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

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