Generating Functions for $a_n:n=i_1+i_2+i_3+i_4$

I'm having an intro combinatorics class and I do understand the theorems but I don't really know how to apply them. It's pretty easy to solve straight forward questions, but I just don't understand how to solve harder questions, can someone explain this question to me? What I got is completely different from the back of the book ... should I drop this class?

Let $a_n$ be the number of ways to write the number $n$ as $i_1+i_2+i_3+i_4$, where $i_1,i_2,i_3,i_4$ are integers such that:

$$00\\i_3=2 \text{ or } 4\\i_4>1$$

How do I write the generating function for this sequence, representing this as a rational function?

The only thing I know to first solve this question is to solve it by using the multiplication rule of power series (perhaps?). I know the generating function is a product of $4$ power series, corresponding to $i_1,i_2,i_3,i_4$.

Help appreciated. Really struggling.