
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 sequence $\{a\_n\}$ with its first term $a\_1=1$, the sum of the first $n$ terms denoted by $S\_n$, and satisfying $2a_{n+1}+S_n=2 (n∈N^*)$, find the maximum value of $n$ such that $\frac{1\,001}{1\,000} < \frac{S_{2n}}{S_{n}} < \frac{11}{10}$.

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