
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  $n \ge  3$  be a positive integer. Find all nonconstant real polynomials  $f_1(x), f_2(x), ..., f_n(x)$  such that  $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$ ,  $1 \le k \le  n$  for all real x. [All suffixes are taken modulo  $n$ .]

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