2013 AIME II Problem 15

Let A, B, C be angles of a triangle with A and C acute and B greater than a right angle satisfying

\begin{aligned} & \cos ^{2} A+\cos ^{2} B+2 \sin A \sin B \cos C=\frac{15}{8} \text { and } \\ & \cos ^{2} B+\cos ^{2} C+2 \sin B \sin C \cos A=\frac{14}{9} . \end{aligned}

There are positive integers p, q, r, and s for which

\cos ^{2} C+\cos ^{2} A+2 \sin C \sin A \cos B=\frac{p-q \sqrt{r}}{s},

where p+q and s are relatively prime and r is not divisible by the square of any prime. Find p+q+r+s.