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Let $\{ a_n \}$ be a sequence defined as $a_0 = a_1 = 1$ and $a_{n+2} = 2 a_{n+1} + a_n - 1$ for $n \ge 0$.

Find the value of this infinite summation $S = \sum_{k=0}^{\infty} \frac{a_k}{3^k}$

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