
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, if $\frac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, calculate the minimum value of $n$ for which $S_n < 0$.

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