What Is This Point?

Geometry Level 5

Let \(\triangle ABC\) be an acute angled triangle with side lengths \(AB= 5, BC= 7, CA= 8.\) Let \(D\) be the foot of perpendicular from \(A\) to \(BC,\) and let \(O\) be its circumcenter. The feet of perpendiculars from \(O\) to \(AB\) and \(AC\) intersect \(AD\) at points \(Q\) and \(P\) respectively. Let \(S\) be the circumcenter of \(\triangle OPQ.\) If \(\cos (\angle CAS) = \frac{ a \sqrt{b} } { c} \) for some square free integer \(b\) and coprime positive integers \(a\) and \(b\), then find \(a+b+c \).

Details and assumptions
- This problem is adapted from an ARO 10th grade geometry problem.

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