
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 the general term formula of the sequence $\{a\_n\}$ as $a\_n= \frac {1}{2n-1}$, and the sequence $\{b\_n\}$ satisfies $2a\_n+b\_n=1$. If the inequality $\sqrt {b\_2b\_3…b\_{n+1}}\geqslant \frac {k}{(1+a\_1)(1+a\_2)…(1+a\_n)}$ holds true for any $n∈N^{*}$, then the maximum value of $k$ is \_\_\_\_\_\_.

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