Thinking face Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠻⠿⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣙⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀
⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣷⣶⣦⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⣼⡿⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⢀⣾⡟⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣧⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⣿⡇⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣿⠁⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣶⣶⣾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢻⣿⣿⠀⢠⣶⡿⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠘⣿⣿⡆⢸⣿⠀⠀⠀⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⢹⣿⣿⡘⣿⡄⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠉⠻⠿⠶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠛⠛⠛⠉⠉⠉⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣷⣶⣶⣶⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣛⣛⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣩⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣍⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⠿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⢊⣭⣍⢺⣿⣿⣟⣋⡥⠾⢿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⡘⠿⠟⣸⣿⣿⣿⡏⢼⣿⡇⢹⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣴⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⠟⣩⢹⣿⣿⡟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠟⣡⣾⣃⣚⣛⣛⣛⣛⡛⢦⡙⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣭⣭⢉⣭⣥⣾⣷⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠷⠶⢆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⣛⣛⣛⣛⣃⣼⣿⣿⣿⣿⣿⣿⡿⠟⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣭⣭⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⠉⠉⠉⠉⠈⠉⠉⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢾⠟⢛⠛⠻⡷⠀⠀⠀⠀⠀⣠⣴⣶⠿⠃⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⣠⣾⠿⠿⣷⣄⠀⠀⠀⠀⠛⠛⢉⣠⣤⣀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣸⣿⠀⠀⠀⠀⠀⢰⡿⠋⠙⢻⣷⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⠘⠿⢷⡾⠟⠃⠀⠀⠀⠀⠀⢻⣷⣀⣀⣼⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⣠⣾⠿⣿⡆⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⣼⡧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⡿⢃⣼⡟⠁⠀⠀⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⢗⣴⡿⠋⢠⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⡀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣿⠇⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⡯⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣈⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⣀⣀⣀⣀⣈⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠻⣵⣦⣤⣤⣀⣀⣀⣀⣤⣤⣶⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠾⠿⠟⠛⠛⠻⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠁⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⢀⣴⠾⠷⣶⡄⠀⠀⠀⠀⢰⡿⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠻⠃⠀⣾⠏⠈⢻⡆⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⣷⡄⠀⠀⠀⠀⠀⠀⣿⡀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠿⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢿⡆⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⡦⠀⠀⠀⣠⣶⠟⠉⢀⣴⡿⠃⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣴⡾⠿⢷⣦⡀⠀⠀⠀⠀⢀⣤⡾⠛⠁⣠⣶⠟⠉⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠙⢷⣄⠀⠘⢿⣦⡀⣀⣴⠿⠋⠀⠠⡾⠛⢻⣦⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⢻⣦⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⣤⣾⣧⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠁⠀⠈⠁⠐⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢰⠂⠀⠀⠠⢄⡀⠤⣀⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠁⠘⠀⠀⢄⠀⠀⠉⠒⣝⠢⣉⠢⢕⣦⡀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠺⠃⠂⠀⠀⢀⠀⠀⢄⢠⡉⠢⢄⠀⠈⠑⠢⢕⣄⠙⢷⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠤⡀⢀⠂⠷⣄⣢⡓⢝⠲⢤⣉⠒⢄⡀⠀⠈⠑⠦⡉⢧⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠃
⠀⠀⠀⠀⠀⢀⠃⡄⠀⠀⢰⠋⠑⠉⠐⠂⠥⠝⢻⠷⢾⣿⣯⡢⣈⠢⡀⠀⠀⠈⠢⣃⠀⠀⠀⠙⠢⢄⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⢸⡀⠇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣆⠈⠂⠀⠀⠠⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠴⠒⠋
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⣇⠴⠒⠲⠶⢤⡄⠀⠀⠀⢠⡴⠶⠒⠒⢿⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠉⠀⠀⠉⢒⡄
⠀⠀⠀⠀⠀⢸⠀⢃⠀⠀⡇⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠑⣄⠀⠀⠀⠀⢷⠀⠀⠀⠳⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⡎⡄⢨⡄⡇⢰⠙⠃⠘⠛⠀⠁⠀⠀⠀⠈⠀⠛⠓⠈⠋⠀⠙⢧⠀⢡⠀⠈⡀⢧⣀⣩⡄
⠀⠀⠀⠀⠀⢱⢸⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠀⠘⡆⠀⠇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣾⠀⢠⡇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⢀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⡿⢀⠏⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⢻⠀⢸⢰⠁⣻
⠀⠀⠀⠀⠀⠀⡇⢸⠀⢿⢻⡄⠀⠀⠀⠠⠀⣤⣤⣤⠀⣀⠀⠀⠀⠀⡏⣼⠀⣼⠃⡰⡃
⠀⠀⠀⠀⠀⠀⡄⠸⠀⠘⡘⣷⡀⠀⠀⠀⠲⢤⣤⡤⠖⠁⠀⠀⢀⢼⠁⡇⡰⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠱⠀⢣⡀⠇⡇⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢸⠀⡱⢁⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠇⠀⢳⠀⠃⡷⠃⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⢀⠁⠆⣾⠁
⠀⠀⠀⠀⠀⠀⢸⠘⡆⠈⡟⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⣼⣿
⠀⠀⠀⠀⠀⠀⡜⠀⣧⠀⢧⠇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢸⠇⢡
⠀⠀⠀⠀⠀⢰⠇⠀⡿⠀⡘⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⡄⠀⢣
⠀⠀⠀⠀⠀⡏⡇⢸⠃⠀⡇⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡄⠀⢁⠁⠀⠁
⠀⢀⣀⠠⠄⠛⢅⡘⠀⡘⠡⡀⠷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣿⠀⢸⢀⠀⡀⠲⠄⣀⣀
⠉⠁⠀⠀⠀⠀⠈⣡⣼⣡⡔⠱⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⢨⠌⡀⠀⠀⠀⠀⠀⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠞⠉⠈⠉⠙⣧⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢹⡿⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡜⠻⢦⣄⠀⠀⠀⢀⣠⠾⠋⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠛⠆⠲⠋⠀⠀⠀⡠⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠞⠁⠀⠀⢣⠀⠀⢸⡆⠀⠀⠀⠀⠀⡰⠁⣀⣀⣀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡞⠀⠀⠀⣠⣸⡆⠀⠀⠇⠀⠀⠀⠀⣠⠗⣩⣄⠀⠈⢳⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢸⠁⠀⠀⢰⣋⣿⣇⠀⠀⠀⠀⠀⠀⣰⠏⣸⠛⣿⡇⠀⢸⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⣰⡏⠀⢹⣾⣿⠇⠀⢸⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⣶⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⡿⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠀⠀⣼⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢹⢀⣠⡤⠾⠦⢤⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠈⠓⢄⣸⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⣀⠴⠖⠒⠉⠉⠉⠉⠁⠒⠢⢼⡀⠤⠤⠤⠤⠤⠏⣀
⠀⠀⠀⠀⠀⣠⡧⠤⠤⢀⡀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠉⠑⠤⡀
⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠈⠲⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠈⢻⡶⢤⣀⡀⠀⠀⠀⠀⠤⠴⠚⠁⠀⣀⡤⠴⠖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⢳⡀⠀⢀⣀⣀⣀⣀⣀⣀⡤⠖⠋⠁⠀⠀⠀⢀⡼⣕⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡅⠀⠈⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠠⠴⠒⠋⠁⠀⢀⡞⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠀⠀⠀⠀⠀⣀⣀⡤⠴⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⢛⣉⣩⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠛⠉⠁

⣿⠛⣿⠛⣿⠛⣿⠛⣿⠛⣿⠛⣿⠛⣿⠛⣿⢛⣿⢛⣿⢻⣿⢻⣿⢻⣿⢻⣿⢻⣿⢻⣿⢻⣿⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠰⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⣸⣷⡾⠷⠾⠧⣼⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⠟⠁⠀⠀⠀⠀⠀⠑⠸⠷⠾⢧⣼⣇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⠸⣿⡼⠟⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠈⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⢠⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⢸⣿⠞⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⣀⠿⣴⣿⣼⣿⣼⣿⣘⣿⣸⣿⣸⣿⢸⣧⢲⣶⢹⣿⢦⣤⣀⣀⣠⡤⢺⡗⢲⡶⢿⡟⢻⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⣀⠟⠊⠁⢁⡉⢩⣭⣍⣻⣿⣿⣿⠿⠷⢶⣯⠻⢿⠿⣼⡿⢸⣿⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠀⣿⣠⠟⠁⠀⣀⣤⠶⠶⠶⠾⠿⣿⣿⡿⢯⡻⣓⠢⢀⠁⠂⠀⠀⠙⠺⣿⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⢀⡟⠁⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⢧⡀⠀⠳⡀⠁⠄⠀⠀⡀⠀⠈⠻⣇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⠀⣿⠏⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠙⣆⠣⡈⠀⠀⠀⠀⠀⠄⠀⠠⠙⣾⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣿⡠⠃⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢣⡰⣄⠀⠀⠀⠀⠀⠂⠀⢠⢹⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⡟⠁⠀⠠⢰⠃⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠾⠿⠗⠶⢬⠢⣀⠹⣎⣧⡀⠀⡀⠀⠀⠀⠸⣾⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⡔⠀⠀⠐⠀⣾⠴⠶⠿⠿⢆⠀⠀⠠⠈⠁⣀⣤⣤⣤⣀⠀⠀⠈⢷⠶⣿⣷⠀⢱⠀⠀⠀⠀⣿⣇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠁⠀⠀⠀⣼⣿⣤⣶⢶⡶⡄⠀⠀⠀⠀⠘⠉⢀⣂⣬⠟⠀⠀⠀⠀⢳⢶⢿⣧⢸⠀⠀⠀⠀⢸⣇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⠀⠀⣸⣿⣿⠛⠦⠮⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⢸⣿⠀⢸⠀⠀⡆⠙⣾⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠦⠀⠀⢿⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠿⡀⢸⣇⠀⢻⡀⢈⠳⣼⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⠀⣿⠀⣷⢄⢘⡾⡇⠀⠀⠀⠀⠱⢤⠀⠢⠄⠠⠃⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢈⣷⡀⠈⠉⠛⠚⠦⡙⢆⠁⠸⣇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⡀⣿⠀⣿⠈⣿⡷⢻⡀⠀⠀⠀⠠⠀⢠⠈⠂⠠⠤⠀⠀⠀⡐⠄⠀⠁⠀⠀⠀⢸⡟⣿⣷⣦⣄⠉⠢⡀⠘⣧⢹⣿⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⡇⣿⠀⣿⠀⡿⠁⡌⢳⡀⠀⡴⢁⣘⡥⠂⠒⠈⣁⠉⠘⠛⠁⠀⠀⠀⠀⠀⢀⣾⢱⡇⢸⡝⢛⠙⡄⢹⠀⢙⣼⡟⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⡇⣿⠀⣿⡄⠀⠰⢀⡞⢹⣦⡀⠈⠩⡉⠁⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⡿⣿⠸⠀⡀⠳⣄⣀⣰⣸⢠⣿⢿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⡁⣿⠀⣿⠇⠀⠀⠊⣰⡏⣏⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⢹⣧⡱⣌⣲⣬⣽⣷⣿⢿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⡇⣿⠀⣿⠈⣷⡀⡀⢿⢟⢻⣷⢭⣽⣦⡀⠀⠀⠀⠀⣀⣤⠴⠛⠉⠀⠀⠀⠀⢸⡏⠙⠲⣿⢿⡟⢻⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⡇⣿⠀⣿⠀⣿⠙⣦⢬⡢⣥⣿⡶⣿⠀⣿⢹⣶⢸⣿⠙⣧⡀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣼⣿⣿⣇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇
⣿⣧⣿⣤⣿⣤⣿⣤⣿⣤⣿⣤⣿⣤⣿⣤⣿⣬⣿⣼⣿⣼⣿⣷⡀⠀⠀⠀⠀⢀⣿⣶⣿⣿⣿⣿⣷⣼⣧⣼⣧⣼⣧⣼⣧⣼⣧⣼⣧⣼⣧⣼⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠰⠛⠉⠉⠈⠉⠉⠛⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠛⠛⠓⠀⠀⠀⠀⠘⣧
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣴⠛⢻⡆⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠛⠛⠳⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⢻⣄⠀⠀⠀⣰⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠙⠀⠀⢠⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⡤⠶⠞⠛⠉⠁⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⣤⠶⠾⠛⠋⠉⠀⠀⠀⣀⣀⣤⡴⠶⠚⠛⠉⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⢳⡀⠀⠀⠠⠶⠞⠛⠉
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠙⣧
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠋
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⣷
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠶⠛⠛⠉⠉⠉⠉⠙⠿⢿⣦⣛⠓⣦⣄⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⠚⠛⢿⣆⠄⣨⠽⢦⡀
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠐⠀⠁⠀⣀⣀⣀⣠⡤⠖⡡⣘⣿⡏⠀⠀⠒⠨⢦⡀⠀⠀⠀⠀⠀⠀⠀⡼
⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣠⠊⠀⠀⢀⠀⠒⠈⠉⠀⠤⠄⠭⠅⠐⠚⢵⣍⣾⣟⡛⠛⠉⠀⠈⠻⣄⠀⠀⠀⠀⣠⠛
⠠⠠⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⡜⠁⠀⠀⠂⢀⠠⠄⠒⠒⣀⣠⣴⣿⣷⡿⠿⠚⠿⢿⣿⣿⣿⢋⠑⠂⠄⠙⣆⣀⠠⠎
⠀⠁⠁⠌⠍⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡞⠁⠀⠀⠀⣀⠄⠒⠒⢉⣛⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣄⠀⠀⠀⠘⢏⠌⠀⠀⠀⠈
⠀⠀⠀⠐⡐⠀⠀⠉⠛⠂⠤⡄⣀⣀⢀⠀⠀⠀⠀⢠⠋⢠⡟⠀⠀⢀⡀⠂⢁⣠⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡂⠀⠀⠀⢘⡔
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠠⠀⠄⣤⠤⢤⠀⢠⡏⠀⠂⠀⠀⢀⣔⠔⣪⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠈⣵
⠀⠌⡄⢡⠡⢠⠀⠄⠀⠠⠡⢡⠉⡼⠀⠀⠀⣷⡾⠀⠀⠀⠴⢀⡾⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡧⠀⠰⠸⣇
⠐⡐⡐⢂⢂⢂⠂⠐⠀⠀⢂⢂⠒⡘⡙⠂⢚⣿⠇⠀⠀⡸⢂⣾⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠑⡀⠂⣧⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠚⠒⣀⠓⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣿⣿⣧⢰⣿⢸⠆
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⡈⠀⢁⠀⠀⣼⠻⠦⠔⠹⣄⣿⣿⠠⠛⠛⠛⠛⠷⠶⠀⠀⠀⠀⠀⠀⠰⠛⢋⣉⣀⣀⠀⠈⠀⠀⠀⠀⠀⢿⣿⣿⣇⢿⣾⡄
⠀⢂⠂⡐⡐⡐⡀⢂⠀⠐⡐⡐⣀⢂⢂⠀⠛⣇⠀⠀⠀⣿⣿⣿⠀⢠⣶⣿⡿⠶⢭⡀⠀⠀⠀⠀⠀⠀⣰⡿⠻⠛⠛⠿⢶⠄⠀⠀⠀⠀⠃⠙⣿⡏⠐⣿⣄
⠀⠂⠂⠰⠐⠐⠀⠀⠀⠐⠐⠐⠀⠂⠀⠀⠲⢿⠀⡄⠀⠃⣿⣿⡆⠀⠈⠑⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⣿⡁
⠀⠀⠀⠡⠁⠀⠁⠀⠀⠀⠠⠁⠉⠈⠀⠀⠠⣼⠀⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠇⢹⡇
⠀⡀⡀⢂⢀⠀⠀⠀⠀⠀⢀⢂⠀⡀⠀⠀⠀⣾⠀⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⠿⠿⠾⣸⣃⡀
⠐⠐⠀⠆⠆⠀⠂⠀⠀⠀⠀⠆⠂⠰⠐⢀⣤⣿⣼⠇⡄⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡀⠀⠀⠀⢹⣿⣿⣿
⠡⠡⠉⠌⠌⠌⠄⠀⠁⠀⠈⣬⣼⣿⣿⣿⣿⣿⣿⡀⠃⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⢰⠀⠀⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠉⢻⣿⣿⣿⣶⡀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠙⡀⠀⠀⢸⠀⢃⣿⡏
⠀⠀⠀⠰⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⠘⣿⣿⣿⣧⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠇⠀⠀⠈⠀⢸⣿⡇⢀
⠀⠌⠄⠡⠡⠡⠁⠈⢼⣿⡟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⢹⢿⣟⣿⣷⡀⠀⠀⠀⠈⠙⠷⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠸⠀⠀⡀⢰⣸⣿⠇⣾
⠀⣈⡀⢡⢁⢁⠁⠀⣼⣿⡇⠀⠀⢠⡀⢱⡄⠀⠀⢿⣿⣿⡇⠈⠀⠈⢻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣤⣿⣿⣿⣼⡟
⠀⣸⡀⠀⠂⠀⠀⠀⣿⣿⡇⠀⠀⠀⢷⡄⢻⣆⠀⠘⣿⣿⣿⡠⠀⠀⠈⡽⣿⣿⣿⣷⡄⠀⣀⠀⠀⠀⣀⠠⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠓⠃⢸⣿⣿⣿⣿⡇
⠀⠽⠁⠄⠌⠄⠀⢠⣿⣿⡇⠀⠀⠀⠈⣿⣆⢿⣆⠀⢹⣿⣿⣧⢂⠀⠀⠀⠙⣯⣿⣿⣿⣦⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⢿⣿⠏⡀⠀⠀⣿⣿⣿⣿⣷
⠀⣳⢃⡘⡘⡈⣀⢠⣿⣿⡇⠀⢠⡀⠀⠘⣿⣾⣿⣆⠘⣿⣿⣿⡞⣄⠀⠀⠀⠹⡺⣻⣿⣿⣿⣾⡀⠀⠀⠀⠀⠀⠀⣴⣿⣀⡀⢰⠐⢿⡼⠃⠀⠀⠘⢻⣿⣿⣿
⠀⢒⠂⠐⠐⠐⠀⢸⣿⣿⡇⠀⢸⣿⣦⡀⠹⣿⣿⣿⡆⣿⣿⣿⣿⣿⣆⢆⠀⠀⢻⣣⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣧⣇⣼⠀⢸⡇⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠾⠄⠀⠀⠠⠀⢸⣿⣿⣿⢀⠸⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⢹⣿⣿⣿⣾⣄⢣⠈⣿⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⡔⠘⣷⡀⠀⠀⡀⠠⠀⣿⣿
⠀⡘⡀⢀⢀⢀⠀⢸⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡜⣧⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⢇⣿⣷⡄⢠⣧⠀⡆⢸⣿
⠀⣰⡐⢂⢂⢂⠀⠸⣿⣿⣿⢸⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⠀⢿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣼⣿⣴⣇⣸⣿
⠀⢸⠁⠄⠆⠄⠀⠰⣿⣿⣿⢸⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⢸⡇⢈⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠻⠇⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣠⣤⣴⣤⣄⡀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⢠⣾⠟⢩⣿⠟⠻⢿⣦⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⣿⡀⢀⣼⣿⡆⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⣽⡿⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠙⠻⠿⣿⣿⠁⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⣇⡀⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⠛⠿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢀⣶⡿⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠸⣿⣄⠀⠈⠛⠿⣷⣤⡀⠀⢀⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠙⠻⢿⣦⣄⠀⠈⠙⠿⣷⣬⣉⠛⠛⠛⠛⠁⠀⠀⣴⡿⠛⠉⢉⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠉⢻⣿⣦⣄⡀⠈⠙⠿⣷⣦⣀⠀⠀⢠⣾⠟⠁⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⣿⡟⠉⠻⠿⠀⠀⠀⠀⠙⠻⣷⣶⡿⠋⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠿⠷⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠛⠛⠛⠳⢦⣄⠀⠀⠀⠀⠈⢿⣇
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⣿⡆
⢀⣿⠇⠀⠀⠀⠀⠀⠐⢤⣄⣀⣀⣀⣀⣠⣤⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣹⣿⣭⣭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣖⠄⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠛⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣄⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣶⣶⣶⣶⣶⣶⣶⣶⠿⠿⠛⠛⠉⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⡄⠀⠀⠀⢀⣾⢻⡆⠀⠀⠋⠉⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣷⠀⠀⣸⠏⠀⢸⣏⣀⣤⣤⡶⠶⠞⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠹⣇⣼⠋⠀⠀⠙⠉⠉⠁⠀⠀⣀⣤⡴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⠶⠶⠶⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠄⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⢠⠠⣤⠀⠀⣤⡤⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠈⠛⠁⣤⣶⣦⣴⣶⡄⠘⠃⢀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⣶⠀⠰⠀⠀⠀⠠⠀⠀⠀⢰⠟⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠂⠀⠀⠰
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⡶⠾⠟⠋⢶⡆⠀⣤⠀⠹⡟⠈⠛⠳⢶⣶⣿⣀⡀⣠⡄
⠀⠀⠀⠠⠀⠀⠀⢀⠀⢤⣤⣦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠛⠛⢶⣤⡀
⠀⠀⠀⠀⣠⡶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠚⢻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⢹
⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠸⠿⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⣸
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢿⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡷⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⡇⠀⣻⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢸⣿⣧⣠⠛⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⡿⣿⡇⠈⢷
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣸⣿⠀⢻⣷⡀⠸⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⣿⣿⠀⢸⣿⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢀⢸⣿⠀⠀⣿⣿⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘
⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⣀⡾⢸⡿⠀⠀⣿⣿⣆⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣿⠇⣼⡇⠀⠀⢹⣿⣿⡆⠀⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⣸⡏⠀⣿⠁⠀⠀⢸⣿⣿⣷⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⣿⠁⢰⣿⠀⠀⠀⠸⣿⣏⢻⣧⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣿⣿⠃⠀⣿⣿⠀⠀⠀⠀⠻⣿⡈⣿⣷⠀⠰⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄
⠀⠀⠀⢸⡀⠀⠇⠀⠀⠀⠀⠀⠛⠁⢀⣾⣿⠏⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⣧⠸⣿⣆⠀⠘⣧⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⠀⠀⢠⣿⣿⡟⠀⠀⣾⡏⣿⠀⠀⠀⠀⠀⠀⢻⡆⢹⣿⣆⠀⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠘⣤⠀⠀⠀⠀⢀⡛⠀⠁⣿⣿⣿⠀⣀⠀⣿⡇⠉⠀⠀⠀⠀⠀⠀⠘⣷⡀⢿⣿⣷
⠀⠀⠀⠀⠙⣦⠀⠀⠀⠸⠃⠀⣰⣿⣿⣿⣴⡟⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠈⣿⣿⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠀⢀⣿⣿⣿⣿⡿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⣿⣧⡀⠀⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⣠⠈⢻⡏⠉⠉⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡈⣹⡿⢀⣀⠀⢀⣄⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠙⣿⣿⣶⣤⣽⣦⣤⣶⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣶⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⡄⢀⡘⠁⠀⠀⠈⠉⠉⠉⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠟⠀⠀⣀⣤⣤⣤⣤⣤⣀⠓⠄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣤⠞⠃⠈⠻⣶⡀
⠀⠀⠀⠀⠀⠀⡆⢀⣾⠏⠀⠀⠀⠀⠀⢻⡆⠈⠳⡀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡼⠃⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠆⠀⡀⠀⠀⠀⠀⣿⠀⠀⠈⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⡀⠀⠠⠀⠰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠉⣀⣤⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠆⠀⠴⠉⠉⠁⠰⠶⠄⠀⠀⠀⠀⠀⠀⠰⠆⠀⠁⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠈⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣴⣤⣤⣴⣦⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⢤⣀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣠⡖⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠙⢦⣬⣉⣛⣓⠓⢠⡀⠀⠀⡚⣒⣘⣡⠶⠛⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠶⠀⠀⠊⠉⠉⠋⠙⢧⠞⠙⠉⠉⠁⠈⠑⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡄⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡀⠀⢀⠞⢉⡑⣴⣜⣧⠐⡀⠢⡀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢦⡀⠨⠢⢼⣤⣨⣴⢇⠀⣼⣦⣍⣠⠇⠔⠠⠠⢻⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣈⣷⣀⠀⠉⠶⣀⢁⠀⠉⠀⠈⠞⢏⡉⣈⡁⢀⡁⣴⠣⢺⣅⣄⠠⠠⠒⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢒⠪⠽⣻⣙⠣⠔⠒⠼⠭⣭⣍⣉⡁⠀⣀⣭⣅⣬⣭⣴⣦⡗⢋⣡⣠⠤⠔⠊⠁
⠀⠀⠀⠀⢀⠄⠂⣀⠤⠶⠇⠀⠀⠈⠉⠑⠒⠂⠤⢤⡰⠿⠙⠎⠅⠙⢉⡠⠔⠾⠛⠋⠉⠁⠀⢲⣄⣁⠀⠐⠂⢀
⠀⠀⠀⠀⠀⢈⡠⠥⢚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠚⠀⠀⠒⠁⠀⠀⠀⠀⠀⠈⢦⠄⣈⡉⠂⠀⡀
⠀⠀⠀⠠⠀⡤⢒⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣈⡁⠀⠁
⠀⠀⠀⠀⢁⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⡆⠠⠤⠀⡀⠀⠄⢀⠀⠀⠀⠀⠂⠀⡀⠤⠔⠒⢉⡽⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⢙⡦⡀⠀⠉⠈⠙⠤⠤⡤⠴⠊⠁⠀⠀⡠⠴⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣬⣴⡦⣐⣦⠤⠤⢀⣀⠀⡀⣀⣠⣤⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠿⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠯⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣤⠤⣤⠤⣤⣤⠤⣤⢤⣖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠈⢳⣿⢃⠎⠀⢀⠀⠑⡄
⠀⣤⡤⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢧⠀⠳⠿⠃⢸⠁⢸⠀⠈⠶⠃⢀⡇⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢤⡄
⠈⢇⠀⠀⠀⠀⠀⠁⠐⠒⠒⠩⠵⠾⢷⣶⣤⣴⣫⣦⠄⢑⣶⣤⠶⠛⠰⠍⠛⠛⠒⠉⠉⠀⠀⠀⠀⠀⡇
⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠘⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠀⠀⠀⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢀⣸⣻⣿⣟⢒⣤⣴⠒⣿⣏⣹⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣈⣉⣀⣀⣀⣀⣀⣉⣀⢈⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠺⠿⠿⠟⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢍⡉⠉⠉⠁⠠⠀⠀⠀⠈⠉⠛⠒⠦⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠤⠀⠂⠠⣄⠉⠠⡌⡠⣤⡤⠤⡐⠒⣉⡽⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⣴⣶⣤⠈⡀⠀⣿⡑⣁⣤⡄⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⣾⣿⡯⠄⡅⢠⠇⠋⠻⡿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠈⠳⣤⣀⡡⠔⠁⠈⠀⠓⠤⠴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠆⠆⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠠⡄⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡌⡙⠤⣴⡴⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⠆⠈⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⢠⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⢻⣽⣯⠟⣶⣲⣶⣶⡶⡐⠶⠛⠉⣿⠙⢤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⣹⠀⢗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣰⡁⢪⢤⣆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣻

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⣀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠄⣁⠀⠀⠀⠀⣀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠉⠒⠤⣀⠀⣤⠀⠒⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠆⢸⡄⠄⡠⠔⠊⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⠀⠀⠀⠉⠃⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠒⠀⠆⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣠⣶⠿⠛⠋⠉⠛⠻⣶⣄⠀⠀⢀⣶⡾⠛⠋⠉⠙⠻⢿⣧⣈⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠞⠉⣿⡆⠀⠀⠀⠀⡄⢲⣦⠀⢠⣿⣿⣧⠀⢠⡐⣶⡄⠀⠀⠀⢀⣿⡷⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡴⠋⢀⠔⠙⢿⣦⡀⠀⠀⠿⣿⢟⣤⠟⠋⠉⠛⢧⣜⠿⠿⠃⠀⢀⣠⣾⠿⡀⠈⢧⣀
⠀⠀⠀⣀⣀⡷⠄⣏⠉⠛⠷⢤⣎⣀⠀⠀⠙⠻⠿⠶⠶⠾⠿⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠛⠁⠀⠑⡄⢈⣻⡵⠶⣆⣤
⠀⠀⠀⠘⡇⠉⠁⠒⠬⢍⣐⠒⠠⢄⣉⠒⠤⣀⠀⠀⠀⠀⠀⡗⢆⠀⠀⢠⠞⠇⠀⠀⠀⠀⠀⠀⡠⠤⠒⣋⡉⠤⠔⣚⣩⠷⣿⣤
⠀⠀⢸⣛⡳⠶⠒⠈⠙⢢⠀⠉⠓⠦⢄⡉⠲⢌⡙⢦⡀⠀⠀⠘⠢⠉⠎⠡⠊⠀⠀⠀⣠⠔⣂⡥⠖⢂⣉⠤⠖⠒⡝⠉⠉⠙⠳⢼⡀
⢀⠒⠉⠁⠀⠀⠀⠀⢀⠼⠦⢄⣀⠀⠀⠈⠓⢤⡙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⢀⠔⣋⠴⢊⣡⠴⠚⢉⣀⡤⠖⢋⣱⣄⠀⠀⠀⠀⠈⠑⠂
⡈⠀⠀⢀⣀⠤⠀⠚⠻⡳⠦⣤⣈⠙⠓⠦⣤⣀⠈⠓⢷⡄⠙⢦⠀⠀⠀⢀⣔⡥⠊⡡⠶⣉⣠⡴⠞⢋⣡⡴⠞⠋⡡⠊⠑⠂⠤⣀⡀
⡗⠊⠉⠉⠀⠀⠀⠀⢀⠇⠀⠀⠈⠙⠓⠦⢤⣈⣙⠓⠶⢬⣦⣈⣧⡀⣠⠟⣁⢐⣩⠽⢦⠈⢣⡴⠚⠋⠁⠀⠀⠀⠱⣀⠀⠀⠀⠀⠉⠑⠂
⡇⠀⢀⣀⢠⡤⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣦⢸⡛⠒⠶⠶⠚⢿⠉⡤⠖⠋⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣱⠶⢤⣀⡀
⠸⡉⠉⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡧⠤⣀⣀⡤⢼⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢠⣾⣧⡀⠀⠀⠉⠙⣲
⠀⠑⣤⣤⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⣤⣄⣀⣤⣼⠀⢸⠀⠀⣤⡀⠀⠀⢸⣿⣿⣇⠀⢸⣿⣀⣽⡲⣤⣄⡰⠃
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⢸⡆⢀⣿⣿⣦⡀⠀⣿⣯⣿⠇⠀⠙⠛⠋⠀⡇⢸⡇
⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⡇⢸⡇⠸⠿⠋⠛⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⡇⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢛⣨⠇
⠀⠀⠀⠀⠳⠦⣭⣙⣛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠒⠶⠶⠶⠒⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢚⣫⡥⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢷⡲⠬⣭⣛⡓⠶⠤⣤⣀⡀⢸⠀⣷⠲⠤⠤⠤⠴⡇⠀⣿⠀⣀⣠⡤⠶⢚⣩⣥⠖⢋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣍⡲⢤⡀⠉⠙⠒⠲⠬⣭⣉⡀⣿⠀⠀⠀⠀⠀⡇⠀⢛⣋⣭⠴⠒⣋⡥⢞⣡⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠉⢏⣻⣿⣿⣶⣦⣤⣤⣀⣈⣉⣛⣦⣄⣀⣠⣤⣿⣚⣩⣥⣴⣦⣿⡷⣾⣉⠎⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠒⠋⡯⠤⠤⣙⣿⡔⠉⢙⡏⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠙⣼⣟⡡⠤⠬⡏⠓⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣠⡜⠀⠑⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠁⠘⣤⣀⡤⠃
⣿⡿⠿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣧⣽⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣭⣥⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣶⣿⣿⠿⣶⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣦⣀⠀⠀⠀⠀⠐⢋⡭⢷⠀⠀⠐⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⡹⢃⠻⠂⠀⠀⠀⠀⣀⣀⣸⣀⠀⢠⣴⣴⠞⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠈⠁⠀⠀⠀⠀⠀⠈⠙⢻⣿⣶⣶⠟⢂⣜⣶⡴⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⣀⣀⡆⠀⠀⠀⠀⡴⣶⢿⢿⣿⣿⡿⠀⠂⠀⠀⠶⠦⡄⠀⠀⠀⣄⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣅⣨⣿⡏⠀⠀⠀⣤⣤⣀⣸⢺⣿⣿⣶⡀⠀⠀⠀⠀⢀⣀⡀⢀⣾⣿⣐⣨⣿⣇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⢿⣿⣿⣿⡇⡄⠀⢠⣶⣴⣤⣼⢸⣿⣿⠀⠀⠈⠀⠀⣤⣤⠄⠀⠘⢿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⡼⠛⢯⣠⣆⠘⢿⣿⣿⠃⡋⠀⠘⠛⠛⠛⠛⢸⣿⣿⠀⠀⠀⡀⠀⠁⠀⡴⣧⠀⠀⠉⡛⠉⠉⣴⡴⣆
⠀⠀⠀⢀⡟⠶⣅⣁⣿⣀⣀⡹⠇⢠⠟⠳⠘⠓⠚⠒⠖⠘⠿⡷⠒⠒⢺⠗⠒⠀⣴⣀⣁⠀⠀⠀⠙⢦⠚⠛⡂⣐⣧
⠀⠀⣠⣶⣷⣴⠋⠉⠙⣤⠀⠀⠀⢨⠗⢺⠀⢴⣶⣶⠀⠀⢀⡀⠀⣀⣤⣤⡀⠀⠉⢿⡋⣀⡀⠶⠶⢾⠶⠶⣿⣟⠉⢅
⠀⠀⣿⣿⣿⠋⠙⠛⠛⡿⠒⠒⠐⢿⡶⣾⠀⠀⣿⠇⠀⠀⣾⣯⡿⣿⣿⣿⠁⠀⠾⣿⣿⡟⠈⠀⠀⣸⠀⢸⣿⣷⣶⡾
⠀⠀⠸⣿⣿⣆⡀⠀⠀⢧⣄⣀⣠⣼⣿⣞⠀⠀⠉⠁⠀⢹⣿⣯⡏⠉⠉⠉⠀⠀⢀⣿⣿⣿⣶⠀⠀⠞⠀⠈⢛⣿⡿
⠀⠀⠀⢽⣿⣿⡿⣿⠀⠀⠻⣿⣿⣿⣿⡿⠀⠀⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⣻⣝⣿⡿⠀⠀⣆⠀⢠⣾⣿⠇
⠀⠀⠀⠈⡈⣿⠦⣞⠀⢈⠀⠈⠛⣟⣫⠅⠀⠀⠸⠿⠿⠿⢹⣏⣿⣿⣿⡟⠀⠀⠀⠛⢟⣿⠀⠀⠀⠈⠻⣿⡋⢛
⠀⠀⠀⠀⠉⡧⣤⡏⠀⠠⠀⠰⠺⣅⣨⡇⠀⠀⢠⣤⣤⣠⢸⡗⣀⣀⣀⡀⠀⠀⠀⢲⡶⢿⠄⠀⠀⢰⡶⡿⣷⣶
⠀⠀⠀⠀⢹⣧⣤⠗⠆⠐⠀⠰⢾⡋⢹⠀⠀⠀⠠⣿⣿⣿⣼⡇⣿⣿⣿⡇⠀⠀⠀⢸⣦⣄⡄⠀⠀⢸⣇⣠⣼⣗
⠀⠀⠀⠀⠸⣅⣨⠷⠂⠐⠀⠤⣔⠉⠓⠀⠀⠀⠀⠛⠛⠛⠓⠓⠛⠛⠛⠁⠀⠀⠀⠀⠫⣿⠁⠀⠀⠈⠙⣿⠅⠉
⠀⠀⠀⠀⠀⢉⣹⠴⠄⠀⢀⣤⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⠀⠀⠀⡟⠟⢷⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⠄⠀⠀⢧⣴⡼⠇

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣤⣦⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢫⡶⢶⣶⡄⣻⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⢋⣤⣾⣿⣿⠋⠀⢻⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣴⣾⣿⣿⡿⠆⠀⢸⡟⠀⠈⡇⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡷⠆⠀⣸⡇⠀⠀⠻⣸⡀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣾⣿⡇⠀⠀⣿⡖⠀⠀⣿⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⠀⣿⡏⠀⠀⣿⠀⠀⠐⠀⢀⣀⡰⡿⣿⣷⡆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡉⣿⣿⣆⠀⢰⣿⣇⠀⠀⣿⡀⠀⠀⠀⢀⠛⠛⠉⠉⠩⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣯⡄⢸⣿⠏⠀⠀⣿⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡿⠀⢸⣿⡇⠀⠀⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡞⣿⣿⣷⡇⢸⣿⠇⠀⠀⣿⠀⠀⠀⢸⠙⢶⡄⠀⠐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠌⠻⠂⠀⠀⠙⠀⠀⠀⠸⠦⠈⠋⠀⠆⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣴⣂⡤⠖⣫⡿⠕⠒⣩⡿⠥⠖⠒⠋⡽⠁
⠀⠀⣠⠔⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⢲⡆⠀⠀⠀⣶⣰⣶⣰⣶⣦⣾⣤⣤⣦
⠀⢸⠁⠀⠀⠀⢀⣠⣆⣀⡤⠀⠀⣀⣀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⣿⠻⣿⠹⣷⡿⠹⣿⣿⡇
⠀⠀⡇⠀⣠⢔⣋⠉⠉⠰⠓⠋⠉⠠⠯⠴⠒⢒⡗⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣧⡞⠑⠁⣀⠉⠂⠀⠀⠀⠀⠀⣀⣀⠀⣸⠀⡼⠀⠀⠀⠸⢿⡿⣴⡿⣷⠀⠀⣿⢷⣼⣯⣽⠃⣾⣦⢻⣶⣤⡷
⢀⡴⢿⡇⣴⡏⠈⠙⡄⠀⠀⢐⣐⣋⣀⡀⠁⣇⣰⡃⠀⠀⠀⠀⠸⠇⠘⠿⠟⠁⠀⠿⠾⠻⠿⠻⠒⠟⠻⠞⠿⠿⠇
⡏⠀⣸⠀⡇⠀⠀⠀⡇⠀⠀⢸⠋⠀⠀⢩⠃⡏⠀⢳
⠙⢤⡇⠀⠘⠢⠤⠚⠀⠀⠀⠀⠓⠤⠖⠁⠀⢸⣀⠏⠀⠀⣼⣿⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠛⠙⠃
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⡼⠀⣀⣀⢀⠀⣀⠠⠄⡀⢀⡀⡀⠀⢠⠀⡀⢀⠀⣀⡀
⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⡜⠁⠀⢹⡋⢹⡶⣿⢰⡆⡿⣼⠀⣷⡋⢰⢀⣷⣾⢰⣏⣤
⠀⠀⠀⠀⠉⢻⠉⠳⣀⡤⠒⣉⢇⡠⠞⠉⠀⠀⠀⠈⠁⠈⠀⠉⠈⠉⠃⠉⠈⠁⠉⠈⠉⠀⠋⠈⠛⠋
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠨⣧⠊⠀⠀⠀⠀⠀⠀⣠⡄⢀⣀⠀⣠⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⡷⠂⣸⢽⡄⣿⣀⢸⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡇⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⡇⢠⣜⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠃⠀⠀⢀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡿⠀⢀⢔⠕⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣼⣶⣿⣿⣶⣤⠘⢷⣦⠴⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠈⠂⢸⣿⣟⢻⣿⣻⠟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⠀⡇⠀⠀⠀⢸⢿⣿⣷⣌⣰⢶⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⢀⡈⣶⠀⠀⠀⠀⠘⣮⣤⣋⣛⣉⣶⡀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠿⠉⠁⠘⠀⠀⠀⠀⠀⠘⣯⡹⡉⢍⠀⠳⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠳⣟⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⠷⠔⠋⠰⣝⠂⠀⠀⠀⠀⠀⠹⡾⡀
⠀⠀⠀⠀⠀⠐⢄⠀⣿⣈⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣎⠠⠼⢿⡀⢻⡴⣆⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⣀⡀⠀⠉⣿⣿⡧⠀⠀⠒⠀⠞⢾⣿⠛⣿⠏⠀⠀⢄⡙⠿⢇⡟⠀⠀⠀⠃⠘⣇⣤⢜⣂
⠀⠀⢀⠼⠫⠁⠟⠶⠽⡯⠁⠀⠀⠀⠀⠀⢸⡿⠴⢹⠀⠀⠀⠄⠙⢦⠀⠁⠰⠧⠿⠚⢹⠟⡠⠚⠛⢠

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠎
⠀⠀⠀⠀⠀⠡⠐⠠⢀⣀⡄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠬⠀⠠⠐⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡕⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⣨⣴⠶⢤⣄⡀⠀⠀⣀⣤⠴⢶⣤⡄⢄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠊
⠀⠀⠀⠀⢀⣀⣀⢀⣿⠋⠀⠀⠀⠈⠻⡄⣼⠋⠀⠀⠀⠈⠹⣇⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠜
⠀⠀⠀⠀⠻⠋⢛⢿⡇⠀⠀⠰⣼⡇⠀⣿⣇⠀⣮⣿⠀⠀⠀⣿⡛⠋⠿⠂⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠔⡵⠁
⠀⠀⠀⠀⣀⡴⠃⠌⠻⣄⡀⠀⠀⣀⠼⠛⠻⢦⣀⠀⠀⣀⣴⢏⠙⣄⢀⠀⠀⠀⠀⡇⠀⠈⢢⡀⡠⡢⠊
⠀⠀⢠⠊⡝⢀⠊⠀⠀⠈⠙⠛⠛⢡⡀⠀⠀⣠⠙⠛⠛⠉⠀⠀⠂⠈⢆⠈⠉⠁⡨⠷⡀⠀⠀⠙⢾
⠀⡰⠁⡜⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡆⠀⠀⡇⠀⠈⠢⡀⠀⠀⠁
⡰⠁⢰⠁⡄⠀⠀⠀⠀⠀⠀⠀⠒⠂⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⣦⡀⠀⠈⠐⠄
⡇⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠇⠝⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⢀⢾⡀⠈⠐⢄⡀
⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⢱⢦⣀⠜
⢃⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠂⠈⠉
⠈⢾⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠈⢧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠇
⠀⠀⠈⢣⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠋
⠀⠀⠀⠀⠉⣪⣗⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣮⣞⠁
⠀⠀⠀⠀⠸⣀⠰⡞⣛⢿⡗⠶⠶⠶⠶⠶⠶⠶⠶⠶⢺⡟⠛⣳⠆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠌⠑⠒⠁⠀⠀⠀⠀⠀⠈⠒⠊⠣⠤⠜

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣁⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠉⠒⢄⣀⠀⢀⡁⠠⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠄⢰⡄⠄⣀⠤⠒⠉⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠈⠉⠻⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠤⠀⠄⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⣠⣶⡿⠿⠛⠻⠷⣦⣄⡀⠀⠀⠀⣤⣶⠿⠛⠛⠻⢿⣶⣔⠐⣢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣠⠟⢿⣇⠀⠀⠀⠀⠀⣀⣄⠀⠀⢸⣿⣿⡆⠀⠀⣠⣄⠀⠀⠀⠀⠀⣼⣿⡉⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠜⣿⣄⠀⠀⠀⢸⣦⣿⡇⣠⣿⠿⠿⣷⡄⢸⣶⣿⡇⠀⠀⠀⣰⣿⠃⠹⣄
⠀⠀⠀⠀⠀⣄⠀⣿⠲⢾⣟⠀⣠⠃⠀⠈⠻⢷⣦⣤⣤⣭⣽⡾⠉⠀⠀⠀⠀⠙⢷⣯⣭⣤⣤⣶⡾⠟⠉⢢⠀⠘⢷⢤⣀⡀⢀
⠀⠀⠀⠰⡖⠿⢥⣟⣢⠤⢤⣉⣙⠒⠦⢤⣀⠀⠈⠉⠉⠉⠁⢱⣄⠀⠀⠀⠀⣠⡆⠉⠉⠉⠉⠁⠀⠀⢀⣀⡵⠴⠚⣋⣩⠷⣿⣠⡀
⠀⠀⢠⣤⣷⣀⣀⠠⠬⣉⠒⠠⢌⡉⠑⠢⢌⡑⠢⣄⠀⠀⠀⠈⢄⠑⢄⡠⠊⡰⠀⠀⠀⠀⢀⡠⠔⣊⡩⠤⠒⢊⣉⢭⠴⠒⠯⣼⢻⡇
⠀⣀⠸⠖⠊⠉⠀⠀⠀⢨⡆⠀⠀⠈⠙⠲⢄⡉⠳⣌⠳⣄⠀⠀⠀⠉⠋⠃⠉⠀⠀⢀⡴⢚⡡⠖⢋⣡⠴⠒⠋⣁⡤⢾⠀⠀⠀⠀⠉⠓⠤⣀
⢰⠁⠀⠀⠀⠀⠀⣀⣴⣎⡉⠙⠲⢤⣄⡀⠀⠈⠓⢬⣳⡈⠳⣄⠀⠀⠀⠀⠀⢀⠔⣡⠖⢉⠤⠚⠉⣀⡤⠖⠋⣁⡤⠞⣳⣤⣀⠀⠀⠀⠀⠀⠹
⣄⢀⡠⠤⠒⠊⠁⠀⠈⢫⠉⠓⠲⢤⣀⡉⠙⠲⢤⣄⣈⠻⢆⠈⢧⠀⠀⢀⣴⠗⢊⡠⢆⠙⢦⡖⠋⣁⡤⠖⠋⠁⠀⢸⠁⠀⠀⠉⠒⠤⣄⡀
⡏⠀⠀⠀⠀⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣉⡙⠲⣷⣖⡓⠀⠞⣡⣴⡥⠖⢋⣳⣠⡽⠏⠁⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠀⠀⠈⠙
⢣⣠⡤⠴⠔⠒⠛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⣿⠈⠉⠉⠉⠁⣿⠰⡏⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠓⠲⠤⣄⣀⣰
⠘⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠒⠒⠲⠖⠒⢿⠀⣷⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⢰⣿⠿⢄⡀⠀⠀⠀⢨⠇
⠀⠈⠲⢶⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠒⠶⠶⠴⠶⢾⠀⣿⠀⠀⣷⣄⠀⠀⠸⣿⡿⣷⡄⠸⣿⣦⣶⡏⣷⠦⡔⠃
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⠀⢸⡄⢸⣿⣿⣷⣄⠀⢻⣷⠿⠃⠀⠈⠉⠉⠀⢹⠀⣷
⠀⠀⠀⠸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⠀⢸⡇⠘⠟⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿
⠀⠀⠀⠀⡇⠿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣛⣥⠿
⠀⠀⠀⠀⠛⠲⠦⣭⣙⣛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠋⠓⠒⠒⠒⠚⡇⢸⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⣛⣩⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠻⣗⠲⠬⣍⣙⡓⠶⠤⣤⣀⣀⣿⢸⡗⠒⠲⠴⠖⠺⡇⠈⣇⢀⣀⣤⠴⠞⣋⡭⣴⠞⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣮⣑⠦⣄⡀⠉⠙⠓⠶⠤⣍⣁⢘⡇⠀⠀⠀⠀⠀⡇⠀⢛⣉⡥⠶⠚⣉⡥⢚⣥⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⢹⣉⣶⢿⣷⣶⣤⣤⣤⣀⣈⣉⣳⣤⣄⣀⣠⣴⣷⣪⣭⣤⣴⣶⣯⠷⣮⣉⠎⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠊⣹⠤⠠⢌⣻⣧⠋⠉⣻⠉⠉⠉⠉⠉⠉⠉⠉⠙⡏⠉⠑⣼⣗⡡⠄⠤⣏⠑⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡠⠋⠘⠲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠃⠹⢄⣀⡤⠟

⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠳⣄⠀⠀⠀⢿⠦⠤⠤⠒⠒⠒⠒⠶⠤⠒⢒⣺⠃⠀⠀⠀⡠⠊⠉⠳⣄
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⢳⠀⠀⠈⡷⣄⠀⠀⠀⠀⠀⠀⠀⣴⢻⡇⠀⠀⢠⠋⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢠⡏⣀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⡇⢺⠷⢤⠀⠀⢀⣰⣾⡅⢸⡇⠀⠀⢸⠀⠀⠀⠀⠘⢶⣄⠹⡄
⠀⠀⠀⢀⡆⠍⠀⠐⠆⠐⣏⣠⠄⠸⢆⠀⠀⡇⠈⠉⢁⠱⣀⠞⡍⣉⠀⢻⠀⠀⢀⠞⡀⢀⠀⠳⠤⠜⠀⠉⠱⡄
⠀⠀⠀⡼⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠘⠦⣼⢀⠀⠀⠀⠀⠉⠀⠀⠀⡀⡄⢇⣠⠋⠀⠑⠈⠋⠀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⢰⠃⠀⡄⠀⣀⢼⠁⠀⠈⠀⠀⠀⠀⡇⠀⠘⠀⢄⠀⠀⠠⡄⠠⠁⠇⠘⠀⠀⠀⠀⢀⠂⠀⠀⠠⠄⠐⠀⡀⢧
⠀⠀⡏⢠⠛⡇⠀⠀⢸⢰⠆⠁⢀⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⣘⡀⠘⠀⠀⠁⠀⠀⠚⠈⢁⠘⡆
⠀⢰⣠⠏⡆⠀⠀⠀⠈⢰⢠⠀⠼⠀⠀⠀⡧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⢐⠈⠒⠀⢀⠀⣀⠀⠃⠀⡀⠀⠀⠈⠀⢻
⠀⠘⠁⠀⠀⡃⢀⢃⠀⠉⢸⠀⢳⠀⡄⡆⢱⠄⢀⠀⢠⠀⠀⠀⠀⠁⠁⡄⡾⢠⠀⢆⠈⡆⣹⠀⣈⠀⠡⡀⠀⠀⠀⠰⡄
⠀⡇⠀⢸⠀⠈⠁⡌⢠⢃⡞⠋⠈⠂⠀⠈⢹⢀⡘⡇⠘⠄⠀⠀⠀⡴⠀⣼⡷⠞⠈⠈⠃⠀⠈⡆⢸⠀⠀⠈⠀⠀⠀⠀⢧
⢠⠀⠀⠀⠀⠀⢀⠇⡜⠉⠀⠀⠀⠀⠀⠀⠘⡄⠀⢈⠀⠰⠄⠃⠐⠃⠨⢰⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⢠⠀⠀⠀⠀⠀⢸
⢸⠀⢠⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠁⠀⠀⠀⠀⠃⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘
⢰⠀⠘⠀⠈⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠲⠀⠀⢠⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠀⠘
⢈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠄⠀⠀⠐⡀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡆⠀⡇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢈⠊⣧⠠⣁⠰⡄⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡁
⡇⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢠⠆⠺⠬⠤⠾⠷⠧⠦⠇⠼⠼⠤⠄⣀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⢧⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⢀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡰⡿⣀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢯⢩⠁⡉⠉⠉⠉⠉⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⡀⠘⣈⣠⠀⠀⠀⢀⡄⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠁⠒⢲⢉⠟⠒⠒⠛⠛⠋⠉⠉⠉⠛⡑⠉⠉⠉⠙⡩⠽⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠀⠀⢺⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠶⠤⢬⠤⠤⣤⣤⣤⣤⣤⣤⣬⣦⡧⠤⠤⣤⢌⣷⣀⣄⣠⣄⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣿⣿⣷⣾⣯⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣏⢁⠈⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⡟⠛⠋⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣾⣿⣿⡛⣾⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡈⢿⣿⠟⠃⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⣠⣴⣖⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⡦⠄⠀⣤⣴⣿⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣾⣷⣿⣿⣟⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣾⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⣿⣏⣿⣽⡏⢉⢁⣀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣷⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣧⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠛⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠉⠉⠉⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀⠈⢿⣿⣷⡀
⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⣀⡀⠀⠙⠿⣷⣤⣄⠀⠀⠀⠀⠀⠿⠿⠟⢛⣋⠉⠉⠁⠀⠀⠻⣿⣿⡄
⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⢹⣿⣷⡀
⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢻⣿⣧
⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿
⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡿⠿⠿⠿⠿⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⢹⣿⣷⡀⠀⠀⠀⠀⠀⣼⣿⡿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏
⠀⠀⢻⣿⣷⡀⠀⠀⠀⢰⣿⣿⠃⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠹⣿⣷⣄⠀⠀⣿⣿⡏⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟
⠀⠀⠀⠀⠙⢿⣿⣧⣸⣿⣿⠀⠀⢸⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡏⠀⠀⠘⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢩⣽⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣻⣿⣿⡟⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣦⣴⣶⣶⣾⣿⣿⡿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠣⠓⢘⠀⠂⠀⠀⠘⠃⠀⠀⠀⠀⣠⠀⡆⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⡁⠁⠀⢠⣪⣈⠀⠀⠀⠀⠘⠦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⡆⢀⠀⠠⠀⠀⠀⠀⠄⠀⠈⠙⠉⠈⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠙⠁⠀⠐⠁⠀⠈⠙⢍⡆⠀⠴⠃⠠⡠⠡⠁⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⠄⠀⠀⠀⣐⡀⠀⠀⠀⠒⢼⣉⠇⡀⣠⣌⡀⠀⠀⣀⠆⠐⠆⠀⢀⣀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢠⣴⣦⠀⠴⠺⣇⡒⠴⡂⠙⠋⠃⠀⢀⢄⠸⠄⠼⣤⠠⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠠⡀⠀⠎⠙⠂⡀⠈⠁⢿⠁⠀⣻⠊⠀⠁⠀⠀⡂⠀⠀⣤⣀⡠⠄⠈⠉⠁⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠉⠀⠀⣌⠄⠐⠀⠈⠳⡀⣿⠔⠺⣄⠀⠀⠀⠀⡀⠙⠋⠁⠰⠂⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⡀⠀⠄⢀⡆⡹⠲⠷⣀⡿⢥⣀⠠⠎⠁⠠⣄⠀⠈⠰⠀⡤⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠙⣑⣾⡁⠁⠀⠀⣀⣹⠀⣀⠰⣾⣷⠆⠀⢢⠶⡄⠀⣠⠁⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠛⢡⠔⠳⣄⣰⡆⠀⢹⠀⠅⠈⡆⠀⢐⠢⢿⠴⠒⠋⠑⡄⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡇⠀⠀⠀⠀⠓⠛⠳⣤⣿⠘⠶⣴⡥⠦⣤⡼⠛⠳⠤⠐⠐⠁⠈⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠛⠀⠀⠀⠀⢠⠖⠀⠀⠙⣿⣷⣾⠃⠀⡀⠈⢓⠀⠀⠀⢰⣶⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠘⣿⣿⡄⠐⠻⢷⠦⠌⡑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠤⠴⡛⢛⠻⣦⡄⢀⣀⣀⣀⣀⣀⣐⣀⣀⠀⠀⠀⣈⠡⣤⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠾⣯⣥⣴⣿⠃⠀⠀⠀⠀⠀⠒⠒⠒⣒⣂⣈⣉⣉⣉⠀⡂⣀⣀⡀
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠉⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠉⠉⠉⠉⠉⠉⠓⠒⠦⠤⣀⡀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⡏⠀⡞⠉⠉⢳
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⠚⠁⣀⡤⠞
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠳⣄⠀⡇⠀⡴⠊
⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠟⣛⣛⠛⠿⣦⡀⠀⠀⠀⠈⢦⣳⢾⠁
⠀⠀⡼⠁⠀⠀⠀⠀⠈⡿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢡⣤⣝⢦⡀⠁⠀⠀⠀⠀⠀⢧⡴⠃
⠀⡸⠁⠀⠀⠀⠀⠀⢸⠁⠀⠈⢿⣿⣷⣤⡤⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣿⣿⣷⣧⠀⠀⠀⠀⠀⠀⠀⢧
⢠⠇⠀⠀⠀⠀⠀⠀⢻⠀⣤⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣟⠸⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠘⡆
⣸⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⠀⠻⣿⣿⡿⠋⡼⠀⠀⠀⠀⠀⠀⠀⠀⣧
⡿⣄⡀⠀⠀⠀⠀⠀⠀⠱⣄⠙⠛⢛⡵⠁⣀⡤⠔⠒⠒⠲⠤⣀⠳⣀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢀⣠⢼
⣧⡀⠉⠓⠦⣄⡀⠀⢀⣀⣈⣉⣫⠭⠔⠋⢁⣠⠴⠖⠒⠦⣄⡈⠙⠺⠭⣍⣉⣀⣀⣀⠀⢀⣠⠴⠚⠉⢀⣸
⢸⠉⠓⠦⣄⡀⡹⠋⠁⠀⣠⠤⠤⠴⠖⠚⠉⠴⠚⠛⠙⠓⠦⠉⠓⠲⠦⠤⠤⣄⡀⠈⠙⢯⣀⣠⠴⠚⠉⡏
⠈⡇⠀⠀⠈⢹⡇⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡏⠁⠀⠀⢰⠃
⠀⠹⡀⠀⠀⠈⡇⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⠇⠀⠀⢀⡏
⠀⠀⠱⡀⠀⠀⠹⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢀⡞⠀⠀⢀⡞
⠀⠀⠀⠙⣄⠀⣀⣹⣄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⣠⣞⣀⠀⣠⠎
⠀⠀⠀⠀⠈⢯⣁⣀⣨⣷⣤⡀⠈⠙⠲⠤⣄⣀⣀⣀⢀⣀⣀⣠⠤⠖⠊⠁⢀⣠⣾⣅⣀⣈⡽⠁
⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠈⠙⠳⠶⣤⣄⣀⣀⣈⠉⠉⢁⣀⣀⣠⣤⠶⠞⠋⠁⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠉⠉⠁⠀⠀⣀⡠⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣦⣤⣤⣤⣤⣤⣤⣴⣖⣚⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣴⡿⠋⠀⠉⠉⠛⠷⠀⠀⢠⠒⠉⠀⠀⠈⠛⢷⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣤⣼⡟⠀⠀⠀⢀⣀⡀⠀⢡⢠⠃⠀⡀⣄⡀⠀⠀⠈⣿⣤⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢈⠝⢻⡇⠀⠀⠀⠻⣾⠇⠀⢸⣹⠀⠀⠷⡿⠇⠀⠀⠀⣿⣏⠁⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠠⠊⢿⣦⡀⠀⠀⠀⠀⠐⠃⠈⠑⢤⡀⠀⠀⠀⣀⣼⢏⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡱⠁⡰⠁⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⢀⠛⠻⠟⠛⠋⠁⠀⠡⡀⠹⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡰⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⢣⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠘⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⣝⣴⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⡿⣞⣩⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠙⠒⠒⠘⠓⠉⠉⠋⠉⠉⠉⠉⠉⠉⠙⠉⠑⠚⠁⠒⠚⠋⠓⠚

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢀⡴⠖⠛⠛⠶⢦⡀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⢻⣆⢰⡟⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢻⡄⠈⢛⡟⢻⡟⠁⣠⡟⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢰⡇⢸⡇⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠀⠀⠀⠀⢻⣴⣧⣼⣦⡟⠀⠀⠀⠀⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣄⡀⠀⠀⣹⠶⠶⠶⢶⣏⠀⠀⢀⣠⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⣤⣌⡛⢶⣄⠈⠹⣟⣻⠛⠁⣠⡶⢛⣡⣤⠶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠶⠶⢤⣬⣙⡳⣽⣷⡀⠈⠁⢀⣾⣯⢞⣋⣥⡤⠶⠶⠶⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⠾⢿⣿⣿⣿⣆⣰⣿⣿⣿⡯⠷⠶⠶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⡶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⢶⡉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣆⡀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢰⢆⡆⡖⢢⢆⢦⡄⡆⠀⡆⢰⠒⣤⠒⡀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠁⠁⠉⠈⠈⠁⠁⠀⠁⠈⠉⠀⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣭⡇
⠀⠀⢠⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾⡿⠋⢡
⠀⠀⢀⠳⣀⠤⠤⠤⣂⠬⠄⣂⡄⠀⠀⠀⠀⣛⣛⢛⣛⡛⠀⠀⠀⠀⠘⠠⠀⠀⣀⠀⠀⠠⠘⠀⠀⠀⢰⣦⣤⣤⣀⡄⡀
⠀⠈⢄⣒⡪⠭⠭⢙⠵⠊⠁⠀⠀⠀⠀⠀⣾⡿⠿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢺⣷⢀⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣙⣺⣹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡏⣿⡭⢿⠺
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠰⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠈⠋⠉⠁⠉⠀⣿⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣴⣿⣷⡦⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣷⡆⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣯⣿⣿⡇⠛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣨⣵⣒⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢩⡄
⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⡶⠿⠷⠶⡿⢶⠾⢾⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣙⣛⢛⣉⣤⣤
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⢳⠒⡗⠒⠗⢺⠒⡺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣫⣭⣿
⠐⠀⠀⠐⣿⣿⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣫⣛⣹⡹⣋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⠿⠟
⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⣤⠖⠛⠛⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠉⠙⠛⠓⠂⠀⠀⠀⠘⣧
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⢷⠀⠀⠀⠀⢠⡟⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠈⢷⡀⠀⢀⡟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠉⠉⠉⣹⠆⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢸⣇⣤⠴⠶⠛⠋⠉⠀⠀⣀⣀⣤⠴⠖⠋⠁⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠙⡆⠀⠀⢠⠴⠖⠋⠉
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣹⠆
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⣯
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⡤⠴⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⢀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠱⡍⠒⠤⣀⠀⣄⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⡎⠀⡠⠔⠊⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠙⣃⣄⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣐⠀⢂⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣴⡿⠋⠉⠉⠉⠛⢷⡄⠀⣰⠟⠋⠉⠉⠉⠛⢷⡔⠊
⠀⠀⠀⠀⠀⣠⣶⣶⣤⣼⡏⠀⠀⠀⢀⣀⡀⠀⢿⣴⡏⠀⡀⣄⡀⠀⠀⠈⣿⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠙⠃⢀⠝⢻⡇⠀⠀⠀⠻⣾⠟⠀⣼⣿⡇⠀⠷⡿⠇⠀⠀⠀⣿⣏⠁⠀⠛⠁
⠀⠀⠀⠀⠀⢀⣴⠋⠠⠊⢻⣦⡀⠀⠀⠀⣀⡴⠛⠙⠻⢤⡀⠀⠀⠀⣀⣼⠯⠀⠳⣄⡀
⠀⠀⠀⣠⠞⡱⠁⡰⠁⠀⠀⠈⠛⠛⠛⠛⢫⡀⠀⠀⠀⣀⠛⠛⠛⠛⠋⠁⠀⠱⡀⠹⡝⢦⡀
⠀⠀⡔⠁⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠢⡀⠈⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠹⡄⠙⣄
⠀⡞⠀⢰⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢱⠀⠘⣆
⣸⠀⢀⡏⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠢⠤⠄⡐⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣇⠀⠸⡄
⡇⠀⣸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⠠⠋⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡇⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠂⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠘⡇⠀⡇
⢣⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢠⠇
⠘⡄⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⡜
⠀⠙⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⡼⠁
⠀⠀⠘⣆⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠟
⠀⠀⠀⠈⢦⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⠏
⠀⠀⠀⠀⠀⠙⢮⣓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣡⠞⠁
⠀⠀⠀⠀⠀⢠⠊⠙⣍⣿⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⣞⣩⠉⢱⡄
⠀⠀⠀⠀⠀⠈⠓⠊⢹⠒⠒⢸⢷⡉⢉⠏⠉⠉⠉⠉⠉⠉⠹⡁⢁⠾⡇⠒⠺⡏⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠛⠋⠛⠛⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣠⣄⣠⣄⣀⣾⠇⠀⠀⢀⣀⣀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⣿⣍⠉⠉⣁⣴⡶⠟⠛⠉⠉⠉⠉⠉⠉⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡟⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢠⡶⠶⢶⣦⡀⠀⢀⣤⡶⠶⢶⣦⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠘⠿⠶⣶⡿⠃⠀⣾⣏⣴⠶⣶⡿⠃⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⣾⠻⣦⠀⠀⠀⠈⠉⣱⡾⣶⡀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠿⠶⠟⠀⠀⠀⠀⠀⠻⠶⠾⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠰⣿⣋⣉⣙⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠛⣷⡈⣿⡀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣦⣤⣤⣀⣀⣤⣤⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢷⣤⢶⠶⠒⢚⣩⣉⡫⠍⠭⣋⣩⣙⠒⠲⢶⢦⣴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣶⠿⠷⣶⣄⢬⣴⣝⢂⢚⣵⣬⢄⣴⡶⠿⢷⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠕⠉⢈⡀⠈⢧⢽⢸⢯⢿⢸⢥⠏⠀⣈⠈⠑⢽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢶⢋⣉⡑⡜⡾⣿⣿⣿⡿⡞⡔⣉⣉⢳⠆⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⡀⠀⢠⡸⣜⣛⣡⠇⢹⣿⣿⣿⣿⠁⢧⣙⣛⡼⣠⠀⠀⡈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡟⣡⠀⠘⠛⠢⠛⠁⢠⢠⡟⠀⠘⣧⢠⠀⠙⠣⠚⠛⠀⢠⡙⡿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣾⡳⡝⠽⡇⡶⠀⢀⡀⢰⣦⣾⣿⣇⠀⢀⣿⣿⣦⣶⠀⣀⠀⠰⡆⡿⠝⡵⣻⣆
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⡘⢧⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣤⠟⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣘⠛⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠛⢛⣠⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠦⣀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠁⠀⠀⢠⡚⠀⠀⣀⠤⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠈⠑⠢⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣘⣃⠁⡋⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⠀⣠⣴⠿⠿⠷⢶⣤⣀⣀⣀⣀⣠⣴⠶⠿⠿⢶⣦⡄⠑⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢨⣿⠏⠁⠀⠀⠀⠀⠈⢻⣆⢀⣾⠋⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠋⢻⣇⠀⠀⠀⢀⡐⣦⡀⢀⣿⣿⣇⠀⣠⢲⣦⠀⠀⠀⢀⣿⢧⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡞⠁⡰⠋⠻⣦⣀⠀⠘⠿⢿⣣⠞⠋⠉⠛⢦⣙⡿⠟⠀⣀⣠⡾⠯⡀⠙⣤⡀
⠀⠀⠀⠀⠀⠠⣤⢽⡢⠷⣈⣙⠒⠶⠤⣀⡀⠈⠛⠛⠿⠟⠛⣇⠀⠀⠀⠀⢀⡟⠻⠿⠟⠛⠋⠀⠀⢑⣀⡸⠿⢒⣦⡆⡀
⠀⠀⠀⠀⠀⣠⣹⡀⢈⣉⣑⠂⠭⣉⠒⠢⢍⡑⠦⣀⠀⠀⠀⠩⡑⢄⢀⠔⡩⠀⠀⠀⠀⢀⡠⢔⣊⠩⠄⢒⣊⡭⠴⣫⡷⣶
⠀⠀⠀⠀⣀⠿⠜⠋⠁⠀⠀⢱⠀⠀⠈⠑⠠⣌⠳⢌⠳⢄⠀⠀⠈⠒⠓⠈⠀⠀⢀⠔⣪⠵⢚⣩⠤⠒⢈⣡⢼⠀⠀⠀⠉⠛⠤⣀
⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⣴⣉⡉⠓⠦⢄⣀⠀⠉⠢⢵⣄⠳⣄⠀⠀⠀⠀⣠⢖⠵⢊⡠⠚⢉⣠⠴⠚⣉⡤⠞⡳⠤⣀⠀⠀⠀⠀⢱
⠀⠀⠀⡧⠤⠔⠚⠉⠀⠀⠈⡆⠉⠛⠲⠤⣌⣉⠓⠦⢤⣙⢢⡈⢧⠀⢀⡶⢛⡡⢐⣍⠙⢞⣉⡤⠖⠋⠁⠀⢸⡀⠀⠀⠉⠓⠢⢄⣀
⠀⠀⠀⡇⠀⠀⠀⣀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢬⡉⡿⠯⠭⠭⠴⡿⢚⡭⠼⢷⠚⠃⠀⠀⠀⠀⠀⠀⠙⢢⣄⣀⠀⠀⠀⢨
⠀⠀⠀⢹⠚⠉⠉⠀⢈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣀⣀⣀⣀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣴⣏⠀⠀⠉⠑⢲⡎
⠀⠀⠀⠀⠣⣤⡤⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⢤⣀⣀⣠⣼⠀⡇⠀⢠⣄⠀⠀⢻⣿⣿⡀⠸⣿⣀⣽⢢⣤⣀⠞
⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⢸⠀⣿⠀⣿⣿⣧⡀⠘⣿⣾⠟⠀⠙⠛⠋⠘⡇⣿
⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⢸⠀⢿⠐⠻⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⢹⠸⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣋⡥⠞
⠀⠀⠀⠀⠀⠀⠈⠉⠓⢲⡬⢭⣛⣲⠦⢤⣀⣀⠀⠀⠀⠀⡇⢸⣈⠉⠉⠉⢹⡆⢸⡄⠀⠀⠀⣀⣠⠴⢖⣋⢭⡶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢬⡉⠓⠲⠭⣭⣛⡒⠶⠇⢸⠉⠉⠉⠉⢹⡇⢸⡧⠶⢚⣫⠭⢒⡿⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢳⠾⣷⣦⣤⣀⣀⠉⠙⠒⠾⣄⡀⠀⠀⣠⣧⠴⢒⣋⣭⣴⣾⡵⢚⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⢻⣋⢍⡻⣿⠙⠛⢻⠛⠛⠛⠛⠛⠛⠛⢻⠟⠛⢍⣿⢋⠍⣹⡟⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⡼⠋⠳⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠹⢄⣀⡤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⣀⣿⠀⣆⠀⠘⡄⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⠀⢧⠀⠈⡇⠀⢹⠁⠀⣰⠁⣠⠤⠤⢄⡀⢀⡇⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣤⢀⡈⢣⠈⢳⡀⢱⠀⢸⠀⡔⠁⡞⢀⡀⣤⠀⠙⣾⠀⠀⠀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠻⠉⢹⠀⡸⠀⠀⡷⢺⠆⢺⠶⠁⠀⡇⢈⠋⢙⠞⢀⡇⠀⠀⣸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢳⣈⣄⡇⠀⠀⣏⣡⠖⢁⣴⠀⠀⢹⣸⢀⡎⡤⢺⠁⠀⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⡡⢃⣧⠤⠤⠽⢯⣉⡭⠿⠶⠤⣿⡹⡝⢿⣅⣸⡀⣰⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠙⠯⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠽⠃⠀⣨⡟⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠉⠛⠻⠿⠿⠿⠛⠛⠋⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⡇⢀⡀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢈⡿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠿⠃⣼⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠁⠀⢠⠇⠀⠑⢦⡀⠀⠀⠀⣀⡤⠞⠁⠀⢹⠀⠀⠉⠉⠙⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣩⠗⠒⠯⣅⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⠖⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣵⣷⣾⣷⡝⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⠋⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡔⣬⣭⢆⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢿⣎⢫⣾⠿⠿⣿⣿⡜⣿⣿⣿⣿⠿⠏⣾⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⠻⣿⣿⡸⢟⣫⣶⣷⡕⢉⢡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢸⣿⣿⣿⠀⢹⣿⣿⣿⣿⠿⠋⠁⠈⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢸⣿⣿⣿⠀⠀⠉⠛⠛⠁⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⢠⣤⣤⣴⣿⣿⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠉⠛




⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣻⣿⣛⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠘⠀⠒⠂⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠲⢤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠐⠀⠀⠀⠀⠀⣆⠓⠀⠀⠀⢀⣠⠴⢊⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠙⠲⠤⣀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠂⢸⠂⠂⠴⠊⠉⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡆⠀⠀⠀⠈⠃⣢⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣠⣔⡲⠒⠀⠸⠀⠠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⣠⣶⡿⠟⠛⠛⠛⠛⠿⣶⣤⠠⠤⠤⢀⣴⡾⠟⠛⠋⠛⠛⠿⣿⣮⡀⣩⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣼⣿⠃⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢻⣧⣿⠇⠀⠀⡠⣤⣄⠀⠀⠀⠀⠀⢹⣿⣴⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠈⢉⡿⢻⣿⠀⠀⠀⠀⠀⢸⣷⣿⣿⠀⠀⣸⣿⣿⡀⠀⠸⣷⣾⣿⠆⠀⠀⠀⠀⢸⣿⡛⠉⠁⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣸⣿⣆⠀⠀⠀⠀⠀⠙⠛⠁⠀⢠⣿⣿⣿⣧⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣾⡟⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⢀⡴⠁⠘⢿⣷⣄⡀⠀⠀⠀⢀⣀⡴⠛⠉⠀⠈⠙⠳⣤⣀⠀⠀⠀⢀⣀⣴⣿⠟⢆⠀⠈⢷⣤⡀
⠀⠀⠀⠀⠀⣠⠖⢉⡞⠁⢀⠎⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⢻⡀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⢣⡀⠈⢳⡉⠲⣄
⠀⠀⠀⢀⡼⠃⢀⡞⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢦⡀⠀⠀⣠⠞⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢳⡀⠈⢦⡀
⠀⠀⢠⠞⠀⢀⡾⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡙⠢⢞⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢷⡀⠀⠳⡄
⠀⢠⠏⠀⠀⣼⠁⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⣧⠀⠀⠹⡄
⢀⡏⠀⠀⢸⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⢄⠀⠀⠀⠀⢀⠔⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡆⠀⠀⢹
⣸⠀⠀⢀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣉⣺⠉⠐⣆⡀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢿
⡏⠀⠀⢸⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢉⠽⠤⠼⡉⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡆
⡇⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡤⠋⠀⠀⠀⠀⠉⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇
⢇⠀⠀⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⢸⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢀
⠀⢧⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⡼
⠀⠘⣆⣷⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⣰⠃
⠀⠀⠘⢿⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣸⡷⠃
⠀⠀⠀⠈⢷⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡟⠁
⠀⠀⠀⠀⠈⢷⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠻⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢟⠾⢽⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣞⣭⠶⢛⠝⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⣸⢶⠟⠋⠻⢟⣿⠛⡿⠿⣷⡶⠶⠶⣶⠶⠶⠶⠶⠶⠶⣾⡿⠿⡟⢻⣟⠟⠋⠛⢶⢮⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣼⠊⠉⠙⢢⡟⣏⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢈⡟⣧⠚⠉⠉⢻⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡋⣠⣴⣤⣈⣿⣿⣿⣿⡿⢀⣿⡿⠋⣉⠙⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣷⣾⣿⠁⣼⣿⡇⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣾⣦⠈⣿⣿⣿⣿⣿⣿⣆⠸⠿⠃⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⡛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⣫⣶⣿⣿⡟⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢟⣛⣛⡻⣿⣷⣶⣶⣶⣿⣿⢟⣥⡾⣿⣿⠟⣭⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡘⣿⣿⣿⣮⡻⣿⣿⡿⣋⣴⣿⣿⣷⣖⣀⠲⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣎⢿⣿⣿⣿⣮⣩⣾⣿⣿⣿⣿⣿⣿⡿⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⡿⠋

⣿⣿⣿⣿⠿⠛⠉⠉⠁⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠉⠙⢿⣏⠿⣿⣿⣿⣿⠟⠋⠀⠈⠻⢿⣿⣿⣿⡿⠿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠢⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸
⣿⣿⠄⠄⠀⠀⠀⠤⠀⠀⠄⠀⠠⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀⠀⠀⠀⣀⠀⢀⡀⠀⣀⠀⢀⣀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠐⠀⠐⠂⠀⠒⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿
⣿⣿⣿⣿⣿⠋⠁⠂⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣶⣤⢀⡀⠀⡀⢀⣀⢤⣶⣿⣿⣿⣿⣿⠟⠑⠂⠀⠂⠀⠂⠙⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠀⢈⠐⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⠀⠀⡀⠀⡀⠀⠈⢿⣿⣿⣿⣿⠋⠀⠀⠰⠐⠒⠢⠀⠀⠘⢿⣿⣿
⣿⣿⡿⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠘⠀⠀⠰⠀⠀⠀⠻⣿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣿
⣿⣿⠀⠀⠀⠀⠈⠓⢴⡤⠠⠉⠀⠀⠀⠀⠟⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⠀⠀⠀⠀⠀⠀⢀⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠓⠐⠆⠘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾
⣿⣿⠀⠀⠀⠠⢀⠠⠒⠒⢄⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣐⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⢠⠐⠐⢄⠀⠂⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡿⠁⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣧⡀⠀⠀⠀⠀⠀⣀⣄⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⣶⣶⣦⣄⡀⠀⢀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠺⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⣀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡞⣩⡍⢳⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢺⡀⠀⠳⢥⡤⠟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠁⠀⢣⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡧⣼⣿⡧⡉⠁⢩⡅
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡙⢰⣿⠟⢹⡜⠀⡲⠁
⠀⠀⠀⠀⠀⠀⡿⢟⣵⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠿⢋⠔⠿⠒⠉⠀⣀⠠⠄⠐⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠙⣟⠵⡉⢢⠣⡀⠀⣀⠀⠀⠀⠀⢀⡌⢀⠔⠁⠀⣠⠔⠂⠉⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⢄⠑⡌⢾⡇⠀⠀⠀⠀⠰⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡈⠢⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡜⢄⠑⢌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠱⡀⠢⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢌⢦⠔⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠑⠤⡀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠄⠠⠾⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠒⠈⠁⠀⠀⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠟⠻⠉⠉⡿⢻⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣿⣿⣷⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣤⡀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠉⠀⠀⠈⠉⠻⠿⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠈⠛⠋⠉⠉⠙⠛⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⢸⣿⡇⢿⣿⡀⠀⠀⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠈⠿⣿⣶⣦⣍⣙⠛⠿⣿⣷⣶⣤⣀⡀⢸⣿⡇⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣤⣀⠀⠀⠀⠉⠙⠻⣿⣿⡶⠀⠈⠙⠛⠿⢿⣿⣿⡇⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⠀⠀⠀⠀⢿⣿⣤⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣶⣤⣤⣴⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉


⣿⣿⡿⠿⣿⠿⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣩⣴⣶⣾⣿⣷⣶⣶⣬⣉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣯⠴⠒⢦⣽⣿⣿⡿⠿⢛⣹⣿⣿⣿⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⡇⢾⣿⡇⣹⣿⣿⣷⡞⣡⣤⡌⢿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣦⣭⣴⣿⣿⣿⣿⣧⡙⠿⢃⣾⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⡿⢋⣴⠇⣼⣿⣿⣤⣤⣍⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣷⣤⡤⠤⠤⠶⠤⠌⣿⡌⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⡓⠒⠂⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣉⡋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣭⣭⣭⣭⣴⠛⠿⠿⠿⠿⠿⠟⢛⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉


⣿⠿⠿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠜⢉⡙⢳⠉⠀⢀⣀⠐⠀⢄⠀⠁⠲⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢤⡀⠈⣄⡀⠉⠉⠂⠃⡁⠀⠀⠂⢀⠘⠀⣸⡀⠈⠰⠀⡀⠃⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢳⠀⠀⠒⠈⠉⠀⠈⠀⠀⡀⡔⠠⡌⠳⢀⣀⣠⢴⠁⠈⠉⠱⣔⢀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⣀⣄⢠⣀⣂⠌⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⢠⠂⣉⠆⡀⠀⠈⢿⠈⠉⠗⠀⡜⠈⠐⠤⢤⠎⢉⣍⠓⠄
⠀⠀⠐⠃⠉⠀⠈⠉⠙⠦⡘⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠈⠀⠀⠀⠀⠈⠲⠤⠤⠒⠀⠀⠀⠀⠴⠄⠷⠌⠓⠛⠒⢤⡀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠣
⡆⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠉
⠘⢄⠀⠀⠀⠀⠀⠒⠠⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠤⠀⠂⠁
⠀⠀⠙⠲⠤⠤⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠲⢂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠃⠀⠀⠂⠀⡇
⠀⠀⠀⠄⠀⠰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⡀
⣀⣀⣀⣈⣀⣀⣀⣀⣀⣲⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠶⠖⠀⠀⠀⠀⠀⠀⠲⠶⠆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⣠⣄⠂⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠟⡀⠀⠤⠀⠈⠻⢃⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠠
⠀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠎⠉⡦⡀⠀⠀⠄⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠠⢄⡀⠀⠀⢀⠀⠀⠀⠀⠒
⠀⠁⠰⠐⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠠⠀⠀⠄⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢀⠈⠀⠀⠈⠀⠐⠃⠀⡄⠀⠀⠈⢀⠐⠁⠀⠀⡀
⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⡇
⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡦⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⢀⡀⠱⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⡀⠊⠈⠉⠁⠈⠁⠐⠚⠆⠀⠀⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠰⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠆⠀⠀⠀⠃
⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘

⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠐⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠖
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀
⠀⠀⠀⣷⣤⣶⣤⣤⣄⠀⠛⠛⠛⠋⠉⣥⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠍⠩⠭⠉⠭⠤⠥⠤⠤⠆⠀⠀⡀⠈⠁
⠀⠀⠀⠟⠉⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠃
⠀⠀⠀⠶⠲⠶⠶⠶⠶⠖⠒⠒⠂⠀⠄⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣀⡀⣀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⡦⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⢩⣉⣉⣉⣉⡁
⠀⠀⠀⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⣤⣄⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠛⠛⠛⠛⠛⢷⣶⣆⣉⠩⠤⠤⠄
⠀⠀⠀⣦⡤⠠⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⡃⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣉⣙⣛⣛⣻⣿⣿⣿⣯⣭⣽⣯⠙⠛⠛⠛⠛⠛⠛⠛⢓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣈⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠴⣯⠉⡠⠀⠀⠀⢄⠉⢿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊⠀⠀⠀⠀⠈⠑⠠⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠉⣢⠀⠀⢀⠀⠊⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠒⠀⠤⠚⢹⠀⠀⡇⠐⠀⠀⠀⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠦⡤⠀⠤⢤⠊⠠⢄⠈⠒⠄⠤⠀⣀⣠⡮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣰⡆⠀⠀⣸⡇⠁⢞⢀⢸⡄⠀⠀⢀⠿⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⡗⠂⠒⢹⠇⣴⣘⠳⡍⢷⠄⠤⠋⠜⠀⠤⠄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⡠⠌⢥⡔⠋⡼⠃⢹⡀⠀⠈⢄⠀⣈⠀⠀⠀⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠻⠧⣄⠀⠘⢦⠀⠀⢸⢧⢀⡠⠄⣍⠀⠑⢄⠀⠈⠑⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣿⠀⠀⠠⠓⠈⠓⠠⠼⠋⠀⣸⠀⠏⠀⠐⢸⠣⠀⠈⠑⢌⠳⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣄⡠⣅⢦⠀⢸⢻⡐⠔⠁⠀⡘⠃⡄⠀⠀⠀⣳⠠⠀⠀⣠⠄⠰⢸⡄⠀⠈⢷⠹⡄⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⢘⣈⠥⠈⡀⡀⠘⡆⠇⠲⣄⡄⠀⠀⠧⠐⠐⠒⢱⠀⠢⠐⠡⠤⣀⢐⡇⢀⡰⣿⠀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠍⢐⡠⠧⠒⠡⡀⢹⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠘⠲⠄⠴⠂⠑⠚⠓⠉⠀⣿⠀⠈⠇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠒⠉⣅⠀⠀⠀⣧⠑⣾⢸⠀⠀⠀⢠⠀⠈⡆⡎⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⡠⠀⠀⠈⢯⠑⠦⠴⣿⠈⢣⠘⡆⠀⠀⢀⠇⠀⣿⡇⢰⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⢁⡀⠈⠒⠈⠑⣦⣠⠇⠀⠈⢆⣧⠀⠀⡸⠀⠀⣿⡇⣸⠀⠀⢰⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠘⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠖⢁⡠⠀⢀⠔⡥⠊⠛⠒⠦⣀⠤⣿⠀⢰⠃⠀⢰⣿⡇⣿⠀⠀⢸⠀⠀⣿⠀⠀⢠⡇⠀⣿⠀⠀⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠎⠀⣠⠞⢸⠅⠀⣀⣀⠠⢬⡂⡈⡄⢸⠀⠀⣸⠸⡇⠘⢦⡀⢾⠀⠀⣿⠀⡇⢸⡇⠀⢻⡀⠀⠀⠙⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡟⠠⡏⠈⢀⡾⢀⡀⠀⢀⠀⠐⣿⠑⣳⠛⠀⠀⡿⠀⢷⠀⠀⠉⢹⡀⠀⣿⠀⠀⠈⣧⠀⠀⠳⣠⠎⠐⠂⠠⠂
⠀⠀⠀⠀⣰⠉⠁⠒⠀⣸⠃⠀⢰⠇⠀⢸⠁⢠⠃⠀⠛⠛⠋⠈⠢⠟⠀⠀⠀⡇⠀⠸⡆⠀⠸⢀⡇⠀⡿⠀⠀⠀⠘⣦⠀⠀⠉⠰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣰⠃⠀⠀⡟⠐⠤⢘⡇⣼⠀⠀⠀⡀⠀⡄⠀⡆⠀⡀⠀⠀⠀⠀⡇⠀⠀⡞⠀⢠⡇⠀⠀⢠⠀⠘⢧⡀⠀⠀⢻⡀
⠀⠀⠀⠀⢿⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⠀⠙⠃⠀⠀⢰⠇⠸⠃⠀⠹⡄⢧⡀⠀⠀⠀⠀⠀⣸⠁⠀⠼⠁⠀⠀⠀⠁⠀⠈⢳⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠷⠀⢀⠞⠁⣀⠤⠀⡴⠂⠀⣰⠏⠀⠀⣠⠞⢉⠁⠀⠳⡄⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢴⡁⠀⠈⠁⠀⢀⡀⠰⠞⣡⠖⠊⠉⠀⠀⢀⡆⠀⠀⠘⠆⠀⠀⠀⠀⠳⠔⠋⠉⠉⠉⠁⠀⠀⠉⠒⠦⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈