
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 $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$. Given that $S_{9}=-a_{5}$.<br/>$(1)$ If $a_{3}=4$, find the general formula for $\{a_{n}\}$;<br/>$(2)$ If $a_{1} \gt 0$, find the range of values of $n$ for which $S_{n}\geqslant a_{n}$.

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