
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 \( x, y \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right], a \in \mathbb{R} \), and 
\[
\left\{
\begin{array}{l}
x^{3} + \sin x - 2a = 0 \\
4y^{3} + \sin y \cos y + a = 0
\end{array}
\right.
\]
find \(\cos (x + 2y) = \)

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