Problem Bee

Time Limit: 2 Seconds
Memory Limit: 65536 KB
Special Judge

**Background**

Imagine a perfectly formed honeycomb, spanning the infinite Cartesian plane.
It is an interlocking grid composed of congruent equilateral hexagons. One hexagon
is located so that its center is at the origin and has two corners on the X-axis.
A bee must be very careful about how it travels in order not to get lost in
the infinite plane. To get from an arbitrary point A to another arbitrary point
B, it will first head from A to the exact center of the hexagon in which A is
located. Then, it will travel in a straight line to the exact center of an adjacent
hexagon. It will move from center to adjacent center until it has reached the
hexagon containing point B. At the destination hexagon, it will move from the
center to point B. In all cases the bee will take a path of minimal distance
that obeys the rules. The figure below demonstrates one possible minimal path
from point A to point B.

**Input**

Input will be in the form of 5 floating point numbers per line. The first number
will be the length, in centimeters, of the sides of the hexagons. The next two
numbers will be the x and y coordinates of point A, followed by the x and y
coordinates for point B. The input will be terminated by a line containing five
zeroes. Neither point A nor point B will ever be exactly on a border between
hexagons.

**Output**

For each line of the input, output the minimum length of a path from point A
to point B, in centimeters, to the nearest .001 centimeters.

**Example**

**Input**

1.0 -3.2 2.2 3.3 0
9 1 4 5 1
0.1 .09 0 .21 0
0 0 0 0 0

**Output**
7.737
5.000
0.526

Source:

**Greater New York 2000**
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