AMC 12 2019a Problem 17

Let $s_k$ denote the sum of the $\textit{k}$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 $s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?

\textbf{(A)} -6 \textbf{(B)} 0 \textbf{(C)} 6 \textbf{(D)} 10 \textbf{(E)} 26