
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
Find the smallest positive integer  $n\neq 2004$  for which there exists a polynomial  $f\in\mathbb{Z}[x]$  such that the equation  $f(x)=2004$  has at least one, and the equation  $f(x)=n$  has at least  $2004$  different integer solutions.

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