
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 a positive integer \( n \), let \( z_{1}, z_{2}, \ldots, z_{n} \) be \( n \) complex numbers each with a modulus not greater than 1. Determine the maximum possible value of \( \left| \sum_{k=1}^{n} z_{k}^{2} \right| - \left| \sum_{k=1}^{n} z_{k} \right|^{2} \).

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