
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 a sequence $\{a\_n\}$ with its first $n$ terms sum $S\_n$, $S\_1=1$, $S\_2=-\frac{3}{2}$, and $S\_n-S\_{n-2}=3\times(-\frac{1}{2})^{n-1}\,(n\geqslant 3)$, find $a\_n$.

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