
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
The sequence $\{a_n\}$ is an arithmetic sequence. If $\frac{a_{11}}{a_{10}} < -1$, and its sum of the first $n$ terms, $S_n$, has a maximum value, when $S_n$ takes the smallest positive value, determine the value of $n$.

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