Problem:
Score : 600 points

Problem Statement

Takahashi wants to grill N pieces of meat on a grilling net, which can be seen as a two-dimensional plane. The coordinates of the i-th piece of meat are \left(x_i, y_i\right), and its hardness is c_i.

Takahashi can use one heat source to grill the meat. If he puts the heat source at coordinates \left(X, Y\right), where X and Y are real numbers, the i-th piece of meat will be ready to eat in c_i \times \sqrt{\left(X - x_i\right)^2 + \left(Y-y_i\right)^2} seconds.

Takahashi wants to eat K pieces of meat. Find the time required to have K or more pieces of meat ready if he put the heat source to minimize this time.

Constraints

All values in input are integers.

1 \leq N \leq 60

1 \leq K \leq N

-1000 \leq x_i , y_i \leq 1000

\left(x_i, y_i\right) \neq \left(x_j, y_j\right) \left(i \neq j \right)

1 \leq c_i \leq 100

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
\vdots
x_N y_N c_N

Output

Print the answer.

It will be considered correct if its absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

4 3
-1 0 3
0 0 3
1 0 2
1 1 40

Sample Output 1

2.4

If we put the heat source at \left(-0.2, 0\right), the 1-st, 2-nd, and 3-rd pieces of meat will be ready to eat within 2.4 seconds. This is the optimal place to put the heat source.

Sample Input 2

10 5
-879 981 26
890 -406 81
512 859 97
362 -955 25
128 553 17
-885 763 2
449 310 57
-656 -204 11
-270 76 40
184 170 16

Sample Output 2

7411.2252

Solution in PyPy3 (2.4.0) (judge verdict: Wrong Answer)

from itertools import combinations

def main():
    EPS = 1e-8

    N, K, *XYC = map(int, open(0).read().split())
    PC = [(complex(x, y), c) for x, y, c in zip(*[iter(XYC)] * 3)]


    def intersection(A, r, B, s):
        D = B - A
        d = abs(D)

        cos_t = (d * d + r * r - s * s) / (2 * d * r)
        sin_t = max(1 - cos_t * cos_t, 0.0) ** 0.5

        R = r * complex(cos_t, sin_t)
        D /= d

        return (
            A + D * R,
            A + D * R.conjugate(),
        )


    def is_ok(T):
        A = [p for p, _ in PC]
        for (p1, c1), (p2, c2) in combinations(PC, 2):
            A += intersection(p1, T / c1, p2, T / c2)

        return any(K <= sum(1 for p, c in PC if c * abs(a - p) < T + EPS) for a in A)


    ng, ok = 0, 100000
    for _ in range(100):
        md = (ng + ok) / 2
        if is_ok(md):
            ok = md
        else:
            ng = md

    print(ok)

main()