Question 35
Find all triples of positive integers $(x, y, z)$, with $z \gt 1$, that simultaneously satisfy
$$
x \text{ divides } y+1, \quad y \text{ divides } z-1, \quad z \text{ divides } x^2+1.
$$
$$
x \text{ divides } y+1, \quad y \text{ divides } z-1, \quad z \text{ divides } x^2+1.
$$