
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and another sequence $\{b_n\}$ satisfies $b_n = 2(S_{n+1} - S_n)S_n - n(S_{n+1} + S_n)$ ($n \in \mathbb{N}^*$).
1. If the sequence $\{a_n\}$ is an arithmetic sequence, and $b_n = 0$, find the general formula for the sequence $\{a_n\}$.
2. If $a_1 = 1$, $a_2 = 3$, and the sequences $\{a_{2n-1}\}$, $\{a_{2n}\}$ 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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