Question 163
Let $\mathbb{Z}_+ = \{1, 2, 3, 4, \dots\}$ be the set of all positive integers. Find, with proof, all functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ with the property that
$$
f(x + f(y) + f(f(z))) = z + f(y) + f(f(x))
$$
for all positive integers $x, y, z$.
$$
f(x + f(y) + f(f(z))) = z + f(y) + f(f(x))
$$
for all positive integers $x, y, z$.