--description--

Let $n$ be a positive integer.

A 6-sided die is thrown $n$ times. Let $c$ be the number of pairs of consecutive throws that give the same value.

For example, if $n = 7$ and the values of the die throws are (1, 1, 5, 6, 6, 6, 3), then the following pairs of consecutive throws give the same value:

$$\begin{align} & (\underline{1}, \underline{1}, 5, 6, 6, 6, 3) \\ & (1, 1, 5, \underline{6}, \underline{6}, 6, 3) \\ & (1, 1, 5, 6, \underline{6}, \underline{6}, 3) \end{align}$$

Therefore, $c = 3$ for (1, 1, 5, 6, 6, 6, 3).

Define $C(n)$ as the number of outcomes of throwing a 6-sided die $n$ times such that $c$ does not exceed $π(n)$.¹

For example, $C(3) = 216$, $C(4) = 1290$, $C(11) = 361\,912\,500$ and $C(24) = 4\,727\,547\,363\,281\,250\,000$.

Define $S(L)$ as $\sum C(n)$ for $1 ≤ n ≤ L$.

For example, $S(50)\bmod 1\,000\,000\,007 = 832\,833\,871$.

Find $S(50\,000\,000)\bmod 1\,000\,000\,007$.

¹ $π$ denotes the prime-counting function, i.e. $π(n)$ is the number of primes $≤ n$.

--hints--

consecutiveDieThrows() should return 653972374.

assert.strictEqual(consecutiveDieThrows(), 653972374);

--seed--

--seed-contents--

function consecutiveDieThrows() {

  return true;
}

consecutiveDieThrows();

--solutions--

// solution required