Let $S$ be the surface of an open box with five faces: the bottom at $z=0$ and four sides, defined over the domain $0 \le x \le 1$, $0 \le y \le 1$, and $0 \le z \le 2$ (excluding the top face $z=2$). If $S$ is oriented outwards, evaluate the surface integral $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ for the vector field $\mathbf{F} = \langle y^2 \cos z, x + e^z, x^2 y \rangle$.