Polyhedron \(ABCDEFG\) has six faces. Face \(ABCD\) is a square with \(AB = 12;\) face \(ABFG\) is a trapezoid with \(\overline{AB}\) parallel to \(\overline{GF},\) \(BF = AG = 8,\) and \(GF = 6;\) and face \(CDE\) has \(CE = DE = 14.\) The other three faces are \(ADEG, BCEF,\) and \(EFG.\) The distance from \(E\) to face \(ABCD\) is 12. Given that \(EG^2 = p - q\sqrt {r},\) where \(p, q,\) and \(r\) are positive integers and \(r\) is not divisible by the square of any prime, find \(p + q + r.\)
(第二十届AIME1 2002 第15题)