Area Under The Curves

If the area of the region \(S=\left\{(x, y): 2 y-y^{2} \leq x^{2} \leq 2 y, x \geq y\right\}\) is equal to \(\frac{n+2}{n+1}-\frac{\pi}{n-1}\), then the natural number \(n\) is equal to ___________.