
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
Consider a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$. Let $A_1$ and $A_2$ be the left and right vertices, and $F_1$ and $F_2$ be the left and right foci of the hyperbola. A circle with diameter $F_1F_2$ intersects the left branch of the hyperbola at point $P$. If a circle with diameter $A_1A_2$ is tangent to line segment $PF_2$, determine the eccentricity of hyperbola $C$.

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