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[ Problem H | 1992 ACM East-Central Regional problem set | My ACM problem archive | my home page]

A particular part of the world has recently been experiencing a sudden and marked proliferation of pirate radio stations. So much so, in fact, that a group of concerned nations have asked for you assistance in pinpointing the source of the illegal transmissions. The respective governments have agreed to provide you with data form a set of mobile tracking units, each capable of determining its distance from a radio signal's point of broadcast. Armed with the information from three of these units and a two-dimensional map of the area, you are to write a program which will determine the distance and the direction of the particular pirate trasmitter from the nearest city in the area.

### Input

The input will consist of a two-dimensional map (using Cartesian coordinates) followed by the number of pirate signals that must be located and a corresponding number of pirate station datasets.

In themap, the data for each city will appear on a separate line. The data for each city will consist of a 15-character name followed by the city's x- and y-coordinates and a radius. The x- and y-coordinates give the city's geographical center while the radius gives the distance from the center of the city to the city limits (all cities map be considered to be circular for this problem). There will be no more than 50 cities and the last city on the map will be located at the origin (0.0, 0.0).

Following the map is a single integer (on a line by itself) which gives the number of pirate station datasets that follow. Each pirate station dataset will consist of nine readings on a single line, separated by spaces. The first three readings correspond to mobile tracking unit A, the next three to unit B, and the last three to unit C. The first two readings in a set of three are the x- and y- coordinates where the mobile tracking unit itself is located while the third reading is the distance the unit determined itself to be from the source of the pirate radio station signal. Note that all coordinate and distance values are given in kilometers and should be stored as real numbers. The locations of the three mobile tracking units reporting information on the pirate signal will not be collinear and will always be at least 10 kilometers apart. No city will be located more than 6000 kilometers away from the city at the origin. You may assume that only valid data will be supplied.

### Output

Your program should process each pirate station dataset using the information supplied by the three mobile tracking units and produce a summary line for each pirate signal which realates the source of the broadcast to the nearest town on the map. Specifically, your output should supply the distance from the illegal transmitter to the city limits (not the geographical center) of the nearest city, the principle compass direction of the illegal transmitter from the nearest city (`North`, `North East`, `East`, `South East`, `South`, `South West`, `West`, or ```North West```), and the name of that city.

The distance should be correct to within ± 0.02 of a kilometer and output to two digits of accuracy. To determine the compass direction of the pirate transmitter from the nearest city, each of the eight principle directions should be considered to be at the center of a 45° arc in that direction. Illegal transmitters which fall anywhere within a particular arc are considered to be in that direction. Thus if due North is at 0°, a transmitter that was at an angle between 22° and 67° (inclusive) would be considered `North East` of the city while a transmitter that was at an angle between 68° and 112° (inclusive) would be considered `East` of the city. When determining the direction of the pirate transmitter, an angle should be rounded off to the nearest whole degree.

Pirate transmitters that are located inside of a city's limits should be reported as originating from that city (no distance or directional information should be output in this case). All pirate transmitter reports should be preceded by a transmitter I. D. number (starting with transmitter 1). Refer to the following example for the acceptable phrasing of your output.

### Sample Input

```Pleasantville  937.8     1277.34     4.9
Avion          494.17    -483.06     12.7
Caniama        -803.24   1351.68     6.53
Kingstons Falls-554.45   -300.0      1.82
Otisburg       0.0       0.0         3.6
5
286.91 1538.6 676.989 1627.84 1450.3 1026.29 1140.4 451.47 705.152
-1021.9 -1064.67 2164.66 1089.23 0.0 1796.91 993.94 -1516.17 2882.78
200.0 -295.6 824.776 -683.94 -1118.64 998.19 474.16 1729.8 2145.37
-173.21 -700.2 695.308 -202.87 191.04 971.421 1407.9 525.65 1369.38
747.02 419.61 628.79 0.0 -582.19 645.469 -987.65 294.3 1300.12
```
According to the above dataset, the town of Caniama is located at (-803.24, 1351.68) and has a radius of 6.53 kilometers, and there are 5 pirate transmitters in the pirate station dataset. Furthermore, in the case of pirate station #2 the mobile tracking units are reporting the following information: Mobile Tracking Unit A, which is located at (-1021.9, -1064.67), has detected the transmission from 2164.66 kilometers away; Unit B, located at (1089.23, 0.0) isreceiving the broadcast from 1796.91 kilometers away; and Unit C, located at (993.94, -1516.17), is picking it up from 2882.78 kilometers away.

### Output for the Sample Input

```Pirate Transmitter 1 is located 354.65 kilometers South West of Pleasantville
Pirate Transmitter 2 is located 524.55 kilometers South East of Caniama
Pirate Transmitter 3 is located 182.27 kilometers North of Kingstons Falls
Pirate Transmitter 4 is located in Avion
Pirate Transmitter 5 is located 275.12 kilometers East of Otisburg
```