P3268 [JLOI2016] XOR Area Union of Circles

Given $N$ circles in the 2D Cartesian coordinate system. It is known that any two circles have no intersection points; that is, two circles are either disjoint or one contains the other. Compute the XOR area union of these circles. The XOR area union is defined as follows: a region is counted if it lies inside an odd number of circles; otherwise, if it lies inside an even number of circles, it is not counted.

The first line contains a positive integer $N$, the number of circles. Each of the next $N$ lines contains $3$ integers $x,y,r$, representing a circle centered at $(x,y)$ with radius $r$. Constraints: $|x_i|,|y_i|\le 10^8$, $0

Output a single integer on one line: the result of the XOR area union of all circles divided by $\pi$.

Translated by ChatGPT 5