
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  $ N $  be a positive integer. A set  $ S \subset \{ 1, 2, \cdots, N \} $  is called *allowed* if it does not contain three distinct elements  $ a, b, c $  such that  $ a $  divides  $ b $  and  $ b $  divides  $c$ . Determine the largest possible number of elements in an allowed set  $ S $ .

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