
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 composite number. For each proper divisor \( d \) of \( n \) (a proper divisor of a positive integer \( n \) is defined as \( d \mid n \), and \( 1 < d < n \)), write the number \( d + 1 \) on the blackboard. Find all positive integers \( n \) such that the numbers written on the blackboard are exactly the proper divisors of some positive integer \( m \).

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