Let $p$ be a prime number, and $G=G L_{2}\left(\mathbb{F}_{p}\right)$, the group of $2 \times 2$ invertible matrices with entries in the field $\mathbb{F}_{p}$ of integers modulo $p$.

The group $G$ acts on $X=\mathbb{F}_{p} \cup\{\infty\}$ by Möbius transformations,

$$\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot z=\frac{a z+b}{c z+d}$$

(i) Show that given any distinct $x, y, z \in X$ there exists $g \in G$ such that $g \cdot 0=x$, $g \cdot 1=y$ and $g \cdot \infty=z$. How many such $g$ are there?

(ii) $G$ acts on $X \times X \times X$ by $g \cdot(x, y, z)=(g \cdot x, g \cdot y, g \cdot z)$. Describe the orbits, and for each orbit, determine its stabiliser, and the orders of the orbit and stabiliser.