Random walk covering a simplex and returning home

An ant starts at one vertex of an $n$-dimensional simplex (which has $n+1$ vertices, every pair joined by an edge). At each step it moves to one of the other $n$ vertices chosen uniformly at random. It stops the first time it is back at its starting vertex having already visited every vertex. What is the expected number of steps?

$n(H_n+1)$, where $H_n=1+\tfrac12+\dots+\tfrac1n$. With $i$ of the $n$ other vertices visited, each step reaches a new one with probability $\tfrac{n-i}{n}$, so covering them all takes $nH_n$ steps on average (coupon collector). From then on each step returns home with probability $\tfrac1n$, adding $n$ more.