⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⣿⣻⢷⣶⣤⡀⢀⣤⣶⡾⣿⣻⡷⣶⣦⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣯⡷⣟⣯⣿⣻⣞⡷⣿⣻⣟⣾⣽⢷⡿⣽⡷⣯⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⡷⣿⣻⣽⢾⣷⣻⢿⡽⣷⢯⣟⣾⢿⣽⢯⣟⣷⢿⡽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⣷⣟⣯⣷⣟⡿⣞⣯⣿⣻⣽⣟⣯⣟⡿⣾⣻⣟⣾⣟⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⣷⣻⣽⡾⣽⣻⢯⣟⣾⣽⣳⣯⣟⣾⢿⡽⣷⢯⣷⢿⣞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣯⣟⣾⣽⢿⡽⣟⣯⣟⣾⢯⣷⢿⡽⣯⡿⣯⣟⣯⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣟⣾⣟⡿⣯⣟⣾⢯⣿⢾⣻⢿⣽⣻⢷⣻⡽⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⡾⣿⣽⢾⣯⡿⣽⣻⣯⢿⣳⣿⠻⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣯⣿⢾⣽⣟⣷⣻⢿⡽⠊⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⢯⣷⢿⣞⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡻⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣷⠻⣦⡶⢈⣴⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣝⢮⠻⣷⣝⢾⡿⠃⠻⣿⣿⣿
⣿⣿⣷⣴⠞⣴⣿⡏⣿⣿⣿⣦⠻⢿⣿⣿⣿⣿⣿⣷⡀⠙⣿⣷⡙⣆⣿⣿⣿⣿
⣿⣿⣿⡟⣼⣿⠿⢃⠻⢿⣿⣿⠰⢇⢽⡟⢿⣿⣿⣿⣿⡄⢿⣿⣷⠘⢿⣿⣿⣿
⣿⣿⡿⢹⢸⣿⢰⡝⡅⡆⢿⣿⣿⢸⣦⠙⠆⠙⢿⡏⠿⣷⡈⣿⣿⡇⠰⣍⢿⣿
⣿⣿⡇⣿⢸⣿⢸⢡⣷⡘⡄⠻⣿⡆⡷⠚⡙⠢⡀⣙⠀⣿⡇⢹⣿⣿⡀⣿⣆⣿
⣿⣿⠃⣿⢸⡿⢎⠴⠚⢿⣮⣬⣦⡉⣇⠠⣷⠂⢹⣿⢠⢇⣴⢸⣿⣿⠁⣿⣿⢸
⣿⡇⡀⣿⠀⠇⢸⢀⣧⠄⣿⣿⣿⣿⣿⣆⣘⣠⣾⣇⡄⣾⣿⠘⠟⠁⠀⣿⣿⣸
⣿⡇⣧⠹⡇⠀⣿⣇⠘⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⣿⡀⡖⠀⢸⡿⢃⡘
⣿⣷⠹⠆⠳⠀⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⡿⠀⣿⡇⠀⠀⠘⠁⣾⣇
⣿⣿⢢⡀⠱⣆⠹⣿⣿⣿⣿⣿⣿⣘⣣⣼⢿⣿⠿⠋⠀⠀⢻⡇⢀⠀⣀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣚⣥⣤⠀⠀⢀⡷⠔⠒⠒⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⣿⠟⠁⡠⠖⣡⣶⣶⣶⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣱⣿⠟⠁⡠⠊⣠⣾⣿⡿⠟⠁⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠉⠀⠘⠛⠛⠁⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠛⠁⠐⠂⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⣾⣷⣦⣠⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣯⡉⠉⠉⠒⠲⢤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠻⣿⣿⡿⡋⠉⠓⠦⡄⠀⠀⠉⢫⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⢢⠤⠤⠜⠀⠀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢇⡀⡖⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⠊⣉⠒⠤⢀⡀          ⡐⢁⠴⢜⢄
 ⡎⢸  ⠉⠐⠢⢌⠑⢄    ⡸  ⡆    ⠣⠱⡀
 ⡇⢸        ⣀⠗  ⠉⠉⠁  ⠙⠢⠤⡀⢃⢱
 ⡇⠘⣄⢀⠔⠉                    ⠈⠁⠘⡄
 ⢇    ⠁                              ⠘⡄
 ⢸            ⢀⣀⣀⡀        ⢀⣀⣀⡀  ⢣
 ⡸        ⢴⣾⡿⠿⠽⠇        ⠘⠛⠛⠛  ⠈⢄
⠰⡁              ⢠⠒⠢⡀⠈⠒⠊  ❤ ⡠⢄  ⡘
 ⠱⣀          ⢀⠜    ⠇        ⢀⠔⠁  ⡏
     ⠑⠤⢄⣀⠔⠁    ⡜        ⠊⠁  ⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⢿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣿⣿⠛⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣄⠄⣼⣿⡿⠋⠄⠄⠄⠄⠄⠄⠄⠛⣿⣿⠄⢀⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣸⣿⠛⡛⢛⣿⠋⠄⠄⢀⠄⠄⠄⠄⠄⠄⠄⠘⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠄⠁⠈⠄⠄⠄⠄⣿⡀⠄⠄⠄⠄⠄⠄⠄⠓⠸⠏⠄⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠄⠄⠄⠄⠄⠄⠄⢠⠿⠿⢻⣦⣤⣠⠄⠄⠄⠄⠄⠄⠄⠄⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠄⠄⠄⠄⠈⠄⠄⢸⣿⣿⣿⣿⣿⢭⡛⡀⠄⠄⠨⠄⠄⠄⠄⠄⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠄⠄⠄⠄⠄⠄⠄⠄⠄⠻⣧⡀⣈⣿⣿⠟⠄⠄⠠⢇⠄⠄⠄⠄⠄⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⢀⢀⣉⡉⠉⠄⠄⠄⠄⠄⠰⠂⠄⠄⠄⠄⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⣿⠿⠁⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣤⣤⣄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀