Circle of candles Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⡶⢆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡰⠃⠈⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢡⣿⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣥⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠛⣫⣿⣿⣿⣿⣷⣾⢠⣶⣮⣭⣭⣍⣛⠻⢿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⢿⣧⣀⠀⠈⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⠟⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠉⠙⠛⠿⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠐⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠂
⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟
⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⢿⣷⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣌⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣝⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠀⠀⠒⠒⠒⡟⠀⠀⠘⠇⢀⠳⠒⠂⠠⠤⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⢀⡀⠀⠀⢸⣰⡄⠰⠶⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⠖⠈⠛⢻⣏⣼⣿⣷⣸⣿⡟⠛⠛⠛⠒⠲⢶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠈⠐⢤
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠄⡀⠀⢀⠈⠙⣿⣝⣿⣷⣦⣀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⣏⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠿⣿⣧⣠⠀⠀⠈⠰⢌⢁⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⣿⣆⢒⣤⣶⣴⣶⣆⣠⣄⡀⡀⠀⠀⢀⣀⣼⣿⣇⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⡀⠀⢀
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⣀⡠⡿⣏⡁⠀⣺
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⡿⠋⠁⣠⡿
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠀⢀⣴⠟
⡏⠀⠀⠀⠀⢲⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣀⡴⠛⠁
⣇⠀⠀⠀⠀⣘⢾⣽⣿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠁⠀⡀⢠⠄
⣇⠀⠀⠀⠀⢙⡷⢿⣟⣿⣾⣯⣿⣿⣿⣶⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠀⣠⣿⣾⣷⡞
⢹⠀⠀⠀⠀⠈⡇⣷⣦⣏⣿⣷⣿⣿⣿⢿⣿⣭⣝⣛⣯⣽⢯⣛⣛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣟⣻⣿⠃⠀⢰
⢸⠇⠀⠀⠀⠀⡇⣿⣿⣏⣿⣿⣿⣿⣯⢹⣿⠟⡏⠹⠋⣉⡛⠛⠛⠛⠒⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠂⠀⢸
⢸⡀⠀⠀⠀⠀⠇⢹⣙⣿⣿⣿⣿⣿⣿⣿⣿⡖⠆⢸⡄⠄⢈⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣿⣿⣿⣿⣿⠿⠀⠀⠀⣿
⠀⣷⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⡛⡇⠰⠄⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣿⢩⣿⣿⣯⣿⣿⣀⠀⢀⣼⣿
⠀⢿⣄⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣍⡃⢸⠒⠒⠠⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⢻⣽⣿⣸⣿⣿⣾⠿⠃⣀⣴⠿⢋⣽
⠀⠘⣷⡄⠀⠀⠀⠀⠀⠹⠎⠋⢻⣿⣿⣿⣿⣿⣏⢸⡁⢀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣧⣿⣿⣿⣻⣿⡿⢛⣿⣠⣴⠟⠋⠀⢰⣾⠏
⠀⠀⠘⢿⣦⡄⠀⠀⠀⢤⣄⣨⠀⠉⠻⠿⢿⣿⣿⣄⡅⢘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⣿⢛⢃⡀⢀⣴⡿⠋
⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠹⢧⣠⡐⡄⠀⠀⠈⠉⠁⠈⠓⠒⠲⠶⠶⠦⠤⠶⠶⠶⠤⠶⠶⠾⠗⢛⠛⠛⠛⠯⠿⠋⠛⢂⣼⣴⣾⣿⠟⡿⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠈⢣⠀⢳⣿⣙⣙⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⣿⣿⢯⣾⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠰⡻⣏⠹⣿⣿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣦⣱⣄⡀⠉⠸⢿⣇⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⢾⣷⣶⣽⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡶⠶⠞⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⢏⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⣶⣶⣶⣿⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣷⣶⣶⣾⠃⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⢸⣿⣿⣿⣿⠀⣀⠻⡿⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⣿⣿⣿⣿⠀⢀⠈⠻⠋⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠻⠟⢤⠀⢸⣿⣿⣿⣿⠀⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠟⠁⡀⠀⣿⣿⣿⣿⠀⢸⣷⣶⣶⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⢿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣻⣽⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦
⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄
⠀⠀⢠⣿⣿⣿⣿⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⢀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⣿⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⣾⣿⣿⡿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡟⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀
⢠⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠉⠁⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣇
⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠀⠂⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⡿⠿⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿
⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⣿
⠸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⣿⣿⡏
⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⣿⣿⡿
⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡰⣡⣿⣿⣿⠃
⠀⠀⠀⠈⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡕⣱⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣤⣤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⢪⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⢀⣴⣿⣫⣵⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠚⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⠿⠿⠛⠿⠿⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠝⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢁⢦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠀⠀⠀⠀⠀⠀⠀⠄⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣗⣀⣀⡀⠀⢀⡀⣐⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣖⡶⠁⠀⢻⣿⣿⡇⠈⠙⠿⣣⣭⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣦⣾⣿⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠄⠒⢂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠑⠊⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⢸⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⣀⠘⡎⠀⠐⢲⡄⠀⠀⠰⠇⠀⠀⠀⠀⠀⢀⡠⠔⠂⠘⠈⠉⠀⠟⠀⠁⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠚⠖⠌⠭⠁⠀⠀⠈⡃⠁⠀⢇⠳⡀⠀⣠⠀⠀⣎⠀⠀⠀⢀⡠⠤⠤⣀⠇⠤⢄⡀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠘⡆⠄⠀⠀⠀⠀⠀⠀⠀⠠⠯⠌⠄⠘⠢⢀⡀⠀⠀⠀⠀⣰⡀⠈⠒⠚⠒⢂⣀⢼⠒⡘
⠀⠀⠀⠀⠀⢠⠧⠄⡠⢊⠀⠂⠊⠀⢀⠀⠀⠀⠐⠊⠉⠀⡇⢐⠖⡰⢨⠆⣷⠂⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡸⡹⠉⠴⢠⣛⠉⢀⠀⠀⠀⠀⠀⠀⡠⣇⠀⡱⢺⠇⠀⠃⠀⡟⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⢀⠟⣿⠈⠀⠁⠒⠀⠀⠰⠈⠀⡡⠘⠁⠀⡇⠀⠀⣀⡇⡀⢀⡬⢠⠀⢀⡀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢃⡴⠁⣼⠁⠀⠤⠤⠄⠀⠀⢀⡀⠀⠁⠀⠀⠇⡇⠀⠀⢸⢂⠡⡂⠈⠧⢁⢤⠰⠄⡀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠠⢁⡜⠀⠰⠀⠀⠀⠁⠀⠀⠉⢉⠀⠒⠁⠀⢀⡜⠀⡇⠀⡠⠆⡀⠀⠈⠀⠀⠚⠒⠤⠉⠉⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠴⠒⠉⠀⠀⠁⠈⠀⠀⢈⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⢀⣶⣄⠀⠈⠻⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣸⣿⠿⣷⡀⠀⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢠⣴⣿⡟⠀⢿⣷⠀⠙⠉⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡄⠀⠀⣿⣿⠀⢰⣿⡿⠛⠀⠀⠸⣿⣧⠀⠀⢸⣿⡇⠀⠀⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠁⠀⠀⠀⠸⣿⠀⣾⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⣸⣿⠃⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠀⢹⣿⡆⠀⣠⣦⠀⢰⣿⠇⣰⣿⡟⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣸⣿⣧⣾⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠉⠀⠉⠀⠘⠟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣠⣄⡀⠀⠀⢀⣠⣤⣾⡷⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣼⣿⡿⠿⠿⠿⠿⠛⠋⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⣠⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢀⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⣿⠃⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣄⣀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⣠⣤⡄⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⡀⠀⠀⢿⣿⣿⡆⢠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠈⣿⣿⣿⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣶⣶⣦⣾⣿⣿⣧⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠉⠉⠙⠛⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣴⣿⣿⡿⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⣴⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⢀⣤⣾⣿⣿⣿⠟⠛⠉⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠉⠙⠻⢿⣿⣿⣿⣦⣄
⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣧⡀
⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠈⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠙⠻⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣬⣉⡙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⣉⣩⣤⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠒⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣿⣶⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣀⣀⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⣀⣸⣇⣀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣯⣤⣶⡿⠿⠿⠿⠿⠿⠿⢿⣶⣤⣽⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠋⣴⣾⠿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠻⣷⣦⠙⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠲⠄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠆⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠓⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣁⡀⠀⠀⠠⠄⠀⠀⢀⣈⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠿⢿⣾⣶⣶⣾⡿⠿⠛⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠅⠂⠀⠀⠀⠀⠑⢼⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡤⠾⡀⠀⠀⠀⠀⠀⣀⣈⣷⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣴⡎⠀⠀⣀⣀⣬⣦⠀⠀⠄⣯⡀⠀⠀⠈⠑⠢⡀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣏⢋⣿⡿⣫⡤⡶⢶⣧⣬⣵⣤⣀⡀⢀⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⢟⡻⠙⡉⠉⠀⠰⠉⠉⠻⢿⣿⡟⠉⠛⢍⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡵⠋⠀⠀⠀⠀⠀⣤⠀⠀⠀⠂⠹⡹⡄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡎⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠑⣽⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠓⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⠁⠈⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠀⠀⠀⠀⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⣀⠁⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡼⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⢧⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣬⡼⡃⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⣯⣢⡤⢀⠀⡀⢀⣀⣀⣤⠖⣋⡉⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡾⠋⠁⠉⠛⠻⠿⠿⠯⢭⣭⢄⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⠋⠉⠁⠀⠀⠀⠈⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢃⣄⠀⣜⡇⠀⠰⠆⠀⠀⠀⠀⠀⠰⡄⠀⣿⡀⠖⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡽⠞⠋⠉⠭⢝⠢⡀⠀⣖⢲⡤⣄⡤⠒⠒⠛⢧⣞⣳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⡤⢒⡶⢋⣀⣈⣽⣖⣧⡀⠀⠀⠀⠀⠉⠛⢄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⢺⢀⡤⣚⣩⠿⠛⠉⠉⠉⠉⠉⠙⠳⢤⣈⢳⣄⡀⠀⠀⢸⡌⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢋⡽⠋⣀⣴⠖⠛⠉⠉⢹⠛⠒⠤⡀⠈⠻⣟⠛⠲⢤⣀⣡⠇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⠏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢣⡀⢠⠙⣅⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢳⡀⢇⠸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣇⢸⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠂⠀⠀⠀⠀⠸⡊⢂⠀⠀⠀⠀⠀⠀⣀⣧⠀⢹⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡄⡇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⡼⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢇⡏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⣜⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⢞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠓⢍⣒⠤⠀⠰⠀⠀⠀⣀⡴⠊⠁⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣲⢤⣈⠉⠛⠒⠛⠋⠉⢁⣀⡤⢞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⢧⡀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣰⠟⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠑⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡰⠋⠀⢀⡴⠟⠉
⠀⠀⠀⠀⠘⠛⠷⠶⢤⣄⠀⢀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠉⠀⡀⢀⡀⠀⠄⠀⡀⠀⡀⠈⠂⠀⠘⡄⢰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⡀⠀⢀⠜⠋⠁⠀⠀⠀⠠⠀⠉⠃⢆⠀⠄⢀⠇⡎⠀⡰⠛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡴⠁⠀⠀⠰⠆⠈⠊⠀⠀⠀⠀⠀⠳⣄⣀⡀⠀⠀⠊⠉⠁
⠀⠀⠀⠀⠠⠠⠠⠀⣀⠀⠀⠤⠾⠁⡀⠘⠓⠀⢄⠀⠀⠀⠀⠀⠐⠀⠀⢃⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠢⢄⠀⠂⠈⡆⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠆⠆⢸⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠰⠀⠂⠀⡄⠀⠀⡞⠀⠀⠀⠀⣀⠀⠀⡈⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠡⡀⠀⠀⠠⠀⠀⡀⠀⢠⠀⠀⠀⡰⠁⠀⠀⠀⡤⠒⠡⠤⠶⠆
⠀⠀⠀⠀⠀⠤⠶⢚⡅⢀⠞⠀⠀⠀⠀⢦⡀⠀⠀⠀⠁⠀⠀⢀⡠⠎⠀⠀⠀⠀⠘⢇⠠⣘⠒⠲
⠀⠀⠀⠀⠀⠠⠴⠋⠀⠃⠀⠀⠲⡄⠐⠀⠀⠉⠀⠒⠒⠂⠈⠁⠀⠀⠀⢀⡄⠂⠀⠀⠐⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡆⠃⡀⠈⠙⢦⠀⣠⠀⢒⣉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢲⡄⠀⢸⡇⠀⢸⡇⣸⢡⠀⡄⠀⣳⢛⠀⢉⡁⠀⢰⠀⠀⡄⢀⡀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡄⠘⠇⣦⠀⢣⠇⠼⡄⠀⠈⠁⢿⡀⠉⢁⠀⢸⠀⠀⡇⢸⠃⢸⠆⣸
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠹⣆⣘⠖⠀⠛⣞⣸⣧⣈⡉⠉⠉⠀⠸⣄⣐⡃⣾⠀⠚⢊⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⠓
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣾⣷⣾⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣌⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⢛⣠⣿⣿⣿⣿⣿⣿⠿⣻⣿⣟⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣯⣶⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡏⣶⢽⣿⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣾⣷⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠻⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣬⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⡛⠉⡿⠋⠉⠙⣿⣿⡿⢿⣿⡿⠟⢿⣿⣿⣿⣿⣿⡟⢡⡌⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡅⠶⣸⣿⣿⣿⣿⣾⡇⠀⠀⢠⣿⡏⠀⠀⠉⡇⠀⢘⣿⠾⣿⣿⣿⣿⣦⣴⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⠉⠙⢿⣿⣦⣤⣿⣯⣴⣶⣶⣾⣷⣶⡿⠋⠀⣠⠜⣩⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⣠⠴⠿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣀⣤⠶⠾⠷⠚⣿⣿⣿⣏⠤⢘⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠟⠋⠛⠋⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣾⡟⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠙⠋⠙⠛⠉⠋⠙⠋⠉⠛⠛⠛⠋⠉⠉⠛⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠻⢦⡄⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠘⡏⢸⡁⠀⢀⣣
⠀⠀⠀⠀⠀⠀⢠⣾⡆⢀⡖⠠⠄⠀⡀⠀⠀⠁⠀⠀⣀⠘⢳⠀⠀⣄⠀⠠⠶⡴⠀⠀⢲⠂⠀⠐⡀⠨⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠃⠀⠀⠴⠀⠀⠀⠀⠀⢦⠹⠁⠈⠚⠣⠈⣃⣄⠀⠀⠀⠀⠘⠀⠀⠀⠱⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠁⠀⠀⠀⠀⡀⣏⠈⠁⠀⢀⠀⠂⠐⠍⣉⢀⠀⠀⠀⠀⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⠀⠉⠀⠂⠀⠒⠀⠀⡂⡄⠀⠙⣼⠀⠀⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡄⠠⡆⠀⠀⠼⠀⢀⠀⡀⢁⡄⠀⠘⣡⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠎⠀⠀⠒⡀⠀⡄⠀⠐⠀⠀⠀⢰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠐⠠⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠂⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠶⠀⠀⠀⠀⠀⠀⡠⠀⠀⢨⠀⠠⠈⢀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⠂⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠀⠀⠀⠀⠀⡈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⣀⡀⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠁⢠⡤⠦⠆⠨⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⠀⠀⠐⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠈⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⡀⠄⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⢄
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢄⡁⠀⠐⠀⠀⢄⡀⠔⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⢀⠀⠰⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⠈⠘⠳⣄⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣤⢁⡄⠀⠀⠀⣠⡶⠂⢠⡄⠁⠀⠩⠄⢠⡀⠀⠀⠀⠀⠲⣀⠨⣷⠀⠀⠀⢀⢄⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡅⢴⠄⠀⠀⢸⡃⠄⠹⢤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠂⡜⠉⠁⢸⠆⠀⠀⣒⠋⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡄⠀⠨⣇⠀⠀⢤⠟⣡⠀⠀⠀⠀⠀⠀⠀⠜⠛⠂⠀⠜⢀⣴⠾⠁⠀⠀⠀⢠⠀⡀
⠀⡃⠀⠀⠀⢀⠈⠀⠀⠀⠀⡔⠹⢃⠀⠈⠠⠤⠔⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⠴⠈⢠⡄⠁
⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⠀⢀⠠⠔⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠭⠐⠀⠋⠒⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠉⠈⢉⡀⠀⢀⠁⠀⠀⠄⠠⠀⡄⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠠⠢⠀⠊⡠⠂⠀⠀⢁⠈⠀⠀⠴⠀⠀⠁
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰⠉⠀⠀⠀⠄⠀⠐⠀⠀⠈
⠀⣾

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⣴⣾⣶⡄⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⠈⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣦⣄⣠⣴⣿⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣶⣤⣀⣴⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣤⣄⣀⣀⢀⡀⠀⢀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠗⠿⣿⡯⠽⠊⠽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⡛⠛⠳⢦⡴⠒⠒⠒⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣟⣿⣛⡿⠿⠙⣿⡟⠻⡉⠙⠛⠒⠶⠷⠶⢶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⡈⢹⡟⠛⠳⣦⠀⠀⣻⣾⡛⠛⠶⢶⣤⡤⠶⢛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣄⠙⠷⣤⣿⡀⠀⢸⣧⣴⠟⠈⠙⡷⠶⠦⣄⠳⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⣷⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⢠⡟⠳⢦⡀⠉⡟⠓⢾⡁⠘⣄⡀⠀⡷⠤⣀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⢿⡻⠛⢃⣼⠃⢼⠀⠀⠀⠘⣦⠇⠀⡟⢳⡀⠀⠈⠐⣇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠂⠀⢣⣠⠏⠹⡆⢸⢦⡀⠀⠀⠘⡳⠮⢵⣄⡳⢤⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡴⠞⠋⠀⠀⢻⡼⠀⠁⠀⠀⠘⠁⠀⣀⣱⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣏⣧⣄⡀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠒⠒⣚⣛⣛⣛⣛⣛⣛⣛⣛⣓⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢠⠚⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢳⡄⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠳⣞⣴⣣⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣝⣮⣳⠚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⡫⠁⠀⠀⠀⠀⠈⣓⡄⠀⠀⠀⠀⠀⠀⣴⣊⠉⠈⠀⠀⠀⠊⠝⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠀⠀⠀⣠⠴⢚⣉⠥⠔⠒⠒⠒⠒⠒⠒⠦⢬⣉⠓⠦⣄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡴⢋⣱⠖⢉⡄⠒⠈⠁⢠⠀⠀⠀⠀⠁⠐⠤⡙⠲⣎⡙⢦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡴⠋⣠⠊⡠⠊⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠑⢌⠱⣄⠙⢆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡜⢁⠚⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⡈⢧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠁⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠂⡠⠄⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠜⠀⠀⠀⠀⠀⠀⠀⢨⡏⢱⠄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢄⠄⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠄⡀⠀⡀⠀⠀⠀⠀⠒⢐⠀⠀⠀⠀⢑⠄⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢲⣌⡘⠀⠄⠀⠀⡀⠀⠀⠀⠠⠄⢂⣡⡖⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣠⠎⠀⠉⠙⠒⠶⠤⠤⠤⠤⠖⠒⠋⠁⠀⠻⣦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣄⠀⠈⠙⢿⣿⣿⣆⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠈⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⠝⠛⠳⠦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠟⠋⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⡘⠀⠀⠀⡄⠀⢠⠀⠀⠀⠈⠀⠈⢢⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢃⠀⠀⠑⡀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠈⡄⠀⠀⠈⢆⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠐⢤⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢆⠀⡠⠒⠘⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠇⠀⣀⢀⣘⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠰⡎⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣦⣠⣿⣀⣼⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠱⣼⣷⣿⣟⣋⣙⣻⣿⣵⢎⣀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣧⠦⠀⢬⣿⣿⠛⠿⣏⣿⣿⠟⢙⣿⣿⣅⡀⠐⢶⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⠀⠚⠋⠙⠂⠀⠠⠴⢟⣿⣿⣶⣖⣲⣶⣿⣿⡍⠁⠀⠀⠛⠈⠋⠀⢤⣄⡖
⠀⠀⠀⠀⠀⠀⠀⠘⣻⡿⠥⠀⠀⠀⠀⡀⠀⢄⠀⠾⡋⣧⡿⣿⣿⡟⢿⠀⠙⠀⡠⠀⠀⠀⠀⠀⠴⠟⣿⠉
⠀⠀⠀⠀⣄⢀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⡵⢁⣉⣈⠃⠀⠔⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⣦⣠⠆
⠀⠀⠀⠠⠾⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⢢⠐⠀⢴⣿⠿⣿⡆⣠⠐⡡⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⡟⠓
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡘⣠⣾⣿⣶⣿⣶⣿⠮⠂⠐⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣠⠄
⠀⠀⠀⠀⢨⡽⣷⠒⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⣠⣿⣯⣿⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠳⠆
⠀⠀⠀⠀⠀⠀⠛⠀⢀⡀⠀⡀⢠⠖⠒⠦⠴⣾⢿⣧⣙⣛⣛⣿⣩⣿⣿⣿⠿⠻⠟⠂⡰⣲⣦⠀⣤⣀⡴⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⣿⣁⠀⠛⠶⠆⢧⣤⣤⡿⣋⣩⢹⣿⡍⣼⣡⠰⡄⠤⢲⡭⠗⠛⠋⠐⠾⢿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⢻⣭⣿⠛⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠉⠒⠢⠴⠾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠂⠀⠀⢠⠀⣤⢤⣀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠐⠐⠂⡰⠁⠀⠀⠀⠀⠠⠞⠀⠡⠴⡶⠶⠶⠶⠶⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠢⠤⡾⠃⠀⠀⠀⣠⠃⣀⠴⢂⡂⡉⠉⠈⡀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⢀⣀⣐⣂⠀⠤⢀
⠀⠀⠀⠀⠀⠀⡈⠀⢀⡠⣠⠞⡵⢊⠀⣪⡥⠂⠒⠒⠒⠂⠶⢄⡀⠌⠂⠀⠀⠀⠀⠘⠣⣷⠈⠙⠛⠻⠿⠿⠿⠷⠮⠶⡑
⠀⠀⠀⠀⠀⢰⡇⠎⣩⠞⢡⠊⠐⣡⠎⠁⠀⠀⡁⠀⠠⠀⢀⠀⠉⠣⡔⠌⢢⡀⠄⣈⠁⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠇
⠀⠀⠀⠇⠰⡀⢓⠜⠁⢀⠇⠀⡼⠁⠀⠁⠀⠀⢂⠀⠀⠀⠀⠐⠀⠀⠸⡆⠀⢣⠊⠀⠈⠀⠈⢦⡀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⢠⠇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⢻⠀⠘⠀⠀⠀⠀⠀⠀⠑⠤⠌⠁⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⢸⠀⠀⠀⠀⠀⠀⠀⡾⠂⠁⠀⠀⠀⢂⠀⢸⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢇⠀⠀⠀⠀⠀⢻⠿⠀⠀⠀⠀⠀⠀⠀⡜⠐⢰⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠑⣆⠀⠀⠄⠀⠨⠀⠀⠀⠀⠀⠀⢀⡼⡁⢠⣧⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠈⡳⢆⡀⠀⠀⠀⠀⠀⠀⡠⡴⠋⠀⡰⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡲⢌⡀⠈⠙⠻⠯⠼⠾⠾⠚⣩⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠙⠒⡶⠒⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠴⠶⠞⠛⠛⠛⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠶⠂⠉⠈⢉⠍⢳⡈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⡔⠉⠳⡀⠠⠆⣄⠀⢐⠹⡉⠉⠉⠉⠈⠁⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠚⠄⢀⠐⠁⢀⡄⠸⠂⠈⢠⣅⣀⠀⣀⣀⣀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢸⠄⠀⢨⠃⠄⢠⠔⠈⠐⠺⡅⢇⠀⠀⠀⠀⢀⣉⣀⣀⣈⣣⡀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢿⠀⠀⠃⠀⢆⠙⠀⠀⠀⠀⡟⠊⠀⠈⠈⡉⠀⠉⠉⠋⢉⠜⠑⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠶⠄⠠⣒⣬⠕⠓⠤⠤⠊⠀⠈⠀⠈⠁⠉⠀⠑⢀⢀⠂⠀⠀⢰⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⠖⠉⢦⠀⠀⠀⠑⠀⢀⣳⣀⣠⡀⠈⠠⠄⠸⠏⡄⠀⠀⠀⢇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣤⡀⠓⠢⡀⠠⣄⠀⠀⠈⡍⠁⠀⠒⠒⣼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠢⠤⢴⠄⠄⠙⢆⠀⠀⢠⠁⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡠⠍⠄⡈⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠝⢠⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⣀⠀⠀⠀⠀⠀⠘⡞⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠌⠓⢜⡆⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⡈⠞⠁⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢌⡯⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⢀⡀⠀⠀⠀⣀⣠⠴⠬⠛⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⣉⣤⣤⣤⣤⣬⣍⡛⠶⠀⠀⣀⣤⠶⠛⠛⠛⠛⠛⠷⣦⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⡤⠶⠶⠶⣦⣬⡁⠀⠀⠘⠉⣠⠴⠞⠛⠛⠛⠻⠶⣦⣅⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢠⡆⣰⠇⢀⠔⠉⠀⠀⠀⠀⠀⠀⠉⠻⣦⡠⠞⠻⢷⣆⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀⠈⣷
⠀⠀⠀⠀⢠⡟⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣷⣄⣠⣠⠿⠒⠛⠶⠶⣤⣀⠀⠀⠈⢷⠀⢸⡇
⠀⠀⠀⠀⢸⠇⣿⠀⡇⠀⠀⠀⠀⣀⣠⡴⠚⠀⠀⠀⠀⣟⣹⣽⣁⣸⣿⣿⣿⣶⣄⠙⢷⡀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⢸⡆⣿⠀⣷⠘⠛⠛⠋⢩⡀⢶⣿⣷⡀⠀⡀⠉⠉⠙⠉⣉⣉⣉⣙⡛⠛⠷⠦⣿⣄⡈⠀⠺⣦⣀
⠀⠀⠀⠀⠈⠃⠹⣦⠘⢷⣄⠀⠀⠀⠻⡌⠻⢟⡥⣊⣤⡴⠾⣻⣿⣿⡿⠿⢿⣿⣿⣿⣶⣶⣭⣙⠷⣦⣀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⡠⠚⠙⠷⢶⣤⡤⠶⢚⣥⣞⣉⡠⠖⠋⠉⠀⢀⡄⢠⣤⡄⠀⠉⠙⠻⣿⣿⣷⣦⡙⢷⣄⠙⢷⡀
⠀⠀⠀⣀⡀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠔⠛⠒⡴⠋⠀⠰⡇⠰⡇⠀⡇⣼⣷⡇⠀⠀⢴⡆⠀⠙⢿⣿⣿⣦⡙⢷⡈⢷⡀
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⡇⠀⠇⡇⢿⡶⠃⠀⢸⡇⠀⢀⡀⠙⢷⡝⣿⡌⢿⡄
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡴⢳⡄⠃⠀⠃⠀⢸⠇⢸⡇⠀⠀⠘⠀⠀⠟⢹⠀⠈⢿⡌⢿⡌⢷⡀
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢸⠀⣇⢀⡇⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⣴⣏⣴⠀⠈⢿⡘⣿⠘⣇
⣿⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⠀⠀⠘⠀⠙⠛⠁⠀⠀⠀⠀⢸⠇⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠸⣇⢹⡇⢹
⠿⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡇⠀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⠀⣿⠈⣇⢸
⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣻⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇⠀⢈⡽⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠶⠋⠀⡿⠀⡿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠛⠁⢀⣴⣆⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⡴⣿⠀⠀⢀⠇⠀⡇⢸
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢹⣇⠀⠀⢸⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣧⣿⡄⠀⠈⠀⠸⠀⠄
⠀⠀⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⢻⣧⠀⢻⣆⠀⢸⡁⣿⠀⠀⠲⠶⣶⠂⠀⣠⠆⠀⠀⣠⠛⠁⠀⠀⣿⠀⠀⠀⠀⠁
⠀⠀⢀⡀⢹⣦⣤⣤⣤⣤⣤⣤⡄⠹⣷⡄⠻⣧⡈⠛⠃⠀⠀⠀⢰⠇⠀⢰⣏⡀⠀⠀⠹⣦⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠈⢿⣄⠙⢷⣤⠤⠤⠤⠤⠤⠤⠜⢿⣦⣙⢿⣦⡀⠀⠀⢀⡟⠀⠀⣾⠙⣧⠀⢠⣴⠏⠀⢀⡤⠚⣲⠎⡀
⠀⠀⠀⠀⠙⢦⡀⠙⢷⣶⠶⠶⠛⠛⠻⣶⡟⠻⣷⣽⣿⣷⣤⣀⡀⠀⠀⠘⠛⠃⠀⢀⣀⣤⠖⣋⡴⢞⣡⠊
⠀⠀⠀⠀⠀⠀⠙⢷⡄⠙⠻⠃⠀⠀⠀⠈⢻⣄⣀⡙⠛⠿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⠿⠞⢛⣩⠖⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣿⣶⣶⠶⠶⠆⠀⠀⠉⠉⠉⠉⢉⣉⣡⡤⠶⠚⠉⠛⠛⠻⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠖⠋⠁⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣢⣲⣭⣵⣽⣦⣶⣯⣭⣖⣢⣤⣐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣾⠿⠛⠋⠁⣼⣿⣿⣿⣟⣿⣷⡈⠉⠛⠶⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢯⣀⠀⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢙⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢮⡑⠲⢤⣀⡻⣿⣿⣿⣿⣯⡞⠀⣀⣠⠴⢚⣭⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠉⠒⢤⢌⣉⣙⠛⠛⣉⣉⢩⠡⠖⠜⠢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠈⠸⠀⠈⠈⠂⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠚⠛⠛⠛⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠛⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠠⠄⢐⣀⠀⠤⢀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣐⣐⣐⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠋⠉⣿⣿⣿⣿⡟⠉⠛⠦⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠴⡛⠉⢿⣿⣿⣯⣿⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛⠛⠚⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠀⠀⠀⠀⠈⠘⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⣀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠉⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠈⠁⠀⠀⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠈⠒⢄


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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⢁⣤⣶⠾⠟⠛⠛⠛⠛⠻⠿⠶⣶⣤⣀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿⠛⠿⢶⣦⣄⠀⠀⡟⠉⢹⡇⠀⢀⣤⡶⠿⠛⠛⠛⠻⢶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣀⣠⡾⠃⠀⠀⠀⠈⠹⣷⠀⣧⠀⢸⡇⢠⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⡂⣻⣀⣸⡇⣺⡏⠉⠛⠛⠷⢶⣤⣀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠉⠛⢛⣷⣦⣄⡀⠀⠀⠈⠙⠷⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⣁⣤⣴⠶⠾⠛⠛⠛⠛⠛⠛⠻⢷⣤⡀⠀⠀⠈⢻⣆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣴⠶⠿⠛⣋⣥⡶⠟⢋⣁⣤⣴⡶⠶⠶⠶⢶⣦⣤⣀⠀⠙⠻⣷⣄⠀⠀⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⣠⡾⠛⢁⣤⠾⠛⠉⠀⠀⣀⣄⡀⠀⠀⠀⠉⠛⠿⣶⣄⠈⠻⣷⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣰⡿⠋⢀⣴⠟⠁⠀⠀⠀⠀⠸⣧⣤⠗⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠟⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠏⢀⣾⠋⠀⠀⢀⣴⠞⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣾⡏⢀⣾⠁⠀⠀⠀⠻⣤⡀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢰⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡘⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⡇⠀⣿⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠀⣹⡆⠀⠀⠀⠀⠀⠐⢯⣨⠇⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠸⣿⠀⠿⣬⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣏⠻⣦⡀⠀⠀⠀⠀⠀⠁⠀⢀⣿⠇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⣇⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠳⣄⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⢿⡀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣙⣷⠀⠀⠀⢀⣼⠏⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢷⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠋⠁⠀⠀⣠⡾⠃⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠙⠄⠈⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠺⣭⡿⠀⠀⠀⠀⢀⣠⡾⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣄⣀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠟⠉⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣀⣠⣤⣶⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠿⠷⠶⢶⣶⣦⣤⣤⣴⣶⣶⠶⠶⠿⠛⠻⣍⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣫⣭⣶⣶⣶⣦⣤⣀
⠀⠉⣵⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢟⣵⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⣄
⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣶⣾⣿⣿⣿⠿⢋⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡆
⠀⠀⠀⠈⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⠽⠟⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⠛⠛⠿⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣃⣀⠀⠀⠀⠀⠉⠙⠻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠞⠁⠈⠉⠉⠛⠲⠦⣤⡀⠀⠀⠙⢿
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠈⠓⠤⣀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠀⣸
⠀⠀⠀⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣫⣥⠴⠞⠉
⠀⣠⣤⡀⠀⠘⢿⣿⣿⠆⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠿⠻⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠹⣿⣿⣄⠀⣴⣿⣥⣄⠈⠻⣿⣄
⠀⠀⠈⠛⠋⠀⠈⣻⣿⠿⠛⠊⠉⠹⣆
⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⢀⣴⣿⠇⠘⢧⡀
⠀⠀⣴⠟⠁⠀⠀⠀⠀⠐⠿⠟⠋⣀⣄⠈⠻⣄
⢀⣾⠋⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⡿⠋⠀⠀⠈⠳⣄
⣼⡇⠀⠀⠀⠀⠀⠘⠿⢿⠿⠈⢹⣶⣦⣄⠀⠀⠀⠈⠳⣄
⣿⣇⠀⠀⠀⠀⠀⠰⣿⣦⣄⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠁
⢹⣿⣆⠀⠀⠀⠀⠀⠈⠻⠿⠇⣰⣿⣷⠄
⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏
⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠖⠂

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡆⠀⡔⢀⡆⠀⡆⠀⠠⠂⠠⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠞⠀⠀⠱⡄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠄⠘⠂⠀⢀⠈⢀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⠀⠀⠀⠐⢀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡰⠃⢀⣘⣠⠀⠀⣀⡆⠀⠀⢆⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢱⠀⠀⡐⠁⠁⠈⠀⠁⠀⠀⠈⠐⠢⣄⡌⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢄⡜⠀⢰⠂⠀⢀⢤⠤⠤⣤⣀⠀⠀⠀⡴⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠜⠢⣤⡇⠀⢠⠶⡟⢩⣝⠳⢯⡁⠀⠀⠃⠀⠱⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠠⣧⣙⣫⣬⡤⠝⠃⠀⠆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⡄⠀⠀⠀⠀⠀⠀⠘⠓⠓⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢤⡀⢀⠀⠀⠀⡀⠀⠀⠁⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠤⠄⠊⠰⣀⠀⠠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠃⠀⠀⠀⠀⠀⠈⠀⠝⠋⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡄⠀⠀⠀⡼⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣀⣀⣁⣀⣀⡀⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠈⢧⠀⠀⡤⠊⠉⠁⡄⠈⠉⠙⠛⢔⠄⠈⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⢠⣄⣼⣃⡉⠀⠀⢃⠀⠰⠃⠀⢸⡆⢀⡀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡱⠀⠫⣯⡶⢪⣷⣾⣿⣿⣏⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠈⠿⣛⣿⣿⣿⣿⣶⣾⣿⣟⣛⢾⣿⣿⣿⣿⣿⢧⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⠈⢊⡿⣫⡵⠞⠛⠉⠹⠛⠻⢿⡻⣧⣽⣿⣝⣛⡘⠀⢀⣀⣤⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣰⢫⣾⠋⢀⡀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣯⠻⣿⡄⠈⠁
⠀⠀⠀⠀⠶⠶⠶⠶⠶⠆⠀⢰⢇⣏⠇⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢷⢹⣇⢻⡿⠀⠀⠰⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⠈⡖⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠶⠒⠋⠉⠀⢸⡸⡼⡀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢹⠇⡿⢸⣧⠁⠈⠉⠓⠲⠦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢧⢳⡱⡀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢀⡾⣸⢇⣿⠇⠠⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⡈⢳⣵⣌⡢⣀⠀⠀⠀⠀⠀⢀⣠⢞⣴⣫⡿⠃⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⢿⣿⣖⣶⣦⣴⣶⣶⣿⣷⡿⠛⠁⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢉⠉⠛⠛⠋⠉⠁⠘⣳⡀⠈⢆⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⡈⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠇⠀⠀⠀⠁⠀⠀⠀⢳⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠈⠠⠠⠊⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⢄⠠⠀⣂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠔⢈⠇⠀⠀⠀⢘⡆⠀⢸⣄⠀⠀⠀⠀⡁⠢⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣉⠳⣌⡠⠔⠁⠀⠀⠀⡔⠁⠙⠋⠉⠀⢱⠀⠀⠀⠀⠤⡀⣊⠴⠓⣄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠂⠀⠀⠠⣄⣀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⢑⠎⠀⠀⠀⠀⠀⢘⠉⠉⠁⠉⠉⠉⠉⡂⠀⠀⠀⠀⠈⠡⢔⠩⠈⠀⠀⠠⠀⠀⠰⠂⠠⠀⠀⠂⠀⠀⠀⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢉⡉⠉⠉⠉⠉⡉⡅⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠀⡆⠐⠀⠀⠀⠀⠀⠈⢠⡌⠀⠘⠁⠌⠃⠀⠀⠀⠀⠀⢀⠂⠀⠀⠖⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢷⠀⠀⠀⠠⠤⠥⠤⠁⠤⠀⠼⠄⠂⠀⠀⠀⡇⠘⠀⢀⡇⠨⠅⢈
⠀⠀⠀⠀⠀⠀⠀⠤⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠣⠄⠄⠀⠒⠚⠒⠒⠒⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠶⠀⠀⠌⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⠔⠉⡄⠀⠀⠀⠀⠠⠈⠂⠈⠀⠀⢰⠅⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⢰⠠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠟⡠⠔⠊⣁⠴⠀⠀⠀⠐⠁⠈⠀⡤⣃⠀⡀⢀⡀⣀⢀⡀⢀⠀⡀⠀⡁⠀⠀⠀⠀⡀⠠⠄⠘⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠱⠖⠂⠉⠂⠀⠀⡐⠀⢀⠀⠠⠄⠁⠉⠀⠀⠀⠀⠁⠈⠈⠉⠈⠁⠈⠁⠉⠉⠁⠁⠀⠆⠰⡀⠀⠐⡀
⠀⠀⠀⠀⠀⢭⣩⣉⣉⡁⠀⠀⠁⠀⠀⠀⠀⡜⢠⠀⠞⠁⠀⢠⢄⡀⠀⠄⠤⠤⠤⠠⠄⠤⠠⠀⠤⠠⢄⠤⠠⢄⠀⠀⠀⠹⠄⠤⠐⡄
⠀⠀⠀⠀⠀⠲⠤⢤⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⠂⠀⠀⠤⠁⢀⡆⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠘⢤⠀⠡⠀⠀⠘⢂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠈⠀⠀⠀⢀⠔⠁⠂⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠤⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⣀⠀⠀⠸⠁⠀⠀⠀⢰⠀⠀⠀⢼⠀⠀⠀⡀⠀⠀⠈⠣⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠀⠀⠐⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⡜⠀⠀⠀⠀⠀⠰⠒⠀⠒⠐⠀⠂⠀⠂⠐⠄⠢⠤⠄⢄⠠⢄⢤⣠⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡀⢀⡜⠀⠀⠀⠀⠀⠀⢠⠤⠠⠤⠤⠠⠠⠤⠄⠀⠀⠄⠤⠠⠄⢤⢠⡤⡘⠀⠀⠀⠀⠀⠀⠀⠐⡤⢀⠉⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢁⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡎⠀⠅⡠⠂⠀⠐⠃⢀⠀⡆⠈⠠⠁⠘⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣦⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠄⠰⢂⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⣼⣿⣿⣧⣶⣀⣶⣢⣀⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣳⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣦⣤⣶⣾⣿⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣴⣾⣆⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠈⢀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣿⡖⠐⠀⠀⠀⠀⣀⣴⣦⠝⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢻⡿⠿⠿⣿⣿⣟⠛⡿⠋⠉⠀⠈⠳⠶⠍⠉⠁⠀⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢸⣷⣀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⡗⢶⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢹⣿⣿⣯⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⣿⣧⠀⢈⠁⠀⠀⠀⠈⣾⣄⠰⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠈⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣷⡿⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣧⣴⣶⣼⣿⣄⣶⣿⣿⣿⠸⣦⠀⠀⠀⠀⠀⣿⣿⠏⢀⡞⢠⢯⣿⣿⣿⣿⣿⣿⣿⢟⡡⠀⠀⠀⠠⠔⢤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢿⣷⣤⣀⣠⣶⣿⣿⣷⣞⣷⣏⡾⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣄⣲⣥⣙⣿⣿⣿⣿⣿⣿⣿⡾⡻⠂⠈⢻⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡋⠀⠀⠊⠁⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠚⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡇⠐⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠄⠀⠀⠊⠹⢿⣿⣿⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣆⠉⠑⢀⡀⠀⠀⠀⠀⠀⠈⠛⢿⡃⠀⠀⣀⣀⣤⠾⢿⣶⣿⠿⠍⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠯⡀⠀⠀⠀⠴⠚⠛⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣟⣯⣥⣤
⠀⠀⠀⢀⡞⠳⣤⢦⡀⠸⣶⡋⠀⠀⣠⠞⠁⢹⣆⠀⠀⠀⠀⢀⣦⡀⣀⠀⠀⠙⠵⣿⣿⣿⢟⣾⣷⣄
⠀⢀⡴⠋⠀⢢⡈⠻⡀⠀⠀⢈⣦⡞⠁⠀⢀⡏⠸⡆⠀⢀⣴⣿⣯⣻⣯⡀⠀⠀⠀⠘⢟⣵⣿⣿⢧⣿⡄
⠀⠘⢧⡀⠀⠀⣹⠖⠃⢀⡴⣫⠎⠉⢢⣠⠞⠀⢀⡇⠀⠘⢿⣿⣿⡷⠉⠁⠀⢀⡴⠋⠀⠈⢿⣫⣿⣿⠇
⠀⠀⠀⠙⠒⢾⢳⢀⡴⣫⠞⠁⠀⠀⠈⠛⠢⡀⡈⠀⠀⠀⠀⠙⠉⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⢻⡟
⠀⠀⠀⠀⢀⡞⣯⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣼⣦⡛⠁
⠀⠀⠀⠀⠈⢧⣛⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠕⠁
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠄⠀⠀⠈⠁⠐⠢⢄⠀⠐⠀⠐⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠚⠃⢱⡄⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠄⠀⠀⠀⠀⠀⡀⠀⢁⠀⠀⠀⡀⠇⠀⢀⡀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠀⠳⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⡿⠀⠁⢀⠉⢀⣀⠴⠞⠛⢢⡈⠓⠈⡄⢱⡎⠀⢀⡀⠀⠀⠀⢦⡈⢻⡄⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⠃⡌⠀⣡⠞⠋⠀⠀⠀⠀⠁⠀⠋⠀⠁⠃⠈⠁⠊⠉⠳⠄⠀⠈⢱⡈⢷⣄⡀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠒⣀⡁⠰⢎⡌⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠛⠦⠯⡝⣦⡀⣠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠔⠈⠀⠀⠐⡉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⣠⣤⣶⢿⡏⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣯⠄⠀⢐⡉⢤⠄⠒⡛⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠂⢀⠀⠀⠠⠀⠈⠻⣆⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⠠⡔⠁⠀⠀⠀⠈⡀⠀⠘⠋⠗⣉⣉⡉⢉⡳⡄⠀⢠⡾⠛⠉⣀⣈⣙⠒⠀⢠⠀⠀⠄⡀⠀⠀⠀⠀⠀⢹⠀⠉⠑⠂
⠀⠀⠀⠀⠀⡠⠄⠚⣿⡇⠀⠀⠀⠀⣀⠀⣴⠇⠀⠀⠐⠪⠿⠿⠇⠐⠀⠀⠀⠸⠇⠐⠹⠧⠿⠉⠛⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⣼⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣎⠦⠀⠀⠫⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠓⠀⠛⠀⠀⠀⠘⠀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠄⢣⡀⠈⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠔⠀⢈⠀⡠⠃⢀⣼⠃⠀⠠⠀⠀⠸
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢸⣐⣴⠇⠀⢻⢾⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⢃⠎⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠆⠀⠀⢠⡔⠶⠛⠻⠋⠚⠻⠋⢀⠇⠀⠀⠀⠀⢀⡤⠀⠈⠁⢤⡔⠁⠠⠀⠀⠀⠀⠀⠃⠀⠐⠀⠐⠸⡦⢤⣜⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠲⣶⣤⣴⠾⠳⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⡄⠐⠒⠿⠋⠙⡆⠀⠈⠀⠀⢸⢠⡖⠺⢋⣉⣉⠝⢳⠀⠀⠀⠀⠀⠀⢸⡆⠀⠤⠤⢤⠤⠀⢈⡙⢦⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢠⠲⡃⠀⠀⠀⣆⡄⠀⠈⠛⠀⠁⡄⠀⠈⠀⠁⠀⠸⡀⠀⠀⠀⠄⠀⠘⠤⡀⠀⠈⠉⠉⠉⠀⠀⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⢆⠀⢀⡣⠀⠀⠀⠈⢥⡀⠀⠀⡀⢠⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢀⡀⠀⢲⡄⠀⠙⢠⣴⡥⠈⠄⠀⠙⠀⠀⠐⢃⠜⠀⢀⡴⠾⠛⠁⠀⠀⠀⡀⠄⣐⠭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⣀⠈⠱⠀⠁⢱⡀⠀⠀⠈⠁⠀⢸⡧⡾⠂⠸⣀⣤⣿⠶⠊⠁⠀⠀⠀⠀⠁⠘⠀⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⢸⡟⠁⠀⠀⣀⠄⢠⠎⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠚⣗⠣⠠⠤⠴⠊⠠⠄⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢀⡴⣍⣩⣿⣾⠿⠋⠉⠒⠀⠤⢀⡁⠀⠀⠀⠀⠐⠤⠤⠉⢀⡀⠨⠅⠀⠀⠀⠈⠳⣤⣴⡾⣷⠀⠀⢀⡼
⠀⠀⠀⠀⠀⢰⣶⠁⠠⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⣠⣾⠟⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⢀⣾⠟⠁⠀⢀⣴⠞⠉⠀⠀⣤⡤⠴⠶⢿⣿⣿⣶⣤⡀⠀⣤⣤⣶⣶⣆⠀⠀⠀⠀⣀⣤⡶⠚⠋⠁⠒⠠⣴⡆⠀⠉⠓⢦⣀⠀⠀⠙⢿⣆
⠀⢸⠇⠀⠀⣰⠟⠁⠀⠀⠀⡔⠉⠀⠀⢀⣼⣿⣿⣿⡿⠛⠀⠙⠉⣿⠉⠀⠀⠀⠀⠘⠻⣿⣷⣦⣄⣀⠀⠀⠀⠱⣄⠀⠀⠀⠙⣧⡀⠀⠈⢿
⠀⠸⠀⠀⢰⡟⠀⠀⠀⠀⡌⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⣀⡠⠤⠴⠛⠒⠒⠶⠦⢤⣄⣀⠈⠙⠿⣿⣿⣿⣦⣄⣀⣿⣧⠀⠀⠀⠘⣷⠀⠀⢸
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢰⣇⢀⣴⣿⣿⠿⣫⣤⡠⠖⣋⣥⣶⣾⡿⠿⠿⠿⠿⣿⣿⣶⣯⣿⣦⣄⣬⡙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⣿⣿⡿⠋⠁⠀⡼⢋⣴⣾⠿⠛⠉⠀⠀⠀⣾⡆⠀⢀⠀⢉⠛⠿⣿⣿⣿⣄⠀⠈⠛⢿⣿⡿⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢠⠎⣴⣿⠟⠁⠀⠹⠂⠀⠀⠀⠉⠁⢀⣿⠀⠚⠁⠀⠈⠻⣿⣿⣷⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⡿⠃⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣾⣿⠁⠈⠛⠀⠀⠀⠙⢦⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⠉⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠀⢸⣼⣿⡇⣴⣶⡆⠀⠀⠀⠀⠀⠀⠈⣶⣿⡄⠀⠀⠀⠀⠀⠀⠸⢿⠷⠀⣿⣿⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⠀⢸⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡄⠘⣾⣿⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⢰⣿⣿⠁⢠⠇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⠀⠹⣿⣿⣧⡀⠋⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⡡⢣⣿⣿⠇⠠⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⢀⣄⠀⠀⠀⢋⡀⠀⠀⠀⢠⡄⠀⢀⠔⣡⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⡈⠁⠀⠀⠀⢸⣿⠀⠀⠀⢀⠡⢂⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣭⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣆⠀⠀⠀⠀⡀⠀⠀⠀⢀⠋⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠤⠤⣄⣀⠀⠀⠀⠀⠈⠢⢌⠙⠃⠂⠸⡏⠐⠘⠛⣉⡴⠊⠀⠀⠀⠀⣀⣠⡤⠄⠄⠀⠤⠄⢀
⠀⠀⠀⢰⣦⡅⠀⠀⠀⠀⠀⠄⠒⠂⠉⠉⡙⠂⣀⡀⠀⠈⠣⠔⢷⣴⣷⡴⠣⠔⠁⠀⠀⣀⠀⠘⡌⠉⠀⠒⠂⠀⠈⠀⠀⠀⠀⣵⣶
⠀⠀⠀⠁⠰⣊⠄⣠⠄⠡⠀⠀⠀⠀⢀⡀⠄⠀⠉⣉⢉⠖⠶⣾⡿⠞⠙⢾⣿⡶⠖⣋⢉⡉⠠⠀⠄⣀⠀⠀⠀⠀⠀⠁⣄⠀⣌⠲
⠀⠀⠀⠀⠀⢱⡾⠖⠀⠀⠴⠖⠂⣈⡵⠚⠁⠐⠊⠀⣉⣉⣻⡿⠂⠔⠓⠀⠻⣿⣋⠉⠁⠐⠂⠈⠙⠲⣭⡀⠲⠦⠄⠀⠲⠿⣖⠈
⠀⠀⠀⠀⠀⠈⠐⢺⣷⡂⢀⠀⠄⠋⣴⠞⠛⠂⠀⡐⡯⡛⠿⢢⣞⠀⠀⢸⢔⠾⢃⡙⣖⠀⠀⠚⠳⢶⡈⠃⠀⢀⠀⣶⡟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠫⠊⡆⠀⠀⣀⠀⠉⣁⡄⠾⠁⠤⠠⣬⡦⢠⣯⣤⠤⠆⠹⠆⣄⡉⠀⢀⠀⢱⡳⠌⠚⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠐⡾⠋⠀⠐⠦⢰⣾⠇⠥⠠⠈⢳⣦⠠⠖⠀⠉⠳⡆⠠⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠊⡠⠉⠜⠀⠆⠀⡄⠸⡍⠁⠘⡒⠂⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢀⢀⠀⡶⢐⠃⣐⠳⣀⢆⡼⢃⠀⣗⠠⠻⠦⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠸⠀⠀⠋⢼⠻⠿⢻⠿⠀⠀⠀⠀⠙⠀⠀⢈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠦⠀⠙⠘⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡐⠀⠐⠀⠀⠀⢀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⢂⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠙⠁⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣿⡿⠛⣻⡿⣿⣿⣿⠿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⠅⠀⠀⠀⠀⢀⡀⠂⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠆⠸⣿⡆⠀⠈⠀⠀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⡘⢄⠀⢻⣿⡀⠆⠀⠀⠉⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢎⢂⠀⢻⣧⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠉⠐⠢⢕⡀⠱⣕⢄⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠶⣤⣌⡐⠌⠓⠤⡙⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠈⠙⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡈⡀⡌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠁⣥⠁⠀⠀⠀⢀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢷⢠⠇⠀⣠⣴⢿⣿⣿⣿⣿⣿⢧⣠⣤⡀⠀⠀⠀⠀⠀⢴⡿⢿⣶⣾⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢸⢠⣾⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠀⢸⣿⣿⣷⣿⣿⣿⣿⡿⠟⢋⣉⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠯⣙⣿⠿⢟⡋⠵⠒⠉⠁⠀⠀⠀⠀⠈⠳⣎⠉⠉⠛⠛⠋⠉⠉⠀⢀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢷⡐⢄⠀⠀⠀⠀⢀⣮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡢⠀⠘⣸⡸⡀⠀⠀⠀⠀⢀⡠⣈⠔⠀⠀⡼⠀⢰⠀⠀⠀⢣⠈⢷⡈⠳⠄⢀⣀⡾⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⢂⠁⠳⣍⣢⣤⢔⡢⠕⠊⠀⠀⠀⡰⠁⠀⢨⠀⠀⠀⠈⡆⠘⣧⠀⠀⠛⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡀⠄⠈⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⢸⣴⠀⠀⠀⠀⠰⡀⠸⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⠋⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡄⡁⠀⠀⠀⠀⠀⠡⠀⠹⡀⠀⢀⣪⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⣀⠠⡈⠳⣄⠀⠀⣀⠠⠔⠊⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠁⠀⠱⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡒⢄⡉⠐⠂⠐⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣤⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀⠀⠀⠈⠒⢬⡓⠤⣈⠁⠒⠒⠒⠁⢸⠀⠀⢈⠉⠐⠒⠂⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⡈⠑⠢⣑⠢⣄⠀⠀⠀⠣⠔⠉⠑⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣺⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠙⠢⢍⡒⠄⠀⠀⠀⠘⠀⠀⢀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⠛⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⡄⣤⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠋⡼⠁⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⢰⠃⣀⡀⢸⠃⠀⣠⠴⠶⠋⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠎⠳⣄⠀⠀⠀⠁⠀⣴⢤⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⣴⣦⠀⠀⠙⠋⠁⠙⠋⣤⣾⠵⢦⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣠⡼⠗⠀⠀⠀⠀⠈⠉⠀⠙⣤⡇⠀⠀⠀⢠⢦⣴⢦⡀⠀⠀⠙⠃⠉⠁⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⣠⠔⠚⠉⠛⠲⣌⠁⢀⡴⠖⠚⠀⠀⠀⣉⣙⣦⠀⢀⡴⠛⠉⠉⠉⠙⢦⡀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⣷⠀⠛⠋⢳⠀⠀⢰⡇⠈⠁⠀⢾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⠧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠼⠃⠀⠀⢾⠀⠀⠀⢀⣠⡤⠖⠋⠀⠀⣀⣸⠀⠀⣸⣇⣀⣀⣀⡈⢙⡶⠠⣤⣀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠛⢯⠀⠱⣶⠚⠋⠉⠹⣭⠄⣴⣾⣅⣀⠼⠉⢩⠿⠁⢰⣇⡈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡇⢀⣠⠆⠀⣀⣀⡀⠀⠀⠀⠀⣀⢀⣀⣀⡳⠄⠨⠂⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠉⣁⠤⠒⠉⠀⠀⠉⠀⠒⠒⠚⠁⠀⠀⠀⠈⠉⠒⠀⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡠⠊⠁⠀⢠⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠜⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⡄⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠎⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⡄⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⠇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠔⠻⣙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠲⢄⠀⠀⠀⡀⠀⠀⡜⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠈⢢⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠥⠀⠀⠋⢀⠞⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠦⣀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠑⠢⣤⣀⡀⠀⠀⠀⡶⠀⠀⠀⠀⢀⣉⠤⠒⠁⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣤⣀⡤⣄⣀⠈⠉⠙⠓⠒⠒⢒⣒⣈⣭⣤⣄⢀⡀⢴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡞⠀⠀⠀⠀⠉⠉⠛⠓⠚⠛⠋⠉⠉⠀⠀⠀⠈⠉⢳⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⡀⠀⠀⠀⠀⢠⠏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⡄⠀⠀⠀⠀⠀⢸⡀⠀⣧⠀⠀⠀⠀⢸⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⡄⠀⠀⠀⠀⠈⣇⠀⡟⠀⠀⠀⠀⣿⢀⡞⠀⠀⠀⠀⠀⢀⡴⠚⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢤⡀⠀⠀⠀⠱⣄⠀⢳⡀⠀⠀⠀⠀⠸⣀⡇⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⢀⡴⠋⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢆⠀⠀⠀⠈⢦⡀⢇⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⣫⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⢧⡀⠀⠀⠀⠙⠾⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣳⡀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠀⠀⠉⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⣠⠖⠉⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠲⠤⢤⣀⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⣤⡖⠒⠒⠓⠒⠚⠓⠲⡄
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠒⠒⠒⠋⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠠⣶⠶⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠀⣀⠴⠋⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣰⠃⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⡶⡀⠀⠀⠀⠀⠀⢧⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡴⠁⠀⠀⠀⢠⠏⢸⠀⠀⠀⠀⢸⠃⠙⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣠⠞⠀⠀⠀⠀⢀⡎⠀⢸⠀⠀⠀⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⠀⠙⠦⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⠀⠀⣼⠀⠀⡞⠀⠀⠀⠀⢸⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡼⠁⠀⠀⠀⠀⠀⢧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡄⠀⠀⠀⠀⠠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⣀⡀⠤⠄⠀⡀⠀⠀⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠘⠢⣄⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠑⠀⠐⢈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⢀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⡾⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⠳⡀⠈⢻⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣴⣴⠟⠁⢠⠜⠠⠖⠀⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠻⠭⣄⣀⣀⣀⣤⣴⠿⠟⠁⠀⠄⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠘⠤⢀⠀⡀⠉⠈⠉⠁⠀⠀⡀⠤⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠩⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠌
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠀⠀⠒⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡨⠀⠀⠀⠀⠒⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢤⣤⣀⡀⣀⡴⠛⠛⠋⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣈⣿⡷⣾⣿⣽⣿⣿⡧⠈⠀⠀⠀⠀⠀⠃⠀⠰⠶⠄⠠⠶⠖⠶⠺⠅⠶⢿⡿
⠀⠀⠀⠀⢰⣿⢿⣭⢋⡀⠐⠀⠧⢢⠙⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠚⢹
⠀⠀⢀⣠⣿⣿⠉⡗⢛⠄⠀⡀⠀⢈⡁⢸⣿⣶⣿⡇⠀⠀⠐⣾⠀⠐⣦⠛⠀⠀⠘⠃⠀⠀⠀⢀⣿
⠀⠠⣿⣿⣿⣟⣶⣧⡀⠄⠚⠡⢀⠀⣡⣾⣿⣿⣟⡇⠀⠀⠀⣿⠀⠀⣿⡇⢠⡀⠀⠀⣠⡄⠀⠸⢻
⠀⠀⡙⠛⢿⣿⢿⣧⡔⠤⣶⢆⣄⡧⠈⢿⣿⠹⠟⡇⠀⠀⡸⡇⢀⠀⣿⡇⠈⠉⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠈⠀⠘⣿⣞⠙⣷⡿⠀⢀⠀⠁⠂⣸⣿⣆⣃⡇⠀⠀⠐⠷⡦⠶⠿⠷⠀⠀⠠⠦⠀⠀⠀⠈⠂
⠀⠀⠀⠀⡀⠹⣮⣗⣻⣧⣀⠐⢂⣇⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⣷⡀⠘⢿⣿⣿⣿⣿⣿⣯⣹⠏⢙⢛⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢿⣟⠠⣀⠀⠉⠙⠛⠋⠉⢀⠀⢸⣋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣺⣿⠿⣿⡿⣦⢶⣶⡶⢾⣏⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴
⠀⠀⠀⠀⢩⣥⣆⢿⣷⣴⣽⣟⢲⡄⢉⠀⢰⣾⣷⡆⠀⠀⡖⢂⣶⢰⠀⡆⡖⡶⣦⣖⠠⡆⠀⢀
⠀⠀⠀⠀⢰⣮⣉⣽⣿⣉⣹⣿⣈⣑⣾⠀⢈⣴⡛⡇⠀⠀⠳⡌⣽⢾⡠⡧⡧⢿⠏⣯⠸⠇⠀⠘⣧
⠀⠀⠀⠀⢸⡿⠿⣿⣷⠿⢻⣿⢿⠋⢠⣤⡀⣿⣿⡇⠀⠀⣀⣀⡀⣀⣀⢀⢀⣀⣀⡀⡀⢀⡀⢠⠹
⠀⠀⠀⠀⢸⡇⣿⢿⣿⣷⣿⣷⢾⡦⠀⣸⠳⢻⣏⡁⠀⠀⣟⣿⣿⡏⣿⢸⢸⣉⣈⡀⠉⠘⠃⠀⢰
⠀⠀⠀⠀⢸⣧⣽⣿⣿⣭⣻⠟⢁⣠⡞⠋⢰⡬⡿⠁⠀⠀⠟⠟⠛⠳⠟⠘⠚⠟⠛⠇⠛⠚⠂⢰⣿⡆
⠀⠀⠀⠀⢹⡟⡿⣿⣷⠟⠁⣴⢿⡿⢆⣀⣰⠟⠁⠀⠀⢠⢤⠄⣤⠤⡤⠠⣤⣤⣤⡤⠀⡤⠀⢸⠻⡇
⠀⠀⠀⠀⢱⢶⣿⣻⡟⠀⣼⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠚⠘⠙⠘⠛⠊⠉⠉⠉⠁⠉⠉⠀⢸⣀⡇
⠀⠀⠀⠀⢸⣷⣶⡿⠇⢀⣅⣀⠁⠀⠀⠀⣲⠆⢐⢒⣖⡒⣲⠒⡒⢲⣶⣶⡖⢲⣂⠀⠀⠰⠆⠾⠟⠃
⠀⠀⠀⠀⢸⡟⣃⠀⠰⠾⠿⠋⣀⡀⠀⣀⣀⡈⡀⠀⠈⠁⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠸⠟⠻⠷⠖⠤⠶⠾⠿⠿⠦⠬⠿⠦⠥⠾⠶⠤⠬⠷⠿⠿⠷⠦⠼⠿⠿⠿⠿⠦⠼⠿⠿⠇

⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢂⠀⠀⠀⠈⠁⠀⠀⠀⡔⠉⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠢⡈⠢⠤⠤⢰⡆⠠⠤⠔⣀⠌
⠀⠀⣤⡆⠀⠀⠒⠐⠀⠄⠩⠑⠂⠄⠀⠀⠈⠑⡠⢤⣸⣄⡤⣀⠎⠁⠀⠀⠠⢐⠚⠭⠤⠐⠀⠀⠀⠀⣂⣄
⠀⠀⠨⢂⡀⣠⠀⠀⠀⠀⠀⡀⠀⠀⠘⡃⠒⠴⣶⣾⠟⠻⣷⡶⠆⣒⡚⠃⠀⠀⡀⠀⠀⠀⠀⣀⡀⢀⡸⠉
⠀⠀⠀⠠⠾⢆⠀⠨⠓⠂⣡⠞⠉⠀⠊⠈⣉⣙⡿⠂⠘⠂⠸⢿⣋⣉⠉⠂⠀⠙⠲⣅⠐⠢⠀⠀⠸⢾⠉
⠀⠀⠀⠀⠀⠹⡯⢠⠀⠀⢁⠞⠉⣁⡀⠼⡎⠙⠻⢮⠀⠀⡷⠛⠋⡗⠄⢀⡙⠓⠷⠈⠀⢀⡠⣷⠏⠉
⠀⠀⠀⠀⠀⠀⠰⠐⠁⠂⠀⠀⠂⢀⡶⠊⡁⠐⣒⡾⠇⠺⣷⡒⠃⡘⠰⣦⣀⠐⠀⠀⠲⠐⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⠜⡱⢿⠀⡁⠈⢰⠿⡈⢣⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⡄⠁⣶⢠⠈⡦⠁⡬⢀⠢⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⢾⢿⡿⡏⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣴⣧⠀⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠋

⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⠀⠀⠀⠀⠀⢀⠀⢤⠄⠂⢀⢀⣄⡀⠀⠂⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⢀⠠⣔⣈⠰⠶⠆⡙⠙⠉⠐⠊⠁⠁⠀⠀⠀⠀⢀⠈⠠⠆⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⢉⠀⠐⠉⠀⠔⡫⠅⠈⢀⠠⠆⠀⠈⠉⠀⠀⠉⠀⠀⠓⠠⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⠀⠀⠉⠀⠀⠁⠐⠂⠈⠀⠄⡠⠔⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⢀⡐⠂⠁⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠛⢀⠀⠈⠁⠀⠄⠀⡀⡠⠚⠁⠀⠀⠀⠀⠀⠐⣀⣠⣄⣠⠀⠀⠈⠑⢄⠂⠀⠀⠑⠀⠀⠀⠁⠃
⠀⠀⠀⠈⠀⠀⠀⠊⠀⡀⠊⠀⠀⠀⣠⠔⠀⠀⠐⠈⠀⣠⠉⠉⣛⡾⣄⠄⠀⠑⡀⠀⠀⠠⢀⠀⠀⠀⠀⠘
⠀⠀⠀⠰⡄⠀⡆⠀⡜⠁⠀⠀⠀⠊⠀⣀⠀⢚⠓⠁⠀⢃⢀⠀⣦⠑⢚⠃⡀⠀⠀⠀⠀⠘⠀⠀⠈⠈⠀⠀⠆
⠀⠀⠀⠀⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠠⡅⠀⠂⠀⡄⢀⢀⡀⠀⠈⢢⠀⠡⠀⠀⠀⢠⣳⠄⡃⠀⠀⠈⡀
⠀⠀⢱⠀⡁⠀⠱⢸⠀⠀⠀⠒⠀⠀⠀⠁⢸⠀⠠⠥⠀⣐⠘⠈⠀⠀⠞⠁⠁⠀⠀⠀⠙⠁⡇⠀⠀⠒⠀⠀⠜
⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⡀⠀⢀⠀⠐⠃⠜⢄⠀⠀⠁⠉⠀⠀⢀⠔⠤⠀⡤⠀⡀⠀⠀⠀⠩⢀⠂⠸
⠀⠀⠀⠀⠀⠀⠁⠀⠐⢄⠀⢀⢄⢀⠆⠀⠈⠓⠁⡂⡠⢀⠠⡈⠴⠀⠀⠠⠁⠀⠂⠬⠀⠀⠘
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠈⢧⡅⣄⠠⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⡐⠈⡳⣶⣁⠆⠠⠄⠂⠀⠄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠁⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⣤⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⠗⠀⠌⠀⠀⠀⠀⠀⠀⠀⡄⠈⠑⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⢂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠃⠠⠀⠀⢀⡁⡄⠀⠀⠀⡀⠠⠆⢂⡀⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠒⠀⠄
⠀⠀⠁⠀⠀⠀⢀⡤⠑⠄⠀⠀⠀⠰⠀⠀⢀⠀⠠⢀⡀⣀⣤⡜⠶⠒⠉⣠⠀⡐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠉⠁⠀⢂⠀⡡⠔⠋⠀⠀⠀⠀⠀⠀⠐⠀⠃⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠘⠁⠀⠻⠄⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠁

⡟⠉⣿⣛⣻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣟⣻⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⡷⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⣀⣀⡽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣀⡤⠴⣖⣺⣭⠷⠞⠛⢋⠁⠀⠀⣴⠶⢦⣤⠀⠀⠉⠋⣀⣀⣀⣀⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⡧⢖⣺⣭⠶⠞⣿⣽⠷⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢶⣿⣀⠀⠀⢿⡝⣏⠉⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠛⠋⣁⣤⠶⠛⠉⠀⢀⣀⡤⠤⠴⠒⠶⠦⠤⢤⣀⠀⠀⠉⠻⢦⣄⠻⣜⢦⠀⠀⠀⠙⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡾⠋⠀⢀⡠⠖⣊⣭⠶⠞⠛⠛⠛⠛⠛⠶⠶⣬⣉⠲⣄⠀⠀⠙⢷⣝⣯⢳⡀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⣠⢔⣯⠶⠛⢩⡀⠀⠀⠀⠸⠇⠀⢀⡾⠰⢶⡍⠻⣦⡱⣄⠀⠀⠹⣿⣷⡱⣄⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⢞⣵⠟⣁⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣀⣄⠛⠿⠁⠀⠈⠻⣎⢳⡀⠀⠘⣦⠻⣜⢆⠀⢰⡇⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡰⢣⡾⠁⢿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣿⠀⠀⠀⠀⢀⣀⠙⣧⠹⡄⠀⠸⣇⠹⣮⣷⡟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣰⢡⡟⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⠁⠀⠀⠀⠀⠈⠁⠀⠸⣇⢳⠀⠀⢿⡀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡿⠀⢰⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⡇⢀⣀⠀⠀⠀⠀⠀⣠⠞⠛⠓⣶⣟⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⡷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡸⡇⠈⠉⠀⠀⠀⠀⠀⠙⢶⠶⠛⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣼⢃⡇⠀⣸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢧⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⢀⡿⠀⠟
⠀⠀⠀⠀⠀⠸⡄⠘⣧⠀⠈⣏⣷⡀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⣷⢠⡟⡴⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠁⠀⢹⣆⠀⠘⢎⢷⡄⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠳⠟⣰⢏⡞⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⠀⠈⠳⣝⢦⡀⠀⢸⡿⣮⠁⠀⢠⡄⠀⠀⠰⡆⠀⠀⢀⣠⠞⡡⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⣌⣷⣄⠀⠈⠳⢽⡷⣬⣍⣁⣀⣀⠈⠀⠀⢀⣀⣤⡴⠞⣫⠵⠋⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡴⠟⠁⠈⠻⣦⣀⠀⠉⠓⠮⠭⠭⢭⣍⣛⣛⣉⡭⠴⠒⠉⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡞⠋⠁⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠙⣿⠚⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠶⠟

⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⣀⡀⠀⠀⠀⠀⠀⠆⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢰⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠐⠃⠀⢠⣀⠤⣄⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠃⢀⠌⣥⡔⣈⡭⣤⡛⢂⠀⢀⣀⡂⠀⢀⠞⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⣂⠠⠄⢋⣠⠤⣌⠡⣆⠘⠁⢀⡀⠂⠀⠔⢠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⠀⡄⡹⡆⠉⢠⡟⢸⠁⠘⠇⠈⠂⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠇⠁⠀⠈⠛⠇⠀⡤⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠁⠀⡐⠛⡀⠛⡠⡠⠞⠀⢀⡈⠂⠀⡠⠐⠁⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⢖⠀⣤⡛⠐⠒⠉⠉⡸⠟⢛⣥⠀⠀⠈⣁⡱⢶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢰⠀⠀⠈⠛⠙⠶⡄⢻⠄⢠⡄⢹⠀⡴⠋⠀⠁⠈⠀⠀⠸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⠐⠀⠀⠆⠀⠀⠀⠠⠀⢴⠀⠀⠀⠀⠈⢀⡅⠀⡀⠁⠀⠀⠀⠀⠆⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠡⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠋⠁⣴⣶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⣠⠄⠀⠀⠀⢄⠐⠗⠌⢁⡀⢡⡀⠀⠀⠀⠰⣦⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢠⡜⠧⠀⠀⠀⢀⠃⠀⠀⠁⠀⠢⠈⣧⠀⠀⠀⠀⠈⠣⠂⠀⢧⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠉⠂⠀⠀⠀⣈⠀⠄⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⢀⣤⠀⣀⢠⣄⠀⢿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠏⠀⡿⠁⠙⠛⣿⠙⠟⢸⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣤⢰⣷⠀⠀⠀⣻⡄⠃⢸⡆⠀⠄⠀⠀⠀⠈⠛⠓⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⢹⢴⠉⢳⡀⠀⠘⣷⡿⢸⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⢀⡼⡀⢀⠳⠀⠀⡻⢷⣸⡄⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⢸⣰⠟⢰⠟⠀⣰⠇⢨⣼⠃⠀⢠⣤⣇⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡀⠀⠼⠀⢀⣠⢄⡴⢊⡙⢠⡟⠀⡰⠛⢀⣾⡿⡂⡀⠫⡀⠀⣠⡀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠊⢅⠀⠰⠦⡄⠉⠠⠈⠰⠋⢀⣼⠀⠸⡇⠀⢸⡇⡷⣇⠀⠀⠈⠀⢻⡀⠠⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⢤⣀⣱⠤⠘⠐⠢⡅⠀⣤⠴⠋⠛⡀⢠⣿⡀⠈⠳⠽⣏⠤⠄⢀⡖⠤⠹⢆⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠃⠀⠐⠋⣠⠔⢣⠀⠀⠛⡀⠠⡐⠀⠀⡇⢸⠷⣴⡂⢠⣀⣤⣤⣶⠚⠀⠀⠰⣄⠲⠟⠀⠡⠤⠴⠋⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠙⠂⡞⠀⠤⠂⠠⣤⣈⠉⣀⣀⠰⠄⢹⠃⠌⡉⠁⠀⠀⠙⠂⠀⠀⠈⣀⠀⠋⠔⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠥⠀⠁⠠⢀⡀⢰⠋⠡⠸⡀⣶⠖⠉⠉⠋⣙⠃⠊⠁⠘⢦⣟⡡⠀⠀⠀⠀⢸⣀⡇⠦⠠⣬⡁⠈⠙⠋⠶⠂
⠀⠀⠀⠀⠀⠀⢆⣰⣆⡀⠀⠈⡠⠂⠘⡒⠲⠞⢡⡞⠁⠁⠀⠆⠑⠒⠒⠀⠀⠈⢄⡘⠧⡀⢈⡀⠁⠤⣦⡔⠉⠤⠄⠀⢂⡰⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⠃⠀⠢⠺⠉⢀⠀⠀⡌⠀⠀⢠⡀⡙⠀⠀⠛⡀⢸⡎⠁⡶⠍⠋⠀⡀⠆⠈⡁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤
⣿⣿⠿⢿⣿⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣥⣬⣽⣿⣯⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣰⡟⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠠⠟⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠚⠛⠛⠶⢦⣄⠀⠀⠀⠐⠃⠀⠒⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣴⠯⠤⣤⣀⠀⣀⠀⢀⡀⣠⠶⠚⢋⣉⣉⣲⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⣿⣿⣿⣷⣦⡭⠇⠛⢾⡟⠁⠓⠤⣾⣿⣿⣿⣿⣿⣿⣷⡄⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡏⢠⡏⢰⠃⣿⣿⣿⣿⣿⠟⣋⣤⡤⠶⣾⣳⣶⣶⣶⣤⣍⣻⣿⣿⣿⣿⡧⢻⠀⢿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠁⣾⠀⣿⠀⣿⣿⡿⢻⡷⢚⣭⣶⡿⠿⠛⠛⠛⠻⠿⢿⣿⣿⣷⣌⡙⠻⢇⡼⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠹⣤⠞⠁⡴⢋⣴⡿⢋⡁⠀⠀⢸⣿⠀⠀⠀⠴⠈⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⣿⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡹⣿⣿⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⢱⣿⠃⠦⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⣷⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡇⢠⡤⠀⠀⠀⠀⠀⠐⣿⡿⠀⠀⠀⠀⠀⠀⠉⠁⣿⣿⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣿⠇⢰⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⣇⠀⢿⣿⣷⡀⡤⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠈⢁⠟⣼⡟⢀⡞⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠘⣦⠈⢻⣿⣿⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣴⢋⣾⠟⢠⠞⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠙⢿⣿⣿⣦⣍⣀⠀⠀⠿⠀⣀⣠⠶⣋⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣿⣿⣿⣿⣯⣽⣶⠿⠛⠻⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠀⠀⠉⠉⠉⠉⠉⠉⣠⠴⠀⠀⠈


⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣦⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢹⣇⠀⠛⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢾⡇⠈⣷⠀⠀⠀⠀⠙⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣴⡏⠀⠀⠀⠀⢀⡿⢸⠇⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠹⡄⠀⠀⠀⢸⡇⣿⠀⢀⣴⣾⣿⣏⣻⣝⣿⠿⣛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⠀⣿⢱⡇⢰⣟⣿⣿⣿⢧⢻⣽⡾⣿⣚⣿⣯⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⡄⢠⣿⣼⢁⣿⣷⣿⣿⣿⡞⢯⢷⣛⢩⣭⣷⣾⡖⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣿⡾⠻⣿⢛⡿⣿⣮⣿⠛⣧⢘⣼⣿⣿⣿⠿⣿⣿⠿⣿⣄⡀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣞⣿⡿⢫⣿⣁⠢⠩⣟⢳⠞⣵⡿⣹⣿⣷⡿⢟⡿⣩⡞⣫⢶⡝⢯⠶⣩⠿⡛⣿⣭⣛⣵⣾⡍⢟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡳⣟⣭⣿⢁⣿⢣⣥⢉⣿⢂⠇⣼⠏⣽⣫⣿⢋⡜⣧⠞⣥⢚⡴⢫⠝⣥⠿⣡⡜⢯⣩⣿⣟⣽⡴⡾⡟⣮⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡼⣱⣟⣾⡧⣜⡏⣗⡮⡙⡙⠾⠽⡟⣼⣳⣿⣋⣏⡸⡏⡶⣹⢎⡸⢇⢿⠙⡤⠹⣧⠚⣇⠴⢻⣿⣷⣻⣿⣱⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣯⣽⣿⣗⣿⣗⣶⣽⡃⢯⣿⣳⣲⢦⣆⣿⣿⣸⠇⣸⣀⢇⡘⢢⡜⢆⣗⠈⡆⠹⡖⢫⣨⣼⡿⣿⣭⣿⣬⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡷⠟⠛⠛⠛⠛⠻⠷⣧⣽⡒⣬⡼⠟⠉⠉⠉⠻⢶⣦⡟⡴⠣⡜⠳⡜⢻⡆⢡⠜⣧⣾⣥⣾⡷⠿⠷⠷⣷⣯⣀
⠀⠀⠀⠀⢀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣤⣑⣮⣉⣮⣽⡾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⢠⡾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷⣦⡀⠀⠀⠀⠀⣠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣆⣤⣤⣴⡏⠀⠀⢈⡷⠀⠀⠀⢀⣀⣤⣤⢶⡶⠶⠶⢶⣤⣤⣀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠙⣦⣀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀⠙⢷⣄⣀⣏⠀⢀⣿⣥⣤⡴⠞⢁⣤⡶⠟⠋⠉⠠⠀⠈⠀⠀⠀⠀⢀⠈⠙⣻⣶⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠉⠻⡟⠛⠛⠉⢉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣶⡾⠛⢹⣿⣿⡙⢷⣤⣶⠞⠋⡁⠀⠀⠃⠀⢀⠀⢠⠄⠀⠂⠀⠁⠀⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠂⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⣿⠋⠀⢀⣾⠋⠙⣷⠀⠻⣧⣀⠀⠄⠀⠐⠂⠀⠈⠀⠀⠀⢀⠄⠀⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠉⠈⣿⣤⣀⣼⠏⠁⠀⢻⣦⠶⠟⠃⠀⠀⠀⢀⡀⠀⠰⠀⠘⠁⠀⡀⢠⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠨⠙⠫⠀⠀⠀⠰⠀⠂⠂⠀⢈⠁⠀⠠⠤⠀⠄⠐⠐⠁⠀⠁⣀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢈⡀⠈⡀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠂⠘⠁⠘⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠄⠀⠒⠀⠘⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣡⣀⠀⡀⠀⠁⠀⠈⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠓⠲⠷⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠉⠁


⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣶⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣤⠤⠶⠶⠦⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠛⠒⠲⠶⠶⠶⠒⠚⠛⠋⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣀⣠⣀⡀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢻⠀⢨⠛⣦⣤⣴⠛⣍⠳⢴⡏⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⡾⠀⢸⡀⡇⠀⢸⠀⢸⠀⢸⡇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⢧⡇⠀⠀⡇⡇⠀⠈⡇⢸⠀⢸⡷⠋⠉⠙⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣤⠞⠁⠀⣿⠀⠀⢸⡇⠀⠀⣧⡇⠀⠀⣧⢸⠀⠠⣧⠀⠀⠀⠈⠉⠈⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⠀⠀⢿⣷⠀⠀⢸⢸⡄⠀⣶⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⠸⡀⠀⢸⣿⠀⠀⢸⡀⣇⣀⣽⣄⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄
⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⣿⠟⡗⢲⠾⢶⠷⣶⠾⠿⡶⢲⡟⠛⣯⢉⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠹⢦⡄⠀⠀⠀⠀⠀⠀⠈⠻⠷⢼⣤⣼⣤⣼⣤⣤⣷⠴⠷⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⡴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠛⠉⠉⠳⣄⠀⠀⢀⣠⠖⠚⠓⠶⣄⡀⠀⢀⡴⠋⠁⠀⠉⠙⠲⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠉⠁⠀⠀⠀⠙⢋⣀⠉⠛⣹⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠈⠁⠈⠁⡀⠀⠀⢠⣄⠀⢡⠀⠀⠀⠀⠀⠀⠨⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢒⠉⠀⠈⡦⢄⡀⠄⠁⠀⠉⠊⠥⠤⢤⣚⡀⠀⣀⠀⡀⢠⡀⡐⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢯⠀⣀⠄⣤⢢⡴⠞⠻⠀⠂⠘⠋⠛⠀⠀⠲⠀⣤⣝⡳⠤⡁⠆⡠⠄⠀⠀⠀⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠁⠀⠉⢈⣴⣣⠶⠀⠀⣸⡦⠀⢀⣂⡀⣶⡖⣀⡶⠀⠀⠀⠈⠱⣌⠢⠀⠀⠀⠰⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢘⠰⠃⡞⠁⠁⠄⢲⣦⡭⠀⠀⠈⠉⠓⢈⡀⠉⠉⠄⠀⠃⠀⠀⠀⠀⠉⢀⢀⠀⠀⢀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⠈⠀⡜⢀⣤⣄⣤⠀⡿⠿⠻⠧⠀⢀⣀⣾⠃⠀⠀⠆⠀⢠⡦⠀⠘⠃⠰⠈⠋⢸⡃⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⢀⠘⢰⡇⠈⠀⠈⢉⣤⡁⠀⠀⠀⠀⠁⠀⢠⡄⢡⡀⠀⠖⢀⡄⠀⠀⠀⠀⣷⠈⡀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⠀⠠⠤⡇⣿⠀⣼⠗⠀⠉⣩⣵⢤⣤⣚⠀⠀⠄⠀⠀⡄⣳⡀⠀⠀⠀⡐⠂⠀⠗⢾⡆⠂⠀⠀⠀⠂⠸
⠀⠀⠀⠀⠀⠀⠀⢐⠦⠌⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠁⠀⠠⠀⠛⠀⠀⣀⠻⢠⡇⠀⠀⠐⠀⠛⠀⠀⢀⣴⡁⠀⠀⢰⡈⡇⠀⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⣧⠀⠀⠀⢡⡜⠃⣿⠀⠉⠀⢲⢀⠀⢤⣀⠀⣤⠛⣿⠀⠀⠀⡀⠀⠘⠀⠀⠉⠉⠀⠸⢯⡀⡇⠀⠀⠊⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣧⠀⢀⠀⠅⡄⣿⠁⠀⠀⠃⣸⣆⢰⠁⠀⠀⠙⣿⢰⡄⠈⠁⣶⣤⠀⠀⠀⠀⠀⠀⠘⣱⡇⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠬⢿⠀⡜⠁⠘⠀⣿⠀⠶⠀⠈⣩⠍⠀⢀⡀⣠⠀⣉⢀⠀⠀⠫⡟⠀⠀⢹⣄⠰⠆⠠⡔⣿⢠⠀⠐⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⢈⠙⡄⢿⡄⠀⠀⠀⠁⠀⠀⣈⠁⠀⠄⢹⠸⠀⠋⣠⠀⠀⠀⣈⣙⠀⡀⠀⢸⠃⡌⣃⠰⠆⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡄⠀⠀⠀⢻⡟⠂⠀⢷⢸⡇⢶⠉⠀⠛⠂⣾⢀⠀⠀⣹⣷⠀⠀⠈⠀⠀⠋⡠⠃⢈⣅⠰⠐⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠒⡄⢺⡀⠀⠀⠀⠀⠰⢃⠀⠄⠑⢠⡼⠘⠖⢢⡧⡁⠀⠀⣀⡼⠟⠋⠀⠁⠀⠀⠴⣢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠩⣀⠛⠛⣂⡄⠀⠂⢀⠀⣘⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠀⠀⠈⠙⠓⠶⣠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠀⣬⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⢢⣤⡄⢀⡄⠑⣤⢬⣤⠶⠓⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄⠂⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢶⠌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠁⡀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⣰⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠉⢲⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠤⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠻⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⠲⣿⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠀⢻⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⡐⠀⠇⠀⠀⢄⣿⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠙⠿⠁⠘⠀⠀⠀⠜⠁⠐⠀⠰⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠛⠀⠀⠴⠷⠦⢀⣀⣀⠤⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣿⠀⠀⣄⠀⠀⢰⣿⣧⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠚⠛⠋⠁⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣶⡄⠀⠀⠉⠀⠀⢀⣠⠴⠶⠚⠛⠛⠛⠉⠛⠛⠛⠲⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⢣⣤⣶⣄⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠘⣷⣄⢸⣿⡇⢀⡞⠀⠀⠀⠀⠈⣿⣻⣛⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡠⠤⡿⢿⡀⠀⠀⠀⠈⣮⠺⠋⠙⣿⠀⠀⠀⠀⠀⠀⣹⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠆⠀⠀⢳⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸⡁⠀⠀⠀⣇⠀⠀⠀⠀⣰⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⢁⠀⡇⠀⠀⢠⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠻⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡷⢷⡟⠁⠀⠀⠀⠙⢶⣤⣤⣖⣸⡇⠀⠀⠀⠀⢻⠟⠲⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣧⠀⠀⠀⠀⠀⢸⠟⠀⠹⣯⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣲
⠀⠠⠤⠠⣤⣤⣬⣤⣤⠤⠤⢤⣤⣬⡶⠶⣿⡦⠒⠀⠀⣤⣬⡤⠤⢨⣼⡤⠤⠤⠤⠤⣷⡦⣤⣤⣤⣄⣤⣀⣐⣀⣀⣀⣤⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣴⣿⡏⠀⠀⠀⠙⣿⣿⣶⣿⣿⣤⡄⠀⠀⠘⣿⣦⣼⣿⣦⡍⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠆⠀⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣀⣤⠤⠴⠶⠶⠶⠶⠶⠦⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣄⢞⣰⠤⠤⠤⣄⡀⠀⠀⠀⢀⠀⠀⠀⡠⢤⣤⣴⣧⣄⣠⣟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡍⠀⠀⠀⠉⠱⠞⣙⡧⠤⠶⠞⠀⢠⣞⡚⠿⣿⣿⡿⠟⠗⠙⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣿⡁⠀⢀⣠⣴⡾⠟⠉⠀⢀⣀⣀⣀⠀⠙⠻⣿⣶⣬⣉⠀⠀⠀⠀⠨⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿⡿⢏⣤⣾⡏⣟⠭⢐⣂⣉⣥⣬⣤⣾⣤⣤⣍⣅⣒⠭⣿⣏⣿⣶⣤⡀⢤⣍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣭⡾⠟⢋⠕⣫⣴⡾⠟⠛⠉⠉⠀⠀⡀⠀⠀⠉⠉⠛⠿⣶⣭⡒⢌⠙⠻⠶⣍⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⢞⣡⡾⠛⠁⠠⠀⠀⠀⠀⠀⡆⠄⠄⠀⠀⠀⠀⠀⠈⠙⢿⣷⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⡶⠋⠀⠀⠀⠀⠂⢰⠀⠀⠀⠃⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⢿⣦⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣰⡟⢁⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⡟⠀⠈⠀⢲⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠫⠀⠈⠀⢻⣇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⠁⠀⠀⠀⠈⠀⠊⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠈⠁⠀⠀⠀⠈⣿⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠻⢷⣄⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠐⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⠟⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣹⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠀⢰⠿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣱⡀⠐⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⢄⣈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡑⣄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡰⢫⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣎⠳⣄⠀⠐⠀⠈⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⠀⠀⠀⣠⢞⣵⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣮⣑⠦⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣀⠴⣋⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⡿⠛⠷⢦⣭⣝⣒⠖⠒⠠⠖⠶⠒⣒⣋⣭⣶⠾⢿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠙⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢶⠶⠿⣛⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⢿⣿⡿⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣠⢿⣿⣿⣿⣿⣭⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢀⡀
⠀⠀⠀⠀⢿⣯⣤⣬⣭⣭⣤⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣯⣿⣿⠷⣆
⠀⠀⠀⠀⣠⣾⠿⢿⠛⣿⠿⢯⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠛⠛⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠽⠖⣚⣩⣆
⠀⠀⢀⣞⠞⠁⠀⢸⠄⣏⡀⠀⠹⣮⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢚⣥⣾⣿⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣻⣏⣭⢴⠶⠭⣄⡀
⠀⠀⣸⡏⠀⠀⢀⢼⡳⣯⢹⡀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⡇⡇⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡇⠀⢸⠀⠀⠀⠉⢦⠀⠀⠒⠂
⠀⠀⢸⡇⠀⠀⡏⣼⣁⠿⣠⠇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠇⠀⠸⣇⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⢞⣧⢤⡞⡗⡄⠀⠀⠈⠆
⠀⠀⠸⣧⠀⠀⠉⠀⠁⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⡟⠀⢀⠀⢘⣀⠙⢷⣤⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠸⣼⢻⢎⠇⠳⠔⠀⠀⠀⡘⠀⠀⠑⠒
⠀⠄⠀⠹⢷⣦⣀⡀⣀⣠⣴⠞⠁⠠⣄⠀⠀⠀⠀⠀⠂⠀⢀⣿⠁⢠⢞⡖⡇⠐⠅⠈⠹⣧⠀⠂⠀⠀⠀⠁⠀⠈⡇⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡰⠁
⠀⢀⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⡄⠈⠃⠀⠀⠀⣀⠀⠀⢸⣿⠀⠀⠈⠁⠁⠀⠀⠀⠀⠹⠀⠀⠤⠀⠀⠀⠀⡤⠈⢦⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠈⠉
⠀⠈⠀⠀⢠⠀⠀⢠⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠦⡀⠀⠀⠀⠀⢀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠑⠒⠒⠒⠉⠁⡀⠀⠀⠒⠤
⠀⠀⠀⠀⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣊⠀⠒⠒⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣂⣒⣶⢶⣶⣒⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡿⠉⠀⠀⠀⠀⠀⠀⠈⢻⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣲⣤⠾⣿⣥⣤⣀⠀⠀⠀⠀⢀⣀⣤⣷⠷⣤⣔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⢀⣠⣬⣿⣿⡆⠿⣿⢿⣿⢿⣤⣄⡀⠀⢈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣳⣿⣿⣾⡟⢯⢀⣼⣋⡀⣠⣿⠀⣙⢦⣸⣿⢛⣷⣦⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣽⠟⣿⣿⣿⣶⡿⠗⣂⣀⣤⣤⣀⣀⠺⠿⣧⣾⣿⡉⠙⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡝⢹⢟⡥⠞⠉⠁⠀⠀⣂⠀⠈⠉⡓⢮⡹⠍⠫⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡴⠋⠀⠀⠀⢀⠀⢸⣩⠀⢀⣾⠁⠀⠙⢦⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡾⠁⢀⣤⡀⠉⠈⠀⠀⠀⢀⠞⠉⠀⢠⡤⠄⢳⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⠁⠀⠈⠛⠛⢖⣀⠀⠀⢀⡎⠀⠀⠀⠘⠃⠀⡈⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⠀⠆⡿⠀⠀⠀⠀⠉⢯⣼⠀⠀⠀⠀⠀⢘⡆⠇⣹⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⣿⡼⠀⠀⡀⠀⠀⠀⢀⠋⠉⢂⠀⠀⠀⠀⡀⠀⠀⣿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠈⠳⡠⠄⠟⠀⣀⢠⠂⠀⠀⠀⠱⣀⠀⠚⣧⣤⠆⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠁⠘⠦⡀⠨⠀⠀⢰⡇⠀⠀⠿⣤⣾⡿⣃⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⣷⣄⡚⢷⡶⣤⣤⣥⣤⣾⡾⢟⣯⣬⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⡿⣿⣿⣶⣾⣿⣿⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠻⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠛⠋⠃⠉⠉⠁⠀⠒⠛⠋⠒⠲⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠶⣤⠤⠶⠮⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠶⣤⣤⣠⢥⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠔⠒⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠑⣖⠁⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠤⠖⣍⠤⠔⠒⠈⠉⠁⠀⠀⠀⠉⠉⠒⠂⠤⣉⡶⣤⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡠⠔⠊⢁⠴⠋⠀⣀⣄⡲⠶⠊⠋⠉⠉⠙⠱⠶⢦⡤⣀⠀⠙⠢⡈⠑⠢⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⢔⡯⠊⣁⠀⠀⠀⢀⡀⠻⣀⠀⠀⠀⢈⠑⠵⡢⡀⠈⠢⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⢔⠕⠁⠀⠀⢹⠄⣶⠀⢈⡇⣀⡋⠀⠁⢠⠘⠁⠀⠈⠪⡢⡀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢮⠋⣀⠀⠀⠀⠸⠁⠿⠀⠀⠀⠀⠀⡌⠀⠸⠄⠀⢀⡀⠀⠙⡵⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣾⠃⠀⠉⣗⢰⢱⡄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢀⠟⠘⠁⠘⣗⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡏⠀⠀⠀⠛⠈⠚⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢹⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⢠⡄⢆⣷⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣤⣤⣤⣤⣤⣤⠀⣗⢀⡄⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡄⠀⠀⠐⠁⠀⠀⢀⣤⣶⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠋⠀⠁⢰⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⣧⠀⠀⠀⢦⡔⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡼⠅⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢻⢃⠀⠶⠃⠉⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢤⠀⡐⡝⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠫⡧⡀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠀⠀⠀⠸⣭⠀⠀⠀⠀⢀⢜⠝⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠙⢟⣦⡀⠀⠦⠀⠀⠀⠀⠏⠗⠀⠀⠀⠀⣤⠀⢀⢴⡻⠋⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠪⢖⡤⢀⡀⠀⠀⠸⠂⠀⠀⢀⣀⢤⡾⠕⠉⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢉⠶⢄⡀⠈⠑⠚⠻⠂⠒⠒⠒⠿⠓⠊⠁⢀⡠⢖⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠋⠀⠀⠀⠉⠒⠰⠤⠤⠤⠤⠤⠠⠖⠒⠈⠁⠀⠈⠣⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⡀⠀⠠⣄⡀⢤⠰⠂⠀⠀⠀⠀⠠⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠃⠀⠀⣀⢘⡲⠖⢃⣠⠶⠞⢀⠀⣀⣸⠥⢔⠒⡀⡈⠀⣸⠆⠀⠀⢀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡥⠀⠀⠀⠙⠀⢄⡀⠋⠡⠂⡏⠁⠐⠊⠌⠡⢈⡀⠂⠀⡝⠻⡄⠀⠄⠀⠒⡀⠁⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⢀⡸⢏⠀⡋⢁⣀⣀⠀⠀⠀⡨⢁⣀⡃⠀⣀⡀⠀⠙⠋⢁⠈⠀⠀⠀⠁⠘⣪⢠⣄⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀
⠀⠀⠀⠀⢀⡴⠟⠛⠉⠀⠁⢸⠀⢩⡶⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠁⠌⡾⠁⠀⠀⠀⠀⠀⠬⠛⠛⠄
⠀⠀⠀⠀⠛⠁⠰⠖⢲⣤⠀⢠⣰⢛⡠⠖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢈⣄⡄⠀⠆⣀⡤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⣼⢇⠀⢀⣊⠁⠀⠐⠀⠄⠀⠀⠀⠈⠀⠀⠀⠂⠀⠄⠀⠈⠀⠀⠀⠈⢢⠠⣜⣷⡥⢻⠅⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠏⢀⡼⠋⠽⢏⣠⣐⠂⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⢨⠅⢸⠀⠀⣠⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢋⠀⣀⢠⠔⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⢸⣧⢃⡿⣰⣄⠰⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣴⢁⣕⣮⣯⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⢈⡎⢳⢿⡌⢿⡤
⠀⠀⠀⠀⠂⡃⠀⢀⡾⢃⡼⣿⣻⣯⣿⠏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣼⠁⠁⠈⡇
⠀⠀⠀⠀⢠⡂⢠⣾⣓⢿⣵⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠁⠀⠀⠎⡬⢿⣄⣠⡴⠁
⠀⠀⠀⠀⠀⢡⣿⣷⣻⡷⣾⣿⣽⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⣀⡀⠀⠰⠀⠀⠀⠀⣰⠻⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⢻⡏⠁⠀⠲⢀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠈⣹⠀⡘⢣⣄⠀⣶⠈
⠀⠀⠀⣴⢩⣽⠟⠋⣠⣏⡸⠙⠂⠀⢣⡀⠐⢀⠀⠀⣀⠀⠄⢠⠀⠀⠀⢀⠀⢂⠀⡁⠰⣄⠀⡁⠞⣩⠀⠠⠀⠀⢀⡀
⠀⠀⠀⣛⠟⠁⠚⠿⣴⠶⢇⠀⠀⠀⢠⣇⣶⡈⣐⡄⠈⠁⣀⡀⣀⡀⣀⣀⠁⠀⠘⢳⡀⠀⠹⠿⠅⠀⠱⡆⠴⡿⠉
⠀⠀⠀⢁⡀⡋⠛⣠⣿⢠⣸⠀⠀⠀⠀⠸⠒⠸⠟⢢⡔⠉⢉⡀⠛⠁⠀⠈⠁⠑⠆⠉⣠⠠⢀⠀⠹⣆⡀⠸⠆⠄⢀
⠀⠀⠀⠀⠉⢀⣸⠟⠛⠙⡉⠀⠀⠀⠀⠀⠀⠀⣦⠞⣿⡛⠎⠠⠦⠀⠀⠐⠀⠄⠀⠀⣀⠀⡀⠄⠀⠁⠀⠀⠀⡀⠀⠒
⠀⠀⠇⠋⡐⢾⠻⡀⢹⣋⣗⠀⠀⠀⠀⢠⣤⣾⣵⡟⡠⠁⠀⠀⠀⠀⢀⢀⠀⠀⠀⠁⡤⠈⠀⠀⠀⠀⠀⠀⣀⣡
⠀⠀⠀⢰⠃⠸⣄⣇⣈⣭⣿⡄⢠⠆⠀⣸⣧⡎⠘⠏⡀⠁⠀⠀⠀⠀⠁⠚⠃⠀⢁⠚⡶⠂⢀⠀⠀⡀⢀⢂⠜⠁⠐
⠀⠀⢀⡃⠀⠀⠛⠛⡛⢺⣯⣿⡾⠰⣦⣟⡿⠁⠀⡔⠁⢄⠀⠀⠀⠠⠆⠀⣠⣠⠀⣀⠀⠀⠀⢀⡀⠀⠀⠊⠄⠀⠀⢀⣴⡧
⠀⠀⠈⢛⠀⠉⠉⠉⣩⠟⣦⣟⡄⠀⠋⡉⠁⢠⡄⠀⠀⢀⠀⠈⢠⠀⠤⠆⣿⠁⠀⠄⠀⣅⡀⡔⢦⣼⣷⢀⡔⠁⠶⡁⠠⠔
⠀⠀⢀⡈⢁⠀⠀⠠⣿⣄⢻⠭⣾⠤⡄⣬⡖⢀⣀⣦⡐⠾⠖⠐⠶⣶⣴⠀⣿⣦⡀⠐⣉⢹⡆⡤⣌⣳⣞⠏⡾⢳⣼⡿⣤
⠀⠀⢉⠀⠛⢡⣤⢤⡄⠙⠈⣿⠹⠛⠀⢪⢥⣆⣽⣭⡘⢶⣟⣞⣬⡵⢇⠼⠃⢠⣟⠖⣧⣶⣼⡗⠻⠉⣯⢻⡼⣿⣿⣎⠙⠆
⠀⠀⠙⡀⡦⣬⠙⠀⠀⠀⠀⣮⢰⡌⠶⠿⠾⠟⠿⠋⠟⠛⠋⢉⠉⠁⠙⠉⠙⠛⠻⢦⣽⣂⣿⠁⠀⠀⠀⠀⡏⣼⠛⠛⡀
⠀⡀⠀⢹⡵⢺⣇⠀⠀⠀⠀⠙⣧⡀⠀⠀⠐⠀⠀⢀⣠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⢈⡽⠁⠀⠀⠀⢀⡐⠀⢨⠶⢼⡇
⠀⠁⠀⠀⠀⢤⠹⠀⠀⠀⣤⡿⠿⢓⡄⠀⠀⠈⠐⠁⠀⡀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢁⠶⣿⡠⡄⢀⢰⠿⣷⡎⢾⠖⢶⠃
⠀⠀⠀⠈⠁⠀⢧⠰⡌⡭⠛⠿⠁⠀⢀⠀⠀⠀⢀⣤⡈⢥⣠⣄⣸⣤⣄⠀⠄⠀⡀⠈⢤⡉⢹⣿⣯⡈⢬⡖⢛⢮⠄
⠀⠐⠀⠈⢉⣄⢁⡀⢻⡀⠀⢛⠀⠀⠇⢀⠤⡄⠼⡯⡃⢾⣧⣤⣧⣴⢋⣁⡀⢤⣴⣀⣠⣧⡄⢹⠟⡰⢃⣠⣴⠦⠈
⠀⠾⠀⠀⠀⠛⠃⢉⣩⠡⣶⡏⡗⠐⢒⣍⠀⣲⣭⢙⣛⢫⡉⠉⠉⠑⣎⢨⢁⡞⢫⣽⣯⣾⡲⠿⠖⠋⠚⢶⠧
⠀⣶⠀⠀⠀⠀⠰⠆⠀⡱⢬⠤⠄⢀⡉⠙⠀⠀⠊⢀⡛⠃⠀⠀⠀⠐⠛⠛⠫⠑⠛⠻⠶⠇⠀⢹⠟
⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⡀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠸⢷⣄
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢞⠛⢶⠹⡊⡴⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⣶⢏⢹⣿⡐⠢⣜⠐⠀⣣⠔⢒⣿⡏⡹⣖⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⠁
⠀⠀⠀⠀⠘⣿⣏⣾⣺⠁⠙⢽⠻⡟⠑⡼⠂⣰⢿⡟⣧⠙⢿⣦⡑⠂⠘⢊⣴⡾⠋⣾⢛⡿⣆⠐⢤⠊⢹⠿⡯⠋⠉⣗⣷⣻⣿⠃
⠀⠀⠀⠀⠀⠙⣿⣡⣿⣭⣒⡬⢄⡁⠠⣕⣻⠃⢾⠧⠟⣤⣦⡍⠛⢦⡴⠛⢩⣴⣤⠷⢾⣷⠘⢟⡫⠄⢈⡠⣥⣖⣽⣿⣬⣿⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣉⡽⣿⡿⣷⣌⠄⠩⢙⡲⠛⠉⢰⣿⣿⡏⠀⠘⠋⠀⢹⣿⣿⡆⠹⠛⠖⢏⠌⠐⣡⣾⢿⣿⢧⣘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣞⢟⡄⢙⣿⡌⢠⡆⢀⡀⠰⠄⢸⣿⡇⣤⣠⣆⣤⢸⣿⡃⠰⠄⢀⡀⢰⠄⢁⣿⡋⢠⡻⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⢰⣷⡇⠀⠁⡿⡏⢠⢒⡟⢢⢹⣾⠳⢞⣷⡏⡔⢳⡒⡆⢹⣿⠈⠀⢸⢿⡄⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣗⡇⠀⠀⡇⢠⠌⠉⢸⡟⣸⠛⠃⠘⠛⣇⢿⡇⠉⠡⡄⠺⠀⠀⢸⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⢿⣳⣶⣾⣧⣿⡀⠀⠀⢀⣿⣼⣷⣦⣞⡿⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⣷⣽⣆⣸⢿⢧⡀⢀⡼⡿⣅⢰⣯⣾⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢦⡀⠙⢷⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡶⠶⠃⠀⠈⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢷⡀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣷⣄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣭⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠋⠉⠉⠁⠀⢀⣤⣤⣄⡈⠉⠉⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠉⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⣱⠀⠀⠀⠀⠀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠴⠖⠒⠒⠚⠋⣁⣀⣠⣤⣶⠶⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠛⠋⠉⠉⠀⠀⠀⢰⣏⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠉⣉⡙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡾⠟⠛⠛⠷⣦⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⢀⣿⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⣿⠸⣧⣤⣤⣶⠟⠋⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⣿⣤⣬⣥⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⣠⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡶⠋⠙⢧⣀⣴⠟⠙⢷⣄⣿⠙⢷⣄⣿⠉⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠉⠁⠀⠀⠙⠋⠀⠀⠀⢘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠉⠛⠳⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⡶⠶⠶⠶⣤⣤⣤⣤⣴⣶⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣴⣶⣿⡽⠿⠛⠛⠛⠋⢉⣉⣉⣛⣛⣛⣛⣛⣻⣿⣯⣭⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣛⠳⠶⠶⣤⣤⣤⣄⣤⣤⡶⠞⣋⣉⣤⣴⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠶⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡀⠀⠀⡠⠔⠒⠋⠉⠉⠛⠒⢤⡀⠀⠀⣠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢀⣁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠉⣀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⠛⠉⠀⠀⠐⢢⡄⠀⠀⢈⠄⡠⠐⠀⠀⠈⠙⠛⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣠⣤⠶⠚⠀⢈⡃⠀⠲⢤⣄⣀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣤⡶⠟⢋⠩⠛⢂⣀⣀⠉⠉⠀⢈⠛⠫⢉⠛⠲⢦⣄⣀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡠⠂⣡⣴⣾⡿⠝⠒⠛⠓⠒⠶⣬⣐⠠⡁⠢⡀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⣫⡯⠋⢀⢄⠀⠀⠇⡇⠀⠀⡠⠈⠻⣮⠢⡈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠰⣱⡟⢀⠀⠀⠑⠂⢠⠈⠁⠀⠐⠀⠀⣀⡈⢷⡑⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⢢⠟⠀⠀⠉⠂⠀⠀⣼⣷⠆⣀⠔⠁⠚⠁⠀⠘⢧⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⡆⢰⣀⣒⠀⠀⠀⠀⢹⡾⠃⠀⠀⠰⠶⢶⠀⢝
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡸⡄⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡹⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢳⡀⠙⡫⠓⢀⡄⠀⠀⠀⠀⠀⠀⠒⠑⠀⣴⢃⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠡⡹⣦⡀⠀⠞⠌⠀⠐⠆⠀⠈⠂⠀⢀⡼⡣⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠪⢑⠦⢄⣀⢀⣀⣀⢀⣠⠬⢞⠩⣪⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡞⠓⠢⢈⡀⠈⠀⠉⠉⠈⢐⣨⡵⠞⠛⠜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⠀⠀⠀⠁⠉⠒⠛⠁⠁⠀⠀⠀⠀⠀⠺⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠋⠉⠉⠉⠀⠈⠉⠉⠉⠙⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⡷⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣡⣤⣤⣤⣤⣤⣤⣡⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣉⣉⣉⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢉⣿⣻⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿