Consider the region $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x} \geq 0\right.$ and $\left.\mathrm{y}^2 \leq 4-\mathrm{x}\right\}$. Let F be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has largest radius among the circles in F. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^2=4-x$.

The radius of the circle C is _____.