Random signals

There are $N$ signal stations on the plane, numbered 1 to $N$. Station $i$ is at $(x_i, y_i)$ and broadcasts into a disk centered at that point. The radius and the strength of a broadcast depend on how much power the station receives.

Station $i$ can send $M_i$ different signals, named $(i, 1)$ through $(i, M_i)$. Signal $(i, j)$ has radius $r_{i,j}$ and strength $s_{i,j}$, and station $i$ always sends it once the power that reaches the station is at least $w_{i,j}$. So when station $i$ receives power $p$, every signal $(i, j)$ with $w_{i,j} \le p$ goes out, and each of those signals covers every point of the closed disk with center $(x_i, y_i)$ and radius $r_{i,j}$.

The strength delivered to a point $(x, y)$ is the largest strength among the signals that cover it. If no signal covers the point, the strength there is $0$. The government measures how far the signals spread with the value $A$ below.

$A = \sum_s s \times (\text{area of the region whose strength is exactly } s)$

Each day the government picks the power for station $i$ among the integers in $[L_i, U_i]$. Every integer is equally likely, and the stations are chosen independently of each other. Compute the expected value of $A$.

The first line contains the number of stations $N$ ($1 \le N \le 20$).

The descriptions of the $N$ stations follow in order, starting with station 1. The first line of each description contains five integers $x$, $y$ ($-10^4 \le x, y \le 10^4$), $M$ ($1 \le M \le 20$), $L$ and $U$ ($1 \le L \le U \le 10^4$), separated by spaces. Each of the next $M$ lines contains three integers $r$, $s$ and $w$ ($1 \le r, s, w \le 10^4$), separated by spaces.

Print the expected value of $A$ on one line, rounded to six digits after the decimal point. Print exactly six digits after the decimal point.

Every input satisfies two guarantees: the expected value is smaller than $10^6$, and it is at least $10^{-9}$ away from the boundary where rounding to six decimal digits switches between rounding up and rounding down.