A circle with diameter \overline{P Q} of length 10 is internally tangent at P to a circle of radius 20. Square A B C D is constructed with A and B on the larger circle, \overline{C D} tangent at Q to the smaller circle, and the smaller circle outside A B C D. The length of \overline{A B} can be written in the form m+\sqrt{n}, where m and n are integers. Find m+n.