
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 second degree polynomial  $d(x)=x^{2}+ax+b$  with integer coefficients, so that there exists an integer coefficient polynomial  $p(x)$  and a non-zero integer coefficient polynomial  $q(x)$  that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

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