Minimizing an Area

Consider two points $A=(4, -4)$ and $B=(8, 0),$ and a third point $P$ on the graph of $y=\frac{1}{2}x^2$ . If $P=(a,b)$ is the point minimizing the area of the triangle $ABP,$ and $S$ is the minimum area, what is the value of $a+2b+4S?$