--description--

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

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 $S(n) = \displaystyle\sum_{p = 1}^n \sum_{k = 1}^n {(a_k + b_k)}^p {(b_k + c_k)}^p {(c_k + a_k)}^p$ for all integers $p$, $k$ such that $1 ≤ p, k ≤ n$.

Interestingly, $S(n)$ is always an integer. For example, $S(4) = 51\,160$.

Find $S({10}^6) \text{ modulo } 1\,000\,000\,007$.

--hints--

rootsOnTheRise() should return 191541795.

assert.strictEqual(rootsOnTheRise(), 191541795);

--seed--

--seed-contents--

function rootsOnTheRise() {

  return true;
}

rootsOnTheRise();

--solutions--

// solution required