= Solution
The <Rao-Blackwell theorem> says that if $U$ is an estimator with finite second moment and $T$ is a <sufficient statistic>, then
$$
U^*=\mathbb E[U\mid T]
$$
has the same expectation as $U$ and satisfies
$$
\operatorname{Var}(U^*)\leq\operatorname{Var}(U),
$$
with equality only when $U$ is already a function of $T$ almost surely.
Here $\widehat\theta=\mathbf1_{\{X_1=1\}}$ is unbiased because
$$
\mathbb E\widehat\theta
=\mathbb P(X_1=1)=p(1-p)=\theta.
$$
Conditional on $T=t$, every weak composition $(x_1,\ldots,x_n)$ of $t$ has the same probability $p^n(1-p)^t$. There are
$$
\binom{t+n-1}{n-1}
$$
such compositions. For $t\geq1$, those with $x_1=1$ correspond to weak compositions of $t-1$ into $n-1$ parts, of which there are
$$
\binom{t+n-3}{n-2}.
$$
The Rao-Blackwell estimator is therefore
$$
\boxed{
\widehat\theta^*(T)=
\begin{cases}
\displaystyle
\frac{(n-1)T}{(T+n-1)(T+n-2)},&T\geq1,\\[6pt]
0,&T=0.
\end{cases}}
$$
For $n\geq2$ and $0<p<1$, $\widehat\theta$ is not determined by $T$; for example, conditional on $T=1$, the sole failure can occur in any coordinate. The variance inequality is consequently strict:
$$
\boxed{\operatorname{Var}(\widehat\theta^*)<
\operatorname{Var}(\widehat\theta)}.
$$
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