⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀⠘⠿⠿⠿⠿⠿⠿⠷⠜⠿⠿⠿⠿⠞⠻⠿⠿⣿⡆⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⢰⣿⠿⠿⠟⠱⠿⠿⠿⠟⠣⠾⠿⠿⠿⠿⠿⠿⠂
⠀⠙⣻⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡥⡙⣿⣷⣝⣷⣄⠀⢀⣤⣿⣿⣿⣿⣦⡀⢀⣠⣾⣧⣾⣿⢋⢬⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀
⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣧⡸⢷⣻⡵⣨⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠟⣅⢮⣟⠾⢇⣬⣭⣭⣭⣭⣭⣭⠭⠁⠀⠀
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣩⠵⣚⡵⣣⣆⡙⡿⢿⢹⣿⣿⣿⣿⣿⣿⡏⡿⢿⢋⣰⣜⢮⣳⠮⣍⣾⣿⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠊⠫⡾⣫⡾⣽⢷⡇⣼⣿⣿⣿⣿⣿⣿⣧⢸⡾⣏⢷⣽⢦⠝⠑⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢫⡾⣹⣿⣿⡿⠿⡿⣿⣿⣿⣎⣷⡝⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⣿⣿⣿⠿⢿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡒⣍⡁⣺⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⢾⡶⣫⢿⡇⣿⢿⣿⡰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡛⠵⠛⠁⣠⣾⡷⠀⠉⠺⣗⣥⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠁⠀⢾⣿⡋⣠⣾⣷⡄⠀⠛⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠸⠁⣶⣦⡀⢹⣿⣿⡋⠙⢿⡦⠨⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣧⠄⠈⠻⣿⡿⠋⢻⣿⣦⠀⠀⢰⣿⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⡱⠄⠀⠈⠀⣴⣿⠟⠁⠀⠤⠡⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣾⡆⡄⣨⣅⢰⠴⣤⠟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠃⠬⠥⠐⠃⠉⠀⠀