Problem: In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.

In the plan developed by the president Takahashi, the trains will run as follows:

* A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.
* In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.



According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.

Find the maximum possible distance that a train can cover in the run.

Constraints

* 1 \leq N \leq 100
* 1 \leq t_i \leq 200
* 1 \leq v_i \leq 100
* All input values are integers.

Input

Input is given from Standard Input in the following format:


N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N


Output

Print the maximum possible that a train can cover in the run.
Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

Examples

Input

1
100
30


Output

2100.000000000000000


Input

2
60 50
34 38


Output

2632.000000000000000


Input

3
12 14 2
6 2 7


Output

76.000000000000000


Input

1
9
10


Output

20.250000000000000000


Input

10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70


Output

20291.000000000000

Solution (in JAVA):

Solution:
import java.io.IOException;
import java.io.InputStream;
import java.util.Arrays;
import java.util.NoSuchElementException;
import java.util.function.IntUnaryOperator;
import java.util.function.LongUnaryOperator;
import static java.lang.Math.min;


public class Main {
    public static void main(String[] args) {
        FastScanner fsc = new FastScanner();
        int n = fsc.nextInt();
        int[] t = new int[n];
        for (int i = 0; i < n; i++) {
            t[i] = fsc.nextInt() * 2;
        }
        int[] v = new int[n];
        for (int i = 0; i < n; i++) {
            v[i] = fsc.nextInt();
        }
        int[] s = ArrayUtil.cumsum(t);
        int sum = s[n];
        double[] mv = new double[sum + 1];
        for (int i = 0; i <= sum; i++) {
            mv[i] = min(i * 0.5, (sum - i) * 0.5);
        }
        for (int i = 0; i < n; i++) {
            for (int j = 0; j <= s[i]; j++) {
                mv[s[i] - j] = min(mv[s[i] - j], v[i] + j * 0.5);
            }
            for (int j = s[i]; j < s[i + 1]; j++) {
                mv[j] = min(mv[j], v[i]);
            }
            for (int j = 0; j <= sum - s[i + 1]; j++) {
                mv[s[i + 1] + j] = min(mv[s[i + 1] + j], v[i] + j * 0.5);
            }
        }
        double ans = 0;
        for (int i = 0; i < sum; i++) {
            ans += (mv[i] + mv[i + 1]) * 0.25;
        }
        System.out.println(ans);
    }
}


class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    private boolean hasNextByte() {
        if (ptr < buflen) {
            return true;
        } else {
            ptr = 0;
            try {
                buflen = in.read(buffer);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (buflen <= 0) {
                return false;
            }
        }
        return true;
    }

    private int readByte() {
        if (hasNextByte()) {
            return buffer[ptr++];
        } else {
            return -1;
        }
    }

    private static boolean isPrintableChar(int c) {
        return 33 <= c && c <= 126;
    }

    public boolean hasNext() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr])) {
            ptr++;
        }
        return hasNextByte();
    }

    public String next() {
        if (!hasNext()) {
            throw new NoSuchElementException();
        }
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }

    public long nextLong() {
        if (!hasNext()) {
            throw new NoSuchElementException();
        }
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) {
            throw new NumberFormatException();
        }
        while (true) {
            if ('0' <= b && b <= '9') {
                n *= 10;
                n += b - '0';
            } else if (b == -1 || !isPrintableChar(b)) {
                return minus ? -n : n;
            } else {
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }

    public int nextInt() {
        long nl = nextLong();
        if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) {
            throw new NumberFormatException();
        }
        return (int) nl;
    }

    public double nextDouble() {
        return Double.parseDouble(next());
    }
}


class ArrayUtil {
    public static int[] map(IntUnaryOperator op, int[] a) {
        int[] b = new int[a.length];
        for (int i = 0; i < a.length; i++)
            b[i] = op.applyAsInt(a[i]);
        return b;
    }

    /**
     * s[i] is the sum of the range [0, i) in a.
     * 
     * @param a
     * @return cumulative sum array of a.
     */
    public static int[] cumsum(int[] a) {
        int n = a.length;
        int[] s = new int[n + 1];
        for (int i = 1; i <= n; i++)
            s[i] = s[i - 1] + a[i - 1];
        return s;
    }

    public static int[][] cumsum(int[][] a) {
        int n = a.length;
        int m = a[0].length;
        int[][] s = new int[n + 1][m + 1];
        for (int i = 1; i <= n; i++)
            for (int j = 1; j <= m; j++)
                s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i - 1][j - 1];
        return s;
    }

    public static void reverse(int[] a, int begin, int end) {
        for (int i = begin; i < begin + (end - begin) / 2; i++)
            swap(a, i, begin + end - i - 1);
    }

    public static void reverse(int[] a) {
        reverse(a, 0, a.length);
    }

    public static void revSort(int[] a) {
        Arrays.sort(a);
        reverse(a);
    }

    public static int[] nextPermutation(int[] a) {
        int[] ret = a.clone();
        int n = ret.length;
        for (int i = n - 1; i >= 1; i--) {
            if (ret[i] > ret[i - 1]) {
                int fail = n, pass = i;
                while (fail - pass > 1) {
                    int mid = pass + (fail - pass) / 2;
                    if (ret[mid] > ret[i - 1])
                        pass = mid;
                    else
                        fail = mid;
                }
                swap(ret, pass, i - 1);
                reverse(ret, i, n);
                return ret;
            }
        }
        return null;
    }

    public static int[] predPermutation(int[] a) {
        int[] ret = a.clone();
        int n = ret.length;
        for (int i = n - 1; i >= 1; i--) {
            if (ret[i] < ret[i - 1]) {
                int fail = n, pass = i;
                while (fail - pass > 1) {
                    int mid = pass + (fail - pass) / 2;
                    if (ret[mid] < ret[i - 1])
                        pass = mid;
                    else
                        fail = mid;
                }
                swap(ret, pass, i - 1);
                reverse(ret, i, n);
                return ret;
            }
        }
        return null;
    }

    public static void swap(int[] a, int u, int v) {
        int tmp = a[u];
        a[u] = a[v];
        a[v] = tmp;
    }

    public static int compare(int[] a, int[] b) {
        for (int i = 0; i < a.length; i++) {
            if (i >= b.length) {
                return -1;
            } else if (a[i] > b[i]) {
                return 1;
            } else if (a[i] < b[i]) {
                return -1;
            }
        }
        if (a.length < b.length) {
            return 1;
        } else {
            return 0;
        }
    }

    public static boolean equals(int[] a, int[] b) {
        return compare(a, b) == 0;
    }

    /*
     * IntArray _________________________ LongArray
     */
    public static long[] map(LongUnaryOperator op, long[] a) {
        long[] b = new long[a.length];
        for (int i = 0; i < a.length; i++)
            b[i] = op.applyAsLong(a[i]);
        return b;
    }

    /**
     * s[i] is the sum of the range [0, i) in a.
     * 
     * @param a
     * @return cumulative sum array of a.
     */
    public static long[] cumsum(long[] a) {
        int n = a.length;
        long[] s = new long[n + 1];
        for (int i = 1; i <= n; i++)
            s[i] = s[i - 1] + a[i - 1];
        return s;
    }

    public static long[][] cumsum(long[][] a) {
        int n = a.length;
        int m = a[0].length;
        long[][] s = new long[n + 1][m + 1];
        for (int i = 1; i <= n; i++)
            for (int j = 1; j <= m; j++)
                s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i - 1][j - 1];
        return s;
    }

    public static void reverse(long[] a, int begin, int end) {
        for (int i = begin; i < begin + (end - begin) / 2; i++)
            swap(a, i, begin + end - i - 1);
    }

    public static void reverse(long[] a) {
        reverse(a, 0, a.length);
    }

    public static void revSort(long[] a) {
        Arrays.sort(a);
        reverse(a);
    }

    public static int[] compress(long[] a) {
        int n = a.length;
        long[] sorted = a.clone();
        int[] compress = new int[n];
        Arrays.sort(sorted);
        for (int i = 0; i < n; i++) {
            int l = 0, r = n;
            while (r - l > 1) {
                int m = l + (r - l) / 2;
                if (sorted[m] <= a[i])
                    l = m;
                else
                    r = m;
            }
            compress[i] = l;
        }
        return compress;
    }

    public static void swap(long[] a, int u, int v) {
        long tmp = a[u];
        a[u] = a[v];
        a[v] = tmp;
    }

    public static int compare(long[] a, long[] b) {
        for (int i = 0; i < a.length; i++) {
            if (i >= b.length) {
                return -1;
            } else if (a[i] > b[i]) {
                return 1;
            } else if (a[i] < b[i]) {
                return -1;
            }
        }
        if (a.length < b.length) {
            return 1;
        } else {
            return 0;
        }
    }

    public static boolean equals(long[] a, long[] b) {
        return compare(a, b) == 0;
    }
}