[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

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

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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 `(i,51)` have data dependence over `(chipCoords,61)` in function `processSmall`? If so, provide a trace.

**Output**:


Ground truth answer: yes