[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, bisect, math, itertools, string, queue, copy
2   from collections import Counter,defaultdict,deque
3   from itertools import permutations, combinations, combinations_with_replacement
4   from heapq import heappop, heappush
5   from fractions import gcd
6   sys.setrecursionlimit(10**8)
7   mod = 10**9+7
8   def inp(): return int(input())
9   def inpm(): return map(int,input().split())
10  def inpl(): return list(map(int, input().split()))
11  def inpls(): return list(input().split())
12  def inplm(n): return list(int(input()) for _ in range(n))
13  def inplL(n): return [list(input()) for _ in range(n)]
14  def inplT(n): return [tuple(input()) for _ in range(n)]
15  def inpll(n): return [list(map(int, input().split())) for _ in range(n)]
16  def inplls(n): return sorted([list(map(int, input().split())) for _ in range(n)])
17  def main():
18      n = int(input())
19      dic = defaultdict(int)
20      dic1 = defaultdict(int)
21      for _ in range(n):
22          a,b = inpm()
23          if a != 0 and b == 0:
24              a = 1
25          elif a == 0 and b != 0:
26              b = 1
27          else:
28              if not (a==0 and b==0):
29                  m = gcd(a,b)
30                  a //= m
31                  b //= m
32          if a < 0 and b < 0:
33              a = -a
34              b = -b
35          if a < 0 and b > 0:
36              a = -a
37              b = -b
38          index = ' '.join([str(a),str(b)])
39          dic[index] += 1
40          dic1[index] += 1
41      ans = 1
42      dic2 = defaultdict(int)
43      for e in dic:
44          a,b = map(int,e.split())
45          if a == 0 or b == 0:
46              continue
47          flag = 0
48          if a > 0 and b > 0:
49              rev = ' '.join([str(b),str(-a)])
50          else:
51              rev = ' '.join([str(-b),str(a)])
52          if dic1[rev] > 0 and dic2[rev] == 0:
53              dic2[rev] = 1
54              dic2[e] = 1
55              flag = 1
56          if flag:
57              continue
58          n1 = pow(2,dic[e],mod)
59          n2 = pow(2,dic1[rev],mod)
60          if n1 == 0:
61              ans *= n2
62          elif n2 == 0:
63              ans *= n1
64          else:
65              ans *= (n1+n2-1)
66          ans %= mod
67      num = 0
68      if dic['1 0'] and dic['0 1']:
69          num = (pow(2,dic['1 0'],mod)+pow(2,dic['0 1'],mod)-1)
70          num %= mod
71      elif dic['1 0']:
72          num = pow(2,dic['1 0'],mod)
73          num %= mod
74      elif dic['0 1']:
75          num = pow(2,dic['0 1'],mod)
76          num %= mod
77      if num != 0:
78          ans *= num
79          ans %= mod
80      print(ans-1+dic['0 0'])
81  if __name__ == "__main__":
82      main()
```


**Question**: Does line `23` have control dependence over line `30` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes