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 \(p + q + r.\)

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

Figure looks like this.

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