Rectangular prisms!

Algebra Level 4

Given a rectangular prism (as shown below) whose edge lengths have a sum of 36, what is the minimum value for the length of the line connecting two vertices which do not share a face?

If the minimum value can be expressed as aba\sqrt{b}, where aa and bb are integers with bb square-free, submit your answer as a+ba+b.


Image credit: http://dooleymath.com/
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