
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 \( x, y \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \) and \( a \in \mathbf{R} \), satisfy the equations
\[
\begin{cases}
x^{3}+\sin x-2a=0, \\
4y^{3}+\sin y \cos y+a=0.
\end{cases}
\]
Find the value of \( \cos (x + 2y) \).

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