Power of the most powerful test between two uniforms

Let $X$ be a random variable having a probability density function $f\in\{f_0,f_1\}$, where

$$f_0(x)=\begin{cases}1, & 0\le x\le1,\ 0, & \text{otherwise}\end{cases}\quad\text{and}\quad f_1(x)=\begin{cases}\frac12, & 0\le x\le2,\ 0, & \text{otherwise.}\end{cases}$$

For testing the null hypothesis $H_0:f\equiv f_0$ against $H_1:f\equiv f_1$, based on a single observation on $X$, the power of the most powerful test of size $\alpha=0.05$ equals