Equilateral \triangle A B C is inscribed in a circle of radius 2. Extend \overline{A B} through B to point D so that A D=13, and extend \overline{A C} through C to point E so that A E=11. Through D, draw a line \ell_{1} parallel to \overline{A E}, and through E, draw a line \ell_{2} parallel to \overline{A D}. Let F be the intersection of \ell_{1} and \ell_{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 C B G can be expressed in the form 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.