Cupid the Caterpillar

Geometry Level 2

A cubical room has edge length 10 feet with A and B denoting two corners that are farthest apart. A caterpillar named Cupid decides that he wants to crawls from corner A to corner B along the walls. In feet, what is the length of the shortest path from A to B that the caterpillar may have taken? If your answer is aba\sqrt{b} find a+ba+b.

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