
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]


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If \( \phi \) is a constant such that \( 0 < \phi < \pi \) and \( x + \frac{1}{x} = 2\cos{2\phi} \), then for integer \( k = 3 \), express \( x^k + \frac{1}{x^k} \) in terms of \( \cos{k\phi} \).

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