Tournament Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣾⠀⣠⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣷⡏⠸⠿⠿⠿⠆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡾⠛⠛⠛⠛⠛⠛⠛⠛⣡⣤⣤⣤⣤⣤⡌⠻⢷⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠻⣿⣷⣦⣤⣀⡘⣷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡀⡿⢃⣀⣤⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⣈⣛⣿⣿⣿⣿⣿⣿⡄⢀⣤⣤⣶⣦⣤⣍⠻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣯⣈⣽⣿⣿⣌⢻⣿⣿⣿⡇⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⢿⡇⣼⣿⣿⣿⣿⠉⢻⣧⣠⣿⣿⡄⣿⣿⣿⡇⣿⡿⠿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⢀⠀⠀⡀⠀⠀⢠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⠀⠀⠈⣆⠀⠇⠰⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⠆⠈⣷⣴⢛⡇⣰⢋⠞⣼⣣⡾⣡⠏⣴⣧⡞⣡⢏⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣴⠚⠁⠙⠋⠉⠑⠋⠚⣡⢏⡴⠋⠚⠋⠈⠚⣹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⠀⠰⠏


⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣰⡈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢙⣿⣟⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣈⡀⢈⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣦⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠈⠃⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠋⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠠⣶⣀⣸⣿⣿⣿⣽⣿⣿⣿⣿⣽⣿⣿⣿⣁⣰⡦⠀⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⠋⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡸⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⣧⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⣿⣿⣿⣿⣿⡿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠛⠛⠋⠉⠉⠛⠛⠓⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⣠⣤⣶⣿⡿⠿⠿⠛⠛⠛⠛⠛⠿⠿⢿⣿⣶⣦⣄⡀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⠀⣀⣤⠤⠤⠤⣄⣀⠀⠀⠀⠉⠛⠿⣿⣷⣄⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⡴⠊⠁⠀⢰⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠤⠤⣄⡀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠀⠀⠸⣆⡀⠀⠉⠳⣄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢣⠀⣰⠟⠙⢳⢦⡀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⣿⡟⣿⣭⣭⣿⣭⣽⣿⣭⣽⣿⣿⢦⡀⠠⠟⠁⠀⡠⠋⠀⣸⢄⣼⣿⣯⣭⣿⣭⣭⣿⣭⣽⣿⠹⣿⣇⠀⠀⢷
⠀⠀⠀⠀⠀⢀⡟⠀⠀⣾⣿⠁⣿⠒⠒⣿⠒⢲⣿⠒⢺⣷⠒⢳⣄⡀⠀⠀⠈⠀⠠⢖⣡⡞⠒⣿⡖⠒⣿⠒⠒⣿⠒⢲⣿⠀⢿⣿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡿⠀⣿⣍⣉⣿⣉⣩⣿⣉⣹⣿⣉⣹⣯⣉⣿⣶⣶⣾⣟⣉⣿⣏⣉⣿⣏⣉⣿⣍⣉⣿⣉⣹⣿⠀⢸⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⣿⠖⠶⣿⠶⠲⣿⠖⢺⣿⠒⢺⣷⠒⢾⡷⠒⢾⡷⠒⣾⡗⠖⣿⡖⠒⣿⠖⠲⣿⠖⢲⣿⠀⢸⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣷⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⣿⡇⠀⠀⡇
⠀⠀⠀⣀⣀⣠⣧⣤⣤⣿⣿⣤⣿⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣿⣤⣾⣿⣤⣄⣸⣇⣀⡀
⠀⠀⠈⠉⠉⠉⠉⣀⡀⢀⣄⠀⣠⣤⣤⡀⠠⣤⡄⠀⢠⣶⠀⠀⢰⣶⣶⣦⣴⡆⠀⣶⡆⣶⣶⣦⡄⠀⢠⣶⡀⠀⣤⡄⠀⠀⣀⡀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⢰⣿⠋⠙⣿⡄⣿⠇⠀⢸⣿⠀⠀⢸⣿⣤⣥⢹⣷⣤⣿⠃⣿⣥⣽⡟⠀⣾⢿⣧⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠇⠸⣿⣄⣠⣿⠃⣿⣀⣀⢸⣿⣀⣀⢸⣿⣛⣋⠀⢻⣿⠋⠀⣿⡋⣹⣷⢸⣿⣾⣿⡆⣿⣇⣀⡀⣿⣇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠛⠛⠁⠀⠛⠛⠿⠛⠿⠿⠿⠺⠿⠿⠿⠀⠸⠿⠀⠀⠿⠿⠟⠃⠛⠋⠀⠙⠛⠛⠛⠛⠃⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣻⡟⣛⢻⡏⡏⢿⠉⡍⣿⠈⠇⣿⡏⠸⣿⠁⡿⠉⣿⢩⣽⡏⢿⢻⣟⠻⣿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣿⣇⢻⣧⠹⢸⣇⢓⣸⣇⣆⣿⣤⣆⣿⣡⣆⣿⣠⣠⣰⣏⠰⣿⢣⡀⣾⡏⣸⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣾⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⢿⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⠀⢠⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠋⢿⣿⡭⠁⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⣼⣶⠠⠶⠄
⠀⠀⠶⠑⠐⣁⣠⡇⢈⡏⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡀⠶⠶⠆⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣸⣏⠉⠉⠉⢱⣷⣶⣷⣭⣇⣀⣤⡶
⠀⠀⠐⠈⠈⠉⠉⠀⠘⠛⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣴⣶⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡟⣴⣾⣟⣷⣍⢿⣿⡇⣿⣿⠷
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠻⠿⠋⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢸⣿⣿⣼⣷⣿⣿⣿⡗⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠶⣶⣶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⢀⣀⣀⣀⣀⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠎⠉⠉⠉⠩⠋⠉⠁⠩⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⠆⣛⡛⢸⣿⡆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢰⡦⠀⠶⠶⠆⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⢈⣍⠀⢿⡇⠈⣭⠀⢿⠇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⣄⣄⣄⡀⠀⠀⠀⢀⣄
⠀⠀⠀⠈⠉⣁⣤⣤⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡂⣶⣶⣆⣴⣦⢶⣶⣦⢰⣄⠀⠀⠀⠀⠀⠀⠘⠿⢇⡀⠀⢀⣨⣽⡿⣭⣅⠀⠀⡐⠿⠋
⠀⠀⠀⠀⣴⣿⣿⠧⡞⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣏⣾⣿⣿⠷⣞⢻⣷⢳⡍
⠀⠀⠀⠀⠿⠿⠛⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⢿⣿⣿⣿⣿⣿⠿⠚
⠀⠀⠀⠀⠀⠠⣤⣴⣤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢀⠦⠿⠤⠦⡶⡶⠴⠶⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⠉⠀⠀⣀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⡀
⠀⠀⠀⠀⠀⠘⠾⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣉⣭⡁⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣿⣿⣿⡇⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⡀⢸⣿⡏⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢧⡘⣿⣿⣿⢱⢇⡼⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤
⣀⡀⠀⠀⣀⡁⠀⠶⠆⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠿⢰⡏⠉⣿⣿⠋⢩⣦⠻⢀⡀
⠈⣿⣿⡆⣿⡏⣾⣿⣿⡞⣿⢱⣶⣿⣯⠀⠀⠀⠀⠀⠀⠀⢈⡃⣶⣘⣀⠀⢹⡇⠀⢀⣇⢶⢈⡁
⠀⠘⠛⡿⠜⠇⣿⣿⣿⡷⠛⠾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣿⣿⣰⣿⡇⣆⣿⣿⠇⠛⠃⠀⠀⠀⠀⠀⠀⠘⠑⠂⠒⠂⠀⠀⠀⠀⠀⠀⠒⠒⠐⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠙⠛⠋⠙⠉⠁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⣀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⣀⣹⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣾⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⣿⡿⠿⠟⡃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠐⣿⣿⣭⣯⣽⣿⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣡⣾⣿⣦⣤⣷⠀⠀⠀⠀⠀⠀⠀⢠⣬⣯⣤
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⣾⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⡟⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏
⠀⠀⠀⠀⠀⢸⡇⣾⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠸⠿⠿⠿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣏⡉⠻⣆⠀⠀⠀⠀⣼⢯⣼⣦⣤⣄
⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣷⣾⡄⠀⠀⠀⠀⣸⣿⡿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣿⡟⣵⣿⢿⣷⣝⢿⣿⣿⣿⣶⣶⣶⣶⢏⣷⡿⢿⣶⡝⣷⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⢀⡀
⠀⠀⠀⠀⠉⠸⣿⣚⢿⣺⣿⠘⠛⠛⠛⠛⠋⠛⠋⢸⣿⡺⢗⣿⡗⠙⠁⠀⠀⠀⠀⠀⣿⣿⣯⣭⣭⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠛⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠸⡈⡇⠀⠀⢸⠀⠀⢀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢢⡀⠀⢇⢹⠀⠀⣇⢳⠀⠀⢸⠀⠀⢸⠀⠀⡟⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢧⡀⠘⡄⢃⠀⢸⠈⡇⠀⣸⠀⠀⡇⠀⠀⣿⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠣⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⢎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⣶⠀⢠⠄⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⣿⣤⣿⣤⣿⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣛⣋⣉⣙⣛⠻⠟⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠁⣠⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⢀⣄⠙⠿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠟⢁⣴⣷⠆⣴⣿⣿⣿⣿⣿⣿⡍⠀⢉⣿⣿⣿⣿⣆⠺⣦⡈⠻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣧⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⣸⣿⣷⣤⣾⣿⣿⣿⠈⣿⡟⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠂⠈⠛⢿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠟⠁⠑⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⢠⠎⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢉⠀⠀⠑⡀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠶⣄⠀⠠⠡⠁⢀⠰⠀⠀⠀⠉⡉⠉⠉⠉⠉⢀⢀⠀⢠⠀⠈⠑⠄⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⡀⠁⠀⠃⠀⠀⢡⠀⢨⢸⠀⠈⠾⠀⠀⠂⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣤⣄⣀⠀⠀⠀⣀⣠⣴⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡄⠀⠀⠀⠈⠈⠀⠀⠀⡀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⡀⠰⡀⢠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⣶⡶⠾⠷⢶⣒⠠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⣰⣾⣿⣿⣿⣿⣷⣬⡻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⡿⢛⣥⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣶⣭⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣛⡻⠿⢛⣥⣾⣿⣿⣿⣷⣝⠻⠿⠿⠿⣛⣛⣛⢸⣿⣿⣿⣿⣇⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⡸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⠿⣛⣁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡎⠿⠿⠿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⢿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣙⣛⣭⣭⣭⣭⢸⣿⣿⣿⣿⠏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣇⢿⡿⢟⣋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡜⣛⣛⠿⠿⠿⠿⠱⣿⣿⣿⣿⣿⣿⠿⣫⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢼⣿⣿⣿⣿⣿⣿⣷⣦⣍⡛⠿⣋⣵⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⡿⠿⠿⠘⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢹⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠌⠻⠛⠛⠻⡟⢉⠻⡛⢟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⠟⠋⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣶⣾⣶⣶⣶⣶⣷⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠉⠙⢿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⢷⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠐⢶⣆⠲⣦⡐⣶⣄⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⡠⠖⠛⠲⣄⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⢀⣴⡞⣠⡶⢂⣶⠞⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠈⠻⣷⡝⢿⣎⠻⣷⡀⠀⠀⠀⢸⡄⢻⠀⢰⠁⠀⣿⠀⢸⠄⠀⡎⢠⠇⠀⠀⠀⠀⣰⡿⢋⣾⠟⣵⡿⠋⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⢘⣿⡎⣻⡷⢹⣿⠆⠀⠀⠀⢳⡸⡀⠀⠳⢤⣉⡤⠊⠀⢠⢃⡎⠀⠀⠀⠀⢾⣟⠱⣿⡋⣾⣟⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⣰⣿⢋⣾⠟⣵⡿⠋⠀⠀⠀⠀⠀⠙⠳⡀⠀⠀⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠈⠻⣷⡝⢿⣎⢻⣧⡀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⢠⣾⠟⣵⡿⢣⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠻⣷⡝⢿⣦⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣶⣴⣦⣴⡟⢉⣉⣉⣉⡉⢻⣶⣦⣴⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠷⠶⠶⠶⠶⠶⠾⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⢻⡿⠻⡿⣿⠿⢿⠿⢿⡟⣿⠿⠿⡿⣿⢿⣿⠿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣧⣵⣧⣽⣶⣼⣷⣬⣧⣾⣤⣦⣧⣿⣦⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣍⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠉⠉⢉⠉⠉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢙⡿⣦⣴⣶⣤⢶⣶⣄⡶⣦⣤⣤⣶⣤⢶⠄⣴⠀⡾⣦⣤⣰⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠙⠿⠿⠻⠟⠛⠘⣿⠿⠟⠿⠿⠛⠏⠸⠞⠿⠿⠙⠿⢻⡿⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠯⡍⠍⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣰⣦⡶⣶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣤⣶⣤⣶⣤⣶⣶⣶⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠷⠀⠙⠛⠻⠖⠀⢸⡇⣿⣿⣤⣥⣅⣽⣷⣥⣍⣡⣦⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⢸⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡠⢄⢤⠀⠀⠀⠀⠀⠀⠀⡀⣀⢔⡤⠔⠢⡀
⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⣀⠼⡓⠒⠂⠒⠛⠛⠻⠛⠛⠌⠁⠀⠀⡇
⠃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡸⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠐⠒⡆
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠠⠤⠆
⠀⠀⠀⠀⠀⢀⡔⠒⠂⠀⠐⡀⠀⠀⠀⠪⢳⡀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⢰⠂⠀⠀⠀⡂⡀
⠁⠀⠀⠀⡡⠌⢃⠀⢀⡀⢀⠈⠀⠀⠀⠀⠋⠀⠂⠤⠄⣠⡵⠂⠙⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠍⣇⠁⠀⠀⠀⢰
⡐⠐⠀⠒⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⡖⡏⠁⠘⠁⠀⠀⠀⢡⢠⠀⠀⠀⠀⠀⠀⠀⣂⢀⣀⣐⡚
⡁⠀⠀⠀⠐⠂⢀⠀⠀⠀⠠⡀⠀⠂⠐⠀⠀⠐⠃⠀⣤⡬⠀⠐⠀⠀⠀⠀⠂⠀⢀⠤⠀⠀⠀⡀⡒⡀
⠈⠉⠉⠉⠁⠠⠴⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣄⠁⠁
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠉⠁⠀⠀⠀⠀⡀⣴⣪⢄⢀⣭⣬⣶⣀⡀⠀⠀⠀⠀⠘⠂⠀⠀⠘
⠁⠀⠀⠀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠉⠉⠙⢻⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠁
⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠒⠂
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠓⠒⠚⠛⠻⠛⠛⠿⠛⠛⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣟⣦⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⣞⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⣦⣾⠟⣡⡤⡤⠤⠤⠤⠤⠤⠤⠤⠤⡤⣤⡙⢿⣦⣞⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⣾⡋⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⣻⣾⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⠟⠻⣿⣿⣦⡀⠉⠉⠉⠉⠁⠉⠉⠁⣠⣾⣯⡾⠛⢷⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠾⠷⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠳⠿⠦⠤⣄
⠀⠀⠀⠀⠀⢠⣤⠤⢼⣿⢍⠀⠀⣀⣀⣠⢤⣠⢤⣤⢠⣤⠀⡄⢠⣤⣤⢤⠤⢠⣀⣀⡀⠀⢨⣽⣿⠤⢤⣤
⠀⠀⠀⠀⠀⠈⢧⠀⠈⣿⣓⠂⠀⢸⡸⣾⠟⡿⠼⠷⠸⠿⠄⠻⠼⠿⠿⢸⠁⣸⣿⢸⠀⠀⢒⣻⡏⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⣧⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠇⢠⡏
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠸⣯⣭⡤⣴⠶⠶⢶⡶⢲⡶⢲⣶⠖⢶⣶⠲⣶⠲⣶⠶⠶⢶⣤⣬⣭⡿⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠸⠤⠶⠶⠒⠛⠁⢻⣄⡴⢋⡴⢋⣴⡟⢁⣴⡄⠙⣷⣌⠳⣌⠳⣄⣼⠃⠙⠓⠲⠶⠦⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡴⠋⣴⡿⠋⠀⢸⣿⣿⠀⠈⠻⣷⡌⠳⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡴⠋⣠⣾⠏⠀⠀⠀⠏⣿⡏⠇⠀⠀⠈⢿⣦⡈⠳⣌⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠏⣠⣾⠟⠙⠿⣦⡀⠀⠈⠿⠇⠀⠀⣠⡾⠟⠙⢿⣦⡈⢳⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⠿⠃⠀⠀⠀⠈⠙⠿⢶⣤⣴⠾⠟⠋⠀⠀⠀⠀⠻⢿⣿⣷⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⠈⠀⠀⠀⠀⠀⠀⠀⢐⡾⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⢤⡐⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠜⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠂⠱⠂⠝⠲⠤⡖⠀⠈⠂⢀⠀⠀⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⠁⠀⠀⠂⣄⠂⡥⠈⠀⠀⠂⠳⠀⠀⢠⠌
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⢯⠂⡌⠂⠀⢶⢂
⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠈⢢⢰⠀⠀⠈⠃⠂⠀⠄⠀⠀⡤⢄
⠀⠀⠀⣰⣄⡀⢀⡀⢀⡀⢀⡊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠡⠐⠀⠲⠄⠀⠀⠀⠀⠉⠉
⠀⠀⢠⠍⡭⠌⠉⠀⢀⡽⣥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢂⠀⠀⠀⠄
⠀⠀⠘⠓⠚⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠰⡀
⠀⠀⠀⢳⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣀⣀
⠀⠀⠀⠀⠉⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠘⣿⣿⣿⣿⣾⣧⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⢼⣍⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣀⠀⠀⠀⡄⠀⢐⣭⣻⡏⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢀⡠⢤⡤⠁⣀⠀⠠⠄⠀⣀⠀⠀⡀⡆⠀⠀⠀⢀⡀⢁⡤⠀⡄⢀⡄⠐⡦⢀⠒⣂⠈⢻⡦⠀⠀⠃⠀⣉⠈⢻⣿⢿⣀⠠⢰⡆⠀⢀⠀⠀⢲⣧⡇⢀
⠀⠎⠀⡰⠀⠔⠁⠀⠀⠀⡠⠋⠈⠀⢿⠀⠀⠀⡀⠂⢀⡔⠁⢠⣄⡞⠀⡼⠁⠀⠀⠙⠁⠰⠗⠢⠄⠀⢀⠸⠂⠸⠿⢾⡟⠀⢼⣄⠀⠊⣠⡶⠹⢳⠃⢈
⣶⠆⠀⠿⢆⢢⠄⢀⡀⠀⣦⡄⠀⠒⠈⠁⠐⠌⠁⠀⢻⣄⠢⠋⢁⠀⠈⠁⠀⠀⠀⢀⣤⡀⢀⠀⠈⣣⠘⢠⠀⠀⠀⡀⢧⢠⡄⣿⣇⣴⣏⢰⢠⣾⠀⠸⣤
⣥⠂⠀⠀⠀⢫⡁⢹⡇⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢀⠀⠂⢺⡅⠀⠈⢩
⠀⠀⠑⣄⠀⢤⡋⠀⠀⢈⡦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠃⠈⠓⠀⡞⠠⣦⡀⢠
⣀⠀⠘⣁⠀⡄⠙⠀⠀⢇⣴⣾⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠂⠈⢢⣀⡀⠰⠳⣆⠀⠀⠈
⠁⠀⢲⣇⠀⠁⠀⢀⢁⡟⢻⣿⢿⣿⣧⠐⠐⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢠⡼⣿⡀⣸⣆⡀⡠⠀⠰
⡤⠔⠟⠁⠀⠀⠀⡈⢠⠟⢿⣿⡆⠉⠙⢋⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⢀⡶⡀⢠⡶⣤⣈⠹⣿⠀⠀⣼⡇⢻⣷⣿⢿⡸⣿⠀⣸
⠀⠀⠀⠀⠀⣠⡾⣀⣀⣤⣶⡤⠀⠀⢀⠀⠐⣦⡄⠐⠀⠀⣻⣦⡤⠤⠀⡙⣿⣥⣤⡤⠤⣤⣤⣤⡼⠁⣴⣿⠀⠀⣉⠷⢀⠄⠊⢿⢹⠘⣿⣌⠙⠳⠞⠉⠉
⠀⠀⢀⣠⣴⣿⣧⣠⢹⣿⡟⠻⣄⢀⡯⢗⣾⡿⠛⠛⠋⠻⡏⠍⡰⠚⢾⣴⡿⠏⠿⠥⠀⠀⠀⠈⠰⠾⠃⠰⡄⠀⠀⣦⠀⠀⠀⡾⣆⢂⢹⣿⣿⣿⠛⠲⣿
⡾⠛⠀⠀⠀⠀⠀⠙⠀⠻⡿⠀⢩⣿⣿⣌⣩⠙⠀⠀⠀⠀⣿⠀⠄⠀⠈⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡄⠈⠿⣀⠀⢸⡇⣿⣄⠀⢿⣿⣿⠀⠀⠈
⠁⠀⠀⠀⠀⢀⣾⣀⣀⣠⡄⠀⣈⢉⣡⣿⠀⠀⠀⠀⠄⠀⠉⠀⠀⠀⡄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠙⣄⣿⣿⣿⣽⣦⠀⠻⣿⣦⣀
⠀⠀⠀⠀⢀⣼⡿⣿⡿⠟⠉⠷⣎⠅⠶⠷⠄⠀⠠⠊⠀⢀⠁⠀⠀⠀⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠴⢾⠃⠈⠻⣿⣿⡿⠟⡋⠀⠁⠀⠙⢿⣧
⠁⠀⢀⡴⣿⡧⠀⠈⠀⠀⠤⠀⢛⠀⠀⠀⠈⠣⠐⠚⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠆⠀⠀⠀⡄⠀⢽⡀
⢡⡀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⠁⠀⠀⠱⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣀⣀⣀⣠⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⣠⡾⠏⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠹⢷⣄⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⣟⠀⠀⠀⠀⢠⡞⡇⡠⣠⡇⡾⠀⠀⠀⠀⠀⣻⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⠌⠁⠙⢷⣄⠀⠀⠀⠳⠏⠑⠉⠋⡗⠁⠀⠀⣠⡾⠋⠀⠩⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣲⣅⣤⣤⣤⣽⣷⣤⡤⠤⠤⠤⠤⠤⠤⢤⣤⣾⣯⣤⣤⣤⣨⣖⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡭⠍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⢭⣹⡇
⠀⠀⠀⠀⠀⠀⢰⡶⠖⠛⢻⣗⣋⠭⠀⠀⠐⣶⢲⢳⣼⢸⣾⣟⡖⣷⡇⣼⡇⣿⣸⡇⣟⢀⣧⡖⢲⠂⠀⠀⠭⣑⣺⡟⠛⠲⢶⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿⣒⡩⠅⠀⠀⢿⠸⣼⠿⣯⠿⠏⠷⠟⠧⠟⠧⠟⠿⠇⢯⢸⠻⡇⣾⠀⠀⠠⢍⣒⣿⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⣒⣒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⣲⣿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢻⣗⣒⣀⣀⣤⣤⣤⡤⠤⢤⣶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡤⠤⢤⣤⣤⣤⣀⣀⣒⣺⡟⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣈⣭⣿⠏⠘⣷⠀⠀⢀⣴⠟⠁⠀⠀⣀⡤⢤⣀⠀⠀⠈⠻⣦⡀⠀⠀⣿⠃⠹⣿⣭⣁⣀⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⣧⣴⡿⠃⠀⠀⠀⡼⠁⠀⠀⠈⢧⠀⠀⠀⠘⢿⣦⣼⠃⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣰⠛⢿⣶⣄⠀⠀⠀⠀⢀⣴⡿⢿⣀⠀⠀⠀⠀⢳⡀⢀⣠⣄⡞⠀⠀⠀⠀⣀⡿⢿⣦⡀⠀⠀⠀⠀⣠⣶⡿⠻⣆
⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣦⣀⣴⡿⠋⠀⠈⠻⣧⣀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⣀⣼⠟⠁⠀⠙⢿⣦⣀⣴⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⢀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠢⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⡠⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⡟⠁⠀⠀⡀⡀⡀⡀⡀⡀⠀⠀⠀⠀⠙⡷⠊⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠳⣄⠈⠢⡀⠀⠿⠯⠵⠇⠷⠿⠤⠄⠀⡠⠊⢀⡴⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣕⢁⠈⢳⣄⣈⣢⣀⣀⣀⣀⣀⣀⣀⣠⣊⣀⣴⠋⢀⢑⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣒⡛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢛⣓⣲⣦
⠀⠀⠀⠀⠀⠀⣤⠶⠶⢾⣿⣓⡂⠀⠀⢤⡄⡴⣤⡆⣦⣶⣴⣦⡆⣶⢰⡆⣶⣴⠂⣦⡤⢤⠄⠀⢐⣒⣺⣿⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠸⣿⣒⣒⠀⠀⠘⡇⣇⡿⣧⡟⡟⡿⠻⠿⠿⣾⢿⠿⣿⡁⡿⡇⣾⠀⠀⢐⣒⣺⡿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⣖⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣾⡇⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢹⣗⣂⣥⣤⣤⠤⠤⠶⣶⣶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡶⠦⠤⢤⣤⣤⣅⣒⣿⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⠭⠿⠃⠸⣇⠀⣠⣾⠟⠁⠀⢀⡤⠤⢤⣄⠀⠀⠙⢿⣦⡀⢀⡿⠀⠻⠯⢥⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠟⠁⠀⠀⠀⡎⢹⠀⢸⠉⡆⠀⠀⠀⠙⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠙⢷⣄⠀⠀⠀⠱⣞⣀⣘⡶⠁⠀⠀⢀⣴⠟⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠈⠻⣷⣄⡀⠀⠀⠈⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠂⠀⠀⠀⠀⣠⣴⡾⠟⠉⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⣠⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠋⠀⢀⣴⣿⣿⣿⣄⠀⠀⢀⣠⣾⣿⣿⠀⠀⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⢿⣿⣿⣧⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠿⠿⠛⠋⠉⠉⠈⠉⢻⣿⣿⣿⣿⣿⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢁⣠⣤⣶⡶⠖⠒⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠁⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⢀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣰⣟⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡀⠀⠀⢀⡼⢦⡀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⠁⠀⠀⠳⡀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠸⠿⠿⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣤⣄⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢳⠀⢸⣿⣿⣦⣀⣀⣴⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢸⠀⠀⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⡟⢏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠛⠋⠉⠀⠀⠙⢦⣀⡜⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠁⠀⠀⠀⠀⢣⠘⡄⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢣⡘⢄⣀⣀⣀⣀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣀⣴⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⠞⠁⠐⠋⠉
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡷⠦⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣿⣇⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡀⠸⣿⣿⣿⡿⠀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢸⣿⣿⣿⠇⠀⢉⣭⣍⠁⠀⢿⣿⣿⣿⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⣿⣿⡿⠀⠀⣿⣿⣿⡇⠀⠸⣿⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⣤⣤⣤⠀⠀⢹⣿⣿⠁⠀⢠⣤⣤⡄⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⠀⣀⣀⣉⣀⣀⠀⢠⣤⣤⠀⢀⣀⣈⣁⣀⡀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣉⡉⢰⣿⣿⣿⣿⡿⢠⣤⣤⣤⣤⠸⣿⣿⣿⣿⣷⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⡆⠀⡆⡆⠀⡆⡆⠀⡆⡆⠀⠆⢠⠠⠀⢰⠀⠀⣄⠀⢰⢰⡆⢰⠀⢰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⡇⡇⠀⠳⡇⠀⡇⠀⠀⡀⠸⢐⠀⠈⠇⠀⡈⠇⢸⢸⡇⢸⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⠀⢀⣠⡀⠀⣀⢀⡀⣀⠀⣀⠀⡠⢄⡄⢀⡄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢁⡾⠀⢠⡧⡾⢹⣏⣴⢋⡞⣰⣏⣼⢋⡾⢡⣾⢵⠏⣴⢋⡾⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠤⣖⠋⠀⠙⠉⠀⠈⠉⠉⢡⡞⣡⠏⠉⠉⠁⠈⠉⣩⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣻⣿⣿⣛⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣸⣦⣄⣀⠜⠀⠀⠀⢠⠖⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣧⣤⣼⡀⠀⠀⠀⠀⠀⣰⣯⣙⡻⠃⠀⢀⣀⣀⠃⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢄⢀⣤⣾⡄⠀⠀⠀⠀⠀⢰⠋⠉⠉⢣⠀⠀⠀⡠⠊⠉⠛⢿⡇⠀⢠⠋⠀⠈⢿⣶⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣡⣼⣦⡀⠀⠀⢀⠇⠀⠀⠀⠀⠣⡠⠊⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣶⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠀⠈⠑⠢⡜⠀⠀⠀⢀⣀⣠⣀⣀⣀⠀⠀⠀⢠⢀⡤⠺⠿⢿⡟⢙⠁
⡠⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡼⠋⠀⣾⣿⣶⡆⢸⠀⠀⠀⠀⠀⠀⣠⠒⠉⠈⠙
⢇⠀⠀⠀⣸⣦⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⢁⠸⠤⠤⠤⣤⣶⠻⣿⡆⠀⢀⡴
⠀⠑⠢⠴⣿⢋⣼⡿⠒⠒⠂⠤⠵⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡧⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠙⢿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡅⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣄⣀⣀⣤⣶⣶⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣯⣷⣦⣬⡽⠿⠟⠛⠉⠁⠉⠉⠛⠛⠳⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠫⠫⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡔⣴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢿⣿⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡛⣿⡇⠀⠀⣀⣀⣈⡙⠀⠀⠐⠶⣋⣥⣄⣀⠀⠈⣿⣵⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡟⠘⠁⠔⠊⠉⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⡧⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⢀⣤⣤⠀⡀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⡔⢀⣀⠄⢈⣀⠈⠢⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢰⣿⣷⣶⣶⢿⣷⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠈⠋⠙⠫⠉⠉⠉⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢉⡋⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠘⣆⠀⠀⠀⠀⠀⡴⠃⠀⣤⣀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣶⣿⣿⣷⠀⠀⠈⠳⠄⠀⣠⠞⠁⠀⢠⣿⣿⣿⣶⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣿⣿⣿⣿⣧⠀⠀⣠⣶⣾⣷⡀⠀⣠⣿⣿⣿⣿⣿⡇⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢻⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠛⠁⠙⠃⠉⠃⠁⠈⠁⠁⠉⠉⠉⠋⠋⠛⠃⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⢤⣤⣤⠴⣄⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⢰⡃⠈⠁⠀⠉⠀⠀⠀⢹⠖⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠴⠶⠶⠤⢤⣤⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⣧⠀⢀⡤⢤⡇⠀⠀⡤⢢⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠁⢀⡞⢀⡟⠀⡜⠉⣧⡤⠀⠀⠀⠀⣤⡞⠙⡏⠙⡖⢢⠹⠴⠁⠀⢻⠀⡼⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢳⡤⢤⠔⢉⠔⠉⢀⠜⠀⡰⠁⠀⠀⠀⠀⠀⠀⢷⠀⢣⠀⢃⠀⢣⣀⡤⣀⡞⠀⡇
⠀⠀⠀⠀⠀⠀⣾⠀⡴⢣⡎⠀⠀⠀⠐⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠃⠀⠂⠀⠻⣀⠎⢻⢀⣇
⠀⠀⠀⢀⣠⠤⢷⠘⣄⠀⡁⠀⠀⠀⠀⢠⠟⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⡇⠀⠀⠀⠀⠀⣿⠘⣯⠜⠉⠛⢦
⠀⢀⡾⠉⠀⠀⠈⢷⣋⡴⠃⠀⠀⠀⠀⡿⠀⣈⠁⠀⠣⣀⣀⠴⠃⠀⠙⢂⡽⠀⠀⠀⠀⠀⠽⠀⣿⠀⠀⠀⠈⢷
⠀⢸⠀⠀⠀⠀⠐⡾⢋⡇⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣿⠲⢿⡉⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠠⠴⠃⠸⡆⠀⢠⠞⠉⣿⠀⡈⡀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⡿⢦⡀⠀⠀⢨⠇⠀⠀⠑⠀⠀⠀⣸
⠀⠘⢦⠀⠀⠀⠀⠀⠀⢧⡴⠋⠀⢰⣧⣀⣈⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣆⠘⢦⣠⠏⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢧⠀⢀⠏⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠉⠁⠀⠀⢈⣧⣔⠁⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠙⢦⣀⠀⠀⢀⣀⡠⠗⢻⡀⠀⣀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠒⠤⠤⣀⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢠⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠁⠘⡆⢹⡛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠙⠒⠒⠶⠒⠒⠚⠋⠉⠁⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡟⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⣄⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠉⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⣀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡸⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣀⣀⣀⣀⡼⠁⠀⠀⠉⠛⠿⠿⢿⣿⣿⡿⠿⠿⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣟⣛⣫⣟⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣭⣭⣽⣿⣯⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠳⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠁⠀⠀⠀⠀⢀⡔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠳⡄⠀⢪⣣⠘⡇⠀⠀⠀⠀⠀⠀⠀⢧⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠢⣀⣻⡁⠀⠙⣿⠿⢦⡀⠀⠀⢠⣤⠒⢢⣦⣾⣄⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠈⣿⠛⠓⢦⡇⠀⠀⠉⢢⡀⠀⢸⣠⣿⠋⠁⠈⠳⡄⢀⡤⠒⢠⡃⡠⠋⠀⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡞⠁⠙⢆⠀⢄⠀⢸⢀⠀⠈⢀⡀⠀⠀⠀⢳⣠⠞⠹⡇⠀⠀⠀⠀⣿⣤⠴⠒⠛⡟⢁⡔⠚⠉⠉⢙⣯⠠⠜
⠀⠀⠀⠀⠀⠀⠙⢶⠄⠈⢧⠀⠑⠈⢎⠆⠀⢇⢣⡀⠀⠀⠈⡏⠀⠀⢳⡀⠀⠀⢸⣿⠁⠀⢀⠄⡇⠘⣿⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢽⣦⣿⣧⣅⡀⠀⠐⠀⡀⢣⠑⠀⢀⣀⡇⠀⠀⠠⠓⢤⣾⢿⣟⠀⣠⢞⠼⠁⠀⣀⠀⢀⣤⡤⢤⡀
⠀⠀⠀⠀⠒⠦⠄⠀⠀⠀⡹⠋⠀⠉⠓⠢⣄⠈⠁⠈⠚⣋⣽⣧⡴⠶⠦⣀⠀⠉⠃⠙⠓⡊⠡⠤⠬⡥⣴⣿⣟⣁⣠⠴⠁⠀⢀⣀
⠀⠀⠀⠠⣄⠀⠀⠐⠤⠀⠳⡇⠀⠀⠀⠀⠈⢣⠠⠀⣪⣿⡟⣿⠀⣀⠀⠀⠙⣗⢀⠖⠡⠖⠊⠀⢀⣀⣸⣏⠁⣀⣀⣀⠔⠊⢁⣠⡿
⢠⡔⠒⠲⠤⣉⡒⢄⠀⣀⣀⣿⢦⠀⠀⠀⠀⠀⢷⣾⣿⣿⢯⡁⠘⠛⠀⠀⠀⠈⣇⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢹⠿⠓⠋⠉⠉
⠃⠳⢶⣶⣦⣤⣹⣿⡟⠀⠀⠉⠑⢥⡀⠀⠀⠀⠘⣿⣿⣿⣼⠋⠀⠀⠀⠀⠀⠀⢸⣄⣠⡏⠁⠀⠀⢀⣀⠀⣀⠔⢑⣫⠴⠒
⠀⠀⠀⠒⢒⣀⣀⣀⣙⣦⡠⡀⠀⠀⠈⠓⢤⣠⡶⡿⣿⢿⣿⣷⣄⠀⠀⠀⢐⣠⡜⠀⠐⠷⠤⠤⠤⠴⠖⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠲⠤⠴⠖⠚⠒⠒⠿⢝⠧⣝⡻⠿⣿⣷⣾⣿⣿⣁⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠩⠟⠁⠒⠒⠒⠂⠀⠒⠒⠿⠶⠭⠽⠿⠽⠾⠵⠂⠀⠀⠈⠩⢭⣭⣭⣍⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣤⣴⣦⣴⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠉⠳⢄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠞⠁⠀⠀⠀⠀⠀⢱⡄⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠔⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⡠⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠻⠤⢶⠆
⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⠀⠀⠀⠀⡠⢄⡀⣀⠔⠁⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⢋⣀⠏⠀⠀⣰⢋
⠀⠀⠀⠀⠀⠀⠀⣬⠽⠋⠀⠀⠀⢸⡀⠐⢢⠀⠀⠀⠀⠈⡗⠒⠲⡊⠁⠀⢀⢀⡀⠀⠹⡉⠉⣇⠀⠻⡅⠀⠀⢈⣉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢒⣶⠞⣀⠀⠙⠲⠤⠲⡀⠀⠀⠀⡌⠀⠀⡕⠀⡰⣓⠦⣹⡄⢀⠇⠀⢏⠀⠀⢸⣀⣀⣀⠰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣽⣋⣠⡤⢦⡤⠞⠦⣄⠀⡄⠀⢀⠄⠀⠈⠁⠀⠈⠀⠀⡣⠀⢸⠀⢀⡞⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠏⠉⠓⠦⣤⡆⠀⠀⠀⠀⠀⠀⠀⣇⣠⡼⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣈⣉⠑⠒⣶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⢰⠞⠉⠉⠉⡆⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⣇⠀⠀⠀⢧⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⠀⠳⣤⣄⠀⠀⣀⡀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⠤⠼⢦⡀⠚⠓⣶⠒⠚⠲⣾⢻⡟⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⠖⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠃⢸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠀⠤⠤⠤⢤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠖⡩⢄⢱⠀⠀⠀⠈⢢⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡔⠁⠀⠀⠀⡎⡠⢍⠲⡀
⠀⠀⠀⠀⠀⢠⠃⠼⣀⠸⣼⡀⠤⣰⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⢆⠤⢀⣧⢇⣀⠧⠘⡄
⠀⠀⠀⠀⠀⡆⠰⠐⠂⠐⠂⠒⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠒⠐⠒⠐⠊⠆⢰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢁⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⡈
⠀⠀⠀⠀⠀⠈⡄⢣⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡜⢠⠁
⠀⠀⠀⠀⠀⠀⠘⡄⠣⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡠⠊⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠑⢄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡠⠊⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡉⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⣉⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠑⠄⣣⠀⠀⠀⠀⠀⠀⣔⣡⠈⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠒⣁⡷⠀⠀⠀⠀⠀⠀⢾⣍⠒⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣤⡄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣶⡖⣿⣇⣿⡏⢸⣿⣭⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⣴⣿⠿⣿⡆⢻⣧⢹⣿⡿⣷⣿⣿⣉⡁⣘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠿⢿⣿⡈⠛⢻⣿⠿⣷⣾⣿⠀⠛⠃⠪⡉⢉⠉⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣤⣤⣻⣷⡰⡀⠻⠇⠈⠁⠀⠀⡀⢣⣀⣇⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢉⣿⡇⠉⠁⠱⢀⡤⠊⢹⣷⠄⡧⠀⡏⠉⠳⣔⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⠀⢀⡜⠀⡧⢀⣇⡀⠀⠘⡎⠀⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢉⡤⢛⠦⡤⠖⠩⠄⠀⠃⡎⢀⡆⢀⣠⣵⢈⢊⠌⡠⠂⠀⡠⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⢄⣙⡦⠭⣴⣇⠠⠀⠈⡄⢠⠋⠘⠛⠂⠘⠮⢝⡫⠀⢊⠜⠁⡠⠊⡠⠔⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⡇⠀⠀⠈⠑⢧⠐⣶⠁⠀⠀⠀⠰⢀⢀⢢⠀⢢⠀⣠⢎⠤⢊⠄⡺⠁⡸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢤⣀⠀⠙⢄⠀⢀⣴⠬⣁⣠⣶⣶⣶⣶⣷⣤⣁⠸⢀⡎⠘⢁⡡⠊⣀⢴⣧⡾⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠲⠈⢭⢵⣆⣙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹⡀⠉⢁⡞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⠽⠷⢾⣁⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢵⠈⡿⠀⠀⠀⣰⣃⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠠⠤⠀⠀⢻⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣤⣤⡼⠀⣷⣤⢀⡴⠋⠈⠁⢀⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⣷⠀⠀⠀⠀⢀⡠⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠡⢤⣄⡩⠛⠈⢉⡉⣄⠀⠁
⠀⠀⠀⠀⠀⠈⢅⣠⠴⠊⠉⠉⠉⠉⣉⠡⢔⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠥⠬⣴⡋⠀⠀⠀⠘⣿⠟⡐⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⠄⠊⢀⡴⠀⠙⠿⣿⣿⣿⣿⣿⣿⣧⡀⢹⡙⠦⠒⠉⠙⢂⠒⠊⠉⠓⠤⣄⡉⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⢀⠐⠋⢀⠞⠀⠄⠀⠉⠙⠛⠛⢉⡁⠈⠋⠉⠛⢿⡆⠀⠀⠈⠓⠀⣉⣲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠔⠁⢀⡔⠁⠐⡔⠀⢀⠀⢠⡏⠀⠘⡆⠐⡄⠀⠀⠀⠙⠂⠀⣠⠔⠋⠁⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠋⠀⠔⠈⠀⠀⡎⠀⠘⡇⠆⠀⢳⠀⠙⢦⣀⣤⡶⣦⠞⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠀⠀⠈⠇⠀⠀⢉⡻⠾⣯⡄⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣀⡴⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠙⠟
⠀⠀⠀⠀⢼⣿⡷⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠈⠛⠿⢿⣦⢠⣄⠀⣴⣿⣶⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⣠⣤⣭⣤⣄⠀⠀⠀⠀⣙⣿⣥⣶⢟⣃⠸⣷⣴⡾⠟⠈⣿⣦⣬⢛⣉⣀⠀⠀⠀⠀⣠⣄⣭⣤⣄
⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⡿⠿⠿⠛⠋⠀⠀⠉⠁⠀⠀⠀⠉⠹⣿⣿⡟⢿⣇⠀⢰⣿⣿⣿⣿⣿⡀
⠀⠀⢸⡟⣿⣿⣿⢻⡇⠀⢀⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠘⠿⠀⣸⡟⣿⣿⣿⢻⡇
⠀⠀⠈⠁⣿⣿⣿⡈⠁⠀⢸⣿⡿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⣿⣿⠀⠀⠀⠈⢡⣿⣿⣿⠈⠁
⠀⠀⠀⢰⣿⠏⣿⡇⠀⠀⣸⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⣿⡄⠀⠀⠀⢸⣿⠹⣿⡄
⣀⣀⣀⣸⣿⣀⣿⣇⣀⣾⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣙⣿⣆⣀⣀⣸⣿⣀⣿⣇⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣀⢀⣠⡄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡽⠀⠀⠀⣸⠀⠈⠉⢰⠃⠘⠛⠀⠀⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⡀⠐⠋⠤⡄⠀⢀⣈⣣⡀⠀⠀⠀⣧⣼⠃⡔⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⢉⣽⣦⡀⠀⠰⡀⠀⠀⠸⡶⠃⢠⠀⣀⠀⠀⣰⠧⣿⠊⠀⢶⠋⠈⡴
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣴⣯⢫⡾⠓⠦⣤⣥⣤⣀⡀⠀⠀⡌⠘⠿⠃⣰⠃⠀⢸⠀⠀⠐⠒⠓⠆
⠀⠀⠀⠀⠀⢀⣠⠜⠳⢤⡄⠉⠻⣅⠀⠀⠀⠈⠉⠙⠻⢷⣤⡃⠀⢀⡜⡏⠀⠀⠸⡇⠀⡠⢀⡥⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢹⣀⣄⣮⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠘⠀⡇⠀⠀⣸⠇⠈⣠⠎⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢠⣴⣮⠅⢀⡉⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢷⣀⣠⠏⠀⣰⠃⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠋⠙⠀⠀⠈⠉⠑⠒⠄⠹⣧⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⣠⠩⠅⠀⣰⣇⠀⠀⠀⠀⣠⠞⢁⡀
⠀⠀⠀⠀⣠⠒⠢⣄⣀⠀⠀⠀⠺⠷⠀⠈⠻⢿⣶⣤⣤⣤⣶⣿⣺⢯⡇⠤⠄⣀⡞⠹⣽⣶⣤⣠⠞⠁⣀⡰⣆
⠀⠀⠀⠀⠈⠲⠤⣾⣩⡟⠒⠒⠒⠒⠲⠦⣄⡀⠈⢉⠛⠛⠿⠿⠗⢋⣠⡴⠞⠻⣻⣦⣀⣩⠟⠁⠀⠀⣹⣀⣰⠋
⠀⠀⠀⠀⠀⢀⣠⠤⠌⠙⢦⣀⠀⠀⠀⠀⠈⠙⢶⣄⠀⡔⠘⣠⠖⠋⠁⠀⠀⠀⠙⠻⢿⠏⠈⠉⠉⠐⠂⠀⠉
⠀⠀⠀⠀⠀⢴⡆⠀⣠⣄⠀⠙⠷⣦⣤⣀⣀⣀⣼⡿⡼⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡰⡄⠀⠈⠉⠉⠉⠉⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠈⠑⣠⣤⠘⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠚⠁⠙⢒⡦⠀⠀⠀⡐⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⢠⣌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣻⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⡀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⣤⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⢿⣧⣀⣀⣀⣠⣴⠾⢛⠁⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠋⠁


⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⢛⣽⣿⣿⣿⣿⣿⣿⣿⠃⣠⡻⣷⡈⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⡇⣿⣿⣿⣿⣴⣿⣿⣿⣿⡟⠛⣧⡇⠀⠀⠀⢀⣘⣿⣿⣿⣿⠽⠏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⣽⣿⡿⠉⠉⠀⠙⠿⠻⡇⠀⠻⠁⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⣻⣿⢃⣿⡿⠁⠀⠀⠀⠀⠀⠀⠋
⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⡿⣧⠀⠉⠁⠼⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣰⣿⣿⠗⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠧⣪⣿⣿
⣿⣿⣿⣿⢯⢿⣿⣿⣿⣿⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⡀⡀⣀⡠⠆⠁⢀⣤⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⠇⡀⠤⢜⣿⣵⣄⠀⠀⠀⠀⣀⡤⢋⢄⢄⡴⣪⢔⣠⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣏⣥⣼⣷⣿⣾⣿⣷⣀⣴⡾⣋⢐⠀⣱⠊⠔⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢛⣿⣻⠙⠋⢞⢁⣠⣞⣴⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡭⠼⠥⣮⣤⣴⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⠟⠀⠀⠀⠀⠀⣠⣀⠀⢠⣄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡞⠁⠈⠉⠛⠛⠋⠓⢮⣷⠦⣀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡾⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢳⡀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⠛⢦⣼⠀⠉⠀⠀⢀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⢹⣾⣧⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡇⠀⢺⣿⠀⠀⠀⢰⠋⠀⢠⣾⡇⠀⠀⠀⣠⠖⢦⠀⡟⣸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢧⠀⠀⠉⠀⠀⠀⢯⣄⣀⡈⢻⠇⠀⠀⢰⠃⣼⣹⠇⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠖⠒⠚⠛⢮⡛⡇⠀⠀⠀⠀⠀⢀⣿⡉⠉⠀⠀⠀⢸⣄⣈⡽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⠟⠁⠀⠀⠀⠀⣀⣀⣀⠙⢧⠀⠀⠀⠀⣾⡆⠉⠀⠀⠀⠀⠀⠀⣿⢎⡅⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡏⠀⠀⠀⠀⢀⡞⠁⠀⠈⠳⡌⢧⡀⠀⠀⠀⢰⣆⣤⣶⣒⣲⣤⡀⣀⠸⢟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢸⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢹⠘⡟⢦⣦⠀⠈⠉⠀⠀⠀⠈⠉⠙⢿⡠⢞
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠙⣌⡻⢍⣳⠶⢦⡤⠤⢤⣶⠚⠉⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⡞⢣⡀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠘⡆⠀⠸⠿⠦⠤⠴⠋⠀⢠⠏⠸⡄⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣾⠀⠀⠙⢦⡀⠈⢻⡖⠒⠒⠒⢒⡟⠁⠀⠀⠀⠀⠀⠀⢀⡴⠟⢶⠞⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡀⣷⠀⠀⠀⠀⠉⠳⣜⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠙⣿⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀⣏⢧⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢷⠒⠛⠉⠁⠸⡞⢆⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣸⣳⡀⠀⠀⠀⠓⠒⠦⣄⣀⡇⠈⢧⡀⠀⠀⠈⠓⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠷⠷⠿⠷⣄⡀⠀⠀⠀⠀⣀⠴⠃⠉⠛⢦⣀⣀⡀⠀⠀⢠⡽⠀⠀⠀⠙⢄⡀⣤⣄⠀⠀⢨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠛⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠼⢧⣯⣤⣤⡿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⢀⣀⣠⣄⠒⠪⠙⠢⢤⣄⠀⢀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚⠓⠲⠟⠓⠉⠉⠉⠀⠀⠀⠀⠁⠀⠉⠉⠘⠛⠓⠲⣄⡈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡴⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠲⡌⠹⣦⠂
⠀⠀⠀⢀⡀⠠⠄⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠐⠀⠈⢳⡀⠀⠀⠀⢀⣀⣀
⠀⡠⠊⠁⠀⠀⠀⠀⠈⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⣱⡄⠀⠁⠀⠀⠀⠀⠁⠢⡀
⡐⠀⠀⠀⢀⠄⠠⢀⠀⠀⡏⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⡤⢘⠹⠟⡀⢀⠤⠐⠠⢄⠀⠀⠈
⡇⠀⠀⠀⠂⠀⠀⠀⠑⣤⡇⢈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⡤⠞⠁⠀⠀⠀⠈
⠇⠀⠀⠀⡀⠀⠀⠀⠃⢨⠷⠸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡇
⠘⡀⠀⠀⠠⠀⠀⠀⠑⢼⣄⠠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠣⡀⣃⠂⠀⠀⠀⠀⢠
⠀⠐⡄⠀⠀⠣⡀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠃⠀⢀⠜
⠀⠀⠈⠢⡀⠀⠀⠠⡀⠈⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄⠊⠀⠀⡠⠂
⠀⠀⠀⠀⠈⠢⡀⠀⠙⢢⡄⠂⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡘⠀⠄⠀⠀⠀⣤⠔⠉⠀⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠹⢌⠄⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠐⠀⡘⠀⠀⠀⢼⠃⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢷⠈⢄⠀⢄⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⡠⠊⢀⠜⠀⠀⠀⠀⡎⠀⠀⠸⡀⠀⠀⣀⢀
⠀⠀⠀⢍⠁⠀⠀⠉⠀⠀⠀⡘⢠⠈⢢⣀⠁⠦⢤⠐⠒⠈⠀⠀⢀⠤⠊⢀⡠⠂⠀⠀⠀⠀⡐⢃⠀⠀⠀⠈⠉⠉⠀⡸
⠀⠀⠀⠀⠑⠠⢀⡀⢀⠠⠊⠀⠀⠳⢅⡀⠖⠰⠦⠤⡭⠬⠭⣥⠐⠖⠒⠉⠁⠉⠀⠀⡨⠛⠀⠀⠑⠠⠀⠀⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠦⢄⣠⡐⠒⠒⢀⣤⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡗⠙⠦⠆⠉⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢾⠠⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠸⠁
⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⢿⣿⣿⣿⣿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣤⣤⣤⣤⣤⣤⣤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠙⢆⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⣀⣤⡄⠀⠀⢠⣾⣶⣤⡀⠈⢳⡀⠀⠀⢎⠀⠀⠱⣈⠀⢀⠔⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣰⠟⠉⠉⠉⠀⠀⠀⠀⠀⠙⠧⠀⠀⢷⠀⠀⠀⡑⠀⠀⠀⠀⠁⠀⠐⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠁⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠐⢤⠀⠀⠀⠀⠀⢠⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⠀⠀⣤⡄⠀⠄⠀⣤⡄⠀⠀⠀⠀⠀⣷⠀⠀⡰⠁⠠⠂⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠈⠁⠀⠀⢀⡀⠀⢹⡔⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠉⠀⢸⣇⡠⠂
⠀⠀⡀⠄⠤⢄⠀⡠⠒⠉⣁⣼⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⣼⠉
⢀⠴⠒⠒⠀⠀⠙⡤⠊⠉⠀⠀⣧⠀⠈⣷⣶⣄⣀⣀⡀⠀⢀⣀⣀⣤⣶⣾⡟⠀⠀⡟
⡇⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⢹⠀⠀⠘⢿⠿⠛⠛⠻⠛⠛⠛⠻⣿⣿⡟⠀⠀⢠⠇
⠈⠒⠟⠹⠀⡆⠀⢂⠁⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠘⠤⠔⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠉⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⢶⠄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢋⠀⠀⠀⠀⠠⡀⠀⠑⣄⠈⠀⠣⡀⠀⠙⣄⠀⠀⠆⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⡀⠙⢦⠀⣈⠃⠀⠀⠈⠀⠀⠈⢣⢐⣦⠀⠢⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢿⠖⣀⠹⣦⠀⠀⢦⡀⠤⣳⣤⣾⠿⠁⠀⠀⠈⠢⡀⢷⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠈⢹⡈⠇⠀⠙⠛⠛⠿⠟⠁⠈⡀⠀⠀⠀⠀⠙⢆⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢳⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢦⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⢀⣀⣀⣀⠀⠀⠀⠀⠀⠰⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⢱⡄⠘⠋⣁⣍⡉⠀⠢⡀⠀⠀⠶⠸⢹⣿⡋⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⢀⠠⠘⠁⣀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠃⠀⢹⠋⢿⡍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠁⢀⡔⠊⠀⠁⠀⢠⡓
⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⠻⡄⠀⠀⣠⠎⠀⠀⠀⠀⣀⠙⢅⠀⣄
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⡆⡆⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠑⢾
⠀⠀⠀⠀⠀⠈⡟⠃⠀⠀⠀⠱⡸⣄⠀⠀⠀⠀⣀⣀⡠⢤⣤⣄⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⡼⠁⠀⠰⠊⠁⠀⢀⠕⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⣆⠀⠀⠐⢿⣟⣿⡿⣽⡿⠃⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠑⣄⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡀⠀⠀⠶⠌⢦⡀⠀⠈⠛⠒⠚⠋⠀⠀⠀⠀⢀⣼⣿⣿⡇⠀⠀⠀⠈⠀⢀⠄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠳⣿⣯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣨⢿⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⠀⠀⠄⠀⠀⠈⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠼⠃⠀⠀⠀⠀⢰⠇⠈⠓⠶⢤⡴⠶⡿⡿⠿⠿⠛⠛⠁⠀⡄⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠚⣂⠴⠋⠀⡀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢷⡀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢐⡂⠀⠀⣀⠀⠀⢴⣋⣵⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣯⡀⠀⣀⣭⠵⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⢀⠋⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡬⠗⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⣀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡽⣦⡀
⠀⠀⠀⢀⡾⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣠⣶⣯⣾⣿⣷⣄⡀
⠀⠀⢀⣼⣿⣶⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⣾⣫⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣀⣠⣴⢞⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⣶
⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠸⣿⣶⣤⡉⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⢿⣿⡿⠀⣠⣤⡄⣠⡄⠉⣻⣿⠘⣿⡄⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁
⠀⠀⠀⣰⣿⣮⣧⣿⣦⠀⠀⠀⢀⣴⣶⣶⣿⣿⣥⣤⠙⠻⣿⡟⠋⠀⠹⠿⢷⣆⣆⣦⣤⣄⡀⠀⠀⣴⣿⣼⣵⣿⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⣿⣿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⢻⣧⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣹⣿⣿⣿⢿⡇⠀⢠⣿⣿⣯⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣧⠈⠛⠀⢸⡿⣿⣿⣿⡇⣿⡆
⠀⠀⠀⠁⣿⣿⣿⣿⠈⠁⠀⢸⣿⡟⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠛⣿⣿⠀⠀⠀⠈⠁⣿⣿⣿⣿⠈
⠀⠀⠀⠀⣿⡿⢹⣿⠀⠀⠀⣸⣿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢿⣿⡀⠀⠀⠀⠀⣿⡇⣿⣿
⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠻⣿⡄⠀⠀⢀⣿⡇⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⠁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣇⣤⣿⠛⣳⣒⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣤⣸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⢹⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢳⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡹⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣛⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⡿⠆

⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⣖⠱⣭⠎⢤⠀⠀⠀⠀⠀⠀⠘⢦⣙⣋⡼⠁⠀⠀⠀⠀⢀⢠⡴⣐⠿⣠⢦⡄⡀⠀⠀⠀⠀⣀⣀⡀⢀⡠⢄⢄
⠀⠸⣿⠃⣿⠰⣿⡇⠀⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠈⢻⡞⢹⣿⠉⣿⡎⠀⠀⠀⠀⡀⠛⠛⢋⡙⠋⣟⢕
⠀⠠⠿⠜⠉⢣⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠿⠻⠃⠻⠇⠀⠿⠗
⠀⠴⠤⠶⠤⠷⠤⠦⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠈⠁

⠀⠀⢠⣴⠶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⢆⡄⠀⠀⠀⠀⠀⠀⢠⣤⠀⣴⡄⢠⡄⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⣼⣿⢈⠁⣿⣥⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣬⣥⣤⣽⣤⣬⣥⡄⠀⠀⠀⠀⣤⣽⣥⣼⣧⣀
⠀⠀⣿⠛⢿⡟⠋⣿⠀⠀⠀⠀⠀⠀⠀⡼⠿⡀⠀⠀⠀⠀⠀⠘⢿⣿⢻⣿⡿⣿⣿⠃⠀⠀⠀⢀⣁⡿⣧⣼⢧⣄
⠀⠀⠘⢷⣾⣷⡞⠃⠀⠀⠀⠀⠀⣏⣉⣇⣀⣏⣀⡆⠀⠀⠀⠀⢸⢿⢸⣿⡇⢿⡿⠀⠀⠀⠀⠤⠭⠤⠬⠭⠤⠬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢾⣿⡟⠟⣿⣿⠂⠀⠀⠀⠀⣀⢀⣤⠀⡿⠫⠿⠇⠀⠀⠀⠀⡀⠘⢀⣁⠀⠃⡀
⠀⠀⢹⣿⣾⣟⠁⠀⠀⠀⠀⠀⠈⢲⣥⠞⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢿⣹⡇⠀⢠⠀⠀⠀⠀⣤⣶⢦⡄⣤⢶⣶⡤
⠀⠀⣿⣏⢙⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣧⠀⠀⡄⠀⠀⠀⠀⠉⠙⠛⠁⠉⠛⠉⠁
⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠂

⠀⠀⢠⣤⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⡀
⠀⠀⠈⣇⠶⠶⣺⠃⠀⠀⠀⠀⠀⠈⠧⣜⣫⠼⠁⠀⠀⠀⠀⠀⢰⡆⢠⣯⡄⢹⡏⠀⠀⠀⠀⠀⣠⣤⣤⡀⢴⣶
⠀⠀⠀⠻⢷⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⡟⠈⣿⠁⡸⠇⠀⠀⠀⠀⠀⠁⠾⠻⠟⠯⠁
⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠜⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠈
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠈⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡶⢶⣤⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠹⣧⡀⠀⠀⠀⠀⢻⣆⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⢷⣄⠀⣀⣀⡀⠉⠛⠛⠛⢋⣠⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠙⠛⠛⠛⠛⠛⠻⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠿⠾⠶⠶⠾⠂⠾⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⣦⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠹⣇⠀⠀⣠⣤⣤⣤⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢷⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠙⣿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠫⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⠁⠀⠀⠀⣀⣠⣄⠀⠀⠀⠸⣇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣴⠶⠛⠋⠁⠹⣦⠀⠀⠀⠹⠷⠾⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⣇⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣷⠶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⠶⠿⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠻⣦⣄⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣸⡿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡼⠋⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⡶⠶⡶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠰⣦⡀⢀⣴⠖⠀⠀⣼⣿⠀⢀⣤⡾⠿⠶⣶⡄⠀⣿⡇⠀⣠⡾⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣏⠀⠀⠀⣿⣿⠀⣾⡏⠀⠀⠀⠘⣿⢸⣿⡇⢸⡟⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠙⠳⠄⠀⣿⣿⠀⠹⣧⣀⣀⣀⣴⡿⢸⣿⠇⠈⢿⣦⣤⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠲⠶⠶⣶⣶⣶⣶⣤⣤⣿⣿⣤⣤⣬⣿⣿⣿⣯⣤⣼⣿⣦⣤⣤⣦⣽⣯⣥⣤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣶⣤⡉⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⢿⣿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠹⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣶⡾⠟⠁⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡶⠞⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠳⢶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣮⣟⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠵⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⠀⠀⠀⠀⠀⠀⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠀⠀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⠀⠀⠾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⣀⣀⣀⣀⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⠛⠓⠚⠛⠟⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢻⣿⢟⡿⢛⣛⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡆⠟⣼⣇⣛⣛⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣴⣿⣯⣭⣭⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡬⢛⣭⢭⣝⢿⣿⣿⣿⠿⣋⣭⣽⡭⢁⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢛⣩⣶⣿⣿⣦⢻⣧⢻⣿⠏⣾⠿⣛⣭⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣛⣭⣦⣙⢿⣼⡿⣸⣿⡸⣿⣿⠿⣛⣥⣮⡙⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣴⣿⣿⣿⣿⣿⣷⣬⡁⠿⠿⠇⣩⣶⣿⣿⣿⣿⣿⣷⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣽⣿⣿⣧⣶⣿⠟⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣛⣻⣿⣿⠇⠀⠀⠀⠀⠈⢿⣿⣛⣛⣛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣄⢾⣿⣶⣾⣿⢇⡄⣀⣀⠀⣀⡀⣄⢿⣿⣶⣾⣿⢆⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣌⡳⢭⣍⣭⠵⢚⣴⣿⣿⣷⣌⢳⣜⠲⢭⣭⣭⠵⣋⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠿⠿⠿⠷⠶⠶⠾⠿⠿⠏⣿⣿⣿⡆⠿⠿⠶⠶⠶⠿⠿⠿⠏⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣾⣾⣶⡾⣛⡛⣶⣾⣶⡆⣿⣿⣿⡧⢰⣾⡞⣛⡻⣶⣿⣾⡆⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⣿⣿⣧⣛⣃⣿⣿⣿⡇⣿⣿⣿⡇⢸⣿⣆⣛⣣⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣬⣭⣭⣭⣭⣭⣭⣭⣭⡅⣿⣿⣿⡟⢨⣭⣭⣭⣭⣭⣭⣭⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢂⠉⠉⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠁⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡠⡀⡤⢌⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣢
⠀⠀⠀⠀⠀⢌⠠⠊⠀⠀⠀⠀⠀⠀⣀⠤⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⢀⠀⠀⠀⠀⠀⣀⠀⢀⡁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠔⠀⠉⠀⠂⠠⠐⠂⠉⠀⢀⣀⣀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠈⢱⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠁⢠⠊⠀⠀⡠⠀⠂⠀⠄⣀⠀⣀⠔⠁⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠇⠀⠀⠜⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠤⢀⡰⠂⠈⠀⠐⠄⠀⠀⢸⠀⠀⠀⠀⠀⠆⡠⠐⠀⠢⡀
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠘⠀⠠⠐⠀⡁
⠀⠀⣀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠄⠀⠀⠀⢀⠂⠀⠀⠠⠃
⡐⠁⠀⠈⠲⠀⠀⢰⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡢⢀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⢘⢀⠠⠐⠁
⠇⠒⠐⠦⠀⠆⠀⠀⠑⠠⢀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⠋
⠐⢄⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠉⠣⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡃⠠⠤⠀⠐⠂⠉⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠑⠠⠄⠤⠓⠄⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⢀⡀⠀⠀⢀⡀⠀⠤⠀⠀⠒⠈⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠠⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠄⠀⠀⠀⠱⠀⠆⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡀⠂⡉⠐⠠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠢⢀⣀⡘⣄⡀⠀⠀⢙⠀⠀⠀⠰⢀⠤⠀⠊⠬⡄⠀⠀⠀⠀⠐⡀⠊⢀⢀⠀⠡⡑⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢀⠒⠉⠀⠀⠀⠀⠁⢢⣀⢄⠀⡔⢻⢅⠀⠀⠀⠀⠈⠂⠀⠀⢠⠂⠉⠉⠀⢀⠇⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠔⡇⠀⠈⡁⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⡌⡀⢴⡆⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⠀⠀⢠⡀⠀⡠⠒⠰⠁⠀⣰⠁⠀⠠⠐⠁⠀⠀⠀⠀⠈⠒⠒⠈⠑⠱⠄⠂⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⡿⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣴⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡾⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠷⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠷⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣾⠟⠁⠀⠈⡇⠀⠀⠀⠀⠀⣠⠔⠻⢷⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⡏⠀⠀⠀⢠⠇⠀⠀⣠⢔⡯⠂⠀⠀⠀⡇⠀⠀⢀⣤⣶⣶⣦
⣠⡾⠉⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢸⣦⣤⣤⠏⠀⣠⣪⣵⠟⠀⠀⠀⠀⠀⢸⠀⢠⠟⠋⠁
⢿⣧⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢰⣷⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⣄⣉⡍⢀⡜⣵⡿⠃⠀⠀⠀⠀⠀⠀⢸⢀⣇⠀⠀⠀⢀⡜
⠈⠻⣇⠀⠀⢀⣾⠧⡀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠋⠉⠉⢣⡜⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡞⠻⢷⣤⠔⠉
⠀⠀⠀⠉⠛⠻⣅⣴⡿⠑⠒⠒⠶⠤⢯⣿⣇⠀⠀⠀⠀⠀⠀⢀⣜⡵⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⣀⡤⠔⠚⠉⠻⢶⠊
⠀⠀⠀⠀⠀⠀⠈⢫⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⢀⢎⡾⠁⠀⠀⠀⠀⠈⡇⠀⠀⡠⢔⣯⡷⠋⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢏⣿⡇⠀⠀⠀⠀⠀⠀⢸⡤⣪⣾⡿⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣿⣦⣄⠀⠀⠀⠀⢀⣷⣿⣷⣾⣿⣷⣶⣦⣤⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢽⡻⢿⣦⣤⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠳⡄⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣶⣿⣆⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣌⣉⣛⣛⣻⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠔⠒⠒⠠⠤⠤⢄⣀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠈⠉⠁
⢈⠉⢆⠀⠀⡤⠂⠉⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠒⢤⠀⠀⠀⡠⠢
⠀⠈⠈⠀⢰⣀⠀⠠⠄⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠤⠀⣀⡀⠀⡃⠀⠰⠄⠊
⡠⠀⠢⠀⡸⢆⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢠⡇⠀⢀⠀⢀
⠑⠊⠀⢠⠃⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣀⣀⣀⡀⠀⠀⠀⠈⠀⠂⠈⠀⢸⡀⠈⠢⠤
⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠉⠑⣦⡀⠀⠀⠀⠀⠀⠀⠉⠳⡀
⠀⠈⠢⢴⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡤⠜
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠙⠉⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠹⠿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠟⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⡨⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣁
⠀⠀⠀⠀⠀⣜⠉⢃⠀⠑⢍⣲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢊⠕⠋⠨⠗⢦
⠀⠀⠀⠀⠰⣁⠀⠁⠀⠀⠀⠀⢀⡀⠉⠙⠒⠶⠤⠤⠤⠤⠤⠴⠒⠚⣩⣀⠀⠀⠀⠀⠐⠀⠚⡄
⠀⠀⠀⠀⠀⠘⠤⠔⠒⠊⠉⠉⣹⣁⡤⠒⢤⠀⠀⠀⠀⠀⢀⠤⠢⢄⣿⠈⠉⠙⠒⠤⣄⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠉⠈⠄⠀⢸⠒⠢⠤⠬⠕⠺⡄⠀⠈⢀⠋⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⢸⡀⠀⠊⠀⠀⠐⡁⠀⠀⠀⠈⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⢃⠧⢀⣠⣄⣀⢼⣱⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢀⡸⠁⠀⠀⠀⠀⠀⠀⠙⢄⠀⣀⠜
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡄⢀⡴⠶⠦⡀
⠀⠀⠀⠀⠀⠀⣴⠞⠓⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣾⠀⣧⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣶⣶⡇⠀⢸⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣠⣤⣀
⠀⢀⠀⠀⠀⠀⠀⢳⣤⣴⣾⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣏⢈⣉⣉⡇⠀⠈⣷⢰⣶⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⡆
⣾⠁⠈⠱⣄⠀⠀⠈⣇⣥⣾⣧⣠⣴⣾⣿⣿⠿⠿⣿⣿⣷⣾⠿⠸⠿⠿⢧⠀⢸⣷⢰⣦⣿⠀⠀⠀⠀⠀⠸⡆⠀⢠⡇⠀⠀⢀⣀
⠙⢄⠀⠀⠙⣆⠀⢀⡟⢈⣵⣿⡿⠿⢿⣿⣇⠀⠀⣽⣿⣿⠏⠀⠀⠀⠀⠀⢇⡾⠉⠈⠉⠉⢇⠀⠀⠀⠀⠀⣧⣤⣼⠁⣠⠞⠉⠉⠛
⠀⠀⠙⢦⡼⢟⣧⣼⢣⣾⣿⣿⡀⠀⣠⣿⡿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⢸⡧⣿⣿⠞⠁⠀⠀⠀⣠
⠀⠀⠀⠀⢻⡞⠋⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠘⣄⣀⣤⣿⡶⣋⢹⣶⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠈⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢿⡷⠟⡏
⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠜⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠹⣜⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣸⠀⠀⢠⠊⠀⠀⠀⡿
⢀⠔⠒⠚⠛⠒⠒⠺⣿⡟⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡟⢀⡴⠁⠀⠀⠀⢰⠇
⣾⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣼⠟⠁⠘⡄⠀⠀⣠⠏
⠘⠷⣤⣤⠤⠶⠒⠚⠋⠉⠉⠛⠳⢶⣤⣄⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢠⡿⠷⡀⠀⠀⠀⣰⠃⠀⠀⠀⠙⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣮⣝⣛⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢻⡿⠃⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡼⣧⢱⡟⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⡄⠀⢿⣷⠁⠟⣼⣿⠀⢠⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⠀⠀⠸⣩⠞⡳⢮⠻⠀⠀⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠆⠀⣿⣿⠀⠀⢀⡻⣌⣇⡿⡀⠀⠀⣿⣿⠀⠠⢾⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣶⢰⣾⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⠀⠀⢰⣶⣶⢰⣶⡦⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⡀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⠀⠀⠠⡄⠈⢢⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠀⠀⠀⠸⠀⠀⠚⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠐⠉⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠈⠀⠁⠀⠀⠀⢠⠄⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠢⢅⣀⠤⠀⠀⠀⠐⠈⠀⢄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠁⠀⠐⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠉⠃⠂⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⠄⣀⣀⣀⠠⠦⠄⠀⠔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⠀⠀⠀⢡⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢂⠀⣅⠣⢠⡝⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⢀⠜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣒⣒⣒⣂⣁⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡰⠀⠀⠀⠘⡝⣿⣿⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡃⠀⠀⠀⠀⠴⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠫⠁⠽⢧⠀⠀⠀⢀⣂⣿⣅⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⠀⠀⠀⠀⠀⢀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢸⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠭⢽⣛⣷⣾⣿⣿⡘⣧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣩⣥⠠⠤⠤⣤⡄⣴⣾⠩⢿⣻⣛⡛⠡⠌⠳⢶⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠂⠀⠙⠛⠛⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⡾⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠙⣧⠀⠀⠀⠀⢻⡄⣴⠏⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⣴⠋⠀⠀⠀⡶⠛⢷⡞⠛⣦⠀⠀⢰⡏⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⣷⠀⠀⢸⣇⡀⠘⠁⢀⣿⠀⠀⠀⣹⠆⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⠀⠸⣏⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⢸⡇⢀⡀⢸⠀⠀⠀⢠⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⣇⣿⠀⠀⠀⢸⣗⣏⣿⣸⠀⠀⠀⢸⣇⣞⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠹⠏⣿⠀⠀⠀⢸⡿⠿⠿⢹⠀⠀⠀⢸⡯⣭⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣤⣤⣿⣄⠀⢀⣼⣧⣤⣤⣼⣄⠀⢀⣼⣧⣤⣤⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣼⠀⢸⣇⣀⣀⣀⣀⣸⠇⢸⣄⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣻⣿⣟⣻⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠟⠁⠀⠀⠠⠄⠀⠀⠀⢠⣤⣄⠈⠛⠷⣦⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠁⣴⣤⠶⠒⠛⠙⠋⠉⠉⠉⢻⣿⡿⠶⣤⡄⠈⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⣷⣤⣤⣤⣤⣴⣶⣶⣶⣬⣿⣧⣤⡴⠛⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡎⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢘⣆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣷⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢸⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⣾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⣈⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⣀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣴⣿⣶⡄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠝⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⠆⠨⠥⠄⠀⠀⣶⣶⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠐⠚⠓⠒⠀⠀⠀⠀⠛⠛⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣿⣋⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠬⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠁⢽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣍⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣁⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣄⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⣿⣿⣿⣿⣿⠇⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠘⢿⣿⣆⠈⠛⠛⠛⠋⢀⣾⣿⠟⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⣴⣶⣶⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠉⢹⣿⣿⡇⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠄⢸⣿⣿⡇⠀⠀⠀⠀⠻⠟⢸⣿⣿⢿⣿⣿⣦⣿⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⣈⣉⣉⣀⣈⣉⣉⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⢸⣿⣿⡛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠛⠉⢹⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠿⠿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣴⣴⣦⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠘⠿⠿⠘⠿⠿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣴⣶⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣭⣭⡉⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠁⢀⣼⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠰⠾⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠲⠤⣄⠀⣴⣦⡼⠛⣋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣴⣎⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⢠⣹⣇⣰⣦⠀⠀⢀⣾⠋⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡜⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣍⢹⣿⣀⠀⣸⡏⠆⡉⠥⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢸⣷⣦⣶⣦⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣜⠿⣿⣥⣽⢀⡬⢂⣠⣤⣤⠀⠀⠀⢾⣿⣿⣿⣿⡆⠀⣿⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣌⢱⡼⣿⣿⣾⣣⣿⣯⡾⣿⣷⡀⠀⠸⣯⡶⠛⠋⠑⠛⠰⣶⣦⣤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⣿⣿⣿⡿⠿⣞⡥⣶⣻⣿⣿⣄⣇⠐⢿⡇⠀⠀⠀⠀⠹⡏⠸⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣻⣷⡻⠋⠉⠛⣿⣿⣿⣿⣿⣷⣾⣿⠶⣄⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡩⢴⣿⣿⣿⠂⠀⠀⢠⣿⣿⣿⣿⣿⣟⣿⡏⠀⠘⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⢿⣿⣿⠢⠀⠀⣿⠈⣿⠟⠋⢉⣉⣀⣀⡙⠂⠀⠀⠀⡗⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⣟⠺⣆⠀⣙⣾⠃⣴⣾⣿⡟⣿⣿⣗⣦⡄⢰⢠⠃⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠿⢿⣷⣧⠹⣶⣿⠟⠛⢛⣁⣐⣾⣿⠃⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠭⣙⣛⣿⣿⣷⡘⣿⠷⠟⠋⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠒⠒⠒⣒⣒⣶⣿⣿⣿⡌⠃⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⢈⣉⣩⡭⠭⠶⣒⣿⣿⣧⣤⣤⣴⣾⣿⡾⣟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⢾⠀⠀⠀⠀⢀⣠⠴⠖⠋⣩⣶⢟⡿⢫⡟⢹⠻⡇⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⠀⠈⠉⠀⢀⣴⡿⠋⣠⠏⢠⠟⠀⠼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠚⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⠄⠒⠀⠛⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠓⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡴⠖⠛⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣉⠙⠳⣼⠃
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠖⠛⣉⣉⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣉⣉⡙⠳⣿
⠀⣠⣶⠿⠛⠛⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠛⣉⣩⣭⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠉⠳⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠋⠉⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⢠⣟⣁⣀⣀⣀⠀⠀⠀⠀⢻⡀
⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢈⣻⣴⡶⠿⠛⠛⠋⠉⠉⠙⠛⠛⠷⢶⣼⣇
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀
⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⣠⡾⠃⣸⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡾⠋⠀⢠⠖⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⣁⣠⠞⠁⣠⡏⢹⡦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⣼⠟⠀⠀⠀⢇⢸⣿⡿⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣠⣤⠾⠋⣠⡟⠁⠈⠛⠶⣤⣀⠀⠀⠀⠈⣿⠀⣾⠏⠀⠀⠀⠀⠀⠉⠉⠀⠀⢰⠋⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣿⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢠⠊⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠸⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⣀⣀⣀⣠⣴⣿⠟⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠉
⣀⢀⣀⡀⣀⣀⡀⣀⣀⢀⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⠛⠛⢻⠛⣋⣁⣀⣀⣀⣀⣉⣛⠛⠷⣶⣶⣶⣶⣶⣶⠟⠛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠶⢸⣏⠀⣿⣿⣿⣿⣿⣿⡗⠶⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⣉⣿⢸⣿⠁⠀⠀⠀⠀⠀⣿⣉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣧⡀⣿⠀⣿⣿⣿⣿⣿⣿⢀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠘⣿⠀⣿⡀⠀⠀⠀⠀⠀⣿⠃⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠹⣇⢹⣿⣿⣿⣿⠏⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⣇⠹⡇⠀⠀⠀⠀⣸⣡⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣟⣛⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠤⠤⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠛⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣤⣤⣤⣤⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣄⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⢤⡀⠀⠀⢸⣃⢀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣦⣶⡄⠀⠘⢿⠢⣧⡄⠀⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⢱⡿⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠀⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⣶⡦⣬⠐⠄
⠀⠀⠀⠀⠀⠀⠀⣤⣰⡀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣟⠉⠁⠀⠀⣈⣉⣁⣬⣤⣴⣤⣤⡈⠙⢛⣛⣿⣿⢇⡤
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠹⣆⠀⠀⠀⠀⠑⠉⡙⠛⠛⠫⠾⠿⢄⣤⣈⣉⢙⣿⣽⣄⡙⠿⠿⠖⠛⠛⣋⣤⣾⣷⡾⠧⠯⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠶⢦⣶⣶⣿⣿⣿⢿⡿⠓⠒⢄⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⣿⣿⣿⣿⢁⡾⠀⠀⠀⠨⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⡄⢿⡃⠀⠀⣠⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⠋⣼⣿⠿⠏⢀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡍⢰⣷⣉⣠⣬⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣦⡈⢤⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣉⣑⠈⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡴⠞⠉⠙⠻⢦⡑⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⡒⢀⣴⣿⣿⣿⣿⣿⣩⠙⠲⢄⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡾⠁⠀⠀⠀⠀⡴⡀⠈⠺⣿⣿⡆⠤⢄⣀⣀⣀⠀⠀⣐⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⠰⠋⠀⠀⢡⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠘⠅⠀⠀⠀⠰⣎⣰⣋⡠⣀⠈⠻⣏⠢⠤⠍⠛⡉⠻⠿⠟⠛⠛⠿⠟⠛⠛⠋⠈⠁⠀⠀⠀⢀⠠⠃⣴⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠓⠆⠀⠀⠙⠲⠖⠃⠀⠀⠸⢆⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⡻⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⢄⣶⣦⣄⣀⣀⢹⠟⠒⠂⠀⠀⠀⠀⠁⠀⠀⡠⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠛⠛⠛⠟⡻⢿⠿⣯⠾⠿⠟⠃⠀⠘⠛⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠉⠛⠋⠋⠙⠉⠛⠻⣻⠿⣿⡛⠁⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠄
⠀⠀⠀⠀⣾⢟⣿⣿⣿⠿⣻⡿⠋⠁
⠀⠀⠀⣰⠋⠞⡹⠋
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣦⣤⣤⣤⣄⣞⣶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣿⢿⣿⣿⣼⣷⣦⣿⣷⣯⣿⣿⣟⠕⣡⡶
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣯⣽⣾⣿⡿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣮⠷⣆⠄
⠀⠀⠀⠀⠀⠐⡁⡀⠀⠀⠀⠀⠀⣠⡿⣫⣽⣿⠟⠋⠁⠀⠀⠀⠘⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣬⣧⣂⡀
⠀⠀⠀⠀⠀⠈⢊⣄⣠⣤⣦⡶⠿⢉⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⢶⣿⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡴⡾⠿⠿⠏⠛⠀⠈⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣾⡿⣧⣾⣶⣤⡉⠏⠻⣿⣿⢶⣯⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠉⠐⠂⠀⠈⠀⠰⢒⣤⠤⣄⣀⠀⠀⠀⡈⠻⣿⣿⣷⣿⣿⡿⠟⠀⣀⣥⣾⣿⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⢀⡼⠄⠘⠈⠁⠡⠛⢓⣶⡶⣶⣤⣷⣶⢶⣶⣾⡿⠿⠿⠏⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠉⠙⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣶⡄
⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣤⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣤⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⣿⠏⠉⣿⡝⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⣽⡏⠉⢿⡝⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣰⡟⠀⠀⠘⣷⡀⠀⠈⢿⣆⠀⠀⠀⢀⣾⠏⠁⠀⣰⡟⠀⠀⠘⣷⡀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢀⣿⠃⠀⣠⡾⠋⠀⠀⠀⠀⠈⢿⣦⡀⠀⢻⣇⠀⢀⣿⠃⠀⣠⣾⠏⠀⠀⠀⠀⠈⠻⣦⡀⠀⢻⣇⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣼⡿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣿⣤⣼⡿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣿⣤⣤⣴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣀⣀⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⣀⣀⣀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁




⣄⣀⣀⣀⣀⣄⣠⣀⣀⣠⣤⣠⣀⣀⣀⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣤⣀⣠⣤⣤⣤⣤⣤⣤⣀⣠⣀⣀⣀⣠⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠔⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠁⠄⠉⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⡀⠀⣤⣠⣶⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣆⣿⠏⠿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠒⠻⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⣿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⡵⡀⠀⡼⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣀⡶⢦⣬⣿⣿⣿⣿⣿⣿⣯⠀⢈⠆⠀⠀⠀⢠⣤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠒⠂⣁⣂⡀⢬⢻⠏⠉⠒⠺⠾⣿⣿⣿⣿⣿⣿⣿⣧⣼⠀⠀⠀⠁⠀⠀⠀⠹⠛
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠱⣿⠥⠤⣍⣚⣻⣿⣿⣿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠛⠁⠰⠟⠛⠿⠿⠿⠟⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⡀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⣠⡄⠠⣤⠀⠚⠀⠀⠀⠀⠀⠀⣾⢠⠂⠀⠀⠀⣠⣽⡁⠀⣦⠸⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⠀⡄⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⢠⣾⡟⠁⢨⠃⠀⠀⠃⠠⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠺⣿⣻⣿⣿⣿⡏⠀⠀⣤
⠀⠀⠀⠲⣦⣤⡼⢿⣿⣿⣿⠟⠋⠀⠘⣥⣤⣶⠆⠀⣠⣤⣤⣶⣶⣿⣦⣀⠉⠙⠻⠿⢃⣤⣶⣥⣶⣶⣦⣤⣤
⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡟⠛⠋⠀⢨⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠙⠛⠛⠉⠉⠁⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣯⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡐⠁⠀⠀⠑⠀⠀⠅⠀⠀⠀⠀⢸⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠘⣤⣤⣤⡌⠀⠀⠀⡟⠛⠛⢛⠀⠀⠀⢣⣤⣤⣤⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠶⠶⢶⠂⠀⠀⣭⣉⣩⡇⠀⠀⢀⠿⠿⠿⠿⡄⠀⠀⢸⣍⣩⣭⠀⠀⠐⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠿⠶⠾⠄⠀⠰⠉⠉⠉⠡⠀⢀⠎⠀⠀⠀⠀⠘⡄⠀⠌⠉⠉⠉⠆⠀⠠⠷⠶⠷
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢡⡌⡰⠀⠀⠀⠀⠀⠰⡌⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⢸⡠⠇⠒⡒⠒⠒⠠⢄⣇⠀⠀⠀⠀⠐⠸⠆⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠰⠀⠀⠀⢀⠔⣡⠖⠉⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠆⡄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⢃⠞⠁⠀⠀⠀⢠⣦⡄⠸⠶⠀⠀⠘⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⡞⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠳⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⠄⠀⠂⠀⢣⠀⠄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠠⠀⡜⠀⠐⠀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡠⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢌⠂⠀⠀⠀⠀⠀⠠⠔⡫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠠⠃⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠲⣀⠀⠀⠈⣄⠈⠀⠀⠀⠰⠆⠀⠀⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢤⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠈⠁⠔⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐⠀⠀⡐⠀⠀⢰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⢰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠑⠀⠀⠀⠀⠰⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠄⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢌⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠁⠊⠓⠀⠀⠀⠀⠀⠀⠠⣤⠤⠶⢦⣤⡤⢤⣀
⠀⠀⠀⠀⠩⠉⠉⠉⠙⣟⠀⠁⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠓⢦⣌⡑⠢⢄⡀
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠘⢧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠈⠉⠲⢄⣈⠒⠤⡄⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⢧⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠁⠉⠢⡀⠈⠒⠤⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢣⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠈⠔⠠⠀⢸⠀⡧⡀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠱⡐⠄⠀⠀⠈⢀⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠰⠃⠀⢁⠀⠀⣇⢀⡄⠀⢀⠐⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠹⡌⢀⠀⠀⠀⠑⠈⠂⠠⢍⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠸⠅⡶⠋⡎⠀⠵⢃⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠂⠀⠀⠀⠈⠂⠀⢀⠁⣀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠠⠄⠣⠀⢀⣀⠈⠀⠀⠊⠁⠀⡘⡀⠦⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠣⠁⡀⠀⠀⠀⠀⢦⡀⠈⠦⣐⠀⠠⠺⠀⠀⠀⠄⠠⠊⠀⠀⠀⠁⢁⠀⠙⡀⠐⠀⠀⠀⡰⢻⠁⣰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠡⡘⠄⠀⠀⠀⠀⠠⡉⠲⠀⠀⠁⠐⢇⠰⡀⢀⠀⡇⠀⠀⠀⢠⠰⠀⠀⢰⣆⣀⣠⢎⠔⠁⠘⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢂⠀⠀⠀⠀⠘⢄⠐⢄⠀⠀⡀⠀⠐⣈⠂⠁⡀⠀⠀⠎⠀⠱⡀⠌⠁⠀⠜⠁⠀⢀⡤⠔⠈⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠈⢀⠢⡀⠀⠀⠀⠀⠁⡀⠈⠐⠇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠔⠘⠀⠀⢀⣌⠐⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠈⠂⠑⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠁⠀⠀⠀⠃⡀⠱⠤⠴⠋⠁⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠈⠠⠀⠠⠀⠀⠀⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⡒⡞
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡥⢒⣳⡀⠀⠀⠀⠀⠀⠀⣰⣓⣒⢬⠇
⠀⠀⠀⠀⢀⡶⠋⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⠵⠒⠳⡄⠀⠀⠀⠀⡴⠁⠀⠀⠙⡄
⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⢦⢀⣀⣼⣥⣤⣤⣀⣀⡇
⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⣱⢄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⢞⣡⠎⡷⣤⡀⠀⠀⠀⠀⡇⠀⣰⠋⠀⢊⣿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡖⣊⡴⠃⠈⠓⢦⣀⠀⢿⣼⠁⠀⠀⠈⠁⠀⢈⣾⠇⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠈⠙⠾⣇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢁⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠾⠛⠉⠙⠲⠤⣤⣄⣀⣤⡤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠖⠒⠊⠙⡆
⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠖⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀
⠀⠀⠀⠀⠀⣼⠀⠀⣀⠤⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠉⠉⠙⢦
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠓⡆⠀⠘⡇
⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠘⡆⠀⠸⡄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠱⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⣄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⠏⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠳⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡴⠃⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠦⠤⣤⠤⠤⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⢿⣀⣀⣀⣀⣀⣀⣀⣸⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⡀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠼⠁
⠀⠀⠀⠲⠤⣐⢆⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⠉⠉⠉⠉⠙⠃⠘⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢊⡷
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠂⠁⠈⠳⣄⠀⠀⠐⠶⠛⠛⠒⠛⠉
⠀⠀⠤⠤⠭⠭⠶⠦⠄⢸⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⠀⠠⠖⠛⠛⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⠀⠰⢦⣚⡛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⠃⠀⠀⠀⠀⠈⠉⠓⠲⠦⢤⣄
⢀⣀⣀⣤⠴⠀⠀⠀⠀⣼⠋⣝⠂⠀⠀⠀⢀⡠⠞⠁⠘⠛⠟⠓⠀⠀⢀⠀⣀⡋⠛⠒⢶⡄
⠈⠉⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠉⣀⠴⠈⠉⠀⠀⠀⠐⠀⠐⡾⠃⠰⠏⡜⢁⡤⠄⠀⠀⠙⠢⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⠁⠀⠀⠀⣠⠌⢻⣛⡷⠀⣠⣤⣤⣀⠀⢏⣰⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠈⢀⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⢀⡾⢱⡏⠉⠒⠶⠤⢤⡄⣠⡾⠃
⠀⠀⠀⢀⠤⣲⠟⠀⣾⠀⠀⠀⠀⠀⢠⠏⣼⠿⢶⣤⣀⡀⠣⠴⠋⣀⡠⠴⠞⠋⠀⠈⠃⠀⣀⣠⣴⠶⠛⠉
⠀⠀⠀⠈⠛⠁⠀⠀⢿⡀⠈⠀⠀⣠⠟⢠⡿⠀⠀⠀⠈⠉⠁⠀⠉⠀⢀⣴⠶⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣢⣴⣾⣥⡶⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⣰⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠻⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⣦
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀
⠀⠀⠀⠀⠀⠀⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡤⠈⢹⣦
⠀⠀⠀⠀⠀⠀⣿⠻⡷⠦⣤⣤⣄⠀⠀⠀⣀⣀⣀⣀⣠⡴⠟⠀⠂⢸⠌⢧⡀
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠙⢦⡈⢉⡉⠉⠉⠉⠁⠀⠀⠀⠀⡿⠀⠀⢹⡆
⠀⠀⢀⡾⠛⠳⠒⡙⠆⠀⠀⠀⠀⠀⠙⢦⡀⠉⠒⠦⣤⣀⠀⠀⣼⠃⣠⡴⠛⣄
⠀⠀⠸⡇⠀⣴⠚⠉⢠⡤⠤⠤⠤⠤⠤⠤⠽⠶⣄⡀⠀⠉⠃⣰⠟⠛⠁⠀⠀⠘⣦
⠀⠀⠀⠹⣆⢻⣆⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠼⢃⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⠳⣍⡻⣦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⣧⠀⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠈⠻⣮⣙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠳⠀⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠛⠶⢤⣄⣀⠀⠀⠀⢀⣼⠏⠙⠛⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣻⣋⣙⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣽⣿⣯⣭⣿⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠄⠀⣀⡀⠀⢀⣤⡤⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢩⣤⡰⠿⣃⡀⠚⠛⣧⣤⣤⠛⠛⢟⡀⠀⣼⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡿⠈⠛⣁⢰⡿⣣⣤⠌⠉⠉⢡⣤⣀⢼⣿⡏⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠄⠰⠿⡷⠘⠿⠃⠸⠛⠉⢶⣶⣦⠺⢿⡏⠀⢨⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣧⡄⢴⣾⠟⠓⢀⣿⣰⣾⡿⠋⢿⠏⠀⣰⣶⠄⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠐⠿⠏⠀⣠⣷⣤⣾⡿⠏⣿⠁⢤⣴⡿⠗⠘⢻⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠻⡃⠀⣸⣷⡦⠻⠿⠷⣀⣤⡆⢠⣶⡄⢾⣶⠆⠚⠿⠟⠁
⠀⠀⠀⠀⠀⠀⣤⣀⢸⣿⡗⠉⠛⢃⣀⣀⡐⠛⢫⣾⠇⢉⣤⡀⣾⣶⠄
⠀⠀⠀⠀⠀⠸⠿⡟⠀⠈⣵⣤⣤⠛⠛⢻⣤⡴⠈⢩⣶⠎⠛⣃⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠚⠛⠁⠀⠈⠉⠀⠐⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣧⣤⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⡰⠁⠀⠀⠀⢱
⠀⢠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠁⠀⠀⠀⠀⢸
⠀⣇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⠀⠀⠀⠀⢠⠇⠀⢀⣠⠄
⠀⠘⣄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡠⢃⡤⠒⠉
⠀⠀⠈⢶⡾⠟⢧⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⣈⣉⡉⠐⠒⠤⡑⠂⠸⣏⠥
⠀⠀⠀⠈⢷⡾⠟⠳⣄⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣷⣦⡈⠳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣧
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡇⡟⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠑
⠀⠀⣀⠤⠤⢜⣆⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠗⠒⠤⠤⠤⠤⣾⡝⣿⡆⠀⢠⡶
⠀⡸⠁⠀⠀⠀⠙⡧⡤⠤⠒⠒⠦⢿⣿⣿⠘⣿⢣⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠟⠛⠊⠉
⠀⠱⡀⠀⠀⠀⢠⣧⡇⠀⠀⠀⠀⠀⠻⢷⠀⠀⡼⠋⠁⠀⠀⠀⠀⠀⠀⢀⡠⠔⣠⠞⠁
⠀⠀⠉⠒⠒⠚⢻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠲⠾⢓⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣓⢤⣀⣀⠀⠀⢀⡴⠃⠀⠈⠒⠒⠒⠒⠒⠒⠒⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣷⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⠥⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠑⣤⡴⢣⣀⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⠰⠶⠶⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠉⠀⠑⣠⠴⣒⣿⠀⢠⠆⠀⠲⡈⠉⢳⠋⠀⠀⢀⣶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠐⠋⠀⠀⠈⢹⠀⠀⣸⠃⢠⡅⠀⠀⠊⠀⠀⠙⠋⠀⠀⢀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⡀⠀⢳⡀⠀⠀⢀⡛⠶⢶⡯⡐⠓⡜⡝⠁⠀⠀⠀⢀⠀⣀⢦⡁
⠀⠀⠀⠘⢦⡀⢀⡞⡵⠃⠀⠉⠀⠀⠹⣦⡀⡈⠉⠛⠛⠎⠀⣼⠁⠀⣠⠴⠚⠉⠁⢀⡨⢷⣰⡄
⠀⠀⠀⠀⠀⠈⠘⠻⠁⠀⠀⠀⠀⠀⠀⠹⣷⣅⢰⡀⢀⠀⣼⣏⡴⠈⠀⢀⣠⡴⠞⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⢰⠹⣿⣿⣧⣿⣼⣿⣿⣡⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡛⢶⣦⣄⣲⣾⡯⠙⠓⠯⣽⡻⣿⣥⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢤⣤⣤⣍⣿⣯⣿⣄⣀⣀⣀⣀⡹⡽⣿⣶⡶⠶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢹⣿⣿⣿⣿⣿⣿⣿⢄⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⣿⣿⣿⣿⣿⠟⠜⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠉⠉⠉

⠀⠀⣀
⠀⠀⠙⢿⣿⣖⡢⢤⣀
⠀⠀⠀⠀⠙⢿⢏⠓⠤⡀⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠑⢌⠢⣄⠑⠢⣄⡀⠈⠁⠒⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣷⣌⡙⠿⣦⣤⣀⠀⠉⠋⠃⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⡻⣿⣶⣌⠙⠻⣫⣀⠔⠒⣦⣀⣴⣿⣶⣶⣄⠀⠀⠀⠉⠑⣒⠤
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⣿⣮⣿⣿⡿⣴⣿⣣⣀⣴⡟⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣉⠇
⠀⠀⠀⠘⠄⢸⡀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣱⣿⣿⢹⣿⣿⡇⣿⣿⠟⢿⣿⣿⣿⡇⠒⠀⠈⢿⣝⡇
⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⠀⠀⠀⠀⠈⢻⣾⣿⣿⣿⣿⣿⢻⣿⣿⣤⣾⣿⣿⣿⡇⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣶⠶⠤⠤⠤⠥⠉⠛⠿⣿⣏⣿⣟⣟⣿⣿⣿⣿⡿⢁⣤⣖⠁⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⠚⠉⠁⠈⠛⠋⠉⠉⠛⠋⠹⡷⠟⠛⠛⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⡤
⠀⠀⠀⠻⢿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡿⠿⣿⡛⢿⣦⠀⠀⠀⠀⠀⠀⠀⣀⡄⢐⣿⣿⢟⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠉⠀⠀⠀⢲⣧⣨⠿⠖⠒⠀⣿⡆⣠⣶⣾⣾⣽⣿⣿⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠔⠁⢀⡀⣠⣶⡚⣿⡇⣿⣿⣿⣿⠻⢿⣿⣾⡻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⣖⣸⣷⣾⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣼⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⡄⢹⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠏
⠀⠀⠀⡖⠒⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡛⠛⣛⡛⠛⠛⡇
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠶⠛⠙⢲⡄⠈⢻⡀⢷⠀⠀⠀⠀⠀⣤⣤⣧⣤⣄⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠈⠉⠉⠀⠀⡷⠶⡆
⠀⠀⠀⢻⡀⢸⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠈⢷⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠰⠖⠛⢳⡄⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠳⣾⢀⡀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⣷⣄⠙⠲⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠰⣞⠁⠀⠀⠀⡇⣸⠃
⠀⠀⠀⠀⠀⠸⣎⢧⠀⠀⠛⠛⠛⠛⠃⠀⠀⢠⡟⠈⠓⢦⣄⣀⠀⠀⠀⠀⠀⣀⣠⠴⢿⣇⡄⠀⠠⣄⣀⣨⠇⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢮⡓⠦⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⡿⠙⠛⠛⠛⠋⢯⠀⠀⠀⢻⡻⣄⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡶⠤⠤⢴⡖⠋⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⠶⣥⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣼⠁⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣤⣤⣤⣤⣬⣧⡄⠀⠀⢀⡴⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⣦⠀⢀⣼⣧⣤⣤⣤⣬⣧⡀
⣿⣿⠷⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⡶⠶⠾⢿⠿⠿⠿⠛⡻⠛⠛⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡧⠸⠿⠿⠿⠿⠛⡿⡟⠛⡻⠛⢟⠋⢙⠉⢉⠉⠹⠉⠉⡄⢠⠃⠘⡀⣘⠀⢑⣀⣰⣃⣠⣵⡤⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⣥⣥⣴⣧⠤⠼⣦⣾⣶⣾⠶⠷⠶⠖⣿⣿⣿⡿⠛⠛⢋⣽⣿⣿⠟⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠁⠀⠀⣴⣿⣿⡿⠁⠀⠀⣠⣾⣿⡿⠋⠀⠀⣠⣿⣿⡟⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⣼⣿⣿⠟⠀⠀⢀⣼⣿⣿⡟⢁⣀⡀⠼⠿⠟⣯⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡏⠀⠀⢰⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⠃⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⡇⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⣿⣶⠉⠉⠉⢳⣶⣶⣶⠉⠉⠉⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⡀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠉⠉⢱⣶⣶⣾⡛⠛⠛⢣⣤⣤⣾⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⢻⣿⣿⣧⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣻⣿⣷⣾⡍⠉⠛⠃⢦⣤⣤⣿⠛⠛⠳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠈⢻⣿⣿⣧⡀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢷⣶⣶⣶⣏⠉⠉⠁⢰⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⣿⣿⣿⣿⣦⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠸⣿⣿⣿⣯⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀⣀⣀⣀⡹⠿⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⢛⣧⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⡀⠀⠀⠀⠀⢀⠔⢢⠀⠀⠀⠀⠀⠀⠀⢘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⢄⠢⡀⢁⡀⠀⠀⠀⠈⡀⠀⠇⠀⠀⠀⠀⠀⠀⠺⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠘⡯⠟⠢⢀⠀⠀⠘⣤⣶⡄⠀⠀⠀⡔⠁⠀⠀⠀⢀⢤⠊⠱⠀⠀⠀⠀⡁⡱⣾⡤⠂
⠀⠀⠀⠀⠀⢀⣀⣼⣲⠀⠀⢁⠀⠀⠀⢣⠀⢸⡏⠀⠈⠢⡀⠘⠀⠀⠀⢠⠀⠃⠀⡇⠸⠀⠀⠀⡠⠋⠀⠀⠁
⠀⠀⠀⠀⠀⢇⠀⠑⢄⡃⠀⠈⢢⠀⠀⠀⢣⠁⠃⠀⠀⠀⠐⡀⠀⠀⠠⠃⣸⣤⡄⠘⠿⠇⠀⡐⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠑⠢⢞⡝⠀⠠⢀⠑⠄⠀⣈⠀⠐⡀⠀⠀⢀⣡⣤⣤⣤⣀⠈⠉⡇⠠⠀⠀⢀⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⢁⠀⠈⠣⠀⠀⠀⠈⠢⠈⠲⠀⠀⠀⡄⣰⣟⣱⣿⣿⣿⣿⣷⡄⣰⣶⠀⠀⠈⠖⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠁⢄⠀⠀⠀⠑⠌⣆⠀⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣡⣾⡆⠀⠀⢸⢇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡺⣻⡿⡻⣷⣴⣖⢧⢮⡩⡉⠀⢿⣿⣿⣿⣿⣿⣿⣿⢏⠛⠉⠈⢀⠀⣸⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠈⠈⠊⠃⡑⠁⠈⠈⠲⣝⢿⣿⣿⡿⢿⣿⣿⣿⡄⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⢀⠈⠂⠀⠀⠀⠈⠻⢷⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢌⠀⠀⠀⠈⠢⡀⢀⠀⠀⢁⠉⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠢⡐⢌⠂⠐⢄⠢⡀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠢⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⡠⣶⣶⣶⢶⣄⡀⠀⠀⠀⠀⠀⣼⣿⡉⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠄⢻⣿⣿⡜⡠⠁⠀⠀⠀⠀⠀⢰⡾⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⢛⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⣃⠀⠀⠀⠀⠀⢀⡖⠀⠀⣿⠀⠀⢲
⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣥⢴⣷⣾⡆⠐⠀⠀⠀⣤⠵⡀⠀⣿⡀⢀⠾⣥
⠀⠀⠀⠀⠀⣼⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⡄⠀⠀⠒⠵⠀⠉⠁⠮⠓
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⢴⡟⢩⡍⢻⡦⠀⠀⠀⠀⠀⠹⣞⣧⢩⡾⠁⠀⠀⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢰⣿⣦⣠⣤⣄⠀⠀⠀⠀⣀⢀⠀⡖⢋⢡⢀⡀⡀
⠀⠀⠀⠀⠸⣧⣀⣃⣼⠓⠀⠀⠀⠀⠀⠀⠙⣸⡣⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠘⢯⣵⡟⢿⣿⣷⠀⠀⠀⠘⢾⡿⠻⣿⡞⣾⡷⠃
⠀⠀⠀⠀⠀⢩⣝⣣⡍⠀⠀⠀⠀⠀⠀⢠⠎⢩⡍⢧⠀⠀⠀⠀⠀⢼⣿⣿⣿⠛⢿⣿⡿⠀⠀⠀⠀⣠⡿⠻⣿⣿⣿⠀⠀⠀⠀⢸⡇⠀⣿⠇⡼⡇
⠀⠀⠀⠀⠀⢸⠿⢿⠇⠀⠀⠀⠀⠀⠀⠘⢦⣘⣣⠞⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⠼⠋⠀⠀⠀⠀⠀⠋⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⡇⠀⣏⡄⠇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢧⣴⢦⠇⠀⠀⠀⠀⠀⠀⠠⣦⣸⣵⡦⠀⠀⠀⠀⠀⢶⢞⣧⣶⣼⢿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣤⠀⠀⠀⠐⢿⡀
⠀⠀⠀⠀⠀⠈⣯⡏⠀⠀⠀⠀⠀⠀⠀⢠⣆⡟⣟⣦⠀⠀⠀⠀⠀⣾⣿⠘⣿⡏⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⢼⣿⣃⣄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃⣿⡇⣿⡏⠃⠀⠀⠀⠚⣹⣿⠀⣿⡇⢻⣿⠙⠀⠀⠀⣀⣀⣤⣀⠀⢸⣿⠳⠀⠀⠀⠀⢰⡟⢋⣹⣤⣤⡀
⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⡏⣧⠀⠀⠀⠀⠀⡟⣿⠀⣿⡇⢸⢹⡀⠀⠀⠀⠙⣿⣿⣿⡆⢸⣿⡆⠀⠀⠀⠀⣼⠁⠘⢻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠉⠈⠀⠀⠀⠀⠀⠁⠉⠀⠉⠁⠘⠈⠁⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠉⠀⠀⠈⠈⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡻⣉⣙⣛⡟⣻⣿⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠱⡔⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢘⣇⠀⢁⠀⠀⠀⠀⠀⠀⠀⣾⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠹⣄⠀⠻⣆⣸⡄⠀⠀⠀⠀⠀⠀⠙⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡑⢤⣷⣅⡀⢸⠟⠛⠢⣄⠀⠀⠚⡏⢨⣾⠿⠷⣄⠀⠀⠀⢀⡀⢠⠚⢳⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠈⡏⠁⠉⠻⠀⠀⠀⠈⢳⡀⢀⣷⣿⠃⠀⠀⠈⣷⡀⠞⢉⣠⣧⡴⠁⢀⣀⣀⣀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠈⢦⠈⠢⡀⢇⢄⠈⢠⢠⠀⠀⠀⠀⣷⡎⠀⢹⠀⠀⠀⠀⣿⡷⠛⠉⠀⢻⠀⣮⠀⠀⠀⣠⠟⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⣧⡠⠀⠈⠪⡂⠈⢆⢣⡀⠀⠀⢸⠀⠀⠈⢧⡀⢀⣠⣿⠇⠀⢠⠂⡞⠀⠝⠢⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⣿⠿⠶⢤⣀⠈⠲⡤⠷⣕⣤⡶⣞⠀⠀⢠⠁⠙⢿⣿⠿⣄⣰⠯⠊⠀⠤⠔⣠⣤⡶⠋⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⣽⠀⠀⠈⠉⠙⢦⠀⠀⢀⢠⣶⣿⡿⠉⠉⠒⢦⣄⠀⢀⡨⢔⣂⠚⠒⠒⠚⣿⠧⠶⠒⠊⠀⣠⠤⠲⢤
⠀⠀⠀⠀⢀⣀⣙⠢⢄⡀⠈⠁⠀⠹⣦⠀⠀⠀⠀⠀⢣⡠⣵⣿⣿⠼⢃⣴⡆⠀⠑⠝⣤⠋⠀⠉⠀⣀⡤⠶⠒⠃⠀⠀⢤⣶⣎⣠⠤⠶⠟
⠀⠀⠀⡼⣅⣀⠈⠙⠲⢬⣷⡤⠒⠺⠿⡳⠀⠀⠀⠀⠈⢿⣿⣿⡏⣷⠄⠈⠀⠀⠀⠀⠸⡄⠀⣴⡾⠁⠀⠀⠀⠀⠀⢀⣴⡯⢄⣀
⠀⠀⠀⠀⠈⠛⣛⡛⠓⠛⠳⣇⠀⠀⠀⠈⠓⢤⣀⠀⢀⣸⣿⣿⣿⣧⡀⠀⠀⠀⠀⡀⢀⠗⠒⣻⡀⠀⠄⣐⣒⣠⠔⠋⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠾⠷⢔⣂⣀⣀⣀⣈⣗⣛⣹⡽⣯⡻⣿⣿⣦⣤⣠⣬⣶⡟⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠠⠤⠤⠭⣍⣥⣄⣀⠀⢈⣉⣪⣝⣲⠭⣽⣿⣿⡿⡛⠒⠒⠒⠒⠶⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠴⠤⡍⠛⠛⠛⠒⠒⠲⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠀⠤⠤⠤⢤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠖⡩⢄⢱⠀⠀⠀⠈⢢⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡔⠁⠀⠀⠀⡎⡠⢍⠲⡀
⠀⠀⠀⠀⠀⢠⠃⠼⣀⡸⣴⡀⠤⣰⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⣆⠤⢀⣧⢇⣀⠧⠘⡄
⠀⠀⠀⠀⠀⡆⠰⠐⠂⠒⠂⠒⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠒⠐⠒⠐⠊⠆⢰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢃⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⡘
⠀⠀⠀⠀⠀⠈⡄⢣⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡜⢠⠁
⠀⠀⠀⠀⠀⠀⠘⡄⠣⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⠜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡠⠊⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠑⢄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡉⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⢉⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠑⠄⡣⠄⠀⠀⠀⠀⠀⣔⠡⠊⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠒⣁⡷⠀⠀⠀⠀⠀⠀⢾⣈⠒⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼