
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 that the ellipse C₁ and the hyperbola C₂ have the same left and right foci F₁, F₂, P is a common point of the ellipse C₁ and the hyperbola C₂ in the first quadrant. The eccentricity of the ellipse C₁ and the hyperbola C₂ are denoted as e₁ and e₂ respectively, and $\frac{e_1}{e_2} = \frac{1}{3}$. If ∠F₁PF₂ = $\frac{\pi}{3}$, find the eccentricity of the ellipse C₁.

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