
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 a point $P$ on the right branch of the hyperbola $x^2-y^2=a^2$ ($a>0$), with $A_1$ and $A_2$ being the left and right vertices of the hyperbola, respectively, and $\angle A_2PA_1 = 2\angle PA_1A_2$, find the measure of $\angle PA_1A_2$.

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