The Chain Rule & Implicit Differentiation · Superchallenge problem

A bead slides on the surface $x^2 + y^2 + z^2 + xyz = 4$, starting at $(1, 1, 1)$ at time $t = 0$, so that its shadow in the $xy$-plane moves along $x = 1 + t$, $y = 1 - t$; its height $z(t)$ varies smoothly with $z(0) = 1$. Find $\dfrac{d^2z}{dt^2}$ at $t = 0$.