[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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 `(start,6)` have data dependence over `(dist,23)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes