
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
Let \(F_{1}(-c, 0)\) and \(F_{2}(c, 0)\) be the foci of the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) where \(a > b > 0\). Suppose \(P_{1} P_{2}\) is a chord perpendicular to the line segment \(F_{1} F_{2}\). Determine the locus of the intersection of the lines \(P_{1} F_{1}\) and \(P_{2} F_{2}\), which is given by \(\frac{x^{2}}{\frac{c^{4}}{a^{2}}} - \frac{y^{2}}{\frac{c^{2} b^{2}}{a^{2}}} = 1\) where \(a > b > 0\).

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