
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with its left and right foci $F_1$ and $F_2$, respectively. Let $P$ be a point on $C$ such that $PF_2 \perp F_1F_2$ and $\angle PF_1F_2 = 30^\circ$. Find the eccentricity of the ellipse $C$.

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