[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.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**: Does line `44` have control dependence over line `52` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes