For all $n\in\mathbb{N}$, define
$$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$
Show that there exists $r>0$ satisfying the following two properties:
\begin{enumerate}
\item for all $g\in\Gamma(n\ln(2))$, there exists $v\in\Delta(n)$ such that $d(gv_0,v)\leq r$,
\item for all $v\in\Delta(n)$, there exists $g\in\Gamma(n\ln(2))$ such that $d(gv_0,v)\leq r$.
\end{enumerate}