Problem:
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.

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.

Sample Input 2

3 5 6 4
3
3 2
5 3
2 4

Sample Output 2

400.000000000000000

Sample Input 3

4 2 2 2
3
3 2
5 3
2 4

Sample Output 3

211.415926535897938

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

// package agc.agc019;

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.*;

public class Main {
    public static void main(String[] args) {
        InputReader in = new InputReader(System.in);
        PrintWriter out = new PrintWriter(System.out);

        int x1 = in.nextInt();
        int y1 = in.nextInt();
        int x2 = in.nextInt();
        int y2 = in.nextInt();
        int n = in.nextInt();
        int[][] fountains = new int[n][2];
        for (int i = 0; i < n ; i++) {
            for (int j = 0; j < 2 ; j++) {
                fountains[i][j] = in.nextInt();
            }
        }
        int minX = Math.min(x1, x2);
        int maxX = Math.max(x1, x2);
        int minY = Math.min(y1, y2);
        int maxY = Math.max(y1, y2);

        List<int[]> availablefountains = new ArrayList<>();
        for (int i = 0 ; i < n ; i++) {
            if (minX <= fountains[i][0] && fountains[i][0] <= maxX) {
                if (minY <= fountains[i][1] && fountains[i][1] <= maxY) {
                    availablefountains.add(new int[]{ Math.abs(x1-fountains[i][0]), Math.abs(y1-fountains[i][1])});
                }
            }
        }

        out.println(String.format("%.15f", solve(maxX-minX, maxY-minY, availablefountains)));
        out.flush();
    }

    static final double HALF = 20 * Math.PI / 2;
    static final double QUARTER = HALF / 2;
    static final double CORNER = 20;
    static final double HALF_LOSS = HALF - CORNER;
    static final double QUARTER_GAIN = QUARTER - CORNER;
    static final double LATTICE_SIZE = 100;

    private static double solve(int W, int H, List<int[]> fountains) {
        int n = fountains.size();
        Collections.sort(fountains, (u, v) ->  u[0] - v[0]);

        int[] a = new int[n];
        for (int i = 0; i < n ; i++) {
            a[i] = fountains.get(i)[1];
        }

        int x = lis(a);

        double base = (W + H) * LATTICE_SIZE;
        if (x == Math.min(W, H) + 1) {
            if (x == 1) {
                return base + HALF_LOSS;
            }
            return base + Math.min((x - 1) * QUARTER_GAIN + HALF_LOSS, (x - 2) * QUARTER_GAIN);
        }
        return base + x * QUARTER_GAIN;
    }

    /**
     * Computes longest increasing sequence.
     * If you want to get actual sequence, see dp[1..ans].
     * <p>
     * O(nlogn)
     *
     * @param values
     * @return
     */
    public static int lis(int[] values) {
        int n = values.length;
        long[] lval = new long[n];
        for (int i = 0; i < values.length; i++) {
            lval[i] = values[i] * 1000000000L + i;
        }
        int ans = 0;
        long[] dp = new long[n + 1];
        Arrays.fill(dp, Long.MIN_VALUE);
        for (int i = 0; i < n; i++) {
            int idx = Arrays.binarySearch(dp, 0, ans + 1, lval[i]);
            if (idx < 0) {
                idx = -idx - 2;
                dp[idx + 1] = lval[i];
                if (idx >= ans) {
                    ans++;
                }
            }
        }
        return ans;
    }


    public static void debug(Object... o) {
        System.err.println(Arrays.deepToString(o));
    }

    public static class InputReader {
        private static final int BUFFER_LENGTH = 1 << 12;
        private InputStream stream;
        private byte[] buf = new byte[BUFFER_LENGTH];
        private int curChar;
        private int numChars;

        public InputReader(InputStream stream) {
            this.stream = stream;
        }

        private int next() {
            if (numChars == -1) {
                throw new InputMismatchException();
            }
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0)
                    return -1;
            }
            return buf[curChar++];
        }

        public char nextChar() {
            return (char) skipWhileSpace();
        }

        public String nextToken() {
            int c = skipWhileSpace();
            StringBuilder res = new StringBuilder();
            do {
                res.append((char) c);
                c = next();
            } while (!isSpaceChar(c));
            return res.toString();
        }

        public int nextInt() {
            return (int) nextLong();
        }

        public long nextLong() {
            int c = skipWhileSpace();
            long sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = next();
            }
            long res = 0;
            do {
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                res *= 10;
                res += c - '0';
                c = next();
            } while (!isSpaceChar(c));
            return res * sgn;
        }

        public double nextDouble() {
            return Double.valueOf(nextToken());
        }

        int skipWhileSpace() {
            int c = next();
            while (isSpaceChar(c)) {
                c = next();
            }
            return c;
        }

        boolean isSpaceChar(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }
    }
}