
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
An equilateral triangle $PQR$ is inscribed in the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, where $Q$ is at $(4,0)$, and $\overline{PR}$ is parallel to the $y$-axis. The foci $F_1$ and $F_2$ of the ellipse are at $(3,0)$ and $(-3,0)$, respectively. Find $\frac{PQ}{F_1 F_2}$.

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