HMMT 2014 G7

Triangle $A B C$ has sides $A B=14, B C=13$, and $C A=15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $A B$, let $B^{\prime}$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $A O$ and $M B^{\prime}$. Find the length of $A X$.