
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 odd-degree polynomials $P(x)$ with integer coefficients that satisfy the following property: For every positive integer $n$, there exist $n$ distinct positive integers $x_{1}, x_{2}, \cdots, x_{n}$ such that for every pair of indices $i, j$ $(1 \leq i, j \leq n)$, the inequality $\frac{1}{2} < \frac{P(x_{i})}{P(x_{j})} < 2$ holds, and $\frac{P(x_{i})}{P(x)}$ is a rational number raised to the power of $d$.

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