
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 a positive common difference, the sum of the first $n$ terms is $S_n$, and it satisfies $a_3a_4=117$ and $a_2+a_5=22$.

$(1)$ Find the general formula for the sequence $\{a_n\}$.

$(2)$ If $\{b_n\}$ is an arithmetic sequence, and $b_n= \frac{S_n}{n+c}$ ($c\neq 0$), find the constant $c$.

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