Let the $\vec{F}=\langle x,y,z\rangle $ be a force vector field in $\mathbb{R}^3$, the curve $C$ is the counterclockwise oriented ellipse given by parametric equation $\vec{r}(t)=\langle \cos t, \sin t,\cos t\rangle , t\in[0,2\pi]$, find the work required to move an object on $C$.