KöMaL 2018 C1486

A regular triangle $ABC$ and a circle $k$ are both centred at point $O$, and have an equal area of $\sqrt{\frac{\pi}{\sqrt{27}}}$. Let the extensions of line segments $AO$, $BO$, $CO$ intersect circle $k$ at points $A'$, $B'$, $C'$, respectively. Find the exact value of the area of the hexagon $AC'BA'CB'$.