Example 2 Given that the complex number satisfies $x^{2}+x+1=0$, find the value of $x^{14}+\frac{1}{x^{14}}$.
From the given, we know that $x=\omega$ or $\omega^{2}\left(\omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right)$, thus $x^{14}+\frac{1}{x^{14}}=\omega^{2}+\omega$ (or $\omega+$ $\left.\omega^{2}\right)=-1$.