
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 an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, and it is known that $a_3=7$, $S_4=24$. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n=n^2+a_n$.  
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Find the sum of the first $n$ terms, $B_n$, of the sequence $\left\{ \frac{1}{b_nb_{n+1}} \right\}$.

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