
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
Do there exist polynomials \( P(x), Q(x), R(x) \) with real coefficients such that the polynomials \( P(x) \cdot Q(x), Q(x) \cdot R(x) \), and \( P(x) \cdot R(x) \) have the same degree, while the polynomials \( P(x) + Q(x), P(x) + R(x), \) and \( Q(x) + R(x) \) have pairwise different degrees? (Assume that the zero polynomial does not have a degree, meaning that these polynomials cannot be equal to it.)

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