A Sliding Equilateral Triangle and a Dot Product

As shown in the figure, an equilateral triangle $ABC$ has side length $2$. Its vertices $B$ and $C$ move along the nonnegative $x$-axis and the nonnegative $y$-axis, respectively, with $A$ on the opposite side of line $BC$ from the origin $O$. Let $M$ be the midpoint of $AB$. Find the maximum value of $\overrightarrow{OA}\cdot \overrightarrow{OM}$.