Airlines
--------
A leading airlines company has hired you to write a program that answers 
the following query: given a list of city locations (latitudes and longitudes) 
and a list of direct flights what is the minimum distance a passenger needs 
to fly to get from a given city to another?

To get from a city to another a passenger may either take a direct flight 
(if exists) or take a sequence of connecting flights (if there exists such 
a route).

Assume that if a passenger takes a direct flight from X to Y he never flies 
more than the geographical distance between X and Y.  The geographical 
distance between two locations X and Y is the length of the geodetic line 
segment connecting X and Y. The geodetic line segment between two points on 
a sphere is the shortest connecting curve lying entirely in the surface of 
the sphere. Assume that the Earth is a perfect sphere with a radius of 
exactly 6378-km and the value of p is approximately 3.141592653589793. 
Round the geographical distance between every pair of cities to the nearest 
integer.

Input
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The input may contain multiple test cases.

The first line of each test case contains three integers N (N <= 100), 
M (M <= 300) and Q (Q <= 10000) where N indicates the number of cities, 
M represents the number of direct flights and Q is the number of queries.

The next N lines contain the city list. The i-th of these N lines will 
contain a string ci followed by two real numbers lti and lni, representing 
the city name, its latitude and longitude respectively. The city name will 
be no longer than 50 characters and will not contain white-space characters. 
The latitude will be between -90 (South Pole) and +90 (North Pole). The 
longitude will be between -180 and +180 where negative numbers denote 
locations west of the meridian and positive numbers denote locations east 
of the meridian. (The meridian passes through Greenwich, London.)

The next M lines contain the direct flight list. The i-th of these M lines 
will contain two city names ai and bi indicating that there exists a direct 
flight from city ai to city bi. Be assured that both city names will occur 
in the city list.

The next Q lines contain the query list. The i-th of these Q lines will 
contain two city names ai and bi asking for the minimum distance a passenger 
needs to fly in order to get from city ai to city bi. Be assured that 
ai and bi are not equal and both city names will occur in the city list.

The input will terminate with three zeros form N, M and Q.
 
Output
------
For each test case in the input first output the test case number (starting 
from 1) as shown in the sample output. Then for each query in the input print 
a line giving the shortest distance (in km) a passenger needs to fly to get 
from the first city (ai) in the query to the second one (bi). If there 
exists no route form ai to bi, just print the line "no route exists".

Print a blank line between two consecutive test cases.

Sample Input
------------
2 1 1
Amsterdam 52.22 4.53
Toronto 43.41 79.38
Toronto Amsterdam
Toronto Amsterdam
2 1 1
Amsterdam 52.22 4.53
Toronto 43.41 79.38
Toronto Amsterdam
Amsterdam Toronto
3 2 1
Toronto 43.41 79.38
Teheran 35.45 51.45
Milan 45.27 9.10
Toronto Milan
Milan Teheran
Toronto Teheran
3 2 1
Amsterdam 52.22 4.53
Toronto 43.41 79.38
Teheran 35.45 51.45
Toronto Amsterdam
Amsterdam Teheran
Toronto Teheran
4 4 1
Amsterdam 52.22 4.53
Toronto 43.41 79.38
Teheran 35.45 51.45
Milan 45.27 9.10
Toronto Amsterdam
Amsterdam Teheran
Toronto Milan
Milan Teheran
Toronto Teheran
0 0 0

Sample Output
-------------
Case #1
5426 km

Case #2
no route exists

Case #3
9123 km

Case #4
9538 km

Case #5
9123 km