Let \(ABCD\) and \(BCFG\) be two faces of a cube with \(AB=12\). A beam of light emanates from vertex \(A\) and reflects off face \(BCFG\) at point \(P\), which is \(7\) units from \(\overline{BG}\) and \(5\) units from \(\overline{BC}\). The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point \(A\) until it next reaches a vertex of the cube is given by \(m\sqrt{n}\), where \(m\) and \(n\) are integers and \(n\) is not divisible by the square of any prime. Find \(m+n\).
(第二十届AIME1 2002 第11题)