Please reason step by step, and put your final answer within \boxed{}.
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 ${a_n}$.
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