Question 6
Segments $A A'$, $B B'$, and $C C'$, each of length $2$, all intersect at a point $O$. If $\angle A O C' = \angle B O A' = \angle C O B' = 60^{\circ}$, find the maximum possible value of the sum of the areas of triangles $A O C'$, $B O A'$, and $C O B'$.