Lines & Planes in Space · Superchallenge problem

Consider the three lines $L_1$: $y = 0,\ z = 0$ (the $x$-axis), $L_2$: $x = 0,\ z = 1$, and $L_3$: $x = 1,\ y = 1$. There are exactly two lines that meet all three of $L_1$, $L_2$, $L_3$ and are parallel to the plane $x + y + z = 0$. Find the product of the $z$-coordinates of the points where these two lines meet $L_3$.