[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**. 

```Java
1   import java.io.*;
2   import java.util.*;
3   public class Main{
4   	static void main() throws Exception{
5   		int n=sc.nextInt(),m=sc.nextInt(),k=sc.nextInt();
6   		int[]start=new int[2],end=new int[2];
7   		for(int i=0;i<2;i++) {
8   			start[i]=sc.nextInt()-1;
9   		}
10  		for(int i=0;i<2;i++) {
11  			end[i]=sc.nextInt()-1;
12  		}
13  		char[][]in=new char[n][m];
14  		for(int i=0;i<n;i++)
15  			in[i]=sc.nextLine().toCharArray();
16  		int inf=(int)1e9;
17  		int[][][]dist=new int[n][m][4];
18  		for(int[][]i:dist)
19  			for(int[]j:i)
20  				Arrays.fill(j, inf);
21  		LinkedList<int[]>q=new LinkedList<>();
22  		q.add(new int[] {start[0],start[1],k,-1,0});
23  		dist[start[0]][start[1]][0]=0;
24  		dist[start[0]][start[1]][1]=0;
25  		dist[start[0]][start[1]][2]=0;
26  		dist[start[0]][start[1]][3]=0;
27  		int[]dx= {0,1,0,-1},dy= {1,0,-1,0};
28  		while(!q.isEmpty()) {
29  			int cur[]=q.poll();
30  			int x=cur[0],y=cur[1],curK=cur[2],dir=cur[3];
31  			int curDist=cur[4];
32  			for(int i=0;i<4;i++) {
33  				int xx=x+dx[i],yy=y+dy[i];
34  				if(xx<0 || xx>=n || yy<0 || yy>=m || dist[xx][yy][(i)]!=inf || in[xx][yy]=='@')continue;
35  				int[]nxt=new int[] {xx,yy,curK-1,(i),0};
36  				if(nxt[2]<0) {
37  					nxt[2]=k-1;
38  					dist[xx][yy][(i)]=curDist+1;
39  				}
40  				else {
41  					if(dir==(i)) {
42  						dist[xx][yy][(i)]=curDist;
43  					}
44  					else {
45  						dist[xx][yy][(i)]=curDist+1;
46  						nxt[2]=k-1;
47  					}
48  				}
49  				nxt[4]=dist[xx][yy][(i)];
50  				q.add(nxt);
51  			}
52  		}
53  		int ans=inf;
54  		for(int i=0;i<4;i++) {
55  			ans=Math.min(ans, dist[end[0]][end[1]][i]);
56  		}
57  		pw.println(ans==inf?-1:ans);
58  	}
59  	public static void main(String[] args) throws Exception{
60  		pw=new PrintWriter(System.out);
61  		sc = new MScanner(System.in);
62  			main();
63  		pw.flush();
64  	}
65  	static PrintWriter pw;
66  	static MScanner  sc;
67  	static class MScanner {
68  		StringTokenizer st;
69  		BufferedReader br;
70  		public MScanner(InputStream system) {
71  			br = new BufferedReader(new InputStreamReader(system));
72  		}
73  		public MScanner(String file) throws Exception {
74  			br = new BufferedReader(new FileReader(file));
75  		}
76  		public String next() throws IOException {
77  			while (st == null || !st.hasMoreTokens())
78  				st = new StringTokenizer(br.readLine());
79  			return st.nextToken();
80  		}
81  		public int[] intArr(int n) throws IOException {
82  	        int[]in=new int[n];for(int i=0;i<n;i++)in[i]=nextInt();
83  	        return in;
84  		}
85  		public long[] longArr(int n) throws IOException {
86  	        long[]in=new long[n];for(int i=0;i<n;i++)in[i]=nextLong();
87  	        return in;
88  		}
89  		public int[] intSortedArr(int n) throws IOException {
90  	        int[]in=new int[n];for(int i=0;i<n;i++)in[i]=nextInt();
91  	        shuffle(in);
92  	        Arrays.sort(in);
93  	        return in;
94  		}
95  		public long[] longSortedArr(int n) throws IOException {
96  	        long[]in=new long[n];for(int i=0;i<n;i++)in[i]=nextLong();
97  	        shuffle(in);
98  	        Arrays.sort(in);
99  	        return in;
100  		}
101  		public Integer[] IntegerArr(int n) throws IOException {
102  	        Integer[]in=new Integer[n];for(int i=0;i<n;i++)in[i]=nextInt();
103  	        return in;
104  		}
105  		public Long[] LongArr(int n) throws IOException {
106  	        Long[]in=new Long[n];for(int i=0;i<n;i++)in[i]=nextLong();
107  	        return in;
108  		}
109  		public String nextLine() throws IOException {
110  			return br.readLine();
111  		}
112  		public int nextInt() throws IOException {
113  			return Integer.parseInt(next());
114  		}
115  		public double nextDouble() throws IOException {
116  			return Double.parseDouble(next());
117  		}
118  		public char nextChar() throws IOException {
119  			return next().charAt(0);
120  		}
121  		public long nextLong() throws IOException {
122  			return Long.parseLong(next());
123  		}
124  		public boolean ready() throws IOException {
125  			return br.ready();
126  		}
127  		public void waitForInput() throws InterruptedException {
128  			Thread.sleep(3000);
129  		}
130  	}
131  	static void shuffle(int[]in) {
132  		for(int i=0;i<in.length;i++) {
133  			int idx=(int)(Math.random()*in.length);
134  			int tmp=in[i];
135  			in[i]=in[idx];
136  			in[idx]=tmp;
137  		}
138  	}
139  	static void shuffle(long[]in) {
140  		for(int i=0;i<in.length;i++) {
141  			int idx=(int)(Math.random()*in.length);
142  			long tmp=in[i];
143  			in[i]=in[idx];
144  			in[idx]=tmp;
145  		}
146  	}
147  }
```


**Question**: Does line `32` have control dependence over line `46` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes