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

Algebra Level 3

The sequence 1,2,3,2,1, 2, 3, 2, \ldots has the property that every fourth term is the average of the previous three terms. That is, the sequence {an}\{a_n\} is defined as a1=1,a2=2,a3=3a_1 = 1, a_2 = 2, a_3 = 3, and an+3=an+2+an+1+an3a_{n+3} = \dfrac{a_{n+2} + a_{n+1} + a_n}{3} for any positive integer nn.

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

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