
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 point $P$ is on the right branch of the hyperbola $x^{2}-y^{2}=a^{2}$ ($a > 0$), and $A_{1}$, $A_{2}$ are respectively the left and right vertices of the hyperbola, and $\angle A_{2}PA_{1}=2\angle PA_{1}A_{2}$, then $\angle PA_{1}A_{2}=$ ______.

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