If the sum

\[ \sum_{n=1}^{2111} \frac{n+2}{n! + (n+1)! + (n+2)!} \]

is written as \( \frac{1}{2} - \frac{1}{a!}\), what are the last three digits of \(a\)?

This problem is posed by Lawrence L.

**Details and assumptions**

The last three digits of \(1023\) is \(023\). You may enter your answer as \(023\) or \(23\).

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