Segments

Little Luka likes to doodle in his notebook during chemistry class. You can think of a page of his notebook as the coordinate plane.

This time Luka drew $N$ segments parallel to the $y$-axis, all at different $x$ coordinates. The segments are numbered from $1$ to $N$ in increasing order of their $x$ coordinate.

Now Luka wants to draw one more segment $AB$ that crosses all $N$ segments, so that his doodle looks like a fish bone.

Luka does not know how to find such a segment, so he decided to pick one at random. He places point $A$ somewhere on segment $1$ and point $B$ somewhere on segment $N$.

During the random choice, every point of a segment is equally likely to be picked. More precisely, Luka picks $A$ and $B$ independently, each one uniformly from its own segment.

Write a program that computes the probability that segment $AB$ touches every one of the $N$ drawn segments.

The first line contains the number of segments $N$ ($2 \le N \le 1000$).

Each of the next $N$ lines contains three positive integers $x$, $y_1$, and $y_2$ ($1 \le x \le 1000$, $1 \le y_1 < y_2 \le 1000$), the coordinates of a segment parallel to the $y$-axis. The segment joins the points $(x, y_1)$ and $(x, y_2)$. The segments are given in order of their numbers, that is, in increasing order of $x$.

Print on a single line the probability that segment $AB$ touches every one of the $N$ drawn segments. The probability is a real number between $0$ and $1$ inclusive. Round it to $10$ digits after the decimal point and print exactly $10$ digits after the decimal point.