Find The Distance Between Their Centers

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.