
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 quadratic polynomials with integer coefficients \( d(x) = x^2 + ax + b \) such that there exist a polynomial \( p(x) \) with integer coefficients and a non-zero polynomial \( q(x) \) with integer coefficients that satisfy
\[
(p(x))^2 - d(x)(q(x))^2 = 1, \forall x \in \mathbb{R}.
\]

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