
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
We have collected 1001 pebbles into a pile. The pile is divided arbitrarily into two parts, and the number of pebbles in each part is multiplied. Then, the operation is repeated on any of the parts containing at least two pebbles: the part is divided into two further parts, and the product of the number of pebbles in these two new parts is calculated. This process continues until all parts contain only one pebble each.

What are the possible ranges for the sum of these 1000 products?

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