# 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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