
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 a right-angled triangle $PQR$ inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, with $Q$ at $(0,b)$ and hypotenuse $\overline{PR}$ parallel to the x-axis. The ellipse's foci, $F_1$ and $F_2$, are such that $F_1$ lies on $\overline{QR}$ and $F_2$ on $\overline{PQ}$, respectively. Given that the distance between the foci $F_1 F_2$ is $2c = 2$, find $\frac{PQ}{F_1 F_2}$.

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