
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 a nonnegative integer  $n$  define  $\operatorname{rad}(n)=1$  if  $n=0$  or  $n=1$ , and  $\operatorname{rad}(n)=p_1p_2\cdots p_k$  where  $p_1<p_2<\cdots <p_k$  are all prime factors of  $n$ . Find all polynomials  $f(x)$  with nonnegative integer coefficients such that  $\operatorname{rad}(f(n))$  divides  $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$  for every nonnegative integer  $n$ .

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