
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 positive integer and  $\mathcal{P}_n$  be the set of integer polynomials of the form  $a_0+a_1x+\ldots +a_nx^n$  where  $|a_i|\le 2$  for  $i=0,1,\ldots ,n$ . Find, for each positive integer  $k$ , the number of elements of the set  $A_n(k)=\{f(k)|f\in \mathcal{P}_n \}$ .

*Marian Andronache*

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