
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 natural number. Find all nonconstant polynomials with real coeficcietns  $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$ , for which
\[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad  1\le k\le n,\]
for every real  $ x$  (with  $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$  and  $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$ ).

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