
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 $C:\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\left(a>b>0\right)$ 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 $C$ at points $A$ and $B$. An isosceles triangle $PAB$ is constructed with $AB$ as the base and the vertex at $P(-3,2)$.

(1) Find the equation of the ellipse $C$.

(2) Calculate the area of the triangle $PAB$.

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