
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
$$
f(x) = x^2 - \alpha x + 1 \quad (\alpha \in \mathbf{R}).
$$
1. For \(\alpha = \frac{\sqrt{15}}{2}\), express \(f(x)\) as a ratio of two polynomials with non-negative coefficients.
2. Find all real numbers \(\alpha\) such that \(f(x)\) can be expressed as a ratio of two polynomials with non-negative coefficients.

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