Let $X_{1}, \dots X_{n}$ be a sample from the distribution with density given by: \begin{equation}p_{\theta}(x) = \theta e^{x - \theta(e^{x} - 1)},\end{equation} where $x > 0$ and $0$ otherwise. Let $\theta$ be an unknown parameter. Determine a most powerful test $H_{0} \colon \theta = \theta_{0} = 1$ against $H_{1}\colon \theta = \theta_{1} = 2$ at level $\alpha = 0.05$.
I intend to solve for $k$ in $\mathbb{P}(\frac{L_{1}}{L_{0}} \geq k \mid \theta = 1) = 0.05$. I've determined the likelihood ratio, for $\theta_{0}$ and $\theta_{1}$. This gave me: \begin{equation} \left(\frac{\theta_{1}}{\theta_{0}}\right)^{n} \cdot e^{\sum_{i=1}^{n} (-\theta_{1} + \theta_{0})e^{x_{i}} + (\theta_{1} - \theta_{0})} \end{equation}
This function must be greater than or equal to some $k$. Simplifying this as the function of a sufficient statistic and inserting the values of $\theta_{0}$ and $\theta_{1}$, means that $\sum_{i=1}^{n} -e^{x_{i}} + 1 = n - \sum_{i=1}^{n} e^{x_{i}}$ must be greater than or equal to some constant $k_{1}$.
I assume I should arrive at a value from the Gamma distribution for $k$/$k_{1}$ but I'm not sure how, or whether I've made a mistake on the way. Would someone be so kind as to help me. Thanks in advance.