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 Rust (1.42.0) (judge verdict: Wrong Answer)

use proconio::*;
use std::ops::*;

const EPS: f64 = 1e-7;

#[derive(Clone, Copy)]
struct Point(f64, f64);

impl Point {
    fn new(x: f64, y: f64) -> Self {
        Point(x, y)
    }
    fn norm(&self) -> f64 {
        (self.0.powi(2) + self.1.powi(2)).sqrt()
    }
    fn scalar(&self, x: f64) -> Self {
        Point::new(self.0 * x, self.1 * x)
    }
}

impl Add for Point {
    type Output = Self;
    fn add(self, rhs: Self) -> Self::Output {
        Point::new(self.0 + rhs.0, self.1 + rhs.1)
    }
}
impl Sub for Point {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self::Output {
        Point::new(self.0 - rhs.0, self.1 - rhs.1)
    }
}

fn run() {
    input! {
        n: usize,
        k: usize,
        p: [(f64, f64, f64); n],
    }
    let p = p
        .into_iter()
        .map(|(x, y, c)| (Point::new(x, y), c))
        .collect::<Vec<_>>();
    if k == 1 {
        println!("0");
        return;
    }
    let mut ans = std::f64::MAX;
    for (i, a) in p.iter().enumerate() {
        for (j, b) in p.iter().enumerate().take(i) {
            // 2個の中間
            {
                let v = b.0 - a.0;
                let g = a.0 + v.scalar(b.1 / (a.1 + b.1));
                let t = (a.0 - g).norm() * a.1;
                let mut cnt = 2;
                for (_, p) in p.iter().enumerate().filter(|p| p.0 != i && p.0 != j) {
                    if (p.0 - g).norm() * p.1 < t + EPS {
                        cnt += 1;
                    }
                }
                if cnt >= k {
                    ans = ans.min(t);
                }
            }
            // 3個
            for (k, c) in p.iter().enumerate().take(j) {}
        }
    }
    println!("{:.7}", ans);
}

fn main() {
    run();
}