
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 that the sequence $\{a_{n}\}$ is an arithmetic sequence, $a_{9}+a_{12} \lt 0$, $a_{10}\cdot a_{11} \lt 0$, and the sum of the first $n$ terms of the sequence $\{a_{n}\}$, denoted as $S_{n}$, has a maximum value, determine the maximum value of $n$ when $S_{n} \gt 0$.

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