Linux distribution Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠻⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢸⠃⠴⠀⠀⠆⢸⡇⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣸⠂⠀⠘⠂⠀⢸⡇⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣁⣀⣀⣀⣀⣀⣉⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠸⣧⡀⠀⠀⠀⠀⢀⣿⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⡻⠶⡶⠶⠾⢟⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠀⠀⠈⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣷⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠉⢻⣿⣿⡿⠉⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢱⣿⣳⠈⣿⣿⡇⢰⣿⣟⡆⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣻⠟⠀⠉⠉⠁⠘⠿⣿⣇⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⣀⣀⣀⣠⠤⠖⣊⡁⣸⣿⡟⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢿⣯⣉⣉⣉⣠⡴⠞⠋⠁⠸⣿⣿⣦⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠻⢿⡿⠟⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣎⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣎⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡘⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣯⣬⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣿⣿⣿⣿⣿⡟⠁⢀⡀
⠀⠀⠀⠀⢀⡤⠶⠶⠶⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢰⣦⣀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⢾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠛⠳⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣾⣷⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠷⣿⣷⡀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⣶⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⣉⡉⠉⠙⠋⠛⠛⠛⠿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠙⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠀⠀⠀⠀⢸⠃⠀⠈⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡾⠿⠟⠋⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⡄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣦⡀⠀⠹⣷⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉
⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋

⣠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷
⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠛⠋⠉⠉⠙⠛⠛⠻⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠈⠻⣿⣿⣦⡀⢀⢤⣾⠟⠋⣁⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡈⠙⠻⣷⣤⡀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣷⡊⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⢙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣔⡹⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠝⢿⣦
⠀⠀⠀⢀⣾⠟⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣷⡀
⠀⠀⢀⣾⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠹⣷⡀
⠀⠀⣼⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⡄⠀⠀⠀⢻⣧
⠀⢰⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢹⣷⠀⠀⠀⠈⣿⡆
⠀⢸⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⡆⠀⠀⠀⢿⡇
⠀⢸⡇⠀⠿⠿⢿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠛⠛⢛⣻⠯⠥⠤⢤⣤⣽⣿⣿⣶⣶⣶⣶⣶⣿⣷⣶⣶⠀⢸⡇
⠀⢸⣷⠀⠀⠀⠸⣿⠀⠀⠘⢿⣿⣿⣿⣿⣫⣴⠞⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⡇
⠀⠸⣿⡀⠀⠀⠀⢿⣇⠀⠀⠀⣈⣭⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⠇
⠀⠀⢻⣧⠀⠀⠀⠘⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⠟⢦⡀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠁⣼⡟
⠀⠀⠈⢿⣆⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠙⠦⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⣰⡿⠁
⠀⠀⠀⠈⢿⣦⠈⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⣴⡿⠁
⠀⠀⠀⠀⠀⠻⣷⣔⠙⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠟
⠀⠀⠀⠀⢀⣴⣿⣿⣷⣅⠉⠻⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⢀⣴⣿⣿⠟⠁⠈⠛⢿⣦⣄⡈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀
⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣯⡇⣔⠆⡔⢺⢸⠤⡇⡖⠆⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⡄⡤⡀⡦⡄⣀⢠⢤⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠑⠂⠙⠋⠈⠀⠃⠑⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠁⠯⠄⠣⠎⠚⠰⠼⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⣧⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⣿⡇⠈⢿⣧⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⢿⣷⡀⢸⣿⠀⢸⣿⠀⠀⠀⣿⡇⠀⠈⢿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⠀⠹⣷⣼⣧⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⡀⢸⣿⠀⠀⣿⠀⠀⠹⣿⣿⠀⠘⣿⣤⣀⣴⡿⠁⠀⣼⡟⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠈⠛⠀⠀⠛⠀⠀⠀⠙⠋⠀⠀⠈⠙⠛⠋⠁⠀⠀⠛⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡇⡤⣄⠠⠄⣆⡄⡄⢠⢠⢠⠤⠀⠤⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠃⠳⠃⠐⠆⠋⠃⠑⠚⠞⠘⠞⠀⠀⠙⠄⢀



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡏⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢟⢿⣿⣿⠟⠛⠛⢭⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⢹⣿⡇⠀⣠⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢿⣇⢤⣭⠤⣀⣿⡿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣡⣾⣿⣿⣿⣿⣿⣿⡷⢦⠙⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠻⠿⠿⠿⠟⢋⣡⣶⠟⢠⣿⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠲⠶⠶⠿⠟⠉⠀⠀⠈⢿⣦⣤⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠰⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠑⣿⣿⣿⡶⠉⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠻⠿⠋⣠⣾⣿
⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣤⣤⣤⣤⣤⣴⣾⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⠿⠿⠟⠛⠛⠛⠛⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⠙⢿⣿⣿⣿⡿⠋⠀⠀⠈⠙⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣴⣶⣄⠈⣿⣿⣿⠃⢀⣴⣶⣦⡀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢿⣿⣿⠤⠿⠿⢿⣄⣸⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡤⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣦⣴⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠋⢁⣰⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠓⠶⠒⠛⠛⠉⣀⣤⡴⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠈⢿⣷⣦⣤⣴⣶⠿⠋⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠈⠙⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⠧⢤⡀
⠀⢀⣀⣀⣀⣠⣼⠇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠿⣿⡿⠿⠋⠁⠀⠀⢰⡇
⣰⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄
⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋
⠻⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁
⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣦⡀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣶⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠉⠙⢿⣿⣿⡿⠋⠉⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⠀⠘⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢰⣿⡇⢰⣿⣿⡄⣾⣿⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡷⠟⠛⠛⠿⢾⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⣦⣄⣤⣤⣶⣿⡿⠟⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠙⠛⠿⠿⠛⠉⠁⠀⠀⠈⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡿⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⡟⠋⠉⠁⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠙⠛⠛⠛⠛⠉⢿⣿⣦⣄
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣤⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠇
⠀⠀⠀⠀⢸⣿⣧⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣶⣤⣀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠟⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⣠⣀⠀⠈⣿⣿⣿⣿⡏⠀⠀⣀⣤⣀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⡄⠀⣿⣿⣿⣷⠀⠘⠛⠛⠛⠀⠀⣼⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣧⠀⢿⣿⣿⣿⣠⣤⣴⣦⣤⣄⣀⢻⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⡷⠀⣹⣿⣿⣍⣼⣿⣿⣦⣹⣿⣿⣿⣿⣯⣀⡀⠘⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣽⣿⡿⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣧⡈⢿⣮⡉⠛⠻⠿⠿⠿⠿⠛⠋⢉⣤⣾⡿⠟⠉⠀⢠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠈⠻⣷⣦⣤⣤⣤⣤⣶⣾⡿⠟⠉⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣤⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡖⠛⠛⠛⠛⠲⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣛⣓⣟⣧⣤⣤⣤⢤⣤⣴⣟⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣯⠯⠤⠷⠴⠷⠶⠾⠶⠶⠶⠶⠷⢤⣯⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣭⣷⣤⣤⡤⣤⠤⢴⠶⠶⠤⠤⠤⠤⠤⣤⣿⣿⣿⣿⣭⣓⣦⣄⣀⡀
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⡶⣶⣶⣾⣿⡷⠻⣿⣾⣿⣶⠟⢶⣿⣿⣿⡟⠛⣿⣿⣿⣿⡛⣿⣿⣶⣯⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢯⣟⣿⣍⡉⠫⡉⠉⠉⡩⠉⡏⠉⠙⡯⢉⡏⠉⡹⠋⣉⣽⢋⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣟⡷⢬⣄⣀⣄⡀⢁⠀⢀⠀⢀⢀⣰⣁⣰⣏⣷⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠚⠀⠀⠙⠛⠿⣻⣿⠛⠛⠛⣻⠿⠛⠋⠀⠀⠚⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⣀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⢉⣽⣿⡿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢹⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠲⡋⠖⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⡀⠀⢩⠀⢀⡀⠀⠀⠧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⡀⠀⠉⠁⢀⠶⡀⠉⠀⠀⠠⠂⠀⠈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠐⠀⠒⠁⠀⠀⠐⠐⠈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡒⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠲⢬⠀⠀⡋⠀⠀⡏⠀⠀⢹⠀⠀⡇⠊⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⠷⠤⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢏⠀⣠⡇⢸⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⠈⠀⠀⠀⠀⠀⢸⠀⣇⠀⢈⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣄⣠⠞⠉⠀⠙⠺⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⡼⠒⠁⠈⠛⢄⠀⡼⢆
⠀⠀⠀⠀⠀⢠⡁⠀⡜⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠹⡄⠀⣠
⠀⠀⠀⠀⢀⠆⠈⡛⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠸⡋⠀⠆
⠀⠀⢠⠊⠉⢉⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡶⠶⠶⠦⠤⠤⠤⠤⠴⠶⠶⠶⠤⠤⠤⠤⠤⠤⠴⠦⢴⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠂⠉⠙⢦
⠀⠠⠚⠈⠁⡈⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠉⠀⠉⠙
⠀⠀⠀⠀⠀⠉⠀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⠀⠀⠀⠀⠀⠀⢸⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠈⠁⠁
⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⢺⠀⠀⠀⠀⠀⠀⢸⠐⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠤⠴⠀⠀⠀⠀⠀⠀⠸⠠⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⡇⠀⠀⠀⠀⠀⡶⠤⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠈⠀⠀⠉⠓⡄⠀⠀⠀⡴⠋⠁⠈⠁⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠈⠀⠀⠩⠆⠀⠸⠁⠁⠈⠉⠉⠉⠁⠈⠹

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠙⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠲⠦⣄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠀⠀⡇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣙⣧⠀⠀⠀⣰⣿⣏⣠⣿⣿⠀⠀⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⡿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⠟⠁⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⣤⣤⠀⠀⠀⣀⡈⢧⠀⠀⠀⢤⣀⣠⠆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⡬⠧⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⣿⣿⠃⠀⢾⣿⣿⠎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⡤⢞⠉⠱⡀⠈⠳⣄⡀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢴⠻⡗⠲⣄⠈⡽⠯⠶⠾⠷⠤⢤⣀⠀⠀⠀⠀⠀⠀⢿⣀⣠⠴⡎⠉⢆⠈⢣⠀⠘⣄⡴⢋⡽⠦⣄⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡧⣄⣀⣈⠙⣷⠦⡈⠻⣷⡖⠶⠦⢤⣄⠀⠀⢙⣦⣠⡤⢴⠒⠛⡅⠑⡄⠘⡄⠀⢣⣀⠷⠚⣡⠔⠋⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⢀⣠⡏⠀⢀⠋⠉⡝⠉⢻⠛⠒⢻⠓⠒⡖⠛⢛⠉⠹⠉⠀⠸⠀⠈⡄⠀⠸⡀⢘⣄⠼⠖⢋⣡⠴⠋⠁⠀⡖⠓⡄⠀⠀⢀⡞
⠀⠀⠀⣠⠖⣉⣞⠀⠀⡜⠀⠀⡇⠀⢸⠀⠀⠘⠀⠀⡇⠀⢸⠀⠀⡇⠀⢀⣇⣀⡼⠴⠒⠋⢉⣠⠴⠚⠉⠳⣄⠀⠀⠀⠓⠚⠁⢀⡴⠋
⠀⡴⠋⠀⠀⠧⣌⠙⠒⠧⠤⣤⣅⣀⣸⣀⣀⣠⣀⣠⣧⠤⠼⠶⠖⠛⠋⢉⣉⣠⠤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡴⠋
⡾⠁⠀⠀⠀⣠⠎⠉⠓⠒⠲⠶⠤⠤⠤⠤⠤⣤⠤⠤⠶⠒⠒⠒⠛⠋⠉⠉⢧⠀⠀⠀⡴⠒⢲⠀⠀⠀⠀⠀⠀⠀⢱⣤⣾⣿⡇
⢣⡀⠀⢀⡜⠁⠀⠀⠀⡴⠒⢦⠀⠀⠀⠀⢰⠇⠀⢠⠴⠤⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠙⠒⠋⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡟
⠀⠉⠓⠾⣀⡀⠀⠀⠀⠳⠤⠞⠁⠀⠀⠀⡞⠀⠀⠈⠲⠖⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠉⠓⣶⣤⣄⣀⣀⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣿⣶⣿⣿⠏⠀⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⡠⠤⠤⠐⠒⠒⠠⢤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⣰⣶⡤⠒⣿⣲⡀⠂⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠁⡘⠀⠀⠀⠁⠀⠀⠀⠁⢢⠋⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡇⢸⡉⠉⠉⠃⠒⡶⢰⢸⠀⠀⢠⡇
⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣑⡢⠤⠤⠔⠃⢸⠈⠓⢺⠋
⠀⢎⠳⡱⡘⣀⡀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠍⠉⠉⠙⠛⢿⡛⠚⢻⡉
⢎⠢⣳⡵⠋⠉⣗⣲⢦⢲⠒⠒⢲⠃⠀⠀⠅⠀⠀⠀⠀⠀⠱⡀⡤⠷⠤⠤⣄⡀
⠀⠱⡏⠀⠀⠀⡏⠀⢸⢸⠤⢤⡏⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⡇⠧⠤⣤⢤⡇
⠀⠀⠈⠒⠤⠤⠣⡀⠈⢸⠀⢰⡇⠀⠀⠀⠀⠀⢄⡘⣸⠀⠀⡇⠀⠀⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠑⢕⣤⡈⠀⢸⠀⠀⠀⠀⠀⠠⠀⠠⠠⡀⠀⡇⠀⠀⢸⢀⡇⡇⠀⠀⠀⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⠀⢸⠀⠀⠀⠐⠀⠉⠁⠈⠒⠁⠀⡇⠀⠀⣿⢸⣇⡇⠀⠀⢀⡨⠖⠛⢻⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡏⠘⠶⠃⠀⠀⠻⣄⡀⠀⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⢄⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢇⠀⠀⠀⠈⠉⠉⠙⠛⠖⡚⠒⠞⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠙⢦⡀⠀⠀⠀⠀⠀⡰⠐⣄⣀⣀⣰⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢧⠀⠀⠀⠉⠒⠲⠦⠤⠖⠓⠒⠉⠁⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠤⢆⡰⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢤⠔⠂⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢘⣀
⠀⠀⠀⠀⠀⣔⡉⠒⠤⣀⢰⣀⠤⠂⢉⡶⠀⠀⠀⠀⠀⢰⠚⠭⣄⣐⠖⢀⡩⠴⠆
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣙⣅⡤⠖⠁⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠈⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣉⠑⠂⢾⠐⠈⣁⡤⠗
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣳⣾⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡴⠀⠀⠀⣎⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⠀⠀⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⣻⣀⣀⣀⣀⣸⣛⣛⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠉⠉⠉⠉⠉⠏⠀⠀⢀⣠⠔⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠓⠒⡒⠒⠒⠒⠒⠶⠶⠶⠶⢶⠚⠉⠀⣀⣀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠉⡇⠶⣿⣷⣲⠀⠀⢰⣶⣾⢷⡆⢸⠀⠀⡼⠀⡼⢙⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣈⡇⠀⠛⠛⠛⠀⠀⠸⠻⠿⠿⠀⢸⠀⠀⢹⡀⣳⣘⠤⠃
⠀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⣇⣀⣀⡀⠀⠐⣶⠂⠀⢸⡀⠀⠀⢀⣠⠞
⢠⡉⢙⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢒⡶⠶⠦⢭⣭⣭⣍⣉⣀⣀⣸⣇⡤⠞⠉⠳⣄
⠣⣈⣩⢿⣉⡷⣶⡀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⡆⠀⠀⠈⢧
⠀⠀⠀⠀⠈⣦⣇⠙⠲⠤⢤⣤⠤⢴⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣠⠚⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠒⠤⢤⣤⣤⠤⡾⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠹⠶⠶⢶⣤⡄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⣀⡂⢀⢐⠸⡇⠀⢀⡠⢼⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢈⣉⣉⡉⠉⠛⠒⠒⠒⠒⠲⠦⠬⠤⠴⡷⢊⡡⠖⠉⣧⡼⠧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⡏⠁⠀⠀⠀⣇⡷⢦⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣤⢫⠟⢿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⠶⡶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⢸⣇⣠⠖⠋⢷⣻⠀⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⠀⠁⡟⠉⢩⡟⠉⠉⠀⠀⠀⠈⠁⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠶⢾⣷⡀⠀⠀⠀⠀⣯⡀⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣚⣉⣀⢀⡴⠋⢀⡇⢀⣠⠔⠋⠀⠉⢻⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠬⠭⠽⠶⠚⠉⢰⣛⡒⠦⠤⢤⣤⠏⣠⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣷⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠛⠋⠉⠉⠙⠛⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠒⠒⠛⠓⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣀⡀⠄⠒⠊⢉⣀⣠⠤⠤⠒⠒⠒⠉⠉⢉⣉⣉⣩⣭⣭⣭⣭⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠥⠒⠉⣀⡠⠔⠒⠋⠉⠀⣀⣠⡤⠴⠶⠚⡛⡏⠉⠉⢩⣇⣀⣀⣠⣇⣀⣠⣇⡀⣹⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡤⠒⠉⢁⣀⡤⣤⠒⠊⠉⠁⢫⣇⣀⠤⠤⠚⠓⠋⠉⠉⠉⡹⠋⠀⣀⠤⡀⠈⢉⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣁⡤⠶⡾⡉⠁⣀⣡⠧⠶⠚⠉⢫⠀⠀⠠⠔⢀⡭⠛⠀⡠⠊⠀⠀⠐⠐⠊⣠⠔⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⡠⠖⢛⣃⢀⣠⠼⠓⠋⡉⠤⠀⡄⠀⠀⢸⠀⠀⠀⣀⣀⣠⡤⠮⠤⢤⣀⠀⡀⡴⣍⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠚⠉⣢⣤⡔⠋⠁⠀⠀⠀⠈⠒⠂⠉⣀⡠⠤⠞⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⡿⢘⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⢶⡴⠚⠁⠀⠈⠉⠓⠒⠒⢀⣤⠴⠚⠉⠁⠀⠀⠀⢀⣀⡠⠤⠤⠦⢤⡄⠀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠯⣶⠋⠀⠰⠥⠤⠛⠀⠀⢀⠔⠋⠀⠀⠀⢀⡠⠔⠒⢉⣀⠤⠔⣂⣤⠶⣛⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣖⡒⠒⠂⠒⢀⠒⡐⢫⡀⠀⠀⠀⢰⣯⣧⣭⣭⡼⢶⣓⣿⠽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠲⠶⠶⠟⠛⠒⠲⠶⠶⠒⠚⠛⠛⠉⠁









⣿⣿⣿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣦⣾⣴⣿⣶⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡿⣁⠀⠀⢠⡄⠀⠀⣈⢿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⠿⠯⠉⠀⠀⠈⠀⠀⠀⠉⠽⠿⠿⡟⣶
⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢻⢉⣁⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣈⡉⡟⠒⠦⢤⣀
⠀⠀⠀⠀⡴⢋⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⢦
⠀⠀⠀⠈⢷⣟⢿⣿⣿⣿⣿⣿⣿⣿⠉⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⠉⣿⣿⣿⣿⣿⣿⣿⡻⣻⡾⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠚⠮⣿⢿⣿⣿⣿⡛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⢛⣿⣿⣿⡿⣿⠵⠓⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣟⢿⣿⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⣿⡿⣻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠤⠤⠤⠤⠤⠤⠶⠚⠛⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣼⣄⢀⡀⡀⡀⢀⡀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠒⠉⡉⠐⢄⠀⠀⠀⠀⡠⠂⢉⠉⠒⡄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⢰⠡⠌⠣⠀⡆⠀⠀⢰⠀⠜⢡⠌⡆⢸⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⠀⠀⠀⠀⠣⡈⠑⠒⢁⡴⠁⠀⠀⠈⢦⡈⠒⠊⢁⠌⠀⠀⠀⠀⡏⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⢀⡀⣀⡀⢀⣀⡀⠀⣀⡀⣀⣀⣀⡀⠠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢤⠤⢠⡄⠤⠤⠤⠤⠤⠤⠀⠇⠀⠀⠸⠀⠤⠤⠤⠤⠤⠤⢤⡤⠤⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⡀⡀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⡆⠀⠀⢸⠦⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠴⠇⠀⠄⢰⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠛⠐⠃⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠘⠀⠛⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠒⠒⠒⠀⢴⡃⠀⠀⢘⡦⠀⠐⠒⠒⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢱⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⢄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣿⠀⠋⢰⡟⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠏⠈⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⡟⠀⠀⢸⠁⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⠻⠦⠤⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠖⠛⠛⢶⣀⡇⠀⠀⣼⠀⠀⣀⣀⣀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢿⡀⠀⠀⣸⢿⡇⠀⠀⡿⢠⡟⠁⠀⠈⢷⠀⢿⠀⠀⠀⠀⠀⠀⠘⢳⠦⣤⣄⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⡈⠛⠲⠚⠉⢸⠀⠀⠀⡇⠈⢧⣄⣀⣠⠟⠀⢸⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢩⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠷⠶⠦⢬⣭⣥⣤⣄⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⠶⠴⠤⠤⠄⠀⣀⣤⣀⣀⡀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⡀⢀⣀⣀⣀⣧⣠⣤⣤⣤⣤⣄⣄⣠⣄⣠⣄⣉⣉⣉⣽⣇⣈⣁⣀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢸⣧⡅⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⢸⠤⢴⠇⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⡏⠀⡆⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠓⠒⠶⣤⡀⠀⠀⠀⠀⠀⣸⡇⢸⠀⡾⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠓⠒⠃⠈⣿⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣿⠃⢸⣺⡃⠀⣀⡴⠋
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡏⠀⠀⠇⠀⢹⣷⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣼⡏⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣾⠀⠀⠀⡀⠀⠀⠹⣷⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣼⠟⠀⠀⢸
⠀⠀⠀⢰⡇⠀⠀⠀⢰⠇⠀⠀⠀⡇⠀⠀⠀⠙⢿⣦⡀⠈⠳⣄⣀⠀⠀⠀⣀⣤⠞⢁⣠⡿⠋⠀⠀⠀⢸
⠀⢀⣤⣾⣁⡀⠀⠀⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣉⣛⣛⣋⣉⣤⡶⠟⠉⠀⠀⠀⠀⠀⢸
⡴⠋⠀⠀⠈⠉⠙⠛⢳⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡇⠀⣿⠙⠒⣶⠄⢰⠏⠀⠀⠀⠓⠂⡄⠐⢶⡶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢶⡶⠢⠞
⠀⢻⣀⣻⡄⣰⠋⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋
⠀⠀⠉⠉⠉⠛⠲⠟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣦⣀⣀⣿⢦⣄⣠⣤⣿⣀⣀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠤⣤⡤⣬⡉⠙⡏⠀⠀⠀⠀⡏⠉⣽⠷⠶⠶⢦⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠾⠃⠀⠀⠀⠈⠷⢶⡇⠀⠀⠀⢀⣷⠞⠃⠀⠀⠀⠈⠓⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠖⠛⠉⠉⠙⠛⠶⣄⣿⠀⠀⠀⢸⣠⠶⠛⠋⠉⠙⠛⠶⣤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⣳⣿⣇⣀⣀⣀⡀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣦⠝⢉⣀⣀⣀⣀⣀⣀⣀⡉⠻⣦⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣟⣻⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠹⣛⡻⠿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣺⣭⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣶⡽⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⣿⣿⣿⣟⣿⣏⣽⣏⣉⣉⣉⣹⣿⣯⣿⣻⣿⣿⣿⣿⣷⣿⣿⣷
⠤⠤⠤⠤⠤⠤⠤⠤⠭⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠋⠉⠡⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢰⣿⣿⣿⣿⡟⣿⢻⣿⣿⣿⣿⡌⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭
⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⣸⢻⡿⡿⢸⠁⣿⠸⠛⡟⢟⠿⡷⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢢⠉⡄⠁⠀⡈⠀⡟⠀⠀⠁⠘⠀⠋⠐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒
⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶
⣶⣌⠭⡭⠭⢭⠭⠭⠭⠭⠭⠭⢭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⠭⠭⠭⠭⢭⠭⠭⡭⢍⣴
⣿⣿⣷⡝⠿⢋⣾⣧⡙⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣡⣿⣎⢻⠟⣥⣾⣿
⣿⣿⣿⣟⣲⣿⣿⣿⣿⡆⠛⣻⡃⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⣾⣿⣿⣿⣷⣚⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⣍⠀⠀⣼⣿⣷⣄⣤⣤⣤⣤⣄⣀⣀⣠⡀⢠⣾⣿⣄⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡏⠉⢽⣿⣿⣿⣽⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣧⡤⠾⠿⠿⠿⢿⣤⣬⣧⣤⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⠿⣿⣟⣻⣿⣿⣯⣽⣿⣿⣿⣧⣾⣿⣿⣿⣷⣾⣿⣿⣿⣷⣼⣿⣿⣿⣯⣽⣿⣿⣟⣻⣿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⠙⠻⠿⠇⣽⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠋⠉⠉⠉⢉⠉⠉⠉⣭⠍⠉⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣯⠹⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⣟⣛⠛⠛⠛⣛⣛⣛⣿⣿⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣳⠺⠋⡿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⣁⡢⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣇⠀⠀⠀⠀⠀⠀⠀⡵⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠷⠀⠀⠀⠼⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⡀⠀⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠀⠀⡀⠻⠀⡈⠀⠀⡓⠐⠴⠆⠙⠀⠸⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡈⡂
⠀⠀⠐⢰⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠋⣾
⠀⠀⠈⠲⡞⡉⠛⠑⠶⣄⢄⡀⠀⠀⠀⠐⠂⠤⠄⠀⣀⣀⣂⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⢀⡤⠊⠀⢔⠏
⠀⠀⠀⠀⠘⢎⠂⠀⠀⠀⠉⠚⠶⣤⡤⠀⠀⠀⠀⠀⠘⠻⠛⠻⠄⠄⠠⠤⠠⣀⣀⡤⠔⠁⠀⢀⢀⡜
⠀⠀⠀⠀⠀⠀⠑⢕⠀⠀⠀⠀⠀⡞⠀⠩⠂⢸⠩⠅⠀⠀⠀⠀⠂⠀⣀⡀⣠⡌⠺⠁⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢄⠀⠀⠃⠀⠭⠂⢸⠀⠀⠀⢀⠀⠂⠀⠀⠶⠬⠰⠂⠀⣀⡠⣔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣀⠀⠬⠄⠸⣀⣐⣒⠓⡦⠒⣀⠀⣉⠥⢜⡁⠀⡇⠱⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⢽⠀⢆⠄⠀⢠⣤⣤⣤⣄⠀⠀⠀⠴⢝⡡⠄⠸⠆⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⢸⠀⠀⠀⠀⠀⠁⠹⠀⡇⠀⠀⠀⠙⠁⠘⠁⠠⢌⣑⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠭⢱⣐⠀⠀⠀⠀⠘⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⠟⠚⠮⠥⠭⠭⠽⢏⣽⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠀⡉⠲⡀⠀⠀⠀⠠⠒⠈⠉⠓⠢⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠒⠼⣀⣀⣂⣁⣧⡵⠃⠀⠀⠀⠳⢬⣐⡀⠠⠀⠈⡷⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠁⠈⠉⠉⠁⠀⠈⠀⠀⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⢦⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠉⠀⠀⠉⠿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠘⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣀⣤⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣾⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢊⡩⠥⠬⠭⠭⢍⡱⠮⢤⣤⣤⣤⣬⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⣀⡤⠶⠿⠀⠀⠀⠀⠀⠀⠿⠷⢤⣌⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣾⣿⣿⡿⠷⠶⡾⠿⢿⠟⠉⠉⠻⡿⠿⠿⢶⠶⣿⣿⣿⣿⡶⡖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⠿⠿⢮⣦⣤⣶⣶⣖⡚⢀⡀⠀⢀⠛⢲⣶⣦⣤⣤⡤⠾⠿⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠀⠀⠙⠛⠿⣿⣛⣋⡉⢉⢉⣛⣿⡾⠻⠛⠀⠀⠀⠀⠐⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠤⠤⠤⠤⠤⠆⠀⢀⠀⠈⠉⠻⠟⠻⠟⠉⠁⠀⠀⢀⡀⠠⠀⠄⢀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠐⢀⡀⠀⡀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣸⡇⠀⠀⣀⣤⣶⣿⣿⣫⣢⣀⣸⣄⣄⡷⠀⠔⠢⣀⢀⣿⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣚⣛⡻⣞⣚⣛⣿⣯⣛⣿⠥⠶⣬⣍⣛⣿⡿⣿⣿⣿⡏⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠩⣯⠕⠒⡖⠖⠒⢲⣿⡭⠭⠤⠤⠄⢰⡧⢐⣺⣿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣿⣄⣸⣿⣄⣤⣚⣿⡇⢰⡄⣼⡄⢾⣿⣬⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠰⠒⠀⠀⠈⠉⠉⠉⡉⠀⠀⢂⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣋⣭⡅⢀⣠⣠⡶⠐⠋⠉⠀⠀⣴⡾⠋⢻⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣤⣾⣿⣿⣷⠀⠁⠀⠀⠀⢀⣄⣛⠛⠛⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⣤⣿⣿⣿⣿⣯⣿⣧⠘⠀⣼⣦⣽⠟⠻⠁⠀⠀⠰⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣧⣼⣿⣿⡟⠟⣱⣷⡿⢿⣿⣷⣔⠩⡍⠀⠀⠀⠀⠀⣨⡇⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢰⣿⣿⣟⢩⡷⠈⠛⠉⠉⠛⠿⢌⡻⣦⡀⠀⠄⡀⠀⠠⣾⠅⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣰⣿⣿⠋⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⢷⠎⢿⣶⣄⡁⠀⣶⠍⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠤⠤⠤⠀⠀⣀⡿⠁⠰⣿⣿⡛⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠙⠻⣿⣇⣴⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⢠⡆⠋⠨⠸⠇⠻⠀⣽⠁⣤⢷⡹⢷⣤⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠀⢄⣼⣎⢹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠀⠛⠀⠁⠀⠀⠀⠀⠀⢰⠃⣼⣿⣟⣴⣼⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⡀⢀⡿⠡⣨⣶⣋⣿⣿⣿⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠁⠁⠀⠀⠀⠀⠀⢠⡤⠤⠀⢁⣾⣿⢯⣎⣼⣿⣿⣿⣷⣬⡵⠆⣤⢤⡄⣼⠷⠊⠀⣠⣨⣹⣿⣿⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⡤⠃⠀⠄⠀⠀⠀⠀⠀⠀⣠⡤⠟⠊⠉⢁⠀⢲⠒⠎⢻⣿⡝⢿⡏⠂⠀⠀⠀⠂⢰⣳⣭⣮⣿⣿⣯⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣴⣤⡦⢶⡖⠒⢁⠴⢋⢁⡄⢤⠸⠘⠀⠀⠀⠁⣠⣟⠿⣧⢀⠀⠀⠀⠀⠀⣀⣠⣇⠞⠹⢋⡡⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠞⠉⠀⠁⠀⣠⠞⠁⡠⡆⠛⠁⡀⢀⢤⡶⠔⣠⠞⠁⠉⠑⢤⣀⠀⠲⣴⣶⣶⣿⠿⣯⡷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⣠⠞⡅⠐⠁⠀⠴⠚⠉⠀⠀⠀⢀⡴⠁⠀⣀⣀⢠⣴⣽⣿⣿⡿⠿⣯⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠂⠀⠀⣠⣤⡾⢿⡷⢀⣀⣰⠋⣲⣶⣿⣿⣿⣿⡿⠟⠋⠁⠐⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡴⢀⣤⣞⣿⣿⣿⣿⣷⡋⠰⣄⠐⢿⣿⣿⡿⠟⠋⠁⠐⠀⠀⠀⢰⣿⡅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣨⣜⠿⡻⣯⣬⡻⣿⣿⣟⣳⣄⡘⠻⠦⣱⠿⡀⢄⠀⠀⠀⠀⠀⠀⠬⠭⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠽⠛⠉⣶⣿⣿⣿⡿⠌⠙⢽⣿⣿⣿⣗⣾⠁⠀⢡⢸⠀⠀⠀⠀⠀⢠⣽⣛⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠤⠖⣡⣾⣿⣿⣿⣿⣦⡀⠀⢀⣙⢿⣿⢵⠇⠀⠀⢸⢸⣤⡀⠀⠀⠀⣴⡿⠃⡸
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⢀⡤⢞⣿⣿⣿⣿⣿⣿⡟⠛⠳⣾⣿⠟⢣⠎⠀⢀⠔⠋⣼⠏⠀⠀⢠⣬⡖⢁⠜
⠀⠀⠀⠀⠀⠀⠀⡱⠃⠐⠁⢉⠔⢫⣿⣿⣿⠿⠟⠉⣠⣤⣞⠁⠈⠀⠋⣀⠔⠁⠔⡠⠋⠀⣠⡦⠔⠈⠐⠁
⠀⠀⠀⠀⠀⠀⣴⠇⢀⡴⠖⠁⠀⠀⠀⠈⠁⠀⣠⣾⡿⠋⠇⠀⠀⠀⠈⢀⠠⠂⠀⠀⠤⠚⡉⠄⠂
⠀⠀⠀⠀⠀⢼⡟⠀⢨⠆⠀⠀⠀⠀⠀⡀⢔⡾⠁⠐⠁⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠁⠒⠉
⠀⠀⠀⠀⡀⠀⠀⢰⠟⠐⣁⢄⡴⠾⠋⠀⢋⣤⠴⠃
⠀⠀⠀⠘⢀⠁⢀⡵⠲⠚⠁⢀⠀⣀⡀⠀⠉⠁
⠀⠀⠰⣶⠇⠀⠈⣀⣠⣤⠖⠁⠀⠁
⠀⠀⢀⣎⣠⡶⠿⠛⠋
⠀⠀⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⢿⣿⠿⠛⠻⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⢿⡞⠉⠀⠀⠀⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠉⠀⠀⢙⡶⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠟⠋⠀⠀⠀⣀⣴⠟⠀⠈⠉⠓⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠃⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡰⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⢀⣄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⢠⣶⠟⠛⢿⣿⠏⠀⠀⠀⠀⢀⣴⡟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⣾⡇⠀⠀⠀⣿⠇⠀⠀⠀⣠⣾⣟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠏⠀⠀⠀⣸⣯⣶⣴⡾⠋⠀⠀⢠⣾⣿⢯⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠞⢦⠀⠀⣰⡿⠋⠀⠀⠀⠀⢀⣴⣿⣿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢠⡦⣠⣿⣏⢳⠊⠑⢤⡁⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠤⠟⣻⣷⣿⡇⠘⢮⡑⡴⡃⠉⡳⠦⢄⣴⣿⣟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣍⡩⠋⠀⢠⣾⢟⣛⣏⣑⣤⣄⠙⠦⣌⣟⢠⠴⣮⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣿⡵⠏⣽⡿⠋⢡⣟⡢⠤⢤⣬⡭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⢺⣿⠁⣰⠟⠀⣠⣟⣽⣷⢾⣻⣿⣒⡦
⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⢠⠃⠈⠃⣰⠏⣠⡾⠛⠉⢩⡼⣸⣿⢿⠧⣄
⠀⠀⠀⠀⠀⠀⠛⢋⡇⠀⡜⠀⠀⢰⢯⡾⠋⣀⣤⣶⣿⠮⠛⠁⣸⠳⢼⠆⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⠃⠀⠀⠷⠋⠀⠀⠈⢁⠜⠀⠀⠀⡴⢻⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣿⠀⡌⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⣀⣴⠋⠙⣆⡇
⠀⠀⠀⠀⠀⠀⢀⡇⡰⠁⠀⠀⣀⠤⠒⠁⣀⡤⢶⢏⡴⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢰⣧⠃⢀⠴⢊⣡⣤⡶⠋⠁⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣼⢃⣴⠵⠚⠉⠈⠁
⠀⠀⠀⠀⠀⣰⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⡄⠠⠤⠀⠄⠠⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠁⠀⠀⠐⠒⠚⠛⠁⠉⠉⠙⠒⡠⠤⢄⣤⡭⠖⠦⠀⠀⠀⠀⠀⠀⠀⠠⠈⢁⠞
⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⡀⠁⠀⡀⠀⠀⠀⠀⠀⠛⠠⡀⠀⠐⠀⠀⠀⠈⠂⠀⢀⣈⠀⠀⠒⠤⠤⡀⡄⠂⠀⠀⡠⠃
⠀⠀⠀⠀⠤⢀⣀⣈⣡⣴⣢⠀⠂⠄⠀⠁⠀⠀⠄⠒⠐⠒⣀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠴⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠨⠄⠤⡄⠠⠀⠀⠐⠖⢈⡀⠀⠠⠐⠤⣄⣂⡱⠖⠃⠀⠘⠻⣋⣀⠀⠀⠀⠀⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠰⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣶⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠎⢿⣿⣿⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⡇⠀⢬⠉⢩⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⡄⠘⣿⣿⣷⣴⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠀⠀⢈⣀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⡇⠀⠀⡘⢩⣿⡿⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⡝⣠⡇⠀⠀⡇⢸⣿⣇⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡴⢿⡭⠒⣿⣿⠂⠀⢸⣿⡿⣿⣿⣿⣿⡟⠛⣿
⠀⠀⠀⠀⢀⣿⠚⠉⡄⠙⣿⣿⣆⠀⠘⠏⠺⠿⠿⣿⣿⠇⠀⣿⡆⠀⠂
⠀⠀⠀⢀⣸⡃⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠉⢹⠂⠀⡟⠁
⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⣽⣿⡄⠀⠀⢀⡀⣴⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⣶⣿⣿⣷⡀⢻⣯⠠⣮⡄⣠⡄⣿⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⡇⠀⢻⣻⡟⠋⠁⠂⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣧⣿⣤⠿⠷⢄⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⡿⡏⠁⠀⠀⠑⣄⠀⠀⠀⠀⠀⠹⢠
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⡏⣷⣄⠀⠀⠀⠀⠈
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿⣷⡆⡟⠀⠙⠒⠤⢀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣙⣋⣹⣟⡏⣿⣿⣿⡿⠉⠇⠃⠀⠀⠀⠀⠀⠉⠲⠤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠛⠛⠻⠿⣿⣭⣉⣉⣉⣉⣛⣿⣿⣟⠒⠒⠛⠒⠒⠲⠶⠶⠶⠤⠤⢤⣤⣤⣬⡇⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈

⠀⠀⠀⠀⠀⠰⡉⠓⠒⢦⡤⣀⣀
⠀⠀⠀⠀⠀⠀⠹⡄⡠⠋⠀⠀⢈⠟⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⡰⠋⢀⡠⠤⠤⣝⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠞⠁⢠⢏⡔⠉⠙⢦⢱⠈⠳⣄⡔⠋⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠘⡌⢇⡀⢀⡼⣸⠁⠀⠈⠳⡤⠦⢤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⡲⠭⠥⠚⠁⡞⠓⠦⣤⡇⠀⠀⠈⠳⢄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠈⢢⡀⠀⠙⠦⡀⠀⡀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢹⠒⠊⠉⠳⢄⡀⠈⠢⡈⠒⢄⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠳⣄⠀⠀⠀⢍⠓⠦⣜⣦⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡈⠳⢄⡀⠀⠑⠢⣀⠉⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠭⢷⣤⣀⠀⠑⠢⢄⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢽⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠒⢚⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡀⠀⢉⡉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠐⠛⠉⠀⠀⠘⠛⠶⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⡀⠀⠀⠸⠾⠇⠀⠸⠾⠇⠀⢠⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣦⣄⣀⡀⠀⠀⢀⣀⣠⣴⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⠿⣿⡿⠛⠛⠛⠛⠻⣿⣿⡿⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠉⠛⠲⠶⠤⢤⣤⣀⣀⡀⠠⠤⠄⠐⠋⠀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡂⠀⠉⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡴⠂⠀⠀⠐⠁⠀⠀⠀⠀⢸⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣏⠘⣧⠞⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⢈⡆⢰⡿⠁⢈⠀⠀⠀⠐⠆⢸⠀⢀⡞⢦⣼⠉⣷
⠀⠀⠀⠀⠀⠀⢠⡇⢨⡟⠁⠀⠈⠙⢶⠀⠀⠠⠄⠀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠙⣞⠁⣇
⠀⠀⠀⠀⢀⡠⠼⢬⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠊⠀⠠⠀⠀⠄⠀⠠⠀⠲⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⢸⣆⣹⣀
⠀⠀⠀⣀⡎⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⡞⠁⠀⠈⢳⡄
⣠⠖⢋⡩⢧⡀⠀⠀⣰⢧⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡼⣇⠀⠀⠀⣰⡗⠲⢄
⠙⢦⡏⠀⡜⠉⢻⠉⠳⡄⣙⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⢰⡋⢀⡜⠓⡖⠚⣇⠙⢦⣀
⠀⠀⠉⠺⣧⣀⣼⠀⠀⠉⠁⠀⠀⠀⠘⠢⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠁⠀⠀⠀⠈⠉⠀⠀⣇⠀⣸⡤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠈⡟⠋⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⣅⡐⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡙⠻⠾⠿⠯⠒⠉⠐⢄⠀⡤⠒⡛⣶⠭⡤⠭⠒⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠉⠀⠀⠘⢦⠀⠀⠀⠀⠀⢹⡔⠉⠀⠀⠳⡄⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⡀⠀⠀⠱⣀⣀⡀⠠⢤⠂⠀⠀⠀⠀⠈⢆⣀⣀⣀⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡇⠈⠀⠀⠀⠁⠀⠀⠀⢸⠀⣰⠁⠑⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⣇⡀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠹⡀⡀⠀⠀⢀⠁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⡆⠀⠀⠈⠀⠀⠀⡜⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⢀⣀⠜⠁⠀⠀⠐⠁⠈⢦⡀⠀⢀⣠⠎⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⢚⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡭⠂⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⢿⣯⡢⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠳⣝⢿⣾⣽⣻⢿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⠀⠀⠈⠑⠯⢿⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠤⣶⠏⡴⠁⢠⡔⣟⣶⣷⣤⡀⠀⠀⣼⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡼⣛⠭⠒⢚⡿⢡⠎⠀⢠⣿⡞⠁⣤⣾⣿⣿⡆⢰⣿⡿⡞⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠙⣏⢒⡞⠈⠁⢠⣶⡿⣡⠏⠀⠀⠸⣽⣇⣾⣿⣿⣿⢫⣿⣾⣿⢻⠃⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣿⡍⠛⢛⡞⠀⢀⣴⣿⣿⢱⠇⠀⠀⠀⠀⠻⣿⣛⠻⣛⣣⡿⣿⣿⢯⡧⢚⡝⠁
⠀⠀⠀⠀⠀⠀⠠⣄⡠⢤⣵⣄⠉⠀⡜⠀⣠⣾⣯⣿⢧⡏⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⢉⣼⣿⢏⣿⠀⠼⠤⠤⢄⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠈⠀⣸⠁⣴⣿⣽⣿⣿⣾⠡⠀⠀⢠⠖⣲⣄⠀⠀⠀⢀⣾⣿⢯⡿⡾⡆⠀⠀⠖⣫⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡁⠀⡇⣼⡿⠋⠉⢉⡟⢿⣰⣀⡴⠋⢠⣿⡳⠃⠀⣴⣟⡽⣳⠟⠀⣷⢷⣴⣒⡻⠴⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢸⢸⡟⠁⠀⠀⢺⠀⣤⣿⡿⠁⣰⣿⣿⣡⣴⣾⣿⢟⣿⠋⠀⠀⣹⢸⠉⠉⠿⠿⠥⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠶⠤⣤⣄⠾⠎⠀⠀⡠⢚⣽⡳⣿⡟⢀⣼⡿⣽⣿⣿⣿⢟⣵⡿⠁⠀⠀⣴⢿⠏⠀⠀⣠⣴⠶⠭
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⡫⢤⡤⠀⠀⡼⢁⡤⠀⠀⡜⢀⣾⢿⣿⣿⣿⢏⠟⢿⡟⠁⠀⢀⣾⣫⠏⠀⠀⠰⢶⣿⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⢠⡇⠊⠀⠀⡼⢡⣿⡿⠋⠉⢻⠓⠋⠀⢸⡆⠀⣴⣿⡷⠁⠀⠀⠀⠙⢥⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⡿⢛⡿⠋⢰⢠⠃⠀⢸⣕⣿⠟⠀⢀⢀⢸⠀⠀⠀⡼⣠⣾⡿⠋⠀⠀⠀⣶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠟⢁⣀⢀⡇⢀⠁⠈⣋⣠⠔⢁⠎⢠⠏⠀⢀⣼⣵⡿⠋⠀⠀⣀⢤⡤⣌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠉⠁⣼⠀⢸⠶⠋⡡⢀⡴⢁⡴⡟⠀⠀⠮⢞⡋⠀⢠⡀⢀⠘⡿⢿⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⢔⡭⠚⣯⠖⠉⢸⣡⠾⢦⡀⠸⡛⡦⣷⡟⠦⡟⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡠⠞⠉⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⢦⡏⠙⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣛⣉⣭⡉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡶⠛⠉⠁⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠶⠖⣚⣛⠛⠓⣾⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⣏⣉⣿⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⠈⠉⠁⡶⠶⣦⠙⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡙⠒⠋⠀⣴⠛⣶⠉⢙⣛⠶⠶⣤⣤⣤⣀⣀⣀⣀⣤⠟⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⡛⠶⣤⣌⡛⠋⠀⣿⣉⣷⠀⡶⠳⣦⠉⢩⣭⣭⡀⢴⡛⠳⣎⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢻⣶⣤⣈⠉⠛⠳⠶⢭⣤⣀⣙⠛⠋⠀⠹⢦⡼⠃⠈⠛⠛⠋⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⣴⠏⠀⠀⠉⠛⣷⠶⣤⣤⣀⣉⣉⠙⠛⠛⢲⣶⣶⠶⠶⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠁⠀⠀⠀⠀⠀⡿⠀⢰⡏⠉⠉⠉⠉⢿⡋⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠾⠃⠀⠀⠀⠀⠀⠀⢿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣶⣦⡤
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⠟⣡⣶⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠁⠿⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡙⠻⠿⠿⠿⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣶⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣤⣴⣦⣤⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠤⠤⠶⠒⠚⠛⠛⠛⠛⠒⠶⠶⢤⣤⣀⡀⠀⠀⠀⠠⣶⠄
⠀⠀⠀⠀⢰⠆⠀⢀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠛⠛⣛⣛⣛⠛⠓⠶⠦⣍⣙⠳⢮⣼⡆⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⣋⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣝⡲⣍⠻⣶⣄⠀⣤⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⣡⣶⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣽⣦⡙⣷⣌⠀⠀⢶⡆
⠀⢀⠀⢀⣴⠟⠉⣠⣾⣿⣿⡿⠟⠉⠁⣀⣠⣤⣤⣤⣤⣤⣄⡀⠀⠉⠛⠿⣿⣾⣿⣿⣿⣿⣎⠻⣷⡄
⠀⠀⣠⡿⠁⣠⣾⣿⣿⠟⠉⢀⣤⣾⠿⠛⣛⣩⣭⣭⣭⣭⣿⣿⣿⣶⣄⡀⠈⠻⣿⣿⣿⣿⣿⣧⡹⣿⡄⠀⠠
⠀⣰⠋⢀⣾⣿⣿⡿⠋⣠⣾⡿⠋⣡⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⣶⡀⠘⢿⣿⣿⣿⣿⣷⠹⣿⡄⢰⡀
⡰⠁⢠⣿⡿⣻⡟⢁⣴⡿⠋⢀⣾⣿⣿⡿⣛⣭⣶⣶⣿⣷⣖⣾⣝⡻⣿⣿⣏⣿⡆⠘⣿⣿⣿⣿⣿⣧⢹⣷⠘⣧
⠀⢠⣿⡟⣹⡏⢀⣾⡟⠀⢠⣿⣿⡿⣫⣾⣿⣿⣿⠿⠛⠛⠛⠿⣿⣿⣮⢻⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡄⠹⡆⣿
⠀⣾⡿⣱⠏⠀⣼⡿⠀⢀⣿⣿⡿⣱⣿⣿⣿⠟⣡⣶⣿⠿⠿⣶⣌⢻⣿⣇⣿⣿⣿⡆⢸⣿⣿⣷⣿⣿⡇⠀⠇⣿⡆
⢸⣿⢇⡏⠀⢠⣿⡇⠀⢸⣿⣿⢣⣿⣯⣼⡏⢰⣿⡿⢁⠆⠀⠈⡿⠀⣿⡿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⠃
⣼⣿⠘⠀⠀⢸⣿⠀⠀⢸⣿⣿⢸⣿⡟⣻⡇⢸⣿⡂⠘⢆⣀⣀⣀⣼⡿⣱⣿⣿⡿⠀⣾⣿⣿⣿⣿⣿⠀⢀⢀⣿
⣿⡟⠀⢀⡀⢸⣿⠀⠀⣹⣿⣿⣸⣿⣿⣿⣿⡀⢻⣷⣔⡠⢍⣉⣉⢭⣺⣿⣿⠟⢁⣾⣿⣿⣿⣿⣿⠏⠀⡀⣸⠇⡀
⢿⣇⠀⠈⠁⠘⣿⡇⠀⢫⣿⣿⣇⢿⣿⣿⣿⣿⣦⡉⠻⠿⣿⣿⣿⡿⠿⠛⣁⣴⣿⡿⣿⣿⣿⣿⢏⡖⡰⢡⡟⢰⠁
⢸⣿⠀⠀⠀⠀⢿⣿⡀⠈⠞⣿⣿⣞⢿⣿⣿⣿⣿⡟⢷⣦⣤⣤⣤⣤⣶⣾⣿⣿⣿⣶⣿⣿⡿⣫⡟⡰⠁⠟⣰⠃
⠀⣿⡀⠀⠀⠀⠘⣿⣷⡄⠈⠚⢿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⢏⠜⠀⢀⡴⠁
⠀⠸⣇⠀⠀⠀⠀⠘⢷⡹⣦⠀⠀⠻⣿⣿⣮⡻⢿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⠿⣫⣶⣟⠕⠁⠀⣠⠏
⠀⠀⠻⡄⠀⠀⠠⠀⠈⠳⣌⢷⣄⡀⠈⠙⠿⣿⣷⣮⣭⣛⣛⣛⣛⣛⣛⣫⣭⣶⣿⠿⠛⠁⢀⣤⠞⠁
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠂⠉⠻⢦⣄⡀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⣀⣤⠖⠋⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⣀⣀⣀⣀⣠⣤⠴⠖⡛⠉⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⣀⡀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠋⠉⠉⠉⠉⠉⠙⠒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠐⠀⠀⠀⠘⢢⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡀⠀⠀⠀⠉⠁⠀⠴⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠉⠢⣄⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⣀⠴⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠉⠒⢢⠤⠤⠤⠤⡄⠒⠘⠁⠀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠔⠁⠀⠀⢸⠁⠀⠀⠀⠀⢠⡄⠒⡄⠀⢰⡏⠉⢲⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⠃⠉⠁⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠀⠀⢷⡀⣹⡤⢾⣀⣠⠎⠈⣳⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠈⣧⠴⠚⠁⣳⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⡤⠛⡿⠿⠛⠛⢿⡟⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⢀⡟⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠏⠀⠀⢸⡁⠀⠀⠀⠀⠀⠈⠁⠀⡝⠉⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣑⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⡜⠀⠀⠀⠀⠙⠦⢄⣀⣀⣄⡠⠴⠚⠁⠀⠀⠀⠹⡋⠛⢤⠀⠀⣀⡤⠚⠅⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠉⢸⠁⠀⢠⠃⠀⣠⣴⣤⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣯⠉⠀⢁⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⢀⡷⠚⢧⢀⡏⠀⠀⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⣽⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠋⠀⠀⠀⢹⠁⠀⠀⠘⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠃
⠀⠀⠀⠀⠀⣀⡤⠯⣼⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡸⠁⠀⠀⠈⢳⠀⠀⠀⢠⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⠴⠚⣉⣧⡀⠀⠀⠀⣼⣄⠀⠀⠸⢇⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⢸⡇
⠹⣤⠞⠁⢠⠛⢲⠶⠻⣄⠘⢦⡀⠀⠀⠘⡋⠲⠦⢤⣄⣀⣀⣀⠘⠂⠀⠀⠀⣀⣀⣀⣠⣤⡴⠖⠊⠁
⠀⠈⠳⢴⡇⠀⢸⠀⠀⠈⠚⠋⠀⠀⠀⠀⣱⠖⠒⠶⢄⠈⠉⠉⠉⠉⠉⢉⡽⠿⠿⢤⡀⢀⠃
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⡶⠒⠠⡀⠳⡀⠀⠀⠀⣴⠃⡠⠄⠲⣾⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⣿⠴⠁⠀⠀⠀⢹⣴⣇⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠓⠛⠓⢾⡁⠀⠀⠀⠀⠀⠀⠀⣹⠶⠒⠛⠛⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣀⣀⣠⠗⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠛⠋⢽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠚⠛⠒⠒⠤⢄⣧⠀⠀⠀⠀⠀⠀⢠⣇⠤⠄⠒⠚⢓⠶⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢠⠈⠁⡠⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⣁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⢤⠤⠤⡤⠤⠤⡶⠴⠤⠤⠤⠤⢾⠀⠀⣾⢠⡤⠤⡤⠤⢤⡤⠤⢤⡤⠤⣤⠄⢾
⣀⣀⣀⣀⣀⣀⣀⣀⣽⣶⣾⣶⣶⣷⣴⣴⣷⣦⣶⣶⣴⣶⣾⣄⣀⣿⣴⣦⣤⣷⣤⣴⣦⣤⣴⣦⣤⣶⣤⣾⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠄⠒⠒⠒⠒⠒⢿⣦⣴⠶⠶⠶⠾⠟⢻⠛⢻⡩⠭⠭⠭⡠⣀⣀⠀⡸⢮⠳⡀
⠀⠀⠀⡠⢔⢚⡺⣿⣿⣯⣥⣤⡤⠄⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡉⠉⠚⠗⣒⣠⠤⣀⡀
⠀⠀⢸⢹⣿⣄⠙⠪⢍⣛⠿⣿⣧⣤⣤⣤⣄⡀⠀⠀⠀⠀⣄⣀⣀⣠⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⣿⣷⣮⠙⢲⣦⣤⣀⡀
⠀⠀⠈⠳⣝⢿⣦⣤⣀⠈⢉⠑⠫⢟⡛⢿⣟⡓⣤⣠⠬⠗⠛⠉⠛⠛⠛⠻⠷⠤⢦⠶⢢⣄⣀⠤⠶⣿⣿⣿⣶⣾⣿⣿⠟⡫⣁⣴⣿⣿⣿⣿⣿⣕⡦⣀
⠀⠀⠀⠀⠀⠙⠺⠿⣿⢿⣿⣿⣷⡶⢽⣾⣿⣿⣟⣉⣀⣠⣀⠀⠀⠀⠀⠀⢙⣿⣿⣿⡏⠀⠀⠻⢧⣤⣙⣛⠿⣿⡁⣴⣶⣾⣿⡿⢿⡟⠉⠉⠛⠻⠿⠿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠿⣧⣱⡻⣿⣻⠏⣛⣛⡶⣳⣿⣿⣷⣶⣖⣲⣶⣦⡬⢥⣄⣚⣻⣿⣿⠟⠿⠁⠀⠈⠉⠚⢿⣿⣄⣀⣀⣀⣠⠤⠤⠖⣢⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⡇⠺⣷⣄⢤⣬⣉⣩⣿⣿⣟⢩⠿⠿⢿⣻⡷⢾⣯⣽⣻⠕⡒⢤⣀⠀⠀⡠⠊⠈⢉⣀⣀⣀⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠳⠮⣉⣻⣿⣬⢿⣇⠀⠀⠀⠀⢀⠺⠏⢙⣿⣷⡻⣿⡿⢻⣷⡱⠚⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡻⠥⠭⢥⣖⢒⣛⣛⣿⣯⡭⠿⠿⠑⠚⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠘⡆⠀⡅⡇⣼⠀⠸⢮⠀⠀⠸⠙⠸⢶⠠⡁⠈⡆⢣⠀⠉⠁⠀⡄⢆⣾⠀⠠⡀⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠐⠁⠀⠀⠢⡀⠀⠀⠀⠀⠀⡔⠁⠀⠓⡄⠀⠀⠀⠀⢠⠁⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡂⠀⠀⢠⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢐⠀⠀⡌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡦⢴⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣷⡾⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠁⠀⠀⠀⠀⠀⠀⠀⠐⡯⠍⠄⠀⠀⠀⠀⠀⠀⠀⢦⣀⡤⠁⠀⠀⠀⠀⠀⠀⠀⡭⢽⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⣷⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠔⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠞⠚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣥⣤⠄⠀⠀⠀⠀⠀⠘⠋⠉⠀⠐⢢⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠀⠀⠀⠀⡄⠀⢀⠃⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠀⡠⠚⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣔⣁⣀⠇⠀⠀⠀⠒⠁⠛⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡒⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⣧⠀⠀⠀⠀⠀⠀⠀⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠗⠀⢤⠀⠀⠀⠀⠀⠀⠀⠰⠰⣿⠀⠀⠀⠀⠀⠀⠱⠞⡀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠈⠝⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠠⢤⡶⠶⢚⣫⡾
⠀⠀⠀⠀⠛⠛⠋⠈⠉⠀⠀⠈⠒⠶⠄⠀⠀⠀⠈⠓⠢⠤⠤⠂⠀⠀⠀⠐⠶⠚⠁⠀⠀⠀⠀⠀⠐⠶⠶⠖⠛⠁⠀⠀⠀⠐⠓⠶⠶⠖⠛⠛⠛⠛⠛⠃


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

⣶⣵⡆
⠈⠉⠁⠄
⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⠁⠀⡀⠈⠅
⠀⠀⠀⠀⠀⢸⡂⠈⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⢠⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠸⣜⠿⢀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡈⠉⠁⠀⠀⣠⠀⠀⠀⠈⠀⠀⠀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠿⠀⢀⡀⠀⠀⠃⠀⠘⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠿⢂⡔⡁⠀⠀⢠⣼⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣀⡤⠐⠀⢞⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣳⣤⡀⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⣼⣥⠀⠈⠙⢿⣿⡇⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⢶⢻⣿⣿⣿⣬⠤⣌⡻⢇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⢳⡘⢻⡿⠋⠀⠀⢹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠈⠛⢿⣿⣿⣿⣧⣀⠉⣇⠀⠀⠀⠀⠀⢨⣦⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠷⢌⣶⣦⠀⡀⠀⠀⢫⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡑⠻⣕⠢⡀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠦⣀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢘⣛⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡠⠖⣲⣬⡻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢐⠁⣸⣿⣿⡷⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠳⠽⡿⠔⣡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠘⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⣰⠃⣛⣆⠀⠀⡰⠋⠀⠀⠀⠀⣿⣿⣿⣿⢫⣣⡀⠀⢠⠏⢘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⣿⣿⡦⠪⢅⠀⠀⠀⠠⠄⢚⣿⣿⡟⣾⠟⠝⠢⡏⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⣿⣿⣴⣊⠴⠚⣦⠀⠀⢸⣿⣻⠟⠻⢄⡒⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣯⠟⠁⠀⠀⡟⠀⠀⢸⣿⣿⡆⠀⠀⠙⢦⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢷⠀⠀⣿⣿⡿⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⡀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⡆⠀⣿⣿⠃⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢧⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⡏⠀⠀⠀⠀⠀⢹⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢈⣧⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠋⠀⠀⠀⠀⠀⠈⢧⠀⣼⣿⠃⠀⠀⠀⠀⠀⠘⠓⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣺⣷
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠯⠤⠶⠶⠒⢒⣒⣛⣛⣛⣛⣛⣛⣛⣓⣒⣒⡲⠶⠦⠽⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⣛⣉⣭⠴⠶⠒⠚⠛⠛⠉⠉⠉⣩⠭⣍⠈⠉⠀⠀⠈⠉⠉⠉⠉⠉⢙⡛⠓⠒⠯⢭⣙⡓⠶⣤⡀
⠀⠀⠀⣠⠴⣫⠵⠚⠋⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠸⠯⠽⠃⠀⠀⠀⠀⠉⠓⠦⣝⠳⣄
⠀⠀⢰⣷⠟⠁⠀⠀⠀⠐⠯⠽⠗⠀⣀⣤⣶⣶⡿⠿⠿⠿⠟⠛⠛⠒⠒⠛⠻⠿⠿⠿⣿⣶⣶⣤⡀⠀⠀⠀⠀⢀⣀⠀⠈⠳⣼⡇
⠀⠀⠘⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⡿⠋⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠻⠾⠃⠀⠀⣿⠇
⠀⠀⠀⠘⢦⡀⠀⠸⣭⠽⠃⠀⠀⢻⡀⠀⠀⣠⠶⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⡄⠈⠉⠛⠲⣄⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡐⠂⠀⣤⣽⠦⣤⡿⠦⡀⡆⠀⠀⡄⠀⢠⠀⠀⠀⢁⠀⠀⠀⠐⢻⠶⠋⠀⠿⢗⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠚⠷⠶⠶⡇⠀⠀⡇⠀⢹⡏⠁⢸⠀⠀⠉⠉⠀⣀⠀⠀⢸⡶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠸⠀⠀⠀⠀⠀⣿⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠘⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⣸⠀⠀⠸⠀⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⣧⠀⠀⠸⠀⠀⢸⠀⠀⠄⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⢿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⠀⠏⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡈⠀⢢⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣋⠹⠉⠉⠉⠉⢹⠙⠋⠉⠉⠉⢙⠛⠏⠉⠁⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⢁⠀⠀⠀⡄⠉⢱⠀⠀⠀⢠⠈⠙⡆⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠀⠀⠀⠀⠸⠖⡎⠀⠀⠀⠀⠒⡖⠁⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣶⡄⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠻⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛⠉⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⡿⢿⣿⣿⢢⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⡿⠿⣿⣷⡾⠛⣷⣿⣿⠿⣾⣼⢏⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⣿⣿⡽⠗⠊⠀⠘⠁⡀⢋⣽⣿⣿⣿⣿⡿⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣾⣿⣯⢼⠋⠀⠀⠀⠠⢄⣂⣴⣿⣿⣿⣿⣿⣿⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠱⣿⡸⡆⠐⢲⣾⣿⣿⣿⣿⣿⡿⣟⣯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⡟⣻⣥⣀⣌⣙⠛⢷⣼⣿⢾⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣾⣁⣿⣿⣿⣿⣟⣻⣿⣷⣿⡿⠿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⢿⣿⣿⣩⣙⠛⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠿⠿⠭⠭⠭⠭⠿⠷⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠥⠤⠶⠶⣲⣒⣶⣶⣶⣶⣶⣶⡶⠤⠬⢷⣄⣀⡀
⠀⠀⠀⠀⣀⡤⣖⣫⡭⠵⠒⠒⠛⠉⠉⠉⢩⣭⣭⠀⠀⠀⠀⠀⠉⠉⠉⣉⣙⠓⠛⠯⢭⣓⠦⣄
⠀⠀⣠⣺⠗⠋⠁⢀⡤⣤⠀⠀⠀⣀⣀⣤⣤⣭⣥⣶⣶⣶⣦⣤⣤⣤⣈⡛⠋⠀⠀⠀⠀⠈⠙⠢⣝⢦
⠀⠀⢿⠁⠀⠀⠀⠀⠉⠉⢠⣾⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⢷⣦⡀⠀⠠⣖⡶⠀⠘⣿
⠀⠀⠘⢦⡀⠀⢾⡿⠆⠀⢸⡇⠀⢀⡤⠖⠚⠋⠉⠉⠉⠉⠉⡉⠉⠓⠶⣄⠀⢹⠇⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⢠⡽⠲⢼⠓⠄⡆⢀⣀⠀⢠⠀⢀⣀⠀⠀⠀⢻⠖⠁⢚⣟⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢻⠀⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⡆⠀⢸⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠿⠀⠈⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⡀⠀⠀⡄⠀⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡄⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⠘⠀⠀⠃⠀⠂⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⠙⣧⠀⠀⠀⠀⢹⠋⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣧⠀⠀⠀⠀⢺⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡎⠙⡾⠃⠀⠀⠀⠀⠙⣷⠀⢶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⠀⠀⠀⠀⠀⠀⡿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠾⠷⠶⠷⠶⠶⠶⠶⠶⠶⠿⠶⠾⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇⠀⠀⡟⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⡗⠀⠀⢸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⡏⠀⠀⠀⢸⡇⠀⢀⣤⣤⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠛⠶⠖⠲⠚⠃⠀⢸⣤⣬⠀⠀⠘⠳⠖⠶⠖⠃⠀⠀⢸⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣧⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢼⠁⠀⠀⠀⠀⠀⠀⣻⠀⠀⢠⡄⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⣤⠤⡤⣤⡀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢀⠀⡀⠀⡀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢻⠁⠉⠉⢻⠀⠀⢸⠋⠉⠉⢹⡇⠀⢸⠉⠉⠉⢹⠁⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⣇
⡶⠛⠓⠒⠒⠒⠒⠒⠒⢿⠀⠀⢸⠀⠀⠺⠤⠤⠤⠾⠀⠀⠸⠧⠤⠤⠼⠃⠀⠸⠤⢤⠄⠼⠀⠀⢸⡇⠀⠀⡿⠉⠉⠉⠉⠉⠉⠉⠉
⣷⢤⡤⢤⡤⠤⢤⣤⠤⣿⠀⠀⢸⡀⢀⡀⠀⢀⣀⣀⡀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⡀⢀⡀⠀⠀⣸⡇⠀⠀⣿⠠⠦⠤⠤⠤⠤⠴⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆⠀⠀⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢉⠀⢀⡀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⠀⠀⢀⡀⠀⡀⠀⢸⡇⠀⠀⡿
⠿⠀⠶⠠⠶⠐⠲⠖⠶⣿⠀⠀⠘⠀⢸⡏⠀⠉⠀⠀⡏⠁⠉⢻⠀⠀⡏⠈⢸⡇⠀⢸⠁⠈⠙⠀⢸⡇⠀⠀⣿⠲⠒⠶⠶⠶⠶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠘⠗⠒⠚⠂⠀⠛⠒⠒⠛⠀⠀⠛⠒⠚⠃⠀⠚⠒⠒⠛⠀⢸⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠺⡟⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠹⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠚⡇⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣾⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣴⣶⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡛⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⡉⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⠋⠙⠿⠁⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⣠⣤⡈⢿⣿⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡿⢿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⣿⣿⠛⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣯⣉⣧⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠘⣿⣿⡦⠀⠈⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣀⣿⣧⣀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣾⣿⠛⢻⣿⣿⠁⠀⢹⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣸⣿⣿⠀⠈⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⡀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⠀⣩⡅⠈⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⣿⣿⣿⡇⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠁⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠛⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⣤⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⠿⢿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣷⣤⣾⣆⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠘⠛⠛⠛⠟⠀⠀⠻⣷⣶⣶⠟⠛⠁⠀⣀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣷⣤⣶⣤⣤⣤⣀⣀⡀⢿⣿⡿⠠⣤⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣿⠋⠁⠈⠀⠀⠙⠁⠀⠀⠉⠁⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠶⠆⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠙⢢⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⠿⠟⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣠⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣶⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠉
⠀⠀⣾⡿⠟⢻⣿⣿⣿⢁⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠂⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣶⣾⣿⣷⣶⣿⣿⣿⡃⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣠⣾⣿⣿⣿⠁⢠⣷⣦⣶⣷⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣶⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆
⠉⠙⣿⣿⣿⣿⣿⣿⣶⣶⣾⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⡛⠛⣁⣤⣄⡀⢀⣠⣶⣶⣶⣦⣄
⠀⠀⠀⠉⠉⠁⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠉⠉⠁⠉⠉⠉
⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣶⣿⣿⣾⣿⣿⣷⣄⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⣀⣠⣽⣿⣀⠀⠀⠀⠀⠀⢠⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡤⣀⣴⠿⣿⣳⣯⢞⡟⢛⠿⣦⣀⢤⣀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡞⣱⣾⣋⣉⣉⣶⣿⠿⢿⣿⣿⣟⣉⣙⣳⣮⢳⣦⣄⠀⠰⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠇⣾⣻⢉⣍⣩⣉⣹⡉⢈⣙⢍⣯⣉⣏⣩⡉⣟⡗⠸⣿⠀⠀⢘⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣬⣶⠾⢿⣛⠻⣟⠻⠏⠙⣋⣥⠺⠷⢮⢉⠋⠹⠟⣹⠟⢛⣻⠷⣶⣤⣌⣀
⡇⠀⠀⠀⠀⠀⣀⣤⣶⣿⡫⣿⣿⣭⣽⣯⣭⣯⣿⣷⣶⣿⣿⣿⣧⣾⣇⣿⢿⣷⣾⣶⣿⣭⣽⣭⣭⣿⣿⢛⣿⠲⢤⣀
⡇⠀⠀⣠⣴⣾⣫⣭⣾⣾⢴⡿⠷⣶⡶⠶⣶⠶⢶⠖⠚⠿⠛⠿⠖⠒⠿⠟⠻⠷⠒⠶⠶⢶⡶⠶⣶⣿⣿⣶⣿⣧⣤⣿⣷⣦⣄
⡇⠀⠀⠙⠛⠿⣿⣿⣚⡛⠛⠛⢿⣿⣁⣤⡤⠴⠦⠰⠃⠀⢻⠃⠈⠉⠀⠙⠁⢰⠆⣤⠖⢥⣤⣄⣸⠟⠛⠛⠛⣛⣿⣿⠿⠛⠋
⡇⠀⠀⠀⠀⠀⣠⣬⣭⣛⡳⠶⢤⣬⣍⣛⣃⠀⠃⠀⠆⠀⠈⠂⠀⠀⠀⠀⠄⠐⠀⢈⠄⢘⣋⣉⣥⡤⠴⢞⣛⠉⠁
⡇⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣷⣶⣦⣭⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣩⡍⣥⣶⣾⣿⣿⣇
⡇⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣴⡿⣿⡟⣿⠿⣿⣿⣿⣿⣶
⡇⠀⠀⠀⠀⠀⠀⠀⢻⢹⣿⣿⣿⣿⣿⣿⣿⡟⢸⠁⠐⣤⣈⣀⠀⠀⢀⣁⣤⣶⣿⡻⠐⠒⠀⠐⠒⠘⣻⣿⣿⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⣿⣿⣿⢿⠛⠸⡇⠸⠀⠀⠻⠿⠏⡠⠀⡀⣼⣿⣿⣿⣷⡄⠀⠀⠀⢀⣴⣿⣿⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠰⣯⣄⠀⢀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠙⣿⣟⢠⣤⣤⢾⣿⣿⡿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣔⡱⣤⡀⠈⠛⢷⣤⣤⣤⣠⣤⠞⢫⠟⠁⠀⠀⠈⢿⣿⣿⣟⠠⠿⡟⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠈⢿⣦⣄⠀⠉⢹⠿⠋⢁⡴⠁⢀⣀⣀⡀⠀⠀⢻⣿⡏⠀⠀⢸⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠢⢀⠹⣿⣷⣤⡀⢃⣴⠋⠀⠀⣛⣹⡋⠀⠀⢀⡀⠙⠃⠲⠶⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⠻⠿⣿⣿⡟⠉⣿⣿⣿⣿⠿⠟⠛⠉⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠘⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠘⣿⣦⡉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣽⣿⡧⢀⣿⣿⣟⣄⣉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢟⣼⣿⡿⣐⠛⣻⣿⣿⣌⠻⣷⣞⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⡿⣹⣿⣿⣟⠋⣰⣾⣿⣿⣿⡳⣘⢿⣷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⣯⣾⣿⣿⣧⠶⢛⣿⣿⣿⣿⣿⣿⣎⢻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠈⠙⡛⠷⠾⢾⣿⣭⣿⣿⣿⡶⠿⠛⢻⠄⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠐⡩⠀⠀⡀⠤⡤⠤⠶⠶⡦⣬⣵⡶⡌⣶⡄⠀⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⢈⠴⠋⢹⣤⣴⣾⣷⣶⣾⡭⠓⣯⣟⠢⢵⣔⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢴⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣶⡜⣯⣝⢦⣄⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣶⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢹⣟⠳⣋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢲⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢹⢯⣕⠛⢀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡅⠀⠘⣷⡚⢿⣿⣿⣿⣿⣿⣿⣿⠻⣵⢯⣝⣲⢺⡬⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢙⡿⠷⠉⠛⠿⠿⠟⢿⣥⣶⣿⣿⣿⣿⠁⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣵⢦⡬⣤⡆⠀⠀⠈⠉⢀⣀⣘⢿⣿⣿⣿⠏⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣿⡘⠃⠿⣡⠀⠀⠀⠀⠐⠛⢿⣿⣟⣛⣃⠀⢀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⣋⣷⢹⣧⠀⠸⢯⡤⢞⣽⣷⠩⣿⣶⣶⠿⠛⠁⢀⣾⣗⢜⠳⢦⣀
⠀⠀⠀⠀⠀⢠⣾⢷⣶⣹⣿⣿⣯⣿⡀⢀⡿⣿⣯⣾⣿⣦⣶⣶⣿⡿⠁⠀⣾⣿⣿⣷⡠⣐⢝⣷⡀
⠀⠀⠀⠀⢠⣿⢝⣿⣾⣿⣿⣿⢻⣿⢷⣾⣻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣸⡿⣿⣿⣿⣿⣷⡏⠜⣿⡄
⠀⠀⠀⢀⣿⡭⢲⣾⣿⣿⣿⣤⣿⣿⣿⣯⢗⣿⡿⢁⣿⣿⣿⣿⠁⠀⢠⣿⣿⣯⣿⣿⣿⣿⣯⠔⢉⣿
⠀⠀⠀⢸⡿⢞⣷⣿⣿⣿⣿⣿⣿⣽⣏⠝⣿⡟⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣷⡄⡀⢺⡇
⠀⠀⠀⢸⣟⣵⣿⣿⣿⣿⡿⢣⣿⣿⡥⢺⡟⣵⣿⣿⣧⣿⣧⣀⣀⣜⣿⣿⣿⣾⡻⣿⠿⣿⣿⣯⢮⣹⡇
⠀⠀⠀⠈⣷⠞⣿⣿⣿⠟⣠⠋⠀⣿⢓⣿⣷⡟⣡⣋⢿⣯⣭⣉⣩⣤⡝⢿⣿⣿⣿⢦⡄⢻⣿⣿⠓⣿
⠀⠀⠀⠀⢹⡟⢻⣿⠋⡴⠁⠀⠀⣯⣹⣿⣿⡟⠉⠈⠀⠙⠿⠍⠉⠁⠰⡌⢿⣿⣿⠈⢷⣨⣿⣇⣹⡏
⠀⠀⠀⠀⠈⢿⣙⣿⣀⠃⠀⠀⠀⡷⣽⣿⣟⠰⠀⠀⠀⠚⠖⠒⠄⠀⠀⢨⡎⣿⣿⠀⠘⣧⣿⣯⡿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⢙⣻⣿⠉⠹⠆⠈⠛⢻⣿⣿⣿⡋⠉⢰⡇⢹⣿⠀⠀⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⢸⣥⡟⠀⠘⠃⠀⠀⠘⠛⠛⠉⠁⠀⠈⠀⠀⠉⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣤⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠋⠉⠉⠉⠉⠉⠙⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢤⣄⣀
⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠖⠚⠋⠉⠀⠀⠀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⢠⠞⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠳⡄
⠀⠀⠀⢸⡀⠀⡞⣉⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣙⡆⢀⡇
⠀⠀⠀⠘⢿⡦⣍⡁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡿⠃
⠀⠀⠀⠀⠀⠉⠓⠯⣽⣒⠦⠤⣄⣀⣀⠘⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⣀⣀⣤⠤⠴⣒⣯⠽⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠾⠭⠭⣿⣛⣒⣒⣒⣒⣶⣶⡦⠤⠤⠤⠭⠤⠤⢴⣶⣶⣒⣒⣒⣒⣛⣿⠭⠭⠷⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠂⠀⠐⠒⠒⠒⠀⠠⠬⠭⠭⠭⠉⠉⠛⢉⣙⣛⠛⠛⠒⠺⠶⢶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⠀⠀⠀⠀⡇⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠉⣐⣒⡒⠠⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⠰⢠⠀⠖⢀⠄⠉⡁⠁⣒⠒⠠⠭⠅⢁⣒⠒⠠⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠤⠐⡠⠄⡀⠐⢂⠀⠙⠁⡄⠈⣠⠒⢁⡠⢄⠒⠤⠀⢉⡉⠁⣒⠂⠤⠭⠭⢁⣒⠂⠴⠃⡇⡀
⠀⠀⠀⠀⢀⡠⠒⠉⡀⠀⢁⠀⠈⠠⠄⠀⠡⠀⠈⠀⠒⠈⠔⠂⠄⠈⢡⠐⠈⠀⠠⠆⠀⠤⠈⢀⠀⠀⠆⠀⠤⠀⣀⢥⣟⣉⠇
⠀⠀⠀⠀⢨⣉⠉⠒⠒⠂⠥⠴⢅⣀⡐⠀⠄⠁⠀⠀⠀⠈⠁⠀⠀⠆⢀⡄⠀⣁⠀⠊⠀⠔⠂⡡⠀⠈⣁⠤⠒⣉⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⣀⣉⠉⠑⠒⠂⠤⠤⢄⣀⡠⠂⠐⠆⢀⠀⠈⡀⠀⠃⣀⠤⠒⣉⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣉⡉⠐⠒⠒⠤⡤⠒⢉⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣶⡶⢦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣿⣿⣻⣿⣿⣿⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣄⠀⢀⠀⠈⠙⠻⢿⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣽⣿⡇⠀⢹⣧⡀⠀⠀⠀⠈⣿⣿⡄⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠙⢿⣿⣿⣿⣆⣠⣾⣿⣿⡇⢀⠀⢀⣿⣿⡇⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣤⣸⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣦⣤⠀⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣮⣙⠿⢶⣤⠤⠀⢀⣠⣴⣿⣟⣻⣿⡄
⠀⠀⠀⠀⠀⠀⣼⡟⣿⣿⡿⣛⠅⠀⠀⠀⠀⠙⠻⣿⣿⣾⣿⣷⣾⣿⣿⣽⣿⣿⣏⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢰⣿⣿⠟⠛⣵⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⣋⠟⢉⡭⡙⢻⣿⣿⠇⢻⣿⣿⣾⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⠃⢉⣶⡾⢟⡽⣪⠴⠹⣿⣦⣼⣿⣿⣟⣻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⠨⢽⣗⣚⡽⣛⡥⢞⠁⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⣠⣾⣟⣿⣏⣽⣿⣿⣦⡩⣵⣾⡈⣑⡾⢋⣼⣿⣷⣿⣿⠛⣰⣿⣿⣿⡿⢿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡷⣦⣭⣭⣭⣴⣿⣿⣿⣿⠟⠁⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣸⣿⣿⣿⣼⣿⢟⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⣣⠁⢜⣽⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⠟⠛⠿⣿⣿⡧⠉⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⡟⠀⠀⠉⢠⣾⣿⣾⣿⣿⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣛⠀⠀⠀⢸⡿⠁⠀⠀⠀⣴⣿⣯⣿⣿⣿⢿⣿⣿⠏⡴⠀⡄⠐⣹⣿⣿⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⡆⠀⠀⢸⠃⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠘⢃⣾⣿⣿⣷⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠷⠄⠀⠚⠀⠀⢰⣿⠿⠛⠛⣿⣿⣿⣿⣿⣿⡤⣀⢀⣴⣿⣿⡟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣿⣿⠋⣹⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣻⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠤⠆⡀⠂⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⣦⣎⣄⠀⠀⠀⠀⠘⠋⠄⣀⢄⣀⣀⣠⠀⠀⣀⡄⣉⡁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢻⡿⠀⠀⢠⠂⣦⡅⠀⡇⠊⣠⠍⢯⣴⢋⠉⢋⠉⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢇⠀⠧⢀⣠⠄⠀⢃⡄⢀⠆⢾⣿⣦⣤⣿⠀⣇⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⣰⠗⠻⣥⣶⠋⣼⣁⠞⣄⠔⣯⣀⣸⣿⠀⢹⡆⣧⡀⢤⡀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⢻⣉⠓⢤⣹⣿⣿⠋⠀⠀⠠⡹⣿⣿⠇⠀⠀⠙⢿⡷⠸⢧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⢿⢸⠄⣙⠢⣿⣏⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⣾⡓⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⡚⠈⠇⢨⣿⣿⣿⡀⠀⢾⡇⠀⠀⠸⠀⠀⠀⠀⣰⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⡴⠖⢻⣧⡿⠦⡄⢸⠏⠛⢿⡷⢴⡿⠿⠿⠿⠛⣷⣶⣤⣶⣿⠇⠸
⠀⠀⠀⠀⠀⠀⠀⡰⢦⠚⣿⠅⠀⠁⠀⣀⡴⢺⠷⣠⠄⠉⢈⢉⣭⣿⣿⣿⡟⢯⡑⠔
⠀⠀⠀⠀⠀⠀⠀⢹⢚⡄⠘⠄⠂⡀⠀⡿⠲⠥⣤⠀⢀⠀⠀⠀⡄⠈⣧⠬⣿⠆⡉⠳⢴⡂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠠⠀⠀⠀⣟⠀⠀⠀⠀⣾⡀⠀⠀⣷⡄⠆⠀⢻⡇⠀⠀⢸⣯⣾⣤
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⠀⣏⠀⠀⠀⣷⣾⣷⠀⢸⣿⡟⠀⠀⠈⣷⡄⢀⣾⣼⣡⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡼⢀⣴⣿⣿⡇⠀⠀⠀⣾⣿⠉⠈⢿⣿⡗⠀⠀⠀⢹⣿⣿⡿⠈⣱⠃
⠀⠀⠀⠀⠀⠀⠀⡇⣰⢏⣼⣿⣿⡿⢻⡄⠀⠂⠀⣿⠇⠀⠀⠀⢻⣿⣄⢀⠀⠀⠹⡟⠫⠆⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠻⣿⠋⢹⡏⠀⠸⣇⠀⠀⠀⣿⣄⠀⠀⠀⣴⣧⣿⣀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣼⡇⠀⠀⣏⣙⢻⣿⢻⣿⠀⠀⡞⣿⠋⡉⢿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢿⢉⣿⣿⣾⡃⠰⣞⣷⣡⡞⠠⡮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣴⣾⡈⠋⣹⣿⣿⣶⣧⠬⢽⣤⣾⣿⣛⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣴⣶⣿⣿⣿⣿⢧⣴⠧⠿⣿⣿⣿⡄⠀⠈⠁⠹⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⢠⣶⣹⣿⣿⣿⣿⣿⣿⣿⠏⠘⠁⠀⠀⠹⢹⣿⣇⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠦
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣤⣀⣀⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣇⣠⣤⣾⣿⣿⣿⣿⠍⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⡀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢎⡳⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠉⠉⠙⢠⡞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠐⠂⣄⣀⣤⠴⠚⢉⡝⠡⡄⠀⠀⠢⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡆⠀⠀⠀⢐⡴⠋⠁⠀⠀⠀⢠⠞⠒⠃⣀⠀⢨⣜⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠿⠆⣢⠖⠉⠀⠀⢄⠀⠀⠀⠀⠀⠚⢁⡤⣖⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠟⠁⠀⠘⢧⣤⡼⠀⠀⠀⠀⠀⠀⠀⣴⣏⡵⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡴⢋⣵⠀⠀⢀⣴⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⡕⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠀⠀⣠⢎⠔⠁⠈⠉⣹⠟⠁⠀⠘⣧⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣠⢞⠕⠁⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⡫⠋
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⣀⣀⣀⠔⠑⠋⠀⠀⠀⢠⡾⠁⠀⣧⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣫⠞⠁
⠀⠀⠀⠀⣠⠏⠀⣠⠴⠚⠋⠉⠉⠉⠕⠛⢷⣦⣄⠀⣰⠟⠀⠀⠀⠊⠉⠀⢀⠴⠊⠝⡢⢀⣤⣾⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⣀
⠀⡠⠒⠒⢡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⠀⠀⠀⠀⣠⠞⠁⠀⢰⣮⣴⣿⣿⢟⡿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⠡⠶
⠀⢷⣦⣤⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣠⠞⠁⠀⠀⣴⣿⣿⢟⣫⣗⡉
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⣜⠁⡀⠀⢀⡾⣛⣭⣶⣿⠟⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠲⣄⢀⣠⣾⣟⠈⢀⣴⣿⣿⡿⢟⢟⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠋⠀⠀⠀⠀⠈⣦⣽⡿⢃⣴⠛⢿⣏⢁⢔⣵⠟⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣄⠀⠀⠀⢀⣾⢃⢜⡵⠋⠁⠀⠘⣫⣶⠟⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢔⡥⡀⢽⠷⣄⢠⣾⢗⡵⠋⠀⠀⠀⠀⠈⣿⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢀⢠⠄⡠⣪⠞⡫⠂⢀⡏⡔⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡼⠃
⠀⠀⠀⡴⣻⠥⣠⣮⢾⣷⣾⠅⢀⠞⡀⠹⣝⣶⡿⠃⠀⠀⠀⠀⠀⢠⠏⠀⣠⠟⠁
⠀⠀⠰⠆⢠⣾⢟⡵⣿⡿⠃⣴⡇⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂
⠀⠀⠀⣴⠟⡡⢊⡼⠋⣠⡎⠿⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠂⢫⡴⠋⠁
⠀⠀⢞⡵⣪⠞⢋⡠⠖⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣦⣴⣾⠂
⠀⢀⡮⠚⠁⠰⠛⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣤⣉⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠠⠀⣿⣛⣛⣻⣿⣟⣿⣛⣟⣛⣏⣿⣟⣹⣻⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠐⣚⣿⡲⡄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄⡀
⠠⢚⣩⣴⣶⣶⣶⣬⣙⠢⠀⠀⠀⠀⢼⡟⢏⠛⠋⠙⣿⡆⠀⠀⠀⠀⠀⠀⠳⡄⢹⢹⠀⠀⠀⠀⠀⢰⣿⠋⠙⠛⠉⢿⣷
⠰⣏⠁⣾⣯⣍⣿⡆⢉⡷⠀⠀⠀⠀⢸⠁⠤⠑⣶⣦⠹⡆⠀⠀⠀⠀⠀⠀⣰⠃⣞⡿⠀⠀⠀⠀⠀⠀⡏⠠⠄⠀⠤⠌⣷⠀⡄
⠀⠈⠓⠾⠿⠿⠿⠖⡋⣀⠀⠀⠀⠀⠈⣇⠀⢰⡈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠁⢨⡞⢡⣀⡄⠀⠀⠀⠀⢳⡀⢀⣆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠣⠀⠀⠀⠀⠀⠈⠳⠤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠦⠴⠋⠀⠰⠛⠆⠀⠀⠀⠀⠀⠑⠦⠤⠔⠁

⠀⢠⢾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷
⠀⠘⣶⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣆
⠀⠀⣿⣿⣷⣦⠹⣆⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠰⠏⣿⣇⠹⣆⠀⠀⠀⠀⠀⠐⠃⣿⣯⠈⠓⠄⠀⢀⣤
⠀⣀⣾⣿⣿⣿⢠⣿⣶⡀⠀⠀⠀⠀⢸⠀⠈⠛⠛⢿⣿⠀⠀⠀⠀⠀⠀⢀⡿⢿⡄⠈⢆⠀⠀⠀⠀⠀⢠⡿⢻⣆⡤⣶⣶⣿⡇
⠀⠯⠿⠉⠉⠉⠉⠹⠯⠇⠀⠀⠀⠀⠈⠳⠤⠤⠞⠙⠉⠀⠀⠀⠀⠀⠀⠾⠃⠈⠷⠀⠈⠂⠀⠀⠀⠀⠾⠃⠸⠇⠸⠁⠀⠹⠁

⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢠⣼⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⠀⠀⠀⠀⠀⢀⣬⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯
⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢀⣴⡿⠂⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⠆⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢨⣿⠟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀
⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⢀⣴⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢖⡲⣶⠖
⠀⣿⣿⣿⣿⣿⢯⣿⠆⠀⠀⠀⠀⠀⠐⢶⣤⣿⣾⣇⣀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣿⣿⣿⣷⠀⣿⠀⠀⠀⠀⠀⣴⢯⢷⡄⠠
⠀⠙⣿⣶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠤⠴⢿⣿⣿⢏⡉⠁⠀⠀⠀⠀⠹⠿⢣⡎⠟⣩⣭⡭⠀⡥⠀⠀⠀⠀⠘⢷⡉⣽⠗⠠⠉
⠀⠀⠉⠛⡽⠁⠀⢐⣴⡂⠀⠀⠀⠀⠀⠀⠿⠁⠽⠀⠁⠀⠀⠀⠀⠀⠉⠉⠻⣿⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡷⡄
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉

⣿⠛⠛⠻⠷⠶⠶⣦⣤⣤⣀⣀
⢻⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⣦⣄⣀
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀
⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠛⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡶⠶⠶⠶⠶⢶⣶⣦⣄
⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠹⣿⣧⡀⠀⠀⣀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠘⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠉⢿⡿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⢻⣦⡙⢷⣄⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠈⠻⣦⣙⢷⣤⡀⠀⢀⣴⣾⣿⣏⡛⠛⠛⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡈⠙⢷⣌⠻⣾⡟⠁⠈⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠙⠳⣾⡇⠀⠀⠀⠀⠀⠉⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⢀⣼⠟⢷⣤⣾⠃⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠉⠻⢶⣤⣀⢠⣾⡃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣧⠀⠀⢠⣀⠀⠀⠀⢻⣶⣤⣀⣻⣿⣦⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⢸⣿⣷⣦⡀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⢸⡇⢻⣿⣦⣼⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣾⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡝⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡇⠀⠈⠻⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⡤⠀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔⠒⠒⠐⢛⣀⣉⣉⡈⠉⠉⠉⠉⠉⠁⠀⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠖⠞⠚⠒⠒⠒⠒⠚⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠒⠀⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠠⢁⠠⠡⡡⡁⡐⢀⢣⣧⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⢁⢁⠂⡄⡑⠡⠀⢀⢂⡴⡷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠂⠂⠌⠘⠡⠡⢁⠂⣼⠂⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠌⠈⢌⠄⠀⠁⠁⠆⣼⠃⠅⢿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠆⠜⡌⢀⠂⠆⢀⠀⠌⣰⢃⡮⠠⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡐⡀⠘⢠⠋⠄⡄⠀⠌⣰⠇⡾⢁⠐⢻⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡐⡰⡐⠠⡇⠀⡐⠌⡘⢠⠏⡸⢁⢂⠂⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠐⡐⠁⡝⡐⠐⠀⡐⢠⠏⡰⢃⢂⠂⠲⢻⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⡐⢀⠛⠀⠁⠐⠔⢠⡎⣰⢃⢊⢎⢮⠡⢹⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠡⠁⠠⠣⠡⠠⠁⠀⢠⡟⣰⢣⢂⠂⢢⢁⠃⢩⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢡⠣⠁⠡⢠⠁⠠⡰⠁⡙⡠⠃⠃⢌⢢⢁⢂⢒⠉⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠃⠃⠴⠡⢃⠶⠠⠁⠰⠴⠕⠌⢎⠊⠆⠀⠂⠞⠄⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡂⣰⣆⣀⢀⣉⣉⣉⣉⣀⢀⣀⣀⣀⣒⣒⣒⣒⣒⣒⣒⣞⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣾⣾⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⢈⠉⠪⣟⠝⢹⣬⡝⣭⣫⣝⣭⣋⢫⣝⣭⣪⡝⣭⢫⣭⣝⣫⣠⡉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠂⠀⠀⠀⢀⣀⣀⣀⣀⣜⣀⣠⣁⣀⣈⣀⣀⣉⣀⣒⣃⣰⣄⣱⡐⣒⣘⣆⣐⣒⣐⣄⣀⣀⣀⠐⢒⡢⠵⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠃⠀⣠⡀⣀⣀⣀⣀⣀⣇⠀⠀⣢⢐⠒⡀⢣⡐⣤⠚⢢⡀⠀⠀⠀⠀⠈⢆⠲⠈⢧⡙⡄
⠀⠀⠀⠀⠀⠀⢀⠎⡔⠀⠀⠀⠀⠀⠈⣹⠋⣉⣉⣃⡈⣨⡁⠈⠀⣉⠀⠀⢈⡀⠀⠈⡁⠀⠀⠈⣊⠢⡙⠀⡙⣬⣥⠛⣍⠀⠜⢏⠨⡠⡂⠀⢳⣜⡄
⠀⠀⠀⠀⠀⢀⠎⣈⠀⠀⠀⠀⠀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣬⡍⠋⠁⠿⡹⢀⡀⢉⣈⡛⣁⣁⣹⡻⡀
⠀⠀⠀⠀⢀⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠥⠍⢭⠁⠉⠈⠀⠀⠙⠬⡑⠭⣋⠉⡋⠧⣉⡋⠙⠛⠦⣳⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⣉⠢⢕⠨⣙⠣⡢⢅⠂⢳
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⡢⢅⠂⢅⠨⡀⢃
⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠀⠚⠢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⣀⣌⡂⢸
⠀⠀⠘⠯⣀⣒⣖⣶⢖⣖⡰⣤⢤⢤⠤⠤⠤⢤⠤⠤⠤⠤⠤⢤⠤⠤⢤⢤⠤⡤⡤⡤⠤⠤⢤⠤⠤⠤⠤⡤⠤⠤⢤⠤⡤⠤⠤⢤⠤⢤⢤⠤⡤⣤⣤⣤⡼⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠘⠁⠙⠋⠊⠘⠉⠛⠋⠋⠚⠋⠛⠋⠚⠙⠛⠛⠉⠛⠛⠛⠛⠛⠘⠑⠃⠛⠉⠚⠑⠛⠛⠋⠋⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣀⢀⠠⢀⠄⠠⠀⠒⠀⢐⠀⠀⠀⠠⠄⠀⠀⠀⡰⠄⡐⠀⠀⠐⢂⡒⠀⡐⠠⢀⠂⠠⣐⠀⠀⠂⠀⠐⠀⠀⠂⠀⠂⠠⢂⠁⢐⡀⠀⡢⠔⡀⢀
⠀⠀⠀⠀⠀⠁⠀⠁⠐⠈⠀⠀⠉⠐⠀⠀⠊⠀⠂⠀⠒⠁⠒⠁⠐⠊⠀⠊⠁⠂⠁⠐⠈⠀⠂⠁⠐⠈⠀⠂⠀⠔⠊⠀⠊⠁⠐⠀⠔⠊⠁⠂⠁⠐⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠑⠒⠂⠀⠤⠤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠒⠶⠤⠤⢤⣄⣀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠤⢈⠑⠂⢅⠂⠄⡉⠲⡍⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⠘⠀⠈⠐⢨⠐⡢⠄⣉⠖⢩⠙⠲⢎⡑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⠀⠀⠎
⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠐⠢⠄⣉⠤⢈⡑⠂⡤⠂⠢⢤⣁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼
⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣁⠘⠠⠐⠋⠤⢈⡰⠆⠀⠀⠀⠠⠗⢀⠄⡀⠀⠀⠀⠀⠀⠀⠤⣀⠠⠇
⠀⠀⠀⠀⠈⠳⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢀⡠⠚⢠⡄⠈⠂⣄⡑⠄⠊⣉⡁⠤⣀⡨⠒⠎⠁⠀⠀⠀⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠠⠠⠂⠈⠀⠄⠒⠀⠄⠐⠒⠤⠀⠒⢠⠄⠂⠁⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⡀⠀⢤⠈⠉⠠⠀⠈⡁⠤⠒⠁⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⢀⣠⠴⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄⢀⣤⣖⣒⣒⣒⡒⣒⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠠⠤⡀⠀⣈⡉⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⢒⡆⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠂⠔⠁⠸⠄⠝⡇⠀⠘⡇⠀⠀⠀⠀⠀⣎⠐⣅⡠⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠤⣄⣀⡠⡤⡀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣼⠲⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠒⠧⠦⠓⠃⢸⢸⣠⠟⠀⠀⠀⠀⡼⠣⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⡟⢻⠛⠓⠛⠋⠀⠀⠀⠀⢀⡞⢤⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⠛⠧⠾⠛⠓⠒⠒⢲⣤⣄⡴⠫⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡅⠈⠉⠉⠉⠉⠓⠒⠒⡖⠉⣯⢺⠹⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢤⡇⠀⠒⡂⡒⠒⠒⢲⠀⡇⠀⣙⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⢤⣷⠀⠟⠲⠃⠀⠀⢸⡗⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣸⢹⠀⢰⣞⡳⠀⠀⢸⣷⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⡼⢸⡄⠘⠒⠒⠒⠒⠚⠁⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣤⡇⠈⡇⠀⠀⣤⣤⣤⣤⣤⣿⠔⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⣤⠗⡄⠓⣒⢲⡖⠒⠒⠒⠲⡿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⢇⣸⠀⣍⣹⠀⠀⠀⠀⠀⡷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠘⠉⠀⠆⢾⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡟⠀⠀⠀⠀⠀⢹⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣇⠀⠀⠀⠀⢠⢾⠚⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠘⠒⠃⡇⠀⠀⠀⢻⣈⡉⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⠒⣒⣒⡭⠃⠀⠀⠀⠈⠒⠬⠭⡩⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉