
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 \(30m\), where \(m\) is 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}
user