A strange problem

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

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.