Time limit : 1 s | Memory limit : 32 mb |
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Submitted : 308 | Accepted : 135 |

There are many scenarios in which we need to know how fast an object is moving.
For example, the air-traffic controller in an airport would like to know the speed
of a descending plane and give warning if it's too fast or too slow. The speed
detection cameras have helped the police to catch the speeding drivers, and so
on.

In this problem, you are supposed to write a program to recognize the speed of
a moving object.

Assume that an object is moving on a two dimensional plane, with a constant speed
and direction. A camera is taking a shot every second. Assume the camera can record
the real situation of the plain - you don't need to worry about the view-angle.

The background is black and the only object is white, except some noise introduced
by the camera. Anyway, you can assume that the largest continuous white area is
always the object. There is always only one largest continuous white area in this
problem.

Given several pictures taken by the camera, calculate the speed of the object.
Output the speed of the object.

A picture is represented by a matrix which contains "." or "x"
only, where"." in the matrix means a black block and "x" means
a white block. There are at least two pictures.

Here are some detailed definitions:

The edge of each block is 1 mm (That is, each "." or "x" in
the matrix represents one 1mm*1mm block.)

"The speed of the object" is defined by the moving speed of the center
of the object as geometric center : .
"""
Anyway, all the objects are made of square blocks. The geometric center of a single
square block is the center of the block. So the geometric center of the object
can be calculated as: .
"""
Where (X[i],Y[i]) is the coordinates of the i-th box's center and N is the number
of blocks in the object.

A white area is a set of white blocks.

A white area is called "continuous white area" if and only if there
is always a path connecting two arbitrary blocks in this area. All the blocks
in the path are within this area. And each pair of sequential blocks along this
path shares a common edge.

The average speed is calculated as described in the physics textbook:
"""
where t is over all the discrete observation time from 0 to T-1. In this problem,
we define T as half of the number of observation points. (There are always even
number observation points in the test suit.) is defined as the observed position
of the object's center in time t, that is, calculated
"""
from the t-th image.

The positive direction of X axle is from the left to the right; the positive direction
of Y axle is from the top to the bottom.

Note that you may get different shape of an object in different pictures.

The picture size is at most 256*256. There are at most 256 pictures in one test case.

For each test case, output a line with two numbers which represent the speed of the object in X and Y direction respectively. The accuracy is up to 2 digits after decimal point. The speeds of the objects are in mm/s.

10 5 .........x .....xxx.x ....xxx... .....xxx.. x......... ---------- .........x .........x ...xxx.... ..xxx..... x..xxx.... ========== 0 0

-2.00 1.00