
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 $d$. The solution set of the quadratic inequality $a_1x^2 + (\frac{d}{2} - a_1)x + c \geqslant 0$ with respect to $x$ is $[\frac{1}{3}, \frac{4}{5}]$. Determine the smallest positive integer value of $n$ that minimizes the sum of the first $n$ terms, $S_n$, of the sequence ${a_n}$.

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