
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 polynomial  $f(x)={{x}^{2}}-\alpha x+1$  with  $\alpha \in \mathbb{R}.$  
a) For  $\alpha =\frac{\sqrt{15}}{2}$ , let write  $f(x)$  as the quotient of two polynomials with nonnegative coefficients.
b) Find all value of  $\alpha $  such that  $f(x)$  can be written as the quotient of two polynomials with nonnegative coefficients.


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