
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
Given the hyperbola $\frac{x^2}{2}-y^2=1$ with its left and right vertices respectively as $A_1$ and $A_2$, and points $P(x_1, y_1)$, $Q(x_1, -y_1)$ are two distinct points on the hyperbola different from $A_1$ and $A_2$, then the equation of the trajectory of the intersection of line $A_1P$ and line $A_2Q$ is ______.

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