Trigonometry · Superchallenge problem

In $\triangle ABC$, $AB = 4$, $AC = 6$, and $\angle BAC = 60^\circ$. Let $O$ be the center of the circle through $A$, $B$, and $C$, and write $\overrightarrow{AO} = x\,\overrightarrow{AB} + y\,\overrightarrow{AC}$ for real numbers $x$ and $y$. Find $x + y$.