
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 cubic polynomials \(P(x)\) and \(Q(x)\) with real coefficients such that the following four conditions are satisfied:
1. These two polynomials take values only 0 or 1 at \(x = 1, 2, 3, 4\).
2. If \(P(1) = 0\) or \(P(2) = 1\), then \(Q(1) = Q(3) = 1\).
3. If \(P(2) = 0\) or \(P(4) = 0\), then \(Q(2) = Q(4) = 0\).
4. If \(P(3) = 1\) or \(P(4) = 1\), then \(Q(1) = 0\).

(Democratic Republic of Germany Math Competition, 1980)

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