
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
Determine all real numbers \(a_{1}, a_{2}, \ldots, a_{n}\) such that for all real \(x\),

\[ a_{1} \cos x + a_{2} \cos 2x + \ldots + a_{k} \cos kx + \ldots + a_{n} \cos nx = 0. \]

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