Consider two opaque planes $$$z = z_1$$$ and $$$z = z_2$$$ ($$$z_1 \lt z_2$$$) in a three-dimensional Cartesian coordinate system. Each plane has a convex polygonal hole, denoted as $$$P_1$$$ and $$$P_2$$$ respectively, which allows light to pass through.
There is a point light source $$$L$$$ moving in the plane $$$z = z_0$$$ ($$$z_2 \lt z_0$$$), in a direction parallel to the $$$xOy$$$ plane. The light source starts at the initial position $$$(x_0, y_0, z_0)$$$ at time $$$t = 0$$$ and moves with a constant velocity $$$\boldsymbol{v} = (v_x, v_y, 0)$$$. Therefore, at any time $$$t$$$, the position of the light source is given by $$$(x_0 + v_x t, y_0 + v_y t, z_0)$$$.
For a point $$$A$$$ in the plane $$$z = 0$$$, define it as illuminated at time $$$t$$$ if and only if the segment $$$LA$$$ intersects the interiors (including boundaries) of both polygons $$$P_1$$$ and $$$P_2$$$. The illuminated area at time $$$t$$$, denoted as $$$f(t)$$$, is the area formed by all illuminated points in the plane $$$z = 0$$$.
Define the average illuminated area over the time period $$$[t_1, t_2]$$$, denoted as $$$\mathbb{E}[f(t) | t \sim U(t_1,t_2)]$$$, as the expected value of $$$f(t)$$$ over the interval $$$[t_1, t_2]$$$, assuming $$$t$$$ is a uniformly distributed random variable over $$$[t_1, t_2]$$$.
Given multiple time periods $$$[t_1, t_2]$$$, your task is to find the average illuminated area for each of these periods.
The first line contains a single integer $$$T$$$ ($$$1\le T\le 10^4$$$) indicating the number of test cases.
For each test case, the first line contains five integers $$$x_0$$$, $$$y_0$$$, $$$z_0$$$, $$$v_x$$$, $$$v_y$$$ ($$$-10^5 \le x_0, y_0 \le 10^5$$$, $$$1 \le z_0 \le 10^5$$$, $$$-10^3 \le v_x, v_y \le 10^3$$$), representing the initial position of the light source $$$(x_0, y_0, z_0)$$$ and its velocity vector $$$\boldsymbol{v} = (v_x, v_y, 0)$$$. It is guaranteed that $$$\boldsymbol{v} \neq (0, 0, 0)$$$.
The second line contains two integers $$$n_1$$$ and $$$z_1$$$ ($$$3 \le n_1 \le 10^5$$$, $$$1 \le z_1 \le 10^5$$$), indicating the number of vertices of polygon $$$P_1$$$ and the value of $$$z_1$$$. Each of the following $$$n_1$$$ lines contains two integers $$$x_i$$$ and $$$y_i$$$ ($$$-10^5 \le x_i, y_i \le 10^5$$$), describing the vertices of $$$P_1$$$ in counterclockwise order.
The following line contains two integers $$$n_2$$$ and $$$z_2$$$ ($$$3 \le n_2 \le 10^5$$$, $$$1 \le z_2 \le 10^5$$$), indicating the number of vertices of polygon $$$P_2$$$ and the value of $$$z_2$$$. Each of the following $$$n_2$$$ lines contains two integers $$$x_j$$$ and $$$y_j$$$ ($$$-10^5 \le x_j, y_j \le 10^5$$$), describing the vertices of polygon $$$P_2$$$ in counterclockwise order.
It is guaranteed that no three or more vertices are collinear for $$$P_1$$$ and $$$P_2$$$.
The following line contains an integer $$$q$$$ ($$$1 \le q \le 10^5$$$), indicating the number of queries. Each of the following $$$q$$$ lines contains two integers $$$t_1$$$ and $$$t_2$$$ ($$$0 \le t_1 \le t_2 \le 10^3$$$), representing a time period.
It is guaranteed that the sum of $$$n_1+n_2$$$ and the sum of $$$q$$$ over all test cases do not exceed $$$10^5$$$, respectively, and $$$z_1 \lt z_2 \lt z_0$$$.
For each query, output a real number representing the average illuminated area. Your answer will be considered correct only if the relative or absolute error between your answer and the correct answer does not exceed $$$10^{-4}$$$.
10 0 3 0 -14 11 03 03 21 24 20 01 01 10 130 101 21 1
0.450000000 1.125000000 2.250000000
For the example, the projections of convex polygons $$$P_1$$$ and $$$P_2$$$ onto the $$$xOy$$$ plane at $$$t=0$$$, and the movement of these projections, are illustrated below. Polygon $$$A_1B_1C_1D_1$$$ is the projection of polygon $$$P_1$$$, and polygon $$$A_2B_2C_2D_2$$$ is the projection of polygon $$$P_2$$$.
