[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 solve(B):
2       if B[0] == 0 or B[-1] == 0:
3           return 'IMPOSSIBLE', None
4       grid = []
5       C = len(B)
6       grid = []
7       for j in range(C):
8           row = []
9           for i in range(C):
10              row.append('.')
11          grid.append(row)
12      shadowed = [False] * len(C)
13      shadowed[0] = True
14      B[0] -= 1
15      B[-1] -= 1
16      i = 0
17      for i in range(C):
18          if B[i] == 0 and shadowed[i]:
19              continue
20          if B[i] > 0 and not shadowed[i]:
21              unshadowed_balls = 0
22              j = i - 1
23              l = 1
24              while j >= 1 and not shadowed[j]:
25                  grid[l][j] = '\\'
26                  shadowed[j] = True
27                  unshadowed_balls += 1
28                  l += 1
29                  j -= 1
30              if B[i] <= unshadowed_balls:
31                  excess_balls = unshadowed_balls - B[i] + 1
32                  j = i
33                  while excess_balls > 1:
34                      j -= 1
35                      excess_balls -= 1
36                      shadowed[j] = False
37              else:
38                  B[i] -= (unshadowed_balls + 1)
39                  shadowed[i] = True
40          if B[i] > 0 and shadowed[i]:
41              r = B[i]
42              j = i + 1
43              l = 1
44              while j < (C - 1) and shadowed[j]:
45                  grid[l][j] = '/'
46                  l += 1
47                  j += 1
48              while j < (C - 1) and r > 0:
49                  grid[l][j] = '/'
50                  shadowed[j] = True
51                  r -= 1
52                  l += 1
53                  j += 1
54      ans = []
55      for index in range(1, C):
56          row = grid[index]
57          include_row = False
58          for cell in row:
59              if cell == '\\' or cell == '/':
60                  include_row = True
61                  break
62          if include_row:
63              ans.append(row)
64          else:
65              break
66      ans = [['.' for i in range(C)]] + ans
67      return len(ans), ans
68  def lay_right_ramp(i, grid, shadowed):
69      pass
70  def lay_left_ramp(i, grid, shadowed):
71      pass
72  def main():
73      T = int(raw_input())
74      t = 1
75      while t <= T:
76          S = int(raw_input())
77          B_raw = raw_input().split()
78          B = [int(x) for x in B_raw]
79          res, grid = solve(B)
80          print "Case #{0}: {1}".format(str(t), str(res))
81          if grid:
82              i = len(grid) - 1
83              while i >= 0:
84                  print ''.join(grid[i])
85                  i = i - 1
86          t += 1
87  if __name__ == "__main__":
88      main()
```


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

**Output**:


Ground truth answer: yes