
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 common difference of the arithmetic sequence $\{a\_n\}$ is $d$, and the solution set of the quadratic inequality about $x$, $\frac{d}{2}x^2 + (a\_1 - \frac{d}{2})x + c \geqslant 0$, is $[0, 22]$. Determine the maximum positive integer value of $n$ that makes the sum of the first $n$ terms of the sequence, $S\_n$, the greatest.

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