
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 the Cartesian coordinate system $xOy$ and the angles $\alpha$ and $\beta$, where the vertices of the angles coincide with the origin $O$ and their initial sides coincide with the non-negative half-axis of the $x$-axis, and their terminal sides are symmetric about the $x$-axis, find the value of $\cos \beta$ if $\cos \alpha = \frac{2}{3}$.

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