
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 the ellipse with foci \( F_{1}(-1,0) \) and \( F_{2}(1,0) \) have an eccentricity \( e \). Consider a parabola with \( F_{1} \) as the vertex and \( F_{2} \) as the focus. One of the intersection points of this parabola and the ellipse is \( P \). If \( \frac{\left|P F_{1}\right|}{\left|P F_{2}\right|} = e \), determine the value of \( e \).

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