
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
\( p(x) \) is a real polynomial with at least \( n \) distinct real roots greater than 1. [To be precise, we can find at least \( n \) distinct values \( a_i > 1 \) such that \( p(a_i) = 0 \). It is possible that one or more of the \( a_i \) is a multiple root, and it is possible that there are other roots.] Let \( q(x) = (x^2 + 1) p(x) p'(x) + x p(x)^2 + x p'(x)^2 \). Must \( q(x) \) have at least \( 2n - 1 \) distinct real roots?

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