Turn one coin and reflip another until four heads

Four fair coins are tossed. Then you repeat the following move: choose any two coins; one of them must be turned over by hand (so its face switches), and the other must be tossed again. You must make this move even when only one coin shows tails, and you stop once all four coins show heads. Every individual hand-turn or toss counts as one coin movement, and the four initial tosses count too. With the best choices, what is the expected total number of coin movements?

$135/16$. Each move costs $2$. With $3$ heads, turn the tail and retoss a head: $e_3=2+\tfrac12e_3$, so $e_3=4$. With $2$, $1$, $0$ heads, turn one tail and toss another tail: $e_2=4$, $e_1=6$, $e_0=7$. Averaging over the $\text{Bin}(4,\tfrac12)$ start gives $4+\tfrac{7+24+24+16}{16}=\tfrac{135}{16}$.