[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 collections import defaultdict, deque
2   from heapq import heappush, heappop
3   from itertools import permutations, accumulate
4   import sys
5   import math
6   import bisect
7   def LI(): return [int(x) for x in sys.stdin.readline().split()]
8   def I(): return int(sys.stdin.readline())
9   def LS():return [list(x) for x in sys.stdin.readline().split()]
10  def S():
11      res = list(sys.stdin.readline())
12      if res[-1] == "\n":
13          return res[:-1]
14      return res
15  def IR(n):
16      return [I() for i in range(n)]
17  def LIR(n):
18      return [LI() for i in range(n)]
19  def SR(n):
20      return [S() for i in range(n)]
21  def LSR(n):
22      return [LS() for i in range(n)]
23  sys.setrecursionlimit(1000000)
24  mod = 1000000007
25  dd = [(1,0),(-1,0),(0,1),(0,-1)]
26  def solve():
27      h,w,k = LI()
28      y,x,s,t = LI()
29      D = [(k,0),(-k,0),(0,k),(0,-k)]
30      y -= 1
31      x -= 1
32      s -= 1
33      t -= 1
34      q = deque([(y,x,-1,0)])
35      d = [[float("inf")]*w for i in range(h)]
36      d[y][x] = 0
37      ri = []
38      le = []
39      do = []
40      up = []
41      c = []
42      for i in range(h):
43          c.append(input())
44          ri.append([0] * w)
45          le.append([0] * w)
46          do.append([0] * w)
47          up.append([0] * w)
48      for i in range(h):
49          ri[i][-1] = 0
50          for j in range(w-1)[::-1]:
51              if c[i][j+1] == "@":
52                  ri[i][j] = 0
53              else:
54                  l = ri[i][j+1]
55                  if l < k:
56                      ri[i][j] = l+1
57                  else:
58                      ri[i][j] = k
59      for i in range(h):
60          le[i][0] = 0
61          for j in range(1,w):
62              if c[i][j-1] == "@":
63                  le[i][j] = 0
64              else:
65                  l = le[i][j-1]
66                  if l < k:
67                      le[i][j] = l+1
68                  else:
69                      le[i][j] = k
70      for j in range(w):
71          do[-1][j] = 0
72          for i in range(h-1)[::-1]:
73              if c[i+1][j] == "@":
74                  do[i][j] = 0
75              else:
76                  l = do[i+1][j]
77                  if l < k:
78                      do[i][j] = l+1
79                  else:
80                      do[i][j] = k
81      for j in range(w):
82          up[0][j] = 0
83          for i in range(1,h):
84              if c[i-1][j] == "@":
85                  up[i][j] = 0
86              else:
87                  l = up[i-1][j]
88                  if l < k:
89                      up[i][j] = l+1
90                  else:
91                      up[i][j] = k
92      while q:
93          y,x,p,l = q.popleft()
94          if (y,x) == (s,t):
95              print(d[s][t])
96              return
97          nd = d[y][x] + 1
98          for i in range(4):
99              if i == 0:
100                  dy,dx = -up[y][x],0
101              elif i == 1:
102                  dy,dx = do[y][x],0
103              elif i == 2:
104                  dy,dx = 0,-le[y][x]
105              else:
106                  dy,dx = 0,ri[y][x]
107              ny,nx = y+dy,x+dx
108              if (y,x) == (ny,nx):
109                  continue
110              if nd < d[ny][nx]:
111                  diff = abs(ny-y)+abs(nx-x)
112                  d[ny][nx] = nd
113                  q.append((ny,nx,i,diff))
114          if p >= 0 and l > 1:
115              dy,dx = dd[p]
116              ny,nx = y+dy,x+dx
117              nd = d[y][x]
118              if nd < d[ny][nx]:
119                  d[ny][nx] = nd
120                  q.appendleft((ny,nx,p,l-1))
121      print(-1)
122      return
123  if __name__ == "__main__":
124      solve()
```


**Question**: Does line `81` have control dependence over line `89` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes