
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
In the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with $c = 2$, the slope of the line passing through points $(0, b)$ and $(a, 0)$ is $-\frac{\sqrt{3}}{3}$.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Let $F$ be the right focus of the ellipse $C$, $P$ be any point on the line $x = 3$. A perpendicular line is drawn from $F$ to intersect the ellipse $C$ at points $M$ and $N$. Find the maximum value of $\frac{|MN|}{|PF|}$.

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