
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 the eccentricity of ellipse $G$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ is $\frac{\sqrt{6}}{3}$, the right focus is at $(2\sqrt{2}, 0)$, and the line $l$ with slope $1$ intersects ellipse $G$ at points $A$ and $B$. An isosceles triangle is formed with $AB$ as the base and $P(-3, 2)$ as the apex.

(I) Find the equation of ellipse $G$;

(II) Calculate the area of $\triangle PAB$.

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