Small loaves eaten before all large loaves are halved

A bag contains $7$ small loaves and $3$ large loaves. Repeatedly, one loaf is drawn uniformly at random from the bag. A small loaf drawn is eaten. A large loaf drawn is cut into two small loaves: one is eaten and the other is returned to the bag. How many small loaves (including halves) are eaten, on average, up to and including the one eaten when the last large loaf is drawn?

$\tfrac{113}{12}$ (about 9.42). Every draw results in eating exactly one small loaf, so we need the expected number of draws until all 3 large loaves have been drawn. Solving the recursion $E(s,l)=1+\tfrac{s}{s+l}E(s-1,l)+\tfrac{l}{s+l}E(s+1,l-1)$ on the counts of small and large loaves, starting from $E(7,3)$, gives $\tfrac{113}{12}$.