(2) (20 points) In the Cartesian coordinate system, a circle with center $C\left(t, \frac{2}{t}\right)$ passes through the origin $O$, and intersects the $x$-axis and $y$-axis at points $A$ and $B$ (different from the origin $O$).
(1) Prove that the area $S$ of $\triangle A O B$ is a constant;
(2) Suppose the line $l: y=-2 x+4$ intersects the circle $C$ at two distinct points $M$ and $N$, and $|O M|=|O N|$, find the standard equation of the circle $C$.