[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 java.util.HashMap;
5   import java.util.Map;
6   public class Solution {
7       public static void main(String[] args) throws IOException {
8           BufferedReader input = new BufferedReader(new InputStreamReader(System.in));
9           int T = Integer.parseInt(input.readLine());
10          for (int i = 1; i <= T; i++) {
11              System.out.println("Case #" + i + ": " + solve(input));
12          }
13      }
14      private static String solve(BufferedReader input) throws IOException {
15          int M = Integer.parseInt(input.readLine());
16          int[] part1 = new int[M];
17          int[] part2 = new int[M];
18          int[] quantities = new int[M];
19          for (int i=0; i<M; i++) {
20              String[] nums = input.readLine().split(" ");
21              part1[i] = Integer.parseInt(nums[0])-1;
22              part2[i] = Integer.parseInt(nums[1])-1;
23          }
24          String[] nums = input.readLine().split(" ");
25          for(int i=0; i<M; i++) {
26              quantities[i] = Integer.parseInt(nums[i]);
27          }
28          Map<Integer, Integer> transmuted = new HashMap<Integer, Integer>();
29          transmuted.put(part1[0], 1);
30          transmuted.put(part2[0], 1);
31          boolean stuck = false;
32          while(!stuck) {
33              int min = Integer.MAX_VALUE;
34              for (Map.Entry<Integer, Integer> pair : transmuted.entrySet()) {
35                  if (quantities[pair.getKey()] / pair.getValue() < min) 
36                      min = quantities[pair.getKey()] / pair.getValue();
37              }
38              quantities[0] += min;
39              for (Map.Entry<Integer, Integer> pair : transmuted.entrySet()) {
40                  quantities[pair.getKey()] -= min * pair.getValue();
41              }
42              Map<Integer, Integer> newTransmuted = new HashMap<Integer, Integer>();
43              for (Map.Entry<Integer, Integer> pair : transmuted.entrySet()) {
44                  int metal = pair.getKey();
45                  int quantity = pair.getValue();
46                  if (pair.getValue() > quantities[metal]) {
47                      if (part1[metal] == 0 || part2[metal] == 0 || pair.getValue() > 1000*1000*1000) {
48                          stuck = true;
49                          break;
50                      } else {
51                          if (newTransmuted.containsKey(part1[metal])) {
52                              newTransmuted.put(part1[metal], newTransmuted.get(part1[metal]) + quantity);
53                          } else {
54                              newTransmuted.put(part1[metal], quantity);
55                          }
56                          quantities[part1[metal]] += quantities[metal];
57                          if (newTransmuted.containsKey(part2[metal])) {
58                              newTransmuted.put(part2[metal], newTransmuted.get(part2[metal]) + quantity);
59                          } else {
60                              newTransmuted.put(part2[metal], quantity);
61                          }
62                          quantities[part2[metal]] += quantities[metal];
63                          if (newTransmuted.containsKey(metal)) {
64                              newTransmuted.put(metal, newTransmuted.get(metal) - quantity);
65                          } else {
66                              newTransmuted.put(metal, -1*quantity);
67                          }
68                          quantities[metal] = 0;
69                      }
70                  }
71              }
72              for (Map.Entry<Integer, Integer> pair : newTransmuted.entrySet()) {
73                  Integer metal = pair.getKey();
74                  Integer quantity = pair.getValue();
75                  if (transmuted.containsKey(metal)) {
76                      transmuted.put(metal, transmuted.get(metal) + quantity);
77                  }
78                  else {
79                      transmuted.put(metal, quantity);
80                  }
81                  if (transmuted.get(metal) == 0) {
82                      transmuted.remove(metal);
83                  }
84              }
85          }
86          return String.valueOf(quantities[0]);
87      }
88  }
```


**Question**: Does line `25` have control dependence over line `81` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: no