
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
Let the complex numbers \( z_{1}, z_{2}, \ldots, z_{n} \) satisfy \( \left|z_{i}-z_{j}\right| \leq 1 \) for any \( 1 \leq i < j \leq n \). Define
\[ S = \sum_{1 \leq i < j \leq n} \left|z_{i} - z_{j} \right|^{2}. \]

(1) If \( n = 6063 \), find the maximum value of \( S \).

(2) If \( n = 2021 \), find the maximum value of \( S \).

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