Let 
$$a_k = 0.\underbrace{00\ldots 0}_{k - 1 \text{ zeros}} 1 \underbrace{00\ldots 0}_{k - 1 \text{ zeros}} 1$$
The value of \(\sum_{k = 1}^\infty a_k\) can be expressed as a rational number \(\frac{p}{q}\) in simplest form. Find \(p + q\).

Let's think step by step and output the final answer within \boxed{}.
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