If $$\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _x\left|\tan \frac{x}{2}\right|+\mathrm{C}$$ where $$\alpha, \beta \in \mathbb{R}$$ and $$\mathrm{C}$$ is the constant of integration, then the value of $$8(\alpha+\beta)$$ equals _________.

Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$$ and the determinant of $$A$$ be 1 . If $$A^{-1}=\alpha A+\beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha+\beta$$ equals _________.

Consider a triangle $$\mathrm{ABC}$$ having the vertices $$\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$$ and $$\mathrm{C}(\gamma, \delta)$$ and angles $$\angle A B C=\frac{\pi}{6}$$ and $$\angle B A C=\frac{2 \pi}{3}$$. If the points $$\mathrm{B}$$ and $$\mathrm{C}$$ lie on the line $$y=x+4$$, then $$\alpha^2+\gamma^2$$ is equal to _______.

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $$\frac{1}{3}$$ and $$\frac{2}{3}$$ respectively. Let $$x$$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $$\mathrm{P}(|x-y| \leq 2)$$ is $$p$$, then $$3^9 p$$ equals _________.