
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
Denote by  $P^{(n)}$  the set of all polynomials of degree  $n$  the coefficients of which is a permutation of the set of numbers  $\{2^0, 2^1,..., 2^n\}$ . Find all pairs of natural numbers  $(k,d)$   for which there exists a  $n$  such that for any polynomial   $p \in P^{(n)}$ ,  number  $P(k)$  is divisible by the number  $d$ .

(Oleksii Masalitin)

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