
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
What is the at most fourth-degree polynomial $p(x)$ that has zeros (and also minima) at $x_{1}=-3$ and $x_{2}=5$? Additionally, we know that the polynomial $q(x)=p(x+1)$ is even, and the value of its local maximum is 256.

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