
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
Suppose that for the sequence $\{a_n\}$, each term is a positive number, and for any $n\in \mathbb{N}^*$, it holds that $4S_n = a_n^2 + 2a_n$, where $S_n$ is the sum of the first $n$ terms of the sequence. Determine the general formula for the $n$-th term of the sequence, $a_n$.

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