[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**. 

```Java
1   import java.io.*;
2   import java.util.*;
3   import java.math.*;
4   public class Main {
5       InputStream is;
6       PrintWriter out;
7       String INPUT = "";
8       static int mod = 1_000_000_007;
9       long inf = Long.MAX_VALUE/2;
10      void solve(){
11          int n = ni();
12          int[] a = new int[n];
13          for(int i = 0; i < n; i++){
14              a[i] = ni();
15          }
16          boolean[] seen = new boolean[n];
17          ArrayList<Integer> ans = new ArrayList<>();
18          int t = 1;
19          int l = 0;
20          int r = n-1;
21          while(true){
22              outer:
23              for(int i = n-1; i >= 0; i--){
24                  if(seen[i]) continue;
25                  if(i<l){
26                      out.println(-1);
27                      return;
28                  }
29                  if(a[i]==t){
30                      seen[i] = true;
31                      ans.add(a[i]);
32                      boolean flag = true;
33                      for(int j = i+1; j < n; j++){
34                          if(!seen[j]){
35                              flag = false;
36                          }
37                      }
38                      if(flag){
39                          break;
40                      }
41                      else{
42                          t++;
43                          i = n;
44                          continue;
45                      }
46                  }
47              }
48              if(!seen[n-1]){
49                  out.println(-1);
50                  return;
51              }
52              boolean flag = true;
53              for(int i = n-1; i >= 0; i--){
54                  if(!seen[i]){
55                      t = a[i+1];
56                      flag = false;
57                      r = i;
58                      for(int j = i; j >= 0; j--){
59                          if(seen[j]) break;
60                          l = j;
61                      }
62                      break;
63                  }
64              }
65              if(flag) break;
66          }
67          for(int i : ans){
68              out.println(i);
69          }
70      }
71      void run() throws Exception
72      {
73          is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
74          out = new PrintWriter(System.out);
75          long s = System.currentTimeMillis();
76          solve();
77          out.flush();
78          if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
79      }
80      public static void main(String[] args) throws Exception { new Main().run(); }
81      private byte[] inbuf = new byte[1024];
82      private int lenbuf = 0, ptrbuf = 0;
83      private int readByte()
84      {
85          if(lenbuf == -1)throw new InputMismatchException();
86          if(ptrbuf >= lenbuf){
87              ptrbuf = 0;
88              try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
89              if(lenbuf <= 0)return -1;
90          }
91          return inbuf[ptrbuf++];
92      }
93      private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
94      private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
95      private double nd() { return Double.parseDouble(ns()); }
96      private char nc() { return (char)skip(); }
97      private String ns()
98      {
99          int b = skip();
100          StringBuilder sb = new StringBuilder();
101          while(!(isSpaceChar(b) && b != ' ')){
102              sb.appendCodePoint(b);
103              b = readByte();
104          }
105          return sb.toString();
106      }
107      private char[] ns(int n)
108      {
109          char[] buf = new char[n];
110          int b = skip(), p = 0;
111          while(p < n && !(isSpaceChar(b))){
112              buf[p++] = (char)b;
113              b = readByte();
114          }
115          return n == p ? buf : Arrays.copyOf(buf, p);
116      }
117      private char[][] nm(int n, int m)
118      {
119          char[][] map = new char[n][];
120          for(int i = 0;i < n;i++)map[i] = ns(m);
121          return map;
122      }
123      private int[] na(int n)
124      {
125          int[] a = new int[n];
126          for(int i = 0;i < n;i++)a[i] = ni();
127          return a;
128      }
129      private int ni()
130      {
131          int num = 0, b;
132          boolean minus = false;
133          while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
134          if(b == '-'){
135              minus = true;
136              b = readByte();
137          }
138          while(true){
139              if(b >= '0' && b <= '9'){
140                  num = num * 10 + (b - '0');
141              }else{
142                  return minus ? -num : num;
143              }
144              b = readByte();
145          }
146      }
147      private long nl()
148      {
149          long num = 0;
150          int b;
151          boolean minus = false;
152          while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
153          if(b == '-'){
154              minus = true;
155              b = readByte();
156          }
157          while(true){
158              if(b >= '0' && b <= '9'){
159                  num = num * 10 + (b - '0');
160              }else{
161                  return minus ? -num : num;
162              }
163              b = readByte();
164          }
165      }
166      private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
167  }
```


**Question**: Which variable instances have information flow over `(t,55)` in function `solve`? List all such variables.

**Output**:


Ground truth answer: yes