
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
(a) Does there exist a polynomial  $ P(x)$  with coefficients in integers, such that  $ P(d) \equal{} \frac{2008}{d}$  holds for all positive divisors of  $ 2008$ ?
(b) For which positive integers  $ n$  does a polynomial  $ P(x)$  with coefficients in integers exists, such that  $ P(d) \equal{} \frac{n}{d}$  holds for all positive divisors of  $ n$ ?

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