One... Two... Three... Infinity!

Algebra Level 3

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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