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

(第十二届AIME1994 第2题)