
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 function
\[ g(z) = \frac{(-2 + 2i \sqrt{3}) z + (-3 \sqrt{3} - 27i)}{3} \]
which describes a rotation around a complex number \( d \). Find \( d \).

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