[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   import itertools
3   import math
4   import collections
5   import functools
6   sys.setrecursionlimit(10000)
7   def inputInts():
8       return map(int, raw_input().split())
9   def solve(N, M):
10      topHigh = None
11      botHigh = None
12      topLow = None
13      botLow = None
14      for i in xrange(N):
15          for j in xrange(i+1, N):
16              a = M[i][0]
17              b = M[i][1]
18              c = M[j][0]
19              d = M[j][1]
20              if a == c or b == d:
21                  continue
22              top = a-c
23              bot = d-b
24              wasNeg = (top * bot < 0)
25              top = abs(top)
26              bot = abs(bot)
27              if b-d > 0: 
28                  if wasNeg:
29                      return None
30                  if topHigh is None or top * botHigh < topHigh * bot:
31                      topHigh = top
32                      botHigh = bot
33              else: 
34                  if wasNeg:
35                      continue
36                  if topLow is None or top * botHigh > topHigh * bot:
37                      topLow = top
38                      botLow = bot
39      if topHigh is None and topLow is None:
40          return [1, 1]
41      if topHigh is None:
42          if botLow == topLow:
43              return [1, 2]
44          elif topLow > botLow:
45              return [1, int(math.ceil(topLow/botLow))]
46          else:
47              return [int(math.ceil(botLow/topLow)), 1]
48      if topLow is None:
49          if botHigh == topHigh:
50              return [2, 1]
51          elif botHigh > topHigh:
52              return [int(math.ceil(botHigh/topHigh)), 1]
53          else:
54              return [1, int(math.ceil(topHigh/botHigh))]
55      a = topHigh / float(botHigh)
56      b = topLow / float(botLow)
57      if a <= b:
58          return None
59      xMin = 1
60      xMax = 1e20
61      while xMax > xMin:
62          mid = int((xMax + xMin) / 2.)
63          low = b*mid
64          high = a*mid
65          if low == int(low):
66              if high > low + 1:
67                  xMax = mid
68              else:
69                  xMin = mid+1
70          else:
71              if math.ceil(low) < high:
72                  xMax = mid
73              else:
74                  xMin = mid+1
75  T = int(raw_input())
76  for testId in range(T):
77      N = int(raw_input())
78      M = []
79      for n in xrange(N):
80          M.append(inputInts())
81      res = solve(N, M)
82      if res is None:
83          print "Case #{:d}: IMPOSSIBLE".format(testId+1)
84      else:
85          print "Case #{:d}: {:d} {:d}".format(testId+1, res[0], res[1])
```


**Question**: Does line `20` have control dependence over line `31` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes