Covering Circle

Description:

Input:

Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases. Descriptions of test cases follow.

The first line of each test case contains three integers $$$n$$$, $$$l$$$, $$$k$$$ ($$$2 \leq k \leq l \leq n \leq 50\,000$$$, $$$k \leq 20$$$).

Each of the next $$$n$$$ lines contains two integers $$$x_i$$$, $$$y_i$$$ ($$$-10^8 \leq x_i, y_i \leq 10^8$$$) — the coordinates of the point $$$A_i$$$. It is guaranteed that all points are distinct and were generated independently from uniform distribution on $$$[-10^8, 10^8] \times [-10^8, 10^8]$$$.

It is guaranteed that the sum of $$$n$$$ for all test cases does not exceed $$$50\,000$$$.

In the first test, points were not generated from the uniform distribution on $$$[-10^8, 10^8] \times [-10^8, 10^8]$$$ for simplicity. It is the only such test and your solution must pass it.

Hacks are disabled in this problem.

Output:

For each test case print a single real number — the answer to the problem.

Your answer will be considered correct if its absolute or relative error does not exceed $$$10^{-9}$$$. Formally let your answer be $$$a$$$, jury answer be $$$b$$$. Your answer will be considered correct if $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-9}$$$.

Sample Input:

4
3 2 2
0 0
0 4
3 0
5 4 3
1 1
0 0
2 2
0 2
2 0
8 3 2
0 3
1 0
0 2
1 1
0 1
1 2
0 0
1 3
5 4 4
1 1
-3 3
2 2
5 3
5 5

Sample Output:

2.00000000000000000000
1.00000000000000000000
0.50000000000000000000
4.00000000000000000000

Note:

In the first test case, we can select subsegment $$$A_1, A_2$$$ and a circle with center $$$(0, 2)$$$ and radius $$$2$$$.

In the second test case, we can select subsegment $$$A_1, A_2, A_3, A_4$$$ and a circle with center $$$(1, 2)$$$ and radius $$$1$$$.