[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
2   from collections import deque
3   from heapq import heappush, heappop
4   import sys
5   import math
6   import bisect
7   import random
8   def LI(): return list(map(int, sys.stdin.readline().split()))
9   def I(): return int(sys.stdin.readline())
10  def LS():return list(map(list, sys.stdin.readline().split()))
11  def S(): return list(sys.stdin.readline())[:-1]
12  def IR(n):
13      l = [None for i in range(n)]
14      for i in range(n):l[i] = I()
15      return l
16  def LIR(n):
17      l = [None for i in range(n)]
18      for i in range(n):l[i] = LI()
19      return l
20  def SR(n):
21      l = [None for i in range(n)]
22      for i in range(n):l[i] = S()
23      return l
24  def LSR(n):
25      l = [None for i in range(n)]
26      for i in range(n):l[i] = SR()
27      return l
28  mod = 1000000007
29  sys.setrecursionlimit(1000000)
30  def A():
31      n = I()
32      b = LI()
33      ans = []
34      d = max(b)
35      i = 0
36      while b:
37          j = len(b)-1
38          while j >= 0:
39              if b[j] > j+1:
40                  print(-1)
41                  quit()
42              elif b[j] == j+1:
43                  ans.append(b[j])
44                  b.pop(j)
45                  break
46              j -= 1
47      for i in ans[::-1]:
48          print(i)
49  def B():
50      n = I()
51      if n%2:
52          ans = []
53          for i in range(1,n+1):
54              for j in range(1,n+1):
55                  if j != n-i and j != i:
56                      ans.append((min(i,j),max(i,j)))
57          ans = list(set(ans))
58          print(len(ans))
59          ans.sort()
60          for i in ans:
61              print(*i)
62      else:
63          ans = []
64          for i in range(1,n+1):
65              for j in range(1,n+1):
66                  if j != n+1-i and j != i:
67                      ans.append((min(i,j),max(i,j)))
68          ans = list(set(ans))
69          print(len(ans))
70          ans.sort()
71          for i in ans:
72              print(*i)
73  def C():
74      return
75  def D():
76      n,a,b = LI()
77      pf = LI()
78      f_ans = float("inf")
79      l = []
80      for i in range(n):
81          l.append(pf[i])
82      l.sort()
83      f = defaultdict(int)
84      for i in pf:
85          f[i] = l.index(i)
86      for c in range(2):
87          p = pf.copy()
88          ans = 0
89          while l != p:
90              d = []
91              i = 0
92              while i < n:
93                  j = i
94                  while j < n-1:
95                      if f[p[j+1]] != f[p[j]]+1:
96                          break
97                      j += 1
98                  d.append(p[i:j+1])
99                  i = j+1
100              m = len(d)
101              ld = []
102              for i in range(m):
103                  ld.append(d[i])
104              ld.sort()
105              fd = defaultdict(int)
106              for i in d:
107                  fd[str(i)] = ld.index(i)
108              if c == 0:
109                  for i in range(m)[::-1]:
110                      if d[i] != ld[i]:
111                          j = fd[str(d[i])]
112                          x = d[i]
113                          if j > i:
114                              k = 0
115                              for o in range(i+1,j+1):
116                                  k += len(d[o])
117                              p = len(d[i])
118                              ans += min(b*k,a*p)
119                              d.pop(i)
120                              d.insert(j,x)
121                              break
122                          else:
123                              k = 0
124                              for o in range(j,i):
125                                  k += len(d[o])
126                              p = len(d[i])
127                              ans += min(a*k,b*p)
128                              d.pop(i)
129                              d.insert(j,x)
130                              break
131              else:
132                  for i in range(m)[::-1]:
133                      if d[i] != ld[i]:
134                          j = fd[str(d[i])]
135                          x = d[i]
136                          if j > i:
137                              k = 0
138                              for o in range(i+1,j+1):
139                                  k += len(d[o])
140                              p = len(d[i])
141                              ans += min(b*k,a*p)
142                              d.pop(i)
143                              d.insert(j,x)
144                              break
145                          else:
146                              k = 0
147                              for o in range(j,i):
148                                  k += len(d[o])
149                              p = len(d[i])
150                              ans += min(a*k,b*p)
151                              d.pop(i)
152                              d.insert(j,x)
153                              break
154              p = [None for i in range(n)]
155              s = 0
156              for i in range(len(d)):
157                  for j in range(len(d[i])):
158                      p[s+j] = d[i][j]
159                  s += len(d[i])
160          f_ans = min(f_ans,ans)
161      print(f_ans)
162  def E():
163      return
164  def F():
165      return
166  def G():
167      return
168  def H():
169      return
170  if __name__ == "__main__":
171      D()
```


**Question**: Does line `110` have control dependence over line `143` in function `D`? If so, provide a trace.

**Output**:


Ground truth answer: no