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)