
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 an arithmetic sequence $\{a_n\}$, if $\frac{a_{13}}{a_{12}} < -1$ and its sum of the first $n$ terms $S_n$ has a maximum value, then when $S_n$ takes the minimum positive value, the value of $n$ is ______.

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