
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 $M\left(2 \sqrt{2}, \frac{2 \sqrt{3}}{3}\right)$ on the ellipse $G$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, and the sum of the distances from point $M$ to the two foci is $4 \sqrt{3}$.
(1) Find the equation of the ellipse $G$;
(2) If the line $l$ with slope $1$ intersects the ellipse $G$ at points $A$ and $B$, and an isosceles triangle is formed with $AB$ as the base and point $P(-3,2)$ as the apex, find the area of $\triangle PAB$.

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