Blocked by an Asymptote

Let $F_1, F_2$ be the left and right foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{4} = 1$, let $A$ be a point on the right branch of $C$, and let $P\left(\sqrt{5}, 2\right)$ be a fixed point. If for every real number $m$ the line $2x + y + m = 0$ has at most one point in common with $C$, find the minimum value of $|AP| + |AF_2|$.