
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 geometric sequence $\{a_n\}$ where $a_2=2$, and $a_2$, $a_3+1$, $a_4$ form an arithmetic sequence; in the sequence $\{b_n\}$, the sum of the first $n$ terms is $S_n=n^2+n$.
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Find the sum of the first $n$ terms of the sequence $\{a_n+ \frac{4}{b_nb_{n+1}}\}$.

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