
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 non-zero common difference arithmetic sequence $\{a\_n\}$, its first term is $1$, the sum of the first $n$ terms is $S\_n$, and the sequence $\{\frac{S\_n}{a\_n}\}$ is an arithmetic sequence.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) Suppose $\lg b\_n=\frac{a\_n}{3^n}(n\in\mathbb{N}^*)$, can $b\_1$, $b\_k$, $b\_m$ ($k,m$ are positive integers and $1 < k < m$) form a geometric sequence? If yes, find all values of $k$ and $m$; if not, explain the reason.

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