
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 none of the terms in the sequence \(\left\{a_{n}\right\}\) are zero. Let the sum of the first \(n\) terms of the sequence \(\left\{a_{n}\right\}\) be \(S_{n}\), and the sum of the first \(n\) terms of the sequence \(\left\{a_{n}^{2}\right\}\) be \(T_{n}\). It is known that
\[ S_{n}^{2}+4 S_{n}-3 T_{n}=0\ \ (n \in \mathbf{Z}_{+}). \]
Find the sum of the first \(n\) terms of the sequence \(\left\{\frac{(n-1)}{n(n+1)} a_{n}\right\}\).

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