[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 areInConflict(x, y):
2       bx = "{0:b}".format(x)[::-1]
3       by = "{0:b}".format(y)[::-1]
4       lx = len(bx)
5       ly = len(by)
6       for i in range(min(lx, ly)):
7           if bx[i]=="1" and by[i]=="1":
8               return True
9       return False
10  def solve(x,y):
11      origX = x
12      origY = y
13      x = max(x, -x)
14      y = max(y, -y)
15      if x%2==y%2:
16          return "IMPOSSIBLE"
17      x1 = 1
18      while x1 < x:
19          x1 *= 2
20      x2 = x1 - x
21      y1 = 1
22      while y1 < y:
23          y1 *= 2
24      y2 = y1 - y
25      if not areInConflict(x, y):
26          bx = "{0:b}".format(x)[::-1]
27          by = "{0:b}".format(y)[::-1]
28          result = [None] * max(len(bx),len(by))
29          for i in range(len(bx)):
30              if bx[i] == "1":
31                  result[i] = "E"
32          for i in range(len(by)):
33              if by[i] == "1":
34                  result[i] = "N"
35      elif not areInConflict(x2, x1) and not areInConflict(x2, y) and not areInConflict(x1, y):
36          bx1 = "{0:b}".format(x1)[::-1]
37          bx2 = "{0:b}".format(x2)[::-1]
38          by = "{0:b}".format(y)[::-1]
39          result = [None] * max(len(bx1), len(bx2), len(by))
40          for i in range(len(bx1)):
41              if bx1[i] == "1":
42                  result[i] = "E"
43          for i in range(len(bx2)):
44              if bx2[i] == "1":
45                  result[i] = "W"
46          for i in range(len(by)):
47              if by[i] == "1":
48                  result[i] = "N"
49      elif not areInConflict(y2, y1) and not areInConflict(y2, x) and not areInConflict(y1, x):
50          by1 = "{0:b}".format(y1)[::-1]
51          by2 = "{0:b}".format(y2)[::-1]
52          bx = "{0:b}".format(x)[::-1]
53          result = [None] * max(len(by1), len(by2), len(bx))
54          for i in range(len(bx)):
55              if bx[i] == "1":
56                  result[i] = "E"
57          for i in range(len(by1)):
58              if by1[i] == "1":
59                  result[i] = "N"
60          for i in range(len(by2)):
61              if by2[i] == "1":
62                  result[i] = "S"
63      elif not areInConflict(y2, y1) \
64          and not areInConflict(y2, x1) \
65          and not areInConflict(y2, x2) \
66          and not areInConflict(y1, x1) \
67          and not areInConflict(y1, x2) \
68          and not areInConflict(x2, x1):
69          bx1 = "{0:b}".format(x1)[::-1]
70          bx2 = "{0:b}".format(x2)[::-1]
71          by1 = "{0:b}".format(y1)[::-1]
72          by2 = "{0:b}".format(y2)[::-1]
73          result = [None] * max(len(by1), len(by2), len(bx))
74          for i in range(len(bx1)):
75              if bx1[i] == "1":
76                  result[i] = "E"
77          for i in range(len(bx2)):
78              if bx2[i] == "1":
79                  result[i] = "W"
80          for i in range(len(by1)):
81              if by1[i] == "1":
82                  result[i] = "N"
83          for i in range(len(by2)):
84              if by2[i] == "1":
85                  result[i] = "S"
86      else:
87          return "IMPOSSIBLE"
88      if x != origX:
89          for i in xrange(len(result)):
90              if result[i]=="W":
91                  result[i] = "E"
92              elif result[i]=="E":
93                  result[i] = "W"
94      if y != origY:
95          for i in xrange(len(result)):
96              if result[i]=="N":
97                  result[i] = "S"
98              elif result[i]=="S":
99                  result[i] = "N"
100      return "".join(result)
101  T = int(raw_input())
102  for t in range(T):
103      x,y = map(int,raw_input().split())
104      print "Case #{}: {}".format(t+1, solve(x, y))
```


**Question**: Does line `66` have control dependence over line `79` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes