
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 all functions  $f \colon \mathbb{R} \to \mathbb{R}$  that satisfy the following property for all real numbers  $x$  and all polynomials  $P$  with real coefficients:

If  $P(f(x)) = 0$ , then  $f(P(x)) = 0$ .

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