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Contest: Task: Related: TaskB TaskD

Score : $900$ points

Problem Statement

In the city of Nevermore, there are $10^8$ streets and $10^8$ avenues, both numbered from $0$ to $10^8-1$. All streets run straight from west to east, and all avenues run straight from south to north. The distance between neighboring streets and between neighboring avenues is exactly $100$ meters.

Every street intersects every avenue. Every intersection can be described by pair $(x, y)$, where $x$ is avenue ID and $y$ is street ID.

There are $N$ fountains in the city, situated at intersections $(X_i, Y_i)$. Unlike normal intersections, there's a circle with radius $10$ meters centered at the intersection, and there are no road parts inside this circle.

The picture below shows an example of how a part of the city with roads and fountains may look like.

1f931bf0c98ec6f07e612b0282cdb094.png

City governors don't like encountering more than one fountain while moving along the same road. Therefore, every street contains at most one fountain on it, as well as every avenue.

Citizens can move along streets, avenues and fountain perimeters. What is the shortest distance one needs to cover in order to get from intersection $(x_1, y_1)$ to intersection $(x_2, y_2)$?

Constraints

  • $0 \leq x_1, y_1, x_2, y_2 < 10^8$
  • $1 \leq N \leq 200,000$
  • $0 \leq X_i, Y_i < 10^8$
  • $X_i \neq X_j$ for $i \neq j$
  • $Y_i \neq Y_j$ for $i \neq j$
  • Intersections $(x_1, y_1)$ and $(x_2, y_2)$ are different and don't contain fountains.
  • All input values are integers.

Input

Input is given from Standard Input in the following format:

$x_1$ $y_1$ $x_2$ $y_2$
$N$
$X_1$ $Y_1$
$X_2$ $Y_2$
$:$
$X_N$ $Y_N$

Output

Print the shortest possible distance one needs to cover in order to get from intersection $(x_1, y_1)$ to intersection $(x_2, y_2)$, in meters. Your answer will be considered correct if its absolute or relative error doesn't exceed $10^{-11}$.

Sample Input 1

1 1 6 5
3
3 2
5 3
2 4

Sample Output 1

891.415926535897938

One possible shortest path is shown on the picture below. The path starts at the blue point, finishes at the purple point and follows along the red line.

c49e52b9b79003f61b8a6efa5dad8ba3.png

Sample Input 2

3 5 6 4
3
3 2
5 3
2 4

Sample Output 2

400.000000000000000
f9090ab734c89424c413f3b583376990.png

Sample Input 3

4 2 2 2
3
3 2
5 3
2 4

Sample Output 3

211.415926535897938
4b76416161f27cad20333a9ac636e00f.png