Easter Eggs

Easter is coming, and the Easter Bunny decided to organise a chocolate egg hunt for the children. He hides two kinds of eggs: blue milk chocolate eggs and red dark chocolate eggs. The field holds blueberry plants and redberry plants, and the eggs go into those plants. A red egg has to go into a redberry plant and a blue egg into a blueberry plant.

The local government issued a permit for the event on the condition that exactly $N$ eggs are hidden. The government does not pay for the dental care of the local children, so the Bunny decides by himself how many eggs of each colour to hide.

By the yearly tradition, the first child to find both a red egg and a blue egg wins a big reward. To make the hunt as hard as possible, the Bunny wants to maximise the minimum distance between a red egg and a blue egg. To keep things fair, he hides at most one egg in each plant. Write a program that computes this value for him. Distances are Euclidean.

The first line contains three integers $N$, $B$, $R$: the number of eggs to hide, the number of blueberry plants, and the number of redberry plants.

Each of the next $B$ lines contains two integers $x$, $y$, the coordinates of a blueberry plant.

Each of the next $R$ lines contains two integers $x$, $y$, the coordinates of a redberry plant.

• $N \le 250$
• $B < N$, $R < N$
• $N \le B + R$
• $-10^4 \le x, y \le 10^4$
• The $B + R$ plants have pairwise distinct coordinates.

Print on the first line the largest minimum distance $D$ between a red egg and a blue egg that the Bunny can achieve. Print $D$ rounded to exactly six digits after the decimal point.

The picture shows the second example input. The eggs are hidden in the four filled plants.