
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 $M(\sqrt{6}, \sqrt{2})$ 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 $A$ and $B$, and an isosceles triangle is constructed with base $AB$ and vertex $P(-3, 2)$, find the area of $\triangle PAB$.

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