Policeman Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣴⠟⠛⢷⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢦⣤
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠈⢹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣤⡄
⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤
⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⣿⣧⢴⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⢺⣿
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣏
⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣦
⠀⠀⠸⣯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣿⠁
⠀⠀⣠⠿⣿⡏⠻⣷⣶⡶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠴⠶⣷⣶⠟⠛⠋⢉⣩⡿⢿⡀
⠀⠀⣿⠀⣨⡿⠖⠊⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠒⣶⣾⣿⡿⠟⠓⢿⣿⣿⡞⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠷⠶⠿⠛⠻⣧⠀⣿
⠀⣤⡟⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄
⣤⠻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉
⣿
⠉⣄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⠀⢀⣀⣼⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠳⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠄⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠛⢻⣦⣄⡀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⣀⣀⣼⡿⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠙⣷⣄⠀⠀⠙⠛⠻⠷⣶⣦⣶⣶⣶⣦⣴⣶⣶⣶⠿⠛⠛⠁⠀⣠⣾⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢀⡽⠛⢳⣤⣀⠀⢀⠀⠀⣰⡏⢹⡟⢻⣆⠀⠀⠀⠀⠀⣠⣤⣾⣿⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠛⠛⢿⣾⠏⠀⢸⡇⠈⢹⣶⡾⠳⣴⠟⠛⠀⠘⢿⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⢸⡇⠀⠀⠉⠁⠀⣸⣷⣆⣠⣄⢸⣷⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢣⣾⡇⠀⠀⠀⢲⣿⠛⠁⠀⠻⣟⠈⢿⡀⠀⠀⠀⠀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠙⠻⢧⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⢹⡇⠀⠀⠀⢸⡧⠀⠀⠀⠀⢸⣿⢸⣷⢀⣤⣴⠟⠋⢹⣷
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⢈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣷⣤⡀⣠⣤⣾⠟⠀⢿⠟⠁⠀⠀⠀⠈⢻⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⣴⡟⠉⠉⠙⠿⠿⠷⢦⣤⣤⣤⣤⣄⣀⣤⣤⣴⣶⣿⣿⣶⣦⣤⣠⣤⣤⣤⣤⠶⠾⠿⠿⠛⠉⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠻⢷⣶⡆⠀⣴⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢿⡏⠀⠀⠀⠀⠀⣻⡏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⣤⠀⢰⣶⡾⠋
⠀⠀⠀⠈⠙⢿⣤⣿⣇⠀⠙⣿⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⢿⡇⠀⣸⣿⣤⡿⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷⣾⣧⠀⠉⠉⠉⠻⠿⠿⠿⠶⠶⠶⠶⠀⣿⣧⣤⣤⣤⣤⣤⣾⣷⠴⠶⠶⠶⠶⠿⠿⠿⠛⠉⠉⠉⠀⢰⣿⠾⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣄⣄⣀⣠⣄⣀⣀⣸⣇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣴⠞⠛⠛⠋⠁⠀⠀⠀⠁⠀⠈⠀⠈⠛⠛⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣣⣤⣤⣀⡀⠀⠀⣀⣤⣤⣤⣛⣿⣼⠿⢧⣝⢻⣤⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣜⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡏⠙⠛⠋⠀⠀⠀⠙⠛⠋⣭⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠳⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⡶⠶⠟⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢶⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⣦
⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡛⠛⠛⠉⠉⠛⠛⢛⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣄⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠙⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣩⣥⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣬⣍⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠛⢿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠻⢍⣭⠉⠉⠛⠻⠿⠿⠛⠛⠿⠿⣿⣿⣿⣿⠿⠛⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢲⡴⠶⠖⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⢦⡖⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⡤⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⡀
⠀⠀⠀⠀⠀⢠⡟⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣻
⠀⠀⠀⠀⠀⠀⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⡀⠀⠀⢀⡀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⣠⠴⠞⠁⠙⣦⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣠⠋⠉⢳⣤⡄
⠀⠀⠖⠃⡀⠀⢻⠁⠀⠀⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠛⠛⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⠉⠉⡁⡶⠀⢀⠜⢢
⠀⠀⠣⣀⡸⠛⠁⠀⣠⠾⡄⠀⢹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⢨⢦⡀⠀⠉⠐⢇⣀⡜
⠀⠀⠀⠈⠙⠲⣆⠛⣡⠞⠛⠶⠎⣈⡛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣉⣵⠶⠟⠣⣌⠓⢶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⣧⡀⠀⠀⠀⠀⠀⠉⠻⡛⠳⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠞⣻⠛⠉⠁⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣤⠀⠀⠀⠈⠳⢤⣀⡀⢀⣩⣿⣿⣿⣿⣍⠀⢀⣀⡤⠚⠁⠀⡀⠀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠤⠄⠤⠤⠀⠈⠑⠋⠀⠈⣿⣿⠁⠈⠑⠉⠁⡄⠒⠀⠀⢹⠀⢿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠘⢀⣀⡀⠐⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⠈⠁⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⠰⠾⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⡅⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⣤⣤⣤⣤⡼⠀⠀⢦⣤⣤⣤⣤⣤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠦⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣹⣿⣿⣿⣿⣷⣶⣾⣽⣓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⣿⡿⠋⠉⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣮⣳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡇⠁⣀⢀⡄⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠸⣌⣿⣿⣿⣿⣧⠀⣈⣠⣷⣶⣶⣶⡶⣶⣯⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣫⣶⣿⢻⣿⣿⣿⢟⣿⡾⠛⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣿⣿⣿⣿⣛⣂⡀⠀⢹⡿⠋⠛⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡙⠛⠟⠉⠉⠉⠀⠀⠀⠀⡰⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠋⠁⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⣇⠀⣭⡽⠄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠉⠉⢁⣀⣀⡀⠀⠀⠀⢀⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠖⢩⡭⠄⠀⠀⠀⠀⢀⣼⠃⠀⠹⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢀⣶⣿⠏⠀⠀⢠⠏⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣤⣤⣴⣶⣿⠿⠋⠀⠀⣴⠋⠀⠀⢀⡾⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠻⠛⠁⠀⠀⣠⡞⠁⠀⠀⢀⣬⢶⣶⣤⣉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠸⣿⡳⡄⢀⣤⠞⠁⠀⠀⢀⡴⠋⢻⣶⣿⣿⣿⣿⣷⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⣙⣿⡟⠋⠀⠀⠀⣀⠴⠋⠀⠀⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⢻⣼⣿⣿⣷⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠸⢇⣿⣿⡙⠿⢶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠟⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠘⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⠶⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡍⠁⣽⣿⣿⣦⠝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣯⣼⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢻⣿⣿⣿⣿⣿⣿⠟⢉⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡷⢿⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⢿⡿⢿⠋⠹⣿⣿⠇⣸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⣠⣾⣶⣶⣧⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣴⠟⠛⠯⠙⠿⠿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⣀⢀⡀⣠⡾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⡍⢿⣿⣿⡿⠟⠛⠛⠀⠈⠃⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠄⠙⠛⠉⠁⠈⠀⣧⠈⢻⠋⠀⠀⠀⢤⠊⠀⡐⠀⠀⠙⠻⢿⡧⡤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠯⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⠏⠙⡑⠒⢀⠺⣄⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⣦⣤⣤⣤⣤⡄⠀⠀⢀⣤⣤⣤⣤⣄⣀⣸⣾⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⡿⠿⠿⠇⠀⠀⠘⠿⠿⠿⣏⣽⡿⣿⣏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣠⠴⠒⠉⠁⢸⡤⠧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠁⠥⠀⢀⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⠁⠀⠀⠀⠀⢠⣆⣀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠠⠊⢀⠀⠀⠀⠀⠀⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠔⠁⠀⠈⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⠠⠤⠄⠀⠀⠀⣀⣤⣶⣿⢿⣿⣶⣌⣁⠶⢦⣄⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣿⡿⠿⠛⠉⠀⠀⠉⠛⠿⣯⣿⣿⣿⣿⠶⠋⢁⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣭⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⣀⡀⠀⢠⣤⢀⣀⣀⣀⣠⣤⣴⣶⣷⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣛⣻⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣭⣭⣴⣿⣿⣿⣿⣿⣟⣣⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⠟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣻⢿⣟⡛⠟⠉⠉⠛⠉⠁⠈⠊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠉⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣸⠿⢿⣧⣼⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠈⠱⠀⠀⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠤⠴⣤⣐⠀⠀⢀⢔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠋⠉⠑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⣞⠃⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠘⠂⠀⠀⠉⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡤⠶⠒⠋⠛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠓⠦⣤⣤⣤⣤⣠⡤⡾⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⢀⡀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠐⢂⣠⡃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⠀⢸⣧⡄⣠⣀⡀⣤⣠⣶⠀⠀⢠⣤⣴⠈⣭⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⡖⠿⠿⠇⣿⣿⣿⠃⠀⠸⠿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⢉⠍⢍⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⣀⠹⣀⡰⠅⣀⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⡩⠤⠐⠒⠒⠒⠒⠠⠬⣑⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠋⠭⠉⠛⠻⠭⠽⠿⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⡅⠀⢶⠆⠀⠀⢴⡶⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢣⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣢⠤⡄⣠⠤⣖⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠋⠉⠠⠁⠱⠔⠉⠚⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⡤⢀⠠⠀⠀⠄⡀⢄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢤⡶⠋⠀⠁⠀⠀⠁⠀⢹⣆⠤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠹⢆⠀⠀⠀⠀⠀⠀⠼⢇⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⣀⣷⠦⣶⢶⡴⣮⡀⣀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⢀⣇⣵⡀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣦⠤⡎⢳⠤⣶⣶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⣈⠾⠁⠘⢾⣁⣀⣸⠄

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠛⣋⣉⣉⣉⡉⠛⠓⠲⠶⣤⣄⣀
⠀⢀⣀⣰⠶⢾⠋⠙⣧⢦⣄⠀⠀⠀⠀⠀⢀⠴⠞⠋⠉⠀⠀⠀⠸⣯⡭⠤⠴⠶⠦⠭⣽⠗⠀⠀⠀⠈⠙⠳⢦⣄
⢰⡟⢹⡇⠀⢸⠀⠀⡇⠀⢻⡄⠀⠀⠀⠸⠧⣤⡀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠈⢛⣦⡀
⢸⡃⠘⠃⠀⠸⠀⠀⠇⠀⢸⣃⣴⠖⢻⣦⠀⠀⠻⣆⡤⠖⣋⣡⣤⣼⣿⣿⣷⣶⣶⣾⣿⣤⣤⣉⡙⠶⣄⡀⣴⠾⠛⠉
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⢰⡟⠁⠀⢠⡶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣸⠇⠀⠀⣾⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣵⣿⠟⡁⢸⡇
⣿⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻⣼⡇⠀⢯⣝⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣋⣥⠀⠀⣷⣸⠃
⣿⡀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⠃⠀⢀⡴⣂⣤⣤⣷⣄⠀⠀⢀⣾⣧⣤⣀⢌⡁⠀⠀⢹⠋
⠹⣧⡀⠀⠀⠘⣧⡀⠀⠀⠀⠀⣠⡿⣤⣀⣀⣀⣼⠀⣠⠟⠋⠁⠀⠀⠀⠙⠛⠛⠋⠀⠀⠈⠙⠻⢿⣦⡀⢸⣦⣤⣤⣀⣀⣀⡀
⠀⠈⠻⣦⣤⡀⠹⠿⠶⠶⠒⠋⠁⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠉⠉⠙⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢿⣦⡀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⢀⣸⡷⠟⠁⢠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⢀⣿⣿⣿⣷⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡆⠀⠀⠀⠀⣰⣾⣿⣶⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠃⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠉⠙⠻⠿⣿⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠻⠦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠼⠃⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠦⠤⠤⢤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⣉⣀⣀⣉⠛⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠟⠋⠉⠀⠀⠀⣽⠟⠻⣯⠀⠀⠀⠉⠙⠻⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⡀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⢀⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣯⣭⣭⣤⣤⣤⣶⣶⣤⣤⣤⣭⣭⣽⣿⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠷⣦⣤⣤⣀⣀⣀⣀⣤⣤⣴⠾⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⡇⠀⠀⣈⡉⠉⠉⠉⠉⢉⣁⠀⠀⢸⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣸⡇⠀⠀⠙⠁⠀⠀⠀⠀⠈⠋⠀⠀⢸⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣇⠀⠀⢀⣤⠀⠀⠀⠀⣠⡄⠀⠀⣸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠙⢷⣦⣴⡾⠟⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⡟⠻⠷⠶⠶⠾⠟⢻⣧⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣭⣭⣭⣍⣽⣷⡀⠀⠀⠀⠀⢀⣾⣯⣭⣭⣭⣩⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠉⠉⠉⠙⢷⣦⣤⣤⣴⡿⠋⠉⠉⠉⠉⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠊⠉⠉⢉⡉⠉⠉⠉⠒⠲⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⢠⡦⠞⠳⢤⡆⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢤⡤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣷⠦⠶⠦⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠰⢿
⠀⠀⠀⠀⠀⣹⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣎
⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡆
⠀⠀⠀⢠⣿⠟⢛⡷⠲⠶⠶⠦⢤⣤⣀⣀⣠⣤⣤⣤⣄⣤⣄⣠⣤⡠⠤⠴⠶⠶⠒⠶⠿⠻⣉⣹⣃⣶
⠀⠀⢀⣾⣋⡴⠋⠀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠰⠖⠀⠀⠲⠍⠲⢬⣙⠒⠤⢤⣄⣀⣀⣀⣀⢠⣤⢾⡷⡄
⠀⠀⣸⡟⠉⢛⠟⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠘⢂⡘⠓⠀⠀⠀⠈⠉⠛⠒⠢⠤⢤⠬⠝⠛⡇⠀⠻⣷
⠀⣰⣿⠀⢀⡞⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡀⠀⢹⣇
⢀⡏⣸⠀⢸⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢷⠀⠈⡿⣃
⢸⠁⡏⠀⣼⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⡇⢻
⡇⢸⠃⠀⣿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⣧⠸
⣷⡟⠀⣰⢿⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⡁⠀⢹⡀
⣿⣴⡞⠁⠘⣇⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠀⠀⣎⠳⣄⠘⣧
⠀⢻⡇⠀⠀⣿⡄⢰⡇⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠋⠁⢸⠀⠀⢰⡇⠀⠈⣿⠟
⠀⣿⡇⠀⢸⡇⢱⣼⠇⠀⠀⠀⠀⣩⣿⠓⠶⠶⠶⠿⠿⠷⠶⠶⠖⠛⣻⡁⠀⠀⠀⠀⠈⡇⣰⠇⣷⠀⠀⣿
⢀⣸⡇⠀⣾⠀⡎⠀⠀⠀⢀⡴⠚⢧⡈⠓⠲⠤⣤⣤⣤⣤⣤⣤⠴⠖⠉⣹⠦⣄⠀⠀⠀⠿⣿⠀⢸⡄⠀⣿
⠘⣿⠁⡼⠁⡼⠁⠀⢀⡴⠋⠀⠀⡞⠙⠲⠤⢤⣴⠋⣯⠹⣄⣄⣠⠴⠚⣇⠀⠈⢳⡄⠀⠀⠙⡆⠀⣷⠀⡏⣦
⠀⠉⠛⣧⠞⠁⠀⠀⣞⠀⠀⠀⣸⠁⠀⠀⠀⠀⢡⣄⣿⠀⠉⣀⡽⢦⣤⢿⠀⠀⠀⠙⣦⢤⣤⣼⣦⣘⣧⡷⠞
⠀⠀⠀⠓⠒⠒⠒⢺⠏⠓⠦⣤⠇⠀⠀⠀⠀⠀⠛⠊⣷⠀⠀⣛⠀⠀⠘⣼⣇⣠⠴⠚⢇⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⠖⠛⠁⢸⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣠⣄⡿⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠙⠋⡿⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠸⣆
⠀⠀⠀⠀⣠⠏⠀⠀⠀⣰⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠘⣆
⠀⠀⠀⠀⣏⣠⠀⡀⠰⡇⠉⠉⠑⠒⠒⠢⠤⢴⡖⠒⠛⢶⡦⠤⠒⠒⠒⠉⠉⠁⢸⡆⢀⡀⣤⣸
⠀⠀⠀⠀⠈⠙⢦⣿⢀⡷⠦⣄⣀⣀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⡤⢾⡇⣸⠧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⡶⠚⠋⠉⠉⠉⠀⠀⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠶⠒⠊⠉⠉⠉⠉⠙⠲⡿⠊⠉⠉⠉⠉⠉⠙⠲⢦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣤⣤⣤⣤⣶⣿⠿⠻⢿⣶⢦⣄⣀⣠⣤⣶⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⠛⠁⠈⠛⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠋⠉⢉⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⠅⣠⣴⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣦⣄⠸⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣾⣿⣤⣾⣿⣿⣿⠇⠸⣿⣤⣾⣿⣿⣿⣷⣷⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣻⡌⢿⣿⣿⣿⣿⠏⠀⠀⠹⣿⣿⣿⣿⡿⢡⣟⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣦⣄⠀⠀⣀⣴⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠁⣠⣶⡿⢣⠤⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣴⣾⣿⠀⣼⣿⣿⢗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣩⣶⣿⣿⣿⣯⣾⣿⣿⣿⣚⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⣻⡿⣿⣿⣿⣿⣿⢻⣤⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⢭⡙⢿⣿⠋⠈⠛⢿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⠀⢡⠀⠁⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠘⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⢘⣿⣆⠀⠀⠀⠀⠀⠘⢫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠃⢻⣦⣄⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⡈⠛⠿⣿⣿⣿⡛⠋⠁
⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣶⣿⣿⡇⠀⠄⠀⠀⢹⡀⠀⡠⢋⣿⣿⣶⣶⣶⠶⣦⡀
⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⡥⠊⠀⣼⣿⣿⣿⣿⡿⠀⢹⣿⣶⣤⣄⣀⣄
⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⣿⣿⠀⢰⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢨⣿⣧⢠⣿⣿⣿⣿⡾⡋⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⡿⣫⡞⠁⢀⡉⠀⠀⠰⣿⣿⡏⣿⣿⣿
⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣾⠟⠀⢠⣯⣤⣀⣤⣾⣿⣿⣇⢹⣿⣿⣆
⣿⣿⣿⡇⣿⣿⡿⣿⣿⣿⣿⣯⣽⣟⣛⣽⢿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⣿⣿⣿⣿⣿⠿⣿⣿⢸⣿⣿⣿
⣿⣿⡿⢱⣿⣿⣇⣿⣿⣿⣿⡿⢿⣿⣿⣿⣶⣾⣿⡟⠛⣿⠏⠀⢶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣭⢻⣿⣿⣷⣾⡇⠀⠀⢾⣿⣿⣿⢿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⢸⣿⣿⣿⠋⠀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣧⣿⣿⣿⣿⣸⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣻⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣻⣿⣟⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠿⢿⡿⠿⠶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠋⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠙⠛⠻⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠻⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠘⢷⣄⣀⣤⣤⣤⣴⣾⣷⣦⣤⣤⣤⣀⣠⡾⠃⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⡏⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠛⢹⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⢀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠙⠿⢷⣶⣶⡾⠿⠋⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠷⣦⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠟⠋⠀⢹⣿⣿⣿⣦⣄⣀⣀⣠⣴⣿⣿⣿⡏⠀⠙⠻⢿⣿⣶⣦⣄⡀
⠀⠀⠀⢀⣴⠿⡿⠟⠋⠁⠀⠀⠀⠀⢸⣿⣿⠿⠛⢹⣿⣿⡟⠛⠿⣿⣿⡇⠀⠀⠀⠀⠈⠙⠻⢿⠿⣦⡀
⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠸⣿⣿⠇⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⣿⠃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠘⣿
⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⡟⣻⣿⣿⣿⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⢛⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠸⠿⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠿⠿⠿⠿⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠑⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⢀⠀⠀⠀⢆⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡳⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡏⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠹⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠰⠶⠶⠶⠶⠂⠀⠀⠀⠶⠶⠶⠶⠆⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣀⢠⠀⠀⠀⣾⣦⠀⠀⢀⣀⠀⠀⣾⣦⠀⠀⠀⡇⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡞⠀⠀⠀⠈⠉⠀⠀⠇⠘⠆⠀⠉⠉⠀⠀⠀⠑⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⢇⠀⠀⠀⠀⠀⠀⢴⠀⠀⡰⠀⠀⠀⠀⠀⠀⠸⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⢀⠀⢀⡠⠄⠂⠁⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡴⠒⠛⡒⢾⠁⠀⠀⠀⡷⢖⡛⠓⠤⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠂⠉⠀⠀⡌⠀⠀⠘⢇⠘⢄⠀⠀⣠⠃⣠⠏⠀⠀⣯⢠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠉⢱⠀⠀⠀⢀⡀⠠⣥⠀⠀⠀⠈⢢⠀⡹⢿⠀⡰⠁⠀⠀⢠⣎⣈⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⠁⣠⣄⠀⣧⡘⠿⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠰⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⣿⠉⡿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡾⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⡯⠗⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠂⠀⠀⠀⠀⢸⡀⣀⠀⣀⣠⣤⣤⣤⣀⠀⣀⣀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⣤⣿⣤⣼⣿⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠧⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⡄⠀⠉⠂⠠⠄⠈⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠁⠀⢀⣀⣀⣥⢰⣿⡷⢠⣄⣀⣀⠀⠈⠁⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣿⣿⣿⣿⣿⣦⣉⣴⣿⣿⣿⣿⣿⣿⣿⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠉⢲⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠆⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠉⠀⡄⠀⣾⣷⠀⠀⠀⠀⠀⣴⣷⡀⠇⠘⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠇⢀⡆⠀⠀⠀⠄
⠀⢀⠄⠰⡄⠀⠀⠀⠀⠐⡀⠀⠀⠉⠀⠀⠙⠋⠀⠀⠀⠀⣀⠘⠋⠀⠈⠁⠀⠀⣀
⡐⡉⠸⢠⠀⢴⠀⠀⠀⠀⠉⠐⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⢨⠓⠂⡉
⡇⡀⠀⢰⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠀⠀⠀⠁⠀⠀⠀⢀⠂⠀⠀⠁
⡇⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠐⠦⢤⠀⠀⠀⠀⡠⠂⠀⠀⢸
⠡⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠑⠠⡄⣀⢀⣀⢠⠄⠊⠀⠀⠀⠀⠈
⠀⢁⣀⠀⠈⠀⠀⠀⠀⠀⡀⠀⢦⠀⢠⠄⠤⠤⡇⠀⠀⠀⠀⡄⠀⢤⡀⠀⠀⠀⠈⡰
⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠀⢦⠀⠀⢃⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠠⡄⠃
⢠⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢂⠀⠀⠑⣤⡄⠁⠀⠰⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠄⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠈⢄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠘⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⡁⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⢸⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡀⠠⢀⣶⣦⠀⠁⠀⠀⠛⠛⠛⠛⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠂⠀⠢⡙⣋⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠢⢄⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⡠
⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⡠⠊
⠀⠘⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠠⠊
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠙⣿⣿⣑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣇⣸⣿⣿⣟⢡⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⣿⣿⣿⣿⣿⠛⣹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠙⠾⢻⠉⢹⠿⠟⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⣴⣿⡿⣿⣶⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠀⠁⠈⣩⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⠹⣿⣿⣷⠶⠾⠏⠀⠇⣴⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠈⠻⠋⠉⠉⠈⡄⢙⡟⠁⠀⠀⠄⠀⡀⠀⠈⠛⠿⣦⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢻⡼⠙⠀⠂⠀⠛⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢸⣷⢿⣿⣿⣷⠀⠀⢰⣶⣿⣷⢶⣶⣾⠄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡠⠤⠤⠈⣙⡋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠉⠹⢲⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⠁⠀⠀⠀⢀⣇⡀⠀⠀⠀⠀⠂⠒⠂⠤⠆⠀⠄⡀⠀⠀⠀⠁⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠐⠉⠀⠉⠉⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⣤⡀⠀⢄⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣀⣀⣀⣠⣤⣶⡿⠟⠛⢿⣷⡦⣽⣵⣦⣄⣠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠛⠋⠉⠁⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣙⣻⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣦⣄⢀⡀⠀⢀⡄⣀⣀⣀⣠⣤⣤⣼⣤⣿⣿⡆
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣟⣿⠇⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣵⣼⣿⣿⡟
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣟⡿⠟⠛⠛⠛⠃⠛⠛⠏⠈
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠉⠑⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠉⠉⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠋⠁⠠⣶⣶⣶⣶⠀⠈⠙⠻⠿⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⠉⠀⠀⠀⠀⠀⢰⣿⡁⠈⣿⡆⠀⠀⠀⠀⠀⠉⠙⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣾⣿⣛⣉⣉⣭⣭⣭⣭⣤⣤⣭⣭⣭⣭⣉⣉⣛⣿⣷⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠛⠿⠷⣶⣶⣶⣶⣶⣶⠾⠿⠛⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⣿⡇⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢸⣿⠿⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⣿⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⣿⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡇⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠙⢿⣦⣄⣀⣤⣾⠟⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣄⣀⣀⣀⣀⣠⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣾⣿⠉⠙⠛⠛⠛⠛⠋⠉⣿⣷⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣤⣤⣤⣼⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⣧⣤⣤⣤⣤⣼⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣤⣤⣤⣤⣾⠟⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡈⠥⡆⠀⠀⠀⢏⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡠⠤⠐⠚⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠺⢄⠒⠢⠭⣀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢋⡤⠀⢸⠂⠀⠰⡁⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠧⠒⢲⣷⠀⣔⠉⠉⢉⣿⣀⣤⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠔⣒⡛⠀⠈⠙⠛⠩⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠈⡓⠄⠸⡈⢁⡀⢂⠀⠀⠀⠀⢹⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠃⡗⡳⠈⠁⠀⠈⡆⠀⠀⠀⠀⢣⠀⠀⠉⠒⠒⠴⠦⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⢐⠁⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⣆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢐⣒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠴⡞⠀⢀⡤⠤⠄⣀⡇⠀⠀⠀⠶⠷⣶⠒⢢⡕⠀⠀⠀⠀⠀⠀⢹⡏⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠑⢴⡷⣄⡇⠀⠀⠀⠀⠀⢻⠒⠊⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠠⡤⡤⣼⡆⢄⠈⠑⠢⢄⣀⠀⡸⠙⡿⠒⡒⢲⡄⣀⣀⣀⣴⡆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⣓⠑⠴⠵⠀⠑⠤⣀⡎⠀⠻⡇⣠⢌⠉⠁⢠⠓⠒⠒⠒⠲⣷⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢤⡤⠀⠀⣏⣡⣍⠉⣆⠀⠈⡉⡟⢇⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡼⠋⠷⡀⠦⠤⠤⠤⠇⠀⠀⠉⠈⠉⣲⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡆⣗⣒⣏⢉⡍⢱⡖⢲⡒⢲⣞⡧⠔⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠦⠼⠤⠃⢢⠉⠣⠤⠤⠼⠀⢈⡏⡏⠀⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠘⡆⠀⡎⡄⠀⠀⠀⠀⠀⢸⢡⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠠⣤⣤⣬⠼⠤⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣝⣁⡐⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠛⠛⠛⠛⠛⠓⠒⠒⠒⠓⠒⠛⠿⠯⠥⠄⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣖⣒⣒⡶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⣾⣿⣿⣿⣿⣿⣷⣾⣝⡲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣤⣙⣦
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⠩⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⡁⠀⠂⢦⣙⣩⣭⣝⣛⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⢟⣥⣶⣿⣿⣿⣿⣟⣽⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⣿⣿⣿⣿⠿⠅⠀⠉⣿⡟⡻⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⠿⠿⠋⠛⠋⠀⠀⠈⢀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠋⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⢸⣀⣬⠽⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣉⣡⠤⠤⠆⠀⠀⠀⣼⠃⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠑⠖⠂⠀⠀⠀⣀⣴⠏⠀⠈⣿⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣀⣴⣿⠏⠀⠀⣰⠃⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡿⠟⠉⠀⢠⡞⠁⠀⠀⣰⣃⣈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⣷⣤⠀⠀⣠⡴⠋⠀⠀⢀⠞⣿⣽⣿⣷⣶⣬⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠻⣯⡷⠚⠉⠀⠀⣠⠔⠃⠀⠘⠿⠿⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⣤⣿⣿⣇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠀⠀⠀⢏⣿⣿⠛⠷⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠋⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠠⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠋⠀⠙⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣀
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠚⠛⠒⠚⠓⠚⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠓⠚⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣄
⠀⠀⠀⠀⠀⠐⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⣀⢿⣽⠟⠛⣷⡶⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⡀⢀⣠⣤⣤⣤⣤⣤⡤⢶⣶⣿⣿⠛⠛⠛⠛⠒⣦⡄
⠀⠀⠀⠀⢀⣿⡿⠁⣠⡾⠋⠀⠀⠀⠀⠀⣀⣰⠟⠋⠉⠉⠛⠛⠛⢛⣉⠛⠛⠛⣉⡻⢿⣉⠛⠿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢛⣷⣏⠁
⠀⠀⠀⢠⣾⣿⣤⡞⠉⠀⢀⣠⣤⣤⣴⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠓⠀⠉⠳⢦⣤⣉⠙⠒⠶⠶⠦⣤⣤⣤⣤⣄⣀⢰⣶⠟⢷⡙⣷
⠀⠀⠀⣸⣿⠋⠉⠛⢿⠟⠉⠉⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠘⠟⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠦⢤⣤⣤⣤⠬⠛⠻⣧⠀⠀⠻⣯⣧
⠀⠀⢸⣿⡋⠀⠀⢰⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢹⡆⠀⠀⠻⣿⡆
⠀⢰⡇⢻⡇⠀⢠⡞⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⡄⠀⠀⣿⣷⡄
⠀⣼⠃⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡿⢇
⠀⡟⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠤⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇
⣠⠃⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇⠈⣷
⡿⠀⡸⠃⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿⠀⢿
⣧⣼⠇⠀⠀⣰⣿⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣻⠁⠀⠀⢻⡀⠘
⣽⡿⠀⣠⡾⠋⢰⣇⠀⠀⠀⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⢀⡿⣧⠀⠀⢸⣇
⠿⢶⣿⡇⠀⠀⠀⣿⡀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡇⠀⠀⠀⢸⡇⠘⠷⣄⡀⢻⡶
⠀⠀⣿⡇⠀⠀⠀⣿⣧⠀⠀⣿⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠋⠁⠀⢸⡇⠀⠀⢠⣿⡇⠀⠀⠈⣿⡿⠃
⠀⢠⣿⡇⠀⠀⢠⡇⠹⣇⠀⣿⠀⠀⠀⠀⠀⠙⠛⠳⣶⣤⣄⣀⣀⣀⣈⣤⣤⣤⣔⣀⣀⣀⣀⣠⣤⣶⠞⠛⠉⠀⠀⠀⠀⠀⠘⡇⠀⢠⡾⢹⡇⠀⠀⢠⣿⡇
⠀⠀⣿⡇⠀⠀⢸⠏⠀⣿⡷⠏⠀⠀⠀⠀⠀⢀⣠⠶⠿⣄⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⢿⢀⣾⠁⠸⣇⠀⠀⢸⣿⡇
⠀⡄⢿⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣠⡴⠛⠻⣦⡀⠉⠛⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠾⠟⠉⠀⣹⣷⣤⡀⠀⠀⠀⠀⠻⠿⣧⠀⠀⣽⡄⠀⠈⣿⡇
⠐⣿⣾⠃⠀⣾⠁⢠⡞⠁⠀⠀⠀⢠⡾⠋⠀⠀⢠⡝⠛⢶⣄⡀⢀⡀⠀⣼⢹⡟⢹⡄⠀⠀⠀⢀⣀⣴⣾⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⢻⣄⠀⠘⣧⠀⢠⡿⢧
⠀⢰⣧⣠⣾⠃⢠⡞⠁⠀⠀⢀⣴⠟⠀⠀⠀⠀⣸⠇⠀⠀⠉⠙⠉⢻⠿⠁⢸⡇⠀⠻⣶⠛⠶⡏⠉⠀⢈⣧⠀⠀⠀⠈⢿⣄⠀⠀⠀⠈⢿⡄⠀⠹⣇⢰⣇⢸⡇
⠀⠀⠀⠉⢻⡶⠛⠀⠀⠀⠀⣸⣁⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣸⡇⠀⠀⠀⣀⣴⠿⢧⣴⡜⢿⡀⠀⠀⠀⠀⢹⣷⠤⣤⣤⣴⣿⣦⣀⣻⣾⡿⠛⠃
⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠾⣿⠛⠷⣦⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⢻⡇⠀⠀⠀⢿⠁⠀⠀⠀⣷⢸⡇⠀⢀⣠⡶⢿⡇⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡀⠀⠀⣠⣼⠋⢹⡶⠋⠉⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣭⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢰⡏⠙⠛⠛⠢⣤⣤⣤⣤⣀⣀⠀⣠⣤⣤⣼⣷⣤⣄⡀⢀⣠⣤⣤⣤⡤⠚⠛⠛⠉⠻⣄⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠈⠷⣾⡇⠀⣶⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇⠀⠀⠀⢈⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡆⠀⣿⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⡄⢘⡿⠶⢦⣤⣤⣄⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠶⠾⣿⢀⣼⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡗⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⢷⣶⣶⣶⠾⠿⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⠤⡤⢤⣀⣸⣧⡠⠶⢶⠶⠶⠶⠶⠶⠶⣤⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣀⣀⠀⠀⠀⠀⢀⣀⣠⠶⣟⣿⣞⣳⠦⣀⡀⠀⠀⠀⠀⠀⣀⣀⡴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠀⠈⠙⠚⢋⣽⣷⣶⣶⣶⣿⣿⡟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⢋⣅⣠⣄⣨⡙⠻⠶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⠶⠞⢛⣉⣥⠴⠀⠀⢻⣿⣯⣧⣿⡿⠀⠀⠦⣬⣉⡛⠳⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣧⠀⠈⠛⠉⠉⠀⠀⠀⠀⣿⣟⠱⡎⣻⣿⠀⠀⠀⠀⠉⠉⠛⠃⠀⣼⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠃⠀⠈⠂⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠘⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣰⠋⠀⠀⠀⠀⢀⣠⡴⠶⣛⣫⣭⡭⠵⠶⠶⠾⠭⣭⣽⣛⡲⢦⣄⡀⠀⠀⠀⠀⠙⣆⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣈⠁⠀⠀⠀⢠⡴⠛⠥⣒⣫⣭⣤⠶⠶⠶⠶⠶⠶⠶⠶⣤⣬⣍⣓⠪⠛⢦⡀⠀⠀⠀⠈⣁⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠉⠉⢰⣯⡴⠶⢛⣉⣥⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⣬⣉⡛⠲⢦⣽⡆⠉⠉⠉⠉⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣿⣸⣷⣿⣿⢹⣷⣿⠀⠀⢸⠇⣾⣿⢸⣿⣿⣿⣷⣾⣿⣿⣿⣾⣇⣿⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢿⣿⣿⣿⡏⣿⣿⡿⠀⠀⣿⢀⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⣿⣿⣿⣅⣻⠿⠇⠀⠠⡟⢸⣿⡇⣿⣿⣿⠿⠿⠿⣟⣻⣽⣿⣿⣁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⡿⠋⠛⢛⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡛⠛⠙⢟⣿⣯⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣻⣿⠃⠀⠘⣏⡿⢿⣿⣿⣏⠘⠁⠈⠃⣹⣿⣿⡿⢿⣹⠂⠀⠘⣿⣟⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢈⣿⠀⢲⡐⠪⠙⣛⠿⢋⠀⣰⠀⠀⣆⠀⡙⠿⣛⠋⠕⢂⡖⠀⣿⡉⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⣿⡘⣌⢳⡀⠀⠤⠚⠁⠀⡿⠀⠀⢿⠀⠈⠓⠤⠀⢀⡞⣡⢃⣿⢑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⣧⠸⠈⡇⢠⠔⣋⣤⡆⠁⠀⠀⠈⢰⣤⣙⠢⡄⢸⠁⠇⣼⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠷⠠⢛⠉⣉⣳⣌⠓⠚⣡⣞⣉⠩⡛⠄⠼⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣬⡀⠀⠰⣵⣟⣽⣿⣿⣿⣿⣿⣿⣯⣻⣮⠆⠀⢀⣥⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡂⣼⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣧⢐⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡿⢯⣩⣽⡿⠿⣯⣍⡽⢿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠿⢯⠀⠀⢐⣚⣻⣟⣓⡂⠀⠀⡽⠿⢿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠭⠗⠒⠒⠺⠭⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⣛⣉⣉⡁⠤⠤⠤⠤⠤⢀⣉⣉⠉⠛⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣩⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣡⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠶⠶⠚⠛⠛⠉⠉⠉⠛⠛⠛⠲⠶⣤⣀⣀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⡾⢡⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢀⡾⢱⣿⠃⠀⠀⠀⠀⠀⢠⣶⣶⠞⠋⠁⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠒⠒⣛⣩⣭⣿⡿⠿⠿⢿⣿⣷⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢸⠇⣿⡿⠀⠀⠀⠀⣰⣾⡿⠋⠀⠀⠀⢀⣠⠴⠚⣿⡿⠿⢛⣛⣩⣥⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⠟⠁
⠀⠀⠀⠀⠀⣾⢰⣿⣧⣾⣦⣶⣾⣿⠟⠀⠀⣀⡴⠺⣯⢀⣠⣤⣶⣾⣿⣿⣿⠿⠿⠿⠿⠛⠛⢛⣛⣉⡩⠭⠤⠾⠿⣟⣛⡛⢧⡄
⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⡟⡴⢶⣏⠙⣷⡴⠿⠟⠛⠋⣉⣁⣠⡤⠤⠴⠒⣒⣊⣩⣭⠭⠶⠶⠒⠛⢛⠛⢙⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢹⡌⣿⣿⣿⣿⣿⣿⣿⣷⣀⣿⣿⣭⠴⠶⢒⣛⣉⣭⡥⠶⠶⠚⠛⠉⠉⠀⠀⠀⣀⣤⣤⣴⣾⣿⣿⣾⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠈⢷⡘⣿⣿⣿⣿⣿⣿⣬⣿⣧⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⢠⣾⡿⠟⢉⣉⣀⢠⡀⣠⣿⡟⢁⣬⣷
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⢀⣠⣤⣤⣾⡿⣿⣿⡏⠁⠀⠀⠀⣼⡟⠿⠭⠿⠿⠿⠷⠿⣟⣿⢃⡞⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⠙⢿⡆⠳⣶⡂⠋⠓⠊⣉⣉⣩⡿⠋⠀⠀⠀⠀⠀⣿⣿⣦⡐⠒⠐⠿⠛⠰⢾⣿⡜⢧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢶⡌⠛⠈⠉⣀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠳⣼⣿⡷⠊⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠂⣽⢰⡀⢸⣿⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢹⡉⢿⣃⣄⡀⠀⠀⣠⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠙⠸⣇⠸⣿⣿⡇⢀⣀⣠⣤⣶⣿⡄⠤⠦⠀⣠⣾⣧⣼⣿⣿⣿⣷⣶⣿⣿⢻⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢻⡄⢻⣿⡁⢸⣿⣿⡿⠋⠁⠀⠀⠀⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⣇⠀⢻⣧⠘⣿⠀⣀⣀⣀⣤⣤⣤⣤⣠⣴⣦⣬⣭⣤⣤⢹⣿⡇⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢻⣧⠹⡆⠙⠻⢭⣅⣈⣉⠉⢉⣉⣠⡴⠞⣩⠄⠈⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠹⣇⠻⡄⠀⠀⠀⠉⠉⠉⠉⠉⠁⣠⣾⠋⠀⣸⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠹⡆⠹⠀⠀⠀⠐⠺⣿⣿⠿⠿⠿⠃⠀⢠⡿⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣠⣤⣴⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠤⠄⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠲⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⡧⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠋⠁⢂⠉⠙⣤⠄⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣈⣥⣨⣤⣤⣌⣤⣊⣁⠀⠀⢠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣟⢸⠷⡿⣷⣿⢿⡟⢼⡯⠀⠀⡏⠈⠻⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠛⡶⢋⠟⠛⢂⠛⢏⠈⠁⠀⠀⣇⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⡵⠋⠀⠀⠀⠣⡀⢳⡄⠀⠀⠙⠦⡀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣥⡶⢚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠥⠤⠤⠤⠤⠤⠽⠄⠙⣦⣀⠀⠀⣸⠃⢀⡖⠉⠀⠁⠂⢄⠀⠀⠀⠀⣠⣾⣿⠟⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⠀⠀⠈⠢⣀⠀⠀⠀⠉⠢⢠⣾⣿⠟⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡉⡇⠀⠀⠀⠈⠉⠑⠒⠒⠒⡶⣶⢶⣇⠀⠀⠀⣀⣀⣁⣠⣄⡀⠀⠀⠉⠁⠤⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⢠⠀⠀⠀⠀⠂⠀⠀⠀⠙⢧⠈⢀⡇⠀⣾⠉⠀⠀⠀⢈⣟⣹⠗⠒⠦⢀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡌⣶⠖⠒⢲⡟⠛⠛⠛⠋⠉⣻⡇⢀⠃⠀⠙⠲⠴⠶⣤⣍⡡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡜⠀⠀⠀⢀⡀⠔⠁⠀⠀⣠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡅⢀⡴⠊⠁⢰⠀⠀⠀⠀⠉⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣇⠎⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠃⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⠻⡄⠘⡄⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣸⠀⠀⢸⡙⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⡇⠀⠀⠎⠀⠸⡆⠀⠀⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠓⢶⡾⢠⠁⡄⢿⠀⠀⠁⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⢸⡇⣸⡾⢱⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣸⠃⠀⠀⠀⠀⠸⡇⣿⢷⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⠁⠀⣿⣾⢱⣿⠇⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⠤⠤⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠆⠀⠀⢻⣏⣿⡇⡛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡜⢻⠀⠀⠀⢸⠀⠀⢀⢸⣟⢧⠁⣷⠀⠀⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣿⠀⠀⠀⢸⠀⠀⠸⠈⠻⡿⢠⠁⠉⠛⠛⠛⣻⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣯⠀⠀⠀⢸⠀⠀⠀⢃⠀⢀⡏⠀⠀⠀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠚⢢⡹⠦⠤⠀⠚⠀⢠⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠗⠀⠁⠚⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣷⣤⡍⠉⠉⢩⣤⣾⣷⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣉⣤⣼⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣧⣤⣉⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⢛⣻⣟⡻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⠇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣮⣵⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠛⠶⣤⣀⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⢤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⢀⡼⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠴⠦⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⢧⡀
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⡴⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣟⣋⣭⣥⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣭⣙⣻⡇
⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠉⠙⠻⠿⠿⠛⠋⠉⠁⠀⠀⠈⠑⠺⠀⠀⠉⠁⠛⠉⠉⠙⠛⠿⠿⠟⠋⠉⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣨⡇⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠀⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⡞⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢳
⠀⠀⠀⠀⠀⠹⠦⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠋⠙⠓⠦⠤⠤⠴⠒⠋⠙⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠒⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠒⠋⠑⠢⣀
⠀⠀⣀⡠⠴⠧⢍⠒⣤⠤⢄⣠⠀⠘⢏⠓⠦⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠴⠚⡻⠃⠀⣄⡠⠤⣤⠖⡩⠼⠦⢄⣀
⠀⡎⠁⡠⣤⡀⠀⢱⠎⠀⠀⡜⠀⠀⠀⠙⠢⢄⣀⡠⠻⣿⣿⠗⢄⣀⡠⠔⠉⣀⡀⡀⢣⠀⠀⠐⠎⠀⠀⣤⢄⠀⢱
⠀⠈⠫⠔⠁⡅⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢊⠁⠀⠀⡀⢸⠀⠀⠀⢨⠂⢠⠈⠢⠝⠁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠘⡀⠀⢀⠄⠀⡆⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠈⠀⠀⠀⠀⠻⣿⣿⠗⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⠈⠁⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣭⣝⡛⠛⠻⠿⠿⠿⠿⠩⠏⠿⠿⠿⠿⠟⠛⢛⣫⣭⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠙⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⣿⣧⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠂⡀⢹⣿⣠⠔⠉⠉⠉⠉⠉⠉⠲⣄⣿⣏⠀⠐⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⣲⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣦⣴⣞⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣦⣴⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣈⣵⣤⣄⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣾⣿⡏⠀⠉⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣿⣷⣦⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⢻⣿⣿⠿⣿⡿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠀⣁⡀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⡀⣄⠀⠉⠀⢀⣴⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⣨⠟⢇⠁⠠⢶⣶⣶⡿⠋⠿⣿⣍⠉⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡠⠒⠀⠉⠀⠀⠈⠆⠀⡠⢿⣿⠷⢀⡴⠋⠈⠉⠒⠒⠤⣁⡄
⠀⠀⠀⠀⠀⢠⠁⠀⠙⠄⠀⠀⠀⠀⠀⠀⠉⢁⣼⣿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠘⠀⠀⠀⣆⣠⡶⠋⠀⠐⢤⡀⢸⣿⣿⣷⠀⠀⠀⠀⡀⠀⣢⢀⡄⠀⢹⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠀⡿⢋⠀⣠⣿⠆⠀⣿⣼⣿⣿⣿⣦⠂⠀⠒⠀⢀⣽⡟⠀⠀⠀⢧
⠀⠀⠀⢰⠃⠀⠀⠀⣸⡁⠁⠂⢊⡎⣦⣾⣿⣿⣏⣙⣿⣿⠴⠦⢤⣤⣁⣀⠇⠀⠀⠀⢸⡀
⠀⠀⠀⡏⠀⠀⠀⡴⠋⢷⢄⠀⢈⣼⣿⠿⠃⣿⣿⣿⡟⠉⠀⣀⣀⠀⣦⣹⠀⠀⠀⠀⢘⡇
⠀⠀⠀⢸⢀⡶⠁⠀⠃⠈⢆⣡⣾⣃⡄⠀⠀⣿⣿⣿⣷⠒⠶⣶⣶⣒⠻⡟⠋⠉⠛⢂⣼⡇
⠀⠀⠀⢸⠋⠃⠀⠀⠀⠀⠀⢙⣏⠀⠀⠀⠀⣿⣿⣿⣿⠀⢈⠓⠒⠂⠀⣿⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠈⠻⣿⠟⠀⠈⣛⣛⣋⢀⣿⢇⡠⠞⠁
⠀⠀⠀⠀⠀⠑⠤⣤⣼⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠘⢶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠙⢋⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣾⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢛⣻⠇⠛⣦⢫⣿⠟⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠆⠀⠶⠛⠛⠛⠷⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⣺⢖⣶⣸⣞⣿⡀⡏⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⣻⡛⠁⡬⢤⣤⣤⣄⣠⣤⣽⣾⣿⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠸⣿⡇⠀⠇⠀⣿⣭⣭⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠈⠓⠒⣀⣀⣿⣒⣲⣿⠟⠋⠙⠺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⢀⠀⢸⣁⢨⣿⣷⡾⠃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠿⠓⢋⣸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠀⢶⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣾⡄⣶⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⡟⠇⢣⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠘⢸⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠄⠈⠉⠙⠋⠉⠉⠛⠉⠛⠉⠋⠙⠋⠛⠉⠛⠋⠉⠉⠉⠠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣴⡻⠿⣟⣤⣄⣠⣄⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⢀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠋⠈⠁⠀⠈⠁⠙⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦
⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣠⠀⠀⠀⠀⠀⣆⠸⠇⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢦⡀⠀⣠⠶⣹⠇⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡘⠛⣡⠔⠁⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⠉⠉⠉⢻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢸⡇⠀⠀⠀⢠⣤⣽⣿⣿⣿⣿⠛⠛⠋⢙⣿⣟⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠋⠉⠙⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠘⣿⡿⠇⠀⠀⢰⡏⢰⠀⠻⣿⠟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣶⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢶⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢙⡄⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣈⣧⣤⠋⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠉⠉⠉⠉⢉⡷⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡤⣤⠤⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⢄⣠⡟⠛⠛⠛⠛⠛⣯⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠟⢦⡀⠀⣠⠞⠛⠃⠀⠀⠀⠀⠀⡛⠳⢆⡈⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠹⠂⠙⡄⠀⠰⠀⠀⠀⠀⠠⠁⠀⡘⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⢀⡸⠂⠀⠘⢆⠀⠈⠡⣦⠈⠀⠀⠄⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⢣⣤⠙⡎⢱⣤⠟⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣀⣶⣄⣀⠀⡆⠈⠁
⠀⠀⡰⠋⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⡏⠉⠙⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⠞⠁⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⣷⣀⣼⡏⠀⡗⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⣠⠾⣌⣃⣀
⠙⢦⡀⠀⠀⠘⢣⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠉⠀⠀⠃⠀⠘⠓⠂⠀⠉⠉⠉⠉⠀⢻⠀⠉⠉⠉
⠀⠀⠙⢷⣄⠀⠀⠈⠶⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⣀⠠⠤⠤⠀⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠈⠑⢦⡀⠀⠉⢢⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⠖⠖⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡸⠟⣢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣁⠀⢻⣾⠃⠄⠀⠀⠀⠀⣶⣶⠷⢶⣶⣶⠄⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣩⣄⣀⣀⣀⣀⣀⣼⣧⣄⣸⣿⣿⣤⣠⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⣽⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠧⡀⢀⣼⡿⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣣⣀⣀⣉⣹⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠉⠋⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⣤⣤⣦⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⢿⡿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢋⣁⣤⣼⣿⠛⠻⣿⣧⣤⣈⡙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⠟⠋⠉⠉⢿⣦⣴⡿⠉⠉⠙⠻⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡻⠷⣶⣤⣤⣤⣴⣶⠿⢛⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠈⠁⠀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣀⣀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠾⢿⣎⠉⠙⠛⠉⣱⡿⢷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠉⠁⠀⠀⠈⢿⣆⠀⠀⣰⡿⠁⠀⠀⠈⠉⠙⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⣀⣀⣀⣀⣀⡀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠛⠛⠛⠛⠁⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⢰⡿⠁⠀⢀⣾⠟⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⢿⣧⡀⠀⠈⢿⣦⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠙⢷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠰⣶⣶⣿⣿⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣿⣶⣶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣿⠁⠀⠈⢻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⠐⠆⢘⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢟⣽⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣉⣿⣿⣿⣿⣿⠿⣿⣟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢿⡟⠻⠏⠉⠁⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠘⠃⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠀⠀⠊⠁⠀⠄⠀⠀⢸⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡃⠀⠀⠀⠀⠀⣠⠇⢈⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣠⡾⠋⡀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣋⠤⠂⠀⢀⣾⣷⣾⣶⡈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡏⠁⠀⠀⢠⡞⠉⢿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⢿⣿⣿⡷⠀⣀⡴⠋⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠓⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠀⠀⠈⣿⡿⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⠀⣀⣀⣸⣿⣿⣇⢀⣀⣀⣀⣀⣀⢀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⡏⠙⠋⠉⠙⠻⠿⠛⠛⠛⠿⢻⣿⣷⣬⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠐⢻⣿⣿⣷⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡀⣻⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠒⢲⣿⣿⣿⣿⡇⠀⠈⠉⠉⠉⠉⠙⢋⣻⡝⠇⢸⣿⠏⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣿⠃⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣤⠀⠀⠂⣹⡿⠁⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⣾⣿⣾⡟⠁⠀⠀⠀⠀⣠⠞⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⡀⢿⣿⣿⠋⠀⠀⠀⢀⣠⡾⢋⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢿⣦⡸⠏⠹⣿⣿⣿⣿⣁⣈⣷⣴⣷⣾⣿⠇⠀⠀⢠⣤⠻⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡘⠻⣷⢲⣶⣤⣿⣿⣿⣙⣿⠋⠉⠁⠀⠉⠀⠀⠀⠂⠉⣲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⢌⡒⠀⡸⡇⣿⣿⣿⣿⣿⡙⠀⠀⠀⠀⠀⠀⢄⠀⢀⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡈⠒⠉⣵⣿⣿⣿⣿⡹⠁⠀⠀⢀⣦⣤⣴⣾⣷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣦⣴⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⠀⠻⣿⣿⣿⣿⠀⠉⠛⠋⠛⠁⠙⠛⠛⠿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣉⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⡶⠿⠛⠋⠁⠀⠀⠀⠀⠉⠙⠛⠿⢶⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⡶⠶⠾⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠷⠶⢶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⠾⠷⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⡀⢀⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠈⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣩⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣉⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠻⣿⠙⡛⠛⠛⠻⠿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣬⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠨⠟⠉⠉⠉⠉⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⠉⠉⠉⠻⠅⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠞⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢶⣄
⠀⠀⠀⠀⠀⠀⢰⣏⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣤⣤⣴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣇⡀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⠳⠶⠶⠶⠶⠞⠋⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣮⣙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣟⣉⣴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡞⠉⠀⠀⠀⠀⠈⠉⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠛⠉⠀⠉⠀⠀⠈⠙⢳⡤
⠀⠀⠀⢀⣠⡤⠺⢧⣤⡉⠳⣤⣤⣤⣤⣠⡄⠀⠈⢫⡙⠳⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠟⢋⡝⠁⠀⢠⣀⣠⣤⣄⣠⠞⢉⣤⠸⠧⣤⣄⡀
⠀⢠⠞⠉⠀⠀⠀⠀⠀⠙⢶⠎⠀⠀⠀⣸⠃⠀⠀⠀⠙⠶⣄⡀⠀⠀⢉⣹⣿⣿⣿⣿⣍⠁⠀⠀⢀⣤⠖⠉⠀⠀⠀⠘⣇⠀⠀⠈⢣⡶⠉⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠘⢶⡴⠞⣙⢿⡀⠀⣠⠞⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⡀⠀⠙⠛⠶⠋⠀⠙⣿⣿⠋⠀⠙⠶⠛⠁⣀⠤⠀⠁⠘⡟⠀⣿⠀⠀⠀⠀⠳⡄⠀⢀⡿⣋⠳⢦⡶⠃
⠀⠀⠀⠙⠛⠁⠘⣇⠀⣸⠀⠀⠀⠀⢀⡏⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⢻⠀⠀⠀⠀⠀⡃⠀⡈⠃⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⡅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠐⠒⠲⠈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⠨⠁⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⢰⡿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠚⢿⠟⠻⠟⠒⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⣤⣤⣤⣤⣤⣤⡾⠀⠀⢷⣤⣤⣤⣤⣤⣤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠋⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣶⣶⣶⣶⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠳⠦⣤⣀⣀
⠀⠀⠀⢠⣤⣤⣤⣤⠤⠴⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠤⢤⣤⣤⣤⣄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠒⠒⠛⠳⠒⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠈⢻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣋⣭⣥⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣭⣍⣻⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠁⠃⠀⠀⠈⠉⠀⠀⠉⠙⠛⠛⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢠⡴⠴⠶⢤⡀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⣤⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄
⠀⠀⠀⠀⠸⣄⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡼
⠀⠀⠀⠀⢀⡈⠙⣇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉
⠀⠀⡠⠒⢏⡈⠦⢼⣆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣞⠠⢞⢉⠗⠤⡀
⠀⢺⠉⢢⡈⠃⠀⣼⠙⢦⡀⠀⠀⠀⠀⠀⠙⠿⢯⣭⣬⠿⠟⠁⠀⠀⠀⠀⠀⣠⠞⠸⡀⠀⠎⢠⠞⢹⠆
⠀⠀⠉⠓⢤⠔⣫⠞⠤⣜⡛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢹⣠⠼⢍⠢⢤⠤⠃⠁
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠉⠹⣙⠲⠦⢤⣤⣄⣀⣀⣤⣤⠤⠶⢚⠟⠃⠉⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⡇⠀⠀⠈⠑⠢⢄⠔⠻⣿⡿⠣⠄⠄⠒⡁⠄⢤⠀⡦⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠄⢀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⠀⠀⠀⠀⠀⠙⢿⠏⠀⠀⠀⠀⠀⢀⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⣿⣿⣿⣶⣶⣶⣠⣴⣶⣶⣾⣿⣿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠒⢲⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣤⣤⣤⡼⠀⠸⣤⣤⣤⣤⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠖⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⡀⠀⠀⠀⣴⣭⣒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⣰⣏⠀⠀⠀⠀⣘⣿⣿⣷⣶⣍⡑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠠⡀⣤⣴⣴⣽⣿⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⢣⣿⡳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣵⣦⣝⣾⣞⢿⣿⣿⡀⠀⠀⣀⣼⣿⣿⣿⣿⣷⣿⣿⣿⣜
⠀⠀⠀⠀⠀⠀⠀⢣⠀⣾⣿⣾⣟⡋⠁⠁⠠⢽⣿⣿⣷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⣿⣿⣦⣤⣄⣰⣿⣿⣿⣿⣿⣷⣯⣯⣽⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⢿⣏⡙⢿⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣦⣾⣿⣿⣫⡿⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢤⣌⠁⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⡀⠀⠀⢈⣻⣶⣦⠀⠀⢠⣴⣶⣟⡁⠀⠀⠀⡒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠁⠀⠀⠙⠛⠉⠛⠃⠀⠘⠉⠘⠛⠉⠀⠀⢸⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠂⠠⠖⠊⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⣀⡀⢀⣤⠤⠄⠀⠀⢀⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠈⣉⣤⠄⠀⠀⣰⣿⠃⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⣤⠾⠃⠀⠀⠀⠀⢱⣤⣀⣀⣀⣀⣀⣀⣀
⠀⢀⣠⣶⣿⣷⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠸⣿⠻⠿⠿⠿⠿⠿⢿⣿⣷⣦⣀⡀
⠰⢿⣿⠟⠋⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣭⣭⣿⣧⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠻⣿⣿⣿⡿⠋⢃⡀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⣹⣿⣿⡟⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⢒⣛⡉⠭⠭⠍⣉⡙⠓⠲⢦⣄⡀
⢀⡴⡶⠛⣿⠀⣿⠛⣦⠀⠀⠀⢠⠔⠉⠁⠀⠀⠀⠈⣷⠂⠈⠉⠙⢲⡋⠀⠀⠀⠀⠉⠳⢦⣀
⢸⠀⡇⠀⠸⠀⠻⠀⢹⢀⣠⢤⡌⠉⢳⡄⣀⠤⠒⣊⣹⣦⣤⣤⣤⣼⣃⣐⠒⢤⣀⠀⣠⡴⠟⠂
⣿⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⡞⠁⠀⡴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣏
⣿⠀⠀⠀⠀⠀⠀⠀⠾⠀⢠⠏⠀⢰⡇⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⡾⠟⡁⢹
⣿⠀⠀⠀⢀⡾⠀⠀⠀⠀⢸⡆⠀⠈⢻⡇⠀⠓⡉⠉⢉⡉⠛⠛⠛⣭⠉⠉⠶⠃⠀⢻⠞
⢻⡄⠀⠀⢸⡇⠀⠀⠀⠀⣸⣇⡀⠀⢸⠃⣠⠾⠒⠛⠉⠙⠶⠴⠞⠋⠛⠳⢷⣦⡀⢸⣀⣀⡀
⠀⠙⢦⣄⠈⠿⠦⠤⠒⠋⠁⠈⠉⠙⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡇⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡼⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⡀
⠀⠀⠀⠈⠻⣦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⠤⣤⣤⣤⣴⣶⣤⣄⠠⠴⠛⠋⠀⣼⣿⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠙⢦⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⠀⠀⠀⠀⣠⣤⣤⠀⣴⣶⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠻⠿⠟⠁⠈⠙⢻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠸⠿⠟⠁⠉⠛⠻⠃⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠖⠉
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠿⣿⠿⠃
⠀⠀⠀⠀⠀⠈⠉⠛⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⡶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄