Consider the region $R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.$ and $\left.y^{2} \leq 4-x\right\}$. Let $\mathcal{F}$ 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 $\mathcal{F}$. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^{2}=4-x$.
The value of $\alpha$ is _____.

Hence, the correct answer is 2.