UMP test for the Pareto shape parameter

Let $X_1,X_2,\dots,X_n$ be a random sample from a continuous distribution with the common probability density function

$$f(x\mid\theta)=\begin{cases}\dfrac{2^\theta\theta}{x^{\theta+1}},&x>2,\\0,&\text{otherwise,}\end{cases}$$

where $\theta(>0)$ is an unknown parameter. Suppose $P(Y>\chi^2_{m,\beta})=\beta$, where $Y\sim\chi^2_m$. For testing $H_0:\theta=1$ against $H_1:\theta>1$, a uniformly most powerful test of size $\alpha$, $0<\alpha<1$, will reject $H_0$ if