Problem: There are N robots and M exits on a number line. The N + M coordinates of these are all integers and all distinct. For each i (1 \leq i \leq N), the coordinate of the i-th robot from the left is x_i. Also, for each j (1 \leq j \leq M), the coordinate of the j-th exit from the left is y_j.

Snuke can repeatedly perform the following two kinds of operations in any order to move all the robots simultaneously:

* Increment the coordinates of all the robots on the number line by 1.
* Decrement the coordinates of all the robots on the number line by 1.



Each robot will disappear from the number line when its position coincides with that of an exit, going through that exit. Snuke will continue performing operations until all the robots disappear.

When all the robots disappear, how many combinations of exits can be used by the robots? Find the count modulo 10^9 + 7. Here, two combinations of exits are considered different when there is a robot that used different exits in those two combinations.

Constraints

* 1 \leq N, M \leq 10^5
* 1 \leq x_1 < x_2 < ... < x_N \leq 10^9
* 1 \leq y_1 < y_2 < ... < y_M \leq 10^9
* All given coordinates are integers.
* All given coordinates are distinct.

Input

Input is given from Standard Input in the following format:


N M
x_1 x_2 ... x_N
y_1 y_2 ... y_M


Output

Print the number of the combinations of exits that can be used by the robots when all the robots disappear, modulo 10^9 + 7.

Examples

Input

2 2
2 3
1 4


Output

3


Input

3 4
2 5 10
1 3 7 13


Output

8


Input

4 1
1 2 4 5
3


Output

1


Input

4 5
2 5 7 11
1 3 6 9 13


Output

6


Input

10 10
4 13 15 18 19 20 21 22 25 27
1 5 11 12 14 16 23 26 29 30


Output

22

Solution (in JAVA):

Solution:
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.lang.reflect.Array;
import java.lang.reflect.Field;
import java.util.Arrays;
import java.util.Collection;
import java.util.HashSet;
import java.util.Iterator;
import java.util.NoSuchElementException;
import java.util.Objects;
import java.util.PrimitiveIterator;
import java.util.RandomAccess;
import java.util.TreeSet;
import java.util.function.IntBinaryOperator;
import java.util.function.IntFunction;
import java.util.function.IntPredicate;
import java.util.function.IntSupplier;
import java.util.function.IntUnaryOperator;
import java.util.function.LongBinaryOperator;
import java.util.function.LongUnaryOperator;


public class Main {
    public static void main(String[] args) throws Exception {
        Field f = System.out.getClass().getDeclaredField("autoFlush");
        f.setAccessible(true);
        f.set(System.out, false);
        execute(System.in, System.out);
    }

    public static void execute(InputStream in, OutputStream out) {
        ExtendedScanner s = new ExtendedScanner(in);
        Out o = new Out(out);
        solve(s, o);
        o.flush();
        o.close();
    }

    public static void solve(ExtendedScanner sc, Out out) {
        int n = sc.nextInt();
        int m = sc.nextInt();
        var x = sc.ints(n);
        var y = sc.ints(m);
        TreeSet<IntPair> lrset = new TreeSet<>((lr1, lr2) -> {
            if (lr1.fst == lr2.fst) {
                return lr1.snd - lr2.snd;
            } else {
                return lr1.fst - lr2.fst;
            }
        });
        for (int i = 0; i < n; i++) {
            int ri = IntArrays.lowerBound(y, x[i]);
            int li = ri - 1;
            if (li >= 0 && ri < m) {
                lrset.add(new IntPair(x[i] - y[li], y[ri] - x[i]));
            }
        }
        var lr = lrset.toArray(IntPair[]::new);
        // for (var p : lr) {
        //     out.writeln(p);
        // }
        n = lrset.size();
        if (n == 0) {
            out.writeln(1);
            return;
        }
        var p = IntArrayFactory.indexToInt(n, i -> i);
        ComparativeMergeSort.sort(p, (i, j) -> {
            if (lr[i].snd == lr[j].snd) {
                return lr[i].fst - lr[j].fst;
            } else {
                return lr[i].snd - lr[j].snd;
            }
        });
        var q = new int[n];
        q[n - 1] = n - 1;
        for (int i = n - 1; i > 0; i--) {
            if (lr[i].fst == lr[i - 1].fst) {
                q[i - 1] = q[i];
            } else {
                q[i - 1] = i - 1;
            }
        }
        var r = new int[n];
        r[p[n - 1]] = p[n - 1];
        for (int i = n - 1; i > 0; i--) {
            if (lr[p[i]].snd == lr[p[i - 1]].snd) {
                r[p[i - 1]] = r[p[i]];
            } else {
                r[p[i - 1]] = p[i - 1];
            }
        }
        // out.writeln(Arrays.toString(p));
        // out.writeln(Arrays.toString(q));
        // out.writeln(Arrays.toString(r));
        final int mod = 1000000007;
        var ma = new ModArithmetic(mod);
        LongSegmentTree t = new LongSegmentTree(n + 1, 0, ma::add);
        t.update(0, e -> 1);
        Deque<IntLongPair> dq = new Deque<>();
        int i = 0;
        while (i < n) {
            while (i < n) {
                long s = t.query(0, p[i] + 1);
                dq.addLast(new IntLongPair(p[i], s));
                // out.writeln(p[i] + " " + s);
                if (p[i] == r[p[i]]) break;
                i++;
            }
            while (dq.size() > 0) {
                var pair = dq.removeFirst();
                t.update(q[pair.fst] + 1, v -> ma.add(v, pair.snd));
            }
            i++;
        }
        out.writeln(t.query(0, n + 1));
    }
}


class LongSegmentTree {
    private final long[] dat;
    private final int n;
    private final long e;
    private final LongBinaryOperator f;
    public LongSegmentTree(final int n, final long e, final LongBinaryOperator f) {
        this.e = e;
        this.f = f;
        int k = 1; while (k < n) k <<= 1;
        this.n = k;
        this.dat = new long[k << 1];
        clear();
    }
    public void build(final long[] a) {
        System.arraycopy(a, 0, dat, n, a.length);
        for (int i = n - 1; i > 0; i--) dat[i] = f.applyAsLong(dat[i << 1 | 0], dat[i << 1 | 1]);
    }
    public void update(int i, final LongUnaryOperator g) {
        i += n;
        dat[i] = g.applyAsLong(dat[i]);
        for (i >>= 1; i > 0; i >>= 1) dat[i] = f.applyAsLong(dat[i << 1 | 0], dat[i << 1 | 1]);
    }
    public long get(int i) {
        return dat[i + n];
    }
    public long query(int l, int r) {
        long resL = e, resR = e;
        for (l += n, r += n; l < r; l >>= 1, r >>= 1) {
            if ((l & 1) != 0) resL = f.applyAsLong(resL, dat[l++]);
            if ((r & 1) != 0) resR = f.applyAsLong(dat[--r], resR);
        }
        return f.applyAsLong(resL, resR);
    }
    public void clear() {
        Arrays.fill(dat, e);
    }
}


/**
 * @author https://atcoder.jp/users/suisen
 */
final class IntLongPair {
    public int fst;
    public long snd;
    public IntLongPair(final int fst, final long snd) {this.fst = fst; this.snd = snd;}
    @Override @SuppressWarnings("all")
    public boolean equals(final Object o) {
        if (this == o) return true;
        if (!(o instanceof IntLongPair)) return false;
        final IntLongPair p = (IntLongPair) o;
        return this.fst == p.fst && this.snd == p.snd;
    }
    @Override
    public int hashCode() {return Objects.hash(this.fst, this.snd);}
    @Override
    public String toString() {return "(" + this.fst + ", " + this.snd + ")";}
}


/**
 * @author https://atcoder.jp/users/suisen
 * 
 * Implementation of deque for generic type, using Ring Buffer.
 */
@SuppressWarnings("unchecked")
class Deque<T> implements Iterable<T>, RandomAccess {
    static final int DEFAULT_CAPACITY = 1 << 6;
    T[] buf;
    int len = 1;
    int mask;
    int head = 0;
    int tail = 0;
    public Deque(int capacity) {
        if (capacity <= 0) {
            throw new IllegalArgumentException(
                String.format("Capacity %d is negative.", capacity)
            );
        }
        while (this.len < capacity) {
            this.len <<= 1;
        }
        this.mask = this.len - 1;
        this.buf = (T[]) new Object[len];
    }
    public Deque() {
        this(DEFAULT_CAPACITY);
    }
    public T getLast() {
        if (size() == 0) throw new NoSuchElementException();
        return buf[(tail - 1) & mask];
    }
    public T getFirst() {
        if (size() == 0) throw new NoSuchElementException();
        return buf[head];
    }
    public T get(int index) {
        if (index < 0 || index >= size()) {
            throw new IndexOutOfBoundsException(
                String.format("Index %d out of bounds for length %d.", index, size())
            );
        }
        return buf[(head + index) & mask];
    }
    public void addLast(T v) {
        if (size() == len) grow();
        buf[tail++ & mask] = v;
    }
    public void addFirst(T v) {
        if (size() == len) grow();
        buf[--head & mask] = v;
    }
    public T removeLast() {
        if (size() == 0) throw new NoSuchElementException();
        return buf[--tail & mask];
    }
    public T removeFirst() {
        if (size() == 0) throw new NoSuchElementException();
        return buf[head++ & mask];
    }
    public T pollLast() {
        if (size() == 0) return null;
        return removeLast();
    }
    public T pollFirst() {
        if (size() == 0) return null;
        return removeFirst();
    }
    public int size() {
        return tail - head;
    }
    public boolean isEmpty() {
        return size() == 0;
    }
    public void clear() {
        head = tail = 0;
    }
    public T[] toArray(Class<T> clazz) {
        T[] ret = (T[]) Array.newInstance(clazz, size());
        Iterator<T> it = iterator();
        Arrays.setAll(ret, i -> it.next());
        return ret;
    }
    private void grow() {
        T[] newBuf = (T[]) new Object[len << 1];
        head &= mask;
        tail &= mask;
        int len1 = len - head;
        int len2 = head;
        System.arraycopy(buf, head, newBuf, 0, len1);
        System.arraycopy(buf, 0, newBuf, len1, len2);
        this.head = 0;
        this.tail = this.len;
        this.len <<= 1;
        this.mask = this.len - 1;
        this.buf = newBuf;
    }
    public Iterator<T> iterator() {
        return new Iterator<T>(){
            int it = head;
            public boolean hasNext() {return it < tail;}
            public T next() {return buf[it++ & mask];}
        };
    }
    public Iterator<T> descendingIterator() {
        return new Iterator<T>(){
            int it = tail;
            public boolean hasNext() {return it > head;}
            public T next() {return buf[--it & mask];}
        };
    }

    /***************************** DEBUG *********************************/

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        sb.append('[');
        Iterator<T> it = iterator();
        while (it.hasNext()) {
            sb.append(it.next().toString());
            if (it.hasNext()) sb.append(',');
        }
        sb.append(']');
        return sb.toString();
    }
}


/**
 * @author https://atcoder.jp/users/suisen
 */
final class IntPair {
    public int fst, snd;
    public IntPair(final int fst, final int snd) {this.fst = fst; this.snd = snd;}
    @Override @SuppressWarnings("all")
    public boolean equals(final Object o) {
        if (this == o) return true;
        if (!(o instanceof IntPair)) return false;
        final IntPair p = (IntPair) o;
        return this.fst == p.fst && this.snd == p.snd;
    }
    @Override
    public int hashCode() {
        int hash = 1;
        hash = hash * 31 + fst;
        hash = hash * 31 + snd;
        return hash;
    }
    @Override
    public String toString() {return "(" + this.fst + ", " + this.snd + ")";}
}


/**
 * @author https://atcoder.jp/users/suisen
 */
final class ExtendedScanner extends BasicScanner {
    public ExtendedScanner(InputStream in) {super(in);}
    public int[] ints(final int n) {
        final int[] a = new int[n];
        Arrays.setAll(a, $ -> nextInt());
        return a;
    }
    public int[] ints(final int n, final IntUnaryOperator f) {
        final int[] a = new int[n];
        Arrays.setAll(a, $ -> f.applyAsInt(nextInt()));
        return a;
    }
    public int[][] ints(final int n, final int m) {
        final int[][] a = new int[n][];
        Arrays.setAll(a, $ -> ints(m));
        return a;
    }
    public int[][] ints(final int n, final int m, final IntUnaryOperator f) {
        final int[][] a = new int[n][];
        Arrays.setAll(a, $ -> ints(m, f));
        return a;
    }
    public long[] longs(final int n) {
        final long[] a = new long[n];
        Arrays.setAll(a, $ -> nextLong());
        return a;
    }
    public long[] longs(final int n, final LongUnaryOperator f) {
        final long[] a = new long[n];
        Arrays.setAll(a, $ -> f.applyAsLong(nextLong()));
        return a;
    }
    public long[][] longs(final int n, final int m) {
        final long[][] a = new long[n][];
        Arrays.setAll(a, $ -> longs(m));
        return a;
    }
    public long[][] longs(final int n, final int m, final LongUnaryOperator f) {
        final long[][] a = new long[n][];
        Arrays.setAll(a, $ -> longs(m, f));
        return a;
    }
    public char[][] charArrays(final int n) {
        final char[][] c = new char[n][];
        Arrays.setAll(c, $ -> nextChars());
        return c;
    }
    public char[][] charArrays(final int n, final int m) {
        final char[][] c = new char[n][];
        Arrays.setAll(c, $ -> nextChars(m));
        return c;
    }
    public double[] doubles(final int n) {
        final double[] a = new double[n];
        Arrays.setAll(a, $ -> nextDouble());
        return a;
    }
    public double[][] doubles(final int n, final int m) {
        final double[][] a = new double[n][];
        Arrays.setAll(a, $ -> doubles(m));
        return a;
    }
    public String[] strings(final int n) {
        final String[] s = new String[n];
        Arrays.setAll(s, $ -> next());
        return s;
    }
    public String[] strings(final int n, final int m) {
        final String[] s = new String[n];
        Arrays.setAll(s, $ -> next(m));
        return s;
    }
}



/**
 * @author https://atcoder.jp/users/suisen
 */
final class IntArrayList implements Iterable<Integer> {
    private int[] a;
    private int tail = 0;
    private static final int DEFAULT_SIZE = 64;
    public IntArrayList(final int capacity) {this.a = new int[Math.max(1, capacity)];}
    public IntArrayList() {this(DEFAULT_SIZE);}
    public void add(final int v) {
        if (tail == a.length) grow();
        a[tail++] = v;
    }
    public int removeLast() {return a[--tail];}
    public int get(final int i) {
        if (i >= tail) throw new IndexOutOfBoundsException("Index " + i + " out of bounds for length " + size());
        return a[i];
    }
    public void set(final int i, final int v) {
        if (i >= tail) throw new IndexOutOfBoundsException("Index " + i + " out of bounds for length " + size());
        a[i] = v;
    }
    private void grow() {
        final int[] b = new int[a.length << 1];
        System.arraycopy(a, 0, b, 0, a.length);
        a = b;
    }
    public int size() {return tail;}
    public void clear() {tail = 0;}
    public void reverse(final int begin, final int end) {IntArrays.reverse(a, begin, end);}
    public void reverse() {IntArrays.reverse(a, 0, tail);}
    public int[] toArray() {
        final int[] ret = new int[tail];
        System.arraycopy(a, 0, ret, 0, tail);
        return ret;
    }
    public void sort() {Arrays.sort(a, 0, tail);}
    public boolean addIf(final int v, final IntPredicate p) {
        if (p.test(v)) {add(v); return true;}
        return false;
    }
    public PrimitiveIterator.OfInt iterator() {return new IntArrayListIterator();}
    private class IntArrayListIterator implements PrimitiveIterator.OfInt {
        private int i = 0;
        public boolean hasNext() {return i < tail;}
        public int nextInt() {return a[i++];}
    }
}

@FunctionalInterface
interface LongComparator {
    public int compare(long i, long j);
    public default boolean eq(final long i, final long j) {return compare(i, j) == 0;}
    public default boolean ne(final long i, final long j) {return compare(i, j) != 0;}
    public default boolean gt(final long i, final long j) {return compare(i, j) > 0;}
    public default boolean ge(final long i, final long j) {return compare(i, j) >= 0;}
    public default boolean lt(final long i, final long j) {return compare(i, j) < 0;}
    public default boolean le(final long i, final long j) {return compare(i, j) <= 0;}
}


final class ComparativeMergeSort {
    private static int INSERTION_SORT_THRESHOLD = 60;
    public static void sort(final int[] a, final IntComparator comparator) {sort(a, 0, a.length, comparator);}
    public static void sort(final int[] a, final int begin, final int end, final IntComparator comparator) {
        for (int i = begin;;) {
            final int j = i + INSERTION_SORT_THRESHOLD;
            if (j < end) ComparativeInsertionSort.sort(a, i, j, comparator);
            else {ComparativeInsertionSort.sort(a, i, end, comparator); break;}
            i = j;
        }
        final int len = end - begin;
        final int[] work = new int[len];
        for (int block = INSERTION_SORT_THRESHOLD; block <= len; block <<= 1) {
            final int twoBlocks = block << 1;
            for (int from = begin, max = end - block; from < max; from += twoBlocks) {
                final int mid = from + block, to = Math.min(from + twoBlocks, end);
                System.arraycopy(a, from, work, 0, block);
                for (int i = from, wi = 0, ti = mid;; i++) {
                    if (ti == to) {System.arraycopy(work, wi, a, i, block - wi); break;}
                    if (comparator.gt(work[wi], a[ti])) a[i] = a[ti++];
                    else {a[i] = work[wi++]; if (wi == block) break;}
                }
            }
        }
    }
    public static void sort(final long[] a, final LongComparator comparator) {sort(a, 0, a.length, comparator);}
    public static void sort(final long[] a, final int begin, final int end, final LongComparator comparator) {
        for (int i = begin;;) {
            final int j = i + INSERTION_SORT_THRESHOLD;
            if (j < end) ComparativeInsertionSort.sort(a, i, j, comparator);
            else {ComparativeInsertionSort.sort(a, i, end, comparator); break;}
            i = j;
        }
        final int len = end - begin;
        final long[] work = new long[len];
        for (int block = INSERTION_SORT_THRESHOLD; block <= len; block <<= 1) {
            final int twoBlocks = block << 1;
            for (int from = begin, max = end - block; from < max; from += twoBlocks) {
                final int mid = from + block, to = Math.min(from + twoBlocks, end);
                System.arraycopy(a, from, work, 0, block);
                for (int i = from, wi = 0, ti = mid;; i++) {
                    if (ti == to) {System.arraycopy(work, wi, a, i, block - wi); break;}
                    if (comparator.gt(work[wi], a[ti])) a[i] = a[ti++];
                    else {a[i] = work[wi++]; if (wi == block) break;}
                }
            }
        }
    }
}


/**
 * @author https://atcoder.jp/users/suisen
 */
class BasicScanner {
    private final InputStream in;
    private final byte[] buf = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;
    public BasicScanner(InputStream in) {this.in = in;}
    private boolean hasNextByte() {
        if (ptr < buflen) return true;
        ptr = 0;
        try {buflen = in.read(buf);}
        catch (final IOException e) {e.printStackTrace();}
        return buflen > 0;
    }
    private int readByte() {return hasNextByte() ? buf[ptr++] : -1;}
    public boolean hasNext() {
        while (hasNextByte() && !(33 <= buf[ptr] && buf[ptr] <= 126)) ptr++;
        return hasNextByte();
    }
    private StringBuilder nextSequence() {
        if (!hasNext()) throw new NoSuchElementException();
        final StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (33 <= b && b <= 126) {sb.appendCodePoint(b); b = readByte();}
        return sb;
    }
    public char nextChar() {
        return (char) readByte();
    }
    public String next() {
        return nextSequence().toString();
    }
    public String next(int len) {
        return new String(nextChars(len));
    }
    public char[] nextChars() {
        final StringBuilder sb = nextSequence();
        int l = sb.length();
        char[] dst = new char[l];
        sb.getChars(0, l, dst, 0);
        return dst;
    }
    public char[] nextChars(int len) {
        if (!hasNext()) throw new NoSuchElementException();
        char[] s = new char[len];
        int i = 0;
        int b = readByte();
        while (33 <= b && b <= 126 && i < len) {s[i++] = (char) b; b = readByte();}
        if (i != len) throw new NoSuchElementException(String.format("length %d is longer than the sequence.", len));
        return s;
    }
    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 = n * 10 + b - '0';
            else if (b == -1 || !(33 <= b && b <= 126)) return minus ? -n : n;
            else throw new NumberFormatException();
            b = readByte();
        }
    }
    public int nextInt() {return Math.toIntExact(nextLong());}
    public double nextDouble() {return Double.parseDouble(next());}
}


/**
 * @author https://atcoder.jp/users/suisen
 */
class Out {
    private final OutputStream out;
    private byte[] buf = new byte[1024];
    private int ptr = 0;
    private static final int AUTO_FLUSH_THRETHOLD = 1 << 17;

    public Out(OutputStream out) {
        this.out = out;
    }

    public void flush() {
        try {
            out.write(buf, 0, ptr);
            out.flush();
            ptr = 0;
        } catch (IOException e) {e.printStackTrace();}
    }

    public void close() {
        try {out.close();} catch (IOException e) {e.printStackTrace();}
    }

    public Out writeln() {return write('\n');}
    public Out writeln(Object o) {return write(o).write('\n');}
    public Out writeln(String s) {return write(s).write('\n');}
    public Out writeln(char[] c) {return write(c).write('\n');}
    public Out writeln(char   c) {return write(c).write('\n');}
    public Out writeln(byte   b) {return write(b).write('\n');}
    public Out writeln(int    x) {return write(x).write('\n');}
    public Out writeln(long   x) {return write(x).write('\n');}
    public Out writeln(double d) {return write(d).write('\n');}

    public Out writeSpace() {return write(' ');}
    
    public Out write(Object o) {
        return write(o.toString());
    }

    public Out write(String s) {
        try {
            Field strValueField = s.getClass().getDeclaredField("value");
            strValueField.setAccessible(true);
            byte[] b = (byte[]) strValueField.get(s);
            int l = s.length();
            if (l > AUTO_FLUSH_THRETHOLD) {
                flush();
                int i = 0;
                while (i + AUTO_FLUSH_THRETHOLD < l) {
                    out.write(b, i, AUTO_FLUSH_THRETHOLD);
                    out.flush();
                    i += AUTO_FLUSH_THRETHOLD;
                }
                ensureCapacity(l - i);
                System.arraycopy(b, i, buf, 0, l - i);
                ptr = l - i;
            } else {
                ensureCapacity(ptr + l);
                System.arraycopy(b, 0, buf, ptr, l);
                ptr += l;
            }
        } catch (Exception e) {e.printStackTrace();}
        return this;
    }

    public Out write(char[] c) {
        int l = c.length;
        if (l > AUTO_FLUSH_THRETHOLD) {
            flush();
            ensureCapacity(AUTO_FLUSH_THRETHOLD);
            int i = 0;
            while (i + AUTO_FLUSH_THRETHOLD < l) {
                for (int j = 0; j < AUTO_FLUSH_THRETHOLD; j++) {
                    buf[ptr++] = (byte) c[i + j];
                }
                flush();
                i += AUTO_FLUSH_THRETHOLD;
            }
            for (; i < l; i++) {
                buf[ptr++] = (byte) c[i];
            }
        } else {
            ensureCapacity(ptr + l);
            for (char ch : c) buf[ptr++] = (byte) ch;
        }
        return this;
    }

    public Out write(char c) {
        ensureCapacity(ptr + 1);
        buf[ptr++] = (byte) c;
        return this;
    }

    public Out write(byte b) {
        ensureCapacity(ptr + 1);
        buf[ptr++] = b;
        return this;
    }

    public Out write(int x) {
        if (x == 0) {
            ensureCapacity(ptr + 1);
            buf[ptr++] = '0';
            return this;
        }
        int d = stringSize(x);
        ensureCapacity(ptr + d);
        if (x < 0) {
            buf[ptr++] = '-';
            x = -x;
            d--;
        }
        int j = ptr + d; 
        while (x > 0) {
            buf[--j] = (byte) ('0' + (x % 10));
            x /= 10;
        }
        ptr += d;
        return this;
    }

    public Out write(long x) {
        if (x == 0) {
            ensureCapacity(ptr + 1);
            buf[ptr++] = '0';
            return this;
        }
        int d = stringSize(x);
        ensureCapacity(ptr + d);
        if (x < 0) {
            buf[ptr++] = '-';
            x = -x;
            d--;
        }
        int j = ptr + d; 
        while (x > 0) {
            buf[--j] = (byte) ('0' + (x % 10));
            x /= 10;
        }
        ptr += d;
        return this;
    }

    public Out write(double d) {
        return write(Double.toString(d));
    }

    private void ensureCapacity(int cap) {
        if (cap > AUTO_FLUSH_THRETHOLD) {
            flush();
        }
        if (cap >= buf.length) {
            int newLength = buf.length;
            while (newLength < cap) newLength <<= 1;
            byte[] newBuf = new byte[newLength];
            System.arraycopy(buf, 0, newBuf, 0, buf.length);
            buf = newBuf;
        }
    }
    private static int stringSize(long x) {
        int d = 1;
        if (x >= 0) {d = 0; x = -x;}
        long p = -10;
        for (int i = 1; i < 19; i++, p *= 10) if (x > p) return i + d;
        return 19 + d;
    }
    private static int stringSize(int x) {
        int d = 1;
        if (x >= 0) {d = 0; x = -x;}
        int p = -10;
        for (int i = 1; i < 10; i++, p *= 10) if (x > p) return i + d;
        return 10 + d;
    }
}



/**
 * @author https://atcoder.jp/users/suisen
 * 
 * 1. DESTRUCTIVE methods for int arrays.
 * 2. methods that receives arrays and return some results (except for int arrays).
 */
final class IntArrays {
    private IntArrays(){}
    public static void swap(final int[] a, final int u, final int v) {
        final int tmp = a[u]; a[u] = a[v]; a[v] = tmp;
    }
    public static void reverse(final int[] a, int begin, int end) {
        while (end - begin > 1) swap(a, begin++, --end);
    }
    public static void reverse(final int[] a) {reverse(a, 0, a.length);}
    public static void sortDescending(final int[] a) {Arrays.sort(a); reverse(a);}
    public static int fold(final int[] a, final IntBinaryOperator op) {
        int ret = a[0]; for (int i = 1; i < a.length; i++) ret = op.applyAsInt(ret, a[i]);
        return ret;
    }
    public static int sum(final int[] a) {return fold(a, Integer::sum);}
    public static int max(final int[] a) {return fold(a, Integer::max);}
    public static int min(final int[] a) {return fold(a, Integer::min);}
    public static IntPair maxWithIndex(final int[] a) {
        int max = max(a); return new IntPair(find(a, max), max);
    }
    public static IntPair minWithIndex(final int[] a) {
        int min = min(a); return new IntPair(find(a, min), min);
    }
    public static int find(final int[] a, final int v) {
        for (int i = 0; i < a.length; i++) if (a[i] == v) return i;
        return -1;
    }
    public static void map(final int[] a, final IntUnaryOperator op) {Arrays.setAll(a, i -> op.applyAsInt(a[i]));}
    public static int filter(final int[] src, final int[] dst, final IntPredicate p) {
        int idx = 0;
        for (final int e : src) if (p.test(e)) dst[idx++] = e;
        return idx;
    }
    public static int filterIndex(final int[] dst, final int beginIndex, final int endIndex, final IntPredicate p) {
        int idx = 0;
        for (int i = beginIndex; i < endIndex; i++) if (p.test(i)) dst[idx++] = i;
        return idx;
    }
    public static int filterIndex(final int dst[], final int endIndex, final IntPredicate p) {
        return filterIndex(dst, 0, endIndex, p);
    }
    public static void accumulate(final int[] a, final IntBinaryOperator op) {
        for (int i = 1; i < a.length; i++) a[i] = op.applyAsInt(a[i - 1], a[i]);
    }
    public static void accumulate(final int[] a) {
        for (int i = 1; i < a.length; i++) a[i] += a[i - 1];
    }
    public static void permute(int[] p, int[] a) {
        int n = p.length;
        boolean[] settled = new boolean[n];
        for (int i = 0; i < n; i++) {
            for (int j = i; !settled[j]; j = p[j]) {
                if (p[j] == i) {
                    settled[j] = true;
                    break;
                }
                int tmp = a[j]; a[j] = a[p[j]]; a[p[j]] = tmp;
                settled[j] = true;
            }
        }
    }
    public static void permute2(int[] p, int[] a, int[] b) {
        int n = p.length;
        boolean[] settled = new boolean[n];
        for (int i = 0; i < n; i++) {
            for (int j = i; !settled[j]; j = p[j]) {
                if (p[j] == i) {
                    settled[j] = true;
                    break;
                }
                int tmp;
                tmp = a[j]; a[j] = a[p[j]]; a[p[j]] = tmp;
                tmp = b[j]; b[j] = b[p[j]]; b[p[j]] = tmp;
                settled[j] = true;
            }
        }
    }
    public static void permute3(int[] p, int[] a, int[] b, int[] c) {
        int n = p.length;
        boolean[] settled = new boolean[n];
        for (int i = 0; i < n; i++) {
            for (int j = i; !settled[j]; j = p[j]) {
                if (p[j] == i) {
                    settled[j] = true;
                    break;
                }
                int tmp;
                tmp = a[j]; a[j] = a[p[j]]; a[p[j]] = tmp;
                tmp = b[j]; b[j] = b[p[j]]; b[p[j]] = tmp;
                tmp = c[j]; c[j] = c[p[j]]; c[p[j]] = tmp;
                settled[j] = true;
            }
        }
    }
    public static void permuteN(int[] p, int[]... as) {
        for (int[] a : as) permute(p, a);
    }
    public static int lowerBound(int[] sorted, int key) {
        int n = sorted.length;
        int l = -1, r = n;
        while (r - l > 1) {
            int m = (l + r) >> 1;
            if (sorted[m] < key) l = m;
            else r = m;
        }
        return r;
    }
    public static int upperBound(int[] sorted, int key) {
        int n = sorted.length;
        int l = -1, r = n;
        while (r - l > 1) {
            int m = (l + r) >> 1;
            if (sorted[m] <= key) l = m;
            else r = m;
        }
        return r;
    }
    public static int countRange(int[] sorted, int fromKey, int toKey) {
        return lowerBound(sorted, toKey) - lowerBound(sorted, fromKey);
    }
    public static int compare(final int[] a, final int[] b) {
        for (int i = 0; i < a.length; i++) {
            if (i >= b.length) return -1;
            if (a[i] > b[i]) return 1;
            if (a[i] < b[i]) return -1;
        }
        return a.length < b.length ? 1 : 0;
    }
    public static boolean equals(final int[] a, final int[] b) {return compare(a, b) == 0;}
    public static String join(final int[] a, final String sep) {
        final StringBuilder sb = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            sb.append(a[i]);
            if (i < a.length - 1) sb.append(sep);
        }
        return sb.toString();
    }
    public static String joinWithPrefixAndSuffix(final int[] a, final IntFunction<String> idxToPre, final IntFunction<String> idxToSuf, final String sep) {
        final StringBuilder sb = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            sb.append(idxToPre.apply(i)).append(a[i]).append(idxToSuf.apply(i));
            if (i < a.length - 1) sb.append(sep);
        }
        return sb.toString();
    }
}

@FunctionalInterface
interface IntComparator {
    public int compare(int i, int j);
    public default boolean eq(final int i, final int j) {return compare(i, j) == 0;}
    public default boolean ne(final int i, final int j) {return compare(i, j) != 0;}
    public default boolean gt(final int i, final int j) {return compare(i, j) > 0;}
    public default boolean ge(final int i, final int j) {return compare(i, j) >= 0;}
    public default boolean lt(final int i, final int j) {return compare(i, j) < 0;}
    public default boolean le(final int i, final int j) {return compare(i, j) <= 0;}
}



/**
 * @author https://atcoder.jp/users/suisen
 * 
 * (NON-DESTRUCTIVE) methods that returns int array.
 */
final class IntArrayFactory {
    private IntArrayFactory() {}
    public static int[] filled(final int n, final int init) {
        final int[] ret = new int[n];
        Arrays.fill(ret, init);
        return ret;
    }
    public static int[][] filled(final int n, final int m, final int init) {
        final int[][] ret = new int[n][m];
        for (int i = 0; i < n; i++) Arrays.fill(ret[i], init);
        return ret;
    }
    public static int[] generate(final int n, final IntSupplier f) {
        final int[] a = new int[n];
        Arrays.setAll(a, i -> f.getAsInt());
        return a;
    }
    public static int[][] generate(final int n, final int m, final IntSupplier f) {
        final int[][] a = new int[n][m];
        for (int i = 0; i < n; i++) a[i] = generate(m, f);
        return a;
    }
    public static int[] indexToInt(final int n, final IntUnaryOperator f) {
        final int[] a = new int[n];
        Arrays.setAll(a, f);
        return a;
    }
    public static int[][] indexToInt(final int n, final int m, final IntBinaryOperator f) {
        final int[][] a = new int[n][m];
        for (int i = 0; i < n; i++) {
            final int ii = i;
            a[i] = indexToInt(m, j -> f.applyAsInt(ii, j));
        }
        return a;
    }
    public static int[] toArray(final Collection<? extends Number> collection) {
        final int n = collection.size();
        final int[] ret = new int[n];
        final Object[] coll = collection.toArray();
        for (int i = 0; i < n; i++) {
            ret[i] = ((Number) coll[i]).intValue();
        }
        return ret;
    }
    public static int[] unique(final int[] a) {
        final HashSet<Integer> set = new HashSet<>();
        for (final int e : a) set.add(e);
        final int m = set.size();
        final int[] b = new int[m];
        int index = 0;
        for (int i = 0; i < a.length; i++) {
            if (set.contains(a[i])) {
                b[index++] = a[i];
                set.remove(a[i]);
            }
        }
        return b;
    }
    public static int[][] transpose(final int[][] a) {
        final int n = a.length;
        final int m = a[0].length;
        final int[][] ret = new int[m][n];
        for (int j = 0; j < m; j++) for (int i = 0; i < n; i++) ret[j][i] = a[i][j];
        return ret;
    }
    public static int[] count(final int[] a, final int max) {
        final int[] ret = new int[max + 1];
        for (int i = 0; i < a.length; i++) ret[a[i]]++;
        return ret;
    }
    public static int[] map(final int[] a, final IntUnaryOperator f) {
        final int[] b = new int[a.length];
        Arrays.setAll(b, i -> f.applyAsInt(a[i]));
        return b;
    }
    public static int[] filter(final int[] a, final IntPredicate p) {
        final IntArrayList ret = new IntArrayList();
        for (final int e : a) ret.addIf(e, p);
        return ret.toArray();
    }
    public static int[] filterIndex(final int beginIndex, final int endIndex, final IntPredicate p) {
        final IntArrayList ret = new IntArrayList();
        for (int i = beginIndex; i < endIndex; i++) ret.addIf(i, p);
        return ret.toArray();
    }
    public static int[] filterIndex(final int endIndex, final IntPredicate p) {
        return filterIndex(0, endIndex, p);
    }
    public static int[] accumulateFromHead(final int[] a, final IntBinaryOperator op, final int e) {
        final int n = a.length;
        final int[] ret = new int[n + 1];
        ret[0] = e;
        for (int i = 0; i < n; i++) ret[i + 1] = op.applyAsInt(ret[i], a[i]);
        return ret;
    }
    public static int[] accumulateFromTail(final int[] a, final IntBinaryOperator op, final int e) {
        final int n = a.length;
        final int[] ret = new int[n + 1];
        ret[n] = e;
        for (int i = n - 1; i >= 0; i--) ret[i] = op.applyAsInt(ret[i + 1], a[i]);
        return ret;
    }
}

/**
 * Minimum modulo operations.
 * 
 * @author https://atcoder.jp/users/suisen
 */
class ModArithmetic {
    public final long MOD;

    /**
     * support modulo p arithmetic
     * 
     * @param p p s.t. p is a prime number.
     */
    public ModArithmetic(final long p) {
        this.MOD = p;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a mod p.
     */
    public static long mod(long a, final long MOD) {
        a %= MOD;
        return a < 0 ? a + MOD : a;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a + b mod p.
     */
    public static long add(final long a, final long b, final long MOD) {
        final long s = a + b;
        return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a - b mod p.
     */
    public static long sub(final long a, final long b, final long MOD) {
        final long s = a - b;
        return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a * b mod p.
     */
    public static long mul(final long a, final long b, final long MOD) {
        final long ret = (a * b) % MOD;
        return ret < 0 ? ret + MOD : ret;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ b * x = a mod p.
     */
    public static long div(final long a, final long b, final long MOD) {
        return mul(a, inv(b, MOD), MOD);
    }

    /**
     * Calculate the value b s.t. a * b mod p = 1.
     */
    public static long inv(long a, final long MOD) {
        long b = MOD;
        long u = 1, v = 0;
        while (b >= 1) {
            final long t = a / b;
            a -= t * b;
            final long tmp1 = a; a = b; b = tmp1;
            u -= t * v;
            final long tmp2 = u; u = v; v = tmp2;
        }
        u %= MOD;
        return u < 0 ? u + MOD : u;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a mod p.
     */
    public long mod(long a) {
        a %= MOD;
        return a < 0 ? a + MOD : a;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a + b mod p.
     */
    public long add(final long a, final long b) {
        final long s = a + b;
        return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a - b mod p.
     */
    public long sub(final long a, final long b) {
        final long s = a - b;
        return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a * b mod p.
     */
    public long mul(final long a, final long b) {
        final long s = (a * b) % MOD;
        return s < 0 ? s + MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ b * x = a mod p.
     */
    public long div(final long a, final long b) {
        return mul(a, inv(b));
    }

    /**
     * Calculate the value b s.t. a*b mod MOD = 1.
     */
    public long inv(long a) {
        long b = MOD;
        long u = 1, v = 0;
        while (b >= 1) {
            final long t = a / b;
            a -= t * b;
            final long tmp1 = a; a = b; b = tmp1;
            u -= t * v;
            final long tmp2 = u; u = v; v = tmp2;
        }
        return mod(u);
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a_1 + 1_2 + ... + a_k mod p.
     */
    public long sum(final long... a) {
        long ret = 0;
        for (final long c : a) ret += c;
        return mod(ret);
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a_1 * 1_2 * ... * a_k mod p.
     */
    public long prod(final long... a) {
        long ret = 1;
        for (final long c : a) ret = (ret * c) % MOD;
        return ret < 0 ? ret + MOD : ret;
    }

    /**
     * calculate the inverse of [1,n] mod MOD in O(n).
     * 
     * @param n
     * @param MOD
     * @return the array that contains the inverse of [1,n]. (array[0] = 0)
     */
    public static long[] invMemo(int n, long MOD) {
        final long[] memo = new long[n + 1];
        memo[1] = 1;
        for (int i = 2; i <= n; i++) {
            long q = MOD - MOD / i;
            long r = memo[(int) (MOD % i)];
            memo[i] = (q * r) % MOD;
        }
        return memo;
    }

    /**
     * calculate the inverse of a[0], ..., a[n - 1] mod MOD in O(n).
     * 
     * @param a
     * @param MOD
     * @return the array that contains the inverse of a[0], ..., a[n - 1].
     */
    public static long[] invAll(long[] a, long MOD) {
        int n = a.length;
        final long[] dp = new long[n + 1];
        final long[] pd = new long[n + 1];
        dp[0] = pd[n] = 1;
        for (int i = 0; i < n; i++) dp[i + 1] = (dp[i] * a[i]) % MOD;
        for (int i = n; i > 0; i--) pd[i - 1] = (pd[i] * a[i]) % MOD;
        long inv = ModArithmetic.inv(dp[n], MOD);
        long[] ret = new long[n];
        for (int i = 0; i < n; i++) ret[i] = (((dp[i] * pd[i]) % MOD) * inv) % MOD;
        return ret;
    }

    /**
     * calculate 0!, ..., n! (mod MOD) in O(n).
     * 
     * @param n
     * @param MOD
     * @return the array that contains 0!, ..., n! (mod MOD).
     */
    public static long[] factorialMemo(int n, long MOD) {
        final long[] memo = new long[n + 1];
        memo[0] = 1;
        for (int i = 1; i <= n; i++) memo[i] = (memo[i - 1] * i) % MOD;
        return memo;
    }

    /**
     * calculate the inverse of 0!, ..., n! (mod MOD) in O(n).
     * 
     * @param factorial [0!, ..., n!] (mod MOD).
     * @param MOD
     * @return the array that contains the inverse of 0!, ..., n! (mod MOD).
     */
    public static long[] invFactorialMemo(long[] factorial, long MOD) {
        final int n = factorial.length - 1;
        final long[] memo = new long[n + 1];
        memo[n] = ModArithmetic.inv(factorial[n], MOD);
        for (int i = n; i > 0; i--) memo[i - 1] = (memo[i] * i) % MOD;
        return memo;
    }

    /**
     * calculate a^0, ..., a^n (mod MOD) in O(n).
     * 
     * @param a   base
     * @param n   maximum index.
     * @param MOD
     * @return the array that contains the a^0, ..., a^n (mod MOD).
     */
    public static long[] powerMemo(long a, int n, long MOD) {
        if (a == 2) return powerMemo(n, MOD);
        long[] memo = new long[n + 1];
        memo[0] = 1;
        for (int i = 1; i <= n; i++) memo[i] = (memo[i - 1] * a) % MOD;
        return memo;
    }

    /**
     * calculate 2^0, ..., 2^n (mod MOD) in O(n).
     * 
     * @param n   maximum index.
     * @param MOD
     * @return the array that contains the 2^0, ..., 2^n (mod MOD).
     */
    public static long[] powerMemo(int n, long MOD) {
        long[] memo = new long[n + 1];
        memo[0] = 1;
        for (int i = 1; i <= n; i++) memo[i] = (memo[i - 1] << 1) % MOD;
        return memo;
    }

    /**
     * calculate a^b (mod MOD) in O(max{MOD, b}).
     * 
     * @param a   base
     * @param b   index
     * @param MOD
     * @return a^b (mod MOD)
     */
    public static long power(long a, long b, long MOD) {
        a %= MOD;
        if (b == 0 || a == 1) return 1;
        long res = 1, p = a, c = 1;
        while (b > 0) {
            long lsb = b & -b;
            for (; lsb != c; c <<= 1, p = (p * p) % MOD);
            res = (res * p) % MOD;
            b ^= lsb;
        }
        return res;
    }

    /**
     * calculate the inverse of [1,n] mod MOD in O(n).
     * 
     * @param n
     * @return the array that contains the inverse of [1,n]. (array[0] = 0)
     */
    public long[] invMemo(int n) {
        final long[] memo = new long[n + 1];
        memo[1] = 1;
        for (int i = 2; i <= n; i++) {
            long q = MOD - MOD / i;
            long r = memo[(int) (MOD % i)];
            memo[i] = (q * r) % MOD;
        }
        return memo;
    }

    /**
     * calculate the inverse of a[0], ..., a[n - 1] mod MOD in O(n).
     * 
     * @param a
     * @return the array that contains the inverse of a[0], ..., a[n - 1].
     */
    public long[] invAll(long[] a) {
        int n = a.length;
        long[] dp = new long[n + 1];
        long[] pd = new long[n + 1];
        dp[0] = pd[n] = 1;
        for (int i = 0; i < n; i++) dp[i + 1] = (dp[i] * a[i    ]) % MOD;
        for (int i = n; i > 0; i--) pd[i - 1] = (pd[i] * a[i - 1]) % MOD;
        long inv = inv(dp[n], MOD);
        long[] invs = new long[n];
        for (int i = 0; i < n; i++) {
            long lr = (dp[i] * pd[i + 1]) % MOD;
            invs[i] = (lr * inv) % MOD;
        }
        return invs;
    }

    /**
     * calculate 0!, ..., n! (mod MOD) in O(n).
     * 
     * @param n
     * @return the array that contains 0!, ..., n! (mod MOD).
     */
    public long[] factorialMemo(int n) {
        final long[] memo = new long[n + 1];
        memo[0] = 1;
        for (int i = 1; i <= n; i++) memo[i] = (memo[i - 1] * i) % MOD;
        return memo;
    }

    /**
     * calculate the inverse of 0!, ..., n! (mod MOD) in O(n).
     * 
     * @param factorial [0!, ..., n!] (mod MOD).
     * @return the array that contains the inverse of 0!, ..., n! (mod MOD).
     */
    public long[] invFactorialMemo(long[] factorial) {
        final int n = factorial.length - 1;
        final long[] memo = new long[n + 1];
        memo[n] = ModArithmetic.inv(factorial[n], MOD);
        for (int i = n; i > 0; i--) memo[i - 1] = (memo[i] * i) % MOD;
        return memo;
    }

    /**
     * calculate a^0, ..., a^n (mod MOD) in O(n).
     * 
     * @param a   base
     * @param n   maximum index.
     * @return the array that contains the a^0, ..., a^n (mod MOD).
     */
    public long[] powerMemo(long a, int n) {
        if (a == 2) return powerMemo(n, MOD);
        final long[] memo = new long[n + 1];
        memo[0] = 1;
        for (int i = 1; i <= n; i++) memo[i] = (memo[i - 1] * a) % MOD;
        return memo;
    }

    /**
     * calculate 2^0, ..., 2^n (mod MOD) in O(n).
     * 
     * @param n   maximum index.
     * @return the array that contains the 2^0, ..., 2^n (mod MOD).
     */
    public long[] powerMemo(int n) {
        final long[] memo = new long[n + 1];
        memo[0] = 1;
        for (int i = 1; i <= n; i++) memo[i] = (memo[i - 1] << 1) % MOD;
        return memo;
    }

    /**
     * calculate a^b (mod MOD) in O(max{MOD, b}).
     * 
     * @param a   base
     * @param b   index
     * @return a^b (mod MOD)
     */
    public long power(long a, long b) {
        a %= MOD;
        if (b == 0 || a == 1) return 1;
        long res = 1, p = a, c = 1;
        while (b > 0) {
            long lsb = b & -b;
            for (; lsb != c; c <<= 1, p = (p * p) % MOD);
            res = (res * p) % MOD;
            b ^= lsb;
        }
        return res;
    }
}


final class ComparativeInsertionSort {
    static void sort(final int[] a, final int from, final int to, final IntComparator comparator) {
        for (int i = from + 1; i < to; i++) {
            final int tmp = a[i];
            if (comparator.gt(a[i - 1], tmp)) {
                int j = i;
                do {a[j] = a[j - 1]; j--;} while (j > from && comparator.gt(a[j - 1], tmp));
                a[j] = tmp;
            }
        }
    }
    static void sort(final long[] a, final int from, final int to, final LongComparator comparator) {
        for (int i = from + 1; i < to; i++) {
            final long tmp = a[i];
            if (comparator.gt(a[i - 1], tmp)) {
                int j = i;
                do {a[j] = a[j - 1]; j--;} while (j > from && comparator.gt(a[j - 1], tmp));
                a[j] = tmp;
            }
        }
    }
}