
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
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, and ${S}_{n}=2{a}_{n}-{2}^{n+1}$. The sequence $\{b_{n}\}$ satisfies ${b}_{n}=log_{2}\frac{{a}_{n}}{n+1}$, where $n\in N^{*}$. Find the maximum real number $m$ such that the inequality $(1+\frac{1}{{b}_{2}})•(1+\frac{1}{{b}_{4}})•⋯•(1+\frac{1}{{b}_{2n}})≥m•\sqrt{{b}_{2n+2}}$ holds for all positive integers $n$.

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