Writing symbol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⠷⢀⡀⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⣷⣄⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣿⣿⣿⣦⡀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠙⢿⣿⣿⣷⣄⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢻⣿⣿⣿⣦⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠙⠿⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⣦⢠⠆⣄⢐⣠⣦⢐⡀⣤⣄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠘⠟⠀⠇⠸⠃⠷⠸⠃⠇⠻⢘⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣧⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⢸⣿⠀⠀⣼⠃⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣵⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⢣⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢡⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿⢣⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢇⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⣼⣿⣿⣭⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢱⣿⣿⣛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⣿⠇⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⡿⣸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢇⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⢸⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⡂⣲⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡽⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡿⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣞⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣴⢟⠊⠀⠉⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢠⠞⠉⠀⠀⢰⣾⢯⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣤⣴⣶⣶⣶⣶⣶⣶⡆⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣼⠀⠀⠱⣴⣿⣻⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⡏⠀⠐⡾⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⡇⢀⡌⣅⠀⠀⠀⠀⣀⣀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠸⣧⣬⡀⣹⣂⡠⠶⠋⠙⠿⠿⠿⣍⣀⣀⣀⣀⠤⠐⠀⠀⢸⡟⠛⢻⡿⠛⠻⡇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣼⣟⡿⣯⢈⡱⠂⡐⢹⠀⢰⣀⣪⣯⣀⣀⣀⣀⣀⡤⠔⠺⠧⠤⠼⠷⠤⠴⠇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⢠⡥⠉⠀⠈⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣄⣀⣀⡀⠀⠀⢀⣀⣀⣠⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⠀⣠⣤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣁⣹⡆⠀⢰⡏⢸⡇⠀⠀⡾⢹⣇⠀⠀⢠⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠈⠉⠉⣿⠉⠙⠛⠛⢺⡗⠶⠾⠷⢤⣿⣤⣤⣿⣀⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⣼⠏⠀⠀⢰⣠⡿⠀⠀⢀⡀⣸⡇⠀⠀⠉⢉⣿⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠋⠀⠀⠀⢷⡾⠃⠀⠀⠘⢁⣠⣶⡟⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⣻⣦⣀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⡾⠋⠀⢙⣷⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⠞⠋⠀⣠⡴⠋⠀⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣴⠞⠁⠀⣠⡾⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⠟⠁⠀⣠⡾⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⠟⠁⠀⣠⠾⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⡴⠟⠁⢀⣤⠞⠁⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⡴⠋⠀⢀⣴⠟⠁⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⠉⢿⣀⣀⣴⠟⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠉⢻⣅⠀⣠⡾⠋⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠻⢦⣀⣠⣤⡶⠟⠛⠉⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠓⠲⠶⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⢠⣞⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦
⠀⠀⠀⠀⠀⠀⡎⠈⡹⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂
⠀⠀⠀⠀⠀⡸⠀⢠⠧⠖⠒⠉⠉⠉⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⣠⣤⣴⣶⣿⣿⣶⣄
⠀⠀⢀⠔⢺⠃⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⣸⢀⠏⠀⣼⡠⠴⡖⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⢸⡟⣾⣿⠳⢟⡋⣭⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⡏⡜⠀⢰⠋⠳⢄⠉⠲⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣾⣹⣿⢇⣷⣝⠿⣾⣝⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⡿⠗⠒⠛⠓⠲⠤⢵⣤⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣯⣭⣤⣭⣟⣛⡘⢿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣀⠜⠁⠈⠉⠓⠒⠒⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠐⡿⠁⠀⠀⠈⠉⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⣠⣾⣿⣿⣦⣴⣦⣦⣴⣶⣶⣶⣶
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠤⠠⠠⠠⠀⣤⣄⣄⠸⢠⡄⢤⡥⣠⢠⣠⠅⡰⢸⡤⠄⠠⠠⠈
⠀⠙⠿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠷⠆⠀⠀⠀⠀⠀⠀⠿⠟⠛⠛⠛⠃⠛⠃⠘⠛⠛⠛⠛⠻⠇
⠀⠀⢀⣠⠠⣤⢀⡀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⢠⠀⠂⠀⠐⠀⠀⠀⠀⠀⡀
⠀⣠⡍⠛⢧⡻⣸⣋⡄⠀⣴⠶⣍⣉⠽⠏⢻⡆⠀⠐⠉⠀⠀⠍⠀⠀⠑⡀⢀⠪⠀⠠⠤⠤⠄
⠀⠋⢙⣉⡽⣧⣼⠷⠶⠀⣷⠀⠉⢀⠒⣿⢛⣼⠂⢠⣠⣄⣤⠀⣤⣤⣤⠅⠀⠂⠀⠀⠀⠀⠀⠰
⠀⠘⢟⢋⣼⠋⣷⠺⠃⠀⠘⢷⣸⡊⠃⣠⣾⠋⠀⠀⠻⣿⣿⣿⣿⣿⠟⠀⠀⠲⠐⠒⠐⠐⠀⠂
⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠉⠉⠛⠉⠀⠀⠀⠀⠀⠉⠉⡟⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢀⠆⡀⠀⠀⢀⡔⠉⠛⠛⠓⣀⠀⠀⢀⣄⠙⠹⡟⠿⢆⡀⠀⠀⣴⠞⠛⠉⢛⠗
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠄⠀⠰⠀⠀⣿⠃⠀⠀⠀⢠⣿⡆⠀⣬⠈⠁⠀⠀⠙⢩⣧⠀⢸⡇⠂⣄⠠⣠⡴⣾
⠀⠀⠀⠀⠀⠈⡀⠐⠈⠃⠐⠀⢀⠀⠀⢏⠀⠀⠀⠉⡘⢿⠇⠀⢹⡶⡶⠖⠂⠀⢾⠏⠀⠘⣷⣑⠌⠚⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠶⠦⠤⠟⠂⠀⠀⠀⠉⠀⠠⠄⠀⠉⠂⠀⠀⢈⣁⢀⠷⢀⣉
⠀⠀⡀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠁⠛⠀⠏⠈⠁
⠀⢀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠔⠁⠀⠀⡈⠀⠆⠀⠀⠀⠉⠉⠸⠿⠿⣦⠀⠀⣠⠀⠀⠉⠉⠁⠄
⠀⢸⡄⠀⠈⠀⠀⠀⣿⠀⠰⠀⠠⠄⠀⠁⠈⠀⠀⢰⡄⠀⠀⠀⠀⠀⢠⡆⢰⣿⠀⠀⠀⠀⠀⣼⡆
⠀⠈⠁⠀⠐⠀⠀⢀⠟⠀⠀⢄⠔⠂⠄⠀⢀⠚⠀⠈⢧⣤⡄⠀⠀⠀⡾⠀⠈⢿⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⠀⠉⠐⠀⠂⠀⢀⣀⣀⣀⣉⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠐⠘⠛⠛⠛⠛⠒⡀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠴⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⣶⣌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⣰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢠⣾⣿⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⢋⠰⣿⣿⠟⣴⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣡⣾⣿⡎⣡⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⢀⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢏⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⡿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣡⣾⣿⠏⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢏⣴⣿⠟⠁⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⡿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠉⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⠿⠿⠛⠉⠉⠉⠘⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⣀⣤⠖⠋⠁⣼⠃⠀⢀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠙⠒⠛⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢧⣀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠒⠚⠛⠛⠓⠒⠲⠦⢤⣤⠴⠖⠒⠒⠒⠒⢲⡞⠁⠀⢈⡿⠋
⠀⠀⠀⠀⠀⠀⢰⠶⡇⠀⣤⣤⣤⣤⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠏⠀⠀⢠⡾⣦
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣤⣤⣤⣤⣤⣤⡤⠀⢸⡇⠀⠀⠀⢀⡼⠃⠀⠀⣰⢻⡇⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠤⠤⠤⠤⠤⠤⠤⠀⢸⡇⠀⠀⠀⡾⡧⣄⣀⡼⠁⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠠⠤⠤⠶⠶⠶⠶⠦⠀⢸⡇⠀⠀⣸⠃⢀⡼⠟⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠰⠶⠶⠶⠶⠶⠶⠆⠀⢸⡇⠀⢠⣏⡴⠋⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⠒⠒⠒⠒⠂⠀⠀⠀⢸⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⢸
⠀⠀⠀⠀⠀⠀⢸⣃⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣼
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢧⠰⢠⠀⡄⠆⠄⡤⢤⢠⢠⠀⢸⢱⢀⠄⣄⡄⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠃⠈⠀⠁⠃⠁⠁⠘⠸⠰⠀⠘⠒⠉⠒⠁⠓⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣦⣄⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⣽⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⣾⣿⣿⡿⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣠⣾⣿⣿⣿⠏⠀⠀⠀⠈⠙⠳⣦⣤⣴⣶⠶⠶⠶⢶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣴⣿⣿⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢠⣿⣿⣿⡿⠃⡰⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⡞⢸⠇⢠⣾⠿⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢼⣥⣮⣴⡿⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⢀⡾⢿⣿⣿⠿⠶⢦⣾⣤⣄⣀⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⠶⢶⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⡞⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠿⠿
⠀⠀⠀⠀⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠺⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⣤⠒⠒⠒⠒⠀⠂⠀⠀⠐⢲⣄⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀⢀⢈⡴⠳⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠊⠀⠀⠀⠑⠕⡄⠀⠀⠀⣀⣀⣀⡀⠀⣀⠀⠀⡸⠋⢀⠞⠉⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡀⢠⡄⢠⡄⢀⠇⠀⠀⠀⠤⠤⠤⠤⠤⠀⡠⠊⢀⠔⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⡄⠉⠉⢠⠊⠀⠀⠀⠀⠤⠤⠤⠤⠀⡜⠑⢔⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠠⠔⢻⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡒⠒⠒⢒⡒⠒⠀⠀⢠⡆⠉⢲⡄⠀⠒⠐⠒⠒⠒⠐⠚⡰





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

⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⣴⢿⡿⢷⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣴⢿⡿⣧⡀⠀⢀⣠⡾⠛⣹⡟
⣤⣤⣤⡾⠋⢸⣿⠀⠈⠉⠉⠉⠉⠙⠛⠛⠻⠶⢦⣤⡀⠀⢀⣠⣴⠶⠟⠛⠛⠋⠉⠉⠉⠉⠉⠀⢸⡇⠈⢻⣶⠟⠉⢀⣼⠛⢷
⣿⠁⢹⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⠛⠁⠀⣴⠟⠁⣀⣾
⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠲⠶⠶⠶⣦⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⡾⠋⣠⡾⠋⠁
⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⡾⠋⣠⡾⠋
⣿⠀⢸⡇⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⣠⡾⠋⢀⣼⠏⢀⡀
⣿⠀⢸⡇⠀⢰⣧⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣤⡾⢻⡇⠀⠈⠁⠀⢸⡇
⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠃⣀⣴⠟⠁⠀⢸⡇⠀⠀⣦⠀⢸⡇
⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⣤⠾⠋⠁⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⠟⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢹⡇
⣿⠀⢸⡇⢀⣾⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡄⠀⣿⠀⢸⡇
⣿⠀⢸⣧⡾⠁⠈⠛⠷⠶⠶⠶⢶⣤⣤⣤⣄⣀⡀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣤⣤⣶⠶⠶⠶⠶⠟⠁⠀⢻⣦⣿⠀⢸⡇
⣿⠀⢸⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣦⣿⣤⡶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⠀⢸⡇
⣿⠀⠘⠉⠉⠙⠛⠛⠶⠶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⣨⣿⣇⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠶⠶⠟⠛⠛⠉⠉⠙⠀⣼⡇
⠛⠛⠷⠶⢶⣤⣤⣄⣀⣀⠀⠀⠈⠉⠙⠛⠛⠷⠶⠾⠋⠁⠙⠷⠶⠶⠛⠛⠋⠉⠉⠀⠀⣀⣀⣀⣤⣤⡴⠶⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⠶⢶⣤⣤⣄⣀⣀⠀⠀⠀⢀⣀⣠⣤⣤⣶⠶⠾⠛⠛⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣷⣶⣶⣟⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣛⣿⣻⣿⣟⢯⠿⢻⢛⢙⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

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

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⣠⠖⠲⣄⠠⣖⠒⠒⠒⢒⡆⠀⠀⠀⡦⠴⠤⣄⢀⠤⣄⣀⢶⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡇⠀⠺⠥⠤⡄⠹⠀⠏⢩⡄⠀⡤⠄⡇⢠⢤⠈⠇⠑⡌⢡⠊⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠙⠛⠒⢲⡀⢈⡀⡇⡆⢣⠱⠴⠃⡄⡆⢸⡤⢀⡆⠀⡇⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣰⡒⢒⣊⡠⠞⠀⠧⠇⠈⠓⠶⠚⠁⠷⠤⠔⠋⠀⠀⠙⠚⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣇⢸⠩⠀⢰⠋⣉⠆⢰⠋⡉⠳⡠⡖⠒⠲⢤⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢅⠀⡆⢸⠀⣉⠘⡌⠀⠋⣴⠃⠀⢘⢢⠀⢣⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠀⡇⢸⠀⡟⡄⢻⠀⡷⡈⢢⡇⠘⠋⢀⡎⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈⠳⠃⠘⠚⠁⠘⠋⠓⠃⠙⠊⠱⠶⠖⠋⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣑⠲⠦⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠚⠒⠒⠒⠒⠲⠶⠄⠀⠤⠤⠄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⣤⣤⣤⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣟⣛⣻⣻⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣉⢉⡀⠀⠀⠁⠀⠐⠀⢀⣨⣤⣤⠀⢸⠀⠄⠁⠀⠸⠿⠿⠿⠷⠠
⠀⠀⠀⠀⡀⠀⠁⠀⠐⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠦⠶⠰⡴⠀⠀⠀⠀⠂⠀⠨⠍⠩⠍⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠴⠒⢀⡄⠀⠠⡀⠀⠀⠀⠀⠐⠋⠈⠋⠀⠀⠘⠀⢸
⠀⠀⠠⠀⠄⣄⣠⣀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠄⠉⢀⡈⠁⠀⢀⡀⠉⠀⠀⠠⠄⠀⠂⠀⠀⠐⠐⠒⠒⠂⠀⠐⠀⠈
⠈⠈⠁⠉⠈⠈⠁⠁⠁⠀⠠⠀⠀⠘⠂⠀⡉⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠒⠅⠀⣤⠁⠂⡠⠂⠀⠀⠀⠂
⠀⢄⣀⡑⢦⡴⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠⠀⠀⠀⠀⠐⠀⢠⢀⠁⠀⠀⠠⠀⠀⠀⠠⠀⠸⢸⠶⡇⡿⡄⠁⠠⡄
⠀⢀⡤⣤⣤⠶⠤⠼⠁⠈⠀⠀⡄⠀⠉⠀⠘⡇⠈⠀⠀⠀⢀⣡⣀⠈⣈⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠂⠁⠀⠋
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢠⠀⠀⠁⠴⠋⢳⣄⣸⠀⠉⠻⠉⠙⢁⠔⠀⠐⠲⢀⠐⠀⠀⠀⠄
⠀⢀⠀⠀⠐⠘⠂⠀⠀⢀⠀⠀⠧⠀⠔⠀⠶⢲⡤⠤⢄⠈⠁⠐⣂⡀⠢⠀⠁⠈⠀⠈⠁⠉⠀⠀⢦⠀⢠⠒
⠀⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠐⡂⠠⠅⠀⠀⢆⠀⠐⠂⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⢐
⠀⠈⠂⠘⠀⠀⠐⠅⠀⢃⠀⠀⠀⣀⠀⠀⠀⠀⠆⠈⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠀⠰⠄⠀⠀⠤⠀⢀⠄⠀⠁⠘⠀⠄⠈⢀⠀⠀⠂⠀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⣤
⠀⣀⠀⠠⣤⠄⡀⢀⠀⠈⠀⠁⠠⠀⢳⠃⠀⠀⠀⠀⠀⠀⠀⣼⠄⠨⢹⠀⡴⠚⠁⠠⢀⠀⠀⠀⢀⣈⡁⠀⠀⢀
⠀⠁⠀⢱⣶⡶⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠁⠀⠀⠡⠀⠘⠓⠒⠋⠀⠧⠀⢥⡀⣤⠁⠀⡠⣶⡒⣠⣔⢤⣤⠀⠠
⠀⢐⠀⠸⣛⡇⠀⠏⠀⣁⢈⠀⠀⠁⠀⠀⠈⠐⠈⠀⠀⠈⠀⠈⠉⠉⠀⢸⢂⡲⢒⡄⠀⠀⠀⠓⢻⡳⢚⠷⣸⢁⡀⢼⣇
⠀⠨⠄⣀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠐⠃⠠⢱⣶⡆⠄⠀⠁⢰⠒⠀⠒⢂⠈⢉⡥⠟⠉⡅⠉⠙⠀⠁⠚⠛⠉⠀⠋⠁⢀⡻
⠀⠀⠀⠀⠡⠀⠀⣀⣂⠋⢀⣠⡀⠀⠀⠀⢸⣿⣷⢀⣤⠀⠀⠠⠀⠀⢀⠀⠾⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠘⢦
⠀⠠⠤⠤⠤⠥⠀⠉⠁⠁⠈⠀⠁⠀⠀⠀⣺⡟⣿⠀⠉⠀⣉⠀⠀⢀⢘⠀⠀⠀⠀⠂⡃⠀⠀⠀⠀⠀⠀⢀⠀⡅⠀⣠⠟
⠀⢠⣤⣤⣤⣤⣄⠀⠹⣦⠎⠀⠀⠀⠀⠠⠉⠁⠁⠠⠀⠀⠈⠃⠉⠁⠁⠀⠀⠀⠀⠀⡅⠀⠀⠚⠛⠀⠴⠿⠂⠉⠀⠉
⠀⠀⠘⢛⠋⣿⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡎⠀⠂⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⡀⠐⠀⠰⠨⡀⠐⠳⢆⡀⠀⣀⣀⣠⣠⡀⠀⠀⠂⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠛⠁⠉⠁⠀⠙⠂⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠖⢉⣀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢴⣂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡐⠆⠀⢠⠃⡴⠉⠈⢦⠀⢶⡀⠀⠀⠀⠈⣿⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⠀⠀⠑⢬⣯⠲⣄⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⣀⠀⠀⠀⠈⡁⠀⢙⠒⣼⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠉⠀⠠⠀⠀⠁⠀⢸⡞⠓⢤⡈⢃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⠀⠀⠀⢀⣠⠔⠣⣀⠀⢀⡧⠃
⠀⠀⣖⣩⡄⢨⣑⣩⣭⣍⡒⡄⠀⡸⠁⠀⠀⠈⠛⠝⠁⠀⠀⠀⠈⢆⠀⠀⢸⣉⣉⣉⣉⣉⣹
⠀⠀⠈⢎⣀⣀⣀⣀⣀⣱⣈⠁⠀⡇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⡇⠀⣟⢀⣪⣆⣀⡀⠀⠀⣀⣀⠤⠤⢄⠇⢀⠋⠉⠉⠉⠉⠉⠹
⠀⢀⣠⠚⠒⠒⠒⠒⠒⠒⠒⣇⣀⣻⣲⠫⠝⠀⠈⡉⡉⠤⠹⢯⣂⣸⣀⣀⠗⠀⠀⠀⠀⠀⠰⣄⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠊⠉⠀⠉⠒⠊⠉⠉⠉⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⣿⣿⣿⣿⡸⣷⡙⢷⣜⣿⣌⢳⣌⠛⠿⣶⣦⣬⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⡿⣛⣵⣿⣵⣾⣿⢟⣠⣾⣣⠟⠁⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠹⣷⣄⠈⢿⣿⣷⣿⣷⣦⣄⡈⠙⢻⣿⠉⣿⣿⣿⣼⣿⣿⣵⠟⣋⣭⣿⣿⣿⣿⣿⣿⡟⢁⣤⣾⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠹⣿⣆⠀⣿⠇⣈⡉⠛⠿⣿⣦⡈⣿⣇⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⠟⠋⢉⣁⢹⣿⣴⣿⡿⠋⣴⡿⢹⣿⣹⣿
⣿⣿⣿⣿⢿⣿⡄⠹⣿⣼⡿⢸⣫⡩⠳⢦⡀⠙⢿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠋⢀⡴⢚⡩⣹⡎⣿⣿⠋⣠⣾⠟⢁⣿⢣⣿⣿
⣿⣿⢿⡿⣧⡻⣿⣆⢻⣿⡇⣿⢸⢹⣞⡎⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡪⡾⡿⠆⡇⣿⣧⣾⠟⠁⢀⣾⣧⣿⣿⣿
⣿⣿⡎⣿⡜⢿⣼⣿⣷⣿⣇⣿⢸⡶⣫⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣭⣻⣿⡆⣇⣿⡿⢋⣠⣾⣿⣿⢟⣽⣿⣿
⣿⣟⢿⣜⣿⣄⠙⢿⣿⣿⣿⣾⣆⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣼⣿⣿⣿⣿⣿⢟⣫⣴⣿⢟⣿⣿
⣿⡛⠿⣿⣿⣿⣷⣦⣽⣯⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿
⣿⣉⣥⣤⣭⣽⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣭⣉⣹
⣿⣥⣶⣶⣾⣿⣭⣭⣽⡳⠆⠀⠀⠀⠀⢸⠀⠀⣿⠊⢳⡀⠀⠀⠀⢠⠏⢸⣇⠀⢈⠇⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣯⣿
⣿⡿⢛⣽⣿⣯⣭⠴⠂⠀⠀⠀⠀⠀⠀⠈⢧⡀⣿⠇⢠⠇⠀⠀⠀⢸⡀⢸⡿⢀⠞⠀⠀⠀⠀⠀⠀⠀⠲⢦⣭⣿⣿⣯⣍⡻⢿
⣿⣴⠟⣫⣿⡿⠁⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠒⠉⠀⡠⡤⠀⠀⠙⠒⠛⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠙⢿⣿⣿⣷⣍⣻
⣿⣷⣿⣿⡟⣠⠞⢋⣽⠿⠛⠉⣿⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⢀⣠⡶⣾⣟⠉⠛⠿⣭⡛⠶⣌⢿⣿⠿⣿⣿
⣿⣾⣿⣿⣿⠃⣰⡟⠁⠀⢀⡼⠛⡏⠀⠈⠙⠳⠦⣤⣀⣀⢴⣿⠤⣀⣀⣤⠴⠞⠋⠁⠀⢻⠻⢦⡀⠀⠈⠻⡄⢹⣿⣿⣾⣿⢻
⣿⡿⡏⢸⣿⠀⡟⠀⠀⠀⠉⠀⠼⡏⠀⠀⠀⠀⠀⠀⠀⢉⡟⡏⢿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣧⣾
⣿⣿⣷⠀⢻⡆⠀⡀⠀⠀⠀⣀⣼⣿⡇⠀⠀⠀⠀⠀⢀⡞⢀⣷⡘⣧⠀⠀⠀⠀⠀⠀⢼⣿⣦⡀⠀⠀⠀⢀⠀⣸⢏⡟⢸⣿⣿
⣿⣿⣿⣧⡈⢻⡀⣷⣄⡐⠋⢉⣴⣿⡇⡄⠀⠀⠀⢀⡾⢠⠏⡏⢳⡈⢧⠀⠀⠀⠀⢠⣸⣿⣮⡉⠁⢀⣠⣿⢰⠟⠈⢠⣿⣿⣿
⣿⣿⣿⣿⣷⣦⣁⣙⣮⣉⣽⠟⠋⠈⣿⡇⡄⠀⠀⡼⣱⠏⠀⡇⠀⠳⡌⣇⠀⠀⢤⢸⡟⠀⠙⢿⣭⣩⣽⣃⣀⣠⣴⣿⣿⣿⣿
⣿⡻⣿⣯⡙⠛⠛⣿⣿⣿⣿⡛⠻⣦⠘⣿⠁⠀⢸⣷⡁⠀⠀⣇⠐⠀⢹⣾⡄⠀⢸⡿⢁⡴⠟⣛⣿⣿⣿⣿⠛⠛⢉⣿⡿⢛⣿
⣿⣿⣎⢻⣧⢠⣾⡿⢿⣟⢿⣿⣷⡘⣇⢸⣧⡀⠘⣿⡇⠀⠄⠉⠠⠀⣸⣿⠀⢀⢾⡇⣾⢡⣾⣿⡟⣿⡿⣿⣷⣀⣾⠏⣴⣿⣿
⣿⣿⣿⠀⢻⣿⣿⣷⣺⣿⣼⣿⢹⣿⣿⣾⠁⠙⣆⢻⣷⠈⠀⠳⠀⠀⣿⡇⡴⠃⢸⣷⣿⣿⡟⣿⣷⣿⣁⣿⣿⣿⠏⢰⣿⣿⣿
⣿⣿⣿⡇⠀⣿⣇⢿⣿⡟⢻⣷⣴⣿⣿⠏⠀⠀⠘⣾⣿⡀⠗⠀⠪⢰⣿⣿⠁⠀⠀⢿⣿⣿⣤⣿⡟⢿⣿⠓⣾⣿⠀⣸⣿⣿⢹
⣿⠹⣿⣷⡀⠹⣿⣿⣟⣁⣐⣬⣽⣿⣿⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⣿⣯⣥⣑⣞⣻⣿⣿⠇⢠⣿⣿⠃⣼
⣿⣷⣾⣿⣿⣦⣬⣝⣛⣻⠿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⣛⣋⣭⣤⣶⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠔⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣿⣇⣀⣏⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⠇⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣏⣌⠥⠀⠈⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⣀⣠⡤⠀⠀⠀⢀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣤⣶⣾⣿⣿⡿⢃⣗⠀⠄⢼⡄⢃⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⢀⣄⣾⠀⠈⠁⣤⡀⣀⡭⠴⣙⡻⣿⣿⡿⡟⢿⠿⣿⡟⠋⠁⢸⣿⠀⠂⣿⡇⢠⡄⠈⠑
⠀⠀⠀⠀⠀⠀⢀⣿⡀⣼⠿⠓⠁⡠⠟⠪⠴⠀⠩⠈⢁⣄⢂⡡⠈⠉⠉⢠⠀⢀⡄⠁⠊⠲⠄⢄⣠⠈⠙⠛⠻⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠸⡟⠁⠡⠀⠀⢊⠑⢠⡆⡀⠠⣀⠀⠈⠁⠀⠀⡠⠐⠈⠀⡄⢸⣇⠀⠀⠀⠀⠀⠘⠩⢂⠀⠀⡉
⠀⠀⡀⠀⡀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠸⡧⠓⠀⡀⠀⠀⠀⣸⡆⣀⠄⠀⣰⣷⣾⠿⠄⠲⢂⠀⠂⠀⠄⢸⠀⠀⠇
⠀⠀⠃⠠⠀⠀⠤⠈⠐⠒⠀⠾⢿⣿⠿⠆⠀⠈⠉⠀⠚⠃⠾⢧⠈⣉⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠲⠄⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠒⢂⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤⣼⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⡾⠋⠙⢷⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠘⣧⡀⢀⣸⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⢸⡇⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣤⠞⠉⢸⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠓⢲⡖⠒⢲⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⠷⠶⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣉⣉⣉⣉⣁⣉⣽⡇
⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢧⡀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣶⣶⣦⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣼⣃⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣄⢠⢰⣎⡄⡆⠄⡠⡀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡪⠑⣺⠇⡇⡿⠀⢘⢹⣷⡇⡖⣧⡟⣦⡦⡠⠀⠀⠀⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣷⢄⣂⡄⠱⠂⡁⠒⠀⢸⠂⠘⠁⠼⠃⠥⠧⠷⠶⠀⠀⠀⠀⠰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⢯⢥⡽⠧⡀⠀⠗⡆⣠⢸⡇⠹⡆⢠⡀⢷⠀⢒⣛⠀⠉⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠰⠂⠌⣁⠬⠉⠀⠀⠂⠁⠐⠀⢱⠀⢻⡄⠁⠘⢆⣨⡥⠢⠄⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣣⠄⠒⠀⠧⢈⠁⠒⡐⠌⠁⠐⠃⠀⠡⠜⠁⠀⠀⠤⠄⠰⠂⠀⠀⠹⢸⠇⠀⠀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⡨⠀⠁⠀⠀⠀⠌⠉⠀⠀⠅⠀⠀⢈⠀⠐⠀⠀⣀⠘⠂⢣⢀⠗⢁⣍⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠰⢻⣼⠊⠀⠉⠀⠀⢠⠀⣤⢀⢠⡌⡤⠱⠀⠆⡈⠰⢈⠐⠀⡞⣭⠤⡀⣀⣠⡿⡿⡄⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠢⢆⠐⣀⢀⡀⠀⠢⠉⠀⠨⠐⠀⠂⢁⠤⠤⣸⠓⢿⠀⠧⣇⣽⣽⣧⢘⡙⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢪⠓⠀⠡⢀⣀⠤⡤⡐⣶⡞⡏⣳⢿⠸⣿⠆⠻⠋⠠⠀⣶⠋⠼⣹⣎⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⢾⡖⢣⢸⢸⡇⣟⠀⢾⣻⣳⣝⠺⣓⡶⠳⣆⠀⠛⡭⠌⠶⣔⠈⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢻⡌⢿⡴⠞⣜⣹⣾⠬⢕⣺⣭⣭⢵⡖⢋⣏⠀⣿⡀⡴⣧⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠈⠁⣴⣺⣟⡭⠳⠂⣻⠋⡍⢃⢈⣀⣧⣉⡽⠶⠶⠛⠛⠓⠀⠀⠠⢖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠊⠀⠀⠀⠈⠁⢁⣈⣷⣣⣐⢋⡶⢰⡞⢻⣣⢭⣶⠮⠎⣠⡀⠐⠀⠀⠀⠀⠀⠉⠊⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣥⢆⢀⣙⣀⣀⡁⣗⣭⣥⡖⣾⡿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⢋⡭⣭⡻⠩⡊⢧⣸⣲⣵⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣝⣓⣝⡃⣀⣁⡁⡤⣵⢛⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣮⣝⣲⣥⣬⠿⠓⡲⠏⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣠⣤⠴⠟⠂⠉⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣀⣀⣠⣤⠶⠞⠋⢁⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡉⠉⣁⣀⣠⡤⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠈⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠄⠂⠀⠁⠀⠃⠨⠐⠀⠀⠒⠚⡑⠈⢤⠊⠡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣈⠀⠀⠀⠐⣄⡄⠀⠀⠢⡀⠀⢀⠀⠀⠀⠔⠀⡀⠁⠐⣰⠃⣀⣀⣡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠉⢁⠃⠠⣌⠉⡀⠠⠀⠈⠂⠤⢠⣤⠤⠔⠊⠀⠐⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⡂⢀⠲⣌⠓⣷⡦⠴⠦⠤⠀⠠⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠊⠡⠄⠀⠀⠀⠈⠁⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⣒⢒⣒⣒⣀⣀⣈⡁⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣟⣁⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡠⠚⠁⠀⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⣭⣭⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠐⠛⣓⡶⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠉⠉⣽⣣⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⠭⣯⡉⠛⠒⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠚⠓⢸⡀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠶⠚⠒⢆⣠⠞⠋⠉⠉⠙⢧⠤⢖⣒⣋⠭⣹⠇⠀⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⣈⠩⠤⠒⠒⠛⠓⠶⠋
⠀⠀⠀⢠⠞⠉⣉⠒⢤⣀⠞⠀⠀⣈⡓⢧⡀⠀⠀⡀⢰⠇⢀⠀⠀⢀⡴⢻⡀
⠀⠀⠀⢻⡀⠈⠑⠛⠆⠋⠀⣀⣼⠉⠉⢲⠙⢦⠤⠟⠋⠓⠿⠶⣶⣋⠀⠀⡇
⠀⠀⠀⠀⠉⠓⠒⠒⠒⢦⡀⠀⠉⠛⢶⡋⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠲⢤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠸⡟⠲⣦⣤⣀⣀⣀⠀⢀⣀⣰⠚⠛⠛⠓⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣄⠘⠶⠳⣀⡝⠳⠏⡙⢯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⢏⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢿⠀⠀⢀⠏⠹⡄⠀⣳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢶⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⠴⠾⠀⠀⣸⡠⠤⢽⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣲⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣧⢸⡄⠀⠀⢠⠿⡀⠉⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠻⣅⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡤⣼⡈⣧⡤⢤⡞⠀⠉⢹⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠏⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⡧⠞⠀⠈⠙⣦⣄⡈⠉⣩⠝⠁⢠⠏⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠁⠢⣀⠀⠀⣰⠛⢻⠙⠛⠁⠀⢠⠏
⠀⠀⠀⠀⢰⡋⠉⠛⣍⣓⣶⡤⢤⣄⣰⣇⠀⢸⣦⠴⠖⣚⠟⠛⠒⡦
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⠏⠀⠉⠙⠓⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠲⡖⠀⡀
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠁⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠓⠒⠒⠒⠿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠐⠆⠀⠖⠈⣦⠠⡄⠶⠴⠲⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⣀⠜⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⢠⠀⠀⡤⠀⢰⡇⠐⠟⢠⠦⡀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⡄⡛⠉⠉⠙⠛⠛⡟⠒⠒⠒⠒⠒⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠂⠀⠘⠀⣈⣤⣤⣶⡄⠘⢧⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠾⣄⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠍⡉⠛⠛⢳⠲⠶⠤⠤⠤⠄⠀⠀⡏
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡔⠉⠙⣿⣿⣿⣿⣿⣿⣇⠐⠂⠈⠁⣀⡠⠂⠑⡀⡀⠀⠀⠀⡀⠃⠀⠘⠀⢘⠁⠀⠀⠀⠻⣦⣄⡘⣄⢂⠀⠀⠀⠀⠀⠀⢃
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡄⠐⠡⠀⠀⠀⠀⠀⢁⠀⢰⡆⠠⢘⡇⠀⠀⠀⢠⡏⠀⠈⠹⠃⠠⠠⢤⡜⣉⠀⠉
⠀⠀⠀⠀⠀⢶⠀⠀⠀⣀⣘⣧⠀⣴⣿⣿⣿⣿⣿⡟⢂⣠⡀⠀⠀⠀⠀⠄⠂⠀⢠⣴⠀⠈⡀⢸⣇⡔⠋⠀⠒⠒⠂⠈⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⣀
⠀⠀⠐⡀⠀⠈⠇⠀⠀⠀⠉⠙⡆⣿⣿⣿⣿⣿⣿⢀⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⢧⡽⠂⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠉⠒⠂⠒⠀⠀⠉⣤
⠀⣴⣿⣿⣦⠀⠈⠀⠛⢲⠀⠀⠡⢈⣿⣿⣿⣿⡇⠈⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠈⠁⠒⠀⣤⣀⡀⠀⢹
⠰⣿⣿⣿⣿⣷⠀⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠽
⠀⣿⣿⣿⣿⣿⣿⠀⡸⢃⣠⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢣⠃
⠀⠟⠛⠛⠛⢻⣉⢱⣧⣬⠛⠋⠉⠉⠉⠉⠛⠟⣿⣷⣶⣾⣿⣾⣿⣿⣻⣿⣶⣶⣸⣿⠀⡄⠚⢼⠀⠀⠰⠃⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣼
⠀⠠⡄⠡⠀⢀⡛⠉⠘⠙⠀⣀⠀⠀⠀⠀⢀⣀⠉⠋⠛⢻⣿⡶⠒⠛⠻⡿⠿⠶⢿⣿⠀⠘⠀⢸⠷⠠⠃⠘⡎⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⣞⠁
⠀⠀⠚⠓⢶⣤⣄⣤⣦⡤⠞⠛⢪⡵⡟⠻⢿⣿⣿⣶⡤⠞⠙⡁⠤⠀⠀⠀⠀⠀⠈⠉⠐⡂⠐⠿⠃⠀⠀⠄⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠿⠁
⠀⠀⠀⠰⠹⢿⣟⣿⡿⠓⠈⠑⠀⣠⣄⠀⠤⠦⠀⠈⠚⠓⢀⠀⠀⠀⢠⣢⠄⠀⠶⠖⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠄⠀⠀⠀⠀⢠⣿⡦⠤⠶
⠀⠀⠀⢢⡀⠏⢡⡏⠵⠀⠠⡄⢀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⡀⠀⠀⠐⠂⠀⣼⡟⠀⣤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠶⠆⣁
⠀⠺⠊⠉⣭⡄⠁⠐⠀⠤⡀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣀⢠⣄⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠻⢿⣯⡛⠦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⢋⣀⣀⡤⢤
⠀⠀⠀⠀⠛⠘⠛⠲⠦⠄⠀⢹⣶⣶⣤⣀⣀⢀⣠⣤⡞⠁⠉⠍⠉⠉⠉⠀⠤⠠⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣶⢤⣄⣠⣶⠟⠿⠶⠶⠖⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⣿⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⢿⣥⢀⠀⠀⠀⠀⣀⣨⠥⠆⠒⠛
⠀⠀⠀⠀⠀⢤⠤⢄⣀⠩⠁⠒⠀⠀⠀⢀⠏⠐⠀⠀⠀⠀⠓⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠀⠈⠙⠖⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⠄⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐

⠀⠀⢀⡴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢸⣯⣿⣁⠀⣀⣨⣿⠉⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣯⡤
⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠿⢶⣶⣾⠛⠛⠛⠶⢤⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⣈⠃⢀⣤⡀⠶⠀⠀⠉⢀⡍⣩⣡⣾⡀
⠀⠀⢸⣿⣿⣿⡿⠿⠶⠤⣄⣀⠈⠉⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠟⢡⡎⢀⡴⠿⠿⠶⣤⣠⢄⡿⣋
⠀⠀⢸⣏⣿⣿⣄⡀⠀⠀⠀⠈⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣿⣇⣀⣤⠀⢀⠀⠀⣀⣀⠀⠀⠀⠄⠀⠙⣤⢈⢠⣤⡀⠀⠀⠘⢘⠛⠁
⠀⠀⣿⣿⣿⠋⠉⠉⠉⠉⠙⠒⠦⢄⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣿⣿⣿⣿⣾⠀⠆⢀⣀⠛⠟⠻⠧⡁⠀⢀⡠⠋⢀⠀⠙⣿⣇⠀⠀⠀⠘
⠀⠀⢻⣿⣿⠷⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢐⣒⣛⣻⣿⣿⣿⣟⣘⢷⣦⠿⠿⠷⣦⣤⣄⠀⠀⠀⠁⠀⠀⠙⠀⠀⣉⠀⠀⢀⣤⡍
⠀⠀⠈⢿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⣿⣿⢿⡿⠋⢀⣠⡀⠀⢘⣿⡂⠀⠐⡿⠷⠀⠀⠀⠀⠠⣇⠰⠟⠁
⠀⠀⠀⠀⠉⠛⠛⣿⣿⣿⣿⣿⣿⡷⠶⠶⠓⠚⠛⠛⠛⢉⣉⣭⣿⣿⣿⣿⣿⣿⣿⠂⠈⠛⠙⢌⣩⣿⣧⣤⡄⣀⢀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠉⠁⠸⣄⣀⣀⣀⣠⠀⠐⠊⠉⠁⢘⣿⣿⣿⣿⡏⠁⠙⢦⣤⣴⣷⡾⠟⠹⡇⠀⠀⠘⠉⠳⣦
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⢈⡭⠋⠉⠀⠀⠀⠀⢀⡴⠿⠿⣿⣿⣿⣷⣀⣰⠞⠛⠋⡿⠛⣷⡀⠈⠠⣴⣶⡄⠀⢿⣄⢲⣿⣿⡄
⠀⠀⠀⣼⣿⣿⠖⠒⠀⠁⠀⠀⣠⠎⠀⠀⠀⠀⠀⣠⡴⠋⠀⢀⠔⣹⣿⣿⣿⣯⠀⠀⢠⣤⡄⠀⡿⣧⣤⠀⠉⠛⡀⣀⣾⣧⠀⢁⠀⠀⡀⠛⢦⡀
⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⣸⡯⠀⠀⢀⣠⠴⠛⠁⠀⠀⠠⠂⠀⠀⣿⠟⣿⣧⠀⠄⠹⠟⢏⡁⢀⣿⣿⣻⣶⠞⠛⠋⢨⡖⠂⠉⣷⣿⣿⣧⢄⣵
⠀⠀⣿⣿⣿⣷⠂⠀⠀⠀⠠⠈⠚⠓⡿⠋⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸⡇⠘⢿⣆⣀⣸⣄⣠⣷⣾⠿⠟⠟⢠⠀⠀⠀⠀⠂⠀⠈⠉⣿⠙⣿⣷⣾⡆
⠀⠰⣿⣿⣿⣧⡀⣀⡴⠊⠀⠀⢀⠊⠀⠀⠀⠀⣀⠄⠛⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⣿⡿⠯⠛⠁⠀⠘⠀⠀⠁⠀⠀⠀⠀⢀⠀⣸⡟⢠⡾⠗⠚⠃
⠀⠀⢿⣿⣿⣿⣿⣿⣆⣀⣀⣤⣾⠀⠀⣀⡠⠂⠀⢠⣤⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠷⣶⡦⢠⠄⠀⠀⠈⠁⠀⠀⠀⠀⣸⣿⣟⢲⣦⠄
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⡾⠻⣥⣟⣳⠘
⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⡶⢾⣟⡻⠿⢵⣦⠜⠿⠇⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⡴⠶⢶⣶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⣿⣿⣿⣿⣶⣾⣿⣿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⢀⣠⣬⣤⣤⣿⡿⠉⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⠀⢀⣫⣥⣤⣶⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋⣁⣠⠤⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⠴⠒⢋⣡⣤⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⡶⠟⢛⣯⣷⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⣋⣥⣶⣿⠿⠟⢋⣽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡝⠟⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⣠⠶⢛⣽⣽⣿⠇⢄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠇⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡹⠟⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣾⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣴⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣟⡿⠋⠁⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠲⢶⣄⠀⠀⢴⣏⠭⠛⢶⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⢍⠢⡈⠀⠈⠻⣦⠀⠀⠀⠀⠈⠻⣶⣄⢀⠄⡠⡘⢶⣄⣀⣈⣿⣷⡄⠹⣧⣀⣤⡿⠀⠀⠸⠟⠶⣆
⠀⠀⠀⠀⠀⣴⣿⣿⢿⡙⠨⠂⠑⠁⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠈⠙⢷⣦⣈⣫⣲⣿⣿⣿⠿⠿⣿⣆⣈⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣏⠢⡕⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢘⡯⣻⣿⣿⣿⠁⠀⠀⠀⣿⣿⣃⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣰⡿⣕⠮⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠀⢫⣾⣿⡯⠽⣿⠀⠀⠀⠀⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⣿⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⢿⣷⣦⣴⣿⣿⣧⢖⣫⡿⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠄⠉⠛⢿⣿⣿⣿⠿⣇⠀⠀⠀⢀⣠⣽⣿⡿⠀⠀⠀⠀⠀⠀⣼⠃
⠠⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡿⢋⣡⠴⢿⠀⠀⢠⣾⣿⣿⣿⣿⣷⣤⣄⣀⣀⡼⠃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠋⠀⠀⠘⣇⢠⣾⡿⠛⠁⠀⠀⠀⠀⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⣿⡿⣆
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⢻⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⣿⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⠉⠉⠑⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⢀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⣷⠒⠁⠀⠀⠈⠁⠀⢄⠙⠋⠢⣄⠐⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣻⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣾⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⣀⠀⠙⡄⠀⠀⠀⠈⠙⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣾⣿⣿⣿⣿⣿⢿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⡿⢿⣿⣿⣿⡇⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡐⠁⠀⠉⠛⠛⠃⣤⣶⣿⣿⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣧⣄⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣠⠴⡚⠛⠉⠉⠁⠀⠀⢠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡿⠻⢿⣿⡿⠿⣿⣿⣿⣷⣾⣿⡿⠿⢳⡀⠀⠉⠅⠀⢠⣦⠄⡀⠀⠀⠀⡴⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⠀⠉⢰⡄⠇⠙⢛⠉⠁⠐⠀⠀⢇⠁⠀⠀⠀⠀⠀⠿⠿⠃⠀⢀⡜⣃⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⣷⡘⢗⣤⣆⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠂⠀⠏
⠀⠀⠀⠀⢤⠤⠶⠂⠀⠀⠈⠁⠹⠇⠸⣾⡿⡌⣿⠸⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⢁⠀⢀⡘⣀⠅⢀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⢀⡄⠘⢣⡭⠀⡈⢡⣯⠄⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⢀⡀⢀⠀⣰⣇⣀⠀⣴⠀⠀⠙⠁⠄⠀⠐⣆⠘⠿⠐⠟⣧⢶⠄⢠⣦⠶⣻⠀⡘⢷⣾⠁
⠀⠀⠀⠀⢀⣾⠟⠁⣼⣿⡟⢁⣶⡟⠀⠀⠀⠀⠀⠀⠄⠈⠀⣘⠛⡒⠃⠀⢴⡜⡙⠎⠏⠀⠉⠂
⠀⠀⠀⢠⣾⠏⠀⢀⣿⡟⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⣿⣟⡄⠀⣼⣿⡏⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⢰⣿⣿⣿⣐⣿⣿⡇⠈⣿⣧⠀⠀⠀⠀⠀⠀⢰⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇
⣦⡀⣼⣿⣿⣿⠙⣿⣿⣿⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⣷⡀⠉⠑⣲⣄
⠻⣿⡘⣿⣿⣿⣦⣿⣿⣯⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢁⣤⣄⣀⠠⠶⠶⡄⠀⠀⠄⢰⠂⠈⠀⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣾⠋⠉⠓⣄⡀⠀⠀⠀⠑⢄⡀⢰⡀⡀⠈⢠⠒⠒⣤⣄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢄⠉⠠⠀⠈⣷⣄⠀⠀⠺⠂⠉⠉⠈⠹⢸⠏⠖⠒⠀⣾⠐⠮⠗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡉⠀⠀⠄⠀⠙⠷⣦⣤⡄⠀⠀⠰⠊⠉⠉⠙⠤⠒⢳⡈⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠻⢄⠀⣄⠀⢀⠠⡤⢬⠤⠶⢾⡦⠤⡲⡄
⠀⠀⠀⠀⠀⠀⠀⣎⡤⠴⢤⡤⠀⠀⠀⠀⠁⠀⠀⡔⠀⠀⠂⢀⣄⠀⠈⢣⣶⣽⣉⠀⢀⡄⣱⠗⠸⢁⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠜⠈⠋⡓⢯⡵⠋⠙⠂⢠⣀⠀⢺
⠀⠀⠀⠀⠀⢠⠞⠛⠳⣶⣤⡄⠀⠀⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢨⢁⡞⠧⡈⠓⣄⣤⠄⢉⢠⢯⡆
⠀⠀⠀⣠⠐⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠘⠀⢀⡀⢐⣦⣀⣀⠀⢀⠀⠰⣈⡂⠑⡌⠲⣄⡉⢠⠄⠀⠈⠉
⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠖⠋⠀⢹⣤⢈⡆⠢⢼⠳⣄⣸⠞⠈⠃⠀⢰⠂⡀
⠀⠀⠀⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣾⡏⣹⠶⠀⠫⣰⡆⠉⡘⢷⡄⢴⠴⠂⠀⠀⠀⠁
⠀⠀⠀⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠈⠀⠐⠏⠀⡀⠀⢸⠄⠈⢰⡃⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⣰⠋⡾⠀⠀⠀⠀⡀⢀⠶⣄⠀⠀⠀⠀⠛⠲
⠀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⢃⡜⠁⠀⠀⠀⢠⠉⠛⠀⠀⠉
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠱⠤⣄⣀⣀⣀⣀⣀⣠⣄⣀⠀⠀⠀⠀⡰⠀⠐⠀⠀⠀⠀⠔⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⣃⡆⢤⣀⡀⠀⠇⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠴⠿⠶⠆⠲⠶⠶⠶⠷⠶⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣴⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⣿⣭⣿⣭⣿⣿⣯⣿⡗⢿⣿⣿⣹⢿⣽⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣛⣿⣿⠿⢿⣿⣿⠿⣿⠿⠈⠀⠉⠙⠛⠿⠷⣿⢿⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣺⣷⣶⣶⣵⣚⣟⢿⣷⣽⡦⣄⠀⠀⠀⠀⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠯⢿⣾⣿⣿⣿⣿⣛⣯⠭⠭⣭⣿⣳⡆⠀⠀⠀⠀⢲⣦⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⣛⣷⣽⣷⣲⣯⠷⢿⣟⣿⣿⣿⡓⠀⠀⠀⣠⣤⣀⣀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⢿⣿⣿⣿⡽⣒⣝⣻⣿⡛⠿⠟⠃⠀⠀⠞⠉⣉⣻⠁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣷⠙⡟⣇⢻⣯⣿⡟⠀⠀⠀⠀⠚⣿⠃⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⣻⣿⣞⣿⣯⠛⢼⠏⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣩⡿⣷⠋⠙⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡽⠛⠁⠀⠀⠈⠳⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⣄⡀⠀⠀⠀⠀⠘⢿⣲⢤⣀⠀⠀⠀⠀⢠⣤⣠⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦⡙⠲⣄⡀⠀⠀⠀⠉⠓⢮⣄⣀⠀⠀⠈⢧⣈⠙⠲⢍⡳⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⠻⣿⣿⣿⣦⡀⠙⠲⣄⡀⠀⠀⠀⣨⠿⠛⠛⢳⡾⣟⠛⢶⡀⠹⡜⢦
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣦⡀⠀⠙⠳⢦⣴⠿⣿⣿⣶⣾⠁⡿⠋⠛⠻⢦⣘⣎⣳
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣆⠀⠀⠀⣿⢳⡹⣿⣿⣿⣟⡴⠋⠓⠶⢤⠈⢹⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣷⡄⠀⢿⣆⠻⠜⢿⣿⡿⣔⢾⡗⠲⠀⢠⡾⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⠿⠿⠛⢻⣷⣤⢠⠄⢤⡧⡌⣿⢁⡽⠟⠁⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠃⣾⣿⣿⣶⣄⢸⡇⠰⣶⣾⠀⠀⠀⢀⣶⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣿⣿⣿⣿⣾⡇⠀⠘⣿⡀⠀⠀⡞⠹⣯⡉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣄⠀⠸⣷⡄⠀⠀⠀⢹⠿⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⣦⡀⡇⣹⣆⠀⢀⠞⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⢿⣿⣿⣿⣧⠿⡽⠗⣁⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣦⣍⠻⣿⣿⣧⣿⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣷⣤⠙⢿⣿⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣣⣾⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣛⡛⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⢹⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣷⢹⣿⣿⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣸⣿⣿⡇⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⣿⣿⢳⣿⣿⣿⡷⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⣿⣿⢸⣿⣿⡿⠁⠀⠀⣿⡇⣿⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢸⣿⣿⣇⣿⣿⣸⣿⣿⡇⠀⠀⡀⣿⣧⣿⣧⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣛⣛⣛⡛⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣻⣿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣛⣦⡤⢤⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣳⣿⠻⢷⣽⠭⣿⠼⣯⣉⠉⠳⣦⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠉⠻⢦⣉⠲⣄⡀⠀⠉⠓⠒⠛⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⠆⠀⠀⠀⠉⠳⣬⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠙⠶⣬⡑⢦⡀⠀⠀⠀⣤⡄⠒⢲⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⠛⡛⢻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣩⣭⣵⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⣽⠀⢃⠸⡇⠙⢧⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⢦⣈⣹⡄⠀⠀⠙⠳⢦⣝⣛⣡⠤⠤⣤⣤⣛⡿⠛⢳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠉⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⣿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⣿⠀⢸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⢸⡇
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡉⠛⠛⣷⠀⠀⠀⠀⠀⢻⡋⠉⣠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡎⠙⠷⣾⣧⣦⣀⣤⣀⣀⣌⣷⡞⠋⡆⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠁⠀⠀⠈⠉⠈⠛⠈⠉⠉⠉⠉⠀⠀⠁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠻⣷⡶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣷⣶⡿⣋⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠲⠶⠶⣶⠛⠉⠁⠀⠀⠀⠈⠉⠑⢲⡖⠋⠉⠁⠀⠀⠀⠉⠉⠓⣶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⢷⡄⠈⢉⣉⣉⣉⡀⠶⢀⡀⠘⠃⢀⡀⠤⣄⣉⣉⣉⡉⠁⢠⣾⠀⠀⠀⠀⠀⠘⢇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠿⠿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠘⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠙⠶⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠶
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢉⡏⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠓⠉⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠓⠚⠿⣦
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣶⣮⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣵⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢄⢀⣤⣴⣯⣶⣿⣿⣿⣧⡄⠂⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠙⣷⣿⣾⢿⣿⣿⣿⣯⣿⣿⣿⣿⣵⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣙⢵⣶⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣁⣾⣭⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⣧⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⡿⣶⠒⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⢠⣬⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⡁⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠓⠛⢷⣞⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣿⠏⠓⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡀⡼⠧⠏⢻⣿⣿⣿⣿⣿⣿⣿⣷⠴⣿⣿⡉⠿⠃⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠛⠚⠋⠿⣿⡯⣉⠗⠙⠾⣁⠙⠖⠂⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⢈⠈⠑⠃⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣶⣦⣴⡁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣻⣟⣏⠿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠻⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⠠⢀⣨⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⢿⠉⣀⡀⢛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⣷⣦⣽⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣷⣴⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠂⠈⠉⠉⠉⠀⠒⠢⢄⡀
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⣀⣤⣤⣀⠀⡶⠤⣀⠀⠈⠲⣄
⠀⠀⡠⠊⠀⠀⠀⣠⠔⠛⠉⠉⠉⠉⠙⠳⢯⣊⣓⣄⠀⠈⠢⡀
⠀⡜⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠙⢯⣾⢧⠀⠀⠑⡀
⣸⠃⠀⠀⢀⠃⠀⠀⠀⠀⠀⡠⢊⢔⡡⠀⠀⠀⠀⠙⣯⣣⠀⠀⢷
⣯⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠔⠁⠉⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡆
⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠀⠄⠀⠀⣧⡇⠀⢸⡇
⢸⣥⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢊⠄⡠⠂⠀⢠⣷⠃⠀⢸
⠀⢳⡅⠀⠀⠐⡀⠀⠀⠀⠀⡠⠊⠊⠐⡡⠊⠀⠀⠀⣾⠋⠀⢀⡟
⠀⠀⠳⡵⣄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠞⠁⠀⢀⣾⣷⢄⡀
⠀⠀⠀⠙⢧⡧⡄⠀⠀⠑⠂⠤⠄⠤⠤⠤⠐⠊⠁⠀⣀⣴⣾⣿⣿⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠙⠻⢥⣅⣄⡦⠀⠀⡄⢀⠀⣤⣤⣶⡿⠟⠉⣟⣫⣃⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠚⠚⠛⠉⠁⠀⠀⠀⠘⢗⢵⠤⠄⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠒⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⡫⡀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡴⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡣⡂⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡢⣂⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡷⣠⡄⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⠇⠀⡀⣫⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡿⠈⢿⣿⣿⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣏⣀⣜⣳⣿⣿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣇⣀⣈⣻⠏
⠀⠀⠀⠀⠘⠿⣿⠿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤
⣿⣿⣥⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣤⡄
⣿⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⣄⣀⣀⣶⣶⣿⣿⣿⣿⣿⣿⢣⠤⡤
⡅⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣦⣄⠀⣀⣠⣶⡄⣀⡀⣠⣶⣿⠃⣹⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠉⠀⠀⣀
⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⣿⣿⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣧⣿⣿⠀⠀⠀⠀⠀⣠⠀⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⡿⠀⠀⢀⣤⣤⣿⣷⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀
⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿
⡿⠟⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣤
⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⡿⠿⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠛⠉⠁⠐
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉⢻⡇
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⣁⡀⠀⢀⣀⣠⣤⠔⠚⠛⣥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣋⣿⣛⣿⣿⣿⣿⣿⠁⠐⠛⢻⡇⠀⠀⠀⠀⢤⣤⣀⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢷⣿⣿⠏⠁⠤⠾⠋⣿⣷⡇⠀⡄⣾⣿⣦⣄⡀⠀⣾⣿⣿⣿⣿⡇⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡁⠀⠀⠀⣲⠏⠀⢻⠛⢋⠀⢀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣨⣿⣿⣿⢿⣿⣿⣿⣿⠋⠀⣾⣀⣤⣦⣤⣴⡿⠛⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⣼⣿⣿⣿⣿⣄⠀⣟⣹⣿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣭⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⡿⠇⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠻⣿⣿⣿⣿⡟⣿⣿⠏⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣴⡄⢻⣿⣧⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡄⢀⣿⡇⠀⢺⣷⣿⠏⠀⣸⣿
⠀⠀⠀⠀⣰⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⠇⠀⠘⠿⠏⠀⠀⠛⠋
⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠉⠉⣿⣿⣿⣷⣴⣿⣿⣿⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠉⠉
⠀⠀⢰⣿⣿⣿⠙⠁⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠿⠟⠀⠙⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣏⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣶⣿
⠄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⢿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣴⣾⣿⣿⢿⡟⢻⡿⠿⣿⣿⢀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠸⣿⣿
⠀⣿⣿⣷⠂⠀⠀⠀⠀⢠⣴⠉⠙⣸⣿⣿⣿⣿⣿⣶⣾⣿⢿⣿⣿⣴⣇⣿⣡⣿⣃⣴⣿⣿⣽⡁⠀⢸⠁⠀⠀⣰⣦⡀⢀⣶⠂⠀⠀⠀⠈
⠀⣿⣿⣿⣿⣙⠋⣿⠃⠙⡋⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⣰⣿⠀⣿⣿⣿⢸⣿⣦⡀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣦⣶⣶⣶⣶⣶⣾⣿⡏⣴⠟⢷⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣍⣭⣍⣭⣭⣽⣿⣿⣿⣧⡘⠳⠛⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠻⣿⠟⢛⣿⣿⡿⠟⢋⣼⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠰⣿⣭⣤⣴⣾⠟⢛⢛⢛⣛⠋⣉⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣴⣿⣷⣬⣿⣿⠛⣉⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣤⠀⠀⠀⢠⣤⡄⠀⠉⠙⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣿⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠠⠄⠛⠋⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠺⠃⠀⠀⠀⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠠⡄⠀⢠⡄⠈⠇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⡼⢣⡀⠀⠀⠃⠀⠀⠀⣠⢶⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠰⡎⠀⠀⠙⠢⡤⠒⠂⠉⠉⡤⣣⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢴⣆⠀⠀⠀⠘⠢⠤⣀⢸⠀⠀⢠⠔⠺⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⢄⣀⣀⡇⠹⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠘⢾⣆⠀⠀⠀⢸⠀⠀⠀⠀⢱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⠧⢤⠴⠊⣉⡳⢦⡀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⢠⠀⡒⠤⢽⣾⡃⠘⣄⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠉⡩⠭⠛⠉⠉⠉⠁⠀⠉⠉⠉⠚⢿⡒⠠⠤⠰⠚⠢⠭⠿⠞⠙⠿⠉⠉⠙⠒⠶⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠇⠛⠒⠤⢤⡤⠤⠴⠒⠒⠒⠒⠒⠤⠤⠬⣦⣀⣤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠬⠷⠄⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡩⠕⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠕⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⢸⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⣒⣒⣒⣒⣒⣛⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠿⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⢆⡜⠱⠔⠶⠀⠀⠐⣿⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠉⠀⠒⠀⣠⠞⠁⡴⡒⠢⠀⠀⠀⠇⣠⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠐⣫⣀⠀⡦
⠀⠀⠀⠀⡠⠛⠲⠎⠁⠀⠀⠣⢀⠔⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⡴⠊⠑⡤⠀⡀⢀⡀⠀⠀⠀⠀⡠⢀⣟⣊⡁
⠀⢧⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠲⠛⠀⠀⢸⠇⠈⢰⠇⠈⠀⡼⢸⢃⡎⠀⣴⣷⣯⠛⠁
⠀⠀⠁⠀⠀⢤⡄⢀⠀⢼⠀⡖⠋⠸⡆⠳⠀⠀⠀⠾⡠⠀⠀⠘⢦⠀⣾⢰⡄⢀⠧⣿⢼⡀⠀⠙⠛⠁⠀⠀⠀⠋⠐⠆
⠀⠘⢶⡄⠋⠁⡇⠈⢣⢸⡄⠒⠃⢢⣇⡐⠒⠖⠃⠀⠀⠀⠀⡀⣀⠀⠀⠉⠉⠙⠚⠁⠦⠀⢠⡖⢙⠒⠦⠀⠀⠈⠉
⠀⠀⠀⡇⠀⠀⢹⠐⣈⠔⠧⠖⠚⠁⠈⠀⣠⣤⣄⠀⠀⠀⠘⠆⠤⠃⠀⠀⣠⢦⣄⣠⠦⡄⢾⡀⠄⠁⡸⠀⠀⠀⢀⡄⡀
⠀⠀⣀⣃⠤⠞⠘⠋⠁⠀⠀⣀⣀⡀⠀⠀⠛⠟⠛⣠⢶⣶⣦⡀⠀⠀⢀⣤⠯⢄⣠⣄⠀⠑⢤⡑⠒⠚⠀⠀⠠⢊⡅⢠⠟
⠀⠀⠉⠀⠀⠀⠀⠀⢀⠔⠈⠁⠈⢻⡿⠦⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⢇⠀⢸⡀⢀⡇⠀⡼⠇⠀⠀⡰⢾⠶⣫⠖⠁
⠀⠀⣰⡆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠛⠛⠛⠁⣀⡀⠀⠈⣗⠀⢉⣉⠀⢠⠁⠀⠀⠀⠛⠾⠋⠁
⠀⠀⠈⠿⣷⡄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢰⡟⠉⠛⣦⠀⠈⠓⠁⢈⡙⡁⠀⠀⠀⠆⠀⠀⠰
⠀⠀⠀⠀⠉⠀⠀⠀⠈⢣⡀⠀⡆⠀⠀⡜⠁⠀⠀⠀⠀⠀⠈⠷⣤⣄⣼⠀⠀⠀⠀⠀⠀⠰⢐⡖⠂⠰⠐⡔⣦⣼⡃
⠀⢠⢤⡔⣖⡤⢠⣤⠄⠀⢧⠀⠳⢀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣀⣀⡀⠀⠈⠀⠉⠈⢀⠈⠀⠁
⠀⠘⣎⢱⡇⡀⠨⡥⠀⠀⣾⡘⢸⣾⠁⠀⢀⣾⡿⠿⢿⣶⡄⠀⠀⣰⠿⠛⠛⠻⠿⠆⠀⠀⠀⢸⣦⣴⣴⣶⣷⣷⣶⣿⡦
⠀⢀⢘⠈⣇⠃⢀⠈⠃⠀⢹⣯⣿⣿⠃⠀⢸⣿⠀⠀⠀⢹⣿⡀⠀⢿⣤⣀⣀⡀⣸⡟⠀⠀⣀⣈⡉⢉⣡⣀⣀
⠀⠈⠁⠀⠉⠁⠈⠉⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣷⣄⡀⠀⣿⡇⠀⠀⠙⠛⠛⣿⡿⢀⣴⠀⢇⢈⡩⠿⠁⣁⠘⠢⡀
⠀⢀⡰⢂⣻⠃⠀⠀⢀⣀⠀⠀⠀⡖⠀⠀⠀⢀⠈⠛⠿⢷⢿⠿⠗⠂⣀⡀⠀⠙⠀⣨⠐⠠⣨⠀⠀⣤⠲⠀⠀⠐⢙⡆
⠀⠹⠟⠉⢡⠄⣤⠤⠿⢨⠀⠀⠸⡧⠈⠒⡆⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃⠐⠦⠀⢰⣇⡀⢀⢸⠁⣸⢡⡈⣨⠃⣨⡜⢰⡄
⠀⠀⠠⠠⠌⠀⠀⠀⠐⠀⠀⠀⠀⢳⡀⢰⡃⠀⠀⠀⡀⣠⢤⠀⠀⡧⠖⠀⠲⠀⠀⣄⠄⠈⢺⣤⢀⢫⠇⡀⠘⠀⠀⣜⠁
⠀⠀⢸⡀⠀⡆⣆⣆⢠⡀⠀⠀⠀⣎⠀⠠⢥⠀⠀⢈⠎⠁⠐⠖⠀⠁⠰⠀⠀⠀⠀⢹⡀⠐⠉⡀⠒⣻⡞⠁⠟⠀⡄⢸
⠀⠀⠨⡿⠠⡿⠽⣿⢼⣧⠀⠀⠀⠈⠉⠁⡾⠀⠀⠈⠐⢋⡠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢏⡙⣍⠪⠇⠱⣇⣠⣊⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠋⠉⠉⠉⠛⠻⢿⣶⣄⠀⠀⠀⢀⡴⢞⣡⡤
⠀⠀⠀⢀⣠⣤⣶⠶⢶⣤⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣠⣿⠟⠋⠁
⠀⢀⣾⡟⠉⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠿⠿⠶⠶⠶⢶⣦⣤⣄⣀
⣰⣿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠋⠉⢁⣠⡀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀
⣿⣿⠀⠀⠀⢤⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠘⢿⣷⡀⠀⠀⠀⠈⠛⠶⠶⣶⡶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠉⠛⠷⠶⣤⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠠⣄⠈⢲⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢀⠀⠀⢠⣿
⠀⠀⠀⠀⢙⣿⣏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣫⠞⠻⣷⣄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡞⠀⠀⡾⢃
⠀⠀⠀⠀⢸⣿⠈⠳⢬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠈⠙⠻⠷⠶⠶⠚⠉⠀⣠⢴⣿⠋⠀⢀⣾⣵⠏
⠀⠀⠀⠀⢾⣿⠀⠀⠀⠙⢮⡳⠄⠀⠀⠀⠀⢠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠖⣫⡴⠋⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠖⠋⠁⠀⠀⢠⣾⣟⣁⣀
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⢻⣷
⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⠀⠐⠉⠘⠃⣠⡶⢷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡦⠀⣿⣷⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣣⣴⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣄⣀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⡿⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢻⡏⢻⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠀⠀⠀⠀⠘⠋⠉⣀⣀⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟
⠀⠀⠀⠀⠀⣿⡄⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠙⠷⣦⣤⣤⣤⣴⣶⠶⢶⣦⣀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⠶⠀⠀⢀⣾⠏
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡄⢀⠂
⠀⠀⠀⠀⠀⠀⠑⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡐⡠⠁⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠆⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠠⠊⢀⠔⠉
⡔⠀⠑⢄⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠔⡠⠊
⠀⠐⢄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣐⠁⠈
⠀⠀⠀⠈⠢⢈⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⢤⡀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠑⠰⣄⠀⠀⠀⠀⣀⠔⠁⠀⣀⣤⣤⡄⠀⠀⠀⢠⣾⣿⣷⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢀⡔⠁⠀⢠⡾⠛⠋⠉⠙⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠟⢀⠄⠀⠀⠀⢀⠀⡠⠐⠂⠐⠠⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣀⣠⡁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢰⣳⡆⠀⢸⠀⠀⣶⣷⠀⠀⡁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠋⠀⠀⠙⠦⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⠘⡀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⢀⠁⢀⡀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡀⠀⡸⠀⠀⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠋⠉⡀⠀⠀⠢⣀⢀⡀⠄⠀⠑⠄⣀⣀⠤⠁⠀⠈⠙⠓⠀⠀⢸⠁⠀⠀⠈⢆⠀⠈⠂⠑⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠃⣄⠀⠀⠀⡈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⡏⠀⠀⢀⠇⠀⠀⢀⡴⢃⠌⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⠀⠘⣿⣿⣷⣄⣀⣀⣄⡀⢀⣀⣄⣠⣤⣾⣿⣿⡿⠀⠀⢀⣾⡒⠒⠊⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣿⢦⡀⠹⣿⡿⠟⠛⠛⠻⠟⠛⠛⠛⠿⣿⣿⣿⡟⠁⠀⡠⠋⠛⠓⠒⠈
⠀⠀⠀⠀⣀⠀⢀⠎⢀⠔⠋⠀⠀⠹⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠀⣠⠊
⠀⠠⠊⠁⠀⠈⠉⢰⠃⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢀⠖⠉⠉⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠎⠀⠀⢀⣴⡂⢠⠀⠀⢅⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠘⠢⠔⠿⠠⠁⠈⠦⡀⣠⠀⠀⠀⠀⠀⡇⠀⠈⠉⠀⠒⠒⠒⠀⠒⠀⣀⠀⠠⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠠⠀⠂⠀⠀⠀⣀⣀⡀⠤⠤⠄⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠒⠂⠈⣉⢁⣀⠀⠤⠤⠄⠒⠒⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠉⢉⣁⠀⠠⠤⠀⠒⠂⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣤⣄⣀⣀⣀⣤⣴⠊

⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣥⣦⣴⣀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣼⣇
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠉⠈⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣿⣿⣿⣄⠀⠴⠖⣂⣂⡀⠀⣿⣿⣿⣿⣯⠿⠁
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠻⠛⠋⠀⢀⡟⠹⠀⠻⠇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡇⣼⡈⣿⠄⠀⠀⣠⣾⣿⣿⠇
⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢷⣾⣧⣤⠀⠀⠀⠀⣠⣿⡇⣧
⣿⣿⣿⣿⣿⣦⡄⢀⣼⣿⣿⣿⣿⣿⣿⣿⠸⠙⢿⠖⠀⢀⣠⣶⣿⣿⡇⢸⡆
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⣋⣶⡀⠌⠛⠙⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡮⠟⣡⠔⠧⠌⢠⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣾⡟⠀⡀⢠⣾⡿⣻⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠉⠀⠀⢠⣿⡟⠀⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡰⢠⣿⣿⡁⢐⣿⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⡿⠋⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⡄⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣷⣤⣄⣀⣈⡄⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⣼⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣾⣧⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣽⠟⠁⠀⢿⣿⡇⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⢸⡀⠀⣿⡿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡿⠛⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠀⣠⣾⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⠀⢠⣄⠀⠙⢿⣿⣿⣆⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣷⠀⠀⢻⣿⣿⣆⠀⢻⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢿⣿⣿⡀⠘⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠇⠀⣿⣿
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡀⠀⢀⣿⣿⡿⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢻⣿⣿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠟⠁⢠⣾⣿⣿⠃⠀⠾⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠹⣿⣿⣷⣤⡀⠉⠉⠉⠉⠉⠉⠀⠈⠀⣠⣴⣿⣿⣿⣿⣿⣦⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣦⣶⣶⣾⠿⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣤⣀⣀⣠⣤⣤⣴⣶⣶⡶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣖⠼⠄
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠈⠛⠲⢦⣄⡀⠀⠀⠀⢸⣿⠆⠀⠀⠈⠁⠀⠀⠀⣀⣀⣠⣤⠤⠴⢖⣚⣟⣛⣻⣭⣭⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠰⠋⠒⠬⣙⠲⢤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣿⣿⣶⣶⣶⢶⣒⣛⣯⣭⣭⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣷⡤
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣈⠑⠢⢄⡀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢖⣚⣯⣭⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠑⠠⢄⣻⣿⣯⣀⣀⣤⣤⠤⣶⣞⣿⣯⣿⡶⠾⠟⠛⠋⠉⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⢈⣉⣩⣭⣿⠷⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⢸⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⣿⣿⣄⣠⣤⣶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣭⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣇⣀⠀⢀⣴⣿⣿⣿⣷⣄⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣤⠀⠀⠀⠀⠀⢀⣰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣷⠀⠀⢙⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⢩⣿⣿⢿⡿⣿⢿⣿⣮⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠰⢻⡟⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣚⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⣤⡄⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣿⣼⣧⣿⣽⣿⡿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠭⣭⣭⣭⡭⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢰⣶⣶⢳⡞⣶⢳⣶⣶⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿⡇⣿⢟⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣧⡰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣶⠀⠀⠸⠁⠀⣘⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢘⢿⣿⣿⣿⣮⡛⠃⠀⠀⠀⠀⠀⠀⣾⢫⣭⣽⣿⡇⠿⠦⠤⠤⢤⠄⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣷⣭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⡿⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡿⠻⠟⠋⠀⠙⠿⠻⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⠋⠈⠐⠀⠀⠀⠀⠀⠂⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠉⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⠈⠉⠁⡀⠀⠀⠀⠀⠹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣿⣄⠀⣠⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠛⠛⣛⣛⣛⣛⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢴⡆⠚⢳⣸⢀⡐⣔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⢀⠘⢕⠋⠅⠰⠜⠟⢡⠟⣹⣫⣭⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠿⢭⣿⡄⠘⢷⢠⠸⡆⢲⠀⡐⡆⢐⣚⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢂⠀⡄⢊⢀⢈⠀⠀⡀⠠⠈⡇⠀⠙⠤⠄⡒⠀⢄⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠞⠀⠀⠉⠀⠀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠠⠌⠄⢠⠌⡼⡄
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠰⣮⠄⠀⠀⠀⠀⢀⠀⢀⢠⠠⠠⠔⠢⢀⠈⡱⣧⠤⠀⢨⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠠⠀⠀⠀⠊⠀⠀⣀⢀⣀⣂⡼⣉⡇⠐⢧⡿⣿⠨⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣁⢄⠐⢶⢳⠏⡻⣧⡸⡳⣝⣧⡉⠠⠤⣛⣈⠙⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⣆⢮⡩⢼⣲⣿⣿⢿⣚⡿⡤⢳⡆⢸⡀⣸⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠁⣴⠩⡭⡓⢈⡛⠍⣁⣤⢵⣬⡗⠲⠛⠙⠀⠀⠐⢔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣤⡾⠓⠬⡜⠘⢋⡡⠶⣛⣨⣴⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡷⢞⣒⣶⠺⠻⠹⣥⣫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡟⣪⣋⡀⣀⣬⠵⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠾⠵⢷⣞⠩⠙⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⢉⣉⣤⠴⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣉⣉⡠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠂⠀⠄⢀⠀⠀⠀⠀⠠⠄⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢺⣿⠆⠀⠀⣀⠐⠓⠋⠀⢀⣀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣤⣌⠁⠂⠄⢀⠀⠀⠀⠁⣴⠏⣷⠀⣴⡄⢠⠀⢠⡲⢶⡿⢹⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡿⠃⠀⠀⢀⠃⠀⡄⠰⣿⣴⠇⣴⣿⡅⠀⣥⠈⢶⣁⣀⣸⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠆⠀⠀⠀⠒⣤⣄⣈⠛⠾⠀⠀⠁⠆⠀⠉⠛⠁⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⢾⠆⠀⣼⣤⢄⡀⢀⡜⠿⠁⠀⠈⠑⠚⠀⠀⠘⡀⢀⠠⣤⣴⠈⠉⠀⠀⠀⠀⡀⠤⠄⠐⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠤⠴⠀⠉⠋⢥⣬⠞⠒⠂⡀⠀⠠⣄⠀⠀⠀⠁⠀⠀⠘⠹⠀⠀⠀⡆⠀⠀⠀⠠⣤⠒⠲⡄⢡
⠀⠀⣀⡀⣀⠀⠀⠠⠤⡀⠀⠐⢐⠢⡀⠀⠀⠙⣍⠠⢀⡀⠀⠈⠠⡈⠆⡀⠀⠀⠠⠀⠔⠁⠀⠀⠀⢰⠀⠄⠰⠆⣀⣰⠒⠻⢧⢄⣀
⠀⠀⠁⠀⣠⠀⣄⠀⠀⠁⠘⡀⢠⣄⡄⠀⠀⠀⠈⠑⠀⣌⢢⠀⠀⠈⠢⠈⢠⠂⣴⠁⠀⠀⠀⠀⠀⣠⡆⠂⠐⠒⠉⠙⠀⡀⠐⠚⢤⢉⠉⠑⠆
⠀⠀⠀⠀⢻⣿⣿⣻⡷⢸⣿⠹⠦⢌⡃⢄⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠃⡞⠛⠡⢀⠂⠤⣂⠅⠛⠙⣄⣀⡀⠤⡆⠀⡇⠰⠶⠄⣗⠀⠘
⠀⠀⠀⠀⠘⠿⠿⠿⠇⢸⠀⠀⠀⠀⠈⣂⠷⠠⡄⣠⠜⢡⡀⠀⠀⠀⠀⢀⡄⠃⣀⠀⠀⠉⠴⡷⣄⠀⠀⠀⠀⠀⠰⠀⠈⢀⣷⣾⠋
⠀⠀⠀⠦⠄⠀⠀⠀⠐⠋⠀⠀⠀⢀⠔⣡⡶⠆⠀⡌⢠⣟⠋⡼⣣⣴⣤⣬⡙⢎⡠⠤⣙⡞⣀⢰⠃⣄⡀⠀⠀⠀⠦⠀⣀⠈⠙⠃⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⠀⠋⢁⠔⡙⠇⢸⠛⣦⣌⠃⣬⡗⠡⣼⣦⣢⡼⢉⠇⡀⡾⠀⠛⠫⠐⠤⣀⠠⠐⡻⠏⠉⠐⠒⠒
⠀⠀⠀⠀⣀⠴⠲⡄⠀⠀⢀⠰⠷⠀⣤⣡⣤⡄⢇⣈⣤⡙⠛⠋⣣⠘⠋⠳⣦⣤⡤⢶⣧⡤⢋⣴⣧⢀⡀⠈⠑⠶⠆⠊⠀⢀⢀⣀⡀⠀⠠⠶⠴⡆
⠀⡠⠐⠉⠠⠀⠀⠀⢦⣾⠄⠂⠁⠒⢄⠙⢮⣻⡊⢹⠋⢉⠟⡟⢴⠀⠀⢠⢏⠳⣚⡺⠷⣝⢯⡹⠁⢆⡆⠂⠀⠀⠀⠀⡏⠀⢀⠒⢄⠀⢀⡀⠀⡇
⠻⡀⠀⢠⠊⠈⢂⠠⠈⢃⠀⠀⠀⠀⠀⠉⠐⢶⣶⣡⢞⣉⠁⡾⢃⣜⠳⢭⣒⡷⢈⣄⢀⣍⣷⠑⣶⣯⣴⣷⠐⠒⠤⠿⠇⠀⠙⠂⠋⡠⢒⢉⠴⡇
⠀⠡⡀⠀⡂⠐⠀⠐⠀⢀⠃⠀⠀⠀⣀⠀⢨⣽⣯⣇⣉⣻⠟⣰⣿⣿⡷⣦⠙⣿⣾⡟⢿⣿⣾⢀⣿⣿⣭⠉⢋⠁⠀⠀⢠⠀⠀⠀⠸⡐⠉⠋⠀⠇
⠀⠀⠁⡀⠀⣁⠤⠒⠉⠀⠀⠀⠀⢠⠁⠉⠒⠩⠙⠛⠛⠿⠾⢧⣼⣏⣴⣿⣰⣿⣵⣿⣤⡿⠧⠾⠟⠛⠩⠐⠉⠀⠀⠀⠸⡀⠀⠰⠶⠧⠤⠄⠀⠚
⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣀⠣⠤⠤⠤⣈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠂⠉⣑⠂⢄⠠⡂⢤⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⢾⠀⠀⠾⠁⠀⠀⠀⢠⡀⠙⠃⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠉⠺⠦⠀⣿⢀⣖⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣦⠠⠀⠀⢠⡾⠛⡢⣼⡗⠻⠳⡀⠀⠀⠀⠀⠀⠀⣰⠾⡖⣶⣶⡦⣶⡄⠀⠅⡻⢬⣶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠋⠀⠀⠀⠀⠑⠢⠤⠐⠊⠠⠐⠁⠀⣄⠀⠀⣦⠀⠁⠦⡀⢋⣛⣂⠤⠃⠀⠀⢵⠠⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⡎⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⢦⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠀⠀⠀⠀⢀⣰⠧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠶⠶⠟⠛⠐⠃⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⣈⣐⡠⠈⠁⠐⠢⡄⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠾⡯⣭⣄⣤⣀⣀⣈⣉⣉⠐⠚⢷⡄⠀⠀⠀⡀⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠰⣀⠙⠳⢬⣭⣡⣽⡽⠟⠛⠛⠋⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⣀⡄⠈⠀⠀⠀⠀⢐⠑⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢆⣀⡀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠋⠉⠀⠀⠀⠀⠀⠀⣀⢾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠚⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠣⢄⣀⠀⠀⠀⠀⠀⣠⡀⠄⢠⡟⠀⠄⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠊⠌⡙⠛⠓⠰⠴⠴⠤⠞⠋⠀⠌⠀⠀⢀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠘⠀⠐⠀⠃⠚⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠠⠀⠐⠖⠀⠞⠑⠞⠳⠶⠳⠞⠰⠗⠿⠞⠷⠋⠞⡑⠟⠺⠒⠷⡺⢞⠶⢃⠾⡚⢟⠳⠛⠚⠛⡛⠟⠻⠟⠛⠀⠆⠀⠂⠀⠀⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢽⣿⣿⣿⣧⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⢿⣿⣿⣿⡿⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢷⢶⣶⣤⣤⣀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⣟⣻⣯⡽⠿⡗⠻⡿⣦⡠⠐⠁⠀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡟⢻⡏⣹⣿⡄⣅⢹⡄⠘⡜⣷⡴⢶⡄⠀⠄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠄⢴⣇⣽⣿⣷⣄⣱⣿⣾⠷⠾⣧⣾⢷⢚⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣸⣧⣶⠼⣿⣛⣿⣯⣿⣿⣶⣿⣿⣿⡋⠌⠌⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣷⡟⢿⣿⢿⣿⣿⣻⣽⣿⡿⢹⢹⣇⢃⢁⢊⡇⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣇⠛⣿⣦⣿⡾⣿⣿⡟⠛⠙⡄⡈⣎⣿⣾⣦⣬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢹⠈⣁⣇⣬⣿⣷⣾⡿⢿⣿⣿⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⢷⣾⣻⣟⣹⣭⠿⠶⢿⠛⠛⠛⠁⢸⣯⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡏⠙⠛⠋⠁⢳⣶⣾⣿⣄⣀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣆⠀⠀⣠⣿⣿⣿⡿⡿⠛⠃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠉⠉⠛⠿⠟⠁⠀⠀⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⣠⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣷⣾⣶⣿⣿⣿⣿⡿⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣶⠞⠳⢶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⢾⡇⠀⠀⠀⡿⠶⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⠀⣴⠛⢁⣀⣀⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠿⢦⡴⠾⢧⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⢸⠇⠀⠈⠁⠀⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠆⠸⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⠹⣦⣀⡀⠀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠉⠉⠉⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡞⠉⠀⠀⠈⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠟⠻⠛⠟⠿⠻⠟⠷⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⡴⠟⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣠⣤⣤⣤⣤⣤⣀⣄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⡖⣦⣰⡆⣟⢦⡄⣲⠆⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠉⠁⠁⠘⠁⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠶⠶⢷⣶⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠋⠉⢉⣉⣡⣤⣤⣀⣀⣀⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣼⡟⠉⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⡾⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠟⠻⡏⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⢰⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⣀⣤⣤⡉⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⠸⡇⡧⢸⣿⡏⠈⣿⡁⠀⠀⠀⠀⠙⠉⢆⠀⢠⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⡆⠳⠹⠘⣿⣿⡄⠈⢀⡀⠀⠀⠀⠀⠐⠂⢀⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⠘⣿⣦⣀⡀⠀⠀⡄⣸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡆⢰⣿⣿⡿⠉⠉⠉⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⠁⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣭⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣭⣭⣿⣏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠚⠛⠛⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣶⣿⡿⠁⠀⠀⢹⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠉⣿⣿⣄⣀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠻⠟⢻⣿⡿⠛⠿⢁⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⣶⡟⠛⢿⡖⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⣠⣼⢧⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⣉⣉⡉⠛⠓⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⣠⠴⠂⠀⠀⠀⠀⠀⢀⣴⠛⣧⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⡴⠋⠁⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠘⣆⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⡴⠋⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⢀⣀⠀⠀⠘⣦⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⠎⢳⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠴⠶⠚⠋⠉⡙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠷⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡏⠀⠀⠀⣀⡻⣤⢷⣶⣤⣀⠀⠀⠀⠀⠀⢀⣤⣶⡾⣤⢟⣀⠀⠀⠈⠙⢿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢸⠃⠀⠀⠀⠈⢻⠃⠸⢿⡿⠹⡆⠀⠀⠀⢠⡏⠻⣿⠇⠈⣿⠀⠀⠀⠀⠀⠈⡇⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡴⢻⡟⢿⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⣸⡇⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡿⢻⡟⢦⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠷⣼⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢹⡇⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣯⠼⠁⠀⣸⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⡶⣶⠿⣶⠶⠶⣶⡶⢶⡶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣐⣄⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠛⠛⠛⣿⡏
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣂⣤⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⠇⠀⣀⠔⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢛⠋⡀⢀⣬⣦⡀⠀⠀⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠒⠂⠈⠀⠀⣷⡿⠋⠄⠀⠀⠀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠉⠁⠀⠀⠀⠀⠐⠈⠀⠀⢈⡄⠀⣀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣷⠀⢀⠀⣶⣦⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣜⣹⣿⢻⡇⠀⠀⠀⠀⣿⡿⠛⠻⣿⠀⠀⠀⠈⠁⠀⣦⣦⣾⡇
⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣼⡇
⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⣁⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⠄⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠘⠿⠛⠛⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⣟⠁⠀⠠⠇⢲⣾⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣙⣛⣛⣛⢸⣿⡇⠀⠀⠀⣿⣿⣿⡇⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⠸⠿⠃⠀⠀⠀⣿⣿⣿⣧⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⠇⣿⣿⣿⡇⢀⣿⣿⠋⠉⠙⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣿⣿⠴⣿⣿⣿⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⠀⣿⣿⣷⣶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡏⠉⣠⣌⠈⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣅⠸⢿⡿⠀⣼⠿⠿⣿⠟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠻⠴⣄⣰⡶⠿⠀⢀⣀⣀⠀⠛⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢠⣶⡄⠙⣿⣄⠀⢰⣿⣿⣿⣷⡀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠈⡛⢁⣨⣿⡏⠀⠘⢿⣿⣿⠟⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⢀⣀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠟⠛⠛⠛⠋⠉

⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠋⠉⠙⠓⠶⣄⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡄⠀⠀⣄⢀⠀⢀
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡔⠮⡘⠙⡟⢄⠎⠉⠒⢻⣤⣀⡟⠀⠀⢀⢤⠆
⠘⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠬⣦⡀⠀⠁⢸⡄⠀⠀⠀⠁⠁⠱⢺⡇⠁⡏
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡤⠖⢶⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⢦⠀⠈⠀⠀⢠⣄⣤⣼⣄⣤⣀⡀⠀⠀⡰⣷
⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠞⠁⠎⠀⢀⠞⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡦⡤⠞⠉⠁⠀⠈⠉⠛⣷⣦⡄⠰⡥⣀
⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠶⠚⠁⠀⠸⣄⡴⠋⠀⠰⠃⢠⠃⢠⠚⡆⢀⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠛⠃⠀⣖⡁
⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠠⣀⣠⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠃⠀⠣⠜⠀⠀⠊⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠥⠄⢀⡼
⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠆⢀⡌⠉⠓⢦⡀⠀⠀⠀⣿⠿⠇⢀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠰⡏⠷⠶⠠⡆⠀⠀⣠⣶⡷⠀⣠⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠛⠒⠲⠞⠁⠀⣜⣡⠴⠦⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠾⡖⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠈⠀⠀⢄⠀⠀⢀⠤⠶⠶⠃⠈⠀⠁⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠰⣄⡀⠀⠀⢣⡀⠀⡀⢈⡄⣠⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠉⠀⠀⠾⠃⠸⣇⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠙⠓⠒⠶⠶⠄⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡤⠶⡋⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢇⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣤⡀⠈⠻⢍⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢿⢀⠃⠀⠀⢀⣼⠓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠊⠻⢄⣀⣠⠾⡀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠸⠀⢢⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⢀⠠⠀⠴⠀⢸⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⣹⣿⣿⣳⡄⠀⠀⠀⠀⠉⠳⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⣌⠙⠿⢷⣮⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣶⣿⡿⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠛⡗⡮⠙⠋⢀⣂⠖⠀⠀⠀⠀⠰⣆⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡚⠀⠀⠀⠑⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⢀⣀⡵⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣎⠀⠙⢿⢢⡀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠤⢟⣭⣥⣶⣟
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢙⣦⣀⣳⡙⠢⠌⠁⠀⠤⠀⠀⣘⠯⠈⠙⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠖⠁⠄⠀⠀⠤⣙⢦⡝⠹⣤⣻
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠑⣝⣆⠹⣟⡤⠀⠒⠒⢲⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢧⢹⡅⠀⠀⠀⢸⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⢄⣀
⠀⠀⡼⠃⠀⠀⢄⠀⠀⠀⠄⠀⣀⡀⢤⣹⠃⡈⡃⠀⠀⠀⠘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑⠤⡀
⠀⡰⠃⢀⠈⠢⢈⢢⡀⠀⠀⣀⣀⣭⣓⠚⢳⣁⣷⣄⠀⠀⠀⡇⠚⣡⣵⡶⢠⠀⠒⠘⢄⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠞⠁⠀⠀⠙⠢⢤⣁⠻⣯⡉⠀⠀⠠⠤⠐⠚⠻⣷⠬⠀⠀⠀⡞⠂⠀⢁⠠⠄⠀⠀⠀⢀⠀⠀⠀⢀⢄⢴⣉⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢳⣄⠀⠀⠀⠀⠀⠀⢘⡇⣀⣀⣠⣁⣀⠡⣌⡃⠀⠀⠐⠀⠀⠀⠀⣀⠴⢧⣎⡀⢀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢞⣰⡿⠀⢠⠀⠀⠀⠀⢫⠢⡄⠳⣄⠀⢀⣢⣛⠆⡀⠀⢠⣄⣤⢀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠴⠚⠁⠎⣠⠔⠁⠀⠀⠀⠀⠈⡇⠁⠀⠈⠐⠋⠠⡙⠴⢀⠄⠔⣭⣭⠢
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⠳⢦⣀⠐⠊⣁⣀⣀⣀⣀⣤⡤⠤⠁⠀⠀⠀⠀⠐⠄⠈⠱⡄⠐⠂⠊⠁⠐
⠀⢾⡒⢒⠀⠀⠀⠀⠠⠠⠄⠀⢁⣀⣀⣀⡀⠙⢍⠣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠢⡀
⢰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠒⢯⡳⠄⠣⡈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣄⡖⣀⣰⣷⡆
⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡷⠲⣾⣧⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⣿⣷⣷⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣷⣤⣿⡟⠛⠙⢿⣿⣿⣿⣿⣻⠇⢀⠸⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠈⡿⢿⣿⠈⣿⠀⠈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⡁
⠀⠀⠀⠀⠈⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⡷⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⡀⠀⠀⠀⡈⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣤⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⢀⠈⠅⢼⠀⣰⠦⠈⠁⠀⢸⣿⣿⣿⣿⣿⣿⠉
⣠⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⡏⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⢩⡇
⣿⣿⡟⡔⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠈⣷⣼⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⠀⠈⡁
⣿⣿⡀⣇⡜⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢄⣀⣀⣤⣤⣶⡶⣶⣶⣶⣶⣶⣶⣾⣿⣷⡆⢹⠀⠀⠇
⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠿⠏⠀⠀⠈⢹⡿⠿⠿⠿⠟⣿⡿⠳⠘⠳⠶⠂
⣿⣿⣿⣿⣧⢺⣦⣦⣇⠀⡀⢿⠀⠀⠀⠀⠀⠻⣧⣀⡿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠑⣄⣸⣷⣦⣤⠤⠯⢥⡄⠀⠀⠀⠄
⠛⠛⠛⣿⣿⣾⣿⢧⣿⠀⣷⣘⢧⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣦⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣁⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢿⣿⣿⣿⣷⣿⡆⣷⣧⠘⣄⠀⠀⠀⠀⠘⠛⠉⢉⣿⣿⣿⣿⣿⣿⣶⣆⠀⣠⠂⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠙⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠿⠻⡿⠛⢻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠛⠛⠿⠛⠛⠉⠉⠉⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣆⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢀⣴⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣷⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣠⣾⣶⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣾⣁⣠⣾⣿⣿⣿⣦⡀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠈⣻⡟⢉⣵⣿⣿⠟⠁⠀⠀⣼⡟⠀⠀⠀⣀⣠⣤⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢻⣿⡿⠛⠉⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠻⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡷⠜⣷⠀⠀⠀⢰⣟⣀⣀⣴⣦⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⠿⠟⠋⠉⠀⠀⠀⢹⣧⠀⠀⠀⠛⠛⠋⠉⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⣴⣶⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⣋⣥⣶⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣿⣿⣿⡿⠟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⠿⠛⢉⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣰⣿⠛⠿⡿⠛⢿⡿⢷⡀
⠀⠀⠀⠀⠿⣧⡀⣠⠀⠀⢴⣄⠀⢈⣶⣶⡶⠀⠈⠙⠛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠈⠳⢉⡀⠀⢠⡟⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⢀⣴⣾⣿⡷⣠⣿⣶⡆⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠰⡄
⠀⠀⠀⢠⣿⡿⠋⠁⢠⠏⠁⠀⠀⢸⣿⡇⠀⠀⠀⣤⣄⠀⠀⠀⠘⠉⠀⣦⠹⣆
⠀⠀⠀⢸⡟⠀⠀⠀⣶⣤⡀⢰⡀⠈⣿⣧⠀⠀⠀⣿⣿⡷⠂⠀⠀⠀⠀⢸⡄⢻⣆
⠀⠀⠀⠈⣥⣤⠀⢀⣿⡿⠁⠈⣷⡀⢹⣿⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⢸⣇⠘⣿⡆
⠀⠀⠀⢀⣿⣿⠀⢸⡿⠁⠀⠀⣿⣧⠀⢿⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⡀
⠀⠀⠀⢸⣿⡇⠀⠼⠁⠀⠀⠀⣿⣿⠀⠸⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣧
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠃⠀⣀⣤⣶⣤⣤⡤⠀⠀⢰⣿⣿⠀⠀⠀⠉⡄
⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⣸⣿⡿⠀⣠⡴⠿⠛⠛⠉⠁⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠈⠙⠛⠿⠦⠀⠴⠛⢩⣥⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠘⠛⠋⠉⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⢀⣠⡴⠋⠀⢀⣤⣶⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣿⣿⣷⡀
⠀⣠⣶⣿⠋⠀⠀⠀⣾⣿⠇⠀⠀⠀⠙⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠉⠁
⣸⣿⣿⡏⠀⠀⠀⠰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⡀⠀⠀⣀⠤⠶⠟⣛⣋⣁⣤⣤⠀⣤⣀⡀
⢰⡀⠀⠀⠀⠀⢀⣴⣷⣦⡀⠀⣷⡀⠀⠀⠀⠛⠛⠿⣿⣿⣿⣿⡄⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠈⣷⡀⠀⠀⠀⠺⢿⣿⣿⣿⣆⢹⣧⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⢹⣷⡄⠀⠀⠀⠀⠙⢿⣿⣿⣎⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿
⠀⠀⠈⣿⣿⣆⠀⠀⠀⠀⠀⠹⣿⣇⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢻⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⢹⣿⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⢻⡆⢹⣿⣿⣿⣦⠀⠀⠀⠀⠀⢹⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠈⡇⠈⣿⣿⣿⣿⣷⡄⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⢟⣦⣄⣀⡘⠀⠘⣿⣿⣿⣿⣿⡦⣀⣤⣴⡆⠸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠀⢀⣼⣿⣿⣿⣷⠀⠀⠀⠉⠉⠛⠛⠻⠇⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠋