
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 $G$ with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b>0$) and an eccentricity of $\frac{c}{a}$, where the right focus is at $(2,0)$. 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 the vertex at $P(-3,2)$.  
(1) Find the equation of the ellipse $G$;  
(2) Find the area of $\triangle PAB$.

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