⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠹⠿⠿⠿⢿⣿⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⣠⣴⣾⠿⠟⠛⠻⠿⠿⠿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄⠀⣀⣤⣶⣾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣶⣤⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣿⣷⣀⣀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⠻⣿⣆⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣾⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣷⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⢀⣴⣾⠿⠛⠋⠉⠉⠁⠁⠀⠀⠀⠀⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠰⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠀⠹⡟⠀⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠿⢿⣿⡇⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⣽⠟⠒⣦⠦⢤⢤⣤⠴⢦⡤⠶⢶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢠⣾⣿⡇⠀⠀⠀⢿⣿⣏⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⠿⣿⠿⠿⠿⣶⠶⠶⠿⡟⠛⠋⣋⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡀⠀⠘⠃⢀⣈⣁⣤⠴⣿⡏⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠙⠀⠀⠀⠈⠀⠠⡤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣽⣿⣉⣉⣉⣉⣀⣴⣿⣿⡁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣤⣶⣿⣿⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⠴⠖⢛⣋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠙⣿⡗⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣷⣶⣿⡿⣷⣿⣿⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠋⠉⠁⢈⣀⣈⣉⣉⣙⣛⡻⡿⠶⠶⠶⢶⣶⣶⣶⣾⠿⠁⠀⠀
⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡇⠀⠿⣿⣶⣿⡇⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣶⠶⠶⠛⠛⠉⢀⣠⣤⠶⠖⠛⠛⠉⠉⠉⠉⠉⠙⠛⢛⣷⣤⣤⣾⡿⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⡿⠁⠀⠀⢿⣇⠀⠀⠀⢹⣿⠃⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣾⣿⣤⣀⠀⠀⠸⣿⡄⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⣿⣿⣟⡛⢿⣷⡄⠀⢻⣿⣄⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿
⣾⡟⠁⠹⣿⣿⡿⠿⢁⣴⡿⢿⣿⡿⢿⣿⣤⠤⠤⣼⣿⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠿⢟⣿⣿⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⣡⣾⡿⠋
⠏⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠛⠿⣿⣿⠁⠀⠀⢸⣿⠛⠿⠿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡿⠿⠛⠛⠉⢉⣠⣴⣿⠟⠁⠀⠀⠈⠙⢿⣦⠀⣠⣴⡿⠟⠉⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣄⣀⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣤⣴⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⠋⠀⠀⣀⣴⣾⠿⠿⠿⠟⠃
⣦⣶⣦⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢸⣿⡟⠿⠿⢿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢷⣀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀
⣿⠿⠿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢠⣾⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣿⡆⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀