[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   from math import sqrt, pow, log, ceil, log10
2   from sys import stdin, setrecursionlimit
3   from sets import Set
4   setrecursionlimit(100000)
5   debug = False
6   def isCycleCorrect(A, C, L, ans):
7       la = len(ans)
8       for i in range(A):
9           pos = 0
10          isThereAWinner = False
11          while (isThereAWinner == False) and (pos < L[i] * la):
12              if C[i][pos % L[i]] != ans[pos % la]:
13                  if debug:
14                      print "->", C[i][pos % L[i]], ans[pos % la], A, i
15                  if (C[i][pos % L[i]], ans[pos % la]) == ('R', 'S'):
16                      return False
17                  if (C[i][pos % L[i]], ans[pos % la]) == ('S', 'R'):
18                      isThereAWinner = True
19                      break
20                  if (C[i][pos % L[i]], ans[pos % la]) == ('R', 'P'):
21                      isThereAWinner = True
22                      break
23                  if (C[i][pos % L[i]], ans[pos % la]) == ('P', 'R'):
24                      return False
25                  if (C[i][pos % L[i]], ans[pos % la]) == ('P', 'S'):
26                      isThereAWinner = True
27                      break
28                  if (C[i][pos % L[i]], ans[pos % la]) == ('S', 'P'):
29                      return False
30              pos += 1
31          return False
32      return True
33  def solve(A, C, L):
34      ans = ""
35      pos = 0
36      isAlive = [True] * A
37      while True:
38          possibilities = {}
39          possibilities['R'] = 0
40          possibilities['S'] = 0
41          possibilities['P'] = 0
42          predR = []
43          predS = []
44          predP = []
45          if debug:
46              print "alive ans", isAlive, ans
47          if not any(isAlive):
48              print ans
49              return
50          for i in range(A):
51              if isAlive[i]:
52                  possibilities[C[i][pos % L[i]]] = 1
53                  if C[i][pos % L[i]] == 'R':
54                      predR += [i]
55                  if C[i][pos % L[i]] == 'S':
56                      predS += [i]
57                  if C[i][pos % L[i]] == 'P':
58                      predP += [i]
59                  if possibilities['R'] + possibilities['S'] + possibilities['P'] == 3:
60                      print "IMPOSSIBLE"
61                      return
62          (r, s, p) = (possibilities['R'], possibilities['S'], possibilities['P'])
63          if debug:
64              print "->", (r, s, p)
65          if (r, s, p) == (1, 0, 0):
66              ans += "P"
67              print ans
68              return
69          if (r, s, p) == (0, 1, 0):
70              ans += "R"
71              print ans
72              return
73          if (r, s, p) == (0, 0, 1):
74              ans += "S"
75              print ans
76              return
77          if len(ans) == 500:
78              print "IMPOSSIBLE"
79              if debug:
80                  print "LEN", ans
81              return
82          if (r, s, p) == (1, 1, 0):
83              for i in predS:
84                  isAlive[i] = False
85              ans += "R"
86          if (r, s, p) == (1, 0, 1):
87              for i in predR:
88                  isAlive[i] = False
89              ans += "P"
90          if (r, s, p) == (0, 1, 1):
91              for i in predP:
92                  isAlive[i] = False
93              ans += "S"
94          pos += 1
95  T = int(stdin.readline())
96  for i in range(1,T+1):
97      A = stdin.readline()
98      A = int(A)
99      C = [""] * A
100      L = [0] * A
101      for j in range(A):
102          C[j] = stdin.readline().rstrip() 
103          L[j] = len(C[j])
104      print "Case #" + str(i) + ":", 
105      if debug:
106          print " -> ", A, C, L
107      solve(A, C, L)
```


**Question**: Does line `51` have control dependence over line `92` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes