
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 a point $P(x_0, y_0)$ on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where $(F_1, F_2)$ are the left and right foci of ellipse $C$, determine the eccentricity of ellipse $C$ when $y_0 = \frac{b}{2}$ and $\angle F_1PF_2 = \frac{\pi}{3}$.

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