HMMT 2011 U9

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