
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
Represent 2006 as the sum of 5 positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Let \( S=\sum_{1 \leq i < j \leq 5} x_{i} x_{j} \). Answer the following:

1. What values of \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) maximize \( S \)?
2. Furthermore, for any \( 1 \leq i < j \leq 5 \) such that \( |x_{i}-x_{j}| \leq 2 \), what values of \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) minimize \( S \)? Explain why.

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