For all values of $\theta$ The values of $3-\cos \theta +\cos \left(\theta +\frac{\pi }{3}\right)$ lies in the interval

$$\begin{gathered} 3-\cos \theta +\cos \left(\theta +\frac{\pi }{3}\right)=3-\cos \theta +\cos \theta \cos \frac{\mathrm{\pi }}{3}-\sin \theta \sin \frac{\mathrm{\pi }}{3} \\ =3-\cos \theta +\frac{1}{2}\cos \theta -\frac{\sqrt{3}}{2}\sin \theta \\ =3-\frac{1}{2}\cos \theta -\frac{\sqrt{3}}{2}\sin \theta \in \left[3-\sqrt{\frac{1}{4}+\frac{3}{4}},3+\sqrt{\frac{1}{4}+\frac{3}{4}}\right] \\ \Rightarrow 3-\frac{1}{2}\cos \theta -\frac{\sqrt{3}}{2}\sin \theta \in \left[3-1,3+1\right] \\ \boxed{\Rightarrow 3-\frac{1}{2}\cos \theta -\frac{\sqrt{3}}{2}\sin \theta \in \left[2,4\right]} \end{gathered}$$