Virginia Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⠏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⣼⣿⣿
⠀⠀⠀⠀⠀⣴⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿
⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣿⣿⣿⣯⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠤⠤⠤⢤⢤⡄⣄⣀⡀⣀⢠⣀⡄⣠⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠲⠎⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠁⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢄⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠳⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠊⠉⢉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠈⠢⠖⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠚⠩⠤⠤⠤⠤⠤⠤⠴⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠄⠤⠤⠤⠤⠤⠼




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣤⠀⠀⠀⣴⣿⢁⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⣯⣣⡄⢸⣿⣷⠀⢾⣿⢳⣟⣹⣆⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣷⡈⠻⡟⠁⠘⠃⠸⠛⠉⣄⠻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣇⠀⡈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠒⠲⣼⣿⣿⣿⠀⠀⠈⣳⣤⣄⣤⡀⠀⣀⣴⣾⣿⠟⠓
⠀⠀⠀⠀⠀⠀⠙⣯⣤⣤⡀⢿⣾⣿⣿⠁⠀⠀⠈⠙⠁⠀⠉⣴⡿⢟⣽⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡟⡘⠉⣿⣿⣧⠀⢠⣴⣾⣻⣿⢿⣷⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⢿⣾⣿⣷⣨⣿⣿⣻⣷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣬⣽⣿⣟⣿⣟⣿⣿⣿⣿⣅⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⡿⠟⠋⣡⣤⣿⣿⣿⠟⢭⣴⣾⣿⣿⠿⡿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⡾⠿⠛⠉⠀⢠⣾⣿⣻⡾⠋⢸⠀⠀⠀⠉⠉⠛⠛⠛⠚⠓⠊⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⢸⡀⠀⠀⠀⠀⠀⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣿⣷⠀⣿⡇⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠠⣷⠀⠀⠀⠀⠀⠹⣿⡇⣿⡄⣶⣎⣹⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢻⣾⣇⣸⣙⣋⣭⣥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠈⠉⣽⡿⣍⠐⠶⣶⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⡄⢀⡀⠀⠀⢀⠞⠉⠀⢹⣿⣦⠀⠙⠻⠿⡿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠘⣷⠀⢻⣷⣴⡋⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠀⠀⠀⠀⠹⣧⡴⠛⠻⣿⣆⠀⠀⠀⠀⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣴⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠋⠙⢿⣏⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡟⣿⢹⠃⣴⡆⢸⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠟⢻⠉⡿⢸⢰⣿⡇⢸⡇⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣿⡇⢸⠀⣿⠀⣸⣿⡇⢸⡇⣿⣿⠿⣿⢂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢦⣛⣥⢸⠀⣿⠀⣿⣿⡇⣾⡇⣿⠃⣾⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⡀⣀⣀⢀⣠⠖⢶⣺⡏⠃⣦⣫⣾⢸⣧⣫⢾⣿⣿⡇⢹⠿⣮⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣏⢻⣆⢘⡀⣧⢰⡏⢴⣶⢨⣭⣥⣾⣿⣿⡷⣀⠞⣸⡀⢋⣆⠸⣉⣀⣀
⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣿⣧⣡⣤⣿⣮⣵⣿⣿⣿⣦⣙⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀
⣿⣿⣛⣛⣝⣻⣟⣽⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⠀⠀⠀⣀⣀⣴⣤⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣀⣀⡀⠀⠀⣀⣀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣆⣴⣾⣿⣿⠈⠛⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⣰⣾⣿⣿⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣿⣿⣿⣿⣛⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢿⣽⣿⣿⡿⠀⠉⠀⠀⠀⠀⠀⠀⢘⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠉⠁
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣶⣾⡇⣤⠛⣽⢡⣎⣟⢰⣿⠏
⠀⠀⠀⠀⠀⠙⢿⣿⢽⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣾⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆⣠⠀⡀⠀⠀⠀⢰⠆⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⣧⠖⣴⠂⢸⠀⠀⢸⠀⠂⡄⣾⢠⡖⢶⡤⢰⡶⢀⡄⣔⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠘⠖⠉⠀⢠⢉⡾⠀⠀⠈⠟⠀⠓⠃⠠⠁⣸⠓⠛⠀⠙⠓⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⢰⣀⡃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⣴⠷⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠃⠀⠀⠀⠀⠀⠠⣊⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠋⠁⠀⠀⠀⠀⠈⠲⢯⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄⡦⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠁⣀⣄⠈⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠣⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠆⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠁⠢⣄⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⢀⡀⠀⠀⠘⠳⣄⠀⠀⢀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢈⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⣀⣀⣀⣀⠦⠿
⠀⠀⠀⠀⠈⠛⠓⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣄⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⠀⢀⣰⡿
⠀⠀⠀⠰⡄⠀⡀⠀⠀⠀⣀⣤⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⠁
⠀⠀⠀⠀⣿⡯⠤⠀⠀⠁⠀⠀⠀⠙⠁⢀⣠⣿⣿⣿⣿⣿⠋⠉⠹⣿⣿⣿⣿⣿⣟⠁⠀⠻⠿⠟⠛⠷⣤⡀⠀⢀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠉⠉⠙⡄⠀⠀⠛⠛⠛⠛⠉⠉⠂⠀⠀⠀⠀⠀⠀⠈⠛⠶⢄⡀⣢⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣼⣦⢦⣤⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⠀⠀⢀⡀⣀⠀⠘⣿⣇⣀⠀⢀⣦⣤⢠⣤⡀
⠀⠀⣀⣀⡀⠀⣤⣼⣿⣷⣤⣼⣿⣿⣶⣾⣿⡷⠙⠛⢸⢿⣿⣄⣀⢻⣿⡛⣿⣿⣷⣆⣀⣶⣆⡀
⠀⠀⣿⣿⣿⣶⣿⣧⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄⣾⣿⣿⣿⣾⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⠆
⠀⠀⠻⠋⠩⢽⣿⣿⠇⠘⢇⠈⠉⣻⠿⠏⠈⠈⠛⠋⠀⠙⠛⠛⠛⢿⠧⠉⠋⠙⠉⠘⠻⠿⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⣦⣦⠀⠀⠀⢀⣤⠀⣄⣀⠀⢹⣿⣿⣀⣄⠀⢀⣈⣧⣶⣦⣰⣶⣦
⢀⠀⠀⠀⢠⣤⣿⣿⣿⣆⣤⣴⣿⣿⣿⣶⣤⣿⣿⣿⠄⠋⠻⠃⣿⣿⣿⣿⡀⠀⠸⣿⣿⠿⣻⣿⣿⣿⣤⣤⠀⠀⣶⣶⡆
⣿⣿⣷⣠⣾⣿⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡤⠀⠀⢁⣸⣿⣿⣿⣿⣇⣿⣿⣿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⡀⣠⣠
⣿⣿⣿⣿⣿⣿⣷⡾⠒⣿⡿⢿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣶⡷⢺⣿⣿⣿⣿⣿⣿⣿⡿⢇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿
⠋⠀⠠⠼⣿⣿⣿⠇⠈⠻⣇⠀⠉⢉⣻⡟⠿⠀⠀⠀⠙⠋⠃⠁⠈⠙⠛⠛⠛⠛⠻⡿⡧⠈⠉⠃⠙⠛⠤⠀⠿⠿⡿⠷⠉⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡼⠛⢿⢿⡇⢀
⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⡘⠿⠷⠀⠀⣾⣿⣿
⠀⠀⠀⢀⣤⣌⠲⠟⠒⠚⠧⣶⠷⠾⣿⣿⣿⣅
⠀⠀⠀⢼⡿⢋⡉⠧⠀⠀⠀⠈⠳⣾⣿⡿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣶⣦⡤⣄
⠀⠀⠀⠘⠃⠈⢁⠀⢰⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⣀⠀⣤⠔⠶⣻⠀⠀⢀⠔⠉⣹⣀⠐⠀⠠⠊⠁⠰⠤⠤⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠠⠩⢘⠏⠁⢀⠏⠃⠀⢹⣦⣴⣿⠃⠀⠐⢠⣴⠃⠀⠀⠀⠀⠀⠉⢫⠙⡀⠀⠈⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⢰⡿⠿⠃⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⢃⢀⡀⠢⠹⣿⣿⣿⣿⣝⡂
⠀⠀⠀⠀⠂⠄⡀⠋⠀⠀⠀⢸⣧⡀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⢡⡀⡇⠙⠌⠻⢿⣿⣿⡧
⠀⠀⠀⠔⢢⠀⠉⠂⠀⠀⠀⠀⠙⣿⠿⠿⠛⠋⠁⠀⢢⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢⠀⠐⡀⠀⠰⣿⣦
⠀⠀⠀⡀⣰⣧⡀⠀⠀⠀⠀⡆⠀⣿⡆⠀⠀⠀⠀⠀⠀⣁⣀⠀⠀⢀⡠⠞⠁⠀⠀⢆⠀⠬⠛⣿⣿⠛⠁
⠀⠐⣀⣿⠿⠁⠙⢦⣀⣀⢴⣁⡀⠘⣿⣄⢀⣀⡠⠔⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠐⠒⠀⠈⠁
⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⣠⡟⠀⠀⠀⠲⢿⣷⣶⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡉⠀⠀⠀⠀⣀⠀⣠⣌⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠉⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣾⣿⡶⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣏⠉⠀⠀⠀⠻⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⢠⡔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣤⣤⣠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠛⢿⣿⣿⣿⣴⣿⣿⣿⣻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠻⣧⠀⠀⣿⣿⣷⣶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⣀⠉⠁⠻⣿⡶⢉⡛⠻⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠈⠀⠀⠀⠙⠿⣏⡀⠀⠀⠁⠀⠀⠈⠁⠈⠛⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⢀⡆⠀⠀⠀⠈⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣀⠀⠀⠀⠀⠙⣦⠀⡆⠀⠀⠀⠙⢦⣄⠙⠆⠀⠀⠀⠀⠀⠉⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⠀⠙⢷⣜⣀⡀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣤⣤⡀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠻⣿⣿⣿⣷⣄⠀⠀⣠⠻⢇⡀⠲⠤⠀⠀⠀⠀⠀⠈⠙⢷⣶⡀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⣿⣿⣿⣿⣷⣄⣽⠀⠎⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠙⢿⣿⡟⠀⠻⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠹⠁⢹⣿⣿⣿⣿⣿⣶⣤⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠴⣶⣤⣹⣷⠶⣤⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣀⣀⠀⠀⠀⠉⠛⠳⠶⢿⡿⠟⠋⠀⠀⠀⠀⢢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇⠀⣿⠟⠛⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠻⣿⣿⠀⠀⠀⠀⠉⣿⠀⠀⠈⠟⠉⠉⠻⣿⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⡄⢀⣶⣿⣶⣤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣿⣷⠂⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⠛⠉⣿⣟⢛⣿⣿⣿⣿⣿⣿⡗⠀⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⢹⠇⠀⢰⣆⠀⠀⠀⠀⠁⢀⣀⣰⣦⡄⢠⡄⠘⠁⠀⠀⠈⠉⢻⡿⠀⢹⣿⠉⠻⣿⡿⠶⢀⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⣤⣴⣿⣿⣷⣀⣀⣠⡄⠀⣸⣿⠀⠀⠈⠁⠀⠀⠀⠀⣴⠟⠛⠃⠀⣸⣿⣧⠀⠈⢻⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢿⣿⣿⣿⠿⠿⠛⠻⣧⢈⡛⠋⠀⠠⠈⠀⣠⣶⣾⣷⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠸⣿⣿⣿⣦⣀⣀⣶⣦⠙⠁⠀⠀⠀⠈⠛⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢻⣿⠁⠛⠛⢻⣿⣿⣷⣾⢿⣶⣶⣤⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣼⣿⣧⣀⣀⠀⠈⠋⠈⠧⠠⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⠷⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⢰⣿⣿⣧⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢠⣶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣾⡿⠀⢹⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⣀⠈⢻⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠰⣿⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠁⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠆⠂⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣅⠀⠆⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿⣄⡀⠀⣠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⢀⡈⠛⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣆⠀⠀⣠⣤⣾⣯⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠀⠘⠋⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠆⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣿⣿⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⣭⣴⣦⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣦⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣷⠀⠀⠀⢰⣿⣿⣿⣿⣹⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⠀⠀⠀⠀⠀⠀⠀⣄⠀⣹⣿⡿⠋⠀⠀⠀⢸⣿⣿⣿⣿⣦⡍⠉⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣰⣶⣶⣀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠛⠻⠿⠷⠄⠠⣤⡀⠀⠀⠀⠉⠻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⠀⠀⠀⠸⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢰⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣶⣶⣶⣶⠶⠶⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀⡹⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⣾⣿⣿⣿⣿⣿⠃⢠⡄⢤⣀⣦⡀⠀⠙⣿⣶⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣂⣿⣷⣀⣙⣿⣿⣦⣤⣿⣿⡆⢻⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣸⣿⣿⣿⡿⠿⠿⠿⠚⠛⠉⣉⣩⣿⣿⣿⣿⡟⠋⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⣿⣌⠛⠛⠛⠛⠓⢀⣠⣤⣶⣿⣿⣿⣿⡿⠋⣿⣧⠀⢀⣴⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⢸⣿⣿⣷⣶⣿⣿⣿⠿⣻⣿⣿⣿⣿⠟⠁⠀⣿⢸⡀⡛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣾⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⠋⠀⠀⠀⣿⠀⣯⣻⡿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⣛⣛⣽⠟⠁⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⠀⢸⣿⡇⢸⡟⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠰⣦⣤⣬⣿⠟⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣴⣶⣿⣿⣿⣿⡟⠉⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⣦⣸⣇⣀⣠⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣄⠀⠀⠀⠈⣿⣿⡟⢻⡁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠻⣦⡀⠀⠸⣿⣿⣦⡀⠀⠀⣿⣿⡇⠀⠃⠀⡀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠈⠻⣦⣀⠻⣿⣿⣿⣷⣦⣸⣿⣷⡇⣀⣸⣷⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢸⡀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⠋⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢻⣿⡇⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠈⣿⣷⣄⠀⠀⠀⣿⣆⠀⠀⠘⣿⣿⣿⡃⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠸⣿⣿⣷⣤⣤⣿⣿⣷⡀⠀⢿⣿⣿⣿⡀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⠻⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⣉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⣿⣿⣿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣤⡄⢠⣀⣴⣿⠟⢉⣴⣿⣿⣿⣻⣷⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣁⣉⠉⣉⣠⣾⡿⠿⣄⣠⣿⣿⣿⠋⠙⢻⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⠻⣿⣿⣿⣿⣶⣶⣶⢿⣿⣏⣀⠈⠻⣿⠏⢿⣿⠋⣠⡄⢀⣿⣿⡿⠿⠞⠛⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣼⣿⡇⢶⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣦⡑⢼⣿⣿⣿⣀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢠⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⠘⠉⠉⠁⢳⣈⣿⣆⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⢘⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣸⣿⣿⣿⡿⠀⡿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⣸⣿⣿⣿⠿⣿⣾⣿⣿⠏⠉⠛⠿⢿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣅⠻⠁
⠀⠀⠀⠀⠀⣠⣾⣿⠟⢁⣼⣿⣿⠟⠁⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢻⣯⣿⠏⠙⣿⣧⡄
⠀⠀⠀⠀⠀⠉⠉⠀⣸⣿⡟⠋⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⣠⣤⣤⣤⣠⣬⣻⣲⣶⣤⣤⣤⣤⣄⡀⣀⢠⣤⣤⣤⣹⣷⣤⣤⣤⣶⣌⣿⡄⡀⣀⣀⡀⣄⡀⣴⣿⣧⣠⣶⣶⣶⣦⣦⣦⣦⣠⣤⡀⡀
⠀⠀⠈⠉⠀⠉⠉⠁⠀⠀⠈⠈⠁⠀⠁⠁⠈⠉⠉⠉⠛⠙⠋⠙⠛⠛⠋⠉⠉⠋⠀⠙⠛⠛⠛⠛⠋⠀⠈⠉⠁⠉⠉⠉⠁⠈⠉⠁⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠈⠉⠭⠭⠍⠉⣱⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⣀⣠⣤⣶⣤⣤⣴⣶⣿⠤⡖⠐⠖⠂⠈⠉⠉⠁
⢶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠐⠈
⠈⣿⣿⣿⡿⠟⠋⠁
⠀⠙⠋⠁

⠀⠀⠀⣀⣀⣀⣀⣠⣤⣄⠀⣀⡄⠐⠛⠋⠠⠤⠄⢁⣹⣿⣶⣶⣶⣶⣾⡿⠿⠾⠷⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⢿⠷⣒⠾⣉⠏⠁
⠀⠀⢸⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⣀⠀⠤⡀⢀⣠⠾⠛⠭⠭⠉⠉⡟⠓⠚⠛⠛⡛⠛⢻⡟⠃⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⠀⠠⡂⠀⠉⠀⠀⠀⠀⠈⣁⠠⢳⣚⡱⠂⠉⠉⠉⠉⠁
⠀⠀⡇⠀⠀⢀⡠⠔⠂⠉⠈⠈⠁
⣤⣤⣧⣴⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⡄⣤⣤⡤⣤⣤⣤⢤⣤⡤⡤⢤⣤⢤⠄⣤⣤⡄⣤⣤⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠬⠤⠥⠬⠭⠭⠭⠭⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠂⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⡷⠀⠀⠀⠀⠀⠰⡿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠈⠻⠀⠀⠀⢏⡇⣿⡧
⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⢺⠆⠀⠀⠀⢧⡟⣡⡀
⠀⠀⠀⠀⠉⠀⠈⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠙⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠻⠛⠦⣀⠀⡼⠀⠘⡆⢀⢠⠄⠙⠃⢀⠁⠈⠢⡀
⠀⠀⠠⣴⡧⡄⣴⣪⣏⣹⣝⣦⠠⢴⡷⠄⠠⠴⣾⠶⠤⢙⠛⠂⠒⢻⠉⠀⠀⠀⠀⡄⠀⠀⠀⠈⠲⣯⣁⠀⠩⢷⠀⡀
⠠⡤⢀⣄⣀⡇⣿⣧⣶⣶⣼⣿⠂⠀⡇⠔⢺⣄⣾⣃⣇⠜⣠⣠⣤⡌⠄⢀⣄⣀⡜⣠⣤⡤⠤⠤⢬⡀⠭⠒⠚⠽⠤⠤⠬⢢⣄
⠀⠁⠈⠛⠛⠋⠻⣿⠙⠋⣛⠟⠁⠘⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣅⠀⠀⠀⠀⠀⠀⠐⠫⠟⠉⠙⣟⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠈⠁⢠⡃⠀⠀⠳⣍⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠘⢸⡆⠀⢶⣿⠀⠉⠓⠺⢷⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢶⣤⣶⣺⣷⣤⠘⣆⣄⠀⠄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⣤⣾⡉⠙⢋⡙⠻⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣿⣦⣀⠉⠻⢿⣿⡆⣀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣶⣾⣿⣿⡏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣧⠠⠒⠂⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣤⡶⠂⠀⠙⣀⠀⢘⣁⠂⠈⠛⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣆⠀⠸⠁⠀⢀⠀⠂⢹⡇
⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠈⠛⠃⠈⢋⣤⡀⢸⣶⣶⣀⣤⡼⣿⣿⡿⣿⠿⠟⠙⠿⠋⠀⠀⠐⠚⠛⠛⠳⢾⡇
⠀⠀⠀⢘⣿⡗⠀⠀⠀⠀⠀⠐⠂⢠⠄⠈⠀⠘⣿⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠞⢷⡀
⠀⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠛⣤⡀⠀⢀⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣍⡙⠳⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠙⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢐⣆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⠑⣄⣀⢀⡀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⣍⢩⡏⢉⡾⢶
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢒⠐⠋⠃⠘⠲⣠⠚⠙⡇⠀⠀⠃⠈⠲⠀⠀⠀⠀⠀⠬⡟⠣⢔⡙⣤⣇⡎⡱⠔⢻⠥
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⡀⠀⠀⠀⠠⡅⠁⠒⡦⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠴⡧⢼⠆⢸⣿⣿⣿⠐⠶⠰⡖⢀
⠀⠀⡀⠤⠄⠐⢺⡂⠀⠛⠉⠪⡅⠀⠈⠙⠚⢉⡃⠀⠖⠀⢈⡟⠉⠧⠒⠀⠙⢓⣳⡴⢺⣿⢋⡏⢿⡝⠓⣼⡈⠉
⠀⠀⠓⠠⠦⣴⠼⠧⠀⡾⠒⠒⠋⠉⠻⠍⣀⢀⡂⠀⢠⣶⣅⡀⠘⠓⠀⠀⠀⢬⠅⠙⣾⣏⣩⣟⣈⣷⠊⠀⠤
⠀⠀⠀⢀⣼⣷⣤⣄⣴⣷⣄⣠⣶⡤⢤⠴⣿⠚⠿⠿⠿⣿⡿⠿⠷⠒⣲⣾⣆⢸⠂⣭⣿⣷⣤⣯⢀⣿⣧⢀⣄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢮⣄⣿⠀⠤⠼⢇⣿⡧⣤⠶⢸⣿⣿⣿⣿⠤⣼⡇⣧⢸⣿⢻⠼⣿⣿⣿⣷
⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠾⣿⣿⣾⢷⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⢿⠟⠻⣿⣿⣿⣭⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⣇⢡⠏⠀⠉⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠆⠸⡅⣾⠀⠀⠀⢀⢀⣀⢿⠿⣷⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⢀⠀⠀⣿⣿⣃⣿⣿⣷⡄
⠀⠀⠀⣾⣿⣿⠿⡿⢿⡿⠿⣿⣿⡛⣶⢀⠸⡏⡆⠀⠀⠘⢿⡉⠉⣟⢺⣿⣿⣿⠟⢉⣉⠉⠉⠻⢿⣿⣿⡿⠿⠅
⠀⠀⢀⣿⣿⣿⣿⡟⡛⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⢂⡄⢸⣿⣿⣷⣿⣿⣿⣿⣿⢂⣾⣿⠎⡉⣿⡟⢃⢀⣠⠳⢀
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡏⣤⣤⣤⠤⣴⣿⣿⣿⣿⣿⠉⠉⠉⣈⡛⠛⣹⡀⣤⣽⣷⣾⣿⣿⣿⠏
⠀⠀⠀⢉⣻⡻⠟⠻⣿⣿⣿⣿⠿⠿⠿⢷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⣤⢸⣷⣿⣼⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⣾⣿⡇⠁⠀⠀⢹⠛⠿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⣥⣽⣿⣿⣷⢨⠉⣙⣭⡟⣿⠿⠛⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠻⠻⢿⣾⣶⢻⢻⠞⣲⣶⣿⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡟⠒⢴⣿⣿⣛⠿⠋⠀⠀⠠⠀⠀⡀⠀⠀⠹
⠀⠀⠀⠀⠘⢯⣭⣿⡏⣸⣾⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡹⠟⠋⠁⠀⠀⠴⠂⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠈⠉⠉⠀⠀⠲⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢾⣤⣤⣶⡾⠒⠂
⠀⠀⠀⠀⠀⠀⠤⠄⠐⠋⠙⠛⠒⠉⠉⠉⠉⠉⠉⠙⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢛⣛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠟⢿⣿⣷⣦⣤⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⠀⠈⠓⠀⣉⠻⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣤⢀⣤⣶⣶⣶⣿⣣⣾⣿⡧⠀⠀⢸⠃⠀⢹⣆
⠀⠀⠀⠀⠀⠀⣴⣘⠷⢎⠸⠋⠀⠀⠺⢯⣿⡿⣿⣿⣿⣿⠀⡄⠘⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⣼⣿⡇⠀⠀⠀⠀⠀⠈⠙⢮⡝⢿⣿⣾⡇⠀⠀⡐⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠢⠙⠿⢁⣤⡴⠃⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣀⣀⠜⢸⡇⠀⠀⠀⠀⠀⠀⢰⡷⠛⠉⠀⠀⢀⣴⡇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⣩⠟⠁⣀⣨⣧⠤⠔⠒⠉⢀⡔⠁⠀⠀⣠⠤⠞⠀⠠⣴⠺⠿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠈⠉⠁⠀⣿⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⢰⠟⠂⠀⠻⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠘⣧⣹⣿⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠉⢻⠀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣾⣿⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡠⠚⠛⠛⠒⠒⠒⠂⠀⢻⡀⠀⠀⠀⢀⣠⣴⠞⠋⠀⢠⠂⠀⢸⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣴⠾⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢰⠋⠉⡄⠀⠀⢰⠏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⢰⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡀⠀⠹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢸⣇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣧⠀⠀⠹⣆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠿⠟⠀⠀⠀⢹⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣦⡀⣀⠀⠀⠀⢀⣴⣾⡿⠋⠁⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣷⡀⢀⣾⣟⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⢰⣴⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡹⠹⠻⡄
⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⠖⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡖⠂⠀⡆⠀⢆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⣜⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠲
⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢰⡆⠀⠸⠀⢀⣇⠀⠀⡁⠀⣿⠃⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠻⠃⠀⡀⠀⠙⢆⢠⠗⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏⠀⠀⠌⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⢀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡅⠀⠐⠄⠀⠀⠀⠀⠀⠀⠒⠀⠂⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠁⠀⠀⡄⠀⢀⠎⠁⡠⠈⡕⢀⣁⠉⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠤⠈⠂⠀⠁⠀⠀⠑⠉⡀⠀⠀⠀⠀⠘⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠒⠀⠀⠐⠈
⠀⠀⠐⠀⠒⠀⠀⠂⠀⠸⠅⠀⠀⠀⠂⢀⣀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠠⠄⠄⣀⣀⡀⠤⠐⠊⠀⠀⠉⠒⠀⠒⠐⠊⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣉⡄⠀⠀⠀⠈⠐⠂⠐⠂⠚⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⢄⡻⡧⠄⣚⠶⡄⠈⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠱⢇⠀⢀⢀⠊⠑⡌⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠂⡇⠀⠾⣻⣷⠀⠀⢰⠰⣂
⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠠⡄⠰⡀⣰⠋⡍⠓⢄⡌⢀⠃
⠀⠀⠀⠠⠀⠚⣓⢒⠂⠉⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢰⠨⠀⢰⢡⠆⣃⢔⢴⡀⠀⠀⠀⠈⢢⣜⡻⠂⠷⢻⣈⡴⢣
⠀⠀⠀⠀⢀⣵⠀⣶⡇⢺⣮⠹⢦⣄⣀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠿⠀⠀⠀⠀⢘⠚⠠⡀⡸⠘⢰⠊⢷⢶⡍⢄⠀⠀⠀⠀⢈⠋⠐⠛⠈⢃
⠀⠀⢤⣶⠟⠁⠴⡿⠁⠘⠿⢗⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠡⠴⠲⢒⠲⠶⢦⡡⠈⠢⠕⠊⠀⠀⠀⠀⠸⠤⠤⠤⠴⠾
⠀⠀⠀⠘⠀⠀⠀⢷⡀⠀⢸⡎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠅⠉⣷⣏⢈⣈⠈⢈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⡀⠀⡄⡀⢸⣆⠀⠀⣧⠀⣼⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠚⠉⠁⠈⠈⠑⠂⡘⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⣔⣓⣚⣳⡄
⠀⠀⠀⠑⠀⠻⠅⠚⠓⠀⠀⠚⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠐⠒⠒⠒⠀⠐⠂⠐⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡠⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡔⠱⢹⡟⡝⢄
⠀⠀⠀⠀⠀⡀⢀⢀⣿⡇⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣠⠊⢀⡇⢸⠇⠸⡈⠂
⠀⠀⠀⠀⢠⠟⠁⠈⠉⢁⡙⠛⠻⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠀⠻⠖⢀⣔⡁⠀⣀⡁⢸⢀⣀⣠⡔
⠀⠀⠀⡠⡋⠀⠀⠀⠀⠢⠀⠰⠀⠑⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁⠀⠀⠀⠉⠀⠀⠛⠁⢸⠙⠃⠈
⠀⠀⠴⢡⠛⠀⢠⠢⠀⠀⠊⠁⢢⡀⠐⠠⢑⣂⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠤⠄
⠀⠀⠀⠀⠈⠛⠋⠈⠛⠛⠀⠀⠈⠛⠒⠂⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡀⠀⠀⠀⠀⠀⠀⢀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠚⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠉⠙⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⢀⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠀⢀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⡙⠋⣹⣿⣿⣿⣿⣿⠛⠂⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣏⠙⢉⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⠁⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⣈⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣁⡆⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣰⢆⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣹⡆⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⢃⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⠸⠶⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣽⣤⠎⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣵⣿⠀⠀⠀⢀⣰⣞⠉⠉⣀⣿⡑⠀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠴⠞⢩⣤⠽⠛⠋⠁⠈⣳⡞⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠻⡟⠉⣿⣿⡧⠞⣠⠶⠊⠁⠀⠀⠀⠀⢸⠛⣷⡐⠶⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⣉⠢⣜⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⢳⡱⡄⢀⣀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠼⣿⣿⣿⣿⣿⣿⡋⠙⠻⠾⠿⠦⠤⠤⠤⠤⠤⢤⣤⣞⣉⣧⣀⣝⡿⢭⣁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⣶⡟⠻⠿⢶⣿⣿⣿⢳⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⢼⣏⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣿⣿⣿⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⡽⠉⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣾⣯⣋⣞⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⢀⡀⠀⠀⢀⣠⡾⠁⠀⣿⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠿⠛⠷⠟⠀⠀⠀⠹⣇⠀⠀⣤⠀⠀⠀⠀⠠⣤⣀⢿⡃
⣘⣏⠛⠋⠈⠁⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣿⡀⠀⠀⠀⠀⠀⠙⢷⣽⣦
⠘⠧⣄⣠⠀⣠⣶⠊⠈⠀⢹⡄⠀⠀⠀⠀⣼⠛⠻⣿⡉⠛⢿⣿⠒⠶⠤⠤⠤⠶⠒⠛⠋⠉⠛⠓⠒⠓⢦⣄⣀⡀⠈⠉⠓⢦
⠀⠀⠙⠛⠛⠁⠁⠀⠀⠀⠀⣷⠀⠀⢀⡾⠁⠀⠀⠀⠻⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠉⣹⢶⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣩⡏⠀⠙⢁⡼⠻⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠎⠀⠀⢹⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡙⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣉⣙⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣉⢹⣆
⡶⠒⠲⡶⠶⡶⡶⡶⢶⡶⣶⡶⡶⣿⢿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣸⣿⣷⡆⣤⡀⠀⢠⡄⢀⠀⣀⡄⣤⣤⣠⠄⡀⢀⣠⢤⡄⣤⢤⡄⠀⢠⢄⢠⡄⣤⣄⣄⡄
⠀⠀⠀⠀⠀⠹⣿⠿⠻⠏⣿⣷⠄⣸⣧⢸⣾⢿⡇⣏⢳⣸⠂⢻⣾⣿⠂⡿⣿⢸⡇⠀⢸⠞⣸⣧⣿⣏⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠟⠀⠀⠋⠙⠂⠃⠊⠳⠋⠛⠛⠂⠚⠛⠛⠒⠃⠛⠘⠃⠀⠘⠀⠋⠙⠟⠟⠁⠉


⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇
⠀⠀⠠⢨⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡜⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠘⣼⠀⠀⡇
⠀⠀⠀⢠⠁⠀⠠⠖⠀⠀⠀⣠⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡞⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠠⡚⠀⠁⠂⢂⡀⠀⠀⠀⠀⡰⠁⠀⠀⢠⠎⠀⠀⠀⣰⠁⢰⠀⠀⠀⠀⠀⢠⡍⠵⠀⠀⣇
⠀⢀⠀⠀⠠⠡⠉⠀⠚⠀⠀⠀⠠⢨⠇⠊⠀⠀⠀⠔⡁⠀⠀⠀⢠⠇⠀⡈⠀⠀⠀⠀⠀⢸⣿⠜⡄⠀⡟
⠀⡟⠀⠐⠀⠆⠀⠀⠼⠀⡀⠀⠀⠀⠂⡤⠄⠤⣢⠎⠀⠀⠀⢠⠏⠀⠀⣿⠀⠀⠀⠀⠠⠳⠶⠶⠿⠀⣇
⢠⠃⢀⡑⢒⡚⣇⢂⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⡠⣤⢀⡄⠤⠿⠤⠤⡤⠤⣐⡀⠀⠀⠀⠀⣿⢀⣀
⠈⠀⢸⡏⠈⠑⡀⠀⠀⠠⠄⠀⢀⡀⠠⠄⠔⠠⠂⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠠⣀⡄⠠⠄
⠀⡤⠾⠂⠀⠀⠙⠀⠠⠰⠋⠀⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠺⠦⠺⠶⠮⠲⠴⢶⠓
⠠⠄⠀⠀⡀⠀⢰⠀⠀⢢⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣀⡖⠀⠀⠈⠃⡁⠀⠐⠂⠈
⢰⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠈⠀⠀⠀⠀⡆⠀⢓⡀
⠀⠀⠀⠀⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠚⠀⠀⠀⠠⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠠⠅⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⢈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⠀⢾⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠑⠾⠀⠀⠀⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠘⠆⠠⠄⠀⠈⠀⠀⠀⢠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣮⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⣿⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠛⠿⠿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣤⠀⣤⣤⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⠻⢿⣿⣿⠿⣿⠟⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢐⡀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢏⣿⣿⣿⣿⣷⠶⠀⠀⠀⣮⣤⣦⣴⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡶⠀⠀⠄⢘⣛⢿⣷⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡞⠛⣛⣋⣙⣛⣿⣿⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣟⠙⠛⠛⢙⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠰⠆⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⢠⡄⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⢠⡄⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠒⠀⠀⠀⠀⢈⣿
⣿⢠⡄⠀⠀⠀⠀⢨⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⡿⢁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣧⢈⡁⠀⠀⠀⠀⠀⠀⠰⠁⠘⠋⠁⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣯⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠹⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣯⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠹⠇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⣿⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣲⡂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⡖⠋⢹⠉⠛⢲⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⢀⡇⣀⡤⡟⠙⢯⣉⡋⢑⣦⢴⡖⠒⠈⢰⠒⠢⣤⡀⠀⠀⠀⠀⠀⠀⠐⢦⣴⡁⣠⠿⡎⢹⠉⣱⠧⡀⢙⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣧⠁⠀⡇⠀⠀⠈⡷⢏⡀⠀⠓⠀⠀⠸⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠛⢻⠁⢹⠓⢤⣽⣾⣶⣧⠴⠛⡅⢸⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠠⡿⡄⠀⡇⠀⠀⠠⡇⠀⠀⠀⠶⠠⠠⠤⢀⠀⠀⢀⠀⠀⠀⠀⠀⣒⡒⢻⠒⢛⠒⢺⣿⣿⣿⣿⡒⠚⡗⢘⣗⣃⡀
⠀⠀⠀⣀⡠⢤⠤⠒⢺⠓⠒⠚⡏⠙⠫⣏⠉⠉⠉⠛⠛⠃⠉⠈⠀⠀⠀⠀⠀⠀⠋⠘⢃⠀⠐⠒⠛⠛⡾⣦⠼⢾⣿⣟⢹⠙⢿⣏⠟⢦⣼⠈⠉⠁
⠀⠀⠸⣅⡀⣰⠠⣄⣸⡇⣀⣠⣧⠤⠤⡿⠷⠶⢶⡗⠋⠁⠈⠇⠀⠀⢰⣤⠞⠰⠀⠶⡄⢀⠀⠀⠀⣒⡏⠙⠦⣴⡿⠓⢻⡖⠚⣿⣠⠞⢀⣉⡀
⠀⠀⠀⠀⠀⢩⣼⡏⠙⠁⠀⣰⡇⠀⠀⣇⡀⠀⠀⠁⣽⣖⣲⣦⣴⣮⣿⣿⣷⣶⣤⣀⡀⢸⢀⣍⣀⣉⡏⢠⢤⣿⣿⠚⢻⡗⠚⢻⣧⠀⠀⠉⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣾⣧⣾⣿⣿⣦⣤⡿⣷⠒⢛⡟⢹⡏⠈⠉⠛⣛⡛⣿⣿⠛⠛⠛⢉⣹⣿⣿⣧⣄⡇⠰⣾⡿⣿⢿⣿⣇⢠⡼⣿⣆⣰⣷⣀
⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⠿⠶⢶⣤⣼⣧⠠⡤⠶⠿⢤⣿⣿⢴⡾⡯⢠⣿⣿⣿⣿⣿⡧⢦⣿⡇⣿⡤⢯⣿⢻⠷⣿⣿⣿⣿⣿⣟
⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡶⠶⢶⣿⣿⣿⣶⢷⣶⣶⣾⣿⣿⣾⣶⣶⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡷⠾⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⠉⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⣸⣧⡀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣼⣿⣿⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣦⣤⡀⠀⠀⣿⣿⣿⣿⣶⣦⣿⣿⣇⣶⣦⣤⣶⣶⣄
⠀⠀⠀⠀⣴⣶⣿⣿⣿⠿⢹⣿⣿⣿⣶⣴⣿⠿⣿⣿⣿⣛⣯⣛⣿⣿⣿⡛⠛⠹⣿⡇
⠀⠀⠀⠀⢘⡏⣽⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡄⠘⠻⣿⣿⣿⠀⣿⣿⣿⣷⣴⣸⣿⣧⣤⣤⣤⣴⣶⣦
⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣮⣉⣩⣿⣿⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣸⣿⣿⣿⣇⣿⣻⣿⣿⣿⣿⢿⣽⣿⡇
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣯⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⡏⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⢻⣿⡇
⠀⠀⠀⢀⣿⣿⣿⢹⣿⡇⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡇
⠀⢸⣷⣿⣿⣿⣿⣿⣿⣏⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⠴⣿⡿⠾⠛⢛⣻⣿⣿⣦⣬⣭⣭⣭⣥⣶⣶⣶⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣬⣂
⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠀⠂⠀⠀⠀⣼⣷⣦⡄⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠀⠀⠀⠀⠀⠠⠟⠿⡿⣿⠁⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡿
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡭⠄⣀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠀⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠳⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠗⢆⠘⠆⢣⣀⠤⣄⣀⠀⠀⠀⡆⡌⠑⠦⠀⠀⡠⠀⠀⠃⠠⡄
⠀⠀⠀⠀⠀⠀⣰⠋⠉⠑⣰⡌⠆⠀⠈⠉⠓⠦⢅⠁⠀⠠⠒⠪⠴⠒⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡇⢀⡔⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⠀⣤⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠈⢃⡞⠈⠁⢸⠇⢀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢄⠘⣆⠀⠘⠆⠀⡀⡀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡠⢞⡭⢭⣉⡓⠲⢖⡂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⣸⣧⠔⠋⡁⠀⠀⠀⠀⠀⠀⠘⠦⠘⢦⠀⠀⠐⠁⠸⡔⢬⣀⠑⠒⠒⠶⢴⣋⠐⠁⠐⠒⠦⢭⣉⣩⠴⠂
⠀⠀⠀⠀⠀⡞⠀⠉⢉⠀⠀⠚⠉⠈⢧⠀⠙⠄⠀⢀⣠⠴⠂⠀⠀⠀⠈⠳⣄⠈⢦⡀⢻⡀⠈⣳⠀⠠⡀⠀⠈⠳⣄
⠀⠀⠀⠀⢸⡇⠺⢀⡝⠊⠀⠀⠀⠀⢸⡆⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠱⠀⢏⠉⠁⠀⠀⢹⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠈⠋⠉⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡄⠀⣀⠘⢦⡀⠀⠀⠀⠇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠁⠤⠤⠤⠄⠀⠀⠀⠀⠀⣸⢠⠇⠀⢹⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠀⠐⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡾⠀⠀⠀⠁⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢠⣀⠀⢳⠀⡇⠀⠈⠉⠁⠀⠀⢀⠞⠀⡇⠀⢀⣀⡤⠴⠚⠁⠀⠀⠳⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⡁⠸⠀⠇⠑⠲⠤⠤⢀⡴⠟⠒⠀⠀⠉⠉⢠⣀⣀⣀⣀⣀⡤⠤⠌⠓⠒⠀⠀⢠⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⢰⠆⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠈⠀⢈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡄⠀⠀⣧⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣈⠀⠠⢿⣀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⡀⠀⠘⣿⣀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⣛⣫⣥⣤⣶⡾⣡⣤⠤⠀⠁⠈⠀⢀⠀⠀⠀⠀⢀⡿⢹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⠀⠘⠡⠭⠭⠵⠶⠶⠦⢤⣴⣟⣿⣶⣤⡀⢀⡴⠋⠀⠀⢀⣀⠉⡰⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠶⠄⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠻⠷⢏⠀⠀⠀⠀⠀⢠⠞⠀⠀⡴⠁⣲⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠟⠛⠉⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢶⠻⠿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⣿⣿⠁⠳⡀
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡆⠐⢲⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠣⣴⣶⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⣿⡟
⠀⠀⠀⡞⢿⣦⣤⣴⡾⠛⣠⡴⠋⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢦⣈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⣻⠀⠀⠈⠙⣧
⠀⠀⣼⣇⣀⣼⢛⡥⠖⠋⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠬⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘
⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣠⠄⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠘⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣂⣠⠴⠖⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠋⠁⣀⣤⠶⠛⠉⠀⠉⠀⠀⠀⠀⠀⢀⣀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⣾⣿⣿⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⢀⣀⠤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢹⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣾⡇⠀⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⢒⣛⣺⣃⡒⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⢞⡩⠖⠻⡈⠀⠩⠀⢈⠟⠲⢍⡳⡄⠀⠀⠀⠀⠀⣠⢎⣣⡵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣴⢣⠎⠑⢄⠀⢱⠀⢀⠀⡎⠀⡠⠊⠱⡜⣣⠀⠀⠀⣴⠣⠵⣏⡀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⢸⢋⠛⠢⢤⣀⠑⢄⣣⣾⣜⡠⠊⣀⡤⠔⢛⡉⣇⠀⠀⣟⢹⡟⠒⠭⣝⢂⠱⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⢰⢄⠀⢀⡠⠶⡊⠉⢑⠶⢄⠀⠀⡠⡆
⠀⠀⠀⣿⢼⠤⠤⠤⠤⠬⣿⣟⠉⣻⣿⠭⠤⠤⠤⠤⡧⡷⠀⠀⢗⣊⣧⠤⠤⠐⠚⠛⢳⣛⡿⠦⠀⠀⠀⠀⠀⠀⠘⡄⠃⠊⠀⠀⠘⠂⠉⠀⠀⠉⠚⢀⠆
⠀⠀⠀⢭⠸⠀⢀⡀⠤⠒⡽⢿⣿⣿⢏⠒⠤⢀⡀⠀⠇⡶⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣏⣩⣯⡉⢙⠉⡏⡉⣭⡉
⠀⠀⠀⠈⡖⢫⡁⠀⡠⠊⢠⢱⣿⡏⡄⠑⢄⠀⢈⡝⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⠀⠀⠈⡵⡄⠀⠀⠀⠀⠀⠀⢀⣀⢬⡿⠊⠰⠷⣬⢤⣿⠟⢱⠾
⠀⠀⠀⠀⠘⢧⡱⢎⡀⢀⠇⣸⢻⣇⠸⡄⢀⡱⢎⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠃⠀⠀⠀⠰⡘⡄⠀⠀⠀⠀⠀⠸⣇⠴⠠⠒⣾⡦⣟⡴⠂⠐⢺⡗
⠀⠀⠀⠀⠀⠀⠙⠶⣍⡟⢠⡾⠶⢼⡄⣻⣩⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣁⣎⣀⣀⣀⣀⣀⣧⣸⣆⠀⠀⠀⠀⣰⠏⠀⠄⠠⠏⠀⠿⠄⠀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⡙⣛⣛⣻⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛⠙⠛⠛⠛⠛⠙⠛⠛⠀⠀⠀⠀⠘⠂⠐⠒⠒⠂⠒⠐⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⡟⢷⡀⠀⠀⠀⠀⠀⢀⣠⣤⢶⡲⣶⡦⣤⣀⠀⠀⠀⡠⣤⠴⣤⢤⡶⢤⡤⡀⠀⠀⠀⢀⡤⠴⠶⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⢚⣝⣀⣇⣘⣌⣢⣄⣀⠀⠀⠘⠒⠷⣞⣓⣚⡲⠷⠒⠃⠀⠈⠓⠺⠶⣛⣚⡳⠾⠖⠃⠀⠀⡜⠁⠀⠀⠀⣠⡶⣦⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢧⣀⣠⣀⣠⣀⣄⣠⣀⣠⣀⣠⡕⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⣧⡴⠒⠫⢍⣈⠄⣿⣈⠭⠛⠒⣤⣿
⠀⠀⠀⠀⠀⢠⠃⠀⡄⠀⡞⡇⡇⢸⡇⠀⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢶⢶⢶⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡗⣶⢤⣼⣧⣿⣤⡤⡶⢾⢩⣿
⠀⠀⠀⠀⢀⠎⠀⢰⠀⢀⠀⡇⡇⠘⢠⠀⢸⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢸⠀⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⣿⢸⡐⡇⣿⢸⡇⣯⣼⠾⠁
⠀⠀⠀⢀⡎⠀⢠⡏⠀⢸⡟⢛⣗⠛⣾⡀⠈⣣⠀⠈⣆⠀⠀⠀⠀⠀⠀⢘⣦⣣⣸⣠⣡⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⡟⠛⠋⠉⠁
⠀⠀⠀⠙⠁⠀⠙⠁⠀⠙⣇⣘⣊⣀⡟⠁⠀⠙⠁⠀⠙⠁⠀⠀⠀⠀⠐⠃⠀⠉⠁⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠉⠉⠙⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⠉⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢸⠀⢀⣀⣀⣀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣾⠿⡿⣿⣶⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣤⣾⣾⠿⢿⠛⠛⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣏⠀⠀⡇⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠛⢻⠀⠀⢸⠀⣰⣿⠃⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⢿⣶⣄⡇⠀⠀⠀⢸⠀⠀⢀⣀⣤⣶⣿⣿⡿⠋⠙⢾⡦⣼⡀⠀⣸⢾⡿⠁⣰⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⡀⠈⠻⡿⣿⣶⣶⣾⣶⣾⣿⡿⠿⠛⠋⠀⢸⠀⢀⣀⣹⣿⣿⣿⣽⣿⣾⡿⠟⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠻⠿⢿⣾⣷⣶⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠟⠛⢻⠉⠀⢸⠀⠀⠀⠀⠀⡁
⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣁⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠘⡇⠀⠀⠈⠃⠀⠀⠸⠁⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠳⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⡖⢰⡆⢰⡆⣶⠐⡆⣶⠀⠀⣶⢰⠆⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣧⢸⡃⢸⡇⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣆
⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⣿⢀⡀⣿⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠋⠀⠁⠘⠁⠙⠀⠀⠛⠈⠃⠛⠘⠁⠀⢀⣀⣤⣶⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠒⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣤⣉⠙⠲⢤⡀⠀⢀⣀⡤⠖⢉⣩⣤⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣃⣤⣶⣞⣉⣡⣴⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠉⠀⠈⢹⠿⠛⡉⠀⠀⠉⠒⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠧⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂⠀⠀⠀⠀⠀⠀⣀⣰⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠐⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠃⠈⡆⡇⢾⠀⠀⡇⢻⢸⠘⡀⠀⡁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠘⠀⠀⠃⠛⠘⠀⠛⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠛⠉⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠉⠛⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢠⠞⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠳⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢰⠇⠀⠀⢠⠀⠀⠀⢦⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠸⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠉⠈⠉⠛⠛⠟⠛⠙⣿⣿⡄⠀⢀⠀⠀⠀⠀⠀⡘⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠁⠢⣶⣾⣿⣿⣿⣦⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡖⠶⠀⠀⡀⠈⠋⠀⠀⠀⡤⠀⠀⠀⣠⠀⠀⢋⠁⢂⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⡩⠁⠈⠀⡄⠀⠀⠀⡀⠀⢀⡀⠈⠀⠀⠋⣰⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢄⠀⠀⠘⢄⠀⠀⢻⣤⣤⡔⠀⢀⠌⠀⢻⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠁⠂⢠⠀⠑⣄⠀⠉⢉⠠⠐⠁⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠬⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⠀⠀⠀⠀⠀⢀⠈⡄⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠿⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣤⣤⣤⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠂⣀⣻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣨⠀⠀⠀⠀⠀⠀⠀⠄⠻⣿⣯⣟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠠⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠔⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢶⣾⣿⠉⠻⢿⣿⣶⣤⣤⣀⣠⣤⣴⣿⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠁⣄⡈⠂⠀⠀⠤⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣤⣤⣶⣿⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠁⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⠄⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠒⠀⠉⠁⠒⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⣿⣧⠀⠀⣀⠠⢆⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠐⠛⠛⠃⠉⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⡰⠀⠀⠀⠀⠠⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⡄
⠘⢿⣦⣤⣄⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡏⠀⠊⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠑⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⣀⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣶⣾⣿⣶⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⢠⡀⠀⠀⠀⢃⠀⠀⠀⠤⢴⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠘⠀⢀⡴⠃⠀⠢⡀⠀⣤⣾⣿⣿⣿⣿⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠋⠀⢠⣤⡄⠐⢄⠁⢿⣿⣿⣿⢿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢠⣠⠄⠀⠀⠀⠈⠀⠀⠐⠾⢤⣈⣙⡋⢤⣤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢿⡟⠀⠀⠀⢠⠰⠷⢆⣀⡀⠄⠺⣯⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠸⢅⠀⠀⢀⣇⢸⡟⠃⣽⣷⣦⣾⣿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢘⣻⢘⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠙⠛⠋⠙⠛⠛⠛⠛⠛⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡄⣦⢰⡄⣦⢠⡆⣴⢠⡄⠠⣤⣾⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⠤⠿⠆⣼⡷⠀⡀
⠀⠀⠀⠀⢸⡇⣿⢸⡇⣿⢸⡇⣿⢸⡇⠀⠈⡏⢹⠀⠀⠀⠀⠀⠀⣀⣀⠄⣠⢞⡗⣄⡠⢄⣀⠀⠀⠀⠀⣀⠒⠲⠀⠘⡄⢰⢀⡄⢀⠖⠒⢀⡀
⠀⠀⠀⠀⠀⣧⣤⡄⠀⠀⢀⣀⣀⡌⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠘⠛⠛⣛⠛⢻⡿⣛⠛⠛⠛⠋⠀⠀⠘⠒⠢⢤⣈⢂⠴⠒⠪⣄⢋⡤⠔⠒⠃
⠀⠀⢀⣀⣀⣿⣿⣇⣰⣿⣾⣿⣿⣧⠀⠀⣿⡄⢸⣷⠀⠀⠀⠀⠀⣀⠀⠉⠁⣸⡇⠈⠉⢀⡀⠀⠀⠀⠪⠽⠀⠠⠄⣍⢰⣿⣧⢨⠄⠤⠀⠾⠽
⠀⠀⠀⠀⠈⢹⢹⢸⢹⣿⡏⡇⡇⡇⠀⠀⠉⡇⢸⠉⠀⠀⠀⠀⢀⣭⣥⣤⣤⣼⣿⣤⣤⣬⣭⡀⠀⠀⢰⣒⡄⠐⢋⠌⡷⠀⢼⡧⣈⠓⠂⣔⣦
⠀⠀⠀⢀⠞⣹⢻⣾⠸⠿⠇⣷⡟⣏⠳⡀⠀⡇⢸⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⢀⣴⡀⠃⢰⠃⠸⠀⢣⠈⠂⣰⡀
⠀⠀⠀⠈⠧⣈⡤⠃⠀⠀⠀⠘⢤⣁⠼⠁⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠶⣄⡀⠀⠀⠀⠀⠉⠁⢬⡷⠀⠼⡀⢸⡧⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠦⢾⣦⠀⠀⠀⠀⠀⢸⡇⠲⠀⠀⢾⠀⡄⠀⣇⠘⠀⠀⠀⠀⠀⠀⠴⠤⠤⠭⠥⠤⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⢱⡀⣼⠈⠀⠀⠀⢴⠀⢸⠃⡏⠈⣦⣸⣤⠀⠀⡿⣀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⠀⠀⠀⢀⣸⣸⣦⡇⡇⠙⢺⢸⠀⣗⡇⣸⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⢰⡃⡇⡇⠀⢸⢸⠉⡇⡏⢹⢘⡅⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣭⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠼⠼⠦⠧⠷⠶⠾⠼⠦⠧⠷⠼⠴⠇⠀⠀⠀⢀⡠⠖⣱⢣⡞⣎⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⠀⣠⣤⡖⠈⠀⠀⣼⢿⣷⣄⣀⡀⠀⠸⣿⣿⣦⣤⡀⠀⠀⠀⠀⣿⢸⡇⢸⠀⣿⢸⡇⣿⢸⡇⢹⠁⣿
⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣀⡀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠉⢸⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⣿⢸⡇⢸⠀⣿⢸⡇⣿⢸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣾⣾⡿⠿⠿⠀⠀⠀⠀⠀⠰⣼⡟⣿⡏⣻⡾⠀⠈⢹⣿⡏⠉⠀⠀⠀⠀⣿⢸⡇⢸⢀⣿⣿⣿⣿⣸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣭⣍⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣈⣹⣿⣿⣉⡀⠀⣀⣼⣿⣇⣀⠀⠀⠀⠀⣿⢸⡇⢸⢸⣿⣿⣿⣿⣿⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠛⠈⠃⠘⠈⠛⠛⠛⠛⠛⠃⠘⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣬⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠻⣷⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠯⣍⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠛⣯⣿⣿⣯⣱⣎⡛⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠱⣾⣤⣀⠀⠀⠀⠀⠀⢠⡾⡿⠻⡉⢠⣿⣿⣿⣿⣿⣿⣽⣶⣏⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⠉⢉⢩⡁⠈⢰⣤⣿⣿⠁⠀⠁⠾⠁⣿⣿⠉⠦⣈⠉⠛⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⢀⣼⣿⣅⠁⢈⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⡀⠘⠀⠀⢠⡇⠀⠀⠀⠉⣙⠿⠟⠻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠁⠀⠀⠀⠈⠙⢿⣾⣿⣟⣿⡗⠐⢾⣷⡦⠀⠘⠑⠽⠿⠿⠏⡈⠉⠛⠛⠒⢾⣗⣒⣒⠒⠌
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⣠⡶⠂⠀⠀⠀⠀⢻⣿⣿⣾⣟⠀⠀⢶⣾⣿⠆⠀⠀⠀⠀⠀⢻⣦⠀⠀⢀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠁⠁⡄⠀⠀⠀⣀⠀⠙⠻⠿⣿⣆⠀⠈⠁⡟⠀⠀⠀⠀⠀⠀⢸⡏⠀⠒⢿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠐⠒⠺⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⠁⠀⢀⣬⣶⣦⣾⣧⣴⡄⣤⡀⢰⢿⡏⠀⣰⠀⣰⠀⠀⠀⠀⠀⢀⣀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⡸⣿⣿⣿⠟⠻⣿⣿⣿⣷⣸⣾⣷⣏⣹⣾⣿⠀⠀⠀⣠⣶⣿⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⡰⢡⡟⠉⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣏⢸⣿⣿⠀⠀⠠⢿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢀⡜⣱⣿⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⠛⢻⡄⠀⠇⠸⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡇⠸⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⡀⠀⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⣻⣦⡀⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢤⣀⣀⣀⣻⣎⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣧⣡⠀⠈⠙⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣯⠇

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚
⠀⠀⠀⠀⢀⣰⣩⠀⣇⢸⢰⣀⠀⠀⠀⠀⠀⠀⣠⢄⠂⡯⢽⠐⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡁
⠀⠀⠀⠀⠀⠖⠒⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠤⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢴⣧⠀⠀⢠⠐⡄
⠀⠀⠀⠀⠀⣥⣤⣤⣤⣦⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡶⠂⢲⡀⠀⠀⠀⠀⠈⢿⣧⢀⡏⠀⢱⣀⣤⣾⡆
⠀⠀⠀⠀⢠⠿⢶⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢾⣿⠞⠋
⠀⠀⠀⠀⠻⣜⣡⠃⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⣄⣒⣚⣓⣒⣠⠄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⣿⠛
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⢉⡉⣿⢹⡇⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⢰⣾⣧
⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⡇⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⡢⢄⣴⡾⢿⣛⣯⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣸⣯⣿⣿⣽⣆⢹⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⣠⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣚⡄
⠀⠀⠀⠀⠀⣠⡾⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣤⣤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠰⠶⡆
⠀⠀⠀⠀⠘⠛⠊⢉⣁⡄⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⢐⠛⠛⡂⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⢾⡛⡏
⠀⠀⠀⠀⠀⠀⠈⢉⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠤⠀⠃⢀⠀⠸⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠐⠂⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁

⠀⠉⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠄⠀⢿⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⢷⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠾⣷⡒⣐⡾⠷⢄⣀⡀⠰⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⣤⢤⡖⣦⣾⣵⣿⢧⡞⢃⡀⠉⣗⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⢀⣾⡀⠘⣌⣧⣻⣧⣿⡿⠿⠷⠿⣦⣏⣼⡀⠉⢾⢱⣄⡀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⢠⣟⣿⣧⣠⠞⠋⠉⠀⢀⣀⣄⣀⣀⣀⠀⠉⢙⠢⣼⣾⣿⣿⡀⠀⠀⢉⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣾⣿⠿⢋⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠙⢧⣬⣁⡄⠀⢠⡟
⠀⠀⠀⠀⢀⣀⣀⣤⣶⣴⣾⡿⠉⠢⣄⠀⠀⠀⠀⡠⠀⠀⠀⢀⣤⡀⠀⠀⣀⡀⢰⡀⠀⠝⣿⣯⣭⣭⣤⣤⢤⡀
⠀⣀⠀⠀⣮⠋⢿⣟⠻⣿⡿⢀⢀⣾⣷⠀⣀⠘⣧⢶⡇⠀⠀⠹⠿⠟⠀⠹⠿⠏⢸⣿⣆⠀⠘⣿⠾⣭⣻⠜⣧⡌
⠰⣏⡀⢈⠃⣠⠘⢈⠰⣿⠁⠁⣾⣿⣗⠀⢼⡆⠸⠋⢀⡜⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⣦⠒⠘⣿⠋⠁⠸⠹⡟⠀⠠⣶
⠀⠈⠉⠀⠀⠀⠀⣤⣶⡇⠀⢸⣿⣿⡗⠈⠾⢯⡀⢧⠞⠀⠀⠀⠀⢀⠔⠈⠢⠀⢸⣿⣿⣿⡈⠀⢿⣠⠂⠀⠈⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠽⢿⠳⠊⢽⣿⣿⡗⠀⠉⠀⣧⠚⡄⠀⢀⣰⡶⠁⠀⠀⠀⢔⣼⣿⠉⢻⡅⠀⢼⡟⠀⠠
⠀⠀⠀⠀⠈⠡⠡⠀⠓⣆⢀⢸⣿⣿⡇⠀⣤⣰⠛⡀⠳⡀⠰⠋⠀⢀⠔⠀⠂⡀⢹⡏⠀⠸⠁⠁⡾⡐⢀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⠒⢻⣧⡇⢠⡾⠛⢠⠟⢧⡀⢅⣄⡗⢁⣤⣀⠀⢹⣼⣿⣦⠈⠀⠹⠁⠁⠈⠀⠈
⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠱⡀⠈⢻⡇⠋⠀⠀⠘⠲⠊⠀⠸⠋⠀⠹⣿⡿⠁⠀⢹⣿⠏⠀⡠⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣸⠀⠁⠀⠀⠀⠀⠱⣴⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠃⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⣉⡀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠻⣿⣿⣷⣦⣀⣀⣀⣤⣶⡿⠿⠛⠁⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣈
⠀⠀⠀⠘⠿⠿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡑⠋⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠈⠐⠠⠭⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘⡆⡆⢰⠀⡄⠀⠀⡀⠀⠀⠀⠀⠉⠀⢀⠀⣀⢀⠀⣀⠀⠀⢄⡄⠰⠀⠂⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠂⠁⠈⠀⠀⠉⠀⠁⠘⠄⠁⠀⠀⠉⠂⠀⠂⠈


⠀⠀⠀⢀⣤⣄⣤⣤⣠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⡀
⠀⠀⠀⢠⣾⡏⣶⣿⢸⡇⡖⢲⢲⢰⡗⢸⢺⣴⠗⢶⣷⢸⠃⡾⠂⡖⠀⠀⠀⣾⠇⠀⢰⢲⣼⡇⡇⣴⠆⡖⡶⢰⠆
⠀⠀⠀⠈⠛⠚⠙⠛⠚⠃⠛⠙⠘⠈⠃⠘⠚⠙⠃⠘⠋⠈⠃⠛⠀⠃⠀⠀⠀⠈⠛⠂⠈⠛⠉⠃⠓⠛⠀⢳⡷⠘⠂

⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⣠⣴⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣾⡟⠁⠾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄
⠀⠀⠀⠀⢸⡿⠀⠀⠀⢳⠀⠀⠀⠀⠀⣠⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀
⠀⠀⠀⠀⠘⠀⠀⢧⣀⡘⡇⢤⣤⣶⣿⠟⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⢱⠸⣷⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠳⠭⠄⠀⡇⣿⡿⠁⢀⠀⢸⠀⣀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣯⠵⢿⣿
⠀⠀⠀⢀⣀⣠⣤⡀⠀⠀⠀⠸⣾⣣⠀⠘⣠⠆⡀⢿⣿⠇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠐⣿⣿
⠀⠀⠀⢸⣿⡛⣗⠪⢔⡀⢄⠀⢹⠃⠀⢀⡿⠀⢱⢸⡟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⣿⣏
⠀⠀⠀⢸⣿⠀⡏⣀⠀⠈⡱⢧⣾⠀⢀⣈⣍⣁⣈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣻⣿
⠀⠀⠀⢸⣿⠀⠁⣿⣿⠇⠈⠃⣿⠀⢸⠛⡏⢻⣿⡟⠀⠈⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⢈⠀⠹⢿⠃⠀⡆⢨⠀⣿⡇⡇⢸⣿⡇⠏⡀⠀⠀⢸⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠂
⠀⠀⠀⣿⣿⣸⠄⠀⢺⠀⠀⡃⠠⢸⣿⣧⡇⢸⠟⠁⠀⣿⣷⠀⢈⣷⣿⣼⣿⣶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⠆
⠀⠀⠀⣿⣿⢿⠂⠀⠈⢁⣠⡄⠛⢺⣿⣽⡏⢹⡏⠀⠐⠻⠨⠀⠀⣿⣹⣫⡆⣻⣤⣀⡀⠂⠀⠀⠀⢼⣿⣿⣿⡎
⠀⠀⠀⣿⡟⢸⢸⣷⣦⠀⠉⠁⠀⢠⣿⣽⡇⣸⡇⠀⠀⠉⠂⠀⡈⣟⣻⣼⣥⣶⣡⣿⣿⡆⠀⠀⠀⣀⣿⣿⡏⢱⡀
⠀⠀⠀⣻⡇⠀⢸⡷⣿⠀⠀⡆⠀⢸⣿⣿⣿⣾⣷⣤⡀⣄⡀⠀⠃⠛⠋⢹⣿⣯⣿⣿⣿⠃⠀⢀⠀⣾⠚⠉⠏⠁⠁
⠀⠀⠀⢸⡇⠈⢸⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⡿⠋⠉⢸⢧⣾⢙⡀⠀⠀⣸⣿⣿⣿⠿⠋⠀⣀⣴⡆⣟⠩⠀⣟⡄
⠀⠀⠀⢸⡗⡆⠘⠁⠛⠀⠀⠀⡄⠀⠀⠙⣿⡇⠀⠀⢸⠀⣧⢈⣷⣤⣰⣛⡟⠿⣿⠡⣤⣾⣿⣿⣷⣷⣶⣴⣷⣾⣦
⠀⠀⠀⠀⠛⠳⠀⠠⠀⠀⢰⣿⠇⢠⣤⣼⠀⠃⣤⠀⠸⠀⠃⢸⢟⣿⣇⣹⣬⣟⣯⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠁⠶⠶⢶⣶⣦⣤⣉⡤⠼⠭⣿⣥⣦⣙⣠⣴⡂⠘⠿⠿⢿⠙⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⡒⣶⣶⢸⣿⠈⠋⠀⠀⠀⠠⠄⠈⠉⠉⣙⠣⡤⣤⠤⣤⣤⠼⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣌⣿⢸⣧⠀⠐⢲⠄⠐⠲⠆⠐⠆⢀⣿⢶⣾⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣷⡦⡄⣶⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠂⠠⣤⣦⠓⠀⠀⠀⠀⠀⠀⣀⣀⡈⢻⡏⠰⣿⣿⣿⣿⣿⣦⠽⣧⠛⠛⣻⡏⠀⠀⢀⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⣀⣀⡄⢉⣽⡇⠀⠠⣤⡾⣿⣶⣿⣿⣯⡾⣾⣿⣿⣿⣿⣿⣿⡇⠀⠇⣰⣿⣿⣇⠀⣀⣸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢹⣿⠇⠸⣿⣧⠀⠈⣿⣧⣿⣉⣇⣸⣿⣷⣿⣿⣿⣿⣿⡆⠹⣇⡀⢠⣿⣿⣿⣿⣋⠈⢙⣿⣿⡟⣯⡄
⠀⠀⠀⠀⠈⠛⠄⠰⠻⢿⣀⣀⣀⠹⡇⢸⠋⢰⣌⣹⣹⡄⣿⣿⣿⣷⣀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣧⣅⣻⡄⠉⠛⠛⠛⣷⢸⣿⣍⣹⣿⠿⠿⠿⠿⢿⠿⣿⣿⣿⡿⣻⣿⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠻⠿⠄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣶⣶⣿⣼⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡛⠋⠁⠀⠀⠀⠀⠀⠀⣀⣶⡿⣫⣴⣿⡿⣿⣿⣤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣅⣀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣄⠠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡄⠀⠀⠀⠀⢲⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣤⡾⡇⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⣈⣿⣿⣼⣿⣿⣿⣿⣿⡿⣯⠉⢙⠿⣿⣿⣹⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆⠀⠀⢺⣿⡇⣿⣿⣿⣿⣿⣿⣿⡜⡗⣩⠞⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⣄⠀⣽⣿⣷⣿⣿⠻⣿⣿⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣾⠿⣿⣿⣿⣿⠞⢛⢿⣯⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⢿⡛⣻⣤⣀⣼⣿⣦⣀⠉⠈⠙⠻⠷⠌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠁⠀⠀⠈⠻⣿⣿⡝⢿⣿⣿⣿⣷⣶⣤⣤⣀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠟⠛⢻⣿⣿⣧⣄⡀⠀⠀⠀⠈⠻⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣤⣴⣶⣿⣿⣭⡉⠙⢿⣶⡆⠀⠀⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡟⢿⣷⡀⢸⣿⣿⣤⣼⣿⡇⠀⣼⣿⡿⣿⣿⣿⢻⣿⣶⣶⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⠘⣿⣿⣧⣿⣿⣷⣼⣿⣿⣿⣿⣿⣥⣼⣥⣽⡿⠛⠁⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣷⣼⣿⣇⣹⣤⣿⣿⣿⣿⣿⣿⣿⣿⡟⢏⠁⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢻⡟⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠈⠛⠿⣿⣾⣷⠀⠀⠀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠈⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣈⣿⣿⣿⣿⣿⣿⡟⢿⡿⠉⠀⠀⠀⠀⠀⠀⠀⣹⣿⡅⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠟⠛⠛⠛⠿⠿⠛⠓⠈⠀⣴⣾⣿⣤⣤⣴⣶⡿⠿⠛⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢻⣿⡿⢿⣿⣿⣿⣿⡆⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢿⣿⠀⠀⠀⣠⣾⣿⣿⣌⣻⣿⣿⣿⣿⠾⠃
⠀⠀⠀⠠⠶⠟⣻⣿⣿⣿⣿⣿⢸⠏⠀⠀⠀⣹⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠃⠘⠀⠀⣶⣿⣿⣿⣿⠿⢿⣿⣧⡀
⠀⠀⣤⣭⣿⣿⣿⡿⢻⡿⠃⠀⠀⠀⠀⢹⣿⣿⠟⠀⠀⠀⠈⠛⣿⣷⡄
⠀⠀⠀⠈⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⢀⠄⣤⠠⣄⢄⡀⠀⠀⠀⣠⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠤⠄⠀⡀⡀
⠀⣼⠃⣼⡇⠀⣿⡆⢹⣆⠀⢀⡙⢿⣬⣥⡄⠀⠀⠀⠀⠀⣂⡃⠀⠲⠂⠀⠛⠃⠀⠲⠆⠀⣒⡂
⠠⣿⠀⣿⡇⠀⣿⡇⠈⣿⠀⠻⢿⠻⣿⣿⡇⠀⠀⢨⣭⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨⣩
⠀⢻⡄⢻⡇⠀⣿⠇⣸⠏⠀⠀⠀⠀⠛⠋⠀⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠙⢆⢿⢠⡏⡜⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠈⠺⢼⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡀
⠀⠀⠀⠀⠀⠄⠀⢀⠀⠐⠰⣄⡀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢺⣗⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⢀⡀⠐⢟⣐⣀⡸⠋⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠋⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋
⠀⠀⠀⠀⢨⠀⠿⠀⠘⠉⠉⣷⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠛⠀⠀⠀⢐⣒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠤⠈⠁
⠀⠀⢀⣐⣿⣿⣿⣿⠀⠀⠀⠆⠰⠀⠆⠀⠀⠐⠂⠠⠄⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣄⠐⠖
⠀⢀⣾⣿⣟⡳⢾⣾⡿⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠖⠀⠤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⠤⠀⠰⠖⠀⢀
⠀⠸⠛⠛⠛⢿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⡍⠉⠀⠀⢻⠏⠀⣭⠀⠀⣈⡀⠀⣭⡅⠈⣿⠁⠀⠀⠀⢸⣿⣿
⠀⠾⢿⣿⡆⣸⣰⣶⣶⣶⣦⠀⢰⠀⢠⠀⠀⢰⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⢸⠉⠉
⣀⣀⣀⣀⣀⣼⣈⣉⣉⣀⣀⣀⣼⣄⣸⣆⣀⣼⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣼⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣯⣿⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡿⠟⢻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣊⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡴⠂⢋⠏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⢀⣴⠟⢀⡴⠇⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡆⠀⠀⠀⢀⣀⣀⠹⠿⠋⢈⣁⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣠⡤⠶⣿⣫⣿⣥⠴⠓⠁⢨⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⡿⠟⠉⠀⣠⠲⠛⠛⠟⠃⠀⠀⢀⣿⠠⠄⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠀⠀⢠⠞⠁⢠⣰⠆⠄⢠⣶⠞⠋⠁⠀⢀⣠⠖⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⣄⣴⣿⣿⠟⠁⠀⠀⠀⢠⡇⠔⠀⣼⠏⠀⢠⡿⠛⠿⠳⠟⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⠀⠀⢰⣠⣾⡏⠀⣴⣾⡟⠐⢦⡿⠁⠀⠀⠀⠀⠀⢀⢀⣀⣄⣿⠁⠀⠀⢠⡆⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⢀⣀⣾⣿⣿⣿⡗⣾⣏⠛⠃⠀⡿⣧⣶⣶⣶⣶⣿⣏⣿⣿⣿⣟⡍⠀⠀⠀⢀⣿⠛⠿⠛⠋
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⣶⣿⡿⠋⠉⠈⣷⣼⡏⠑⠲⡄⠘⠛⠻⠿⢿⣿⡿⠿⢿⣿⣯⣿⡇⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠸⣿⠃⠐⠋⡙⢏⠀⠀⠀⢠⣿⣿⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣾⣾⠋
⠀⠀⠀⠀⠹⠿⣿⠖⠋⠉⣿⣺⣆⠀⣠⣾⡿⡩⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⣇⠚⢿⣿
⠀⠀⠀⠀⠀⠀⠙⣷⣶⣞⣽⣿⡶⠖⢻⣿⠃⢷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣧⠀⠀⣿⡄⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠀⣼⠇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣷⡍⠀⠀⢸⣷⡀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠷⠙⢿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠈⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⢼⣿⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠂⠀⠸⢋⠅⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⠄⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣶⠶⠦⣤⣤⠆⣤⣤⣤⣤⣶⡷⣿⠿⢗⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣠⠤⣖⣨⢹⡏⣝⣳⠦⣄
⠀⠀⠀⣠⢞⣵⠋⠙⡄⠀⡆⢠⠋⠙⡢⡛⣄
⠀⠀⢰⢧⣏⡀⠑⢄⠸⣸⣇⠎⡠⠊⠀⣙⡼⢆
⠀⠀⣆⣚⣀⣈⣑⣢⣽⡿⢿⣾⣔⣊⣁⣀⣱⣸⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣏⢭⠀⣀⡠⠔⡻⣷⣾⢿⠢⠄⣀⠀⢸⢸⠀⠀⠀⠀⢀⡼⠛⢧⠀⠀⠀⠀⠀⠀⢀⢖⠄
⠀⠀⠸⡞⣏⠀⡠⠊⡰⢹⣿⢷⠳⣄⠀⢩⢓⡟⣠⠔⢲⢾⣉⣁⣉⣈⢉⣁⠆⠐⢀⡀⡇⡯⡀⢰⣤⡀
⠀⠲⠲⠟⢬⠺⣗⢳⠃⣿⣿⠼⣦⣼⠗⡧⠎⢀⣧⡴⠾⢲⣧⡌⠀⠣⠀⣼⣆⣂⣅⣍⣛⣹⡇⢠⠏⢏⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠒⠛⠢⢏⣰⣻⣟⣷⣽⣴⣫⡥⠖⠋⢙⣾⣴⣯⢼⡗⣶⢲⡀⢹⡓⢿⣽⡿⠽⠚⠁⠀⠀⠈⠀⢀⠛⣡⠚⠃
⠀⠀⠀⠒⠛⠓⠂⠐⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠋⠓⠐⠀⠐⠓⠀⠚⠀⠀⠀⠀⠒⠂⠀⠐⠛⠛⠛⠛⠛⠛⠒⠛⠒⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠖⠆⣀⣀⡀⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣟⢧⡀⠻⠝⢿⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣻⣿⣿⣀⣦⣷⣽⣽⣽⣧⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⠟⠉⠉⠈⠀⠀⠈⠛⠹⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠶⠦⠀⠘⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢺⣶⣤⡀⠀⠘⣿⣯⣽⣶⣶⠀⠀⠀⠀⢻⣿⣿⣻⡿⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠯⠿⣷⠀⠀⠛⣟⠋⠉⠙⠀⠀⠀⠀⢸⣿⡷⡿⠂⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠸⡗⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠤⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠐⠊⣉⣻⡿⠋⠀⠀⠀⠀⠀⣠⣾⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠈⡟⠀⠀⣼⣿⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣶⠲⡞⠛⠋⠉⠀⠀⠀⠈⠀⠀⣰⠟⡏⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡟⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢰⠇⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡞⠀⣰⣿⠁⠀⢸⣧⣄⡀⠀⠀⠀⠀⢀⣠⡞⠁⠀⡞⢠⡿⢛⡁
⠀⠀⠀⠀⠀⠀⢀⡤⣶⢚⡉⣥⣶⣿⣠⡞⢹⡯⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣾⠃⠘⣠⣿⡇
⠀⠀⠀⠀⢀⣴⢫⣾⣿⢟⣾⣿⢯⣾⣫⠔⣸⣧⣶⡷⠀⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡏⢀⣾⣿⣿⡇
⠀⠀⠀⠀⣹⡟⢿⡽⣫⣿⣟⣉⣟⣻⣟⡲⣿⢿⣿⠃⠀⢸⣧⣿⡟⣵⣦⠀⠀⠀⢀⡞⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢙⣵⣿⣶⡿⣿⣿⣿⣿⣿⣿⠇⣴⣾⠃⠀⠀⢸⣿⣿⡇⢿⣿⣧⡀⠀⡞⢠⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⡇⢹⣿⣿⣏⢹⣧⢧⠙⢿⣷⣄⠀⢸⠹⣯⣷⢸⡇⢿⣿⡾⣀⣈⠋⠉⣰⠿⠿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣇⣿⣞⣧⣘⣿⣿⣇⠀⠀⣿⣿⡄⠀⠘⣼⣿⣮⢻⣿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⣿⡏⣿⡏⢘⣿⣿⣧⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠰⣿⣿⣷⠀⡰⠁⡉⣫⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠿⠁⠿⠰⠿⠿⠿⠿⠿⠿⠿⠿⠆⠻⠿⠿⠿⠀⠀⠿⠿⠿⠠⠷⠿⠽⠿⠃⠻⠋⠸⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠂⠢⡀⠀⠀⠀⠀⢀⡴⢟⡍⢀⣠⣴⣾⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠁⠐⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡼⠋⠀⠀⠙⢿⣷⡌⢧⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠰⢠⡾⠁⠀⠀⠀⠀⠀⠹⣿⡎⢷⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠤⡀⠀⣀⠀⠈⢀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⡇⠀⠀⠀⠀⠀⠒⢄⣀⠔⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠀⣧⠀⠀⠀⠀⢸⣷⣄⠀⠀⢸⠇⡿⢠⠐⠂⡀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⡄⠀⠀⠀⣼⢿⡆⠙⢦⡉⢰⡇⠸⣀⡠⠃⠀⠀⠀⠑⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⢷⡀⠀⠀⡏⠘⣿⡄⢈⡇⠸⡇⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⢧⣠⣄⠹⣄⣀⡇⣿⣿⣷⣼⠁⠀⢷⠀⠀⠘⠢⢀⡀⠁⠀⠀⠀⠁⠊⠦⡄⡀
⠀⠀⠀⠀⢋⠀⢈⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣮⣉⣓⣏⣩⢴⡏⠀⠀⠸⡄⢀⠓⠋⢒⡄⠀⣈⣀⠐⠀⠀⠄⠀⡉⡠
⠀⠀⢀⢷⠀⢹⠁⠀⠀⠀⡐⠈⠀⢀⣼⡠⢾⡿⣿⣯⣷⠉⡼⠁⠀⠀⠀⢳⡀⠀⠀⠀⡇⠉⢀⠈⠋⠉⠒⠒⠦⠀⠥
⠐⢀⡞⠈⣧⢸⢈⢆⠈⢇⠜⠀⠄⠿⠿⢗⣿⣩⢻⡷⣟⡿⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇⠀⠈⠀⠀⠀⢠⡁⠁⠀⠑
⣀⡼⠀⠀⠘⣿⠹⠐⠤⠀⡁⣴⠮⢠⡧⢀⡟⢢⣻⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠁⠀⠀⠀⠀⠀⠘⠃⠓⠴⠂
⢹⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡆⠈⠷⢚⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⡦⠄⢒⠀⠀⠀⠀⠐
⠘⢦⣀⣀⣠⠆⠀⠁⠀⠀⠀⠐⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⢀⡀⠀⢠⡄⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡛⠟⠛⠔⢁⠀⠀⠀⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠄⠀⠠⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠦⣤⣴⣤⣶⣶⡯⢴⣶⣶⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠴⠿⠿⠿⠿⠿⠷⠾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⣤⢀
⠀⠸⣷⣾⡿⠀⠀⣄⠀⠀⣄⡀⠀⠀⠀⢀⠀⠸⠂⠀⢸⣹⣃⣴⣄⠀⠀⢀⣤⣶⣿⣿⣾⣿⣿⣿
⠀⠀⣾⣿⣧⠀⠘⣏⠠⠿⠿⠗⣰⣿⡅⢸⣷⠶⠃⠠⡺⠀⠉⠙⠿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠈⠁⠈⠀⡠⠈⠀⣀⠠⠘⡛⠻⣿⣿⣿⣄⡀⢀⠀⠠⢠⠀⡀⠀⠀⠹⣿⣿⣿⡿⠻⠿⠟⠛
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠋⠀⡔⠀⠀⠀⠹⣿⣿⣿⣮⠏⡇⠢⠄⠄⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠔⣈⡀⠙⢦⢐⡥⠚⠁⡄⢀⠀⠀⠈⢿⣿⣗⠓⡿⠊⠱⣶⣤⣀⠒⠚⠻⠿⣁
⠀⠀⠠⣤⣈⣡⠆⠈⣧⣤⡤⢾⠀⠀⠀⠀⠀⠀⠻⣿⣶⣷⡄⠙⣿⣋⠈⠙⢦⠀⠀⠀⠙⢄
⠀⠀⠀⠰⢦⣅⠎⠻⢄⡀⠉⠚⡦⢄⣀⣀⣀⣀⣀⠼⠋⠹⣷⠀⢻⣿⣆⠀⢸⡇⢐⡄⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⢷⣄⣹⣧⢀⡾⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠀⠀⠀⠀⢀⠔⠀⠀⠹⣦⣽⣿⣿⣿⡀⠀⠀⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡀⢹⡇⣷⣿⠏⠀⣰⠁⠁⠀⡸
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⠦⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡇⣼⠇⠉⠉⣀⠔⠳⡄⠀⠰⠁
⠀⠀⠀⠀⠀⠀⢈⣿⡟⠻⡟⠀⠠⠀⠀⠀⠀⠀⡔⠂⠀⠀⡄⣰⣟⡠⠔⢺⠁⠀⠀⢸⢀⣠⠀⢀
⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠘⢆⠀⢀⡴⠤⠷⢤⡇⠀⠂⢸⣿⣿⣿⣶⣿⣾⣦⡀⠀⢸⣼⣿⣋⠀⠈
⠀⠀⠀⠀⠀⠈⣿⣿⣀⠀⠀⠀⠉⠉⠀⠀⠀⢀⠇⠀⣠⣿⣧⡀⠉⠉⠙⠛⠛⠁⠀⠈⠻⠛⠁⠱⠀⢄
⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡎⣠⡜⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠈⠀⠐⠂⠒⠀⠘⠀⠀⠀⠀⠒⠂⠀⠐⠒⠀⠘⠀⠃⠈⠀⠀⠀⠀⠀⠁⠂⠐⠒⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣤⣿⣻⣿⣿⣶⣷⣶⣶⣶⣦⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⣠⢸⣿⣿⣿⣿⣿⣷⣽⣿⣿⢿⣿⣿⣿⣙⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡟⢇⢾⣿⡿⣿⣿⣽⠻⣿⣿⡙⢶⡙⣿⣧⢻⣷⣹⣿⣿⣷⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣾⣺⣷⣿⣏⣻⣿⣷⣙⣿⣿⣦⣉⡾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣝⠿⢿⣿⣿⡿⢽⡿⡿⠋⠛⠟⠛⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣻⣿⣿⣿⣿⣿⣶⣿⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⠿⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠝⠛⣛⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣍⢿⣧⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⠇⠀⢀⣀⣤⠶⣶⣶⣤⣄⠀⠀⠀⠀⠐⠛⢛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢟⠇⠀⠰⣿⣏⣀⠤⣒⡛⠛⠿⠀⠀⠀⠀⠀⠀⠄⢺⣿⡿⠓⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡆⠀⠀⠰⠿⣿⣿⠛⠿⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣼⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⠟⠃⠈⠋⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⡄⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢉⠿⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠀⠀⢸⣆⡀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏⠹⡇⠀⢸⣿⣷⡀⠀⠌⠀⠰⠟⠋⠉⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠸⣿⣿⣿⣆⠀⠀⠈⠙⠂⠈⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣶⣤⣤⣴⣶⣶⣶⣶⣶⣶⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⡀⠀⢠⣄⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⢀⡄⠠⡄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡅⣸⡶⡄⠹⣶⣤⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠰⣾⣦⣿⠟⣛⢿⣦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⣯⠁⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⠃⠀⠼⢫⣽⣷⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⣤⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣽⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡿⠆⣉⠽⢛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣾⢁⠀⠈⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⢨⡟⠀⠀⠀⠀⢶⣶⠏⠀⠀⢸⡜⠛⡄⠙⠋⠲⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣀⣼⣷⡦⠄⠀⢀⣸⡿⠀⠀⠀⡈⠀⡀⠹⣆⣈⠀⣠⡶⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢇⠀⠀⠀⢘⡂⠻⣇⠀⠀⠀⠟⡃⠘⢶⣾⣏⡼⠋⣀⠳⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠸⣿⣶⡉⠦⡀⠀⠀⠡⡀⠉⢢⡙⣷⢲⡏⢋⡁
⠀⠀⠀⠀⠀⠄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠈⠳⣤⡀⠁⢠⣤⠿⣁⡷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⡴⣯⣤⡀⠀⠀⠀⠀⠀⣿⡽⢻⡇⣀⠀⠀⠀⠉⣉⠉⠉⠀⠈⣤⣀
⠀⠀⠀⠀⠀⠀⡄⠀⠹⣦⡀⠀⠀⠀⠘⣻⠞⠁⠙⣇⠐⠡⡀⠙⠋⡁⠉⢳⠀⠈⣠⠀⠁⢦
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣷⠅⠀⠠⠊⡁⠀⠀⠀⣿⠀⠀⠈⠃⠀⠉⠈⡌⠀⠀⠁⠀⡀⠀⠣
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡷⢈⣴⠀⠀⢇⣀⡀⢸⣿⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠘⠊⡀⣡⠀⠀⠀⢀⠀⠀⠅
⠀⠀⠀⠀⢠⠄⣶⠀⠠⡟⠉⢘⠁⠀⢸⡏⣿⣿⢿⢁⠀⠀⠙⠀⠀⠀⠴⢞⠂⣀⠉⠂⠀⢀⠀⠁⠤⠀⣠⠜⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣀⢷⠀⣆⠀⠀⣸⣧⣿⣿⠈⣆⠁⢀⡀⠀⠰⠁⠀⠈⠀⠈⠁⢀⢀⡎⠐⠀⠀⡠⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠤⠘⠼⠿⣿⣿⣿⣿⣧⣿⣿⡄⡘⣦⠈⣧⡀⠀⡄⠚⠳⡌⢂⣤⠀⣄⡴⠄⠴⠋⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣵⢞⣿⣿⢣⡾⠁⠈⠑⢬⣁⢀⠘⠢⢀⠘⣮⡧⣽⠃⠀⠀⢀⠴⠓⠊⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣥⠀⠆⠀⣿⣿⢉⣠⣐⡛⠲⢄⠉⠛⠦⡂⠀⠀⠈⠲⡀⠀⠀⠀⢨⡐⢤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠶⠀⠀⠀⢿⠛⠤⠀⠀⠀⠀⢀⣽⣷⣶⢶⣿⡷⠆⡀⠀⠐⠀⠀⠀⠁⠁
⠀⠀⠀⠀⣀⡀⠀⢀⡞⠀⠀⠀⠡⡄⠀⠈⠀⠀⠀⠀⠀⠐⠋⠐⣿⣿⡿⠟⠁⠀⠀⢬⠃⠀⡄
⠀⠀⣠⡞⠛⣻⣿⣿⡇⠀⠀⠀⠈⡛⢶⢶⣤⢤⡶⠀⠀⣠⣀⣮⠔⠉⠀⠀⠀⠀⠀⠀⠙⢦⠠
⠀⠀⢿⣤⣿⣧⣉⡻⣧⠀⠀⠀⠠⠀⣜⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢣
⠀⢠⣾⣿⣿⣏⣉⣿⣿⡇⠀⠀⠂⠀⣸⡿⠻⣿⣿⣿⣿⡟⠋⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢱
⠀⠀⠈⠛⠛⠛⢶⣄⣮⣇⢀⠀⠀⠀⣼⠁⠀⠈⢻⣿⣿⠇⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸
⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⡆⢀⠀⢀⡟⠀⠀⠀⢸⠗⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣀⣃
⠀⠀⠀⠀⢐⣿⠛⢙⣧⣾⠇⠀⢠⡎⠀⠀⠀⠀⠈⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠈⡹
⠀⠀⠀⠀⣺⣿⢿⠿⠛⠏⠀⢠⡆⠀⠀⠀⠀⢠⡘⠀⠸⡀
⠀⠀⠀⠀⠻⡏⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠑⠡⡂⠱⣄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠂⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠤⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠘⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⠿⢿⠟⢻⠿⢿⠿⠻⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣷⣿⣶⣿⣷⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⡔⠂⠀⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠂⣄⣀⡀⠘⢆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠄⠄⠒⠛⠛⠿⠇⠀⢂⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡟⠓⠀⠀⠀⢀⡖⠂⠀⠘⠘⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⢠⠀⣼⣅⣾⡟⣀⠀⠀⣀⢶⡛⡻⢣⡀⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠆⠀⠀⠀⠀⠀⠈⠄⠠⠉⠁⠀⡁⡇⠀⣅⣉⣄⣋⠋⢃⠘⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡖⠀⠐⠂⠀⠀⠠⠀⠋⠀⠀⠀⠀⠉⠉⢙⠃⠃⠛⠛⠻⠛⠃⣨⡁⢦⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⠀⠀⠀⣠⠖⠀⠀⠒⠂⡀⠀⡀⡄⠀⠺⠛⠃⠟⠛⠸⠺⠗⣇⡀⠀⠉⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣂⣰⣶⢂⠁⠀⠀⠀⠶⠀⠜⠅⡄⣡⡤⠸⠛⠇⠿⠹⠨⠿⠆⠆⠀⢀⡀⠣⡀⢉⡁⠀⣴⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢅⣹⣿⣿⡜⢃⣀⡀⠀⠦⢷⡗⡶⠇⣼⠶⣾⡿⡓⠛⢾⡼⡛⢢⣬⣧⣤⣤⣁⡷⢤⡄⢼⡍⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⣿⣿⣿⡏⠠⠀⠁⠀⠀⠐⠂⢀⡐⢰⠀⣶⢶⢀⠀⠀⠀⠀⠁⠆⣤⣇⡸⠤⣄⣌⠀⠈⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢰⡇⢰⡿⠀⠋⢸⢳⣿⣾⡆⠀⠀⠀⢨⡏⢰⡾⠙⣆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⢻⠁⠁⠀⠐⠄⠖⠿⠗⢿⡷⡾⡗⡿⠿⠿⠿⠇⠶⠶⠿⠿⠿⠾⠿⠟⠿⠻⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⣿⣻⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⢄⣀⠀⠀⠀⢀⣿⣄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⡁⣀⣀⡼⢿⣿⣿⠟⠛⠛⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣾⣿⣿⢿⣇⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⢧⣸⣿⣌⠀⠀⣱⣄⣀⣀⣀⡀⠴⣿⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣾⣿⣿⣿⣿⠃⠀⠝⣿⣿⣷⣾⡉⣉⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣁⣸⣿⣿⣿⣿⠏⠀⠀⠀⠹⢮⢻⣿⣷⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⠛⠋⠙⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣶⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣄⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⡏⢸⠿⠀⠀⠀⠀⢐⣃⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣿⣿⣿⡹⣆⠈⠆⠀⠔⠈⢫⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⢻⣷⣽⡳⠄⣤⢰⣿⣾⠟⡋⣸⣯⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡅⠈⠹⠟⠉⠋⣠⣇⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠿⠛⣷⣄⡀⠀⠀⢠⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⡆⠀⠀⣾⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣾⣻⡀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣿⣿⠁⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⠟⢀⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣶⣶⡶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢶⣛⣩⢭⡯⢭⣋⣳⠦⣄
⠀⠀⠀⠀⢠⡼⣧⠔⠋⢳⠀⠀⡇⠀⢰⠋⠓⢬⡽⢆⡀
⠀⠀⠀⢰⢏⡼⠉⠳⣄⠀⢧⠀⡇⢠⠇⢀⡴⠋⠹⣎⢣⡀
⠀⠀⢀⡏⡝⠓⠦⢄⣈⠳⣌⣦⣧⣟⡴⢋⣀⠤⠒⠛⡍⣧
⠀⠀⢸⣥⡧⠤⠤⠤⠬⢽⣾⡟⠙⣿⡾⠭⠤⠤⠤⠤⣧⢼⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⢸⡄⡇⠀⣀⡠⠔⢚⡽⣿⣾⡿⣟⠒⠤⣀⡀⠀⡇⢼⠀⠀⠀⠀⠀⢀⢴⠃⠙⡦⡀⠀⠀⠀⠀⠀⠀⢀⢜⡵⢆
⠀⠀⠈⣷⠺⡉⠁⢀⡴⠋⣰⢹⣿⣽⡾⢷⣄⠀⠉⡹⢳⠇⢀⣤⠶⣶⣾⣁⣃⣘⣀⣙⣘⣂⡢⠄⠒⠒⠒⣎⣮⡄⠀⢠⣀⡀
⠀⢠⣤⣬⣷⣙⣴⣯⠴⣴⠃⡿⣿⣇⣧⣀⣹⣷⣜⣡⠏⠀⡟⢠⣠⣦⣼⣇⡸⠈⠘⡇⠘⣸⡤⠴⢂⡋⡇⠷⠿⣸⠀⢉⡟⣟⠂⠀⠀⠀⠐
⠀⠀⠀⠉⠀⡙⣦⣝⢳⠯⣠⣇⣿⣯⣨⢿⢛⡥⠞⢁⣠⡶⢿⣉⠍⣀⡿⢹⣧⣀⣠⣡⠀⢿⣿⣷⣶⢿⣷⣶⣿⡿⠀⡚⠀⠘⡄⠀⠀⣦⡆⢰⡤⡀
⠀⠀⠀⠀⢀⣀⣀⠈⠛⢲⠞⠼⠧⣾⣿⣛⣿⣷⣿⣋⣉⣤⣤⢿⠟⠿⠒⠻⡇⠸⠆⠿⠀⠸⠆⠙⠟⠛⠛⠋⠁⢀⣀⣁⣀⣀⣀⣀⡧⠁⠞⡇⠀⠆
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣤
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢻⠟⠈⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⡴⣶⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⣄⣀⠀⠀⠿⠃⠀⠀⠀⠀⢰⣴⡯⢤⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣾⡖⣢⣤⡀⣀
⠀⠈⢱⣿⠁⠹⣄⣀⡶⠀⠀⠀⠀⠀⠸⣿⣿⠀⠁⠀⣆⡀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣶⣶⣿⡄⠀⠀⠀⠀⢚⣿⣾⣟⣿⣽⣭⠇⠉
⠀⢀⡾⠙⣇⢀⣠⢿⡙⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠈⢛⣋⣩⣍⣋⡑⠃⠀⠀⠀⠀⠐⠾⠃⣉⣙⣟⣀⡀
⠀⠘⠃⠀⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛
⠀⠀⠀⢀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⢉⠄⠀⠀⡉⠶⡀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⢠⠶⠲⡆
⠀⡋⢀⠀⢠⣶⡄⠀⡀⣱⠀⠀⠀⢀⠀⡤⢤⣶⣶⣀⢀⠈⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⣿⣷⣧⠀⠀⠀⠆⠀⠀⣈⢔⠊⢲⡀⣠⠒⢢
⠀⢣⠐⢀⡜⠉⢦⠀⠂⡌⠀⠀⠀⠈⠘⠂⠶⣿⣿⡥⠙⠰⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢈⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢆⠀⠈⠁⠀⡸
⠀⠀⠳⣼⢁⣀⢈⣦⠎⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠞⣃⣻⣿⣈⡓⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊
⠀⠀⠺⠀⠀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠖⠒⠒⠒⠲⠒⠲⠲⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⡰
⠀⠀⣧⣏⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢢⣷⠞⣿⣁⣀
⠀⠀⣿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⣼⡿⣙⠯⣍⠁
⠀⠀⣧⣀⣈⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡾⠳⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣷⣿⣷⡤⣀⣀⠀⠀⠀⠀⠀⢀⣾⣛⣕⡩⠒⠉
⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⡞⢳⡀⠀⠀⠀⣖⢸⡇⠀⠀⠀⢀⣸⣛⣿⣭⣤⣬⣿⣥⣄⠀⠀⠀⢨⣿⡿⠋
⠀⠀⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠚⠛⠛⠛⠀⠀⠀⠘⠛⠛⠛⠛⠒⠒⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡄⠀⠀⠀⠀⠈⠀⠀⣾
⢦⣀⣿⣀⣀⡀⠀⠀⠀⡀⠀⠀⢀⡤⣄⡻⠃⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣾⣤⡤⠤⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠡⢿⡇⠤⠤⠤⠤⠤⠤⠀⠀⠀⢰⣟⠻⣄⡀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠻⣿⡀⠃⣸⣿⠇⠀⠀⠀⠀⠀⢰⣶⣀⣴⣦⣶⡆
⠀⠀⣸⠟⣆⠀⠀⠀⠀⠀⠀⠀⣠⠟⢹⠀⣠⣤⣴⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠈⠉⣻⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇
⠀⠀⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⣈⣁⣈⣀⣀⣀⣊⣁⡀⠀⠀⠁⠈⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠛⠉⠛⠋⠛⠛⠛⠛⠛⠓⠐⠒⠂⠒⠄⠉⠉⢁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡠⠀⠐⠀⠀⢀⠀⠀⠠⠀⠀⡌⠀⠀⠀⠀⠃⠁⠀⠀⠀⠟⠛⠻⠛⠒⠒⠲⠒⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⡴⠶⠿⠟⠁⠀⠐⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⡙⠒⠛⢑⠊⣳⣶⣿⣒⣚⢒⡒⣒⠒⠃
⡆⠀⠂⠐⠈⠉⠑⠰⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⡦⠰⠀⠤⢀⠀⢀⠄⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠔⠚⠉⠀⠁⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⢀⡀⢶⠟⠋⠙⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠈⠀⢈⣧⠴⠚⠋⠉⠈⠀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠀⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠄⡖⣿⢐⠺⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠀⢀⠀⠀⣰⡊⠀⢸⣇⠐⠂⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡈⢿⠿⡇⠀⠀⠀⠀⠁⢀⣴⠉⠀⠀⢠⡎⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠘⠂⠈⡀⠀⡀⢸⡋⠀⠀⠀⢸⠻⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡙⠒⠲⠾⡀⣟⡄⠀⠀⠀⠈⢧⣿⣤⡴⠛⠙
⠀⠀⠀⠀⢀⠄⠠⠳⠀⠸⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸⠀⠀⢸⡇⠀⠀⠀⠠⣄⣘⣿⠤⢀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠒⢈⠀⡆⠄⠘⣧⠀⠀⠀⠀⠀⠀⠁⠁⠌⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠰⠀⠎⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡄⢀⠀⠀⠹⠀⠀⠠⣀⡠⠆⠐⣶⠀⠀⠤⠀⠈⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⠍⠀⢰⣶⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣼⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣿⡀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢹⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠀⠁⠀⠀⠀⠁⠀⠀⠉⠉⠁⠂⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⡀⠀⠐⠊⠀⠀⠀⠴⠉⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠰⠀⠀⠉⠰⠂⠀⠀⠀⠀⠀⠠⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠚⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢷⡓⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣀⡢⢒⡈⢳⡄⠀⠀⠀⣀⡀⠀⠀⢠⠃⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣽⢧⣄⣧⡔⠁⢉⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣾⡿⡁⠀⠠⢀⡇⠀⠀⠈⠀⠀⠑⢦⡏⠆⢀⡼⡆⢰⠁⠄⠀⠀⠀⠀⠀⣀⡰⢻⡏⠉⠻⠇
⠀⠀⠀⠀⠀⠀⢸⣄⠉⡞⠀⠈⠄⢀⠼⠁⠀⠀⠀⠀⠀⢠⣿⣏⣟⣿⠀⣇⣿⡀⣸⠀⠀⠀⠀⠀⠙⠃⠘⣿⠒
⠀⠀⠀⠀⠀⠀⠠⠿⠶⠶⠦⠴⠾⠟⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠰⠿⠆


⠀⠀⢀⣤⣶⣤⡐⣿⣿⣿⡿⣗⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣾⣿⣶⣶⣤⡀
⠀⠀⠀⠻⠿⡽⢻⣯⣿⡓⠋⠁⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⣷⣶⠀⠀⠀⠀⠀⢀⣀⣖⠰⢾⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠠⣶⣿⣿⡶⣿⠿⠇⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠠⢆⣠⠚⠫⠝⠿⣿⣿⣿⣿⠿⠋⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⡷⡇⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠐⠂⠀⠀⣠⣿⣿⠃⠀⠀⢀⢴⣗⠒
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⣿⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⣒⣒⠒⠒⣶⣶⣶⣶⣤⣤⣤⣴⣤⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣤⣴⣾⡧⣴⠂⠴⠂⠀⠀⠈⠉⠉
⠀⠰⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉⠁
⠀⠀⢻⣿⣿⠿⠛⠉
⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢤⣤⣤⣴⣶⣶⣶⣦⣤⣶⠈⣁⣀⣀⠻⠧⠤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠛⠋⠁⠛⠒⠁
⠀⠀⠀⠸⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡶⢖⣒⠒⠒⡤⠤⠤⠤⠤⠤⠤⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⡄
⠀⠀⠀⡄⠀⠂⠀⠈⠀⠈⠀⢉⠡⡴⢒⠄⠒⠒⠃⠒⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⢁⠀⠀⠀⡀⠤⠒⠈⠁⠂⠁
⠀⠀⠀⠘⠀⠂⠁