Conic Sections · Superchallenge problem

The ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$ has foci $F_1 = (-1, 0)$ and $F_2 = (1, 0)$. A line through $F_1$ meets the ellipse at $A$ and $B$. Find the largest possible area of triangle $ABF_2$.