[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**. 

```C
1   #include<stdio.h>
2   #include<string.h>
3   #include<stdlib.h>
4   #include<stdbool.h>
5   #include<time.h>
6   #define inf 1072114514
7   #define llinf 4154118101919364364
8   #define mod 1000000007
9   #define pi 3.1415926535897932384
10  int max(int a,int b){if(a>b){return a;}return b;}
11  int min(int a,int b){if(a<b){return a;}return b;}
12  int zt(int a,int b){return max(a,b)-min(a,b);}
13  int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
14  int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
15  int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
16  int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
17  int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
18  int nHr(int a,int b){return nCr(a+b-1,b);}
19  int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
20  int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
21  int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
22  int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
23  int sankaku(int x){return ((1+x)*x)/2;}
24  long long llmax(long long a,long long b){if(a>b){return a;}return b;}
25  long long llmin(long long a,long long b){if(a<b){return a;}return b;}
26  long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
27  long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
28  long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
29  long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
30  long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
31  long long llnCr(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
32  long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
33  long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
34  long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
35  long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
36  long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
37  long long llsankaku(long long x){return ((1+x)*x)/2;}
38  double dbmax(double a,double b){if(a>b){return a;}return b;}
39  double dbmin(double a,double b){if(a<b){return a;}return b;}
40  double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
41  int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
42  int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
43  int llsortfncsj(const void *a,const void *b){if(*(long long *)a>*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
44  int llsortfnckj(const void *a,const void *b){if(*(long long *)a<*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
45  int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
46  int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
47  int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
48  int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
49  int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
50  int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
51  void shuffledget(int x[],int n){
52      srand(time(0));
53      int i,b[524288],p,c;
54      for(i=0;i<n;i++){
55          b[i]=i;
56      }
57      for(i=n;i>=1;i--){
58          p=rand()%i;
59          c=b[i-1];b[i-1]=b[p];b[p]=c;
60      }
61      for(i=0;i<n;i++){
62          scanf("%d",&x[b[i]]);
63      }
64  }
65  int dx4[4]={1,-1,0,0};
66  int dy4[4]={0,0,1,-1};
67  int dx8[8]={-1,-1,-1,0,0,1,1,1};
68  int dy8[8]={-1,0,1,-1,1,-1,0,1};
69  int search(int x,int a[],int n){
70      int st=0,fi=n-1,te;
71      while(st<=fi){
72          te=(st+fi)/2;
73          if(a[te]<x){st=te+1;}else{fi=te-1;}
74      }
75      return st;
76  }
77  void prarr(int arr[],int n){
78    int i;
79    for(i=0;i<n;i++){
80      if(i){printf(" ");}
81      printf("%d",arr[i]);
82    }
83    printf("\n");
84    return;
85  }
86  typedef struct{
87  int val;
88  int node;
89  }sd;
90  int sdsortfnc(const void *a,const void *b){
91  if(((sd*)a)->val < ((sd*)b)->val){return -1;}
92  if(((sd*)a)->val > ((sd*)b)->val){return 1;}
93  return 0;
94  }
95  int main(void){
96      long long i,j,w,fl=0,dg,st,fi,te;
97      bool mem[16][2]={0};
98      char s[8],t[8];
99      for(i=9;i>=0;i--){
100        w=llpow(10,i);
101        printf("? %lld\n",w);
102        fflush(stdout);
103        scanf("%s",s);
104        if(s[0]=='Y')
105          {mem[i][0]=1;}
106        else
107          {fl|=2;}
108        w=llpow(10,i+1)-1;
109        printf("? %lld\n",w);
110        fflush(stdout);
111        scanf("%s",t);      
112        if(t[0]=='Y')
113          {mem[i][1]=1;}
114        else
115          {fl|=1;}
116      }
117      if(fl==1){
118        for(i=9;i>=0;i--){
119          if(mem[i][1]==0){
120            dg=i;
121            break;
122          }
123          if(i==0){
124            printf("! 1\n");
125            fflush(stdout);
126            return 0;
127          }
128        }
129      }
130      if(fl==2){
131        for(i=0;i<=9;i++){
132          if(mem[i][0]==0){
133            dg=i;
134            break;
135          }
136        }
137      }
138      if(fl==3){
139        for(i=0;i<=9;i++){
140          if((mem[i][0]&mem[i][1])==1){
141            dg=i;
142            break;
143          }
144        }
145      }
146      st = 1;
147      fi = 1;
148      for (i = 1; i <= dg; i++) 
149        st *= 10;
150      for (i = 1; i <= dg + 1; i++) 
151        fi *= 10;
152      fi -= 1;
153      while(st<=fi){
154        te=(st+fi)/2;
155        printf("? %lld\n",te*10);
156        fflush(stdout);
157        scanf("%s",s);
158        if(s[0]=='N'){
159          fi=te-1;
160        }
161        else{st=te+1;}
162      }
163      printf("! %lld\n",fi);
164      fflush(stdout);
165      return 0;
166  }
```


**Question**: Does line `112` have control dependence over line `105` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no