New england Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣷⣶⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣒⣀⣶⣶⣶⠖⡀⣐⣲⣶⡶⠂⠀⢠⠀⢰⠐⠂⢀⣠⡀⠀⠐⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢈⣿⣿⣿⣿⠁⠀⠀⢰⣿⣿⠇⠀⠀⠊⠀⡘⠀⠀⢾⠿⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣾⣿⣿⣿⠩⠀⠠⢀⣿⣿⣿⠀⠐⠒⠒⠐⠁⠤⠨⠉⠁⠠⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢽⣿⣟⡏⠀⢠⠀⢠⠅⠀⠁⠁⠐⠀⠀⠋⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢯⣾⣻⣾⣿⣿⣿⣿⣿⣿⠇⠀⡄⠀⣾⣿⣟⢀⡀⠈⠀⡈⠀⠀⠀⠀⢸⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣰⣿⣿⣿⡇⢻⣿⣿⣿⣿⠀⢰⡅⠀⡎⡏⣽⡗⠀⠁⠀⠉⠛⠒⠒⣠⣤⣤⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⡀⣼⣿⣿⣿⡇⠨⣿⣿⣿⣿⠀⢸⣇⠀⣥⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠁⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣶⣟⣿⣿⣿⡇⠀⣿⣿⢿⣷⠀⢸⣿⠀⣗⠀⠸⣿⠀⠀⠀⠀⢠⣧⣤⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣹⣿⣿⡟⠁⠀⢹⣿⠘⠹⣧⢸⣿⡀⣿⡄⠀⢿⣦⠀⠀⠀⠀⠙⢿⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⣿⣻⣿⢿⠀⠀⠀⢻⣆⠀⣽⣿⣿⣇⢸⠹⣦⣞⢿⣷⣄⠀⠐⣶⠀⣿⡟⠁⣾⠁
⠀⠀⠀⠀⠀⢀⣴⣻⡿⠀⣼⠟⣿⣿⣬⣤⣤⠀⠀⠌⠧⠀⢽⣿⣿⡄⠆⠈⠛⢾⣿⣿⣷⣤⣹⣾⣿⡇⢠⡟
⠀⠀⠀⠀⣠⣯⣿⡟⢁⣾⡿⢲⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠲⠶⠿⠿⠿⠿⠿⠿⠿⠷⣼⡇
⠀⣀⣤⣴⣿⣿⠏⢠⣿⡿⢀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠙⠿⣿⣿⡿⣿⣢⣿⠧⠀⣾⣿⠋⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠈⠓⠚⠉⠉⠂⠤⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠁⠄⠀⠀⠀⣄⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⠀⠀⠀⠀⡠⠔⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠐⠒⠺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠁⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠀⠀⡜⠀⠀⠀⡀⢐⠀⠀⠀⢀⠤⠤⠄⠒⠂
⠀⠀⡀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠈⠀⢰⡇⠀⠀⠀⠀⠈⡀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⣶⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⢸⠁⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠄⠐⠒⠰⠂⠼⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢸⣀⣀⣀⣾⣀⣀⢀⣀⣀⣠⡴⡇
⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⣸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠋⠢⠀⠀⠀⠀⠀⠀⠀⢀⠋⠈⠉⠉⠋⠉⠋⢹⠛⢳⣆⠀⠀⢓⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣇⢝⠀⠀⠀⠀⠀⣀⣰⣟⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠘⡂⠀⠐⠀⠀⠀⠀⠸⠂⠀⣻⣤⠖⡯⠨⠀⠉
⠀⠀⠀⠀⠀⠀⣾⣿⢛⠻⠄⠀⠀⠀⠀⣸⠈⠁⠱⠦⠀⠼⠛⠂⠀⠀⠀⠀⠀⠀⠑⠂⠀⡀⠀⠀⢀⠀⢀⡠⠤⠤⠲⠖⠚⠓⡚⠉⠀⠘⠴⠂⠤
⠀⠀⠀⠀⠀⠀⠈⠻⡵⠂⢂⠀⡅⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠒⣤⣿⡿⢡⠀⠠⠴⠛⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠖⠛⠂⠀⠀⠚⠒⠠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣠⣞⣉⣁⡠⠤⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣿⣷⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⢋⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣤⢤⣀⣀⢀⢤⣤⣦⣠⣤⣴⣴⣦⣠
⠀⠀⠀⠀⠠⣶⣶⣄⣀⢬⣿⣤⣿⣿⣿⣻⣿⣿⣷⣽⣿⣿⣿⡏⠓
⠀⠀⡠⢀⣠⡝⠻⢿⣿⣶⣯⠙⢿⠿⣿⡿⣿⣻⣿⣿⣿⣍⣻⣿⠤
⠀⠀⡀⡀⠀⣀⡀⢠⣉⣹⣾⣶⣤⣤⣃⡁⣭⣿⣿⣿⣿⣟⡿⠿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠦⣽⣳⣺⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⣷⣾⣏⣻⣿⣿⣿⣿⣎⠉⠀⠀⠀⠀⠀⠀⠠⢤⣼⣿⣤⣢⡤
⢈⣿⣿⣿⣿⣉⡾⣿⡏⠉⣛⡻⢁⣿⣾⣿⢾⢿⣷⢿⣿⣿⣿⣿⣿⡾⠦⠀⠀⠀⠀⣉⣷⣾⣿⣿⣷⣏⡉⣄⣄
⠈⢻⠹⣿⣶⣾⡚⣫⣯⣶⣻⣥⣾⣿⣿⣾⣿⣿⠷⣒⣿⣿⣿⣿⣿⣿⡿⡦⠀⠀⠉⣥⣮⢿⡹⣿⣿⣿⣷⣿⣭
⠀⢃⡶⣻⣿⣿⣛⣿⠊⣻⣯⣿⡏⣹⣷⣿⣿⣿⣿⢟⣿⣿⡃⠉⠋⠙⠟⠀⠀⠐⢚⣮⣝⣧⣮⣋⢻⣿⣿⣇⠉
⠀⢀⣷⣿⣷⣿⣿⣿⣿⣿⣁⣽⣷⣯⣻⣿⢿⣿⣿⡿⣏⣿⣿⢿⠀⠀⠀⣀⠀⠀⠈⠱⣿⡿⢿⣷⣾⣿⣿⣿⠀⠀⣄
⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣠⣌⣿⡿⣿⣾⣿⣟⢿⠎⠉⠙⠃⣀⣤⣴⡿⠋⢡⠀⢼⡷⣴⣻⣿⣬⣿⣿⣟⣂⠀⠸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⡟⣼⣿⣿⣿⣯⠿⣧⠀⠀⣶⣧⡾⡿⠃⣀⣀⣀⠈⠿⣿⢻⣿⣿⣿⣿⣿⣿
⠈⢿⠿⠿⠟⠛⠛⠉⢻⣿⣿⣿⢸⣷⣿⣿⢻⣿⣿⣄⣤⣠⣄⠟⠋⢙⣱⣶⣾⠿⢿⣷⣆⡙⣼⢻⣿⣿⣿⡿⣿
⠀⣞⢺⣭⣧⣅⣱⣮⣸⠿⣿⠼⢱⡏⢿⢿⢾⣿⣿⣿⣿⣿⣿⠀⠤⢩⠿⢻⡿⣿⣿⣿⣿⣧⠀⣤⠀⠙⢻⣿⠷⣸⡟
⠀⣼⣿⣿⣿⣿⣹⣏⠉⠠⢦⡤⣾⡆⣄⡈⢓⣌⠚⠛⢛⣛⣓⣀⣀⢀⣤⣬⣥⢮⣤⡬⠁⠁⢠⣉⣤⣤⣼⣿⠀⣿⣿
⠀⠸⡻⣿⣿⣿⣿⣿⣿⣿⣿⢴⣿⣧⠶⣖⠀⣠⣛⣒⣮⣍⠙⣻⡿⠸⠟⠛⠻⠛⠛⣛⣿⣭⠉⠉⢀⣀⡀⠀⠈⠉⠉
⠀⡣⣧⣙⣿⡿⣿⣿⣿⣿⡷⢸⣿⡛⢯⣤⣤⡿⣿⠙⢹⣥⣀⣈⢱⢴⣤⣤⣄⣀⠲⠻⣿⣾⣤⣐⣶⣽⣧⢀⣠⠔⢀
⢠⣿⣿⣿⡿⠿⠋⣢⣴⣿⠀⣼⣿⣷⠾⣯⢼⣿⣧⠀⠀⠛⡉⣽⣿⣿⣳⣬⣯⢿⣿⣷⣿⡿⢛⢧⣿⡿⣿⡟⡭⠻⠇
⠀⢙⣿⣿⡇⠹⢾⣿⣿⠏⢰⣿⣿⣿⡾⣿⣟⠛⠻⣿⣄⣾⣩⣿⣉⠁⠺⢿⣿⣷⣝⣻⣿⣽⣿⣿⣾⣡⢿⡮⢄⡰⠆
⠀⠀⠈⠙⠃⡐⠫⠟⠋⣠⣿⣿⣿⣿⣿⣿⣿⣭⣷⣤⣌⡛⠛⠛⠿⠿⠶⠤⠈⠛⠻⣿⠷⣯⢹⢻⡟⠃⡴⢿⣧⡋
⠀⠀⠀⠀⠀⠀⠈⠖⠑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠖⠒⠀⠀⠀⠀⠈⠀⠂⠘⠻⢄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠒⠀⠐
⠀⠀⠀⠐⢰⡶⠒⠀⠤⠠⠤⠾⠿⠿⠯⣿⣷⣤⣤⣀⠸⣀⣰⣄⣀⠠⠤⠠⢠⠤⠤⡄
⠀⠀⠰⠶⣉⣦⣴⣴⣶⣶⣶⣤⣥⣧⡥⣼⣽⣿⣽⣿⣼⣿⣯⣿⣝⣀⡀⠀⠀⣀⡤⣤⣤⣀⣀
⠀⠀⠀⣠⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⠛⣤⣾⣯⣦⡾⢿⣿⣿⡳⢒⠦⣄
⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣽⡮⣽⠋⢹⣝⢋⣩⣽⣿⣿⣿⣿⣵⣮⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙⠛⢻⠛⠚⢃⡈⡉⢛⡋⠉⠉⠉⠉⠋⠉⢛⣭⠉⣁⡈⠁⠛⠿⣯⣼⣷⣿⣝⣿⡺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠉⡉⠉⢁⣀⣘⣛⣝⣛⣒⣠⣀⣀⣪⡉⡔⣿⣿⣷⣶⣦⡄⠀⠻⠭⣤⣄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠐⠈⠙⠓⢿⣂⣟⣧⡌⠉⠙⢿⠦⠆⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣀⡀⡰⠷⣾⣿⣿⡤⢀⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⢀
⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣸⣿⣿⣷⣾⣐⡧⠘⠃⣤⡀
⠀⠀⠀⠛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛
⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣷⣿⡟⢿⣿⣿⣿⣿⣿⣍⠉⠁
⠀⠀⠀⣈⣉⣿⣿⣿⣿⣿⣟⡿⠙⠿⠿⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠸⢿⣿⣿⣿⣿⣿⡃⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣓⣖⣒⣒⠒⠒⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣦⣤⣤⢼⣿⣿⣿⣿⣵⡗⡶
⠀⠀⠠⡽⢿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠙⠛⡛⠟⠲⠦⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣷⣾⣈⣳⣯⣑⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣴⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣚⣿⣿⠿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣹⣿⣷⡯⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣴⡄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣭⣿⣿⣿⣂⣬⡉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣤⣤⣤⣄⣸⣷⣬⣭⣿⣽⣻⢿⡏⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠋⠉⠙⠛⠛⠛⠛⠛⠛⠟⠿⠛⠛⠛⠿⠛⠛⠛⠛⠿⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣴⢈⡀⠀⢀⠀⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡆⢀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⢠⡿⢁⣼⣷⣸⣧⣼⣿⡆
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣷⠈⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣂⡊⢀⡠⣶⣾⣧⠸⣿⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠄⠀⠀⠀⠀⠘⢿⢿⣇⣈⣿⣿⣆⠠⢠⣥⡬⠤⢥⠀⠠⢤⣴⠏⣰⣦⣯⣿⡙⣿⠠⢿⡿⣿⣿⣿⣿⣿⡀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⢾⣿⣿⡄⠈⠀⠀⠈⠁⡸⣿⣿⡇⠹⢿⡃⢈⣰⣛⣶⣀⣀⣀⣂⣋⣹⡟⠻⣿⣿⣿⣷⣿⡗⣿⣿⣿⣿⢿⣿⣽⣧⡇⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣸⡇⣈⣽⡿⣿⢻⡟⠿⠉⠊⡹⣟⣸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠜⠃⢀⢹⣃⢸⡇⠃⠐⠈⠀⣠⣠⣮⣦⣼⡿⠷⣟⠫⢌⣱⣮⣯⣬⣿⣟⣻⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⡆⠘⠻⣿⣠⣷⠀⣟⠒⠀⠀⠒⢀⢠⢬⠉⠍⠉⠉⠉⠉⠉⠉⠋⣋⡉⠉⠉⠉⠐⡾⣶⣶⡿⠿⣿⣿⠿⢿⣾⣻⣿⣿⣿⣷⠖⡀⢸⡇
⠀⠀⠀⠴⡯⡄⡁⢪⡆⠁⣿⣿⡿⠂⡋⢀⠐⠀⢁⠈⠙⣨⣴⣤⣆⣰⣾⣿⣿⢇⣿⣟⣆⣀⣠⣖⡓⣾⣿⢡⠒⠛⠛⣤⠐⣶⣾⣿⣿⣿⡿⠃⢘⠈⢘⠅
⠀⠀⠀⠀⠀⢰⣦⡶⡇⠆⣿⣿⣿⠇⡅⣀⢀⣐⡻⠀⠤⠤⠭⠭⡭⠭⣭⣭⣭⣭⣭⣭⢭⣭⣭⣭⣭⢡⣼⠯⠐⡄⠈⠀⠂⣿⢹⣿⣿⣿⢹⠀⠨⠀⢰⠂
⠀⠀⠈⠀⠀⢀⣘⣀⡗⠀⣛⣿⣿⠀⠿⠂⠀⠋⣣⡄⢲⣧⠤⣥⣥⣤⣿⣿⣿⣿⣯⣥⣤⣼⣍⣹⣟⣼⣿⢆⢿⠋⠀⠀⣈⠿⢲⣏⢻⣿⢸⠀⠀⢀⡘
⠀⠀⠈⠀⠈⠙⣿⡯⡏⢰⣿⡟⡍⠀⠁⠀⠀⣴⣍⣴⣼⣯⣿⣷⣷⣿⣿⣿⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠺⠀⠀⣀⡙⠏⢀⣿⣿⣿⢁⣹⣶⣿⣎⡁
⠀⠀⠰⠀⠀⠈⠃⡥⣿⢀⠟⢃⠘⣶⡀⠒⠀⠁⠘⣿⣿⣆⢸⡧⣿⣿⢿⣿⣾⡏⣽⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⣅⣷⡈⢯⣿⣿⣼⡟⠀⠿⠷⠀⡣
⠀⠀⠀⠀⢴⠀⣧⣤⣇⡾⡆⠀⠀⡇⢱⠀⠀⠀⠀⣿⣛⡙⢸⡿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣸⣿⣯⣿⣯⣧⣿⡝⢙⡓⢢⠤⡀⠁
⠀⠀⠀⠀⡀⠰⠮⠛⠉⠁⢼⠀⣴⣿⠃⡱⢢⣤⣴⡿⢾⣿⢸⣷⣿⣿⣧⡀⠀⠀⣦⢹⣿⣿⣛⢻⢺⣿⣿⠀⠈⢹⣏⡙⠛⠉⢿⣿⣇⣿⡇⠉⠘⠃⠂
⠀⠀⠀⡀⡿⣾⣷⣂⣀⠀⢈⡸⣿⢿⣷⣿⠀⠈⢛⣷⣾⣿⣼⣸⣿⣿⣿⣷⣾⣶⣿⣷⣿⣿⣿⣶⢹⣿⣿⠀⠂⠘⣿⣿⣷⣮⣍⣉⣥⡤⠶⠤⠀⠛⠃
⠀⠀⠀⠀⣷⣼⣿⣿⡿⣿⣿⣇⠙⣯⣅⣿⣆⠞⣿⣯⣿⣿⡇⢘⣛⢩⣭⣽⣉⠛⠿⠿⡿⠿⠛⠛⠚⠋⣛⣁⡀⢀⣿⣽⣿⠍⠻⢿⣷⡶⡿⢶⠟⠒
⠀⠀⠀⠀⣧⣿⣿⣿⣿⡟⠿⠿⠧⣶⣹⣿⢿⡇⣿⣿⣿⣿⡿⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⠾⠷⠀⠀⢰⣿⢉⠉⠉⡁⠾⣟⣤⣴⣶⣬⣷⢺⡓⡖⣒
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⢸⠀⣷⣤⣼⣿⢩⣏⠛⣹⣿⢩⣥⣭⣭⡕
⠀⠀⠀⠀⣿⢿⣟⣻⣿⣿⣿⣷⠀⠀⠀⣼⡟⣿⠼⣷⠆⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣇⣿⡿⣿⢸⡈⣿⣿⣿⣷⢸⣿⣷⣬⣿⣸⣿⡿⢿⠅
⠀⠀⠀⠀⡿⠤⣿⡀⣼⣿⣿⣿⡄⠀⠘⠻⣷⡏⠀⠈⢷⣸⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⣀⣀⣿⣿⣷⣿⣋⡇⢹⣿⣿⡉⠿⠿⠿⣯⣿⡸⠒⠆⠀⠒
⠀⠀⠀⡀⣿⢘⣏⣍⣯⣿⣿⣿⣷⣤⣶⣿⣿⣷⣦⢄⡈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠻⠿⠛⠛⠛⠾⠿⠟⠻⠇⠀⠢⡀⠀⠁
⠀⠀⠀⠀⠛⠹⡟⠻⠿⢿⣿⣿⣿⣿⡿⢿⡟⠻⠯⠤⠍⠉⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⢀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⣦⣦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠸⣿⡟⠀⠀⠠⠠⡀⠀⠀⡤⢀⣤⣿⢿⡆⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣧⣤⠀⠀⠀⠀⢹⣿⣎⠻⣟⢀⣘⣣⣉⣁⣄⠛⡷⢺⣿⣿⣼⣧⣾⣿⣯⣿⣿⣧⡆⠂
⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠹⡏⠀⣿⢨⡿⠿⠻⠛⠉⠩⢫⣼⣿⣿⢿⠟⣿⢿⣿⣿⣿⣿⣧⠁⠆
⠀⠀⠀⠀⢀⠀⠿⣿⢠⡀⡀⠀⠀⠀⢀⡰⠷⠾⠧⠶⠒⠶⠾⠟⠿⠏⢹⣬⣴⣿⣿⣿⣷⣾⣿⣿⣷⡄⠀⡄
⠀⠀⠰⠆⠀⢴⠒⣿⣿⠂⠛⠐⠀⡀⠙⢩⣤⣄⣰⣾⣷⣶⣿⣄⢀⣖⢲⢿⡹⠐⢃⡄⢾⣼⣿⣿⡟⢀⠀⠇
⠀⠠⠀⠀⠳⢻⡈⠿⣿⡇⣤⠀⠬⠁⢒⣒⠒⠖⢲⣾⣿⣷⠗⠓⡶⢶⡄⣾⢡⡤⠀⠀⣼⣹⢿⡿⡇⠐
⠀⠀⠀⠐⣶⣾⠆⣾⢟⠁⠀⠀⣠⣚⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠁⣀⠺⢨⣾⣾⠋⣧⣶⡐
⠀⠀⠀⠀⠉⢳⣣⠛⠄⡶⡄⠀⠁⢙⣿⡄⣿⣿⣟⣿⣿⡼⣿⣿⣿⣿⣿⣿⠀⠀⣬⣧⣹⢿⣿⣏⠙⠓
⠀⠀⠀⠀⣛⠞⠊⢁⠀⣥⠆⠀⣀⣸⣿⣄⣿⣿⣿⠛⠛⢀⢻⣿⡿⣿⣿⣿⠀⠾⡿⠿⠿⣿⣿⣾⠐⠲⠆
⠀⠀⢰⡶⣷⣄⣀⢈⣸⣿⣶⡆⠉⣻⣶⣿⢇⣿⣿⣿⣶⣾⣾⣿⣿⣮⣿⣷⠀⠀⢿⣶⣶⣛⣫⠭⠤⠀⠒
⠀⠀⠀⣷⣿⣿⣿⣿⣥⠑⣼⣷⡮⣿⣾⣿⡠⣤⣶⣼⣭⣭⣹⣟⣉⠉⠉⣽⠤⠤⠛⣿⠃⣙⡻⣟⣻⣋⡁⠁
⠀⠀⠀⣿⣿⣿⣿⣬⡉⠉⠻⣿⣷⣿⣿⡿⠀⠈⠉⠉⠉⠉⠉⠉⠁⣀⣀⣿⠂⣠⣀⣿⣻⡛⢻⣷⣃⣒⣓⠄
⠀⠀⠀⡿⣿⠿⣿⣿⡇⠀⢀⣿⡿⠾⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⣿⣿⣸⢹⣿⡿⣸⣿⣦⣿⠿⠟⠟
⠀⠀⠀⡏⣿⠲⣿⣿⣷⣀⣠⣿⣇⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠿⣿⡿⣾⣼⣿⣷⡌⠉⠙⠓⠂⠉
⠀⠀⠘⠓⠿⠻⠿⣿⣿⡿⠿⠿⠻⠧⠁⠙⠆

⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣤⣤⠀⠀⠀⢀⣠⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣚⣛⣿⣿⣟⣿⣿⣿⣾⣗⡀⠀⠀⠀⠀⠈⠀⠀⣠⣤⣤⡤
⠀⠀⢠⣴⣿⡿⢿⠿⠿⣿⣿⡆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⡡⠄⢛⣁⣀⡙⠊⠻⠿⣿⣯⡁⠀⠀⠀⠀⠀⢣⣥⠿⠛⠓⠲⠾⣷
⠀⠀⣿⣿⡏⠀⡛⢐⠄⢰⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢲⡖⠚⢿⣷⠀⢐⣿⠿⣶⡤⠀⠀⠀⠠⣄⠸⢃⣀⠀⠀⠀⡆
⠀⣸⣿⠟⠸⠏⠋⠜⢣⣈⡀⠇⠀⠘⠁⡀⠄⠠⠤⠀⢀⣤⣶⣼⣧⣤⣿⣿⡧⠀⠛⠀⠳⣶⡀⠀⠀⠀⢨⡶⠨⡉⠉⣀⣀⠊⠙
⠟⣿⠋⣦⠀⠀⠰⠁⠰⠟⠛⢄⣀⡀⠀⡜⡀⠠⡔⡄⣸⠟⢻⣿⣿⣿⣿⣿⡄⢠⣘⢃⠀⢿⣍⠀⠀⡠⠀⠁⠀⠀⠸⣿⡟⢀
⢈⣹⣿⡟⠀⢀⠀⠁⠀⣀⠀⠀⣴⠁⠀⢀⡿⠀⠀⣀⠠⠋⣿⡟⠉⠀⣹⣿⠇⠈⠉⣴⠷⠲⡿⠀⠀⠐⠤⡠⠈⢁⠁⠙⠐⠛
⢸⣿⡿⠁⠀⠋⠀⢀⣂⠋⠘⠀⡁⡌⠃⠾⣧⠄⣴⠛⢠⡦⠰⢦⠀⢰⣿⠏⠀⠀⠀⣯⠀⠀⣬⠓⡶⠟⡢⠄⠀⠸⣧⡟
⢀⠼⠂⠀⠀⠀⠀⠘⠷⠋⠓⠀⡜⠁⢈⡄⢻⠀⡽⠀⣼⠀⠀⠈⠀⠀⠈⠀⣀⣀⡀⠙⠓⠚⢿⣬⣄⢀⣭⣴⡄⠀⠹⠗⢄⢀⡀
⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⢀⠁⢸⢶⡇⠆⠀⠀⠁⠄⠀⠀⠹⠀⠨⣿⠏⠀⢀⠆⠀⡆⡟⢿⣿⠇⠀⠀⡤⠀⠀⢸⠟
⠀⠁⠀⠀⠀⠀⠀⣰⠂⠀⢘⠸⡏⢷⣄⡀⠨⡈⡅⠀⠀⠀⢠⣀⡀⠄⠀⠀⠐⠓⠨⠷⠄⠟⠀⣿⠇⠀⣿⠀⠉⠋⠀⠈⢓⣸⣆
⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠸⠀⠀⢸⡍⢹⣿⢿⣿⣿⣿⣿⣿⣿⡍⢓⠋⢠⠐⠛⠀⠒⠒⠣⠌⢹⠀⣰⡏⠀⢴⡗⢒⣾⠿⠻⡟
⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢾⡅⠈⡏⠀⠀⢺⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⡀⣿⠇⠀⡾⣇⡛
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢸⣶⠀⡗⠀⠀⡀⠈⠕⡿⢿⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⡿⣿⠀⠉⠀⠙⠄⡀⠃
⠀⠀⠀⠀⠀⢀⣤⣄⠀⡀⠀⠀⠀⠈⠀⠈⡇⠀⠁⠅⠒⠀⠁⠜⠀⣀⣂⠐⠀⢀⠀⠀⠀⠀⠱⢿⠀⠀⠀⢁⣄⠛⠢⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠸⠦⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠸⡆⠀⠀⢀⠀⠻⠆⠀⠈⠀⣠⣶⠀⢰⡟
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠒⢤⡀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⢿⡿⢰⡄⠀⠀⠀⠀⢀⡟⢀⣴⣶⣿⣿⣶⢀⣾⣿⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⢁⣿⣿⡆⠀⠀⠀⡾⣠⡞⠁⠀⠀⠙⠃⣼⣿⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⢀⡾⢻⣿⣿⡀⠀⣼⢱⣿⡇⠀⠀⠀⠀⣼⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⡤⠤⣤⣿⣏⣀⠀⠀⠀⣀⡤⡼⠁⠈⣿⣿⡇⣸⠃⣾⣿⡇⠀⠀⠀⣰⣿⡿⠁
⡇⠀⠀⠀⠀⢀⡴⠞⠛⠉⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠈⠉⠉⣽⣹⠃⠀⠀⢹⣿⣷⠏⠀⣿⣿⡇⠀⠀⢠⣿⣿⠃
⡇⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⢀⣼⢻⠃⠀⠀⠀⢸⣿⡏⠀⠀⢹⣿⡇⠀⠀⣾⣱⠏
⡇⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣾⢿⢧⠏⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⣿⣧⠀⢀⡿⠁
⡇⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⢠⣏⡞⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⡇⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⣠⠐⠓⠶⠀⠀⠀⢠⡞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⡇⠀⠀⠀⠘⢦⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣀⣴⠏⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⡇⠀⠀⠀⠀⠀⠙⠳⢶⣤⣬⣀⣠⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⣤⣀⣀⡀⢀⣀⣀⣠⣤⠶⠟⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡇⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢠⠸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠀⠀⣀⠀⣀⠀⣀⢀⡀⠀⠀⠀⣠⠿⠲⠀⢀⡀⢀⠀⠀⢀⣀⠀⣼⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡟⠀⣠⠏⠘⢡⠏⣴⠏⡘⠀⠀⠀⣾⠃⠰⢒⠄⣼⢇⡾⣡⠞⣱⠏⣼⠃⡴⢋⡼⢁⡿⢠⠏⣰⠊⡸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠀⠁⠛⠀⠉⠈⠀⠀⠀⠀⠙⠆⠐⠈⠸⠃⠘⠃⠻⣸⡟⠀⠃⠘⠗⠙⠁⠞⠀⠟⠀⠃⠘⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⢰⣸⣶⣾⣧⡄⠀⠀⠀⠀⣤⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢤⡄⢀⣠⣴⣿⣿⣿⣿⡿⣦⣄⡀⣴⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣷⣄
⠘⢿⡿⠿⠛⠛⣍⠫⡹⢌⠉⠙⠛⢿⡿⠟⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣿⣿⣧⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠈⠿⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠻⢋⣋⣉⣙⣁⣿⣏⣻⣉⣩⡹⡝⢻⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⢠⣤⣤⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠂⠀⠀⢀⠀⠋⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣷⡇
⠀⠀⠀⠀⠀⢸⣿⣧⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⣾⣦⠀⠘⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡗
⠀⠀⠀⠀⠀⣼⡿⣿⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠉⠉⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇
⠀⠀⠀⠀⠀⡟⡱⡷⡒⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠹⣿⣿⣷⣶⡦⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠋⠘⡟⠐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠟⠻⠋⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⣿⣿⣿⡇
⠀⠀⠀⣶⣾⣦⣴⣥⣤⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠐⠛⠉⠁⢰⡿⠆⢀⣀⡀⠀⠀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣿⣿⠃⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣦⢤⣤⣤⣤⣄⣀⣠⣭⣭⡥⢤⣶⣿⣿⣅⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣿⣿⠀⣿⣿⣿⣷
⠠⣀⣘⣿⡿⣿⣿⣿⣿⣿⣿⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠿⣧⣼⣀⣠⣠⣄⣿⣇⣣⣠⣤⣼⡿⠿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⢿⣿⣿⣶⣿⠿⣿⣿⣿
⠀⠀⠉⠉⠀⠀⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣶⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣧⣿⣿⣼⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⣾⣿⡟⣾⡿⣽⡿⣻⣿⠇
⠀⠠⠖⣤⣬⣿⣿⣶⣛⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⡇⣿⣿⣷⣿⡇⣿⢻⡏⣮⣠⣾⣧
⠀⢀⣀⣀⣾⣿⣭⣭⣭⣽⣿⣯⣭⣭⣦⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⣿⡇⣿⣾⣇⠿⠻⣿⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠠⣄⣀⡿⢿⣿⣿⡿⢿⣀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣸⣿⣧⢿⣷⣻⣷⣽⣷⡄⠉
⠀⠈⠉⠉⠉⠉⣉⣉⠉⠉⣉⣍⠉⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⣠⡄⢠⣌⣥⢀⡄⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡆⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠙⠛⠉⠉⠛⠛⠉⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡬⢁⣄⣡⡈⣌⢩⣈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣎⣿⣞⡿⣯⡍⢱⣧⡲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⣞⣯⣷⠻⡁⢣⣿⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣭⣧⢿⣶⠟⢀⠐⠟⡀
⠀⠀⢀⢐⠀⢢⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣘⢩⠜⡟⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄
⠀⠀⠘⠀⣈⠠⠬⢭⡇⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢐⣿⣿⡷⡆⠀⠀⠀⠀⠀⠀⠀⡞⠈⣾⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆
⠀⠀⠀⠺⡛⣀⠞⠴⣇⠀⠀⠀⠀⠀⢠⡜⠀⠀⢰⣦⠀⠈⠉⢹⠹⡇⠀⢰⣦⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⡀⠀⠀⠀⢠⡆⢀⡴⠛⠾⢑⣋⡽⣻⣿⣧
⠀⠀⠀⠀⢳⡁⠀⣀⣀⣀⣀⣩⣍⠀⠈⡰⠄⠀⠹⠿⠖⠀⠀⣤⢀⣁⠀⢀⣠⠀⠒⢒⡗⠒⠒⢸⡇⢸⣴⣷⣬⡇⠀⠸⢷⡏⠁⠙⣿⠻⣾⣨⣽⣿⣿⡆
⠀⠀⠀⠀⠘⢛⡞⠛⠛⢓⢎⠉⠐⠒⢒⠓⠐⡇⢈⡱⠒⠀⠀⠉⠈⣙⠓⠛⠛⣓⡄⠼⠷⠦⠴⠼⠿⠿⠿⠿⠯⠧⠔⠶⠸⣇⣀⣀⡀⣠⣿⢙⣿⣿⡽⣳⠖
⠀⠀⠀⠀⠒⠚⠃⠀⠀⢰⣠⣦⣴⣤⣿⡆⠀⠣⠀⣀⢀⡄⠀⢀⢠⣿⠦⠤⠀⠚⠀⠒⠂⢒⣀⡂⠔⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⣋⠙⠙⠉⠊⠐⠛⠓⠙⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⡛⠋⠀⠀⣻⠀⢀⢄⠘⠉⠘⠉⠁⠈⠁⡸⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠇⠀⠀⠠⠋⠀⠈⠀⠀⠀⠂⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢡⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡍⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠟⠀⠁⠀⠀⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⣠⢦⡴⠀⠈⠀⠀⠲⠄
⠀⠀⠀⡀⠀⠀⠀⢀⣜⠧⠤⠤⠴⠐⠐⠶⠀⣁⢀⠃⠀⡄⠀⢹⣼⠧⠞⠢⡠⠀⠱⠇
⠀⠀⠠⣅⠀⠀⠀⠼⣋⠤⠠⠠⠄⠀⠄⠀⠘⠃⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠤⠄⡀
⠀⠀⣠⠎⠀⠀⠀⣠⠈
⣰⠉⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢂⠀⢀⣀⣀⠦⠠⠿⠦⣄⠀⠀⣠
⠁⢡⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡄⢈⠉⣷⢾⣿⡀⣄⣠⡴⠸⡽⣉
⡄⠈⠉⠀⠀⠀⠐⠒⠚⠃⢤⠤⢀⠀⠀⠀⠀⠔⡀⢠⣤⡀⣠⣄⠐⠀⣢⡤⣀⣠⣀⣤⣤⣄⣥⣼⡿⣷⡶⣆⠶
⠁⠀⠀⠀⠀⠀⡄⠀⠀⢀⣼⣧⣼⢴⡎⣤⣤⣋⡇⢨⣫⣧⠒⣴⣶⠀⣸⠳⠈⡿⢙⠶⠻⠛⠀⠀⠀⠀⠈⠻⠓
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠉⣩⣭⣯⣶⣀⠛⠃⠁⠃⢈⣤⠤⠤⢰⡆⣶⣧⠀⣆⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠉⠩⣝⠘⠀⣾⠙⣿⣿⢀⠻⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⡁⣤⠁⢼⠛⡇⡏⡏⠍⡿⠂

⠀⢀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣀⣀⣀⣀⣀⣀⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡍⠀⠀⠀⠀⠀⢀⣰⡀
⠀⣿⣿⣿⡿⣿⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠛⠂⢰
⠀⣿⣳⢿⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠧⣄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⢧⡞⣿⣿⠃⠀⠀⠀⠀⠀⢠⣾⣿⠋⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡂⠀⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢤⡔
⠀⠻⣼⣽⣿⡏⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠴⠋⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⣏⡁⢀⡏
⠀⣾⣿⣿⣿⠇⠁⠀⠀⢀⣠⣾⣿⣦⣤⣶⣿⡛⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠤⣰⣿⣿⣀⠀⢷⡄⠀⠀⠀⠀⣾⡢⠀⡀⢠⣶⣾⣷⣤⣾⣗⣏⡀
⠀⣽⣟⣻⡿⠀⢀⣠⣾⣿⣿⣿⣿⠛⣿⣿⣿⣿⣏⠀⠀⢀⡞⠛⠁⠀⠀⢀⣀⣀⣸⣿⣿⢿⠀⠻⣷⠀⠀⣿⣿⣿⣷⣾⠀⢸⠦⢘⣿⣿⣿⣃⣿⣷
⠀⢾⣿⡟⣤⣶⣿⣿⣿⣿⣿⣿⣿⡀⠨⣿⣿⣻⣿⣆⣤⣼⢃⣻⣿⡯⡀⢼⣿⣟⣻⣿⣿⡿⠀⡀⣿⠀⠀⢸⣿⣿⣿⣯⣇⣸⣇⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠆⣸⣿⡿⢿⣿⣿⣿⢋⢿⣿⡒⢈⣾⣾⣟⠉⠹⣿⣿⡀⡇⢹⠁⠀⢸⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⡿⢿⠿⠁⠀⢹⣿⠕⣫⡾⠿⢿⠘⡿⣿⣿⡇⠇⠸⠀⣀⡸⣿⣿⣿⣿⡿⢈⣿⣛⣨⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠆⠸⣿⣟⠷⡀⠀⢀⢸⣿⣾⣿⠇⠀⢶⡄⠀⢘⣿⡇⢀⡅⢀⠀⠄⣿⣿⣻⢽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠟⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⣸⣿⣤⠈⠀⠀⠈⠻⡗⠀⢻⠈⢀⣀⣼⠉⢻⣼⣿⡿⢿⠿⠟⠿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⠄⢰⣾⢟⣟⢿⢷⣤⣶⣿⣿⣿⣿⣿⣿⣤⠀⡀⣄⡀⠀⠀⠀⠀⣾⣿⣿⠀⢳⣿⠋⣱⡋⢸⡀⠀⢸⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠅⢰⠳⠁⣼⣄⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⢀⠙⣿⣿⣧⡀⠀⠇⠿⣿⣿⡜⢺⣿⠆⡻⣧⡼⠳⠾⠿⣿⣿⣿
⠀⣿⣿⢿⣿⣿⣿⣿⣫⣿⣿⣿⣿⡂⠈⠀⠀⡉⣿⡶⠤⢶⣦⡤⢉⢹⠿⠛⠛⠋⠀⠀⡘⠠⢿⡄⠀⠈⠀⠘⢿⣿⡈⣿⡁⣿⣧⣴⣿⣶⣄⠀⠀⠘
⠀⢹⣿⣷⣿⣍⣿⣷⣾⣿⣿⣿⣿⡿⠄⠀⢸⠇⠉⠙⠉⠉⠀⠀⠀⢀⣤⣶⢿⣶⣦⣄⢠⣄⣘⣷⣀⢀⣧⣀⠘⣧⡴⠬⠟⣿⣿⣿⣿⢿⣟⡀⡀
⠀⢸⣿⣿⣿⣯⣴⣤⣉⣉⠉⡀⠈⠷⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⡋⠀⢶⣿⣿⣿⡸⠿⣟⠛⠻⠁⢀⣬⣭⣿⣶⣄⣾⣿⣿⣿⣿⡿⢻⣿⣧⡄
⠀⠸⠿⠛⠁⢉⡙⠋⠉⠩⠙⠛⠻⠲⣄⢀⣠⡴⠀⠀⠰⣆⠀⠉⠈⠉⠁⠀⠈⠙⠛⠿⠀⠀⠀⠁⠀⠀⠈⢉⣉⣉⣉⣻⣿⣿⢿⠿⠿⢷⣾⢸⡿⠃
⠀⢸⣷⣿⣟⠤⠿⠖⠦⠀⣀⠀⠀⠀⠙⠻⢿⣤⢀⣀⣴⣿⢀⠀⡖⠀⠀⠀⠀⠠⠄⡀⣄⣀⣀⠀⠀⠀⠀⠈⢹⠿⣿⣷⣮⣿⣶⣤⣾⡿⢟⡾⠁
⠀⢸⣿⣿⣿⣷⣶⠆⠰⢳⣟⣿⠛⢂⣀⡀⠀⣹⡟⢻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⢿⣏⠛⠃⡉⠀⢢⡄⠀⠉⠙⠻⠛⠿⠷⠾⢿⣿⣦⣀
⠀⠸⣿⣿⣿⣿⣿⡶⢃⠀⣉⣶⣋⣛⣷⣶⣴⣿⣯⣿⣋⣽⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⡿⣛⡁⠀⠀⠠⠄⠀⢀⠀⠀⠀⠈⠙⠿
⠀⠀⠛⠛⠿⢿⢿⣿⣯⡀⠝⠿⣷⣿⣿⣿⣿⣿⣆⣂⣿⣩⣽⢻⣿⣦⣍⠁⠻⢟⣃⣿⣟⠛⠻⡴⢤⣴⣶⣿⣶⣭⣷⢾⣤⣄⠀⠀⡀
⠀⠀⠀⠉⠁⠶⠦⠌⠟⠛⠷⠶⠈⠹⠿⠁⠿⠿⠿⠍⠛⠛⠻⠿⠟⠛⠛⠿⠿⠿⠯⠿⠻⠿⠿⠯⠴⠅⠩⠍⠛⠟⠛⠋⠛⠛⠛⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠶⠶⠚⠛⠛⠉⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⢸⡁⢛⢻⣆⠐⠆
⠀⠀⠀⠀⠀⢀⣤⡤⠶⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⢸⢸⠈⡅⢸⠸⣿⡆⢠
⠀⠀⠀⢠⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⣄⠀⣾⡿⣲⠌⣧⠀⠀⢀⡿⠀⠀⢸⣾⠀⡆⢸⠀⣿⡇⢸
⠀⠀⠀⢸⡻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⣤⣀⣒⣚⠩⠍⢳⡟⠁⢹⡆⣿⠂⢸⡆⠸⣧⠀⣸⠇⠀⠀⢸⡟⠰⡇⢸⠀⢿⡇⢸
⠀⠀⠀⢸⣿⡏⠀⠀⢀⡀⠀⡴⠄⢰⣷⢀⣾⠟⠛⠉⠃⠛⠈⢉⠉⠀⠀⣼⠁⠀⠈⡇⣿⣁⡾⠃⠀⠹⣧⡟⠀⠀⠀⢸⡇⢰⡇⠈⢀⢸⡇⢸
⠀⠀⠀⢸⠋⡇⠀⠀⢙⡇⠀⠄⡆⠸⣿⠘⠧⠶⢖⣠⣄⠀⠀⠘⢀⠀⠀⣿⠀⠀⢀⡇⣿⠛⢷⡄⠀⠀⢻⡃⠀⠀⠀⢸⡇⡞⢠⠀⢸⢸⡇⢸
⠀⠀⠀⢸⣠⡇⠀⠀⠈⣷⠀⡄⡇⠀⣿⠀⠀⠀⠀⠀⠙⣳⠄⠀⢸⠀⠀⢻⡀⠀⣼⡇⣿⠀⠈⣷⠀⠀⢸⡇⠀⠀⠀⢸⡇⡇⢸⠀⠸⢸⡇⢸
⠀⠀⠀⢾⣿⡇⠀⠀⡤⢛⠭⢤⡃⠀⣿⠀⠀⠀⠀⢀⠠⣠⠄⢀⣸⠀⠀⠘⢷⡾⠋⠀⣻⠀⠀⢻⡆⠀⢸⡇⠀⠀⠀⢸⡇⠁⣼⠀⢰⣾⡇⢸
⠀⠀⠀⢾⣿⠇⠀⠀⡇⡏⠀⢸⡇⠀⣿⠀⣦⣶⠶⠿⠛⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠇⠀⢸⡇⠀⠀⠀⢸⡇⡇⠉⠘⢸⢿⡇⢸
⠀⠀⠀⢸⣯⡀⠀⠀⡇⡏⠀⢸⡇⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⡇⣿⠀⢸⠸⢸⡇⢸
⠀⠀⠀⢸⣾⡀⠀⠀⡇⡃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢇⣿⠸⠀⢸⡇⢸
⠀⠀⠀⢸⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⡰⢶⣦⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠀⡄⢸⡇⡞
⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⡠⠄⠉⠀⠀⠀⡆⢀⠀⠐⠋⣿⡄⠀⣱⣨⣿⡄⠀⠀⣯⣀⣀⢀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠀⡇⣾⡇⡇
⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⣰⠈⢦⠀⣗⣻⡀⠀⠟⢀⡀⢹⡆⠉⠻⠾⣧⠘⠁⡀⠀⠀⠀⠀⠘⣿⠀⢹⠀⡇⣿⡇⠃
⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠁⢠⡀⠀⢀⡐⠂⡿⢙⡆⠰⣶⣷⢬⡁⠀⢠⡼⠟⠻⣆⠀⠓⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⡏⠀⢸⠀⡯⢻⡇⢸
⠀⠀⠀⣸⣼⡇⠀⠀⠀⠀⠀⠀⢨⠗⢤⠁⣤⣾⡄⠉⠀⢀⣟⣀⡀⠙⣷⣬⣽⡶⡆⠈⠳⣶⣂⣤⠞⠀⠀⠀⠀⠀⠀⠀⡟⢰⠀⠀⠀⣸⡇⢸
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⣶⠋⠀⠰⢾⠏⠙⢿⡶⢦⣬⣟⡋⠹⣧⡈⠁⠀⠹⡷⢄⣠⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣀⠀⡀⢿⡇⢸
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠈⠒⢤⡀⢾⣴⢶⣏⢣⠈⠉⠋⠉⠀⠘⢳⣀⣀⣠⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⡇⢸⡇⢸
⠀⠀⠀⣿⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠿⠧⠀⠰⠶⠒⠒⠛⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⡇⢸⡇⢸
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠘⣃⠀⡇
⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⠀⡇⡇⢼⠀⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠁⡇⢸⠀⡇
⠀⠀⠀⢻⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⡀⠃⢸⠀⡇
⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⡇⣆⢸⠀⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⡇⣿⠸⡆⡇
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠇⣿⠀⡇⡇
⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡆⠀⣿⠀⡇⡇
⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡇⡀⡿⢠⠃⠇
⠀⠀⠀⠀⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠥⣇⣯⣴⣄⡔
⠀⠀⠀⠀⡟⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣤⠿⠛⠉
⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠴⠖⠒⠒⠀⠀⢰⣶⣶⣶⠿⠿⠿⠿⠿⠿⠛⠛⠛⣛⣛⣁⣀⣤⠞⠋
⠀⠀⠀⠀⠻⣽⣇⣀⣀⣛⣓⣒⣋⣉⣉⣉⣉⣀⣤⣤⣤⡤⠤⠤⠴⠶⠶⠶⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⠀⠀⣠⠔⠚⠛⠛⠒⠦⣤⣀⣀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⢻⡇⢰⠁⡤⠶⠶⠦⢤⣀⣀⣀⣤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣘⣇⠘⢛⣥⣦⣄⣠⣀⠀⠀⠀⠉⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⢻⢠⣟⠉⠀⠀⠙⠳⢮⡵⠶⠛⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⠖⠀⠂⠀⠀⠀⠀⠀⠀⠬⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⣠⣷
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⡆
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣧
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡃⡏
⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡴⢠⠇
⠀⠀⠀⠀⠀⠀⠈⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⡽⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣾⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠆⠀⢀⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠤⢤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠄⠠⠁⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠤⡤⠥⠤⣤⣀⡀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⢪⡐⠠⠐⢀⠌⢉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⠬⠤⠴⠁⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠛⠓⡲⠤⠧⠐⠒⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⢀⡀⣠⠏⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠀⠀⡇⣧⠴⠒⠛⠉⠉⠉⠉⠓⢦⡀⠀⠀⡘⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠲⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠀⠀⢀⡴⠒⠋⠉⠉⠛⠒⠒⠼⠤⠀⡲⠒⠒⠲⢤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠙⣦⠰⠊⠀⠀⠀⠀⠀⠘⢆⠀⠀⢀⠏
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡜
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⠁
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠂⠀⠀⠀⠀⠀⣰⠗⠁
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣀⠈⠳⢄⡀⠀⠀⠀⠄⠉⠐⠄⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠉⠁⠀⠀⠀⠀⠈⠑⠛⠒⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣟⣻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣩⡽⠿⢾⣍⣿⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡶⠿⢿⣟⣻⣿⣽⢻⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⢟⣻⣿⣿⣿⣿⡇⠀⠀⠀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⢿⡀⢀⡞⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⡼⠒⢤⣤⣻⣾⣶⡿⠍⣀⢀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣤⣄⢀⠈⠉⠉⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣤⣿⣽⡿⠆⠀⠀⠀⠀⠀⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⡄⣤⣶⣶⣾⣧⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠙⡇⣌⣃⠈⠀⣿⢿⣋⣽⣽⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣸⢰⡇⡟⠟⠀⠀⠃⠎⠿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡌⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡻⣿⣿⣿⣿⣿⣿⣿⣏⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣹⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠿⠀⠀⠂⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢿⣿⣿⣿⣿⣿⣷⣸⠀⣀⣴⣄⣑⣴⣧⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣽⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⣛⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠛⠻⠿⣿⡿⢿⠿⣶⣿⠿⣿⣿⠯⠅





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⢹⠆⠀⠀⢀⣤⣠⡏⠉⠃⠀⠈⡗⠒⠀⠐
⠀⠀⠀⠒⠶⡶⢼⡶⣶⣦⡀⠀⠀⠀⠂⠀⠀⠀⠂⢤⠤⣘⣤⡟⠲⡁⠠⠒⠱⠋⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡌⢿⠄⠈⠀⠀⠀⠀⠀⠀⡸⢐⠛⠀⠀⢀⣄⠀⠀⣤⠐⠉⠁⠀⢠⣾⢻⣻⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣣⠀⡇⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠀⠀⠏⠈⠲⠒⠊⠉⠙⠋⠁⠀⠀⠀⠈
⠈⠺⡄⠀⠀⠀⠜⡿⠾⡟⠦⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠄⠀⠀⠉
⠀⠀⠈⢩⠖⠀⠀⡅⠀⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠀⠀⠀⠋⠀⠀⣠
⠀⠀⠀⠈⠀⠀⠤⢰⡤⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠶⠀⠃⠆⠀⠇⠶⠘⠰⠆⠖⠒⠓⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⢠⡴⠶⣶⠉⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠄
⢀⠔⠚⠂⠀⠈⠀⠀⠜⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣤⣾⡛⠛⠯⣃⢄
⢸⠄⠀⠀⠀⢀⣤⣤⡄⠀⠐⠋⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⣴⡜⠛⠦⠀⠙⠦⡈⢳⣃
⢀⠀⠀⢀⣨⣿⣭⣷⠁⠀⠀⠠⢄⠀⠀⠂⠐⢀⡀⠉⠀⠀⠀⠀⠆⠠⢤⣿⣤⣔⠉⠀⠇⠀⠀⠑⠄⠀⠱⠄⣿
⠈⠟⠉⠁⠀⠈⠉⠀⠀⠀⢀⣀⠈⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢡⠈⢇⠀⠀⠀⠀⠀⠈⠆⠀⣧⡟
⠀⠀⡀⢀⠄⠀⠀⢠⠀⠀⠀⠁⣠⠎⠀⠀⠀⠠⠀⠀⠀⠀⠢⣀⡀⢈⠀⠀⡰⢀⣿⠄⠀⠐⠀⠀⠀⢸⡴⠋
⣤⣤⣥⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣥⣤⣽⣶⣶⣿⣬⣥⣤⣵⣤⣤⣵⣦⣥⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⢀⣩⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠨⠉⠀⠀⠀⠀⠀⠈⠉⢀⣥⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⠢⢠⡄⣠⣤⣄⢴⣷⣾⣿⠇⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢛⠙⠃⠀⠀⠀⠀⠠⠈⠀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠐⠃⠨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶
⢀⡀⠀⠠⠴⠤⠶⠤⠶⠶⠶⠶⠾⠶⠶⡿⣿⣿⣿⣥⣿⣿⣶⣶⣤⣼⣥⣬⣤⣴⣷⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⡀⣀
⠩⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⡀⢀⣶⢾⠁⠀⠀⠐⠛
⣛⣏⣋⣀⣀⠀⣾⣷⣦⣴⣶⣶⣿⣤⣤⣤⣤⣀⡀⡶⣾⣶⣶⣄⣠⣴⣤⣄⣠⣄⣤⣤⣬⣤⣤⣿⣿⣷⣶⣶⣆⣲⣄

⠀⠀⢠⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⣼⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⣿⠿⣾⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⢿⠀⣀⠀⠀⠀⠀⠀⠈⠛⠲⠶⢶⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠉⠹⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠠⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⣏⠇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⢠⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⢠⣄⣄⣀⣀⡴⠃⠀⠀⡔⠒⠒⠒⠒⣄⣤⡤⠀⠀⠀⠀⣤⣶⣶⣶⣄⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢘⣿⣋⣽⣿⡃⠀⠀⠀⡇⠀⠀⢠⣾⠋⠁⠀⠀⠀⠀⣾⣿⣿⡇⣽⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢈⣿⣯⣿⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣿⣟⠓⢻⣿⡿
⠀⠀⠘⠛⠛⠻⠿⠟⠋⠀⠀⠀⠴⠋⠈⠛⠛⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠐⡍⡏⡄⣌⣟⣝⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛
⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⡇⡇⡇⡿⣿⣿⠀⠀⠀⠀⠸⣧⣀⠀⣀⣼⠇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠹⠿⠿⠏⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⢀⠆⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣠⣿⣥⣤⣤⠄⠀⠀⠀⠀⣰⣿⠃⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣥⣤⡝
⠀⠀⣘⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢿⠿⢿⣿⣿⠁⠀⠀⠀⠀⡟⠅⢠⣤⣤⠀⡸⠀⠀⠀⠀⠀⠀⡨⠋
⠀⠀⠛⠒⠒⠲⠒⠒⠚⠃⠀⠀⠀⠸⣷⣶⣶⣶⣾⠏⠀⠀⠀⠀⠘⠦⣈⣙⣻⠴⠁⠀⠀⠀⠀⠀⢠⣶⣷⣶

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

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈
⠀⠈⠀⢈⠀⠀⠄⠁⠀⠀⠀⠀⠀⠠
⠀⣠⠀⠀⠇
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠰⡀⠀⠀⠀⠀⠀⢸⣃⡁⠈⡄⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠒⠒⠚⠀⠀⠂⡀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⢠⠀⠠⠄⡄⢤⡄⢀⡄⡄⠀⠇⠀⠀⠀⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⢹⢀⢣⢰⡆⢇⠜⢷⠃⠁⡇⠀⠀⠀⠀⠀⠠⠀⠀⠠⢐⠠⢴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⢠⣠⠀⣤⠄⣤⡄⣄⡄⠠⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠸⠐⠙⠠⠛⠢⠏⠇⠚⠧⠀⠇⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠆⢘
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢂⣀⡀⢀⠐⠂⠀⡀⠀⠀⠿⠀⠄⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠈⠉⠉⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊⠑⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠔⠉⠐⠛⠂⡀⠀⣨⠛⠘⠋⠒⡪⠬⠭⡕⠚⠛⠚⢢⢄⠀⠀⠀⠀⢠⠀⠈⠐⢀⠀⠁⠀⠀⡄⠤⠤⠌⢳⠖
⠀⠠⡀⠀⠀⠀⠀⠄⠀⢿⠀⠀⡀⠀⡟⠀⠸⡇⠀⠀⠀⢸⠖⠆⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣖⢰⣤⣿⠃
⠀⠀⠀⠑⠠⠖⠂⠀⠀⠈⠒⠲⠀⠒⠁⠀⠀⠐⠛⠒⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⠀⠀⣠⠞⠁⣠⣶⣶⣲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣦⣾⡔⢀⠀⠀⠀⢆⠉⠀⣶⡇⢸⣷⣯⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⣿⣷⢮⣤⢄⣄⢻⡿⠘⢿⣼⣿⡏⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠠⠛⢿⠾⢀⣉⣺⡇⣸⢃⡨⠷⡴⢊⠡⢒⡨⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠶⢮⡓⢿⡺⢃⡔⣻⣿⢁⡿⢛⣴⣨⠌⠱⡾⠷⠦
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡾⢺⣥⣤⣘⣯⣑⣅⣌⠲⣿⣿⣼⡗⣫⠞⠚⠋⠉⢀⣀⣀⣀
⠀⠀⠀⠀⡀⣦⡙⣿⣷⠃⡀⠈⠱⣿⣿⠟⢿⣼⡎⣹⡻⢯⡿⢋⣠⢴⣲⡾⠛⠀⠀⠉⠁
⠀⠀⠀⣀⣼⠻⣧⣽⣿⣿⣯⢉⡉⠩⣶⢍⠢⢿⣷⣿⣽⣦⣳⣾⢗⢏⠍
⠀⠀⣴⡿⢿⣟⣿⡿⣿⣿⣝⡉⣀⣀⢹⡃⠕⣬⢻⢿⣿⡏⣤⠝⣲⣍⠤
⠀⠀⠁⠀⣙⣿⣿⣿⣿⡋⣀⡈⠅⠛⠉⠻⣾⣿⡟⢶⡛⢿⣿⡼⠞⣙⣳⡴⠊
⠀⠀⠀⠀⡉⣿⣙⣻⢿⡟⢡⠀⠀⠀⠀⢤⡟⡍⢻⣾⣿⡜⡬⣴⣾⡏⡈
⠀⠀⠀⢠⣸⠛⣿⣳⣿⣿⣏⡤⠀⠀⠀⢼⠐⢿⣼⡿⠻⣷⣷⣞⣿⢿⠴⣯⠒
⠀⠀⠀⢈⣿⣶⣿⣿⣿⣏⣧⠄⠀⠀⠀⣀⣽⣼⣿⣷⡀⣹⣟⣿⣷⣿⡄⠃
⠀⠀⢀⣾⣟⣿⣻⣾⣿⣿⣿⣆⠶⠦⣾⣃⣱⣟⢻⣿⣎⣿⣿⢏⣇⣾⡟⠂
⠀⠴⠟⠛⠿⠿⠿⢿⣿⠿⠿⠫⠍⠀⠄⢈⣭⣿⣾⣿⣿⡸⢿⠚⠋⣼⣇⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡠⢾⣼⣧⣿⣿⠿⣿⣇⠈⣸⣅⢹⣿⢰⢲⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣞⣿⢱⣿⠇⣟⣶⢸⢸⡇⢃⣿⣌⣿⡆⣠⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣻⢻⡿⡀⢻⢿⢔⢿⣇⣜⡹⣿⣿⣷⣿⣧⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢽⣾⢿⣾⣿⣿⣩⢀⣿⣾⡏⡟⡟⣿⣿⡿⣯⢹⡟⣿
⠀⠀⠀⠀⣄⣀⣀⠠⠤⢤⣤⣰⢾⣿⣷⣿⡟⣰⢹⣧⣸⣿⣷⣿⣧⣻⣿⢹⣿⣾⣷⣿⠆
⠀⠀⠀⠘⠼⡗⡡⣠⠖⣃⠿⠾⣿⡟⢿⢿⣷⡿⢸⣿⣿⢸⡿⣿⢿⣿⣟⣿⣿⣿⣿⣿⠾⠖⢒⠶⢶⡴⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⣿⣷⠁⡾⠁⠀⢀⡇⢹⠀⡇⣻⡏⠻⢿⣿⣴⣾⣽⡸⣿⣻⡻⣿⣿⡟⠁⣠⠊⢠⠂⠁⢹⡟⣏⠹⡙⣦⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠁
⠀⠀⠀⢠⠄⠉⡻⢺⡇⠀⣤⣾⠧⠿⠾⣷⠋⢭⣄⣸⡏⡿⣿⣿⣿⣿⣿⣽⣿⣿⣇⢀⡇⢰⡇⠀⠀⣼⠁⣼⢰⣇⣿⠄⣼⣅⣤⣶⡉⢱⣌⡢
⠀⠀⠀⠐⡆⣐⣬⣿⣷⣼⣫⡼⠶⢅⠐⠠⡀⣠⣿⠿⣿⡿⢿⢌⠻⣿⣿⡟⡿⢙⣻⠾⠟⣙⣧⣶⡞⢻⣾⣿⣾⣿⡟⠺⠯⠿⣍⡱⣶⢀⡈
⠀⠀⠀⠀⠈⠿⣹⡞⢡⢠⡏⢀⠀⠈⢳⡄⣻⣶⣺⡦⣌⣥⡀⣄⢠⣙⠛⠡⣶⣯⠈⠑⠚⠪⣿⢑⢔⢿⣛⡿⢻⣻⣾⣿⡿⠤⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣇⢸⣀⣧⡘⣧⣄⣸⣿⣉⢻⣷⣅⣤⣴⣯⣥⡿⠏⠛⠒⠋⠁⠀⠀⠀⠀⠈⠉⠉⠈⢉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢽⣿⠿⠶⠞⠄⠼⣟⣽⠋⠹⠁⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠄⣰⣾⣟⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⣸⡕⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⠟⠁⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⠆⠀⣒⣦⣌⡤⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⣗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠀⠀⠀⠀⠈⠁⠲⢖⣀⣀⣀⣴⡶⣿⢻⣿⣿⢿⣾⣭⣿⢿⠟⣃⡅
⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠈⠉⢹⣷⣶⠠⠐⡖⣶⣾⡖⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⢍⣰⣴⣿⣿⠾⢋⣽⣿⣿⣟⣿⣿⣯⡷
⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠁⠀⠛⢿⢃⢈⠓⠁⠀⠙⢆⠘⢀⠀⠓⢦⡀⠄⠀⠀⠀⠀⠀⠀⢀⠤⣾⣳⣛⠯⠀⠀⣻⣷⣼⡿⣿⣶⣻⣿⡇
⠀⠀⠀⠀⠀⢀⢠⠉⠀⠀⣀⣲⠃⠀⠀⠀⢷⡤⠀⠀⠀⠀⡈⠛⢴⣿⡋⣲⡦⣦⠐⣦⣴⣤⣤⣿⣿⣯⡽⣦⠔⠒⠩⢿⣿⡿⣿⣧⣿⣽⡇
⠀⠀⠀⠀⠀⠈⠉⠀⠡⠏⠙⠁⠀⠀⠀⠀⠀⢋⠀⠀⠀⣰⠂⢨⠄⠈⢀⡟⣁⣾⣾⣿⣿⠙⠛⢣⣠⣵⠦⣀⡀⠀⢠⣔⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⢀⢜⣱⠔⢂⡀⠨⠄⣂⠏⢠⡤⢛⣛⣹⣦⣴⣿⣶⣶⣿⣿⣼⢿⣿⣿⣾⣻⣿⣿⣿⣿⣿⡿⣷⡇
⠀⠀⠀⠀⠀⠨⠀⠘⠛⠁⠀⠈⠀⠉⠔⠨⡆⠡⠀⠈⣄⢀⡀⢴⣿⣿⣛⣿⣽⣿⣿⣿⠾⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣋⢻⣿⢿⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⢠⣤⣄⠀⡘⢤⡄⢀⢀⣻⣟⣻⡟⠻⢿⡿⢿⣡⣶⢏⢇⣲⣳⡿⣟⣾⢷⣾⣿⣿⣿⣶⣳⣶⣿⣿⣿⣼⡿⠇
⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠃⠀⠀⢡⠚⠉⣠⢚⣯⣾⠿⢝⠷⢛⡀⣊⣤⡿⢿⡿⢿⠟⠟⠛⢋⠗⢿⡟⣻⠟⣻⣿⣩⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⢰⡄⠀⠀⡄⢐⣂⣾⡱⡷⠇⠿⠅⣺⡷⣾⣏⢀⣘⣟⡻⠳⣺⣿⣽⣇⣴⢞⣣⣾⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣖⣴⣢⠻⠜⡉⢁⢏⠀⣲⠾⠛⠻⣟⡿⠻⠏⣱⠗⡙⣛⢞⣂⣿⣛⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠉⠁⠛⠘⠁⠐⠣⠸⠦⠈⠑⠆⠘⠏⠤⠶⠄⠣⠏⠶⠷⠿⠾⠿⠟⠿⠟⠿⠟⠿⠿⠿⠿⠿⠻⠟⠿⠿⠏⠹⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢴⡶⠤⢽⣫⢭⡷⢍⣉⡓⠢⠄
⠀⠀⠀⣴⢓⠦⣄⠀⠀⠀⢀⠀⢠⡀⢤⡰⢟⣒⣉⣉⠓⠦⣉⠓⠦⠬⠷⠈⠑⠀⠉
⠀⠀⢰⠏⠀⠀⠈⠙⠶⣄⠈⠓⢶⣞⠓⢬⣓⢤⡒⠶⠿⠂⠀⠁
⠀⠀⣾⣅⠀⠀⠀⠀⠀⠈⠙⠦⢄⡈⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄
⠀⠀⢸⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠏
⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠖⠛⠋⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠉⠻⣧⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠻⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⣠⣤⣶⡶⠟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠂⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣠⠀⠀⠀⠀⠄⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⡀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠛⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⢿⡇⢀⣀⡤⢔⠚⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠚⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣀⣤⣥⠤⣀⣀⣀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣲⣠⠴⠒⠋⠙⠧⣄⣀⡀⠀⠀⠀⠀⠉⠓⠒⠦⢀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣽⣦⠈⣧⠤⠶⠶⠶⠤⣤⣍⣉⣉⣛⣛⡛⠛⠒⠶⠦⣄⡑⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠶⠶⠤⠬⠧⠷⠄⠒⠒⠒⠒⠉⠉⠀⠀⠀⠈⠉⠝⠢⣄⠀⠀⠉⠳⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠀⠈⠉⠓⠲⠤⣤⣤⣤⠤⠶⠤⠴⠶⠞⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣟⣟⣛⣟⣯⣛⣏⣻⣋⣛⣛⣋⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⡀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢦⠤⡄⠀⠀⠀⠀⠀⠀⠄⠰⢠⠄⠀⡀⣀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⡋⢰⡄⠀⠄⠀⠄⠀⠀⠀⣀⠰⢊⡋⠀⠀⠀⡀⠐⠄⢀⠀⢐⠒⠳⠀⠸⠾⠶⢧⠀⠠⡀⠀⢂⠐⡄⠀⠃⠘⢃⣀⠈⡃⣀⠀⢰⢽⣳⠀⠇⠀⠀⣎⣷⠄
⠀⠀⠀⠉⣡⠟⠀⠠⣤⠲⠐⠀⠈⠤⠜⠂⣡⠤⠒⠒⠓⠒⠢⠄⠀⠀⢢⠄⠀⠐⠀⡐⠀⠀⠀⣈⠉⠋⠉⠀⡰⠄⠉⠀⢈⠀⠀⠉⣤⠈⠋⠀⢀⠀⠂⠐⠁
⠀⠀⠀⠀⠈⣀⢤⣅⠈⠏⢀⡀⠀⠀⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⠧⠄⠀⠀⠘⠙⠃⢀⣠⠈⠢⠠⠥⡂⠀⠵⠋⣀⠀⠆⢄⡀⠐⠂⠐⠂⠁
⠀⠀⠈⠑⢤⣿⡬⠅⠀⠀⠨⠁⠀⣴⠤⣤⣄⣀⣠⣤⡀⣀⣀⡀⡀⢤⣴⠀⠀⢈⣍⠓⢀⣀⢰⣥⣤⡆⠈⡤⠀⠀⠀⢠⠀⠤⠀⠀⠈⠀⡄⠀⢃⠀⠀⠀⢀⠆
⠀⠀⢀⠀⠀⠁⠀⢀⡒⠆⠀⠀⠀⠀⢠⣦⢀⠉⠀⢨⡇⠈⠁⠀⡀⠀⠀⣄⠱⠦⠀⠀⠀⠄⠀⠉⢉⢀⡀⠀⣎⡁⠰⣿⠀⠀⠀⢀⠤⡈⠀⢀⠊⠀⠦⠆⠈⢦
⠀⠀⢸⣦⣴⠆⢀⠠⠤⣤⣄⠀⡠⠀⠀⠉⠆⡀⠀⢸⡇⠀⠰⠀⠓⠀⠀⡈⡄⠀⢰⢻⢢⠀⠀⡠⢄⢈⡀⠀⡯⠅⢩⣿⠀⠀⢠⠃⠀⠀⢢⠏⡀⠀⠀⠀⠀⠈⢃
⠀⠠⠽⠯⠇⠠⠁⠀⢰⠁⠀⠀⣤⠄⠀⠈⢈⡴⠀⠘⡇⠀⠃⣔⠒⠲⠀⠰⣧⠀⠘⠨⡃⠄⠀⢧⣀⠬⠁⠀⢳⢨⠼⠿⣝⡆⢀⠀⠀⠀⡌⠀⢁⡀⠀⢠⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢒⢚⡇⠙⠂⠐⠢⠂⠀⠀⡇⠀⠒⠃⠄⢀⠀⠐⡂⠣⠀⠡⣷⠃⠀⣀⠀⠀⡀⠀⠘⠂⠸⠴⠛⠁⠈⠀⠀⠀⠃⠀⠈⠀⠊⠃⠘
⠀⢀⣀⡠⢃⠔⠂⠐⠀⠀⠪⢹⡍⠑⠀⠀⠚⠁⠸⠥⠁⠀⡀⡄⠀⠈⠀⠀⠀⢄⢃⠀⢿⠀⠀⡏⠄⠀⣙⠄⠀⣀⠀⠈⠃⠀⠀⠈⠠⣀⢀⠀⠀⠀⢹⠀⡀
⠀⠈⠁⠰⠑⠖⠃⡀⠀⠀⢈⡐⠃⡄⠀⠀⢸⠂⠀⢠⡎⡃⢈⠤⠁⢰⠀⠀⠀⠀⠲⠀⠈⠀⠀⢃⠀⢊⠜⠀⠀⠙⠀⠀⢂⠀⡆⠀⣴⡀⠉⠃⠀⣁⡙⣨⠁⠈
⠀⠀⠑⠀⢤⠄⠈⠀⠀⣀⡀⠀⠀⠀⢀⠰⠋⠀⠀⠈⠓⠀⠀⡀⠐⠈⠀⠀⠀⢀⢶⡄⢀⡀⠀⠈⠉⠉⠀⠀⠀⠦⠁⠀⠾⢯⠃⢀⠆⠀⡀⠐⠆⠀⠀⠀⠀⣤⣠
⠀⠰⡀⠀⠀⠁⠀⢴⡀⠀⠧⠀⡄⠀⠉⠁⢢⠀⠀⠒⠉⠁⠈⠉⠎⠋⠀⠀⠀⡽⢀⡁⣀⠁⡔⠄⠀⢀⠔⠀⠢⠀⠐⠠⢦⠀⠀⠸⡀⢀⠄⢈⣀⣦⣀⡀⠀⢠⡀
⠀⠀⠷⠀⠖⠀⠀⠀⠉⠉⠁⠀⠐⠀⠀⠠⠎⠠⡂⣀⠊⠀⠈⠠⠿⡆⠀⣀⡌⠁⢮⡷⠶⠒⢂⣤⣞⢣⡄⢀⡄⠀⠀⡄⠀⢹⠀⠀⠀⡠⢖⠥⠎⡐⣁⣌⡴⡄
⠀⠀⣸⠀⢀⠀⡀⢠⠖⠀⠀⣤⣶⣴⡶⡆⠀⢰⡟⡄⠀⣀⡨⠤⠤⠤⠀⠀⠀⠈⠀⠠⠤⠐⠚⢼⠉⠹⡀⠀⠃⠀⠀⢀⠀⡀⠀⠀⣜⠽⡳⡴⢟⠴⡹⣾⣪⡵⡆
⠀⠀⠀⠀⣀⠀⣿⡶⠀⠀⠈⠉⠉⠀⠇⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣧⡄⡀⠀⠘⠑⠃⢼⡀⢠⠶⢮⠒⢺⡿⢿⣾⠖⢻⠟⢿
⠀⠀⢠⢀⠿⠶⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠻⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡽⡷⣜⡖⠸⡀⠀⠀⠀⠸⠀⠀⠀⠀⠃⠘⠸⠀⠀⠀⠘
⠐⣂⠸⠿⢤⠀⠄⠀⠀⠀⠀⡀⠀⠀⠉⠠⡭⠀⠀⢰⡴⢦⡔⠁⠁⢀⠈⠀⠀⠀⠀⠉⠳⣄⣤⣠⠀⠘⢓⠈⠉⠀⠈⡏⢁⠀⠀⣤⣤⣤
⠀⠙⡧⠤⠀⠄⣄⡰⢞⠀⢀⢉⡁⠉⠀⠈⠀⣀⣀⠐⠂⠑⢁⢀⠀⠦⡤⡤⠤⣤⠐⠁⠀⠛⠀⠊⠀⢀⡉⣄⠀⠀⣠⠴⠋⡀⠀⢳⠒⠻⠀⠀⠀⠀⠀⠷⣤⡤⣼
⠀⠀⠀⠀⢠⣀⠙⠓⠀⢰⡓⣭⠘⣦⠀⠀⠎⠀⠂⡀⠀⠤⡀⡀⠀⠀⠉⠙⠛⠉⠀⠤⡀⠀⠀⠐⠂⡀⠀⠀⠅⠀⡱⠚⠋⠀⠀⣈⠀⣰⣄⠀⣆⡄⠀⠀⠐⠂
⠀⠀⠰⢷⡐⠻⠂⠀⠀⠀⠛⠰⠼⠋⡀⣀⠐⠒⢎⢀⠀⠀⠠⣤⣠⣖⣆⠀⣠⠀⣈⡀⠅⠠⠈⠁⠀⡐⢀⠐⠀⠂⠈⠄⠀⣀⠀⢤⣀⣈⣹⠆⠀⡀⠀⠐⠦⠴⠃
⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⢠⣟⠂⠀⣾⡟⠁⠀⠀⠟⣿⠾⢿⡇⠀⠀⠠⡄⠀⠃⠀⢠⠐⠠⠞⠀⢢⠂⠀⠂⠀⠨⠁⠀⠀⡀⠀⠀⣿⣿⡆⠀⣠⡀⡀
⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠉⡀⠀⠈⠷⠆⢀⢀⢀⣷⠀⢰⠃⠀⠀⢠⡄⠀⢡⣤⡈⠀⠀⠀⠀⠈⠀⠀⣈⠀⠀⢠⣀⣁⣀⢠⣀⠈⠉⠀⠘⡳⢮⠜
⠀⠀⠀⠈⠀⠁⠸⠀⠲⠁⠇⠐⠈⠄⠀⠁⠀⠀⠀⠀⡀⢳⣟⠛⠁⡀⢸⡄⠀⠀⠘⠃⠀⡟⢀⣷⡃⠀⠄⠀⠀⠀⢉⠁⠀⠄⠈⠀⠁⠈⠀⠁⠀⠀⢔⣠⢉⡠
⠀⠚⠀⠒⠀⠐⠂⠙⠒⠀⡐⠊⠰⠚⠀⠀⠁⣘⢳⡖⠱⣀⠈⠉⠉⠉⢋⢅⠀⠀⣀⠀⠀⣓⢬⡧⠀⠀⠀⠀⣀⡀⢀⣥⡄⠈⠃⠤⢄⣠⠤⠘⠁⠀⠀⠁⢠⡛
⠀⠀⢠⡆⢰⡄⠀⠸⠀⢠⢇⠀⠫⣞⠃⠘⠁⠼⢏⠀⣀⠣⠄⠀⠀⢮⠸⠐⠀⠀⢧⡠⠀⡅⠘⠟⢂⡄⠐⠀⠋⠇⡀⠀⠀⠀⠀⠀⣠⠔⢂⠲⠄⠀⣄⠀⡀
⠀⢴⣆⡬⢤⠆⠀⠘⠀⠈⠀⠀⠀⠓⠀⠀⠀⠛⠈⣶⠚⢀⠃⠄⠀⠀⠈⢲⠐⡳⠀⠙⢀⠀⠈⠈⠀⢠⣶⠀⠀⠸⠀⣈⡄⠀⠀⡰⡁⠁⠈⠂⠀⠄⠀⠀⠀⠈⠙
⠀⣀⣈⢡⡄⠀⠀⢸⠀⠀⠀⠀⠁⡁⡀⠀⠀⠘⠦⣀⠀⠂⠀⠀⡖⠀⠂⠈⠀⠀⠃⢀⡏⡁⠀⠀⠀⣾⡇⠀⢀⡀⠛⢾⡇⠀⢰⢀⠀⠀⠀⠇⠘⠀⠀⣀⣱⠂
⠀⢈⢡⡤⠇⠀⠇⡸⠀⠀⣄⣀⡀⠈⠀⢸⠂⠘⢦⣡⠄⠂⠀⠈⠁⠀⣰⠀⠀⠀⠾⣘⡗⠁⠀⠀⠀⢱⠂⠀⠣⣠⢺⣦⢵⠀⠸⡆⠀⠀⢀⢇⠸⠀⠀⡠⡝⡀
⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠈⠉⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠉⠈⠉

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⡶⣜⠷⣼⡯⢰⠛⠛⡟⣷⠛⢺⠐⠛⡏⠀⠀⠀⠀⠀⠀⡯⢾⠤⣷⠙⡞⢳⡯⠁⢸⡿⣿⠀⢶⡸⠄
⠀⠠⠄⠌⠤⠭⠁⠬⠽⠤⠡⡥⡇⣬⢹⠠⣁⣤⡄⢠⣠⣤⠠⣧⡏⢰⠛⡞⢎⠉⠁⠀⠌⠁⠯⠎⠶⠷
⠀⢀⣉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠈⠀⠁⠭⠌⠱⠨⠌⠉⠠⠈⠀⠈⠀⠁⠈⠀⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⣶⡀⢀⣀⢰⣀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠄⣀⣀
⠀⠈⠛⠋⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠆⠀⠉⠓⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡄⠈⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠁⠀⠀⠀⠀⠐⠐⠰⡆⠀⠀⠀⠀⢀⣀⠀⠀⠠⠀⣄⠀⠐⡓⠀⠁⠀⠀⠠⠤⠤⠄⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⡀⢠⣤⡦⠁⠷⠎⠐⡐⣀⣚⡾⣆⠀⠀⢩⣤⣂⣿⣍⠚⠒⠀⠀⠀⡤⣤⣤⡤⣄⠰⠇
⠀⠀⠀⢘⠀⠀⠁⠒⠀⠀⠀⠚⠛⠁⠛⠋⠙⠛⠁⠀⠀⠘⠛⠉⠃⠒⠊⠁
⠀⠀⠀⢸⠀⠀⣀⣀⣀⡀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣴⡆⠆⠀⠀⢀⠀⢀⡀⠀⢀⡄
⠀⠀⠀⠐⠐⠀⠉⠈⠻⡟⠋⠈⠁⠀⠀⠀⠘⠀⠀⠐⢢⠤⠀⠉⣋⠙⠻⢣⠸⠇⠀⠰⠤⠼⠶⠠⡌⢤

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠂⠈⠀⠀⠃⠀⠀⠠⠀⠀⠀⠀⡁⠀⠁⠀⠀⠀⠀⡀⠂⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⡏⢉⣏⣻⢶⣀⢄
⠀⠀⢀⠀⠀⣀⠀⢀⣀⠀⠬⠅⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⢀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠐⠆⠄⠄⣘⠀⠞
⠀⠈⠂⠀⠈⠋⢹⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠦⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠰⡲⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⢠⡤⠁
⠀⠀⠀⠠⠀⠄⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⢀⠀⠑⠂⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⡄⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠁
⠀⠐⠨⣿⡱⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢶⡀⠀⠠
⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠀⠂⢠
⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡆⠀⢰⠀⢠⠖⠓⠀⠀⠒⠒⡖⠒⠃⢀⠖⠛⢢⡀⢰⠛⢢⠀⠰⡄⠀⠀⡴⠀⠀⠠⠀⠰⠀⠟
⠀⠀⠀⢀⠸⠍⠯⠑⠀⠀⣇⣀⣀⡇⠀⢸⠀⠘⢦⣀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⡇⢸⣀⡸⠀⠀⠘⣦⠞⠀⠀⠀⠀⡖⢢⠀⢀⡀
⠀⠀⠀⠘⠐⠀⠐⠃⠀⠀⡏⠉⠈⡇⠀⠸⡆⠀⠀⠈⢳⡄⠀⠀⡇⠀⠀⢿⠀⠀⢀⡇⢸⠉⠳⡄⠀⠀⢸⠀⠀⠀⠀⢀⠃⠹⠁⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇⠀⠀⠇⠘⠦⡤⠜⠁⠀⠀⠇⠀⠀⠘⠦⣠⠜⠁⠸⠀⠀⠹⠀⠀⠸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢲⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠇⠀⠀⣾⠏⠀⠀⠀⠀⠒⠓⠒⠛⡐⠓⠒⠃⠐⢿⡦⠀⠼⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠐⠙⠂⠀⠙⠃⠞⠁⠀⠀⠀⠀⠀⠀⢰⣶⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢁⣀⢀⢤⠄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⠀⠀⠀⠂⠀⠉⠏⠁⠀⠙⠀⠘⠛⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠠⠀⠠⠀⠀⠀⠀⠒⠠⠖⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠹⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⢀⠀⢈⠀⠠⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠑⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠼⠀⠈⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠘⠸⠀⠀⠂⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢄⡤⠴⣦⡀
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢶⣀⠻⢍⡙⠚⢿⡺⢿⣒⠦⠌⠑⠂
⢀⡾⠈⠝⡲⣄⡀⠀⢤⣀⠐⣬⣓⢭⣓⠚⡒⠦⣌⠑⠢⠌⠙⠛⠋
⣸⡕⠀⠀⠀⠀⠉⠳⢀⠈⠙⠯⣍⠓⠮⠓⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢺⡗⠀⠀⠀⠀⠀⠀⠀⠘⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡍
⠘⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠛⠛
⠀⠀⠈⠙⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⣀⣀⣤⣶⡶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⠟⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠃⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⠆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡘⣿⡇⠀⠀⠀⣀⣤⡤⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⣤⣴⡲⠯⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠒⠊⠉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⠀⠘⡄
⢠⠿⠗⠈⠭⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⣀⣁⠰⡀⠀⠀⢀⠤⠷⢄
⠈⢻⠀⠀⡄⠀⡀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⠁⠀⠀⠑⠷⠀⠀⠁⠀⡀⠈⢳⠀⢀⠄⠀⠰
⠀⢸⠀⠀⠀⠂⡀⠀⢀⠼⠄⣲⢾⢗⠒⠒⠒⠒⠒⠚⠌⠋⡢⣀⢈⣿⡄⢡⠀⣀⠐⠀⠀⠀⠳⣆⡀⠐⢀
⠀⠀⢣⠀⠀⠀⢠⢞⣧⢖⠏⣯⢃⠾⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠄⡀⠂⠀⢢⠈⠀⠀⠀⠀⠐⢏⠁⠀⠁
⠀⠀⠀⠑⠤⠔⡫⠎⡇⢸⢠⣇⠎⣸⠂⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠉⠂⠄⠱⣄⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢠⡇⠇⡇⢸⡜⣡⢞⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⡠⠔⠀⠁⢄⠀⠀⡈⣆
⠀⠀⠀⠀⠀⢸⣇⠸⡰⡼⢋⢔⠥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡠⣴⢄⠀⠀⠀⠑⣠⠋⠈⢱⣀⣀
⠀⠀⢀⡠⢄⡀⢾⣷⣳⡯⢓⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣩⡤⡴⢸⠗⣟⢋⠀⠀⢫⠀⠀⠀⠀⠀⢸⡀
⠀⠠⠋⠀⢀⡨⠛⠳⡷⡬⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠤⣔⢫⠔⠁⠀⠑⠄⠫⠬⡤⠚⠠⠄⠠⢍
⢀⠃⠀⠆⠀⠀⡇⢀⠇⠀⠀⢠⠤⠤⠔⢢⡀⢀⡤⠳⢄
⠀⡆⠀⠀⠀⠀⣠⠎⠀⠀⠀⢸⠀⠠⡀⠀⠑⠋⠀⠂⠘⢄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠸⠤⠒⠂⠈⠁⠀⠀⠀⠀⠰⡄⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⡀⢐⠁⠀⣠⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⡼⢉⠉⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⢀⠄⠀⡼⠯⢭⠤⠤⠤⠤⠤⢀⣀⡀⠀⠀⢀⠇⢄⠡⠢⡆⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⢏⣀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⡄⠄⣄⠡⣠⠈⢸
⠀⢀⣀⠀⢀⡤⠊⢉⡀⠀⠀⠀⠀⠈⠁⠀⡼⠹⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⡈⠉⣽⡖⠁
⠀⢻⠸⡶⠷⠒⣋⣉⡽⠀⠀⠀⠀⠀⠀⡰⢃⡄⢧⠀⠀⠀⠀⠀⢸⢹⠆⠀⠀⠀⠀⢀⣀⠈⠉⠈⠿⢦⡀
⢯⣉⡀⡶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⢧⢸⡀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠙⣄
⠞⠀⡇⢧⠀⠀⠀⣀⡤⠴⣄⠀⠀⢀⠟⡸⠁⢸⡀⡇⠀⠀⠀⢀⡇⣿⠀⠀⠀⠀⠀⠘⡇⣇⠀⠀⠀⠀⠀⠈⢆
⠀⠀⣻⠸⠶⢚⣉⣥⠴⠒⠃⢴⠒⠚⠐⠧⠤⣬⡇⢿⠀⠀⠀⢸⢡⡇⠀⠀⠀⠀⠀⠀⣇⢹⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢳⡀⡞⠉⠀⠀⠀⠀⠀⠀⡹⢁⠟⠓⠒⠒⢦⠀⣽⠆⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⢾⢸
⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⢰⢃⡞⠀⠀⠀⠀⢸⡆⡇⠀⠰⡇⡏⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆
⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠈⡇⢻⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇
⠀⠀⠀⢸⢸⡄⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⣤⣌⣉⣉⣉⣛⣛⡆⠀⠀⣧⠹⠤⡤⠴⠖⠚⣙
⠀⠀⠀⠘⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠻⠴⠒⠒⠚⠉⠉⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⠴⠚⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣠⣴⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡞⠁⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠦⠼⠷⠾⠿⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⠤⠖⠲⢤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠛⠿⢿⣿⠿⠟⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀
⠀⠀⠉⠓⠒⠒⠒⠒⠒⢲⣲⢶⡚⠋⠛⠧⠤⠤⠤⠴⠦⠀⠀⣀⡤⠊⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢑⡄⠀⠀⠀⢠⢉⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠤⠤⠀⢨⣉⠁⣳⣾⡀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣄⠀⠀⠀⠠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⡌⢠⠁⠀⠀⠸
⠀⠀⠁⠀⠈⠉⠈⠉⠀⠉⡙⠉⡟⠋⠉⠛⠓⠒⠛⠓⠒⢠⠻⠈⠀⣃⠀⠑⢦⡀⡼⠁⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄⢠⠁⡜
⠀⠀⠀⠀⢀⢤⣄⠀⠀⠸⠁⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⢹⡁⢈⣤⠟⠃⠀⠀⠘⠣⠤⠔⠒⠦⢄⣀⣀⡠⠔⠊⠀⡈⢀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⣹⡠⢀⡌⡁⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢑⠢⡀⠀⢈⡽⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⡁⣸
⠀⠀⡴⡾⣋⢵⢏⡸⠃⡤⠋⠙⠃⠠⠲⠂⠀⠀⢀⠄⠫⠠⡢⣑⠞⠉⠁⠀⠀⢀⣀⡀⢈⠻⠎⠀⠀⠀⠀⢄⡤⠙⣉⠞⠁⢀⢃⠇⠀⠀⢠⠀⣀
⠀⠀⠧⠬⠕⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣘⠄⢺⡇⠀⠀⠀⠀⠀⠈⠉⢋⣁⠔⠀⠀⢐⡤⢟⠳⢀⢗⠚⠃⠀⣴⡼⠀⠀⠈⠛⣗⠻
⠀⠀⡠⣤⣤⣀⣀⢎⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⡁⠀⠆⠀⠀⠀⣠⠴⣉⠀⠴⠢⠉⠀⠠⠀⠄⠁⢇⡀⠀⠀⠁⣅
⠀⠀⢱⠛⠛⠛⠚⠐⠢⠭⠭⢶⣒⣒⡀⠀⠀⠒⠒⠒⠒⠒⢒⣦⠴⠆⠀⠉⠁⠀⠀⢀⣤⡾⠗⠋⠁⠀⠀⠀⠀⠀⠀⠚⠂⠠⠂⣬⡹⠀⣖⡾⠇
⠀⠀⠀⠑⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⣀⣤⣠⣤⣄⣄⣀⣀⣀⣀⣉⣄⠀⠀⠀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠃⠀⠹⠀⠀⠀⡀
⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⢀⣤⠄⢠⢀⠀⠀⠀⠀⠀⠀⠸⡅⠠⠆⢀⠆⢠⠆⠀⠀⠀⠀⠁⠈⠙⠲⢬⣆⠀⠀⠀⡀⠀⠇⠀⢀⠆⠀⢠⢀⠀⠀⢀⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠂⠀⠐⠀⠀⠀⠀⠙⣆⢀⠀⠀⠫⣤⣤⣤⠉⠁⡄⠀⡂⠀⠀⢍⠳⣦⡀⠱⡀⢀⠄⠊⠀⠀⠀⠘⠃⠀⠀⠤
⠀⠀⠀⠀⢀⡀⣀⠀⢀⠀⠀⣀⡀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠘⢦⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠈⠂⠙⢦⡙⠁
⠀⣴⠚⠟⠌⠁⠍⠀⡌⠀⢐⠃⠀⠏⠉⢐⠂⢤⣄⡀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⣀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠹⣾⢠⠄⡘⠀⠰⠀⢀⡁⠀⠘⠀⠀⠠⠀⡘⠀⠻⠑⡶⣀⡒⠒⢂⠀⠉⠓⠒⠒⠒⠒⠦⠤⢬⣥⣤⣔⣒⣂⣐⣢⣼⠃⠀⠀⠀⠀⢠⠀⢒⣤
⠀⠀⠈⠻⣄⡂⠀⡒⠀⠠⠀⠀⠉⠀⠀⠂⠀⢤⡄⢧⠤⢤⣌⡛⢦⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠂⠐⠒⠃⠐⡄⣲⠀⣄⣘⡄
⠀⠀⠀⠀⢎⠙⠒⠮⠄⣒⠀⠠⠅⠀⡚⠀⡴⠀⡬⠉⣒⠒⡤⠏⡀⣺⡆⠀⠀⢀⣠⠤⠤⠐⠀⠉⠉⠀⠀⠀⠀⠀⣀⠠⢄⠒⠒⠉⣄⠘⠠⠷⡀⠃
⠀⠀⠀⠀⠀⢈⡁⠐⠒⠋⠀⠉⠐⠖⠂⢚⡤⠚⠃⢲⡓⠻⣛⣛⡻⠁⡄⠙⡥⠉⠔⠀⢁⢀⠌⢢⠐⠄⠒⠂⠈⠀⠀⠂⠁⠀⠀⠀⠀⢘⠀⡧⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠅⠈⠓⣴⠃⠂⢠⠟⡝⢧⠠⠄⡇⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⢣⠐⡄
⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠠⣤⠤⠤⠄⠀⡀⠀⠀⠠⠀⠀⠛⡄⢃⠈⠦⠥⠋⣀⢁⠇⠀⠠⠂⠀⠁⠀⠀⠐⠌⢆⠀⠀⠀⠘⠀⠐⡀
⠀⠀⠉⠂⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠙⢌⠁⠼⠄⠐⣠⠞⠀⠀⠆⠄⠀⠠⠠⡀⠀⠀⢸⠀⠀⡄⠀⡇
⠀⠀⠀⠄⠀⠤⠜⠄⠠⠀⠄⠠⠤⠄⠀⠄⠀⠀⠄⠀⠀⠀⢀⠀⠀⣀⣀⣀⣀⠰⣉⠑⠂⠉⠀⠀⠀⠀⢃⠀⠀⠑⠐⠀⡀⠄⠄⠀⢠⠰
⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠠⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⣀⣈⣈⡀⠀⣀⡀⣀⠀⠓⠀⡀⠀⡀⠤⠊⠀⠀⣚⠀⣇⠀⠀⠀⠁
⠀⠀⠀⣄⠀⠠⠀⣤⠄⠒⠒⠖⢤⡤⠒⠒⠀⢠⣔⠂⠀⠐⠒⣴⡂⠀⠒⠒⠠⣀⡀⣉⣀⡀⠀⠀⠀⠀⠀⠙⠒⠤⡄⡀⠀⢠⠚⠙⠀⠠⡄⠈⠰
⠀⠐⠋⠉⢹⠒⠂⠙⠛⢶⣶⠞⠛⠟⠒⢶⡶⠚⠛⠳⢷⡖⠛⠛⠛⢶⡶⠒⠚⠭⣇⠚⠿⠤⣄⡀⠀⠀⠀⠀⠀⠀⠃⠂⢠⢧⣀⠖⠤⠔⠀⠀⠉⠂
⠀⠠⢞⠑⠈⠉⢻⡏⠉⠉⠙⠳⣶⠖⠞⠛⠛⢲⡆⠚⠛⢳⣶⠖⠂⠉⠓⢲⣤⠒⠺⠥⢤⡔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠚⢧⡄⡈⠀⢀⣠⡀⠤⠄⠘⠄
⣀⣀⣀⣈⣉⣉⣁⣉⣐⣒⣈⣉⣈⣐⣛⣛⣂⣉⣙⣒⣚⣉⣉⣁⣚⣛⣒⣈⣑⣂⣀⣀⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣈⣉⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣀⡀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⢠⣦⣅⠀⠀⠄⠀⡩⠒⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠎⠠⠄⠀⠦
⠀⠀⠀⠐⠮⠂⠜⠷⠀⡀⠒⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠂⡄⠀⡁⢅⠀⠂⠄⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⡠⠄⠢⠀⠀⡂⡸
⢤⡾⡤⡄⠀⢀⡀⠰⠀⠐⠀⠔⠼⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠀⠀⠴⠂⠀⠀⠂⠁⠀⠀⠀⠀⠀⠟
⠃⠂⠱⣭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⣠⠄⡄⠀⠣⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠄⠀⠀⠀⡄⠀⠤⠀⠀⠀⠃⠀⠀⠘⠃⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠠⠀⠸⠀⡀⠀⠄
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡠⠀⠰⣄⡀⠠⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⠈⠁⣀⠀⠐⠀⠀⣁
⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠂⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⢈
⠠⠠⠀⠘⠋⠀⠀⠀⠀⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠐⠀⠦⠠⡄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⡅⠀⠀⣀⢀⠀⠆
⠀⠦⡠⢀⢄⠂⠀⢀⠀⠑⠀⠀⠘⠁⠀⠀⠤⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠚⠁⠀⠀⡀
⠀⠀⠹⠀⠀⠑⠄⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠀⠀⠈⠈⢣⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⢡⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡲⢿⠒⠐⠉⠠⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠪
⠀⠀⠀⠀⠀⢦⣬⡻⠂⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐⠁⠈⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠓⠐⠀⠀⠀⠀⠲
⠀⠀⠺⢭⢻⡄⣗⡽⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣈⣁⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⣀⣀⠀⠒⢡⣄⣀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠉⢄⠠⢠⠤⠐⠤⠎⠀⠀⠝⠛⠃⠤⣉⠒⠄⡀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠐⠀⠀⠀⠀⣈⣤⠆⠪⡵⡀⢀⠃⠃⠈⡆⠀⠀⠀⠉⠦⢌⠒⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⠀⠀⠀⠸⣀⠀⠀⠀⡠⠭⢋⡘⢤⡵⡇⠀⠀⠀⡰⠒⠀⢱⢄⠑⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠁⠢⠜⠀⠀⠚⠋⢿⠀⠳⣄⣢⠞⠀⠀⡀⢄⠌⠓⣄⠢⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠋⠀⠀⠀⠀⠀⠀⢀⠅⠎⠌⢀⡟⢂⠐⡀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⠖⠲⠶⢄⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠂⠈⠀⠸⢤⡾⡄⠰⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢣⠀⠀⢰⠋⠀⠀⠀⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⠕⠁⡀⠐⡠⡥⠻⠿⡀⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢹⠆⠀⠀⢮⣤⠀⠀⠀⠀⠀⠀⢀⡠⠊⢔⠕⡁⡠⠀⢊⢘⠅⠀⠀⠀⠅⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣄⠂⢸⡆⠀⠈⠃⠀⢳⡀⠀⠀⡠⢔⠡⣊⠔⡡⡈⢈⢄⡑⠋⠁⠂⠂⣀⣀⡀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠱⠦⣀⣀⠀⢀⠠⢒⠥⢂⠥⢊⠔⡀⢊⢔⠅⠈⢇⠀⠀⠀⣸⠉⠌⠆⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⢮⡔⠨⢀⠨⢐⠁⣈⠔⡡⠂⠠⠈⣓⣴⣾⢡⣤⠶⡆⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⡂⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠐⡨⠔⣈⠄⣈⢔⡡⠢⡾⠋⠀⠀⠀⠀⠠⠀⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡐⠀⠤⠄⢂⡨⠅⡞⠀⠀⠀⠀⠀⠀⠀⢰⢒⡊⠅⡐⠅⢒⠩⡠⢑⠠⣺⠁⠀⠀⠀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠌⠩⠀⠢⢕⡃⠀⠀⠀⠀⠀⠀⣠⠭⢀⠀⠁⡀⠈⡐⠅⢊⠤⣪⠌⠀⠀⢀⡀⢀⠀⠠⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠂⠀⠐⡄⠀⠀⠀⠀⡀⠔⠓⠈⠄⠒⠥⠐⠉⠄⡪⢔⠨⢈⢗⢶⡀⡗⠠⠃⡠⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣒⣨⠆⠀⣠⠊⠉⠀⠒⠨⠄⡒⠬⢐⡈⠄⡂⠥⡂⢔⠡⣾⣁⠖⠀⠊⠀⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⣔⣀⠠⠄⠂⠈⠅⢒⠈⠅⡂⠅⠂⠅⣪⡅⠜⢋⠡⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⢉⠘⠃⠬⢍⣉⠒⠒⠀⠶⠒⠀⠒⠛⢂⣉⠠⠔⠂⠁⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣙⣒⣶⠶⡆⠐⠐⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠹⠾⠮⣕⡉⠂⢀⠩⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⣔⡃⠤⠄⠀⢤⣀⣀⣅⣈⣄⣪⢕⣬⣢⣽⢦⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠛⠩⠒⠲⠒⠲⠖⠶⠆⡶⠼⠼⠶⠞⠺⠛⠁⠊⠀⠂⠀⠊⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⡨⠋⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢠⠂⠀⠀⢀⠃⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⢢⣠⠖⠩⢠⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠦⠤⠖⠒⠒⠢⡀⠰⡇⣀⡴⠤⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⣴⡟⠁⢥⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢆⠀⠀⢀⣀⣸⣿⣿⣤⣮⣀⡀⢀⡤⠤⠤⠤⠖⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠉⠰⠒⠚⡶⠞⠋⠉⠙⣿⣿⡏⠉⠙⠛⠛⠒⠒⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣄⣄⣤⣤⣤⣀⣀⣀⣦⣠⣀⣀⣀
⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⡤⠤⢤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡞⢁⡇⠀⡼⠁⣠⡇⠀⠀⢈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡅⢸⡃⣸⣧⠼⣿⣃⡉⠙⠒⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⠀⠀⠀⠀⠀⣰⡇⢸⣧⢿⡗⠙⠿⣦⣀⡉⠙⣛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠳⡀⠀⠀⢰⣹⣿⡛⠷⢤⣙⠳⠶⣄⣨⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢀⣴⣟⣲⣬⣿⣷⣶⣜⣋⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠉⠉⠉⠙⡝⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⣉⣑⡒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢍⣤⣶⣶⣷⣄⡠⠤⣴⣿⣼⣷⣿⣷⣶⣶⠷⠖⠂⣩⣿⣿⣿⣯⣽⣄⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢩⡿⣫⣿⠟⡩⢟⠔⢆⠔⣠⣤⣤⡄⣦⡄⢠⠀⡀⢨⠫⡙⠝⠕⠂⠙⢧⡀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠐⢁⠞⠉⠀⠄⡴⠣⠂⠈⠄⣾⡟⣿⡇⠻⠈⠄⢷⡘⡔⡑⡘⢎⢂⠀⡀⠀⠙⣆⠀⢢⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠂⠂⠂⠀⠀⠀⠀⠀⠀⠐⠘⠇⣿⡟⡇⠀⠀⠈⠘⣧⣸⡄⠐⡈⠆⠀⠀⠀⠀⠘⣧⠀⠂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡇⠇⠀⠀⠀⠀⠸⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⢀⠀⢸⠇⠀⠀⠀⠸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠰⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠀⠀⠀⠀⣸⢀⢸⠀⠀⠇⠀⡇⠀⢰⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡇⢇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⡆⠀⠀⠀⢰⡸⡘⠈⠀⠘⡀⢠⠁⠀⢸⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣸⠀⠀⠀⢀⡆⠀⠀⠀⢀⢠⠀⡄⠀⠀⠀⠀⢰⢸⠀⡇⡇⢰⠀⠀⡇⡇⡇⠀⠀⢠⠇⡌⡘⠀⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣧⠀⠀⠈⣿⡀⠠⠀⢸⢸⠀⡇⢸⠀⠀⠄⢸⢸⢶⠃⢣⡞⠠⣼⣼⢸⠁⠀⢀⡾⠐⢰⠁⡸⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⢹⣥⢠⢀⣜⣯⣤⡇⣸⡀⢰⡆⣧⡇⢸⢰⣼⣇⢷⣗⡇⣾⢠⣧⣯⣷⢣⡃⢰⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⢷⣰⣿⡎⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⡟⡎⣿⣾⣿⣿⢠⣧⣿⣻⣿⢣⠏⣱⢫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣯⣻⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⡇⣾⣿⣷⣿⣷⢣⡞⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⠿⠛⠁⠀⠀⠉⠉⠉⠉

⠀⠠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡄
⠀⠀⠙⢾⡂⢰⠁⠂⡆⢲⡄⠠⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⢤⣄⢠⡤⠠⣴⠒⣴⠄⠀⢸⡍⡳⠇
⠀⠀⠉⠂⠰⠯⠵⢥⠃⡘⢑⠀⠳⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⡡⠋⣘⢓⡐⡙⠐⠉⠀⠁⠄⠒⠉⠈
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠸⠌⢣⠙⠌⣢⠻⣯⡀⡗⢺⢇⠀⠱⣗⠀⠸⠀⠇⠈⠂⠈⠀⠀⡀⠀⠀⠄
⠀⠀⣀⡀⣄⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂⠒⠀⠂⠐⠂⠂⠀⠀⠀⢠⠀⠀⠀⠀⡄⠃
⠀⠀⠈⠉⠚⠑⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⢀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⣄⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⣐⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡇⠐⠂⠀⠀⠀⠀⢠⠤⠀⠀⠤⠒
⠀⠀⠀⠀⠀⣀⣀⡀⠒⣢⡄⠀⠀⠤⠐⢒⣀⣄⠀⠁⠠⠇⠀⡠⢐⣠⠄⠠⢄⠀⠀⠀⠀⣀⣀⣠⣤⠤
⠀⠀⠀⠈⠀⠉⠙⠉⠀⣈⣥⡤⠐⣤⣤⢤⣟⣱⠆⠀⠄⣰⣲⣴⠮⠕⠢⠤⠄⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠈⠈⠀⠀⠁⠀⠈⠉⠀⠀⠀⠉
⠀⠀⡀⠀⡶⣦⣶⣐⢀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠀⠀⠀⠀⣤⢤⣶⣶⡇⢸⠀⠀⠀⡀⠀⢀⣦⣀
⠀⠀⠀⠀⠉⠈⠛⠙⡒⠛⠿⠀⠀⠀⠀⢸⠇⠆⠀⠿⣄⣰⠀⠍⠹⣛⠻⢻⣷⣮⠏⠀⠀⠛⠉⠉⢙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠐⠂⠛⠙⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣢⢤⣄⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢤⡀⠀⠀⢠⡐⠄⡀⠀⠀⠀⠸⣁
⠀⠀⠀⠀⠀⠀⢀⠴⠤⠀⠘⢒⠒⢢⣬⣯⡕⠀⡄⠀⠠⣀⡐⠀⠬⠒⠀⠀⠀⠀⠀⠈⠳⢮⡆⠇⠳⣆⠄⡀⢄⢀⠙⣆⠁
⠀⠀⠀⢀⢀⡀⣠⠀⢦⡴⠼⡾⠳⠋⢉⠃⠀⠀⠆⠀⠀⠀⠉⠙⠂⠀⠸⡄⠀⠀⠈⠂⡀⠀⢙⣀⠀⠘⣄⣮⣺⡆⢀⣻⣼⡄
⠀⠀⠈⠁⠒⠀⢄⣀⠀⠀⠀⠀⡰⠂⡡⠀⠰⠶⠿⠶⠄⠤⢶⣦⠈⠁⠀⠐⠀⠀⠀⠀⠉⢒⠋⢝⣓⣤⡬⢿⡙⣿⠂⣋⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⠀⠠⣀⡒⠆⠉⣼⣱⠂⣰⢋⠆⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⢉⡠⠖⠫⠰⣀⡴⠋⠀⢠⠃⢼
⠀⠀⠀⠀⠀⠀⠀⠒⠦⢤⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠈⡠⠀⠴⠂⠀⠀⠀⢁⠃⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣄⣀⣀⣀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠛⠋⠉⠉⠀⠀⠀⠠⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⠔⠀⠂⠠⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⣤⠒⠇⠋⡡⠔⠉⠀⠀⠄⠀⠀⠶⠶⠿⠦⠄⢠⣦⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠒⠒⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⢰⣿⡄⠀⠀⠰⢿⠇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠙⠛⠛⠛⠋⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠘⣿⣷⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠈⠻⣷⣦⣤⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠣⣄⡀⠀⠀⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡐⠀⠄⠀⠀⠀⠘⢦⣤⣀⡀⠀⠀⠀⣀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠃⠐⠈⠂⠀⠀⠀⠙⠿⠛⠛⠻⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠉⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠰⣄⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠀⠙⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠋⠀⠀⠀⢀⠀⠀⠀⠀⢀⣦⣶⠂⠀⢰⣿⠆⠀⠀⠀⣼⠃
⠀⠈⣆⠀⠀⠀⠀⠀⠠⣴⣾⡖⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠈⠀⠀⢀⣾⡟⣠⣀⣠⣼⠏
⠀⠀⠈⠢⣀⠀⠀⠀⠀⡸⢿⡼⠁⠀⠀⠀⠀⠀⠀⠲⢴⣇⠀⠀⣀⣠⣾⡿⠂⠀⠀⠐⢠⣼⣄⣀⣴⣿⡟⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠑⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⠤⠴⠶⠶⠶⠦⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⢤⠀⣤⠀⡤⠤⠤⣀⠀⣀⡀⢀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠈⠤⠌⡆⠃⢂⢰⠚⢸⡁⠠⢾⡘⠦⡄⢀⠤⣷⢊⣉⠑⣖⣩⣳⡞⢦⠜⡆⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⠀⠶⡄⡇⠀⢸⢸⣀⠀⠉⣙⠆⢸⠀⠀⢸⠀⡇⢾⢀⡇⣭⣬⠳⡘⣦⡞⠁⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⠀⠀⠀⠈⣤⡄⣇⡠⠶⠭⣤⡤⠗⢯⣥⠤⠋⠀⢀⣘⠇⠱⢤⣭⠴⠻⠇⠱⠟⣟⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠙⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠍⠉⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢉⣛⣓⣒⣒⣒⣒⣒⣒⣒⡀⣀⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣭⣉⣉⣉⣙⣓⡒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⢀⣀⣀⣀⣠⣄⣠⣤⣤⣤⣭⡭⠭⠭⠥⠬⠉⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠔⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⠠⢀⣀⣀⡠⢞⢅⠀⠀⠀⠀⡎⠁⠀⣺⠋⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄⠀⠀⠀⡇⠸⠀⠀⠀⢠⢇⣀⣰⡗⠂⠀⠈⠛⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠃⠀⠀⣤⠀⢰⠉⠀⡐⠀⢀⡾⡠⢒⠟⠂⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡔⠤⡀⠹⣧⠸⢘⡐⠠⢠⣻⡉⠤⢊⡆⠀⠠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠓⣟⣇⠀⠡⢔⠊⣏⠡⢲⡞⠉⠐⠲⠀⠀⠇⠀⠤⠤⢤⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠂⢁⢻⡖⢃⠲⣄⣁⠴⣉⡏⠨⠀⠈⠀⠂⠀⠀⠀⢠⠞
⠀⠀⠀⢄⣀⣀⡤⠞⢛⢾⡻⢁⡆⠃⠼⠋⠽⣦⠖⢅⡈⢢⡏⢠⠄⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠈⢷⢯⣒⢌⢆⠱⣎⢩⣷⣭⣭⡭⡼⢿⡂⠨⣤⡆⠁⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠘⢤⠈⠛⣻⣦⣈⠢⣝⢦⡀⠀⠀⠀⠙⢶⣄⣥⠁⠀⠀⠀⠀⠀⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠇⢈⠃⠀⠉⠛⠾⢿⣿⣶⣶⣶⢶⡶⣿⣵⡷⠖⠐⠐⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠙⡄⠻⠄⠐⠈⢉⡽⢫⠟⠉⠝⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣁⡐⠊⠀⠀⢠⠀⢘⠔⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⡤⠥⠄⠐⢢⡠⠔⠊⠁⠛⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠁⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠁⠀⠀⠀⣀⣀
⠀⠀⠀⢀⡠⠤⠒⣰⠇⠀⠀⠠⢴⡎⣥⢞⡇⠶⠶⠶⣦⣄
⢀⡠⠒⠁⠀⠀⠀⠉⠓⢢⡀⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠂
⠛⠲⢍⣉⠛⠙⠒⠒⠒⠓⠿⠦⡀
⠀⠀⠀⠈⠉⠉⠛⠑⠒⠒⠂⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⡡⠚⠀⢠⠶⠅⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣒⣠⠥⠖⠂⠉⠀⣠⠔⠋⣠⠔⠉⠀⣀⠄⠚⠐⣠⠔⠁
⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⠒⣊⠤⠔⠒⠈⠉⠁⠀⠀⠀⢀⡠⠊⡠⠔⠉⢀⡠⠔⠈⢀⡠⠔⠋⠁⡠⠔
⠀⠀⠈⠉⠉⠭⠤⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⣀⡤⠴⢊⠡⠖⢉⣠⠴⠚⣁⠤⠔⠊⢁⢀⠤⠒⢉⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠤⠖⣊⣭⠤⢒⣫⠥⢔⣚⠭⢔⣊⣩⠤⠄⠒⠚⠛⠂⠠⠬⢁⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⢉⣈⣉⡉⣁⡉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣂⠁⣀⠒⠐⠀⠀⠠⠠⠄⣀⠀⠉⠀⠀⠒⠀⠠⣠⠤⠀⠈⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠈⠀⠈⠁⠀⠉⠉⠉⠀⠐⠀⠀⡀⠀⠐⠂⠠⢤⠀⠀⠀⣐⠯
⠀⠀⠀⠀⠀⠀⢀⣠⣄⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠤⢄⣀⠍⠥
⣀⣀⣀⣀⣐⣎⣩⣤⣤⣤⣦⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⣬⣤⣔⣒⣛⣋⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠑⢤⣀⠐⠀⠠⡀⠀⠀⠀⠠⠌⢀⠀⠀⠀⠀⣀⠐⠀⢀⠀⢠⠖⠀⡓⠀⠠⠀⠀⠂⡢⣰⠀⠀⠀⠁⢀⠄
⠀⠀⠉⠢⣄⠐⢅⠐⠀⠀⠀⠀⡄⠙⡀⠀⠀⠀⠀⠠⡜⠀⠆⠀⡌⠀⠀⠀⠄⠁⠀⡴⠩⡊⠁⠀⠂⠀⠰
⠀⠀⢀⠄⠈⠂⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠠⠈⠀⠀⢀⠀⠐⠄⠐⠐⠀⠀⠀⠀⠀⠀⠡
⠀⠀⡀⠀⠒⠁⠀⠀⠀⡤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⠀⠀⠄⠊⢠
⠤⢠⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⢀
⠀⢀⠀⠈⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⠖⠀⠀⣰⢢⢠⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⡈⠀⠐⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠏⠀⠀⣿⡜⣸⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐
⢁⣐⠋⠄⠀⠀⠁⠀⠀⠀⠠⠔⠋⠀⢸⡷⣻⢠⡖⢀⡟⠀⣿⠁⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠑⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠿⠜⠓⠁⠷⠊⠻⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠠
⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠽⠃⠀⠀⣾⢱⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣽⠖⠀⠀⠀⣇⠏⣿⡜⠠⠴⠤⠄
⡀⠴⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣴⠃⡇⣠⡏⢀⣿⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠇⠙⠚⠋⠀⠛⠁⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢄⡊
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢠⠧⠈
⢀⠆⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠠
⠁⠠⠀⠐⡉⢀⡂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣌⢂⠬
⠀⠀⠀⠀⠀⠁⠸⢀⣘⠆⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡈⠠⢾⠈⠀⢀
⠀⠠⠀⢠⠀⠀⡀⠠⠀⠰⠀⠀⢃⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄⠈⣀⠈⠐⠀⠀⠀⠀⠀⠀⠈⡆⢀⠁
⠈⠀⠀⠈⠂⠐⠀⠂⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡒⠒⢦⣄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠢⡈⢷⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠠⣦⠀⠀⠀⠀⠀⠀⠙⢦⣘⠂⢻⣴⠁
⠀⠀⢀⡠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⣋⡭⠭⠉⠉⠙⢷⣤⡏⠀⠀⠀⣀⠴⠒⠛⠛⠛⠳⣾⠷⠦⣄⡀
⠀⠀⠈⠙⢆⠱⡀⠀⠀⠀⠀⠀⢀⡴⠋⠤⣤⠞⠁⠀⠀⠀⠀⠀⠀⢙⣗⠶⣴⠊⣀⢤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠑⢱⢀⣠⠤⠤⠶⠾⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⡄⣹⠋⠁⡸⠀⠀⠀⣠⡤⣈⠁⠉⠒⠀⣈⣧⣀⣀
⠀⠀⠀⣀⡴⠶⢾⡿⠃⠀⠀⢄⠀⠀⠀⠙⢦⡀⡀⠀⢀⡠⠀⢀⣀⣀⣸⣿⠁⣀⡴⠃⠀⠀⠀⠈⢙⣮⣳⡄⠂⠉⠁⠀⢀⡤⠋⠙⠲⣄
⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠈⢳⡙⢿⡯⠖⠋⠉⠁⠈⠉⠻⡏⠁⠈⠉⠓⢦⣀⠞⣩⠖⠛⠉⠒⠢⠀⠀⠞⠐⠢⢀⠓⠌⢳⡀
⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡄
⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷
⠈⣿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⣄⡀
⠀⣽⠻⠿⣥⣢⣠⣀⡀⠀⠀⠀⠀⠀⠀⢀⡼⣱⣟⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠐⡄⣦⡆⡄⠆⣀⠀⠀⣀⣀⣀⣀⣰⡯⠟⠁⠉⢷
⠀⠹⣦⣄⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⢿⣥⣽⣯⣿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⠿⠴⠶⠧⠶⠶⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⣀⢠⣀⣄⣼
⠀⠀⠀⠹⣿⣻⣶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠼⢷⡟⠛⠋⠉
⠀⠀⠀⠀⠈⠻⢷⣮⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠾⠶⣷⣾⣼⣶⣤⣄⣀⣴⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⠀⡀⡀⢀⣀⡀⠀⡀⠀⢀⣠⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠾⠷⠷⠾⠾⠶⠾⠾⠿⠾⠿⠿⠷⠾⠿⠿⠟⠛⠛⠋⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠢⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢨⠀⠉⢁⠀⣠⠀⠀⠀⠃⠘⠍⠠⠄⢀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⠁⠀⠀⠃⡉⢀⠨⠵⠖⣐⠶⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐⠁⠀⠒⠠⠀⠀⠰⠂⠈
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⠠⠀⠄⠐⡡⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠤⢀⡀⠀⠈⠀⠀⢀⠀⢄⡰⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠢⡠⠃⢀⠎
⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠜⠉⠀⠰⠁⢀⠜⠀⠀⣀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⡊⠀⢻⠀⠀⡈⠡⢈⣀⢁⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠲⢄⠠⢀⠙⡃⠈⠀⠀⠀⡀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠐⠂⢠⠀⠛⠒⠈⠢⠤⢀⣀⡀⠇⣀⢀⣀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⡀⠀⢂⡀⠜⠢⠤⠴⠠⠄⣀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠓⢀⡇⠀⢀⢀⠠⣀⠤⠀⠄⢀⣀⠀⠀⠠⡤⠤⠀⠄⠠⠤⠤⠤⠄⠒⠊⠁
⠀⠀⠀⠀⢀⠘⠉⠀⠀⠀⠀⠰⠽⠒⡥⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠀⠀⠁⠀⠈

⢐⣦⠲⣦⢶⣂⣶⠂⢶⠐⠖⢂⠆⠐⣄⢀⡴⠄⠐
⢸⣸⠛⣿⠀⢹⣘⣆⣾⣙⡻⣼⠈⡖⡲⠈⠀⢘⣀⠀⠀⡀
⠀⠙⠀⠙⠎⡦⠀⣏⣩⠏⡼⣗⣴⣂⠈⠡⣿⣿⣏⣽⡦⣿⠆⡀
⢈⠛⣤⠐⣿⠃⠸⣏⠮⠐⡼⢏⣸⡇⠭⣠⣿⡿⠙⢙⡀⠛⠂
⠠⠀⠈⠀⢿⡐⢌⡿⣀⢱⣋⠭⢛⡭⠟⣹⣿⣿⣀⣚⠇⠀⠁⠀⡄
⠠⠔⠸⠐⣼⢿⢸⢟⠓⣾⠵⣾⣮⡴⠋⠸⢿⡇⠀⣤⠍⠁⣤⣠⠿⣀⢀⡀
⢀⠂⠘⢇⣿⣹⡿⠗⣻⢃⣴⠟⡡⢟⡖⠲⠶⠤⢤⣬⣁⣌⣱⣾⣤⣤⠁⣀⡁⠈⠀⠀⠀⠀⠀⠠
⢠⡂⣈⡏⣿⣾⡿⣽⠟⠹⠿⣀⡤⣾⢻⣿⣿⣾⣶⠒⠀⠀⠈⠭⠶⠈⠉⠉⠩⠤⠻⣛⠟⣲⠷⣶⢤⣄⣀⣀⣠
⢸⣿⣿⣧⡟⣿⢰⠏⠈⠀⢻⠍⡇⣿⢸⣿⣿⣿⣿⠀⠚⠿⠆⠀⠀⣿⡇⢈⣿⠐⣳⠘⣖⣾⣛⣿⠄⠠⠀⢁
⢸⢻⣿⣿⣧⣿⣯⣶⡄⠚⠘⡇⡇⣿⢸⣿⣿⠉⠀⠀⢀⣄⠀⠀⠀⣀⡀⢈⣉⠀⡈⣀⣽⣿⣿⣿⣷⣈⣀⣏⠀⠀⠀⠀⠠
⠰⠈⡯⣿⣿⣿⣿⣿⣿⣿⡀⡷⡇⢻⠠⡄⢹⠐⣶⡶⣿⣿⡷⣷⡦⠀⠴⠜⣱⡚⣛⠛⠃⠉⠩⠩⢽⡇⠇⠹⠍⠈⠁⠤
⠀⣶⢻⢹⣿⠟⣿⠛⢻⣿⣧⡥⡔⣤⡀⢳⡆⣙⡟⡑⠀⠀⠠⠠⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠂⢀⡈⡇⠅⠈⠀⠀⠐
⠀⢠⠘⠀⡆⠀⣍⡀⠈⠉⠉⠉⠙⠈⠃⠈⠭⠭⠣⣅⣀⣂⣀⣀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠉⠀⠉⠉⠃⠀⠓⠒⠒⠂⠐⠒⠚⠃⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⢈⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣘⣁⣀⣀⠀⠀⠀⠈⠀⠉⢩⡉⠁⠄⣠
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠷⠁⠤⠀⠀⢤⣀⣽⠆⠈⠲⠊⠻⠷⠀⢶⠏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⠀⠀⠐⠂⠐⣢⣼⣿⣿⣿⡄⡀⠀⠀⠙⠲⢾⡟⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢿⠇⠹⠧⠀⣤⣶⣿⠗⣉⣥⣶⠟⠓⣽⣷⢦⣄⡀⠀⠀⠀⠀⠀⣦⡀⠞
⠀⠀⠀⠀⢀⡀⠀⠿⠶⠴⠶⠿⠋⠁⠐⠛⠛⠛⠻⠿⠗⠚⠻⠦⠼⠿⠶⠄⠀⠀⠀⠀⡏⠳⣄⣀
⠀⢀⡀⠀⠀⠹⠿⣦⣤⣤⡤⣤⠦⠤⠠⠠⠄⠀⠀⠀⣀⡤⠀⣀⡀⣴⣤⣴⡤⣬⣬⣿⣿⣿⡟⠁⢀⡀⢰
⠀⠈⠁⣾⠤⣤⣾⣿⣗⣦⣴⣿⣿⣿⣁⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠤⠿⠛⣿⣿⣿⡽⣵⣄⢘⠀⢸
⢀⣄⣠⣷⣿⣯⣼⣷⣶⡴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣶⣾⣿⣿⣿⠶⠎⠻⣦⣾
⠀⠙⠻⡿⣿⣿⣟⣻⣯⣐⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟⣹⡿⠟⢁⣆⢿⣭⡁⡈⣭⣭⣋⡁
⠀⠀⢠⣭⣭⣿⣯⣿⣿⣿⣿⣴⡎⠻⣿⣿⣿⣿⣿⣿⣽⢞⢿⣷⣿⡾⠟⠀⢰⠈⠀⢸⣟⡿⡻⣟⣫⡇
⠀⠀⠐⣿⢿⣿⣽⣹⣿⣿⣗⣿⢹⣿⠶⣇⢽⣿⣿⣷⣿⡟⢀⣩⣿⡗⢞⠁⢸⠀⠀⣸⠀⠃⡇⠹⠀⠇
⠀⠀⢀⣿⣻⣷⣿⣿⣿⣾⣯⣿⣸⣿⡟⢳⣸⣿⠿⢿⣿⡿⠛⢉⣿⠁⠩⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⠄
⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⣽⣿⢠⣾⠿⠇⠀⠀⣽⠀⠇⡆⢸⠀⡆⣼⠀⡁⡇
⠀⠀⠨⣿⢸⣿⡿⠋⢭⣿⡟⣿⣿⣿⡏⠼⣿⣿⣛⠋⠀⠀⢰⣌⠯⠁⡇⡇⢸⠀⠉⣙⠲⠀⠇⠀⢰⡆
⠀⠀⠰⣿⢸⡟⠷⠲⡁⢸⡧⣿⣿⣿⡇⠀⠛⠀⣀⡀⢀⡄⠀⠑⠀⠀⡇⠇⢸⠀⠀⢸⣶⣦⡀⡇⢸⡇⠠
⠀⠀⠀⣿⠀⠀⡄⡀⣼⢺⡗⣿⢺⡿⡧⠀⠀⠀⢈⣢⣬⡗⣤⡀⠀⠇⡇⡄⢸⠀⠉⢠⣿⠋⢿⣇⢸⡿⣿
⠀⠀⣰⡞⠀⣼⣴⣷⢇⣿⣷⣿⠾⠍⣹⣶⠶⣤⠼⣿⡏⠁⠈⣷⡄⠀⠑⠆⢨⠎⣀⠸⣿⡆⠸⡿⠙⠓⠛
⠀⠀⣼⠄⡀⠙⠯⠣⣀⣄⣾⡇⠷⣢⠍⣼⢢⡀⢢⣾⣿⡆⢀⣻⡟⣄⠀⠀⠐⠚⠋⢘⣭⢳⣴⣿
⠀⠀⠋⠀⠀⢈⠀⣂⡼⣽⣿⠟⠇⠃⢸⠟⠀⢷⣎⣶⡫⢲⣿⣻⣧⠘⠆⠀⠀⠀⠀⠨⢳⣤⢨⣿⡆
⠀⠀⡀⠀⠀⠰⣦⡀⢹⣿⣿⢸⠀⢀⡜⠀⠀⣸⠇⣿⡅⢸⣿⡟⢿⠂⠺⣆⠠⡄⠀⢸⠸⡏⣿⣿⣇⠠⡄
⠀⠒⠁⠀⠀⢠⡇⢉⡟⢻⣿⣼⠀⣒⠃⡰⠀⠿⠀⠻⠆⢸⣿⡇⠘⣗⠄⠰⣆⡐⠀⡏⡄⣷⠸⣿⡇⠲⡄
⠀⣰⠏⠀⢀⣼⣣⠟⣠⣿⡟⠛⣠⠅⢠⠁⣰⠁⠀⠀⠀⠘⣿⣿⡇⢹⣗⡀⢚⣷⠀⢷⣧⢹⡅⣿⠙⢆⡀
⠀⢠⠀⠀⠜⠙⣡⣾⠻⠟⠀⠘⠗⠀⠖⠀⠸⠀⡆⠀⠀⠈⣿⣿⡧⠀⢶⣏⡀⠘⠀⠘⢻⠘⣧⡽⣶⠄⠃
⠀⡓⠀⣀⣰⠋⣠⡛⠀⠀⠘⣻⠂⢰⡛⠁⠈⢓⣇⣐⣄⣈⣿⡿⣿⡖⠰⣟⡁⠈⠠⣟⣰⣧⢻⣿⡿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣄⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠢⠀⢂⠀⠀⠀⢀⢀⣈
⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠤⣀⠀⠀⠀⠀⡠⠀⠀⠀⣉⠁⠆⠀⠀⠀⠚⠤⠀⠀⢀⡚⠟⢩⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠅⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠈⠀⠀⠑⠉⢳⣦⠀⠗
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢠⡀⠀⠀⠠⡀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⡐⡰⢰⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡂⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠄⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠸⠃⠀⠀⠀⠘⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠄⠘⣿⣴⣥⣤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⡄⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠩⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⢄⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣀⣀⣀⡰⠄⠈⠀⠂⠀⠀⡖⠒⢆⠀⠀⠀⠀⠰⠂⢀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠅⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⢂⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣦⣤⣤⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠉⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠉
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠛⠋⠉







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⡀⢀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠁⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠁⠀⠀⠀⠁⠈⠀⠁⠁⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢶⣄⡁⢠⡄⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢂⠀⠀⢰⣄⣠⣤⠤⡄
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢹⣦⠀⠀⠀⠂⡾⠿⠿⠄
⠀⣠⣄⣄⢀⣀⣠⠀⣀⣤⣀⢄⣀⣢⡤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢁⣴⡟⠃⠄⠀⠒⠀⠀⠛⣻⣶⡧⠈⠁
⢸⡷⠿⠛⠉⠉⠀⠒⠛⠁⠒⠈⠁⢈⣉⡉⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⡀⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⢠⠙⠛⢯⡁⢐⠛⠛⠳⠁⠀⡞⠛⠛⠄⠀⠀⠀⠀⠿⠙⠙⡄⠀⠛⠛⠁⠰⠾⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠁

⣶⣶⡶⣶⠶⢶⣿⡶⣶⣷⢶⣷⣾⣶⣷⣶⣶⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣷⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢤⣤⣴⣄⣀⠤⠠⡄⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⢟⣟⡫⣽⣛⠿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠈⣹⣿⣿⣿⣶⡆⣇⡐⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠳⢾⣿⣦⡈⠙⠓⢾⡿⣷⢦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣶⣿⣿⣟⠓⡦⠀⠸⠛⢈⠚⣦⡀
⠀⠀⠀⠀⣤⣽⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⠋⢁⠀⠀⠀⠀⠀⠠⣬⣷⡄
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⢿⡄
⠀⠀⠀⢀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠀⢻⣿⣿⣿⣿⠟⠉⠀⠘⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢦⣸⣷
⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠀⠈⠋⢿⣾⣿⣄⣠⡆⠈⠙⠓⢠⡤⠀⠀⠀⠀⠀⠰⠦⢸⣿⣿⡇
⠀⠀⢀⣠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠬⠉⠉⠛⠷⣶⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⣽⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⣿⣿⣿⣿⣿⣦⡄⠀⠀⢸⣿⣿⡇
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣽⣿⣿⡇⣀⡀⣤⠔
⠀⠀⠐⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠷⢦
⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⡁⠀⣀⢀⢀⣤⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣛⣈
⠀⠀⠀⠀⢰⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣾⡿⣛⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢠
⠀⠀⠀⠀⠀⠉⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣟⣫⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠻⠟⠋⠉⠊⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠻⠿⠿⠛⠋⠈⠉⠁⠉⠉⠙⠁

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠑⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⣴⣶⣄⠀⢀⣠⣤⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⢀⣤⣄⠀⠀⠀⠇⠀⢀⠁⡀⠀⠀⠀⠉⠛⠁⠀⢾⣿⡿⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠽⣀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠒⠀⠸⡐⡼⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢱⣇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⡄⠃⢠⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠥⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⠤⠂⠠⠤⠀⠀⠀⠀⠀⢀⣰⠛⠛⠒⠛⢹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢖⠉⠀⢠⢤⣀⡀⠁⠀⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠠⠆⢀⣾⡇⣰⣷⠄⠀⠀⠀⠀⠀⣇⠜⠀⠀⠀⠈⠢⠀⠢⠄⠀⠀⠀⠀⠈⢹⠇⠀⠀⠀⠸⠏⡃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⢙⡌⠀⠀⣄⡀⠀⠀⠀⠂⠀⠀⠃⠀⠸⠾⠷⠮⠶⠴⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⢀⠉⠀⠙⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠈⠧⢤⠀⠀⠎⠁⠀⠐⠀⠀⠸⠟⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣯⡁
⠀⠀⠀⠃⠄⠀⠀⠀⠠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⣾⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢂⢄
⠀⠐⠚⠞⠓⠲⠒⠐⠲⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣶⡄⠀⠁⠉⠂⠀⠀⠀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠲⠙⠀⠀⠀⠀⠘⣿⠿⠀⠀⣀⣠⣀⣀⣂⡈⠛⠁
⠀⠀⠀⠀⡀⠀⠐⠀⣀⠴⠃⠀⠀⠀⠃⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⣼⣤⢀⠛⢿⣿⡙⣿⣿⣿⠆
⠘⢧⡀⠀⢉⡠⠔⣊⡀⠀⠈⠤⢀⠀⠀⡄⠀⠛⠁⡴⠀⠐⠠⢀⠸⠛⠁⠀⠀⠸⠛⠁⠘⢿⠟⠀⠀⢰
⠀⠀⠙⠛⠉⢀⢴⣿⠗⠀⣶⠀⠀⢙⠾⡡⠀⠀⢄⡀⠁⠀⠀⠀⠀⠘⠂⠀⠀⠠⣀⡀⠀⠨⠖⠐⠒⡿⠒
⠀⠐⣩⠙⢢⠀⠘⠿⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⢀⣠⠜⠀⠀⠂⣄⣀⢘⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⢁⡄⠘⠻⢇⠀⡆⢰⡀⡀⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠖⠢⣌⠀⡗⠚⡇⡆⠺⡁⢹⢰⠒⣴⡖⢹⣼⠀⠸⣇⣘⡶⠖⠛⠲⡄
⠠⠀⠀⠐⠀⠀⠀⠀⠀⡏⠀⠀⢸⠂⠇⠀⠇⠃⠘⠁⠈⠈⠉⠈⠁⢀⡟⣶⣷⠀⠀⠢⠀⠹⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠒⠒⢿⣄⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠉⣸⢿⠇⠀⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⡀⠈⠀⠉⠀⠁⠀⢀⠀⢰⣏⡎⠀⠈⢻⡀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⣉⠽⡆⠀⠀⠢⠄⠒⢤⢐⠀⠻⠞⠀⢀⠀⠀⠁⠀⠛⠻⣿⡀
⠀⠀⠀⠀⢠⠄⠐⡀⠆⠀⠀⠀⠀⡇⠙⠀⢳⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠉⠓⡀⠀⠙⠆⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⡀⢣⣠⡷⠾⠆⠀⠣⠀⠊⠄⠘⠀⢀⡀⠀⠀⠀⠀⢀⣀⣠⣼⡟
⠀⢀⠀⠀⢀⠀⠀⠄⠀⠠⠀⢀⠀⠈⡤⠀⠀⠀⠠⠄⠠⠼⠀⠀⡄⠀⠀⠀⠐⣠⣾⣿⣿⣿⣿⡄
⠀⠈⠉⠉⠋⢠⠄⣄⣀⣀⣀⡀⢠⣀⡠⡤⠀⠶⠐⠐⠀⠀⠒⠃⢧⠀⠀⠐⣼⣿⣿⣿⣿⣿⣿⡇⠀⠄
⠀⣠⣄⡄⠀⣟⠀⠈⡇⠀⣿⠀⠀⣇⣀⢡⡀⣀⠀⠘⣿⡿⠇⠀⢈⣛⠖⠚⠋⠉⠉⣿⣿⠋⣿⡇⢰⠖
⠀⠙⠛⠛⠀⠉⠀⠀⠃⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠋⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠛⠋⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣦⠀⠀⠀⠀⠀⢀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀⠀⠐⣤⠞⢀⣀⣨⣬⡀
⣿⣿⣦⠀⠀⠀⠀⢼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡖⠻⠟⣯⣤⠐⠚⠉⣫⠴⠛
⣿⣿⣿⣇⡀⠀⡙⣻⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠄⢀⡃⠠⠈⣉⣯⢥⡦⡄⠀⠸⠠⠄
⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡐⠋⡙⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠉⠀⢳⠀⠀⡀⠀⠰⠠⡎
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⢐⢦⠀⡀⠈⡡⢄⣶⡀⠈⣧⡗
⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠁⠀⠉⠉⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠚⠐⣿⣵⠃⠺⠿⠀⠁⢠⢈⢩⣿⢮⡿⠇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣴⣿⣿⣷⣦⠀⠀⢀⡀⢸⠰⠀⢸⠠⠼⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⢹⡇⠽⠋⠁⠀⠀⠙⠓⠀⠻⠀⠘⠃⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢀⡀⣠⢂⣀⠀⣠⣾⠷⢳⣰⣶⣦⣶⡀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠛⢿⣟⠋⠉⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⣿⣿⣿⡿⣿⢲⣿⠁⢀⣴⢾⣿⡧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣿⣆⠀⠙⡿⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢠⡟⣹⠻⠛⡇⠸⣿⡏⢠⡌⠉⣭⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣇⢀⣀⢴⣤⣀⠁⠘⣶⡀⠀⢀⡾⠛⠀⠀⠀⠀⣾⠟⢡⡶⢶⣌⠛⣿⣋⠛⢽⠾⡛⠉⢧
⠀⠀⠀⠀⠀⢀⣿⣿⣟⢹⣗⠐⠀⠈⠀⢸⣿⣧⣴⠟⠁⠀⠀⠀⠀⢰⣋⢠⣾⣄⣸⣿⣶⣿⣟⣀⡀⢀⣀⣀⡾
⠀⠀⠀⠀⠀⣾⣿⣿⣽⠀⢹⣣⣄⠐⠂⣾⠿⣻⣧⠀⠀⠀⠀⠀⠀⣾⡟⢙⣿⣿⣿⣿⣿⣿⡏⣹⠹⠋⠉⢼⡇
⠀⠀⠀⠀⠘⣷⣿⣿⣷⠄⠀⠻⣿⣦⣾⡿⢿⢿⠇⠀⠀⠀⠀⠀⢠⣏⣁⣿⣿⢻⣿⣿⣿⣿⡇⢡⣶⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠙⢻⡅⡀⣀⣲⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣆⣠⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠌⠉⠁⢈⢿⣿⠏⠀⠼⡟⠀⠀⠀⠀⠀⠀⠀⣾⠃⣷⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠃⢀⡰⣿⣿⣿⡄⠀⡁⢸⡄⠀⠀⠀⠀⠀⢀⡇⠀⠁⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣉⣰⣿⣿⣿⣷⠀⠄⠈⠁⠀⠀⠀⠀⠀⡸⠁⡾⠀⣸⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣶⡟⠚⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢺⣿⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠉⠀⢉⡏⠉⠙⠁
⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⠀⢸⡟⢛⣗⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠰⣿⣼⡏⠀⠀⠋⢸⣯⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢀⡄⡌⠙⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠨⣴⢷⡀⠘⣣⠃⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠐⠊⠀⠋⠉⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠚⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣀⡠⠰⠂⣂⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠉⠄⣼⣿⡿⠃⠀⠀⠁⠀⠀⠉⠢⢎⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⢐⣾⣿⣏⣀⣀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠶⢄⠀⠀⣾⡿⠟⠛⢻⡟⠓⠒⠊⠙⠿⠷⣶⣆⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣠⡑⡾⠉⠀⠀⠀⠀⠙⢤⣀⣀⣀⣴⣶⣶⣷⣿⣿⣿⣿⣿⣿⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠠⠻⣷⣧⠐⠠⠤⠄⠠⣄⣀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡟⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⣀⡀⣀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣄⠀⠀⠀⠀⠀⢻⠘⠟⣿⣛⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⠀⠀⠀⠀⠠⢪⣿⣶⣄⠀⠀⠈⢧⠀⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣿⠿⠋⠁⠀⠀⠀⠀⠀⠔⢿⣿⣿⣷⣄⠀⠀⠑⠤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣴⣶⣶⣿⣿⣿⣿⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢄⠄⣄⢄⢀⢠⠀⢀⠀⡀⢀⢀⣰⣴⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠊⠎⡞⣾⡞⣎⡟⣷⣾⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⢈⠀⢛⢃⢻⢹⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢣⡃
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠸⠸⠈⢸⢹⣿⢻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡟⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⠀⡀⠈⠀⢀⠀⢸⢸⣿⡞⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⢸⢠⢸⠸⣹⢡⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⣿⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⢸⢸⣾⠀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠈⢸⣿⣤⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠈⠀⠀⠀⠀⠈⢸⣯⡷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⡆⡆⠀⠀⠀⠰⢸⡇⣇⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⡆⡇⢸⣾⣧⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠈⡐⠐⢠⠀⢀⢰⢠⠘⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠃⠞⡄⡜⣸⣸⢒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣁⠀⠘⡠⢰⣣⢡⢫⡏⡞⣾⣿⣿⣹⣿⣽⣾⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣷⣣⣯⣾⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠟⠛⠛⠛⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢨⡷⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⡀⠀⢀⣀⡤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢹⠀⠀⠀⠀⠀⣀⣀⣠⢶⡇⠀⠀⠀⠀⠀⠈⠙⣶⣏⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠘⠧⢤⡄⠀⣸⠉⠉⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⡇⠈⢻⣄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠀⠳⠖⠃⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⢐⠉⢶⡀⠀⠀⠘⡆
⠀⠀⠀⢠⡀⠈⠙⡆⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢯⣠⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠓⠄⠀⠀⢻⡀
⠀⢀⣠⠞⠉⠙⠉⠁⠀⠀⠀⢣⠀⠀⠀⠆⠀⠀⠀⠈⠳⢤⡤⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠉⠳⡄⠀⠀⠀⠀⠤⢄⠀⠈⢳⡀⢸⡀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠷⠖⠚⠛⣳⠀⠀⠈⠲⢦⣳⣘⠀⠀⠀⢀⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⢸⠇
⠀⠀⡀⠀⠀⣶⠚⠁⠀⠀⠀⠀⠀⠈⣿⡤⣤⠤⠤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠞
⠀⠀⣿⠀⠀⣿⣀⡀⢀⡤⣤⠀⣀⢠⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡴⢄⣀⣤⣄
⠀⠀⠸⡆⠘⠉⠉⠉⠋⠀⠿⠋⠉⠋⠀⢸⣇⠀⢀⡤⠤⡤⠤⣤⣗⡆⠀⠀⠀⠀⠀⢀⣀⡀⢸⠃⠀⠉⠀⠀⠁⠀⢸
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣿⢀⡡⠄⠠⡀⠈⠛⢦⡀⠀⠀⠀⡞⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⢸⢿⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠙⠒⠴⡀⣁⢈⠹⡄⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻
⠀⠀⠀⠘⡇⠳⡌⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠓⣦⡴⠃⣴⠒⠃⠀⠀⠀⠀⠔⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⣇⠀⠘⣆⡀⠀⠀⠈⠙⠒⠦⣄⠀⢿⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣿⠀⠀⠀⠀⡤⠎⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢸⡄⠀⠀⠹⣀⠀⠀⠀⠀⠀⠈⢧⠘⢧⡀⠀⠀⣀⡤⠞⠁⠀⠀⣠⠟⠀⠀⣠⠞⠀⠀⢀⣠⡤⠴⠋
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⠢⣄⠀⠀⠀⠀⠈⣧⠀⠉⠉⠉⠁⠀⠀⠀⠀⢸⠁⠀⣠⠟⠁⠀⣀⣀⣈⡷
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠞⠁⠀⣀⡟⠉⠈
⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠐⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣠⡴⠟⠉⠉⠉⠉⠁
⠀⢸⡓⠦⣄⠀⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣤⡼⠇⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⣤⣤⣄⡀
⠀⠸⡇⠀⠀⠙⠲⣄⠀⠀⠀⠀⢀⡤⢶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣲⣤⣾⠇⠀⠀⠙⠲⠳⡄⣀⣤⠶⠖⠲⠶
⠀⠀⠻⡀⠀⠀⠀⢸⠀⠀⢠⠞⠁⠀⣼⠀⠀⠀⠀⣤⣠⠶⠞⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⢯⣀
⠀⠀⠀⠙⠲⠤⠴⠋⡀⠀⣿⠀⠀⣠⠇⢀⣦⡀⠀⠀⠩⣶⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⢉⣿⡇⠀⠀⠛⢲⡄
⠀⠀⠀⠀⠀⣀⣤⢤⣈⠢⣸⠗⠚⠁⢠⠎⠀⢳⠀⠀⠀⠈⣙⡷⢤⡄⠀⠀⠀⠀⠀⣠⠋⠘⠇⠀⠀⠀⠀⣘⣦
⠀⢀⣤⠖⠋⠁⠀⠀⢸⠏⠉⠳⣄⠀⡏⠀⠀⣸⠀⠀⠀⠀⠀⠀⣸⠓⠀⠀⠀⢀⠜⠁⠀⠀⠃⠀⠀⢰⡖⠛
⠀⠀⠈⠙⠒⠶⠴⠖⠋⢀⣀⣀⣈⢦⡳⣦⠴⠃⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉⠀⠀⠌⣷⠻⣇⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⢀⣀⡿⣄⣀⠀⠀⢲⡷
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠤⠴⠃⠀⠙⣆⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣠⢤⠏⠉⠀⠈⠙⠲⠷⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠾⠷⢺⠶⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⠀⢰⠦⡄⢄⡤⣐⣶⣾⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⡆⠀⡆⠀⠀⠀⠀⢠⠃⢠⣿⣷⣤⣤⣀⣤⠄⠈⠉⡹⠳⣄⠀⠀⣌⠜⢢⡄⢬⣿⣿⠇⠀⠀⠀⠀⢾⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋
⠀⠘⡄⠹⠄⠀⠀⠀⡞⢀⣿⠁⠀⠀⠉⠉⢹⠀⢠⢠⠃⢰⢾⢛⠒⣅⠋⣻⡛⣷⣿⣿⡄⠀⠀⠀⠀⠈⢻⠳⣀⢀⠀⢀⣠⠶⠞⠋⠁
⣀⣤⣴⣾⣿⣿⣷⣤⣇⢸⡧⣶⣶⣶⣤⣀⡃⠀⠆⡎⠀⠘⢉⣡⠒⡈⣤⠱⢼⣿⣿⠃⠙⣦⡀⠀⢀⠀⢠⣟⣯⡶⠛⠉⠁
⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣝⣿⣿⣿⣽⣛⢿⣶⣶⣤⣤⣴⣉⣿⠀⠈⣶⣼⣿⣿⠀⠀⢣⡈⢑⡶⢞⣵⡿⠋
⠀⠀⠀⢹⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣤⣤⣴⣥⣿⣾⠟⠋
⠀⠀⠀⠸⡀⠘⢧⣸⠉⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠉⠣⣄⡀
⠀⠀⠀⠀⢣⠀⠀⠀⠀⣼⠇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠣
⠀⠀⠀⠀⠈⡂⠀⠀⢐⡏⠀⠀⠈⠀⢠⠙⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣻⣿⣿⣿⣶⣄⠐⢄
⠀⠀⠀⠀⠀⢸⡀⠢⣼⣷⠀⢰⠀⠀⠀⢠⢀⢻⣿⢿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢧⠀⠑⢿⣦⠆⠀⠀⡀⠈⠈⣌⣿⡞⢻⣿⣿⢻⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡷⢄⣀⣀⣠⣤⣤⣤⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠀⠀⠠⠈⢇⣼⣾⣿⣿⣾⡯⠙⢿⣿⣏⣝⣫⣭⠙⣿⣿⣿⣿⣿⣿⣿⣟⣋⣙⠋⠟⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⢀⣸⣴⠿⣿⣿⠛⠻⣶⣧⠘⠛⡟⠫⠙⠁⠊⢩⠋⣛⠉⢛⡿⠿⠏⠙⢻⣿⢟⣧⣿⣿⣿⡄⠈⢙⠐⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡀⠙⢷⡤⢘⡡⠚⠸⣿⣿⣿⣾⣆⣤⣤⣀⣶⡈⣴⣼⣶⣾⡁⣀⠀⢀⠀⠉⠩⡍⣯⣽⣿⣷⣄⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡔⣿⣿⣿⣿⣿⣦⣿⡿⠺⢷⣾⠋⢛⣿⣿⣿⣿⣿⣿⣿⡷⠶⠿⠿⠛⠛⠛⠓⠛⠉⠉⠉⣿⡽⠟⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡆⡄⠀⠀⠀⣠⣿⣿⣿⣿⢿⢟⣛⢩⠇⠲⠔⣿⣴⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢿⣿⣿⣿⣯⡖⠀⣙⠈⠞⣹⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢨⣮⣿⣯⠉⢹⣯⡿⣅⢀⣴⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⡿⣿⢻⣿⡷⠿⠻⣧⡑⣡⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣼⠃⣿⣿⣿⠿⢠⡖⠞⣴⣿⢿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⡿⢸⠧⣿⢿⣄⢼⢷⣾⣿⣿⣿⠟⣼
⠀⠀⠀⠀⠀⠀⡸⠁⠀⠎⠇⢸⠀⡟⠈⡿⣸⣿⣿⣿⣿⠏⢨⡏
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⠀⡇⠀⠀⠀⣶⢿⣟⣿⣿⢋⠀⢼
⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⣤⣤⣤⣄⣤⣠⣄⣤
⠀⠀⣿⣿⡿⠷⠿⣿⣯⣍⡉
⠀⠀⣿⣾⣷⣟⣿⢿⠏⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⣀
⠀⠀⡟⡽⠟⠉⠉⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠘⠃⢀⣿
⠀⠀⣏⢠⣀⡀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡀⠀⠀⠀⢲⣶⣦⣼⣿
⠀⠀⣟⣿⣿⠄⠀⣠⣼⠃⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣤⡀⣶⠀⣤⣸⡟⠛⣿⣿
⠀⠀⣏⢀⡌⠀⠀⠿⡋⠀⣸⠄⠀⢀⠉⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠁⣀⠾⠿⠇⠀⠀⠛⠘⠇⠀⣿⣿
⠀⠀⠟⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣶⡀⠈⢿⡿⠄⠀⠚⠠⠶⠀⢰⡀⢀⣿⣿
⠀⠀⣛⣟⣳⣶⣿⡁⠀⠀⠠⢐⣅⣤⡄⠀⣾⣿⣿⣧⡀⠀⠀⣠⢸⣿⣿⠟⣀⡞⠀⠀⣷⠒⢀⠀⠒⢀⠇⠘⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣧⣀⣀⣴⣿⣿⣿⠀⠀⢻⡿⣿⣯⣿⠀⠀⠿⡿⠿⠱⠔⠋⠀⠀⢰⣷⣷⠀⡀⠲⠾⠇⢀⡝⠛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠦⢼⡇⣿⣿⣿⣦⡀⣠⡥⢈⣛⣿⠄⣀⢀⣾⣿⣿⠿⠿⡿⠷⠀⠾⠿⠃
⠀⠀⡟⠛⠿⠋⠉⠉⠀⠉⠉⠙⠒⠀⠲⠤⠿⠛⠛⣿⣻⣿⣿⠟⠿⠛⠉⠽⠯⠽⠯⠌⠏⠉⠒⠐⠈

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⣸⡀⠀⢀⣀⡀⣾⣿⡇⠀⠀⣵⣶⣄⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢦⣿⠄⠀⠘⢿⣷⣿⣿⣥⣶⡆⠙⠻⣿⠅⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡈⠰
⠀⠀⠀⠐⣾⣧⠄⠀⠻⢿⣿⢿⣿⣿⣍⠀⣤⣼⣿⣿⣀⣤⣠⡌⢻⣿⠀⠀⠀⠀⠀⠀⠌⠀⠀⠄⠀⢬⠔⠂
⠀⠀⠀⠀⠈⠉⢀⣴⠀⠈⠁⠈⠛⠉⣉⣠⣀⣬⠀⠈⠛⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⡀⠔⠈⢠⡼⠀⠀⡠
⠀⠀⠀⣿⣆⢠⣿⡟⠀⠀⠀⢀⣰⣾⣿⣿⡿⠇⢀⣤⣾⣇⠀⢰⣷⡄⠀⠀⠀⢀⠀⢡⠂⠘⢦⡀⣸⠧⠖⠊⠀⡴⠈⠁
⠀⢀⣀⣹⣿⣿⣿⣴⣶⡤⠀⠺⠿⢿⠿⠟⠋⣰⣿⣿⣿⣿⣧⠘⢿⣿⠀⠀⠀⠀⠁⠺⢄⠀⡀⠹⡏⠀⣀⣜⣁⣀⣀⡀
⠀⠀⠉⠉⠛⠻⣿⡉⠉⠀⢰⣶⣦⣠⣴⡆⠀⠘⠛⠿⣿⣿⠟⠠⡄⠀⠀⠀⠀⠀⠀⠐⠲⠛⠳⣼⣿⠟⠡⠀⠈
⠀⠀⠀⠀⢀⣀⡈⠛⠀⠀⢈⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⡀⠀⣿⣏⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⢠⣾⣿⣿⣷⣆⠐⠿⢿⣿⣿⠟⠛⠉⠀⠀⠀⣿⡟⠈⣿⣗⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠛⠿⠿⠟⠂⠀⠸⠿⠃⣀⠀⠀⠀⠀⡀⢛⣁⠀⠹⠿⠹⠍⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⣠⣾⡷⣀⣤⣤⠀⠀⠀⠀⢠⣿⣿⣷⠀⣼⣿⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠈⠁⠀⠙⠋⠙⠿⠛⠛⠀⠘⢿⣿⠟⠀⠻⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠙⠛⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢤⠀⠀⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠬⠷⣄⡀⣇⠀⠀⠀⠀⠻⣆⠀⢀⣴⢿⣏⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠙⠻⠶⣤⣀⠀⠀⣿⣠⡾⠁⠀⠙⠳⠶⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⣀⡤⠞⠛⠳⣄⣹⡟⠁⠀⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⢙⣿⠁⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⣻⠇⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠋⠉⠉⠛⠛⠛⠶⠶⣤⣸⡇⠀⢠⡟⢀⣠⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣇⠀⣼⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣿⣤⣿⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿
⣿
⣿
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣤⣤⡴⠾⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣬⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣟⣛⣙⣿⣈⣛⣛⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣭⣭⣽⣿⣿⣭⣭⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠶⢿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣯⣻⠿⢿⣿⣯⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣹⣿⣿⣿⣿⢭⣿⣿⣿⣟⠿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⡖⠁⠀⠁⠁⠀⢀⣴⣤⣤⣴⣤⣷⣿⣿⣦⣭⣿⡿⣲⣿⢿⣿⠯⣝⡿⣟⣿⣿⡿⣿⣿⣿⠿
⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣦⣾⣿⣿⣽⣿⡿⠟⠛⠛⣫⣿⣿⣿⣧⠀⢿⣗⣻⣷⡎⡙⢬⣻⣿⡟⣻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠈⠉⠈⠛⠀⠁⠀⠀⠒⠀⣿⣏⣾⣯⣿⣯⠘⣟⣿⣏⠹⢓⣾⣾⠫⠯⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠽⠿⢿⣿⣿⡡⢍⡰⢾⢳⡏⠤⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠏⠁⣀⣰⡿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠀⠀⠀⠀⠐⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶