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 ___________.

Let the point $$(p, p+1)$$ lie inside the region $$E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$$. If the set of all values of $$\mathrm{p}$$ is the interval $$(a, b)$$, then $$b^{2}+b-a^{2}$$ is equal to ___________.

Let $$a \in \mathbb{Z}$$ and $$[\mathrm{t}]$$ be the greatest integer $$\leq \mathrm{t}$$. Then the number of points, where the function $$f(x)=[a+13 \sin x], x \in(0, \pi)$$ is not differentiable, is __________.

Let $$\mathrm{A}=\{1,2,3,4, \ldots ., 10\}$$ and $$\mathrm{B}=\{0,1,2,3,4\}$$. The number of elements in the relation $$R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}$$ is ___________.