
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 odd positive integers less than  $ 30m$  which are not multiples of  $ 5$ , where  $ m$  is a given positive integer. Find the smallest positive integer  $ k$  such that each  $ k$ -element subset of  $ S$  contains two distinct numbers, one of which divides the other.

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