
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
\(a_{1}, a_{2}, a_{3}, \cdots\) is an arithmetic sequence where \(a_{1} > 0\). Let \(S_{n}\) represent the sum of the first \(n\) terms. If \(S_{3} = S_{11}\), and the maximum value among \(S_{1}, S_{2}, S_{3}, \cdots\) is \(S_{k}\), then \(k =\) ______.

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