Suppose $X$ is a single observation from a $\text{Poisson}(\lambda)$ distribution. Determine the uniformly most powerful size $\alpha = .05$ test for the hypothesis test $H_0: \lambda = 5$ and $H_1: \lambda = 10$.
My thoughts: The Neyman-Pearson Lemma tells us the UMP level $\alpha$ test is the likelihood ratio test. The likelihood ratio function is $\Lambda = \frac{5^x e^{-5}}{x!}/ \frac{10^x e^{-10}}{x!} = 2^{-x}e^5$. We reject the null hypothesis if $\Lambda < k$ which is equivalent to rejecting if $x > k^*$.
Our hypothesis test will be of the form $$\phi(x) = \begin{cases} 1 \hfill &\text{if $x > k^*$} \\ \gamma \hfill &\text{if $x = k^*$} \\ 0 \hfill &\text{otherwise} \end{cases} $$ To make a size $\alpha$ test we want to solve for $\gamma$ in the following: $$\alpha = .05 = \text{E}_{\lambda=5}(\phi(x)) = P_{\lambda=5}(X>k^*) + \gamma P_{\lambda=5}(X=k^*).$$
Can someone offer some confirmation/guidance on this? I am not confident in my method.