Die with the first rolled face removed

You roll a fair six-sided die and then remove the face that came up, leaving five numbered faces. You keep rolling until one of those five remaining numbers appears. What is the expected total of the first roll and that second recorded value?

$7$. If the first roll is $i$, the second value is uniform on the other five faces, with mean $\tfrac{21-i}{5}$. Averaging $i+\tfrac{21-i}{5}=\tfrac{21+4i}{5}$ over $i=1,\dots,6$ gives $7$. Equivalently, the second value is marginally uniform on $1$–$6$ by symmetry, so each roll averages $3.5$.