
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 5 positive integers $x_1, x_2, x_3, x_4, x_5$. Let $S = x_i x_j$. The questions are:  
⑴ When do $x_1, x_2, x_3, x_4, x_5$ take values such that $S$ reaches its maximum value?  
⑵ Further, for any $1 \leq i, j \leq 5$ with $|x_i - x_j| \leq 2$, when do $x_1, x_2, x_3, x_4, x_5$ take values such that $S$ reaches its minimum value?  
Explain your reasoning.

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