Ache Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⣭⣽⣿⣷⣾⣿⣿⣿⣿⣶⣤⡠⠴⠃⠀⣠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠉⠀⠈⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡆⠀⣠⡤⠴⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⠀⠀⠀⠀⠀⠙⠿⣿⠿⡎⠓⠉⢻⣿⣿⣿⠀⠤⠤⣄⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠈⠉⣿⣿⡇⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡎⠀⠠⠤⠒⠊⠀⠀⠀⠈⠒⠚⠦⡀⠀⠀⣠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠐⠒⠁⠀⠀⠀⠀⠑⠒⠀⠈⡆⠈⠁⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢱⠒⠚⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣧⠘⠃⠀⠀⡠⠤⠤⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⡷⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢱⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⣱⠈⢹⠙⢯⣩⠭⠋⡜⠀⠀⠈⢇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠃⠀⠈⠚⠁⠈⠑⠾⠀⠀⠀⠀⠈⠣⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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



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

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

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⣠⢢⡀
⠀⠀⠀⠀⠀⠀⠸⠇⠛⢀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢺⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠙⡄⠘⣤⠛⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠋⠠⢶⣄⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⣠⡄⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠲⠶⠂⢀⠈⠯⠌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡀⠈⠧⠀⢠⣶⠀⠐⠀⠀⠀⠁⠀⠊⠁⠀⢠⣿⡄⠐⠂⠀⠀⠴⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠉⢁⠀⢀⣀⠤⠔⠤⢀⣀⠀⠀⢀⠁⠀⠀⠶⢏⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠶⠶⣶⠖⠁⠀⡀⢀⠀⡀⠀⠙⠲⣍⠉⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠰⠃⠀⠀⢈⠄⠀⠀⢀⣀⣶⣠⡈⢆⠀⠀⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡇⠀⠀⠀⠈⠀⠠⢀⡎⠁⠀⠈⠛⣴⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⢆⠀⠀⠀⠀⠑⠈⠇⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠄⠀⠀⢀⣴⢿⡇⠀⠀⡤⠈⠃⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠙⠀⠀⠘⠁⢄⡀⠀⠀⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣜⠀⢶⡶⠤⠄⠥⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⡾⠉⠳⣤⣠⣀⠀⣰⠀⠘⠳⠄⢹⣍⠁⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⣗⠀⠊⠀⠄⠀⠀⣿⠀⠀⠲⣤⠈⢿⡆⠀⠀⣄⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠘⠃⢀⠿⠀⠀⡘⢂⠀⣄⣹⠀⠐⠶⠌⡁⠘⠇⡀⠱⠾⣧⠘⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠷⠀⠸⠀⠀⠸⠁⠘⢀⠌⢛⠀⠀⠰⠤⣉⠀⢸⠁⠃⠰⠁⣴⠂⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠖⠀⣠⡀⠘⠀⡠⠂⠠⡀⠀⠻⣅⠀⠀⠂⠈⠀⠀⠀⠌⠀⡴⠋⢀⠄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠍⠃⢈⡠⠊⠀⠀⠀⠙⡀⠐⢯⡀⠀⠀⠀⠀⠐⠃⣠⠊⡀⠖⠁⡤⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠴⠚⠁⠀⠀⠀⠀⠀⠀⢡⡄⠸⢷⠀⠀⠀⠀⡠⠞⠟⠁⡠⠖⢁⡠⠊⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠓⠄⠋⣀⠠⠔⠃⡠⠐⢁⠠⠔⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠂⠤⠀⠂⠀⠄⠂⠀⠤⠒⠉⡤⠔⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣨⠄⠂⢉⡀⠐⢈⡀⠐⠂⠁⢀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠰⠀⣉⠀⠄⢉⡀⠠⠂⢈⠁⠦⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢘⡁⠤⠈⣃⡀⠤⠀⢃⠠⠤⢀⣁⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⠀⠠⠐⠂⠄⠒⠈⠁⠀⠂⢨⡀⡤⠀⠀⠀⠀⣠⠬⣀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠈⠉⠉⠉⡉⠙⠋⠁⠀⠀⠀⠀⠆⠰⠀⠀⠧⢠⠗⠀⠀⠔⠚⠉⣄⠒⣄⠓⠀⣀⢀⣀⣤⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠶⢄⡉⠲⢄⠱⠄⠀⣠⡟⠀⠂⡀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠐⠀⠀⠘⠁⠀⣀⣤⣤⡟⠀⠀⣈⡉⠲⢄⡈⠳⡈⠃⣠⣾⢋⣉⣉⡉⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠂⠰⠦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠈⠀⠀⣀⣠⣾⣷⡭⠉⢹⠁⠀⠀⣈⡙⠓⢤⠙⠂⣠⡜⢋⣈⠉⡁⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠐⠀⠀⠀⠐⠒⠐⠒⠒⠒⢺⠄⠀⠒⠂⠀⠀⠀⢸⠛⠋⠛⠛⠓⠒⢸⡆⠀⠀⠈⠍⠀⠄⠀⢰⣍⣀⣤⣤⣤⠤⡤⣠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠐⠀⠀⡟⠓⠒⠂⠀⠀⠀⠀⢡⠀⠀⠠⠀⠐⠒⠀⠈⣯⡀⣀⠀⠀⠀⠀⠀⠀⢦⡜⠃⠘
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⣀⠀⢀⣏⢉⣉⠁⠀⠀⠀⠀⠘⣄⠀⠠⠀⠈⢉⡀⠀⢹⡄⣀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠠⠤⠀⠸⠇⠈⠉⠀⠀⠀⠀⠀⠀⠛⡀⠀⢛⣀⠠⠤⠀⠀⡧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣄⣀⡀⠒⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠠⠄⠐⠀⠀⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠁⠀⣠⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠉⠐⠀⣉⠀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⢈⣀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡘⠀⠀⠀⠀⣠⠔⠛⠑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠠⠄⢠⡞⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠁⠀⣠⠴⠋⠁⠀⠛⠛⠋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⢀⡴⠯⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋
⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠈⠒⠈⠁⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣋⣉⡉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠃⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⠶⠶⠖⠒⠒⢛⣛⡛⠛⠛⠛⠛⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⢠⣤⠤⣤⡤⠀⠀⠈⢀⣮⠄⠀⠄⠠⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠚⠛⠛⠛⠛⠒⡒⢐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

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

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

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

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

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

⠀⣜⣔⣾⣿⠀⡰⣡⣾⣿⣿⣴⡼⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠀⠄⣿⣿⣿⣿⣾⣦⠀⣷⠈
⠀⡟⢻⣿⣿⣴⣿⣿⣿⣿⣿⠾⠀⠀⠀⠀⠀⡎⠁⢸⠀⠀⠆⠂⠀⡖⠀⠀⠀⠀⡶⣿⡟⣻⣿⣿⣿⣷⣿⡆
⠀⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⡴⠀⠀⠀⠀⠉⣧⠉⢹⡀⠉⣁⠀⠀⣇⠋⠉⣡⠞⣶⡏⣹⣿⣿⣿⣵⣿⢹⣷⡀
⠀⣨⣿⣿⡗⠋⢿⡟⠛⠿⠿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣥⣾⣿⣷⠿⠟⠋⢿⠙⣿⡏⢻⡇
⠀⣙⣧⣿⣷⠀⢸⠇⠘⣟⠛⠛⠛⢛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠛⠙⠙⢻⠆⠀⢸⢠⣿⠇⢼⡇
⠀⢻⡇⢆⢻⡄⣼⠀⠀⠘⠦⣄⡀⠀⠀⢀⠈⣻⣿⡟⠋⠁⢈⣿⡿⠋⠀⠀⠁⠀⣀⡴⠋⠀⠀⢸⣾⣿⢀⣴⡇
⢀⡘⢧⢨⣷⣷⡿⡄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣤⡄⠀⠀⢘⣯⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⠏⣤⡏
⣽⡷⠚⢿⡿⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡁⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣤⣟⠁⢀
⠛⠀⣀⠈⣻⢶⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡁⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣰⣿⡟⢶⣄
⢀⡤⠀⠀⡏⠀⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⢹⡝⠀⠀⠉
⠋⠀⠀⣼⣥⡆⠀⠸⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣾⢦⡁
⡀⠀⣰⠿⣏⢹⢀⢦⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢸⣿⣷⣯⡀⠀⣀
⡿⣷⣤⡀⠿⣣⣿⣿⣧⢸⣿⣷⡀⠀⠀⠀⠀⢉⣁⠀⢀⣀⣀⡀⢈⠉⠀⠀⠀⠀⢀⣾⢿⣟⣾⣿⣿⣶⣿⣿⣿
⣿⠾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠙⠀⢀⣀⣤⣴⣿⠾⣿⣻⣿⣿⣿⣿⣿⣿⣿
⣏⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡿⣦⡀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣧⣄⣼⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠛⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠺⠿⢿⡟⠿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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



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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣯⡙⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠢⢤⣀⡄⠀⢀⣀⠀⠀⠀⠀⣻⡟⠷⣌⡙⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠖⢁⡠⣦⠞⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠞⠁⠰⣮⣿⠏⡠⣸⣿⣎⣉⣉⠉⠉⠙⢻⡿⠿⣿⣿⣿⣿⣶⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣼⠀⢰⢸⣿⠋⡀⢠⣿⣿⠿⢻⡿⢿⣿⣿⣿⣿⣶⣦⡄⢈⣿⠉⢹⣇⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢠⡟⠀⣬⣾⡏⢠⠃⣾⣿⠋⢀⠘⠀⢀⡶⣿⣿⣿⡟⠈⠙⣿⣿⡀⣼⣿⣏⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⢸⠁⢸⣿⠟⠀⣾⢰⣿⣿⣧⣄⠀⠀⢸⣿⣩⢼⣿⡇⠀⠰⣿⣿⣷⣿⣿⡟⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢧⠀⡇⠀⣾⣏⠀⢀⡧⢸⣿⣷⡋⢨⣷⡆⠀⢧⠀⢸⣿⡇⠀⢻⣿⣿⣿⣿⡿⡇⢸⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⠀⠘⡄⢻⡀⣿⣿⣷⠾⠀⢸⣿⣿⠀⠺⠋⢿⡄⠀⢤⣹⡼⣷⣤⣠⣿⣿⣿⡿⣱⠇⣸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡀⠀⢿⡄⠀⠰⠈⣧⢹⣿⡏⠀⠀⠀⣿⡷⠐⠃⠀⣸⣧⠀⠈⣟⣻⣾⣿⣿⡆⢸⣿⣵⠋⣰⡏⠀⣿⡃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣿⠀⠀⢱⣿⣿⣿⣧⠀⠀⢠⣿⣿⣄⠀⠀⠿⠾⣷⠊⢻⠀⣿⣿⣿⣷⣿⣿⠃⣼⡿⠀⣸⣿⠀⠀⠀⠀⢸
⡄⠀⠀⠀⠀⠀⠀⣿⡹⣄⠛⢀⠀⠈⠛⠿⠋⠁⠀⣰⣿⣿⣿⣿⣶⣦⣄⠀⢀⣀⣼⣶⣿⣿⣿⣿⣿⠇⣼⡟⠀⢀⣿⡇⠀⠀⠀⠀⡬
⡄⠀⠀⠀⠀⠀⠀⠸⡇⢹⡆⢸⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⡟⠀⠀⣾⡏⠀⠀⠀⠀⣼⡇
⡆⠀⠀⠀⠀⠀⠀⠀⠃⢸⣧⢸⡇⠀⢄⠀⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣼⣿⠁⠀⣰⣿⠁⠀⠀⠀⣼⣿⡇⠀⠀⠆
⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣧⠀⠈⠦⢠⠛⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⠇⠀⢠⣿⠏⠀⠀⠀⣸⡿⠁⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠸⡆⠀⠀⣼⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣾⡟⠀⠀⣼⡏⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⢿⠀⠀⡿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⠀⠀⢰⣿⠁⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢰⢸⣇⠀⡇⠀⢂⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠁⠀⣾⠇⠀⣰⣿⠇⠀⠀⢠⣾⠋
⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠘⣿⠀⡇⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⠀⢸⡟⠀⠀⣿⡟⠀⠀⣠⡟⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣹⣧⢸⡆⣿⠈⠳⣠⣇⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⣾⠃⠀⣸⡿⠁⢀⣴⡏

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠻⠶⠶⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⢶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠻⣦⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣠⣤⡶⠶⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣠⡟⠉⢀⣠⣤⣤⣿⣋⢉⣙⠻⣶⣄⡀⢀⣴⠟⠀⣰⣶⣄⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏
⢰⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠁⢀⣴⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣶⣾⣿⠟
⣿⠁⢰⢻⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⠀⣼⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢿⡄⢿⣽⣷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠃
⠸⣧⢸⣿⢻⣧⠀⠀⠀⠀⠀⠈⠛⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⢻⣦⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠙⠻⠷⠿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠶⠶⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠻⣦⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣦⡀⢠⡀⢀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⢀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡼⡅⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠟⢠⡟⣧⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠀⠀⢠⣿
⠀⠀⠀⢀⣤⡶⠶⠶⠶⣦⣠⣤⣄⡀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢩⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⠷⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⣰⡟⠁⢀⣤⣶⣶⣿⣍⣈⣉⣙⢷⣦⣄⣠⡾⠋⠀⣼⣦⣄⣀⠈⠀⢀⣀⣠⣤⠶⣶⣷⣾⣿⣶⣾⣿⣇⠀⠀⠀⠀⠀⠀⣀⣴⡟
⠀⢰⡟⠀⣴⣿⣿⣿⣟⣻⣿⣿⣿⣿⣷⣽⣯⣉⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣶⢿⣿⡟
⠀⣼⠇⢠⢻⣿⠋⠉⠻⣿⣮⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⣿⡀⢸⣹⣏⠀⠀⠀⠈⢿⣮⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠁
⠀⢹⡇⢹⣾⣿⡀⠀⠀⠀⠀⠙⢿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠟⠉
⠀⠘⣷⡸⡷⣿⣷⡄⠀⠀⠀⠀⠀⠙⠻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⠟⠋⠁
⠀⠀⠘⣷⣝⣵⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣿⣿⣯⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢰⣷⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠶⣿⣟⣉⠁⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⣠⢾⡆⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠈⠙⠛⠶⣦⣄⡀⠀⠀⠀⠈⠙⠷⣴⠋⠘⣇⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣿⠀⢀⣠⢴⣶⣶⣦⡀⠀⠀⢀⡾⠋⠉⠛⢶⣄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠗⠀⠀⠀⠀⠘⣧⠀⠀⠀⣠⠞⣹⡷⠋⢡⡟⠁⠙⠛⣷⣠⡤⠞⠁⠀⠀⠀⠀⠙⢳⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣤⣤⣤⣤⣤⣤⣿⠄⢀⡾⠃⠀⠉⠀⠀⣾⠀⠀⠀⠀⣿⠋⠀⠀⣠⡴⠶⣦⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠁⠀⠀⣼⣿⡷⢠⡟⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠡⠟⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢿⡁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠈⠻⣦⡀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠰⢶⡇⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⠻⣦⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠸⣧⠀⠀⢠⡞⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⢨⠿⣦⣀⡀⠀⠀⠘⣷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠙⣶⠋⠀⠀⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠁⠀⠀⠀⠀⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣦⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣿⣿⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣾⣿⣿⣆⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣼⣿⣿⣿⣦⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⣿⣿⣿⡿⠟⠋⠹⣿⣿⣿⣿⣿⣿⡿⠀⠒⠒⡄⠀⠙⢿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⢁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠆⠀⣾⣿⣿⣿⠿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠂⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⠀⣿⣿⣿⠃⠀⠀⠈⠙⠻⣿⣿⡷⠶⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣿⣿⣷⡄⠀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⡀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠿⠿⠟⠁⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠃⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣾⣿⣷⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⣠⠃⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣍⢀⣤⣇⠀⠆⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡀⠀⠈⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⢀⠘⢿⣿⣷⣄⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠈⠋⠀⠈⢻⣿⠿⣦⡀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠘⣿⣿⣿⣿⣿⡃⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⠀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠈⠛⢿⣿⣍⢦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣤⣄⣀⡙⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣇⠀⣾⣿⣿⣿⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⡇
⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠁⠨⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠉⠉⠉⠉⠁⠐⠒⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠉⠀⠀⠀⠀⠀⣀⣠⡄⠲⠦⠐⠒⠒⠒⠒⠒⠪⠽⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣤⡦⠓⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠐⠀⡀⡄⠈⢠⠀⠈⢢⡀
⠀⠀⠀⠀⡠⠀⠀⠀⢠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠐⡆⢀⠁⡀⠀⡃⣀⠇⠁⢠⡌⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠁⠀⢠⡾⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢀⡀⣣⡜⠤⠓⠈⣀⠯⡠⠤⠊⠳⣄⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢀⣜⠀⠠⠀⠀⠀⢀⠈⠀⠀⠀⢀⠀⡇⠀⢀⣾⠗⠋⠡⠄⠂⠉⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠸
⠀⠀⡄⠀⠀⣼⠃⡕⠀⠀⠀⠀⠃⠀⠄⠀⢀⡨⠀⣀⡁⠎⠏⠀⠀⠆⢊⠂⠁⠀⠀⠀⠀⠀⢀⠌⣣⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡧⣌⠀⠀⠀⢀⠀⠀⠀⠀⣠⡴⠊⢱⠛⠉⠀⠀⠀⠀⢀⠀⡠⠀⠈⠀⠀⠀⠐⠉⠀⢹⠀⠀⠀⢀⡇
⠀⠈⠀⠀⣷⠀⠆⠀⠀⢀⣀⣀⣀⠀⢠⠇⠚⢀⣏⠀⡄⠀⠀⠀⠀⡌⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸⡇
⠀⠛⣦⠀⣏⢀⠀⡤⠊⠁⠙⠀⠙⠶⢶⣴⠚⠋⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⡠⠖⠒⠒⠢⢄⢀⡼⠁⠀⡰⡄⣾
⠀⠀⢸⡇⣿⢽⡐⠄⠠⠗⠂⠀⠀⠀⠀⠩⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡰⠊⠀⠀⠀⠈⢋⠀⠀⢠⡑⢸⠇
⠀⠀⠀⢁⠇⠈⣇⠀⠈⠛⠶⠄⠀⠀⠀⠀⡆⡟⢊⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⣠⠀⠀⠀⠀⢸⠃⠀⠁⢀⡟
⠀⠀⢰⠏⣰⣿⣼⣆⠀⠀⠀⠀⠀⠈⢀⠀⣧⡸⡅⠈⠀⠀⣀⣀⠠⠤⠚⠃⠀⢉⣀⠀⠀⠀⢸⠀⠀⣠⠎
⠀⠀⠛⠘⠳⠴⣏⠁⠹⣄⡀⠄⠀⠀⣤⣾⠏⠀⠈⠉⠉⠁⠀⠀⠀⣠⠔⠲⠀⣾⠅⠀⠀⢠⠇⠀⣠⠃
⠇⠀⠀⠀⠀⠀⠈⢷⡀⠀⢹⡶⠒⢞⡉⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⣿⠀⡜⠀⠁⠀⢀⡴⠋⣠⣴⠃
⠀⠙⢢⣴⠲⣤⣤⡏⣇⣀⢾⣿⢿⣿⠗⢂⣤⡀⠀⠀⠀⠀⠄⠰⠙⠹⡰⢆⣀⣤⠖⠉⠆⢠⢈⡏
⠀⠀⣸⣧⣴⢀⠀⠈⢫⣤⣯⠔⣁⡿⢺⡙⢦⠱⣄⠀⠀⠀⠀⢈⢆⠀⢃⡎⡸⢣⠁⠈⠂⡼⡼⣇
⠀⠀⠩⣄⣀⣁⠚⠅⠐⠀⢠⡐⢩⣀⠀⠤⠤⠱⡘⠆⠀⠀⠀⠀⠀⢰⣾⠳⡏⡄⠀⠀⢠⡇⢣⢻⡄
⠀⠀⠰⣀⢀⣠⡤⠋⠀⣠⠊⣯⠆⠀⢀⣀⣤⠴⢣⠄⠀⠀⠀⠀⢀⡾⡣⠸⠰⠈⠀⠀⢸⠑⠌⢦⠽⣄
⠀⠀⠀⢹⣟⠇⢁⡬⣾⠁⢈⣹⡂⠋⠉⠁⠀⠀⠀⠘⠛⠛⢒⠝⡰⣆⢵⠇⠀⠀⠀⠂⣼⡄⠀⠸⣧⣜⢷⡀
⠀⠀⠀⡸⡉⣿⡄⠂⢸⣦⠸⠟⠛⠂⠀⣀⣀⠤⠤⠒⠚⡻⣫⣊⠼⠁⡜⠀⠀⠀⠀⠀⣿⠀⢄⠀⠘⣷⡅⠋⠳⢄⡀
⠀⠀⠀⠀⠳⣜⣏⣰⣭⠤⠖⢒⣖⣛⣿⣿⣵⣒⠲⠪⠟⢽⠯⠁⠀⢰⠇⡀⢤⠀⠀⠠⡟⡄⠈⠳⡄⠘⡎⠄⠀⠠⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⢻⡍⣉⠀⠒⠀⠺⣧⣧⠀⠀⠀⡞⠀⠁⡀⠀⠀⠀⡇⢡⠀⠀⢻⣦⠙⡀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣠⣷⡄⠀⠈⠻⣆⠀⢸⠁⠀⠂⠃⠀⡀⠀⡇⡄⢢⡀⠀⠞⢦⣡⠆⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠘⣷⡑⠤⠄⠀⣨⠃⠀⠈⣦⠀⠀⡁⠀⣷⠸⠘⡧⠀⢨⡄⢿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢀⠈⢣⠐⠀⢳⠏⠀⠀⢸⢹⠀⠀⡅⠀⣩⠈⠀⠃⠀⠀⠹⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⡉⠓⠦⠄⡁⣀⣋⠀⠀⢠⣿⠹⠀⠀⢃⡀⢻⣞⡉⠶⣖⢲⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠐⠡⠋⠀⠔⠉⠻⠛⠛⠟⢛⠞⠋⠘⣩⠞⡉⢀⠈⠛⠝⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠙⠒⠤⢄⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⠏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⡀⠘⣧⠀⣷⠀⠀⠀⠀⠀⠛⠶⢤⡤⠴⠶⠚⠋⠉⠉⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣼⣧⡄⣿⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠶⠶⢦⣄⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⡄⣿⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣼⣧⡄⣿⠀⠀⠈⣷⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⡄⣿⠀⠀⠀⠸⡇⠀⠀⠀⣼⢁⣠⡤⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⣧⡀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠏⠁⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠚⠋⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⡀⣿⠀⠀⠀⠀⢸⡆⠀⠀⣧⠀⠀⠀⠀⣠⣤⣀⣤⡀⠀⠘⣳⠀⢀⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣸⣇⡀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣴⠾⣃⣄⡉⢀⡻⢶⡌⠉⠀⢸⡇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠸⠇⢀⡿⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⢀⡿⠀⢿⣼⠇⠸⠇⠸⣇⠀⠀⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢰⡟⠁⠀⠀⠀⢀⡿⠀⠀⢠⡇⠀⠘⢧⡄⠀⢰⣿⠆⠀⣠⠿⠀⣼⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣠⡞⠁⠀⠀⣾⠁⠀⠀⠘⠧⠤⣄⣀⡤⠤⠟⠀⣰⠃⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣀⣠⡴⠞⠉⠀⠀⠀⠀⣿⠀⠀⣴⠂⠀⠀⠈⠉⠀⠀⢀⡼⠃⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⠃⠀⠀⠀⠀⢀⣰⠟⠁⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣠⡴⠛⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠲⠶⢦⣤⣤⣀⣀⣀⣀⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠙⠲⠦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠋⠛⠛⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣋⡀⠀⢀⣀⠀⢻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠶⠀⠀⠀⠶⠀⠀⠛⠛⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⣀⣠⡾⠀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠉⠛⠛⠛⢿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢀⣀⣀⠀⠀⢾⡇⠀⢀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉⠙⠛⠷⣤⡶⠛⠛⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣤⠞⠂⠀⠀⠀⠀⠀⠀⠠⠄⢀⢀⠕⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣀⢠
⠀⠀⠀⠞⠁⠂⠖⠀⠀⠀⠀⠀⠀⠘⠈⠇⠢⡀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠋⢠⢛⢱⢤⠀⠀⠀⠀⠀⠀⠀⠀⠁⣏⢸
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⣿⣷⣷⡆⡰⠀⠀⠀⠀⠀⠉⡀⠀⠉⡋
⠀⠀⢘⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⡆⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢉⡙⠀⠀⠀⠀⠀⠀⠀⠈⢀⠆⠐⢋⠰⡆
⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⠀⢰⡿⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢸⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠈⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠂⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠀⠊⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⡠⢗⠀⠺⢄⡀⠀⠀⠀⠀⠀⠀⢀⠄⠙⠿⠂⡀⠀⠀⠀⠀⠀⢀⢉⡓⠀⠚⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣹⠁
⡎⢠⡆⠀⣿⢐⠁⠉⢱⠀⠀⠀⢠⣦⡀⠀⠠⠢⠞⢿⣶⠀⠀⠀⠀⠀⠈⠄⠠⠠⢃⣀⠁⠀⠀⠀⠀⢠⣄⠀⠀⠉⠁⠈
⠣⠀⠄⠸⠿⠏⠃⠈⠘⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢸⠀⡆
⠀⠀⠀⡦⣄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢈⠀⠃⠀⡀
⠀⠉⠉⠀⣸⠠⡚⡠⠀⠀⠀⠀⠀⠀⢸⠉⠛⠉⠟⠀⠀⠀⠀⠀⠀⠀⠐⡯⢇⠈⠠⠈⠀⠀⠀⠀⠀⠀⠄⢀⠀⡀
⠀⠁⠀⠇⠈⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⣰⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢴⣤⡦⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⠶⠿⣿⠀⠀⠀⠀⠀⠀⠀⢹⢟⣛⣻⡇⠀⠀⠀⠀⠀⠀⣶⣶⡆
⠀⠀⢰⡖⣒⢶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⢶⣿⠁⠀⠀⠀⠀⠀⠀⢸⣞⣿⣇⣿⠇⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠸⠧⠭⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠋⠿⠀⠀⠀⠀⠀⠀⠀⠀⢷⡶⣾⠋⠀⠀⠀⠀⠀⣘⣻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠿⠀⠀⠀⠀⠀⠀⠛⠽⠛⠛⠛⠻⠿⠿⠿

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠈⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠤⠄⠂⠑⠂⠒⠀⠐⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠃⣠⣮⠤⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣠⣎⣠⡴⠒⠒⠒⢦⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢡⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⠃⠀⠢⡀⠀⢀⡀⢀⠀⠊⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠰⠀⠀⠀⢘⠘⢇⠀⠡⡀⠀⠀⠀⠀⢀⠔⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠢⠀⢱⠤⠨⠶⠐⠂⠈⠀⣠⠜⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⡠⠚⠦⣀⡠⠤⠤⠒⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡎⠀⠀⡠⠆⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠰⠅⠀⠀⠌⠀⠀⠀⠀⡴⠖⠀⡇⠀⠀⠀⠀⠀⠀⢣⠀⠐⡁⢣⡀⠀⠀⠘⠢⣀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠄⠊⠁⢐⣳⠀⠈⡱⠶⠦⠀⠊⢱
⠀⠀⠈⠁⠀⢀⠤⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠑⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠑⠛⠉
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠠⡤⠀⠀⠄⠤⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠤⠤⠀⠉⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣉⣉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⠓⠶⠶⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣄⡀⠀⠀⣀⠀⠀⠀⠀⣠⠋⠀⢀⠠⠒⠊⣉⠉⠉⠉⠄⠀⢀⡀⠈⠉⠲
⠀⠀⠀⠀⢠⠔⠋⠁⠀⠀⠀⠀⠀⠉⠋⠉⠈⣷⠴⠖⠊⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠁⠀⠀⠁
⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠐⠁⠙⠂⠤⡀⠀⠈⠀⠀⠀⠀⢠
⠀⠀⣸⠃⠀⠀⢀⡠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢀⠇⠀⠀⢠⠎⠀⠀⠀⠀⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡨⠓⠔⠒⠒⠲⡀⠀⠀⠀⠀⠀⢰⠃
⡔⠉⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢠⠏
⢳⡀⠂⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠉⠲⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡠⠊⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡠⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⡠⠊⠀⢀⡴⠿⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠉⠀⢀⡴⠉⠀⠀⠀⠈⠉⠙⠛⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⣀⠞
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠚⠦⡑⢦⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⠬⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⢀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡠⠀⡔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⡇⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⣇⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⣆
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠠⠤⠤⢴⣶⠈⠃⠀⠀⠀⡰⠁⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠠⠘⠇⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠔⠒⠉⠀⠀⠀⠀⢰⢁⠊⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠱⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡩⠁
⠀⠀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣬⣤⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠰⠆⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⡸⣀
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢟⠠⢃⣴⢃⡿⠀⡀⢠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣋⣵⡿⢃⣸⠖⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⡟⢳⣿⣿⢃⣴⣿⠶⠞⠁⠀⡀
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⣿⣿⣿⣿⣷⣿⣿⣇⣾⡏⣴⣿⣿⣿⣿⡷⠋
⠀⠀⠈⠈⠉⠉⠙⠛⠛⠒⠶⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠘⣻⣿⣿⣿⡿⣟⣿⣷⣾⣿⣿⡟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠙⠛⠛⠛⠛⠻⠿⠛⠀⠀⠀⢰⠀⠀⢀⣹⣿⣟⣵⣿⡿⠛⣰⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⡐⣿⣿⣿⣿⣿⣷⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⡏⢈⣿⣿⣟⠘⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣴⣿⣟⣩⣿⣿⣷⠐⣠⡿⣿⠀⠘⠋
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣴⣾⠟⠉⣼⣿⣷⣿⣿⣿⡆⠡⠘⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⡟⠀⠐⣽⣹⡏⢹⣿⡿⣷⠀⠐⢈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠙⢿⣿⠇⠀⣰⣼⣹⣿⣷⣾⣿⣷⣼⡆⠀⠀⠐⢆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡟⢿⣷⣄⡙⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣠⡾⣿⡿⠁⡀⣾⣿⡟⣽⣿⣿⣿⢻⣿⣿⣿⣿⡀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣹⠋⢰⡿⢁⣼⣿⣿⠏⣼⣿⣿⣿⡿⡟⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣹⣟⣇⠘⠃⢞⣿⣿⠋⢠⣿⣿⣿⣿⣷⣿⣿⡿⢿⣿⣼⣧⠀⠀⠄⠀⠀⠙⢢⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣿⡙⠋⣿⠟⠐⢠⣾⡟⢁⢀⣾⣿⣿⡿⢧⡙⢿⣿⣷⣸⣿⢾⡿⣧⠀⠀⠀⠀⠀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⢳⣇⠛⠀⢰⣿⠟⢀⣿⡼⣿⣿⣿⡗⠒⠛⠿⣿⣻⣿⣿⡯⠻⠿⣧⠀⠈⠀⡀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢣⡾⠁⣀⣰⣿⡙⣀⢠⣁⠀⠌⣰⣿⣴⣿⣿⣷⣻⣿⣟⢙⡷⠻⠀⢙⣷⡈⠄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠃⠘⠉⠀⠿⠋⠟⠈⠁⠀⠙⠒⠀⠸⠿⠶⠛⠛⠛⠋⠉⠁⠀⠀⠀⠉⡀⢈⠁⠀⠑⡀⠠⠀⣾⣿⣦⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣿⣾⣿⣷⣶⣿⣯⣿⣧⠀⠀⠀⠀⣿⣿⡯⠿⠛⠛⢄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡈⠳⣦⡀⠀⠙⠒⠈⠈⠉⠈⠉⠉⠉⠉⣛⣿⣿⣿⣿⣿⣟⣿⠗⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⠙⢦⡀⠀⠀⠀⠀⠀⠒⠀⠠⠚⠉⠀⠩⠛⠿⠷⠋⢁⣀⢠⡆⠀⠐⡀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠳⣌⡀⠠⠠⠀⠀⠀⠈⠀⠢⠤⠀⠀⠀⠀⢀⣽⣿⠃⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠓⠤⣈⣙⣧⡀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠒⠛⠫⠤⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠒⠒⠋
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢷⠀⡀⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⣀⠼⣦⣀⣀⣀⣀⣠⣤⠤⣶⣶⠶⣦⣄⠀⠀⢰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠐⠉⠀⢿⠉⠁⠈⠁⣨⣶⣿⠿⠛⢷⡌⠉⡙⠂⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠔⠁⠄⠒⢄⠀⠀⠣⠤⠾⠛⢋⠀⠀⠀⠈⠇⡇⢡⠀⠀⠀⢠⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠄⠀⢒⣒⣲⣤⠃⠀⠀⣀⣁⣀⡑⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠄⣀⠀⠈⠉⠀⠀⠠⠀⢠⠀⠀⠀⠀⠐⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣰⠁⠀⠀⠀⠈⠑⠆⠙⠂⢁⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡇⠀⠀⠀⠢⣀⣀⠀⠀⠁⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢸⣠⡤⠀⢤⡀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢉⠀⠀⠀⠀⡇⠀⢸⠀⠘⠂⠃⠀⠀⠀⠀⠀⠀⠀⠆⠙⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣄⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡠⠊⠁⡉⠒⢄⡁⠀⠈⠀⠀⢀⠆⠀⠀⠀⠀⣤⢀⡠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠳⢄⠈⠁⠀⠀⠘⠂⠠⠄⠔⠁⢠⠖⣀⣴⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⠶⣤⣀⠀⠀⠀⢢⡀⠀⣠⠞⣋⠞⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⡴⠊⡇⠠⡄⠀⠀⣩⠛⠂⠎⢇⠈⠉⡠⠚⠁⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠚⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡔⠁⠀⣸⠆⢰⢃⠔⠠⢃⣼⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠈⠓⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⣑⠖⠉⠈⠹⡏⠰⠋⠀⠀⠐⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢧⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠉⠀⢀⡜⠁⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⡴⠋⠀⠀⠀⠰⠀⣠⠞⠀⠄⠀⠄⠀⢈⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⠀⠀⠘
⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠰⠏⠂⠈⠉⢢⣠⠂⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢳⡀
⠃⠀⠓⠒⢲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡠⠔⠒⠈⠁⠀⣀⢾⠁⠀⠀⠀⠀⢰⠉⠉⠑⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠙⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢏⡁⠘⠀⠀⠀⠀⠀⠀⠳⢄⠀⠈⣆⠐⢦⣀⠀⠀⠸⠖⠊⠉⠁⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⢸⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢣⡀⠀⠀⢀⡼⠀⠀⢀⣠⠀⠉⠈⠙⠃⠀⠲⠶⠶⠂⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⠓⡄⠀⠀⠀⠀⠠⢄⣀⠀⠠⠾⠄⠒⠂⠉⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠖⠣⡀⠀⠀⠐⠂⠤⣇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⢠⡄⣀⠤⣄⣠⡤⠤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠙⠢⣄⡉⠀⠒⢛⠁⠒⠒⠊⠉⣀⠀⠀⠀⠀⠀⡼⠃⣴⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢹⡀⠐⠀⣉⣡⡴⠂⢀⡤⠚⠳⠞⠉⠛⠛⠛⠛⠛⠋⢩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⣏⣉⣩⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⣀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠆⠀⢠⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⠟⠛⡛⠭⢞⣩⠟⣳⢶⡤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠴⠶⡶⠚⠋
⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢈⠡⠖⢋⡅⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠠⠐⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠋
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣶⣆⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⢀⣀⣀⠀⠀⣀⣀⠀⠊⠁
⠀⠀⠀⠀⣤⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡀⣿⡇⠀⣰⣿⣿⡿⢠⣿⡿⠿⠀⠀⢀⡤⠾
⠀⠀⠀⠀⠿⠃⠘⠿⢷⣶⣦⣄⠀⣿⣿⠸⣿⣿⣿⣿⡿⠁⣿⣿⠃⠹⣿⣧⡀⠈⠻⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣿⡆⣿⣿⠀⣿⣟⢻⣷⡀⠀⣿⣿⣀⣀⠈⢻⣿⣦⢀⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠿⠛⠀⢿⣿⠀⣿⣿⠈⣿⣷⣄⡛⠛⠛⠛⣀⣸⣿⣿⢺⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠃⠈⠉⠉
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠒⠶
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣾⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢈⣩⣿⡿⠁⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⣻⣿⠀⠂⠀⠀⠀⠈⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠱⠀⠀⠀⠀⣈⣻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⢋⡁⠀⣸⠀⠀⠀⠀⠀⡸⠄⣈⡙⣛⣿⣦⡀
⠀⠀⠀⠀⠀⣠⠟⠛⠁⠀⠀⠀⠀⠉⠢⣄⣀⣀⡃⠀⠀⣩⡷⢤⣭⣿⣿⣆
⠀⠀⠀⢀⠖⠿⣠⠔⠒⠖⡊⠩⠙⡆⠀⠀⠀⢈⡇⠀⢀⣿⣶⠛⠿⢻⣿⣿⠄⡀
⠀⠀⠀⣼⠄⡱⠀⠀⠂⠔⠁⠁⠘⢧⡀⠀⡖⠉⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⡄⠀⣁⢀⡀
⠀⠀⢀⡟⢰⠁⠀⠄⠀⠀⠠⠖⢂⡌⢷⠀⡟⠀⠀⠀⠀⠙⢿⣽⣿⣿⣿⣿⣧⣼⢽⠋
⠀⠀⢸⣷⣼⠀⠀⠀⣧⠀⠀⠺⠉⠁⡌⢷⣷⣸⣿⠀⠀⠀⠐⠛⣿⣿⣿⣿⡿⢛⣧⣷⡄
⠀⠀⠸⣿⡀⡇⠀⠠⠁⠀⢦⡀⣀⣀⣀⣀⣽⣿⣿⣴⣼⣷⣤⣴⣿⣿⣿⣿⣷⣿⣿⣿⣯
⠀⠀⠀⢻⣿⣧⣀⡞⡀⠀⠀⠉⠉⣻⢮⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠻⣿⣯⣿⣦⣴⣄⡴⠞⣽⢦⡰⣾⣿⣿⣿⣿⣿⣿⣿⣧⣄⣨⣭⣽⣯⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣽⡆⠘⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠅⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⢟⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣦⣿⡇⡆⣿⣿⣿⣿⣿⡟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⢁⡏⣿⣿⣿⣿⣿⠿⡆⢲⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠿⠧⠔⡾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡗⣿⣿⣿⣿⣿⣿⣿⣦⣙⢿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣠⢷⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣿⣿⣿⣿⣿⣼⣿⣿⣧⡻⣿⣿⡄
⠀⠀⠀⠀⣠⠮⠈⠉⠛⠉⢸⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣌⢿⣷
⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣿⣿⡟⠁⣠⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⢿⣿⣿⠻⠿⠿⣿⣿⣿⣿⣞⢷⡽⣆⠹⡆
⠀⠀⢰⠂⠀⣀⠀⢠⣠⣿⣿⣿⣧⠀⣿⣿⣿⣿⣯⢸⣿⠿⣿⢿⣿⣿⣿⣿⣿⣶⣤⣀⠙⢿⣿⣿⣜⢿⣿⣆⢄
⠀⠀⠉⢄⣀⣼⣗⣿⣿⣿⣿⠁⠀⢘⣿⣿⣿⠿⣵⣿⣿⣾⣇⠀⣿⣿⠋⠙⡏⠙⢦⣝⢷⣦⠻⣿⣿⡾⣿⣽⣎
⠀⢠⠀⠀⠀⠀⣿⢻⣿⣿⣿⣆⠀⣿⣿⣿⣽⣿⡀⣿⠏⡙⢿⡆⠉⠀⠀⢰⠃⠀⠀⠈⢷⣿⣷⡀⢹⣧⡘⣿⡉⠸⡀
⠀⢸⠀⢇⢀⣼⣿⡜⣿⣿⣿⠟⣦⣽⣿⣿⣿⣿⣴⣷⣿⣿⣿⡃⠀⠀⠀⠘⡆⠀⠀⠀⠀⢻⡿⣷⡈⢹⡇⢻⣧⠄⠃
⠀⠸⠤⣸⣿⣿⠟⢿⣿⣿⣿⡄⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⢨⣷⠀⠀⢀⣼⣁⠀⠀⢀⣤⣘⣿⣿⡇⢘⣿⠐⠛
⠀⠀⠀⠀⠉⠉⠀⠙⠺⠿⢿⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣶⣷⣿⡿⣯⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠛⠛⠋⠀⠀⠀⠈⠉⠉

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⢸⡄⠀⠀⠈⢧⡀⠀⠀⠀⢀⣀⣤⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣤⠀⣿⠈⣧⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⣀⣀⣀⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⡁⣿⠀⠸⡆⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠁⠉⠉⠳⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⡅⣿⠀⠀⢻⡀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠄⣿⠀⠀⠈⣇⠀⠀⣸⣡⡴⠶⠶⠦⣄⣀⠀⠀⠀⣀⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣿⠒⣿⠀⠀⠀⢹⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠉⢉⡉⠁⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠃⣿⠀⠀⠀⢸⡀⠀⢹⠀⠀⣠⠞⡙⠚⠳⣄⡐⠋⠀⡟⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠛⠁⡿⠀⠀⠀⢸⠁⠀⢸⠀⢸⡇⠸⣽⠆⠾⠀⣇⠀⢰⠇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄⡞⠁⠀⠀⢀⡞⠀⠀⡟⠀⠘⢦⠀⠰⠿⠀⢠⠞⢀⡏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⢀⡠⠏⠀⠀⢸⠁⠀⢀⠈⠛⠲⠴⠓⠋⢠⠞⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠶⠶⠖⠛⠉⠀⠀⠀⠀⠸⡆⠀⠻⠀⠀⠀⠀⢀⡴⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠴⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠶⠶⠶⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⠲⠤⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠



⡏⠙⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡏⠉
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⠄⠦⡀⡤⠀⠤⠀⠠⡄⠀⡤⠠⡄⣤⠀⢠⠄⠀⠀⢠⠄⠀⠠⡆⠀⠤⢠⠀⡠⠀⢄⢠⠄⡤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⠀⢀⡇⡇⠀⠸⠀⡞⠸⡄⠛⠘⡄⣽⠀⠐⠋⠀⠀⠀⠑⡆⠀⡇⠀⡗⢺⠀⡅⠀⣸⢸⠛⢄⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⣠⣄⠀⠀⠿⠟⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠋⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⡿⠋⣡⣤⣀⣤⣄⣠⣤⣄⣤⣤⣠⣤⡄⢠⣿⣿⡿
⠀⠀⠀⢰⣿⣿⣿⣿⡟⠉⠀⠈⠙⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⣉⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⡟⢠⣾⣿⣿⠃
⠀⠀⠀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⢴⣶⣶⣶⣿⣿⣿⣿⣿⣯⣀⣸⣿⣿⣿⣿⣿⣷⣿⡿⢋⣴⣿⣿⡿⠁
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡈⠻⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⠟⢋⣴⣾⣿⣿⠟
⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣌⡙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣤⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⡀⡀⠁⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠠⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠆⠀⣠⣀⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⣀⠄⠁⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡠⠈⠀⠠⠆⠀⠀⠀⠀⠀⠈⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠊⠁⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠄⠀⠀⠀⠀⠀⠂⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⡀⠀⠀⠢
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠑⠄⠀⠊
⠀⠀⠀⠀⠈⠢⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠄⠐⠁⠀⠀⠜⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡁⠀⡠⠂⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⢀⠀⢈⠀⠀⣀⠤⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⡰⠊⠐⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠸⠁⠀⠀⠀⠁⠀⠒⢄⠀⠀⠀⠈⠂⠤⠐⠂⠀⠀⠀⠀⠠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣉⣁⣁⣁⣈⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⢀⠜⠋⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢠⠞⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢾⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠠⠄⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⡀⠀⢀⠀⣀⢄⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣍⡉⠀⠙⠁⠀⣀⣤⠞⠯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⡶⠋⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠙⢦⠾⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠸⠀⢄⢈⡦⠂⠀⠀⠀⢀⡀⠀⣀⠤⢄⠀⢀⠜⠁⠀⢀⠇⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢣⡤⠏⠀⠀⠀⠀⡄⢁⡀⠀⠈⠢⠜⠊⠁⠀⠀⢠⠎⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⠊⢀⡀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣀⢀⠔⢠⠖⣇⠀⠀⠀⠀⠀⠈⠉⡏⠀⡀⠀⠀⣀⠎⠀⠀⠀⣠⢤⣀⡤⠤⡀⠀⣸⠃⠀⠀⠀⠀⠀⢀⠀⣀
⠀⢰⠁⢨⡀⠘⢤⢽⡀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠁⢠⠞⠉⡟⠀⣰⠃⠀⠀⡿⢠⠏⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀
⠀⠈⢳⠂⠀⠀⢀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠓⠀⠈⠀⠀⢸⡵⠋⡜⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠈⡆
⠀⠀⢸⡄⠀⢄⠼⠀⠀⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠂⠉⢂⠀⠈⡇⠀⢇⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢧⡀⠀⠀⢀⣠⠴⠋⠉⠓⠶⠤⠤⠤⠤⠴⣶⣦⣤⣼⠀⠐⠤⠤⠋⠀⠀⡧⠀⠈⠳⢄⡐⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠙⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠖⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣄⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠖⠛⠉⠛⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠛⠲⣄
⣼⠁⠀⣠⣖⣃⡤⢤⣤⣉⣙⠒⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈
⣧⠀⠀⢧⠬⣭⡽⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⢘⡆⠀⠈⠛⠺⠛⠒⠶⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣾⠀⠀⠀⠀⠀⠉⠓⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠚⠉
⢻⡀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠉⠉⠛⠒⠲⡶⠦⠤⠤⢤⡤⠤⠤⠶⠖⠒⠛⠉⠁⠈⣇⠀⠀⣠⢤⡀⠀⢀
⢰⠇⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣹⠶⣄⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡏⣿⢸⡙⠆⠸
⣿⠀⠀⠀⠀⠀⠀⠀⢸⣆⣤⣤⣀⠀⢀⡼⠁⠀⠈⡇⠀⠈⠉⠙⢦⡀⠀⠀⢀⣠⣴⡇⠈⣿⠻⣞⢹⡆
⠹⡄⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣷⣀⣀⡀⠀⠈⠓⠶⠖⠋⠁⠀⢷⠀⣿⠀⡿⢸⡇⢰
⣸⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣠⡤⣤⣤⣤⠴⠃⠈⠉⠉⢳⡄⠀⠀⠀⠀⠀⢀⡏⠀⠟⠀⠀⠈⠃⠈
⢿⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⡶⠋⠻⣄⣀⣠⡴⠖⠲⠦⠤⠤⠶⠛⠲⠤⠤⠶⠚⠙⡇
⢨⡇⠀⠀⠀⠀⠀⠐⣏⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠉⠙⠛⣀⣀⡀⠀⠀⠀⢺
⣾⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠓⠦⢤⣤⠤⠶⢦⡀⠀⣠⠞⠁⠀⠀⠀⢠⣤⠴⠖⢚⡿⠶⢛⣶⠀⠀⠈
⢹⡄⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠲⠖⢛⣓⡟⣭⠇⠀⠀⢀
⠀⠙⠶⢤⣤⣤⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠠⣄⣀⠀⠞⠉⠉⠁⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠙⣟⠓⠲⠶⠶⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣰⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣯⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠎⠹⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠋⠈⣋⠿⠂⠀⠀⠀⠀⣴⡀⠀⠀⠀⣹⢂⣄⣰⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡰⢦⣹⡄⠀⠀⠰⡆⠴⠉⠧⢄⡀⠀⠟⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠄⠀⠀⢸⣠⣄⡞⠋⠁⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⢀⠀⠀⠃⠀⠀⣀⣀⣤⣟⣁⣘⣃⠀⠀⠁⠀⢉⣀⣴⠏⠧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣖⠋⠘⣓⡶⠂⠈⠁⠀⢀⡴⠚⠻⠤⣄⣀⡀⠈⢉⣻⣦⡀⠀⠈⣳⣠⣄⢾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠞⠳⣯⠀⣀⠀⠀⢀⡾⠲⢤⣄⡀⢀⣨⠿⠛⠉⢁⣦⡽⣄⠀⠟⠁⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⣯⣤⣀⣀⡤⠟⠉⣀⣠⠴⠚⠉⠀⠣⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣀⡴⠋⢁⣠⣴⣚⣉⣀⣀⣀⣀⣀⣈⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⢀⢀⢠⠰⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠰⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣬⠼⠖⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣄⠉⠓⠛⠛⠋⣉⡉⠉⠉⠉⠡⣤⣀⠀⠀⠀⠘⠓⣋⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣼⠃⠀⠀⠴⠻⠟⠀⠀⠀⠀⠀⠘⠼⠛⠆⠀⠀⢺⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⣠⠀⠀⠀⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣗⠀⢰⠲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣄⠀⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣹⠖⣶⠤⣤⠤⣴⢋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠾⠀⠚⢡⠞⢡⠞⡡⠋⡼⠓⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢾⡉⠈⢧⡙⢦⡀⠀⠀⠐⠃⠞⠡⠎⢚⡵⠋⣰⢋⡿⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠛⢷⡂⣹⠦⣍⡓⠶⠤⠤⠴⠶⠚⣉⡤⢺⡧⠟⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠉⠓⠶⣮⣩⣟⣳⣖⣒⣟⣩⡿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠴⢃⡴⠂⡀⠀⠀⠀⠀⠈⢀⠉⠉⡀⠀⢀⠀⢠⠄⢀⡴⠁⡴⠃⣠⠆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋⡠⠊⣠⠞⢀⡴⠃⡰⠋⣠⠊⢁⡴⠃⡴⠋⣠⠋⢠⠞⢀⡞⢁⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠠⠞⠠⠞⠁⠠⠋⠀⠚⠡⠞⠁⣰⠋⢠⠞⠀⡼⠁⡴⠃⢠⠏⢠⠋⡴⠂⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠃

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠸⢿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⢰⣾⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣁⣤⣶⣶⣿⣿⣿⣿⣷⣶⣄⠙⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢨⣽⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⠘⢻⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠁⣼⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⣾⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⠰⠋⣡⣴⣶⣶⣶⣶⣤⣈⡙⠛⠿⠿⢿⣿⠿⠿⠛⡁⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⡁⢘⣹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣴⣶⣾⡿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿⠛⠻⠿⠟⠻⢿⣿⣿⣦⠈⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠘⣿⣿⣿⠿⠛⢀⣾⣶⣶⣾⣧⡈⠛⢿⣤⣴⣿⣿⠇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⡅⢨⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣇⠰⣿⠏⢡⡄⠹⣿⠋⣹⣷⠀⣿⣿⣿⡿⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⢰⣿⣿⠋⣠⣿⣧⣈⣁⣼⣿⣦⣽⣿⡀⢻⣿⣿⠃⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⣸⣿⣿⣄⠙⢿⣿⣿⡇⠀⠈⣿⣿⡟⢁⣰⣿⡏⢰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⢀⣿⣿⣿⣿⣇⠘⠿⠿⣿⣶⡾⠿⠿⠀⣿⣿⡟⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⡿⢿⣷⣶⣦⣈⠋⣠⣶⣶⣾⣿⠏⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⡏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⡿⠃
⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⣿⣿⣿⠟⠁
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣌⣉⣉⣉⣉⣤⣴⣾⣿⣿⣿⠟⠁
⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿
⠈⢿⣿⣿⣿⣿⣧⡀
⠀⠈⢻⣿⣿⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⠶
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠢⡀⢠⠔⠀⠐⢄⡀⠀⠲⠀⠀⠀⠰⡄⠀⢠⠀⠀⢀⠆⣤⠀⡖⠀⠐⠄⠀⠄⠀⠀⠀⠀⠔⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡤⠁⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠃⠰⠀⠀⠆⠀⡘⠀⣿⠀⡧⠀⢠⠆⠀⠀⠀⠀⠀⠀⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠑⢄⣀⣀⠔⠃⠀⠀⠀⢀⠋⠀⠀⠓⠀⠐⡆⠁⠀⠻⠀⡇⠀⠈⠆⠀⢆⣀⣠⠆⠠⢄⣀⡠

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

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

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

⠀⠀⠀⠀⢀⠠⠀⠀⠄⠀⠀⠀⠀⠀⢀⠢⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠄⠤⠀⠠⡀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠈⠀⢈⡁
⠀⠀⠠⠀⢀⠀⠀⠠⠀⠀⠀⠀⠐⠠⢀⠀⢁⠂⡀⠀⠀⠀⠈⠤⡀⠀⣡⣩⠀⠀⠀⠀⠀⠠⠀⠰⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠠⠀⠐⠀⠄⠀⠀⠀⠀⠀⢀⡁⠐⠀⠄⠀⠀⠀⠀⠀⢂⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⡑⠠⠀⠤⠀⠀⠀⠀⠀⠀⠁⠡⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠠⠁⠀⠂⠐⠀⠁⠀⠀⠀⠐⠁⢀⠀⢀⠔⠀⠀⠀⠀⠀⠀⢀⠀⠰⠕⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢄
⠀⠀⠀⠀⠠⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠄⠀⠀⠀⠀⠀⢈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠨⡂⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠂⡆⠀⠀⠀⢁⠡⠀⢀⠠⢂⠅
⠀⠀⠠⠡⠄⠀⢈⠁⠁⠀⠀⠀⠐⠠⢄⠀⢄⠂⠆⠀⠀⠀⢀⠀⠁⢤⠊⠀⡀⠀⠀⠀⠀⠂⠂⠁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡰⠒⠲⡴⠊⠙⢦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠈⠃⠀⠀⠀⠀⠀⠘⠋⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠈⠁⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁⠀⠀⣀⣀⡶⠤⠞⠙⠋⠈⠑⢻⠀⠀⠀⠀⠿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠄⠀⠀⠀⣼⠉⠀⡀⠀⠀⠀⠀⠀⠀⠙⣇⣀⠀⠀⠾⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠀⣾⠁⠀⠋⢉⠃⠀⠀⠀⠚⠓⠂⠈⢧⣠⡀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡬⠗⠀⣸⠦⠞⠀⠀⠋⠉⠉⠀⠀⠛⠉⠙⠂⠀⠀⠩⡇⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠘⡦⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⣀⣿⠀⠀⠀⠀⣠⠤⠤⠤⣤⡀⠀⠀⠀⢸⠉⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⢧⠀⠀⠞⠁⠀⠀⠀⠀⠙⠦⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⠗⠒⠒⠒⠒⠚⢯⣀⣀
⠀⠀⠀⢤⡀⠀⡀⠀⢀⡴⠚⠉⠉⠉⠉⠉⠉⠓⠶⠦⠤⠴⠒⠋⠀⠈⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠙⠲⠟⢦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡠⢶⡜⠁
⠀⠀⠀⠀⠀⠀⢀⡞⢳⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡎⠁⠈
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠉⠈⢦⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣤⡠⠚⠁⢳⠀⢀⡀⠀⢀⡤⠖
⠙⠢⣀⣦⡀⢠⡇⠀⠀⠀⠀⠀⣿⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⠋⠈⠀⠀⢀⣸⡖⠉⠷⠚⠁
⠀⠀⠈⠁⠙⣾⣵⣦⡀⠀⠀⢰⠃⠀⠈⢧⣤⡀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠈⣧⠶⣦⠖⠋⠀⣇
⠀⠀⠀⠀⢀⡏⠈⠁⠙⠺⢳⣿⡀⠀⠀⠀⠀⠙⣄⠀⠀⣠⠞⠁⣀⠀⣠⠞⣿⠖⠁⠀⠀⠀⢹
⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⢸⢸⠙⠾⠓⠢⢄⠀⠈⠁⠈⢁⡤⠊⠙⠋⣡⠜⠁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⣸⣀⠀⠀⠀⠀⢾⣒⠲⠶⠯⠤⠴⢶⠋⠁⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠉⠓⠿⠙⠛⡶⠶⠏⠛⠚⣾⡿⠉⠉⢩⣖⣒⣛⣛⠀⠀⠀⠀⠘⣆⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⠀⠀⠰⣯⣭⡤⠴⠖⢂⠀⠀⣀⡤⠟⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠀⠛⠒⠺⠯⠽⠒⠋⠁⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢉⠙⡆⢳⠀⠀⠀⠙⠦⠤⠤⠶⠒⠛⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢿⠂⡇⠈⣆⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠢⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣸⡁⡇⠀⠸⡄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠈⣆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠄⡇⠀⠀⢷⠀⠀⣞⡤⠖⠲⠦⢄⣀⠀⠀⣀⣼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢻⡂⡇⠀⠀⢸⡄⠀⡏⠀⠀⢀⣀⣀⡈⠉⢭⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣾⠅⡇⠀⠀⢸⡇⠀⣧⠀⣴⢚⣬⡁⡙⢢⠉⠀⡟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⡴⠃⠀⠀⢸⠃⠀⡇⠀⣏⠈⢛⣥⠁⢸⠆⣰⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⡰⠋⠀⢸⠁⠀⠘⠧⢤⣡⠤⠇⣠⠃⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠬⠤⠒⠋⠀⠀⠀⢸⡀⠸⠅⠀⠀⠀⢀⡴⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣠⠴⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⣠⠤⠤⠤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⡖⢲⡄⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠒
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡆⠐⡄⡆⠐⡂⢀⢦⠀⣆⡠⢰⠀⠀⡂⠀⠀⢄⠀⠀⡆⠀⡂⡢⡠⠂⢢⢰⡠⠂⣔
⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣇⣀⣂⣇⣀⣂⣜⣈⣂⣇⣘⣈⣀⣀⣅⣀⣀⣀⣙⣀⣇⣀⣉⣡⣑⣄⣜⣸⣉⣆⣣⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡤⠂⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢟⡠⠀⠀⠀⠂⠴⠁⣠⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠳⠞⠅⣠⠶⣠⠀⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠲⢶⣿⡄⠀⠉⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣬⣥⣀⣹⣿⣦⠀⣌⣃⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢼⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣿⣿⡏⠻⢯⣉⣥⠟⠀⡘⢻⡄⠀⠀⠀⢠⣧⡀⠀⣀⣄⣈⠰⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⡏⢸⣷⣔⣤⠤⠤⠄⣠⣴⡿⢧⡀⠀⠀⠈⠻⢍⣺⡇⠈⠀⠀⠀⠀⣤⣆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⡓⣼⠃⣸⣿⠟⠻⠛⣿⢿⠟⠋⢀⣴⡷⣶⢶⣔⢷⠌⢈⡹⠀⠀⠀⠀⠀⣿⣿⣿⣶⣰⣀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⣿⡌⣴⠟⢀⣤⣾⣧⡇⠀⣴⠏⠈⢳⣿⣿⢿⡏⠀⠾⠃⠀⢀⠴⣷⠆⠛⠙⢿⣯⣿⣿⣦⣤⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣧⣹⡆⣾⣿⣿⡿⢡⡿⠁⠀⠀⠀⢹⡟⣼⣧⡀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣈⣙⠿⡿⣟⣡⣿⣧⡆⠠⠀⠀⢻⣿⣿⣿⡋⠉⠀⠀⠀⠀⠐⠀⠀⠀⢶⣏⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⡄⢱⡍⠁⢉⣘⣿⠴⣶⣶⣾⣿⣿⡿⠁⢀⡳⢀⠀⣾⢯⣴⣶⡆⠀⢩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣥⣶⣿⣿⣄⠈⢿⠁⠀⠉⠙⢉⣽⣿⢃⣼⡇⠀⠂⣼⢿⣿⣿⣨⡇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⣿⠛⣿⡽⠋⠀⢀⣀⡤⠤⠔⠋⠙⠉⠉⢻⡇⠀⣐⣾⣿⣿⣼⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠽⣯⣾⣿⣶⣞⢿⡻⣍⠁⠀⠀⣀⠀⠀⠀⣬⣷⣤⣸⣿⣿⣿⣻⡇⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣦⠀⠈⠀⠀⠀⣿⠀⠀⠀⣿⣿⠉⣿⣿⣻⣿⡟⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢠⡀⣦⠈⠀⠀⠀⣿⡇⢠⡞⠉⢿⡟⠁⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢘⣷⠿⠀⠀⠀⣸⣿⠃⢨⠃⠀⢠⡷⡀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣌⠿⡆⠀⠀⠀⢿⣿⢶⣼⠆⠀⣼⠃⢱⣄⡀⢰⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣄⠀⠀⢸⣿⡇⢠⡛⠀⠀⠙⠀⠀⠘⣧⡉⠝⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣡⢨⠟⠃⢠⢿⣿⣎⠁⠀⠀⣄⠀⠀⠀⠙⠛⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢨⠄⠀⢀⢸⣾⣧⡙⣄⠀⢀⡅⠀⡀⠀⢀⠀⠈⠻⣶⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠈⢯⣿⣿⡁⠈⠂⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡇⠀⠀⠀⠠⣾⣿⡿⢿⡟⠒⢺⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⠉⠙⢈⠻⠝⠺⡃⠀⠒⢠⠄⠀⢀⣈⣢⣀⣠⡀⣸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢠⣄⣀⠀⢀⣀⡀⣀⡑⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠉⠁⠙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⢿⣿⡆⠀⠉⠁⠉⠙⢿⣿⣿⠂⠠⡶⠶⠠⠄⠂⠀⠀⠄⠀⠐⠆⠀⠠⠀⠼⠟⠛⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⢻⣧⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣷⣴⣤⣦⣉⣤⣀⣠⣦⣀⣀⣠⣷⣦⣄⣀⣶⣄⣠⣄⢶⣶⣶⣿⣿⣿
⣀⣀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⠙⣛⣻⣛⣋⣉⣉⣋⣉⡉⣉⣀⣈⡉⣉⠙⠋⠉⢉⣉
⣿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠺⠛⠛⠛⠛⠛⠻⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢀⣴⠾⠛⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠹⣟⠛⠻⢦⡄
⠀⠀⠀⠀⢀⣼⡇⠀⠀⣼⠃⠀⠀⠀⢠⡄⠀⠀⣼⠃⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⢻⡀⠀⠀⣤⠀⠀⠀⠀⢿⡀⠀⠈⣿⣀
⠀⠀⢠⡾⠋⠉⠀⠀⠀⣿⠀⠀⠀⢠⡟⠀⠀⢀⡏⠀⠀⠀⢿⡘⣇⠀⠀⠀⠀⠀⢀⡟⣸⠇⠀⠀⢸⡇⠀⠀⠘⣧⠀⠀⠀⢸⡇⠀⠀⠉⠉⠻⣆
⠀⠀⣿⠃⠀⠀⠀⠀⠈⣿⠀⠀⠀⣾⠁⠀⠀⠀⠷⠀⠀⠀⠈⠃⠉⠀⠀⠀⠀⠀⠉⠀⠋⠀⠀⠀⠸⠇⠀⠀⠀⢹⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡆
⠀⠀⢹⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠙⣿⡆⠀⠀⣴⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠂⡟⠀⠀⠀⠸⠇⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⠟⠛⢷⡄⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠘⠳⣦⣴⡇⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠠⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠄⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢨⣷⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣼⠇
⠀⢀⣀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣸⡇⠀⠀⠀⢀⣀⣀⢀⣬⣽⣶⠀⢀⡤⢠⡄⣀⠀⣴⣾⣥⣄⢀⣀⣀⠀⠀⠀⠀⣿⣀⣠⡾⠃⠀⠀⠀⠀⠀⢀⣀
⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠈⠉⢻⡇⠀⠀⣾⠋⣽⠋⠀⠀⠀⠀⠠⠞⣱⠏⡼⠋⠀⠀⠀⠀⠀⠈⢻⡉⢻⡆⠀⠀⣿⠋⠉⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⣸⣷⡀⠀⠈⠛⠁⠀⠀⠀⠀⣀⣤⡀⢀⣀⣀⠀⣠⣀⡀⠀⠀⠀⠀⠙⠋⠀⠀⣸⣷⡀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⣶⡟⠋⠀⠀⠀⠀⢰⡏⠈⠳⣦⣀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠋⠁⠉⠛⠁⠈⠁⠀⠀⠀⠀⠀⢀⣤⡾⠋⠘⣷⠀⠀⠀⠀⠈⠛⣷⡄
⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⣼⠇⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠶⠟⠋⠁⠀⠀⠀⢿⡄⠀⠀⣠⡶⠟⠉
⢀⡀⠀⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⠀⠀⠀⠀⣶⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⣰⡆⠀⠀⠀⠀⣸⡇⠀⠘⠋
⠉⠛⠛⠳⢶⣶⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡶⠾⠛⠛
⠀⠀⠀⠠⣾⣧⣤⣤⣤⡤⠀⠈⢷⡀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣼⠏⠀⠠⣤⣤⣤⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠈⠙⢷⣤⣀⣠⣤⡀⠀⠀⣤⣤⣀⣠⣴⠟⠉⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⣠⣴⠶⢻⣇⣠⡶⠗⠀⠀⠙⢷⣤⣀⠀⠀⠀⡟⠉⠉⠉⠾⣧⡀⣠⡿⠎⠉⠉⠹⡇⠀⠀⠀⣠⡴⠟⠁⠀⠘⠷⣦⣀⣿⠷⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣶⣧⣄⠀⠀⠀⢸⣿⣏⠀⠀⠀⢀⣴⣧⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠄⠑⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⡂⠀⢀⠤⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⣾⣿⢿⣛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⡇⠠⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⢺⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠈⠀⠈⠒⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣿⣷⣷⣿⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠁⢐⡖⢲⡀⠀⠄⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣟⣵⡍⣿⣯⣞⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠢⡀⠉⠛⢀⡀⠄⠊⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠉⠙⠷⠶⠶⠿⠟⠋⠀⠀⠀⠑⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠌⢉⠉⢻⠉⢙⢫⡿⢽⣿⣿⠿⣿⠿⣿⣿⡿⢿⣿⣿⡿⣿⡟⢻⢿⣿⣿⣿⠋⣉⠻⣿⣿⣿⣿⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡉⠱⠀⢸⠃⢰⡄⢹⠀⣶⠀⣴⠀⡃⢰⡄⢸⠁⣴⣴⡇⢰⡄⢹⣿⣿⠀⣉⣰⠁⣶⠄⣿⠀⡇⢰⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⡤⣾⣦⣼⢶⣤⣤⣾⣤⣿⣤⣿⣦⣷⣤⣤⣼⣶⣤⣤⣧⣾⣧⣼⣿⣿⣴⣿⣿⣶⣤⣤⣿⣴⣧⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠈⠉⠁⠀⠁⠁⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡞⡇⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠈⣳⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣼⣷⢙⣾⣯⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢇⢀⡶⠛⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⡜⡟⣵⢱⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢈⡟⣇⣘⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⡿⣛⢛⣧⣷⣭⣼⢹⣟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣖⣒⡉⢀⣬⡉⣼⠏⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣟⡿⢾⡁⠆⣸⣧⣼⣣⡿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡁⢺⣇⣸⡇⠘⢶⡷⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣷⣭⣭⣭⣶⣿⣹⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠻⠿⠿⠓⠛⠉⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠰⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠟⠛⠛⠛⠛⠿⢿⣷⣦⡀⠀⠀⠀⠀⢠⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣾⡿⠋⠁⣀⣤⣴⣶⣶⣦⣤⡀⠈⠻⣿⣧⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡦⠀⢠⣿⡟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠀⢰⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠛⢻⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣿⣟⠉⢹⣿⣿⠁⠀⣶⡆⠀⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣥⣄⣀⣀⠀⠀⠀⠀⢠⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣧⣀⠀⣀⣴⣿⣿⣇⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠁⠀⠀⢠⣿⡟⠀⣠⣿⣿⡿⠋⠀⣀⠈⠹⣿⣿⣿⡿⠿⣿⣿⣆⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢠⣿⣿⣿⣿⠀⠘⠟⠀⢠⣿⣿⣿⣧⣤⣿⣿⣿⡄⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣀⣸⣿⣿⡿⢿⣿⡇⠀⣾⣿
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⢿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣄⣠⡿⠁⢠⣿⡟
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⡄⠈⢿⣿⣿⣿⡟⠀⣠⣄⠀⢹⣿⣿⣿⣿⠟⠁⢠⣾⡿⠁
⣰⣿⣿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣶⣿⡿⠟⠁⢠⣿⣿⣿⣿⣧⡀⠙⠋⠀⣸⣿⣿⠟⠁⢀⣴⣿⠟
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠁⠀⠉⠉⠉⠁⢀⣠⣶⣿⣿⣿⣿⠿⢿⣿⣶⣶⣿⣿⡿⠃⢀⣴⣿⠟⠁⠀⠀⠀⢠⣶⣦⣤⣀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⠿⠋⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⠗
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⢀⣀⣴⣿⡿⠋⠀⠀⠀⠀⣀⠀⠀⠀⠠⣾⣿⣿⣥⣄⣀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⡿⠿⠛⠁⠀⠀⠀⠀⠀⠐⠿⣿⣦⣄⠀⠀⠈⠉⠛⠻⠿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣶⠄
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⣿⣿⣉
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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