
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 the sequence of positive numbers $\{a_{n}\}$ be defined by
$$
\begin{array}{l}
a_{1}=\sqrt{2}-1, \\
a_{n+1}=\frac{2 n+1}{S_{n}+S_{n+1}+2} \quad (n=1,2, \cdots),
\end{array}
$$
where $S_{n}$ represents the sum of the first $n$ terms of the sequence $a_{n}$. Determine the general term of the sequence.

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