
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]


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Given a sequence $\{a\_n\}$ with its first $n$ terms sum $S\_n$, let sequence $\{b\_n\}$ satisfy $b\_n = 2(S_{n+1}-S\_n)S\_n - n(S_{n+1}+S\_n) (n \in \mathbb{N}^*)$.

1. If sequence $\{a\_n\}$ is an arithmetic sequence and $b\_n=0$, find the general term formula for sequence $\{a\_n\}$.
2. If $a\_1=1$, $a\_2=3$, and both sequences $\{a_{2n-1}\}$ and $\{a_{2n}\}$ are geometric sequences with a common ratio of $2$, find the set of all positive integers $n$ that satisfy the inequality $b_{2n} < b_{2n-1}$.

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