[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   import bisect,collections,copy,itertools,math,string
2   import sys
3   def I(): return int(sys.stdin.readline().rstrip())
4   def LI(): return list(map(int,sys.stdin.readline().rstrip().split()))
5   def S(): return sys.stdin.readline().rstrip()
6   def LS(): return list(sys.stdin.readline().rstrip().split())
7   def main():
8       n = I()
9       b = LI()
10      ans_lst = []
11      while len(b)!=0:
12          lst = collections.deque([])
13          mx = 1
14          phase = False
15          while len(b)!=0:
16              nxt = b.pop()
17              mx = max(mx, nxt)
18              if mx==1:
19                  lst.appendleft(nxt)
20              else:
21                  phase |= nxt == 1
22                  if phase and nxt != 1:
23                      b.append(nxt)
24                      break
25                  else:
26                      lst.appendleft(nxt)
27          ctn = True
28          up = True
29          now_mx = 0
30          now_mn = mx
31          lenth = 0
32          for x in lst:
33              if up:
34                  if x>=now_mx:
35                      if x==mx:
36                          up = False
37                      else:
38                          lenth += 1
39                  else:
40                      ctn = False
41                      break
42              else:
43                  if x<=now_mn:
44                      now_mn = min(now_mn, x)
45                  else:
46                      ctn = False
47                      break
48          if not ctn or lenth+1<mx or lst[0] != 1:
49              print(-1)
50              exit(0)
51          gr = [[k,len(list(g))] for k, g in itertools.groupby(lst)]
52          com = len(lst)
53          phase = False
54          down = False
55          now_len = 0
56          i = 0
57          while com!= 0:
58              nxt = lst[-1]
59              if not phase:
60                  if now_len+1 < nxt:
61                      if not down:
62                          if gr[i][0] == mx:
63                              down = True
64                              i -= 1
65                          else:
66                              ans_lst.append(gr[i][0])
67                              now_len += 1
68                              com -= 1
69                              gr[i][1] -= 1
70                              i += 1
71                      else:
72                          if gr[i][1] == 0:
73                              i -= 1
74                          else:
75                              ans_lst.append(gr[i][0])
76                              now_len += 1
77                              com -= 1
78                              gr[i][1] -= 1
79                  else:
80                      ans_lst.append(nxt)
81                      com -= 1
82                      lst.pop()
83                      phase |= nxt == mx
84              else:
85                  if nxt == mx:
86                      ans_lst.append(nxt)
87                      com -= 1
88                      lst.pop()
89                  else:
90                      if not down:
91                          if gr[i][0] == mx:
92                              down = True
93                              i -= 1
94                          else:
95                              ans_lst.append(gr[i][0])
96                              com -= 1
97                              gr[i][1] -= 1
98                              i += 1
99                      else:
100                          if gr[i][1] == 0:
101                              i -= 1
102                          else:
103                              ans_lst.append(gr[i][0])
104                              com -= 1
105                              gr[i][1] -= 1
106      print(*ans_lst, sep="\n")
107  main()
```


**Question**: Does line `57` have control dependence over line `77` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes