Colorful consecutive (Part 10)

\[ \Large{S=\color{green}1\color{red}2\color{blue}3!+\color{red}2\color{blue}3\color{orange}4!+\color{blue}3\color{orange}4\color{magenta}5!+\ldots+\color{red}7\color{blue}8\color{orange}9!+\color{blue}8\color{orange}9\color{magenta}0!} \]

Find \( S \) \( \text{mod}\) \( \color{green}1\color{red}2\color{blue}3\color{orange}4\color{magenta}5\color{green}6\color{red}7\color{blue}8\color{orange}9\color{magenta}0 \)

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