Consider the sum, \[ S = \sum _{n = 0} ^{\infty} \frac{a_n}{5^{2n}}\]

where \(a_n\) is a sequence defined by recurrence relation

\[ a_{n+2} = 2a_{n+1} + a_n \quad \forall \, n \in \mathbb Z^* \]

and \(a_0 = a_1 = 1\). If \(S = \dfrac{p}{q}\), then find \(p+q\).

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