[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   def solve(R, C):
3       moves = []
4       if R > C:
5           flippedmoves = solve(C, R)
6           for move in flippedmoves:
7               moves.append((move[1], move[0]))
8           return moves
9       if R == 2 and C == 2:
10          return moves
11      if R == 2 and C == 3:
12          return moves
13      if R == 2 and C == 4:
14          return moves
15      if R == 3 and C == 3:
16          return moves
17      if R == 3 and C == 4:
18          moves.append((1, 1))
19          moves.append((2, 3))
20          moves.append((3, 1))
21          moves.append((2, 4))
22          moves.append((1, 2))
23          moves.append((3, 3))
24          moves.append((1, 4))
25          moves.append((2, 2))
26          moves.append((3, 4))
27          moves.append((2, 1))
28          moves.append((1, 3))
29          moves.append((3, 2))
30          return moves
31      if R == 4 and C == 4:
32          moves.append((1, 1))
33          moves.append((2, 3))
34          moves.append((3, 1))
35          moves.append((4, 3))
36          moves.append((2, 4))
37          moves.append((1, 2))
38          moves.append((3, 3))
39          moves.append((1, 4))
40          moves.append((2, 2))
41          moves.append((4, 1))
42          moves.append((3, 4))
43          moves.append((2, 1))
44          moves.append((4, 2))
45          moves.append((1, 3))
46          moves.append((4, 4))
47          moves.append((3, 2))
48          return moves
49      if R == 2:
50          for c in range(3, C+1):
51              moves.append((2, c))
52              moves.append((1, c-2))
53          if C != 6:
54              moves.append((2, 1))
55              moves.append((1, C-1))
56              moves.append((2, 2))
57              moves.append((1, C))
58          else:
59              moves.append((2, 1))
60              moves.append((1, C))
61              moves.append((2, 2))
62              moves.append((1, C-1))
63          return moves
64      if R == 3:
65          for c in range(3, C+1):
66              moves.append((2, c))
67              moves.append((1, c-2))
68              moves.append((3, c-1))
69          moves.append((2, 1))
70          moves.append((3, C))
71          moves.append((1, C-1))
72          moves.append((2, 2))
73          moves.append((1, C))
74          moves.append((3, 1))
75          return moves
76      prevmoves = solve(R-2, C)
77      for c in range(3, C+1):
78          moves.append((2, c))
79          moves.append((1, c-2))
80      if C != 6:
81          moves.append((2, 1))
82          moves.append((1, C-1))
83          moves.append((2, 2))
84          moves.append((1, C))
85      else:
86          moves.append((2, 1))
87          moves.append((1, C))
88          moves.append((2, 2))
89          moves.append((1, C-1))
90      for move in prevmoves:
91          moves.append((move[0]+2, move[1]))
92      return moves
93  def invalid(pt1, pt2):
94      if pt1[0] == pt2[0]:
95          return True
96      if pt1[1] == pt2[1]:
97          return True
98      if pt1[0] - pt1[1] == pt2[0] - pt2[1]:
99          return True
100      if pt1[0] + pt1[1] == pt2[0] + pt2[1]:
101          return True
102      return False
103  T = int(raw_input())
104  for i in range(T):
105      stuff = [int(j) for j in raw_input().split()]
106      R = stuff[0]
107      C = stuff[1]
108      moves = solve(R, C)
109      if len(moves) == 0:
110          print "Case #" + str(i+1) + ": IMPOSSIBLE"
111      else:
112          print "Case #" + str(i+1) + ": POSSIBLE"
113          for move in moves:
114              print str(move[0]) + ' ' + str(move[1])
```


**Question**: Does line `31` have control dependence over line `55` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes