Snowpix Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣴⣶⣴⣶⡀⢀⣴⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣆⣻⣷⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠙⢿⢣⣶⢈⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢀⣬⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠟⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⣿⣿⢿⣿⣟⣆⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⣿⣿⣽⣿⠛⠿⠟⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣤⣤⣾⣷⣾⣿⣧⡈⡟⣩⣦⣾⣿⣦⠀⣳⡷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⡴⢿⣿⣛⣿⣯⣭⣿⡷⣠⣝⠛⠿⠛⠋⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠿⠛⢻⠿⠟⢻⣽⣿⣿⣿⣷⠿⠗⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣹⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠛⠁⠿⠇⠀⡌⢙⣿⠟⢛⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡭⠖⠛⠉⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣾⣿⠿⠋⢀⣠⣤⡤⢤⣤⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⣿⠃⠀⣷⡿⠋⠀⢠⡄⠠⢮⠻⠷⢀⢀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⠒⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢷⡄⠀⠀⠀⣀⣀⠴⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⣾⣶⣶⠆⢸⣿⣿⡇⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⠿⢿⣀⣸⣿⠿⠗⠀⠀⢸⣇⠀⣶⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣾⣿⡄⣄⡀⠀⠀⣿⠿⠟⠂⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣦⣠⠀⢸
⠀⠀⠀⢀⣼⠟⠁⠙⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣠⣾⡿⠁⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⡀⠸⠀⢠⣤⠀⢀
⠀⠀⠀⣼⠏⠀⠀⡀⠀⠈⠉⠛⣻⣧⣤⣴⣶⣿⠟⠋⢀⢀⡀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠈⠉⠰⠿⠇
⠀⠀⢰⣿⠀⠀⠀⠈⠈⠉⠀⠐⠀⣹⠿⠿⢿⡇⠀⠬⢡⣥⢐⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⣿⣿⣼⣿⡿⢆⣠⣤⡄⣀⣠
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠩⠡⠜⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⣈⠻⠟⣷⣿⣿⢿⣿⣟⣿⣭⡦⠆
⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⢿⠟⣠⠑⢷⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣨⣝⣁⣴⡾⣻⣷⡈⠛⠁⠉⠁
⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠋⠉⠛⠛⠛⠛⠀⠀⢀⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢾⣻⣟⣿⣿⣾⣿⡟⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣶⣶⣶⣶⣶⣶⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣤⣶⣴⣶⣦⣶⣶
⣿⣽⣭⣭⣽⣵⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣶⣿⣾⣿⣶⣶⣾⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡠⢄⡈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⡸⠔⠛⠒⠢⢄⠀⢠⠃⠀⡈⠓⢊⡀⠀⡧⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⠀⠈⠲⢄⠀⡎⣠⠔⡲⡁⠒⢤⢱⣃⠴⠊⠉⢺⡁⠤⠒⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠂⠁⠐⠃⡕⠒⠢⠤⣙⡄⣅⠴⡁⡈⠒⢚⠞⠁⠀⠀⠀⡔⠀⠀⠀⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢄⣈⣉⣹⣄⠀⠀⠀⠀⠉⠑⣆⠶⣉⣜⡜⢂⠀⠀⡠⠊⢀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⢆⡉⠉⠁⣠⠾⣀⠞⠁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⢸⠉⡄⠀⠟⡆⠀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠒⢤⠀⠀⡇⠀⠀⠘⡀⡇⠀⢸⠘⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇⠀⠀⠀⡇⠳⠤⠎⠀⠱⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠸⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡆⠀⠀⠀⠀⢸⠸⡀⠀⢀⣀⢤⠊⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠣⢄⣀⣀⢀⡼⠀⠑⠚⡏⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⣈⡄⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠈⠫⢅⣈⡱⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⡰⠋⣩⠏⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⢀⣀⣑⣉⣀⠠⠤⠤⠐⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣿⣠⡤
⠀⠀⠀⠀⠲⣤⣀⣻⣂⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⣿⣬⣿⡗⠒
⠀⠀⠀⠤⣤⢬⣾⡉⣹⣯⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠈⠉
⠀⠀⠀⠀⢠⡶⠊⣽⠉⠺⣦
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣶⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣿⣉⣿⡷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⢄⢸⡃⢤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⢿⡋⣉⣷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠒⠎⠈⢳⠒⢢⡤⢀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢹⡉⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠈⠁⠸⠦⠤⠤⠂⠈⠁⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠂⠛⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠁⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⢐⣶⡼⣧⣶⠂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢤⣀⣾⣀⡤
⠀⠀⠀⠈⢉⡿⢷⡾⣿⡉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒⢺⣿⣭⣿⡗⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠀⠛⠈⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡇⠞
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⠒⠖⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠐⠲⠒⠒⠛⡄⠀⠀⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠂⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣠⣎⣁⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠢⡀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢠⠔⢱⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠉⠉⠉⠉⠉⠳⣤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⢀⠃⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡌⠀⠘⡀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠐⠤⡄⠈⢁⡸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⢀⡠⠄⠖⠀⠀⠀⠒⠆⠨⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢡⠄⠄⠠⢤⢀⡀
⠀⠀⣠⠖⠓⠀⠂⠐⠒⠠⠤⣀⠀⠀⠀⠈⠢⣄⠀⠀⢀⡀⠤⠤⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⢀⡠⠤⠐⠒⠂⠨⠵⢦⣄
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⠖⢉⠠⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⣿⡇⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⣿⡇⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠙⣿⣿⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⣿⠋⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠈⣿⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⡟⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣽⣧⠀⢸⡇⠀⠀⠀⠀⠀⠸⣷⣤⣿⣠⣶⠇⠀⠀⠀⠀⠀⢀⣿⠃⣰⡟⢀⣠⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⠶⠾⠿⠿⣦⣼⣇⠀⣿⡀⢠⣶⣦⠈⢻⣿⠋⢀⣤⣄⠀⢠⡄⠀⣾⣇⣰⣿⣿⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⡶⠾⠿⠿⣶⣿⡇⠈⠛⢿⡀⢸⣿⠀⣸⡿⠛⠀⣿⣧⣼⣿⣿⣋⠉⠛⠛⠻⠷⢶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠛⠻⢷⣤⡈⢷⣸⣿⣼⠏⣀⣤⣾⠿⢿⣥⣄⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤⣍⣻⣿⣿⣿⣿⣾⣛⣉⣀⣤⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠛⠁⣈⣭⣿⣿⣿⣿⣿⣯⣉⠉⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣀⠙⢛⣷⣶⡾⠛⠉⣰⠟⣿⡏⢿⡍⠛⢷⣦⣤⣴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⢶⣦⣤⣤⣀⣩⣿⣿⡟⢹⡿⠀⣴⣾⡏⠀⣿⡇⠈⣿⣦⡄⢸⣿⠿⣶⣶⡶⠾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⠏⢹⡿⠀⠘⠃⠀⠙⠛⠁⣠⣿⣧⡀⠛⠟⠁⠘⣿⠀⢹⡟⠻⣶⣶⣶⠶⠿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁⣾⠏⢠⣿⠃⠀⠀⠀⠀⠀⢰⠿⠋⣿⠛⢿⡆⠀⠀⠀⠀⠀⢸⡇⠀⢻⣟⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⣼⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⡀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⣠⣿⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⣿⣄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢸⣿⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⢸⣿⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠃⠈⠀⠄⠒
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠡⡌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⡈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠰⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡇⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠠⠦⠠⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⡆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⣠⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⠀⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠈⣀⠁⠤⠂⠁
⠀⠀⠀⠀⠐⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⠉⠀⠂
⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠀⠘⡀
⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠘⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠴⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠂⠀⠀⡀⠀⠂⠀⠀⢄⡴⣤⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⣿⣓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠦⠤⢄⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠐⠠⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠐⠀⠐⠵⠶⢄⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠈⢀⡀⠛⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢔⡠⠆⡠⣀⢄⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠞⢁⡄⠏⢠⢄⢤⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡆⣠⢠⡀⣉⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢰⠇⠀⠠⠸⠟⠽⣈⡀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣱⢀⡁⢍⣬⣙⢢
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠽⠡⠠⣤⡀⡀⠀⢀⢀⠀⠀⠑⡴⠊⠐⠔⠂⠀⣷⣭⡥⠄⡛⠚⠿⠻⠤⠐⠂⠀⠂⠐⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⣀⣒⠓⠧⠗⠀⠾⠿⡇⠀⡧⣿⣏⣷⠁⠈⠃⠀⠿⠓⠚⣇⢀⠠⠀⠦⠔⠐⠒⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠸⠒⠀⠀⠐⠒⠸⠔⠒⠢⣼⠀⠐⡆⡠⠴⠊⠋⠉⠉⠐⠀⠀⠀⠓⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠺⣄⠤⢄⣀⣀⠀⠀⣀⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⠀⠀⠀⠀⡤⢤⠊⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⡡⠀⠀⢀⡤⠖⠋⠉⡀⠄⠊⢹⠀⠀⠀⣠⠓⠀⠀⢉⡝⠀⠀⢠⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠢⠃⠀⢴⡁⠤⠀⠀⠉⠀⠀⠀⢺⠤⠤⡄⠈⠙⣅⠴⠤⠇⠀⠀⢗⡀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢀⠠⠊⡠⠾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢠⠂⢠⠞⠑⠐⢂⠈⠀⠀⠁⠀⠰⡆⠙⣆⠀⢀⡤⢄⡠⠖⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⡗⠈⠉⠀⡀⠘⠁⠄⡐⠆⠀⠀⠘⡆⢸⡇⠈⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠐⡈⠀⠀⠜⠀⠀⠀⢠⠇⡞⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠲⠄⠀⠀⠀⡜⠦⢀⠀⠀⠸⠷⠖⠀⢀⠀⡠⠾⠚⠁⠠⡈⢀⡶⠉⠀⠆⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣠⡛⠀⡠⠖⠓⢄⠀⠉⠀⠒⠀⠒⠣⡄⠀⠈⢆⠀⠀⠀⠀⡼⠁⠀⠈⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠑⠠⠤⠀⠀⠄⠰⡄⠀⠈⠑⢄⢀⡜⠁⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡴⠠⠄⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⢻⡀⢀⠶⠉⠀⠚⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠈⢀⠀⠀⠀⠈⠣⡀⠀⠀⠈⠂⠀⢀⢴⠃⠀⠲⡂⣠⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠂⠀⣟⠀⠁⠀⠢⢨⡀⠀⠀⠀⠀⡄⠀⠀⠈⢀⣈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠊⠀⢹⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊⠀⠀⠀⠀⠻⠟⠀⡘⠛⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣤⡀⠀⠑⢄⣀⠠⠀⠀⠤⢀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⡤⢊⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠄⠀⠀⢴⠁⠀⡞⠁⠀⠀⠀⠀⠀⠑⣤⠤⠔⠋⠙⠦⠤⠴⠚⠉⠰⠶⠙⠋⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠀⠖⠂⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⡀⠀⠄⠀⠀⠀⠀⠜⡖⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠋⠉⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠰⠀⢀⠀⢀⠄⠀⠀⠰⠙⠙⠄⠀⠀⡈⠀⠀⠂⠀⠀⠠⡀⠰⠀⠀⠀⠀⣵⡮⠀⠀⠀⠒⢝⠟⠦⠀⠀⠀⠰⡷⠀⠀⢀⢀⠥⠤⠁
⠀⠰⠌⠃⠈⠰⢃⡀⠰⠀⠀⢀⡀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠤⢆⡁⡙⠁⠀⠀⠊⠁⠀⠀⠀⠄⠈⠀⠀⠀⠉⢀⡀⣀⠀⠀⠀⡀⠢⠞
⠐⠒⣂⠰⠱⠀⠥⠄⢀⠀⠀⠘⠊⠀⠀⠀⠀⠄⣌⡐⡀⠄⠀⠐⠀⠈⠀⠤⠀⠀⣄⢀⠂⠌⠀⠀⠀⠀⠀⠠⠄⠴⠀⠂⠀⠀⠁⣀⠈
⠀⠉⡴⢠⠀⡌⠲⠀⠀⠀⠀⠀⢠⠀⠀⠠⠄⠘⡛⠣⢀⠀⠀⢀⣢⣔⡃⠀⠀⠠⠐⠀⡰⠤⠀⠀⠐⠀⠀⠀⠈⠂⠈⠀⠐⠂⠀⠉⢀⡀
⠀⠊⠀⠈⠀⠨⠀⠀⠀⠈⡓⠐⡈⠄⡀⠀⠀⠀⠁⠀⠈⠀⠀⠑⡿⡾⠃⠁⠀⠠⠀⠀⠲⠒⠁⠠⠐⠀⠀⣤⣾⣤⠂⠀⠴⢀⠀⠀⣀
⢶⣁⡀⠀⣁⣤⣀⠀⠀⠤⢈⠜⢛⢒⠂⠀⠀⠀⠀⠀⠀⡄⠀⣀⠁⠀⢁⠀⠀⠀⠀⠀⢀⠂⠈⠈⠃⠀⠀⠼⠻⡟⠃⠀⠀⠀⠃⠀⠉⠀⠤
⠘⠀⠀⠀⠿⣿⠿⠀⢀⡄⠀⠸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⣁⠂⠄⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡧⠀⠀⠚⠀⠐
⠀⠀⠇⠀⠀⠉⠠⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡀⠻⠷⠂⡃⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠄⠁⠀⠀⠒⠀⠀⠀⠀⠒⡘⢠⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠃⠂⠡⠀⠀⠀⠀⠨⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⡄⠂
⠐⠀⠂⢤⠀⠰⠆⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢁⠈⠀⠈⠉
⠘⠁⠀⠀⠀⢀⢀⠀⠸⠿⠀⠀⠠⠂⡄⢀⠀⠀⠀⠀⠀⠠⢼⣾⣁⠀⠀⠈⠛⠁⠀⠀⠄⠀⢀⠠⠀⠀⠁⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠛
⡀⠀⠠⠀⠀⠐⠂⠄⠀⢀⠀⠀⠃⣃⠀⢨⢀⠀⠀⠀⠀⠀⠋⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠
⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠁⠀⠀⠁
⠀⠀⠀⠀⠰⠗⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⡀⠀⠀⠌
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠁⠀⡀⠀⠀⠀⠀⠂⠀⠀⡀⠃⠠⠀⠀⠀⠠⣶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠩⠀⠐
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣼⠛⣷⠤⠄⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠂⠀⠀⠀⠀⠀⢠⣀⢶⠀⠀⠀⢠⠀⠀⠀⠶⠀⠈⠀⠀⠁⠐

⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡐⣤⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡆⠐⠀⠂
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⣠⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠂⠀⠀⠀⠀⠀⠀⢠⠌⠒⠊⣠⡀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠡⢬⠈⠀⠁⠀⠀⠀⠀⠀⡰⠅⠴
⠀⠀⠀⠀⢙⢂⡈⡤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰
⠀⠀⠀⠈⠉⡰⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⢁⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠺⠆⠠⠄⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠃⠠⠠⣃⠐⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⢂⠐⠀⠒⡠⠀⠀⠀⠀⠀⠀⠔⠙⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢐⠦⡨⡰⠚⠀⠀⠀⠀⠀⠀⠄⡤⠀⠌⣅⠁⠀⠀⠀⠈⠁⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢬⠘⠂⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠒⢠⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡠⠀⠂⠈⠅⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⣈⠀⠁⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠿⠢⠂⠐⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠡⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠱
⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠱⡀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⡀
⢊⣰⠄⠤⠤⠀⠀⠀⠈⠉⢉⡤⣄⠓⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⢚⠃⡉⠁⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⢀⠄⠉⠀⠀⠀⠀⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠄⣀⣀⣣⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢦⠘⡠⢀⠀⠀⠀⠀⠀⠈⢨⢶⡠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠁⠁⠁⠈
⠀⡀⢘⠀⡠⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠬⠀⠀⠀⠀⠠⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠈⠤
⢀⠄⢫⠣⣄⠀⠀⠀⠀⠀⠈⠡⠉⠄⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠂⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣄⣀⣂⣀⣀⣀⣀⣀⣊⣀⣀⣀⣂⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣁⣀⣈⣀⣈⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣿⢠⡿⠇⠀⠀⣰⣌⣿⣵⡧⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣎⣉⡂⣖⣉⡇⠀⠀⠀⠀⠉⢉⡉⡍⠉⠀⠀⠀⠀⠀⠀⠀⡥⠨⠇⠯⠀⡇⠀⠀⠀⣿
⠶⠀⠀⠀⠀⠀⠀⠘⢣⢾⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⡄⡤⠀⡇⠀⠀⠀⠀⢀⣄⠘⣧⡠⠀⠀⠀⠀⠀⠀⠀⡓⠐⠂⡖⠒⡇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠧⠧⠔⠁⠀⠀⠀⠾⠀⠿⢄⠹⠧⠠⠀⠀⠀⠀⠀⠀⠈⠒⠓⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⣄⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠂⠀⠀⠀⣠⡄⢦⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⡆⠀⠀⠀⣹⡮⢯⢴⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣻⢯⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⣩⠉⠁⠀⠀⠀⠘⠱⠿⠳⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠉⠁⠀⣠⣤⣤⣤⣤⡀
⠀⠀⢤⡄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣉⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣃⣀⣀⠀⠀⠀⠀⠀⢠⢄⢤⡤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⡛⣛⠛⠃
⠀⢨⣉⡷⣛⡓⠀⠀⠀⠀⠀⠀⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇⠀⠀⠀⠀⡸⠊⡅⢨⠕⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠓⠀⠀⠄
⠀⠀⠙⠀⠛⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣼⣃⢀⠀⠀⠀⠀⣿⢀⡃⢈⣀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢹⣯⠈⣿⡆⡂⠀⠀⠀⠡⣀⣤⣠⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣠⠀⠀⠀⠀⠀⠀⠈⠉⡛⠟⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢄⡤⠠⡀⠀⠀⠠⣷⢼⠵⣯⠄⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣁⣀⣤⡀
⠀⠀⠀⢠⣶⡄⠀⠀⡸⠮⡁⣨⠵⡇⠀⠀⠈⠱⢺⠳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇
⣴⣶⣶⣶⣷⠀⠀⠀⣗⢒⡂⢓⡒⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠉⡾⠁
⠈⢩⡍⡍⠉⠀⠀⠀⢧⣀⣁⣇⢀⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡠⠤⢄⠤⠤⡀⠀⠨⠟⣴⣋⡉⠀⠀⠀⢀⡆⢹⣇⠘⣿⡀⠀⠀⠀⠰⣾⣘⡾⠤
⣦⠘⣯⡄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣻⡋⠀⠀⠀⡕⠫⡁⡩⠲⡅⠀⠈⠿⠇⠹⠓⠀⠀⠀⠀⠀⠈⠉⢤⣭⡭⠁⠀⠀⠨⢤⡟⢷⠏
⡟⠸⠿⠇⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠇⠀⠀⠀⣗⣒⡃⣓⣒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈
⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠿⠜⠀⢀⠀⠀⠀⢧⣀⣇⣃⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣧⢸⣾⠄⠀⠀⠀⣿⠇⣿⡿⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡩⣭⡛⣶⡒⠀⠀⠀⢩⣤⣭⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⡿⢿⠷⠀⠀⢘⠬⡂⠑⡃⢜⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣴⡂⠀⠀⠀⢸⠤⠼⠸⠤⢼⠀⠀⠀⢀⣠⣀⣙⣟
⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢤⣤⠄⠀⠀⠀⢸⡀⢹⢸⠉⣹⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⣠
⢾⣙⡂⠀⠀⠀⢀⡀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠄⣨⣷⣉⡀⡀⠀⠀⠠⢭
⠊⠓⠀⠀⠀⠀⢰⠼⡊⢑⠣⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⣿⡇⢰⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡷⠠⠇⠸⠄⢶⠀⠀⠀⢻⣟⣀⣠⣀⠀⠀⠀⢀⡚⢿⣰⣥⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢨⠉⠛⡉⠀⠀⠀⠆
⠀⠀⣠⣤⠀⠀⢋⠀⣃⢘⠁⡙⠀⠀⠀⠹⠿⠿⠿⠿⠃⠀⠀⠐⣻⢩⢷⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣙⣟⣡⠀⠀⠀⠀⠂⠀⠀⠐⠂⠈
⣦⣤⣼⣏⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡀⠀⢣⣟⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⣩⣯⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣛⢛⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣿⡆⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠁⠁
⢜⣂⢀⣠⡀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠉⠁⣬⣭⡍⠀⠀⠀⠀⢔⢠⡒⣢⢖⡆⠀⠀⠀⠀⣽⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢪⡱⠝⠵⣫⠃⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣀
⠀⠀⠀⠀⠀⠰⣿⡆⠀⡇⠀⠷⠛⢀⡀⠀⠀⠀⠀⠴⠀⢀⡓⣦⣀⡬⢀⡄⠀⠀⠀⠀⠀⠀⢤⠀⠒⠀⡠⠀⠀⠀⠀⠀⠀⠀⢆⡔⢰⠂⠫⠴⢂⢠⡤⡀
⠀⠀⠀⠀⣠⣄⠀⠈⣶⠓⢦⠤⠒⠛⠋⠀⠀⠀⠤⠒⠪⠕⠀⠁⠵⡋⠀⣀⠀⠀⠀⠀⠀⢁⠀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣌⠙⠀⣄⠀⠚⢍⠂
⠀⠀⠀⠀⠙⠋⠉⠉⠙⠲⠛⠠⢄⣠⡄⠀⠀⠀⢀⡤⢐⡫⢴⡤⡟⠮⣁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⢐⡬⡖⠀⡀⠀⡦⣎⡄
⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⡤⣄⠀⠙⠃⠀⠀⠀⠀⠀⡨⠀⢠⡑⠦⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢋⣗⠒⠃⠁⠘
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠐⢶⠀⠀⠃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⣴⡆⠀⠀⣠⣀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠂⠄⠀⠀⠂⠀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⣤⠀⡇⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠂⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢾⡳⠀⠀⠀⠀⠀⠀⠀⢠⢢⠤⠤⠤⢴⣷⠤⠤⠤⢄⡀⠀⠀⠀⠀⠄⠄⠄⠠⡆⠡⠠⠠⠀⠄
⠀⠀⠀⠀⠴⡲⠀⠀⠏⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠁⠀⠀⠶⠀⢾⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡄⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡔⠆⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢷⠃⠁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⣼⡦⣴⠛⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣴⣄⠤⡮⢿⣿⢹⡗⠒⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣢⣧⠞⣁⣂
⠀⠀⠀⠀⠈⠀⠀⠀⢀⠃⡀⠀⠀⠀⠀⠀⠀⠀⢉⠀⣠⡽⠿⡶⠛⢄⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣤⠊⡛⢬⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⠀⢙⡁⢠⣧⠀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢚⠡⢼⣗⠮⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠢⡅⠀⡀⢠⠀⠀⠀⠀⠀⠀⠛⢀⣀⣤⢄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠿⢆⠀⠁⠀⡠⠟
⠀⠀⠀⠀⠀⠤⠾⢦⡀⡏⣠⡾⠗⠁⠀⠀⠀⠀⠀⠀⠩⠕⠃⢠⡅⠑⠋⠉⠁⠀⠀⠀⢀⢄⠀⢴⠁⠀⠈⢡⡀⢀⡀⠀⠀⠀⠀⣤⣄⣀⣡⠶⣤⡡⣴⠄
⠀⠀⠀⠀⢀⢀⣠⡤⠞⡟⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣈⠱⠖⠌⡁⠰⠶⣂⡀⠀⠀⠀⠀⠀⠀⠸⠢⡠⡰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠌⢻⠑⢄
⠀⠀⠀⠀⠀⠀⠚⠀⢰⣧⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠊⡗⠄⢀⠸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠶⠁⠀⢸⠀⠀⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣿⡇⠀⠀⢸⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠻⣿⣿⡇⠀⠀⢸⣿⣿⠟⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠈⢿⡇⠀⠀⢸⡿⠁⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠈⢿⣷⡀⠀⠀⠁⠀⠀⠈⠀⠀⢀⣾⡟⠁⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄⢰⣿⠋⠀⠙⣿⣆⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⣼⣿⠋⠀⠙⣿⡆⣠⣤⣶⣤⣄
⠀⠀⠀⠀⢰⣿⠋⠉⠙⠻⢿⣿⣧⠀⠀⠘⣿⣆⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⣰⣿⠃⠀⠀⣼⣿⡿⠟⠋⠉⠙⣿⡆
⠀⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠈⠛⠧⠀⠀⠘⣿⣆⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⠀⣰⣿⠃⠀⠀⠼⠛⠉⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⢉⣻⣿⣶⣤⡀⠀⠀⠀⠀⠀⠘⣿⣆⡀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢀⣰⣿⠃⠀⠀⠀⠀⠀⢀⣤⣶⣿⣟⠉
⠀⠀⠀⠀⠀⢠⣿⡟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣀⣿⡇⠀⠀⢸⣿⣀⣤⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣿⡀
⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠈⠙⠻⣿⡇⠀⠀⢸⣿⠟⠋⠁⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠿⣷⣦⣀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⣠⣴⣾⠿⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⠖⠀⠀⠀⠀⠀⠀⠲⣿⣿⣏⡁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣶⡿⠟⠉⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠉⠻⢿⣶⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢠⣿⠟⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠁⠀⠀⢀⣠⣴⣿⡇⠀⠀⢸⣿⣦⣄⡀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⢻⣿⡄
⠀⠀⠀⠀⠀⠈⣿⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⣿⡇⠀⠀⢸⣿⠉⠛⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣽⣿⠿⠛⠁⠀⠀⠀⠀⠀⢠⣿⠏⠁⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠈⠹⣿⡄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣯⣄
⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⣀⣤⡖⠀⠀⢠⣿⠏⠀⠀⠀⠀⢀⣿⡇⠀⠀⢸⣿⡀⠀⠀⠀⠀⠹⣿⡄⠀⠀⢲⣤⣀⠀⠀⠀⠙⠻⣷⡄
⠀⠀⠀⠀⠸⣿⣄⣀⣠⣴⣾⣿⡟⠀⠀⢠⣿⠏⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠹⣿⡄⠀⠀⢻⣿⣷⣦⣄⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠉⠛⠿⠛⠉⠸⣿⣄⠀⣠⣿⡏⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⢹⣿⣄⠀⣠⣿⠇⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⢀⣾⡿⠃⠀⠀⡀⠀⠀⢀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠙⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢀⣾⡇⠀⠀⢸⣷⡀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣴⣿⣿⡇⠀⠀⢸⣿⣿⣦⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⡇⠀⠀⢸⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⢿⡿⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣀⠀⠙⢿⡿⠋⠀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢻⣿⣿⡄⣿⢸⡇⣿⢠⣿⣿⡟⠀⠀⠀⠀⢠⡄
⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⣬⣭⣿⣿⣸⣇⣿⣿⣭⣥⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⢀⣀
⠈⢿⣿⣿⣦⢹⡇⠀⠹⣿⣿⢠⡦⢸⣷⣄⡀⠀⠈⢻⡟⠁⠀⢀⣠⣶⡇⢴⡀⣿⣿⠏⠀⢸⡇⣴⣿⣿⡿
⠀⠀⠙⣛⣛⣿⣿⣜⡻⢾⣿⡿⠁⠘⣿⣿⣇⢠⡆⢸⡇⢰⡄⣸⣿⣿⠃⠈⢿⣿⡷⢟⣣⣿⣿⣛⣛⠉
⠀⠀⠀⠉⠉⠀⢶⣭⣛⢶⣽⣇⠀⠠⣬⣝⣿⣿⠁⢸⡇⠈⣿⣿⣫⣥⠄⠀⣸⣯⡶⣛⣭⠶⠀⠉⠉
⠀⠀⠀⣠⣾⣿⣶⡾⣿⠟⠛⠙⠻⣦⣄⡈⠉⢿⡄⢸⡇⢠⡿⠉⢉⣠⣴⠞⠋⠙⠻⣿⢷⣶⣿⣷⣄
⠀⠀⠉⠛⠛⠛⠋⢼⠏⢀⣀⠐⢶⣄⠉⠛⢶⣤⣻⣾⣷⣟⣤⡶⠛⠉⣠⡶⠂⣀⡀⠹⡧⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡷⣿⡷⠶⠶⢾⣿⣿⣿⣿⡷⠶⠶⢾⣿⢾⣿⣿⣿⡷⠆
⠀⠀⣀⣤⣤⣤⣄⢺⣆⠈⠉⠡⠾⠋⣀⣤⡶⠛⣿⢿⡿⣯⠛⢶⣤⣀⠙⢷⠌⠉⠁⣰⡗⣠⣤⣤⣄⣀
⠀⠀⠈⠙⢿⣿⡿⢷⣿⣦⣄⣠⣴⠞⠋⢁⣀⣾⠃⢸⡇⠘⣷⣀⡈⠙⠳⣦⣄⣠⣴⣿⡾⢿⣿⡿⠋
⠀⠀⠀⣀⣀⠀⠶⢛⣭⡾⣿⡏⠀⠐⠛⣫⣿⣿⡀⢸⡇⢀⣿⣿⣝⠛⠂⠀⢹⣿⢷⣭⡛⠶⠀⣀⣀
⠀⠀⣀⣭⣽⣿⣿⢋⣵⢾⣿⣷⡀⢰⣿⣿⡟⠸⠇⢸⡇⠸⠇⢻⣿⣿⡄⢀⣾⣿⡷⣮⡙⣿⣿⣯⣭⣀
⢀⣼⣿⣿⠟⣹⡇⠀⣠⣿⣿⠘⠗⢸⡿⠋⠁⠀⢀⣼⣧⡀⠀⠈⠙⢿⡇⠺⠃⣿⣿⣄⠀⢸⣏⠻⣿⣿⣧
⠉⠉⠉⠁⠀⠈⠀⠀⢹⣿⠟⠁⠀⠀⠀⢛⣛⣿⣿⢹⡏⣿⣿⣛⡛⠀⠀⠀⠈⠻⣿⡏⠀⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣼⣿⣿⠇⣿⢸⡇⣿⠸⣿⣿⣷⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠉⠀⣠⣾⣷⣄⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣾⣷⡙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀⣤⣀⣤⣤⣤⣄⣀⣀
⠀⢀⣀⣤⣤⣤⣤⠶⠶⠶⠶⢶⣶⡶⠿⠛⠛⠋⠉⠛⢻⣿⣿⠿⣿⣿⣋⣉⣉⣹⣿⣭⡿⠟⢋⣩⠿⣷⣶⢾⣿⣿⣿⣷⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⣟⠿⠿⢷⣤⣤⣤⣾⣯⣀⣀⣀⣀⣀⣤⡴⣾⣧⣤⣶⣿⣿⣿⣭⣭⠷⣾⣿⣶⠾⠉⠁⢀⣸⣷⠿⠛⠉⢸⣿⣶⠟⢻⡿⠁⠀⠀⠀⠉⢻⣷⣦⣄
⠀⣿⡟⣉⠛⠿⣾⣷⣌⣁⣽⣿⡏⠉⠉⠉⠉⠉⠀⠀⠈⣿⠁⠀⢀⣀⠉⠉⢀⣲⠄⣹⣿⣶⡶⠾⠛⠋⠁⠀⠀⠀⢸⣿⡁⠀⠘⣧⠘⠀⠀⠀⢠⡾⠃⠀⠈⠳⣄
⠀⣿⠠⠉⠀⠀⠀⠀⠉⠉⠙⠛⠻⠷⠶⣶⣶⣶⣤⣤⣼⣷⣤⣤⣬⣵⠶⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢵⢀⣠⣿⣷⣤⣄⣤⡟⠁⠀⠀⠀⠀⢹
⢰⣿⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣤⠔⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⠀⢸⣿⡷⠛⠉⠀⠀⠀⠈⢻⡁⣄
⢸⡏⠈⠀⣴⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠞⠁⠀⠀⠀⣤⣄⡀⠀⢄⠀⡈⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠟⠛⠒⠚⠻
⢸⡇⠀⠀⠋⠈⠁⠀⠀⠐⣿⡅⠀⠀⠀⢀⠏⢁⣤⣾⣿⠀⣿⣿⠚⣦⠐⣿⣷⢤⣀⣤⠀⠈⠉⠛⢪⣿⣦⡂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣹⡐⠁⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣼⢷⡀⡜⠀⣿⢁⣸⣿⠀⢻⡏⠻⠟⠁⣿⡇⢸⡇⢸⠀⠀⠀⠀⢡⠉⢿⠉⠿⣧⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀⡀⢀⣾
⢸⡇⠀⢀⣀⡀⠀⠀⠰⠆⠀⡿⠇⠈⠿⠇⠀⠉⠉⣹⣿⠀⠿⠿⠀⠀⠀⠿⠧⠘⠃⠘⠀⠀⠀⠠⢧⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⠗⠎⠉⠙⠻⣿⠇
⢸⡇⣰⡿⠋⢻⡄⠀⠀⡀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢠⡎⠉⢳⠄⣿⡇⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢠⠟
⢸⡇⠘⠳⠶⠟⠀⠘⣿⡇⠀⠀⢨⡋⠻⠛⠀⣴⣶⣀⠀⣼⣿⠧⠀⢀⣴⢦⠄⠀⣾⣧⠴⢶⡄⠀⠀⠘⠷⠖⠋⠀⣿⠁⠀⠀⠀⠀⠀⣰⠿⣦⡀⠀⠀⣰⠏
⢸⡇⣤⠀⠀⠀⠀⠀⢻⣧⢀⡄⢸⠀⣿⡟⠀⣾⡿⢹⠇⣿⣇⡴⡄⣾⡟⠉⢠⠀⣿⡏⣶⡤⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠏⠀⠀⠙⢷⣴⠏
⣸⡇⠀⠀⠀⠀⠀⠀⠈⣿⡏⠹⣾⡀⣿⡏⠀⣹⡟⢸⡀⢿⣇⣃⠇⠸⠿⠾⠃⠀⣿⡇⠀⠀⠀⠀⠐⣶⣧⠄⠀⠀⣿⠀⠀⠀⣠⣾⠡⠀⠀⠀⢀⣼⠏
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠀⠙⠉⠈⠃⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⢀⣠⣀⠀⠀⠀⠹⠄⠀⠀⣿⠀⢀⡼⠋⠙⠷⣤⣀⣠⠞⠁
⣿⡇⠀⠀⢠⣄⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⣠⠀⣶⠄⣴⠞⠙⢳⣴⡶⢀⣴⠻⡆⣼⡏⢠⣽⠇⠀⠀⠀⠀⠀⠀⣿⣴⡟⠀⠀⠀⠀⢨⡿⠃
⢻⡇⠀⠀⠠⣤⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⡏⢠⣤⡄⣿⣆⠓⠋⢹⡇⢸⣿⠞⡁⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣌⠀⠀⢀⣴⠋
⢸⡇⠀⠀⠀⠈⣃⠀⠀⠀⠀⠈⣿⣀⢿⡅⢸⠀⠀⣿⠀⠈⢿⣧⠀⣿⣧⣄⠛⠛⣁⡀⢹⣿⠀⡀⠀⠀⠀⠀⠀⢸⣿⠀⠉⣷⣤⠞⠁
⢸⡇⠀⠀⠀⢠⣴⣞⠀⠠⣤⣤⣿⡏⠀⢻⣧⠀⢀⣿⡌⢿⣤⡿⠀⣿⡇⠈⡇⠀⠹⠶⠾⠃⡀⠀⠀⠀⠀⠀⠀⢸⣯⠀⣠⡾⠋
⢸⡇⢤⢿⠒⠃⠉⢻⡧⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐⠛⠋⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋
⢸⣷⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⢀⡶⠛⠛⢆⠀⠀⠀⠀⠀⠀⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⣿⡄⠂⠀⠀⠀⠀⠟⢉⡀⠀⠀⠀⠸⣧⣤⣴⠏⠀⠀⠀⠀⠀⠀⠙⠁⠄⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄⠀⢀⣼⣿⠁
⠀⠙⣿⣄⡋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠞⣁⣴⣾⣿⠃
⠀⠀⠈⠻⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⡶⠿⠛⠻⢿⡿
⠀⠀⠀⠀⠈⢻⣯⡟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣭⣉⣉⣅⢀⣀⣀⣸⣵⣿⠇
⠀⠀⠀⠀⠀⠘⠻⠿⠿⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠾⠶⠿⠟⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡬⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣑⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠯⠭⣭⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠤⠽⠟⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⣀⠀⠀⠀⠠⠤⠬⠭⠭⠭⠿⠿⠿⠿⠷⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⡏⢰⣼⠇⣿⢼⣷⢸⡏⣷⢰⣿⢰⣷⣤⣸⡟⠃⢿⡶⣿⠂⡐⣿⡄⢾⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣤⠔⣛⣘⣛⠘⣛⣓⢘⣛⢘⣚⣛⡙⢃⣀⣊⡁⢛⣂⣃⠹⣷⠛⣺⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⢾⣽⠽⣿⡄⢿⡟⢸⣿⢨⣿⡿⣿⣸⣿⠤⠴⣿⣦⡙⠀⠙⠳⠤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣧⢸⠃⢘⣛⣘⠻⠗⠛⠙⢛⡒⢂⣉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⣼⣿⣏⠉⢉⡢⠐⠀⠨⠟⠸⣧⣡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠟⠛⣉⠀⡄⠐⠐⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠻⠟⠃⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠬⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢉⣉⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠲⠤⣄⡀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣞⡏⠀⠀⠀⠀⡦⠀⠀⠰⡆⠀⠀⠀⠀⡷⠓⠋
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢰⠋⢒⣒⣶⠄⠀⠀⡇
⠀⠀⠀⠲⠶⠶⢿⣓⠦⣀⢀⡟⠀⠀⠀⠹⡄⠀⠀⠀⠀⢤⣉⠅⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⢀⣀⣀⣀⣉⡛⣽⣇⡀⠀⠀⢰⠋⠒⠢⠤⠤⣤⣤⣤⡤⠤⠒⠋⠈⢣⠀⠀⠀⠀⢹⣀⡠⣖⡫⠽⠛⠛⠛
⠀⠀⠀⠀⠉⠉⣩⡿⠿⠚⠛⠣⠽⣒⠤⣎⣳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⢼⣿⣫⣥⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠤⠤⠞⠛⠉⠀⡠⡟⠉⢹⠒⠒⠤⠤⠤⠤⠤⠤⠔⢿⣒⡪⠕⠚⠋⠙⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣀⠴⠊⡼⠁⠀⢸⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡖⠉⠁⢀⠜⠁⠀⢀⠏⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣀⠴⠋⠀⠀⢠⠎⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠉⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠚⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠐⢿⠆⠀⡀⣀⣀⠀⠀⠀⢀⣐⣆⠙⢿⡿⠋⣴⣂⠀⠀⠀⢠⣀⣀⠐⣤⢹⡏⣤⡆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣤⣿⠿⢿⣧⠷⡟⠁⠀⠀⠀⣬⠿⠿⣦⣼⣧⣴⠿⢿⣥⠀⠀⠀⠹⠟⢦⣽⣿⣿⣿⡴⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣧⣀⣸⣿⢃⡀⠀⠀⠀⠀⠶⣦⣤⡾⢿⡿⢷⣤⣴⠶⠀⠀⠀⢐⠛⢻⣿⣿⣿⣿⣟⠛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠻⠉⠟⣻⠻⠃⠟⠾⣤⠀⠀⠀⠈⢛⡟⢀⣼⣧⡀⢿⡛⠁⠀⠀⢠⣾⢿⠋⣿⢻⡟⣿⡍⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢾⣷⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⢠⣙⣟⣡⡀⣀⣠⡄⠀⠀⢤⣄⡀⠀⠀⢺⠊⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⣦⣈⣏⣸⣶⠛⠛⠀⠀⠀⠀⠨⠛⢢⣖⣹⣑⣦⠚⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⣤⠿⡿⢷⣴⣋⡀⠀⠀⠀⠀⠀⣀⣠⡾⢿⠿⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠏⠱⠔⣟⠲⠏⠻⠿⠆⠀⠀⠴⠞⠛⠁⠈⣹⡉⠀⠉⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠈⢾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣤⡀⠀⢹⡏⠀⢀⣤⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢦⣼⣧⠴⠛⠛⠁
⠀⢠⢠⣀⢀⣀⡀⣀⣠⣀⡀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠖⢻⡟⠲⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣸⣇⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂
⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡶⠄⠀⠀⠀⠀⠀⠀⠨⠄⠀⢉⡀⠀⠀⠀⠀⣠⠾⣶⠟
⠀⠀⠀⡸⢴⡒⠒⠒⢢⠀⠀⠀⠀⠀⢨⣭⣿⣿⣭⡄⠀⠀⠀⠀⠠⠄⠤⠄⠀⣀⡀⠀⠀⠀⣴⡯⢖⣺⠟⠁
⠀⠀⠸⣁⡠⠬⠽⠛⠥⣄⠀⠀⠀⢀⣒⣛⣿⡷⠶⠒⠀⠀⠀⠠⠤⠀⠀⠀⠀⢀⣀⡀⠀⠈⣡⠖⣋⣤⠶⣺⠇
⠀⢠⠊⠉⠑⠒⠢⠤⠤⠚⠀⠀⠠⠭⠭⠿⣿⣿⣭⣭⣥⠄⠀⢀⣀⠀⠀⣀⠀⠀⠀⡀⢠⣾⠷⣚⡭⠖⠋⠁
⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠚⠛⠛⢻⠛⠒⠒⠒⠒⠊⠁⠀⠀⠀⣼⠀⠀⠀⠀⣠⣶⣯⠥⠖⣾⠉
⠀⠀⡤⠭⠿⠾⠿⠶⠶⡤⡤⠤⠤⠤⠤⠤⠿⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠿⠶⠦⠤⠤⠤⠤⠤⠤⠴⠧⡄
⠀⠀⡇⠀⠀⠀⡠⠊⠁⠀⡌⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⢠⠊⠀⠀⠀⠈⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⢀⠀⡮⠲⡀⢴⠖⠐⠠⡂⠀⢲⠄⡰⠃⡴⠐⠆⢲⠀⣴⠰⡆⠀⣦⠀⠂⠀⡇
⠀⠀⡇⠸⡀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡜⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⡀⢰⠁⠀⠀⢠⠎⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡰⠃⠰⠁⢠⡾⠼⠇⠰⡂⠠⣺⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⢰⡏⠉⠻⠾⠉⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣰⣧⡴⠖⠛⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠛⠲⢦⣿⡀⠀⠀⠀⠀⠀⠙⠶⠤⠾⠋
⠀⠀⠀⠀⠀⢠⡖⠲⢶⡟⠉⢹⡆⠀⢀⡾⠋⠀⣀⣀⣤⣤⡤⠴⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣼⠃⢀⣼⠷⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠾⠁⠀⣿⡀⢀⣠⣤⠴⠶⠒⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⣀⣽⠀⠀⠀⣠⡴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⡄⠀⠀⡟⠀⠀⠹⣦⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣷⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣴⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣄⠀⠀⠀⠀⠀⢻⡀⠘⢧⣀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣿⣤⣿⠇⠀⢀⡴⠖⠶⣄⠀⠀⠀⢸⣧⣼⡿⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡉⠉⠀⠀⢸⡅⠀⠀⣹⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠙⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⣠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣀⢾⣷⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣄⡀⢸⡿⣦⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣾⣥⡿⠛⠳⠦⣤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⢦⣄⠀⣀⡴⠞⠁⠀⣈⣿⣾⣻⣮⣳⡾⠋
⠀⠀⠀⠀⠀⣾⠋⠛⣦⣾⡏⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠛⢦⣠⠴⠖⠶⠦⣼⠋⠀⠀⠀⠀⠈⢿⠉⠀⠀⢀⡾⠉⠀⠈⠻⣄⠉⠁
⠀⠀⠀⠀⠀⣻⣄⣀⣸⡟⣷⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠻⠿⠿⣿⠀⠹⡆⠀⠀⠀⠀⠀⠀⠻⢦⣿⣻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠸⣞⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⣄⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠳⣤⣀⡀⠀⠀⣀⣠⡼⠿⢿⡶⣤⣀⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠰⠟⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⣄⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠙⠳⣄⠀⣠⣿⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⠉⠙⠛⠃⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⣀⡴⠟⠋⠉⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠤⠶⠶⠚⠛⠛⠲⠞⠋⠛⠛⠛⠒⠒⠒⠲⠖⠒⠒⠒⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠋⠉⠛⠛⠓⠶⠶⠶⠚⠛⠉

⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⢟⡻⡿⢿⣿⢿⣻⣿⡿⠿⣻⠿⣟⢿⢟⡻⢿⣿⡿⣿⡿⡿⢛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣨⣠⣙⣣⣿⣿⣿⣜⣻⣏⣇⣜⣛⣤⣛⣻⣜⣣⣞⣿⣩⣼⣇⣿⣘⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡯⣿⣟⡩⣟⡯⣫⣿⢻⣿⡟⢿⠛⡿⢻⣟⢻⢻⣛⣿⠛⢻⠟⣿⢻⡿⣿⣿⡟⣟⠛⣿⣿⢛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣦⣵⣦⣧⣼⣧⣽⣼⣤⣶⣦⣵⣦⣧⣿⣽⣧⣭⣾⣤⣦⣿⣧⣿⣿⣿⣿⣧⣿⣿⣯⣿⣿⣿
⢸⣿⣿⣛⣩⣭⣶⠀⠀⠐⠒⠒⣶⡶⢭⣭⣛⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠛⠛⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿
⠈⠛⠛⠋⠉⠛⠛⠓⠚⠓⠒⠖⠛⠛⠛⠋⠉⡟⠛⡛⠛⠛⠃⠀⠀⠙⠛⠛⠛⢛⡛⠋⠁⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀⠠⠀⠀⠀⠀⢀⠇⠀⠯⣴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡻⢿⠿⠛⠁
⠀⠀⠀⠈⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠲⠶⠚⠛⠛⡋
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠶⠂⢀⣶⡶⠀⠡⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢀⠐⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠿⠤⠼⠿⠖⠘⠋⠁⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠉⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠈
⠀⠀⠀⠀⣠⡄⢶⠀⠀⢁⣀⣸⠿⠛⠀⡄⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⣀⠌⠀⣤⠀⣀⡀
⠀⠀⠀⠀⠉⠙⠛⠁⠈⠉⠁⠀⠀⠀⠀⢻⠀⣀⣤⢔⣐⣶⣤⠉⠙⢯⣭⣭⣐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠛⠉⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠃⠀⠀⢠⣪⣵⣦⣀⣤⣀⣹⣿⣶⣶⣾⣿⣿⠿⠀⠀⠀⣀⠀⠀⠀⠀⢀⠸⣧⠁⠀⠈⠹⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠄⠀⠖⢀⣀⡿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⣁⣀⣴⠀⠀⠈⠁⠀⠢⠀⠀⠘⠓⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⡀⠀⣾⣿⠟⢷⣤⣤⣤⣤⣴⣶⣶⣶⣿⠿⠟⠋⢉⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣧⠀⠁⠉⠉⠉⠉⠉⣉⣁⣀⣠⣤⣴⡆⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⠿⠿⠿⠿⠿⠿⠿⠛⠛⢉⡁⣴⠀⢸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣤⣤⣶⣶⣶⣶⣿⣿⠿⠇⠛⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⢋⣉⣉⣉⣉⣁⣤⣤⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠠⠀⢀⡀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠡⠂⠒⠒⠩⠬⠉⡆⠀⠀⠠⠆⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠓⢢⠉⢓⣐⣊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢈⢁⠟⠁⡀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠜⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠒⠀⠀⠄⠈⠀⠈⠒⠒⠂⢀⠀⠠⠀⠠⠀⠒⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠
⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠔⠒⠀⠐⠂⠀⡀⠀⠀⢀⣡⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠇⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡄⠀⠀⠀⠀⢀⡄⠤⢠⠈⠀⠀⠀⠉⠐⠂⠀⠀⠠⠠⠄⣼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⡀⠀⣀⠀⠀⢀
⠀⠀⠀⠐⠀⠀⢀⠠⡄⠀⠀⠀⠀⡠⠋⠀⢀⡀⠀⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⡆⣸⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⡐⠁⣀⠄⠒⠊⠩⠹⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⡜⢡⡋⣀⡤⠞⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡀⠴⠼⠁⠀⠀⠂⠀⡦⢀⣀⠀⠀⠘⠀⠀⡔⠈⣀⠒⠒⠲⠦⠴⢡⣾⡞⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠔⠈⠉⢉⡴⠄
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠰⠋⠁⠀⠀⠀⠀⠀⢠⠁⣼⠾⠺⠭⠶⡲⢂⣤⢤⢈⣀⠐⠂⢠⣀⠤⠑⠀⠀⠀⠉⠀⡀⠤⠊⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠌⠀⠀⢀⣀⠖⠂⢓⠋⠁⠂⠀⢀⣀⡀⠀⠀⡌⢠⡟⢀⠀⠀⠀⠀⣰⠏⠉⠐⠒⡯⠇⢰⡏⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠔⠃⠀⠔⠁⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠈⠑⢦⣠⠋⠠⠀⢀⠀⠀⣰⠃⠈⠀⠀⠀⠀⠀⣼⡟⠒⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠈⠳⠔⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠁⠀⠂⠈⠘⢁⠈⠀⢀⠀⠀⠆⢠⡿⠀⠤⠒⠀⣠⠤⠄⠀⠈⠐⠤⠤⢄⠀⠀⠀⠀⠀⠀⠐⡹
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠶⢄⠀⠀⢀⠀⠤⠀⠀⠀⡀⠀⠀⢀⠌⢠⣿⢗⠊⡴⠒⠚⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⢤⡀⠀⠀⡖⠉⠁
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠒⠢⢅⠀⠀⠀⠀⠀⠈⢲⡠⢄⡴⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⢶⡁
⠱⢀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠳⠦⣄⠀⠀⡠⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵
⠀⠀⠀⠉⠒⠒⠒⢦⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠳⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⢀⠀⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢈⠢⠤⠄⠐⠒⠤⢤⣠⡀⠠⠤⠔⠊⠐⠠⠐⠃⠒⠤⠤⠰⠒⠁⠩⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠶⡤⠤⣤⡤⡤⠠⠶⠤⠠⠤⠤⠒⠋⠈⠑⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⡺⣏⢖⣞⠀⠀⠀⠀⢀⡘⢽⢨⠝⣀⠀⠀⠀⠀⢠⣆⠹⠁⣤⠀⠀⠀⠀⠀⡄⡔⠭⡠⠤
⠀⠀⠀⠘⢻⢜⡿⡣⡾⠃⠀⠀⠀⠸⠟⣸⢻⡘⠟⠀⠀⠀⠀⢠⡤⠛⠃⣤⠀⠀⠀⠀⠀⣌⡔⣛⢄⣁
⠀⠀⠀⠀⠊⠙⠱⠩⠙⠀⠀⠀⠀⠀⠘⠚⠘⠚⠀⠀⠀⠀⠀⠀⠁⠚⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⡀⣿⢂⣀⡀⠀⠀⠀⠠⢦⡨⡯⡠⠦⠀⠀⠀⠀⠀⢦⡼⣥⠗⠀⠀⠀⠀⢠⣠⠀⣛⢀⣀⡀
⠀⠀⠀⠀⠙⣹⠤⣏⡉⠀⠀⠀⠀⢠⡤⠾⡿⠦⣤⠀⠀⠀⠀⠙⣹⢦⠼⣉⠃⠀⠀⠀⠀⠁⡹⠤⣁⠁
⠀⠀⠀⠀⠘⠋⠀⠙⠃⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠁⠘⠂⠁⠀⠀⠀⠀⠀⠐⠉⠀⠉⠂

⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡦⣱⠼⣌⢴⡆⠀⠀⠀⠘⠶⢄⣇⠴⠎⠀⠀⠀⠀⠛⠢⣺⣢⠚⠃⠀⠀⠀⠀⠳⠶⣿⠴⠞
⠀⠀⠀⠀⢐⡸⠶⢏⣀⠀⠀⠀⠀⠠⠳⢉⣏⠰⠧⠀⠀⠀⠀⠶⠊⢲⡊⠲⠆⠀⠀⠀⠀⠖⠛⣿⠛⠷
⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⢀⣀⠀⣿⠀⣀⡀⠀⠀⠀⢀⠀⢘⣕⠀⢀⠀⠀⠀⠀⠀⢀⠰⢂⠀⠀⠀⠀⠀⠀⢠⡀⠶⢠⡄
⠀⠀⠀⠀⢉⣻⠉⣙⣉⠀⠀⠀⠀⠛⠻⢢⢧⠾⠻⠀⠀⠀⠀⠛⣢⡟⣳⡒⠃⠀⠀⠀⠀⠦⢸⣓⡅⠴
⠀⠀⠀⠐⠋⠁⣿⠌⠙⠃⠀⠀⠀⠀⢞⡛⠘⣟⠆⠀⠀⠀⠀⠀⡴⠉⠑⠄⠀⠀⠀⠀⠀⠐⠃⠤⠸⠂
⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡏⠉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⣀⡼⡴⢃⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣛⣻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⣠⠏⠀⣏⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡤⠤⢤⣤⡼⠛⢩⣏⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣦⣶⣶⣶⣶⣶⣖⣒⣒⣒⣖⣾⣵⣾⣿⣶⣶⣬⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠁⠈⡇⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⣀⠀⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⢳⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠦⣀⣿⣠⠔⠃⠀⠀⠀⠀⠀⡗⠋⠀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠶⠒⣺⣿⣗⠒⠶⠀⠀⠀⠀⠀⣇⣀⡤⠏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡜⠁⣿⠈⠧⠀⠀⠀⠀⠀⠀⣇⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⡤⠖⢻⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠦⣤⣈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠁⣴⠋⠀⠀⠀⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⣠⠏⠀⠀⡈⠙⡆⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⢄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⣡⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠉⠓⠲⠦⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠊⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣄⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⡤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠤⠀⠀⠀⠀⠀⠐⠂⢠⠀⠀⠀⠀⠺⠧⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠈⠀⠀⠛⠀⠀⠀⠀⠀⠌⠡⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⣄⠀⠀⠐⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠀⠳⠀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠶
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⡀
⠀⠀⠀⠂⠈⠀⠂⠀⠀⠀⠂⠁⠐⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠈
⠀⠀⠀⠸⣤⡤⠀⠀⠀⠀⠰⣤⠇⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⡈⠀⡉
⠀⠀⠀⠰⠆⠰⠆⠀⠀⠀⠰⠄⠰⠄⠀⠀⠀⠺⠂⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙

⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠂⢀⠀⠀⠀⠀⠀⠄⠒⠀⠄⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠈⢀⣀⠈⠀⠀⠀⠀⠄⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⢀⠀⠀⠄
⠀⠁⠀⠂⠀⠂⠀⠁⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣰⠦⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⢀⡄⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠘⣁
⠀⠀⠀⠛⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠔⠀⠀⢦⠀⠀⠀⠀⠀⠴⠦⠴⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠄
⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢉⠄⠀⠀⠀⠀⢀
⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠐⠘⠁⠀⠀⠀⠂⠀⠀⠀⠈⠛
⠀⠀⠀⣄⠀⣤⠀⠀⠀⠀⠀⢠⡄⠠⣤⠀⠀⠀⠀⠀⠀⠰⠆⠐⠶⠀⠀⠀⠀⠀⠀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⢼⣧⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡁⣸⣇⢈⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⢸⡇⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠹⠇⣭⡀⠆⢸⠀⣷⠀⠀⠳⠦⢧⣄⠐⣿⣾⣷⣿⠆⣀⡼⡰⠖⠀⠀⣾⠀⡗⠰⢀⣬⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠝⠟⣶⣤⡁⣠⣄⡀⢸⡦⣟⣷⡖⣡⣌⢢⣾⣻⢵⡇⢀⣠⣄⢘⣤⣴⠿⠏⡅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀⣬⠉⣻⢾⡾⣿⠼⠿⡟⠋⡃⣿⣿⢈⠙⢻⠿⠧⣿⢷⡷⢟⠛⣥⡐⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠻⠾⠟⡄⣾⣷⣬⡲⣼⡾⢷⣧⢔⣥⣾⣷⢠⠻⠷⠟⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⠴⣶⣟⣷⠂⠬⣝⣿⠉⢀⡀⠉⣿⣫⠥⠐⣾⣻⣶⠦⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠺⠿⣯⡿⠄⢒⣫⣿⣀⠈⠁⣀⣿⣝⡒⠠⢿⣽⠿⠖⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⣴⢶⣦⠃⢿⡿⢛⠕⢻⢿⡾⡟⠮⡛⢿⡿⠘⣴⡶⣦⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠈⢛⣄⣵⢿⢷⣟⢲⡶⣧⣄⡅⣿⣿⢘⣠⣼⣶⡖⣿⡾⡿⣮⣀⡛⢁⠴⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢘⣢⣾⠟⠛⡅⠙⠋⠁⢸⢗⣯⡿⠧⡙⢋⠼⢿⣽⠺⡇⠈⠙⠋⢈⠛⠿⣶⣔⡂⢄
⠀⠀⠀⠀⠀⠀⢀⣴⡆⡛⠅⠆⢼⠀⡿⠀⠀⠴⠎⡞⠉⠰⣿⢿⡿⣿⠆⠙⢣⠑⢦⠀⠀⢿⠀⡆⠰⢨⢛⢴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⢸⡇⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡁⢹⡇⢈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡠⢻⡗⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠘⠃⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠛⣷⠀⠈⠉⠉⠛⠛⠛⠓⠒⠒⠒⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⢠⡟⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠻⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡼⠃⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠉⠉⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠞⠉⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠻⣷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣄⣠⣾⣿⠃⣠⠖⠂⠛⠑⢦⠀⠀⣿⣄⣀⣼⣿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⠻⠿⠛⠁⠀⠙⠲⣤⣀⣀⠜⠀⠀⠈⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⢀⡀⠀⣤⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠈⣻⣶⣿⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⢸⣧⣴⠆
⠀⠀⠀⠙⠛⠛⠛⢷⣦⡀⠀⠀⠀⢸⡿⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢻⡆⠀⠀⠀⣀⣴⠿⠛⠻⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⣠⡾⣧⡀⠉⠛⢶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠛⠁⠀⣼⢷⣄⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠀⠈⠷⣦⣀⠀⠀⠉⠙⠛⠛⠿⠶⠶⠶⠶⠶⠾⠟⠛⠛⠉⠁⠀⠀⢀⣤⠾⠃⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠁⠀⠀⠈⠙⠻⠶⣦⣤⣤⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣤⣤⣴⠶⠿⠋⠹⣆⠈⠋⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⣿⠀⠀⠀⠀⢧⠀⠘⣧⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧⠀⠘⣧⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢹⣇⣤⡾⠃⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⢿⣀⣠⣤⣴⠶⠟⠃⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣤⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣤⣤⣶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡶
⠀⠀⠀⠀⠀⡀⠂⣀⢀⣠⣄⠀⠀⠀⠀⠀⠀⢁⢹⡀
⠀⠀⠀⠀⠸⢷⣤⠀⠉⠟⢡⣤⠻⡇⠀⠀⢠⣬⣆⠿⠒⠒⠂⢀⣰⢠⡄
⠀⠀⣠⠒⠐⡆⠀⠑⠻⠏⠙⠉⠘⠇⠀⠀⠘⡟⠙⠓⠂⠀⠐⠚⢣⠀⠀⠀⠀⠀⠀⣄⠀⠐⢄⠉⢀⡀
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠀⠉⠉⠁
⠀⠀⠈⠓⠦⠴⣆⣀⣀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣯⣗⠶⠂⠀⠀⠀⣀⣀⢀⠈⠂⣉⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡛⠻⢄⣀⠟⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠝⠈⠁⠀⠀⠀⠀⠉⠈⠀⠉⠁⡁⠈⠁⠀⠀⠀⠀⠀⢀⣀⠙⠣⠤⠶⠶⠬⠴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠀⠄⣤⡤⣾⠋⠘⡆⠀⠀⠀⠘⠁⢱⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⡀⠀⠄⡂⠀⠀⠀⢠⡀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠃⣿⠀⠀⡇⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⣿⡑⣦⡍⢉⡅⣠⣾⡇⠀⠀⠀⠀⢸⣇⠀⢀⢨⠀⣿⡤⠀⠀⠀⠀⠀⠈⢀⣘⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⠋⣽⣩⠄⠀⠀⠀⠈⠻⣦⣬⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠤⠀⠀⠸⠾
⠀⠀⠀⠀⠀⠈⠓⠄⠠⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠸⡥⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣐⣺⡛
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠓⠚⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠦⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠄⠀⠀⠀⠁⠀⠐⠀⠀⢀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠤⢆⡑⠀⡴⠜⢃⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⢰⠸⣿⠒⠒⣶⠂⣄⣤⡖⡇⠀⡀⣄⠀⠓⢴⡚⢚⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠚⠓⠛⠀⠀⠁⠒⠕⠊⠇⠀⠠⡅⠹⡤⠀⠀⡠⠞⠒⠓
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠀⠈⠟⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠴⠖⠒⢲⠀⠀⠀⢀⠆⠀⠈⠙⠒⡐⠈⠱⢲⡷⠄⠀⢢⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⡀⢀⡀⠀⠠⢜⡁⠀⠀⠁⠀⠀⣀⠀⠀⣗⠀⠀⠁⠀⠀⠀⠓⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶⠀⠈⢄⠀⠄⡀⡀⠀⠀⠀⡰⠁⠉⠀⠀⣀⡠⠴⠒⢉⢄⡂⢼⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⠀⠉⠒⠢⠅⠀⠀⠀⣇⣄⣠⠶⣻⢶⠦⢖⠋⠁⠀⣤⡁⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠸⡍⠉⠒⠶⣄⠁⠂⣬⡇⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⠀⠈⠢⠄⠀⠀⠙⠢⡈⠑⢤⣀⠀⠀⠙⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣉⣠⣤⡠⡼⠶⠂⠀⠈⠓⠦⣈⣉⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠠⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣠⠤⡤⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠁⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⠉⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠰⠂⠑⠆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠃⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠂⠠⠤⡀⠀⢘⠀⢸⠀⡐⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠃⠠⠇⠀⠀⢀⠀⠀⠈⠀⠈⠀⠈⢠⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣄⡀⣀⠀⠈⠆⠀⣸⠏⣔⠀⢀⣸⠆⢠⠁⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣜⢋⠁⡀⠂⠁⠈⢲⡁⠄⠤⢍⠐⠍⡆⠈⠼⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⡉⠀⠀⠹⠦⠀⢤⠀⢨⡀⠀⠀⠁⠀⠱⠾⢦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠄⠐⠐⠆⢊⡀⡔⢁⡘⠀⠀⢨⠒⣄⡴⠂⠸⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⢀⠀⡄⢀⠀⠀⠠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠠⠈⠀⠂⠠⢂⠄⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠠⠁⠱⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡊⠴⠂⡅⠀⠒⢌⡒⢀⠀⠁⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠆⠂⠀⠀⠀⠇⠍⠀⠀⠀⠂⠋⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠶⠀⠀⠀⠀⠠⡀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⢀⣤⣤⡄⠀⠀⠡⠀⢤⠆⠀⢀⡈⠾⣦⡤⣄⠳⣄⡆⠈⠀⣼⣬⢧⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣰⣿⣿⣿⣷⣤⡀⠀⠀⠀⡀⠀⠀⠡⠒⠒⠛⢿⣇⣀⡳⠀⠘⢛⣿⡀⢱⡄⢀⡆
⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢀⡀⠓⠦⣸⣟⢀⠀⠹⣾⠁⠀⠄
⠀⠀⠀⠀⠀⢀⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠰⠄⠀⠀⠀⠀⢀⢀⡚⠦⠤⣹⣿⡈⢷⡀⢿⣒⢀⡗
⠀⠀⠀⢠⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠲⠦⠰⠆⠀⠀⠀⠀⠀⠀⠈⣿⣷⠾⣧⠸⣧⣸⣇⡀
⠀⠀⠀⣾⣿⣿⠿⠋⣿⣟⣛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠋⠛⣦⢤⣤⣤⣴⣾⡟⣻⣿⠇⠈⣿⣿⣿⡏⠄
⠀⠀⠀⣸⠟⠁⠀⠀⠀⢈⣻⣧⡀⢸⣻⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⡀⠚⣡⠴⠏⠉⠈⠉⠙⠻⣿⣇⠐⣿⣿⠟
⠀⠀⠠⠯⠤⢶⣷⣠⣶⣾⣿⣿⡇⠸⠿⡁⠘⣿⣿⠿⣿⣿⣿⣿⠿⠷⠷⠇⠬⠧⠀⠡⢼⡆⠰⣼⣯⢿⠀⡏⠫
⠀⠀⠀⠀⠀⠀⣹⣷⣤⣬⡍⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠒⠛⠛⠛⠛⠒⠓⠑⠀⠀⠀⣀⣿⣷⣿⣷⠀⢿⡥
⠀⠀⠀⠀⢀⣠⣠⡝⠋⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⣶⣷⣶⣶⠿⢿⡟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠉⠛⢙⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⡉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠈⠀⠀⠀⢠⣤⠀⠀⠀⢼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡀⠁⠉⠄⣈⠀⠀⠀⠀⠀⢰⠀⢰⠀⠈
⠀⠀⠁⠒⠀⠃⠛⠀⠈⠃⠐⡷⠀⢀⣤⣄⣤⡄⠂⡀⠀⠤⠀⠇⠂⠀⠀⠉⠂⠀⠃⠀⠀⢸⠀⠶⠀⠈
⠀⡖⣳⠀⠀⠠⠮⠀⡀⠀⠀⡄⡀⠈⠀⠹⠁⠀⠀⣴⡑⠀⠀⠀⢀⠸⠀⠸⣀⠀⠘⠋⠀⠀⣾⠆⡀⢀⣀⣀⠀⠀⠁
⠀⠛⠋⠀⠀⠀⠀⠀⠈⠉⠁⠁⠃⡀⠐⠀⠠⠀⠸⣿⠧⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠁⠈⠁⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⠀⠈⡀⢀⣤⡤⣤⣠⠀⠀⢤⠀⢈⡀⠀⠐⠙⠀⠈⠈⠀⠀⠀⠀⣀
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠤⠀⢉⡉⢉⡉⠀⠰⠖⠂⠘⠘⡄⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠓
⠀⠀⢀⣤⣤⡀⠸⠃⠃⢎⠁⡄⠀⠀⠀⠀⠤⠠⠠⢠⠀⢀⡆⣀⡠⠢⣀⡀⠈⢀⣾⣿⣿⣿⣧⠀⠀⠒⢶⢶⠀⠀⡀
⠀⠀⡯⠄⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⣤⡀⠔⠀⠀⠀⠠⠀⠄⢀⣤⢀⣤⡄⠀⣼⠛⠿⠿⠟⠻⡆⠀⡄⠸⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠛⠀⠀⠀⣸⣿⣷⠀⠀⠂⠂⠄⠀⠀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠄
⠀⠀⢀⣀⣀⡀⠀⠀⠄⠀⠀⠀⢀⣿⣿⣿⡆⠀⡀⠀⠀⠀⠀⠀⠹⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠁
⠀⠀⠸⠻⠻⠁⠀⠈⠐⠟⠛⠀⠀⠉⠹⠉⠁⠀⠐⠂⠂⠲⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠴⠆⠠⠄⠀⠀⠉⠀⠎⠀⠄⠀⠀⠀⠀⠛⠀⠀⢀⠆⡴⠐⠦⣤⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⡀⠀⠂⠀⡈⠁
⠀⢠⣀⣤⣤⣀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢁⠀⠈⠉⠙⠉⠁⠀⠀⠤⠘⠛⠡⠀⠀⡀⠀⡀⡂
⠀⠀⠋⠸⠷⠉⠸⠱⢠⠀⠀⣄⠀⠀⠀⠀⠀⠀⢠⣞⣳⡁⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⡀⢀⡀⢀⡀⣠⠀⡄⠄
⠀⠀⠠⡄⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢸⣼⣧⡇⠀⢘⠀⠀⠈⢠⣤⣤⣤⣤⡄⠘⠓⠋⠀⠀⠀⠠⢶

⠀⠀⡀⠂⡄⠒⠈⠀⠉⠈⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠈⠁⠉⠐⠂⠄⡀⢀
⠀⢀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠄⠉⡀⠠⠀⠂⠐⠀⠂⠐⠀⠄⠠⢀⡀⠑⠄⡠⡄⠀⠀⠀⠀⠀⠀⠀⠈
⡉⠀⠀⠀⠀⠀⠀⠠⢀⣡⠌⠓⠈⠀⠀⠀⠀⣀⢀⡀⣀⠀⡀⠀⠀⠀⠀⠉⠰⠅⣎⡃⡄
⠄⠀⠀⠀⡀⠴⡈⠴⠈⠁⠀⢀⡀⠄⡐⠨⠁⠂⠐⠀⡂⠐⠂⠊⠠⠂⠤⢀⠀⠀⠈⠃⣦⠁⢄
⠂⢀⡠⢂⠁⢢⠉⠀⠀⣀⢐⠠⠀⠁⠀⠀⠀⠀⠈⠄⠐⠀⠀⠀⠀⠀⠁⠒⢀⠂⡀⠀⠀⠺⡠⠑⢄
⠀⠫⠐⠀⡐⠀⠀⢀⠐⠔⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡐⢂⠂⡀⠀⠉⠌⠠⢐⡔
⠀⠀⠀⡌⠀⠀⠠⠠⠉⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠁⠀⠰⠁⠄⠀⠀⢂
⠀⠀⠐⠀⠀⠀⠂⠂⠀⠀⠀⠀⡠⠆⠒⠐⠂⡀⠀⠀⠀⠀⠀⡠⠂⠂⠐⠂⡀⠀⠀⠀⠀⡐⡀⠀⠈⡄
⠀⠀⡁⠀⠀⡄⡅⠀⠀⠀⠀⠈⢡⠁⢩⠀⠀⠐⠂⠀⠀⠀⠈⢨⠁⢩⠀⠀⠐⠀⠀⠀⠀⠠⠡
⠀⠠⠀⠀⠠⠰⠀⢀⡀⠀⠀⠰⠀⠈⠀⠀⠀⢀⠁⠀⠀⠀⠰⠀⠈⠀⠀⠀⢀⠁⠀⣀⡀⠐⠐⠀⠀⢘
⠀⠐⠀⠀⢘⢘⠈⠄⠄⠀⠀⠰⠨⢀⢃⢈⠘⡈⠀⠀⠀⠀⠐⡨⠀⢃⢉⠈⡈⠀⠈⠄⠐⠈⠀⠀⠀⠸
⠀⠈⡀⠀⠀⡄⡀⠀⠀⠀⢀⠒⠈⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠠⠀⠀⠀⠀⢘⢘⠀⠀⢘
⠀⠀⠄⠀⠀⠂⠂⠀⠀⠀⠨⡀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠆⠀⠀⠀⠄⡌⠀⠀⡄
⠀⠀⠐⡀⠀⠈⢃⢂⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⠤⠴⠤⢤⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠐⠀⠀⠰
⠀⠀⠀⠈⡄⠀⠀⠢⠠⠀⢌⡘⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠉⠂⠀⠀⠌⢑⠂⠠⠌⠌⠀⠀⡠⠁
⠀⠀⠀⠀⠐⢂⠀⠀⠉⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢘⡁⠃⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠉⠨⠀⡂⢀⠀⠀⠀⠈⠄⠔⠀⠀⠀⢀⠀⠄⠨⠐⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⡀⠀⠀⠀⠐⠀⠅⠥⠐⠀⠀⠨⠅⠥⠐⠀⠁⠀⠀⡀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣶⣶⣶⣶⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠟⠛⠛⠛⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣥⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣋⣛⣛⣭⣽⣷⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⠯⠭⠿⠧⠤⠛⠂⠈⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣾⣿⣿⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⣤⡀⠀⠀⠀⠀⠀⢀⣴⠗⠂⡄⠀⠀⠈⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣧⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⡤⣾⡇⠀⠀⠀⠀⠀⠘⣿⣶⣾⣏⢙⣆⠀⠀⠀⠳
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⢧
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠃⠀⠀⡠⠄⠀⠀⢰⠆⠹⠛⠛⠉⠁⣰⠆⢀⣀⣀⣠⣴⠖
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⢸⡁⢰⡿⣴⡏⣤⣠⣤⣞⣷⣾⣿⣶⠿⠛⠋⢩
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣇⢻⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢸
⠈⠙⠛⠛⠛⠛⠉⠉⠉⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷⡽⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣤⣤⣴⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢦⡈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠟⠙⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠈⠓⠀⠙⠻⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⠋⠀⣴⠁⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⣦⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⡿⠿⠛⠁⠀⢀⣼⠋⠀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠋⠙⠛⠃⢸⡿⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣤⣤⣀⣀⣸⠀⠀⠀⠀⠀⢨⣧⣠⣤⣤⣶⣶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠀⠀⠀⠀⢸⡟⠉⠁⢹⣿⣷⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⡀⠀⠀⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏⠈⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢸⡍⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⡇⠀⣸⣯⡁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣧⠈⣿⣿⣿⡇⠀⠀⠀⠳⣤⣀⠀⠀⠀⠀⣀⣠⣴⠾⢛⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⡇⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢻⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠓⠛⠛⠛⠉⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⠶⠶⠾⠛⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠸⡀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣀⢿⣿⣶⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⣩⣼⣿⣿⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣤⣤⣤⡶⠛⣿⠟⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⣀⣾⣀⣴⣶⣶⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣤⣤⣀⣀⣀⡀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣀⣤⣤⣴⠶⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠟⠻⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣠⣄⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣨⠜⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉⠉⠁⠓⠒⠓⠛⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠠⠆⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡻⠶⢾⠶⢄⣀⡠⠤⠴⠀⠉⠤⠼⢀⣢⡴⠀⡆⠀⠀⠀⣀⠀⠀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⢞⢦⣀⡠⢂⠆⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠄⠀⠈⠉⠉⠉⠁⠉⠉⢫⠷⠓⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠁⠀⡀⠀⠀⠈⠐⠲⠤⣍⣁⠐⠂⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡞⠀⠠⣒⠄
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢧⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠙⠳⢶⠴⠀⠀⠄⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⢀⢉⡁⠦⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣁⡀⠀⠀⠀⠀⠀⠀⢀⢵⡀⠀⠄⠄⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣐⡵⠃⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢅⢀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠠⠤⠤⢠⠞⢢⠀⠀⠀⠀⠀⢔⠤⠋⠀⠀⠀⠀⠀⠀⠀⠑⠐
⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠈⠣⢑⢀⡠⠂⠙⠲⠦⣀⣀⠀⠠⠠⠠⠄⢀⣸⢄⠀⠀⣠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡀⢀⠀⠁⠉⠹⡝⠉⠉⠉⠄⠀⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠁⠀⠀⠀⠀⠴⣿⠆⠀⠀⠀⠈⣆⠀⠀⠀⠀⠋⡈⠄⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⡎⣆⡄⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⣂⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠅⠀⠄
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠔⢃⠀⠀⠀⠀⠔⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⠀⠀⠘⢬⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠢⠤⡄⠐⠀⠀⠂⡀⢀⣀⣠⡤⠞⠀⠘⢧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠊⠀⠀⠀⠘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⠄⣤⣀⣀⣊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠎⠀⢀⣠⠞⠢⠀⠀⠀⡀⠄⠀⡄⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠐⡀⠀⠉⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉⠲⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⡤⠼⡄⠀⠀⠀⢋⠀⠀⢀⣹⣴⡊⠉⠉⠙⣆⣀⣀⡀
⠀⢀⣤⠒⣉⣁⣠⣤⠤⡴⠳⣀⢀⣠⣎⣀⡔⠉⠀⠀⠹⡄⠀⠀⣵⡤⠤⣉⣉⡉⠉⠒⠒⢤⡀
⠀⢸⠈⠉⠓⠒⠾⠤⠤⠧⠤⠬⢧⣀⣀⣀⣱⣄⡀⠠⠤⠿⠭⠭⠤⠷⠴⠒⠒⠛⠛⠉⠉⠉⡇⠀⣀⣀⣀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⠀⠀⠈⠙⠢⣄
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠉⠒⠦⡀⠀⠀⠀⠱⡀
⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⡇⠀⠙⢦⣧⠎⠀⠀⢳⠀⣰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⢧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣳⣼⠀⠀⠀⡇⠀⠀⠀⣵⠾⠥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠑⢄⠀⡇⢀⡴⠊⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⣤⣄⡬⢶⠤⠤⠤⠤⢵⣷⠯⠤⠤⠤⢄⣴⣊⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠸⠀⠀⠀⠀⠀⠀⠀⠉⢀⡔⠁⠀⠀⢀⠔⠁⡇⠑⢄⠀⠀⠀⠘⢦⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠀⠀⠀⣸
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣴⠋⠀⠀⡇⠀⠈⠳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠊⠉⠀⠀⠀⠀⣠⠃
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⢸⠀⠀⣰⣇⠀⠀⠀⡏⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡠⠞⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠠⠊⣸⡀⠳⡄⠀⠃⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠒⠒⠒⠊⠉
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈⠢⣸⢁⠔⠁⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠠⢦⣠⠀⠀⢀⠀⠀⠙⣄⠀⠀⢸⠃⠀⠀⣠⠋⠀⠀⡀⢀⠀⣄⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠱⠧⣀⡄⠀⠀⡿⢝⠦⣼⣤⠴⡻⣻⠀⠀⢠⣀⠼⠎⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠒⢄⡟⢸⢷⢼⠤⡾⡇⢿⡠⠒⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⠴⢚⣩⠟⢬⣶⣼⠔⠻⣍⡓⠦⣄⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠙⠲⢬⣓⣤⢔⢿⢻⠢⣤⣋⡥⠖⠋⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠤⠊⣧⢸⡮⠼⠦⢵⡇⣾⠑⠤⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡰⡖⠈⠃⠀⠀⣷⣪⠖⢻⠛⠲⣵⣼⠀⠀⠸⠁⢲⢆⣈
⠀⠀⠀⠀⠀⠀⠀⠐⠖⠉⠀⠁⠈⠀⠀⣴⠋⠀⠀⢸⡀⠀⠀⠙⣄⠀⠀⠁⠀⠀⠋⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢀⠔⢩⡈⠢⡀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⣤⣄⣾⣿⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢉⣄⠀⢰⡆⠀⠀⡠⣿⡏⠀⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡓⠀⢠⠤⡀⠀⠀⠀⢨⣧⠀⢸⠿⣷⣏⠀⣿⡇⠀⣼⡷⢻⣇⠔⣿⠆
⠀⠀⠀⠀⠀⠀⢀⡠⣿⠀⠀⠘⢷⣷⡾⢿⠀⢸⣿⣀⡞⢰⢹⣿⠀⣿⡇⢀⣿⠃⣼⡏⣰⠏⢀⡴⠶⣤⠀⣀⡀
⠀⠀⠀⠀⢀⠔⠁⠀⢻⣇⢠⢦⠀⠈⡇⢸⣧⠞⠙⠛⠀⠻⠼⠋⠀⠙⠗⠋⠛⠀⠉⣼⠃⢠⡟⠀⣰⠏⢰⣿⠃⠀⢀
⠀⠀⠀⢰⠁⢰⡟⠑⠈⣿⡇⠘⣧⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡠⠊⠛⢺⣯⠴⠊⣿⡤⠒⠁
⠀⠀⠀⠸⠀⢸⣿⠀⢀⠛⣿⡄⠹⣆⢸⡆⠀⠀⠀⠀⢀⣀⣀⠀⠀⢠⣶⠞⡢⠀⠀⠈⠀⠀⠀⠀⠀⡠⠊⣿⣷
⠀⠀⠀⠀⠀⠀⢻⣧⡜⠀⠈⠃⠀⠉⠛⠀⠀⠀⠀⠈⠀⡀⠀⠈⢡⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢘⡛⠀⠀⣼⡏⠀⠀⠀⣀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⢠⡏⠀⢀⣿⡇⠀⢰⣿⣁⡀⠀⣾⠏⡇⠐⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⡿⠐⢹⣿⠃⣠⢾⡿⠀⣿⢸⣿⠎⢀⠜⢁⠝⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡼⠁⠀⠸⣿⡴⠁⢸⣇⣴⠗⠊⠿⠖⠁⠀⣏⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠉⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣀⡀⠀⣀⣤⣤⣴⣶⣶⣾⠂⠀⠀⠀⠀⠀⡠⢆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⣭⡶⠿⠛⠛⣉⡻⣧⡀⠀⠀⠀⠀⠀⠀⢠⣾⢞⣩⣖⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣏⠀⠀⠳⠂⣦⣝⡃⣼⡇⠀⠀⠀⠀⢀⠔⣸⠟⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣤⠀⠀⠀⣠⣽⠟⠁⠀⠀⠀⢀⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠁⠀⠀⠀⢀⣄⠀⠉⠑⢦⡀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣿⠄⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣦⠤⠀⠀⠀⠠⠶⠶⠾⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠋
⠀⠀⠀⠀⠀⠤⠶⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠲⠶⠾⠿⢷⣤⣤⣤⣤⣄⡀


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠰⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠁⣀⠀⠀⠀⠀⠀⢀⠛⠷⠺⢏⣠⣠⡀⠀⠀⠀⠀⠚⠶⢀⢈⠀⠴⠟
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠛⠀⠀⠛⠀⠀⠀⠀⠚⠻⠟⢧⣠⣜⡙⠉⠀⠀⠀⠀⠀⣀⣤⠐⠛⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠚⠀⢠⣄⠀⠓⠀⠀⠀⠀⠀⠀⠀⢸⡝⠂⠻⠷⠀⠀⠀⠀⠀⠀⠉⠀⠀⣾⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢇⠀⠰⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⢀⡀⠀⠀⠀⠀⠀⡤⣠⠘⣝⠗⣄⣤⠄
⠀⠀⠀⠀⠀⠠⢶⡦⢠⢽⡜⣥⢀⣄⡄⠀⠀⠀⠀⠈⠁⢼⣾⡆⠈⠁⠀⠀⠀⠀⠀⢙⠒⢣⠐⡔⠓⡋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⠘⣀⠀⠁⠀⠀⠀⠀⠀⠰⠀⠐⠉⠂⠐⠆⠀⠀⠀⠀⠀⠮⢯⢁⠭⡈⠭⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠐⠹⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⢘⡾⢠⠒⠔⡦⢼⣖⠀⠀⠀⠀⠐⢂⣇⡀⡏⢀⣦⡘⠀⠀⠀⠀⠐⢂⣤⡀⠿⢀⣤⡐⠂
⠀⠀⠀⠀⠀⠀⢠⣀⢣⠀⠀⡬⢇⡀⠀⠀⠀⠀⠀⠤⣀⠽⠿⠥⣀⠄⠀⠀⠀⠀⠀⠀⣠⡅⠻⢫⣄
⠀⠀⠀⠀⠀⠀⠚⠙⠈⣫⣝⠃⠿⠓⠀⠀⠀⠀⠘⠃⠃⡠⠶⡀⠃⠘⠁⠀⠀⠀⠐⠂⠉⠀⠿⠄⠈⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣦⣷⣦⠀⠀⠀⠀⢀⢠⡀⡀⠀⠀⡇⡇⠀⠀⠀⢠⢀⠀⠀⠀⠀⢠⣶⣦⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⢿⡀⠀⠀⠀⠬⣻⣦⣵⢖⢤⠁⣥⢲⠶⡜⣼⡋⠄⠀⠀⠀⣸⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠣⡙⢦⡀⢶⠉⠲⢟⣁⢀⢗⡀⡜⡅⠀⡏⡳⠚⠛⠆⡠⢊⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣢⠙⠹⡤⢄⣈⣾⣇⣾⣇⣿⣜⣾⣿⡀⢤⡶⠋⣤⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣆⣢⢳⡀⢛⢦⡙⢿⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⢏⣴⠺⠃⢰⣏⣎⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣛⠯⢕⣾⣼⣿⣿⣿⣿⠿⠛⠋⠛⠿⢿⣿⣿⣿⣏⣶⣶⠮⢿⣹⡭⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠠⡾⢄⢀⠠⢸⣿⣿⣿⡿⠶⠦⠀⠀⠀⠀⠴⢿⣿⣿⣿⣟⠤⣀⣀⢜⠅⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⣲⣾⣿⣿⣗⣒⣒⣒⣚⡊⢑⣛⣛⣿⣿⡿⢧⠀⠀⡴⠁⠀⠀⠀⢐⡿⣿⣿⣟⣛⡃⠉⣛⣒⣒⣒⣒⣿⣿⣿⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠚⠿⠛⠁⠀⠀⠀⣌⡜⠑⢭⣻⣿⣿⠿⠛⠉⠉⠻⠒⢢⠞⠙⠒⠿⣿⣿⣟⡩⠓⠑⣝⡄⠀⠀⠀⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣒⣭⢿⣼⢳⡟⠁⠀⠀⠀⠀⠈⣷⡟⠀⠀⠀⠀⠀⠹⣿⣧⢡⣾⣽⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠞⡞⠈⣧⣧⡀⢀⣀⣀⣤⣴⣿⣷⣀⡀⠀⠀⠀⢀⣟⡜⠈⠧⠧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡶⢋⣠⠯⠝⢻⣿⡟⣿⡟⣿⢟⠿⣿⠟⠾⣯⡉⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠋⣽⣤⢔⡯⣇⠘⣾⠃⢻⠎⠀⣕⣷⣦⣦⡍⠢⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣵⠋⠀⠀⠀⢊⣽⠛⡺⣖⠵⠀⠮⣁⡴⢣⢻⣯⠂⠀⠀⠈⢪⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⣿⡿⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⡁⡃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⢿⡿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠂⠉⠉⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠈⠉⠀⠂⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠮⠤⠤⠤⠤⠤⠦⢄⡀⠀⠀⠀⠀⠀⠰⠚⠒⡢⠀⠤⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡧⢀⣠⠁⠀⠀⠀⠀⠐⠁⠀⠀⠉⠉⡰⠒⠀⠢⠤⠠⣀⣰
⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⠀⠎⠀⠀⠀⠉⠉⠒⠒⠂⠦⠤⠀⢀⣀⢠⡀⠀⠀⠀⠀⢀⠊⠀⠁
⠀⠀⠀⢀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠈⠉⠉⠀⠐⠚⠒⠐⠇
⠀⠀⠀⡸⠤⢤⡤⠀⠐⠒⠒⠰⠂⠉⠉⠉⠈⡘⠁⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠢⡀
⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⣀⠀⡀⢀⣀⣀⠧⠀⠠⠤⠤⠄⠺⡒⢺⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⢋⠋⠉⠈⠀⠀⠀⠉⠁⠀⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡼⣀⠒⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣪⣱⠀⠀⠀⢠
⠀⠀⢎⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡹⠙⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠉⠐⠂⠐⠒⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠒⡇
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣽⠒⠂⠠⡴⠶⠆⠀⠒⣊⠁⠀⡀⠀⠀⡰⢤⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⢇
⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⣿⡀⠀⢰⠄⠀⠠⠤⠤⢆⠀⠀⠈⢢⡜⢁⠆⡆⠀⠀⠀⠀⡇⠀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠰⢏⠀⠉⡙⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠑⡌⠀⡇⠀⠀⠀⠀⠇⠀⢨⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠲⡀⠀⠀⠁
⠀⠀⠀⠀⠌⠀⠀⠀⢀⠃⠈⠉⠈⠁⠥⠕⠀⠀⠀⠀⠘⠀⠀⠀⠀⢡⠀⠱⠤⢀⡀⠤⠃⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢲⢄
⠀⠀⢀⡎⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡔⢄⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢱⠤⠀⠔⠁⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠑⠤⣀⣀⣀
⠀⢀⠊⠈⠙⠐⠂⡼⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠀⠀⠀⠘⣄⠤⠐⠐⠈⠉⡄⠀⠀⠀⠀⠀⢈⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠈⠢⡀
⢀⠎⠉⠐⠂⠤⣀⠂⠀⠀⠀⠀⠀⠀⠣⠜⠀⠀⠀⠀⠀⠀⢰⣀⡠⠤⠐⠊⠙⡄⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡌⠀⠀⠀⠀⠀⠈⠢⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣠⠜⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢇⠀⠀⠀⠀⠀⡠⣀⡸⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢆⡀⣤⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠰⡁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠈⠒⠤⠀⠔⠊⠀⢸⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠑⠢⠄⠀⠔⠋⠀⠀⠀⠀⠀⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠉⠁⢹⡁⠀⠀⠀⠈⠉⠉⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⢈⠁⠀⠉⠉⠀⠀⠀⢁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⢀⡠⠒⡇⠀⢀⣀⣀⣀⡀⢸⠀⢀⣀⣀⣀⣀⠀⢸⠃⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⢀⡴⠛⠈⠉⠉⠉⠑⠢⢄⡀⠀⡙⠈⠀⠀⠀⡀⠤⠈⠛⠒⠀⠤⠌⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠐⠻⢦⣄
⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠂⠀⠤⠄⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣁⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠋⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠈⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡄⠀⠀⠀⠀⠀⠈⣳⠒⠤⠂⠙⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⣄⡀⠀⠀⠀⠀⢀⠔⠂⡄⠀⠀⠀⠀⣀⣰⣟⣇⢸⡀⠀⠀⠀⢠⣞⡁⠀⠀⠀⠀⢀⣵⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢦⠲⠶⣏⠀⠀⠀⠀⢰⣀⣄⡀⠀⠀⠀⠀⠇⠚⠓⠛⠒⡇⠀⠀⠀⣤⠛⠙⠂⠀⠀⠀⠁⠈⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⢀⡅⢠⠘⠁⠀⣘⢢⠘⠛⣹⣷⡄⣃⣀⠀⠀⠀⡀⣠⣬⣷⣮⠆⣠⠊⠁⠀⢀⣀⣀⡀⢀⡤⠾⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⢶⣄⣠⣤⣄⠈⠒⠀⡉⠉⠘⠷⡇⣰⠋⠙⣿⠀⠀⠼⠿⣸⠃⠈⣿⣦⣤⣧⡿⠾⢿⡷⠛⠙⢻⡷⠚⢉⡀⠈⢻⡏⠀⠰⣄⣴⣿
⠀⢀⣀⣀⣀⣀⡠⠞⠁⣠⣶⠀⠀⣽⠃⠀⣿⣦⣌⠠⠓⠂⠉⣰⣄⣿⣤⣴⣿⠶⠶⢧⣴⠋⠀⠀⢹⠃⠀⢁⡄⠀⢀⣴⠀⠀⡞⠁⢠⡿⠁⢀⣿⠷⣄⠀⠈⣻⡇
⡰⠋⠁⠉⠈⠉⠀⠀⠚⠋⠁⣀⣼⡟⠀⢠⡿⠿⣷⡶⠲⠶⠛⠙⡟⠉⢹⠏⠀⢠⣄⣰⡷⠀⢠⣾⡟⠀⢀⣿⠁⠀⣾⡏⠀⢰⠁⠀⠛⠁⠀⢸⣇⡀⢈⣀⣴⣿⠃
⣇⠀⢠⣾⡗⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⣈⠀⠀⣸⠃⠀⣠⣀⣼⠃⠀⣾⠧⣄⠀⠈⣿⠃⠀⣼⣿⠁⠀⣾⡇⠀⣸⣿⠀⠀⣿⣦⣤⣾⣿⣷⣿⣿⣿⡿⠿⠛⠷⣤
⢻⣶⣬⣿⡇⠀⠀⣿⣿⣿⣿⣿⠇⠀⣼⡏⠀⢠⡏⠀⢰⣿⣿⡇⠀⢸⣃⠀⠉⢀⣠⣿⣤⣴⣿⣿⣶⣾⣿⢿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣀⣀⣀⠘
⠀⠉⠛⢻⡇⠀⠀⠻⣿⠿⠛⡟⠀⢰⣿⠀⠀⢾⣄⣀⣿⣿⣿⣷⣴⣾⠿⣿⣿⣿⠟⠛⠛⠛⠋⢀⡽⠟⠛⢶⣄⠀⢿⣄⠀⠙⠿⢿⠿⠛⢉⣠⣴⣾⣿⠿⠿⠿⣷
⠀⠀⠀⠈⢿⣦⣄⣀⣠⣴⣾⣷⣶⣾⣿⡀⠀⠈⢉⣽⡟⠁⠀⠈⢉⣩⠖⠛⠛⠶⣦⣀⠀⠀⣠⠋⠀⣠⡄⠀⣿⡀⠈⠿⣷⣶⣶⣶⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠀⠉⠉⠙⢿⣷⣾⡿⠟⠁⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠉⠻⠾⠁⢀⣾⣿⠇⢀⣿⠁⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡏⣠⣾⣿⣿⣿⣿⣶⣤⣄⠀⠀⠈⢉⣁⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠙⢷⣿⣿⠀⠉⣿⠟⠻⣿⡟⠀⢰⣿⠿⠿⠛⠁⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⣾⠀⠀⣾⣿⡏⠀⢠⣇⣀⣰⣿⠁⠀⣿⢧⣄⢀⡴⠚⠛⠛⣶⠋⠁⣀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⣿⡏⠀⢰⣿⣿⠃⢀⡞⠙⢻⣿⡏⠀⢀⡄⠀⣿⠋⠀⡼⠃⣰⣿⣄⠀⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣬⣿⠀⠀⣾⣿⠃⢀⣾⠃⠀⣼⡿⠀⢠⡿⠀⢀⡇⠀⠀⠴⠞⣿⠁⠈⠃⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠇⠀⢰⡿⠃⣠⣿⡏⠀⢸⣿⠇⠀⠘⠁⣠⣾⣧⣀⣀⣤⣾⡿⣷⣶⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⣠⣾⡿⢻⣦⣤⣿⠻⣶⣿⣶⣿⡿⠋⠙⠻⠟⠛⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⠋⠀⠈⠙⠛⠋⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⡄⠀⠀⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠹⠁
⠀⠀⠀⠀⠀⠛⠀⡀⠒⠬⠀⠄⠢⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣶⡴⣽⣲⣤
⠀⠀⠀⠀⠀⠐⡛⠲⠶⠤⠦⠴⠶⠚⢣⠄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠶⠶⠒⠛⢣⠀⢀⠀⠀⠀⠀⠀⡠⣼⠈⠀⠉⢹⠹⠅⢘
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡶⠣⢀⣀⡀⠀⠀⠀⣿⣰⣧⡀⠀⠀⠀⠐⠄⢸⠀⠀⠀⠘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⣠⠎⠁⠀⠆⠀⠀⠀⠀⠀⢀⡿⠀⢰⡿⠀⠉⣿⠛⠛⣿⠁⠄⠀⠀⠀⠀⠑⠺⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠘⠶⣶⣾⣿⣿⠿⠿⠿⠖⠚⠁⠀⠀⣷⠐⠐⣶⠉⠳⢄⣸⣇⣀⣀⣿⣀⣤⡟⠔⠁⠀⠀⠀⠀⠒⠿⠷⠶⠤⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠿⠒⠛⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠸⡗⠀⠀⢀⠀⠀⢈⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣥⠐⠂⠉⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢘⠛⠛⠛⠛⠛⠛⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡁⠀⠀⠂⠈⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⣶⣶⣾⠷⢶
⠀⠀⠀⠀⡌⠀⠟⠉⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠉⠉⠁⠀⢰⣄⠠⠀⠃⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢻⣄⠀⠀⠀⣐⣴⡀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠂⠀⢈⡀⠠⠀⠄⠠⠡⡈⠀⡡⠀⠀⠀⠀⠀⠀⠀⡇⠱⣄⠀⠀⠀⠀⢸⡸⣦⠀⠤
⠀⠀⠀⠀⠀⠙⢣⡀⠀⠘⠉⠂⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⠠⠜⣆⠉⠀⠀⠈⢠⠿⠎⡀⠀⠀⠀⠀⠀⠀⠂⢹⠀⠀⠉⠐⠒⠒⠚⠛⡹⠄⠂⡤
⠀⠀⠀⠀⠀⠀⢈⣏⡓⠒⠒⠓⣻⠃⠀⠶⠀⠀⠀⠀⢠⣥⣀⡐⠨⠷⠦⠤⠶⠋⢀⣀⠬⠀⠀⠀⠀⠀⣈⣄⠈⠢⣀⡀⠀⠀⠀⣀⣠⣁⣀⡎
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣛⣭⡶⠾⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣼⠟⠁⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⠀⢸⣇⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣄⠈⠢⣤⣤⡴⠿⠋⠉⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣆⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣷⡘⢷⣄⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣦⡙⢷⣄⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⠻⣶⡿⣧⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠚⠉⠙⢿⣎⠻⣯⣿⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡟⠁⠀⠀⠀⠀⠀⢿⡄⠈⢻⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⣷⣄⡀⠀⠀⠀⠀⠘⣇⠘⣦⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣿⣿⣷⠶⠖⢀⣈⠀⠸⣷⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠶⣛⣛⣛⡻⣦⢹⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠉⠻⣿⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠾⣇⡀⠀⠀⠀⣿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠁⠀⠀⢠⡿⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢦⣤⣄⡀⠀⣼⠃⠐⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠑⠒⠀⠛⠀⠀⠈⣠⡤⠶⣦⣬⣙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⣤⣤⣤⣤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⢷⡌⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠸⣷⡀⢿⠉⠀⠈⠉⠉⠓⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⣿⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣽⠟⠉⠁⠀⡛⢷⣦⣀⡀⠀⠸⣧⣄⣀⣀⡀⠀⠀⣹⡇⠀⠀⢠⣿⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢼⡇⠀⠀⠀⠀⣿⠘⢮⣍⣛⠻⠶⣤⣭⣉⡉⠶⠾⠛⠋⠀⠀⣠⠾⣫⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣤⡀⠀⠀⢹⡄⠀⠈⠙⠛⠷⢶⣮⣭⣝⣓⠦⠀⠀⠠⢊⣵⡾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⢶⣄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣦⡀⠰⠋⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣡⠔⠀⠀⠀⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣉⠙⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡾⠁⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠉⠛⢦⡈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣷⡀⠀⠀⠀⠀⠀⢠⡀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣴⡟⠁⢢⠀⠀⠀⠘⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠻⢶⣤⣀⠀⠀⠀⠻⢦⣄⣈⠉⠛⠛⠿⠿⠟⠛⠋⠁⢀⣠⠟⠀⠀⠀⣰⡟⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣬⣙⡛⠷⣶⣦⣤⣌⣙⠛⠿⠶⢶⣶⣶⠶⠾⠟⠋⠁⠀⠀⢀⣴⠟⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⡷⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠿⣿⣶⣄⠀⠀⠀⡼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠄⠀⢰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⠉⠉⠉⠉⠉⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣀⠀⠀⠀⠀⣠⣄⠀⠈⠁⠀⣀⠤⠴⣀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠋⠀⠀⢠⠞⠡⠈⢳⣴⠋⠉⠁⠀⣀⣀⡱⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⠤⣀⠀⢀⡴⠛⠲⠚⠲⠋⢸⢃⣞⠉⢉⣥⠛⠢⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⣏⠀⠀⢑⣏⡀⠀⠐⠖⠀⠀⣨⣌⢀⣟⣉⡈⠶⠲⠟⢆⠐⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⡴⠁⠈⠛⠑⠋⠀⠙⠦⡀⠀⣰⡎⢧⡼⠉⠀⠀⠙⣄⠂⠀⠈⠣⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠎⠀⠀⠃⠀⠀⠁⠀⠀⣠⢬⡞⠁⢧⠀⠀⠀⣀⣀⣠⢬⠇⠀⠶⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡍⠓⢄⣀⣠⠤⠴⠶⠶⠤⠮⣀⣻⢩⣿⢹⠒⠚⠉⢱⣳⣄⣸⢑⣦⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒⠓⠚⠒⠚⠉⡉⠀⠀⠊⠁⢀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠀⠀⠐⠃⠀⢀⣀⠤⠤⠴⠦⠤⣀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⣡⠔⠛⠋⠉⠁⢛⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠐⠀⠁⠀⠀⠁⠀⠀⡀⠀⠈⠀⠀⠀⠄⠀⠀⠉⢀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠶⠒⠒⠒⠚⢉⣉⡠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠉⠉⠙⠛⠒⠒⠒⠒⣒⡒⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠐⢏⣀⣉⡱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢬⣓⣀⠀⠠⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⢄⣀⡠⠤⠒⣋⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠤⠬⣭⣉⣉⣙⣛⣛⣛⣛⣛⣛⣛⣛⣁⣈⣩⡭⠥⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⠀⠀⠀⠀⢠⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣄⠆⠀⠀⠀⠀⠀⠐⠬⡄⣸⠿⠁⠀⠠⠀⠀⠀⠰⡀⢀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⡛⠀⠀⠂⠀⠀⠀⣂⣢⡄⢤⡤⣀⡀⠀⠀⠀⠈⠀⢑⠺⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠂⠂⠈⠌⠨⠩⠽⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠘⠛⢍⠀⠀⢀⢀⠔⢀⠀⠀⠀⣠⠖⠀⠀⠀⢀⠀⢠⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⢠⠊⠳⡄⡀⣠⣞⡠⠆⠀⣠⠚⠁⠀⠀⠀⠐⠀⢀⠀⢉⢉⡑
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⡀⡀⠀⣀⠀⠁⠀⠀⠂⠌⠀⠀⡇⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠐⠰⠦⠦⢜⢀⢉⠉⠉
⠀⠀⠀⠀⠀⡛⠒⠚⢂⢈⣉⡍⠛⠁⠀⠀⠀⠀⠀⠀⣠⠄⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠤⠦⠤⢦⢝⠋⠊⠛⠓⠀⠀⢀
⠀⠀⠀⠀⢘⠃⠛⠑⠎⡉⢉⠉⠉⠙⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⢠⡤⡠⠤⡤⣼⠟⠜⠚⠳⠎⠁
⠀⠀⠀⠀⠐⠛⠒⠛⠫⠁⠓⠊⠂⠙⠄⠀⠀⠀⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⣤⣠⡀⣤⣀⣈⢹⡆⠀⠐⠀⠁⠀⠁
⠀⠀⠀⠀⠓⠛⠊⠚⠯⠶⠤⠦⠴⠴⡶⠁⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠐⠈⠀⠀⠀⠀⠀⠠⠀⢄⠀⠀⠈⢙⡇
⠀⠀⠀⠀⠁⢱⠇⠘⠔⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠐⠃⠀⠀⠀⠒⠢
⠀⠀⠀⠀⠀⠘⠃⠀⠄⠀⣀⢻⠈⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣌⠄⠀⠀⠐⠠⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠘⠷⢄⡀⠀⠄⠈⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠠⠮⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠠⢆⡴⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⠀⠈⢱⠊⠀⠀⣈⣀⡤⠤⠔⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⢀⣀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠚⠉⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠚⠋⠉⠉⠉⠀⠀⠀⠀⠙⣍⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣀⡤⠔⠚⠉⠉⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣄⣠⠶⠋
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⢴⠊⠁⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⣀⠀⠄⢀⡀⢾⣧⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠈⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠅⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⢷⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢦⣈⠉⠉⠙⠒⠒⠒⠒⠋⠉⠉⣠⣴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⣲⠖⡶⡶⠶⠒⠒⠚⠋⠁⠈⢧⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣰⠃⣸⠓⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢠⠏⠀⡏⠈⣧⠈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⢀⡇⡼⠀⠈⡿⠴⠚⠂⠄⠆⠀⠀⠀⠀⠀⠀⣇⣰⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⠒⠊⠀⠧⠴⠥⠷⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠋⠁⠈⠙⢦⠀⠀⠀⠀⠀⡴⠋⠀⠀⠉⠳⡄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣀⣠⠟⠛⠋⠉⠉⢧⣄⣀⣀⣀⣠⠟
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠂⢠⣾⣿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠂⠀⠀⠀⠀⠐⠄⠀⠁⠀⠰⠇⠀⠀⠔⠙⣿⣷⡀⣸⣿⠁⠀⠀⣠⣶⠗⠆
⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⡀⠀⠀⣴⣦⠀⠀⠈⠓⢰⣶⠆⣀⡰⠀⠀⢸⣿⠀⣿⡇⠀⠀⢸⣿⠇⠐⢡⡶⠲⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠔⠀⠀⠀⢿⣿⡀⠀⢀⠃⢸⡏⢈⣿⣿⠀⠀⣾⠏⠀⣿⡇⠀⠀⢸⣿⡀⠀⣿⣇⠠
⠀⠀⠀⠀⠀⠀⡠⢿⣇⠀⠀⠀⢀⠀⠻⣧⢀⠄⠀⠈⠃⠀⠘⠿⠂⢰⡟⠀⠀⠸⣇⠀⠀⠀⠈⠉⠀⠙⢿⣄
⠀⠀⠀⢀⠔⠁⠀⠈⢿⣆⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⢰⣾⠆⠀⠀⡾⠀⠀⢀⣀⠙⢦⠠⠂⠀⠀⠀⠀⢀⣿⣿⠆
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠙⠳⠆⠀⠀⠠⠀⠀⢠⡆⠀⣸⣿⠀⠀⠀⠁⠀⣰⣿⠏⢑⠀⠀⠀⣀⡤⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⣇⣀⣠⡄⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣾⣿⡀⠸⠀⢸⡏
⠀⠀⠈⠂⠀⠤⠊⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠠⠸⣿⠀⢀⣀⣀⡠⢻⣿⡀⠀⠀⠀⠈⢻⣿⣦⡀⠈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⡀⠀⠀⠙⠇⠂⠀⢿⡠⠁⢀⢿⣷⠀⠙⠷⠦⠀⠀⠀⠀⠄⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⡐⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠤⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣽⣿⣿⣭⣭⣭⣯⣭⣭⣭⣿

⠀⠠⠀⠀⠀⠀⡰⠆⠀⠀⡰⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠈⠐⠁⠘⢰⢎⢉⢴⡘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⡀⢤⡞
⠀⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣼⢫⠀⠁⢀⠿⣆⡂⠀⠀⠈⠁⠀⢀⡀⣀⣔⡖⣫⣜⢳⢂
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⢀⡾⠙⠈⠀⠀⠈⠈⠝⣦⠀⠀⠀⠀⠀⠀⠉⠉⠻⣯⡻⣫⣻⠙
⠀⠀⠀⠀⠀⠠⡀⢀⠄⠀⠀⠐⠂⠀⠀⠠⣆⠄⢀⠀⢠⡄⠀⠀⠀⠈⠀⠀⠀⠂⠁⠀⠀⠀⣠⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⡺⠷⠉⢱⣧
⠀⠀⠀⠀⠠⠤⠰⠆⠴⠆⠀⠀⠀⠀⠀⣀⡘⠆⢘⠀⠛⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠄
⠀⢀⣄⠀⠀⠰⠃⠈⠢⠀⠀⡀⠀⢀⣠⣠⠉⣮⣷⣯⣐⢉⣄⣄⡀⠀⠀⠀⠈⠀⠀⠀⢘⣧⠀⢠⠀⣀⠀⣀⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠢⢟⠧⠐⠦⡄⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⣻⣧⠏⠀⠘⣦⡟⠃⠀⠀⠀⠀⣤⡄
⠄⠀⢰⡀⢠⡔⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣴⠂⢸⠀⢳⡄⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠈⠹⣆⠀⣵⠏⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠄
⠀⠀⠀⢧⣜⣁⡄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⣌⣦⣠⣤⣌⣹⣼⣁⣤⣄⣠⣞⣄⡄⣀⠀⠀⠀⠀⢀⣈⣉⣤
⠎⠀⢧⣎⡉⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⢉⠟⠀⠉⢡⡞⢹⣤⠉⠁⠘⢯⡉⠁⠉⠀⠀⠀⠀⠀⠀⠀⢌
⣷⣲⣮⠀⣠⠂⡴⢁⡄⠀⠀⠀⠀⠀⠀⡶⢄⡀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⣀⣀⣠⡟⠁⠀⠛⣧⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢳⡀⢤
⣿⣫⡿⠉⠻⡉⠻⣍⠋⠁⠀⠠⠀⠀⠀⢹⠀⠙⡆⢰⣆⢀⡴⠋⢹⠃⠀⠀⠁⠀⠀⠀⣰⡎⢳⠀⠀⡼⠹⣦⡈⠁⠀⠀⠀⠀⠀⠐⠛⢸⠟⡿⠉
⡅⠁⣷⠤⠄⠀⠀⠈⠓⠀⠀⠀⠀⠀⣠⣀⠳⣄⢰⠘⠇⡼⢀⡠⠋⡀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠁⠀⠀⠀⠉⠆⠀⠀⠀⠒⠀⠀⠀⠀⠋⠀⠠⠴
⠙⠐⢁⢷⠒⠒⠒⠀⠀⠀⠀⢀⣀⣠⣀⣉⠓⢬⠟⠉⠑⢻⡡⠖⠏⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣚⡻
⠃⠀⠎⢠⢿⠁⠀⠀⠀⠀⠙⠣⢤⣀⣀⡤⠽⢻⡀⠀⠀⢀⠷⢍⣉⡉⠉⣒⣦⠀⠀⠀⠀⠰⠄⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠱
⠀⠀⠀⠀⠀⠁⠀⠀⢠⡀⠀⠀⠀⠀⣠⣤⠻⣩⢷⢲⠒⡯⣜⠲⢦⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡉
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⠞⠁⡜⣸⣇⢧⠈⢣⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠈⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⢸⢀⠜⠁⠘⠃⠈⠳⢄⡇⠀⠀⠀⠀⠀⠀⠀⠛⣠⠐⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠎⠀⠀⠀⠀⢀⡀⢁⣀⢁⣀⡀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⣀⣠⣠⡄⣤⡉⠀⢡⣤⢀⡄⣠⢀⡀⠀⠀⠤⠀⠀⠸⡗⡀
⠀⠀⠠⠀⠀⠀⠀⡌⠩⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⡷⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⡔⢀⢠⣼
⠄⡀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠘⠃⠀⠀⠀⠳⡄⠄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⢀⢀⢀⢄⠀⡒⠛⠀⠻⡤
⠺⡟⠒⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⣠⡠⣽⣶⣧⢡⡤⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠉⠋⠛⢩⣏⡀⢀⠐⡓
⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡙⢇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠑⠚⣇⣀
⠀⢀⡀⠀⠀⠈⠴⢠⠀⠀⠄⠴⠃⠀⠀⠃⠀⠀⠊⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⠹⡆
⠀⠄⠀⠀⠀⠈⠀⢃⡀⢠⡜⠁⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠆⠀⠀⠀⠨⠳⣆⠆⠀⠀⣀⡾⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⣠⢠⡑⡄⣄⢀⠁⠘⣤⡄⢠⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣙⣤⠚⣤⡏⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠘⠀⠁⢠⠒⠚⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⡦⡷⣵⠤⡇⣴⣷⡆⣼⢥⡦⡦⣤⡄⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢠⡞⢳
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡟⢠⠇⠀⢀⡀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠈⣿⠉⠉⠉
⠀⠀⠀⣴⠻⡆⢸⣧⣏⣀⠀⣿⣿⠀⠀⠀⣹⡁⢀⡶⡦⠀⢸⣿⠀⠀⠀⣀⠀⢀⣀⠀⠀⣴⠖⢶⣿⠀⢠⠖⡆
⠀⠀⢸⡏⠀⠀⢸⣿⠏⣿⠀⢸⡷⢶⡄⠀⣿⠀⢸⡶⠃⠀⢸⣿⠀⠀⣿⠟⣧⠏⣿⠀⢸⡟⠀⢸⣿⠀⣿⡞⠁⠀⢀⡄
⠀⠀⢸⣇⠀⢀⣿⡟⠀⣿⠀⢸⠇⣾⡇⠀⣿⠀⡾⠿⣦⣤⢻⣿⠀⢠⣿⠀⡿⠀⣿⠀⢸⣧⠀⣼⡇⢠⡟⢷⣦⡴⠋
⠀⠀⠈⢿⣤⠞⠙⠃⠀⣿⠀⡼⠀⢻⡇⣸⠻⡴⠃⢠⡞⣿⠈⣿⠀⣼⠿⠠⠇⠐⣿⠀⡟⠙⠛⠹⣇⣸⠀⢠⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡴⠃⠀⠈⠛⠁⠀⠀⠀⠸⠶⠋⠀⠻⡶⠃⠀⠀⠀⠀⢿⡼⠁⠀⠀⠀⠙⠁⠀⠻⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢠⡶⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⣿⠃
⠀⠀⢰⣶⡆⠀⣾⠃⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⣿⣿⠀⢀⣾⠋
⠀⠀⢸⣿⡇⠀⢿⡄⠀⠀⢸⡇⠀⣴⣶⡄⠀⢸⣿⣇⣴⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⢸⣿⣧⠀⠘⣷⡀⠀⣼⡇⠀⣼⣿⠄⠀⣾⣿⣿⠟⠋⠛⣷⣄⠀⠀⣰⣾⣏⣩⡿⠀⠀⣴⠟⢩⡿
⠀⠀⠀⣿⣿⠀⠀⠈⠻⢦⣿⣀⣴⣿⣿⠀⠀⣿⣿⡏⠀⠀⢀⣿⣿⡶⢶⣿⡿⠉⠁⠀⠀⢸⣿⣶⠋⠀⠀⠀⠀⣀⡴⠆
⠀⠀⠀⢹⣿⣇⠀⠀⢀⣾⠏⠀⠘⣿⣿⠀⢠⣿⣿⣃⣴⠞⢻⣿⡿⠀⠘⣿⣿⠀⠀⠀⣠⡟⠙⠻⢿⣶⣤⣴⠞⠋
⠀⠀⠀⠀⠙⢿⣦⣤⡾⠋⠀⠀⠀⠻⣿⣴⠟⠙⢻⣟⣀⣠⣿⠟⠁⠀⠀⠙⢿⣧⣤⠾⠋⠀⠀⠀⣴⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣴⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠄⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢸⣿⣿⡿⠋⠙⠁⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠋⠙⢿⣿⣿⡇
⠀⠀⠀⠘⣿⢿⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠨⡿⣿⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⢧⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡰⠁⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⢾⣿⣿⠏⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⠄⠀⢀⣤⣤⡄⠀⠀⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⡿⢣⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣌⣽⣞⢿⡶⣍⡡⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⡭⣾⣿⣿⢽⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠜⢿⣝⣟⣯⡿⢧⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠐⠻⢈⣏⠩⠟⠀⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⣄⣤⢠⣄⢠⣀⣤⣄⣠⡤⡄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠿⠁⣿⢠⡟⢿⠀⣼⡇⢸⣟⡤⣿⢻⡄⡄⡤⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠟⣆⢀⣀⣀⣼⣻⠏⠉⠉⠙⠷⣠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⠞⠉⢻⣀⡤⣧⣬⠶⢛⠉⠛⢀⢀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣀⡁⠄⠚⠑⠀⠀⠀⢰⣼⢅⣀⠀⠸⣀⠘⣠⡷⠄⠠⠾⠉⣟⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣴⠐⡆⠀⠀⠐⡄⠱⣦⡀⠐⠁⠘⠶⢴⣦⣀⡈⠢⠚⠁⠀⣈⣳⣄⡋⠀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⡇⠀⠀⢀⠏⢢⠰⡁⢠⠐⠉⠉⠉⠀⠀⢷⠘⣀⠀⢦⠁⠉⠉⠠⡁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡾⠈⣀⣴⠎⠀⠀⠈⡇⡹⠸⡆⠂⠀⠀⠀⠐⠘⡇⠃⠈⠾⣇⠠⠀⠀⠈⠀⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⠲⣴⠳⣄⠀⠀⠊⠰⡧⠀⢻⡀⠀⠀⠀⠀⠀⢷⢷⠀⢠⠏⠀⣁⣴⡿⠖⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡶⠶⠤⣄⣸⠃⠀⠀⠘⣗⣨⢭⣁⡒⠶⠘⠆⣀⣳⡄⢠⠤⠶⣤⣀⡶⠶⣻⡼⣿⠉⠀⡎⢂⢰⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢞⡽⠛⢩⠽⠿⠯⢭⡓⠀⣽⠀⠀⠀⠀⠈⠁⠀⠀⢹⡶⣤⡤⢦⡁⠀⣡⠋⠛⠉⠁⠙⠞⠋⠀⢸⣷⣼⠁⠈⡏⢸⡇
⠀⠀⠀⠀⠀⠀⣰⠃⢀⡴⠖⠚⠛⠛⠓⠦⣅⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⠈⣿⢷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⠀⠀⣧⣿⡇
⠀⠀⠀⠀⠀⢠⡏⢀⡞⠁⠀⠀⠀⠀⠀⠀⠘⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡇
⠀⠀⠀⠀⠀⢸⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠘⣇⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠐⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢹⡄⠈⠳⣄⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⠞⠓⠚⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠈⠉⠙⠛⠒⠒⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠂⠠⢀⣄⣁⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠶⠛⠓⠋⠉⠉⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠤⠤⠴⡶⢶⡴⠶⠦⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠨⠀⡰⠃
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠀⠀⠄⠂⠀⡠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣠⣄⠔
⠀⠀⠈⠎⢡⡀⠂⠀⢣⠀⠀⠁⠀⡀⠀⠈⠀⠀⠇⠀⢀⠴⠃⠂⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠂⠀⢀⠀⢨⠁⡀⠀⠀⠀⡰⢁⠀⠂
⠀⠀⠀⠤⠤⠅⠢⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⣘⡀⠒⠂⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⣀⠀⠉⠃⣀⡄⠄⠁⠁⡄⢀⢀⡀⠢⠌⠤
⠀⠀⠀⠀⠠⠤⠤⠤⠀⠑⠂⠠⠀⠃⠁⢰⠀⠂⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠓⣤⢙⢥⡄⠈⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡉⠀⠎⣌⠢⢄⠉⢁⠄⠁
⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⡄⠞⡀⠄⡄⠙⠐⠐⠄⠀⠂⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣐⠢⠔⠈⠀⠂⡀⠀⠄⡙⠣⢀⠄
⠀⠀⠀⠠⠤⠀⢀⡠⠄⠁⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠧⠂⣉⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠌⠀⠐⠀⠀⢔⠀⠀⠁⠀⠉⠘⠀⡀
⠀⠀⠀⠀⠠⠔⢀⠀⠀⠆⠀⡠⠂⠀⠁⠀⡀⠀⠰⡀⠀⢄⠀⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠯⠁⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠀⡸⠀⠀⠀⠊⠀⠀⢀⡇⠀⠙⠄⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆⢀⡃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⡠⠀⢀⠀⠀⠠⡇⠀⠀⠀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠤⡀⠀⠈⡃⠀⣰⠇⠀⠀⢸⠀⠀⠀⠃⡀⠤
⠀⠀⠀⠀⠀⠀⢴⠂⠀⠀⠀⢄⠀⠍⠔⢀⡀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠙⢶⠆⠣⠀⠘⠀⡎⠠⢇⣤⡴⠂⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠨⠁⠄⡄⠈⠁⢀⠠⠒⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢆⠀⠠⡠⢀⠀⠀⡡⠁⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠚⠂⠀⠀⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⠂⠠⠀⠁⠀⠀⠈⠠⠐⠐⠋
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⢰⠂⠀⠀⠰⠂⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⡠⠖⠛⢉⠂⢳⠀⠀⡐⠀⢠⠆⡀⠀⠀⣀⣔
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⣄⠐⠃⠃⢀⡀⠃⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠁⡄⠀⠀⡮⠐⠁⣠⡀⠁⢄⡀⠀⠙⠰⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠁⠀⠈⠅⡀⠁⠀⡄⠰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠈⠀⠀⠀⢸⠇⠀⠀⠑⠀⠀⡆⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⢈⠆⢡⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠢⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣡⣏⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢅⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠦⣀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢹⣶⣸⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠀⣰⡏⠁⠀⠀⢺⣇⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⢈⠽⠀⠀⠀⠀⢸⠿⠄⢺⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠀⠀⠀⡀⠀⢰⠊⣠⣾⠀⠀⠀⠈⠆⡡⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡆⠀⠂⠀⠋⠁⠀⠀⠀⠀⡀⠄⢰⣤⠰⢉⣠⣾⣿⣿⡇⠀⠀⠀⢀⡑⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄⠀⠀⠀⣀⡂⠰⣼⠀⠙⣃⣠⣤⣶⣿⣿⣿⣿⣿⡇⠀⠀⢀⡜⡃⣸⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⡀⠘⢿⣿⣿⣿⣿⣿⣷⣶⣤⣭⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡀⠙⢡⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⡇⠀⡂⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠐⠺⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠁⢀⠈⠂⠄⠀⣈⠙⠛⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠠⠵⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⡟⠁⠀⠀⢌⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢀⠀⠁⠐⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⣀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉⠙⡿⠿⠟⠛⡁⢀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢀⡀⠀⠂⠤⠀⠨⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠀⠀⠀⠀⠤⠀⠂⠉⢀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠻⠿⡿⠛⠛⠋⠁⡠⠀⠀⠑⠀⠀⣀⠀⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠁⠐⠂⠀⠉⣀⠀⠤⠒⠀⠉

⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠖⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠁⠀⠃⠀⠀⡸⠦⡧⢶⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⠀⠐⠀⢠⠀⠁⠀⠈
⠀⠀⠠⠀⠀⢀⠄⠀⠄⠀⠄⠀⠀⡀⠁⠀⠀⠈⠀⠠⠑⠘⢗⠸⠄⠀⢀⠔⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠄⠂⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡈⣶⠄⠀⠁⠀⠀⠁⠀⠠⠀⠤⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠘⡍⠃⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠄⠀⠀⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡄⠀⠀⠀⠀⡀⠑⠷⠠⠀⠀⠀⡀⠀⠀⠀⢸⠁⠀⠀⢰⠀⠀⠀⠃⠊⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠠⣄⠆⡇⢆⡤⠀⠀⠁⠀⠀⠀⢀⡠⢂⡀⠀⠀⣀⠤⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢠⡟⠛⣧⠸⣤⠀⠀⠦⠀⠀⠋⠀⠀⠘⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠠⠐⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⡛⡟⡣⣄⠀⠀⠀⡀⠀⠀⢦⡀⢀⡴⠁⠀⠀⠀⠀⠐⠁⡀⠀⠀⠲⠀⢀⣩⢠⠀⠴⠀⠀⠈⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠐⡨⣼⠶⣧⢅⠂⣀⡀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⢀⢻⠶⡟⡔⠀⠉
⠀⠀⠀⠬⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢤⠇⠀⠀⠀⠀⠠⠄⠀⠀⠼⠂⠈⣈⠘⠀⠲⠀⠀⠸⠀⠐⢆
⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠄⠀⡂⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⠀⢠⡄⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣀⠬⡬⠢⡭⢅⡀⠀⠈⠁⠀⠀⠈⠓⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣰⣑⢒⡃⡂⠀⠀⢀⣀⣶⣀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢹⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⢠⠏⠙⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⠤⠤⠤⠤⠤⠶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⣀⣀⣀⣀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠣⣤⣀⣀⣀⣀⣀⣠⠼⠉⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠒⠒⠶⠤⠤⠤⠤⠤⠶⠖⠒⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣴⡄⠀⠀⢠⣦⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢠⡄⣀⣴⢤⡄⡀⠀⠀⡏⠛⠻⢍⣉⠛⠛⣦⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⣼⣧⣿⡁⣼⠟⠁⠀⡀⢳⡀⠀⠀⠈⡝⠒⣣⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐⢾⣇⠙⣿⢸⣇
⠀⠀⠀⠀⠀⠾⠋⠉⠻⣿⣄⣦⣠⡇⠀⢳⣄⠀⠀⠈⠉⠁⠀⠀⠀⢀⣠⣾⡀⠀⢸⡀⣠⢀⣿⡾⠛⠻⣯
⠀⠀⠀⠀⠀⠀⠀⢛⣿⠛⠛⢿⣿⣧⠞⠉⠈⠉⠛⠒⠒⠒⡖⠚⠉⢹⠀⠀⠙⣟⢻⣿⣿⠿⠿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣹⠿⢤⣀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠘⣧⠴⠚⢉⣶⡟⠁⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠙⠓⠒⠒⠒⠺⡄⠀⠀⢹⡀⠀⠉⠁⢻⡓⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣞⣹⠆⠀⢻⡀⠀⠀⢳⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⢳⡀⠀⣀⡷⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠆⠀⠀⠀⠓⠊⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠳⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⢸⣿⣶⣠⣾⣿⠀⠀⠀⠀⠀⢠⣀⡆⠙⣿⡟⠁⣆⣠⠀⠀⠀⢀⣀⣄⠈⢻⡟⠁⢠⣀⡀
⠀⠀⢀⣠⣬⠏⠲⣰⢏⢫⣤⣄⠀⠀⠀⠠⠟⢻⢾⣽⣽⢾⠻⠧⠀⠀⠀⠀⠞⠛⣶⣾⣷⣶⠛⠻
⠀⠀⠈⠻⠷⡈⣵⠻⡍⣱⠿⠛⠁⠀⠀⠐⣶⣼⣞⣻⣛⣿⣴⡖⠀⠀⠀⠠⢤⣤⠿⣿⣿⠿⣤⡤⠄
⠀⠀⠀⠀⢸⣿⣶⠰⣾⣿⠀⠀⠀⠀⠀⠈⠉⠇⣠⣿⣧⡀⠏⠉⠀⠀⠀⠘⠉⠋⠀⣼⣧⡀⠙⠉⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠁

⠀⠀⠀⠀⢀⣀⠀⣶⡆⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣦⣀⡀⠀⠀⠀⠀⠀⠀⢰⣆⢳⡞⢀⡤
⠀⠀⠀⠀⠻⣜⣶⡟⣿⢾⡿⠃⠀⠀⠀⠀⢠⣿⣿⡟⠿⢛⣹⢷⡆⠀⠀⠀⢬⣧⣀⠻⢾⡇⡾⣁⣾⡅
⠀⠀⠀⣾⣻⣛⡏⠀⠈⣿⣿⣿⡆⠀⠀⢐⣿⣿⡄⠀⠀⠀⢠⣿⣿⠀⠀⠠⣤⣤⡭⢛⣾⣷⠋⣥⣤⣄
⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⡷⡄⠀⠀⠀⠀⢻⣿⡤⡄⣀⢠⡤⠿⡟⠁⠀⠀⠤⣴⢖⣭⢹⣏⣟⠦⣤⠄
⠀⠀⠀⠀⠈⠋⠀⣟⡿⠈⠙⠁⠀⠀⠀⠀⠈⠹⠷⣿⣿⣿⠮⠏⠁⠀⠀⠀⠈⠡⠟⠁⣸⡇⢻⡆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙

⠀⠀⠀⠀⠀⢠⣄⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⣆⠀⣠
⠀⠀⠀⣀⠀⡼⢿⠒⡿⢤⣠⣶⠀⠀⠀⠀⢸⡀⣿⠷⣶⡾⣿⢀⣇⠀⠀⠀⠀⣶⣬⣿⣾⣿⣾⣏⣤⡄
⠀⠀⠀⠉⡟⠓⢼⣷⣶⡉⢧⠀⠀⠀⠀⠈⣡⣽⠿⣶⠟⢶⠿⢿⣌⡑⠀⠀⢶⣴⣿⡿⠉⠉⠻⣿⣯⣴⠄
⠀⠀⢀⣤⡽⡎⢀⠉⡆⢨⠟⢾⠄⠀⠀⢤⣉⣻⣶⠟⣶⠞⣷⣟⣋⡤⠀⠀⣴⢾⣿⣆⠀⠀⢀⣿⣿⣦⡄
⠀⠀⠀⠛⠁⢹⣿⠒⢻⣏⠀⠀⠀⠀⠀⠀⢸⠁⣿⠾⠿⠷⣿⠀⡇⠀⠀⠀⢀⡾⠟⣿⣷⣿⣿⡿⢷⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠸⡿⠘⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣀⡹⠶⠖⠒⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠓⠒⠦⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠈⣆⠀⠀⠀⢀⡴⠃⠀⠀⣀⢞
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠴⠊⠁⠀⠀⢀⠀⠈⠑⠢⣄⣀⣀⣽⣀⠀⡼⠁⢀⡀⠀⡼⡇⠈⡇⠀⡰⠃⢸
⠀⠀⠀⠀⠀⠀⡎⠀⡴⠁⠀⠀⣠⠖⠛⠣⣌⠉⠲⢄⡀⣠⣠⣀⢸⠀⡷⠀⡇⠈⢺⡀⡇⠀⡇⢠⠃⠀⡆⢀⠤⢤
⠀⠀⠀⠀⠀⠀⢧⠜⠀⠀⢀⠾⠷⠖⠆⠀⠈⠦⠤⢀⡜⡱⠢⠈⢺⣠⠃⠀⠘⢄⠀⠳⠅⠀⠳⠎⠀⢸⠔⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⢾⡀⠀⢰⣒⡄⠀⠀⠀⠀⠀⠀⠀⣜⣰⠁⠀⠀⢠⠞⢦⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠉⠁⠀⠀⣀⠀⠀⠀⠀⢸⠁⣠⠔⢒⠧⠤⠴⠋⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣖⠃⠐⢇⣀⣀⡤⠤⣀⣀⣠⡆⢀⠴⢏⠉⠙⠢⡀⠀⠀⠀⡔⠚⠚⢕⣬⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⢴⡡⠀⠗⡄⠀⠀⠠⠤⠤⣤⠭⣑⣒⣒⡶⠏⠀⠀⠢⡇⠀⡴⠚⠳⡄⠐⠢⡑⢿⡦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢄⣤⠐⠉⢀⣠⢴⠎⠲⠖⠁⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠁⠀⡇⠀⠀⠀⠀⠀⢹⠎⠀⠀⠀⢘⣆⣀⣈⢢⠙⡍⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡝⠀⠘⠤⣤⢔⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡤⠞⠁⢠⠞⠀⠀⠀⠀⠀⣀⣀⣀⡠⢴⠾⠃⠀⠀⠀⠀⢀⡯⠤⠒⠉
⠀⠀⣀⡠⠖⠒⠒⠒⢢⠖⠒⠚⠉⠀⠀⠀⠀⡠⢴⠋⠀⢀⡴⠳⠤⠤⠴⠖⠚⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣠⠋
⢀⠞⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⡤⠞⠟⡽⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⠀⣠⠞⠁
⠘⠦⠖⠊⠉⠢⢄⠀⢠⠝⠦⠤⠤⠒⠊⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⢆⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠉⠉⡩⠛⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢒⠁⡀⠀⠀⠜⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠳⡦⠤⠤⠴⠒⣖⣚⣻⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⣉⣉⣉⣉⣉⡠⠤⠤⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠢⡄⠀⡀⠀⠀⠀⣰⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⣀⣀⡀⢠⠏⠀⠀⠀⠀⠀⠀⢹⣎⢀⠄⠀⠤⣿⣿⣷⠤⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡟⠑⠂⠤⢀⣀⣀⠤⢻⠀⠀⠀⠀⣰⣿⣿⣿⣗⠀⠀⡜⠀⠀⠀⠈⣭⣿⣿⣿⣥
⠀⠀⠀⠀⠈⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⡍⠀⠀⠀⠉⡏⠀⠀⠀⠐⠉⠹⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡔⢦⢸⣂⡄⢰⡆⢠⠂⠐⠈⡇⠀⢺⣠⣴⠃⡴⠑⡞⢠⡖⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠇⠸⣇⠘⠲⣄⡀⠧⡀⢼⠃⠃⢀⢿⡄⢧⡈⠳⢤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡠⠞⠘⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⣹⠆⠀⠀⠀⣴⡤⠀⠀⠀⠀⢰⠊⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠉⠁⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⡀⠀⠀⠀⠀⢰⠇⠀⠀⢠⠖⠒⣆⠀⢀⣶⣿⡿⣿⣿⣷⠀⠀⣀⣠⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢷⠀⠀⠀⠀⡿⠀⠀⠀⡿⠀⠀⣼⠀⠈⢽⡿⣿⢿⡿⠿⠖⠀⣬⣍⣀
⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⢠⡄⠀⣇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⢠⡄⠀⣻⠞⣡⣄⠀⣴⡷⠀⣾⠹⠀⢀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣸⠃⠈⣧⣰⠛⣿⠃⣸⡟⠻⣤⠞⢉⣧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠙⣰⢻⡇⠀⠀⢠⡟⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠙⠛⠁

⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠶⠀⠀⠀⠀⢴⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⡠⠤⢄⠀⠁⠀⣀
⠀⠀⠰⠾⠀⠀⢀⢼⡗⡀⠀⡊⠑⠦⡀⠀⠀⠀⠀⠀⠾⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⢎⠆⣉⠢⢱⠀⢌⠀⠈⢆⠀⠀⠀⠁⠀⡀⠀⣀⡀⠀⠀⠀⠀⢠⡄
⠀⢀⠀⠀⠀⢀⣎⠜⠱⣱⠀⠑⠲⠤⠬⢑⡒⢤⠀⠀⢀⠀⠀⢀⢾⢧⠀⠀⠛⠀⠀⢸⢸⠘⠟⢁⠇⡇⠈⠒⠦⢤⡉⢢⠀⠀⡼⣧⠀⠧⣈⣑⠢⡀
⠀⠀⠀⠀⠀⡞⠁⠰⠆⠈⡇⠀⠀⠀⠀⣀⡈⢦⡇⠀⠀⣀⣠⣎⣎⣈⡤⣀⠀⠀⡖⢒⡡⠁⠀⠘⡘⠒⢢⠀⠂⠀⢧⡌⠀⣰⠃⠚⡄⠀⢀⢀⣱⢦⠀⢀
⠀⠀⡄⠀⠀⡇⠰⡆⠰⠄⡇⣠⢾⢿⣾⣿⠏⢿⡇⠀⠀⡏⠈⣉⠉⢁⡁⠈⣁⡀⣹⢀⢶⣄⢀⢴⢌⠙⣁⠔⡨⠒⠿⣹⢀⣧⡤⣄⣥⢚⠕⣽⠪⣣⠀⠈
⢀⠔⣿⡄⠀⠀⠐⠂⠐⠂⡿⠙⠓⠛⠛⠊⠠⡄⢱⠖⠲⠃⠘⠛⠀⠘⠛⠀⠓⠚⢻⠸⠽⠍⠸⠽⠽⢸⠕⠉⠀⣶⡆⠈⠺⡟⠧⠗⢛⠉⠀⠉⠀⠙⠁⣀⣴⣄
⠠⢊⣟⢪⠀⠀⠘⠃⡎⡆⠇⢠⡄⢠⢦⠀⠠⠄⢸⠀⣶⡆⠀⢀⣀⣀⠀⠀⣶⡆⢸⠀⠀⢰⠛⡆⠀⢸⠀⣠⣄⠉⢁⣤⠀⣵⠶⡖⣧⠘⠃⢀⠘⠃⡿⠍⠀⠒⢳
⢘⠕⡿⡱⠀⣿⡘⢃⢙⠃⢀⠈⡁⠈⡉⠀⡠⣦⢸⠀⠉⠁⡔⠁⢈⠀⠙⠆⠀⠀⣸⠀⠀⣸⠤⡇⠀⢸⠀⣻⣿⠀⢸⣽⠅⣟⣛⣛⣹⠀⢀⢾⡇⢠⣇⠸⣿⡇⢸
⠰⠊⡟⠮⠀⢿⡂⣿⣹⠀⢼⠰⠕⠸⠇⠀⣇⣼⢸⠀⠀⢀⠁⠒⢸⡔⠆⢰⠀⠠⣿⠅⣠⣏⣈⣳⡀⢸⠐⠏⠿⠀⠾⡙⠇⡻⣯⣥⢿⠀⠐⢻⠓⢐⣗⠀⡄⠄⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⣤⣄⣴⠞⠉⠁⠀⠀⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣤⣼⡋⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠶⠶⠶⢤⣄
⠀⠀⢀⣠⣴⠛⠉⠉⠛⠁⠀⠈⠿⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⣦⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠈⠳⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠞⠋⠉⠛⢷⡴⠶⠶⠶⣤⣸⡇
⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢶⣄
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⡶⡶⡶⢾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢟⡟⣟⠇⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢽⡮⣥⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠃⠀⠀⠀⠀⢼⣿⡧⠀⠀⠀⠀⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠼⢳⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠸⣺⢿⡺⠂⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣴⣷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⠻⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣰⣿⣇⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣸⣿⡇⠀⠀⠀⠀⣠⣤⣸⣿⡏⣠⣤⠀⠀⠀⠀⠀⣿⣿⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣿⡇⢠⣶⡄⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⣴⣶⠀⣿⣿⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣿⣾⣿⡇⢺⣿⣦⣸⣿⡏⣠⣾⣿⡆⣿⣿⣾⡿⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣧⣿⣿⠻⣿⣿⣿⡿⢻⣿⣧⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣿⣿⣿⡀⢸⣿⡏⠀⣸⣿⣿⣿⣴⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣦⡀⠸⣿⣦⡈⠻⣿⣯⡉⠉⠻⣿⣾⣿⣧⣾⡿⠋⠉⣩⣿⡿⠋⣠⣾⡿⠀⣠⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣾⣿⣿⣶⣾⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⢡⣿⡿⠋⣡⣿⣿⠋⠁⣨⣿⣿⣿⡿⣿⣯⡁⠈⠻⣿⣧⡉⠻⣿⣧⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠸⢿⠿⣿⣿⣿⣿⠋⢸⣿⡇⠈⢻⣿⣿⣿⠿⠿⠿⠀⠈⠋⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣴⣾⡿⢿⣿⣠⣾⣿⣿⣦⣸⣿⡿⣿⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣩⣿⣿⣿⡅⢸⣿⡿⢻⣿⡟⠻⣿⣿⡁⣾⣿⣿⣿⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⡏⢸⣿⠃⠈⠉⣠⣾⣿⣿⣦⡈⠉⠀⢿⣿⠈⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢻⣿⡇⠀⠀⠀⠀⠸⠿⢻⣿⡟⠻⠿⠀⠀⠀⠀⠀⣿⣿⠻⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢻⣿⡟⠻⡿

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡄⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⣀⣀⠀⢀⠔⠉⠀⢁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠰⠄⠀⢸⡠⢇⡀⡸⠷⢒⡠⠂⠁⠀⣀
⠀⠀⠀⠚⠀⠀⠀⠀⠀⢠⠂⠐⢢⣀⣀⠚⠐⠣⡀⠠⠃⢦⠀⣸⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠒⠚⠒⠒⠑⠀⠠⠦⠶⠀⠤⠤⢈⡐⣇⣈⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⢐⡄⠀⠀⠀⠀⠀⠀⠀⢸⡔⠂⠲⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠐⠛⣥⣰⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢲⢦⣁⠉⢀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠌⠀⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡰⠀⠁⠒⣛⠋⠄⠀⠀⠀⠀⡜⠁⠈⢀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠓⠒⢲⣲⠒⠀⠉⠀⠀⠀⡜⢁⣉⡀⡠⣐⠀⠀⠐⡦⢀⠠⠀⠄
⠀⠀⠀⢚⡚⣦⠀⢀⠠⠒⠠⡀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⢶⣀⢁⣀⠌⠁⠈⠍⠈⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠊⠉⠛⠉⠕⠛⠂⠄⣀⡘⠉⢽⣖⠋⠠⢬⣋⡀⠂⢈⠉⠉⠀⠈⠉⠋⠉⠙⠚⠉⠁⢀⠙
⠀⠀⠀⠠⠀⠀⠤⣀⡂⣤⡠⢅⡠⠍⠠⣅⣄⡀⣀⠁⢀⠀⠀⠠⠍⠍⠀⠐⢖⠂⢠⣠⣄⣐⣡⡄⠀⠁
⠀⠀⠀⠭⠨⠙⠁⠀⠊⠀⠀⠈⠃⠐⠀⠀⠀⠈⠉⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠑⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠥⠢⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠔⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⢀⣀⡤⠶⠖⠢⢄⢨⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠐⠒⠊⠉⠀⠀⠈⠀⠀⢠⠇⠀⠀⠀⠀⠀⠂⢈⠑⠢⠄⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢿⣋⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠾⡟⠉⠀⠀⢀⠕⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠰⣤⣴⣶⠶⠾⠿⠿⠿⠟⠛⠋⠉⠛⢦⠀⠣⠤⠀⠂⢁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⢏⠁⠀⠀⠀⢀⡠⠔⠊⣁⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣁⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⡹⠛⠛⠀⠀⠈⠙⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⣖⡒⠒⠤⠄⠂⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡟⠛⡿⠿⠿⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣄⣀⣀⠈⠉⠀⠒⠒⠂⠤⠤⠤⠜⠁⠸⣀⠀⠀⠀⠈⡇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠦⠤⣤⣤⣀⣀⣀⣀⣀⣀⣉⣀⣈⣁⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢠⣔⡉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠰⠒⠒⠒⠀⠤⢤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠠⢀⣃⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢨⠋⠀⠒⠂⠰⠄⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠈⠉⠁⢒⡚⠂⠤⠤⢤⣄⣤⣀⠀⠀⠀⠀⢐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠉⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠜⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⢀⣀⠤⠂⠁⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢓⠒⠒⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠈⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⢀⠆⠀⠀⠀⠀⠀⢸⠀⢀⠊⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠉⠉⢀⠔⠡⠁⠀⡘⠀⢠⠀⠀⠀⠀⢸⠀⠈⠢⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⠢⡀
⠀⠀⠀⠀⠀⢀⠔⠃⢠⠁⠀⠀⠡⣄⠀⠁⠒⡲⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡏⠒⢄
⣀⣀⡀⣠⠔⠁⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠔⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡅⠀⠀⠙⠲⡒⠒
⠀⠠⠔⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣵⢏⣿⣿⡇⢹⣿⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⣾⣿⡟⢿⠈⣿⣧⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣠⣿⣿⣇⠸⡇⢿⣿⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⣿⠀⢻⣾⡼⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠐⣓⣦⣿⢹⠃⢻⡄⠈⢿⡇⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢭⡽⠇⣿⠀⠈⢷⡀⢸⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⢛⡿⠀⠀⠈⠁⠀⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⣋⣱⠆⡀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢾⠋⣢⡄⠀⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠅⢰⠟⠧⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠈⠛⠦⣂⠤⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠚⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠀⠀⠈⠀⠀⠀⠀⠀⠁⠐⠤⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠀⠂⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠀⠀⠀⢀⣀⣀⠀⠠⠤⠀⠀⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣻⡛⠶⣤⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠧⣬⣯⣙⡷⣎⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡛⠶⠶⣄⣠⠛⠧⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣭⣋⡛⠶⣮⠙⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠿⣭⡙⢶⣍⠻⡆⠄⠂⠀⠀⠀⠀⠀⣀⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠉⠁⠀⠀⠀⠀⠀⣀⡴⠞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠞⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⠨⠐⠁⠀⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢀⣀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⡀⠠⠄⠀⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣶⣧⣽⣼⣷⣶⣧⣽⣽⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣷⣾⣿⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠈⠀⠀⠀⠤⢡
⠀⠀⠀⠀⠀⠀⠀⠐⣁⣀⠤⡀⠀⠔⠒⢷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠘⠐⠒⠚⠐⠢⠄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⡎⠁⡁⠀⢱⡀⣈⣄⣈⣀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⢄⡀⠀⣰⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠠⣀⠠⠀⠀⠀⠀⠀⠉⠐⠁
⠀⠀⠀⠀⠀⢀⡧⠆⡠⠖⠚⠉⢁⣀⣀⠀⠀⠁⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣯⣾⢤⡀⠀⠀⠀⠀⠀⢀⣠⠗⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⡀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⣞⠁⠀⠆⠀⠠⠀⠀⢀⠀⢀⡉⡢⡽⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠁⠈⠙⠫⣗⢤⡀⣠⢔⣋⢀⣀⠀⠀⢀⠂⠐⣠⠀⠀⠀⠀⠈⠢⣀⠀⠈⢂
⠀⠀⠀⠀⠀⠈⢦⠤⢤⣠⠉⠁⠀⠁⠐⠀⠠⡀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠵⢮⣳⡁⠘⣌⠆⠳⣄⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠋⠀⠀⢃
⠀⠀⠀⠀⠀⠀⡇⠰⠀⠁⠀⠀⠀⠀⠈⠈⠁⠸⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣴⣶⠀⣠⣤⠐⡄⠀⠈⠉⠀⠸⣀⡀⡀⠙⠣⣄⠀⠀⠀⠀⠀⡠⠋⠀⢀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⠓⠚⡇⠀⠁⢀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡌⠀⠀⠀⠀⢀⠨⢅⠀⠉⠁⠀⠈⠢⠤⣤⠴⣀⢈⠁⠈⠢⣀⠀⠙⠁⠀⠘⠻⠛⠃⠂⠥⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⠀⠀⠀⣀⡤⠊⠀⠀⠠⠂⠁⠂⣀⣉⣀⡀⠀⢀⡇⠀⠀⣀⡀⠀⠀⠈⡇⠊⠥⣀⠀⠀⠀⠉⠳⢄⣀⢀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡸⠈⠢⠄⠀⠸⠅⠐⠀⠀⢠⠀⠀⠉⡂⠀⠀⡇⢴⠆⢀⣤⢠⡶⠉⠀⠀⠀⡰⠃⠀⠀⠀⠙⠲⢤⡀⠀⠀⠉⠁⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠐⡨⠊⡩⠓⠀⠤⠀⠀⠠⠀⡧⠄⠊⠀⠱⡀⢀⡕⠀⠊⡀⠀⠀⠈⠁⠀⠀⠀⠀⡠⠊⠠⡀⠀⠀⠀⠀⠀⢀⡉⡳⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⡀⠊⡐⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠇⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢯⡀⠔⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠸⡤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠘⢤⡀⠀⢔⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⡓⠠⠤⠤⠤⠤⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠃⠙⢀⠀⢀⡴⠀⠀⠀⠤⠎⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠶⠒⠂⠀⠀⠀⠀⠀⠀⠘⠀⢨⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢱⠀⠀⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠊⠀⠀⠀⠀⠀⢠⡠⠞⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢏⠀⠀⠀⠀⠀⠀⡆⠀⠰⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠑⠒⠙⠒⠋⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⢀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⢀⣀⡀⠀⠀⢀⣸⡤⠄⠀⠀⠀⣄⣀⠵⢠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠑⠞⠉⠙⣎⢙⠢⡴⠤⠤⠤⠐⡟⠣⢄⣀
⠀⠀⠉⠙⠋⠁⠀⡇⠀⠀⠀⢀⡀⡇⠐⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⢀⠇⠀⠀⠀⠀⠸⣯⠄⣧⠒⠀⠀⠀⢇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠠⡇⠍⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⠟⠁⢹⣉⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠧⢄⣀⠀⠉⠑⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢣⠀⠀⡠⠚⠁⣀⠴⠋⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠐⠧⠤⠤⣀⣀⣀⣀⣀⣈⣉⠿⠁⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣠⣟⡒⠯⣠⣔⣊⣁⣀⡠⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠒⠉⣑⠲⢤⠉⠉⡉⠑⠂⠀⠀⣠⠤⠤⠾⠯⣄⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡣⠄⠀⠀⠀⠲⠆⣤⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠾⠰⠀⠀⠀⢀⡄
⠀⠀⢀⣃⠀⠀⠀⢨⡑⢣⠀⠀⠠⠄⣄⠳⠆⠀⠀⢓⡀⠀⠀⠀⢀⡀⠀⠀⠀⣐⡁⠀⢠⠀⠀⡭⠋⡲⠍⡀⠀⣐⠓
⢠⠀⢐⠊⠀⠀⠐⠌⠠⢃⣀⢀⣤⠴⠆⣄⡃⠀⢰⠰⠂⠀⠀⠀⠸⡇⢀⡆⢘⠂⠌⠀⡬⠀⣄⠐⠫⠅⣨⣁⠠⢄⠊
⣸⠀⡄⠠⢁⣳⠖⠪⡥⢠⡠⠀⠤⠀⢀⣈⠁⠀⣚⠓⠾⠀⣶⡀⠯⡀⣸⡔⠢⠌⣒⣠⡚⡂⠰⠚⠀⠀⣀⣬⡐⢀⠉
⡠⡧⠄⣁⢀⣺⠻⡤⠊⠠⠜⠃⠀⡛⠀⠉⠐⣧⠖⡈⢊⡥⢉⣟⠓⣃⢿⣨⠁⢢⠤⠴⣪⢳⣤⡄⠻⠃⠀⢀⡴⢐⠉
⢌⣬⡃⢁⡛⠧⠆⠶⠀⠔⢘⠛⠉⠉⠷⠂⠶⠘⠚⢀⡈⠡⢵⡮⠉⢰⢮⢅⡡⠄⠛⡤⢓⠽⣯⢄⣠⡄⠛⠛⠀⠛⢤
⡥⠂⡄⠂⣼⡛⠁⠶⠐⠲⠛⡶⠖⠃⣀⠀⢠⣄⣨⡵⡀⢁⣯⠙⠉⠀⠉⠩⠹⠄⠦⠐⠛⠓⢀⢈⣁⢠⣤⡄⠙⠓⡞
⢽⡉⠏⣨⠞⠛⣁⠐⠐⢿⡆⠀⢀⣀⡈⠟⠀⠀⢈⣙⣧⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠉⠁⢨⡄⣤⠶⣤⡿
⠀⠀⠁⠉⠻⠶⢖⡂⢶⠂⠚⠃⠈⠉⠁⠦⣀⣀⡌⠁⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠻⠖⠠⠔⠂⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠄⠴⠛⢦⡴⣉⡈⢤⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠚⠞⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣦⣀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⠟⠛⠿⣿⣿⣿⣷⣄
⠀⠀⠀⣴⣿⣿⣿⠟⠋⠉⠈⠉⠻⣿⣷⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠈⠙⣿⣿⣿
⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⣿⣷⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠙⣿⣆⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠈⢿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⡾⠋
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⢿⣟⠙⠿⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡛⠃⠀⣿⣦⠀⢠⣴⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣥⣄⡀⠉⠋⠀⠺⠛⠁⠀⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿
⠀⠀⠀⠀⢸⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⣀⣀⡀⠀⠙⣟⣿⣿⣿⣿⣿⣿⣿⠁⢀⣤⣤⣶⣤⣤⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣾⣿⣿⣿⣿⣷⣄⣼⣷⣾⣿⣿⣧⣿⣇⣰⡿⠿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣁⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣡
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⡇⠼⠈⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣘⠂⢦⠀⡀⠂⠀⢀⠂⠠⠀⡤⢒⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠐⠊⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠌⠐⠂⡀⠈⠀⢀⠤⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡀⢀⠂⠉⠀⠇
⠀⡀⠀⠀⠈⠁⠀⠔⠂⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣀⠠⠀⠀⡄⠉⠒⠂⠄⡰⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠒⠋⠀⠀⠀⠀⠀⠒⠉⠀⠙⠐⠄⢴⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⡇⣹⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡮⠵⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠰⡆⠉⠀⠀⠉⢰⡄⢀⠀⠀⣠
⠀⠀⠀⠊⢲⠆⡀⢀⠀⠰⢸⠆⠀⡀⠀⠠⡤⠕⠂⠀⠀⠀⠀⠀⠐⠀⠘⠓⢬⡁⠁⠀⠀⢀⡇⠀⠀⢀⡀⠌⠚⢭⡑⠃⠐
⠀⠀⠀⠀⠀⠀⠠⠟⠂⠴⠀⠤⠀⠥⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠐⢤⣀⠘⣇⣠⠟⠉⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣂⠀⡀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⠙⢙⡄⣔⠒⠳⣖⠀⠘⠓⠂
⠀⠀⠀⠀⣠⠀⠊⠑⠀⢀⡀⠁⠈⠗⠠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢀⠤⠖⠛⠀⢸⠆⠀⠈⠁⠀⣇
⠀⠀⠀⠹⠁⠁⠀⠀⠀⠈⠇⠁⠀⠀⠀⠈⠉⠪⠄⠀⠀⠀⠀⠀⠀⠀⠐⡡⠒⠑⠆⢠⡄⠀⠀⠺⠀⢠⡀⠴⠸⠆⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠑⢒⡄⣤⠐⢈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠈⠢
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⢠⠄⠢⡄⠀⢰⡀⠀⠀⢰⡀⡟⠉⠉⡇⠀⠀⢸⢸⠋⠁⠀⠀⠀⠉⢹⡏⠉⠉⠁⠀⣤⠀⠀⢀⣄⠀⠀⠀⠀⠀⢾⠀⠀⠀⡏⠉
⠀⢀⡏⠀⠀⠀⠀⢸⣇⣀⡀⣸⡇⣿⣠⠜⠁⠀⠀⢸⠈⠳⠤⢤⡀⠀⠀⢸⡇⠀⠀⠀⠀⡟⢧⢀⠎⢹⠀⠀⠀⠀⡜⢸⡇⠀⠀⠻⠦⠤⣀
⠀⠸⡇⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇⢿⠙⢦⣀⡀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⢀⡇⠘⠟⠀⢸⡆⠀⠀⡜⠛⠛⢷⡀⠀⠀⠀⠀⠈⡇
⠀⠀⠙⠦⣤⡤⠶⠸⠁⠀⠀⠘⠁⠸⠀⠀⠈⠉⠉⠸⠀⠳⠤⠤⠞⠀⠀⠀⠇⠀⠀⠀⠸⠃⠀⠀⠀⠀⠳⠄⠞⠀⠀⠀⠈⠳⠄⠙⠶⡤⠴⠃
⠀⠀⣠⡴⠶⠶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⢠⣾⡟⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣀⣴⣀
⣾⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⢰⡟⠀⠀⠀⢹⠿⣏⠁⠀⠀⠀⠸⠟⠏
⠻⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢀⣴⣶⡆⠀⠀⠀⣼⣿⠃⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠈⠛⠛⠁⠀⠀⣼⣿⠃⣠⡟⠁⠀⠀⠀⢶⣷⠆⠀⠀⠨⠿⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⡀⠀⠀⠀⠀⣸⣿⢇⣴⠏⠀⠀⠀⠀⠀⠊⠉⠂⠀⠀⣀⣠⣤⣀⠀⢺⣿⡖⠀⢰⣿
⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣰⣿⡟⠀⠀⠀⢰⣿⡿⠟⠁⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⢸⣿⠀⠉⠈⠁⢀⣿⣿
⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⣼⣿⠟⠀⠀⠀⢀⣿⡿⠁⠀⣾⠋⢻⣷⠀⠀⠀⠀⣴⣿⠏⠀⠀⣠⣾⠏⠀⠀⠀⣠⠞⠘⣿⣇
⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢰⣿⠏⠀⠀⠀⠀⣸⣿⡇⠀⠸⣿⠀⠀⣿⠀⠀⠀⣼⣿⠃⣀⣴⡾⠟⠁⠀⠀⣠⡴⠋⠀⠀⣿⣿⡀⠀⠀⢀
⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠻⣆⣸⡟⠀⢀⣴⣿⡟⠋⠉⠁⠀⠀⠀⢀⣼⣿⠙⢦⡀⠀⢸⣿⡇⠀⣠⠋
⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣿⣿⣀⣀⣠⠾⠁⣿⣿⠀⠀⠀⠀⣸⠟⠓⠚⠉⠘⣿⣧⠀⠀⠀⠀⣀⡴⠟⠹⣿⡄⠀⠉⢓⣿⡿⠓⠉
⠀⠀⠀⢿⣿⣇⣀⣤⠞⣿⣶⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠘⢿⣷⣤⣴⠞⠋⠀⠀⠀⠀⠀⠈⠛⠻⠶⠚⠛⠉⠀⠀⠀⠘⠻⠷⠶⠿⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠀⠀⢻⣿⣧⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⠀⠀⠀⣀⣠⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣷⣀⡀⠀⠻⣿⣷⡀⠀⠀⠀⢾⣿⣷⠀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠘⢿⣿⣦⣀⠀⠀⠀⠈⠙⣛⣿⣿⣷⣶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠿⠟⠛⠛⠿⠿⢷⣾⣿⣿⣦⣄⣀⣈⣛⣿⣷⣤⡀⠀⠀⠙⠿⣿⣦⣄⣀⣀⣀⣩⣿⣿⣿⣶⣾⠿⠿⢛⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠛⠋⠉⠛⢛⣻⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡿⠛⠛⠛⢉⣽⣿⡿⠋⠀⠀⠀⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠰⣿⣿⠟⠁⠀⠀⠀⠠⣾⣿⡿⠉⠀⠀⠀⠀⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣏⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⠋⠉⠉⠉⣹⢦⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠶⠒⠚⠛⠛⠋⠙⠛⠛⠛⠒⠒⠶⠤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠿⠃⣠⠤⣄⡀⠻⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢧⣀⣀⣉⡓⢦⡀⠀⠀⡼⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠝⠉⠉⠉⠀⢀⣼⠁⠀⠀⠀⠀⣸⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⣄⡀⠀⢀⣴⡋⠳⢤⣀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠈⢳⡀⠀⢀⣠⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⣳⢯⣀⠙⠶⣄⡈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢳⡞⠉⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠈⠛⠶⢤⣉⣓⠲⠦⠤⣤⣤⣤⣤⡶⠚⠻⡏⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⣍⠉⠉⠉⠀⣷⠀⠀⢹⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠀⠀⠀⠀⢸⡄⠀⠀⢷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠘⡇⠀⣀⣸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠟⠀⠀⠀⠀⠀⠛⠋⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠘⠲⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⣀⣠⣤⠴⠖⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⣹⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡷⠁⡿⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢹⡴⠁⠀⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠀⠠⢷⠒⠒⠒⠶⢤⠀⠀⠀⡰⢻⣸⠁⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠉⣍⡷⠒⠚⢛⢢⠀⡠⠞⠓⠒⠒⢧⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡶⠛⢋⠹⡄⠀⠀⠀⡇⠀⠀⠈⢸⣴⡇⠀⠀⠀⠐⢻⡟⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠘⢬⣓⠶⠤⣭⣙⣛⣩⣓⡒⠂⠤⠾⢋⣓⣒⣈⣚⡭⠭⠗⣚⡽⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠓⠒⠶⠦⠬⠭⠭⢭⠭⠭⠭⠭⠥⠶⠖⠒⠚⠉⠉⣇⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀⠀⠪⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⡇⠀⠀⠀⠱⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⣿⣹⠿⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢴⡶⢾⣿⣗⠚⠇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⣠⠇⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢲⢏⣿⡈⠿⠂⠀⠀⠀⠀⠀⣀⢸⡏⠉⠉⠉⢀⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⠚⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠛⠛⠋⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⣼⣦⡀⠀⣠⣄⢤⠀⣔⣦⣀⡠⠀⠠⠀⠁⠀⣠⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢒⠀⠀⠀⠀⠀⠀⠀⠐⣺⡘⣿⣿⣶⡟⠉⠈⠀⠁⠈⢻⣇⠀⠄⣠⠀⠘⢣⡀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠠⢻⠑⠶⠆⠀⠀⠀⠀⠀⠀⣠⠁⠛⠹⠋⠙⢧⣐⣘⡀⠀⣀⣉⡉⣐⣴⣧⠀⢀⣸⣿⠟⢁⣺
⠀⠀⠀⠀⠀⠀⠈⣹⠆⣰⣦⠀⠀⠀⠁⠀⡄⠙⠿⠅⢿⠇⠞⣉⠈⠉⠉⠞⠻⣷⣾⣿⣾⢿⣤⣿⡿⠃⠀⢸⣿
⠀⠀⠀⠀⠂⠀⠻⡟⡀⣠⠛⠀⠀⠀⡼⠗⠀⠠⠤⡥⠀⣻⣿⢯⣰⢻⡄⣿⣦⣈⣭⣏⣿⣿⣿⡿⠀⠀⣠⣼⣿
⠀⠀⠀⠀⠖⣠⢼⡃⣸⣿⣢⣬⣤⢤⣠⣄⣀⡄⠀⢁⣮⠟⢡⣠⡌⠈⠀⠀⢽⣿⣿⢛⡉⣽⠛⠁⡀⠀⢙⣛⣿
⠀⠀⣼⠀⢢⣿⣿⢁⣈⣹⡟⠋⠀⠀⠛⠁⡈⣲⡎⠀⢁⣴⣿⠋⠀⠀⢸⣾⣮⠙⣿⣿⣿⣿⣶⣾⣷⣶⣿⡷⠏
⠀⠫⣟⠀⠛⣹⢟⣹⣿⣾⣷⣸⣾⠃⣤⣼⡃⢈⣿⣿⣿⠛⡛⣴⡆⣾⣼⣿⡟⠳⣿⣿⣾⠿⣿⣥⣾⠋⠁⣀⣸
⠀⣿⣟⠔⢛⣿⣸⣿⣿⣿⠟⣉⣷⠈⠛⡉⡷⠊⢻⣿⣿⣿⣿⡏⠀⠀⣹⣿⡇⠠⠙⣿⣿⢳⣿⣁⣽⣧⣼⣽⣿
⠀⣿⠯⣾⣿⣿⣿⣿⣿⣧⣾⣿⡇⠀⢀⣿⣿⣦⣼⣿⣿⣛⣿⡵⣾⣶⣟⡟⠀⣰⠆⣿⣿⣾⣿⣿⣿⣿⣿⠿⠟
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣦⣾⣿⣿⣿⣿⣻⣭⣴⡾⣿
⠀⣿⣿⣿⣿⡿⣿⣿⣏⡄⣰⣷⣧⣸⣯⣿⣻⢛⠻⣿⣟⣿⡿⠉⢻⣿⣿⣿⣿⣿⠷⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⣿⣿⣿⣿⣴⣿⣿⣿⣧⠈⣽⣿⢃⣾⣷⣌⣿⢦⣿⣿⣿⠃⠀⢺⣿⣿⣶⣿⣯⣦⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⢈⢹⣷⣶⠏⢹⣯⣿⣸⣿⣿⣿⠄⢠⣴⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠠⣿⣿⣿⣿⣿⣿⣿⡟⠁⣮⣿⡏⠁⢰⣾⣿⣿⣿⣿⣿⣧⣾⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣾⣿⣿⡟⠈⣩⣻⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠉⠉⠈⠁⠀⠉⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠉⠛⠉⠓⠋⠀⠉⠉⠉⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠐⠁⠀⠀⠒⠂⠀⠀⠀⠐⠂⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠒⠂⠀⣀⣀⣁⣁⠀⣀⣀⣈⣁⡀⠀⠀⠒⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠔⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⡠⠔⠒⠘⠉⠉⠉⠁⠀⠀⠈⠉⠉⠙⠓⠒⠠⢄⡀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⢀⣄⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠻⠿⡣⠖⠉⠀⡐⣱⠀⠘⠿⠇⠀⠀⠀⠀⡸⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣔⠦⣀⡈⢹⠀⠀⠀⢘⣄⠀⢀⠄⠀⢝⠀⠁⠙⣏⡩⠀⠀⠀⡠⠀⢠⡀⠀⣠⡃⠀⠀⠀⡏⢁⣀⠴⣢
⠀⠀⠀⠀⠀⣠⠕⣀⣅⠸⢄⠀⢀⠋⠈⠣⣈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⣀⠔⠁⠙⡀⠀⣠⠇⣨⣀⠪⣄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠲⢍⡲⢆⠀⠀⠀⠉⠒⠤⠄⢀⣀⣀⣤⠒⠒⠒⢤⠒⠈⠀⠀⠀⡠⢖⡩⠖⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠑⡄⠀⠀⠑⡄⠀⣠⠔⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⠦⢤⡄⣀⡀⣠⠃⠀⠘⡄⠀⠀⠘⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⡀⡠⢧⡀⢀⣠⠇⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠈⠀⠀⠀⠀⢀⠔⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⠘⢅⡠⠇⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⠤⠤⠤⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠽⡶⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠒⠒⠚⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠖⠒⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠰⠒⠊⠒⠘⠁⡀⠀⠀⠀⠀⠀⠀⣀⠂⡀⣉⢀⡠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣥⣶⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷⢦⠰⠔⠆⠢⠈⠒⢁⠀⠀⠀⠀⠀⠀⠄⣥⡤⠤⠄⢀⡀⢁⡀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⠿⠟⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⡀⢄⡠⣀⠤⡰⠢⠈⢃⠀⠀⢀⡄⠠⡄⠢⠔⠢⠈⠒⠁⢊⡉⠀⠀⠀⠀⠀⠀⢺⣧⣴⣤⣴⣶⣶⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢈⠀⡀⣀⠠⢄⠉⠚⠁⠀⠀⡠⢀⠀⢀⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⣍⣩⣭⣽⣿⡿⣿⣿⣿⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠐⠁⠘⠋⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠑⠈⡗⠊⠀⠐⠂⠁⠀⠀⠀⠀⠐⢶⣶⣿⣿⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡒⢛⣲⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⢿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⡞⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠖⠒⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢸⣯⠞⠀⠀⠀⢾⡼⠀⠀⠀⠿⠇⠀⠀⣀⣑⠒⠚⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠀⣾⡇⣰⢻⣧⠀⡸⠛⣶⡆⢀⣶⡆⠀⢸⣿⠈⠃⢀⣿⡇⠀⣿⣇⣴⡄⣴⡇⠀⣰⡟⠙⣶⡆⠀⣿⡇⠙⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢿⣇⢀⡜⢹⣷⠇⣿⡇⡰⠁⢸⣿⢀⠞⣿⡇⣰⠃⠻⣧⣀⠞⣿⣇⡜⣿⠟⣿⡿⣿⡇⡰⣿⡇⣸⣿⠁⡜⠉⢿⣆⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⢿⡷⠁⠀⠈⠿⠋⠀⠻⠷⠃⣶⠒⢻⡇⠀⠙⠋⠀⠀⠀⠉⠁⠛⠟⠁⠈⠛⠁⠻⠋⠀⡖⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣩⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠿⠇⠀⢠⣿⡵⠃⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣧⡇⠀⣴⡆⠀⢸⣿⢀⡾⢿⡆⠀⣼⣏⣸⠀⣸⣿⠘⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣸⠓⠊⣿⡇⢠⠟⣿⣸⠁⢸⡷⠚⢻⣏⠀⡰⠉⠻⣧⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠹⡷⠋⠀⠈⢻⠞⣽⠃⠀⠈⠛⠋⠁⡖⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠃⠀⠀⠀⠀⠀⠀⠈⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢻⣏⢋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡕⢆⠁⠈⣉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣋⣀⣻⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⣩⣽⣿⡿⢿⣿⡏⢻⡏⠉⠉⠉⠙⠛⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⣸⠟⠉⠉⠀⢘⡿⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣿⣿⣿⣶⠀⠀⠀⠀⠀⣰⡟⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣠⣤⡤⠤⠤⣤⣄⡀⠉⠙⠿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣴⣿⠄



⣶⣶⣶⢷⣷⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⢷⢶⣶⣶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠐⠂⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⣀⢈⠀⡠⢀⠄⡠⢀⠈⠐⠤⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠈⡉⠁⠀⠈⠉⠙⠒⠦⣠⠂⠄⠀⢢⡄⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠆⠀⠀⠀⠀⠘⢖⡠⢀⣲⢐⢀⣠⣠⣁⣀⣀⢁⢀⠂⠄⡀⠙⢢⣼⠘⣃⣶⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣈⠛⠥⣀⠀⠀⠠⠈⠉⠙⠚⢶⣄⡠⠀⠹⣸⠀⠀⡹⠈⠣⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠳⣤⣀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⢄⡋⠀⠀⠃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡷⠟
⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⡭⠃⠀⠀⠀⠐⠂⠀⠀⠀⢀⠘⠦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⢎⣢⡀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠉⠐⠒⠒⠒⠈⠁⠀⠀⠌⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠇⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠄⠉⠒⠤⣤⣤⣤⢀⡀⠀⣀⣠⣾⠤⡄⠀⠀⠀⠀⠀⢔⡢⠀⠀⢀⢔⡡⠗⠖⠟⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣦⡀⠀⠀⠘⣠⢀⠀⠀⡀⠀⡄⠀⠈⠻⣿⠛⠋⠠⣨⡄⠀⠀⠀⠀⠀⠀⢀⣔⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠄⠀⠀⠀⠀⠈⢻⡢⡀⡰⠉⠛⠲⠦⢥⣄⣡⠀⠀⠀⢸⡁⢶⡵⠟⢦⡀⠀⠀⠀⢀⠀⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠀⢀⠤⡄⠀⠀⠀⠙⠀⠀⡀⠀⣯⠀⠆⠀⠠⠙⣄⢀⣴⠗⠋⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈⠙⠃⠀⠀⠀⠐⠀⠀⠂⠀⣿⠄⠘⢄⠀⠠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣇⠐⡀⠈⠑⠠⠅⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢠⣼⠆⠀⠀⠀⠀⢸⠀⠀⠕⢀⣋⢧⡀⠁⠄⠀⣬⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⢸⡅⠁⠉⠀⢤⠼⠃⢘⠁⠀⠀⠐⠯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢸⠐⠀⢼⣣⠀⠀⠈⠋⠱⠈⡉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠢⠟⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠘⢃⠋⠀⠀⡐⠋⠉⠙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠔⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡼⢄⡤⠴⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣈⣆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠐⢨⠀⣨⢄⣄⣦⠖⠊⠉⠉⠉⠛⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠔⠉⠀⠀⠈⠢⣉⠂⢀⣀⢠⣀⢄⡄⡤⣀⢠⣤⡂⠀⠀⢈⣰⠃⠀⢈⠁⠀⠀⠀⣠⡤⣤⣀⠀⢀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠁⠀⠀⠀⠀⠂⠀⠄⠁⠅⠬⠟⠁⠀⠀⠉⠋⠀⠀⠀⠀⣀⣀⣁⠀⠀⠈⠉⠒⢤⢂⠁⠃⡗⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⡠⠊⠁⠀⠀⠉⠣⠔⠈⠁⠀⠀⠑⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⣀⣀⠀⠀⢀⠀⢄⢀⠄⠀⡀⡀⡀⢀⠀⠀⠀⠀⡀⠀⠀⠈⠌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢀⣀⢚⣟⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣼⣿⣾⣊⣿⡵⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⣞⣿⣎⢆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠉⡿⣪⢶⢤⣰⡦⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿⣧⡀⣰⣿⣾⣿⣿⣿⣦⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⣿⡏⠉⠉⠉⠛⠻⠯⣽⣿⣿⣿⣿⠿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⡛⣧⣤⣤⣤⣤⡄⣀⣼⡧⣤⣤⣤⣤⣤⣽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣛⣛⣛⣛⣛⣛⠙⡛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢠⠊⠀⠀⠀⡔⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠒⠂⣄⠀⠴⠊⠁⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠈⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠒⠀⠒⠚⠤⡒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢰⡀⢰⠊⠀⠀⠀⠀⠈⠂⠀⢠⠔⠒⠀⠑⠒⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⢸⢀⡄⠈⠡
⠀⠀⠀⣀⠠⠄⠒⠒⠀⢼⠀⠀⠀⠉⠑⠒⠠⠤⠄⢀⡀⣈⡀⡁⠀⠀⠀⠀⠀⠀⡁⢀⣤⣀⡀⠠⠤⠄⠒⠋⠁⠀⠀⠀⠇
⠀⣠⠊⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢰⠁⠀⠀⠀⠀⡠⠄⠀⠤⣘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⡇⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⢡⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠈⣆⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⢸⠀⠀⢀⠄⠀⠄⠄⡠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⢦⡀⠀⠀⠀⠳⣄⠀⠀⠀⢸⠀⢠⠑⣀⠂⢐⡀⠀⣌⠀⠀⢀⣀⠀⢠⠂⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠑⢄⡀⠀⠀⠀⠙⠦⣄⣸⠀⠀⡕⡁⠀⡉⠀⠠⠊⡠⠊⡔⠀⠇⡇⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⡹⠀⠹⠭⢀⠔⠡⠀⠑⢂⢅⠀⠓⡀⠫⡀⠘⢂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣠⠇⠎⠁⠈⠉⠀⠀⠉⠈⣡⣬⡁⠉⠀⠀⠉⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⠔⡨⠂⠅⠠⠀⠀⠀⠀⠀⢀⣀⠀⢀⢠⠠⡀⢀⠀⡀⠄⠰⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠠⠃⡐⠏⠈⠔⡡⢂⠏⠨⠠⢊⠲⠤⠗⢡⠏⡀⡰⢁⢊⠄⢠⣡⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⡆⠀⡇⠀⢀⠜⠀⡩⣀⢷⠁⠀⢀⢮⠀⢒⠥⠙⣄⣨⠷⡤⠬⡁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⠱⠤⠤⠴⠒⢕⠒⠚⠀⠒⠛⠈⠉⠬⠭⠉⠀⠀⠒⠒⠀⣀⣨⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠒⠲⠶⠶⠦⠤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⢀⡠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣀⡠⠤⠒⠉⠀⢀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠉⠀⠀⠀⣀⡠⠖⠉⡸⠉⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠒⠉⠁⣀⡤⠚⣡⠔⠋⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⠒⠚⣉⣥⣶⡉⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲⠒⠊⠉⠀⠺⣾⠟⠀⢰⣽⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢰⡀⢴⠊⠒⡄⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣈⣊⢦⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠸⡷⣌⣑⣚⡁⠀⠀⠀⣸⠁⠀⠀⠀⠀⢰⠶⡆⢀⡤⢶⡆
⠀⠀⠀⠀⠀⠀⠘⠓⠤⢍⡀⠣⣀⠀⠀⠀⠀⠀⣇⡞⠒⢤⣀⡑⠦⠬⢛⡥⠴⠂⢪⣱⠀⠀⠀⠀⠀⡜⠐⢋⡡⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢜⠷⣦⡀⣀⠴⠚⠧⡀⢘⠀⢨⠉⠉⠏⠱⡖⠒⠉⢻⣤⣀⠀⡠⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠞⠁⠀⠀⠀⠈⠙⠒⠤⠧⠤⠼⠤⠼⡄⠒⠉⢈⡫⡙⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠒⠉⠀⢈⣎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡧⠀⠀⠀⠀⠘⡤⠔⠚⠁⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⣁⣀⠤⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠨⣺⠗⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣛⣻⣷⣾⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠟⢻⣿⣿⣿⣟⡁⢠⣤⣤⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⢿⠷⠶⣾⣽⣿⣿⣧⣿⠟⢛⣗⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠁⠘⢁⣿⠇⠘⣛⣿⣴⣼⣿⣷⢟⣏⣉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⣴⡴⠂⢀⣬⣷⣈⣿⣿⣿⣟⢿⣿⣾⣷⣿⡾⢛⣿⡛⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⣀⠀⠀⠀⠠⣶⡇⠀⣼⡗⠛⣋⠌⢳⠹⣿⣿⣿⣿⣿⣿⣛⡛⢿⡿⢻⡓⠆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⢀⣽⣇⢸⣿⡟⠚⢻⣇⣸⣼⣿⣿⣿⣿⡟⠻⣿⠟⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣏⣄⣠⣼⣿⣿⣿⡿⠰⢶⠖⣾⣿⠛⣿⣿⣿⣿⣿⣧⣤⣤⣴⣏⣠⣴⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⢿⡻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⠀⠻⠋⢩⡄⡲⠻⢻⣿⣿⣿⣿⣧⣼⣫⡽⠿⠿⠿⢭⢯⠁
⠀⠀⠀⣀⣄⠀⠀⠀⠚⠁⡿⢿⣿⣿⣿⣿⣿⠋⠀⣴⣏⢩⠟⡿⢿⣇⣇⣠⣿⣿⣿⣿⣿⣞⣉⣭⣈⣯⣄⣀⣀⣉
⠀⢴⣾⣿⣿⣷⣄⠀⡀⠀⠀⠸⣿⣿⣿⣿⣿⡀⢰⡋⣽⢷⣦⣁⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⠙⠿⠉
⠀⢿⣿⣿⣿⣿⣿⠟⢿⡆⢠⣤⣞⣿⣿⣿⣿⣧⠠⡝⣛⠟⢻⣿⢻⡿⣿⣿⣿⣿⣿⣿⠁⢰⠃⠠⠽⠓⢀⣤⣦⣤⡄⠀⢀⣀
⠀⠈⠻⢿⣿⣿⣿⣶⣸⣿⣾⡍⣸⣿⣿⣿⣿⣿⠚⡃⣽⠿⢦⣭⣤⣴⣿⣿⣿⣿⣿⣿⣶⣾⣶⢾⣗⢠⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⡇⠙⢀⣘⣿⣿⡟⣿⣿⣿⡿⠛⣻⠿⠿⣿⣿⣿⣥⠄⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠈⣿⣿⣿⣿⣿⣿⣴⠞⠽⣫⣿⠛⣿⣿⣿⡇⠀⣿⠀⠀⣿⣿⣿⣟⣫⠄⠉⠉⠉⠛⠋⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⡶⣿⢻⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⣯⣖⡘⠁⣠⠚⠻⠻⠿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⠿⣏⣰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⢿⠀⠀⡀⠄⣠⣦⣴⠾⠿⠛⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⢻⡀⡀⢰⣄⣤⠉⢀⣴⣆⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⠟⠋⡘⣇⣻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣷⡾⠿⢿⣷⠾⠏⠿⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⢿⣿⡇⠀⢸⣿⠃⢻⣿⠀⡆⣭⢟⡿⠛⢋⡖⠃⠀⢼⣿⡟⣻⠋⣿⣿⣿⣿⣿⡏⠁⠀⢰⠉⠈⠈⡇⢳⠐⠀⢸⣿⣿⠹⣿⣿⡀
⠀⠀⠀⠀⠀⣿⡿⣼⣿⠀⢀⣼⣿⠀⢈⣿⡾⡿⠀⠀⠀⠀⢸⡇⠀⠀⡎⣁⠁⠛⣴⠁⠀⠈⢶⠟⠃⠀⠀⢸⠀⠀⠀⡆⠸⠀⠀⢠⣿⣿⡄⣿⣿⣷
⠀⠀⠀⠀⠀⣼⣷⣿⣿⡀⠈⢻⣿⠀⠙⢿⣧⠀⠀⠂⠀⠀⢸⡇⠉⠀⡟⠉⠈⠁⣿⡀⠀⠀⠈⠁⠀⠀⠀⠼⠀⠀⠀⠂⠀⠀⠀⢈⣽⣯⠀⢉⣹⣿
⠀⠀⠀⠀⠀⠈⠁⠻⠛⠿⠆⠈⠛⠂⠀⠈⠛⠂⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠐⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣒⡒⠒⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡩⠉⠀⠀⡩⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠴⠆⠖⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣋⣁⣀⣘⠀⠀⠀⠈⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠁⠒⠀⠄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠈⠂⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⠹⣿⣿⠇⠀⠀⣼⣿⣦⡀⠀⠀⢳
⠀⠀⠀⠀⠀⢀⡔⠉⠉⠀⠴⠱⠀⠀⠀⠀⠀⠀⢹⡴⠀⠀⠀⠀⠀⢀⠀⠰⡌⠉⠁⠹⡄⠀⢸⠀⠀⠀⠀⠀⠀⢀⠈⠦⠊⠁⠉⢢
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢡⠤⡀⠀⠀⠀⠀⠳⢦⡀⠀⠘⢦⡈⠣⠄⡃⠀⠀⠀⠡⠀⠏⠀⠀⠀⠀⡠⠤⡊⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣰⡣⣆⡌⠀⠀⠀⠀⢠⠂⢙⢦⣀⢀⡁⠀⠐⠀⠘⠀⣂⣠⠀⠀⠀⠀⠀⠀⢰⡰⣜⡄⢀⣀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠷⠋⠙⢦⣆⠀⢀⣼⠆⢠⡎⠉⣀⠴⠀⠀⠠⡀⠉⠡⠄⠛⢦⡀⠀⣠⠾⠉⠙⠶⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⢋⠟⠀⣘⠈⠫⠥⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠙⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠇⡴⠰⠇⢸⠁⠀⠀⢀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⡆⡜⠀⠀⠀⠘⠦⠔⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠒⠒⠒⠲⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣄⣀⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢉⣉⠉⠁⣈⡆⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠒⠒⠂⡆⠀⠀⠀⠀⠸⣀⡸⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠒⠒⠲⠿⢄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠸⠤⠆⡆⠀⠎⠀⠀⢐⣈⢉⡁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠐⡆⠘⡔⠀⠀⠀⡠⠄⢘⡛⢦⠧⠔⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⡠⠎⠆⠀⠀⣀⠱⠴⠊⢉⡠⠐⠈⠋⢐⡬⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠌⠀⠀⢦⠄⠐⡞⠉⡀⢠⡄⠀⠀⠤⠌⠙⡷⠞
⠀⠀⠀⠀⠀⢀⠄⠂⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠄⣤⣌⡆⡆⢀⠁⢈⡄⠀⠄⠀⢇⠀⢠
⠀⠀⠀⠀⠀⠈⠔⠈⢀⠠⠀⠀⠀⠬⢀⡄⠀⠀⠄⠀⠀⡔⠀⠈⡆⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠐⠀⠦⠤⠼⠤⠼⠄
⠀⠀⠀⠀⠀⠈⢀⠊⠀⢀⠀⠀⢀⡠⠀⡘⢆⠀⠀⠀⡠⠒⡀⠉⠉⢀⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⠂⠘⠋⠀⠒⠙⢦⠀⢀⡼⡨⠄⣂⠀⠀⠀⢐⡈⢦
⠀⠀⠀⠀⠀⠠⡄⠸⠁⠀⠼⠆⠀⠀⠿⠀⠰⠀⢈⢊⠔⡡⡔⡤⠀⠠⢆⡐⠨⠄
⠀⠀⠀⠀⠀⠀⠙⠄⠄⠀⠀⠰⠀⠂⠀⢀⠀⠀⢋⠀⠌⠐⢠⡆⠀⠀⣀⡀⠈⠁⠀⠀⡔⠋⠀⠈⠑⢄
⠀⠀⠀⠀⡀⠠⠄⠼⡃⠶⠤⠤⡤⠄⠂⠁⠀⠀⠘⡌⠀⠀⠀⠁⣀⠀⠛⠀⢠⠀⢀⠮⠤⠀⣀⠀⠀⣤⠒⡄
⠀⠀⠀⠘⠓⠀⠀⠒⠚⠒⢶⢰⡆⠀⠀⠀⠐⠀⠀⢀⠀⢀⣀⠀⠒⠀⠀⡠⠚⡰⠁⠨⡄⡐⠂⠉⢲⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠌⠀⡀⠐⠚⠘⣸⣲⠀⠀⠀⠀⠀⢀⠗⡼⠬⠉⣁⠼⠁⣴⠀⢻⣀⡐⠬⠄⢂⣀⡸⠀⠀⢀⠀⠀⠦
⠀⠀⠀⢀⡀⣀⡃⠄⠉⠈⣺⢺⢨⠁⠀⠀⠀⠀⠀⡘⡜⠀⠀⢄⣤⢀⠑⣾⣆⠬⠄⣉⣀⣀⣀⡄⡸⡀⣀⣢⣤⠴⠃
⠀⠀⠀⢺⡀⠀⠈⡢⠤⠔⢇⡈⠀⠱⡀⠀⠀⠀⠀⠙⢃⠀⠒⠐⠜⠨⠁⠀⡏⠀⠀⠰⠀⣜⣰⠀⠃⠈⡄
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠁⡴⢋⠝⠀⠀⠀⠀⠀⠈⠖⠒⢒⠒⢚⠀⠀⡄⠀⠀⠂⠘⠊⢛⣤⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣾⠒⣎⠀⠀⠑⢄⠀⠀⠀⠀⠘⠚⢃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠋⠉⠈⠀⠀⠀⠀⠙⠂⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣤⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣛⣩⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣉⣭⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣴⠶⢖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⠾⣋⣥
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠛⣥⡾⢂⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⡟⣱⡦⢀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣁⣶⢠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣭⣉⣙⣛⣛⣛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠙⠛⠓⠶⣬⣝⡛⠿⣿⣿⣿⣿⢛⣛⢿⣿⠿⠿⠟⢛⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣭⣥⣬⣤⠴⠶⠞⠛⠛⢛⣯⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣆⠀⠀⠀⠀⣀⣠⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠈⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡀⠀⢁⠆⠀⠀⠁⠆⠀⠀⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠤⠀⠀⠀⠀⠀⠀⠀⠀⢑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⡠
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⡆⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠌⠢⡀⠀⠈⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⡏⠀⠀⡠⠂⠀⠀⠀⠈⢆⠀⢀⡀⡨⠓⠀⠀⢀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆⠀⡀⠀⠀⢠⣮⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣰⠀⠀⠀⠀⠈⠸⡁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠈⠉⠀⠸⠀⠠⠀⠀⡀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠠⠀⣀⠀⠀⠰⠀⡠⠀⣀⡠⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⢱⡁⠉⠀⢀⣀⡀⠄⠁⠀⣠⠂⠀⠀⡀⢠⠀⠀⡈⢁⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠠⠆⠀⠀⠀⡀⠀⢀⠀⢅⠊⠓⠤⣄⣀⣀⡠⠤⢚⠀⠠⠒⠉⠁⣈⢀⠀⠘⢇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠁⠈⠐⠐⠀⠉⢄⡰⠁⠀⠀⠑⠂⠤⠀⠤⠤⠵⠒⣉⠒⠤⠤⠤⠚⢉⠀⠄⠀⠀⠉⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡣⣀⠐⠁⠎⠐⢀⠂⠈⠀⠀⢠⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠊⡑⠒⠒⠒⠚⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠧⢄⡠⠤⠤⢀⣰⡦⢄⣰⡖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠔⠁⠀⠀⠠⠤⠤⠀⠢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⠟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣧⠞⣶⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢠⠇⠀⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠲⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣰⡇⢸⠏⠀⠀⢿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣤⣄⠀⠁⣟⣁⠀⠈⠉⠓⡄⠀⠀⢰⠃⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠋⠁⠀⠰⢼⠋⠉⠉⢨⢳⠀⡴⠋⠉⠉⠉⡙⣆⣟⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡛⠋⠭⠸⣆⠀⠀⢀⣸⠀⠀⠀⠀⢸⠼⡇⠀⠀⠀⠀⠃⣿⡉⠙⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠳⣌⡓⠲⠤⢭⣍⣙⣉⣘⣓⡒⠀⠒⠚⣋⣙⣒⣉⣉⣭⠭⠷⢒⣋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠙⠓⠲⠶⠦⠬⠭⣭⣭⣭⣭⣭⣭⡭⠭⠤⠶⠶⠒⠛⠉⢹⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠩⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢸⠉⠉⠉⠉⠳⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠢⡦⢶⣴⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣦⣧⣏⣴⡦⠀⠀⠀⠀⠀⠀⠃⢸⠆⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⣉⡝⡟⢢⣀⡁⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⠤⠤⠖⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠷⠌⠉⠀⠀⠀⠀⠀⠀⠀⡆⢸⣄⣒⣐⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠶⠶⠶⠶⠒⠒⠚⠛⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⣠⡤⣿⣯⢦⣠⣀⠀⠀⠀⣀⣀⠀⢸⣿⡆⢀⣀⡀⠀⠀⢀⡀⡄⠒⢾⠖⠂⡄⣀⠀⠀⠀⣀⡀⠀⣸⣿⡀⢀⣀
⠀⠀⠀⢛⡿⢧⣸⣅⡼⢿⡛⠀⠀⠀⠹⢿⢿⣶⣿⣦⢿⠿⠁⠀⠀⠀⡽⠳⢤⣸⣠⠴⠻⡅⠀⠀⠀⠙⣿⢧⣤⢧⡬⢿⠛
⠀⠀⠠⣜⣶⡖⢻⠛⢶⣶⣧⠀⠀⠀⢠⣴⣾⡿⣿⡿⣶⣦⠀⠀⠀⠐⢦⣤⠖⢻⠓⢦⣤⠖⠀⠀⠀⢠⣶⡴⠻⡞⠳⣴⣠
⠀⠀⠀⠙⠛⠷⣺⣷⠜⠚⠋⠀⠀⠀⠉⠙⠁⣺⣿⡂⠙⠋⠁⠀⠀⠈⠉⠇⣠⢾⢤⡐⠏⠉⠀⠀⠀⠉⠃⠁⣺⣗⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⣄⢰⠙⢾⡶⢣⣆⣠⠀⠀⠀⣤⡄⣶⢦⣯⢶⡇⣤⡄⠀⠀⢠⣄⡄⠈⣻⡋⠀⣄⣤⠀⠀⠀⣄⣄⠀⣸⣟⠀⣠⣀
⠀⠀⠀⠼⠟⢳⡞⢳⠞⠻⠤⠀⠀⠀⢉⡽⠿⢦⣧⠼⠿⣭⡁⠀⠀⠀⠛⢻⡿⣽⡽⣿⠛⠃⠀⠀⠀⠈⠛⡿⣮⣧⠿⡟⠋
⠀⠀⠀⣩⢷⠞⢳⡟⢳⡾⣉⠀⠀⠀⣤⡽⣿⢉⣏⢹⡿⣥⡄⠀⠀⢀⣶⡾⠻⣽⡽⠻⣶⣆⠀⠀⠀⣰⣶⠿⣩⣯⠿⣷⣦
⠀⠀⠀⠀⠘⠴⢻⡛⠎⠃⠀⠀⠀⠀⠈⠁⠛⢋⣯⡙⠇⠉⠁⠀⠀⠈⠁⠁⠰⣾⡶⠀⠁⠁⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⣿⣿⡿⠛⠛⠛⠛⠿⠿⣿⣽⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠙⠳⠮⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡇⣿⣿⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠂⠉⠛⠒⠶⠶⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⢐⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡿⢸⡆⠀⠤⣜⣿⡿⠿⠛⠉⠍⠁⠀⠉⢙⣛⡻⠿⠿⠿⠿⠟⢋⡰⢃⡾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣧⢸⠁⠀⠀⠀⠈⣿⣦⣄⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⡇⡿⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣶⣶⣷⣶⡄⠀⠀⠀⠀⡠⠛⢹⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⡿⣱⠃⠀⠀⠀⢀⠀⡇⠘⠰⠌⠙⠻⡿⠿⠿⠟⠀⠀⠀⠀⢰⣧⣀⢸⡇
⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠃⠀⠀⠀⠀⠸⠶⠃⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣀
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿
⣿⣿⣷⡄⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠶⣒⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢚⣋⡉⠉⣓⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣶⣯⣥⣿⣻⣿⣯⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠀⠀⢀⣴⣦⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠁⣸⣿⣿⡷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠏⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣁⣴⠀⠀⠀⣴⣿⡿⠋⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⣿⠀⠀⣼⣻⠟⠀⠀⢀⣴⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⠏⠀⢰⠟⠁⢠⣄⠀⡼⣿⠉⢹⡭⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠀⢀⡾⠀⠀⠀⠙⠾⠓⠹⣒⢯⣠⡿⢹⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠲⣎⣾⠃⠀⠀⠀⠀⠀⠀⠀⠙⠇⣼⣃⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿