
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 \( S=\sum_{1 \leqslant i<j \leqslant 5} x_{i} x_{j} \).  

1. When \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) take what values, does \( S \) reach the maximum value?
2. Given that for any \( 1 \leqslant i, j \leqslant 5 \), we have \( \left|x_{i}-x_{j}\right| \leqslant 2 \), when \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) take what values, does \( S \) reach the minimum value?

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