Find the positive integer value of \(n\) such that

\[ \frac{n^3 - 3}{n^3} + \frac{n^3 - 4}{n^3} + \frac{n^3 - 5}{n^3} + \ldots + \frac{4}{n^3} + \frac{3}{n^3} = 169.\]

This problem is posed by Jose N.

**Details and assumptions**

The pattern in the above sequence is that the numerator decreases by 1, while the denominator is constant.

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