It is well known that for a discrete random walk on the integers with a fair coin, the expected distance of the walker from the origin after $N$ time steps is $\sqrt{\frac{2N}{\pi}}$ if $N$ is large. For example, Wolfram Mathworld has a thorough explanation here.

Consider instead the following situation. We have two types of weighted coins. Type A coins have a probability $1/3$ to land heads, and a probability $2/3$ to land tails. Type B coins have a probability $2/3$ to land heads, and a probability $1/3$ to land tails. At each integer location, we place one of the coin types there at random, so that each integer location has exactly a 50% chance to get a type A coin, and a 50% chance to get a type B coin (coins are independently chosen at each integer location).

When the walker is at a particular integer location, she flips the coin placed there and goes one step right if it lands heads, and one step left if it lands tails. I believe in the literature this is known as a "Random Walk in a Random Environment" (RWRE)

What then is the expected value of the distance of the walker from the origin after $N$ steps for large $N$? I have done numerics that suggest that this value is still $\sim \sqrt N$, but the constant term is probably not $\sqrt{\frac{2}{\pi}}$.