Let the set of all values of $$p$$, for which $$f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$$ does not have any critical point, be the interval $$(a, b)$$. Then $$16 a b$$ is equal to _________.

Let the set of all positive values of $$\lambda$$, for which the point of local minimum of the function $$(1+x(\lambda^2-x^2))$$ satisfies $$\frac{x^2+x+2}{x^2+5 x+6}<0$$, be $$(\alpha, \beta)$$. Then $$\alpha^2+\beta^2$$ is equal to _________.

Let $$\mathrm{A}$$ be the region enclosed by the parabola $$y^2=2 x$$ and the line $$x=24$$. Then the maximum area of the rectangle inscribed in the region $$\mathrm{A}$$ is ________.

Let the maximum and minimum values of $$\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$$ be $$\mathrm{M}$$ and $$\mathrm{m}$$, respectively. Then $$\mathrm{M}^2-\mathrm{m}^2$$ is equal to _________.