
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
Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$, where $a_{1}=1$, and $S_{3}=3S_{2}-a_{2}$. Define $b_{n}=\left(n-1\right)a_{n+1}+\left(1-2n\right)a_{n}$. <br/>$(1)$ If $\{|b_{n}|\}$ forms an arithmetic sequence, find the general term of the sequence $\{|b_{n}|\}$; <br/>$(2)$ If $\{|b_{n}|\}$ forms a geometric sequence, let $T_{n}=|b_{1}|+2|b_{2}|+3|b_{3}|+\ldots +n|b_{n}|$, determine $T_{n}$.

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