Eccoci.

For a positive integer $n$, define $f(n)$ to be the number of non-empty substrings of $n$ that are divisible by $3$. For example, the string "2573" has $10$ non-empty substrings, three of which represent numbers that are divisible by $3$, namely $57$, $573$ and $3$. So $f(2573) = 3$.

If $f(n)$ is divisible by $3$ then we say that $n$ is $3$-like.

Define $F(d)$ to be how many $d$ digit numbers are $3$-like. For example, $F(2) = 30$ and $F(6) = 290898$.

Find $F(10^5)$. Give your answer modulo $1\,000\,000\,007$.