
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let's denote $\sin x = a$, $\sin 2x = b$, $\sin 3x = c$.

Then $a^3 + b^3 + c^3 = (a + b + c)^3 \Leftrightarrow a^3 + b^3 = (a + b + c)^3 - c^3 \Leftrightarrow$

$$
(a + b)\left(a^2 - ab + b^2\right) = (a + b)\left((a + b + c)^2 + (a + b + c) \cdot c + c^2\right) \Leftrightarrow
$$

$$
(a + b)(a + c)(b + c) = 0 \Leftrightarrow
$$

$\Leftrightarrow\left[\begin{array}{c}\sin x + \sin 2x = 0 \\ \sin 2x + \sin 3x = 0 \\ \sin x + \sin 3x = 0\end{array} \Leftrightarrow\left[\begin{array}{c}\sin x \cdot(1 + 2 \cos x) = 0 \\ \sin \frac{5x}{2} \cdot \cos \frac{x}{2} = 0 \\ \sin^2 x \cdot \cos x = 0\end{array} \Rightarrow\left[\begin{array}{c}\sin x = 0 \\ \cos x = -\frac{1}{2} \\ \cos x = 0 \\ \sin \frac{5x}{2} = 0\end{array} \Leftrightarrow\right.\right.\right.$

$\Rightarrow x = \frac{k\pi}{2}; \quad x = \pm \frac{2\pi}{3} + 2\pi m; \quad x = \frac{2\pi n}{5}$.

Present the answer in LaTex format: \boxed{Your answer}
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