If $$\alpha$$ denotes the number of solutions of $$|1-i|^x=2^x$$ and $$\beta=\left(\frac{|z|}{\arg (z)}\right)$$, where $$z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1}$$, then the distance of the point $$(\alpha, \beta)$$ from the line $$4 x-3 y=7$$ is __________.

In the expansion of $$(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$$, the sum of the coefficients of $x^3$ and $$x^{-13}$$ is equal to __________.

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=1,|\vec{b}|=4$$, and $$\vec{a} \cdot \vec{b}=2$$. If $$\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$$ and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\alpha$$, then $$192 \sin ^2 \alpha$$ is equal to ________.

If the integral $$525 \int_\limits0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\operatorname{Cos}^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$$ is equal to $$(n \sqrt{2}-64)$$, then $$n$$ is equal to _________.