Labor Emojis & Symbols

⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣻⡿⣿⡿⡟⣿⢿⣿⡟⣿⠻⣿⡿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣾⣞⣇⣿⣇⡇⣭⣻⣿⡇⣿⣣⣿⣿⣹⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣬⣭⣽⣦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣿⣿⣟⡛⢛⣛⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⣿⠻⣿⢸⣿⠟⠿⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠿⣿⣿⣿⣿⣿⣇⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⣾⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢻⠟⢿⠛⠛⠟⠛⢟⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣦⣼⣤⣼⣤⣾⣇⣶⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⣿⠛⢻⣿⠛⢛⠛⢿⠟⠋⠛⢿⡟⠛⡛⠻⣿⣿⠛⠛⠛⢿⣿⠛⢻⣟⠛⡿⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⠇⠐⠀⢿⠀⣨⠀⢺⠀⢾⡇⢸⡇⢈⠁⢰⣿⣿⠀⢸⠇⢸⡇⠐⠀⢿⡆⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣿⣦⣼⣤⣤⣤⣾⣷⣤⣤⣾⣧⣼⣧⣬⣿⣿⣤⣤⣴⣾⣤⣿⣦⣼⣷⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠁⡀⠀⠀⢀⣀⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣨⣤⣴⣶⣶⣶⣶⣦⣤⣄⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⠋⢉⣀⣠⣤⣤⣤⣤⣀⣹⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⠻⠿⠿⠟⠉⠉⠻⠿⠟⢋⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠛⡀⠀⠀⣤⣤⣤⡤⠀⠀⣄⣿⠟⢁⡴⠋⠀⣠⠟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠈⠆⠀⠀⠭⠭⢄⠀⠘⢿⡇⠈⠉⠀⠐⠛⠁⢀⡴⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠛⠛⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠈⠋⢀⡄⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⠳⢶⣦⣤⣶⣶⣿⠟⠉⠑⠢⠀⠀⠀⠀⠀⠉⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣥⠖⠛⢉⣉⣙⣳⣄⠀⡙⢿⣿⠿⠋⠀⠀⠢⣄⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⢁⣠⣤⣭⣿⡷⠞⠋⠁⠀⠀⠀⠀⠀⠈⢳⡀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣱⠆⢀⡴⠛⠉⢡⠞⠁⢀⠄⠀⢀⡔⠁⠀⠀⠀⢀⣾⡷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⢠⡟⠀⠀⠐⠏⠀⣰⠃⢀⡴⠋⠀⠀⠀⠀⣠⡿⢿⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⡄⣾⡇⠀⠀⠀⠤⠴⠋⠴⠋⠀⠀⠀⠀⠀⣰⣿⡃⠀⢻⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣷⢻⣇⣴⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢀⣴⣿⣿⣧⣤⣼⣷⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣄⠀⠀⠀⣀⣴⣫⣴⣾⡋⠀⢻⣿⣿⡆⠀⢹⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣖⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⣠⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠤⠤⠒⣶⠀⠀⠀⠀⠸⡟⠀⢀⡤⢖⠒⣿⠗⢄
⠀⠀⠀⠀⠀⠀⢠⠟⣰⠞⣶⢇⣼⠟⢻⣤⣾⠟⣆⣼⠂⠻⠾⢛⣼⠏⠀⣸⣴⠟⣾⢢⡗⢠⡇⠀⢠⡀
⠀⠀⠀⠀⠀⢠⠏⢰⡃⢺⠇⡼⣃⡴⠋⣟⣠⠞⣿⠅⠀⠀⠀⣼⣃⠴⠊⢽⠁⣾⠇⣟⣴⡟⣠⠴⠋
⠀⠀⠀⠀⠾⠋⠈⠒⠒⠦⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣯⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⡠⠴⠒⢲⣶⣶⣬⠉⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⣼⠀⢠⣤⣾⠽⠿⣿⣿⣷⠆⠈⢓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⢤⣦⢀⣼⠋⠁⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⣀⣠⣤⠀⢛⣿⠛⠃⠀⠀⠀⡀⢨⠀⠀⠀⠀⠀⠀⣄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣤⢶⣿⠋⠉⠐⠛⠀⠈⠿⡟⣧⡀⠀⠚⠛⠂⠀⠀⠀⠀⠠⠀⠁⠆⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠉⣿⡇⠀⠀⠀⠀⣀⣴⡁⠈⠙⢶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⢉⣻⠠⡴⢶⢿⡟⢻⣇⠀⠀⠈⢀⣽⣉⣿⠓⠶⢄⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠁⡲⢍⣧⣡⠈⣶⣇⠠⡙⢑⠒⢺⣿⢏⠸⣿⡆⠀⢠⠉⢦⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡅⢐⣬⣶⣿⣇⣆⣻⣿⡄⠈⠲⠉⠛⠁⠀⠀⣿⢿⡀⡾⠀⠈⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣴⡆⠀⢋⣽⣿⣿⣏⢹⣥⡇⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⡇⠀⠀⢸⡆⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⡾⠀⠀⢀⣿⣼⣿⣿⣿⣿⣤⣇⠀⠀⠀⠀⠀⠀⢀⡈⣤⣿⣇⡠⠔⠛⣹⣦⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠟⣿⢿⣤⣴⡾⠿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣾⠇⠀⠀⣿⡟⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡉⢻⠀⢹⡧⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣅⣀⣄⠀⠀⣿⠃⣼⠻⠃⠀⠀⠀⠀⠀⠀⠀⠘⣇⣸⣤⣟⠀⠀⢀⡇⢀
⠀⠀⠀⠀⠀⠀⠐⢆⠘⣤⠊⠉⢹⠉⠙⠲⢿⣻⣧⣤⠂⠉⠀⠀⠀⠈⢱⣖⣠⠿⠛⠋⠉⡗⠀⠉⢢⡜⡠
⠀⠀⠀⠀⠀⠀⠀⠾⠀⠙⢆⠀⢾⡀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣹⠀⡠⠞⠤⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠒⠊⠉⠉⠉⠓⠦⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢧⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢀⡀⣀⣤⡤⠤⠶⠶⠖⠒⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⣀⡠⠤⠄⡶⠒⠂⠉⠉⠉⠉⠉⠉⠓⠄⡉⠉⠉⠉⠉⠒⠒⢢⣤⠴⢇⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢛⡁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⢀⠆⢏⠰⢦⠙⢆
⠀⠀⠀⠀⠀⠀⢰⠁⡴⠃⠈⡗⠀⠐⠁⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠡⠀⠀⠀⠀⠑⠀⠀⠌⠀⡦⠀⠀⣇⡼⠦⠶⢄
⠀⠀⠀⠀⠀⠀⠈⢆⠳⠤⡄⢷⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠁⠀⢠⠀⠀⠀⠈⠁⠀⠀⢷⣰⡞⠁⠀⠀⠀⠀⠙⠤⠔⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⠼⢆⠀⠀⠀⠀⠴⢯⣀⠀⠀⠀⠀⠀⠀⣠⠞⢤⠀⠀⠀⠀⠀⡸⠖⣾⠀⠀⠀⠀⠀⠀⢀⣒⠒⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢷⡀⠈⢣⡀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⠷⠚⠋⢹⠀⡆⢸⠋⠉⣇⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⡇⢹⠋⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⠀⢣⠀⡇⢸⠇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡇⠸⡄⠀⠀⡼⠁⠉⣒⣤⡤⢤⡤⠤⠤⠤⣴⠲⡞⠛⠒⠒⠢⠤⠤⠤⠤⠤⣤⡼⠤⠥⣴
⠀⠀⠀⠀⠀⠀⢘⠁⢰⡋⠀⠀⡵⠀⢀⣱⠖⠋⢀⠏⠀⠀⢳⡄⠀⡰⠁⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⡼⠀⣠⠇⠀⢀⣃⠴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠹⡞⠁⠀⠀⠀⢹⡄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢸⠀⢸⠁⠀⣠⠞⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⣀⡷⠙⢧⡀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢷⠢⣄⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠿⣀⠀⠀⠙⠲⣄⠀⢀⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⠙⠺⣧⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⡼⠉⠓⢤⡀⠀⠈⠙⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠙⠢⢄⣀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠑⠢⠤⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢀⣠⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠒⠶⢤⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠴⠖⠒⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠸⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢘⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠑⠒⠒⠢⠤⠤⠤⠤⠤⠒⠒⠒⠊⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⣼⠂⢠⡶
⠀⠀⠀⠀⣸⣃⣀⣿⠁⣀⠀⠀⢀⡄⢠⣤⠀⣠⠀⣤⡄⠀⡀
⠀⠀⠀⢸⡏⢉⣿⠁⡾⣡⣿⢀⣾⡱⢡⣿⢴⣏⠎⣴⡧⣾⠁⣰⣗⡤
⠀⠀⠀⠾⠀⣼⠃⠀⠛⠁⠷⢻⡟⠰⠋⠀⣾⠃⠞⠁⠀⠛⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡔⣹⠃⠀⠀⡴⠃⢠⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠒⢃⠀⢠⠞⠠⠔⣡⠔⡤⢀⡼⣣⠀⣠⢶⠀⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠒⠒⠀⠀⠒⠀⠀⠒⠀⠀⢠⢀⡷⡃⠀⠀⢸⠁⠺⠁⠾⠀⠉⠸⠡⠂⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠐⢉⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠉⠀⠀⠀⠀⠴⠀⠁⡔⢃⠄⡀⢀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡾⡱⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⠀⠐⠃⢀⠈⠀⠇⠸⠀⠇⠈⢀⠀⠀⠀⠀⣼⡟⠁⢠⡟⣰⢏⣷⠆⣰⠇⢀⡶⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠔⠀⠀⠀⠉⠀⠘⠀⠰⠀⠰⠀⠈⠀⠀⠀⢠⡿⢀⣠⠟⠁⠻⠊⢿⠔⠻⠐⣹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡰⠃⠀⢠⠀⠀⢀⠀⠀⢄⠀⠀⢃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠎⠀⠀⢸⠀⠀⠸⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⠀⢀⣶⡄⠀⣶⣲⠄⢰⣖⡦⠀⢦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠈⠉⠉⠀⠉⠀⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣤⣤⡀⠀⢀⣤⣤⣄⡀⠀⢠⡀⠀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⢹⣆⠀⠀⣿⠶⢾⡇⠀⢾⡇⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⣉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⡤⠀⠼⠋⠉⠹⠆⠀⠛⠛⠛⠃⠀⠈⠛⠛⠛⠁⠀⠘⠷⠾⠋⠀⠀⠿⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡀⠀⠀⣀⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⢸⡏⠉⣷⠀⢀⣟⣻⡄⠀⠹⣶⠏⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣶⠖⠀⠀⠀⠘⠷⠚⠋⠀⠺⠉⠉⠻⠀⠀⠛⠀⠀⠀⠀⠀⠺⣶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣍⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⣄⠀⠀⠀⠀⠀⠀⢠⣤⣩⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣛⣩⡄⠀⠀⠀⠀⠀⢀⠀⠰⡇⠀⣹⠀⠖⠈⠁⢆⠀⠀⠀⠀⢠⣝⣛⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⠃⠀⠀⠀⠀⠊⠙⠀⠀⠃⠀⢣⡄⢀⣠⡀⠸⡄⠀⠀⠀⠘⡿⠛⠋
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢸⣆⠀⢠⣀⣸⣤⠜⠛⠻⢯⠀⠀⡇
⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢉⡟⠉⢿⠁⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⠠
⠀⠀⠀⠀⠀⠀⠈⠙⢲⣾⣶⡤⠀⢠⠄⡀⠀⠘⡌⠀⠀⠀⡀⡀⠀⢀⡼⠁⠀⠀⠀⢀⠀⠤⠀⠤⠀⠈⢀⣡⣤⡴
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⡋⢀⣀⡘⠀⠁⠐⠂⠘⣶⣶⠦⠼⢾⣷⣿⠁⠀⠀⠐⠂⠀⠀⢀⣀⡀⠱⠊⠉⠙⠉
⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠂⠘⠋⠉⢺⡶⡤⠺⠀⢸⠁⠀⠀⠈⠑⠀⠐⠂⠀⢐⢤⠶⡗⠉⠉⢃⡀⠀⠀⠀⠈⢿⣤⣦⡄
⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠙⣄⣀⡀⡀⠀⠃⠀⢸⠀⠀⢠⡏⢀⣀⣴⡞⠈⠈⢀⣀⢀⣀⣠⠊⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⣿⡏⠉⠙⠇⠀⠀⣼⠀⠀⢸⡇⢨⣭⡅⠀⠀⠀⠸⠋⠁⢹⣿⠀⠀⠀⠀⠀⣚⠻
⠀⠀⠀⠢⣀⡼⣯⠋⠀⠀⠀⠀⣿⠇⠀⠀⡀⠀⠀⣿⠀⠀⠈⡇⢸⣿⣷⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣸⡏⡀
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢠⣿⢰⠀⠀⡀⠀⠀⣿⠀⠀⠀⣿⣼⣿⣿⡆⠀⠀⢠⠀⠀⡇⣿⡆⠀⠀⢠⣿⣰⠁
⠀⠀⠀⠀⠀⠀⢀⣘⣆⠀⠀⣾⣿⣿⣤⣀⡇⠀⠀⣿⣠⣤⣶⣿⣿⣿⣿⣷⠀⠀⢸⣀⣤⣿⣿⣷⡀⢀⣾⣛⣃
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⠿⣿⠿⠿⡿⠿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡅⡟⣀⠹⠀⣥⡇⣠⣼⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⣿⡿⢿⣿⡿⠿⠿⣿⣿⣿⠿⢿⣿⡿⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⠿⣿⡿⢿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⡿⢀⠀⢿⡇⠀⠆⢀⡟⠀⣴⡀⢹⡇⠰⠆⠀⣿⣿⡇⠀⣦⠈⢻⠇⡀⠸⣧⡀⠋⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠛⠻⠃⣠⡄⠘⡇⠐⠒⠀⣧⡀⠛⠁⣸⡇⢰⡄⠘⣿⣿⡇⠈⠋⢀⡞⢀⣤⠀⢻⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⡠⠔⠒⠈⠉⠀⠒⠲⢤⡙⢿⣿⣿⣿⣿⣿⠟⢁⡠⠔⠒⠒⠀⠠⢄⡀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣼⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣉⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣻⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⢟⣿⠿⠋⠉⠉⠉⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠹⣷⠆⠀⠀⢈⣿⡟⣩⢍⢻⣿⡏⣾⣿⣿⣿⣟⣵⣿⣽⣿⡿⣱⡿⢗⡯⠀⠀⠠⠤⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⠘⠛⠁⠨⠈⣼⣿⢳⣿⣿⣿⣿⣻⢟⣿⣿⣿⣿⣯⡞⠁⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣲⣾⢿⣿⢻⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⣠⢠⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢫⣿⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⣤⡤⠖⠋⢁⣾⣯⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⡿⣿⣿⣾⣿⣿⣆⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⣾⣿⠿⠟⠛⠛⠛⠻⣿⣿⣿⣷⣭⣿⣶⣿⣿⣯⣿⣿⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢭⣀⡤⠖⠒⣛⣒⣒⣈⣹⣿⣿⣿⣿⡿⠤⠬⣍⣉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡴⢚⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⠙⠦⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⡚⠃⠃⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠄⠠⠀⢀⠘⠀⠀⠂⠀⠀⣠⣦⣰⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠰⠀⠁⢀⠄⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⠂⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠈⡌⠙⠈⠉⠁⠀⠀⣀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⠮⠴⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠀⡐⠋⠙⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣤⣤⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡚⠉⢹⣍⡉⠀⠀⢉⣳⡄⠀⠀⡞
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⠀⠀⠀⠀⠀⠀⠀⡤⣒⠹⠦⠉⠠⠁⠀⠀⠀⠤⠀⠿⠃⠧⠤⠔⠃
⠀⠀⠀⠀⣼⣛⣷⣿⣿⣿⣿⣿⣿⣿⣛⣿⡿⢻⠂⠀⠀⠀⠀⡂⠈⠚⠁⠀⠠⠀⠀⢠⠈⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠁⣽⣿⣧⣈⣻⣿⣿⠿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⢀⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⢀⣀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⠾⢺⣷⠴⠚⢡⣾⣗⡖
⠀⠀⠀⠀⠀⣠⠔⢢⡿⣄⢐⣾⣄⣰⣿⣿⣛
⠀⠀⠀⡀⠺⣥⣬⣄⢰⣮⡭⠟⣛⣿⣿⡿⠃
⠀⠀⠀⢙⣴⡏⠉⠻⣿⣿⡄⠺⠍⢿⡏
⠀⠀⡠⡌⢻⡷⣄⠠⣿⣿⢿⢦⣀⣼⢗⢦⡀⢰
⠀⢀⠀⠀⡘⠁⡡⠀⠹⣧⣬⠙⠫⠰⠶⢲⣸⠿⣆
⠀⠀⠈⠋⠡⣬⡷⠀⢀⣧⣼⢄⡶⠂⠐⠿⠗⢀⡈⣣⢀⠀⠀⠀⠀⣤⣄⠀⠤⣀⡀⢀⠤⣄⠴⡄⡔⠲⡖
⣀⡀⠤⣦⣄⡼⠀⠀⣼⣿⣿⣏⣴⣶⣿⣿⢶⣿⡿⡧⡟⢻⡊⢑⢲⣀⣽⣁⣥⣧⣼⡽⣆⢷⣦⡷⠚⠈⢓
⠄⠀⠀⠹⠟⠅⠐⣣⡿⣯⣾⣮⣄⡩⢉⣿⣿⣿⣿⢷⡿⠾⢿⡾⠉⠃⣿⣶⡿⣿⡇⣧⣤⡾⣿⡳⡀⠀⣼
⠀⠀⠀⡀⠀⠐⠛⠁⠢⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣾⣿⣹⣯⣡⣥⣿⣿⣷⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⣤⢀⡇⠀⠀⡤⢀⣒⣂⡴⡤⠐⠾⣠⢿⡟⠀⠈⢾⣆⡐⠒⡰⢦⡴⠒⢈⡧⠃⠀⢀⣠⠴⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠢⡙⠣⢄⡀⠀⠀⠉⣿⣉⣁⢲⢀⣿⣿⣷⣿⣿⣾⣿⣿⣰⣧⣤⡵⠾⢿⣵⣦⠔⡋⠁⠉⠙⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢻⠀⠠⣼⠶⣦⣄⣀⡈⠙⢳⠛⠉⣤⣤⣤⣤⣤⣐⡚⠉⢻⣾⣿⣫⣵⣶⠇⠴⡀⠀⠀⠀⠀⢸⡄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣀⢈⣇⠀⠈⢷⣧⣄⡑⣾⣇⡙⣶⣒⣹⣽⣿⢿⠿⠓⣃⠻⠿⠿⢿⣿⣟⠳⢿⣿⣿⢻⠶⠼⣛⡀⠀⠠⠾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⣻⡀⠀⢱⡙⢦⣩⣇⣠⡴⣿⣵⢟⣯⣵⠾⠟⠛⠛⠛⠻⠷⣴⣬⡧⣼⠓⢿⣷⡌⠀⢒⡾⣿⠋⢹⣿⢿⠿⠂
⠀⠀⠀⠀⠀⠀⣎⠉⠉⢲⣿⡇⠀⠀⢳⣾⣓⣯⢵⣿⣿⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢮⣷⣤⢹⠣⡀⣿⠟⠋⠀⣼⣯⡾⠄
⠀⠀⠀⠀⠀⠘⣿⡆⠀⢈⣿⣷⣤⣴⣾⡗⠙⢐⢾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠀⠈⢻⣄⠄⣸⣿⡿⣶⠦
⠀⠀⠀⠀⠀⠀⢨⣭⡇⢸⠟⠛⣿⡿⣿⣤⢴⡸⢻⡇⠀⠀⠀⠀⠀⣀⠀⣿⢶⣻⡆⢀⠀⠀⠀⠀⠀⢘⣿⡆⠀⠀⡿⠲⠿⢭⡶⠷⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠇⢸⠀⣴⣿⣄⣬⣷⣺⣿⢸⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⡟⣧⢐⠐⢷⡶⢦⡄⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⣼⠂⢸⡴⣋⣥⣤⣄⠀⠀⠙⠺⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢡⣟⡐⠶⣌⠳⣦⡆⢸⣧⡟⢠⠁
⠀⠀⠀⠀⠀⠀⠨⠸⠀⢸⣿⣿⣿⣿⣻⣿⣿⢶⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣘⣿⡗⠳⠞⠛⢉⣹⠈⠋⢀⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣏⠻⣷⣿⣿⠷⣶⣰⣿⠏⠻⣽⣆⠀⠀⠀⠀⠠⣶⡆⠀⠆⠀⠀⣰⣿⣽⣿⣿⣿⡛⠶⢺⡿⠿⠯⢴⣶⣋
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠛⢦⣼⣯⣴⣾⡿⠋⠁⠲⡀⢌⣿⣿⢦⣄⣀⠀⠉⠁⢀⣠⣴⢟⣿⣭⣿⠿⠙⣿⣇⢀⣼⠳⣶⠶⣶⣶⢴⠆
⠀⠀⠀⠀⠀⠀⠀⣴⠂⢸⡄⣸⣿⡿⠟⠁⠀⠀⠀⣠⣴⣾⡿⢡⠾⡏⡿⠿⠛⢿⣿⠙⣾⣷⣯⡿⠇⠀⠀⠈⣿⣮⣾⢰⡆⠀⣧⠥⢿⠖
⠀⠀⠀⠀⠀⠀⠋⡟⡀⠈⡿⣿⠟⢀⠀⠀⢀⣠⣾⣿⣷⡟⢡⠃⢸⣼⣧⣦⣧⣼⣽⡷⣾⠋⠉⠑⡀⠠⠈⠄⠈⠻⢿⡾⣧⢀⡟
⠀⠀⠀⠀⠀⠀⢸⡟⣽⢳⡛⢛⣷⠀⣡⣴⣿⣿⡉⠙⠻⣡⣧⣠⣿⡥⠶⠞⠛⠉⢹⣷⠈⠀⠀⠀⡁⢄⠀⠀⠀⢀⡈⢛⣨⣸⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠐⠈⣉⣯⣁⣴⠿⠛⣈⡵⠛⣦⣀⠀⠙⠋⠉⠑⠒⠺⠅⠤⠴⢟⣓⠶⣄⣰⡟⢂⣠⠴⢂⣉⠟⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠭⡴⠞⠉⠻⠙⠉⠿⢾⠿⣶⣦⣭⡴⠀⠀⠀⠀⠀⢤⣤⣍⣟⣷⣀⡸⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡿⣆⠀⢸⣏⣹⡇⢰⡟⠉⠹⣆⢸⡏⢙⣷⠀⠀⣿⠉⠙⣦⠀⢠⡿⡆⠈⢳⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⢀⡾⠓⢻⡄⢸⣏⣉⡷⠸⣇⣀⣠⠏⢸⡏⢻⣅⠀⠀⣿⣀⣠⡿⢀⡿⠒⢿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠀⠀⠁⠈⠉⠁⠀⠀⠈⠉⠁⠀⠈⠀⠀⠈⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠁⠀⠈



⠀⠀⠀⢀⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠔⠒⡖⠒⠒⠒⡖⠒⠒⢲⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠙⠛⠻⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣰⠥⠖⠊⢣⢰⠀⠀⠀⢇⠀⠀⢸⠀⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⡞⠈⡄⠀⠀⢸⠀⠀⠘⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠛⠿⠶
⠀⠀⢶⣤⣤⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡜⠀⠀⡇⠀⠀⠸⡀⠀⠀⡇⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⢿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⢸⣧⡀⠀⠈⢿⣿⣿⣿⣤⣤⣴⣧⣶⣶⣿⣿⠿⠠⣿⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⠀⠀⠠⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⣀⡀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⣿⡇⠀⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⡇⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠃⠀⠀⠀⠀⠀⢸

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⢀⡀
⠀⠀⢸⡇⠀⠀⢰⡿⣷⠀⢸⣟⣻⡆⣴⠟⠛⢻⣆⢸⡟⢻⡧
⠀⠀⢸⣧⣀⣀⣿⠷⢿⣇⢸⣏⣹⡇⢻⣦⣀⣼⠏⢸⡟⢿⡅
⠀⠀⢈⣉⣉⣀⡁⠀⠀⠁⣈⣉⠉⣀⡀⠈⢉⣁⠀⠈⠁⠈⠁
⠀⠀⢸⣿⠛⠛⢿⣆⠀⣸⡿⣿⡄⠘⣿⣄⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⠀⠀⣸⡿⢠⣿⣧⣽⣷⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠸⠿⠿⠿⠛⠁⠾⠏⠉⠉⠿⠇⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢








⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⢰⣿⣿⣿⣿⣗⢸⣿⣿⡇
⠀⠀⠀⠉⣿⣿⡏⠉⠈⢹⣿⣿⠉⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠈⢹⣿⣿⠉⠉⢹⣿⣿⡇⣉⣻⣿⣿⠉⠉⢻⣿⣿⠈⠉⢿⣿⣿⡁⢈⣿⠋⠁
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠀⢠⣿⠀⣿⣿⣧⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢿⣿⣧⣼⠇
⠀⠀⠀⠀⣿⣿⡟⠉⠉⢹⣿⣿⠀⠀⠀⣼⡟⠛⢻⣿⣿⡄⠀⠀⢸⣿⣿⢋⣿⠟⣩⡽⢛⣥⣾⣿⣿⠋⣹⡿⣽⡆⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⣶⣿⣿⣷⣶⢰⣾⣿⣿⣶⢠⣶⣿⣷⡄⣾⣿⣿⣷⣤⢰⣾⣿⣿⣾⣥⡾⢋⣴⠟⣷⣿⣿⣿⣶⢉⣴⣿⡇⠀⠀⢰⣶⣿⣿⣷⣶
⠀⠀⠀⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠘⠟⠛⠛⠃⠛⠛⠛⠛⠛⠸⠟⠻⣿⠿⢃⣴⠟⣡⡶⢛⣿⠟⢻⡿⠛⣡⣿⡇⠀⠀⠘⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠟⣡⡾⢋⣴⠟⣡⡾⢋⣴⠟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠞⣫⡴⠟⣡⡾⢋⣴⠟⣡⡾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⠀⠀⠀⠈⣿⡾⢋⣴⠞⣋⣴⠟⣡⡾⢋⣴⣿⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⢻⣷⡟⣡⡾⢋⣥⠾⢋⣴⢟⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠻⣿⣿⣴⣟⣡⣶⣯⣴⣿⠟⠁⠀⠀⠀⠀⠀⣀⣤⣾⡿
⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠟⠋⠁⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣇⣀⡀
⠀⠈⠛⢿⣿⣿⣿⡿⣫⣶⠾⣭⣛⡿⠃⠀⠀⠀⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠘⢿⣻⣭⡷⢶⣝⢿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠈⠙⣫⣾⣿⠋⣴⣾⣿⣿⡿⢶⣶⣤⣤⣿⣿⣇⣸⡂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣧⣤⣤⣶⣶⡿⢿⣏⣁⠳⢾⣿⣷⡙⠋⠁
⠀⠀⠀⠀⠀⠘⠿⣷⣤⣘⠛⣹⠿⣋⠆⢸⣿⡏⢉⣍⠙⣿⠿⠛⠻⢿⡿⠿⠿⠿⢿⣿⣿⣿⡋⢩⣭⠙⢿⡇⢆⡙⠿⢷⣀⣩⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣯⣄⣸⡛⠁⠶⠄⢰⡏⠀⣿⡇⢈⣿⠀⣛⡁⢸⣿⣿⣿⣿⠀⠻⠃⣸⣇⣬⡷⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⣶⣿⣿⣶⣤⣶⣿⣧⣤⣼⣷⣦⣾⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡆⠘⡅⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠈⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⢠⡆⢀⣇⠀⠀⠀⣀⣀⠀⠈⠉⠁⠐⠲⣶⣤⡀
⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣴⣿⡿⠁⠀⢀⣠⡄⠀⠘⢿⡉⠉⠙⡆⢸⣉⡟⠁⠀⣠⣀⠀⠀⠈⢿⣿⣦⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⣹⣿⣿⣿⠁⠀⠀⠈⣿⣿⣦⣤⡾⠿⠶⢾⡃⢸⡿⠦⠤⢶⣿⣿⠀⠀⠀⠈⣿⣿⣿⡏⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣅⣀⣀⣀⣀⣿⣿⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⣧⣄⣀⣠⡗⢺⣄⣀⣠⣼⣿⣿⠀⠀⠀⢀⣿⣿⣿⣧⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣦⣤⣼⣿⣿⣿⣿⡏⠉⠉⠻⡄⢠⠟⠉⠉⢻⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⠃⠘⣷⣶⣾⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣧⣄⣀⣀⣀⣀⣤⣤⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⢀⣿⣿⣿⣿⣿⣿⣧⣀⣀⣼⠇⢸⣇⣀⣀⣼⣿⣿⣧⡀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠉⠉⢻⣿⣿⣿⡟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣹⡿⠻⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡋⣠⠞⢁⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠈⢧⣠⢾⡀⢷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠠⠞⠲⠇⢘⣿⢿⣦⡀⠀⠀⣤⣦⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⡀⢀⣴⡿⣷⣄⠙⢿⣶⡿⠋⠈⠻⣷⣀⠀⠀⠀⢀⣤⣤⣶⣄
⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢀⣴⠟⠛⠛⠉⠀⠈⢻⣷⡾⠋⠀⠀⠀⣠⣾⠟⠀⣠⡆⢸⣿⣿⣿⣿⠀⣦⡀
⠀⢠⣾⠟⠁⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⣤⡿⠋⠀⠀⠀⣠⣾⣿⡁⢠⣾⣿⣿⠀⣿⣿⣿⡏⢰⣿⣿⣦
⠀⠘⣿⣄⠀⠀⠀⠀⠀⢻⣧⣤⡀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣠⣾⣿⣥⡿⢃⣿⣿⣿⣿⣤⣿⣿⣿⣧⣼⣿⣿⣿⣇
⠀⠀⠈⠻⣧⡀⠀⠀⣠⣾⠟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠈⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣦⣾⠟⠁⠀⢀⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠸⣿⡁⠀⢀⣴⡿⠋⠀⠙⣷⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣶⣿⠋⠀⠀⠀⠀⣿⠀⣤⡶⢶⡄⢸⣿⠶⣦⡀⢠⡴⠶⣤⠀⣶⣴⠆⠀⠀⢠⡶⢦⣿⠀⣤⡶⢶⡄⢴⡆⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠀⠀⠀⣿⠀⣴⠖⢾⡇⢸⣿⠀⢸⡇⣿⡇⠀⣿⡇⣿⡇⠀⠀⠀⣿⡇⠀⣿⠀⣴⠖⢾⡇⠈⣿⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠓⠛⠛⠘⠛⠛⠛⠁⠈⠛⠛⠋⠀⠛⠃⠀⠀⠀⠘⠛⠛⠛⠀⠛⠓⠛⠓⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⡄⣸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠞⠁⠟⠹⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠀⠀⠂⠀⠀⠀⠀⠃⠉⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⢦⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠶⡡⢿⡞⡏⠀⠀⠀⠀⢰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⢀⠟⡉⠀⠁⣘⢳⢿⡄⢸⠀⠀⢨⠻⠻⣤⣇⢀⣖⠦
⠀⠀⠀⠀⠀⠀⣴⣥⠄⡀⠀⣹⡗⣡⣾⣿⠃⣰⡇⡸⠀⣿⠿⣄⣠⣿⢣⡀⠈⠹⣾⠁⠀⠈
⠀⠀⠀⠀⠀⠠⠋⠉⠛⠳⢾⣯⣾⣻⢳⠃⢠⣿⠇⡇⠀⢿⣾⡘⡈⢻⣇⢻⣤⠬⢶⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠉⠭⠙⠒⠋⠛⢻⣿⡄⠚⠛⠓⠓⠘⠻⡏⢁⡤⠔⠈⣷⣀⠄
⠀⠀⠀⠀⠀⠢⡀⠀⠀⡀⠘⠇⠀⠀⠀⠀⠀⠀⢸⢸⡇⠖⠂⠀⠀⠀⠀⣹⣀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠷⠞⢀⣀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⢀⠌⠀⢠⠋⠱⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠉⠀⠉⠉⢉⡉⠉⢩⡀⠴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠀⠀⠀⢀⠀⢐⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣝⡀⠀⢀⠁⡊⠀⠀⠈⢻⢦⠀⠴⢂⡿⡄⠀⠀⠹⡦⡐⠐
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠓⢍⠓⢤⣁⠀⠀⠀⠀⡟⢀⠀⢀⠞⠀⡟⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⢧⠀⠈⠳⡀⠐⣦⡙⢮⠢⢸⠁⢠⠃⠀⡠⠋⠀⠈⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢖⠈⣀⣀⡀⠀⠀⢸⡤⠀⠀⢧⢠⣇⠹⡌⢳⠘⣆⡿⢀⡞⠁⠀⠀⢀⠆⢀⠎⢣⠀⡀
⠀⠀⠀⠀⠀⠈⠀⣈⠉⢳⡎⢢⡀⢸⠀⠶⡀⠈⠻⠂⠀⢱⠀⠀⡿⢽⣼⠀⢠⠀⢀⡌⢀⠠⡀⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠙⠸⡇⢀⠱⣾⡄⢁⠈⠢⡀⢠⡄⡞⠀⢀⠇⡄⠋⡠⢁⢣⠏⠀⢸⢣⢰⠇⢀⢦
⠀⠀⠀⠀⠀⢀⣆⢺⡊⠀⠹⣇⠰⠠⢄⠀⠂⠀⠰⠂⢠⠃⢀⡞⡐⢸⠐⠀⢠⠃⠀⡰⠃⣸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⡏⢦⣀⠀⠀⠈⠳⣝⢤⡈⠲⠀⠀⡄⢸⢠⠟⠐⠀⢾⠀⠂⡇⣠⠏⠀⣀⡎⠀⣠⢀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠈⠑⠢⢤⡀⠘⡄⠱⠀⠀⠀⠆⠈⠿⠀⠀⠀⠀⠀⡆⢹⠇⡄⢠⠎⢀⡰⣻
⠀⠀⠀⠀⠈⢻⠆⠈⠲⢄⠀⣄⡙⠶⣗⠀⠁⠀⠀⠘⢄⠈⠂⠰⠀⠀⠀⠐⡤⠊⢄⣮⠖⢋⡜⠁⣦⡀
⠀⠀⠀⠀⠀⠘⠒⠠⠱⠀⠱⡀⠂⠑⠄⠀⡄⠀⠀⠀⠀⠉⠂⣤⠀⠀⠀⠀⠄⠐⠎⠥⢈⠏⡀⠀⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⡀⡇⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢐⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠓⢤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⠀⠀⡀⠁⠀⠔⠀⡀⠀⡀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠀⠈⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠦⢤⡀⠀⢀⡤⠴⢤⡀⠀⠀⡠⠄⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠈⡆⣾⠀⠀⣹⢹⢠⠂⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⢀⣴⡆⠀⠀⠀⠀⠀⢠⡟⠘⡶⠬⣵⡇⠸⣆⠀⠀⠀⠀⠀⠀⢰⡀⠈⠑⢦⣀
⠀⠀⢀⣠⠴⠊⠁⢀⣤⠞⠉⢈⣷⣀⠀⠀⠀⠐⣏⡀⢀⡇⠀⢸⣇⢀⣹⠆⠀⠀⠀⠀⣠⣟⠙⠶⣄⠀⠈⠓⠦⣀
⠐⢾⣉⣀⣠⡴⠞⠋⢀⣤⠞⠁⣠⠟⠛⣶⠀⠀⠀⠉⠉⡇⠀⢸⡍⠉⠁⠀⠀⣾⠛⠻⣏⠈⠳⣄⡈⠛⢦⣄⣀⠀⠉⣒⡶
⠀⠀⠀⠉⠉⠒⠲⠾⣯⣤⣴⣞⣁⣤⠞⠉⠳⣄⣀⣀⣀⡇⠀⢸⣇⣀⣀⣤⡾⠋⠳⣄⡘⠳⣤⣈⣙⣶⡤⣬⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢧⡀⠀⣠⢭⠭⢭⡇⠀⢸⡍⠉⡉⠅⡀⠀⣠⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⣤⣀⣠⠼⡇⠀⢸⠧⣄⣀⣤⠈⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠘⢿⡑⢤⡇⠀⣿⠄⢀⡽⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠓⠳⠤⣿⡒⠉⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⣦⣄⡀⠀⠀⠉⠓⢾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⣽⡿⠷⣦⣄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇⢡⠀⣿⢹⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠳⢾⣀⣿⠎⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠈⠙⠦⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⢷⣤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢰⣿⠹⣷⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⢾⠀⣿⠇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⠳⣄⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⢶⣤⡀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⢻⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠙⢼⡇⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣦⣄⡈⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⠛⣿⢿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣄⣿⣸⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⡏⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠈⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠙⠍⠩⡍⠭⠉⠉⠉⠉⠉⠉⠉
⡇⠀⡀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠁⢠⠚⠈⢀⠤⠚⠑⠆
⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⠀⢠⡄⡀⠄⠀⠁⠀⠀⠀⠈⠄
⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠁⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰
⡇⠀⠀⠣⣀⠀⠠⡀⠀⠀⢠⠀⠀⠁⣠⣁⡠⠂⠀⠀⠀⠀⢨⡑⠠⠄⣈⣁⠛⠛⠛⣀⣉⠁⠤⠀⣽⠀⠀⢠⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠈⠇⠓⠛⠂⠆⠚⠶⠒⠚⠛⡅⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠠⠈⠀⡁⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⣠⠄
⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡁⠀⠀⠀⠀⠀⠯⠀⢠⡏⠅⠀⡉⠈⠁⠀⠠⡀
⡇⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠃⠀⠀⡐⠲⣌
⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠁⠀⠈⠷
⡇
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⠀⠂⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠁⠀⡀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⣀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠐⠀⠁
⡇⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠒⠀⠀⠀⠃⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠀⠀⢰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⡀⢄⠂⠀⠀⠠⠀⠒⠀⠈⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁
⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠬⠀⠉⠐
⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠄⠂⢤⠀⠄⠊⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠃⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠠⠐⠈
⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠈
⡇⠀⠀⠀⠈⠂⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡀⠄⠂⠁
⣇⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡐⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣛⣘⣃⣴⣠⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⣿⡟⠛⣿⠛⡛⢻⠟⣛⠻⡟⢛⠛⢿⣿⠛⡛⢻⡿⠛⣿⠛⠟⣻⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⠿⢁⡀⢹⠀⠆⢸⡀⠻⢀⡇⢠⠀⣿⣿⠀⠟⣰⠃⣀⠸⡇⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠀⠀⠀⠀⢈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢣⣴⠾⠟⠛⠛⠛⠛⠲⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷⢶⣿⡟⠛⠿⢿⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⠃⠀⠀⣤⣤⡤⠀⢠⣹⠟⣁⠞⢁⡸⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠐⠀⢀⣛⡂⠀⠸⣯⠈⠁⠀⠉⢀⡔⠋⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷⢤⣀⣀⣀⣠⣾⠟⠤⡀⠀⠀⠀⠀⠂⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⡥⠖⢒⣒⣲⣄⠉⢻⣿⡿⠃⠀⠠⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⠀⣠⣤⣬⡿⠗⠋⠁⠀⠀⠀⠀⠘⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠎⢠⠞⠁⣠⠏⢀⠄⠀⡠⠂⠀⠀⢀⣼⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣀⣿⠀⠀⢀⣠⠎⡠⠊⠀⠀⠀⢀⣾⠋⢻⡄⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⠀⠀⠀⠀⠀⢀⡤⢀⣠⣾⣿⣦⣤⣷⠾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣴⣯⣶⣿⣁⣘⣿⣿⣀⣸⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠘⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡀⣰⣿⡿⠃⠀⢠⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡼⣿⠁⠀⠀⠸⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣦⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⢤⡀⠀⢀⣿⣷⡀⠀⠀⣸⣿⣿⡟⠀⢤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⡇⠀⠀⠀⢱
⠀⠀⠀⢤⡀⠀⣴⡿⠀⡀⠙⣿⣿⣿⣿⡆⠀⠀⢸⣷⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠈⣿⣷⡄⠀⠀⠙⠻⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⣿
⠀⠀⠀⠈⣿⣶⠟⠀⠀⣿⣦⣾⣿⣿⣿⠇⢀⣴⣿⠟⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⠉⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣇⠀⢀⣿⣿⠀⡀⠀⠘⢿⣿⣿⣿⢟⣴⣿⣿⠋⠀⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⣇⣀⣽⣷⡀⠈⠙⠿⢋⢸⣿⣿⣿⣷⣤⣾⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣿⣌⣿⣿⣿⣿⣿⡿⢡⣾⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⠀⠙⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣶⣿⣿⣿⣿⡟⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣘⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠙⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣀⠈⢷⣄⣠⣼⠃⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣭⣭⣭⣭⣤⣤⣤⣤⣬⣍⠙⣷⠐⡇⢸⡇⣴⠏⢉⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⡩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⣤⢤⣤⣤⣤⣤⣤⠤⢽⡆⢹⣤⡇⢸⣇⡟⢀⣯⣤⣤⣭⣭⣍⣉⣉⣉⣩⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣶⣤⣤⠶⠶⠶⠶⠿⣄⣈⡇⢸⣏⣠⠾⠧⣤⣤⣤⣬⣭⣭⣭⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢶⣶⠶⠖⠒⣛⣻⣷⢸⣟⠓⠶⠶⠶⢶⣤⡴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⡛⠉⠉⣿⢸⡏⠉⠛⢛⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠙⠛⣉⣷⠀⣿⢸⡇⢰⣟⡙⠋⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣴⠾⠟⠋⠀⣿⢸⡇⠀⠙⠻⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠈⠓⠦⢤⣤⣿⣸⣇⣤⡴⠞⠁⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣦⣤⣄⣀⡈⠉⠛⠶⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣼⠏⣿⢹⠛⣶⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠙⠷⢿⣼⣴⠟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣦⣤⣄⣀⡉⠛⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣞⣻⢹⣻⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠙⠻⢧⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⢷⣦⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣻⢸⣸⣸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⢸⣿⣿⡇⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠛⢿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠉⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⣼⣿⠻⠿⠈⣿⡿⣿⣿⣧⠁⠀⠀⠀⠀⢀⣤⡙⠿⢿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠫⣾⣷⣄⠀⠀⠀⠀⠀⠙⣿⠀⠀⠠⢽⣦⣾⡿⠋⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠘⣧⡈⠉⠿⣿⣿⠃⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⡴⣏⠙⢻⣿⣿⣿⣦⠀⢠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⢉⣤⡞⠁⢹⣶⣾⣿⢟⣹⣿⣧⣤⣍⡁
⠀⠀⠀⠀⢾⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣾⣿⣧⠾⢿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣖⣻⠟
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣶⣤⣀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠃⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣇⢠⣤⣴⣿⣿⣿⣿⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⠀⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣈⣛⣛⣿⣿⣿⣿⣿⠀⠀⢻⣿⠘⠛⢉⣭⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠀⣿⣿⣿⣿⠛⠋⠉⣛⣻⡿⠿⠟⠛⠋⠉⠉⠛⣿⣀⣉⣹⣿⠀⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠀⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣁⠀⠘⣿⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢠⠾⢿⣷⣦⣀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣷⣾⣵⣶⣵⣾⠟⠉⠀⠈⠻⢿⣿⣿⣶⣿⠀⠀⣀⣠⣤⣠⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣁⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣈⣻⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠉⠙⠙⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⣇⠈⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇⣦⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠰⡆⣨⢛⢥⡶⠀⠀⠀⠀⠀⠀⢸⢸⣄⣧⣠⠀⠀⠀⠀⠀⢀⠀⣲⠛⡏⠀⠀⠀⠀⠀⠀⡾⡿⠜⡄
⠀⠀⠀⣷⣼⢀⡾⣤⠦⡼⠀⠀⠀⠀⢸⣾⠈⡟⠙⢻⠀⠀⠀⠀⠀⠘⢟⢫⣿⣀⡀⠀⠀⠀⢀⠠⢇⠠⣤⢷⡄
⠀⠀⠀⠘⠦⠾⠇⠘⠃⠁⠀⠀⠀⠀⠀⠁⠰⠛⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⠐⢯⠃⠀⠀⠀⠘⣷⠛⠘⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠘⡿⡏⡅⠀⠀⠀⠀⠀⢀⢀⠀⣿⢣⣀⡀⠀⠀⠀⠀⠀⠀⠎⣼⣿⢰⡀⠀⠀⠀⠀⠀⡄⣾⡼⣃⡋
⠀⠀⠀⠀⠸⣧⠛⣇⠀⠀⠀⠀⠀⠈⠉⢷⠾⡌⠛⢛⡁⠀⠀⠀⠀⢰⣜⣇⢹⡾⣧⡀⠀⠀⠀⠀⡁⢏⢁⢹⠠⠇
⠀⠀⠀⠸⣆⣿⢶⣼⠀⠀⠀⠀⠀⠀⣻⡛⠀⡿⣆⠛⠇⠀⠀⠀⠀⠸⣿⠹⢰⡀⠻⠀⠀⠀⠀⠠⣼⡾⢿⡘⢾
⠀⠀⠀⠀⠀⠰⠏⠋⠀⠀⠀⠀⠀⠀⠈⠃⠻⠅⠙⠋⠀⠀⠀⠀⠀⠀⠙⠘⠋⠃⠀⠀⠀⠀⠀⠀⠈⠃⡎⠻⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠁⠀⠈⠐⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠠⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠁⠈⠈⣀⣐⠁⠀⠀⢠⣤⠃⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡄⠀⠀⠀⠀⢄⠀⠰⠄⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⠀⠸⣄⠀⠀⠀⠠⠀⠠⠀⠀⠀⣠⠇⠈⣿⡏⠀⠀⠒⠒⠒⠐⠂⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢻⠇⠀⠀⠀⠉⠁⠀⠀⢹⡟⢀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡜⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠢⢀⠀⠀⡀⠔⠚⠛⠃⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢲⠀⠀⡶⠲⣄⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⢀⠶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⠔⢼⠀⠀⠃⠀⠈⢠⠀⠴⣶⣄⢀⠀⢈⠽⡇⠀⢸⠿⡁⠀⠀⠀⠀⠀⠀⠐⠃⡼⠉⢶⣆⠄⠀⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠈⢠⠀⠀⠀⠈⠦⡈⠀⠀⠀⠀⢨⠀⠃⠀⠲⠤⠘⠀⡎⢂⠀⠀⢀⣠⡦⠖⠛⢄⠂⠂⡉⠀⠀⢃⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠋⠀⠀⠁⢸⠀⡄⠀⠀⠠⠠⠊⠀⠄⠆⠐⠀⠚⢀⣧⡤⠖⠛⠉⠀⠀⠀⢀⡡⡠⡑⠈⠢⠀⠆⢙⠀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⡁⠘⡀⠀⡼⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠃⠰⠀⢐⣰⡶⠖⠚⠉⠀⠀⠀⠀⠀⠄⠀⠈⠀⠘⠁⠀⢙⡂⡉⢀⡉⢀⡐
⠀⠀⠀⠀⠀⠀⡐⠰⠄⠹⠸⠁⠀⠀⠀⠀⠈⢏⣒⣂⣤⣤⠶⠖⠚⠉⠀⠀⠀⠀⠀⠄⡊⠈⢡⢠⠠⠐⠐⡎⢁⠈⣉⡀⢡⡉⢄⠹⣄⠁⢄
⠀⠀⠀⠀⠀⠌⠠⢀⡞⢀⢀⠄⠀⠀⠀⠀⣋⠭⠟⠛⠉⡀⠀⢀⢀⡠⠴⠖⠂⠀⠀⠹⠁⠀⠀⡐⠐⢄⠀⠃⠉⠈⠛⠋⠈⠉⣄⠀⠈⠓
⠀⠀⠀⠀⠈⣠⠁⡼⠀⠄⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠵⠀⠈⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠄⣀⠤⠂⠀⠀⠀⠄⠄⠀⠀⠁⠠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣴⡆⢰⣤⣶⣴⣤⠀⠀⠀⠀⠀⣤⣤⣧⣴⠀⣧⣴⣤⣤⠀⠀⠀⠀⠀⣤⣴⣦⣴⡄⢰⣤⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⢷⣄⠀⢀⣠⣄⡀⠀⣠⡾⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⣿⣿⣿⣿⡞⣿⡄⣿⣿⣿⣿⢀⣿⢳⣿⣿⣿⢿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣴⢾⣻⣽⣾⣿⣯⣿⣿⣿⣿⣽⡿⣟⢷⣼⣿⣟⣧⡾⣫⢿⣯⣿⡿⣿⣾⣿⣿⢷⣯⣟⡷⣶⣤⣀⣀
⠀⠀⠈⠙⠻⣭⣿⣿⣿⣻⣯⣿⣿⣿⣟⣯⣿⣿⣟⣿⣿⣯⣷⣯⣿⣿⣽⣶⣿⣿⣿⣻⣿⣿⣽⣻⣿⣿⣿⣽⣟⣿⣿⣿⣭⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⣿⣿⣟⣿⣟⣽⣯⣻⣿⣻⣿⣿⢿⣷⣽⣻⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣩⣭⣭⣉⠀⢸⡇⠀⣉⣭⣭⣍⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡉⠛⠻⠗⢸⡇⠺⠟⠛⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣦⣼⣧⣴⣶⣾⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠟⢻⡍⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣘⣧⣴⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⡛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⢸⡇⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⣧⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣜⣧⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣫⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⣤⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⢠⠟⠙⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⡀⢠⡆⠀⠀⠀⠈⣧⠄⠐⠸⠀⡤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠀⡰⠉⠷⠀⠐⠑⠀⠃⠀⣤⡠⠞⡷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠸⢤⡇⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠐⡁⠘⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠈⠁⡐⠂⡰⠁⠀⠀⠀⠤⠂⡳⠀⠇⠆⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣦⡱⡄⠲⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠞⠀⠔⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠊⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠈⡤⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢥⡀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠮⠴⠖⠖⢤⣼⢶⡿⣷⣾⣷⠋⢻⣿⣷⡖⠦⢤⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣤⣤⣴⠞⠋⠀⣸⠁⢀⣿⢻⣆⠀⠙⢯⣟⠉⣷⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣁⣀⣠⣶⡇⠀⠸⣿⠄⣻⡆⠀⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⣿⠷⠏⠛⠉⠘⠶⠶⠞

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣠⡇⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠋⠀⢀⣾⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣴⣿⡏⠀⠀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢳⣿⢀⠠⠾⢻⣿⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⣿⡿⣱⣿⡿⠃⠀⠀⣼⣿⣼⠠⠄⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣽⣿⣿⣿⣿⣆⣠⣾⣿⢿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣿⡿⣿⣿⣿⣿⣿⢣⣼⣿⡎⢷⣇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⢏⣾⡟⣸⣿⣿⣿⣿⡇⡾⣿⣿⢻⡌⠻⣆⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⢠⠁⣿⠟⣿⡿⣰⣿⣿⣿⣿⣿⡘⣇⢸⣿⣿⣿⠀⠙⣷⣄⠠⡱⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⣾⣿⡇⣿⢸⣿⣿⣿⡿⠧⠀⣼⡈⣿⣿⡇⢢⠘⣿⣧⠙⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡀⣼⢿⣿⣿⠀⣿⣼⠳⠏⢠⣷⢀⣼⣿⠃⣸⢹⡇⣸⡄⠸⣿⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣧⢹⡜⣿⣿⡄⢹⣿⠀⠀⢸⣾⡈⠻⣿⢀⡟⣹⠇⣿⣥⠀⣿⣷⠀⢻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣿⣿⣦⡿⣷⣄⡈⠻⠏⠀⡹⣾⣱⠏⣼⣿⣧⠀⣿⣿⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣻⣯⣛⣿⣿⡝⢷⣂⠠⣼⣷⣿⢏⠼⠁⢸⡇⢰⣿⡟⠀⣸⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠈⠋⣉⣿⣿⣿⣷⡙⠿⠿⠯⠁⠀⢀⠴⠋⣴⣿⠟⠁⣰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣿⣿⣿⣿⣷⣦⣤⣤⣤⣷⣼⣿⣿⠗⢀⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣿⣿⣿⣿⣿⠿⢛⣡⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣧⣴⣿⡭⠤⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⠟⢻⠟⢿⠿⢻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠀⢀⣀⡤⠴⠶⠦⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⣀⣸⣄⠀⠀⠀⠀⠈⢟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⣀⣀⣀⣀⣸⣅⣉⣛⣆⣠⣶⠄⠀⢸⡎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⣴⢟⣥⡶⠟⠋⠉⠉⠉⠉⣿⡟⣅⣠⢼⣿⡁⢀⣀⣾⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣦⣬⡵⠟⠻⢿⣦⡙⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⢀⣾⣿⠋⣾⡄⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⡀⠀⣀⣀⡟⠛⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⣼⣿⠃⢸⣿⣷⡀⠀⠀⠀⠀⣆⠀⠀⠀⠈⠳⠴⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢰⣿⠃⠀⢸⣿⠋⠓⠀⠀⠀⠸⣿⡀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣾⠃⠀⠀⢸⣿⣇⠀⠀⠀⠀⢀⣿⣿⣋⣠⣶⠋⠀⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣿⠀⠀⠀⢸⣿⣿⠛⠛⠓⠒⣿⣿⣿⡿⣿⣿⣀⡠⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⠏⠀⠈⠻⣏⠀⠀⠺⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠙⢧⡀⠀⠀⠀⠹⣦⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⣷⣦⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠻⢦⣄⡀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠈⣿⣿⣿⣿⣿⣿⣿⡟⠻⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠲⣄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣼⡆⢀⣄⠈⢻⣟⢳⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⢹⠀⠀⠀⠉⠢⣀⣴⣶⣾⣯⣤⣽⣞⣽⣡⢾⣌⣿⠂⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣨⣧⠀⠀⠀⠀⢹⣹⠁⢉⣿⡿⠿⠿⣿⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇⠀⠀⢸⢫⠾⣇⣆⣄⣄⣸⡏⠀⣸⠿⠁⠀⠘⠟⠁⡿⠶⠶⠶⠶⠶⣶⣤⣾⣯⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠀⠻⠬⠔⣛⣛⣒⣲⣿⠁⠀⠈⠀⠀⠰⠤⠦⢶⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠒⠒⠞⠀⢈⣩⡽⠟⠉⠉⠙⠛⠛⠛⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠶⠞⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⢯⣍⠉⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢻⡄⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣦⡀⠤⠾⠃⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣠⣤⡤⠶⠶⠶⠶⢶⣶⡶⠆⠀⠀⠀⠀⢢⢉⠄⡄⢀⣀⣱⣤⠤⠤⠶⠒⠒⠛⠉⠉⠉⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣟⣻⡇⠀⠐⠛⠆⠟⢺⣁⣀⠠⠤⠤⠐⠒⢦⠸⠀⡁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠼⣧⣁⠀⠀⠀⠄⠀⠤⡸⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣸⡟⣄⠀⠀⠐⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣄⣤⠤⠤⡾⠹⡇⠈⠙⠲⠦⢤⡔⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁⠀⠀⠀⠀⣇⠀⠳⡀⠀⠀⠀⣸⠇⠀⢏⠉⠛⠒⠒⠀⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣀⡙⡖⠒⡶⣃⡤⢄⣼⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡏⠙⡟⠒⠲⠦⢤⣤⣄⣀⣀⠀⠀⣿
⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠊⣾⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠀⠀⠀⢸⠀⠀⠉⠉⠙⠛⠛⠛⠛
⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸
⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼
⢧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⢶⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠐⠲⠶⠤⣤⣄⣀⣤⠞⠁
⠘⣧⠀⠀⠀⠀⠀⠀⢿⠉⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣦⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⢆⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠹⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⢣⣀⣀⣤⠶⠚⠻⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠉⠁⢳⠀⠀⠀⠘⢆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢣⣷⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⡀⡈⣿⢿⡟⡛⢷⡶⠶⠶⠶⢶⠶⠶⢶⠚⠛⣛⠛⣋⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢁⡁⠖⣿⣤⣿⣛⣿⣇⣀⣀⣀⢸⣶⣶⣾⡄⠀⣿⣧⣿⣤⠽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⢸⠶⠭⠷⢤⡄⠀⠀⣠⣤⣼⣿⣿⣿⣷⡆⠀⠀⣀⣀⣠⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⣀⣀⣀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⠃⠀⢸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣷⣿⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⠛⠻⡆⠀⢹⠀⠀⠞⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠞⠛⠉⡀⠄⡀⢽⣀⣀⣸⣀⣀⣨⢄⡢⢄⡉⠙⣻⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠈⠙⠻⠷⣖⣩⣵⣒⡭⢖⡩⢂⡄⠉⠁⠁⢹⣿⣳⣕⠮⣓⠮⣗⣶⣭⣒⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠚⠻⠒⠊⠀⢸⠀⠙⠛⠻⠛⠚⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠚⠀⢸⠀⠛⠤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⣄⡀⠀⢸⠀⠀⠀⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣦⣾⣶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⣀⡀⠀⠉⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⠋⢻⠙⢻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⢷⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢮⢤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡞⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣭⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢾⢣⣤⠀⡄⠰⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣘⢦⣤⠤⠜⡁⠀⠀⠀⠐⢀⣤⢞⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢬⣳⣿⣿⣇⠀⣴⡀⢀⣴⣿⣾⡫⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠺⣿⣿⣿⣷⣿⣿⣿⡟⢿⣞⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⢠⡀⠈⠉⡿⠿⣿⢶⣶⡄⠀⡅⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠀⠀⠛⢻⣿⠟⣿⡇⠴⠒⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⣀⣤⣀⣲⡶⣿⢼⣴⡿⠠⠆⣤⡤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠠⠟⢹⢿⣿⠭⢿⡓⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣀⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⡆⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⣿⣿⣤⣶⣶⣶⣯⣴⣷⣭⣿⡆⠀⠛⢲⡀⢛⣒⣶
⠀⠀⢀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⡏⢿⣿⣿⠀⢀⢀⡐⢾⢈⣦⣠⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣰⣶⠟⣤⢿⡏⢈⡷
⠀⢀⣾⣿⣿⣧⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠋⠀⢀⡌⣦⣠⢷⣏⣿⣯⣯⣧⡐⠀⠀⠉⢈⣍⣉⣩⣍⣭⣥⣤⡄⠀⣜⣿⣶⣼⠟⢹⣏⣸⡅
⠀⣼⣿⠏⠀⢿⣿⣇⠀⠀⠸⣿⣿⢿⢿⣿⣿⠃⠀⠀⢀⣜⣾⣷⣿⣿⣿⣛⣿⣿⣼⣿⣿⣷⣦⡰⣼⣿⣍⠓⢺⣟⡋⠉⠉⠀⠘⠿⠞⢿⣶⣿⠿⠛⠁
⠀⠻⢧⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣁⣸⣿⣿⡟⠛⠈⠑⠀⠀⠀⠀⡀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡀⣴⢠⡀⠀⠀⠀⠀⠀⠀⠆⡐⠂⡆⠐⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⣸⢼⣠⢧⣟⣳⢼⡰⣅⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠁⡄⣑⣘⣄⠀⣀⣙⣦⡴⣃⠀⠀⠀⢠⡀
⠀⠀⠀⢸⡆⠀⠀⣯⣻⣾⣾⣿⣿⣿⣾⣷⣿⣷⣿⡄⠀⠀⠀⠀⠀⢀⣼⣧⣶⡤⣄⠀⠀⠀⠀⢀⠙⠐⠀⠑⠿⠟⠀⢠⣞⣿⡻⣞⡽⠁⠀⢠⣤⣽⠆
⠀⣰⢠⡿⢁⡀⠀⠈⠛⠛⢉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⢠⣼⣽⣿⣿⣿⣿⣿⣤⠀⠀⣀⡴⠛⢦⡞⠀⠀⠀⠀⠀⣹⣟⣿⣏⠀⠀⢰⣛⣦⣹⣦
⠀⢿⡈⢷⣼⣄⠀⠀⠀⣠⢿⣆⣤⠀⠀⠀⠀⣦⣷⡄⢣⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣞⠀⡄⡟⢷⡜⣮⣷⣴⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⢈⣽⣿⠁
⠀⢸⡿⢸⣿⣿⠇⠀⣰⣯⢾⣿⣏⣦⠀⠀⠲⣬⡟⣷⣦⡙⣷⠀⠁⢸⣿⣿⣿⣿⡟⠃⠸⣧⣿⣿⣵⣿⣟⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣿⡟⠋
⠀⣾⣷⣾⣿⣿⠀⠀⣿⡇⠀⠋⢹⡿⠀⠈⣧⣿⣷⣿⣼⣷⣡⣧⠀⠻⣿⡟⢽⣿⡅⠀⠀⣽⣯⣿⣿⣿⣿⣻⡇⠀⠀⣀⣤⣴⣆⡀⠀⠀⠀⠉⢻⣿⣦
⠀⠈⠛⠛⠛⠋⠀⠀⠈⠛⠀⠐⠋⠁⠀⠀⠙⠛⠉⠋⠋⠙⠛⠋⠀⠀⠘⠀⠘⠋⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠃⠀⠾⠿⠿⠿⠿⠟⠛⠗⠀⠚⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀⠀⠀⠀⣤⢠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⣿⣾⣿⣶⣶⣾⣿⣿⡟⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⣿⣶⣿⣿⣿⣿⣷⣾⡆⢫⣿⣝⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣅⠀⣿⣿⡗⣿⣿⠷⣿⣟⣿⢎⣟⡻⠿⢿⡿⠇⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣝⡿⡷⢏⣵⣿⡿⣻⢯⣾⣿⣾⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⣾⣟⣿⣵⣿⣿⣿⣿⣿⡿⠧⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣯⢩⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠀⠀⢹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠂⠉⠉⢹⣿⠉⠁⠲⠆⣤⣀
⣀⠀⠠⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⣿⠇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡟⠑⢤⡀
⡇⢹⠆⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠙⢦⡀
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠱⡀
⠇⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠀⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⢰⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢰⠋⠉⠙⠷⣌⡉⠒⠒⠢⠤⠤⠄⣀⣇⣀⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⢸⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠈⣳⠆⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠁⠛⠉⠛⣉⣥⡾⠿⢶⡄
⢸⠀⠀⡘⢸⡇⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣸⠃⠉⣹⠿⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⡶⠾⠛⠋⠁⠀⠀⢻
⢸⠀⠀⡇⢸⣷⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣶⠉⠀⢀⡤⠴⢄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠙⠶⠀⢣⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠇⢸⣿⠀⠀⠀⠀⠀⠀⡜⢹⣆⠀⠀⢸⠀⠀⠈⣾⣷⣾⣧⠀⠀⠀⠀⠀⠀⢴⣯⣶⠙⡆⠀⠀⠙⡦⠀⠀⠀⣠⠞
⠀⡀⠀⠰⢸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠻⢦⡴⠋⠀⠀⠰⣽⣿⣿⠏⠀⢀⣾⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⢸⠃⠀⠀⢸⡇
⠀⡇⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠈⠒⠀⠘⣧⠀⠀⠀⠀⠈⠉⠁⠀⣴⡟⠁⠀⠀⠀⠘⠋⠛⠋⠀⠀⠀⠸⣄⣀⣠⠎⠑
⠀⣧⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠋⣀⠔
⠀⡟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠐⠶⠴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠉⠉⠁
⠀⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⢀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣤⣄⣀⣀⣀⣠⣤⣴⡞⠋⠁
⢰⡯⣤⣼⢻⣉⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠟⠁⠀⠈⢭⠀⠀⢀⡜⠁⠀⢀⡟⠒⠤⣀
⠈⣗⠒⣻⠀⢹⣿⢀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⢸⣀⠀⠀⠀⠈⠳⡔⠋⠀⠀⠀⡎⠀⠀⠀⠈⠑⠦⡀
⠀⢹⡗⠋⠀⢸⣧⠀⠉⠒⠠⠤⠠⣎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠉⠘⠇⠒⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢸⡇⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠉⠒⠒⠠⠤⠤⢴⠦⠠⠀⠄⠤⠷⡀⠀⠀⠀⠀⠀⠀⠀⠈⡳
⠀⠀⠁⠀⠀⢸⣿⠈⠒⠤⢀⡀⠀⠀⠱⡀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⣠⠞⠙⣄
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⠉⠐⠚⠳⠤⠐⠎⡱⠺⡓⠠⣤⠤⣀⣀⣨⣀⣀⣀⣀⠀⢠⡤⢮⠈⢤⣀⣤⠊⠁⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠉⣮⠀⠀⢀⣂⣀⡀⣨⠀⠀⠀⣸⠀⠀⠀⡴⢆⠀⠀⠀⠱⣄⠀⠀⠀⠘⡄
⠀⠀⢰⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠡⠄⠈⢀⠠⠈⠀⠉⠁⢹⠀⠀⠀⠉⣰⡤⠤⠆⠈⢂⠀⠀⠀⠈⢆⠀⠀⠀⢡⡀
⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⠛⠢⠄⣀⠀⠀⠀⡜⠓⠾⠤⠲⠀⠀⠘⠠⠒⢀⡠⠜⠀⠀⠀⠀⢨⠃⢀⠀⠀⠈⠳
⠀⠀⢹⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⢠⠃⠀⠀⢠⠏⠀⠛⠛⠀⠀⠀⠀⠀⠶⠤⠴⢴⣇⡈⠁⠀⠀⠀⠀⠀⠘⠶⠛⡆⢆⠂⢺
⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⡏⠀⠀⠀⠀⢰⣄⣀⡠⠄⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠞⠛
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡄⣸⡿⠀⠀⠀⠀⠀⢠⢼
⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡟⣟⠀⠀⠀⠀⠀⢠⢸⡆
⠀⢠⠀⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⢸⡇⣿⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠰⠀⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠇⢹⡄⠀⠀⠀⠀⠀⢀⢿⡀
⠀⢰⠀⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣘⣇⠀⢷⠀⠀⠀⠀⠀⢀⣬⣷⡀
⠀⠈⠀⡁⢀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠒⠒⠒⠋⢹⡄⠸⡷⠴⠒⠒⠚⠉⠁⠀⢧
⠀⠀⠀⠁⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢘⡇⠀⢳⠀⠀⠀⠀⠀⢀⣠⣼⠂
⠀⢀⠀⢠⡈⢦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡖⣲⠒⠒⠒⢻⡁⠀⠈⢷⡤⠖⣒⠚⠉⠁⠘⡄
⠀⠐⠀⢸⢸⠈⠄⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠐⠘⠳⢌⠀⠀⠀⠈⢧⠀⠀⣠⠿⠲⠿⠀⠀⠀⠀⢱⡀
⠀⠰⠀⢸⠊⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠸⠀⠸⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⣀⣀⣀⣀⣤⣴⠿⠋⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠍⠛⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⣀⣀⣀⠀⢀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⣿⡇⣿⡏⣿⡇⣿⡟⣿⡄⣿⢻⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣬⡉⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣻⡇⣿⡟⣿⡄⣿⡇⣿⡇⣿⢿⣯⠀⣴⣿⣿⠋⣠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣸⡿⢻⣇⣿⣧⣿⠇⢿⣧⣿⠃⣿⢸⣿⠀⠈⠉⠛⢶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠁⠈⠉⠁⠀⠀⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣷⢰⣿⣿⡆⣿⣴⣿⠀⠀⠀⠀⣿⣿⣿⣿⣯⡉⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⢹⣿⣿⢿⣿⣿⣄⠛⠋⠛⢶⣄⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡿⢸⡿⢿⡇⠀⣿⠀⠀⠀⠀⢀⣾⣿⡿⠀⢻⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠈⠉⠀⠉⠀⠀⠀⢀⣾⣿⡿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠈⠙⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⢰⣷⣿⠇⠀⢻⣷⡆⠀⠀⢰⣷⣿⠿⣿⣿⣆⡄⠀⠀⠀⠀⢻⣿⣿⣿⣶⣇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣶⠀⠀⠀⠀⡄⣿⣿⠏⢿⣿⡆
⠀⠀⠸⣿⠁⣰⣦⠈⣿⡇⠀⠀⢸⣿⠿⠀⠘⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠐⣶⡿⠁⡀⠛⣿⠀⠀⠀⠀⣿⡿⠁⠀⢸⢿⡿
⠀⠀⠀⠙⠦⡙⢛⠴⠋⠀⠀⠀⠀⠻⠄⠀⠀⠠⠟⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠹⠄⢾⡿⠀⠋⠀⠀⠀⠀⠙⠧⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣆⢸⣿⡄⡄⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⢠⡄⣾⣿⢿⣿⣀⡄⠀⠀⠀⠀⣶⣻⣿⣷⢀⡄
⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⢈⣿⣿⠏⠻⣿⣿⣶⠀⠀⠀⣼⣿⣿⠋⢻⣿⣧⠀⠀⠀⣿⣿⣿⠟⠈⣿⣿⣿⠀⠀⢰⣦⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡇⢀⣦⠀⣿⠇⠀⠀⠀⢻⣿⠁⠀⣼⢿⡟⠀⠀⠀⠸⣿⠉⢀⣆⠉⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠈⠁⠘⠟⠁⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠛⠟⠀⠉⠀⠀⠀⠀⠈⠙⠚⠛⠛⠋

⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⢀⢀⣼⣿⣷⡄⠀⠀⠀⠀⡀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⠀⡀
⠀⠀⢀⣿⣿⣿⢿⣿⣿⡄⠀⠀⢀⣿⣄⣽⣿⡟⢀⡀⠀⠀⠀⡀⢻⣿⠛⢿⣿⣆⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⡀⣿⣿⣿⣾⣿
⠀⠀⠸⣿⡟⠉⠀⠹⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣾⠋⢠⡀⢻⣿⠀⠀⠀⢠⣸⣿⡏⠿⣷⣾⡆⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿
⠀⠀⠀⠻⣧⡀⠀⣰⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢿⣄⣿⣧⡼⠃⠀⠀⠀⠀⠻⡏⠁⠐⢻⡟⠀⠀⠀⠈⢿⡿⠷⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣦⣄⢸⣿⣿⣿⣿⣇⣠⣶⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠙⠿⣿⠀⠀⠀⠀⣿⠿⠋⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⢿⡿⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣷⡦⢼⣶⣾⣿⣿⠿⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣄⠀⠀⠈⠉⢙⡋⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢺⡗⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠟⠋⢹⣿⣿⣿⣿⡇⠙⠻⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣠⣶⣿⠀⠀⠀⠀⣾⣶⣄⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⡆
⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⠀⠀⠀⠀⣿⠿⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣷⣦⣨⣿⣿⣿⣿⣇⣴⣾⣄
⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣠⡀⢠⣄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠛⠻⠟⠛⢿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣦⣄⠀⠀⠈⠉⢙⡫⠉⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡗⢺⣿⣿⣿⣿⠋⠁
⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢺⡗⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡦
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⢿⣿⣷⣾⣿⡿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⠀⠀⠀⠀⣿⣶⣄⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⢐⣿⣿⣿⣿⡇⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠰⠳⣄⡎⠀⠀⠁⠠⠀⡜⢆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠈⠆⠀⠀⢀⠀⠰⠁⢸⠈⠰⠃⢰⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡏⠲⢦⠀⠀⠀⠀⠀⠈⢄⠁⠀⢸⠀⠀⠀⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⡘⠘⡐⢚⡇⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⡀⠀⠳⠉⠀⠌⡰⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣄⡀⠀⠀⠀⠀⠂⠄⠐⠂⠀⠀⠁⠄⠂⠀⠀⠤⠊⠉⠈
⠀⠀⠀⠀⠀⣼⠹⣭⢍⣠⢴⠚⢠⣀⣤⡤⡖⠢⣤⢼⣐⠒⣦⠀⢲⢢⡀⠘⠛⣻⣻⡄⢠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠻⠼⠛⠛⠊⠳⣾⡴⠃⠰⣌⣠⡼⠷⣤⣸⡖⠲⠮⠯⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠁⡠⠊⠣⢠⠴⣞⡏⢳⢰⢋⡙⡄⡇⢠⣍⠱⡏⢨⣍⢻⡍⠹⣄⡞⢠⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⠉⠱⡀⠀⠈⣇⠘⠃⠈⣿⠘⠃⠹⡇⢈⡥⠼⡇⢈⣁⠼⢹⡄⢉⡴⠃⠀⠈⢣⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠱⡀⠀⠸⡄⣻⣄⡽⠼⠉⠣⠛⠚⠃⠀⠓⠚⠀⠀⠸⣅⣼⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠱⡀⠀⠉⠁⠀⢀⣠⠤⠤⣀⠀⡴⠠⡄⡤⢄⣠⣄⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣡⠀⠛⠐⠀⠈⢑⠼⡀⡖⢢⠈⣿⢁⡆⢷⠘⡜⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⡀⠸⠈⢇⡄⠀⠱⡈⠁⠀⣇⠹⠚⣀⣏⡠⢡⡘⡆⡇⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠓⡚⠀⠌⡇⡤⠔⢇⠀⢀⠞⢸⣉⡁⡙⠀⢩⠉⠉⠉⣚⣠⠄⠂⠐⢄⢀
⠐⠶⠆⠀⠀⠀⠀⠀⠀⠹⠤⣂⠷⠶⠖⡇⠐⠀⠈⠆⠸⡄⡀⠀⠀⠐⡡⠊⢀⠔⠋⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢦⡀⠘⢆⠀⠀⠘⡄⢱⢀⢀⢰⠋⠐⡴⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⢦⢀⠀⠘⣆⡀⠇⢸⡉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢣⡀⠀⠘⡅⢸⢸⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣠⠀⢪
⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠁⠄⣣⠀⠀⠹⡀⠀⠁⢸⠀⡂⠀⠀⠉⢫⠀⠀⠐⠋⡙⠓⠤⣀⠈⡅⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠐⠀⠀⠈⣧⠀⠀⢙⠀⠀⢻⠀⡁⠀⠀⠀⡾⠩⠄⢰⠎⠀⠀⠀⠀⠀⠁⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣥⠈⠑⠀⡀⠀⠈⠣⠀⠚⡄⠀⣼⠀⠁⠀⠀⡀⠁⠈⠐⢷⣄⠔⠁⠰⠐⠄⢀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠤⡁⡆⠐⠓⠂⠏⠀⠊⠓⠘⠒⠀⠃⠀⠐⠖⠀⠈⠀⢆⠠⠀⠁⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠂⠀⡇⠀⢰⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⢸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠨⠀⠐⡇⠀⠘⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⠀⠀⢀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣴⡶⠉⢋⣹⣿⡶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡴⠃⠀⣴⡿⠋⠁⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⠿⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⣼⡦⠄⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⠁⠀⠀⠀⠀⠈⠉⠲⢆⣀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣤⣄⣀⣀⣤⣤⣴⣶⡾⣿⣿⠿⠿⠿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⣽⣿⣭⣩⣿⣠⣶⣿⡐⢦⡀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠁⠈⠁⠀⠈⠉⠀⠙⠓⠈⠁⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢉⠀⢸⣿⠀⠀⠀⢼⣷⠀⠀⠀⣿⠋⠉⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢸⡇⠀⣱⠶⠒⢢⡈⠁⠀⠀⠀⣥⠀⢠⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣽⣦⣘⣈⡁⣤⠀⠀⠀⣸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣤⣤⣄⠀⠀⠀⠀⠀⢸⡇⠀⠿⣄⡈⣁⡼⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⡍⠁⣿⡏⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⢀⣤⠔⠠⠒⢢⣤⠜⣃⣤⡷⠷⡼⠃⠀⠀⠀⠀⠀⠀⠘⠣⣤⣀⠀⠀⣀⣤⣤⠚⠋
⠀⢠⡏⠀⠀⣠⠶⣌⣵⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⠉⠀⣗
⠀⠈⣱⡤⢋⡤⠛⠁⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⢛⠞⠛⣤⣶⣦⠐⠛⢳⡟⣳⣄⡀
⣴⢮⡤⠚⢯⡄⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠋⢸⠆⣾⣠⠖⠁⠀⠈⢦⣀⣼⠃⣿⠋⠙⢦⡀
⠈⠋⠀⠀⠀⢹⠀⠈⢣⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⠀⡎⠀⡿⠁⠀⠀⠀⠀⠀⠘⡟⢀⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠘⢃⠀⠀⠀⠀⣠⠎⠁⠀⠀⢸⡀⠃⢠⡁⠀⠀⠀⠀⠀⠀⣀⠇⢸⡟⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠹⡀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣷⠀⢀⣉⣉⡉⣉⣉⣉⣉⠉⠀⢸⠁⡆⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠻⣄⠴⣇⠀⠀⠀⠀⣠⣾⡇⠀⢸⠁⠀⠀⠀⠀⠀⣹⠀⠀⠸⠀⢳⡀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠙⣆⠀⣠⠞⠁⢸⠀⠀⠘⣄⠀⠀⠀⠀⢠⡟⠀⠀⠀⣶⠀⢹⣦⡀⠀⠀⢠⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣠⠞⠛⠁⠀⠀⢸⠀⠀⠀⠈⠓⠶⠦⠔⠋⠀⠀⠀⠀⠙⡀⢸⡇⠙⢶⠛⠁⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣠⠞⠉⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⡇⠀⠸⠇⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⣸⣇⢶⡀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣿⡆⠁⠘⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣿⣇⠀⠀⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠋⠙⠂⠀⠀⠀⠀⠀⠀⠐⠂⠙⡇⠀⠀⠈⠛⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠀⠀⠘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢻⠀⠀⠀⣿⠀⠀⢀⠄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⣰⠁⢠⡜⠋⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠏⡀⢹⡇⠀⠀⠀⠀⢸⡏⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡧⢄⣀⣀⡠⠾⠛⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣸⠀⣇⢸⡇⠀⠁⡈⠁⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⣉⡉⢉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠀⠀⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣟⠀⠈⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⢸⡆⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠈⠛⠒⠺⠋⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⢡⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣼⠀⠀⣼⠀⠀⠀⠉⠙⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢿⠀⠀⢸⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⠀⠀⠈⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⢿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣸⡇⠀⠀⠀⠀⠀⢁⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠈⠉⢙⡇⠀⠀⠀⠀⠀⡘⠋⠉⠉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡈⣁⣀⣀⣤⣦⣤⣼⠇⠀⠀⠀⠀⠀⢿⣦⣤⣤⣄⣀⣀⡉⢳⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⣠⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢠⠞⠋⠙⠳⣄⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠀⠀⠀⠙⢆⠀⣏⠀⠀⠀⠀⢸⠀⢠⠋⠁⠀⠀⠉⠑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⡫⣗⠀⠀⣀⣀⣤⠤⠀⠀⠈⣆⠘⢦⣄⣠⣴⠋⢠⠃⠀⠀⠠⢤⣀⣀⡀⠀⣸⠍⣒⢤⣀
⠀⠀⠀⢀⣀⣀⣠⠤⠖⣚⡩⢴⣊⣁⣀⣤⠽⠋⠁⣀⠷⠶⢶⣶⠂⠘⢦⣸⠀⠀⣺⡠⠋⠐⣶⡶⠶⠾⣀⠈⠙⠻⢥⣀⣀⣑⣢⢍⣓⠲⠤⢄⣀⣀⡀
⠲⠯⣭⣭⣭⡤⠖⠚⠉⠁⢀⣀⡬⠟⠙⠥⣤⣶⠋⢁⣠⠔⠋⢈⡷⠒⢦⣽⠀⠀⣿⡴⠖⠺⣁⠈⠲⢄⡀⠙⢶⣤⡤⠟⠛⠦⣄⡀⠈⠉⠓⠲⠤⣬⣭⣭⡭
⠀⠀⠀⠐⠳⠤⢤⡴⠖⠚⠉⠀⣀⡤⠖⠋⠀⢈⡩⠛⠣⢖⡾⣏⡤⣴⣫⣼⠀⠀⣿⣜⣶⣤⣸⠷⡶⠼⠛⢭⡉⠀⠉⠒⢤⣀⡀⠉⠓⠒⢦⡤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠒⢒⡟⠋⠁⣀⣠⡴⠖⠉⠀⣠⠔⠉⠀⣠⣞⣠⣎⢸⡇⠀⣿⢑⣄⣑⣄⠀⠈⠢⣄⠀⠈⠓⠦⣄⣀⠀⠙⢻⡖⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣥⣤⠤⢶⠟⠁⣀⡴⠚⢉⠁⠒⣈⣽⡇⠀⣿⣉⠐⠂⢉⠑⢦⣀⠈⠙⣶⠦⣤⣬⡯⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⣟⠀⠀⣏⣙⠋⠁⢸⡇⠀⡏⠈⠙⣛⣹⠆⠀⢹⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠉⢉⣉⡿⠷⠒⠛⠋⠉⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠛⠉⠀⠀⣀⣀⣤⡤⠤⠖⠶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡴⠛⡏⠀⡏⠓⣆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠘⠲⡇⢀⣧⠴⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠴⠚⠋⠉⠀⣀⣤⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡴⠚⣯⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣻⡷⠴⠛⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠁⢀⣠⡴⠶⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣯⡰⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠞⠉⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢾⡽⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⢤⡈⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⢸⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⢿⢸⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢸⣷⢠⣤⢉⣥⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣼⣛⠋⠉⠛⠿⣿⣿⣠⣴⠞⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⣴⣦⠩⣿⡛⣳⣦⣴⣽⣿⠯⠀⠀⢀⣀⣤⣤⣤⣤⣄⢀⡀⠀⠀⠀⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠘⢹⡇⣹⡧⠖⠾⠷⣌⣼⣿⡿⣧⣨⣶⠿⠿⠿⠟⣛⡿⠛⢿⣿⣿⣶⣤⡀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠸⣿⣷⡟⣡⣤⠖⢳⣾⣿⡟⣬⠿⠁⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠙⣻⣿⣿⣧⡀⠀⠀⠸⢆
⠀⠀⠀⠀⠀⣼⠀⣿⣻⡇⣏⠷⣶⡟⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠿⣿⣷⡄⠀⢰⣾⡆
⠀⠀⠀⠀⠀⠸⠷⣿⣿⠛⣠⡠⡏⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣟⢀⣼⡟⣡⠆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⠟⣿⠃⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣰⢾⣟⢿⣿⣿⡿⣽⢿⢓⠉
⠀⠀⠀⠀⠀⠀⠸⢻⣿⢦⡄⣿⠀⣿⡟⣷⢀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣶⣿⣿⣿⣿⡟⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠈⣌⣿⡿⠛⢿⡐⣿⣿⠁⠀⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⢻⡦⢸⡿⣿⣿⣿⠿⣿⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣦⣼⣷⢿⣇⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣼⣿⣿⣿⣿⠀⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣩⣿⣮⣏⢻⡤⠂⢸⡆⠀⠀⠀⠀⠀⠀⠀⣸⡿⢋⣠⣽⠿⠿⢄⢸⡗⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣐⣿⣿⣷⣶⡀⠸⣧⣀⠀⠀⣀⣀⣴⠞⠉⠠⢌⣤⣤⣤⡄⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠿⠿⢿⢀⠀⠙⣿⣿⡟⠈⣻⡇⠀⠀⠀⠀⠠⣤⣿⣷⠄⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠘⣿⡇⠀⢸⣇⠀⠀⠀⠀⠀⢸⠿⣿⡄⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⠄⠀⠘⣧⠀⠀⣿⠀⠀⠀⠀⠻⠀⠀⠹⣇⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠋⠀⠀⢿⡀⠀⠀⠀⠀⢃⠀⠀⠈⠂⡀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠁⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢷⡄⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠈⠳⣦⣄⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣄⢀⠀⠀⣠⣴⠾⠋⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠂⠹⠇⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣦⣤⣶⣶⠟⠋
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠛⠻⠟⠻⢿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠈⠫⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⢿⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣤⣤⣾⠟⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡹⠋⠉⠁⠀⠀⠈⠁⠀⢀⣤⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⡀⣀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⠾⠛⠛⠛⠉


⣦⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠉⠛⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠟⠛⠛⠋⠉⣉⢍⣙⠛⠛⠻⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠋⠁⠀⠀⠀⠀⢀⡔⠙⡆⠀⡞⠑⣄⠀⠀⠀⠈⠙⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢥⣤⣬⣭⣩⣭⣤⣤⠽⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡹⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢿⠘⠳⣿⣿⣿⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠈⡳⡄⢻⡟⠃⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠰⠊⠁⢱⡿⠈⠙⠲⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⣰⣇⡀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⣀⣀⣀⣘⣆⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⢰⠇⠀⠉⢷⣤⠞⠁⢸⠀⠀⢀⡶⡶⢶⣶⣶⣦⣦⠀⢀⣿⣽⠾⠿⢋⣤⠴⠿⢦⡀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠸⡆⠀⣸⢳⣦⣾⣿⣿⣿⠛⠒⠋⣁⣤⠴⠚⠉⠀⠀⠀⢸⡷⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⢷⠀⠀⠀⡇⠀⠀⠀⣷⠾⠋⣘⣿⣿⣿⣿⣏⣤⡶⠛⠉⠀⠀⠀⠀⣀⣤⠴⠞⠁⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠈⢧⠀⠀⢿⠀⢀⠌⠀⠀⢀⠋⢹⣿⡿⠟⠋⠁⠀⢀⣀⠤⠖⠚⠉⠁⠀⠀⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠈⣇⠀⠀⠩⣏⡄⠀⠀⠠⠴⠂⣿⣇⣠⣤⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⣙⠀⠀⠀⢠⣽⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡗⠦⠖⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠂⠶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⠀⠀⠀⠉⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠠⠤⠀⠀⠀⠀⠤⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣴⡶⢶⡶⠦⣤⣤⣀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠿⣿⠺⠿⠿⠿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⠀⣄⣀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡘⠳⣬⡥⠔⠂⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⡀⠀⢀⡾⢿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣠⣽⡟⢿⡟⠻⠛⠁⠈⢻⡋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠔⠺⣏⠈⣷⡿⠉⢿⣦⣥⣀⠤⠔⠁⣸⣿⠆⡄⠀⡀⠀⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠹⡆⠙⣷⡄⠈⢇⠀⠀⠀⠀⢰⡿⠋⣄⠀⡰⠀⠀⢠⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢿⡙⠆⠀⠀⠀⠀⣰⡟⠁⣸⠏⣰⠃⠀⠀⡞⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⣇⠀⠐⠀⠀⣰⠏⠀⢰⠏⢀⠟⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣖⣸⡇⠀⠟⠀⠀⠈⣦⠁⠀⠀⡞⠀⡾⠀⠀⠀⣬⣠⣯⣤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⠀⣀⣤⠴⠞⢻⡿⢯⣘⣧⣄⣷⣤⣤⣀⡋⠤⠤⠴⣧⣴⣇⠀⠀⢀⣿⠋⠙⢷⣦⣤⠀⠀⢀⣀⣀⠀⠈⢳⡀
⠀⢸⠁⠀⢀⠀⠠⠤⢄⣼⠚⠋⠀⠀⠀⠸⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢿⡏⠀⠀⠀⠀⠈⢻⣿⣿⣥⡁⠀⠀⠀⣻
⠀⠸⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠶⠾⠛⠛⠙⠛⠛⠃⠘⠃⠁⠙⠉⠲⠤⢄⣤⣼⠁⠀⠀⠀⠀⠀⡞⠉⠙⠒⠤⠀⠀⣠⠏
⠀⠀⠑⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡐⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠈⢢⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⡔⠋⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠙⢄⡀⠀⢄⠀⠀⠀⠀⠀⠢⢼⣏⠀⠐⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠠⣈⠁⠀⠀⡀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠰⡆⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣴⣧⡀⠠⠀⢀⣸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡄⠀⠀⠀⢻⡀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣿⣄⠀⢈⣾⣷⣶⡖⡿⣿⣿⡿⢲⣶⡶⠴⠷⠛⠉⠃⢠⣠⣤⠄⠛⠦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠴⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⢿⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⢦⣌⢷⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⣆⠙⠷⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡓⠦⡀⠈⢷⡀⠀⠀⠀⣿⠀⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠻⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⣠⠟⠀⠀⠀⠹⣄⡀⢻⠀⠸⣇⠀⠀⠀⠀⠀⢰⡇⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠟⢰⡏⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⡉⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠳⢦⣄⠀⠀⠈⠳⡄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠈⠉⢻⠓⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠃⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢠⡞⠀⠀⠀⢀⣤⠟⣿⠃⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠀⠀⢹⡆⢀⡤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⡖⠉⠀⣀⡴⠞⠋⠀⣰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠘⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠘⢧⣠⠞⠁⠀⠀⠀⠘⠁⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⢸⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠁⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⣰⠋⣠⡾⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠃⢀⣠⠄⣀⣴⠆⢀⠀⠀⠀⠞⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠰⣶⣤⣛⠀⠈⢁⣾⣿⣷⣿⣿⠃⠀⠀⣀⠀⠖⢀⣀⡀⠀⠀⠀⠀⠀⣰⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣍⣻⣦⣄⡀⠀⣀⠐⣀⣈⠛⠻⣟⣻⣿⣿⡿⠟⣋⠀⠀⣠⣤⣤⣤⡶⠿⠛⠁⠀⠀⠀⣀⡼⠯⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠶⠛⠻⢶⣤⣄⣈⣁⣰⣿⡿⣯⣥⣤⣀⡛⣿⣿⣶⣬⣍⠀⠻⠷⠶⢶⣆⣤⣴⣾⣿⣶⣶⣶⠤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣤⡤⠤⠶⠶⠚⠋⠉⠉⠉⠉⠁⣀⡤⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⠲⠶⣿⣿⣭⢤⣭⣍⣛⡛⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠋⢑⣻⡿⠟⠋⢁⣀⡈⠉⠉⠉⠋⠉⠛⠒⠾⠭⣅⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣯⣉⣀⠤⠖⠚⠉⠉⠀⠠⠖⠛⢉⣀⡙⠚⠙⠛⠶⠶⢦⣄⣠⣤⡴⠶⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢰⣿⡀⠠⠄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣾⡟⠀⢁⣼⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢓⠀⡷
⠀⠀⠀⠀⢰⡟⡥⣰⣾⣿⣿⣿⣿⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡙⣷⣿⠛⠣⡀
⠀⠀⠀⠐⡿⡅⣼⠿⣻⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣇⢀⠗⠸⢽
⠀⠀⠀⢰⣟⡺⢃⣾⡿⢟⣛⣙⣛⡻⠿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠛⣹⠏⠀⠀⢸⡇
⠀⠀⠀⠈⢏⣧⣾⢯⡾⡿⡛⠉⠩⢿⣷⡌⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠛⠀⣿⢧⣀⠀⠲⡌⢧⡀
⠀⠀⠀⠀⢹⣻⡁⠉⠁⠠⣿⣀⠂⢆⢾⣷⣾⡓⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠹⣿⠋⠀⠀⠀⢀⠔⠁⠀⠙⣥⠀⠈⠈⠓
⠀⠀⠀⠀⠀⠸⢿⠀⢠⠛⢟⣽⡃⢱⣿⣿⣿⣿⣶⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠐⡀⠘⠀⠀⠀⠈⣹⢿⣿⣽⣿⣿⠿⠿⡷⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠂⠀⠒⣶⣾⡏⠀⠀⠀⠀⠀⠻⣷⡄⠀⢠⣶⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⢛⣿⣿⡁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣹⣶⣦⣤⣤⣠⣤⣤⣤⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡍⠀⠀⠈⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠾

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡀⠈⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢶⡄⢀⡫⠬⠥⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠰⠤⡴⠋⠀⠄⢀⡄⢸⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢡⠀⢃⠀⣙⠁⠀⠀⠈⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⣇⠘⠇⠘⣯⠁⠀⠀⠀⣒⣎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠖⠉⢀⠛⢀⣂⡉⠀⡇⠉⠙⠒⠦⠤⠤⠀⢀⡀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈⠫⣸⡌⠀⠀⠇⢠⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢓⡒⠠⣾⠁⠀⠀⢸⠀⠀⠀⢸⡎⠉⠉⠙⠋⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣄⡾⠁⠀⠀⠀⢸⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠤⠄⠀⠀⣀⣼⠤⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠐⠀⠀⠀⡶⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠤⣄⣀⣀⣧⡤⠄⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣶⡁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡼⠈⡇⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢰⠃⠀⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⡎⠀⠀⢸⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠦⠠⠤⢸⡀⠀⠀⢸⠒⠒⠒⠾⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡠⠤⠬⠀⠀⠀⠠⠤⠤⣀⣀⣀⠸


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⢿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠂⠉⠉⠉⣿⡟⠉⠑⠦⠀⣀⡀
⣀⡀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣽⡟⠳⢤⡀
⡏⠰⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢰⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠙⢦⡀
⡇⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠱⣄
⠇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣤⠴⠾⣷⡂⠤⣤⢀⡀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⢀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠈⠻⣶⣄⠀⠉⠉⠁⠒⠒⠒⠳⠄⢤⣤⣤⡤⣼⡅⠀⣀⠀⢀⣀⣀⡀⣀⣤⣤⣤⣴⣄⡀
⢸⡇⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⠀⠉⢻⡶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠋⣽⠆
⢸⡇⠀⣼⠈⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡞⠀⢠⠽⠟⠛⠻⠷⠶⢶⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⣿⠛⠉⠁⠀⠀⠀⠀⢻⡀
⠈⡇⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠰⡏⠀⠀⢠⢐⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠓⠀⠹⣄⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠁⠀⢹⠀⣿⡇⠀⠀⠀⠀⠀⠀⡴⠫⣷⠀⠀⠀⡿⠀⠀⡃⣾⣿⣾⣷⠀⠀⠀⠀⠀⠀⢀⣺⣵⣶⡈⡆⠀⠀⠘⣷⠄⠀⠀⢀⣤⠟
⠀⠀⠀⠈⠀⣿⡇⠀⠀⠀⠀⠀⠀⣄⠀⠹⠷⣤⠞⠁⠀⠀⢰⣻⣿⣿⠟⠀⠀⣸⡆⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⢰⡏⠀⠀⠀⣾⠁
⠀⢠⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⢿⡄⠀⠀⠀⠀⠉⠀⠁⠀⢠⡿⠋⠀⠀⠀⠀⠙⠽⠟⠋⠀⠀⠀⠘⣦⡀⢀⣰⠟⢢
⠀⢸⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠛⠁⢀⡜
⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠑⠛⠛⠁
⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠛⠁
⢀⡼⠦⠤⣤⡼⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠬⠛⠛⢟⠛⠛⠚⠛⡻⠛⠛⠥⣤⣀
⠸⣟⣛⣻⡎⠱⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠨⠇⠀⠀⠀⠙⢆⠀⣠⠞⠁⠀⠀⣴⠋⠀⠙⠳⢄⡀
⠀⢹⡄⣼⠃⠀⣿⡟⠆⣀⠀⠀⠀⠀⢀⡄⠴⠚⠉⠀⠀⠀⠀⡶⠤⠤⣀⣀⣀⣀⣻⡁⣀⠀⣀⣰⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠈⡟⠁⠀⠀⣿⡇⠀⠀⠈⠁⠐⠆⢾⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⡇⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢸⠀⠈⠙⠛⠓⠒⠰⠶⠶⠶⠶⠶⠶⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⡇⠀⠀⠀⣿⡇⠈⠓⠦⣀⣀⠀⠀⠀⡆⠀⠀⠀⢀⣠⡎⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣀⠴⠃⠘⣀
⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠙⠦⠖⠛⢉⡶⢛⠁⠲⢦⡄⢀⣀⣨⣅⣀⣀⣀⣠⠀⣤⡔⠿⠈⠶⣀⣴⡎⠁⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢱⡎⠀⠀⢨⡅⠀⠀⢈⡃⠀⠀⠀⣤⠀⠀⠁⠰⢆⠀⠀⠀⠙⢦⠀⠀⠀⠀⠘⡀
⠀⠀⢠⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⢤⠌⠸⠀⣠⠋⠈⠉⠉⢙⠋⠉⠉⠉⠹⠂⣤⡄⠦⠈⢧⠀⠀⠀⠀⠱⣄⠀⠀⠀⢇⡀
⠀⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡰⠶⣄⠀⠀⠀⠀⠀⡰⠶⣤⣄⡤⣄⠀⠈⠫⣄⠠⠂⠀⡰⠃⠀⠀⠀⠀⢹⠄⠀⠀⠈⠉⠲
⠀⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠰⠁⠀⠀⢩⠷⠒⠒⠐⠃⠀⠀⠀⠀⠺⡀⠀⣤⣤⡆⠘⠈⠀⠀⠀⠀⠀⠀⢿⣤⠷⡄⡀⢀
⠀⠀⢸⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⢠⡟⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢈⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣬⣼
⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⢉⣶⠞⠁⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠘⠃⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠃⣼⢿⠀⠀⠀⠀⠀⠀⠛⢸⠄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢲⡞⢹⡄⠀⠀⠀⠀⠀⠒⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠞⣧
⠀⠀⠀⣸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢼⡇⠈⣷⠀⠀⠀⠀⠀⠀⠰⠿⡆
⠀⠀⠀⢸⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣤⣷⡀⠘⡆⠀⠀⠀⠀⢀⣀⣴⠿⣆
⠀⠀⠀⢸⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⠛⠛⠛⠛⠛⠉⠉⣧⠀⢿⠓⠚⠛⠛⠉⠉⠀⠀⢻⡀
⠀⠀⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⣿⠀⠸⣇⠀⠀⠀⠀⠀⢀⣀⣼⠇
⠀⠀⠀⢸⣀⠱⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡖⣶⠴⠶⠛⠻⡅⠀⠀⠻⢦⡶⠖⠚⠛⠋⠉⠘⡇
⠀⠀⠀⠀⠈⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠺⠧⡉⠀⠀⠀⠀⢻⡀⠀⠀⣸⡷⠖⠿⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⡴⠀⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡗⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⡗⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠲⣦⣤⣤⣤⣤⣤⡶⠟⠋
⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠋⠙⢻⡏⠀⣙⠛⢙⣏⠟⢹⣿⠟⣏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⢸⣿⣿⡷⠀⢨⣽⠁⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢈⣻⣟⣋⡀⠸⣖⡲⠯⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠀⠆⠈⣭⡥⠔⠀⢿⣛⠿⡖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠛⠛⠋⠐⠶⠒⠃⢰⣮⠵⠀⠨⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⢠⢄⠀⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠘⠏⠇⠚⠓⠘⠁⠘⠋⠠⠏⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⣠⡖⠓⡆⠀⠀⢀⣀⣀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣶⣴⠇⣾⡏⠀⣠⠃⠀⣰⠏⠀⣸⠀⣿⡇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡉⠁⢸⣿⡠⢖⣁⠀⢸⣏⡤⠖⢁⡤⣹⠷⣤⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠇⢰⣟⣹⡿⣿⡥⢲⣦⣯⡄⢈⡧⠊⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣉⣷⣏⠀⠀⡿⢿⡧⢻⠤⣺⡿⢿⣏⣼⣁⢸⣿⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⢋⣁⣀⠐⢧⡴⠟⠁⠀⠀⢿⣛⠽⠻⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⠛⠳⣄⢀⣴⠗⣤⡄⣠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢁⣿⡇⠀⠀⣹⡞⣿⡴⣿⢳⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣀⠤⠊⣿⡇⠀⢀⣻⢯⣛⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⣹⡿⠣⡀⠀⣼⡿⠁⡴⠁⠀⢠⣾⡇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠙⠛⠋⠀⠀⠳⣄⣀⣀⣭⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡀⢪⣦⠘⠃⢙⠃⠐⠟⢀⣼⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡈⠉⠀⠉⡉⠸⣇⣸⠰⣾⡀⡌⣋⠸⡇⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⡄⠘⠛⠁⢸⡷⠀⣿⡇⣹⣳⠀⣉⠀⡇⢩⢞⡄⢈⡅⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣙⡛⣁⢆⠸⡷⠀⠀⢀⣄⡀⣿⣠⠀⠙⠃⣽⢃⡀⢰⣈⣵⡸⠟⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⣤⣿⠓⡾⠷⣄⡸⠇⠀⠛⠃⡟⠙⡀⠘⠓⡇⣈⣳⠚⣏⡈⠛⣄⠸⠯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠿⠂⠃⠀⢸⠛⢠⣤⡳⣄⠐⠿⠇⣧⠺⢇⠛⢸⣧⡼⠃⣶⢉⠥⠐⢻⣶⡃⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠀⣰⣷⢀⣠⡄⠀⢹⠅⢉⡝⢦⠒⣿⢀⡸⢇⣿⠷⠆⠈⣤⠘⣷⣴⡯⠴⣓⠂⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⣴⡄⠲⢾⣥⡤⠜⠃⣬⠳⠤⣉⢡⠸⣆⣿⣈⡵⠻⡅⢠⡶⢠⠿⢒⣛⢱⡶⠀⠛⠃⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⣻⣍⠑⠟⠂⠐⠉⢠⣾⠎⠙⢶⣿⣿⣿⣷⠚⣛⠓⣻⡅⠀⠛⣿⢆⢈⣤⡰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠋⠀⡸⠷⠘⡷⠀⠋⠀⡾⠃⢿⣿⣿⠃⡆⠙⠀⠙⠗⠀⠀⠸⠃⠀⠈⠇⠈⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣿⣿⣿⣷⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣳⠖⣋⣉⣷⠞⢋⡉⠛⠛⣿⣿⣿⣿⠛⠛⢉⡉⠳⣾⣉⣙⠲⣾⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⣶⠞⢉⣀⣐⣚⣭⣿⣶⠞⠉⣿⣿⠉⠳⣶⣿⣍⣗⣂⣀⡈⠳⣶⠬⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣴⢋⡽⠋⣤⠞⣩⠖⣶⣴⢿⣿⢦⣶⠶⢍⢳⣄⡙⢧⡙⣦⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⢤⡟⢠⠏⣿⢰⢃⣴⡛⠁⢸⡏⠈⢻⣦⡈⡇⢿⠙⡆⢹⠤⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠞⣡⢞⣴⣷⡏⣾⠈⣿⠀⣼⢿⡀⣿⠁⢿⢹⢾⣦⡻⣌⠳⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠏⢰⢇⡴⢿⠀⣏⠀⢸⡄⠇⣹⠀⡿⢦⡸⡆⠹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠼⢰⠟⣧⢘⡇⣼⠻⡆⠧⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣹⡟⠀⠀⠀⢠⣄⠀⠀⠀⢀⡔⠋⠉⠉⠉⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⡟⠀⠀⠀⠀⣿⣿⣷⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠁⠘⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⠶⠶⠋⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢰⡇⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢻⡀⠀⠀⣧⠀⠀⢸⠀⠀⠀⣸⠁⠀⢀⡞⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠘⣦⠀⠀⠈⣷⡀⠀⢹⡀⠀⢸⡆⠀⠀⡟⠀⠀⣾⠁⠀⠀⣰⠋⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢧⡀⠀⠈⠀⠀⠘⠃⠀⠈⠁⠀⠈⠃⠀⠘⠃⠀⠀⣼⠃⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⠀⠀⠈⠻⠆⢀⡈⠃⣀⠀⠀⣀⡀⠀⢀⣀⣀⡀⢀⣀⣀⡀⢀⡀⠘⢁⡀⢠⡾⠃⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⢸⡇⠀⣿⡃⢠⣿⣿⠀⢸⡟⠙⣿⢸⣿⠙⣿⠘⣿⠀⣿⠃⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢸⣷⣤⣿⠂⢸⡇⣿⡆⢸⣧⣤⡿⢸⣿⣤⣿⠃⢹⣷⡏⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡄⣿⣧⣾⡇⢸⡏⠉⠀⢸⣿⠉⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠻⠀⠟⠀⠘⠟⠘⠗⠀⢀⠘⠟⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣼⠇⠀⠀⠀⠀⠀⢠⡾⢁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠁⠀⣠⣄⣀⠀⢀⣿⠗⠋⠀⠀⢀⣰⠊⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡀⣼⢃⣼⣿⢁⣼⣏⠔⢩⡷⢴⡟⠉⣴⠛⠘⣳⡖⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠁⠀⠙⠉⠘⠷⠊⠙⠃⠚⠋⠀⠈⠑⠋⠀⠀⠐⣿⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⣤⣤⠀⣤⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇⠀⣿⢿⡇⢹⣧⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⢸⣇⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⣿⣄⣼⡇⣼⡟⠻⣿⠀⢸⡇⠀⠀⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⢀⡔⠀⢠⠉⠉⠉⠀⠉⠀⠀⠉⠀⠈⠁⡀⠐⢆⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⡰⠃⠀⡖⠀⡆⠀⠀⠀⡆⠀⢆⠀⠹⡄⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠁⠀⡸⠁⢠⠃⠀⢺⠀⢸⠀⠈⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠈

⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⢶⠶⠶⠶⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⡏⢿⣧⡀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡇⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢸⠁⠈⣿⣿⣀⢀⣄⣀⣼⣿⣿⣿⠿⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡟⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢠⡇⠀⠀⠀⠀⢠⣞⣽⣿⣿⣿⣿⣿⡷⡀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢶⢄⣀⠀⠀⢀⣿⣾⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣬⣽⣶⣾⣿⣿⣿⣿⣿⡟⠉⢻⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⢺⡿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⠿⠟⢻⠏⢿⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⡏⠁⠀⠀⠀⢏⡜⠀⡄⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣃⣿⣿⣧⠀⠀⠀⢰⠏⠀⠀⠇⠀⢀⣾⣥⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⠛⡄⠀⠸⠀⢀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⠀⡇⠀⠑⠒⠁⠀⢠⣿⣿⣿⡿⠖⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠁



⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀
⣿⡟⡉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣧⠖⠲⢤⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡞⠈⠙⠲⠦⠽⡶⠦⣄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡁⠀⠀⠀⠀⠀⢻⠀⠘⡆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⢻⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⣿⡄⠀⠀⠀⠀⠘⣆⠀⠀⣧⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠇⠙⠶⢶⠛⢦⣀⣈⣳⣤⡟⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢟⣀⣀⣠⠞⠀⠀⠀⢈⡽⠋⠉⠳⣄⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠏⠁⠀⠀⠀⠀⣠⢶⡏⠀⠀⠀⠀⠘⡆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠻⣄⣀⣀⣠⠴⠚⠁⠸⡇⠀⠀⠀⠀⠀⢻⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⢩⣳⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡁⣴⠶⢭⣓⡮⣽⣦⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠈⠓⠲⠞⠛⠉⠁⠉⠉⠓⠲⢤⡀⢸⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢰⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⡴⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣴⠶⠤⢤⣀⡀⣠⠞⠉⠀⠀⢀⣠⠴⠊⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣾⠭⠿⠆⠀⠉⠁⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡗⠒⠶⠶⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⢠⠖⠉⠉⠀⠉⠉⠛⠓⠲⠦⠤⠤⠤⠤⠤⠆⠀⠀⠈⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⢠⠌⠙⠉⠙⠃⠴⠶⠀⠀⢀⠀⠀⣀⣠⠤⠦⠤⠴⢆⣠⣄⣤⡀⣤⣠⣄⡠⣦⣀⣀⣀⣀⣀⠀⠀⠀⣀⣠⡀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠋⢁⠉⠓⠁⠀⣠⠀⠁⠙⠢⠆⠀⠄⠒⠐⠀⠉⢉⠐⣄⡉⠙⠛⠯⠉⣙⠛⢚⣿⡧⡉⠑⠛⠖⠋⢩⠉⣩⡿⢺⡿⢄
⠀⠀⠀⣰⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⡄⠀⠀⢀⠀⠰⠂⠀⠀⠁⠌⠑⠒⢀⡂⠀⠀⠤⠌⠛⠛⠂⠒⢤⡈⣉⠉⠙⠋⠀⠀⠠⠀⠸⠀⠹⡶⢋⡴⣻⡄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠆⠀⠀⠰⠄⠀⠀⠈⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠸⠂⠁⠀⠩⠄⡀⠀⠈⡀⠀⠀⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⣉⡀
⠀⠀⠀⠀⣞⡁⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠓⣦⠔⠒⠉⡗⠆⠀⠀⠀⠀⠀⠀⠀⣀⢴⠟⠭
⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠆⠀⠀⠀⡀⠀⢀⠀⠀⠘⠀⠀⠈⠀⠀⠈⠀⠁⢠⠀⠀⠁⠀⢀⠀⠀⠛⠀⢀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠀⠀⠂
⠀⠰⠄⠀⠀⡄⠀⠀⠁⠀⠀⢀⠀⠀⠨⠄⠀⠺⠀⠀⠀⠀⠄⡰⠶⡄⠐⠠⣄⠈⠀⡄⠀⠀⠉⠀⠀⡄⠀⠈⢸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠃⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⢇⣀⣀⣁⣉⣉⠀⠀⠁⠀⠀⠀⠀⠀⠁⢀⢀⠇⠀⠀⠀⠀⠀⠃⡀⠀⠀⠁⠸⡆
⠀⢀⡆⠀⠂⢀⡄⠀⠓⠀⢠⠀⠀⢰⠀⠀⠘⠀⠐⠀⡀⣳⣾⣿⡟⠹⡗⠐⢭⣥⣆⣄⠀⠀⠀⡀⠐⠀⠀⡜⠀⠀⢀⠂⠀⠀⠀⡀⠀⠀⠀⡀⡇⠀⠇
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⣠⠖⠉⣴⡟⠀⠉⠣⡀⠀⠐⢂⠼⠿⠖⠁⠀⠀⠇⠀⠈⡸⠁⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠙⠀⠀⢰⠆⠀⠶⠀⠀⠈⠀⠀⠀⡼⠃⠀⠌⡟⠤⡁⠒⠀⠱⣄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡀⣀⡠⠄⢠⠀⠀⠀⠀⠂⠀⠀⡜⠀⣆
⠀⢰⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠂⠀⠀⡼⠁⠀⠀⠢⡳⣴⣌⡑⠦⢀⣈⣦⣌⠀⠀⣌⣀⣀⣠⣲⡛⢳⣚⡍⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⡇⠀⠀⢀
⠀⠈⠀⠈⠀⢀⣀⢀⣄⠀⠀⠀⠀⠀⢰⠃⠀⠂⠀⠀⠀⠘⠻⣯⣖⠠⣀⠀⠁⠉⠙⠃⠟⠋⠉⠉⢉⣺⣟⠀⠀⠈⠘⠀⠃⠀⠀⠃⠀⠀⠀⡇⠀⠀⠆
⠀⠀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣬⠀⣀⠀⠀⠐⣶⣖⣶⣷⡏⠈⠒⠬⠩⠉⣐⡲⠤⢤⣤⣤⡾⠉⠀⠀⡀⠀⠀⠀⠀⠀⠌⠀⠀⡈⠀⣧⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢼⠀⠀⢯⡤⣙⡄⢻⡗⣰⣭⣽⣡⠷⣒⣒⡓⠒⢄⠀⢁⠀⡶⢹⠋⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠐⠁⠀⣸
⠀⠀⢀⠤⢄⣀⣀⣀⠀⠈⠂⠀⠀⠀⢸⡏⡯⠖⠻⡾⠁⠈⠛⢷⣿⣿⣫⠄⠁⠈⡇⠀⡼⠁⠃⠘⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠁⠠⠬⢅⠀⠀⠈⠉⠉⠉⠉⠹⣷⣝⡳⢦⡴⢒⠀⠀⢀⠈⠻⣿⣿⠵⢦⠁⡸⠁⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠀⠀⠀⠀⢸⡇⠀⢠⠃
⠀⠀⠀⠀⢤⠀⠀⠴⠀⠀⠤⠤⠀⠀⠀⠀⠙⠻⢿⣾⣥⣶⢟⡥⠂⢀⣾⣷⡿⡷⠾⠇⣀⣠⣦⢒⡆⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈
⠀⠀⠒⢠⣄⠀⠀⠀⠀⠀⠐⢲⣦⣄⣀⡀⠀⠤⠤⣍⣿⣿⡋⡴⠁⢸⢸⣿⠛⠛⠃⢄⡀⠈⠉⠙⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠒
⠀⠀⠑⠖⠛⠓⠛⠛⠛⠛⠓⠒⠶⠛⠿⠿⠾⢿⣿⣿⣿⣿⡈⣴⡀⢸⣾⣿⣿⣶⡦⠤⠤⠀⠀⠒⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠂⡰⡀⠰⢻
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣹⡥⠈⣁⡿⠺⠿⣖⡴⣾⣖⠥⢲⣤⣀⡀⠀⠀⠀⠀⠀⠂⠈⠈⠛⠣⡀⢸⠄
⠀⠀⠊⠀⠱⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡼⢇⣰⣽⡣⠃⠀⠀⠎⠙⠳⠭⣻⠖⠉⢣⡉⢉⡀⠉⡆⠀⠈⠀⠠⠀⠀⠢⠘⠃
⠀⠀⠀⠀⠈⠀⠀⠀⡤⢤⡀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⣁⣭⣴⣦⣽⠀⠀⡜⠀⠀⠀⠀⠈⠙⠲⢅⡊⠓⠿⣶⠄⣀⠀⣄
⠀⠀⠀⠁⠀⢁⠀⠀⠀⠀⢃⠀⠀⠁⠀⠒⠐⠒⠢⠀⠀⠀⡜⠁⠀⠘⠻⣄⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢔⡢⣝⠍⠳⠦⠸⡌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠓⠢⣀⢀⣀⣤⡾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠨⠛⡁⠢⠐⠄⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠸⠦⠒⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⠀⢆⠤⡀⠩⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠓⠚⠛⠓

⠀⠒⡖⠓⠀⠀⠀⠂⠀⠀⠐⢠⠀⠀⠲⣶⠢⢒⢺⠖⡖⠒⠒⠓⠒⠒⠂⣲⢶⠒⠲⢺⠓⠚⠒⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢸⢰⡝⠀⣆⠀⠀⠀⠀⠀⠀⢠⣿⠶⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠈⢸⡅⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠘⣷⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣴⣿⡑⠀⠀⠁⠈⣽⡆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡅⢀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⣸⣿⣿⡇⠀⠀⠼⠰⡞⠃⠀⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⠀⠀⠀⠀⠲⠀⠀⠁⠀⢸⡿⣿⣿⣿⠀⠀⠘⠀⠁⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⡇⢻⣿⡟⡀⠀⠀⠀⠀⠀⠀⡧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠘⠃⠀⠀⠀⠀⠀⢰⠀⠀⠉⠀⠀⡧⠀⢻⢻⡇⠀⠀⠀⠀⡄⠀⣷⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣬⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠋⠀⢠⡏⡽⠄⠀⠀⠀⠁⠀⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠰⠀⠀⣿⠃⠀⠂⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠀⢰⡄⢀⣿⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠀⡿⡁⡂⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠦⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡦⣄⠰⠦⠤⡤⠐⣶⣶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡄⠆⣸⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⣬⠀⢀⠀⠃⠀⠉⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⣸⣛⢻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠁⠀⡁⠀⠀⠀⠀⢸⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣿⠀⣿⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⢸⡋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣹⠇⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⢰⠀⡇
⠐⠒⠒⠒⠒⠂⠀⠀⠂⠂⠀⠀⠛⠘⠓⠤⠤⠤⠴⠤⠤⠶⠤⠤⠲⠂⠓⠚⠦⢄⣀⠀⠀⠰⠿⠛⢸⠻⠟
⠀⠀⠀⠀⠀⠀⠤⠟⠷⠤⠶⠦⠤⢤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣿⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠆⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⣼⡇⢠
⠀⠀⠀⢦⡠⣾⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⣿⡆⣰⠇⠀⠀⠀⠀⠀⢻⡌⢿⣌⣧⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⣰⡟⣰⡏
⠀⠀⡀⠘⣷⣿⣷⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣿⣿⠏⢀⡀⠀⠀⠀⠱⣌⢻⣾⣿⣿⣧⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣾⣿⣿⣴⠟⣴⠃
⠀⢢⡙⣦⣬⣿⣿⣯⣿⣿⣿⡿⣿⡄⠀⠀⣾⡿⣿⣿⣿⣯⣿⣿⣯⣴⣞⡴⠁⠀⠀⠠⡙⢷⣽⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣵⠞⣁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣈⡳⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡷⢞⡡
⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⣛⣉
⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠉⠁
⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⡇⠀⠀⠀⠛⠁⠀⠀⢀⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⡿⣿⣿⣷⣦⠘⠃⣴⣾⣿⣿⢿⡿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢐⠀⠀⠐⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢞⡽⠛⠋⠉⠈⠀⠉⠙⠻⢏⠣⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⠄⠃
⠀⠀⢀⡀⣆⣶⣤⡀⠀⠀⠀⠀⠀⠐⢁⠐⠀⠀⠀⠀⠀⠀⣤⣰⣴⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡅⠂
⠀⠠⣌⣷⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣣⡄⠀⠀⠀⠀⠀⠀⢠⣠⣿⣠⣾⣴⣾⣴⣶⣤⠈⢀⣴⣶⣼⣶⣽⣦⢸⣧⣤
⠀⠠⣭⣻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣟⣫⡄⠀⠀⠀⠀⣤⣲⣾⣿⣿⣿⣿⡿⠛⠋⠉⠁⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣿⣶⣠⡄
⠀⠠⣘⣿⣿⣿⣿⣿⣶⣾⣶⣷⣆⠀⠀⠀⢠⣶⣾⣶⣶⣾⣿⣿⣿⣿⣏⡤⠀⠀⠀⢠⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡤
⠀⠠⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⠤⠀⠀⠀⢦⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣵⠆
⠀⠀⠀⠲⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠶⠀⠀⣹⣿⣿⣿⣿⣿⣿⠿⠖⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⡇⠀⠠⠀⠰⣿⣿⣿⣿⡿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⢀⡀⠀⠀⣀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠾⠂⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣟⣿⣿⣿⣄⣀⠀⢰⠀⠀⣀⣼⣿⣿⣟⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡻⣿⢿⣿⠟⢁⡄⠰⠀⣄⠙⢿⣿⢿⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠚⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠘⠃⣘⠀⠋⠀⠀⠁
⠀⠀⢠⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠠⠇
⠀⠀⠈⣧⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣾⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠
⠀⡀⢦⡹⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⡟⣰⠃⡀⠀⠀⣀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢂⡀
⠀⠹⣌⢷⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣣⡞⠀⠀⠠⣝⢷⣽⣦⣀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⢀⣠⣾⣷⣛⡄
⠀⠀⢌⣿⣿⣿⣿⣿⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣳⣿⣿⣿⣿⣫⠄⠀⠀⠠⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡤
⠀⠀⠀⠙⣿⣿⣿⣿⣭⣿⣿⣷⣤⠀⠶⠀⣠⡶⣿⣿⣯⣽⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⠙⢛⣛⣿⣿⣿⣿⣿⣿⡃⠀⠀⠶⠂⠀⠈⣻⣿⣿⣿⣿⣿⣟⣛⠛⠉
⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⢻⡧⠀⢸⣿⢾⣿⣿⣻⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠩⠾⣻⠿⠻⠿⠃⠀⠀⢰⠀⠀⠀⠻⠿⠻⢿⠻⠮⠉⠉⠁
⠀⠀⠀⠈⠙⠻⠿⢟⣿⡿⠟⠛⢻⡅⠈⠈⣟⠛⠛⠿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡨⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⢸⠇⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠿⠀⠀⠀⠀⡆⢸⠇⠀⠀⢠⡀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⠀⠀⠀⢰⠃⠉⠀⡼⠀⠀⣇⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠁⠀⢠⡏⠀⠙⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⣶⠀⠀⢰⡀⠈⠁⢠⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⡟⠀⠀⠈⣷⠀⠀⢸⡄⠀⠇⠀⠀⣀⢠⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠸⠇⢄⠀⡄⠀⠀⠀⠈⠁⠀⠀⢠⠀⠛⠈⠀⣧⣰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⢀⠈⠀⠁⡀⠀⠀⠚⠀⠀⣆⠛⠀⠀⡇⢸⢿⠋⠀⡀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⠋⢸⠃⠀⠀⠃⠀⢰⣆⠑⡆⣼⠘⠀⠀⠙⠀⠀⠄⢠⠀⠈⠀⠠⡇⠸⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠈⣷⢈⡁⡀⠀⠀⢀⠘⠛⠀⠁⠻⠀⠀⢠⡀⠀⠀⣄⠀⠰⠀⠀⠀⠃⡿⠀⣆⠸⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⠐⡋⢻⠃⣷⠀⠀⠚⡀⠀⠀⠀⠀⠀⢰⡌⡅⠏⢠⠟⡄⠀⠄⡀⠀⠀⣀⠠⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡸⠇⠑⠴⡄⢡⣲⠐⣄⡇⠀⢀⠀⡄⢈⠀⡀⠿⠀⡾⠠⣠⡇⠸⠇⢀⡜⠃⠀⠀⢰⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⠂⠓⡆⣠⡄⠀⢧⣴⣾⡿⣶⡔⡌⠇⠀⠘⢠⡇⠀⠀⠀⠀⠋⡠⡖⣠⣸⣧⡘⠁⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡦⠘⣄⠀⠈⠛⠈⠃⠙⢟⠃⣿⠿⣦⠀⠶⠀⠈⠓⣦⠀⠀⠐⠈⣷⠏⣻⢁⣼⡿⠋⢀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣤⣌⠀⡮⡄⢈⣗⣦⡼⢆⢠⡄⠻⠃⠀⠀⠊⡰⣿⠄⠈⠀⠰⢷⠠⠉⠈⠈⣴⢒⡊⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢈⣛⣠⠤⣿⣄⡧⢴⣾⣿⡹⠇⢸⣧⢱⣄⡶⠀⠆⠁⠁⠁⢠⣄⠀⢁⣈⡀⢻⣧⢀⣿⡌⠃⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⠁⠀⠀⠀⢹⣿⣆⡈⠯⠀⣀⣀⠀⠈⠛⢡⣿⣧⠤⠓⠠⢾⣿⣦⡈⠿⠃⢘⠡⢜⣿⡛⠁⠀⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣠⣾⡾⠁⠀⠀⠀⣨⡿⠂⠒⠒⢿⠏⠀⠀⠀⠀⠀⠙⢿⡟⠈⠁⠙⣄⠀⠈⠙⢿⣯⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⡽⠯⠽⠿⣵⣠⢤⣄⣾⡿⣇⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⣠⣿⣷⣤⣠⢔⣿⡆⠀⣠⡴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠐⠾⠶⠛⠿⠧⢯⡥⠿⣿⣓⠠⠄⠂⠾⠛⣋⠿⠷⠿⠿⠳⢿⠶⠾⠾⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣷⠄⠼⠇⢸⣦⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠛⢀⡄⠀⣠⡦⠀⣡⡐⠿⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⣉⠁⠀⠾⠇⠀⠙⠃⡘⠿⠀⣀⡸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⢿⠆⢀⣶⠀⣶⡄⠀⠳⡦⠰⣿⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣶⣶⠘⠛⠀⢰⣀⠈⣋⠀⡄⠀⠴⣷⠀⣄⠀⢰⣤⠉⢡⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣦⡀⠀⠁⠖⣤⣬⡈⠉⢐⣛⣀⡛⢀⣀⣤⠈⠉⢀⡌⠁⡠⠻⠟⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡄⢤⣆⠀⠉⢋⣀⣹⡃⠀⠀⣻⠄⣩⠖⢋⣀⣛⡃⠘⠃⢀⣾⠈⠉⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⢁⠈⠹⣆⣀⣀⠉⣷⢰⡟⢸⡇⣼⠉⣀⣀⣀⢾⠇⡈⠩⠘⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠃⢾⡇⠀⢽⠋⠉⠈⠉⢿⡟⠘⣿⣾⠃⢻⡿⠉⠉⠉⡙⣻⠄⠿⠄⠸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⢠⣈⡓⢦⠈⠷⣤⣀⠘⢿⣄⢹⣿⢠⡾⠃⣀⣤⠼⠁⢛⡆⠠⡾⢰⡶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠉⢠⣌⠷⠶⠞⠛⠳⣼⣯⣸⣿⣾⣧⠞⠛⠛⠶⠶⠋⠶⠂⢰⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⠖⢄⣩⣤⡾⠿⢿⣶⣿⣿⣿⣿⣿⣿⣴⠾⠿⢷⣤⣤⠖⢱⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠃⠴⠇⠰⡿⠘⢿⣿⣿⣿⣿⣿⠋⠾⠂⠻⢦⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉



⣿⣏⢽⣟⡻⣿⣛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠀⠀⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⡶⣌⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢷⣶⡂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠠⠐⡾⠤⠀⠀⠀⠀⠰⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠉⠙⠢⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⡄⠀⠀⠀⠀⡠⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⣀⡴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠉⠣⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠛⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠛⣿⡇⣄⡀⠈⠑⠠⢄⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠈⠛⠳⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⠿⣰⣦⢀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠇⣀⠀⠀⠀⠈⠛⠉⣿⣿⢸⢿⣍⢸⢿⡄⠀⠀⠈⠙⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠀⠈⠁
⠀⠈⠰⠆⠀⠀⡀⢀⡄⢸⠀⡟⣴⣠⢀⠀⠀⠀⠈⠹⠿⣸⡉⣸⣾⠁⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠑⠀⠀⠴
⠀⠀⠀⠀⠀⠀⡖⠈⡁⠘⠏⣧⢻⣿⣿⠷⣦⡀⠀⠀⠀⠈⠛⠎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠔
⠀⠀⠀⠀⠀⢀⣋⡀⠀⠀⠀⠈⠘⠿⣜⡇⡟⢹⣷⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢨⡏⠐⠀⠀⠀⠀⠀⠀⠙⠃⡇⢸⠿⡏⢼⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢼⣇⣸⡏⠹⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣧⡄⢸⠁⡟⣴⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠀⣀⡠⠤⠐⠀⠈⠁⠀⠀⢁
⠀⠀⠀⠘⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠼⢀⣿⡇⣹⣼⡇⡄⠀⠀⢸⠀⠀⠀⠀⠀⢸⡟⠓⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢠⠀⠀⢄⠑⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣇⠀⠀⠀⠀⠘⠉⣧⡿⣾⣿⡇⠀⡦⠀⠀⠀⠀⠀⠐⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠁⢤⡀⠀⢄⠁⠄⠀⠀⠀⠀⢠⡟⣀⢹⠀⠀⠀⠀⠀⠀⠈⠁⠟⣿⠃⢰⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡆⠀⠀⠈⢿⣦⡀⠁⢄⠀⢀⠀⢠⡟⠀⠹⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠴
⠀⠀⠀⣇⠀⠀⠀⢴⡟⠁⣆⠈⠢⠀⠀⠈⠳⣄⠲⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠰⠀⠀⠀⠊⠈⠃⠠⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⣤⣤⠎
⠀⠀⠀⢨⡄⠀⠀⢾⠁⠀⢸⣷⣄⠈⠢⡀⠁⠈⠳⣄⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠺⣿⣿⠉⠉⠠⠖⢣⠎
⠀⠀⠀⢸⡓⠓⠀⠘⠄⠀⠀⣿⣿⣦⡀⠘⠢⡀⠑⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠒⠒⣦⠀⠇⢸⡇⠀⠀⠀⠀⠈⠻⣀⠀⠀⢠⠏
⠀⠀⠀⠘⣯⠤⠀⠀⡄⠀⠀⠾⣿⣿⣿⣄⠀⠙⢦⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣤⡾⠁⢠⠀⣿⠇⠀⠀⠀⠀⠀⠀⣼⡷⣠⠃
⠀⠀⠀⠀⢡⡄⠀⠘⢡⠀⠀⠀⠘⣿⣿⣿⡷⠀⠀⠑⢄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣾⠉⠀⠀⣀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡿⠱⠁
⠀⠀⠀⠀⠘⣤⣤⣴⣷⠀⠀⠀⠀⠘⣿⣿⡇⠈⢂⠀⠀⠑⠀⠀⠑⢀⠀⠀⠀⠀⢀⠁⠀⠀⡇⠀⠀⠀⠻⠀⣿⠀⠀⠀⠀⠀⠠⠏⣡⠃
⠀⠀⠀⠀⠀⠰⠆⠈⠁⠀⠀⠀⢀⠀⠘⣿⡀⠀⠀⠑⠄⠀⠀⠳⡄⠀⠑⠄⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠰⠀⢻⣦⡸⠶⢶⣶⣶⣀⠃
⠀⠀⠀⠀⠠⣄⣠⠴⠀⠀⠀⠀⠈⡄⠠⠼⡇⠀⠀⠀⠈⠢⡀⠀⠈⠦⡀⠀⠁⠀⠃⠀⠀⠸⠀⠀⠀⠀⠀⠳⣈⣠⠶⣶⠾⠻⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⠀⠀⠀⠀⠑⡄⠐⠀⠛⣦⣄⡰⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠛⠂⠒⠂⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠰⣇⡀⠀⠈⢻⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⣾⠓⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⣆⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣠⣄⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣔⣡⣈⣟⣿⣀⣠⣄⣀⣠⣀⣇⣠⣤⣄⣀⣀⣀
⣿⣛⣛⠻⢛⣿⢫⡙⡛⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣏⣿⣿⣿⣿⣿⣛⣟⣛⣿⣛⣛⣻

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⣿⡏⠀⠀⠀⠀⡀⠀⠀⠀⣸⡿⣿⡀⠀⠀⠀⠀⣿⡏⠉⠙⠻⣷⡄⠀⠀⣠⣾⠟⠋⠉⠛⢿⣦⡀⠀⢸⣿⠋⠉⠙⠻⣷⡄
⠀⣿⡇⠀⠀⠀⢘⣟⢀⡤⢺⣿⠁⢻⣧⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⠇⠀⣸⣿⠁⠀⠀⠀⠀⠀⢻⣷⠀⢸⣿⠀⠀⠀⢀⣿⡇
⠀⣿⣇⠀⠀⠀⣿⣿⠉⢀⣿⠇⠀⠈⣿⡆⠀⠀⠀⣿⡿⠿⠿⢿⣥⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣶⣶⣶⣿⠋
⠀⣿⣿⣤⣠⡄⢻⣿⠀⣼⡿⠛⠛⠛⠻⣿⡄⠀⠀⣿⡇⠀⠀⠀⢻⣷⢀⢹⣿⡀⠀⠀⠀⠀⠀⣸⣿⠀⢸⣿⠀⠀⠀⢻⣷
⠀⣿⣧⣹⣿⣁⣸⡟⢰⣿⠃⠀⠀⠀⠀⢻⣷⠀⠀⣿⣇⣀⣀⣵⣿⣏⣹⡇⠻⣷⣤⣀⣀⣠⣼⡿⠁⠀⢸⣿⡄⠀⠀⠈⣿⡇
⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠛⠛⠋⠉⠀⠉⣹⡇⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠨⠉⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⢰⡀⣀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣏⣿⢩⠇⠀⠀⠀⠀⠐⠀⠀⠘⢧⣿⡁⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⢤⣄⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠈⠉⠙⢿⣦⡀⠀⠀⠀⢀⣿⠻⣷⠀⠀⠈⢻⣧⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⠁⠀⠀⠀⠀⢹⣷⢀⣀⠀⣼⡏⠀⢻⣇⠀⠀⢈⣿⣇⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣃⣾⡏⠀⠀⠀⠀⢀⣾⣿⣜⠛⢰⣿⣥⣤⣽⣿⣦⣄⣹⡏⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢻⣿⠁⠀⠀⠀⠀⣰⣿⣿⡟⢀⣿⠏⠀⠀⠀⠹⣿⡉⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⡄⠘⠛⠷⠶⢶⠶⠿⠛⢹⠟⣆⠸⠟⠀⠀⠀⠀⠀⠻⠇⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠒⠒⠢⠤⣀⠀⠀⠀⢀⡤⠖⠉⠑⠒⠒⡒⠒⠒⠒⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⡄⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠙⠦⠘⠉⠀⠀⠀⣠⠴⠋⠀⣠⡀⠀⠀⠸⡄
⠀⠀⠐⠒⠛⠋⠁⠒⢺⣷⡖⠒⠂⠉⠀⠒⠒⠂⠈⠀⠒⠒⠂⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⡆⣈⣉⣁⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠠⡀⠠⡀⠀⠀⠳⣄⠀⠀⠁⠀⠈⠯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠐⢄⠱⡌⢳⡀
⠀⠀⠀⠀⠀⠀⣰⠒⡉⠓⢮⣇⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠃⠀⣇
⠀⠀⠀⠀⠀⠀⢿⠀⡡⠃⣸⠛⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠚⠗⠢⠽
⠀⠀⠀⠀⠀⠀⠈⠑⢲⡾⠁⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀⠀⣀⠴⠒⠂⠀⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⠤⠛⠀⠀⣀⣠⣄⣠⣤⠤⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠠⣇⡀
⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣤⠤⠤⢤⣄⣀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣤⠴⠇⠀⠀⠀⠀⠀⠙⠒⠒⠒⠲⠤⠴⠚⢧⠀⠀⠉⠛⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⣀⣀⡴⠦⣄⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣉⣙⣓⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠄⠐⠒⢀⣠⠤⡀⠀⠃⡀⣼⠇⣀⣤⣤⠄⠄⠀⠀⠀⠀⠀⠀⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠙⣯⣟⢻⣍⡓⢦⣶⡶⢦⣄⠉⠊⢢⠐⣯⣿⡐⢁⡥⢶⢖⡶⣐⡶⢖⡲⠂⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣊⠝⡳⢶⢭⣗⣟⣿⣯⣷⣪⠻⣿⣔⣟⣛⣿⣯⣍⡶⡚⢌⡂⢄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠴⣄⠌⡩⢉⡉⣙⣍⡀⣯⢌⣄⡩⠀⡀⣀⣑⣪⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡩⠩⠄⣤⣖⣷⡀⠤⠩⡙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡞⠈⠁⢸⡇⠈⠐⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠧⡀⠀⢸⡇⢀⠔⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣰⠞⢋⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡵⠾⣯⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠁⠐⡇⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠢⣸⡧⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⣡⡴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡸⢻⡿⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠱⣼⡷⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣖⣵⡶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢸⣿⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⢟⣣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣵⢻⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⢭⠉⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀⠀⣸⠄⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⣉⣤⣤⠤⠤⢤⣤⡤⠀⠀⠀⠐⠀⠈⢀⣉⣦⡠⠤⠤⠔⠒⠒⠊⣿⠙
⠀⠀⠀⠠⣶⣦⡀⠀⠀⠀⠀⠀⠈⠛⡏⠿⠀⠘⠛⠘⢸⠄⠄⠀⠐⠒⠣⣠⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⣄⢙⣷⡄⠀⠀⠄⠐⠂⠀⢻⢦⠀⠀⠡⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣈⣿⣿⡛⢿⡦⣅⠦⣀⡀⣴⠻⡄⠓⠦⠤⣤⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡾⠀⠈⢉⣻⠷⣬⡙⠷⣬⠶⠛⠲⡱⣄⠀⢠⠏⠀⣿⣿⣿⢖⠀⠢⢤⡄⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣃⡴⢚⣉⣀⣠⣤⠽⠋⠀⠀⠀⠀⠙⣿⡿⣫⣾⣶⣿⣿⡏⢸⠀⠀⠀⠙⡇⠘⠇⠉⠉⡗⠒⠲⠦⠤⠿⠤
⢠⠞⠋⠉⠉⠀⠀⠀⡇⠀⣀⣠⣴⣎⠉⠉⠘⢧⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⢷⠀⠀⠀⠀⡇
⡏⠀⠀⠀⠀⠀⢠⡐⣻⡿⠁⠀⠀⠙⣷⢦⣸⣿⣿⢿⣿⣿⣿⠀⢸⠤⣄⣀⡀⠀⠀⠀⠀⢠⠇
⠹⣄⡀⠒⠲⣾⣿⠿⢻⡇⠀⠀⠀⠀⡏⢳⣝⢝⠉⠳⢍⡿⢷⡲⠾⡄⠀⠀⠉⠉⠓⠒⠚⠁
⠀⠀⠉⠉⠉⠉⠀⠀⠸⣇⠀⠀⠐⠉⠀⢀⡿⣧⡑⠄⠀⠀⠀⠹⡄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠈⡟⢹⡆⠠⢰⠓⣤⣇⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⢀⡎⢠⠀⠈⠀⢰⣧⢧⠘⣼⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣼⣇⣿⠀⢀⣷⣸⣿⣆⠀⠈⠛⢛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠛⠛⠷⠶⠾⠿⣿⣿⣿⣿⣶⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣷⣷⣧⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠌⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⠂⠄⠀⠀⠀⣴⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡆⠀⠀⠀⠰⡟⢻⠳⠀⠓⠀⢶⠀⠘
⠀⠀⠀⠠⠄⢀⢄⠀⠠⢠⣀⠔⠒⠂⠠⠄⠀⠀⠀⣿⡇⠀⠀⠐⠐⠀⠀⠡⠡⠀⠐⠛⠂⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠸⠈⠀⠠⠠⠀⢄⠀⡠⠈⠐⠰⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠃⠀⢰
⠀⠀⠀⠠⠘⠃⠴⠀⠀⠀⠀⢨⠀⠀⠀⠘⠆⠀⠀⡻⣇⠀⠀⠨⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠁⠉⠀⡄⠁⢀⠑⡀
⠀⠀⢀⡰⢐⢀⠜⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⢈⠀⠀⢸⡀⡇
⠀⠀⠃⢑⠂⠀⢖⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⡈⠀⠄⠁⠱⠄⠀⠀⠀⠀⠀⠔⠁⠱⠄⠀⡀⠀⠩⣀⣠⡤⠄
⠀⠊⠠⠁⠀⠀⠀⠠⡀⠂⠀⠀⠀⠀⠀⡞⡆⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⡇
⡀⠀⠁⠀⠀⢀⠀⡀⢀⠀⡄⢠⠀⠀⠀⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⡀
⡇⠐⠀⠀⠀⡀⠒⠰⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠄⠤⠀⠄⠀⠀⠀⡇⢸⠉⡇
⠐⠈⠀⡀⠀⠨⣾⠁⠀⠐⠀⢐⠀⡀⠀⠿⠄⠀⠀⠉⢠⣀⡀⣀⢼⠈⠁⠋⢸⠉⠁⢹⢀⡀⣀⡈⠈⠁⠀⡇⢸⠤⠃
⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⡀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⡇⢸
⠀⠀⠀⡇⠀⢘⣿⠀⠀⠈⠀⠀⠀⠉⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⡇⣸
⠀⠀⢸⣗⠀⢈⠿⡀⠀⠀⠀⢀⡤⠀⠀⢠⠀⠀⠀⠐⠒⠒⡒⠒⠒⡒⠓⠓⠚⠒⠒⠒⠚⠒⠓⠒⠒⠀⠀⠻⠟
⠀⠀⢸⣷⠀⡁⠒⢰⠀⠀⠀⠀⡄⠀⠀⠀⠀⡈⠀⠃⢠⠁⢈⠀⠋⢈⠀⠃⠀⡀
⠀⠀⠸⠇⠀⠈⠀⠁⠀⠐⠀⠄⠀⠀⠀⠠⠀⠣⠐⠣⡾⠀⠾⢼⠀⠸⠜⠁⢰⠗⠁
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⡝⠀⠀⠐⠁⠎⠠⡀
⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠘⠁⠀⠐⠃⠀⠀⣄⠘⠀⠄⠓⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⠄⢠⠇⠲⢤⠂⢰⣠⠄⠀⠄⠙⠂⠀⠀⠀⡄⠀⠀⠀⡤⠐⡆⠠
⠀⠀⠀⣀⣀⠊⢀⠴⡀⢠⡀⢾⣀⠀⠠⣀⠨⠁⢀⢠⡠⠁⢀⠀⢀⣀⠀⠀⡠⠂⠅⢀⡄⣤⠀⣰⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣆⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠇⢠⠀⡁⠀⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠈⠀⡀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣃⠀⡀⢄⡄⡀⠀⠈⣠⠠⠀⠄⠀⠀⠀⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠑⠀⠀⡀⢀⡈⠀⠀⠀⠀⠀⠠⠀⢈⠀⠀⠀⠀⠀⠘⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠠⢐⡬⡄⠀⠀⠂⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⣤⠈⠀⠀⠤⠀⠠⠀⠀⠈⠁⠀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⠀⠁⠀⠻⠇⠀⠀⠀⠀⠀⢀⠁⠀⠐⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠰⠖⢶⠂⠀⠐⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⣀⠄⠀⠂⢄⠈⠁⠐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠨⠁⢂⠒⠀⠀⠀⠂⠋⠡⠀⠀⠄⢁⢀⡀⡄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠣⠀⡈⠀⠀⠈⠀⠀⣀⣤⠆⠁⠀⠀⡰⠀⠌⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⢰⠂⠀⠀⠆⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⠧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠁⡀⠈⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⡀⠀⠀⢀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⡐⢀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣠⠤⢠⠄⢌⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⡔⣉⢀⠄⠀⠼⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠐⢀⠃⠀⠀⠐⠀⠀⢀⠳⠀⠀⢁⠀⠎⠀⠀⢆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠓⠀⡌⠀⠀⠀⢀⠀⠈⠡⠵⠀⠀⠨⠌⠮⠤⠤⠤⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠀⠀⢀⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⡀⠀⠀⠄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠃⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠄⠀⠐⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡜⠀⠈⠄⠀⠐⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠈⠈⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣿⣋⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡷⠄⠀⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣡⣴⠛⢤⡖⢈⣥⣒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣵⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠿⣿⣿⣿⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢹⣫⣤⠀⠈⠃⠈⢣⣿⠛⠻⣆⠀⠀⠀⣀⣀⡤⠴⠖⠒⠒⠶⠤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⣿⠀⣶⢬⡗⣀⣼⣿⠋⣰⠊⣠⡂⠀⢐⡀⠀⠠⠄⢈⣍⢑⡲⠦⠴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⢀⣾⣏⣿⣧⣎⠙⣿⣿⡿⢲⠇⣠⣍⢀⣈⣉⢁⣟⠃⣒⢶⣶⢉⡠⠒⠋⠙⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠚⠉⢿⣿⣯⡞⢡⣴⣿⣿⣿⢸⡆⢬⣄⠀⣤⡤⢼⣏⢨⣿⣿⠻⡏⠀⠀⣠⣾⣆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡯⣤⠟⠏⢠⣿⣿⣿⣿⣿⣳⣶⣶⣶⠖⠛⠋⠘⠛⢿⡿⠀⠀⠀⢀⡼⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢋⣿⢉⡠⡖⠿⠈⢻⣿⣿⣿⣝⣿⣿⣿⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡝⢸⣿⡟⣁⣴⣴⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣷⡀⠀⠀⠀⠀⠀⢀⡴⢋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣦⣥⣿⣿⣿⠿⠻⠟⣻⢷⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣨⣿⢻⡗⣶⣾⡿⠙⢳⠙⣿⣿⣿⣿⣿⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢳⣿⣿⣾⠇⢹⣷⣴⣿⣥⣶⣿⣿⣿⣿⣿⣦⢶⣶⡚⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣿⣻⣿⠿⣿⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⢩⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⣭⡿⣟⠳⣿⠀⠀⠈⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⡿⣿⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⢡⠁⠉⢻⡽⠀⠀⠀⣼⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣳⣏⣾⣫⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣾⣿⠟⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⣿⠟⣁⣴⣿⣿⣿⣿⣿⣿⣿⣏⢿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠁⠀⠰⠏⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢆⣿⡸⣿⣿⡿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣦⡃⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣷⣿⢟⣿⡸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢫⣃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢸⣧⠈⠙⣽⣿⣻⡇⠘⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⢠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⣿⣆⠀⣹⣿⣿⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣢⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢹⣿⡷⡉⠁⠁⠀⠀⠀⢨⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⣿⣗⢱⡄⣤⣄⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣛⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⣾⣿⣷⣾⣟⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣟⡹⣿⣤⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⡤⠁⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⣿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣧⣤⣤⣼⣯⣭⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⣠⡆⠠⠸⠂⠉⠉⠉⠉⠑⡆⢀⠀⠳⣤⠀⠀⠀⠈⢆
⠀⠀⠀⢀⠀⠀⠘⠀⢰⠁⠀⢀⠠⠄⠊⡹⠋⠃⡇⠀⢧⠀⠀⠀⠀⠀⠀⠱⠀⠣⡄⠙⢞⠀⠀⠀⠈⡆
⠀⠀⠠⠁⠀⠀⠀⡀⡸⢠⠊⠀⠀⢀⠜⠁⡆⢠⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⣆⠀⠈⢢⠀⢳⡀⠀⠀⢱
⠀⠠⠂⠀⠀⠀⠀⠉⣠⠃⠀⠀⣰⠏⠀⠀⠇⣾⡇⠤⠀⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠱⡄⢃⠀⠀⠈⡆
⠀⠐⠀⡀⠠⠀⠀⠘⡇⠀⠀⢀⡏⠀⠀⠀⡎⢸⡇⠀⠀⡄⢆⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠈⢾⡀⡄⢀⢱
⠀⡄⠘⢄⠠⡀⠀⠀⢱⣄⢰⡿⠁⠀⠀⢸⠀⢀⡇⠀⠀⡇⠈⡄⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠹⡖⢔⣿
⠀⡄⠀⠀⠀⠈⡆⠀⢸⣿⡾⠁⠀⠀⠀⢀⠀⠘⡇⠀⠀⢇⠀⠘⡄⠀⠀⠀⠠⣔⣀⠀⠹⡄⠀⡀⢸⡄⠉
⠀⢱⠀⢰⠀⠀⢳⣤⣲⡟⠁⠀⠀⠀⡠⠋⠠⡾⠇⠀⠀⠳⣤⢤⠱⢤⠀⠀⠀⠠⠄⠁⠀⠙⠢⣅⣃⡇
⠀⠈⠒⡇⠀⠀⠀⠉⠓⠃⠀⠀⠀⠈⢠⡎⠀⡗⠀⠀⠀⠀⠡⣈⡘⠁⠠⠀⣁⠤⠀⢀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠀⢸⠃⠈⠁⠈⠙⠀⠂⠀⠤⢀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⠀⠀⠀⢀⣌⠀⠀⠀⠀⠠⣯⣈⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⢀⣽⢧⡼⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠊⠉⠚⠣⠴⠊⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⣸⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢨⠓⠉⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠇