Eccoci.

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:
(1,1,5,6,6,6,3)
(1,1,5,6,6,6,3)
(1,1,5,6,6,6,3)
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 $\pi(n)$.¹
For example, $C(3) = 216$, $C(4) = 1290$, $C(11) = 361912500$ and $C(24) = 4727547363281250000$.

Define $S(L)$ as $\sum C(n)$ for $1 \leq n \leq L$.
For example, $S(50) \bmod 1\,000\,000\,007 = 832833871$.

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

¹ $\pi$ denotes the prime-counting function, i.e. $\pi(n)$ is the number of primes $\leq n$.