
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 $A_{1}$ and $A_{2}$ be the two endpoints of the major axis of the ellipse $\dfrac{x^{2}}{9} + \dfrac{y^{2}}{4} = 1$, and $P_{1}$, $P_{2}$ be the endpoints of a chord perpendicular to $A_{1}A_{2}$. Determine the trajectory of the intersection point of lines $A_{1}P_{1}$ and $A_{2}P_{2}$.

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