
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 $\{ a_{n} \}$ with a common ratio of $2$, and $a_{2}$, $a_{3}+1$, $a_{4}$ form an arithmetic sequence.

$(1)$ Find the general formula for the sequence $\{ a_{n} \}$;

$(2)$ Let $b_{n} = \log_{2}a_{n+1}$, $c_{n} = \frac{1}{b_{n} \cdot b_{n+1}}$. 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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