
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
A sequence $\{a_{n}\}$ where each term is an integer satisfies $a_{2}=-2$, $a_{7}=4$, the first $6$ terms form an arithmetic progression, and from the $5$th term onwards, they form a geometric progression.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Find all positive integers $m$ such that $a_{m}+a_{m+1}+a_{m+2}=a_{m}a_{m+1}a_{m+2}$.

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