
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
For given positive integers  $n$  and  $N$ , let  $P_n$  be the set of all polynomials  $f(x)=a_0+a_1x+\cdots+a_nx^n$  with integer coefficients such that:

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(a)  $|a_j| \le N$  for  $j = 0,1, \cdots ,n$ ;
(b) The set  $\{ j \mid a_j = N\}$  has at most two elements.
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Find the number of elements of the set  $\{f(2N) \mid f(x) \in P_n\}$ .

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