
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 that the sequence $\{a_{n}\}$ is a geometric sequence with the first term being $2$, common ratio $q \gt 0$, and $a_{4}$ is the arithmetic mean of $6a_{2}$ and $a_{3}$. <br/>$(1)$ Find the general formula for $\{a_{n}\}$; <br/>$(2)$ Let the sequence $\{b_{n}\}$ satisfy $b_n=\frac{1}{\log_2a_n \cdot \log_2a_{n+1}}$, find the sum of the first $2023$ terms of $\{b_{n}\}$ denoted as $T_{2023}$.

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