
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
Find the smallest positive integer  $k$  such that, for any subset  $A$  of  $S=\{1,2,\ldots,2012\}$  with  $|A|=k$ , there exist three elements  $x,y,z$  in  $A$  such that  $x=a+b$ ,  $y=b+c$ ,  $z=c+a$ , where  $a,b,c$  are in  $S$  and are distinct integers.

*Proposed by Huawei Zhu*

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