Tricky Quadrilateral

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 $AD$ ?