Limb Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠈⠛⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⠶⣤⣹⣿⣿⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⣦⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠿⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⣿⣿⡁⠙⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣾⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢻⣿⠛⣠⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣿⣶⣿⣷⣶⠆⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⠀⠸⠿⢿⣄⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣄⣀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠛⠻⠂⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⣾⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠛⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⣶⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠿⠿⠿⠟⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⢿⣶⣿⣷⣤⠀⠀⣀⣀⣀⣀⣴⣦⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣄⠀⠉⠀⠈⠉⠀⠀⢘⣿⣿⡏⠁⠀⣠⣤⣤⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⣿⡿⠟⠿⠛⠓⢶⠖⠈⠻⠿⠃⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⡀⠀⠀⢀⠀⠀⠉⠉⠉⠀⠉⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠙⠷⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠉⠉
⠀⠀⣶⡿⠖⢶⣶⣿⡿⢠⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠠⠀⠀⠀⠀⠀⠲⠗⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀
⣶⣾⣿⣶⣶⣾⣿⣿⢁⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⣶⣷⣾⣿⠁⢠⣶⣤⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⣷⣶⣶⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇
⠉⠹⣿⣿⣿⣿⣿⣷⣶⣶⡟⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⡛⠛⣁⣤⣄⠀⢀⣠⣶⣶⣶⣦⡀
⠀⠀⠀⠉⠉⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠁⠀⠀⠉⠉⠁⠉⠉⠁
⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⠀⢀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⡽⢟⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢋⡵⣻⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣟⢶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡵⠋⣴⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⢟⠟⡛⢟⠟⣿⡄⠙⢿⣿⣏⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣡⠞⣡⠞⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡿⢋⢋⢊⠎⠈⠰⠟⠀⠹⣿⡄⠰⠙⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡴⢋⡞⣡⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣯⡟⣸⠁⠇⠆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠹⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣴⣏⡼⢃⡜⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢿⣿⡆⢠⣿⡈⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣠⠏⡼⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢁⢃⠋⢿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡾⢡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠧⢸⢸⢰⣌⣿⣶⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⠛⠋⠀⠀⠀⠀⠀⣄⠰⠀⣷⢻⠸⢸⢸⢰⢽⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢻⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠘⢣⣵⣿⡿⣦⠶⣾⣾⣾⣾⣿⣿⣿⣿⣿⣿⠓⣿⣿⡿⠿⠿⠿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣼⡃⠀⢸⣿⠿⣯⣤⣿⣿⡿⣿⠃⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢨⠏⣿⣿⡇⠀⠈⢿⣿⣦⣌⣉⣉⣿⣧⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢠⡿⢇⣿⣆⠀⠀⠉⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠀⠀⠀⠸⠇⡸⣎⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢱⣹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠐⢯⡆⠀⠀⠀⠀⠀⢰⡿⣾⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⣷⣳⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠸⡇⠀⠀⠀⠀⡰⢳⢳⢣⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠐⠿⠀⡀⢀⠞⡵⣣⢣⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⣼⠗⢡⢎⡔⠡⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⠀⠡⢊⡴⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⠠⠋⠠⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⡀⠀⠀⠀⠐⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⠀⠀⢀⡟⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢀⠇⢈⢎⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⢠⢪⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠸⢀⡏⠀⡣⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠂⠘⠀⢠⡗⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠸⡙⢎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⢻⡄
⠀⠀⠀⠀⠀⣀⣤⣶⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠘⡿⠁⢾⣷
⠠⡾⠶⠿⠟⠛⠛⠉⠁⠀⠀⠀⢀⣠⣤⣔⣶⣾⣷⡶⢤⣄⣀⠀⠚⡀⣘⣿⡇
⠀⠙⠿⠷⢶⣤⣤⣠⣄⣎⣽⣾⠿⠟⠋⠉⠉⠉⠙⠛⠛⠻⠿⢿⣿⣶⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠘⣈⡔⠁⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡞⢡⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠖⢄⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠓⠂⠁⠀⠀⠀⠀⠆⠀⡀⠀⠀⢀⢂⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⡐⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢀⠊⠀⠀⢀⠄⠘⢠⠂⠀⡀⠀⠎⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡔⠁⠀⠀⠀⡌⠐⠁⡌⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠎⠀⠀⠀⠀⡈⠰⠂⠀⠃⠀⠀⠀⡀⠐⠒⠀⠀⠀⡄
⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⡏⠀⠀⠀⠀⠰⠀⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡰
⠀⠀⠀⠀⡇⠀⠰⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐⠰
⠀⠀⠀⠀⡇⠀⠀⠘⢐⠀⠀⠀⠀⢠⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⢰⠀⠀⢸⠀⠀⠀⠀⠀⡀⠉⠄⠀⠀⠘⠁
⠀⠀⠀⠀⡇⠀⠀⠐⠠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠆⠀⠀⠀⠀⡁⠀⠀⠀⠘⡄⠀⠀⠀⠌⠐⠀⠀⢀⡜
⠀⠀⠀⠀⢸⡀⠀⠘⠀⠁⡄⠀⠀⠠⠀⠐⠀⠀⠀⠘⡄⠀⠀⠀⡆⠀⠆⠀⠀⢇⠀⠀⢰⠂⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢧⡀⠀⠀⡀⠉⢀⠀⠀⢠⠀⠀⠀⠀⠀⠘⣄⠀⠀⢘⠀⠀⡀⠀⢸⡄⠀⢀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠧⠠⠀⠁⠐⠄⠀⠀⠀⠀⠀⠲⠄⠀⠀⠈⠢⠀⠀⠂⠀⠇⠠⠀⠻⠄⠸⠀⠀⠀⠠⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⣿⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣸⠀⠀⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⡀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⣾⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡄⠘⣿⣿⣿
⠀⠀⠀⢰⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⡄⢿⣿⣿
⠀⠀⠀⣼⣿⣇⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢾⠇⠅⠸⣿⣿
⠀⠀⠀⢹⣿⡟⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⡄⠀⢸⣿⣿
⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿
⠀⠀⠀⠀⠸⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢌⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣠⢞⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣾⣿⣿⣦⠀⠀⠀⠀⢠⣾⣿⣷⣆
⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡀
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⣠⣼⡿⠟
⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣾⠟⠁⠀⢀
⠀⠘⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠉⠀⢀⣈⣷⣄⡴⠋⠀⠀⢀⢔⠝⠁⠀⠀⡰⠋
⢠⠞⠻⠿⠿⠟⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡠⠒⠁⠀⠐⠋⠀⠀⠀⢐⠕⠁⠀⢀⡴⠋
⠸⡄⠀⠀⠀⠀⣠⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡜⠁⠀⠀⠀⠀⠀⠀⣠⠖⢁⣠⠴⠚⠁
⢀⡏⠉⠀⠈⢉⣀⣀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠔⢋⡤⠚⠉
⢸⢀⠀⠀⠈⠀⠀⠉⠉⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠿⠊⢁⡤⠚⠁
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁
⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠙⠻⠷⣖⣶⠖⠐⠖⣒⡮⠽⠚⠁
⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣆⣀⣀⣀⣀⣀⣀⣀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⣿⣧⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣤⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⠀⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡌⠀⢢⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣛⠹⠉⠉⠉⠉⢹⡙⠋⠉⠉⠉⢹⡛⠏⠉⠁⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⢃⠀⠀⠀⡖⠉⢱⠀⠀⠀⢠⠉⠙⡆⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠀⠀⠀⠀⠙⠖⠎⠀⠀⠀⠀⠖⠾⠁⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣶⡄⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠉⠻⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⢿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛⠉⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠐⡆⠱⠀⠀⡶⢡⠐⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣠⠄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢏⠁⠈⣾⢁⡞⠀⣷⡟⠀⠀⠀⠀⠀⢀⣠⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠔⠂
⠀⠀⠀⠀⠸⣧⣀⣾⠋⠀⣠⣿⠇⠀⠀⣠⣶⣾⣯⣭⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀
⠀⠀⠀⠀⣀⣿⣿⠿⠟⠛⠋⠁⠀⠀⠀⠉⠀⠀⠉⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⠆
⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⠴⠶⠒⠃⠈
⠀⠀⠀⠀⠀⠀⠀⣠⠾⢶⡚⠁⡠⠂⠀⠀⠀⠀⣞⡀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠀⡀⢠
⠀⠀⠀⢀⣠⣴⡿⠏⠀⠀⠈⠙⠒⠊⠀⣀⣠⡴⣭⣄⣀⠀⠀⠀⠀⠀⠀⣴⠀⢰⣧⡷⠂
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡶⠟⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣤⣺⠿⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⣋⣤⡶⢶⣶⡶⠶⠤⠞⠒⠒
⠀⠀⣀⣀⣾⡿⠛⢿⣍⣛⣡⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣾⣿⣶⣋⣉⣀⠀⠀⠉⠙⣖
⠀⠠⠟⠛⠉⠀⠀⠀⠙⠛⠒⠒⢀⣀⠔⣀⣤⠚⠀⠀⠀⠀⠀⠈⠀⠀⠈⠩⢳⠦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣶⡯⠕⠊⠉⠀⠉⠀⠒⠯⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⣴⣶⠿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⣤⠴⠚⠀⠀⠀⡀
⠀⠀⣿⣷⣦⣴⡾⠟⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣄⣠⣤⣾⣿⣇⠀⠈⠀⢀⣠⠶⠖⣁
⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣀⣀⣀⠾⠀⠀⠀⠙⠟⠉⠀⠀⠈⠛⠓⠶⣿⣁⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠄⣴⠋⠉⣀⠈⠉⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠈⠉⠁
⠀⠀⠀⢀⣠⣴⣾⣿⢷⣦⣤⣤⣴⣿⡶⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠃
⠀⢤⣶⣿⡿⠋⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⣶⣦⣤⣼⠷⠶⠤
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢋⣽⣿⣿⣿⣿⣿⣿⡿⣀⣸⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠺⣮⣽⣿⣿⣿⣿⣿⠟⠉⠀⣀⣀⠄⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣿⣦⣉⠛⠻⢿⣿⠟⠁⢀⡀⢺⣿⣿⣿⡦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⢠⠏⠠⠀⠘⠀⠀⠙⠟⠟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⢋⣁⣀⠈⡟⠀⠀⠀⢿⣦⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢿⣿⣟⣿⣣⡇⠀⣀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡛⠃⠉⢠⢧⠐⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⢐⠇⠀⠑⠈⢧⠀⠀⠀⠀⢈⠉⠀⠈⠛⠉⠻⡿⠛⠛⠉⠉⠉⠉⣁⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠐⢴⣸⠀⠉⡹⠛⠒⠒⠤⢄⡀⠀⠀⠁⠈⠙⠻⡏⠉⣀⡙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣽⣦⠎⠀⠀⠀⠀⠀⠘⠁⣲⡄⠀⠀⠀⠘⢤⣿⡿⢟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⠚⠉⠉⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠐⠀⠆⠀⡾⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢠⠔⠉⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡿⠀⢀⠀⠀⣀⢀⠈⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠆⠈⠀⢸⣃⡴⣢⠀⠠⣿⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⢀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢈⡁⠀⠀⣠⣶⣿⠿⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡂⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠖⡦⣄⣿⣿⣏⠀⠀⠊⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠧⢸⡄⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠀⣿⡿⠛⠁⠀⠀⡌⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⡄⢉⠅⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⢈⠒⠋⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣶⡀⠸⣆⠙⢦⡀⠀⠀⠀⠀⠀⠀⠒⣩⠉⢢⠁⢐⡏⠀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣼⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠇⠀⠀⠀⠀⡤⠀⣈⣙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢛⣿⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣤⣤⣤⡀⠀⠙⠚⠋⢙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⡓⠋⠁⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣿⣿⡆⠀⣠⣶⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢼⣧⣤⣴⡓⠋⠀⠀⠀⣸
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣠⡶⠛⣲⣶⣿⣿⣿⣶⣿⣿⡅⣀⣤⣶⣶⣶⣦⡀⠰⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣼⡿⠋⠉⣹⣷⠀⠀⠀⠀⣿
⠀⠘⠷⣶⠛⠛⢿⣷⡄⣸⡿⠷⠀⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣻⡿⠛⢻⣿⣿⣶⣶⠟⠀⠀⣀⣠⣶⣶⠤⠀⠀⠀⠀⠀⣴⠋⣴⣿⣟⣀⣴⣾⣿⠿⢀⣀⣔⣶⣿
⠀⠀⠀⠘⣷⣤⣾⣿⣷⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣦⣀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣦⠀⢿⡀⠀⢷⣼⣿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿
⣶⣾⣶⣴⣿⣿⣿⣿⣼⣿⣇⠈⠛⠚⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⠻⣿⣷⣶⣿⣿⣿⣉⣙⣻⣿⡿⠷⢟⣉⡳⠀⢹⣿⡀⢲⣿⣿⣿⢿⣿⣿⣷⣿⣿⡻
⣿⣈⠙⠛⢛⣿⣿⣿⣿⣿⣿⣷⣤⠀⠙⢛⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⣎⣻⡿⣿⣿⣷⣿⣿⣽⡿⡇⢰⡏⣉⣡⣤⣨⣿⣷⣾⣿⣷⣻⣹⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⡿⢋⣿⣿⣛⣿⣿⣿⣿⣧⣠⣿⣿⣿⣻⣷⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⢋⣿⣾⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣾⣿⣿⣇⣀⣥⣤⣿⣿⣛⡛⢿⣿⢿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣻⣿⣿⣿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡻⠛⢿⣿⣿⣿⣿⡿⢩⡇⣠⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣯⣬⣿⣿⣿⣿⣿⣿⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣉⠻⣦⡴⢠⣐⢮⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠗⠻⡄⠘⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠁⠀⠲⢄⡾⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⢀⡴⠋⠱⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡇⡀⣤⣮⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⢏⡽⣋⢀⠀⢤⣄⠀⠀⣀⣶⢲⣴⣶⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⣴⡿⠷⣯⣾⡃⡴⠟⣷⡶⣷⣿⣿⡿⢻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠧⠀⣯⠀⠀⠛⠛⢶⣛⣶⡞⣿⣾⣿⠏⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠘⣧⠐⠀⠊⠛⠻⣿⣿⣹⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣟⡠⠈⢷⣄⠀⠀⠒⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠹⣷⣯⣿⣷⡀⡙⠻⢦⡀⢀⡹⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣳⣤⣽⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢣⣄⠀⠀⣦⣼⣿⣿⣿⣿⣿⣿⣿⣄⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢿⣦⣝⣿⡿⠙⣿⣿⣿⣿⣿⣿⣆⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣿⣿⣂⡼⠉⢿⣿⣿⣿⣿⢯⣴⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⢿⣿⣿⣿⣿⣿⡄⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢞⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣣⣄⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠇⠘⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡖⢠⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡐⠀⠐⢀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⣀⣴⠏⠀⠀⠀⠀⠀⠀⣀⠤⠀⠤⢤⡀
⢀⠠⠤⠤⢀⣀⠀⠀⡀⠀⠀⠀⠀⠔⠊⠁⡠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⡸⢀⠤⠤⢄⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠈⠻⣁⠈⠢⠀⠐⠁⠀⣰⠝⠀⠀⠀⠀⡠⠤⠤⢄⡀⠀⠀⠰⣇⠀⠀⠀⠙⢦⠀⢱
⡀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣰⠃⠀⢜⡁⠀⠀⠀⠀⡜⠀⡰⢶⡤⠴⠄⡀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠈⠻⡿⢶⣦⡀⠀⠀⠀⠱⣿⠀⠀⣾⡇⣄⠀⠀⢀⠇⢠⠋⠁⠀⢀⡀⠀⠐⠄⣀⠀⣀⡤⠋⢀⡞
⠀⠀⠘⡄⠈⠹⡦⡀⠀⠀⠸⠀⠀⢸⠃⠀⠉⠈⠁⢀⠏⠀⠈⣸⡟⠀⠢⢀⠀⠈⠁⠀⠀⡠⠊
⢢⠀⠀⢷⠀⠀⡀⠈⢦⡀⢸⡇⠀⢰⣿⡤⠤⠴⠚⠉⠀⣤⣰⡿⠁⠀⠀⠀⠉⠙⠒⠈⠁
⣾⡀⠀⠘⣄⠀⠁⠀⠀⠙⠸⣧⠀⠠⣿⣶⠀⠀⠀⢀⣠⡾⡟⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⣀
⢧⣹⡄⠀⠙⢄⣄⠀⠤⢤⣀⡘⠀⢠⢿⡹⠓⠛⠉⠀⢸⠘⣇⣀⣼⠆⢠⠞⣁⡶⠒⠛⠳⣮⠳
⠺⣿⡙⣆⡀⠈⠙⠂⠴⠶⠚⠁⠀⣼⡟⡥⣀⠀⠀⠀⠀⠓⠦⠕⢊⡴⣁⡼⠋⠀⠀⠀⠀⡾⣠⠁
⣤⣦⣽⠽⣭⣦⠠⣄⣀⣤⣀⣔⣾⢛⠯⣄⣠⠀⠁⠐⠢⠤⠔⠚⢁⣴⠟⠁⠀⠀⠈⠃⠛⠒⠁
⣿⠭⠛⠁⠚⠫⠹⠿⢛⠲⠛⠉⠉⠀⠀⠀⠉⠙⠙⠆⠶⠴⠲⠛⠋

⠀⠀⡀⠄⠒⠀⠈⠉⠙⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⡀
⣴⡏⠀⠀⠀⠀⠀⢀⠀⣀⣀⠤⠼⠖⠂⠀⠤⢀⡀⠀⠀⠀⡠⠊⠁⣔⡧⡱⠎⠄
⣿⣶⡶⣂⣺⣏⣶⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⣰⠃⠀⡾⠁⠀⠸⣽
⣿⠦⠄⠉⠙⠛⠁⠀⠀⠀⢀⡀⠰⡎⢤⣙⡟⡀⢶⠤⠀⠈⠓⠼⣇⣀⣠⠤⠟⢛⡈⣭⢨⣅⣤⣀
⠟⠀⠀⠀⠀⠀⠀⠀⣀⢐⠻⡜⢳⢾⠶⣷⡦⠰⢯⡁⢼⣳⡒⣦⠠⣤⢀⣤⣮⡽⠾⣿⡝⠉⠙⠻⣷⣶⣤⣤⠶⠃
⣗⠀⠀⢀⡠⢤⡴⡘⣣⣍⠞⠟⠛⢿⡀⢓⠀⡀⠀⠑⢤⣇⣉⠿⠗⠋⢉⠟⢙⡁⠀⠈⠻⣗⠀⠀⠀⠈⠁
⣾⢁⣈⣋⣐⡠⠷⠤⣝⣙⣮⣠⠷⠀⠹⣝⠂⢚⠀⡀⠀⠉⠀⠀⣀⠴⠁⠀⠈⢇⠀⠀⠀⠻⣄⠀⠀⠀⠀⡀
⠀⢉⣅⡞⠉⠀⠀⠀⠀⠹⣶⢡⣆⣰⠦⠡⡈⠙⠤⢡⣤⡦⡶⣺⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠛⠒⠒⠊⠁
⣷⢨⠏⠀⠀⠀⠀⠀⠀⠀⠈⠹⡼⠕⣆⠰⠐⠦⠴⠊⠀⠾⢰⠃
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⠏⣸⠖⣀⠚⡀⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠉⠉



⣀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⢀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⡀⢀⣀
⣿⣿⢛⠿⢛⠛⢿⢟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠀⠀⠐⠒⠒⠀⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡡⠊⢀⠠⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠢⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠈⠄⠈⠀⠀⠀⠀⢀⠠⠀⢀⢄⠀⠀⠀⠀⠈⠢⠀⠱⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠖⠁⠀⠑⢄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡁⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠑⠠⢈⠒⠠⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⢈⡗⣹⡠⡄
⠀⠀⠀⠀⠀⠀⠀⡤⣘⢣⡞⠀⠀⠀⠀⠀⠀⠀⢸⢉⢉⡇⠀⠀⠀⠀⢴⠄⠀⠀⣀⠀⠀⠀⠶⠀⠀⠈⠁⠀⣶⡚⠧⠀⠀⠀⠀⠀⠁⠀⠁⠉⠮⡄
⠀⠀⠀⠀⠀⠀⢠⢴⠘⠈⠁⠰⠘⠆⠀⠀⠀⠀⢳⠈⠒⢇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠠⠀⠀⡰⠂⢁⠀⠀⠀⡠⠊⠑⢄⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⠏⠑⡈⣆⠀⠈⠢⠋⠀⠀⠈⠛⠋⠀⠀⠀⠘⠐⠀⡰⠇⣀⠀⠈⠶⠊⠀⠀⠀⢀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠒⠉⠀⠑⠄⡀⠀⠀⡙⠀⠠⠟⠉⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢾⠢⡤⠜⠢⡀⠀⠣⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠈⠁⠚⠠⠀⡇⠐⡎⠸⡁⠈⠁⠲⠂⢤⠠⠀⡔⠒⡼⠁⠀⢀⡆⠈⠀⠀⠘⣄⠐⣣⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⢠⡒⢤⡞⠀⠀⢠⠀⠈⡷⠀⢤⠱⣀⠋⠈⣣⠌⢰⠠⠂⠉⡆⠀⠃⠀⠀⢀⣯⠐⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢠⡄⠉⡘⠁⠀⠀⠸⠀⠀⠇⠀⠘⢄⠙⡄⢠⠋⡠⠊⠀⠀⠀⠃⠀⣤⠠⠊⠁⠀⠢⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠔⠋⠲⠀⠀⡆⠀⢠⠀⠀⠀⠀⠑⡜⢣⠎⠀⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⡄⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠰⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⢠⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⣠⢃⡀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⢀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡇⠀⠀⠀⠈⠉⠉⠉⢉⡍⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡸⠉⡲⠒⡦⠀⣤⢀⣄⠀⡤⠠⠴⠲⠆⠸⠉⢏⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⡰⠁⡰⠁⢠⠁⠀⡄⠀⡇⠀⡇⠀⡀⠀⠇⠀⢃⠈⢆⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠜⠀⡜⠀⢠⠃⠀⣤⠀⢀⠇⠀⡇⠀⣷⠀⢠⠀⠈⡄⠈⢆⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡰⠋⢀⠞⠀⣠⠃⠀⢰⠆⠀⢸⠀⠀⠁⠀⣿⠀⠘⠀⠀⢡⠀⠈⢳⡀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⣀⡰⠃⠀⡴⠁⠀⢀⠇⠀⠀⡸⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡆⠀⠀⠱⡀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠓⠠⠾⣀⠀⢀⠎⠀⠀⠀⡇⠀⢸⠀⠀⣸⠀⠀⢰⠀⠀⠸⣀⣀⠤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠐⠊⠀⠀⠀⠀⠀⡨⠋⠑⠒⠀⠴⠤⠤⠤⠤⠤⡼⠤⠀⠚⢲⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡇⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢄⡄⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⠂⠠⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠚⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⢻⣛⣟⣋⡜⣛⣿⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣍⣉⣿⣟⣿⣿⣟⣛⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⢡⠀⠀⠀⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⣒⠂
⠀⠀⠀⠀⠀⢠⡰⠐⠑⠚⠚⠔⡄⡄⠀⠀⠀⠀⠀⠀⣾⠀⠔⡢
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠈⠐⠅⢅⠈⠀⠀⢴⠂⠠⠁⡂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢘⣆⠀⢐⠃⣀⠍⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠐⡢⠃⠈⠩⠀⠀⢈⡀⠀⠀⠀⢀⡠⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠁⠀⠂⠀⠈⡋⠀⠀⢐⠀⢀⣂⠀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⡸⠀⠀⠀⠀⠀⢐⡒⢐⠂⠔⡀⠐⠀⠀⠀⠂⠄⡐⠁
⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⢐⠀⢨⠀⠁⠆⠀⡂⠀⢈⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠀⠅⠈⠅⠀⠄⠀⠀⠐⠀⠀⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡆⠀⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡆⠠⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠃⠀⠠⡀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡀⠀⡄⠀⠀⠠⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠄⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠰⠌⠂⠀⠀⠀⠀⠈⠊⢢⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠊⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⡓⠀⠀⠀⠀⠌⠰⣀⠱⢐⣚⠂⣑⣂⠀⡀⠆⠀⠘⠈⠉⠀⠀⠀⠀⠀⠊⠠⠀⡖⠀⠀⠀⠀⠀⣀⠐⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡀⠀⠀⢀⡐⠹⠉⠀⠀⠈⣀⣄⠂⢈⢣⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠈⠀⠀⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⢰⠁⢀⠴⠂⢿⠂⠀⠀⡴⠁⠀⠈⠡⡈⣶⠃⠀⠠⠄⠁⠀⣤⠀⠀⠀⠀⢰⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⣸⠎⡙⠀⠀⢋⠁⠀⠀⣗⢟⣻⠊⠀⠀⡁⢀⠀⠀⠀⣰⠁⢺⠀⠀⠀⠂⢠⢂⠄⠀⠀⡄
⠀⠀⠀⠀⠀⢀⠔⠂⠁⠀⢀⣅⠔⠈⣀⠁⠀⢾⢥⠀⡄⠀⢸⠀⡄⢠⠀⠠⡃⠂⠀⠀⠠⢓⠄⡠⡀⠀⠀⠀⢸⠈⠀⠀⠀⠁
⠀⠀⠀⠀⢠⠃⠀⠠⠄⠀⡹⠃⢰⣤⠋⠀⠠⢈⠣⢧⠀⠀⢰⠏⠀⡤⠒⠉⠀⠀⠀⠀⢠⠟⠀⡲⠥⠀⡀⣴⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠁⠀⢤⣏⠀⡜⡱⠁⠀⠀⠀⢀⡜⠀⢸⠀⠀⠀⠀⠠⠂⡤⠏⠀⠐⢁⣼⠏⠩⠉⠐⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠈⡀⠀⠐⠀⢸⠀⠀⠐⢯⠀⡁⢁⠀⠀⠀⠰⡏⠀⠀⡘⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⠋⠐⠀⠠⠀⠉⠀⣠⠖⠉
⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⢧⠀⠀⠀⢷⣇⣫⠀⠂⠀⠀⠹⡄⠀⡇⠀⠀⠀⠀⠀⠞⠁⢀⠀⠘⡆⠘⠀⠀⠀⠀⡞⠋⠀⠀⠀⠀⡄⠀⠀⡀
⠀⠁⠀⠀⠀⡇⠀⠀⣀⠀⠈⣧⠀⢌⠢⠙⣾⢰⠀⠂⠀⠀⠈⠳⣧⠀⠀⠐⠀⠄⠘⢷⠂⠠⡜⠀⠰⠑⠀⠀⠀⡇⠀⠀⠀⠂⠀⠁⠀⠀⠐⠄
⠀⠀⠀⠀⢀⠇⢸⠀⠀⠂⠀⢸⠀⠈⠆⠃⠘⣮⡀⠀⠠⡀⠀⠀⢸⡀⠀⠀⠀⠁⠀⠾⠀⠀⣻⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠐⣫⡄⠀⢰⠀⠀⠀⣳⢀⠀⠀⢄⠀⠈⡇⠀⠀⠀⠀⠠⠈⠳⡄⡞⠀⠀⠀⠈⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⡌⠀⠘⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠳⣧⡀⠀⠀⠠⠀⠘⡄⠀⠀⠀⠀⠀⠀⠣⣩⠀⠘⠀⠀⠀⠀⠀⡇⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⣶⣾⣿⣿⣷⣾⣍⣽⣿⣿⣿⣯⣽⣿⣷⣶⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢤⣄⢠⠴⠃⣠⢾⣿⢿⣿⡧⣿⣽⢿⣿⣿⣾⣿⢿⣿⣋⠻⢷⣿⣿⣦⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡏⡤⠉⠀⡆⠀⠿⣟⣘⢿⣿⡿⠟⠛⠛⠛⠰⠂⠈⡁⠈⠛⠯⣙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢀⠀⠀⠀⠉⠉⠀⠐⠂⢀⠁⠀⢀⠀⠂⠀⠸⡄⠀⠀⠀⢉⠛⠉⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣀⠴⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠤⠄⡀⡆⠀⢠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢸⠁⠀⠀⠒⡐⠒⠒⠒⠒⠒⠚⢦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠋⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⢤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⣍⡓⠶⢤⣀⢰⠏⠀⠀⠀⠙⠓⠚⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠃⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠀⠀⠀⠀⠀⠀⠀⡀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⠀⠀⣈⠀⠙⢀⠀⠰⠀⢠⡞⠀⡀⠘⠂⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⢀⡀⠐⠂⠀⣀⡴⠃⠀⣠⠣⣼⠀⠀⢀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⣷⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⠀⡀⠘⣀⣠⡤⠴⠖⠛⠁⠀⢀⡴⠃⠀⠙⣄⠀⣾⠀⠀⠛⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠐⠉⠀⢀⣀⣈⡤⠶⠛⠉⠀⠀⠀⢀⡤⠶⠚⠋⠀⠀⠀⠀⠈⢧⣿⠀⠀⡴⠀⠄⠀⠈⠀⠀⡀⠉⠀⠀⢠⡄⢻⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢤⣄⣀⣀⡤⠶⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢠⡇⠀⠀⠐⠂⠀⠀⠁⠀⠁⠀⠀⠀⢸⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣷⠀⠀⠀⢀⠈⠃⠀⠀⢀⠁⠘⠃⢸⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢀⡇⠀⠐⠂⠀⠈⢀⡀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣘⡇⠀⠀⠀⠀⠀⠈⠁⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⣠⡤⠶⠒⠒⠀⠀⠠⡄⠀⠀⠈⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⠋⠁⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠘⠂⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠔⠋⠀⣼⠃
⠀⠀⠀⠀⠀⠀⡼⠁⢠⡞⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⣄⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠴⠚⠁
⠀⠀⠀⠀⠀⢰⠃⠀⡾⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠀⠀⠀⠈⠁⠀⢠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⣿⠒⠦⢤⣄⣀⣀⡴⢞⡭⡍⠳⣄
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠋⠀⠀⠰⠄⠀⠀⠀⣴⠃⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠐⠒⠀⠀⠀⠀⢚⣀⣠⠤⠀⠀⠀⣸⡇⠛⠛⠓⠒⠲⣦⠰⣄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠈⠐⠒⠒⠶⠶⠶⠤⠤⠤⠶⠶⠶⠶⠖⠛⠛⠉⣁⣠⣤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⢷⣌⡙⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠤⠤⠤⠤⠤⠤⠶⠶⠶⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⢛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣾⣥⡀
⠀⠀⠀⢀⣾⣻⣟⣍⣿⡇
⠀⠀⠀⠸⣧⡈⠈⠉⠉
⠀⠀⠀⠀⠹⣿⣾⣄⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣷⣦⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⡿⣫⣶⣿⣿⣿⠿⠿⣿⣿⣷⣄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢰⣫⣾⣿⣿⠋⠁⠀⠀⠀⠈⠻⢿⣾⣐⣀⣶⣼⠆
⠀⠀⠀⢀⣴⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣠⣤⣴⣿⣿⣿⣿⡿⡟
⣿⣿⣿⣿⣽⣿⣿⡇⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⣙⠛⠛⣋⣽⣿⣿⣧⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⣿⣧⡀
⣀⣼⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧
⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰
⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⣿
⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣠⣤⣤⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣭⣭⣯⣽⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠐⢋⣠⣶⣿⡿⠁
⠉⠁⠀⠙⠻⣿⣿⣭⣿⣯⡼⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠉⠉⠓⠻⡿⠟⠻⠟⠉⠀⠀⠀⠀⠀⠀⠙⠛⠿⣽⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠰⢤⣀
⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⢸⠈⠙⠦⣄⡀
⠀⣤⣤⣤⣤⢸⠀⠀⠀⡏⠙⠲⢄⡀
⠀⣿⣿⣿⣿⣼⣤⣤⡄⡇⠀⠀⢸⠉⠳⢤⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣇⣇⣀⣀⢸⠀⠀⠀⣿⠓⠦⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣀⣀⡀⣿⠀⠀⠈⡟⠦⣄⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⢹⠲⢄⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡇⠀⠀⢸⠀⠀⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡆
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⠶⣦⣄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⢠⡾⠛⣿⣦⠀⠀⠀⢠⣶⠉⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠙⢀⣿⡿⠀⠀⠀⠈⣿⣧⠀⢀⣴⡾⠟⢁⣤⣶⣶⣦⣤⡀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣴⣿⠁⠀⠀⠀⠀⢠⣴⣿⠋⠀⠀⠀⢀⣼⣿⠏⢠⣿⠋⠀⠀⢸⣿⣿⠁⠀⢻⣿⡆⢸⡅⠙
⠀⠀⢠⣶⣻⣭⣽⣫⠖⢿⣿⠟⠋⠀⠀⠀⠀⠀⡏⣿⣿⣧⣤⣤⣶⣿⡿⠋⠀⢸⣽⣧⡀⠀⠘⣿⣿⣦⣀⣸⣿⣇⣸⣿⠄
⠀⣨⣼⡿⠛⠛⠻⣿⡴⣎⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠛⢿⣿⣯⣳⣮⣛⠿⢿⣿⣿⡿⠿⠋
⠀⣿⣿⣧⣀⣠⣤⣼⣧⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠛⠉⢿⣿⣿⢛⠀⠀⠀⠈⠛⠿⣿⣽⣷⣾⡿⠃
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣬⢯⡛⡱⢖⡠⠤⢤⣤⡀⠀⠈⣿⣿⣦⣴⣿⣿⢇⡞⠀⠀⠀⢀⣴⣾⡿⠿⠛⣯⢳⡄
⠀⠀⠀⠉⠉⠉⠉⠘⣿⣿⣶⣦⣭⣬⣓⣹⡛⢙⣗⠟⢲⣿⣿⢿⣿⣿⣿⠛⠀⠀⠀⠀⣼⣿⠿⠁⢀⣤⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⢿⣿⣿⣷⣞⢻⣿⡿⠟⠹⢾⡛⣿⣦⡀⠀⠀⢻⣿⣿⢹⡟⢹⣿⢏⣽⠇⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣠⣾⣿⣿⣿⠇⣿⣿⣤⣤⣤⣤⣿⣛⣿⣷⠀⠀⠀⠉⢛⣿⣷⣾⣿⣼⣷⠾⠟⠁
⠀⠀⠀⠀⡠⢴⣿⣿⣿⣿⣿⣿⣛⣙⣽⣿⣿⡿⠿⣿⣿⣻⡟⢛⡿⣿⣿⣿⣿⣿⣦⣄⠀⣰⣿⡟⢿⣿⣿⣿⣦
⠀⠀⣠⠾⡖⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⢀⣀⣑⣾⣿⡹⢶⢧⡚⣻⣿⣿⣿⣟⠈⢻⢿⣿⡀⠀⠹⣿⣿⣿⣧
⠀⠈⣕⣰⠀⣿⣿⣿⡿⠛⠉⠁⠀⠀⢀⡴⠚⣿⣿⣿⣿⣿⣿⣿⣿⠽⣾⣿⣿⣿⣿⣀⣸⣦⣿⣿⣶⣶⣿⣿⣿⣿⣀⣀
⠰⡺⢩⡏⢰⣿⣿⡟⠁⠀⠀⠀⠀⣴⣏⣴⣼⣿⣿⣿⠟⠛⠉⠁⠀⢀⣼⣿⣿⣿⣿⣿⡟⢀⣽⣿⣿⣿⣿⣿⣿⣯⣽⠿⣗⡤⡀
⠠⣙⣬⣿⡀⢿⣿⣧⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⣿⡿⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠏⢠⠋⠀⣿⣿⣿⣿⡿⢿⡿⢿⣿⠾⣶⠣
⠀⠩⡃⠠⡹⣾⣿⣿⣦⣀⠀⠀⣿⣿⣿⣿⡿⡌⣷⢳⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢠⣯⣴⣾⣿⣿⣿⡟⠀⠈⣇⣨⣟⠈⣸⠁
⠀⠑⠴⣷⠖⢻⡿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣘⢻⣝⢫⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⠁⠀⣸⣿⣿⣿⡟⣿⡍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣞⡋⠉⡉⠩⠉⡭⠉⠉⠉⣐⠀⢰⠄⠠⢠⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡖⠋⠉⠀⠐⠀⠂⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⠟⠉⠋⠅⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⢌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⢣⠀⠀⠀⠃⣧⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠰⡈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢰⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠘⠠⠀⠀⢸⡄⣾
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠂⠀⠀⣢⡿⢱⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⢀⡐⣶⣶⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⣱⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣬⣼⡟⠁⠀⠹⣷⠀⠀⣀⣀⣀⢀⣈⠥⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠻⣷⣞⣥⠶⠖⠏⠀⠊⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣯⢦⡄⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢀⡟⣀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠼⠿⠿⠚⠉
⠀⠀⠀⠀⠀⢀⣞⣵⠃⠀⢸⣿⡏
⠀⠀⠀⠀⣰⠿⠟⠉⠀⡀⢾⣿
⠀⠀⠀⠀⣧⠈⠀⠰⣶⡇⣀⣿
⠀⠀⠀⠀⠈⢳⠶⢶⡠⢠⡿⠃
⠀⠀⠀⠀⠀⠈⠙⠓⠒⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢚⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣒⣲⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢠⠾⠛⠋⠉⠉⠛⠓⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠋⠉⠉⠙⠛⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡚⠋⠀⠀⠙⠶⣦⣤⠄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⣮⣝⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⣥⣾⣷⣿⡿⣄⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⢃⡾⣿⣏⢘⣿⣿⡛⣮⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⢛⣿⣿⡋⣹⣿⣷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠁
⠀⠀⠀⠀⠀⢀⡼⠷⠿⢽⣟⣿⣻⡿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣵⠿⠿⢿⣟⣿⣽⡧⠿⠾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡽⠖⠛⠉⠉⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠶⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢤⠴⠖⠚⠛⠛


⠀⠀⢠⣶⡖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣆
⠀⠀⠸⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠋
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡀⠀⠀⣠⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠈⢻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡛⠲⠦⣤⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠛⢉⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⣀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣀⣀⣠⠖⠻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠛⠚⠛⠓⠾⣆⣀⣀⣤⣀⣀⡼⠛⠛⠋⠉⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⣀⣤⣴⠃⢀⡴⠒⠚⠿⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣴⠋⣴⠃⢠⡟⢀⣤⠤⠾⠯⠤⢬⡝⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢸⡃⡼⠃⣠⠟⢀⡾⠁⠀⠀⠀⣀⣀⡹⣆⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠸⡇⢻⠖⠻⠶⡾⠁⣠⠟⠻⡿⠉⠉⠙⠻⢾⡇
⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣠⣤⣀⣀⣀⣀⣀⣠⣤⣤⣄⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⡞⠀⠀⠀⠙⠛⠁⠀⠀⠀⣰⠷⢦⣄⠀⢳⡀
⠀⠀⠶⣿⡁⠀⣹⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣍⣀⠙⠃⠉⠋⠉⠁⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠈⢻⠞⠁
⠀⠀⠀⠀⢹⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠤⢤⣤⣤⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⠀⠀⠀⣠⣤⠴⢾⣅⡀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠈⠉⠛⠲⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣾⣿⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣀⣰
⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠿⠿⣯⡉⠙⠻⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠁⠀⣠⣿⣿⣿⣻⣿⠀⢀⢀⣴⣷⣿⣿⣿
⠀⠀⠀⢠⣿⣿⠏⠸⡁⠀⠀⠀⠘⣿⡀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀⢀⣴⣿⣿⣿⣿⡿⣁⣴⣿⣿⡿⢿⣯⣭⣿
⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣿⣿⢟⣿⣿⣥⣾⣿⣬⣾⣿⣾⣾⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣦⣤⣤⣤⣴⣾⠟⠁⠀⠀⣠⣿⣟⣿⣃⣀⣀⣀⣀⣀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠟⢩⣿⣿⣿⣿
⠀⠀⢀⣀⣀⡈⠉⠋⠉⠉⠉⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣿⣿⣿⠟⣻⣿⡿⡿⠋⠁⢠⣿⣿⣿⣿⣿
⠀⣴⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣿⡿⣟⣸⣿⣿⡿⠋⠀⠀⣴⣿⣿⣿⣿⣿⣿
⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣿⣿⣿⣅⠀⠀⣀⣀⣤⣾⣿⡟⣛⣿⣵⣿⣿⣽⢿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⣿⣿⣿⣟⣿⣿⢿⣿⣻⣿⣧⣾⣿⣿⣟⣟⢾⣟⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣻
⠘⣿⣿⣷⣦⣤⣤⣤⣴⣾⣿⣿⡿⠟⠁⠀⠘⠿⠿⣿⣿⣿⣿⣿⣟⣻⣿⠍⠊⠀⠉⠻⡿⣿⣿⣟⣻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿
⠀⠈⠛⠿⢿⣿⢿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣿⡟⠁⠀⠀⢀⣠⡶⠚⠛⠛⡟⠛⠻⠛⠛⠛⠋⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⣤⡶⠟⠉⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠉⠉⠉⠉⠉⠉⢛⡆⠀⠀⢀⣀⣈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⠤⠒⢉⡠⠖⠊⠁⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠄⣀⣀⣀⡀⠀⠀⠘⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣠⠔⠋⢁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡼⠁⠀⡰⠋⠀⠀⠀⣀⣀⡀⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⡸⠁
⡇⠀⢰⢀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣰⠁
⡇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠲⠢⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠱⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠒⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡇⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⢣⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠦⡀⢳⡄⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠙⣄⠙⣄⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣆⠈⠳⢄⡘⠲⢄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠣⣀⠀⠈⠉⠉⠀⠈⣣
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠂⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣦⣤⡀⠀⠀⠈⣯⡞⠄⠀⡽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣿⣦⡀⠀⢻⠒⣃⣄⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠸⣿⡧⢶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢿⣉⡟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢸⣦⠼⡙⣀⠀⠀⠀⣠⣴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⢀⣼⣟⣿⠃⠀⣼⣿⠖⣫⡉⣀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣶⣿⣿⣿⠃⠀⠖⣪⣽⡷⠦⢎⣉⡘⣿⣿⡀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⣿⣧⠚⣚⠤⣾⣿⠨⠤⡈⢁⡙⢿⣿⣶⣿⠋⠉⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠼⢇⣩⣭⡟⠁⢠⣷⠛⢉⢭⢴⡏⢻⡶⠂⣌⠉⢃⢨⣿⣯⣿⡷⣄⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡼⠾⣉⣤⣽⣃⣴⣿⡿⠛⡁⠂⣾⠇⢸⡦⠴⣀⠋⠛⣿⣿⠈⢳⣍⡁⠒⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣸⢿⡷⠾⣹⣭⣿⣿⡽⠻⢗⠪⠠⣥⣿⠀⣸⡤⠥⠋⠓⠚⣿⣿⡄⢈⡧⠄⢊⢃⢸⡇
⠀⠀⠀⠀⠀⠀⣰⡟⢸⠶⠺⣱⢤⢿⡟⠀⡷⠶⢊⢄⣨⣿⠀⣿⣄⣠⠂⠒⠒⣿⣿⣷⣼⠷⠂⢀⣉⠸⣇
⠀⠀⠀⠀⠀⢰⢹⠇⢸⡶⠚⢋⡶⠛⢷⣰⠶⠴⠉⡀⣀⣿⣸⣏⡀⠀⠐⠦⠄⡆⢿⣿⡟⠓⠂⠠⠤⠀⢿⡄
⠀⠀⠀⠀⠀⢨⡈⣇⠈⣿⡶⢋⠊⡭⣉⣿⠤⠤⡢⡀⠀⣿⡟⠁⢀⠀⢆⣀⢠⡅⣿⣟⢉⡈⠀⠀⠶⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⢊⠈⠳⢼⣿⣷⠏⠩⠐⢭⠀⠀⠄⢉⣽⠟⠋⠉⠁⡄⠀⢀⡙⡰⡏⠁⠀⠀⠐⠒⠃⠀⢰⠣⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣾⣶⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⣛⣟⣙⣛⣟⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⡇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⣷⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠸⡆⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣷⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡀
⢀⠀⠀⡀⡀⠀⢀⢸⡆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡟⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⢰⠇⢀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢰⡟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠄⠀⠀⣺⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠂⠀⠀⠀⠳⡄⠀⠀⢸⠟⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⢝⠒⠢⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣸⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⡀⠐⣄⠑⣄⢱⣄⣱⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠋⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢠⣶⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣀⠀⠈⠳⡿⣟⠧⢸⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠁⣠⠤⣤⡖⣒⠛⠳⣄⣧⡈⢹⡟⠋⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⡇⠐⠂⠀⠀⠶⠀⠈⢀⡗⠀⠳⣄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢺⣿⡆⠀⠉⢣⠈⠋⢀⣠⠔⢦⠀⡞⢀⡠⠞⠙⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣓⡜⠀⠀⠈⢣⣀⠈⠻⠤⠞⡴⡏⠹⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠛⢻⣀⡤⠞⠻⢹⣛⣿⡟⣯⡴⠙⣄⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠃⢠⡴⠒⠺⠟⢋⣽⠏⢳⡞⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣌⣲⣤⣤⠤⣴⡟⠲⠴⣶⣟⡥⠖⠛⠦⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠉⢏⠀⠀⠀⡼⡋⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡀⠀⠀⠀⠀⠘⣆⣀⡜⠤⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠙⢒⣤⣀⣀⡈⠙⢧⣀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡟⠋⣠⠞⠉⠳⡄⠈⠉⠓⠒⠒⠚⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⡾⠋⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡤⢯⠀⠈⠳⠄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⡄⠀⠀⣀⣀⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠉⣻⠉⠁

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠎⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡯⠀⠐⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡺⠁⠀⠈⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣢⠀⠀⡹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠛⠛⠻⣷⣦⣄⠀⠀⠀⢻⡾⢗⡂⠀⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠸⣷⣻⣷⣄⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⢿⣿⡷⢾⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣧⠀⠸⣟⢒⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⠀⠀⢿⣁⣙⡖⡆⠀⠀⠀⠀⠀⢀⣠⣤⣤⠶⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⢀⢽⣯⣉⣧⠴⠆⠀⠀⣠⣾⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣄⣀⣀⢀⣴⡿⣯⣿⠃⢀⡈⣽⣿⡕⣺⠒⣴⡆⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⡁⢀⠘⠩⡧⣿⣿⠉⡱⢤⣤⡘⣿⣿⣧⡀⢀⣠⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⢀⣤⣿⣿⣻⣿⡟⠉⣿⣿⠞⢛⡟⢴⣿⣿⠏⡐⢧⣈⣱⡘⢿⣿⣿⣻⡿⠋⠁⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢠⠼⢏⣘⣭⣽⠏⠀⣰⣿⠿⠓⣍⣯⣼⡏⣻⣷⠞⢡⠊⠝⢇⢠⣿⣿⣿⣿⢷⣄⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣷⣶⠋⣉⣹⡏⢀⣴⣿⣿⠗⢋⠱⢠⣿⠇⢸⣤⣤⣌⡹⠒⠿⣿⣿⡆⠹⣯⡓⠒⠿⡄⠀⠀⣤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⣤⡴⢗⡛⣻⣿⣿⣿⣿⡧⡤⠊⡈⢘⣿⠀⢸⣃⠀⣨⢓⣦⡴⣿⣿⣇⠀⢹⣏⠁⡴⢶⠄⣼⠁
⠀⠀⠀⠀⠀⠀⢀⣾⠇⣏⣑⣸⣞⠹⢿⣿⡿⠁⣌⣐⣨⠆⠊⠽⡷⠀⣿⠏⠁⢁⢢⣀⣀⣿⣿⣿⡄⢸⣋⣁⡔⠲⠇⣿
⠀⠀⠀⠀⠀⠀⣼⣿⠀⣟⠁⣉⠾⡉⣟⣿⡀⢸⠋⠍⣠⠜⡒⢾⡇⢠⣿⠷⠶⠇⢊⠀⠉⢛⢿⣿⣿⣿⣙⡈⠐⠒⠆⢹⣇
⠀⠀⠀⠀⠀⢸⢕⡇⠀⣿⡋⠍⣡⢿⣉⣋⣧⡾⠙⠛⠁⠢⠠⢴⣧⣾⣧⡄⡔⠂⠘⠓⠪⡿⢸⣿⣿⣏⣁⠀⠚⠒⠆⠈⣿⣆
⠀⠀⠀⠀⠀⢓⠇⢳⡀⠸⣿⡟⠩⢉⡼⢭⣸⣷⠦⠶⣃⢇⢀⣸⣿⡏⠀⣂⡀⢰⣤⣀⢠⡆⣼⣿⣏⣀⡠⠀⠂⢸⠃⠀⢹⡙⣆
⠀⠀⠀⠀⠀⠀⢂⠀⠹⠦⣿⣿⣿⡟⢩⠙⠂⣧⢀⢀⡴⢀⢩⡿⠛⠉⠉⠉⣀⠀⠐⠀⡙⢰⣿⣁⣀⠀⠄⠚⠒⡟⠀⠀⣼⠸⣗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣙⣵⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠟⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⡉⠙⣌⡋⠉⣿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⢟⣫⣴⣿⣿⡿⢁⠱⣄⡿⣷⠧⠟⢉⠹⠽⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣛⣭⣷⣿⣏⡀⣧⣼⠁⣬⠀⣾⣿⠋⠙⡇⠀⠀⠐⠁⠘⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⡿⢟⡛⣋⡇⣫⣩⠌⠙⠃⣈⢴⣟⣻⣿⠂⣤⠀⠀⠀⠀⠈⠀⠀⠀⠉⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠋⠁⣹⠿⣋⢸⠁⠋⢀⡦⣞⡿⣻⣯⣭⣿⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣩⡀⠦⢨⠟⠐⣯⠈⣠⡄⢸⠘⣿⣿⡿⣿⣧⣿⣷⡄⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⢻⢶⣤⣤⡤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢿⡟⣯⣅⣼⢄⣠⡐⠫⡬⠟⠟⢀⣼⡷⠃⢀⣤⣿⣾⣿⣿⣿⣶⣄⢀⣄⠄⡀⣄⣴⣾⣤⣀⡁⢉⡤⡅⠀⠁⢀⣴⣿⣻
⠀⠀⠀⠀⠀⠀⠤⠂⠠⠁⠂⠀⠻⣤⣍⣶⣾⣿⣣⢒⡖⠆⠚⢺⢷⠇⢠⣿⠋⠋⠛⣿⠿⠿⠿⠛⠿⠿⠿⠿⠛⠛⠛⠻⢿⣷⣦⣼⣻⢖⣶⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠚⠀⠆⠀⠈⢠⠀⠈⣿⡛⠑⣟⡻⣿⣿⣿⠷⠦⣿⣤⣿⡏⠄⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠉⠀⠀⢀⡴⠃⠀⠀⠀⢿⣷⣤⣜⣃⣿⣦⡄⣀⣼⣿⡿⠛⠀⠀⠀⠀⡸⠀⠀⢀⠔⠃⠀⠀⠀⠀⢠⠅⠠⠤⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠙⣄⠀⠀⠣⣿⡿⢯⣼⣭⣿⣿⡟⢩⠀⠀⣴⣦⡤⠖⠁⠀⠀⠁⠀⠀⢀⠄⣀⠔⢁⣠⠄⢀⣀⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣴⣟⠆⠀⠀⠈⠛⠿⣟⣯⣯⣷⢶⣿⣦⣪⠖⠋⠀⠀⠀⠄⡀⠀⡠⠔⣉⡼⠷⡾⣋⣛⣋⣉⡁⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣁⣀⣤⣤⣤⣄⡠⠈⠉⠛⠿⠿⠿⠟⠋⠀⠀⠀⢀⣎⣜⠵⠚⢋⣛⣁⣀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠈⡁⣸⣿⠃⠀⠀⠠⢦⣄⠀⠀⡀⠀⠀⠀⠼⠿⣟⣛⣿⣿⠿⠿⠷⠈⠙⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠀⠰⠆⠤⠀⠠⡄⠀⣁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠋⠙⠛⠒⠶⠦⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠓⠒⠦⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠶⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠢⠤⠤⣀⣀⡤⠴⠖⠚⠋⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠉⠑⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣏⠉⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡩⠕⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⣤⠴⠿⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡴⢞⣋⠉⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣧⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⢸⡇
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⢀⡠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⢸⢀⣠⡴⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠷⣦⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠰⣍
⠀⢀⡾⠀⢀⠀⠀⠀⠀⠀⠀⠘⣿⣋⣀⠀⣤⡿⠀⠀⠀⢸⣿⢀⣴⣶⣄⢠⡶⠛
⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢸⣛⠉⠉⢻⡄⠀⠀⠀⠠⣿⡇⣼⠇⠈⣿⠏⢳⣄⡀
⠀⠀⠀⣸⡇⠠⠀⠀⠀⠀⠀⢸⣊⡀⠀⣸⡇⠀⠀⠠⣼⡟⠀⠘⠂⢀⡿⠅⠀⠈⢭⢂⡰⠞⠛⠛⠿⣶⣄
⢠⣶⣶⣿⣄⠀⠀⢠⣴⣶⣶⣾⣿⣴⣊⠿⡁⠀⢀⢡⡟⠀⠀⠀⠠⡈⠊⢠⡾⣿⣛⣸⠄⠀⠀⠀⠀⠈⣿
⠀⠉⡹⡻⣿⠇⢠⣿⣿⠟⠛⠛⠉⠉⠻⣮⡝⢷⡂⢸⡇⠀⠀⣈⠳⡄⢠⣟⡳⠋⠉⠁
⡰⢩⣉⠶⠋⠀⢒⡻⣿⣄⠀⠀⠀⠀⢀⣾⠏⣸⣃⠐⢇⠀⠶⢭⡿⠀⡿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠨⣝
⢿⣿⣴⡀⡀⠀⠈⠓⣮⡽⠿⣦⢀⣴⡏⣡⠞⢣⠘⠗⡻⡔⢛⡚⣷⠀⣇⣿⢰⡀⠀⠀⠀⠀⣀⠀⣄⢲⣾
⠀⠹⣿⡷⢶⢀⠀⠀⣡⡹⡞⠻⣿⣿⠤⣯⢠⡄⣟⠻⡞⢻⡉⣭⠼⣧⢰⡜⣦⠳⡀⠄⠀⣾⣍⢓⣴⠟⠁
⠀⣰⣿⢉⣟⡡⠀⠀⣼⠀⣻⠙⣿⣿⡶⠻⡖⢳⣼⠛⣷⠛⢳⡤⢶⠛⣿⣧⣸⣧⡜⢦⣠⡦⣼⢺⣇
⣾⣿⣤⣿⣉⣧⣴⣾⣿⡇⣿⢠⣿⣿⣿⣎⢹⡆⣿⣗⣻⣗⣹⣷⣞⣻⣼⣿⣷⣾⣻⣮⣿⣷⠶⣆⣻⣦⣀
⣿⣷⣺⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡉⣿⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⡶⠶⠶⡶⠛⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡤⠴⠶⢖⠚⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠯⠁⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⣠⣾⠟⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⢀⣴⡟⠁⠀⠀⢠⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⢀⣰⣶⣿⠋⠀⠀⠀⣰⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣶⣶⣿⠿⠛⠩⠤
⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⢰⣿⣿⡿⠃⠀⠀⢠⡾⠁⠀⠀⢠⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⣊⣵⣿⡿⠛⠉
⠀⠀⠀⠀⢀⣴⠋⠀⢀⣰⡿⠋⡞⠀⠀⢀⣴⠏⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⣀⣀⣄⣠⡀⣠⣀⣠⣤⣾⣶⣾⠿⠟⠋⠁
⠀⠀⠀⢀⡾⠁⠀⣠⣾⠏⠀⣼⠁⠀⣾⣿⠃⠠⠄⣀⡴⠋⠀⠀⢀⣤⣶⣿⡿⠿⣿⡿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⣰⠟⢃⣄⣴⡿⠁⠀⣸⠃⠀⢸⣿⠃⠀⢠⣿⠟⠀⠀⢠⣴⣿⠟⠉
⠀⠀⢿⣤⣿⡿⠋⠀⠀⢠⡏⠀⠀⣾⠏⠀⢀⣾⠏⠀⢠⣴⡿⠋⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⡞⢀⠀⣼⠏⠀⠀⣾⠃⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⡟⠀⢀⣾⠃⢂⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠢⢠⡿⠒⣤⣾⡧⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⡿⢧⡨⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠃⣇⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⡼⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⡤⠀⠀⠀⣸⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⠇⠀⠀⢰⠃⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠃⠀⠀⣏⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢰⠃⠀⠀⠀⠈⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢸⡇⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠳⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣃⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣈⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡯⣛⣻⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣛⠳⠶⣭⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣓⠨⠭⣛⣶⠏⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣟⡋⠵⢶⣬⣁⣾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣓⠂⠬⢉⣒⣢⠿⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢏⣐⠢⠬⣑⠒⣼⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠯⢁⡒⠠⠭⣒⢢⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⢍⡐⠒⠤⢄⣉⠒⢢⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡉⠒⠢⢌⡉⠒⠠⢄⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⠠⢌⣉⠒⠤⣈⡉⠒⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠫⢅⡈⠑⠢⠤⣉⠒⠤⣨⠵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠭⣑⠢⢌⡁⠒⠤⣀⠉⣲⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠫⢍⡑⠢⢍⡒⠬⢍⡒⠤⣹⠁⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣬⣅⡂⠬⣑⠢⢌⣿⠁⠀⠀⣀⣠⣴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣍⣾⣡⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠐⠂⢢⠀⡀⠀⠀⠀⠀⠈⠢⡌⠉⠉⠀⠀⡐⠀⠀⠀⠰⠀⠀⠀⠉⠉⠉⠉⡇⠂⠒⠂⠀⢐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⡂⠀⠀⡂⢀⠈⠀⢀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⡀⠀⠀⠰⡀⠀⢢⠠⡄⠀⠀⠒⠒⠂⠠⠄⠀⠀⠴
⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⡠⠀⠈⠀⢠⠀⡀⠀⠐⡇⠀⢰⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠐⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⢸⠀⠀⢰⠀⠀⠘⢀⡔⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂⠁⠀⠀⡀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡁⡀
⠀⠀⠀⠸⠄⠀⠄⠐⠄⠀⠀⡄⠀⠔⠀⠀⠀⢂⠀⡀⡁⠀⠠⠄⠐⠈⡃⠀⢀⠆⠄⠀⠀⠀⠀⢀⡀⢠⠇⠀⢈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⡀⠀⠀⠈⠁⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡠⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⢀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠄⠐⣀⣱⠆⠄⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⢀⠀⠠⠀⢀⠈⠀⠀⡀⠀⠀⠀⠀⠀⢑⠃⠀⠀⠲⠒
⠀⠈⠀⠀⡀⢀⠂⠀⠀⢀⠄⠀⠀⠈⠁⠐⠀⠀⠀⡀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⡀⠀⠀⠘⠈⠢⠀⡄⠓⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠁⢀⠈
⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠂⠄⢀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠉⢁⠠
⠀⠀⠀⠠⠀⠃⢠⠀⢀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄
⠀⠀⠑⠀⠜⣉⠃⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠢⢀⡠⠁⠀⠀⠀⠀⡄
⠀⠈⠀⠊⢀⢅⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠠⠀⠠⠂⠀⠀⠀⠀⡐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡊⠀⢁⠆
⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠠⠐⠀⢄⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠐⠰
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤⠤⠤⠄⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠄
⠀⠀⠀⠊⠀⡠⠀⠀⠀⠀⠣⡀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠄⠊⠀⠀⠀⠀⠃⡀⠃
⠀⠀⠀⠀⠀⣠⠂⠄⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢀⣠⣀⣠⠀⠀⠈⠄⠀⡐⠀⠀⠦⠀⠀⠀⠈⠀⠀⡀⠀⢀
⠀⠀⢀⠐⠂⣀⠀⠤⢆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⡔⠈⠀⠀⠀⠀⠑⠂⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⣐⠐⢄⡙⠀⠒⢰
⠀⠈⡌⠡⡔⠁⠀⠀⠀⠙⠈⢀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠄⠂⠀⠄⠀⠀⠱⠀⠀⠀⠌⠀⠀⠀⠀⠀⡨⠌⢤⡀⠡⢠⠋⢀⠐⠢
⠀⠀⡤⢠⡇⠀⡄⠀⡄⠀⠋⠠⡄⠀⠀⠀⠐⠁⠀⠈⠀⠀⡈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠈⠢⠀⡀⢌⠀⠀⣁⠉⠁⢃⠄
⠀⠈⠠⠂⢈⠦⠀⣁⠀⠚⠂⠤⠀⠀⠀⠈⠢⠀⢀⣀⠀⠜⠡⡀⠀⠈⠐⠀⠀⠀⢀⢜⡀⠀⣈⠀⠀⠀⠀⠀⠀⠳⠂⢀⡤⠉⢣⠀⠉⡉⡁
⠀⠀⠀⠈⠁⢄⠀⡄⢠⠀⠏⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠅⠢⣀⠦⣐⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡿⢿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⢻⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⡆⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣰⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⢢⡐⣅⡦⡵⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠲⠶⣖⡒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠤⠤⣀⣀
⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠢⣤⡤⠋⠙⣆⠀⠀⠀⣀⠤⠲⡄⠈⣉⡷⠶⣄
⠀⡠⠚⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⣻⠽⠛⠉⠀⢀⡠⠟⠊⠁⠀⠀⢘⡝⠢⡀
⡼⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⣋⠤⠛⠉⠀⢀⡠⠔⠊⠁⠀⠀⠀⣀⣤⡶⢭⡄⠀⠹
⣇⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡤⠖⠋⠁⢀⡠⠔⠊⠁⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⣇⠀⢀
⢻⣦⡀⠀⠀⠈⠓⠦⠤⢤⣤⡤⢲⡟⠋⠁⠀⠀⣴⠊⠁⠀⠀⠀⣀⠤⠶⠯⠤⠤⠤⢤⠤⠴⠒⠒⠋⢀⣾
⠀⠳⢝⠢⣄⠀⠀⠀⠀⠀⠀⠉⢹⡿⣄⣠⠴⠚⠙⣆⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡤⢚⠵⠃
⠀⠀⠀⠙⠲⢬⣑⡲⠤⢤⣀⣀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⢒⣒⣒⠶⠾⣒⣫⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣟⡒⠲⣬⠭⣍⣉⣛⣒⣒⡒⠒⠒⢒⣒⣒⣛⣉⣭⠭⠴⡖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢬⣑⢤⡈⠳⡄⠀⠙⢦⡀⠀⠀⠀⠀⣠⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣌⢦⠀⠀⠉⠒⠦⠤⠚⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡠⠔⠋⠁
⣄⣀⣄⣠⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣞⣉⣀⣀⣀⣀⣀⣠⣀⣤⣠⣀⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠆⠀⠈⠉⣉⣉⣉⠉⠁⠐⠝
⡇⠀⠀⠀⣗⣝⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⡇⡄⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⡔⡂⠀⠒⠂⠠⠤⢀⣀⣴⠒⢄⠀⢠⠆⡈⠉⢉⣥⣤⣤⡀
⠇⠀⠀⠀⢀⣠⡷⢿⣀⠀⠀⠀⠻⠿⠯⠽⠽⠴⣶⣤⣤⡉⠀⠀⣑⣇⠀⠀⠠⣟⢽⠛⢮⢻⡄⠀⢠
⡄⣀⣀⣀⠘⠛⠒⠒⠚⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢩⣵⠶⠛⠁⠻⢰⠀⠀⠙⢉⣠⠚⣹⠁⠀⠸
⠻⠮⠾⠯⠯⠯⠯⠯⠭⠯⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⢀⠸⠀⠀⠀⡾⢹⣞⣉⣁⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⣁⣠⣤⠷⠀⠀⠀⣇⣄⣡⣨⡴⠿⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠔⢈⣠⣴⠿⠛⠋⠁⠈⢀⠧⠤⠤⠤⠔⢒⢒⢒⣒⡛⠁
⠀⠀⡀⠚⠉⠁⣀⣀⣀⣀⣀⣀⠈⠍⣣⠐⠀⢀⣹⡉⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠇⠀⠀⠀⠷⠵⢴⡅⣣⠏⠀⠀⡿⠿⠟⠛⠛⠁⠀⠀⠀⠀⢀⣀⡀⠀⠤⠄⠀⠀⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣼⣁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡔⡃⠀⠀⠀⠀⣠⣤⠤⡄⠀⠀⠈⢸
⠀⠀⠘⠀⠀⠀⠓⠛⠳⢶⣯⢷⠀⠀⣃⡄⡀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⢀⣞⣩⡖⢍⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣴⣲⢤⠾⡁⠙⠀⠀⣟⢦⣄⣈⠐⠂⠠⠞⠱⡅⠀⠀⣠⠋⢹⢯⡯⠴⢧⣤⣄
⠀⠀⠀⢠⣔⣀⣈⣉⣛⣛⣁⣀⣐⣀⠉⠉⠙⠻⠿⣓⡶⢀⣀⣽⡄⠀⢿⣀⣄⣁⡄⡢⣀⣥⡾
⠀⠀⠘⠓⠻⠾⠮⠾⠾⠷⠷⠿⠾⠋⠀⠀⠀⠀⠀⡼⠿⠟⠛⡇⠀⠀⠀⠀⠀⠀⣯⣕⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣀⣀⣀⣀⣀⡉⡉⡁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⠦⠤⠬⠕⠵⠵⠥⣕⣵⣡⠝⠁
⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠁⠊⠐⠀⠀⠀⠐⠀⠊⠀⠑⠨⠀⠉⠂⠐⠁⠈⠔⠅⠠⠊⠔⠁⠪⠊⠀⠕⠀⠪⠀⠕⠠⡢⠠⠀⠕⠄⠄⠄

⢀⣤⣶⣦⡤⠶⡦⠀⠦⠀⠀⠀⠀⠀⢠⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣤⣤⣤⣤⢀⡄
⠸⣿⣿⡿⠀⠂⠀⠀⠸⠧⠀⠀⠀⠀⣸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡆⣾⠀⣼⠆⣿⣿⣿⣿⠋⠀⣴⡀
⠀⢻⣿⣷⣿⣿⡇⠀⠀⣶⢄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣏⠀⣿⠀⢸⠃⢿⣿⣿⣿⡆⠀⣾⡆
⠀⠀⣿⡉⣸⣿⣼⠀⠈⢷⡄⠀⠀⣧⡅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⡿⠀⢹⠀⢸⠀⢸⡿⣿⠸⠇⠀⢸⣁
⠀⠀⠈⢉⡿⣿⡉⠀⠘⠟⠀⠀⠀⠋⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣌⢿⡇⠀⣿⠀⠘⠀⢈⠁⢫⣶⣤⠀⢸⣿
⠀⠀⠐⡿⣿⣿⣿⠀⡰⠛⠆⠀⠀⠀⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠀⠐⣤⠛⠀⠀⠇⠀⠀⠀⠻⣿⡀⠈⢾
⠀⠀⠀⠛⠹⢟⣿⣠⠋⠀⠀⠐⣁⣴⢾⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⢹⡇⡀⠀⠠⠀⠀⠀⠀⠹⡇⠀⢿
⠀⠀⠀⠘⢷⡌⣿⣧⣤⣀⣰⣿⣿⣿⠟⡧⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡿⠀⠋⠀⠲⠈⠀⠀⠀⠀⠁⠀⠸
⠀⠀⠀⠀⠀⢣⣼⣿⣿⣿⣿⣷⣿⣿⣷⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡘⡁⠀⠂⠀⣤⣾⠆⢀
⠀⠀⠀⠀⠀⣾⡟⣽⣿⣿⣿⣿⡏⠹⣿⠂⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⣾⣧⣿⡄⠀⣈⠻⣯⡄⠙⡇⠀⢀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠻⣿⡇⢨⣿⠉⣁⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠺⣿⠉⣿⠇⠀⣾⡗⣿⡇⠀⢿⠀⢸
⠀⠀⠀⠀⠐⢿⣍⠛⣿⣿⠁⣙⢃⠈⣿⡆⡟⣸⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣸⣿⢿⡄⠀⢯⠌⠉⠱⢤⡀⢠⣼
⠀⠀⠀⠀⣠⣿⣿⠇⣿⡇⢸⡇⠋⢰⣿⣷⠀⢹⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⢿⡇⣿⡇⠀⡇⠀⠄⠀⠀⠀⢸⣾
⠀⠀⠀⠀⡿⣾⣿⢸⣿⡇⣄⢰⠷⠘⣿⣿⠀⢘⣿⡄⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠹⣿⣇⢠⡏⠀⠀⣠⣀⣀⣸⡿
⠀⠀⠀⠀⣷⣾⣿⣦⣿⣷⠛⠸⠆⣰⣿⣿⡷⢈⣿⣷⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣶⣤⡘⠃⠰⠇⠀⠀⢨⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣥⣶⡄⣿⣿⣿⣷⣾⣿⣿⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠀⠁⠈⢻⡿⠻⠒⣶⣞⣠⡙⠟⠙
⠀⠀⠀⠀⢰⣿⣿⡿⢻⣿⣿⣿⡇⣾⣿⣿⣿⣿⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⢠⣿⣤⣴⠀⢁⠻⣧⣽⠂⠁
⠀⠀⠀⠀⠈⢿⣿⣧⠈⣿⣿⡀⡄⣿⣿⣿⡄⢻⡷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠘⣿⡁⣿⠀⣼⡆⠘⠏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⣿⣷⡞⢿⣿⣿⡇⢈⣷⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⢻⣯⣿⡇⢸⣿⣶⡆⢠
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⡇⣿⣿⢻⠀⢠⡗⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⢺⣽⢟⡁⣸⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣬⣿⠉⣿⣿⣿⣿⠀⣻⢻⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣼⡿⣻⡇⢻⣏⣿⣯⣽⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⣿⠀⣇⣿⣷⡟⠀⣿⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢹⣿⣏⡇⢸⣿⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠀⡏⣿⡿⠃⢰⣇⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠸⠛⠙⡇⢸⣿⢶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡟⠀⣸⡏⠀⠀⢸⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⣀⡇⢸⣿⣿⣿⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡿⠀⠈⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⡇⢸⣿⣿⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠐⠀⢀⠀⠀⢿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣷⠘⡇⠀⢩⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠋⠠⡇⠈⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣆⣥⠈⣸⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣼⢻⡆⠀⢰⡿⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⡿⠂⢹⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠀⠛⠘⠃⠀⣸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢉⡀⣼⣿⣥⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣠⠖⠀⠀⠀⠙⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠘⠃⠻⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⣿⠀⡖⠀⠀⠂⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⣸⡀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠘⣧⣄⠀⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠚⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⡇⢸⡝⢿⣿⣦⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⢀⣿⣧⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⠣⣜⣂⣠⣿⠛⣦⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠈⢉⣿⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⡔⢫⣿⣟⣠⡄⠙⣷⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣟⡀⠀⢸⣿⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⣿⣿⣾⣩⣶⢰⣿⠟⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣅⠀⠂⣿⡇⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⣿⢿⡀⣧⠸⡇⢻⣀⢄⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡿⣿⣁⡀⢁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⡟⣼⠃⢻⡄⢿⡄⣿⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢨⠋⣠⡌⣿⠟⣻⣀⣷⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣷⡏⠀⢬⠀⢼⠀⣽⠷⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠏⠛⠁⠄⣸⣿⠄⠀⠀⠛⢻⡿⢫⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠘⠿⠁⠀⣸⠀⢀⠀⢹⠆⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠟⠓⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠻⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⣶⣇⠀⣿⠀⣾⡁⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢽⡧⠀⠀⠀⠀⠀⣯⡝⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢉⣿⠀⢸⡇⠸⣷⡌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⣤⠀⠀⢀⡀⢾⣷⣶⣦⣀⣰⣦⣧⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠸⢿⣦⣸⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠿⣷⣄⡙⢿⠛⠛⠛⠛⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣦⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣴⣦⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣶⣶⣶⣆⠀⠀⠀⠀⢴⡶⣶⣶⣶⡄⠀⠀⢰⣦⡀
⠀⠀⠿⠋⠉⣿⡆⠀⠀⠀⢸⡇⠹⣿⣿⣗⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠘⣿⠀⠘⢿⣿⣧⡄⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢈⣿⠀⠀⠀⠙⢿⣷⣆⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠈⣿⡆⠀⠀⠀⠀⠙⢿⣷⡄⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡆⢹⣧⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣀⠨⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣽⣦⡀⣻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣟⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡷⠙⠛⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⢧⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⠹⣶⣾⣦⣤⣙⠿⣷⣦⣤⣴⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡟⠻⢿⣿⣷⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠁⠀⠈⠉⠙⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠻⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⠙⢦
⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣈⣛⣛⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣴⢄⠀⢠⣽⣄⡀⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⡷⢛⣉⡻⢷⣒⢾⡈
⠀⠀⠀⠀⠀⢀⠠⢖⣯⠗⠋⠀⠺⣇⡽⠀⠉⠛⠶⣧⣦⠄
⠀⠀⠀⠀⠀⠀⠁⢁⠈⠀⢠⣤⣤⡤⠤⣤⡄⣶⢶⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠐⠀⣿⠉⠄⠈⣸⣷⣿⣒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠐⠀⣿⡷⠆⡒⣿⠛⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣢⣡⣿⣟⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠲⢭⣽⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠶⣒⠓⢲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢡⠀⠁⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠐⢿⡖⠒⠒⠒⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠠⢨⡧⠤⠤⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠰⣷⣷⡿⠓⠒⠒⠒⢲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡀⠀⠀⣀⣂⣦⣀⣈⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡓⠿⣇⠀⣠⡤⠡⠤⠤⠠⢤⣬⣩⣭⣩⣄⣀⣀⣀⣀
⠀⢀⣶⠂⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠛⠛⠛⢻⡶⠾⠯⣭⡉⢿⡟⠛⠛⠛⠛⠛⠒⠂
⠀⡸⢿⣧⡄⡀⠄⠀⠀⠀⠀⠀⢰⠚⠛⠂⠀⠒⠒⠶⢤⣴⡿⣿⣮⡪⣝⣿⠒⠒⠂⠀⠀⠀⠀⣀⣀
⠙⠒⠋⠉⣿⠃⠀⠀⠀⠀⠰⠒⠋⠀⠉⠀⠠⠤⠀⣀⣾⣧⡝⢮⡻⣟⢎⣿⠒⠂⠀⠀⠈⣁⠠
⠀⠀⠀⠀⣮⡀⠀⠀⡤⢒⣘⠒⠀⠤⠤⠄⠀⠀⣀⣾⢠⣝⡿⣷⣽⣿⡿⠇⠀⠐⠪⠛⠋⠁
⠀⠀⠀⠘⠉⣱⣤⣰⣄⣀⠀⠀⠀⠁⠀⠀⠀⣶⢯⡃⣍⣿⡿⠟⠫⠄⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠤⠀⠿⠶⠿⣯⣹⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠳⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢠⠃⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⣁⡁⠀⠀⢀⠈⠡⠀⢠⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠢⢀⡸⠀⢀⡤⠇⠀⢡⠀⠀⠀⡀⠀⡆⠀⢨⠀⠅⠀⠀⡀⠁⠐⠀⠂⠂⠀⠀⡈⢰⠉⠁⠀⡀
⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⠑⠊⢀⠀⡀⠒⠀⠀⠑⠚⠀⠌⢂⠙⣠⠄⢸⠁⡴⠈⠀⠲⠄⠀⠀⠁⠈⠀⠀⢀⠇⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⢌⡄⠀⠀⠀⠀⠠⠀⠀⢀⠎⠀⠀⢀⠇⢸⡇⠀⠀⠀⢂⡃⠀⠁⠀⢀⠀⢸⡀⡠⠀⠀⠈⠀⠀⠘⠄⠂
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⢧⠈⠁⠀⠁⠈⡀⠀⠀⠀⢀⠀⠡⠈⠁⠀⠄⠰⠉⠁⠀⠀⠀⠰⡀
⠀⠀⠀⠰⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠑⠈⠀⠀⠀⠈⢷⠈⠀⠀⣇⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⡇⢀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠈⠀⠀⠀⠈⡂⠀⢀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠃⠸⠀⠠⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⢠⠀⠀⠌⠀⠃⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⢰⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⠀⠀⢀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⠴⠚⠉⠁⠀⠀⠀⠈⠉⠒⢤⡀⠀⠀⡴⠁⠀⠀⠉⢦⠀⠀⠀⡴⠋⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⢀⠔⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⢦⡀⢳⠀⠀⠀⠀⠀⢧⠀⡼⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⢠⠏⠀⠀⠀⠀⠀⠠⡄⠀⡠⡄⠀⠀⠀⠀⠀⢳⠈⢳⡀⠀⠀⠀⠈⠶⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⡼⠀⠶⠀⠀⠀⠀⠻⠏⠀⠻⠃⠀⠀⠀⠀⠓⠀⡇⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⣀⣀⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰⠃
⠀⡜⠁⠈⢳⠀⣿⠀⠀⠀⠀⠀⢸⠀⢳⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⢳⠀⠀⠘⠓⠋⠀⠀⠀⠀⠀⣸⠀⠈⢧⠀⠀⠀⠑⠢⠄⢀⣀⡀⠠⠔⠉⠀⠀⢀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠳⣄⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠈⠓⠤⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣼⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⢾⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⢻⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⢘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⢠⠶⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⠁⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠃⠶⠤⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠧⠄⠀⠐⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡟⠕⢁⡴⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡄⢠⣴⠲⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠺⠵⠷⠤⠤⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠧⠼⠦⠗⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨⢙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢄⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⢘⢻⣄⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢚⣳⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣀⣤⣬⣽⡇⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠀⢄⣀⣴⣮⠾⠛⠉⢩⣿⣿⡃⠔⡠⠖⡄⠀⠤⠀⠀⠄⠀⠀⠀⡿⠀⣴⡒⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢁⣼⡿⠋⠀⠀⠀⠀⠀⣿⣿⣷⡺⠀⠓⠘⠤⠐⢀⡐⡆⠀⢀⣾⠁⠀⠻⢉⣒⡶⠚⠁
⠀⠀⠀⠀⠀⢸⣿⣄⣠⣄⣤⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⢻⣧⠆⣀⣠⠐⠅⠔⣰⡎⢿⣽⡇⠀⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠛⢁⢀⡀⠀⠰⣾⡆⣿⣷⣦⢦⡀⠹⣿⣿⣿⣿⣿⣾⠿⠟⠋⠉⢀⠤⠄⣤⣕⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡞⢻⣾⣧⣴⣾⡿⣳⠃⢹⣽⢃⢳⠀⠙⠛⠛⠋⠁⠀⠀⣀⡀⢠⣜⣛⣋⣩⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣝⣛⣛⡫⠊⠈⣶⡿⢣⠇⡸⠀⠀⢀⡴⠶⢾⣛⢺⣯⣕⣒⣻⣟⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀⢆⣀⠓⠚⠛⠋⠀⠀⠀⠠⠲⠿⠋⠐⠖⠋⠉⠉⠥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠁⠆⠉

⠀⠀⠒⠀⢲⡆⠀⠀⠀⠀⠀⣀⠐⢁⣁⠀⠀⠀⠀⢀⠀⡂⠭⠈⠃⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢰⣷⠒⡾⣇
⠀⠀⢈⠈⢹⡇⠀⠀⢀⠔⠊⣀⡐⠊⠀⠀⠀⠠⠪⠖⠈⠀⠠⠀⡀⠀⠀⠀⠀⢠⡖⠊⠀⠀⠀⣟⣙⣿⠇⠸⡀
⠀⠀⠰⠀⢼⡇⠀⠀⢀⡠⠚⠀⠀⠀⠀⠀⠀⠀⡀⡐⠀⠁⠁⠀⠀⠀⠀⢠⢼⠉⠀⠀⠀⠀⣼⣁⣁⣿⠀⠀⢳
⠀⠀⠒⠀⠘⠃⠀⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠘⠂⠀⠀⠀⠀⠐⠂⠀⠒⠚⠀⠀⠀⠀⠀⠐⠃⠀⠘⠀⠃⠀⠈


⠀⣦⣀⣶⠀⠀⣄⣨⠀⠀⠀⠀⣧⣤⣼⠁⠀⠀⠡⠀⡀
⠀⣧⣀⣽⠀⠀⣁⣀⡇⠀⠀⢸⣇⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⡄⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⣧⣤⣼⠀⢰⡄⢠⡇⠀⠀⢸⣁⣀⡿⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⢀⠯⠤⣼⢡⠀⠀⠀⠀⠀⠰⢷⠠⠈
⠀⡧⠤⢼⠀⢸⠤⠤⡇⠀⠀⣼⠉⠉⡇⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣸⡀⡆⡏⠸⠀⠀⠀⠀⠀⠄⢠⡀⠀⠆
⠀⡧⠤⢼⠀⢸⠀⠠⡄⠀⠀⡏⠉⢻⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠇⠀⢹⡇⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠰
⠀⡧⠤⢼⠀⡠⠤⠤⡇⠀⢠⡟⠛⢾⠁⠀⠀⠐⠀⠘⠀⠀⠀⠀⡰⠉⠐⢺⠅⠀⠈⠀⠀⠀⠉⠀⠀⠁⠀⠀⠂
⠈⠁⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘

⠀⠀⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⢠⠄⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⡀
⠀⠀⣿⠠⢼⠀⠀⠀⠀⢀⠄⡎⣁⣴⣦⠀⠀⠀⣠⣴⠞⡏⢸⡇⠀⠀⠀⠀⠀⠀⣀⡶⠒⠀⠀⡼⢵⣤⡎⢡
⠀⠀⣿⠒⢺⠀⠀⠀⠚⠀⣠⣾⣿⣿⣿⠀⠀⠘⠉⢸⠀⣧⣼⣿⡆⠀⠀⠀⢠⠤⠏⠀⠀⠀⢰⠧⠤⣷⠀⠀⡇
⠀⠀⣿⠀⣹⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⢠⣶⣿⣿⣿⣿⣿⡇⠀⣀⣶⠚⠀⠀⠀⠀⢀⠗⠒⢲⠟⡄⠀⠸
⠀⠀⠁⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣩⣿⣿⣽⣿⣟⡉⡩⡟⣻⢶⡋⠉⣀⣤⢉⡟⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⡁⠀⠘⣿⣇⠁⠀⣹⣿⣶⣧⣷⠾⣟⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠀⠣⠇⠀⠠⡈⢿⣷⡋⡏⠁⠸⣿⡆⣤⣾⢇⣳⣬⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣠⡀⠑⢄⡻⣶⣇⠀⠀⡻⣷⡞⠛⠛⢻⣯⡀⠈⠋⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⢷⣿⣿⠦⣶⣤⣿⣷⣼⣦⠀⠐⠻⣷⣄⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢠⣶⣦⡈⠷⣠⡜⠙⠿⣷⡈⠙⢿⣻⣿⣿⣇⡀⠂⢀⣽⣆⣘⣿⠆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⠃⠀⠀⣀⡢⡦⢘⣿⣿⣿⣦⡙⠿⣦⠀⠘⣾⠆⠹⣿⣍⣇⠢⡟⢶⣿⢿⣿⣟⣏
⠀⠀⠀⠀⠀⠀⡾⠍⠉⠀⠿⡠⣄⣱⡾⢿⣿⡿⠿⠿⠿⠿⠿⠿⣮⡓⢬⣉⣁⣶⣏⠸⣿⠇⣼⣿⣿⠛⠲⣷⡿
⠀⠀⠀⠀⠀⠸⣅⣈⡁⢀⣶⣾⡿⠋⠀⠀⠀⠀⢀⣀⢢⣀⣤⠀⡈⢯⠑⢶⡄⢈⣽⣿⡷⣾⠛⣿⢿⢦⣤⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡾⠿⠋⢀⣠⢤⣤⠐⠈⣻⣿⢏⡿⣿⡿⡞⢦⠣⡀⢱⡌⢻⡌⢳⣷⣞⣵⣋⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣹⣿⣿⣶⣾⣿⢟⢼⠁⢸⣿⢧⠈⡇⠈⠙⠿⣼⣇⠀⠊⠛⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⣟⣛⡫⠕⠋⠈⠾⡿⢯⠞⣠⠇⠀⣤⡀⠈⠙⠻⢶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠖⠶⣄⣀⡀⠀⠀⠀⠀⠀⠓⠚⠛⠋⠁⠶⠤⠀⠡⠀⢉⣀⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢷⠻⢮⡻⣗⠶⡤⣀⣠⣔⠠⣈⠱⠀⠀⠀⠘⠀⠰⣾⣆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠿⣕⣳⣿⣦⣍⡓⠾⣷⣥⣄⣀⠀⣀⠀⠙⠷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⡧⠶⠖⠛⠛⠛⠋⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣀
⠀⠀⠀⠀⠀⠰⢋⡉⡆⢀⣴⠞⠉⢀⣀⠀⠘⠢⠀⠀⠀⠀⢠⣤⣠⡀⠀⢠⣤⣤⣤⢤⣀⠤⢀⡤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠊⠉⠛⠢⣦⡄
⠀⠀⠀⠀⠀⣏⠘⣏⢠⣾⡏⢀⡜⠉⠈⢹⡄⠀⠁⠀⠀⠀⢯⣹⣟⣇⠀⠀⠈⠁⠈⠉⠘⠃⠛⠚⠂⠵⢬⣅⣤⡀⠀⠀⠀⠀⠈⠀⠀⡠⠤⢄⠀⠀⠀⡹⢆
⠀⠀⠀⠀⠀⠸⣦⠈⢻⡷⡂⢸⠀⠀⣀⠜⠀⢤⠃⠀⠀⠀⢾⣙⣿⣺⠃⠀⠀⠀⠒⠒⠑⠉⠉⠈⠁⠀⠠⢄⠘⢟⣆⠀⠀⠀⠁⡀⠘⠴⠛⠀⣱⠀⠀⣷⡼⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⣧⠘⢯⣉⢀⣄⠴⠋⠀⠀⠀⠀⢸⣯⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣹⡆⠵⠁⠀⠀⠀⠈⠑⢉⡠⠾⠃⠀⣰⢣⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡿⢤⡙⠛⠠⠔⠂⠁⠀⠀⠀⣼⣽⣽⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠥⢀⢛⡄⠀⠀⠀⠀⠒⠋⠀⣀⣖⣋⣩⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠯⣽⣷⣲⣖⣋⡇⠀⠀⢀⣧⣿⣿⣯⠆⠀⠀⠀⠀⡴⢋⣭⡙⢦⠀⠀⠙⠒⠲⠓⠃⠀⠀⠀⠀⢀⣤⠜⠕⡷⣺⠛⠋⠁
⠀⠀⠀⠀⠀⡠⢒⡶⡽⠀⠀⠀⠉⠉⠙⠛⠎⠛⠀⠀⢀⡼⢦⣽⣿⡍⠀⠀⠀⠀⠀⠇⣏⠾⣃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣷⠿⠿⠏⠈
⠀⠀⠀⢠⡎⣹⠙⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⣿⣷⡿⠇⠀⠀⢀⠀⠀⠀⢣⠈⠛⢯⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠰⠊⣩⠱⡞⠃⣶⠃⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣻⣿⠛⠀⠀⠀⠀⠈⢀⡀⠀⠀⠳⢄⡀⢬⣸⣭⡟⠦⣄⡀⠀⠀⠀⠀⠀⠀⢄
⠀⢀⠜⢛⡡⡾⢃⣼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠃⣔⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⠀⠙⢶⢤⣙⠳⢟⢛⡟⡦⡀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠰⡀
⠀⢠⠶⡋⣤⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⣽⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣾⡁⠀⠀⢸⡄⠈⠳⡌⢷⠡⠽⣶⡀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠸⡄
⠀⢘⡶⢃⡴⢂⡇⡐⠂⠋⠐⠀⠀⠀⠀⠀⠀⠑⠦⣍⡉⠛⠶⢄⠀⠀⠀⠀⠀⠀⠈⢍⣿⡀⠀⠀⠳⡀⠀⠘⡆⠅⠼⠻⢇⠀⠀⠀⠀⠀⠀⠘⢂⢀⠀⠙⡄
⠀⠘⢦⡕⢄⠘⣇⠀⣿⡝⢢⠐⡀⠀⣎⠃⣀⠀⢀⡀⠈⠑⠦⡈⢱⡄⠀⠀⠀⠀⠀⠈⢉⣿⡀⠀⠀⠘⠤⠜⢁⡃⢶⠛⡟⠀⠀⠀⡠⠒⠒⠦⡈⣅⠀⢀⢹⡀
⠀⠀⠘⢯⣂⡦⡈⠳⠯⠠⠈⢈⡇⠀⠈⠉⠉⠙⠲⡈⢢⠀⠀⢹⢠⣷⠀⠀⠀⠀⠀⠀⠈⢏⠱⣦⣀⢀⡀⢠⠋⢠⡴⢟⠇⠀⠀⢼⠀⣠⡖⡄⣱⡪⠀⠀⣾⡇
⠀⠀⠀⠀⠘⠻⣿⣶⣤⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⠱⣄⠁⠐⢊⣼⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⢌⣽⠛⣛⣡⣾⣔⡯⠞⠀⠀⠀⠰⡀⠹⣮⡵⢋⡘⠂⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⣀⣤⡤⣤⣤⣀⡀⠀⠀⠙⠉⠙⠋⠉⠀⠀⠀⠀⠀⠈⢇⠀⠉⠙⠉⠁⢀⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡤⣲⣮⣴⣤⣌⡉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠴⠤⠤⠔⠋
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡇⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢀⡼⠃⢰⡮⠛⠉⠑⠚⢯⣟⣯⢷⡦⢬⡙⠓⢂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠄
⠀⠀⠀⢀⠔⠁⡄⣶⠈⡄⠀⠃⠀⠈⠀⡌⠀⠀⠀⣆⣀⣀⡴⢋⣠⠿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠒⠯⣿⡷⢶⢤⢼⡷⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⢠⠚⠀⢈⠣⡤⠤⠁⢀⠀⠀⡀⣠⠁⠀⠀⠀⠀⠈⠃⠒⠉⠁⢀⡤⢶⡖⠒⠒⠲⢤⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⠀⣀⣤⣤⣤⣀
⠀⠀⢀⠑⡀⠸⡀⠀⠀⠀⡸⠀⠠⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠁⠀⣀⢀⡐⢂⠤⣌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣴⡟⠛⠷⡀
⠀⠀⠀⢁⠰⠀⠈⠒⠒⠋⠀⠐⠁⡄⠁⠀⠀⠀⢤⣀⣀⡤⠞⠁⠀⠤⠈⠋⠀⢡⠏⣉⡝⣾⣳⠈⣇⠀⣀⠀⠠⢄⡀⠀⠀⠀⡰⣋⢨⠀⠐⢯⣉⠆⢲⡇
⠀⠀⠀⠀⠡⣌⠓⠠⠴⠐⠃⠰⠈⠀⠀⣾⣶⠀⠀⠈⠁⠀⢠⡀⠋⠀⠀⠀⠀⠸⣄⠹⢶⢵⡽⢀⡏⠀⡁⠻⢉⣷⣮⣱⣶⣾⣷⡿⣮⣲⠀⠀⣀⢶⠇⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠉⠀⠀⠀⠈⢇⢩⡛⢡⣍⡹⠓⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⠴⠖⠋⠀⠀⠈⠛⠉⠈⠙⠟⠿⠟⠛⠉⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⢿⢿⣟⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣛⣟⣿⣿⣛⣻⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣴⣶⠟⠛⠲⡄⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⢺⣿⡇⠀⠀⠀⠀⢿⣿⣤⠀⠀⣘⣀⡤⠖⢻⣿⡿⠁⠀⠀⣠⣿⣿
⠀⠀⠀⠀⢼⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⣀⡤⢾⣿⡟⠛⢻⠋⠁⠀⠀⢸⣿⠁⠀⡠⠘⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣿⡿⠶⠚⠋⠁⠀⠘⢿⣧⠔⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⣼⣿⠏⠀⠀⠀⠀⠀⣀⣤⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠒⠒⣾⣿⠟⠒⠒⠒⠒⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⡀⠀⠀⠈⠉⠉⠁⠉⠀⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⣉
⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⠓⢯⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⣿⣼⡵⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠻⠅⠉⢧⡀⠈⠳⢾⡯⣳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡂⠀⠀⠙⠻⣿⣽⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣆⠀⠠⠶⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢀⣹⠿⠯⢛⣿⢦⡀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞⢮⣷⡛⢀⣿⠀⣬⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⣴⣯⣤⠴⠶⠌⣛⣒⣚⢶⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⣲⣙⣿⣦⣿⡧⣁⠄⠿⠂⣈⠑⠶⣤⣴⣤⠔⠒⠀⠀⠀⠀⠀⣀⣤⡾⠾⠽⡲⢦⣿⣃⡽⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠈⢿⠒⡛⢿⢿⣿⣿⣀⠉⠒⠵⣩⠄⠶⢸⠛⣌⢦⠀⣀⡠⣴⢶⠉⣿⡄⠘⠛⠀⢀⣹⠞⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡌⠁⢸⠌⣿⡿⣿⣷⣦⡀⠀⠉⠢⡟⢸⠈⢦⡹⣟⢻⡦⠚⠃⠠⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠛⠘⡇⢹⣿⢻⡝⢿⣿⣷⣄⠀⢠⠇⠀⠀⠑⢮⡳⣤⣶⠗⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⠘⠂⣿⠸⣿⣸⡇⠀⠙⣿⣋⣆⡞⢀⠀⠤⢤⣜⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡄⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡁⠦⢸⡀⢿⣿⠇⠀⠀⠘⢯⡞⡡⢖⠚⣿⡏⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⣅⣠⣄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢃⢠⠈⡇⢸⡟⠀⠀⠀⠀⠀⢿⠀⢙⡃⢹⡇⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣼⣿⡄⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⣀⢡⢼⡅⠀⠀⠀⠀⠀⠸⡄⠘⡛⠸⡇⢘⣿⠀⠀⠀⠤⠐⠒⢈⣥⡟⣰⣀⣾⠿⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣥⣁⢸⣾⡇⠀⠀⠀⠀⠀⠀⢇⠀⢋⠠⡇⢀⣿⡎⠀⠀⣀⣔⣮⡟⢽⣧⠟⠻⣅⣀⠘⢿⡆⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠛⣏⢯⠉⠉⣿⣾⣾⡾⡁⠀⠀⠘⡀⠛⡵⡣⢿⠿⡇⣶⢽⣿⣭⣻⠿⠋⠀⠀⠀⠀⠉⠁⠊⣿⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⢀⡴⢐⣁⣤⣾⡿⠾⣷⣾⣿⠇⠁⠃⠀⠀⠀⠀⣖⡴⣪⢀⡿⣀⣇⠘⣿⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠆⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢠⣿⠃⢉⡹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠘⠻⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣦⣤⣀⡀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⠾⠟⠛⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⠲⠚⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣠⠴⠶⠴⠶⠲⠶⣄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢹⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⠀⠀⠒⠒⠀⠀⠈⠉⠃⠀⠈⣧⠀⣀⣸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⡄⠀⠀⣶⠀⠀⠀⠰⠆⠀⠀⠘⠛⡉⢹⡆⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⠁⣸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣇⠀⠀⠀⢈⡀⣀⠀⠀⠀⢀⡾⣶⠞⠋⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠀⠀⢀⣠⠞⠁⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡙⣳⠶⠤⠴⠖⠋⣿⣀⣀⣸⣆⣀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠉⠉⠉⠉⠹⣆⡀⠀⣀⣤⠞⠉⠉⠁⠀⠈⠉⠉⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠲⣤⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠋⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣴⠋⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⣠⡾⠁⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣧⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⠀⠀⠀⠀⠘⣆
⠀⣠⠞⠁⠀⠀⠀⠀⣴⠛⠻⠦⣤⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠴⠚⢻⡆⠀⠀⠀⠀⢻⡀
⣰⠏⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠑⠭⣛⢶⣄⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⢷⡀
⣿⢠⢀⠀⡆⢰⡇⣸⠀⠀⠀⠀⠀⠀⠀⠙⠺⢝⡦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⡟⢀⣦⠀⡀⠀⠀⠘⣧
⣿⡈⢸⠀⡄⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⠓⠲⠶⣶⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⣿⠀⠇⢠⠀⡆⢹
⠈⠙⠶⠦⠷⢾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⢸⠀⡾⢀⣅⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⢒⡟⠙⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⢀⣠⡴⠚⠋⠁⠀⠀⢐⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⢷⣤⡤⠶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣂⣀⣀⣀⣤⡴⠶⠚⠋⠉

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

⠙⠒⠤⣄⡀
⠀⠀⠀⠀⠉⠙⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢦⣤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⡿⣿⢟⣷⣷⣿⣷⣦⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢏⣉⣩⣁⣥⣬⣬⡿⠂⠚⠋⠙⠙⠛⠋⠛⠒⠁⠘⠛⠉⢁⣀⣈⣤⣤⡧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣁⣋⣁⣨⡥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⢟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣿⡟⢻⣟⣟⠶⠆
⠀⠀⠀⣀⣀⣀⣤⡤⠤⠶⠶⠶⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⡀⠀⠀⠀⢨⡇⠈⢿⣾⣷⡆
⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠛⠓⠚⠯⢀⣨⣤⡤⠤⠶⠒⠛⠛⠛⠳⠿⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣥⣬⠥⠤⠌⠀⠠⠬⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⣶⡶⢾⠿⠿⠿⠻⠟⠛⠛⠳⠒⡒⣛⡛⠉⠉⠻⢿⠿⠿⠿⠿⢿⡟⣿⠿⡿⡿⠟⠛⠛⠛⣒⣢⣄⡀
⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣴⡶⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢸⣿⠿⢷⠙
⠀⡆⠤⠀⠀⠀⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠶⣶⣤⢬⣷⣦⡦⠞
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣻⣁⠀⠀⡛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠉⠉⢹⡟⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣄⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⡀⢂⣁⡈⠀⠀⠀⠠⢀⣋⠠⠌⠉⠛⠓⣶⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⠗⠚⠉⠁⠀⠀⠈⠁⠀⠈⠉⠙⠛⠻⠿⢧⣭⣃⣀⣈⣂⣠⣽⣉⠉⠛⠓⣺⠲⠖⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⣀⡴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣤⣤⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡟⢻⣿⡿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠉⠁⠀⠉⠏⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣉⠛⠛⣛⡛⠛⠻⣿⣿⣿⣿⣶⣶⣶⣴⣶⣶⣶⢿⣿⡿⠃⠉⣤⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡉⠀⣤⣶⠆⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⠿⣿⡁⠀⠀⠀⠈⠉⠉⠙⠃⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣷⣶⣶⣾⣿⣷⡀⠿⢿⣿⠂⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⢃⣈⣉⣩⣿⣿⣿⣧⢈⡿⠃⣀⠸⣿⠄⢀⣼⣿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢋⢳⣿⡏⠉⠛⠋⠉⠉⠉⠀⠁⠀⠙⠁⠀⠀⠀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢛⣿⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣶⣾⣿⣄⣸⣿⡿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠁⠀⠉⢁⣴⣿⠿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠀⠉⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠁⣶⠖⠉⢠⣤⣟⣉⣤⣤⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⠀⣤⠿⣿⠛⣿⣿⣿⣿⣦⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⡛⢷⣦⠀⣾⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣱⣶⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠃⠙⣿⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⡟⠿⠁⠀⠉⠀⠀⣿⣿⡇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠧⢨⠿⠁⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠁⠀⠀⢀⣀⣀⣀⣀⣀⣀⣐⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢹⣿⣿⣿⡿⠿⣿⡿⠟⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢀⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣾⣿⣿⣿⡇⢀⡀⠀⠛⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣿⣿⣿⣿⡀⠘⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿⣏⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⣿⣿⣿⣿⡟⢀⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣼⣿⣿⣿⣿⡟⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣶⠿⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⣤⣶⣿⣷⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣠⡾⠋⠁⠀⠀⠀⠀⢹⣦⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⢱⡆
⠀⠀⢤⣤⣤⣀⠀⢀⣀⣀⣀⣀⣤⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢿⠁⢠⣾⣿⣿⣷⣤⣤⣤⣤⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⢀
⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠀⠉⠛⠿⠿⣿⣿⣿⣿⡀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⣀⣀⠀⠘⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠷⣤⣶⣿⣾⣿⣿⣿⣿⣉⣷⣄⠀⠀⠀⠈⠉⠉⠛⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣹⣿⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠁⠙⠋⠉⠋⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠱⣠⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢤⡈⣑⡄⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⣷⣆⡸
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⢠⣼⡗⠁⠀⠀⠀⠀⠀⠈⢻⡧⠀⠀⠀⠀⠀⠓⣿⠛⠁⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⡟⢿⠀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⢸⢻⡄
⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠉⠀⢛⠀⠀⠀⠀⠀⠀⠀⡝⠸⠀⠀⠀⠀⠀⢨⠁⢇⠀⠀⠀⠀⠀⠨⠂⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠠⣀⢀⠀⠀⠀⠀⢀⣰⡠⠊⠀⠀⠈⡅⣀⠀⡰
⠀⠀⢤⠀⢠⡄⠀⢀⠀⠀⠑⢴⣦⣀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⣿⠊⠁
⠀⠀⠀⠉⣿⡏⠉⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⢻⡄⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠐⡏⠿⠀⡀⠀⠀⠨⠁⠇⠀⠀⠀⠀⠀⠸⠉⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣶⣶⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⣍⢀⣄⡀⠀⠀⢰⠬⢻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢀⡄⠠⠆⠰⡀⠒⠈⡛⢆⡈⠃⢨⡈⠲⢄⠈⠁⢤⠀⠀⠀⠱⣗⣈⡟⠃⠀⠀⠀⢊⠀⢻⡷⡀
⠀⠀⠀⠀⣀⣆⠀⠘⢛⣃⠈⢠⠀⠴⠆⢻⠀⠰⠀⠈⠙⣦⡠⠁⢴⠺⠳⡄⠀⠈⢂⠀⠀⠀⣠⣧⡐⣤⡄⠈⠀⠈⠤⢻⡕
⠀⠀⠀⠠⠢⠁⢰⡶⠁⠘⣆⠁⠄⢢⡄⢸⣧⡀⠈⠒⡄⠈⠑⣄⠈⢁⣘⠸⣆⠀⠈⠂⠀⠀⠸⣏⡀⢇⠁⠀⠀⠀⠘⢘⣷⢆
⠀⠀⡘⠁⢀⣰⠞⠀⠀⠀⢻⠀⠆⢠⡄⢸⣿⣷⣄⠀⠈⠁⡘⢻⣂⠀⠑⠂⢨⣇⠀⠈⢆⠀⠀⠀⣚⠇⢰⣃⠀⠀⠀⠐⣺⣞⠀⣠⠴⠚⠛⠛⠲⡄
⠀⠀⠓⡈⠯⠤⣖⡳⢀⠀⢸⠀⡀⢀⠀⢸⠀⠙⢿⡆⣠⠈⡡⢘⡗⠀⠀⠀⠀⠹⠀⠀⢸⠀⠀⠀⠐⠒⣌⠂⡀⠀⠀⠀⡚⣿⢰⣷⡊⠀⡞⢫⣃⠈
⠀⠀⠰⣄⠀⠁⡀⢀⣾⢃⡎⠠⠀⣈⠀⣿⡦⠲⠞⠀⠉⢁⡂⢊⣿⠀⠀⠂⢰⣲⢀⠀⠈⡄⠀⠀⠀⠀⢇⠀⢁⠀⠀⠈⢰⠙⡀⣿⣧⠀⠡
⠀⠀⠀⠈⠓⠿⢷⡞⣥⠞⠀⠂⠠⠀⢠⣿⣄⠁⠒⠐⢀⣁⡴⠋⣿⡄⠀⢰⠀⣿⡆⠀⠀⠙⠢⢄⡀⠀⠀⠈⠄⠸⠀⠀⠐⢄⢲⣿⡿⠀⢠⡅
⠀⠀⠀⠀⣴⠿⢚⢛⡁⠐⠀⠀⣠⣴⣿⠛⠛⠿⣷⠿⠛⠉⠀⠀⢸⣿⡀⠀⢈⢚⢿⠟⠷⠒⠶⢬⡁⠱⡄⠀⠈⢹⠂⣆⠀⠈⡆⠛⠀⠀⣠
⠀⠀⠀⠆⠁⡠⢪⣥⡌⣶⠶⠛⡩⠼⠛⠒⠂⠡⢤⣀⡔⠂⠀⠀⠂⣙⣷⠄⡀⠀⠀⠀⣀⣀⠀⢸⡎⢂⠀⠀⠀⠀⠈⢟⢒⣀⢀⢁⣁⢠⠟
⠀⠀⠀⠀⠀⢧⣠⠟⢡⡟⠀⡼⠀⡴⠒⠒⠦⣄⠀⠀⠉⠈⠁⠀⢀⣿⠛⠷⡖⠞⠉⠁⠉⠀⠁⣸⠓⠚⠀⠀⠀⠀⠀⠀⠁⠉⠍⠥⡤⠚
⠀⠀⠀⠠⣀⣀⣀⡴⠋⠀⠀⢿⣦⢙⢓⠻⠂⢸⠿⠷⠶⠷⠶⠚⡛⣢⣀⠀⢁⡶⠿⠶⣶⠴⠚⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠮⠦⠖⠋⠀⠀⠀⠀⠀⠀⠱⠬⡥⠴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿