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 $s_{k+1}=as_k+bs_{k-1}+ck_{k-2}$ for $k=2$, $3$, $\cdots$. What is $a+b+c$?