If the domain of the function $$f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$$ is $$[\alpha, \beta) \mathrm{U}(\gamma, \delta]$$, then $$|3 \alpha+10(\beta+\gamma)+21 \delta|$$ is equal to _________.

Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha+1)$$ in the first quadrant is __________.

In the figure, $$\theta_{1}+\theta_{2}=\frac{\pi}{2}$$ and $$\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})$$. If the area of $$\triangle \mathrm{CAB}$$ is $$2 \sqrt{3}-3$$ unit $${ }^{2}$$, when $$\frac{\theta_{2}}{\theta_{1}}$$ is the largest, then the perimeter (in unit) of $$\triangle \mathrm{CED}$$ is equal to _________.

Let the equations of two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ be $$2 x-3 y=-23$$ and $$5 x+4 y=23$$. If the equation of its one diagonal $$\mathrm{AC}$$ is $$3 x+7 y=23$$ and the distance of A from the other diagonal is $$\mathrm{d}$$, then $$50 \mathrm{~d}^{2}$$ is equal to ____________.