[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.util.ArrayList;
3   public class Main
4   {
5       public static void main(String args[])
6       {
7           Scanner in=new Scanner(System.in);
8           int t=in.nextInt();
9           while(t-->0)
10          {
11              int n=in.nextInt();
12              int a[][]=new int[n][n];
13              for(int i=0;i<n;i++)
14              {
15                  for(int j=0;j<n;j++)
16                  {
17                      a[i][j]=in.nextInt();
18                  }
19              }
20              vestigium(a,n);
21          }
22      }
23      public static void vestigium(int a[][],int n)
24      {
25            int k=0;
26            int r=0;
27            int c=0;
28            int b[][]=new int[n][n];
29            for(int i=0;i<n;i++)
30            {
31                k+=a[i][i];
32            }
33            for(int i=0;i<n;i++)
34             {
35              for(int j=0;j<n;j++)
36              {
37                 b[i][j]=a[j][i];
38              }
39             }
40             for(int i=0;i<n;i++)
41             {
42                 Arrays.sort(a[i]);
43             }
44             int repeat=0;
45             int count=0;
46             for(int i=0;i<n;i++)
47             {
48                 for(int j=0;j<n-1;j++)
49                 {
50                     if(a[i][j]==a[i][j+1])
51                     {
52                         repeat++;
53                     }
54                     count=Math.max(count,repeat);
55                 }
56                 if(repeat>0)
57                 {
58                     repeat=0;
59                 }
60             }
61             for(int i=0;i<n;i++)
62             {
63              Arrays.sort(b[i]);
64             }
65            int count1=0;
66            int repeat1=0;
67               for(int i=0;i<n;i++)
68            {
69                for(int j=0;j<n-1;j++)
70                {
71                    if(b[i][j]==b[i][j+1])
72                    {
73                        repeat1++;
74                    }
75                    count1=Math.max(count1,repeat1);
76                }
77                if(repeat1>0)
78                {
79                    repeat1=0;
80                }
81            }
82           r=count+1;
83           c=count1+1;
84          int counter=1;
85            if(count1==0 && count==0)
86            {
87                System.out.println("Case #"+counter+": "+" "+k+" "+count+" "+count1);
88                counter++;
89            }
90            else if(count1!=0 && count!=0)
91            {
92              System.out.println("Case #"+counter+": "+" "+k+" "+r+" "+c);
93                counter++;
94            }
95            else if(count1==0 && count!=0)
96            {
97              System.out.println("Case #"+counter+": "+" "+k+" "+r+" "+count1);
98                counter++;
99            }
100              else if(count1!=0 && count==0)
101            {
102              System.out.println("Case #"+counter+": "+" "+k+" "+count+" "+c);
103                counter++;
104            }
105      }
106  }
```


**Question**: Does line `46` have control dependence over line `75` in function `vestigium`? If so, provide a trace.

**Output**:


Ground truth answer: no