Circular Sectors

Bobo has drawn $n$ circular sectors on the plane. He would like to know the area of the union of all the circular sectors.

The input contains zero or more test cases, and is terminated by end-of-file. For each test case:

The first line contains an integer $n$, the number of circular sectors ($1 \leq n \leq 500$).

Each of the next $n$ lines contains five numbers $x\_i$, $y\_i$, $r\_i$, $s\_i$ and $\theta\_i$ ($-100 \le x\_i, y\_i \le 100$, $1 \le r\_i \le 100$, $0 \le s\_i \le 6$, $0.1 \le \theta\_i \le 6$). Here, ($x\_i, y\_i$) is the coordinate of the circle center, $r\_i$ is the radius of the circle, $s\_i$ is the starting angle in radians (counter-clockwise from the positive direction of the $x$ axis) and $\theta\_i$ is the central angle in radians (this means that the sector arc goes from angle $s\_i$ to angle $s\_i + \theta\_i$ where the angle is measured counter-clockwise from the positive direction of the $x$ axis). Also, $x\_i$, $y\_i$ and $r\_i$ are integers, and $s\_i$ and $\theta\_i$ are real numbers with exactly $3$ digits after the decimal point.

It is guaranteed that the sum of $n$ does not exceed $500$.

For each test case, output a real number denoting the answer. Your answer will be considered correct if its relative or absolute error doesn't exceed $10^{-6}$.

The image below shows the third test case.