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

$(1)$ Find the general formula for the sequence $\left\{b_n\right\}$;

$(2)$ Let $C_n= \frac{(a_n+1)^{n+1}}{(b_n+2)^n}$, find the sum of the first $n$ terms of the sequence $\left\{C_n\right\}$, denoted as $T_n$.

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