
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 hyperbola $C: \frac{x^2}{a^2}-\frac{y^2}{a^2}=1(a>0)$, with left and right vertices $A_{1}$ and $A_{2}$ respectively. Point $P$ is a point on the hyperbola $C$ other than the vertices, and $\angle A_{1}PA_{2}=3\angle PA_{1}A_{2}$. Find $\cos 5\angle PA_{1}A_{2}$.

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