
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 an arithmetic sequence $\{a\_n\}$ with the first three terms being $a-1$, $4$, and $2a$, let $S\_n$ denote the sum of its first $n$ terms.
(1) If $S\_k=30$, find the values of $a$ and $k$;
(2) Let $b\_n=\frac{S\_n}{n}$, find the value of $b\_1+b\_2+b\_3+\ldots+b\_n$.

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