
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
An arithmetic sequence $\{a_{n}\}$ with a non-zero common difference satisfies $a_{15}=a_{3}a_{5}$ and $a_{1}=-2$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let $S_{n}$ denote the sum of the first $n$ terms of $\{a_{n}\}$. Find the largest positive integer $n$ such that $S_{n} \lt a_{n}$.

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