Function $f$ is defined by $f(x)=(x-5)(x-1)(x+2)$. Function $g$ is defined by $g(x)=f(x+1)$. 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 $a+b+c$?