123 Numbers

We define 123-numbers as follows:

• 1 is the smallest 123-number.
• When written in base 10 the only digits that can be present are "1", "2" and "3" and if present the number of times they each occur is also a 123-number.

So 2 is a 123-number, since it consists of one digit "2" and 1 is a 123-number. Therefore, 33 is a 123-number as well since it consists of two digits "3" and 2 is a 123-number.
On the other hand, 1111 is not a 123-number, since it contains 4 digits "1" and 4 is not a 123-number.

In ascending order, the first 123-numbers are:
$1,2,3,11,12,13,21,22,23,31,32,33,111,112,113,121,122,123,131,…$

Let $F(n)$ be the $n$-th 123-number. For example $F(4)=11$, $F(10)=31$, $F(40)=1112$, $F(1000)=1223321$ and $F(6000)=2333333333323$.

Find $F(111\ 111\ 111\ 111\ 222\ 333)$. Give your answer modulo $123123123$.