Clatter Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⣀⣄⡀⠈⠋⣴⠷⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡿⠀⣀⣴⡾⠿⢾⣿⣽⡿⢻⣷⠀⠶⠿⠂⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡟⣡⡶⠷⣶⡿⣫⣶⠟⣻⡇⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠘⣿⠀⠙⣿⣟⣡⣾⣿⣅⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣿⠀⢀⣿⠀⠀⢸⣿⣋⣤⢈⣿⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣠⡾⠋⣀⠿⠃⠀⠀⠀⠉⢉⣥⡾⠋⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢁⣾⠟⠀⠀⣤⣴⠶⠟⠛⠉⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠁⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠶⠦⢠⣄⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⣀⡤⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠉⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡤⠶⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⢁⣀⣀⣠⣀⣀⣀⡈⠉⠑⠦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠶⢄
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣠⠴⠎⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠱⠦⣄⠉⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡴⠊⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠎⠀⠀⠀⠀⠰⠋⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⡄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⣰⠇⢰⠇⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠸⡆⠀⠀⠀⠀⠠⣎
⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠇⢀⠇⠀⠀⠀⠀⠀⠀⠈⠓⠖⠰⠊⠁⠀⠀⠀⠀⠀⠀⣠⠶⠶⠦⡀⠀⠸⠀⢺⠀⠀⠀⠀⠀⠀⠙⠦⡀
⡔⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠘⡄⠀⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢀⡴⠀⠀⡇⠀⡇
⠑⢦⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⣇⠀⢇⠀⠀⠀⠀⠀⠀⠰⠉⠁⠈⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡺⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠘⢇⠘⢧⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡸⠃⠀⠀⠀⠀⠈⢣
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢳⡄⠀⠀⠀⠈⠲⠆⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡄⠉⠲⠦⣤⣀⡀⠀⠀⢀⣠⣤⠴⠖⠉⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⣤⣤⣀⡈⠀⣀⣤⣤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⣠⠄⠀⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠔⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢢⣄⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠒⠠⠄⠒⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⣤⣤⣤⣤⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣶⣶⣶⣶⣶⣦⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⡆⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⢆⠀⠀⠈⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠢⡀⠀⠀⠘⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⢒⡀⠀⠀⠙⢤⡀⠀⠀⠀⠸⡀⢀⣀⡀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⡀⠀⠀⠀⠀⢀⣿⡀⠀⡳⢶⡄⣤⠀⠉⠢⣀⠀⠀⠉⢉⠆⢇⣀⡤⣄⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢇⡊⡁⠈⠻⣦⠀⠀⠀⢸⡇⠻⣄⠀⠈⠳⠶⢶⣀⡄⠈⠙⠦⣄⡀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠳⣄⣀⡎⠀⠀⠉⠑⠋⠒⠮⢷⣄⣰⢏⠘⢄⡈⠷⣤⡀⠀⠙⠒⢲⡆⣶⣀⠀⠉⢁⣰⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠨⠁⠀⠀⠀⠀⠀⢀⠄⠀⠈⠑⠁⣀⠀⣉⡿⠮⠿⠛⠲⠶⠦⠭⠥⠾⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡴⠖⠊⠁⠀⠃⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠖⠒⠛⠛⠛⢛⡿⠛⣄⠀⠀⠀⠀⡠⠂⠘⢝⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣼⡟⡂⠘⠁⠀⠀⠀⠘⣖⡝⣧
⠀⠀⠀⢀⣠⣤⡤⠤⣤⣤⣀⠀⠀⢀⡴⠋⠀⢀⣴⠟⠁⡗⡇⠀⠀⠀⠀⠀⠀⠹⡐⣸
⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠉⠙⠲⠋⠀⠀⣰⠞⠁⠀⢰⡇⣇⠀⠀⠀⠀⢀⣀⠤⠞⠃
⣰⠋⠀⣠⠞⠋⠉⠉⠉⠛⠲⣄⠀⠀⢠⡞⠁⠀⠀⠀⢸⡇⠈⠉⠉⠉⢻⡉
⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠻⣆⠀⠀⠀⠘⣇⠀⢀⡶⠤⠤⠬⠗
⢧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢿⡄⠀⠀⠀⠈⠳⣬⡇
⠘⣆⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇
⠀⠘⢦⡀⠈⠳⣦⣀⠀⠀⠀⠀⣠⡞⠀⢀⡿
⠀⠀⠀⠙⠦⣄⠀⠉⠛⠛⠛⠛⠁⠀⣠⠟⠁
⣀⣀⣀⣀⣀⣈⣙⣲⣦⣤⣤⣤⣖⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠚⠛⠛⠛⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢱⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⢇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠀⠀⠈⢣⠀⠀⠀⠑⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠄⡀⠀⠀⠱⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡅⠀⠀⠀⠘⢆⠀⠀⠀⠑⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⢹⣀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠡⣀⡤⠲⡄⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡛⠳⢦⣀⠀⠀⠀⠀⠀⡾⢷⡀⠠⠛⢹⡆⠶⠀⠀⠉⠢⣄⡀⠀⠀⠘⠁⠣⠤⠞⢢⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡞⠑⢌⣀⠄⠀⠙⢷⡀⠀⠀⢠⠷⠈⠳⣄⠀⠀⠙⠛⢻⡤⠴⠀⠀⠙⠓⢤⣀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⠤⣄⡾⠀⠀⠀⠀⠉⠉⠐⠒⢿⣦⣰⠏⠀⠓⣄⠈⠳⣦⣀⠀⠀⠛⠋⣇⢹⡧⡀⠀⢀⣈⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡇⠀⠀⠀⣠⠄⢹⣷⠄⠀⠀⠁⠂⣀⡀⢄⣩⠿⠶⠟⠛⠛⠲⠶⠶⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠖⠋⠁⠀⠐⠀⣄⣠⣥⣠⡾⡇⠀⠀⠀⠀⠀⠀⠹⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠶⠒⠛⠛⠛⢛⡿⠳⡀⠀⠀⠀⠀⢀⡔⠈⠩⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣸⡗⠒⠒⠊⠀⠀⠀⠀⢱⠀⠹⣆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⢹
⠀⠀⣠⡶⠛⠉⠉⠀⠈⠉⠉⠛⠲⢤⡴⠋⠀⠀⣠⠞⠁⠀⢠⡇⡄⠀⠀⠀⠀⠀⠀⠀⣈⡶⠏
⢀⡾⠋⠀⣀⡤⠶⠶⠶⠶⢤⣄⠀⠀⠀⠀⣨⡾⠃⠀⠀⠀⢸⠃⠙⠂⠀⠀⢰⡖⠚⠉⠁
⣾⠁⠀⣼⠋⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠻⣄⠀⠀⠀⠀⠸⡇⠀⠀⣀⡀⠀⠙⠦⣀
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⡆⠀⠀⠀⠀⠳⣄⠀⡏⠉⠉⠉⠉⠁
⢹⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠈⠓⠛
⠀⢳⡀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿
⠀⠀⠙⢦⡀⠈⠛⢶⣤⣀⣀⣀⣀⣴⠟⠁⢀⣼⠃
⠀⠀⠀⠀⠙⠦⣄⡀⠈⠉⠉⠉⠉⠀⠀⣠⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡗⣠⣤⣶⣶⣬⡭⣍⣋⣀⣠⣄⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣷⡖⠒⠶⢿⣿⣿⣿⣿⢿⣿⣷⣬⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠯⢏⣹⣯⣷⣾⣿⣟⣻⢿⣶⣬⢛⢝⡿⣿⣷⣽⣿⣿⡿⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠭⠻⢷⣼⠿⣛⣫⢭⠬⠭⠿⣿⣿⣿⣷⣝⢿⣿⢿⣿⣿⣿⣿⣮⣷⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡋⠔⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡻⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⢯⣿⡜⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢸⡇⣠⡤⣤⠀⠀⠀⠀⡤⠤⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣿⣿⣿⣷⡙⣿⡆⠇⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⡇⢸⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣽⣿⣧⢿⣧⡄⠸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⣧⢸⠖⣤⡀⠒⠀⠀⠀⠀⠀⠐⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⢿⣿⠟⡇⡍⣹⡏⣿⣷⠈⠀⢀
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢻⣾⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣼⠟⠁⢀⠿⢀⣿⡇⣿⣽⡇⠀⢸
⠀⠀⡄⠙⢄⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠏⠀⢀⣾⠀⡜⢹⢀⡏⢻⣻⡄⢸
⠀⠀⠘⢄⠀⠑⠄⠀⢦⠀⠀⡿⣿⢸⣿⠀⠀⠀⠈⠒⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢋⠏⠀⠀⣼⠃⡼⠀⣾⣼⠀⠈⣿⡇⠘
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠃⠀⠃⡇⣾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠈⠀⠀⢰⠃⢰⡇⣰⣿⡇⠣⠀⠸⣷⡏
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢵⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⡠⢁⡟⢠⣿⣿⣇⠀⠈⢦⡿⢁
⠀⠀⠀⠀⠑⢄⠈⠃⠀⠀⠀⠀⠘⠇⢸⡄⠀⠆⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠃⣼⢡⠿⠋⠙⠿⠷⣄⠘⡧⢸⡄
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣷⡀⠘⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢶⡀⠀⠙⢷⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢸⣷⡀⠘⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠈⠐⢄⠀⠹⣆⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⣿⣷⡀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣼⡄⠀⠀⠠⡀⣀⠇⠀⢹⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠹⣿⣧⠓⠆⠀⠀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣰⣿⠓⠀⠀⠀⢻⣧⣤⣶⢟⣼⠃
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⠋⣿⠀⢠⠀⠀⠸⠀⠀⠒⠛⠁
⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡰⢏⡴⠋⠀⢠⠃
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠁⠀⠀⠀⠀⠀⠀⢠⣕⠪⠃⠀⠀⣠⠴⠃
⠀⡞⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⡤⠤⠒⠋⠁⠀⢀⡄
⠀⡇⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠈⠑⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠃
⠘⢇⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠉⠺⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡴⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡖⡫⣍⢰⠍⡀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣾⣟⣳⣤⣋⣀⣉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⡴⢏⣼⡷⡟⠖⠀⠀⠀⠀⠀⠤⡉⢆⠈⣷⡀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠙⡋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠻⡄
⠀⠀⠀⠀⠀⣸⠃⣤⣿⣋⣄⠀⠀⠀⠀⠀⠀⠀⠰⢤⡛⠀⢳⠀⠀⠀⠀⢀⠏⠀⣁⠉⢀⣀⣀⣀⣤⣀⣄⣀⣀⣀⣀⡀⢹⡄
⠀⠀⠀⠀⠀⣯⠐⣿⣜⣏⠀⠧⣀⣠⠂⠀⠀⠀⠀⠄⠐⠂⠘⡆⠀⠀⠀⣾⣴⣿⣿⣿⣿⣽⡯⠷⠊⠀⣀⣠⠂⠀⡜⠉⣏⣧
⠀⠀⠀⠀⢸⠃⣾⡾⢳⠎⢡⠠⢮⡀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢠⡟⢿⣿⣿⠉⠁⠙⢷⠳⣆⠚⠁⠀⠙⣦⢶⣶⠟⢺
⠀⠀⠀⠀⢸⠤⣮⣡⡞⣧⠛⢄⡸⠁⠀⠀⡠⠄⠀⠀⠀⠀⠈⣽⠀⠀⢸⡇⣼⣿⣿⠀⠀⠀⢸⡳⢍⣆⠀⠀⠀⣷⢬⡏⠀⢸
⠀⠀⠀⠀⢸⡶⠛⠟⠣⠔⠤⠀⠀⠀⠀⠀⣀⠐⢃⡤⠀⠀⠀⡟⠀⠀⠘⣧⣿⣿⣿⣧⣤⣴⣿⡶⢒⡽⣦⡤⢴⠋⠸⣇⡀⢸
⠀⠀⠀⠀⠘⣗⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢨⠀⢦⡠⠈⠀⢠⡇⠀⠀⠀⢿⡿⠿⠿⠿⠿⠿⠿⠛⠚⠓⠚⠲⠾⠶⠶⢼⣽⣿
⠀⠀⠀⠀⠀⢻⠉⣿⣵⡶⠋⡴⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠹⡄⠸⡙⢂⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠈⣧⠸⣿⡟⡞⠱⠲⣅⠀⠀⠀⢀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢳⣖⣿⣿⣿⣿⣿⣿⣿⣯⠏⢻⢷⠲⢦⣤⡞
⠀⠀⠀⠀⠀⠀⠈⢧⣹⣿⣇⡰⢦⠛⠀⠀⠀⢄⠐⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢻⡉⠉⢻⣿⣿⠋⠀⠈⢧⠀⢠⠟⢹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡼⣷⣶⡖⠀⠀⠀⠐⠂⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣸⣿⣿⣦⣀⣠⣟⠲⡇⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢏⡶⣦⡛⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⣿⡇⠾⠟⠻⠧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⢷⡼⠓⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠤⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣟⡧⠦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣶⠖⠒⠲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠈⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠀⠀⢀⠀⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠠⠄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣙⣛⣿⣛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣏⣩⣭⣽⣿⣿⣛⣛⣻⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠩⡍⢑⢶⡦⠄
⠀⠰⢀⠉⣁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⡀⠀⣀⠀⠀⠸⠀⠀⠂⠀⠀⣼⠁⡖⠓⣁⠼⠂
⠀⠀⠀⠡⡀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⢠⠎⠁⠀⠀⠀⠑⠤⣀⣀⣀⣀⡠⠖⠁⢐⣋⡥⠵⢶⡖⢉⡇⠦
⠀⠀⠀⠀⢡⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢐⢠⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡏⢠⣀⠤⣥⣤⢾⡄
⠀⠀⠀⠀⠁⠀⠐⠠⠋⠀⠀⢰⡀⠀⠀⠀⠈⠊⢁⢀⠙⢆⠈⠐⢤⠀⠀⠀⡰⠁⡴⠀⠁⢏⣂⡤⠔⠳⣆⠚⠀⠀⠀⠀⠀⡾⠿⠟
⠀⠀⠀⠀⠀⠐⢄⠄⠈⠠⠂⠠⠂⣠⡤⢄⠀⠀⢠⡆⠀⠀⢄⠀⠀⠁⠤⠊⠀⠀⠀⠀⠀⠀⠉⠲⠋⠑⠤⠁
⠀⠀⠀⠀⠀⠀⠈⡆⠳⠀⠀⠀⢠⠀⠀⢠⠓⠃⠀⠂⠀⠀⠀⠉⠉⠉⢙⡿⢶⣤⣍⠲⣔⠤⣈⡁⠀⠀⠀⠀⠀⡤⢢⢄
⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⡄⠀⠀⠀⢏⡀⠤⡠⠇⣀⣤⣴⣶⡆⠀⠈⠲⣨⣟⣿⣷⣼⠀⢬⣿⡇⠀⠀⠈⠣⠤⣤⣬⣑⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠤⠀⠄⠡⠀⢇⣾⢩⠁⡄⣸⠇⠀⠀⠀⠈⠝⣭⡛⣿⢦⣶⣿⠖⠂⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠛⠃⠀⠀⠐⠈⠂⢷⠘⠣⠿⠋⠀⠀⠀⠀⠈⣆⣿⣧⠯⣈⠿⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠄⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢅⠤⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⣧⠘
⠀⠀⠀⡀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠀⠀⠀⠀⢀⣶⠉⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⣷⠀⠀⠀⠀⡀⠠⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠚⠃⣦⡀⠀⢠⠆⠉⠁⠀⠀⠀⠙⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⢠⠷⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠃⢀⠋⠀⠀⠀⣴⣆⠀⠀⠀⠈⡃
⠀⠀⠀⠀⢀⡎⠀⠀⠀⢀⣀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢂⠀⠀⢀⠀⢁⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠐⠀⠀⢺⡄⠀⠀⠀⡀⠄⠀⠈⠭⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⣀⣀⠀⠀⠂⠈⠀⠃
⠀⠀⠀⠀⢸⠁⠊⠉⠁⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡬⠀⢸⡆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣠⠶⠒⠀⢀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⣿⠄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⣀⠄⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⣶⠟⠿⣦⡉⢠⡄⢤⠀⠀⠀⠐⢀⠀⢀⣤⣈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣀⣤⣴⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠘⣷⠀⢻⡌⠃⠀⠀⠀⠀⠀⣿⣿⣿⠇⡸⠀⠀⠀⡀⠀⠠⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡇⠀⠀⢀⣀⡀⠸⡀⢶⣷⠠⠀⠀⠀⠀⠀⣤⣽⣿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣀⡾⣿⣿⢿⣓⡇⠈⢁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢿⣧⣥⣥⡷⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠋⠉⠉⠙⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣄⡀⠀⠘⠻⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢀⣴⣿⣿⣿⠿⠿⠿⣿⣿⣿⣦⡀⠀⠀⠀⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡗⠀⠀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠙⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⢸⣿⣿⡇⠈⢻⣿⣿⣷⣄⠀⢸⣿⣿⡇⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⣿⣧⠀⠀⠙⠻⣿⣿⣷⣼⣿⣿⠇⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠹⣿⣿⣷⣤⣀⣀⣈⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡿⣿⣿⣦⣀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣤⣠⣾⣿⣿⡿⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠙⣿⣿⣷⣦⣀⣀⣀⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠛⠙⠛⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠿⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠉⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣤⣀⣀⣤⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣀⠀⠀⠀⣀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠛⠋⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⢿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠈⢿⣿⣿⡄⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣾⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣤⣤⣄⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠻⣿⣿⣿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏
⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡟⠁⠀⠈⠻⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣄⡀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⠿⠛⠉⠀⠈⠛⠿⣿⣿⣿⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠹⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿⣶⣷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⣠⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡴⠋⠁⠀⣿⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⠋⠀⠀⠀⠀⢻⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠀⠀⠀⠀⠀⢸⣿⣸⡄⠀⢀⡀⠄
⠀⠀⠀⡀⠀⠀⣀⠠⠠⠴⠖⠋⠀⠀⠀⠚⠀⠀⠀⢀⠀⢀⣀⠼⠛⢋⣉⠄⠈⠁⠂
⠀⠀⣀⡼⠖⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠴⠛⠉⠙⣿⢟⡙
⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡈⡇⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡇⢳⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠈⠁⠘⠶⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿⢿⣿⣻⣿⣿⣿⣯⣿⣿⣿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⣿⣿⣟⣿⣶⣾⣿⣻⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡠⠀⠈⠙⠻⢿⣿⣿⢿⣿⣿⣿⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢠⣤⣁⡐⠀⠀⠀⠀⠀⢻⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿⣿⣿⣿⣿⣷⡄⠀⣠⣤⣿⠇⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠄⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣿⣿⣾⣿⣿⣟⣿⣿⣿⣿⡟⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠠⣸⣿⣿⣿⣽⣿⢿⣮⣿⣹⠏⠠⣿⡇⠀⢀⣀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⣹⣿⣿⣿⣿⣿⣾⣿⡿⠁⠀⢁⣼⣧⣾⣟⠋⠁⠀⠛⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣧⣶⣿⣿⣿⣿⣿⣿⣿⠞⠉⡠⢊⣀⣿⣿⣟⡛⠀⠀⠀⠀⠀⠀⠱⠒⠢⣀⠔⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢼⣾⣿⣿⣿⣷⣿⡿⠟⢋⣁⡔⡉⣴⣿⣿⠍⠉⠉⠁⠀⠀⠀⠀⠀⠠⣤⣄⣀⣹⣤⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⣀⣙⡾⢿⣿⣿⣿⣭⣤⣤⣾⢿⡿⠿⠿⠻⠿⢿⡶⠶⣤⣬⠭⡽⡿⣲⣡⠀⠿⠯⠽⠇⠙⠛⠿⠛⠛⠛⠠⣤⣤⠴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆
⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢸⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⠀⠀⠀⢸⣮⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣤⡀⠀⠀⠀⠀⣠⣼⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⢫⣾
⠀⠀⠀⠀⢿⣛⣦⣾⣤⣤⣀⠀⠀⠀⢠⡾⡡⡸⠓⠒⠢⣝⣦⣄⣤⢞⡥⡒⡲⡻⠤⠿⣦⠀⠀⠀⢀⣠⣤⣽⣦⣟⣻⠄
⠀⠀⠀⠀⣴⢛⡿⠷⠒⠦⢭⣛⣦⣴⠋⣪⡶⠞⠛⠻⢶⣄⢙⢿⢟⢕⣴⠾⠛⠛⠶⣦⠈⢷⣤⣾⣫⠥⠒⠢⡿⣟⣳⡄
⠀⠀⠀⣼⡡⣢⣴⠶⠷⠶⣥⡉⠛⠁⢠⣿⠀⠀⠀⠀⠀⢻⣔⠕⠁⣾⠃⠀⠀⠀⠀⢸⣷⢐⢝⠉⣁⡴⠾⠾⢶⣮⡈⢹⡄
⠀⠀⣼⡡⣠⣿⡃⠀⠀⠀⠈⢿⠠⠪⠺⣿⢦⣀⠀⢀⣠⣾⢇⠀⡠⢻⣦⣀⠀⢀⣠⢾⠟⢕⠁⣴⠏⠀⠀⠀⠀⣿⣷⢄⢻⡄
⠀⠀⢷⡈⠸⡘⢧⡀⠀⠀⣀⣾⡷⠤⠪⠮⠦⢍⣉⣩⢟⠕⠌⣅⡀⡠⡻⠭⣙⣯⢅⠞⢔⠕⠕⢹⣆⡀⠀⠀⣠⠟⣙⠐⣰⠇
⠀⠀⠈⣧⠀⠙⢦⣉⣛⣫⠽⠃⠊⢂⣂⣠⣤⣤⣶⣶⣶⡄⠴⠊⠉⡛⣾⣶⣾⣧⣥⣤⣁⣀⠄⠪⠻⢍⣛⣋⡥⣞⡑⢡⡏
⠀⠀⠀⠈⠛⠶⣅⣑⣑⣠⣴⣶⠿⢛⣫⠍⢕⢖⠜⢝⣉⣁⣀⣀⣊⣊⣉⠉⠙⢒⢖⠽⠏⣛⠻⢷⣦⣬⣊⣀⣠⡦⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⠏⠲⠫⡊⢢⣤⠶⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠶⣵⣇⠠⡪⡀⠈⣷⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠨⢪⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡯⡨⢸⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⣠⢀⠛⢭⣛⠶⠦⣤⣤⣀⣀⣀⣀⣀⣠⣤⡤⠶⠞⣫⡽⠫⡪⠸⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣤⣀⡐⠕⠝⠝⠓⠒⡲⡲⡠⠼⡷⢔⠖⢖⢖⠝⠉⠀⠠⣢⣪⣮⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠦⠬⣬⣤⣼⣤⣤⡥⠴⠷⠶⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⣉⡉⣉⣉⣓⠦⣄⠀⢀⡤⢖⣋⣉⡉⣉⣉⡒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⢟⡫⠿⠟⠳⢯⣝⢿⣮⣳⣫⣾⢟⡭⠿⠛⠻⢯⣛⢷⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣠⢏⣾⣣⣿⠖⠛⠉⠉⠙⠻⣷⡽⣿⡿⣵⡿⠛⠋⠉⠉⠛⢿⣷⢹⣦⢷⣀⣀⣄⣠⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡠⢞⣩⣥⠤⠶⠶⣾⣭⣾⡇⣿⠃⠀⠀⠀⠀⠀⠀⠘⣧⣿⣧⡟⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣮⣭⡶⠶⠦⢤⣬⣙⠢⣄
⠀⠀⠀⠀⠀⢠⠞⣴⢟⡽⢚⣉⣭⣽⣶⣝⢻⣷⢻⡀⠀⠀⠀⠀⠀⠀⢠⢏⣿⣇⢧⠀⠀⠀⠀⠀⠀⠀⣸⢣⣿⢟⣵⣚⣩⣍⣙⣶⣝⢷⣌⢦
⠀⠀⠀⠀⠀⡟⣼⢯⣟⡷⠋⠁⠀⠀⠉⠻⣷⢻⣷⣙⢦⣀⣀⣀⣠⠴⣫⣾⠿⢿⣮⡳⢤⣀⣀⣀⣠⢖⣱⣿⢳⡿⠋⠁⠀⠀⠉⠙⣿⣧⢿⡞⣇
⠀⠀⠀⠀⢸⡇⣿⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣷⣶⣯⣽⣶⠿⣛⢵⣿⣷⣽⡻⢷⣾⣿⣿⣶⣿⣿⣿⢾⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⣿
⠀⠀⠀⠀⠀⣷⢻⣟⢧⡀⠀⠀⠀⠀⠀⢀⣌⣼⣿⣿⢛⣽⠶⡖⢚⣋⡵⠞⠁⠙⠻⣿⣻⠒⠒⣿⣽⡻⣿⣿⣞⣆⠀⠀⠀⠀⠀⠀⢠⢏⣿⢳⡇
⠀⠀⠀⠀⠀⠸⣷⡻⣷⣝⡶⠤⢤⠤⢖⣫⣾⣿⣿⢣⣿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣷⢻⣿⣿⣮⣗⡦⠤⠤⠴⣞⣵⡿⣣⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣽⣟⣿⣿⣿⣛⣿⣛⡿⣿⣇⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣾⡿⣟⣻⣟⣻⣿⣿⣿⣿⣽⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⢿⣿⣿⠿⣿⠿⣿⡎⣿⣷⣝⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣡⣿⡟⣼⡿⠿⡿⢿⣿⣿⡿⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣩⣍⣋⣴⣿⣿⣿⣷⣬⣙⡲⠦⠤⠤⠤⠤⠶⢚⣩⣴⣿⣿⣿⣷⣌⣛⣭⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣝⡻⢿⣿⣿⣿⣿⣿⠟⣠⣶⠟⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣟⡚⢯⠽⢏⣅⣘⣛⣁⣀⣾⣛⣛⣛⣛⣯⣭⣵⣒⣩⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⠤⢬⣭⣭⣭⣽⡿⠿⠿⠟⠛⠋⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠷⠷⠄⡀⠀⠀⡀⠴⠷⢇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠤⠠⠀⠠⢀⣄⠲⠆⣬⠁⣩⡄⠲⢂⣄⠡⠀⠀⠠⢴⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡡⠀⠜⠳⢆⠟⠀⠀⠀⠈⡏⡏⠀⠀⠀⠘⣧⠶⠛⠄⠠⡉⠄
⠀⠀⠀⠀⠀⠀⠀⢰⠘⣿⠀⠀⠀⠀⢗⣄⡀⠀⢀⣿⣇⠀⠀⣀⣄⠇⠀⠀⠀⢸⡗⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠄⣀⣀⢄⣾⡻⠋⠛⠏⠁⠉⠟⠋⠻⣻⣆⢄⣀⡀⢜⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠟⠫⢭⡻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣫⠭⠛⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣽⣿⣦⣄⣀⣀⣀⣠⣴⣿⣿⣅⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠉⠛⣻⣷⣿⡛⠋⠉⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣘⠀⡃⠀⠀⡀⠀⠀⠀⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳⠀⣡⣉⠴⢠⣄⠁⣿⠀⠀⠀⠀⠁⢢⢠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡿⠉⡗⠒⠿⢿⡷⠀⠀⠀⠀⢁⣤⣀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠷⣶⣶⠶⠶⡼⠁⠀⠀⠀⣰⣿⣿⣅⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢠⠄⢀⠀⢀⡠⠦⡁⠀⢶⠏⢰⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣀⠀⠘⣚⡉⠀⠐⠋⣰⠎⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢋⡿⠹⣿⣿⣿⣷⣦⣿⠠⡤⢖⣧⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢴⡋⡂⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣷⣭⣿⣿⣿⣿⣿⣿⠷⣿⡿⠇⠀⠊⠀⠀⠀⠀⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡤⠀⠀⠀⠀⠈⠛⠛⠫⠑⠻⠙⠋⠉⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠃⠀⢾⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠤⢄⠄⠈⠑⠐⣀⠀⠀⠀⠐⠚⠋⣼⠿⠟
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠐⠁⠉⠐⠀⠀⠁⠤⠀⠀⠀⠐⠶⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⣤⠠⡀⠀⠀⠴⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠻⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠠⡬⠴⠂⠂⠀⠀⠀⠀⠂⠚⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⡄⠀⠀⠀⠀⠠⠀⠀⠙⠋⠀⠐⡄⠀⠀⡈⠀⠀⠀⠀⠀⣠⠊⠉⣝⠻⠶⡟⠢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠀⠀⠈⠉⠀⠠⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⢀⢠⣿⣟⡔⡾⠘⠉⠳⣄⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡀⠀⠀⠀⠀⣛⠉⡛⡟⠽⡻⠢⡄⠀⠀⠘⣷⣤⡄⠀⠀⠁
⠀⠀⠿⣦⣤⣤⣴⣾⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⢱⡀⠀⠀⠛⠈⠓⠴⠀⠐⠳⠤⠉⠀⠀⠀⠘⢿⠕
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠒⠂⠁
⠀⠀⠀⣀⡀⠀⢀⡴⠀⠀⢀⡄⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠨⠀⠀⠀⠰⠀⠀⠀⠀⠂⠈⠀⠀⡠⠴⠲⡆
⠀⠀⠀⢹⡯⢽⡿⠀⠀⠀⠘⠛⠙⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⡲⢉⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠁⡜⠀⠀⠀⠀⡤⠒⠀⢀
⠀⠀⠀⡀⠀⠀⢀⡂⠀⠀⠀⢀⠀⡆⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⢈⠇
⠀⠀⠀⠉⠙⠋⠉⠁⠀⠀⠀⠘⠒⠐⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠙⠒⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠠⣤⣤⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡆
⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠈⢠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠋⢉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣬⣿⢩⣬⡇⣬⣽⢪⣤⣍⣍⣡⣤⣤⣴⣦⣴⣶⣶⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡤⠠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⡿⠛⢿⠟⠹⣿⠛⠙⠇⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡶⠿⠖⠛⠒⢲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢳⣶⠖⣶⡖⢂⡖⠒⠀⠐⠒⠒⠒⠒⠤⡀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠱⣿⠉⢺⡏⡁⠀⠀⠀⠀⠀⣠⣤⠈⣇⠀⠀⠀⠀⠀⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⣴⠩⠶⠥⣤⡤⢤⡔⣠⣬⣥⡤⣬⣤⠬⣤⡬⢤⣤⣬⠭⡭⣽⠦⣥⣿⠹⠶⣶⡶⠦⡀
⠀⠀⠀⠀⣠⠖⠂⠐⠢⡀⠀⠈⠳⣄⣀⣀⣘⣶⣶⠶⠶⠶⣿⠰⠶⣾⠃⠐⣏⡈⠲⡄⠘⠛⠻⠟⠛⠛⢛⡛⢿⡿⠘⠛⠛⠛⠋⠁
⠀⠀⠀⠀⣛⡄⠀⠂⣰⡇⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⢸⡀⣀⣿⠀⠀⡇⠈⣷⣛⡛⣻⡟⠛⠛⠛⠛⠛⢛⡅
⠀⠀⠀⠀⣿⠆⠀⡷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠥⠤⠤⠤⠤⠤⠤⠿⠥⠼⠦⠤⠶⠖⠚⠉
⠀⠀⠀⠀⡿⢶⡤⠟⣿⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠲⡀
⠀⠀⠀⠀⠇⠈⠁⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣆⣆⢀
⠀⠀⠀⠀⠈⠀⠈⣠⣿⡀⢢⡀⢀⣀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠉⠁⠉⠭⠟⠙⡟⠈⠁⠛⢏⠉⠉⠛⠛⠛⠒⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁⠈⠇⠀⢏⢀⣇⠀⢀⡀⣀⣀⣀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠃⠇⡀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⢻⠀⢿⡀⠛⠉⠉⠉⠙⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠗⠋⠀⠁⠀⣈⠠⠀⠀⠸⠀⢸⠀⠸⠄⠄⠀⠀⠀⠀⠀⢀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠰⢲⠒⣼⠂⣸⣀⣴⠀⣀⠀⢆⠀⠀⢰⠁⠀⠀⠀⢹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠆⠀⢸⠀⢿⠀⢹⣿⢸⠀⡇⢩⣾⡇⠀⢸⡀⠀⠀⠀⡠⠈⠳⣦⡄⠀⣀⡤⣄⣀⣀⣀⣀⣀⣐⣗⣀⣀⣉⣠⣼
⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠰⠀⢸⠀⠀⣿⣼⠀⣇⢎⡏⡇⠀⠀⠈⠐⠐⠊⠀⠀⠚⠁⠀⣼⠫⠊⢙⣽⣿⣿⣛⣛⣙⣛⠛⠛⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠈⣎⠆⠈⠲⣄⣀⢸⡀⠀⣿⠀⢀⡡⢊⢨⠃⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⡅⢐⠛⠀⢠⣾⣿⣦⠀⠈⡆⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢯⡢⡀⠀⠀⢹⡃⠁⠿⠉⠈⢁⡼⠃⠀⠀⠀⠈⠉⠉⠀⠀⠛⠀⠄⠀⠘⢿⣥⣼⣿⣿⣶⣾⣿⣟⣉⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⢀⣰⡇⢐⣣⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀⠀⢀⣀⠀⣠⣤⣤⡤⠖⠚⠉⠀⣿
⠀⠀⠀⠀⣼⠋⢹⡏⠉⣿⠉⠉⠉⠉⠉⠉⠉⢹⡿⣿⠿⢿⡿⠿⢿⡿⠿⠿⠿⠿⠿⢿⣟⣿⣿⢤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢿⡀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢸⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠠⠤⠀⠄⠀⢠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠆⠀⠀⠆

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠙⢿⡟⠻⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣴⣄⠀⣠⡶⢻⡇⢠⡞⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⡇⠉⠙⢿⣄⠁⢀⡾⠠⣋⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⠉⠀⣴⢦⣄⡈⠳⠾⣥⡞⠁⢀⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡿⠀⢀⣀⡲⠈⠁⣠⠔⣚⢿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⡴⠀⠀⡀⠉⠉⠀⢸⡃⠺⠏⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⠀⡞⠉⡀⠛⠒⠼⢧⡀⠀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠁⡄⢷⣍⣁⣲⠖⠛⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣄⠀⢃⠀⠩⠭⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⡄⢈⣿⣾⣄⣠⡀⣰⡇⠈⠢⣀⠀⠀⠈⠉⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠐⣹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠑⣢⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠈⠙⠻⡆⠀⠀⠛⠋⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⡀⠀⠹⣍⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠹⣦⠀⠐⠀⠀⣻⠀⠀⠀⣠⠋⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢀⣻⢀⠰⡎⢰⠃⣰⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠹⣷⡐⢯⡉⣾⠿⣿⡉⠛⢧⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠹⣿⣴⠹⡇⠄⠸⣵⣤⠴⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠹⠇⠐⢻⢿⣶⡟⣿⣶⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠈⠓⠛⣠⠟⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⡀⠀⠀⠀⡞⢬⡦⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠤⢀⠀⠀⠘⣼⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⣪⣶⡷⢾⣿⣷⡾⠤⠤⢤⣟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠻⣥⣭⣴⣾⢧⡀⠀⠀⡟⠳⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠠⣦⠄⣤⡶⠞⠻⣿⣿⡿⣶⣽⣷⡶⣥⣤⡼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠀⠈⠃⢀⣀⣈⡛⠻⢶⣤⣬⡴⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠟⢳⣦⣀⠀⠉⢿⣿⣿⣼⣿⡿⣤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⠁⠈⠳⣤⣀⡙⠛⠻⠛⢀⡍⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣀⣸⣿⣃⡀⠀⠀⠀⠀⠉⠉⠒⠒⠉⠀⡀⣤⡐⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣥⣬⣥⣴⣥⣽⣮⣴⣯⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠂⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡾⠋⣿⢠⢻⡇⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣠⡟⠀⡀⡿⡀⢸⡇⣿⠀⠀⠀⣾⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⢰⡟⠀⠛⠉⣤⢀⡇⡇⡇⢸⢣⡏⢀⣤⡾⢣⡶⢋⡔⣼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⡛⠷⣤⣉⠘⣿⠈⡄⠀⠀⡾⢱⡇⡇⢸⢸⣧⠞⣭⡾⣿⢀⡾⣰⠇⠀⠰⠟⢀⡶
⠀⠀⠀⠀⠀⠀⢀⣿⠇⣿⣿⣯⣛⣿⠀⣧⠀⠀⠇⣿⣩⡇⢸⣈⣥⣾⣿⣷⠏⡼⢡⡿⠀⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⢰⣿⠋⠘⢿⡹⡟⢿⣿⠀⡟⠀⠀⢸⣯⡿⣵⡿⠛⠉⢹⡇⡏⢸⠇⣾⠃⠀⢀⣴⠟⣴⢸
⠀⠀⠀⠀⠀⡼⠃⡀⠛⠘⣧⣷⠈⣿⢀⡇⠀⠀⣿⣿⣿⣿⠁⠀⠀⢸⢿⠇⣾⢰⡿⢀⣴⠟⢁⣾⠃⣼
⠀⠀⠀⠀⢀⡀⠀⣿⠀⡀⢸⣿⡄⣿⠘⣧⡀⠂⣿⡾⣻⠇⠈⠁⠀⡜⠄⣠⡟⢸⡧⡞⠁⣤⠞⡳⠀⠛
⠀⠀⠀⠀⠸⣷⡄⢹⣆⠀⣼⣿⡇⢻⣴⡿⢁⣠⡿⢠⡏⢀⠀⠀⠀⠁⢠⣿⡇⣸⠇⣠⣾⢯⡞⠑⠀⣰
⠀⠀⠀⠀⠀⢼⣿⠈⣿⠰⣿⣿⣷⢸⡏⢧⡟⠁⠇⡿⡔⠀⠀⠀⠀⠀⠀⣿⢠⡿⣰⢿⣿⣿⠀⣰⠛⢁
⠀⠀⠀⠀⠀⠘⣿⡆⢸⡇⣿⣿⣿⢸⡇⣾⠁⡞⣸⣷⠃⠀⢠⡄⠀⠀⢠⡇⣾⠱⣯⣿⡿⢣⡾⣣⣴⠏
⠀⠀⠀⠀⠀⠀⢸⣷⠸⣇⣿⣿⣷⠸⢇⡿⠀⣧⣿⠟⠀⢀⠀⢀⡄⣤⢸⢱⡏⢰⣿⡿⣳⡟⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⣿⣿⢀⣾⠃⣸⣿⣿⠀⠀⠈⠀⣾⢱⡇⡄⣸⡇⣿⣟⣵⢟⣼⡿⠁
⠀⠀⠀⠀⠀⠘⠁⢻⡇⢿⣿⣿⡃⣿⡇⢠⢿⡿⠋⠀⠀⡄⢠⢧⡟⢠⣧⣿⡇⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⣿⣿⣇⣿⠁⢸⣿⠇⠀⠀⣴⢃⡟⣼⠁⡞⣽⣿⡇⢟⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢹⣿⣿⡿⠀⢸⠃⠀⠐⣶⢇⡿⣸⠃⠈⢰⠇⣿⠁⢚⣿⠃⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⡞⢿⣿⡇⠀⡟⠀⣠⣞⣡⡾⣱⠏⡄⢠⠏⣼⠏⢠⡾⠃⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠛⣁⢸⡇⣼⢹⡇⠸⠁⠈⢹⢿⡟⣵⢋⡜⢀⠏⣸⢋⡴⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠉⠀⢰⡏⣾⠇⢴⠀⣠⡟⣼⡿⣧⡞⢀⠊⣼⠃⠚⠁⡀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡞⣰⠏⡀⢀⣴⠟⢠⠟⣰⡯⠀⢂⣼⠃⠀⠀⠾⠿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⣼⢡⡏⡼⠁⣿⠏⢠⠏⣼⢿⠁⢠⡾⡁⠀⢂⠀⢠⠀⢤⡀
⠀⠀⠀⠀⢀⣴⠿⠋⣴⢃⡞⣼⠃⢰⡟⡀⡞⣼⢃⣤⡴⠛⢀⡇⠐⣼⡆⠹⣆⠀⢹⡆
⠀⠀⠀⣠⠟⠁⢰⣾⢃⡾⣱⠇⢠⡟⡘⡼⣰⣷⠛⣁⡔⢡⢸⡧⠘⢹⣇⠀⢹⣇⣼
⠀⠀⣼⠋⢀⣤⣾⡏⡼⢡⠏⢀⣿⡴⣽⡷⠏⣱⣿⠋⠀⡼⣸⣇⠀⣆⢻⡆⠀⠛⠋
⢀⡾⢁⣤⣿⠟⠋⢹⣥⣾⠀⣼⡿⣱⢟⡴⢿⣿⣿⡔⣸⠇⣿⣿⢰⡏⢸⡇
⢊⣤⣧⣤⠞⠁⣠⣾⢹⡇⢠⣿⠟⠁⢸⡇⢸⣿⡇⡇⣿⠀⣿⣿⡌⢿⡌⣷⡀
⠀⢀⡜⠁⣠⣴⣿⣇⡾⣡⡟⠁⠀⠀⢸⡇⢸⢿⡇⣇⣿⢸⡟⠛⢿⣆⠻⡿⠃
⠀⣨⠶⠛⠉⠰⠏⣼⠱⠋⡀⢠⡆⠀⠀⢷⠸⣇⠻⢸⣿⢸⡇
⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠃⣉⢳⡘⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠓⢸⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⠿⣿⠛⠿⠿⣿⠿⠿⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣶⣷⣶⣿⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⠉⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢄⣐⣤⣤⠀⠀⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠀⠀⠀⢨⠙⢶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠀⠀⠀⠁⠀⠓⠀⠄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠒⣒⠒⠀⠀⠀⠀⠀⠠⠀⠀⢈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣘⡂⠀⠰⢖⣐⣈⣩⣆⠐⣼⠊⠇
⠀⠀⠀⠀⠐⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣽⠃⠀⠀⠈⠀⠈⠁⠃⠘⡤⢲⠊
⠀⠀⠀⠀⠈⠿⠿⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠁⠁⠀⠀⠀⠀⠀⠀⢸⡦⠋⣖
⣏⠩⢭⢹⡇⣖⣶⢺⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠑⢇⠀⠀⠄⣐⡂⠔⡞⠉⠁⢀⣜⠀⠑⡄
⣿⣀⢨⣚⡇⠇⠿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⠲⡀⠢⡤⢥⡞⠀⠐⠈⠁⠀⠀⡨⠝⢳
⢩⣽⣯⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠬⣦⠛⠛⢛⢯⡟⢀⠤⢠⠀⠀⡤⠋⠱⡀⠌
⣸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⢀⠓⠂⠒⠐⠏⠀⠀⠠⠋⠀⠜⠀⠀⠀⠹⢄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⢸⣠⣤⢀⣰⠀⠀⢠⠁⣠⠈⢆⠀⡀⡠⠴⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡇⣣⡎⣠⡐⢁⣁⣄⣈⣄⠀⣋⣼⠋⠔⠳⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⣭⣙⡟⢙⣻⣥⣴⣷⣽⠟⣧⣀⡠⠤⠲⣋⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠈⠉⠠⡯⠀⠑⠺⠲⣫⠁⠀⠀⠀⠀⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠀⠈⣿⠁⠀⠀⠀⡃⠂⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⡓⢎⠐⣟⠋⠛⠚⠉⢉⢷⠈⠄⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢼⠥⢼⠀⠸⣆⡁⢈⠀⠀⠘⣆⠀⢠⡔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢿⡂⢀⡻⢤⢦⡴⠤⠒⠛⣯⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡣⢘⡅⠀⠀⠁⠛⡶⠖⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⣓⣿⡆⠀⠀⠀⠀⢰⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠵⠼⠚⠓⠃⠀⠀⠀⠠⠛⠤⠚⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⣀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⣾⢟⣿⢓⢀⣾⡡⠖⠀⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠰⠀⠦⠘⡄⠀⠀⠀⠀⠈⠈⣻⠶⡈⢉⠀⣰⣶⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢦⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⣅⡺⠟⣸⠛⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣂⣠⢤⣄⣜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠙⢄⣠⠃⠀⠀⠔⠀⠉⠀⠈⠙⠂⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⣸⠇⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⢥⠄⡀⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠁⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣾⣿⡿⡼⡟⠀⢀⡟⢂⡇⡔⡁⠀⠀⣀⠀⠀⠀⠚⠁⠈⢊⢱⣾⠁⠀⠀⠀⠀⣠⣤⣾⣿⣿⠿⠛⠃⠀⠠⠀⡠⡀⣀⣀⣀⠁
⠀⠀⠀⠀⠀⢸⣿⣿⢇⡟⣷⠀⢸⠀⠌⠁⠵⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢾⣶⣶⡿⠿⢿⣛⠉⠀⠀⠀⠀⡤⣵⡡⠔⠿⣿⠿⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡼⠀⣿⡀⠘⡞⠲⢦⡴⠀⠀⠀⠒⠒⠒⠒⠒⠳⡆⠀⠀⠀⠀⠀⠀⡷⠾⠗⠀⠀⠿⠉⠐⠂⠀⠈⡆⠄⠀⡠⠾⡀⠀⠀⠈⠹⠟
⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠸⣇⠀⢇⢠⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⡎⡵⠴⠚⠁⠐⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣆⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠤⠂⠉⠑⡀⠢⠄⠈⠢⠠⠈⠀⡂
⠀⠀⠀⠀⠀⠀⢘⠾⣿⡀⠀⠀⠙⡿⡉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠸⠠⣀⠈⠀⠰⢀⡔⠉⠉⢦
⠀⠀⠀⠀⠀⠰⣷⡘⠈⢷⣤⣤⣼⣿⣽⣿⣽⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠃⠀⣿⠟⠀⠜⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⡀⠀⡴⠃⠀⠀⣱⣿⡀⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣶⣤⣤⣤⡠⣶⡟⠀⠀⣠⠚⠉⡝⠂⡀⠀⠉⠉⠆⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠛⢋⠣⡀⠈⠢⡀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⢠⠁⠀⢰⡇⠀⣱⣦⣤⠠⠄⠠⠄⠀⠇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁⠈⠢⠀⠐⠄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠔⢄⣀⠠⠃⠴⠿⠿⠿⠷⠀⠈⠀⠀⠀⠀⠀⠸⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠠⠀⠀⠠⠀⠀⠀⠠⠀⠀⠤⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣉⡬⠦⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⢀⠞⠀⠀⠀⠙⢦⡀⠀⠀⠀⣀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡔⠊⠁⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠙⠲⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠊⠀⠀⠠⢤
⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠘⣿⡿⠃⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠎⠀⠀⠀⠘⠲⠦⠤⣀⣀⡄
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⡔⠒⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢨⣀⡢⠒⡄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⣠⠶⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⣠⣴⣿⡿⢗⡽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⢐⣥⡿⢟⡩⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢠⡶⠒⠒⠒⠒⠒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⡆⠀⠠⠊⠀⣈⡭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠓⠦⢤⣈⡀⣰⠆⠀⠀⠉⠑⠒⢤⠀⠀⠸⣄⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠑⠒⣲⠶⠲⠶⢞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢠⠁⠀⠀⠀⠀⢀⡐⠀⠀⠀⢠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⣸⠋⢋⠁⠀⣀⣤⣤⣤⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠋⢈⢀⣴⠶⠶⠷⢄⡀⠈⠑⢦⡀⠀⠈⣁⠉⠀⠸⣴⠋⣁⡸⠖⠲⠶⣾⠶⠶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⠐⠒⠒⠊⠓⠲⢊⠥⠒⠈⠁⠈⠉⠒⠤⡶⠶⠒⠒⠒⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⠃⡠⠔⠒⠀⠀⠀⠀⠀⠈⢄⡀⡀⣀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠍⡇⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡸⠉⠉⠛⠲⠒⢒⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠊⢀⢹⠄⠀⠠⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⢀⠌⢹⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠀⢢⠀⠈⠀⡜⠁⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠈⣷⠥⠨⢭⡇⠐⠄⠀⠀⠀⢀⠄⠀⢸⡗⠠⣄
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠑⠀⠜⠀⠀⠀⠉⠲⢤⣄⣀⣀⣤⡴⠾⠉⠠⠎⠈⠖⠨⠷⠃⠀⠈⠂⠀⠀⠈⠒⠶⠶⠶⠾⠟⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⠄⢀⠀⠀⠀⣶⠚⠒⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⢠⣿⣇⣴⡏⠀⠀⠀⠨⠀⠀⣼⣿⡷⢤⡀⠀⠀⠀⠀⠀⠀⠈⠚⢤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⢘⣿⣿⣿⡇⠀⢠⠀⠈⠀⢀⣿⠏⠁⠀⠑⣄⠀⠀⠑⢦⡀⠀⠀⡈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢭⠿⠿⠖⠀⠉⠻⣿⠃⠀⠀⠀⠀⡠⠋⠀⠀⢀⣤⣾⣿⣧⠀⠀⠀⠈⢶⡀⢛⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⣄⠀⠙⣧⢂⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠢⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣦⣤⣬⡳⡀⠘⠧⢠⠀⣠⣴⣿⣿⣿⠛⠋⠙⢻⣿⠇⠀⠀⠀⢀⠀⠰⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠃⠤⠀⠀⠀⠀⠀⠃⠀⠤⣅⠀⣀⣼⣿⡿⠟⣉⢔⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠀⠄⢠⣿⣷⢆⣸⣾⡀⠀⠀⠀⣠⣶⠟⢷⣮⡻⣿⡀⠻⢛⣭⣿⣿⣿⠻⠿⠀⠄⠀⠀⠀⠀⠀⠀⠃⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠺⠇⠀⠙⢷⣄⣴⠀⣟⠁⣠⣾⡿⣳⢌⣵⠞⠛⣿⡉⠉⠙⠷⠀⠀⡀⠀⠀⡀⢀⠀⠀⠈⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⣀⠹⣿⡿⢿⡙⣏⠸⣏⡳⠀⠀⢾⡠⣀⠈⠀⡼⠃⠀⠀⠀⠀⠱⢠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⠀⠀⠀⠀⢀⣠⠞⠋⢀⣴⡿⢟⣦⠹⣟⠈⣷⣴⡷⡝⠿⠌⢆⠈⢷⡷⡶⠋⠀⠀⠀⠀⠀⠀⣀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠀⠀⢀⡔⠛⠁⣠⣶⢟⣩⡴⠛⢟⠷⡙⠈⡿⢮⢿⠊⢄⡲⡼⣅⠀⠻⣦⠀⠀⠀⠀⠀⠁⠘⠡⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣴⡿⢉⣴⠟⠁⠀⠀⠋⠙⢷⣤⣄⣴⡱⣾⣦⠳⢶⡊⣦⠀⠹⣧⢸⡄⣆⢶⠀⠔⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⢠⣿⠁⠀⣀⠀⣶⠀⠻⣀⣈⣉⡑⢶⠋⢅⠀⠀⠀⠘⣧⡠⡘⢿⣹⣿⡻⣓⡶⠲⠦⢤⣤⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠸⣿⠀⠿⠂⣂⣭⣴⣶⢾⠿⠙⠻⠟⠙⣺⣷⡈⠧⠀⣀⡴⢿⡝⡇⠈⣿⣧⣿⣉⣻⣛⡿⠂⠈⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⠘⣁⣠⡴⢶⡾⠿⢛⣋⢈⡲⠠⣏⡖⢛⡴⠀⢈⣩⣴⡶⢏⠁⠀⠀⣻⢸⡆⢸⣟⠿⣿⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⡾⠛⠉⠈⡃⢒⣶⢭⣵⠤⠃⠭⠶⠚⢉⣩⠴⠚⣉⡤⠓⣿⡇⣗⣤⣞⠥⠚⢀⣼⠋⣾⢎⡓⢦⣀⠀⠀⢂⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠐⣿⠙⠀⠀⠀⣖⣈⡡⠴⠾⣿⣦⢀⣤⡶⣟⢉⣤⠔⠋⠁⣀⡠⠖⠚⠿⠀⠤⠤⠖⢿⠻⠛⠾⠟⢿⡤⠙⢿⣶⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣟⣿⠁⠘⠾⣧⡤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠗⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠒⠒⠒⠂⢼⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠞⣽⡏⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⢸⣿⡏⠈⠑⠢⢄
⠀⠀⠀⠀⢀⠃⠀⣿⡇⠀⠀⠀⠀⢻⣿⡄⠀⢠⠊⠁⠉⢢⠀⠀⠀⣴⣶⡄⠀⢸⠀⠀⠉⠀⣾⣿⠃
⠀⠀⠀⠀⠈⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠸⣀⣴⣶⣌⠀⣠⠊⠝⣿⣷⠀⢸⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⣿⠃⠀⠀⣽⣿⠟⡾⠊⠀⠀⣡⠹⣿⣇⡎⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣾⠏⠀⠀⠀⢿⣏⡰⠁⠀⠀⠀⠘⠀⢻⣿⠇⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢀⡼⠃⢀⣼⡟⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣴⡶⢦⠀⠀⠀⠀⣠⣴⡦⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠁⡼⠀⠀⠀⣸⠟⠁⣰⣶⠀⠀⠠⠔⠒⠊
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠀⠀⠀⢸⣿⠀⡠⠒⢿⣦⡀⢀⣼⣿⠋⠀⠀⠀⠀⣿⠀⢀⢿⡟⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⢰⠁⠀⠈⣿⣿⠁⠸⣿⡀⣠⠎⠀⠀⠻⠤⠊⢸⡇⢀⠎⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⣀⠀⠀⢿⣾⠀⠀⡎⢹⣿⠀⠀⠉⠉⠁⠀⠀⠀⢀⣀⡀⠘⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣇⠀⠈⠉⠀⠸⠀⢸⣿⠀⢀⣠⡤⣄⠀⠀⢠⣿⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⡀⠀⠀⠀⠀⠙⠛⠋⠀⣿⡟⠀⢸⠀⠀⢸⣟⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣶⠦⡀⠀⠀⢿⣿⣄⠎⠀⠀⢸⣿⡰⠁⡠⠒⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⠏⣠⠃⠀⠀⠀⡹⣿⣷⠂⠀⠀⣿⡇⡰⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣿⡇⠀⢸⣿⠉⠀⠀⠀⡠⠚⠁⣿⠃⠀⢠⠎⠹⣷⡏⠻⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣠⠟⠀⠀⠀⠉⠓⠒⠒⠉⠀⠀⠀⢻⣀⠔⠁⠀⠀⠀⠀⠀⠈⠑⠢⠤⠔⠃

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⢈⡆
⠀⠀⡉⠒⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⣀⠀⣀⠀⠴⠋
⠀⠀⠀⠁⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠦⠤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠠⠤
⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠦⡀⠀⠈⠂
⠈⠀⠄⠀⠀⠀⠴⠤⠀⠀⠁⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠙⠂
⠀⡂⣂⠀⠀⠀⠀⠑⢸⠄⠀⠀⢀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣶⠄
⠀⠸⢀⠡⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠘⠐⠀⠀⠀⡆⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⡤⠚⢀⠀⠀⠈⠊
⠇⡀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠋⠉⠉⣀⠴⠚⠂
⠰⠂⠀⠈⠀⠀⠀⠠⠀⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⠍⠉⠄⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠈⠁⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠘⢏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠆⠤⣄⠀⠀⢀⣤⣁⣄⡀⡀⠀⠀⠀⠀⣾⣿⡝⣲⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⣰⠀⡿⠿⠀⠀⠄⠉⠩⠛⠁⡇⠁⡀⠀⢠⡀⣾⡓⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣿⠂⡴⠇⢘⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⡠⠀⠀⠘⢽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣟⢛⣮⣸⡅⠀⠀⠀⠀⠣⣸⢃⠔⠁⠀⢂⠀⠁⠘⠆⠀⠚⠙⣛⠛⠢⡀⡀
⢶⠀⠠⣆⠀⠀⠀⠀⢉⠻⠟⠡⢤⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⢨⠀⠀⠀⢀⠐⣠⠶⠟⠓⠋⠁⠈⠢⣄
⢈⡇⣐⠉⡄⡀⢤⢒⡎⢙⡏⡴⢠⣟⣷⣸⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⢸⡇⢹⢽⣿⡇⡟⢸⠀⢸⢠⢧⡧⡸⡿⣿⠀⠀⠀⠀⠀⠀⢀⣽⡣⠎⢻⣿⣿⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠘⣫⣦⢹⠸⡄⡟⢸⢸⡇⠝⡜⡶⠇⠀⢢⠀⠀⢠⠎⣼⠱⣀⣿⣿⣯⣞⠠⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢀⠀⠀⠉⠚⡎⣇⣇⢧⢸⢸⢡⡲⠔⠀⠘⠄⢲⡆⢤⠀⠀⠇⢠⣾⣿⣿⣿⣯⣾⣶⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠇⠀⠠⠆⢄⣸⣹⡸⠸⢼⢸⣼⠁⠘⢄⠀⠀⠉⠀⡀⠓⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⡄
⠈⠢⠄⠀⠀⠤⢤⣷⠿⣼⣏⡼⠻⠀⠀⠈⠳⠤⠴⠈⠤⠠⠠⢼⣿⣿⣿⣿⠟⢿⣿⣿⣿⣯⣀⠀⠀⠀⠀⡰⡆
⠀⠀⠀⠈⠉⠁⢸⣿⣿⣿⣿⣿⣿⣧⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠉⢀⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣬⡄⠘⢄⠈⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣾
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠑⢄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⢶⣶⣦⣴⣶⣤⣤⣾⣿⣿⣦⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠶⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡼⣿⡿⠛⠿⡿⠋⠈⠀⠀⠀⠀⣴⠛⠛⣿⣿⣿⣿⣿⣿⡿⠟⠙
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⠀⣸⠿⡇⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⢰⣴⡆⡆⠀⡦⣤⢸
⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠘⠂⠐⠂⠀⠀⠠⠀⠀⠀⠀⠈⠙⠀⠘⠉⠃⠃⠀⠃⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⣀⠀⠀⠠⠀⡀⠀⠀⠐⠐⠐⠐⠒⠒⠒⠂⠀⠐⠋
⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠈⠂⠂⠄⠀⠂⠀⢙⡆⠀⠆⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⠀⠿⠇⠘⠃⠐⠁
⠀⠀⣀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢰⠀⠀⢸⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠈⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⡈⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠞⠉⠃⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠉⠀⠀⠀⠀⠄⠀⢈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠉⠈⠁⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⡀⡆⢸⢸⠂⠀⠀⢸⢸⡇⡇⠀⠀⠀⡇⣤⢸⠀⠀⡀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⡀⠀⢀⠀⡆⡇⢰⠀⠀⡄⣿⢰⢸⡇⡇⢸⢸⡆⠀⣤⢸⢸⡇⡇⢰⢠⠀⡇⣿⢸⠀⠀⡇⣤⢰⠀⡇⡇⢰⢀⡀⡇
⠀⠀⠀⠀⡇⣶⢸⠀⡇⡇⢸⢸⡆⡇⣿⢸⢸⡇⡇⢸⢸⠀⡇⣿⢸⢸⡇⡇⢸⢸⠀⡇⣿⢸⢸⡇⡇⣿⢸⠀⡇⡇⢸⢸⡇⡇
⠀⠀⠀⠀⠇⠉⠸⠀⡇⡇⢸⠘⠁⡇⣿⢸⢸⡇⡇⢸⢸⠄⠃⣿⢸⢸⡇⡇⢸⢸⠀⡇⣿⢸⠘⠃⡇⣿⢸⠀⡇⡇⢸⠸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠈⠀⠀⠀⠿⠈⠘⠇⡇⢸⢸⡀⠀⠀⢸⢸⡇⡇⠈⠀⠀⡇⣿⢸⠀⠀⠇⠀⠈⠀⡇⠇⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠘⠁⠀⠀⠘⢸⡇⠃⠀⠀⠀⠃⠁⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸

⢐⢢⠀⠢
⠢⣆⣆⠠⠓⣻⣗⠠
⠐⢺⡄⠢⠺⣿⣿⣷⣄
⠀⠁⠴⠀⠀⠘⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣿⣿⣦⡀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣿⣿⣿⣷⣏⠉⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠚⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠘⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⠀⠀⣀⡘⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠢⣄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡽⠖⢪⢳⡌⠓⠶⠤⠴⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⠑⣤⠤⣄⡀⠀⠀⢠⢤⡀⠰⠉⣠⠤⠚⠀⠻⢦⣐⣲⠆⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣇⣀⣀⡙⢦⡀⠸⣄⠀⠰⠊⠁⠀⢀⠀⣠⣄⠈⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢢⡱⡄⠈⢧⡀⠀⠀⠀⢠⡥⠚⠁⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⢠⣌⡇⠀⠀⠀⠈⠢⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣄⠀⠉⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⢣⡀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣆⠀⠀⠀⠀⠀⡠⠖⠒⠓⠒⠦⡀⢳⠙⢦⢀⡠⠊⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠘⣆⣷⠚⢋⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠳⢦⣸⣄⡀⠀⢀⣀⣤⡴⠞⠋⢈⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡈⠉⠉⠉⠉⠉⠀⣀⡴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡐⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢿⣿⣿⡟⣰⣿⣷⣄⢶⣶⣶⣬⣭⣙⣛⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⣝⡿⢸⣿⡟⣩⣴⡌⣿⣿⣿⣿⣿⣿⣿⣶⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⠛⠛⢻⡇⣼⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⡿⣇⢸⣿⡘⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣫⣾⣿⡜⣿⣿⣦⡭⢰⣿⣿⣿⣿⡿⠿⢛⡉⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣛⣋⣀⣋⣩⣭⣥⣶⢰⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⠟⣡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠶⢈⣿⣿⠰⠆⣹⣿⡇⠶⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⢛⣛⣫⣾⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠇⣼⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣤⢠⢠⣄⢠⢀⡤⣄⢠⠀⡄⣤⡀⡄⣠⠤⢠⡤⢠⣄⢠⡄⣤⠄⣤⡀⡤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⡄⡏⢿⠸⠏⢿⠸⣤⡾⠸⣤⠇⠿⠹⡇⢧⣄⢸⣯⠸⠻⠟⡇⣿⡅⠿⠹⡇⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣧⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣜⢿⣦⡘⢧⢻⡄⠘⠓⠀⠀⢀⣠⣤⣶⣶⣶⣶⢦⣤⣀⠀⠀⠀⠀⠸⠏⠀⣠⣾⣵⣫⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠘⢿⣮⠻⣿⡌⢿⡟⠀⠀⠀⢰⡿⣻⣉⣍⡙⠉⣀⣼⣿⣿⣷⡀⠀⠀⠀⠀⠀⢟⣽⣿⠿⢋⣠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⢷⡀⠻⣷⡘⣿⡄⠀⠀⠀⠀⣬⠘⠁⠙⠛⠓⠘⠻⠟⠛⠩⣿⣿⡄⠀⠀⢀⣤⡿⣋⡵⠶⢋⣡⡴⠞⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⣿⣦⡙⢦⡈⠳⠌⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡇⠀⠀⠀⠿⢛⣡⠴⣟⣫⣥⣶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣝⠻⣿⣿⣦⡻⠆⠀⠀⠀⠀⠀⢀⡋⠐⢳⠦⠀⠠⡶⠶⠦⠀⠀⣻⣿⠁⠀⠀⠀⣼⣿⣿⣿⣟⣯⣥⠶⠾⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⣬⡙⢷⣌⠻⣿⣿⣆⠀⠀⠀⠀⠀⢸⠁⠀⠛⠓⡀⠀⠘⠷⠀⠀⠀⠛⣭⠀⠀⠀⠈⠁⠚⢛⣋⣭⣴⠶⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣉⡂⠉⠳⣬⠛⠿⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⣀⡙⠛⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠃⡀⠙⠁⢀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠰⠶⠶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣭⣙⡛⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠒⠈⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⠛⠻⠷⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣄⡠⡀⠉⠁⠀⢀⣴⠏⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠆⠀⠀⠀⢀⣤⣤⣴⣶⣶⠿⣿⣿⡇⠈⢻⣶⠟⠛⠁⡈⠀⠀⡏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⠀⠀⠀⣴⠋⠀⠀⠈⠛⠁⢠⣿⡟⢁⣠⣿⣿⡄⠀⠐⠀⠀⠀⡇⠀⠀⠈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠾⠛⠋⠀⠀⣠⠞⠈⠁⠀⠀⠀⠘⣦⡘⠛⠀⠛⣡⣿⠁⠘⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⢠⣼⠄⠀⠀⠀⡀⠈⠁⠀⢠⣾⣿⠉⣄⠌⠀⠀⠰⠂⠒⠁⠀⠀⠀⠀⠀⠠⠖⠚⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠉⠀⠀⢸⣿⠀⠀⠀⠀⣷⠀⠀⠀⢸⣿⣴⣾⠋⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠘⢷⣶⣤⡀⢸⣿⠀⠀⠀⠀⣹⣧⡀⠀⣻⣿⡿⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣵⠖⢛⡿⠟⠛⢿⣭⡄⠉⣀⡠⠖⠉⡀⠀⠀⠉⠛⢋⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⢠⡇⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣴⠂⠀⠀⠀⠀⢀⣀⣠⣾⠋⢠⣾⠟⠻⠀⠀⠀⠁⠈⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠈⠻⠦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣼⠇⢠⡾⠿⢦⡈⢛⣴⣦⠴⣀⢻⡿⠦⢄⡔⢀⠄⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡸⠂⠀⢠⣰⠏⣠⣤⡒⣭⣼⡟⠐⠊⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢳⡀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣾⠖⠁⠀⡐⠻⠀⠙⢂⣽⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠻⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠞⠢⣼⠟⣛⢈⣹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⡄⠀⠀⠀⠙⣦⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⠇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣿⣿⣿⣷⣷⠀⠀⠀⠀⠈⢳⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⢹⣿⠀⠀⠀⢸⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠎⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢰⠀⠀⠈⠁⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⡜⠁⠉⢀⣌⠻⣿⣿⡟⡿⠶⠿⠶⣶⣄⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠘⠀⠀⠀⠀⡴⠋⠀⣠⣴⠟⠁⢸⣿⣟⠛⠀⠀⠈⠉⠉⠉⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣡⣴⠿⠋⠁⠀⢀⣤⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⠀⢸⠀⠀⠀⠀⠀⠀⢸⡿⠉⠻⠟⠑⠀⠀⠀⠀⠀⠶⠟⠋⠀⠀⠀⢠⡾⡋⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⢋⣱⠟
⠀⠀⠀⠀⠲⢤⠀⠘⠒⠀⠀⢸⣿⠀⢸⢀⠂⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠒⠚⠓⡙⣦⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⢸⢸⠁⠐⠃⠀⣀⡀⡠⢀⠀⠀⠀⠀⠀⢀⠰⠀⣀⠄⠀⠀⠀⠀⠀⠀⢰⡼⡇⠀⠀⠀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠈⢠⡀⠀⠐⠿⡵⢀⣀⣈⡠⠄⠐⠒⠉⠉⠁⣀⣀⠀⠀⠀⠀⠀⠀⣸⡇⣧⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢼⡀⠀⠁⠀⠀⠀⠁⠈⠛⠉⠀⠀⠀⠀⠀⠌⡋⠿⣯⣄⠀⢀⣀⣤⣾⣯⣿⣿⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠬⣙⠿⣋⣹⣿⣿⣿⣿⣿⣏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠲⣍⡛⢿⣿⣿⣿⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣙⠫⢄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢌⡑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣂⣀⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀
⠀⠀⠻⣍⢃
⠀⠀⠀⢹⣎⠄⠀⠀⠀⣠⠴⣒⡫⠍⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⢻⡖⠀⠠⢊⠕⠋⠀⣀⣀⠀⠀⠀⠀⠀⠉⠒⣍⡲⣄
⠀⠀⠀⠀⠈⢿⡁⢰⠛⢴⣤⡾⢧⣉⣷⣤⡀⠀⠀⣰⠛⠉⢻⣮⡳⡄⠀⠀⠀⢠⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠘⠁⣏⠀⢠⡿⣡⢀⣽⢋⡿⠃⠀⢀⡿⢀⣠⡼⠃⠱⡹⡄⠀⠀⠀⣿⡟⠛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣦⠹⠁⢀⡿⣡⣾⣥⡄⢠⡾⣡⡟⠉⠀⠀⠀⢷⢳⠀⠀⠀⣜⠀⡴⢷⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠰⡘⣿⡆⠀⠘⡤⢴⡫⠹⠿⠟⠀⣘⣳⣤⣀⠀⠀⣼⣾⠀⢀⣜⣛⢆⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠳⡙⣿⣤⡟⢡⡼⠛⣳⣞⣡⡬⠶⢮⣇⣯⢑⣶⠣⢃⣴⡿⢟⡿⢳⡄⠀⠀⣸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣷⣿⣠⣜⣏⠉⠁⠀⠀⣀⣿⡽⢟⣡⡴⠟⢋⣴⣋⣤⣾⠷⠿⠟⢟⠿⠟⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣛⠁⠠⣭⣭⣭⣭⡽⠶⠞⠛⠉⣠⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠙⠒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠶⠴⠶⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣦⡀⠀⣀⣴⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣟⣭⠿⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⡿⣻⡗⠉⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⠀⠈⠙⢿⣿⣿⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠈⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠙⢾⣿⣿⡆
⠀⠀⠀⠀⢾⣿⣛⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣀⠀⠀⢀⣠⢾⣽⣿⣿⣧⣀⠀⠀⠀⢀⣤⢿⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⡿⣷
⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣿⢷⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⡏⠛⡏⣷⣺⣶⣖⣿⢹⡛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣾⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠙⠋⠉⠀⠈⠉⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣼⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠿⠿⠟⠛⠉⠁





⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣏⢽⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣠⣶⣁⣀⣀⣠⣄⡀⠀⢀⣀⣴⣿⣧⣤⣄⣠⣿⢀⡀⠂⡃⠀⠀⠀⠀⢀⡀⣀⡇⠀⢠⣄⣠⢀⣈⣦⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠋⠀⠁⠀⠈⠍⠁⠀⠀⢱⡆⣷⠀⠠⠙⢂⣨⠁⡻⠁⠀⠀⠉⠀⠀⠀⠉⠉⠙⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠃⠀⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⣈⠰⡈⢭⣯⣟⠦⡄⡀⠀⠠⢒⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⠴⣛⣿⣫⢉⠐⡀⠜
⠀⠀⠀⠀⠀⠀⠈⢻⢤⠂⠈⠉⠙⢻⠏⠶⠮⠬⠢⠁⠈⠁⠸⠉⡶⢄⠀⠀⠀⢠⠶⡏⢹⣗⢛⣦⣿⠶⠜⢿⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡾⠀⠀⠀⣠⡿⠀⠀⠀⡐⢁⡀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡂⠀⢸⣿⡀⣿⡎⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⡗⡇⢀⡤⡞⠃⠀⠀⠀⠀⠙⠟⡦⣞⣀⡄⢴⠩⠶⣾⡶⣿⠶⢍⡠⣌⠙⣤⣿⣧⣤⠀⠀⠈⢳⡤⡄⣠
⠀⠀⠀⠀⠀⠀⢰⣱⣄⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠹⠷⠂⠀⠀⣸⠀⣇⠀⠀⠶⠟⠍⣿⡗⠀⠀⠀⠀⢀⣀⣝⠅⣿⡧⠤⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣯⠷⣽⣶⣦⡀⠀⠀⢸⣿⠀⠀⠀⠀⠹⠿⣿⣛⣛⣻⠟⠻⠿
⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⢸⠀⠖⢲⡖⠒⣶⠒⢲⡖⠲⣤⡶⠛⠉⠀⠀⠀⠀⠀⠈⠛⢶⣄⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⢹⠁
⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣧⠤⣤⠤⣤⠤⣿⡆⡼⣦
⠀⠀⣀⠀⡤⡿⢿⣇⡠⢊⢿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⠀⡈⠀⠈⠀⠈⠁⢳⠃
⠀⢸⢠⡍⢿⣛⣛⡆⢀⠃⢸⠀⣤⣼⣧⣤⣿⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣿⠀⠸⠤⠿⠤⣾⠇⢸
⠀⢸⠀⠀⠀⠀⠉⢠⢸⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠋⠈⠳⡀⡀⠀⠀⣿⠀⢸
⠀⠁⠀⠰⣓⡒⣢⡞⠀⠇⣸⠀⠀⠀⠀⠀⣻⠀⠀⠀⢰⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠋⠀⠀⠘⠁⠐⠄⠀⣿⠀⣼
⡆⠀⠀⢄⣉⣉⡿⣠⠲⢺⠿⢧⠀⠀⠀⣰⣿⣇⠀⢠⠏⠀⠈⠻⣶⣤⣤⣀⣤⣤⣶⠟⡋⠀⢀⠀⣀⠀⢠⠀⠘⣰⠿⣷⠟⢷
⠑⣄⠀⠈⠉⠉⠀⠦⢙⡌⠀⠈⢧⠀⣰⠃⠀⠘⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠀⠈⠁⠀⠀⣰⠇⡠⠟⡦⠀⠧⡀
⣀⣀⣉⣒⣦⣀⣒⣀⣉⣀⣀⣀⣈⣀⣈⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣠⣤⣤⣤⣄⣊⣉⣀⣈⣀⣈⣀⣈⣀⣀
⣿⣿⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⠶⢦⣄⡀⠀⢀⡴⠟⢷⡀⢠⣤⡀⠀⠰⡇⢸⡇⢤⡴⣶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇⣴⠏⠀⠀⢈⣷⠘⡟⢳⣄⠀⡇⠘⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠶⠛⠉⠘⢧⣤⣤⡴⠞⠁⠀⡇⠀⠙⢷⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣿⡿⣿⣦⡀⠀⠀⠀⠀⠘⢇⡶⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⢰⣿⡟⠈⠻⣿⣿⣏⣀⣸⣿⣷⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⣾⣷⣤
⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⠛⢿⣿⡿⠋⣹⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⡆⠀⣾⣿⠁⠿⢿⣿⡿⠿⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠸⣿⣷⡙⢷⣾⣿⣷⣶⠿⣣⣿⣿⠀⠀⢸⣿⡏⢿⡀⠈⠁⠀⠸⠃⢸⣿⠇⢸⣿⣿⣿⣶⣿⣉⣷⣶⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣮⣍⣛⣋⣥⣾⣿⣿⡟⠀⠀⠘⣿⣇⢀⢁⠀⣀⡀⡀⡀⢸⣿⠀⠸⣿⣿⣿⣭⣛⡙⣉⣩⣽⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠹⢿⣿⣿⣶⣿⣷⣿⣿⣿⠏⠀⠀⠙⠿⠿⠛⠿⡿⣿⣟⣛⠛⠛⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⠙⠷⣄⣀⣴⠦⠀⠀⠀⣀⡀⠉⢙⣉⣀⠉⠉⠀⠀⣀⠀⠀⢠⣄⠀⠀⣀⣴⢿⡋⠉⠛⠶⣶⠏
⠀⠀⠀⠀⠀⠀⠰⢦⣄⣠⡟⠀⢿⡀⠀⠈⠉⠀⠀⠀⠀⢸⣭⣽⣄⢨⣯⠥⠀⠀⠀⡜⢙⡄⠀⠀⠉⠛⠛⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠊⠘⠦⠖⠂⠀⠸⠋⠉⠳⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⣶⢒⡂⢶⠒⢲⣴⣆⣤⠆⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠋⠉⢹⡆⠀⠀⠀⠀⠀⠀⣤⣀⡗⣿⣉⡥⢸⠶⣏⣸⡿⢤⣀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠛⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣷⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⢈⡀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠛⠉⠉⠹⡆⠀⠀⢠⣤⡤⢀⡖⠋⢻⣠⣤⣤⠄⠀⠀⠀⢸⡏⠛⠓⠶⠃⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡶⣆⠀⣄⠀⠀⣦⣠⣤⣤⣤⢠⣤⣤⡄⣤⣤⣤⣤⡀⣄⠀⠀⣶⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣀⣻⡀⣿⢷⡀⣿⠀⠀⣷⠀⢸⡧⠶⠆⢸⣇⣠⡼⠃⣿⠳⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠶⠛⢰⡏⠉⠙⠇⣿⠀⠙⠿⠀⠀⠇⠀⠸⢧⡤⠖⠺⡟⠛⠲⠖⣿⠀⠙⠟

⠀⠀⠀⣠⣤⣀
⠀⠀⣰⣿⣿⣿⣿⣶⣤⣄⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠶⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣶⠲⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠻⢿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⡀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣷⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠈⠉⠉⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⢻⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠉⠻⣿⣟⠉⠉⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣀⠀⠠⠤⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣷⣦⣄⣤⣦⡀⠀⠀⠀⠀⠀⢀⣴⣿⠇⠸⣿⣆⠀⠀⠀⠀⣼⡏⠀⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣼⣿⠟⠁⠀⢰⣿⡏⠀⠀⠀⣾⠏⠀⠀⢹⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣷⣄⠀⠉⠛⠛⢻⣿⣿⣧⠀⠐⠀⠙⠁⠀⠀⠀⣼⣿⠁⢀⣴⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣿⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠹⢿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⠏⠀⠾⠋⢀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⣸⣿⠋⠀⣀⣴⣾⠿⠋⠉⠉⠉⠉⠉⢹⣿⣿⣿
⢺⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⣿⠁⠀⠀⠀⠀⣀⣴⣶⣶⣶⣄⡀⠀⠀⠀⠈⠛⣷⣶⣿⣏⣠⣾⡿⠋⠁⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⠄⠀⠀⠀⠀⠙⠛⠛⠉⠙⠻⢿⣷⣶⠀⠀⠀⠀⠉⠉⠛⠛⢿⣤⣄⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣶⣿⣿⣿⠟⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⡿⠿⠿⠻⠏⢹⣿⠃⠀⠀⠀⢀⠀⢀⣤⡄⠀⣄⠀⠀⠀⠀⠀⣤⡾⠃⠀⠉⠁⠉⢹⣿⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠋⠙⠻⢿⣿⣶⠃⠀⠀⠀⢀⣾⡏⠀⢀⣀⠚⢻⡧⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⡇⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⡏⠀⠀⠸⣿⣷⡆⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣷⡄⠀⠘⠛⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠉⠁⠀⠀⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣤⠛⠁⠀⠀⢴⡦⢴⣿⠋⠠⣦⢀⣠⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⢀⡿⠀⠀⠀⢠⡄⠀⠀⠀⠉⠀⠀⠘⠛⠉⠙⣷⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⣿⠃⠀⢀⠀⢨⡅⠘⠛⠃⣤⣤⣤⠀⠀⠀⠀⠈⠻⣿⡆⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⣿⠀⠀⢸⠀⠈⢿⣷⣤⣀⠀⠀⠀⠀⢀⣼⠿⠃⠐⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠀⠀⢸⠀⠀⠈⠻⢿⣿⣿⣿⣶⣾⠿⠁⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣉⣉⣉⣉⣀⣾⠇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⢶⣀⠀⠀⠘⣇⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⡀⠀⠓⡄⠛⣷⣄⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠈⠻⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⢿
⣿⣿⣿⣿⣿⡿⠃⠀⡇⠀⠀⠀⠛⣤⡝⢿⣶⣾⣆⡀⠀⣤⣤⠀⠀⠀⣴⣿⠁⠀⠀⠀⠘⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⡿⠟⠁⠀⠀⢀⡄⠰⣿⣶⣬⣿⣿⣿⣿⣿⣿⣦⣤⣥⣤⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠣⣄⡀
⣿⠿⠋⠀⠀⠀⠀⠀⠈⠃⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠉⠛⠻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡈⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋
⠿⠿⠿⠿⠿⠿⠿⠷⢦⡀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⢀⣠⠶⠈⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠁
⡄⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⢀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⡶⠏⠀⣠⣤⠤⣤⣀
⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⠿⠦⠀⠀⢸⡿⠀⠀⠀⠘⣷⣤
⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⢀⣤⣾⣏⠀⢠⣴⠉⢃⣤⣄⡀⢸⣧⣄⠀⢀⣼⠿⠀⠈
⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⣾⡟⠉⠉⠙⢆⣤⣷⣶⢦⣤⡝⠃⠛⠋⢩⣤⣼⡇⠀⠀⢰
⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠻⣧⣄⠀⠀⢸⣿⡉⠻⠿⣿⣿⣦⠀⠀⠀⠉⠹⢿⣿⡷⠾
⡆⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠈⠛⣷⣆⣈⠿⠃⠀⠀⠈⣿⠋⣷⡆⠀⠀⠀⠈⢻⣷
⡁⠀⠀⠈⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣀⣼⠇⠈⣿⣶⠀⠀⠀⠀⠈⣶⡏⠀⠀⣀⣤⡤⣾⣿⣶
⡇⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇⠀⠀⠹⣇⠀⢀⣦⠀⠀⢿⣷⣾⡇⢹⡏⠀⠈⠙⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠀⠀⣶⣶⣿⣶⣾⣿⣧⡀⣸⣿⣿⠁⠘⡇⣠⣤⡀⢘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠈⠉⢀⡈⠹⣿⣿⣿⣿⡟⠁⢀⣾⣷⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢻⣄⣀⣸⣿⡿⠃⠀⣼⣿⣯⣿⠿⠁⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠁⠀⢀⡹⠟⠉⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣶⣀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⢿⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠉⠀⠀⢀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠀⢺⣿⣿⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⣿⡿⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠆⢀⣈⠁⠀⠀⠀⠀⠈⠁⠀⠉⢢⡄⠀⠀⠀⠀⠀⠀⢀⣿⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠁⢄⡀⢈⣉⣁⣿⣿⣿⣿⣿⣿⠀⣰⣿⡃⠀⣀⠀⠀⠀⠀⢸⣿⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣶⣾⣿⣿⠻⣿⠛⣿⣿⡛⠛⠿⣶⡟⠛⠋⠙⠛⠛⠀⢶⣶⣶⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡉⠉⠁⠀⠀⠀⠈⠉⠋⠀⣀⣀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡤⣤⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣦⣤⣤⣤⣶⣶⣾⣿⣿⡏⢸⣦⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠹⢷⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠻⣷⣤⠀⠀⠀⠀⠸⠟⠀⠀⠀⠈⢹⣶
⣿⡆⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣄⣀⡀⠈⠛⢿⣶⣦⣴⣾⠀⠀⠀⠀⠀⢠⣿⠆
⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⠗⠀⠶⣿⣿⣿⡿⠋⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⠇⠸⠿⢿⣿⣶⣆⣠⣿⠉⣉⠁⠶⢿⣿⣿⣿⠿⠋
⣿⠀⠀⠀⠀⠀⠀⢳⣿⡏⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣶⣦⣀⠀⠈⠉⠈⠙⠉⠁⠉⢀⣴⠆⠉⠁
⣿⠀⠀⠀⠀⢰⡆⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠙⠛⠛⠻⣿⣿⣶⣶⣶⣶⣄⣠⣴⠟⠁
⣿⠀⠀⠀⠀⢸⡇⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⠀⢻⣿⣿⣿⣋⣿⠿⠁
⣿⠀⠀⠀⠀⣾⡇⠘⣿⠿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠈⠉⢉⣥⡀⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣤⣤⡀
⣿⠀⠀⠀⠀⣿⣧⠀⢀⡀⠀⣸⣿⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠹⣿⣄⡙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠉⣛⠉⠁
⣿⠀⠀⠀⠀⢻⣿⣶⣾⣿⣿⣿⣿⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠈⠻⢿⣶⣮⣍⣽⣿⣿⣿⡿⠀⠀⠹⣿
⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⣀⡀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⢶⣶⣶⣤⣀⡀⠀⠀⠀⢀⣠⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢠
⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠧⣤⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⢠⡄
⣿⣷⣶⣶⣶⣤⣤⣿⣿⣯⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣴⣤⣀⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠟⠙⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣦⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠶⠶⠶⠶⣶⠶⠶⠶⢶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣿⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣄
⢀⠤⠤⠤⠶⡤⡴⠶⠦⠤⡤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠴⠶⠶⣿⠶⠶⢶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⣄⣀⣀⣀⣇⣇⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣼⣠⣄⣀⣀⣿⣀⣀⣀⣸⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⡟⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⢀⣁⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣾⣿⣿⣷
⢰⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣾⣿⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠙⠛⠛⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣸⣿⢺⣿⢺⣿⣇⣀⣠⣤⣤⣤⣤⣦⣄⣀⣀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠉⠻⡿
⠀⠀⠀⠀⠀⠈⠿⣦⣄⣀⣀⣰⣟⣛⣛⣛⣛⣛⣻⣿⣾⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣸⣿⣻⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠀⠈⢁⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠸⠿⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⠶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⣀⣠⠤⠤⠴⠒⠒⢋⡿⠛⠉⠀⢀⣰⣆⣶⣰⣎⣻⡙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠛⠛⠻⣷⣆⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠁⠀⠀⠀⠀⠀⣸⢿⡶⠾⢿⣷⣋⣉⣉⣉⣙⣉⠉⢹⡝⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⡞⢡⣴⡞⢿⢿⣿⠻⣿⠟⣿⡗⢺⡗⠲⣾⣿⣷⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⡄⠀⡄⠀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣹⢻⡟⢻⡗⢾⣿⣼⣾⣾⣿⡷⣷⢾⣤⠶⠿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⡇⢰⣶⣰⣷⣰⣶⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣶⡆
⠀⠀⠀⠀⠀⠀⢹⣿⡞⣇⠘⠷⠟⣿⢼⢼⣿⣿⡷⡏⢸⡏⣇⢀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣷⢿⣿⢹⣿⣽⣿⣷⣼⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⣿⣷⡈⠓⠶⠦⢿⢾⢾⣾⣿⣷⡇⢈⣷⣇⣼⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡗⣾⣿⢺⣿⢻⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷⣄⣤⡴⣿⢸⠈⠉⠸⡿⣧⣼⢻⡿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠟⠘⠛⠈⡟⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣝⡓⢿⣾⠓⠒⠚⢿⣿⡶⠋⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣟⢻⠋⠉⠉⢹⡏⣿⣿⣷⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠿⠟⠻⠶⠶⠛⠛⠉⢉⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡄⠀⠀⠘⣧⣿⣿⡏⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠛⠛⠛⣩⡟⢸⡇⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣟⡛⠛⢛⣉⣴⠟⠁⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡏⠉⠛⠋⠉⠁⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡼⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣄⡀⠀⠀⠀⣀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⢠⣿⣴⣷⣠⡄⣀⡀⠀⢀⣀⣾⣄⣤⠀⠀⠀⢠⣶⣠⣄⣠⡀⠀⢰⣧⠀⠀⣀⡄⣀⢰⣿⣴⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⠷⠾⣾⣿⣿⣿⣿⣿⠺⣿⣿⡷⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⢿⡟⢻⠁⠋⠀⠉⠀⠀⠘⠻⡏⠈⠀⠀⠀⠘⠻⠏⠙⠁⠁⠀⠘⢻⠀⠀⠀⠃⠀⠹⢿⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⢰⡖⣉⠁⠈⠁⠀⣀⣀⡀
⠀⠀⠀⠀⠲⣾⣿⣿⡿⣦⠖⠊⢡⡀⠈⠀⠀⠀⠀⠀⠀⣠⡶⠀⣠⣴⠶⠆
⠀⠀⠀⢶⣾⣿⣿⡟⠉⠯⠰⠄⠘⠁⠀⠀⠀⠀⣠⣾⠟⠋⣁⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠰⢆
⠠⡾⣾⣿⣿⣿⠟⠃⠀⠀⠀⠀⠠⠄⠂⠀⠒⠦⠙⣷⣤⣾⣿⡿⠗⠀⠀⠀⣀⣠⠴⣂⡤⠶⠶⣶⡣⢄⠂
⠀⠀⢈⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⣀⢀⢀⠠⠖⣋⡥⠖⠛⠉⠀⠀⠀⠸⠃⠀⠀⢡
⠀⠀⠂⣨⡇⠀⠀⢀⡀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣆⠁⠀⢫⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣰⣷⣿⡿
⠀⢸⡾⡧⠗⣈⣩⡤⠴⠟⠛⡈⠀⠐⠀⠀⠀⠀⢼⣿⣿⣿⣿⡟⣭⣼⡟⣴⡅⡄⢄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷
⢘⠳⢀⡔⠋⠁⠀⠀⠀⠀⠀⢨⣄⣀⠀⣀⣤⡤⠘⣿⣿⣿⡿⠛⢉⣤⠶⠛⣠⡿⠃⢆⠀⠀⠀⠀⢀⡀⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡩⠠⠍⠙⠷⠶⢿⣿⠋⡂⡀⠰⠶⢚⣛⣉⣤⠞⣱⡀⠀⠀⠀⠨⠀⠙⠘⢿⢿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠇⠞⠃⠀⠀⠙⡘⣃⣾⠃⡌⣫⣭⣜⡲⢛⣋⣄⠀⠀⠀⢘⣡⠃⣄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⡼⢋⣿⡏⢻⣿⣟⡁⠫⠄⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠁⠨⠉⠛⢠⡭⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠛⠀⣾⣦⣀⡼⠃⣻⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣤⣀⣘⠏⠙⠳⡾⠟⢋⣁⣠⣼⣿⣶⣦⠀⢀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣷⣾⣿⠿⠿⠛⢷⣶⣶⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠋⠈⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠅⣀⢔⣀⠛⠿⠿⢿⣿⣿⣿⣷⣾⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠋⠉⠛⠟⠙⠿⣿⣇⠀⠊⠑⣿⡿⠃⠀⠀⠀⠈⣿⣿⠉⠛⠛⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⡻⢿⣷⣦⡄⠀⠀⣤⣤⣤⠈⢠⡁⢀⣴⣿⣷⣶⣄⠀⠀⠀⣿⣿⠂⣀⣤⡄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠊⣠⣾⡄⠀⠻⠟⣛⡀⢰⣿⣿⡇⢰⣾⣶⣾⣿⣿⣿⣿⣿⣷⠀⠀⠘⠿⠿⠛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⣩⠞⢈⠒⣬⣭⠀⠰⣶⣶⣶⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠀⠀⠀⠀⠀⠀⠻⠿⠞⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⢙⣧⣴⡿⠋⣀⣤⡿⢿⣿⡿⠛⢀⡿⡟⢿⡿⣿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣠⣤⣤⣾⣿⠟⠃⢰⡾⠛⠉⠀⣈⠕⠀⠁⠻⢤⠇⠈⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠋⠚⠛⠁⠀⠀⣘⣓⢂⠀⠀⢡⠴⠶⠞⠁⠋⠀⠀⠀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠈⡈⣴⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⠁⣀⣺⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣴⣷⣶⣄⡑⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣖⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠩⠉⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣿⣿⣿⡿⢿⣿⠧⠀⠀⠀⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⠿⣿⡟⠉⣠⡌⠟⠀⠈⠓⠀⠋⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠛⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⣠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡆⠀⠀⢶⡶⠀⠀⢰⣶⣦⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⣠⣾⣷⣀⣰⣿⣿⣿⡇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣹⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⢠⡄
⠀⠀⠀⣰⣴⡏⠀⠀⠀⣷⠉⠙⣿⣿⣿⡶⣶⣄⣠⠤⠠⢀⡿⠇⠀⢀⠀⠀⠀⣈⣁⣐⣀⣀⣀⡀⡠⢄⣘⣛⣿⣦⡤⠤⠤⠠⡤⢤
⠀⠀⢀⣿⣿⡏⠀⠀⢐⣿⡂⠀⢿⣿⣿⣻⣷⣯⠉⠙⢿⢿⠓⣾⣶⣿⠀⠀⠸⠋⠁⠀⠸⣤⢀⣶⣷⣾⣿⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠸⣿⣿⣗⠀⠀⠐⣿⠇⠀⠀⠙⠻⠛⠿⠜⠀⠀⠀⠈⠋⠙⣻⠃⠀⣿⡀⢀⣀⡀⠀⠀⠈⠉⠉⠉⠙⠿⠻⠉
⠀⢀⣀⣿⣿⣗⠐⠀⠪⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠘⣧⠀⠈⠉⠙⠛⣷⠄⠀⢠⠴⠒⡲⣶⠒⠒⠦⡀
⠀⠸⣿⣿⣿⣇⣀⣈⣨⣿⣿⠇⠀⣲⣖⣒⣒⣒⣒⣒⣡⣬⣦⣤⣤⣤⣄⠀⢿⡀⠀⠀⠀⠀⣼⡂⠀⣂⠀⣀⡂⠁⣀⣂⠀⡂
⠀⠀⢻⢹⣿⣿⣿⣿⣿⡏⢿⠀⠀⠀⠀⠁⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢼⡇⠀⠀⠀⠀⣿⣇⡀⢿⣿⣿⣶⣶⣶⣿⣿⠇
⠀⠀⢣⡀⠻⠿⣿⡿⠟⢀⡌⠠⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⢠⡄⢀⣴⠀⢹⣿⠢⡀⠀⠀⣿⣧⡁⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠙⠲⢦⣥⡴⠞⠋⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢸⠇⢸⡇⠀⢸⣿⡀⠀⠀⠀⣿⢭⣇⣀⠀⠀⠀⣠⡶⠗⠲⣄
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠉⠛⣿⠿⣿⣿⣿⣿⣿⣿⣿⣀⠘⠀⠈⢻⡆⡂⣿⣷⠀⠀⢰⢿⣽⡟⣼⢿⠃⣸⡿⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⢀⣀⡇⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠋⠉⢻⣟⠁⡿⠀⠀⣠⠞⢃⠇⢻⡿⠔⠆⠘⣿⣺⣱⣿⡆⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠂⠉⠛⠻⠗⡆⠀⠀⠀⠀⠀⠃⠀⣠⣿⣀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠸⠀⣻⣩⣴⣿⡇⠀⢹⣴⣦⣤⣀⣾⣿⡁
⠀⠘⠶⣦⣤⣤⣴⣿⣿⣷⠾⠃⠀⠀⠀⠀⠈⠁⠀⠸⡇⠀⠀⠀⠀⠘⠆⠀⠀⢻⠁⠀⠁⠈⢹⢢⣿⡿⠃⠀⠸⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⣀⣄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⡫⡪⡪⡻⡻⡿⣷⢷⡻⡪⡮⣺⣯⢮⡼⣟⡽⢏⡚⢯⡻⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⡪⣪⣫⣪⡻⡺⡻⣳⢵⡱⣵⣕⢿⠻⣛⠛⣏⢻⡏⣷⣸⢡⢻⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⣿⣾⡿⣯⣾⣾⡪⣺⡳⣝⣞⡎⢻⡳⡹⡆⡏⡎⡇⡇⣧⢿⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⡻⣿⣿⣿⡽⣽⠕⣭⣿⢘⢯⢣⡣⣇⢇⣷⣷⣳⡿⣾⣮⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢿⢺⣹⣿⢾⢹⣧⡿⢆⣮⣄⢌⡪⡳⡝⣎⢮⢮⢫⣺⢮⢾⣝⣗⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣸⡜⣇⡘⣣⣯⢦⣛⣴⣶⣝⢝⢜⡮⡺⡳⡱⣷⣵⣵⣷⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣙⡿⢫⢯⣵⣭⣟⣺⡻⡯⣳⣿⣟⢾⣿⣾⡪⣪⡫⡪⣪⣪⣪⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣢⣊⢋⢶⡏⣗⣟⣿⣻⡺⣿⠷⡽⣵⣿⣿⣿⣿⣿⣯⣾⣾⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢶⡔⢿⣷⣽⢥⣿⣮⡾⣦⣺⣻⣿⣿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⣽⢕⢱⢵⢿⣽⣿⣧⠻⣿⣿⡿⣯⡪⣮⡯⡫⣻⡿⣿⣿⣿⡾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣕⣿⣷⢿⣿⣿⣷⣽⣮⣾⣻⣮⡻⣿⣮⣮⡯⡿⡿⣯⣿⣫⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣮⡮⣾⣿⡿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠁⠁⠁⠁⠉⠉⠉⠉⠈⠈⠈⠈⠈⠈⠈⠈⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠘⠂⠒⠐⠂⠐⠒⠒⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⠁⠀⠀⠙⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣤⣀⣠⣴⣿⡏⠉⠉⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠈⢻⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣆⣶⣾⣿⡟⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣽⣽⣭⣽⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⢃⠀⢸⢣⠀⠀⡀⠀⠀⣤⠀⠀⡜⢆⠀⡰⠆⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠆⠘⡀⠇⠀⠣⠞⠙⣄⡼⠈⠆⢰⠀⠈⡤⠁⠘⡀⠌⠁⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠿⠀⠀⠀⠀⠀⠈⠁⠀⠘⠏⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⡀⠀⢸⠀⡄⠀⠀⢠⠀⠀⡄⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⢠⠸⠀⠀⡀⠀⡄⠆⠀⠘⠀⠀⠀⠀⡆⡄⠀⠀⠀⠀⠀⠀⢠⠘⠀⡀
⠀⠀⠀⠈⠰⢰⠀⠀⠀⡆⠁⠀⠂⠠⠠⠘⠀⢀⠀⠃⠆⠇⠀⠀⠀⢸⢠⠘⠀⠁⠃⡆⠁⠀⢀⢠⠰⠈⠀⠀⠇⠄⠃
⠀⠀⠀⠀⠁⠀⠐⡀⠀⠘⠀⢂⠀⠀⠂⠀⡆⣼⠀⠸⢠⠀⠀⠀⠀⡆⡀⠆⠀⢠⢀⠀⠀⠀⠘⡀⠆⡄⣄⠀⠐⢠
⠀⠀⠀⠀⠀⠇⡄⠁⢀⠀⢠⠈⠀⠀⠀⡆⠁⠁⢠⠀⠘⢠⠀⠀⠀⠃⠃⠀⣤⠸⠈⢰⠀⠀⠀⠁⠀⠃⠉⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠈⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⣀⡇⣿⣄⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⠀⡇⡀
⠀⠀⠀⠀⠀⣴⣀⢠⣾⣦⣿⢠⡀⠀⣰⣿⣇⣿⣇⢸⣧⣠⡀⠀⢀⣿⣇⣷⡀⢀⠀⠀⣰⣿⡀⢠⣾⡀⣿⣿⣤⡀
⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⡷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⣿⣿⣿⣿⣿⢾⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶
⠀⠀⠀⠀⠀⠻⠉⠘⢿⠿⣿⠘⠁⠀⠹⣿⡏⣿⡏⢹⡟⠹⠁⠀⠈⣿⡏⠿⠁⠈⠀⠈⠹⣿⠁⠘⢿⠁⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠙⡇⣿⠋⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠸⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⡄⠀⠁⠉⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠧⠤⡤⣄⣄⠀⠀⠄⠀⠀⠀⠀⣀⢠⠀⠈⠈⠁
⠽⠧⠅⡶⠋⠉⠳⡈⠀⠘⠀⠀⠠⠀⢨⠶⠄⠄⠐⠀⢠⠀⠀⠈⠓⠂⠀⣀⣄⣔⣚⡿⠿⠬⢠⠟⠉⠙⣆
⠀⠀⠀⢷⣄⣀⡴⠃⠼⠉⠢⣄⣀⠤⣎⣤⠀⠹⡄⠀⠘⢷⡀⠀⡄⠉⠁⠡⣀⠀⠀⠀⠀⠀⠸⣦⣈⣠⠞
⡶⠦⠶⠀⢀⠀⠀⠂⡠⢀⢀⠀⠀⠀⢨⡟⠀⠀⠻⣄⠀⠀⢳⡀⠀⠀⠀⠤⠶⠷⠶⢶⠶⠴⠆⠀⡀⠀⠐
⠉⠙⠦⠀⠊⠀⢀⠀⠐⢤⣍⢑⢠⣄⠀⠀⠀⠀⠈⣿⣇⠀⠀⢻⡄⠀⠀⠀⠀⠀⡀⠈⠉⠳⠄⠐⠁
⠀⢀⣀⡀⢁⡀⠈⠆⠀⠈⠁⠈⠓⠎⠀⠀⠀⠀⠀⠚⠫⠃⠀⠄⢻⣶⠀⠀⠀⠀⠀⠁⠀⣀⣀⡉⣀⠀⠱
⠀⠄⠈⡿⠥⡭⠄⣦⠛⠃⠀⠀⠀⡄⠠⠀⢀⠠⠔⠂⠀⠀⡄⠀⠀⢹⣶⡀⢀⠀⢸⠆⠠⠠⢹⠯⢌⠧⢰
⠒⡆⠀⠙⠃⢈⠉⠏⢨⠀⠀⠘⡇⠀⠀⠀⠐⠀⠘⠁⠈⠀⠈⠀⠀⠀⠉⠀⠂⠀⠘⠓⢲⠀⠈⠛⠀⡉⠙
⠰⠀⠀⠀⣠⠌⣠⡖⣞⢃⠀⠂⣷⠀⢸⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠁⠆⠀⠀⢀⡤⣁⣴
⡀⠈⠀⠈⠁⠀⠀⡠⠄⠈⠀⠀⢸⠀⠈⡇⠀⠀⠠⠤⠏⣷⡔⠂⠀⠀⠐⠀⠀⠀⠀⢀⠀⠁⢀⠉⠀⠀⢀
⣈⡢⡀⠀⠀⠈⠀⠀⠘⡀⠀⠀⠘⡆⠀⣧⠀⠿⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣑⢄
⠉⠉⠙⠻⡒⠦⢄⠀⠀⢻⡄⠀⠀⠁⠀⣿⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠉⠈⠉⠛⢗⠲⠤⡀
⠠⠀⠀⠀⠈⠳⡀⠠⠃⠈⣱⣄⠀⠠⠀⢼⡄⠀⠀⣤⣬⣁⠐⠂⠠⢰⠀⠂⠈⠀⠀⠀⢄⠀⠀⠀⠙⠆
⠀⣞⠿⠍⠉⠗⠈⠂⠀⠀⢹⣿⡤⠀⠀⢚⡏⠀⠀⠀⠈⠉⠛⠳⠶⠂⠀⢀⠀⠀⠀⠀⢰⡻⠏⠙⠹⠂⠐
⢀⣿⣿⣿⣿⠂⠀⠀⠀⠀⠘⠓⠂⠀⠀⢘⠾⡀⠀⠀⠠⠀⠀⠈⠅⠀⠀⠀⠁⠀⠀⡆⣼⣿⣿⣿⡗
⠼⠿⠿⠿⡟⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠂⠁⡠⠔⠀⠐⠒⠀⠀⠀⠀⠐⠀⠀⠡⠿⠿⠿⠿⡇
⣀⡆⡀⢀⣀⣀⡀⠀⠀⢀⠐⠂⠀⠀⢀⠀⠀⠀⠀⠓⠲⠞⠶⢦⡀⠀⠂⠀⠀⠀⢀⣀⣰⢀⠀⣀⣀⣀
⠚⠓⠂⡏⠁⠀⠹⡄⢀⣩⠀⠀⠐⠀⣰⠛⠂⣂⠈⠀⢰⡀⠀⠀⠉⣁⣀⡤⠦⠼⠽⠟⠛⠒⢸⠋⠀⠈⢧
⠀⠀⠀⠻⠦⠥⠞⠁⠚⠀⠱⠦⠤⠒⢧⣶⠀⠘⣄⠀⠈⢻⡄⠀⠆⠀⠀⠐⠤⡀⠀⠀⠀⠀⠘⠷⠬⠴⠋
⣟⣓⠓⠀⢠⠀⠀⠁⠖⠠⣀⡄⠀⠀⠰⠏⠀⠀⢙⣦⡀⠀⠹⣄⠐⠀⠀⠒⠛⠛⠛⢻⣛⡚⠂⠀⡄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣒⣒⣺⡿⠿⠛⠓⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣿⣿⡁⢀⡀⠀⡀⠒⢀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢼⡌⢷⡀⠀⠀⢀⠀⠤⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣔⣒⣬⡽⠃⠀⠀⠀⠈⣧⠈⠻⣄⠀⠀⠀⠀⠀⠀⡀⠈⢻⣽⣤⣔⡲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣒⠭⠔⠊⠉⠤⢄⡼⠁⠀⠉⠁⢀⠀⢼⡆⠀⠻⣦⠀⠀⠄⢀⠄⠀⠀⠀⠹⣶⣤⡈⠙⠒⠮⢭⣒⠤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⢒⡩⠖⠉⣀⣀⠀⠀⠐⢒⡞⠀⠀⠀⠐⠀⠈⠀⠀⣷⠀⠈⢈⣷⡀⠀⠈⠀⠒⠀⠀⠀⠘⣎⠻⠷⣖⡒⠒⠀⠙⠢⢍⠢⢄⡀
⠀⠀⠀⠀⢀⠔⢉⢔⣉⠀⠘⠓⢂⣤⣤⠴⣿⠏⠀⠀⢀⡀⢄⠀⠀⠁⣼⢻⡄⢀⠀⠀⢷⡄⠁⠠⠀⠄⠄⠀⠀⠈⢦⡧⠈⠛⢷⣀⡉⠉⠙⠛⠢⡙⠢⡀
⠀⠀⢀⡴⣅⣜⢓⡒⣤⡤⢾⡿⢫⠏⠐⣴⠋⠀⠀⠀⠀⠀⣀⠉⠀⣼⠃⠸⣧⠀⠀⠀⡀⢻⡄⠈⠁⡀⠀⠀⠀⠀⠀⠱⡄⠠⡈⠟⢿⣭⣍⣁⣀⣈⢢⡘⢦
⠀⢲⠞⢤⡮⢶⡿⢫⠏⢠⠎⠀⠋⢠⡾⠁⠀⠀⠀⠈⠀⠀⠀⠀⣼⠃⠀⠐⢻⠀⠈⠁⠀⠁⢻⣄⠀⢀⠀⠀⠁⠀⠀⠀⠘⢦⠉⠸⠄⠙⠻⣤⣤⣀⣀⣱⡀⢳⡀
⠀⡎⢠⡎⢠⠏⠀⠋⠀⠉⠀⠀⢠⠞⠠⠀⢀⠐⠂⠀⠀⠁⢀⣼⠋⠀⠀⠄⢸⡆⠀⠀⠀⠀⠄⠹⣆⠀⠀⠀⠀⠊⢀⠀⠀⠀⠳⣄⠀⠀⠣⢩⠹⣶⣦⠤⢷⠄⢣
⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠐⠂⠈⠀⢠⡞⠡⠄⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠀⠈⣁⠀⠀⠀⠀⠈⠳⣄⠀⠈⠃⢱⠈⢻⠺⡶⠜
⢸⡀⠸⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠐⠀⠈⠁⠀⠀⣴⠏⠀⠀⠴⠀⠀⠀⡼⠹⡇⠈⠀⠀⣈⠠⠀⠀⠹⣦⠀⠀⠀⠀⠐⠀⠄⠀⠀⠈⠓⢄⡀⠈⠃⠈⣶⠀⢠
⠈⣇⠀⢣⠀⠀⢀⠔⠁⠂⠀⠄⠀⠀⠀⠀⠀⠁⢠⡾⠁⠀⠀⠀⠀⠀⠀⣰⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠈⠄⠀⠀⠀⠀⠒⠂⠀⠀⠙⢄⡀⠀⡜⠀⣼
⠀⠹⣆⢈⡧⠖⢉⡀⠀⠀⠀⢀⠀⠀⠰⠀⢀⣴⠋⠀⠀⡠⠁⠀⠀⠂⢀⡿⠀⢀⠉⣧⠀⢀⡀⠀⠰⠀⠀⠀⠹⡌⠦⣀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠉⣹⠆⣰⠇
⠀⠀⠹⣮⡳⣄⠀⠀⠠⠄⠀⠄⠀⠀⢀⡴⢫⠃⠀⢀⠀⠀⠀⢀⠀⠀⡼⠁⠀⠁⠀⠸⣧⠀⠂⠀⠀⠈⠀⠀⠀⠙⣄⠈⠳⣄⠀⠀⠐⠀⠈⠂⢠⣠⠔⢋⣾⠋
⠀⠀⠀⠈⢿⡮⡙⠲⢤⣀⠀⠀⢠⠔⠋⢀⠏⠀⠀⠠⠀⠀⠀⠈⠀⡼⠁⠀⠉⠀⠀⠀⠘⣦⠀⠒⠀⠀⠀⠉⠀⠀⠘⣆⠀⠀⠙⢦⣀⣤⣤⠖⠋⣀⢔⠕⠁
⠀⠀⠀⠀⠀⠈⠪⣑⠤⣈⠙⠫⣍⣀⡠⠊⠠⠀⠀⡀⠀⠠⠀⠀⡽⡇⠀⠀⠈⠀⠀⡀⠀⣸⢧⡀⠀⠀⠀⠀⠁⠀⢀⠘⢦⡀⢀⡠⠔⠊⣁⠤⣪⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⢝⡢⢤⣘⠷⣤⣀⣀⡈⠁⠀⠀⣀⡾⡽⠁⠚⠁⠐⠂⠀⠀⢀⡟⠀⢳⡀⠀⠁⠀⠐⠀⠀⢀⣠⡿⣁⠤⢖⡩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢐⡒⠬⠭⢉⣉⠙⠓⠚⠷⠶⠦⠤⠴⠶⠶⠾⠥⠤⠔⠒⠓⠒⠦⣤⡶⠶⠞⣛⡫⠥⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠬⠭⠍⢙⣁⣐⣒⣒⣒⣒⣒⣒⣂⣙⡉⠭⠭⠅⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⣿⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢸⡇⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⡿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠸⡇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠉⣀⣄⡀⠉⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠁⠀⠀⠈⠁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣩⠞⠁⠀⢹⡎⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣴⣾⣷⡆⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠾⠥⠤⠴⣤⠚⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣛⣛⣛⠛⣡⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢠⠇⠀⣠⠿⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⣸⣿⠟⣀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⣶⡖⠛⠁⣠⠞⠁⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⠙⢫⣭⣶⠟⣡⣾⣿⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠉⠓⢿⣲⡞⠁⠀⣠⠞⠉⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣼⣨⣛⣡⣾⣿⡿⣫⣶⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⠳⢤⡞⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣼⣼⣿⠽⣝⡿⣫⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠣⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢲⣿⣿⡛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠚⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣡⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⢦⣄⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⡴⢫⠕⣲⣒⣒⣒⣲⣶⡶⡶⣶⣒⣟⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠙⠷⣦⣄⡀
⠀⠀⢸⠀⡇⢸⠔⠒⠒⠚⢃⣀⣿⣷⢿⣿⣷⡂⠀⢨⠀⡰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠩⢗⡭⡻⢶⣄
⠀⠀⠸⡄⢧⣹⡓⠒⠒⣒⣳⣶⣦⣼⢿⣻⠷⠶⠶⠗⠊⣠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣉⠤⠼⠀⠙⢷⠄
⠀⠀⠀⠘⠢⠤⢌⣉⣀⣀⣀⣀⣀⣀⣀⣙⣄⡀⠀⢴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⢞⠕⠊⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠙⠢⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⣵⡧⣠⠤⢤⣖⡉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⠀⠃⠀⠀⢀⢦⠀⠀⠀⠀⣜⡜⣸⡳⢿⢸⢠⢸⢦⡿⢟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⣀⣀⡠⠴⠋⢺⡆⠀⠀⢠⢿⢰⠁⢱⣸⢸⡇⣿⢸⠤⣄⠑⡜⡄
⠀⠀⠀⠀⠀⡠⠴⠒⣒⣒⣒⣂⣉⣉⠉⠩⢽⠩⠭⠭⠭⠤⠤⠤⡴⣚⣿⡇⠀⠀⠸⣼⠣⠭⣟⢸⣸⣇⡿⣸⣏⣁⣃⣱⣧
⠀⠀⠀⢀⢎⡔⢋⡡⢤⣤⣤⣤⣤⣤⣤⡤⠼⣦⠤⢤⡀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⢯⢿⣸⣝⡸⣻⣞⣷⣛⣷⡔⢣⢿⡇
⠀⠀⠀⢸⡜⢰⢻⣶⠾⢿⣋⡇⣖⣿⠿⣿⣿⣛⣻⡟⡍⠉⠉⠉⠒⢌⣿⡇⠀⠀⠀⠈⠫⡻⣻⡶⠺⠰⠸⠒⠛⢻⢟⠟⠀⣀⣀⡠⡄
⠀⠀⠀⠸⣧⠸⣟⣿⣀⣞⣿⣿⢻⣿⣎⣀⣀⠇⠀⡸⡇⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠈⠛⠿⢾⣮⣼⡶⠾⠓⠉⠉⠉
⠀⠀⠀⠀⠹⢷⣌⡺⠷⠶⠶⠶⠶⠶⠶⠶⠯⢽⣋⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡂⣂⣠⣤⣤⣤⣤⣤⠤⡒⠒⢯⣾⡇
⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠒⠒⣵⣿⠀⠀⠀⠉⠁⢹⠉⢉⣉⣉⢋⡏⢇⣀⣀⣀⣀⠀⠀⣹⠀⠉⠢⣽⠁
⠀⠀⠠⡋⠀⢸⠁⡇⠀⡆⠀⠀⢸⢸⠀⠀⠉⠉⠉⠉⠉⠻⡟⠀⠀⠀⠀⠀⠈⠉⠈⠒⠂⠙⠂⠐⠒⠒⠒⠚⠉⠉⠁
⠀⠀⠀⠑⠒⠚⠒⠓⠒⠓⠚⠛⠛⠿⠶⡤⠤⡤⠤⠤⠤⢤⣁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠒⠛⠒⠒⠒⠒⠦⠤⠤⠤⠤⠬⠉⣉⣉⣉⣉⣉⣑⣦
⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣷⣭⣿⣥⣵⣯⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠋⠉⠉⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢓⣶⣛⣉⣠⡄⠀⠀⠀⠘⡶⠖⠒⠚⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠒⢶⠲⠦⣄⡀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡞⠀⢀⣾⡴⠓⠦⡾⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⢁⡴⠚⠉⠙⣆⣡⠞⠀⠀⡾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠓⠒⠒⢶⣼⠋⠈⠀⠀⡴⠋⠀⠀⢀⡼⠽⠧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠶⠤⠤⠬⠷⠦⠤⠴⡇⠀⢀⡤⠚⠁⠀⠀⠀⠘⢐⣛⣛⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣠⡼⠂⠀⠙⢉⡭⠷⣄⠐⡺⠒⠂⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠋⣩⠋⠀⠀⠀⠀⢻⣀⣤⠿⠋⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⢄⣀⠀⠀⠀⠈⣝⣳⢲⠒⠚⠋⠉⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣪⣕⣨⣔⣶⠽⠧⠴⠤⣄⣈⠉⢻⡀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⢠⣴⠶⠾⠭⢭⣵⣶⣖⡶⠒⣒⠚⢛⡏⠉⠁⠀⢳⠀⠀⠀⠀⢠⠗⣶⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠚⠯⠭⡥⢸⣁⠀⠀⣤⣞⠙⡇⠀⢀⣠⠿⡚⠡⡾⢭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣫⠷⠚⡟⠀⡠⠒⡿⠛⠓⡶⠻⡆⣀⠴⡎⠁⠀⠎⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⣴⡡⢊⣴⡞⠁⠀⡼⠁⢀⠛⠁⣸⠀⠀⠀⡸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡿⢊⢟⡞⠀⠀⡼⠁⠀⡟⢰⢣⡇⢠⡄⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⢋⡞⠀⠀⡰⠁⠀⣼⣠⠇⡞⣽⡶⠴⠿
⢿⣿⣍⣉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠦⢤⣴⣁⡞⠀⠀⢰⠃⠀⣰⠃⣽⢴⡗⠌
⢸⠉⠙⠛⠿⣶⡦⢄⡀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣳⣦⡎⣠⢺⣧⠴⣫⣾⠌
⠀⠑⢦⣀⢠⣆⡿⣷⠮⡗⠈⠭⠭⠤⠔⠒⠒⢉⡟⢨⠁⠉⣳⡀⣠⠞⡏
⠀⠀⡼⠍⠉⣻⢦⣽⣆⠀⠀⠀⠀⠀⠀⣀⣀⡼⠀⠛⣉⣉⣬⡽⠻⣢⢇⡔⣲
⠀⠀⣇⣀⣠⢽⡒⠀⠉⡟⡲⡟⠏⣉⣙⡿⢿⡿⢋⡹⡟⠋⠉⠓⣄⠈⠻⣰⡁
⠀⠀⣿⡇⠈⢱⣜⡿⠋⡇⡏⠉⢲⡶⣶⠲⣿⣿⣭⡾⠟⠀⠀⠀⢾⡱⢲⠦⢌⡱
⠀⠀⠀⠙⠷⠚⠶⠶⠿⣷⣟⠦⠵⣾⡓⠿⢯⡏⠁⠀⠀⠀⠀⠀⠳⠥⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⢻⢫⡏⠉⣉⠉⡏⢿⣍⠟⠻⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⣉⣀⣹⣘⣛⣟⣃⣈⣠⣇⣟⣛⣃⣺⣃⣉⣛⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⣠⠤⠤⠤⠤⠤⠤⡄⢀⡞⢯⠀⣠⠤⠤⠤⠤⠤⠤⠤⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⣇⢘⣋⣛⣙⠛⣋⣉⢃⡾⣹⢏⢧⠙⣒⣒⣒⣒⣒⣒⡂⢸⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠹⠗⣿⠸⠴⠦⠼⠯⠷⢀⡼⣱⢃⡘⣎⢷⡈⠘⠧⠤⠬⠥⠟⢸⠤⠖⢻⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣶⡇⡇⠰⠶⠶⢶⣏⠁⣾⣱⢃⣾⣷⡘⣎⢧⠐⠒⠶⢯⣭⣅⢸⣻⣯⣍⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡭⠤⢭⣧⢘⣲⡶⠶⠆⣼⢳⢏⣼⣸⣟⣷⡸⣞⢧⠘⣛⡛⠒⠒⣸⠘⢃⠂⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠒⠀⡟⠰⠛⠛⠃⣼⢳⠏⡠⠤⠤⡤⣤⢄⠹⣎⣧⠸⠛⠛⠛⢹⣄⣨⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠤⠤⠦⡇⠨⠿⠏⣼⢣⡏⢈⢯⣿⢠⣇⡇⡿⠀⢹⡞⣧⠩⠿⠝⢸⢺⣶⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⣚⣛⡇⢶⡆⣼⢧⡟⠀⠬⠭⠥⠭⠬⠥⠭⠥⠄⢻⡜⣧⣶⡶⣼⠨⠭⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠀⡧⠖⣼⢏⡟⢸⣿⡏⣧⣇⣿⢻⣽⡏⣿⣿⡅⢻⡜⣧⠰⢻⠙⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣇⣼⢏⣾⣠⣈⣉⣩⣭⣭⣍⣉⣉⣩⣍⣩⣥⣌⣿⡜⣧⣼⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡶⠶⠶⡶⡶⡶⣶⡶⡶⠶⢲⢶⡶⡶⣶⠒⠲⢶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣋⡃⠿⣿⣵⡂⠒⢸⢘⡏⣇⣹⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠶⠷⠶⠶⠶⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣄⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⢀
⠀⠀⢀⣴⡄⣀⣸⣿⣧⣾⣿⣿⣶⣸⣿⡀⠀⠀⠀⢠⣼⣿⣾⣼⣆⠀⠀⢀⡀⣴⣄⠀⠀⠀⠀⢀⣴⣄⣄⣠⡀
⠀⠛⠻⣿⡟⠿⢿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠟⠒⠒⢻⣿⣿⣿⣿⡿⠟⠻⠿⡿⣿⡿⠛⠓⠀⠐⠻⣿⣿⣿⢿⠟
⠀⠀⠀⠈⠀⠀⠘⣿⡇⢹⠿⣿⠋⠘⡟⠀⠀⠀⠀⠀⠹⡿⠹⠹⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⡤⢤⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡐⣃⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣇⣛⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⠀⢸⠀⠀⠚⠚⠓⠒⠚⠆⠀⠀⠀⣈⣍⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⢿⣮⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀⣾⠡⡆⣳⢸⠀⠀⠉⠀⢰⠉⠁⠁⠀⠃⢰⠋⡴⢹⡆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇⠘⠳⠖⠃⢸⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠓⠶⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠾⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠲⠖⠲⠶⠲⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠀⠀⠀⠀⠀⠀⡄⡀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠇⣿⠂⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠃⡰⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠇⣴⠄⠀⠀⠀⠀⠀⠀⣿⠀⡏⢸⡟⢠⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣴⡿⢠⠄⠀⠀⠀⠀⠁⢿⠇⠀⠀⠀⠀⠀⠀⣿⢰⠇⣼⢀⠃⢀⠂⢠⠁
⠀⠀⠀⠀⠀⠀⢈⡃⠀⠀⠀⠀⠀⠈⠀⣠⣴⠟⡠⠂⡠⠀⠀⠀⠀⠀⠃⣼⠆⠀⠀⠀⠀⠀⠀⣿⢸⢠⡇⠄⠀⠂⢀⠇
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠴⡾⢂⡅⢊⠄⠺⣇⠀⠀⠀⠀⠀⠥⢹⠁⠀⠀⠀⠀⠀⠀⣿⡟⣼⡇⠀⠌⢀⠾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠀⣄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⡤⠶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⠋⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⣰⢫⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣔⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣟⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠶⣦⣤⣤⣤⣤⡶⠾⣿⠁⢸
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣀⡙⠶⠞⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡇⠀⢸⡇⠀⠀⢸⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠛⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣸⡇⠀⢸⡇⠀⠀⢾⠀⠀⣿⠠⠊
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠸⠇⠀⠠⠾⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⠀⠀⠀⢸⣭⡛⠀⠀⠀⢠⡾⠒⢷⠀⠀⠀⠈⣩⠋⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠓⠚⠃⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠰⣦⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣁⠶⠶⡚⠉⠉⠐⢋⡬⠥⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⢹⡇⠛⠀⣸⠀⠀⢰⡏⠀⠠⣦⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⢼⡋⠉⠳⠟⣁⣀⣀⡹⠝⠒⠒⠚⣿⠥⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⢻⡄⠠⠀⠀⢁⡜⠁⠉⠀⠀⠀⣰⣿⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡯⠷⠲⠮⠫⠷⡤⠴⡲⠶⠯⠓⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠖⣾⢰⣗⡶⠶⠶⠮⠷⠵⣶⡤⣄⡀⠀⠘⣤⠓⡤⢄⠀⠀⠀⢀⡠⠖⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡄⠹⠁⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠵⣾⣍⡁⠀⢞⠀⠀⠀⡞⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣄⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⢆⠀⠀⢻⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠈⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡎⠀⠀⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⠴⠋⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠤⠖⠚⠉
⠀⠀⠀⠀⣰⠋⠳⠛⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠛⡞⢳⣄
⠀⠀⠀⠸⢌⣁⣀⣀⣁⣀⣀⠠⠤⠴⠲⠶⠤⠤⠤⠤⢤⡤⠤⠤⠤⠴⠶⠖⠒⠒⠦⠤⢤⣄⣀⣀⣈⣀⣀⣨⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠱⠄⠈⢁⣀⣵⡙⠳⡿⠄⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⢀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢕⣤⠀⠀⣠⠊
⢀⣴⣕⠄⠛⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠉⠀⠛⠀⢀⡀⠀⠀⠀⡔⢠⠀⠀⠀⠘⠁⠀⠜⠁⠸⣆
⣝⠂⠙⢀⣴⠄⠓⠀⠀⠀⣠⠰⠲⠲⠆⠁⠀⠀⠴⠀⠒⠠⠀⠀⠈⠀⠀⠀⠀⠀⠁⠤⡄⠀⠀⠀⠀⠠⠀⠀⠈
⡙⠂⠀⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡚⠶⠄⠀⠀⢀⠥⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠘⠉⠁⢢⠀⠀⠀⠈⠀⠀⢸⠒⠒⢀⡀⠀⠀⠀⠁⠀⠔⠀⠀⠀⡠⠚⠁
⠀⠀⢠⠀⠤⠰⠇⠀⠀⠀⢰⠃⠚⠁⠠⡄⠀⠀⢁⠇⠰⠀⠀⠀⢰⠤⠐⣀⠸⠂⠒⢀⠰⡄⠀⢀⡀⠀⠀⠀⠠
⡰⠈⠙⠂⠀⠈⡱⠔⠶⠤⠌⢠⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠸⡀⠀⠉⣠
⠳⠦⠀⠀⠀⠀⣁⠀⡤⠬⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠄⠀⠠⠅⠀⠠⠒
⣦⠀⠀⠀⢰⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⠂⠀⠀⠉⠉
⠀⠈⡁⠀⠘⠠⠂⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⡤⠤⠄
⣤⡦⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣴⡶⠖⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣼⡷⠖⣋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⢴⡶⠟⣋⠄⠀⠀⣠⠔⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠢⣌⠙⠿⣿

⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⠈⠟⠁⢹⠋⠈⠿⠀⠀⡴⠶⠄
⣿⣶⡶⠶⠾⠿⠛⠻⠿⢿⣟⠛⠛⠀⢐⣠⣤⣶⠾⠞⣛⣿⢛⣷⠟⢷⡾⢻⣄⡀⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⢀⣴⠟⠉⢀⣠⠴⠛⠉⠑⠚⠛⠖⣾⡶⠚⠉⠛⠷⢦⣄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣿⠃⡀⠔⠉⠀⠀⠀⠀⣴⣤⢠⣆⡿⣧⣤⡀⠀⠀⠀⢹⡷⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡿⢁⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⠏⢡⡿⠈⣇⠀⠀⠀⠈⡷⠈⢿⣧⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⡀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠀⣀⢠⣅⠄⣀⡸⠿⠀⠀⠀⢀⠃⠀⠈⢿⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⣀⣤⣦⠤⣴⠤⢤⠀⣤⣶⣿⢴⣾⢍⡦⡿⢹⡾⣶⠐⣦⣤⣰⣦⣀⠈⣿⣷⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣰⣏⠀⠈⢳⡀⠀⢸⠶⠀⠀⡿⢲⡗⢨⠃⡇⠜⠁⠀⣤⠄⠀⠀⠀⠉⠹⣌⢯⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠻⣿⢧⡀⠸⡇⠀⢸⠀⠀⠀⣿⢰⠁⣘⢘⡧⠀⠀⠀⠆⡏⣦⡄⠀⠀⠀⢸⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣝⣲⣧⣀⡈⢀⠀⣀⡧⢸⣸⢯⣀⠁⠀⠀⠀⠆⡃⠁⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠄⠀⠀⠈⣹⠛⠿⣇⠸⠿⠿⡇⠘⣥⣌⠀⠀⠀⠀⢀⠀⡅⠀⠀⠀⠀⠀⣿⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⡀⢠⡴⠋⣀⠀⣧⢀⠀⢀⡇⢠⠀⣀⢀⠀⠀⠀⢸⠂⠀⠀⠀⠀⣀⣼⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠙⠚⠓⠒⠛⠒⡧⠘⠛⠚⡇⣿⠙⢻⢹⠁⠿⠟⢿⠀⢷⡶⠿⠤⠍⠘⢁⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡆⠀⠀⠀⠀⣿⡆⠀⠀⡗⣿⠘⠋⢸⣤⠀⠀⢸⣤⠀⣷⠀⠀⠴⢁⠎⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠹⣄⠂⢊⠀⠛⠂⢰⡚⠃⠀⠀⣸⠓⠀⡟⠀⢊⡴⠋⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠈⠀⠀⠘⢦⡀⠉⠁⠀⢀⣡⠠⠞⣡⠔⠋⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⠀⠀⠀⠉⠐⠚⢛⣉⣴⠶⠋⠁⠀⠀⠀⠜⠁
⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢉⡉⠥⠞⠉⠀⠀⠀⠀⡀⠄
⠀⠀⠀⣻⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢏⣠⠔⠚⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢘⣯⣻⢍⡙⠛⠻⠶⠶⠦⠦⠶⠤⠖⠲⠚⠐⠊⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠘⣷⡌⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⡹⣿⡿⠿⠿⠟⠳⠒⠒⠘
⠀⠀⠀⠀⠀⠘⣩⠁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠑⡆⣀⡀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⡤⠄⠙⠄⠀⡀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⠈⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⣴⡏⠠⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠅⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠇
⣶⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⢲⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠚⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠤⠤⠴⢾⠀⠀⠀⠀⠀⢰⣄⣠⡤⠴⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⡤⣄⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣞⠀⠀⠀⠀⠸⣀⡠⠴⠒⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠈⠉⠒⠢⠤⣀⡀⢘
⠀⠀⠀⢸⢀⡀⠀⠀⠀⠉⠃⠉⠉⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠦⠴⠦⣾⠋⠉⡉⠉⠙⠒⡦⣤⣀⣉⡿
⠀⠀⠀⢸⠞⠓⠒⠒⠶⠦⠶⠒⠋⠀⠀⠉⠙⠲⠤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⡽⠛⠂⠀⠀⠈⠓⠚⠳⠦⠤⣄⡇
⠀⠀⠀⠈⠲⠤⢤⣤⠤⠤⠤⠤⠤⠤⠤⢄⣀⡀⠰⢤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⡠⠤⠖⠒⠒⠒⠲⠤⠤⠤⣤⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⠒⠚⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢄⣀⡀
⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧
⠀⠀⠀⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢰⠧⠤⣄⣀⡀
⠀⠀⠀⠀⠀⡇⠀⣀⣠⠤⠖⠊⠉⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢶⠤⢤⡀⢸⠀⠀⠀⠀⠉⡶
⠀⠀⠀⠀⠀⣟⣉⣁⣀⣤⠖⠚⠋⠉⠙⠻⣦⠤⠤⠤⠤⣄⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠉⠓⠚⠋⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⠀⠉⠀⣜⣀⡠⠤⠤⠔⠚⠁⠀⠀⠒⠶⣒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⠴⠚⠉⠁⠉⠓⠤⠤⠤⠤⠔⠲⠦⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⣀⣀⣠⡤⠤⠦⠤⠤⠤⢤⣈⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⠥⣤⠄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡾⠛⠀⠀⠀⠀⡜⡔⠙⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠋⢀⠃⠀⠀⠀⠀⡇⡇⢀⡿⣾⡀⠀⠀⠀⠀⠀⠀⣴⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡇⣼⠀⠀⠀⠀⠀⣿⣫⠜⣿⠉⡇⠀⠀⠀⠀⠀⣴⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣤⢹⣿⡇⣿⠀⠀⠀⠀⠀⢹⣇⠀⣰⠒⣇⠀⠀⠀⠀⣼⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡲⠈⢧⢻⣧⠹⡄⠀⢀⠀⢄⢸⣋⡥⢻⣾⢹⠀⠀⢀⡾⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠁⠙⢷⣜⢦⡈⣧⢟⠘⣇⣀⠸⢻⡸⡆⢀⡞⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⣻⠏⡘⣇⡴⠚⣿⡇⣧⠊⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⡻⣼⠀⢻⣷⠛⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣴⣶⠿⠛⢉⣉⣀⣈⠒⠦⣄⡀⠀⠀⠀⢉⠙⠲⣅⡺⢷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠠⠟⣩⠴⠒⠉⠁⠀⠀⠈⠉⠒⠦⣙⠲⢤⡄⢸⠀⠀⠀⠙⠲⣝⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢁⡴⠋⠁⠀⠀⢀⠄⠂⢉⣀⠀⠉⠒⠤⣍⡒⠆⢸⣦⣄⡀⠀⠀⠀⠙⠪⣑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⢠⠁⣠⠊⠁⠀⠀⠈⠐⠠⢄⣉⠃⣬⢿⣷⣯⣗⣤⣀⠀⠀⠀⠁⠈⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⢀⡛⣏⡛⢤⡠⢉⠒⠌⡙⠀⠽⣿⣿⣦⢄⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠱⣄⠀⠀⠠⡬⠁⣤⣛⡌⠃⢄⡈⠐⢢⣑⢤⣈⠈⠋⢝⠿⣶⡀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣈⠹⣲⠤⣀⠀⠠⡶⢅⡀⠀⠨⠕⡢⢁⡀⣿⡟⣶⠤⡈⠢⡙⢦⡀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢍⡓⠵⣖⡿⣶⠬⣑⡢⠀⠀⣹⠀⡿⣧⠸⡆⠈⢢⠈⡆⠹⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⣦⣭⣓⠺⢾⡿⢗⣦⣍⣂⠿⢿⣀⣇⠀⣸⠃⡇⢀⠇⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡾⣿⠃⠈⠉⠒⣤⣄⡉⠉⠂⠿⣶⡤⠭⡍⠥⠊⣠⠎⠀⠀⢘⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⣿⣷⡆⠀⠀⠀⠀⡍⠃⢹⠷⣦⣤⣀⠈⠉⠛⠛⠋⠁⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⢋⣴⣿⣿⣹⠃⠀⠀⠀⠀⠁⢸⠘⢃⡟⣴⠌⠛⠓⠶⠶⣶⣞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠿⣿⣿⡏⣿⠀⠀⠀⠀⠀⢸⠸⡠⣟⣀⠥⣤⡶⠶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢿⡄⠀⠀⠀⠀⠈⡀⡇⢹⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣇⢦⡘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⢀⣀⣀⢀⠀⠀⠀⣀⡀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠙⢻⣿⣿⡿⠉⠉⢹⣿⠉⠉⠉⠉⠉⢹⡿⠉⠉⣿⣿⣿⣿
⠀⠀⣠⣶⣶⣶⣦⡀⢸⣿⣧⠀⠀⠀⣀⣀⣀⡀⠀⢿⣿⣿⣿⣿⡟⢻⡖⠒⠒⢶⣾⣿⣿⣿⣿⡆⠈⠙⠉⠁⠀⠀⠀⣿⣇⢀⣤⣤⢀⣿⠇⠀⠀⠘⠛⠛⠋
⠀⠀⣿⣿⣿⣿⣿⣷⣾⣿⣿⣤⣤⣼⣿⣿⣿⣿⡆⠈⠛⠛⠛⠛⠁⢸⡇⣀⣀⣾⠟⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣼⡿
⠀⠀⠹⢿⣿⣿⡿⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠉⠛⠛⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠉⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡄⣼⣿⣄⠀⠀⠀⠀⠀⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⣴⣿⣆⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⣷⡀⣴⣶⣶⣶⣄⣼⣿⣿⣿⣿⣧⣀⣿⣿⣿⣀⣀⣀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣇⠛⠛⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡄⣿⡇⠀⠀⠀⠀⠈⠛⠿⠿⠿⠃⠘⣿⣿⣿⣿⠀⠀⠙⠿⠿⠿⠟⠀⠉⠛⠉⠉⠀⢀⣿⣿⣿⡟⠈⠉⠉⠉⠉⠉
⠀⠀⣰⣿⣿⣿⣿⡆⢹⣿⡟⢻⣿⣿⡇⣴⣶⣶⣶⡄⠀⠀⠀⠀⠀⢰⣿⠇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣶⡄
⠀⠀⣿⣿⣿⣿⣿⣿⢺⣿⠃⣿⣿⠛⠻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⠛⣿⣿⠟
⠀⠀⠘⠿⢿⣿⡿⠃⢸⣯⢠⣿⠁⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⢠⣿⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢼⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠛⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠐⠈⠉⠉⠑⠢⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠙⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠂⠀⠀⠤⠄⠀⠀⡄⠀⠀⠀⠑⠖⠖⢀⣀⠄⠀⠀⢀⣄⣀⠄⠀⠀⠈⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠒⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⢀⡉⢣⠀⠀⠀⠀⠐⠦⠤⠄⢀⣀⡀⠀⠢⠤⠔⠊⠒⠃⠀⣠⢀⠄⠀⠀⢠⠀⠀⠀⠈⢢
⠀⠀⠀⠰⠊⠁⠀⢀⠀⠀⣈⣧⣀⠉⡉⠀⠙⠓⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡜⣠⠀⠀⠀⠀⠁⢳
⠀⢀⡤⠂⠀⠐⠊⠹⣄⡀⢀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⡀⠀⢀⡜⠀⠀⡀⠈⠹⠄⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠀⡴⠀⠀⠈⠀⢤
⠀⠊⠀⠀⠒⣆⠀⠀⣀⡍⠀⠀⠀⠑⢪⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠀⠠⣄⡀⠐⠆⠀⠒⠁⠒⠒⠋⠀⢀⠦⠔⠄⠈⠇
⡀⠀⠀⠀⢤⡈⠁⠀⠰⡁⠀⠸⣄⠀⡀⠑⠂⢄⠀⠰⣄⡀⠄⠀⠀⣀⠀⠀⠣⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢄⣰⠤⠂⠈⠋⠁⢀⠄⠀⠀⡀
⢇⠱⠄⠀⠀⣉⠁⣠⠀⠉⠀⠀⢀⠉⡀⠀⠀⠀⠉⠁⠀⢀⠀⣉⡉⢀⣀⡠⢄⣈⠉⣉⣁⡀⠠⣄⣠⡄⠀⠀⢤⡀⠀⠀⠈⠉⠁⠠⠖⢊⡁
⠈⠢⠀⡄⣄⠈⠀⠻⠀⠀⠙⠤⠈⠉⣁⡜⠐⠋⠈⡐⠊⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⣀⣜⠃⠀⠀⠀⠠⠴⠒⠊⠀⠄⠈⡄⠊
⠀⠀⠀⠈⠢⡁⠀⠤⡀⠄⠐⠂⠀⠀⠀⠈⠋⠉⠉⠀⣿⠀⠀⠀⢰⠀⡇⠰⠀⠀⠀⠀⡼⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⣀⠀⠚
⠀⠀⠀⠀⠀⠉⠒⠂⠐⠤⡍⠉⢀⣐⡊⡐⠒⠀⠀⠖⡿⠀⠀⠀⢸⠀⢇⡇⠀⡈⠠⢸⠃⠒⠓⠒⠂⠀⠀⠉⡈⣁⣀⡠⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠑⠄⠀⢺⡀⢈⠸⠈⡇⠸⠀⢃⠇⡆⡿⠖⠤⠒⠐⠊⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡆⡀⡇⡄⠠⣸⢰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⡇⠁⠀⡇⡏⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡼⢇⡠⠴⢧⣇⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠌⠉⠁⠀⠀⠀⠀⠀⠁⠘⠋⠳⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⢠⣴⠦⣴⠠⢤⣤⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠲⠄⠠⣿⡇⣿⠀⢸⠀⢸⢣⠔⠢⠤⠆⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⡿⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⡇⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢳⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠴⢦⠀⠀⢈⠤⢤⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠣⠀⠀⠀⠀⠋⠑⡀⠀⠀⠉⠀⠀⠁⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠀⠀⠔⢢⡤⡈⠂⢀⡀⢩⠤⠒⢤⢤⡀⠙⠀⠐⠳⡄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⠠⠄⠀⢤⡀⠀⠀⠀⠀⠀⠈⠀⠐⠆⠀⠀⠑⠲⠀⠈⠓⠀⠈⠈⠋⠂
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢆⡁⢨⡀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⠁⠐⠂⠐⣌⠷⣈⡀⠀⠒⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣾⡗⡏
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⢀⠀⠳⠀⠀⢃⣧⣶⡄⠈⠢⣀⠀⠀⢀⣿⡾⠻⡄⠀⠀⠀⢀⣴⡿⠋⢸⠁
⠀⠀⠀⠀⢀⠔⠢⠀⡃⡐⢦⡁⠀⢲⠀⠀⢹⣿⣇⠀⠀⠀⠉⠒⠛⠋⠀⠀⠙⠦⠤⠶⠟⠋⠀⠀⡾
⠀⠀⠀⠀⠀⠐⢄⠀⠤⠈⠈⡁⠛⠀⠀⠀⢸⣿⣿⠀⠀⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⣠⣴⣧⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠰⠀⠀⠀⠀⠀⢠⠸⣏⠣⡈⡢⠌⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠒⠛⠉⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⡉⠈⠢⠦⣄⠂⠀⡠⠀⢄⠄⠶⠂⣀⠠⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣧⠀⠀⢠⠀⠀⠀⡆⠀⠀⠘⣄⠀⠀⠘⢮⠀⠢⠄⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠈⠦⠀⠀⠀⢄⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠀⠀⢰⢇⠀⠀⠀⡇⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠙⠲⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢛⣋⣥⣦⣤⣄⡀⠀⠀⢸⡀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠿⣿⡆⠀⠀⣧⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⣠⣤⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡿⠟⠛⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠺⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⠟⠉⠀⠀⠀⠀⠀⢿⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⠿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠈⠱⣄⠀⠀⠀⣀⠤⠔⠒⠀⢉⣉⣀⣀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠈⣢⠔⠋⢀⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣴⠞⠁⠐⠒⠠⣀⠉⠓⢦⣀⠀⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠇⢸⣧⡀⠀⠀⠀⢀⡴⠉⠲⢀⠉⠳⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣇⠈⢝⠻⣷⣦⣔⠋⡠⠊⠀⠀⠙⢦⡈⠳⣄⠀⠀⠀⠀⠀⣠⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠣⣿⡿⣋⣶⡄⡀⠊⠀⠀⠀⠉⢆⠘⢷⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣯⠞⣿⡅⡀⠀⠀⠀⠀⡤⠳⡀⠹⣦⡀⠀⠀⠀⢠⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⢫⡪⢿⣦⣠⡊⠀⠔⠀⠘⢆⠘⣷⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣿⢿⣿⡿⣦⣠⠔⡀⠈⣆⠸⣧⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠼⣿⡟⣷⣈⠀⢀⣼⠀⣿⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡉⠛⠣⠴⠅⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠂⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣐⠂⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⢁⣤⣶⠟⣯⠁
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⣢⠘⣧
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣤⡶⠟⠉⠀⠀⢀⡤⠚⢁⣠⠞
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣏⠁⠀⠀⣀⠤⠞⠉⣀⡴⠋⠁
⠀⠀⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠛⠉⠙⢦⡘⠦⠖⠋⣁⣤⠶⠋⠁
⠀⡾⠁⢵⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠙⠶⠚⠛⠉
⠘⡇⠀⠀⠓⢤⣀⣀⣀⣀⡴⠟⠋
⠀⠈⠢⢀⣀⣠⣴⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡀⠀⢀⡤⠀⡀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⢽⣂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣼⠋⣀⣘⣷⢾⣃⡀⢣⣬⠟⣀⣀⣀⣠⣤⡶⡊⠀⠼⠀⠀⢿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⢚⡩⡵⣤⣀⣀⢽⢾⢄⣉⣭⡿⣍⣉⣉⡁⢀⣀⣀⣀⣠⠄⠀⢀⣾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣏⣴⣞⣿⣛⣿⡗⢛⡟⠛⣿⡛⢻⣿⣾⡟⠋⣿⡿⠿⠿⠛⠿⠿⣟⣀⣤⣤⣼⣻⠄
⠀⠀⠀⢈⣄⣷⣠⡀⠀⠀⢀⣠⣾⡾⠿⡭⠿⠅⠀⠀⠿⠷⠾⠴⠠⠿⠷⠹⠤⠿⣧⣾⣯⣴⣖⡀⠠⢀⠀⠀⠩⢉⣛⣿⠹
⠀⠀⠀⠈⠙⣋⢙⡯⠉⠉⠉⣭⣭⣭⣭⣯⣭⣩⣭⣿⣿⡟⢻⣿⢿⡿⠿⣷⢖⡒⠚⠛⠳⠲⠟⢿⣳⣦⣄⡀⠀⠀⠉⠛
⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⡇⢸⣇⣇⡔⠉⢀⣀⠀⡼⠧⠾⠷⠸⠷⠶⠿⠸⠷⠼⢿⡗⠶⠦⣄⠈⠻⣧⠉⠳⣤⠲⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⡿⠊⠀⠉⣀⠀⣤⣾⣤⣟⣛⣂⣂⣒⣂⣀⣀⣀⠀⠐⠖⠿⢳⡀⢺⡆⠀⠘⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢿⠃⠀⣰⣧⣀⠀⠸⠿⠿⠥⠿⣾⣟⣸⡽⡏⠀⢹⠀⠱⠀⠰⣾⡇⠀⣇⠀⠀⢹⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠀⢀⡇⠉⠙⣦⠀⠉⠩⢁⠈⢉⣉⣉⡍⠊⠀⢸⠀⠀⢀⢠⢹⡇⠀⠉⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡜⡇⠁⠘⠀⠰⠨⠬⠤⠭⠬⠭⠤⢀⣠⠀⠙⣗⣚⣥⣫⡞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣝⠦⢄⡀⠀⠲⣶⣶⣦⣿⣚⣛⣛⣛⣿⣃⣈⣭⠭⠁⢂⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠏⠙⠛⢛⣛⠉⣉⢉⣉⣩⣭⢡⣀⣀⡤⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠉⠉⠁⠀⠈⠁



⡖⠒⢲⠖⠶⠶⠶⠶⡖⠶⠶⠶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⡶⡶⢶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⢀⡀⠀⡀⠀⠠⡀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣀⠤⣤⣀
⠀⠀⠀⠠⢂⣴⠆⠀⠀⠈⡱⡘⠛⢶⣼⣿⡷⣤⣢⡤⠄⠀⢀⣀⢐⣿⡏⠀⠀⠀⠀⠘⠉⠒⠓⠒⠒⠛⠛⠛⠛⠋⠀⠀⠈⠑⡌⠁⠉⠀⢰⣿⣿⡆
⠀⠀⠀⢻⣿⣿⠆⠑⠀⠠⣿⡅⠀⢘⢿⣿⣿⢷⣮⣻⠛⠓⢶⡦⣾⢅⣰⣤⣾⡇⠘⠻⢿⣿⡶⠶⠶⣶⣾⣾⣿⣿⣿⣿⣿⣿⠷⠖⠲⠲⢿⣿⣿⡷
⠀⠀⢀⣾⣿⣿⡥⠪⠐⢐⣿⣳⠀⠀⠛⣿⣿⣟⣿⢇⠄⠀⠀⠁⠻⠤⠿⢿⡿⠁⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡷⠶⠶⢦⣤⣼⡿⠋
⠀⠀⠀⢹⣿⣿⣂⢕⠀⡀⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⣀⠜⠁⠀⠈⣿⡄⠐⠲⠦⠤⣤⢤⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡈⠉
⠀⢠⡤⣾⣿⣿⠤⠠⠄⠊⣿⣧⣤⠀⠈⠂⠀⠀⠀⢀⣀⣀⣀⣶⣶⣀⣀⣀⣀⣀⣀⠘⣇⠀⠀⠀⠀⠀⠉⣟⠀⠀⠔⠉⠁⠄⡛⢃⠄⠉⠙⠆
⠀⠈⢻⢻⣿⣿⣤⣤⣤⣴⣿⢻⡏⠀⠺⠉⠁⠸⣩⢠⣴⣴⣿⣿⣿⣿⣄⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢲⡧⡄⠀⣥⣀⣤⠥⠤⠤⢬⣄⣠⡅
⠀⠀⢘⠇⣿⣿⣿⣿⣿⣿⡟⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠐⣿⡆⠀⠀⠀⠀⢸⣷⡡⡀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠘⡄⠈⠿⠿⢿⢿⠟⠀⡰⠀⢶⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣤⠀⣠⠆⠀⣿⣏⠐⠀⠀⠀⢸⡷⣥⠁⠀⠀⠙⠉⠛⠛⠙⠉
⠀⠀⠀⠈⠳⠤⣤⣅⣤⠴⠛⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⢸⡷⠠⣿⠀⠀⢸⣿⠀⠈⠀⠀⢸⣿⣹⣅⡀⠀⠀⠀⢀⣴⡶⠶⠦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠀⠸⠀⠀⠙⣦⠀⠌⣿⣧⠁⠀⠀⡼⣟⣺⡟⣽⣶⠆⢠⣿⠏⠀⠀⠀⠹⣶⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⣿⡟⠛⠻⣿⠿⠉⡟⠀⠀⢀⡠⡿⢠⠆⣿⣿⡡⠄⠀⡧⣯⡽⢰⡿⠌⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⡠⠤⠶⢶⣾⣇⡁⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡇⠀⠀⣿⠁⠀⠁⠀⠀⢸⡇⠀⠁⠀⣼⡇⠀⠀⠀⢉⡗⠾⢸⡇⡇⠀⢸⣋⣀⠀⠀⠀⣠⣿⡇
⠀⢠⣄⠀⠀⠀⠈⢠⣧⣤⣀⣀⣤⠀⠀⠀⠀⠚⠻⠶⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣿⠃⠀⠸⠀⣄⣷⠹⣸⣿⡇⠀⢈⣿⣿⣿⣿⣼⣿⣿⡇
⠀⠀⠙⠛⠿⠷⠶⠾⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠻⠀⠀⠀⠀⠀⠁⠛⠿⠟⠀⠀⠈⠻⠿⠿⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⡤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢈⡷⣕⡗⠀⠀⠀⠀⠀⠀⠀⣨⠻⡟⠀⠀⠀⢠⠎⠄⠌⠤⡀⠹⡄⠀⠀⠀⣼⣻⢟⣷⣿⡿⣆
⠀⠀⠀⣤⢄⡴⠋⠀⠀⠀⠀⠀⠀⢀⣄⢠⠞⠁⠀⠀⠀⠀⠀⢸⡀⠀⡘⢀⠄⢄⡇⠀⠀⠀⣿⢿⡻⣤⣧⣇⡿
⠀⠀⠀⠫⣳⡵⠀⠀⠀⠀⠀⠀⠀⠸⢿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣔⣂⡤⠞⠀⠀⠀⠀⠘⠿⣿⣙⣽⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣠⣆⣀⢀⣀⣀⡀⣤⡄
⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⢠⣼⡄⡄⣽⡅⣤⢰⣦⠀⠀⣾⡇⣶⣸⣧⣶⢸⣷⠀⠀⠘⠛⠸⣴⣷⡼⠃⠟⠃
⠀⠀⢠⣴⢰⡿⢿⢿⣀⣦⡄⠀⠀⠉⠁⠉⣿⣏⠉⠘⠉⠀⠀⠀⠁⠀⣽⣿⠈⠈⠀⠀⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠘⠻⠈⠊⠈⠚⠈⠟⠃⠀⠀⠀⢸⡟⠉⠉⣿⠀⠀⠀⠀⠀⠀⢀⡟⢹⡄⠀⠀⠀⠀⠀⠀⠀⣼⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠙⢿⣆⣀⣾⡿⠃⠀⠀⠀⢀⡩⠅⣶⣯⣿⡆⠀⠀⠀⠀⣸⡑⡷⢠⣾⡤⠀⠀⠀⢶⣧⣶⠐⣖⣸⠋⡿⠇
⠀⠀⠀⠾⢠⢟⣛⡣⡐⠇⠀⠀⠀⢇⡀⢠⠉⡠⠼⡻⠂⠀⠀⢠⢿⢿⢀⣀⣚⡠⠄⠀⠀⠈⣉⣉⢹⣿⠁⠒
⠀⠀⠀⠀⢻⡻⠤⢇⠇⠀⠀⠀⠀⢠⠁⠀⢷⣤⣾⡁⠀⠀⠀⢸⠘⠈⠥⠤⠈⢤⠀⠀⠀⠀⠉⠁⣼⢻⡄⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠋⠀⠛

⠀⠀⠀⠠⠄⢠⣯⡀⣲⡀⠀⠀⠀⣴⠒⠒⠒⠒⠒⢲⠀⠀⠀⢸⠛⢛⡛⠛⢛⣛⡇⠀⠀⠀⡏⢩⣭⠉⣉⠉⢹
⠀⠀⠀⢿⡟⠈⣿⠀⣿⠃⠀⠀⠀⣿⢸⡗⠒⠂⣿⢸⠃⠀⠀⢸⢀⢈⣁⢀⢈⣉⡇⠀⠀⠀⣇⣈⣉⣉⣟⣀⣸
⠀⠀⠀⣘⣃⣶⣷⣦⡭⣤⠀⠀⠀⣿⣀⣀⠀⢀⣀⣸⠀⠀⠀⢸⢈⢀⢁⡀⢈⣈⡇⠀⠀⠀⣿⠹⡏⠙⣛⢻⣿
⠀⠀⠀⠛⠚⠛⠛⠛⠓⠛⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠘⠒⠒⠒⠒⠚⠒⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠹⢂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠂⠀⡔⠉⠃⠀⠀⠀⠀⡔⠀⠢⡀⠀⠀⠀⠀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠘⣄⠀⠰⡀⠀⠀⡀⠸⢄⡀⣅⢀⠠⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠔⠒⢬⣦⡀⠈⢲⠉⠀⢀⡠⢋⠉⠀⠀⣀⡠⠔⠫⠀⠀⠀⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢶⡀⠀⠑⠀⠈⠀⠀⠀⡏⠀⢀⣠⢤⠶⡀⠀⡇⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠚⢓⡄⠀⢠⡄⠀⠀⠀⡘⠀⠀⣧⠬⢆⣇⠱⠠⠇⠃⣀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⡆⠈⠉⠂⠀⡐⠀⠀⠀⠦⣀⠂⢀⡗⠀⠒⠀⠠⠄⡀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣱⠀⠀⠀⡰⠁⠀⠀⠀⢠⠀⡇⡇⡇⠀⠀⠀⠀⠀⠑⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢬⡀⠀⣠⠁⠀⠀⠀⠀⠀⠂⡇⡇⠠⣀⡀⠀⠀⠄⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠉⠦⡷⠳⠀⠀⣀⠀⠀⣠⠁⡇⠀⠀⡀⣉⠐⠐⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣨⣂⣀⣥⠀⡄⢠⣋⠀⠥⠐⠂⠃⠀⠀⡇⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠁⠀⡋⠀⡇⢹⡀⢙⠸⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢤⡤⠆⠀⠰⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢈⢁⠀⡠⡄⢃⢸⣆⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡸⠀⠒⣔⣁⡨⡅⠃⡅⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢳⢀⢠⠤⠵⢶⠒⢰⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠥⠓⡶⣖⣐⠿⣉⠁⠀⠀⠀⠀⠀⠀⢸⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⢿⡡⣿⡾⡾⠿⠭⢟⡉⠁⢠⡈⠁⠚⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠁⠀⢀⠀⠀⠀⠀⠚⠒⠲⠽⠧⠼⠦⠤⠴⠞⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠉⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⢉⡠⢴⠟⠈⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠠⠀⠀⢐⠐⠋⠀⠀⠀⠀⠀⣳⢴⣷⠆⠄
⠀⠀⠀⠀⠀⠀⣞⣁⠔⠊⠁⡴⠋⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠂⣏⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⢀⣀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇
⠀⠀⠀⠀⠀⠀⢀⡷⢌⡀⠀⠀⠀⠀⠀⣀⣠⡴⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡔⠁⢴⢧⠄⣤⡂⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠮⣸
⠀⠀⠀⠀⠀⠀⠈⣮⠀⠀⠹⡋⠋⠛⠙⡯⠁⠀⢸⡿⠀⠀⠀⠀⠠⠖⡀⠀⠀⢸⢀⡆⡸⣰⢠⠱⢸⠘⠠⣼⠀⠀⠀⠀⠀⡌⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠅⠀⠀⠀⣭⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠘⠛⠄⡗⡷⢾⢭⣩⡧⢠⠁⡀⠀⠀⠀⢀⠅⠀⠀⠌⢈⠇⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠹⢛⣿⣿⠏⠀⡰⠋⡔⠀⠀⠀⠀⠈⢀⠀⠐⠄⠼⠀⠀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠓⠀⠈⠀⠊⠀⠀⠀⠀⠀⣠⠎⠀⠀⢀⠀⢠⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢛⡵⣣⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡞⡩⢊⡴⣁⢊⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡔⠀⠀⠀⠇
⠀⠀⢲⡀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠠⣴⢾⠋⢽⠤⠿⣜⠕⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠐⢀⠆⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⡄⠌⠀⠀⠀⠀⡆
⠀⠀⠘⢶⣢⠉⠀⠀⠀⠀⠀⠀⡠⠖⠩⢀⣽⣳⠆⣀⡼⠂⠀⠀⠀⠀⠀⢀⡜⠁⢄⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠈⡙⡀⠀⠀⠀⢀⠀⠜⢗⠔⠠⠏⠁⠠⠒⠟⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠛⠁
⠀⠀⠀⠀⠁⢡⠀⡀⠐⠈⠀⠜⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠆
⠀⠀⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡂⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡼⠶⡦⠀⠀⠀⠀⢒⡡⢒⠛⢋⡀⢀⣠⣄⠀⠀⠀⠠⠏⠀⠀⢠⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠀⠀⠀⠀⢂⡖⠒⠛⠂⠀⠀⠒⠂⢴⠋⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⣄⡇⠠⢁
⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢯⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢉⣇⢀⠂
⠀⠀⠀⠀⠀⡴⠊⠀⠙⣆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢴⠀⠊⠀⠀⠀⠀⠀⣀⡀⣠⠾⠋⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠣
⠀⠀⠀⠀⠀⠻⠇⢶⢚⣃⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠒⠲⠶⠈⠉⢋⠓⠚⠉⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠘⠁⠀⢀
⠀⠀⠀⠀⠀⢠⠃⠀⢻⣵⣶⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠨⠿⠗⠂⢀⠠⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠇⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠲⢁⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠬⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣀⠄⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠻⣦⠀⠈⠁⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⠀⠘⣇⠀⠩⠤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⢖⠋⠀⢸⣄⣀⣀⠀⢸⠀⠐⡒⠚⠈⣁⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⠿⠋⢹⠀⠉⠁⠁⠀⠸⣿⣿⠿⠀⢸⡇⠀⠤⠤⢄⣀
⠀⠀⠀⠀⠀⢸⡁⡟⠉⠁⠀⠀⠀⠸⡄⠀⠀⡀⠀⠀⢏⠁⠀⠀⢸⡇⠀⣈⠑⠂⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠈⠓⠸⢤⣶⣶⣦⣴⠖⠛⠀⠀⠒⠚⠲⠼⣦⠀⠀⢸⠇⢀⠀⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⢠⡀⠑⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖

⣬⠋⢷⣿⣿⣿⣿⣷⡄⠀⠀⠈⠙⠒⠦⡀⠈⣴⣶⡻⢧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡱⠃⠀⠀⠀⠀⣴⣾⠿⡻⢿⣿⣷⣿⣿⣿⠀⠀⠀⠹⡄⣰⢿⢒⡰
⢋⣦⢸⣎⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠚⠛⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⣿⣾⣛⣼⣿⣿⡿⠟⠁⠀⠀⠀⠀⠘⣟⣸⣟⣧
⣁⡿⢫⣾⣿⠿⠛⠛⠛⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⢿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢿
⣥⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⠀⢀⡄⠀⠀⠀⣀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣀⣠⡖⠀⠀⠀⠙⢷⣤
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣠⠟⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠈⢿⡅
⠀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⢠⣿⣮⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡄⠀⠀⠀⠀⣰⣿⠁⠄⠀⠸⣿⡄
⣤⣶⣿⣿⡿⠟⢉⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣾⢿⣩⠽⠛⢿⠀⠀⠀⢠⣿⣿⢰⣶⣶⠄⣿⣷
⣿⣿⣿⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣀⣀⣄⡤⠖⠃⠀⠀⠀⠻⢄⡀⠠⠀⠸⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⡻⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣻⡭⠿⠿⣿⠇⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠨⢿⡷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣡⣤⣼⣧⠀⠀⠈⠛⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢛⡛⠛⢳⣦⡀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⣠⢴⡶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣾⣿⢥⡾⢉⣼⢷⣿⡯⢻⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢻⠀⠀⡻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⣀⣠⣾⣿⣯⣾⢹⣖⡀⢙⢷⠗⣬⡏⠳⣄⠀⠀⠀⠀⢹⣆⠀⠀⠀⠈⢧⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣿⣉⢛⣿⣿⣿⣿⠟⠀⠀⠀⠐⣿⣿⣿⣿⣿⣧⣽⣄⠈⡟⠶⠆⢒⡿⠁⠀⠈⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⢧⠀⠁⢿⣿⣿⣿⣿
⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠎⠓⠷⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣾⠿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⣶⣼⣶⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⠀⠀⠀⠀⠀⣠⡄⣤⡤⠞⠃⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⣄⣀⡀⠀⢀⣀⣤⣤⣶⣿⡿⠂⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠰⢻⡿⠈⠀⠀⠀⠀⠘⠁⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⣀
⣿⣿⣿⣤⣿⣿⣿⣿⡟⠩⣤⣶⡤⠄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⣠⣴⣾⣿⠂⠀⠀⠀⠀⠀⠀⢨⠙⠛⢿
⡟⢻⣣⣿⣜⣻⣿⣿⣥⣄⡆⠀⠀⠀⠀⠀⠀⠘⠿⣿⣭⢬⣝⣻⢿⣧⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠻⢿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⣠⣤⣬
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡇⠐⠀⠀⠀⠀⠀⢠⢿⢄⣠⡲⢞⠛⢆⢹⣧⠀⠀⣼⣻⠿⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠙⢿⣷⣌⢻⣿⣷⣧⠀⠀⠀⠀⠀⡀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠑⢡⣔⣷⠞⢸⡟⠀⠀⠙⢿⣿⡿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⡇⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⣤⢦⣾⡀⠀⠀⠀⠀⠈⢧⣣⡸⠷⢿⣿⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠀⠀⠤⠇⢈⠋⠉
⠛⠿⣿⣿⣿⠿⠛⡄⠈⠀⠄⢹⠰⡀⠀⠀⠀⠀⠀⠉⢓⡶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣶⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠈⢻⣭⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⠶⠶⣾⡿⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢠⣝⣿⣿⢀⠈⠲⠆⢤⣄
⣶⣶⣶⣶⣦⡄⠀⠀⢀⣾⣿⡜⠟⠉⠁⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠁⠀⠀⠀⣰⠃⠀⠀⠙⠧⣤⡀⠘⠀⡠⢲⣿⠀⠀⠀⢠
⣶⣿⣿⣿⣿⠏⠀⠀⠀⠈⠉⠁⠀⣠⣤⣄⡀⠀⠀⠙⣾⡿⠀⠀⠀⠀⣰⣿⡇⠘⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣠⡾⠅⠀⠀⠀⠀⠀⠀⠹⣶⣾⣷⣿⠋⠀⠀⠀⠙
⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⡟⣻⠟⠷⡀⠀⠀⠈⠉⠁⠀⠀⢠⣿⣿⠁⣤⣤⣤⡄⣿⣿⣇⠀⠀⠀⠸⡻⠿⠀⣰⣾⣿⣿⣦⡀⠀⠈⠉⠉⠁
⢲⣤⡀⣠⣶⣤⣄⣀⣀⢀⣀⣺⣟⡊⡁⣠⢀⡙⢲⣄⡀⣀⣀⣀⣀⣸⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣀⣀⣀⣀⣀⣀⣼⣿⣿⣯⣻⣿⣿⣆⣀⢀⢀⣀⣀⣀⣀⣤⠖
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠄⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⡀⠀⣀⣠⣅⣀⠀⡀
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠋⠈⠙⠚⠉⢉⣻⣿⣿⡟⣻⠛⠛⠛⢛⠓⠒⠲⣶⡇⠀⠀⠀⡜⢠⡟⠉⠀⡀⢹⢳⡌⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠈⠲⠾⠶⠿⣿⣿⡿⠿⠿⠶⠷⠚⠀⠀⠀⠈⠁⠀⠀⠰⠁⢾⣀⢴⡜⢩⣷⠈⡇⡇
⠀⣾⣿⣿⣷⣄⠈⠙⠉⠰⣄⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⣸⡿⣿⣿⣿⣟⣆⠇⡇
⠀⣿⡿⠿⠛⢿⣶⣾⣿⣶⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⢰⠀⢹⣇⢾⣿⣟⣷⣿⣆⡇
⠀⠘⡇⠀⠀⠀⠙⠿⠿⠻⠉⠙⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠖⣒⣺⣓⠺⢗⢣⠀⠀⠀⠀⠀⣼⠀⢸⣿⣾⣿⣼⡿⢰⡏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⣮⣼⣧⣽⣆⣀⣷⡀⠀⠀⢄⣀⡇⠀⢸⣿⣯⣽⣿⣳⠿⡇⡅
⠀⠀⠀⠰⠓⢲⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⡀⠁⢹⣿⣿⣿⣿⡿⣿⡐⣕⡁⠀⠉⠀⡏⠉⢹⣿⡿⠘⠙⠛⠀⢠⠁
⠀⠀⠀⢠⠞⡫⠅⠒⢘⣯⠥⡶⢶⣤⣀⠀⠀⠀⣶⡀⠀⠀⣿⠖⣺⣿⣛⣻⣿⡿⠟⠸⢳⡇⠀⠀⠀⡇⠀⠀⣿⡑⠤⣤⣠⠵⠋
⠐⠀⡰⠡⠂⠀⠀⠀⠀⠉⠀⢈⣦⣿⣿⣆⠀⠀⠀⠈⠲⢦⣀⣺⢄⣀⢸⠀⣨⠁⠄⡡⠋⠀⠀⠀⣼⠁⠀⠀⢻⣇
⡄⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣽⠟⠀⠀⣆⣀⣠⣶⣤⣭⣬⣭⡭⠭⠽⠒⠋⠀⠀⠤⠄⠼⠧⠀⠀⠴⠧⠿⠷⠦⠄
⢧⠀⣇⠀⠀⠀⠀⠀⢰⡾⣛⢛⣿⣿⢻⠀⠀⠀⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠘⣆⠘⢦⡀⠀⠀⠀⢼⣦⣷⡈⣱⢃⡞⠀⠀⠀⡟⠀⠀⠀⡤⢤⠤⢶⡦⠤⠤⠤⢤⣤⠤⠤⡤⣤⣤⣬⣡⡄
⠀⠈⠳⣌⡛⠦⣄⣀⣀⣻⡵⠟⣡⠎⠀⠀⠀⠀⠇⠀⠀⢀⠀⢸⠀⢰⡀⠀⠀⠀⠀⡏⣿⠉⣻⠉⠈⠛⠛⠟⠒⠒⠒⠒⠒⠂⠒⠒⠖⠲⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⢶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢟⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢍⠉⠃⠀⠀⠀⢀⠄⠒⠒⠂⢀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣺⣯⣶⠄⢀⠤⠄⠂⠀⣀⠀⠀⠔⠃⠀⠐⠤⡀⠐⠄⠀⢀⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡝⠟⠉⡀⠎⠉⣶⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⢀⡈⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣫⡞⠀⢼⠀⢀⡀⣷⡀⠀⢄⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡇⠊⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣾⣦⣼⣶⣤⣽⣿⣿⣶⣾⣷⣤⣿⣿⣦⣄⣀⣀⣤⣄⡀⢀⣀⣁⡀⠀⠀⠁⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣝⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⡿⠿⠶⣶⣦⣴⢀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠳⣭⣿⣥⣝⣿⣷⣶⣿⣿⣿⣿⣁⡀⠀⣀⡀⢠⣿⣿⣿⣾⣿⡿⠿⣿⣫⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⢸⠿⢿⣿⣷⣿⣏⠉⢻⡟⣿⣇⠉⠋⠀⠀⡟⠃⣼⠟⠉⣹⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣬⡉⢿⡙⢿⣆⠀⠁⣿⣿⠀⠀⠀⢠⡇⢰⠏⢀⣾⣿⠟⣡⣾⣿⠁⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠲⣝⠆⠙⢧⠀⣿⣿⠀⠀⠀⠈⡇⠀⣠⡿⢋⡴⡚⠱⠂⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣭⣙⡛⠂⠁⠀⠀⢰⣿⣿⠀⠀⠀⠀⢱⡐⠋⠀⠑⠊⠁⠔⢛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣭⣽⣛⣛⠟⠃⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⢧⠀⠀⠀⠴⠒⢛⣛⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣾⣿⣿⣿⡄⠔⠒⢂⣀⠜⠆⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠁⡀⠀⠀⠈⠺⣿⠟⠛⠻⣿⣦⣴⣈⡀⠀⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣖⣁⣀⣀⣀⣈⣙⢿⠶⣶⣀⣀⣠⡤⠈⠛⠉⠙⠛⢿⣿⣁⣙⣒⣲⣷⣶⣶⣀⣾⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⣟⣹⢛⣟⣻⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⡠⢨⣥⠀⠀⠈⠐⣄⠲⠦⣤⣸⣿⣣⡤⣀⣀⣀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⠘⠃⠅⠬⠠⠤⠴⠶⠿⠿⠷⠞⠀⠀⠐⠳⡁⠒⠐⠊⠁⣴⣿⣷⡀
⠀⠀⠀⢰⣼⣿⣆⠁⠁⠀⢀⡾⡆⠀⠸⣿⣿⢻⣭⣝⠳⡶⠬⣭⣍⣉⣾⠋⢁⢀⣠⡆⠠⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣧⣤⠤⡤⣴⣿⣿⣿⣇
⠀⠀⠀⣬⣿⣿⡷⢈⠀⡑⢨⣿⣇⠀⠘⢼⣿⣺⢯⣿⡷⠃⠀⠀⠙⠕⣏⣉⣿⣿⣿⠇⠀⡀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣉⣩⣀⣁⢈⣼⣿⠟⠁
⠀⠀⠀⢻⣿⣿⣏⠀⠁⠪⢒⣏⡟⠀⠀⠀⠑⠛⠛⠺⠥⠆⠀⠀⠀⠀⠉⠉⠈⢹⠏⠀⠐⣿⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠉⠉⠉⠛⠻⠟⠁
⠀⠀⠀⢨⣻⣿⡗⠈⠂⠕⠨⣯⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡑⠒⠒⠁⠀⠀⠀⠻⣷⠀⠀⠉⠉⠙⠛⢷⣤⠀⠀⢀⡤⠶⠒⢶⣶⠒⠶⠤⣀
⠀⢸⣿⣿⣿⣿⡏⠐⠀⠈⠐⣿⣏⣿⠀⣀⣤⠤⠤⡤⠤⠤⠤⢠⣬⣭⣭⡤⠀⠀⠐⠒⠲⠀⢿⡀⠀⠀⠀⠀⠀⣸⡷⠀⠀⠅⠀⠀⠨⠐⠐⠈⠄⠀⠈⠄
⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⡿⣹⠀⠀⠈⠀⠀⠀⠓⠃⠿⠿⠿⢿⣿⣿⠿⠧⠤⠄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡞⡄⠀⣷⣦⣵⣶⣒⣒⣲⣾⣥⣾⠇
⠀⠀⢀⠃⠹⣿⣿⣿⣿⣿⣿⠃⠌⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣹⣿⡀⠀⠀⠀⠀⣿⣷⠰⡀⠙⠻⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠈⢧⣄⠘⠻⠻⠻⠟⢃⣐⠔⠀⢻⣿⣿⣿⣿⣿⣿⣇⣀⠀⢀⣀⣀⠀⢰⣶⠀⣴⡗⠊⢸⣿⡄⠰⡀⠀⠀⣿⣟⣧⡀⠀⠀⠈⠁⠈⠁⠀⠁
⠀⠀⠀⠀⠙⠲⠦⡤⢥⠶⠛⠁⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠀⢸⠟⠀⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡷⢽⣦⣄⠀⠀⠀⢀⣴⡿⠟⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠉⠙⢻⣿⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠀⠸⠀⠀⠉⣷⡄⡄⢹⣷⡄⠀⠀⢠⣿⣏⣿⡇⣽⢿⡗⠀⣾⡟⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢹⣿⠛⠛⢿⣿⠟⠙⡇⠀⠀⢀⣠⡿⠁⡇⢺⣿⣇⠤⡀⠘⣾⡧⡜⢱⣿⡼⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠤⠴⢶⣶⣿⣘⠀⠀⠀⠄⡀⠀⠀⡇⠀⠀⢸⡗⠀⠀⢸⡿⠀⠀⠃⠀⠀⢸⡏⠀⠀⠁⣸⣿⠄⠀⢡⠀⢹⡷⠿⢸⣿⡁⠀⠘⡟⣁⡀⠀⠀⠠⣸⣿⡇
⠀⢠⣀⠀⠀⠀⠈⠃⢼⣄⣀⡀⢀⣨⠀⠀⠀⠀⠐⠿⠷⠆⠀⠘⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⡟⠀⠀⠌⠀⣸⡖⢺⢸⣿⡇⠀⠀⣿⣿⣿⣷⣦⣾⣿⣿⡆
⠀⠀⠙⠿⠷⣶⣶⣶⣾⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠸⠇⠀⠈⠀⠀⠘⡉⠿⣻⠿⠁⠀⠀⠻⢿⣿⠿⢿⣿⣿⡿⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣫⣾⣿⣿⠿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⣡⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⡾⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⡾⣫⣴⣿⡿⠿⢿⣷⣄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣾⠟⣡⡾⣫⣾⡿⠋⠁⢀⣠⣀⢹⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣄⠀⠀⣰⢁⡴⠋⣡⡾⣫⣾⡿⠋⠀⣠⡾⠋⢁⣿⣸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣶⣜⣿⡛⠦⣿⠋⣠⠞⣡⣾⡿⠋⠀⣠⠞⠁⢀⣴⣟⣵⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣽⣯⠉⠳⣄⠀⠺⣡⣾⡿⠋⠀⣠⠞⠁⢀⣴⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠳⣄⠈⠳⣄⠈⠉⢀⣤⠞⠁⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣡⣄⠈⠳⣄⠈⠳⣄⠈⠁⢀⡴⣻⣽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⣿⣿⣿⡗⠀⠈⠳⣄⠈⠳⣄⠈⠺⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡵⢋⣼⡿⣿⣝⢿⣦⣱⣄⠈⠳⣄⠈⠳⣤⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣵⡿⠋⠀⠈⠻⣷⣿⣿⣿⣷⣄⠈⠳⣦⣽⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠹⢿⣦⣤⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁
⠀⠰⢾⣿⡏
⠀⠀⠀⠙⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠸⡋⠻⣿⣿⣧⣴⣶⣶⣶⡄⠀⠀⠀⢀⠰⠒⠒⠀⠀⠀⠛⠉⠢⠄⠀⠀⢠⣿⣿⣿⣿⣿⣤⣴⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠘⡻⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢧⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⡀⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠈⠙⠻⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠿⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣄⠀⠀⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣸⡿⠶⣦⣀⢀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠍⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⡠⢤⡴⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠋
⠀⠀⠀⠀⠀⣀⣤⠶⠟⠉⠁⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀
⣀⣠⣴⣶⠟⠁⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒
⣿⣫⣭⣴⣶⣶⣿⣿⠿⠟⢛⢫⣿⣷⣦⣤⣄⣀⡈⠉⠙⢿⣿⣷⣄⣀⣠⣤⣴⣶⣞⣫⣭⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⣀
⣿⣿⣿⠿⠿⣿⣯⣤⣄⣀⠈⠰⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿⠋⠉⣿⣿⣿⠛⠉⠉⠉⠛⠻⣶⣦⣤⣤⣶⡶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠉⠀⢄⣀⡀⠸⠿⣿⣿⣿⣿⣾⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣸⣿⡿⠁⠀⠀⠀⢀⣴⣾⣿⣿⠟⠋⠉
⠀⠀⠀⠉⠛⢦⣤⡂⡨⠝⠹⡛⠹⢛⠻⠛⠋⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣤⣤⣶⣿⣿⣿⣿⣾⣾⣿⣿⣶⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠂⠂⠀⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠁⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⢷⣄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⠶⢦⣤⣤⣤⡴⠾⠛⢻⡄⣿⠀⠀⠀⠀⠀⠀⢀⣾⠛⣷⠀⠀⠙⣧⡐⠛
⠀⠀⠀⠀⠀⠀⠀⣰⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡆⠀⠀⠀⠀⣠⡾⠁⠀⢹⣆⠀⠀⠈⢷⡀⣠⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇⠀⠀⣠⣾⡏⠀⠀⠀⠀⢻⡟⠛⢷⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣴⠟⠋⢸⡇⠀⠀⠀⠀⠀⢻⣄⣾⠃⠈⣷⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢰⡟⠁⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣴⡿⠋⢰⣿⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⣀⣀⣘⣷⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣛⣉⡁⠀⠀⠘⢿⣿⣦⣀⣀⢀⣀⣀⣽⣶⠞⠛⠉⠉⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⢿⣍⣩⡟⠙⠛⠶⣦⣀⠉⠉⠉⢻⣟⠉⠻⣷⣽⡆
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⢀⡿⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣦⠀⠙⣿⡄
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣼⠃⠸⣧⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠙⣷⡀⢘⡿
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠛⠛⠁
⠀⠀⠸⣧⣤⣤⣤⣀⣀⣀⣀⣀⣀⣈⣻⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁