VTRMC 2005 Problem 4

A cubical box with sides of length $7$ has vertices at $(0,0,0)$ , $(7,0,0)$ , $(0,7,0)$ , $(7,7,0)$ , $(0,0,7)$ , $(7,0,7)$ , $(0,7,7)$ , $(7,7,7)$ . The inside of the box is lined with mirrors and from the point $(0,1,2)$ , a beam of light is directed to the point $(1,3,4)$ . The light then reflects repeatedly off the mirrors on the inside of the box. Determine how far the beam of light travels before it first returns to its starting point at $(0,1,2)$ .