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 [0,…

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