$2 \cos \frac{\pi}{2 k}+2 \cos \frac{\pi}{k}=\sqrt{3}+1$
$\begin{aligned} & \cos \frac{\pi}{2 \mathrm{k}}+\cos \frac{\pi}{\mathrm{k}}=\frac{\sqrt{3}+1}{2} \\ & \text { Let } \frac{\pi}{\mathrm{k}}=0, \cos \theta+\cos \frac{\theta}{2}=\frac{\sqrt{3}+1}{2} \Rightarrow 2 \cos ^2 \frac{\theta}{2}-1+\cos \frac{\theta}{2}=\frac{\sqrt{3}+1}{2} \\ & \cos \frac{\theta}{2}=\mathrm{t} \quad 2 \mathrm{t}^2+\mathrm{t}-\frac{\sqrt{3}+3}{2}=0 \\ & \mathrm{t}=\frac{-1 \pm \sqrt{1+4(3+\sqrt{3})}}{4}=\frac{-1 \pm(2 \sqrt{3}+1)}{4}=\frac{-2-2 \sqrt{3}}{4}, \frac{\sqrt{3}}{2} \quad \because \mathrm{t} \in[-1,1], \cos \frac{\theta}{2}=\frac{\sqrt{3}}{2} \\ & \frac{\theta}{2}=\frac{\pi}{6} \Rightarrow \mathrm{k}=3\end{aligned}$