Question 171
Find, with proof, all functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ with the property that
$$
f(x + f(y) + f(f(z))) = z + f(y) + f(f(x)) \quad \text{for all } x, y, z \gt 0.
$$
$$
f(x + f(y) + f(f(z))) = z + f(y) + f(f(x)) \quad \text{for all } x, y, z \gt 0.
$$