I am trying to compute the integral $\iint_{\partial D} \langle\operatorname{curl}F,n\rangle \,dS$ where $D=\{(x,y,z)\in\mathbb{R} : \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}<1\}$ and $F(x,y,z)=(\cos(x\sin(z))+y^3, xy, \cos(x\sin(z\textrm{e}^y))$.
I am given the hint that applying Stokes Thm twice would simplify the surface of integration.
I found that $\operatorname{curl}F=(-xz \textrm{e}^y \cos(z\textrm{e}^y) \sin(x\sin(z\textrm{e}^y)), -x \cos(z)\sin(x\sin(z)) + \sin(z\textrm{e}^y) \sin(x\sin(z\textrm{e}^y)), y).$
I'm not sure where to go from here though; I know I need to find a normal vector and probably a paramaterization of $D$ but I'm not sure how. I'm also unsure how I would apply Stokes twice?