[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 sys
2   def Q(n,N):
3       n = int(n)
4       if (n<=N and str(n)<=str(N)) or (n>N and str(n)>str(N)):
5           return "Y"
6       else:
7           return "N"
8   def main():
9       ans = ""
10      print("? "+"1"+"0"*10)
11      sys.stdout.flush()
12      A = input()
13      if A=="Y":
14          n = 1
15          print("? "+"9"*n)
16          sys.stdout.flush()
17          A = input()
18          while A == "N":
19              n+=1
20              print("? "+"9"*n)
21              sys.stdout.flush()
22              A = input()
23          print("! "+str(10**(n-1)))
24      else:
25          l = 0
26          r = 10
27          S = [""]*11
28          S[0] ="Y"
29          S[10] ="N"
30          while 1:
31              m = (l+r)//2
32              print("? "+str(10**m))
33              sys.stdout.flush()
34              A = input()
35              if A=="Y":
36                  if S[m+1]=="N":
37                      K = m+1
38                      break
39                  S[m] = "Y"
40                  l = m
41              else:
42                  if S[m-1]=="Y":
43                      K = m
44                      break
45                  S[m] = "N"
46                  r = m
47          l = 1
48          r = 9
49          S = [""]*10
50          S[9] ="Y"
51          ashi = "9"*10
52          print("? "+ans + str(1) + ashi)
53          sys.stdout.flush()
54          A = input()
55          if A=="Y":
56              ans += "1"
57          else:
58              S[1] = "N"
59              while 1:
60                  m = (l+r)//2
61                  print("? "+ans + str(m) + ashi)
62                  sys.stdout.flush()
63                  A = input()
64                  if A=="Y":
65                      if S[m-1]=="N":
66                          ans += str(m)
67                          break
68                      S[m] = "Y"
69                      r = m
70                  else:
71                      if S[m+1]=="Y":
72                          ans += str(m+1)
73                          break
74                      S[m] = "N"
75                      l = m
76          for i in range(K-1):
77              l = 0
78              r = 9
79              S = [""]*10
80              S[9]= "Y"
81              print("? " + ans + str(0) + ashi)
82              sys.stdout.flush()
83              A = input()
84              if A=="Y":
85                  ans += "0"
86              else:
87                  S[0] = "N"
88                  while 1:
89                      m = (l+r)//2
90                      print("? " + ans + str(m) + ashi)
91                      sys.stdout.flush()
92                      A = input()
93                      if A=="Y":
94                          if S[m-1]=="N":
95                              ans += str(m)
96                              break
97                          S[m] = "Y"
98                          r = m
99                      else:
100                          if S[m+1]=="Y":
101                              ans += str(m+1)
102                              break
103                          S[m] = "N"
104                          l = m
105          print("! " + ans)
106  main()
```


**Question**: Does line `13` have control dependence over line `39` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes