
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
\[ 
8.451 \frac{1 - \cos 2x + \cos^2 2x - \ldots + (-1)^n \cos^n 2x + \ldots}{1 + \cos 2x + \cos^2 2x + \ldots + \cos^n 2x + \ldots} = \frac{1}{3} \tan^4 x, \quad |\cos 2x| \neq 1 
\]

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