2013 AMC 12B Problem 24

Let ABC be a triangle where M is the midpoint of \overline{AC}, and \overline{CN} is the angle bisector of \angle{ACB} with N on \overline{AB}. Let X be the intersection of the median \overline{BM} and the bisector \overline{CN}. In addition \triangle BXN is equilateral with AC=2. What is BX^2?

\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad \textbf{(B)}\ \frac{2}{9} \qquad \textbf{(C)}\ \frac{5\sqrt{2}-3\sqrt{3}}{8} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad \textbf{(E)}\ \frac{3\sqrt{3}-4}{5}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.