
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 $n (n \geqslant 2)$ be a positive integer. Determine all real-coefficient polynomials $P(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}$ such that $P(x)$ has exactly $n$ real roots that are all less than or equal to -1, and satisfies the equation $a_{0}^{2} + a_{1} a_{n} = a_{n}^{2} + a_{0} a_{n-1}$.

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