Change of Variables & the Jacobian · Challenge problem

Let $u = x + y + z$, $v = xy + yz + zx$, $w = xyz$. Near the point $(x,y,z) = (1,2,4)$ this map can be inverted. Find the Jacobian determinant $\dfrac{\partial(x,y,z)}{\partial(u,v,w)}$ of the inverse map at that point.