
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
Let the common difference of the arithmetic sequence $\{a_n\}$ be $d > 0$. If $a_1 + a_2 + a_3 = -9$ and $a_1a_2a_3 = -15$,

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

$(2)$ Find the sum of the first $n$ terms of the sequence $\{a_n\}$, denoted as $s_n$;

$(3)$ Find all positive integers $m$ such that $\frac{a_ma_{m+1}}{a_{m+2}}$ is a term in the sequence $\{a_n\}$.

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