⠲⣿⠓⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⣆⠀⢾⣿⡎⢻
⠀⠀⠀⠀⡀⣴⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡜⣶⣧⠀⠒⠀⣶
⠀⠀⢀⣾⢱⣿⣿⣿⡇⣼⣿⣿⣿⡿⡟⢿⣿⣿⣿⡽⣿⣿⣷⢰⣿⣿⣆⠀⠀⣿
⠀⠀⣾⣿⣿⣿⢹⡿⠀⣿⣿⣿⡿⡇⢿⢀⠻⣿⣿⡇⢋⡟⣿⠂⣿⣿⣿⡄⢀⣿
⠀⡐⣿⣿⣿⡇⠈⠃⣁⢸⣿⠿⡇⠀⡘⢸⣆⠀⢶⡆⣄⢄⢻⠀⢸⣿⣿⣧⢸⣸
⠀⡇⣿⢻⣿⠀⣦⠀⣾⡌⠛⣀⢁⠀⣧⢸⠿⠂⢈⣀⠉⠘⠘⢼⢸⣿⣿⣿⠈⣿
⠀⢱⠈⡆⡿⠄⠋⣀⡨⠉⡄⠹⣎⢰⡻⠿⣰⣿⠟⠍⠉⠲⡀⠀⢸⣿⣿⣿⡇⣿
⣄⡈⠀⠁⠀⢠⣾⡏⠀⠀⢹⠶⣌⢸⣿⣿⣘⢿⠸⠧⣤⠆⣷⠆⢸⣿⣿⣿⣷⡽
⣷⣔⠀⠈⠄⢑⣙⣃⣄⣭⣜⢺⣿⣿⣿⣿⣿⣿⣿⣿⣆⣻⡟⠀⣾⣿⣿⣿⣿⣷
⣿⣿⣷⡀⠀⢠⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠀⣿⣿⣿⣿⠟⢸
⢿⣿⣿⠃⠀⠘⣻⣿⣿⣿⣏⠓⠚⢛⡛⢒⡋⠛⣿⣿⣿⡿⣡⣸⣿⣿⣿⠏⡀⠘
⣾⣿⣿⠀⢠⠀⠘⣿⣿⣿⣿⡆⡰⡟⢁⡼⢁⣴⣿⡿⣫⡼⢣⣿⣿⣿⠃⢀⡐⠂
⢻⡿⢿⠀⡎⠀⠀⠌⠻⣿⣿⣿⡔⡗⢀⣴⣿⣿⣷⠿⠋⠀⢞⣛⠿⠁⠠⣿⣳⠀
⢸⡇⠈⢀⣿⣿⣤⣾⡘⠊⠻⢿⣿⣶⣿⣿⠟⡛⢁⡤⠀⣨⣴⣶⣿⣷⣦⣝⢋⡠
⢸⣷⠐⣿⣿⣿⣿⣿⣿⣧⠀⠂⠉⠋⢉⡶⢆⠺⡟⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢻⣿⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠸⣿⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢾⣿⠿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣧⡌⠋⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠅⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠃⠀⠀⠀⠀⠀⡀⣴⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⠀⠀⣈⡿⠟⡿⠿⠿⣿⢿⠶⠶⠶⠶⣶⣶⣶⣶⣤⣤⣤⣴⣿⣟⡁⠀⠀⠀⠀⢀⣀⣼⣿⣿⠀⠀⠀⠀⢸⣿⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⠀⠀⣿⣦⣻⣯⣾⣇⠀⣠⠞⠉⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠐⠚⡼⠁⠀⠀⠂⠀⠀⠨⠻⣇⡀⠀⠀⢀⣿⣿⣿⡿⡿⠀⠀⠀⠀⣾⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡇⠀⠀⢹⡿⢻⠿⣳⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠹⡀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢾⣄⠈⣿⣿⠟⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⠀⠀⢸⡕⣻⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⠀⠓⢀⡴⠋⢰⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣹⣯⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣦⣠⣴⢟⣵⣴⣥⣤⣤⣤⡀⠀⠀⠀⠰⣼⣽⠇⠀⠀⠈⠀⠉⠸⣿⣿⠀⠀⠀⡠⣤⣤⣤⣤⣀⣀⢠⣀⠈⠻⢷⣐⣦⠀⠀⣽⡿⠄⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠡⣼⡿⢱⣿⢿⣿⣿⣿⣷⣄⡀⠈⠊⠁⠀⣠⠖⠋⠳⣄⠀⠙⠑⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠙⠻⣶⣾⣿⠃⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠀⢰⣿⡇⣿⣿⣸⣿⣿⣿⣿⣿⣇⡀⣀⠴⠋⠀⠀⠀⠀⠋⠫⡀⡀⣾⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢹⣿⠧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠒⢲⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⡁⠀⠀⠀⠀⠀⠀⠀⠀⡕⣧⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⠀⠀⠀⡀⣈⣿⠁⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⢈⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠛⠉⠉⠋⢹⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⢯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡇⠀⠈⠐⠡⣿⣿⠿⢿⣿⣿⣿⠿⠟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠼⢿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⣤⣄⣤⡼⠛⠉⠀⠀⠙⠊⠀⠠⡄⠆⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠷⣤⡠⣤⣤⢶⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠋⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣒⢲⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠙⢿⣿⣿⠟⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⣷⣶⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣦⣾⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣝⡷⠶⣄⣀⠀⠀⠀⠀⠀⠀⢀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⡶⠞⣻⣿⣿⣿⣟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣟⢋⠉⠉⠁⠀⠈⠉⠀⠀⠀⠀⠠⡘⠿⠉⢽⠁⠀⠀⠀⠀⠀⠀⠀⢶⡶⠚⠛⠋⠙⠁⠀⠰⠿⢿⣵⡻⡿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⢻⣷⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣰⣿⠇⠈⠈⠨⣿⣧⡐⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢰⣶⣼⡿⠋⠀⢀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢾⣏⡕⡁⠀⠀⠀⠉⢛⣿⣕⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡟⠻⠙⡿⣶⠾⠀⠀⠀⠀⢀⢤⡶⣻⣀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠹⣧⡅⠀⠀⠀⠀⠘⢙⣿⣎⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠈⡼⠿⢻⣟⢢⣤⠀⢣⣦⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⡶⣿⡏⠀⡼⠦⣤⡤⠟⠿⣿⣆⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡟⠀⠂⣡⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡫⠙⢿⠘⠇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢹⣿⡁⠀⠉⠉⠀⠀⠀⠀⠀⢠⣿⣿⡝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡖⣁⣬⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣺⡿⠀⠀⠀⠀⠀⠀⠀⡄⠀⢨⢳⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣽⢻⠀⠀⠀⢰⢰⡜⠀⠀⡏⠷⣌⣀⣡⣿⡶⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠚⣼⢹⠦⠴⠃⠀⠈⠉⠘⢿⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⡾⠃⠀⠀⠀⠀⢄⡄⠀⠠⣿⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠂⣽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠉⠀⠀⢀⣀⠀⠀