Orbit

In this universe every celestial body is a point of a plane, and the Earth is the point $(0, 0)$. The universe holds $N$ stars. Star $i$ sits at $(x_i, y_i)$ and its luminosity is $L_i$. No star ever moves, and the universe holds nothing besides the bodies named here.

Two deep space sensors will be placed on the orbit of radius $R$ around the Earth, so each sensor sits at distance $R$ from $(0, 0)$. A star of luminosity $L$ at distance $r$ from a point produces the brightness $F = \frac{L}{4 \pi r^2}$ at that point, and the brightness a sensor observes is the largest of the brightnesses that the $N$ stars produce at its position. The distance between $(x_i, y_i)$ and $(x_j, y_j)$ is $r = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}$. The sensors, the Earth and the stars are points of negligible size.

The two sensors must observe the same brightness, and among the placements that meet that condition the distance between the sensors must be as large as possible. Two points of the orbit are never farther apart than $2R$, and a pair at distance exactly $2R$ with equal observed brightness always exists, so the two sensors face each other across the Earth.

Write the first sensor at $(R \cos \theta, R \sin \theta)$ with $0 \le \theta < \pi$. The second sensor is then $(-R \cos \theta, -R \sin \theta)$. Several values of $\theta$ can satisfy the condition, so report the smallest one.

The first line has two integers $N$ and $R$, the number of stars and the radius of the orbit, separated by one space. Each of the next $N$ lines has three integers $x_i$, $y_i$, $L_i$: the coordinates and the luminosity of star $i$. $R$, $x_i$ and $y_i$ use the same unit.

Print four real numbers $X_1$, $Y_1$, $X_2$, $Y_2$ on one line, separated by single spaces: the position of the first sensor and the position of the second sensor. Round every number to exactly 6 digits after the decimal point. Here $\theta$ is the smallest angle in $[0, \pi)$ at which the two sensors observe the same brightness, $(X_1, Y_1) = (R \cos \theta, R \sin \theta)$ and $(X_2, Y_2) = (-X_1, -Y_1)$. Print a rounded value of zero as 0.000000, never as -0.000000.

• $1 \le N \le 100000$
• $1 \le R \le 10000$
• $|x_i| \le 10000$
• $|y_i| \le 10000$
• $1 \le L_i \le 10^9$
• Every number in the input is an integer.

Let $b(\theta)$ be the brightness observed at $(R \cos \theta, R \sin \theta)$ and let $d_i(\theta)$ be the distance from that point to star $i$. Then $b(\theta) = \frac{1}{4 \pi} \cdot \frac{1}{\min_i d_i(\theta)^2 / L_i}$, so comparing $b(\theta)$ with $b(\theta + \pi)$ is the same as comparing $\min_i \frac{d_i(\theta)^2}{L_i}$ with $\min_i \frac{d_i(\theta + \pi)^2}{L_i}$. The difference of the two minima is continuous in $\theta$ and takes opposite values at $\theta$ and at $\theta + \pi$, so it is zero for at least one $\theta$ in $[0, \pi)$.