Equilateral \(\triangle ABC\) is inscribed in a circle of radius 2. Extend \(\overline{AB}\) through \(B\) to point \(D\) so that \(AD=13,\) and extend \(\overline{AC}\) through \(C\) to point \(E\) so that \(AE = 11.\) Through \(D,\) draw a line \(l_1\) parallel to \(\overline{AE},\) and through \(E,\) draw a line \(l_2\) parallel to \(\overline{AD}.\) Let \(F\) be the intersection of \(l_1\) and \(l_2.\) Let \(G\) be the point on the circle that is collinear with \(A\) and \(F\) and distinct from \(A.\) Given that the area of \(\triangle CBG\) can be expressed in the form \(\frac{p\sqrt{q}}{r},\) where \(p, q,\) and \(r\) are positive integers, \(p\) and \(r\) are relatively prime, and \(q\) is not divisible by the square of any prime, find \(p+q+r.\)

(第二十四届AIME2 2006 第12题)