--description--

An integer $s$ is called a superinteger of another integer $n$ if the digits of $n$ form a subsequence 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 $c(10) = 18$.

$$\begin{align} & \{p(i) : i ≥ 1\} = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots \} \\ & \{c(i) : i ≥ 1\} = \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, \ldots \} \end{align}$$

Let $P^D$ the sequence of the digital roots of $\{p(i)\}$ ($C^D$ is defined similarly for $\{c(i)\}$):

$$\begin{align} & P^D = \{2, 3, 5, 7, 2, 4, 8, 1, 5, 2, \ldots \} \\ & C^D = \{4, 6, 8, 9, 1, 3, 5, 6, 7, 9, \ldots \} \end{align}$$

Let $P_n$ be the integer formed by concatenating the first $n$ elements of $P^D$ ($C_n$ is defined similarly for $C^D$).

$$\begin{align} & P_{10} = 2\,357\,248\,152 \\ & C_{10} = 4\,689\,135\,679 \end{align}$$

Let $f(n)$ be the smallest positive integer that is a common superinteger of $P_n$ and $C_n$. For example, $f(10) = 2\,357\,246\,891\,352\,679$, and $f(100)\bmod 1\,000\,000\,007 = 771\,661\,825$.

Find $f(10\,000)\bmod 1\,000\,000\,007$.

--hints--

superinteger() should return 775181359.

assert.strictEqual(superinteger(), 775181359);

--seed--

--seed-contents--

function superinteger() {

  return true;
}

superinteger();

--solutions--

// solution required