
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 angles $\alpha$ and $\beta$ have their vertices at the origin of coordinates, and their initial sides coincide with the positive half-axis of the x-axis, $\alpha, \beta \in (0, \pi)$. The terminal side of angle $\beta$ intersects the unit circle at a point whose x-coordinate is $-\frac{5}{13}$, and the terminal side of angle $\alpha + \beta$ intersects the unit circle at a point whose y-coordinate is $\frac{3}{5}$. Then, find the value of $\cos\alpha$.

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