Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.