Cultivar Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⠟⠿⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠙⢷⠟⢉⣤⡾⠦⣈⣉⣉⣉⣉⣡⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡴⠋⢀⣰⠏⢹⣆⠰⠋⠉⠉⠙⠉⠉⠉⢀⣈⣙⡛⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⠃⠀⣿⡇⠀⠀⣀⡴⠶⠶⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠰⠖⠂⣄⡀⠀⠀⠀⠀⠙⢧⣴⡇⠀⠀⢸⣦⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠶⠂⠉⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢠⣤⠘⠋⠀⢀⣠⣤⣤⣤⣤⣤⣄⢠⣤⢠⣤⠀⠀⠀⠀⢀⣄
⠀⣠⣤⡴⠛⢠⣤⡄⠒⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠚⠛⠒⠒⠀⠀⠀⠀⠠⢤⣤⣤⣤
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠈⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠿⠁⠈⠳⣄⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠚⠛⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⠀⠀⠀⠀⢸⣿⣀⡾⠀⠀⠀⠀⠈⠛⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠘⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠴⣄⠀⠀⢠⡀⠀⠀⠀⢿⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣼⡇⠀⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣆⠀⠀⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡌⠉⠉⠳⣄⠀⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢢⣤⠀⠀⠀⢠⡎⠁⠀⠀⠀⠈⠻⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠶⠀⣀⡀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣶⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⣀⠁⠲⢦⡀⠀⠀⢀⡘⠀⠀⠛⠀⠀⣀⠰⠖⠂⣀⡹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⠻⣆⡀⠸⠃⠀⢶⣀⠰⠂⠁⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡆⠀⢀⣿⣿⡶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣡⡄⠀⢀⡘⠛⠓⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⡀⠳⣴⣿⠃⢠⠀⢸⣿⣶⡾⠋⣠⣤⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠃⣠⠟⢀⣾⠏⠀⢸⡆⠘⡁⠙⢷⣆⠉⢷⣾⠿⣦⣤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣅⡘⠟⢀⣼⠋⣀⣶⣭⡁⠀⡁⠀⠀⢿⣇⠈⣿⡀⠈⠙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠃⠀⠈⣁⡾⠻⠈⢹⡇⢰⡇⠀⠀⠀⠙⠀⢻⣷⠀⠀⠹⣿⡻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣠⣿⠃⠀⠀⠀⣿⡇⠀⠀⠘⠁⢀⡵⣦⣄⠀⠀⠀⠀⠻⣦⠀⠀⢸⣷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣴⡿⠁⠀⠀⢠⣾⢿⣧⠀⢰⡆⢠⡿⣷⣞⢿⣷⡀⠀⠀⠀⠉⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠐⠛⣩⣿⠃⠀⠸⠃⠈⠷⣿⡿⣿⡉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡾⠋⠀⠀⢰⡆⢰⡆⢰⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠁⡄⠀⠸⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡇⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⣇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⢀⣀⣀⣀⡀⢀⣀⢀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠛⠛⠛⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣏⠀⢀⡤⠀⢸⣿⠁⣼⡧⠙⠁⢿⡇⠉⠉⠀⠈⡟⢹⣷⠀⣼⠋⠁⠀⢹⠏⢠⡿⠃⠛⠁⠈⠁⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣀⣠⣶⣿⣿⣶⣿⣷⣿⣶⣾⣷⣾⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣾⣿⣿⣿⣗⣿⣯⣟⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⣤⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣀⣠⣤⣶⣶⣤⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣉⣙⣻⠿⣿⣿⣿⣿⣿⣿⡿⠿⣋⣁⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡻⢶⣿⣿⣿⣿⣿⡷⢋⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⡇⠹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⡿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠻⠿⣿⠿⠿⡟
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠉⠉⠉⠉⣭⡍⢉⣭⣉⣭⣧⡽⣿⠛⠿⣯⣥⢭⣉⡉⣉⣉⡉⠉⢉⣽⡯⠉⣿⠉⠉⠉⢉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠈⠛⠁⠀⠀⠻⠈⠁⠀⠠⠜⠛⡩⠰⠟⠃⣠⡴⣇⣀⣈⠛⠉⠓⢠⡉⠛⠿⠥⠀⣀⠀⠀⠀⠙⠗⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠀⠀⠀⣶⠶⠀⠀⠀⠀⢰⣶⠄⢊⡽⠃⠈⣿⠻⣶⡄⠠⣤⣀⡀⡀⠀⠀⠿⠇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠠⠀⠉⢀⠀⢊⡟⠀⠀⠈⠐⠋⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⢀⣀⡀⠘⠇⠀⢀⡀⠀⠀⠀⠂⢾⣿⠀⣤⠖⣿⠿⠗⠀⠀⠀⠀⢰⡆⠀⠀⠀⠶⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠛⠋⠀⠀⢠⣶⠈⠀⠀⡄⠀⢹⣄⠠⠤⢠⣤⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠼⠀⠀⠸⠃⠀⣀⡏⠱⠆⠼⠿⢆⡀⠀⠀⢿⡇⠀⠀⠀⠶⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠛⠁⠘⠃⠀⠀⢰⡾⠇⠀⠀⡞⢻⠋⢠⡄⠀⠀⠈⠃⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠰⣾⠇⠘⡄⠉⠁⠀⠸⠿⠂⠀⢀⣀⡀⠈⠁⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠘⠛⠀⠠⡦⠀⠀⠀⡰⠁⠐⠷⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠀⠀⠀⠀⢨⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄
⠀⢸⢿⣦⡀⠀⠀⠀⢸⠘⢲⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠘⢲⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⢢⠙⢿⣦⠀⠀⠸⠄⢿⡄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠈⠁⠇
⠀⠀⠀⠱⣤⣙⡇⠀⢸⠁⣟⡇⠀⠀⠀⠀⠀⠀⠀⠇⠂⠀⠄⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⣄⡀⠘⠗⠀⠀⡀
⠀⠀⠀⠀⠀⠸⡇⠀⢸⢰⡦⡇⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠘⠢⠗⠂⢠
⠀⠀⠀⠀⠀⠀⠇⠀⢸⠈⡷⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣧⣿⠘⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠳⠀⠀⡀⡘
⠀⠀⠀⢀⡴⠂⠠⠠⠰⣰⠋⠙⠀⠸⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡼⢠⡊⢀⢠⣄⣀⡉⠂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠇⠀⠀⠀⠑⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣿⡏⠀⢠⡤⣶⣋⣷⣰⣶⣄⠀⠀⠀⠀⠀⡄⢀⢀⠀⠀⠀⠀⠀⡰⣗⠠⢤⣄⠀⠀⠀⣤⣶⣶⡾⣴⣶⣿⡶⣾⢻
⠀⠀⠀⠀⠀⠀⠀⣿⡘⢿⣿⣿⣄⢀⣩⣝⣿⣿⡏⣉⣁⣀⠀⠀⠀⠀⠀⠂⢀⠑⠀⢤⡀⠈⢰⣿⢛⣶⣮⣽⡄⢈⡶⢮⣽⣯⣿⠷⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣸⠿⣒⣛⣻⣿⣾⣭⣭⡅⠐⣙⣻⣿⣿⣿⣷⡀⢼⣦⠀⡀⠁⠀⠀⠹⠲⣭⣹⣤⣿⣿⢿⣷⣿⡧⢾⡿⠞⠉⠁⠁⠀⠡
⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠋⠉⠀⠀⠀⠳⡀⠀⢼⣿⣿⣿⣿⣿⠌⣣⣞⠋⠁⠀⠀⠰⢿⣷⡭⣿⣽⡿⣿⣿⣽⣿⠻⠭⠀⠀⡤⠖⠉
⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⠁⠀⠀⠠⠀⠰⠀⢉⣢⣼⣿⣿⣿⣿⣿⡗⠉⠉⠛⠦⣄⡀⠈⢆⣩⢦⣼⣷⡇⣼⣮⠿⠥⡄⠀⠄⢀⠁
⠀⠀⠀⠀⠀⠀⢹⠟⠿⠟⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⠋⡽⢯⡴⣲⠤⠀⡈⢙⢦⡈⠐⢲⢋⣿⣿⣹⣿⣨⣄⡒⣈⢠⣈⡀⢀⢉⢥⡀
⠀⠀⣀⣄⣀⠀⣖⣀⠀⠀⠀⠀⢀⠀⠀⣸⢿⣿⣿⣿⡟⠻⠿⠀⣾⣾⢳⣿⠿⠁⠰⡼⣆⡙⢦⡀⠙⠟⢿⣿⡿⠛⠍⠉⠀⠀⠀⠀⠀⡴⠋
⠀⠻⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠈⠙⡀⢸⣿⣿⠇⡼⠋⠅⠁⠶⡷⢿⠇⣶⠂⠀⡆⠚⠛⠿⣶⣿⣦⡠⢸⡟⢻⡄⠠⠈⠀⠐⠀⢠⠎
⠀⠀⢀⣈⣋⣽⣿⣿⣶⡤⠒⠀⠀⠀⠀⣼⣟⣀⣀⠀⠠⠔⠂⠀⠑⠬⡀⠃⢀⠼⡟⣶⣶⡦⣤⠽⣿⣿⣼⣇⠀⠁⠁⠀⠒⠐⠈⠁
⠀⠠⡿⠽⠏⠀⢸⣿⡯⡉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⢻⠿⠋⠉⠉⠉⠙⠛⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⣢⣼⡇⠄⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣴⣈
⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠂⠠⢄⢠⣴⡿⠊⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠎⠀⢰⠀⢄⡀⠀⠀⠀⢿⡦⢼⣿⣿⣾⣇⡐
⣄⣀⣀⣩⣽⡢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢚⣷⣾⣿⣿⣍⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣄⠀⠀⠈⠀⠚⣻⣿⣿⣿⡥⠤
⠀⠈⢿⣿⣿⢽⡶⣝⡃⠀⠀⢁⣠⡴⠖⠂⠀⢀⡴⡟⣹⠟⣿⣿⣿⠒⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢿⣷⢼⡿⠉⢻⠌⠁
⠀⠀⠈⢿⣿⢰⡸⠹⢷⣀⣶⣿⠏⠀⠀⠀⣴⠋⠀⠿⢋⣫⣿⣿⣿⠀⠈⠛⢃⣠⣖⣆⣅⣬⣡⣈⢢⣄⡀⠑⠢⣘⣿⡟⣡⣤⣶⣀⣄⡀
⠀⠀⠀⠈⢿⣯⣿⠀⣠⣿⡿⠁⠀⠀⢠⡞⠁⠀⠀⠃⠈⣹⣿⣿⡟⠀⣠⣾⣻⣿⣻⣿⣎⣽⣷⣿⣿⣿⣷⡄⠀⡘⣻⣿⠃⠀⠉⠛⠉⠁
⠀⠀⠀⠀⠸⠿⣷⣠⣾⢟⡃⠀⠀⣰⡟⠀⠀⠈⠀⡄⢀⣿⣿⣿⠁⠀⠈⢇⢪⡍⣋⣭⡿⢟⢻⣿⣿⣿⣿⣿⡞⠑⣉⣩⢌⠉⢙⣢
⠀⠀⠀⠀⠀⢸⡇⢸⣿⡀⠀⠀⢠⡟⠁⠀⠐⠂⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠉⠫⠤⠶⣘⣿⣧⣿⡿⢹⠿⡙⣿⣿⡴⠶⣶⢿⣽⠻⠆
⠀⠀⠀⠀⠀⢘⣿⣿⡟⠁⠀⢠⡏⠡⠀⣰⠁⠀⠀⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠐⢀⣰⣿⣷⣿⣋⣿⡼⡞⢧⡀⢸⠹⣶⣶⣄⣌⣠⢀
⠀⠀⠀⠀⠀⣼⠇⣼⣷⠀⠀⣿⠀⠀⣴⠅⠀⣴⢰⣿⣿⣿⠁⠀⠀⠀⠀⡠⠉⠁⣠⣴⣫⣿⠣⣏⠚⢿⣪⣷⣿⣸⣮⣼⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⣿⠄⣽⣿⡀⢰⡇⠀⣰⠃⠀⠰⣿⣿⣿⡟⠀⠀⠀⠀⠀⢰⠱⠛⣭⣥⣴⡯⡏⠙⣟⣡⣼⣭⣿⠁⠀⠀⢻⣿⢎⣷⣶⡄⠁
⠀⠀⠀⠀⠀⣿⣧⣿⣿⠁⣼⠀⠙⠏⢀⣚⣽⣿⣿⡿⠃⠀⠀⠀⠀⠀⢮⠀⣟⢩⡿⢻⠀⢷⠀⠿⢇⢘⣭⠟⠀⠀⠀⢸⣿⡀⠘⠛
⠀⠀⠀⠀⠀⠹⣿⣿⠏⠀⢺⣴⣻⢢⣤⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⡇⠷⠜⢁⡴⠖⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢸⣿⣟⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣷⣦⡀⠀⠀⣀⣀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡃⠠⣿⣿⡿⠃⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⢀⣷⣄⠀⠀⣶⣿⡇⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣑⣾⣉⡍⠀⢠⣄⡀⠀⠘⣿⣇⣼⠀⠀⠀⠈⣿⣿⢀⡀⢿⡟⢠⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠁⠙⣦⣄⣸⠿⠋⠀⠀⢹⠟⣫⡄⠀⠀⠀⠈⣇⢘⡷⢼⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠉⠀⠀⠀⠀⠈⠛⣇⡀⠀⠀⠀⢸⠛⠟⡀⢀⡰⣀⣤⡟⠁⠀⠘⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⡿⢇⠀⠀⠀⠀⣠⡟⠁⣹⡗⠋⠀⢿⣿⣆⠀⠀⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁⠘⣆⠀⠀⠀⡟⢁⡴⠛⠿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠀⠀⠀⠈⢆⠀⠀⣷⠞⠀⠀⠀⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠈⢧⣀⡏⠀⠀⠀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⡟⠒⠶⢦⡀⠀⠀⢸⡇⠀⠀⣾⣿⣟⣿⣯⣿⣿⡿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣬⣧⣟⠀⠀⠙⣦⠀⢸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⣰⣿⣿⣿⣷⠀⠀⠈⢳⣾⠃⠀⢸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⡤⢼⡧⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⢦⣿⠀⣰⠞⠛⠁⠈⣧⣾⡀⠀⢀⣨⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣀⡿⠀⠀⢙⣶⣿⣿⡁⠀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠇⠀⠀⣿⣿⣿⣿⣿⡆⠹⠿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠶⠶⠶⢶⡿⠂⠀⠘⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡟⠉⠁
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣿⣇⣀⣀⣀⣀⣸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠻⠿⣿⣿⠿⠿⡿⠻⡿⢿⠿⢻⣿⠿⠛⠛⠛⠛⢛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠺⠟⠀⠀⢴⠄⠀⠀⠀⠘⠃⠀⠀⣿⡆⠀⣾⠱⡶⢴⠿⠃⣸⡏⠱⡾⠇⠀⠀⠙⠋⠀⠀⣤⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢠⣤⠀⠀⠀⠀⢠⣤⣿⠀⠘⠀⢠⡇⢀⡄⠀⣿⠁⠀⠀⠀⠀⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⣠⠀⠈⠋⢁⠴⠐⠛⢹⡀⢈⠁⠀⠟⠃⠀⢲⡶⠀⠈⠀⠀⠀⠀⠀⠠⠾⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠶⠀⠀⡀⠀⠉⠀⠀⣴⡿⠀⠐⠁⣰⣿⠧⠀⠀⠀⠀⣄⠀⠀⠀⢰⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠺⠟⠀⠀⢠⣤⠀⠀⠀⠀⠔⠙⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣠⠁⠀⠀⠿⠇⣀⠀⠀⢸⣿⠀⠀⠀⢰⠄⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠁⠘⠃⠀⠀⢠⡾⠆⠀⠀⠈⠟⠃⣠⠀⠀⠀⠙⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣀⠀⠀⢀⣀⡀⠀⠀⠰⣶⠆⠀⠀⠈⠀⠀⠿⠇⠀⠀⣀⣀⠈⠁⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠘⠋⠀⠀⠶⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⣶⡖⠚⠛⠉⠉⠓⣲⠶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠙⢦⡀⢀⡤⠞⠣⢄⡀⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⡆⠀⣠⠀⠀⠀⢀⡀⠀⠙⢿⠀⠀⠀⠀⡈⠳⡀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡰⠀⠀⡇⢠⠃⠀⠀⢠⠎⠀⠀⠀⢸⠂⠀⡀⠀⡆⠀⠘⠢⡀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⢤⡀
⠀⢀⣀⣀⣀⠀⢠⠃⠀⠀⡇⠀⠀⢸⡏⠀⠀⡰⠋⠀⠀⠀⢀⡏⠀⠀⢠⢰⠃⠀⠀⠀⣱⣀⣀⣀⣀⠀⢸⣀⠴⠋⠀⠀⠸
⢠⠞⠁⠀⠈⠉⠻⣄⠀⠀⣇⠀⠀⠘⡇⢀⡎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠃⠀⣠⠖⠋⠉⠀⠀⠀⠈⠉⢿⠁⠀⠀⡠
⡏⠀⠀⠀⠀⠀⠀⠈⠱⡀⠸⡀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡆⣠⠞⠁⠀⠀⠀⡀⠀⠀⠀⠀⢈⡄⠀⡰⠁⡸⢠
⡇⢸⡀⠀⠀⡆⠀⠈⢆⠙⢆⡇⠀⠀⠘⣾⠀⠀⠀⡠⡜⠁⢸⠀⠀⠀⡴⠁⠀⠀⠀⠀⡜⠀⠎⠀⠀⡀⢸⠃⢀⠇⡴⠁⡞
⣇⠀⠱⡀⠀⢸⠀⠀⠘⡆⠈⢻⣀⠀⠀⠸⡀⡠⠊⠘⣇⠀⠀⢃⠀⣸⠁⠀⠀⡔⠀⢸⠀⠸⠀⠀⡴⠀⡞⠀⣼⠞⠀⣸⠁
⢹⡀⠀⠈⠲⢼⡄⠀⠀⣧⠀⠀⢯⠱⡄⠀⡟⠀⠀⠀⢸⡄⠀⠘⡆⡏⠀⢀⠞⠀⠀⠘⡀⣇⡤⠊⠀⡼⠡⣰⠁⠀⣰⠃
⠀⢧⠀⠀⠀⠀⠙⢆⠀⢸⠀⠀⠈⠳⡘⡄⠱⠀⠀⠀⢸⡇⠀⠀⢿⠃⠀⡏⠀⠀⠀⠀⡿⠉⠀⠀⡜⠁⢀⡇⠀⡴⠃
⠀⠈⠳⣄⠀⠀⠀⠈⠳⣼⠀⠀⢀⠀⠙⠺⣦⠱⡀⠀⠀⠳⡄⠀⡞⠀⠀⡇⠀⠀⢀⠞⠀⠀⢀⡞⠀⢀⢞⣠⠞
⠀⠀⠀⠈⠳⣄⠀⠀⠤⣈⣓⣄⣞⠀⠀⠀⠙⣷⡑⡄⠀⠀⢹⣼⠁⠀⠀⣁⣀⣴⣡⠤⠔⢂⡞⢠⢔⠵⠋
⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⠀⠈⠉⠒⠦⢄⣸⡳⢼⣄⠀⣸⠃⠀⢀⡔⠋⠁⠀⠀⠀⢀⣼⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⢤⣄⣀⣈⣙⣲⣽⣷⣧⠤⢴⣥⣤⣤⠤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢻⡖⠒⢂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠭⠅⢀⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣯⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠒⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⣼⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠆⠀⠜⠃⠀⢀⠄⠊⠡⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⣷⠁⠀⠀⠀⠀⠀⢀⣄⡠⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⠙⠛⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠀⠀⠀⢁⣀⣴⣶⣶⣶⣐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣅⡤⠠⠋⠁⠉⠀⠀⠉⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡌⠁⠀⠀⣀⡴⠚⠀⠐⠒⠀⠀⠀⣀⣴⡖⠂⠀⠀⢠⡄⠀⠀⡀⠁⠈⣦
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣐⢧⡃⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠋
⠀⠀⠀⠀⠀⢘⡇⠀⠀⠴⠁⠀⣼⡿⠆⠀⠀⠀⠀⢘⠉⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡴
⠀⠀⠀⠀⢀⡜⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠈⠅⠀⠀⣀⣀⠀⠀⣠⠀⠀⠀⠀⠀⡄⢀⣀⣤⠈⠻⣏⡀
⠀⠀⠀⠀⠋⣰⣾⣇⠄⠀⠀⠘⠁⠀⣾⠖⠀⣼⡿⡆⠀⢀⡴⣾⢸⠔⠂⠿⠚⠋⠀⠀⠼⠻⠛⠋⠉⠀⠀⠀⠹⢶⡀
⠀⠀⠀⠀⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣿⠗⠀⡠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡀⠀⠛⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠂⣈⡄⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠏⣠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣬⣩⠤⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣈⣽⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠐⠐⠂⠈⠑⠚⢿⣿⣿⣯⣙⡁⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢚⣿⣷⣌⠙⡛⠿⠿⠃⠀⠀⠀⠀⠀⠐⠀⠈⠀⢠⣿⣿⣿⣿⣿⣿⡀⠀⠸⠀⠀⠀⠀⠀⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣷⣌⣙⣛⠙⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣮⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⣉⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠈⠻⠉⣻⣿⣿⣿⣧⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣷⢄⠐⠌⠻⠿⣿⣯⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛⠻⠿⣿⡾⠷⠦⠬⠻⢾⣽⣿⣁⣤⡀⢠⢠⡦⣀⣐⡋⡠⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠋⠉⠉⠈⠩⠉⠀⠀⠀⠀⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⢀⠸⢄⡰⡀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠠⢀⢰⠀⡸⠀⠀⠈⠀⡤⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠹⠂⠀⡀⠄⢠⠟⠑⠚⠁⠀⣾⣁⡤
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠤⠼⠣⣌⠃⠀⠘⢻⡆⠀⠀⢳⠏⢀⠈⠀⠀⠀⣨⣀⡠⢤⠤⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠤⣾⣧⣀⠀⢀⢀⠀⢀⢙⣦⣀⠀⢋⠀⣠⡟⠐⠀⠀⢁⣴⣻⠧⠄⠀⠀⠀⠀⢀⣬⡀
⠀⠀⠀⢀⣊⣤⠭⠙⠛⠒⠒⠒⠛⠓⠚⠷⠿⣿⣼⣾⡿⣠⣦⣢⡾⠻⠶⢿⡄⠀⢀⣤⣴⣂⠉⣱⠋⠀⠀⠀⠀⠀⢻⣦⡄
⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣾⣯⡾⣶⠀⠈⠄⠀⢍⠀⢀⣠⠤⣷⡄⠠⣀⣨⣻⣿⠿⠋
⠀⡴⣿⠤⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣯⣙⢟⡃⠘⣤⣀⡴⣻⢿⣧⣨⠟⠋⠁
⣸⠗⠁⠀⠀⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠾⠿⠿⠿⢿⣟⡁
⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣉⠑⠢⣀
⢿⡷⡀⠐⠠⣀⠀⠀⣠⡖⠀⢀⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠀⠀⠁⣢⣈⣷⣄
⠈⢷⣿⢄⠀⠀⠙⣶⡿⠧⠀⣼⠄⣎⠀⢀⡤⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠶⣶⣿⡟⠃⢀
⠀⠈⠻⢶⣾⣦⣼⣿⣇⠀⠀⠸⣞⣿⢿⣷⣶⡒⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡏⠁⠀⠠⣤⣙⡼
⠀⠀⠀⠘⡟⠋⠁⠀⢙⣢⣤⣀⣘⣿⣏⡏⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⣿⣇⠐⠾⣿⣿⣿⡷
⠀⠀⠀⠀⠈⠢⣀⠲⣦⣿⠋⠉⠁⢈⣉⣙⠣⣄⣙⡿⣿⣥⣤⣀⣄⡀⣤⠀⠀⠀⠀⠀⡀⠀⢀⣤⠤⣶⣻⣿⡿⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢩⣿⠿⢤⡿⠿⢿⣭⣭⣽⣿⣉⣼⣿⣏⣩⣠⡽⠟⠋⠙⠛⠒⠲⠶⠾⠶⠶⠶⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⢧⣀⢀⣈⠉⠉⠉⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣠⠔⠋⠉⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡆⠀⣴⣶⣶⣾⣷⠀⠀⣠⣤⣶⣾⣿⣿⣷⡀⠀⠀⣠⣶⣶⣶⣄⠀⢠⣶⣶⣦⣀
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢻⣿⣿⣿⠟⠀⠈⠿⣿⣿⣿⡧
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⢻⠟⠉⠉⠀⠈⠻⠿⠿⠿⠿⢿⣷⡀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⢹⡟⠛⠃
⠀⠀⠀⠀⠈⢿⣿⠿⢿⣿⠟⠁⢀⣀⣠⣄⣀⠀⣠⣶⣶⣶⣤⡀⠀⢀⣶⣤⣄⡀⠀⢀⣤⣶⣶⣶⣦⡀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣿⣿⣿⣿⡇⠻⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣶⣶⣤⣄⣀⠀⠈⠛⠿⢿⡿⠋⠀⠀⠀⠈⠉⠛⣷⠀⠉⠻⡿⠿⠃⠀⠻⣿⣿⣿⣿⠿⠋⠈⠛⢿⠿⠛⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠆⠀⢀⣩⣄⡀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠁⣀⣤⣤⣀⠀⠽⠟⠁⢠⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠻⣿⡿⣿⡟⢀⣾⣿⣿⣿⡗⢠⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⣿⣧⠀⠀⢠⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⡿⠿⠿⠀⠀⠻⠿⣿⣿⡿⠟⠃⠀⠘⢻⣿⡿⠿⠃⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠁⣤⣶⣶⣦⣤⣤⣄⠁⠀⠀⠈⠙⠛
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣧⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠙⠛⢿⡏⠁⠀⠀⠈⢻⣿⠿⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠿⢿⣿⣿⣿⣿⡿⠀⢻⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⢿⠉⠁⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⣆⢤⡀⠀⢀⠞⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠶⢤⣄⡐⠿⠷⣤⡏
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⣀⣠⡤⠶⠞⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣥⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣻⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠲⠦⢤⣤⣾⡇⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣶⣤⣀⠀⠀⠀⠀⢰⣿⠁⠸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⣿⠁⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢸⣿⠀⠀⠘⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⠷⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠈⠙⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⡄⠀⠀⣀⠀⠀⠤⠀⢶⣎⣉⠽⢛⡶⠛⠛⣩⡟⠀⣿⡿⠻⣅⠂⢶⣶⡆⠀⠀⠾⠟⠀⠀⣀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁⠀⠀⠀⠉⢀⡀⠀⠀⠀⠁⠀⠈⠈⠁⠀⢺⠏⠹⣶⣦⣄⡀⠁⠘⠈⠁⠉⠁⣴⡄⠀⠀⠀⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠗⠀⠀⠀⠻⠋⠀⠀⠀⠀⢸⠿⠃⡀⠴⠋⢀⠠⡿⠿⠉⠣⣶⡈⠃⣤⡄⠀⠛⠃⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠿⠀⠀⣠⣌⠀⠀⣈⣡⣾⡧⡄⠀⠀⠀⠀⠠⠛⠁⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣤⡄⠈⠁⠀⣠⣤⠀⠀⠀⠀⢛⠛⠋⠀⡉⠟⢻⡁⢀⡀⠀⠀⠸⠆⠀⠀⠀⠙⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠘⠿⢫⠀⢀⡀⠇⠀⠈⡟⠪⣴⣷⡀⡀⠀⠀⣤⣄⠀⠀⠀⢀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⢠⣤⡄⢠⡀⠀⠀⠀⢀⡀⠈⠃⠀⣴⣾⠀⠀⠘⠋⢡⡄⠀⠀⠻⠋⠀⠀⠀⠙⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠐⠟⠃⠀⢀⡲⠉⢹⡰⠖⠀⢀⣀⡀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣤⡄⠀⠀⢠⣤⠀⠀⠀⠀⠿⠃⠀⠸⠁⣀⠀⠘⠛⠁⠀⠀⣤⡄⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⠁⠀⠛⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⢦⡀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⣆⡛⠉⣷⣼⣿⡀
⠀⠀⠀⠀⠀⢀⣶⡟⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡾⣉⠈⢠⠘⣿⣿⣽⡇
⠀⠀⠀⠀⢠⣿⢏⠂⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢇⢿⣿⡏⠃⣸⣿⣿⡇
⠀⠀⠀⣰⣿⠏⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⣀⢀⣴⣴⣾⣷⣷⣧⣗⣿⣷⡄⠀⢣⠨⢸⣿⠃⠹⢹⣿⣿⡇
⠀⢀⣾⡁⠁⠀⠀⠀⣿⣿⠁⠀⢀⣤⣼⣿⣡⡾⢻⡿⣟⠉⠙⢟⢿⣟⡘⢡⣦⠀⠻⢦⣵⡿⠶⠿⠟⠋
⠀⢸⣿⣧⣶⣴⣣⣾⣿⠏⠀⣤⡿⢿⣻⣿⣏⡠⢾⢃⠈⠳⣆⣰⣵⣱⡜⣷⡟
⠀⠈⢻⣿⣿⣿⣿⡿⠋⠀⠀⡿⠀⣿⣿⠁⠀⠀⠀⢩⠞⣦⣈⣿⣿⣽⢃⣿⣶⢠
⠀⠀⠈⠋⠉⠉⠉⠀⠀⠀⠀⢸⣀⡛⢿⣇⠀⠀⢀⢸⣢⣾⣿⣿⡿⣫⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣿⣿⣦⣤⡎⣸⣟⣿⣟⢊⣼⣿⣿⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⣽⣿⣿⣿⣿⣿⣵⠿⠟⣩⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣾⡗⠂⠀⠈⠛⠛⠻⠅⣴⣦⣶⣿⣿⠟⢃⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠲⣿⡿⢹⣟⡷⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠈⢉⠑⠦⣤⡄
⠀⠀⠀⣀⣠⣤⣀⢫⡝⣷⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣳⣶⣿⣦⣾⣄
⠀⠀⠀⠙⢻⣽⣿⣿⣾⣿⣿⣿⣧⣀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠍⠀⠀⠉⣝⣅⢈⣿⡄
⠀⠀⠻⠷⠞⠋⢡⣴⣶⣿⣿⣿⣿⠃⣰⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣦⣤⣤⣤⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣟⠁⠀⢧⣾⣿⡿⠿⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣷⣿⣧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣟⣛⣟⣛⣻⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⢠⣤⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠋⠀⢤⣤⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⣰⠞⢿⣥⢹⣤⣤⡍⢢⠐⠛⡷⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⠉⠁⠀⠀⠀⠈⠳⠼⡃⠀⠀⠾⠿⠋⠀⠀⢾⣿⣿⣿⣤⡌⠁⠈⣹⠀⠛
⠀⠀⢰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠹⣄⡇⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⡏⠉⠉⠀⠀⠉⠁
⠀⠀⠈⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⠀⡀⢀
⠀⠀⠐⠀⠀⠀⠀⣿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⣀⢰⣧⠏⠀⠀⣠⣄⠀⠀⠀⠀⢠
⠀⠀⠠⠀⠀⠀⠀⢀⡴⠞⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⢤⣤⡆⠀⠀⠀⠹⡌⠁⡀⠀⠘⠉⠉⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠲⣄⠉⠀⠀⠀⠀⠀⠀⠠⠴⠶⠙⠀⠋⠀⠀⠀⠀⠀⢀⡀⢰⡀⠂⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⢈⣉⡷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⠃⠀⠀⠀⠀⠠⠚⠋⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁
⠀⠀⠀⠐⠀⠀⠀⠀⢠⠄⣼⣫⡅⠠⡀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢄⠀⠀⠀⠀⠀⠀⡼⠉⢝⠋⠒⠈⠀⠀⠀⢹⣈⡤⠀⠀⠀⠀⣴⡄⠐⠿⠖⠀⠀⠠⣼⠆⡀⠀⠀⠀⢸⡆⠀⠘⡰⠎
⠀⠀⠘⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⡼⠁⠀⠀⣠⡄⠈⢻⣃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡇⢠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⡰⠀⠀⠀⠠⢀⣀⡐⠀⠀⠋⠀⠀⠈⠉⠀⠀⠉⢠⡀⠀⠀⠀⢀⠀⠀⠀⣧⠋⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡇⠀⠀⠀⠀⣦⡀⣾⢿⣗⡀
⠀⠀⢀⡧⠆⠀⢀⡄⢀⠀⠀⠀⡎⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢀⡇⠀⠀⠀⠀⢹⣿⢆⣼⣷⠤
⠀⠀⢈⣳⡇⠀⣾⠁⠀⠂⠀⢐⡇⠀⠀⠓⡧⠀⠀⠀⠀⠀⠀⠓⠀⠀⣤⢠⠀⠒⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⠈⠻⣶⣶⣤⢈⣭⣤⣄
⠀⠀⣈⣙⡀⠀⢠⠀⠘⠀⠀⠈⠀⠀⠹⠆⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠂⠀⢐⠁⠀⠀⠀⠟⠀⠀⠀⠀⠹⣿⣿⡿⠛⠛⠉
⠀⠀⣉⣋⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣇⡠⠀⠘⠠⠛⠀⢠⠄⢠⣾⡀⠀⢀⡀⢸⠀⠀⢠⡀
⠀⠀⠙⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡛⠓⠒⠓⠀⠀⠀⠛⠀⣼⡿⣡⡾⠋⠀⠘⠀⠸⢸⣷
⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠹⡄⠀⠀⠀⠀⠈⡁⣀⠀⠀⠀⠀⣤⣤⣤⡌⠑⠟⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⡋⠁⠀⠀⢠⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠉⠁⡀⠀⠀⡆⣰⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⣇⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⢰⣶⣶⡄
⠀⠀⠀⣴⡿⠀⠀⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣅⠀⠀⣀⣤⡍⠸⠿⠃⠀⢸⣷⣌⠉⠉
⠀⢀⣤⡏⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⣤⡁⠀⠈⠻⠿⣿⠃⠀⠀⠀⠘⠛⠉⠁
⠀⠟⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⡀
⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⡌⡆⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂⠀⠀⠀⠠⣷⠀⠀⠀⠀⠀⣀
⠀⠠⡿⡟⢦⡀⠀⠀⠀⠤⠇⡇⠷⠆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡧⡂⠀⠀⣑⠿⠀⠀⠀⠀⠍⡚⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂
⠀⠀⠳⡈⠢⣝⢦⡀⠀⠭⢁⡯⠭⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡏⠅⠀⠀⠲⡪⠁⠀⠀⠀⠁⡇⠇⠀⠀⠀⠀⠀⢀⡔⠛⢣⡏⠀⡠⠀⠂
⠀⠀⠀⠙⣆⠈⠑⠑⠀⠉⠈⡭⠅⣭⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⡃⠀⠀⠼⡂⡅⠀⠀⠀⠄⡇⠅⠀⠀⠀⠀⠀⠈⠀⠐⢀⣃
⠀⠀⠀⠀⠈⢳⡶⣴⡆⠀⠀⡇⣵⠯⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡇⡅⠀⠈⢱⡏⠁⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⡠⠶⣶⡀⠀⡇⠀⡀⠐⡦
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠒⡇⣺⣯⣹⠀⣠⠀⠀⠀⠀⠀⠀⠀⠂⡃⠃⠀⠀⠩⡀⠄⠀⠀⠀⠀⣆⠘⠀⠀⠀⢰⠇⡈⠀⠛⠲⡟⠋⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠠⡏⠿⣿⣸⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠘⠰⠃⠻⡆⠀⠷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⡷⣶⣟⣻⡟⢠⡄⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⣠⠀⠠⣇⡄⠚⠁⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠂⡇⡗⢻⣻⣾⣿⠀⢻⡀⠀⠀⠀⠀⠀⠀⣰⢄⠀⠀⠀⡇⠀⠁⠀⠀⠐⡇⢸⡀⠉⠈⠀⠰⠃⠀⠀⣟⠀⠀⢲⠸⠎⠁
⠀⠀⠀⠀⢀⣴⢋⡁⠐⠦⠦⢿⣽⠃⠀⠋⠀⠈⣇⢀⠄⠀⠀⠀⠀⡏⠀⠁⠀⠀⡇⠈⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⠃⡨⠃⣀⣀⣀⣀⡀⠉⠒⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡇⠀⢀⠇⠀⠁⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⣷⠪⠋⠁⠉⠪⡻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣉⠉⠀⠋⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⣤⡧⣤⡦⢄⠀⢀⣀⢠⡄⠀⠀⠀⣄⣸⠹⡄⡤⠤⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⣠⣍⣿⣴⣡⡴⢋⡄⢀⠀⢀⣷⣙⠳⡷⢷⡒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠉⠉⡏⢠⣹⢳⣴⢁⢦⣼⡲⡿⣷⣻⣽⡝⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠴⢾⣋⡭⠟⢧⠀⠲⣷⠟⠉⠚⠋⡡⡄
⠀⠀⠀⠀⠰⡄⠠⠤⣀⣀⠀⠀⠀⡔⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢀⡀⠀⣈⠳⣤⠇⠀⡏⡆⢸⠀⢰⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠘⠢⡀⠀⠈⢱⢀⠞⠀⣼⢀⠔⡆⠀⢠⠋⠉⠹⠟⠑⠌⡟⣇⢲⡆⠈⣆⡞⢀⢸⡇⢉⠿⣀⡸⢠⠊⠁⠉⠁⢀⡲
⠀⠀⠀⠀⠀⠀⠀⠉⠙⣲⠟⢦⣦⠘⡗⠉⣰⠀⠀⢠⠀⠀⠀⠀⠀⡄⠣⡘⣿⠁⣠⢿⢡⡇⡆⣑⣿⡒⢙⡋⠙⣗⠒⠒⠊⠁⡠⠤⠰⠂⣠⡀
⢤⠤⠤⣄⣫⢳⡠⢟⣉⣀⣀⡴⣃⣑⣞⠤⢋⢦⠀⠈⠒⠤⠤⠤⠊⠀⠀⠈⢿⠉⡰⡋⢀⣹⠯⡘⣤⠈⡎⠑⠤⣈⣢⣀⣠⣼⢖⠖⣵⠋⡔⢀⠤⠒⠆
⠀⠑⠦⠬⣛⢶⣱⠶⣥⡌⠛⡿⠀⣤⣏⣳⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⢱⠖⠉⠈⠓⣬⢾⣦⣳⠶⠶⠿⠿⠭⢬⡷⢎⣘⣤⠔⡷⠁⢠⠎
⠀⠀⠀⢠⡡⠞⣧⠟⠛⠁⠼⠐⠫⣞⣡⠽⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠂⢀⠠⡪⠑⠈⠁⠀⠁⠀⠀⠀⠀⠀⡇⢱⡾⠈⠣⢩⠻⢗⣨⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠔⡵⣧⣤⣤⣄⣠⣤⡴⡾⠛⢶⡖⠢⢌⠋⡇⡔⠁⢿⠀⢸⠳⡀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣤⡑⢕⢄⠀⠀⠀⠀⠀⠀⠀⡅⢨⠊⠀⡠⠬⢿⡤⠝⣧⣰⠁⠀⠀⠀⠀⠈⡆⠉⠀⠀⠈⢆⡀⠀⠈⠑⠢⠌
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢢⣀⣴⣉⣦⢵⣜⣹⣼⠈⠢⡑⢤⡀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠁⠀⢸⠻⣦⢄⡀⠀⡠⠜
⠀⠀⠀⠀⠀⣤⣊⣀⣀⡀⠴⡟⠀⣰⠾⡉⢸⡔⢹⠋⠓⠚⠷⢌⠢⡀⠀⠀⡇⡇⠀⠈⠢⣀⠀⢀⡠⢪⠔⠻⡷⠋⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠔⠁⠀⢃⡘⠓⠋⠀⠀⠀⠀⠀⠑⢌⠢⣀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢥⣤⣴⠒⠒⢲⣤⡀⠀⠀⠱⡈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⠉⠀⠈⢦⡟⣟⠙⠳⣄⠀⢰⠈⡄⠀⠀⠀⠈⡗⣀⠀⠀⠈⠁⠒⠈⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⢀⡾⠀⢻⡆⠒⢌⠙⡇⢰⠀⠀⠀⢠⢚⢷⢈⡆⣶⣉⣶⣤⡮⠠⠀⠉⠑⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣤⣖⠁⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣹⣾⠒⡟⠓⢷⣤⡀⠐⢏⠁⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠑⠾⠁⠀⠙⡧⡀⠀⠀⢀⠜⠰⠀⢀⣠⠴⠊⠉⢧⡙⣿⡀⡖⠀⠙⠦⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠉⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠉⠉⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠆⢠⣴⣶⠄⠀⠈⢿⣄⠀⠀⠀⠀⢀⣄⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⢢⡜⣛⠋⢀⡀⠀⠘⣿⣆⣄⠀⠀⠸⣿⡇⠀⢿⡟⣠⣾⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⢣⣄⡸⠟⠃⠀⢹⣟⣧⠀⠀⠀⢹⡁⡳⢼⣶⣍⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠙⣧⡄⠀⠀⠈⠛⣫⢀⡔⣤⢶⣉⡀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠏⢂⠀⠀⠀⡾⠃⣸⣿⠀⠀⣿⣷⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⢆⠀⠀⣇⠜⠁⠉⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⠀⠀⠀⠀⢣⣀⡏⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢀⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⢹⢉⠒⠢⡀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⣠⣿⣿⡄⠀⠙⢦⣸⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⢻⣿⣿⠟⠀⠀⠈⣿⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠉⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⢀⣠⣤⡤⡿⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠑⣿⢀⡞⠉⢠⣈⣷⣧⠀⢠⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢸⣾⠃⠀⣠⣿⣿⣷⣄⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣬⣿⠀⠀⢿⣿⣿⣿⡿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠈⢹⠟⠋⠁
⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⣤⣼⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠈⠉⠉⠉⠉⢉⣉⠉⠉⠉⠉⠉⠉⣭⠉⠹⣿⠟⢛⣟⡉⣹⣏⢹⡟⢉⣉⠉⠉⣽⠏⠉⠉⠉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠘⠃⠈⠁⠀⠀⠋⣀⠀⠀⠀⠀⢀⣿⠁⠈⢨⡏⠙⠁⢸⠇⠈⠉⠀⣤⡄⠀⠀⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠈⠋⠀⢀⡀⠘⠛⠃⠀⠶⢺⠁⠘⠀⠿⠇⠀⣴⠄⠘⠁⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⣤⠀⢀⡀⠰⠆⠀⣀⠈⠁⠀⠴⠗⠀⠁⣸⠿⠄⠀⠀⢀⡄⠀⠀⠰⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⠋⠀⠀⠶⠀⠀⣀⠀⠀⡀⠀⢠⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢤⡄⢠⡄⠀⠀⣀⡀⠉⠀⣴⠄⠀⠈⠁⢤⠀⠀⠛⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⡄⠀⠀⠓⠀⢤⡄⠀⠀⠀⠘⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠃⠠⠆⠈⠀⠀⠀⠲⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡴⠀⠀⠐⠖⠠⠀⠀⠀⡄⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡤⠀⠀⢿⡇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠄
⠤⠄⠀⠀⢠⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡤⠤⠤⣰⠰⠂⠀⠀⠀⠀⠀⠀⠀⠠⣂⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠉⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⢘⠀⠀⢴⡆⠀⠀⠀⢠⡾⠃⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠐⠓⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⢠⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣀⠠⢄
⡤⠄⠂⡀⠀⠈⠓⡀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠌⡑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⢤⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠋⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠄⠁⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢏⠈⠐⠊⠁⢈⠂⠀⠀⠀⠈⠲⠐⠊⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⢀⡀⡀⠀⠀⠀⠅⡀⢀⠠⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢠⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠆⠐⠊⠀⢁⠐⠁⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠢⡀⠀⠀⢨⠀⠀⠀⠀⠠⠂⠰⠀⠀⡀⠀⠀⠨⠙⠀⠀⠘⠠⠰⠊⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠊⠁⠀⠀⡀⠀⠄⠀⠀⠰⡿
⠛⠳⠢⣄⠀⠀⢈⠆⠀⠀⡆⡀⠐⠈⠐⠈⠀⠀⢀⠃⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠁⡀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠈⠀⢙⡀⠀⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠐⡇⠀⠀⡮
⠀⠀⠀⣸⠁⠀⠁⠀⠀⠀⠀⠌⠀⠑⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠁⠆⠀⠧⡈
⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠁⠀⠈⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠰⠀⠀⠈
⠀⠀⠀⢀⠀⠀⡀⠀⠒⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⢿⣆⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠐⠢⢀⠀⠘⢿⣷⣄⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠈⢻⣆⠀⠀⢀⠀⠀⠀⠦⠈⠁⠈⠁⢠⠀⢠⠚⠁⠀⠀⠁
⠀⠀⠘⠁⠀⠈⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠗⠀⠀⠈⠟⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⠀⢴
⠀⠀⢠⠀⠠⡄⠀⠠⠀⠀⠀⠀⠀⠈⠀⢀⡀⢿⣿⣦⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⡀⠀⠄
⠀⢠⠚⠀⢀⡄⢀⡀⠐⠀⠀⠀⠀⠀⠀⣼⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⢀⠀⠄⠀⠀⠐
⠀⠈⠀⠀⠇⢤⣌⠁⠀⠈⠀⠀⠀⠴⢀⠏⠀⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡀⠈⠐⠡⠁
⠀⠀⠀⠀⢀⠀⠆⠀⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠴⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠐⠃⠀⠗⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠶

⠉⠂⠀⠘⣇⢀⣌⢘⠠⠆⠀⠁⠸⠀⡀⠸⡄⠿⡾⠓⠀⢀⡴⡶⣶⡻⠃⠐⢿⠆⠀⠀⠀⠓⢀⠇⢀⠈⠉⢇⣤⠉⠙⢶⡞⠃⠠⠂⢨⡮⠨⡅⠀⠀⠉⠁⣠⣡⡀
⣌⠒⠂⠈⢡⠷⡄⢠⡄⠘⡁⠀⠄⢠⡀⠈⠀⠩⠐⠀⠁⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠊⠀⠀⠹⠤⢦⡼⠓⠄⢀⠢⠂⠛⠃⠀⠀⣠⠐⢾⠙⣏
⠤⣱⢄⡄⢄⠐⢱⣤⣁⣠⠢⠀⠐⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠠⣀⡡⠀⠄⣴⠅⠠⣌⡰⠁
⠒⢯⡜⢔⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠆⢀⢘⠇⠀⠀⠂
⡤⢄⠙⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡄⡄⣠⠄⢠⢤⢀⡤⣄⢀⠤⡀⡠⢄⢀⠤⠀⢠⡀⢠⡀⡀⡄⣠⠄⣄⠀⣠⢠⠄⠀⠀⠀⠀⠀⠀⠈⠉⣀⠤⠔⢦⣀
⠧⠜⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠸⠱⠊⠄⠇⢧⡄⠸⠻⠸⢄⠼⠻⣤⠌⠜⠢⠰⣄⠦⠟⠳⠸⠑⠇⠇⢉⠗⠏⠖⠱⢨⡷⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣶⢾⠏
⠀⠀⢐⡠⠒⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠠⢴⠏⠀⡀⠔⠟⠁⠀⣄⠀⡦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢖⡚⠺⡄⠘⠃⠁
⠀⠘⠛⠋⣥⠎⠀⠀⠀⢸⠇⠛⠛⣧⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢔⠿⢧⠄⠀⠐⠝⣁⡁⠷⠂⢆
⣠⠎⠀⠊⢀⠄⠠⠀⠙⠋⡠⠲⠒⠉⡦⡄⠈⣄⡠⠉⣀⠀⠀⢠⠖⡬⠀⠀⠀⠀⣇⡀⢤⠧⢀⣀⡄⠀⠰⢎⠙⠄⠜⢸⡶⠀⣄⡤⠂⡂⣯⡆⠀⠁⠴⡂⢿⠆⡕
⣷⣦⣤⣤⣷⣶⣾⣧⣤⣼⣧⣤⣴⣾⣶⣤⣤⣤⣷⣮⣤⣼⣦⣬⣿⣧⣤⣤⣤⣾⣷⣴⣦⣴⣿⣧⣤⣴⣷⣴⣧⣤⣴⣾⣧⣼⣧⣤⣴⣷⣿⣤⣤⣤⣬⣴⣶⣾⣦
⣿⣿⣇⣀⣈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⠠⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡓⠃⡠⠀⠀⠈⠑⢄⢈⡀⠀⠀⠀⢄
⠀⠀⢀⡀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⢄
⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠢⠄⠀⠀⠀⠀⠲⡀⠀⠆⠀⠀⠀⠀⢁⡀⠀⠀⣀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢂⣀⠀⠀⠀⠐⠀⡀⠠⠶⠶⠶⢦⠄⠂⠀⠀⠀⠀⠠⠀⠀⠸⣁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠈⠀⠀⠿⠿⠇⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠉
⠀⠀⣠⠂⠀⠀⡀⠠⠀⠀⠀⠀⢠⡀⠰⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠠⡦⠀⠀⠀⠀⠈⠀⠀⢰⢀⠀⠀⠀⠀⠀⠑⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⢀⣄⠀⠀⠁⠠⡀⠉⢂⠘⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠠⡀⠄⠀⢀⠄⠀⠈⠁⠘⡆⠀⢶⣤⣄⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⢀⡀⠀⢇⣀⡸⠀⠄⠀⡇⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⡤⠦⡄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠟⠁⠀⠐⠘⡆
⠀⠀⠀⠀⢠⠀⠱⠴⠃⠀⠀⠀⠀⠀⠀⠀⠐⠷⣤⡀⠀⠘⣇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠃⠀⠈⠒⠆⠔⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠹⠤⣠⠖⠀⠀⠀⠀⠀⠀⠶⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠂⠀⠀⠀⣀⡀⠀⠀⡠⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠄⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⣠⠊⡠⠒⠂⣉⡁⠀⠀⠀⠉⣤
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠑⠊⠀⠀⠀⠈⠉⠙⠒⠲⠤⡄
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⡶⠉⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⣠⣤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡔⠁⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣰⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣂⠀⠀⠃⠀⠀⠙⠛⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠨⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢁⣄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠈⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠁⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⡼⠀⠀⠀⠈⠀⠐⠂⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠐⡸⡀⠀⠀⠈⠉⠀⠀⠀⠃⠧⣛⡫⡰⠀⣠⠁⡃⠸⠃⠀⠀⠀⠠⠜⢢⡑⠊⠐⡸⡀⠀⠀⠈⠉⠀⠀⠀⠃⢣⣛⡫⡸⠀⣠⠁⠇⠺⠃⠀⠀⠀⠠⠜⢢⡑
⠀⡁⡉⢶⠀⠀⠀⠀⠀⢀⣧⡠⠤⠞⠁⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠈⡉⡈⢶⠀⠀⠀⠀⠀⢀⣧⡤⠤⠞⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⢨⣶⣎⠀⡯⠄⠀⠀⣠⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⡡⡠⡍⢦⠀⠀⣉⡏⢨⣶⣎⠀⠧⠄⠀⠀⣠⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠞⡡⡀⡍⢦⠀⠀⣉
⢤⣉⡡⢜⠑⠂⠀⠀⡏⠀⠀⡔⢉⡠⠤⣉⠆⠀⠀⠘⣌⢜⢋⠆⢃⠀⠀⠁⡱⢤⣉⡡⢜⠁⠂⠀⠀⡏⠀⠀⡔⢉⠠⠤⣉⡆⠀⠀⠘⡄⢜⢛⠆⣸⠀⠀⠁
⠃⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠁⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠚⠁
⠀⠀⠀⢤⢀⠀⠄⠀⠀⠀⠀⣀⡤⠦⠦⣌⡠⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⡀⠀⠀⠀⢤⢀⠀⠄⠀⠀⠀⠀⣀⡤⠦⠦⣌⡠⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡄
⢂⠀⠀⢢⡏⢙⡦⠄⠀⠀⠠⢜⣶⣕⣢⠀⢉⣧⠀⠀⠀⢜⢀⣹⡆⠀⠀⠳⡌⢢⠀⠀⢢⡏⢹⡦⠄⠀⠀⠠⠜⣶⣗⣢⠀⢉⣧⠄⠀⠀⢜⢀⣙⡆⠀⠀⠳
⠤⠀⠀⠀⠹⠙⠈⠃⢀⠀⠀⠀⠊⠝⡦⣶⡿⠻⣦⠀⠀⠈⠑⠉⠀⠀⠀⠀⠘⠦⠂⠀⠀⠹⠙⠈⠃⠀⠀⠀⠀⠊⠕⡶⣶⡿⠻⣦⠀⠀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠠⡀⢨⠀⠀⠀⠀⠀⠁⠁⠈⠀⢹⢆⠀⠀⠀⠀⡄⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢸⠀⠀⠀⠀⠀⠁⠀⠈⠀⢛⢆⠀⠀⠀⢀⡤⠂⢢
⢀⠂⡄⠀⠀⠀⠤⣱⠼⢤⡎⡠⠔⠁⢠⠠⡀⠀⠀⠑⠓⠀⢀⠎⠀⢀⡟⠀⠀⢀⠒⡄⠀⠀⠀⠤⣱⠼⢤⡎⡠⠔⠁⢀⠠⡀⠀⠀⠑⠓⠀⢀⠎⠀⢀⡟
⡎⡸⠀⠀⠁⠀⣰⢱⣶⡆⢸⡖⠊⠀⠀⢧⠰⡀⠀⠀⠀⠀⡌⠀⢠⠎⠀⠀⠀⡌⡼⠀⠀⠁⠀⣰⢱⣷⡆⢸⠖⠊⠀⠀⢧⠐⡀⠀⠀⠀⠀⡜⠀⢰⠎
⠷⠃⠀⠀⢀⡡⠘⡦⣄⣨⡶⢍⡁⠀⠀⠸⣄⡇⠀⢀⣀⣼⡀⣤⠂⠀⠀⠀⠀⠧⠃⠀⠀⢀⡡⠘⡦⣄⣠⡶⢍⡁⠀⠀⠸⣀⡇⠀⢀⣠⣼⡀⣠⠂
⠀⡀⠀⠈⠁⠀⠜⠀⠂⢃⠘⠀⠈⠂⠀⠀⠉⠀⠶⠡⢊⢊⠍⡅⠀⠀⠰⡀⡐⠀⣀⠀⠈⠁⠀⠞⠀⠆⢡⠘⠀⠈⠂⠀⠀⠉⠀⡶⢡⢊⢊⠍⠁⠀⠀⠰⡀
⡛⡥⠆⠀⠀⡠⢤⡀⠀⠘⠀⢀⣀⣀⡀⠀⠀⢀⠔⣡⢃⠃⡰⠀⠈⢱⢊⣅⠭⡛⡥⠆⠀⠀⠠⢤⡀⠀⠘⠀⢀⣀⣀⡀⠀⠀⢀⠔⠡⢋⠁⡰⠁⠈⢱⢊⢍
⢃⢙⠉⠀⠀⠢⠾⠁⠀⠀⡔⡽⣷⡖⠸⠀⢀⠇⠔⣡⡎⠐⠃⠀⠁⢹⡘⢝⡣⢃⢻⠉⠀⠀⠢⠶⠃⠀⠀⡔⡑⣷⡶⢱⠀⠀⠆⠔⣡⡎⠐⠃⠀⠁⢹⡘⢟
⠠⠎⠦⣀⠀⠀⠀⠀⠀⠀⡄⠉⠒⣪⠎⠀⠚⢠⠃⠈⠀⠀⠀⠀⠈⠀⠜⠲⡄⠠⠮⠦⣀⠀⠀⠀⠀⠀⠀⡄⠉⠒⣊⠎⠀⠚⢠⠂⠈⠀⠀⠀⠀⠈⠁⠜⠢
⠠⣂⡢⠙⡄⠂⠀⠀⠀⣴⠟⠊⠉⠁⠀⠀⠐⠃⠀⠀⣠⠄⠀⠤⡀⠀⠀⠤⡆⠠⣂⡢⠙⡄⠀⠀⠀⠀⣴⠟⠋⠋⠁⠀⠀⠐⠃⠀⠀⣠⠄⠀⠤⡀⠀⠀⠤
⠘⠿⠑⢠⢏⡁⠀⠀⡾⠁⠀⢀⠤⠄⠀⠤⡀⠀⠀⢠⠁⢪⣦⡃⡙⠄⠀⠖⢆⠘⠿⠗⢠⠍⣁⠀⠀⡾⠁⠀⢀⠤⠀⠀⠤⡀⠀⠀⢠⠁⢪⣦⠃⡹⠀⠀⠖
⡝⠒⢊⠱⠀⠀⠀⠀⠃⠀⠀⠣⠘⠈⠉⠓⠃⠀⠀⠀⠣⢑⣐⣡⠜⠀⠀⠀⠊⡙⠒⢂⠱⠀⠀⠀⠀⠃⠀⠀⠣⠚⠊⠉⠓⠁⠀⠀⠀⠣⢑⣐⣁⠞
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠠
⡀⠀⠀⡙⡼⣴⣁⠀⠀⠀⠀⢲⡋⢍⡉⠓⠮⡔⠀⠀⠀⢠⠒⢢⡀⠀⠀⢆⠢⡀⠀⠀⡙⡼⣤⡁⠀⠀⠀⠀⢲⣋⣍⡉⠓⠪⡔⠀⠀⠀⢠⠒⢦⡀⠀⠀⣇
⠘⡀⠀⠘⢧⢴⠣⡅⠀⠀⠈⠐⡻⢗⢛⣄⣴⣿⡁⠀⠀⠱⢄⠾⠃⠀⠀⠈⢣⠘⡀⠀⠘⢧⢼⠣⡅⠀⠀⠈⠐⡻⢗⢛⣤⣴⣿⡁⠀⠀⠱⢴⠶⠃⠀⠀⠈
⠉⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠁⡏⠟⠫⠈⠿⡀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠁⠏⠟⠫⠈⡿⡀⠀⠀⠀⠀⢀⡀⡀
⠀⡀⠀⠀⠀⠀⡈⢆⢸⠀⡠⠀⢀⠄⠀⠀⠀⠀⠐⢕⣄⠀⠀⡰⠋⠀⣴⠀⠀⠀⣀⠀⠀⠀⠀⡈⢆⢘⠀⡀⠀⢀⠄⠀⠀⠀⠀⠘⢕⣄⠀⠀⡰⠋⠀⣴
⠰⢠⠃⠀⠄⠀⢉⢞⣍⡙⢣⡊⡥⠀⠸⡌⢂⠀⠀⠀⠀⠀⠰⠁⠀⣠⠃⠀⠀⠰⢠⠃⠠⠄⡀⢙⢞⣍⡑⢣⣂⡥⠀⠘⡍⢂⠀⠀⠀⠀⠀⢰⠁⢀⡰⠃
⡇⡎⠀⠀⠀⠄⢺⡘⠛⠃⣘⠧⠄⠀⠀⢸⠀⡆⠀⠀⠀⢠⠃⠀⡘⠁⠀⠀⠀⡇⡎⠀⠀⠀⠄⢺⡘⠛⠃⣨⠧⠄⠀⠀⢹⠀⡆⠀⠀⠀⢠⠃⠀⡸⠁
⠉⠀⠀⠠⠔⠊⣠⠋⡓⠞⢫⠑⠢⡄⠀⠈⠧⠃⣀⠔⡪⡻⠮⠟⠀⠀⢀⠀⢀⠉⠀⠀⠠⠔⠊⣠⠋⡓⡞⢣⠑⠢⡀⠀⠈⠶⠃⣀⠔⡺⡻⠮⠟⠀⠀⢀
⣤⠆⡀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢋⠌⡘⡔⢁⠅⠠⢄⡠⠆⠢⣤⠆⡀⠀⠀⠀⠁⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⢉⠌⡴⠕⢁⠆⠠⢄⡨⠆
⡆⢯⠥⠀⠀⡊⣹⠆⠀⠀⢀⢴⣖⢒⢢⠀⠀⡔⢁⠞⡰⢀⡎⠀⠄⢼⢰⣷⡌⡆⣯⠥⠀⠀⡉⣙⠆⠀⠀⢀⢴⣖⣒⢆⠀⠀⡔⢁⠎⡰⢀⡎⠀⠄⢼⢰⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡔⠂⡆⠀⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢈⢧⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢌⠿⣤⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⡙⠻⣻⡆⣲⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡈⠻⢆⣻⣴⠀⢀⠇⠀⢣⡀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑⢦⢹⣿⠀⡜⠀⠀⣿⢣⠀⠀⠀⠀⢤⠀⠀⣼⣩⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠠⢀⠙⠻⣣⠇⠈⡸⡎⠸⠀⢰⢀⠀⢸⠀⢰⣯⠕⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⠠⠒⠤⠔⣿⣧⠰⡟⢀⠆⠄⢸⠘⠀⢸⣀⣼⡡⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠰⡀⠀⠀⠑⢤⡐⠶⢌⡳⡷⢁⠎⡘⠀⡼⡀⢀⡶⣿⣋⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢦⠀⠉⠓⢷⣷⣁⠀⠀⠀⡀⠉⠐⠘⣷⡎⠀⡇⢀⣃⢶⣽⣫⣡⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢀⠀⠈⠀⠈⢷⡆⢄⡀⠀⠀⠀⠀⠀⢣⠀⣧⣚⡵⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠻⣕⠂⠀⠀⠀⠀⠊⢲⠟⠁⠈⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠐⠒⠒⠪⠷⣧⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⢀⣀⣈⣻⡂⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢸⢣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡙⣅⡸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡇⣚⢄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠇⠐⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡁⠴⠖⠷⠼⠋⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣤⣤⡄⠀⠀⠀⠀⠀⠈⠓⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣦⣬⣏⡿⣧⣝⣛⠂⣀⢀⠀⠀⡁⠑⢾⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⣵⣿⣟⢻⣯⢲⢾⢧⣸⠻⠟⢀⡈⠖⠀⠉⠁⠀⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣉⣽⣉⣶⣷⡶⣉⣽⣿⡿⠿⡟⢇⢸⡷⠺⠉⣲⠒⣀⠀⠆⠠⠄⠈⢅⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉⠙⠛⠲⠌⠽⠿⠿⠛⠛⠻⠋⠉⠟⠲⠖⠻⠯⠭⠞⠁⠉⠁⠈⠉⠉

⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⢠⠀⡀⠀⠀⠀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠁⠀⠀⠀⠀⠀⢂⠀⠨⣀⠀⠀⠀⠁⠂⠀⠀⠀⠈⠀⠁⠀⠀⠰⠊⠁⠉⢡⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢠⠀⠌⠀⠂
⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠂⠀⠀⠐⠈⠀⠀⠘⣀⠀⠀⠔⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠂⢀⡀⠁
⠀⠀⠘⠀⡄⠐⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠒⠀⠐⢄⠀⠂⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⡀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠄⠁⠈⠂⠀⡃
⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠈⠈⠀⠀⠉⠀⢀⢀⠤⠞⠀⠀⠀⠠⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⣀⣀⣤⣤⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠘⠀⠄⠀⠀⠀⣠⣾⠟⠛⠫⠛⡛⡛⢉⣙⡻⢯⡀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠐⠢⠀⠈⠀⢠⣿⠘⢸⡄⡄⣌⣳⣜⣈⣯⣩⣼⠿⡿⡏⠉⣿
⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠁⠒⠂⢀⠁⠀⠸⡿⠶⢾⠛⠻⡏⣭⣭⡩⡇⠒⢓⡐⢠⡀⢰⣽⠄
⠀⡠⠂⠀⠢⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⣌⠌⠂⢦⣀⣝⣉⣷⣹⣌⢉⣣⣠⣥⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠐⠀⠄⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠙⠉⠋⠛⠛⠛⠉⠁⠀⠀⢀
⠀⠀⠁⠀⠁⢀⠀⠀⠀⠀⠀⢍⢈⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⠱⠃⠚⠁⠀⠀⠀⠄
⠀⡊⠀⠀⠌⠀⠀⠀⢀⠀⠂⢀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠐⠠⠛⠁⢀⡀⡀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠂⠀⠀⠀⠀⠈⠀⠀⠐⠰⠖⢲⢸⡣⢈⠄⠂
⠀⠈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠘⠁⠁
⠀⢀⠂⢐⠈⠀⢃⠂⠀⠐⢈⠄⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡠⠐⠁⠀⢀⠃⠀⠀⠀⠊⠐⠆⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠢⠀⠀⠀⠀⢄⠀⣀⢂⠐⠀⡜⡠⠐⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠀⠄⠐
⠀⠈⠀⠡⠠⠐⠀⠊⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⣲⠒⣲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢫⡅⣀⣤⣴⣚⣉⡉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣸⠞⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣠⠾⠥⠤⠶⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣀⣠⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠓⢲⣦⣄⣀⣹⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⢈⣩⠟⠛⣹⠋⢈⡏⠉⠉⠙⠻⣷⣄⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡞⠱⣲⠖⢻⠇⠀⠐⠃⢀⠞⠀⠀⠀⢀⡤⠼⣿⡻⡿⠯⠗
⠀⠀⠀⠀⠀⣼⢠⣾⠁⠀⠈⠀⣀⣠⠴⠋⠀⠀⠀⠀⣞⣸⠛⠁⢹⡙⡆
⠀⠀⠀⠀⠀⣿⣃⠬⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢳⢹⡀
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣿⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣠⠶⠶⢚⠿⠶⢧⣺⣅⡽⠶⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⠉⠦⠀⠉⠻⠿⢽⣇⠀⣴⠤⢷⡶⠋⣤⡄⠀⣦⠈⢧⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠲⣄⡉⠒⠒⠤⢤⢄⣈⢁⡼⠋⠁⠀⠈⣉⠀⠈⠉⠢⠀⠐⠚⠉⠁⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢈⣁⠀⠀⠀⠀⠀⠀⣉⠚⣾⡁⠛⠰⠆⢀⡀⠀⠀⡤⢄⠶⠄⠀⠛⠓⠃⠒⢂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⣦⡬⣄⣘⣥⣬⣉⣠⠵⠞⢿⣙⡒⠶⢦⣤⣅⣀⣀⣠⠤⢤⣴⣚⣭⡉⣯⡍⠩
⠀⠀⠀⠀⠀⠀⠈⠣⠈⢓⣒⡚⢃⠤⠶⢾⡿⠚⠆⡴⠿⠁⢀⡉⠩⡍⠓⣦⠰⢯⣉⣁⣀⣾⣂⡤⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠒⠛⠋⠘⠃⠀⠳⠶⠲⠿⠷⠒⠛⠛⠁⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⠤⠠⢤⣀⡀
⠀⠀⠀⠀⠀⢀⠖⣡⠴⠖⠒⠒⠒⠒⠒⠲⠶⠶⠶⠤⠤⣤⣀⣀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⣴⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀⠀⠱⣄
⠀⠀⠀⣸⠏⢀⣀⣠⣤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢣⡀⡏⢧
⠀⠀⠀⣯⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠦⣄⠀⠀⣧⡇⣸
⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡙⣾⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠄⠤⠤⠤⣄⣀
⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣶⣏⢻⠁⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⢀⣀⡀⠀⠀⠈⠑⢄⣀⡀
⠀⠀⠀⠸⣄⠉⠓⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⢁⡟⢸⠎⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⡀⡴⠖⠉⠉⢉⡉⠉⢶⡀⠀⠈⠳⣇
⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠋⠛⠋⠉⠉⠉⠁⠀⣠⠞⢀⠏⠀⠀⠀⣠⠔⠉⠀⠀⢀⣴⠞⠀⠀⠀⡄⡘⠸⠸⠈⡀⢃⣄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠈⠙⢽⣓⠲⠤⠤⢤⣤⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣠⠴⠚⣁⠴⠋⠀⠀⣠⠞⠁⠀⠀⣠⡶⠋⠀⢀⢀⠓⠰⠀⠆⠈⢴⠌⠊⠀⠌⢣⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠤⠤⠤⠤⠬⠭⢉⣉⣉⡭⠥⠤⠒⠋⠁⠀⠀⠀⡼⠁⠀⠀⡠⠾⠂⠀⠐⠂⠢⠠⠀⠁⠠⠶⠋⠁⠐⠀⠭⠅⡸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⡾⠃⢠⠀⠉⢀⠀⠃⣀⣴⠕⠁⠠⡀⠌⠠⠁⠁⣰⠇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠋⠀⠂⠀⠀⠄⢀⠄⠊⠋⢀⠀⠒⠐⠀⠂⠀⢀⡾⠁⠀⠀⡠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣦⡉⠄⠉⠀⡒⠕⠁⠀⠄⢠⠁⡄⠈⢀⣤⠖⠋⠀⠀⢀⢔⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠩⢔⡀⠡⠊⠠⠀⡌⣈⠀⠂⠀⢠⡴⠊⠁⠀⠀⣠⢔⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢻⡄⠌⠐⠐⠂⢀⣠⠔⠛⠁⠀⠀⢀⠔⡪⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠁⠙⠂⠒⠋⠉⠀⠀⠀⢀⠠⣊⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⠀⠀⣀⠠⢔⣪⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠰⣿⣷⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠈⠃⠀⠀⢀⡀⣶⡀⠠⣴⠀⠀⢰⡶⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣴⡀⠀⠀⢀⣐⣶⣷⣒⣿⣶⣿⣿⣿⣿⣦⡈⠁⣀⠈⢷⣶⣶⣦
⠀⠀⠀⠀⠀⠤⠀⠀⠀⣿⣿⣷⣀⠁⠘⠿⠴⠀⠀⠀⠉⠏⠀⠀⠀⣰⣾⡿⠿⠿⠿⠛⠛⠿⠻⠟⠻⡁⢀⣨⣷⣾⣿⣿⣿⣶⣶⡶⢀⣤
⠀⠀⠀⠀⢰⣴⠀⠀⠤⢭⣽⣉⣤⠀⡴⣄⣀⣀⣀⣤⣤⣴⣾⢏⡀⠈⠀⠀⡀⡠⠀⠀⢠⠄⠀⡄⠀⠄⠀⠁⢰⡆⠀⠁⠀⣠⠄
⠀⠀⠀⣌⠉⠉⠉⠉⠉⠉⠉⣉⡉⢩⣏⡉⠀⠀⢈⠉⠉⢉⡉⠙⠛⠛⣩⣀⣄⠃⢃⣤⡏⢃⡜⠿⣾⡄⠀⢠⣮⣀⣤⠀⣴⠛⠁⢀⣀
⠀⠀⢰⣿⠀⠀⣾⣧⣿⣿⣿⡿⠗⠘⡯⠁⠀⠀⣸⣃⠱⠖⢻⣾⣿⣿⡇⠀⠉⢘⣨⣭⣇⠀⠐⠀⠘⠀⠀⠛⣹⣿⣿⢹⡋⠁⠈⠀⡍
⠀⠀⢸⢥⣄⡤⣏⠇⣈⠉⡛⠁⠀⠄⢠⣶⣽⣿⣿⡷⠰⠒⠛⣿⣿⣿⠓⠶⡶⠺⣿⣷⣿⡇⢠⣿⡖⠛⠃⠀⣿⣿⡿⢾⠿⠇⠀⠀⡆
⠀⠀⣠⣴⣶⣿⣷⣶⣿⣿⠟⠀⠀⠀⢸⣿⣿⡍⠹⣿⣄⢈⠈⢸⣿⣿⠀⠁⠁⠂⠉⡿⣟⠋⢿⣇⣀⡦⣄⠀⣿⣻⡃⣸⡿⢷⠀⠀⡇
⠀⠀⠀⠉⠉⠉⠉⢈⣉⣀⠀⠀⠀⠀⣸⡟⠛⠃⠀⠈⠀⠙⠀⢸⣿⡏⠀⠄⠰⠀⠛⠟⠸⡄⢸⣿⣿⡏⠀⠀⠉⢭⣥⣿⡆⠀⠀⢀⣀⡀
⠀⠀⢀⣀⣘⠁⠠⠘⠃⡉⢉⠀⠀⠀⠿⠇⠸⠧⢴⡦⠀⠆⠰⠄⣨⡀⠀⠀⠀⠒⠀⠀⠀⠑⣤⡟⠛⡧⠀⢤⣾⣿⠿⢿⡇⢀⠈⠁⡃⠉
⠀⠀⠙⠛⠑⠀⠀⠀⠉⠀⢀⣀⣠⣡⣦⡒⡠⠄⠁⠀⠀⠀⠀⠀⣀⣤⣤⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⡟⢿⡟⠛⠿⠟⠺⢧⡍⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠋⠛⠁⠉⠉⠉⠉⠉⠉
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠂⠀⠉⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⣌⠚⢄
⠀⠀⠀⠀⠀⠀⠀⢈⣨⡔⠀⠀⡠⠀⠉⠐⠀⠀⠀⠀⠀⠤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⢋⣉⣉⣤⣤⣤⣤⣉⣉⡙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⣁⣤⣶⠿⠟⠛⠋⠉⠉⠉⠉⠙⠛⠻⠿⣶⣤⣈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠉⠻⢷⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡟⢁⣴⡿⠋⠀⣿⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠹⣧⠀⠀⠀⠀⠙⢿⣦⡈⢻⣧⡀
⠀⠀⠀⠀⢀⣾⠏⢠⣾⠏⠀⠀⢀⣿⣿⣷⠶⠓⣠⣤⣤⣄⠀⠀⠹⣦⣼⠋⠀⢀⣤⣤⣤⡀⠙⣿⡄⠹⣷⡀
⠀⠀⠀⠀⣾⡏⢠⣿⠃⠀⠀⢀⡿⣹⡏⠀⠀⠀⢻⣆⣀⣹⢷⣴⠾⠋⠙⠷⣦⡾⣃⣀⣼⠇⠀⠘⣿⡄⢹⣷
⠀⠀⠀⣸⡿⢀⣿⠇⠀⠀⠀⣼⢷⡟⠀⠀⠀⠀⠀⠉⠋⠙⣷⠀⢈⣀⠲⠂⠀⣾⠋⠉⠁⠀⠀⠀⠸⣿⡀⢿⣇
⠀⠀⠀⣿⡇⢸⡿⠀⠀⠀⢰⣏⣾⠃⠀⠀⠀⠀⠀⣀⣤⣤⡿⠀⣏⣍⣳⠀⠀⣿⣤⣤⡀⠀⠀⠀⠀⢿⡇⢸⣿
⠀⠀⢀⣿⠀⣿⡇⠀⠀⠀⣾⢛⡟⠀⠀⠀⠀⠀⣼⠏⠀⣱⡟⠻⢾⡍⣥⡶⠛⢷⣍⠈⢹⡆⠀⠀⠀⢸⣿⠀⣿⡀
⠀⠀⠈⣿⠀⣿⡇⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⣸⠇⢻⣄⠀⠀⠙⠛⠛⠁⠀⠀⠀⢸⡿⠀⣿
⠀⠀⠀⣿⡇⢸⣷⠀⠀⠀⢻⡌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿
⠀⠀⠀⢹⣷⠈⣿⡆⠀⠀⢸⡟⢿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣾⡏
⠀⠀⠀⠀⢿⣇⠘⣿⡄⠀⠀⣿⣼⡇⠀⠀⠀⠐⣶⣼⡷⢦⣤⣤⠶⠾⠛⠛⠛⢷⣴⡟⠀⠀⠀⢠⣿⠃⣸⡿
⠀⠀⠀⠀⠈⢿⣆⠘⢿⣆⠀⢸⡇⣿⠀⠀⢘⣻⡏⢀⠰⠖⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠃⠀⣰⡿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣧⡈⠻⣷⣌⠛⠛⠀⠀⠉⣹⣷⣬⣀⡐⠀⠀⣀⣀⣠⣴⠾⠋⠀⠀⣠⣾⠟⢁⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠻⢷⣦⣀⠀⠀⠉⠰⠟⢹⠟⠛⠛⠉⠉⠁⠀⠀⣀⣴⡾⠟⢁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡉⠛⠿⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣶⠿⠛⢉⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣌⣉⣉⠛⠛⠛⠛⣉⣉⣡⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣹⡿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⢏⡶⣿⣾⣿⠿⠿⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣟⣋⣅⣀⣤⣤⣬⣽⣭⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠛⠛⠿⢿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⣿⡷⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣛⣿⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⢿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡧⠄⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣄⣰⠃⠈⠉⠐⠀⠀⡀⠀⣀⠠⠂⠀⠀⣰⡖⠀⠀⠀⠀⠘⡦⠙
⠀⠀⠀⠀⠀⠀⡀⢯⠀⠀⠐⠀⣠⣤⠀⣿⣤⠤⠂⣤⠄⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡔⡎⠆⠕⠛⠓⠾⢿⡀⢰⣿⡄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⡀⠈⠀⠀⣷⡶⠠⠀⠀⠀⠀⠒⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢀⣀⠀⠈⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠈⠑⠉⠿⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡈⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠊⠐⣷⣄⡀⠀⠘⠀⠀⡐⠀⠀⠀⠀⠀⡠⡤⠂⡠⠗⣼⣧⡄⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⡿⠆⣹⠟⠉⠀⣀⣶⣾⣷⣿⣷⣦⡽⠟⠛⠘⠁⣴⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠂⠀⠀⠀⠀⢼⣿⣿⣷⠶⠋⠀⠀⣤⠿⠏⠉⠉⠀⠀⠀⠀⠀⠀⢠⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠛⢻⣶⣿⡀⠀⠀⠀⢺⣿⢿⣤⣆⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⣤⡠⠤⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣏⠳⢄⠀⠈⢿⡀⠀⠀⠀⠂⠀⢀⠀⢄⠀⢤⠾⠿⠿⠿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠑⡀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠻⣿⣷⣴⣿⡀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠺⢤⣀⠀⠀⠀⠉⠛⢻⣧⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠶⢦⣄⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣀⣀⡀
⠀⠀⠀⠀⠠⠂⢠⣤⠀⠘⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⠉⠉⠲⣀
⠀⠀⠀⠀⡇⠀⠀⠈⢁⠀⠀⠀⠸⠀⠀⠀⠀⠴⠦⠶⠠⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠸⡇⠀⠙⠶⣄
⠀⠀⠀⡤⠀⠀⠀⠄⠘⠛⠻⢧⣾⣄⣀⡌⣀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠋⡀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀
⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠻⢏⠙⢶⣄⠀⢠⡴⠒⠶⠶⢀⡀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠿⠀⢋⠰⣀⣀⠀⣈⠁⠀⠀⠀⠀⠈⠛⠛⠛⠃⠈⠉⠀⠀⠄⠀⠠⠀⠀⠈⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠘⠋⠰⢋⡈⢠⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⡧⠀⠀⠀⠠⠀⠐⠖⠉⣻⠀⠀⠀⠀⠀⠀⣠⠿⣉⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢀⠀⠀⠁⠰⣿
⠀⡀⠀⢀⣤⠐⠀⠀⠸⠃⠀⠀⠀⠁⣤⠛⠁⠀⠉⢀⣄⣠⣷⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠀⠘⣿⡀
⠘⠇⠀⠉⠀⠀⠀⠀⡶⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠋⢈⡿⠁⠀⠉⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠰⠂⠰⢸⡇
⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⢁⠀⡇⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⢠⡄⣴⣂⠀⠀⠀⠀⠀⠀⠄⠀⢘⡦⠀⠠⠈⢤
⠀⢸⡀⠀⠰⠀⠀⠀⠀⠀⠠⣤⠾⠧⠤⡀⠀⠀⠀⢸⣿⡆⠀⠀⠀⡇⠘⢠⣿⣷⡄⠀⠀⡤⠀⠀⠀⢨⣧⠀⠀⠀⣤
⠀⠈⣧⠀⢰⠀⢠⠀⠀⣾⣇⠀⠀⠀⠀⠀⠉⢦⡀⠸⣿⡇⠀⠀⡄⠀⠀⣿⣿⡟⠀⠛⠳⢤⡶⠶⣄⠘⠃⠀⠀⠠⣙
⠀⠀⠈⣶⠋⠀⠘⠀⣠⠛⢿⣿⣷⣦⣘⣦⡀⠀⠐⢦⡙⣇⡀⠐⢁⣠⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⡀⢠⡟
⠀⠀⠀⠻⣀⠀⣿⠆⠁⠀⠀⠙⠋⠉⠙⢻⣟⣇⠀⠈⣿⣿⠟⠛⠋⣽⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣱
⠀⠀⠀⠈⢿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⡞⠉⢷⡾⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⢀⣾⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣰⡿⠛⠛⠙⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠓⠻⣀⠈⢿⣷⣶⣦⣄⣀⣬⣷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠂⠀⠀⠀⠀⠻⡷⣼⠿⣦⡉⠙⠻⠶⠈⠀⣉⡙⠛⠳⠾⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠏⠀⠀⠀⠀⠀⠀⠀⢀⠛⢤⡈⠻⣦⡀⠀⠀⡀⠈⣿⠀⠀⠀⠈⠻⡖
⠀⠀⠀⠀⠀⣠⠞⠉⠉⠀⠀⠀⠀⠶⣠⡶⣌⠑⠈⠀⠀⣀⠶⢻⣧⠀⠀⣹⠀⢹⡆⠀⠀⠀⠀⠀⢰
⠀⠀⢀⡶⢺⡇⠀⠀⠀⠀⠀⠀⠙⣠⠛⠀⠈⠀⢠⣤⠞⠁⠀⢸⡏⠀⢼⠁⠀⢈⡄⠀⠀⠀⠀⠀⠙⠃⢀
⠀⠀⠚⠀⠘⠇⠀⠀⠀⠀⠀⠈⠁⠛⠀⠀⠀⠀⣠⠏⠂⠀⠀⣸⣷⡀⣸⠃⠀⠘⡃⠀⠀⠀⠀⠀⠈⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡾⠁⠀⠀⠀⢤⠿⡇⠸⠁⠐⠄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡦⠀⠀⠀⢐⡿⠃⠀⠀⠄⠁⠈⢠⠘⠀⠼⠃⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉⠀⠀⠀⠀⡸⠁⠀⠀⢈⢠⠄⠀⠸⢇⠀⠀⠁⠀⠀⣶⠀⠀⠸⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠃⢸⡀⠀⠀⠀⠀⠀⠘⠀⣿⣤⡄⠀⠀⠀⠿⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠞⠳⡄⠀⠀⠀⠀⠀⠀⡄⠠⣾⠀⠀⠀⠀⠸⠗⠶⠒⢢⣄
⠀⠀⠀⠀⠀⠀⠘⠎⢻⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⠀⠀⣼⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⣜⢦⠀⣿⠊⠀⠀⠀⣀⡤⠔⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠌⢷⣠⠃⠀⠀⡴⠋⢁⣠⠎⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣠⡴⠋⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠋⣉⠉⠙⠳⢄⡀⠀⢸⠀⣠⠞⠓⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠴⠥⢤⣌⡉⠉⠉⠉⠙⠛⠦⣵⡀⢸⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠤⠤⠤⠖⠋⠙⢼⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣰⡫⠍⠉⢹⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⢤⡟⠁⠈⠛⠊⠁⠀⠀⢰⢟⣩⣍⠳⡄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣸⠇⣿⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⡏⠓⢤⡀⠀⠀⠀⣸⠃⣠⠞⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣕⡉⢹⡍⠑⡄⠉⠲⢄⣠⠏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣤⢼⣌⣹⣏⣹⣿⣉⣦⣠⢞⡥⠾⢥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠡⡞⠘⠐⢧⡞⢢⣉⠴⡏⣩⠗⠋⠀⠀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣔⠉⠓⠒⠋⠧⠞⠈⠳⣇⡙⢲⣄⣀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣷⠈⡉⠻⡭⢇⠀⣀⣀⣠⣿⣍⢩⠿⠞⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢯⣀⣈⡿⣤⣓⡾⠴⢋⡙⠷⣯⠱⣞⡆⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠐⡏⣝⢒⢆⠔⢒⡆⠓⠊⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠁⠈⠹⠆⠙⠉⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⢸⡀⠀⠀⣀⣀⡤⠴⠖⠋⠉
⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⢀⡀⢀
⠀⣠⣿⡆⢄⣰⣾⠂⡀
⠰⠚⣏⡑⡀⠾⣁⣠⠁
⣨⣓⡦⠙⠛⣊⣧⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣤⣤⣤
⢀⣠⡄⠸⢿⠙⢳⣍⣿⡏⣠⣄⠀⠀⠀⠀⠀⠀⢰⣄⣾⣿⢿⣧⣠⠀⣀⣀⠠⠀⠀⠀⡀⠤⠠⠀⠄
⠺⣿⣅⠐⠲⣾⣿⣿⣼⡿⣯⣿⣤⣀⠀⠀⠀⠁⠙⠿⣿⣿⡾⠋⠀⣭⣬⡏⣀⡀⣶⡎
⡶⠋⢫⣗⣴⣟⣋⣙⠛⣹⣿⢿⣟⠛⠀⠀⠀⠀⠀⠰⡟⠀⠀⠈⣀⡍⠀⡔⠛⠣⠾⡃⠀⠀⡰⠟⠂
⣾⣿⢷⣯⣧⡦⣟⣯⡀⠙⢿⡽⢯⠀⠀⠀⠠⠀⢠⠈⠁⣾⣿⠀⠀⣀⠞⠀⣾⣷⠀⠀⠀⡜
⣿⣿⡧⢞⢉⡤⠿⢻⡇⠀⢹⠀⠈⠀⠀⠂⠒⠀⠄⠀⡀⢠⣥⣴⣻⠋⠀⠀⡙⢿⣠⣴⢯⣀⣀⡀
⢿⣧⠤⣾⣿⣿⣾⣤⣁⣀⣿⠐⣀⡀⢤⣤⣿⣿⣶⢊⢳⣿⣷⣟⣡⣤⠋⢺⣿⣿⣿⣿⣷⣿⠟⠃
⣿⡇⡽⣿⡟⣿⣿⣿⣿⣿⢹⣤⠷⢛⣃⣩⣿⣿⡇⠸⢸⣿⣿⣿⣼⡏⠀⢾⣿⣿⢼⣿⣿⡁⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⢀⢠⣆
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢻⠿⣷⡄⣶⡿⣿⡿⠃⠰⣼⣿⣿⣿⣿⠃⠀⢲⣆⣿⡼⠿⢟⡷⣀⢀⣴⣿⡆⠀⠀⢠⣧⣀⡈⢰⣿
⣿⣾⣿⣿⣿⣿⣿⣿⡿⢛⢚⣼⣿⣇⡀⠃⠛⠃⠀⠴⢴⣿⣿⠙⠙⠂⣶⣿⣿⣿⣷⣶⣾⣶⣾⠍⠘⣿⣧⣤⣤⣧⣿⣿⣿⣿⣿
⣿⣭⣓⣽⣯⠐⣶⣶⣰⣶⠟⢊⣛⣻⣿⣿⣾⣿⣿⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣽⣥⡶⠀⠒⣄⣮⣤⣩⣬⣍⡽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⣽⣿⣿⣿⣧⣷⣶⣶⣦⣬⣭⣿⣩⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣥⣬⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢡⠟⠛⠻⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣾⣿⣿⣶⣶⣿⣷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢋⣤⣾⡟⠛⡟⣿⣿⡿⠛⢿⣿⡟⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⡠⠚⣿⣟⣻⡟⠀⣿⣿⣄⠀⠀⠹⣇⡀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣡⣶⠁⠈⠉⠏⠀⣿⣿⠛⠿⠀⠀⠸⣿⠇⠀⠀⠘⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠄⠀⠀⣠⡟⠁⣐⣆⠀⠀⠀⣿⠛⡆⠀⠀⠀⠀⠘⣷⣿⡀⠀⠀⢳⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠀⠀⣰⣿⠁⠀⠉⠉⠀⠀⠀⢻⣦⠀⠀⠀⣀⠀⠀⠻⠛⠟⠀⠀⠀⠘⢀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠁⠀⠀⢀⠛⡿⠀⠀⠀⠀⠀⠀⢀⣾⡄⠀⠀⠀⠈⠀⠀⠀⢿⣤⠀⠀⠀⠀⠘⣦⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡅⠀⠀⠀⡘⢠⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠈⢇⠀⠘⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠘⢧⣿⠀⠀⢀⣄⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⣧⠀⠘⣄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠰⣷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⠙⡄⠀⠻⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠐⢁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣸⣿⣿⣿⣿⣯⠉⠛⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠻
⠀⠀⠀⣿⡇⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⣵⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆
⠀⠀⠀⣿⣟⣻⣿⣿⣷⣦⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠇
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⡀
⠀⠀⢸⣿⢸⣿⣿⣿⣿⠿⠃⠀⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⡇⢈⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣼⣧⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣿⣿⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⣿⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣿⣿⢸⣿⣿⠉⡿⠁⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢹⣿⠸⣿⣯⢸⣦⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢸⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⠂⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢸⣿⡄⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⣿⡇⣿⣿⣿⢁⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⣿⣿⢹⣿⣿⣞⠛⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⢸⣷⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⣿⣿⣼⣿⣿⣿⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⣿⡉⣿⣿⣿⡧⡄⠀⠀⠀⢜⣿⠀⠀⠈⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢻⣿⣽⣿⣿⣿⣆⡀⠀⠀⢸⣿⠄⠀⠀⠠⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⢀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⣠⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣼⣿⣥⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣀⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣻⡰⠶⣀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣠⣾⡇⢀⣴⣨⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣴⣾⣧⣼⣤⣤⡼⣶⣿⣿⣳⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠟⠛⠿⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠋⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣤⣈⢳⡾⣁⣤⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢷⠀⠀⠀⠀⣧⠀⠈⢻⡟⠁⠀⣼⠀⠀⠀⠀⡾⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸⠃⠀⠀⠀⣈⣓⡶⢾⡷⢶⣚⣁⠀⠀⠀⠘⣇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣜⣶⣋⡤⠤⡄⠀⣏⠀⠙⢾⡷⠋⠀⣹⠀⢠⠤⢤⣙⣶⣣⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⣿⠁⠀⣠⠇⠀⢈⡳⠤⢼⡧⠤⢞⡁⠀⠸⣄⠀⠈⣿⠁⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣽⡒⣿⢒⣫⢥⡄⠀⣏⠉⠙⢾⡷⠋⠉⣹⠀⢠⡬⣝⡚⣿⢒⣯⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⣿⠋⠀⣠⠇⠀⠙⠦⣤⣼⣧⣤⠴⠃⠀⠸⣄⠀⠙⣿⠋⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣓⡒⣿⠖⣚⣁⡀⠀⡏⠉⠳⣼⣧⠞⠋⢻⠀⢀⣈⣓⠲⣿⢒⣚⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠙⣿⠋⠁⢀⡇⠀⠙⢦⣀⣸⣇⣀⡴⠋⠀⢸⡀⠈⠙⣿⠋⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣿⠤⠖⠋⠀⠀⡞⠛⠲⣼⣧⠖⠛⢳⠀⠀⠙⠲⠤⣿⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣄⣀⣸⣇⣀⣠⠏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⠀⠀⡿




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠟⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠟⠛⠋⠉⠉⠉⠉⠉⠛⠻⢿⣶⣄⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣰⠶⠛⠉⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⣰⡿⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⣴⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠁
⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠶⠞⠛⠛⠿⠶⣦⣤⣄⡀⠙⠁⣾⠋⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠟⠁
⣿⣿⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⣀⣤⣴⣶⡾⠿⠛⠋⠁
⠉⠉⠉⠉⠙⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣦⣀⠈⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡿⠿⠛⠉⠙⠿⣷⣦⡀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⠿⠟⠛⠻⢿⣿⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠰⢶⣦⣼⣿⠀⠀⣼⠟⠿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣾⣿⠀⠀⠀⠉⠻⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠉⢻⣿⣿⣿⣾⠿⠿⢿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣼⡿⠀⠀⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⣿⡟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠈⠛⠿⣿⣿⣷⣶⣶⣶⣿⡿⠿⠿⠿⠿⢶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣿⣦⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣰⣿⠛⣿⡿⢿⣿⠿⠟⠛⠛⠋⠉⠛⠛⠻⠿⢿⣶⣤⡀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢠⣿⠇⠀⣿⡇⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⡿⠀⠀⠻⣿⡀⠹⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⣇⠀⠀⠀⠙⢿⣦⣀⠉⠙⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠸⣿⡆⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣤⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠛⠻⠿⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⡀⠀⠀⠀⠉⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⣶⣿⣿⣷
⣶⣷⣶⣿⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣾⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣿
⣿⣿⣿⣿⣿⠛⠋⠻⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⠟⠿⠿⢿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣍⡻⣿⡿⢸⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣭⣴⣿⡿⢛⣛⣿⠿⠿⣿⣿⣿⣿⣶⣄⣀⠀⠀⢨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣶⣿⣿⡏⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⣫⣥⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⢸⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠠⠀⠈⢿⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⡙⣿⣿⣿⠿⠿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠛⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⣻⣿⣿⣦⣀⣠⣿⣿⣿⡿⢃⢿⣯⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣧⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣯⠀⠀⢀⣿⣿⠟⠉⡉⠙⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⢿⣿⡀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠂⠀⣿⣿⣷⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⢀⣹⣿⣿⣷⣶⡀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣷⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⣀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⢦⣶⠛⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠘⠀⠀⠀⠙⠘⡀⠀⠀⠀⠁⠉⣦⠂⠆⣔⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⢎⣱⣿⠷⣶⠉⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⢻⣄⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠄⠈⢋⣿⠢⠁⢄⠈⢸⣏⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢠⣧⠘⠺⠅⠀⠀⠔⡀⠀⠄⠀⠀⠀⠀⠀⠀⠘⡛⠃⠀⠐⠚⠻⠏⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⡟⠀⠀⠠⠀⡗
⠀⠀⠀⣀⠀⠐⠈⠀⠀⢀⣔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢄⡂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⡀
⠀⠀⢸⣿⠀⠀⠀⠀⣤⡼⠟⠒⠒⠀⠰⠂⠀⠀⠀⠀⢰⡀⠚⠀⠀⠀⠀⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⡇
⠀⠀⢸⣿⡄⠀⠀⢤⡏⠁⡀⠀⠀⠀⠀⠘⠗⠁⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⠀⠀⡆⠀⠀⢂⣀⠃⢉⡽⡀
⠀⠀⢸⣿⡇⠀⠀⠈⠛⠚⣁⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣆⠀⠀⣤⣦⣞⡜⠁
⠀⠀⢸⣿⣷⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠿⠟⠻⢿⡿⠇⠀⠀⣽⠝⣸⣃⠄⠀⡀
⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⠄⠒⠢⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⢬⠀⣴⠞⠶⣿⡿⣿⣶⠋⠀⣿⣿⠆⣸⠃
⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠀⠀⢰⡟⡰⠁⣄⡠⠄⠀⠀⠀⠀⠀⠀⢁⡂⠀⠈⠙⠻⠆⠠⣈⠋⣈⢫⣤⣤⣿⣿⣿⣇⣆
⠀⠀⠈⢿⣿⣿⣿⣄⣆⠀⢀⣤⣶⠀⠀⣿⣷⡇⠀⡀⣰⡷⠀⠀⠀⠀⣤⣦⣄⢀⠀⢀⡄⠀⠀⠀⢻⡿⠿⠿⣿⡿⠿⠿⢛⡠⠝⠁
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣧⣡⠍⠀⢸⣿⡇⣠⡘⢛⣻⣿⡀⡀⡄⠍⠁⠉⠻⠀⣣⡜⡂⠀⠀⠀⣾⡇⠀⠠⣶⡄⠒⠒⠁⠐⠀⣀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣛⢙⣇⣼⢻⡟⣀⣸⣿⠛⠉⠀⠀⠙⣯⠀⠀⠀⡀⠸⢿⣿⣷⣶⠄⡀⣿⣧⣾⣦⡛⠂⠀⠀⢠⡶⠋⢁
⠀⠀⠀⠀⠀⠉⣶⣿⣿⡏⠋⠻⢿⣷⣾⠃⠀⠈⣷⠀⠀⠀⠀⢰⣧⣤⣀⣤⡇⠀⣤⣷⠚⡛⠊⢳⣿⡿⠟⢻⣿⡆⠀⠀⡎⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢀⢸⣿⣿⠈⣷⣤⠏⠀⠀⠀⠀⠈⢻⠿⢫⠥⠁⠀⠛⠉⠀⢛⣛⣤⣶⣷⠆⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠈⠘⠛⢿⡈⠿⠿⠦⣀⣤⣤⣤⡤⠜⠀⠀⠀⠀⠤⠴⠶⠿⠿⠛⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠤⠤⠄⠀⠐⠒⠒⠒⠒⠲⠦⠤⠄⣀⠑⠲⠤⡀⣀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⠤⠴⠶⠒⠒⠒⠒⠒⠒⠤⠤⠄⠤⣀⡀⠉⠑⠄⠈⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠄⠀⠠⠤⠤⠐⠒⠒⠒⠒⠒⠒⠂⠤⢄⣀⠈⠁⠃⠀⢀⠀⠠⡄⠐⠈
⠀⠀⠀⠀⠀⢀⠀⢤⠀⠀⠀⠀⢀⡑⠒⠀⠀⣀⡀⠀⠒⠒⠒⠒⠒⠒⠠⢄⣀⠀⠀⠉⠂⢌⠀⢄⢀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⠀⠀⠈⠑⢄⠀⠀⠈⡃⠀⠈⡆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⡆⠀⡸⠁⠀⢀⡇⠀⠀⢸
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠃⡐⠁⠀⠀⠜⠀⠀⠀⢸
⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠂⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢈⠰⠃⠀⠐⠀⣙⠌⠂
⠀⠀⠀⠀⠀⠀⠠⠡⢀⠀⠀⠀⠀⠈⡉⢦
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⡗⠀⠀⠠
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⡁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣶⣾⣷⣼⣦⣶⣿⣿
⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣒⡿
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣿⣶⡟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠁
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣛⣿⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⡴⡫⣾⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣤⠀⣰⢶⡫⣮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⢀⣄⣴⣿⠵⣫⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⣿⣿⣿⡿⠛⠃⠈
⠀⡀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣚⣿⣻⡿⠁⠈⠁⠈⠀⠀⠀⠀⠀⠂
⠠⣭⣑⠪⢭⣛⣓⠶⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣲⡦⣶⣻⣿⡿⢹⠂⣤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠓⠲⠯⣭⣛⣒⠦⠤⣄⣀⣃⣀⠀⠀⠤⠤⠄⠒⠛⢽⣿⣷⣶⠟⣽⣻⣯⣿⠻⢶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⢤⢭⣽⡿⠏⢠⣤⣾⡖⢋⣽⠗⣂⡀⢸⣧⣿⠟⠓⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣤⣌⣛⣺⣿⣿⣭⣾⠟⠁⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠤⠄⠀⣀⠀⠀⠀⠈⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠠⡀⠀⠀⠀⠁⠀⠀⠀⢻⡷
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠂⠁⠰⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠐⠒⠊⠉⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⠀⢠⢨⡗⠀⠀⠀⠀⠀⠤⠥⠄⡀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠛⠋⢉⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠰⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠉⠁⢀⣀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⡉⠙⠛⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⣀⠤⠒⠉⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡀⠀⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡠⠊⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠉⠑⠢⣄⡀⠀⠙⢷⡀⠀⠀⠀⣸⠃⣠⠊⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠲⣄⠀⢻⡄⠀⢀⣏⡔⠁⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⣄⢷⠀⣸⣟⣠⣤⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⣀⡈⢿⡆⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠖⠋⠉⠁⠀⠀⠀⠉⠙⠲⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠆



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⣠⡀⣀⠀⣀⡀⣠⢀⠀⡀⢀⣀⢠⠠⣀⢀⠀⣀⠀⢄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠁⠈⠀⠀⠁⠈⠈⠀⠀⠀⠁⠀⠈⠁⠁⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡆⢀⠀⡀⠀⡄⠘⣆⣘⣄⣸⣤⣧⣴⣥⣎⠎⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⢰⠸⡀⢠⠀⣇⢈⣆⣡⣤⣧⡼⠶⠷⣚⠛⠻⠉⡉⢩⠄⠀⠀⠀⠈⢝⢿⣴⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢱⠀⢇⣀⣱⣬⣦⡴⠶⢷⡛⠛⠋⠉⠉⠑⡀⢀⠦⢠⠄⠀⠀⠶⠀⠁⡄⠀⠀⠀⠀⣘⠃⠁⢹⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⣼⠞⠛⠩⠉⣅⠀⠀⣈⣙⡤⣄⡀⢀⢀⠁⠀⠁⠀⠀⠄⠀⠙⠀⢠⠄⠀⠤⠀⠀⢠⡀⠀⠠⠆⠀⣷⣒⠂⠈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣹⡞⡑⠙⠆⠀⠀⠀⠀⠀⠐⠢⣌⠁⠮⠋⡣⠑⠀⠘⡃⠀⠀⠁⠀⠀⠀⠀⠋⢀⡀⠀⣀⡀⠀⠖⠐⣼⡓⠤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠤⡤⣿⠀⠸⠃⠛⠀⠀⠀⠀⠀⢸⣀⣼⣧⠶⠠⠃⡜⠀⢀⠀⠠⠠⠀⠀⠀⠀⠖⠀⠆⠀⠲⠈⠠⣐⣥⢾⡓⠬⠁⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠐⠒⣿⡀⡠⡄⠉⢀⡄⠶⠀⠀⠀⠓⠀⠉⠤⠍⠙⠁⠤⠈⢐⡀⠫⠄⣀⣀⣀⣤⡤⣶⠶⡖⢛⠻⡙⠄⠣⠉⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡨⢷⣜⠀⢃⠀⠀⠶⡀⠀⠀⠘⠀⣀⣒⣠⣤⣤⡴⠶⡖⠺⡟⢻⠉⠋⠃⠘⠀⠉⠀⠁⠈⣠⣴⡶⠶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢊⡝⢻⠶⡶⡷⢶⠶⡞⢻⠛⢏⠹⡉⢣⠸⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⠀⠃⠘⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢶⡶⠾⠋⠀⠀⠀⠀⠀⠂⠀⠂⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈

⣴⣶⣶⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣀
⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣀⢀⠀⠀⠀⠐⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣧⣤⣤⣄⣀⣀⣤⣄⡀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂
⠀⣠⠀⠛⠉⠹⠟⠛⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠉
⠀⠿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠈⠀⠀⠀⠀⠘⠃⠀⠀⣄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣠⡤⠴⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠛⠿⠆⠀⠀⡄

⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⡏⠀⠘⢿⡏⠲⠂⠀⢠⣴⣄⣠⡿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⣧⣄⠀⢸⡇⠀⠀⠀⠘⣿⣿⣿⠀⣿⠃⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢀
⣿⣿⣿⣿⣷⣶⣄⠀⠀⣿⢿⣯⣴⡏⠐⣯⣀⣠⣶⣤⣴⣶⣤⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⢰⣿⡿⢿⠷⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣶⣶⣦⣐
⣿⠿⣿⣿⣿⣿⠿⠶⡄⣹⡿⣿⠿⠿⡿⢿⠿⢿⣿⣿⣷⣾⣭⣹⣿⡏⢩⣿⣿⣷⣶⣦⡄⣀⣨⣿⠇⠈⠰⡇⠻⣿⠓⣶⡶⢀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿
⣿⣿⡀⣿⣿⣿⣿⣿⠟⣨⣷⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⢙⣿⣿⡿⠿⣽⣟⣿⣷⣾⠟⠲⡀⣠⣿⣏⣁⣀⣴⢏⣹⣿⣿⣿⣥⣤⣴⠖⠒
⣿⣿⡇⣿⣿⢻⡿⢫⣾⣿⣿⣿⠈⠢⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠙⢿⣷⣶⣶⣿⣫⣿⣽⣿⠋⠐⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⢿⣿⡇⣿⣇⡟⣵⣿⡻⣿⡬⣿⡦⠀⠈⢆⠀⠀⠀⠿⠉⠉⢉⣻⠟⣿⠟⢛⣿⣿⠟⠛⣻⣷⣦⡀⠋⠁⢠⣉⣯⣴⣿⣧⣤⡾⠿⠿⠿⠿⠿⠛⠛⠛⠛⢻
⠈⠻⠃⢸⠃⣾⣿⣿⢹⣿⣷⣿⣷⠀⠐⠆⠁⢀⡀⠀⠀⠀⠈⠋⢲⣟⣴⣿⣿⣷⣌⣿⠛⠿⠿⣿⣿⣄⢸⣿⣿⣿⣿⣿⣿⠉⠁⠀⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⢉⠛⢻⢾⣿⣿⣿⣿⣁⣀⣤⣤⣈⡓⢄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣾⣧⣄⣠⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣅⡀⠀⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣿⣙⠉⠻⢿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⡘⠿⠿⢿⣿⣶⣶⣾⣿⠈⣿⡿⣿⣿⣿⣿⣿⣷⣤⣛⢻⣿⣿⣿⡃⣾⣿⣿⣿⠉⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣄⣿⣿⣿⣿⣎⠛⣿⣟⠛⣿⣿⣿⣷⠼⠿⣿⣿⣶⣦⣬⣨⣿⣿⡀⢿⣷⣾⣿⣿⣿⣿⣷⣌⠻⣾⡿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣸⡿⠿⣿⡿⢻⣿⠃⠈⡿⣦⢛⡋⠙⣿⣴⣿⣿⣿⣿⣿⡛⣿⣿⣿⠃⠈⣧⣽⣿⣿⣿⣿⣙⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿
⢿⡇⠀⠹⣷⣼⡏⠐⠀⠀⠁⠈⠀⠃⠛⠟⠀⠹⣿⡿⠻⣿⣿⣿⣿⠁⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠻⣿⣿⣿⣿⣿
⠘⣇⠀⠞⢻⣿⡇⢀⠀⠀⡀⠂⠀⢠⠀⠀⡄⠀⣿⠿⠈⠙⢿⣿⣿⣷⠀⠈⢿⣉⠹⣿⠙⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢼⡄⠀⢸⣿⣇⣾⣧⡇⠉⢀⠀⣿⣇⡄⢿⣀⡧⠀⡔⠀⠸⣿⣱⠀⢃⣀⣾⠁⠀⣿⠰⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣧⢠⣿⣿⠏⠸⣧⢶⡄⣸⡆⣿⣿⣧⣼⣿⣇⡾⢀⡀⣠⣿⣯⣄⠈⠉⣿⣄⢠⡏⠀⢻⣿⣿⠙⢿⡏⠩⣿⠀⡿⣿⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⣿⣿⣿⣿⡀⣻⡏⢸⣧⣿⠁⣿⣿⠟⢸⡿⣿⠁⡀⡇⣵⢟⡘⣿⢀⠀⢸⣿⡼⠇⠐⠀⢹⣏⠀⠀⣿⠀⣿⠀⢃⠈⢿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣷⣠⣿⣿⣿⣿⣿⣿⠃⣾⢻⣿⣴⣿⡏⢀⣿⡇⣿⣴⡟⢁⣡⠞⢹⡏⠛⠀⣼⢻⣧⢠⡔⢲⣄⣿⠀⢸⣿⣷⠟⠀⣻⣆⠈⣿⣟⢻⣿⣿⣿⣿⣟⣿⡿⢋
⢿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣷⣾⡟⣽⣿⣿⡇⢀⢿⠋⣾⠇⠀⢰⡇⠀⣯⠀⠁⢸⣟⣃⣴⣿⣻⡏⠀⢠⣿⣿⣤⣿⣿⣿⡿⢹⣿⣿⠁⣸⠃⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣜⢸⢸⣿⡄⠀⣸⣷⣾⡇⢨⡇⣿⡿⠋⣸⣿⣿⡇⢲⣮⣿⣿⣿⣿⣿⣏⣷⣾⣿⣿⣦⠁⢠⣿
⢻⣿⣿⣿⣿⣿⣿⡉⢹⣿⣿⡏⣿⣿⣿⣿⣷⣿⣿⣿⢰⣿⣿⣿⣿⢡⣾⣿⣿⣿⣧⣾⢃⣿⠀⢠⣿⡏⣿⣿⣾⣿⣿⣯⠹⣿⣿⣿⣿⣿⣿⡟⢻⣦⣾⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⡄⣹⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡋⣾⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⠻⣿⣿⣟⡛⣛⣿
⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣻⣿⠀⣿⣿⣿⣿⣿⣿⣧⢈⣿⣿⣿⣆⡸⣿⣿⣿⣿⡿
⣾⣿⣿⣿⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⡟⢿⡛⠋⠉⠹⢻⣿⣿⣽⣧⣿⣿⣿⣧⣿⣾⣯⣿⠿⣿⣿⣧⣿⡏⣸⢻⡏⠹⣧⣾⣿⣿⣿⣿
⢿⣿⣿⣿⡿⠃⠀⣿⣿⣿⡟⠋⢩⣿⣿⣿⣿⣿⣿⣿⣿⡷⠚⣻⣶⣤⡴⢻⣿⠀⢹⣿⡙⣿⣿⣿⣿⠋⢹⡇⢰⡟⢿⣿⠏⠀⠉⢘⡇⠀⢻⠉⡄⣸⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⡀⠀⠀⠀⠀⣀⣾⣥⡠⡀⠄⠠⣀⣠⠒⠀⠐⠒⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠌⠉⠉⠓⢀⢸⣧⡄⠀⣰⠟⣁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⣥⣾⣧⠞⠛⠋⠐⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠔⠭⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠉⠣⡀⠀⠀⠀⠀⠀⠀⠈⠆⢀⠀⠀⠈⠉⠒⠄
⠀⠀⠀⠀⠀⢀⠀⢠⠔⢣⠖⠀⠀⣠⠋⠀⠜⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣄⠀⠀⠀⠀⠀⠑⡀⠀⠄
⠀⠀⠀⠀⠀⢁⠐⡙⠋⠓⠀⣠⡪⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠷⡀⠀⡄
⠀⠀⠀⠀⢀⣏⠄⠀⠀⠀⣴⠉⢀⠀⠀⡴⠀⠀⠞⠁⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⢱⠀⠙⡀
⠀⠀⠀⠀⢸⣩⠀⠀⠀⡜⠁⠀⠬⠀⠰⠀⠀⠀⠌⠀⡜⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⣾⢻⠀⠀⢰⠃⠀⠂⠊⠀⠀⠀⠀⠀⠀⢠⠇⠐⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠂⠀⠀⠈⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⣴⡄⡄
⠀⠀⠀⣵⠏⣫⡄⠀⡟⠶⢃⢰⡧⠀⠀⠀⠀⠂⠀⡬⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⡀⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⠀⠀⠀⣤⡸⡄⢀⠄⠀⠈⠿⠿⡃
⠀⠀⠀⣿⣸⣿⣼⣾⡇⣸⠃⣼⠀⠄⠀⠀⢠⠀⢰⡇⠀⢠⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⠁⠀⠈⡀⠀⡀⢀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠳⡀⠀⠀⠈⢠⣼⡇
⠀⠀⠀⢹⣿⡿⢿⣿⡇⢿⣦⢿⣷⣀⠰⠀⠈⠀⢺⡇⡇⠘⡄⠀⠀⠀⠀⠀⡆⠀⢰⡄⠀⠀⠀⢸⣶⡄⠇⡘⠄⠀⡀⠀⠀⠑⠀⠐⠀⢀⡇⠐⠀⠀⢺⢧⡇
⠀⠀⠀⢸⡇⡅⣻⣾⠁⡈⠹⢚⣄⠀⢠⢀⠀⠀⣿⡇⠁⠀⠙⡆⠀⡀⠀⠀⡝⠀⠐⠂⠠⠀⠀⠀⢩⣷⡆⠀⢀⠀⠀⠀⠆⠀⠉⠁⠀⢘⠃⠀⠀⠀⠸⠸
⠀⠀⠀⢸⡙⣿⢸⡿⡇⢿⡇⢰⠀⡄⠀⠈⠃⠀⣘⣧⡀⢀⡆⠀⠈⡄⠰⢠⡄⠛⢀⠀⣤⠀⠀⠀⠀⢹⡇⠀⠀⠁⠈⠀⢀⠀⠀⠀⠀⢸⡆⠀⠰⠂⠁
⠀⠀⠀⠀⣿⣏⣾⣇⣧⢸⡁⣆⡆⠛⢀⠂⢦⣀⣿⣏⢷⡸⣿⡄⠀⢳⠀⠬⠍⠓⣶⣦⡌⠉⣤⠀⠀⢸⣧⡀⢸⡀⠀⠀⠸⠀⠀⠀⠀⣼⢁⠀⡁⢠⡖
⠀⠀⠀⠀⠘⡟⠟⢯⣇⣯⡻⢷⣗⣴⣨⠛⢳⣞⣿⡧⢸⣧⡀⣎⠀⠸⢸⣄⣀⠀⢠⣉⠆⠀⠉⠀⠀⣿⣯⢛⡈⠷⢠⠀⠀⠀⢰⠁⣠⠏⢎⡞⣧⠃⠃
⠀⠀⠀⠀⠀⠘⢦⡳⢼⣿⢿⡰⢹⣿⡜⣷⢿⣴⣿⣿⠺⢿⣛⣾⣧⣅⠀⣈⢉⡀⠈⠹⣦⡀⢦⣠⠀⣿⣏⠊⢷⢂⡌⢁⣶⢠⠆⠀⢠⣠⢼⡃⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠿⢤⣮⣿⣤⡠⡟⢭⣮⣼⣽⣿⡄⠰⣇⡼⣇⠺⣾⡥⢨⣛⣆⣀⣩⣥⡀⠄⣺⣿⣟⣡⠴⢋⡁⠀⣤⢛⡴⢰⠟⠓⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠮⣷⣾⣯⣭⢻⣌⠙⠺⣞⠲⠶⣶⣌⢉⡿⢥⡤⣷⣴⢽⣿⡗⡭⣰⣾⣽⠟⡍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠘⠟⠾⣶⣬⣿⣿⠻⣿⣳⢠⠤⡘⢛⠹⠿⣯⣭⡥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⢠⡄⢤⣤⢤⣤⡄⣤⣤⢠⣄⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣇⣾⣿⣿⣿⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢸⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤⣬⠀⠀⠤
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠄⣿⡇⢸⣿⠀⠀⠀⠀⠀⢸⡏⠉⠉⣿⠀⠀⠀⠀⠀⡄⠀⠠
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣗⣿⡿⠀⠀⠀⠀⠀⠀⢰⡆⠈⣿⠀⢻⠇⠘⡟⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣠⣾⣿⣷⣄⣿⣿⡧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠉⠀⠈⠀⠀⠉⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡡
⠀⠀⠀⠀⠀⠀⢠⡿⣟⠽⠋⢹⣿⣿⣿⠟⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⡂
⠀⠀⠀⠀⠀⢀⡿⢊⠁⠀⢀⣿⣿⣿⠃⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⢀⡤⠒⠒⣀⠒⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠐⠻⠃
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⣴⣿⣿⣿⣿⣷⣼⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣥⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠃⠀⠀⠀⠀⠀⢰⡿⣛⠿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠸⣇⡁⢰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⡷
⠀⠀⠀⠀⠻⠥⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠈⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣤⢠⣀⡄⣀⣠⢀⡀⡄⡀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠰⠇⠎⠼⠰⠡⠆⠮⠠⠠⠠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣴⣶⣿⣶⣦
⠀⠀⠀⣰⣋⡘⣘⣁⢋⡋⣘⢁⢀⢀⠈⠀⠐⠁⠀⠀⢀⡔⠄⠠⠁⠀⠀⠀⠉⠀⠠⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⢰⠆⡴⠰⠤⠆⡶⠴⠠⠀⠄⠤⠠⣀⠁⠀⠀⠀⠈⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⡐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠈⣉⣬⣽⣿⣿⡏⡋⠀⠀⠀⠀⡤⠀⣼⡟⢉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣿⣿⣿⡆⠀⠘⣿⣿⣿⣿⡇⣹⠀⠠⣶⣆⠉⠙⣿⣿⡿⠟⢃⡀⠄
⠀⠀⢀⡄⢂⣀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⢼⣿⣿⣿⣧⣤⡴⢛⣿⣿⣿⢿⣿⠆⠀⢸⣿⡗⡖⢾⣿⣿⡿⠿⠷⢶⣀⣀
⠀⣀⣾⡶⠜⢃⣠⡀⠤⠤⠖⠂⠠⠤⠤⠠⠤⣿⣿⣿⣿⢽⣿⣿⢽⣿⡿⠿⣿⣶⣶⡾⡿⠇⠇⠾⠿⢷⣗⢈⡐⠛⠛⠙
⠀⠉⠉⠙⡋⠨⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠄
⠀⠀⢠⡘⠸⣆⢓⡢⢸⡕⠶⣈⡿⣤⡎⠛⣤⡕⠊⡊⢄⠀⠀⡰⠜⠤⣴⠐⡀⠄⣠⠨⢥⡆⡤⢁⠄⠀⣆⠠⠀⠘⠀⢠⡤
⠀⠀⣈⡏⣳⢠⢏⠅⠐⠿⡃⢁⡟⡟⣡⠝⢾⠠⢕⡿⣾⣌⡌⠢⢠⢔⣦⣾⢠⡞⢯⡵⠲⣀⣼⣇⢶⡏⢩⣉⠁⣀⡛⠡⣷⡶⠚
⠀⠀⢼⡇⣿⡼⡇⣊⣹⢾⡇⣾⡝⣧⢽⡄⣺⡀⣞⣁⠿⣈⢱⡶⡌⢸⡵⠆⣠⠄⠛⡏⠷⣷⠙⣹⠋⠐⠪⠵⠛⠑⠁⣀⡿⡤⠌
⠀⢀⣸⡇⣿⠘⡗⣾⠉⢺⡇⠹⣝⣧⢹⡇⣿⢙⣿⠂⣻⢐⠏⡓⠱⢛⢿⠆⢾⠀⣶⣄⠀⣿⣋⣽⣷⠸⡷⣻⣟⡤⢣⣯⠶⢀⠁
⠀⠀⠤⢯⣍⣉⡇⠋⠒⢺⠃⠴⠲⡧⠼⠇⣸⡄⣏⣁⣿⠈⠸⡝⡞⠉⢻⠃⢞⠠⠶⠏⠡⣿⡁⢎⣿⣘⡗⣩⣟⠗⢲⠇⢾⠐
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠚⠷⠦⠤⢼⣤⣤⣀⣿⠀⠉⠨⣿⠀⠉⠓⢚⠡⠹⠄⠀⣿⡀⣀⣻⣬⡏⣹⠟⠁⣹⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠛⠲⠶⠤⢼⣄⣀⡀⠀⣿⠀⠀⠁⢸⡏⠁⠑⠒⢹⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠒⠆⠧⠤⠀⠀⣺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⠿⣿⠿⠉⠉⠙⠛⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⡸⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢖⢆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠱⡀
⠀⠀⠀⠀⠀⢠⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⡏⠰
⠀⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⣸⡗⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢺⠀⠆
⠀⠀⠀⠀⢘⣽⡇⠀⠀⢀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⢀⠃
⠀⠀⠀⠀⠈⣬⣧⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠱⡟⡉
⠀⠀⠀⠀⠀⢸⣿⣏⡀⠀⣿⣄⠀⠀⢀⠀⠀⢁⠀⠀⠀⢀⣿⠀⠀⢂⣴⣯⠁
⠀⠀⠀⠀⠀⠀⠛⣿⣴⣰⣙⣿⣆⠀⠀⠀⠀⠸⠀⠀⠀⢼⡿⢢⢄⢢⡿⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣀⡻⣿⣧⣀⠀⠀⣰⡆⠀⠀⣿⣇⣼⣯⠟⠁⢀⣠⠄⠒⠉⠉⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣅⣾⣼⣧⢨⣾⣿⣿⡾⢋⠔⠊⠁⠀⠀⠀⣀⠤⠒⡶⠄⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠻⠿⠿⠿⠟⠛⠋⠁⠐⠁⠀⠀⠀⠀⠐⣯⡽⢗⣦⡴⠒⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⣻⢻⣗⣏⣽⠿⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣶⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣔⠂⠐⠒⠀⠒⠂⠀⠀⠀⠉⠀⠀⣽⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⡀⣠⣀⣄⣤⣤⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠲⡎⠱⢠⡔⢤⡔
⣧⠍⠈⠀⠔⢲⣆⡀
⠛⠳⢄⡂⣰⠟
⠀⠀⠞⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⣀⢀⣀⣤⠀⡀⠀⢀⡀⠀⣀⢀⣄⣄⢀⡀⠀⢀⢀⣄⣠⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡿⣄⣿⢸⣧⠤⢸⡇⠀⢸⡇⠀⣿⢸⡧⠄⢸⡿⣆⣸⠀⣠⡟⠁⢠⡛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠃⠈⠟⠘⠇⠀⠘⠷⠦⠈⠳⠾⠃⠸⠷⠦⠸⠃⠈⠛⠐⠿⠶⠀⠟⠁⠙⠇
⠀⠀⠀⡀
⣰⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣢⡀
⠎⢳⣿⢲⡆⣠⠎⠀⠀⠀⠀⠀⠀⠙⠋
⠠⠈⢀⡁⢠⣿⡄⢀⡤
⠠⠄⠈⣀⠉⠉⣘⣡⠄

⡆⠠⡄⠀⠀⠴⣯⡅
⠀⠠⡄⠀⠑⢆⢀⡋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢲⠀⣄⠀⣴⣠⡿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⢅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⣤⣄⣹⡰⢿⡉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢁⡄⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠙⠃⠸⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀⣠⣜⠓⠂⠀⠀⠀⠀⠐⠶⢤⣴⢧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠿⢹⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣷⣶⠾⣶⣶⠜⠀⠀⠀⠀⠀⠀⠠⣄⠈⢛⠀⢸⠃⢰⡆⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠛⠋⠁⠈⣦⡄
⠐⠾⠇⠀⠀⠀⠀⠀⠙⣹⣧⣤⣿⡛⠳⠆⠀⠀⠀⠀⣤⠀⣘⡶⠾⠛⠛⣟⠿⠃⢴⡎⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⡿⠟⢋⠀⠀⠀⠀⠠⣄⣩⠾⠇⠘⢗⠀⠀⠉⠀⠀⠰⠞⠛⣶⡞⠉⠀⣀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠂⠀⠈⠀⠀⣀⡀⠀⢨⡟⠘⠳⡄⠈⢀⠀⡿⠁⢀⠴⠃⠶⠟⢷⠖⠛⠁
⠀⠀⢀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⢙⣋⠀⢈⡁⠈⠳⠈⠀⠰⠂⢀⣄⠰⠶⠿⢶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣨⣄⣠⣃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡋⠀⡈⣁⢀⡀⠀⠷⠀⡀⠀⠀⡀⠀⠠⣴⡇⠄⠀⠀⠀⠀⠀⠀⠀⣀⢰⡭⢱⡄⠀⠀⠀⠈⠷⠘
⢻⣌⠁⠈⠀⠿⠄⡶⢀⢀⡤⠚⠁⠀⠀⠀⢠⣤⠄⢿⡴⠟⠉⠀⠀⣠⠀⠀⢁⠐⠶⡉⠳⣶⡾⠷⠄⣠⠄⠀⠀⠀⠀⠀⠈⠳⠴⠛
⡀⠙⠀⢀⡔⢀⠘⣧⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣀⣰⡆⠀⠉⠀⣤⠀⣧⠄⠀⣦⡸⠷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠲⠀⠈⠀⠀⠀⠀⢤⠿⡞⠁⠀⠀⠀⠀⠀⠀⠚⠋⠀⣘⡿⢅⣠⣜⣇⠀⢀⣼⣀⠴⢾⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠛
⢀⠤⠀⠀⠀⠉⠛⠰⠖⢀⡷⠄⠀⢤⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠿⠉⢻⠛⡛⠙⠇⠀⠀⠻⠖⠀⠀⠀⠀⣿⢀⠤⢤⡄⢻⡁⠀⠀⠀⠀⠀⠘⢋⡰⠖
⠋⠠⠄⠸⡏⠐⣤⠔⠶⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠁⠀⠀⠷⠀⠀⠐⢷⣆⡀⠀⠀⠀⠈⡷⢠
⡄⠦⠀⣤⠤⠦⡁⢠⠤⡀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⢀⠁⠀⠀⠀⠀⠀⣠⠀⠠⢈⣡⡞⠃⠀⠈⠁
⠱⠶⢆⡻⠀⠄⠉⠲⠀⠈⠣⡤⠀⠀⢠⣄⠀⣱⣄⡄⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠳
⠇⠀⠘⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣁⢈⡁⠶⠈⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠈⠴
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠓⠻⡧⠌⣠⠀⠴⠒⢴⣧⣀⠀⠀⠠⠼⠃⠀⠀⠀⠀⠘⣭⠀⠀⠴⠀⠀⠀⠰⠆⠀⠀⠈⠀⠄⡴⠀⠀⠀⠋⠓
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣶⡀⠰⠂⡰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡵⣄⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⠁⡴⢤⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠉⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠿⢧⠄⠐⡦⠀⣀⠀⢠⠿⠾
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠠⡬⠁⠐⠈⠁⢡⡄⠀⠀⠀⠀⠀⠀⡀⠀⢠
⣰⢦⡗⢀⣸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣧⠸
⠉⠉⠁⠁⠩⠶⠒⢄⣰⠃⠀⠀⠀⠀⠀⣎⣷⠀⠀⠀⠀⠐⢢⡄⠀⣀⢨⡄⠈⠀⠉⠀⠙⣿⠶⣆⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁⠘⠆
⠀⠀⢌⠁⠀⠀⠈⠙⣃⣢⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠋⢠⡀⠁⠀⠐⡂⠀⠀⠁⠀⠉⣱⣖⠁⠀⠀⠀⠀⠀⠀⣀⡱⠀⠀⠈⠉⣻⡉⢁⡄
⠀⠀⠀⢀⠈⠋⠀⠀⡡⠙⡶⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠛⢀⠀⠀⣠⠁⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡯⣤
⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣦⣤⣶⣤⣴⣶⣶⣶⣶⣾⣶⣤⣶⣶⣶⣶⣶⣶⣶⣴⣾⣷⣶⣷⣶⣿⣶⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣦⣶⣿⣦⣶
⣿⣿⣿⠻⠍⠛⠟⠉⠙⠛⢻⡟⠏⣹⡿⢻⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠤⠠⠀⠀⠀⠀⠀⠂⠋⠍⠀⠀⠛⠐⠀⠀⢀⠂⠈⠀⠀⠀⠀⠠⠀⠀⢺
⢀⡀⣀⡀⠀⠀⡀⠀⠀⠄⠠⠀⡀⣀⠀⠀⠄⢀⠄⢀⠐⠀⠀⠀⠀⠀⠐⠀⢀⠋⡀⢋⠀⠀⠲⡄⠐⡐⢠⡴⠂
⠿⡅⢂⠹⠀⠄⠁⠈⠠⠀⠠⠀⠄⠀⠄⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐⠀⠀⠀⠀⠠⡶⠁
⠀⠡⠈⠄⠁⠄⠠⠀⠠⠈⠀⠁⠀⠁⠀⠠⠐⠀⠂⠈⠀⠀⠀⠀⠈⠠⠈⠤⠀⠆⠠⠀⠴⠀⠄⠡⠤⠄
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠀⢀⡀⠀⠀⠁⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠒⠙⣤⣀⣙⢠⣆⡥⠀⡀⠂⠀⠨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠈⢹⣏⣰⠞⠳⠚⢉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⣀⣀⣘⣿⠀⠀⠀⠀⠉⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣀⠀⠲⢤⠀⣄⠐⢋⡟⠁⠀⠀⠀⠛⠀⠀⠀⠙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠈⢧⠀⠀⠀⣤⣈⡀⠀⠊⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢂⣤⠀⠀⠀⢀⣴⣾⣹⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⡝⠛⢿⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠰⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⢈⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠈⠢⠔⠀⠀⠀⡂⠀⠈⠀⠀⠀⢠⠀⢀⠀⢀⣠⠆⠁⠀⠀⠼⣏⠀⣺⠃
⠀⠀⠀⠀⠀⠀⠲⠘⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠀⠀⠀⠀⢸⣇⠀⣠⣸⠃⠀⡀⠀⠀⢠⣧⣼⣻
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠚⢣⠤⣄⡀⠀⠀⠸⡇⠈⠀⠀⠀⠀⠠⠈⠑⣿⣷⡃⣸⣇⠤⠐⠁⠀⠈⠻
⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠐⠀⠉⠀⠀⠀⠈⠳⠶⡀⣁⠀⢁⠒⢦⡀⠀⠀⣈⠙⡿⠛⣿⣏⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠿⢇⠀⠓⠀⠁⠀⢹⡇⠀⠀⠐⠈⠁⠆⠠⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢣⢀⡐⠢⣀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠱⢴⣆⠁⠐⢻⣿⣄⠀⠀⠀⠀⠁⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⣈⠁⠀⠛⠀⠃⠘⡰⠋⠉⠈⠁⠐⠂⣹⣆⠀⠹⣿⣿⡧⠐⠁⠀⠀⡞⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⡐⠒⠁⠀⠀⢀⡀⠄⡴⠂⠈⠛⠄⠈⣿⣠⡶⣄⣀⠀⢨⠷⢲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠈⢁⡐⠀⠀⠀⠀⢌⡤⢹⣿⣤⣆⡁⢠⡿⡠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠳⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⢤⢙⣯⡟⠻⣮⢄⡽⡰⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡷⣄⣀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⢯⡉⠡⠆⣌⡃⣼⣣⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡏⣸⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⡀⠀⠀⠀⠀⠀⠀⢀⠞⢠⣿⡿⠀⠀⠀⠀⢠⣦⠀⠀⢀⡤⠂⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣤⠏⣰⣿⠏⠀⠀⠀⠀⢀⡛⠄⣰⢞⣁⣴⡾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠜⢁⣾⣿⠋⣠⡴⡾⢃⣴⣬⣀⣖⣩⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⢤⠋⣠⣿⡟⣷⠿⡉⣵⡟⢻⣿⣿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠉⢠⣾⣿⠋⢶⢁⣾⣾⣿⡴⢿⡿⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⡁⢀⡁⣼⣿⣥⢶⣿⡾⣿⠿⣽⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣂⣵⠖⠉⢀⢈⠁⠈⠠⣿⠚⢻⣄⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢂⣢⠼⠖⠋⠉⣀⣤⣤⠴⠶⢿⠞⠶⢶⠿⣯⡭⣭⣭⣭⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⢋⡉⡡⢆⣠⡴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠁⠉⠐⠀⠈
⡠⢴⠊⣥⠄⠞⠉⣅⣰⣮⣿⠶⣾⡟⠏⠃
⡁⢀⠳⢀⣰⠶⠿⠛⠛⠂⠈⠀⠀⠁
⠡⣌⣴⠟⠁
⣿⣿⠟⠛⣿⡿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣈⣷⣶⣦⣴⣶⣶⣦⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣖⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠐⠻⠛⠂⠰⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣈⣿⣦⠀⣿⡇
⠀⠀⠀⠀⣈⡙⠛⠿⢶⣶⣦⣼⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣿⣷⡶⠶⠒⣀
⠀⠀⠀⠀⢀⡙⠳⣶⣤⣄⣉⠛⠿⢿⣿⣿⣷⣶⣤⣀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡿⠟⠋⣡⣤⣶⣿⠋⣰
⠀⠀⠀⠀⢀⠙⢶⣄⡙⠻⣿⣿⣶⣤⣈⠙⠻⣿⣿⣿⣿⣶⣌⡙⠿⣿⣿⣿⡿⠛⢉⣤⣶⣿⣿⣿⡿⢁⣼⡟
⠀⠀⠀⠀⠘⣷⣄⠹⣿⣶⣄⠙⢿⣿⣿⣿⣶⣄⡉⠻⢿⣿⣿⣿⣶⣌⠛⢁⣴⣾⣿⣿⣿⣿⣿⠋⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠘⣿⣧⡈⠻⣿⣷⣤⡙⠻⣿⣿⣿⣿⣷⣄⡉⠻⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⠟⢁⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡈⠻⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣶⣄⠙⢿⣿⣿⣷⣄⠻⠟⢁⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠙⢿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣷⣄⡙⠿⣿⣿⣷⡄⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄⠻⣿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⢿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⡿⠿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠶⠶⠶⠶⠶⠦⢤⣤⣀
⠀⠀⠀⢀⣀⣠⡤⠶⠖⠚⠛⠛⠛⠛⠓⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⢀⠀⣀⣀⣤⠤⠄⠀⠀⣩⠿⠛
⠀⠀⠀⠈⠛⢦⣄⠀⠀⠲⠤⣤⣤⣀⣀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⡼⠃⠀⣀⣠⣶⠞⠋⠉⠈⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠁⠀⠈⠙⠻⣦⣄⡄⠀⠻⡄⠀⠀⠀⢸⠃⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⣷⠀⠀⠀⢺⠀⣰⠏⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⣀⠀⠀⠀⠘⢷⡀⣽⠀⠀⠀⢸⣶⣇⣤⠴⠶⠒⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠶⢤⣷⢯⡀⠀⢠⣞⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣄⡾⡾⠁⠀⠀⠀⢀⣠⡤⠴⠶⠶⠶⠶⠶⠤⣤⣄⣀
⠀⠀⠀⠀⣀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠛⠒⠶⢤⣀⠀⠀⠀⢳⣻⢿⡇⠀⠀⣠⠞⠉⠀⠀⢀⢀⣀⣀⣠⠤⠀⠀⢀⣩⠿⠃
⠀⠀⠀⠀⠙⠻⢦⡀⠀⠀⠦⠤⣤⣤⣀⡄⠀⠀⠈⠳⣦⠀⠈⣯⢸⠃⢀⡞⠁⠀⢀⣤⡶⠟⠋⠁⠙⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠙⡷⣦⣀⠀⠈⢳⡀⣿⢸⡀⣼⠀⠀⣸⠟⠉⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠈⡇⢹⢸⡇⣯⠀⡼⠇⠀⠀⠀⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⡀⠀⠀⠀⠘⢧⠀⡟⢸⣿⡇⢻⣼⣁⡤⠶⠶⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⣼⣷⣇⢸⣿⣷⢯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⢿⡿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣧⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣷⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⢻⣦⡀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣷⣹⣇⣠⣬⣡⠴⠤⠶⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠛⠻⢿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀
⠀⠀⠀⢸⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡀⠀⠀⠠⠊⡷⠈⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⢀⡽⠞⠂
⠀⠀⢀⣼⣧⣠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣤⣀⣤⡷⠤⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠇⢺⣿⣶⢀
⠀⠀⢸⣿⣿⡿⠀⣰⢣⣤⣶⣟⡉⢉⡉⠁⠒⠒⠚⠛⠉⠁⠀⠀⢀⣴⣶⣶⣶⣟⣀⡙⠙⣿⣓⠦⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠻⣿⣿⡇
⠀⠀⢸⣿⣿⣷⡯⠁⠉⠀⢤⣿⣛⠻⢯⣍⡛⣶⣶⡶⠋⠀⢀⣼⣿⣿⣿⣿⡏⣏⡍⡙⠻⢦⣰⡄⠺⣦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⡇
⠀⠀⢸⣷⠦⣀⡀⠄⠀⠀⠘⣿⡁⠀⠀⠈⠉⠛⠟⢿⣷⣦⣿⣿⣿⣿⣿⣿⡛⠛⠿⣷⣰⣀⡉⠳⣄⠀⠑⠀⠀⠀⠀⣷⠀⣤⣿⣿⢂⣀
⠀⠀⢸⣟⢷⣄⠙⡦⡐⢥⡀⠈⠻⣷⣤⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣟⢿⡗⢆⣀⠉⢛⢷⣄⠘⢦⠀⠀⠄⠀⣀⣼⣷⣽⣿⣿⣿⣿⠁
⠀⠀⠀⢻⣌⠙⢿⣾⣌⡓⠀⠀⠀⠀⢹⣿⣿⣦⡀⠀⣼⣿⣿⣿⣿⣿⣿⣻⣶⣿⣦⣕⡀⣿⢿⣻⣧⠘⣇⠀⡤⢈⣿⢟⣛⣏⣹⣿⣿⠿
⠀⠀⢀⣤⣽⣷⣶⣤⣭⡉⡑⠈⠂⠈⠀⠹⣏⠈⠛⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣿⡹⣿⣷⡿⢿⣮⣿⣿⣿⣭⣍⡑
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⠀⠀⢹⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢧⡏⣿⣧⣬⣱⣿⣷⣿⣿⣿⣟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⡿⠃⠨⠻⠧⠔⣸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣂⣀⠿⠿⠿⠫⠭⠯⠭⣽⣭⡉⠉⠁
⠀⠀⠀⠙⣿⣿⣿⣿⣶⣾⣧⣀⣀⣀⡄⣠⠟⠴⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠂⠀⠀⠚⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠠⠿⠿⢿⠿⢻⣿⠿⠋⡟⠿⣴⠿⠀⠀⢾⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⢀⣤⣤⣄
⠀⠀⠀⣠⠖⣨⣵⣾⡿⠃⠀⠈⢇⠀⠈⠛⣶⠤⢨⣉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣦⡌⠀⠀⠉
⠀⠀⠀⣡⣞⣟⣾⡟⠀⠀⠀⠀⠸⠂⠀⠈⢮⠻⣿⡻⠃⠈⠉⠹⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠁
⠀⠀⠀⣻⣿⣿⡟⣁⠀⣴⢻⣿⣷⡀⠀⠀⣠⣤⡟⠋⠀⠳⠀⠀⣠⠉⠁⠈⠙⣉⠉⠙⢻⡿⠿⠿⠿⠿⠿⠛⠛⠛⠷⠤⠀⣷⢮⢯⣤⢬⡆
⠀⠀⠀⠿⣿⣿⣥⣼⡿⠿⠿⠿⠿⠿⠆⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠚⠩⠁⠀⠄⠌⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⠀⠀⠀⠀⣤⣖⣲⣴⣲⣰⣒⣦⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠶⠊⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⡟⡏⠙⢹⣩⢹⣿⢹⡇⠀⠀⠀⠀⠂⣾⣿⣿⡆⠀⠀⠀⠀⠈⢧⡚⠟⠂⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠
⠀⠀⠀⢀⣎⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⡇⠻⠟⠸⠿⠘⠿⢸⡇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⢷⣄⠀⠀⠀⠀⠀⠀⠃⠀⢨⡇⠀⢀
⠀⠀⠀⠸⠏⠻⠇⠿⠸⠇⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠀⠨⠇



⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⢀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠉⢸⣿⣿⡎⢒⠂⠀⠀⠀⠀⠀⠂⠘⣿⣿⣿⠂⠐⠀⠀⠀⠀⢠⣾⢻⢯⣽⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠳⡀
⠀⠀⠀⠀⢰⡾⠿⠿⡁⠂⠃⠀⠀⠀⠀⠀⠀⠄⠻⠿⠟⠀⠀⠀⠀⠀⠀⠈⢿⣯⣾⣹⣾⠷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣷⢳
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉



⠀⠀⠀⢀⡴⠚⠉⢉⡓⢤⡀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠛⠢⣄⠀⠀⠀⠀⠀⣠⢖⠛⠉⠛⠢⣄⠀⠀⠀⠀⠀⠀⣆⠶⢆⡤⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⡀
⠀⠀⠀⡏⣀⣄⠀⠿⠟⠀⢳⠀⠀⠀⠀⣸⢁⣾⣿⣯⣠⠄⠘⡆⠀⠀⠀⢸⠁⠸⡅⢄⢠⠴⠚⡆⠀⠀⠀⠀⠀⠈⣓⡺⣅⠀⠀⠀⠀⠀⠀⠀⢿⠟⠛⢻⡿
⠀⠀⠈⡇⠻⠟⢀⣤⣶⡄⢸⠀⠀⠀⠀⢻⠀⣻⠿⣿⣿⡄⢀⡇⠀⠀⠀⢹⠀⠀⢸⠴⠋⠀⠀⡇⠀⠀⠀⠀⠘⣆⠣⠇⠘⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡾⠁
⠀⠀⠀⠙⢦⣀⠈⠻⣛⡵⠃⠀⠀⠀⠀⠈⠣⣄⠀⠟⢉⣠⠞⠀⠀⠀⠀⠈⠳⣄⡘⠦⢀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠍⠙⣆⠀⠀⠀⠀⠀⠀⠈⢘⣻⣯⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⡤⠠⡄
⠀⢄⣀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢇⡿⣴⠀⡄⠀⠀⠀⠀⢀⠀⣧⢷⣃⡀⠀⠀⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣾⣿⣿⣶⣀⠆⡀⠀⠐⠈⣞⣿⣾⡟⢠⢠⣼⣾⣟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⢿⣿⣏⣏⡰⠁⠀⡄⣶⢿⡿⣿⡕⠸⢸⣿⢿⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠱⣿⣿⣿⣿⣼⣥⠁⠀⢡⢿⣿⡿⣿⡰⣆⣿⡷⡟⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠢⣼⣿⣿⣿⣱⣧⢣⢗⣾⣦⣿⣿⣿⡄⣿⣹⣾⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣯⣿⣧⠞⡀⣿⣿⣿⡽⢁⣣⡻⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣫⣏⠀⠣⣾⣿⣿⣷⢾⢹⣿⣿⣃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣽⣿⡟⠍⠀⠀⣿⢿⡿⡿⢀⣿⢹⡿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣎⠀⠀⠀⠙⢿⡿⠁⠘⣝⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡄⠀⠀⠀⢸⣿⠀⠀⠈⢿⡏⠀⢀⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣴⡀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⣾⡜⣉⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⣿⠈⢯⠱⡀⠀⢸⣿⠀⠀⠀⢸⡇⠀⣿⡟⠁⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣷⠀⠀⣿⠀⠈⢷⣧⠀⢸⡟⠀⠀⠀⢸⡇⢀⣿⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⠁⢿⣇⠀⣿⠀⠀⠈⢿⡄⣸⡇⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡟⠀⠘⣿⡄⡇⠀⠀⠀⠘⣷⣿⠃⠀⠀⠀⢸⣧⣿⠇⠀⠀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠘⣿⠀⢰⡆⠀⢸⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢻⡇⠀⠀⢠⣶⠀⣿⠀⣼⠀⢄⢸⡟⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⡴⣼⠇⠀⠀⢸⡿⢀⣿⠀⢻⡇⢸⢸⡇⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢸⠃⣿⠀⡞⠀⡞⢁⢸⣏⠀⢾⡇⢸⣿⡇⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⣿⢰⡇⣸⠇⢸⡇⣾⢹⣷⢸⡜⣇⣼⣿⡇⡿⢸⠇⠀⣴⡏⠀⠀⡠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡇⣆⣿⡖⡟⣸⠱⢻⠀⢸⣇⣿⣾⣿⣼⠁⣿⣿⣿⣿⢇⣞⣼⣿⣟⡄⣠⡞⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⣄⣇⣿⣿⣷⣷⣿⡇⣿⢆⣼⣟⣿⣿⣿⡟⣠⣿⣿⣿⠇⢺⢸⣿⣿⣿⣿⣿⠁⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠟⠟⠿⠻⠛⠿⢿⣿⣿⣞⠛⢿⣿⣹⡿⠛⢿⡾⠿⠿⠸⣿⠸⢸⣿⣿⣣⡟⣸⡿⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⠿⠛⠛⠁⠀⠘⢇⢦⣄⣠⡀⠶⠿⠿⠟⠿⠿⠻⠃⠋⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀⠄⠀⠀⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣴⣦⢀⡴⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⡍⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠾⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠂⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠱⠁⠀⠀⢌⢄⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠂⠀⡷⠈⠆⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣇⡠
⠀⠀⢀⠄⠀⠐⠶⠆⠀⠀⠀⠱⠀⠀⠀⠐⠋⢧⣘⣇⣀⠁⠉⠀⠒⠠⣀⡀⠀⠀⠰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠣
⠀⠀⠀⠀⠒⠰⠮⡇⠀⠀⠀⠀⠙⠤⣄⠀⠐⠀⠈⠸⠁⠀⠀⠈⠙⠂⠀⠈⠀⠀⠀⣀⡤⠚⠀⠉⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡷⢀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⢠⣀⣄⣠⣠⠄⠄⠰⠐⠐⠉⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣡⡿⠅⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡾⠀⡀⠀⠀⠀⠈⠁⢀⣴⠟⠁⠀⠀⢠⠤⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⣀⣈⣸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⡼⠁⠐⠀⠀⣀⣀⣤⣶⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣠⣾⣿⣾⣿⣿⡿⠿⠛⢻⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠚⠜⠃⠀⠴⠛⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⣁⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣶⣶⣶⣶⣶⣶⡶⠿⠛⠛⠋⠉
⣿⣿⡟⠛⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢺⠓⡀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⢀⢬⠝⠀⠀⠀⠀⠀⠀⠤⠤⡀⣃⠀⢠⠔⢓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠾⡄⠀⠀⠀⠀⠀⠀⠈⠢⢻⣟⡔⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡚⣇⠴⠃⠀⠀⠀⠀⠀⠀⢀⣧⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⢏⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣖⣲⠖⢒⣒⡒⠲⣲⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠲⣆⠀⠘⢆⠹⣤⠤⠤⠤⢤⣤⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠿⠃⡸⢀⣠⡈⠓⢌⡻⣦⠼⢛⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⠛⢧⡏⡹⠁⠀⡴⢛⡴⢎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠤⣿⠃⢠⠞⣴⡯⠠⢤⣵⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⢼⣤⠋⣼⢉⠉⠉⢉⣉⣙⣆⢹⡀⠀⠀⢀⣨⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢨⠓⠛⠛⠒⠒⠒⠚⢻⡄⡗⢦⣀⠜⣡⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡇⠤⠒⠨⠀⠀⠲⣾⠃⡇⠸⢡⣾⣩⣾⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠱⣤⠤⠤⠤⢤⣴⠇⡼⡡⠖⠣⡀⠳⠵⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡙⢮⠽⢋⡤⠞⠊⢱⡂⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡥⢆⠛⢄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣠⠿⠅⠨⠷⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⣓⣲⠖⠒⢲⡾⣆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⠀⠀⠀⠀⠀⠀⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢲⠾⡀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣦⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠦⠦⡍⠃⠀⠀⠀⠀⠀⠀⠀⠘⢢⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣂⠓⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⠖⠻⣗⣶⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢿⠁⠰⣆⡀⠈⡉⠀⢹⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠉⠉⠀⠀⢳⡬⠛⠉⠠⠘⣧⣧⡀⠀⠀⠀⠀⣠⡄⠀⠀⢠⢦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⡖⢤⡀⢀⣄⡈⠀⢀⠉⣏⠁⠀⠀⠰⣒⡋⡷⠖⠓⠚⣇⣩⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡛⣅⠀⣸⠎⠀⣸⡤⠚⠛⣂⣰⣧⡀⣤⡀⣴⠉⠉⠁⠀⠀⠀⠈⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠟⢦⣤⢀⠄⠛⠀⢀⣴⣧⠎⠀⠀⡱⠙⣯⠀⢀⡤⠜⡇⠀⡼⡂⠀⢬⠶⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠟⠣⢵⡖⠚⠋⠉⠛⠀⠀⠚⠊⡿⠁⣠⣄⡈⠙⠁⠀⠓⠚⠃⢸⡟⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠸⣅⠌⣱⡼⠀⡰⣩⠵⠆⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡤⢼⣢⠉⡼⡳⠦⣄⠀⢠⡄⠀⠀⠀⠘⠷⣍⣄⠄⢙⠁⣠⣰⠻⢥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠸⡏⠀⢊⡁⠙⠀⣀⠀⡑⣷⠎⠧⣀⡀⠀⠀⣴⡫⡸⡋⠉⠉⠉⠸⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠋⠁⣀⡜⢹⠀⢰⠙⢦⡄⠈⡺⣯⠉⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡏⠀⠀⠉⠑⠋⠀⠀⠉⠁⡄⠠⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢈⡽⠀⠾⣁⢒⠄⠀⡔⢒⡦⠀⠀⣤⠿⡲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⣤⢠⡈⠚⠀⠀⠷⠁⠀⠃⠀⣿⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠚⣦⣄⠀⣊⣵⢶⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢣⡇⠉⠉⠉⡝⡤⠽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠀⢀⣠⡾⠷⠶⠶⣻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢰⣇⣠⠶⠛⠿⠶⠶⠶⢠⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡼⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣰⠿⠤⠴⠶⠶⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠒⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢶⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣉⣉⣀⣈⣩⣽⡷⣦⣤⣬⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠈⣡⠴⠛⠉⣰⠏⠀⣸⠇⠀⠀⠀⠙⠿⣷⣦⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠴⣶⡶⠋⡏⠀⠀⠐⠋⠀⣰⠏⠀⠀⠀⠀⢀⣤⠬⣿⡻⣿⠷⠭⠗
⠀⠀⠀⠀⠀⠀⢰⠇⣠⡞⠉⠀⠘⠃⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣰⠏⣴⠶⠋⠻⡌⢷
⠀⠀⠀⠀⠀⠀⣾⣴⠋⢇⣀⣤⡤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠹⠞⠋⠀⠀⠀⢻⡈⣧
⠀⠀⠀⠀⠀⠀⣿⡷⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢸⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣟⣀⡏⢡⡞⠛⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠖⠛⠓⠶⣦⣤⡴⢿⡅⠀⠐⡚⠩⣍⣙⣶⠟⠛⠉⠀⡈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣁⡀⠙⠃⠀⠀⠙⠋⡙⡛⢦⣄⡿⠟⠋⢽⣻⠁⠻⣷⡀⠈⠳⠀⠘⢳⣚⡛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣈⠻⢤⡀⠉⠉⠁⠲⠦⠄⣀⡁⢠⠞⠉⠀⠀⠀⠀⣀⠀⠀⠈⠘⠳⠀⠈⠉⠉⡀⡀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣈⡀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠋⡟⠀⠟⠀⠦⠄⢀⡀⠀⠀⢀⣤⣠⠾⡇⠀⠐⠛⠱⠃⠖⠒⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣤⣉⣋⠀⠰⠖⠀⠒⠒⢀⣙⡴⢺⣏⠳⢦⣤⣤⣀⡘⠋⠀⠀⠈⢀⣉⣀⣠⠴⠶⠒⠒⡖⣒⡚⣉
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠄⠸⣧⣤⣉⣿⠏⠉⣉⣩⠿⣟⣿⠋⢁⣤⡶⠛⠛⠲⠶⠶⠾⣭⣉⠛⣻⠿⠭⠭⠍⣻⣟⡏⢙⡇⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠢⣶⠤⣞⣋⣩⣭⠿⣁⡀⠃⣾⠉⠉⣀⣶⣆⣐⣛⣃⣠⡿⠀⠙⠒⠶⠶⠶⠞⠒⠚⠋⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠀⠀⠉⠛⠋⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⢀⡤⣾⣭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⣴⣿⡻⠿⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟⠻⠋⠁
⠀⠀⣒⡫⢒⣒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⣬⣾⡿⠯⢀
⠀⠀⠀⠉⠑⠒⠫⠥⣒⣲⠤⠤⠀⣀⣀⡠⠄⠐⠒⣽⣿⠶⠎⣏⣿⣾⠳⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⢿⣁⡐⠺⢏⣱⣿⡤⣄⡾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡖⠛⠛⠛⠉⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛
⠀⠀⠀⠀⠠⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⡖⡶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢿⣛⡢⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⣛⠤⢖⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⣾⢿⡿⡕⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣍⠳⠺⣌⠙⠂⢀⠀⠀⠀⠀⠀⠹⣮⢿⡌⡈⢂⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣗⠯⡢⡀⠕⢄⠀⠀⠀⠀⠹⣧⡻⡜⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⡍⢰⠀⢿⠀⠃⠀⠀⠀⠀⠀⠀⠀⣼⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⡀⢄⠑⣅⡀⠀⠀⠀⢹⣿⡜⠌⠈⠀⢀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣧⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠃⡊⠉⠀⠀⠀⠀⢯⣧⢸⡀⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣅⠠⠉⡀⠀⠀⠘⣻⠄⢇⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡿⠎⠃⢱⡇
⠀⠀⠀⠀⠀⠈⡄⢀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⠈⢂⠀⠀⢣⢁⠈⡼⡄⢸⠀⠀⠀⠀⠀⠀⡰⠒⠙⠈⠋⠒⣀⠀⠀⠀⠀⢻⣃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⠘⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠆⠀⠀⠀⠀⢈⡁⢰⣡⡼⠀⠀⠀⠀⠀⠀⠀⠀⢦⡼⠆⠀⢙⡀⠀⠀⠀⣸⣬⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠀⠀⠈⠳⣄⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⢀⠾⠀⠀⠀⠀⣿⠟⡀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠘⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⣇⡆⢠
⠀⠀⠀⠀⠀⠀⠠⠌⠑⠰⣀⠀⠀⣤⣈⣠⣤⡄⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣹⣵⢿⡇⢾
⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⢸⠃⢸⠏⢁⠀⠀⠙⢦⠀⠀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⢀⡀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⣯⠀⠃⠈⡄
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠈⢺⣦⣼⡞⠉⡉⠙⠂⡤⠁⠀⣿⣿⢇⠀⠘⠀⠀⠀⢀⢐⡅⠀⡀⣠⡤⢤⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⣿⢠⠀⠀⢰
⠀⠀⠀⠀⠁⢀⠀⣢⠔⠋⠉⢹⠛⣄⠀⠀⠰⣤⠁⠀⠀⣿⣿⠂⠀⠀⠀⠀⡀⠱⠉⠀⣪⠞⠁⠀⠀⠈⠻⡐⢆⠀⠀⠀⠀⠀⠀⣼⡄⡏⣸⢠⠸⢸
⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⢠⠛⣇⠈⠢⠄⠀⠘⡄⠀⠀⣿⡏⠠⠂⠀⠀⠌⣡⢈⣀⡼⠁⠀⠀⠀⠀⠀⠀⣻⠘⣰⠀⠀⠀⠀⠀⠉⡇⠣⢰⢸⠄⣾
⠀⠀⠀⠀⠀⣼⠁⠀⡰⡁⢾⠀⠙⣄⠀⠀⠀⠘⠃⠀⠀⠉⠉⠀⠀⠀⢐⠞⠃⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠉⢴⠀⠀⠀⠀⠀⠀⠈⡿⣤⠈⡼⠸⡏
⠀⠀⠀⠀⠀⠸⠀⣸⠁⠀⡈⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠼⣸⢀⡟⠀⠀⠀⢠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⣁⣷⠡⠁
⠀⠀⠀⠀⠀⠀⢠⣏⠀⣼⡧⠀⠠⣄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⡄⠴⠀⠏⡜⠀⠀⠀⠀⢼⣇⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⢹⣄⠀⠈⢦⣀⠐⠀⠀⠀⠀⠀⠀⠀⠘⢁⢀⢧⡄⣗⠀⠀⠀⠀⠈⢽⣾⢦⡀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠈⠈⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⢮⢸⠃⣹⡀⠀⠀⠀⠀⠈⢪⢳⡙⢦⡀⠠⠊⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢀⡿⢰⠏⠁⠀⠀⠀⠀⠀⠡⡳⣝⢦⡉⡇⢀⣤⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡳⣟⡇⢨⡁⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣝⣆⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠁

⢀⠀⠠⠀⠂⡄⢀⠀⢀⠔⠒⠒⠤⢀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢀⣠⠠⣄⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⠀⠒⠂⡀
⠀⠈⠀⠀⠀⠈⠤⠄⠘⠂⠄⠤⣀⠀⠀⠈⡀⣀⣀⠀⡸⠀⠀⣌⠀⠀⠀⢃⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⢀⠇⠱⠄
⠠⠀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠦⠠⠄⡃⠀⠀⠀⣀⠀⠀⢠⠂⠀⠀⠀⠐⢶
⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠁⠄⠠⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠤⢀⠀⠀⠈⠁⠉⠁⠂⠀⠐⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠈⠀⢈⠄⡀⡐⠀⡀⠀⢠⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠨⠠⠀⠀⠀⠄⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⢀⠀⡀⠁⠀⠀⠈⠢⡀⠈⠓⢦⠀⠀⠀⡀
⠀⠁⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠀⠀⠀⠀⠠⠀⢀⡴⠀⠇⠀⠰⠀⠁⠀⠀⠀⠂⠈
⠈⠤⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀⠤⠐⠁⢠⠞⠀⠀⠈⢀⡗⠀⠀⠀⠀⠀⢀⢠
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠒⠈⠉⡁⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⡶⣶⠶⠶⣾⣿⡟⠛⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⡀⡀⠁⠠⠀⠀⣾⢹⢡⠀⠈⠰⠎⠦⢰⡶⠎⣻⣮⣼⠟⢿⡆
⠀⠀⢠⠀⠀⠀⠀⡀⠀⠘⣀⠸⠀⠠⠀⢸⣿⣀⣼⠦⣶⠗⢾⡒⢺⠛⣻⡏⠍⡈⠆⣼⡇⠀⠀⠈⠈⠀⠂⠐⠀⠀⠄
⠀⡀⠤⢄⠀⠀⠀⠀⠀⠄⢀⡀⠐⠂⠀⠈⢯⠀⡹⠀⡌⠘⠿⢿⡜⣇⠰⠶⠀⡓⢀⣿⡏⠀⠀⠀⠀⠀⠠⠀⠄⠀⡀
⢐⠀⠀⠀⢱⠀⢀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠓⠧⠶⠿⠾⠷⠾⠿⠿⠶⠶⠿⠿⠟⠋⠀⡀⠀⢢⠀⠀⠐
⠈⠢⠄⠤⠃⠀⠀⠀⡀⠁⡐⠒⡄⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡐⣰⠈⠾⠄
⠀⠀⡄⢠⠊⠉⠁⠢⠀⠤⠈⠂⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢄⠀⠀⠀⠄⢠⠂
⠈⠒⠁⢂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⠁⠑⠀⠀⠀⠘⠠⠜⠀⠀⡠⣤⣠⡔⢶⢰⡖⠢
⠀⠂⡀⠀⠑⠒⠂⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠝⠛⠀⠁
⠀⠠⠠⠔⢀⠀⠀⠆⡡⡀⡂⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢠⠢
⢄⠁⠊⠁⠈⠀⠈⠀⠀⠈⡢⠅⠀⢀⡤⠒⠉⢑⡄⠀⠀⡀⠒⢠⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⢈⠀⠀⠀⠀⠄⠀⠀⡀
⢂⡀⣀⠀⠀⠀⠀⠀⢀⠈⠐⠀⡰⢃⠤⠒⠂⠈⡀⠠⠂⠀⣀⠜⠀⠀⠈⠀⠀⠀⠀⠂⢠⠀⠔⠀⠀⠀⠈⠀⠄⠀⠂
⠀⠂⠂⡁⡉⠂⠊⠀⠁⠀⠁⠀⠈⠁⠀⠀⠀⠈⠒⠒⠂⠈⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠀⠐⠀⠀⠀⠂
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡇
⢀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣄⠀⠀⢀⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠀⣠⣶⠞⠁⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢠⣄⡀⠀⠀⠀⣿⣿⠀⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠆⠀⠹⠟⠀⢴⠟⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⣿⡄⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⡀⠀⠸⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣾⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⢶⣤⣤⡀⣻⡇⠀⠀⢸⣿⠃
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣀⢠⣿⣿⣦⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡏⠉⠻⣿⣦⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠙⢿⣷⣄
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⡀⠹⣿⣧
⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⣠⣤⣶⣿⠿⠿⢿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠃
⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣶⣦⣄⣀⡀⢸⣿⡿⠿⢶⣦⣄⣀⣀⣀⣀⣀⣠⣤⣶⡿⠿⠋⠁
⣠⣿⡏⢀⣤⣤⣶⣿⡿⣿⣿⣿⣶⣤⣄⡀⠉⠙⠛⣿⣿⣿⠟⠀⠀⣼⠏⠉⠉⣿⡿⠛⠛⠋⠉⠁
⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⡶⠟⠛⠉⠀⢀⣠⡾⠋⠀⣀⣾⡿⠁
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⢀⣴⣾⣿⣿⣶⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢠⣿⡟⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⡀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣶⣦⣤⡀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠈⠙⠻⣿⣷⡄⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁