A $(0_x, 1_y, 2_z)$-string is an infinite ternary string such that: If there is a $0$ in position $i$ then there is a $1$ in position $i + x$, if there is a $1$ in position $j$ then there is a $2$ in position $j + y$, if there is a $2$ in position $k$ then there is a $0$ in position $k + z$. For how many ordered triples of positive integers $(x, y, z)$ with $x, y, z \leq 100$ does there exist $(0_x, 1_y, 2_z)$-string?