
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
Determine the polynomial \( q(x) \) such that the identity \( p(x) = x^2 + x + 1 \) along with \( p^2(x) - 2p(x) q(x) + q^2(x) - 4p(x) + 3q(x) + 3 \equiv 0 \) is satisfied.

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