[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.lang.*;
2   import java.io.*;
3   import java.util.*;
4   class Solution {
5       public static void main(String[]args) throws IOException {
6           BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
7           PrintWriter pw = new PrintWriter(System.out);
8           int T = Integer.parseInt(br.readLine());
9           for(int i=1;i<=T;i++) {
10              pw.println("Case #" + i + ": " + solve(br));
11          }
12          pw.flush();
13      }
14      private static String solve(BufferedReader br) throws IOException {
15          int N = Integer.parseInt(br.readLine());
16          List<P> paris = new ArrayList<>();
17          for (int i = 0; i < N; i++) {
18              StringTokenizer st = new StringTokenizer(br.readLine());
19              int x = Integer.parseInt(st.nextToken());
20              int y = Integer.parseInt(st.nextToken());
21              paris.add(new P(x, y));
22          }
23          String res = getRes(N, paris);
24          return res;
25      }
26      private static boolean check(String res, List<P> paris) {
27          if (res.equals("IMPOSSIBLE")) return true;
28          StringTokenizer st = new StringTokenizer(res);
29          int x = Integer.parseInt(st.nextToken());
30          int y = Integer.parseInt(st.nextToken());
31          int[] v = new int[paris.size()];
32          for (int i = 0; i < paris.size(); i++) {
33              int z = paris.get(i).a * x + paris.get(i).b * y;
34              v[i] = z;
35          }
36          int[] v2 = new int[v.length];
37          System.arraycopy(v, 0, v2, 0, v.length);
38          Arrays.sort(v);
39          boolean r = Arrays.equals(v, v2);
40          if (!r) {
41              debug(v, v2);
42          }
43          return r;
44      }
45      private static String getRes(int n, List<P> paris) {
46          TreeSet<F> pfracs = new TreeSet<>();
47          TreeSet<F> nfracs = new TreeSet<>();
48          boolean hasDup = false;
49          for (int i = 0; i < n; i++) {
50              P p1 = paris.get(i);
51              for (int j = i + 1; j < n; j++) {
52                  P p2 = paris.get(j);
53                  int d1 = p1.a - p2.a;
54                  int d2 = p1.b - p2.b;
55                  if ((d1 == 0 && d2 == 0) || (d1 > 0 && d2 > 0)) {
56                      hasDup = true;
57                      break;
58                  }
59                  if (d1 != 0 && d2 != 0 && (d1 > 0 || d2 > 0)) {
60                      if (d1 < 0) {
61                          pfracs.add(new F(Math.abs(d1), Math.abs(d2)));
62                      }
63                      if (d2 < 0) {
64                          nfracs.add(new F(Math.abs(d1), Math.abs(d2)));
65                      }
66                  }
67              }
68              if (hasDup) break;
69          }
70          if (hasDup) {
71              return "IMPOSSIBLE";
72          }
73          F above = pfracs.isEmpty() ? null : pfracs.first();
74          F below = nfracs.isEmpty() ? null : nfracs.last();
75          if (above != null && below != null && above.compareTo(below) <= 0) {
76              return "IMPOSSIBLE";
77          }
78          if (above == null && below == null) {
79              return "1 1";
80          }
81          if (above == null) {
82              long z = below.num;
83              long r = below.denum;
84              long y = 0;
85              if (r % z == 0) {
86                  y = r / z;
87              } else {
88                  y = r / z + 1;
89              }
90              return "1 " + (y + 1);
91          }
92          if (below == null) {
93              long z = above.num;
94              long r = above.denum;
95              long x = 0;
96              if (r % z == 0) {
97                  x = r / z;
98              } else {
99                  x = r / z + 1;
100              }
101              return (x + 1) + " 1";
102          }
103          int gcd = gcd(above.denum, below.denum);
104          long b2 = above.denum / gcd;
105          long lcd = below.denum * b2;
106          long a = above.num * (lcd / above.denum);
107          long b = below.num * (lcd / below.denum);
108          long z = a - b;
109          long r = 2 * lcd;
110          long x = 0;
111          if (r % z == 0) {
112              x = r / z;
113          } else {
114              x = r / z + 1;
115          }
116          return x + " " + (x + 1);
117      }
118      private static int gcd(int a, int b) {
119          return (b != 0) ? gcd(b, a % b) : a;
120      }
121      private static final class P {
122          int a;
123          int b;
124          public P(int a, int b) {
125              this.a = a;
126              this.b = b;
127          }
128      }
129      private static final class F implements Comparable<F> {
130          int num;
131          int denum;
132          public F(int num, int denum) {
133              int gcd = gcd(num, denum);
134              this.num = num / gcd;
135              this.denum = denum / gcd;
136          }
137          @Override
138          public int compareTo(F o) {
139              int gcd = gcd(o.denum, denum);
140              long b2 = o.denum / gcd;
141              long lcd = denum * b2;
142              long a = num * (lcd / denum);
143              long b = o.num * (lcd / o.denum);
144              return Long.compare(a, b);
145          }
146      }
147      public static void debug(Object...args) {
148          System.out.println(Arrays.deepToString(args));
149      }
150  }
```


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

**Output**:


Ground truth answer: yes