Slant Drilling

A survey map marks an oil pocket at $(0, 0)$, exactly $0$ metres above sea level. The map draws the ground with contour lines. Each contour line is a closed polygon that does not cross itself, and it comes with two heights: the height of the ground immediately outside it and the height immediately inside it. The ground between contour lines is flat, so the height at a point is the inner height of the smallest contour line containing that point. A point that no contour line contains has the outer height of a contour line that no other contour line contains, and all of those contour lines have the same outer height.

A drill starts at a point of the ground surface and reaches the pocket in a straight line. That line does not have to be vertical. Starting from a point $(x, y)$ where the ground height is $h$, the drilled length is $\sqrt{x^2 + y^2 + h^2}$.

Find the shortest length that has to be drilled.

The first line contains one integer $N$, the number of contour lines.

Each of the next $N$ lines describes one contour line. The line begins with three integers $H_0$, $H_1$ and $M_i$ ($0 \le H_0, H_1 \le 10^6$, $3 \le M_i$). $H_0$ is the height of the ground immediately outside the contour line and $H_1$ is the height immediately inside it, both in metres. $M_i$ is the number of vertices of this contour line. The rest of the line holds $2 M_i$ integers, the coordinates $x_j$ and $y_j$ ($-10^6 \le x_j, y_j \le 10^6$) of the vertices in order along the contour line.

The vertices of one contour line are distinct and a contour line does not cross itself. No two contour lines touch, and no contour line passes through $(0, 0)$. If a contour line is contained in another one, its outer height equals the inner height of the containing contour line. The number of vertices over all contour lines is at most $10^5$.

Print the shortest drilled length, rounded to exactly six digits after the decimal point.