
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 the first term is $\frac{1}{2}$, and the common ratio is not equal to $1$ of the geometric sequence $\{a\_n\}$, the sum of the first $n$ terms is $S\_n$, and $S\_3$, $S\_2$, $S\_4$ form an arithmetic sequence.

$(1)$ Find the general term formula of the sequence $\{a\_n\}$;

$(2)$ Let $b\_n=n|a\_n|$, the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $T\_n$, find $T\_n$.

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