[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.util.*;
2   import java.io.*;
3   public class Solution {
4   	public static void main(String[] args) {
5   		Solution run = new Solution();
6   		run.run();
7   	}
8   	public void run() {
9   		Scanner in = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
10  		int numCases = in.nextInt();
11  		StringBuilder build = new StringBuilder();
12  		for (int c=1; c<=numCases; c++) {
13  			build.append(String.format("Case #%d: ",c));
14  			int n = in.nextInt();
15  			double p = in.nextInt();
16  			double[][] dims = new double[n][2];
17  			double minP = 0;
18  			for (int i=0; i<n; i++) {
19  				int first = in.nextInt();
20  				int second = in.nextInt();
21  				dims[i][0] = Math.min(first,second);
22  				dims[i][1] = Math.max(first,second);
23  				minP += 2 * (first + second);
24  			}
25  			boolean small = true;
26  			smallCheck: for (int i=1; i<n; i++) {
27  				for (int j=0; j<2; j++) {
28  					if (dims[i][j] != dims[0][j]) {
29  						small = false;
30  						break smallCheck;
31  					}
32  				}
33  			}
34  			if (minP == p) {
35  				build.append(String.format("%.7f\n",p));
36  			}
37  			else {
38  				double maxRem = p - minP;
39  				if (small) {
40  					int minHalf=0;
41  					for (; minHalf<n; minHalf++) {
42  						if (maxRem >= dims[0][0] * 2) {
43  							maxRem -= dims[0][0] * 2;
44  						}
45  						else {
46  							break;
47  						}
48  					}
49  					if (maxRem == 0) {
50  						build.append(String.format("%.7f\n",p));
51  					}
52  					else {
53  						int maxHalf=0;
54  						double maxAdd = Math.sqrt(dims[0][0] * dims[0][0] + dims[0][1] * dims[0][1]) - dims[0][0];
55  						for (; maxHalf<minHalf; maxHalf++) {
56  							if (maxRem >= maxAdd * 2) {
57  								maxRem -= maxAdd * 2;
58  							}
59  							else {
60  								break;
61  							}
62  						}
63  						if (maxHalf < minHalf) {
64  							maxRem = Math.max(0.0,maxRem - maxAdd * 2);
65  						}
66  						build.append(String.format("%.7f\n",p - maxRem));
67  					}
68  				}
69  				else {
70  					build.append("\n");
71  				}
72  			}
73  		}
74  		System.out.printf("%s",build);
75  	}
76  }
```


**Question**: Does line `41` have control dependence over line `54` in function `run`? If so, provide a trace.

**Output**:


Ground truth answer: yes