
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 common difference $d < 0$, $a_6$ and $a_8$ are the two roots of the equation $x^{2}+10x+24=0$.

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

$(2)$ Find the sum of the first $n$ terms $S_n$ of the sequence $\left\{ \frac{a_n}{2^{n-1}}\right\}$.

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