
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
For which real numbers  $k > 1$   does there exist a bounded set of positive real numbers  $S$   with at
least  $3$  elements such that  $$ k(a - b)\in S $$  for all  $a,b\in  S $  with  $a > b?$ Remark: A set of positive real numbers  $S$   is bounded if there exists a positive real number  $M$  such that $x < M$   for all  $x \in  S.$ 

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