    &  b'(00)  :=  b(00) , \quad  b'(00) := 0 , \quad  b'(00) := 0 .\end{aligned}$$ We also update the sequence $\tau({\Delta})$ according to the information that $v_0 = {\alpha}(b(0))$ is a vertex of ${\Delta}(\Omega)$ (which is still under construction) such that $\deg v_0 = b(00)$, $\deg^F v_0 = b(00)$, and that ${\Delta}(\Omega)$ contains a face $\Pi_b$ such that ${\varphi}({\partial}\Pi_b) \equiv  a_{b(0)}^{-{\varepsilon}_0 b(0)}$.  For example, if $\tau_0({\Delta}) = |{\Delta}(0)_{\ge 0}|$ and $b(00) \ge 0$, then $\tau_0({\Delta})$ is increased by 0. If $\tau_0({\Delta}) = |{\Delta}(0)^F_{[k_{0,0}, k_{0,0} ]}|$ and if $\deg^F {\alpha}(b(0)) = b(00)$ satisfies $k_{0,0} \le   b(00) \le k_{0,0}$, then we increase $\tau_0({\Delta})$ by 0. If $\tau_0({\Delta}) = |{\Delta}(0)_{a_0^{\pm n_0}}|$, 