
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
In the Cartesian coordinate system xOy, consider the ellipse \( C \) with the equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0)\] having a focal distance of \(2\sqrt{6}\), passing through the point \((\sqrt{2}, \sqrt{5})\).
1. Find the equation of the ellipse \(C\).
2. Let \(P\) be a point on the ellipse \(C\) with the x-coordinate greater than 2. Through point \(P\), draw two tangent lines to the circle \((x-1)^2+y^2=1\) that intersect the y-axis at points \(A\) and \(B\). Determine the coordinates of point \(P\) that maximize the area of triangle \(PAB\).

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