On the tail distribution of the solution to some law equation

We consider a distribution equation which was initially studied by Bertoin \cite{Bertoin}: $$M \stackrel{d}{=} \max\{\widetilde{\nu}, \max_{1\leq k\leq \nu}M_k\}.$$ where $\{M_k\}_{k\geq 1}$ are i.i.d. copies of $M$ and independent of $(\widetilde{\nu}, \nu)\in\mathbb{R}_+\times\mathbb{N}$. We obtain the tail behaviour of the solution of a generalised equation in a different but direct method by considering the joint tail of $(\widetilde{\nu}, \nu)$.