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