E. Alley
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Igor, Ira, and Veronica go for a walk outside every day. They especially like to walk along the alley near the house. Veronica especially likes to walk in a stroller and look at everything around her. It is important for Igor and Ira that Veronica walks as long as possible in the fresh air; that's why, when walking with a stroller, Igor and Ira prefer to go in the shade, which is made by the crowns of trees, so the sun shines less on Veronica.

One day Igor thought about a serious question, what is the area of the shade of the trees? Igor, Ira, and Veronica live in the Mathematicians' district; there are $$$n$$$ trees growing in the alley. All the trees are located on one straight line, and each $$$i$$$-th tree is defined by a center coordinate $$$x_i$$$; also, every $$$i$$$-th tree has a perfect shadow in the form of a circle with a radius $$$r_i$$$. It is reliably known that the distance between $$$i$$$-th and $$$j$$$-th trees is not less than the radiuses $$$r_i$$$ and $$$r_j$$$.

Help Igor determine the shadow area that all the trees leave.

Input

In the first line, there is a given integer $$$n$$$ $$$(1\leq n\leq 10^3)$$$ – the amount of trees.

Then in each following line, there are given integers $$$x_i$$$, $$$r_i$$$ $$$(-10^3\leq x_i\leq 10^3, 1\leq r_i\leq 10^3)$$$ – coordinates of $$$i$$$-th tree and radius of its shadow.

It is guaranteed that all the trees have different coordinates. It is guaranteed that $$$|x_i-x_j|\geq max(r_i,r_j)$$$ is true for each $$$i$$$, $$$j$$$.

Output

Output the area of the shaded part of the alley with an accuracy of at least $$$4$$$ decimal places.

Example
Input
3
0 1
2 2
8 3
Output
42.57923071