Sign of the cross Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠛⠛⠳⢶⣤⡀⠀⠀⠀⠀⣠⣤⠶⠞⠛⠛⠛⠳⠶⣤⣀
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠈⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⢸⡇⠀⠀⣴⣶⣶⣶⡿⠀⠀⢸⣷⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠸⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⢿⡄⠀⠛⠛⠛⠻⣷⠀⠀⢸⡟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⢹⡏⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣤⣾⡇⠀⢸⣿⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⢸⣿⣿⠟⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠶⠶⠶⣶⣦⣶⣶⣾⣿⠶⠿⣿⣿⣶⣶⠶⠶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣼⣿⣿⠀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⣿⣿⠀⣿⡿⠿⣿⣿⡟⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣾⣿⣆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠿⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣀⣤⣀⡀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⣾⣿⣿⣿⣿⡆⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠘⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠙⢿⣷⣦⠀⠀⣿⣿⣿⣿⣿⡇⠀⢠⣶⣿⠟⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠿⣿⣷⣦⡀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⣠⣶⣿⡿⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠋⠁⢀⣀⣀⠀⠉⢻⣿⣶⣶⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⣷⣶⣾⣿⠋⠁⢀⣀⣀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⣠⣾⡿⠛⠁⠀⣼⣿⣿⣿⣿⡄⠀⠉⠉⠉⠉⠉⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⠉⠉⠁⠀⣼⣿⣿⣿⣿⡄⠀⠙⠻⣿⣦⡀
⠀⠀⢰⣿⡟⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠘⣿⣷
⠀⠀⢸⣿⡃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣽⣿
⠀⠀⠸⣿⣧⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢠⣿⡿
⠀⠀⠀⠙⢿⣷⣤⡀⠀⢻⣿⣿⣿⣿⠃⠀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⣀⣀⡀⠀⢻⣿⣿⣿⣿⠇⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣄⡀⠉⠙⠉⠀⣀⣼⣿⠿⠿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡿⠿⢿⣿⣄⡀⠈⠉⠉⠁⣀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣶⣿⡿⠟⠁⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⠙⠿⣿⣷⣶⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠠⣿⣿⣿⣿⣿⡧⠀⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣉⣉⣹⣈⣿⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢈⣛⣛⣩⣿⣿⡇⠀⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⣿⣿⣿⣿⣿⡇⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠋⠀⢀⣿⣿⣿⣿⣿⣇⠀⠈⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠻⣿⣿⣿⡿⠁⠀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠈⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠿⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⠀⠀⠉⠙⠻⣶⣄⣠⣶⠟⠋⠉⠀⠀⠀⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⡇⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⠀⣿⣇⣀⣀⣀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⠀⢈⣉⣉⣉⣿⠀⠀⣼⠇
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣿⠀⣿⡏⠉⠉⠉⠁⢠⡿
⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⡇⠀⢸⣷⣶⣶⣤⣤⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠸⡇⠀⢸⣿⣿⣿⣿⣿⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⣧⠀⠘⣿⣿⣿⣿⣿⣿⠀⢸⡀
⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣿⠀⠀⠛⠛⣿⣿⣿⣿⠀⠈⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣄⣀⣀⣹⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠛⠲⠶⠶⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣶⣶⣶⣶⠖⠀⠀⢸⣿⣿⣿⣷⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡇⠀⢀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⡇⠀⠀⠀⢸⣿⣿⣿⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⣿⣿⣿⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⡇⠀⠀⠀⢿⣿⣿⣿⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⣧⠀⠀⠀⢸⣿⣿⣿⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠘⣿⣿⠿⢿⠆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⠀⠀⠀⠀⣿⣧⣤⣤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⡆⠀⠀⠀⢿⣿⣿⣿⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⡇⠀⠀⡀⠸⣿⣿⣿⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣇⠀⠀⠀⠀⠈⠹⣿⣿⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢄⡀
⠀⠀⠀⠀⠀⠻⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣰⣷
⠀⠀⠀⠀⠀⠀⠈⠢⠹⢦⡀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠛⠉⢻⠆⠀⠀⢀⡀⠀⠀⠰⣟⠉⡉⢉⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣤⠀⣨⡾⠟⢦⡀⠀⠀⠀⠀⠀⢰⣼⡇⣶⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⣿⣿⣿⣿⣿⢿⣿⡻⣽⣿⣳
⠀⠀⠀⠀⠀⠰⣿⣿⣷⣅⣀⠈⠙⢦⡀⠀⠀⠀⠸⠋⠃⠛⠀⠀⠀⣠⠟⠛⢦⡄⠀⣀⣁⣥⣤⣦⠀⠀⠀⠀⠀⠀⣾⡁⢰⡆⠀⠈⠙⠙⢻⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢈⡙⠿⢸⡿⠀⠀⢈⣿⣦⡀⠐⠟⠾⠛⠛⠻⡧⠚⠁⢠⣆⠀⠈⠓⢿⠀⠀⠀⢰⠀⠀⠀⠀⠀⣻⢹⡇⠘⠃⠐⢛⡀⠀⢸⡃⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣃⠀⠀⠀⠀⢰⣿⠙⡏⢀⣀⣀⣀⣀⣀⣀⣶⣆⣸⣧⣠⣶⣦⣼⣶⠶⠶⠶⠆⠀⠀⠀⣰⠃⠈⢠⠀⣤⢸⣿⠀⠀⢸⡇⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣮⠁⠀⠀⠀⠸⠏⠀⠀⠈⠉⠉⠉⠀⠈⡿⠉⠉⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⢰⡏⣄⣠⢸⡇⣿⢸⣿⠀⠀⢸⡇⠜⠉⠥
⠀⠀⠀⠀⠀⢀⡀⣿⣿⢀⠀⠀⠀⢲⠦⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡶⡆⠀⠋⠁⣿⣿⢸⡇⠟⠈⠁⠀⠀⢸⡇⢠⣀
⠀⠀⠀⠀⠀⢸⠶⠿⡿⢾⡏⠀⠀⣸⢹⡇⡀⠀⠀⠀⣠⠀⠀⠂⠀⢰⣿⢿⣷⡄⠀⣀⠀⣰⡆⠀⡟⠉⠁⢲⠆⡿⠋⠈⠄⠀⢺⡇⠀⠀⢸⡇⢘⣿
⠀⠀⠀⠀⠀⠈⠒⠌⡳⠦⠀⠀⠀⣾⣸⠃⠇⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⣼⣟⠁⠀⡏⠀⢿⡇⠀⣷⣶⢻⣿⠀⢀⠀⡶⡇⠀⢸⠟⠀⠀⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢁⠀⠃⠀⠀⠀⠿⠆⢀⡀⣠⡖⠉⣍⠈⠓⡤⣡⡀⠈⠁⠀⠼⢿⣦⢻⡌⣿⠀⡯⠟⠀⠻⣏⠁⠀⠈⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡂⠀⠀⠀⢹⡯⠀⢀⣴⠒⠚⠛⠛⣻⡿⠷⠶⠐⠛⠒⠒⠓⣘⡿⠁⠀⠀⠀⢸⡏⢻⡄⢋⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡟⠀⢸⣿⡆⠀⣀⡀⠀⠏⠀⢉⣽⠷⠷⣦⡀⠀⣇⠸⠥⠃⠀⢰⣧⣾⣇⣤⣿⠀⣄⠀⠐⠜⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠛⠓⠲⠂⠀⠘⠉⠀⢾⣿⡇⠀⠀⠀⠀⠃⠀⣭⢳⣿⣿⡌⣷⠀⣇⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⠀⠀⢠⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⣴⣿⣶⡷⠸⠿⠃⠀⠀⠀⠀⠱⣤⣿⡆⣭⣭⢸⡿⢶⣇⠀⡇⠀⠀⠀⠉⠙⠃⣿⣿⠀⠀⢸⣿⠄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣿⣟⣸⠇⠀⢀⡾⠀⠀⢰⠆⠀⣿⣶⡇⣿⣿⠠⣷⡾⢿⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⢤⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣫⡇⠀⠀⠙⠉⠀⠀⠀⡈⠀⡇⠀⢸⣶⠀⣯⡻⠇⠉⠉⠁⢻⡕⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠿⠀⠰⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⢼⠟⢻⡇⠀⠘⠟⠀⢿⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⢦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣸⠳⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣀⣴⣄⣾⣶⡴⣿⣿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⢀⣸⣿⢹⣿⣿⣿⣾⡅⣿⣷⠿⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠖⠒⠒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⡟⣿⣿⣯⣿⠻⣿⣿⣷⣶⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣷⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢳⣿⣿⣿⡿⢷⣹⣿⣿⡙⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠥⠤⠾⣿⣿⣿⣟⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣼⣟⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣃⣀⣉⣉⣉⣤⣤⣭⣭⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣩⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠶⠶⠶⠖⠒⠒⠒⢒⡒⣒⣒⠒⠒⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⢷⣄⣿⠛⢻⣿⣷⡿⡿⠿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⢉⣩⣯⣽⢿⠻⣿⣭⣭⣭⣭⣭⣥⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿⣄⡿⠐⠋⠞
⠀⠀⠀⠀⠀⠀⠉⠉⣿⡶⠲⢶⡾⣡⣿⣦⠘⢿⣿⣶⠶⠶⠶⠦⠀⣾⡇⠀⢰⣿⡆⠀⣸⣿⡀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣩⣿⡽⣣⣿⣿⣷⡈⢿⣯⣭⣍⣉⣉⣛⣿⡇⠀⢸⣿⣯⠀⣿⣿⡇⢸⣿⡗⠀⠀⢸⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⡟⣰⣿⡟⣉⠉⣷⡈⣿⣛⣲⣶⣒⣒⣿⡇⠀⢸⣿⣿⠀⣿⣿⡇⠘⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⣆⣰⣄⣿⣿⣾⣧⣿⡟⣿⣽⣷⣿⡇⣿⠤⠤⠤⠤⢤⣿⠇⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣸⣿⣿⣿⣯⠿⣿⣿⣿⣿⢹⣿⡇⣿⣿⡏⣿⠄⣿⢭⣭⠛⠋⢙⣿⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣯⣿⣿⣶⣾⣷⣿⣿⠧⣿⣾⣿⣇⣿⣿⢧⣿⠂⣿⣿⣿⣛⠛⢻⣿⠀⠀⠸⠿⠿⠀⠿⠿⠇⠀⠿⠟⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⣉⣿⡿⠿⣬⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣒⡒⠶⠖⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⢀⣾⡟
⠐⠛⠻⠿⠿⠿⠿⣿⣿⣟⣛⡒⢒⡀⢠⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣶⡶⠿⠶⠾
⣀⣀⣢⣤⡄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠃⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠐⠒⠂⠒⠈⠉⠉⠉⠁
⠀⠀⠑⢀⠶⠊⠀⠀⠀⠀⠀⣤⣀⡁⡀⢀⠀⢤⣍⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣨⠽⠵⠀⠀⠀⠀⠀⠲⠂⠠⠄⠤⠄⠀⠄⣀⡀⠀⠤⠰⠂⠉⠉⠉⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠒⠒⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣤⣄⣠⣤⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠻⣿⠟⢻⠿⢿⠿⢻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣶⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄⢀⣀⣴⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠻⠟⠙⠁⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⣤⣷⣄⣈⣉⣃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠙⣤⣠⠌⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠔⠀⠐⠧
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠈⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⡛⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⢆⣾⡿⣫⡌⢻⣦⡀⠀⠀⠀⠀⣸⣶⡙⠻⡇⠀⠀⢀⣴⣦⠀⢀⠀⠀⢰⣿⡀⢀⡀⢀⣴⣿⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠍⠀⣮⡀⣥⠙⢿⣶⡆⠀⠀⢸⣿⡇⠀⠅⢨⠀⢸⣿⣿⠀⠘⠓⠀⣿⣿⡇⠈⠉⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠙⠃⠻⠁⠨⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⢩⡅⠀⠀⢀⣶⠀⡀⢠⣿⢺⣇
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡜⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⡟⠀⠒⢸⣿⠀⠉⢸⣷⢸
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⢿⣿⡇⠀⠰⣆⢹⣿⠀⠀⠀⠈⢻⣏⠀⠀⢸⣿⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠸⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢘⠁⠈⠀⢸⣿⣿⠀⠀⠘⠓⢸⣿⡆⠀⠀⠀⠀⣿⠄⠀⠀⠀⢸⡏⠀⠀⢈⣿⠀⠀⡠⣿⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡿⢿⡇⣸⠀⠀⠀⢀⣼⣿⡇⠈⠀⢘⠀⢸⣿⣿⡄⠀⠀⠀⠈⠏⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠⡰⠆⣻⠀⡆⠁⢂⣸
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠐⠇⠀⠀⢹⠀⠀⠀⠀⠀⠉⠉⢰⠀⢠⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠆⠂⢸⣿⢸⡇⠀⠹⠂⣿⡄⢈⠱⠂
⠀⢰⡆⡆⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⢘⢁⠰⠿⣦⡄⠀⠉⠙⠃
⠀⠀⠁⠀⣠⠞⣻⣄⠅⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⣿⠀⢘⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠐⠚⠀⠐⠂
⠀⠀⢤⣴⠗⣫⠭⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣸⣿⣷⠀⠂⠀⠀⠉⠛⠊
⠀⠀⠀⠀⠉⠓⠯⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠒⠿⠀⢠⡄⠀⠐⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠃⠀⠓⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡇⠀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⠤⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡤⠚⠁⠀⠀⠘⠢⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢉⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⡔⠋⠀⠀⢀⡴⣺⠿⢭⠿⢿⣦⡀⠀⠀⠙⣦⡈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⢹⡆⠀⠀⢠⣯⡟⠁⠀⢐⠀⠀⢹⣿⡀⠀⠠⡇⡿⠦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⡇⠀⠀⠸⡿⡇⠉⠉⠙⠀⠉⣨⣿⠃⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⢼⠀⡇⠀⠀⠀⠙⠮⣒⡶⠶⠶⣚⡿⠋⠀⠀⢸⢸⣯⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣉⠤⠒⠉⠁⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢸⢸⠀⠈⠓⠦⣍⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⡥⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⠀⠀⣀⡀⠙⠳⢬⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢺⣥⢴⡛⡇⠀⠀⣠⡶⠿⢷⣦⡀⢸⡄⣇⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⡿⣿⠀⣰⡿⠻⠛⢷⡄⠀⠈⢳⢦⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣧⠀⠐⣿⠀⠲⠆⣿⡇⢸⡇⣿⠀⠀⢠⢞⡿⠋⢹⠉⠻⣿⣦⠀⠀⠀⡇⣿⠀⢿⡇⢙⠃⣻⡗⠀⢀⡏⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡀⠀⠙⠧⣴⣶⠿⠁⠸⡇⢸⠀⠀⡟⣸⠁⠀⢸⠀⠀⠘⣿⣧⠀⢠⡇⡏⠀⠈⠛⠾⠟⠋⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⣿⠤⠤⢼⠤⠤⠄⢿⣿⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⡗⠀⠀⢸⠀⠀⠀⢸⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡄⠀⠀⠀⠀⠀⠀⠀⣧⢸⡄⢸⡇⡧⠤⠤⢼⠤⠤⠤⢼⣿⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣴⠒⠒⡖⠛⠛⡆⣿⢸⡇⢸⡇⡇⠀⠀⠸⠀⠀⠀⢸⣿⠀⡇⢸⢰⠖⠒⠶⠒⠲⡆⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣹⡀⠀⠁⠀⠀⡇⢿⢸⡇⢸⡇⡇⠀⠀⢀⡀⠀⠀⢸⣿⢰⡇⢸⢸⠀⠀⠸⠀⢰⣷⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣏⠛⠛⠛⠛⠛⠁⢸⢸⡇⢸⡇⡏⠉⠉⠉⠉⠉⠉⢹⣿⢸⠃⢸⠈⠙⠛⠛⠛⠋⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⠀⠀⡀⠀⠀⢸⡏⢸⠀⢺⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⢿⣿⠿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣯⣋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣶⠞⠝⢶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠅⠄⠀⠤⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠠⣏⢘⠃⢱⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⡈⠔⠃⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣆⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣏⣿⡀⠀⠉⢱⠀⣠⣤⣤⠠⣾⠿⢽⠿⣿⢿⣿⣷⠄⢠⣄⣀⠀⢰⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢠⣿⣏⣿⣷⡀⠀⢸⠀⣿⣿⠉⠀⠀⢀⡔⢂⢐⡒⣄⠀⡇⢸⣿⡿⡄⢸⠀⣤⣤⠄⢠⣤⣤⠄⢸
⠀⠀⢰⡿⣟⣯⣿⣯⡁⠀⢸⠀⣿⠋⠰⠀⠀⣮⠛⠛⠛⠛⠋⠀⣧⢸⠟⠀⠇⢸⠀⣿⣃⠀⠘⣇⣋⡆⢸⠀⠀⢀⡆
⠀⠀⠀⣴⣿⣷⣿⢿⣷⡄⣸⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠂⢸⠀⡿⠀⠀⠠⡇⠀⠇⢸⠀⠀⣼⣿
⠀⢀⣾⣿⣿⣏⣿⣿⣿⡄⢿⠀⠋⣶⠐⣠⠖⣯⠀⠀⠀⠀⠀⢠⣿⡎⢠⠆⠀⢘⡀⠃⠀⠀⠀⠃⠀⠠⢼⠀⣸⣿⣿⣧
⣠⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⠀⠀⢼⠋⢠⠤⠯⠤⠤⠤⠤⠤⠴⣿⡿⠦⣄⣀⣸⣁⣉⣉⣙⣉⣿⣿⣿⣏⣰⣿⣿⣿⣿⣆
⠉⠉⠉⠉⠉⠱⢿⣿⣿⣿⣿⣶⣶⣸⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⡘⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠿⠛⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡄⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⡿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠋⠉⢷⣄⠀⠀⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡟⠁⠀⠀⠀⠙⢧⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡇⠀⠀⠀⠀⠀⠈⠗⠲⠶⠶⠦⢽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⢰⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⡩⢾⣿⣦⣸⠀⠀⠀⠀⢸⣯⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠊⠀⠀⠈⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢣⠞⠁⠀⣀⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⡴⠃⠀⠀⢸⡅⠀⢱⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣋⣠⡏⠀⠀⠀⠀⠘⢧⣀⠞⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠉⠉⢦⠀⠀⠀⠀⢸⠀⠀⢀⠤⣄⠀⠈⠉⠉⢉⡙⠛⠒⠒⠶⠤⢤⣤⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡟⠛⢻⠛⠻⡀⠀⠀⠀⡜⠀⢠⠃⠀⢸⠄⠀⠀⡰⠉⢹⡆⠀⠀⠀⢀⡾⠋⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⢸⠀⠀⡇⠀⠀⠀⡇⠠⠞⠒⠄⠾⡄⠀⡤⢇⣀⣼⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡀⠀⠘⡇⠀⢸⠀⠀⡇⠀⠀⢸⠁⠈⠛⠓⠒⠒⠁⠀⠓⠂⠤⠤⠜⠀⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠘⠷⠶⢦⡷⠤⠾⠤⣤⣇⣀⣀⣼⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⠋⠉⠉⠛⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢺⣯⣀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣦⣾⣋⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⣴⡾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠒⠂⠀⠰⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠠⠦⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⡀⠠⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⣿⠡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⣿⢏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⢀⣀⣸⣇⠀⠀⠹⠴

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⢝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠊⣏⣸⢀⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⣹⡍⢹⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠭⠤⠼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡐⢀⡠⠾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣴⣿⣷⣦⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢫⠟⠱⢽⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⠔⠀⠀⠀⠄⠂⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⠅⠉⠁⠀⠀⠀⠀⠀⠊⠁⡝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢟⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⠅⠁⣂⣀⠀⠀⢀⢀⣠⣄⡀⠐⠀⠈⠀⠀⣨⣙⢝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⠕⠑⠂⠀⠀⠀⠀⠀⣠⣶⡫⠟⢧⣄⠀⠀⠀⠀⠀⠀⠀⠐⢙⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠕⠋⠀⠀⠀⠀⠀⠰⣤⡾⠫⠊⠀⠀⠑⢝⢷⣄⠀⠀⠀⠒⠀⠀⠀⠙⢿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣷⠁⠀⠀⠈⡀⠀⠉⣩⡾⡡⠊⠀⠀⠀⠀⠀⠂⠑⢝⢷⣅⠀⠀⠀⠀⠀⠀⠀⠗⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠄⠀⠀⣴⣶⡤⠘⠛⢺⡧⠭⠥⠍⠁⠀⠀⠀⠀⢽⡿⠛⠁⠀⣢⣖⠀⠐⠂⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢹⣽⣟⠀⠀⢸⣟⠀⢠⡤⠤⠬⠥⣄⠀⣿⡇⠀⠀⢘⣿⣿⣃⠀⠁⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠠⠴⢺⣿⡷⠀⠀⢹⡷⠀⢸⡅⢠⡄⡂⡇⠀⣾⡇⠀⠀⠰⣿⣿⠂⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠂⣀⣸⣿⣟⠀⠀⢸⣗⠀⠘⡄⢸⡇⡇⡇⢀⣻⡇⠀⠀⢺⣿⣿⡂⠐⠆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠄⠀⠠⣾⣿⣯⠀⢀⢸⡯⠀⠠⠀⠸⠇⡈⡧⠄⣽⡇⠀⠠⢴⣿⣿⡇⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡂⠀⠀⠛⠛⠛⠂⠀⢸⡷⠀⢰⡇⠐⠐⠆⡇⠀⣿⡇⠀⠀⠘⠉⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣁⡀⠈⡉⠉⣉⣁⣹⣹⡶⠈⠉⢉⠩⠿⢽⣿⣏⣏⣀⣁⣉⣀⣉⣉⣀⣁⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⢀⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠘⠛⠋⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⣶⣶⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁⠀⠈⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠙⠦⠄⠹
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠈⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠘
⠀⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣤⣤⣤⠐
⠀⠀⠀⠀⠈⠁⣠⣶⣄⠀⠀⠀⠀⠙⢿⣿⡿⠿⠿⠿⠿⠿⠟⠻⠟⠛⠛⠛⡛⠻
⠀⠀⠀⠀⠚⠸⠟⡷⠸⢷⣄⡀⠀⢸⣧⠙⠇⠀⢠⣿⡄⢠⠀⣼⣷⠰⠄⣼⡇⠐⠢⣤⣤⣤⣤⣴⣶⣶⣶⡞
⠀⠀⠀⠀⠀⠀⢀⠻⠸⠂⡋⠁⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣽⡇⠀⠀⠛⡋⠉⢉⣭⢉⢨⣧⡇
⠀⠀⠀⠄⠀⠀⠈⣻⣿⡇⡄⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⣻⣇⠀⠀⢽⡇⠀⠀⣼⡏⠀⠘⡇⠈⢸⡇
⠀⠀⠀⠀⠀⢠⠀⢻⣿⡇⡆⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠦⢹⡿⠀⠈⠘⣇⠀⠀⠀⡇⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠀⠘⠻⠇⡇⠀⠀⢼⣿⠀⠀⠀⢸⣿⠇⠀⠀⠘⠛⢀⠀⠀⠀⠀⠀⠀⣗⢀⠂⡇⢠⠀⢡
⠀⢀⡀⠀⠀⣠⡦⠈⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⡆⠀⠀⠐⠁⠀⠟⢸⣄⠈⠸⠦⠀⠁
⠀⠈⠀⣠⣾⡭⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⣿⡆⠀⠘⠀⠂⠀⠀⠈
⠀⠀⠺⢽⣇⠀⠀⠀⠀⠀⠇⠀⠂⠀⠀⢸⠀⠀⠀⠀⠠⠾⠿⠄⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⣋⣉⣡⣦⣴⠍⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⠈⠣⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣯⣽⣿⣿⣾⣷⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⡌⠳⣄
⠀⠀⠀⠀⢀⣼⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣦⡙⠦⡀
⠀⠀⠀⣠⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡀
⠀⠀⣼⡯⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠻⣆⠀⠀⣀⣀⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠋⠉⠉⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡄⣻⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⠂⠀⣀⣠⣾⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣶⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⣿⣿⣿⣿⣿⣧⠀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡿⠿⠿⠿⠿⠷⠤⠤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⣾⠁
⠀⠀⠀⠈⠉⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⠛⠛⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠦

⠀⠀⠀⠠⠤⣤⣤⣤⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠃⡀⠀⢀⣰⣿⣩⣿⣟⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠽⣿⣿⣟⣻⠃⠀⠀⠀⢣⢻⣟⣿⡿⡟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠟⠆⠀⠀⠀⡼⠸⠫⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠟⠟⠟⠂





⣶⡶⠶⠶⠶⢶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠘⠂⠖⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡴⠑⠦⠾⠟⠁⢀⣠⣶⣶⣾⣁⠀⠀⠀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠤⠤⡄⠀⣠⣮⡙⠛⠉⠉⠙⠋⠀⠽⣾⠛⠋⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡙⣿⠟⠋⠀⢠⣦⡀⠀⢀⣨⡇⠀⣆⢀⠀⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⢻⣷⡶⠆⠀⠛⠉⠉⣶⣼⡏⠁⠀⢘⠿⠇
⠀⠀⠀⠀⠀⠀⣠⠈⠉⠀⠀⠁⠀⣴⣴⣿⣷⠐⢻⣿⣶⣶⠏⣸⣷⣄⢰⡿⠶⡆
⠀⠀⠀⠀⠀⠘⠋⠁⠀⣀⣶⠆⠸⡿⣯⣉⠻⣦⠘⣿⣿⡟⠴⣿⣿⠽⠒⠈⠀⠙⠲⣄
⠀⠀⠀⠀⢠⡾⢷⣦⠀⢼⠁⠀⢀⣀⠈⠻⣿⣿⠟⣛⠽⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠋⠠⠿⢿⡀⢀⣰⣶⣿⣿⡿⢿⡟⠀⠀⠑⢄⠀⠀⠀⠀⠀⠐⢶⣤⡤⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⣿⣿⣿⣿⣿⡏⠀⠸⡀⠀⠀⠀⠈⠱⢄⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⡴⠟⠛⢛⣩⣿⠉⠀⠀⠘⣖⠃⢤⣀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠖⠉⣉⡿⠃⢀⡀⢸⣿⡆⠀⣬⠆⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣊⣥⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠁⠀⢉⡁⠈⣿⣧⣶⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣀⡤⢖⣫⣵⠶⠛⠉⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⣀⣠⣌⠁⣤⣼⣿⣿⣿⡀⠄⠂⠀⠀⠀⠀⠓⠤⠖⣩⡵⠾⠛⠁⠀⠀⢀⣀⣄⣷⠾
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠙⠃⠐⠖⠉⠃⠛⠀⣨⣿⠿⠏⠙⠃⠀⣇⠀⢠⠀⠀⢀⡴⠛⠉⠀⠀⣀⣠⣴⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠾⠿⢿⣷⣄⡰⠄⣀⡶⠙⣦⣤⠄⢀⣾⠁⠀⢀⣸⣧⡙⢿⣿⣿⣒⠒⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⡴⠿⠋⠀⠀⡿⢃⣠⠴⠾⢶⡸⣇⣾⣶⣿⣿⣿⣷⣌⣇⠉⠉⠉⠙⠒⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢯⢀⡟⠁⠀⠀⠀⠀⠛⠉⠈⠉⠛⢤⡉⠙⠻⢷⣖⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠤⠀⠀⠄⠒⢉⣉⠒⠠⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠖⣋⣉⠀⡀⠐⠒⢒⣈⠤⠤⣈⣀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢁⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠠⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⡦⠔⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠹⡒⠤⢀⣀⣀⣤⠤⣤⣀⡀⠻⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠼⣦⠤⠵⠖⠚⠉⠑⢦⡀⠀⠉⢿⣿⡷⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣠⣤⠘⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣷⠀⠈⠳⣄⡤⠤⣄
⠀⠀⠀⠆⠀⠀⠀⣼⠙⠟⠋⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⣠⡿⢿⣦⠔⠊⠉⡆⠀⠀⠑⢄⢰
⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠘⣆⠀⡀⠀⠀⢇⠀⠀⢹⠀⠈⡇⠀⠀⠀⡇⠀⢠⣶⠀⢙⣦⡀
⠀⠀⠀⠀⠀⣰⣿⡏⢷⠀⣠⠀⠀⢀⡼⠐⠁⠀⠀⠀⠉⠀⠀⣇⠀⢻⠀⠀⠀⠙⢦⣄⣛⣠⠞⠀⠙⠀⡀
⠀⠀⠀⠀⢠⣿⣿⡇⠘⠋⢀⡤⠒⠁⠀⠀⠀⢀⣠⠤⠤⠄⣀⠈⠀⠘⡄⢀⠀⠀⢀⠈⣿⣿⡀⠀⠀⠀⠑
⠒⠀⠀⠀⣼⠋⠹⣏⡤⢞⣉⡀⠀⠈⠉⢰⡾⠉⠀⠀⣀⠀⠈⠳⡀⠀⠀⡶⠒⣂⣶⣦⣿⣿⠀⠤⡄⠀⢠
⠀⠀⠀⣠⡟⣠⠖⣏⠀⠀⠀⠈⠓⢤⠀⢸⡇⠀⠀⠈⠁⢹⠀⠀⡇⠀⢸⠃⠸⡿⠋⢻⣿⡏⠀⠀⡃⢠⠇
⠀⠀⢐⣧⡏⣴⣆⢹⠀⠀⠀⠀⠀⠀⠳⣄⠙⠦⢄⣀⡠⠞⠀⣰⠃⠀⡿⠀⠀⢻⠀⠀⣿⠁⠀⠀⢃⠊
⢀⡴⠋⠹⣧⡘⠏⢸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⠤⠊⠁⠀⠀⠃⣄⠀⠘⠂⢠⠏⠀⠀⢠⡇
⢸⠀⠀⠀⢹⣿⣿⣾⡇⠀⡀⠠⠤⠤⠴⠒⠒⠒⠒⠂⠀⠈⠉⣉⡒⠦⢧⠈⠀⠀⣠⠞⠀⠀⠀⠈
⢻⠀⢁⡤⢰⣿⣿⣟⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠼⠌⠀⠀⠀⠙⠀⡴⠋
⢸⠀⡞⠰⣿⠋⢻⡟⠀⠈⠛⠲⠦⠤⢤⣄⣀⠀⢀⣀⣙⣛⣀⡀⠀⢀⡀⢴⠞⠀⠀⠀⠐⠂
⠸⣸⠁⠀⢻⡀⢸⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣹⣿⣿⠏⠉⠉⠁⠀⠈
⠰⡏⠀⠀⠘⡧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣈⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠉⣉⣩⡵⠞⠁
⣶⣶⣶⣶⢶⣶⣶⣶⣶⢶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⢰⣿⣛⣿⠃⠀⣀⡀⠀⠀⠀⠀⠈⠉⢻⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠛⠋⠀⢸⡏⢙⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠛⠈⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⣳⢉⡵⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⢀⣠⡿⠀⠀⠀⠀⠀⢀⣀⡀⠛⠃⠰⠏⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠹⢶⡾⠟⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡇⠀⠀⠀⢰⣶⠀⢠⣤⣤⣾⣽⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⠀⠀⠀⣾⡇⠀⠀⣻⡏⠀⠀⢈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢹⣿⣿⠀⠀⠀⢻⣿⡀⠰⠷⠿⣶⢶⡾⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠉⠛⠀⠘⣿⣿⠀⠀⠀⢸⣿⣷⣤⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⢀⣤⡀⠀⠀⢿⣿⠀⠀⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⢸⡟⠀⠀⠀⠸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣥⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣘⣿⣁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣾⡟⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡛⣷⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠹⣇⠿⢤⣸⠓⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣤⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⣿⡧⠀⠀⠈⢳⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠲⠷⠶⢤⣤⡔⣂⣀⣀⡐⠲⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢸⠀⠀⢰⣿⠀⠯⢭⣭⡭⠭⢽⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠿⢿⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡟⡇⠀⢸⣿⠀⠘⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡯⡇⠀⢸⣿⠀⠀⣏⡙⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢤⣇⣶⣾⠀⠀⠀⠀⢸⠀⠀⠘⠓⠀⠀⢸⣿⠇⠀⢸⣻⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣴⡏⣇⣾⢻⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⣼⣟⡁⣿⡞⠁⢰⠀⠀⠀⠀⠘⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⣀⡀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠃
⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢧⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡆⠀⠀⠈⢧⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢳⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⢸⡄
⠀⠀⠀⡀⠀⠀⠙⣆⠀⠀⠀⠙⢦⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠀⠀⣧
⠀⠀⠀⢠⠀⠀⠀⠈⢷⣄⠀⠀⠈⠳⣄⠀⠹⣄⠀⠀⠀⠀⠀⠐⡇⡆⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠉⠳⣤⡀⠀⠈⠑⠀⠈⠳⣦⡀⠀⠀⠀⣇⠸⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠈⠻⣦⣄⠀⢻⡀⠳⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠻⢿⣾⣧
⠀⠀⠀⢆⠀⠀⠀⠈⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢶⣤⣀
⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⢀⡾⠁⠀⠘⠟⠸⠿⣶
⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡻⣦⡀⠀⢀⡞⠁⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⢷⡔⠋⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⠀⣀⣀⣀⡀⠠⠤⠐⠊⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⡎⡧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⠶⠞⠋⠁⣀⡴⠂⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠖⠚⠉⠀⣀⡽⠖⠛⠉⢀⡠⢋⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠠⠴⠖⠋⠁⠀⠈⣩⠟⠉⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠶⠛⢁⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠠⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠱⡀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⣠⢴⣶⢀⣠⣤⣾⠛⠁
⠀⠈⠢⣤⣀⣀⣀⣀⣀⣠⣄⠤⠤⠐⣂⡤⠆⡀⠀⠀⠠⠤⠠⠤⠔⠚⠛⠉⠁⠀⠛
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣤⡶⢟⣩⠖⢋⠠
⠀⠀⠀⠀⠀⠂⠤⠤⠖⠛⠉⢀⡴⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣠⠏

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡔⠉⠀⠈⠑⣦⡖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⣬⣢⣀⠔⠢⣀⣞⣵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣶⡿
⠀⠀⠀⠀⠈⠒⠤⠔⠊⠁⠉⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠒⡺⠛⠱⣾⡷⠎⠻⣗⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣧⣤⣤⣤⡀⠀⠀⠚⠒⠒⢲⢲⡖⡖⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣼⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⣰⢟⡻⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⠐⣷⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣾⡦⠀⠀⣶⣷⣦⠀⠀⠀⠀⣿⡤⠀⣼⡇⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⣰⡟⠀⠁⢻⣦⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⢸⣿⢿⣷⡆⢶⣿⠿⣿⠀⠀⢰⣾⠿⠁⠀⠙⢿⣦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⢳⡆⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠻⠿⠛⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄
⠀⠀⠀⠀⢹⣧⣾⣷⣼⡟⠀⠀⠀⠀⠀⢀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣤⣶⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀⣤⣶⣿⣿⣿⣵⣶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡇⢸⣿⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠻⢿⣿⣿⠿⠋⡡⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⢸⢿⡇
⠀⣷⠈⣧⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣍⠲⣄⡀⠀⠘⣿⠁
⢠⢿⣆⡿⣷⠀⠀⠀⠀⠀⢸⡈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡷⡄⠙⢦⡀⠈
⡟⠀⠟⢠⣿⠀⠀⠀⠀⠀⠈⢷⡄⣿⠀⠀⠀⠀⠀⣠⠀⠀⣾⣿⡄⠀⢻⣆⠀⣀⠻⣦
⢧⠀⢠⣿⠋⠀⠀⣀⠀⠀⠀⠈⠻⠃⠀⢀⡀⠀⢀⣿⠀⠀⢸⠁⢸⠀⠈⣿⡆⠈⠻⣿⠳⡄
⠈⠳⡿⠁⠀⢀⣼⠃⠀⠀⣀⣠⣤⢶⡾⣿⡄⠀⠘⡟⣧⠀⠀⢣⡾⠀⠀⢻⣿⠀⠀⠈⢷⣽⡄
⠀⠀⠀⠀⣠⢣⢻⠀⠀⠀⠈⢳⡍⢁⠁⠘⡇⠀⠀⢻⡌⢳⡀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣰⠃⣾⣾⠀⠀⠀⠀⠀⠹⣾⠀⡀⠙⣄⠀⠀⠙⢦⣷⠀⠀⠀⠀⣿⡇⠀⠀⣄⠀⢹⣿⡇
⠀⠀⢰⠇⢠⢿⣿⠀⠀⠀⠀⠀⠀⠘⣧⡇⠀⢈⣳⣄⠀⠀⠹⠇⠀⠀⣸⣿⡇⠀⠀⠸⣷⣼⣿⡇
⠀⢀⣿⠀⠈⢸⣿⣧⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⠀⠉⠳⣄⠀⠀⠀⢠⠏⣿⠃⠀⠀⠀⣿⠻⢹⠃
⠀⢸⡇⠀⡄⠘⣿⢿⣦⡀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠈⢷⣀⣴⠏⠀⠋⠀⢰⡆⠀⣿⢠⡟⠀⠀⢠⡇
⠀⣿⡇⢰⡇⠀⢻⡘⢿⣷⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣻⡅⢀⡄⠀⣠⣿⡇⣼⣷⠏⠀⠀⢠⢛⡿
⠀⣿⣧⣾⡇⠀⠀⠃⠈⠻⣿⠻⣶⣤⣀⡀⢹⡄⠀⠀⠀⠀⠈⠙⠻⣿⣥⡾⢋⣺⠿⠛⠛⠢⣄⠀⢿⠞⠁
⠀⢸⡜⡏⢿⠀⠀⠀⠀⣀⠈⠣⠀⠙⠻⢿⡟⠷⠀⠀⠙⢦⡀⠀⠀⠈⠛⠻⠟⣶⣚⠓⢦⣀⠘⡆⠘
⠀⠀⢷⡀⠸⣧⡀⠀⠀⢹⣦⣀⠀⠈⠲⣤⣀⣀⡀⠀⠀⠀⠙⢷⣦⣄⣀⣀⣀⣈⡩⠴⠒⠋⢁⡇
⠀⠀⠈⠻⣆⠙⣷⣄⡀⠈⢿⡻⢷⣦⣄⣈⣙⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠫⣄⡀⠀⣀⡠⠞
⠀⠀⠀⠀⠈⠻⢮⣿⣿⣶⣬⣿⣶⠄⢉⣩⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁⠀⣀⣀⣀⣘⣛⠻⣍⠳⣄⠀⢀⣀⡠⢾⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣭⣽⠿⣂⡬⠿⠘⡆⣬⣽⣶⣭⣥⡥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠙⢿⣧⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣟⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠌⠈⡇⢨⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⢸⣀⣠⠤⠔⠂⠁⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣀⣀⠻⠋⠉⠰⠀⠁⠀⠀⠀⠀⠀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⢻⠉⠁⠸⠁⠀⠀⢠⠀⠄⣀⣠⣤⣴⣶⣶⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣾⡏⠹⢐⡇⠃⠘⡄⠀⠘⠀⠀⣴⣷⣾⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠇⢹⣟⣺⣷⣴⣾⣿⣷⠀⠀⠀⢹⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⡿⠿⠟⠛⠋⣟⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⢧⠤⠒⠒⠼⠤⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠶⣿⣷⣶⣀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠾⣍⠻⣿⣧⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⠾⢦⡇⣿⣿⡅⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⣶⣶⠆⡿⣿⣷⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢾⣿⠃⢸⡇⢹⣿⣿⣷⠀⠀⠁⢻⡗⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡄⠀⠅⢸⣿⠛⢯⡀⠀⠀⢸⣿⡨⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⢸⣿⣦⣄⠑⢄⣦⡞⣿⡇⡀⠀⠀⠀⠗⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢘⠀⠈⢹⣷⣦⣽⠇⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⡟⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⢛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣹⠟⠀⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣟⣻⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠧⠽⠈⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢵⣿⠀⠀⣿⠀⢿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣌⣛⣀⣀⣛⣃⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡏⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⢰⣯⣿⣦⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢒⡭⠚⣿⡀⠘⠾⠽⠋⠀⣿⡓⢯⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡮⠕⠊⠁⠀⠀⣹⣧⣦⡀⢀⣴⣄⢿⡇⠀⠈⠑⠺⢵⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⡵⠚⠁⠀⢀⡀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣸⡇⠀⠀⢀⡀⠀⠈⠓⠮⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⡅⠀⡴⢻⣿⣯⠳⡄⠀⡏⣿⣿⡇⢸⣿⣿⢹⡇⢠⠞⣹⣾⡝⢦⡀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡇⢸⢁⣀⠉⢁⡀⢸⠀⢧⡟⠉⢃⡘⠉⠛⣿⡇⡇⢀⡈⠉⣀⡀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⢸⣾⣿⣧⣿⣽⢸⠀⣼⠇⣴⠋⡏⢦⠀⣿⡇⡇⣿⣿⣴⣯⣷⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣷⣿⣦⣾⣿⣼⠀⢿⠠⡇⢰⡇⢸⡆⣾⡇⡇⣷⣗⣽⣻⣆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣺⡇⢸⣿⣿⡟⣿⣽⢻⢰⣿⡶⡇⢸⡇⢸⢷⣿⡧⡇⣯⣯⣿⢽⡏⡇⢰⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠸⠽⠿⠥⠿⠯⠼⠀⡛⣶⡇⢸⡇⢸⡆⣻⡇⠧⠿⠿⠭⠿⠯⠇⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⣤⣧⣤⡤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⢤⣤⣴⡧⢧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡿⠛⠃
⠀⠀⢠⠄⡤⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⢀⡀
⠀⢸⢸⡆⣧⢸⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⢸⡇⡇⣸⠀⠀⠀⠀⠀⠀⣾⣸⣿⣷⣶⠀⠀⠀⣿⡇⠀⠀⠀⣶⣿⣼⣷⡇
⠀⣸⡎⣇⢻⠸⡄⠀⠀⠀⠿⠃⠀⠀⠀⡟⣸⢰⡇⣏⡀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡄⠀⠀⠻⠇⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠹⡄⠀⠤⠤⠤⠴⠦⢄⠀⡠⠤⠶⠤⠤⠤⠈⢀⠜⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣒⡢⡄⡠⢖⣒⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠈⠰⣿⣿⣿⣿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡄⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠀⠀⢱⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⢀⠀⠀⠀⢐⠃⠈⠀⡀⠐⢂⠒⢂⣀⡀
⠀⠀⠀⠀⠀⠀⣴⢢⣨⣓⡩⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⡀⠀⠀⠀⠀⠁⢹⡷⢤⣀⡟⢀⣶⣴⣮⢖⡋⠴⠞⠋⠁
⠀⠀⠀⠀⢀⣼⣥⣾⣉⣥⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡿⠊⠉⠉⠋⠀⡀⠀⠈⢀⣤⡀⡿⠛⠛⠛⠻⠾⢽
⠀⠀⠀⠉⠀⠀⠀⠀⣠⣤⣤⠀⠀⢸⠀⠀⠀⠀⣠⡤⢀⡀⠀⠀⠀⠐⣦⠈⠲⠂⢱⣯⡅⠁⠀⠚⠷⣶⣤⣈⡤⠴⠤
⠀⠀⠀⠀⠀⠀⢧⣠⣼⡏⢸⡰⣤⣤⡄⠀⢠⣤⡜⠀⢸⣇⣀⡀⠀⠀⠈⠳⠒⠚⠛⠉⠻⣦⣄⡀⣼⠏⡁⠉⠀⠀⠀⠀⠀⠠⠉⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠱⠘⠺⠇⢀⡤⠊⠔⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⠙⠛⠧⣄⣀⣀⠈⠉⠛⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡄⠰⡀⠀⠀⠠⠤⠔⢊⢆⠞⠀⢰⠀⣀⠀⠀⢀⣎⡗⠐⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋
⠀⠀⠀⢀⣄⡄⠀⠀⡇⠁⠻⣄⡈⠲⠤⠄⠠⠼⣫⠞⢸⡆⠈⣇⠉⠀⠠⠞⠾⣇⠀⢰
⠀⠀⠀⠼⢉⠃⠀⠀⡧⠀⠀⡏⠃⢰⣶⣶⡆⠀⣏⡀⣸⠁⠀⠘⣆⢠⢠⠀⢠⡙⢧⣤⠀⠀⠀⣀⡤
⠀⠀⠀⠀⣾⡆⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠈⠁⠀⠀⠀⠀⠀⠙⠳⢴⢷⣿⠶⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⣠⢀⣄⠀⣀⣄⣢⡄
⠀⠀⠀⠀⠞⡉⠂⢧⡀⢄⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠞⢁⡉⠢⡽⠂⡄⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢤⣉⠘⢻⠿⠁⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠂⠢⠀⢂⣸⠃
⠀⠀⠀⠀⠐⠛⠛⠶⠁⠀⡥⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡤⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠹⢦⣬⣤⡴⠃
⠀⠀⠀⠀⠀⡀⠀⡘⣦⣼⠃⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡀⠀⠀⠀⠀⠀⠘⣿⠾⠃
⠀⠀⠀⠀⠀⡁⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠁⠀⠀⢀⣘⢾⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢋⣥⣴⡀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⣴⡁⡐⢤⣤⣿⡷⠄⠀⢸⠀⠀⠀⠀⠀⣴⠟⡛⠲⠿⢿⣿⣷⣄⠀⢀⡠⠊⠁⠒⣈⣉⣉⣙⣃⠀⠀⠀⠀⢖⡫⠘⣖⠂
⠀⠀⠀⠀⠁⠛⠛⢃⣬⠏⠀⠀⠀⢸⠀⠀⡆⠀⣰⣿⣷⣿⣭⣭⠭⢽⣶⢭⣵⣆⣤⣴⣶⣿⣿⣿⣷⣾⣿⣧⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠀⣤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣽⠀⢿⡤⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡼⠹⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢰⠁⠀⢰⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⠃⠀⠀⠀⣦⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⠃⣀⣀⡀⠀⠀⡆⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀⠁⠀⠒⠂⠀⠀⠋⠀⠀⢳⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⠁⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⣦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣰⠁⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠸⠇⠀⠲⠖⠒⠐⠒⠒⠒⠲⠶⠶⠃⠀⠰⠆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⡷⠀⠤⠤⠴⠦⠤⠤⠦⠤⠤⠴⠆⠠⠶⠶⠾⣀⣸⠧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣀⠠⢠⡄⢀⣀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡋⠉⠉⠉⠀⠀⠀⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣉⣉⣉⣉⣀⣁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⣾⠁⠀⠀⠀⣨⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⡀⠀⢀⡀⢁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⠐⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⠶⠋⠉⡄⠑⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⣠⡴⠋⠀⠀⠀⢀⡇⠀⠈⠛⢤⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠊⠉⠀⠀⠀⢀⡴⠞⠻⡄⠀⠀⠀⠙⢦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀⡰⣿⠛⠋⠉⠉⢻⠶⢆⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⢀⠏⠀⠉⠀⠀⠀⢰⡆⠀⠀⢳⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢠⠀⠀⠀⠟⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡠⠀⠀⠀⣴⠚⠻⢆⡰⠏⠀⠀⢸⠶⡀⠀⠀⠃⠀⠀⠇⠀⠀⢀⣴⣿⠀⡆⠀⠀⣀⡀⠙⢆⡀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠘⣦⣠⠴⡟⠀⠀⠀⠀⠛⠠⠖⠋⠁⠀⠀⠀⠈⠀⠹⣄⠀⢸⡆⣨⠀⢀⡴⠋⠀⠀⠀⠹⢀⠘⠋⣁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣤⣧⣇⡴⠋⠀⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠀⠀⠀⠙⣤⡀⠀⠀⡰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡆⠙⠋⠙⠑⡆⠀⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠤⠦⠄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀⢀⣀⣀⣀⠀⠀⢸⡏⠀⠀⠈⠉⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⡾⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢹⡇⠀⢸⠃⠀⠀⠀⠀⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⡄⣤⠀⣿⠀⢀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⣯⠐⢯⣤⠃⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠆⠀⠀⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⠀⠀⠀⠐⣷⣄⣸⡇⠀⢠⣦⣀⡀⠀⠒⠒⠒⠒⠒⠂⠘⣿⠀⠐⠒⠒⠒⠒⠒⠛⠶⠖⠀⠀⣠⣄⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⠀⠤⠀⠉⠁⠉⠉⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠛⠋⠀⠀⠠⠶⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠛⠉⢹⡆⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⣤⠉⠑⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣴⠊⢳⠀⠀⠉⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⣤⢀⣈⣄⠀⣼⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠋⢀⡴⠞⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢰⡏⠀⢀⡀⠴⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠓⠄⠺⢆⡀⡨⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠘⠛⡆⠀⠀⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠘⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢄⣀

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⢠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢹⣿⣿⣆⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡋⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡩⢤⢥⣬⠄
⢠⠄⢶⣦⡐⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠖⢒⣀⣴⣆⡀⢀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡖⣒⣶⣶⢲
⢸⣀⣈⠛⠛⣳⣤⣈⣀⢒⡠⠤⢀⣀⣀⣀⣀⣰⣽⣿⢿⣛⣷⡈⢀⣀⣀⣀⣀⣀⣀⠀⠀⠤⢊⣁⣤⣥⣤⣴⣾⣿⣿⣟⣼
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣷⣎⠉⠙⠛⠛⠻⣧⣀⠲⣠⡿⠛⠛⠻⠊⠉⢉⣩⣀⣠⠤⠖⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠿⣆⠨⡟⠿⢯⡇⠘⠀⠀⢠⣴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⡀⠀⠤⠶⢚⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⣁⡀⢠⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠀⠀⠈⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⢢⣄⡀⠀⠀⠀⠀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⡑⢤⠤⠂⠊⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⢍⠻⠲⠆⢚⡴⠖⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⢦⣀⣌⠽⠒⠉⣼⡇⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⡿⠁⠀⠀⣴⣿⣳⣺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠻⠁⠀⢀⣴⠻⣿⠟⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢻⣤⠀⢭⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠙⣆⢚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠄⢸⣼⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⢱⠀⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠮⢠⡏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⣿⠇⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠙⠿⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⡠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⠄⠀⠀⠀⠀⢀⠎⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠐⡀⠀⠀⡔⠁⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⡠⠊⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣔⠁⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣻⣧⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠯⠞⣷⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢚⡛⢿⢴⣯⣿⣧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡇⠈⢄⠰⣿⣿⣇⡨⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢹⡇⠀⢀⠠⠀⣷⣿⢻⠀⡆⠄⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣿⠃⠀⠸⢀⡾⠁⢉⢀⢠⣿⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⣸⣿⠀⠀⠀⠘⠁⠀⠈⡀⣾⣿⣷⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⣼⡿⠋⠀⢀⠀⠄⠁⠐⠀⢀⢹⣿⣿⣷⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⣠⣿⠟⠀⡆⠀⣼⠀⠀⠀⢦⠀⠀⠢⣹⣿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣼⡿⠃⠀⠀⠀⠠⢻⡀⠀⠀⠈⠀⠀⠀⣍⢻⣿⣿⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⢸⣇⠘⡀⣀⣨⡶⣾⠟⠄⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⣼⡿⠁⠀⠀⠀⠀⠄⢀⠀⣸⣿⣿⣿⡿⠁⣸⣿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠄⣰⡿⠁⢠⡿⠁⢰⠃⢿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠠⠁⠀⢠⡟⠀⢀⡿⠁⢠⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢀⠂⠀⣠⠋⠀⠀⡼⠁⠀⠺⣿⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡜⠁⠀⠀⡘⠁⠀⡔⠀⣿⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠊⠀⠀⠀⠄⠀⠀⣸⡇⠀⠘⡀⣤⣬⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⣠⡮⠤⠠⠔⠉⠁⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⣀⣠⣴⣶⣿⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⣀⣤⣶⣿⡿⢿⣿⣿⠁
⢠⣴⣿⡿⠟⠋⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⢿⡟⠁⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣾⡿⠉⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⠁⠀⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣠⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠰⠶⠶⣿⣿⣤⣤⠀⠀⣸⣿⡇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣿⣿⠇⣴⠟
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠉⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⣴⣾⣿⠿⠋⢹⣿⡇⠀⠀⢀⣴⣿⣿⠿⠿⢿⡆⠀⠀⣼⣿⠃⠀⠀⠀⣿⣿⠇⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠉⠋⠁⠀⠀⣿⣿⡇⠀⠀⣾⡿⠋⠀⠀⢠⡆⠀⠀⢸⣿⡟⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠸⣿⣿⠋⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⠁⠀⠀⣰⣿⠃⠀⢀⣾⣿⡇⠀⠀⠀⣸⣿⡇⠀⠀⠀⢀⣼⣿⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢿⣦⣴⣾⢿⣧⣠⣴⠏⠹⣿⣧⣤⣴⠞⠉⢻⣷⡀⣀⣠⠞⠁⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠙⠉⠀⠈⠛⠋⠁⠀⠀⠉⠋⠉⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠛⠃⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿
⣿⣿⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢢⠄⠀⠒⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠙⢏⠉⠉⠀⠀⠀⠀⡀⠄⡤⠄⠀⠐⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠠⡀⠀⠓⠄⡀⠀⠈⢳⡄⠠⠄⠀⡘⠁⣾⠆⠀⠀⢀⣀⣤⠞⣻⣏⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠑⠠⡄⠙⠄⠀⠀⠋⢰⡋⠀⠀⢀⣀⣶⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢳⣄⠀⠀⠉⠘⣲⣦⠀⠊⣥⢤⣴⣾⣭⣴⣾⡿⠛⠩⠿⣟⠻⣽⣿⣿⣿⡀⠀⢸⠀⠀⠀⠀⠀⢀⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠿⣦⡀⠀⢰⣻⣿⣿⣿⣶⣿⢟⣡⠌⠻⣿⣄⣀⣀⡀⠈⢳⣾⡞⢿⣿⠇⠀⢸⠀⠀⠀⣠⠞⠉⠀⡜⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠙⠻⣶⣿⣿⠷⢿⣷⣀⠀⠙⢮⣭⣿⣿⣦⣼⣿⣷⣤⣤⣄⠀⣻⣧⣴⠟⠁⠀⠀⠀⠀⣠⠾⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣟⣿⡗⢠⣽⡿⠋⠀⠀⠀⣠⠊⢀⣴⠟⣡⠞⢁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣿⢿⣿⣶⣁⣀⠀⠀⠀⠰⣿⣿⣿⡟⣹⠛⠛⣻⡿⠋⠀⠀⠀⢀⣾⠁⠀⠈⣠⡾⢃⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣟⠉⠁⠀⣦⣴⣿⠈⣮⡛⢌⣻⣦⣭⣀⡀⠀⠀⠀⣼⣿⠀⣠⡾⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣿⢛⠻⣿⡐⣀⢻⣼⣿⣛⠛⠉⠙⠛⢻⣿⣿⠉⡜⣦⠀⢠⣿⣿⠟⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡻⡆⢻⣿⣿⣷⣼⠟⠋⠁⠀⠀⠀⠀⢰⣟⣿⡦⢳⡟⠀⣾⠟⢁⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣾⢰⡿⠃⢸⡧⠒⠀⠀⠀⠀⠀⠀⣿⣿⡿⢡⡟⣠⠞⠁⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣼⡀⢈⠿⠓⠋⠀⠀⠀⠀⠀⠀⢿⣼⢀⣿⣷⠋⠀⣸⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠛⠋⠻⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡏⠀⢠⠇⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠁⠘⠛⠛⡋⠁⠀⠒⠂⠀⠰⠶⠟⠃⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⣯⠀⠀⠀⠓⠤⣀⣀⣰⠦⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡀⠀⠀⢶⡀⠀⠙⠶⣄⠀⠀⠀⠀⢀⣠⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⢇⠀⠀⢠⡉⠐⠢⡀⠀⠑⠒⠒⠒⠋⠁⢠⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠸⡀⠀⠈⠻⠄⠀⠈⠀⠄⠀⢀⡀⠀⢠⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠁⠀⡄⠰⣄⠀⢄⡀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢰⠀⢇⠀⠙⠶⢤⣍⠁⠀⠀⠀⣾⣿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣸⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⡇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⢠⡟⠰⣿⠀⠀⠀⠀⠀⢀⣀⣀⠈⣿⣿⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⣠⣬⡿⣄⠀⠸⠀⠀⢀⡠⠞⣩⣤⣿⣷⣿⣿⣿⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣴⠟⠋⢉⣩⣽⠟⠀⠀⢙⣋⠁⠲⡾⠿⠿⠛⠛⢿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠞⣁⣴⢞⣿⣷⣤⣖⣒⣿⠟⠛⠉⠁⣠⡶⠋⠀⢀⡀⠻⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⠿⠶⣉⠛⠷⣤⠍⠉⠀⠀⣀⣴⣾⣧⠶⠛⠻⣿⣿⡆⢸⡗⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⢷⣄⠈⠛⠶⣤⣀⣠⣴⠾⠛⢋⣁⣠⣄⡀⠀⠈⢿⠇⣾⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣙⣷⣄⣀⣀⣉⣉⣤⣶⣾⣛⣉⣁⣈⣙⣷⣦⣀⣀⣱⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣹⣏⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣐⣂⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⠀⠒⠒⠒⠒⠒⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⡶⠶⣦⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠄⡀⠈⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠀⠀⣠⠶⠛⠋⠁⠀⣀⠀⠉⠛⠳⡦⠀⡀⠀⢠⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⢴⠖⠛⠉⢁⠀⠀⠉⠀⠈⡇⠀⠀⠀⢠⠊⠀⠁⠀⠀⡀⠀⡄⠀⠁⠀⠈⠀⠀⠘⠀⠈⠉⡟
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢰⠀⠀⡆⠀⠀⠀⡄⠀⠐⠀⠂⠀⢰⠀⡇⠀⠇⠀⠰⠂⢠⠁⠀⠀⠀⡇
⠀⠀⠂⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠒⠒⠲⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⠶⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⡂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⡛⢛⡟⣡⣾⣿⣮⡢⡛⢛⡲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠈⢿⣿⣿⣿⣿⣿⠃⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡌⣿⣿⣿⣿⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢹⠋⠉⢻⡀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⡛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣄⠻⠶⠤⠤⢤⣤⣤⣄⣀⣠⣿⣿⢸⠀⠀⢸⠀⠹⣦⣀⣀⣀⣠⣤⠴⣶⣿⠿⢊⡅⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢟⣵⣿⣿⣷⣶⣤⠤⠤⠶⠦⠬⠵⠿⠯⠼⠀⠀⠘⠦⠤⠤⠭⠭⠥⠤⠤⢬⣽⣶⣿⣿⣷⣅⢝⢷⣄
⠀⠀⠀⠀⠀⠀⢾⡋⢄⠱⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡷⠱⣸⠗
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣙⣿⡿⠟⣛⣭⣭⣭⣭⣭⣭⣭⣭⣍⢻⠀⠀⢰⢋⣭⣭⣭⣭⣭⣭⣭⣭⣭⣛⡻⢿⡟⡁⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡜⠩⢚⣯⠿⠛⠋⠉⠉⠉⠉⠛⣯⣿⢸⠀⠀⢸⢸⣯⠞⠋⠉⠉⠉⠉⠉⠙⠻⣝⣷⡄⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⢸⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣾⣄⣀⣸⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢃⣿⣿⣿⣿⡔⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⢁⣼⣿⣿⣿⣿⣷⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣼⣧⣬⣍⢿⣿⣿⡟⢭⣭⣴⣂⣽⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣷⣝⡋⢀⡼⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣿⠀⠀⠀⣶⠶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⣿⠀⠀⢠⠏⠀⠀⠉⠙⠛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁⠀⠀⢠⡿⠀⣠⠏⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠛⠛⠉⠁⠀⠀⠀⠀⠀⣼⡷⠞⠁⠀⠀⣀⡴⠚⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣖⣋⣁⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠛⠛⠛⠋⠉⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⣤⣀⣤⣠⡀⡀⠀⠀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠶⠾⣿⠿⡟⣿⣿⣿⣀⠞⠁⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⠁⡾⠴⠾⢯⡅⠀⢠⠀⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣰⡇⠀⢀⣾⡆⠀⢰⡀⠙⠿⣦⣤⣀⣠⣤⠶⢋⡁
⠀⠀⠀⠀⠀⣠⢾⣿⠁⣼⡿⠀⠀⡼⣼⡇⠀⣿⣧⠀⠀⢀⡌⠉⠀⣄⠀⠀⣷⣤
⠀⠀⠀⢀⢞⡕⣿⠇⡜⡽⡇⠀⣸⣹⢻⡇⢠⣿⢻⡀⠀⢸⣧⠀⠀⡿⣄⠀⢸⣹⣣⡀
⠀⠀⢀⠏⣯⣷⠏⢸⢻⣿⠃⠀⡇⣏⣾⠇⢸⡸⣦⠃⠀⣏⢿⠄⠀⡇⡟⡆⠈⢿⠿⢳
⠀⠀⠀⠓⠚⠁⠀⠘⠒⠋⠀⠀⠙⠒⠋⠀⠀⠣⠤⠃⢸⢸⣾⠀⠀⡇⢇⣧⠀⠈⠑⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠶⢠⣶⣿⣿⡿⠿⠿⠓⠲⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠐⡽⣱⡞⣿⣿⣿⣛⣛⣛⣿⣿⠲⣾⣿⣷⡀⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⢰⡇⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻⣿⣿⣧⣌⡻⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡂⢸⠏⣿⣿⣿⣿⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣧⡟⣻⣾⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣶⣿⣿⣿⡇⡷⢸⣷⠈⠻⢿⣶⣾⡏⣴⣿⣿⣿⣿⣿⣿⣿⣿⠣⣻⣿⣿⣿⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣻⣿⣿⣧⣙⠦⠈⣴⣿⣷⣮⡌⢉⣷⠿⠟⣿⠿⣿⣿⣿⣿⣦⣿⣹⣿⣿⣿⠻⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣭⣭⣥⣤⣤⣴⣿⣿⣿⣿⢹⣿⣷⠠⣌⣙⠛⠿⠃⠸⣿⣿⣶⣤⣄⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡆⢧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣈⣛⣿⣭⣿⣿⣿⠿⣻⣾⣿⣿⡄⡭⢽⣃⠓⠚⢶⣬⡻⡿⢿⣷⢹⣿⣿⣿⣡⣿⣾⣿⣾⠋⠖⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢿⣿⣿⣽⣮⣭⣭⠶⠚⣿⣿⣟⣯⣤⣶⣎⣻⠿⣦⡲⣍⠃⢶⣿⣿⣿⠿⣿⣽⣿⣷⡿⢟⡉⠃⢈⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠹⣿⣿⣿⡿⣿⣷⣶⣽⣯⣭⣴⠿⠛⠛⠛⢓⣈⣥⣬⣤⠰⣮⣉⣩⣅⠙⠻⢟⣡⣾⡟⣡⡖⣸⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠙⢷⣄⣩⣵⣾⣿⠿⢿⣿⡟⠋⠁⠠⣤⣤⣀⡉⠁⠀⠨⠉⠁⢠⢤⣬⣍⡀⡹⠧⠛⣛⣉⣫⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⡏⠁⠀⠀⠀⠠⣌⠉⠻⢦⣄⡀⠀⠀⠈⠀⠉⠉⢣⠉⠛⢻⣿⣿⣿⣿⠃⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⢰⣿⣿⡧⠀⠀⠀⠀⠀⠈⠛⠶⠒⠏⠛⠓⠀⠀⠀⣀⣀⣠⣤⣤⣼⣿⣿⣿⡏⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣉⣀⡠⣠⣿⣿⣿⣿⡿⣷⣿⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠉⣴⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣾⡁⣀⣤⣾⣿⣿⣿⣿⣿⣿⡇⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡾⠿⠿⣿⣷⣷⣿⣄⢀⣾⣿⣿⣿⡿⣿⣿⢿⡽⣹⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠠⣤⣈⠙⡛⢿⣿⢿⣿⣿⠏⣼⣿⣷⠟⣴⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡄⠀⠀⣿⣿⠛⣩⡟⠁⠘⣿⡏⣧⣿⣿⣏⣾⠟⣰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣶⣿⣯⡴⠋⠀⠀⢰⣿⣧⢻⣿⣿⣿⣏⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠉⠉⠀⠀⠀⠀⣿⣿⣿⢸⣿⡟⣹⣿⣿⣿⣿⠏⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣹⣿⡿⢸⣟⣿⣿⢟⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⣿⡟⢠⣿⣿⣿⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠟⢀⣿⢿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣾⠀⢸⣿⡾⣾⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠈⠻⠽⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇
⠀⠀⠀⠀⢠⣄⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⠁
⠀⠀⠀⠀⠈⢁⡙⠛⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡛
⠀⠀⣤⠐⠻⠿⣿⣷⡦⢤⣤⠀⠀⠀⠀⠀⠀⣀⠠⢤⣴⣶⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⢀⣁⣤⠄
⠀⢀⣼⣴⣶⡄⣀⠈⠻⣏⡀⠀⢀⡀⠂⠉⢁⣤⣖⣋⣩⠄⠀⣿⣿⡇⠸⠀⠀⠐⢤⣾⣟⠋⣽⠷⠉
⠀⠈⠉⣹⣵⣄⠄⣭⢀⡈⢉⣰⣾⣶⣶⣿⣿⣿⣿⣿⡟⠀⠀⣼⣯⡄⢰⡄⠓⠒⠛⣋⡉⣿⠨⠟⢲⣮⣿
⠀⠀⡽⢿⣿⣿⣿⣿⠁⢠⣆⣿⣿⣿⢸⣿⣿⡇⢶⣿⡇⠀⣤⢸⡿⠇⢸⠀⢸⠀⣾⣇⣸⣧⣤⣌⣺⣿⣿⣀
⠀⢶⣷⣾⣿⣿⣿⢋⣰⣿⣿⡿⠟⣻⣿⡿⣿⣿⣿⣿⣧⠀⣉⣼⣿⣷⡘⠶⠌⣴⣝⢿⣿⣟⣟⣿⣿⣿⣿⣿⡇
⠀⣭⣭⣭⣥⣿⣿⡿⠏⠉⠉⠈⠛⠛⠂⠒⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⡀⣀⣀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⡀⠀⢀⡀⠀⣀⣀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⢀⡛⠀⠈⠁⠈⠁⡉⠉⠉⠉⠉⠉⠃⠀⠈⠉⠀⡀⠉⠁⠀⠈⠉⡉⠉⠉⠀⠈⠉⠃⡀⠉⠈⠉⠉⠉⠉⠉⠁⠉⠉⠀⠉⠉⠉⠉⠉⢋⠀⣀⠘⡆
⠀⠀⣄⠀⠰⡶⠠⠒⠛⠐⠒⠚⠓⠚⠛⠛⠒⠐⠚⠑⠚⠷⠊⠒⠚⠋⠀⠀⠔⠋⠚⠒⠛⠐⠛⠂⠁⠈⠚⠋⠓⠓⠚⠓⠒⠋⠙⠒⠀⠚⣿⡇⢸⠇
⠀⠀⡍⠀⠠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⡇
⠀⠀⢧⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢉⣤⣁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠤⡄
⠀⠀⠆⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁⠂⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠟⢠⡟⢚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡅
⠀⠀⡗⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⢇⡄⠠⠀⠈⠂⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢘⡀⢤⡄⠋⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⡇
⠀⠀⣃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⡠⠁⢤⠀⢀⠈⠀⡀⡄⠀⠀⠐⠁⠈⢔⠗⠥⠀⠀⠐⠖⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⣾⠀⠠⡁
⠀⠀⣿⠂⢠⡇⠀⠀⠀⠀⠀⠀⠘⣶⡀⠀⠀⠄⢁⠀⠀⠈⡄⠀⠱⠁⠀⠀⠙⡆⠀⠀⠀⠀⠀⠐⠑⠦⠄⠀⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⣿⡂⢀⡆
⠀⠀⡴⠀⠈⡇⠀⠀⠀⠀⣄⠀⠀⠘⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢿⠄⠀⡁
⠀⠀⢯⠀⠐⡇⠀⠀⠀⠀⠋⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⣏⠀⠐⠇
⠀⠀⣾⠂⣈⡝⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢀⡇
⠀⠀⣿⠀⠀⢱⠀⠀⠀⠀⢀⠌⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣉⢙⠧⡉⣉⠈⠃⠀⠐⠂⠀⠀⠀⠀⠀⠈⠀⠀⣘⠀⠀⠀⠀⠀⠀⢿⠄⠀⡀
⠀⠀⠟⠀⠐⡏⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⡇⠀⢁⣎⠁⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⣮⠀⠀⠃
⠀⠀⡟⠀⣀⡅⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠊⢀⠀⠈⠙⠀⠀⠀⣀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⣳⠆⠀⡇
⠀⠀⣷⠀⠘⡇⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠦⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⣿⡃⠈⠃
⠀⠀⡿⠀⠲⡇⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠶⠀⡀⠀⢃⠀⠀⠀⠄⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡏⠀⢀⡆
⠀⠀⣸⠦⢀⡼⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠃⠄⣠⣅⡠⠂⡸⠂⢀⢠⣀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣧⠀⠈⠃
⠀⠀⢹⡀⠈⣿⠀⠀⠀⠌⢀⠀⢁⡀⠀⠀⠸⣇⠀⠀⠐⠄⢠⢊⣵⡎⢻⡔⢧⠞⢁⣴⢦⡐⡄⢂⢀⣠⡏⠀⠀⠊⠀⠀⠀⣀⠁⠀⠀⠀⡿⠀⠰⡄
⠀⠀⣾⠁⠰⡞⠀⠀⢀⠄⠁⠀⠀⠁⠀⠀⠀⠛⢿⠛⣿⣳⡆⠛⠀⠈⣧⢷⡟⠠⠎⠀⠀⠈⠠⠮⡍⠛⠁⠀⠀⠀⡀⠀⠐⠃⠀⠀⠀⠀⣿⠀⢀⡇
⠀⠀⣰⡀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢴⠷⠾⡃⠚⠂⠤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠨⡇
⠀⠀⢻⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢁⠰⠆⠈⠀⢠⡄⠈⠀⠠⠦⢤⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠠⡇
⠀⠀⡜⠑⠤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠉⠂⠭⠩⠀⠀⠆⣠⠀⠀⢀⡀⠀⠀⠐⠐⠀⠁⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡃
⠀⠀⠰⠄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⡋⠑⠆⡀⡀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⢀⠒⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠊⡇
⠀⠀⢧⠀⠈⢻⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠘⠤⡀⠠⡄⠀⠈⠆⠀⠀⠂⠑⡀⠀⠀⠂⠰⠀⠀⠀⠊⠀⠂⠀⡀⠁⠀⠀⠀⠀⠀⠀⢀⡋⠀⠠⠃
⠀⠀⣞⠂⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⡇
⠀⠀⣱⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⡇
⠀⢠⣿⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁⠀⡠⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⢘⠇
⠀⠀⣿⠄⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⣁⢲⣄⢰⣀⢀⢠⡀⣀⣠⢦⡄⠘⠋⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⠂⢨⡄
⠀⢀⣽⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡴⠞⠺⢥⠟⠷⢻⡏⢻⠿⠳⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⢘⡄
⠀⠘⣳⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⢸⡃
⠀⠰⣝⠃⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠩⠩⠍⠯⠍⢠⠄⠱⠀⠀⠀⠤⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⡆
⠀⢸⡧⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⠁
⠀⢸⠁⠀⣲⣿⠟⠛⠓⠛⠓⠐⠒⠛⠛⠘⠉⠙⠓⠲⠗⠒⠒⠏⠀⠒⠚⠛⠓⠐⡛⠙⠓⢾⠁⠀⠐⠃⠀⠐⠋⠉⠈⠓⡉⠀⠒⠆⠀⠚⢷⡆⠀⢁
⠀⠨⠁⠘⠏⠈⠙⠦⣀⠀⠤⠤⠤⠔⠂⠤⠄⠦⠄⣀⠰⠀⠀⠀⠆⠀⠀⠲⠤⠄⠈⠦⠄⠠⠦⠀⠀⠀⠦⠤⠀⠴⠤⣀⠶⠤⢄⣼⠦⠀⠈⠳⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⢀⣠⠤⠖⠚⠋⠉⠉⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣀⣾⢸⡇⠀⠀⢀⡀⠀⠀⢀⣾⡿⠁⢀⡇
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣀⣠⡤⠤⢬⣿⣼⣹⣻⢸⠃⠀⢠⣿⡇⠀⢀⣾⡿⠁⢀⠞
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣁⡀⠀⠀⢸⡟⠁⠀⢀⠀⠀⠈⠻⣿⠟⠓⢲⣿⣿⣄⣀⣼⣿⠥⠚⠁
⣧⠀⠀⠀⠀⢀⣀⠤⠴⠒⣻⣿⠏⠉⢻⡆⢀⣴⡶⢢⣰⣿⠃⠀⢀⣴⡏⠀⠀⣼⣿⠁⠀⢠⣿⢇⣴⣶⡆
⠙⢦⡀⠀⠀⠉⢉⠉⠓⢦⣿⡟⠀⠀⠀⠀⣾⡟⠀⢸⣿⡟⠀⠀⣸⣿⠇⠀⠀⣿⡏⠀⠀⣼⣿⡞⣸⣿⠃
⠀⠀⠙⠳⠶⡶⠟⠀⠀⢸⣿⢧⠀⠀⠀⠀⣿⡇⣠⢻⣿⠃⠀⢠⣿⡿⠀⠀⣼⣿⠀⠀⢠⣿⡟⠀⣿⡟⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⡆⠀⠀⠀⠈⠉⠁⢸⣿⠀⡰⠃⠹⣧⣠⠞⠘⣿⠀⡰⠃⠙⠁⠀⣿⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠈⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⠀⠄⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⡜⢠⠀⠀⠊⠀⣴⠀⠀⠀⠀⠀⠀⠈⢡⣿⢷⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⡘⢠⠏⢀⠤⢤⡾⠋⠀⠀⠀⠀⠀⠀⠐⣻⣿⢻⣆⠀⠀⠀⠀⡇⠀⠀⢾⣡⣄
⠀⠀⠀⠀⢰⢀⡿⠀⠇⢠⡾⠃⠀⠀⠀⠀⠀⢀⠀⠐⠀⢿⠸⠙⢷⣄⠀⠀⠸⣦⠶⠀⠋⠙⣆
⠀⠀⠀⠀⡎⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢧⠢⠙⢧⡀⠀⠈⢶⡙⡀⠀⠈
⠀⠀⠀⠀⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢀⣿⠀⢘⡇⠀⡀⣨⠷⣤⠈⠙⠃⠀⠐⡀
⠀⠀⠀⠀⠁⣾⣿⡇⠀⠀⠀⠀⡀⠰⢿⣤⣦⠀⢀⠀⢼⣿⡆⡌⠻⣿⣿⣷⡀⢸⡇⠀⠀⠈⣿⡗
⠀⠀⠀⠀⠁⠿⠛⠁⠀⢀⠀⢀⠀⠀⠀⢙⠟⠀⣸⠙⣿⣿⣇⠀⠒⠛⠻⣿⣷⣼⡇⠀⠀⠀⠛⠃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢈⣧⣏⠀⠀⠉⠛⠀⠀⢆⠀⠈⢋⣿⡇⢀⣵⡄⢸⣿⣿⡟
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⢸⣇⠀⠀⡸⠀⠀⠀⠠⠈⠠⠀⢹⠤⠤⠏⠀⠘⣿⣿⡏⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⢁⠀⠀⠀⡀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡄⠀⠀⠀⢀⣏⡀⠰⠒⠿⠀⠀⣼⣿⢠⣿⣿⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⢸⡇⢀⠀⠈⠀⣄⣸⠏⠁⣤⣤⡴⠗⠚⣫⣡⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⠀⠀⢰⠀⠀⠉⠙⠓⠺⣭⣤⣤⡤⠾⠟⠟⠻⢿⣿⠀⠀⡗⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣧⡀⠈⠁⡸⠀⠀⢠⣔⠻⣿⣶⣖⠂⠁⣀⠀⠀⣿⠀⠀⠂⠀⢰⡧
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣸⣿⡇⠀⣧⡇⡀⠀⠈⢿⣦⠘⣻⣉⣤⡾⠃⠀⣿⣿⠃⠀⠀⠀⢸⡗
⠀⠀⠀⢠⠀⡀⠀⠀⣾⡎⢸⣻⡇⠠⣤⢇⡁⠀⠀⠘⣿⣷⣼⣿⠟⠀⠀⢸⣿⣿⡇⠀⠃⠀⢼⡇
⠀⠀⠀⠠⠀⡀⠀⢀⣿⣷⣾⢲⡇⠐⣿⢸⠀⡆⠀⠀⠹⣿⣿⡏⠀⠀⠀⣼⣿⣿⣧⣰⠀⠀⣿⡇
⠀⠀⠀⢰⠈⢣⠜⢻⣿⣿⣿⡾⢴⣤⡏⣿⣣⡿⡀⠀⠀⠹⢿⠁⠀⠀⢀⣿⣿⣿⣿⣿⡀⣄⣿⡇
⠀⠀⠀⢨⠀⠤⠂⠤⠿⠛⠛⠀⠀⠙⡇⠔⠉⠀⡀⠀⠐⠢⠌⢷⠒⡀⠚⠿⣿⡿⠟⠁⠉⠉⠙⠃
⠀⠀⠀⢸⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠓⠂⠤⠂⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠰⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠄⠀⠀⠀⠘⠛⡇⣀⣰⠀⣠⣆⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣎
⠀⠀⠀⠀⢤⡘⡣⣷⢶⡍⢠⣴⣶⣯⣟⣛⠠⠼⢟⢤⣷⣥⣔⣒⠺⡟⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠛⠘⠀⠁⠀⠀⠈⣨⡟⠈⠉⠉
⠀⠀⠀⠀⣶⢿⢷⠉⣱⡿⢧⣬⠙⡞⠓⠎⠓⠒⠾⠳⠿⠿⠓⣟⢳⠟⡋⠁⠀⠀⢀⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⣀⣀
⠒⠓⠒⣰⠻⡈⣧⡶⣫⠚⣍⣁⡈⠀⠴⠆⣤⢤⠔⠦⠤⡤⣤⣼⢿⢏⡉⠂⠀⠀⡞⠠⠆⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣴⣿⠟⢻⡇
⢠⣀⡀⠀⠙⣤⢡⡾⠃⢀⠾⠷⣴⡟⣋⣭⢭⣌⣙⣓⠘⠖⣦⠹⠏⠉⠋⠂⠀⠀⡇⠀⠀⠀⣀⣠⣴⣾⠟⠋⠁⠀⠀⠀⢳⣿⡀⠩⡇⠀⣠⣲⣶
⠀⠈⣹⠟⠴⠿⣄⠴⠭⢒⡿⠗⠶⠶⠶⠒⡶⠁⠈⠋⠙⠋⢹⣄⠞⠙⠋⠁⠀⠀⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢷⡾⠟⠉⠈
⠀⠀⠀⠀⠀⡰⡻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣯⠿⠀⠈⠀⣀⠴⠚
⠀⠐⠒⠒⠯⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠖⠖⠻⠻⠈⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⣤⣶⡾⠿⠋⠁⠀⢠⡀⢸⡀
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⢶⣄⠀⠀⠀⢧⢻⣷⠀⠀⢀⠴⣿⡇⠈⡇
⠀⠀⠀⠀⣠⡶⠻⠿⢶⣤⡟⠋⠀⠀⠀⠙⢷⡄⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡴⠖⠖⠖⢦⣄⠀⠈⢻⣧⠀⠀⠘⡆⣿⣆⡴⠃⠀⡿⡇⠀⢹
⠀⠀⠀⣼⠁⠀⡴⡄⠈⣿⠃⢰⠟⢶⡀⠀⢘⣷⠀⠀⠀⠈⡆⠀⢀⡾⠋⣠⣶⣟⡿⣷⣦⢘⡷⢖⣿⠟⠛⢦⡀⠳⠾⠋⠀⠀⠀⣿⣷⠀⠈⡇
⠀⠀⠀⡇⠀⠘⣧⣄⣴⠟⢧⣈⢃⣸⠃⠀⢠⣿⠀⠀⣠⢾⣧⠀⡼⢠⣾⣿⣿⣯⣴⣿⣿⢋⣴⠟⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⡀
⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁⠀⠠⣽⠇⠀⣼⠁⠀⠉⠀⠃⣿⣿⣿⣿⣿⣿⣿⡏⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢈⣧
⠀⠀⠀⠘⣧⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⡏⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⡇⢿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣿⣿⣄⣠⡿⠋
⠀⠀⠀⠀⠈⠛⠿⠶⣧⣶⣴⣠⡄⡀⢸⠃⠀⠀⠀⣷⠀⠀⡀⠀⣧⠈⠛⠻⠿⠿⠛⢋⢳⣬⣳⣄⣀⣀⣴⡏⠀⠀⠀⠀⠀⠀⠀⣷⣿⠿⠋
⠀⠀⣔⡒⣤⣄⡀⠀⠀⠀⠀⠉⠙⠻⣿⢠⡖⢦⡀⢹⣦⡦⠷⠶⠿⠿⣮⣦⣴⡶⠾⠾⠛⠋⠉⠉⠛⠛⠋
⠀⠀⣯⠹⣎⢫⡹⣄⠀⠀⠀⠀⠀⠀⠈⠧⣭⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⣤⣄⣀⡀⠀⣀⣀⣀⠀⠀⠀⣀⣤
⠀⠘⣾⠳⡎⢷⡱⢌⠳⣔⡟⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠉⣉⡩⣤⠀⢲⠂⢠⡏⣹⡷⠚⢩⡉⠉⣍⣉⠩⡃⢸⣶⣄
⠀⢠⢏⣧⠻⡄⠻⡌⢧⡘⢧⡸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢁⡆⠀⠉⠉⠋⠁⣈⡀⢸⡀⠨⣇⠀⠘⠋⠀⠛⠙⢂⠁⢸⡏⡏⢻⡆
⠀⣾⢸⡟⣇⠘⠦⠙⠆⠉⠀⠙⠎⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⢘⠃⠀⠈⢛⠀⠀⠒⠒⢸⡇⢹⡇⠀⠈⠐⠆⠀⠈⠐⠀⠈⣇⡇⢸⡇
⠀⢹⡄⠻⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠀⠈⠐⠀⠀⠐⠒⠀⣏⢸⡧⠀⠀⠤⠅⠀⠀⠈⠀⠀⣿⡇⠈⡇
⠀⠀⠻⡄⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡌⠀⠀⡀⠀⠀⠀⠀⠀⢀⡇⣸⡯⠀⠀⠐⠐⠀⠀⠀⠂⠀⣿⡇⠀⡇
⠀⠀⠀⢧⠀⠘⣷⣂⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⠇⠀⠀⠠⠀⠀⠀⠒⠒⢸⠇⢸⣟⠁⢀⣁⣀⣀⣀⠈⢁⣠⠿⣇⠀⡇
⠀⠀⠀⠘⣧⠀⠈⢷⣖⡀⠀⠀⠀⠀⠀⠈⢙⡦⠀⠀⠀⠀⠀⡿⣾⣿⣷⣦⣤⣤⣶⣶⣾⣿⣿⣿⣶⣬⣷⣾⣿⣿⣷⣿⣿⣿⣯⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠈⢧⠄⠀⠈⢿⣧⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣿⣫⠞⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢼⡁⢀⣠⡾⠋⠈⢷⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠘⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣿⢿⠿⠿⡿⡟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣿⣟⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡯⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠋⢏⠏⠻⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡇⣿⠀⢸⡇⠙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣧⣿⣤⣬⣇⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣤⣄⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⣯⡇⠸⣿⣿⠇⢨⣷⣢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⠮⠒⠉⠀⡈⣇⣄⠀⢀⣄⠈⡇⠈⠑⠪⢗⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣨⠗⠊⠀⣀⡀⠀⠀⡇⣿⣿⠀⢸⣿⠇⡇⠀⠀⢀⡀⠈⠓⢯⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⠀⡴⠻⣿⡏⠢⡀⡇⣿⣿⡀⣸⣿⡇⡇⣠⠊⣷⣯⠓⡄⠀⡏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⣇⣤⠀⣤⡀⡇⡇⡏⢉⣥⣍⠉⠁⡇⡇⣠⣄⢠⣄⠀⠀⡃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⣿⣿⣿⣻⡇⡇⣇⡇⡞⢸⠈⡆⢠⡇⡇⣿⣿⣟⣟⠇⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⣿⣿⣿⣿⡟⣇⣽⣇⡇⢸⠀⣇⣼⣇⡅⣿⣿⣿⣿⢲⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⣿⡿⠛⣿⣇⠇⡏⡏⡇⢸⠀⡇⠉⡏⣇⣿⣟⠻⣿⡰⠈⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡧⣼⣤⣤⣤⣤⣤⣤⣤⣧⣷⣷⣾⣶⣷⣦⣧⣤⣤⣤⣤⣤⣤⣤⡤⢧⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡇
⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿
⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣷⡀⠀⠀⣶⣤⣀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⣷⡀⠀⠹⣿⠻⣷⣦⣄
⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠈⢿⣆⠀⢻⣧⡀⠙⠻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠙⢷⣄⠻⣷⡀⠀⠀⠉⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠙⢷⣜⣿⣄⠀⠀⠀⠀⠉⠛⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⡀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡛⠷⣦⣝⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠙⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⢿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠙⢷⣄⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠷⣦⣄⠙⣷⡀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⠙⠷⣌⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠈⠻⣄⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠹⣆⠀⠀⠈⠻⠀⠀⣶⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣵⣿⠃⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣿⡄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣼⣧⢿⣧⠀⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠈⠛⡿⠋⠀⣀⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢦⣾⣿⣿⣦⡀⢀⠴⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⣸⣿⣿⣾⣿⣿⣿⣿⡟⠁⠀⠹⢿⣿⣿⣻⣿⣇⣀⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣶⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣤⣽⣿⣷⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠿⠿⠿⠶⠶⠶⠦⠤⠤⣀⣀⣀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣶⣾⣿⡿⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣧⡙⢿⣿⣿⠀⠀⡼⠋⣼⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⠀⠜⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡩⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⣿⣿⣿⣿⣶⡧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⡽⠿⠞⠋

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

⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⡟⣿⣿⡇⠀⠀⠀⠀⠀⢹⣧⡀⣠⣷
⠀⠀⠀⠻⢿⣿⠿⠃⠀⠀⠀⠀⠀⡀⣤⣤⣤⡄⡀⠀⠀⠀⠀⣿⣿⣭⡅⣶⣾⡇⠀⠀⠀⠀⠀⠘⣿⣷⣿⡿
⠀⠀⠀⠀⢘⡏⠀⠀⠀⠀⠀⢠⣾⣧⣿⣿⣿⣧⣿⣦⠀⠀⠀⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⢠⣆⢦⣼⣿⣟⣵⣶
⠀⠀⠠⠶⠾⠿⠶⠦⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠻⠿⠿⠟⠛⠛⠃⠀⠀⠀⠙⠉⠈⠙⠋⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠒⠲⣄⡴⠒⠒⢦⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⡇⠀⠀⢀⣴⣿⢻⣷⣄⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣧
⠀⣠⡴⠞⠛⠳⢦⣤⡞⠁⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⡇⠀⠀⢸⠿⡿⠾⡿⣿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸
⠀⠉⠓⠶⠤⠶⠞⠉⢷⡄⠀⣇⣀⣀⣀⣿⣇⣀⣀⣀⡇⠀⠀⢸⣤⡇⠀⣧⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣸

⠀⠀⢀⡀⢠⣤⠀⣀⠀⠀⠀⠀⡄⠀⢀⢠⢀⠀⠀⡄⠀⠀⠀⠀⣼⡷⢲⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣼⣇⣼⣿⣄⣿⣄⠀⠀⠀⡇⣷⣾⢸⢸⢰⡇⡇⠀⠀⠀⢰⣿⣷⢰⣿⣧⠀⠀⠀⠀⠀⠀⠶⢾⠶⠆⢀
⠀⢨⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⡇⣛⢹⢸⢸⢘⡃⡇⠀⠀⠀⢸⣿⣿⢸⣿⣿⠀⠀⠀⠐⠚⡃⠀⢸⠀⠐⡛⠒
⠀⠈⠉⣿⣿⣿⣿⡏⠉⠀⠀⢰⠛⠉⠉⠛⠋⠋⠙⢱⠀⠀⠀⣸⣿⡿⠸⣿⣿⡀⠀⠀⢀⣤⣷⣶⣾⣶⣶⣧⣄
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠁⠀⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠴⠦⢼⣷⠴⠴⠄⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⠶
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠓⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣇⣀
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠓
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣟⣻⣻⣟⣿⣟

⠀⠀⠠⣤⣤⣤⠄⢠⡄⢠⣤⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⣤⣤⣴⡄
⠀⠀⢐⣀⣄⣿⠶⠎⠹⠾⠉⠀⠀⠐⠃⡜⠻⠀⢤⠀⠀⠀⠀⠀⠃⢳⣄⠀⠀⠤⠉⠠⠄⠉⢢⡀⠉⠠⠴⠀⠈⣠⠿⢿⠃
⠀⠀⠈⠒⢶⣶⢀⣀⣀⣀⠀⠀⢀⠋⣼⣛⠁⣈⠀⢀⠀⠀⠀⠘⢷⡀⠆⠀⠀⠀⠀⠀⢘⡛⣛⡛⠀⠀⠀⠀⠀⠙⠖⢿⣷⡀⠤⢠⠤⡄
⠀⠀⢠⣖⠛⢁⡾⠄⠷⠬⠄⠀⡈⢠⣛⣃⣀⡈⢠⠃⠄⠀⠀⢠⡃⠉⡙⠀⠀⠀⠀⠀⢀⣇⣇⣛⠀⠀⠀⠀⠀⣾⣻⣿⡿⠱⠞⠹⣤⣘⣆
⠀⠀⠰⣎⣏⠿⠀⠀⠀⠀⠀⠤⢠⣎⣹⣿⣿⣿⣃⠛⡴⡄⠀⠀⡝⠶⠶⠀⠀⠀⠀⠀⣿⠉⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⡏⢈⣿⡆
⠀⠀⠘⣽⣬⡵⠀⠀⠀⠀⠐⢂⣾⣉⠋⠉⠉⠉⠁⡿⢣⢇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢈⣄⣰⣿⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⣷⣿⡉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣡⣶⣌⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣾⣿⣿⣿⣧⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿⣿⣿⣿⣿⡿⢟⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⢸⣿⣿⣿⡇⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⣇⡞⠿⣿⣿⡇⣸⡇⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣩⣌⠙⢷⣤⣤⣤⣾⣿⣿⣿⣿⣿⡇⣿⣧⣤⣤⣤⣾⡿⣦⣌⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣄⣼⣿⣿⣮⣤⣤⣤⣤⣴⣼⣽⣿⣿⣿⣧⣥⣤⣤⣭⣭⣭⣼⣿⣿⡷⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⠆⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡩⢿⣿⣿⠿⠟⠛⠛⠛⠛⠛⢛⣿⣿⣿⡟⠛⠛⣛⣿⣿⣟⡻⣿⣿⡟⢵⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣉⣁⣠⡾⠋⠉⠉⠙⡇⢸⣿⣿⣿⡇⢸⡏⠉⠉⠉⠻⣷⣎⣉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡇⢠⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⣿⣿⠿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⣸⣿⣿⣿⣗⡈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠉⢿⣿⣾⠓⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣤⠾⠃


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠉⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠚⠁⠒⠀⢄⠀⠀⢀⡬⠀⠀⠀⢠⣠⣀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⣠⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠡⡀⠘⠀⠀⠀⠀⡘⠉⠃⠀⠀⠀⡴⠀⠀⠀⢀⡴⢋⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠴⠂⠀⠀⠀⠢⠀⠀⣷⠀⠀⠀⡀⣦⡄⢀⣻⣀⡑⠀⠿⠀⢐⠀⠀⠀⠀⠀⠀⣀⠃⡠⠀⠀⠈⠁⠀⠈⢢⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠰⠀⠠⢍⡩⠅⠈⠄⠉⠀⠑⠀⠀⢀⣀⢀⠀⣈⡀⠀⠡⠀⠀⠀⠀⠈⠋⠀⠈⠀⠃⠀⠆
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠀⠀⠄⠠⣦⡀⣰⠚⠆⠀⠀⠤⠀⠈⠀⡄⢁⠀⣨⠼⠋⠀⠀⠀⠀⠰⠀⠼⢦⡀⠀⠌⠀⠀⠤⠐⢠⠖
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠠⠈⠀⠘⢠⠖⠀⠀⠐⠀⠰⠂⡠⠀⢠⠆⠀⠀⠀⠀⠰⠃⠀⠈⢀⡀⠀⠑⠦⢦⡀⠠⠤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣤⠋⡀⠀⠀⡀⢳⡆⠀⠀⠀⢦⠖⠀⠀⢡⡀⠂⠀⡟⠀⠀⠀⠁⠀⠀⠀⠀⠄⠈⠀⢀⠔⠉⡀⠁⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢹⠀⠀⠚⣀⠇⠀⠀⠀⠀⠇⠠⡆⠀⠀⢀⢀⠀⣰⢇⣤⠂⠀⠡⠄⠀⠄⠱⠀⢠⡲⢆⣤⣤⣁⡀⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠐⣿⠀⠀⠈⠀⠄⠙⠊⠀⠀⠀⠀⡇⠀⡟⠁⢠⢢⠄⠀⠀⠀⠂⢀⠐⢡⠊⠀⠄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⡃⠀⡀⠀⠀⠀⠀⠑⡇⠀⣆⠀⡟⠀⡇⠀⠄⣀⡀⠈⠀⠀⡰⠃⠠⠃⠀⠑⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⢢⡀⠀⠀⠀⠀⠆⡃⠀⠹⡀⢾⠂⠈⠡⠆⠈⠉⠉⠀⠀⢀⠞⠀⠀⠀⢀⡠⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⢈⠧⡀⠀⢰⠂⠂⠘⠆⣘⠀⣨⠀⠀⡸⠃⠀⠄⣠⠌⠀⠨⡀⣠⣴⣷⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠉⠈⠃⠤⣄⡆⠀⢹⣶⠁⠀⠀⣧⡠⠔⠂⣀⡀⠔⠐⢠⡏⠀⢸⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠇⠠⠄⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠎⠁⠀⠀⠀⣾⠀⠀⣾⠁⠀⠀⠀⠀⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠰⢄⣀⡀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⡀⠀⠀⣀⠆⠀⠀⠀⠀⢠⡏⠀⡼⠃⠀⠀⠀⠂⠀⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⣀⠙⢆⠆⠀⠀⡀⠀⣶⣶⠶⡶⣶⣤⣤⡀⠐⠇⠀⠀⠂⢀⣤⠟⠀⠀⠙⢳⠀⠀⠀⠆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⢠⡌⠁⢠⠀⠈⠑⠲⠞⠉⠉⢳⣀⠀⠀⠀⠈⢙⠻⡿⠒⠶⠖⠒⠋⠁⠀⡄⡀⠀⢾⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⡄⠀⢹⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠋⢠⠙⠂⠀⢸⡀⠠⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠃⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠀⠃⢀⣤⣴⣶⠟⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠘⠦⠘⠀⠀⠁⠀⠀⠁⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠞⣧⠀⠀⠇⠠⠄⠀⠀⠀⠀⠀⢠⡀⢀⠈⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠓⠈⠠⠂⠀⠀⠀⠀⠠⣤⠄⠀⢀⠀⠀⠂⠈⠂⠀⠀⡾⠂⠀⠰⢿⡶⠒⠚⣦⣠⣄⠐⣴⡆⠤⣶⣻⣦⡲⣐⠔⠢⣶⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠊⢁⠀⠀⠀⢀⣠⠃⠸⠀⠀⠀⠤⡀⠀⡄⠀⠀⡅⣄⢀⣀⣤⠀⠀⢀⣈⡁⠀⠀⠀⠀⠈⣹⢏⣾⢻⡜⢿⠓
⠀⠀⠀⠀⠀⠀⢀⠄⠼⠆⠈⠀⢀⡔⠈⠁⠠⠊⠀⠐⠂⠀⠐⠄⠀⠠⠐⢖⣀⣀⣈⣀⡄⢀⣀⣤⣀⠀⠀⣀⡀⠀⢡⠻⠟⠋⠻⢦⡑⣤⣦⣀⢀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢰⡏⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢷⡴⠿⠟⢛⠀⠤⠁⠀⠁⠀⠀⠈⠁⠀⠁⠀⠈⡋⢁⠀⠱⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢉⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠄⠀⠀⠐⢂⠀⠰⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠒⣨⠀⠂⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠽⣇⠈⢷⡀⠆⠘⠆⠀⠰⢃⠤⠀⠀⠀⡀⠀⠀⠄⠈⢤⣀⣀⠤⠤⠀⢉⠀⠉⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⠋⠳⣤⠛⠾⢆⡀⠀⠐⠃⠴⠀⠀⠀⠃⠈⢁⠚⠀⠀⠀⡀⠀⠀⠀⠊⠈⠛⠉⠓⢄⠀⠀⢰⠛⡉⠑⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠤⠄⢨⡧⠄⣀⡀⣔⡀⢀⠀⠀⢤⠀⠀⠈⠁⠚⠉⠀⠀⠀⠆⠀⡈⠀⠀⠀⡄⢀⡆⢀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣈⠀⢠⠄⠁⠀⠀⠀⠀⠀⠉⡁⠀⠓⠀⠠⠘⠀⠑⡀⠀⠀⠠⠐⠒⠒⠄⠀⠁⠈⠂⠸⠀⠀⠙⠁⡀⠀⠀⠠⢉⡘⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣈⢀⠘⡁⠤⠌⠻⠀⠀⠉⠁⡈⠉⠀⡄⠀⠰⠃⢠⠀⢠⡐⠂⠀⠂⣄⠀⢠⡅⠀⠀⡜⠀⠃⠀⠀⠀⠀⠄⠀⡘⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠜⠁⠀⠀⠀⠀⢀⠀⠀⠐⠒⢢⠈⠉⠐⠒⠛⡄⣠⠕⠚⠉⠉⠓⠛⠁⠀⠀⡈⠉⠀⠀⠐⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠂⠉⠂⠁⠀⠤⠀⠀⠀⠤⠂⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠶⢦⣤⡏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠀⣰⠋⠉⠛⢶⡇⣠⠴⢤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⣠⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠛⠁⠀⠀⠈⣷⠀⠀⠀⠀⣰⣿⣿⠟⠉⠀⢹⣦⣾⣿⡟⠋⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣈⣳⣄⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠳⠤⠤⠖⠋⠁⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣸⠇
⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⢸⣿⣿⠇⠀⠀⣠⡾⣿⣿⡏⠀⠀⣀⡼⠋
⠀⠀⣼⣿⣿⡇⠀⢀⣤⣄⠀⠀⠀⢸⡇⠀⠀⣀⣤⣤⣤⣄⣼⣿⣿⣿⠀⠀⣿⣿⣿⠶⠖⠛⠁⠀⣿⣿⡷⠶⠛⣩⣶⣶⣦
⠀⠀⣿⣿⣿⠀⢠⣿⣿⣿⠀⠀⠀⣸⠁⠀⣼⣿⣿⠟⠉⢹⡟⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⠀⠀⠰⣿⣿⡇⢀⡾⠁⣽⣿⣿
⠀⠀⢻⣿⡏⠀⣼⣿⣿⡿⠀⠀⠀⡿⠀⠀⣿⣿⡏⠀⢀⡿⠀⣿⣿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣧⣟⣀⣼⣿⡿⠃⠀⣤
⠀⠀⠈⠻⢿⠶⠋⣿⣿⡇⠀⠀⢰⡇⠀⠀⢿⣿⡇⢀⡾⠁⠀⣿⣿⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⢸⣿⣿⡿⣟⠛⠛⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢀⡾⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⢻⣿⠀⠀⢠⡇⠈⠻⣷⣄⣀⡴⠏⠀⠉⠀⠹⣄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠄⠒⠃
⠀⡎⠈⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠠
⠀⠡⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⢠⠋⠁⠀⠙⡄⠀⠀⠀⠀⢸⣿⡟⠛⠻⣷⡆⠸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⠋⠉⠉⢳⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⠺⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡿⠀⢿⣷⣄⣼⣿⠏⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠋⠁⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢸⣿⡇⣀⣼⡿⠃⠀⠀⠉⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠒⠒⠋⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠿⣷⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣾⠿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⢠⣶⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⢀⣠⣤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣠⣿⣿⠛⠛⢷
⣼⣿⣿⠁⠀⠀⠀⠀⣴⣤⡀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠦⠤⣿⣿⣿⣀⣀⣠⣿⣿⠟⠀⠀⢸⡇
⢿⣿⣿⡀⠀⡿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣀⡀⠀⠀⣿⣿⡇⠀⠀⣽⣿⡇⠀⠀⠀⣼⠁
⠈⠛⠿⠿⠞⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⣿⣿⡇⠀⠀⣿⣿⠃⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⣴⣶⡄⠀⠀⣿⣿⡇⠀⠀⣿⣿⠗⠛⢁⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣤⠶⢾⣿⣷⠤⠤⠤⠶⠃⢀⣴⣿⡿⠷⣾⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡀⢠⡞⢰⣿⣿⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣾⣿⡟⠀⢰⡇⢿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⣧⡞⠀⢸⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⢻⣿⡇⠀⡾⠀⢸⣿⡇⠀⠀⣿⣿⣇⠀⣰⢿⣿⡇⠀⠀⡟⠻⠟⠀⠀⠈⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠛⠷⠞⠁⠀⠈⣿⡇⠀⢰⠇⠈⠛⠛⠋⠈⢿⣧⠀⣸⠃⠀⠀⠀⠀⠀⢿⣿⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣧⣿⣿⠿⣿⢿⡿⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⢾⣿⣿⣴⣮⣝⣿⣧⠞⠉⣾⠀⠀⢀⡆⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⡞⣿⣿⣿⡿⣿⣿⣿⣿⣧⡤⢚⡛⠶⣤⣾⣁⣤⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠃⣴⣿⣧⢻⣿⣦⢾⣿⢿⣿⣿⣿⣿⣿⠟⣀⠛⠋⢁⠀⠀⢻⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⡿⠿⠟⠛⠋⣭⠄⣩⡔⠻⠷⣿⢹⣿⠿⣻⣥⣾⣿⢃⡿⠃⠴⠠⣾⡀⣀⣀⡠⢤⣄⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠛⢃⣀⠀⠰⠶⢶⣦⣀⠘⠻⢿⣷⣶⣤⠘⠁⠜⠛⠟⣀⣠⡘⠁⣠⣆⡄⠈⣫⡵⠏⠙⠁
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⢌⡛⡞⠒⣀⣶⣦⡄⠈⠛⠛⠒⢈⠉⣀⠠⣤⣶⣶⡀⠈⠉⠁⣀⣘⣉⠉⠑⣠⡿⠀⢤⣄⣠⡄⠀⠀⠀⣀⡀
⠀⣼⠿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣾⣿⣿⣯⣾⣿⣷⣿⣿⣿⣿⡶⠿⠷⠞⣻⢏⣾⣧⣤⣀⡀⠉⢀⣀⣶⢦⣙⣿⣿⡟⣦⣸⣇⣙⣉⣉⣥⠤⠶⠞⠛⠉⢻⡄
⢰⣿⣷⣌⢿⣿⣿⣿⣟⣽⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠟⠉⠻⡇⠀⠀⠀⣰⡿⠋⠻⡟⠁⣀⡉⠙⠛⠟⠿⣧⣀⣟⣿⠛⠿⣭⣁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⣷⣿⣿⣦⣿⣿⣿⣿⣷⣿⣿⣿⣹⣾⣿⣿⣿⠿⣿⡏⠀⠀⠀⠘⠀⢀⣼⠏⠀⠀⠀⠛⠈⣭⠗⠀⠀⠄⠀⠀⣹⣿⣟⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠀⠀⣿⢿⣿⣽⣿⣿⢳⣿⣿⣾⣿⣿⢿⣿⠇⢹⣯⣿⣷⣾⣿⣿⣿⣿⣧⣶⣄⣠⣤⣄⠀⠀⠘⠂⢀⣴⣶⣾⣿⣯⠉⠷⣤⣀⣀
⣿⠸⣿⣿⣮⣉⣩⣿⣯⢾⣿⡟⣿⢿⣿⣿⣿⣄⠰⠟⠛⢻⡭⠉⠉⠉⠉⠉⠉⠻⠿⠿⠇⠀⠀⠀⢠⣾⡟⠻⣿⣿⠉⠁⠈⠪⠉⣁
⢿⣷⡟⠛⣿⠟⣿⣿⣿⣾⣿⣿⣿⣾⣿⣯⡏⠀⠀⠀⠀⠻⣷⣦⣤⣤⣴⡾⠀⠀⠲⠶⠀⠀⠀⠀⠟⢻⠁⣰⣿⠋⠓⢦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⣿⣿⣴⣿⣴⣿⣿⣿⣸⣿⣿⣿⣾⣏⣩⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠀⠃⢹⡀⠀⠈⠷⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢻⣿⣿⣿⣿⣿⢻⡾⣿⡟⢻⣿⣟⣿⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠷⠀⠀⢠⠟⠉⠉⠉⠛⠛⠓⠲⠤⢤⣀⡀⣰⠇
⠀⠀⠙⠉⠉⢹⣿⡈⠀⣿⣷⢸⡿⠛⠛⢿⣿⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠛⠻⠃⠀⢹⣿⣧⠀⠀⠀⠀⠻⣿⣿⣏⠘⠒⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢃⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⢻⣷⣶⠀⠀⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠄⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠈⠛⣶⣾⣶⣶⣤⣤⠠⢼⣿⣿⠟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠿⠛⠛⠛⢿⠋⠁⠀⠀⢸⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣶⣖⣶⣾⡿⡄⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣿⣿⡇⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⢤⣀⣀⣀⣀⣀⣤⡤⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣦⡀⠀⢀⣴⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣤⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢿⣿⣿
⣿⣿⣿⣿⠟⠙⠛⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣷⣤⣄⣀⡀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠉⠻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡅⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠗⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣳⣀⠀⠀⠀⠀⠀⣈⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠇⠀⠀⠀⠐⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣶⣶⡀⠀⠀⣴⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⠀⢠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⠃⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⣻⣿⢫⡔⣂
⠀⠀⠀⠀⣄⣰⡎⠟⠁⢀⡀⠀⠀⠀⠀⣀⠀⣀⡉⠻⠂⢀⢀⡀⠀⠀⠀⣴⣧⣭⣭⣿⣿⣥⣽⣿⣦⡄
⠀⠀⠰⠶⠟⠋⠷⠤⠀⠿⠦⠄⠀⠀⠐⠿⠸⠿⢀⠀⠨⠿⠷⠖⠀⠀⠀⠛⢟⡭⣭⢹⣿⢹⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⡄⠀⠀⠀⠀⠀⠀⠀⢰⠃⢹⢸⣿⢸⡎⡂
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣼⢸⣿⢸⡀⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⠈⠁
⠀⠀⠀⠀⠀⠠⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⡐⢠⡄⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⡞
⠀⠀⠀⢀⠐⡥⢿⢪⡐⠷⣁⠢⠀⠀⠀⠀⠀⠀⢦⣮⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧
⠀⠀⣖⣋⠶⣀⡘⣽⠩⣤⣘⢟⡑⠂⠀⠠⣦⠄⢀⠈⠰⠀⠠⣴⠄⠀⠰⣀⣀⣶⣆⣸⣅⣀⣴⣆⣐⡄
⠀⠀⢇⣹⡷⠉⢻⣬⣿⠜⠋⣯⣉⠀⠁⠰⣞⠤⠘⢃⠀⠒⠲⣻⠇⠈⠀⠑⠀⠉⠀⢸⠁⠀⠈⠛⠂⠑
⠀⠀⠀⠉⠀⢀⡼⢠⠈⣤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠸⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠙⡶⢾⡟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤
⠀⠀⠀⠀⠀⠀⡧⢸⡅⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢠⠁⣾⠷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠮⣵⠻⡧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠁⡄
⠀⠀⠀⠀⠀⠘⠐⠋⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡿⣻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⣽⡗⢙⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠀⢸⣷⣼⣧⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢲⡒⠲⢾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⢸⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣁⠀⣸⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢉⡀⠀⠸⠟⣿⡿⠳⡧⢈⡟⣱⠟⢿⣿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠰⢜⠛⠒⠀⡷⣤⡴⢡⣦⡌⠻⣟⡛⠓⠒⠨⠁⠤⣁⣊⣉⣈⡉⡙⠫⠿⠿⠗⠶⣲⣶⣦⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⡃⢀⠀⢛⣺⣷⢀⣘⠿⣏⣀⠘⠛⠁⠀⠈⢤⣀⣊⣍⣑⣂⣀⣀⣈⣉⡉⡉⠁⠤⠄⠠⠀⣀⣀⣉⣑⣀⢀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⡆⢐⡁⠉⠂⣻⡇⠐⣿⡃⢠⣶⡖⠀⠐⢻⡏⠉⣩⣿⣯⡍⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠳⠀⠘⡷⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⡟⢺⢞⠀⠀⣿⡄⠀⣿⠇⣾⣿⡯⠀⡀⢺⡄⣼⣿⣿⣿⣿⠂⠀⢱⠀⢀⣴⣾⣿⣆⠀⠀⢠⠀⢠⣴⡀⠀⠀⢀⣾⣏⡇
⠀⠀⠀⠀⠀⠀⢤⣶⠰⠿⣿⠿⡷⢸⠟⣁⠿⠿⢆⠀⣿⠀⢿⠿⣧⠀⠀⢸⡃⢻⢿⣿⡟⢿⠀⠀⢸⡄⢸⣿⣿⣿⣿⡄⠀⢺⡉⣿⣿⠇⠈⣀⢀⠛⣿⡥
⠀⠀⠀⠀⠰⠤⠀⠁⠉⠂⠀⠀⠀⠸⠿⠥⠺⠿⠟⠂⣠⡤⠤⠂⠀⠀⠀⢸⡧⠍⠀⢀⡀⢸⣿⠂⠸⠇⠿⠿⠿⠛⠟⠃⠀⠀⣗⠛⠛⠂⠨⠉⠈⠀⡁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠈⠛⠀⠀⠀⠀⠀⠁⠀⠀⠛⠖⠒⠒⠀⠤⠀⠙⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣻⣻⢿⣻⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠩⠞⠀⠀⠠⠀⠔⠲⠀⠠⠠⣤⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠛⠛⠻⡖⠉⠃⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣀⠲⡄⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⢀⣤⠔⢈⡀
⠀⠀⠀⠀⠠⣾⡺⢶⣥⣀⣈⣽⣓⠿⣷⡀⠀⠀⠀⠀⢾⢀⠀⠅⠀⠀⠀⣠⡾⠧⠒⣾⠁⣈⠤⠞⢉⣠
⠀⠀⠀⠀⠀⣬⣥⣧⠶⠦⢽⡿⢋⣓⠀⢫⠀⠀⠀⢀⡴⠘⠁⢄⠀⠀⢊⡜⠀⠂⠀⠾⠭⠤⠖⢛⣋⡀
⠀⠀⠀⠀⠀⠀⠔⠉⠃⠀⠹⣻⣽⣣⣦⠄⠳⠤⡤⠎⠀⠀⠀⠈⢦⠤⠊⠀⠀⠀⢀⠀⠞⠯⠭⠭⠁
⣀⣶⣶⣦⣤⣤⣤⣤⠞⠛⠉⢉⣈⣴⢷⢶⣶⣦⣤⡀⠀⠀⠀⠀⣤⢀⣀⢠⠤⣉⡍⣙⣛⠶⢦⣬⣥⣤⣤⣤⣴⣶⣶
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠚⠉⠀⠁⠟⠟⠉⠈⠃⢀⠀⣤⠰⠋⠛⠛⠛⠓⠚⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⡠⠃⢰⠂⠖⠐⢄⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠰⠄⠀⠂⠀⠀⠀⠈⠀⠀⠀⠈⠳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡛⠉⠉⠛⣿⣿⡷⠶⠦⠍⡻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⢛⣥⣶⠚⠉⢙⣿⣿⠒⠦⣄⡐⢬⣻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣫⣾⣿⠟⢡⣶⣿⠟⠁⠈⠻⣷⣍⠻⣶⣭⡛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⠿⠛⣁⣴⡿⠟⠁⠀⠀⠀⠀⠈⠙⢷⣦⣉⡛⠷⠝⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⢋⣠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣬⣝⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣴⣾⠿⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣡⣾⣿⠁⠀⠔⠛⣿⣿⣿⣶⡄⠀⣰⣾⣿⣿⡟⠳⢆⢠⣿⣯⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣼⣿⣿⡟⠀⠀⣠⠿⣷⠾⡟⠛⡇⠀⣿⡟⡭⢶⡶⢤⡨⣿⣿⣿⡄⢈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⣿⣿⣿⠇⠀⠀⠉⠐⠋⠀⠀⠀⠁⠀⣿⣷⠀⠀⠀⠘⠁⠈⠹⣿⣧⠘⣯⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡈⠳⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡟⣸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣟⡃⠀⠀⠀⠀⠀⠀⣸⢣⣿⣿⣿⣦⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣥⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣠⣽⣿⢿⣿⣭⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢣⣿⣿⣿⣧⠀⠀⢠⣶⣿⣿⣿⣥⣠⣽⣿⣿⣷⣄⠀⢠⣾⣿⣿⣿⣿⣿⡿⢁⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢣⣾⡏⣿⣿⣿⣷⣾⣿⡿⠉⠀⢠⣤⣤⣤⠀⠉⠻⣿⣷⣾⣿⣿⣿⣿⣿⡟⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⣱⣿⣿⢀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠋⠉⡉⠉⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣴⣿⣿⡇⢸⣿⣿⣿⠻⣿⣿⣿⣧⡄⢀⣼⣿⣄⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣿⣿⣿⣇⢸⣿⣿⣿⠀⠈⠻⣿⣿⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣦⠙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢨⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠁
⠀⠀⠀⠀⠀⢀⣾⡏⢠⣶⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠋⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⡄⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢠⣿⡿⠛⠁⠀⠈⠙⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢖⠭⠭⡓⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡔⡂⠁⠀⠀⠈⣰⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢳⣶⢿⣶⣾⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡏⢸⣿⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣾⠏⢱⠘⣿⡋⢴⠙⢦⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢏⠉⢉⣓⣒⠾⠿⡏⢇⠣⠁⠹⠇⠀⣑⡤⠵⢖⣒⡋⠹⡸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠡⠁⠀⠀⠉⣿⣿⣷⣿⣆⠀⣠⣄⠀⠀⠼⠺⣿⣿⡟⠀⠀⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣄⠀⠀⣻⣿⣿⡿⢿⣛⠄⡓⠌⠃⠋⢀⣾⣿⣿⣿⠂⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⣄⣪⠭⠴⠒⢧⡄⠀⣤⣤⣤⠤⠔⢿⣿⠟⠲⠮⢤⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⢶⢸⣿⣿⠀⡦⠚⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣏⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠟⠉⠿⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢻⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠈⠃⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⠿⣷⣶⣶⣤⠾⢀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢦⣈⠻⢛⣁⠴⠒⠚⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣄⣀⣰⣾⣨⠿⠤⠤⢤⢯⣹⠴⣤⣀⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⢿⣿⠟⠀⠀⠀⠐⠁⠀⠀⠀⠈⠿⣿⣦⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣴⣾⣿⣥⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣽⣿⣦⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣏⠉⠉⠉⠉⡋⠀⠀⠙⡏⠉⠉⠉⠉⢙⠻⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣢⠀⠀⠀⡇⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢀⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢑⣄⡀⣇⡀⠀⠀⡆⠀⡤⠁⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠹⣿⠧⢶⣷⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠰⡇⠀⢀⣠⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⣿⠀⠀⢰⣿⣿⣿⠿⠛⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠚⠚⠛⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⣯⡟⣽⣿⣿⣿⣟⢝⡲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⠊⢉⣄⡖⡆⠆⡈⢠⠀⣹⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣦⣝⠻⣿⣾⠛⢩⣿⣿⣷⣽⠿⣋⣭⡷⣄⠀⠀⠀⠀⢀⣔⡿⣋⣼⠷⣽⡯⠳⣶⡿⣇⢣⣶⣏⣁⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢽⣷⢫⣿⣷⣮⠇⢠⡿⢝⡿⣟⣷⣻⣿⡿⠿⢛⢧⠀⠀⣰⡛⢿⣾⢟⣵⣾⠛⠛⠒⢬⡤⠒⢼⠛⣫⣟⣡⠖⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⣽⡧⢾⡿⣝⡿⠀⢛⣿⣾⠟⣹⢟⣫⣷⣴⣺⣭⡿⣧⣰⣿⣿⣞⣿⣞⡼⠏⠀⠀⠀⠀⡱⠀⠀⢳⢛⣽⣖⡉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡧⣾⣟⢽⣧⡞⡜⠀⠈⠋⣙⣿⣻⣿⣧⣿⣽⠟⠛⠛⠛⢿⣿⣶⣽⣷⠇⠀⠀⠀⠰⣱⣀⣀⣸⡶⢷⣿⠅⣲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣷⣿⠿⢷⣼⣿⡁⠀⠀⠸⣧⣆⢿⣸⣿⣻⡇⠀⠀⠀⠀⢸⡟⢛⣭⣾⡷⣦⣤⣴⢾⢿⣿⣿⣿⣿⣦⣽⣧⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣶⣒⣛⢷⠟⢩⢧⣤⣀⣠⣾⣾⣿⣯⣿⣿⢱⠀⠀⠀⠀⠸⣿⣿⣿⣭⣾⣷⣿⠷⡷⡷⢶⣧⣽⣻⣷⢫⢏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⣷⣜⠿⣷⣿⡿⡽⣿⣻⠋⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠙⢿⣳⣣⡻⣼⣷⣿⣭⣟⣛⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢊⣕⣪⡏⣧⢻⣼⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣟⣱⣿⠿⣿⣿⣒⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡭⢭⣧⡿⣟⣿⣻⣿⣄⣀⣀⣀⣀⣤⣄⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣠⣼⣿⢿⣯⣯⠧⣟⣛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⢀⣿⡿⣟⣿⣷⣛⣻⣾⣽⣾⣿⠀⠀⠀⠀⣧⢻⣷⣟⣿⣽⣿⡿⣫⡫⢿⣷⣱⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣮⣷⣿⡿⢟⣿⣿⣿⡿⣿⣻⣿⠀⠀⠀⠀⢻⣷⣿⡿⣿⣻⡭⢮⣟⣕⣵⣹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⣿⣿⣯⢎⣯⢿⣷⡏⠀⠀⠀⠀⢸⣿⣿⡡⠙⣟⣺⡯⠾⢟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢷⣿⢟⣾⢼⣟⣿⠇⠀⠀⠀⠀⠈⣟⣯⣷⣿⢿⡿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⣏⣩⣿⠀⠀⠀⠀⠀⠀⢫⢯⣱⢟⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣥⢥⣿⠀⠀⠀⠀⠀⠀⣸⡿⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⡦⣄⣀⣀⣠⠔⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣞⢏⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠁⡀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡠⠊⠈⠀⣿⠀⠀⠀⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠔⠁⣴⠶⠋⠈⠓⣄⠀⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠒⡟⠑⠒⠂⠀⠘⡛⢇⠳⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡄⡃⠀⠀⠀⠀⠀⠀⠜⡆⢠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡼⢋⠔⠙⡄⠀⠤⠄⠀⠄⠐⠀⡰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠊⠀⠈⡀⠚⠀⠀⠀⠀⠚⠤⡀⠈⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⠂⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡁⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠐⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡆⠇⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠇⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠎⠀⢸⡇⠀⢃⠀⠀⣇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡄⠀⠀⠠⠀⠣⠤⡤⠀⠀⢠⠎⠁⠀⠈⠠⠄⠒⢦⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠇⠀⠀⠀⠀⢠⠊⠀⣄⢰⡄⠀⠀⠀⡀⠂⠀⠆⠀⢱⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⡎⠀⢠⠏⠀⡄⠀⠀⠀⠱⢄⠀⢸⣀⠀⠱⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣠⢴⠂⠀⢰⠀⡘⠀⠀⠀⠀⠀⠱⡀⠈⣿⠀⠀⠱⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⣸⠀⣰⠁⣠⠁⠀⠀⠀⠀⠀⠀⠙⢄⣹⡄⠀⠀⡏⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⢶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⢳⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⢸⣇⢀⣿⡄⣼⡇⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠸⣿⡾⠟⠛⠛⠛⠛⠿⣿⣀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣤⣀⣹⠟⢉⣀⡴⠶⠒⠲⢤⣀⠈⢻⣧⣤⠤
⠀⠀⠀⠀⠀⢀⣤⣄⣠⣠⣤⣤⣄⣤⣤⢤⣤⣿⡏⠀⡼⠃⠀⠀⠀⠀⠀⠘⣆⠀⣿⣥⣤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⠻⠿⣿⣿⣿⣀⣈⣻⠀⢸⠃⢠⣾⣿⡿⣷⡀⠀⢹⠀⢸⣯⣤⣤⣼⣿⣿⡿⠿⠟⣛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣻⣥⣠⣴⣶⣤⣌⠉⠈⠉⠻⠤⠼⠄⢸⡃⠉⠉⠻⣧⡀⠼⣤⠼⠟⠁⠀⠈⣉⣠⣶⣾⣷⣤⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠦⠤⢠⡀⠀⠁⢀⡀⢸⡏⠀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢼⣿⡿⠀⢰⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣠⠭⡀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠃⠀⠘⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠄⠀⠀⠐⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠐⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠘⠶⠂⠜⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⠀⠀⡰⢼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠁⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡸⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⣀⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡗⢂⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡇


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣟⣋⠍⢹⢿⠀⠀⢀⠀⠀⠙⡉⠀⠀⠀⢨⡄⠀⡤⠠⠄⠀⠀⠙⢶⠾⣹⢏⣾⡇⠉⠻⠁⠄⣸⣟⣶⣮⡹⣿⣿⣦⣀⡄⠱⡐⠒⢫⣷⣼⣟⣾⣟⠫⣽⣿⣏
⣷⢿⣿⣻⢿⣧⢀⡃⣀⣠⣎⡁⠀⠀⢸⣸⡃⣼⠇⣉⠀⠑⠄⠀⠀⣸⣿⣿⢻⡷⠦⠄⡅⠞⠉⠙⢾⣿⠿⢤⡹⣷⣿⠀⢀⡷⣤⣤⡿⣱⣿⡼⢿⡛⠉⠉
⠇⠋⡞⣽⣿⣿⣿⣤⠈⠁⠘⢣⡀⠀⠂⠐⢀⠁⠀⠀⠀⠀⠐⡠⢠⣿⣻⡿⣿⣥⣤⣦⡂⢳⣴⣶⣿⣿⡶⠿⠿⢾⣿⣞⣟⣳⣻⡯⣞⡙⢤⡖⣔⣳⡀⣀
⣟⣶⣴⠤⡡⢋⡿⣻⣷⡀⠀⠀⢹⣄⠀⢾⡾⣀⠀⢤⠴⠦⠯⠀⠀⠈⢻⣿⣽⣿⣿⣗⣋⣿⡿⢛⠽⠚⠋⣉⣉⠙⠓⠮⡛⢿⡏⠀⠀⠣⡈⡍⠀⢄⠉⠐⠦
⣝⣟⠃⠠⠒⣩⢘⡽⢿⣷⡀⠀⠠⣟⣶⡸⠈⠁⠐⢛⡖⠦⢴⡀⢀⠀⠐⠹⣟⣿⡻⢙⣷⢟⡶⠁⠐⠃⠀⠀⠀⠈⠁⠀⠈⢢⡳⣄⠀⠀⢹⡇⠀⠈⢓⠄
⠢⣙⠂⣠⣴⢾⡥⣈⡭⠿⣳⠀⠀⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⢱⣼⠄⠠⢤⡹⣿⣿⣾⡟⡞⢀⡗⠁⠂⢄⠀⠢⠀⠀⠀⣇⠀⢧⢿⣧⠀⠐⣇⠀⠈⠸⡊⠓
⠀⠙⣧⢻⡾⢜⣽⡴⣘⣯⣿⡇⠀⠀⠀⠀⠈⠻⣆⠔⠀⢠⡾⠉⠀⠠⠁⠀⠀⢈⠋⢹⡇⡇⠀⣿⠀⣀⠀⠑⠅⠊⠒⠠⣿⡄⢸⣸⣿⠀⠀⣿⠀⠀⢠⠏⠂
⢈⠃⢸⡓⡟⠙⢉⡟⣂⣒⣿⣯⠀⠀⠀⠀⠀⢐⠃⠀⠀⡈⠀⠀⠃⠀⠀⠀⠀⠀⢀⣬⣧⢧⠀⠇⠀⠈⠀⠀⠙⡆⠃⢈⣸⣧⡜⣿⣿⠀⢀⣿⠁⠀⢠⠁⠨
⠤⠁⣼⢪⣤⣆⣀⣗⠊⠭⣽⡟⠀⠀⠀⠀⢸⣥⠐⠒⠀⠀⢀⣀⠘⠀⢄⠠⠀⡤⣾⡿⢿⣮⠣⡄⠀⠀⠀⠀⠀⠀⠰⣿⡈⣯⣷⣿⡻⠀⢰⠏⠀⢀⠍⠢⡝
⣠⡾⢃⣾⠼⠬⣿⡯⢭⣝⣿⠇⠀⠰⠦⠴⠂⠀⠑⠒⠒⠒⠚⢻⠓⢶⡧⠂⠈⠀⣿⡟⠻⣿⡷⣬⣓⠤⣈⣦⠀⠀⢺⡿⠛⠻⢿⠛⡁⠄⠉⠀⡠⠈⠀⣞⠁
⣩⣌⣁⢉⠉⣻⡃⢝⡳⢾⡿⠀⠀⠘⢀⠀⠀⠀⠀⡀⠀⠀⢀⡾⠀⣼⠁⠀⠀⢀⢸⡓⢀⢈⡀⣀⣽⠿⠛⢯⠂⠀⠀⠀⠀⠀⠈⢃⣀⡠⢔⠎⢡⡴⠉⢀⣠
⣯⢽⣆⢠⡫⣝⢬⣳⣿⡟⠁⠀⠀⠀⠀⠑⠶⡒⢂⣈⠁⠀⠉⠶⠾⣇⣀⠀⠀⢀⣾⣇⣦⣀⡽⠿⢡⡆⢀⠞⠂⣀⠀⠀⠀⠀⠀⠀⢧⢤⠶⡛⠅⠆⣤⠚⠁
⠞⡏⠏⢦⡱⣝⡿⡿⠋⠀⠀⠀⠀⠀⠀⠰⢦⡛⠔⠲⠴⠶⢠⡆⠀⠈⠀⠉⠐⠊⠛⠷⠿⢥⠴⠖⢛⠋⠣⡀⠉⠀⠀⠀⠀⠀⠀⡀⠈⣤⡤⠶⡊⠉⠀⠀⢀
⢰⢹⣼⣧⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡟⢞⣍⣯⡐⢤⡀⠀⠀⠀⠀⠀⠀⡄⠀⣦⠀⠀⠁⠀⢱⡄⠀⠀⠁⠀⢆⠀⢃⠀⠀⢀⣂⣠⡱⠴⢪⢹
⣾⠝⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠸⣄⣝⢷⣎⡁⠐⠠⣀⡀⠀⠀⠰⠈⠀⠀⠀⠀⠀⣳⡀⢀⠀⠐⡈⢆⠘⡄⠀⢟⠉⡌⢠⢠⢸⡆
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠄⠸⣏⠁⠂⠩⠶⣦⣄⣉⠅⢰⠶⣶⡶⣶⢖⠋⠉⡝⢣⠀⠀⠀⠘⡌⠀⢃⠀⠘⡀⢰⢸⡆⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣠⠟⠀⠀⡀⠘⢿⠀⠀⠀⠀⢸⣿⣿⣧⠀⠃⢿⠡⣹⠸⡇⠀⠙⠀⢧⡀⠀⠀⡘⡀⢸⡄⠀⢹⣄⣦⣷⡿⣿
⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⡩⠁⠒⠃⠈⢰⠁⠈⠀⢠⠄⠀⠀⢀⣾⡿⠿⣿⣆⠀⢾⠄⡿⢀⡠⠤⣄⣠⢌⢇⠀⠀⠠⡐⠈⣷⠀⠈⣿⣿⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⠙⢒⠀⠐⠛⠿⡓⢤⣾⠀⠀⢀⣾⣿⣷⣙⣿⢾⣆⠀⡄⠻⣿⠀⠀⠰⠁⢠⠈⢧⡀⠀⠙⢴⡘⢆⠀⠘⢻⣿⣶⣿
⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡼⣍⠀⣸⠀⠃⠀⠂⠈⡦⠊⠓⣠⣿⣷⣿⡿⠁⢹⡷⣿⣆⠐⠀⢿⡌⠀⠀⡆⠀⠠⠘⣆⠂⠀⠀⠱⡌⢢⠀⠂⠹⣿⣭
⣿⣦⣄⣀⣀⣿⠭⠑⠾⠗⠒⡻⢉⠄⠈⠉⢿⣳⣈⠃⠀⠀⠀⠀⢈⣹⣿⡀⠀⠳⠀⠜⠛⠻⣿⣆⠀⠈⣷⠀⠀⡇⠀⠃⡀⠈⢢⠀⢀⠀⠘⢆⠣⡀⠀⠈⢿
⡉⠒⠏⠻⠛⠷⢟⢢⠬⣍⡃⠀⠤⢼⠶⡀⠀⠙⡖⠤⠀⢀⠀⠢⠞⢻⣿⣿⠟⢂⣁⠰⣶⣾⣼⡟⢧⡀⠸⡆⠀⡇⠀⠉⠀⠀⠀⢳⡀⠑⠄⠀⠁⣑⣄
⢿⡿⠿⠛⠛⡻⢩⠟⠃⠠⢉⣄⡙⠚⠐⠠⠴⠚⠉⠑⠒⢲⡤⢶⣊⠉⢹⡟⠛⢿⢷⠟⠛⠛⠻⢷⣾⣿⣄⢱⡀⢠⡆⠀⠀⠀⠀⠀⠙⣄⠀⠈⠢⣉⠂⠑⡂
⢀⡅⠒⣠⠄⢠⠄⣴⠀⠂⢀⠈⢙⣦⠀⠀⠀⠀⠂⣠⣶⡇⠍⠉⣉⣽⣿⠇⠀⣿⣻⠇⣐⢨⢀⡆⢀⠉⠙⠿⢷⡰⣥⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠓⢄⡈
⣸⡇⠀⢀⡴⠃⠀⡿⡀⠁⠀⠀⠨⢡⢠⣤⣠⣴⡎⣿⡰⡇⢰⡞⠿⣙⣟⠀⠀⠈⠣⢔⡵⢓⠞⢠⣤⣤⣴⣦⣴⣶⠜⣧⠀⠀⠀⠀⠀⠐⢄⠳⡀⠀⠀⠀⠙
⠿⣤⡔⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⠂⣿⣿⣿⣿⣇⣿⡇⢸⡇⡦⢉⢩⣀⡄⠀⠀⠈⠳⣋⣰⣿⣷⣿⣿⡻⣿⠹⣧⡘⣇⠀⠀⠀⠀⠀⠀⠁⢬⣢⣄
⠀⠸⠂⠀⠀⠀⢀⣠⣷⡈⠤⡤⢷⠿⠼⣿⣿⣿⠧⣿⣄⡇⣼⣇⣌⣬⡥⠵⢗⡄⠀⠀⠀⠙⣧⣯⣾⣇⢼⣿⣿⣰⣟⢿⣌⢧⠀⠀⠀⠀⠀⠀⠀⠉⡽⡏⠐
⣤⣥⣤⣶⣶⣢⡟⢿⣯⣿⣶⣾⣿⠀⠀⣾⣟⣇⢀⠂⡟⠷⣿⣻⡄⢀⠀⢄⣿⣻⣿⣶⣤⣀⠀⠙⡿⢷⣿⣿⣯⠀⣿⣿⣿⣮⣃⠀⠀⠀⠀⠀⢀⣨⣃⠻⠆
⣝⢿⣮⢫⣵⣷⣾⢮⡷⢏⠙⢯⣿⠀⠀⠉⠟⡡⣎⠜⢁⡾⡁⠙⢿⡄⠠⠈⣿⣿⣩⠿⠿⢯⣿⣦⡀⠄⠀⣩⣧⠀⣿⣽⢻⣻⣿⣧⡀⠀⠀⠀⠀⠀⠸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⠛⠀⠈⠓⠲⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⣶⠀⢰⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠒⠺⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⣠⠤⣄⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣀⡴⠋⠁⠀⠈⠑⢦⡀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢀⡤⠞⠉⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠳⣄⡈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣠⡞⠋⠀⠀⠀⠀⣠⢞⣫⠽⠷⡶⠿⣭⣳⣦⠀⠀⠀⠈⢻⣦⡀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡴⣾⢹⡇⠀⠀⠀⠀⡾⣱⠏⠁⠀⠀⡇⠀⠀⠙⣿⣧⠀⠀⠀⢸⠉⡿⢦⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⢸⡇⣿⠒⠒⠒⠲⡗⠒⠒⠒⣿⣿⠀⠀⠀⣼⢰⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⠀⡇⠀⠀⠀⠀⢷⡹⣄⡀⠀⠀⡇⠀⢀⣴⢿⡟⠀⠀⠀⣿⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⠶⣿⠀⡇⠀⠀⠀⠀⠀⠙⠲⢭⣙⣛⣛⣛⣭⠵⠋⠀⠀⠀⠀⡇⢸⠷⣌⡛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⠴⠚⠉⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇⣿⠀⠀⠙⠳⢦⣍⡛⠶⣤⡀
⠀⠀⠀⣠⡴⠞⢉⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠈⠙⠶⣤⣙⠳⢦⣀
⣠⠶⠋⣁⣤⡞⠋⠁⠀⠀⣀⣤⣤⣄⡀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣠⣶⣶⣶⣦⡄⠀⠉⠳⢦⣌⠛⠶⣄
⠹⠶⠟⢻⠘⡇⠀⠀⢀⣾⡿⠛⡟⠻⣿⣆⠀⣿⠀⣿⠀⠀⠀⠀⢀⣤⢶⣾⣿⣿⣓⠶⣄⠀⠀⠀⠀⠀⢸⠃⡏⠀⣰⣿⠋⣹⡉⢹⣿⡆⠀⠀⢀⡿⣻⢦⣤
⠀⠀⠀⠘⡆⣿⠀⠀⢸⣿⠖⠒⡶⠒⢾⣿⠀⢹⠀⢸⡀⠀⠀⢠⡞⣱⠏⠁⠈⡇⠉⠻⣮⢷⡄⠀⠀⠀⣸⠀⡇⠀⢹⡿⠉⢹⠉⣹⣿⠇⠀⠀⣼⢣⡇
⠀⠀⠀⠀⢿⢸⡆⠀⠀⠻⢵⣤⣷⣶⡿⠋⠀⢸⡆⢸⡇⠀⠀⣿⢠⡏⠀⠀⠀⡇⠀⠀⠈⣷⢻⡄⠀⠀⣿⢸⡇⠀⠀⠙⠷⠾⠾⠟⠋⠀⠀⢰⡏⣼
⠀⠀⠀⠀⢸⡆⣇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⢸⡃⠀⠀⠀⡇⠀⠀⠀⢸⡎⡇⠀⠀⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡇
⠀⠀⠀⠀⠈⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⢸⠉⠉⠉⠉⡏⠉⠉⠉⢸⡇⣿⠀⢰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾
⠀⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⢸⡇⣼⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⡇
⠀⠀⠀⠀⠀⠸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠇⣿⣤⣤⣤⢤⣧⠤⣤⣤⢼⡇⣿⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾
⠀⠀⠀⠀⠀⠀⣷⢸⡆⣤⠤⠤⠴⠶⠶⠶⢶⠀⣿⠀⣷⠀⢸⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⣿⠀⣿⠀⡟⣴⠶⠴⠦⡤⠤⠤⣤⠀⡿⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⣧⢻⠀⠀⠀⡇⠀⠀⢸⡆⣿⠀⣿⠀⢸⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⡇⢠⡇⠀⡇⢸⠀⠀⠀⡇⠀⠀⣿⢸⠇⡟
⠀⠀⠀⠀⠀⠀⠀⣇⢹⣸⣇⣀⣀⣧⣀⣀⣸⠇⣿⠀⣿⠀⢸⡄⣧⠤⠤⠤⢤⣧⡄⢠⢀⣸⡇⡇⢸⡇⢰⡇⢻⣀⣀⣀⣇⣀⣠⣧⡟⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡌⣧⠉⠉⠉⠉⠉⠉⠁⠀⢹⠀⣿⠀⢸⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⡇⢸⠇⢸⡇⠀⠉⠉⠉⠉⠉⠉⣸⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⢸⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⡇⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁