
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 a quadratic function \( p(x) = ax^2 + bx + c \), there exists some integer \( n \) such that \( p(n) = p\left(n^2\right) \). Provide an example of a function \( p(x) \) for which the number of such integers \( n \) is the greatest. What is this greatest number of integers \( n \)?

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