Function $f$ is defined by $f(x)=(x+1)(x-5)(x+7)$. Function $g$ is defined by $g(x)=f(x-2)$. The graph of $y=g(x)$ in the $xy$-plane has x-intercepts at $(a,0),(b,0)$, and $(c,0)$, where $a,b$, and $c$ are distinct constants. What is the value of $a+b+c$?