Dual exponential tests with flipped hypotheses

Let $X_{1},X_{2},…,X_{n}$ be a random sample from an exponential distribution with probability density function $$f(x;\theta)=\begin{cases}\theta e^{-\theta x},&x>0\\0,&\text{otherwise},\end{cases}$$ where $\theta\in(0,\infty)$ is unknown. Let $\alpha\in(0,1)$ be fixed and let $\beta$ be the power of the most powerful test of size $\alpha$ for testing $H_{0}:\theta=1$ against $H_{1}:\theta=2$. Consider the critical region $$R=\left\{(x_{1},x_{2},…,x_{n})\in\mathbb{R}^{n}:\sum_{i=1}^{n}x_{i}>\frac{1}{2}\chi^{2}_{2n}(1-\alpha)\right\},$$ where for any $\gamma\in(0,1)$, $\chi^{2}_{2n}(\gamma)$ is a fixed point such that $P\left(\chi^{2}_{2n}>\chi^{2}_{2n}(\gamma)\right)=\gamma$. Then, the critical region $R$ corresponds to the