[INSTRUCTIONS]

You are a program-analysis assistant. Please perform a **static** analysis of “information flow” on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.

## 1. Information Flow Definition

An **information flow** from variable `x` to variable `y`, written `x -> y`, exists whenever information stored in `x` is transferred to or used to derive information transferred to `y`. 

We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` is **defined or updated** at `lineNumber`.


### Types of Direct Flows

- **Direct Explicit Flow**  
  A direct explicit flow occurs when the value written by one variable instance is directly used in computing the value of another, through operations like assignments.

  Formally, if `(varA, lineA)` writes a value and `(varB, lineB)` reads that value to compute its own, then there is a **direct explicit flow** from `(varA, lineA)` to `(varB, lineB)`, written as: `(varA,lineA) -> (varB,lineB)`. 


  **Example 1**:
    ```
    1  x = 5
    2  y = x + 1
    ```
    `(x,1) -> (y,2)` is a direct explicit flow.


  **Example 2**:
    ```
    1  x = 5
    2  if cond:
    3      y = x + 1
    ```
    `(x,1) -> (y,3)` is a direct explicit flow. 

    Under static analysis, we treat **each conditional or loop as potentially taking any outcome**, and we **do not perform semantic pruning**. Therefore, even if the statement at line 3 is conditionally executed, we still recognize `(x,1)` as used to compute `(y,3)`.
    


- **Direct Implicit Flow**

  A direct implicit flow from `(varA,lineA)` to `(varB, lineB)` occurs when `(varA,lineA)`’s value directly influences whether `lineB` executes. In other words, it corresponds to a **direct control dependence**, where the condition has **at least one branch** where `(varB, lineB)` must run and another where it may not. All conditional structures (e.g., if, while, for, switch, etc.) generate these flows.

  Formally, if `(varA,lineA)` is **directly read** in a condition at `lineC` and that condition **directly** decides whether `lineB` executes, then we say there is an **direct implicit flow** from `(varA,lineA)` to `(varB,lineB)` (or `(varA,lineA) -> (varB,lineB)`).

  **Example**:
  ```
  1  z = x + 5
  2  if z > 1:
  3      y = 10
  ```
  `(z,1) -> (z,2)* -> (y,3)` is a direct implicit flow through the condition at line 2. Line 2 is a conditional statement and the value of `(z,2)*` here is not redefined, therefore marked with "*".
  
  The asterisk * indicates that the variable is used for the conditional statement but not redefined at that line (e.g., used in a condition).



### Information Flow Between Variables

An **information flow** exists from a variable `(varA, lineA)` to another variable `(varB, lineB)` if there is a **transitive chain of direct flows** — explicit, implicit, or both — connecting them.

That is, there exists a sequence of variables:
`(varA, lineA) -> ... -> (varB, lineB)`



## 2. **Output Format**  

When asked "Which variable instances have information flow over `(targetVar, targetLine)`? List all such variables.", respond:, respond with a JSON object:

```json
   {
    "InfomationFlowSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

If you believe there is no variables have information flow on `(targetVar, targetLine)`, respond:

```json
{
  "InfomationFlowSources": false
}
```


## 3. **Intraprocedural Data 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  def example_func():
2      status = 0
3      flag = False
4      balance = 1000
5      balance += 500
6      if balance > 1000:
7          status = 1
8          flag = True
9      limit = status * 5000
10     transaction = limit * 0.2
11     return flag
```

#### Example Question 1.1:
Which variable instances have information flow over `(transaction,10)`? List all such variables.


**Analysis**:
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`. 
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from `(status,7)` (or `(status,2)`) to `(limit,9)`. 
- Line 7: `(status,7)` is updated only if line 6’s condition is true, so has a direct control dependence on line 6.
- Line 6: checks `balance > 1000`, reads from `(balance,5)`, so there is a direct implicit flow from `(balance,5)` to `(status,7)` through `balance` at line 6.
- Line 5: `(balance,5)` reads from `(balance,4)`, so there is a direct explicit flow from `(balance,4)` to `(balance,5)`.
- Line 4: `(balance,4)` initializes the variable balance.
  

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["limit", 9],
      ["status", 2],
      ["status", 7],
      ["balance", 4],
      ["balance", 5],
    ]
   }
```

 
#### Example Question 1.2
Which variable instances have information flow over `(flag,3)`? List all such variables.

**Analysis**
Line 3 initializes `flag` without any data dependency or control dependency. 


**Output**:
```json
{
  "InfomationFlowSources": []
}
```



### Example 2
```python
1.  val = 5
2.  size = 3
3.  arr = [0] * size
4.  i = 0
5.  j = 1
6.  total = 0
7.  if val > 2:
8.      arr[j % size] = j
9.  while i < size:
10.     arr[j] += 1
11.     score = arr[j+1]
12.     total += score * 2
13.     diff = total - score
14.     j = (j + 1) % size
15.     i += 1
16. last = arr[-1]
17. summary = diff + j
```

#### Example Question 2.1:
Which variable instances have information flow over `(summary,17)`? List all such variables.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
    ]
   }
```


**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. `diff` is from line 13. Next, we focus on the path through `j`. `j` is updated from line 14 and line 5. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. When line 9 never evaluates to `true`, `(summary,17)` reads `j` from `(j,5)`. When line 9 evaluates to `true`, line 14 will be executed. 
- Line 14: `(j,14)` is updated inside the `while i < size` loop, so its update is control-dependent on the loop condition at line 9. There is also an explicit flow from `size` to `j`.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`. It also reads `i` from line 15 or line 4.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have information flow over `(score,11)`? List all such variables.


**Analysis**:
- Line 11: `(score,11)` reads `arr[j+1]`. There is a direct data dependence on both `arr` and `j`. That `j` may come from either `(j,5)` (initial) or subsequent updates at `(j,14)`. `arr` may come from line 3, line 8, or line 10 given the index `j+1` is with an unknown value. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. Therefore, `j` could be read from either line 5 (e.g., when it is the first iteration of the while loop) or line 14 (e.g., when it is the not the first iteration of the while loop). Similar for `arr`, it could be read from line 3, line 8, or line 10.
- If `arr` is read from line 10, there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.
- If `arr` is read from line 8, there is an explicit flow from `(j,5)` and `(size,2)` to `arr`. There is also implicit flow from `val` (as in the condition in line 7) read from line 1.
- If `arr` is read from line 3, there is an explicit flow from `(size,2)` to `arr`.
- Line 11 is control-dependent on line 9, therefore there is a implicit flow from `i` (possibly from `(i,15)` or `(i,4)`) and `size` (possibly from `(size,2)`) to line 11.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```



#### Example Question 2.3:
Which variable instances have information flow over `(last,16)`? List all such variables.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8, by the loop updates in line 10, or by initialization at line 3. Under static analysis, multiple update sites may contribute. 
- If from Line 3: there is an explicit flow from `(size,2)` to it.
- If from line 8: `arr[j % size] = j` executes only if `val > 2`, so `(arr,8)` has a direct control dependence on `(val,1)` via the `if` at line 7. There is also an explicit flow from `(j,5)` and `(size,2)` to `arr`.
- If from line 10:  there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.


**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```

---

[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**: Which variable instances have information flow over `(w,155)` in function `main`? List all such variables.

**Output**:


Ground truth answer: yes