
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
The polynomial \( P(x) \) with real coefficients has degree \( 10^5 \) and its leading coefficient is equal to 1. Find the smallest possible degree of the polynomial

$$
R(x)=P\left(x^{1000}+1\right)-P(x)^{1000}
$$

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