
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 natural number. A whole number \( a > 2 \) is called \( n \)-factorable if \( a^{n} - 2^{n} \) is divisible by every number of the form \( a^{d} + 2^{d} \), where \( d \) is a natural divisor of \( n \) distinct from \( n \). Find all composite natural \( n \) for which there exists an \( n \)-factorable number.

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