
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
Given the equation \(a \cos^2 x + b \cos x + c = 0\), where \(a\), \(b\), and \(c\) are real numbers. Construct a quadratic equation using \(a\), \(b\), and \(c\) whose roots would be the corresponding values of \(\cos 2x\). Compare the given and the constructed equations for the case \(a = 4\), \(b = 2\), and \(c = -1\).

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