
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$. Let $F\_1$ be its left focus, and let $P$ and $Q$ be the points at which the vertical line through $F\_1$ perpendicular to the $x$-axis intersects the ellipse. Denote the right focus by $F\_2$. If $\triangle PQF\_2$ is an equilateral triangle, find the eccentricity of the ellipse.

A) $\frac{\sqrt{2}}{2}$
B) $\frac{\sqrt{3}}{3}$
C) $\frac{1}{2}$
D) $\frac{1}{3}$

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