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)