I have a question regarding the expansion of product notation in the picture below. Equation 3.1 in the attached picture is
$$\prod_{n=1}^{\infty}(1+zq^n)(1+z^{-1}q^{n-1})=\frac{1}{\prod_{n=1}^{\infty}(1-q^{n})}\sum_{m=-\infty}^{\infty}z^{m}q^{{m+1}\choose{2}}.$$
$\varphi(z)$ refers to the left-hand side of the above equation.
In the attached picture, I am unsure of the logic following ", exactly as often as one finds pairs of terms...".
How did the writer come to the conclusion that $q^Nz^0$ arises as often as one finds pairs of terms $q^{a_{1}+a_{2}+\ldots+a_{m}}z^m$ with $a_{1}>a_{2}>\ldots>a_{m}\geq1$ from the first factor (1+zq^n) and the same logic for the second factor? 