355. Inside an angle with vertex $O$, there is a point $A$ such that $O A$ forms angles $\varphi$ and $\psi$ with the sides of the given angle. Find points $M$ and $N$ on the sides of the angle such that $\angle M A N=\beta(\varphi+\psi+\beta>\pi)$ and the area of the quadrilateral $O M A N$ is minimized.
355. Given the result of the previous problem and reasoning in the same way as in problem II. 351, we get that if $\varphi>90^{\circ}-\frac{\beta}{2}$ and $\psi>90^{\circ}-$
$-\frac{\beta}{2}$, then the quadrilateral of the smallest area exists and for it

$|M A|=|A N|$. If this condition is not met, then the desired quadrilateral degenerates (one of the points $M$ or $N$ coincides with vertex $O$).