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

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

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

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