
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
For what constraints on the integers $p$ and $q$:

a) does the polynomial $P(x) = x^{2} + p x + q$ take even (or odd) values for all $x \in \mathbf{Z}$?

b) does the polynomial $Q(x) = x^{3} + p x + q$ take values divisible by 3 for all $x \in \mathbf{Z}$?

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