Let a_k, b_k, and c_k represent the three solutions (real or complex numbers) to the expression 1/x = (k/x)²(k+x²) - kx.

For instance, for k = 5, we see that {a₅, b₅, c₅} is approximately {5.727244, -0.363622+2.057397i, -0.363622-2.057397i}.

Let S(n) = Σ (a_k+b_k)^p(b_k+c_k)^p(c_k+a_k)^p for all integers p, k such that 1 ≤ p, k ≤ n.

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

Find S(10⁶) modulo 1 000 000 007.