Below is a diagram of a large cube constructed from eight smaller cubes. An ant wishes to travel from point Q to point S along the route of shortest distance. The figure also shows the route QRS. This route is, indeed, the shortest possible path length if travelling along the surface of the cube. However, it is not the only possible such route.

How many routes are there in total, from Q to S, that have the same path length as the route QRS?

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