Spherical Mirrors

There are $N$ spherical mirrors in three-dimensional space.

A laser is fired from $(0,0,0)$ in the direction $(u,v,w)$. The laser always travels in a straight line.

Suppose a laser fired from a point $I$ is reflected at a point $Q$ on a spherical mirror. Let $N$ be a point that lies outside the mirror and on the straight line joining the mirror's center and $Q$. Then the laser is reflected in a direction $R$ that satisfies the following condition:

(1) $R$ lies in the plane determined by $I$, $Q$, and $N$, and $\angle IQN = \angle NQR$.

Write a program that finds the position of the point at which the laser is reflected for the last time.

The first line contains the number of test cases $T$. For each test case, the first line contains the number of spherical mirrors $N$. The second line contains the direction $u$, $v$, $w$ in which the laser is fired, separated by spaces.

Each of the next $N$ lines contains four integers $x_i$, $y_i$, $z_i$, $r_i$ describing a spherical mirror whose center is $(x_i, y_i, z_i)$ and whose radius is $r_i$.

$1 \le N \le 100$

$-100 \le u, v, w \le 100$

$-100 \le x_i, y_i, z_i \le 100$

$5 \le r_i \le 30$

$u^2 + v^2 + w^2 > 0$

The distance between any two spherical mirrors is at least $0.1$. The point $(0,0,0)$ lies outside every spherical mirror and is at least $0.1$ away from each of them.

The laser is reflected by the spherical mirrors at least once and at most five times. Moreover, the angle of reflection $\theta$ (that is, $\angle NQR$, which equals $\angle IQN$) is always smaller than $85^\circ$.

For each test case, output the coordinates of the last reflection point, separated by spaces. Every coordinate is rounded at the fourth decimal place and printed to three decimal places.