
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
A sequence $\{a\_n\}$ is defined as a "delicate sequence" if the ratio of the sum of its first $n$ terms ($S\_n$) to the sum of its first $2n$ terms ($S_{2n}$) is a constant. Given that a non-zero common difference arithmetic sequence $\{b\_n\}$ with first term 1 is a "delicate sequence", find the general term formula for $\{b\_n\}$.

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