Finding all $x+y+z$ for positive integers satisfying $\frac{13}{x^2}+\frac{28}{y^2}=\frac{z}{85}$ [closed]

Let $x, y, z$ be positive integers such that$$\frac{13}{x^2}+\frac{28}{y^2}=\frac{z}{85}$$Find all $x+y+z$.

I observe that $(x, y)=(1, 1)$ or $(1, 2)$ will give some solutions. However I can neither find any more solutions nor prove that there are only these solutions. Can anyone help me out.