
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 sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=3n^2+8n$, and $\{b_n\}$ is an arithmetic sequence such that $a_n=b_n+b_{n+1}$.

1. Find the general term formula for the sequence $\{b_n\}$.
2. Let $c_n=\frac{(a_n+1)^{n+1}}{(b_n+2)^n}$. Find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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