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 $1\le n\le L$.
For example, $S(50)mod1000000007=832833871$.

Find $S(50000000)mod1000000007$.

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