Spigot Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠛⢿⣿⣶⣾⣿⠟⠉⠉⠛⣿⣷⣶⣿⡿⠛⠛⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⢀⣨⣭⣭⣅⠀⠀⠀⠀⣨⣭⣭⣥⡀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣴⣿⠟⠛⠻⣿⣦⡄⢀⣴⣿⠟⠛⠿⣿⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣠⣶⡿⣿⣧⣼⣿⣿⣷⣄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣀⣀⣀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⠏⠙⢿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠙⣿⣷⣤⣤⣤⣤⣤⣤⣤⣾⣿⠋⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⣿⠋⠉⠻⣿⣷⣄⣀⣀⣀⣀⣠⣶⣿⠟⠉⠉⠉⠉⠉⠉⠋⢻⣿⣦⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢰⣿⣏⣀⣀⣀⣀⣘⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣤⡀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠁⠀⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠈⠉⠉⠉⠉⣿⡏⠀⠀⢹⣿⠉⠉⠉⠉⠀⠀⠀⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣶⡶⠶⠶⠶⢿⣷⣤⣤⣾⡿⠶⠶⠶⢶⣦⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⣿⠉⠉⣿⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣤⣴⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣤⣤⣤⣤⣤⣤⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁⠁⠈⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣷⣄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⣀⣀⣀⣀⣸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠁⠒⠒⠒⠒⢠⠊⠉⠑⡄⠠⠤⠤⠄⠀⢐⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⢀⣄⣠⣤⣼⡰⢯⣺⣤⣠⣀⠤⠄⠀⢸⡅⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⣿⠀⠈⠉⠉⠙⠛⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠙⡔⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣅⣠⣠⣧⡬⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⢤⣼⣧⠦⠶⠿⠷⢾⡿⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡱⠉⠀⠀⠁⠀⠀⠀⠺⣧⠀⠀⠀⠀⡖⠺⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⡠⠤⠔⠒⠒⠂⠈⣿⡽⠯⠒⠁⡠⠒⠂⠀⠀⠈⠐⠩⢜⠟⠖⠖⡞⢷⠀⣧⣰⠀⠀⠀⣀⠀⡀⠤⠠⢼
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠀⠀⠀⢀⢸⡗⣿⢹⠀⠁⠚⠛⠛⠓⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢠⡄⠐⠀⡀⠀⢀⠀⠀⠰⡂⣰⠆⠀⠀⢻⠀⠀⡟⢸⣿⣿⣾⠀⠆⠀⢀⣠⠄⠀⠤⢠
⠀⠀⠀⠀⠀⠀⢀⢀⡦⠞⠓⠛⠛⠮⣀⠀⠀⠈⠀⠁⠀⡐⣤⡶⠚⠓⠛⢧⢺⣿⡿⡇⠠⠈⠠⠄⠀⠐⠒⠒⢒
⠀⢰⠀⠀⢀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠈⠛⠦⢶⣰⣶⠼⠟⠁⠀⠀⠀⠀⠀⠙⢿⣽⠃⣂⠉⠀⠀⠀⠀⠀⠀⣼
⠀⢰⠀⠀⢠⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣖⠖⠴⠒⢚⣃⣀⣤⣼⠋
⢸⢸⠀⠀⢈⣈⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⠙⡏⣹
⡇⣙⠒⠀⠈⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⡅
⠱⠭⣀⣀⣐⠭⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠆⠉

⠀⠀⠠⢓⣄
⠀⠠⡁⠀⡌⠢⡀
⢠⡷⠁⠀⡀⢔⡷⡀
⠰⠆⠀⢠⣤⣼⣟⠇
⠘⢤⢀⣉⣙⣉⠍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠉⠉⠓⠲⡶⠾⢟⢻⠉⠉⠀⠀⠉⢹⢛⠿⠶⠶⠒⠊⠉⠉⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠡⠀⠀⠀⠀⠀⢀⣀⡇⠀⣀⠞⡄⠀⠀⠀⠀⡞⢸⡀⠀⣀⡀⠀⠀⠀⠀⠀⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣐⣒⣒⣩⠥⠖⠛⠋⢻⡀⣇⠀⠀⠀⢠⠃⣸⠋⠙⠳⠮⢍⣒⣒⣂⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢷⣼⣤⣤⣦⣞⣴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⡷⠶⠶⠶⠶⢷⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⠶⣾⠒⠒⠒⠒⠒⠺⣷⠸⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡟⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⡷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⢀⡖⢛⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⠒⣚⣛⠩⠅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⡇⢸⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⡤⠞⢋⣡⠤⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢸⠀⠀⡆⡇
⠀⠀⠀⠀⡴⠋⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢸⢸⢸⢠⡆⡇⣿
⠀⠀⠀⡼⢁⡜⠁⠀⠀⠀⡠⠂⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⢛⣛⣿⣿⣿⡇⢸⢸⢸⢸⣿⡇⣿
⠀⠀⢸⠃⡼⠀⠀⠀⠀⢸⠀⢀⡴⠋⠉⠉⠉⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⣤⣤⣄⣒⣨⣭⠴⠖⠛⠉⠁⠀⠀⠈⣧⣸⢸⣿⣸⣿⣧⣿
⠀⠀⡿⢰⠇⠀⠀⠀⠀⣸⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⣭⣉⣉⣩⡽
⢀⣶⣇⣼⠀⠀⠀⠀⠀⣿⢀⣏
⢸⠅⡀⠈⠩⠭⠭⠭⠭⠭⠭⣾
⠘⠢⢴⣋⣩⡿⠿⢿⣯⣭⣿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣞⠻⣿⡙⠉⠉⠉⠉⠉⠉⠉⠉⡉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠃⣸⣿⣰⣷⣿⣿⣿⣿⣷⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠛⠋⠉⠙⣿⠟⣿⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠷⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⣽⣦⣤⣼⣷⣀⣿⢷⡀⢀⣠⡶⠛⠳⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⠿⠻⠟⣿⣿⡿⠟⣼⡿⠽⣿⡆⠀⠘⢿⣿⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⠶⣤⣤⣤⣤⣿⡏⠀⠀⠀⢨⣿⣷⣴⡿⠃⠀⣿⣿⠀⠀⠸⣿⣿⣿⣭⣟⢻⠟⡟⢿⡟⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⣿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⡀⠀⠠⠀⠀⢠⠀⠄⠀⢀⡀⠀⠀⢸⣿⠀⠀⠁⣺⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣔⣽⠾⠷⠾⠶⢶⣽⣦⣶⡶⠶⠷⠷⠾⠶⢤⣕⡐⣦⣿⡿⠀⠀⢠⣿⡿⢿⣾⣧⣷⣯⣼⣭⣯⢿⣼⣷⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣼⣿⣇⣀⣀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡧⢀⣠⣠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠘⣻⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣿⣶⣦⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠁⠈⡟⣿⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠾⣿⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⢿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⣀⡀⠠⠴⠃⢰⣦⢶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⣀⠰⠊⢀⣤⣶⣾⣿⣆⠀⠰⢂⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⠀⠈⠁⣀⣀⡀⢸⢿⣯⣿⣿⡇
⠀⠀⠀⠀⢠⠞⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⠃⢀⣾⣿⠏⠁⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠿⡿⠿⠿⠿⠿⠇
⠀⠀⠀⡾⠀⣾⣿⠃
⠀⠀⠀⠃⠸⠿⠇


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⢀⣤⣤⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠛⣛⡛⠛⠛⠛⡛⠋⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠛⠉⢹⡿⠛⣿⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣟⣛⣻⣷⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⡉⠻⡍⠉⠉⠋⠉⠉⠙⢶⡴⢶⡶⠶⠶⠶⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠈⢻⡆⢹⡀⠀⠀⠀⠀⠀⠈⣿⠈⡇⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⡟⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡀⣰⡟⣸⠀⠀⠀⠀⠀⠀⣰⡟⣰⡃⠀⠀⢀⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡿⠿⣶⣧⣤⣤⣤⣤⣤⡾⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢻⣄⠀⠀⠀⢀⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠢⢭⣛⣛⣛⣫⠵⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠦⠴⡶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⣦⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⢸⠀⠀⠀⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⣿⠀⢸⠀⢀⡀⢸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⡇⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠘⠂⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⣿⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⡇⡇⠀⠀⣿⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣿⢠⡇⠁⠀⠀⣿⠀⠛⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⠀⣿⠈⠀⠀⡆⠀⣾⠀⠀⣉⡀
⠀⠀⠀⠀⢀⣀⣀⠀⠀⢠⡼⠛⠶⠀⣿⠀⠀⠘⠃⠀⢻⡄⠘⠋⠃⣀⠀⠀⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠓⠀⣠⠄⠀⠀⣀⠀⣀⠀⠋⠀⠒⠆⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠙⠂⠀⠸⠉⠀⠈⠁⠘⢤⠀⠀⠀⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⢿⣄⠀⢠⡦⠀⢶⣀⣠⡿⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠞⠛⠛⠷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠉⣿⢻⠒⠒⠒⠒⠛⠉⣿⠁⠀⠀⠀⣼⣿⡏⣿⠓⠲⠦⠤⠶⠒⣿⠉⠉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣴⣿⣿⠟⢛⠻⠷⣶⣤⣿⣀⡄⠀⣈⣱⣿⣿⣿⣷⣶⣶⣿⣶⣶⣿⣠⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠛⢽⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⣸⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣠⣰⣤⣼⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⡙⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠦⣭⣶⣿⡿⠯⣭⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⣿⣿⣇⠀⠀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠓⠒⠒⠒⠂⠀⠐⠾⣿⣿⣿⣯⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠍⠉⠉⠉⠉⠙⢋⢻⣿⣿⣿⡟⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠱⢄⡀⠀⠀⠀⠀⠀⠀⣰⠶⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠉⠓⠶⣤⣤⡤⢾⣿⠀⢠⢴⣴⡤
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⠤⡄⠀⡀⡄⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣿⠀⣿⢸⡇⡇
⠀⠀⢠⡾⠁⠀⠀⠀⣠⣴⣾⣿⣿⣷⣿⣿⣿⣄⣇⣧⡀⢀⢰⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⠀⣿⢸⡇⣇
⠀⢠⠏⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢾⡷⣿
⢀⡟⠀⠀⠀⢠⣿⣿⡿⣿⠯⠁⠀⠉⢝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠹⣿⠛⡏⠛⠙⠋
⣸⠇⠀⠀⢃⣿⣿⣿⣱⠃⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠃
⣿⠀⠀⠀⠸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉
⣏⡄⠀⠀⢷⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠉⠙⠻⣿⣿⣿⡿⠋⠀⠀⠉⢿⣿⣿⣿⠿⠋⠉⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠅⠀⠀⠀⢀⠰⠶⣶⣄⠀⠀⠀⠀⣠⣶⣶⣦⡄⠀⠀⠀⠈⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⢹⣷⡄⢠⣾⣿⠟⠉⠑⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣼⠟⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠋⠀⠌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣾⣿⠛⠉⠀⠀⠀⠀⠀⠀⢾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠉⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢰⣿⡿⠻⠻⢿⣷⡄⠀⠀⠀⠀⠀⢀⣠⢎⠻⠿⠿⠿⠿⠿⠟⠿⠿⣿⣷⡀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⢸⣿⡇⠀⠀⠈⠁⠀⠄⠀⠀⠐⣴⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢓⣀⡀⠀⠀⣀⡚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣭⣽⣷⣶⣶⣾⢹⣿⣿⣶⣶⣶⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁⠀⠀⠈⣻⣛⠃⠀⠀⠀⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⣿⣿⡶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣷⣤⡄⣀⡀⣀⣠⡿⣿⣿⣿⣿⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣼⣯⠏⣿⣽⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢰⣿⢿⡇⣿⣿⣿⣿⣇⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠁⠛⠙⠺⣿⠿⠇⠋⠁⠀⠉⠉⠛⠛⠛⠛⠉⠁⠉⢳⡝⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣁⣀⣀⠀⠉⠀⠀⢀⡀⣴⠋⠉⠉⠉⣳⡄⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣽⣟⣻⣻⡻⣛⣚⣶⠟⠧⣤⣤⠄⠉⣁⣀⣀⣀⣀⣀⣀⡀⡈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⠛⠻⢿⣿⣆⠀⠙⠃⠀⣠⣷⣿⣶⣭⣭⣝⣛⣉⣉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⠋⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠾⠿⠿⢿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠧⠤⠤⣤⣼⢧⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣼⣧⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣶⣒⣻⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠿⠿⢿⣟⠛⣻⣟⣛⣧⣄⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡋⢙⠙⠋⠉⠉⠉⠉⠉⠉⠀⢨
⠀⠀⢠⠠⠤⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠀⠀⣀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠠⠤⠄
⠀⠀⠐⠐⠲⠛⠚⠇⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠻⠷⣦⣀⡀⠀⠠⣀⡀⠀⠰⢀⠃⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣡⢡⣆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠸⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⢠⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⠒⣦⣰⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠸⢀⣀⢠⣄⡇⡇⠀⠀⠀⢀⡀⣢⢼⣮⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣼⡿⡅⢻⡀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⢸⣿⣿⣿⣿⣇⣿⣴⣤⣴⣶⣶⣷⣬⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣖⡂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠻⣿⣷⣦⡀⠀⢠⣀⡀⠀⠀⠀⣰⡽⠛⠁⠈⠹⣿⣿⣽⣶⣆⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢧⣾⠧⠤⠤⠶⠛⢳
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡬⠿⠷⢒⣒⡓⠠⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡯⣤⣴⣶⣞⣛⣛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⠶⠞⠓⠶⠶⠟⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⢀⡞⠁
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢰⡇⣀⣀⣤⣤⣄⣀⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠦⠶⠶⠖⠛⠉
⠀⠀⢸⣏⠉⠀⠀⠀⠀⠈⢉⣿
⠀⠀⠀⠉⠙⠛⠒⠒⠚⠛⠋⠁
⠀⠀⠀⠀⠀⣠⠞⠙⢧⡀
⠀⠀⠀⢀⡼⠃⠀⠀⠀⠻⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠘⣦
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠙⠶⢤⣀⣀⣀⣀⣠⡤⠞⠁
⣶⣶⡶⠶⠶⠾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡖⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠋⠻⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣿⣿⢿⣷⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⣿⣤⣤⣤⣤⣴⣿⣿⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠶⠶⣦⣄⠀⠀⠀⠀⣿⡏⠉⠉⠉⠙⢷⣿⠿⠉⠉⠉⣿
⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠻⠃
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⣀⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⡄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣠⣾⠟⠻⠟⠀⠀⣿⡏⠉⠉⠛⠀⠀⠀⠀⠀⠠⣿⠋⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠿⠟⢿⡇
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠻⠿⠾⠿⠷⠶⠾⢶⣶⠀⠀⠻⠿⠾⠷⠿⢶⣦⡀⢸⣷⣶⣶⣿⣷⣶⣶⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢿⣧⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣸⣿⠀⠀⢠⣤⣀⣀⣀⣠⣿⠃⠀⠁⠀⠀⣿⡇⠀⠀⠁⠀⢻⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠀⠠⣶⡦⠀⠀⠀⠛⠙⠉⠁⠀⠀⠘⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠐⠿⠗⠀⠀⠀⠀⡏⢹⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣶⣶⠈⠀⠀⠀⠀⠀⣤⣄⣀⣸⡷⠀⠀⣼⣧
⠀⠀⠀⠀⣠⣤⣤⣄⣶⣤⣤⣿⡇⠀⣾⡟⠻⠿⠀⠀⠀⢻⡟⠀⠀⠀⠀⢸⡿⠛⠛⢿⡇⠀⠀⠀⠀⠛⠉⠉⠉⠁⠀⠀⠉⠉
⠀⠀⠀⠀⣿⡏⠉⠉⡏⠉⠉⠉⠀⠀⣿⡇⠀⠀⠀⠀⣠⣼⣷⣄⠀⠀⠀⢸⡇⠀⠀⠿⠷⠀⠀⢴⣴⠄⠀⠀⢠⣤⣤
⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠰⣾⡟⠁⠈⢻⡿⠆⠀⣸⣇⠀⠀⠀⠀⣀⢀⣼⡟⠀⢠⣤⣀⣿⣇⣠⣤
⠀⠀⠀⠀⠀⣠⡀⠀⠈⣿⡟⠀⠀⠈⠛⠋⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣠⣴⠼⣀⠟⠋⠀⠀⠘⠛⠙⣿⡏⠙⠛
⠀⠀⠀⢀⣴⡿⠛⠀⠀⡻⣃⠀⠀⢠⣄⣀⡀⠀⠀⠀⢿⠿⠿⠿⢿⡦⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓
⠀⠀⠀⢿⣯⡀⠀⠀⠘⠦⠜⠀⠀⠸⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⢧⠀⠛⠛⠀⢀⣀⡀⠀⢰⣶⠆⠀⠀⠀⠀⠰⣶⡆
⠀⠀⠀⠈⠙⣿⣶⠀⠀⠸⣿⣄⣀⣀⠀⠀⢿⣦⣴⡆⠀⠀⠀⠀⠀⣹⡖⠁⠀⠀⠀⠀⣈⣿⠃⠀⠰⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠙⠻⠿⠀⠀⠀⠀⠈⠁⠀⠀⣿⠾⠿⠋⠀⠀⠀⣤⣤⣸⣁⡕⠀⠀⠘⣿⣀⣤⣤⣤⣤⣀⣿⠂
⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⢀⣶⣾⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⣄
⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⣀⣸⣿⠀⠉⠉⠉⠀⣿⣇⣀⣄⠀⢸⣿⣿⣿⠀⠀⣶⣤⣤⣾⣿⣿⣦⣤⣴⡄⠀⢰⣤⣤⣴⡿⠛⠁
⠀⠀⠀⠀⠘⠛⠃⠀⠀⠻⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠻⠟⠀⠈⠿⠿⠃⠀⠀⠛⠉⠙⠛⠛⠛⠋⠉⠙⠃⠀⠘⠋⠉⠉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣷⣬⣿⣵⣥⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⢛⣭⣙⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣛⣥⣾⣿⣿⣿⣷⣮⣙⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣛⣵⣾⣿⡿⢛⣛⣛⣛⡛⢿⣿⣿⣮⣝⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⣵⣾⣿⣿⣿⣿⣿⣶⡆⣶⢰⣶⣿⣿⣿⣿⣿⣿⣮⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢟⣫⡅⣶⣶⣶⢨⣝⣛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢏⣾⣿⠿⣃⢿⣿⡿⢘⠻⣿⣷⡝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⢣⣿⣿⣶⣭⣶⣿⣿⡎⣿⣷⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡇⣿⣿⢟⣘⣛⣈⢿⣿⣿⣿⣿⣿⣿⢃⣛⣋⡘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠁⣿⣿⣷⢰⣶⢰⣿⣿⣿⣿⣿⣿⣿⣶⡶⢶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠿⣿⡿⢸⣿⠸⣿⣿⣿⣿⣿⣿⣿⠟⣼⣎⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⡄⣀⣴⣿⡿⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢀⣀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⠇⡇⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘⠻⣿⣶⣄⠿⠿⠿⠿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣀⣀⣀⣀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠉⠙⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⣽⡍⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣠⣿⣧⡀
⠀⠀⠀⠀⣠⣄⡀⠀⣿⡇⠀⠀⢐⣿⡿⠟⠋⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⡀⠀⣽⣿⠀⣿⡇⠀⠀⠈⠁⢠⡄⠸⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⢿⣿⣿⠟⠛⠻⣿⣿⣿⣄
⠀⠀⠀⠻⠿⠿⣿⣷⣌⠃⠀⠀⠀⠀⠈⢀⣀⠸⠇⠈⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠈⠉⣀⠀⡶⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡻⣿⣷⣄⣀⣀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢻⣿⡟⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠻⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⡄⠀⣠⡄⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠻⠟⠀⠿⠟⠈⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⣴⡄⠀⣰⣄⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠻⠋⠀⠻⠟⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣾⣷⠀⢀⣶⡄⠀⣠⣦⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠛⠃⠀⠙⠟⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠖⠚⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡶⠚⠛⠉⠉
⠀⠀⠀⢀⣠⣴⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠉⠻⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠻⠦⠶⠴⠶⠴⠶⠦⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⡄⠀⠀⠀⠀⠀⠀⢀⡆⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⡅⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢸⡅⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⡀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠁⢀⣠⡤⠶⠷⠶⠧⣀⠼⠶⠶⠦⢤⣄⠀⠈⠻⣎⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣀⠎⠁⠙⣦⣤⣤⣤⣤⣤⣤⡤⠤⠖⠑⢎⠛⢦⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⣤⣤⡤⠴⠁⠀⢠⡚⠀⠀⠀⠀⢠⠒⠋⠀⠀⠀⠀⠸⠢⡀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⣿⣤⣤⣀⣀⣈⣉⣩⣭⣿⣷⣶⣦⠌⢦⠀⠀⣀⠀⠀⠀⣠⡇⠀⠘⠓⠒⠒⠒⠒⠲⠖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡴⣉⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⠤⠞⠈⠷⠤⠤⠤⠤⠤⠤⢤⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡇⠀⠈⢟⠛⣅⣈⡗⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢯⠕⢠⠝⡟⠻⠀⠀⣸⣤⣬⣥⣤⣄⣀⠤⠲⠞⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠱⣀⣠⣏⣸⣍⠇⢠⠋⠉⠀⠀⠀⠀⠘⠲⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠆⠈⢿⣛⡛⠛⠛⠳⡀⠀⣠⠀⠀⠀⣠⠇⠀⠀⣀⣠⡤⠶⠛⠉⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡲⠀⠀⠳⠚⠓⠲⠶⠾⠿⠼⠦⠤⠾⠷⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠤⠤

⠀⠀⠀⡘⢉⡑⠂⠤⠤⠄⣀⣀⣀
⠀⠀⢰⢥⣤⣄⣀⡀⠀⡀⠠⠤⠤⠀⠀⠀⠈⠋⠙⣽⣷⣄⡀
⠀⢠⡃⠀⠈⠉⠉⠁⠩⠹⠗⠶⠖⠢⠤⣤⡴⣦⣤⢹⣿⣿⣿⣦⣄⡀
⢀⠎⠉⠛⠓⠶⠶⠾⡤⣤⣄⣠⣄⡉⠀⢀⡀⠀⠸⣾⣿⡫⢿⠛⠛⠿⠂
⠾⠶⣤⣤⢦⣀⡀⠀⡀⠀⠀⠀⠉⠉⠹⠙⠛⢛⢷⣿⠃⣤⣤
⠀⠀⢸⣿⢝⣿⡏⠛⠛⠓⠲⠶⠶⣾⠶⣿⣭⡿⠁⣿⢰⣿⢽
⠀⠀⣼⣿⠻⢻⣷⢷⣶⡶⣦⢤⣤⣠⣄⣀⡀⠀⠀⢸⠸⠛⢻
⠀⠀⢙⣿⣿⣿⣷⣻⣿⣿⣿⣇⣾⣏⣽⡞⣉⠉⠘⠺⣠⡲⣿⡀⠀⠀⣀
⠀⠀⢘⣿⣿⣿⡁⠀⠈⠉⠉⢹⣿⠛⠻⠿⠛⠛⠿⠆⢣⣿⣷⠾⠽⣿⣸⣣
⠀⠀⢸⡟⢹⠟⡥⠤⣤⡤⠤⢸⣿⠤⠤⣤⣀⣀⣀⡀⣷⣿⡇⠀⠀⠅⡿
⢀⠔⢺⣧⣨⣤⣷⣖⣾⣟⣿⢿⢿⣟⠉⠛⡗⠒⠲⡇⣽⣿⣷⡄⢀⢰⠇
⣼⣤⠤⠤⠽⣏⣎⣿⡿⣿⣟⣻⢸⣿⢛⠒⠧⣄⠀⣣⠘⣿⣶⣼⡌⡾
⢻⡗⢄⠀⢀⡨⠛⠓⠿⠿⢿⣿⣼⣿⣿⣯⡦⠟⠯⡛⠶⠿⢻⣟⢧⡇
⠸⡇⠀⣽⠗⠤⠤⣀⡀⣠⡾⠃⠀⠀⠀⠇⠀⠀⠀⢈⡱⣶⠮⠿⠞⠣⠄
⠀⠙⣦⣻⠀⠀⠀⠀⠈⣿⠓⠒⢒⡒⠲⡗⣒⣚⣭⡝⣬⡧⣤⠄⠊⠐⠰⢶⣉⠄
⠀⠀⡏⠛⠷⣤⣤⣀⣸⡏⠀⠀⢸⢿⢭⡀⡿⣪⣿⣿⢿⣿⠃
⠀⠀⡟⠶⣤⣸⣏⠉⠋⠻⠟⠛⣗⢛⣯⡿⡫⡋⣻⣗⣼⣿⠀⠀⠀⡀⠀⢀⠀⣀
⢠⡦⢸⣄⠀⠉⡟⠛⠲⢶⠷⠦⣵⣵⡧⣾⣾⣾⣿⡻⣻⢟⠇⢧⠻⡏⣼⡟⢔⡟⢏⢰⢄⢀⢀⡀
⠘⢐⢻⣌⡟⢶⣧⣄⣀⣸⣀⠀⢿⡹⣿⣕⣣⣪⣿⣗⣿⡧⡿⢜⣆⡇⡔⣿⣸⣔⢻⠓⡆⢀⢯⡫⢟⡆⣶⣴⠀⢀⣀
⠀⠀⠈⠙⠻⣬⡄⢩⣏⡍⣉⠙⡿⣻⢫⡛⣿⣼⡟⣻⠇⣄⠳⡸⢎⣹⣅⣄⣨⠎⣱⢢⢥⡄⢸⢩⡅⠰⠴⢀⢣⠎⠛⡥⠟⡟⣂
⠀⠀⠀⠀⠀⠈⠈⠏⠙⠗⠻⠗⠿⣼⢿⣾⢿⡾⣿⡏⠦⡜⣏⡷⠎⣯⢘⡸⡤⡒⡝⢴⡀⠦⠎⠆⠏⠌⠐⠆⠻⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣸⠀⣿⣀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠋⢻⠀⣿⠉⣻⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷
⠀⠀⠀⠀⠀⠀⣤⠴⠶⠶⠶⠶⠶⠶⠿⠷⠾⠶⠿⠶⠿⠶⠶⠶⠶⠶⠶⠶⣤⡀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣞
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⢿⣿⠀⠰⠶⠀⠀⠀⠰⠆⠀⢸⣹⠆
⠀⠀⠀⠀⠀⠈⠛⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣾⠁⠀⣶⢦⣄⠀⠹⣆⠀⡀⠀⠀⠀⢀⡀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢹⡆⠀⢻⡀⠙⣆⠀⠙⢦⣍⡛⠛⠛⣉⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡀⠀⠳⠖⠛⠁⢀⣿⣾⠛⣿⣿⠉⠋⠉⣿⣿⠛⣿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠞⠁⠀⠈⠳⣤⣠⡤⠶⠚⢻⡅⣿⠀⣿⠹⣦⣤⣴⠏⣿⠀⣿⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠈⠹⣯⡭⠭⢭⡿⠉⠀⠀⠀⠀⠀⠀⠀⠸⣥⣤⠶⠚⠛⣷⣼⠀⢉⣙⣉⣉⣙⣉⠉⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠈⠹⣆⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠴⠞⠛⢙⣿⠀⢸⡏⠉⠉⠉⢹⡁⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠇⠀⢠⡞⢳⡄⠀⢻⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠖⠛⢉⣹⠀⢸⡇⠀⠀⠀⢸⠄⠀⣿⡄⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡇⠀⢸⠀⣿⠀⠀⢸⡇⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⣹⠶⠖⠛⠉⠙⢷⣼⣇⡀⡀⠀⣸⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡇⠀⢹⣄⣿⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⢰⡏⠀⣠⠴⢶⡆⠀⣿⠉⠉⠉⠉⠉⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⢠⡾⠦⠤⠼⣷⠀⢷⡀⠀⠈⠀⣠⡤⠾⠦⠤⠤⠾⠧⠤⠤⠿⠤⠤⠼⠧⠤⣼⡆⢸⣿⣠⡟⠀⠀⣶⡆⠀⠀⠀⣼⡟⠚⡇
⠀⠀⠀⠀⠈⠛⢛⣿⡛⠛⠀⠀⠙⠲⠦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⠀⠀⠀⠀⢻⢸⡏⠉⠀⠀⠀⣿⡇⠀⠀⠀⢹⣇⣠⡇
⠀⠀⠀⠀⠀⣰⠟⠀⠙⢦⡀⠀⠀⠀⠀⣿⠀⠀⡶⠶⠶⠶⢶⡄⠀⣶⢾⡇⢸⢶⡆⠀⠀⢸⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇⠀⣿⢸⡇⢸⢸⡇⠀⠀⢸⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⠛⠛⣧⣀⣀⣀⣼⠟⠛⣿⣸⡇⢸⣤⡟⠛⠛⢻⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣸⣁⡀⠀⠀⢸⣸⣇⣀⣀⣀⣀⣿⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠙⠳⢤⣤⡤⠶⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⣉⣉⣹⣇⡀⠀⢸⡟⠉⠉⠉⠉⣿⠉⣿⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣀⣀⣴⠋⠉⠉⠉⠉⠉⠉⠻⣦⡾⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠷⠶⠶⠶⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡹⠦⠤⠤⢤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⠤⣶⡚⠁⠀⠀⢀⡰⢋⡰⠃
⠀⠀⠀⠀⠀⠀⠊⠀⢸⠀⠋⠉⠒⠒⠤⢀⡀⠀⢰⠙⠃⠀⠀⠀⠋⠉⠉⡛⡉⠉
⠀⠀⠀⠀⠀⢘⠀⢠⠚⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣑⣤⠀⠀⠀⠀⠀⠂⠃⣍⠉⠉⠉⠉⢉⠉⠉⢉⣉⡉⠁⡲
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠘⡇⠀⠜⠋⠻⠉⣉⠠⠐⠫⡉⠃⢀⠀⠀⠉⠀⣠⠋⠀⠀⠈⣆⠈⢁⠍⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠧⠤⠤⠤⠜⠨⠊⠀⣀⠀⠰⠀⢢⡈⣶⡖⠊⢇⠉⠒⠠⠞⠁⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡠⠋⣿⠀⠀⡇⠀⠀⠛⠃⠀⠘⣦⠀⡀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠏⠑⠚⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠒⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢐⣁⠤⠒⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠐⢄⠠⠁⠀⠀⢀⡠⠖⠂⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⠀⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⠀⠊




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠤⣤⠴⠦⠀⠀⠀⠤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠻⣅⠀⠀⣀⣀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⠋⠉⠉⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣉⣀⣀⣀⣤⣶⣶⣿⣿⡿⢛⣛⣫⣉⠉⣿⣿⠿⠿⢿⣿⣿⣿⣭⣿⠿⠿⠶⣶⣤⣤⣶⠋⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣉⡉⠀⠈⠉⠙⠒⠛⠓⠻⡿⠿⠿⢥⣿⣿⣷⣶⣤⣬⣭⣿⣯⣿⣶⣶⠾⠿⠛⠛⢯⣄⣀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠛⠛⠶⠶⠶⣤⣤⣤⣎⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⠀⠀⣀⣀⣀⣀⣤⣤⣶⡶⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠈⠉⠉⠉⠉⠩⢿⡉⠉⢻⡟⠛⠛⠋⠙⠛⠛⠉⠉⠀⠀⢺⡇⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣃⡠⢄⣀⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢤⣄⣁⡀⠀⠀⠀⣘⠀⠤⠀⠀⠀⢀⣀⡀⠀⠀⢸⡧⠤⣀⠤⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⢿⡟⠓⠶⠿⣶⣤⣬⣐⣂⣀⣀⣀⣀⣠⣼⣧⣄⣀⣠⣤⣤⣶⣶⡿⠋⠡⡴⢯⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⢸⡗⠒⠀⠀⠀⠀⠉⠉⠉⠉⢿⡿⠛⠛⠻⠋⠉⠭⠭⠕⠒⠶⠘⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡧⠀⢀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢮⠙⠛⠛⠻⠿⢶⣤⣤⣄⣀⣀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠤⠀⠀⢀⣀⣤⠴⠛⢛⣭⣥⡌⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢦⠄⠀⠀⠀⠀⠀⠈⣿⠉⠉⠙⠿⣛⠛⠛⠛⠛⠛⠛⠿⣿⡟⠉⠉⠉⠠⠐⠚⠋⠁⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⠀⠀⠀⠀⠉⠁⣿⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣃⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣀⣿⡄⠀⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠂⠀⠀⠀⠀⠀⠲⠒⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠚⠋⠉⠉⠛⠳⢶⠶⠛⠛⠛⠓⠒⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⠒⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⣤⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣤⠤⠲⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⢠⡞⢻⠇⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣷⡇⢀⠀⠀⣴⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⠓⠛⠋
⣿⠀⣸⠀⠀⠀⠀⠀⢨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠹⠃⠘⢀⣾⠁⠀⠀⠙⠶⠶⠶⣄⣀⣀⡴⠋⠀⠀⠀⠀⢠⣤⣄
⠙⠶⠋⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢀⣤⠤⠶⠦⠶⠞⠋⢹⡄⠀⡀⠀⣼⠃⠀⠀⢀⠀⠀⠀⠀⠈⠉⢁⡀⠀⠀⠀⠀⠀⢸⡄⢸
⠀⠀⠀⠀⠀⠀⡴⣶⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⢰⠁⢰⠇⠀⠀⠀⠸⡿⢦⡀⠀⠀⠀⠈⡟⠳⣦⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟⠀⠀⠀⠀⠀⣇⢀⡵⠀⠀⠀⠀⠻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣴⣧⣤⣼⣿⠂
⣿⣿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀⠀⣀⣬⣿⣄⡀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀
⠿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠿⠸⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠀⠿⠿⣿⠿⠇⠀⠿⢿⣿⣦⠀⠀⣴⣿⠿⠿⠸⠿⠧⠾
⠀⠀⠈⠛⣡⣶⡀⠀⠀⠀⠀⠀⢀⣦⣘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠸⣿⣯⠀⠀⠀⣀⣀⣀
⢸⣶⣶⣶⣿⣿⣿⣶⣾⠀⣷⣶⣿⣿⣿⣶⣶⣶⡆⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⣰⣾⣿⣿⣿⡿⠿⠋⠀⠀⠙⠿⠿⣿⢸⣿⣿⣿
⠈⠉⠉⠉⠉⣉⠉⠉⠉⠀⠉⠉⠉⣉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢐⣷⣶⣦⡀
⢠⣀⣀⣀⣾⣿⣦⣀⣠⠀⣄⣀⣴⣿⣷⣀⣀⣀⡄⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠙⣿⣷⠀⠠⣤⣤
⠸⠿⠿⠿⣿⣿⣿⠿⠿⠀⠿⠿⣿⣿⣿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⢀⣤⣶⣶⠠⣶⣾⣿⠏⢰⣦⣿⣿
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠉⠈⠉⠉⠀⠀⠈⠉⠉⠉
⢸⣶⣶⣶⣶⣶⣶⣶⣾⠀⣷⣶⣶⣶⣶⣶⣶⣾⡇⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⢿⣿⣦⣤⣤⣤⣄⠀⠀⢠⣤⣤⣤
⠈⠀⠀⣈⠹⣿⠟⠁⠉⠀⠁⠈⠙⣿⢟⣅⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠙⠛⠛⠛⣿⣷⠀⠘⢛⣿⣿
⣤⣀⣴⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣄⣤⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠀⣄⣠⣿⡿⠀⠀⠀⠉⠉
⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠋⠀⠀⢠⣦⣤⣤
⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿
⢹⣿⡇⠀⠀⢹⣿⠁⠀⠀⢹⣿⠁⠀⠀⢹⣿⠀⠀⠀⢹⣿⠀⠀⠀⠀⣠⣶⣿⢸⣿⣶⡄⠀⣿⣿⣿⡇
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⣼⣿⠀⠀⠀⠐⠛⠛⠀⠀⠘⠛⠓⠀⠀⠐⠛⠃⠀⠀⠀⠀⣠⣄⠀⠀⠀⢠⣤⣿⣿
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠈⣿⡏⠀⠀⠈⣿⡏⠀⠀⣀⣴⣿⠟⠀⠀⠀⠘⠛⠛⠛
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⣸⣿⠀⠀⣄⣿⣿⣠⠀⠀⣀⣿⣿⠀⠀⣀⣿⣧⡀⠀⠀⣿⣇⠠⣾⣿⠟⠁⠀⠀⠀⠀⠀⣤⣤
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠉⠉⠀⠀⠿⣿⣿⠻⠀⠾⣿⣿⣿⠀⠾⣿⣿⣿⣿⠆⠀⣿⣿⡿⠎⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠀⠀⠀⠈⣿⣿⠀⠀⠈⣿⡟⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠉
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠉⠉⠁⢀⣀⡀⠀⠀⠀⢨⣿⣿⡀⠀⠀⢈⣉⣉⠀⠀⢈⣉⣉⠀⠀⢈⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠
⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢀⣾⣿⠿⣿⣷⡄⠀⢈⣿⣿⠁⠀⠀⢀⣿⣿⠀⠀⢠⣿⣧⡀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⢸⣿⡇⠀⠀⠛⠛⠃⠀⠀⠀⢸⣿⡇⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⠀⢶⣿⣿⣿⡿⠂⠀⣿⣿⡿⠂⣀⠀⠀⠀⠀⠀⠀⠀⣉⣩
⢸⣿⡇⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣿⣧⡀⢀⣽⣿⠀⠀⠀⠉⣿⣿⠀⠀⠉⣿⡏⠀⠀⠀⣿⡏⠀⠚⢿⣷⣄⠀⠀⠀⠀⠀⣿⣿
⠚⠛⠃⠀⠀⣿⡏⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠃⠀⠀⠀⠠⣿⣿⠀⠀⢠⣿⣧⠀⠀⢠⣿⡇⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠿⠿
⢹⣿⡇⠀⢠⣿⡇⠀⠀⣴⣦⠀⠀⢤⣶⡄⢀⣠⣤⣤⣄⠀⠀⠀⠀⠠⣶⣦⡀⠀⢠⣶⡦⠀⠀⠰⣶⡆⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡶⣶
⠈⠿⣿⣿⣿⠟⠁⠀⠀⢻⣿⣦⣤⣾⡿⠀⣾⡿⠛⠛⣿⣷⠀⠀⠀⠀⠻⢿⣿⢸⣿⠟⠁⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⣭⠂

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢁⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠀⠀⠛⣇⠀⠀⣤⠦⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⢠⡶⠿⠿⠗⠇⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠙⢶⠾⠃⠀⠀⠀⠀⢀⣤⠶⠋⠼⠛⣫
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠸⢛⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣤⣤⣯⣽⣤⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡛⠇
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⠂⠀⠀⠡⡄⠀⠀⠀⠀⠸⣧⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣹⠿⠋
⠀⠀⠻⠟⠛⠛⠛⠟⠀⠀⠀⠀⠈⠢⠀⠠⠊⠁⠀⠀⠀⠀⠀⠠⣿⣿⡅⠀⠀⠀⠀⠀⠰⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⠤⠤⠤⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠒⢆
⠀⠀⣠⣴⣿⣦⣀⠀⠀⠀⠀⠀⠀⡴⣊⠔⠊⠁⠀⠀⠀⠀⣀⣤⣾⣦⣄⡀⠀⠀⠀⠀⢸⠉⣩⠲⢼
⠀⠀⠉⠉⠉⠙⠉⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⠏⠉⠀⠀⠀⠀⠀⢸⣀⣻⣃⣸

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠴⠄⢴⡆⠰⠄⠀⠀⠀⠀⣴⡤⠔⠀⢀⣄⠀⠀⠀⣴⣶⣴⢤⣤⣤⣄
⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⡇⠀⠀⠀⠀⣿⣦⡴⠊⠉⠀⠀⠀⠀⠈⠈⠀⣦⡈⠉⠁
⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⠀⠀⠈⠛⠋⠛⠋⠁⠀⠀⠀⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣐⣬⣲⣾⣿⡿⣶⢶⠷⠖⠒⠒⠒⢺⢶⣶⣶⣶⣶⣶⣶⣶⠶⠛⢉⣙⣋⣈⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠙⠛⢿⡿⡟⢀⡀⠀⠀⠀⢸⣮⣿⣿⣿⣿⣿⡿⣿⡴⠛⠉⠉⠉⠉⠙⠢⡄⡝⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⡁⠀⠀⢶⢾⣿⣿⣿⡿⢿⣿⠋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠎⠁⠀⠀⢀⣬⠀⣿⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠦⣄⠘⢿⣴⢿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠛⢅⠀⢙⣺⠢⠈⣻⣿⣽⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⠀⠀⠑⢿⣿⣀⢄⠈⠳⣿⣿⣷⣥⡀⠀⠀⠀⠀⠀⠀⠞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠁⠀⠀⠀⠈⣿⣿⣧⡀⣹⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⡸⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠭⠭⠭⠠⠤⢄⢤⠤⠄⡤⠀⢽⢿⡿⠿⢯⣭⠭⢭⠍⢭⠹⠭⠩⠶⠄⣧⣷⣥
⠀⠀⠀⠀⢀⣤⡖⢉⡀⠠⢶⣀⡠⡀⠀⡠⢈⣠⠴⠋⡄⠀⣸⠾⡉⢢⠀⠈⢷⡄⢡⠀⡄⠀⢀⡔⠶⠀⠈⠙⠳⢄
⠀⠀⣠⡶⠋⠯⠙⠯⢹⣾⠋⠙⠡⠃⡐⢻⡟⠁⠀⠀⠄⣼⢻⠀⡈⠀⢳⠦⠀⢻⡘⣿⡇⠀⠈⠻⣶⡶⣾⣷⣤⣤⣁⠄
⠀⡴⠻⠇⠀⠀⠀⡼⢿⠿⠇⠀⠀⡼⢟⡟⠀⠀⠠⡞⣿⠷⢾⠚⠁⠀⠀⡄⠀⠘⡇⣿⣿⠀⠀⠀⢙⣿⣿⡿⢿⣟⠛⠿⣦
⠰⢁⡀⠠⢲⠄⠀⠀⠀⠀⠀⠀⣼⣶⡟⣀⣀⣀⠀⡸⠉⠆⢸⣀⠀⠀⠀⡇⠀⠀⡇⢿⣿⠀⠀⠀⢘⣿⣟⢠⣽⡿⠷⣦⡘
⡆⠈⣹⣾⡯⠄⡰⠀⠀⠀⠀⢰⣿⣿⠃⣽⠃⢈⣷⠃⠀⠀⣸⣿⠀⠀⢠⠁⠀⢰⡅⢸⣿⡂⠀⠀⢸⣟⢧⣽⢫⣾⣿⣌⣳
⠇⠀⣤⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⡟⢠⡿⠀⠉⡾⠀⠀⡤⢽⣿⠀⢀⡎⠀⢀⡿⠁⢸⣿⡇⠀⠃⣾⣿⣣⣿⠞⢿⣿⣿⣷
⢸⡄⠋⠙⠋⠀⠀⠀⠀⠀⠀⣿⣯⡇⠚⠃⠘⣹⣦⠴⠞⠀⠸⠛⢢⡎⠈⢐⡿⠁⠀⣸⣿⣷⣾⣶⣿⣯⣷⡻⣷⣿⣤⣿⡟
⠀⠹⡀⣰⢀⠀⠀⠀⠘⠀⢀⣿⣿⠁⡤⠞⢁⡬⢙⡓⠃⠀⢸⣤⢂⠀⣰⠟⢁⣠⣌⣽⣿⣿⣿⣿⣿⣿⣿⣯⣠⣌⠉⢛⡵
⠀⠀⠹⣧⡤⢀⠢⣀⣠⣤⣾⣻⠏⠀⢡⠄⠀⡷⠦⢜⣠⣴⣻⣧⡽⠚⣁⣤⣶⡿⠿⣿⣟⡋⣽⡟⢿⣵⡿⢟⠛⠁⢀⠝
⠀⠀⠀⠈⠻⢿⣿⡏⢠⣞⣋⣥⣴⠞⢀⣀⣤⣴⠶⣿⣿⠾⢛⣩⣴⣾⣯⣿⣿⣿⣿⣿⣿⠿⢿⣿⣾⣿⣛⣩⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡿⠿⢿⡿⠿⠿⠿⡿⣿⠟⠻⣍⡉⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣿⣟⣛⣛⡷⢿⣷⣶⣶⣶⣶⡲⡶⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠛⠛⠛⢻⠿⣿⣿⣿⠏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢨⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠇⠀⠀⠀⢰⠐⣦⣿⣿⣧⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠶⠶⢶⣶⣾⣿⣿⣿⣿⣧⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣤⣀⠂⠸⣿⣿⣿⣛⡟⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⣡⣤⠀⠻⣿⣿⡿⠁⢠⢿⣿⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠉⠉⠱⠀⠱⠞⠀⠆⢹⡟⠐⠆⠹⠼⢫⣿⣆⠙⠻⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢿⣿⠿⠿⢯⡯⠉⠇

⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠦⠤⢤⣄⣤⣠⡤⠤⠶⠶⢦⡀
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣠⣤⠤⢾⣤⡿⠤⣤⣄⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⣹⣏⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⢇⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⠒⠒⠒⠚⢧⡀⢀⡴⠒⡆
⠀⠀⠀⠀⢀⠔⠈⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⡇
⠀⠀⠀⠀⠈⠀⢐⠉⠉⠉⠉⠉⠉⠙⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠯⠤⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠃
⠀⠀⠀⠀⠠⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠒⢒⢒⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠭⠿⠁
⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠙⣆
⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠴⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⢀⣤⣤⣄⡀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣛⣛⣛⣛⣛⣋⡉⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢉⣿⣛⣿⡉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣟⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣄⠈⢦⠉⠉⠉⠀⠀⠉⢳⡖⢾⡟⠛⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠘⣷⠘⡆⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣽⠀⡇⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⣴⡟⡰⠁⠀⠀⠀⠀⢀⣼⣧⣼⣥⣤⣤⣤⣤⣤⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⣿⣙⠛⠓⠶⠶⠶⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣌⠷⣤⣤⣤⣤⠿⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣙⡒⠒⠒⢒⣋⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡍⠉⠉⠙⠉⢉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢸⡇⠀⡄⠀⠀⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⡇⠀⡇⠀⠀⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠇⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢸⡇⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⢸⢸⠀⠀⢸⡇⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⢸⡇⣼⠈⠀⠀⢸⡇⠘⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠃⢸⡇⠁⠀⣰⠀⢸⡇⠀⣈⡁
⠀⠄⠀⠀⢀⣀⡀⠀⠀⣠⠞⠛⠆⠸⡇⠀⠀⠋⠀⠘⣇⠀⠛⠛⣀⠀⠀⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠃⢀⡴⠀⠀⢀⡀⢤⡀⠚⡀⠚⠲⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠓⡀⠀⠇⠀⠀⠈⠈⠣⠄⠀⠀⢈⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠴⠾⢧⣀⣀⡾⠀⠸⣦⣤⠾⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⢾⡇⢸⡷⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡤⢾⡇⢸⡷⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠾⠧⠾⠧⠦⠤⣄⠀⠀⣠⡴⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡏⠀⠈⣧⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⠀⢿⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⢏⠇⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠡⠄⠐⠒⣒⣒⣒⣒⣒⡒⠂⠀⠤⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣅⢀⢚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠠⢂⠭⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡆
⠀⠀⠀⠀⠀⢀⢁⠂⠀⠏⠭⠀⡀⠀⠀⠀⠠⠤⠀⠠⠤⠀⠀⠐⠀⡂⠨⠍⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠌⠈⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁
⠀⠀⠀⠀⢀⢀⠁⠀⠉⠀⠂⠀⠂⠉⠡⠤⠀⠀⠀⠀⠀⠠⠤⠉⠁⠂⠐⠀⠈⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠄⢠
⠀⠀⠀⠀⡄⠁⠀⡁⠔⢢⠀⠀⠠⠒⠀⠀⢀⠤⠀⠀⠀⡀⠀⠀⠀⡠⠄⠀⠀⠦⡁⠀⠀⠀⠀⡈⠈⠘
⠀⠀⠀⢀⠘⠀⢠⠁⠀⠀⢄⠔⠁⠀⢃⡠⠀⠀⢃⢀⠎⠈⠄⠀⠔⠁⠘⣀⠎⠀⢡⠀⠀⠀⠀⠯⠀⣚
⠀⠀⠀⠘⢜⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠠⠀⠩⠀⠐⠒⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⠒⠀⠀⠁⠄⢠⠀⠀⠰⠠⢠⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡋⠉⠹
⠀⠀⠀⠀⠀⢒⠀⠀⠤⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⢐⡀⠜⠊⡴⠃
⠀⠀⠀⠀⠀⠠⠀⠈⠁⠒⠀⠀⠀⠀⠠⠤⠤⠀⠭⠥⠤⠤⠄⠀⠀⠀⠀⠒⠀⠉⠀⢸⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠄⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡴⠶⣷⡤⢤⣼⣆⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣲⡸⠏⠀⢠⣸⣧⡄⠒⢻⣷⡆⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠛⢋⣵⣾⣷⡐⣧⠀⠈⠶⢿⣷⣚⣩⠟⠂⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⣿⣿⣿⡇⠀⠐⠿⣿⣿⣾⢄⠀⠀⣾⣿⣉⡽⢧⣤⣶⣤⠀⠉⠓⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠸⣿⣿⣿⣿⣾⣇⠀⠘⢻⣿⣿⣮⢆⠀⠈⣿⠟⠿⣷⣿⣭⣷⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⢿⣿⣿⣿⣿⣆⠀⠘⢿⣿⣿⣦⠃⠀⠹⠀⠀⠀⠀⠀⠉⠛⠿⣟⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄⠀⠘⢿⣿⣿⣷⡰⣸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⣷⡽⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⡄⠀⠀⣻⣿⣿⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣅⠀⠈⢱⣿⣿⣷⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣆⡀⠀⣩⡿⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⣚⣭⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢠⡀

⠀⠀⠀⠀⠁⠀⠁⠳⠂

⠀⠀⠀⢠⠋⠑⠒⠠⠤⢠⣀⣀⣀⡀
⠀⠀⢀⣇⣀⣡⡀⠀⠀⠀⠀⢀⡠⢄⡉⠉⠉⠉⠉⠛⠏⣿⣦⣀
⠀⠀⡎⠀⠙⠓⠉⠙⠉⢉⣳⠶⠤⠴⠤⢤⣀⣠⣦⣀⣸⣹⣿⣿⣗⣤⡀
⠀⡜⠛⠲⠦⢤⣤⣤⣆⣈⡀⠀⢀⠀⠂⠂⠈⠁⠈⠉⣇⣿⣿⢳⣿⠿⢿⣷⠄
⣼⢤⣀⡀⣠⠀⠀⠀⠀⠉⠉⠉⠉⠛⠒⠲⠴⠷⠤⢴⣼⡿⠃⢁⠘
⠈⠉⠙⣿⡾⢿⣶⠶⢴⣦⣤⣄⣀⣀⣠⣌⣠⠄⣢⡳⢹⡇⣈⣿⢻
⠀⠀⢀⣿⣯⣾⣿⣠⣀⣀⣀⡈⠉⠉⠉⠉⠉⠛⠛⠁⢸⣇⢫⣟⡿
⠀⠀⢿⣿⣭⣡⣿⣫⢿⢻⢯⣿⡽⢳⣾⢷⢖⡶⠤⢀⣸⢦⠈⣀⣤
⠀⠀⠀⣿⣿⣿⡿⠻⠿⠼⣿⣿⣶⡟⣧⣾⣿⣄⣤⣤⣜⢸⡎⣮⡿⠤⢄⣀⠤⡄
⠀⠀⠀⣷⣽⣿⡧⠀⠀⠀⠀⠀⢸⣿⠁⠈⠛⠉⠉⠉⢛⢈⢳⣿⡟⠙⠓⢿⢰⡿⠂
⠀⠀⢀⣿⠨⠟⢻⠄⠲⣶⣒⠒⣺⣿⠒⠒⢶⡤⠤⠤⢬⢸⣻⣿⡇⠀⠀⡂⡿
⢀⠔⠉⣿⣤⣤⣾⢶⣾⣻⣿⣿⢿⢻⣯⢍⠉⠛⠉⠑⢺⢸⣽⣽⡷⣄⢀⢡⡇
⣼⣤⠤⠤⠘⢿⣵⣳⣿⠿⣷⣏⣹⢸⣿⣛⡓⠲⢤⡀⢸⣀⠹⣷⣶⣼⡎⣸
⢻⡗⢄⠀⠀⣀⠜⠛⠺⠿⠿⣾⣿⣸⣿⣿⣿⣵⠼⠾⢝⠻⠾⠟⢻⡟⠇⡏
⠸⡇⠀⢹⡾⠤⢄⣀⡀⠀⣠⡾⠃⠀⠀⠀⠍⠀⠀⠀⠀⢉⣦⣴⢪⣿⡼⢅⣀⡀
⠀⠙⣄⢸⡇⠀⠀⠀⠈⢹⡿⠓⠦⠤⠤⠤⡶⠲⠶⢒⣎⢏⣦⣧⣏⠿⠟⠶⢦⣬⡟⣢
⠀⠀⠿⠿⣷⣄⣀⠀⠀⢸⡇⠀⠀⢸⢕⡕⡇⢸⣋⣿⣯⣾⣿⣼⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢻⣤⣀⠉⣿⠟⠿⠞⢷⡲⣦⣬⣵⣷⣳⠾⠾⠫⣿⡵⣼⣯
⠀⣀⢸⡀⠉⠛⠿⠷⣤⣄⣸⣁⣠⣊⣄⣿⣺⣖⣬⣼⡿⡿⣿⣿⡄⡆⣦⣧⢠⢶⢆⣼⣶⠔⡄
⢸⠪⣸⡟⣦⣄⣸⡀⠀⠈⢹⠉⠉⣿⡿⣯⢝⣿⢿⢿⣿⣡⣟⠎⣤⢹⡄⣿⢌⣿⠉⠚⣨⣶⣘⡈⢢⡶⡦⣤⢀⢀⡀⡀
⠀⠐⠌⣷⣿⡌⡟⣻⢷⡶⠾⠶⢤⣬⣮⣿⣿⣧⡗⣿⣿⣽⢻⢩⡏⢳⣣⢧⡎⠘⠜⢞⠌⢇⠸⠰⠐⢱⡯⡅⠣⠋⡞⢇⢄⢶⡄⢄⣄⣄
⠀⠀⠀⠀⠈⠹⠾⢤⡾⣼⣃⣶⣀⣏⠼⣽⢏⣽⡫⣿⡡⣟⠰⣇⡏⡌⠱⡼⠧⢷⡖⠋⡰⡅⢓⠳⡂⢸⡸⣦⢊⠁⡑⣎⡎⠬⠴⠎⠹⠆⠖
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠉⠛⠻⣟⢿⠘⡿⠹⠿⠘⢛⡉⡦⠛⢣⠷⣂⠇⢋⢩⡵⡁⠧⠐⠓⠃⠁⠃⠈⠀⠙⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠉⠉⠓⠲⠦⢤⣤⣤⣀⣰⡋⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⣤⠤⠴⠒⠋⠉⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⢀⣀⣤⠴⠖⠚⠋⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⢀⡷⠃⠈⠉⠙⠛⠲⠦⢤⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡶⢶⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠷⠾⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠋⠉⠉⠉⠉⠉⠉⠻⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⣹⣉⣋⡏⢹⠛⡟⢻⠛⡟⢻⠛⡏⢻⢙⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠋⠉⠉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⡴⠞⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⢠⡿⠁⠀⠀⠀⠘⣷⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⣼⠃⠀⠀⠀⠀⢰⠏⠀⢈⡿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡖⠲⠶⠤⢤⣤⣀⣀
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⢸⡇⠀⠀⠀⠀⠸⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠈⢿⡀⠀⠀⠀⠀⢻⡤⠖⠛⠛⠛⠒⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢷⣄⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠦⠴⠶⠖⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⠲⠤⣤⣄⣀⣀⣀⣤⡤⠶⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⣤⣤⣤⡴⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠲⠦⠤⠀⣀⣤⡎⠉⠉⠉⠉⠉⠉⠉⢳⡄⣀⣠⡤⠶⠶⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⢟⣋⣭⠤⠤⠼⢿⡇⣤⣤⣤⣀⣀⣀⣀⠸⡇⠀⣀⣰⠶⢶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠓⠚⠛⣿⣿⠿⡿⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿⠀⠁
⠀⠀⠀⠀⠀⡴⢃⠎⠀⠀⢀⠞⠀⠐⠁⠔⠁⠐⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠈⠀⠁⠟⠀⠀⠐⠀⡀
⠀⠀⠀⢠⠊⡔⠁⠀⠀⡀⠁⠀⠂⠀⠊⠀⠀⡠⠂⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⡇⠋⠀⠀⠀⠀⠐⠈⠙⠉⢹
⠀⠀⠀⣡⠊⠀⠀⢠⠎⠀⠀⠀⠐⠁⠀⠀⠔⠀⠀⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⠄⡄⠀⠀⠀⠀⣿
⠀⠀⡔⠁⠀⠁⡐⠁⠠⠂⢠⠂⠀⠂⡠⠂⡀⠀⡔⠁⣰⠁⠀⠀⠀⠀⠀⠀⣴⣧⣄⣿⡇⣁⡀⠀⠀⠀⣀⡀
⠀⠜⢀⠂⢀⠈⠀⡀⠁⡰⠃⠀⠀⡔⠁⠐⢁⠊⣠⠊⣠⠀⠀⠀⠀⠀⣠⠚⠻⢭⣋⣙⡈⠭⠁⠠⠍⠛⠋⠀⣿⣀
⠀⠀⠁⠠⠂⢀⠎⢀⡜⢁⠄⢀⠎⠠⠂⡰⠁⡴⢁⠌⢘⠀⠀⠀⠀⠀⣿⣿⣯⣄⣀⡉⠛⠃⠀⠤⠴⠶⠦⠭⠖⠛⡇
⠀⠀⠀⠀⡠⠃⢀⠞⢀⠈⣠⠂⠐⠁⠌⠀⠜⠰⠁⠂⢸⠀⠀⠀⠀⠀⠙⢻⣿⣿⣟⣛⣛⣀⣀⣀⣀⣀⣀⣀⣤⡶⠃
⠀⠌⠀⡔⠁⢠⠋⡠⠃⣰⠃⢬⣠⣤⣤⣬⣤⡴⡳⠒⣻⣆⡀⠀⠀⢀⣠⠾⣗⡿⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⢦⡀
⠀⢀⠎⠀⡠⠃⡰⢁⣼⠋⠀⠀⠀⠀⠀⠀⠊⣴⠁⣴⠉⠈⠉⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⡀
⢀⠃⠀⡜⢁⡜⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⢀⠎⢀⠞⠀⠊⣿⢀⡀⡀⢀⡄⡀⠀⠀⡂⡇⠸⢣⣤⣤⣠⣤⣤⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢠⠊⢀⠀⡠⠁⡰⣿⣼⣿⣿⣿⣿⣿⣿⣾⡇⢃⠀⣿⣿⣿⡿⠿⠿⠻⢿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣿⣿⣞⣡⣀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠁⠀⢁⡼⠀⡐⠀⡸⣿⡿⠛⠻⠻⠹⠿⠟⢿⡘⡄⠙⠛⠈⣀⣤⣤⣤⣄⡀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠿⠿⠿⣿⣿⣿⣯⡖⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠘⠀⡜⢀⡜⠀⠈⠛⣿⣶⣶⡶⣶⣶⡞⣿⣷⣀⣴⠟⠉⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠚⠛⠒⢦⣌⠘⠻⣟⣭⣴⡀⠀⠀⠀⠀⠀⢿⡀
⠀⠈⠀⡜⢀⠎⠀⡠⠁⡘⠛⣿⢟⣼⣟⣟⣼⡿⣱⠃⣠⠂⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⣤⣤⡤⠾⠃⠀⠀⠀⠀⠈⠻⣆⠈⣿⣿⡯⠀⠀⠀⠀⠀⠘⣧
⠈⠀⠜⢀⠎⠀⡰⠁⡔⠀⡜⠁⡾⢋⣿⢞⡟⣴⣿⣟⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠨⣻⣿⡥⠀⠀⠀⠀⠀⢹
⠀⠈⠀⠎⠀⡼⠁⡼⠀⠜⠀⡔⠀⠜⢉⠞⡼⣳⣿⡟⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⣿⣿⣏⠕⠀⠀⠀⠀⠈
⠀⠀⡎⠀⡰⠁⡜⢀⡎⠀⡼⠀⠈⠀⠈⡼⣵⢟⡟⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢻⣿⠗⣫⠄
⠀⠜⠀⣰⠁⡜⠀⠈⠀⠸⠁⠘⠀⠀⡜⠁⡞⠈⣼⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢾⣿⣟⡅⣾
⠠⠀⢠⠃⡸⠁⠄⠀⠰⠁⠐⠀⠀⠰⠁⠜⠀⡼⠉⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣥⣻⣏⣀⣀⣤⡾
⠀⢠⢃⡔⠀⡄⠀⢠⠁⠀⠀⠄⢠⠃⡰⠀⡸⠁⢠⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠊⡸⠁⡔⠀⢀⠇⠠⠁⡘⠀⠇⡰⠁⣰⠁⢠⠃⡰⣿
⡄⠰⠁⠀⠀⢀⡎⠀⠀⠀⠀⡞⠰⠁⢠⠇⢀⠃⣸⠃⡏
⠰⠂⡰⠀⠀⡜⠀⠄⢀⠂⠘⢠⠃⠀⡎⠀⠌⢠⠃⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣘⣛⣛⣛⣛⣛⣛⣛⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣛⣛⣛⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢠⣬⣭⣭⣭⣿⣯⣭⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢰⣶⣶⣶⣴⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠼⢛⣩⣭⣿⣭⣽⡟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠏⠀⠀⠀⠀⠀⢀⣴⢋⣥⣶⣶⣶⣬⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⣿⣿⣿⣿⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢿⡸⠿⠛⣛⣛⣻⡟⠿⢇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⢸⣿⣿⡜⢧⢸⣿⡟⢿⣿⣿⣿⡎⢣⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠸⠟⠈⠉⠉⠉⠀⢸⣿⣿⣸⣿⣿⣿⡇⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢏⣿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⢸⣿⡟⣸⡿⢸⣿⡏⣠⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣇⢿⣧⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣭⣭⣤⣭⣿⣿⣿⣿⣿⣮⣿⣿⣾⣿⣜⣿⣷⣽⣿⣭⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣤⣭⣭⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣬⣭⣥⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡄
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢻
⠀⠀⣸⠙⢦⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢸⡇⢰
⠀⠀⣿⠀⠀⠉⠓⠶⠖⠋⠁⠀⠀⠀⠀⣸⡇⠸⠸⠁⠀⠀⠀⡠⠠⠔⠐⠤⠐⢄
⠀⠀⢹⡀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠱
⡀⣄⠈⠳⣤⣀⠀⠀⣸⠇⠀⠘⢶⡛⠁⠀⠀⠀⠀⠀⠀⠈⢒⠀⠀⠄⢂⠀⠀⠀⠐⢀⠄
⠳⡈⠑⠀⠈⠉⠉⠉⢹⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠙⠀⠘⠆⠉⡠⡀⠤⠂⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠒⠛⠓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠲⠶⠶⠶⡆⢠⠶⡶⠶⠒⠒⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣠⠀⠀⢊⠄⠠⡀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢟⠋⠀⠀⠀⡇⠆⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢤⣤⠤⠤⠖⠚⠉⢷⡄⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡌⠈⢆⠀⠀⠀⠰⠦⠔⠂⠀⠀⠀⠀⠀⠀⢸⡗⠢⠤⠤⠒⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⠦⣌⡻⣦⡀⠀⠀⠀⠈⠳⠶⠤⣤⠄⠀⠀⠀⠀⠈⢤⠠⠊⠀⠈⢐⠆⠠⠀⣀⣀⣀⣀⣠⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣤⠐⠁⠀⣀⠔⠁⣠⣞⣁⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠬⠓⠓⠤⣀⣀⣀⣀⣠⢤⡀⠀⠀⠐⠳⠤⡀⠈⠁⡩⠀⠀⣰⠏⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠙⠒⠉⠉⠙⠢⠤⠴⠒⠛⠧⢄⣀⣀⣀⣀⡀

⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⡿⣻⣿⡟⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢺⡿⠒⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣬⠟
⠀⠀⠀⢿⣿⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣾⣽⣿⣿⣶⣾⣦⠀⠀⠀⠀⠀⢠⣤⣀⣀⣼⣷⣄⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠘⠿⣕⣻⠿⠿⠿⢯⣓⡟⠿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠛⠁
⠀⠀⠀⠀⢼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠙⠁⠀⠉⠁⠀⠀⠀⠀⠀⠰⠭⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣀⣰⣍⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣶⣶⣶⣾⠂⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡞⠁
⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁


⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣯⡙⢦⡀⠀⢠⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢟⡻⣾⡇
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠛⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡇⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢠⡶⢏⣴⡟⢿⣮⡻⣦
⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠄⢘⣿⣿⠈
⠀⠀⠀⠈⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠷⠿⠿⠿



⠀⠀⠀⡴⠋⠙⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⢲⡄
⠀⠀⠰⡇⠀⠐⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣤⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⣛⣛⠿⠿⠿⠆
⠀⠀⠀⠙⢦⣀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢋⢉⠉⡙⡙⠁⠀⠀⠀⠀⠀⠀⠸⠿⠋⠉⠁⠉⠉⠉⠻⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣬⣥⣾⣿⣿⡆
⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠂⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠉⠙⠓⠶⠖⠛⠋⠛⠓⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠒⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢶⣄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⣠⣤⡄⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⡟⠋⠉⠛⢛⣦⣤⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡤⠞
⣼⠁⣸⠀⠀⠀⠀⠙⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢿⡇⠰⠀⣰⠏⠀⠸⣄⣀⣀⡀⠀⠀⠀⣸⠃
⢷⣀⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⡼⢿⡀⠈⠀⠀⣰⠏⠀⠀⠀⠀⠉⠉⠙⠲⠖⠛⠁⠀⠀⠀⠀⢸⡟⢳
⠀⠉⠀⠀⠀⠀⣠⡄⠈⠳⠦⣤⡴⠞⠋⠉⠉⠉⠁⠀⠸⡇⢀⠆⢠⡏⠀⠀⠀⢶⣤⡀⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠈⢷⣼
⠀⠀⠀⠀⠀⢸⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⡾⠀⠀⠀⠀⢸⡏⠹⡄⠀⠀⠀⢸⡄⠉⡇
⠀⠀⠀⠀⠀⠸⢦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢰⡇⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣩⣇⣀⣸⣷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⡿⢿⣿⣿⣿⣿⡿⢿⢿⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⢸⠀⢸⣿⡇⠀⢸⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣼⠀⢸⣿⡧⠄⢸⢼⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣿⠀⢸⣿⡧⠄⢼⢸⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⡿⠀⢸⣿⡇⠀⢸⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⡇⠀⢸⣿⡇⠀⢸⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⣥⡇⠀⢸⣿⣧⡄⣸⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣸⡇⠀⢸⣿⣇⡀⣸⢠⡇⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⠁⠀⢸⣿⡇⠀⢸⠀⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⢸⠀⠀⢸⣿⡇⠀⢸⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣸⠀⠀⣾⣼⠀⠀⢸⣿⡧⠀⢸⠀⣷⠀⠀⡇
⣤⣄⣀⣀⣀⣀⣿⣀⣀⣿⣿⣀⣀⣼⣿⣧⣤⣾⣄⣿⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣷⣴⣆⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠙⡿⣿⠉⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣷⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢠⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠉⠙⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠛⠁⠀⠀⠀⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⢀⣀⣤⣄⠀⠀⣀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠓⠒⠖⠀⠉⠉⠀⣿⣯⣙⣿⣿⡆
⠀⠀⠀⠀⠀⢀⡴⠊⢁⣴⣾⣿⣿⣿⣿⣧⣤⣭⣴⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣶⣤⣤⣴⣶⣶⣀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡰⠃⢀⣾⣿⡿⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢰⠃⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠉⠁
⠀⠀⠀⣾⠀⣾⣿⠃
⠀⠀⠀⠈⠈⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⣀⣠⣀⡀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⢻⣿⣿⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠘⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣤⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣹⣇⠀⣹⣋⣉⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣿⢙⡇⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⢤⡀⠀⠀⠀⠀⢸⡧⠶⠶⠦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡟⠛⠛⠛⢻⠏⠀⠀⠀⠀⠈⢻⡛⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡇⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⣤⠟⠒⠒⠒⠛⢧⣀⠀⠀⢀⣠⠞⠓⠒⢲⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⢀⣇⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢸⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣹⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⢀⣶⣿⡟⠃⠀⠀⠀⠀⠀⠷⢶⣶⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣻⡇
⠀⠀⠀⠀⣩⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣴⡿⠃
⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆


⠀⠀⠀⠀⠀⣠⡴⠶⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠞⠆
⠀⠀⠀⣀⡾⠋⣠⠞⠁⠀⠀⠀⠀⠀⢠⡾⠛⠛⠻⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣭⣄
⠀⠀⠺⣍⢳⠜⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⣀⡾⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⢰⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡠⣤⣤⣤⣤⡄
⠀⠀⠈⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⢶⣶⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢰⣾⣷⣿⣧⣤⣿⡇
⠀⠀⠀⠀⠈⠛⠑⡨⠁⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠂⠈⠀⠀⠀⠀⠀⠘⠚⠉⠉⠉⠙⠚⠁


⠀⠀⠀⢀⣀⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢸⣿⡇⠠⠧⠀⠀⠀⠀⠀⠀⣀⣴⣞⣭⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣆⣴⡄
⠀⠀⠀⠸⣿⣿⣶⣶⠂⠀⠀⠀⠀⠀⣽⣛⣻⣥⣾⠿⠃⠀⠀⠀⠀⠀⠀⢀⣏⠉⠁⠀⠀⠀⠀⠀⠀⠿⠟⠛⠛⠛⠻⠃
⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠀⠉⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠒⠈⠉⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣐⠁⢠⣤⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠀⠀⠀⣠⠁⠀⠀⢜⣀⡀⡅⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⣡⠖⠛⠉⠀⠀⠙⠑⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢠⠆⠁⠂⡀⠢⠀⠀⡀⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣂⠸⠀⠀⠸⠏⠀⠀⠀⠣⠛⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠠⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⣁⠀⠀⠘⠢⣄⠀⠠⠤⠰⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠄⡀⠀⠀⠨⣑⠨⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⢤⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⠒⠒⠂⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠢⡀⠀⠺⡀⠀⠀⠈⠁⠉⢉⡁⠀⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢃⠀⠀⢸⢦⠀⠈⠲⠀⠤⠀⠤⠤⢋⢆⠀⠀⣌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠈⠀⠀⣼⠀⢣⠀⠀⠓⠒⠒⠒⠊⠁⠈⡄⠀⠘⢆⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠑⡄
⠀⠀⠀⠀⡠⠐⠒⠂⠒⠀⠀⠀⠀⠀⢰⠁⣄⣘⡄⠀⢸⠠⠤⠤⠄⠐⠒⡇⠀⢠⢸⠀⠀⠑⠢⡀⠀⠀⠀⢀⠂⠐⢄⡀⡠⠀⢀⠜⡄
⠀⠀⢀⡔⡡⠊⠀⠀⠀⠁⠊⠀⠀⠀⠘⠃⠀⠀⠈⢃⢸⠀⠀⠀⠀⠀⠀⠧⠤⠞⡎⠀⠀⡠⠄⠱⠄⡀⢀⡄⠀⠀⠈⠶⠒⠊⠁⣀⠶
⠀⠀⢨⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠁⠤⠤⠤⠤⠄⠒⠂⡃⢠⡇⢀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠆⠒⠒⠉⢁⣴
⠀⠀⢸⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⡀⢊⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠈⠁⡰
⠀⠀⠀⠣⣈⠢⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠈⣩⠂⠐⠂⠀⠀⠀⠀⠀⠐⠂⠃⠈⠚⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠳⠒⠊⠉⡃
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠑⢄⡀⠀⠀⠀⢂⡩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢘⣷⣴⠶⠶⠶⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠛⠛⠛⢻⠀⠀⢰⡟⢛⡄⢠⡉⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⠶⠞⠃⠀⠘⠛⠛⠀⠈⠙⣣⠀⠀⢀⣴⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢄⣸⣄⣀⣠⣄⣀⣀⣀⣤⣤⠴⠿⠗⠛⠋⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠚⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⢀⣰⠟⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⣠⡾⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠶⠶⠶⠶⠄⠀⠂⠀⠀⠉⠉⠉⠉⠒⠶⠶⣄⠀⠀⠈⢷⡀
⢠⡟⠀⠀⠀⢀⡠⢤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢷⡀
⣸⠇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠛⠀⠀⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⡇⠀⣠⡄⠀⠀⠀⠀⠊⠀⠀⡀⠠⡄⠀⠀⠉⠀⣀⠰⢆⠀⠀⠳⡄⠙⠀⠀⠠⣄⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⡇⠀⠀⠋⠀⠀⢀⡀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⢀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠠⣄⢤⡀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⢹⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠢⠌⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⡄
⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠃⠀⠀⠀⠀⢸
⠀⣇⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⠀⠀⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⣿⠀⠀⠰⠀⠱⠄⠀⠀⠀⠈⢦⠀⠙⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⣧
⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢸⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⣇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠲⠄⠙⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠘⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢹
⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⢸⡀⠀⠀⠀⢸
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠋⢀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠸
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠
⠀⠀⠸⡇⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠞⠁⠀⠀⠀⣸
⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠙⠶⠶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⢶⡶⠶⠶⢶⡶⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⣇⣀⣀⣸⡄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⢋⣿⣭⣭⣿⡙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣋⣁⣀⣀⣈⣙⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢛⠛⠛⠛⠛⠲⣤⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸
⠀⠀⠀⢠⡏⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣼⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⣾⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⢿⣤⣤⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⢠⡞⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⣠⠏⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⣴⠋⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣿⠀⣶⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⢿⠀⠘⣧⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⠆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⣸
⠈⢧⡀⠈⠙⠛⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⣰⠏
⠀⠀⠙⠳⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣒⣚⣋⣁⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⡀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡇⠀⠀⠀⠀⡆⠀⢸⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠟⠻⠛⠀⠀⠀⠙⠃⠀⠀⠀⠛⠛⠿⠿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠐⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠒⠒⠒⠒⠒⠲⠒⠒⠂⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⡀
⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠆⠀⠀⠀⠙⠷⣶⣤⣤⣤⣴⠾⠋⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⣀⣤⣾⠶⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠀⠀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⢰⣿⡿⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁⣀⣴⡾⠟⠋⠀⠀⢀⣠⣶⡶⠆⠀⠀⠀⢀⣤⣶⠿⠛⠁⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⠁⠀⠀⠀⠰⠿⠋⢁⣠⣶⡿⠛⠉⠀⠀⢀⣤⣶⠿⠛⠁⠀⠀⣀⣴⡾⠟⠋⠁⠀⢀⣤⣾⠿⠋⠁⠀⠀⣀⣴⡆⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠀⠀⣀⣴⣾⠟⠋⠁⠀⢀⣠⣶⡿⠟⠉⠀⠀⣠⣴⡿⠟⠋⠀⠀⢀⣠⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠛⠋⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⣀⣤⣾⠿⠛⠁⠀⢀⣠⣶⠿⠛⠁⠀⠀⢀⣤⣶⠿⠛⠁⠀⠀⣀⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠁⠀⠀⣀⣴⡾⠟⠋⠁⠀⣀⣤⣾⠿⠋⠁⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠾⠟⠋⠁⠀⢀⣠⣴⠿⠛⠉⠀⢀⣠⣴⡿⠟⠋⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⢀⣤⣶⠿⠛⠁⠀⣀⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁⠀⢀⣤⣶⠿⠛⠁⠀⠀⣀⣤⣾⠿⠛⠁⠀⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢰⡾⠟⠋⠁⠀⣀⣤⣾⠟⠋⠁⠀⠀⣠⣴⡾⠟⠋⠁⠀⢀⣠⣴⡾⠟⠋⠀⢀⣤⣶⠿⠛⠉⠀⣀⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⢀⣠⣶⡿⠛⠉⠀⠀⢀⣤⣶⡿⠛⠉⠀⣀⣴⣾⠟⠋⠁⢀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢠⣶⠿⠛⠁⠀⠀⣀⣤⣾⠿⠋⠁⠀⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡾⠟⠉⠀⢀⣤⣶⡿⠛⠉⠀⣠⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⣠⣴⡾⠟⠉⠁⠀⢀⣠⣴⡾⠟⠉⠀⢀⣤⣶⠿⠛⠁⠀⣀⣴⣾⠟⠋⠁⢀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠀⠀⢀⣤⣶⠿⠛⠁⠀⣀⣴⣾⠟⠋⠁⠀⣠⣴⡾⠟⠉⠀⢀⣤⣶⠿⠋⠁⠀⣠⣤⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠘⠋⠁⠀⠀⣀⣴⣾⠟⠋⠁⠀⣠⣴⡿⠟⠉⠀⢀⣠⣶⠿⠛⠁⠀⣠⣴⡾⠟⠋⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠉⠀⣀⣤⣶⠿⠛⠁⠀⣀⣴⡾⠟⠋⠁⢀⣠⣶⡿⠟⠉⠀⣀⣤⣾⠿⠋⠁⢀⣠⣴⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣴⠿⠛⠁⠀⣀⣴⣾⠟⠋⠀⢀⣠⣴⡿⠟⠉⠀⢀⣤⣾⠿⠋⠁⠀⣠⣴⡾⠟⠋⠀⢀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠀⢀⣠⣶⠿⠛⠁⠀⣠⣴⡾⠟⠋⠀⢀⣤⣶⠿⠛⠁⠀⣀⣴⡾⠟⠋⠀⢀⣠⣶⠀⠀⠀⠀⠀⣿
⢹⣿⠀⠀⠀⠀⢴⠿⠛⠁⠀⣀⣴⡾⠟⠋⠀⢀⣠⣶⡿⠛⠉⠀⣀⣴⣾⠟⠋⠁⢀⣠⣶⡿⠟⠉⠀⣀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⣿
⢸⣿⡆⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⣀⣤⣾⠿⠋⠁⢀⣠⣴⡿⠟⠋⠀⢀⣤⣾⠿⠛⠁⠀⣠⣴⡿⠟⠋⠀⢀⣤⣶⠀⠀⠀⠀⢀⣿
⢸⣿⡇⠀⠀⠀⣼⠿⠛⠁⠀⣠⣴⡾⠟⠋⠀⢀⣤⣾⠿⠛⠁⠀⣀⣴⡾⠟⠋⠀⢀⣠⣶⠿⠛⠉⠀⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡿⠛⠉⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡿⠟⠉⠀⣀⣴⡶⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⢾⠿⠋⠁⢀⣠⣴⡿⠟⠉⠀⢀⣤⣾⠿⠛⠁⢀⣠⣴⡿⠟⠋⠀⢀⣤⣶⠿⠛⠁⢀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁⢀⣠⣴⡾⠟⠋⠀⢀⣤⣾⠿⠛⠁⠀⣀⣴⣾⠟⠋⠁⣀⣤⣶⠿⠛⠁⣀⣤⣶⡿⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⢾⠟⠋⠁⢀⣠⣶⡿⠛⠉⠀⣀⣤⣾⠟⠋⠁⢀⣠⣴⡿⠛⠉⠀⣠⣴⡾⠟⠋⠁⣀⣤⣾⠟⠋⠁⣀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⢀⣠⣴⡾⠟⠉⠀⢀⣤⣶⠿⠋⠁⢀⣠⣴⡿⠛⠉⢀⣠⣴⡿⠟⠉⢀⣠⣶⡿⠛⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠼⠟⠉⠀⢀⣤⣶⡿⠛⠁⠀⣠⣴⣾⠟⠉⠀⢀⣤⣶⠿⠋⠁⢀⣤⣶⠿⠛⠁⣀⣤⣶⠿⠋⠁⣀⣤⠀⠀⠀⠀⢸⣿
⢸⣿⣧⠀⠀⠀⠀⢠⣴⣾⠟⠋⠁⢀⣠⣴⡿⠛⠉⠀⣠⣴⣾⠟⠋⠀⣠⣴⣾⠟⠋⠁⣠⣴⣾⠟⠉⠁⣠⣴⡾⠟⠃⠀⠀⠀⠀⣼⣿
⠈⣿⣿⡀⠀⠀⠀⠈⠉⠀⣀⣤⣾⠿⠋⠁⢀⣤⣶⡿⠛⠉⢀⣠⣴⡿⠛⠉⢀⣠⣶⡿⠛⠉⢀⣤⣶⡿⠋⠁⣀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢻⣿⣧⠀⠀⠀⠀⠀⠘⠟⠉⠀⣀⣤⣾⠿⠋⠁⢀⣤⣶⠿⠋⠁⣀⣤⣾⠿⠋⠁⣠⣴⣾⠿⠋⢁⣠⡶⠟⠁⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⢠⣴⡾⠟⠋⠀⣠⣴⡾⠟⠋⢀⣠⣴⡿⠟⠉⠀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣠⡶⢟⣛⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠳⡄⠀⠀⣠⠞⠉⠚⣫⣽⣿⣿⣿⠿⠶⠄⠀⢀⣠⠄⢀⣀⢀⣤⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⡶⢷⠀⢴⣄⠀⢿⣿⣿⣿⣿⡄⠀⣚⣉⠀⢰⠋⢉⣭⣿⣦⣄⡀⠙⣆
⠀⠀⢀⣀⠀⠀⠀⠠⠤⠤⣤⣄⠀⠀⡷⠃⠀⠶⠉⠀⠨⣿⣟⠻⡿⠁⢈⣉⣀⠀⠈⠓⢼⣿⣿⣿⣿⣿⣦⡏
⣶⢋⣉⠘⡆⠀⠀⠀⠀⠀⠀⠀⢠⡆⣓⣂⣠⣼⡶⠄⠀⣀⣽⡷⠃⠀⠀⢀⣩⣅⢀⣀⣀⠈⠙⠛⠛⠛⠋⣀⣀⡀
⠘⠋⣹⣤⠷⡄⠀⠀⠀⠀⠀⠀⠘⣇⠙⠶⠾⠋⠀⠀⠀⢨⣇⣠⡴⠶⢺⡏⢠⡏⠉⠉⠉⠉⠓⠲⠔⠋⠙⡿⠿⣯⡝⠳⣄
⠀⢿⣥⢴⠲⡇⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⣤⠴⣶⣦⠔⣻⣿⠁⠀⢀⣈⣇⠘⣧⣤⣤⡶⠶⠶⢾⣠⣴⣾⡟⣴⠟⠀⠀⢈
⠀⠀⢸⣿⡆⢧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠖⣿⡧⡞⢦⡇⡇⠀⠙⢻⠋⠉⠉⠉⠀⠀⠀⠀⠀⢺⣿⣥⣸⡵⠃⠀⠀⢠⠞
⠀⠀⠈⠻⢧⢸⣄⡀⠀⠀⣀⡾⢯⡈⠙⣄⣀⣇⠒⠚⠉⠁⠘⢦⣀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⣰⠋
⠀⠀⠀⢠⣞⣏⡄⢹⠖⠋⠁⠀⣠⣿⠟⠋⠉⠉⠙⠲⢦⣄⠀⠀⣈⣉⡉⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠟⠁
⠀⠀⠀⠘⣿⣛⣷⣰⡶⠾⠟⢻⡿⠁⠀⡆⢲⣄⠀⠀⠀⠈⠳⠞⠿⢭⣍⠀⢸⣷⠀⠀⠀⠀⠀⢦⡀⠀⠈⠉
⠀⠀⠀⠀⠘⢷⡏⣏⠀⠀⠀⢸⡇⠀⠀⠹⣄⡙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢀⡤⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠙⠁⠀⡀⠀⠈⢷⡀⠀⠀⠈⠙⠂⣦⢙⡶⣤⣤⣤⣤⣴⣾⠿⠃⠲⡗⠞⠰⣋⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷⣒⣒⣒⣒⠦⢼⡈⣇⠀⠀⠉⢁⣼⢢⡄⠀⠀⠸⣜⣹⡌⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢰⡒⠻⠿⣛⡋⠹⣾⠁⠹⣦⡀⠖⠛⠁⠸⠃⠀⠀⣠⡿⠙⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠙⠓⠲⠖⠻⣔⣺⠆⠀⠈⠻⢶⣦⣤⣤⣴⡶⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣮⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⠄⠄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄
⠀⠀⠠⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣜⣛⣛⠀⠀⠀⠀⠀⠀⠋⠛⠛⣿⣿
⠀⠀⢠⣤⡤⣾⣿⣦⣤⣤⠀⠀⠀⠀⠀⠄⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣦⢸⣿⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀
⠀⠀⠸⠿⠷⢿⣿⠿⠿⠿⠀⠀⠀⠀⠀⠡⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠟⢸⣿⣿⠹⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠇

⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠧⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠐⣁⣀⣈⡂⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣄
⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠁

⠀⠀⢀⡀⠠⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣤⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡄⡀⠀⠀⠀⠀⠀⠙⣿⣟⣿⣟⣻⡟⠁
⠀⠀⠀⢸⣿⣹⣾⣹⣿⣿⠀⠀⠀⠀⠀⠀⡿⡉⠗⠀⠀⠀⠀⠀⠀⠐⣼⣿⡿⣹⣿⡄⠀⠀⠀⠀⠀⣸⡇⢸⡏⣼⡇
⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡧⠁⢤⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣿⣿⡗⠀⠀⠀⠀⠀⣹⡇⢾⡇⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈

⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⢼⡿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⣀⣀⠈
⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡐⢺⡗⠲⣿⠂⠀⠀⠀⠀⠀⡠⠒⠀⠒⢄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠋⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢺⣷⣶⣿⡆⠀⠀⠀⠀⣿⡏⢉⠉⡉⢉⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠶⠀⠆⠰⠆⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠇⠀⠀⠀⠀⠀⠈⠪⠅⠅⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠟⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠰⣙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⠞⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠛⠻⠻⣧⡀⠀⢀⣤⣤⣄⣀⣠⣤⣄⣀⣀⣤⡾⠋⠘⠇⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣛⣿⡅⠉⠙⠛⠛⢸⣿⣿⠿⣤⣤⣀⠀⠀⢀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣶⡶⠛⠉⠉⠉⠉⣷⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠿⣶⣤⣷⣄⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣿⠀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣶⣶⣾⣶⣶⣶⣶⣶⣿⣷⡟⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⠯⠭⠽⠿⠟⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣭⣵⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⣶⣿⠺⢿⣿⣒⠀⠀⠀⠀⠀⠀⣲⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⠛⠻⠿⢶⣶⣤⣤⣤⣴⡶⠿⠟⠛⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⡿⠀⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣾⠿⠏⠀⠉⠉⣉⢉⣉⣉⡉⠉⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠚⠉⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆⡀⠀⠐⣒⣲⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠛⠻⣛⡛⠿⣉⠉⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢿⡇
⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⢸⡇
⠀⠀⠀⠀⠸⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡶⠟⠛⠿⣷⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣸⡇
⠀⠀⠀⠀⠀⣿⣁⠀⠀⠀⠀⠀⠀⠀⡄⣀⣠⣿⠋⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣰⠛⢿⣯⣀⣀⠀⠀⠀⡆⢐⣂⣛⣛⣿⡄⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠹⣦⣤⡀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠈⠙⠻⠷⠶⣦⣤⣤⣤⣤⣤⣾⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢻⠀⡄⢧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠘⣇⢳⣀⡁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⡇⠀⢻⡄⠉⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣷⣴⠘⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠂⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣦⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣀⡀
⣤⣤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣠⣤⠀⠀⠀⠀⠀⠀⠙⣶⠀⠀⢹⠿⡏⠀⢀⣶⠋⠀⠀⠀⠀⠀⢀⣤⡄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠⣤
⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣸⣿⣦⡀⠀⠳⠾⢿⣧⡀⠀⠀⠀⢀⣼⡿⠷⠟⠀⢠⣶⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣻
⠛⠛⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢙⣛⣛⣛⡀⠀⠀⠀⠙⢿⣦⢀⣴⡿⠋⠀⠀⠀⢀⣛⣛⣛⠙⠛⠃⠉⠉⠉⠉⠉⠉⢉⣉⠉⠉⠉⠘⠛
⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⢿⣶⣄⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⢀⣵⡿⣫⡀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣿⡷⣾⣿⡆
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠘⣿⡆⠀⠀⣿⣿⣿⠀⠀⣠⣶⣶⡿⠋⠀⠙⢿⣶⣶⣄⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣿⠀⠀⠉
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⣧⠀⠀⢿⣿⣿⠀⠀⠁⢀⣿⠃⠀⠀⠀⠸⣿⡀⠈⠀⠀⣿⣿⡇⠀⠀⠀⣀⣀⣠⣤⣀⣀⣿⣿⣇⣀⣀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣩⡉⠀⢰⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣻⣿⠀⠀⠀⠿⠿⢿⣿⡿⠿⣿⣿⡿⠿⠿⣿⡆
⠰⣿⣿⡷⠀⣿⡃⢰⣿⣿⡆⠀⣿⡗⠀⠀⢱⣶⣶⡆⣶⣶⣠⣤⣤⣤⣤⣤⣤⣤⡄⣶⣶⣴⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣿⣧
⢀⣹⣿⣵⣿⣿⣿⣦⣿⣟⡀⠀⠀⠀⠀⠀⠀⠙⠿⠇⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⢠⡄⠀⠀⢩⡉
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣠⠄⠀⡀⠻⣟⠦⣾⠟⠀⠀⠠⣿⡇
⢀⣤⡴⢶⣶⡶⣾⣿⣿⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⢸⣿⣦⣼⡇⠀⣸⡿⢾⣷⡀⠀⣠⠈⢁
⠈⠁⣴⣿⡿⠞⠋⠉⠉⠙⢷⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠙⣿⡟⠁⢀⣛⠁⠀⢛⡃⠀⢿⣦⡾
⠀⣼⢻⡟⠁⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⣰⣿⣷⣦⣿⢻⣿⣦⣶⣿⣷⠀⣾⣿⣶⣴⣿⢓⣵⣶⣾⣿⡆⠀⠀⣿⡇⠀⢸⡿⠀⠀⢸⣿⠀⠘⣿⠁
⠀⢻⣼⣧⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⠀⠈⠛⠋⠀⠙⣿⡟⠁⠉⠛⠋⠀⠙⠛⠉⠈⢻⣿⠊⠈⠙⠛⠁⠀⢰⣿⡇⠀⢸⡇⠀⠀⢸⣿⠀⠀⣿
⠀⠈⢿⣿⣷⣄⡀⠀⠀⣀⣼⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⠀⠀⢸⡏⠀⠀⣿
⠀⠀⠀⠉⠻⢾⣿⣿⣟⣯⡵⠋⣼⠃⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⢀⣀⠀⣀⡀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⣸⡇⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⢾⣿⣷⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡿⡇⠀⠀⣿⡄⠀⣿⠀⢰⡟⠛⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠺⠽⠿⠯⠇⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠈⠛⠁⠀⠀⠛⠀⠀⠙⠁⠀⠁⠀⠉
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠰⢶⠀⣰⡆⠀⠀⢀⣴⣶⣿⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢶⣶⣶⣶⣶⡄⠀⠀⠈⠛⠃⠀⠀⠴⠯⠜⠻⠟⠀⠀⣼⣿⣿⣿⣿⣿⡝⠛⠛⠿⣶⣄⠀⠀⡼
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣶⡀⠀⠘⠛⢛⣛⡛⠛⠀⠀⢠⡀⢀⣄⠀⣤⠀⠀⠀⠀⠀⠙⢷⠀⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠈⠛⠁⠀⠀⠀⠀⣼⠃⠀⠀⣤⠈⢡⡌⠁⣤⠉⣠⡄⠀⠀⠀⠀⢸⡆⣗
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣀⣉⣁⣀⣀⣀⣀⣸⣿⣿⣿⣿⡇⢠⣤⣤⣤⣀⣀⣀⣀⣀⣁⣀⣀⣀⣉⣀⣈⣁⣀⣁⣀⣈⣤⣤⣤⣤⠀⢸⠀⢸⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⡏⠀⠀⢧
⠀⠀⠀⣼⣿⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⠁⠀⠀⠸
⠀⣠⣶⣿⣿⣷⣦⡀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣸⠀⠀⠀⠈
⣰⡿⣿⡿⣿⣿⣿⣷⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠛⢳⡿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢺⣿⠛⠋⠀⠀⠀⢸⡀⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠤⠶⠶⢦⡄⠄⣠⣄⣤⣤⣀⢀⣀⣀⣤⣴⣷⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣿⡿⠛⠋⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠤⠄⠤⠤⠤⠤⠤⠀⠀⠀⠀⠬⣿⣿⣿⣥⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⢀⣁
⠀⠀⠀⠀⠀⠁⠶⠷⠤⠴⠶⠤⠭⢯⠶⠶⠶⠶⠶⠶⠶⢶⡶⣶⣿⣿⣿⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠷⠶⠖⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠐⠒⠒⠒⠒⠲⠿⠛⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⣿⡿⠷⠶⠶⠖⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣈⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣤⠤⠤⣠⣤⣤⣄⣀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠀⠀⠀⠉⠉⢒⠚⠿⣹⠀⠀⠀⠀⢀⠏⠛⠓⠒⠉⠉⠀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⠀⠀⠀⣀⡠⠤⣼⣠⣤⠃⡆⠀⠀⠀⣸⠸⣥⣀⡤⠤⢄⡀⠀⠀⠐⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠒⠚⠉⠀⠀⠘⣆⢁⣀⣀⣀⡇⡼⠁⠀⠈⠙⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣏⣉⣉⣉⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢥⣴⠥⠤⠤⠤⠬⣧⠸⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⢞⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣄⣀⣤⣤⡏⢁
⠀⠀⠀⠀⠀⠀⣠⠤⠖⢚⣉⡉⠭⠤⠄⠐⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡴⠋⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⢀⡄⢸
⠀⠀⢀⡞⢠⠊⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⣾⣿⣶⡇⢸⣿⡏⣿⡷⣿
⠀⠀⡼⢠⠇⠀⠀⠀⢸⠀⢠⠖⠋⠉⠉⠉⠙⠓⠲⠦⢤⣄⣀⣀⣀⣀⣀⣒⣬⠥⠖⠚⠉⠉⠀⠉⢇⡸⣹⣇⣿⣏⣿
⠀⢠⡇⡾⠀⠀⠀⠀⡜⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠬⠭⠭⠭⠽
⢰⠾⠡⡇⣀⡀⠀⢀⣇⡸⡇
⢸⠀⠀⣥⠀⣒⣒⡒⢦⣬⡇
⠀⠉⠒⠓⠾⠿⠿⠟⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸
⣀⢀⡀⠀⠀⠀⠀⠀⠙⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⠶⣦⣤⣄⣀⡀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠉⠉⠛⠻⣿⣿⠿⢇⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢼⣿⣿⣷⣦⣬⣙⣿⣯⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⠉⠉⣿⠏⢩⡴⣦⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢸⣿⣏⣠⣤⣿⠀⠈⢻⣮⡹⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⣿⣿⣿⣕⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡳⠄⠀⠀⣰⡆⠈⠻⣿⡿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⢰⣿⡇⠀⠀⠈⠁⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠀⢐⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠟⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠂⡀⢰⣿⣿⣿⣿⣿⣄⣤⣤⣈⣿⡇⠀⠀⠀⡀⢠⡆⠀⠀⠀⠀⠀⠀⣀⠀⣻⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⣿⣿⣿⣿⣿⠏⠛⠉⠋⠉⢃⣀⢀⡀⠈⢻⣷⠀⠀⠀⠀⠀⠀⣹⣷⣿⡿⠛⠛
⠀⠀⠀⠀⠠⣾⣷⡄⠀⠀⠀⠀⢀⣾⣿⡇⠀⢰⢀⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⣿⣟⣯⠙⡇⢸⠁⠀⠄⠀⠀⠀⡾⠛⣿⣷⠇
⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⢀⣴⣿⡏⣹⡃⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢫⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⡏
⠀⠀⢲⣄⣠⣤⣼⠏⢀⣤⣾⣿⡿⢷⣌⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡇
⠀⠀⠀⢹⣟⣿⠁⡀⠜⢛⠉⠁⠀⠀⢋⠀⢀⡘⢿⣿⣿⣿⣿⣿⡅⠀⠀⠀⢀⣾
⠀⠀⠀⠈⢿⣿⣿⣷⣤⣿⣦⣄⢠⣀⣌⡠⠟⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⣤⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣿⣿⣿⣿⡇⠀⣠⣾⣿⣴⣾⣦⣴⡶⢄⣾⣯⣴⣶⣿⣿⣿⣅⣭⣧⣤⣀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡶⠊⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⠋⠈⡿⢿⣴⣀⠀⡄⠈⡀⡀⠈⠁⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⠇⠀⠀⢰⣦⡀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢻⣦⣝⣿⣧⢻⣦⣵⣲⣀⠀⠀⣀⣰⡶⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⡴⠚⠋⠀⢁⠀⣿⣿⣿⣿⣿⣿⡀⠀⢺⣶⣾⣻⣿⣿⠇⠙⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣏⠁⠀⠀⢀⡸⠀⠘⣿⣿⣿⣿⣿⣟⠀⠸⣿⣿⣿⣷⣟⡀⠀⠀⠠⠤⠠
⠀⠀⣀⠁⠀⣠⠞⢻⣿⠿⣿⣷⣉⠀⣄⠀⣼⠁⠀⠀⠸⣿⣿⣿⣿⣿⡆⠀⡿⣿⣿⣿⣿⠖⠂⠀⠀⠀⠐⠀⠠⠀⠸⠀⠀⠀⢀⢀⣤⣴⡄
⠀⠠⡿⣧⢸⣯⣄⣭⣍⡁⢹⣿⣿⣶⣿⡿⠁⠀⠀⠀⠀⠛⠋⠋⠙⠻⢿⣀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠉⠉⠻⢇
⠀⢐⢿⣯⣼⣿⣿⡂⢉⣹⡾⠿⠋⠉⠉⠀⠀⠀⡘⠀⠀⢀⣾⣿⣿⣷⣾⣿⣷⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⠟⠒⠒⠒⠀⢸
⠀⡬⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⢀⠂⢐⠃⠀⠀⠀⣿⣿⣦⣭⣙⣛⠛⢿⣿⣿⡿⠷⠶⠶⠶⠶⣦⣤⣄⣀⡀⣀⠀⠀⠀⠀⠀⠀⣸
⢾⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣦⣦⠜⠃⠀⢨⠀⠀⣰⣄⣼⣿⣿⣿⣿⣿⣆⠀⠹⣿⣿⣷⣶⣤⣶⣶⣦⣤⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢻
⣽⣿⣸⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⡿⠏⠀⣯⣼⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⡟⠀⠀⠀⡿⠉⠀⢱⣦⡀⠀⣀⣀⠀⠀⡀⢠⠆⡨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⡿⢿⣏⡑⠀⣙⣧⣿⣿⣿⣿⣿⣿⣿⡆⣿⡟⠀⠀⠀⣀⠀⢀⡀⢀⣿⣿⣀⡍⣋⣤⠀⠸⠇
⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣉⣶⡿⠅⠀⠈⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⣀⣤⣤⡀
⠈⣳⣿⣯⣙⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣷⢖⣟⠾⣿⣿⣽⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟⢟⠛⠟
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣥⣤⣼⡿⢻⣿⣿⣿⣿⣿⡍⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⠼⠟⠞⠓
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⠻⠿⢿⣿⣿⣦⣬⣭⣿⣍⡻⠿⣿⣿⠿⠿⠿⠧⠄
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⣉⡉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⢀⡴⠖⠶⣄⠀⢀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡶⠦⣤⡀⠀⠀⠀⠀⣼⠁⠀⠉⠉⠉⢻⠁⠀⠀⢸⡏⠉⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠘⢦⠤⠴⠶⠒⢻⠀⠀⠀⢸⠓⠲⠶⠦⢤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣀⣼⣤⣤⣤⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⣧⠀⠀⠀⠀⠀⢠⡷⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠻⢤⣤⣤⡼⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⣶⣀⠀⣀⣤⠾⠛⠀⢀⣀⠀⣠⡀⢠⣄⡀⢀⠀⠀⠀⢀⣠⣤⢤⣠⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠘⠉⠉⡁⠀⠀⠀⡜⣦⡼⡿⢷⢿⡿⠆⠘⡾⡶⣡⢿⢷⡇⣾⠇⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣡⡶⠶⣦⣴⡙⣦⣄⠀⠈⠈⠀⠀⠈⢀⡀⠀⣀⠀⢠⣀⡀⢨⡆⠀⣀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⠣⠂⢀⣘⠛⠈⠉⠀⠀⠀⠀⠀⠀⡛⣤⡼⡟⣷⢿⡿⠀⡞⢸⣳⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⠄⣾⠁⢳⣤⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠉⠈⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢠⣶⢾⣾⢿⣤⠀⡄⠀⢀⣤⣴⣿⠟⠷⢦⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⢦⣿⠘⣿⣯⣙⣲⡇⢚⣫⣴⠟⠃⠀⠀⠰⠆⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣿⣌⠫⣳⣼⣿⣭⣭⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⣿⣦⣬⣽⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠃⠀⡸⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣀⣼⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣄⣠⡤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⣧⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢦⡀
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣹⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⢀⣀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡶⠒⠓⠖⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠋
⣰⠋⣿⠀⠀⠀⠙⢶⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢹⡇⡔⢀⡼⠁⠘⣆⣀⣀⡀⠀⠀⣸⠋⠉
⠹⣤⠏⠀⠀⠀⠀⢺⡀⠀⠀⠀⣀⣠⣄⣀⡤⠟⡇⠀⠀⠀⡾⠁⠀⠀⠀⠉⠉⠛⠒⠛⠁⠀⠀⠀⠀⡏⠳
⠀⠀⠀⠀⠀⣠⡶⠀⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⣿⢠⠃⣸⠃⠀⠀⢸⡶⣄⠀⠀⠀⢸⠷⢤⠀⠀⠀⠙⠞
⠀⠀⠀⠀⠀⣇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⠸⣇⣸⠂⠀⠀⠘⢧⣤⠇
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⣼⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣻⣀⣸⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠒⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢶⠶⠶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⢻⠇⠀⢀⣰⣀⡀⠀⢻⡛⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣤⣼⡄⠀⠉⢹⠉⠁⠀⣸⣧⣤⣄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠻⣦⣀⣀⣀⣠⡴⠃⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢸⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠴⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⢂⣶⡐⡖⢰⣶⢰⣶⡆⠰⣖⣰⡆⣶⣶⣲⢰⣲⢶⡆⢲⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠘⠉⠃⠃⠈⠛⠈⠉⠃⠈⠛⠁⠙⠁⠉⠀⠘⠀⠈⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠤⡷⠀⠀⠀⠀⡄⡤⣼
⠀⠀⠀⠠⠤⠇⢠⡄⢸⣤⣤⢤⣸⠀⠀⢠⠔⠀⡖⠀⠐⠒⠓⠐⢲⠀⡗⡶⣶⠒⠚⠃⠓⠛⠀⢤
⠀⠀⠀⠴⠶⠖⠒⡇⢸⠻⠗⠆⢸⠀⠀⣼⣠⡏⠙⠉⠙⠛⡏⠉⠉⢆⠈⠛⢻⠛⠛⠋⢛⡿⠱⠈⡇
⠀⠀⠀⠀⠈⠀⠀⡏⢻⡀⢀⡻⣿⠀⠀⢸⠙⡇⠀⠀⢀⢴⡗⠀⠀⠀⡉⠉⠉⠛⢻⡏⠉⠀⠀⠸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠣⢀⣉⣹⣍⣉⣀⣀⣈⣀⡀⢀⣀⣈⣀⣉⣀⣀⣘⣯⣀⣉⣉⣉⣁⣀⣀⣉⠀⣉⣈⣧⣄⡉⠱
⠀⠀⠀⣶⠀⡖⡒⠀⠀⡄⠀⠁⠙⢦⠦⢤⠸⡇⠀⠠⠄⠀⡧⠐⠶⡖⠀⢄⠀⡀⢸⡇⠀⠀⢰⢤⡧⠀⠀⠀⣃⣴⡆
⠀⠀⠉⠋⠀⠛⠛⠹⠄⢸⠀⠈⠐⠚⠓⠂⠀⡗⠐⠚⢻⢀⡟⠛⠙⠛⣦⢸⡀⢙⠙⢇⣀⣀⣀⠀⢇⣀⡀⠀⠉⢹⡇
⠀⠀⠀⠀⢀⠀⠀⠨⠐⣿⠀⠀⢀⣀⣤⡴⠴⠇⠀⣤⠈⠀⠇⠀⠀⠀⠀⢻⠃⠘⠦⠤⢼⡦⠿⠀⠶⡄⢹⡄⠀⢸⡇
⠀⠀⠀⢠⠨⡗⠀⢀⣀⣿⠀⠀⡇⢠⠶⢧⣤⡄⠀⠷⠀⠂⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠸⠀⢉⡙⡉⣉⠉⢀⣸⡗⣿⣆⣀⡉⠁⠀⠀⠀⣈⣽⠀⠀⠀⠀⢈⣁⣀⣀⣀⣹⣵⣿⠀⣟⣁⡰⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⡏⠀⠈⠀⠀⠀⠉⢽⡅⠀⠀⠀⠉⠉⠀⠀⠀⠁⢹⠁⠀⠀⠀⠉⠉⠙⢀⡌⠀⠀⠀⠷⢾⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡄⣥⠀⠀⠀⡀⠀⠋⠘⠦⠶⠠⠤⠴⢶⠄⢀⡀⡤⢸⡄⠀⠀⠀⡆⡆⣶⠀⣗⢲⣴⠐⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠘⡏⠻⣤⣤⠀⢠⣤⡤⠀⠛⠓⠒⠛⠛⠛⠳⠾⠧⠅⢹⠀⣠⠀⡞⠛⠛⢻⣸⣿⠙⠋⠁⣂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠮⠷⠀⠸⠶⠶⠐⠀⠠⠤⠒⠢⠦⠶⣶⠶⡆⢘⠀⠹⠚⡏⠀⠀⢨⠉⡇⠀⠀⠀⠟⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⣿⠂⠀⠀⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣗⣾⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠢⣄⣠⠖⠓⠢⣀⣠⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡴⠚⠓⢄⢀⡴⠛⠳⢤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠚⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠽⠙⠯⣅⠀⠀⠀⠀⠀⢀⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠚⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⢾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡇⠀⠈⠓⠦⠤⠴⠊⠀⠀⠀⠀⠀⠘⠦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠤⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠙⣞⠉⠉⠉⠉⠉⠑⠢⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢺⠁⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠁⠀⠘⣆⠀⠀⠀⣠⠇⠀⠀⠀⠀⣠⠦⢄⣀⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠙⠒⠛⠉⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⠀⠀⠀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⡠⢴⠿⠋⠀⠀⠀⡈⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠖⠋⢰⠃⠀⠀⢀⡠⠚⠳⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⢷⣶⣤⣤⣤⣶⠿⠋⠁⠀⠀⠈⠛⢿⣶⣤⣤⣴⣶⡾⠿⠛⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠾⠿⠟⠛⠛⠉⠉⠉⠉⠛⣷⣦⣤⣤⣤⣤⣴⡿⠛⠉⠉⠉⠉⠛⠛⠻⠿⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⣤⣤⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡿⠛⠉⠉⠉⠉⠉⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡛⠛⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠻⠿⠿⠿⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⣤⣤⣤⣤⣶⣶⣶⣶⣶⡶⠶⠾⠿⢿⣿⣦⣄⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⣶⣦⣤⣀⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣤⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠿⢷⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣸⣿⣴⡶⠿⠿⠿⠿⢶⣶⣤⣤⣄⣀⣀⣀⣤⣤⣤⣶⣶⠾⠿⠿⠟⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣤⣤⣴⣶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⣴⡾
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⡀⠸⡉⢦⠀⠣⡀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠸⡀⣠⠔⠊⠀⠑⢄⡘⡄⠣⡀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡴⣿⣿⣶⢦⣽⣶⣞⣀⡀⠀⠐⠀⠈⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣽⣿⣿⣿⣇⣀⣸⣿⣿⣿⣿⣧⡀⠀⠀⣠⡟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⢻⠿⠿⣿⣿⣿⣿⣿⣷⢄⡼⠋⠀⠀⠀⠃⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠈⠉⢻⡿⣣⣟⣅⠀⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⢠⣶⣼⢫⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⡿⠃⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡴⢉⡿⣟⠀⠀⢻⣿⣿⣿⣆
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠞⣠⠎⣠⠏⠀⠈⠣⣤⠖⣻⣿⣿⣿⣧⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⡴⢁⡼⠁⠀⠀⠀⠀⢸⠊⠁⣨⡿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠎⣠⠏⠀⠀⠀⠀⠀⠀⠈⠑⠉⢢⠑⡜⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡜⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢎⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣠⠋⣰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠚⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉

⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⣶⣶⣶⡶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⠶⢶⣦⣤⣶⣿⣿⡿⣷⡶⠾⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⡶⢶⣦⣴⣶⣶⣶⣤⣄⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣠⣿⣉⣹⣧⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣙⣻⣿⣷⣤⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣏⡀⠻⡎⠙⠛⠛⠉⠁⠈⠙⢿⣶⡶⣶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠟⠋⠁⠀⠙⣷⡆⠹⡄⠀⠀⠀⠀⠀⠀⠀⢹⡆⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠸⣷⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⣤⣦⣾⢃⡾⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣿⣤⣤⣤⣤⣴⣶⢶⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⢼⣏⠛⠿⠷⠶⠶⠶⠶⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⣿⠉⣿⣄⣀⣀⣀⣀⣸⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡎⠛⠦⣍⣛⣛⣛⣉⡭⠞⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣤⣤⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⢰⡆⠀⠀⠀⠀⠀⢀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⡇⠀⢰⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⣏⠀⢸⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⣿⠀⢸⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⣸⠀⠀⣀⠀⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢹⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠺⠀⠀⡇⠀⣿⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⡇⠀⡀⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⢸⠇⡇⠀⠀⢸⣿⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⢸⡇⢸⠇⠃⠀⠀⠀⣿⠀⠛⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢸⡇⠸⠀⠀⣤⠀⠀⣿⠀⠀⠰⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⣲⣤⣄⡀⢸⡇⠀⠀⠀⡇⠀⠀⣿⠀⢠⡶⣶⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠰⠶⠶⠄⠀⢀⣾⡏⠉⠉⠁⠘⠃⠀⠀⠀⠁⠀⢀⡿⠀⠈⠁⠀⣶⣦⠀⠀⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠁⠀⣴⠒⠂⠀⢀⣤⠀⢤⣄⠉⡀⠈⠉⠓⠀⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠘⠛⠀⠀⠿⠀⠀⠀⠈⠀⠹⠦⠀⠀⠀⠈⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣴⣿⣇⠀⠀⢰⡦⠀⢸⣆⣀⣴⣿⣦⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠻⠾⠟⠁⠀⠀⠉⠉⠉⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠙⠿⠟⠙⡟⠻⣟⠻⢹⣻⠟⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠶⠶⠶⣤⡾⠛⠓⠶⠶⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣴⠀⠀⢀⡤⠔⠢⠤⠤⠴⠶⠶⢦⠀⠀⢠⡀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠉⠉⠉⠉⠙⠣⣄⣤⠞⠉⠉⠙⠛⠛⠋⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠱⢤⡄⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⢠⡟⠀⠀⠀⢀⣤⢤⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠈⣆
⠀⣾⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠸⡀
⢰⡇⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣷
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠀⠀⢹⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡇
⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣷
⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢼
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣀⠀⠀⠀⣿⡆⠀⠀⠀⣺
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸
⠀⠈⣷⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿
⠀⠀⠘⢧⣄⠀⠀⠀⠀⠙⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠁⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠻⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢀⣠⣀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢀⣠⠖⠋⠀⠈⠓⢦⣀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⢀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⢀⣠⠞⠋⠀⠀⠀⢠⠶⠶⠶⠶⠶⣦⠀⠀⠀⠈⠙⢦⣀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⣤⡞⠉⠀⠀⠀⠀⠀⠀⠈⠉⢻⠉⢻⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⣦⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⡤⣼⠛⠛⠛⣷⠤⣤⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠉⠀⠀⣸⠀⠀⠀⣿⡀⠀⠈⠻⣆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡏⠀⢀⡴⠋⠙⢦⣄⣤⠏⠉⠳⡄⠀⠘⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⡇⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡇⠀⠠⣞⣓⣒⣛⣶⠀⠀⠀⠀⠀⠀⠀⢰⣟⣓⣒⣻⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⡖⠀⡇⠀⠀⢸⡏⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠙⠛⠉⢸⡇⠀⢸⠀⠉⠋⠉⠙⠛⠛⠛⠁⢿⡀⠀⣽⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠹⣄⣀⣀⡀⢀⡼⠁⠀⣸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠙⠋⠁⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢹⠉⡟⠉⠀⠀⣰⠃⠀⠀⠀⠀⠀⣞⠁⠉⠳⢤⣠⠞⠛⠋⠉⠉⠉⠉⠀⠉⢳⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣼⣈⣧⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠙⠦⠤⠤⠶⠶⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢸⠀⠠⠀⠁⢠⢣⠀⢤⠀⠀⠠⠀⠀⠂⠀⡏⢸⢴⢠⠀⠦⠴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠐⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠈⠑⠂⠀⠀⠀⠘⠀⠂⠃⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠒⠐⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠤⠤⡀⠀⠀⡀⣠⠤⢴⠃⠀⠂⡶⢶⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣱⣀⣤⣿⣶⣶⣦⣤⠀⠘⠓⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠈⢲⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣠⣀⣄⡀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠉⢹⣿⣿⣿⣿⠟⠀⠀⠈⠑⠀⢀⢾⣿⣿⡇⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣀⣤⣿⣿⣿⣿⠁⠀⠀⠠⠀⠀⠀⠀⠄⢿⣿⣷⠀⠀⠑⢦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣀⠆⣴⣿⣿⡏⢈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡽⠘⢿⣿⡧⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠃⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⡎⠙⠁⠀⠀⣢⣤⣼⣿⣿⣿⢟⠁⠈⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡐⠀⠀⠀⠈⠙⣿⠛⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠐⠉⠲⡖⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⢀⣀⡀⠀⢀⡠⠤⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢤⣀⣀⣠⢤⢤⠈⠀⣀⣾⢈⣍⠁⣀⡀⠀⠀⠀⢈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠤⠍⠑⠓⠃⠛⠑⣜⣛⡀⠀⠁⠛⠓⠻⠭⠭⠏⡛⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐⠀⢀⣀⡛⣛⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⠁⠨⡉⠉⠍⠭⣛⡻⠇⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠄⠀⠉⠀⠀⠅⠀⡐⠄⠀⢱⠈⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⢀⣀⠀⠀⠀⠀⠀⠄⠀⠄⡂⠀⡾⠁⠂⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡤⢚⠉⠀⢀⢨⡀⠀⢀⠀⠀⠀⠀⠀⡇⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁⠀⡐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠎⠀⠀⢐⡨⠌⠈⠀⠀⠀⠀⡈⠐⠀⠀⣗⠪⠀⠁⠀⠀⠤⠤⠤⠀⠀⠀⠀⠰⠊⡀⠢⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡴⠁⠀⠀⠔⠁⠀⠀⢠⠀⠀⠠⠀⠢⠁⠀⠀⠧⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⢤⣤⣶⣶⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠄⠀⠠⡀⠀⠀⠀⠀⢆⠁⡂⡀⠠⠀⠐⡃⣤⢤⠠⠄⠤⠤⠤⠄⡤⢠⠠⢤⠤⠴⠂⢾⠀⡇⠰⠶⠶⠶⠦⢸
⠀⠀⠀⠀⢀⠀⠀⠈⡄⣰⡀⣀⢀⡔⢀⡘⡀⠀⠀⠀⠈⠐⠨⠀⠀⠠⠈⢐⠔⠕⡡⡪⡨⡪⢋⡪⠢⢂⢂⢀⡂⠠⡀⠈⠀⡇⣘⣛⣻⣿⠗⢘⠅
⠀⠀⠀⠀⠀⡀⠀⠀⡇⢻⡇⣿⠈⡇⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠂⠀⠁⠀⢄⠁⠰⢀⣹⣾⣭⡭⠁⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣅⢈⣁⣛⣀⣓⠘⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⣐⡲⠶⢒⣂⠼⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠕⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣊⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡫
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡟⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢆⡐⠠⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⡀

⠀⠀⢀⣶⣶⣄⠀⠀⠀⠀⠀⠀⣤⣾⠄⠀⠀⠀⠀⠠⣄⣢⠄⠀⠀⠀⠀⠀⣂⣀⣆⠀⠀⠀⠀⠀⢀⣲⣞⣀
⠀⠀⢸⣿⣨⣿⠀⠀⠀⠀⠀⣨⡿⠋⠀⠀⠀⠀⠀⢲⣿⣿⡷⠀⠀⠀⠀⠸⡁⠀⣹⠀⠀⠀⠀⠰⡟⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⠶⡄⠀⠀⠀⠀⢀⡀⢐⡖⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⣶⣶
⠀⠀⣝⣓⣚⡯⠄⠀⠀⠀⢠⠾⡡⠊⠀⠀⠀⠀⠀⠸⣧⣴⠇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⣭⡽
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠛⠃

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣤⣄
⠀⠈⠋⢹⣶⢖⠀⠀⠀⠀⠨⠿⠶⢺⠆⠀⠀⠀⠀⠡⣦⢀⣦⠀⠀⠀⠀⠀⠘⠀⣻⡇⠀⠀⠀⠀⣿⣿⡍⡇
⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠙⠁⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄
⠀⠀⢸⡇⣠⣤⠀⠀⠀⠀⠀⠟⠛⠻⠀⠀⠀⠀⠀⣶⣿⣿⣶⠀⠀⠀⠀⠉⠛⢤⣀⠀⠀⠀⠀⠀⠟⠛⠛⠇
⠀⠀⠸⠇⠿⠿⠀⠀⠀⠀⠀⠱⠶⠆⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⣿⡿⠂⠀⠀⠀⠀⠀⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⣖⣲⣦⠀⡀⠀⠀⠀⢠⣶⡶⣶⡦⠀⠀⠀⠠⣭⣤⣤⣤⠄⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⢸⣿⣳
⠀⠀⠫⠟⠛⠫⠃⠀⠀⠀⠐⠿⠿⠿⠋⠀⠀⠀⠀⠿⠛⠛⠿⠀⠀⠀⠀⢸⠿⠷⠈⠀⠀⠀⠀⠀⢸⣤⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠉⣿⠀⠀⠀⢀⣠⡶⠛⠉⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⢻⠋⢀⣠⠶⠛⠁⠀⢀⣠⣀⡀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⡟⠁⠈⢻⠄⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⢀⣤⣤⣀⣀⣠⡶⣦⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣀⡀⠀⠀⠸⣇⣠⣬⣭⣽⡇⣿⣭⣭⣤⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠀⣠⡼⠷⠿⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠶⠶⠶⣿⠉⠁⠀⠀⠀⠀⠉⢹⡷⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠻⠶⣤⣀⠀⢀⣠⡴⠾⠃⠈⠛⢷⡀⠀⠀⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⢀⣼⣧⣤⣤⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣴⠶⣴⠾⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠾⠃⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣆⢠⠀⢄⠠⡄⢀⢄⢀⡄⠀⢴⡂⢀⠀⡀⡀⣄⢀⢠⡀⢆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠎⠰⠽⠀⠇⠻⠌⠸⠀⠀⠠⠽⠸⠴⠆⡧⠞⢸⠴⠃⠇⢳⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠦⡄⠀⠀⢰⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢿⠀⠀⣼⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣤⣀⡼⠃⢀⡼⠖⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠰⢯⠀⠀⠀⠀⠀⡞⠛⠛⠛⠒⠒⠲⠶⠤⢤⣄⣀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⠀⠹⠤⠴⠦⠴⠾⣃⣠⣤⠤⢴⣇⠏⠉⠉⠙⠒⠒⠒⠤⠤⠤⡌⢹⠆⣀⣴⣋⣀⣀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⣈⣳⠀⠀⠀⣠⠖⠋⢁⣀⡤⠤⡾⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣄⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⣡⣤⣤⡾⠁⣠⠖⣋⡁⠀⣰⣣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣯⠘⣖⣒⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠉⢩⡇⡼⠁⠀⠙⠃⢀⡏⡼⠑⠒⠒⠲⢤⣀⣀⣀⡤⠴⡿⢼⠏⣧⢹⠉⠉⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣀⣈⣷⢧⣀⠀⠀⠀⣸⢱⢣⡖⣆⠀⠀⠀⠤⠥⠤⠄⢠⡇⣾⣠⣿⣿⣀⣠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠷⣼⠃⠈⠉⠓⠲⠷⠿⠤⣭⣥⣄⣀⣠⣤⣤⠤⠶⠚⠛⠉⠁⠸⡦⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢰⠛⣯⡟⣷⣦⣤⣀⣠⡄⠀⠀⠀⣶⣀⢀⣀⣤⣶⣞⣿⢸⠻⡄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢠⠏⠀⢻⣷⣿⣍⠙⡏⣧⠙⠿⢯⠿⠟⠹⢿⣿⡞⣁⣡⠟⡿⠀⢻⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠉⠛⠿⢿⣃⣻⣷⣶⣶⣶⡶⠿⢛⣘⣫⡭⠗⠋⠁⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⢠⣞⣁⣀⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣯⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣿⣀⣀⣀⣀⣀⣀⣀⣀⣳⡄
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠉⠉⠙⢳⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣟
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⣠⡴⣶⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣏⡿⠉⠀⠈⠉⣽⠷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⡴⠛⠁
⢰⠏⠀⡟⠀⠀⠀⠀⠉⠻⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢻⡇⢰⠃⢀⡾⠁⠀⠹⣦⣀⣀⣠⠀⠀⠀⢀⣼⠃
⢸⣄⣰⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⡾⢿⠀⠈⠀⠀⢀⣾⠁⠀⠀⠀⠀⠉⠉⠙⠳⠶⠖⠋⠁⠀⠀⠀⠀⠀⣿⠛⢦
⠀⠉⠉⠀⠀⠀⠀⢀⣀⠈⠻⢦⣄⣀⣤⡾⠛⠛⠋⠉⠉⠉⠀⢸⡇⠀⡔⠀⣼⠃⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⠀⠀⠀⢻⣄⣼
⠀⠀⠀⠀⠀⠀⢰⡟⣯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠁⢰⠇⠀⠀⠀⠀⢹⡉⢳⡄⠀⠀⠀⠀⣿⠉⠻⣆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⣠⠇⠀⠀⠀⠀⠙⠷⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠔⢸⠇⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡟⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⣼⣧⣤⣼⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⣀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⡀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⡤⠤⠔⠒⠒⢢⠀⠀⠀⠀⡔⠒⠒⠢⠤⢤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠑⠂⣶⠒⠒⠖⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠛⠤⠤⠒⢖⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠔⠁⠀⠀⠀⠀⠈⠑⠢⠄⣀⣀⣠⠄⢤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢘⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠴⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡰⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢼⡀⣰⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠉⠑⠒⠒⠂⠐⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⠤⡤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⢼⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢆⠀⠀⡔⠐⡄⠀⡸⠀⠀⢰⠀⠈⠉⠁⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢦⡤⡗⠒⠁⠀⡇⠀⠀⢸⠀⠀⠀⡰⢋⠷⠴⠃
⠀⠀⠀⠀⠀⠀⡠⡀⢰⠢⣱⡀⠀⢢⠜⠉⢲⣇⠔⠁⠈⠳⡀⡰⢁⠏⡤⠒⠠⡀
⠀⠀⠀⠀⠀⠈⠢⠊⠈⢣⡈⠓⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠳⠁⠮⠊⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⢤⣀⡀⠀⠀⣀⣀⡤⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠛⠒⠦⢤⣄⣀⣀⡀⣰⠃⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣤⠤⠖⠛⠉⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢰⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⢀⣀⣤⠴⠖⠚⠉⠉⠉⢻⡀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠙⠓⠲⠤⣤⣀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⠓⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠶⠶⠶⠶⠶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢾⢿⠥⡤⡤⣤⣤⣤⣤⣤⣽⡶⡦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠚⠒⠶⠶⠶⠶⠶⠷⠾⠶⠶⠷⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⡶⠒⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠏⠀⠀⠀⠈⢳⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⣿⠀⠀⠀⠀⢠⡏⠀⢠⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠒⠲⠶⠤⢤⣄⣀⡀
⢸⡇⠀⠀⠀⠀⣼⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠸⡇⠀⠀⠀⠀⢿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⣿⠀⠀⠀⠀⢸⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⢹⡄⠀⠀⠀⠀⢧⣀⣀⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢻⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⠤⣤⣄⣀⣀⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠓⠦⣤⣀⣀⣀⣀⣀⣀⣤⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⣤⣤⣤⡴⠶⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣠⠄⢠⡄⢀⡤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⣿⠀⢼⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠻⡄⠙⣦⠘⢧⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡟⠀⡿⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋⠀⠘⠁⠘⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣝⠻⠿⠟⣩⣭⣭⣙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣘⠿⠿⢋⣼⡏⣴⣮⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣛⠻⣷⣬⣥⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡘⠿⢃⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⢀⣀⣀⡀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠸⠿⠿⠄⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⣠⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⣿⠙⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⣿⣇⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠶⠶⠶⠶⠦⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇
⠀⠘⢿⣷⣦⣽⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⠟
⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⡆⡶⢰⢶⠐⢲⠖⢰⣖⠂⣶⢲⠄⢠⣖⠂⢰⠀⣶⢰⡖⣦⢰⠒⡆⣶⠀⢶⣰⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣹⣏⣿⣃⣿⣚⣇⣸⣀⣸⣭⣄⣟⣹⣄⣀⣬⣗⣸⣤⣟⣸⣏⣁⣸⣋⣁⣿⣤⣈⣏⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⡈⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡥⠻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⢀⠀⡑⢈⡐⠉⢂
⠀⠀⠀⠀⠀⠀⡄⢶⠎⡄⢉⠁⠀⣈⡤⠅⠀⠈⠰⠄⡄
⠀⠀⠀⠀⠀⠀⠈⡡⠀⠇⡐⠓⠞⢉⠄⠀⢦⢄⠄⠄⠠⠄
⠀⠀⠀⢠⠒⢢⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠙⠉⠓⢆⡀⡄⠀⠀⠀⠀⢀⡀⠀⠤⠤⠠⠤⠤⠤⣀
⠀⠀⠐⠁⢠⡮⠠⠚⡠⡂⠀⠀⢧⠀⠀⠀⠀⠀⠀⡂⠠⣇⠜⠀⠀⢀⢼⣡⡦⣤⠴⠤⡦⢤⣠⢤⣼⣦⡀
⠀⠀⠈⠫⠜⢀⠒⢀⠐⠄⠀⣤⠿⠀⠀⠀⠀⠀⢈⢮⠀⡸⠀⠀⢀⠂⠏⠀⠀⠀⢀⠀⡀⠀⠀⠐⠀⢁⢳
⠀⠀⠀⠀⠀⠜⠀⢀⢏⠀⠼⠃⠀⠀⠀⠀⠀⠠⡀⠩⡥⠁⠀⠀⢠⢸⠀⠀⠀⠀⢣⣲⠞⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠣⢀⠉⢘⡁⠀⠀⠀⠀⠄⠠⢀⠀⠈⠢⡐⠄⠀⠀⠀⢄⢇⣠⠐⠊⠉⠚⠛⠒⡤⠄⣀⡱⠁
⠀⠀⠀⠀⠀⠡⠠⠑⠄⠝⡆⠠⠒⡀⠉⠉⣰⡷⢢⣢⠮⢊⣷⡀⢀⢴⠏⠀⠀⠀⠈⠉⠉⠀⠀⠀⠢⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠙⠒⠉⠀⠈⠈⠁⠈⠀⠉⠁⠈⠛⠿⣧⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⣴⣶⣼⣶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⢛⠺⠁⠀⠀⠀⢀⣀⢀⠀⣀⣛⢀⣉⣛⣛⣋⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠁⠈⠈⠉⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠂⡀⠀⠄⠐⠐⡈⠀⠀⠀⠀⠀⠀⠀⠘⡘⠄⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠢⢄⣀⢂⠀⠌⠊⠌⢄⢀⢄⠀⢂⡢⠡⡀⡸⣡⣷⠚⠁⠀⠀⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣂⣠⣀⠨⠉⠀⠂⠒⠲⠲⠖⠖⠒⢲⢒⣫⣽⠋⠀⠐⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠒⠛⠚⠛⡉⠉⠁⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⢷⣷⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠆⠀⠀⣿⠀⠀⠐⢶⠒
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣉⣀⣀⣀⣈⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣀⣀⣀⣀⣀⣀⣛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⠛⠛⡘⠛⡛⠛⠛⠛⠛⠛⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⠀⠀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣵⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⢱⣆⠀⠀⢀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣼⣿⣿⠛⠉⠉⠻⣿⣿⣇⠀⠀⠀⠀⢸⡌⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣶⠈⠉⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⢻⣿⣧⠹⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣷⣦⣤⣶⣿⣿⡟⢠⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠀⢠⣶⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⠶⡾⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢀⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⡀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢀⣿⠁⢀⡴⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣦⡀⠈⣿
⠀⠀⠀⠀⠀⢸⡇⢀⣾⠁⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⣷⡀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠸⣇⣸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠻⠷⣶⣶⣴⡶⠿⠋⠀⠀⠀⠈⠙⠿⣶⣶⣶⡶⠾⠟⠛⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀⣠⣤⣤⣴⣶⣦⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣤⣄⣀⣀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠀⠀⠈⠻⣿⠿⠿⠿⢿⣿⠋⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠿⠿⠿⠿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣽⣿⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣤⣤⣴⣶⣶⣶⡶⠶⠶⠾⠿⠿⢿⣿⣦⣄⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠶⣶⣦⣤⣀⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣦⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⣿⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠷⠶⠾⡿⠛⠉⠁⠀⠈⠉⠙⠛⠛⠿⠷⠾⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣀⣀⣤⣤⣴⡶⠿