<s>[INST] Let $\overrightarrow{CA}=\mathbf{a}$ and $\overrightarrow{CB}=\mathbf{b}$ be such that $|\mathbf{a}|=4,\,|\mathbf{b}|=2,$ and the angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta=\frac{2}{2\sqrt{3}}$. Find $\overrightarrow{AD}.$[/INST]