
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
The hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b < 0)$ has its left and right foci as $F_1$ and $F_2$, and its left and right vertices as $A_1$ and $A_2$, respectively. A line passing through the focus $F_2$ and perpendicular to the $x$-axis intersects the hyperbola at a point $P$. If $|PA_1|$ is the geometric mean of $|F_1F_2|$ and $|A_1F_2|$, calculate the eccentricity of this hyperbola.

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