Rao-Blackwellizing a crude guess for a uniform ceiling

You have i.i.d. observations $X_1, \dots, X_n$ from Uniform$(0, \theta)$ with unknown ceiling $\theta$. A crude unbiased estimator uses only the first observation: $T = 2X_1$, which works because $\mathbb{E}[2X_1] = 2\cdot(\theta/2) = \theta$.

Rao-Blackwellize $T$ by conditioning on the sufficient statistic $S = X_{(n)} = \max_i X_i$, and identify the improved estimator.