
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, $S\_n$ represents the sum of its first $n$ terms, and $S\_{12} > 0$, $S\_{13} < 0$, find the smallest value of $n$ that makes $a\_n < 0$ true.

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