
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]


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The common difference of the arithmetic sequence $\{a_n\}$ is $d$, and the solution set of the quadratic inequality $dx^{2}+2a_{1}x\geqslant 0$ regarding $x$ is $[0,9]$. Determine the maximum positive integer value of $n$ that makes the sum of the first $n$ terms of the sequence, $S_{n}$, maximum.

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