
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 sum of the first $n$ terms denoted as $S_n$, it holds true that $S_n = \frac{a_n}{n}$ for any $n \in \mathbb{N^*}$. If $1 < S_k < 9$ for some $k \in \mathbb{N^*}$, then the value of $k$ is _______________.

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