
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
The vertices of an equilateral triangle lie on the ellipse given by the equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$, and the centroid of the triangle is at the origin (0,0), which is also the center of the ellipse. Calculate the square of the area of the triangle.

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