
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
Find the number of positive integers  $n$  satisfying  $\phi(n) | n$  such that
\[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right]  \right) =  1992\]
What is the largest number among them? As usual,  $\phi(n)$  is the number of positive integers less than or equal to  $n$  and relatively prime to  $n.$ 

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