An integer $s$ is called a superinteger of another integer $n$ if the digits of $n$ form a subsequenceA subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. of the digits of $s$.
For example, $2718281828$ is a superinteger of $18828$, while $314159$ is not a superinteger of $151$.

Let $p(n)$ be the $n$th prime number, and let $c(n)$ be the $n$th composite number. For example, $p(1)=2$, $p(10)=29$, $c(1)$ = 4 and $c(10)=18$.
$\{p(i):i\ge 1\}=\{2,3,5,7,11,13,17,19,23,29,\dots \}$
$\{c(i):i\ge 1\}=\{4,6,8,9,10,12,14,15,16,18,\dots \}$

Let $P^D$ be the sequence of the digital roots of $\{p(i)\}$ ($C^D$ is defined similarly for $\{c(i)\}$):
$P^D=\{2,3,5,7,2,4,8,1,5,2,\dots \}$
$C^D=\{4,6,8,9,1,3,5,6,7,9,\dots \}$

Let $P_n$ be the integer formed by concatenating the first $n$ elements of $P^D$ ($C_n$ is defined similarly for $C^D$).
$P_{10}=2357248152$
$C_{10}=4689135679$

Let $f(n)$ be the smallest positive integer that is a common superinteger of $P_n$ and $C_n$.
For example, $f(10)=2357246891352679$, and $f(100)mod1000000007=771661825$.

Find $f(10000)mod1000000007$.