In \(\triangle ABC,\) \(BC= 5, CA= 7, AB= 8.\) Let \(\omega\) and \(\Gamma\) denote the circumcircle and incircle of \(\triangle ABC\) respectively. A circle \(\delta \) centered at point \(P\) is externally tangent to \(\Gamma\) and internally tangent to \(\omega\) at \(A.\) Another circle centered at \(Q\) is internally tangent to both \(\omega\) and \(\delta \) at \(A.\) The length of \(PQ\) can be expressed as \(\dfrac{a}{b\sqrt{c}}\) for some coprime positive integers \(a,b\) and a prime \(c.\) Find \(a+b+c.\)

**Details and assumptions**

This problem is inspired by an old USAMO problem.

This diagram is not mine. I took it off from the AoPS thread.

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