Suppose $X\sim\mathrm{Poisson} (N)$ where $N\sim\mathrm{Poisson}(\lambda)$, What is the P.G.F of $X+N$?

Question

I have tried conditioning on $N$ and I obtained that:

$$G_X(Z) = \exp\left(\lambda\left(e^{Z-1}-1\right)\right).$$

I then used the property that states that $G_{X+N} = G_X(Z)*G_N(Z)$. But I was unsure as to whether you could do this as it is not clear in the question that $N$ and $X$ are independent.