
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
Given the complex number \( \mathrm{z} = \mathrm{t} + 3 + \sqrt{3} \mathrm{i} \), where \( \mathrm{t} \) is a complex number such that \( \frac{\mathrm{t} + 3}{\mathrm{t} - 3} \) is purely imaginary:
1. Find the equation of the locus of the point \( \mathrm{P} \) representing \( z \) in the complex plane.
2. Determine the range of values for \( \arg z \).
3. Find the maximum and minimum values of \( |z - 1|^2 + |z + 1|^2 \).

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