
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 \(a, b, c\) are known real numbers. Derive a new quadratic equation in terms of $\cos 2x$. Compare the given equation and the derived equation under the condition where \(a = 4, b = 2, c = -1\).

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