[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, math, os
2   from fractions import Fraction, gcd
3   LocaL = "MATE_DESKTOP_SESSION_ID" in os.environ
4   def mle_assert(x):
5       if LocaL:
6           assert x
7       elif not x:
8           x = range(10**8)
9   def tle_assert(x):
10      if LocaL:
11          assert x
12      elif not x:
13          for i in range(10000):
14              for j in range(1000):
15                  for k in range(1000):
16                      pass
17  sys.setrecursionlimit(10000)
18  ssr = sys.stdin.readline
19  ssw = sys.stdout.write
20  ssf = sys.stdout.flush
21  sew = sys.stderr.write
22  def rdline(): return ssr().strip()
23  def rdstrs(): return ssr().split()
24  def rdints(): return map(int, ssr().split())
25  def rd1int(): return int(rdline())
26  def do_one_case(cnum):
27      N = rd1int()
28      XY = []
29      for i in range(N):
30          x, y = rdints()
31          XY.append((x,y))
32      XY.sort()
33      A = {}
34      for j in xrange(1,N):
35          x2, y2 = XY[j]
36          for i in xrange(j):
37              x1, y1 = XY[i]
38              a = x2-x1
39              b = y2-y1
40              g = gcd(a,b)
41              a /= g
42              b /= g
43              if (a,b) in A:
44                  A[(a,b)].append((i,j))
45                  A[(-a,-b)].append((j,i))
46              else:
47                  A[(a,b)] = [ (i,j) ]
48                  A[(-a,-b)] = [ (j,i) ]
49      m = 1
50      for a,b in A:
51          E = A[(a,b)]
52          I = {}
53          O = {}
54          for i,j in E:
55              if i in O:
56                  O[i].append(j)
57              else:
58                  O[i] = [j]
59              if j in I:
60                  I[j].append(i)
61              else:
62                  I[j] = [i]
63          NI = set(range(N)) - set(I)
64          C = []
65          for x in NI:
66              if x in O:
67                  Q = [ (0,x)] + [(len(I[y]),y) for y in O[x] ]
68                  Q.sort()
69                  C.append([y for x,y in Q])
70              else:
71                  C.append([x])
72          C1 = [ q for q in C if len(q)==1 ]
73          C2 = [ q for q in C if len(q)&1==0 ]
74          C3 = [ q for q in C if len(q)>1 and len(q)&1==1 ]
75          while len(C3)>1:
76              C2.append(C3[-2]+C3[-1])
77              del C3[-2:]
78          C3 = (C3 + C1)[:2]
79          l2 = sum(len(q) for q in C2)
80          l3 = sum(len(q) for q in C3)
81          if l3:
82              l = l2 + l3
83          elif len(C2)>1:
84              l = l2-1
85          else:
86              l = l2
87          m = max(m, l)
88      print "Case #%d: %d" % (cnum, m)
89  def main():
90      T = rd1int()
91      for i in range(T):
92          do_one_case(i+1)
93          sys.stdout.flush()
94  if __name__ == "__main__":
95      main()
```


**Question**: Does line `54` have control dependence over line `71` in function `do_one_case`? If so, provide a trace.

**Output**:


Ground truth answer: no