Given an equilateral triangle $ABC$ with side length equal to $a$. Let $M$ and $N$ be two points respectively on the side $AB$ and $AC$ such that $\vec{AN} = K\ \vec{AC}$ and $\vec{AM} = \frac{\vec{AB}}{3}$. If $\overrightarrow{BN}$ and $\overrightarrow{CM}$ are orthogonal, then the value of $K$ is equal to