A k-1 form on a compact, orientable, k dimensional manifold

Hi I am stuck on the following problem: Let $M$ be a $k$-dimensional, orientable, compact manifold. Without using directly Stokes theorem, show that for every differentiable $k-1$ form $\omega$ $$\int_{M}d\omega=0$$ Conclude that if M is additionally connected, $d\omega$ admits a zero in $M$. First I am wondering about the phrasing "not using directly Stokes", second we got also an hint which is stated the following: $M$ is the union of sets $G_1$ and $G_2$ with smooth boundary, where the boundaries seen as sets are the same. I am really confused about this tip, because I don't see how this decomposition could possibly work out in general. Is somebody able to help me please?

"Not using directly Stokes" probably means that you're supposed to use a different argument than: $$ \int_Md\omega = \int_{\partial M} \omega = \int_\emptyset \omega=0. $$ I interpret it as okay to use Stokes on $G_1$ and $G_2$ though. You can get $G_1$ as a smoothly embedded closed ball, and $G_2$ as the compliment of the interior of $G_1$.