
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
A set  $ X$  of positive integers is called *nice* if for each pair  $ a$ ,  $ b\in X$  exactly one of the numbers  $ a \plus{} b$  and  $ |a \minus{} b|$  belongs to  $ X$  (the numbers  $ a$  and  $ b$  may be equal). Determine the number of nice sets containing the number 2008.

*Author: Fedor Petrov*

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