
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 $b_n= \frac{1}{\log_{2}a_{n+1}\cdot \log_{2}a_{n+2}}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. If $T_n > 0.99$, find the minimum value of $n$.

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