The sequence $1, 2, 3, 2, \ldots$ has the property that every fourth term is the average of the previous three terms. That is, the sequence $\{a_n\}$ is defined as $a_1 = 1, a_2 = 2, a_3 = 3$, and $a_{n+3} = \dfrac{a_{n+2} + a_{n+1} + a_n}{3}$ for any positive integer $n$.

This sequence converges to a rational number $\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Find $a+b$.

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