
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 $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ with an eccentricity of $2$, where the left and right vertices are $A_{1}$ and $A_{2}$ respectively, and the right focus is $F$. Point $P$ lies on the right branch of $C$ and satisfies $PF\bot FA_{2}$. Find $\tan \angle A_{1}PA_{2}$.

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