Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_0 = 3$, $s_1 = 5$, and $s_2 = 9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \cdot s_k + b \cdot s_{k-1} + c \cdot s_{k-2}$ for $k = 2$, $3$, $\ldots$ Find the value of $a + b + c$.

Let's think step by step and output the final answer within \boxed{}.
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