
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 point $M(\sqrt{6}, \sqrt{2})$ is on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, and the eccentricity of the ellipse is $\frac{\sqrt{6}}{3}$.

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

$(2)$ If a line $l$ with slope $1$ intersects ellipse $C$ at points $A$, $B$, and an isosceles triangle is formed with $AB$ as the base and vertex at $P(-3,2)$, find the area of $\triangle PAB$.

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