Largest Index Before a Sequence Becomes Non-Integer

Let $a_1$ be a positive integer less than $200$. Define a sequence $\{a_n\}$ by $3a_{n+1}-1=2a_n$ for $n\geq1$. Let $A$ be the set of all indices $m$ such that $a_m$ is an integer but $a_{m+1}$ is not. What is the largest possible element of $A$?