Table Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠛⠛⡻⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⠛⠛⢟⠛⠛⠛⠛⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⡾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛
⣇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠉⠉⢹⡏⠉⠉⠉⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⠀⣼
⠀⠀⠸⡇⠀⠋⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⣿⣿⠛⢻⢻⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⡄⠀⣿⣿⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⣿⣿⠀⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⣿⢿⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⣿⢸⠀⣽⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⢻⢸⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣧⠀⠀⠀⢸⣿⣄⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣇⣀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⢹⠀⠀⠀⢸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⡟
⠀⠀⠀⢸⣄⣀⣀⣨⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⣤⣤⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠤⠀⠛⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣾⡀
⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⠛⠛⠛⠛⠛⠛⠛⢳⡄
⠀⢸⣿⣿⡯⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠉⠀⠉⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣛⣛⣛⣛⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣤⣀⣀⣀⣀⣤⣤⣤⡀⠀⠀⠀⠠⠤⠿⠤⢴⣷⣼⡿⠧⠴⠶⠤⠀⠀⠀⣼⣯⣭⣽⣯⣭⣽⣿
⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠺⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠚⠉⠉⡀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣲⡿⠟⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⢒⣛⡭⠷⠒⠋⠉⠀⠀⠀⠠⠀⠀⠈⢙⣲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣚⣩⡥⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠽⠓⠋⠉⠉⠓⠦⣄⡀
⠀⠀⣀⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣚⣩⠥⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡹⢦
⠀⠀⡟⠳⢤⣀⠀⠀⠀⠀⠀⢀⣀⣤⢴⣚⣫⠭⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠁⢀⡼
⠀⠀⠳⣄⡀⠈⠙⢦⡤⣶⡾⠿⠓⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠁⢀⣠⣤⠖⠋⠉
⠀⠀⠀⠀⠙⡗⠦⣄⣇⡏⠓⠦⣄⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⢀⣀⣤⠖⠛⠉⠀⣸
⠀⠀⠀⠀⠀⡇⠀⠀⠉⢳⠦⣄⡀⠙⠲⢤⣀⢀⣀⣤⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠁⢀⣀⡤⠶⠚⠉⠀⢸⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⢲⠦⣝⡲⢤⣀⠈⣿⡿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠀⠀⣀⣠⢴⣚⠉⠉⠙⠲⢤⡀⠀⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⠀⠀⠉⠳⢬⣙⠻⣇⣀⠈⠙⠲⢤⣀⣀⡤⠶⠚⠉⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠈⠛⠦⣄⡀⠀⠈⠓⡏⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠦⣍⠓⣦⣄⡀⢹⠁⠀⣀⡤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⢠⠞⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠉⠛⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢻⠀⠀⠘⢦⣀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠟⠲⣄⡀⠀⠈⠙⠲⢤⣀⡇⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠉⠓⠦⣄⠀⠀⢹⡇⠀⠀⠀⠀⠀⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣼⡇⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠴⠚⠁

⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠟⡇
⠀⠀⣀⣴⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠔⠋⠀⠀⡇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡠⠖⡇
⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡧⠖⢻⠀⠀⡇
⠀⠀⣿⠀⢸⡇⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⢸⠀⠀⡇
⠀⠀⣿⠀⢸⡇⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠀⢸⠀⠀⡇
⠀⠀⣿⠀⢸⡇⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢸⠀⠀⡇
⠀⠀⢹⠀⢸⠁⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠀⢸⡆⠀⡇
⠀⠀⢸⠀⢸⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⡇⠀⡇
⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⡇⢸⡇
⠀⠀⢸⠀⢸⠀⠀⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⡇⢸⡇
⠀⠀⢸⠀⢸⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⡇⢸⡇
⠀⠀⢸⠀⢸⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠸⡇⢸
⠀⠀⢸⠀⢸⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠉⠉
⠀⠀⠸⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡼

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣽⣯⣮⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡭⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⣁⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢉⣠⣴⡞⣿⡇
⠀⠀⢀⣠⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣤⡶⠛⢹⣿⡇⣸⡇
⠀⠀⣿⡏⠛⠒⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⢁⣠⣴⠾⠛⠁⠀⠀⢸⣻⡇⢼⡇
⠀⠀⠈⠛⢻⣷⢶⣤⣄⣉⠙⠛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⣀⣤⡾⠛⠉⠀⠀⠀⣀⡤⢶⡏⢸⡇⣹⡇
⠀⠀⠀⠀⢸⡇⠀⢸⣿⡟⠛⠷⢶⣤⣤⣀⠉⠙⠓⠶⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢁⣠⣶⡿⠋⠉⠀⠀⢀⣠⠴⠚⠁⠀⢸⡇⢸⡇⢹⡇
⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠉⠙⠛⠿⢶⣦⣤⣀⡈⠉⠛⠲⠶⢤⣄⣠⠴⠛⠉⣠⣴⡾⠟⠋⠁⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⠀⢸⡏⢧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢶⣦⣤⣀⣸⣧⣤⣶⡿⠛⠉⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣹⡇
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠈⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⢹⣿⠁⠀⠀⢀⣠⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢼⡇
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣤⣀⡀⠀⣿⠀⠸⡇⢸⣿⣀⡤⠞⣿⡝⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢺⡇
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠀⠀⡇⢸⣿⠁⠀⠀⣿⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⣹⡇
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢸⣿⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⢺⡇
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣨⣧⡿⠃
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠠⣿
⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢨⣿
⠀⠀⠀⠀⠈⠻⠶⣼⣧⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣤⣷⡶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠆⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣥
⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡴⡇
⠀⠀⠄⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠲⡶⢁⡴⠋⠀⡇
⠀⠀⢄⣀⣠⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀⡷⠋⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⣀⡀⣤⠤⠤⡄⠀⡇⠀⠀⠀⢀
⠀⠀⠀⡏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠉⠉⠀⠀⡀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢰⡧⠄⠀⠀⠂⠂⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢸⠀⢸⠀⠀⠀⠘⠜⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠸
⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠄⠠⠄⠀⠂⠒⠒⠒⠒⠃⢸⡀
⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠤⠒⠖⠒⠒⠒⡶⠀⠸⡄⠀⠀⠀⢀
⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⢠⣦⠀⠀⠀⡇⠀⠰⡇⠀⠀⠀⠸
⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⣀⣈⣁⣀⣀⣀⡇⠀⢸⡇⠀⠀⠀⢰
⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡩⠷⠶⠶⠶⠤⠤⠤⡄⠀⢸⠃⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢀⣤⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠸⡄
⠀⠀⢸⣀⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣇⣀⣤⡤⠥⠤⠶⠦⡇⠀⢸⡤⠴⠊⠧⠴⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⢧⣠⡼

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⣿⠿⣍⡉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⣿⣷⣬⣙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⣿⢹⢮⣙⠷⣦⣉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣶⣤
⠀⠀⠀⠀⣿⢸⢀⡏⠓⢮⣙⠻⣦⣌⠓⢦⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⠴⠶⠶⠖⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣿
⠀⠀⠀⠀⣿⢸⢸⡇⠀⠀⠈⠓⢦⣙⠻⣦⣼⣏⣉⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠶⠶⠶⠾⠿⠟⠛⠛⣛⣛⣛⣻⣿⠙⣿
⠀⠀⠀⠀⣿⢸⠈⡇⠀⠀⠀⠀⠀⠈⠳⡏⣿⡏⢹⣯⣭⣭⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣿⢿⢻⡟⠛⠛⠛⠛⠋⢻⣻⡇⠀⣿
⠀⠀⠀⠀⣿⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⢸⡇⠀⠀⠀⠀⠀⢸⣿⣇⠀⣿
⠀⠀⠀⠀⣿⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⠀⠀⠀⠀⠀⢸⡟⣿⠀⣿
⠀⠀⠀⠀⣿⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣸⣇⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⣿⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣝⡯⣝⠷⣤⡀⠀⠀⢸⡷⣿⠀⣿
⠀⠀⠀⠀⣿⣘⢿⣛⢦⣀⠀⠀⠀⠀⠀⡇⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣮⣟⢷⣦⣼⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠙⠷⣬⡓⢮⡓⢦⡀⠀⠀⡇⣿⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣭⡛⢬⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣍⠳⣍⡳⣄⣷⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣌⣹⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡙⠾⡿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠢⡀
⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡉⠳⣄
⠀⠀⣠⠴⠏⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⣴⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠙⠛⠛⢛⢢⠀⠀⢸⠟⡇⠀⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋⠉⣹⢟⡋⠉⠉⡟⠛⠛⠋
⠀⠀⠀⠀⢸⠈⠀⠀⣿⡀⣧⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⠀⠀⡏⣼⡇⠀⡀⢻
⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡆⠀⢠⠃⣿⠁⠀⢠⢸
⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⠸⡈⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠁⡸⠀⣿⡇⠀⠈⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠳⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠃⠀⢹⡇⠀⠀⠀⡇
⠀⠀⠀⠀⣸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠂⡇
⠀⠀⠀⠀⢏⣀⣰⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⡀⢀⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠂⠠⠤⠤⠤⠄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⡡⠊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡉⠖⢹⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠔⠊⠀⠀⢸⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡡⠚⠁⠀⠀⢀⢴⡯⠽⠭⢿
⠀⠀⠀⢀⣠⢒⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢉⠔⠁⠀⠀⠀⣠⠔⠁⢸⡇⠀⠀⢸
⠀⠀⠀⠸⠆⣭⠉⢩⣍⠉⣅⢈⡙⣛⣓⣂⡐⠒⠒⠒⠂⠶⠶⠦⠀⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⡴⢊⠤⠊⠀⠀⠀⢀⡴⠊⠀⠀⠀⠈⡇⠀⠀⢸
⠀⠀⠀⠀⠀⣽⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠐⠒⠒⠒⠒⠠⠤⠤⢤⡤⠄⣀⣿⡔⠁⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠆
⠀⠀⠀⠀⠀⣽⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⡇⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢽⣄⣰⡏⡶⠒⠐⠒⠒⠠⠀⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡇⠀⣷⠀⣇⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡼⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢹⠉⠉⠙⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡟⠈⠹⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠊⠀⠀⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢘⡁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠓⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡴⠃


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠤⠔⠒⠒⠂⠈⠀⠉⠉⠉⠉⠑⠒⠒⠒⠒⠶⠤⠤⠤⢤⣤⣀⣀⣀
⠀⠀⣶⠿⠯⠍⠉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠶⠤⠤⠤⣤⣄
⠀⠀⠈⠉⢹⡗⠶⢶⣤⢤⣭⣍⣉⣐⠒⠒⠤⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠔⢒⣊⣭⠭⠟
⠀⠀⠀⠀⠸⣧⣀⣀⡀⠀⠈⠊⠋⠙⠙⠛⠻⠶⣶⡦⣤⣤⣭⣉⣉⠓⠒⠒⠦⠤⠤⣄⣀⣀⣀⣠⠤⠤⠒⢚⣉⣩⣥⠴⠶⠾⠿⠛⢻⡇
⠀⠀⠀⠀⠀⡏⢹⢹⠉⠉⠉⠓⠒⠲⠤⠤⣤⣀⣀⠈⠀⠁⠙⠛⠻⠻⠟⠝⠳⠶⣶⡦⣤⣌⣧⣤⠖⠚⠛⠛⠋⠋⠉⠁⣀⠀⢤⣶⣾⠁
⠀⠀⠀⠀⠀⡇⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣟⣶⣶⠶⠤⠤⣤⣀⣀⠀⠀⠀⡇⠉⠀⣀⣀⠤⠶⠒⠋⠉⠉⠀⠀⢸⢻⢸
⠀⠀⠀⠀⠀⡇⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⡟⡗⠀⠀⠀⠀⠀⠈⠉⢹⠿⡷⣞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⣸
⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⣀⣤⣤⡖⠒⠊⠉⠛⠻⠿⠶⢦⣇⣇⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⢸
⠀⠀⠀⠀⠀⡇⠀⢸⣋⣁⡤⠭⠿⠛⠛⠛⠓⠶⠤⣤⣄⣀⠀⠉⠉⠙⠓⠒⠦⠤⠤⣸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿
⠀⠀⠀⠀⠀⡇⢰⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⠤⣄⣀⡀⢸⠀⡇⡟⠛⠒⠲⠤⢤⣀⡀⣀⣠⣤⣶⠿⣿⡏
⠀⠀⠀⠀⠀⠙⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⡆⣷⣤⣤⣀⡀⣀⣤⣶⣿⡿⠟⠺⣏⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣷⣶⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠶⠤⣤⣀
⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿
⠀⠀⠙⢮⡳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣫⡞⠁
⠀⠀⠀⠀⠉⡷⢭⣛⠲⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠴⢖⣛⡭⡖⠛⡇⡇
⠀⠀⠀⠀⠀⡇⢸⠈⣿⠓⠒⠾⠭⠭⣽⣿⣿⣿⣛⣛⣋⣩⣭⣭⡭⠭⠶⠒⠚⠋⠉⠀⠀⣿⠀⡇⡇
⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⢻⢸⣿⠇⠀⣷⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇
⠀⠀⠀⠀⠀⡇⢸⢠⡇⠀⠀⠀⠀⠀⠀⠘⠾⠟⠀⠀⣿⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇
⠀⠀⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠻⠾⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡉⠙⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⣩⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣀⡴⠋⣡⠴
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⡴⣞⣷
⠀⠀⠀⣀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣠⠞⠉⣠⠟⣿
⣠⣖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⡴⠋⢀⡴⠞⡇⠀⣿
⣿⣈⠙⠶⣄⡀⠀⠀⠀⠀⠀⢀⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⡴⠞⠁⣠⠶⠋⠀⠀⡇⠀⣿
⣿⠉⣳⣦⣄⠙⠳⢤⣀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣠⠖⠋⢀⡤⠞⠁⠀⠀⠀⠀⡇⠀⣿
⣿⠀⣿⣿⠈⠙⠦⣤⡉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣁⡴⠋⠁⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿
⣿⠀⣿⠈⣿⠦⣄⡀⠙⠲⣤⡈⠙⠲⣤⡀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⡤⠞⠁⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⣿⠀⣿⠀⡿⠀⠀⠙⠲⣤⡀⠉⠳⢦⣀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣠⠴⠋⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿⣠
⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠉⠳⢤⣀⠈⠛⢦⣄⠈⠛⠶⣄⡀⠀⣠⡴⠋⢁⡴⠚⠁⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⣯⠀⣿⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠈⠙⠲⣄⡀⠙⡟⢁⣠⠞⠉⠀⣠⠖⠋
⣿⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣸⡏⠳⡿⠋⣿⢀⡴⠚⠁
⣿⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⡇⠀⣿⠋
⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⣇⡀⣿⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡇⢀⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣿⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⣛⣟⣹⠛⣛⢟⣿⢻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⢀⡀⠀⣀⣀⠀⠀⣀⡀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⡀⠰⣾⣦⣿⡇⠀⣿⣿⠀⢰⣿⣿⣀⣶⡄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠘⠿⣿⣿⡿⢿⠇⠀⠿⠿⠀⢾⡏⢹⣿⡿⠁⣀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣃⣀⢈⣃⡀⡈⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠉⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣫⡛⣻⣙⣛⣟⣛⣿⣛⡛⣻⣙⣛⣟⣙⣛⣋⣙⣋⣛⣛⣛⣛⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣾⡀⠀⣰⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⡇⠀⢹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠁⣄⢀⠀⠀⠂⠀⠀⢀⣀⡀⠀⠀⠀⠂⢀⠀⠀⠘⡇⠀⠘⣷⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡟⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠃⠀⣧⠀⠀⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠲⡿⠀⠿⠇⠀⠀⢰⣼⠄⢼⡷⠀⠀⢠⣶⠄⣴⡶⣿⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠠⠠⠀⠀⠉⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢰⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⠀⠀⣤⣤⣤⠀⣤⡄⠀⠀⠀⠀⣤⣤⠀⢀⣤⡀⢀⣤⡀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢀⠀⣿⣯⣿⠇⣿⡷⠀⠀⠀⢸⣿⣿⣇⠀⠙⣿⣿⠋⣀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⡏⠁⠀⣿⣿⣶⡆⢠⣿⠛⠛⣿⡄⠀⣿⡧⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣧⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⡣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢌⢉⢀⢨⠁⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡌⠀⠀⠀⠀⠀⢀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠉⠁⠈⠈⠁⠉⠁⠈⠉⠉⠉⠉⠉⠀⠀⠉⠁⠉⠉⠈⠙⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠠⠠⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠆⠰⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣀⠀⢀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠐⠉⠀⠈⠋⡀
⠀⠀⠐⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠀⠀⠀⠁⠁⠒⠦⠴⢤⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠐⢿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⣤⢬⣉⡅⡉⠽⠉⠿⠯⠶⠷⠳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠘⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⠽⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠄⠀⠀⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢄⠀⠁⡀⢠
⠀⠀⠀⠐⢪⣷⣻⠯⡑⠀⠳⠟⣓⡃⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠹⠧⠃⡁⠤⠚⡤
⠀⠀⠀⠀⢸⡋⠚⠧⢵⠒⢲⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡜⣒⠙⠉⣀⢐⠸⡆⣀⡚⡀
⠀⠀⠀⠀⠀⣷⠋⠠⢈⣵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡟⠻⡶⢄⠻⣯⠑⣾⡟⢘⡟⣾
⠀⠀⠀⠀⠀⠘⣆⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢛⢯⠟⣾⣠⣿⣯⣽⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠚⠾⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠘⣛⣟⣀⡂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣾⣟⣸⠃⠀⠀⠈⠙⠻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣴⡀⢩⡟⠋⠉⠀⠉⠛⠏⡍⠘⣆⠀⠀⠀⠀⠀⠘⣮⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⣽⠚⣿⡿⠜⠷⣄⠀⠀⠀⠀⠠⡈⢂⣬⡷⣄⡀⠀⠀⠀⣸⠘⠇
⠀⠀⠀⠀⢀⣠⣤⣀⡠⢿⣶⡌⢷⡀⠈⠈⠑⠀⢨⣽⣿⣒⢶⠢⠴⠾⣿⣿⣶⡝⠛⠒⢚⣩⣄⡀
⠀⠀⠀⢐⡈⢱⡏⠀⠀⠘⣿⠫⡀⠙⠓⡶⠤⠤⠾⠿⠿⠿⣿⢒⠒⢒⣈⡿⠿⠉⠐⠂⢸⣿⠏
⠀⠀⠀⢾⣿⠎⠳⢦⣤⣀⣈⣐⣤⣁⣸⣿⣿⣿⡗⠀⠐⠀⠐⣠⢰⡍⠀⠂⢑⠲⢢⠀⠀⡶⡶⠀⠀⠙⢑⣂⡀
⠀⠀⠀⠈⠃⠀⠀⠀⣾⣷⣿⣶⡆⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⡟⠀⠂⠈⢸⢕⡈⠰⠀⠀⠑⣼⠉⠁⠀⢀⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠦⠀⠸⠈⠀⢰⠄⠀⠒⠀⠀⠀⢠⡈⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⣀⠀⡤⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⡴⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⢀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠤⠄⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠘⠀⠀⠀⠀⠀⠈⢄⠀⠰⠁⢀⠀⠉⠆⠀⠀⠌⠠⠊⠠⠀⠟⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⠡⣈⢀⡄⠐⠰⣲
⠀⠀⠐⠀⠐⠀⠀⢉⠣⠀⠀⠣⡀⠈⠀⠈⠆⠈⢠⠴⡀⠊⠀⠀⠀⠂⠀⠠⠀⠄⠀⠁⠀⠀⡠⠁⠀⠒⠄⠀⠉⠉⠀⢐
⠀⢀⠀⣋⡀⠀⠀⠀⠈⣀⢀⢀⠈⠒⠈⠄⢀⡴⠀⡯⢚⠁⠀⠀⢠⠐⠂⡐⠠⢀⢀⠀⢀⠜⠀⢣⢄⣀⡀⠀⠀⠀⠤
⠀⠘⠀⠀⡆⠀⡠⠊⡁⠀⠃⠂⢌⡑⢄⠀⠸⠋⠀⢸⡵⠀⠀⢸⠃⠀⢀⡅⡀⠘⠄⠐⠁⢀⠀⠠⣀⡀⠀⡀⠠
⠀⢠⠖⢦⠀⢄⣠⠒⡂⢜⣊⠼⠓⢊⠁⢠⠒⠒⠤⡀⠃⠀⠀⠀⠁⠨⠉⠁⠀⠀⠀⠀⠠⠂⢤⠀⠠⠀⠀⠁⠀⠈
⠀⠀⡆⣈⢂⠀⠀⠀⠀⠒⠒⠒⠈⠁⠀⡌⠀⠀⢰⠀⠰⢠⠀⠀⠂⠆⠀⢀⣴⣶⢻⣟⣳⡆⠀⡀⠀⢀⠀⢀⡰⠉⡘
⠀⠀⡄⠂⢠⠀⠀⠀⢖⠒⠀⠠⠀⢲⠆⠑⠀⠲⠈⠀⠀⠁⠃⠀⠀⠀⠀⢀⢀⠽⠉⠉⠉⠉⠀⠀⡀⠐⠀⠀⠉⠉
⠀⠀⠈⠉⠁⠔⢄⠀⡄⠉⠀⠒⠀⠁⢀⠀⣴⡢⡄⠀⠠⠀⠀⠀⢀⠖⡆⠀⠀⠀⠀⠍⠀⠁⠀⡀⠃⠀⠀⡔⠉⢆⠁⢠
⠀⠦⠀⠀⠈⠀⠀⠀⠀⢀⠀⢈⠈⠡⠏⠓⢉⠠⠣⢶⠠⢅⢤⡤⠀⠀⠀⡀⠈⣁⠀⠈⠈⠀⠂⠀⠀⠀⠨⠀⡠⠊⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠂⠈⢀⣐⣃⡠⠤⠗⢉⠀⠀⠀⠀⠰⠀⠀⡀⠼⢑⠀⠉⡂⠲⢀⠑⢀⡱
⠀⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠈⠉⣤⠀⠀⢘⠭⠁⣀⠀⠐⣁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠎
⠀⠐⠂⠀⠀⡠⠄⠀⠀⠊⢟⠀⠀⢠⠶⠛⠉⠐⢹⣶⠄⠈⡠⠀⠀⠀⠀⠈⠀⢠⠄⠀⠀⠀⠀⠀⠀⣡⠀⠀⠘⠀⠂⠄
⠀⠰⠠⠍⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢈⠠⠒⠀⠀⠀⠀⠉⠉⠉⠈⠐⠡⠄⢀⣀⠀⠂
⠀⢠⠀⠀⡀⠄⠈⡆⡕⠀⠈⠀⠐⠀⠀⠠⠒⠃⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠄⠠⠏⠀⡀⠀⢀⠠⠀⠀⠙⠇
⠀⠀⠀⡀⢀⡄⠀⢡⠁⢀⡀⠌⠀⡄⠂⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠒⢅⡀⢀⠰
⠀⠠⠐⠊⠸⠈⠆⠈⢀⠡⠀⠀⠦⠠⢖⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⡄⠀⢠⠂⠉⠀⠀⠀⠈⠀⢀⠜⡀⠀⠀⢻⠧⠂⠁
⠀⠀⠠⠨⠄⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠄⠀⠀⠀⠠⠒⠃⠀⠀⣀⠀⢌⢀⠢⠤⠬⠔⡀⠤⠔⢊⠈⠃⡀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠠⠂⠀⣀⡀⡔⠒⠒⡆⠀⢢⠀⠐⠀⠀⠁⠀⠈⠀⠠⡈⡀⢄⠀⠠⠀⠈⠀⠀⠀⠀⠐⠁
⠀⢀⡀⠐⠀⠀⠀⠀⠂⠄⠁⠀⠀⠘⠄⠂⢘⣀⣸⠀⠀⡀⠀⠐⠄⢤⠆⠀⠀⠈⠚⠁⠡⠊⠈⠠⠁⠀⠤⠀⠐⠀⠒⢤
⠀⠀⣠⡶⢀⡶⠀⢀⠀⠃⡀⠂⠈⠠⠀⠀⡀⠄⠠⣀⠀⠀⠘⠀⢀⠀⠀⠀⠃⠀⠠⣀⣠⣤⠄⠀⠀⠀⢄⠀⠀⠀⠀⠔
⠀⢸⣆⠅⠊⠀⡀⠁⠀⡂⠀⠀⡀⢊⠀⠌⡠⠊⢉⡉⠠⠀⠈⠠⠊⠀⠀⠀⠁⠀⢀⠜⠈⠡⡀⠀⠀⠀⡈⠀⠈⠁⠁⢠
⠀⠀⠁⠂⠀⠀⠃⠀⠀⠈⠀⠉⠀⠈⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⣸⣿⣿⣿⣷⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⠉⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡵⠟⣫⣭⣭⡛⠭⣉⠉⠙⠻⢿⣿⣿⡄⢀⡾⠁⢰⣶⣶⣤⣤⣤⣶⠶⠾⣛⠿⢟⣛⡻⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⣿⣿⣿⣾⣿⣮⡳⡑⡦⠄⣙⣿⣿⣿⡇⠀⣿⣿⡿⠿⠛⠛⠁⡠⠊⠁⣿⣩⣭⣝⣿⣿⣿⢿⣆
⠀⠀⠀⠀⠀⢀⡿⣴⣲⣶⣶⡟⢿⡿⣿⣿⣿⣇⢱⢱⣤⣤⣿⣿⣿⠏⠀⠻⣆⣠⠒⠒⡲⣰⠁⠀⣀⣻⠛⠛⠙⠽⣿⠿⣦⢿⡄
⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⠗⣀⠸⣿⣿⣿⣾⢈⢶⣭⣽⣿⣿⣷⠀⠀⣸⡏⠛⣿⣿⡃⡇⣾⢩⣿⣍⢸⡉⡱⠸⣿⡿⣸⢸⣷
⠀⠀⠀⠀⢀⣿⣿⡸⣿⣷⣾⣿⣿⣷⣿⣿⢗⣋⣘⡺⢟⢿⣿⣿⣿⠆⠀⠘⣧⠀⠳⠞⠋⣣⠹⣌⣿⣧⣦⢨⣄⣸⣿⣿⡿⣼⣿⣇
⠀⠀⠀⠀⣸⣿⣿⣿⣜⡻⢿⣿⣿⣯⠟⣵⠟⠁⠀⢨⣷⡹⣹⣿⡏⠀⠀⠀⢸⣧⠀⡔⣫⡶⠒⠚⣭⡻⣿⡻⢿⣛⢿⢟⣽⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⡇⠈⠛⠿⣷⣶⣶⣶⡄⣇⠀⣀⣴⡿⢋⡇⣿⣿⣿⠀⠀⠀⠻⣏⢸⢼⠋⠀⠀⣼⡿⣣⢨⣭⣭⣥⣶⠿⢿⣿⣿⣿⢿⡀
⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣧⡹⢿⣿⣷⣶⢟⣴⣿⣿⡟⠀⠀⠀⠀⢹⣷⡸⣦⣠⣾⣟⣵⡟⣼⣿⣯⠉⠀⠀⠈⢿⣿⣿⡞⣧
⠀⠀⢠⡟⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⢿⣟⠲⠦⣴⣶⣿⡿⠁⠈⠀⠀⠀⠀⠀⠸⢿⣷⣬⣛⣛⣛⣩⣾⣿⣿⣿⣇⠀⠀⠀⠈⢿⣿⣿⠹⡆
⠀⠀⢸⠧⣿⠁⠀⠀⠀⣾⣿⢟⡽⠃⠀⠙⠛⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣉⣹⣿⠿⠋⠘⣯⡻⣄⠀⠀⠀⠈⢿⣿⡇⣿
⠀⠀⠸⣧⡙⠀⠀⠀⠴⣟⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠚⠋⣠⠟
⠀⠀⠀⠈⠛⠓⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠠⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄
⠀⠀⠀⠓⠄⡐⠓⠈⠁⠢⡀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡄⠀⢀⠔⠈⠁⠚⢃⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠣⣀⡸⣄⢀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⡀⣀⢇⢀⠜⠁⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠹⣿⠀⠀⠀⠀⣲⠀⠀⢰⡀⠀⠀⠀⠀⣿⠟⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠋⠀⠀⠀⢰⡇⠀⠀⠀⣷⠀⠀⠀⠀⠉⠒⠒⠊⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣴⠟⢀⠀⢂⡄⢢⡴⢊⠉⢡⣄⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠉⠁⠰⠁⠶⠰⠆⠼⣀⡙⠁⠀⣁⠀⡄⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⡜⢃⠴⡇⠸⠆⠀⠋⠸⠹⣦⣸
⠀⠀⠀⠀⠀⣴⠀⠀⣿⡇⣀⡹⣿⢇⠀⣠⣶⠀⠀⠀⠀⠀⠀⠑⢆⣧⠀⠨⠛⢴⣶⢋⣙⢃⣀⣙⠟
⠀⠀⠀⠀⣀⣬⡙⠍⠓⠈⠃⠙⠀⠋⠻⠙⠉⠀⠀⠀⠀⠀⠀⠀⢀⣉⣥⣤⣤⣤⣭⣉⡉
⠀⠀⠀⠀⠉⠂⠀⣀⠄⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠛⣿⡟⢻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣴⣿⡿⢿⣷⣈⣿⡶⡿⡧⣾⣏⣰⣿⢿⣿⣦
⠀⠀⠀⠀⠀⣤⣤⣄⣠⣤⡀⠀⠀⢉⡉⠁⠀⠀⠀⣼⣿⠿⢿⣶⢿⣿⠞⠛⠋⠺⠲⣜⢿⣥⢂⣹⣿⣧
⠀⠀⠀⢰⣶⡋⢐⣴⣦⡈⢳⣴⠀⢿⡿⢸⣿⠄⢸⠋⣽⣷⣶⣯⣟⡁⠀⠀⠀⠀⠀⠈⢣⣻⡿⠟⢻⣿⡆
⠀⠀⠀⢸⣤⣿⡦⠀⢤⣴⣅⣼⠀⠀⠀⠀⠉⠀⢸⠀⣧⣤⣿⣺⡁⣿⣌⠐⡾⠤⣀⣠⢘⡿⡿⠿⢿⣿⡇
⠀⠀⠀⢸⣺⣿⣗⠒⢺⣿⡿⣺⡀⠀⠀⠀⠀⠀⢸⡀⢫⣡⣽⣾⣧⠻⣿⣏⣶⣶⣿⠏⣼⠫⡿⣶⡾⢛⡇
⠀⠀⠀⠘⢺⣿⣗⠀⢸⣿⣥⠟⢛⠒⠺⢶⠀⠀⠀⢣⠈⢟⣉⣽⣷⣷⣦⣭⣉⣩⢴⠪⡣⢿⣦⡼⠁⡜
⠀⠀⠀⠀⠀⠉⠛⠀⠘⠋⢹⣄⠘⠛⠃⢀⠷⡇⠀⠀⠳⡈⠻⢏⣠⣿⠛⣿⡿⣿⣟⠻⣿⣤⠛⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠁⢘⣿⠧⠀⠁⠀⠀⠀⠈⠲⢄⡉⠛⠺⠿⠧⠼⠟⠒⢉⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠊⠉

⠀⢀⠔⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠢⢄
⠰⠁
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣤⣀⣼⠿⢿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣿⣿⣇⣀⣨⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣹⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛
⡇
⡇
⡇
⠰⡀
⠀⠈⠢⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠴⠒⠒⠋⠉⠉⠓⠒⠒⢤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢱⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡞⡸⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⢣⠹⡀⠀⠀⠀⠀⠀⠀⡼⣰⠃⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⢧⢳⡀⠀⠀⠀⠀⣰⢡⠃⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢣⠀⠀⠀⢰⢃⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱
⠀⠀⠀⠀⠀⢀⡖⠦⢍⣓⠦⢄⣀⠀⠀⠀⠀⠘⡎⢇⠀⢠⠇⡞⠀⠀⠀⠀⢀⣠⠤⢒⣫⠥⠒⢢
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠒⠮⣍⡒⠦⢄⡀⠹⡜⣆⠎⡜⢀⣀⠤⢖⣊⠥⠖⠊⠉⠀⠀⠀⢘⡆⣠⣤
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠬⣝⡓⠈⠘⣋⡭⠔⠚⠉⠀⠀⠀⠀⠀⢀⡴⣺⠗⠋⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⣚⡭⡤⢀⠠⡬⣙⡲⠤⣄⡀⠀⠀⠀⠀⠈⠙⠯⣶⢤⡀⢿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡠⠴⣒⡩⠴⠒⠉⠁⣰⢡⠋⢧⢳⠀⠉⠑⠲⢭⣑⠲⢤⣀⡀⠀⠀⢸⠃⠈⠋
⠀⠀⠀⠀⠀⠀⠷⢒⣩⠥⠒⠋⠁⠀⠀⠀⠀⢰⢃⠏⠀⠘⣆⢧⠀⠀⠀⠀⠀⠉⠓⠢⢭⣓⠢⠞
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⠀⠀⠘⡌⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢀⡎⡼⠀⠀⠀⠀⠀⠹⡘⡆⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡜⡸⠁⠀⠀⠀⠀⠀⠀⢳⡸⡄⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡼⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢶⣶⣤⣤⣤⣤⣤⣴⣶⡾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠙⣄⠀⣠⠖⠁⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠘⣶⠃⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⢠⣤⠀⠀⠀⠀⢹⠀⣤⡀⠀⠀⠀⠀⠀⡟⠓⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⣧⡇⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢷⡀⠀⠈⠛⠀⠀⠀⢀⠷⡀⠙⠁⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠱⢄⠀⠀⠀⠀⣠⠎⠀⠙⠤⣀⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⢀⡾⣡⡴⠂⠀⠀⠀⠉⠃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⢠⡞⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⢠⡟⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⢀⡾⠀⠀⠀⠀⠀⡟⠢⣄⡀⠀⠀⠀⠀⣀⣀⠄⢀⣤⠶⠚⠛⠛⠛⠛⠓⠲⠶⣤⣄⣀⠀⠀⢠⡏
⠀⠀⣼⠁⠀⠀⠀⠀⠀⣇⠀⠀⡏⠈⠉⠈⠁⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠟
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠸⡄⢸⡀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⢉⣩⠟⢹⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠲⠶⢤⣀
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠲⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣤⣤⣤⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠦⣤⣤⣤⡤⠤⠶⠶⠞⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠉⠙⠲⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡎⠂⣾⡁⢸⠤⣤⠖⢂
⠀⠀⢀⠴⠒⠂⠠⢀⡀⢀⠔⠁⢀⡤⠖⠒⢤⠀⢹⠀⠀⠀⠀⣀⠉⢳⡼⠀⢸⣷⣀⠓⣇⠀⠀⠊⠠⠁
⠀⢠⡏⠀⣀⣀⣀⠀⠈⠃⠀⡰⠋⠀⠀⠀⠈⡇⢸⡇⠀⡠⢼⣄⠑⠌⠑⠂⠀⡿⣷⠀⠈⡄⠀⠀⢦
⠀⢸⡀⠐⡅⠀⠈⠉⢻⠀⠰⠁⡄⠀⠀⠀⢰⠁⢸⠇⠀⠑⢦⡙⠇⠀⠀⠀⠀⠐⡽⠄⠀⠀⠀⠀⠈⡄
⠀⠈⣇⠀⠰⡀⠀⠀⠘⡄⠀⢠⡀⠀⠀⠀⡌⠀⣸⠀⠀⠀⠀⠉⢢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠘⢆⠀⠹⣄⢠⠖⠀⣠⡂⠉⠒⢄⡰⠁⢠⠇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⠳⡀⠈⠁⠀⡴⠁⠈⠢⡀⠀⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠘⢆⣠⠊⠑⠢⠤⠖⠉⢂⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⡠⠤⠴⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡠⠄⠲⠊⠉⠉⠉⠩⠉⠉⠛⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠑⠒⠤⣀⡀
⠀⠀⢠⠏⠉⠈⠉⠈⠉⠁⠁⠈⠉⠉⠉⠈⠉⠁⠀⠀⢳⠀⠀⢠⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⢨⠧⢀⠄⠀⠀⠀⢀⠀⠀⣀⠀⢀⡀⣀⣀⠀⠀⢺⡀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠠⠤⠀⠀⠀⠈⢧
⠀⠠⡚⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢀⠀⠀⠀⢹⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠧⡄⢀⡠⠬⡈⢀⡀⠀⠀⠁⣀⣁⢈⣐⠀⣠⠌⣹⠋⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠑⠤⣀⣀⣀⠀⠀⡀⠀⠊⠀⠀⢀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠦⠀⠀⢘⣺⣄⡠⢀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠄⣠⡁⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠋⠀⠀⠐⠀⠀⠀⠀⠀⢀⠘⠛⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⢀⠀⠀⡆⠀⠠⠘⢥⠽⠀⠀⠀⡀
⠀⡀⠀⠐⠀⢠⠀⠒⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠠⠄⠀⠀⠀⠀⢀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠰⣷⢄⠀⠀⠂⠀⠁⠀⢠⡇
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣊⢄⠀⠀⠀⠐⢜⢃⡀⢀⣀⣐⣤⡤⢊
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠑⠈⠉⣓⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⢊⠀⠀⠀⠀⡊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⡠⠀⠀⠀⠉⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠐⡀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡉⢉⡒⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⣄⡠⠉
⠀⠀⠀⠀⠀⠤⢤⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠀⠀⠈⢶⠀⠀⠈⢱⠂⠐⠒⠒⠋⠁⠄⠀⡇
⠀⠀⠀⠀⠀⢠⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⣄⡀⠐⢆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⡠⡧⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠉⠘⣆⡀⠀⢀⠀⠲⡄⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠄⠼⠀⠀⠀⠀⠀⠀⠀⢌⠢⣄⠐⠀⢀⡴⠤⢛⢽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠐⠒⠂⠈⡼⠀⠀⠀⠀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣠⠀⠑⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠞⠁
⠀⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠫⠄⠀⠀⠁⠀⠐⠤⢀⣀⣀⡠⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠀⠀⠘⠋⠀⠠⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⣰⠀⠀⠀⠀⣠⠄
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⣧⠶⠛⠛⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⡅⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⢻⠛⡟⢻⢻⠛⠙⢹⡇⢀⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡶⡟⣿⢷⣤⡾⣿⠀⡇⢸⠘⡀⡇⢸⣈⣀⣠⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣴⢿⠉⡏⢻⡇⣧⢸⢸⡈⡇⢸⣀⣷⡾⠟⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠾⠿⠟⠛⠁⠀⠀⠀⠘⠃⠀⠀⢠⣴⣿⣾⡄⢇⢸⣧⣼⣼⣤⣇⣧⡾⠛⠁
⠀⠀⢠⡶⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⢸⣿⣧⣼⣼⣿⠁⠈⠙⠛⠋
⠀⢰⡟⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢿⡜⢿⡏⠀⠈⠉
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣠⢾⣷⠟⠋
⠀⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣼⣷⢸⣿
⠀⢸⣷⢹⠛⡿⢿⡿⡷⢶⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡿
⠀⠀⠻⣿⣄⣧⠘⡇⡇⢸⢼⣧⠀⠀⠀⠻⠇⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⡿⣶⣤⣤⣀⣤⣴⡿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡇⠀⡏⢹⠉⢻⠈⡇⢣⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣷⣬⣦⣼⣤⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⠀⠀⠈⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⢀⠀⠀⣄⠀⠈⠈
⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢄⠀⠀⠀⠀⠈⠀⠀⠀⠰⠄⠀⠀⠂
⠀⠀⠘⠀⠀⢀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠂
⠀⠀⠠⡀⠀⠈⠿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠁⠒⠂⠐⠀⠀⠀⠠⠤⠤⠄⠀⠀⡀⢀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠓⠦⠀⠀⠀⠰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⢀⠁
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠂⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠔⠐⠐⠇⠒⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢅⣀⣴⣀⣀⣀⣐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠉⠙⠉⠉⠉⠙⠛⠛⠉⠁⣼⣿⣟⠛⠛⠋⠙⣿⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⣈⣤⣸⣿⣿⣽⣶⢿⣥⣴⣶⡀⠉⠉⠟⠉⠉⢿⡏⣿⣏⣴⡿⠹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⡟⠉⠀⠙⠍⢛⣗⣤⣹⣿⡟⠁⠀⠀⠀⠀⠀⣸⣿⣿⡿⣿⡛⢻⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗⣩⠁⠀⠀⠀⠃⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⣼⣿⣿⣿⣇⣿⣿⣼⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠁⠀⠀⠰⠈⢳⡀⠀⠀⠀⠀⡀⠀⠀⢠⢀⣾⣿⣿⣿⣿⣿⣿⣾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣾⣶⣿⣿⣷⣶⣼⠇⠀⠀⠀⠀⠘⠀⢻⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⣿⣥⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⣄⢺⠿⡼⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢏⡁⢀⣈⣹⣿⣿⣷⡀⠀⠀⣌⣻⡿⠂⢱⣇⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠿⠿⠟⡿⠛⢁⡀⢰⣿⣿⣧⣳⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣰⣴⣠⣿⡗⠀⣰⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⢿⣿⣿⠿⠿⢶⣾⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡿⢟⠋⠃⠈⣿⣶⣤⣦⣴⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⣯⡪⣻⣿⣿⣿⣿⣿⣿⣿⡿⣟⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡿⡫⡞⠡⠃⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⠔⠁⣻⢇⣿⣿⣿⣿⣿⢫⣼⣿⣷⣬⢻⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⢇⠛⡕⡘⣠⢂⡔⢠⠂⢸⢧⡹⠋⡉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣖⣉⣤⣾⣥⣾⣿⣿⡿⠋⣱⣾⡿⣵⣿⢻⣷⣿⡿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⡼⣿⠇⢨⣾⣶⣄⣏⣘⣜⣾⠀⣿⣾⡀⠀⠀⢫⣕⣬⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⡿⠟⠋⢀⣴⡿⣋⢞⡽⡳⠿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣧⣿⣰⣿⣿⣇⣽⢸⣿⣿⣷⣦⣿⡇⠀⠀⡀⡀⣍⢻⣿⣾⣿⣿⣿⣯⣉⣩⡭⢉⣀⣀⣶⣿⣿⣿⣾⠷⣣⣜⣇⡴⣹⣟⣿⣿⣷⡿⣿
⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠁⠀⠀⠐⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠰⠦⠶⠴⠆⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠆⠀⠀⣀⣀⡀⠀⠠
⠀⣠⣴⣾⠟⣿⡇⠀⠀⠀⠈⠀⠀⠆⠀⠑⠒⠚⠂⠚⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐
⠀⣿⣬⡼⠶⠛⠃⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢀⣠⣞⡇⠈⠀⠫⠩⠥⠼⠃⡀⠀⣶⣳⠐⠂
⠀⠉⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠔⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣣⠟⠁⠀⠀⠀⠰⠇⠀⠀⠀⠀⠘⠋
⠀⢀⠀⠀⠀⢀⡀⠀⠀⣰⣂⣂⣲⣀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⢸⠟⢁⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠻⠭⠿⠭⠝⠇⠀⠰⡏⣋⣹⣷⣽⠇⠀⠀⠀⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠉⠁⠐⠀⢀⠀⠀⠀⢀⠀⠀⠦⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠩⢐⠀⠀⢨⠀⠀⠀⠀⠸
⠀⠀⠴⠀⠀⠀⣈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⠀⢨⣲⠀⢨⠀⠀⢄⣀⠈
⠀⠠⠀⠀⠀⠀⢀⣈⠔⠀⠀⠀⢰⡶⠶⠄⠀⠀⠀⠀⠀⢠⡤⣤⣤⠀⠀⠀⠁⠀⠈⣤⡤⠆⠀⠀⠀⠀⠀⠈⠐⠀⠐⠌
⠀⠀⠀⠀⠀⠁⢉⠄⠐⠀⠀⠀⣼⣇⠀⠲⠤⠤⠄⠀⠤⠘⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣽⣉⣁⣄⣀⣀⣀⣀⣀⣁⣀⣀⣙⠀⠀⠀⠀⠀⠀⡈⢲⣠⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⡏⣠⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠂⠀⠤⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢿⡟⠃⡀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⡶⡢⡀⠀⠁⠂⠐⠒⠀⠀⢀⢀⠀⠀⠀⡀⡀⡀⠀⣰⠖⠀⠒⠀⠐⠐
⠀⠈⠀⠀⠀⠝⠂⠀⠀⠀⠀⠀⠀⡎⣭⢉⣑⢂⣜⣣⡥⠏⠀⡄⡀⠀⠀⠘⠀⠛⠘⠀⠂⠀⠃⠓⠃⣼⡏
⠀⡀⠀⠀⠀⠀⠀⠁⠀⢀⡆⠀⠀⠛⠾⠿⠿⠛⠋⣽⣀⣠⠀⡁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀
⠀⣀⠀⣀⣀⣀⣐⣒⣀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢡⠐⡴⣆⠀⠍⠠⢂⢠⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠄⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⡂⡤⡒⠂⠀⠂⠄⠀⢂⢢⠀⠂⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠈⠐⠀⠠⠀⠀⠀⠤
⠀⠀⠀⠀⢠⡀⠤⠀⠀⠀⠀⠀⣀⠁⠀⠁⠁⠀⠁⠀⠀⠁⢀⡀⠀⠈⠂⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠓⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡴⠶⢶⡄⠆
⠀⠀⠀⠠⢠⠀⠀⡠⠀⠀⠀⠀⠀⠀⣾⣿⣭⣽⣽⣽⣿⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠂
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⡀⢀⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠐⠀⠀⠁⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⢐⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡐⢀⠁⠀⠀⠀⠀⣨⠑⠒⣂⠐⢀⡚⠒⢉⡀⠄⠀⠀⠀⠀⠛⠛⠀⠀⠀⠄⠀⠀⠀⠀⢠⡠⠀⠀⠀⠀⠀⢄⠠⡀
⠀⠀⢀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠐⢁⣂⣉⠀⢈⣁⡢⠓⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢠⡔⠇⠐⠂⠂⠀⢄⠠⡄
⠀⠀⠘⠐⠁⠦⠤⠤⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⢤⡤⠀⢀⢠⠁⠀⠀⠀⠀⠀⠀⠤⠁⠀⠈⠠
⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⠤⠄⠀⢠⠀⠄⠅⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠴⠰⠤⠤⠤⠤⠤⡴⠴⠄⡀⠀⠀⠀⠈⠀⠈⠁⠁⠈⠀⠃⠀⠀⢠⠈⠁⡄⠀⢀⠀⠤⠐⢈⡁
⠀⠀⢘⠁⠀⠅⠀⠀⠀⠀⠀⠀⡇⡀⠘⠄⠀⠀⠈⠀⠔⠢⠈⠉⠁⠉⠉⣀⠒⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠒⠈
⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢀⠈⢘⠀⠀⠄⠑⠄⠐⠀⢁⠀⢀⠠⠀⢒⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠠⠂⠂⢀⢀⠀⢀⠀⠀⠁⠺⠗⢈⠀⠀⠈⠀⠀⠒⠀⠉⠁⠉⠁⠐⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⢤
⠀⠀⠄⠀⠐⠀⠉⠈⠁⠈⠈⠁⠐⠀⠀⠄⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠘⠄⠀⠀⠀⠠⢀⠀⠀⠀⠠⢊⠠⠀⣤⠀⣤⠀⢴⢵⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⢄⠐⠀⡀⡀⠘⡁⠂⠂⡁⠀⠀⠀⢀⠤⡀⠀⠀⠄⠀⠆⠀⢀⠀⠒⠀⢀⣤⣤⣤⡀⠈⠊⠀⡀
⠀⠐⠃⠃⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠳⠀⠀⣂⠐⠀⠐⢡⠉⠁⠁⢄⠁⢄⠄⢁⠀⠀⠀⠐⠤⠈⠀⠉⠄⠁
⠀⠀⠖⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⠢⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⢙⠀⠀⠁⠀⠀⠀⣠⣄⠘⠀⠃⣀⣄⠀⠀⠐⠀⢀⠀⠀⠁⠀⠈⠀⠀⢀⠀⠁
⠀⠀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠄⣀⡀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡁⢈⠂⠀⠀⠀⠀⠖⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉⠀⠀⠉⠠⠂
⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠄⠉⠀⠀⣄⡀⣠⢠⠀⠜⠃⠈⠀⠀⠸⠿⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢠⠒⢢⠄⠀⠀⠀⠀⠀⠀⢀⡔⢢⡄⠀⠀⠀⠀⠈⠀⡼⢀⢀⠀⢨⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠀⠉⠁⠀⠤⠀⠄
⠀⠀⠀⢑⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⡁⡡⠀⣉⢘⠁⠀⠀⠀⠀⠂⠔⠀⠒⠀⠒⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠃⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠶⠶⠀⠀⠀⠲⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⠦⠨⠁⠀⠀⠀⠀⠀⠀⠈⠡⡄⠶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣀⢀⣠⣼⣦⣶⡄⠀⠀⠀⢰⣬⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⠀⠈⠀⢀⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⡿⠿⠀⠀⠀⠀⣸⣿⣿⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣍⣙⣷⣶⣦⣤⡄⠿⠟⠛⠻⠿⠿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇⠈⠉⠙⣻⣥⣴⣶⡄⠀⠀⢀⣿⣻⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⣀⢀⣨⣿⣿⣿⣻⣷⣶⣒⣿⣛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣙⢛⣿⠃⠀⠙⢿⣟⣽⣿⣿⣿⣿⣷⡼⣷⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠻⢿⡏⠀⠀⢀⣎⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⣔⣾⣿⣶⣶⣿⠛⣿⣿⣿⣿⣿⣿⣿⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⠿⢿⠛⣿⣿⣿⢿⣯⣼⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⠀⠀⠀⣹⣷⣾⠟⠚⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠾⠿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⡀⢠⠀⠀⢄⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠠⠐⠀⣀⢡⡀⠆⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⡄⠀⠸⠀⠔⠘⠘⠊⠄⠀⠉⠀⠀⠡⠀⠀⠀⠀⠀⡀⡀⣧⡄⣧⡗⡠⡐⠳⡘⠆⡅⠄⣦⢄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡈⠁⠀⠀⠀⠀⠀⠁⠆⠋⣁⠘⡰⢇⠉⠆⠙⢸⠀⠄⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠳⠾⣄⣅⠣⠈⣤⠆⣻⡼⢢⡄⢠⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⡀⢠⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠴⠥⡌⠀⠄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣷⠀⠀⠀⠀⢸⠀⠐⠠⠄⠀⠔⣠⠀⡄⣶⠀⠘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠐⠲⠶⠶⠞⠴⠟
⠀⠀⠀⡈⠓⠠⠤⢄⠀⣀⣀⣀⠀⠀⠰⠞⠃⣈⣾⠀⠀⠀⠀⠀⢇⠀⠁⠀⠁⠈⠉⠉⠁⠒⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⡖⠒⢤⣀⡀⢀⡀⠀⠀⠀⠀⢀⡄⠉⠭⣲⠆⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⢠⠉⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣰
⠀⠀⠀⡗⢦⡈⠉⡀⠀⠀⠁⠈⠉⠊⠉⠀⠀⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⢀⠘⠷⠤⠀⠀⠀⠠⠴⠔⠒⠀⡠⠖⠁⠀⠀⠀⠀⠀⡄⠈⡀⠀⠀⠄⣄⠀⠠⠃⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⠠⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠃⠂⠘⠂⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠄⣠⣄⠐⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢀⠙⠲⠄⠀⠢⠴⠀⠀⢫⣙⢃⠀⠑⠒⠶⠆⠀⠀⠢⠐⣦⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠓⠋
⠀⠀⠀⠀⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠐⠂⠈⠠⠔⠠⠄⠩⠗⠠⠔⠰⠒⢠⡀
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠊⠀⠒
⠀⠀⠀⠀⠀⠀⠉⠾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠐⠁⠀⠀⠀⠠⠘⠁⠀⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢰⠋⠀⡀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠁⠀⠀⣠
⠀⠀⡀⠀⠀⡀⠂⢀⡀⠈⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣻⠿⠃⠀⠰⡄⡀⡀⡠⠎⠀⣶⠄⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠚⢡⠀⢞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠘⠛⠠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠐⡄⠀⠀⠠⠀⠀⠀⡀⠐⠀⠆⠴⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄⠀⠀⢁⠈⠦⠂⠀⠀⠀⠀⠁⠘⠠⠒⢄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠂⠉⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠓⠀⡈⠀⠒⠀⠀⠄⠀⠐⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⠆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡆⠛⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⢐⢦⢴⡦⣴⡃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠃⢴⠀⣀⣸⠇⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠀⠀⣈⣯⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣌⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢶⣦⣄
⠀⠀⠀⠀⠆⠀⣤⠀⠀⠰⠆⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣄⣀⣀⡄⠀⠀⠀⢀⣀⠘
⠀⠀⠀⡘⢤⠐⠆⠀⠀⠰⠆⡁⠲⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⣀⡈⢘⣻⠇⠀⠘⠠⠄⠙⠃
⠀⠀⠠⠅⠀⣙⠀⠀⠀⠠⡄⠉⠴⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢄⠠⣠⠀⠀⢀⡆
⠀⠀⢀⠄⣤⢤⣤⢤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⢤⡄⠀⢠⣄⡐⣠⣾⡀⠀⠀⠀⠤⣠⡄⠀⠀⠈⠻⠇⠀⠀⠀⠂⠙⠐⠛⠀⠀⠈⠁
⠀⠀⣷⠔⠊⠙⠈⠀⡧⠐⠎⠋⠉⠈⡇⠁⠀⠀⠀⢠⡇⠀⠀⢙⡿⢿⣿⡇⠀⠀⠘⠶⣋⡐⠃⠊⢀⠆⠀⠀⠐⠗⠐⠒⡀⠀⠀⠐⢀
⠀⠀⡇⠐⠐⣲⣲⠀⡇⠒⠔⣒⡖⢀⡅⠒⠔⣂⡖⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⢛⣀⢀⣘⠃⠠⠀⠀⠈⠳⠄
⠀⠀⡇⠀⡐⣒⡏⠀⡇⠁⡀⣰⡇⠸⠃⠁⡀⣺⡇⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠄⣀⣄⣀⣀⡀⢀⠈⠉⠈⠁⠀⠀⠈⢀⡀
⠀⠀⡇⠀⢀⠤⠀⠀⡇⠀⢀⠤⢀⠘⡇⠀⢀⠀⡀⡈⡇⠀⠀⠀⠀⢠⠴⠚⠋⠉⠉⣀⣤⡀⡉⢉⣉⠙⠛⠳⢮⣔⠢⡀⠀⠀⠀⠀⠁
⠀⠀⠓⠆⠀⠤⠶⠶⠷⠦⠤⠤⠶⠒⠗⠾⠾⠟⠾⠶⠃⠀⠀⢠⠞⠁⡀⠶⣤⡻⠧⠙⢫⣹⡇⠸⠏⡾⢛⣤⣄⠈⠻⣌⠢⠀⠀⠠⣀⣠⠄
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢲⠀⢬⣝⣓⡂⠀⠀⠀⠀⠠⠀⠀⠀⠈⢻⣶⣿⣃⠀⠙⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠒⠰⠠⣡⡀⠀⠠⡀⠢⡘⠳⣖⠃⣀⡀⣠⣴⣶⣤⣀⠀⣄⠒⡮⢿⠉⠀⠀⠀⠀⠀⠈⢱⠐
⠀⠀⠈⠁⠜⠂⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠈⠀⡀⠏⠀⠀⠀⠀⠀⠠⠈⠙⠘⠛⠶⠿⠥⠿⠇⠽⠷⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙
⣿⣿⠿⢿⡿⢿⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⡔⠂⢶⣀⡶⠔⠢⠄⠀⠀⠀⠀⢀⠄⠤⢤⠀⠄
⠀⣀⠁⠀⠀⢀⢀⠀⠀⠉⡀⠀⢈⣁⣁⢀⣁⣀⢀⣀⣉⠀⠀⠀⠀⠄⠃⠐⠀⠀⠀⠂⠘⠠⡀⠀⡎⠀⠀⠀⠀⠂⠀⠄
⠀⠭⠀⠑⠉⠈⠀⠉⠀⠀⠤⠀⠀⢭⠅⠀⢬⠍⠀⢬⡍⠀⠀⠀⠀⠀⣀⡀⠈⣀⠁⢁⣀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⡀⠜⠐
⠀⣛⠀⠀⢰⠀⠀⡄⠀⠀⡓⠀⠀⣊⣀⢀⣊⣀⠀⣚⣀⠀⠀⠀⠀⠀⠙⠁⠈⠛⠁⠈⠋⠀⠂⠀⠀⠄⠠⠀⣃⣀⣀⠀⢀
⠀⢉⠀⠀⢈⠁⠈⠀⠀⠀⠁⠀⠄⠁⠀⠀⠁⠉⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠁⠀⠀⠠⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⣠⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⡠⠀⠀⠀⠀⠀⠀⠠⣀⠌⢠⠌⠄⡂
⠀⠀⠀⠈⠉⠉⠹⠏⠋⢢⡄⠀⠀⠐⡒⡲⠖⠲⡒⡲⠢⡀⠀⠀⠀⠀⡤⡁⠂⠁⠀⠈⢤⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠉⢂
⠀⠀⠀⡁⠀⠀⠀⠁⣠⠟⠀⠀⠙⠍⠹⡉⠙⠉⡹⠉⡩⠃⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠉⠀⠐⢪⡕⠂⠀⢙
⠀⠀⠀⠃⡐⠀⠔⠀⠇⠀⠀⠀⠀⠀⠂⢐⣀⡰⡡⠊⠀⠀⠀⠀⠀⠂⡄⠄⠀⠀⠀⠠⢠⠔⠀⠀⠀⠀⠚⠀⠀⠈⠀⠀⢐⢆
⠀⠀⠀⢡⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠐⠀⠪⠀⡠⠀⠀⠀⠀⠀⠁⠀⠛⠀⠈⠃⠑⠁
⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠄
⠀⠀⠀⡠⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠑⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠨
⠀⠐⠄⠀⡴⠦⢀⠠⠂⠀⠀⠀⠀⠄⠀⠰⠋⠆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠔
⠀⠀⢀⡔⡁⣈⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠁
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⢄
⠀⠀⡀⢋⠠⠀⠀⠀⡐⢀⠀⠀⠢⢤⡆⠀⠈⠀⠉⠀⠀⡆⠀⠀⠀⠀⠉⣤⠈
⠀⡀⠀⠀⠁⠀⠀⠈⢊⠀⢠⠀⠀⢺⡇⢆⡅⢰⡀⢰⣸⡄⠀⠀⠀⠀⠃⠀⠘⠠⢀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠤⠠⠬⢤⠄⠠
⠀⠔⠀⠀⠨⢠⡄⠄⡐⠀⠠⠀⠀⠸⡗⠀⠃⠂⠊⠂⠚⡅⠀⠀⠀⠀⠀⠀⠀⢀⠁⠙⠀⠂⠀⠀⠀⠀⢄⡰⠆⠆⢀⠨⣀
⠀⠀⠤⡀⠀⠠⠄⠈⢈⠤⠀⠀⠈⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⡀⢈⠁⠂
⠀⠀⠀⠁⠂⠀⠀⠐⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡶⠖⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠖⠛⣏⠉⠙⣇⣀⣀⣷⣤⠴⠿⣦⡀
⠀⠀⢀⣀⣤⢶⡞⠛⢻⠁⠀⢹⣀⣤⠼⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠙⣦⡀
⢰⡟⠋⢯⠀⢀⣧⣤⠼⠗⠛⠋⠁⠀⠀⠀⠀⠀⡀⣄⠰⣄⠀⠀⠀⠀⠈⠻⣄
⠈⣷⡴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠉⠈⠂⢀⣀⡤⠴⠒⠛⣦
⢰⠿⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠸⣆
⣿⠀⣿⠈⢳⡄⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⣿⠀⠸⡇⠀⢻⣄⡠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⣿⠀⠀⢿⠀⠀⢷⠀⢠⠞⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣧
⢸⡄⠀⠸⡇⠀⠘⣧⠀⠚⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⡀⢹⡄
⠈⣧⠀⠀⢿⡀⠀⢹⡆⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢷⣤⠀⣿⣁⠀⢷⡀
⠀⢹⡆⠀⠘⣇⠀⠀⢿⡀⠀⣟⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠏⠁⠈⣿⣦⠀⢉⡇⠘⣧
⠀⠀⢷⠀⠀⢻⡄⠀⠈⣧⠀⠈⠓⢦⣄⠀⠀⢀⡀⠀⣼⠋⢹⣆⠘⠷⠤⠆⠈⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠸⣇⠀⠈⣷⠀⠀⠸⡆⠀⠀⠀⠉⢷⢹⡏⠻⣆⠻⠦⠾⠙⠀⠀⠀⣀⣠⠴⠒⠀⠀⠀⠀⢿⡀
⠀⠀⠀⢻⡀⠀⢸⡆⠀⠀⢻⡀⠀⠀⠀⣸⠇⠓⠀⠙⠀⢀⣀⡤⠖⠚⠉⠁⠀⢀⡀⢠⠀⠀⠀⠈⣧
⠀⠀⠀⠈⣧⠀⠀⢻⡀⠀⠈⣧⠀⠈⠛⠉⠀⠠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⢸⠇⠀⠀⠸⡆
⠀⠀⠀⠀⢸⡆⠀⠈⣧⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠘⣇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠹⡇
⠀⠀⠀⠀⠀⠈⣧⠀⠀⢹⡆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢻⡀⠀⢻⡀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⣷⠀⠘⣧⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠸⣇⠀⢻⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢴⡖⠛⣏⢁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢻⡄⢸⡇⠀⣓⢌⠑⠆⠀⠀⠀⣀⣀⣤⢶⡞⠛⠹⡅⢀⣠⣷⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⣷⢸⡇⠀⠈⢁⣠⣤⢶⠞⠛⠙⣇⠀⢀⣧⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡤⢶⡟⠙⡇⠀⠀⣧⣠⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣠⣷⠴⠾⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⠶⣤⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣤⡄⠐⡄⣤⢰⠀⠀⠁⠶⠀⠂⣈⠀⠀⠀⠀⣧⠀⣷⠀⠙⡆⠀⠐⠉⢃⠀⡀⠈⠀⠀⠀⣴⣻⢞⡽⣻⠎⠀⠀⠀⠀⠀⠀⣤⠚⣶
⠀⠀⠀⠀⠀⠀⠓⠃⠘⠀⠀⠈⠛⠋⠛⠉⠀⠀⠀⠀⢹⣁⠀⢂⡼⠃⠀⠀⠒⠊⠉⠀⡂⠀⠀⠈⠐⠫⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢀⣤⣤⣠⣦⣤⠀⠀⠀⣀⠐⠄⣈⠀⠀⠀⠀⠀⢠⣄⣠⣄⣤
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡒⠦⠀⠀⠀⢠⣤⠙⠃⠀⡼⠂⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣴⠆⠀⠑⢿⡄⠀⠀⠘⠛⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣤⠐⠂⠛⠁⠠⠀⠀⠀⢀⠈⠀⠔⠊⠀⠀⠀⠀⠁⠀⠀⡀⠀⣦⠀⠠⠟⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⣶⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠀⠁⠒⠀⠀⠀⢤⡀⠈⠹⠇⢤⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠈⠿⠀⠀⠀⠀⠀⠠⠁⠀⠀⡙⣷
⠀⠀⠀⠀⠠⣶⣤⣤⠀⠀⢾⡀⠁⠀⠀⠀⠰⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⠀⠀⠀⠠⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁
⠀⠀⠀⠀⠀⠿⣮⠗⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢠⠦⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠉⠛⡂⠀⠀⠀⠀⢰⠀⢃⣠⠖⠀⡀⠀⠀⠄⠀⠰⠀⠀⠀⣴⣶⣶⣶⡶⢤⣀
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⡣⠀⠠⢤⡄⣤⠖⠉⠀⠀⠀⣅⢠⠄⠀⣀⡀⠀⠀⠀⠳⢄⠀⠀⡴⠖⠋
⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠹⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠡⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⢾⡿⠁⠀⢀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⣄⠁⠠⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠁⠀⠀⠀⠀⢀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣷⢄⡇
⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠄⠀⠉⠑⠀⠄⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀
⠀⢆⣀⣀⠀⠀⣀⣠⠀⠀⠸⣿⣍⣶⣍⣻⡀⠀⠀⠀⠰⠒⠆⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⠀⢀⡜⠁⣴⣴⣀⡶⢤⡀⢠⣾⣛⢛⣿
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢶⣤⣄⣤⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠠⠀⠃⣜⢁⣸⡧⡴⠃⠀⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠘⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢤⣤⠀⠀⠀⠀⠘⢿⡿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡄⠀⠀⠀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠄⣀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠀⠀⠠⠀⠀⠀⠘⠳⠶⠶⠟⠉
⠀⠀⠀⠀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⠒⠀⢀⡇⠀⠀⠀⠀⠀⠤⢤⠐⠂⠀⠀⠀⣶⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠾⠷⠂⠐⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⢠⡜⠛⣅⠀⠀⠉⠉⠉⠀⠀⠃⠀⠀⠀⣷⡾⡿⠻⣆⠀⠀⠀⠀⠀⣾⢀⡀⠟⠀⠀⠀⣤⣤⠀⠀⢀⠀⠀⠀⢀⠸⠋⠙⠉⠙⠁
⠀⠀⠀⠉⠀⠉⠂⠉⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠳⠊⠐⠓⠀⠛⠁⠀⠀⠠⣅⠈⢨⠅⠀⠀⠀⠀⠉⠠⣦⠆⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠂
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠀⣤⣤⡀⠀⡀⠀⠀⠀⠿⡇⠀⢀⣠⠤⢄
⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠠⠀⠀⠘⠇⠀⠀⠀⣴⠾⠟⠉⠁⠀⠐⠀⠀⠁⠀⣘⣦⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠡⠀⠀⠀⠁⠐⠀⠀⠿⠁⠟
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠟⠁⠀⠀⢈⣿⡷⠒⠾⠟⠂⠀⠀⠀⠀⠀⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⣿⠋⢻⠛⢛⡛⠟⣛⣿⢛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣾⣿⣶⣾⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡆⠀⠀⢀⠀⠀⠀⡾⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠰⠀⢠⠚⠻⣷⡄⠀⣇⠀⠀⢸⠀⠀⠀⠃⠀⠀⠸⠁⠀⠀⣰⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⠀⢠⠀⠀⢀⠀⠀⠀⠰⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣄⠀⠀⢸⡄⠀⠰⣿⠀⠀⣀⣀⡘⠀⠀⡿⠀⢀⡟⢠⠖⠳⣦⠀⢀⡴⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⡀⣼⠉⠈⠉⣷⡴⠶⢦⣄⠀⡏⠀⢀⣿⠀⠋⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⣴⠾⠛⠛⠛⠋⠉⢉⣭⣭⣭⣭⣭⣭⡇⠀⠀⠀⣿⠀⠈⠁⣿⣭⣭⣭⣭⣍⠉⠉⠙⠛⠛⠻⢶⡄⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠖⢂⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢀⡀⢸⣿⠀⡀⢰⣿⣿⣿⣿⣿⣿⠀⠀⣠⣶⡒⢦⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡾⠰⢡⣄⡸⡟⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣏⣀⣿⣿⣿⣿⣿⣿⣿⢠⣤⡝⠛⣡⣤⡱⠸⡆⠴⠒
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⡇⠀⠘⠛⣩⣥⠛⠛⣸⣿⣟⣩⠴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⣣⣦⡙⠛⠁⠀⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⡇⠀⠀⠀⠙⠛⣰⠟⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣙⠛⠁⠀⠀⠀⡇⠐⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢹⡀⠀⠀⠀⠉⣿⠀⠀⠀⠛⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠀⠈⠉⠁⠀⠀⠀⣸⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⢧⣀⡀⢀⠀⢹⡀⠀⠀⠀⢸⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⣀⣀⡀
⠀⠀⠀⠐⠒⠛⠋⠉⠉⠀⠀⠀⠀⢉⡛⠛⠛⠛⣧⠀⠀⠀⠈⠉⠉⠙⠛⠿⠛⢿⠿⡿⠿⠿⠿⠿⣿⣿⠛⠛⠛⠛⠛⠛⢛⡋⠁⠀⠀⠀⠈⠉⠉⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠖⠛⠉⠁⠀⣀⡄⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⠈⠙⠦⣄⡀⠀⠉⠀⠀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⣀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⠀⠑⠂⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠠⠚⠁⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠁⠀⠀⠀⠢⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⣠⠆⠀⠀⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠶⣿⡟⠁⠘⠆⠀⠀⢦⡀⠀⠈⠳⠀⠀⠀⠀⠀⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣁⣀⣀⢀⣞⣁⣀⣠⣆⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣧⣀⣀⣀⣦⣀⣀⣙⣆⣀⣀⣀⣙⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⠾⠉⣉⣉⣩⣉⣉⠉⠿⠿⠉⣉⣉⣩⣉⣍⠉⠿⠉⣉⣍⠙⠟⢉⣉⣍⠉⢉⣉⣉⣉⣍⣉⣩⡉⠉⢉⣉⣉⣉⣉⣍⡉⠹⣶⣀
⠀⢰⡶⠶⠶⠶⠾⠂⣶⣿⠉⠉⠉⠻⠿⠀⠀⣾⣿⠉⠈⠉⣿⣶⠀⠠⣿⣿⣧⣴⣾⣿⡿⠀⢸⣿⡏⠉⠉⠀⠉⠁⠀⢸⣿⡏⠉⠁⢹⣿⡆⠈⠻⠶⠶⠶⣶⣀
⠀⢸⡇⠸⠿⠿⠂⠀⣿⣿⠀⠀⠺⣿⣿⠀⠀⣿⣿⣤⣤⣤⣿⣿⠀⠈⣿⣿⠘⠟⢸⣿⡇⠀⢸⣿⡿⠾⠿⠿⠂⠀⠀⢸⣿⡿⠿⠷⢿⣯⡀⠀⠺⠿⠿⠀⠈⣿
⠀⠘⢻⣦⣤⣤⣤⡀⠛⣿⣤⣤⣤⠟⣿⠀⠀⣿⣿⠁⠀⠈⣿⣿⠀⢀⣿⣿⠀⠀⢸⣿⣇⠀⢸⣿⣧⣤⣤⣤⣤⡄⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠠⠀⠀⠀⠠⠀⠃⠀⠀⠀⢐⣆⢁⢁⣔⠀⠀⠀⡀⣀⠀⠠⡄⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⡀
⠀⠰⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢲⣽⢼⡵⣏⠀⠀⠀⠀⢉⡐⠆⣤⠀⠀⠀⢳⠀⠀⠀⠀⣶⡀⢀⡡⣮⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠈⠉⠋⠛⠉⠀⠀⠀⠀⠈⠁⠶⢀⡀⠀⠈⠋⠀⠀⠀⠀⠁⡐⠂⡀⣧⡀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡆⢀⣶⣄⠀⠀⠀⠀⠛⠀⠀⠉⠀⠀⠀⠨⠀⠀⠀⠀⠁⠀⠡⠾⠂⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢄⣤⠈⣀⡀
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⣄⠈⠉⡀⠛⠇⠀⠀⠀⠀⠀⡀⠀⣖⡒⣀⢀⡀⣀⡀⠐⢦⡀⠄
⠀⠀⠀⠐⣿⣾⡷⠀⠀⠀⠀⠀⠈⠛⠀⠸⠟⠀⠀⢀⢄⡴⢊⣁⡰⠦⠫⠹⠋⠈⠬⢫⣴⣶⣀⢃⡜⠀⢀
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠶⠠⡶⠀⠀⠀⠁⠏⢠⣬⣭⠁⠀⠀⠀⠀⣀⣀⢶⡶⠚⠁⠋⢀⣠⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠋⠓⡄⠀⠀⠀⠀⡈⠡⠴⠿⠱⠿⠼⢛⡊⠩⠀⠀⠀⠀⠀⠿⣯⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡌⠀⢈⠇⠈⢦⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢧
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⡃⠀⠈⠉⠀⠀⠀⠠⠯⠀⠀⣸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠎⠢⣁⡐⡆⠰⣂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣺⠟⠁⢰⣄⢀⡈⡉⣤⠀⣀⣈⢱⠐⣀⣀⠀⢸
⠀⠀⠀⠰⠂⢈⣶⠏⣿⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣿⠈⡍⠀⣿⡇⢨⠁⠘⠀⠠⡁⠀⡎
⠀⠀⠂⢄⠀⠠⡯⠀⠲⠀⠀⠀⠈⠢⠅⣰⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠶⠦⠳⠶⠖⠆⠱⠴⠶⠒⠗
⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡂⢄⠀⡀
⠀⠀⠀⠆⠀⠀⠀⠠⠆⢩⠁⠀⠀⠀⢠⣴⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡤⢀⠈⡄⠀⠀⠀⠠⢠⣉⣩⠝⠀⣤⠤⠛⠂⡀
⠀⠀⠀⠈⠃⠁⠀⠀⠀⠗⠀⠁⠀⠀⠀⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⣟⣀⣴⡾⠃⠀⠀⠤⠾⢖⡽⠇⠀⠌⠇⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃

⠀⢀⣤⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣡
⠀⠈⢯⣽⣿⠏⣹⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣤⡁⣤⠀⣴⡀⢠⡤⣄⢠⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢫⠸⠀⣾⣿⡏
⠀⠀⠈⢻⠙⠛⠓⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣝⡷⣿⠻⣿⣰⠿⢷⠻⣤⡤⢸⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢹⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢺⢿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⡀
⠀⠀⠀⠀⠈⠁⣀⣀⢤⣶⣀⠀⠀⠂⢀⡤⠀⠐⢯⠿⠀⠶⡆⠀⠀⠀⠘⠛⠈⠉⠉⠃⡀⠈⠓⠢⠬⠅⠀⠀⠠⠄⠐⠀⡀
⠀⠀⠀⠀⢸⠯⠛⢟⠋⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂
⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠀⣠⣤
⠀⠀⠀⠀⠘⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⠄⠠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠴⠷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢘⡃⠃⠙⠈
⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡤⣠⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣍⣀⣀⣩⡄
⠀⠀⠀⠂⠀⠓⠒⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠘⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠭⢀⡉⣤⣄⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡈⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣠⡄⢠⣿⣿⣿⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⡄⣴⣿⣷⣿⣿⡟⢿⣿⣿⣿⣿⣿⣇⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣿⣿⣿⡟⠈⢻⠇⠀⠘⠋⢻⣿⣿⣿⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⣀⣠⡏⣙⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣤⣾⡸⣆⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢡⠱⢻⡀⣼⣿⣿⡇⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⡇⣿⣿⣧⣀⡟⠎⡌⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢫⡳⣄⠰⢦⠘⣀⣄⣿⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣿⣿⢰⣀⠃⡴⠂⣠⢞⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠃⠀⠱⣭⣾⣿⣿⣇⣝⠷⢹⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠟⣻⣿⣿⣷⣭⠎⠀⠘⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣳⣌⢻⣿⣿⢿⣹⣷⣌⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡿⣿⣿⣟⣡⣞⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠗⠛⣛⣛⠛⠛⢾⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⣿⣿⡛⠺⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⡿⠞⠁⣀⣀⣀⠀⢀⡀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠶⠏⠀⢀⣀⡀⠀⣀⠀⠈⠳⣿⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⢀⡯⠟⢹⡀⠈⠉⠀⠀⠀⠀⢀⣀⣈⡀⠀⠀⠀⠠⠋⠉⠀⣞⠉⣹⠀⠀⠈⠻⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⡀⢸⣏⡉⠳⣤⠛⢉⣽⡇⠀⢀⣀⣀⠻⠭⠭⠟⣀⣀⡀⠀⢠⡞⠓⡎⠛⢡⠞⠳⡄⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢰⠇⠘⠓⠳⡞⠈⢳⠞⠚⠃⣴⢫⠀⠈⢳⡄⢀⡞⡁⠀⠙⣦⠈⠓⠞⣡⠶⢬⠳⠞⠁⠀⠈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏⠸⠀⠀⠀⠀⠷⠷⠾⠀⠀⠀⠻⣜⠳⢂⣼⣿⣿⣧⡛⠦⣠⠟⠀⠀⠀⠻⢤⠞⠀⠀⠀⠀⣆⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⢱⠀⠀⠀⠀⠀⠀⠠⠄⣠⣴⠆⣈⣭⣭⣤⣭⣭⣤⣭⣭⣁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢾⡿⣼⠟⢡⢞⣁⣀⢀⣀⡀⣀⠀⣀⢀⣈⠳⡌⠻⣦⣄⠀⠀⠀⠀⠀⠀⠐⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣷⠀⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⢸⡇⣤⣸⣭⡗⣿⣾⣿⢸⡧⠀⠙⢦⡘⢿⣷⠀⠀⠀⠀⠀⠀⣠⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⣿⠀⠀⠀⠀⠀⠰⠏⣠⠞⠀⠀⠘⠳⠟⠘⠀⠛⠛⠋⠛⠘⠳⠂⠀⠀⠳⣄⠹⠀⠀⠀⡀⠀⠀⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣷⣦⣀⣀⠀⢀⣾⠁⠀⠀⠀⠀⢉⡿⢸⡟⣿⣿⣶⡇⣿⣉⠀⠀⠀⠀⠈⣳⡀⠀⣘⣿⣩⣾⣿⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡙⠛⠟⠛⣉⣴⣻⣦⢳⡀⠀⠀⠀⢼⣧⢸⣧⡿⢿⠹⡇⣿⡤⠀⠀⠀⢀⡞⣵⣟⣦⣈⠛⠻⠛⢋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⢿⣿⣿⣧⣙⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣋⣼⣿⣿⡿⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠉⠉⠀⠀⠁⠀⠒⠂⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠃⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣀⣠⡶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣉⣀⣀⣀⣠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⠉⠝⢻⡇⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠙⠛⢶⣄⠀⠀⠈⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡇⠀⠀⠠⠙⠿⠷⢶⣄⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠶⠃⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠾⠶⠶⣶⣶⡶⠶⠶⠟⠛⠉⠉⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠁⢰⣿
⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠤⡦
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣖⣤⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠦⠀⠀⠀⡄⠀⠀⢀⡎⠀⠁⢀⡖⠀⠀⠀⠀⢸⠃
⠀⠀⢸⣶⣤⡀⠀⠀⠀⠀⠀⠉⠛⠒⠳⠶⢢⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠄⠀⠉⠀⠀⠀⣀⡤⠖⠉⠀⢀⣤⢸⡂⠀⠀⡘⠋⠀⠀⠀⠐⣿
⠀⢀⠉⠉⠛⠛⠽⠆⣶⣶⠀⢸⡗⠒⠛⡿⠟⠁⠀⠀⠀⢀⣀⣀⠈⠉⠑⠂⠀⠀⠀⢀⣀⠦⠚⠉⠀⠀⠀⠀⢰⠃⠉⠀⡻⢣⡞⠁⠀⠀⠀⠀⠜⠁
⠀⠈⠻⢿⣶⣤⡄⢀⣀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠉⠀⠀⣀⡠⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⢁⡌⠁⠀⠀⠀⢶⣾
⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠀⠁⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⢀⠠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣇⡄⣴⠉⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣀⣤⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣿⣤⠍⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠉⠉⠓⠶⠤⣀⠀⠀⠀⠀⠀⠀⡾⣿⡁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠛⠒⠒⠚⠛⠋
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡄⡀⠀⠀⠀⠸⣿⣿⣿⢷⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠤⠦⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠉⠛⠿⣿⣮⣭⡙⠳⢦⣤⠔⠒⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⠠⡶⠋⠉⠋⠉⠓⣆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣷⣀⣠⡦⠔⠋⠉⠀⠀⠀⠀⠠⠎⠈⠐⢦⣄⣀⣀⠀⠀⣸⠷⣦⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠰⠛⠙⢸⡟⢛⣏⠀⠀⠘⡇
⠀⠀⠀⠑⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⣠⠀⠀⠀⠀⠈⡿⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠈⠙⠿⠿⢿⣏⣀⣚⢛⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⡀⠑⢤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⢉⡁⠀⡄⡖⠀⣆⣖⠈⣝⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣠⡧⠼⠓⠓⠛⠛⠛⠋⠉⠙⠻⢾⣧⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⣀⣀⣈⡀⠀⠀⠀⠀⡙⠷⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⡤⢾⣿⣿⣿⣛⣛⢫⡝⣛⡷⣮⡂⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⢞⣭⣭⣭⣹⣗⣲⣶⣮⣽⣶⣶⡶⣟⣳⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣲⣿⠿⠿⢿⣿⠷⣚⠯⣽⣿⣿⣷⣾⣿⣿⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢙⣿⠓⢤⡿⢷⣾⣿⣿⣻⠟⢿⠿⣷⣾⣦⠽⠿⢻⣯⢮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠠⣟⠀⠀⠀⠀⠛⢛⠍⠈⠀⠘⠠⠄⠻⠼⠁⠀⠈⡃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢩⢀⣧⡀⠀⠀⠀⢀⠄⠀⠀⢀⣀⠀⠀⠀⣰⢠⡕⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢿⠈⠑⢣⠀⠀⡈⢙⣻⣶⣞⣿⣉⠁⠀⠀⢡⣾⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣯⣧⠀⠀⣼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⣼⠋⢁⡒⣶⣶⣶⡆⠀⠑⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⣤⣀⡀⠛⢛⠋⠀⣀⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣸⡏⡏⡛⣿⡾⣫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢸⢹⣋⣧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⢀⠀⠀⣤⣤⢀⢀⢀⠀⡀⠀⠀⣤⡄⢠⠀⠀⢀⣀⡀⡀⢀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠾⠀⠯⠸⠁⠀⠻⠟⠾⠟⠒⠘⡏⠀⠀⠛⠳⠸⠸⠇⠻⠚⠸⠁⠾⠻⠸⠏⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⢀⡠⣤⡤⢀⡟⢩⠉⠛⢲⠎⢈⠉⠳⢦⣀⠘⠋⢠⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡀⣀⣀⣠⡍⣰⣧⡾⠛⠛⣾⢁⣋⠃⢰⠏⣀⠹⠀⠀⠀⠈⠉⠲⢾⣹⣧⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠛⠻⠱⣹⠋⣭⢻⣦⣻⣅⡛⣹⠇⠀⠛⣰⠏⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡄⠀⠻⢿⠄⠀⢀⣀⢠
⠀⠀⠀⢀⡔⠋⠑⢦⠘⠻⢮⡵⣿⣴⠶⢿⣯⡟⠀⠀⣸⠃⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣰⠯⢀⣒⣲⠭⣽⣖⠚⠂⣇
⠀⠀⠀⢸⡁⠠⡤⡟⠀⠀⠘⠱⣿⢡⢔⣤⣼⠃⠀⣸⠃⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⠀⠀⣰⠿⢿⣯⡁⠀⠈⣩⣄⡀⠀⢹⡀
⠀⠀⠀⠀⠉⠒⠋⡖⠉⣉⡰⣿⠿⣧⣄⣠⡿⠀⣸⠃⠀⠀⠀⠘⠻⣿⣿⣿⠀⠀⠀⠀⣰⣯⠠⠀⣿⣷⠀⠀⠛⠛⠀⠀⡈⣇
⠀⠀⠀⠀⠺⠗⠘⣇⢸⠁⣸⠉⠉⠉⠉⠉⠓⠺⠇⣀⡀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⣴⢿⣿⡄⣨⣿⡿⠀⢀⣀⣠⠤⠶⣲⣼
⠀⠀⢀⣀⣀⣀⣀⡀⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⢠⣤⠀⣼⢃⣀⣿⣏⠉⠉⠙⠛⠯⢤⣀⣀⡀
⠀⠀⠙⢅⠀⠀⠀⠈⢻⣯⣄⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠮⢥⣼⣯⠭⢿⡇⠀⠉⠙⣶⡄⠀⠀⠀⠀⢈⡽⠃
⠀⠀⠀⣼⠁⠀⠀⠀⠘⠿⣄⡀⠀⠸⡇⠉⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠛⣻⣿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠋⠉⠙⠓⠢⠤⣄⣀⡈⢻⣿⡇⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠲⠤⠤⠤⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠤⠄⣀⣀⣀⣠⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢄⡀⠀⠀⠀⠀⣾⢍⠀⣁⣪⢶⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡟⣉⠻⣏⡀⠀⠀⠀⣏⢀⢹⠁⢐⡏⡰⠁⠀⢉⡇⠀⢠⠏⡄⢡⠃⡄⠙⠒⠤⣀⠀⠐⣯⠿⠄⠐⡶⢶⣀
⠀⢀⠄⣴⡆⢄⠈⠟⣧⣍⣵⠟⠃⠀⠀⠀⠈⠱⠧⠴⠊⠁⣇⠀⠂⣞⠇⠀⡜⠐⢢⠇⠚⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠚⠻⡏
⠀⢿⡟⣉⠍⣿⢣⠀⠉⠐⣋⠁⠀⢀⡤⡤⠤⡄⠀⢀⡴⡀⠈⠉⠓⠋⠀⢰⠃⢠⠏⠀⢀⣠⣶⠀⠀⠀⢠⠏⠀⣀⣠⣦
⠀⠺⡿⢦⡴⣻⠏⡰⣿⠖⡻⣦⣄⠈⠦⡀⣰⠃⠀⣏⢀⡇⠀⠀⣀⠀⠀⡾⢠⠏⠀⠀⠻⢿⣿⠀⠀⢠⠏⠀⠀⣸⢯⣿⠒
⠀⠀⠈⠉⠉⠀⠀⢷⣧⣘⣃⣼⡁⠀⠀⠈⠃⠀⠀⠘⠋⠀⡴⠊⠈⢣⢰⠇⣎⠀⠀⠀⠀⠀⠀⣀⢠⠏⠀⣠⣄⠀⠀⠉⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠏⠙⠁⠀⠀⠀⠀⢰⡉⡣⠤⡀⠙⠲⠖⠞⠈⠳⠾⠿⠦⢄⣀⠀⢠⢁⠞⠀⠀⠙⠉⠀⠀⠀⠤⣿⡿
⠀⣄⣤⣤⣄⣰⣤⣤⣤⣤⡤⠶⢶⣦⠶⠲⠀⢎⣁⢴⡖⠁⠀⡤⠒⠀⠐⠒⠀⠀⠀⡀⠈⠁⠒⠊⠀⠀⢠⣔⢀⠒⠂⠤⠤⠀⣠
⠀⡅⠀⣀⡀⠀⣾⣯⠀⢹⡇⠀⣠⣤⠀⢸⠀⠀⢀⡠⠤⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠸⠆⢈⢽⡆⠀⠀⢰⡁
⠀⡇⠀⠉⠁⠀⠸⣷⣠⣿⠇⠀⠀⣁⣀⣸⠇⠀⠀⢱⠀⠀⠈⠿⣉⣉⣻⠉⠐⠒⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠂⠄⠀⠓
⠀⠁⠉⠭⠉⠌⠉⠉⠉⠉⠀⠈⠉⠁⠉⠛⠀⠀⠀⠈⠉⠉⠀⠂⠀⠘⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠀⠀⠀⠀⢄
⠀⠈⠒⠤⢤⠤⢄⣀⣁⡀⠀⠀⠀⠠⠤⠄⣀⢣⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⡀⣑⠀⠀⠀⠘⠆⢧⡄⠀⠀⡱⠊⢍⡉⠉⠐⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠠⡀⡀⡀⠀⣄⢀⠀⠠⣀⠀⡀⠀⠃⠔⢱⢸⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⡄⠐⠒⠁⠇⠀⠠⡀⢠⠀⣀⠀⠠⠂⠉⠻⣄⠀⠀⡸⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⠚⠀⣬⡀⠀⢀⣱⢀⢀⣀⠞⣧⣤⣤⣴⠏⡷⠊⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⢾⠈⠙⠿⠍⠉⣼⠄⢠⡀⠀⠈⠉⢀⣼⡅⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣠⣤⣶⠗⠛⠻⣌⣐⣓⣫⠽⠷⣄⠠⡀⠀⠀⠀⠀⠀⠈⣇⠄
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⠉⠉⠀⠄⠀⠀⠀⠘⠛⠒⠒⠒⠾⢷⣾⣦⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⢥⠀⠀⠀⠀⢻⠿
⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠠⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠘⢎⠓⠤⡄⠤⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⢉⣹⠧⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠴⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠀⠀⠀⠀⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡠⠤⣄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢱⠊⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣀⠀⠈⡆⣀⠀⠀⢸⠒⠒⠲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡀⠀⠙⠀⢀⣇⠁⠀⢀⠜⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠒⠒⠒⠋⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠞⠉⠆⠇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠰⠙⡆
⡔⠒⢌⣂⠘⠃⣰⡤⠒⢦⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣼⣿⣷⣤⣀⡀⠀⢀⡴⠚⠉⠉⠉⠙⠒⠢⢤⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠤⣀
⠈⢒⢤⠀⠀⠀⠁⠀⡔⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢻⣿⣿⣿⠏⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢉⡥⣼⣁⡇
⠸⣁⠤⠖⢤⠔⣲⠊⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠸⡟⠛⠛⡄⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠈⠁⢀⡠⠃
⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠒⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠘⠢⢤⣀⣀⣀⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⠶⢄⣀⡀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠉⠉⠒⠢⢄⡀⣰⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠉⠉⠙⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠉
⠀⠀⠀⠀⠀⠀⠠⢴⣶⣶⣶⣶⣦⣤⣀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣷⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠙⠛⠻⠷⣶⣦⠀⠀⣰⡟⠁⠀⠀⠙⣷⡀⠀⢠⣶⡶⠿⠛⠛⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣶⣟⠁⠀⢰⣿⣶⣶⠀⢰⣶⣾⣷⠀⠀⢙⣷⡆⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣶⣦⡈⠻⣷⣄⠀⠀⠀⣿⠀⢸⡇⠀⠀⢀⣴⡿⠋⣠⣶⣄⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠉⠻⣦⡈⠻⡷⠀⢀⣉⣤⣌⣁⠀⠰⡿⠋⣠⡾⠏⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⡇⠀⠀⠙⠻⡆⢠⣾⠟⠉⠉⠉⠙⢿⣆⠀⠾⠛⠀⠀⠀⣿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠀⠸⠷⠶⠶⠶⠶⢀⣿⠁⠀⠀⠀⠀⠀⠀⢹⣇⠰⠶⠶⠶⠾⠿⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⢠⣤⣤⣤⣤⣤⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⣤⣤⣤⣤⣤⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣼⡏⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠈⣿⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣴⠶⠿⠦⠈⠙⠻⠿⠿⠛⠉⠀⠾⠷⢶⣤⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⣷⣤⣴⡇⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣾⡏⣿⡆⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡿⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣿⣅⠀⢀⣽⣧⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⠟⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢧
⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢾⢾
⠀⠀⠀⠀⠀⠘⡆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⣟⣠⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣤⣵⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢦⡀⠸⡅⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⣬⢽⡀⠀⠀⣠⣤⡴⠦⡿⣅⠈⢸⡇⢠⡇⠀⡇⢀⣴⣶⣾⡿⠿⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢰⠃⠀⠰⣩⣯⡉⠰⣇⣼⣇⠀⢳⣸⣿⣦⡷⠋⢩⣴⣶⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣤⣷⣤⣀⣿⣿⣷⠀⠻⢿⣿⣦⣤⠽⣯⣷⣶⣶⡿⣋⣵⢖⠛⣻⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣭⠾⠟⣛⣭⣴⣶⣮⣭⣿⣦⡶⠞⣿⣷⣤⠀⢿⣿⣿⣿⣵⣿⢿⣼⣾⣿⢟⣛⣻⣿⣦
⠀⠀⠀⠀⠀⠀⡞⣀⣰⣿⣻⣿⣟⣛⠿⣿⣿⣿⣷⣦⠈⠻⣿⣷⣦⣿⣿⣿⣿⣴⣼⣾⡿⠿⣿⣶⣿⡟⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣯⣭⢻⣿⣿⣮⢿⣿⣿⡿⠿⠾⠿⠋⢻⡉⠻⠿⢿⣾⣿⣿⣿⣿⣼⣿⡿⣱⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⣙⠿⣵⣦⡌⣿⣿⣿⣿⣿⣶⣤⣶⣤⡿⠤⢤⣤⣄⣀⡏⠀⠟⢻⣿⣿⣵⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣗⢿⣿⣷⣿⣿⣷⣿⣿⣿⡿⢛⣛⢻⣿⣟⢴⣿⣷⣿⣿⠟⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡙⣆
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣇⣿⣻⣷⣿⣿⣾⣿⣿⠟⠁⢀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣭⣭⣭⣭⣾⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⡯⠀⢀⡎⠀⠀⠀⠙⠮⣽⣿⡿⠿⠿⣿⠟⡼
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⠻⣿⣿⠛⠋⠀⣽⣿⢤⣤⣂⡼⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⡿⢣⢻⢦⣀⣹⣧⣀⠀⢜⣻⠁⣀⣽⠇
⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣶⡇⠀⢠⠃⠈⡷⠤⢤⣾⣟⣛⠛⠛⣋⡩⠏
⠀⠀⠀⠀⠀⠀⠙⢬⣛⠿⠿⣁⡴⠋⠀⠀⠉⠋⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⢀⠠⠄⢀
⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⢉⠑⢆⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡁⠀⢁⠀⢀⠀⢠⡀⠠⠀⠈⠁⢆⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠢
⠀⠀⠀⠘⠀⠈⠀⠀⡀⠠⠀⠐⠄⠁⡀⢨⡀⠀⠀⠀⣄⠀⠈⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠸⢀⠈⠀⢀⣀⢀⣠⠊⠓⢀⠀⠜⠁⠀⠀⠀⠈⠂⢀⡀⠁⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠈⠀⠉⠀⠉⢄⡀⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⢄⡂⠐⣈⣖⡓⢼⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠉
⠀⠀⠐⠀⠀⠒⠀⣱⠠⢆⡡⢉⣡⡀⠃⡴⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠈⠀⠈⡉⠂⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⢈⠂⠀⡘⠁⠀⠀⠹⠄⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⠼⠄⣀⣀⡠⡀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠈⠉⢀⠣⡀⠀⠒⠂⠀⠀⠀⠃⠠⠆⠀⣠⣬⡄⠀⢥⣶⠆⡞
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣅⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠁⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠔⣀⡀⢈⣙⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠺⠶⠷⠶⠤⠿⠤⠀⠔⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⢸⠀⠀⠀⠀⠀⢈⠕⢢⡀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢡⠀⠀⠀⡀⠀⠐⢹⠄⠀⠀⠀⠀⠀⠀⡠⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠢⠀⠀⡄⠀⠂⠁⠀⠀⠀⠀⠀⠀⢰⠐⣐⠨⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠠⢈⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠉⠁⠀⠀⠀⢉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠚⢇
⠀⠀⠀⠀⣎⡀⠉⣱⠄⠀⢠⠠⠀⣀⠀⠀⠀⠀⠀⣀⢤⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⠤⠄⣀⡀⠀⠀⠀⠀⢸⠉⠉⢀⣠⡤⣤⠀⡼
⠀⠀⠀⠀⠜⠁⢠⠏⠀⠀⢸⢀⢀⣸⡃⠀⠀⣀⡮⠒⠙⡇⣀⣀⣠⠄⠀⠂⠑⠊⠉⠛⢛⣳⣼⣷⠀⠀⠀⠀⢧⡼⠯⠀⢀⠎⢸⡁
⠀⠀⠀⠸⠈⠁⡎⠀⠀⠀⡈⢀⢸⡏⠀⠀⠀⠚⠁⠀⠀⢉⠿⠋⠀⣀⣠⡤⠤⠶⠶⠶⠶⠤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⡟
⠀⠀⠀⡇⠘⠀⢧⡀⠀⡠⢃⡀⣸⡇⠀⠀⠀⢦⣀⣀⠤⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⢠⡇⣘⣇
⠀⠀⠀⠘⣆⡘⠀⠈⠩⠄⣈⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣄⠙⡋⢉⢁⣀⣤⠀⠀⠈⠿⠛⠋
⠀⠀⠀⠀⠈⠙⠶⠶⣶⠿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢮⡳⢤⣀⠀⠀⠀⠀⠄⡈⠋⠀⣬⠪⡭⠉⢀⣂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠰⠀⠀⠐⠝⢷⣮⠉⢲⡄⠘⢮⡛⠀⣀⠉⡰⢐⣣⣬⠃⠀⢀⢿⡀⢓⡀
⠀⠀⠀⡠⢐⡩⠔⣲⢆⡥⠾⠶⡆⠐⡄⠱⡄⠀⠀⠄⠀⡄⠀⠀⠀⣈⠢⢿⣲⡾⠀⠀⠀⠉⠢⣛⡰⢱⣨⣿⡃⡶⠀⣠⢢⡶⠌⢥⡄
⠀⠀⠀⡧⣀⣑⠤⠉⠈⠡⠼⠛⣓⣉⠶⢺⡇⠀⢸⡀⠀⠷⡖⣀⣥⠴⠖⠛⠁⣀⠀⣄⣀⣠⣄⣌⠉⠉⠀⠀⢳⡀⠀⢛⡔⠵⡏⢀⡴⠏
⠀⠀⠀⠓⠦⣭⣀⣀⣀⣉⣈⣀⣤⠤⠞⠋⠀⠀⠀⠈⠉⠉⠉⠁⠀⢀⡠⠚⠋⢱⡄⢀⠠⠦⠀⠈⠉⠉⢙⠆⠀⢣⡠⠊⡀⣾⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠲⠆⠐⠀⠀⡈⠀⠀⣠⣄⠀⠀⢸⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⠤⠤⣤⡀⠄⡀⠀⠀⠀⠀⠀⠀⢇⠲⠂⢀⠀⢰⡆⠸⡇⠀⠺⢿⢿⠇⠀⡜
⠀⠀⢰⠉⡔⠉⢩⣾⣿⣟⣛⡻⠛⠿⣟⠀⣸⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢦⠀⣇⣸⠀⠀⠀⠉⠁⠀⠀⡇
⠀⠀⢺⣷⢮⡤⣄⡀⠸⠿⠿⠿⠗⠀⠀⢹⡾⠀⠀⡠⠤⠶⠰⠴⣤⡀⠑⢄⣧⡻⣛⣤⠤⠤⣄⣀⣛⣡⠃
⠀⠀⠀⠉⠓⠧⣄⣁⡉⠉⠐⠒⠒⠶⣒⡿⠀⠀⠣⢄⠀⠀⣶⠚⠛⡇⠀⠀⠙⠵⠟⢁⢀⡀⠠⠤⠀⠒⠂⠀⠀⠀⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⡘⣠⡻⣿⣍⡉⠉⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡇⢀⣠⣤
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠴⠦⣤⠴⢾⣏⠁⢰⡃
⠀⠀⠀⣄⠘⠛⠻⠚⣀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣈⡛⠓⠒⠒⠒⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣧⣶⣃⣀⠀⢙⣦
⠀⠀⠀⠸⢷⠶⠦⠬⠭⣶⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⡤⠽⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉
⠀⠀⠀⢸⣬⣙⣛⣛⣛⣿⡃⠀⢀⣀⣀⠀⢀⠀⢸⡷⣀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠒⠢⠤⢄⣀⠀⠀⠀⣶⣶⠦⠤⡤⣀
⠀⠀⠀⠘⠯⣽⣛⣛⣿⣿⣿⣘⠒⠿⠿⢆⣠⡇⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠻⠭⠿⣦⣀⣠⡷
⠀⠀⠀⠸⣿⠶⠿⠷⠶⣾⣧⣽⣦⣭⣭⡼⣿⠀⠀⠙⠓⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣼⠃
⠀⠀⠀⠀⠈⠉⠉⠛⠉⠙⢯⣱⣂⢠⣰⣉⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠤⣄⣀⣀⡀⠠⣬⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠂⠀⠀⠀⠀⠀⠈⠈⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⢿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⢀⡠⠐⢊⣱⡀⠀⠀⠀⠀⣀⡾⠤⣄⠀⠀⠻⠋⠀⠀⠀⠀⠀⣠⠖⠈⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⡛⠀⠤⠊⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠠⠂⠉⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠸⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡿⠇⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⠀⡠⠴⣚⣵⡀⠀⠀⠀⢀⣠⡾⠤⣄⠀⠀⠳⠏⠀⠀⠀⠀⠀⣠⠔⢘⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⠛⡁⠐⠂⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠤⠊⠉⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠘⣄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⠀⠀⠀⠀⣖⡒⢢
⠀⠀⠀⠀⠈⠂⠀⣁⣀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠚⠀⣀⣀⠀⠛⠋⠊
⠀⠀⠀⠀⣸⠟⡉⢉⡉⠉⠻⣆⠀⠀⠀⠀⠀⣸⠝⠉⠉⠉⠉⠻⣇⠀⠀⠀⠀⠀⣸⠟⠩⠉⠉⠉⠻⣇
⠀⠀⠀⢸⠸⡆⠀⠬⠅⠀⢰⢋⠆⠀⠀⠀⠰⡙⡆⠀⢌⠉⠈⢰⠇⡇⠀⠀⠀⠰⡙⡆⠘⠌⠥⠋⢰⠇⡇
⠀⠀⠀⠀⠑⠳⠦⠤⠤⠴⠓⠋⠀⠀⠀⠀⠀⠙⠚⠦⠤⠥⠴⠞⠊⠀⠀⠀⠀⠀⠙⠚⠦⠤⠬⠴⠞⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡴⠷
⠀⠀⠀⠀⢠⣤⠤⠤⠤⠄⣤⡀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⡤⡀⠀⠀⠀⠀⠀⢠⣤⠤⠶⡶⠬⣤⡄
⠀⠀⠀⢠⢾⡁⠀⢀⡀⠀⢘⡷⡀⠀⠀⠀⢀⢾⡃⠀⢠⡄⠠⢘⡷⡀⠀⠀⠀⢀⢾⡃⠄⢠⡤⠠⢈⡗⡄
⠀⠀⠀⠘⢌⣇⠀⠨⠅⢀⡸⡰⠁⠀⠀⠀⠘⢌⡏⢡⠀⠀⠀⣸⡡⠃⠀⠀⠀⠈⢎⢇⡀⠀⠀⠀⣸⡡⠃
⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣦⣰⣦⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣤⣤⣽⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡿⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣷⠟⠑⠂⠀⠀⠀⠀⠀⣤⣀⡀⢀⣀⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣤⣎⣁⣠⣤⣤⠠⢬⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⣿⣩⢹⡿⣿⡟⢡⣾⣿⣫⣿⣥⣼⢿⣿⣿⣧⣀⢹⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⣿⣿⣿⡟⢻⣇⣿⣦⡿⣼⣿⣿⣾⣿⣿⣿⡿⣿⣛⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢻⣿⠉⣿⠀⢸⣿⣎⢿⡇⣿⣿⣿⣷⣤⣼⣿⣿⠉⢵⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠘⣿⣧⣿⣄⢠⣿⣿⣷⡇⣿⣿⣏⡻⠉⣿⣿⣿⡇⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣯⣿⣿⠛⢻⣿⣿⣃⣴⣶⣶⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⢿⣿⣿⡉⢸⣿⣿⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣾⢿⣿⣤⣬⣏⣖⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⣿⠛⠛⠉⣿⣤⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⣿⣷⣿⣶⣶⣿⣿⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣾⣿⡿⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⢰⣦⡀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⣼⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠉⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢀⣠⠤⠒⠉⠉⠒⠤⣄⡀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⢀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⢀⠀⠉⠓⠦⢄⡀⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠖⠋⠁⠀⠀⠀⢾⣉⡇⠀⠀⠀⠰⣏⣹⠄⠀⠀⠀⠈⠑⠲⣤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠉⠒⠤⣄⡀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⠴⠒⠉⢸⠀⢸
⠀⠀⠀⠀⠀⢠⠶⠷⠶⠷⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠶⠶⠶⠾⠶⠾⠶⡄
⠀⠀⠀⠀⠀⢸⠀⠀⢠⠴⢦⡀⠠⡄⠀⣤⠀⢠⠴⢤⠀⢠⣄⠀⡄⠀⣠⠶⢤⠀⢠⡶⠶⠆⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢀⡷⠶⣿⠀⣿⢤⢼⠁⢸⡏⢆⡏⠀⣿⠀⠀⠀⢸⡷⠶⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠲⠚⠁⠘⠃⠀⠛⠀⠛⠀⠘⠃⠘⠃⠈⠃⠀⠙⠲⠚⠀⠘⠓⠒⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠶⡶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⢶⡶⢶⠶⢶⠶⢶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⡇⠀⠀⣠⣄⠀⠀⠈⠓⠚⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⠢⢄⡀⠀⠀⠀⢇⣸⠂⠀⢸⡇⠀⠘⠧⠼⠀⠀⠀⢀⡠⠔⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡤⠖⠋⠁⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠉⠒⠤⣸⣇⠤⠒⠉⠀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⢀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄⠈⠢⠀⠀⠀⠀⠀⡀⡀⡠⠂⠀⠘⠁⠀⢀⡴⠤⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠱⡀⠀⠀⠀⠀⢀⠈⠀⠘⡀⠀⠀⢸⠋⠹⠇⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈⢢⣼⣄⡀⠀⠰⠆⠀⢸⡀⠀⡆⠀⠀⠈⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠗⢾⠁⠀⠈⢧⣠⣴⡦⠤⠾⠀⠀⣹⡄⢀⠔⢀⠀⠦⠸⣦⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣀⣤⣴⣾⡟⠟⢿⠐⠀⠙⠢⣀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢰⠀⠀⢘⣦⣤⣴⣾⣿⣿⣿⣿⣿⣄⠴⣬⣧⡀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄⡼⠀⠂⠀⣸⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣄⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠒⠊⠀⣿⣇⣸⣦⣿⣿⣿⣿⣽⠿⠛⠛⠉⠀⠀⠀⢀⣾⣿⣷⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡅⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠙⢿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⡀⢀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⠌⢂⠀⢸⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣏⣍⣹⣏⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣶⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠁⣀⡀⡱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢹⡧⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣷⠋⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡚⢛⡛⠻⠟⠛⢓⠔⢻⠀⢸⣿⣿⠿⢟⢃⣤⣾⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠰⣇⠎⡠⡀⢃⡄⠀⠈⠀⠈⣃⠈⠅⠀⠀⠠⡏⠁⠀⠀⣤⣴⣶⡶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⡀⠐⢻⠀⠏⠦⠬⠤⠿⠧⠶⠉⠁⣦⡏⠦⠤⠦⣠⣤⣥⣤⣄⠐⣨⠒⠠⠦⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠑⠀⣠⣶⣿⣿⣿⣷⣦⠀⠀⢀⣀⣀⡷⣧⣀⠔⡿⠚⣿⢄⠀⠊⡗⠐⠒⠀⠠⣼⠐⣅⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⠀⣠⣵⣿⣿⣿⣷⡜⣿⣴⣾⣿⣿⣿⣿⢩⠏⢰⠁⠀⠈⣿⡂⣤⢣⠶⠿⠄⠀⣿⣏⢉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⠛⠉⠻⢿⣿⣿⡿⣿⠀⠀⠁⢸⣿⣿⠿⠘⢄⣿⡀⠀⠀⣼⣇⣏⣨⠃⠒⠒⢰⡿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠘⠒⠒⠒⠛⠛⠀⠀⠀⠀⠈⠉⠉⠉⢉⣀⣈⠩⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣺⣿⣿⡏⢻⠿⣶⣶⡖⣶⣖⣶⡾⢿⢻⣿⣷⣿⣶⣶⠶⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⡟⢹⠀⠈⠉⢓⢺⡀⠀⢹⣯⣿⣿⣃⡒⠠⢸⡇⠀⠉⠙⢏⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠚⠓⢺⡒⠦⠠⠾⠎⠉⠁⠈⠀⣌⡇⠀⠉⣉⣉⣑⠒⠠⠸⠟⠛⠿⠋⢩⣉⣉⣭⣽⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠇⠀⠉⢘⠛⢛⣀⠀⠀⠄⢀⣀⡈⠓⢛⡿⣿⣿⣿⣿⠛⢷⣶⠛⠻⠿⣿⠿⠟⢻⣷⣾⠧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⡀⠀⠀⢨⡾⠥⣀⣀⠀⠀⢀⣀⣨⡆⣸⣼⣿⣿⣿⣿⣷⠈⣿⡃⠀⠀⠠⠀⣀⣇⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠭⢍⣛⣉⠓⠤⢀⣀⠈⢉⣁⣀⡽⠿⠛⢿⣿⣿⣿⣿⣿⣾⣋⡈⣁⣂⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠶⠶⠶⣦⣤⠦⠤⠉⠑⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡾⠵⠥⠤⠴⢲⡎⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⠿⣤⣤⣤⣤⣤⣀⠢⠭⢛⣩⣿⣷⣦⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠀⠐⠒⠒⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠙⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠰⢇⣠⣴⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⠛⠿⢋⣤⣤⣤⣤⣶⣶⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣟⢿⣿
⠀⠀⠀⠀⠀⣴⣿⡶⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⢻
⠀⠀⠀⢀⣼⣿⡿⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⠛⠛⠒⠲⠶⠶⣶⣶⡄⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀⣙⠿⠿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠙⠿⠟⣋⣀
⠀⠀⠀⣀⡀⠀⠀⢰⣿⣿⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣽⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⡞⣿⡿⣷⠀⠀⢀⣀
⠀⠀⠀⣿⣇⠀⠀⢸⣿⢿⣧⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢹⣿⣇⣿⠀⠀⣸⣿
⠀⠀⠀⢸⣿⠀⠀⣾⣿⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⢿⣿⣿⣿⡟⠀⢀⣤⣤⣶⣶⣶⣿⣿⣷⣶⠦⣼⣿⢹⣿⡀⠀⣿⡇
⠀⠀⠀⠘⣿⡇⢰⣿⣿⣾⣛⣻⣷⣶⣖⡀⠀⠀⠀⠀⠀⠛⠿⢛⣎⡛⠟⢋⣀⣀⠈⠙⠛⠛⠻⠿⢟⣹⣶⣖⣛⣛⣫⣾⣿⡇⢸⣿⠃
⠀⠀⠀⠀⢿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⣼⣛⣛⣛⣛⣛⣛⣛⢷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣼⡿
⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⠀⢸⣿⣿⢰⣷⡄⠀⠀⢠⡇⣿⣿⣿⣿⣿⣿⣿⢸⡆⠀⠀⠀⠈⣿⣿⡇⠀⣴⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢻⣿⠀⣸⣿⡏⠸⣿⣿⡄⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⢹⣿⣇⠀⣿⡟⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠘⣿⠀⣿⣿⠃⠀⠘⢿⣿⣆⠀⠉⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⠃⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠿⠐⢿⠟⠀⠀⠀⠀⠛⠛⣰⣷⣿⣿⣿⣿⣿⣿⣿⣸⣆⠀⠀⠀⠀⠀⠻⡿⠂⠿⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣿⠛⠛⠛⠛⠛⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣀⠀⠀⠀⠀⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡀⠀⢀⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠿⠇⠀⠸⠿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⣶⣶⣦⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠸⠛⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⣀⣤⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠙⠿⠿⠟⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⡴⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣺⠇⠀⠀⠀⠀⠀⠉⠫⠐
⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠈⠰⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠒⠒⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄
⠀⠁⠀⠀⠀⢈⠀⠀⠀⠀⡄⠀⠠⠀⠰⠟⠛⠚⠆⠀⠀⠀⠀⠠⠅⠀⡤⠈⠀⠐⠤⠀⠀⠀⠀⠀⠠⠆⠁
⠀⣤⣶⢿⢻⣻⠀⠀⠀⠀⠉⠑⠚⠀⠘⠛⠛⠃⠛⠀⠀⠀⠎⠀⠀⡀⠈⠀⢀⡀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠠⡶
⠀⣿⡮⠿⠛⠛⠀⠀⡈⠁⠀⠀⠀⠀⠀⣀⢀⠀⡀⠈⠀⠀⠀⠀⠀⠀⣠⡶⣿⡇⠁⠉⠃⢻⠿⠋⠀⠀⣼⣽⡀⠂
⠀⠁⠄⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠰⠀⠇⠐⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⠾⠋⠀⠀⠀⠈⠛⠄⠀⠀⠀⠉⠉⠀⠐
⠀⣤⣀⣀⣀⣠⠀⠀⢠⣯⣿⣿⢧⡄⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠘⠁⠴⠄⠀⠀⠂⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰
⠀⠉⠚⠛⠒⠹⠀⠀⠻⣽⣷⣿⣿⠟⠀⠀⠀⠚⠉⠀⠁⠀⠀⠠⡀⠀⢀⠠⠀⠀⢀⠀⠀⢀⢤⡀⠀⠀⡄⠀⠀⠀⢈
⠀⠀⠀⠀⢀⠤⠰⠀⠂⠄⠀⠀⠠⠐⠀⠂⠄⠀⠀⠠⠂⠀⠉⠐⠀⠦⠀⠀⠀⠄⠂⠄⡀⠀⠣⠅⠀⠰⠀⠀⠀⠀⠘
⠀⠀⠉⢀⠀⠄⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠄⠀⢀⡀⠀⠈⠭⠂⠰⣲⠉⢑⢠
⠀⠘⠐⠀⠠⠀⠴⠞⡈⠀⠀⢨⡿⠉⢙⠀⠀⠀⠀⠀⢸⠛⢿⣿⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⠁⠉⡀
⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠸⡿⠤⠈⠉⠉⠉⠀⠉⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠜⡑⠐⠀⠖⠀⠤⠠⠤⠠⠄⠐⡇
⠀⠀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠘⠓⠖⠚⠒⠒⠒⠒⠒⠶⠳⠒⠂⠀⠀⢀⡀⠀⠐⠀⠳⠋⠀⠀⢄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⣾⣥⠋⢀⠀⠀⠆⠀⠀⠀⢀⣀⣠⣤⡀⠀⠀⡄⠀⡀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⣀⠠
⠀⠠⠀⠀⢀⣿⠡⠐⠀⠀⠀⠃⢠⢳⠻⠛⡝⠫⢯⠖⡷⠀⡃⠀⠀⠀⢐⠰⡆⠄⢠⠀⢸⠰⡸⢀⣾⠋
⠀⠠⡄⢠⠀⠠⠀⠄⠀⢠⠀⡀⠹⣭⣶⣾⣶⠿⢟⣋⣁⠀⡇⠂⠒⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠒
⠀⠅⠦⠀⠀⠤⠴⠤⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⢃⠠⣭⡢⠖⡲⣠⡆⡆⠀⠀⠀⠀⠀⠀⢌⣒⠀⢱⢱
⠀⠋⠉⠉⠉⠉⠉⠉⠉⠓⠀⢀⠠⢀⠠⢄⠀⡤⠀⠐⠤⢄⠈⠄⠙⠛⠐⠛⠒⢁⠃⡀⠀⠀⠐⠂⠄⢀⡀⠁⠀⣀
⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⠈⠈⠀⠈⠊⠀⠑⠑⠀⠊⠈⠀⠀⠐⠄⣀⡠⠒⠁⠨⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠌⠈⠤⠀⠀⠀⠀⠈⠉⠁⢀⣀⣀⣀⣀⡀⠀⠈⠁⠀⠧⠀⠀⡀⢴⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡤⣤⣄⡄
⠀⠀⠀⢄⡀⢀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣟⣽⣻⣭⣿⠀⠀⠀⠀⡄⠁⠀⠁⠐⠀⠀⠀⠀⡀⠀⠀⡀⡎⠛⠶⠞⠃⠂
⠀⡀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⢀⠄⠀⡀⢀⣛⠃⢰⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠀⠀⢀⡅
⠀⠣⠤⠤⠤⠤⠤⠤⠤⠽⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠃⠀⠀⠄⠀⠄⠊⠀⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀⠀⠀⣤⠤⣴⡟⠉⢙⣶⠾⠛⠲⣄⠀⣾⠟⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣁⣀⣀⡀⠈⠓⢾⣇⣀⣿⠁⠀⢸⡇⠀⠀⠀⠘⡄⣿⣀⣤⠴⠗⠒⠀⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠚⠉⣥⣶⡍⠙⢷⣄⠀⠉⠉⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⢻⠉⢀⠴⠞⢛⣛⡛⠶⣌⣆
⠀⠀⠀⠀⠀⠀⢀⡟⢀⣤⡀⠙⠿⠃⣀⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢿⣿⠇⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⣸⠁⠻⣿⠏⠀⠀⠐⣿⣿⠀⡇⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡇⠀⠀⢠⣿⣿⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢦⡀⠀⣾⣿⡆⠀⠀⠘⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠁⣴⣶⡀⠙⠋⠰⢻
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠙⠲⠾⠭⠤⣠⠖⣂⣀⡀⠀⣷⠀⠀⠠⠈⠠⡀⠀⢀⠤⢒⡂⠙⠦⣄⣻⠟⢁⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠁⡾⠉⠉⢻⡄⣿⡆⠀⠀⠀⠀⠘⣆⠀⡼⠋⠙⢷⠀⡄⠀⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⡾⢡⣏⣿⡄⠀⠀⠀⠔⠉⠳⣷⣀⣀⡾⢡⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣸⠲⠬⠤⠶⠋⠀⠈⠻⣤⡀⠀⠀⠀⠀⠙⢏⣩⣴⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠈⣧⡈⣿⣿⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢀⡜⠀⢿⣇⠀⠀⠀⠀⠀⠀⣰⣿⢿⡇⠀⠀⠀⠀⠚⠁⣿⠏⢿⣷⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡤⠴⠋⠀⠀⠸⣿⡀⠀⠀⠀⠀⢠⣿⠃⠀⢻⡀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣄⡀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⡠
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠡
⠂⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⢶⣾⡄⠀⠠⠶⠂⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠆⢠⡄⠀⠀⠀⠈⠈⠀⣍⠝⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠁⠀⠀⢀⣴⣤⠀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠄⢀⠁⠂⠀⣤⣄⣠⠀⠀⠑⠚⠁⠀⠁⢀⢦⣤⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠈⠛⠇⠀⡀⠀⢀⠀⡀⠉⠀⠀⠀⠈⠉⠉⠀⠀⠒⠶⠋⠀⠀⢀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠒⠠⠀⠀⠀⣀⡤⠤⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⣀⡤⠤⠤⠤⠤⣤⣀
⠀⠂⠀⠠⠊⣠⠶⠚⠻⠆⠛⢶⣬⡙⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣠⠶⠚⠻⠆⠛⢶⣬
⠀⠀⠀⠀⠈⢻⣂⣀⣤⡥⣀⣄⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢂⣀⣤⡤⣀⣄⡿
⠀⢠⠃⠢⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⣈⠁⠁⢀⢀⠀⢠⠀⠀⢩⡇⢠⠀⠀⠂⠀⠐⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠁⢀⢀⠀⢠⠀⠀⣹⡇⢠
⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣠⠀⠀⠠⠏⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂