4. Let the two branches of the hyperbola $xy=1$ be $C_{1}$ and $C_{2}$, and let the three vertices of the equilateral triangle $PQR$ lie on this hyperbola.
(1) Prove that $P$, $Q$, and $R$ cannot all lie on the same branch of the hyperbola;
(2) Suppose $P(-1,-1)$ is on $C_{2}$, and $Q$, $R$ are on $C_{1}$. Find the coordinates of vertices $Q$ and $R$.
(1997, National High School Mathematics Competition)
(Hint: (1) Grasp that the perpendicular bisectors of the three sides of an equilateral triangle all pass through the center of the triangle, and use proof by contradiction to derive a contradiction; (2) Note that the inclination angle of the line connecting point $P$ and the origin is $45^{\circ}$, thus, the inclination angles of $P Q$ and $P R$ can be determined (or by the symmetry of $P$ and $Q$). $(2+\sqrt{3}, 2-\sqrt{3}),(2-\sqrt{3}, 2+\sqrt{3})$.)