
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 an ellipse $C: \frac{x^2}{9} + \frac{y^2}{b^2} = 1 \left(b > 0\right)$, the minimum distance from a moving point $P$ on the ellipse to another focal point is $3-2\sqrt{2}$.

$(1)$ Find the equation of the ellipse $C$.

$(2)$ If a line $l$ intersects the ellipse $C$ at points $M$ and $N$, and $A$ is the right vertex of the ellipse, with $\overrightarrow{AM} \cdot \overrightarrow{AN} = 0$, find the maximum area of $\triangle AMN$.

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