The sum

\[\sum_{k=0}^{\infty} \left[ \frac{1}{(13k)!} - \frac{1}{13(k!)} \right]\]

can be expressed in the form \(\displaystyle \frac{1}{13} \sum_{n=1}^{12} e^{\rho_{n}}\) where \(\rho_{n}\) is the nth complex root of a polynomial of degree \(12\) with distinct roots that is irreducible over the integers. (Whew, that was a mouthful.)

Find the value of

\[\sum_{1 \leq i < j < k \leq 12} \rho_{i}\rho_{j}\rho_{k}\]

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