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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