Now connect $\{v_{\alpha\beta}^{-1}, x_{\alpha\beta} \}$ to $\{v_{\alpha\gamma}^{-1}, x_{\alpha\gamma}\}$ with a hierarchy in ${\mathcal{P}}(S \backslash v)$. Note, to make our notation cleaner, we will refer to this as the hierarchy between $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$; similarly later on we won’t necessarily specify the second curve. By the Bounded Geodesic Image Theorem the geodesic connecting $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$ in ${\mathcal{C}}(S\backslash v)$ has length at most $2M$. Now consider any curve, $w$, in the geodesic $[v_{\alpha\beta}^{-1}, v_{\alpha\gamma}^{-1}]_{S \backslash v}$ contained in the hierarchy connecting $\{v_{\alpha\beta}^{-1}, x_{\alpha\beta} \}$ to $\{v_{\alpha\gamma}^{-1}, x_{\alpha\gamma}\}$. Assume $w$ is not a domain separating curve in $S$ and let $w^{-1}$ and $w^{+1}$ be the two curves before and after $w$ on $[v_{\alpha\beta}^{-1}, v_{\alpha\gamma}^{-1}]_{S \backslash v}$. Then the geodesic connecting $w^{-1}$ to $w^{+1}$ in ${\mathcal{C}}(S \backslash (v \cup w))$ has length at most $4M$ by using the Bounded Geodesic Image Theorem on $[w^{-1}, v_{\alpha\beta}^{-1}]_{S \backslash v} \cup [v_{\alpha\beta}^{-1}, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}^{-1}]_S \cup [v_{\alpha\gamma}^{-1}, w^{+1}]_{S \backslash v}$; note $w$ cannot be on this path because $w$ is distance $1$ from $v$, so if it was anywhere in the path it would be violating the assumption that we have geodesics. Therefore the hierarchy between $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$ has length at most $8M^2$. Similarly 