[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   from collections import defaultdict,deque
2   from heapq import heappush, heappop
3   import sys
4   import math
5   import bisect
6   import random
7   def LI(): return [int(x) for x in sys.stdin.readline().split()]
8   def I(): return int(sys.stdin.readline())
9   def LS():return [list(x) for x in sys.stdin.readline().split()]
10  def S(): return list(sys.stdin.readline())[:-1]
11  def IR(n):
12      return [I() for i in range(n)]
13  def LIR(n):
14      return [LI() for i in range(n)]
15  def SR(n):
16      return [S() for i in range(n)]
17  def LSR(n):
18      return [LS() for i in range(n)]
19  sys.setrecursionlimit(1000000)
20  mod = 1000000007
21  def A():
22      n = I()
23      a = LI()
24      ans = 0
25      for i in a:
26          ans ^= i
27      if not ans:
28          print("Yes")
29      else:
30          print("No")
31      return
32  def B():
33      n,m = LI()
34      v = []
35      for _ in range(n):
36          v.append([])
37      f = [0]*n
38      dic = defaultdict(lambda : 0)
39      for i in range(m):
40          a,b = LI()
41          a -= 1
42          b -= 1
43          v[a].append(b)
44          v[b].append(a)
45          f[a] += 1
46          f[b] += 1
47          dic[(a,b)] = 1
48          dic[(b,a)] = 1
49      ans = []
50      d = [0]*n
51      q = deque([0])
52      bfs = [1]*n
53      bfs[0] = 0
54      while q:
55          x = q.popleft()
56          s = []
57          for y in v[x]:
58              if dic[(x,y)]:
59                  s.append(y)
60              if bfs[y]:
61                  q.append(y)
62                  bfs[y] = 0
63          if s:
64              if len(s)%2:
65                  if d[x]:
66                      for y in s:
67                          ans.append((x,y))
68                          dic[(x,y)] = 0
69                          dic[(y,x)] = 0
70                      d[x] = 0
71                  else:
72                      for y in s[:-1]:
73                          ans.append((x,y))
74                          dic[(x,y)] = 0
75                          dic[(y,x)] = 0
76                      d[x] = 0
77                      y = s[-1]
78                      d[y] ^= 1
79                      ans.append((y,x))
80                      dic[(x,y)] = 0
81                      dic[(y,x)] = 0
82              else:
83                  if not d[x]:
84                      for y in s:
85                          ans.append((x,y))
86                          dic[(x,y)] = 0
87                          dic[(y,x)] = 0
88                      d[x] = 0
89                  else:
90                      for y in s[:-1]:
91                          ans.append((x,y))
92                          dic[(x,y)] = 0
93                          dic[(y,x)] = 0
94                      d[x] = 0
95                      y = s[-1]
96                      d[y] ^= 1
97                      ans.append((y,x))
98                      dic[(x,y)] = 0
99                      dic[(y,x)] = 0
100      if sum(d):
101          print(-1)
102      else:
103          for i,j in ans:
104              print(i+1,j+1)
105      return
106  def C():
107      n = I()
108      p = [(1<<i) for i in range(100)]
109      if n in p:
110          print("No")
111          quit()
112      if n+1 in p:
113          print("Yes")
114          for i in range(1,2*n):
115              print(i,i+1)
116          quit()
117      ans = []
118      for i in range(1,3):
119          ans.append((i,i+1))
120      ans.append((3,n+1))
121      for i in range(1,3):
122          ans.append((i+n,i+n+1))
123      u = 1
124      d = 1
125      for i in range(2,n//2+n%2):
126          ans.append((u,2*i))
127          ans.append((d,2*i+1))
128          ans.append((2*i,2*i+n+1))
129          ans.append((2*i+1,2*i+n))
130          u = 2*i+n+1
131          d = 2*i+n
132      if n%2:
133          print("Yes")
134          for i,j in ans:
135              print(i,j)
136      else:
137          ans.append((n-1,n))
138          for i in range(n):
139              if p[i]&n:
140                  break
141          ans.append((p[i+1]-2,2*n))
142          print("Yes")
143          for i,j in ans:
144              print(i,j)
145      return
146  def D():
147      n = I()
148      return
149  def E():
150      n = I()
151      return
152  def F():
153      n = I()
154      return
155  if __name__ == "__main__":
156      B()
```


**Question**: Does line `83` have control dependence over line `96` in function `B`? If so, provide a trace.

**Output**:


Ground truth answer: yes