
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
Let  $a,b,c$  be real numbers. Consider the quadratic equation in  $\cos{x}$  \[ a \cos^2{x}+b \cos{x}+c=0.  \] Using the numbers  $a,b,c$  form a quadratic equation in  $\cos{2x}$  whose roots are the same as those of the original equation. Compare the equation in  $\cos{x}$  and  $\cos{2x}$  for  $a=4$ ,  $b=2$ ,  $c=-1$ .

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