Conditional Probability, Independence & Bayes · Superchallenge problem

An ant starts at a vertex of a cube. Every second it walks along an edge to one of the $3$ adjacent vertices, chosen uniformly at random and independently of its past. Given that the ant is back at its starting vertex after $6$ seconds, what is the probability that it was at the vertex diametrically opposite the start after $3$ seconds? Express your answer as a common fraction.