[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.OutputStream;
2   import java.io.IOException;
3   import java.io.InputStream;
4   import java.io.PrintWriter;
5   import java.util.StringTokenizer;
6   import java.io.IOException;
7   import java.io.BufferedReader;
8   import java.io.InputStreamReader;
9   import java.io.InputStream;
10  public class Main {
11      public static void main(String[] args) {
12          InputStream inputStream = System.in;
13          OutputStream outputStream = System.out;
14          InputReader in = new InputReader(inputStream);
15          PrintWriter out = new PrintWriter(outputStream);
16          TaskC solver = new TaskC();
17          solver.solve(1, in, out);
18          out.close();
19      }
20      static class TaskC {
21          public void solve(int testNumber, InputReader in, PrintWriter out) {
22              int h = in.nextInt();
23              int w = in.nextInt();
24              if (h % 3 == 0 || w % 3 == 0) {
25                  out.println(0);
26                  return;
27              }
28              int offset = 2000;
29              int minSum = Integer.MAX_VALUE;
30              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
31                  if (a <= 0 || a >= h)
32                      continue;
33                  for (int b = w / 2 - offset; b < w / 2 + offset; b++) {
34                      if (b <= 0 || b >= w)
35                          continue;
36                      int s1 = a * w;
37                      int s2 = b * (h - a);
38                      int s3 = (h - a) * (w - b);
39                      int max = Math.max(Math.max(s1, s2), s3);
40                      int min = Math.min(Math.min(s1, s2), s3);
41                      minSum = Math.min(minSum, max - min);
42                  }
43              }
44              int x = h;
45              h = w;
46              w = x;
47              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
48                  if (a <= 0 || a >= h)
49                      continue;
50                  for (int b = w / 2 - offset; b < w / 2 + offset; b++) {
51                      if (b <= 0 || b >= w)
52                          continue;
53                      int s1 = a * w;
54                      int s2 = b * (h - a);
55                      int s3 = (h - a) * (w - b);
56                      int max = Math.max(Math.max(s1, s2), s3);
57                      int min = Math.min(Math.min(s1, s2), s3);
58                      minSum = Math.min(minSum, max - min);
59                  }
60              }
61              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
62                  if (a <= 0 || a >= h)
63                      continue;
64                  for (int b = h / 3 - offset; b < h / 3 + offset; b++) {
65                      if (b <= 0 || b >= h)
66                          continue;
67                      if (a + b >= h)
68                          continue;
69                      int s1 = a * w;
70                      int s2 = b * w;
71                      int s3 = (h - a - b) * w;
72                      int max = Math.max(Math.max(s1, s2), s3);
73                      int min = Math.min(Math.min(s1, s2), s3);
74                      minSum = Math.min(minSum, max - min);
75                  }
76              }
77              x = h;
78              h = w;
79              w = x;
80              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
81                  if (a <= 0 || a >= h)
82                      continue;
83                  for (int b = h / 3 - offset; b < h / 3 + offset; b++) {
84                      if (b <= 0 || b >= h)
85                          continue;
86                      if (a + b >= h)
87                          continue;
88                      int s1 = a * w;
89                      int s2 = b * w;
90                      int s3 = (h - a - b) * w;
91                      int max = Math.max(Math.max(s1, s2), s3);
92                      int min = Math.min(Math.min(s1, s2), s3);
93                      minSum = Math.min(minSum, max - min);
94                  }
95              }
96              out.println(minSum);
97          }
98      }
99      static class InputReader {
100          StringTokenizer st;
101          BufferedReader br;
102          public InputReader(InputStream is) {
103              BufferedReader br = new BufferedReader(new InputStreamReader(is));
104              this.br = br;
105          }
106          public String next() {
107              if (st == null || !st.hasMoreTokens()) {
108                  String nextLine = null;
109                  try {
110                      nextLine = br.readLine();
111                  } catch (IOException e) {
112                      throw new RuntimeException(e);
113                  }
114                  if (nextLine == null)
115                      return null;
116                  st = new StringTokenizer(nextLine);
117              }
118              return st.nextToken();
119          }
120          public int nextInt() {
121              return Integer.parseInt(next());
122          }
123      }
124  }
```


**Question**: Does line `51` have control dependence over line `55` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes