Soreness Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣒⣒⣮⣭⣽⣿⣿⣿⣿⣷⣶⣤⡀⢀⠤⠏⠀⠀⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣾⣿⣿⠿⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣀⣠⣄⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⢿⡙⠿⣿⣿⣿⣧⠀⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡏⠛⠀⠀⠀⢻⢿⣿⣿⡆⠉⠓⠛⠒⠳⠖⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠹⡄⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡇⠀⠐⠒⠊⢁⡀⠀⠀⠀⠀⡉⠙⠛⢦⡀⠀⢀⡴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠁⠀⢹⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠃⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⡗⠒⠋⠁⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣧⠀⠇⠀⠀⢀⣠⠤⣄⡀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣻⣿⣷⣤⡀⠀⠈⠀⠀⠀⠁⠀⠀⠀⢀⡴⠉⡆⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢀⡝⡝⠒⡖⢶⡶⠶⢒⡶⢺⠏⠀⠀⢹⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡸⠁⠀⢳⡤⠟⠛⢥⣀⡏⠀⠀⠀⠀⠳⡄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣷⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠋⠉⠉⠉⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⣿⣿⣇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣠⣿⣿⣷⡀⠀⠀⠀⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⣿⠿⠛⠉⢿⣿⣿⣿⣿⣿⠇⠐⠒⠄⠀⠙⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡏⡠⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠀⣿⣿⡿⠛⠻⢿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣅⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⠀⣿⣿⡇⠀⠀⠀⠈⠙⠿⠏⠉⠑⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡄⣿⣿⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠇⣿⣿⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⣴⣶⡄⠸⣷
⠀⠀⠀⠀⠀⢠⠇⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⣌⢀⣼⣇⠰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⢻⣿⣿⣿⣿⣷⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠙⠿⣿⣿⣿⠁⠀⠀⣸⣿⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⣠⠘⢿⣿⣷⡀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢿⣿⣿⣿⣿⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡗⠀⠀⠁⠀⠀⠻⢟⣿⠄⢸⣿⣿⣄⠀⠀⠀⠀⢸⣿⣿⡿⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣁⢀⠁⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠈⠻⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠠⡄⠘⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⡂⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠙⠿⢿⡑⠄⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⡏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣶⣤⣤⣁⡄⢀⣀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣗
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⡶⠂
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⢆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢈⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣄⣀⣀⣀⣀⣀⠀⠀⣿⣿⠿⠿⢀⣤⣤⣄⠀⣀⣤⣤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⣆⠀⣰⣿⡿⠿⠿⠿⠿⣿⣿⠿⣿⣿⡿⢿⣿⣿⡀⣿⣿⡀⢠⣿⣿⡿⠇⢸⣿⡿⠛⠀⠀⠀⡴⠞⠛
⠀⠀⠀⠀⠼⠛⠁⠹⠿⢿⣶⣶⣄⠀⣿⣿⡆⢻⣿⣿⣿⣿⠿⠁⣿⣿⠛⠈⢿⣿⣦⡀⠈⠻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣼⣿⣿⡇⢿⣿⡇⢸⣿⡏⣿⣷⡀⠀⣿⣿⣀⣀⡀⠙⣿⣿⣆⢀⣀⣀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠁⢻⣿⡇⢸⣿⡇⠘⣿⣷⣄⠛⠛⠿⠿⠇⢀⣼⣿⣿⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠙⢿⣿⣿⣿⣿⣾⣿⣿⣿⠿⠃⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠉⠀⠀⠀⠀⠀⠤⣤⠄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠊⢡⠀⠀⠀⠀⠀⠀⠐⠿⣷⣶⡶⠀⠀⠠⠀⡰⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠃⠀⠀⠀⠀⠀⠀⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠓⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑

⣿⡿⣟⢻⡟⢿⠻⡟⢻⡛⡟⢻⡛⣟⢻⠛⡟⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⢻⠛⡟⣻⢛⡟⢻⠛⡟⢻⠟⡿⢻⡟⣻⠟
⣿⣷⡘⣆⠹⡌⢧⠙⡄⢣⠘⠄⠳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠞⠠⠃⡜⢠⠏⡼⢡⠏⣰⢃⡾
⣿⣇⠳⡌⢧⠙⡄⢣⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⡜⢠⠋⡼⢡⠟⣰
⣿⡏⢧⠙⡄⠣⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠜⢠⠋⡼⢁
⣿⡛⣄⠣⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠛⠋⠉⡉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⣠⢋
⣿⡷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣴⣶⣶⣤⡀⠹⣷⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠞
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⣿⣿⣿⣿⣿⣿⣦⣿⣷⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠻⣿⣿⡿⠿⠿⢿⣿⣿⡞
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⣆⡀⠀⠀⠀⠘⣽⣿⣇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡼⠟⠁⠀⠀⠀⠐⠋⠛⠛⠲⠶⣤⣀
⣿⡇⠀⠀⠀⠀⠀⢀⡴⢚⡭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⠛⠳⠶⢦⣄⡀
⣿⡇⠀⠀⢀⣠⠶⣫⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠙⢶⣄
⣿⠇⣠⠾⣫⣶⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣿⡟⣡⣾⡿⠁⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⣷⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠳⢦⣄
⣿⣿⣿⠓⠀⠀⠀⠀⠀⣠⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣿⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣶⣦⣤⣄⢄⠀⠈
⣻⡿⠁⠀⠀⠀⢀⣤⡾⠋⠉⠙⡇⠀⠀⠀⢀⣤⣾⡿⠃⠀⠀⠀⣿⠀⠀⠈⢻⣶⣦⣤⣤⣀⣠⣴⡟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣕
⣿⡇⠀⠀⢶⣞⡉⠁⠀⠀⠀⠀⣷⢶⠀⠀⠸⡿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿
⣿⣷⣄⠀⠀⠈⠙⠦⣄⠀⠀⠀⢻⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠿
⣿⣿⣿⣷⣤⣀⠀⠀⠈⠓⢦⣄⣘⡇⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⢴
⣿⡟⠾⣽⣟⡿⠿⣶⣤⣠⠀⠀⠉⠙⠻⣷⡄⠀⠀⠀⠀⠀⡄⠀⢿⡇⠀⠄⠀⠀⠀⠀⢿⢿⣿⡇⢀⣀⣠⣴⠾⠋⠀⢀⣴⣾⣿
⣿⡇⠀⠀⠈⠉⠙⠒⠲⠭⢭⣀⡀⠀⠀⠈⠻⣄⠀⠐⣤⡀⢷⡄⢰⣧⡼⠀⣠⠆⠀⠀⢰⢿⣿⠿⠛⠉⠁⣀⣠⣶⣾⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢈⠓⠲⣌⠃⠈⠀⢰⣿⡇⠀⠃⣴⠞⠁⣰⠏⠁⢀⣹⣿⣿⠿⠿⠛⠛⠋⠉⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙⣍⣉⡁⠀⠀⠀⣾⣿⣿⡄⠠⣶⣚⣛⡁⢠⣦⣿⡏⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⢣⣿⣿⣟⠛⠓⠒⠚⠛⠛⠛⠛⢋⣭⠭⠟⠁⠈⢻⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠳⣾⣍⠉⣩⡀⠀⠀⠀⠲⢶⣶⣾⣿⣷⠖⢛⣁⠀⣼⣧
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⠛⠋⠁⣴⠆⡀⠀⠀⠀⠉⢻⣿⣿⣿⣏⣡⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⣿⡇⠄⡀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠊⠁⢰⠏⢰⡇⢸⡆⠙⡆⠙⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈
⣿⣇⡔⡠⢀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⠀⠰⣚⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀⢀⢢⠱⡘
⣿⠏⡼⣡⢃⠎⡐⠀⢀⠀⣸⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⢿⣿⣿⣿⣿⣿⡆⠀⢀⠡⡐⡄⢦⠳⡘
⣿⡟⡴⢣⢏⡜⡰⢡⢆⠆⣿⡇⠀⠀⠀⠀⠀⠦⣌⡛⠶⣤⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡗⡌⢆⢣⠱⡘⣎⢳⡹
⢿⣿⣤⣧⣾⣴⣧⣯⣾⣴⣿⣧⣤⣤⣤⣤⣼⣶⣬⣽⣿⣶⣾⣿⣿⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣧⣼⣼⣦⣷⣽⣼⣧⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣻⣿⣿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣭⢎⡏⣽⢉⡹⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣾⢇⣾⣿⢻⡾⣷⣾⣿⣿⣿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠓⢿⢸⣿⣯⡿⢿⣿⡿⠟⠿⣿⠋⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⠈⠋⠁⠀⠀⠀⠀⠀⠈⠡⠜⡹⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠀⠒⠒⠒⠒⠒⠒⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣌⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢃⣨⠭⠈⢀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⣠⠎⢻⠀⠁⠀⠀⠀⠈⡆⠙⠀⠀⠀⠀⢀⣀⣤⠖⠃⠀⠀⠀⠉⠙⠓⠶⠶⠤⠤⠀⣀⡴⠋
⠀⣠⠖⠂⠈⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉
⣸⠁⠀⠀⠀⠀⠀⠀⠻⣷⡆⠀⠀⠀⠀⢀⠀⠀⠀⢀⡰⣄⠀⢀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⠀⠀⠀⠈⣷⠀⠘⠉⠀⠘⣦⡼⠀⠀⠒⠉⠉
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⡄⠀⠀⠀⢹⡄⠀⠀⠀⠀⠘⣧
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣸⠃⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⣦
⠀⠸⡷⣄⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢹⣤⣀⣠⣤⡶⠛⠘⣧
⠀⠀⠃⣈⣦⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⢾⡉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⠀⠀⠀⠀⠀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠽⣖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠳⡬⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⢄⡀⠀⠀⠀⠀⡸⢧⡀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢰⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠈⣆⠈⠁⠀⢀⣀⣀⣀⡀⠆⠀⠀⠀⢠⠃⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⢀⠐⠀⠀⢣
⠀⠀⠀⠀⠀⠈⠶⣀⠁⠀⠀⠈⠉⠃⠀⠀⢀⠔⠁⠀⠀⠀⠞⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠤⠴⠒⠁⠀⠀⠀⡐⡠⠁⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠠⠊
⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤
⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⣶⣶⣿⡆⠀⠀⠀⠀⠀⣴⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣠⣤⡄⣾⣿⡇⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⢹⣿⣿⡟⠻⠶⠀⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠈⠉⠀⠀⠀⠀⢀⣤⡜⠻⠛
⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⢀⠀⠀⠀⠀⣿⣿⡃⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⡧⠀⢀
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠇⠀⠀⠀⠀⠀⣾⡿⠁⣿⡇⠀⠀⠀⠀⠀⣴⣼⣿⡧⣿⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⢀⡄⣰⣶⣦
⠀⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣶⡄⠀⠀⠀⠀⠀⠈⠀⢀⣙⣛⣁⠀⠀⠀⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⠀⢸⣇⣘⣛⣋
⠀⠀⢀⣤⣜⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠀⠀⠀⠀⠘⠿⣿⣿⣿⡀
⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⠀⠀⠠⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃
⠀⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⢸⣿⣻⣿⠀⠀⠀⠀⠁⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⢠⣭⣭⣭⢻⣿⡈⠙⠁⠀⠀⠀⠀⣼⣿⢻⣿⡛⠛⣿⡟⢻⣷⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡟⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣼⣿⣿⣿⡆⢿⠇⠀⠀⠀⠀⠀⠀⢿⡿⠀⢿⡇⢸⣿⠁⢸⣿⠀⠀⠀⠀⠸⣿⢻⡿⠀⠀⠀⠀⠀⠀⢿⡟⣿⠇⠀⠀⠀⠀⠀⠀⣿⠿⣿⠃


⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢠⡾⠻⣿⠟⢳⣆⠀⠀⠀⠀⠀⠀⣀⣿⣿⡿⠀⠀⠀⠀⠀⠀⣰⠟⠿⠟⢷⡄⠀⠀⠀⠀⠀⣴⣶⣆⣠⣤
⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⠷⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠰⡿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣽⣿⣿⣿⠃
⠀⠙⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⢀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿

⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣄⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣷⣶⣄
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣿⣿⣿⣿⣯⠀⠿⠿⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⣿⣿⣿⣷
⠀⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⠄
⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠊⠛⠶⢦⣄⣀⡀⠀⢲⡀
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⡌⠙⣦⠈⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣷⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢼⠀⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠹⡷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣧⢸⢴⠃⢸⣿⣦⢠⠀⣀⣀⣤⠶⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣹⡀⠈⣷⣄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣓⠄⠀⠙⠃⠒⠖⠐⢂⣠⣤⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛⠋⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀
⣿⢛⣿⣻⢛⡟⣫⣙⣛⡟⡛⠛⣿⣿⣿⣿⣿⣿⣿⣹⣿⣉⣉⣉⣉⣿⣿⣟⣛⣛⣻⣟⣻⣛⣋⣙⣛⣹⣛⣋⣛⣻⣙⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠘⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠉⠉⠙⢻⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣷⣄⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⣁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣦⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠆⠀⠀⣠⣤⣤⣤⣤⡄⠀⠀⠸⣿⠄⠀⠀⢀⣿⡇⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠇⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠉⠉⠉⠁⠀⠀⠀⠉⠀⢀⣠⣾⡿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠿⢋⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠁⠀⢀⣤⣶⣶⣶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⢸⡿⣡⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣰⣿⢋⣿⡟⠁⠀⠀⠀⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⠀⠀⠀⠁⢿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡿⠋⣰⣿⢃⣾⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⣿⡀⠀⠀⠀⠈⠙⠛⠛⠛⣹⣿⠟⠙⠛⣋⣉⣉⣉⠀⠀⣰⣿⣧⣬⣯⣹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣤⣤⣤⣤⣤⣶⡿⠟⠁⠀⠀⣰⣿⢛⣿⡟⠀⠀⠈⠉⢉⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣰⣿⢃⣾⣿⣀⣀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣬⣟⢻⣿⡟⠁⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢙⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⣧⠀⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⠶⠶⠶⢶⡶⢶⡶⢶⡶⢶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣿⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣶⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢶⣄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣽⠆⠀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⣳⠆⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣿⣿⡿⠁⢠⡴⠖⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠠⠶⣄⠙⢿⣿⠟⠀⠀⠀⠀⠀⠀⢰⣿⣁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⢠⣿⣿⠃⠀⠈⠙⠛⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⠀⣸⣿⠏⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣬⠋⠀⠿⣯⡀⠀⠘⣧⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠙⠛⠁⠀⠀⠒⢸⣷⠀⠀⠈⠻⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠻⠿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠷⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⡴⠟⠋⠁⠀⠀⠀⠀⣰⡟⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠈⣱⠟⠛⢷⡄
⠀⠀⠀⠀⣠⡶⠟⠉⠉⠉⠉⠀⠀⠀⢀⣠⣶⣦⣴⣟⡀⠀⠀⠙⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠋⣿⠀⠘⢳⣾⠋⠀⣠⡞⠻⣦
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠠⣾⠛⠉⠀⠀⠈⣹⡿⠛⠂⠀⠀⠙⠻⢶⣭⣙⣛⣻⣿⡿⠛⠛⠛⢿⡍⠀⠀⢀⣿⠆⠀⠀⠙⠳⢾⡋⠀⣰⡟
⠀⠀⠀⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⡆⠀⠀⣻⣦⣴⠟⠁⠀⠀⢀⣤⣀⣈⣿⠿⠋
⠀⠀⠀⠀⠈⠻⣦⡖⠀⠀⣴⠂⣴⠟⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡾⠃⠉⠁⠀⠀⠀⣠⡾⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡴⠟⠛⠉⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢶⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣴⠾⢶⣄⠀⠀⠀⠀⣿⡷⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⠹⣧⡀⠀⠀⢹⣇⠀⠀⠈⠉⠙⠛⠳⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⣠⡾⠋⡠⠋⠀⠀⣰⡟⠻⢦⣄⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢻⣶⣦⣤⣾⠃
⠀⢀⣴⠏⣠⠞⣡⣜⠳⣾⠋⠀⠀⠀⠙⠷⣿⣍⡛⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⠻⣧
⣰⠟⣡⠞⠉⡞⣡⢘⡶⠻⣦⡀⠀⠀⠀⠀⠀⠈⠛⢦⣬⡿⠛⠻⠶⢶⣤⣤⣤⣿⠁⠀⠀⠀⢹⡇
⠛⢶⣧⣤⣴⠿⠋⠉⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠈⣽⠏⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣦⣀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⡀⠀⠀⠉⡿⠛⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠓⢦⣀⠉⠿⣤⡾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣌⡑⠲⣎⡁⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡙⠢⣼⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣷⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⣤⣤⣤⣬⣹⣯⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠥⣶⠶⠿⢥⡈⠙⣆⠘⠟⢾⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⡾⠛⠉⠙⠀⠀⠀⠈⠀⠀⠀⡿⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡼⣿⣫⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡼⣱⣿⡏⠀⠀⠀⠀⢀⡴⠀⠀⠠⡄⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡇⣿⢻⡗⠀⠀⠀⠀⠉⣀⠀⠀⢀⠙⠂⣼⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⢡⣿⢸⠃⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⡟⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⢸⠙⣿⡤⣤⠀⠀⠀⠀⠀⠀⠜⠀⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡏⣸⠀⡟⠀⣾⢳⣄⠀⠀⠸⡽⠀⠀⢀⡼⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣸⢁⡏⣸⠃⢀⡏⣸⢋⡟⠲⣤⡴⣶⣺⠋⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠞⡴⢃⡤⠟⠛⠛⢛⡶⢦⣇⡇⣿⣿⡷⢧⡈⢧⢀⡤⠖⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⡞⠀⠀⠀⠀⢻⡹⡾⠘⠑⠻⢿⠇⠀⢻⢀⡟⠀⣀⣠⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⡞⠀⣴⠋⢹⣷⢻⠏⡏⣿⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢿⠀⠀⡼⠀⠀⣠⠟⠀⠀⠘⣦⣠⡏⢸⢰⠇⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⣠⢾⣁⠀⣰⠇⠀⠀⠀⠀⣽⠈⣇⠀⠀⠀⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠸⠃⠀⠈⠛⡿⠀⠀⠀⠀⣼⢁⣰⠏⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿⠈⠙⠦⠤⠴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢴⣶⠤⠀⠙
⠀⠀⠀⣤⣤⣄⡀⠒⠾⠶⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⣀⡀⠀⣀
⠀⠀⠀⣿⣿⣷⡂⣠⣶⡶⠂⣠⣶⡶⠀⣠⣶⡶⢀⣠⠄⠀⠀⠀⢸⣿⣿⣿⠂⣠⣿⡟⢀⣨⣿⡟⢀⣨⣿⡟⢀⣨
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢰⠲⢄⡖⢦⣰⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠒⠊⠓⠋⠘⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢸⡇⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⠇⠀⠸⠇⠀⠀⠀⠀⠰⠆⠸⡇⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠖⠛⠒⠶⢦⣤⡀⢠⡴⠶⠶⠶⠶⠶⠶⠶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⢰⡄⠀⠀⠀⠀⠀⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⣹⡇⠀⠀⠀⠀⣤⠀⠀⢸⣇⣈⣀⣀⣀⣀⣀⣀⣸⡷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣈⣉⣀⣀⣀⣤⡴⠿⣦⣀⡀⢹⡏⢉⡉⢉⡉⢉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⠉⢉⣉⠉⠉⠀⣀⣀⠀⢹⣧⣼⡇⠘⠁⠘⠇⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⠀⠺⠻⠀⠀⠀⠻⠟⠀⢸⡇⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⠀⠀⠀⣀⣄⣀⠀⠀⠀⣸⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠿⣧⡀⠐⠋⠉⠙⠃⢀⣴⠿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⢸⡟⠷⣶⣶⣶⠶⢻⡿⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⢸⣧⡾⣏⠀⣹⡷⣾⡇⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠛⢿⡄⠈⢁⣀⣀⣀⣀⣀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⢸⣏⣉⣉⣉⣽⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣈⣷⣀⣈⣉⣉⣉⣉⣉⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣛⡽⠋⠀⣀⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣻⣿⣿⣿⣿⣥⠶⠛⢛⡋⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣋⣴⣾⣿⣟⣯⣞⡉⠀⣋⡼⠟⠻⣄⡤⠂⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣋⣴⣾⣿⢮⠟⣻⣿⣿⣿⣶⢦⣭⣶⣥⠂⠁⢪⣷⠀⠀⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠋⠁⠀⣠⣴⣿⡿⠋⢀⣹⣿⠋⠝⣿⣧⣿⣿⣿⠟⣶⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣯⠿⡿⠟⠒⣰⢛⣿⣿⣏⣴⣿⣿⣷⣾⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣠⠞⢝⣭⣿⣿⣿⣿⣿⣿⣿⠇⠒⠛⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣫⡾⢿⠿⠛⣯⡵⢟⣿⡟⠉⠀⢀⠀⠄⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⠚⠋⠀⢀⡴⠚⠁⣶⡿⠋⠙⠛⠾⡚⢀⣀⣀⡀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠁⠀⠀⣸⠋⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠲⡆⠀⠈⣿⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠉⠐⠂⠤⣀⣤⠤⠖⠚⠋⠉⠉⢦⠀⠀⠀⡏⠀⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣽⣆⠀⠀⠀⠑⠀⠈⢷⡄⠀⢿⠿⡟⠉⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⢠⡿⠈⢷⣀⠀⠀⠀⠀⠈⢷⡄⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠛⠛⠿⠿⠿⠿⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⣾⠃⠀⠈⢻⣆⠀⠀⠀⠂⠈⢻⣆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠃⠀⠀⠀⣸⡟⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠙⣦⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠃⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⢀⣼⣿⡉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⣦⣀⣾⠏⠀⠀⢀⣰⣿⣁⠀⠀⠀⠀⠀⢀⣀⣀⣈⣻⣆⣴⡏⠁⠀⠑⠦⠀⠀⠀⠀⣀⢀⡠
⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠠⠄⠀⣀⣀⠀⠠⠊⠘⢿⠿⠟⠋⢸⣿⣅⣀⢀⣠⣿⣿⣿⣿⢷⡄⠐⠈⠉⠀⠀⠀⠀⠈⠉⠻⡇⠀⠀⠀⠀⠀⠙⠀⣼⣽⡍⣃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⣷⠀⠀⠀⠈⢳⣀⣤⠾⠿⢷⣍⡀⢈⣿⠉⠉⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣶⣦⣤⣤⣤⣼⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣿⠀⠀⠀⢹⡀⠀⠀⢴⡟⠋⠀⠀⠀⡀⠈⢳⣿⠁⠀⢀⡼⠃⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠭⠉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠙⢶⠆⣾⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠽⣶⡞⠻⠿⡿⣿⠀⠀⠀⠀⠀⢀⡴⢻⡟⢠⡶⠛⠳⡄⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣰⠟⠁⢰⡿⠛⠀⢀⣼⠟⢳⡄⠀⠒⠊⢁⣴⠿⠟⠋⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⣾⠀⣿⣿⡇⠀⠀⠀⠀⠀⣷⣤⡄⠀⠀⣿⣤⣤⣾⣿⣅⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⣴⡟⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⢿⣟⢿⠿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣔⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⡈⠁⡀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠯⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠱⡎⠁⠈⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠈⠁⠰⠶⢈⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠶⡀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⣀⠀⠀⠀⠀⠈⠙⠛⢀⡀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠘⢢⡤⠾⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠰⠿⠆⠈⣁⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡄⠀⠀⠀⣀⠴⠒⠉⠉⠀⠶⣄⡀⢠⣀⡀⠀⠛⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠛⣻⠟⠁⠀⠤⡀⠀⠠⠀⠀⠘⢧⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠃⠀⢀⠉⠀⠀⠠⠀⠀⠀⣠⠤⠛⠰⣦⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡀⠀⠀⠀⠀⢄⠠⣏⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩⣆⠀⠀⠀⠀⢀⣩⡄⠀⠀⠀⢸⡇⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡭⡄⠀⢰⠾⡙⠃⠀⠀⠛⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣀⡀⠘⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢀⠾⢤⡈⠉⠉⠉⢙⠀⠠⣄⡘⢷⠦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢸⠀⡠⠛⡛⠛⠀⣾⠀⠀⣀⡁⠸⣷⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣤⠀⢸⡆⠁⠠⠄⠀⠀⢸⠀⠠⣄⡙⠂⠸⡇⠀⢌⣱⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣤⡀⠀⡎⠀⢀⠆⠀⠄⠘⠾⠀⠀⢠⡁⠲⠀⢰⠒⠄⢰⠃⣠⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⠄⠀⡀⠀⠆⠀⡠⢀⠀⠁⢼⡆⠀⠀⠌⠓⠀⠘⠀⡐⠀⣴⠇⢀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣐⠾⠉⠂⡠⠊⠀⠀⠱⡀⠠⣿⠀⠀⠀⠀⠀⠰⠆⢀⠜⢁⠔⠃⡀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠉⠀⠀⠀⠀⠀⢋⡀⠈⣧⠀⠀⠀⠀⣀⠴⡗⠂⢀⠔⠈⣠⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠄⠀⠦⡠⠞⠁⢀⠐⠊⣁⠔⠉⣀⠔⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣁⠀⠒⢉⡠⠀⠈⡀⠔⠉⣀⠴⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣈⠠⠀⢈⡀⠀⢈⡀⠤⠀⠁⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠠⠀⣀⠀⠄⢋⡀⠤⠄⣀⠠⠄⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠐⡂⠤⠐⢃⠀⠤⠐⡂⠠⠄⠐⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡀⠠⠔⠂⠄⠒⠈⠁⠀⠂⢩⠀⡄⠀⠀⠀⢀⡄⠥⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠉⠉⠉⠉⠉⠋⠉⠀⠀⠀⠀⠆⠰⠀⠀⠇⠨⠃⠀⠠⠔⠋⠡⡄⠢⡌⠂⢀⡀⣀⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠁⠄⠁⠀⠐⠲⢌⡉⠢⣈⠳⠀⢀⣼⠃⠀⣀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⢀⠀⠆⠘⠀⠃⠀⣠⣤⣶⡟⠀⠀⣈⡉⠲⢤⡈⠣⠈⢂⣴⣏⣉⣁⣉⠁⠉⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠘⠁⠐⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⡀⠀⠀⣀⣴⣾⣿⡎⠁⠸⠁⠀⠀⣀⣈⠑⠦⠙⢀⣴⣟⣁⡈⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⠈⣀⡀⠀⠀⠁⠉⠉⠉⠙⡇⠀⢈⠁⠀⠀⠀⢸⡋⠉⠉⠉⠉⠁⢸⡆⠀⠀⠠⠄⠀⠀⠀⢾⡤⠤⠤⠤⠤⠤⠤⠤⠀⠀⣀⢨
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣟⣉⣉⡁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠘⡧⠠⠄⠀⠀⠀⠀⠀⠙⣶⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠀⠠⠄⠀⣯⢀⣠⠀⠀⠀⠀⠀⠈⣆⠀⠐⣀⠀⠤⠀⠀⢱⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⡆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠒⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⡡⠤⠐⠂⠀⢸⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠦⠤⠈⠁⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠐⠢⠅⠀⢸⡍⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠂⠀⣰⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⢉⠀⠂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣠⠁⠀⠀⠁⠀⣠⣾⣉⢉⡉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⣀⡴⠋⣡⣄⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠈⢀⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢆⣀⠴⠞⠁⠀⠀⢀⣤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⣴⣛⣛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⠒⠒⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠐⠁⠐⠒⠒⠒⠒⠒⠒⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠛⢘⠓⠒⠒⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠁⢠⣴⣶⠾⣯⡤⡭⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⢠⡄⠀⠀⠠⢀⡀⠀⠀⠀⢠⡄⠀⠘⢿⣯⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⡆⣦⣼⡇⠀⠀⠈⠀⢲⠀⠀⠀⠠⠄⠈⠉⣿⣶⣿⠘⠉⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠨⡇⠀⠀⠀⠀⠀⠠⠄⠄⠄⠀⣀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠠⣾⣿⣛⣻⣿⣻⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠀⡀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠈⣿⡟⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠴⠾⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀
⠀⠀⠀⠀⢰⠶⠿⢷⣓⣚⣯⣇⠀⠀⠀⠀⠸⠿⠿⢿⡯⢸⡇⠀⠀⠀⣰⣟⣋⣛⣉⠉⣉⡉⣈⣡⣀⣀
⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣤⣴⣶⣴⣶⣾⡶⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠹⠛
⠀⠀⠀⠀⠀⠈⠘⠡⠟⠛⠻⠟⢻⣿⠿⠟⠛⠛⠿⣿⡿⠋⢀⣴⣿⠋⠃⠀⠀⠈⠈⠁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡋⣝⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢷⣤⡀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣴⣿⠛⠛⠓⠂⠈⠓⠀⠃⠀⠘⠃⠀⠀⠂
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣤⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠹⣆⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⣷⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠰⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⢶⣶⣄⠀⠀⠀⠀⣴⡄⢀⠀⣤⡀⡀
⠀⠀⠫⠻⣶⣦⡀⠒⢟⣿⣤⣆⣼⣿⣿⣆⡑⢀⣰⣤⡴⣤⣁⡐⣤⣴⡖⠰⣤
⠀⠀⠘⣷⠞⣿⢗⡛⡌⣹⣿⣿⠿⣯⠈⢹⠞⠿⢻⣴⣟⣿⣿⠐⢿⡿⣻⠃⠘⣣⣰⡆⠶
⠀⢤⡠⢄⡀⢺⠾⣷⣿⢿⣿⣏⢀⡩⡙⢷⠄⠀⠀⠈⠙⠉⠛⠀⢸⡇⢷⣤⡼⢫⡽⢃⠀⣴⠀⣄
⠀⠀⢙⢻⣯⡾⢷⣿⣾⣿⣿⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢿⡀⠱
⠀⠀⣄⠱⣾⠃⢺⣿⣿⠋⠁⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢢
⠀⠀⣼⣄⠀⠉⣾⣿⡇⢼⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃
⠀⠀⠘⣿⣶⣷⣿⡟⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⢰⣾⡿⣿⣿⣿⡟⡟⢡⣄⣤⣀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠘⣿⣿⣿⣿⢋⡤⠸⣧⠄⠀⠀⠉⠙⠛⣷⣶⠶⠚⠂⠐⠂⠤⢀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣼⣄
⠀⠀⡈⢛⣿⢿⡇⣾⠗⠀⡿⠂⠀⠀⠀⠀⠀⠈⢷⠶⢦⡂⠀⠀⠀⠀⠠⢠⣄⣤⠾⠋⠀⠀⣠⡾⠈⣠
⠀⠀⠀⢈⣱⣾⡇⢿⢀⣾⠓⠀⠀⠀⠀⠀⠘⣀⣈⢠⣿⣦⣈⣿⠟⢣⡰⠿⢟⢻⣶⣦⣀⠈⢤⠛⠀⡿
⠀⠀⢀⠺⡟⢿⣇⡀⢸⣿⡆⢠⣴⠀⠀⠈⠛⢻⣮⠻⢯⠚⣿⢹⣴⣯⠿⠀⣾⢈⡿⠷⠹⣿⢣⡄⡘
⠀⠀⢺⠒⢿⣾⣿⢡⠈⠻⣇⠀⠀⠀⠀⠀⠠⠜⢻⠀⠀⢹⡏⠙⠁⡇⠀⠀⢩⡆⠰⠑⡆⠚⢇
⠀⠀⠙⠀⢨⣽⣿⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠁⠀⠀⣸⡏⢼⡂⠙⢦⠈⢥⠀⠆
⠀⠀⣤⠀⣾⣿⠳⣷⣄⣀⣼⣏⠀⡤⠀⠀⠀⢀⠢⡄⢰⠃⡐⠀⠀⢀⠀⢠⡟⠙⠾⠹⡆⠀⢣⢈⣆
⠀⠀⠈⣻⡉⢿⠀⢹⣿⡏⢙⡿⠀⢼⠀⠀⠀⡻⡄⠀⢸⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⡆⠘⠀⢀⡤⠡⡄
⠀⠀⠸⠿⣷⣿⠐⢈⣿⣿⣀⠀⠀⣼⡀⠀⡸⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⠘⣧⠀⢠⠘⢃⡀
⠀⠀⠶⢀⣿⣟⢠⣼⣿⣿⡏⠀⠀⠸⣀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠐⡏⠀⠂⣸⡇⠀⠑⡀⠘⡇⠀⠇
⠀⠠⢰⡇⣿⣧⠋⢹⣿⣿⡇⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⡄⠀⠳
⠀⢀⣾⣿⣿⣿⠆⠈⢿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠟⠀⠀⢈⡤⠀⠀⠰
⠀⠸⣿⣿⣿⣿⠀⢐⢾⡟⢿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠈
⠀⠀⣿⣿⣿⡿⠀⠠⠛⡁⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⣶⡏⠀⠀⠀⠀⠀⢀
⠀⠀⡿⢻⣿⠗⠰⢷⡄⠐⣀⣤⡅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄⣸⠇⠀⠀⠀⢠⣶⠟⠇
⠀⠀⠛⣼⣏⣤⣤⣤⣼⣶⣿⣿⣟⠦⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣻⣿⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠈⠀⢨⡄
⠀⠀⠚⠛⣿⡿⢻⡿⣷⠶⠿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⡤⠄⠀⠀⠀⠀⠀⠀⠂⠻⣤⣄⡀
⠀⠀⠀⠠⠘⣷⢿⣿⡗⠀⠈⠚⢀⠐⣧⢴⣄⣄⠀⠈⠉⠁⠀⠁⡈⢠⣡⣗⠌⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠂⢸⡾⠟⣏⣀⠠⠀⣈⡀⠀⠀⠈⢿⡻⣎⠊⣡⣅⡘⢰⠈⡳⣿⣄⠀⠀⠀⢀⠀⢀⣾⡟⢳⡶⣤
⠀⠀⠀⠀⠀⠋⠀⠀⠸⢿⣿⠲⠁⠙⠚⣿⠻⢦⡗⠸⡇⠀⣿⡇⢠⣤⢠⡿⢏⠂⣤⣄⠎⠈⡳⡈⠢⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣠⣸⢷⣬⣄⢸⡇⠀⠀⢠⠈⠃⢈⢹⠈⠀⠘⠿⣿⠏⢀⡀⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⢀⠍⠞⠘⢃⣀⣄⣀⣀⡶⢀⣈⣠⢰⣤⢰⡄⠚⠿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⢠⡖⠳⠃⠀⣿⠋⠠⠁⠘⠏⠈⠙⢻⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⠄⠀⠀⣴⠟⠀⠸⣶⠄⠀⠀⣾⠛⢹⡆⠀⠀⣼⡟⠛⢻⡆⠀⠀⢠⡥⠷⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⡀⢀⡁⠀⠀⢻⣇⠀⣸⡇⠀⠀⠘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠯⢿⢻⠟⠀⠀⠀⡉⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⣏⠃⠀⠀⠀⠁⠐⠋⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⣴⠲⣦⠀⠀⠀⢠⡾⢷⡆⠀⠀⠀⢰⡟⠻⢷⠀⠀⠀⢠⣾⡿⢦
⠀⠀⠀⠀⠀⠉⠁⠈⠃⠀⠀⠀⠁⠀⢈⡀⠀⠀⠨⣀⣀⡟⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠙⠛⢁⠀⠀⠀⠀⠄⣀⠤⠀⠀⠀⢠⠀⠀⠠⡄
⠀⠀⠀⢠⣾⣿⣿⣿⣷⣆⠀⠀⠐⠀⠂⠈⠁⠀⢘⠂⠀⠚⠀⠀⠀⢀⠃⠀⠚
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠂

⠀⠀⠀⠀⠀⡾⠿⢿⡆⠀⠀⢠⣾⡟⢷⡄⠀⠀⠀⠞⢛⠳⠀⠀⠀⢀⣶⣿⣦⠀⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢸⡉⠀⢉⡇⠀⠀⠘⠏⠉⠙⠃⠀⠀⠸⡟⠛⢻⠀⠀⠀⣾⡟⠈⢛⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡿⠀⠺⡇⠀⠀⠀⠐⠠⠒⠀⠀⠀⠀⠻⠿⠏⠀⠀⠀⠹⠧⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣌⠀⣡⣼⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠑⠿⠿⠇⠀⠀⠀⠈

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢐⡭⣗⢦⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣦⡀
⠐⣩⡴⣶⣶⣶⡶⣬⡑⠀⠀⠀⠀⣿⠛⢍⠉⠉⢿⡇⠀⠀⠀⠀⠀⠈⢣⢸⣿⡇⠀⠀⠀⠀⢸⡟⠉⠉⠉⠹⣿
⠸⣅⡀⣿⣷⣼⡟⢀⡽⠂⠀⠀⠀⢟⠐⠆⠻⣷⣌⡇⠀⠀⠀⠀⠀⢠⠏⡸⡽⠀⠀⠀⠀⠀⢸⡃⠲⠀⠰⠂⣿
⠀⠀⠉⠛⠛⠛⠋⢡⡤⠀⠀⠀⠀⠈⢇⠀⠧⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣦⡆⠀⠀⠀⠀⠱⡀⠸⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠁⠉⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⢸⣹⣅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣅⠀⠀⠀⠀⠀⠀⠀⣠⣬⣟⠁
⠀⠘⣿⣿⣙⡻⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⢸⢿⣿⠻⣆⠀⠀⠀⠀⠀⢸⢿⣿⡟⠷⣄⠀⠀⣀
⠀⠀⣽⣿⣿⡇⣽⣦⡄⠀⠀⠀⠀⣼⠀⠻⠿⢯⣿⡀⠀⠀⠀⠀⠀⢸⡿⣇⠈⢆⠀⠀⠀⠀⠀⢸⡿⣧⢀⣤⣤⣴⡏
⠀⢯⡟⠛⠛⠓⠛⢯⡗⠀⠀⠀⠀⠘⠦⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⡾⠁⢻⠄⠈⠦⠀⠀⠀⠀⠾⠁⡿⠉⠟⠉⠹⠇

⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣴⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠂⠀⠀⠀⠀⠀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄
⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠈⠿⢿⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣾⡟⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠤

⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠
⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢤⣀⣸⣧⡜⠉⠀⠀⠀⠀⠀⣴⣿⣷⣤⣾⣿⣦⠀⣷⠀⠀⠀⠀⢈⣿⢿⡁⠀⡀
⠀⠿⣿⣿⣿⡷⠛⠁⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⠛⠒⠀⠀⠀⠀⢿⣿⢡⢻⢟⣛⣛⡁⣋⠀⠀⠀⠀⢸⡗⢙⣿⠀⠒
⠀⠀⠛⢛⡟⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠿⢩⡇⠉⠀⠀⠀⠀⠀⠒⠲⣿⣆⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣽⡾⣅
⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⢰⢂⢄⠀⠤⣤⣄⣀⣴⣉⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠣⢙⡆⠻⢦⡀⠘⡴⠊⠉⢳⡄⠀⠀⠈⠺⢺⡗⡀⠀⢨⠻⠋⠙⡟⢢⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠁⢀⣀⣅⣸⡀⠀⠰⣅⡴⠄⠀⠀⠀⠀⠀⠀⢠⠃⡷⠀⠢⠀⠸⡇⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣦⡀⠀⢀⣸⣿⡧⠝⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⢧⠤⡤⣤⣼⣤⣤⢤⣤⠀⠀⠀⣠⠤⠴⠶⢾⠀⠉⢀⡰⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠉⠉⠉⠉⠉⠉⠉⢉⡏⠉⠉⠁⠀⠀⢀⣎⣡⡤⢲⣶⠋⠀⣠⣏⠀⠀⠀⠀⠈⠉⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣀⣤⣶⣿⡿⠋⢴⣿⣿⣾⡿⢋⣩⣷⡄⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣼⠋⠀⣠⣔⣊⣀⡿⠿⡏⠀⠀⠀⠈⠙⠁⣰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠤⡾⠓⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠅⣼⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠁⠀⠀⠀⢠⣾⣿⣿⡿⠋⢹⠉⠙⡄⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣾⣿⣟⣫⠤⠤⠔⠒⠚⠉⠉⠀⠀⠀⠀⠀⢀⡠⠾⠟⠛⠛⠒⠈⠉⢇⠀⢩⠑⡄⡜
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣠⣤⣄⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡾⠋⢛⠿⣿⣿⣿⣿⣿⡉⣉⣙⡛⠻⠿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠎⠀⠂⠉⠻⣿⣿⡇⠀⠀⣀⣉⡉⡚⠈⠑⡄⠀⠀⠀⠀⠀⠀⠸⠠⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡴⠁⠀⠀⠦⠤⡔⠁⢹⣗⣒⣂⣁⣭⣉⡇⠀⢀⠯⠽⠀⣒⣒⣒⣐⠁⢰⠤⠄⠬⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⢢⠎⠀⣠⠎⠑⠒⠩⠭⣉⡉⡓⠈⠁⠀⢀⣀⡠⠤⠀⠘⠒⠊⠹⢹⢹⠉⠉⠐
⠀⠀⠀⠘⠮⠶⢌⣁⠎⣤⡀⠀⣀⠔⢁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⡏⠒⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣳⣤⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⣈⡇⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⣛⣛⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⡶⠋⢉⣟⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⡏⠀⠀⠘⠋⠁⠀⠀⠰⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣠⡴⠖⣿⠁⢀⡴⠋⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⡟⠓⠒⠒⠲⠶⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡉⠀⠀⠐⠓⠚⠉⣀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠈⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⡾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣧⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡆⠀⠉⠓⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⣉⣥⣤⠀⣠⠞⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢶⠀⠀⠀⣴⠖⠛⠉⠡⠄⠒⠛⠁⠁⠀⢹⡞⠁⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠘⠀⢀⣀⣤⣤⣤⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⣠⠞⢻⣀⣠⡀⠴⠒⣀⣉⡤⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣯⣿⣿⣿⣷⣤⣄⣀⣀⣀⣀⣠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⣠⠞⠃⠀⢈⣩⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣿⠋⠛⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⠶⢶⣶⠶⠶⡶⠶⠶⠶⠶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⡴⠟⠉⠉⠉⠙⠃⠛⠙⠙⠓⣦
⠀⠀⠀⠀⠀⠀⢰⡏⣷⠀⣀⣄⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠸⣏⠉⠁⣿⠀⠀⠐⣠⡀⡆⠀⠀⡠⢠⠋⠀⠙⣿⣷
⠀⠀⠀⠀⠀⢠⣯⠀⡀⢻⡀⠀⡋⠀⠀⠃⠎⣻⢦⣴⣌⡴⠀⠀⡠⠖⣽⣯
⠀⠀⠀⣠⡤⢾⡇⠳⠃⣾⡇⠀⡇⢀⠴⢼⡾⠓⡶⣿⡿⡅⠀⠀⣴⠋⢸⣿⠆
⠀⠀⣴⣷⢞⢻⣇⢀⠀⣿⣧⠀⢣⡆⢲⣾⠁⠰⠆⠈⠁⠀⠀⠀⠀⡀⠘⡟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣠⣤⣄⡀
⠀⠀⣿⣿⠸⣼⣿⠀⠁⢽⡏⠐⢿⣯⣾⡿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢠⡗⠀⠀⠀⠀⠀⠀⣟⠨⣷⣽⣿⡇⠀⠈⡿⠛⣷⣤⣄
⠀⠀⢹⣿⡇⡁⣿⠀⠀⢸⣿⣿⣾⣿⣿⣿⣧⠄⠐⢶⠄⢀⣀⣀⣀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠉⢳⣿⣿⣿⠛⠒⠚⠣⠀⠏⠀⡸⢶
⠀⠀⢸⣿⣯⣟⣻⡀⢐⣿⣿⣿⣿⡿⠉⣹⣿⡷⡀⠀⠀⠉⠉⠉⠁⣰⣿⢦⣤⣴⣶⠶⣦⣄⠀⠀⠛⢻⡇⠀⠀⠖⢳⣶⣤⣄⣤⣟
⠀⠀⠀⢿⢿⡶⣾⡇⡌⣿⣿⣿⣿⡇⠀⣿⢿⣿⣿⣷⢔⠀⠤⠄⢸⣿⠿⣀⢸⣿⢸⡇⠀⠉⢻⠒⡶⣤⣿⣦⠀⠀⠀⠛⠛⢿⣿⡟
⠀⠀⠀⢀⣬⣷⣈⣯⣷⣿⣿⡿⠿⠷⢿⣿⣾⣷⣿⣿⠀⣱⠀⠀⠀⠁⠐⠈⠛⣿⡛⠿⣤⣤⣼⡧⠽⣿⣿⡿⠑⠒⠒⠀⠀⢺⡄
⠀⠀⣰⠏⠀⠈⠉⠉⠉⠉⡛⠁⠀⠀⠀⢻⣧⢄⢉⣿⣤⣽⣷⠠⠤⠐⠀⠀⠀⠈⢷⡄⣿⠈⢸⡇⢰⠷⣿⡇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⢸⣿⣦⣾⣿⣟⣿⡉⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⡶⢾⡧⣼⢰⣿⡇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠈⣧⣀⣀⣀⡀⠤⢜⠀⠀⠀⠀⡝⢰⡯⣽⡇⠀⠈⢽⢿⣿⡀⢀⠄⠂⠀⠐⠢⡀⠘⢿⣼⣷⣿⣸⣿⣧⠀⠀⠀⠀⣀⣴⠏
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⢧⣿⣅⣀⠀⢻⡿⠶⣷⠁⠀⢡⠀⠐⡆⠈⢆⠈⣿⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠁
⠀⠀⢸⣇⠀⠀⠀⠀⠀⣰⡀⠀⣀⠌⠀⢿⡄⠘⣧⣿⣿⣿⡛⠒⠻⡀⠈⣾⠀⠀⣻⠀⠘⡄⢸⡀
⠀⠀⠀⢹⡇⠐⠒⠂⠉⠁⠀⠀⢹⠀⠀⠈⣯⣀⣿⢲⠀⢹⡇⠀⠀⢱⠀⠰⠀⠀⠀⠐⠀⠃⢸⡇
⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⢀⠔⣳⠀⠀⣸⢧⡿⢿⡉⠛⠓⢷⡶⣶⣾⡆⠁⠀⠐⠥⡀⠀⡀⠸⣷⡀
⠀⠀⠀⠀⠈⢻⠀⠀⠀⠈⠀⣠⢛⣶⢾⣿⣿⠟⢸⡇⠀⠀⠘⣧⣹⣷⡇⠀⠀⠀⠀⠀⠐⠁⠀⠟⢷⣄
⠀⠀⠀⠀⠀⠘⣧⣀⣤⣴⣾⣿⢰⡿⢀⣿⠙⠒⣾⢧⣤⣴⣶⡯⠿⠿⠿⠀⠠⠀⠂⠁⠀⠀⠀⠸⣾⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣼⠏⠀⢀⡟⢺⠖⢢⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣜⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⢈⣿⡿⣦⣀⣼⠠⠀⠂⣸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡿⢡⠺⢹⠏⠛⠲⠶⣽⣀⣀⣀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣗⢈⡁⣿⠀⠀⠀⠀⣿⢾⢯⢏⣹⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠈⠓⢶⣇⡀⠀⠀⢰⣟⠄⠚⡟⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠿⠻⣇⠀⠀⢰⡏⣟⣷⣶⢾⠯⣥⣧⣇⣀⣨⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣼⣛⣤⠀⠾⢸⠀⠀⠀⠀⠁⣽⠤⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠶⠶⠶⠾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡴⠋⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢳
⠀⠀⠀⠀⠀⠀⠀⣠⠋⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢠⡔⠋⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡖⣲⡆
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠶⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢯⡤⠃
⠀⠀⠀⠀⠀⠀⠈⠓⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠰⠧



⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣀⡚⠉⢠⠀⠃⠃⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⠋⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣉⣈⢱⣀⣀⡀
⠀⠀⠀⠀⡜⢁⡷⠀⠸⣄⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⢰⢱⠒⠂⠀⣄⠀⠀⠀⢹
⠀⠀⠀⠀⠹⡼⣄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠻⣕⡄⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠋⠀⣠⠀⢸
⠀⠀⠀⠀⠀⠘⠮⣓⠚⢉⡹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⢯⠉⢁⡇⠀⡤⣳⠃⠀⠀⠀⠀⠀⠀⠀⠸⡉⠉⠉⠉⠁⠈⠉⠟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠊⠁⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⢠⠔⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣒⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠃⣴⠛⣦⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡞⠁⠀⠹⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠬⠭⠭⠭⠭⣍⠉⢹
⠀⠀⠀⠀⠀⠀⢧⠈⣛⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠳⢄⣠⠜⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⣶⢆⢰⠆⢸⢀⣸
⠀⠀⠀⠀⡀⣴⠚⠉⠉⢽⣶⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢰⣦⣖⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢤⣠⠤⠤⠤⠼
⠀⠀⠀⠀⠹⣼⠄⠊⠉⠓⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠟⠛⢞⣛⣭⣭⣭⣶⣶⣮⣵⣶⣭⡭⢛⡻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣵⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣶⣥⣘⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣡⣶⡟⠛⠉⣙⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢷⣮⣛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣼⣿⣿⣿⣿⠾⠟⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⣿⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡷⢸⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⣽⣿⣿⣯⣥⣤⣴⣶⣿⡿⠿⣿⢟⣛⠧⠝⠣⠂⠀⠀⠀⠀⠀⠀⣀⠔⢺⣿⢿⣧⢻⡄
⠀⠀⠀⠀⠀⠀⢰⡏⣮⡿⣿⣿⣿⣿⣏⠁⠀⠀⢠⡶⠚⠋⠈⢉⠄⠀⠀⠀⠀⠀⠄⠀⢀⠞⠁⠀⢸⣿⠘⣿⡇⢹
⠀⠀⠀⠀⠀⠀⣸⢣⣿⡇⢹⡏⢻⣿⣿⢆⣠⠖⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⢻⣦⣭⡇⢸⡇
⠀⠀⠀⠀⠀⠀⣇⢸⣿⣧⣼⣿⣾⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠛⠛⠳⡄⠀⠀⠻⣿⡇⣸⣧⣤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⡏⠈⢿⣿⣿⣿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⣠⠛⠛⠂⠀⠀⠀⠀⢿⣆⣥⣶⠶⢶⣬⣙⢷⡀
⠀⠀⠀⠀⠀⠀⢹⡆⣼⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠛⢠⣤⠀⠀⠀⠀⠀⢸⠟⢁⣤⡒⢤⠀⢹⣇⢿
⠀⠀⠀⠀⠀⣠⠾⢓⣼⠋⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⡆⠀⠀⠀⠀⢬⡉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠹⡆⠀⢸⣿⢸
⠀⠀⠀⠀⡼⢣⡾⠛⠉⠳⣌⠉⠉⠁⠀⠀⠀⠀⠀⣀⡠⠶⠛⣩⠞⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠐⢂⠙⢦⡀⠀⠀⠀⣼⢇⡟
⠀⠀⠀⢸⡇⣿⡁⠾⠉⡆⠈⠁⠒⠂⠐⠲⠶⠎⣩⣤⠔⠂⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠊⠋⢁⣛⣤⠾⠃⡴⠃⠀⠀⣼⢏⡾⠁
⠀⠀⠀⢸⡟⣿⠃⣶⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢢⡄⠀⠙⢦⠜⠋⠉⠀⠀⠀⠑⢀⣠⡾⢣⡞⠁
⠀⠀⠀⠈⢷⡹⣦⡀⠂⠈⠳⣆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠴⠴⠤⠠⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣫⠶⠋
⠀⠀⠀⠀⠈⠳⣌⠻⣦⣀⣀⠘⠀⠻⠀⣡⡄⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⡾⣣⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠦⣭⣛⣛⡛⢿⣷⡉⣻⠌⠳⣤⡞⢉⡦⠀⠀⠀⢠⣒⣈⣶⣤⠀⠀⠀⠀⠀⠀⣠⣾⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣎⢻⣄⠁⠀⠲⣄⡴⣏⠀⠀⠀⠘⠿⠿⠻⡿⠁⠀⠀⠀⠀⢀⣾⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡛⢶⣤⣀⡔⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢬⣛⡻⢤⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣝⢷⣤⣀⠀⠀⢀⣀⣀⣴⡿⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⠁⣤⡉⠉⠁⣼⠙⣧⡘⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣼⣇⠀⠈⣹⠖⠚⠓⣤⠹⣿⣿⣶⣭⣝⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣴⣿⣿⠹⠀⣠⠻⣆⠀⠀⢻⡛⠿⣿⣿⣿⣿⣿⣷⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣩⡞⣿⣿⣿⣿⡀⣷⠉⠀⠹⣧⣀⣾⡇⠀⢹⣿⡿⣿⣿⣿⣷⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⣿⣿⡘⣿⣾⣿⡇⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⣿⢗⣽⣿⣿⣿⣿⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⣿⣷⠈⠿⣿⣧⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⢸⣿⢻⣿⢻⣿⣷⡿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢻⢿⣿⣿⣧⠹⣷⣿⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⡟⣾⣿⢸⣿⣿⣶⢷⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣿⢀⣿⣿⣿⣆⠸⣿⣇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣽⣿⣿⢼⣿⣿⣿⣿⣎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣏⣾⣿⣿⣿⣿⣆⠹⣿⡄⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣼⡿⠟⠋⠁⢻⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣆⢨⣗⠀⠀⠀⢸⣿⣿⣿⣤⣤⠿⠋⠁⠀⠀⠀⠀⠀⢸⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠉⠉⠁⠸⣿⣿⣿⣆⠻⣷⣀⡀⠼⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⣷⣦⣤⣤⣤⣤⣿⣿⠷⠀⠻⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⠅⢒⡀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⣛⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣀⣼⣦⡶⠶⠶⠶⣛⣛⣛⣯⠭⠷⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⣛⣛⣫⣭⠴⠶⢮⣭⡵⠶⠞⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠠⠐⠉⠀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠡⢀⣀⠴⡵⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢐⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢄⣶⠠⡄⣀⣀⠀⠉⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣅⣀⠀⡌⠩⢕⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠦⣌⣉⣩⡀⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢱⡀⠀⠀⠀⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⠀⢃⠀⢠⠟⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠈⣄⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠙⢓⡒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⠈⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣽⣧⠀⢢⠀⠀⠀⠀⠈⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢂⢀⡾⠘⡷⢤⠗⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠈⠀⠁⣑⠿⠴⠚⠤⣀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠈⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢺⠋⠈⢠⣤⠐⠒⠀⠙⠃⡁⠀⠀⠀⠀⢀⠄⠒⠉⠁⠁⠀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠉⠀⠂⠇⠀⠀⠀⠀⠀⠈⣦⠀⣠⡶⣥⣦⢄⣸⠇⠐⢾⠻⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⢾⣖⠉⡛⡀⠀⠀⠢⢄⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠙⢄⠙⣧⠀⠀⠀⠀⠀⠣⣙⠢⣀⡀⠙⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠑⢌⠺⡄⠀⠀⠀⠀⠈⠓⢌⠃⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⣨⠤⠇⠀⠀⠀⠶⣀⠳⡈⢢⡀⠀⠀⠀⠀⠈⠂⣸⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢞⠉⠉⠉⠉⠉⠉⠉⠐⣷⠛⠉⢛⡻⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠛⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠆⠰⠂⠀⠐⠀⠐⠐⠀⠀⠐⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠐⢷⣬⡑⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢁⣤⠾⣳⣷⣶⣶⣶⣶⣶⣾⣧⣌⠳⢤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡜⠁⢲⣿⠏⢄⣿⣯⣭⣉⣉⣙⣿⡻⠿⠿⣿⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠃⡸⣿⠃⡄⣼⡿⠋⠙⠉⢻⣿⣿⣿⠳⣤⣿⠀⣿⡦⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⢀⣿⠏⢸⢣⣿⣧⣈⠀⠀⣿⣋⣿⡇⠀⣸⣿⣶⣿⡿⡆⠀⢹
⠀⠀⠀⠀⠀⠀⢸⡀⢻⠐⡁⣸⣇⠀⡼⢸⣿⡎⢩⣷⡀⠩⠀⣻⡇⠀⢿⣿⣿⣿⡗⢹⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⢳⢸⣿⡟⠁⢸⣿⡃⡘⠈⣧⠀⢲⣷⣻⣶⡞⢻⣿⣷⢇⡞⢠⡏
⠀⠀⠀⠀⠀⣧⠀⢹⡀⠀⣼⣿⣿⡆⠀⢸⣿⣆⠀⠰⣿⣦⠜⡏⣿⣿⣧⣼⣿⢃⡾⠁⣼⠇⠀⠀⢸
⠀⠀⠀⠀⠀⢹⡳⡜⢁⠀⠈⠻⠋⠁⢰⣾⣿⣿⣷⣦⣀⣀⣠⣷⣿⣿⣿⣿⢣⡾⠁⢰⡿⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠈⡇⣿⠸⡄⠀⠀⣄⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⠁⢀⣿⠃⠀⠀⣰⣯
⠀⠀⠀⠀⠀⠀⠀⣿⡀⣇⠀⢢⢙⠆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣾⠇⠀⣾⡏⠀⠀⣰⡟⠋
⠀⠀⠀⠀⠀⠀⢀⣿⣧⢹⡀⠀⢸⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⡟⠀⢰⡟⠀⠀⢠⡟⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣧⠀⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠁⢀⣿⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⢸⡄⡇⠀⡆⠀⢸⣿⣿⣿⣿⣿⣿⢿⡇⢸⠇⠀⣿⡏⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣷⠸⢸⡇⠣⣼⣀⣠⣾⣿⣿⣿⣿⣿⡿⢸⡇⣿⠀⣰⠟⢀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣀⡀⠀⠈⠉⠛⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⢀⣀⢀⠀⢠⡆⠀⠀⠐⠶⠶⢶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒
⠀⠀⠀⠀⢀⠀⣀⠀⡤⠀⣄⠀⢀⠀⡐⠀⢀⠀⢄⡞⠃⠀⢀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠛⠻⢛⡋⠂⣾⣤⣟⠲⣿⣈⣿⠦⣿⡲⣾⣁⢰⣏⢺⣌⣹⢤⡶⢠⡆⢠⠀⡆⢀⠁⠀⠀⠀⣦⢀⣱⡦⣄⠲⡈⠲⢌⡑⠢⡈⠓⣦
⠀⡀⣴⣿⣷⣿⡹⣶⡿⣾⢯⣲⠿⠿⠧⠙⠻⡿⠿⢏⡙⠶⠿⢍⠻⣾⠯⣹⣾⠷⢿⡤⣤⡰⠦⡐⡈⢦⠐⣆⠣⡀⠰⣄⠙⢦⡈⢦
⠀⠵⡘⢀⠀⠀⢣⡑⡀⠀⠀⠹⣄⠆⡀⠈⠳⣕⢦⡀⠈⠦⢌⣠⡈⠺⣑⣄⡉⠳⣤⡐⢯⣷⣄⡑⠛⠨⠇⢸⡆⣹⣀⣙⣀⠈⠋
⠀⠀⢠⠈⢲⣇⠀⢽⣌⠣⣀⠀⠨⢳⣙⠠⡄⢈⣟⣦⣱⣂⠾⠛⣿⣷⢿⡛⠛⠓⢾⣿⡏⠛⢉⣯⡉⢱⣿⡆⢹⡇⢸⡏⡸⠁
⠀⠀⠀⠙⣆⠉⠓⠲⠿⢷⣬⣵⡆⣿⡿⣷⣶⣦⣤⣼⡼⣿⡍⢉⣛⣿⢻⣿⣤⣤⣿⣺⣇⣤⣸⣿⠧⢬⠟⠓⠚⡋⠈⠁⠘
⠀⠀⠀⠀⠈⡗⡶⣠⣀⡀⠁⢿⡇⣿⣿⣀⠈⠅⢉⢙⣵⣿⡿⣿⣿⣿⣿⡇⠉⠙⠛⡏⠀⠈⠉⣀⠀⠀⠒
⠀⠀⠀⠀⠀⣷⠄⠉⢹⠷⣷⣼⣽⣏⣿⠙⠛⠻⠿⣿⣿⠿⢹⠈⠙⠉⣿⠀⠀⠀⠀⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢈⠈⠀⠋⠘⠙⡟⠉⠛⠀⠀⠄⠈⢸⠀⠀⠈⠑⠂⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡆⠐⠀⡇⠀⠀⠀⠀⠀⠘⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠉⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣶⣶⣶⣾⡿⠷⠶⣤⣄
⠘⠿⣿⣶⣦⣤⣀⠀⠀⠀⠀⣾⠀⠀⠀⣠⠶⠶⠷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠙⢷⡀
⠀⠀⠀⠉⠙⠛⠛⠃⠀⠀⠀⢻⡀⢀⠎⢁⣤⠤⣤⣀⠙⢿⣷⣤⣀⣀⡀⣀⣀⡀⠀⠀⠀⠘⣿⡄⠀⣈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⡄⠁⠀⠀⠀⠙⢷⣄⡙⠿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢰⣿⣷⠀⣿⣷⣌⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣤⣤⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⢠⣿⣿⣿⣧⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠛⠋⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣏⢈⣿⣿⣿⣿⣿⣿⠃
⠀⠈⠙⠛⠉⠉⠁⠀⠀⠀⣾⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠀⠻⡀⠸⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣡⠀⠀⢸⡄⠀⢇⠀⠛⠃⣽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣦⣀⠀⢠⡀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⢸⡃⠉⠆⠀⢸⡇⠀⢸⠀⢰⡀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⠛⢛⡼⠃⠀⠀⠀⠉⠙⠛⠻⢷⣦⡀⠀⠘⡇⠀⠰⡀⢸⠇⠀⢸⠀⠘⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣠⡴⠋⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠙⢦⠀⢳⡀⠀⢳⣾⣓⡒⢾⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠹⢷⣶⣤⣤⠀⠀⠀⠀⠸⣇⠀⠀⠛⠉⠙⠛⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠙⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⣷⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡇⣠⣤⣧⡀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠈⠐⠪⢽⣷⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⣶⡶⠶⠒⠛⠛⠋⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣄⣤⣤⡴⠾⣿⢦⣤⣤⣤⣤⡶⠞⠋⠁⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣳⣶⣶⣾⣟⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠂⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠂⠠⢀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠐⡀⢤⠀⠀⠀⠀⠀⠪⠀⠄⠀⠧
⠂⠀⠀⠈⠀⠀⠀⠁⠀⠝⠦⠤⢀⠀⠀⢠⠊⠉⠠⡆⠉⠀⠠⠤⠀⠀⠷⣾⢛⣻⠇⠀⠀⠀⠀⣦⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⢈⠁⠀⢸⡀⠀⠀⠁⠂⠠⠀⠀⠀⠠⠀⠀⠋⠙⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠤⠤⠀⠀⣀⡀
⠀⢠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠛⠢⠦⠀⠠⠤⠠⣄⠠⠠⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠔
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢤⣔⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⣂⣀⣀⣀⣀⣀⣀⣀⣘⣾⣃⣙⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣌
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠙⠢⡰⠿⠿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡄⣀⣉⣹⡦⡤⡭⢴⠶⢄⠙⠛⠉⣠⠪⢀⣾⣿⣿⣿⣿⡛⡀
⠀⠀⠀⠀⢴⠁⣸⠟⠋⣡⠞⠁⠀⠁⣠⣾⣭⣐⡚⠥⣄⠸⠋⢹⣿⣿⣿⠗⠁
⠀⠀⠀⢡⠏⠈⠀⠀⠀⢁⣀⡄⡈⢠⡿⣿⣷⢶⣤⣤⣀⣙⣃⡟⠋⠘⠃
⠀⠀⠀⠈⡌⠁⢠⠁⠀⠀⢰⠀⠆⠘⣇⢹⣿⠀⠀⢈⣉⣁⣠⣤⡆⠀⠢
⠀⠀⠀⢠⡇⠀⡾⠀⠀⠀⠈⠀⢠⠀⢻⣄⢻⡄⠀⠈⠉⠉⢡⠟⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢾⠇⢠⡇⠀⠀⠀⠀⠀⠘⣄⠀⠙⢷⡱⠀⠀⠀⠀⠎⠀⣰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⣸⣷⣶⣤⣝⣦⡀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣶⣶⡄⠀⣤⡴⡇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⠃⠀⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠸⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⢀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠘⠢⠄⣀⣀⣘⣤⣀⠀⣤⠾⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⣿⣿⣿⣿⣿⣿⠁⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠁⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⣼⡧⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡟⠁⣍⢨⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠈⠁⢞⣡⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠿⣿⣿⠿⠿⠟⠛⠛⠛⠋⠉⠁
⠀⠀⢇⠀⠀⠀⠀⠀⠘⡱⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠐⠶⢦⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣾⠫⠭⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣭⣭⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣽⣿⣿⣿⡿⠿⣿⣿⣿⣿⠏⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣿⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⠉⠰⣶⣶⣶⡗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣿⣿⣿⣿⣿⡿⠤⢄⠀⠀⠀⠀⠀⢉⣉⠛⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣶⣌⠻⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣛⡉⠂⠈⠛⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠤⠀⠀⣤⣶⣭⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣿⡿⠋⠁⣴⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⣿⠿⠛⣃⣸⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⢰⣿⣿⣷⣿⣾⣿⣿⣿⣿⢻⠟⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡿⠻⠈⣠⣞⡈⢻⣿⡀⠀⢀
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣠⣾⣿⣿⣅⡀⠘⢧⠀⣸⣷⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠟⠻⠿⣿⣿⣿⣿⣿⣾⣿⣿⣏⠛⠓⠀⠀⢐⡀
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠉⠙⠛⠻⠟⣿⣿⣦⣀⡀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠉⠉⠉⠛⠋⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣼⣿⣽⣻⣤⣶⣶⣄⠈⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⡀⠀⣠⡈⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⠛⠟⣋⣭⣧⣤⣦⣶⣶⣷⣾⣭⣽⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢛⣯⣴⣦⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠻⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⠀⠀⠀⠀⢪⣀⣀⣈⣲⡀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣔⣉⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⢠⠋⠉⣩⠷⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣋⣡⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣷⠀⠸⡇⠀⠀⠀⠺⣍⡉⠙
⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⢠⡞⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⣿⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⣘⡧⠀⠀⠀⠀⠀⠀⣾⣁⣀⡈⣷⠀⢸⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠘⣷⠶⠶⠚⠛⠻⠏⢹⠇⠀⠀⠀⠀⠀⠀⠸⣄⣉⣩⠟⠀⢸⣧⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠙⢦⣀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣤⠀⠀⠀⠉⠁⠀⠀⢸⣷⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠻⢿⡿⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠞⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠞⠋⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣬⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣸⣃⠹⢿⣿⣿⡏⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢀⡟⠉⣻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠉⠙⠙⠿⡿⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡶⠛⠓⢲⡇⠀⡸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣛⡿⠿⢿⣶⣤⢤⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣇⠀⠤⣾⢧⣴⠁⢰⠏
⠀⠀⠀⣀⡴⠶⠛⠛⠛⠉⣠⡶⠋⠁⠀⠀⠀⠈⣧⣄⣀⣙⣷⣀⠀⠀⠀⠀⠀⠀⣠⣴⣟⠁⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢁⣀⠨⠛⠁
⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡧⠀⠙⢛⠿⠶⠶⠾⠟⢹⣄⣽⣧⡄⠀⠀⠠⠄⣠⡴⢋⣤⡾⠃⠠
⠀⠀⠙⣶⠃⠀⡀⠀⢀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⠻⣷⣶⣶⣶⣿⣯⠾⠛⠁
⠀⠀⠀⠘⠷⣾⠁⢘⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠖⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣦⠀⠀⠀⠀⠀⠀⢰⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⠙⣧⠀⠀⠀⠀⠀⣿⡯⠛⠛⠻⠶⠶⣤⣤⣀⣀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⢀⣴⠟⣱⠋⢀⣴⢻⣆⠀⠀⢀⣠⡿⣶⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢻⡟
⠀⠀⠀⣠⡾⢃⠜⠁⣼⣿⢷⡎⢻⡄⢀⣿⣏⠀⠙⢷⣄⣀⠀⠀⠀⠀⠐⠶⢦⣴⣟
⠀⢀⣼⠏⡰⣏⠀⢀⡿⠃⠈⢷⡀⠙⢋⣿⠟⢦⡴⠛⠉⠙⠻⢶⣤⣀⣀⣠⣴⣿⣷⣤⣀⡀
⣠⡿⢡⠞⠁⣸⣷⠟⠁⠀⠀⠈⠻⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠙⠻⢦⡌⠙⣷
⠛⠷⠿⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⢲⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠙⠛⢿⠿⠟⠉⠛⢛⣷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣉⣓⡒⠶⠒⠦⠤⣤⣤⣏⣀⣀⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠶⢶⣶⣦⣬⣭⣍⣹⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣾⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢹⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣌⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⢟⣿⣷⠶⠤⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⡟⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣶⠀⠀⠈⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⢿⣿⣿⠙⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠿⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀

⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠙⠛⠋⠛⠛⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⢰⠈⠛
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠀⠀⠀⠀⠀⣀⠀⢤⣌⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⣎⡀
⣶⣶⣆⠀⠀⠀⣾⠁⠀⠀⡌⠀⠀⠀⠀⠀⠀⠻⣤⡴⠶⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣉⣉⣀
⣿⣿⣿⣷⡆⢸⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡛⠻⠎⠻⢿⣷⣆⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠂⠀⠀⠊⠃⢀⡤⠀⠀⣠⣄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠛⢳⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡝⠻⠛⣆
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⣠⡳⠖⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦⣀⡀⠁⢹⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠑⠀⠀⠀⠀⠀⠀⠿⠟⠐⠾⠋⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡅⠀⠀⢻⣿⣷⣦⣝⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⣼⡟⢿⣿⣿⣿⣧⣐⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⣾⣿⣿⣿⣷⣿⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣙⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢋⣽⣿⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣿⣟⠻⠟⠋⠙⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⡙⣿⠃⣼⡏⣠⡴⠀⣀⣴⣟⡟⠉⠉⠀⠐⠿⠿⠿⣿⣿⣿⠿⠿⠛⢛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣻⣿⣰⣿⣿⣿⢁⣠⣟⡿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠛⣛⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⡃⣯⣴⣶⣤⣶⣦⣤⣶⣦⣤⣄⣀⢀⣀⣀⣤⣄⣤⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡧⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢖⡊⠀⢩⠟⠃⠀⠀⠀⠀⣶⠀⠀⠀⠀⣝⡠⣄⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠲⢼⡆⠀⠀⠐⣖⠚⠀⠳⣤⡄⠘⠋⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠂⠀⠀⣸⡴⢦⡏⠁⢀⡄⠀⣀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣄⠀⠀⠀⣄⠀⠁⠀⢀⣠⡤⠤⠭⠤⣬⣅⡀⠀⠀⠀⣤⡤⠞⠘⢓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡍⣀⢩⠟⠋⠀⠀⠀⢀⡞⠋⠙⠒⠶⢤⣄⣀⣠⠽⠷⣄⠀⢀⣏⠴⣜⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠻⠀⠠⠄⠀⢠⠞⠉⠓⠦⣄⣠⠴⠋⠉⣀⣤⠷⡛⢦⡈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⡟⠒⣦⡤⠞⠉⣀⡤⠖⠋⠁⠀⠀⢉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣞⣁⣤⡴⠿⠥⠤⠴⠶⠶⠶⠶⠶⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣒⣩⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⣄⣁⣤⣤⣤⣤⠤⠤⠤⠶⠖⠛⠉⠁⠀⢸⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⣤⠀⠀⢀⣤⣴⠂⠀⠀⠀⠈⢻⢲⣄⠀⠀⠀⡴⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠂⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠱⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⡆⠀⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⢠⡆⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠃⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠖⠛⠉⠉⠉⠉⠃⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢲⣤⣄⣀⣀⣀⣠⠞⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡌⠴⢋⡩⢋⡽⢋⠞⣹⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⠖⠛⢧⠻⣄⠀⠀⠈⠠⠎⡴⢣⠞⠵⣠⠏⣹⠓⡶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠳⣎⡌⠳⣌⡓⠦⢤⣄⣀⣤⠤⠶⢚⣡⢾⣡⠞⠁⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠙⠻⣤⣝⣛⡶⢲⡶⠶⡶⣚⣿⠵⠋⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢠⠖⣀⠄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⢀⠀⢀⠀⢀⠔⢀⡴⠀⡠⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⠞⢁⠔⢁⡔⠁⣠⠆⣠⠎⣠⠞⢁⡴⠋⡴⠃⡰⠋⣠⠊⣠⠎⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠴⠃⡴⠋⠀⠞⠁⠞⢁⠞⠁⣠⠋⢠⠞⢀⡞⠁⡼⠁⡰⠃⡴⢁⠔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠇

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠸⣿⣧⣀⣠⣤⣴⣾⣿⠐⢀⡤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⠂⠠⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⢿⣿⣿⣷⡄⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣶⣿⣃⡀⠀⢸⠟⢹⣿⣿⣿⣿⡆⠀⣵⣦⢀⣤⢹⠿⣿⣿⡏⡅⢩
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣸⠛⠉⢉⣹⣆⣤⣤⣤⠙⠛⠛⠿⠿⠈⣦⣶⣎⣛⣵⣤⣿⣠⣷⣶⠂
⠀⠀⠀⠀⠀⢻⡟⢋⣉⣻⡟⠛⢃⣌⠉⠹⢿⣿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⡿⠙⠿⣿⣏⣤⣤⣀
⠀⠀⠀⠀⠀⠈⣿⢿⣟⡛⣅⣺⣿⣾⠏⠀⠀⠀⠀⠐⠂⠠⢄⣤⣼⣶⣴⣶⣦⠈⠁⠐⣾⣿⠋⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠈⣺⣏⣾⣿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⠻⠟⠛⣿⣿⠛⢀⠀⢸⣟⠃⠂⠀⠆⠁
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣯⡅⠀⠈⢹⣷⡀⠀⠀⢀⣀⠀⠀⠙⡀⠀⠄⢸⣷⠿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠋⠁⣘⢿⣿⠁⢀⣤⣼⣿⡦⠀⢰⡿⠿⠿⣤⣄⢀⣀⣀⡀⡟⠄⠀⠄⠚⠳⣤⣤⠖
⠀⠀⠀⠀⠀⠰⠀⡀⠀⠙⣀⣿⣿⣿⢁⠠⣟⡛⡀⠀⣀⢸⣿⠻⠿⣿⣿⣿⣶⣷⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠶⠶⣴⣶⣤⣶⣿⡀⠀⢻⣿⠿⠉⠉⠻⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣈⣉⢉⠉⠛⠉⢾⢿⣷⡦⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⢹⡁⢀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣶⡾⣿⣿⣿⣿⣷⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣯⡿⣶⠋⠋⠈⠁⠻⠣⡟⣿⣿⣿⣿⣿⣿⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡆⠐⣿⠼⣿⣿⣿⣿⡟⠋⡟⠁⠟⠀⠀⠀⠀⣊⣸⡇⡟⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⡀⠀⢁⡇⠉⠘⢾⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣁⢀⣨⣸⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡟⠙⠀⠈⠀⠀⠀⠀⠙⠘⣿⡇⠀⠀⠀⠀⠎⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢠⡄⠠⠐⠉⢁⡤⠤⠀⠈⠙⠻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⢠⡀⢀⣠⣄⣀⣤⣄⠀⠀⠄⠀⢠⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⢫⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠸⠈⣿⣿⣿⣿⡿⠇⠀⢠⡤⡤⡸⠃⠀⠀⠀⠀⢀⣄⣤⣤⣦⣿⣅⣿⢠⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⠦⠰⢠⣿⣿⣿⡋⠀⠀⠀⠘⢻⣷⠇⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢻⣿⣿⡄⠈⣿⣟
⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣄⠀⠀⣠⣶⠿⣷⣄⣠⡤⠄⠀⣸⣿⣿⣿⡀⢿⣿
⠀⠀⠀⠀⠀⢸⢇⣴⣿⣿⣿⣿⣿⠉⠀⡀⠀⠀⢻⠃⣁⡀⠀⠾⠋⠈⣽⡿⠟⠀⠀⠀⣿⣿⣿⣿⣷⣿⡿
⠀⠀⠀⢴⣤⣼⣿⣛⣿⣿⣿⣿⣿⣶⣶⡷⠶⠶⠞⠛⢓⣦⣤⣤⣴⣦⣤⣧⣄⣀⣀⠀⠾⣿⣿⣿⣿⢱⡇
⠀⠀⠀⠙⠛⠋⠭⣭⣿⣿⣷⣤⣤⣤⣼⣿⣶⣴⣿⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣷⣽⣶⣾⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣿⢫⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣼⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠁⠹⠛⢻⡿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠿⠻⠿⠛⠉⠹⠋⠉⠉⠋⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⣿⣿⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠇⣰⠃
⠀⠀⠠⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⢠⡟⡰⠁
⠀⠀⠀⠈⢆⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡿⠋
⠀⠀⠀⠀⠀⠙⢇⢂⠀⠀⠀⠀⢀⣠⡤⠞⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠺⠇⣀⠀⡴⠛⠁⠀⠀⠀⢀⣔⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠛⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢰⡷⠤⠤⣄⣈⠉⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠤⠀⠀⠀⠀⠀⢴⡉⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠛⠛⠛⣾⣅⣠⣶⣴⣄⡀⠀⠀⠀⠀⠀⢀⠀⣈⣀⠶⢶⣤⣶⣶⠂⢈⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠹⣿⠖⠀⠀⠀⣷⣷⡛⠉⠙⠣⢰⠏⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢧⡀⠀⠀⣠⣽⣼⣿⠀⢀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠖⢠⡞
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⣄⢀⣼⢻⣿⣿⣷⣝⠢⣤⣤⡤⠀⠄⠒⠐⠂⠀⠀⢰⣶⣠⣴⡟⠉
⠀⠀⠀⠀⢀⣈⣽⣦⣶⣦⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣏⠀⢻⡿⢿⣿⣿⣦⡿⠤⠀⠂⠀⠀⠐⠒⢃⣼⣿⣿⡏⠀⣀
⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣮⣠⠙⠆⢹⣿⣿⣤⣀⣀⣠⣿⣷⣾⣿⣿⣿⣿⣿⣁⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠚⠉⠻⣟⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣯⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡆
⠀⠀⠀⠀⠀⣰⣄⣀⣠⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐⠿⠟⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠻⢿⡟⢷
⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⢁⣽⠿⠛⣻⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣿⣷⡙⢚⡇⢸⣿⣷⡙⢿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠹⢖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⢹⣷⣻⢋⠠⣼⣿⣿⣷⣤⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⢦⣉⣉⡠⣽⣦⣈⢻⣿⣿⣿⣿⣿⣷⣦⣤⣤⣄⣀⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⠿⣿⢻⣿⣿⣿⣿⣿⣷⣶⣿⠁⠛⣿⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⡻⠛⠙⠲⢶⣶⡾⠿⢿⣯⣿⠛⠻⢿⣿⠁⠀⠙⠻⣿⣿⡏⠙⠉
⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣤⣮⠿⣾⣤⡘⠿⣿⣿⣶⣩⠿⡗⠀⠀⠀⣹⣿⣿⣦⠄
⠀⠀⠀⠀⠛⠛⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠈⢹⣿⣿⣿⣿⡇⣿⣿⣥⡀⠈⠿⣝⣷⣾⣼⣻⠿⣿⣯⣤⣤⣴⣿⣿⡈⢿⣄
⠀⠀⠀⠀⠀⠀⠉⠁⢠⣿⣿⣿⣿⠛⠛⢿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⢨⡿⠛⣿⡏⠀⠙⠛⠿⣿⣿⣷⣴⣿⡿⣿⣿⣿⣿⣷⡄⠙
⠀⠀⠀⠀⠀⢀⣠⣶⠿⣿⣿⣿⣿⡄⢀⣾⡿⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡏⢀⡀⠹⣟⡀⠀⠀⠑⢄⠈⠙⢿⣿⣧⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⣿⡟⠂⠛⠉⣿⣿⣷⠀⠃⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢟⣿⣿⣧⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠛⡋⠍⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡯⢿⣿⡏⠈⠙⠳⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠈⢿⣤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠋⠉⠻⣿⣿⣯⣀⣼⡿⠿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠰⡌⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣼⡅⠀⠈⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣿⡏⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣼⣿⣧⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢛⣿⣿⣿⣿⣿⣿⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡿⠣⠞⣩⣷⣦⣽⡿⠁⡉⢻⡿⣿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢠⡞⠁⣤⠾⠿⠿⣿⣿⣿⣦⣸⣌⢿⡼⣷⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⣷⡿⠃⢈⣵⢺⣿⣾⣿⣿⣿⣿⣿⣿⡄⠐⢭⣿⣦⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡀⠋⠀⠀⠀⠛⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣾⡇⠀⢀⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣹⣛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⡞⢀⣻⢃⡀
⠀⠀⠀⠠⣴⣶⣶⣴⠀⠀⣀⣴⣦⡆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣰⡟⠀⢢⡟⡾⠀⠂
⠀⠀⠀⣠⣿⡿⠿⠿⢿⣿⣿⣿⣿⠇⢀⡈⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⡟⠀⢀⡟⠂
⠀⠴⣾⣿⠇⠀⠀⠀⣤⣿⣿⣿⣿⡆⠘⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠞⠛⠀⣦⣿⣷⣤⣀⡤
⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⡆⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠘⣿⣿⣶⠀⣼⠟⠛⠿⣿⣿⡇⠀⠀⢻⡄⢻⣿⣿⣿⠟⠁⠀⣠⡞⠀⣸⣽⣿⣿⡿⠿⠟⢿⣿⣿⣷⡀
⠀⠀⣦⠼⣿⣿⡄⠀⢀⠀⠀⠻⣿⣧⠀⠀⢻⣧⢸⣿⣿⠏⠀⡠⠾⣿⢀⡼⢻⣿⣿⣿⡧⠀⠀⣿⣿⣿⣿⡇⠐⠆
⠀⠀⠀⢠⠈⢿⣿⣷⠘⠦⠀⠀⢸⣿⠀⠀⠈⣿⢸⣿⠙⠂⠀⠁⠘⠋⣿⠃⣾⡿⠿⠛⣁⣼⣷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠠⣽⣿⣇⣴⢠⠀⢸⣿⡀⠀⢀⣿⣿⡇⠀⣀⠀⡀⠀⣾⡏⣸⡿⠦⠴⢛⣉⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣎⠻⣿⣤⣀⠀⠛⠃⣴⡿⣻⣿⣷⣤⡉⠠⠶⣦⠈⠈⠃⠄⢠⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡮⠻⣿⣧⣄⠀⣩⠷⢿⣿⣿⣿⣿⣷⣦⣤⣤⣾⣿⣶⣶⣿⠿⠋⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⣝⣻⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⢀⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡏⢻⣿⣿⣿⣿⣿⣿⡿⢸⣏⢻⡇
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⡇⢸⣿⣿⣿⣿⣿⣿⡇⣼⡏⠈⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣻⡇⣀
⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠁⣤⠉
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠉⠉⠉⠉⠛⠓⣀⣛⡛⠿⢿⣿⣿⡿⠿⢛⣃⣤⣚⣋⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⣤⣶⣿⣿⡿⠿⣒⣤⣤⣐⠻⢿⣿⣿⣿⣿⣿⣿⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠃⢠⣿⣿⠟⣋⡥⢶⣿⣿⣿⣿⣿⣿⣶⣮⣝⡻⢿⣿⡁⠘⠉⣠⣤⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢘⣩⣶⣿⣿⣿⣮⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⣲⣬⡃⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⠋⠸⣿⣿⣿⣿⡿⢋⣾⣿⣶⣬⣭⣭⡉⢴⣶⣿⣿⣿⠇⠈⠳⣄⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣨⣭⡍⣴⡞⡻⠿⢿⣿⣿⡿⣛⡛⢷⣦⢉⣭⣥⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠆⠸⣿⣿⡇⣿⡔⢧⣻⢆⣿⣿⠸⣥⢝⣆⣿⢸⣿⣿⡇⠐⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣧⣤⡄⠀⠘⢿⣷⣿⣶⣵⣾⣷⣤⣾⣿⠃⠀⠀⠀⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠃⠀⠀⠀⠀⠀⣰⣷⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣐⢾⣿⣿⣿⣿⣿⣿⡷⢂⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣝⠿⢿⡿⠿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⡤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣁⠈⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣁⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡄⠰⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢤⣄⠈⠉⢳⡖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠰⠆⠀⢦⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠄⠀⢠⡄⠀⠠⠄⠒⢢⣤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⢃⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢀⣤⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡦⠀⠀⣠⠤⠘⠛⠋⠛⠓⠦⣀⠴⠤⠤⠄⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⠋⠀⠀⡄⠂⡐⠀⢀⠈⢠⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠰⠆⠀⠀⠀⢀⠀⣠⠶⠛⠻⣦⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡆⠀⠀⠀⠠⠢⢌⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠟⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⣹⠦⠀⠀⡴⢋⠁⠀⢀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⣀⣀⣈⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠉⠉⠉⠁⠆⠰⡆⠸⡷⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠀⡘⠋⠛⠀⠀⠀⠠⣀⠀⢻⡇⠀⠀⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⣾⠂⢀⡆⠀⠀⠀⠀⠀⡙⠳⠀⢻⠀⢴⢣⣴⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠛⠀⠙⡀⠨⠿⠀⢰⠇⠀⠀⠙⢢⡄⠀⡐⠂⡸⠃⣄⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠀⡀⠃⢀⠤⡀⠈⠰⡇⠀⠙⠦⠀⠀⠀⠴⠁⣴⠆⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⠄⠈⠀⠀⠆⠐⢧⠀⠀⠲⠀⠀⠒⢀⠐⣀⠔⢁⡄⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠶⠉⠀⠀⠀⠀⠰⠀⢺⣇⠀⠀⠀⠀⣠⡎⠀⢁⠴⠉⡠⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠌⠠⢄⡰⠊⢀⡠⠀⢁⠤⠈⣠⠆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄⠁⠤⠒⠁⠄⠚⠀⠔⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡁⠒⠊⡁⠒⢉⡀⠒⠈⢀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠄⢈⡁⠄⢉⡀⠠⢄⡉⠢⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠒⠤⠐⠂⠀⡤⠐⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⣁⠰⠂⠁⠀⠀⢉⣀⠀⠈⠐⠂⣤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠛⠛⠛⠻⠛⠁⠀⠀⠀⡀⠀⠀⢴⠀⡼⠁⠀⡠⠛⣁⠤⠘⢣⠀⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⠐⠀⠠⣤⡐⠂⣄⠐⡜⠁⢀⡾⠀⠉⠁⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⡄⠈⡀⠀⠀⠀⣠⣴⠎⠀⠐⠲⢄⡈⠳⢄⠲⣀⣰⠟⠓⠂⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣴⣾⠛⠻⡟⠀⠀⠘⠓⠤⣈⠢⠈⣠⠾⠷⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠀⠀⠐⠒⠒⠒⠛⠀⠀⠐⠆⠀⠀⣾⠛⠛⠛⠃⠀⠇⠀⠀⠈⠒⠀⠁⠀⢸⠟⠒⠒⠒⠂⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠰⠀⢈⡁⠐⠀⠀⣏⠉⠃⠀⠀⠀⠀⡄⠀⢈⣁⠀⠐⠀⢺⡦⠤⠀⠀⠉⠉⠉⢉⣁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢘⠉⠁⡀⠀⠀⠀⠀⢁⠀⠀⠤⠀⢀⠀⠘⡦⠠⠀⠀⠀⠀⠀⠈⠛⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⡄⠀⢺⠉⠉⠁⠀⠀⠀⠀⠘⠀⠀⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⢺⣁⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣉⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠀⠰⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠈⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⣼⠓⠐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂⠀⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠁⠀⠀⢀⡴⠋⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⠂⢰⠶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡠⠞⠉⠀⠸⠛⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡴⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡦⠀⠀⠉⠉⠀⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠤⠴⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣈⢀⡤⠤⠤⠤⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⡦⠤⠤⠴⠶⠆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠒⠒⠛⠛⠋⠉⣉⣁⡀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢀⣀⣀⢀⠀⠀⠀⠃⠸⡇⠀⠒⠒⠒⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢉⣉⣁⣀⣀⣈⣁⡹⠲⠶⠶⠿⠷⠶⠶⣤⣤⠆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⡿⠛⠉⠀⠀⠀⠀⠀⠀⣰⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣰⡫⠊⠀⠀⠀⠀⠀⣀⣤⠴⢊⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡞⠁⠷⠦⠴⠖⠒⢋⣉⠤⠔⠊⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠓⢒⡒⢒⡛⠉⠉⠀⠴⠒⠂⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⢿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣏⡀⠀⠀⠀⢀⣴⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣥⣿⣿⣿⣿⡟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠖⠈⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠈⠁⠀⠀⠄⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣭⣿⣿⣿⣷⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣇⠘⢿⠟⠁⢸⡟⣿⣿⢻⣟⠻⢿⣷⣶⣤⣄⣀⡀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠾⣿⣷⣼⣇⠛⣣⣾⡟⢱⣿⢋⣿⢏⣭⢻⣿⣿⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢈⠉⠛⠛⠿⠿⣷⣿⣿⣼⣧⠙⠿⣰⣿⣃⠸⠟⣻⣿⣿
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⡆⢻⣷⠈⣴⡆⠀⣦⣍⠉⠛⠻⠿⣾⣿⣧⣿⣦⣼⣿⣿⠇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⡷⢀⡎⢀⡞⠛⠀⣠⠀⢀⡉⠛⠛⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡄⠀⣤⡀⠀⠀⠀⠈⠻⠶⠢⠿⠷⢾⠇⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⢠⣇⢀⠁⣤⠀⢰⠄⢠⡶⣀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⣟⠃
⠀⠀⠘⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠉⠉⠀⠻⠶⠃⠀⣶⣦⡟⠃⠀⠀⠀⠀⠀⠀⠈⠑⠞⠉⠀⢀⠀⣄
⠀⠰⣆⡀⠀⠀⠀⢠⡷⢦⡾⠁⣾⣛⡉⠀⢀⣀⠀⠀⠁⠘⠓⠀⢠⡜⠃⣤⠀⡆⠀⠀⠀⠀⢀⡘⠁⠀⣙
⠄⠀⠈⠁⠐⠒⠀⢀⣀⠘⠁⢰⣭⣍⠁⢀⣾⣿⡀⠐⣶⣄⠀⠀⠈⡇⠰⠂⡀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠁⠀⠤⠀⠉⣀⠹⠇⢠⣁⣘⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢲⣤⣉⠉⠁⠀⠀⠀⠀⠦⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠹⡦⠈⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⠟⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣡⣼⠷⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⢀⡰⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⠁⠀⠀⠦⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⢀⣠⠆⠙⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠉⠁⠘⠁⠈⠁⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢀⣤⣀⣀⣀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠘⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡷⣀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠻⠟⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢀⠀⠀⠀⠀⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡏⠐⣁⠀⠀⢘⡂⠀⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⣀⠀⠁⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠠⠒⠂⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡟⠒⠀⣴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢇⠀⠀⡑⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⠀⠀⣀⣀⣀⣤⣠⣄⣧⠄⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠲⣞⡰⣿⣿⠛⠈⢉⣉⣀⡀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠁⠂⠀⡆⣀⣀⠤⠜⠛⠒⠚⠛⠉⣀⡇⠀⠀⠈⠙⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣿⣿⣿⡆⠀⠀⠈⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣶⣶⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⠟⡛⢋⣥⣬⣵⣷⣦⣴⣦⡙⣋⣛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣡⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣍⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢈⠀⠈⠲⣄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠉⠉⠈⠻⣿⣿⣿⣿⣿⡄⠀⠀⢀⡔⠉⠀⢈⡆
⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⢰⠋⠀⣰⠖⠀⠀⠘⣿⣿⣿⣿⣿⠏⠀⠀⢠⡀⠀⠈⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣶⠀⢿⣿⣿⡇⠀⠀⠀⠐⠳⣈⠉⢳
⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⢀⣀⣉⡶⠋⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠀⣼⣋⡀⠙⣯⡄⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠸⣧⠴⠶⠒⠛⠿⠉⡿⠀⠀⠀⠀⠀⠀⢫⡉⠙⣛⠃⠀⢿⡁⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣧⠀⠀⠀⠀⠹⣄⡀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⢸⣦⣾⣃⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠟⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⠛⠋⠹⡆⠀⠀⠀⠀⠀⠀⣠⣼⡶⠾⣇⡑⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣰⢿⣿⣿⣷⣾⠃⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠧⠤⣝⣿⣿⡏⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣼⡁⣸⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⣴⠟⠀⠀⠀⠀⢸⠁⡀⢈⣿⠀⡸⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣹⠿⢷⡟⠙⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠘⣇⠀⠙⠋⠛⣧⠔⠃
⠀⠀⠀⠀⢀⡄⠀⢤⣀⣀⣀⣰⠃⢰⠃⠀⢠⡟⠛⣾⡟⣿⠶⢦⣤⣀⣠⡤⡶⣿⢷⣄⠀⠀⠀⠀⠀⠀⣰⠋⢀⡤⠀⠊⠁
⠀⠀⠀⠀⡌⠀⡽⠋⡀⠀⠈⠁⠀⠘⠒⠦⣼⡗⢀⣿⠇⡗⠀⠲⣟⣨⡷⠂⠀⢻⣾⣿⣦⣄⣀⣠⣴⣋⡥⠞⠉
⠀⠀⠀⢰⠁⠀⡿⠃⢁⡀⢀⡴⠞⠳⣤⣀⣼⣧⣼⡟⣸⠃⠀⣼⠋⢉⡗⠀⠀⢸⣿⡇⠈⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢳⡔⠉⣰⠏⠀⠀⠁⠘⠭⣝⣿⠿⠛⠁⠀⣸⠃⠀⠀⣳⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣞⠀⢠⡏⠀⠀⠀⠀⠀⢰⣩⡿⡀⠀⠀⠀⢻⡄⠀⠀⡼⠀⠀⢸⡿⠁
⠀⢠⡄⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⣮⣼⣏⠛⠳⠶⢦⣤⣼⣟⣋⡀⠀⠀⢸⡇
⠀⠐⠀⠈⢡⣦⣀⡀⠀⠀⠀⠀⠀⠀⢰⣵⣿⣤⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢳⡿
⠀⠈⠀⣴⠟⡡⠋⢱⣶⣄⡀⠀⠀⠐⡞⣾⠇⠙⢷⣄⣄⠀⠀⠀⠀⠒⠦⢤⣿⠁
⠀⣰⡾⢋⠼⠅⠀⣸⠟⠀⠹⣇⠂⣶⡿⠟⠶⠖⠋⠈⠙⠻⠶⣤⣄⣠⣴⣾⣿⠦⣤⣀
⣴⡟⡴⠋⢀⣼⠞⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⢳⡆⠉⣿
⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⡾⠃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣡⣿⣥⣤⣤⣤⣀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣀⣀⣉⠙⠛⠃⠀⠈⡽⠛⠳⣤⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⣾⣥⣭⣍⣍⣛⣓⣚⠳⠶⠤⠬⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢀⡀⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠘⢡⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠀⠀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⡍⢀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠈⠀⠀⠐⠋⠉⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⠙⠃⠀⠀⠀⠤⠀⠀⠀⡠⠶⠀⣀⣀⣀⣀⣀⡠⠤⠄⠀⠒⠒⠛⠛⠚⠒⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠢⢄⣀⠀⣀⣔⣀⣂⣱⡄⠀⠀⠀⠀⠀⠈⠳⢀⠀⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠒⠊⠉⠁⠁⠀⠉⠉⠉⠳⡄⠀⠀⠀⠀⠀⠟⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡜⢃⠀⠀⠀⠈⠁⠀⠀⠠⠏⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠤⠤⣀⣀⣀⣀⠀⣀⠆⠘⠀⠁⣂⣸
⠀⠀⠀⠀⠀⠀⠙⠁⣀⠀⡝⠃⠀⠠⠄⠀⢀⠂⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⠒⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⢂⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠙⠂⠀⠀⠀⠀⠀⠘⠀⠐⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣣⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠦⣴⠶⠖⠓⣞⠳⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⣤⠰⡎⠛⠫⢾⡱⢄⠈⠲⣕⡤⠀⠈⠳⣄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⣠⠄⡺⢏⠍⠃⠴⠀⠈⣠⠚⠝⢦⣐⠀⠈⠑⠠⣄⠀⠀⠀⠀⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⠔⠊⠉⠁⢠⡚⠗⣀⣊⣠⠔⠋⠀⠀⠀⠀⠈⠷⢦⡀⠀⠀⠙⠢⣔⠤⣠⠱⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡠⠞⠁⢀⠠⣔⠊⠃⢁⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠨⡓⢄⠲⣼⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡠⠊⠁⢀⠤⠊⠐⣋⡁⠀⠀⠀⠀⠀⠀⠀⢀⣉⠉⠁⠒⠂⣌⠳⡀⠀⠀⡔⢀⠱⣩⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠎⡠⣐⠕⢁⠐⣵⢛⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⢛⣷⣤⡀⠈⠀⠱⠀⠐⢠⠀⠆⢡⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡎⠰⣉⠊⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⡇⠀⡇⠠⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢇⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⡇⠀⡀⠀⠰⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠄⠀⣞⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡇⠀⠁⢠⢣⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣷⢽⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠒⠒⠒⠒⠢⠀⠀⠀⠀⢠⣿⠀⠀⠀⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⢸⡅⠚⣷⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡗⡄⠀⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⢸⡂⠝⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢲⡭⣻⢱⠀⠘⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⡘⡥⢚⡵⠂⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⣛⠭⣋⣿⢸⣿⡇⢸⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡇⡃⢛⡡⢚⠴⢪⠖⢛⠛⠲⣦⣄⣀⠀⠀⠀⢀⣀⢤⡶⡟⢣⡗⠊⢕⠺⠵⠟⡐⠛⠓⠀⡇⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠃⡇⠀⠔⡡⠒⡡⠒⣩⡆⠀⢠⠈⡩⠋⡰⠋⠙⠉⠔⠄⡐⢲⣋⠁⡈⠔⠛⠀⠹⠀⠄⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢠⠃⠀⢊⠄⠊⠔⠉⢐⡇⠀⠀⠈⠀⠊⡠⠊⠔⡁⠀⠊⡄⢪⠖⠩⠐⠀⠀⠀⠂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡈⡄⠀⠠⠖⡡⢊⠀⣈⠇⠀⠀⠀⠀⠈⠁⠊⠔⠄⢀⢂⠄⠘⠃⢐⠌⠀⡀⠀⠀⡀⠁⢀⠃⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠎⣴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠒⢤⡤⣄⡈⡔⣀⠀⡌⡸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡠⠊⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠙⢶⡇⠀⡰⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠄⠀⠀⢀⠀⠀⢠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠑⢆⠘⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠃⢀⠠⢂⡀⠐⣀⠄⢀⡀⠀⡀⠀⡀⠀⡀⠀⣀⠀⠀⠄⠀⡠⢂⠄⡀⠀⡀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠐⢊⠠⠂⢁⠔⠊⡠⠒⠁⠔⢈⠔⢊⠔⢉⡴⠊⡠⠊⢀⠔⡡⠊⡡⠊⡡⠊⡤⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠈⠠⠊⠠⠂⠡⠀⠁⠔⠁⡠⠊⠀⠖⡡⠊⠠⠊⡀⢈⠴⢁⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠤⡈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢉⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠡⠇⠀⠀⠀⡘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠳⣆⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠤⠄⣀⠤⠚⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⣠⠶⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⡾⠁⠀⠀⠀⠙⣀⣀
⠀⠀⠀⠀⠀⠀⣀⠼⣄⠀⠼⠁⢀⠄⠂⢀⣀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡤⠠⡀⠀⠀⠀⠁⠀⠀⠃⠀⣀⠀⠀⣞⠁⠈⣷
⠀⠀⠀⠀⠀⠐⣇⣀⡠⠇⠀⢀⡴⠚⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠖⠚⠛⠛⠒⠦⣁⠀⠈⢠⠞⠉⠳⡄
⠀⠀⠀⠀⠀⠘⣁⣈⡵⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠡⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡱⠒⠚⣅
⠀⠀⠀⠀⠀⢸⡅⠈⢀⡞⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠉⠙⠸⡇⠀⠀⠀⠀⠀⠈⠑⠒⠋⠁⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠙⠲⠤⠶⠂⠀⠀⠀⠀⠀⢸⡤⡤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣁⠰⠁⠀⠘⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠶⣄⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣠⠶⠋⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⢷⡀⠀⢀⡴⠟⠛⠛⢶⣄
⠀⠀⠀⢀⣤⡶⠶⠶⠶⣄⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⡟⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢠⠟⠁⠀⠀⠀⠀⠈⢷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡶⠶⠶⢦⣤
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠏⠀⠀⠀⠀⠘⣷⡄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⣷
⠀⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠲⠶⠶⢦⣤⣿⡀
⢰⡇⠀⠀⣠⠶⠋⠈⣿⠀⠀⠀⣤⣀⣤⣀⣠⣄⣀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣲⠀⠀⠀⢹⡇
⢸⡇⠀⢰⡏⠀⠀⠀⠉⣷⠀⠀⢉⠉⠉⠀⠛⠁⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣉⣉⣀⣀⣀⣸⡇
⢸⡇⠀⠀⠻⢤⠶⢦⡤⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣤⡤⠴⢺⡇⠀⠀⠉⠉⠉⠉⠉⢹⣧
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠶⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠃⠀⢀⣠⣼⣇⡀⣀⠀⡄⠀⠀⠀⠀⣿⡀
⠀⣿⠀⠀⠀⠀⠀⣾⡖⠻⡷⠚⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣤⡤⠖⠛⠉⠁⢸⡏⠙⣿⠛⡿⠖⠀⠀⠀⢹⡇
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣧⣠⠤⠴⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢰⡟⠁⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠈⠙⢶⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠋⠉⣹⠿⢦⣄⣀⣠⣤⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⡄⠀⣸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⣦⣀⣠⣿⡶⠖⢿⠋⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠚⠋⠹⠇⠀⠈⠃
⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⢻⡆⢀⡴⠞⠳⣤⣤⠞⠁⠹⣦⡀
⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠉⠻⠷⣶⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢠⡶⠛⠿⠀⠀⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠳⠤⠿⠳⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀
⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠈⠀⡀⠀⡄⠁
⠀⠀⠀⠀⠀⢠⡘⣄⢳⣰⡆⡆⠇⡀
⠀⠀⠀⠀⠀⠀⠓⠘⣀⣫⡽⣤⣾⡥⠀⠀⠀⠀⡤⠠⠇⠴⣄
⠀⠀⠀⠀⠀⠀⢉⣐⠛⠁⠀⣼⣿⣷⡀⠀⡄⠂⠀⠀⡀⠀⣥⣾⣧⡇
⠀⠀⠀⠀⠀⠀⠿⠷⡷⠶⣿⣿⣿⣿⣶⣶⣶⡷⢶⣿⠿⣶⣿⣿⣿⣿⣷⣤⣀⡀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⣾⡿⠋⠋⠁⣀⠁⠈⢁⣂⡙⣿⣿⣿⣯⣝⠛⠻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠂⣠⢄⣪⣿⠋⠀⠀⠀⣿⣿⠀⠀⠀⠙⠛⠻⢿⣿⣿⣯⡀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⡀⡘⢋⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣴⣄⣀⣀⣼⡇⢸⣿⠟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠀⢀⣠⣿⠿⠛⠛⢣⢎⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠁⠀⠀⣤⣾⠟⠁⠀⠀⢀⠛⢀⡿⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⢀⡶⠋⠀⠀⠀⢰⠀⠘⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢏⠣⠀⣀⡾⠋⠀⠀⠀⠀⠀⠈⠂⠀⢇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⣴⠟⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣏
⠀⠀⠀⠀⠀⠀⠐⡀⠉⠐⣿⢏⣀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢀
⠀⠀⠀⠀⠀⠈⠰⣿⣿⣿⣷⠞⢿⣿⣷⣴⣆⢁⣅⡓⡀⣀⠀⡀⠀⠀⣀⣹⣿⡀⢀⢀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠋⠀⠚⠓⠘⠘⠛⣿⣭⣿⣏⣻⣿⣶⣴⣶⣯⠙⣧⢁⡒⢥⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠽⠛⠙⠙⠛⢿⠿⠆⠻⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣾⣿⣿⣶⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢈⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⢆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣠⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣿⡿⢟⣛⡍⠹⣿⠟⣛⡻⢿⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣏⡘⠻⣨⣿⣿⣷⣦⣵⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢇⣿⣿⣿⠿⢿⣧⣶⣿⣿⣿⣿⣿⡟⣼⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠉⢸⡟⣡⣶⣿⣶⣶⣭⣭⣛⣛⣫⡅⠆⠈⠉⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠈⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠘⣿⣷⣶⣾⣿⣿⣿⣶⣾⣿⠇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣭⣭⣽⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⡔⢶⣶⠖⣶⣶⡶⢢⣾⣿⣐⣛⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡶⠡⣦⡍⣩⣶⣿⣿⣿⣿⣟⣛⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣶⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠟⠛⠛⠛⠛⠛⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡀
⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣼⡟⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢻⣷⣾⣿⣿⣄⠀⣀⣤⣶⠿⠛⣿⡄⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢠⣿⠉⠛⠛⠛⠋⠁⠀⠀⠘⣿⣤⣿⠋⣤⡙⣿⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⣿⡀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢿⣿⠀⣿⠇⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠻⣷⡘⠟⠛⠛⠿⠀⠀⠀⠀⠀⠉⠀⢀⣼⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⣿⠇⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⢿⣧⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⢰⣿⡁
⠀⠀⠀⠀⠀⣸⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⣴⡿⠛⣿⡄⠀⠀⠀⢀⠟⠀⠀⢀⣼⡷
⠀⠀⠀⠀⠀⠻⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⢿⣦⠀⠀⠈⠀⠀⢀⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣦⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠁⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠴⠾⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣴⣶⠿⠿⠿⠿⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣐⢅⡪⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡊⠅⡂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⠒⠒⠛⠚⠓⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⡨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⠉⠉⠉⠀⠉⠉⢉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⢲⠤⠤⣍⣀⡂⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⠈⠀⠀⣠⠔⠋⢍⢻
⠀⠀⠀⠀⠀⢀⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡠⠤⠒⠉⠉⠉⠉⣑⢺⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢱⠤⠏⠀⢀⠸⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⣗⣒⡻⠟⠛⠉⠛⠛⣲⠾⠀⠀⠀⠀⠀⣀⠈⢒⠶
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⢲⠦⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣤⠴⠒⠭⣑⡇⠀⠀⠄⠀⠀⠨⠖⠺⠄
⠀⠀⠀⢀⡔⠑⢲⠂⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⣀⣀⡠⠓⡋⠻⡀⠀⠉⠀⡀⠀⠅⠀⠀⠐⠂⠀⠀⢠
⠀⠀⠀⠀⠨⠎⠘⠂⠀⠁⠂⠀⡴⠊⡏⠉⡇⠈⣇⠈⠣⡀⡞⢄⠈⠢⣄⠀⠀⠲⠀⠀⠀⠠⠄⠐⢆⠁⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣼⠘⣄⣸⠞⡽⡄⠘⢄⠀⠙⡇⠈⡧⡀⢈⠓⢍⣠⠀⠀⢄⠀⠀⠀⠚⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠘⢄⡜⠁⡜⠀⡟⢄⠈⠳⣔⡇⠀⡇⠈⠢⡑⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢳⣼⣠⣴⣤⣇⣵⣾⠬⢷⡥⢷⡭⠴⣚⠫⢋⠠⢑⠢⢐⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠁

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠐⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⡠⠊⣷⠤⠤⠤⢖⣲⣿⣯⡛⠶⣄⠀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡠⠀⠠⣄⠉⠰⣀⡴⠿⠋⠁⠀⢻⠀⢂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠀⢀⠒⢪⡤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠆⠀⠀⠣⣀⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠞⠀⠀⢀⠀⠈⠀⠄⠒⠐⠒⢄⠀⠀⠀⠀⠀⠰⢀⠀⡀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢀⣀⠈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢈⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠹⣄⢨⡀⠐⠀⠑⠄⠀⠀⡠⠂⡠⢀⡠⠚⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⠚⠓⠭⣀⠀⠀⠀⡄⠀⡠⢚⠰⠛⠉⠐⢄
⠀⠀⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠉⣆⠀⠀⠀⠈⠃⠀⠀⢀⠔⢉⠀⢂⠀⢀⠉⠀⠙⠀⢁⠔⠁⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢧⣠⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠼⡠⠇⠤⡼⢠⠇⠊⠈⠐⠖⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠓⢢⡀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠀⢀⠟⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢠⠊⠁⠀⡀⠂⣠⠃⠀⠀⠄⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⠀⠈⠃
⡔⢳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠈⠉⠀⡀⠷⠂⠀⣼⠀⠀⠀⢀⡤⠠⣀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠈⠓⠢⢤⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠔⠁⠀⠀⢰⠚⠉⠀⠀⠀⠈⠡⡀⠈⠀⠀⠀⠀⠀⠿⠤⠶⠄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢸⠀⠀⠀⠀⠀⠉⠋⠈⢁⠀⠀⣀⠼⠀⠠⢤⠈⠉⠛⠃⠒⠖⠖⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣘⣱⠀⠀⠀⡀⠀⠆⠀⠛⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⣀⠀⣀⡀⣀⣀⡤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⢄⡀⠀⠈⠷⠠⠤⠴⠒⠂⠀⠀⠀⣠⢃⡀⠀⠀⠀⠀⢀⡠⠞
⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠑⠒⠋⢆⠀⠐⣈⠥⠀⣠⠔⠳⠚⠛⠓⠒⠒⠚⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠥⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠲⠶⠞⠛⢝⡻⢖⢤⡤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⡶⠖⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠂⣠⣾⣿⣯⣽⣻⢦⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠹⣿⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⢏⡼⠃⠘⣧⠀⢹⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⣙⣿⣿⣿⣿⣿⣿⣿⠟⢃⡾⠁⢀⠀⣻⡇⠀⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⢾⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣽⠇⢰⡞⠁⢠⣿⠀⠘⡇⠀⠂⠀⣿⣿⣿⣿⠱⣾⣿⣶⣦⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠻⣿⣿⣿⣿⣯⣿⣿⣿⣧⣾⣿⣿⠀⢸⠁⠀⣿⡏⠀⠀⠀⠀⢀⠞⣿⣿⣿⢋⣿⣿⡀⠁⡙⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠀⠀⣿⠇⠀⠀⠀⠀⠈⢠⣿⣿⣿⣿⣾⣿⣾⡿⠓⠻⡿⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠄⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣿⠧⠀⠀⡠⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣄⢀⠔⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠹⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⡇⠉⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⡀⠛⢷⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⣿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡶⠤⣌⣹⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣄
⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣷⣾⣷⣿⡦⠀⠀⠀⠀⡀⠀⠀⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢸⣿⢿⣶⣿⣿⣿⣿⡇⠀⠀⠀⢺⡿⠇⠀⠐⣹⣦⡆⢸⣆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀⠀⠀⠀⢀⣦⡤⠀⠀⣌⣉⡈⠙⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠒⢿⣿⣷⣄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡍⠀⠘⠀⠀⠀⠠⣤⣀⣤⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡃⠒⠀⢉⣽⣿⣏⢙⣉⣡⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣠⣄⠀⠀⢤⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠹⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠐⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⡟⣿⣿⣿⣿⣿⣻⣿⣿⣿⡏⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠹⠿⠿⠿⠿⣿⣿⣿⣿⣅⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠲⠆⠀⠀⠀⠀⣠⣶⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠛⣿⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣧⣭⣭⠯⣭⣥⣥⠤⠥⠽⠿⠿⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⣐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⠀⠀⡞⠀⣤⣤⣌
⠀⠀⢀⣀⣠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡧⠰⠜⠛⠋⠉⠀⠀⢀⠗⢾⣿⣿⠿⠇
⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⠀⠻⠟⠁⠀⠀⠀⣶⣾⣧⡀⠠⠈⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠘
⠀⣿⣿⡟⠙⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⡏⠀⠀⠀⠀⠀⠸⠟⠹⢿⣿⣷⠾⡧⠀⢀⣠⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡄⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣤⡟⣀⠀⠁⠀⠀⡀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢦⣄⠂⣠⣤⣤⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣶⣦⡀⣴⠟⢀⡤⢠⣶⣆⣶⣇⣤⣤⣴⣲⠦⣽⣿⣿⡿⠀⢷⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡿⣿⣿⣿⣿⡏⣿⣿⣷⣼⣿⠽⢿⡿⢻⣿⣛⣿⣸⣭⣟⣿⡛⣿⣷⣿⡃⢀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠞⣥⣾⣿⣻⣿⣿⡷⣧⣿⣿⡟⠀⠒⠀⠉⠀⠑⠛⣿⣿⣷⣏⠻⣿⣿⣼⣿⣿⠩⠖⠠⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⣬⢼⣿⣿⣿⣿⡃⣱⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡿⣛⡽⠀⠠⣬⣻⣿⣿⣿⣿⣄⠀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣔⣵⣿⣿⣿⣿⡗⡦⣾⠿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⠌⣿⣞⣇⠻⠘⣿⣿⣹⡕⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⡽⡿⠋⠲⣿⣻⣾⣷⣒⡞⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣬⠣⣍⠛⠅⡾⣿⢻⣿⣧⣼⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⣜⡞⢠⢴⡿⢿⣨⣿⣿⡿⣦⣦⣄⡀⠀⡄⠀⠀⠀⣠⣿⣿⣿⡿⣷⣶⡄⣾⣶⣿⣾⣿⣿⣗⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠾⠀⠈⣸⣿⣿⣟⣛⣻⣿⣿⣶⣧⣴⣿⣿⣿⣿⠛⣛⢹⣿⣿⣿⣯⣯⡿⣿⣿⣿⣿⣏⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢙⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⣿⣿⠿⠿⢻⡿⠟⣻⣯⠟⢇⣞⢷⣿⡿⣿⡿⣿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⡶⣜⢹⣿⣿⣿⣿⣿⡿⡏⠀⠀⣽⠏⠀⠛⠦⣤⣾⣿⠏⠀⠻⣿⣧⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠸⢸⡇⣍⠄⠈⠙⢋⢟⠉⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠉⠉⠀⠘⡇⠙⣿⣿⣿⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠴⢓⠻⡄⠀⠀⠀⠀⣠⡴⠁⠀⠀⠛⠀⢢⡀⠀⠀⠀⠀⢴⣶⠙⠀⡀⢨⣿⠏⠉⠁
⠀⠀⠀⢀⣠⠴⢠⠂⠈⠁⠝⠓⠶⢼⣳⡄⠀⠀⢸⣧⣏⠕⠂⣠⣯⣶⡾⢧⠠⣀⠀⠀⠁⠀⠀⣰⣛⣿⡁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡜⠰⢛⢀⣋⠛⠻⠛⠟⠉⠁⢀⡹⢤⠀⠠⠀⠀⠀⣿⠟⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠈⢿⣻⣿⠟⠛⠛⠿⠓⠗⢂⣶⣷⣤⣾⡿⢗⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣶⠀⠀⠃⡁⢹⣧⠀⠀⠀⠈⠃⠰⠀⠀⠀⢱⢛⠋⣀⠀⢳⡇⡈⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⢄⠀⠉⠺⣙⠆⠀⠀⠀⠀⠈⠀⠀⢠⡀⠈⣰⠋⢠⣯⠛
⠀⠀⠀⠶⠆⣠⡀⢐⠶⠂⠂⠄⣀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠁⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠀⡠⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢳⡦⠈⠉⠁⠠⢀⡁
⠀⠀⠠⠄⠾⠋⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠈
⠠⣖⣤⠚⠉
⢸⢛⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠠⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠀⠀⢰⠂⠀⠀⠂⠀⢀⠀⠀⣀⠠⡈
⠀⠀⠀⠀⠀⠐⠠⠊⣄⡀⠇⠐⠀⠀⠀⢀⡤⠀⠀⡆⠀⠀⢀⠀⣦⡀⠀⡎⠀⠀⢈⠀⠀⢁⣀⠎⠀⡀⢀⡘⡀
⠀⠀⠀⠀⠀⠰⡠⡞⠀⠀⠢⡤⠒⣞⠉⠈⡃⠆⠀⢦⡀⠀⣠⡷⠦⢴⠞⠀⠀⠀⢸⡀⠀⠀⠀⣴⠀⠀⠀⣧⢰⡀
⠀⠀⠀⠀⠀⠰⠎⠢⠤⣵⠞⠀⣤⣀⣀⣰⣧⣊⣀⣪⡴⠛⠁⠀⢒⡉⠀⠀⢀⡠⠃⠈⠓⡲⠋⠐⢀⡠⢷⢒⣲⡇
⠀⠀⠀⠀⠀⠀⠑⢔⣆⡸⠦⠤⣠⠀⣩⢟⡫⣉⡿⣻⡴⢴⣖⡤⠤⠖⠒⠒⢉⣇⣀⣀⣉⡵⣢⡼⠋⢠⢮⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣶⣶⡾⣳⣾⠞⣥⡔⢫⠀⣺⡟⡦⣤⡐⢢⣚⣉⣀⣠⣤⠀⢱⣥⣤⣴⣯⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⣷⠿⣾⣆⣿⢶⠷⠸⣞⣀⣋⣿⣀⣭⣍⣛⣛⣛⣯⣵⣿⣛⣉⣽⣚⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢿⣿⡻⣥⣟⣒⠒⡿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠛⣿⠿⣿⢿⣿⢾⢿⠋⡇⢸⢿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⡟⢠⡧⣼⡇⠘⢸⠀⡇⠈⠘⣿⠉⠿⡿⠿⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢸⡇⠀⠈⠀⠇⠀⠀⣇⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡄⢠⠀⠀⡄⠰⢀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⡐⢀⠇⠂⡐⠁⠀⡰⠃⠞⠠⠂⢀⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠰⠊⠁⣿⣈⡠⠀⢢⡜⡑⢁⡜⢁⡐⢡⠞⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⣿⢏⡱⢀⠃⠜⡀⠋⣴⠏⣰⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠆⠀⠀⠀⡀⢠⠇⡊⠀⠉⠀⠞⢁⢜⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠄⠠⢠⣯⠏⠀⠘⠀⠀⡀⢫⠋⡠⠂⠀⠀⠀⢈⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠹⡀⠀⠀⢰⠀⠀⢸⠀⠀⡏⢠⠎⢀⣿⠃⠀⠀⠀⢠⡾⠧⠋⠀⡄⠋⠀⡰⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡄⠰⠀⢀⠀⢷⠀⠀⣼⠆⣆⣾⣀⡸⣡⣿⠞⣼⠃⡀⠈⣠⡑⣤⠈⠀⠀⠄⠀⣠⠞⢁⡐⠀⠠
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⢧⡿⣄⢡⡌⣦⢸⡄⣶⡿⠀⣿⣿⣯⣿⣿⠋⣾⣯⠆⢠⢈⠁⣉⠴⣡⠎⣀⡴⠞⠁⠐⢁⠄⠈
⠀⠈⠒⢦⣄⠀⠀⠀⠀⠀⠈⠳⣤⣿⣽⣿⡄⠛⠘⣿⡿⣿⠃⠀⠻⠟⢿⡿⠁⢰⣿⠇⣰⠟⢠⠚⣉⠼⠛⣉⠥⠒⠁⠄⠊⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠛⢦⣀⠲⢤⣄⣠⣙⣿⣟⢿⣧⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠃⣸⣥⡜⣡⡴⣞⣭⣿⡋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢉⠛⢦⣍⡛⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠾⠟⠛⠛⢁⣾⣟⣉⣩⡁⠤⠄⠑⠲⠦⣡⡴
⠀⠀⠀⠀⠀⠀⠀⣀⣈⣛⣿⣷⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠐⠂⠤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠉⠅⠀⠐⠒⠲⣿⣿⡷⠴
⠀⠀⠀⠐⠒⠤⠤⠤⣤⣝⣛⡛⠛⠀⠀⢠⣾⡿⠂⠀⠀⣀⣤⡴⠷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣁⠤⠤⠈⠉⠁⠉⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⠛⠀⢀⡿⠋⠀⢀⣴⠟⠋⠉⢀⣠⡀⠙⠳⣤⡀⠑⠀⠀⠀⠀⠀⠀⣠⣤⣤⡔⠀⠀⠀⠀⣀⣀⡚⠛
⠀⠀⠀⠀⠀⠉⠙⠿⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠀⣀⡤⠞⠛⠋⠉⢠⣄⠈⠑⠂⠀⠀⠀⠀⢀⣈⣉⣉⣉⢉⠉⠉⠉⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠐⢶⣶⢷⣶⣶⠶⠀⠀⠀⠀⠀⠠⠖⠋⢀⣠⣴⡶⢆⠉⠛⠿⣦⡀⠀⠀⠀⠀⡘⠛⠛⠛⠻⠥⠤⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⣋⣭⣶⡶⠀⠀⠀⠀⠀⠀⠀⠒⠉⣀⣤⣤⣾⣿⣦⣀⠈⠁⠀⠀⠀⠀⢠⣿⣿⣿⣷⡲⠒⠒
⠀⠀⠀⠀⠀⢀⣠⠶⠾⣛⡉⢉⣁⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣴⣶⣬⣤⢉⣓⡢⠤
⠀⠀⠠⠀⣉⣡⣤⠶⢟⣭⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠸⣿⣿⣿⡟⠋⠭⣉⠁⠒
⣀⠀⠄⠊⡙⠋⠡⠼⢋⣽⣿⢿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣿⣿⡇⠀⠀⠐⣬⣿⠿⣿⣔⠲⢤⣄⠀⠐⠒⠆
⠀⠀⠒⠀⠀⠁⣠⣴⣿⣿⣽⣿⡇⠀⠀⠀⢸⣿⡟⣫⣭⠘⠿⠟⠁⠈⠀⠀⣀⣤⣼⣿⡇⠀⠀⠀⣿⣍⠳⣦⣀⠀⠠⡈⠀⠀⡀
⠀⠐⠉⢀⠐⣨⠋⣴⣿⣿⣿⣿⡆⠀⠀⠀⢼⣿⣾⣿⣿⣏⠀⣴⣴⣷⠀⣶⣿⣿⣿⣿⡇⠀⠀⠀⢹⡛⠻⣤⢈⠓⠠⠈
⠀⡀⢚⡡⠊⠀⣨⡾⢋⣿⣿⣿⡇⠀⠀⠀⢸⣿⡟⢴⣿⡟⢠⡿⣿⣿⣆⣿⣿⢿⡟⠙⣿⠀⠀⠀⢸⣟⠢⣀⠁⠄
⢀⠄⠈⠀⢀⡾⠁⠀⡳⠟⢉⣾⡇⠀⠀⠀⢺⣿⣻⠀⢿⣷⡼⠇⢹⡉⠿⠈⠁⠀⠹⡔⣽⠀⠀⠀⠘⣯⠙⢤⡁⠄⠀⠀⠐
⠊⠀⠈⣡⠊⢀⡶⠃⠁⡴⡾⣿⡇⠀⠀⠀⢸⣿⣿⣶⣾⣿⣷⢀⡄⢰⠀⡆⣀⠰⠴⠟⠁⠀⠀⠀⠀⣙⠣⠀⠙⢦⡀⠀⠐
⠀⣴⠟⠁⠀⡜⠂⠀⠀⡄⢰⣿⡇⢴⠀⠀⠈⠛⠛⠛⠿⠿⣿⣿⠧⠾⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠈
⠀⠀⠀⢀⠞⠀⠀⠂⠈⢠⣾⣿⡇⣼⣄⣀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣟⠃⠀⠀⠀⠀⠀⠂⢀⠀⠐
⠀⠀⣤⠎⠀⠀⢠⠎⢠⠋⢸⣿⣷⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢺⣿⣶⣿⣿⣿⣿⢿⡿⠿⠉⠋⠀⠀⠀⠢⡀⠀⠑⠄⠁
⠀⠐⠁⢀⠆⢀⠋⢀⠎⢀⣿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠀⠉⠻⣿⠇⠘⡆⢳⠀⠀⠀⠀⡀⠀⠀⠀⠢⡄⠀⠑⠂
⠈⠀⠀⠈⢠⠂⠀⠈⢀⡞⠀⠿⣹⣟⣿⣿⡟⠀⠂⠀⠀⠀⢀⣴⣿⣿⡆⠀⠀⠀⠁⢦⡀⠁⠀⡆⠀⠀⢀⠀⢀⠀⠀⠀⠘⢶⣄
⠀⠰⠃⠀⠃⢠⠀⠀⡾⠀⠘⠀⠛⠈⢹⣟⣡⠀⢀⠀⣄⢀⣾⣿⣿⣿⡆⡀⠀⠀⠀⠈⠃⠀⠀⢻⡀⠀⠀⠆⠀⠱⡀⠀⠀⠀⠹⣷⡄
⠀⠀⣴⠃⠀⠀⠀⡰⠃⠐⠀⠘⠀⠀⣸⣿⣟⣠⡏⣸⣷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠈⠻
⠀⠐⠁⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡄⠀⠀⠈⢄
⠀⠀⠀⢀⠎⠀⠀⠀⠄⢀⡇⠀⠀⠀⣿⠋⠛⣿⡟⣿⣿⣿⠿⣿⣿⠘⣿⣷⠀⣤⠀⢀⠀⣦⠀⠂⢹⡀⠀⠐⠠⠀⠀⠀⠀⠢
⠀⠀⢠⠟⠀⠄⠀⢠⠀⣼⠡⠀⠀⠀⠏⠀⠀⠙⡇⣿⣿⣿⠀⣿⣿⡄⢹⣿⡀⢸⡆⠘⡄⠸⣇
⠀⢠⡟⠀⠀⠀⠀⡄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⢻⣿⠄⢈⣿⡀⠘⣿⣷⣾⣧⠀⢷⡀⢿⣖⠇
⠀⡾⠀⠀⠀⠀⢰⠃⢸⠃⠀⠀⠀⢀⡀⢠⠀⠀⠀⠻⠀⠋⠀⡏⠛⠁⠀⠛⠛⠉⢻⡶⢾⡟⠛⠋
⠀⠀⠀⠀⠀⢀⡌⠀⠈⠀⠀⠀⠀⠨⡇⢸⠀⢸⠂⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠛⢿⣟⣓⣶⣶⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⢀⣙⡛⣶⠟⣉⣭⣿⡟⠻⣥⣅⡉⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠐⠶⠒⣶⠮⡿⢡⣾⣟⡳⠮⣍⠛⠢⠈⠁⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢉⣩⣭⡁⠛⢧⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠲⠶⠶⠶⢤⣤⣿⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠙⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠟⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢦⡀⠙⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⠀⠰⠶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣭⠛⠛⠛⢶⡄⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡷⠶⢶⣄⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡂⠀⠀⠀⠀⠀⠀⠈⠛⠶⣾⡟⠀⠀⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠖⠀⠀⠀⠀⠀⠀⣀⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⡇⠀⠀⠀⠀⠀⠛⢁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⣞⢻⣦⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⣾⡛⠛⠹⣇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣆⠹⣦⠀⠙⠷⠶⣦⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⡷⠦⣤⣤⣤⣤⣤⣤⣤⣤⠴⣾⣿⣏⡙⠷⣦⢹⡇⠈⣷⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢤⣀⠀⠀⠀⠀⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡏⠙⢳⣿⣿⠃⢀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠙⣿⠀⠀⣸⠃⠙⢦⣀⣀⠀⠀⠀⠀⠀⣀⣠⡴⠃⠸⣇⡀⠀⠈⢻⣄⡼⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠛⢀⡼⠁⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠈⠻⢦⡶⠀⣿⢣⡟⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢿⣾⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⢸⡏⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸⡇⢸⣷⣄⣙⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣿⠟⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⢆⡘⢣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠀⠀⢠⣄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣿⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠙⡍⠁⠒⠒⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠄⡄⢀⣀⣘⡄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠞⠃⠀⣠⠀⠀⠀⠠⠀⠀⠀⡿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠗⠠⠀⠀⠀⠀⢀⠖⢁⠤⢒⣾⣒⡦⣄⠑⣄
⠀⠀⠀⠀⠀⠀⣿⣽⠀⠘⠀⠛⠛⠁⠀⠈⠁⠀⠇⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠙⣿⣶⣤⣀⠀⠀⠊⢰⠃⠀⠀⠻⣿⣿⣮⣿⠉⡆
⠀⠀⢦⣿⠶⠆⠒⢤⣤⣤⡼⠷⠿⠶⠷⠷⠷⠶⠶⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠈⢙⡿⠟⡿⡿⠂⡀⢺⢀⡢⠠⣤⡟⠈⠀⢸⠀⡀
⠀⠀⠀⣷⠀⠀⠀⠰⠀⣶⣅⣀⣐⣶⣦⣤⣤⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢿⣿⣿⡋⠀⠀⢠⠇⠀⢣⠘⢇⠛⠇⢸⡇⡀⡠⠶⢤⠃
⠀⠀⣐⣷⣴⣾⣧⣀⣀⣹⣇⣙⣉⣹⣿⣟⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣽⣿⣥⣤⣴⣉⣀⣀⣀⣑⢀⡀⣃⣠⣿⣋⡀⠔⠁
⠀⠀⡯⢿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⢿⣷⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⢿⡏⠀⣿⣿
⠀⠀⠁⠉⠉⠉⠉⠙⠻⣿⣿⣿⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣻⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠀⠒⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠂⠉⠉⠉⠑⠢⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⣀⣀⡰⢀⡜⠉⠁⠀⠀⠀⠈⠢⣀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠳⠖⠘⠃⠀⠀⠈⠀⠀⣀⠖⠉⠀⠀⠀⠸⢘⢴⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⢀⡀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠉⠀⡰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⡀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⡦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠙⠁⠀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⡐⠉⢠⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣀⠀⠀⠀⠀⢀⡀⢶⠁⡑⠦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢠⠗⠒⠉⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⡀⠀⠀⢸⠀⠠⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⣡⠀⠀⠀⡀⠀⢐⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠃⡹⡄⠀⠀⣿⠒⠼⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠀⠠⠠⣤⢤⠞⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠀⠀⠈⢹⠧⠭⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠈⠀⡞⠀⠀⠐⠒⠒⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠰⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⡀⠀⠀⠀⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠲⠲⡆⠦⠀⣀⢀⡀⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠎⠉⡏⠛⣷⣤⠀⣷⠰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠁⠀⢀⠀⣖⡴⠄⠀⠲⠙⠦⠉⣈⣉⣁⡼⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠉⠁⠈⠖⠀⠣⠤⠠⠴⣄⡆⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠭⠕⠳⢆⣀⣀⣀⣀⣤⢦⣄⣤⠴⠾⠛⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡛⢶⣄
⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣆⢻⣧⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⢶⣆⣚⠛⠛⠻⠉⠙⢻⣿⣿⣿⡇⢸⣿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣷⡌⣷⡄
⠀⠀⢸⣿⣿⣿⣤⣾⣼⣿⣿⣿⡇⢸⣿⠀⢠⠋⠻⠿⠿⠿⠶⢶⣶⣶⣶⣤⣤⣤⣤⣀⣤⠚⢿⣿⣿⣿⣿⣿⣷⠸⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣧⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡁⠀⢯⣀⣈⣿⣿⣿⢀⣿⣦⣤
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣡⣿⠏⠀⡇⠀⢳⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⡿⣸⡟⠉⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣯⣵⣾⠟⠁⠀⠀⢸⠀⠀⢳⡄⠀⠀⠀⣾⣿⡟⠁⡤⠙⡄⠀⠀⠀⠸⠁⠀⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⣄⡀⠀⣿⠋⠁⠀⠁⠈⣧⠀⠀⢀⣇⡀⠀⡝⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⣄⣘⡇⠀⣀⡀⢈⣧⣠⡔⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣀⡀⠉⠑⠲⣤⡙⠛⢷⣄⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⠛⠆⠀⠀⠸⣷⡀⠀⠙⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⡀⠀⣿⣿⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⠟⠉⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⢰⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣾⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠰⣤⣀⣀⡀⠀⢠⠛⠙⠒⠉⠉⠀⢸⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⡸⣇⠀⠀⠉⠗⠁⠀⠀⠀⠀⢀⡴⠻⣓⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡏⠉⠉⠳⢤⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⠿⠟⠉⠉⠉⠉⠉⠓⠶⠶⠊⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣻⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣴⠶⣄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠻⢤⣈⣱⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⡞⠉⢉⣹⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠛⠿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⢹⣿⣿⣿⣿⡟⠀⣠⡴⠶⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠛⠆⠘⠟⠁⠀⠀⠀⠀⠀⠸⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⢸⣿⣿⣿⡿⠁⠀⠈⠛⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⡷⠀⣾⣿⣿⡿⠁⠀⢴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠻⠟⠁⠀⠭⢹⡟⠁⠀⠀⠈⢧⡀⢹⡆
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡼⠃⠀⠀⠀⠀⠀⠀⢰⣤⡾⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣦⣄
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⢁⣀⣈⣻⣄⠀⠀⠀⠀⠀⠀⠀⠘⣥⣈⣉⣹⡿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠉⢙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠙⢯⡿⣿⣿⡆
⠀⠀⠀⠀⣠⡶⠟⠛⠛⠋⠁⠀⠀⣀⣤⣀⣾⠁⠈⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡟⠀⢦⣴⠟⠀⢠⠿⣧
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣴⠞⠋⠀⠉⢛⡷⠲⠀⠀⠙⠿⣭⣛⣛⣳⣶⡶⠶⠞⢿⣏⠁⠀⢈⣷⠀⠀⠛⠦⣴⠋⢀⣾⠁
⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠰⡄⠀⣹⣧⣴⠟⠁⠀⢀⣠⣀⣨⡷⠛⠁
⠀⠀⠀⠀⠙⢷⡆⠀⢠⣎⣴⠏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⠁⠀⠀⣠⡾⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣄⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⢀⡼⢋⡴⠛⢷⡀⠀⠀⢿⡀⠉⠉⠛⠛⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⢠⡟
⠀⠀⢀⣴⠏⡠⠋⠀⢠⡾⠛⢦⣄⠈⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⣶⣤⡾⠁
⠀⣠⠟⣡⡎⡠⣎⢱⡟⠀⠀⠀⠙⠻⠃⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⣾⣁⠞⠁⣉⣧⠼⠋⠙⢦⡀⠀⠀⠀⠀⠀⢀⣴⠏⠉⠙⠛⠲⠶⣦⡄⠀⠀⠀⢸⡇
⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠙⠷⣤⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⣯⣀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃⠈⠉⢻⠞⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣍⠒⢤⣈⠺⣅⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣌⡓⠮⣁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣑⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⡠⢆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢣⢠⠛⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⡄⠀⠀⠀⠀⠀⣄⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⣷⣄⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠒⠠⠤⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠒⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠁⠐⠒⠃⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⡈⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢶⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢶⠃⠀⠸⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣄⠀⢶⣄⣀⡸⠛⢢⣤⣠⣴⣛⡦⣤⣤⣼⢳⣦⣤⠟⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⢀⠀⠀⠀⠀⢸⠙⠋⢠⡤⠔⠊⢁⣀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠉⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⡉⠭⢦⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠉⠉⢱⠖⠚⠁⠀⠐⠮⡠⢄⡀⢀⡀⠔⠀⠀⠢⠤⠀⠀⠀⣠⠖⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠈⠁⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣈⣵⡤⠇⠀⠀⠀⠀⠀⣀⡼⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁⣄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⡴⠛⣇⠀⠀⠀⣰⣧⠞⠀⠙⠒⠦⢬⣭⣉⣉⣻⠉⠉⠉⠉⣉⣀⣀⣀⣠⡼⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠈⠛⠛⠛⢡⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⢰⡖⠋⠉⠉⠇⢚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢧⠀⠀⠀⢸⡼⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⣇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣨⣦⡀⠘⠓⢦⠤⠼⠷⢤⡾⢻⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡤⠚⠉⠀⠀⠀⠙⠛⠛⠋⠙⠛⠒⠚⠁⡞⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠂⠉⠉⠉⠁⠒⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⡀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⣤⢮⠁⠓⠀⠀⠀⠀⠘⠃⢉⠶⣤⡀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣸⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⢻⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⠖⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠆⠀⠀⠈⢧⠀⢠⠊⣡⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠽⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⠰⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣿⠀⠀⠀⠀⠀⠀⡠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢿⠀⠀⠀⠀⠀⡼⢲⢶⣦⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣅⠀⠀⠀⠀⡟⢁⣎⡀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠻⣄⠀⠀⠈⠛⢋⣉⣉⡙⠛⠀⠀⠀⠀⠀⠀⠀⢠⢺⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣧⡀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠀⠁⢸⠦⡀⠈⠀⡇⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢩⠆⠀⠀⠀⠈⡿⠀⢠⣄⠠⠤⠔⠋⠁⠀⠀⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⢛⡁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⠜⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢸⡞⢣⠀⠀⠀⠀⠀⡠⠔⣁⠔⠁⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢸⢣⡈⢆⠀⣀⠴⢊⡡⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣴⠴⢇⠀⠀⠀⠀⠈⡄⠳⡈⠋⣠⠔⠋⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⢰⠁⠀⠘⣃⠀⠀⠀⠀⢱⡄⠙⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⡀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡤⠾⡁
⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠾⣄⡀⠈⠦⡀
⠀⣴⠁⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⣠⠞⠁⠙⣄⠀⠱⡀⠀⠀⠀⠀⠀⣰
⠰⠁⠀⠀⠀⠀⠀⠀⢀⡾⠀⡇⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⢶⠉⠀⠀⠀⠀⠘⢃⠀⢹⡀⠀⠀⠀⢠⠋
⡇⠀⠀⠀⠀⠀⠀⢀⠾⠃⢰⠀⣀⢴⠃⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⠀⣡⠀⠀⢠⡎
⡇⠀⠀⠀⠀⢀⡴⠟⠾⠦⠛⠁⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣀⠞⢱
⠀⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣛⣠⠞⠁⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠉⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠈⠛⠛⠛⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂