Finding the residue of an infinite series

Question

I want to find the residue of $$ \sum_{n=1}^{\infty} \frac{z^n}{n!(1-z^n)} $$ at $z=1$. I've tried $$ \sum_{n=1}^{\infty} \frac{z^n}{n!(1-z^n)} = \sum_{n=1}^{\infty} \left( \frac{1}{n!(1-z^n)} - \frac{1}{n!} \right) = 1-e +\sum_{n=1}^{\infty} \frac{1}{n!(1-z^n)} $$ but I don't really see how to get the summand into a multiple of $(z-1)^n$ after this.