
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
Let \( X = \{1, 2, \cdots, 2001\} \). Find the smallest positive integer \( m \) such that for any \( m \)-element subset \( W \) of \( X \), there exist \( u, v \in W \) (where \( u \) and \( v \) can be the same) such that \( u+v \) is a power of 2.

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