estimating a parametric function I am working on this problem and am stumped. Can anyone take a look at it?

$X_1, \ldots ,X_n$ are distributed Bernoulli$(p)$ where $p$ is unknown.  Consider the parametric function $\tau(p)=(p+(1-p)e^3)^2$.
1.) Find a suitable unbiased estimator of $\tau(p)$.
2.) Since the complete sufficient statistic is $\sum_{i=1}^n X_i$, use the Lehman-Scheffe theorem and evaluate $E_p[T|U=u]$.

There were two ways I looked at it.  Immediately apparent is that the MGF of Bernoulli looks a lot like it, taken at $t=3$. But the problem there is that $p$ and $(1-p)$ are switched around. So I wasn't sure where to go from there. 
The other way I looked at it is distributing it out, and I got $p^2+2p(1-p)e^3+(1-p)^2e^6$ 
I could find an unbiased estimator of this, but that conditional expectation looks nasty...could anyone start me out? I really have no clue about this problem.
 A: Since the probabilities are flipped you can look instead at $1 - X_i$.  An unbiased estimator then could be
$$
T = e^{3(1 - X_1) + 3(1 - X_2)}
$$
since we have
\begin{align}
\text{E}(T) &= \text{E}(e^{3(1 - X_1) + 3(1 - X_2)}) \\
&= \text{E}(e^{3(1 - X_1)})^2 \\
&= (p + (1 - p) e^3)^2.
\end{align}
Finding the conditional expectation is the challenging part but it's not impossible.  We just use the fact that conditional on $S = \sum_{i=1}^{n} X_i$ all $\binom{n}{S}$ combinations are equally likely, and consider how they affect $X_1$ and $X_2$.
There are three cases to consider: when both $X_1$ and $X_2$ are zero, when both are one, and when one is zero and the other one.  There are $\binom{n-2}{S}$ combinations corresponding to $X_1 = 0$ and $X_2 = 0$, $\binom{n-2}{S-2}$ corresponding to $X_1 = 1$ and $X_2 = 1$, and $2 \binom{n - 2}{S - 1}$ consistent with $X_1 = 1$ and $X_2 = 0$ or $X_1 = 0$ and $X_2 = 1$. Collecting everything together we get
\begin{align}
\text{E}(T \mid S) &= \frac{\binom{n-2}{S} e^6 + \binom{n-2}{S-2} + 2 \binom{n - 2}{S - 1} e^3}{\binom{n}{S}} 
\end{align}
where we set $\binom{n - 2}{k} = 0$ if $k < 0$.
