\begin{equation}
v \leq \frac{k}{N+1} \left( 1 - \frac{1}{2 \sqrt{k}} \right)
- \frac{\sqrt{k}}{N+1} \left( \ln \frac{12}{\beta} + \ln \left(\frac{\beta}{6}+k+1 \right) \right),
\label{bound-on-v}
\end{equation}
