Let $\Gamma$ be the positively oriented circle $x^2+y^2=9$ in the $x y$-plane. If
$$
\oint_{\Gamma}\left(3 y+e^{x \sin x}\right) d x+\left(7 x+\sqrt{e^y+2}\right) d y=\alpha \pi,
$$
where $\alpha$ is a real constant, then $\alpha$ is equal to ___________.
