
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
We fix a natural number \( k \geq 1 \). Find all polynomials \( P(x) \) that satisfy

$$
P\left(x^{k}\right) - P(k x) = x^{k} P(x)
$$

for all values of \( x \in \mathbb{R} \).

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