
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 two foci of the conic section $C$ be $F_1$ and $F_2$, respectively. If there exists a point $P$ on curve $C$ such that the ratio $|PF_1| : |F_1F_2| : |PF_2| = 4 : 3 : 2$, determine the eccentricity of curve $C$.

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