Roots on the Rise

 Published on Saturday, 6th September 2014, 10:00 pm; Solved by 1150;
Difficulty rating: 25%

Problem 479

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

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 $1\,000\,000\,007$.