Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the center, angles of $\frac{\pi }{k}$ and $\frac{2\pi }{k}$, where  $k>0$, then the value of  $\left[k\right]$ is
[Note:  $\left[k\right]$ denotes the largest integer less than or equal to $k$ ]

Let  $\theta =\frac{\pi }{2k}$

$\cos \theta =\frac{x}{2}$

$\Rightarrow \cos 2\theta =\frac{\sqrt{3}+1-x}{2}$

$$\begin{gathered} \Rightarrow 2{\cos }^2\theta -1=\frac{\sqrt{3}+1-x}{2} \\ \Rightarrow x^2+x-3-\sqrt{3}=0 \\ \Rightarrow x=\frac{-1\pm \sqrt{1+12+4\sqrt{3}}}{2} \\ =\frac{-1\pm \sqrt{13+4\sqrt{3}}}{2} \\ =\frac{-1+2\sqrt{3}+1}{2}=\sqrt{3} \\ \therefore \cos \theta =\frac{\sqrt{3}}{2}\Rightarrow \theta =\frac{\pi }{6} \end{gathered}$$

$\therefore$Required angle $=\frac{\pi }{k}=2\theta =\frac{\pi }{3}$

$\Rightarrow k=3$