
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 polynomials of degree three with real coefficients \( P(x) \) and \( Q(x) \) such that the following conditions are satisfied:
1. Both polynomials take values 0 or 1 only 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 \( Q(1) = 0 \).

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