[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.BufferedReader;
2   import java.io.IOException;
3   import java.io.InputStreamReader;
4   import static java.util.Arrays.*;
5   public class ProblemB2 {
6       BufferedReader rd;
7       ProblemB2() throws IOException {
8           rd = new BufferedReader(new InputStreamReader(System.in));
9           compute();
10      }
11      private void compute() throws IOException {
12          start();
13          int n = pint();
14          for(int i=1;i<=n;i++) {
15              out("Case #" + i + ": " + solve());
16          }
17      }
18      int[][] v = new int[123][123];
19      private void start() {
20          for(int[] q: v) {
21              fill(q, -1);
22          }
23          int[][] rb = new int[111][2];
24          int p = 0;
25          for(int i=1;i<=6;i++) {
26              for(int r=0;r<=i;r++) {
27                  rb[p][0] = r;
28                  rb[p][1] = i-r;
29                  p++;
30              }
31          }
32          int s = 1<<p;
33          for(int i=0;i<s;i++) {
34              int u = i;
35              int r = 0;
36              int b = 0;
37              int cnt = 0;
38              for(int k=0;k<p;k++) {
39                  if((u&1) > 0) {
40                      r += rb[k][0];
41                      b += rb[k][1];
42                      cnt++;
43                  }
44                  u >>= 1;
45              }
46              int cur = v[r][b];
47              if(cur == -1 || (cur < cnt)) {
48                  v[r][b] = cnt;
49              }
50          }
51      }
52      private int solve() throws IOException {
53          int[] a = intarr();
54          int r = a[0];
55          int b = a[1];
56          if(v[r][b] != -1) {
57              return v[r][b];
58          }
59          if(r < b) {
60              int t = r;
61              r = b;
62              b = t;
63          }
64          int res = 0;
65          int nxt = 7;
66          while(v[r][b] == -1) {
67              r -= nxt;
68              nxt++;
69              res++;
70          }
71          res += v[r][b];
72          return res;
73      }
74      private int pint() throws IOException {
75          return pint(rd.readLine());
76      }
77      private int pint(String s) {
78          return Integer.parseInt(s);
79      }
80      private int[] intarr() throws IOException {
81          return intarr(rd.readLine());
82      }
83      private int[] intarr(String s) {
84          String[] q = split(s);
85          int n = q.length;
86          int[] a = new int[n];
87          for(int i=0;i<n;i++) {
88              a[i] = Integer.parseInt(q[i]);
89          }
90          return a;
91      }
92      private String[] split(String s) {
93          if(s == null) {
94              return new String[0];
95          }
96          int n = s.length();
97          int start = -1;
98          int end = 0;
99          int sp = 0;
100          boolean lastWhitespace = true;
101          for(int i=0;i<n;i++) {
102              char c = s.charAt(i);
103              if(isWhitespace(c)) {
104                  lastWhitespace = true;
105              } else {
106                  if(lastWhitespace) {
107                      sp++;
108                  }
109                  if(start == -1) {
110                      start = i;
111                  }
112                  end = i;
113                  lastWhitespace = false;
114              }
115          }
116          if(start == -1) {
117              return new String[0];
118          }
119          String[] res = new String[sp];
120          int last = start;
121          int x = 0;
122          lastWhitespace = true;
123          for(int i=start;i<=end;i++) {
124              char c = s.charAt(i);
125              boolean w = isWhitespace(c);
126              if(w && !lastWhitespace) {
127                  res[x++] = s.substring(last,i);
128              } else if(!w && lastWhitespace) {
129                  last = i;
130              }
131              lastWhitespace = w;
132          }
133          res[x] = s.substring(last,end+1);
134          return res;
135      }
136      private boolean isWhitespace(char c) {
137          return c==' ' || c=='\t';
138      }
139      private static void out(Object x) {
140          System.out.println(x);
141      }
142      public static void main(String[] args) throws IOException {
143          new ProblemB2();
144      }
145  }
```


**Question**: Does line `26` have control dependence over line `29` in function `start`? If so, provide a trace.

**Output**:


Ground truth answer: yes