
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 positive geometric sequence $\{a_{n}\}$ where $a_{n}a_{n+1}=2^{2n+1}$ for all $n\in N^{*}$.<br/>$(1)$ Find the general term formula for $\{a_{n}\}$;<br/>$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=\log _{2}a_{n}$, find the sum of the first $2n+1$ terms of the sequence $\{b_{n}\}$ denoted as $T_{2n+1}$.

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