
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
Let \( n (\geq 2) \) be a positive integer. Find all polynomials with real coefficients
\[ 
P(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0}, 
\]
such that \( P(x) \) has exactly \( n \) real roots less than or equal to -1, and \( a_{0}^{2} + a_{1} a_{n} = a_{n}^{2} + a_{0} a_{n-1} \).

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