
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
Let $q$ and $p+1$ be divisible by 3. Represent the polynomial as

$$
Q(x) = x^3 - x + (p+1)x + q = x(x-1)(x+1) + (p+1)x + q
$$

The first term is divisible by 3 for any integer $x$. The second and third terms are divisible by 3 by the choice of $p$ and $q$.

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