
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
Consider the polynomial \( a_{0} + a_{1} x + \cdots + a_{n} x^{n} \) whose coefficients \( a_{0}, a_{1}, \cdots, a_{n} \) take only the values +1 or -1, and such that the equation \( a_{0} + a_{1} x + \cdots + a_{n} x^{n} = 0 \) has all real roots. Find all such polynomials with the described properties.
(Putnam Exam, 29th Competition)

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