
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 an ellipse intersects the x-axis at points $A$ and $B$. For any point $P$ on the ellipse other than $A$ and $B$, the product of the slopes of the lines $AP$ and $BP$ is always $-\frac{1}{2}$. Find the eccentricity of the ellipse.

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