Determine the value of the line integral $\int_C \vec{F} \cdot d\vec{r}$ where $\vec{F} = \langle 2x e^y, x^2 e^y + \cos(z), -y \sin(z) \rangle$ and $C$ is the path from $(0, 0, 0)$ to $(1, 1, \pi)$.