
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 vertices of angles $\alpha$ and $\beta$ are at the origin, their initial sides coincide with the positive semi-axis of the $x$-axis, and $\alpha, \beta \in (0, \pi)$. The terminal side of angle $\beta$ intersects the unit circle at the point with an abscissa of $- \frac{1}{3}$, and the terminal side of angle $\alpha + \beta$ intersects the unit circle at the point with an ordinate of $\frac{4}{5}$. Find $\cos \alpha =\,$\_\_\_\_\_\_.

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