Pulley Belts
------ -----

You have been asked to determine the length of a belt
that connects two pulleys from the XY coordinates of
the pulley centers and the radii of the pulleys.


Input
-----

For each of several test cases, first one line con-
taining just the test case name, and then one line
with six floating point numbers in the format:

	X1 Y1 R1 X2 Y2 R2

where the first pulley has center at (X1,Y1) and radius
R1, and the second pulley has center at (X2,Y2) and
radius R2.  

    -10,000 <= X1, Y1, X2, Y2 <= +10,000
    0 < R1, R2 <= 1,000
    distance between (X1,Y1) and (X2,Y2) > R1 + R2

Input ends with an end of file.


Output
------

For each test case, first a copy of the test case name
line from the input, and then one line with the belt
length as a floating point number with exactly 3 decimal
places.  The belt may NOT cross itself.


Sample Input
------ -----

-- SHOULD BE PI + 10 --
0 0 0.5 0 5 0.5
-- SHOULD BE PI + 10 --
8 -7.5 0.5 4 -4.5 0.5
-- SHOULD BE NEAR PI --
0 0 0.5 0 0.500002 0.000001
-- RANDOM SAMPLE 1 --
8.56 -0.23 3.289 -5.738 17.290 7.387
-- RANDOM SAMPLE 2 --
356 782 256 753 -219 74


Sample Output
------ ------

-- SHOULD BE PI + 10 --
13.142
-- SHOULD BE PI + 10 --
13.142
-- SHOULD BE NEAR PI --
3.142
-- RANDOM SAMPLE 1 --
79.512
-- RANDOM SAMPLE 2 --
3221.263


Tips
----

Input consists of lines read from the standard input.
Input ends when an end-of-file is read from the standard
input.  Output consists of lines written to the standard
output.  For example input/output code see

    ~/demos/solutions/summer/summer.EXT

where EXT is c, cc, or java.

To solve the problem you, of course, draw a diagram,
draw one or more extra lines, and apply the fundamental
ideas of plane geometry and trigonometry to compute a
bunch of line segment lengths and angles.  Thus you can
add debugging code to print these.  Some of what is out-
put when we run `make debug' on the judge's solution is:

-- SHOULD BE PI + 10 --
... |BD| = 5 CAB = 1.5708 ACD = 1.5708
13.142
-- SHOULD BE PI + 10 --
... |BD| = 5 CAB = 1.5708 ACD = 1.5708
13.142
-- SHOULD BE NEAR PI --
... |BD| = 0.00173205 CAB = 0.0034641 ACD = 3.13813
3.142

where A is the center of pulley 1, B is the point where
the belt is tangent to pulley 1, C is the center of
pulley 2, D is the point where the belt is tangent to
pulley 2, and ... denotes output we omitted because it
might tell you where we drew the extra lines.


File:	   pulleys.txt
Author:	   Bob Walton <walton@seas.harvard.edu>
Date:	   Tue Oct 15 20:33:08 EDT 2013

The authors have placed this file in the public domain;
they make no warranty and accept no liability for this
file.

RCS Info (may not be true date or author):

    $Author: walton $
    $Date: 2013/10/16 00:37:35 $
    $RCSfile: pulleys.txt,v $
    $Revision: 1.7 $