Consider the curve $C_1$ given by

$$ y=e^{-x} \quad \text { for } x \in[0,10 \pi], $$

and the curve $C_2$ given by

$$ y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] . $$

Let $n$ be the total number of points of intersection of the curves $C_1$ and $C_2$.

Suppose that $\alpha_1, \alpha_2, \ldots, \alpha_n \in[0,10 \pi]$ are the $x$-coordinates of the points of intersection of the curves $C_1$ and $C_2$ such that

$$ \alpha_1<\alpha_2<\cdots<\alpha_n . $$

The value of $n$ is $\_\_\_\_$ .

Consider the curve $C_1$ given by

$$ y=e^{-x} \quad \text { for } x \in[0,10 \pi], $$

and the curve $C_2$ given by

$$ y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] . $$

Let $n$ be the total number of points of intersection of the curves $C_1$ and $C_2$.

Suppose that $\alpha_1, \alpha_2, \ldots, \alpha_n \in[0,10 \pi]$ are the $x$-coordinates of the points of intersection of the curves $C_1$ and $C_2$ such that

$$ \alpha_1<\alpha_2<\cdots<\alpha_n . $$

Let $\beta$ be the area of the region enclosed between the curves $C_1, C_2$, and the lines $x=\alpha_1$ and $x=\alpha_4$. Then the value of

$$ -\frac{1}{\pi} \log _e\left(\beta-2 e^{-\frac{\pi}{2}}\right) $$

is $\_\_\_\_$ .

Consider the ellipses given by

$$ x^2+4 y^2=1 \quad \text { and } \quad 4 x^2+y^2=1 $$

Let $P$ be the point in the first quadrant where the given ellipses intersect. If $\theta$ is the acute angle between the tangents to the given ellipses at the point $P$, then the value of $4 \tan \theta$ is $\_\_\_\_$ .

Consider the ellipses given by

$$ x^2+4 y^2=1 \quad \text { and } \quad 4 x^2+y^2=1 . $$

If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot \alpha$ is $\_\_\_\_$ .