Let $\omega=\operatorname{cis}\left(\frac{2 \pi}{3}\right)=\cos \left(\frac{2 \pi}{3}\right)+i \sin \left(\frac{2 \pi}{3}\right)$ and $f(x)=x^7-2 x^4-4 x^3+8$. Which of the following options is correct?

• A
  $\left\{2^{\frac{1}{3}}, 2^{\frac{1}{3}} \omega, 2^{\frac{1}{3}} \omega^2\right\}$ is a subset of the solution set of $f(x)$.
• B
  $\left\{2^{\frac{1}{2}},-2^{\frac{1}{2}}, 2^{\frac{1}{2}} i, -2^{\frac{1}{2}} i\right\}$ is a subset of the solution set of $f(x)$.
• C
  $\left\{2^{\frac{1}{3}}, 2^{\frac{1}{3}} \omega, 2^{\frac{1}{3}} \omega^2, 2^{\frac{1}{2}}, -2^{\frac{1}{2}}, 2^{\frac{1}{2}} i, -2^{\frac{1}{2}} i\right\}$ is the complete solution set of $f(x)=0$.
• D
  $\left\{2^{\frac{1}{3}}, \omega, 2^{\frac{1}{2}} i, -2^{\frac{1}{2}}\right\}$ is a subset of the solution set of $f(x)$.