
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 all polynomials of the form \( x^{n} + a_{1} x^{n-1} + \cdots + a_{n-1} x + a_{n} \) (where \( a_{i} = \pm 1 \) for \( 1 \leq i \leq n \)) such that all roots of each polynomial are real numbers.

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