12. (10 points) When $N$ takes all the numbers in $1,2,3, \cdots, 2015$, how many numbers of the form $3^{n}+n^{3}$ are divisible by 7?
【Analysis】Through analysis, we can know: $3^{n}$ modulo 7 has a cycle of 6, $3, 2, 6, 4, 5, 1$; $n^{3}$ modulo 7 has a cycle of 7, $1, 1, 6, 1, 6, 6, 0$; thus, the total cycle is 42. When the sum of the remainders is a multiple of 7, it can be divided by 7. List as shown in the figure:

Now let's see how many such cycles are in 2015, and calculate accordingly.
【Solution】Solution: As shown in the figure:
$3^{n}$ modulo 7 has a cycle of 6, $3, 2, 6, 4, 5, 1$;

Then the total cycle is 42:
$$
\begin{array}{l}
2015 \div 42=47 \cdots 41 \\
47 \times 6+6 \\
=282+6 \\
=288 \text { (items) }
\end{array}
$$

Answer: There are 288 items that can be divided by 7.