
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  $ p$  of one variable with integer coefficients such that if  $ a$  and  $ b$  are natural numbers such that  $ a \plus{} b$  is a perfect square, then  $ p\left(a\right) \plus{} p\left(b\right)$  is also a perfect square.

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