[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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 `(l,87)` have data dependence over `(up,89)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes