
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
In the real-coefficient polynomial \( p(x) = x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 \), for any given root \( \alpha \), \(\frac{1}{\alpha}\) and \(1 - \alpha\) are also roots of \( p(x) \). Find \( p(x) \).

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