[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   import sys
4   import math
5   def LI(): return [int(x) for x in sys.stdin.readline().split()]
6   def I(): return int(sys.stdin.readline())
7   def LS():return [list(x) for x in sys.stdin.readline().split()]
8   def S(): return list(sys.stdin.readline())[:-1]
9   def IR(n):
10      return [I() for i in range(n)]
11  def LIR(n):
12      return [LI() for i in range(n)]
13  def SR(n):
14      return [S() for i in range(n)]
15  def LSR(n):
16      return [LS() for i in range(n)]
17  sys.setrecursionlimit(1000000)
18  mod = 1000000007
19  def A():
20      n = S()
21      for i in n:
22          if i == "1":
23              print(9,end = "")
24          else:
25              print(1,end = "")
26      print()
27  def B():
28      n = I()
29      for i in range(n,1000):
30          if i%111 == 0:
31              print(i)
32              quit()
33      return
34  def C():
35      return
36  def D():
37      n = I()
38      p = []
39      for i in range(n):
40          p.append(LI())
41      b = []
42      for i in range(n):
43          x,y = p[i]
44          for j in range(i):
45              s,t = p[j]
46              if (abs(x-s)+abs(y-t))%2:
47                  print(-1)
48                  return
49      ans = []
50      for i in range(n):
51          f = 0
52          x,y = p[i]
53          if not (x+y)%2:
54              x -= 1
55              if not ans:
56                  ans.append(1)
57          if x < 0:
58              x = -x
59              f |= 1
60          if y < 0:
61              y = -y
62              f |= 2
63          bx = list(map(int, bin(x)[2:]))
64          by = list(map(int, bin(y)[2:]))
65          bx = bx[::-1]
66          by = by[::-1]
67          for _ in range(39-len(bx)):
68              bx.append(0)
69          for _ in range(39-len(by)):
70              by.append(0)
71          b.append([bx,by,f])
72      for i in range(39):
73          ans.append(1<<i)
74      v = [[] for i in range(n)]
75      for i in range(n):
76          x,y,f = b[i]
77          if x[0]:
78              p = 1
79              v[i].append(0)
80          else:
81              p = 0
82              v[i].append(1)
83          for j in range(1,39):
84              if p:
85                  if x[j]:
86                      if y[j]:
87                          v[i][-1] ^= 3
88                          x[j+1] = 1
89                          v[i].append(1)
90                          p = 0
91                      else:
92                          v[i].append(0)
93                  else:
94                      if y[j]:
95                          v[i].append(1)
96                          p = 0
97                      else:
98                          v[i][-1] ^= 3
99                          v[i].append(0)
100              else:
101                  if y[j]:
102                      if x[j]:
103                          v[i][-1] ^= 3
104                          y[j+1] ^= 1
105                          v[i].append(0)
106                          p = 1
107                      else:
108                          v[i].append(1)
109                  else:
110                      if x[j]:
111                          v[i].append(0)
112                          p = 1
113                      else:
114                          v[i][-1] ^= 3
115                          v[i].append(1)
116          for j in range(39):
117              if v[i][j] in [0,3]:
118                  if f&1:
119                      v[i][j]^=3
120              else:
121                  if f&2:
122                      v[i][j]^=3
123      p = ["R","U","D","L"]
124      print(len(ans))
125      print(*ans)
126      for i in v:
127          if len(ans) == 40:
128              print("R",end = "")
129          for j in i:
130              print(p[j],end = "")
131          print()
132      return
133  def E():
134      return
135  def F():
136      return
137  def G():
138      return
139  def H():
140      return
141  if __name__ == "__main__":
142      D()
```


**Question**: Does `(f,51)` have data dependence over `(y,76)` in function `D`? If so, provide a trace.

**Output**:


Ground truth answer: yes