Let $$f:(0, \pi) \rightarrow \mathbf{R}$$ be a function given by $$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0< x<\frac{\pi}{2} \\ \mathrm{a}-8, & x=\frac{\pi}{2} \\ (1+\mid \cot x)^{\frac{\mathrm{b}}{\mathrm{a}}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.$$

where $$\mathrm{a}, \mathrm{b} \in \mathbf{Z}$$. If $$f$$ is continuous at $$x=\frac{\pi}{2}$$, then $$\mathrm{a}^2+\mathrm{b}^2$$ is equal to _________.

If $$\alpha=\lim _\limits{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$$ and $$\beta=\lim _\limits{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$$ are the roots of the quadratic equation $$\mathrm{a} x^2+\mathrm{b} x-\sqrt{\mathrm{e}}=0$$, then $$12 \log _{\mathrm{e}}(\mathrm{a}+\mathrm{b})$$ is equal to _________.

The value of $$\lim _\limits{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots . \sqrt[10]{\cos 10 x}}{x^2}\right)$$ is __________.

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f:[0, \infty) \rightarrow \mathbf{R}$$ be a function defined by $$f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$$. Let $$\mathrm{S}$$ be the set of all points in the interval $$[0,8]$$ at which $$f$$ is not continuous. Then $$\sum_\limits{\text {aes }} a$$ is equal to __________.