
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 a geometric sequence \(\{a_n\}\) where the sum of the first \(n\) terms is expressed as \(S_n = 2^{n-1} + k\). Find the maximum value of the function \(f(x) = x^{3} - kx^{2} - 2x + 1\).

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