Let \(a, b, c, d\) be rational numbers with \(a > 0\). If for every integer \(n \ge 0\), the number \(an^{3} + bn^{2} + cn + d\) is also an integer, find the minimal value of \(a\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Let's think step by step and output the final answer within \boxed{}.
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