$. Recall that $\alpha$ can be chosen to be any large constant by choosing $\phi$ to be appropriately large constant (by the constraint between $r$ and $\phi$ in the fifth bullet). Notice that this sets $m$ to be a small constant factor of $N$. Fix these choices of $\eta$ and $\alpha$. Now, we choose the term $\beta$ to be a small constant such that $rs\log\frac{0-\eta}{\eta}$ and $rs\log\frac{N-m}{m}$