You are given a convex quadrilateral \(ABCD\).
\(E\) and \(F\) are points on \(BC\) such that \(\angle DEB = \angle AFB = 90 ^ \circ\).
\(G\) is a point on \(AF\) such that \( \angle DGA = 90 ^ \circ\).
You are also given that \(AB=CD=26\), \(BC=82\), and \(BF=DE\).
If the lengths of \(AD\), \(DG\), and \(AG\) are positive integers, then what is the value of \(AD\)?