Recursions in Recursions - 2

Algebra Level 4

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

For natural number \(n\), consider the two recurrence relation above subjected to conditions, with \(a_1 = 0, b_{1} = b_{2} = 1 \).

If the value of expression \((a_{b_{2015}} + 3)(a_{b_{2016}} + 3)\) can be expressed as \(p^{b_{q}} \dfrac{r}{s}\), where \(b_{q}\) is one of the term 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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