[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   def examC(inf):
2       L = LI()
3       ans = inf
4       for i in range(4):
5           if i==0:
6               for j in range(2):
7                   ansC = []
8                   if j==0:
9                       cut = L[0] // 3
10                      ansC.append(L[1] * cut)
11                      L2 = [ L[0] - cut,L[1]]
12                      cut = L2[0] // 2
13                      ansC.append(L2[1] * cut)
14                      ansC.append(L[1]*L[0] - sum(ansC))
15                      ans = min(ans, max(ansC) - min(ansC))
16                  else:
17                      cut = L[0] // 3+1
18                      ansC.append(L[1] * cut)
19                      L2 = [ L[0] - cut,L[1]]
20                      cut = L2[0] // 2
21                      ansC.append(L2[1] * cut)
22                      ansC.append(L[1]*L[0] - sum(ansC))
23                      ans = min(ans, max(ansC) - min(ansC))
24          elif i==1:
25              for j in range(2):
26                  ansC = []
27                  if j==0:
28                      cut = L[0] // 3
29                      ansC.append(L[1] * cut)
30                      L2 = [ L[0] - cut,L[1]]
31                      cut = L2[1] // 2
32                      ansC.append(L2[0] * cut)
33                      ansC.append(L[1]*L[0] - sum(ansC))
34                      ans = min(ans, max(ansC) - min(ansC))
35                  else:
36                      cut = L[0] // 3+1
37                      ansC.append(L[1] * cut)
38                      L2 = [ L[0] - cut,L[1]]
39                      cut = L2[1] // 2
40                      ansC.append(L2[0] * cut)
41                      ansC.append(L[1]*L[0] - sum(ansC))
42                      ans = min(ans, max(ansC) - min(ansC))
43          elif i==2:
44              for j in range(2):
45                  ansC = []
46                  if j==0:
47                      cut = L[1] // 3
48                      ansC.append(L[0] * cut)
49                      L2 = [ L[0],L[1]-cut]
50                      cut = L2[0] // 2
51                      ansC.append(L2[1] * cut)
52                      ansC.append(L[1]*L[0] - sum(ansC))
53                      ans = min(ans, max(ansC) - min(ansC))
54                  else:
55                      cut = L[1] // 3+1
56                      ansC.append(L[0] * cut)
57                      L2 = [ L[0],L[1]-cut]
58                      cut = L2[0] // 2
59                      ansC.append(L2[1] * cut)
60                      ansC.append(L[1]*L[0] - sum(ansC))
61                      ans = min(ans, max(ansC) - min(ansC))
62          else:
63              for j in range(2):
64                  ansC = []
65                  if j==0:
66                      cut = L[1] // 3
67                      ansC.append(L[0] * cut)
68                      L2 = [ L[0],L[1]-cut]
69                      cut = L2[1] // 2
70                      ansC.append(L2[0] * cut)
71                      ansC.append(L[1]*L[0] - sum(ansC))
72                      ans = min(ans, max(ansC) - min(ansC))
73                  else:
74                      cut = L[1] // 3+1
75                      ansC.append(L[0] * cut)
76                      L2 = [ L[0],L[1]-cut]
77                      cut = L2[1] // 2
78                      ansC.append(L2[0] * cut)
79                      ansC.append(L[1]*L[0] - sum(ansC))
80                      ans = min(ans, max(ansC) - min(ansC))
81      print(ans)
82  import sys
83  import copy
84  import bisect
85  from collections import Counter,defaultdict,deque
86  def I(): return int(sys.stdin.readline())
87  def LI(): return list(map(int,sys.stdin.readline().split()))
88  def LS(): return sys.stdin.readline().split()
89  def S(): return sys.stdin.readline().strip()
90  mod = 10**9 + 7
91  inf = float('inf')
92  examC(inf)
```


**Question**: Does line `4` have control dependence over line `69` in function `examC`? If so, provide a trace.

**Output**:


Ground truth answer: yes