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Contest: Task: Related: TaskA TaskC

Score : $200$ points

Problem Statement

There are $N$ people numbered $1, 2, \dots, N$ in the $xy$-plane. Person $i$ is at the coordinates $(X_i, Y_i)$.
$K$ of these people, Persons $A_1, A_2, \dots, A_K$, will receive lights of the same strength.
When a person at coordinates $(x, y)$ has a light of strength $R$, it lights up the interior of a circle of radius $R$ centered at $(x, y)$ (including the boundary).
Find the minimum strength of the lights needed for every person to be lit by at least one light.

Constraints

  • All values in input are integers.
  • $1 \le K < N \le 1000$
  • $1 \le A_1 < A_2 < \dots < A_K \le N$
  • $|X_i|,|Y_i| \le 10^5$
  • $(X_i,Y_i) \neq (X_j,Y_j)$, if $i \neq j$.

Input

Input is given from Standard Input in the following format:

$N$ $K$
$A_1$ $A_2$ $\dots$ $A_K$
$X_1$ $Y_1$
$X_2$ $Y_2$
$\vdots$
$X_N$ $Y_N$

Output

Print the answer as a real number.
Your output will be considered correct if its absolute or relative error from the judge's output is at most $10^{-5}$.

Sample Input 1

4 2
2 3
0 0
0 1
1 2
2 0

Sample Output 1

2.23606797749978969

This input contains four people. Among them, Persons $2$ and $3$ will have lights.
Every person will be lit by at least one light if $R \ge \sqrt{5} \approx 2.236068$.

Sample Input 2

2 1
2
-100000 -100000
100000 100000

Sample Output 2

282842.712474619009

Sample Input 3

8 3
2 6 8
-17683 17993
93038 47074
58079 -57520
-41515 -89802
-72739 68805
24324 -73073
71049 72103
47863 19268

Sample Output 3

130379.280458974768