
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 the sequence $\{a_n\}$, satisfying $a_1=2$, $a_{n+1}-a_n=3$, and the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$, satisfies $S_n=2b_n-b_1$, and $(b_1, b_2+1, b_3)$ forms an arithmetic sequence.

$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.

$(2)$ Let $c_n=(a_n+1)b_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, $T_n$.

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