[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 Solution {
4       private static Scanner in;
5       public static double roundToDecimals(double value, int places) {
6           double scale = Math.pow(10, places);
7           return Math.round(value * scale) / scale;
8       }
9       public static void xuly(){
10          double a = in.nextDouble();
11          double[] centerX = {0.5,0,0};
12          double[] centerY = {0,0.5,0};
13          double[] centerZ = {0,0,0.5};
14          double[] x0 = {0.5, 0.5, 0.5, 0.5,-0.5,-0.5,-0.5,-0.5};
15          double[] y0 = {0.5, 0.5,-0.5,-0.5, 0.5, 0.5,-0.5,-0.5};
16          double[] z0 = {0.5,-0.5,-0.5, 0.5,-0.5, 0.5,-0.5, 0.5};
17          double i = 0;
18          boolean isFound = false;
19          while (i<=45){
20              isFound = false;
21              double rad = Math.toRadians(i);
22              double[] xr = new double[8];
23              double[] yr = new double[8];
24              double[] zr = new double[8];
25              for (int j = 0; j < 8; j++) {
26                  xr[j] = x0[j] * Math.cos(rad) - y0[j]* Math.sin(rad);
27                  yr[j] = x0[j] * Math.sin(rad) + y0[j]* Math.cos(rad);
28                  zr[j] = z0[j];
29              }
30              double x1 = 1,y1 = 1,x2 = -1,y2 = -1;
31              for (int j = 0; j < 8; j++) {
32                  x1 = Math.min(x1,xr[j]);
33                  x2 = Math.max(x2,xr[j]);
34                  y1 = Math.min(y1,zr[j]);
35                  y2 = Math.max(y2,zr[j]);
36              }
37              double dt = (x2-x1) * (y2-y1) ;
38              double diff = Math.abs(a - dt);
39              double[] cX = new double[3];
40              double[] cY = new double[3];
41              double[] cZ = new double[3];
42              for (int j = 0; j < 3; j++) {
43                  cX[j] = centerX[j] * Math.cos(rad) - centerY[j] * Math.sin(rad);
44                  cY[j] = centerX[j] * Math.sin(rad) + centerY[j] * Math.cos(rad);
45                  cZ[j] = centerZ[j];
46              }
47              if (diff <= 0.000001) {
48                  for (int j = 0; j < 3; j++) {
49                      System.out.println(cX[j] + " " + cY[j] + " " + cZ[j]);
50                  }
51                  break;
52              }
53              else{
54                  double[] xr0 = new double[8];
55                  double[] yr0 = new double[8];
56                  double[] zr0 = new double[8];
57                  for (int j = 0; j < 8; j++) {
58                      xr0[j] = xr[j];
59                      yr0[j] = yr[j];
60                      zr0[j] = zr[j];
61                  }
62                  double ii = 0;
63                  while (ii<=45) {
64                      isFound = false;
65                      double rad0 = Math.toRadians(ii);
66                      for (int j = 0; j < 8; j++) {
67                          xr[j] = xr0[j];
68                          zr[j] = yr0[j] * Math.sin(rad0) + zr0[j] * Math.cos(rad0);
69                      }
70                      x1 = 1;
71                      y1 = 1;
72                      x2 = -1;
73                      y2 = -1;
74                      for (int j = 0; j < 8; j++) {
75                          x1 = Math.min(x1, xr[j]);
76                          x2 = Math.max(x2, xr[j]);
77                          y1 = Math.min(y1, zr[j]);
78                          y2 = Math.max(y2, zr[j]);
79                      }
80                      dt = (x2 - x1) * (y2 - y1);
81                      diff = Math.abs(a - dt);
82                      if (diff <= 0.000001) {
83                          double[] cXn = new double[3];
84                          double[] cYn = new double[3];
85                          double[] cZn = new double[3];
86                          for (int j = 0; j < 3; j++) {
87                              cXn[j] = cX[j];
88                              cYn[j] = cY[j] * Math.cos(rad0) - cZ[j] * Math.sin(rad0);
89                              cZn[j] = cY[j] * Math.sin(rad0) + cZ[j] * Math.cos(rad0);
90                              System.out.println(cXn[j] + " " + cYn[j] + " " + cZn[j]);
91                          }
92                          isFound = true;
93                          break;
94                      }
95                      ii=roundToDecimals(ii+0.1,6);
96                  }
97              }
98              if (isFound) break;
99              i=roundToDecimals(i+0.1,6);
100          }
101          System.out.flush();
102      }
103      public static void main(String[] args) throws IOException {
104          in = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
105          int numCases = in.nextInt();
106          for (int test = 1; test <= numCases; test++) {
107              System.out.print("Case #" + test + ": ");
108              xuly();
109          }
110      }
111  }
```


**Question**: Does line `82` have control dependence over line `80` in function `xuly`? If so, provide a trace.

**Output**:


Ground truth answer: yes