[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 solution(number_of_signs):
2       a_to_b = []
3       for i in range(number_of_signs):
4           sign = raw_input()
5           d, a, b = sign.split()
6           d = int(d)
7           a = int(a)
8           b = int(b)
9           a_to_b.append((d + a, d - b))
10      N = None
11      M = None
12      maximum_set_size = 1
13      number_of_maximum_set_sized = 0
14      current_set_size = 1
15      a_for_comparision, b_for_comparision = a_to_b.pop(0)
16      is_finish = False
17      while not is_finish and len(a_to_b) > 0:
18          for index, tup in enumerate(a_to_b):
19              if index == len(a_to_b) - 1:
20                  is_finish = True
21              a, b = tup
22              if N is not None and M is None:
23                  if a != a_for_comparision:
24                      M = b
25                      b_for_comparision = b
26                  current_set_size += 1
27              elif M is not None and N is None:
28                  if b != b_for_comparision:
29                      N = a
30                      a_for_comparision = a
31                  current_set_size += 1
32              elif M is not None and N is not None:
33                  if b != b_for_comparision and a != a_for_comparision:
34                      if current_set_size > maximum_set_size:
35                          maximum_set_size = current_set_size
36                          number_of_maximum_set_sized = 1
37                      elif current_set_size == maximum_set_size:
38                          number_of_maximum_set_sized += 1
39                      current_set_size = 1
40                      N = None
41                      M = None
42                      a_to_b = a_to_b[index:]
43                      break
44                      a_for_comparision = a
45                      b_for_comparision = b
46                  else:
47                      current_set_size+=1
48              elif a == a_for_comparision and b == b_for_comparision:
49                  if (N is None):
50                      a_for_comparision = a
51                  if (M is None):
52                      b_for_comparision = b
53                  current_set_size += 1
54                  continue
55              elif a == a_for_comparision and b != b_for_comparision:
56                  if (N is None):
57                      N = a
58                      a_for_comparision = N
59                  b_for_comparision = b
60                  current_set_size += 1
61              elif b == b_for_comparision and a != a_for_comparision:
62                  if (M is None):
63                      M = b
64                      b_for_comparision = b
65                  a_for_comparision = a
66                  current_set_size += 1
67              elif b != b_for_comparision and a != a_for_comparision:
68                  if len(a_to_b) > index:
69                      if a == a_to_b[index+1][0]:
70                          N = a_to_b[index +1][0]
71                          a_for_comparision = a_to_b[index +1][0]
72                          M = b_for_comparision
73                          current_set_size+=1
74                      else:
75                          M = a_to_b[index +1][1]
76                          b_for_comparision = a_to_b[index +1][1]
77                          N = a_for_comparision
78                          current_set_size+=1
79                  else:
80                      current_set_size += 1
81      return maximum_set_size, number_of_maximum_set_sized
82  def read_input():
83      number_of_tests = raw_input()
84      for test_number in range(int(number_of_tests)):
85          number_of_signs = raw_input()
86          try:
87              print_output(solution(int(number_of_signs)), test_number + 1)
88          except Exception as exc:
89              pass
90  def print_output(answer, test_case):
91      print "Case #{0}: {1} {2}".format(test_case, answer[0], answer[1])
92  read_input()
```


**Question**: Does line `3` have control dependence over line `42` in function `solution`? If so, provide a trace.

**Output**:


Ground truth answer: no