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

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