Poly-Poly...Here I come!

Polyhedron $\text{ABCDEFG}$ has $6$ faces. Face $\text{ABCD}$ is a square with $\text{AB} = 12;$ face $\text{ABFG}$ is a $\text{trapezoid}$ with $\overline{AB}$ $\parallel$ $\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $\text{CDE}$ has $CE = DE = 14.$ The other $3$ faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is $12$ . Given that $EG^2 = p - q\sqrt {r},$ where $p, q,$ $r$ $\in$ $\mathbb{I}^+$ , find $p + q + r.$

Try my set

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( $r$ is not divisible by the square of any prime) Hint:

Figure looks like this.