Let $$Y=Y(X)$$ be a curve lying in the first quadrant such that the area enclosed by the line $$Y-y=Y^{\prime}(x)(X-x)$$ and the co-ordinate axes, where $$(x, y)$$ is any point on the curve, is always $$\frac{-y^2}{2 Y^{\prime}(x)}+1, Y^{\prime}(x) \neq 0$$. If $$Y(1)=1$$, then $$12 Y(2)$$ equals __________.

The number of real solutions of the equation $$x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$$ is _________.

Consider two circles $$C_1: x^2+y^2=25$$ and $$C_2:(x-\alpha)^2+y^2=16$$, where $$\alpha \in(5,9)$$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $$C_1$$ and $$C_2$$ be $$\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$$. If the length of common chord of $$C_1$$ and $$C_2$$ is $$\beta$$, then the value of $$(\alpha \beta)^2$$ equals _______.

Let a line passing through the point $$(-1,2,3)$$ intersect the lines $$L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$$ at $$M(\alpha, \beta, \gamma)$$ and $$L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$$ at $$N(a, b, c)$$. Then, the value of $$\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$$ equals __________.