Combinatorial dice game with re-rolling

Alice and Bob are playing a game where each player rolls $$x$$ dice and they score a point for each $$5$$ or $$6$$ rolled. They are playing with standard 6-sided, fair dice, but Alice gets to re-roll a die whenever she rolls a $$1$$. My question is:

How much is Alice expected to win by as $$x$$ increases?

Clearly, as $$x\rightarrow\infty$$, they each score infinitely many points. I'm curious about the finite behavior. Bob is expected to score $$\frac{x}{3}$$ points, but how many is Alice expected to score? In the case where $$x=1$$, she has a $$\frac{2}{5}$$ chance of scoring, but what happens when $$x>1$$?

As long as she is rolling ones she won't be scorning any points. The first time she rolls something else she is equally likely to obtain any of the results: $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, two of which give her one point so she gets $$\frac{2}{5}$$ points on average from a single die. Since there are $$x$$ dice, by linearity of expectation the total numer of points is just $$\frac{2}{5}x$$. For Bob the average score is by a simillar reasining $$\frac{2}{6}x=\frac{1}{3}x$$. So the expected numer of the difference is again by linearity of expectation $$\frac{2}{5}x-\frac{1}{3}x=\frac{1}{15}x$$.