢷⣼⠏⠀⠀⠀⠀⠀⠀⠙⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡿⠉⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣤⠈⠁⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣤⠀⠐⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢀⠴⠁⠀⠀⠈⡄⠀⠀⠀⠘⢁⠀⠀⠀⠀⠠⠂⠀⠀⢀⠠⠄⠀⠂⠈⠁⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠆⢆⡀⣀⠬⠗⠀⠀⠄⠀⠀⠃⠀⠂⠀⠀⠈⠘⡇⠈⠈⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠄⠠⡀
⠀⠀⠀⠀⡠⠚⠋⠀⣻⣅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠃⡘⠀⠀⠀⠀⢀⠀⠠⠀⠠⠀⠎
⠀⠀⠀⠊⠁⠀⠀⠀⠀⡯⠷⠤⠤⢀⡀⠁⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠠⠄⠀⠃⠀⠀⠀⠀⠈⠀⠀⢀⠀⡀⠀⠠⠀⠐⠁
⠀⠀⠀⠀⠀⠆⠀⠀⣠⠃⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠀⠀⠀⠀⢰⠃⠀⣠⡶⠲⢄⣀⠁⠀⠀⠀⡄⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⡇⠙⢆⢠⡙⢶⣄⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠿⣏⡀⠀⣀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⡎⠀⠀⠉⠙⠔⠁⢀⠀⠀⠤⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡇⢰⠀⠀⠀⠀⠉⠀⢠⠃⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡏⠀⠀⢰⠂⠀⠌⠁⠀⠀⠑⠂⡀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠁⠀⣠⠃⠀⢀⡤⠖⠛⠀⠂⠄⣋⣠⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢠⢿⠀⡰⢁⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡆⢸⣰⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢃⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⠻⠿⣿⠟⣿⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠠⠁⣀⣠⣂⠀⡤⠃⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⢀⠈⢸⡆⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠃⠄⠀⠁⠀⠁⠏⠀⠀⡤⠂⣠⠾⠅⠀⠀⠊⠭⠋⠀⣈⠉⠛⢤⡀⣧⡀⠀⠘⣿⢃
⠀⠀⠀⠀⠀⠀⠀⢠⢦⡟⠀⢁⠞⠁⠀⣢⠎⠀⣂⣴⡤⠊⠁⠀⠈⠈⠁⠀⠀⠀⣸⣿⣿⡧⠀⠙⢿⡄⠑⢀⣿⡈⣞
⠀⠀⠀⠀⠀⠀⡰⠃⡿⠀⢸⠃⣠⡶⢋⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⡄⠘⠀⠉⣷⣾⣿⣧⡿⡄
⠀⠀⠀⠀⠀⣰⠁⢰⡇⠀⣿⣟⠁⣰⣿⠡⠄⠀⠀⠀⠀⠀⠀⠒⠠⠤⣭⡽⠦⠤⡿⣿⣿⣿⣧⠀⠀⠀⣌⣿⣿⣿⡇⠘⡄⠀⡀
⠀⠀⠀⠀⢠⣿⠀⢸⣧⣰⡿⠈⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⢦⡀⠀⠀⣿⣿⣿⡇⠀⢸⣠⠃
⠀⠀⠀⠀⠘⣿⠀⠀⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⣿⣿⡇⢀⡾⠁
⠀⠀⠀⠀⠀⢻⡄⠀⣿⣿⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣼⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⠃⠀⡊⣉⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⣧⠀⢀⠀⠀⠀⠀⠀⢈⣽⡇⠀⣿⠀⠀⣄⡟⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠛⠻⠿⡿⠶⡷⠶⠶⣤⣬⣾⣿⣸⣿⣶⡶⠿⡿⣿⠟⠋⠉⢙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⣧⣤⣤⣴⣷⣤⣧⣦⣀⠠⠬⣿⣿⠛⡲⣗⡂⢠⢠⣿⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣨⣿⣾⣿⠁⠀⠀⠴⠿⠿⢭⣭⡭⠀⠼⠿⠷⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠶⠒⠾⠖⠛

⠀⢀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠀⢀⡄⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢮⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⢣⡄⠀⠀⠀⣤⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⢀⠀⠈⠁⠠⠂⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀⣀⡴⡼⠁⠀⡠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⢀⠊⢹⣷⠻⡆⠸⣴⠏⠠⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⡄⠰⠏⠀⠀⡀⠀⠀⠀⢀⠀⠀⣿⢷⣿⣿⣱⢿⣾⢷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠆⠀⠀⠀⠂⠀⠀⠁⠀⠀⢀⠈⠀⠀⡹⣾⣿⡿⠁⣾⢃⠌⣰⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⢀⡀
⠀⢸⣀⡆⠀⡀⠀⠀⠀⢀⠤⡀⠀⠑⠀⠀⠁⢿⣿⡇⢠⠛⡏⣴⠿⠒⠀⣴⠁⠀⠄⢠⠈⠀⠀⠈⠙
⠀⠐⠀⠀⠈⠇⠀⢀⠔⣋⡀⠄⠁⠷⠛⢷⡉⠷⣼⣷⡶⢻⣠⡿⣢⠴⠆⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠖⠈⣩⠖⠈⠀⠀⠀⠀⠀⠀⠚⠳⣦⣴⡬⠛⠛⠿⣵⠄⠀⠀⠛⠓⠚⣁⡴⠆⠰⢤⠤⠄
⠀⠰⡄⢰⠀⡔⠁⠀⠐⣦⡠⡄⠀⠀⠀⠀⠐⠉⣙⡕⠊⠉⠑⣖⠓⣞⡤⡶⠐⣊⠁
⠀⠀⠄⡌⡀⠀⠀⠀⠀⠈⠻⣶⣃⡀⠀⠀⠀⠀⡿⠀⠴⠀⠛⠹⡆⣉⢷⡄⢀⢈⠙⠢⣀⠀⠀⠀⣠⠄⠶⠀⡄⠀⠀⠀⠄
⠀⠰⡎⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠇⠀⠀⢀⡝⠳⡄⠰⠟⠀⢿⣸⠎⠷⠆⢘⣋⡄⢹⠗⠂⠉⠐⠖⠒⠈⠀⢠⠾⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠀⠉⠒⣰⣷⡶⠟⠊⣠⣔⣻⣧⢈⣁⣙⣧⠳⣄⢦⣄⠠⠀⠉⣸⢇⣀⡀
⠀⢰⠆⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⣼⠆⣠⣤⡀⢠⠞⢢⣴⢇⣼⠿⠿⡄⣉⡻⣹⡆⠈⠴⣉⣦⡰⠞⠁⠀⠈⢉⠆
⠀⢨⡂⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠴⠇⠀⠳⢶⣢⣶⣶⣿⣷⠄⣀⠿⢿⠙⢦⣛⡶⡘⠗⣤⠸⣄⠀⠰⠆⠈
⠀⢠⠍⠀⠀⠈⠷⢀⠀⠀⠀⠀⠀⠈⢉⠅⡲⣓⢛⠶⠹⠃⠌⢹⡄⣀⠙⠶⡫⠄⠰⢇⠠⡏⢳⠚⣤⠀⠙⢿⡓⠳⣄
⠀⢠⠀⣆⠐⠀⣄⠀⠙⠙⠢⠄⣠⣖⣁⡴⢉⠋⡽⢛⢄⣼⢳⣷⡾⠙⠥⠀⡿⣤⡚⠁⠀⠿⠀⠃⢻⡉⠩⣈⠣⢤⣁
⠀⠈⠀⠉⠠⡀⠈⠀⠀⢀⠀⠉⠉⣠⠞⣴⠋⣼⠳⠟⣘⡣⠙⠁⢀⡐⠂⠤⠭⢙⣡⡴⠂⠀⠀⠙⢮⣹⠶⠺⢿⠂⣉⠓⡀
⠀⠀⢀⠀⠰⠁⠀⠀⠴⣿⠀⠀⠀⠀⠘⠀⠢⠙⠒⠎⠁⠴⠣⠾⣦⡀⠀⠀⠀⠈⠳⠶⢞⡉⣤⡄⡀⠈⢷⡒⠦⠀⢉⡜
⠀⠀⠈⠂⠀⠀⠀⠀⢠⣄⠙⠆⣰⡆⠀⠂⠀⠀⠈⠀⠀⠀⢰⡞⢻⣿⡧⠀⠀⠀⠀⠀⠈⡡⠟⠐⣻⣿⡀⠘⢦⠁⠈⠓
⠀⠈⠀⢀⠄⢰⡆⠀⠰⣿⠀⠘⠉⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠⡿⢇⣈⣿⣀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠏⠿⣄⠈⠼
⠀⠀⠀⠈⢀⣈⠹⠀⡀⠁⣾⠀⠀⠃⠐⠀⠀⠀⠃⠀⠀⠀⢷⣄⣤⠙⣟⠻⡶⣦⣤⣄⣰⡁⠀⢸⣿⣿⠛⡇⢠⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠟⠀⢠⡁⠢⠘⠄⠀⠀⠐⣦⠠⡀⠀⢀⠀⠀⠀⠀⢠⡁⣘⣇⠠⡏⡹⢏⡿⢻⢦⡏⠀⠀⠀⡀
⠀⠀⠁⢀⡈⠀⣀⣾⠑⠊⠀⠀⠀⠀⠈⠈⡠⡏⠀⠘⠀⠀⠀⠀⠀⣉⣹⠟⢻⡷⠇⠈⠱⡟⢬⣿⣶⠟⣿⣀
⠀⠀⠀⠸⠓⠁⠙⢉⠀⠀⠠⢦⠀⢠⡀⠄⠡⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⣤⣼⢁⢦⡀⠆⢻⡆⣿⠻⣿⣽⣯⣶⡖⠀⠠
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠈⣇⠄⠻⢠⠀⢈⡇⠀⠐⠀⠀⠀⠀⠀⠀⠤⠜⣡⡜⠀⠈⠀⡌⢛⣦⡁⠈⠉⢻⡎⣠⠀⠇
⠀⠀⠀⠀⢳⡤⠤⣄⡈⠀⠀⠈⠀⡀⣦⠈⢀⣊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠁⠻⡄⠀⠀⠸⡇⠙⠻⣦⣡⣤⢹⡗⠀⡀
⠀⠠⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠞⠉⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠙⠄⠀⠀⣷⢀⡀⠟⣀⠀⠀⠀⠀⠁
⠀⢰⣤⠐⠒⣋⠟⣚⣡⡾⠲⣟⣵⠞⣍⣓⣶⣤⣼⡵⠦⠠⠀⢠⠀⠠⠄⠠⠤⠆⠤⠤⠔⠉⣉⣠⣾⠷⢦⠟⠷⣦⡄
⠀⢸⡟⠛⠛⠟⠛⠉⠏⠑⠛⠋⠉⠀⡁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠙⠋⠴⠁
⠀⠸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⣀⡀⢘⡀⢀⠐⠀⡀
⠀⢠⡀⡀⢀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠁⠁⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠒⠒⠂⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤



⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣿⣛⠛⣻⡆⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡄⠰⠝⠿⣾⣿⡆⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣠⣤⡼⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠋⠈⣿⣿⣦⣀⡀⠀⠀⠀⢸
⠀⠀⢦⡀⠀⠀⠀⡀⢸⣿⣻⠿⠋⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢹⣿⣿⣦⡀⠀⢸
⠀⠀⠀⠈⠀⠀⢾⣟⡛⠉⠁⣿⣷⣄⡀⢤⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣄⡀⢀⣠⣶⣿⣿⣿⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⠟⠛⠛⠛⠿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢻⣆⡿⣶⢿⡿⠿⠛⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠀⠀⠀⠈⣷⠀⠀⠀⠀⢸⣧⡛⣿⣾⠻⣿⣷⣶⣶⣶⣄⠀⢠
⠀⠀⠈⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⣾⣷⠀⡄⠀⠀⠐⢿⣄⠀⠀⠀⠈⠛⠉⣿⣿⡇⠋⢝⠻⣿⣿⣿⡆⠈
⠀⠀⠀⠀⠀⠈⠛⠛⠶⠖⠛⠷⣄⠀⠀⠀⠀⠀⡔⢿⣲⣿⣿⣿⣿⣟⠉⠁⢈⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠻⡿⠇⡀⠐⢦⡘⣿⡿⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠈⠷⣄⠈⢻⣿⣯⣥⣶⠦⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢄⠀⠈⠛⣆⠻⠷⠈⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡼⠃⠀⠀⠀⠈⠀⠀⠀⠈⠳⠄⠀⠈⠛⠂⠀⠀⠀⢀⠞⠁
⢰⠀⠀⠀⠀⣴⡿⢳⣶⡾⠂⡀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⣶⣷⣖⣤⡀
⢠⠀⠀⠀⠀⢹⣿⣿⠯⠄⠂⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⣿⡇⣀
⢸⠀⠀⠀⠀⣻⣯⢁⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠚⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣶⣦⣄
⠘⠀⠀⠀⠀⠉⣀⡁⠀⠀⠠⠄⠀⠀⠀⠀⠠⢄⡀⢰⠀⢠⣏⣞⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢧⣿⠋⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⢠⢎⣾⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⢠⡏⢼⣿⣿⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡾⠀⠈⠻⠟⠋⠀⠘⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⠵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠐⠀⡀⢠⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢋⣬⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣋⣴⣿⣿⣿⣿⣿⢟⣿⣯⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣗⣿⣽⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣇⠐⢿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣒⣸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⣿⣟⠋⡽⠋⠉⢻⣿⡿⠿⣿⡿⠛⣿⣿⣿⣿⣿⡟⣭⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣓⣽⣿⣿⣿⣿⣇⠀⠀⣼⡿⠁⠀⢈⡇⢀⣼⡷⠿⢿⣿⣿⣶⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠋⠈⠙⣿⣷⣾⣿⣷⣿⣿⠿⠿⠿⠋⢀⡴⢊⣽⣿⣿⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⣠⠴⠚⠙⠻⣿⣿⣿⣿⣿⣷⣶⡶⠋⠉⠉⠁⢿⣿⣿⣆⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⣿⣿⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⡌⠙⠐⠀⠀⠙⠿⣯⠄⠀⠀⠀⠀⠀⠈⠓⠶⠏⠀⠀⠀⠀⠸⠂⠀⠀⡀⢠⠛⠂⠆⠀⠀⠐⠲⠄⠉⠀⠀⢀⠀⠀⠐⠀⣀⡐⠏⠁⠙⠀⠀⠀⠠⠈⠡
⠁⠆⠀⠀⠀⠀⠀⠈⠙⠦⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠑⢠⡄⠀⢰⡆⠠⠀⠀⠀⠀⠀⠈⠉⢀⡀⠁⠞⢁⣤⡟⠣⠬⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣀⠀⠀⠀⠀⠂⠄⠀⠀⠀⢀⠿⣠⠤⢰⡆⢔⡀⠀⠀⠀⠀⣀⠀⠈⠷⢲⣶⡝⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠳⢶⣄⠀⠀⠀⠀⢀⠀⠀⠀⡉⠀⢸⡇⠀⠙⡂⠄⠀⠀⠠⠄⣀⡾⠋⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣀⠸⠇⠀⠀⠐⠒⠘⠛⠆⠒⠿⠃⢠⢶⣿⠛⠛⢀⠀⢀⠀⠀⠀⠀⠐
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢴⡏⠀⠰⠄⠁⠀⠘⢀
⡞⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⣸⠃⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸
⠟⠛⠛⠗⠐⠒⠒⠒⠒⠒⠺⠷⠾⠟⠓⠛⢷⡿⠃⠀⠀⢀⡔⠉⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠈⡷⠛⢀⠂⠐⠻⠿⠓⠒⠒⠀⠀⠑⠶⠟⠛⠻⠿⢻
⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣦⡏⠀⠀⢠⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡀⠀⠀⢸⣿⣿⡉⣹⣿⣦⡀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⡅⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣻⣦⣀⣠⣧⠀⠀⠀⠀⠸⠐⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠰⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠟⠉⢻⡀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⠀⠚⢷⡂⠀⠀⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⣦⡴⢏⣱⣄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⠀⠀⠰⠓⠈⠈⠑⢀⠀⣄⠀⠄⠀⠀⠱⠂⠀⢰⠀⠀⠈
⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⠃⣴⢿⣯⣝⣳⣦⠀⣀⣾⣿⣿⣿⣿⣿⣄⠀⠀⡀⠁⠂⠀⠈⠀⡄⡀⠟⠿⢲⣤⡰⡇⠀⠀⠀⠀⠐⠶⣷⢀
⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⢿⡿⡾⠛⠔⠳⢾⡿⢿⣿⠿⠿⠿⣿⣻⣿⣿⣿⣿⣦⣾⡟⠀⠀⠀⠀⢱⣾⣶⣶⣏⣭⢱⣧⣀⠀⠀⠠⠄⠀⠃⡈
⠛⠀⠀⠀⠀⠀⠀⢀⣤⠾⡩⣄⠰⡇⠸⢃⠀⡾⠃⠀⡀⢈⠃⠀⠀⠀⠀⠉⠻⡟⣉⣏⡹⢿⣟⣳⣦⡤⣿⡞⣻⣿⣿⣿⣿⣎⢿⣿⣿⣄⠐⢶⡀⡀⡇
⡀⠐⠃⠀⠀⢰⡶⠊⠁⠀⠀⠀⠢⣜⣠⡟⠁⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⢀⡰⠿⠻⣷⣶⡦⢿⣿⢿⣏⣿⣿⠻⣽⣿⡿⡿⣿⡿⣿⣿⣹⡷⣶⣶⣴⡀
⡆⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣢⠼⠸⢡⡄⠀⢀⣾⣷⣷⣀⠀⣀⣤⣆⣠⠀⣤⣾⠋⣩⣿⣿⣿⡿⣧⣿⣙⣿⣿⣋⣿⣸⣿⣴⣏⣿⣿⣿⣿⣹⡷⠇
⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠳⠃⠆⠀⣶⠄⠀⠀⠀⠻⣿⡷⠟⣽⢿⣿⣵⡿⠚⠉⢿⣿⣿⠿⢿⣿⣟⣿⣿⠸⣿⣯⣿⣜⡿⠛⣿⣿⣿⣿⣿⣽⢻⣏⣳
⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠜⠁⠀⠀⣟⠀⠀⠀⠀⢶⣞⣴⣮⠻⣿⡿⠉⠀⢀⣇⣸⣿⣿⣷⣜⣛⠓⠋⠉⠀⠀⠀⠈⠙⢿⣾⣽⣿⣿⣗⣾⡿⣸⣿⡏
⠀⢀⣤⠀⠀⠀⠀⠀⢠⠂⡒⠀⠀⠀⣧⡖⠿⢟⣴⡿⠉⢉⣽⣶⢿⡤⠀⢀⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⣤⢾⣿⣟⣿⣯⣼
⢀⡹⠟⠛⠃⠀⠀⠀⠀⠀⠷⠀⠀⢀⣟⣪⣷⣿⣭⠑⢉⡾⠋⢹⠿⣧⣠⣿⣿⠿⠾⣿⣿⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢿⣿⣫⠿⣿⣿⣧⣿
⣸⣇⣠⣄⠀⠀⠀⠀⠀⣀⡽⢀⠀⠿⣿⣧⣞⣡⠄⠰⠃⠀⠀⢰⡆⢸⣿⣿⠃⠀⣶⣾⡏⠀⠀⠠⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠈⣸⣿⣿⣿⡏⣹⣿⣿
⢋⣿⡆⠁⠀⠀⡀⠀⠈⠙⠚⠻⣶⣄⠉⠉⠋⠁⠀⠀⠀⠀⢀⢀⣒⡤⣽⢯⣧⣰⡿⠟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣟⣿⠿⢿⣿
⠂⠀⠀⠀⠄⠀⠈⠠⠀⠀⠀⢁⡜⠿⠀⢀⢠⣤⣤⣄⢠⣶⡯⠏⠛⠁⣄⠙⠛⢿⣿⣿⣷⡀⠀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⢮⣻⣷⣶⣿⣿⣿⣟⣻
⠀⠀⠀⠀⢀⡰⠁⢠⡄⠀⢉⡼⠋⢠⣶⣿⡿⢿⣿⣽⣍⠘⠁⠐⠀⠰⣀⣞⣭⣄⠀⠀⠘⢿⠖⠀⠀⠀⠙⠑⠀⠀⠀⠀⢠⣞⢿⣿⣿⣯⣏⢻⣿⣏⢛
⠀⠀⠀⠀⠛⣥⢺⡟⢠⣴⠶⣶⢿⣦⣿⡼⢟⣻⢾⢎⣹⠆⣀⠄⣐⣈⣍⣿⣏⣻⣾⣿⡗⠀⣿⣿⡇⣤⣤⣄⣴⣶⣷⣷⣿⣿⣿⣿⣿⣭⣯⣿⣿⣿⣿
⠀⠀⠀⢀⠺⠭⠊⠹⠿⢛⢺⣿⣼⡇⢯⣀⣸⣿⢀⣼⢻⣬⣟⡀⠿⠙⠻⣾⣻⣿⣿⠳⡄⣠⢿⣿⡇⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⡅⠠⣴⡈⡄⢸⣏⣿⣽⡅⢸⡟⢾⡅⠻⣿⠿⣿⣏⣿⣶⣶⣷⣿⣿⣿⣙⡗⣿⣏⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠀⠀⠱⠿⢼⣿⣿⡿⠿⠅⣠⠻⣄⣰⣿⣆⡿⣿⡿⣿⣿⣿⣷⣉⡳⠬⢿⣽⢟⣿⣧⣝⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠂⠀⢃⠐⠲⢀⣀⣷⡽⢇⣤⣶⣏⣻⣿⠗⣻⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⠛⢿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣿⣾⣿⣿⣷⣼
⠀⠐⡀⠄⢓⣾⣿⡟⢺⣧⣬⣗⣿⣿⠿⣿⢷⣼⠾⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣉⣨⣿⣿⣿⣿
⣤⣊⢠⣶⡾⠋⠿⢸⠛⠛⠹⢇⢿⣮⣭⣹⢒⣿⣲⠞⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⣿⣿⣷⣾⣿⣭⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢠⡐⠂⢰⠐⠄⢠⠀⡆⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣐⠀⡀⠀⢱⣎⣈⠐⠡⣀⡨⡅⠀⠉⠠⡀⢀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠃⠠⡆⠀⡂⠙⣏⡥⢀⠛⢿⢏⠀⠃⣤⡀⣿⡏⡄⠧⢀⢀⠁⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢠⠀⣜⣠⢾⠛⡏⢫⠿⣦⣼⠒⡿⣠⡁⠀⠷⣷⠂⠜⡠⠁⠖⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣠⢉⣽⡈⢛⣶⣬⣏⠘⢁⣍⠿⢻⣿⡴⠛⢻⣿⡅⡲⢎⣽⣇⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢘⡟⢃⣞⣱⣧⣿⣵⣾⣷⣞⣿⣿⣽⡤⣿⣬⣤⣤⠜⢃⡈⡛⠂⠊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠚⢆⠈⣋⣀⡍⢫⣯⣽⡍⠥⣿⣿⣿⣿⣿⣷⣿⣞⣷⣾⣿⢂⣻⣶⣎⡹⡇⢀⢦⡆⠁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠹⠀⢠⣿⠈⡙⠘⣬⣿⣷⣶⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⣿⣿⢻⣏⣻⡓⢻⠮⣨⣁⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡾⢗⣁⣈⢓⣻⣿⣿⣿⣯⠿⣾⣷⣾⣯⣈⣿⣿⡶⣟⣻⡿⣿⠛⢫⣶⡄⢞⠣⢙⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡈⢠⡏⠉⣴⣡⣽⣯⣷⣮⣿⣴⣯⣿⣿⣿⣿⣿⣟⣛⣿⣿⠟⡿⣋⣿⣵⣦⢎⠟⣈⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⡴⢣⡍⢟⣲⠼⣿⠿⣿⣯⣾⣿⣿⣿⣟⣺⣿⣿⣇⡟⠸⣧⡟⢬⠉⢁⡀⠤⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⢊⠋⠉⠷⣬⣿⢶⣸⣻⡿⣿⣿⠻⣿⣿⢛⣽⣿⣛⣿⡅⠿⣞⢃⠁⣸⣻⠃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢂⢃⡀⢽⣯⣀⢾⣽⢮⣤⡤⣿⣿⡟⣿⢹⣿⡷⣼⣿⣟⠻⠟⢟⡰⠅⠱⠒⠊⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌⡙⠳⠆⡸⠚⠫⠙⢻⣾⢗⣾⠟⣟⠱⢦⡽⢻⢷⣬⡸⢋⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⠠⠓⠃⠒⠉⣼⣠⣥⡴⢦⠐⣀⣼⢿⡐⢻⣯⡴⣤⠐⠴⠻⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⠠⠒⡀⠉⢻⠳⠛⠊⢁⡱⠥⠚⠙⠂⡁⠀⠀⢀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠋⠁⡀⠁⡀⠈⠴⢆⢉⡆⠹⠄⠋⠒⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠤⠄⠂⠐⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⣉⣴⣆⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠲⣯⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠅⠀⠀⢂⠀⠀⠆⣿⣿⡟⢳⠀⡃⣀⢀⡀⠀⠀⠀⡀⡰⠀⣴⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠄⠀⠈⠯⠡⠄⠁⠐⡟⠛⠺⡋⠁⠀⢸⡅⡇⢀⠘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⣤⣄⠀⣀⠀⠀⠀⢁⣀⠀⠐⡀⠀⠈⣿⣷⠋⢀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢀⣼⡮⣿⣎⡡⢿⣛⠋⠐⠸⣸⠁⠀⠁⠀⢰⢱⠉⠁⠺⠲⠤⢤⢀
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠠⣟⣽⠀⠀⠙⠄⠉⠀⠈⠀⡏⢹⡇⠐⢆⣀⡀⢸⠀⢠⢃⣤⣶⠏⡏⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⡠⣴⡖⠒⠲⣾⡝⣼⡟⠀⠀⣻⣶⣾⣣⣿⣿⣿⣿⣴⡷⠀⡀
⠀⠀⠀⠀⠀⠇⠀⠀⢀⡞⠁⠀⠀⣁⢴⠛⠏⠁⠀⠀⠀⢻⣿⡿⠄⠀⠘⠿⡟⠁⠀⠉⠛⢿⣿⡭⠎
⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⣀⡴⠃⠘⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠛⢋⡤⠀⠀⠀⠈⢢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠅⡜⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⣖⠂⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣆⣃⣀⣠⣤⣴⣶⡿⠟⠉⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⡉⠂⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠿⣟⡿⢿⡿⢿⣿⣿⣿⣦⡀⢿⡿⣻⣿⣿⣶⣄⣀⠀⠀⠀⠀⠉⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠈⠉⠉⠉⠙⠛⠊⠛⠛⠛⠛⠙⠛⠋⠙⠓⠒⠦⠼⠛⠁

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠐⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠍⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⡀⠀⢋⣀⢀⠢⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⢢⡔⢣⠑⢋⢲⣧⡀⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⡘⠟⢯⣜⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⢿⣿⣧⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛
⠀⠸⣄⡙⢄⣅⠒⡤⠛⠥⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢰⡈⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣿⠼⢧⡀
⠀⠀⠈⢷⣄⡲⠭⢤⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠤⠄⣼⢩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠷⠊⠁⠀⠀⡄
⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢈⠈⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⢀
⠀⠀⠀⠀⠀⠀⣀⣴⢠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠗⠀⠀⠀⠀⠀⠀⠀⠧⢶⡒⠀⢀⣠⠒⠉⠂
⠀⠆⠀⢤⢜⠘⡏⡉⣱⡄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡆⠀⠀⡀⠀⠀⠀⡎⠠⠀⠈⢠⣿⠾⣕⡿⠁⡀
⠀⠀⠠⠀⠁⡀⠀⠀⠀⣠⠅⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡐⠚⣧⣀⣸⠇⣠⠄⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⡞⠂⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠆⢁⠔⠈⠥⢄⣀⡀⠀⠀⠀⠀⠀⠀⢦⣀⠈⢙⣾⣿⢟⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⡃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⢀⣻⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⢋⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠉⠉⢳⣀⠀⠀⠀⠀⠀⠑⠀⠐⢄⡠⠜⠀⠀⢠⡠⠂⠀⠐
⠀⠀⠀⠀⠈⣝⡿⠠⣄⣤⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡀⡄⣸⣿⡞⠀⣨⠒⠀⠀⠀⠀⠀⠀⠱⠀⠁⠀⢪⠀⡜⢉⡕
⠀⠀⠀⠀⠉⣉⣇⠀⢹⠁⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠀⢰⣜⠉⡀⣀⠉⠄⠀⠀⠀⠀⠀⠀⠀⣀⢺⠀⢺⠰⣧⠆
⠀⠀⠀⠀⠈⡿⠋⡄⠈⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⡵⣅⣾⣷⡥⠔⠈
⠀⠀⠀⠀⠀⠁⠂⠃⠀⠟⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣛⣻⣻⣫⣝⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣭⣿⣯⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⠀⣴⣶⣶⣶⡄⠉⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣰⡀⠻⠿⠿⠟⠃⠸⠂⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⢿⠟⠀⠀⠀⠰⠆⠀⠀⠘⠁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣾⣿⠟⠀⠃⠀⠈⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⠻⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⢰⠄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠘⠋⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠺⣿⠗⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣭⣥⣤⣶⠾⠃⠀⠀⠀⠈⠻⠿⠋⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⡥⠒⠉⠀⢀⡤⢲⠎⢀⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⣟⠀⠀⠙⣆⠠⡇⡇⢀⡔⠁
⠀⠀⠀⠀⡀⠀⣸⠃⢀⠀⢰⠁⢀⣀⣰⣋⣠⣤⡴⠧⠒⠀⠀⢀⡠⠤⠒⣀⠀⠀⠐⠂⢀⡂⠄⠀⡀⠀⠀⣿⠀⠀⠀⠘⣿⣷⢷⣞⡀⠀⢀
⠀⢘⡐⢰⠃⡰⠃⢀⠈⡠⠃⠀⠘⠁⡝⠁⠀⢀⣄⠤⠀⢀⡴⣋⠀⠀⠀⠰⠋⣩⠟⠉⠉⠉⠉⠛⠧⣄⠀⢸⡇⠀⠀⠀⠸⣿⡾⡁⠀⣠⠋
⠀⢸⡇⣿⠞⠁⠶⢉⡼⠀⠀⣈⠀⡼⠁⣄⠐⣠⠃⣀⡶⠫⠤⠅⡦⠤⠖⢊⡉⠀⠀⠀⣰⣷⣶⣶⡊⠈⠳⣌⣇⠓⠠⣄⠈⣿⡇⢧⡴⠁
⢠⡞⣸⡏⠀⣀⡰⠋⠀⠀⣀⡵⠊⠁⣀⣭⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⠃⠀⠀⡈⠻⣦⣀⠉⢠⣿⣧⠘⡇
⠏⢰⡿⠀⠀⣽⠁⢀⣴⠟⢉⣴⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠈⠉⠐⠀⣹⣷⣿⣿⣿⣆⡿⣄
⠀⢸⡇⠀⢰⣿⡾⠋⠀⣰⣿⢋⣤⡁⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⣉⣉⣉⣙⣳⣤⣀⣘⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡈⢹⣿⣿⣿⣿⠇⠙⣆
⠀⢸⡇⠀⣼⣿⠱⠆⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣤⠤⠤⠤⠞⠋⠉⠁⠈⠉⠉⠛⠻⣿⣿⣿⡀⠀⠀⠀⠉⠹⣿⣿⣿⣿⠀⠀⠸
⠀⢸⣿⣰⣿⡏⢀⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠙⢶⣄⠀⠀⢠⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⣿⣿⣿⣿⡆⠀⣸
⠀⢸⣿⣿⣿⣿⡏⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣇⣰⡟
⠀⠈⣿⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⡄⠀⣿⣿⣿⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡀
⢷⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠍⢸⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣃⡤⠾⡟⣻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢀⠀⠀⠀⠀⢀⡞⠀⠀⡇⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⣿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢛⣻⣧⠀⢰⣯⠄⠀⠀⡞⣼⠃⣠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣧⣤⣦⣤⣀⣀⠀⢀⢒⣻⣷⡒⣶⣻⣧⣤⣤⣿⣿⣾⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⣿⠟⠉⢹⣿⡏⣿⣿⡃⠀⠈⠉⠉⡿⡉⠉⡏⢹⠛⠛⠻⠷⠭⣽⣯⣷⣿⠿⣿⠟⠛⠛⡛⢻⡿⠁⠀⠀⠀⣠⣿⣿⠿⠛⠛⠛⠻⠿⠿
⠛⠉⠀⠀⠀⠀⢿⣷⣜⢿⣷⣦⣤⣤⣴⣷⣷⣶⣷⣾⣤⣀⠀⠶⠶⣿⣿⡛⢛⡛⣟⣋⡁⢰⡇⣼⣇⠀⠐⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠀⠀⣿⣷⣾⣿⡏⠀⠈⠉⣉⣽⣿⣿⣝⣋⣛⣏⡁⢀⣼⣷⢿⣿⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣤⢀⣀⣠⡀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠚⠟⠛⠛⠉⠉⠙⠛⢶⣤⡀⡀⠀⡄⢀⠠⠚⠉⠁⠀⠀⠀⠈⠉⠛⢿⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠤⠀⠉⢿⣏⠀⣿⣏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠛⠚⠻⣷⡟⠊⠑⠢⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⢀⡱⡄⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠂⠀⠀⢋
⠀⠀⠀⢀⣀⠀⢀⠀⠄⢀⣀⠀⠀⠀⠙⣿⣿⣿⣧⡤⡄⠀⢈⣥⣄⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⣥⠀⠀⠀⢰⣿
⠀⠀⠛⠛⠿⢷⣾⣣⣶⣿⠏⣆⠀⠀⡤⢈⣻⡛⢻⣆⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⢀⡰⣿⡇⠶⠀⠸⣿⠀⠈⠃⡀
⠀⠀⠀⠐⠀⠛⠛⢻⣿⠳⡖⣯⡀⠠⠱⡘⣿⣷⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⠀⣆⣠⣶⡇⠀⠀⠀⠸⢦⡀
⠀⠀⠀⠀⢠⣴⣶⣾⢿⣦⡹⣿⣆⠒⠄⣠⣿⣿⣏⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠁⠹⢿⠿⠷⣯⣿⣼⢿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠁⢈⣼⣿⡇⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⣮⣙⠿⠞⠋⠿⢷⣿⣇⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⣼⠟⠋⠉⠀⠘⢻⣿⣷⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣿⣿⣽⣿⣿⠿⠿⠷⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠧⠀⠀⠀⠀⢸⣿⣿⠿⠷⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⣬⢄⣴⠆⠔⠀⢸⣿⢏⠛⣷⡀⠀⠀⠀⠀⠀⣠⠶⣿⠿⢥⠀⠀⢀⠄⡄⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣻⠿⠿⢷⡀⠀⠀⠘⠀⢠⡶⢿⣯⡑⠶⡄⠀⣠⣶⣦⣤⣀⣬⣒⡤⠞⠻⠉⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⣀⠀⢁⣄⣠⡄⠀⠈⠛⠛⢉⣟⣴⢻⣿⣿⠛⢿⡏⠉⠁⠈⠀⠀⠀⠘⠓⠀⢀⣾⠟
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠘⠿⢷⣮⣻⠀⠀⠀⣶⡿⠁⣿⢹⣿⡙⠷⠾⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⣄⠸⡏⠀⠀⣿⠸⣿⣷⣀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⣜⣣⣔⣤⠼⠃⢘⣿⣿⣿⣿⠿⠛⠛
⠀⠀⠀⠀⠞⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠁⠀⠉⠙⠻⢦⣀⡀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠄⠙⠂⠀⠀⠈⠐⠢⢄⣀⠀⠀⠀⠀⠛⠷⣤⣀⡀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⠤⣄⣉⠳⠶⠤⣤⣀⣤⣼⣿⠿⠛⠓⠒⠶⠦⠽⠿⣿⣿⣟⣹⠇⡸⠏⡁⠀⠀⠀⠀⠀⠀⣠⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠟⠚⠣⠴⠾⢦⡤⠴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢍⠁⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠴⠃⠀⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡶⠦⠾⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⣄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⢠⡤⠠⣄⡜⠋⠹⡀⠀⠀⢠⡆⢀⠦⣄⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⢸⠷⡄⠀⠀⢀⡇⠀⢳⡀⠀⣰⠃⢸⡀⠀⠀⠀⣀⠀⠀⠀⢀⡖⠲⡆
⢀⡠⠊⠁⠀⠀⠀⠀⠁⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠙⢦⡀⡼⠀⠀⠀⠻⡴⠃⠀⠈⡇⠀⣠⠞⡻⠰⡖⠂⠋⠀⠀⠻⢠⣤
⠉⠒⠠⣤⠀⠀⢀⢀⣀⡀⠠⠖⠋⠀⠀⠀⠀⠀⣟⠒⠦⣄⡀⢸⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⡇⠀⢈⠆⠀⠀⠀⠂⣾
⠀⠀⠀⢸⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣄⣀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣸⣤⣴⡷⠤⡤⠟
⠀⠀⠀⠀⣧⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠃⠀⢈⠻⡀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠈⣹⠛⢦⡀⠀⠀⠀⠘⠛⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⡴⠤⣄⠀⠀⠀⢰⡖⠒⠛⠳⡆⢠⡤⠐⠋⠀⠀⡠⢶⣉⠁⠀⠀⠀⢀⠔⠁⠀⠀⠈⠲⢄⠀⠀⠀⠀⣴⣃
⠀⠀⠀⠀⠀⡜⠀⠀⠽⠦⢤⣤⡀⠙⠢⣄⠀⠛⠚⠁⠀⠀⠀⡜⠁⠀⠈⠉⠒⠤⣤⠋⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠉⠉⠒⠒⢤⣤
⠀⠀⢶⡐⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣈⣳⡄⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⡀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⡦⠀⠀⢀⡀⠀⠀⠸⣞⠉⠉⠉⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⡸⠁⠰⣎⠁⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠓⣻
⠀⠀⠀⠀⣇⣀⡠⠞⠉⠉⠘⠚⠉⠑⢤⡀⠀⠀⠀⢀⠇⠒⠀⠀⠤⠤⠀⣀⠀⠀⠀⣃⠀⠠⠔⠊⠁⠀⠀⠀⠀⠰⠁⠀⠀⠈⠉⠲⢄⠀⠀⠀⠀⠛⠢⡄⢀⣀⣨
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣠⠶⠒⠒⠋⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣀⣉⡶⠀⠀⠀⠀⣧⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⢛⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⡴⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠎⠋⠀⣠⠔⠉⢇⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⡰⢻⣄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣀⡞⠉⠳⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⠉⠓⢄⠑⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⡶⠦⠿⢦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣀⣀⠀⠉⠂⢵⣼⡆⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠙⡄⠀⣸⣠⡟⠲⠛⠁⠀⠀⠀⠀⡴⠋⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠁⢈⠉⠉⠉⠁⠀⢠⣦⠀⠀⠀⠀⠀⠀⠑⡄⠀⣷⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⢍⡁⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠉⣇⠀⠀⢀⡴⠋⠸⡆⠀⠀⢀⣶⡀⠀⠈⢦⣿⣮⣀⠀⠀⠀⢸⡟⠒⠦⢤⣀⠹⡄⢀⢀⠀⠀⠀⠀⠀⠀⢸⣀⡶⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣰⠏⠀⠀⠀⢳⡀⢀⡞⠀⢳⡀⠀⣼⠋⠀⠈⠛⠲⢤⣘⡆⠀⠀⠀⠈⠙⢋⠏⠈⢳⡀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠳⠟⠀⠀⠀⢳⡀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠠⠤⠄⠚⠀⠀⠀⠛⠒⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠙⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢠⣀⠀⢁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠊⠀⠀⠀⠀⠀⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⣀⡔⠁⠉⠳⠾⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠟⠒⠤⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠐⣤⣼⣿⣧⣿⣇⠃⠉⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢷⣾⣿⣿⣿⣿⣿⣦⣖⢈⣛⡖⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠛⠈⣿⣿⣿⣿⣿⣎⡥⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢑⣶⡞⢹⣿⡿⣿⠛⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⢒⠾⠛⠉⢿⣾⣿⣿⣶⣶⣦⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠠⣶⣿⣿⣿⣿⢿⣿⣿⣿⣦⠀⢀⢀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣗⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣼⣿⡟⠛⠛⣛⡛⠻⠿⢿⡗⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⢀⡾⢋⣠⣴⣿⣿⣿⣿⣦⣀⠈⠳⣄⠈⢢⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣬⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠙⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠻⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠇




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

⠀⠀⠀⠀⠀⠀⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡴⠚⠓⣶⡼⢿⣾⣷⣾⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣶⠏⠀⠀⠀⡰⣿⠃⢋⣴⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⣠⠋⠀⠀⢺⡂⠁⠀⠐⠛⠛⠛⠿⠛⠶⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢽⠀⠀⡁⠀⠠⢔⣾⡅⠀⢀⣻⣛⣻⡟⡟⢛⡟⠉⠉⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢀⡾⠏⠀⡇⠀⢀⡼⣻⢸⣷⣦⣽⣏⢙⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⣇⠐⠡⣰⡇⠻⠛⣧⡈⣻⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⢀⠀⠀⠀⢀⣠⣤⣆⣀⣨⣿⣿⣧⠶⢻⣿⣿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠲⢶⡿⠟⠛⢉⣽⠟⠉⠁⢈⣿⡿⠛⣧⣠⣼⡿⠿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠋⠌⣿⠂⢤⣾⡟⠊⠁⢉⣿⢻⡗⢂⡾⣻⡟⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⢀⣁⣈⣆⠔⠋⠀⠀⢠⣿⣿⣿⣴⠪⣭⣬⡙⢦⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢚⣿⣿⣏⣀⠀⣠⣟⣯⡿⠋⠁⢚⣃⣨⣤⣤⣤⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣥⣽⣿⣿⣿⡿⠿⠿⠿⠿⠿⠻⠛⠛⠋⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠
⠀⠀⠀⢠⢏⣽⣥⢮⡄⠀⠀⠀⠀⡀⠀⠀⠀⠠⠁⠀⣒⠤⠀⠀⢰⣿⣷⠂⠀⢀⣿⡶⠀⠀⠀⠀⠀⠒⢫⠆⡀
⠀⠀⠀⠀⠀⠹⡏⠀⠀⠀⠀⣼⣿⣿⣷⡀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠛⠃⠀⠀⠉⠉⢀⠀⠀⠀⠀⠀⠀⠸⣣⢃⡠
⠀⠀⠀⠀⠀⠀⠃⠆⠀⠀⠀⠸⣿⣿⣞⠆⠀⠀⠀⡆⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠠⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣰⠀⡹⠶⠃⠀⠐⢰⠼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡼⡾⠇⠀⠠⠸⣀⡦⠢⠀⠀⠘⠀⠀⠀⠀⠊⠉⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡅⠀⠀⠀⠀⠀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡌⣉⡀⠀⢰⣶⠀⠀⣀⡃⢸⠋⠁⠀⢀⠀⠀⠀⢈⣁⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠤⢸⢀⡀⠀⢠⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣹⢯⣦⠀⠀⠀⠀⠀⠀⠀⠘⡿⠃⠀⠘⢻⡂
⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠘⠁⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠰⡷⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠈⠀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⠂⠤⣴⢟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⡶⣽⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠂⠤⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠌⢛⠉⢀⠠⠀⠀⠀⠀⠀⠔⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠆⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢦⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡎⠁⢀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠈⢡⠊⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⠀⠘⠆⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⢠⠤⠐⠦⡆⠀⠠⠸⣄⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢴⡿⠦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠂⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠈⠀⠀⠈⠹⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⡀⠀⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡚⠄⠹⡀⠀⠾⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠤⠠⠀⠂⠄⡧⠂⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠐⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀


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

⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣰⠀⢰⠀⠀⠀⢀⠀⠶⢀⡀⠀⠀⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⢰⡀⠀⠀⢠⡀⠀⢠
⠀⠀⡅⠉⡄⠈⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠃⢠⠞⠉⠂⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢉⠀⠠⡐⠋⠘⠃⠈⠀⠏⠀⠈⠒⢀⠀⢨
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠾⠒⠲⣄⠈⠳⡀⠀⣠⡄⠀⠀⠀⠀⠀⣄⠀⠀⠉⣤⣀⡂⠐⢤⣄⠀⠀⠘⠋⠁⠘
⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⡾⠃⠀⣤⢀⠁⠀⠀⢿⡆⠘⠀⠀⠀⠀⠠⠒⠉⠀⢠⠻⠋⣉⣹⣶⠀⠙⣷⠀⠀⠋⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠐⣏⠀⠀⠀⢱⠀⠀⠀⠀⠈⢡⠃⠀⠀⠰⠃⣠⠜⠀⠀⠚⠀⠷⠀⢷⡀⠀⢠⠀⠰⠛⢐⠤⠀⠀⠀⠀⠉⠀⢸⣧⠀⢸⡇⠀⠖⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⡤⠂⡲⣘⠛⣗⣤⠘⠣⠀⠙⠀⢀⣄⠙⠃⠀⠠⡀⠀⣠⡄⣠⡏⠀⣸⠃⠀⢤⡄⣩
⠀⠀⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣀⠀⠘⣠⠂⣠⠟⠁⠈⢁⣈⠁⠐⢀⡆⠀⠀⠛⢁⠑⡶⠀⠀⣿⡛⠿⠿⠋⢀⣰⠏⠀⠀⣀⡸⢿
⠀⠀⡇⠉⡆⠀⠀⠀⠰⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣥⠀⠐⠃⢰⣯⠀⠀⠀⠀⠀⢀⣶⡄⣥⠀⠀⠀⠈⠣⡀⣀⠀⠀⠑⠒⠲⠚⠛⠁⠀⠤⡀⠸⣀⣈
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠦⠀⡄⠀⠀⠀⢀⠀⣀⣾⠇⠀⠛⠇⠀⠐⠿⣿⡇⢸⣇⠀⣿⢆⣠⠀⠉⣀⠴⡂⢀⠂⡄⠀⣷⠀⡜⠋⠛⠉⡸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡖⠁⠀⠀⠀⠀⣰⣧⡟⢰⠄⠀⠀⡀⠄⠠⣿⢠⡇⣿⠀⠑⠈⠉⠂⡀⠋⣤⠈⣸⡇⣧⡜⠀⠐⠶⣠⣶⣾⢷
⠀⠀⡇⠀⠀⣰⠃⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠀⠘⠀⠀⢠⡟⠁⢃⣼⡀⠀⠀⠀⠀⣸⣷⡌⢃⠘⡆⢻⣏⣀⣀⡀⠚⠛⣦⠀⣤⡈⠁⣦⣽⣦⠟⢻⡇⢘
⠀⠀⡅⠀⠀⠈⠃⠀⠀⠀⠘⠄⠀⣠⣶⡏⠀⠀⠀⠰⡄⠁⣰⣶⠚⢅⠱⢦⣀⣤⣴⡉⠏⣛⠉⠴⣣⠀⢉⠉⠀⣄⢹⠀⠘⢦⠛⡧⠘⠃⠙⠃⢴⣬⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡶⠀⢻⣿⡌⠛⠂⠀⡀⢾⠀⣀⠀⠘⠋⢼⣿⠀⣉⣤⢷⡆⢸⡍⠛⣀⠙⢆⡀⠁⠀⢿⣥⡄⠀⡀⠀⠇⠘⠀⠀⢀⠾⢋⡀⢸
⠀⠀⣯⠀⠀⠀⠀⢈⡷⠀⡴⠀⠀⣀⣤⠐⠒⠋⢤⡈⠀⡀⠉⠀⠀⠀⢻⡇⠀⣨⡼⣷⣾⠕⠊⡙⠰⠚⠉⢂⡀⠀⠀⠰⣶⡅⡀⠀⠀⠀⠀⠈⢰⡾⠃⢸
⠀⠀⠉⠀⠀⠀⠀⠈⠁⠸⢧⠄⠒⠂⢠⠀⠳⢄⠀⠹⣆⠀⠀⠀⠈⠂⢸⣇⠀⢈⠀⢻⡃⠀⣹⡍⠂⢀⡤⠚⠡⣄⠀⠘⣶⣴⡍⠆⠀⠀⠀⠀⢩⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡼⠖⠀⠀⠀⢋⣶⡄⠓⡀⠉⠳⢦⡁⢀⠀⢸⡟⠀⠈⠀⣸⠁⡂⢹⣇⡀⠂⣠⡈⣰⠛⣀⠀⠿⣤⠀⢄⠀⠀⠀⣠⠞⠁⡄⣠
⠀⠀⡄⠀⠀⠀⠀⡾⢠⣾⠃⢲⣶⣟⠀⠈⠁⣿⠀⠀⣠⣤⣄⠙⢦⡀⢨⡗⠂⠔⠸⢷⣈⣱⡞⢋⠀⣤⠼⠈⢉⡴⢇⠀⠂⠀⠀⢸⠁⠀⣐⠁⠀⠈⠓⠹
⠀⠀⡧⠤⠔⠀⠤⢇⣼⠟⠀⣿⢿⣿⠚⣤⠀⠙⣧⡀⢀⠋⠀⠛⠲⠷⣌⡛⠛⠛⠛⠂⡉⢡⡀⣹⣶⠏⠠⢠⡾⡇⠀⢣⡾⠦⣀⠈⣦⠀⡉⠠⠤⠤⠄⣸
⠀⠀⠃⠀⠀⠀⣴⣾⠇⠀⠀⣹⢿⣿⠐⢸⡄⠀⠈⢷⡀⠻⣷⣄⡀⠀⠈⠙⠛⠧⠼⠶⠷⠾⠿⠿⠟⠓⣐⠿⡄⠈⠒⡋⠀⠞⠛⠛⡤⠟⠀⠀⠀⠀⠀⢿
⠀⠀⡆⠀⠀⠈⢻⠋⠀⣤⠀⠇⢸⣿⠀⠈⠀⠀⠀⠈⠳⣄⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠚⠟⣇⠈⡖⠀⠀⣾⠀⠘⣾⠤⠀⠦⠀⠀⠀⠀⣸
⠀⠀⡇⢠⠀⠀⠇⢀⡴⠉⠀⠃⣼⠇⠀⠀⢸⡀⠀⡀⠀⠻⣆⣀⠈⠛⢿⣶⣄⡀⠀⠀⠤⠄⢀⣀⣀⡀⠀⡈⠓⠀⡀⢸⣿⣧⣀⡀⠀⠋⠀⠀⠀⢀⠀⢨
⠀⠀⡁⠘⠆⠀⣀⡌⠁⠀⠁⠀⣿⠀⠀⠄⠀⣷⠀⠹⡀⠀⠙⢾⡦⠀⠈⠻⣿⣿⣷⣤⣤⣤⣴⡾⠋⣡⡞⠃⣠⡾⠃⢸⣿⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⢼
⠀⠀⠟⠃⠘⠃⢸⠀⢀⣿⣠⡄⢩⣷⡄⠀⠀⠘⣧⠀⠀⡀⠀⠈⠷⡼⢆⡀⠈⠙⠷⠾⠟⠉⠁⢀⣼⡟⠀⣠⠁⠘⠃⠸⣿⣿⠏⠠⠖⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠎⠀⠈⠁⠀⠶⠾⢻⣿⢀⡄⢻⣷⠀⠀⠀⢹⣄⠀⠘⠀⠀⠀⠈⠺⢏⠂⡀⠀⠀⠀⡀⠀⢠⡿⠀⡄⢙⡂⠀⠀⠁⠈⢡⣤⡀⢀⠀⠀⠀⠞⠁⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣹⣿⣆⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠈⠀⠉⠀⠈⠀⣈⣡⠏⠱⠀⠀⠀⣠⣌⠟⠻⢮⣧⣆⣰⠇⠀⢀⣿
⠀⠀⠇⠀⠀⠀⠠⠀⠀⣰⠊⠙⢿⣿⣿⡟⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⡏⠀⢠⠀⢠⡎⠁⢈⣅⢠⣼⠋⠙⢿⡎⢉⠉⢻
⠀⠀⠄⠀⠀⢠⠆⣠⠎⡉⠁⠀⠀⠙⢿⡇⠀⢆⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⠒⠄⠀⠠⠒⠋⣠⣼⠟⠀⢠⡞⠂⠺⣁⠀⠸⢧⣸⣧⡀⢀⠸⢅⣸⣇⣤
⠀⠀⠆⠀⢸⡅⠀⠹⡄⠁⠤⠀⠘⠇⣠⠀⠀⠘⢄⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠠⡾⢂⠀⠛⣷⣤⣤⣠⡄⠀⠀⠀⢤⡄⠀⠆⢸
⠀⠀⠂⠀⠃⠰⣦⡄⠈⢑⣆⣠⣤⡄⢸⠀⠀⠀⠀⢂⠀⠀⠀⠄⡀⠀⠀⠀⠀⠈⠉⠿⠿⠛⠉⠀⠀⣠⠀⣷⠈⡆⠀⠸⠁⢘⡁⢀⠀⠄⠀⠀⠉⡀⠀⢘
⠀⠀⠁⠐⠂⠄⠘⢷⠟⠛⠋⠉⠀⠀⢸⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⢱⡄⠀⣦⠎⠁⠐⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⢾⣧⣀⡀⣀⣄⠀⠀⠸⢂⡀⠀⠀⢮⠀⣤⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⢀⡜⠀⠀⢸⠖⢲⠆⠀⠐⠛⠁⠀⠀⠀⣦⡀⠀⡤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠉⠁⠀⢠⡆⠈⠷⠀⠀⠆⠀⡏⠑⣄⣀⢀⠀⠀⠀⠢⠀⠀⠀⣠⡞⠀⠀⠀⡞⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⡄⠀⠀⠸⠁⠀⠀⢆⠀⠐⠀⡇⠸⠟⠉⠉⠻⠷⣤⣀⠀⠀⠈⠀⠀⠀⢀⡰⢿⠄⠀⡅⠀⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡄⠀⠀⠀⠐⡀⠀⠀⣀⣴⡄⠀⡇⠀⣿⡟⠳⠦⣤⣤⡾⠛⠙⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⣁⣀⣀⠀⠠⠤⠤⠤⠀⠀⠀⢰⣿⣟⠀⢱⡄⣠⠀⠀⢹⣿⠀⢀⠀⠀⠿⣿⠀⠀⠘⠁⡇⠀⢸⡆⠘⣷⠀⠀⡀⠤⠛⠀⠓⠀⠀⠤⠤⠤⢀⡀⠀⢨
⠀⠀⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠈⠁⠈⠁⠀⠦⠀⠀⠛⠀⠸⠀⠀⠂⣿⠀⠀⠀⠀⢱⠀⢸⡇⠀⢻⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠄⠀⠀⠀⢳⣄⠀⠀⠄⡇⠀⠀⢹⡀⠀⠀⠀⢸⡀⠈⣷⠀⠀⢻⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠒⣰⣤⡀⠀⠀⢸⣿⠟⠉⠀⣷⠀⠰⣿⣧⠀⠀⠀⢸⠃⠀⣹⣦⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠞⠛⠁⠙⠁⠶⠚⠛⠦⠤⠴⠿⠀⠀⠿⠗⠀⠀⠐⠚⠓⠘⠛⠹⠶⢶⡴⠆⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⡀⠐⡄⠀⠀⢤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢎⠉⠀⠀⠀⠀⠈⠻⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⣤⣴⡖⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠙⠿⠃⠀⠀⠀⠀⣠⣤⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠂⠀⠙⠛⠛⠋
⠀⠀⢸⣿⣦⠀⠀⠀⣠⠔⠚⠉⠉⠉⢉⣑⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡶⠶⡄⠀⠀⠀⠄
⠀⠀⠀⠉⠋⠀⢠⠞⠁⠀⠀⠀⠀⠶⣿⣿⠟⠀⠀⠑⢄⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠑⠀⠀⠀⠠⡟⠀⠀⠀⢠⣴⡶⣦⡄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠉⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⠻⡧⠀⠀⢻⡆⠀⢀⡄⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⠈⠻⠿⠇
⠀⢀⣤⣶⡄⠀⠳⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⣼⣿⡿⠃⠀⠀⠙⠢⢤⣄⣤⡤⠞⠁⠀⠀⠀⠀⠀⢸⣿⣧⠀⠇⣸⣷⡀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⣲
⠀⠘⠉⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢸⠻⡿⠃⠀⢀⡠⠤⠐⠓⠂⠀⠐⠒⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡤⠒⠁⠀⠀⢶⣾⣶⣄⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠇⠐⠋⠀⠀⠠⡀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⣀⣀
⠀⠀⠘⠂⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣶⡄⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠳⣦⠀⠀⠀⠘⡆⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡞⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠈⠺⠿⡟⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣴⡿⠀⠀⠀⢠⡇⠈⠛⠻
⠀⢀⣠⣤⡄⠀⢷⡀⠀⠙⠂⠀⣹⠀⠀⠀⠀⠀⠀⢀⠀⣰⠃⢀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣀⡾
⠀⣾⣿⡿⠃⠀⠀⠢⢄⡀⣀⠴⣣⣤⣤⣀⡀⠀⠀⠀⢢⡏⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⠽⠋
⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠋⠁⠀⠀⠀⣸⠁⠀⠛⠉⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠔⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⢣⣰⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⢸⠁⠀⠿⠀⠀⣸⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢀⡠⠃⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢲⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿
⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⣴⣿⣿⣷⣾⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣼⠟⢒⡆⠀⠀⠀⠀⣀⣀⣄⣀⣤⣾⡷⢿⣿⣿⡿⠟⠉⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣄
⡟⠀⣿⣵⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣄⣨⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⢿⡿⣷⡀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣟⠛⠛⠛⢙⡃⠀⠀⢁⣿⣿⣿⣶⣶⠶⠆
⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣟⠁
⣴⡾⡗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀
⣽⠟⢿⣶⡆⠀⠀⠀⠀⠀⠀⢸⣟⣹⣿⣟⠁⠛⣿⡏⠉⠻⣿⣿⣿⣧⣀⡴⢾⣿⣷⣴⣾⢿⡿⣿⣿⣿⡿⠛⠹⢿⣟⢾⣿⣿⡇⠀⠀⠀⠀⠀⢦⡻⠄
⡟⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⣾⣽⣶⣤⣭⣳⣠⣼⠺⣿⡟⣹⣿⡦⣴⣿⣴⠿⠇⣯⣿⣿⣫⣯⣤⠠⣤⣶⣛⡮⣿⡻⠀⠀⠀⠀⠀⠀⢾⣷⡀
⠤⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⢠⣽⣿⣿⡿⣏⣾⡟⢿⣧⣿⣻⣾⡀⢈⣛⢿⣿⣿⣿⣿⠿⢫⣙⢿⣿⡿⢛⣿⣷⣼⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⡿⠂⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣟⣺⡿⢧⣌⢁⡀⠀⢿⢿⣿⣿⣶⣾⣿⣿⣟⡍⣤⣴⣿⣿⢛⣅⣀⣿⣧⣆⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠓⠀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡽⠧⢥⠡⣟⢢⣙⣿⣿⣿⣿⡿⣿⣿⣿⠿⠯⢽⣿⣿⣿⣿⣿⣈⣮⣿⠵⠼⣿⠵⠶⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣌⣸⣶⣠⣿⡶⠿⣿⠿⠛⢛⣉⡉⢰⣿⣿⣿⣿⣿⡷⠀⠀⠙⠻⠿⠷⣷⣿⣾⣮⣷⣾⣁⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣿⣿⡿⢀⣴⣇⠀⠀⠀⠀⢙⣿⣿⣋⣉⣁⣸⣟⡏⠀⠘⣛⡏⠀⢀⣉⣭⣿⣿⣿⣿⣿⣶⣄⢠⣀⣈⣿⣿⣻⣿⡿⣿⣾⣿⣿⠓
⢸⣿⣿⠇⣾⣿⡿⠀⠀⠀⠀⠀⠉⠛⢿⣟⠻⣿⣷⣶⡶⣶⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡾⠿⠋⠀⣭⡉⠉⠉⢁⣘⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀
⣾⣿⣿⣸⣿⣿⠇⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⢿⣿⣷⣗⣷⣾⣿⣿⣿⣿⣿⡿⠏⠀⠀⢀⣤⣤⣤⣤⣤⡿⣿⣿⠋⠟⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⣿⢿⣯⢹⣿⣿⣿⣿⣿⣿⣶⣴⣤⣴⣾⡿⠿⣿⡟⣹⣶⣿⠟⠁⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⣿⠀⡙⢾⣿⠟⣿⣿⠁⡘⣿⢿⠋⣼⣯⣴⣷⡟⠩⣿⣿⣿⠀⠐⠋⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⣠⠀⣿⠀⣀⣼⣿⣿⣿⣿⣾⡇⢹⠘⡀⣸⣽⣿⣿⣿⡿⠋⢹⣿⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠤⠀⠻⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣱⣿⣿⣿⣿⣿⠄⠀⢸⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣶⣿⡄⠈⢹⣿⣿⣿⡿⣿⣿⣿⠀⢿⣿⣿⠟⠉⠃⠀⠀⣸⣿⠈⠛⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿
⣿⣿⣿⣿⡇⢀⣴⣦⡀⠀⠀⠀⠀⠀⠃⠀⠀⠻⡿⠃⠀⠘⣿⣿⣿⡇⠘⣿⣿⣶⣾⡿⠁⠀⠀⢀⠀⠀⣻⡿⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟
⣿⣿⣿⣿⣧⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⣿⣿⣿⡇⠀⣿⢿⣿⠏⣹⣷⣷⡞⠁⠀⠀⠿⠁⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⣿⣿⡇⠀⣿⡇⠁⠀⠹⠟⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠉⠉⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣴⡀⠀⠀⢰⣞⡿⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⢾⣝⢦⡆⢸⡿⣿⡄⠶⣿⡟⣵⣟⣹⣆⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⣿⣦⠘⢻⣿⠃⠀⠛⠀⣿⠟⠋⡈⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠉⠉⠙⠋⠓⠀⠡⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣈⣿⣧⣴⡆⠀⠀⠰⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⡿⡭⠙⠂
⠀⠀⠀⠀⠀⠀⠰⢶⡆⠀⠂⠀⢹⣞⣿⣿⡳⠄⠀⠀⠸⢿⡿⠋⠻⠄⢠⣾⣿⢿⣿⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣷⣶⡈⣿⣿⣿⣷⡀⠀⠀⠀⢀⣤⣤⣄⣰⠟⢋⣵⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⢸⣿⢿⣧⡀⠰⣿⣿⣦⣿⢟⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣻⣿⣄⣿⣿⣯⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣮⣿⢿⣏⣟⣟⣿⣿⣿⣿⣿⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣿⡿⠛⢉⣠⣤⣿⣿⣿⡿⠭⣥⣴⣾⣿⣿⠿⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠋⠁⠀⣠⣾⡿⢿⣵⠟⢹⡇⠀⠀⠉⠙⠛⠛⠿⠻⠖⠲⠒⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣿⣿⡿⠋⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⣿⣦⠀⣼⣧⠀⠀⠀⠀⠀⢀⣴⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣿⣆⢻⡟⢀⣀⠤⢦⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣬⣻⣿⠸⡷⣬⡿⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠙⠿⣿⣦⣧⣽⣴⠶⠷⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣄⡈⠻⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢸⣧⡄⢀⡀⠀⠀⢀⡜⠁⠀⠀⢹⣿⣦⠀⠈⠛⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⢿⣦⠈⢿⣷⣤⡏⠀⠀⠀⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁⠀⠀⠀⠈⢿⣇⣴⠟⠿⣿⣆⠀⠀⠀⠀⠈⠙⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢸⡍⠁⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⣸⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡏⢻⡆⠀⡀⢰⣷⣀⠀⣰⠇⠈⠃⠀⠀⠀⢠⡀⠀⠸⠿⣿⡧
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⡿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣽⣇⣴⠂
⠀⠀⠀⠀⠀⠀⠛⠭⢽⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡔⠀⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣵⣀⡀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠘⠛⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠿⣿⣿⣿⣿⡷⠶⣎⡿⠇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⠁⠀⢸⡿⠃⠱⡞⢿⣿⣷⠆⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⢠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠾⠾⠶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠁⠀⠀⠀⠔⢂⣈⡭⠥⠤⢤⣀⡐⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡷⠋⠀⠀⣀⣤⠴⠄⠒⠋⠉⠁⠀⠀⡄⠀⠀⢉⠛⠆⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡴⠖⠒⠉⣁⣀⣀⣤⣴⣦⣾⣴⣿⣿⣿⣷⣷⠟⠋⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠯⢱⣤⠐⢠⡉⢹⣿⠟⠉⠉⠀⠉⠉⠋⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣷⠂⠀⠀⡜⠁⠀⣀⠤⠴⠤⢄⠀⠀⠒⢓⣛⣒⣒⣚⣛⡒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣒⣸⡆⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠛⠋⡩⡭⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⢿⠞⢫⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠦⠰⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡿⠀⢀⡀⢀⣠⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⡄⠀⠀⠀⠀⢀⡾⠀⢰⣧⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣾⣿⡿⠟⠿⠷⠷⠼⢼⠟⠁⠈⢨⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠂⠀⠀⠀⠀⠀⠘⠀⠈⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⡀
⠀⠀⠀⡀⢠⣤⣤⣶⣮⡽⣿⣿⣿⣿⣿⣧⣿⣶⣦⣤⡀
⢀⣴⣾⡽⣾⣟⢿⣿⣿⣿⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⣿⣿⣿⣿⣯⠻⣷⣿⣿⣯⣿⢿⣿⣿⡿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣆
⣫⣩⣯⣶⣭⡀⠿⣟⢱⣿⣷⡜⣿⣟⣵⢿⡿⢟⣿⣿⣿⣟⣿⡿⡯⢾⣫⡷⣦
⣿⢋⣦⡿⣿⣷⣶⣬⣧⡈⢿⠃⣽⢈⡇⣼⣿⡿⠯⢞⣝⣻⣵⡶⠛⣿⣿⣿⣞⣢⣄
⢟⣾⣿⣿⣿⣷⣿⣿⣿⣿⣾⢃⣿⡟⣴⠛⣠⣶⣿⣿⣿⣿⣿⣿⣄⠘⣿⣿⣿⣷⣬⣦⣤⣀
⢀⣿⣿⣿⣷⣾⡏⢉⣿⣿⡏⡄⡏⡛⡇⣾⣿⣿⣿⣿⣿⣦⣴⣿⠏⡀⣿⣿⣻⣿⡿⣿⣧⣽⣧⣄
⢸⣿⠉⠉⠉⠁⢀⣾⣿⣿⣧⡉⣆⡧⠆⢸⣿⣿⣿⣏⠻⣿⢿⣤⣠⡄⢸⣿⡿⣿⣿⣿⣿⡿⣷⣮⣤⠤⡀
⣸⡿⢡⢠⣷⣦⣼⣿⣿⣿⣿⣠⣿⣿⣧⣌⠽⡿⢿⣶⣤⣶⡿⣏⣛⣷⣾⡿⣛⣻⣯⣿⣿⡿⠛⠋
⣉⡻⣿⡿⢿⣿⣿⡿⣴⣷⡿⣼⣿⣿⣿⡯⣾⣻⣦⢽⠛⠋⠴⣿⡿⡋⢿⣛⢽⣡⣿⣿⡟⠁
⢿⡃⣿⣾⡿⣝⣿⢿⣫⣼⡟⢿⣿⣿⢛⣶⡺⠧⠀⠈⢀⣴⣴⣟⠛⠤⣾⣧⡍⣵⠛
⣯⡾⣿⠿⣷⠟⣻⣮⡻⠾⣧⡿⣛⠿⢿⡽⠏⠉⣀⡌⣻⣤⡭⢹⡷⣷⣘⡦⠽⠁
⢻⣿⡇⣮⢴⣶⣇⢙⢽⣿⡟⣹⣾⢷⡖⣋⣴⣸⡷⡱⣛⣛⣣⣿⡵⠟⠋
⠀⠃⠠⣌⡛⣋⡼⣋⣛⣟⣿⣍⢛⣟⣤⢟⣿⣭⣽⠴⢿⡯⠛⠉

⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣏⣿⣿⣥⢿⣿⣻⣿⣿⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣍⢳⣿⣿⣿⣿⣾⣯⣷⡝⣿⣷⣿⡿⣿⣿⣟⡿⣽⣿⣿⣿⢹⣿⣿⣿⣿⣿⣻⣿⣾⣿⣿⣿⣯
⡀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣸⢋⣽⣧⢻⣝⡿⣛⣳⣭⣛⢇⢸⣿⣿⢃⣿⣿⣿⡜⣿⣿⢿⣃⣾⣿⣿⣿⢿⣿⣻⣿⣿⠿⣿⣿⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⢻⣯⣽⣿⣿⣷⣶⡿⢉⢹⣿⣿⣿⣟⡙⠿⣇⠛⢿⣿⡇⢸⡏⢿⣿⣡⣿⣭⣾⣿⢟⣿⢿⣿⡾⣻⣗⣯
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣭⣙⣶⣾⣿⣿⣿⣿⣿⣿⣠⣿⣿⣽⣿⣿⢿⣿⣿⣷⣷⡀⢻⡅⢸⡏⣰⡿⣻⣿⠿⠛⢷⣋⣡⣯⣼⣿⣿⣯⠅
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠙⣡⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⢿⣿⣿⣿⣿⣼⠃⣾⣿⡟⣴⠿⢁⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣝⣛⣿⣿⣿⣿⣿⣿⣿⣿⠃⠨⣿⣿⣿⣿⣿⣷⣿⡏⠙⢿⣿⣿⡽⡇⢻⡻⡿⣯⢰⣿⣿⣿⣿⣿⣿⣿⣇⣈⣠⣿⣿
⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡾⣿⡟⠛⠛⠛⠛⠉⠀⣼⣿⣿⣿⡗⠿⣾⢹⡇⣹⠸⣿⣿⣿⣿⣿⠛⣿⣿⣿⣟⣋⠁
⡀⣶⡆⠀⠀⠀⠀⠀⠐⠛⠛⠛⢻⣷⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠀⢹⣿⡅⠀⢰⣀⡄⠀⣴⣿⣿⣿⣿⣿⡆⢿⣷⣟⣻⣌⣿⣿⣿⣿⡿⠇⠈⢛⣛⣻⣿⣷
⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣝⣿⣧⣾⣿⣭⣿⣿⢐⣿⣿⢁⡖⢸⣿⣿⣼⣿⣿⣿⣿⡿⣿⣤⣾⣿⣿⣿⣿⣿⠿⣿⠿⣿⣿⣶⣿⣿⠿⣏⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢽⣿⣯⣿⣿⡯⢻⣋⡙⣿⣽⣿⣾⣿⣿⣿⣿⡿⣥⣾⣷⣿⣫⣿⣿⣿⣿⣿⣧⣾⢿⣿⣌⣿⠟⢿⢃⣴⣿⣿
⣦⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣯⣿⢿⢰⣿⣿⣿⡇⢹⣟⣷⣿⣿⣿⣿⣷⡽⠛⣹⣿⣿⣿⣿⣿⣿⣿⣬⣛⣟⣉⠅⣾⢄⢁⣄⣨⣿⣿
⠋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡣⣿⣮⣟⠉⢛⣹⢧⣾⣿⣿⣟⣿⣞⣿⢹⣾⣿⣿⡆⣹⣿⣿⣿⣱⣾⣷⣮⣏⣁⣀⣠⣤⠿⢿⡿⣟⣄
⣄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢸⠛⢙⡿⣻⣷⣟⣿⠟⣛⡿⠝⢍⣿⣷⣛⡛⢻⣷⡿⣫⢭⠿⠚⣷⣿⠛⠀⣠⣴⠀⣽⣿⣷⠳⣌⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨⠁⠹⣿⣟⡏⢰⣧⣶⣾⣾⡀⢙⣿⣿⣿⡟⣹⣷⣿⣿⡗⠻⢁⣾⣨⣿⣾⢹⡛⣛⠿⢳⣿⣫
⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⡟⠀⡉⠻⢿⡷⠋⣱⠿⠿⣿⣿⣷⡾⢷⠿⡿⠃⣽⣿⣿⡙⢷⣾⣠⣿⣿⣿⠿⠟⠃
⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⢙⢿⣶⢴⣿⣇⣿⣿⢻⡾⣿⣿⣦⡾⣿⣿⣯⣶⣿⣿⡿⣿⣹⠟⡋⠁
⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉⠉⠉⡟⢮⣻⡷⠘⡗⠛⠉⠛⠛⠻⠁⠀⠁⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⣟⢹⡇
⣾⣞⣇⡀
⣿⣿⣿⣇⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣄⠀⠀⠀⠀⣼⣶⣦⣴⣷⣤⣿⣴⣷⡴⢶⣾⣧⢴⣷⣷⣤⣤⣮⣶⣿⣶⣶⣴⣦⣦⣤⣶⣴⣶⣾⣦⣶⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢀⠀⠤⠈⠑⠒⠒⠦⢆⡀⠘⢂⡀⣤⠈⠫⡒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣰⣆⠀⢀⠀⢈⣡⣴⡴⠂⢀⣠⡅⣈⣳⣀⠉⠀⠈⠐⠐⢆⠀⠲⠿⢦
⠀⠀⠀⠀⠀⠈⢧⡃⣿⢂⣡⣶⠾⢁⡥⣄⠀⠈⠀⠀⠀⠈⢥⠀⠒⠠⡄⠈⠺⣆⠐⠀⠨⣷⡇
⠀⠀⠀⠀⠀⡸⠂⢻⡿⢻⢋⣀⡠⠁⠒⠐⠆⠉⠀⠀⠀⠀⠈⢣⡆⣀⣧⠐⠂⢼⠙⣦⠀⡼⣇⡰⣆
⠀⠀⢸⡀⢠⠀⡸⢟⡷⣟⢩⣀⣀⣠⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠴⢼⣄⢹⡁⡄⣿⣧⡟
⠀⠀⡄⢧⣊⢲⡟⡿⣼⠟⠉⠑⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⠊⠦⣘⡁⠻⢷⣿⣸⣿⢠⠄
⠀⢣⠰⠞⣧⣟⡽⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⡀⣿⠹⣿⢲⡂
⠀⠈⣷⡀⢸⣿⡡⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣙⣿⡇⣿⣔⠇
⢶⡀⣏⢿⣾⣿⣼⣯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⢹⡟⣿⣿⠅
⠀⢻⣸⡎⣿⣿⠋⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⢾⣿⣿⠏
⠀⠈⣿⣧⣿⣿⢸⠟⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠈⠫⣼⣿⣿⡄
⠠⣿⡼⣧⣿⣿⡅⠀⣻⣟⡆⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡶⠷⣾⣿⡇⣷
⠀⢺⣿⡹⣿⣿⣧⠀⢊⣿⣷⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡿⢈⣿⣿⡇⡇⠀⠄
⠀⠀⠩⣾⠘⣿⣿⣧⠘⣿⡏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣽⡗⣾⣿⣿⣷⠁⠈
⠀⠀⠀⠙⣧⣿⣿⣿⣷⣜⣾⣿⣭⣶⣼⡆⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⣰⣶⢽⡟⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣯⣄⣻⣺⣿⣿⣦⠀⠀⠀⠀⠈⣻⠻⠾⠷⢧⣧⣯⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠙⢻⠿⢄⣀⢹⣿⣋⡉⠉⢉⠉⠙⠛⠻⣥⠋⢁⡔⢺⣓⣰⣿⣿⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⢿⣶⡞⣛⣶⣤⡔⣀⡤⣷⡺⠛⠉⠉⢉⣉⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠻⡟⠉⠝⠉⠂⠨⠃⠀⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠠⢀⣠⣤⣤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⣴⣿⠿⢙⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⣴⣦⣖⣁⠀⣀⠀⠀⠀⠀⠠⠀⠀⢀⣡⡀⡀⠀⠀⢠⠀⠀⢀⣀⠄⢀
⠀⠀⠀⡀⠀⢹⣿⣀⣤⣼⣿⡄⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠈⣹⣿⠃⠁⠀⠀⠀⠀⠐⠀⣰⣿⡟⢿⢿⣿⣦⠀⢄⡀⠙⣽⣷⣿⣤⡤⠜
⠀⠀⠀⠈⠀⣄⠙⠿⠟⠙⠉⠁⠀⠃⠈⢪⡀⠀⠀⠀⣠⣴⣃⠀⠠⢻⢿⡿⢿⠿⠘⠁⠀⠀⠀⠀⠀⢠⠻⣿⣯⣵⣾⠿⡇⠀⠈⠀⠀⠈⠛⠿⠛⠇
⠀⠀⠀⠀⡀⠀⠑⠶⠆⡘⢃⠀⢠⠀⠐⠀⠉⠛⠃⠀⠸⠔⠉⠑⠄⠀⠀⠤⠂⠀⣠⡶⠄⠈⠀⠄⠀⠑⠀⠙⠟⠛⠋⠀⡉⢸⠀⠠⠄⢂⠤⠤⠶
⠀⠀⠀⠀⠀⢈⣀⢲⣸⣧⠠⡡⠴⠒⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠔⢈⣥⠀⣀⣀⠀⠔⠂⠀⠀⠀⠀⠁⠘⣶⠸⡷⢤⣠⠊
⠀⠀⠀⠀⠀⠀⠈⢙⠻⣆⠄⠠⠀⠐⠁⢀⣅⠀⠀⠀⠀⠀⠀⠀⠨⢉⡉⢦⣶⢹⣥⡄⡀⢅⡀⠀⠀⠀⠀⠄⠀⠀⡀⠙⠠⣀⠈⢁⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠖⣿⢀⠃⣠⠔⠉⠁⠀⠀⠀⠀⠀⡀⠠⠄⠀⠉⠸⠛⢽⡄⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡁⠒⢠⣿⡁⠟⣋⠙⠦⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠘⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⣀⠛⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠟⠛⠛⠛⠉⠉⠙⠛⠛⠿⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠉⠀⠀⢀⠄⠂⠉⠉⠉⠙⠒⢄⠀⠀⠀⠉⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⣰⠏⡀⡠⠀⠀⠀⠀⠠⡀⠑⡄⠀⠀⠀⠀⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠃⠀⠀⠀⠀⠀⢠⠛⣼⡟⠁⠀⣠⠀⢠⠀⠙⣧⣿⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⡾⢀⡟⡰⢀⣸⣥⣶⣶⡅⠀⢻⣏⠇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⢇⣾⣿⣿⣿⣍⣰⣊⣭⣿⣶⣮⣿⣾⡀⠀⠀⠀⠠⠀⠀⠄⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡞⠚⢸⣾⡿⠿⠛⣶⡞⠻⡿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⣿⣤⣶⣮⠸⢿⠿⢈⣽⣧⣼⡇⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⡀⢾⡿⢰⣾⣿⣿⣿⣧⡀⠀⠀⢨⡿⢹⣿⣷⡄⣿⠛⢆⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⢀⣄⡒⠻⣿⡇⣾⢿⣿⣿⣝⠙⠛⠒⠚⠛⢁⣿⣿⢿⣿⣶⡶⠖⠛⣒⣀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⣠⣍⡉⠉⠁⣼⣷⣿⣦⠹⣿⣿⣷⣀⣀⣀⣰⣿⣿⠏⣼⣿⢿⣡⠂⠈⠉⣩⣑⣀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣏⣻⡭⠍⠁⠈⣍⠈⣿⡿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⢯⣿⡯⢀⣭⠁⠈⠭⣽⣏⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡽⣿⣿⣿⣮⣏⠳⡈⢻⡌⢻⡿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⣿⠏⠠⢋⣽⣴⣿⣿⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢿⣿⣿⣿⣿⣷⣄⠀⢻⣿⣾⢿⣿⠁⠙⢻⡿⣿⢿⣿⣿⠃⢀⣴⣿⣿⣿⣿⣿⢧⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢨⣿⢿⣿⣿⣿⣿⣷⣄⢻⣷⡈⣿⢠⠘⢷⢆⡏⣾⣿⠃⣴⣿⣿⣿⣿⣿⢿⣧⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠏⠱⢸⣟⣻⢽⣿⣿⣿⣦⣵⠏⣸⠀⠐⣿⣸⡇⣿⣇⣼⣿⣿⣿⡫⣏⣽⢳⠇⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡘⢻⡌⠃⠈⠈⠙⣿⣿⣿⣯⣿⡆⢈⠻⠿⣯⣿⣿⣟⡿⠋⠀⠀⠉⣼⠛⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠁⠀⠀⠀⠀⠈⠙⠹⢿⣿⡇⠠⠏⢠⡿⢿⠼⠋⠀⠀⠀⠀⠀⠹⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⢹⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠘⠛⠻⣿⣿⠿⠛⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠠⠤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠿⣿⣿⡿⠿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠁⠚⠆⠀⠀⠠⡸⡄⢠⢇⠄⠀⠀⠰⠓⠈⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⢀⣀⡀⢌⡒⣄⣱⡇⢸⣎⣠⢚⡡⢀⣀⡀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠠⠐⠒⠢⠬⢕⣼⣦⡼⠀⠀⢧⣴⣧⡪⠥⠔⠒⠂⠄⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠐⠈⣉⠬⢿⣯⡇⠀⠀⢰⣼⡿⠧⣉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀
⠀⠀⠀⠀⠀⢀⠉⢉⣹⡦⢄⡀⠀⠀⠀⠀⣀⠀⠒⢈⣠⣾⣏⣲⣖⣹⣷⣌⡑⠒⠀⣀⠀⠀⠀⠀⢀⡠⢴⣏⡉⠌⡄
⠀⠀⠀⣠⠔⢚⠲⣮⠘⡟⠀⠑⠒⢒⠖⠉⠀⠀⡪⢒⠝⡟⡟⠁⠈⢛⢻⠻⡒⢕⠀⠀⠉⠲⡒⠒⠊⠀⢻⠋⣵⢖⡓⠲⢄
⠀⠀⠀⠘⢄⠈⣫⣉⣫⠿⠛⠉⠫⡏⠀⠀⠀⠀⡔⠁⠘⠀⠁⢀⡄⠘⠀⠃⠈⢢⠀⠀⠀⠀⢹⠝⠉⠓⠿⣽⣉⣝⠁⡠⠃
⠀⠀⠀⠀⠀⠉⠉⣮⠁⠀⠉⣉⠅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⣉⠉⠀⠈⣵⠉⠉
⠀⠀⠀⠀⠀⢀⠎⡠⠴⡄⢈⠀⢸⠣⡀⠀⠀⠀⠀⢀⡴⡖⠺⡸⢇⠗⢲⢦⡀⠀⠀⠀⠀⢀⠜⡇⠀⡁⢠⠤⢄⠱⡀
⠀⠀⠀⠀⠀⠸⡆⢠⠊⠀⠇⢀⠘⡠⡿⡵⢲⠒⡈⠁⢂⠟⠓⠁⠈⠛⠘⡐⠉⢁⠒⡖⢮⢿⢄⠃⡀⠸⠀⠑⣄⢰⠇
⠀⠀⠀⠀⠀⢸⡀⢁⠤⣄⠔⠁⠉⠀⢱⠐⢁⣀⣡⠎⠋⠀⠀⠀⠀⠀⠀⠙⠱⣌⣀⡈⠂⡎⠀⠉⠈⠢⣠⠤⡈⢀⡇
⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠈⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠁⠀⠀⠀⠀⠀⠀⠈⠱⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢺⡛⠉⠉⠉⠉⠉⠈⠉⠁⠀⠀⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢲⣟⠛⠙⠋⠛⠛⠉⠉⠀⢀⠀⠁⠐⡇⠀⠀⠀⢠⣤⢠⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡴⠖⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⢀⣶⡀⠀⠀⢀⡇⠀⠀⠀⢸⣿⡜⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢰⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡜⢀⡀⠀⣶⠀⢠⣇⠀⠀⣸⠹⣧⠀⠀⣸⣷⠀⠀⠀⡞⣿⣧⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⣠⡄⠀⣸⣇⠀⢠⢿⣷⢠⡇⢸⣇⠀⣿⣆⢸⣿⡄⠀⡅⠀⣿⡇⢀⠅⣿⠀⠀⠀⡇⣿⣿⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣧⡀⡿⣿⡀⣼⠈⣿⣼⣷⣼⣿⢸⢻⣿⣾⣿⣇⠀⡇⠀⡿⣿⢸⠀⣿⡇⠀⢰⠃⢻⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⠀⡇⢻⣷⡏⠀⢻⣿⡿⠛⣿⡿⠀⢻⠁⣿⣿⢀⠇⠀⡇⠹⠟⠀⢹⡇⠀⢸⠀⠀⠀⣇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⣿⡇⠘⢷⡇⠀⠀⠿⡇⠀⠙⡇⠀⢸⡆⣿⠹⣾⠀⠀⡇⠀⠀⠀⣼⡇⠀⣼⠀⠀⢀⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡏⣿⠋⣷⠀⠀⠁⠀⣰⠀⡇⠀⠀⡇⠀⢸⠇⠻⠰⢹⠀⠀⠁⠀⠀⢰⣿⡇⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠸⡀⣸⡆⠀⠀⠀⣿⠀⠟⠀⡆⢹⠀⢸⠀⠀⠀⣼⡇⠀⠀⠀⢀⣿⢻⡇⠀⣿⠀⠀⡌⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡄⠀⣿⣿⠀⠀⠈⣿⡆⠀⠀⠀⣿⡆⢸⠀⠀⠀⣿⣧⠀⠀⠀⣼⢹⢸⣿⢀⢿⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⣿⢹⡇⠀⠀⣿⡇⠀⠀⡀⣿⣇⣸⡄⠀⠀⣻⣿⠀⠀⠀⡇⢸⠸⡏⢸⠘⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⡘⣿⠀⠘⡟⣿⡀⠀⠀⡏⢹⣿⡇⠀⠀⡏⣿⠀⠀⢰⠃⢸⢰⣷⡜⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡏⣿⢿⠀⢰⡷⣿⠀⠀⢠⡇⣼⢹⡇⠀⠀⡇⣿⠀⠀⢸⠀⢸⣸⠻⠇⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⢸⡆⢸⣟⢻⠀⠀⢸⣇⣿⠸⡇⠀⢸⠁⣿⠀⠀⡏⠀⢸⣿⠀⠀⠀⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⢸⣾⠄⠀⢸⢿⠟⡀⡧⠀⢸⠀⣿⡄⢠⠇⠀⠈⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⢿⡇⠀⡾⠈⠀⠃⣟⠀⢸⠀⣿⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⣿⠀⢸⡄⡇⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⢹⠀⣿⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠘⣦⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡎⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠤⠠⠄⠀⠀⠀⠀⠀⡎⣴⢄⡰⠊⠁⠀⠀⠀⠁⢀⡤⠠
⠀⠀⠀⠀⠀⠠⠚⠃⠀⠀⢀⡀⠀⠀⠀⠠⠀⠀⢨⢠⣿⣽⠡⠄⣀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠑⡀⠀⣠⠔⢂
⠀⠀⠀⠀⠀⠈⠢⢤⠖⠀⠀⠀⠈⠛⠛⠓⢂⡄⠈⡴⠉⠀⠀⠀⠀⠑⢶⡏⢀⣀⠀⠀⠀⠀⠀⠂⠔⢁⡡⠘⣆⠔⠂⠂⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠀⠤⢠⣴⣦⡌⠘⡃⠀⠀⠀⠀⢀⣾⣮⠧⣿⣿⠿⠤⠀⡀⠀⣯⣸⣿⣯⣽⣿⣿⠇⢶⣥⢴⡖⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠤⢴⡛⢮⡿⡾⢷⡦⢰⠀⠀⠀⠀⢚⠟⣿⣿⣿⣷⠶⠶⠆⠙⡶⢁⠤⠤⢤⣙⣋⠋⠑⠀⠹⣇⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⣄⣀⣀⠀⠂⣀⣤⣯⢾⡳⣆⡀⠤⠊⠾⢽⠻⣇⠈⠙⠀⠀⢀⠳⠁⠀⢠⣤⣽⣿⣧⣔⠁⠀⠙⠤⡀
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠉⠉⣿⡞⠻⣿⣿⣆⢱⣦⡈⠳⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡤⠾⠿⣿⣷⣧⣯⠙⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡤⢀⣤⣤⠴⢻⣿⡘⠿⠉⠹⠀⠻⣿⣦⣭⢿⣶⣦⣄⣠⣔⣠⠄⠀⠀⢊⠀⠀⠀⠙⠛⠛⠿⠀⠀⠀⠇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠃⠈⣃⠀⠀⠀⠀⣼⠁⣾⣿⠞⠻⠀⠉⢻⣟⣿⣿⣶⣦⡉⠂⠀⠀⠀⠀⣀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠺⣟⠛⠉⠁⠀⢽⠀⠋⣁⠔⠃⠀⠀⠸⣟⠹⠛⠿⠉⠁⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑
⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠰⢶⠱⣾⡄⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⢸⣆⠀⠠⡄⢰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢠⣦⡈⡵⡞⣧⠀⠀⠀⠌⠉⠉⠀⠀⠀⠀⠀⢸⠛⠀⠘⠃⠸⠛⣧⠀⠐⠀⠀⠀⠀⠀⢀⣆⠀⠐⣇⠀⠀⠀⠀⡆⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⢈⢶⠀⠀⢠⣏⠀⢡⠞⣷⣿⡆⠀⠐⣢⣤⣴⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣤⠛⠀⠀⠈⣄⢀⣀⣠⢼⣿⣆⠠⠀⠀⣷⣇⠀⢃⣷⣿⣿⣿
⠀⠀⠀⠀⢘⡷⠰⠁⡁⠀⢸⠏⣷⣌⣘⠻⠂⢙⠀⡀⣹⣷⣯⡼⣦⠀⠀⠀⢸⠀⠀⠀⠀⠘⡯⠄⠀⣷⢦⣿⡏⠏⡌⠉⣿⠉⠟⠚⠈⢉⣵⠀⠈⠙⢻⣿⣿
⠀⠀⠀⠀⠉⠱⠀⢀⣿⠀⣿⣀⠫⠹⡿⣏⣼⣿⠀⣶⣿⣟⢁⡀⣻⣅⠀⠀⢸⠀⠀⠒⠒⢀⣭⣦⡴⠁⣴⡶⡀⠀⠁⠈⠀⠀⠀⣀⡠⣾⣿⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⢍⡀⠈⠀⠡⠿⣽⢻⠀⠿⣿⠛⡏⠻⠿⣿⣷⠀⢸⠀⣴⡂⢰⡏⠉⣾⣋⡁⣿⣿⡇⠀⢸⣆⠀⠀⠀⠀⢠⣿⢿⠀⠀⠀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠉⠃⢀⠀⠉⠆⠀⠠⢠⠤⢥⢰⠀⠀⠀⠛⠙⠀⢸⡀⠛⢁⠈⠃⣴⠅⢸⣽⣿⣿⡿⠀⠀⠈⢷⡀⢀⣺⠖⣏⣿⡇⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⠇⠀⡀⠋⠀⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⣼⡤⠆⠀⠠⠙⣯⣾⣿⡄⠀⠀⠀⣸⣿⡆⠀⠀⢸⣿⠅⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⠀⢸⡇⠈⠀⣤⣄⣴⣶⣾⣿⣿⣿⣀⢀⣤⣾⠿⠋⠀⠀⠀⠈⠁⠀⠰⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠎⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠐⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠹⠁⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⠂
⠀⠀⠀⠀⠀⠈⠓⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⢠⠀⠀⠀⢸

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⡀⠀⠘⣄⠀⡀⢀⠀⢀⠜⢀⠄
⠀⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠊⢧⠑⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢄⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠐⠒⠐⠂

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

⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⢉⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠃⠀⠀⣉⣀⢀⠢⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⢶⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⢣⡄⢣⡣⢋⢲⣆⡁⣧⠀⠀⡄⠀⠀⠀⠀⠀⢀⡘⠳⢯⣜⠑⠀⠀⠀⠠⠀⠀⠀⠀⠈⣿⣿⣇⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠃
⠀⠸⣄⡙⢤⣌⠒⡼⠻⠥⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣜⡈⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⢠⣼⡧⠿⣦
⠀⠀⠈⢿⣄⡲⠭⢤⡂⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠤⢄⡼⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠖⠉⠂⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡁⠊⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⡀
⠀⠀⠀⠀⠀⢀⣀⣴⠠⠁⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⡘⠴⢞⠊⣀⣀⡔⠊⠑
⠀⠖⠀⢶⣨⠙⡏⡉⣴⠤⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠙⠟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣠⡇⡀⢀⡀⠀⠀⢀⠁⠄⠀⠀⣸⣗⢯⢺⡛⢁
⠀⠀⠰⠂⠁⣀⠀⠀⠀⣠⠄⣄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡐⠹⣦⣀⣸⠂⡠⠀⠀⠀⠀⠀⠀⠐⠐⠿⠿⠄⠀⠀⠀⠀⠀⠀⢀⡛⠈⠈⠙⠉
⠀⠀⠀⠀⠐⠆⣡⠖⠬⠤⠄⠂⠀⠀⠀⠀⠀⠀⠐⣦⡀⠘⢙⣾⡿⣫⡥⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⢀⣀⣽⡥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠙⢃⡀⠀⢀⠀⠀⠀⡀⠀⠂⠀⠄⡈⠈⠉⢢⡀⠀⠀⠀⠀⠈⠳⠀⠲⣀⠄⠇⠀⠀⣄⠔⠀⠀⠂
⠀⠀⠀⠀⢀⣭⡿⠠⣤⡴⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠒⠀⠁⠂⠀⠀⠠⣀⠀⠀⢈⢠⡄⣿⣿⠞⠀⣝⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠀⡇⢠⠏⣡⠂
⠀⠀⠀⠀⠉⣉⣄⠀⢹⠁⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢣⣙⡁⢀⣀⠁⠀⠀⠀⠀⠀⠈⠂⠤⡀⣇⠀⣇⢾⡔
⠀⠀⠀⠂⠈⡿⠁⡆⠌⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣻⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂⢨⡾⢰⣿⣮⠤⠂
⠀⠀⠀⠀⠀⠁⠂⠁⠀⠛⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⡟⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⠛⠉⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠒⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡼⠃⠀⠀⠀⢿⣿⣿⣆
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠀⣼⣷⣄⡀⠀⠀⠈⢯⣉⣿⣷⠄
⠀⠘⠉⣀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢰⠁⣿⡀⠀⠀⠀⠀⠀⣰⣶⡷⡀⠈⢿⠀⣿⠿⠈⣿⣦⠄⠀⠛⠛⣿
⠁⠀⢠⣿⠷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⠏⠀⢀⠀⠌⠁⣷⠿⠂⢿⣄⠀⣶⡆⠈⣿⣷
⠀⠀⣘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠈⣱⣠⡄⠀⠀⠀⢸⣿⣿⣷⣦⣠⣾⣏
⠀⠀⠈⢻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⠀⠀⠀⠀⠈⢁⠹⡟⠉⠠⢴⡷⠈⠹⢿⣿⣿⡏⣹⣿⣦⡀
⠀⠘⠋⢩⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠆⢠⡾⢻⡦⢻⣄⠀⠘⠧⠀⠀⢸⣿⠀⠠⢽⡟⣯⡉
⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⢿⠀⠀⠗⠈⡿⠇⠹⢿⡄⢠⣾⡏⠀⠀⠀⢠⠸⠇
⠀⠀⠀⢀⢠⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠻⠶⠀⠂⢠⣶⣦⠀⣸⣿⣦⠿⢿⣆⡀⠀⠀⠀⠐⠂
⠀⠀⢀⣸⣿⣿⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠘⠻⢿⡀⠛⠀⢸⣷⠎⢻⡁⠀⠀⠀⣤
⠀⠀⠀⠉⠸⢿⣷⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⡦⠀⠀⠸⡇⠀⢸⣅⠀⠀⠀⠻
⠀⠀⠀⢠⡀⠘⢛⢻⣿⣦⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⢹⣤
⠀⠀⠀⣿⠃⠀⠈⣼⣿⣿⣿⣷⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠘⠣⣦⣤⣿⣝⠋⠹⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣶⣾⡿⠻⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⡛⣿⡃⠀⠉⣿⡿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⡿⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠛⠟⠁⠀⠰⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⡀⠀⢻⣿⣇⠀⠀⠀⢹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣴⠆⢸⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠈⠁⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⢃⡀

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⣧⡉⠙⣀
⠀⠀⠀⢸⠃⠁⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠠⣽⠁⠰⡅⣄
⠀⠀⠀⠸⡃⠐⢶⠄⠶⠐⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⣸⡅⠈⠀⠀⠀⠀⠠⡴⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣧⠦⠀⠀⢤⣦⡀⣼
⠀⠀⠄⠀⠘⠆⠰⣦⠀⠀⢄⠀⠀⠀⠀⠀⠀⠻⡀⠀⣾⣟⠁⣀⠂⢀⣀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⡁⣰⣾⣿⣷⡟⠈
⠤⠤⢍⠀⠘⡇⠀⠛⠀⠀⡎⠒⠀⠀⠀⠀⠀⠀⠱⡀⠹⡿⢠⣄⡈⢋⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣴⢷⠿⠟⠀⠈⡀⢠
⠀⠠⣴⣆⡄⠇⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⣀⣤⣴⣄⠀⢈⢋⡅⠙⡿⢷⠒⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⢸⡟⠍⠀⠀⠀⠀⠘⠂⠙
⠙⣷⢬⢽⣿⡀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⢙⡿⠛⠙⠢⣹⣿⣀⠴⣷⠞⣷⣤⠀⠀⠀⠀⠀⠘⠁⢀⣠⡌⢷⠢⢂⠀⠀⠀⣠⣤
⣀⣾⣿⠈⠄⠁⠀⢀⡸⠃⠀⠀⣠⡾⢇⣠⠈⢋⡀⠀⠲⠀⡟⣷⡍⠁⠀⠀⢘⣦⠀⠀⠀⡆⢠⣠⣶⣿⠟⢁⢈⡀⣸⠀⢀⠀⠹⠟⢰
⢨⣿⡿⠀⠈⠀⠀⠋⠰⠃⠀⠀⣿⣿⣾⣿⠀⠏⢀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠠⢿⠀⠀⠀⢃⣾⣷⣭⣤⣤⡴⣻⡆⢸⣶⡘⠆⠁⠀⠴
⣻⣿⣷⠖⠖⠃⠀⠀⡘⠀⢴⣾⣿⣿⡟⣿⠀⣶⣿⡀⠀⣼⡇⣿⠀⠀⢰⠀⠀⣾⠀⠀⢀⣿⣿⣿⣿⡇⣭⠀⠈⣷⣆⠹⠿⢿⣧⣄⣐
⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⣴⣿⡟⠉⠁⢹⠀⣻⡿⠻⠾⠿⠗⠛⠀⠀⠀⠀⠀⠿⠀⠀⣿⡏⢻⠟⡏⠃⢹⡇⠀⠶⣿⡇⠀⠀⡈⠙⢻
⣿⣿⠿⠃⠀⠀⠀⠃⠀⠸⢻⣿⡗⡞⠀⢸⠀⠂⢀⠀⠀⣰⡄⢲⣄⠀⠀⠀⠀⢆⠀⠀⣿⠥⠄⠀⠁⠀⠈⠀⠀⢰⠈⢻⡆⠀⠁⠀⣸
⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠉⢼⡇⠀⢀⣾⠀⠁⠀⢠⣸⣿⠀⠸⣏⣶⡀⠀⠀⢸⡆⠀⣿⠀⠀⠀⠀⠀⠀⠺⠂⠘⠷⠸⢷⡀⣀⠀⣻
⣀⠀⣀⣀⠀⠀⠀⢀⠀⣀⠀⢰⡗⠀⠰⠸⠀⠀⣦⠈⢹⣿⡀⣶⣿⡇⢱⡆⠀⠸⣷⠀⠃⠀⠀⠀⠀⣰⡃⠙⠢⡄⢠⣄⣤⣿⠟⠋⣭
⣿⠀⣿⠀⣀⠋⠉⠄⠀⠛⠘⢋⠀⠀⠀⢠⠀⡀⠘⠆⢠⣿⠁⣦⢹⡇⠈⠀⠀⠀⢻⠀⣠⣠⣤⡤⢰⣿⠏⠀⠀⠀⣬⡖⠙⣷⣶⣂⠙
⣽⠰⢩⠀⠂⠀⠀⠀⠀⠀⠀⠈⣷⣀⡄⢸⢘⣿⢓⣄⢨⡿⡄⢹⣿⠁⢀⡀⠰⡔⢆⠀⠀⣥⡤⠄⣾⡏⢀⣾⠁⠀⠸⠃⠀⠀⣄⢿⠠
⣿⠉⢺⡬⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠈⣿⠺⣿⡾⣧⢣⡾⠃⠀⡀⢿⣆⠙⢄⠁⠀⢸⡇⢸⡟⠀⣸⣟⠀⠀⠀⡄⠀⠀⢀⠈⠐
⢿⢀⣸⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠄⢀⠈⠀⡈⣁⣿⣿⣜⡾⠃⠀⢧⢧⢧⠻⣄⠀⠀⠀⠀⣿⠸⠀⠀⣿⡏⠀⠀⣶⡇⠀⠄⠛⠃⠈
⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁⠈⠸⣧⢹⣿⣸⣿⡇⠀⣄⠀⠘⣯⠃⠙⢦⡀⠁⣁⣤⠀⠀⣴⣾⣷⣾⣿⣿⣤⣀⣤⠘⠷⢒
⡀⠘⣿⡇⠀⠀⠀⠀⢷⡀⠀⠀⠈⠃⠀⠀⢰⣿⡆⢻⣿⣿⡇⠀⠹⣆⠀⠘⣧⠀⠀⠉⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠘⠏⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⠀⠈⣿⣿⣄⠹⣿⠇⣴⡀⠹⣦⠀⠈⢷⣄⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢻⣿⠋⠀⣠
⣦⠀⠀⠀⠀⠀⠀⠀⣠⠘⠀⠚⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠸⣿⣷⡆⠘⢷⣄⠀⠙⢦⡅⠀⠀⣤⠀⢻⣿⣿⣿⣿⣿⠁⠀⢹⣧⠀⠟
⢻⣆⠀⠀⠀⠀⠀⢠⡷⡀⠄⠁⠀⠀⠀⠀⠀⢻⣿⣻⣿⠀⣆⠈⠛⠀⠀⠈⢻⣦⡀⠀⠁⠀⠀⠁⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⢻⣶⣀
⠀⢻⡄⠀⠀⠀⠀⠸⡅⠟⠁⢰⠇⠀⠀⠀⠀⠀⠹⣿⣿⠸⣿⣷⣄⠀⠀⠙⢦⡙⣿⣶⡆⠀⠀⠰⠃⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿
⠀⠀⢻⡆⠀⠀⠀⠀⠙⢮⡷⠋⠀⠀⠀⠀⠐⣇⠀⣹⢛⣿⣛⢿⣿⣷⠀⢲⣤⡈⠨⡛⠇⡀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠈⣿⣶⡏⢸⣿⣿⣷⣏⠈⢶⣄⠉⠙⢂⡈⡅⠀⠀⠀⣴⡀⢽⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⣶⠖⠉⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣄⣸⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⠔⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣻⣿⣿⡝⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣙⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⣀⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢠⠀⠀⠀⠀⠀⢀⠄⠁⠰⣴⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃⡄⠀⢀⠀⢀⢀⣄⣤⣾⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠀⠀⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⡉⢢⣤⢠⢀⣈⣿⣿⢿⣏⡑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠋⠀⢠⡤⠈⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠜⠝⢭⠘⠛⠿⠿⠿⠛⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠐⠜⠁⠀⣴⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⣀⠀⢸⣿⣤⣤⣔⠲⣄⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠵⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡏⠀⠘⠁⢨⣯⣿⡇⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢺⣇⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⠀⢀⠀⢁⣦⣿⡇⠀⠀⠀⠏⠀⢹⣿⣀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢏⣴⣿⣧⠀⠀⠠⢦⡀⣆⠛⠻⠛⠋⠑⣄⠀⠀⠀⠀⠁⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⣠⣿⢠⣿⡦⡄⠀⢀⣧⣌⠓⠦⣤⣀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⣀⡴⣾⣿⡽⢛⣿⣿⠶⡄⠈⠀⠀⠀⠀⠀⠈⠉⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣖⣺⣯⣷⣾⣿⠟⢀⡜⢻⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠴⠁⠂⠉⠀⠀⠹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠺⢿⣷⣾⣿⣿⠟⢀⣀⣀⣀⣀⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣶⡒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣦⡴⠂⣐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠐⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣠⡿⠂⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⡀⠀⠀⠘⠁⠀⢀⣽⠿⣿⣿⣿⣿⣿⣿⣿⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⢰⠊⠀⣈⣿⣽⣿⣿⣿⣿⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠻⣿⣿⣿⣿⡿⢿⣿⣿⡿⠛⠋⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣇⡀⠉⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⠈⠁⠸⣻⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢋⣿⣿⣿⣿⡆⠀⠀⠀⠿⠿⠿⠿⠂⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⣿⣿⣿⣿⡀⠀⣠⣶⣶⣶⣶⣄⠀⠀⢠⣾⣿⣿⣿⣿⡿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠆⣾⣿⣿⠋⠃⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡏⠙⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⣿⣿⣿⣙⠛⡋⢉⣙⣧⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠋⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⡿⣿⣿⢿⣿⣿⣿⡾⠃⠀⠀⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⢿⠏⠸⡏⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⣀⡠⠴⠒⠒⠶⠖⠒⢤⠄⠐⠒⠒⠒⠂⠀⠀⠐⠒⣠⣄⢀⡤⠖⠒⠒⠒⠒⠒⠒⠒⢦⣤⡀
⠀⠀⠀⠀⠀⠻⠤⡀⣀⡀⣐⠤⠖⠚⠛⠋⠀⠀⠠⣀⡀⢀⡀⠀⠀⠀⠈⠉⠉⠉⠩⠽⢯⣄⣀⣀⣀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢃⡏⢙⡧⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⣻⣦⠶⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⠦⢤⠀⣠⣄⠀⢀⠀⠀⠀⠑⠦⢤⣄⢠⣶⠶⣸⢟⠀⠀⢲⢸⣿⡞⡏⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⢏⠘⡇⡟⠀⠀⠀⠀⠀⠀⠹⢸⡇⠀⣏⣌⠀⢠⡿⣼⣿⡇⣁⣸⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⣷⡆⢠⠖⠀⠀⠀⠀⣾⡁⠀⣼⣿⠀⢸⣇⢻⢸⠀⢻⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡏⣿⡇⢸⡆⠀⢰⡄⠀⣿⡇⠀⡿⢹⠀⣿⣿⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣷⢸⣿⠀⣼⡇⢠⡇⡇⢰⡇⢸⢘⣿⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣲⢿⢸⢸⣿⠀⣿⠇⢸⡇⡇⣸⡇⣷⢸⣿⣿⡀⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⢸⣿⡇⣿⢰⢸⣧⡇⠏⡇⣇⢸⣿⣿⡇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⢸⡇⡇⣿⢸⠸⣷⡷⡆⡃⡟⢸⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⣇⢹⣼⠀⣿⣿⡄⠀⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡇⣿⣾⣿⡆⣿⣿⡇⠀⡿⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇⣿⢻⢿⣧⣿⣿⡇⠰⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢃⡟⣼⣿⠷⠿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠛⢻⠀⠀⣿⠇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡆⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣉⠛⠲⣤⡀⠀⠀⠀⡟⢀⣇⢠⡇⣠⠴⠚⠛⠛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠙⣷⣄⡞⠛⠻⡏⢻⣟⣡⠞⢁⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡝⣿⣟⣰⣇⣠⣾⠙⡇⡴⠿⠷⠶⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣿⠿⣮⡛⠈⡵⣶⡶⣤⡐⣛⣉⣉⣉⣀⣀⣀⡙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣩⣴⠟⠡⠞⡥⣞⠓⢦⣄⠙⢦⠙⣯⠵⣤⠀⠀⠉⠉⠉⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠊⣡⣿⣄⡴⠊⠀⠈⢧⡀⢹⡻⢶⣤⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣠⠶⠛⠁⠀⠀⠀⢀⡴⢶⡌⢷⡀⣧⠀⣿⠀⣾⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⢀⣸⡇⠀⢹⣌⣷⣸⡆⢻⢸⡇⠀⠀⠀⣠⠟⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠈⣧⠀⡀⣿⣍⡁⢷⡾⢸⡇⠀⢠⡾⠋⠀⣼⠃⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣠⠴⠛⣿⠀⣷⠸⡇⢳⡀⠀⡼⢃⡴⠋⠀⢀⣼⡧⠖⠛⠉⢉⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡴⠛⠁⠀⢀⣿⣠⡷⠤⣷⣄⣷⣀⣷⠚⢁⡤⠖⠋⠁⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⢠⠞⣻⠟⠡⣎⠙⡆⠀⢘⣯⡉⠉⢻⣏⣠⠖⢋⣠⠶⠋⠁⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⠈⠉⠙⢿⣛⢲⣶⣛⢀⣹⡖⢀⡤⢴⡷⠒⠛⣧⠤⢶⡟⠉⢷⡶⠋⣡⡴⠖⠛⠋⢉⣩⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠢⢬⣹⣇⣨⣿⣀⡉⣿⠀⣞⠆⢸⡶⣏⣀⠸⠇⠀⣼⠶⠚⣛⠲⠶⠶⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣾⠁⣀⣽⢠⣀⣹⣿⣉⣼⠯⠉⢻⠀⣸⣛⡛⠋⣉⣀⣀⣤⣤⠽⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣥⣿⣀⣹⡇⠀⡽⢡⡀⣸⣷⣀⣼⣯⡶⠻⣬⡙⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢁⣀⡤⠶⣲⠟⣩⡝⠳⣶⠻⠶⠋⠁⣀⣥⠞⣿⠙⢳⣦⣉⠀⠈⢳⡄⠀⠀⠀⠀⠀⢀⣴⠞⠋⢉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠞⠋⠁⠀⣰⠏⢈⡥⣠⠞⠉⠷⢤⡞⣿⠉⠀⢠⡟⢧⠸⡇⠉⢳⣦⡀⠹⣆⠀⠀⢀⣴⣋⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⠟⣱⠋⠀⠀⠀⢸⣿⡇⡏⠀⣼⢧⠈⢧⣿⠀⢸⡄⠙⠻⠟⠀⢠⡾⠋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢋⣼⠃⠀⢀⣤⡤⢾⡏⠀⠃⢰⡷⠬⣧⣈⣿⡇⠀⡇⠀⠀⠀⢠⡟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠾⠁⢀⡶⠋⠀⣀⣼⡇⠀⠀⣿⣦⣄⠀⠈⠙⠷⣤⣿⠀⠀⠀⣾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣠⠴⠛⠁⠀⢿⣄⣸⠇⠀⠉⠛⢶⣄⡀⢨⡻⣧⡀⠀⣿⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠙⢶⣽⣮⣷⡄⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣼⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢠⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⣾⢷⡀⠹⡀⢸⡘⡆⠀⠀⡟⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄⠀⢹⡈⣧⠀⠀⠀⢷⡇⠀⠀⢀⣀⠀⠸⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣶⣤⠀⠱⡀⠈⢧⢸⡄⠀⠀⢸⡇⠀⢰⠃⣿⠀⠀⢰⢻⠀⠀⢀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠙⣷⡄⠀⠀⠈⣆⣇⠀⠀⠘⠁⢀⡏⢠⡏⠀⠀⣞⡞⠀⣰⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡓⢦⣄⠀⠀⢻⡀⠀⠀⠈⢿⣄⠀⠀⠘⣿⠀⠀⠀⠀⣸⠁⣾⠁⠀⢸⡟⠁⠀⠁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢝⣷⣄⠀⠻⣄⠀⠀⠀⢻⣆⠀⠀⠘⣧⠀⠀⠀⣿⣼⠇⠀⢠⠟⠀⠀⣠⡾⢛⡇⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⢷⣄⠀⠀⠻⣦⠀⠀⠈⠀⠀⢀⣿⡏⠀⠠⠋⠀⣠⡾⠋⢀⡞⠁⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠉⠳⣦⡀⠹⣧⠀⠀⠀⠀⢸⡟⠀⠀⠀⣠⣾⠟⢁⡴⠋⠀⢠⠞⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⠭⠷⠶⠤⠀⠀⠀⠉⠳⢼⣷⡀⠀⠀⠟⠀⠀⣠⡾⢋⣡⡴⠋⠀⠀⠀⠀⢀⡤⢞⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠖⠂⠈⠑⠀⠀⠀⠴⠿⠟⠛⠉⠁⠀⠀⠀⣀⣴⣞⣭⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠀⠀⣀⣀⣤⣤⡶⠚⠁⠀⠀⣰⠂⠀⢠⠀⠢⣀⠀⠀⠀⠐⠒⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⢛⣋⣩⠅⠂⠁⠀⠀⠀⣠⣾⠁⠀⠀⠀⣧⠀⠈⠳⣦⣄⡀⠀⠀⠀⠠⣤⣤⣤⣴⣚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢀⡀⠀⠀⣀⠜⡱⠃⠀⡄⠀⠀⠸⣷⡀⠀⠈⠻⢍⡓⠶⣤⣀⠀⠤⣄⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⣠⠞⢁⡼⠁⠀⢀⣧⠀⠀⠀⢳⠙⣄⠀⠀⠀⠉⠓⠦⢌⣹⣶⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⠀⣴⡟⢁⣴⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠣⡈⢧⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⢠⡆⠀⠀⢼⢸⡇⠀⠐⣄⠀⠀⠾⠆⠀⠈⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠸⣿⠁⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⠶⠦⣤⣤⣄⠀⢀⠀⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢤⡀⠀⠀⠀⠀⠀⠉⢰⣧⣶⡟⡩⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣤⣶⡶⢶⣦⠀⠀⢸⡟⣤⡐⡘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⢰⣹⠋⡭⠆⠸⢰⣆⠀⠀⠐⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠘⢁⡓⣆⠒⣰⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢫⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⣀⣀⠀⠀⠚⠋⠻⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⡇⣿⠀⠀⠀⠀⢠⣿⡿⠛⢆⣤⣾⡏⠡⡆⢩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⢸⣿⡇⣿⡀⠀⠀⢀⣿⠋⠀⠀⢸⡹⣿⠁⣁⡰⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⢸⣏⠃⢸⡇⠀⢀⡾⠁⢀⣀⠀⠀⢻⡇⢩⠬⣥⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣿⡀⢸⡇⠀⣾⡿⠛⠉⠉⣛⣀⡀⠃⠰⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⢸⡟⠀⢀⡤⠴⠶⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⣿⠁⢠⣦⡰⡌⢡⠄⠂⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣧⢈⣿⠈⠀⠀⠀⢹⣷⣤⡶⣶⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⢸⣿⠀⠀⠀⠀⠈⠁⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⣿⠀⠀⠀⣠⣴⣾⣿⠇⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⣿⠀⣠⠞⣿⠟⠡⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠹⢸⣿⠀⣡⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣾⢏⡼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢋⣤⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡤⠁