[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   #include<assert.h>
7   #define inf 1072114514
8   #define llinf 4154118101919364364
9   #define mod 1000000007
10  #define pi 3.1415926535897932384
11  int max(int a,int b){if(a>b){return a;}return b;}
12  int min(int a,int b){if(a<b){return a;}return b;}
13  int zt(int a,int b){return max(a,b)-min(a,b);}
14  int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
15  int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
16  int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
17  int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
18  int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
19  int nHr(int a,int b){return nCr(a+b-1,b);}
20  int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
21  int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
22  int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
23  int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
24  int sankaku(int x){return ((1+x)*x)/2;}
25  void swap(int *a,int *b){int c;c=(*a);(*a)=(*b);(*b)=c;}
26  long long llmax(long long a,long long b){if(a>b){return a;}return b;}
27  long long llmin(long long a,long long b){if(a<b){return a;}return b;}
28  long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
29  long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
30  long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
31  long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
32  long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
33  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;}
34  long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
35  long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
36  long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
37  long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
38  long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
39  long long llsankaku(long long x){return ((1+x)*x)/2;}
40  void llswap(long long *a,long long *b){long long c;c=(*a);(*a)=(*b);(*b)=c;}
41  double dbmax(double a,double b){if(a>b){return a;}return b;}
42  double dbmin(double a,double b){if(a<b){return a;}return b;}
43  double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
44  void dbswap(double *a,double *b){double c;c=(*a);(*a)=(*b);(*b)=c;}
45  void chswap(char *a,char *b){char c;c=(*a);(*a)=(*b);(*b)=c;}
46  int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
47  int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
48  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;}
49  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;}
50  int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
51  int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
52  int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
53  int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
54  int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
55  int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
56  void shuffledget(int x[],int n){
57      int i,b[524288],p,c;
58      for(i=0;i<n;i++){
59          b[i]=i;
60      }
61      for(i=n;i>=1;i--){
62          p=rand()%i;
63          c=b[i-1];b[i-1]=b[p];b[p]=c;
64      }
65      for(i=0;i<n;i++){
66          scanf("%d",&x[b[i]]);
67      }
68  }
69  int dx4[4]={1,-1,0,0};
70  int dy4[4]={0,0,1,-1};
71  int dx8[8]={-1,-1,-1,0,0,1,1,1};
72  int dy8[8]={-1,0,1,-1,1,-1,0,1};
73  int search(int x,int a[],int n){
74      int st=0,fi=n-1,te;
75      while(st<=fi){
76          te=(st+fi)/2;
77          if(a[te]<x){st=te+1;}else{fi=te-1;}
78      }
79      return st;
80  }
81  void prarr(int arr[],int n){
82    int i;
83    for(i=0;i<n;i++){
84      if(i){printf(" ");}
85      printf("%d",arr[i]);
86    }
87    printf("\n");
88    return;
89  }
90  void getperm(int a[],int n){
91    int i,p;
92    for(i=0;i<n;i++){
93      a[i]=i;
94    }
95    for(i=n-1;i>=1;i--){
96      p=rand()%(i+1);
97      swap(&a[p],&a[i]);
98    }
99  }
100  typedef struct{
101  int val;
102  int node;
103  }sd;
104  int sdsortfnc(const void *a,const void *b){
105  if(((sd*)a)->val < ((sd*)b)->val){return -1;}
106  if(((sd*)a)->val > ((sd*)b)->val){return 1;}
107  return 0;
108  }
109  void coordinate_comp(int a[],int n){
110    int i,c=0;
111    sd dat[524288];
112    for(i=0;i<n;i++){
113      dat[i].val=a[i];
114      dat[i].node=i;
115    }
116    qsort(dat,n,sizeof(dat[0]),sdsortfnc);
117    a[dat[0].node]=c;
118    for(i=1;i<n;i++){
119      if(dat[i-1].val!=dat[i].val){c++;}
120      a[dat[i].node]=c;
121    }
122  }
123  int main(void){
124    int cs;
125    int i,j,n,m,k,a,b,c,h,w,l,t,sn;
126    int r=0;
127    int fl[64];
128    scanf("%d",&t);
129    for(cs=1;cs<=t;cs++){
130      scanf("%d",&n);
131      sn=n;
132      printf("Case #%d:\n",cs);
133      w=0;
134      for(i=1;i<50;i++){
135        w+=(1<<(i-1));
136        if(w>=n){
137          l=i;
138          break;
139        }
140      }
141      for(i=l;i>=1;i--){
142        if(n>=(1<<(i-1))+(i-1)){
143          fl[i]=1;
144          n-=(1<<(i-1));
145        }
146        else{
147          fl[i]=0;
148          n--;
149          }
150      }
151      a=0;
152      w=0;
153      for(i=1;i<=l;i++){
154        if(fl[i]==1){
155          w+=(1<<(i-1));
156          if(a==0){
157            for(j=1;j<=i;j++){
158              printf("%d %d\n",i,j);
159            }
160          }
161          else{
162            for(j=i;j>=1;j--){
163              printf("%d %d\n",i,j);
164            }
165          }
166          a=1-a;
167        }
168        else{
169          w++;
170          if(a==0)
171            printf("%d %d\n",i,1);
172          else
173            printf("%d %d\n",i,i);
174        }
175      }
176      while(w!=sn){
177        l++;
178        if(a==0)
179          printf("%d %d\n",l,1);
180        else
181          printf("%d %d\n",l,l);
182        w++;
183      }
184    }
185    return 0;
186  }
```


**Question**: Does line `154` have control dependence over line `155` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes