First Problem in 2016

\[\displaystyle\sum_{k=2}^{\infty}0.\overline{01}_{k}=S_{4}\]

If \(S=\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

Note: \(S_{4}\) represents \(S\) written in base 4.

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