
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 an equilateral triangle whose vertices lie on the hyperbola \( xy = c \) and on the hyperbola \( x/y = c \) for a constant \( c \). Assume the centroid of the triangle is located at the origin (0, 0) which also serves as the center for these hyperbolas. What is the square of the area of this triangle?

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