Function $f$ is defined by $f\left(x\right)=\left(x+6\right)\left(x+5\right)\left(x+1\right)$. Function $g$ is defined by $g\left(x\right)=f\left(x-1\right)$. The graph of $y=g\left(x\right)$ in the xy-plane has x-intercepts at $\left(a,0\right)$, $\left(b,0\right)$, and $\left(c,0\right)$, where $a$, $b$, and $c$ are distinct constants. What is the value of $a+b+c$?