Recursions in Recursions - 2

Algebra Level 4

an=2an1+3bn=bn1+bn2 { \begin{aligned} a_{n} & =& 2a_{n-1} + 3 \\ b_{n} &=& b_{n-1} + b_{n-2} \\ \end{aligned} }

For a positive integer nn, consider the two recurrence relations above subjected to the conditions a1=0a_1 = 0 and b1=b2=1b_{1} = b_{2} = 1 .

If the value of the expression (ab2015+3)(ab2016+3)\big(a_{b_{2015}} + 3\big)\big(a_{b_{2016}} + 3\big) can be expressed as pbqrsp^{b_{q}} \frac{r}{s}, where bqb_{q} is one of the terms in the recurrence relations sequence above and (p,r)(p, r) and (r,s)(r, s) are pairwise coprime integers.

Find the value of (p+q+r+s)mod673(p+q+r+s) \bmod {673}.

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