
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 the real numbers \( x \) and \( y \) satisfy the equations \(\left\{\begin{array}{c} x^{3} + \cos x + x - 2 = 0 \\ 8 y^{3} - 2 \cos^{2} y + 2 y + 3 = 0 \end{array}\right.\), determine \(\cos (x + 2y)\).

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