Particle

Time limit1sMemory limit256 MB

Summary

Decide whether a point bouncing off rectangle walls with fixed velocity reaches target A before target B or neither.

Level: Medium7 of 10
Topics: Number theory, Geometry
Solved: No attempts yet

Problem

A particle moves inside a W by H rectangle with constant velocity (a,b). It reflects off walls like light on a mirror, and at a corner it reverses direction. Starting from (x0,y0), decide whether it reaches A=(x1,y1) before B=(x2,y2), the opposite order, or neither ever.

Input

The first line has T test cases. Each test case is one line with W, H, x0, y0, x1, y1, x2, y2, a, b.

Output

Print A if A is reached first, B if B is reached first, or O if neither is reached.

Examples4

Example 1

Input

3
5 5 1 3 4 2 2 1 1 1
5 3 1 2 0 2 3 0 3 2
5 3 2 2 2 1 3 2 1 3

Expected output

A
B
O

Example 2
Input
```
1
6 6 1 1 5 5 3 3 1 1
```
Expected output
```
B
```
Example 3
Input
```
1
4 4 2 2 3 3 1 1 1 1
```
Expected output
```
A
```
Example 4
Input
```
1
7 5 3 2 6 4 1 1 2 3
```
Expected output
```
B
```