
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 that an arithmetic sequence \(\left\{a_{n}\right\}\) has a common difference \(d (d \neq 0)\) and the sum of the first \(n\) terms is \(S_{n}\). If the sequence \(\left\{\sqrt{8 S_{n}+2 n}\right\}\) is also an arithmetic sequence with a common difference \(d\), then find the general term \(a_{n}\) of the sequence \(\left\{a_{n}\right\}\).

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