Problem: This is an interactive problem.

This is a hard version of the problem. The difference from the easy version is that in the hard version 1 ≤ t ≤ min(n, 10^4) and the total number of queries is limited to 6 ⋅ 10^4.

Polycarp is playing a computer game. In this game, an array consisting of zeros and ones is hidden. Polycarp wins if he guesses the position of the k-th zero from the left t times.

Polycarp can make no more than 6 ⋅ 10^4 requests totally of the following type: 

  * ? l r — find out the sum of all elements in positions from l to r (1 ≤ l ≤ r ≤ n) inclusive. 



To make the game more interesting, each guessed zero turns into one and the game continues on the changed array. More formally, if the position of the k-th zero was x, then after Polycarp guesses this position, the x-th element of the array will be replaced from 0 to 1.

Help Polycarp win the game.

Interaction

First, your program must read two integers n and t (1 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ t ≤ min(n, 10^4)).

Then t lines follow, each of which contains one integer k (1 ≤ k ≤ n). It is guaranteed that at the moment of the request the array contains at least k zeros. In order to get the next value of k, you must output the answer for the previous value of k.

After that, you can make no more than 6 ⋅ 10^4 requests in total.

Use the following format to output the answer (it is not a request, it doesn't count in 6 ⋅ 10^4): 

  * ! x — position of the k-th zero. 



Positions in the array are numbered from left to right from 1 to n inclusive.

After printing t answers, your program should exit immediately.

In this task, the interactor is not adaptive. This means that within the same test, the hidden array and the queries do not change.

In case of an incorrect query, -1 will be displayed. When this value is received, your program must immediately exit normally (for example, by calling exit(0)), otherwise, the testing system may issue an arbitrary verdict.

If the number of requests is exceeded, the verdict wrong answer will be displayed.

Your solution may get the verdict Idleness limit exceeded if you don't print anything or forget to flush the output buffer.

To flush the output buffer, you need to do the following immediately after the query output and the end-of-line character:

  * fflush(stdout) or cout.flush() in C ++; 
  * System.out.flush() in Java; 
  * flush(output) in Pascal; 
  * stdout.flush() in Python; 
  * see the documentation for other languages. 



Hacks

Use the following format for hacks:

On the first line print the string s (1 ≤ |s| ≤ 2 ⋅ 10^5), consisting of zeros and ones, and an integer t (1 ≤ t ≤ min(|s|, 10^4)) — hidden array and number of requests, respectively. In the next t lines output the number k (1 ≤ k ≤ |s|).

The hacked solution will not have direct access to the hidden array.

Example

Input


6 2

2

2

1

1

0

1

0

Output


? 4 6

? 1 1

? 1 2

? 5 5

! 5

? 2 2

! 2

Note

In the first test, the array [1, 0, 1, 1, 0, 1] is hidden. After answering the query k=2, the array changed to [1, 0, 1, 1, 1, 1].

Solution (in JAVA):

Solution:
import java.io.DataInputStream;
import java.io.IOException;
import java.io.OutputStreamWriter;
import java.io.PrintWriter;

public class F2 {

    private static int n;

    private static void run() throws IOException {
        int k = in.nextInt();

        int left = 1;
        int right = n;
        while (left < right) {
            int mid = (left + right) >> 1;
            int sum = query(mid);
            int count = mid - sum;
            if (count >= k) {
                right = mid;
            } else {
                left = mid + 1;
            }
        }
        out.printf("! %d\n", left);
        out.flush();

        update_range(left, n, 1, 1);
    }

    private static int query(int right) throws IOException {
        int ans = query(right, 1);
        if (ans != -1) {
            return ans;
        } else {
            out.printf("? %d %d\n", 1, right);
            out.flush();
            ans = in.nextInt();
            update(right, ans, 1);
            return ans;
        }
    }

    static Node[] nodes;

    static void init(int left, int right, int p) {
        nodes[p] = new Node();
        nodes[p].left = left;
        nodes[p].right = right;
        if (left == right) {
            return;
        }

        int mid = (left + right) >> 1;
        init(left, mid, p * 2);
        init(mid + 1, right, p * 2 + 1);
    }

    static void update(int pos, int value, int p) {
        if (nodes[p].left == pos && nodes[p].right == pos) {
            nodes[p].flag = true;
            nodes[p].value = value;
            nodes[p].plus = 0;
            return;
        }

        push_down(p);

        int mid = (nodes[p].left + nodes[p].right) >> 1;
        if (pos <= mid) {
            update(pos, value, p * 2);
        } else {
            update(pos, value, p * 2 + 1);
        }
    }

    private static void push_down(int p) {
        if (nodes[p].plus != 0) {
            nodes[p * 2].plus += nodes[p].plus;
            nodes[p * 2 + 1].plus += nodes[p].plus;
            nodes[p].plus = 0;
        }
    }

    static void update_range(int left, int right, int value, int p) {
        if (nodes[p].left == left && nodes[p].right == right) {
            nodes[p].plus += value;
            return;
        }

        push_down(p);

        int mid = (nodes[p].left + nodes[p].right) >> 1;
        if (right <= mid) {
            update_range(left, right, value, p * 2);
        } else if (left > mid) {
            update_range(left, right, value, p * 2 + 1);
        } else {
            update_range(left, mid, value, p * 2);
            update_range(mid + 1, right, value, p * 2 + 1);
        }
    }

    static int query(int pos, int p) {
        if (nodes[p].left == pos && nodes[p].right == pos) {
            if (!nodes[p].flag) return -1;

            nodes[p].value += nodes[p].plus;
            nodes[p].plus = 0;
            return nodes[p].value;
        }

        push_down(p);

        int mid = (nodes[p].left + nodes[p].right) >> 1;
        if (pos <= mid) {
            return query(pos, p * 2);
        } else {
            return query(pos, p * 2 + 1);
        }
    }

    static class Node {
        int left, right, value = -1;
        int plus = 0;
        boolean flag = false;
    }

    public static void main(String[] args) throws IOException {
        in = new Reader();
        out = new PrintWriter(new OutputStreamWriter(System.out));

        n = in.nextInt();
        nodes = new Node[4 * n + 10];
        init(1, n, 1);
        int t = in.nextInt();
        for (int i = 0; i < t; i++) {
            run();
        }

        out.flush();
        in.close();
        out.close();
    }

    private static int gcd(int a, int b) {
        if (a == 0 || b == 0)
            return 0;
        while (b != 0) {
            int tmp;
            tmp = a % b;
            a = b;
            b = tmp;
        }
        return a;
    }

    static final long mod = 1000000007;

    static long pow_mod(long a, long b) {
        long result = 1;
        while (b != 0) {
            if ((b & 1) != 0) result = (result * a) % mod;
            a = (a * a) % mod;
            b >>= 1;
        }
        return result;
    }

    private static long multiplied_mod(long... longs) {
        long ans = 1;
        for (long now : longs) {
            ans = (ans * now) % mod;
        }
        return ans;
    }

    @SuppressWarnings("FieldCanBeLocal")
    private static Reader in;
    private static PrintWriter out;

    private static int[] read_int_array(int len) throws IOException {
        int[] a = new int[len];
        for (int i = 0; i < len; i++) {
            a[i] = in.nextInt();
        }
        return a;
    }

    private static long[] read_long_array(int len) throws IOException {
        long[] a = new long[len];
        for (int i = 0; i < len; i++) {
            a[i] = in.nextLong();
        }
        return a;
    }

    private static void print_array(int[] array) {
        for (int now : array) {
            out.print(now);
            out.print(' ');
        }
        out.println();
    }

    private static void print_array(long[] array) {
        for (long now : array) {
            out.print(now);
            out.print(' ');
        }
        out.println();
    }

    static class Reader {
        private static final int BUFFER_SIZE = 1 << 16;
        private final DataInputStream din;
        private final byte[] buffer;
        private int bufferPointer, bytesRead;

        Reader() {
            din = new DataInputStream(System.in);
            buffer = new byte[BUFFER_SIZE];
            bufferPointer = bytesRead = 0;
        }

        public String readLine() throws IOException {
            final byte[] buf = new byte[1024]; // line length
            int cnt = 0, c;
            while ((c = read()) != -1) {
                if (c == '\n') {
                    break;
                }
                buf[cnt++] = (byte) c;
            }
            return new String(buf, 0, cnt);
        }

        public int nextSign() throws IOException {
            byte c = read();
            while ('+' != c && '-' != c) {
                c = read();
            }
            return '+' == c ? 0 : 1;
        }

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

        public int skip() throws IOException {
            int b;
            // noinspection ALL
            while ((b = read()) != -1 && isSpaceChar(b)) {
                ;
            }
            return b;
        }

        public char nc() throws IOException {
            return (char) skip();
        }

        public String next() throws IOException {
            int b = skip();
            final StringBuilder sb = new StringBuilder();
            while (!isSpaceChar(b)) { // when nextLine, (isSpaceChar(b) && b != ' ')
                sb.appendCodePoint(b);
                b = read();
            }
            return sb.toString();
        }

        public int nextInt() throws IOException {
            int ret = 0;
            byte c = read();
            while (c <= ' ') {
                c = read();
            }
            final boolean neg = c == '-';
            if (neg) {
                c = read();
            }
            do {
                ret = ret * 10 + c - '0';
            } while ((c = read()) >= '0' && c <= '9');

            if (neg) {
                return -ret;
            }
            return ret;
        }

        public long nextLong() throws IOException {
            long ret = 0;
            byte c = read();
            while (c <= ' ') {
                c = read();
            }
            final boolean neg = c == '-';
            if (neg) {
                c = read();
            }
            do {
                ret = ret * 10 + c - '0';
            }
            while ((c = read()) >= '0' && c <= '9');
            if (neg) {
                return -ret;
            }
            return ret;
        }

        public double nextDouble() throws IOException {
            double ret = 0, div = 1;
            byte c = read();
            while (c <= ' ') {
                c = read();
            }
            final boolean neg = c == '-';
            if (neg) {
                c = read();
            }

            do {
                ret = ret * 10 + c - '0';
            }
            while ((c = read()) >= '0' && c <= '9');

            if (c == '.') {
                while ((c = read()) >= '0' && c <= '9') {
                    ret += (c - '0') / (div *= 10);
                }
            }

            if (neg) {
                return -ret;
            }
            return ret;
        }

        private void fillBuffer() throws IOException {
            bytesRead = din.read(buffer, bufferPointer = 0, BUFFER_SIZE);
            if (bytesRead == -1) {
                buffer[0] = -1;
            }
        }

        private byte read() throws IOException {
            if (bufferPointer == bytesRead) {
                fillBuffer();
            }
            return buffer[bufferPointer++];
        }

        public void close() throws IOException {
            din.close();
        }

    }
}