Collect die faces until a four or higher

You roll a fair die and are paid the number shown on every roll. After a $1$, $2$ or $3$ you roll again; after a $4$, $5$ or $6$ the game is over (you still get paid that roll). What is the expected total payout?

$7$. Let $E$ be the expected total. One roll pays $3.5$ on average and with probability $\tfrac12$ the game restarts, so $E=3.5+\tfrac12E$, giving $E=7$. (Equivalently, the number of rolls is geometric with mean $2$, and Wald's identity gives $2\times3.5$.)