[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**. 

```Python
1   import sys
2   def input(): return sys.stdin.readline().strip()
3   def Z(): return int(input())
4   def ZZ(): return [int(_) for _ in input().split()]
5   def main():
6       N = Z()
7       A = []
8       for _ in range(N):
9           A.append(Z())
10      A = sorted(A)
11      used = [0] * (N)
12      output1 = output2 = 0
13      i1 = i2 = 0
14      j1, j2 = N-1, N-2
15      f1 = f2 = True
16      used[i1], used[i2], used[j1], used[j2] = 1, 1, 1, 1
17      while f1 or f2:
18          if f1: output1 += (A[j1]-A[i1])
19          if f2: output1 += (A[j2]-A[i2])
20          while used[i1]:
21              if i1 == N-1:
22                  f1 = False
23                  break
24              i1 += 1
25          used[i1] = 1
26          while used[i2]:
27              if i2 == N-1:
28                  f2 = False
29                  break
30              i2 += 1
31          used[i2] = 1
32          if f1: 
33              output1 += (A[j1]-A[i1])
34          if f2: 
35              output1 += (A[j2]-A[i2])
36          while used[j1]:
37              if j1 == 0:
38                  f1 = False
39                  break
40              j1 -= 1
41          used[j1] = 1
42          while used[j2]:
43              if j2 == 0:
44                  f2 = False
45                  break
46              j2 -= 1
47          used[j2] = 1
48      i1, i2 = 0, 1
49      j1 = j2 = N-1
50      f1 = f2 = True
51      used = [0] * (N)
52      used[i1], used[i2], used[j1], used[j2] = 1, 1, 1, 1
53      while f1 or f2:
54          if f1: 
55              output2 += (A[j1]-A[i1])
56          if f2: 
57              output2 += (A[j2]-A[i2])
58          while used[j1]:
59              if j1 == 0:
60                  f1 = False
61                  break
62              j1 -= 1
63          used[j1] = 1
64          while used[j2]:
65              if j2 == 0:
66                  f2 = False
67                  break
68              j2 -= 1
69          used[j2] = 1
70          if f1: 
71              output2 += (A[j1]-A[i1])
72          if f2: 
73              output2 += (A[j2]-A[i2])
74          while used[i1]:
75              if i1 == N-1:
76                  f1 = False
77                  break
78              i1 += 1
79          used[i1] = 1
80          while used[i2]:
81              if i2 == N-1:
82                  f2 = False
83                  break
84              i2 += 1
85          used[i2] = 1
86      print(max(output1, output2))
87      return
88  if __name__ == '__main__':
89      main()
```


**Question**: Does line `81` have control dependence over line `84` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes