A military maneuver is going on a two-dimensional Cartesian plane, and $$$n$$$ enemy targets are hiding somewhere on the battlefield, whose locations are known to our headquarters.
Our headquarters will airdrop a beacon in a rectangular region with sides parallel to the coordinate axes uniformly at random to expose all the enemy targets to our troops on the battlefield so that our troops can surround all the enemy targets. The bottom-left corner of the region is at coordinate $$$(x_l, y_l)$$$ while the top-right corner is at coordinate $$$(x_r, y_r)$$$.
After being dropped, the beacon will firstly receive two parameters $$$r$$$ and $$$R$$$ that satisfy $$$0 \le r \le R$$$ from our headquarters, then scan an annulus region, that is, the region lying between two concentric circles, where the radius of the inner circle is $$$r$$$ and that of the outer circle is $$$R$$$, and finally mark those enemy targets hiding in the scanned region (including the boundary).
However, the beacon can only scan a unit area in a unit of time, and the commander would like to know the expected minimum time for the beacon to scan the designated annulus region so that it can mark all the enemy targets.
The first line contains four integers $$$x_l, y_l, x_r$$$, and $$$y_r$$$ ($$$-10\,000 \le x_l, y_l, x_r, y_r \le 10\,000$$$, $$$x_l \lt x_r$$$, $$$y_l \lt y_r$$$), denoting the coordinates of the bottom-left and the top-right corners of the rectangular region where the beacon will be dropped.
The second line contains a single integer $$$n$$$ ($$$2 \le n \le 2\,000$$$), denoting the number of enemy targets on the battlefield.
Each of the following $$$n$$$ lines contains two integers $$$x$$$ and $$$y$$$ ($$$-10\,000 \le x,y \le 10\,000$$$), denoting an enemy target located at coordinate $$$(x,y)$$$.
It is guaranteed that no two enemy targets share the same locations.
Output a single real number, indicating the expected minimum time for the beacon to scan the designated annulus region.
Your answer is acceptable if its absolute or relative error does not exceed $$$10^{-6}$$$. Formally speaking, suppose that your output is $$$a$$$ and the jury's answer is $$$b$$$, your output is accepted if and only if $$$\frac{|a - b|}{\max(1, |b|)} \le 10^{-6}$$$.
0 0 2 2 2 3 1 1 3
8.377580409572781970
0 0 2 2 2 5 1 1 3
37.699111843077518863
In the first sample case, if the beacon is dropped to $$$(0.5, 1.5)$$$, the minimum time as well as the minimum area of the feasible annulus region is $$$4 \pi$$$. The expected minimum time when the beacon dropped in the rectangular region uniformly at random is $$$\frac{3}{8} \pi$$$.
Figure: The feasible annulus region for the beacon at $$$(0.5, 1.5)$$$
