I am studying on UMVUE, and I'm struggling to find that conditional expectation
Let $X_1,\ldots,X_n$ random sample of $X\sim U[0,\theta]$. i) Show that $2X_1$ is a unbiased estimator for $\theta$ and use the Rao-BlackWell Theorem for found the UMVUE for $\theta$.
ii)Calculate $E[X_{(n)}]$ and explicitly find UMVUE for $\theta$
I already show that $2X_1$ is unbiased and also found that $X_{(n)}=\max(X_1,\ldots,X_n)$ is a complete and sufficient statistic for $\theta$, but I am having trouble finding the conditional, how I can calulate $$E[2X_1\mid X_{(n)}]=2E[X_1\mid X_{(n)}]$$ How do I calculate the conditional distribution and the expectation?