
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 $\{a_n\}$, the first three terms are $a-1$, $4$, $2a$. Let the sum of the first $n$ terms be $S_n$.  
$(1)$ If $S_k=30$, find the values of $a$ and $k$;  
$(2)$ Let $b_n= \frac {S_n}{n}$, find the value of $b_3+b_7+b_{11}+\ldots+b_{4n-1}$.

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