
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 $A_{1}A_{2}$ be the major axis of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. Let $P_{1}P_{2}$ be a chord perpendicular to $A_{1}A_{2}$. Then the locus of the intersection point of the lines $A_{1}P_{1}$ and $A_{2}P_{2}$ is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$.

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