
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 the following conditions, determine the sequence \( \left\{a_{n}\right\} \):

1. \(\left\{2^{a_{n}}\right\}\) is a geometric sequence.
2. The sum of the first \(n\) terms \( S_{n} = n^{2}+1 \).
3. \(a_{1} > 0\), and \( a_{k} = \frac{2}{k-1}\left(a_{1} + a_{2} + \cdots + a_{k-1}\right) \) for \( k \geqslant 2 \).

Which of the above conditions ensures that \( \left\{a_{n}\right\} \) is an arithmetic sequence? Fill in the appropriate condition number.

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