Let $S$ denote the surface of the box $[1, 2] \times [-1, 3] \times [2, 6]$ with the outward pointing orientation, and let \begin{align*} \mathbf{G} = (x^2 y + e^y, z^2 - 2x^2, x^2 + 4y^3 + 3z) \end{align*} Evaluate $\iint_S \mathbf{G} \cdot d\mathbf{S}$