Let  $S=S_1\cap S_2\cap S_3$ where $S_1=\{z\in C:|z|<4\}{S}_2=\{z\in C:\mathrm{Im}\left[\frac{z-1+i\sqrt{3}}{1-i\sqrt{3}}\right]>0\}$ $S_3=\{z\in C:\mathrm{Re}z>0\}$ If the area of S  is $k\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.$  then  $k=$

$$\begin{gathered} S_1\Rightarrow C(0,0)r=4 \\ {S}_2\Rightarrow \frac{(x-1)+i(y+\sqrt{3})}{1-i\sqrt{3}}\times \frac{1+i\sqrt{3}}{1+i\sqrt{3}} \\ \frac{(y+\sqrt{3})+(x-1)\sqrt{3}}{1+3}>0 \\ \sqrt{3}x+y>0 \\ {S}_3\Rightarrow x>0 \end{gathered}$$

Area of sector $=\frac{1}{2}{r}^2\theta =\frac{1}{2}\times 16\times \frac{5\pi }{6}=\frac{20\pi }{3}$