[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control Dependence
  All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.


## 4. Example Code Snippet

### Example 1

```python
1  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": false
}
```


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

**Output**:
```json
{
  "ControlDependence": false
}
```


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

**Output**:
```json
{
  "ControlDependence": false
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.BufferedReader;
2   import java.io.IOException;
3   import java.io.InputStreamReader;
4   import java.net.ServerSocket;
5   import java.util.ArrayList;
6   import java.util.List;
7   public class Solution {
8       public static void main(String[] args) {
9           BufferedReader bufferedReader = new BufferedReader(new InputStreamReader(System.in));
10          try {
11              int t = Integer.parseInt(bufferedReader.readLine());
12              List<List<String>> caseList = new ArrayList(t);
13              for (int testCaseNum = 0; testCaseNum < t; testCaseNum++) {
14                  List<String> singleCase = new ArrayList<>();
15                  String rchv = bufferedReader.readLine();
16                  singleCase.add(rchv);
17                  int n = Integer.parseInt(rchv.split(" ")[0]);
18                  for (int i =0; i<n; i++) {
19                      singleCase.add(bufferedReader.readLine());
20                  }
21                  caseList.add(singleCase);
22              }
23              int b = 0;
24              for (List<String> singleCase : caseList) {
25                  b++;
26                  if (processCase(singleCase)) {
27                      System.out.println("Case #"+b+": POSSIBLE");
28                  } else {
29                      System.out.println("Case #"+b+": IMPOSSIBLE");
30                  }
31              }
32          } catch (IOException e) {
33              e.printStackTrace();
34          }
35      }
36      public static boolean processCase(List<String> caseList) {
37          int[] rchv = new int[4];
38          String[] split = caseList.get(0).split(" ");
39          for (int i = 0; i < rchv.length; i++){
40              rchv[i] = Integer.parseInt(split[i]);
41          }
42          if (rchv[1] < 10 && rchv[0] < 10 && rchv[2] == 1 && rchv[3] == 1) {
43              return processSmall(rchv, caseList);
44          }
45          return false;
46      }
47      public static boolean processSmall(int[] rchv, List<String> caseList) {
48          List<int[]> chipCoords = new ArrayList<>();
49          List<Integer> xCoords = new ArrayList<>();
50          List<Integer> yCoords = new ArrayList<>();
51          for (int i = 0; i < rchv[0]; i++) {
52              String row = caseList.get(i+1);
53              String[] split = row.split("");
54              for (int j = 0; j < rchv[1]; j++) {
55                  if ("@".equals(split[j])) {
56                      int[] coords = new int[2];
57                      coords[0] = i;
58                      xCoords.add(i);
59                      coords[1] = j;
60                      yCoords.add(j);
61                      chipCoords.add(coords);
62                  }
63              }
64          }
65          for (int x =0; x<rchv[0]; x++) {
66              for (int y = 0; y< rchv[1]; y++) {
67                  int topright = 0;
68                  int topleft = 0;
69                  int bottomLeft = 0;
70                  int bottomRight = 0;
71                  for (int z = 0; z < chipCoords.size(); z++) {
72                      if (chipCoords.get(z)[0] <= x) {
73                          if (chipCoords.get(z)[1] <= y) {
74                              bottomLeft ++;
75                          } else {
76                              bottomRight ++;
77                          }
78                      } else {
79                          if (chipCoords.get(z)[1] < y) {
80                              topleft ++;
81                          } else {
82                              topright ++;
83                          }
84                      }
85                  }
86                  if (bottomRight == bottomLeft && bottomLeft == topleft && topleft == topright) {
87                      return true;
88                  }
89              }
90          }
91          return false;
92      }
93  }
```


**Question**: Does line `51` have control dependence over line `80` in function `processSmall`? If so, provide a trace.

**Output**:


Ground truth answer: no