
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 $\{a_{n}\}$ where each term is not equal to $0$, and the sum of the first $n$ terms is $S_{n}$, with $3a_{n}-2=S_{n}$. Another sequence $\{b_{n}\}$ satisfies $b_{1}=-1$ and $b_{n+1}=b_{n}+n$. <br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$; <br/>$(2)$ Let $c_{n}=\frac{2a_{n}b_{n}}{n+1}$, find the sum of the first $n$ terms of sequence $\{c_{n}\}$ denoted as $T_{n}$.

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