
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 $\{a\_n\}$ is an arithmetic sequence, $a\_1+a\_2+a\_3=156$, $a\_2+a\_3+a\_4=147$, and the sum of the first $n$ terms of $\{a\_n\}$ is $S\_n$, determine the value of $n$ that maximizes $S\_n$.

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