USAMO 2018 Problem 2

Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that $$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1$$ for all $x,y,z >0$ with $xyz =1$ .