
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  $X=\{1,2,\ldots,2001\}$ . Find the least positive integer  $m$  such that for each subset  $W\subset X$  with  $m$  elements, there exist  $u,v\in W$  (not necessarily distinct) such that  $u+v$  is of the form  $2^{k}$ , where  $k$  is a positive integer.

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