MVUE for Bernoulli Random Variable

Question

Let $X_1, X_2$ be a random sample from a Bernoulli distribution with $P(X = 1) = p$ and $P(X = 0) = 1-p$. I want to find a MVUE for $p$.

$E[X_1]=p$ and $(X_1,X_2)$ is complete and sufficient for $p$.

Therefore, by the Lehmann–Scheffé theorem, $E[X_1|(X_1,X_2)] = X_1$ is the MVUE for $p$.

I am almost certain this argument is invalid, but I don't know why. Help please.

Answer 1

$T=X_1+X_2$ is complete sufficient statistics(can you prove?). Hence, by the Lehmann-Scheffe Theorem, if we can find a function of $T$ whose expectation is $p$, it is an MVUE.

Can you find it ?