Differential Equations

Let \(y=y(x)\) be the solution of the differential equation \((x+y+2)^2 d x=d y, y(0)=-2\). Let the maximum and minimum values of the function \(y=y(x)\) in \(\left[0, \frac{\pi}{3}\right]\) be \(\alpha\) and \(\beta\), respectively. If \((3 \alpha+\pi)^2+\beta^2=\gamma+\delta \sqrt{3}, \gamma, \delta \in \mathbb{Z}\), then \(\gamma+\delta\) equals _________.