Let The Mouse Go

Probability Level 4

At a famous scientific laboratory, a mouse is stuck in a (infinitely large) three dimensional lattice. As scientists have injected it with experimental drugs, the mouse is extremely groggy and thus has a uniform probability of moving to any of the 6 adjacent cells (up, down, left, right, front, back).

What is the probability that after its sixth move, the mouse is back at its starting cell?

Give your answer to 5 decimal places.