
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 the first term $a_1 = \frac{3}{2}$, and the sum of the first $n$ terms denoted as $S_n$, it satisfies $2a_{n+1} + S_n = 3$ for $n \in \mathbb{N}^*$. Find the sum of all $n$ such that $\frac{18}{17} < \frac{S_{2n}}{S_{n}} < \frac{8}{7}$.

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