Triangle \(ABC\) has side lengths \(\overline{AB}=2\sqrt{3}\) and \(\overline{BC}=2,\) as shown. \(D\) is the midpoint of \(\overline{BC},\) satisfying \(\overline{AD}=\sqrt{7}.\)

Let \(E\) be the point of intersection between \(\overline{AB}\) and the bisector of \(\angle ACB.\) \(\overline{CE}\) meets with \(\overline{AD}\) at point \(P,\) and the bisector of \(\angle APE\) meets with \(\overline{AB}\) at point \(R.\) An extension of \(\overline{PR}\) meets with \(\overline{BC}\) at point \(Q.\)

Given that the area of \(\triangle PQC\) is \(a+b\sqrt{7}\) times larger than that of \(\triangle PRE\) for some rational numbers \(a\) and \(b,\) find the value of \(ab.\)

*This problem is a part of <Grade 10 CSAT Mock Test> series.*

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