
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
Suppose P is a point of intersection of the hyperbola C₁: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ and the circle C₂: $x^2 + y^2 = a^2 + b^2$, and ∠PF₂F₁ = 2∠PF₁F₂, where F₁ and F₂ are the two foci of the hyperbola C₁. Then, the eccentricity of the hyperbola C₁ is \_\_\_\_\_\_.

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