Water cooled Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⢹
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠙⠋⠉⠙⠓⠒⠒⠲⠶⠒⠒⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠄⠀⢠⢤⢄⡀⠀⠀⠉⠉⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠠⠁⠀⢆⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⢄⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⡟⠒⠒⠉⠉⠈⠁⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠙⢦⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠎⠀⠀⠀⠈⠦⠀⠀⠀⠐⠖⠊⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡑⠠⠤⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠖⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣀⠀⠉⠉⠉⠉⠉⠉⠉⣀⣠⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠈⠉⠙⠻⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣷⣦⣄⡀⠀⠀⠈⠛⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣾⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⣄⠀⠀⠀⠙⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⠛⠛⠛⠓⠀⠀⠀⠀⠠⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⣿⠇⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣼⡟⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢠⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣼⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣏⣠⣤⣤⣤⣤⣼⣿⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⣿⠿⠛⠋⠉⢩⣿⡇⠀⠀⠀⠈⣿⡟⠛⠛⠿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣟⠉⠁⠀⠀⢹⣿⣇⣀⡀⠀⠀⢰⣿⣿⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⠁⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⣿⠟⠛⠿⠿⣿⣿⣿⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠠⣾⣿⣟⣹⡟⠀⠀⠀⠀⠈⠻⣿⣦⣤⣤⣀⣸⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣄⣠⣤⣴⣶⣿⡄⠀⠀⠀⢰⣿⣿⠇⣠⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣾⣿⡏⢰⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⡿⠀⠻⠃⠀⢰⣿⠿
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠻⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⣀⣼⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣿⣷⣶⣶⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⡟⠛⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠻⣿⣷⣤⣤⣤⣤⣤⣠⣿⣷⣄⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠉⣹⣟⠛⠛⠿⠿⢿⣿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣤⣄⣀⠈⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠟⠛⠛⠛⠛⠷⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⢀⣤⣶⣿⡿⢦⣤⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢠⣿⠇⠀⠀⢠⣶⠿⠋⠉⠀⠀⠘⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠘⣿⣿⣤⡼⠋⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⣀⠀⠛⠛⠛⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣾⣿⠀⣰⣶⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⢸⣿⡏⢰⣿⡇⠀⣿⠀⢼⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⢿⣿⡇
⣾⣿⠀⣾⣿⠀⢸⣿⠀⠀⠉⠙⠻⠿⣿⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⡿⠟⠋⠉⠀⠀⠘⣿⡇
⣿⠇⢰⣿⠏⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿

⠀⠀⠀⠀⠀⣠⡄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣤⣾⡿⢀⣀
⠀⠀⠀⠀⣤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠆⠀⢿⣿⠇⠈⠙⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⢠⣧⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣾⡄
⠀⠀⠀⠈⠛⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⡀⠀⢨⣿⠃
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣠⡿⠇⠀⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄
⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧
⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⢠⣧⠀⣠⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣾⡄
⠀⠀⠀⠈⠻⣤⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡟⠁⢨⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣀⡼⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣇⣀
⠀⠀⠀⠀⣀⣹⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⢋⣀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷
⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟
⠀⠀⠀⠀⠀⠙⢫⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠛
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣤⣜⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣣⣤
⠀⠀⠀⠀⣤⣽⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⣿⣦
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠀⠀⣿⣿
⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣿⣿⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛
⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿
⣿⣦⣠⣤⣄⣠⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣴
⣿⠃⠀⠈⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠄⠀⣤⡀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢀⣤⡀⠀⠀⠀⠀⠀⠀⢸⡿
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠀⠀⠉⠁⠀⣾⣿⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁⠀⠀⠀⠈⣿⡇⠀⣛⠁⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠻⠃⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠿⠃⠀⠘⠃⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠶⠆⠀⢹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⣿⣾⣿⠇⠰⣿⠆⠀⠰⠄⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⣶⣦⣄⣀⣤⣤⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⡀⠀⢠⣄⣀⣠⣤⣶⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡚⠋⠙⣿⠟⢻⡟⠉⠉⠉⠉⠉⠉⠛⣿⠋⠙⣿⠛⠉⠉⠉⠉⠉⠉⠻⣿⣿⠋⠉⠛⠃⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⣷⣤⣤⣤⣤⣤⣤⣤⡌⠀⠀⢱⣤⣤⣤⣤⣤⣤⣤⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⣿⠀⠀⠉⠹⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡟⠁⠀⠇⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⣤⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡆⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠀⠀⣿⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡄⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⣄⣀⣤⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⣿⣇⣀⣤⡀⠀⠀⢀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⢠⣴⠛⠛⠛⠻⣷⣶⣴⣶⣶⡾⠛⠛⠛⠛⠛⠿⣶⣶⣶⣶⣶⡿⠛⠛⠛⢷⡀⠀⠀⠃⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⣿⡅⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣷⣶⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠋⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠙⠋⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⣿⣄⣀⣀⡀⢀⣼⣇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣨⣧⣀⣀⣀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⣿⠿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⠿⠿⠿⠿⠿⠿⠿⠷⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠙⠛⠿⢶⣦⣤⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣤⣴⣶⠿⠛⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⢀⣠⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠿⢷⣶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⡶⠿⠛⠋⠁⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠻⠿⠷⢶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠿⠿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣄⠀⢀⣠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢈⣽⣿⣿⣏⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠛⠋⢸⣿⣋⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣴⡾⠟⠛⠋⠉⠉⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣾⠏⠀⠀⣤⣤⣤⣶⣾⣿⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠿⣿⠁
⠀⠀⣿⡄⠀⠀⣿⡅⠀⠀⢸⣷⠀⠈⠉⠛⠛⠿⠷⣶⣶⣶⣤⣤⣤⣤⣴⣶⣶⡶⠿⠟⠛⠋⠉⠀⠀⢰⣿
⠀⠀⢿⣧⣤⣤⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⢈⡉⠉⠉⠉⠁⠀⠀⢈⣿⣶⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⠿⣿⣿⡆
⠀⠀⣾⡇⢀⡀⠀⠀⠀⠀⣾⡟⣿⣌⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⡿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁⠀⣸⡟⠘⣿⡀
⠀⠀⠛⠀⢹⣷⠀⠀⠀⣼⡟⠀⠈⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠀⠀⢹⣧
⢀⡀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠈⠛⠻⠿⢶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⢿⣇
⣾⡇⠀⢠⣶⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠈⠉⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⢰⡦⠀⠀⠀⣸⡟⠀⢾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠿⠃⠀⠀⢰⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⢤⡦⣀⠀⠀⠀⠀⠀⠺⠂⠀⢉⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⢈⣤⡬⢹⠀⠀⠀⠀⠀⠀⡠⠥⠎⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡉⡀⠉⠈⠁⠀⢂⠀⠀⠀⠂⠀⠱⠖⠺⠳⡄⠀⢸⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡃⠥⡤⠀⠀⠁⠳⣴⠀⠀⢧⡆⠀⠀⠀⠀⣷⠀⠟⡗⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠙⠓⠀⠀⠋⠴⣟⠈⢇⠀⠰⢂⠠⠏⠆⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠘⡆⠀⢉⠂⠈⠶⡀⠑⢦⣠⣤⡃⠐⡓⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠲⣄⠀⠠⢀⠀⠖⠁⠐⣜⠄⢀⠀⣻⣦⠀⠙⢟⣧⠈⠉⠀⢬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠛⢆⠀⠑⠀⠀⠀⠘⠿⠷⢛⡙⠘⢦⡀⢸⣿⡄⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡿⠉⠉⡞⣷⣮⠍⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠀⠀⠀⠀⠢⠀⠀⡘⠀⢀⠇⠀⢀⡇⣿⣿⣀⠻⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣦⡀⠀⠠⠀⠀⠀⠀⢘⠂⠈⢘⠗⠉⠉⠉⢀⠀⠸⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠙⠿⠷⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠿⠿⠏⠉⠑⠒⠻⠀⠒⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣖⣉⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠾⠷⠒⠚⠋⠙⠟⠒⠛⣻⣃⡠⠤⢶⠀⠘⠻⢭⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢸⡀⠀⢲⡈⠉⠀⠀⠀⠀⠀⣠⠤⠶⠶⠶⠶⠶⠦⠤⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣀⣨⡭⠭⠤⠤⣜⠀⠀⢸⢾⠋⠚⠃⠀⠀⠀⠀⣠⢎⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣌⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠑⣢⠴⠚⠉⢁⣀⣀⣀⣤⡤⠏⠀⠀⡾⣿⠀⣀⣀⠀⠀⠀⢠⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠈⢧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠒⠒⠶⢺⣟⣊⣉⣉⣠⠽⠶⠚⠉⠀⠀⠀⢀⡧⠛⢉⣉⡉⢹⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡩⠷⠒⣋⡁⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠞⠉⠉⣡⡾⠁⠀⢸
⣀⠤⠤⠐⠒⠒⠚⠋⠉⠉⠀⢀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⠒⠋⠁⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠓⠒⠒⠛⠒⠶⠋⠙⠒⠒⠑⠒⠒
⠱⢤⣀⣀⣀⣂⣀⣡⠤⠶⠚⠉⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢛⠙⡆⠀⠀⠀⠀⠀⠀⠀⣠⢋⠟⢶⠀⠀⠀⠀⠀⠀⠋⠀⡈⠁⠀⠀⠀⠠⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⡐⠧⡀⠀⠀⠀⠀⠀⠸⠏⠘⢀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠀⠐⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠞⣳⣝⢆⠀⠀⠀⠀⢻⠀⡎⢩⠇⠀⠀⠀⠀⠀⡄⣲⠀⠶⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠚⢡⡞⡎⠆⠀⠀⠀⠸⡈⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⡆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣥⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠉⠁⣀⠴⠺⠥⢿⢸⣾⣄⣀⡀⠷⠁⣿⢤⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠠⠎⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⢹⡇⣠⠴⢚⣿⣚⠋⠉⠉⠁⠀⠀⡠⠔⠋⠁⠀⠀⠀⢸⠀⣤⠤⠔⠲⠶⢤⣸⣄⣀⡀⠀⠀⠀⠐⠚⠁⠀⠀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⢐⠿⣁⠀⣠⢶⠿⡛⢷⣮⠗⠊⠉⠳⡄⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⡤⠒⠩⠤⠌⠉⠩⠭⠭⠭⠭⣭⣭⣭⣤⣍⣑⠲⡄
⠀⠀⣀⠔⠊⣡⠔⢋⡟⠁⣺⣑⡼⠋⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⠃⡇⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⠀⠀⠀⢸
⠀⠀⠙⠒⠤⣍⠗⠁⢀⡶⢭⠭⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⣍⠙⠃⡇⠀⠀⠀⠀⣷⠀⠀⠘⣿⠀⠀⢀⣖⣦⠀⠀⢈⣿⠀⠀⠀⡸
⠀⠀⠀⠀⠰⣧⣄⡴⢛⣒⢩⡭⠽⡉⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⡀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⡆⠀⠈⡅⣸⠀⠀⢸⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⢻⡭⠤⠔⠒⠒⠈⣳⡆⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⡟⠁⡆⠇⠀⠀⠀⠀⢸⡇⠀⠀⢹⣇⠀⠀⠓⠋⠀⠀⣼⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⢣⠤⣶⣶⠛⡏⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡇⡄⠀⠀⠀⠀⠀⣇⠀⠀⠸⠿⠶⠶⠶⢶⣶⣶⡿⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠇⠀⠀⣰⠓⢹⡄⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠘⣦⠽⠛⣳⡿⠒⠒⠛⠋⣹⠃⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠘⡜⣄⠀⠀⠀⡠⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠋⠐⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠐⢌⡒⠐⢊⠔⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠠⠀⠀⠀⠀⠀⡴⢻⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⠀⣠⠞⠠⣆⣀⣀⣀⣠⠤⣧⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡽⠁⣸⠁⠀⠈⠉⠙⠚⠛⠲⠶⠖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⡤⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡄⠀⠀⠀⠀⠈⠁⠀⠃⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠢⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠄⡠⣀⡖⠂⠰⡠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠑⠒⠒⠈⢑⡄⠀⠀⠠⢊⠀⠀⠀⠀⠀⠀⠀⠐⠽⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⡄⠉⠀⡠⠄⠀⠀⠉⡄⠀⠀⠀⠠⠒⢐⡠⠺⠀⡐⠀⠒⠒⢀⢀⠀⠀⣀⡓⠀⠈⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠠⠄⠀⠀⠀⠞⠠⢑⠂⡐⡌⠁⠀⠀⠀⡒⠄⡈⠁⠀⠙⢏⢀⡀⠀⠆⠀⠂⢩⠀⠉⠀⡠⡑⠊⠀⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰⢹⡕⠂⠀⠀⠀⠈⠢⢰⡄⠄⣡⠋⠯⠝⢠⠀⢀⠁⢀⢣⣸⠀⠂⠇⠀⠀⠠⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠐⢸⡥⡀⠀⠀⠀⠀⠀⠝⠉⠅⠀⢀⠁⢀⡣⠀⠨⢀⠌⡐⢁⠏⠠⠃⠀⠈⠁⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⣀⠀⠀⠑⠄⠑⠠⡉⠀⡠⢡⢯⠝⢠⡀⢀⠕⣡⡎⠄⣁⡂⠀⠑⣡⠊⢀⠠⢐⠥⡾⠂⣤⠀⢈⣀⡐⠊⠀⠃⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡤⣁⣠⣐⠒⠁⡹⠮⠤⠔⡧⡀⡔⠀⢪⣁⣨⠉⠴⠒⠲⠐⠺⠛⠛⢟⠛⢻⡖⢶⠗⣆⣒⠆⠡⢒⡒⠂⠁⠉⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠨⡲⢭⣈⠈⠀⠁⢀⡖⠦⣠⠬⡤⠯⠀⠀⠀⣅⢄⣄⠀⠀⡈⣄⢀⣵⡊⢀⡄⢧⠏⢼⡿⠛⠲⠥⠔⠁⠈⠄
⠀⠀⠀⠀⠑⠁⢀⡀⢀⠀⢀⠀⠍⠳⡄⣠⣾⠄⠒⠉⡳⡂⠀⣀⠒⠚⢁⡀⡙⠚⠋⢀⣙⢁⣢⠟⠀⠀⠀⠀⠠⠁⢄⠀⢠⡄
⠀⠀⠀⠀⠓⠈⠒⡁⠈⠁⠀⠀⠀⠁⠘⡑⠄⠠⣖⣯⣁⣀⠠⢥⣰⠪⢒⣀⠭⠞⠉⢀⣀⠠⠄⢀⠂⠀⠣⠄⢄⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠂⠀⠀⠀⠒⠃⠘⠀⡇⠀⠀⢀⡀⠀⢀⠃⢀⠀⠀⠀⠀⠁⡀⠠⢞⠿⠕⠚⠋⠀⣘⠀⡐⠀⠀⠀⠀⠀⠀⣉⠀⠀⡀
⠀⠀⠐⠀⡀⣘⠄⢻⡍⠒⡀⠄⠐⣡⡎⠀⠁⠀⠀⢄⡇⢨⣑⣼⠆⣀⣀⡠⠔⠁⠑⠀⢅⠐⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠅⠾⠷⡏⠀⠀⠉⠙⠉⠱⠉⠀⠀⢀⠕⢊⠔⠋⠉⠀⠀⠀⠀⠀⠀⠃⢀⠁⢪⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⢂⢸⠀⠀⠀⠀⠭⠖⢀⠠⠤⠐⠈⠀⠠⡄
⠀⠀⢀⠀⠀⠀⠀⠎⢳⠤⠄⠀⠀⠰⠁⠀⠀⢐⡠⠀⠀⠀⠐
⠀⠀⠀⢄⡀⠀⢀⡵⢻⠎⠴⠀⠀⠘⠄⡀⠀⣠⢠⢀⠠⡈
⠀⠀⠀⡠⢠⣴⠁⠄⣀⠠⠄⠈⠀⠀⣠⠤⠈⣀⠤⢀⠢⠁
⠀⠀⠘⠂⠂⠈⠁⠀⠀⠀⢂⠀⠄⠄⣀⠐⠀⠀⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⡉⠉⠉⡉⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣛⠋⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⣦⣦⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡿⠆⠸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣦⠴⠖⠛⠋⠉⠁⠀⠀⠀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠉⢁⠘⣀⣀⣀⠠⠴⠶⠞⠛⠛⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢴⣄⣴⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣤⣤⣤⣄⣒⠒⠶⢶⣶⣶⣶⣶⣶⡶⠶⣖⣒⠒⠶⠤⢤⣤⣄⣠⣟⠀⠀⢀⡾⠁
⠀⠀⠀⣀⣤⠶⠖⣩⣭⡽⠷⠶⠖⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠲⠶⠶⣴⣿⣉⡙⢶⣾⣅
⠀⣴⠞⢉⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠈⢹⠿⣦⡉⠳⣄
⢸⣇⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠋⠀⠈⣷⠀⣿⡆
⡧⠉⠳⢦⣀⡙⠳⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡰⠃⣀⣠⡾⢋⣴⠟⣧
⡇⠀⠀⠀⠉⠙⠛⠶⢤⣍⣉⡛⠛⠓⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⣛⣛⣻⠷⠿⠛⠋⠁⠀⣇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠲⡶⠛⠛⠋⣿⠉⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠛⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡿⢠⠚⠁⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⢰⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣧⠘⣧⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣿
⣿⠀⠀⠉⠙⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⢦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠒⠲⡾⠁⠀⢀⣠⣤⠶⠛⠉⠀⠀⠀⢀⡏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠲⠶⠶⠶⠶⠶⣶⠷⠶⠶⠶⠷⠂⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢀⡠⠎⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢰⡇
⢰⡄⠀⠀⠤⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠉⠛⠶⠖⠋⠀⢀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣼⠁
⢸⡇⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡿
⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠀⠀⠀⠀⢀⡠⠖⠉⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣀⠠⢋⡼⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠰⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠁⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⡼
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣴⠁⠀⠰⢃⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠰⠞⠉⠁⠀⠈⠻⣆⠀⠀⠀⠐⠀⠐⠒⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⡆⠀⠐⠒⠤⣤⣀⣀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠈⠉⠙⠛⠛⠒⠶⠶⠤⠤⠤⠤⠤⠴⠒⠒⠋⠉⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⠶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣄
⠀⠀⠀⠀⣰⣿⠟⢁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⢰⣿⣏⣾⡿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠻⠿⢿⣷⣦⡀⠀⠀⢿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡀⠀⢸⣿⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⣿⣿⣿⡇
⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣧
⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠛⠛⠉⠉⠉⠉⠙⠛⠛⠿⢷⣦⣄⡀⢸⣿⣿⣷⠀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⢸⣿⣿⣿⠀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢀⣌⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⢸⣿⣿⣿⠀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣇⣤⣴⣶⣾⣿⠃⠀⢻⣿⣿⡏⠉⠉⠉⠙⠉⣿⣿⣏⠀⠀⠀⢹⣿⢸⣿⣿⣿⣦⣍⠛⠻⠿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⢸⣿⣿⡿⠿⢿⣿⣿⣷⣄
⠀⠀⣼⣿⣿⣿⠟⠋⠉⢹⣿⣿⣷⣄⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣾⣿⣿⡛⠁⠀⠀⢹⣿⣿⣿
⢸⣷⣿⣿⣿⣯⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡿⠟⠋⠀⠀⢀⣾⣿⣿⡟
⢸⣿⣿⣿⢿⣿⣷⣄⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠘⣿⣿⣿⠀⠈⠙⠿⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠟⠉⠰⣿⣿⣿⡇
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⡾⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣶⣶⣶⣦⣤⣤⣤⣶⣿⣿⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠿⠿⠟⠋⢻⣿⠏⠙⠻⠿⠃
⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣰⣿⡄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣶⣶⣶⣶⣾⣿⣿⠷⠶⣦⣄⡀
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣥⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢹⣿⣷
⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠉⠁⠀⠀⢙⣿⣆⠀⢸⣿⣿
⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⢸⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠸⡿⠿⠿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣨⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢴⣾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠇
⠈⠻⢿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣠⣤⣶⣾⣿⣿⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⢠⠆⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⣉⣉⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣉⣈⣑⡄
⠀⠀⠀⠀⠀⢸⡤⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⠣⢞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭⠭⠭⢭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭⠭⠭⢭⠇
⠀⠀⠀⠀⠀⢸⡴⢦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⠶⠴⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠶⠶⢍
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎⠱⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠘⠦⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣃⠂⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⠤⠼⡄


⠀⠀⠀⠀⢀⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠘⡖⡈⠁⢲⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠹⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡵⢷⡀⡇
⠀⠀⠀⠀⠐⣇⣁⣀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠯⠭⢕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⣀⡸⡇
⠀⠀⠀⠀⠘⢫⡁⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⣀⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠀⡇
⠀⠀⠀⠀⢸⢠⣿⢿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⡇
⠀⠀⠀⠀⢘⠀⡛⠛⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣶⣦⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠁⠀⡇
⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⢘⡖⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⢠⠗⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⠐⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠉⠉⠉⠉⠉⠹⠁
⠀⠀⠀⠀⢰⣃⣒⣂⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣙⣒⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠸⣩⡿⢯⡁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡤⠤⣤⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠦⣤⣤⠾⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠴⠶⠒⠚⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠶⠦⢤⣤⣀⡀
⠀⢀⡤⠖⢚⠛⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⣤⡀
⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻
⣏⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟
⣷⠀⠀⠉⢙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠚⡋⠀⢠
⢿⠀⠀⠀⠀⠉⠑⠒⠬⠍⡙⠛⠲⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠒⠒⠛⠛⠛⠉⠉⠀⠀⠈⠁⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⡇⠀⠀⠀⠀⣀⡠⠤⠒⠂⠀⣠⠶⡶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⠀⠀⠐⠈⠀⠀⠀⠀⠀⡴⠃⠀⠘⣆⠈⠙⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢸⡇⠀⠀⠐⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣞⣁⣀⣀⣤⡤⠤⠤⠓⠆⠁⠀⠀⠀⠀⠀⣸
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⢸⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠘⠀⠀⠀⣿
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠈⠒⣤⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠂⠀⠀⢀⣠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠊⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠀⠀⠀⠀⠀⢿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠔⠞⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠃
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠉⠲⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠲⠶⠶⠖⠚⠛⠛⠛⠉⠉⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢰⡆⠀⡐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣧⡤⢧⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣇⢠⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⡷⢺⡄⠀⠠⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠟⠛⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠓⠒⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠐⠛⠛⢩⠉⠁⢀⣀⣠⣤⣴⣾⠏⡄⠀⠀⣤⢠⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡀⠀⢠⠀⠀⠀⣆⠀⠰⠀⡀⠀⠐⣀⠰⢶⣾⣾⣿⣿⣿⣿⣿⡿⠋⢸⣷⠀⢠⣿⠀⢚⠛⠃
⠀⠀⠀⠀⠀⠀⣸⡇⢰⣼⠧⠊⢙⠯⢆⡄⢘⡓⠊⡉⠄⢛⠸⠛⠋⠉⠀⢁⠈⠥⠦⣀⡈⢉⠒⠼⠿⡆⠋
⠀⠀⠀⠀⠀⠀⣿⠓⢈⠀⠀⠔⠃⢠⠀⠀⢀⡀⠀⠋⠉⠁⠀⠄⠂⢉⡀⣿⠀⠠⠄⠀⠈⠿⠲⠀⠀⢈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡜⠂⢀⠀⠀⠀⢰⡆⣀⠄⠁⣸⣧⡿⠧⠠⠖⠀⠀⠀⠀⠆⠀⠸
⠀⠐⠀⠤⢀⠀⠀⠀⠐⠀⠺⠿⠯⠿⠆⠀⠉⠁⠐⠃⠸⠧⠈⡉⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠂⠈⠰⣶⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠲⢢⢤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠛⠀⠀⠀⠀⠒⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⠲⠴⠿⠷⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡝⠋⢙⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢰⡄⠀⠀⠑⣾⠃⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠎⡀⢰⠄⢸⠿⠛⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢢⡀⠀⣆⡀⠀⠀⠀⠀⠒⠀⣺⠏⠀⡉⡉⣁⠀⢠⣤⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡌⢠⣡⠜⠀⠀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⡶⢀⡹⢷⡀⢻⡇⠀⠀⣀⣤⣤⣄⡉⣤⣼⣿⣡⣿⣰⣾⠁⣤⣄⣀⡀⣷⢾⣤⠀⠀⣠⡿⠈⣁⠤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠆⠙⢬⣥⠀⠀⠀⢠⠿⣯⣽⣯⣿⢸⡟⠛⢻⣿⣿⠛⠛⡟⢙⡏⢹⣿⣷⣾⣶⢀⡿⠀⠂⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣀⣤⡄⠀⣠⡄⠈⠃⠁⢠⠉⠉⣁⡴⣠⣷⠀⣿⠟⠁⢹⣷⣤⣴⡟⠿⠆⢀⣵⣤⡵⣿⣿⣟⣿⢋⣼⣷⣿⠞⠋⣀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠛⠒⠀⢷⡌⠠⣽⡇⠘⠉⠠⣾⡿⠇⣿⠟⢠⡉⣿⣶⣶⢻⡟⠉⠀⠀⢾⡉⠙⠿⢿⣿⣾⣿⣭⣿⣍⣆⣴⠄⠀⠀⠀⢀⣀⡐
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠄⠠⠀⠀⠛⣶⣿⣽⡾⣿⡿⣾⠀⠀⠈⢲⣦⡿⠿⠿⣿⣶⣿⢦⣶⣶⢿⣿⣶⣿⣿⣿⡏⠛⠛⣿⣿⣟⣭⣤⣴⠆⣼⠋
⠀⠀⠀⠀⠈⠁⠀⠈⡉⠀⠀⠰⡄⣠⢆⠡⢈⣿⣿⣿⣿⡞⣧⣀⢠⣾⣁⡀⢀⡀⢉⣻⡇⠘⠋⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠶⣤⠀⠄
⠀⠀⠀⣤⣴⣿⣿⣶⠈⠠⢉⡀⠠⠛⠛⣦⣜⠛⣿⣿⣋⠀⠛⣿⠀⠙⠉⣹⣾⣷⣶⡿⢃⣠⣾⣿⠛⢿⣿⣗⣾⡟⣼⣏⣛⣿⣿⣿⣿⣧⣤⠲⣏⠠⡄
⠀⠀⣾⡗⠿⠺⠿⣏⠀⠠⣿⣷⣤⣈⣷⣾⠷⣿⣿⣿⡇⠀⢰⡖⠀⠈⠢⡘⢿⢿⣟⣳⣾⡁⠉⠀⣀⣾⠟⠿⠿⠊⢘⣿⣿⣿⠿⡛⠿⠿⣿⡷⢾⡇
⠀⠀⢻⣾⣿⣶⠀⠹⣷⡀⠁⢀⣼⡏⠀⠰⠤⣾⡿⠿⢧⡀⠀⢰⠀⠀⠁⠐⣦⠐⠛⣋⣽⣿⣿⣿⣿⡇⠀⠐⠀⢈⣿⡟⠁⣤⣶⣴⣷⣤⠌⡥⠨⠇
⠀⠀⠈⠙⠛⠁⠤⠐⣿⡴⣶⣾⣿⡿⠃⠠⠀⠀⣄⡀⠀⠙⢶⡄⠀⠀⠀⠀⠀⢶⣾⠟⢀⣿⣭⣿⣿⣦⠀⠀⠀⣨⡿⣦⣼⣏⣀⣽⣿⢦⠈⣇⣀⣴⠇⠀⠀⠂
⠀⠀⠀⢀⠄⠀⠀⢰⣿⢧⡩⠶⡟⠠⠏⡄⠀⠀⠨⠛⠻⣧⠀⠁⠀⢀⣠⡄⠚⠁⠀⠀⡼⢋⡉⠉⠉⠹⣿⣿⣿⣿⣧⣼⠛⠛⠛⠛⢻⠟⣠⡿⠿⠷⢀⡀
⠀⠀⠀⠆⠀⠰⠶⠂⣹⣬⠀⢠⣿⡷⣦⡖⠀⣀⠖⣻⣷⠈⣥⠙⣤⡀⣈⣻⡶⠂⡶⠛⣦⡠⣤⣴⣶⣶⠾⣀⡼⢿⣿⣿⣶⣤⣬⣤⣤⣾⣿⢿⠷⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠲⠟⣿⣿⠀⣾⣿⠠⣿⣿⣿⢋⣰⢻⡿⣄⣠⣶⣻⣿⣿⣿⡁⢸⠀⢀⣯⡀⢰⠿⠟⢧⣀⣍⣡⣚⣿⣿⣿⣿⠛⣶⣿⣿⣿⣿⡶⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⡉⢱⣠⣿⣯⡶⣀⣈⡉⠘⢛⡌⠁⣸⢏⡿⠉⡉⠙⢿⣷⠄⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣷⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣟⠁⣼⡉⠛⠞⢉⣴⠿⠿⣿⢧⣤⣤⠐⠚⠁⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣠⣤⣤⣶⣄⣹⣷⣾⣿⣿⣿⠋⣽⠗⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣄⣸⠷⠈⠉⣻⡿⣿⠏⣰⣆⣼⡇⠀⠀⣈⣤⠀⠀⠀⣀⣤⣶⣻⠉⠀⠀⢉⠉⠉⢨⣿⣿⣿⠟⣻⣿⣿⣿⣿⡵⣄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢞⣿⣿⣿⡂⠀⢀⣿⡷⣽⣶⣿⡿⠋⠁⠚⠛⠻⠟⣠⣴⣿⣿⠻⢿⣿⣄⣀⣴⣿⣦⡀⠀⣿⣿⣏⣼⣿⣍⣁⠄⢸⣿⣿⣷⣄⣼⡇⢘
⠀⠀⠀⠀⠀⠀⣡⣴⣿⠁⢻⣯⡓⢀⣾⣿⡇⢄⡸⠋⠀⠀⠀⠀⢀⣴⣾⣿⡿⣿⡿⢁⣾⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣉⣠⣤⣿⣉⣉⡉⠙⢳⢈⡁
⠀⠀⠀⠀⠀⠉⠹⠿⢉⡇⠈⠿⣧⡾⠉⢿⡷⠊⠀⠀⠀⠀⢀⣴⠟⠛⣽⣿⡇⢸⣿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⠿⢿⡿⠃⠀⢻⡿⢿⣿⣿⣿⣯⣤⣴⠀⢴⠃
⠀⠀⠀⠀⠀⠠⣄⠲⣎⡀⢤⠄⢀⣵⠆⠁⠀⠀⠀⠀⣠⣶⣿⣏⡤⠿⠟⠻⣿⣿⡿⠟⠉⠙⡛⣿⣿⣿⠋⠁⠀⠀⢨⣿⡄⣤⣿⠃⣸⣿⣿⣿⣏⠻⣿⣿⠈⡇
⠀⠀⠀⠀⠁⣀⡶⢚⡇⢲⣀⡴⠊⠁⠀⠀⠀⠀⣠⣾⣿⣿⣙⣋⠁⠈⠀⠓⠀⠷⣆⠐⠊⠉⠙⣿⠯⣤⣀⣠⣄⣠⡿⠋⠀⢈⣷⣼⣿⠋⣹⣿⣿⣷⣬⣿⡗⢷
⠀⠀⠀⠀⠈⠉⠑⠛⠳⢯⣥⣄⡀⠀⢀⣤⣶⡿⢟⠃⠘⠣⣬⣍⣙⠛⠚⠳⠖⢶⠷⣤⣤⣴⣶⣿⣤⡜⠉⠏⠀⠈⣷⣤⣤⣾⡟⠀⣿⣶⣿⣿⣿⣿⣿⣿⡅⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠟⠋⠈⠙⠁⢀⣼⣷⣦⠀⠈⠉⠉⢳⡆⣹⣟⣶⠶⠛⠉⠉⠙⠛⠛⠳⣶⠟⠰⠾⠟⠁⠀⢨⣷⠞⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠍⠃⣠⡄⠀⠀⠺⠿⠧⠋⠀⠀⠀⠀⣾⣿⣿⡟⠁⣀⣠⠆⠀⠀⠀⠀⢀⣿⣷⣄⣠⠷⣤⢴⣿⠟⠀⢀⣾⣿⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⡿⠶⠋⠒⠈⠵⠷⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⢸⣿⡿⠀⠀⠀⠀⠀⣴⣿⡿⢿⣯⣀⣀⢀⡀⠀⠀⣿⠉⢻⣦⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⠀⠀⠀⠀⢀⡄⠀⢀⣿⣿⣿⣽⣆⡈⢿⡇⠀⣶⣦⣀⣀⣿⣿⣿⠟⠛⣬⣿⣿⣷⣤⣴⣿⢤⣾⠟⠉⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠷⣦⣠⣤⣶⠟⣠⣴⣿⣿⣿⣿⣿⢿⣿⠿⠿⣤⣄⣘⣿⣿⣿⠿⣻⣿⣧⣾⣿⣿⣉⠻⣧⣤⡴⠛⢶⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⡿⠿⢿⡿⠿⠿⠙⠛⠋⠉⠌⢿⡂⠀⠠⢉⣉⣉⣽⠋⢺⣯⣿⣿⠟⠡⠌⢻⣴⠿⠋⠐⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣱⠞⢶⡶⣦⣤⣄⣤⡶⠾⢻⠟⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠛⠛⠋⠑⠻⠟⠛⠯⣹⣿⠄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢿⠏⠩⠁⠀⠀⠀⠀⠤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢰⢠⠄⡄⠀⠂⠈⣄⡉⠉⠫⠿⠫⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡠⠄⠰⢏⣟⣯⣒⣂⣀⡀⠘⠃⠤⢠⣤⡤⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠒⠊⠉⢠⠤⣞⣠⢟⣎⣞⣿⣶⡶⢄⠂⣠⠀⢀⠀⠀⠌⡀⠹⢎⣳⡽⢂⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⢀⣀⠐⠄⠿⠟⣻⡿⡿⡿⢿⣞⢴⣏⠀⢣⢀⡒⣆⢠⢦⠁⢀⠀⢁⡵⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢠⣾⣽⣿⣷⡁⢪⢼⡟⣙⣗⣯⣿⣄⠛⢀⡀⢾⠘⠢⠌⠻⣃⠂⢉⠟⠒⣚⢁⢀⣠⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣭⣾⣼⣿⢾⡿⠿⣷⡽⢃⢋⡟⣉⡤⢻⠧⠈⢳⣄⣠⡀⣥⡄⠁⠀⠀⣁⣠⢿⠁⢐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣮⡹⡾⣿⣾⣏⣯⠸⣿⣣⢁⣸⡟⠡⠆⠀⠲⡀⢸⠛⠾⣛⠚⣷⢀⡀⢦⠘⣧⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⡿⣷⣿⣷⡿⣳⣾⢋⡔⢁⣀⠤⠒⠒⠒⠶⣭⣿⠀⣃⣼⣙⣿⠟⢦⣈⡿⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣭⢷⣺⢹⠟⣷⣿⡷⣯⣴⣿⠁⠀⠀⠀⠀⠀⠘⣷⣦⣮⡘⠺⣏⡔⡐⣾⣛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡭⢽⣋⠿⣬⣻⠿⣿⣽⡿⠀⠒⠀⠀⠀⠀⠀⢻⢋⣢⡑⢾⡋⠞⢻⠯⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣵⣿⢿⣿⣿⠾⣾⢽⡼⣿⡏⠁⠀⠁⠀⠀⠀⠀⢀⣿⡘⠷⣻⠮⢯⡟⢛⠒⣅⣐⣋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣟⣿⣟⣛⡾⢺⣿⠇⠀⠀⠀⠀⠀⠀⣀⣼⡿⠉⣻⠆⡜⣦⣌⡿⢉⠀⢿⣧⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣤⠀⠀⠟⡷⣸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠋⢀⣼⡅⢾⣷⢄⣿⡷⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠿⣾⡇⢆⣱⢸⣿⠀⠀⠀⠀⠀⡠⠂⠀⠐⠿⢿⣿⣾⣻⡫⠫⡧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢹⢟⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⠘⣷⡿⣿⣧⠧⢘⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⠸⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣏⣔⣁⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣉⣻⡟⠦⣄⣀⠀⠀⠀⢀⣷⡀⣀⠔⢹⢹⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢄⡿⣉⣙⣿⣙⡿⢋⣻⣿⠤⠧⢜⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⡟⠁⣠⠸⢿⡮⣌⠈⠑⣦⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡼⠁⠄⣮⡀⠁⠳⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠁⠀⠉⠒⠒⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡆⠠⠀⣀⣤⣤⣶⣶⣶⣦⣤⣀⡀⠠⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣾⣿⣿⣿⣿⣿⠟⣻⢿⣿⣿⣶⡐⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⣿⣿⣿⣿⣬⡟⠻⣬⣶⢳⣿⣿⣿⣿⡸⣷
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⣀⠀⢸⣿⣿⣿⣿⣿⡷⢠⣼⡿⣻⣿⣿⣿⣿⠇⢨⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠿⠆⡼⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⡿⠀⠼⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⣿⣷⡹⣿⠟⢛⣉⠉⠉⠯⠉⠛⠻⢿⡿⣣⢷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣩⣏⣅⡀⠉⠀⠈⠁⠀⠀⠙⠛⠂⠒⠉⠀⠈⠙⢉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡙⢿⣿⠓⠶⣤⣀⣀⠀⠀⠀⢀⣀⣂⣴⠴⢾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠂⣬⣿⣶⣶⣦⣬⣭⣤⣤⣬⣥⣤⣶⣾⣿⡁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠧⠤⠌⠻⠋⠅⠈⣿⣿⣿⣿⣿⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣄⠀⠻⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠛⠇⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡀⢸⠃
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢿⣿⠇⠀⠀⠀⠀⠀⠘⠛⠋⠛⠛⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠹⠟⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠁⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⡟⠁⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⣴⣿⠫⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⡀⠀⠀⠀⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⣷⠀⠀⠀⠀⠀⣿⣬⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣯⣀⣀⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣿⠃⠀⠀⠀⠀⢀⣾⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢸⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣖⡒⠛⢻⣿⣿⣷⣼⣷⣶⣶⣶⣶⣤⣴⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣧⡄⠤⡄⠀⠀⠀⢨⣭⣿⣿⡟⠀⠀⠈⠉⢉⣿⣛⡙⠿⠛⢿⣿⣦⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣴⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⣿⣿⣿⣇⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣗⠾⣿⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣻⣿⣿⣿⣯⣿⣿⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡇⣽⠟⠛⠛⢿⣿⡿⠛⠉⠙⢿⣿⡿⠛⠋⠉⠛⢿⣯⡀⢀⠈⣿⣾⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⣿⣦⢳⣄⣾⣿⠨⣿⣤⡄⢻⣿⡄⢺⣤⡄⠀⣾⣿⣷⣥⣼⣿⠿⣏⠉⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣾⣯⠉⠉⢻⣿⣷⣿⣿⣿⣧⣷⣿⣷⣿⣿⣿⣦⣭⣤⣾⣿⣿⣿⡿⢿⡁⠀⠘⣦⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⣿⣂⣤⣿⣯⡉⠉⠀⢻⡟⢻⡿⠟⠋⠉⠉⢻⡟⠋⠉⢹⡏⠀⠀⠀⣿⣤⣾⡟⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣾⣷⣤⡀⣀⣀⣼⣷⣤⣴⣾⣿⣦⣶⣼⣿⣷⡿⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠛⠛⠟⠿⣿⠿⡿⢿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠙⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⡿⠿⠿⠟⠻⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⣀⣀⣀⣀⣀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⡖⠂⠀⠀⠈⠀⠠⠉⠓⠠⣀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⣽⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣄⠉⠉⠙⠻⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⡏⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡸⠉⠙⠻⠿⠇⣠⣤⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⣴⣶⣶⣾⣿⣿⣿⣿⣿⡄⠀⠀⣀⣠⠤⠖⠒⠚⠉⠉⠀⢄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡿⠛⢿⡿⠿⢿⣿⣿⣷⡿⠿⠟⣳⠀⠈⠉⠀⢠⣆⣂⠀⠀⠀⡔⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣀⢠⡀⢀⠇⠃⠈⠩⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠛⠋⠀⢠⠞⠅⠸⡇
⠀⠀⠀⠀⢀⣀⡀⠀⠐⠖⢻⡌⢷⣾⡼⣷⢠⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠈
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠈⠻⠗⠙⠁⠠⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡁⠸⣃⡇⠀⢃⣞⠀⠀⠀⠀⠀⠀⡀⠀⠉⢀⠘
⠀⠀⠀⠀⢀⡀⡀⢀⣾⣀⠀⣶⠀⠀⠋⠄⠀⠰⡄⠻⣃⠛⣧⣶⠀⣴⠴⣹⢀⡸⣶⠃
⠀⠀⠀⢀⣾⠏⠀⣿⡿⢁⣾⡏⠀⠀⠀⠀⠀⠀⠁⢨⠋⠉⠁⠀⠾⠉⠋⠃⠀⠈
⠀⠀⢀⣿⠏⠀⣸⣿⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⣿⣦⢠⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⣄⠀⣿⣿⣿⠿⣿⣿⡆⠸⣿⡀⠀⠀⠀⠀⠘⣆⠐⠒⢤⣀⠀⠀⠀⠀⠀⠀⠁
⢻⣇⢻⣿⣿⣦⣿⣿⡇⠀⢹⡇⠀⠀⠀⠀⠀⠙⣷⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠆⠀⠀⢀⣀⠀⠀⠀⡀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠈⠀⣈⢰⠦⢶⠋⣱⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⣀⠀⢀⠁⣾⢀⡴⢋⢀⠀⠠⠀⠐⣮⠛⢶⣦⣿⡉⢿⠁⢸⣇⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⢿⣿⠀⠀⠀⢰⣾⣱⣸⣿⣆⠀⠀⠘⢿⢦⣈⣿⣛⣿⢯⡵⠎⠉⣤⠥⡼⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣇⠀⢡⣌⣿⡻⣿⣿⠏⠀⠀⠀⢘⣦⣴⡎⠉⢳⠀⣖⡀⠈⠀⠸⠯⣬⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟⠨⡭⠝⣻⣦⡀⠓⠌⠛⠾⠿⠿⠀⠀⣠⣾⢿⠷⢾⠲⣦⣤⠃⠋⠀⠀⠰⢇⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⢈⠀⠀⢀⡴⢋⣿⡟⠦⣄⣀⣀⣠⠤⠔⠚⠁⠀⠀⠀⠀⢱⢦⣿⡆⢳⡀⠗⣀⡪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢴⣍⣠⡤⣂⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡽⠈⢠⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠉⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠄⣀⡈⠁⠐⢁⡐⠃⠀⠀⠀⣀⣀⣀⠀⣶⠀⠀⠻⠇⠀⠀⢘⣤⣬⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠏⠀⠀⠀⠀⢏⠙⠦⠴⠿⠉⠀⠀⠀⠀⠀⠀⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⢁⡄⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠞⡀⢀⣀⣀⠀⣀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⣠⣾⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡏⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠀⠀⠀⢰⠋⢠⡎⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣦⣨⣄⠘⠁⣊⠳⣄⠀⠀⠀⠀⠀⠀⠹⣦⣤⣀⣀⣀⣀⣀⣤⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠩⠿⢳⣽⡷⠀⠁⠄⠀⠐⢶⣤⠀⠀⣀⠀⠘⠛⠋⠁⠈⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢻⣥⡀⣖⢘⣧⣉⠁⠘⢷⡄⠀⡀⢹⠀⠘⡿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⠿⠋⢹⠆⠊⢻⡶⠺⠫⣴⠀⣤⡤⠀⠀⠀⠀⠀⠉⠙⠛⠶⠤⢠⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠌⠀⠀⢄⠸⠋⠑⠆⡐⠂⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⣱⠂⡤⠠⢣⠀⠲⠀⠈⠘⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⡾⠁⠀⢰⠀⡿⡷⠦⠖⢶⠦⠀⠴⡀⠀⠉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣤⡆⣼⠆⠠⠆⠈⠉⠄⣀⣨⠟⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣈⡝⠀⣤⠄⠴⠛⢤⠻⣥⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄

⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⣤⣀⠀⠀⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⡀⣈⡉⠻⣄⣰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠆
⠀⠀⠀⣸⣧⠀⠻⣿⢋⣉⣩⠽⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠮⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⣿⣿⡇⣶⣫⢘⣟⢒⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢩⠟⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢿⠛⣷⣿⡿⣆⣻⣏⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠄⠀⠀⠀⠠⠀⣀⠀⠠⠈
⠀⠀⠀⢈⣧⡀⠘⡗⠊⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣗⡀⠀⠀⠀⣁⡌⢱⡄⠩⣐
⠀⠀⣠⣿⡗⢿⡳⢿⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠂⠤⢼⣿⣿⡛⣡⠗⡣
⠀⠀⣿⣿⣿⣄⠸⣿⠢⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠴⠖⢀⠀⠉⣉⣨⠅
⠀⠀⣿⣿⣿⣿⣦⣿⣧⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⣶⣯⣍⡙⠿⠃⠂⠀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠛⠋⠉⠉⠀⠀⠀⠁
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠄⠀⠀⡌⢆⠀⠀⠀⠠⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠈⢀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⠒⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠉⠀⠀⠈⣿
⢸⣿⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⡠⠊⠉⠉⠉⠈⠑⠂⠤⠤⠄⢄⡤⠤⠔⠒⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⢐⣿
⢸⣿⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡤⠞⡻⠟⠛⠛⠛⠒⠒⠒⠀⠀⠀⠀⠰⣿
⢸⣿⠀⠀⠀⠀⠀⢸⠃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠛⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⣏⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠂⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠸⡠⠀⠀⠠⣄⠘⡄⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠘⢦⠹⡄⢀⡠⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⠈⠳⣿⣌⢷⡘⠾⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⣌⡻⢷⣌⠂⠜⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⣿⣟⣦⣌⡻⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣯⢯⡛⠯⡛⠭⠙⠒⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠂⠤⠤⠄⢀⣀⣠⠤⠚⠃⠙⠆⠈⠀⠀⠀⠀⠀⠈⠣⣅⡢⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⠀⠀⠀⠠⡐⠈⠀⠀⠐⡄⠈⠻⣦⡑⡀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠀⠢⠀⠀⠀⠁⠠⢆⢿⡟⣎⠆⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢮⠟⢧⠇⠰⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠄⠀⠁⠀⠁⠀⠎⠀⣿⢂⠂⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⡀⠀⠀⠀⠈⠁⣸⡯⠂⠀⠀⣿
⢸⣿⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡗⠁⠀⠠⠀⠀⠀⠀⡄⢪⡟⠁⠀⠀⠀⣿
⢸⣿⠀⠈⠐⠢⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⣀⣐⣡⠈⠀⠀⠀⣀⡔⢀⡠⠈⠀⠀⠀⠀⠠⢗⠏⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠁⠒⠀⠤⠀⠀⠈⠉⠀⠈⠀⢐⣫⠭⠙⠒⠀⠀⢀⣾⣫⡶⠃⠀⠀⠀⢀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣿
⢸⣿⣦⣦⣤⣤⣴⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣾⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⠁⠀⠀⠀⠀⢊⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠀⠀⢰⡀⠀⠀⠀⠀⠀⠶⢇⠆⠀⠀⠀⡇⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡂⠀⠀⠀⢀⡀⠠⡜⠀⠀⠀⡌⠁⠀⠀⠀⣤⠀⠀⠘⠃⠀⠀⠂⡇⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢈⠀⠀⠈⠧⢤⠀⠀⠀⠀⢀⠂⠀⠀⠠⢻⡆⠀⠀⠂⠀⠀⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⡌⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢐⠳⠋⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠂⠀⠀⡀⠀⠀⠀⠀⠛⠁⠀⠂⠀⠀⠀⠁⠀⢠⢠⠀⠀⠀⡄⠨⢧
⠀⠀⠀⢀⠀⠀⠀⣸⠀⡄⠀⠀⠀⣮⢀⠀⠀⠀⠘⠀⠀⠀⠑⠂⠀⠀⢠⡐⠂⢠⢸⠀⠀⢀⠀⠀⠈⣸⠀⠀⠀⠁⡀⢹⡇
⠀⠀⠈⠀⠆⠀⠰⠉⠀⢀⠀⠀⠀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠒⠟⠉⠙⠁⠉⠀⠀⠀⠛⠀⠀⠀⢈⡄⠺⣷⠀⠀⠀⢸⠿⡆
⠀⠀⠀⠠⢭⣶⣶⣂⣀⢾⠀⠀⠀⢨⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠄⢫⣿⣄⠀⠀⠸⡦
⠀⠀⠀⠐⢉⠛⠻⡉⠫⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣿⡇⠀⠀⠀⡃⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⢸⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⣘⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠅⠀⠀⠀⢇⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠈⡩⡟⢿⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⡄⠀⠀⠀⢀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡀⠀⠀⠉⠍⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄⢱⠀⠐⠈⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡈⣱⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠍⠁⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠝⠛⢄⠀⠀⠀⠀⠀⠀⠀⠠⢤⠦⣰⣇⠀⠒⠀⠀⠜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠁⠈⡄⠀⠀⠀⠀⠀⠀⠀⠻⢛⠛⠻⠀⢀⡄⠛⢰⠹⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡅⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠈⠃⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠶⠀⠁⠀⠘⡆⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡠⠌⠁⠀⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⠀⠠⡄⠀⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⣴⡄⢆⠀⠀⠀⠆⠀⠰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢥⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⠀⣤⠸⠃⠀⢀⣀⠀⠀⡀⢈⡄⠀⠀⠀⠀⠀⠀⠈⠋⠈⢆⢰⠠⠆⠀⣎⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⡄⠐⠂⠀⠈⠝⣢⡀⢹⣆⢻⡀⠀⠀⠀⠀⠘⢻⡤⠌⠐⢧⡄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢃⠌⠀⠀⠠⠀⢀⣀⠉⠙⠶⣽⣦⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠁⡠⠀⠘⣟⣧⡆⡀⠀⠉⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠘⠱⠂⠈⠹⡿⠱⠄⠀⠐⠂⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣷⣿⣿⣶⣶⣶⣤⣲⣞⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡾⡻⣿⡁⢿⣿⣿⣿⠿⡌⠙⠉⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⣤⣠⣏⣳⣾⣇⠈⠁⠱⢷⣶⠃⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⡇⠻⢿⣿⣿⣷⣵⣤⣀⣀⠀⣴⣿⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠢⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⣄⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⠛⡿⣿⣿⣷⡄⠀⠀⠈⠛⠻⢿⠿⠿⣿⠘⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠘⠀⠀⢸⣿⣿⣿⣶⣷⣶⣿⡆⣠⣶⣾⣿⠰⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠁⠀⠛⠈⠉⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣳⣷⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⠟⠷⣿⣿⣯⣽⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢸⡟⡿⠿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⠀⢘⡀⢠⣴⠟⠋⠀⠑⠆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣿⣷⣄⣈⣛⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠒⠿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠐⠂⡀⠰⠀⠀⠀⠀⢰⠁⣴⣿⣿⣯⡙⠿⠿⣻⣤⣙⣲⣦⣄⡀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠂⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⢿⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⡿⢛⣦⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢷⢽⣿⢇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠷⠋⠀⠀⠹⠠⠄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢘⢉⡉⣻⣟⠉⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⢿⡷⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣐⠛⠿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠂⠀⢀⣈⠁⠀⠀⠀⠂⠀⠀⠀⠄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠶⠿⠟⠁⠳⠶⢶⣀⡤⠀⠀⠛⢷⣦⣭⣼⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠴⢧⠠⣤⠀⢸⣷⣽⣿⣿⣿⣷⡀⢃⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⠷⠢⠀⠀⠠⠿⠶⠼⠿⠿⢿⣿⣿⣿⣷⡀⢷⡄⠀⠀⠀⠀⠀⠈⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣉⡀⠀⠀⠀⢠⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣿⢁⠡⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠀⣴⡿⢙⣿⡯⠈⠀⠈⠻⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣟⠀⢀⣿⣿⣷⣿⣿⣆⠀⠀⠂⠀⡀⠀⠀⢠⣴⣏⠀⠀⡀⣀⣴⡞⠁⠰
⠀⠀⠀⠀⣀⣠⡖⠤⠀⢀⣴⡏⠐⠲⣾⣿⣿⡏⢰⣿⣿⡛⣿⡿⣿⣿⣦⣄⡀⠀⠄⠀⠀⡴⣿⠇⠀⣈⠀⢰⡞⠁
⠀⠀⠀⢸⣿⣿⣇⢀⣼⢿⣿⡁⡀⠀⡀⢿⡿⣿⠉⠿⠛⣍⠁⠐⠉⠉⠻⢿⡿⣦⣀⠐⠦⢘⢋⠰⢳⡇⠀⡄⢁⡀⣷⡄
⠀⠀⠀⠀⡉⣁⡅⣾⣿⣼⣜⣧⣷⣆⣇⣠⣧⣩⣼⣇⠀⣿⡆⡀⠀⠀⠀⠀⠈⠛⣙⢿⡆⡂⣸⡇⢺⣿⣀⣿⣸⣧⣿⣷⡆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣿⣀⡀⠲⠶⢶⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⣿⣿⢿⣿⣻⡃
⠀⠀⠀⠀⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠦⠀⠈⠉⠛⣿⣿⣾⡏⡋⡏⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠿⠟⠛⠛⠋⠉⠉⠀⣴⠀⢠⣀⣧⣫⠋⢻⣤⠷⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⣿⢾⣿⣿⠸⣽⣿⣿⡿⠛⢉⣅⠀⠀⢠⡆⠀⠀⠀⠀⣶⣤⣿⣷⣿⣿⣏⣿⣇⣄⠀⢰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡇⠛⣄⣻⣿⣿⠿⢃⠸⣿⠀⠠⣽⣧⡄⠀⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣌⣻⣿⣿⣟⡯⢸⣷⠟⠀⣻⣿⣣⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣾⢇⣽⣿⣵⣆⡄⠀⠀⢰⣿⣴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡶⠀⠘⣿⣿⣷⣟⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣟⡋⣝⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠃⠛⠛⠋⠉

⣶⡶⢲⡶⣶⣶⡶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⣿⣿⣿⣷⢻⣎⠻⣮⣿⣆⠳⣌⠛⠷⣶⣶⣽⣿⣿⣿⣾⣿⢃⣴⣿⡿⢟⣫⣾⣧⣾⣿⢟⣠⣿⣵⠏⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡄⢻⣧⡀⠹⣿⣿⣿⣿⣶⣄⡈⠙⣿⡎⢻⣿⣿⣾⣿⣯⡾⣫⣭⣿⣿⣿⣿⣿⣿⠏⣠⣶⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⢻⣷⠀⣿⢇⣤⣌⡙⠻⢿⣦⣹⣧⣾⣿⣿⣿⣿⣿⣴⣿⡿⠛⢉⣡⣤⣸⣿⣾⡿⠋⣠⡿⢣⣿⣹⣿
⣿⣿⣿⣿⢿⣷⡀⢻⣷⣿⣸⢳⣎⣙⢦⡀⠙⠿⠛⠋⠉⠉⠉⠙⠛⠿⠋⢀⡔⣋⣥⠸⡇⣿⡟⣠⣾⠟⠁⣼⢣⣿⣿
⣿⣯⢻⡟⣧⡹⣿⣌⣿⣏⣿⢸⣙⡾⣧⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡰⣿⣿⡃⡇⣿⣿⠟⠁⣠⣾⣿⡿⣿⣿
⣿⣿⣎⢿⣌⠻⣿⣿⣿⣿⣾⡘⣛⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⢣⣷⣿⣯⣴⣿⣿⢟⣩⣾⣿⣿
⣿⣷⣿⣾⣿⣦⣌⣻⣿⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢟⣿⣿⣯⣶⣿⢟⣱⣿⣿
⣿⠶⢈⣙⣻⣿⣯⡉⠉⠁⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠈⠉⣉⣽⣿⣿⣿⡿⠛⢻
⣿⣫⣭⣭⣿⣟⣛⣛⡶⡄⠀⠀⠀⠀⡞⠀⢰⡏⠳⡄⠀⠀⠀⢠⠞⣽⠀⠈⡧⠀⠀⠀⠀⠀⢾⣛⣿⣿⣿⣿⣿⣿⣻
⣿⡿⢟⣿⣿⣫⣭⠍⠁⠀⠀⠀⠀⠀⠸⡄⢸⡇⠀⡇⠀⠀⠀⢸⠀⢿⡇⣠⠃⠀⠀⠀⠀⠀⠈⠭⣭⣛⣿⣯⣟⡻⢿
⣿⣴⢟⣿⣿⠟⠁⢀⣀⣀⣀⠀⠀⠀⠀⠈⠚⠓⠊⠀⠀⡤⠀⠈⠑⠚⠛⠁⠀⠀⠀⠀⣀⣀⣀⡀⠈⠻⣿⣿⣷⣭⣻
⣿⣶⣿⣿⢇⡴⢋⣭⠿⠛⠉⣿⡷⢦⣄⡀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⢀⣠⡴⣾⣿⠉⠙⠻⣍⠛⢦⣹⣿⡿⣿⣿
⣿⣾⣿⣿⡟⢀⡾⠁⠀⢠⠾⢫⡇⠀⠈⠙⠳⠦⠤⣤⢰⣿⣂⡤⠤⠴⠚⠋⠁⠀⢸⡙⠧⡀⠀⠈⠳⠀⢿⣿⣿⣿⢻
⣿⣿⡇⢹⣧⠸⠁⠀⠀⠀⠀⢚⣇⠀⠀⠀⠀⠀⠀⢀⡏⡇⢯⡀⠀⠀⠀⠀⠀⢠⣼⡃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣾
⣿⣿⣧⠈⢿⡄⢤⠀⠀⣀⡤⢟⣿⠀⠀⠀⠀⠀⢀⡞⣠⣿⡌⢧⠀⠀⠀⠀⠀⢸⣿⠳⢤⡀⠀⠀⡄⢠⡟⠟⢸⣿⣿
⣿⣿⣿⣷⡈⠳⠸⡷⠤⣤⣴⡿⢻⣷⠀⠀⠀⢀⡞⣰⠃⡇⠹⣌⢧⠀⠀⠀⢸⣾⠟⢷⣤⣤⠴⣾⠇⠟⠀⣠⣿⣿⣿
⣿⣿⣿⡿⢿⣷⣶⣿⣷⡿⢯⣤⡀⢿⣴⠀⠀⣼⡼⠁⠀⡇⢀⠘⢮⣧⠀⠈⣾⡟⢀⣤⡽⣿⣾⣷⣶⣶⡾⢿⣿⣿⣿
⣿⣮⠻⣷⡀⢠⣾⣿⣿⣿⣶⣌⢻⡌⣿⡀⠀⣿⣇⠀⡀⠟⠀⠀⣼⡿⠀⢀⣿⢠⡟⣡⣾⣿⣿⣿⣷⡄⢠⣿⢏⣵⣿
⣿⣿⣧⠘⣷⣿⣿⡸⣿⣼⣿⢿⣾⣇⣿⠙⢆⢸⣿⠀⠀⣤⠀⢀⣿⠇⡴⠋⣿⣾⣾⡿⣿⣷⣿⢡⣿⣿⣾⠃⣾⣿⣿
⣿⣿⣿⠀⢸⣿⠹⣿⣿⢿⣿⣸⣿⣿⠇⠀⠈⢿⣿⡆⡦⠀⠄⢸⣿⡾⠀⠀⢻⣿⣿⣅⣿⡿⣿⣿⠫⣿⡇⢠⣿⣿⢻
⣿⢻⣿⣇⠈⣿⣷⡿⣃⠘⣭⣿⣿⣿⠀⠀⠀⠸⣿⡇⠀⠀⠀⣸⣿⠁⠀⠀⠀⣿⣿⣿⣭⠁⡛⣿⣾⡿⠀⣼⣿⠏⣸
⣿⣶⣿⣿⣷⣬⣝⣛⣻⠿⣿⣿⣿⣿⡀⠀⠀⠀⣿⣷⠀⠀⠀⣿⣿⠀⠀⠀⢠⣿⣿⣿⡿⠿⣛⣋⣭⣤⣾⣿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⢡⢡⢡⢡⢡⢁⢃⠃⠇⠀⢰⠁⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠆⠀⡆⡆⡀⡀⡈⠸⠸⢸⠀⢀⠇⡰⢁⠞⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢠⢠⢡⢁⢣⢣⢣⢣⠣⢁⠎⡴⣡⢊⠔⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⠆⡆⡆⢆⢆⢪⣢⡓⢡⢋⠞⡴⣡⢫⢞⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⡐⡐⡜⢌⢢⢻⠱⢠⢃⠎⡜⡰⢡⢇⡞⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢂⠢⡱⡑⡌⡌⠆⠇⠇⠃⡎⡸⢸⢡⠇⡎⡼⢰⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢀⢀⢖⠔⡕⡔⡜⡜⡘⠸⢸⢸⠀⡇⡇⡘⢸⢸⠁⡇⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢁⠄⡀⠕⡡⢂⠎⡜⢠⠀⠇⡇⢀⢸⠀⡄⡇⡇⢇⢸⢸⢸⠁⡇⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⢀⠁⠀⠀⠀⡀⠤⠀⣂⠥⠐⡈⠔⢈⠔⢁⠜⢀⠆⢰⠀⡇⢸⠀⡇⠀⢱⢰⢸⢸⢸⡘⡀⡇⡇
⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⢀⠔⠁⠀⡠⠂⡁⠄⢂⠡⠔⢂⠡⠐⢊⠡⠔⢁⠴⠂⡠⠂⡠⠃⡸⠀⡶⠈⠈⡌⡌⡍⡇⡇⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢈⠄⢊⠠⢒⡩⠄⢒⠩⠤⢒⡩⠤⢒⡉⠤⢊⡠⠔⢁⠔⠁⠀⡰⢡⢣⢣⢧⢇⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠠⡢⠀⠀⡀⢁⠔⡡⠒⡡⢂⡡⢒⡡⢐⡩⠔⣊⠭⢐⣢⠭⢕⣒⡨⠥⠐⠊⢁⣀⡤⣪⢕⡵⣣⢊⢞⡜⡼⣹⢱
⠀⠀⠀⠀⠀⠀⣪⡿⢀⠎⡐⢡⠊⡠⢊⠔⡩⢐⡡⢒⠡⢒⡩⢔⡪⠕⠒⠉⣀⡠⢤⣒⡾⢯⣻⡿⣝⡺⢕⡪⢖⡵⣪⢞⡵⣡⢏
⠀⠀⠀⠀⠀⢠⡾⢁⢂⠌⡔⡡⢊⢔⡥⣪⢔⡡⣊⠕⡪⠕⠊⢁⡠⢔⣒⠭⣒⡮⣕⣒⠭⣗⣲⢭⣒⡮⢕⣺⠵⣪⠵⣫⢞⡡⢋
⠀⠀⠀⠀⠀⢱⠁⠂⠎⡌⡐⡱⡡⢪⢞⠕⡡⣊⠔⠉⣀⠴⡪⢕⡪⣕⡺⣽⣲⣽⣿⣾⣿⣷⣿⣿⣶⣯⣷⣺⠭⣖⡯⠖⡋⢐⡡
⠀⠀⠀⠀⠀⢸⠀⠀⢠⠰⠰⠱⢡⢡⢣⢎⢔⠁⡠⢊⢕⡩⣚⢕⡪⣗⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⣛⣛⣉⣡⣤⣴⣚⠭⣚
⠀⠀⠀⠀⠀⠈⣇⠀⠘⢀⠀⠀⡆⢆⢃⢂⠊⡜⡰⢣⠎⡜⡱⢫⢞⡕⡩⢻⢟⡿⠛⠋⢁⣤⠴⢒⣫⠭⢟⣻⡿⢿⣿⣿⣿⣯⣗
⠀⠀⠀⠀⠀⠀⠘⢀⢤⡌⢀⣠⡇⢸⠸⡌⢸⢀⡇⡞⢸⠀⡇⡟⢸⢰⠇⡏⣼⠀⠁⢰⠋⢠⠞⢡⠖⢋⡭⠖⣋⡭⢖⣚⡭⢝⣛
⠀⠀⠀⠀⠀⠘⠀⠀⢧⢱⣼⡍⢿⠈⣷⠀⡌⢸⠀⡇⢸⠀⡇⣧⢸⠘⡇⣇⢹⠈⡀⢸⠀⢿⠀⣿⠀⡞⠀⡞⠁⡴⠋⡴⠚⣩⠴
⠀⠀⠀⠀⡀⠁⠘⠀⠸⢃⣯⡗⣿⣾⡿⠀⡇⢸⠀⣇⠸⡀⣇⢸⠈⡆⢳⢸⡜⣇⢣⠈⣇⠸⡄⢹⡀⢷⠀⣇⠀⡇⢸⡇⢰⠃⢰
⡀⣀⣀⡈⠀⣊⢀⣤⣲⣮⣾⣿⡿⣿⡟⠀⡇⠘⡀⢸⠀⡇⢸⡀⣇⢹⡈⣆⢳⠸⡌⢇⠸⡆⢻⡀⢧⠘⣆⠸⡄⢳⠀⣇⠘⡇⢸
⠍⠉⡁⠀⢁⠪⢑⠫⡫⡫⠪⠫⡻⡽⠇⠀⡇⠀⡇⢸⡀⢱⠀⣇⠸⡀⢧⠸⡌⢧⠹⡌⣆⠹⡄⢳⡈⢧⠸⡄⢳⡈⢧⠘⡆⢹⡀
⠊⡡⠐⡀⠔⡈⠔⡩⢊⠔⡱⡱⠱⠹⠀⢰⠃⠀⡇⠀⡇⠘⡄⢹⠀⢧⠘⡆⢱⡈⣇⢱⡘⣆⠹⡄⢳⡘⢆⠹⡄⢳⡈⣇⠹⡄⢳
⠊⡠⠊⡠⠊⡠⠊⡴⢁⠎⡰⢡⢣⠃⠀⡸⠀⢰⠃⠀⡇⠀⡇⠈⡇⠘⡄⢹⡀⢧⠘⡆⢳⡘⣆⠹⡄⢳⡘⣆⠹⡄⢳⠘⣆⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⡿⢻⡿⠀⠀⠀⠀⠀⣀⠀⣠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢧⠚⠁⢠⢄⠀⠐⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢢⠉⠀⠀⠀⡠⠔⠒⢂⠎⣸⡖⠁⣴⢦⣄⠈⠙⣿⣿⣤⡀⠀⡌⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠩⣐⣚⡁⠀⡀⠠⠘⠁⠘⡄⠙⠚⠿⣿⠦⣼⠛⠿⠛⠀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣹⠇⡀⠄⢇⠀⢠⡁⣤⠔⠊⠀⢡⠈⡟⢂⠀⠀⡉⠀⠀⢠⡾⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠾⡻⢃⡀⣴⡆⠱⡀⠀⠸⣉⠦⠒⠋⠺⠁⠺⣦⡐⠸⠀⢀⣩⣴⣿⣛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠲⠆⠻⢯⣬⣿⣾⣏⢀⣀⠃⠀⡀⡀⠀⢠⡀⠈⠈⢇⠀⠀⠀⠈⠻⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣂⢠⣠⡄⠐⠒⢩⠛⠉⣔⡄⠀⣉⠲⡄⠃⣀⣠⣤⠀⡠⢀⠉⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡁⡨⢣⡀⢰⠁⠀⠀⡽⠧⠛⠃⠀⡎⢁⡮⠛⠃⠪⢄⣨⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠨⠅⠀⡋⠉⠲⠏⠉⠋⠀⣼⡀⡠⠒⠉⠉⠁⠲⡷⢃⢠⡕⡀⣧⠕⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠁⡀⠙⠢⢭⣳⠀⡄⠀⠰⠆⠀⢹⠀⠰⡆⡴⠶⠖⠘⠇⡸⠱⢡⣏⣀⡊⠑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠂⢐⡡⠚⡇⠀⢠⣤⣴⡄⠀⠀⡀⠄⠠⠀⣤⠀⠀⡟⠀⢀⠸⡃⡐⢍⢣⡊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢔⣈⣏⣀⣈⣑⣉⣀⣀⣀⣇⣑⣘⣘⣛⣀⣀⣃⣀⣈⣀⣇⠠⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢸⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⣀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⠀⠈⠁⠀⠀⢀⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⠀⠘⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠐⠒⠒⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣠⣤⣿⣯⣶⣿⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣶⣤⣦⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⢿⠂⠀⢀⠀⢰⣿⡛⣷⣶⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠸⡿⠀⠀⠾⡗⠿⠻⢶⡆⣄⡀⠀⠀⡄⡅⠀⢹⡟⠀⠀⠐⠚⠻⡿⠭⠻⠏⠙⢺⣿⡅⢱⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣧⠀⠀⠀⠀⢀⣤⡈⠉⠀⠁⠀⠀⡀⢡⠀⢸⣇⠀⠀⠀⠀⠀⠁⠀⢀⠀⠂⠈⠛⠘⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣧⣹⣿⣿⣈⣀⣤⢾⠣⠀⠀⠀⠀⠀⡓⡀⡇⠈⣿⣦⣀⣀⣀⢀⡀⠀⠐⠾⠿⢦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡘⢿⣿⣿⣿⣏⠙⣶⠀⠀⠀⠀⠀⠀⣒⢨⢿⣀⣻⣿⣿⣿⡇⡽⢁⣀⣴⣶⣿⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢐⣏⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣤⡉⠁⠺
⠀⠀⠀⠀⠀⠀⠐⠒⠶⣤⣀⠀⢹⣷⡹⣿⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠩⣆⠘⢿⣿⣿⣿⣿⣿⣿⣤⡈⣻⡟⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠙⢷⡽⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡹⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢹⡀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⣿⣧⡘⢿⣿⣿⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠻⣿⡄⠀⠈⠹⣿⣷⡌⣿⣿⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⠉⠉⠁⠀⠉⠉⠉⢙⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣷⠀⣿⠛⠛⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⢿⡄⠀⢸⡇⠀⢰⡟⠀⠀⢀⣴⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠿⠟⠛⠛⠻⠿⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠠⠶⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣠⡤⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠀⣿⠀⠀⠀⠀⣿⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠈⠉⠋⠀⠀⠀⣰⡟⠀⠀⣀⣠⣿⣄⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢻⣷⢶⣤⣤⡶⣾⡛⠉⠻⣦⡀⠀⠀⠀⣼⠏⠀⢠⡾⠋⠉⠉⠉⠛⢷⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠾⠃⠀⢸⡇⠀⠘⠷⠀⠀⠈⠻⠆⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⠻⣦⣤⣀⣠⣴⠟⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠉⠉⠉⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⢸⣿⡆⠀⠀⣠⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⣦⠀⠻⠛⠇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣤⣀⠈⢉⣤⣶⣿⣿⣿⣶⣤⡉⠀⣠⣴⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⠟⠀⡉⠻⢿⣿⣿⣿⡿⠟⢋⡀⠛⠻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠀⢠⣤⣤⠀⠺⣿⣧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡀⠠⠄⠀⠀⣠⡿⠋⠀⠀⠘⣿⡇⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠑⡇⠀⠀⣀⠄⠊⠀⠀⠀⠀⠹⣤⠊⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠟⠀⠀⠀⠀⠈
⠀⠀⠠⡀⠀⠀⠀⠱⣄⡎⠀⠀⠀⠀⠀⠸⡀⠀⣿⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⢳⣠⠁⠀⠀⠈⠙⠶⣤⣠⠀⠀⠀⠙⣦⡏⠀⠀⠀⢠⠎⠀⢀⡠⠂
⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠹⣦⠀⠀⣠⡟⠀⠀⢀⣠⠞⠉⠉
⠀⡀⠀⠀⠀⣀⡤⠞⠋⠉⠉⠉⠙⢿⣧⣼⣯⣴⠶⠿⠻⠷⢤⣀⠀⠀⠀⠀⣀⠠⠄
⠀⠈⠉⠈⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⣏⠛⠛⠉⠉⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠋⣿⠉⠛⠶⣤⣤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠸⠀⠀⠀⠀⠙⢆⠐⠀⠀⠀⠉⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣟⣛⣿⣻⣿⣿⣶⣶⡆⢀⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⡯⠡⠏⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢛⣵⣿⣿⣿⡟⠿⡧⠴⠶⠿⠛⠻⠟⠟⠋⠁⠀⠀⠘⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠿⠟⠿⣿⠀⢿⣾⣿⣿⠉⠀⠠⠤⠀⠠⠄⠀⠀⢀⣶⣿⣷⣤⣶⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⣟⣿⣶⡘⣦⣌⠛⢃⣀⢀⣠⣀⣄⣀⣀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢹⣿⡿⠟⠁⣨⣿⠁⠰⠿⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡏⠀⠀⠀⣠⣶⣿⢠⡇⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣰⣾⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣿⡗⣾⣿⢟⣀⣈⣠⣾⣿⣛⣟⡃⠿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠆⠈⢻⣿⡟⠋⠉⠀⠍⠻⠉⠙⠻⢿⣷⣦⣄⡀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡁⢰⢃⣼⡟⠀⠐⠲⠦⠄⠀⠀⠀⢊⠄⣩⡙⠟⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡷⢸⠿⡃⠀⠘⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⣡⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣷⡖⠛⢀⠀⠀⠀⠄⡀⠀⠀⠀⠄⠀⠀⠀⠒⠀⠀⠀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢛⣿⠀⢸⣿⡦⡀⠀⠀⡐⢦⣄⡀⠀⠀⢀⣠⣄⠀⠁⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠀⢸⣿⡅⣄⠀⠀⠀⠀⠉⠙⠛⠉⣈⣻⡏⠧⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⢸⣿⣗⡙⣺⣶⠶⠦⣀⡀⢀⣈⡿⢻⣿⣀⠀⠋⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣿⢿⣿⣿⣿⣿⣿⢇⡠⠟⠋⣿⣿⣿⣾⣿⢏⠠⡈⡇⡔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡟⠾⢿⡧⠿⠟⢽⡿⣿⡿⠛⠙⠙⣷⣾⣿⣿⡷⣾⠀⠫⠐⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣦⠤⣤⠀⣌⠉⢮⡉⠁⠀⠈⠙⠛⠛⠻⠇⢿⡄⢆⣀⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⠛⠉⡰⠛⠂⠛⠀⠀⠘⠛⢶⠀⠀⠀⠀⠀⢺⡏⠉⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣇⣀⣄⣀⠀⠀⠀⠀⢠⡀⣷⣶⣤⣶⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠛⠻⠏⠙⡟⠣⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠈⠑⠒⠀⠀⢀⣀⣠⡤⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡼⢻⠀⠀⠀⢠⣃⠔⠋⠀⠀⠀⣀⡀⠠⠤⠂⣥⣾⠥⠤⡶⠒⠒⠈⠁
⠍⠁⠂⢤⡀⠀⠀⠈⠁⠘⠉⠓⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠼⠁⠈⠀⠀⠀⡼⠁⠀⠀⠀⡀⠀⠀⠉⠑⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠠⠤⠬⢭
⠀⠈⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⠐⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢄⠀⠀⠀⠶⡄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣒⣭⡄⡠⠖⠁⠀⠀⢀⡔⠀⢀⡤⠒⢁⡤⢒⠝⠁⠀⠀⠀⠀⠨⡀⠄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢑⠢⠀⠀⠈⠢⡐⣮⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡱⠋⠊⠀⠀⠀⠀⢠⡿⡲⠉⠀⢀⢖⣵⢆⠈⠀⠀⠀⠀⠀⢀⠀⠙⠲⣄⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢿⠢⡄⠀⠀⠈⠜⡎⠓⡀⠀⠀⠀⠀⠀⣠⡵
⠀⠀⠀⠀⠀⠀⡀⢉⠊⠀⠀⠐⡵⢡⠏⠀⠀⠀⠀⠀⠀⠀⠘⡗⢤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⢊⢧⠈⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⢨⠊
⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣄⢨⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠀⠀⠘⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⡄⠀⠇⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢠⠀⠘⢆⢿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣧⢃
⠀⠀⠀⠀⢣⠀⠀⢠⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣇⢆⠀⠈⡎⣇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣸⡎⡄
⠀⠀⠀⠀⢳⢣⠀⠘⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⡜⡄⠀⠘⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠠⠹⡇⢣
⠀⠀⠀⢦⡈⢧⢣⠀⠆⡏⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡹⠇⡇⠀⠀⠙⠀⠀⠀⠀⢀⠋⡄⠀⠘⣆⠀⠀⢳⡀⡆⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠀⠤⠤⠤⢔⣠
⣬⠢⡀⠀⡙⢾⡆⣇⠰⢱⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢱⡀⡇⠀⠀⠀⠀⠀⠀⠀⡘⠀⠇⠀⠀⡏⡆⠀⠀⣧⠃⠀⠀⡠⠊⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠈⢁⠙⡄⠈⢢⠁⢹⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⡼⠀⠀⠀⠀⠀⠀⢀⢜⡴⠀⠀⠀⣰⠁⢱⠀⠀⠋⠀⢀⢎⣴
⠀⠀⠑⢼⡀⠀⢃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⠀⠀⡀⢔⡵⡿⠃⠀⠀⡠⢣⡾⠈⠀⠀⠀⢀⣠⢻⠇
⠀⠀⠀⠈⠁⠀⢸⠁⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⡇⣸⡇⠀⠀⠀⠀⠀⢀⡨⠚⢁⠞⠁⠀⠀⡔⡵⢫⠇⠀⠀⠀⢀⢲⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⡞⠀⠀⢠⣧⠇⠀⠀⠀⠀⡠⠘⠁⠀⠐⠁⠀⠀⢠⣪⠎⠀⠎⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠒⠠⠤⠤⣬⡭⢥⣴⢂⠜⠀⠀⠀⡼⢋⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠗⠁⠀⠈⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⠁⠀⢀⡴⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠯⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣞⠍⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⣟⣴⣫⠟⠀⣠⠀⠀⠀⡀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠬⣄⣒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢉⠙⣛⡩⠊⠁⠀⢀⣙⡀⠀⠀⡳⣄⠹⣿⢦
⠢⣄⡀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠛⠛⠂⠈⠒⢄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠯⠡⠤⠔⠋⠁⠀⠀⠀⠀⡜⠀⠇⠀⢀⠀⠈⠓⢮⣷
⠡⠤⠬⠥⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢵⡄⠀⠀⢀⠀⢮⠄⠀⠀⠀⠀⠀⠀⢀⠄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣴⠂⠀⠀⠜⣴⡇⠀⠀⠈⠓⣄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣒⠒⠒⣀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠳⣜⣎⢆⠀⠀⠀⠀⠀⠀⠈⠑⠒⠯⠭⣭⣭⠍⣩⠔⠁⠀⠀⣪⠎⠸⠁⠀⠀⠀⠸⢈⠂
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠉⠲⢌⠻⣝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⠳⣄⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠄⠈⠁⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧
⠳⣄⠀⠀⠀⠀⠀⠀⠁⢄⠀⡀⠀⠀⠙⠪⣓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠕⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠁
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⣙⢪⣑⠤⣀⠀⠀⠀⠉⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠈⢜⢷⢄⠀⠀⠀⠀⠀⠈⠉⠙⠓⠀⣉⡒⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠎⠀⠀⠀⡀⠔⣠⡴⣚⣁⣀⣀⠠⢄⠢⠁
⠀⠀⠀⠀⠈⠉⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡰⣡⡎⠀⠀⠀⢀⠞⠐⣛⡩⠭⠉⠂⠀⠀⠁⠀⠀⢀⠠⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠺⢿⣛⡀⠀⢀⣤⣤⠄⠀⠘⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⣉⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⣏⣼⡇
⣿⣿⣿⠿⠛⠉⠁⣨⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⡄⠀⠈⠉⠉
⣿⣿⣅⣀⣀⣤⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⣿⡟⢿⣿⣥⡘⠳⣮⡻⣿⣧⡀
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡻⣿⣆⠈⢿⡜⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡅⠈⠉⠉⠁⠒⠤⣀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠠⣤⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣦⣶⣄⣸⣷⣶⣤⣈⠙⠷⢏⠀⠀⣠⣀⣤⣤⣄⡙⠿⢵⣦
⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣭⠀⠀⠐⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⢈⣅⣄⣸⣿⣿⣿⣿⣷⣒⢤⣄
⠀⠘⢿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠙⠻⠖⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠢⠀⠀⠻⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⡛⠒⠄⠢⡄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣠⣤⠄⢠⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣄⣠⣤⣤⣤⣤⣦⣶⣤⣤⣤⣄⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⣦⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢹⡇⠀⣿⡇⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⣶⡀⢸⡇⠀⠈⠁⠀⠰⠿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣾⣿⣶⣤⡀⠀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡇⠀⠾⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⡿⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠻⠿⠛⠋⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣠⡄⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⡇⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢀⣿⣿⡇⠸⣧⡀⠀⢀⡀⢀⣀⣤⣤⣬⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣾⣿⣿⣷⣆⠙⣿⡀⠀⠙⠛⠛⠛⠁⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠲⢶⣶⣶⣿⡿⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠘⠿⣿⣿⠿⠋⣰⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⢀⣈⣋⣀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣾⠟⠛⠛⠛⢿⣦⡉
⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⢀⣀⣀⢠⣿⠃⠀⠀⠀⠀⠀⢻⣷⢀⣀⣀⡀
⠀⠀⠸⠿⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⢸⣿⠀⠀⠀⠈⠉⠉⠘⣿⡆⠀⠀⠀⠀⠀⣸⡿⠈⠙⠉⠁
⠀⠀⢀⣀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣙⢿⣦⣤⣤⣤⣾⠟⣁
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⠀⢸⣿⠀⠀⠀⠀⠀⢰⡿⠋⠀⠈⢉⣍⠉⠁⠈⠻⣷
⠀⠀⢸⣿⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⢸⣿
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⠀⣼⣿⡀
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⣾⡟⠹⣷⡄
⠀⠀⢸⣿⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⢛⣿⡏⠀⠀⠹⣿⡄
⠀⠀⢸⣿⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⣼⣿⡀⠀⠀⠀⣿⡇⠀⠀⢀⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⢿⣶⣶⡾⠟⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⡟⠀⠀⠀⢀⣾⡏
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠘⠿⠿⠿⠿⠿⠟⠀⠀⠘⠿⠿⠿⠿⠿⠟⠀⢸⣿⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢻⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢠⣿⠏⠀⠀⠀⢠⣿⠏
⠀⠀⢸⣿⠀⢰⣿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠸⠿⢿⣷⠀⢸⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⣻⡧⠀⠀⠀⠀⣹⣷⠀⠀⠀⠀⢹⣷
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠋
⢀⣤⣼⣿⣦⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⣿⣷⡀⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣯⣴⣿⠛⠟⠛⠁⠒⠀⢀⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠻⠀⠀⠀⠀⠀⠀⣀⡒⣒⠚⠁⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠤⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⣀⣀⣠⡄⠀⠀⠀⠨⣥⡄⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⣤⠀⠀⠁⠈⠀⠀⠒⣂⣿⡇⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠂⠀⠀⠀⠀⢦⣹⣃⠀⠀⠀⠀⣠⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡌⠀⠠⠀⠐⢾⣻⡷⠀⠀⠄⠰⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⡄⢀⠀⠀⢙⣷⣄⡀⠄⠀⣹⣿⡇⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠙⢄⠀⠶⠻⣿⣿⡆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠆⠻⢶⣿⣿⣿⠃⢀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⡀⠈⠉⠛⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣰⠓⠚⡶⠿⢋⡶⣯⣷⣟⣣
⠒⠒⡾⠿⠿⠻⠿⠻⠿⠻⢟⠳⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣿⡿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣯

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⡿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠙⢋⣴⣟⢫⢻⠀⡏⣸⠋⣽⣦⣀⡁⠁⠙⠋⢉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠋⠁⠀⠀⣿⣷⢰⡿⣿⣸⢧⠏⣸⠏⢘⣿⡄⠀⠀⠀⠉⠀⠉⠛⢿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣋⡁⣠⡶⠿⠛⠁⠀⠀⠀⠄⠀⣿⣿⣾⢳⣷⣿⠟⣴⠋⣴⠟⣽⣿⠂⠀⠀⠀⢄⡀⠀⠀⠚⠛⠷⣦⠐⣉⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⠟⠄⠀⠀⢄⠁⠀⠀⢀⣿⣿⣿⣿⡿⣫⡞⣱⢞⣥⣼⡿⢿⣄⠀⠀⠀⠀⠀⠲⠄⠀⠀⠞⢯⡟⠛⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠏⠀⣀⣈⢈⣀⣀⣀⣀⣀⡑⠲⣶⣾⣟⠿⣽⣿⣿⣿⣿⣡⠾⣛⣽⣶⣾⡟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢸⣦⢀⠘⠿⣿⣿⣿⣿
⣿⣿⣿⣍⢀⣴⡟⠀⠀⠀⡀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣾⣷⠀⣶⣿⣦⣄⠀⠀⠀⠑⠀⠀⠀⠀⠈⣷⡄⢈⣽⣿⣿
⣿⣿⠏⠊⡍⠋⠉⠀⠠⠂⠀⠀⠀⠀⠀⠀⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡏⠀⣠⡿⠸⣿⣿⣶⣄⡀⠀⠀⠀⠢⠀⠈⠉⠫⡈⠋⢿⣿
⠿⢿⣤⣾⡀⣀⡀⠊⠀⠰⣤⣴⣄⠴⣵⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡿⠉⠀⠀⣿⡇⢰⣿⣯⣙⡿⣿⣦⡀⠀⠀⠈⢂⣀⡀⣸⣦⣼⠿
⠐⠉⠉⠉⠚⠛⠑⠀⠀⠀⢸⠿⠏⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⡇⢀⣤⣿⣿⢁⣾⣿⣛⣿⣷⣶⣽⣿⣶⣄⠀⠀⠉⠛⠋⠉⠉⠰
⡀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠆⢀⣠⣶⣾⣿⣇⢿⡿⠿⢿⣿⣿⣿⣷⣿⣿⣟⠀⠀⣿⣿⠟⣾⣿⣿⡿⠛⠁⠀⠈⠉⣿⣼⣷⣤⡀
⠀⠉⠒⠠⠀⠀⣀⣴⣟⣯⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣿⣿⣻⣿⣤⣀⣿⠇⣠⣾⣿⣇⣀⣀⡦⣶⣶⣧⣿⣿⢛⣙⢻⣄⠀⠐⠚⠁
⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⢟⣉⣨⣿⡄
⠀⠰⢾⣻⣉⣁⣀⡀⡀⠀⠀⠀⣿⣁⣀⡀⠉⠙⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣧⣼⡿⠋⣙⡇⠀⠐
⠀⢰⠌⣉⣥⣤⣤⠼⣤⠤⠀⢀⠹⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢿⣻⣿⡟⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⠟⣿⣿⣿⣧
⠀⠘⣷⣬⣼⣶⠿⢿⣿⣿⣯⣠⣴⡖⠉⠉⢸⣷⡀⠀⠀⢀⣠⣾⣿⣿⡃⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣥⡾⣯⣶⣿⣿⣿⣿⣬⣦⣀
⠀⣨⡿⣽⣿⣁⣀⣀⣾⣟⣃⣼⣿⡇⠀⠀⠀⢿⢷⡀⠰⣟⣿⣿⣿⣿⢃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣡⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⠿⠟
⠀⠹⣿⣓⣛⣚⣛⣛⣛⣚⣓⣳⣷⣶⠀⠀⠀⠈⠃⠁⠀⠀⠉⠉⣡⡿⣯⣯⣝⣿⣿⠀⠀⠀⠈⠹⣿⣿⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠸⠁⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠒⠒⢆⡀⣀⠀⠀⠀⠀⠴⡀
⢸⣄⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⠀⠃⠉⠠⡀⠈⠓
⢻⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠀⠃⠀⠠⡀
⠘⢡⣶⣳⡄⠠⠂⠀⠀⠀⠤⣴⣶⣶⠚⠛⠉⠁⠀⠀⠀⠐⣿⣷⢀⡀⠀⠀⣂⡃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠒⠀⣄⣀⠀⠀⠀⠈⠉⠙⠛⠲⢾⣿⣶⣴⣦⡄⠀⠀⣀
⠴⠂⠀⠀⠀⠀⠀⣀⣀⣠⣄⣐⣦⡤⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠁⠀⠀⡀⠀⠀⠀⠀⠒⠂⠄
⢀⣠⣤⣶⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡞⠋⠁⠀⠀⠀⢀
⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣿⣷⣦⣤⡀⠀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⣤⣤⣶⣷⣶⣿⣶⣶

⠀⠀⠀⠀⡆⢀⡇⠀⡆⠀⠀⠀⠀⠀⠀⢀⠄⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⡀
⠀⠀⠀⠈⡇⠀⣇⠀⣇⠀⠀⠀⠀⠀⠀⡏⠀⡏⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠰⣇⢠⡇
⠀⠀⠀⠀⢻⡄⢹⡄⢻⡄⠀⠀⠀⠀⠀⠹⣆⠹⣆⠈⠛⢦⡀⠀⠀⠀⠀⠀⢸⠃⣼⠃⡷⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡆⣷
⠀⠀⠀⠀⠈⣇⠀⣧⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⣾⢡⡟⢠⡏
⠀⠀⠀⠀⠀⡏⠀⠟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢳⠀⠇



⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⡞⠀⡏⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢳⡠⡏⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼
⠀⠀⠀⠀⢷⡀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠈⢷⠈⢷⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⢯⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡄⠻
⠀⠀⠀⠀⠀⢳⠈⢷⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⠈⣧⠘⠇⠀⠀⠀⠀⠀⠀⠹⢀⡞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠹⣆
⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡼⠁⣼⠀⢻⣄⡀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⢰⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⡀
⠀⠀⠀⢧⡀⠘⠷⣄⠈⠙⠆⠀⠀⠀⠀⢻⡄⠸⣆⠀⡼⠁⠀⠀⠀⠀⠀⠰⡜⣆⠳⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡷
⠀⠀⠀⠈⠻⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⡆⠁⠀⠀⠀⠀⠀⠀⠀⢳⣌⢷⡙⣆⠇⠀⠀⠀⠀⠀⠀⢸⡀⠙⣆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠸⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⡀

⠰⢶⣦⣤⣤⠄⠀⠀⠀⠀⢢⡑⢄
⠀⠀⠈⠙⠳⣤⡀⠀⠀⠀⠀⠙⢦⡁
⠀⠀⠀⠀⠀⠈⠙⢦⣌⠻⣶⣤⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⡶⠘⢧⡀⠀⠀⠀⡠⢀⣠⠀⢼⣏⡁⠇⠦
⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢻⣄⠘⠁⢀⢠⠊⠐⠓⠳⠀⠀⠈⢉⠁⡂
⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠳⡙⣦⠀⠛⠒⣒⠀⠉⠭⠄⢠⡴⢋⣾⣯⡍⡄⣸⠆
⣿⣿⣿⣿⣦⣄⠀⠀⢀⣴⠋⠘⠛⠂⠈⢎⢧⡀⢹⡄⠀⣀⣀⣀⡜⣹⠟⠋⣹⠟⣰⠏⠀⢱⡄
⣉⠻⢿⣿⣿⣿⣷⣤⡀⠁⠀⠀⠀⠀⠀⠀⠣⣳⡨⠋⠈⠈⣹⣿⡟⣡⡄⣰⣫⠞⠁⣤⣧⡈⠳
⠭⠭⠶⣬⣙⠛⠿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢣⢢⣴⠾⣛⡿⠿⢉⡾⡿⠁⣾⣀⠛⠻⠯⣳⠄
⣶⣶⣶⣤⣤⣬⣅⣀⣉⠛⠛⠿⢷⣦⣄⣠⡴⠖⠀⢏⠀⢰⣿⡠⠞⠩⠾⢧⣤⣤⣌⡉⠓⢦⡈⠁
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠻⠶⠶⠮⠥⠶⠦⠄⠈⠀⣾⣫⣴⣶⠤⠀⢰⣤⠈⢹⣿⣦⠢⢹⡄
⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⡀⠀⠀⠀⠐⠒⣶⠾⣭⣥⣴⣾⡿⠟⡀⢠⡇
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠝⠋⠀⣰⠁⢸⡀⢀⣤⣾⣿⣿⡶⠂⠀⠤⠤⠤⠤⠼⠁
⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⢰⡇⠀⢰⡇⠈⢻⣿⣿⠟⠀⠀⠀⢰⠂⣀
⣿⣿⣿⡶⠦⣖⣢⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⢻⡇⢀⡾⠁⠀⢈⢿⣿⡀⠀⠀⠀⠀⢃⡿⢷⠁
⠤⣴⡶⠖⢛⡭⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢌⣀⠁⠀⠀⠀⠀⠀⡆⢿⠛⠷⠶⠦⣤⡤⠉⠂
⡋⣅⠶⠛⠉⠀⠀⠀⠀⠀⢀⣴⣿⣇⠀⠀⣀⢤⠀⠀⡰⣄⠀⢀⣷⢸⢧
⠊⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡷⢚⡵⡏⠀⠀⢳⡙⠷⡿⠏⢀⣇⢇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⢀⡴⢋⠎⠀⠀⠀⢈⢻⣤⡀⠀⣸⣿⣬⣆
⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⣠⣾⠟⡴⢋⡾⠀⠀⢸⡜⣆⢹⣅⣘⣀⣼⠄⣿⣄
⠀⠀⠀⢀⣴⣿⡿⠃⠀⣠⣾⠟⣡⠞⣱⡿⠁⠀⠀⠀⣷⣹⡶⠈⠷⠶⠏⠀⡤⣦⠤
⠀⢀⣴⣿⡿⠋⠀⣠⣾⠟⢁⡴⢋⣼⠯⠁⠀⠀⡇⡆⢿⣷⢿⣄⢄⠀⠀⢠⡇⣿⢀
⣶⣿⠟⠋⢀⣤⣾⠟⢁⣴⠏⣴⡿⡓⠁⠀⠀⠀⡇⣇⠸⡿⣎⢿⣦⡙⠻⠛⠀⣿⡙⡀
⠋⠁⣠⣴⡿⠋⣡⡴⠛⣡⣾⣟⠅⠀⠀⠀⠀⠀⣿⡹⠀⢻⡹⡤⡹⣿⣶⣤⣼⣿⠀⡇
⣴⡿⠟⢁⣤⡾⢋⣤⣾⡿⡫⠂⠀⠀⠀⠀⠀⠀⡏⢰⠀⠘⣧⠹⡜⢌⠻⣿⣿⠟⢸⡇
⢁⣤⡾⠟⣡⣴⣿⣿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⠀⠀⠸⣧⠹⣮⠳⣤⣀⣠⡾⠁⢠⡀⠟⢂⣀⣄
⠛⣉⣴⣾⣿⣿⢟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡇⠀⠀⠻⣧⠈⠳⣌⡙⠛⢁⡖⢸⡇⠚⠉
⣾⣿⣿⣿⢟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⢸⡀⠹⣷⣄⠈⠛⠿⠛⠁⣸⠁⢴⡫⢭⣭⣿⣷⣶⣶⣶⣦⡀
⣿⡿⡋⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⢸⡆⠀⢷⠀⠹⣿⣷⣄⡀⠀⣠⣿⠀⢠⠞⠛⠛⠛⠻⠿⣿⣿⣶⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢠⠄
⠀⠀⠀⠀⠀⢦⠀⣇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⡃⢸⠀⢻⡀
⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠻⣦⠙⢷
⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡟⣸⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡞⠁⡼⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠜⠁⠜⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠡⡎⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡇⠀⢆
⠀⠀⠀⠀⠀⢸⡄⢠⡏⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⢋⡵⠲⣄⠀⠀⠀⠀⠀⠀⠃⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⣼
⠀⠀⠀⠀⠀⠘⣇⠸⡇⢰⡏⠀⠀⠀⠀⠀⠀⣠⠶⣄⠈⢳⠈⢯⠒⠃⠀⠀⠀⠀⠀⠀⢰⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠧⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⡄⠳⠘⠁⠀⠀⠀⠀⠀⠀⠙⠾⠈⠇⠈⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢠⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄
⠀⠀⠀⠀⠀⡏⠀⣸⠈⣧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡇⢀⡄
⠀⠀⠀⠀⠀⢻⡄⠁⠀⣸⠀⣹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠏⣠⠞
⠀⠀⠀⠀⠀⠀⢹⠄⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢈⡇⠈⡇⠀⠀⠀⠀⠀⠀⡇⠀⡾⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠣⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠿⢿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣬⣿⣵⡶⢾⣷⣶⡯⣿⣿⣶⢛⣿⣷⣶⣼⣷⣾⣿⣷⣶⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⢛⣫⡍⢉⡙⢻⣿⣽⣾⣽⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢿⡿⢯⣝⠛⠛⢋⢹⡃⣨⠑⣷⢩⡿⢉⢻⡟⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠙⠀⢸⢟⠿⢫⢹⢸⡇⢸⠀⠻⠾⣷⣴⢸⡁⢏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢉⣌⡇⢰⣦⡆⢰⣝⣼⣽⠴⠛⠁⣠⡄⠻⠿⡂⢚⣗⢛⣛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⣛⣀⣤⡭⠁⣈⣁⣀⣠⣴⡾⣶⣿⣿⣿⣿⡷⡞⣿⣦⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⠃⠊⠻⣿⣿⣯⣵⣦⣙⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣏⣿⣿⣿⣿⣿⠿⣿⣿⣿⢠⠀⣸⣿⣿⡿⠿⠿⣟⣳⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟⣿⣿⣿⣿⡏⠙⠺⣿⣿⡼⢀⡙⢿⣿⣷⣾⣧⠊⠑⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⡇⠀⡦⣩⣙⣷⣦⣄⠀⠈⣿⣿⣿⠊⢦⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣧⣄⣗⣿⣿⣿⣿⣿⣿⣶⣦⡍⣿⠀⣉⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣟⡯⣿⣿⣽⡟⡛⠿⣿⣿⣿⣷⣌⣄⢶⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⣿⠉⢩⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣧⣿⣿⣿⣿⠿⣮⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢦⣽⣋⣾⣸⡟⢸⣷⢿⣿⣿⢿⣿⣇⣿⣿⣿⢿⡿⣿⣳⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡉⠉⣹⡰⢤⢊⣫⡜⢸⣽⢸⣿⣿⣷⣾⣾⣧⣴⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠛⠈⠀⠁⡼⢴⣿⣶⡎⠀⠀⠀⠈⠉⠉⠉⠉⡙⢻⣯⠟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⠄⣀⠀⣀⢙⠸⠛⠋⠁⠀⠀⠀⠀⠀⠀⠠⠈⠀⠊⠁⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⣿⡿⣿⡿⢿⡿⡟⠚⢻⣇⣴⣶⠴⡶⢒⠿⢴⡁⢀⡠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠠⢉⡇⣰⣏⠛⣠⣧⣾⣏⣿⢾⣤⣿⣽⣿⣷⣷⣯⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠟⠻⠛⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠒⠀⠀⠀⠀⠀⠀⢀⡠⢄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⢀⡀⠀⠀⠰⠀⠠⠂⠀⠀⠀⠀⡐⡡⠊⠁⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠀⠉⠉⠀⠀⠈⠒⠤⠄⣀⣀⡀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⠟⠙⢧⣀
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⠀⢂⠂⠀⠀⠆⠀⠀⠀⠀⣽⣄⠱⣄⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠉⠐⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢰⠀⠀⠀⠛⢦⡜⠀⠀⠀⢠⠞⡡⠊⠳⣤⡑⠢⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠓⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡀⠜⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠞⠁⠀⡠⢊⡽⠓⠚⠁⠀⢀⠄⠠⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⢀⢞⡴⠋⠀⠀⠀⠀⠀⠸⡀⢀⠆
⠀⠀⠀⢰⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀⠀⠀⣀⡴⢋⠔⠁⠀⠀⡐⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠓⠐⠁⠀⠀⠀⠐⡇⠀⠀⠒⠒⠒⠂⠀⡦⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠉⢁⡰⠁⠀⠀⢠⢊⡞⠁⠀⠀⠀⢀⠤⠀⠐⠂⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⢄⡀⠀⠀⢀⡠⡺⠁⠀⠀⢀⡼⠃⠀⢀⠔⠒⠒⠒⠒⠒⠁⠀⠀⠀⡔⢡⠏⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠑⠈⠀⠀⠁⠊⠀⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡠⠒⠁
⠀⠀⠀⢀⠔⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢷⡀
⠀⠀⠰⡁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⠎⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢳
⠀⠀⠀⠈⠢⠄⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠉⠢⡀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⡠⠀⠀⣸⠁⠀⠀⢀⢐⠌⠁⠀⠡⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡠⠃⠀⠀⠀⠈⠑⠠⢄⡀⠀⠀⠀⠀⠀⠑⣤⠚⠀⢀⡴⠋⠀⠀⠀⡑⠁⠀⠀⠀⠀⡀⠿
⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⢀⡾⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡨⠋⠀⢀⡴⠋⠀⠀⠀⠀⢰⡇⢀⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠸⡀⠀⡐⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠉⠈⠀⣠⠞⠁⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠋⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠟⠢⡀⠀⠀⠅⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡴⠋⠀⠀⠀⢀⠃⠀⠀⠀⠈⠢⡀⠀⠀⢰⠀⠑⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣠⡴⠋⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢠⠁⠀⠀⠸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⡄⠈⠐⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠀⠉⠁⠀⠀⢀⠜⡄⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⢄⠀⠀⠈⠢⠀⠀⠀⢀⠔⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⠄⡀⠀⠁⣀⠔⠁⢀⡴⠞⠁⠀⠀⠀⠀⠀⢰⠒⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠈⠁⠈⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⠀⠉⠐⣠⠄⠀⠀⠀⠀⠀⠣⠊⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠉⠀⠀⠀⠀⠔⠈⠐⠀⠀⠀⠈⠢⡑⠀⠀⠠⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁

⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣶⠶⡤⠒
⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢐⣈⠅⠀⣠⣄⡩⠿⠯⠊
⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⠀⠀⠀⣆⢰⠸⠛⠾⠦⠄⡢⡽⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠂⣀⠀⢋⠏⠀⠀⣤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠳⣆⠀⠈⠃⡀⢠⣿⠁⠀⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⡀⠀⠀⠀⠹⡇⣠⣸⠟⠈⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠛⠁⠀⠀⠐⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠀⠀⠀⣔⣠⡄⢀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠠⠾⠁⠀⠀⠀⠀⡠⠒⠊⠉
⠀⠀⠀⠀⠈⠳⠇⠀⠀⢰⣦⡄⠳⠀⠀⠤⠊⠀⠐⠒⠾⠍⠉⠁⣉⠁⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠛⠛⠷⠄⠴⠿⠳⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⡙⠓⠂⠒⠫⠁⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣈⣉⣉⣉⡑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡡⠲⡙⡄⠀⡄⠇⠀⠉⢢⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡞⠀⠀⠱⡜⢦⡁⠆⡠⠔⡡⠊⢣⠘
⠀⠀⣀⣀⣀⠠⠤⠤⠤⠄⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠰⡇⢘⠀⠀⠲⠤⠸⠎⠁⠉⣤⣈⡀⣀⠀⡆
⡴⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠈⠉⠉⢙⢖⣠⢦⡀⡶⠦⠤⠨⠀⡇
⡇⠀⠉⠛⠒⠶⠤⠤⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⣧⠼⡤⢴⠞⡛⠛⠁⢸⡗⡌⢦⠀⠀⣰⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠠⡀⠀⠀⢢⢸⡁⠈⠢⣳⠔⢁⠜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢈⠑⠂⠘⢼⡇⠒⢊⣡⠔⠉
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢺⠃⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⡆⡞⠙⠲⠶⠶⠖⠚⠋⠉⣉⢳⡦⣄⡀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠋⠉⠛⠲⣄⠀⡇
⠀⡇⣇⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠘⡄⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⢇⢸⠀⠀⠀⠀⡎⠀⣠⠔⠚⡄⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁
⠀⢸⠸⡀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠆⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾
⠀⢸⡀⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢈⢶⠋⠉⠉⠉⠀⠈⠱⣄⠀⠀⠀⠀⢸⡇⡏
⠀⠀⡇⢇⠀⠀⠀⠀⢦⠢⡀⢀⡠⠔⠊⢸⠀⠀⢰⠒⠒⠒⠀⠈⢣⠀⠀⠀⢸⢸⡇
⠀⠀⣇⢸⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⣸
⠀⠀⢸⠘⡀⠀⠀⠀⠀⠀⠀⡴⠋⠒⠤⣈⢆⡄⠀⠀⠀⠀⠀⢰⢸⠀⠀⢠⡇⡿
⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⡼⠀⠀⣦⡀⠀⠉⢻⠦⠄⠀⠀⠀⠨⠞⠀⠀⢸⢡⡇
⠀⠀⠀⡇⢳⠀⠀⠀⠀⠞⠀⠀⡜⠀⠈⠓⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⢱⠸⠀⠀⠀⠀⠼⠀⠈⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡇⡿
⠀⠀⠀⢸⡄⡇⠀⠀⠀⠈⠧⡀⠀⠀⠀⢀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡇
⠀⠀⠀⠀⡇⢷⠀⠀⠀⠀⠀⠈⠑⠢⠄⠡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⢷⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⢸⡌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠃
⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼
⠀⠀⠀⠀⠀⢹⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢠⡇
⠀⠀⠀⠀⠀⠈⢧⣌⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⢋⣠⡞
⠀⠀⠀⠀⠀⠀⠸⡏⠓⢦⣄⣉⡉⠙⠛⠛⠛⠋⣉⣉⣠⠴⠖⠋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣲⣂⣤⣤⣤⣤⣤⣤⣲⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⡀
⣿⣻⡻⢿⣿⡳⣶⢦⡲⣤⠠⣤⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣤⢤
⢿⣿⣿⣷⢮⡪⢶⣕⣿⡷⣯⣴⣻⡙⢶⣚⠯⣅⡻⣍⡲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⢝⢕⣵⣵⢟⣵⣿
⡕⡵⡽⡵⡵⡝⡕⡝⣝⢮⡳⣌⠳⣍⠳⣌⡳⢮⣙⠦⣝⢮⡳⣕⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⣚⢕⣗⢿⣵⣿⡿⣿⣻⣮
⡳⣵⣵⣕⡑⢕⣽⣝⣮⣳⣝⣦⡽⣊⣽⣶⣟⡥⢖⡯⢗⣵⡽⡮⣝⡪⠭⣽⣖⡒⣶⣖⢒⡺⢭⣝⡳⣽⣿⣻⡧⠒⠈⣻⢽⣿⣿⡾⣿
⠽⣟⣴⠭⣽⣒⣲⠾⢭⣙⠯⣝⢪⡹⡳⣽⣿⣪⢕⡪⢕⡿⣫⣺⢵⣻⣯⣽⣿⣿⣧⡿⣯⣿⣖⢿⢛⣷⣾⣷⢾⣮⡮⣖⣵⣝⣯⣾⡫
⣭⡓⠮⣭⠓⣶⠌⣿⠂⣿⠃⣿⢸⡇⣇⣗⡾⣿⢕⣪⠵⣫⣟⣷⣯⣿⠷⠶⢾⣿⣿⣻⣿⣿⡝⣵⢿⡿⡻⢞⠵⣫⠞⣵⢋⡴⢋⡴⢫
⡆⢹⢆⡾⣡⠏⣴⠃⣸⡇⢸⡇⢸⣇⢻⡘⢜⡾⣯⣾⣿⣹⣽⣿⡿⣷⣛⣛⣛⣻⣷⣿⣿⣿⡿⢗⡽⣫⠖⣡⡾⢃⣾⠃⣼⠇⣼⠁⡿
⡱⣫⢎⣼⠃⢸⣿⠀⢿⣧⠈⢿⣄⡙⠦⣝⣒⣯⠷⠾⣉⣭⣕⣋⣴⣿⣵⣒⣻⣯⡿⡿⣝⠯⣚⡵⢚⣥⠞⣋⣴⠟⢁⣼⠟⢠⣿⠀⡇
⢱⣇⠈⢿⣦⡈⠻⢷⣤⣉⣛⣲⣦⣭⣭⣷⣚⣛⣣⣭⣽⢷⠶⠾⠥⠼⠭⢿⣛⣿⣛⣒⣚⣋⣳⣯⣭⣒⣚⣩⡤⠶⠛⣁⣴⡿⠃⣰⡇
⣄⡛⣷⣤⣍⣙⣛⣒⣒⣒⣒⣒⣒⣒⣒⣚⣛⡿⠿⠿⠿⠿⠯⠭⢭⣭⣭⣭⣵⣖⣒⣒⣒⣶⣶⣿⣶⣶⣖⣒⣒⣛⣩⣭⣥⣶⠞⢋⣠
⣟⠉⣹⣋⣛⣛⣛⣻⣟⣟⣛⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣻⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⣱⢸⠀⠀⠀⠀⣱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣋⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⡟⠀⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⢈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣽⣆⡀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⢺⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣿⣧⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢹⡾⣿⡿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣗⡿⣾⣿⣄⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣿⡿⣿⣆⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠈⠛⣷⣞⣛⣿⣿⣿⣛⣶⣤⡀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠻⣷⣄⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠈⠀⠉⠀⠈⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡗⠄⠀⠀⠒⠀⠒⠀⠀⠀⠀⠀⢘⢙⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣧⠐⠂⠀⠀⠀⠀⠀⠀⠀⠃⠂⠂⣳⣿⠿⠛⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⢠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢠
⠀⠀⠀⡇⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣧⠀⠀⡠⠃⠀⠀⠀⡰
⡀⠀⠀⠈⠢⣀⡀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉⠁⠀⣿⠤⠊⠀⠀⠀⡠⠊
⢣⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⡇⠀⠀⠠⠖⠋
⠀⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠠⠶⠒⠒⠒⠈⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢀⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⣀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠦⠄⠀⠀⠀⠀⠁⠀⡟⠀⠀⠸⠀⠀⠀⠀⠒⠒⠒⠒⠒⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠋⠁⠀⢀⣠⠴⠒⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⣤⣄⡀⠀⢀⣀⡀⠀⠉⠛⠻⣦⡀
⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠠⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⠞⠋⠀⠀⠀⢀⣀⣤⣶⠿⢃
⠀⢀⠙⠳⣶⣶⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠾⣿⠟⠛⠉⠀⠀⠈
⠀⠈⠉⢉⠱⠿⢤⣭⣉⣉⣙⠛⢛⣛⢛⠓⠛⠛⠋⠀⠀⣓⠒⣛⠛⠛⠛⠛⠛⠋⣉⠉⠉⠉⠁⢀⡄⠀⠀⠀⠈⠀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⣾⠀⣀⠀⠂⠈⠉⠉⠘⠛⠛⠛⠛⠋⠉⠁⢀⠛⠉⠉⠉⠀⠀⠀⠀⠔⠊⠉⠑⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠚⠛⠋⢹
⠀⠀⠀⠸⠀⠈⠙⠛⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠾⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠶⢀⢸⡇
⠀⠀⠀⠀⠀⣿⣎⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠀⠀⠀⠀⠀⣠⡴⠶⠾⠶⢤⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢻⡏⠓⠤⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠈⠀⠀⢐⡒⣤⠶⢷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠖⠒⠒⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠈⠃⠌⠉⣉⠝⠋⠉⠉⠉⣉⡉⠉⠙⠛⠒⢤⡀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡼⠃⠀⠀⠤⠀⠀⠈⠉⢄⡀⠀⠀⠀⠉⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠸⠀⠀⠀⣧⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣤⣤⡤⠐⠂⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⡠⠤⠐⠛⠀⣀⣉⠤⠤⠤⠤⡴⠒⠓⠀⠀⠀⠠⠠⠀⠈⢁⡈⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⠀⢾⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⣶⣄⣀⠀⠈⠛⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠘⠷⣤⣤⣀⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠦⠤⢠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠗⠂⠈⠒⢀⣦⡀⣠⣄⣀⠀⣀⣤⠤⠁
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡵⢂⣨⢷⡞⠉⠇
⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠳⢬⡷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠛⢳⡤⢀⡐⢢⣠⣾⣶⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠟⣿⣿⣿⣿⣦⠞⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣾⣿⡉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠟⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠠⠈⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣯⣿⠀⠀⠀⠀⠀⠀⠀⢤⠶⠃
⠀⠀⠀⠀⠀⢀⣀⡠⢠⢀⢠⣶⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣀⡼⢶⠂⠀⠀⠀⠀⣴⣶⣤⡴⡍⢲⡄
⠀⠀⠀⠀⠠⠊⣞⠐⠋⠉⠉⠁⠚⠉⡟⠿⠿⣿⣿⣿⠷⣿⠉⠀⠀⠀⠈⠋⠁⠀⢀⠀⠀⠀⢀⢀⢸⠏⠉⢼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠟⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠳⠤⠼⠃⠀⠺⠏⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⣀⣀⡀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡾⠉⠀⠙⣧⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠆⠀⢀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣇⣀⣀⣀⣸⠀⢸⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡷⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⡿⠇⠀⣿⣿⣿⣿⣿⠀⠸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠆⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣼⣿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠘⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣷⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣄⡀⠉⠉⠉⠉⠉⣀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣶⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⣠⣶⡦⠀⠀⠠⠄⠠⠤⠠⠀⠤⠠⠄⠀⠄⠆⠀⠀⠄⠀⢠⣦⡠⢀⣤⣀⣀⡀⡀
⠀⠻⣿⡆⠀⠀⠠⠠⠀⠀⠀⠠⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠻⠐⠟⠱⠿⠯⠿⠿⠶⢶⣤⣆⣀
⠀⠀⠈⠀⠀⠈⢰⢲⣠⢦⣀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⢀⠀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠒⠒⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠆⠈⠀⠀⡀
⠀⠀⠀⠀⢀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⣆⣰⣀⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢛⠈⠁⠁⠁⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠟⠻⠘⠃⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⡤⠀⠀⠄⠈⠀⠀⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡶⠩⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠒⠞⠀⠀⠀⠠⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠋⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠗⠒⣀⠀⠀⠀⠀⢀⣾⣄⠄⠀⣀⠀⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢤⡤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠈⠀⠈⠉⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣶⣶⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⣀⣠⣤⣤⣤⣤⣴⣹⣿⣿⣿⣿⣿⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡏⠉⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣧⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣛⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⠛⠒⠒⠶⠶⠶⠒⠚⣛⣿⣿⣾⣿⣿⣿⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠐⣿⡏⠉⠀⠀⠀⠀⠀⠉⠉⠙⠲⢽⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠼⡇⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠢⡰⢶⣦⣀⡀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡶⠹⠿⢁⣤⢷⡀⢀⣀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠷⡄⠛⠋⢁⢀⡝⣮⣽⠟⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⡀⠀⠹⣿⣃⠁⣭⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢿⡇⣚⣷⣎⣍⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢶⢸⡧⠙⢃⣰⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣚⣓⣇⣈⣉⣁⣈⣉⣑⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠈⠉⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⡭⣽⣖⣺⡿⠿⠭⠭⣁⣀⣀⣬⣭⣽⣶⣮⢝⡿⣦⣀⠀⠙⢷⣦⣀
⠝⠉⠁⠀⠀⡀⠀⠀⢠⡄⠠⣄⠀⡈⠉⠛⠷⣭⡪⡻⣷⣄⠀⠹⣿⣄
⠀⠀⣁⣩⣤⣴⠶⠶⠶⠶⠶⠶⠦⣤⣤⣀⠢⠈⠸⣙⣜⢿⣷⣄⠈⢿⣧⡢⡀
⢛⣉⠭⠤⠔⠒⠒⠒⠒⠒⠲⠶⣦⣄⠈⠙⢷⡄⢀⠸⣫⢣⡹⣿⣷⣦⣿⣿⣔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⣴⣶⣶⣿⣿⣿⣷⣶⣄⠀⠀⢈⣿⠀⠀⣸⣿⠀⣆⠱⡑⡙⢮⡻⣿⣿⣿⣿⣮⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⡇⠀⣼⡟⣫⣼⠸⡄⠙⣎⢢⡙⢮⡙⢿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣵⡫⢖
⡀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣄⠈⢧⠀⢿⣸⠁⠀⠀⣧⠀⠈⢳⡝⢦⡙⢦⣙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣪⠞⡩⠞⣉
⣿⣿⣿⣶⣤⣄⡈⠙⠻⣿⣿⣿⣷⣌⠣⡘⣿⠀⣿⣶⣿⠀⠀⠀⠙⢦⡙⠳⣌⠳⣝⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⢔⠐⠁⣨⠖⣫⡭
⠀⠀⠈⠉⠛⠛⠻⠿⠶⢦⣬⣉⣛⠻⣷⡁⠹⡄⡇⠀⠀⠀⠀⠀⠀⠈⢿⣦⠈⢳⣌⠳⣝⢿⣧⠀⠀⠀⠀⢀⣴⠗⢁⣴⣫⣢⡽⠶⢮
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣬⣭⣙⡲⣽⣆⢃⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠀⠙⠷⣌⡳⣝⢷⡀⠀⠀⠈⣡⣶⠿⢛⣩⣥⣤⣭⡀
⣿⡿⠟⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⢪⢻⡈⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠒⠂⠙⠒⠑⠀⠀⠀⠀⠒⠓⠁⠐⠒⠒⠦⣌⠻
⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠹⣇⢛⢧⡀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠶⡟⠋⠉⠁⠀⠀⠀⠀⣖⠯⣒⠲⢦⣄⡀⠀⠈⢣
⠙⠻⠷⣶⣤⣤⣤⠤⠤⠔⠒⠒⠒⠒⢄⢻⡆⢹⡘⣌⠻⠶⠦⠔⣒⢍⣢⢖⣡⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠻⣦⡈⠻⣄⠀⠈
⣭⣓⣒⣒⣒⣂⣤⡤⠶⠖⣛⡉⠉⢉⡲⡀⣿⡀⢷⡈⠙⠒⢚⣩⡴⢟⣵⠏⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣿⡄⢹⡆
⠉⢙⣛⣉⣉⣩⠤⠔⢊⡩⣴⡶⠿⠿⢆⠐⢸⣇⠪⠛⠷⣿⡿⢋⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣿⣿⠀⣷
⠉⠉⠉⠉⣀⠤⢐⡮⠗⣋⣵⣶⣿⣿⣦⠀⠈⣿⣦⣀⣈⣥⣾⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣿
⡖⢒⣋⣭⠴⠚⡡⣔⣿⠟⠋⠁⠀⠀⠙⡆⠈⡜⣿⣿⣿⠿⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏

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