[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 "Which variable instances have data dependence over `(targetVar, targetLine)`? List all such variables.", respond:
   ```json
   {
    "DataDependenceSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

  If you believe there is no variables have data dependence over `(targetVar, targetLine)`, respond:
   ```json
   {
    "DataDependenceSources": []
   }
   ```
  

## 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:
Which variable instances have data dependence over `(step,9)`? List all such variables.


**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
   {
    "DataDependenceSources": [
      ["step", 9],
      ["step", 3],
      ["value", 2],
    ]
   }
```





#### Example Question 1.2 
Which variable instances have data dependence over `(total,7)`? List all such variables.


**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads from `total` at line 1 or 7 (itself), and `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)`



**Output**:
```json
   {
    "DataDependenceSources": [
      ["total", 1],
      ["total", 7],
      ["value", 5],
      ["value", 2]
    ]
   }
```



### 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:
Which variable instances have data dependence over `(arr,6)`? List all such variables.



**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)`.




**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["arr", 4],
      ["arr", 1],
      ["i", 5],
      ["x", 2]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have data dependence over `(temp,7)`? List all such variables.


**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed. And `arr[i]` can only refer to `(arr,6)` as they have the same index.
- Line 5: `i` is initialized and iterates over the range 0 to 3.


**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["i", 5],
    ]
   }
```

#### Example Question 2.3:
Which variable instances have data dependence over `(result,9)`? List all such variables.

**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).
  - If it is from line 6: `arr[i] += 1`, `arr[i]` is modified based on the value of `i`. Line 5: `i` iterates over 0 to 3.
  - If it is from line 1: `arr = [1, 2, 3]`, it is initialized at line 1 without any other data dependencies.

**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["arr", 1],
      ["i", 5],
    ]
   }
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.BufferedReader;
2   import java.io.Closeable;
3   import java.io.IOException;
4   import java.io.InputStream;
5   import java.io.InputStreamReader;
6   import java.util.StringTokenizer;
7   class Solution {
8   	public static void main(String[] args){
9   		try(Scanner sc = new Scanner(System.in)){
10  			final int T = sc.nextInt();
11  			LOOP:
12  			for(int tt = 0; tt < T; tt++){
13  				final int N = sc.nextInt();
14  				System.out.printf("Case #%d:\n", tt + 1);
15  				System.out.printf("%d %d\n", 1, 1);
16  				long rest = N - 1;
17  				long depth = 1;
18  				long position = 0;
19  				while(true) {
20  					final long lines = depth + 1;
21  					final long line_sum = 1l << depth;
22  					if(rest < line_sum) {
23  						break;
24  					}
25  					if(line_sum * 2 < rest) {
26  						System.out.printf("%d %d\n", depth + 1, position + 1);
27  						rest -= 1;
28  					}else {
29  						if(position == 0) {
30  							for(long i = 0; i < lines; i++) {
31  								System.out.printf("%d %d\n", depth + 1, i + 1);
32  							}
33  							position = lines;
34  						}else {
35  							for(long i = position; i >= 0; i--) {
36  								System.out.printf("%d %d\n", depth + 1, i + 1);
37  							}
38  							position = 0;
39  						}
40  						rest -= line_sum;
41  					}
42  					depth++;
43   				}
44  				while(rest > 0) {
45  					if(position == 0) {
46  						System.out.printf("%d %d\n", depth + 1, position + 1);
47  					}else {
48  						System.out.printf("%d %d\n", depth + 1, position + 1);
49  						position++;
50  					}
51  					rest--;
52  					depth++;
53  				}
54  			}
55  		}
56  	}
57  	public static class Scanner implements Closeable {
58  		private BufferedReader br;
59  		private StringTokenizer tok;
60  		public Scanner(InputStream is) {
61  			br = new BufferedReader(new InputStreamReader(is));
62  		}
63  		private void getLine() {
64  			try {
65  				while (!hasNext()) {
66  					tok = new StringTokenizer(br.readLine());
67  				}
68  			} catch (IOException e) { 
69  			}
70  		}
71  		private boolean hasNext() {
72  			return tok != null && tok.hasMoreTokens();
73  		}
74  		public String next() {
75  			getLine();
76  			return tok.nextToken();
77  		}
78  		public int nextInt() {
79  			return Integer.parseInt(next());
80  		}
81  		public long nextLong() {
82  			return Long.parseLong(next());
83  		}
84  		public double nextDouble() {
85  			return Double.parseDouble(next());
86  		}
87  		public void close() {
88  			try {
89  				br.close();
90  			} catch (IOException e) { 
91  			}
92  		}
93  	}
94  }
```


**Question**: Which variable instances have data dependence over `(rest,51)` in function `main`? List all such variables.

**Output**:


Ground truth answer: yes