
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 following equation:

\[ 
(2 \cos \alpha - 1) x^{2} - 4 x + 4 \cos \alpha + 2 = 0 
\]

where 

\[ 
\alpha < 90^{\circ} 
\]

1. For which values of \(\alpha\) are the roots of the equation real?

2. What are the signs of the roots for these values of \(\alpha\)?

3. Transform the product of the roots for logarithmic use.

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