Error probabilities for a chi-square test

Let $X_{1},X_{2},X_{3},X_{4}$ be a random sample of size $4$ from a $\chi^{2}_{m}$ distribution, where $m\in\mathbb{N}$ is an unknown parameter. To test $H_{0}:m=1$ against $H_{1}:m=2$, the critical region $\sum_{i=1}^{4}X_{i}>6$ is being used. If $\alpha$ and $\beta$ denote the probabilities of Type-I error and Type-II error, respectively, then which one of the following statements is true? [You may use $\chi^{2}_{1,0.0143}=\chi^{2}_{2,0.0498}=\chi^{2}_{4,0.1991}=\chi^{2}_{8,0.6472}=6$]