
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
A regular equilateral triangle with center at the origin in the complex plane has a side length of \( \sqrt{3} \) units. One side of the triangle is parallel to the real axis. Let \( T \) represent the region inside the triangle, and let \( U = \{ z^2 \ | \ z \in T \} \). Find the area of \( U \).

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