[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.io.IOException;
6   import java.util.InputMismatchException;
7   import java.io.InputStreamReader;
8   import java.util.ArrayList;
9   import java.util.StringTokenizer;
10  import java.io.BufferedReader;
11  import java.util.Collections;
12  import java.io.InputStream;
13  public class Main {
14      public static void main(String[] args) {
15          InputStream inputStream = System.in;
16          OutputStream outputStream = System.out;
17          InputReader in = new InputReader(inputStream);
18          PrintWriter out = new PrintWriter(outputStream);
19          TaskD solver = new TaskD();
20          solver.solve(1, in, out);
21          out.close();
22      }
23      static class TaskD {
24          public void solve(int testNumber, InputReader in, PrintWriter out) {
25              int n = in.nextInt();
26              long W = in.nextLong();
27              ArrayList<Long>[] lists = new ArrayList[4];
28              for (int i = 0; i < 4; i++) {
29                  lists[i] = new ArrayList<>();
30              }
31              long w1 = in.nextLong();
32              long v1 = in.nextLong();
33              lists[0].add(v1);
34              for (int i = 1; i < n; i++) {
35                  long w = in.nextLong();
36                  long v = in.nextLong();
37                  lists[(int) (w - w1)].add(v);
38              }
39              for (int i = 0; i < 4; i++) {
40                  lists[i].sort(Collections.reverseOrder());
41              }
42              long[][] sum = new long[4][];
43              for (int i = 0; i < 4; i++) {
44                  int m = lists[i].size();
45                  sum[i] = new long[m + 1];
46                  for (int j = 0; j < m; j++) {
47                      sum[i][j + 1] = lists[i].get(j);
48                  }
49                  for (int j = 0; j < m; j++) {
50                      sum[i][j + 1] += sum[i][j];
51                  }
52              }
53              long ans = 0;
54              for (int i = 0; i <= lists[0].size(); i++) {
55                  for (int j = 0; j <= lists[1].size(); j++) {
56                      for (int k = 0; k <= lists[2].size(); k++) {
57                          for (int l = 0; l <= lists[3].size(); l++) {
58                              if (w1 * i + (w1 + 1) * j + (w1 + 2) * k + (w1 + 3) * l > W) continue;
59                              ans = Math.max(ans, sum[0][i] + sum[1][j] + sum[2][k] + sum[3][l]);
60                          }
61                      }
62                  }
63              }
64              out.println(ans);
65          }
66      }
67      static class InputReader {
68          BufferedReader in;
69          StringTokenizer tok;
70          public String nextString() {
71              while (!tok.hasMoreTokens()) {
72                  try {
73                      tok = new StringTokenizer(in.readLine(), " ");
74                  } catch (IOException e) {
75                      throw new InputMismatchException();
76                  }
77              }
78              return tok.nextToken();
79          }
80          public int nextInt() {
81              return Integer.parseInt(nextString());
82          }
83          public long nextLong() {
84              return Long.parseLong(nextString());
85          }
86          public InputReader(InputStream inputStream) {
87              in = new BufferedReader(new InputStreamReader(inputStream));
88              tok = new StringTokenizer("");
89          }
90      }
91  }
```


**Question**: Does line `43` have control dependence over line `50` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes