
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 the set \( N = \{1, 2, \cdots, n\} \), select \( m \) numbers such that no number is a multiple of another by \( p \). Find the maximum value of \( m \).

The maximum value of \( m \) can be given by the formula:
\[ m_{\max} = \sum_{i=0}^{\infty} (-1)^{i} \left\lfloor \frac{n}{p^i} \right\rfloor, \]
where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to the real number \( x \).

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