A strange problem

Algebra Level 4

\[ \sum_{i=1}^{n} \dfrac{1}{a_{i} ( a_{i} + a_{i+1})( a_{i} + a_{i+1} + a_{i+2} ) \cdots (a_{i} + a_{i+1} + \cdots + a_{i+n-2}) } \]

Given are the real numbers \( a_{1}, a_{2}, a_{3}, \ldots , a_{n}\) whose sum is 0. And \(a_{n+m} = a_{m} \) where \(m = 1,2,3,\ldots, n\).

Determine the value of the sum above.

Assume all the denominators are non-zero.

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