The number of solutions of the equation $\sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0$ for $\theta \in [-3\pi, 2\pi]$ is:

The number of solutions of the trigonometric equation $1+\cos x \cdot \cos 5 x=\sin ^2 x$ in $[0, 2 \pi]$ is

If $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$,then $\sin \left( \theta + \frac{\pi }{4} \right)$ equals

If $0 < \theta < \frac{\pi}{2}$,then the solution of the equation $\sin \theta - 3 \sin 2 \theta + \sin 3 \theta = \cos \theta - 3 \cos 2 \theta + \cos 3 \theta$ is

The general solution of $\sin x + \cos x = \min_{a \in \mathbb{R}} \{1, a^2 - 4a + 6\}$ is

The sum of the solutions in $(0, 2\pi)$ for the equation $\cos x \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right) = \frac{1}{4}$ is