Let $ABCDEF$ be a regular hexagon with side length 1. Denote by $X,$ $Y,$ and $Z$ the midpoints of sides $\overline{AB},$ $\overline{CD},$ and $\overline{EF},$ respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$ ?

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