
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
Let an arithmetic sequence $\left\{a_{n}\right\}$ have a common difference $d$ (where $d \neq 0$) and the sum of the first $n$ terms be $S_{n}$. 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}
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