
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
Find all polynomials  $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$ with the following properties

(a) all the coefficients  $a_{1}, a_{2}, ..., a_{n}$  belong to the set  $\{ -1, 1 \}$ ; and
(b) all the roots of the equation  $f(x)=0$  are real.

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