[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 <stdlib.h>
3   #include <math.h>
4   int max1(int a, int b) {
5   	if (a > b)
6   		return a;
7   	else
8   		return b;
9   }
10  int main()
11  {
12  	int N = 0;
13  	int *A,*B,*C;
14  	int SUM = 0;
15  	int sumA = 0, sumB = 0, sumC = 0;
16  	char ref = 0;
17  	int N_limit = pow(10, 5);
18  	if (scanf("%d", &N) <= 0) {
19  		exit(1);
20  	}
21  	if ( N <= 0 || N_limit < N)
22  		exit(1);
23  	A = (int *)malloc(N * sizeof(int));
24  	B = (int *)malloc(N * sizeof(int));
25  	C = (int *)malloc(N * sizeof(int));
26  	for (int i = 0; i < N; i++)
27  	{
28  		if (scanf("%d%d%d", &A[i], &B[i], &C[i]) < 3) {
29  			exit(1);
30  		}
31  		if (A[i] <= 0 || pow(10, 4) < A[i])
32  			exit(1);
33  		else if (B[i] <= 0 || pow(10, 4) < B[i])
34  			exit(1);
35  		else if (C[i] <= 0 || pow(10, 4) < C[i])
36  			exit(1);
37  	}
38  	int ref_A = 0, ref_B = 0, ref_C = 0;
39  	for (int i = 0; i < N; i++)
40  	{
41  		if (i - 1 <= 0)
42  		{
43  			sumA = A[i];
44  			sumB = B[i];
45  			sumC = C[i];
46  			ref_A = sumA;
47  			ref_B = sumB;
48  			ref_C = sumC;
49  			continue;
50  		}
51  		sumA = A[i] + max1(ref_B, ref_C);
52  		sumB = B[i]+ max1(ref_C, ref_A);
53  		sumC = C[i] + max1(ref_A, ref_B);
54  		ref_A = sumA;
55  		ref_B = sumB;
56  		ref_C = sumC;
57  	}
58  	if (sumA >= sumB)
59  	{
60  		if (sumA >= sumC)
61  		{
62  			SUM = sumA;
63  		}
64  		else
65  		{
66  			SUM = sumC;
67  		}
68  	}
69  	else if (sumB >= sumA)
70  	{
71  		if (sumB >= sumC)
72  		{
73  			SUM = sumB;
74  		}
75  		else
76  		{
77  			SUM = sumC;
78  		}
79  	}
80  	else if (sumC >= sumB)
81  	{
82  		if (sumC >= sumA)
83  		{
84  			SUM = sumC;
85  		}
86  		else
87  		{
88  			SUM = sumA;
89  		}
90  	}
91  	printf("%d", SUM);
92  }
```


**Question**: Does line `31` have control dependence over line `77` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no