
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 with the first term $23$ and an integer common difference. The sixth term is positive, and the seventh term is negative.
(1) Find the common difference;
(2) Find the maximum value of the sum $S\_n$ of the first $n$ terms;
(3) Find the maximum value of $n$ when $S\_n > 0$.

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