
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
In the sequence $\left\{a_{n}\right\}$, $a_{1}=1$. When $n \geqslant 2$, $a_{n}$, $S_{n}$, and $S_{n}-\frac{1}{2}\left(S_{n}\right)$ (where $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$) form a geometric sequence. Find $\lim _{n \rightarrow \infty} n^{2} a_{n}$.

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