[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   #include <math.h>
3   int N;
4   typedef struct _mole {
5   	int c;
6   	int j;
7   }Mole;
8   Mole arr[305];
9   int Abs(int a) {
10  	if (a < 0)
11  		return -1 * a;
12  	else
13  		return a;
14  }
15  int gcd(int a, int b) {
16  	if (b == 0)
17  		return a;
18  	else
19  		return gcd(b, a%b);
20  }
21  int main(void) {
22  	int t, T;
23  	int i;
24  	int j;
25  	int ans;
26  	int flag;
27  	double max = -1;
28  	int ans_c, ans_j;
29  	int flag_big;
30  	scanf("%d", &T);
31  	for (t = 1; t <= T; t++) {
32  		ans = 1;
33  		scanf("%d", &N);
34  		for (i = 1; i <= N; i++)
35  			scanf("%d%d", &arr[i].c, &arr[i].j);
36  		flag = 0;
37  		flag_big = -1;
38  		for (i = 1; i < N; i++) {
39  			j = i + 1;
40  			if ((arr[i].c <= arr[j].c && arr[i].j <= arr[j].j))
41  				continue;
42  			if (arr[i].c >= arr[j].c && arr[i].j >= arr[j].j) {
43  				flag = 1;
44  				break;
45  			}
46  			double a = arr[j].c - arr[i].c;
47  			double b = arr[i].j - arr[j].j; 
48  			if (flag_big == -1) {
49  				if (a > 0)
50  					flag_big = 1;
51  				else
52  					flag_big = 0;
53  			}
54  			if (flag_big == 1) {
55  				if (a < 0) {
56  					flag = 1;
57  					break;
58  				}
59  				if (max < b / a) {
60  					max = b / a;
61  				}
62  			}
63  			else {
64  				if (a > 0) {
65  					flag = 1;
66  					break;
67  				}
68  				if (max < b / a) {
69  					max = b / a;
70  				}
71  			}
72  		}
73  		if(flag)
74  			printf("Case #%d: IMPOSSIBLE\n", t);
75  		else {
76  			if (flag_big == -1) {
77  				ans_c = 1;
78  				ans_j = 1;
79  			}
80  			else if (flag_big == 1) {
81  				ans_j = 1;
82  				ans_c = floor(max+1);
83  			}
84  			else {
85  				ans_c = 1;
86  				ans_j = floor(max + 1);
87  			}
88  			printf("Case #%d: %d %d\n", t, ans_c, ans_j);
89  		}
90  	}
91  }
```


**Question**: Does line `31` have control dependence over line `52` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes