
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]


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Given that the eccentricity of an ellipse is \( e \), with foci \( F_1 \) and \( F_2 \). A parabola \( C \) has \( F_1 \) as its vertex and \( F_2 \) as its focus. Point \( P \) is an intersection point of the parabola and the ellipse. If \( e|PF_2| = |PF_1| \), find the value of \( e \).

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