
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 that $F$ is the right focus of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a moving point on $C$. A line passing through $F$ and perpendicular to the $x$-axis intersects $C$ at points $M$ and $N$. If the length of $MN$ is three times the minimum value of $PF$, determine the eccentricity of $C$.

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