Turning gears (hard)

Jano the mechanic has many gears of different sizes. He took $n$ of them and assembled a complicated machine. He wants to know whether gear number $n$ rotates when he turns gear number 1, and if so, how fast.

The gears are numbered 1 to $n$, and all of them lie in the same plane.

Gear $i$ is a circle with center $(x_i, y_i)$ and radius $r_i$. All coordinates are integers. The center of each circle lies strictly outside all other circles.

When two circles intersect, their gears are linked. A single point of contact counts as an intersection. The size of the overlap does not matter.

A rotation is described by a speed in full revolutions per second and a direction, either clockwise or counterclockwise. If two linked gears have radii $r_1$ and $r_2$, and the first turns at $v_1$ revolutions per second, the second turns in the opposite direction at $v_2 = \frac{v_1 r_1}{r_2}$ revolutions per second.

This equation applies to every linked pair in the machine. If a gear would have to turn at two different speeds or in two different directions, it does not rotate at all.

Jano turns gear number 1 clockwise at 1 revolution per second. How does gear $n$ rotate?

The first line contains an integer $t$, the number of test cases. Each test case is preceded by a blank line.

Each test case starts with $n$ ($2 \le n \le 1\,000$), the number of gears. The next $n$ lines describe gears $1$ to $n$. Line $i$ contains three integers $x_i$, $y_i$, and $r_i$. You may assume $0 \le x_i, y_i \le 50\,000$ and $1 \le r_i \le 1\,000$.

Each test case also satisfies these guarantees. There are at most 200 gears. All gears are directly or indirectly linked to gear 1. There are exactly $n-1$ pairs of directly linked gears.

For each test case, print one line with the speed of gear $n$ as a reduced fraction $p/q$, where the greatest common divisor of $p$ and $q$ is 1. Follow it with the direction, either clockwise or counterclockwise. If gear $n$ does not rotate, print does not rotate instead.

In the first test case, gears 1 and 2 touch at a single point, and that still counts as linked. The answer is printed as 1/1. The denominator is always printed, so 2/2 and 1 are both rejected.

In the second test case:

• Gear 1 turns clockwise at 1 revolution per second.
• Gear 3 is linked to gear 1, so it turns counterclockwise at 6/7 revolutions per second.
• Gears 2 and 4 are linked to gear 3, so both turn clockwise. They both have radius 5, so each turns at 6/5 revolutions per second.
• Gear 5 is linked to gear 2, so it turns counterclockwise. Gears 2 and 5 have the same radius, so gear 5 also turns at 6/5 revolutions per second.

The third test case shows one of the two reasons the answer can be does not rotate.