Suppose that $ (X_1,X_2, \ldots, X_n) $ be a random sample from Bernoulli (p). Find the UMVUE of
$$\tau(p)= (1-p)+ e^{2}p $$
My approach:
I know $T(X) = \sum_{i=1}^n X_i$ is a complete sufficient statistic for the Bernoulli distribution. And by Lehmann_Scheffe theorem,
$$\phi(T)= \mathbb{E}(W(X)\mid T(X)) $$
where W(X) is an unbiased estimator. Can one define $W(X)= (1-X_1)+ e^{2}X_1$?
How do I continue from here?