[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**. 

```C
1   #include <stdio.h>
2   int A[1005];
3   int ori[1005];
4   char res[1005][1005];
5   int cnt[1005][1005];
6   int max(int a, int b){ return a > b ? a : b; }
7   int main(){
8       int T;
9       scanf("%d", &T);
10      for(int tc=0; tc<T; tc++){
11          int C;
12          scanf("%d", &C);
13          for(int i=1; i<=C; i++) {
14              scanf("%d", &A[i]);
15              ori[i] = A[i];
16          }
17          int mx = 0;
18          for(int i=1; i<=C; i++) 
19              mx = max(mx, A[i]);
20          printf("Case #%d: ", tc+1);
21          if(A[1] == 0 || A[C] == 0){
22              printf("IMPOSSIBLE\n");
23              continue;
24          }
25          int R = 0;
26          for(int i=1; i<=C; i++) 
27              for(int j=1; j<=C; j++) 
28                  res[i][j] = '.';
29          int cur = 0;
30          int cons = 0;
31          for(int i=1; i<=C; i++){
32              if(cur >= 1){
33                  int fi = 0;
34                  for(int j=1; j<C; j++) 
35                      if(A[j]){
36                          fi = j;
37                          break;
38                      }
39                  for(int j=i; j>fi; j--){
40                      res[i-j+1][j] = '/';
41                      R = max(R, i-j+1);
42                  }
43                  A[fi]--;
44                  cur--;
45                  cur += A[i];
46                  cons = 0;
47              } else{
48                  cur += A[i];
49                  if(cur >= 1){
50                      cur--;
51                      A[i] = cur;
52                      cons = 0;
53                  } else{
54                      cons++;
55                      for(int j=1; j<=-cur+1; j++) 
56                          res[j][i] = '\\';
57                      R = max(R, -cur+1);
58                      cur--;
59                  }
60              }
61          }
62          for(int i=1; i<=R+1; i++) 
63              for(int j=1; j<=C; j++)
64                  cnt[i][j] = 0;
65          for(int i=1; i<=C; i++) 
66              cnt[1][i] = 1;
67          for(int i=1; i<=R; i++){
68              for(int j=1; j<=C; j++){
69                  if(res[i][j] == '.') 
70                      cnt[i+1][j] += cnt[i][j];
71                  else if(res[i][j] == '\\') 
72                      cnt[i+1][j+1] += cnt[i][j];
73                  else if(res[i][j] == '/') 
74                      cnt[i+1][j-1] += cnt[i][j];
75              }
76          }
77          int valid = 1;
78          for(int i=1; i<=C; i++){
79              if(res[R+1][i] != ori[i]){
80                  valid = 0;
81              }
82          }
83          if(!valid){
84              printf("IMPOSSIBLE\n");
85              continue;
86          }
87          printf("%d\n", R+1);
88          for(int i=1; i<=R; i++){
89              for(int j=1; j<=C; j++) 
90                  printf("%c", res[i][j]);
91              printf("\n");
92          }
93          for(int i=1; i<=C; i++) 
94              printf(".");
95          printf("\n");
96      }
97      return 0;
98  }
```


**Question**: Does line `83` have control dependence over line `29` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no