
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
For a point \(P=\left(a, a^{2}\right)\) in the coordinate plane, let \(\ell(P)\) denote the line passing through \(P\) with slope \(2a\). Consider the set of triangles with vertices of the form \(P_{1}=\left(a_{1}, a_{1}^{2}\right)\), \(P_{2}=\left(a_{2}, a_{2}^{2}\right)\), \(P_{3}=\left(a_{3}, a_{3}^{2}\right)\), such that the intersection of the lines \(\ell\left(P_{1}\right)\), \(\ell\left(P_{2}\right)\), \(\ell\left(P_{3}\right)\) form an equilateral triangle \(\Delta\). Find the locus of the center of \(\Delta\) as \(P_{1} P_{2} P_{3}\) ranges over all such triangles.

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