
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
Express 2006 as the sum of five positive integers \( x_1, x_2, x_3, x_4, x_5 \). Let \( S = \sum_{1 \leq i < j \leq 5} x_i x_j \).

1. What values of \( x_1, x_2, x_3, x_4, x_5 \) maximize \( S \)?
2. Given \( |x_i - x_j| \leq 2 \) for any \( 1 \leq i, j \leq 5 \), what values of \( x_1, x_2, x_3, x_4, x_5 \) minimize \( S \)? Explain the reasoning.

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