Find the value of $a+b$, if the principal value of ${\cos }^{-1}\left(-\frac{1}{2}\right)=\frac{a\mathrm{\pi }}{b}$

If the principal value of ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$ is $-\frac{\mathrm{\pi }}{k}$, find the value of $k$.