[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.util.*;
2   import java.io.*;
3   public class Solution {
4       public static void main(String[] args) {
5           int[] dx = { 0, 1, 0, -1 };
6           int[] dy = { 1, 0, -1, 0 };
7           Scanner sc = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
8           StringBuilder ans = new StringBuilder();
9           int T = sc.nextInt();
10          for (int t = 1; t <= T; t++) {
11              int X = sc.nextInt();
12              int Y = sc.nextInt();
13              String M = sc.next();
14              int[][] g = new int[2001][2001];
15              int px = 1000;
16              int py = 1000;
17              List<List<Integer>> tour = new ArrayList<>();
18              List<Integer> te = new ArrayList<>();
19              te.add(px);
20              te.add(py);
21              tour.add(te);
22              g[py][px] = 1;
23              for (int i = 0; i < M.length(); i++) {
24                  char d = M.charAt(i);
25                  if (d == 'N') 
26                      py--;
27                  if (d == 'S') 
28                      py++;
29                  if (d == 'E') 
30                      px++;
31                  if (d == 'W') 
32                      px--;
33                  g[py][px] = 1;
34                  te = new ArrayList<>();
35                  te.add(px);
36                  te.add(py);
37                  tour.add(te);
38              }
39              boolean[][] visited = new boolean[2001][2001];
40              int myx = 1000-X;
41              int myy = 1000+Y;
42              LinkedList<Pair> q = new LinkedList<>();
43              q.offer(new Pair(myx,myy,0));
44              int tmp = -1;
45              int nearX = -1;
46              int nearY = -1;
47              while (!q.isEmpty()) {
48                  Pair now = q.poll();
49                  if (g[now.y][now.x] == 1) {
50                      tmp = now.dist;
51                      nearX = now.x;
52                      nearY = now.y;
53                      break;
54                  }
55                  visited[now.y][now.x] = true;;
56                  for (int v = 0; v < 4; v++) {
57                      int nx = now.x+dx[v];
58                      int ny = now.y+dy[v];
59                      if (nx < 0 || visited[0].length <= nx || ny < 0 || visited.length <= ny) 
60                              continue;
61                      if (visited[ny][nx]) 
62                          continue;
63                      q.add(new Pair(nx,ny,now.dist+1));
64                  }
65              }
66              ans.append("Case #"+t+": "); 
67              if (tmp == -1) {
68                  ans.append("IMPOSSIBLE\n");
69                  continue;
70              }
71              int ti = 0;
72              for (int i = 0; i < tour.size(); i++) {
73                  List<Integer> ll = tour.get(i);
74                  if (ll.get(0) == nearX && ll.get(1) == nearY) {
75                      ti = i;
76                      break;
77                  }
78              }
79              boolean[][] visited2 = new boolean[2001][2001];
80              if (tmp <= ti) {
81                  ans.append(tmp+(ti-tmp+1)/2).append('\n');
82              } else {
83                  boolean success = false;
84                  for (int i = ti; i < tour.size() && tmp-ti+i < tour.size(); i++) {
85                      int mfx = tour.get(tmp+i-ti).get(0);
86                      int mfy = tour.get(tmp+i-ti).get(1);
87                      visited2[mfy][mfx] = true;
88                      int ffx = tour.get(i).get(0);
89                      int ffy = tour.get(i).get(1);
90                      if (visited2[ffy][ffx]) {
91                          ans.append(i).append('\n');
92                          success = true;
93                          break;
94                      }
95                  }
96                  if (!success) ans.append("IMPOSSIBLE\n");
97              }
98          }
99          System.out.print(ans);
100      }
101      private static class Pair {
102          private final int x;
103          private final int y;
104          private final int dist;
105          Pair(int x, int y, int dist) {
106              this.x = x;
107              this.y = y;
108              this.dist = dist;
109          }
110      }
111  }
```


**Question**: Does line `10` have control dependence over line `56` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes