
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]


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Given that the sum of the first \(n\) terms of a sequence \(\left\{a_{n}\right\}\), denoted by \(S_{n}\), satisfies: \(S_{n} + a_{n} = \frac{n-1}{n(n+1)}\) for \(n = 1, 2, \ldots\), find the general term \(a_{n}\).

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