Description

You are given $N$ celestial bodies with their positions $(x_i, y_i, z_i)$ in space, initial velocities $(v_x, v_y, v_z)$, and masses $M_i$. It is known that, under the influence of the "Singer" civilization, time in the three-body world does not flow continuously (it updates every $0.01$ seconds). All celestial bodies are treated as point masses. Compute the positions of all celestial bodies at time $t$.

The universal gravitational constant $G$ is $6.67408 \times 10^{-11}$. In code, you may write:

#define G 6.67408e-11

If your answer’s relative error does not exceed $0.5\%$ compared to the standard answer, you will receive AC on that test point. In other words, you may choose how many decimal places to keep. The standard answer keeps $12$ decimal places. This problem uses SPJ to check correctness.

Input Format

The first line contains a positive integer $N$ and a nonnegative real number $t$, representing the number of celestial bodies and the time, respectively.

The next $N$ lines each contain $7$ real numbers $x_i, y_i, z_i, M_i, v_x, v_y, v_z$, representing each body’s position, mass, and initial velocity (the velocity components in the three directions).

Output Format

Output $N$ lines. The $i$-th line contains the coordinates $x_i, y_i, z_i$ of the $i$-th body at time $t$, separated by single spaces.

3 100
0 10 0 10000000 0.006207480877613 0 0
8.660254037844 -5 0 10000000 -0.003103740438807 -0.00537583613352 0
-8.660254037844 -5 0 10000000 -0.003103740438807 0.00537583613352 0

0.620349511786 9.980741705470 0.000000000000
8.333401109655 -5.527609289167 0.000000000000
-8.953750621441 -4.453132416303 0.000000000000

3 100
0 10 0 10000000 0.06207480877613 0 0
8.760254037844 -5 0 10000000 -0.03103740438807 -0.0537583613352 0
-8.660254037844 -5 0 10000000 -0.03103740438807 0.0537583613352 0

6.204092324054 9.982347016794 0.000000000000
5.642963405596 -10.364100727695 0.000000000000
-11.747055729651 0.381753710901 0.000000000000

Hint

Constraints: $3 \le N \le 30$, $0 \le t \le 100$, $-100 \le x_i, y_i, z_i \le 100$. $M_i$ is within the range of long long.

Translated by ChatGPT 5