Free Throws, Given a Make

Tom makes his first free throw and misses his second. From attempt 3 on, the chance he makes attempt $n$ equals the fraction of his first $n-1$ attempts that were makes.

As $n$ grows this fraction converges to a random limit $P$, and that limit is Uniform(0,1). Conditional on $P$, far-apart attempts behave like independent coin flips with heads probability $P$.

In the limit, what is the probability that Tom makes attempt $n+1$ given that he made attempt $n$?

Answer format: round to 3 decimal places.