
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
In the geometric sequence $\{a_n\}$, the common ratio $q > 1$, $a_{2}=2$, and the sum of the first three terms $S_{3}=7$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Let $b_{n}=\log_{2}{a_{n}}$, $c_{n}= \frac{1}{{b_{n+1}}\cdot{b_{n+2}}}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_{n}$.

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