
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 the sequence $\{a_n\}$ have a sum of the first $n$ terms $S_n = \frac{3^{n+1}-3}{2}$, and the sequence $\{b_n\}$ satisfies $b_n = \frac{1}{(n+1)\log_{3}a_n}$, and the sequence $\{c_n\}$ satisfies $c_n = (2n+1)a_n$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Find the sum of the first $n$ terms $B_n$ of the sequence $\{b_n\}$.  
$(3)$ Find the sum of the first $n$ terms $C_n$ of the sequence $\{c_n\}$.

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