HMMT 2016 G10

The incircle of a triangle $A B C$ is tangent to $B C$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle A B C$. The $B$-mixtilinear incircle, centered at $O_{B}$, is tangent to lines $B A$ and $B C$ and internally tangent to $\Gamma$. The $C$-mixtilinear incircle, centered at $O_{C}$, is defined similarly. Suppose that $\overline{D H} \perp \overline{O_{B} O_{C}}, A B=\sqrt{3}$ and $A C=2$. Find $B C$.