Sufficient but not complete statistic

Let $X_1$ and $X_2$ be a random sample from a continuous distribution with the probability density function

$$f(x)=\begin{cases}\dfrac1\theta e^{-\frac{x-\theta}{\theta}}, & x>\theta,\ 0, & \text{otherwise,}\end{cases}$$

where $\theta>0$. If $X_{(1)}=\min\{X_1,X_2\}$ and $\overline X=\frac{X_1+X_2}{2}$, then which one of the following statements is TRUE?