
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 $\{a\_n\}$ with its first $n$ terms sum $S\_n$, where $a\_1=2$ and $a_{n+1}=S_n+2$.

1. Find the general formula for the $n$-th term of the sequence $\{a\_n\}$.
2. If $b_n=\log _{2}a_n$, find the sum $T_n$ of the first $n$ terms of the sequence $\{\frac{1}{b_nb_{n+1}}\}$.

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