
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  $S$  be the set of all the odd positive integers that are not multiples of  $5$  and that are less than  $30m$ ,  $m$  being an arbitrary positive integer. What is the smallest integer  $k$  such that in any subset of  $k$  integers from  $S$  there must be two different integers, one of which divides the other?

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