[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.lang.reflect.Array;
2   import java.util.*;
3   public class Solution {
4       public static void main(String[] args) {
5           Scanner input = new Scanner(System.in);
6           int T = input.nextInt();
7           for (int turn = 0; turn < T; turn++) {
8               int n = input.nextInt();
9               ArrayList<Point> points = new ArrayList<>();
10              for (int i = 0; i < n; i++) {
11                  Point point = new Point(input.nextLong(),input.nextLong());
12                  points.add(point);
13              }
14              ArrayList<Line> lines = new ArrayList<>();
15              for (int i = 0; i < n; i++) {
16                  for (int j = i+1; j < n; j++) {
17                      lines.add(new Line(points.get(i),points.get(j)));
18                  }
19              }
20              HashMap<Double,ArrayList<Line>> map = new HashMap<>();
21              for (int i = 0; i < lines.size(); i++) {
22                  if (!map.containsKey(lines.get(i).k)) {
23                      ArrayList<Line> curr = new ArrayList<>();
24                      curr.add(lines.get(i));
25                      map.put(lines.get(i).k,curr);
26                  }else {
27                      map.get(lines.get(i).k).add(lines.get(i));
28                  }
29              }
30              ArrayList<Integer> results = new ArrayList<>();
31              Iterator mapIterator = map.entrySet().iterator();
32              while (mapIterator.hasNext()) {
33                  int result = 0;
34                  Map.Entry mapElement = (Map.Entry)mapIterator.next();
35                  HashMap<Double,ArrayList<Line>> bmap = new HashMap<>();
36                  double k = (double)mapElement.getKey();
37                  ArrayList<Line> blines = (ArrayList<Line>) mapElement.getValue();
38                  for (int i = 0; i < blines.size(); i++) {
39                      double b = blines.get(i).b;
40                      Line l = blines.get(i);
41                      if (!bmap.containsKey(b)) {
42                          ArrayList<Line> curr = new ArrayList<>();
43                          curr.add(l);
44                          bmap.put(b, curr);
45                      } else {
46                          bmap.get(b).add(l);
47                      }
48                  }
49                  ArrayList<Integer> resultarray = new ArrayList<>();
50                  Iterator bmapIterator = bmap.entrySet().iterator();
51                  while (bmapIterator.hasNext()) {
52                      Map.Entry bmapElement = (Map.Entry)bmapIterator.next();
53                      ArrayList<Line> gongxian = (ArrayList<Line>)bmapElement.getValue();
54                      HashSet<Point> currp = new HashSet<>();
55                      for (int i = 0; i < gongxian.size(); i++) {
56                          if (!currp.contains(gongxian.get(i).p1)) {
57                              currp.add(gongxian.get(i).p1);
58                          }
59                          if (!currp.contains(gongxian.get(i).p2)) {
60                              currp.add(gongxian.get(i).p2);
61                          }
62                      }
63                      resultarray.add(currp.size());
64                  }
65                  int sum = 0;
66                  for (int i = 0; i < resultarray.size(); i++) {
67                      sum+=resultarray.get(i);
68                  }
69                  if (sum%2 == 0) {
70                      if (n-sum >=2) {
71                          result = sum+2;
72                      }else {
73                          result = n;
74                      }
75                  }else {
76                      if (n -sum >=1) {
77                          result = sum+1;
78                      }else {
79                          result = n;
80                      }
81                  }
82                  results.add(result);
83              }
84              int max = 1;
85              for (int i = 0; i < results.size(); i++) {
86                  if (max < results.get(i)) {
87                      max = results.get(i);
88                  }
89              }
90              System.out.println("Case #" + (turn + 1) + ": " + max);
91          }
92      }
93  }
94  class Point {
95      long x;
96      long y;
97      Point(long x, long y) {
98          this.x = x;
99          this.y = y;
100      }
101  }
102  class Line {
103      Point p1;
104      Point p2;
105      double b;
106      double k;
107      Line(Point x, Point y) {
108          p1 = x;
109          p2 = y;
110          k = ((double)(p1.y-p2.y))/(double)(p1.x-p2.x);
111          b = (double)p1.y - p1.x*(k);
112      }
113  }
```


**Question**: Does line `56` have control dependence over line `57` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes