Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3-4x^2+7x-9$. In particular, $s_0=3$, $s_1=4$, and $s_2=2$. 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 $k=2,3,\dots$. What is $a+b+c$?