[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis 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. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

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

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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**: Does `(r,20)` have data dependence over `(t,55)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: no