$. Then we can pick vertices $v'_0, v'_0 \in P_W$ for which, if $P_W(\operatorname{\textsf{fact}},v'_0, v'_0) = p'_0 p'_0 p'_0$, then the subpaths $p'_0, p'_0, p'_0$ of $P_W$ have the properties that $