I have checked that the following expression

\begin{align} \sum_{i=1}^N\sum_{j=1\\ j\ne i}^N\frac{A_iA_j(A_i+A_j)}{(A_i-A_j)^3}\prod_{k=1\\ k\ne i\\ k\ne j}^N\frac{A_i A_k}{(A_i-A_k)^2} \end{align} is zero for $N$ up to 7. I would like a general proof that it is zero for $N\in \mathbb Z_{+}$. I tried induction but I didn't find any inductive structure that I could make use of.

This was previously asked on math stack exchange ( https://math.stackexchange.com/questions/4770113/prove-that-the-expression-is-identically-zero ) and since there was no answer, it was suggested that I post the question here.