Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation $\frac{1}{x}=(\frac{k}{x}{)}^2(k+x^2)-kx$.

For instance, for $k=5$, we see that $\{a_5,b_5,c_5\}$ is approximately $\{5.727244,-0.363622+2.057397i,-0.363622-2.057397i\}$.

Let ${\displaystyle S(n)=\sum _{p=1}^n\sum _{k=1}^n(a_k+b_k{)}^p(b_k+c_k{)}^p(c_k+a_k{)}^p}$.

Interestingly, $S(n)$ is always an integer. For example, $S(4)=51160$.

Find $S({10}^6)$ modulo $1000000007$.