Hitting Time on the 3-Dimensional Hypercube

A random walk moves on the $3$-dimensional hypercube graph $Q_3$: the $8$ vertices are binary strings of length $3$, and two vertices are adjacent if they differ in exactly one coordinate. At each step, the walker picks one of the $3$ coordinates uniformly at random and flips it. Starting at vertex $000$, what is the expected number of steps to reach vertex $111$ for the first time?