
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
Consider the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) where \(a > b > 0\). Given that \(ab=2\sqrt{3}\) and the eccentricity \(e=\frac{\sqrt{6}}{3}\):
1. Find the equation of the ellipse.
2. There is a line \(l\) with slope \(k\) passing through the right focus \(F\) of the ellipse and intersecting the ellipse at points \(A\) and \(B\). Point \(P\) is on the line \(x = 3\). Does there exist such a line \(l\) such that triangle \(\triangle ABP\) is an equilateral triangle? If so, find the equation of the line \(l\); if not, provide reasoning.

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