Let \( y=y(x) \) be the solution of the differential equation \( \left(x+y+2\right)^{2} \, dx = dy, \; y(0)=-2 \). Let the maximum and minimum values of the function \( y=y(x) \) in \( \left[0,\tfrac{\pi}{3}\right] \) 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 ____________.
