Level pending

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 $$AD$$?

×