**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Boss, Thirsty
time_limit_ms: 2000
memory_limit_mb: 256
Description: Pak Chanek has a friend who runs a drink stall in a canteen. His friend will sell drinks for $n$ days, numbered from day $1$ to day $n$. There are also $m$ types of drinks, numbered from $1$ to $m$.

The profit gained from selling a drink on a particular day can vary. On day $i$, the projected profit from selling drink of type $j$ is $A_{i, j}$. Note that $A_{i, j}$ can be negative, meaning that selling the drink would actually incur a loss.

Pak Chanek wants to help his friend plan the sales over the $n$ days. On day $i$, Pak Chanek must choose to sell at least one type of drink. Furthermore, the types of drinks sold on a single day must form a subarray. In other words, in each day, Pak Chanek will select $i$ and $j$ such that $1 \leq i \leq j \leq m$. Then all types of drinks between $i$ and $j$ (inclusive) will be sold.

However, to ensure that customers from the previous day keep returning, the selection of drink types sold on day $i$ ($i>1$) must meet the following conditions:

  * At least one drink type sold on day $i$ must also have been sold on day $i-1$.    * At least one drink type sold on day $i$ must not have been sold on day $i-1$.

The daily profit is the sum of the profits from all drink types sold on that day. The total profit from the sales plan is the sum of the profits over $n$ days. What is the maximum total profit that can be achieved if Pak Chanek plans the sales optimally?

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 1000$). The description of the test cases follows.

The first line of each test case contains two integers $n$ and $m$ ($1 \leq n \leq 2 \cdot 10^5$; $3 \leq m \leq 2 \cdot 10^5$; $n \cdot m \leq 2 \cdot 10^5$) — the number of rows and columns in a grid.

The next $n$ lines of each test case contain $m$ integers each, where the $i$-th line contains the integers $A_{i,1} A_{i,2}, \ldots, A_{i,m}$ ($-10^9 \leq A_{i,j} \leq 10^9$) — project profits of each drink type on the $i$-th day.

It is guaranteed that the sum of $n \cdot m$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, output a single integer: the maximum profit that Pak Chanek can achieve.

Examples:
input:
1
3 6
79 20 49 5 -1000 500
-105 9 109 24 -98 -499
14 47 12 39 23 50
output:
475

Note: Here is Pak Chanek's optimal plan:

![](CDN_BASE_URL/4277bb4b771ce557c9df4ba84c32fcb1)| ![](CDN_BASE_URL/9da16237201d15894364e851ed195f46)| ![](CDN_BASE_URL/3b7487c42d65f64db69a0a6a263bbf48)   ---|---|---        * On day $1$, Pak Chanek sells drink types $1$ to $3$. Generating a profit of $79+20+49 = 148$.    * On day $2$, Pak Chanek sells drink types $2$ to $4$. Generating a profit of $9+109+24 = 142$    * On day $3$, Pak Chanek sells drink types $1$ to $6$. Generating a profit of $185$.

So, the total profit of Pak Chanek's plan is $148 + 142 + 185 = 475$.

Buggy Code:
/*
*Author: @Jemin.s
*/
import java.io.*;
import java.util.NoSuchElementException;

public class TRY {

    public static void main(String[] args) {
        fastScanner sc = new fastScanner();
        int t = sc.nextInt();

        while (t-- > 0) {
            int n = sc.nextInt();
            int m = sc.nextInt();

            int[][] profit = new int[n][m];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < m; j++) {
                    profit[i][j] = sc.nextInt();
                }
            }

            System.out.println(maximize(n, m, profit));
        }
    }

    private static long maximize(int n, int m, int[][] profitGrid) {
        long totalMaxProfit = Long.MIN_VALUE;
        long[][][] dp = new long[n][m][m];

        for (int i = 0; i < m; i++) {
            for (int j = i; j < m; j++) {
                long sum = 0;
                for (int k = i; k <= j; k++) {
                    sum += profitGrid[0][k];
                }
                dp[0][i][j] = sum;
            }
        }

        for (int i = 1; i < n; i++) {
            for (int j = 0; j < m; j++) {
                for (int k = j; k < m; k++) {
                    long cp = 0;
                    for (int l = j; l <= k; l++) {
                        cp += profitGrid[i][l];
                    }

                    for (int prevJ = 0; prevJ < m; prevJ++) {
                        for (int prevK = prevJ; prevK < m; prevK++) {
                            if (isValidTransition(prevJ, prevK, j, k)) {
                                dp[i][j][k] = Math.max(dp[i][j][k], dp[i - 1][prevJ][prevK] + cp);
                            }
                        }
                    }
                }
            }
        }

        for (int i = 0; i < m; i++) {
            for (int j = i; j < m; j++) {
                totalMaxProfit = Math.max(totalMaxProfit, dp[n - 1][i][j]);
            }
        }

        return totalMaxProfit;
    }

    private static boolean isValidTransition(int prevJ, int prevK, int j, int k) {
        boolean common = false;
        boolean different = false;

        for (int l = prevJ; l <= prevK; l++) {
            if (l >= j && l <= k) {
                common = true;
                break;
            }
        }

        for (int l = j; l <= k; l++) {
            if (l < prevJ || l > prevK) {
                different = true;
                break;
            }
        }

        return common && different;
    }
}

class fastScanner {

    private final InputStream in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    public fastScanner(InputStream in) {
        this.in = in;
    }

    public fastScanner() {
        this(System.in);
    }

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

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

    private 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 int nextInt() {
        return Integer.parseInt(next());
    }

    public long nextLong() {
        return Long.parseLong(next());
    }

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

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that maximize the failure probability. The key aspects to consider are:

1. **Transition Validity**: Ensure that the transitions between days meet the conditions of having at least one common and one different drink type.
2. **Profit Distribution**: Distribute profits in a way that the algorithm might incorrectly calculate the maximum profit due to suboptimal transitions or incorrect handling of negative profits.
3. **Small Scale**: The input should be small enough to be manageable but large enough to highlight the bug.

Let's construct a test case that meets these criteria:

1. **Number of Test Cases**: Set to 1 for simplicity.
2. **Number of Days (n)**: Set to 3.
3. **Number of Drink Types (m)**: Set to 4.
4. **Profit Grid**: Design the grid to ensure that the transitions are challenging and the algorithm might make incorrect decisions.

Here is a small-scale test input that meets these criteria:

```plaintext
1
3 4
1 2 3 4
-10 5 6 -7
8 9 -10 11
```

Explanation:
- **Day 1**: The profits are 1, 2, 3, 4. The optimal choice is to sell all drink types, giving a profit of 10.
- **Day 2**: The profits are -10, 5, 6, -7. The optimal choice is to sell drink types 2 and 3, giving a profit of 11.
- **Day 3**: The profits are 8, 9, -10, 11. The optimal choice is to sell all drink types, giving a profit of 18.

This test case ensures that the transitions between days are valid and challenging, and the profit distribution is such that the algorithm might make incorrect decisions if it doesn't handle the transitions correctly.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Boss, Thirsty
time_limit_ms: 2000
memory_limit_mb: 256
Description: Pak Chanek has a friend who runs a drink stall in a canteen. His friend will sell drinks for $n$ days, numbered from day $1$ to day $n$. There are also $m$ types of drinks, numbered from $1$ to $m$.

The profit gained from selling a drink on a particular day can vary. On day $i$, the projected profit from selling drink of type $j$ is $A_{i, j}$. Note that $A_{i, j}$ can be negative, meaning that selling the drink would actually incur a loss.

Pak Chanek wants to help his friend plan the sales over the $n$ days. On day $i$, Pak Chanek must choose to sell at least one type of drink. Furthermore, the types of drinks sold on a single day must form a subarray. In other words, in each day, Pak Chanek will select $i$ and $j$ such that $1 \leq i \leq j \leq m$. Then all types of drinks between $i$ and $j$ (inclusive) will be sold.

However, to ensure that customers from the previous day keep returning, the selection of drink types sold on day $i$ ($i>1$) must meet the following conditions:

  * At least one drink type sold on day $i$ must also have been sold on day $i-1$.    * At least one drink type sold on day $i$ must not have been sold on day $i-1$.

The daily profit is the sum of the profits from all drink types sold on that day. The total profit from the sales plan is the sum of the profits over $n$ days. What is the maximum total profit that can be achieved if Pak Chanek plans the sales optimally?

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 1000$). The description of the test cases follows.

The first line of each test case contains two integers $n$ and $m$ ($1 \leq n \leq 2 \cdot 10^5$; $3 \leq m \leq 2 \cdot 10^5$; $n \cdot m \leq 2 \cdot 10^5$) — the number of rows and columns in a grid.

The next $n$ lines of each test case contain $m$ integers each, where the $i$-th line contains the integers $A_{i,1} A_{i,2}, \ldots, A_{i,m}$ ($-10^9 \leq A_{i,j} \leq 10^9$) — project profits of each drink type on the $i$-th day.

It is guaranteed that the sum of $n \cdot m$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each test case, output a single integer: the maximum profit that Pak Chanek can achieve.

Examples:
input:
1
3 6
79 20 49 5 -1000 500
-105 9 109 24 -98 -499
14 47 12 39 23 50
output:
475

Note: Here is Pak Chanek's optimal plan:

![](CDN_BASE_URL/4277bb4b771ce557c9df4ba84c32fcb1)| ![](CDN_BASE_URL/9da16237201d15894364e851ed195f46)| ![](CDN_BASE_URL/3b7487c42d65f64db69a0a6a263bbf48)   ---|---|---        * On day $1$, Pak Chanek sells drink types $1$ to $3$. Generating a profit of $79+20+49 = 148$.    * On day $2$, Pak Chanek sells drink types $2$ to $4$. Generating a profit of $9+109+24 = 142$    * On day $3$, Pak Chanek sells drink types $1$ to $6$. Generating a profit of $185$.

So, the total profit of Pak Chanek's plan is $148 + 142 + 185 = 475$.

Buggy Code:
/*
*Author: @Jemin.s
*/
import java.util.*;
import java.io.*;

public class TRY {
    static final int MAX_SIZE = 200005;
    static final long LONG_INF = 1000000000000000000L;
    static long[] prev = new long[MAX_SIZE];
    static long[] curr = new long[MAX_SIZE];
    static long[] val = new long[MAX_SIZE];
    static long[] presum = new long[MAX_SIZE];

    public static void main(String[] args) throws IOException {
        FastScanner scanner = new FastScanner();
        PrintWriter writer = new PrintWriter(System.out);

        int testCases = scanner.nextInt();
        while (testCases-- > 0) {
            int rows = scanner.nextInt();
            int cols = scanner.nextInt();

            Arrays.fill(prev, 0);
            long max = -LONG_INF;

            for (int i = 1; i <= rows; i++) {
                for (int j = 1; j <= cols; j++) {
                    val[j] = scanner.nextLong();
                    presum[j] = presum[j - 1] + val[j];
                }

                Arrays.fill(curr, -LONG_INF);

                if (i == 1) {
                    for (int j = 1; j <= cols; j++) {
                        curr[j - 1] = Math.max(curr[j - 1], val[j]);
                        curr[j] = Math.max(curr[j], val[j]);
                    }
                }

                long maxpre = -LONG_INF;
                long temp = -LONG_INF;

                for (int j = 2; j <= cols; j++) {
                    maxpre = Math.max(maxpre, -presum[j - 2]);
                    temp = Math.max(temp , maxpre + prev[j - 1]);
                    curr[j] = Math.max(curr[j], temp + presum[j]);
                }

                maxpre = temp = -LONG_INF;

                for (int j = cols - 1; j > 0; j--) {
                    maxpre = Math.max(maxpre, presum[j + 1]);
                    temp = Math.max(temp, maxpre + prev[j]);
                    curr[j - 1] = Math.max(curr[j - 1], temp - presum[j - 1]);
                }

                if (i == rows) {
                    for (int j = 0; j <= cols; j++) {
                        max = Math.max(max, curr[j]);
                    }
                }

                System.arraycopy(curr, 1, prev, 1, cols - 1);
            }

            writer.println(max);
        }

        writer.flush();
        scanner.close();
    }
}

class FastScanner {

    private final InputStream inputStream;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int bufLen = 0;

    public FastScanner(InputStream inputStream) {
        this.inputStream = inputStream;
    }

    public FastScanner() {
        this(System.in);
    }

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

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

    public boolean hasNext() {
        while (hasNextByte() && (buffer[ptr] < 33 || buffer[ptr] > 126)) {
            ptr++;
        }
        return hasNextByte();
    }

    public String next() {
        if (!hasNext()) {
            throw new NoSuchElementException();
        }
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (b >= 33 && b <= 126) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }

    public long nextLong() {
        if (!hasNext()) {
            throw new NoSuchElementException();
        }
        long num = 0;
        boolean isNegative = false;
        int b = readByte();
        if (b == '-') {
            isNegative = true;
            b = readByte();
        }
        if (b < '0' || b > '9') {
            throw new NumberFormatException();
        }
        while (true) {
            if (b >= '0' && b <= '9') {
                num = num * 10 + (b - '0');
            } else if (b == -1 || (b < 33 || b > 126)) {
                return isNegative ? -num : num;
            } else {
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }

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

    public long[] nextLongArray(int length) {
        long[] array = new long[length];
        for (int i = 0; i < length; i++) {
            array[i] = this.nextLong();
        }
        return array;
    }

    public int[] nextIntArray(int length) {
        int[] array = new int[length];
        for (int i = 0; i < length; i++) {
            array[i] = this.nextInt();
        }
        return array;
    }

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

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1
3 3
1 2 3
4 5 6
7 8 9
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
#include <iostream>

int main() {
   int t;

   std::cin >> t;

   while (t-- > 0) {
      int l = 2;
      int r = 1000;
      int answer;

      while (r - l > 1) {
         int m = (l + r) / 2;

         std::cout << "? " << 1 << ' ' << m << std::endl;
         std::cin >> answer;

         if (answer == -1) {
            return 1;
         }

         if (answer > m) {
            r = m;
         } else {
            l = m;
         }
      }

      std::cout << "! " << l + 1 << std::endl;
   }

   return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1

9999
```

### Explanation:
1. **Test Case Count**: The input starts with a single test case, which is the smallest non-zero value for `t`. This ensures the loop runs at least once.

2. **Query Response**: The response to the query is `9999`. This is designed to test the edge case where the missing number `x` is very close to the upper limit of 1000. Specifically, if `x = 1000`, a query for a `1 x 999` rectangle would result in a measured area of `999 * 1000 = 999000`, which is much larger than `9999`. However, if `x = 999`, the measured area would be `999 * 1000 = 999000`, which is also much larger than `9999`.

3. **Bug Exposure**: The buggy code assumes that if the measured area is greater than `m`, then `x` must be less than `m`. However, in this case, the measured area is `9999`, which is much larger than `m` (which would be around 500 in the first iteration). This causes the code to incorrectly update `r` to `m`, leading to an incorrect final answer.

By using this test input, the buggy code will fail to correctly determine the value of `x` and will output an incorrect answer, exposing the bug in the logic.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
#include <iostream>
using namespace std;

const int MAX_X = 999;

void solve() {
    int t;
    cin >> t;

    while (t--) {
        int l = 2, r = MAX_X;
        int result = -1;

        while (l+1 < r) {
            int m = (l + r) / 2;

            cout << "? "<<l<<' ' << m << endl;
            fflush(stdout);
            int area1xm;
            cin >> area1xm;

            int area = area1xm - l * m;
            if (area == l||area==l+m) {
                r = m;
            }
            else {
                l = m;
            }
        }

        cout << "! " << r << endl;
        fflush(stdout);
    }
}

int main() {
    solve();
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1

18
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
/*
File: G1. Ruler (easy version).c++
Author: Adrian
*/

#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <map>
#include <set>
#include <string>
#include <vector>

inline void solve() {
    /* Code Here */
    int lower = 0, upper = 1001;
    while (upper - lower + 1 - 2 > 0) {
        int umid = upper - (upper - lower) / 3;
        int lmid = lower + (upper - lower) / 3;
        std::cout << "? " << umid << ' ' << lmid << std::endl;
        int input;
        std::cin >> input;
        if (input == (umid + 1) * (lmid + 1)) upper = lmid;
        else if (input == (umid + 1) * lmid) lower = lmid;
        else lower = umid;
    }
    if (upper == 1001) std::cout << "! " << upper << std::endl;
    else std::cout << "! " << upper << std::endl;
}

int main() {
    // Code for checking if ADRIAN_ENV is defined
#ifdef ADRIAN_ENV
    if (std::freopen("input.txt", "r", stdin) == nullptr) {
        std::cerr << "Error opening input.txt" << std::endl;
        return 1;
    }
    if (std::freopen("output.txt", "w", stdout) == nullptr) {
        std::cerr << "Error opening output.txt" << std::endl;
        return 1;
    }
    if (std::freopen("error.txt", "w", stderr) == nullptr) {
        std::cerr << "Error opening error.txt" << std::endl;
        return 1;
    }
#endif

    // Optimization:
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    std::cout.tie(nullptr);

    int t = 1;
    std::cin >> t;
    while (t-- > 0) solve();

    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3

12

20

10000
```

### Explanation:
1. **Test Case 1:**
   - Query: `? 3 4`
   - Expected Response: `12` (if `x = 4`, the ruler will measure `3 * 5 = 15`)

2. **Test Case 2:**
   - Query: `? 4 5`
   - Expected Response: `20` (if `x = 5`, the ruler will measure `5 * 6 = 30`)

3. **Test Case 3:**
   - Query: `? 99 100`
   - Expected Response: `10000` (if `x = 100`, the ruler will measure `99 * 101 = 9999`)

These test cases are designed to:
- Check the boundary conditions (e.g., when `x` is close to 2 or 999).
- Ensure the code handles the maximum number of queries correctly.
- Verify the correctness of the logic in handling the ruler's behavior for different values of `x`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
def ruler():
    t = int(input())
    while t:
        q = 10
        l = 1
        r = 1000
        err = False
        while q and l < r:
            m = (l+r)//2
            a = 1
            b = m
            print('? '+str(a) + ' ' + str(b))
            area = int(input())
            if area == -1:
                print('wrong answer')
                err = True
                break
            if area != a*b:
                r = b-1
            else:
                l = b+1
            q-=1
        if not err:
            print('! ' + str(l))
            t-=1
        else:
            break
ruler()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
2

18

9999
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
def ruler():
    t = int(input())
    while t:
        q = 10
        l = 1
        r = 1000
        err = False
        while q and l < r:
            m = (l+r)//2
            a = (l+m)//2
            b = (r+m)//2
            print('? '+str(a) + ' ' + str(b))
            area = int(input())
            if area == -1:
                print('wrong answer')
                err = True
                break
            if area == (a+1)*(b+1):
                r = a-1
            elif area == a*(b+1):
                l = a
                r = b-1
            elif area == a*b:
                l = b
            q-=1
        if not err:
            print('! ' + str(r+1))
            t-=1
        else:
            break
ruler()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
2

20

20
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
import java.util.Scanner;

public class test1 {
    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        int cases = in.nextInt();

        for (int _i = 0; _i < cases; _i++) {
            sol(1, 999, in);
        }
    }

    public static void sol(int l, int r, Scanner in) {
        System.out.println("? " + (l + r) / 2 + " " + r);
        System.out.flush();
        int n = in.nextInt();
        if (l == r) {
            System.out.println("! " + l);
            System.out.flush();
            return;
        }
        if (n == (l + r) / 2 * (r + 1)) {
            sol((l + r) / 2 + 1, r, in);
        } else {
            sol(l, (l + r) / 2 - 1, in);
        }

    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3

12

10000

2016
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
import java.util.Scanner;

public class Main{
    static Scanner sc = new Scanner(System.in);
    public static void main(String[] args) {

        int t = sc.nextInt();
        for (int i = 0; i < t; i++) {
            int start = 1;
            int end = 1000;
            dfs(start,end);
        }
        sc.close();

    }
    public static void dfs(int start,int end){
        if (start==end || start == end+1){
            System.out.println("! " + end);
            return;
        }

        int e1 = (start+end)/2;
        int s2 = e1+1;

        int res = ask(start,e1);
        if (start * (e1+1) == res){
            dfs(start,e1);
        }else {
            dfs(s2,end);
        }

    }
    public static int ask(int start,int end){
        System.out.println("? "+start+" "+end);
        System.out.flush();
        return sc.nextInt();
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
2

9999

10000
```

### Explanation:
1. **Test Case 1:**
   - The secret value of \( x \) is 100.
   - The query \( ? 99 100 \) will return \( 99 \times 101 = 9999 \).
   - The buggy code will incorrectly assume \( x \) is 100 and output \( ! 100 \).

2. **Test Case 2:**
   - The secret value of \( x \) is 101.
   - The query \( ? 100 101 \) will return \( 101 \times 102 = 10302 \).
   - The buggy code will incorrectly assume \( x \) is 101 and output \( ! 101 \).

This test input is designed to expose the bug in the buggy code where it incorrectly handles the edge cases near the upper limit of \( x \). The code fails to correctly identify the value of \( x \) when it is close to 100 or 101.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
#include<iostream>
#include<algorithm>
#include<vector>
#include<set>
#include<unordered_set>
#include<string>
#include <string.h>
#include <bitset>
#include <queue>

using namespace std;

const int MOD = 1e9 + 7;

int x = 4;
int test(int a, int b) {
    if(a >= x) a++;
    if(b >= x) b++;
    return a * b;
}

int ask(int a, int b) {
    cout << "? " << a << " " << b << endl;
    int val;
    cin >> val;
    return val;
}

void solve() {
    int l = 2, r = 999;
    int ans = -1;
    int cnt = 0;
    while(l + 2 <= r){
        int a = l + max(1, (r - l) / 3);
        int b = l + max(1, (r - l) / 3) * 2;
        int val = ask(a, b);
        cnt++;
        if(val == (a + 1) * (b + 1)) { // x <= a, x <= b
            ans = a;
            r = a - 1;
        } else if(val == (a * b)) { // x > a, x > b
            l = b + 1;
        } else if(val == a * (b + 1)) { // x <= b, x > a
            ans = b;
            l = a + 1;
            r = b - 1;
        }
    }

    if(l <= r) {
        if(l == r) ans = l;
        else if(l + 1 == r) {
            int val = ask(l, l);
            if(val == (l + 1) * (l + 1)) ans = l;
            else ans = r;
        } else {
            int val = ask(l, l + 1);
            if(val == (l + 1) * ((l + 1) + 1)) ans = l;
            else if(val == (l * (l + 1))) ans = l + 1;
            else ans = r;
        }
    }
    cout << "! " << ans << endl;
}

int main() {
    int t = 1;
    std :: cin >> t;

    while(t--) {
        solve();
    }
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
2

18

25
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
/**
 you can ^_^ (•‿•)
( إن الله وملائكته يصلون على النبي يا أيها الذين آمنوا صلوا عليه وسلموا تسليما )
**/

#include <bits/stdc++.h>
#include <cstdio>
#include <cstring>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
//#include <ext/pb_ds/detail/standard_policies.hpp> // get all type of pd |

using namespace std;
using namespace __gnu_pbds ;

#define line "\n"
#define ll long long
#define pr pair< ll,ll >
typedef tree<int, null_type, less<int> , rb_tree_tag, tree_order_statistics_node_update > ordered_set;

void fast(){ ios_base::sync_with_stdio(false); cin.tie(0); cout.tie(0); }

const ll N = 2*1e5+5;
ll mod = 1e18+7;

void solve()
{
  int n = 0 ,mid=1000,ans=0;
  int l = 0 , r = 1000;
  while (l<=r)
  {
    mid = (l+r)/2;
    cout <<"? " << 1 << " " << mid << endl ;

    cin >> n ;
    if (n>mid)
    {
      r = mid-1;
      ans = mid ;
    }
    else
      l = mid+1;
  }
  cout <<"! " << ans << endl ;

}

bool f = 1;
int main()
{
  fast();

  int t = 1;
  if (f) cin >> t;
  while (t--)
    solve();
}

/**

 you can ^_^ (•‿•) .
( إن الله وملائكته يصلون على النبي يا أيها الذين آمنوا صلوا عليه وسلموا تسليما )
**/

/*

*/

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
2

10

25
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>

using namespace std;
using namespace __gnu_pbds;
#define vi vector<int>
#define int long long
#define pb(x) push_back(x)
#define vvi vector<vector<int>>
#define vvc vector<vector<char>>
#define vii vector<pair<int, int>>
#define pii pair<int, int>
#define vc vector<char>
#define ss second
#define ff first

#define MOD 1000000007
#define MOD1 998244353
#define INF 1e18
#define mp make_pair
#define PI 3.141592653589793238462
typedef tree<int, null_type, less<int>, rb_tree_tag, tree_order_statistics_node_update> pbds;

void solve() {
    int low = 2, high = 1000, mid;

    while (low <= high) {
        mid = low + (high - low) / 2;

        cout << "? 1 " << mid << endl;
        cout.flush();

        int ans;
        cin >> ans;

        if (ans == mid) {
            low = mid + 1;
        } else {
            high = mid - 1;
        }
    }

    cout << "! " << mid+1 << endl;
    cout.flush();
}

signed main() {
    ios::sync_with_stdio(0);
    cin.tie(0);

    int t;
    cin >> t;

    for (int p = 0; p < t; ++p) {
        solve();
    }

    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3

2

1000

999
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
from sys import stdout
import random

def solve():
    r = 500
    l_r = 2
    r_r = 999
    turn = 10
    # ans = random.randint(2, 1000)
    # print('目标值','',ans)
    while turn > 0:
        print('? ', 1, ' ', r)
        s = int(input())
        if s > r:
            r_r = r
            r = (l_r + r)//2
        else:
            l_r = r
            r = (r+r_r)//2
        stdout.flush()
        turn -= 1
    print('! ', r+1)

for _ in range(int(input())):
    solve()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3

10000

10000

10000
```

### Explanation:
This test input is designed to challenge the buggy code by providing a consistent large response that the code might misinterpret. Here's how it works:

1. **Multiple Test Cases**: The input starts with `3`, indicating that there are three test cases.
2. **Consistent Large Response**: For each test case, the response is `10000`. This is a large number that the buggy code might not handle correctly, especially since the code uses a binary search approach and might not properly adjust the `l_r` and `r_r` values based on the response.

### Why This Input is Challenging:
- **Edge Case Handling**: The large response `10000` can be a result of measuring a rectangle where one side is just below the missing number `x` and the other side is just above `x`. For example, if `x = 100`, a `99 x 100` rectangle would be measured as `99 x 101`, giving an area of `10000`.
- **Binary Search Misinterpretation**: The buggy code might incorrectly adjust the search range based on this large response, leading to incorrect final values for `x`.

By providing this input, you can expose the logic flaws in the binary search implementation and force the code to make incorrect assumptions, ultimately leading to an incorrect determination of `x`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
import sys

def main():
    t = int(input())

    for _ in range(t):
        result = query(3, 5)
        if result == 18:
            x = 4
        else:
            x = 100

        print(f'! {x}')
        sys.stdout.flush()

def query(a, b):
    print(f'? {a} {b}')
    sys.stdout.flush()
    return int(input())

if __name__ == '__main__':
    main()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided buggy code, we need to focus on the logic that determines the value of `x`. The code currently assumes that if the result of the query `? 3 5` is `18`, then `x` must be `4`. Otherwise, it assumes `x` is `100`.

However, this logic is flawed because it does not consider other possible values of `x` that could also result in the same output. Specifically, the code fails to handle cases where `x` is between `5` and `999`.

Let's generate a test input that exposes this bug:

1. **Case 1: `x = 5`**
   - For `? 3 5`, the ruler will measure the sides as `3` and `6`, resulting in an area of `18`.
   - The code will incorrectly assume `x = 4` and output `! 4`.

2. **Case 2: `x = 100`**
   - For `? 3 5`, the ruler will measure the sides as `3` and `5`, resulting in an area of `15`.
   - The code will correctly assume `x = 100` and output `! 100`.

To ensure the test input is small-scale and complete, we will include both cases:

```plaintext
2

18

15
```

This test input will:
- First, test the case where `x = 5`, which should result in the buggy code outputting `! 4`.
- Second, test the case where `x = 100`, which should result in the buggy code outputting `! 100`.

By including both cases, we can clearly demonstrate the flaw in the logic of the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
import java.io.*;
import java.util.*;

public class Test {
    static BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
    static PrintWriter pr = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
    static StringTokenizer st;
    static int MOD = 998244353;
    public static void main(String[] args) throws IOException {
        int T = readInt();
        for (int t = 0; t < T; t++) {
            int lo = 2, hi = 999, ans = -1;
            while (lo <= hi) {
                int mid = lo + hi >> 1;
                System.out.println(String.format("? %d %d", mid, mid));
                System.out.flush();
                int area = readInt();
                if (area == mid * mid) {
                    ans = mid; lo = mid + 1;
                } else {
                    hi = mid - 1;
                }
            }
            System.out.printf("! %d%n", ans + 1);
        }
    }
    static String next() throws IOException {
        while (st == null || !st.hasMoreTokens())
            st = new StringTokenizer(br.readLine().trim());
        return st.nextToken();
    }
    static long readLong() throws IOException {
        return Long.parseLong(next());
    }
    static int readInt() throws IOException {
        return Integer.parseInt(next());
    }
    static double readDouble() throws IOException {
        return Double.parseDouble(next());
    }
    static char readCharacter() throws IOException {
        return next().charAt(0);
    }
    static String readLine() throws IOException {
        return br.readLine().trim();
    }
    static int readLongLineInt() throws IOException {
        int x = 0, c;
        while((c = br.read()) != ' ' && c != '\n')
            x = x * 10 + (c - '0');
        return x;
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on scenarios where the binary search logic fails to correctly identify the missing number \( x \). The bug in the code is that it assumes the area reported by the ruler is always \( mid \times mid \) if \( x \) is not found, and then it adjusts the search range accordingly. However, this assumption can lead to incorrect results, especially when the ruler's measurement is off by one due to the missing number \( x \).

### Step-by-Step Reasoning:
1. **Binary Search Logic**: The code performs a binary search to find \( x \). It queries the area of a square with side length \( mid \).
2. **Measurement Error**: The ruler reports \( y + 1 \) if \( y \geq x \). This means if \( mid \geq x \), the area will be \( (mid + 1)^2 \) instead of \( mid^2 \).
3. **Misleading Result**: If the ruler reports \( (mid + 1)^2 \), the code incorrectly updates the search range, leading to potential misidentification of \( x \).

### Test Case to Expose the Bug:
We need a test case where the ruler's measurement is off by one, causing the binary search to incorrectly identify \( x \). A good candidate is when \( x \) is close to the middle of the search range, and the ruler's measurement for \( mid \) is \( (mid + 1)^2 \).

### Example Test Case:
Let's choose \( x = 5 \) and see how the code behaves.

### Test Input:
```plaintext
1

25
```

### Explanation:
- **Test Case 1**: The missing number \( x \) is 5.
- **Query 1**: The code will query the area of a \( 5 \times 5 \) rectangle.
- **Response**: The ruler will measure the side length as \( 6 \) (since \( 5 \geq 5 \)), and the area will be \( 6 \times 6 = 36 \).
- **Code Behavior**: The code will see that the area is not \( 25 \) (which is \( 5 \times 5 \)), and it will incorrectly update the search range, leading to a wrong conclusion.

### Expected Output:
```plaintext
? 5 5
! 6
```

### Actual Output (Correct):
```plaintext
? 5 5
! 5
```

This test case should expose the bug in the binary search logic, where the code incorrectly identifies \( x \) due to the measurement error.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;

public class G_Ruler_Easy_Version {
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter pw = new PrintWriter(System.out, true);

        int t = Integer.parseInt(br.readLine());
        int l = 2, r = 999;
        int ans = 0;
        while(t-- > 0) {
            l = 2; r = 999;
            while (r - l > 1) {
                int mid1 = (2 * l + r) /3;
                int mid2 = (l + 2 * r) /3;
                pw.printf("? %d %d\n", mid1, mid2);
                pw.flush();

                int response = Integer.parseInt(br.readLine());

                //both give correct values -> x is after mid1 and mid2
                if (response == mid1 * mid2) { //because a*b/a = b and a*b/b = b | if we get any value gt a or b -> we got back wrong area
                    l = mid2 + 1 ;
                } //both give wrong value -> x is before both mid1 and mid2
                else if (response == (mid1 + 1) * (mid2 + 1)) {
                    r = mid1;
                } //one gives wrong values and another correct -> between mid1 and mid2
                else if (response == (mid1) * (mid2 + 1)) {
                    l = mid1;
                    r = mid2;
                }
            }

            pw.println("! " + r);
        }
        pw.flush();
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1

10000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
int n,m,t;
int N=2e5+10;
signed main(){
	int t;
	scanf("%d",&t);

	while(t--){
		int l=333,r=666,m=999,z=2;
	while(r-l>=3){
		cout<<"? "<<l<<" "<<r<<endl;
//		printf("? %d %d",l,r);
		fflush(stdout);

		int s;
		cin>>s;
		if(s==l*r){
			int tl=l,tr=r;
			z=r;
			l=tr+(m-tr)/3;
			r=tr+(m-tr)/3*2;
		}
		else if(s==l*(r+1)){
			z=l;
			int tl=l,tr=r;
			l=tl+(tr-tl)/3;
			r=tl+(tr-tl)/3*2;
			m=r;
		}
		else {
			int tl=l,tr=r;
			r=z+(tl-z)/3*2;
			l=z+(tl-z)/3;

			m=tl;
		}

	}
	int s=0;

	while(s!=z*(z+2)){
		z++;
//		printf("? %d %d",z,z+1);
		cout<<"? "<<z<<" "<<z+1<<endl;
		fflush(stdout);

		cin>>s;

	}

//	printf("! %d\n",m);
//	printf("! %d\n",l);
//	printf("! %d\n",r);
//	printf("! %d\n",z+1);
cout<<"! "<<z+1<<endl;
	fflush(stdout);
	}

	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that can maximize the failure probability. The code uses a ternary search to narrow down the value of \( x \). The key points to consider are:

1. **Initial Range and Search Strategy**: The code initializes the range with \( l = 333 \) and \( r = 666 \), and uses a ternary search to narrow down the range. This can fail if \( x \) is outside this initial range or if the ternary search logic is flawed.

2. **Ternary Search Logic**: The code updates the range based on the measured area, but the logic for updating the range might not handle all cases correctly, especially near the boundaries.

3. **Final Search for \( x \)**: The code uses a linear search to find the exact value of \( x \) once the range is narrowed down. This can fail if the range is not correctly narrowed or if the final search logic is incorrect.

Given these considerations, a challenging test input would be one where \( x \) is close to the boundaries of the initial range or where the ternary search logic might not handle the updates correctly. Here is a test input that can expose such bugs:

```plaintext
2

9999

10202
```

Explanation:
- The first test case has \( x = 100 \). The initial range \( [333, 666] \) does not include 100, so the ternary search will not correctly identify \( x \).
- The second test case has \( x = 101 \). This is also outside the initial range, and the ternary search will likely fail to find \( x \).

This input will challenge the code to handle cases where \( x \) is outside the initial range and will expose any issues in the range update logic.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ruler (easy version)
time_limit_ms: 1000
memory_limit_mb: 256
Description: This is the easy version of the problem. The only difference between the two versions is that in this version, you can make at most $$$\mathbf{10}$$$ queries.

This is an interactive problem. If you are unsure how interactive problems work, then it is recommended to read the guide for participants.

We have a secret ruler that is missing one number $$$x$$$ ($$$2 \leq x \leq 999$$$). When you measure an object of length $$$y$$$, the ruler reports the following values:

- If $$$y < x$$$, the ruler (correctly) measures the object as having length $$$y$$$.
- If $$$y \geq x$$$, the ruler incorrectly measures the object as having length $$$y+1$$$.

The ruler above is missing the number $$$4$$$, so it correctly measures the first segment as length $$$3$$$ but incorrectly measures the second segment as length $$$6$$$ even though it is actually $$$5$$$.

You need to find the value of $$$x$$$. To do that, you can make queries of the following form:

- $$$\texttt{?}~a~b$$$ — in response, we will measure the side lengths of an $$$a \times b$$$ rectangle with our ruler and multiply the results, reporting the measured area of the rectangle back to you. For example, if $$$x=4$$$ and you query a $$$3 \times 5$$$ rectangle, we will measure its side lengths as $$$3 \times 6$$$ and report $$$18$$$ back to you.

Find the value of $$$x$$$. You can ask at most $$$\mathbf{10}$$$ queries.

Input: Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

Output: None

Examples:
input:
2

18

25

9999
output:
? 3 5

? 4 4

! 4
? 99 100

! 100

Note: In the first test, the interaction proceeds as follows.

SolutionJuryExplanation$$$\texttt{2}$$$There are 2 test cases.$$$\texttt{? 3 5}$$$$$$\texttt{18}$$$Secretly, the jury picked $$$x=4$$$. The solution requests the $$$3 \times 5$$$ rectangle, and the jury responds with $$$3 \times 6 = 18$$$, as described in the statement.$$$\texttt{? 4 4}$$$$$$\texttt{25}$$$The solution requests the $$$4 \times 4$$$ rectangle, which the jury measures as $$$5 \times 5$$$ and responds with $$$25$$$.$$$\texttt{! 4}$$$The solution has somehow determined that $$$x=4$$$, and outputs it. Since the output is correct, the jury continues to the next test case.$$$\texttt{? 99 100}$$$$$$\texttt{1}$$$Secretly, the jury picked $$$x=100$$$. The solution requests the $$$99 \times 100$$$ rectangle, which the jury measures as $$$99 \times 101$$$ and responds with $$$9999$$$.$$$\texttt{! 100}$$$The solution has somehow determined that $$$x=100$$$, and outputs it. Since the output is correct and there are no more test cases, the jury and the solution exit.

Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

Buggy Code:
#include <bits/stdc++.h>

using namespace std;

int main() {
    int tt;
    cin >> tt;
    while (tt--) {
        int x = 499;
        int res, iter = 252;
        bool g = false;
        if (iter < 2) iter = 1;
        for (int i = 1; i < 11; i++) {
            if (x > 999) x = 999;
            if (x < 2) x = 2;
            cout << "? " << 1 << ' ' << x << '\n';
            cout.flush();
            cin >> res;
            if (res == x) {
                x += iter;
                g = false;
            }
            else {
                x -= iter;
                g = true;
            }
            if (iter >= 2) {
                if (iter % 2 == 0) iter /= 2;
                else iter = (iter / 2) + 1;
            }
        }

        if (x > 999) x = 999;
        if (x < 2) x = 2;
        cout << "! " << x << '\n';
        // 571
    }

    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
1

18

25

9999
```