The number of solutions of $$\sin ^2 x+\left(2+2 x-x^2\right) \sin x-3(x-1)^2=0$$, where $$-\pi \leq x \leq \pi$$, is ________.

Let $$S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\left(\sin ^6 \theta+\cos ^6 \theta\right)=0\right.$$ has real roots $$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3\left((\alpha-2)^2+(\beta-1)^2\right)$$ equals __________.

If m and n respectively are the numbers of positive and negative values of $$\theta$$ in the interval $$[-\pi,\pi]$$ that satisfy the equation $$\cos 2\theta \cos {\theta \over 2} = \cos 3\theta \cos {{9\theta } \over 2}$$, then mn is equal to ____________.

Let $$\mathrm{S = \{ \theta \in [0,2\pi ):\tan (\pi \cos \theta ) + \tan (\pi \sin \theta ) = 0\}}$$. Then $$\sum\limits_{\theta \in S} {{{\sin }^2}\left( {\theta + {\pi \over 4}} \right)} $$ is equal to __________.