[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   long long int gcd(long long int a, long long int b)
3   {
4   	long long int r = a % b;
5   	while (r > 0)
6   	{
7   		a = b;
8   		b = r;
9   		r = a % b;
10  	}
11  	return b;
12  }
13  int main()
14  {
15  	long long int n;
16  	scanf("%lld", &n);
17  	long long int i, j, k;
18  	char s[100];
19  	long long int a[200005];
20  	for (i = 0; i < n; i++)
21  	{
22  		scanf("%s", s);
23  		a[i] = 0;
24  		k = 0;
25  		for (j = 0; s[j] != '\0'; j++)
26  		{
27  			if (s[j] != '.')
28  			{
29  				a[i] = 10 * a[i] + (long long int)(s[j] - '0');
30  				if (k > 0)
31  					k++;
32  			}
33  			else
34  				k = 1;
35  		}
36  		if (k == 0)
37  			k = 1;
38  		for (; k < 10; k++)
39  			a[i] *= 10;
40  	}
41  	long long int q = 100000;
42  	for (i = 0; i < n; i++)
43  	{
44  		if (a[i] % q > 0)
45  			a[i] = -1;
46  		else
47  			a[i] = a[i] / q;
48  	}
49  	q = 100000000;
50  	long long int b[10004];
51  	j = 0;
52  	for (i = 1; i * i <= q; i++)
53  	{
54  		if (q % i == 0)
55  		{
56  			b[j] = i;
57  			j++;
58  		}
59  	}
60  	i--;
61  	if (i * i == q)
62  		i--;
63  	for (; i > 0; i--)
64  	{
65  		if (q % i == 0)
66  		{
67  			b[j] = q / i;
68  			j++;
69  		}
70  	}
71  	long long int min, mid, max;
72  	long long int c[10004];
73  	for (i = 0; i < j; i++)
74  		c[i] = 0;
75  	long long int ans = 0;
76  	long long int l;
77  	for (i = 0; i < n; i++)
78  	{
79  		if (a[i] < 0)
80  			continue;
81  		l = q / gcd(q, a[i]);
82  		min = -1;
83  		max = j;
84  		while (max - min > 1)
85  		{
86  			mid = (max + min) / 2;
87  			if (b[mid] < l)
88  				min = mid;
89  			else
90  				max = mid;
91  		}
92  		ans += c[max];
93  		for (k = 0; k < j; k++)
94  			if (a[i] % b[k] == 0)
95  				c[k]++;
96  	}
97  	printf("%lld\n", ans);
98  	return 0;
99  }
```


**Question**: Does line `84` have control dependence over line `86` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes