
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 an eccentricity of $\frac{\sqrt{6}}{3}$, a right focus at $(2 \sqrt{2},0)$, and a line $l$ with a slope of $1$ intersecting the ellipse $G$ at points $A$ and $B$. An isosceles triangle is constructed with $AB$ as the base and $P(-3,2)$ as the vertex.

(1) Find the equation of the ellipse $G$;
(2) If $M(m,n)$ is any point on the ellipse $G$, find the maximum and minimum values of $\frac{n}{m-4}$;
(3) Find the area of $\triangle PAB$.

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