
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
Call a two-element subset of  $\mathbb{N}$  *cute* if it contains exactly one prime number and one composite number. Determine all polynomials  $f \in \mathbb{Z}[x]$  such that for every *cute* subset  $ \{ p,q \}$ , the subset  $ \{ f(p) + q, f(q) + p \} $  is *cute* as well.

*Proposed by Valentio Iverson (Indonesia)*

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