Taco mushroom Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢀⡔⠁⠈⠉⠁⠐⠐⢌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢢⡤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢈⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⢰⡶⠒⠲⣄⠀⠀⣀⡀⠀⢸⡀⣸⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠀⠀⠀⠀⠀⡷⠤⢴⡏⢀⣞⣉⣹⣆⣸⡃⣹⡇⠐⢋⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⣇⣀⣠⡿⠈⣯⣉⣉⣍⣿⡇⢸⡇⢸⣉⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠂⢀⠈⠉⠉⠀⠉⠀⠈⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠠⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⢠⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢺⡄⠀⠀⡾⠀⠓⢀⣴⣈⠁⣐⣤⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢷⡀⣸⠃⠀⡷⠈⣿⢍⠐⢛⣉⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠘⣷⠿⠄⠀⣿⠀⣿⡀⠸⣏⣀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣧⠀⡀⢠⣌⠀⠀⠈⠁⠠⢈⠁⠀⠈⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠤⣀⠀⠀⢀⡀⢤⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⢰⣷⡀⠀⠀⣼⡇⠀⡀⠀⢉⠁⠀⣀⣀⠀⢸⡧⣀⡈⠐⢀⢀⡀⠀⣈⣃⠀⠀⠀⣀⡁⡀⢀⡀⣀⡀⢀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⡏⣧⠀⣼⢻⡇⢸⡇⠀⢸⡇⢸⣏⣙⠓⢸⡟⠉⢹⡄⢸⠋⠁⡾⢋⠙⣷⢱⡿⠙⠙⣇⢸⡏⠉⢻⠋⠉⣷⠀⢿⣉⡙⠂
⠀⠀⠀⠀⠀⢸⡇⠘⣷⠇⢸⡇⠘⣧⣀⣼⡇⢴⣌⣹⣿⣸⡗⠀⢸⡇⢸⠃⠈⢿⣕⣠⡿⠘⢧⣀⣱⡏⢸⡇⠀⢼⠀⠀⣿⠀⣦⣈⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠠⠉⠉⠀⠈⠁⠀⠈⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠁⠀⡀⠄⠈⠙⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠘⢀⠀⠀⠕⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢣⠊⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠙⠛⠛⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣁⣠⣠⣾⣿⣿⣿⣦⣠⣀⡉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⣼⣿⠟⢿⣿⡇⠀⢿⣿⠟⢿⣿⡄⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣠⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⡀⢸⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⣟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣷⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⢠⣿⡟⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣀⣾⣿⠃⠀⢿⣿⣦⣤⣀
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣦⣄⡀⠉⠛⠻⢿⣿⣦
⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣄⠀⠀⠙⣿⣷
⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⠀⢹⣿⣧⣀⡀
⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⢼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠙⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿⡆⠀⠀⠀⢼⣿⡇⠀⠀⠀⢾⣷⡄⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⢀⣹⣿⣷⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⠿⠛⠛⠛⠛⠛⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣀⣀⣶⣿⣿⣷⣤⣤⣤⣈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠇⠀⠀⢾⣿⡄⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⢿⡿⠛⢻⣿⡆⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⣼⣿⠇⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠉⠀⠀⠀⠀⠀⣿⣿⣿⣦⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⡁⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡄⠀⠀⠀⣾⣿⠃⠀⠀⠀⣿⣿⣠⣶⡄⠀⣾⣷⣦⣾⣿⠃⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣶⣿⡿⠛⠋⣡⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⣿⣿⣄⣀⣉⣉⣉⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⢸⣿⡟⠿⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠂⠀⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⠠⠀⠐⠊⠐⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠉⠀⠉⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡩⠀⠀⠀⠀⠀⠙⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⣀⣤⣤⢶⡶⢶⣶⡶⡶⠶⢶⣤⣀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣴⣛⣩⢎⢦⢺⡧⠀⢸⣿⢜⣜⢿⣿⣿⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠻⣵⣳⣽⣎⣎⡏⠀⠈⣿⣼⣼⡿⠿⣋⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⠀⠀⠙⠳⠦⣬⣭⣭⣽⡇⠀⠀⣿⠥⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠁⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⡟⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠭⠉⠀⠀⠀⢀⡹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢸⠇⠀⠀⣀⣤⡶⠟⠃⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣀⣤⣤⣤⣄⣀⣄⣀⡀⠉⠻⣆⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⠀⢠⡾⡯⡱⠁⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⠿⢷⣷⣿⠅⣸⣯⣯⣿⡷⠀⣼⠀⠀⠀⠀⣰⠇⠀⠀⢰⠇⠀⢹⡑⠄⠀⠀⠀⠀⠀⠀⠀⠈⠝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⡏⠀⢹⡭⠭⠥⠶⠚⠁⠀⠀⠀⢠⡏⠀⠀⢀⡟⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠘⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢈⡿⢛⣶⠶⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⡞⠁⢠⡏⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⡄⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡟⠀⠀⠀⠀⠀⣠⠟⠀⢀⡞⠀⣠⠴⠶⠟⠋⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢻⡄⠀⠀⠀⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⠀⣰⠏⠀⢀⡾⠁⢸⡇⣤⡦⠤⣤⣀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⠳⣄⡀⠈⢧⠀⠀⠹⣆⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠙⢮⣙⣾⠏⢨⣟⡶⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠉⠛⣶⡌⢧⡀⠀⠘⢷⡀⢸⡿⠀⠀⠀⣿⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⠀⠀⣼⠋⢀⡿⢭⣝⣩⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠊⠀⠀⠀⠀⠀⠀⣪⠟⠈⢷⡀⠀⠀⢻⣼⡇⡄⠀⢂⣿⠀⠀⠀⣾⠃⠀⣰⡏⠀⠀⠀⢀⣤⠞⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⣤⠤⢶⡖⠋⠁⠀⠀⠈⢷⡀⠀⠐⣿⣇⠇⡠⣺⣻⡇⠀⢰⡟⣇⡇⣿⡇⠀⣠⡶⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⢳⡀⠀⠀⠀⠀⠘⣧⢂⡜⡼⣿⢸⢰⢹⢻⣷⠀⢸⣷⣼⡇⣿⣧⠞⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢶⣤⣤⣤⣄⢹⡼⡰⠐⣻⡏⠎⣇⢏⢻⣇⢸⣿⡿⡇⣾⣿⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠨⣒⣫⣭⣽⡏⣷⠇⠃⠩⢿⣜⢎⠎⢢⡟⢸⣯⠰⢃⢻⣷⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠛⠁⢹⣶⢸⠀⣼⡟⠛⠛⠋⠀⠀⢻⣜⣌⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠾⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠚⠂⠀⠔⠉⠉⠙⠛⠛⠛⠿⠿⣿⠿⠿⠿⠿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⡀
⠀⠀⠀⠀⢀⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⣴⡿⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⢻⣿⣶⣿⣤⡆
⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⡀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣷⠀⠂⠘⣿⣿⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⢿⣿⣿⡄⠀⠀⠛⠋⠀⣶⣾⡧⠀⠀⠀⠀⠀⣠⢶⣬⠿⣿⣿⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⠀⣴⠦⣽⣿⣿⠟⠋⠁⠀⠀⠀⣴⣶⣀⡀⠿⠻⣿⣥⡼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⢀⠿⣿⣿⠀⠀⠀⠀⠰⢿⡶⢯⡈⠁⠀⠀⠈⠁⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⡆⠀⠀⠀⠰⣶⠄⣈⡁⠀⠸⣦⡀⣰⣦⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣹⣿⣿⠀⠀⠀⠀⠀⠈⠉⠀⢠⠄⠈⠃⠙⠃⣹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠿⣶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠐⠦⣤⡀⠀⠀⠐⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠉⠛⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣹⣿⣇⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡼⣿⣿⡄⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⣿⣿⣧⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⣿⣿⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣆⢾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⢫⡸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢶⠟⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠟⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣈⣽⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠖⠦⣤⣤⣤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠂⣀⣀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢀⠴⠟⠋⠁⠙⢻⣦⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢀⠰⠒⠀⢠⠀⠀⠀⠀⠀⠈⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡴⠞⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⢴⣾⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⢀⠀⢀⣰⠶⣆⠀⣠⣾⣾⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠒⠙⠛⣛⣻⣿⣇⠀⢸⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠛⠸⠀⠸⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢁⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⠃⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡼⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠈⠛⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣤⣤⣤⠬⡁⠀⣀⠀⡀⠀⠈⠀⣼⠃⢀⠀⠀⠀⠈⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢾⠀⢹⠾⠋⠑⠛⠀⠀⢷⠿⠊⠀⣀⣄⣈⠈⣻⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠽⣿⠇⠀⢻⡯⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣌⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠹⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠧⡄⢠⡼⠃

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⣚⠿⠛⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣤⣲⡟⣿⣷⣿⣿⡿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠀⣠⠤⠒⠟⡠⠖⠉⠀⢀⢀⣀⣀⠀⠈⠆
⠀⠀⠀⣠⣴⣿⣟⣿⢏⣿⣿⣟⣽⡿⣿⣿⣿⣿⣿⣾⣶⡝⣿⠆⠀⠀⠀⢠⠖⠋⠀⠒⠒⡼⠳⠁⡠⠊⢀⠄⠀⠁⠀⠀⠁⠈⢱⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⢀⠄⠉⠀⡈⠀⢀⡀⠀⡀⠀⠐⠁⠀⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠰⣶⠄⠈⠓⠀⢀⠀⠈
⠀⠀⠀⠻⣿⣽⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠝⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠀⢒⣰⢤⢀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣴⢀⣰⣦⣄⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⠀⠀⠉⠝⡿⠻⠐⠢⢤⢤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⣾⣯⣿⠻⢿⣿⣿⣿⣿⣿⣏⣭⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠘⠇⠁⠁⢸⠉⢢⢄⠀⠀⠀⠐⠒⠞⠛⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣭⣿⡿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠉⠒⠠⣀⣀⣀⣀⣀⡀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣷⠶⠓⠋⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠑⠢⡄⠀⠀⢀⠀⢀⠀⣈⠈⠡⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠇⠀⠠⢠⠂⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡧⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢸⣿⣿⣿⣿⣿⣿⣷⣰⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⡀⣇⠀⡄⠀⡇⠀⠐⢌⡛⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡜⡏⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⢙⠻⢦⣻⣴⣇⣄⠀⢰⡿⠛⠋⡉⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠙⠿⠒⠃⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣰⡷⢀⣶⣿⣧⣜⣃⣀⢀⣤⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣠⣶⠖⠂⠠⣤⣄⡀
⠀⠀⠀⠀⠀⢈⣾⠃⡿⢽⣿⣷⠿⠃⠻⣉⣽⡱⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⡉⢁⣀⠀⠀⠀⠘⢭⣿⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣼⣿⣧⠛⢀⣾⣿⣿⣶⣂⣨⣽⣿⣦⣌⣉⠛⠶⡄⣾⠉⠀⣠⣿⢶⡄⠘⠀⠮⣿⣉⣬⣴⢤⣈⣿⣷⣤⣄⠀⢻⣶⣄⡀
⠀⠀⠀⠀⣴⡉⣹⣷⢢⣤⡌⠙⣩⣼⡿⠃⠀⠈⢙⠃⠉⢩⣀⡀⢻⠀⣿⠻⠃⠘⠃⠀⣀⣀⣤⣤⣭⣿⣧⣿⣯⣝⢻⣿⡆⢀⠈⠈
⠀⠀⠀⢰⣍⣿⣿⣿⣿⠿⣴⣟⣻⠁⠀⠀⠀⠀⠈⣁⡠⡄⠈⡉⣀⠀⢀⣴⡶⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣦⣾⣏⣹⣿⣷⣾⣷⣶⣶⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣶⣿⣄⣀⡜⣿⠈⠛⣦⠀⠁⠙⠿⢶⡉⠐⣇⣠⣄⠠⠘⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⡶⠛
⠀⠀⠀⠙⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣞⣯⡞⡀⠀⠀⣈⠁⢰⣿⣿⣿⣷⣶⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠐⣀
⣴⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣶⣶⠦⣀⡴⢶⣤⡑⢄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣴⡆
⣿⣧⠀⠀⣶⣿⣧⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⡀⢹⣷⣄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡇⠒
⣿⣿⣦⠀⠈⡛⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣅⠸⣿⣿⣧⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡦
⣿⣿⣿⣷⠀⣧⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢸⣿⣿⣿⣷⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢺⣁
⣿⣿⣿⣿⡄⠙⠂⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⡿
⣿⣿⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣼⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿
⣿⣿⣿⣿⣿⣿⣆⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⣿⠷⡶
⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠹⣿⣿⣿⣿⣿⣿⢿⣿⣯⣶⣿⣌⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡙⠛⠻⠛⠉⢹⣿⣇⠀⠈⠉⠛⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⡏⠠⠤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠈⡙⠛⠛⠻⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣌⡀⠺⠶⣮⣝⠷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡀⠀⠀⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⢷⣯⣙⣻⣶⣬⣉⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⣿⣿⣿⣿⣷⣷⡻⠿⣿⠶⣌⡙⠗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⢿⣿⣿⣷⣦⣅⡀⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⡈⠨⠙⠻⢿⣿⣷⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠐⠄⠀⠠⠒⠒⠒⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠩⢀⣀⠂⠀⢀⢁⠒⠈⠠⠄⠤⠤⣌⠙⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⢈⠀⡉⠉⠉⡀⠀⠀⣀⠀⠉⡰⠂⠀⠂⠀⠀⠀⠐⠦⠋⠀⠒⠄
⠀⠀⠀⠀⠀⡠⠊⠀⠐⡈⠀⠀⠀⠠⡝⠉⠀⠬⠁⠀⡀⠤⠐⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠜⠀⠀⠀⠄⢣⠀⠀⢈⠥⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⡀⠀⠊⡄⠉⠀⠀⢤⠀⠈⠀⢀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠈⢄
⠀⠀⢰⠁⠗⠈⠁⠀⠀⡀⠐⠠⠂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⡄⠀⡀⠀⠀⠀⠠⠄⠀⠠⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⢰
⠀⠀⠃⠂⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠰⠀⠀⡀⣔⠡⠒⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢃⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠⠤⠤⠀⠀⠐⠒⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠑⠢⠤⠤⠄⠒⠋⠉⠉⠉⠉⠉⠉




⣿⣟⣛⣟⣛⣻⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠾⠿⣿⠶⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠠⡦
⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⢀⣴⡾⠋⠉⢹⡆⠀⠀⠀⠙⢧⣤⣤⠿⠉⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣄⣠⡼⠃⠀⣴⠞⠛⢶⡄⠀⠀⢸⡏⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⣄⠀⠀⠀⠀⢀⣀⣘⣧⣀⣀⣀⣟⠀⠀⠈⠳⣤⡤⠞⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣠⣽⡶⣾⠛⢿⠛⠙⡏⠉⢹⠋⠉⢹⠛⠛⣿⠶⢶⣤⣄⡀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠙⢆⠈⠂⠈⠁⠀⠁⠀⠈⠀⠀⠈⠀⠀⠁⠀⠞⠁⣸⠟⠷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⡀⠀⡠⠀⠀⡀⠀⢠⡄⠀⠀⠀⢠⡄⠀⠀⠀⢀⠀⠠⣄⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⣴⣧⣤⣼⡇⠀⠀⠀⢸⣧⣤⣼⣦⣾⡷⠿⠛⠉
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠃⠀⠀⠘⡌⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⢰⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡌⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⡸⠀⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⠃⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣴⡟⠀⠀⠀⠈⢻⣿⣿⣿⣇⣠⡄⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢨⣿⣿⣶⣄⠀⠀⣸⣿⣿⣿⣿⡿⠃⣠⣤⣾⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣶⣾⣿⡿⠟⠛⠛⠁⠈⠛⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡟⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣧⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⡟⠋⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⣿⣿⣿⣿⣿⣯⣿⣿⣭⣭⣍⣍⣉⣉⣉⣉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣽⣷⣀⡀⠀⠀⢀⡀
⣿⠿⠿⣿⣿⡿⠿⠛⢋⣉⣴⣾⣟⡩⢛⣶⣿⢦⣷⣾⣷⣿⣿⣿⣿⣷⣾⣿⣿⡻⠧⣤⠀⠘⠛⠂
⠿⠅⠤⠀⠀⠀⠀⣺⣿⢭⣝⠉⠁⠀⠀⣶⣿⣿⣿⡟⢿⣿⡿⠛⠉⣿⣿⣿⣿⣿⠆⠁⠀⢠⣤⣶⣶⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣶⢀⡀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣟⠁⣀⣰⣿⣟⠋⠀⠀⠀⠘⢲⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠆
⠀⠈⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣦⣤⡀⣽⠟⢩⣿⣿⣿⣷⣿⡿⠛⠀⠰⠀⠀⠘⢿⣾⣿⠟⣿⣍⣹⣿⣿⣿⣿⣿⣿⣿⢁⣴⣆
⠀⠀⠀⠀⠀⠀⠀⢾⡗⢛⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⡿⠟⠁⠀⠀⠰⣦⡠⠤⠀⠐⠛⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣾⣿⣿⣧⣄
⠇⠀⠀⠀⠀⠀⠀⠀⠉⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠀⢀⣀⣀⣴⠊⠀⡠⢰⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦⣽⣿⣿⣾⣿⣿⣿⠟⠂⠀⣀⠀⠀⠀⢤
⠁⠀⠀⠀⠀⠠⠄⢀⣴⡆⠘⣻⣧⠀⢨⣿⣿⣿⡿⠁⠀⢀⣴⣿⣿⣿⠁⢀⣴⣥⣸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣛⣛⣛⠿⣿⣿⣛⢻⣷⣷⣽⣄
⠀⠀⠀⠀⠈⠁⠒⢿⣿⣷⣶⣿⣿⡿⠏⢹⣿⡟⢠⣾⣿⣿⣿⣿⣿⣿⣶⣿⣯⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣻⣽⢿⣯⣿⡁⣠
⠛⠓⠂⠀⠀⠀⢠⣤⠎⠛⠿⠿⠟⠀⢠⣿⡿⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣽⣟⡶⠿⠞
⠀⠒⠦⢤⣀⠀⠀⠀⠒⠶⠶⠶⣾⣿⣿⣿⣿⣼⣿⣟⡛⠛⠻⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠟⠛⠁⠀⢠⡶⢐⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠻⠿⠿⠿⠶⠶⠛⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⢟⡿⠛⡇⡀⢰⣤⡀⠆⣴⠷⣾⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠁⠀⠈⠑⠀⠀⡀⠀⠈⠁⠸⠀⠀⠉⠰⡄⠘⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⡇⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠐⠀⠀⠀⠠⡁⠀⠀⢹⡄⠀⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢸⣿⣧⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢻⡄⢃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⢈⠈⠉⡟⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠷⠁⠀⠀⠀⠀⠀⡆⣸⡀⣀
⠀⠀⠀⠀⠀⠀⣤⢠⠐⠏⠀⠁⠀⣶⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢧⡟
⠀⠀⠀⠀⠀⠀⣹⣟⠀⠀⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢖⡀⠀⠀⠀⠀⠠⠀⠀⢸⣧⣄
⠀⠀⠀⠀⠀⣴⣿⡏⠃⡀⠈⡂⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠰⠆⠀⠀⣀⣴⣷⣶⣾⡖⠒⠒⠀⣤⠀⡆⠀⢘⣿⣿⡀
⠀⠀⠀⠀⠀⣿⢿⠃⢀⠁⠀⠃⠀⠀⠀⠀⢠⣶⣤⠤⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣾⣟⢿⢯⣻⠛⠁⠀⠀⠀⡠⠁⠀⣸⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣾⠀⢈⠀⠀⠀⠀⠀⠀⢀⣾⠁⠛⣀⡴⢾⣿⣯⡙⢥⣸⣼⣿⣿⣿⣿⠕⢊⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡄⢀⠀⠀⠀⠀⢀⣴⣿⣷⣶⣛⣋⣭⣼⣿⣿⣿⣿⣿⣿⡿⠟⠃⢲⣾⣷⣿⣿⣷⣾⠖⠁⠀⠀⠀⠀⠀⠇⢧⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⢠⠀⠀⠀⢠⣿⣿⣿⣿⠏⠚⠉⠁⠐⠒⢩⡾⣫⣤⣤⣤⣀⣼⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣷⠈⠁⠀⠀⠘⠻⢿⣿⣿⣿⣶⡀⠀⣀⣾⣿⣿⣿⣿⣿⠁⠙⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁⠀⠀⠀⠀⢹⡔⠀⠀⠀⠀⠐⣌⠀⠂⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠤⣀⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⢠⠈⡇⠀⠀⠀⢀⡠⠎⠀⠉⠀⠀⠀⠀⠅⣠⢸⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⢈⡓⢀⠀⠑⠄⠐⡄⣀⠀⠀⠀⠀⠀⠠⡆⠀⣇⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠱⠞⢉⡿⠋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡅⠀⣾⠆⣀⠠⢡⣷⣆⠀⠀⠀⠀⠀⠇⠊⢻⣦⡀⠹⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡞⢶⣂⣘⣧⣼⣿⣷⣶⢿⣿⡦⣀⠂⠀⠀⣿⠀⠀⠈⠊⣹⠿⣿⡻⠒⠒⣢⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢉⣽⣟⣿⣵⡿⢿⣧⠆⠉⣙⣻⣄⠀⠜⣁⠔⠈⠀⣸⡏⠠⣿⣧⣬⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣰⠈⠁⣶⣶⣿⣿⣿⡞⣧⡀⣤⣴⣿⣧⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠻⠿⠿⠟⠛⠛⠓⠛⠉⠁

⣾⣿⣧⣾⠄⠀⠀⠀⠀⠀⠀⠠⣿⣮⠙⠃⠀⣠⣾⣿⡟⢟⣿⣷⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⠀⢉⡁⠀⠀⢰⣿⣿⣭⣶⣿⣯⣶⣾⣏⠓⠚⣿⣿⣦⠖⣰⣦⣤⣀⣀⠀⠀⠀⢶⣶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦
⠉⠉⠛⠀⠀⠀⠀⠀⠀⡶⠿⠃⠈⣫⣤⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣿⣿⣥⢠⣿⣿⣽⣟⢻⣦⡄⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣿⣷⠀⣈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣏⣿⣿⣿⡏⠉⣦⠀⣶⠈⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣻⣷⣍⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⢸⣿⣿⣿⣿⣿⣧⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁⢹⡿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⢁⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⣷⠞⠁⠀⠀⣸⣧⣤⠀⡀⠐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⢻⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⡝⢿⢿⣦⡄⣤⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢸⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⢿⣿⠿⢿⣷⡂⣹⣿⢀⢤⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⣈⣛⣋⣭⣽⡷⢶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣟⣿⣿⣷⡶⢤⡄⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠿⡿⠯⠽⣶⣶⡾⡿⣯⡛⢛⠷⣾⣿⣷⣶⣤⣭⣉⣍⣖⣢⣤⣤⣤⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠉⠀⠀⠀⠀⠀⠀⠙⠧⠤⣤⠈⠓⣿⣷⣮⣿⣿⣿⣿⣿⣿⡿⠛⠛⠃⠉⠉⠉⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⢿⣷⠀⠀⢀⡀⠀⢀⣤⣶⠀⠺⣇⣤⣿⣿⣿⣿⣿⣿⣿⡟⣛⣧⣀⣴⣶⣿⣤⡤⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃⠈⢹⣿⡿⣿⣿⣷⣯⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⠗⠉⠉⠛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣠⣾⣿⣿⣷⣾⣿⣛⢻⣿⣿⣿⣿⣿⣿⣿⠟⠛⣋⡉⠛⢳⣿⣿⣿⠃⠀⣶⣿⠇⠻
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⣁⣴⣿⣿⡿⠿⠲⠶⣿⠋⢀
⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⡿⣿⣿⣿⣏⠁⠀⠀⠀⠈⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢾⠇⠀⠀⢴⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⢛⡝⠧⠀⠀⠀⠀⠀⠀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⠿⣿⡷⠤⠀⠀⠀⠀⠀⠀⠂⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣶⣿⣿⡆⢀⡀⠀⠀⠀⠀⠴⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣛⣦⡀⠠⣚⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣱⣿⣿⠿⠻⠈⠀⠀⢀⣀⣤⣴⠿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠙⣿⣿⣷⣦⣿⡟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⠿⠟⠙⠿⠇⠀⣀⣀⢀⣼⠟⠉⠁⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠛⢿⣿⣏⣠⣤⣾⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⢀⠀⢺⣿⡿⠋⠁⠀⠈⣰⡜⠉⢿⣾⣄⠀⠀⠀⠀⠀⠀⠘
⠷⠀⠐⠀⠀⠀⠀⠀⠘⣣⡴⠟⠃⠀⠀⠀⠈⠻⠿⠿⢿⣿⣿⡿⠋⠋⣁⣠⣤⣤⡄⠀⣀⣹⠿⠛⠁⠀⠀⣠⣶⣿⡟⣡⠀⣼⣿⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣯⠌⣀⣤⣿⣿⣿⣿⣿⠛⠉⠀⣀⣤⣴⣶⣿⣿⣿⣿⣼⡿⢾⣻⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣴⣾⢃⣴⣿⣿⣷⣿⡏⠙⢡⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⡿⢿⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣥⣤⣬⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⣿⣿⣿⣿⣤⣤⣭⣤⣄⣤⣬⣭⣿⣿⣿⣿⣷⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢉⠿⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣠⣽⣿⣿⣿⣿⣿⠋⠛⡛⠿⠶⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠉⠀⡜⠳⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠟⠛⠛⢻⡿⠛⠛⠛⠋⠈⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠦⣄⡀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠉⠓⠀⠤⠴⠶⠶⠶⠦⠠⠄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⢸⠛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠛⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠖⠛⠉⠀⠀⢀⡀⠀⠀⠀⢀⡄⠀⠀⠀⣀⠀⠀⠐⠀⠤
⠀⠀⠀⢀⡴⠚⠋⠉⠀⠀⠀⠀⠀⢀⡀⠀⠘⠁⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⠃⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠓⡲⠋
⠀⠀⠀⠀⠀⠉⠻⢿⣓⡶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⢤⠤⠤⠀⠒⠒⢉⠥⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠿⣦⣬⣉⣯⣍⣉⣉⣩⡉⠀⣤⣤⠶⠶⠒⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠋⠉⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣄⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⣧⣠⣄⣄⠰⢶⡃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⢠⠀⣿⠇⠀⠀⠀⠀⢧⠀⢀⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⠈⣧⠀⣿⠀⠀⠀⠀⠀⠈⠄⢸⠀⡄⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⢸⣇⢹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠸⢀⣃⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡿⣼⣾⡿⡇⠀⠀⠀⠀⠀⠀⢟⡗⣺⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡏⣿⠀⡇⠸⠀⠀⠀⠀⠀⣼⢡⠇⣸⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣖⣷⣿⡘⣧⡇⢠⠀⡄⠀⡄⣿⣯⣴⢛⠉⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣧⣸⣄⣇⣤⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠈⠉⠓⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⠉⠙⢿⣿⡿⠋⠉⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡜⠁⠀⡠⠓⠛⠺⡄⠀⡸⠚⠓⠣⡀⠀⠘⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣿⠄⢸⣿⣿⣿⣷⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⡜⠀⠀⠀⣷⣿⣷⠀⣸⠰⣿⣿⡧⠀⡇⠀⠀⠘⡆
⠀⠀⠀⢀⣏⢀⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣵⣶⣿⠟⢻⡏⠙⡀⠀⠀⢰⠱⣳⠀⠀⠈⠛⠋⠒⡅⠀⡝⠋⠐⠊⢀⢤⠀⡴⣼
⠀⠀⠀⢸⠋⠙⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠊⠙⣿⡇⠀⠀⡇⢰⢢⠘⠝⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⢩⡒⡌⠀⡇
⠀⠀⠀⠛⠓⠚⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠓⠚⠛⠛⠀⠀⠧⠬⠥⠤⠤⠤⡤⢤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠭⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠋⠒⠉⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡿⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠋⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣂⣤⣾⣿⡿⠿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠿⠟⠛⠉⠀⢀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠾⢿⣿⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⠀⠀⢠⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⡾⠀⠐⠢⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⡇⠀⠀⠀⡷⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⠷⠤⠤⠞⠁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣦⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁⣀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣤⣤⣤⣴⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣰⣠⣀⢀⣀
⠀⠀⠀⣰⠊⠀⠐⠃⠀⢀⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢹⣿⣿⡿⠿⠟⠛⣿⣿⠀⢠⣤⡄⣀⣀⣀⣀⡻⠿⠿⣿⣿⢀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⣤⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢸⣿⡇⠀⣀⣤⠀⠘⣿⡀⢸⣿⠇⠻⠿⢿⣿⡗⠀⣸⣿⠃⢸⣿⠁⠘⣿⡇⠀⣿⣿⣧⠀⠀⣿⡇⢀⣴⣿⠟⢻⣿⡇
⠀⠀⠀⠈⣿⡇⠀⢻⣿⣶⣾⣿⡇⢸⣿⠀⠀⢀⣿⡟⠀⣰⣿⠃⠀⢸⣿⠀⠀⣿⡇⠀⣿⣿⣿⣆⢰⣿⡇⣼⣿⠃⠀⠀⠉⠁
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⢻⡇⣸⣿⠀⠀⣾⡟⠀⢠⣿⠃⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⡏⢻⣿⣼⣿⢣⣿⡇⢰⣿⣿⣿⡟
⠀⠀⠀⠀⢸⡿⠗⢲⣿⣧⠀⢸⣿⣿⡏⠀⣼⡟⠀⠀⣿⣷⣶⣶⡄⢸⣿⠀⠀⣿⡇⠀⣿⡇⠀⢿⣿⣿⠸⣿⣇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠈⠋⠉⠀⣸⣿⣿⡇⢸⣿⣶⣶⣶⡎⠉⠉⠉⠀⢸⣿⣤⣤⣭⣄⠀⣿⡇⠀⠘⣿⣿⠀⢿⣿⣄⢀⣼⣿⠃
⠀⠀⠀⠀⠀⣿⣷⣦⣤⣶⣶⣿⡿⠃⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠉⠀⠀⠀⠈⠉⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣀⣤⣄⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⢿⣧⢰⣿⡏⢻⣷⡀⣿⡟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠘⣿⡆⢻⣿⠈⣿⡇⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⣸⣿⢣⣿⠏⢰⣿⢃⣾⡟⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⠃⣿⡏⢰⣿⠏⣾⡟⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⣿⡆⢿⣧⠘⣿⣆⢻⣿⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡿⠃⠈⠿⣷⡿⠏⠈⠻⣷⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠋⣉⣠⣤⠴⠶⠶⠶⠒⠲⠶⠶⠶⢤⣤⣈⡉⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⠁⢀⣠⠶⠛⢉⣡⣤⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⢤⣌⠙⠳⢦⣀⠀⠉⠛⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠋⠀⠀⢀⡴⠋⠁⣠⡶⠋⠁⠀⠀⢶⣴⠂⠀⠀⠻⣦⡖⠀⠀⠈⢳⣄⠀⠉⠳⣄⠀⠀⠈⠙⠿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠋⠀⢀⣀⣀⣴⠏⠀⣠⣾⣿⠁⠀⢦⣀⡴⠋⠙⢷⣄⣠⠞⠉⠻⣦⣠⡴⠈⡏⠻⣦⡀⠈⢷⣤⣄⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣰⠟⠁⣸⠃⠀⣴⢻⡏⢿⠀⢀⡴⠛⢦⡀⠀⣠⠟⢷⣄⠀⢀⡼⠻⣦⠀⡇⠀⠈⢳⡄⠀⢻⡀⠙⢷⡀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⡟⠀⢰⡇⠘⣇⠘⣧⡀⠀⠀⠀⣹⣾⡁⠀⠀⢙⣷⣏⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠘⡇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⣇⠀⢸⡇⠀⠻⣄⠈⠛⢦⣀⠀⠁⠀⠙⣦⡴⠋⠀⠙⠃⠀⠀⣼⠀⠀⠀⠀⣿⠀⢀⡇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⢻⡀⠀⢻⡀⠀⢷⡀⠀⠉⠳⣤⡀⠉⠳⣤⠀⠐⠋⢙⣤⣀⠀⠀⠀⢠⡏⠀⠀⠀⣰⠏⠀⣸⠃⠀⣼⠃⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠙⠳⠾⢷⡀⠈⠳⣄⠀⠀⠀⠙⠷⣄⠘⣇⠀⠠⣏⠀⢸⡇⠀⣠⡟⠀⠀⣀⡼⠋⠀⣰⠿⠶⠚⠁⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠹⣦⡀⠈⠛⢦⣄⣀⠀⢻⡄⠙⢦⣀⠉⠛⢋⣠⣴⠟⣀⣤⠞⠋⠀⣠⡾⠁⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠙⠶⣄⡀⠈⠉⠛⠚⠷⣤⣀⠉⠙⣛⣿⠟⠛⠋⠉⠀⣀⡴⠞⠁⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣄⣈⠉⠛⠶⠦⣤⣄⣀⣉⣙⣋⣉⣀⣤⡤⠶⠞⠛⢉⣀⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣷⣶⣦⣤⣤⣬⣭⣉⣉⣥⣤⣤⣤⣶⣶⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁


⣿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣧⣶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠁⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡾⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠖⢶⡀⠰⢖⡓⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠉⠉⠀⠀⠈⠁⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣀⠙⠃⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢄⠀⠈⠙⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣾⠐⣤⣀⣀⡠⠀⢸⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠤⠞⢿⣟⠢⡐⠉⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠈⠛⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠒⠉⠻⣇⢰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣖⣈⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢿⣂⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⠟⠋⠁⠑⢆⢻⣷⡂
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢻⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⢈⣠⡏⠉⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡄⢀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠉⠉⠛⠶⣄⠄⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⣱⠞⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣼⡏⠀⠀⠀⠀⠀⠀⢀⡠⠰⠿⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠄⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⢷⣦⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠻⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢹⡇⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣠⡾⠃⠀⠀⠀⣰⡟⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠘⣷⡀⠀⠀⠀⠻⣦⣀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⣿⠁⠠⣿⣉⣻⡇⠀⠀⣿⣋⣹⡧⠀⢹⡇⠀⠀⠀⠀⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣦⠀⠀⣿⡄⠀⠙⠻⠛⠁⠀⠀⠙⠻⠛⠁⠀⣼⡇⠀⢀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠸⣷⡀⠀⠀⣤⡀⠀⠀⣠⡄⠀⠀⣰⡿⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠈⠛⠛⠛⠋⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⣿⠓⠀⠉⠛⠿⠶⠶⠶⠾⠟⠋⠀⠐⢻⡿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠿⠀⠀⠀⣤⣀⠀⠀⠀⢀⣤⡄⠀⠀⠸⠇⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣤⣤⣤⣤⣤⣬⣽⣷⣦⣶⣿⣭⣤⣤⣤⣤⣤⣄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⣿⣦⣼⠿⣧⣤⡿⢿⣤⣼⠿⣧⣤⡿⢿⣤⣾⣿⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⡿⠷⠶⠾⠿⠶⠶⠿⠷⠶⠾⠿⠶⠶⠿⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⡾⢻⣷⠟⠻⣶⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⣠⡾⠟⠁⠀⢘⣿⣀⠀⠀⠙⠿⣦⡀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣿⠿⠋⠀⠀⣀⣴⠟⠋⠛⢷⣄⡀⠀⠈⠻⢿⣇⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⠛⠁⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠙⠻⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣾⣷⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣶⣶⣶⠆





⣿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣟⣿⣟⣛⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂⠐⠀⠀⠁⠈⠀⠁⠉
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠻⣿⡟⢻⣿⡛⣻⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢀⣄⠀⢠⡄⠀⣴⠄⠠⠦
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⡀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⡿⢿⠈⡏⣰⣷⡄⢻⣿⠈⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⢀⣈⣿⢰⡏⠉⢻⡆⠀⣿⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣶⣾⠀⣧⡘⢿⢃⣼⣿⣄⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠉⠉⣿⠘⢧⣄⡼⠃⠀⠿⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡷⠟⢿⢛⡛⠟⢻⢛⣝⣿⢩⢽⣿⣿⣿⣿⠀⠀⠀⠀⠀⢘⠀⠀⠀⢈⣠⡀⣬⢤⣠⣄⡴⠶⠄⣖⣒⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣩⣼⠨⣅⢻⢸⣘⣟⣿⣘⣛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⠶⠋⣷⠺⣝⡏⠧⠠⠆⠧⠤⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⡿⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⣄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣠⣿⣧⣼⣿⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠄⠀⠀⠐⠟⠀⠘⠃⠀⠙⠁⠀⠉⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⢀⢀⠀⠄⠠⠠⠀⠀⠀⠐⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡋⠀⠀⠀⢼⣿⡏⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠈⣻⣿⡆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⣸⣿⡟⠀⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠘⢿⣿⡄⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⠀⠀⢰⣿⡿⠁⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⣄
⠀⠀⠀⢀⣾⣿⠏⠁⠀⢀⣠⣴⣶⣶⡆⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⢀⣿⣿⠃⠀⢀⣴⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⣸⣿⡇⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⣿⣿⠁⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣶⣶⣶⣿⡿⠛⠙⠻⣿⣧
⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⣿⣿
⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠿⠿⠿⠿⠿⣿⣷⣄⠀⠀⠀⣻⣿⡆
⣠⣤⣤⣤⣤⣽⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣶⣤⣴⣿⣿⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⢹⣿⣇⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣰⣿
⠀⢻⣿⣦⡀⠙⠻⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⠿⠋⢀⣴⣿⡟
⠀⠀⠙⢿⣿⣦⣄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡐⠉⠁⠀⠀⠀⠀⠀⠱⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢀⠄⠒⠀⠚⡆⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⠄⠀⠀⢀⡄⢠⠤⠄⠒⠀⠘⠛⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⡰⢋⡄⠃⠀⡐⠀⠀⢠⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡀⠐⠈⠔⢣⡘⡀⠀⠸⠇⠀⠀⠘⢞⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠀⠀⡇⠉⠀⢀⠀⠀⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠀⠀⠈⠲⣆⠀⠐⠂⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠄⡀⠀⠐⠡⠅⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠑⠤⡀⠀⣠⢔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠈⠀⠀⠀⢠⣄⠀⠐⠤⢌⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢀⠀⠀⡟⠀⡴⣇⣀⠘⡄⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣀⠤⠒⠂⠉⠉⠀⠀⠉⠉⠙⠒⠠⢄⡃⠀⠀⠀⠃⠀⠈⠑⠢⡀⠀⠀⠀⠘⠀⠀⡐⠊⠀⡠⠚⡀
⠀⠀⠀⠀⠀⡠⠐⠂⢒⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡾⠀⠀⠀⠐⠉⢀⣄⠠⠀⠀⠀⢐⠒⠈⣀⠤⢑
⠀⠀⠀⢀⠌⡀⠈⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡇⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡊⣉⡀⠄⠊⠄
⠀⠀⠀⢀⠈⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣤⠀⠄⠊
⠀⠀⢀⡼⠞⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠚⢆⠀⠀⠀⠀⠁
⠀⠀⢎⠀⠀⢪⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠐⠂⠀⡀⠔⣸⠃⠀⢠⠃
⠀⠀⠈⠳⢤⣀⠑⠢⠤⣀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⣐⣠⠥⠖⠋⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠭⢭⣽⣿⣟⣛⣛⣛⣛⣻⣿⡭⠭⠭⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣠⠖⢲⠔⠊⢹⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⢀⣀⢀⡜⠁⡰⠃⠀⠀⢸⠀⣠⠀⡰⠊⠉⠲⣄
⣴⡖⠤⠋⠈⠋⠀⠰⡁⠀⠀⠀⣸⡤⠄⠠⢇⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠓⠊⠉⡆⢀⡠⠤⣈⢧⠀⢼⡡⠤⠄⠒⢶⣸⢶⡖⣒⡇⠀⣀⣀⠤⠤⠤⠔⠲⠤⢄⡀⠀⠀⠀⠀⢾⡵⠒⣷⣄⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠁⠀⣀⣈⣯⣷⠀⠉⠓⠒⠚⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠸⡅
⠀⠀⠀⠀⠀⢀⣸⡁⠠⠶⠮⠁⠀⠀⢀⡀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡔⠛⢤⠀⠀⡟⠲⡁
⠀⠀⠀⠀⠀⠈⢧⢈⣲⠶⠒⢢⠀⠈⠙⡧⣄⠀⠀⠀⠀⠉⠂⠄⢀⡀⠀⠄⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢱⠸⠊⠀⠀⠈⡯⣉⣱
⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⢀⡜⢣⠀⠀⠹⡜⢳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⠤⢤⣤⣀⠠⠄⣶⠤⠒⠁⣀⠷⠖⠒⠚⠁
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠑⢤⣣⣀⡀⠈⠃⠀⠙⠂⠀⠀⣀⣀⣀⣀⡤⢄⣈⢻⠉⠓⠤⢀⠀⡇⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⡝⠀⠙⠢⠤⣄⣈⣉⠉⠉⠉⠁⠀⠀⡆⠀⠀⠉⠳⠝⠂⠀⠃⠉⠛⣢⣄⡀⠀⢀⣀⢠⣤⠤⠒⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠂⠃⠀⠀⠀⠀⠀⠀⢀⣠⠒⠉⠀⠤⠋⠁⠀⡠⠊⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠒⠌⡒⢄⠄⠀⢀⡞⠁⠀⣀⠄⣠⣞⠾⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⡛⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⠞⠛⠂⠤⢾⣴⠊⢉⣼⡶⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢦⣕⡢⢄⡀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠑⠻⣁⠀⣀⡠⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⠉⠓⠮⣕⣢⢄⣠⢎⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠃⠀⠀⢀⣀⠜⣹⠙⠓⠦⠭⣕⣢⣤⣄⣀⣀⣠⡯⠛⠉⡷⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠤⠖⣊⠭⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢩⢹⠀⠀⠀⡰⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠒⣹⠁⠀⠀⣘⠤⠢⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠻⠔⣒⡺⠅⢴⣉⠦⢶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢃⣴⣿⣷⣾⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡿⠉⡀⠛⢠⡤⠿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣦⣣⣤⠒⠀⠀⠀⠀⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣋⣤⠖⠀⢀⠀⠀⠲⢶⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⢀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⠠⠶⠾⢿⣭⣭⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⠓
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠻⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡻⣿⣦⡝⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⣯⣍⠉⠉⠀⢠
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠌⠛⠃⠠⣰⡆
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⢠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⡇⠀⡰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠋⠁⣆⠀⠈⠀⢰⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡟
⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠁⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡇⠒⣿⠇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣿⣿⠀⠘
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠃⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⣿⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⡇⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⠃⠀⠀⢃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢠⣿⡏⠀⠀⠀⣼⣿⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣼⡿⠁⠄⠀⢠⣾⣿
⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⠁⠀⠀⢀⣾⣿⡇⠀⠀⠆
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⣼⢟⣿⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠜⢁⣾⠋⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣶⡾⠋⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣾⡿⠋⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⠁⠀⡀⠀⠀⠛⠁
⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠒⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⠀⠀⠀⢰⠇
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠐⠲⡄
⠞⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠉⠀⠐⢁⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⠿⠿⠿⢿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠐⠀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣍⣁⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣦⣔
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠟⠋⠀⠁⠀⠂⠀⠀⠁⠀⠀⢀⠀⠉⠛⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⠟⠁⠀⣜⠐⢶⣶⡄⠀⠀⠰⠿⠛⠀⠇⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀⠀⠰⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠇⠀⠄⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⣷⣉⠛⣛⣿⣿⠿⠱⠿⣿⣍⣉⣤⣴⣶⣶⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⠠⠄⢻⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⠏⢻⣿⣿⣿⣿⡧⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠈⢿⠿⠷⠀⠀⠀⠀⠐⠛⠛⠿⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠀⠀⠀⠀⠀⡄⠀⡇⢀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⢷⠀⠀⠀⢰⡿⣄⠐⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠜⠀⠀⠀⠈⡧⢈⠑⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣧⠀⠀⠀⠀⠀⠘⢖⠁⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⢸⠀⡼⠀⠀⠀⠀⠀⠑⣤⠂⢀⠙⣆⠀⠀⠀⢀⠞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⢀⠨⣸⠀⢇⠳⡀⢤⣀⠀⠀⠈⢢⠀⢐⡘⡆⠀⢠⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⡂⠑⠄⢳⠀⠀⠁⠺⠌⡮⠳⡄⠀⢸⠂⡁⠁⢵⠀⠘⣡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠘⢄⠈⠢⡀⠀⠀⢠⠇⠂⣻⢠⠏⠀⠇⠠⡍⠀⢰⢻⡀
⠀⠀⠀⠀⠀⢯⠢⣄⡀⠀⠀⢳⠅⠘⡀⠑⢄⠙⣆⣠⠋⣠⠀⠿⠉⡠⠁⠂⢠⠃⠀⡼⠐⣇
⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠱⡄⠑⡀⠀⠳⡈⠃⡐⠀⡄⢀⠎⠀⡐⢠⠏⢀⣴⠇⠀⢻⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠰⣤⣅⡚⠀⠀⠀⠘⣦⠐⡀⠀⠱⡐⠀⠀⡖⠁⠀⡐⢠⣮⡴⠛⢁⡀⠀⣼⠀⠀⢀⠜⣹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⡌⠓⢤⠀⠀⢸⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠛⣀⠄⠒⡉⠀⣴⠏⠀⢐⡡⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⡀⢳⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⢀⠎⢀⡤⠁⠀⠀⢈⠔⢫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⡄⢷⠟⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⡜⢁⣴⠏⡴⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠐⠀⢠⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠘⣵⠯⢹⢰⠃⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠃⢦⢘⠟⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠈⢦⡰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠀⣀⣀⠀⠠⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⠐⠈⠐⠠⠀⠠⠀⠤⠂⠄⠂⠀⠠⠀⠠⠤⠆⠑⠀⠀⠁⠒⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠚⠛⠛⠛⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠉⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠠⠤⠒⠊⠉⠀⠀⠉⠉⠓⠒⠒⠒⠒⠘⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠤⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠄⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣔⣂⡀⠀⠀⠀⢀⠤⠈⠉⠑⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡡⠀⠀⡞⠁⠙⡀⠀⠀⠐⢋⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣍⠤⡃⠀⠀⢣⡔⡄⠀⠀⠀⠀⠋⠂⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀⢀⡇⠀⢀⠞⠁⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢣⠀⠀⣀⣀⠀⣀⡠⠀⠈⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⡀⢠⠓⠤⣀⠀⠣⡀⠣⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣧⠱⡼⣄⠀⠈⠳⡄⠙⠦⡀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠒⠓⠢⡈⢳⡀⢀⠋⠀⢀⡜⡖⠀⣀⡴⠊⢣⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⢀⣠⡤⠚⢦⡴⢀⡌⠀⠀⠈⢧⠘⢾⡉⢀⡠⠃⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠴⢮⠠⡜⠀⠀⠀⠀⠀⠉⠀⠉⢟⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠤⠤⢤⠴⠚⣄⢣⠀⠀⠀⠀⠀⢀⣀⣠⠦⢖⠴⠂⣸⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⣸⠋⡆⢣⣀⡤⠖⠚⠉⢁⡠⠖⠉⠀⠀⢻⣇⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⠞⡄⠁⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⢸⡟⢦⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⡴⠒⣬⣇⡼⠋⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠎⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠋⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠇⠀⠈⠳⡄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠚⠋⠉⠀⠀⠀⠀⣠⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡈⠙⠒⠶⠦⠤⠴⠿⠶⢶⣶⠶⢒⣀⣠⠄⣠⠞⠉⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠉⠉⣉⣉⡳⠿⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣷⣶⣶⣾⣶⡿⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣾⣷⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⡾⢿⠿⠿⠿⠾⣷⡶⠶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⠴⠃
⠀⠀⠀⢀⡔⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠈⠧⣤⣀⣱⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢦⣀⠀⠀⠀⠀⣠⠶⠒⣦
⣰⠒⠒⣦⡀⠀⠀⠉⠃⢀⣤⠶⠛⠉⠉⠁⠀⠀⠉⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⣠⠴⢊⠩⠷⣄⠀⠀⠷⠶⠖⠋
⠙⠶⠚⠉⠁⠀⠀⢀⠖⠋⢀⡠⠤⠒⠒⠒⠒⠒⠂⠤⢄⣀⡀⠈⠉⠓⠲⠶⠦⠴⠶⠒⠛⢉⣁⠠⠐⠊⠁⢰⠊⠁⡠⢖⡩⢷⡀
⠀⠀⠀⠀⠀⠀⣰⠋⡠⠊⠁⢀⡠⠤⠄⠒⠒⠒⠒⠂⠤⠤⣈⠉⠓⠒⠒⠒⠒⠒⠒⡋⢉⣤⠤⠖⠊⠉⢀⣡⣴⡿⢊⣥⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢧⣤⠞⠁⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠈⠳⠤⠚⠁⠉⠉⠀⠀⡀⠀⡠⠤⠆⠉⠉⠩⠟⠁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⡠⠗⠀⠀⠀⢁⣀⣀⢉⣉⣁⣒⣒⣒⣂⠀⢉⣁⡠⠴⠒⠒⠋⠁⢀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠊⠁⠀⣀⠤⢒⣩⠵⠶⠒⠒⠛⠓⠶⠦⢤⣤⣀⣀⣀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣊⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⠉⠩⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠈⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠾⠳⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⣀⣠⡤⠶⠖⠛⠉⢁⣀⡤⠤⣄⠀⠘⢧⡀⠘⡜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡥⠴⠚⠋⠉⢀⡠⠤⠔⠚⠉⠉⠀⠀⠀⠈⢆⠀⠀⠻⣄⠈⠒⠡⠙⠡⢐⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠁⠀⡤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣆⠀⠀⠀⠀⠀⠙⡜⡄
⠀⠀⠀⠣⡀⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⣠⠏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⢀⠎⠀⠀⡴⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡤⠚⠁⣷⠀⠀⠀⠘⣧⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠘⠀⢀⡞⠀⣠⣤⡀⠀⡇⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⣨⣧⠔⠋⠁⣀⡤⠚⠁⠀⠀⠀⣠⠾⡆⠀⠀⠀⢇⢇
⠀⠀⠀⠀⠀⠀⡀⢀⠊⢀⠄⢀⡞⠀⠀⠀⠈⢻⣤⡷⠤⣄⣀⣟⠶⠶⠶⠶⢒⣫⠕⣃⡤⠖⠋⠁⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⠈⠢⣕⡂⠤
⠀⠀⠀⠀⠀⠔⠠⠁⠰⠁⢀⡞⡴⠾⣯⠞⠉⠙⣿⢧⣀⣀⣀⣈⣩⠭⠭⠿⠛⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⠀⠱⠀⠀⠀⡞⠰⡁⢿⡏⠀⣾⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠉⠳⣤⣬⣿⣡⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠑⠒⠒⠲⠦⠤⠤⠤⢤⠶⢶⣶⣖⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⡠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣠⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣸⠃
⠣⣈⠢⢄⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠶⠚⠋⠓⠚⠋⣡⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⣉⣠⠴⠊⠁
⠀⠀⠑⠢⠬⣙⡒⠲⠦⠤⢤⣤⣤⣤⣤⣤⣤⣬⣿⣿⠥⠤⠶⠒⠚⠋⣉⣁⡤⠴⠖⠋⠉
⣀⣀⣀⣀⣀⣀⣈⣉⣙⣒⣲⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣖⣒⣚⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣙⠛⣛⢏⣿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⢜⡅
⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⡤⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢇⠀⠀⠀⠘⣆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠤⣆⠀⠈⠉⠁⠀⠀⠠⣀⠀⢷⡄⠀⠀⠀⢸⡄⠀⠀⠀⠀⢻⠀⠀⡠
⠀⢀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠈⠳⣬⡗⠈⠀⠀⠀⣇⠀⠀⠀⠀⠘⡤⠚⠁
⠀⠈⠉⠃⡀⠀⣄⡀⠀⠀⠀⠀⠀⠟⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡀⠘⣁⣠
⠀⢀⡄⢀⠓⢀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠄⢈⡋⠀⠀⠀⠀⠀⠀⢸⣿⠀⡆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⢸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⠀⢸⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣏⡍⠢⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢸⡇⡟⠠⠄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠀⠀⠀⠀⠀⢸⠁⠀⣿⠃⠀⣌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⡟⠠⠚⠁⠨⠙⢤⡀⠀⠀⠀⠀⠀⠐⠈⣷⠀⣄⢀⠀⡟⠀⠀⡟⠀⠀⢸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠘⢧⠹⡌⢸⠃⠀⢠⢧⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⣇⣧⡞⠀⠀⡇⢠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⡼⠀⢬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣸⠁⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢐⡐⠁
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣛⣿⣛⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠒⠂⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⡾⢺⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠒⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⣟⡾⣿⡀⠀⠀⠀⠀⠀⣠⡾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠀⠀⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⠈⠁⠀⠠⢀⠀⠀⠀⠀⠈⢺⣯⠈⠌⠛⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢂
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠄⠀⡀⠀⠀⠀⠀⠐⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡤
⠀⠛⠤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠒⠉⠱⠀⠀⠀⠀⠉⠀⠀⣤⢺⠈⠲⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠐⠀⠀⠀⠰⠐⠋
⠀⠀⠀⠀⠈⠈⠙⢛⠲⣴⠢⣄⡤⢀⡀⠀⣀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠀⠐⠘⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠄⢀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠓⠈⡍⢀⠐⠟⡙⠛⠛⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢬⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠓⠀⢿⠗⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠈⠄⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠁⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⣓⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠄⠀⠀⠐⠉⣇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢹⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡐⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠊
⣿⣿⡟⠻⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠐⡐⢠⠀⣀⠀⠈⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⠄⠀⡀⠐⠆⠀⠀⣄⠀⣠⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⠁⢀⠀⠀⠀⠆⠀⠄⠈⠃⠈⠀⡄⠀⠈⡄⠀⢀⡄⠈⠀⠒⠰⠃⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠠⠐⠀⠀⣀⠐⠈⠀⡀⠐⠄⠀⠀⠈⠀⡀⠄⠃⠀⡀⠠⠆⢢⠄
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠆⠀⠀⠘⠈⠀⠀⠀⠈⡀⠀⣀⠀⢈⠈⡇⠙⢁⠀⠁⠈⡡⡯⡻⢀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠰⠀⠂⠂⠄⢀⢌⠁⠀⠀⡀⣀⣰⢨⣄⠀⣅⠊⠀⠃⠀⣂⣀
⠀⠀⠀⡀⢀⠄⠠⠘⢀⠀⠀⢐⠂⡠⠌⢠⡶⠐⣐⡃⡄⠁⠠⠃⠄⠊⡄⠀⢍⡀⡄⡏⠿⡄⠤⣵⢌⢡⠆
⠀⠀⠀⠀⠀⠃⢠⠑⠈⠀⠰⢤⠀⠄⡀⠀⠁⠠⠈⣅⠀⣬⠠⢏⠌⠀⣻⡀⢘⢿⡍⠁⡌⠃⠃⡀⢠⡃
⠀⠀⠈⠋⠀⢠⠀⠄⠀⢠⠤⡐⠘⢠⡉⣀⣤⠳⠗⠂⣡⠉⠐⡄⣯⠈⡠⢶⢤⠈⣛⠛⡶⠀⢹⠿⠚⢁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⢶⣤⠙⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣶⣬⣿⠟⠦⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣼⠋⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⠆⣠⡅⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠐⣋⣽⡇⠀⢾⣭⡉⢸⡿⢤⠖⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⠀⠉⠀⠀⠀⠀⠸⠿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⠀⠓⠒⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⢓⠂⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⢀⣴⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⢿⣿⣿⣿⠋⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣍⡋⠁⠀⣠⠎⠀⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣌⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⣤⠴⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣢⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⡰⡁⠀⠀⠀⡀⠐⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠁⢀⡸⠇⢀⠀⠀⠀⠸⡆
⠀⠀⡼⠁⠈⠀⠀⡌⠀⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣤⣾⣿⣿⡟⢻⡗⣦⠙⢷⣌⠀⡀⠀⠀⣿
⠀⢠⡇⠤⠒⠘⣼⡃⠀⠀⠀⢥⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠋⣾⣿⣿⣿⣧⣸⣧⠀⠂⠀⠻⣷⡘⣆⠀⢸⡀
⠀⢸⡇⠀⠀⠀⠹⣁⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠃⠠⣿⣿⣿⣿⣿⣷⠟⠂⠀⠀⠀⠻⣖⠘⡆⢸⡇
⠀⣜⠁⠀⠀⠀⢀⣻⣴⣶⡞⠁⣇⠀⠀⠀⠀⠀⠀⢸⡇⢰⣴⣿⣿⣿⣿⣿⣿⣶⡶⠀⠀⠀⠀⠀⢹⡇⣷⢸⡇
⠀⠘⣦⡀⠀⠀⠸⣴⣿⡟⠀⠀⠏⠀⠀⠀⠀⠀⠀⠘⠣⣾⣿⣿⣿⣿⣿⣿⣿⡉⢻⠀⠀⠀⢔⣀⢴⠂⠀⠸
⠀⠀⠈⠻⣄⣀⣀⣻⠟⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠦⠶⣄⡐⠀⠒⠀⢁⡾⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠷⠲⠾⣽⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢸⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⢘⠗⠀⠀⠀⠀⠀⠌⣹⡄
⠀⠀⠀⠀⠀⢸⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠁⠀⠀⢠⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠘⠃⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠀⢁⡀⣰⠆
⠀⠀⠀⠀⢲⠆⢀⠀⠘⠁⡄⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⠀⠀⠔⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣰⠋⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢠⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡇⠀⠀⠀⠀⠀⠀⠃⢸⠃⠀⠀⢸⢠⠀⠀⠀⠀⡄⠀⠀⢀⡄⠀⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡀⡆⢠⠀⠃⣾⢸⠀⠇⣾⣸⠄⠀⠀⡌⣼⠀⢀⡿⠀⠀⠀⢀⠜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢹⡀⠀⣧⡇⢸⠀⡄⡇⢸⢸⠀⢸⡿⠀⠀⠰⡼⠘⡀⣼⠇⠀⠀⡰⢃⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠈⡇⠀⢹⣧⢸⡆⡇⡇⢈⢰⠀⢸⡇⠀⠀⢇⡇⢀⣷⡇⠀⠀⣼⡡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⢻⠀⢸⣿⢸⡇⠇⡇⠀⠀⠀⢸⡇⠀⠸⢸⢀⠎⡏⢀⠀⣰⡟⢀⠃
⢀⣀⡀⡀⡀⡀⠀⡀⢸⣗⠀⢸⡄⢸⡏⣿⣓⡀⣷⡀⣀⣀⣸⣿⣀⣂⣾⡎⣰⣇⢘⣠⡏⡇⣸⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣉⣿⣿⠉⣿⡟⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢻⡶⠋⢿⠏⠸⡿⠁⣰⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣁⢿⡇⣦⣄⠀⢀⣤⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢇⣼⣇⠿⣿⣧⠸⠟⠀⣹⣡⣿⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⢻⡏⢸⣿⠉⢀⡈⠁⠀⣶⡄⢿⡟⣯⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣸⣿⣿⢸⣿⣿⣷⡀⠘⣿⣧⠀⢿⣿⠀⢰⣿⢛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⡿⠚⢿⡏⠉⣷⡀⠈⣿⡇⠘⠛⠀⠛⠁⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠇⢠⡜⢻⣆⠻⠇⠀⢹⣿⡀⢀⣶⡖⢀⣀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⢸⡇⠘⠟⢀⠀⣲⢸⣿⠀⠈⠛⠁⣿⣿⠁⢰⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⣿⠀⠐⠿⢀⡟⠀⠀⢠⣿⣦⠀⣙⠋⠀⣾⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢻⣷⣿⡇⠀⣴⣿⠃⣿⡇⠸⣿⣿⡄⢸⣷⠀⠉⢰⣿⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢸⣿⣿⢧⣼⣿⠟⠁⠛⠀⠀⣉⠹⠇⠀⠛⢀⣀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠈⣻⡟⠈⠉⠛⠀⣴⣦⠀⠀⣿⣷⣴⣿⠆⢾⡏⣾⡇⢠⣿⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣽⣿⠀⣿⡁⢰⣿⣦⠀⣿⣿⠀⠀⢻⣿⣿⡿⠀⠀⠀⣿⡇⢈⡁⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⣿⡏⢹⡀⢿⣷⠀⠉⠁⠀⠘⠋⠀⠀⠀⠹⣿⠃⠀⣶⡄⢿⡇⠸⣧⠚⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣿⣿⣿⠃⣸⣷⣦⠉⣼⣿⣿⡆⣶⢰⣶⣦⡀⠀⢠⣦⡀⢻⣧⠀⠀⣤⡀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣸⣿⡟⣿⣿⣯⠘⠋⠀⠈⠙⠛⠁⣿⠘⣿⣿⣷⡀⢹⣿⠇⣄⠛⢀⡀⢻⣧⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣽⣿⣧⣿⢉⣟⢰⣶⣆⠀⣿⣷⡾⣿⠀⣿⣿⣿⣷⡀⠀⣸⣿⠀⢿⣿⢸⣿⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣿⣿⡿⢻⣿⡟⠸⣿⡟⠀⠹⠟⠃⠹⣆⠹⣿⣿⣿⠇⢀⣿⣿⠀⠸⣿⠘⠿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣷⣿⡋⢷⣾⣿⡄⠀⠀⣠⣤⡀⠻⣧⠀⠉⠁⠰⠿⠋⠻⠆⠀⢀⡀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⢿⣿⣿⡏⢷⣬⣿⠛⢃⡀⠘⣿⣿⣷⠀⣀⣤⣦⡄⢠⣾⣷⣰⣿⠄⣾⣇⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣏⢱⡈⣿⣿⣷⡼⣿⡆⠹⠿⠃⠘⠿⠿⠟⠁⠀⠛⠛⠈⠋⠘⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣽⣿⣿⣦⢹⣷⣜⢿⡇⢿⣅⡀⠀⣶⡀⠀⣴⡶⢀⣼⣿⣤⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣷⣾⣷⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠙⠀⠁⠀⠀⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⢸⣿⠀⢰⣆⡄⠀⡀⠀⠀⣴⡿⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢸⣿⡀⠘⣿⣧⠸⣷⣆⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡍⣿⣷⣘⣷⢱⣤⣿⣜⣶⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣌⣿⣿⣿⣧⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⢠⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⡀⡜⠀⠐⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠉⣐⡉⠈⠀⠄⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠂⠀⠀⠀⠀⠀⠂⠀⡉⠃⠀⠐⢀⠀⠴⠁⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣯⣤⣬⣉⣁⣂⡐⠀⠀⠤⠿⠽⠛⠂⠴⢂⣄⣀⣀⣐⣷⣃⠟⠀⢀⣀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠉⠙⠋⠙⠉⠉⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⡏⢹⡅⢰⡊⠁⣀⣠⣄⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⣒⣚⣻⣿⣽⡂⠨⡂⣼⡠⠊⠁⠀⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣚⣋⡭⠭⣖⠀⠆⣼⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠈⠥⠀⢒⣒⣛⡭⠅⢘⠁⡇⠀⠀⠀⠀⢀⠜⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠍⠩⠭⠵⣖⣒⠀⠐⠀⡇⢀⠠⠤⠒⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⣨⡭⠭⠽⣒⠈⠥⢠⡗⢀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⣒⣺⠭⠭⠗⢀⠅⡼⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢂⣴⣾⣿⡟⠡⣡⠞
⠀⠀⠀⢀⣀⡠⠤⠤⣶⠒⠒⠒⢒⣈⣹⣦⣅⡀⠀⠀⠐⠄⠶⠀⢶⣾⣿⡟⣇⣡⣾⣿⣿⣿⣖⠒⠶⠶⠦⢤⣄⣀⡀
⠀⠀⣞⢉⢀⡀⠀⠠⡂⠀⠀⠀⠀⢄⠀⠉⠫⢉⣀⣀⠀⠀⠘⡛⢫⣭⣭⣍⣭⠽⠋⣀⠄⠈⡉⠉⠉⢓⡦⠀⣀⠙⡯⣷
⠀⠀⠈⠓⠿⣟⣒⠶⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣅⣉⣀⣀⣴⣶⣶⣦⣤⡤⠬⠴⠖⣚⣫⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠦⠤⣤⣤⣤⣤⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣤⣤⣤⠤⠴⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⡈⠁⠀⠀⠀⠀⠀⠀⡞⠠⡊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢽⠀⠀⠀⠀⠀⠀⢀⣇⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡄⠀⠀⠀⠀⠀⢸⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣾⡆⠀⠀⢀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢃⠀⠀⠸⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⡆⠀⠀⠇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⡴⠀⠀⠀⠀⡇⠀⣼⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢤⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⡤⠀⠄⡀⠠⠀⠀⠀⠘⠀⢸⣆⠄⢼⠀⠀⣀⠄⠂⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢈⠁⡄⠀⣄⠀⠈⡢⣀⡀⠀⢀⠆⠀⠀⠀⢀⠱⡊⠠⠒⢁⠀⠱⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡠
⠀⠀⠀⠀⠀⢠⡕⠀⠀⡇⠀⠘⢄⠀⡈⠪⣻⡦⠉⠀⠀⠀⢰⠀⡄⢇⡀⠈⠁⠔⠀⣱⠔⠂⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⠁
⠀⠀⠀⠀⠀⡞⠈⡓⠦⡘⡄⠀⠈⠳⡈⠂⠀⠙⣇⠀⠀⡀⠀⠀⡀⢸⠂⠉⠐⢂⡠⣤⠀⠈⠓⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠐⠬⠒⠤⠉⠒⢤⣀⡈⠢⣄⠀⠸⣆⠀⠡⡀⠀⠁⣼⠋⠩⠭⠅⠉⢀⡑⢦⣠⠔⠉⠀⠀⠈⢢⡀⠀⠀⠀⣀⠄⠀⢀⠜
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⠀⢨⠃⠀⠱⠾⢋⡳⠢⢼⠶⢞⠡⠔⣒⣉⠩⠠⡨⢉⠓⠨⡀⢄⠀⠠⠖⠀⠧⣀⠤⠊⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢠⠃⠀⡞⠀⠀⠀⠀⠀⠀⢹⠃⠒⠴⠲⠅⢀⣀⢁⡀⡁⣁⡀⠁⠜⢦⡙⠄⣀⡈⢂⡡⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠰⡉⠉⠢⣄⡎⠀⢰⠁⠀⠀⠀⠀⠀⠀⡆⠀⢀⠇⠀⠀⢀⡀⠈⠉⠉⢨⣞⠁⡤⠂⠂⠒⠁⠰⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠈⠳⢬⠁⠀⡧⣀⡀⢀⣀⣀⣀⣸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⢠⠃⠈⠉⠉⠉⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠫⡑⢾⠀⢸⢦⣀⣀⣀⣀⣀⢀⡏⠀⢠⣧⡀⠀⠀⠀⠀⠀⣀⣠⠃⠀⡜⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠀⡞⠠⢀⣀⣀⣀⣀⢸⢁⠀⣼⣀⠈⠑⠒⠂⠒⠉⠀⣼⡀⢠⠃⠒⢄⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⡟⡌⢠⣇⠈⠙⠒⠶⠶⠖⠒⢐⢇⠁⡼⢢⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⡀⢰⠇⠁⢸⠀⠉⠓⠶⠶⠖⠒⠀⣸⣠⠴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠚⠶⠶⠶⠶⠖⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⡀⠘⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠁⡀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⡄⠀⢠⡞⠁⠱⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠐
⠀⠀⠀⠠⠞⠉⠀⠈⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⡔⠀⠈⠉⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠃⣠⣴⠶⠶⠤⣀⠀⠈⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠟⠋⠀⠀⢄⠀⢸⠀⡰⠁⠈⡣⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡀⢸⠀⠀⠀⠈⡆⢸⢀⡇⢀⣎⢼⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠉⠉⡾⠀⣿⠙⠾⣦⠈⢆⠀⣷⠀⢸⡇⣸⢁⡞⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠃⢇⢹⡀⠀⣿⡆⠸⡄⣸⣧⢸⡇⣿⣾⡇⢠⠃⡀⠀⠀⢀⡜
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠰⣜⡄⢧⠐⣿⢹⠀⡇⣿⢸⢸⡇⣿⢸⠀⡞⡸⠁⢀⡔⣿⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠠⡀⠀⢹⣧⢸⡄⢻⡀⢧⡇⡏⢸⢸⣿⢿⢸⢠⢧⠇⢰⡟⢠⡇⠀⠀⢀
⠀⠀⠀⠰⡄⠀⠀⠀⠀⢱⣰⢸⡿⡄⡇⣿⢧⠘⣇⡇⢸⢸⡇⢸⠸⣾⡟⣼⡾⠀⣸⠀⠀⢠⠃
⠀⣀⡀⠀⠈⢂⠀⠀⣀⠀⣻⣎⡇⣇⣿⣋⢸⣄⣿⡀⣼⣸⣇⢘⡀⣿⣃⣿⣁⢀⣿⡇⢠⡏⠀⢀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⣭⠞⣋⡴⢛⡽⢟⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣷⠞⣩⠶⣫⠶⣋⣴⠞⣹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢦⣤⣤⣴⡷⢛⡵⠞⣡⠞⠉⠉⠙⣧⠞⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣫⡴⢛⣥⠞⣩⠶⢋⣠⠞⣻⣄⠀⠀⢀⣿⠶⢋⡴⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠞⣡⠶⢋⡴⠞⣁⡴⠋⣡⠞⢉⡽⢛⣾⠟⣡⡴⢫⡞⠋⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⢛⡵⠞⠛⠛⠾⣏⡴⠟⣡⡶⢋⣴⠿⣥⡞⣡⡶⣿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⣿⠁⠀⠀⠀⠀⢸⣧⠞⣡⡶⣿⠀⠀⠈⣿⣡⡶⣫⡷⣤⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⣀⠀⠀⠀⠀⣸⣧⠞⢁⡴⢛⣶⠶⣾⠟⣡⠾⣫⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⣾⣿⣷⣶⣯⣴⣿⣤⡾⠷⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠻⠿⠿⠿⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠖⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣠⠞⢀⡀⠀⠀⠀⠀⣸⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠊⠀⠀⠀⠀⠀⠀⢰⢡⣄⠸⡇⠀⢇⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⢤⡀⠀⠀⠀⠀⠀⡇⢯⡔⢘⡇⠀⠀⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠆⠀⠀⠀⠈⢆⠱⠀⠀⠀⠀⠀⢸⢠⣬⠞⠀⢀⣴⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡡⠂⠀⣤⠀⡠⣣⠆⠘⠃⠀⠀⠀⡎⢼⠁⠀⠀⡇⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⡑⡄⣤⠀⢸⢱⠁⠀⣤⠀⠙⠀⢸⢨⡽⠀⠀⠀⢱⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⡇⣼⣇⣀⣀⣀⣈⡋
⠀⠀⠀⠀⡠⠔⣚⡭⡵⢶⠶⢶⠳⡳⣚⡛⡚⡛⣚⠉⠉⢹⢱⡟⣒⣶⢒⠒⠒⠭⢭⡁⠒⢄
⠀⣀⠴⡺⣄⡀⠳⢼⣜⣺⠾⠷⠓⠛⠛⣻⠛⠻⠟⡻⣟⠻⠰⠟⠍⠛⢹⡏⠛⣓⣤⠼⠀⣸⠄
⡞⠁⡾⠛⣟⢝⡲⢴⣤⣨⣩⣉⣉⡉⣉⣉⡉⡉⡉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣭⣤⠦⠾⣚⣽⠀⠀⣰⢒⡆
⠹⣶⡙⢤⡟⠭⡂⠀⠙⠋⠊⠉⠉⠻⢿⣛⣟⣉⢉⣙⡋⠉⠛⠛⠛⠒⣒⣒⣒⣋⢭⣭⣷⡟⣳⢞⢡⡿⠃
⠀⠈⠛⠓⣿⡷⣢⠀⠀⠀⠀⠀⠀⠨⠭⠀⣐⣒⣒⡒⠂⠀⠀⠀⠀⠀⢐⣒⣒⡦⣭⣰⡾⡋⣅⠾⠋
⠀⠀⠀⠀⣗⠬⣒⠀⠀⠀⠀⠀⠀⢈⡻⠥⠶⠲⠄⠒⠂⠀⠀⠀⠀⠀⢀⣒⣾⠋⣬⣠⣴⡟⠁
⠀⠀⠀⠀⢳⡭⡒⠅⠀⠀⠀⠀⠀⠀⠨⠭⠄⣐⣒⣒⠂⠀⠀⠀⠀⠀⢐⣒⣛⣿⣯⣒⢽⡇
⠀⠀⠀⠀⢸⠭⣒⠅⠀⠀⠀⠀⠀⠠⠭⠍⢂⣒⣒⠶⠆⠀⠀⠀⠀⠀⢀⣛⡒⠭⠝⣊⢽⠁
⠀⠀⠀⠀⢸⡪⢐⠅⠀⠀⠀⠀⠀⠨⠭⣒⣒⣒⣒⡒⠀⠀⠀⠀⠀⠀⢐⣒⡀⠭⢕⣊⢽
⠀⠀⠀⠀⢸⡫⢖⠭⠀⠀⠀⠀⠀⠀⡯⠴⢒⣂⣀⣐⠂⠀⠀⠀⠀⠀⢒⣒⣒⠩⠕⣒⣽
⠀⠀⠀⠀⠘⣟⢕⠭⠀⠀⠀⠀⠀⠀⣪⠭⠒⣒⣒⣒⠀⠀⠀⠀⠀⠀⠂⠒⣒⡂⠭⣂⣿⠶⢶⣀⣀⣀
⠀⠀⠀⢠⢾⣟⢵⡭⡀⠀⠀⠀⠀⠀⡨⢭⣐⣒⣂⣀⠂⠀⠀⠀⠀⠀⠭⠅⣐⠢⢭⠺⣗⣒⠒⢒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠸⡝⢭⠷⢯⣒⠀⠀⠀⠀⠀⣻⣭⣉⣉⣂⣈⠀⠀⠀⠀⢀⣀⣵⣶⣶⡿⢷⡿⠳⣏⢷⡀⠀⣀⣈⣉⣉⣉⣉⣉⣁⣐
⠀⠀⠀⠀⠈⠒⠭⣒⠤⣍⡓⠲⠦⠤⣤⣍⣉⣉⣉⠈⠀⠉⠉⣉⣟⣛⣫⡭⠷⠮⠭⣭⡭⢯⡽⠉⠩⠍⠉⠭⠉⠉⠭⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠭⣝⣒⡒⠲⠶⠶⠿⣿⣟⣻⣿⣟⣛⣛⡧⠶⠶⠚⢛⣯⠭⠿⠭⠭⠭⠥⠬⠭⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢰⡟⢻⡆⢀⣤⣄⠀⠀⢀⡴⢶⣿⣛⠳⢶⣤⡀⠀⠀⠀⠀⣠⣤⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣖⣶⣤⠤⢤⣀⠀⠐⠯⠿⠀⢀⣴
⠀⠀⠙⠋⠀⠈⠛⠛⢀⣴⡯⠩⠄⠀⠀⠄⠀⠈⠙⡲⢤⣀⠀⠀⠀⠀⠘⠛⠁⠀⣴⣶⡄⠀⡠⢊⡽⠛⢓⡾⢳⠤⠬⣇⠀⢻⡛⢦⡀⠀⠀⠘⠛
⠀⠀⣴⢶⡄⠀⠀⣠⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠊⠔⡀⠩⡳⢦⣤⣀⠀⠀⠀⠀⠛⠛⣠⠾⠒⣻⢠⠋⠉⣳⢸⣤⣀⡼⠓⠺⡅⣞⠙⢦
⠀⠀⠛⠞⠃⢀⡼⠛⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡆⠀⠀⠀⠀⡴⣧⠤⣞⣽⠃⢙⠮⣵⠃⠀⠈⣷⣤⡤⡿⠾⣗⢮⠳⡄
⠀⠀⠀⠀⢀⣞⣡⣬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠀⠀⠀⢀⣾⠗⠋⢹⣆⠎⠀⣇⡀⣸⠶⠤⠖⠡⡞⠙⢳⡀⠈⣿⠓⢾⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⡿⠿⠛⠛⠉⠀⠀⠀⢀⣴⡧⠻⠤⠖⢋⣾⣀⣀⣠⣭⣥⠤⠤⠤⠤⠽⠶⣶⣿⣶⣛⣦⡼⢇
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢻⡋⠋⠛⠉⠉⠛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⠶⡒⠛⡋⠉⠉⣁⣠⠤⠐⠒⣒⣂⣒⣒⠂⠤⣄⡀⠀⠉⢙⠉⠛
⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠐⠂⠀⢰⡏⠀⠀⡀⠋⣄⠊⠰⠸⡇⠀⠰⠋⣉⣀⣀⡉⡱⠀⢨⠎⡆⠈⡄⡁⡄
⠀⠀⠛⠛⠀⣤⣦⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡇⢶⣶⣾⠇⠀⠀⠀⠀⠈⠳⣄⡀⠐⠀⠈⠒⡠⠈⠘⢓⣒⣿⠛⠛⠛⣿⠳⠛⠓⠩⠐⠊⠐
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡇⠀⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⡉⠛⠒⠲⠶⠿⠿⠶⠶⠦⣿⠀⠀⠀⣿⠤⠤⠤⠤⠦⠴⠶⠖⠚
⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⢴⡷⢹⡀⠀⠀⠀⣷⠀⠀⠀⢀⣴⣿⠏⠉⠁⠈⠉⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠀⠀⠀⢀⣠⣄
⠀⠈⠉⠁⠀⠀⠀⢀⣀⡤⣄⣀⣀⢳⡀⠀⠀⠘⡆⢀⡴⢫⣿⠃⠀⠀⠀⠀⢀⠤⠐⠚⠙⢷⡄⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⣅⣹
⠀⠀⠀⠀⠀⣠⢶⣉⡿⠦⠾⣄⣀⣿⢧⠀⠀⠀⣹⠟⡣⡟⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠉⠁
⠀⢀⣀⢀⡴⢛⣇⣠⠞⠂⠘⢧⡀⠀⢹⡆⠀⣴⠋⠀⠁⠀⠀⠀⠀⠀⡀⣀⣀⣀⣤⣤⣶⣾⠿⠋⠀⠀⢰⠇⠀⠀⠀⣿⠀⠀⠀⣀⡤⠖⣾⣩⠿
⠀⠈⣹⣿⣩⣏⣀⣹⡀⢀⡀⠮⣨⠛⣍⢹⠘⢧⣄⣀⣃⣠⣤⣶⠶⠿⢿⡿⠿⠟⣻⠟⠋⢳⡀⠀⠀⠀⡼⠀⠀⠀⠀⡇⢠⣴⣾⣵⡒⠏⢤⢞⣒
⠀⠀⠻⣾⣾⣧⣦⣬⣭⡉⢠⢤⣷⣾⣿⣾⡆⠀⠹⣹⡏⠉⠉⢻⠀⠀⢸⡇⣤⡚⢁⠢⠀⠀⢻⡄⠀⢰⠃⠀⠀⠀⢠⡇⠈⠉⠛⠻⠿⣿⣿⣶⣤
⠀⢀⡀⠀⠉⠙⠛⠛⠛⡆⠀⠀⠟⠛⠉⠁⡇⠀⠀⢿⡇⠀⠀⠘⡇⠀⢈⡟⠻⠿⢿⠿⡿⢷⣦⠗⠀⡞⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⡍⠉⢻
⠀⠈⠘⠀⠀⠀⠀⠀⠀⡉⠀⠀⡄⠀⠀⠀⡇⠀⠀⢸⣇⠀⠀⠀⣧⠀⠸⡇⠀⠀⠸⠀⠇⠀⠀⠀⢠⡇⠀⠁⠀⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⢀⠇
⠀⠀⠀⠀⣀⣤⢤⣀⠀⡇⠀⠀⡇⠀⠀⠀⡇⠀⠀⠘⣿⠀⠀⠀⡏⠀⢠⡇⠀⠀⡇⢰⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠸⣇⣴⠟⠉⠉⠽⣳⡄⢸
⠀⠀⠀⣼⠃⠀⠈⠪⢳⣄⠀⢀⡇⠀⠀⠀⡇⠀⠄⠀⣿⠀⠀⢰⠃⠀⢸⡇⠀⢀⠇⢸⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣰⣟⣁⠂⠁⠀⠀⠀⠙⢾
⠀⠀⢸⣏⠃⠀⠀⠀⣠⡼⠆⢸⡇⠀⠀⠀⡇⠀⠀⠀⢹⡇⠀⡞⠀⠀⢸⡇⠀⢸⠀⢸⠀⠀⠀⠀⡏⠀⢠⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢿⣵⠧⣶⡾⢻⡸⠀⠀⠀⢇⠀⠀⢸⡇⠀⠀⡇⠸⡷⣰⠃⠀⠀⠸⡅⠀⡾⠀⠈⡄⠀⠀⣸⠁⠀⠸⠀⠐⣏⠂⢤⡀⢀⣒⡒⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⢸⠉⠀⢇⠀⡀⣄⡼⢀⠀⠘⡇⠀⠀⠈⠀⢳⡟⢰⠀⠀⠀⢣⡐⠧⠂⣤⣿⠀⣵⠃⠀⠀⠀⠀⠀⠈⠛⢲⣾⣦⠤⠭⢴⡖⠚⣩
⠀⠀⠀⠀⠐⢾⠀⠀⠘⡷⢿⣿⣦⠒⠻⢶⡹⡐⡄⢇⢸⡆⠗⢁⡄⠠⠀⠀⠙⢿⡿⡿⢿⡟⠁⠀⠀⠀⠀⢠⡀⠶⠄⠘⣻⡇⠀⠀⠸⣸⠎⠠⣊
⠀⠀⠀⠀⠙⢾⣤⣀⢀⡼⠇⠀⠀⠁⠀⠒⠿⢿⣷⣌⣼⣷⠖⠨⢶⣷⠋⠚⠁⠊⠁⠁⣿⠀⡀⠀⢀⣠⠀⢛⠕⠀⣰⣶⡟⠀⢀⣀⠀⠹⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠈⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠈⠛⠉⠀⠛⠁⠀⠀⠁⠀⠛⠛⠶⠛⠀⠁⠘⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠜⠁⠁⠀⠀⠀⠀⠆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⣃⠠
⠀⠀⠀⢀⠀⠀⠀⠌⠀⢀⠔⠱⠋⠅⠀⠀⠀⠈⡁
⠀⠀⠀⠈⠢⢀⠀⠄⠠⢀⠘⠀⠰⠇⠀⠀⠰⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠈⠧⠠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠤⡀⠁⠁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠔⠀⢢⠑⠀⠀⢀⡀⠁⢈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡔⠀⠸⠤⡄⠄⠀⠀⠈⢄⠀⠀⢀⣤⠄⠀⠐⠀⠢⣞⠆⠀⠀⡠⡄
⠀⡀⠀⠀⠀⠀⡀⠀⠀⣆⠄⢨⣣⠀⠈⡄⠀⠀⠀⢀⠈⠀⠨⣄⠈⡁⠀⠄⠐⠀⢀⡀⡄⠀⠀⠈⠡⣄⠀⡆⠈
⡌⡀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⢸⠂⢀⠃⠀⠐⠀⠾⣄⠀⠛⢁⠓⠀⠂⠀⠀⢀⡀⣄⠀⠂⡄⣊⠀⠐⠋⢁⣠
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠊⠁⠀⠀⠀⠀⠀⠾⡙⡶⡤⢤⣈⣉⣉⣉⡀⠉⣉⣉⣉⣀⣠⣤⣴⠾⠛⠉
⠘⠤⠀⠠⠀⣄⠀⠀⠀⠀⢀⢠⠉⠉⠉⠉⠉⠉⠈⠉⠀⠈⠉⠉⠉⠉⠉⠋⠙⠛⠻⠛⠟⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣉⣢⣤⣤⣔⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣩⠔⠚⠓⠢⣍⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⡥⠒⠉⢀⣠⣼⣦⣄⡀⠉⠲⢌⡙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⣈⠳⢤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠴⢋⡠⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠑⠢⣌⡙⠿⣿⣿⣿⡿⠟⢉⠤⠒⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠉⠒⠤⣉⣡⠔⠊⠁⠀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⣧⣠⣄⡀⠀⢸⠀⠀⠀⢠⠿⢄⡀⠙⠲⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⣿⣿⡄⢸⠀⠀⠀⠀⠳⢄⡈⠑⡶⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⣀⣘⠀⢁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢿⡿⣿⣶⣿⠿⣶⣄⡀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣿⠟⠁⠀⢸⠀⠀⠀⢠⠛⢅⣉⠀⢸⠀⣿
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣛⣻⣿⡿⢟⣿⣻⢿⣿⣟⣿⡿⣷⣤⣀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⢄⡉⠀⠸⠀⣿⣶⣤⣄⡀
⠀⠀⠀⢰⣿⡿⣷⣾⣯⣿⣿⣯⡺⠿⠿⠿⢿⣪⣿⣿⣾⣟⣿⣿⢿⡆⠀⠉⠓⠦⣄⡀⠀⠀⢸⠀⠀⠀⢀⣠⣴⣿⣷⣿⣾⣿⣿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣷⣭⣻⢿⣭⣿⢿⣷⣾⣶⢿⣿⣭⣿⠿⣻⣽⣾⣿⣷⠀⠀⠀⠀⠀⠙⠳⢤⣸⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢸⡏⠻⣿⣿⣿⣷⣯⣛⠿⣿⣟⣻⡿⢟⣫⣷⣿⣿⡿⠿⠋⣿⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⠟⠋⠁⠈⠁
⠀⠀⠀⠘⠳⣄⡀⠉⠙⠻⢿⣿⣿⣾⡍⣽⣾⣿⡿⠛⠁⠀⠀⣀⣤⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀⠀⠙⠻⢿⣧⡿⠟⠁⠀⢀⣠⣴⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⢸⠀⣀⣤⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠿⠿⠃⠀⢾⣿⣿⡇⠀⠀⣩⣽⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣵⣶⣄⠀⣴⣶⣶⡄⠉⠉⠀⠀⠘⢿⣿⡿⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣏⠀⠛⠿⠟⠁⠀⣾⣿⣿⡆⠀⢀⣀⠀⢰⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⡟⠻⢟⣛⠻⣿⣿⣿⣦⣀⣴⣿⣷⡌⠛⠛⠁⠰⣿⣿⡇⠈⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⢀⣠⡄⠀⠀⠀⢸⡁⠙⠒⠦⢤⣉⣘⣛⣿⣿⣿⣿⣿⡿⠁⣰⣾⣶⡄⠈⠉⠁⣰⣶⣶⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣌⣯⣼⣿⣿⠁⣀⠀⠀⠸⡏⠓⠒⠢⠴⠤⠤⠤⠶⣻⣿⣿⣿⣿⣶⣿⣿⣿⣇⣴⣶⣦⠘⠛⠛⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⣿⠟⠛⣵⣾⣿⡆⠀⠀⠹⣖⠒⠒⠒⠒⠚⠋⠉⢁⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣷⣆⣹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡸⣆⠸⣿⣿⣿⣅⣀⣀⠀⠈⠓⢶⡄⠶⠒⠂⣉⣴⠇⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣆⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠉⠓⢾⣍⣡⠟⠀⠀⠀⠀⠀⠉⢛⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢯⣀⣈⣭⣭⣥⣤⣤⣀⡀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷⡈⢦⠙⢦⡙⣮⣱⡤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢿⣿⡟⠁⠉⣿⣿⣿⣦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠚⠛⠋⢉⣿⣅⠀⠀⣠⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣄⣀⣰⣿⣿⣶⣶⠟⠛⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠘⢿⣿⠇⣸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⣿⣿⠋⠉⢻⣿⣿⣀⣀⣼⠟⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣤⣤⣤⣤⣀⣀⣽⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⣀⣾⣿⣄⣠⡼⠿⣿⣿⣿⣿⣦⣠⣾⣷⡀⠀⠀⠀⠀⢀⣴⣾⠛⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⣿⡀⠀⢀⣿⣿⡁⠀⣹⣿⣿⣿⣦⠀⠀⢰⣿⣯⣨⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⡈⠠⠐⢊⢝⣿⢻⠿⠿⣿⣿⣿⣿⡿⢿⠛⠛⢛⡞⠀⠀⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠐⢊⡵⢋⠼⣿⠀⠀⠀⠀⠀⢧⣁⠂⣬⡽⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢧⣼⡿⠀⠀⠀⠀⠀⠈⣿⠋⠁⠀⢀⣼⠃⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⣰⢠⣿⡏⠀⣀⢀⣞⡴⠋⠙⠻⠿⠿⢿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣸⣿⣾⠃⢀⠀⠀⠀⣄⠀⠰⣿⣿⣾⣿⣇⣾⣿⡾⠋⠀⠀⠀⠀⠀⠀⣸⣥⣴⡶⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣿⣿⣿⣿⣿⣿⣿⡀⢸⣷⣀⠘⣿⣄⢀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡆⣀⡤⠀⣀⣴⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠳
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣰⣿⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⣿⣿⣿⡿⠿⠷⠾⣿⣿⣿⡟⠻⢿⣇⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠸⠛⠉⠁⠀⠈⠙⠿⠙⠷⠀⠙⢿⠃⢸⡿⠋⠁⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠈

⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⠋⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⢀⡴⠒⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢠⡀
⠀⠀⠀⠈⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⣀⡀⠀⠀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡶⠟⠋⠁⠉⠲⠀⢀⠈⠹⠄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢩⡀⠀⠀⠐⡀⢸⠀⡆⠀⡀⢫⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠤⠴⢤⡖⢹⡷⣤⣇⠀⢄⠀⣷⠘⢰⡇⢸⠗⡿⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⣴⠀⠀⠀⠸⢡⠸⡇⠀⢻⡇⠘⡄⢸⣄⢸⡇⣿⣼⠃⢀⠃⡀⠀⠀⢀⡎
⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⣄⠆⢱⠀⢸⠹⡀⡇⢸⢹⢸⡇⣿⢹⠀⡞⡰⠀⠀⣤⡾
⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠘⣿⠈⡆⢸⡆⢣⣷⢸⢸⢸⣷⣗⢸⢰⢣⠃⣠⠎⢸⠃⠀⠀⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠘⡄⡄⣯⣇⢳⢸⢧⠈⡇⡇⢸⢸⡏⢸⢸⣼⡟⣸⡟⠀⡟⠀⠀⡰⠁
⠀⠀⠀⠀⠳⠄⠀⠀⠀⠸⣧⣿⢸⢸⣸⠘⡆⣿⠃⢸⢸⡇⠸⠘⣿⢳⣿⠁⠀⣷⠀⣰⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣻⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡒⠓⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢸⡴⠒⢲⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡜⢀⠎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡴⢹⠇⢀⡞⠀⢠⠇⣰⠉⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡜⠀⡘⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡼⢱⡇⡀⠺⠤⠤⠋⠀⠓⠚⣁⡤⠤⠤⢴⢀⡞⠋⠉⠉⡉⢰⠁⡸⠁⡌⢉⠇⡰⠋⠃⠒⠲⡄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠰⡇⠀⠻⣞⡆⠀⠀⡰⠉⡲⠀⠧⣤⠀⢀⠎⢀⡽⠋⡠⠞⢡⠃⢰⠃⡸⠀⡜⢰⠁⡰⢺⠃⣸⢁⠴⠋⣀⣀⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠙⠦⣤⡃⠀⠀⢰⠃⢠⠃⢠⠞⢁⠔⠁⡰⠋⠀⠾⢰⠀⡞⢠⠇⢠⠃⡸⢀⠃⣰⢃⡏⢠⠇⡏⠀⣞⡡⠦⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠹⡄⢠⠏⢠⠏⡰⠁⠀⢉⡹⠐⠧⠀⠒⠒⠋⠀⠉⠉⣀⣀⣉⣀⣀⣉⠁⠈⠙⠊⠀⠣⣤⣀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⠀⠀⡇⢼⣀⠞⠀⠗⠒⠋⠉⠀⠀⠀⠀⠀⠀⢠⣎⣉⣉⣭⣄⣤⣤⣤⣍⣉⣉⣙⠓⠒⠤⠤⠃⣰⠁
⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⢠⠟⢤⣸⠁⠀⣴⣶⣶⣶⣦⠀⣶⣶⣶⣶⣶⢰⣶⣶⣿⣶⠄⠀⢀⣶⣶⣦⠀⠀⢠⣾⣿⠙⣲⣶⡖⣡⣶⣶⣶⣦⡄
⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⣠⠏⠀⣸⡧⡄⢰⣿⣿⠋⣿⣿⠀⠛⣻⣿⡟⠋⢸⣿⣿⠛⠛⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣰⣿⡿⠀⣿⣿⡟⢻⣿⡇
⠀⠀⠀⠀⣟⠀⠈⠧⠤⠚⠁⢀⡴⠃⡷⠇⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⢸⣿⣿⣿⡿⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠘⠲⢤⣀⣀⡠⠔⠋⠀⡞⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⢻⣿⣇⠀⢸⣿⣿⣿⣇⠀⠀⠻⣿⣷⣄
⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣽⣿⡿⣿⣤⣤⡿⡿⣧⣤⣼⠿⠿⠿⠿⣦⣴⡿⣿⣼⣿⣿⣤⣼⣿⢿⣿⣿⣤⣤⣤⣬⣿⣿⣿⣦
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⡇⠉⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⡏⠁⠘⠛⠈⠉⣿⣿⠀⢸⣇⠉⢻⣿⣿⣿⣿⡯⠛⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢱⣿⣿⣿⠛⢻⣿⡇⠀⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⣴⣶⡄⠀⢻⣿⠀⢸⣿⡀⠘⣿⡟⠛⣿⡇⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣮⢻⣿⣿⡀⠈⠉⠀⢀⣿⣿⡇⠀⣿⣿⣿⠀⠈⠉⠉⣿⠀⢰⣿⣿⣿⠀⢸⣿⠀⢸⣿⣇⠀⢻⡇⠀⠉⠁⠀⣼
⣿⣿⣿⡿⠿⠿⠛⡟⢻⣿⠿⠿⠿⠟⠘⢛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⣿⣿⣿⣿⣿
⣿⣿⣿⢁⠒⠗⢨⣥⣾⣶⣿⣿⡿⣇⣙⡋⢭⣿⣿⣿⣿⣶⣶⢦⣼⣕⡂⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⡿⠣⢿⡟⣸⣿⣿⣷⡿⢿⡏⠜⣽⣿⣿⣿⣿
⣻⣿⡧⣄⠱⠄⣹⢿⡏⣛⠛⣂⣭⣶⣶⢜⠻⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡧⡪⣳⣿⣿⣿⡿⢛⣉⡉⢀⣶⡏⠸⠉⡿⢱⠏⠼⣋⣴⣌⣵⣿⣿⣿⣿
⣼⡟⢘⡕⣪⣧⡼⢤⡑⢀⣈⣩⣭⣭⣭⣼⣿⣿⢿⣛⣙⠀⣽⣿⣿⠛⠛⠺⡴⠁⢻⣿⣿⣿⣿⡿⢡⣿⣿⣀⣾⣤⣶⣶⣿⣿⢿⣿⣿⣿⣿⣿⢿⣿
⣿⣷⡘⣻⠞⣱⢱⢇⡻⣿⣿⡟⣿⠏⠛⢛⣋⣙⣉⣢⣩⣤⣾⠿⠄⠄⠐⠊⠀⢀⣿⣿⣿⣿⡿⣷⣾⣿⢟⣭⠛⣿⣿⣿⣿⠿⡿⣫⡭⢋⡼⢁⣾⣿
⣿⣿⣿⣦⣙⠾⠈⠺⢇⣉⣉⠉⠉⠉⠉⠉⠉⠘⠛⢛⣻⣿⠷⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⠣⣶⣾⣿⣇⣼⡏⣰⡏⡼⠱⣋⣸⣤⣭⣾⠶⠂⣘⣻⣿
⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⢁⣾⠍⣽⣿⢟⡋⣰⣿⣷⣿⣿⣿⣿⣿⣟⣰⣫⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣵⣾⣿⡧⢞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠙⠶⠦⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⢀⣰⡩⠂⠀⠀⠀⠀⢠⣞⣠⣴⡂⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢧⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⣇⡀⠀⠀⠐⠒⣞⠉⠁⠀⠀⠀⠀⢀⣠⠤⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⢹⠁⠀⠀⠀⠊⢈⢦⡀⠀⠀⣠⠖⢋⠀⠀⠀⣂⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠸⣄⠀⠀⠀⠀⠁⣠⣷⣄⣼⣅⠀⠀⠶⠀⠀⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠈⣷⣶⣦⣴⣶⣿⣿⣿⠟⠿⠂⠀⠀⠀⠄⠚⠀⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣌⠓⠀⠀⠸⢆⡀⠨⠄⠺⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣛⣛⣟⣿⣛⣿⣿⣛⣦⣤⣤⡀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠀⣾⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠾⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠟
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠙⠲⠶⠶⠚⣹⣤⠼⠷⠛⢻⠈⠉⠙⠛⠿⣿⣿⣿⣶⣶⣶⡀⠐⠶⠶⢶⡶⠶⠒⠒⢒⡶⠒⠊
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠐⢧⣀⡀⠀⠀⣠⠶⢿⣁⠤⠀⣠⠟⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠠⣴⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠉⠉⢹⠉⠶⣄⣹⠆⠉⣀⠴⠊⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⡏⠛⠷⣦⣤⣤⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠑⠒⠲⢶⠶⣶⠺⡘⢆⢸⢹⠀⠀⠈⠢⡀⠀⠀⠀⣈⡆⠀⠀⢱⡀⠙⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢰⠙⣆⠰⡑⢷⡀⡸⣺⠀⠀⠀⢀⡜⠀⠀⠀⠘⢷⡆⠀⠀⠇⢠⡏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡄⠀⡜⠠⡿⢤⣀⠈⠛⣥⠇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠸⣿⣿⣿⣿⡿⢁⡴⠀⠀⠀⠈⡿⠚⠧⠰⠧⡀⠈⠉⠒⣾⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⢿⣿⣿⡿⣡⠏⢀⣤⠤⢶⡞⠉⠀⠀⠦⢀⢹⠀⠀⠀⣿⠊⣀⡤⠤⠀⠀⠀⣤⣄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏
⠀⠈⢿⣿⣷⠃⠀⠁⠀⣠⠾⠀⠁⠀⠠⢀⢈⡇⠀⠀⠀⡸⢿⡁⠰⡶⠀⠀⠀⠈⠡⠀⠀⠀⠀⢹⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠈⢿⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠢⢤⠟⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠘⡇⠀⠢⣼⣷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣴⡄⠀⠀⠀⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠱⣄⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣶⡆⠀⠀⠈⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣿⣷⣶⣶⣶⣾⣶⣶⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣴⠏⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢿⣆⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠈⢿⣦⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⢻⣇⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠋⠀⠀⠤⠤⠫⠤⠀⠀⠀⢉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢂⣉⣩⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣌⣉⣁⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⡷⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⢸⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠊⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⠐⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠡⠤⠤⣔⣂⠁
⠀⠀⠀⠀⡤⠒⠊⠉⣁⣤⣴⣶⣶⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣶⣶⣶⣦⣤⣈⠉⠑⠒⢤
⠀⠀⠀⠀⠑⠢⠤⣀⣉⠛⠻⠿⠿⢿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⣉⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢖⡐⠒⠀⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠄⠀⠒⢂⡲⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⡆
⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠒⠛⠒⠶⠶⣤⣀
⣠⠴⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣤⣶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳
⣿⠦⣤⣀⣀⣀⣠⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⠀⢀⣺⣿⡦⢾⡇⣿
⢿⠀⣀⡈⠉⠉⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠋⠉⠁⠀⠀⣾⠶⠿⠶⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⠞
⠸⡇⠈⠉⠙⠓⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠛⠋⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠙⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠟⠉⠉⠀⠈⠃⠉⠻⡟⠟⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠋⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⣿⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢜⡵⠋⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠔⡵⠃⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⠁⠈⢳⡀
⠀⠀⠀⠀⢀⠎⠎⠀⠀⠀⠀⠀⣇⠀⠀⣀⣠⣄⣠⣄⣠⣠⣴⡦⣤⣶⠴⢦⠴⠦⠴⣧⠤⢴⣼⡟
⠀⠀⠀⠀⢸⣴⣾⣶⢶⣶⣾⣛⣛⣋⣛⣿⣿⣿⡟⠛⠻⢿⣿⣿⣷⣦⣤⣠⣤⣄⣤⣶⠿⠛⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠛⠛⠿⠿⠿⠿⠿⢿⠁⠀⠀⢸⣿⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡤⣀⣀⣿⠟⣩⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣟⠛⢛⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠉⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠿⢿⡻⣶⣄⠀⡇⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⢻⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣸⠿⣾⡇⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⢏⠀⠘⣷⡀⡞⡇⢠⠀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠽⠟⠻⠻⠮⠼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠂
⠀⠀⠀⠀⠀⣠⣤⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⠀⡄
⠀⠀⠀⠀⡺⢽⣅⣀⠍⠁⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⢐⠄
⠀⠀⠀⠈⠃⠒⠀⠓⠲⡐⠰⠄⠀⠀⠀⠐⠛⠀⠀⠀⡄⠀⠀⡠⠔⣀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠠⠀⠀⡀
⠀⠀⠀⠈⠀⠂⠠⢏⣠⡶⠆⠀⠀⠀⠀⠀⠘⠶⠄⠠⠄⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠳⢆⠀⠀⠄⠽⠃
⠀⠀⠀⠀⠀⢀⠤⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠊⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⠟⠉⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣾⣿⠻⡆⠘⢿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣼⣿⣿⣿⡿⠋⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⢱⣿⠿⠁⠀⠁⠀⠀⠻⣿
⣿⣿⣿⣿⣿⡟⣿⣿⣿⡟⢸⣿⣿⣿⣿⣇⣾⣿⣿⣿⠟⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⢠⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⠁
⣿⣿⣿⡿⣿⣎⣿⣿⣿⣾⣿⡟⣸⣿⣿⣿⣿⠃⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡟⠻⡟⠀⠀⠀⠀⠀⡀⣤⡀⠠
⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠃⢀⣴⠏⠀⠀⠠⠂⠀⢀⣤⣠⣴⠞⣽⣿⣿⡏⠀⣴⡇⠀⠀⠀⠀⠀⡇⢹⡇
⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⢠⡿⠃⠀⠀⠀⣠⣴⣾⣿⣿⣿⣣⣾⣿⣿⣿⠀⣼⣿⣧⣤⠀⠀⠀⠀⠁⢸⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⡆⡀⠀⠀⠀⢸⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⡿⣧⠃⠀⠀⠀⠸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣷⣼⣿⣿⣿⡿⢨⣿⢿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣧⠟⠀⠀⠀⡀⠀⠁
⣿⣿⣿⣿⣿⣿⣻⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣯⣾⣿⣿⣿⣿⡟⠁⢸⡞⣿⣿⣿⣿⣧⠀⠀⢠⣇
⣿⣿⣟⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⠿⣿⠟⠀⣿⣿⣿⡿⣿⠛⣿⣧⡄⠀⢇⠈⠻⢿⡇⠀⠀⠀⠘⣿
⣿⣿⣿⣿⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡟⠀⣠⢿⣿⣿⣿⣿⠀⠈⠻⣷⡀⠘⠆⠀⢸⣷⡀⠘⠀⠀⢻⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠇⠀⡇⠈⣿⣿⣿⢹⡆⠀⠀⠹⣧⠀⠀⠀⢸⣿⣷⡤⠀⠀⠈⣿
⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⣾⣿⣯⠇⠀⠀⠀⠙⡆⠀⠀⠀⢿⣿⣧⣇⠀⠀⢹⡆
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢰⣿⠃⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⠀⠀⠀⠀⢰⡄⠀⠀⠈⢻⡿⣿⡀⠀⢼⠁
⣿⣿⣿⣿⣿⣿⣿⣟⣁⡀⠟⠁⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⠳⠀⠀⠀⠘⣇⣿⣧⠀⢸⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣾⣿⡟⢸⣷⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣷⣀⠀⣀⣤⣴⣄⣀⣤⣴⣿⣿⠀⠸⡇⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⣿⡗⠀⠀⣿⠟⠿⠛⠛⠻⠿⠿⠛⠛⠛⠉⠁⠀⢸⣿⡟⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⠸⠛⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣴⣟⠋⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⠻⢦⣀⣀⣀⣀⣴⣿⣿⡧⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡷⣰⡌⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠻⣿⡟⠛⠛⠉⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⡟⣿⣿⣿⡿⣟⡿⠛
⣿⣿⣿⣿⣷⡜⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡚⢿⣿⣿⣶⣶⣾⣿⣶⣶⣶⣶⠂⣠⣤⣾⣿⣿⣿⣿⣿⠇⠿⣿⠿⠚⠉
⣿⣿⣿⣿⣿⣿⣞⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣾⡖
⣿⣿⣿⣿⣿⣿⣿⣯⠻⣆⠈⠛⠿⣿⡿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣉⣉⣭⣿⣿⣿⣿⣿⠿⠟⠫⠐⠋⠁⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣷⣦⡀⠀⠉⠲⠬⣝⣛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⢀⡀⠀⠀⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⠀⡰⠁⠀⠐⠀⠀⠈⠀⠀⠙⠒⠚⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠉⠛⠋⠀⠀⠀⠈⠲⠴⠊⠀⠀⣀⡀⠀⠀⠀⡸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣎⠁⠉⡃⠀⠀⢄⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⡹⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣠⠀⢠⠖⠒⢦⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠐⠆⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠚⠁⠀⠰⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⠀⠘⠦⣤⡤⠂⠀⣠⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠠⠒⠙⡆⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⡷⠶⣶⣶⣤⣧⣀⣀⠀⠀⠀⠀⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡀⠈⠒⢊⣤⣴⠾⠟⠟⠙⠉⠃⠉⠀⠀⠈⠻⣧⠀⠀⠀⠀⠉⠋⠻⣿⠷⣦⣄⡓⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣙⣠⣾⠛⠫⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⣇⣤⣼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠁⢠⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣇⣘⣀⣀⣀⣴⣿⠟⠁

⠀⣦⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠠⢞⡁⠀⠊⠀⣦⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⢞
⡄⣇⢸⡏⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠐⠃⠀⠀⠑⠀⠀⡔⡇⢸⠏⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠹
⠁⢹⠀⠞⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡠⠀⠈⡀⠁⢹⠀⠞⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡠⠀⠈
⠀⠀⣀⣤⠞⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡄⠀⠀⠀⠀⢀⠁⢀⣴⠞⠂⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⡀
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠁⠀⠀⠃⠀⠀⠀⢀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡔⠀⠠⢉⠀⠀⠃
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠋⠀⠀⠀⢸⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢀⠀⠀⠀⠸⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠸⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠸⠷⢠⡄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠹⠖⢠⡄
⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠦⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠆
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠈⠐⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢦⡀⠀⠈⠀⠂⠠⢀⠀⠈⠒⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢢
⡀⠀⠀⠔⡀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠊⠀⠀⠑⠀⠀⠀⠴⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠁⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢑⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡑⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠘⠁⠀⣠⠤⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠒⣶⠀⠁⠘⠁⠀⣀⠤⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⡀⠀⠀⠙⠘⣶
⢀⣤⠞⠁⠀⠀⠀⠑⡶⠀⠀⠀⠀⠀⠀⠀⢀⣆⠰⠋⠀⠀⠀⠀⠀⠀⢀⡠⢀⡤⠞⠁⠀⠀⠀⠑⡶⠀⠀⠀⠀⠀⠀⠀⢀⡄⡰⠋⠀⠀⠀⠀⠀⠀⢀
⠞⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠠⣰⠄⠈⠀⠊⠀⠀⢀⡄⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⢸⡴⠋
⠀⡷⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡠⠄⠡⢄⠀⠁⠀⡗⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡠⠘⠫⢄
⡇⣧⠸⢃⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠘⢠⠀⡎⣧⠸⢃⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⢠
⠀⠘⠀⠋⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠤⠀⡸⠀⠋⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠌
⠀⠀⣰⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡜⠀⠀⡆⠀⠀⠀⠀⠀⠀⣰⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⠁⠀⡆
⡀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣠⠎⠀⠰⢠⡀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⠆⠀⠐⢠⡀⠀⠁
⡵⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠁⠀⠀⠀⢸⠇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠆⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⢛⠐⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠘⢛⠰⠂
⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠡⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⠁⡀⠠⠄
⠀⠀⠀⠀⠀⣠⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠀⠀⠀⠀⠀⠈⣠⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠑
⠢⠀⠀⠘⡄⠛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠢⠀⠀⠐⠄⠛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠰⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⠀⠀⠀⡤⠀⠠⠀⠀⠀⠀⠀⠰⠀⠐⠁⠀⠀⠠⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠄
⢂⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠈⣠⠁⠈⠁⡌⢂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣁⠁⠈⠁
⠈⠀⢀⡴⠒⠀⠀⠈⡐⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡰⠀⠀⠀⠀⠈⠟⠁⠀⠈⠀⢀⡴⠒⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⣰⠄⠀⠀⠀⠈⠛⠁
⢀⠶⠋⠀⠀⠀⠀⠐⠿⠀⠀⠄⠀⠀⠀⠀⠰⠦⠙⠁⠀⠀⠀⠀⢀⠀⣠⠖⣠⠾⠋⠀⠀⠀⠀⠐⠿⠀⠀⠄⠀⠀⠀⠀⠰⠆⠙⠁⠀⠀⠀⠀⠀⠀⢠
⣦⣤⣤⣴⣶⣤⣶⣤⣤⣤⣶⣦⣤⣤⣤⣴⣶⣤⣤⣴⣶⣦⣤⣶⣾⣶⣧⣶⣷⣤⣴⣴⣷⣴⣶⣤⣦⣤⣶⣦⣤⣤⣤⣴⣶⣤⣤⣶⣶⣤⣤⣶⣶⣶⣦
⣿⣿⣿⠋⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠛⠛⠛⠓⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠛⢛⡛⣛⡛⠛⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⢛⣩⣭⣯⣿⠿⣿⣭⡉⠉⠉⠛⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⡭⠥⢬⡀⠁⣸⠀⠐⠂⠀⠀⠒⡂⢹⡖⠀⠀⢠⠖⠒⠻⢿⣟⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⠞⠉⠀⠀⠀⣿⠀⢸⠀⠀⠀⠂⢀⠀⠀⢸⡇⣆⠀⣿⠀⠀⠀⠀⠉⠳⣌⠳⣄
⠀⠀⠀⠀⠀⣠⠏⣠⠞⠁⠀⠀⠀⠀⢀⡿⠀⡿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠀⣛⠀⠀⠀⠀⠀⠀⠈⠳⡌⢷⡀
⠀⠀⠀⠀⣰⠃⣴⠷⠶⠶⠶⠶⠶⠶⣏⠀⠀⡿⠒⠒⠒⠒⠒⠶⠟⠳⢾⠀⠐⣿⠀⢀⠀⠐⠄⠀⠀⠀⠙⣆⢳⡄
⠀⠀⠀⢰⠇⣸⠃⠀⠀⠀⠀⢀⠀⠀⠈⢷⠀⡏⠀⠀⠀⡀⠀⠁⠨⠅⠈⡇⠀⣿⠀⠁⠀⠀⠀⠄⣀⠠⠡⢹⡄⢷
⠀⠀⠀⡾⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢼⡇⠸⡆
⠀⠀⢀⡇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣒⣂⣼⠀⣇⣐⣠⣤⣤⣤⣀⣀⣀⣠⠇⠀⢹⣄⣀⣀⣀⣀⣀⣀⣀⣊⡼⢻⡄⢳⡀
⠀⠰⢯⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⡭⠭⠥⠤⠬⠭⠭⠭⠭⣭⣭⡭⠭⠥⣤⣤⣤⠭⢭⣭⢭⡭⡭⠭⢭⠍⠠⠤⠥⣤⠷⢀
⣎⣲⠼⡀⠀⠛⠛⠛⠛⠛⠛⠲⠶⣄⠀⠀⣶⡇⠀⠐⡟⠛⠋⠉⠛⠻⢦⠘⠋⠉⢹⠋⠉⠉⢉⠛⠀⠀⠀⠀⢀⣆⢸⠲⣉
⠈⠁⠀⣇⠘⣇⠀⠀⠀⠀⠀⢸⠀⣽⠀⢸⡟⡇⠀⠀⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣸⣇⡏
⠀⠀⠀⠘⡄⢻⡀⠀⠀⠀⠄⠀⠀⣿⠀⣼⠁⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢹⠀⠀⠀⠀⢠⡿⣼⠁
⠀⠀⠀⠀⢻⡀⢳⣀⣈⣀⡀⠀⣀⡟⢠⡏⠀⣻⠀⢀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⡇⠀⠠⠸⡄⠀⠀⢠⡿⣱⠇
⠀⠀⠀⠀⠀⢳⡄⢻⡉⠉⠉⠉⠉⠀⡼⠛⠋⢻⡀⢸⡷⢶⣖⠶⡶⠒⠋⢀⠀⠀⠀⡇⠀⠀⠀⠘⠶⢶⡞⣱⠏
⠀⠀⠀⠀⠀⠀⠻⣄⠱⣄⠈⠀⠀⢰⠇⠀⠀⠸⡇⢸⡇⠁⠙⢷⡁⠐⠂⠡⠀⡀⠀⡇⠀⠀⠀⢀⡴⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣜⢦⠀⠀⡟⠀⠀⠀⠀⣷⢸⡇⠀⠃⠀⠹⡄⠀⠀⠀⠀⠀⡇⠀⣤⠔⢋⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣘⠃⠀⠀⠀⠀⠹⢸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠁⣠⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠗⠲⢦⡤⠤⠌⣁⣀⣀⣠⣠⠤⠤⣤⠶⠒⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅⣀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣴⣿⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠐⠲⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⠇⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⢀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠤⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⡀⠀⢀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡃⢻⡋⣿⣖⢒⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⣩⠿⣧⡼⠻⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣞⢙⣶⡋⣳⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣤⠿⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡉⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠀⡰⠚⠉⠉⠩⢭⣉⡙⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠁⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢰⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠑⠆⠀⠀⠰⠚⠉⠉⠁⠀⠀⠀⠀⠀⡎⠀⠠⠴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⢰⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣇⠀⠐⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠶⠒⠛⠛⠉⠙⠛⠛⢻⡇⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁⢀⡤⠤⠀⠀⠠⠖⠲⠄⠘⡇⠀⠀⠀⠘⢦⠀⠀⠀⢰⠑⢮⠱⢄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⠀⠈⠁⣠⣤⣤⣷⡌⡦⢀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡄⢹⡀⣠⣴⣾⣿⣶⣿⣿⣿⣶⣾⣧⣴⡗⠀⠀⠀⣴⣋⣀⣄⠈⢡⠃⢰⠟⠀⣱
⠀⠀⠀⠀⠀⢰⠀⠀⠀⣾⠀⠀⠘⢷⣾⣿⣿⡿⠟⢉⣉⣉⠉⠙⠛⣿⠛⠋⠀⠀⠀⢰⠛⡇⠀⢸⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠁⠀⣀⡼⠃⠠⡄⠀⠀⠀⢸⡀⠙⠤⠎⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⡆⠀⠀⠀⠀⡎⠀⠈⠙⠓⠶⠚⠛⠉⣀⠀⢀⡇⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳⡀⠀⠀⠀⣹⠖⢋⣩⣤⣷⠀⢸⣧⠤⠈⠻⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠱⡄⠀⢰⠇⠀⠈⣈⠅⠛⠒⠋⠑⠢⢄⣀⣈⡷⢰⠚⢲⣀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠈⠢⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡶⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣄⣀⣀⣀⣀⣀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣻⣉⣟⣻⣋⣿⣫⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠤⠀⠰⠶⠴⢿⠛⠋⠙⠂⠋⠀⠀⠀⠀⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠣⠆⠀⠀⠀⠀⠠⢀⣰⠒⠛⠃⠀⠀⠀⠔⠒⠀⠚⠀⣠⣄⣬⠀⠀⢂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣹⠇⠀⠀⠚⣿⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠛⠆⢿⣷⠀⠀⢀⡀⠜⠳⢄⣀⡴⠖⠲⣄⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠠⠏⢠⣦⡄⠀⠀⠈⠀⠻⠿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣻⣍⣁⠀⠀⠀⠀⢀⣰⡆⠠⠆⠀⠀⠀⣤⣀⡀⠈⠉⠉⠠⣴⠐⠀⠠⣤⡤⠖⠒⠚⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⣠⠀⠀⠀⠀⢀⠀⠉⠈⢡⠉⠀⠀⠀⠀⠈⠁⣾⠤⠀⢠⡖⠛⠉⠛⠀⣁⣀⠀⠀⠁⠈⣷⣶⡏⠀⠀⠀⠀⠀⠙⡷
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢻⣶⣾⠋⠀⠀⠀⠠⣾⡥⠄⣤⠈⡴⢄⠀⠀⠀⠈⢃⡹⠂⢀⡸⠇⢀⠀⠚⢿⣿⠛⠀⢀⡸⠛⢀⣠⣴⣿⠐⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠋⣸⣆⠀⠀⠀⣤⠃⠘⣷⡘⠧⠀⠀⢦⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⣦⡟⠁⠀⠈⠁⣭⣤⣌⣀⠀⠀⠚⡂
⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⢀⣄⣈⣉⠇⠠⠾⡇⢠⡄⠘⠁⡈⠃⠀⠐⠊⠁⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠘⠛⠀⠀⠻⣄⠀⠀⠀⠀⠈⣹⣙⠉⠀⠐⡀⠉⠲⠶
⡀⣀⣰⣆⠀⠀⠀⠀⠁⠈⠿⢧⣄⠀⠀⠀⠀⠀⠀⣻⢾⡇⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠘⢻⠀⠀⠀⠀⠀⢳⣤⡄⣸⣷⠶⠖⠂⠈⠛⢀⣾⣿⣦⠀⠀⢀⡀
⠀⠀⠙⠟⠀⢀⠀⠀⠈⠀⠀⠀⣹⡄⠀⠀⠀⠀⢀⠣⠈⠁⠘⠲⣄⠙⣿⣶⠶⠖⠲⡇⠀⠀⠀⠈⣇⠀⠀⠀⠀⣾⠃⠀⠈⢙⠀⠀⠀⠀⢢⡈⠻⠛⠁⠀⠀⣼⡇
⠀⠀⠀⢦⣤⠾⣳⠴⢀⡀⠀⠀⠉⠁⠂⠐⠆⠀⠀⠐⠀⠀⣀⣼⣿⡄⠀⠀⠀⠀⠀⢰⡿⠰⢀⣠⣄⡄⠂⠐⠂⠛⢃⣉⠀⠈⠁⡟⠀⠀⣤⡄⠸⠻⠇⢠⡔⠉
⠀⠀⠀⠸⡇⠀⠀⠀⠘⠁⠀⡀⠀⠰⠓⠀⠀⣀⢠⠀⢤⣠⡈⢉⣿⣥⡀⠀⡀⠐⣦⠈⠀⠀⠀⢹⣿⣤⡀⠀⠀⢰⣦⠙⢶⡌⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣇⠀⠀⠂⠀⠄⣀⣽⠿⠇⠀⢰⣾⡟⢣⠀⠀⠉⠛⢀⣄⡙⠟⠈⢀⣸⣿⡇⠀⣠⣀⣠⣷⣄⢡⡀⣤⡈⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠛⢇⣸⠀⠀⠀⠀⠸⣿⣇⠀⠀⢿⣿⡄⠀⠀⠉⠹⠾⠿⠛⡛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⠣⢀⣠⣿⣄⡀⠘⠀⠀⠀⠈⡃⢀⠶⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠙⣻⡀⢀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠄⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠔⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢄⡁⠀⡉⠠⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣗⠀⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣿⠀⠀⣀⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠄⠔⣀⡠⠤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁⣀⡨⢄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠠⠊⠀⢰⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⡀⠸⠈⡁⡇⡠⠀⣠⣔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃⣀⢵⣄⠀⠃⢞⡈⢉⡽⠠⠔⠁⠀⠉⠉⠉⠉⠉⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⣚⠳⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠉⠀⠒⠤⠄⡀⠀⣀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢑⠃⡠⠃⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⡀⠑⢌⡠⢁⣳⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠁⡀⡈⣿⣛⡠⠊⠁⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣉⠀⠐⠲⠖⠀⠉⠀⣰⢿⡵⡂⢠⠆⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⣔⠀⠀⠀⠀⢀⡷⢛⡗⢞⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⢄⡀⠅⠋⠘⠤⠊⡶⢯⡶⠛⠹⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠦⣙⣀⢠⠛⢄⠊⢃⠸⢷⠆⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣿⠀⠆⡡⢚⠀⣆⠤⠢⠊⡜⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⠀⠠⠔⠂⣡⣂⣤⣤⣾⣽⡾⠷⠔⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠛⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢄⣀⠀⠀⠐⢴⡟⠆⠀⠰⣾⠧⠀⢰⢦⠀⡀⢂⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠤⠄⠞⠻⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠚⠉⠐⠒⠲⠤⣀⠀⠀⢀⠄⣠⠄
⠀⠀⢸⡅⢸⡁⠀⠀⠀⢸⠀⠒⢒⠢⡀⠀⠀⠀⡠⠺⠷⠶⠀⠀⠀⠀⠀⠀⠐⠛⠺⡗⢦⡀⠐⠁⢀⡀
⢾⠀⢺⠀⢮⢠⠆⠀⠀⠈⠦⠼⠷⠤⡐⠀⢀⠞⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠑⡄⠀⠹⠛⠂
⢻⠀⣻⠄⡿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣷⠶⠸⠀⣰⠀⠀⠀⠸⠿⣆⣸⠀⠀⠀⠸⠀⠠⣤⣄
⣹⠀⢻⠀⠗⣧⠀⠀⠀⠀⠂⢤⣀⠤⠀⢸⠀⠀⠀⠀⡀⠀⢎⡀⢤⠄⣤⡀⠀⠀⠉⠒⠀⠀⠀⡇⠀⠛⠁
⢹⠂⡠⠲⡄⢁⣀⠀⠀⠑⢄⣤⡕⠀⠀⢸⠀⠀⡠⠚⠀⢀⡀⠀⠀⠀⣀⣙⠦⠤⢤⠀⠀⠀⠀⡇⠀⢠⢄
⠈⠁⢈⠟⠓⠚⢤⡠⢦⠀⠀⠀⠀⠀⠀⠈⡆⠀⠁⠈⠣⠤⣀⣀⣀⣀⣀⠤⠜⠃⠄⠀⠀⠀⢠⠃⠀⠲⠄
⠀⢰⡃⠀⢰⡀⠀⢣⠀⠀⠒⠤⣀⣀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠩⢂
⠀⠈⣧⠀⠈⣧⠀⠈⡆⠀⠐⢄⣤⡶⡀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⠽⠇
⠀⠀⠸⡆⠀⠺⡏⠀⠸⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⡀⠀⠔⠊⠁⠀⠀⠀⣱⠒⠒⠄
⠀⠀⠀⢻⡀⠀⢻⡆⠀⢣⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠑⠂
⠀⠀⠀⠈⢷⠀⠀⢷⠂⠈⢆⠀⠀⡠⠉⠀⡣⠀⡐⡀⡀⠀⢀⠀⠠⠂⠀⠀⢠⡀⠰⣄⣀⣛⡃⢹⠀⠈
⠀⠀⠀⠀⠘⣇⠀⠘⣧⠀⠘⡄⠂⠀⠀⠉⠀⠀⡧⠁⢣⠀⡷⠷⣧⠀⠸⡿⠂⠃⠀⠀⠀⠀⠙⠈⣄
⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⢱⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠄⡀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠐⠄⠀⠑
⠀⠀⠀⠀⠀⠀⢷⠀⠸⠿⢇⣀⠇
⣀⣀⣀⣀⣀⣀⣘⣧⣴⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣷⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⢠⣈⢳⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⣼⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⡇
⠀⠀⠀⠀⠈⢻⣏⡉⣯⣶⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡏⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⡇⣾⡄
⠀⠀⠀⠀⠀⢾⣿⣇⠉⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣟⣶⢋⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢷⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢕⣮⡯
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠟⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣤⣄⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣽⣾⣿⣿⣿⣿⡇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿⣞⢦⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⡟⣭⡿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢸⣿⡇⣿⡇⣿⢱⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢿⠇⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢞⡇⠀⠀⣿⠇
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⣿⣿⣿⣦⡀
⣿⣟⠀⣸⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣶⣷⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⡟⣿⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠈⠛⠛⣁⡀⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡿⠓⠂⠀⠀⠀⣠⡾⠃⣀⡀⠀⠃⠀⠛⠃⠈⠛⡋⠠⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣴⠟⠀⠀⠛⠁⢀⣤⡀⠀⠆⠀⠘⠟⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠖⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠆⠀⠀⠀⠀⠉⠀⠀⠀⢀⡄⠀⠀⠀⠀⠉⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣏⣤⣴⣶⠾⢛⣫⠟⣻⢿⠟⢹⣿⣿⣿⢹⡙⢯⡛⢭⡙⠢⢭⡉⠓⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⣼⡟⣩⠖⠉⣠⠞⠉⡴⠋⢰⣋⣀⣼⣿⣿⣿⠾⣷⣤⣷⡀⠹⣄⠀⠙⢦⡀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠀⠀⠀⠻⢿⣧⣤⣾⣶⣶⡾⠿⠟⠛⠋⠉⣽⣿⣿⡿⠀⠀⠈⠉⠛⠷⢾⣦⣄⣀⣳⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣀⣤⣤⣀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡟⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡾⢻⣿⣿⠏⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢩⠀⢿⣿⠟⢈⡉⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⢶⡆⠉⣵⡄⠁⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⡴⠶⠶⣤⣄⡀⠀⠀⢀⢸⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⡀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠽⠒⠚⢋⣩⠿⣿⣿⣦⡀⠈⠘⣇⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⣁⣠⣤⣶⣶⣾⣿⣿⣟⡛⠓⠶⢬⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⣠⠖⠋⣠⠎⣿⡟⡟⣿⣦⡀⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⠟⣫⠟⢹⣿⡟⣆⠙⢎⠙⠫⣝⠢⣄⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣞⣁⢀⣰⣃⣼⣿⢃⠇⢹⢻⠻⣾⠀⠀⠀⣸⣿⣿⣿⣿⣷⣃⣼⡷⠶⢿⣿⡟⠛⠶⣤⣷⣀⠈⠳⣌⠳⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢻⣿⣿⠛⠶⠾⠾⠶⠛⠀⠀⣰⣿⣿⣿⣿⡟⠛⠉⠁⠀⠀⢸⣿⡇⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⢺⣷⡄⠀⠻⣇⠀⢺⡆⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⢠⣤⠀⢰⣶⠀⠀⣴⣦
⠀⠀⠀⠀⢴⡍⢧⡀⠀⠸⡄⢸⠃⠀⣼⣿⡿⣿⣿⣿⣧⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⡏⢀⡞⠁⠀⣸⠟⣃
⠀⠀⠀⠀⠈⠙⣄⠳⣄⠀⢷⡏⢠⡄⠻⢿⡇⢟⢻⢻⢿⣄⠀⠀⣿⣿⡇⣿⣿⣿⡟⣿⣿⡆⠀⢻⡿⡟⢻⣿⡟⠿⠂⠀⢀⡀⢳⣼⠁⠀⡴⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠈⠳⢽⣷⣌⢧⡈⢳⡀⠈⣿⡸⡌⣦⠳⣝⢧⡀⠛⣿⢃⣹⢻⡟⢣⣸⢸⣧⠀⣿⢁⢇⡜⢻⣧⠀⠀⠀⣸⠇⣸⠏⣠⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⡙⠻⣿⣶⣽⣷⣾⣷⣷⢨⠳⡈⠳⡿⣆⢹⣾⡇⠸⣻⡈⣿⡀⢿⡄⣿⡏⡌⡇⣸⢻⣆⣤⣞⣡⣴⣿⡾⠛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠀⠈⠻⣿⣦⣽⣄⣿⡟⠘⣿⢻⠀⢿⡇⠙⣷⡍⣷⢻⣄⣇⣣⣏⣼⠟⠛⠛⠛⠛⠒⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠘⠿⣧⣼⣇⣰⣿⣰⡿⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠒⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠦⠄⢸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡏⠉⠑⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⡀⠀⢸⠃⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣧⠀⠘⠀⣤⠀⠀⠀⡘⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⠁⠀⠀⠀⢹⠀⠀⠀⢸⡄⠀⢀⠇⡜⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡀⠀⠀⡜⡄⠀⠰⣸⣗⠀⢸⢸⠇⠀⢸⠀⠰⠀⡠⠂⠈⠀⠰⠇⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⡀⠀⣧⡇⠀⢢⣿⢸⠀⡇⣾⠀⠀⢸⡀⣠⠿⡀⠀⡀⠀⢰⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢱⠀⢸⡇⠀⣿⣸⣼⡄⣷⢹⠀⠀⢸⡿⠁⠀⢿⠛⣷⣤⣼⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠘⡆⠀⣿⢠⡏⢿⢻⣿⣿⢸⠀⢠⢾⠇⠀⠆⠈⣦⢀⡾⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡇⠀⣿⣸⡇⢸⢸⡿⡏⠘⣠⠏⣸⠀⣼⠀⠀⣿⠛⣾⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢡⠀⢸⣿⡇⢸⠀⡇⡇⢠⡏⠀⣿⠀⡏⠀⡾⠈⣤⡏⣸⠏⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⣷⣼⠀⣸⣿⡃⢸⡄⡇⢡⡞⣁⠀⣿⢀⣇⣸⢃⠀⣻⣷⡟⣠⢃⡆
⣿⣏⢹⣟⡻⣿⣟⢟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠋⠉⣿⣿⣿⡇⡀⢀⢍⢉⡻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠸⣿⣟⠕⡔⡰⡰⡀⢻⣿⣿⡟⠌⢎⢢⠱⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⡀⠙⢿⣮⣤⣁⣃⣃⣼⣿⣿⣿⣼⣼⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠉⠙⠙⣿⣿⣿⡏⠉⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⠛⢻⣿⣿⠛⠻⠿⣿⣿⣦⠀⠀⠀⢠⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡠⢊⢀⢀⢻⣿⣿⡆⠄⠄⡔⢸⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢶⣤⣼⣿⣿⣮⣬⣼⡾⠟⠁⠀⠀⣸⣿⣿⣿⡿⡛⡛⠛⣻⣿⣿⠛⠻⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⡀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣷⣬⣼⣰⣼⣿⣿⣐⣂⣢⣼⣿⠏
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠉⠉⢛⣿⣿⡿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⢠⣿⡿⠟⢛⣿⣿⡏⠛⠛⢿⡆⠀⢹⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠈⠻⢤⣤⣤⣿⣿⣤⣤⠷⠟⠁⠀⠀⢿⣿⣿⣧⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⣰⣿⣿⣿⠀⣀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⣿⣿⣿⣿⡇⠀⠀⢀⢼⢿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⣠⣿⣶⣶⡶⠇⠺⢿⡛⠻⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣀⣾⣿⣿⣿⡿⠃⠀⠀⠈⢉⣿⣿⡿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⣿⣿⣿⣿⢿⣿⣿⠏⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣠⡀⣄⢠⡄⣠⣄⠰⢤⣄⢠⡄⣀⣤⣠⠆⣀⠀⡀
⠀⠀⠀⠀⢀⡀⣀⣘⣇⣼⡿⣯⣾⠋⢿⣾⠟⣷⡾⠋⠙⢷⠿⠛⠷⠿⠛⡿⠾⠛⣿⣤⣭⡦
⠀⠀⠀⣀⡈⢃⣼⠿⢿⡿⣤⣤⣤⣤⣠⣤⣤⣴⣶⣤⠄⢰⡲⠤⣶⣶⣢⣶⢖⣶⠿⠀⣾⢁⣀
⠀⠀⠀⠀⠺⠿⣷⣜⢛⣛⡛⢿⣝⠓⢿⣛⣛⠙⣛⣮⣁⠛⣫⣽⣭⣍⣉⣬⣭⣉⢠⠾⡛⠓⠈
⠀⠀⠀⠐⠂⠐⠛⡿⠉⠹⠟⣫⠛⠛⢫⠍⠙⠛⠉⠰⠙⠛⠉⢰⠂⠉⠉⠐⠮⠉⠛⠀⠉⠃
⠀⠀⠀⠀⠦⠲⠖⠤⠶⠶⠂⠓⠂⠘⠒⠒⠓⠒⠂⠀⠐⠒⠂⠂⠚⠀⠐⠓⠀⠀⠓⠀⠑⠋
⠀⠀⠀⠀⠔⠞⠚⠂⠘⠚⠒⠂⠚⠒⠛⠒⠒⠁⠘⠃⠘⠚⠚⠛⠓⣚⣀⠉⠓⠚⠓⠚⠐⠓
⠀⠀⠀⠀⢀⡴⠛⢉⢉⠑⠖⠋⠉⣧⣤⣤⠔⠒⠒⠖⢚⠳⣄⣤⣼⠋⢭⣙⠶⠒⠶⢤
⠀⠀⠀⠀⢿⡈⢂⣛⣭⡆⢟⣁⣤⡤⠜⣇⢰⠶⠶⠿⠈⢀⣉⠉⣦⠒⠄⠈⢙⠓⠶⢆⣟⡛⠲⢄
⠀⠀⠀⠠⣏⠘⢉⣉⠰⠞⠛⢉⣡⢶⠎⠰⠶⠖⠂⠘⠓⠙⣫⢡⡏⠘⡃⠤⣬⣉⡂⢶⣬⣙⡳⢚⠇
⠀⠀⠀⢰⣏⢘⣋⡡⠞⠉⠉⠉⠀⠈⠓⢲⠀⢰⠷⢷⠿⠧⣈⡰⢷⡄⠛⠆⣤⣈⠉⠐⠶⣉⣁⠏
⠀⠀⠀⠀⠉⠉⠁⠀⢻⣎⣉⡉⠉⡉⢉⣉⠉⠛⠙⠒⠒⠚⠉⠉⠉⠉⠛⠒⠒⢥⣘⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠙⠙⠀⠁⠈⠩⠭⠥⠌⠩⠭⠭⠄⠑⠘⠙⠊⠁⠀⠀⣇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣤⠤⠴⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⡀⠀⠒⠒⠒⠒⠢⢄⡀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆⠠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠭⠍⠉⢩⠭⠑⢦⣸⡧⠯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠀⠘⠃⠀⠸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣟⣀⡀⠀⠀⠀⠀⠀⡇⢜⣁⣀⠒⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠉⢗⢦⡀⠀⠀⢣⠈⠻⠛⠋⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⡄⠀⢠⡗⠢⠄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢗⡦⣟⠣⣄⣀⠤⠖⠋⢡⠷⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠙⠦⢭⣙⣒⣒⡒⠒⣲⣶⣶⣾⣿⡿⠟⠃⠘⢦⡈⡇⢀⣠⠔⠁⠀⠀⠀⠈⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⡏⢿⡉⣹⢹⠃⠀⠀⢰⠃⣬⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠁⠘⢦⣄⠀⡞⢀⠁⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣦⣄⠤⠞⠉⠳⣅⠘⠃⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⠤⡀
⠀⠀⠀⠐⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠠⠀⠠⢀
⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡠⠤⠒⠂⠁⠀⢀⠔⠁⠀⠀⠀⡀⣀⠑
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠒⠒⠒⠒⢚⠉⠀⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡦⠄⠀⠀⠀⠀⠀⠆⢀⠂⠀⠀⠀⠑⠘⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡀⠠⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⡀⠤⠀⠒⠀⢊⡀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢉
⠀⢀⠐⠈⣀⡠⠤⠀⠀⠚⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆
⠀⠁⢀⠀⡉⠃⠀⣀⠠⠄⠁⠀⠀⢀⠐⠀⢠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣀⠀⠀⢀⠀⠠⠈⠀⠀⠐⠂⠀⠀⠀⠀⢰⠌⠠⢖⣁⠤⠄⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠰⠀⠄
⠀⠘⠠⠀⠈⠂⠀⠰⠕⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⡁
⠀⠀⠀⠀⠠⠐⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠂⢰⠀⠀⡀
⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠯⠐⣠⣂⡰⢠⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠒
⠀⠀⠀⠠⡘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠉⠀⠉⠁⠋⣰⣥⠅
⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⠀⠒⠈⠉⠐⠲⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡊⢠⠄⠠⣄⠀⢰⡄⡄⢠⠂⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠣⣜⢀⢀⡙⠧⠀⢁⣷⣃⠀⠀⠴⣾⡧⡠⠶⠆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡇⢀⠜⠋⠀⠀⠀⠀⠀⠀⠁⠘⠃⠾⠋⣨⣔⣥⣤⡄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠯⢷⣶⣦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⣤⣻⠶⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⠴⠒⢛⣛⠩⠉⠉⠉⠛⠒⢦⣄⠀⠀⠀⠀⣼⣀⣒⣋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⠛⣡⠶⠶⢦⠀⠀⠀⠀⣿⣿⣦⣀⣰⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⣽⠛⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠶⠄⠀⠀⡀⠠⢒⠤⠀⠐⠂⠀⠀⠀⠀⣷⢈⡇⠹⢻⡄
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣇⠀⠒⡧⠀⠓⠐⠁⠂⠀⠈⠁⠀⠀⠀⠀⠸⡌⠰⣡⡏
⢀⡀⠀⡎⠠⠄⠄⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠸⠞⢻⠃⠀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡶⣿
⢀⠹⡀⡇⠀⣠⠀⠐⣋⡔⠒⠶⢄⡀⠀⠀⠀⡜⠀⢸⣄⡰⣶⣬⣜⠩⠉⠀⡇⠀⠀⠀⠀⠀⣸⠁⣯⡀
⠈⠄⠡⣡⡀⠙⣆⡀⠛⠀⣰⠂⠀⣉⠀⠀⠀⢧⠠⡇⢾⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⢃⠔⣵⠟⠉⢹⡆
⠀⠐⣄⠘⠓⠂⠀⠀⠁⠀⠘⡆⠀⣯⠀⢧⠀⠘⣆⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠑⢱⠞⣩⡤⠀⠈⣿⠲⠖
⠀⠀⠀⠢⡀⠙⠄⡀⠉⠀⠀⠁⠀⣼⠀⡼⠀⠀⠈⢢⠸⡇⠀⠀⠀⡠⠈⠀⠀⣠⠎⢂⠄⠈⠀⠀⠀⠀⠰⠾⠿
⠀⠀⠀⠀⢧⠀⠀⠈⢦⠀⠀⠀⠀⠘⢦⠃⠀⠀⠀⠀⠑⣌⠐⠒⠈⠀⢀⣠⡞⡡⠂⠄⢒⡁
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⡀⠀⣠⡧⢄⡀⠀⠀⠀⠀⢘⣶⣤⣴⠚⣉⢂⣪⠔⠒⠚⠉
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡠⠚⠓⠒⠓⢌⠒⠃⠉⢩⡶⢚⠿⡠⣞⠻⠛⠄
⠀⠀⠀⡔⢿⠁⠀⠀⢀⠀⡌⠀⠀⠀⠀⠀⠀⠑⠦⠤⠗⠋⠁⠀⠀⠈
⠀⠀⢰⠀⠈⠀⠀⠈⠀⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣏⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠄⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠉⠉⠀⠀⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣂⡤⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠭⢥⣴⠞⣣⠈⠢⡀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠀⣀⡴⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⡊⠀⠧⠀⠈⢂⠀⠀⠀⠀⠈⠑⣶⠀⠈⠉⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠿⠻⣷⡆⠀⢧⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⣀⠀⠐⢲⡍⠻⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⢣⡔⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠙⢶⣄⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠁⠀⠀⠀⠀⠀⠀⢠⡖⡾
⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⠐⢶⣶⠶⣎⠈⠛⠻⣿⣿⣿⣿⠿⠒⠛⣛⠃⠈⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⢀⣴⡟⣸⠃
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣂⣸⡿⠇⠿⠧⠀⠘⢛⣩⣥⣦⡀⠸⣿⡹⣶⡗⠀⠄⠀⣼⢸⣿⣦⡀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⣴⡿⠏⢠⡏
⠀⠀⠀⠀⠀⠀⢰⠘⣿⣿⣿⣿⣿⣿⣟⠁⠐⡆⢀⣿⡇⠘⠁⠀⠈⠻⣧⣞⣶⠀⠀⡿⠀⠉⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠿⣿⣿⣿⣿⣿⡒⣠⠀⠸⠟⠃⠀⠀⠀⠀⠀⢸⣿⡇⣠⣴⡇⠀⠀⠸⣿⣿⣷⣆⣀⣠⣶⣿⣿⡿⠏⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣴⣿⣿⠁⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣛⠿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠐⠂⠀⠀⠁⠀⠄⠸⠂⠀⠀⠀⠶⠰⠀⠀⠀⠀⠠⠰⠀⠄⠈⠐⠀⠀⠀⠐⠢⠒⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢡⠀⠈⠆⠀⠀⣶⢠⠀⢸⡂⡤⠀⢀⡒⡔⠀⠀⠀⠐⡄⠀⡠⠂⢀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠅⢰⡀⠀⠀⣾⠈⠀⠘⣇⣩⠀⡟⠛⠀⡸⠤⠄⡀⠹⠰⠃⢰⠀⢧⠐⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⢸⡇⠀⠀⢿⠀⠆⠀⢸⡎⠀⡀⣆⠀⢱⡀⠀⠐⠄⠀⠀⠈⢧⠘⠀⠀⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⢠⠁⠀⠀⠀⠃⢠⠆⠈⠀⠈⠀⡌⡗⠠⠁⣸⠀⠈⡄⠀⠀⠘⡀⠀⠀⠈⢃⡞⠀⡀⠀⢀⡤⠆
⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⡌⠀⢀⠀⠰⢠⣃⠾⢰⠂⢸⡄⢹⡏⢀⠁⠜⠉⠀⢷⡄⠀⢀⠁⠀⠘⠀⡟⠀⠀⢸⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢸⡀⣤⣾⠁⠀⡜⠀⡀⣹⡊⢇⠈⢀⠀⠀⠂⢡⡇⢰⡏⠀⠀⢀⢠⣷⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⣷⢿⣿⣆⠀⢷⣼⣟⡁⠳⣈⠳⣜⠀⠰⣦⡏⡇⠸⠃⢀⣀⡆⡎⣿⢠⣦⡄⣿⣴⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⠇⠉⠙⠙⠛⠈⠘⠋⠉⠀⠀⠉⠀⠉⠀⠀⠈⠀⠀⠈⠁⠈⠉⠉⠉⠛⠉⠁⠉⠙⠙⠛⠁

⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡦⠌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⡔⠆⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠼⠂⠁⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⠤⠀⠄⠀⣀⠀⠀⠀⠀⣀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣟⣁⣀⣤⣤⣤⣤⣠⣀⣀⣀⣀⠈⠀⠒⠿⡞⢷⠺⡶⢷⡼⡴⠄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⠦⣄⡀⣽⣏⣹⣍⣻⣛⣟⣆
⠰⡒⠀⠠⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣿⣿⡿⠟⠹⡃⠈⣻⣿⣿⣷⡷⠤⢧⡬⣭⢆
⠡⠁⠀⠄⡀⠀⠻⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⡫⠋⠁⣿⣷⣿⣖⢳⡚⠶⠧⡀
⠀⠢⠀⡐⢀⠀⡀⠙⢧⣈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠟⣹⣾⢟⠕⠀⠚⠻⣿⣿⣿⣿⣿⡘⢓⠓⡄
⠀⠀⠑⠀⠤⣤⣤⠬⠿⠏⠑⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣛⠉⠀⠀⢶⡀⠀⠼⠿⣫⣾⣍⣹⣉⣉⣉⣄
⠀⠀⠀⠀⠀⠀⠙⠷⢤⡀⠀⠀⠀⠀⠐⠂⣠⣤⣤⣾⠿⠿⠟⢛⠯⠟⡫⠙⠁⠐⣹⣯⠽⡆⢷⡤⠵⡤⠭⡤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⢶⣶⠾⡿⠉⠉⠀⠀⡀⠤⠤⠒⠈⠀⠀⠀⠀⠀⠀⠚⠻⠿⠛⠛⠛⠛⠒⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⠿⠋⠉⠉⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠙⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠐⣿⣏⠀⠀⠀⠽⠿⠁⠀⠀⢠⣴⡀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⢀⣶⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠈⠺⣻⣷⡄⠀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠘⠀⢀⣠⣶⠾⠟⠛⠉⠛⠛⠙⠛⠛⠷⣶⣄⡀⠀⠀⠀⢀⣀⣤⣼⣿⣿⠿⠟⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⣶⡧⠄⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⣶⡾⠿⠟⠛⠋⠉⢀⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⠿⠿⠛⠋⠉⢀⠀⣠⣴⣦⡐⣿⡛⠃⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣸⡟⢁⣀⣠⣤⣴⣶⠿⠿⠛⠛⠉⠁⢀⣀⣤⡄⣶⡀⢹⡇⢿⡇⠹⠃⢹⣧⡶⠄⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣀⣠⣤⣴⣿⡿⠿⠛⠋⠉⢁⢀⣀⠀⠀⣶⠟⣷⡌⣿⡁⣀⢻⣿⣎⣿⡘⣿⠀⣠⡌⣿⣀⣀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⣀⣀⣼⣿⣷⡾⠿⠛⠛⠉⠁⣀⣠⣄⠀⣶⡀⢻⡇⢿⡆⠀⢻⣇⢬⣧⢹⣿⠛⠈⣿⡘⢿⣧⠹⣧⣼⡇⠘⠛⠛⠀⢀⣀⣠⣤⣿⣷
⠀⢾⣿⠟⠛⠉⠁⠀⣀⡀⣤⡀⢸⣇⢻⡏⠹⣧⢹⣧⠈⣿⡘⣷⠀⠘⣿⡄⢻⡆⢿⣤⣶⠸⠇⠈⠛⠀⢀⣀⣠⣤⣶⣶⣿⣿⡟⠋⠉⠁
⠀⠘⣿⡄⠀⠀⠀⠀⢹⣧⢻⣿⣌⣿⡘⣿⠀⢿⡆⣿⡄⢻⡇⢻⣧⣶⠘⠿⠟⠋⠈⠁⣀⣠⣤⣴⣶⣿⠿⠛⠋⠉⠁⠀⣼⣿⡇
⠀⠀⢹⣷⠀⠀⠀⠀⠀⢿⡎⢿⡙⣿⣇⢹⣧⣼⡷⠘⠻⠟⠃⠀⠉⣀⣀⣤⣴⣶⡿⠿⠛⠛⠉⠁⣽⡟⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠘⣷⠸⡧⠈⠛⠈⠉⠉⣀⣀⣤⣴⣶⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢠⣾⠟⠀⢀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠉⣀⣀⣤⣤⣶⣾⢿⣿⣟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠁⠀⠐⢾⠿⠀⢀⣼⣿⡏
⠀⠀⠀⠀⣿⣧⣤⣤⣶⣾⣿⡟⠛⠉⠉⠀⠀⠀⠈⠛⠿⣦⣦⣤⣀⣀⣀⣠⣤⣴⣶⠾⠛⠁⠀⡀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠘⠛⠋⠉⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⣄⡀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⢻⢿⠁⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣦⣄⡀⠀⠉⠟⠇⠀⠀⢀⣦⣖⠀⠀⠀⣱⣿⠄⠀⠀⠀⢀⣠⣶⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣯⣰⣤⣀⡀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⢀⣀⣄⣰⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣾⣿⣶⣶⣶⣤⣤⣶⣶⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁


⡿⢿⠿⢿⠿⡿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⡿⣿⣿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣰⢾⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢰⠟⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⠁⠈⠀⠀⠀⠀⣀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠁⠀⠀⠀⢀⠀⣠⢤⣀⡴
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠻⢦⡀⠀⡠⠖⣷⠞⠙⠿⠛⠋⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠒⠉⠟⠁⠈⠉
⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⣀⣀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⡀
⠀⠀⠀⣰⢿⠋⠉⣷⣠⡴⠞⠋⠉⠁⠀⣹⠀⠉⠻⠷⠄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠉
⠀⠀⠘⢷⡈⢷⣾⠟⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢷⡞⢁⣠⠶⠛⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡿⢿⣿⠿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠀⠀⠀⠀⣶⠀⠀⠐
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢰⡀⠀⠀⠀⢀⠀⢸⣰⠤⠀⠀⠀⠀⠀⠀⢠⣾⣷⠚⠀⠀⠀⠀⠀⣿⠀⠄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⡤⠲⡆⢁⡞⠀⠀⠀⠀⠈⣦⡠⣬⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⢟⡈⠀⢀⠁
⠀⠀⡄⠀⠀⠜⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠁⠀⢹⢸⣇⠀⠀⠀⠀⠀⠀⠁⠔⣰⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏
⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠈⣾⡟⠀⠀⠀⣀⠀⠀⠀⠀⠋⡀⡀⡀⠀⠀⢀⣶⠗⠂⢀⣀⣀⡀⠀⠸⣇⠜⠀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠑⠋⠛⠛⠃⠀⠀⠋⠀⠚⠛⠟⠛⠓⠃⠀⠀⠑⠛⠛⠐⠑⠂⠈⠁⠀⠚⠛⠛⠛⠛⠀⠀⠀⠀⠊⠋⠋⠉⠁

⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠣⡀⠤⠀⠀⠀⠀⣀⣁⢦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⢐⠀⡀⠀⠀⠀⠀⠘⠟⢠⡀⢀⡀⠀⠀⠀⠀⠀⢦⠏⡰⠁⡴⠂
⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠆⠀⠀⠀⠀⠀⠀⠀⢊⣷⡟⠀⠀⠀⠀⠈⢦⠎⠀⡰⠋
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠙⠆⠛⢏⣠⠀⠀⠀⠀⠀⢨⠀⠔⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢡⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠺⠺⠟⠳⠓⠀⠀⠀⠈⠀⠠⠶⠆⠶⠦⠀⠀⠀⠀⠀⠀⠠⠚⠓⠾⠖⠒⠀⠀⠀⠈⠀⠀⠰⠶⠳⠒⠀⠀⠀⠀⠀⠀⠁⠓⠛⠊⠂
⠀⠅

⠀⠀⠰⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣤⡀
⠀⠘⢣⠀⠀⠀⠃⢀⡒⠀⠀⠀⠀⠀⣄⣀⠀⣀⠀⢀⠠⠀⡀⠀⠀⠀⣀⣴⣟⣋⢭⢤⡀⠝⢋⣦⣜⠳⠄
⢀⠀⠾⠀⠀⢀⢡⡦⠁⠀⠀⠀⠀⠚⠀⠀⠌⠁⠀⠈⠀⠰⠀⠀⠀⢰⣿⣿⣿⣿⣥⡈⠳⢦⡀⠀⠫⢷⡀⠰⡀
⠂⠀⠀⠀⡄⠈⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⠈⠻⣶⠹⢆⠀⠹⢆⢄⢙⡇⠀⢷
⠀⢠⣦⡾⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣤⣤⣴⡿⡀⠰⢄⢀⣻⣆⠢⣇⠀⠈
⣾⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⢴⠉⣷⠀⢳⢹⣿⣆⠹⠀⠘
⠑⠙⢾⠃⡀⠀⠈⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⢸⡀⣇⡇⣠⠘⣿⢜⠇⡆⢀
⠀⠀⢐⡀⢀⣟⡄⠱⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠐⠶⠖⡿⠒⠒⠒⠛⠃⣻⢿⡌⠁⣝⠄⠀⡷⣷
⠀⠀⢈⡁⠀⡜⣆⠏⣄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣷⠄⠀⠀⠀⠀⠻⣈⠟⢷⢈⣇⡀⢰⣟
⠀⠀⠈⢥⠂⠀⠈⠘⠀⢀⠀⠀⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⠀⠀⠎⠰⡀⢹⠀⠀⠀⢏
⠀⠘⠣⡀⠀⠀⠀⠁⠀⠳⠆⠀⠀⡀⠀⠀⠀⢘⣤⡀⠀⠀⠀⠀⠉⢠⠆⡠⢤⡴⢂⡜⣶⠀⠀⠇⣸⠀⠈⠐⠺⡄⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠐⢋⠆⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠏⠁⣾⡀⠀⠀⣀⠋⣸⣿⢒⣧⣤⣷⡾⢄⠄⠀⠃⠀⢂⡄⠐⢄⣂⡀⠀⠀⠘⠂⠼⠄
⠀⠀⠀⠈⢆⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠃⠁⢰⡾⠁⣴⠟⢿⡟⣼⣄⣈⠠⣀⠆⠀⡄⢳⠀⡀⢟⠣⠀⠙⠒⠂
⠀⠀⠀⠀⠈⠀⠀⣶⠀⠀⠈⣄⠀⠀⠄⠀⠀⠀⡀⠁⢀⠞⠃⠀⣼⠶⣷⡄⣿⣠⠼⣿⠓⢹⡆⢤⠀⠸⡆⠁⠈⠑⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⣋⠀⢠⡴⠁⠠⠃⠀⡀⠀⠠⠀⠰⠥⠀⢠⣀⣸⣿⣽⣧⣞⣹⡾⢷⣿⣄⠐⢸⠇⠸⣧⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠁⠀⠞⠀⡀⠁⠀⠀⠀⠀⠀⠀⠐⠁⢠⠀⠁⠠⠘⣯⠿⠛⣿⣿⣽⣯⡶⢿⣿⣼⣿⠊⠇⡀⠡⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣷⡆⢠⡾⠁⠀⠀⠀⠀⠀⠁⠐⠁⠀⠈⠀⠀⠟⢛⡛⣟⣋⡿⢻⡟⢻⣶⣿⣯⣧⣿⠀⣀⠁⡄⠀⠀⠀⡀⠀⠀⠀⠿
⠀⠀⠀⡨⠁⠠⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣌⡀⢻⠛⢤⣼⣷⣿⣩⣧⣼⣿⣬⠀⠿⣦⠀⠠⢠⣄⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠐⠎⠀⣧⡀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠀⠈⠀⠠⠈⢻⠲⢮⣴⡂⠠⢔⣵⣻⣿⡷⣾⣿⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠉⠁
⠀⢀⠀⠐⠒⣋⣅⠈⠙⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠡⠀⠲⢟⡤⢽⣶⣿⣋⣭⠃⣵⡇⢿⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣳⣾⠟⠛⣿⣷⣶⠿⢳⣿⡆⠺⣿⡇⠀⠀⢤⣶⡆⣦⡛⠰⠀⢃⠒
⠀⠀⣄⠀⠀⠀⠀⠔⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣻⣧⣄⢳⣾⣿⠷⣿⣶⠏⢰⡿⣄⠈⠁⢸⣷⣤⡉⠃⠉⠐⠆⠉
⠀⠀⢉⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠻⣿⢿⣿⡏⠜⠙⢧⠘⣇⡿⢦⡀⠈
⡀⠀⣬⠛⣤⠧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠾⠁⠀⠁⠀⢠⣬⣧⡉⣶⣿⣿⣵⠾⣥⣶⡌⢠⣿⡇⣀⡽⠦⣀⠀⠀⠀⠲⠄⠀⠖⠦⠤
⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⡔⢠⠀⠃⠀⠀⠁⠀⠀⠀⠀⣾⡁⢿⣷⣝⣟⣿⣟⠒⡟⠹⢧⣼⣿⣿⠿⠖⠛⢻⡻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠄⠂
⠀⢅⠐⠊⢀⠗⠁⠀⠀⢀⣾⣼⡳⠋⠀⠀⠀⠀⠀⠀⠀⢘⡫⠿⠻⣷⣻⣏⣹⡿⣿⡦⢠⣼⣏⣀⠋⡀⠒⠦⣾⠁⠈⠻⣝⢳⠠⢀⡀⠀⠀⠀⠲⠀⠀⢰⠁⡄
⠀⠋⠠⢖⣯⠁⠀⡀⢠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣦⣴⡆⢰⡟⢡⣽⣆⣼⡇⢾⡇⢠⣿⡿⠋⣆⢸⠇⠀⠀⠀⠈⠻⣧⡀⠀⣉⡠⠖⢶⣤⣀
⠀⠀⠀⠀⠃⣀⠰⠦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠉⠿⠛⢡⣿⣯⣿⡍⠛⠑⢁⡼⠋⢁⢰⡏⣿⠴⠒⠶⠶⠶⠶⢞⣷⣮⣄⡀⠀⠸⡄⠉⠓⢦⣁
⠀⠀⠀⠀⢰⡁⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣀⣤⣤⣄⠀⢉⡀⠘⣠⣤⣴⠟⠇⢠⣄⣾⣇⢸⡆⠀⠀⠀⣀⡴⢋⢀⢻⡆⠉⠓⠤⣿⣀⣠⠴⠞⠛
⠀⠀⢀⡀⠈⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⣹⢿⣹⡟⣷⣀⣡⡏⣴⣦⣄⣿⣿⡏⢼⣦⣹⣄⡤⠞⠉⠠⠀⠾⠀⢿⡀⠀⢠⠼⡿⣄⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⡟⠠⡟⢁⣿⣋⠙⠿⣯⡄⣼⠋⢸⡷⡀⠹⣿⢳⣄⡀⠄⠀⠀⡾⠀⠈⣳⡴⠉⠀⠁⠈⠑⢤⡇
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡀⠀⡇⠀⢩⡜⠻⣿⣢⣄⣼⢷⣿⠀⣏⠀⣿⠗⠻⠘⢯⠙⡦⠀⠀⣀⡠⠐⠛⢧⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⣚⠛⣽⠋⠙⣿⠛⣦⠟⠚⠻⢦⣄⠀⣾⣞⡴⠾⠛⠛⠳⢄⡀⠈⢧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢠⠀⠀⠀⠈⣿⠞⠹⣃⣼⢅⠀⠙⠣⣄⡙⣿⡏⠀⠋⡉⠀⡰⠆⠀⠀⠀⣉⣳⣬⣷⣄⡀⠤⠀⢒
⠀⠀⠀⠀⠀⠀⠀⢀⣐⠀⣿⠀⠀⠀⠀⠀⠖⠀⠀⠀⢉⠀⠀⢿⡆⠀⠀⠀⢻⡳⣄⢻⡜⠻⣤⣿⠶⠈⢫⠘⠷⠂⠀⢠⠐⠀⢀⣤⣄⠀⠀⠈⠁⠉⠉⠁⠀⡠⠃
⠀⠀⠀⠀⠀⠈⠀⠈⡍⣄⣸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⠀⠀⠀⠵⢜⠳⣿⡆⢸⣇⣠⣀⣄⠋⠀⠂⠀⠉⡠⡄⠀⠀⠙⠆⠀⠀⠀⠀⠀⢀⣶⠁
⠀⠀⠀⠀⠀⠀⢀⠒⠀⣄⡜⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠙⢿⣿⠈⣳⠊⠘⣦⠀⠀⠚⠋⠶⢧⣤⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⢳⠀⠀⠂⠀⠀⡀⣀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠢
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⠀⠀⠨⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠐⠓⠀⠀⠈
⠀⠀⠀⠀⠀⢨⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⠀⠑⠀⡴⠂⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠋⠀⠠⡀⠐⠀⠠⠂⠠
⠀⢁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠒⡄⠀⠀⢸⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⢣⡀⠀⠀⢁⠀⠀⠀⠠⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠋
⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰
⠀⢡⠀⠀⠈⠲⣄⢀⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⣿⠀⠀⠀⠰⣜⡜⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠆⠀⠆⠀⠀⡀⡸
⠀⢿⠀⠀⠀⡆⢹⠃⠐⡄⠀⡆⠀⢰⠀⠀⣧⠀⣀⠀⠀⠀⡀⢠⡄⢸⡀⡇⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡟⠀⡸⠀⡀⠀⡷⠁
⠀⢸⡆⡄⠀⡇⣸⣇⠀⣷⢰⡇⠀⢸⣸⢼⣿⠀⡿⢀⡆⡀⣷⣼⠀⡯⢣⢱⣿⡄⡀⣸⢹⠀⡀⠀⠀⡆⢀⡇⢀⡇⢀⡇⢸⠃⠀⠀⡄
⢰⢸⡇⡇⢠⠃⣿⢹⡀⡘⣼⡇⡀⢸⢸⣾⣿⠀⣇⠈⡇⣿⢸⠛⣄⡇⢸⡼⡏⡇⠃⡏⢸⡇⡇⠰⠀⡇⢸⡇⢸⢷⢾⡇⣾⠀⠀⢀⣷
⢸⡌⣷⣧⢸⢸⢻⠘⡇⢃⣿⠂⡅⢸⢸⣿⣿⡆⣿⠀⣿⡟⣼⣼⣼⡇⢸⣿⣷⡇⣰⡗⢸⡇⡄⡇⠀⠇⡟⣷⡆⠻⣼⡇⣿⢰⠀⡇⡇
⠈⡇⢸⣿⣼⣾⢸⠀⣧⢘⢸⠀⠱⣼⣸⣿⣿⡇⣿⠀⡏⣿⣿⣿⢹⡇⢸⢸⣿⡇⣿⡇⢾⡇⡇⣾⢸⢸⠃⣿⣇⣇⢻⡇⣿⢸⢀⣧⣧
⠀⡇⠘⡇⢻⣿⣼⠀⢻⢸⢸⠀⢸⡟⣿⣿⢿⠁⣿⠀⡇⣿⡹⣿⣾⢹⢸⢸⡏⡇⡿⡇⠈⡇⠇⠘⣸⣿⠀⣿⡿⢨⢸⠃⢿⣼⢸⡇⡿
⠀⣿⠀⡇⠸⣿⢹⠀⠸⣾⢼⡆⢸⡇⣿⣿⢸⠀⣿⠀⡇⣿⣇⡏⣿⢸⢨⠈⡇⣧⡇⡇⠀⣇⡄⠀⢻⣿⠀⣿⡇⠈⢸⡀⢸⡇⢸⡇⡇
⠀⢸⠀⣇⡆⣿⣼⠀⠀⣿⢸⠇⢸⣇⣿⢸⢸⣠⣿⡀⡇⣿⣿⠁⡏⡞⣾⠘⡇⣿⡇⡇⠀⣿⢁⠀⢸⣿⠀⣿⡇⠀⢸⡇⣸⡇⡟⡇⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣀⣤⣤⠶⠶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠏⠡⢀⡼⠋⠀⢠⡖⠋⠉⣩⡟⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡿⢰⠁⣿⣧⣤⣤⣤⣹⣦⣀⣿⣃⠀⠐⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⣜⣦⣀⠉⣡⣤⣼⣟⠿⠿⢿⡹⢻⣦⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣦⠄⠈⠛⡦⠶⠟⠛⠛⢟⢻⣦⠀⠈⣦⢹⡆⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⠞⠁⠀⠀⠀⠀⠀⢻⣿⠆⠀⣽⣿⠁⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡔⠋⢉⣽⠃⢸⡀⠀⣚⢂⠦⠀⢠⣾⡟⠀⣠⣿⠉⡀⠡⠄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣰⡿⠡⢹⣾⣧⡈⠉⠉⠀⠀⢸⡟⠀⢰⣿⡿⣷⡙⠶⣾⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢼⠛⣿⡿⠀⠀⠀⠀⠜⢣⡆⢸⡟⢀⢉⢻⣶⣤⣄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣿⠷⣄⠀⠀⢶⠆⢈⣧⣀⠑⠀⠠⣤⣝⢿⣍⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⠒⠲⠶⠶⠾⠛⢹⡆⠈⠂⠀⢀⣤⣿⣿⣿⣿⣷⣦⠠⡙⢷⡈⠙⢦⠘⠆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠲⣤⡀⢿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠄⣸⡇⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⢀⠀⠙⠛⠉⠁⠀⢸⣿⡯⠀⣰⡟⢠⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⠁⠠⠟⠀⠻⣿⡏⠑
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⣄⡑⢀⠠⠀⢉⠛⠳⣎⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣯⡙⠀⠸⡇⠀⠛⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣄⡀⠀⠀⠀⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⡀⢷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠉⠓⠤⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⠤⠖⠒⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡾⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣿⡁⠀⢸⠟⠛⠉⠀⠀⠀⠀⠀⠐⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⠳⢶⡖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠰⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢀⡀⠀⢻⣷⣄⡀⠘⣷⣤⣀⠀⠀⢀⣀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠘⢉⣛⡋⠀⠉⣛⣋⡛⠈⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢺⡇⠀⠙⠛⣷⠀⢸⡟⠛⠉⠈⢿⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡗⠀⠀⠀⡼⠀⠸⡄⠀⠀⠰⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢶⡶⣿⠶⢶⡷⡶⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⡀⠀⠀⣀⡐⠀⠀⣠⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡉⠁⠀⠈⢲⣄⣀⣀⣤⠊⠁⠀⠈⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠉⠳⠤⣄⣭⡃⠀⠀⣸⣅⡤⠴⠋⠁⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⡔⠁⠘⢁⡼⣄⠙⠀⠙⡶⢤⡀⠀⠀⠀⠀⠈⢹⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢹⡀⠀⠀⠀⠀⠀⢀⣜⣀⡤⠖⠋⠀⠈⠓⠤⣄⣘⣄⢈⡗⠀⠀⠀⠀⡼⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠲⡀⠀⠀⠀⠀⣀⣀⡞⠀⠀⠀⠀⢠⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣇⣀⠀⠀⢀⠉⡞⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠀⠀⠈⠉⠳⣄⠋⡼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⣐⣼⠿⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠸⠀⣿⣸⠃⠀⠀⣀⣠⡞⠉⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⣠⣀⣠⡄⠀⢠⠿⠛⠲⠶⣿⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠉⠉⠓⠚⠋⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢻⡀⠀⠀⠀⢀⡀⣀⡤⠞⠓⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣹⣦⣤⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⣈⣷⡶⠦⠤⠶⠿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠠⠀⠀⠀⠀⣀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⠃⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠓⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠠⣄⠀⠀⠀⣀
⠀⠢⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⡆⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣠⡀⣼⠋
⠀⠀⠀⠀⠀⠀⠠⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⡀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠙⣀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⠚⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠰⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠎⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠉⠙⢲⠖⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠁⠀⠀⠀⣠⠞⠉⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⢀⡠⠞⠁⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠙⣆
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣟⣟⣻⣿⣛⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡦⠀⠰⠆⠀⠷⠀⠘⠂⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⣠⡤⠄⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣠⡬⡇⢾⠁⢉⡇⠀⣗⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠃⡀⠛⣈⡃⠚⣁⠀⣩⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⡏⢹⣮⡗⠂⢸⡿⣇⢹⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠋⠀⠉⠉⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠶⠀⠰⠆⠀⠻⠀⠘⠃⠀⠙⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣈⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣦⠼⠟⠉⠙⠻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄⠻⠥⠂⣀⣔⣀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠶⠈⠚⠀⠉⠉⠉⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠋⣀⣀⣠⣾⠟⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⠀⡄⢀⡀⣀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠙⡋⠁⠀⣰⢸⡤⣿⢛⣆⢹⣆⣿⢸⠩⡗⢺⡇⡿⣾
⠀⠀⠀⠀⠉⠉⠸⣇⣀⡀⠈⠘⠃⡿⢿⡿⢿⣿⣿⣧⣦⣴⣶⠂⣧⡼⠀⠀⡀⡀⢀⣤⣤⡴⠃
⠀⠀⠀⠀⢄⡐⠊⡀⠈⠀⢀⠀⢸⡇⣼⡇⣾⣿⠏⣻⢿⡏⢉⣶⣿⠇⠀⠀⡀⠁⠈⠈⠉
⠀⠀⠀⢰⢈⡄⠘⢳⣃⠀⣾⡄⢺⣿⣿⡷⡚⣡⣾⠇⠀⣼⣾⣿⡏⠀⠀⠀⠀⣀⠀⠲⡆
⠀⠀⠀⠀⢰⣿⡀⣼⣿⡠⣿⣧⠜⢛⣁⣮⣿⣿⣿⢴⣧⣹⡿⠟⠀⠀⠀⠀⢀⠻⠗
⠀⠀⠀⠀⢾⣿⣇⣽⣿⡷⠛⣡⣾⣿⣟⣋⣉⠉⣿⣻⣿⡿⠃
⠀⠀⠀⠀⢿⡋⣻⡾⢿⣷⣿⡿⠯⠶⠿⠓⣆⣠⡽⣽⣿⠁⠀⣀⠀⣀⠐⠋
⠀⠀⠀⠘⡋⠁⠃⣶⣿⣿⣿⣦⣤⣶⣦⣼⡏⠈⠀⠈⠉⠈⠈⠉⠉⠉
⠀⠀⠀⠀⠅⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠄⠇⠰⠆⠐⠀⠸⠀⠂
⠀⠀⠀⠴⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠃⣴⡶⠒⢶⡶⠀⠠⠤⠐
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠁⠈