Since $x+x^{-1}$ is a root, $$\begin{aligned} 0 & =\left(x+x^{-1}\right)^{3}+\left(x+x^{-1}\right)^{2}-2\left(x+x^{-1}\right)-1 \\ & =x^{3}+x^{-3}+3 x+3 x^{-1}+x^{2}+2+x^{-2}-2 x-2 x^{-1}-1 \\ & =x^{3}+x^{-3}+x^{2}+x^{-2}+x+x^{-1}+1 \\ & =x^{-3}\left(1+x+x^{2}+\cdots+x^{6}\right) \end{aligned}$$ Since $x \neq 0$, the above equality holds only if $x$ is a primitive seventh root of unity, i.e. $x^{7}=1$ and $x \neq 1$. Therefore, the only possible value of $x^{7}+x^{-7}$ is $1+1=2$.
Let $x$ be a complex number such that $x+x^{-1}$ is a root of the polynomial $p(t)=t^{3}+t^{2}-2 t-1$. Find all possible values of $x^{7}+x^{-7}$.