[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.IOException;
2   import java.io.InputStream;
3   import java.util.ArrayList;
4   import java.util.Arrays;
5   import java.util.Comparator;
6   import java.util.LinkedList;
7   import java.util.List;
8   import java.util.NoSuchElementException;
9   import java.util.PriorityQueue;
10  import java.util.ArrayDeque;
11  import java.util.Queue;
12  import java.util.HashSet;
13  public class Main {
14      public static void main(String[] args) {
15          FastScanner fs = new FastScanner();
16          int n = fs.nextInt();
17          boolean flag = true;
18          int[][] m = new int[n][n];
19          int x = 1;
20          int count = 0;
21          List<ArrayList<Integer>> graph = new ArrayList<>();
22          boolean[] exist = new boolean[n*n];
23          boolean[] possible = new boolean[n*n];
24          Arrays.fill(exist,true);
25          Arrays.fill(possible,false);
26          HashSet<Integer> set = new HashSet<Integer>();
27          Queue<Integer> queue = new ArrayDeque<>();
28          for(int i = 0; i < n; i++){
29            for(int j = 0; j < n; j++){
30              List<Integer> near = new ArrayList<>();
31              graph.add(new ArrayList<>());
32              if(i == 0 || i == n-1 || j == 0 || j == n-1){
33                set.add(x);
34              }
35              if(x - 1 > i*n){
36                graph.get(x-1).add(x-1);
37              }
38              if(x + 1 <= n*(i+1)){
39                graph.get(x-1).add(x+1);
40              }
41              if(x - n > 0){
42                graph.get(x-1).add(x-n);
43              }
44              if(x + n <= n*n){
45                graph.get(x-1).add(x+n);
46              }
47              int p = fs.nextInt();
48              queue.add(p);
49              m[i][j] = x;
50              x += 1;
51            }
52          }
53          while(queue.size() > 0){
54            int a = queue.poll();
55            exist[a-1] = false;
56            if(set.contains(a)){
57              possible[a-1] = true;
58            }
59            if(possible[a-1] == true){
60              for(int w : graph.get(a-1)){
61                possible[w-1] = true;
62              }
63            }
64            else{
65              for(int w : graph.get(a-1)){
66                if(possible[w-1] == true && exist[w-1] == false){
67                  possible[a-2] = true;
68                  possible[a-1-n] = true;
69                  possible[a] = true;
70                  possible[a-1+n] = true;
71                  possible[a-1] = true;
72                  count -= 1;
73                  break;
74                }
75              }
76              count += 1;
77              if(a-2 >= 0 || a-1-n >= 0 || a <= n*n || a-1+n <= n*n){
78                if((exist[a-2] == false && possible[a-2] == true) || (exist[a-1-n] == false && possible[a-1-n] == true) || (exist[a] == false && possible[a] == true) || (exist[a-1+n] == false && possible[a-1+n] == true)){
79                  possible[a-2] = true;
80                  possible[a-1-n] = true;
81                  possible[a] = true;
82                  possible[a-1+n] = true;
83                  possible[a-1] = true;
84                }
85              }
86            }
87          }
88          System.out.print(count);
89    }
90  }
91  class FastScanner {
92      private final InputStream in = System.in;
93      private final byte[] buffer = new byte[1024];
94      private int ptr = 0;
95      private int buflen = 0;
96      private boolean hasNextByte(){
97          if(ptr < buflen){
98              return true;
99          }else{
100              ptr = 0;
101              try{
102                  buflen = in.read(buffer);
103              }catch(IOException e){
104                  e.printStackTrace();
105              }
106              if(buflen <=0){
107                  return false;
108              }
109          }
110          return true;
111      }
112      private int readByte(){
113          if(hasNextByte())return buffer[ptr++];
114          else return -1;
115      }
116      private static boolean isPrintableChar(int c){
117          return 33<=c && c<=126;
118      }
119      public boolean hasNext(){
120          while(hasNextByte() && !isPrintableChar(buffer[ptr]))ptr++;
121          return hasNextByte();
122      }
123      public String next(){
124          if(!hasNext()) throw new NoSuchElementException();
125          StringBuilder sb = new StringBuilder();
126          int b = readByte();
127          while(isPrintableChar(b)){
128              sb.appendCodePoint(b);
129              b = readByte();
130          }
131          return sb.toString();
132      }
133      public long nextLong(){
134          if(!hasNext()) throw new NoSuchElementException();
135          long n = 0;
136          boolean minus = false;
137          int b = readByte();
138          if(b == '-'){
139              minus = true;
140              b = readByte();
141          }
142          if(b < '0' || '9' < b){
143              throw new NumberFormatException();
144          }
145          while(true){
146              if('0' <= b && b<='9'){
147                  n*=10;
148                  n+=b-'0';
149              }else if(b==-1 || !isPrintableChar(b)){
150                  return minus ? -n : n;
151              }else{
152                  throw new NumberFormatException();
153              }
154              b = readByte();
155          }
156      }
157      public int nextInt(){
158          long nl = nextLong();
159          if(nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE)throw new NumberFormatException();
160          return (int) nl;
161      }
162      public double nextDoutble(){return Double.parseDouble(next());}
163  }
```


**Question**: Does line `53` have control dependence over line `55` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes