[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   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 line `50` have control dependence over line `61` in function `D`? If so, provide a trace.

**Output**:


Ground truth answer: yes