\[\begin{array}{l}
4\sin x\sin \left( {x + \frac{\pi }{3}} \right)\sin \left( {x + \frac{{2\pi }}{3}} \right) + \cos 3x = 1\\
\Leftrightarrow - 2\sin x\left( {\cos \left( {2x + \pi } \right) - \cos \left( { - \frac{\pi }{3}} \right)} \right) + \cos 3x = 1\\
\Leftrightarrow - 2\sin x\left( { - \cos 2x - \frac{1}{2}} \right) + \cos 3x = 1\\
\Leftrightarrow 2\sin x\cos 2x + \sin x + \cos 3x = 1\\
\Leftrightarrow \sin 3x - \sin x + \sin x + \cos 3x = 1\\
\Leftrightarrow \sin 3x + \cos 3x = 1\\
\Leftrightarrow \sqrt 2 \sin \left( {3x + \frac{\pi }{4}} \right) = 1\\
\Leftrightarrow \sin \left( {3x + \frac{\pi }{4}} \right) = \frac{1}{{\sqrt 2 }}\\
\Leftrightarrow \left[ \begin{array}{l}
3x + \frac{\pi }{4} = \frac{\pi }{4} + k2\pi \\
3x + \frac{\pi }{4} = \frac{{3\pi }}{4} + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{{k2\pi }}{3}\\
x = \frac{\pi }{6} + \frac{{k2\pi }}{3}
\end{array} \right.
\end{array}\]
