[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<string.h>
4   #include<math.h>
5   #define sl(x) scanf("%ld", &x)
6   #define ss(x) scanf("%s", x)
7   #define pl(x) printf("%ld\n", x)
8   #define pd(x) printf("%lf\n", x)
9   #define ps(x) printf("%s\n", x)
10  #define INF 1000000000000000000
11  #define MOD 1000000007
12  #define MIN(a,b) (((a)<(b))?(a):(b))
13  #define MAX(a,b) (((a)>(b))?(a):(b))
14  int compare_long(const void *a, const void *b) {
15  	return *(long*)a - *(long*)b;
16  }
17  int main() {
18  	long N, M;
19  	long a[300001];
20  	char s[100010];
21  	long i, j;
22  	char ans[100010];
23  	long pre, next;
24  	scanf("%ld", &N);
25  	scanf("%s", s);
26  	for(i=0;i<N;i++) 
27  		ans[i] = 'S';
28  	ans[N] = '\0';
29  	for(i=0;i<N-1;i++) {
30  		pre = (i + N - 1) % N;
31  		next = (i + 1) % N;
32  		if(ans[i] == 'S') {
33  			if(s[i] == 'o' && ans[pre] != ans[next]) {
34  				ans[next] = ans[pre];
35  			}
36  			if(s[i] == 'x' && ans[pre] == ans[next]) {
37  				ans[next] = ans[next] == 'S' ? 'W' : 'S';
38  			}
39  		} else {
40  			if(s[i] == 'o' && ans[pre] == ans[next]) {
41  				ans[next] = ans[next] == 'S' ? 'W' : 'S';
42  			}
43  			if(s[i] == 'x' && ans[pre] != ans[next]) {
44  				ans[next] = ans[pre];
45  			}
46  		}
47  	}
48  	if(ans[N-1] == 'S') {
49  		if(s[N-1] == 'o' && ans[N-2] == ans[0]) {
50  			printf("%s\n", ans);
51  			return 0;
52  		}
53  		if(s[N-1] == 'x' && ans[N-2] != ans[0]) {
54  			printf("%s\n", ans);
55  			return 0;
56  		}
57  	} else {
58  		if(s[N-1] == 'o' && ans[N-2] != ans[0]) {
59  			printf("%s\n", ans);
60  			return 0;
61  		}
62  		if(s[N-1] == 'x' && ans[N-2] == ans[0]) {
63  			printf("%s\n", ans);
64  			return 0;
65  		}
66  	}
67  	for(i=0;i<N;i++) 
68  		ans[i] = 'W';
69  	ans[N] = '\0';
70  	for(i=0;i<N-1;i++) {
71  		pre = (i + N - 1) % N;
72  		next = (i + 1) % N;
73  		if(ans[i] == 'S') {
74  			if(s[i] == 'o' && ans[pre] != ans[next]) {
75  				ans[next] = ans[pre];
76  			}
77  			if(s[i] == 'x' && ans[pre] == ans[next]) {
78  				ans[next] = ans[next] == 'S' ? 'W' : 'S';
79  			}
80  		} else {
81  			if(s[i] == 'o' && ans[pre] == ans[next]) {
82  				ans[next] = ans[next] == 'S' ? 'W' : 'S';
83  			}
84  			if(s[i] == 'x' && ans[pre] != ans[next]) {
85  				ans[next] = ans[pre];
86  			}
87  		}
88  	}
89  	if(ans[N-1] == 'S') {
90  		if(s[N-1] == 'o' && ans[N-2] == ans[0]) {
91  			printf("%s\n", ans);
92  			return 0;
93  		}
94  		if(s[N-1] == 'x' && ans[N-2] != ans[0]) {
95  			printf("%s\n", ans);
96  			return 0;
97  		}
98  	} else {
99  		if(s[N-1] == 'o' && ans[N-2] != ans[0]) {
100  			printf("%s\n", ans);
101  			return 0;
102  		}
103  		if(s[N-1] == 'x' && ans[N-2] == ans[0]) {
104  			printf("%s\n", ans);
105  			return 0;
106  		}
107  	}
108  	ps("-1");
109  	return 0;
110  }
```


**Question**: Does line `43` have control dependence over line `68` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no