Recursions in Recursions - 2

Algebra Level 4

\[ { \begin{eqnarray} a_{n} & =& 2a_{n-1} + 3 \\ b_{n} &=& b_{n-1} + b_{n-2} \\ \end{eqnarray} } \]

For a positive integer \(n\), consider the two recurrence relations above subjected to the conditions \(a_1 = 0\) and \(b_{1} = b_{2} = 1 \).

If the value of the expression \(\big(a_{b_{2015}} + 3\big)\big(a_{b_{2016}} + 3\big)\) can be expressed as \(p^{b_{q}} \frac{r}{s}\), where \(b_{q}\) is one of the terms in the recurrence relations sequence above and \((p, r)\) and \((r, s)\) are pairwise coprime integers.

Find the value of \((p+q+r+s) \bmod {673}\).

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