
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 quadratic equation in terms of $\cos x$: \(a \cos ^{2} x + b \cos x + c = 0\), where \(a\), \(b\), and \(c\) are known real numbers, construct a quadratic equation having $\cos 2x$ as the root. Compare the given equation and the newly constructed equation for the case where \(a = 4\), \(b = 2\), \(c = -1\).

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