Evaluating $\sum\limits_{n \ge 0} \frac{1}{x^{2^n}-y^{2^n}}$ where $x, y \in \mathbb R^{+}$ and $x \ne y, x>1.$

Question

I reduced a competition problem involving Fibbonacci numbers to the evaluation of this simple sum. I've tried telescoping, factorization, and even rewriting the product as

$$S = \frac{1}{x-y} \sum_{n \ge 0} \prod_{m=0}^{n-1} \frac{1}{x^{2^m}+y^{2^m}},$$

but none of these approaches led anywhere. I'm wondering if there is an elementary method for finding $S.$ I can't even solve the special case $y=1.$