
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
Georgina calls a  $992$ -element subset  $A$  of the set  $S = \{1, 2, 3, \ldots , 1984\}$  a **halfthink set** if 

    
- the sum of the elements in  $A$  is equal to half of the sum of the elements in  $S$ , and
- exactly one pair of elements in  $A$  differs by  $1$ .

She notices that for some values of  $n$ , with  $n$  a positive integer between  $1$  and  $1983$ , inclusive, there are no halfthink sets containing both  $n$  and  $n+1$ . Find the last three digits of the product of all possible values of  $n$ .

*Proposed by Andrew Wu and Jason Wang*

(Note: wording changed from original to specify what  $n$  can be.)

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