357. Find the radius of the largest circle that can be covered by three circles of radius $R$. Solve the problem in the general case when the radii are equal to $R_{1}, R_{2}, R_{3}$.
357. The radius of the largest circle is equal to the radius of the circumscribed circle around an equilateral triangle with side $2 R$, i.e., $2 R / \sqrt{3}$. (Let's take such a triangle and construct circles on its sides as diameters.) For any circle with a larger radius, if it were covered by the given circles, there would be an arc of at least $120^{\circ}$ covered by one circle, but such an arc contains a chord greater than $2 R$ - a contradiction.

In the general case, if there exists an acute triangle with sides $2 R_{1}, 2 R_{2}, 2 R_{3}$, then the radius of the circumscribed circle around it will be the desired one. In all other cases, the radius of the largest circle is equal to the largest of the numbers $R_{1}, R_{2}, R_{3}$.