
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 conic section \( C \) have two foci \( F_1 \) and \( F_2 \). If a point \( P \) on \( C \) satisfies \( |PF_1| \) : \( |F_1F_2| \) : \( |PF_2| = 4 \) : \( 3 \) : \( 2 \), then find the eccentricity \( e \) of \( C \).

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