Minimum of convex sum

Question

Let $X$ be a random vector distributed uniformly over the centered sphere of radius 2 in $\mathbb{R}^n$ and let $Y$ be a random vector distributed uniformly over the centered sphere of radius 1 in $\mathbb{R}^n$. Define $Z$ to be the minimum norm of a convex sum, i.e. $$ Z := \min_{\alpha + \beta = 1, \alpha, \beta \geq 0} \|\alpha X + \beta Y \|_2 $$
What is the expectation of $Z$? My intuition is that when $X$ and $Y$ are pointed in roughly the same direction then all the weight should be on the smaller vector, but for more general layouts it's not clear to me.