Triangular Grid

There is an infinite grid in the Cartesian plane composed of isosceles triangles, with the following design:

prb785

A single triangle in this grid is a triangle with vertices on intersections of grid lines that has not other triangles inside it.

Given two points P and Q in the Cartesian plane you must determine how many single triangles are intersected by the segment $\overline{{PQ}}$ . A segment intersects a polygon if and only if there exists one point of the segment that lies inside the polygon (excluding its boundary).

Note that the segment $\overline{{PQ}}$ in the example intersects exactly six single triangles.

Specifications

Input

The problem input consists of several cases, each one defined in a line that contains six integer values B, H, x₁, y₁, x₂ and y₂ (1 ≤ B ≤ 200, 2 ≤ H ≤ 200, -1000 ≤ x₁, y₁, x₂, y₂ ≤ 1000), where:

B is the length of the base of all isosceles single triangles of the grid.
H is the height of all isosceles single triangles of the grid.
(x₁, y₁) is the point P, that defines the first extreme of the segment.
(x₂, y₂) is the point Q, that defines the second extreme of the segment.

You can suppose that neither P nor Q lie in the boundary of any single triangle, and that P ≠ Q.

The end of the input is specified by a line with the string "0 0 0 0 0 0".

Output

For each case in the input, print one line with the number of single triangles on the grid that are intersected by the segment $\overline{{PQ}}$ .

Problem information

Time Limit: 4 seconds
Memory Limit: 64 MB
Balls for the passed test: 50
Complexity: 75% 1/4

Example

Example input

100 120 -20 -100 160 160
10 8 5 5 5 4
10 8 5 5 10 5
10 8 5 5 10 10
0 0 0 0 0 0

Example output

← Burger Time?

Problems

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