$. This makes sure that $r\log r$ dominates $(n-r)\log\frac{\ell}{\ell-m}$. Recall that $\alpha$ can be chosen to be any large constant by choosing $\phi$ to be an 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 