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)