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Evaluate the infinite series

$S= \sum_{k=0}^{\infty} \frac{(-1)^k~(\log_{2017} 2018)^k}{k!~(\log_{2017} 2016)^k}$

If $\ln(S) = -\log_a(b)$, where $a$ and $b$ are integers, find the smallest possible average of $a$ and $b$.

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