Question 14
Let $A = (0, 0)$ and $B = (b, 2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB} \parallel \overline{DE},$ $\overline{BC} \parallel \overline{EF},$ $\overline{CD} \parallel \overline{FA},$ and the $y$-coordinates of its vertices are distinct elements of the set $\{0, 2, 4, 6, 8, 10\}.$ The area of the hexagon can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n.$