
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
Given \( x \cdot y \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \), \( a \in \mathbf{R} \), and the system of equations \( \left\{ \begin{array}{l} x^{3} + \sin x - 2a = 0 \\ 4y^{3} + \frac{1}{2} \sin 2y - a = 0 \end{array} \right. \), find \( \cos(x + 2y) \).

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