Legacy Screensaver

On a very old operating system, a screensaver consists of two rectangles flying around the screen. The screen is $W$ pixels wide and $H$ pixels high. The origin is the top-left corner of the screen, the $x$-axis points right from the origin, and the $y$-axis points down.

Rectangle $i$ ($i = 1, 2$) is $w_i$ pixels wide and $h_i$ pixels high. Its top-left corner starts at $(x_i, y_i)$, and its initial direction of movement is $(\delta x_i, \delta y_i)$, where each of $\delta x_i$ and $\delta y_i$ is $-1$ or $1$. At the end of each second, the top-left corner of rectangle $i$ instantly changes by $(\delta x_i, \delta y_i)$.

When rectangle $i$ touches the left or right border of the screen, $\delta x_i$ changes sign before the next second. Likewise, when it touches the top or bottom border, $\delta y_i$ changes sign before the next second. If rectangle $i$ touches two borders at the same time, which can only happen at a corner of the screen, both $\delta x_i$ and $\delta y_i$ change sign.

The rectangles always stay fully within the screen. In short, collisions with the screen borders are perfectly elastic. Movement is still discrete, though: at the end of each second, each rectangle moves 1 pixel in both directions at once.

The two rectangles overlap when their intersection has positive area. Let $f(t)$ be the number of integers $\tau = 0, 1, \ldots, t - 1$ such that the rectangles overlap during second $\tau$ (second $0$ is the time before the rectangles start moving).

Find the limit of $\frac{f(t)}{t}$ as $t \rightarrow \infty$, as an irreducible fraction. It can be shown that this limit is rational.

The first line contains the number of test cases $T$ ($1 \le T \le 1000$). Each test case follows.

The first line of each test case contains two integers $W$ and $H$, the width and height of the screen ($3 \le W, H \le 4000$).

The next two lines describe the two rectangles. Each rectangle is given by six integers $w_i$, $h_i$, $x_i$, $y_i$, $\delta x_i$, $\delta y_i$, which are its width, height, top-left corner coordinates, and initial direction. The constraints are $1 \le w_i \le W - 2$, $1 \le h_i \le H - 2$, $0 < x_i < W - w_i$, $0 < y_i < H - h_i$, and $\delta x_i, \delta y_i \in$ $-1$ or $1$.

The sum of $W + H$ over all test cases does not exceed 8000.

For each test case, print a non-negative integer $p$ and a positive integer $q$ separated by a slash (/) with no spaces, so that the limit of $\frac{f(t)}{t}$ as $t \rightarrow \infty$ equals $\frac{p}{q}$. The fraction must be irreducible, so the greatest common divisor of $p$ and $q$ is 1.

In the second test case, the state of the rectangles during the first few seconds is shown in the pictures below. The rectangles overlap during seconds $\tau = 0$ and $\tau = 6$. So, for example, $f(8) = 2$.