Example 3 Real numbers $x, y, z$ satisfy
$$
x+y+z=4(\sqrt{x-5}+\sqrt{y-4}+\sqrt{z-3}) \text {. }
$$

Then $x=$ $\qquad$ ,$y=$ $\qquad$ ,$z=$ $\qquad$ .
(10th Five Sheep Cup Junior High School Mathematics Competition)