2002 AIME I Problem 11

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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.