[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.lang.*;
2   import java.io.*;
3   import java.util.*;
4   class Solution {
5       public static void main(String[]args) throws IOException {
6           BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
7           PrintWriter pw = new PrintWriter(System.out);
8           int T = Integer.parseInt(br.readLine());
9           for(int i=1;i<=T;i++) {
10              pw.println("Case #" + i + ": " + solve(br));
11          }
12          pw.flush();
13      }
14      private static String solve(BufferedReader br) throws IOException {
15          int N = Integer.parseInt(br.readLine());
16          List<P> paris = new ArrayList<>();
17          for (int i = 0; i < N; i++) {
18              StringTokenizer st = new StringTokenizer(br.readLine());
19              int x = Integer.parseInt(st.nextToken());
20              int y = Integer.parseInt(st.nextToken());
21              paris.add(new P(x, y));
22          }
23          String res = getRes(N, paris);
24          return res;
25      }
26      private static boolean check(String res, List<P> paris) {
27          if (res.equals("IMPOSSIBLE")) return true;
28          StringTokenizer st = new StringTokenizer(res);
29          int x = Integer.parseInt(st.nextToken());
30          int y = Integer.parseInt(st.nextToken());
31          int[] v = new int[paris.size()];
32          for (int i = 0; i < paris.size(); i++) {
33              int z = paris.get(i).a * x + paris.get(i).b * y;
34              v[i] = z;
35          }
36          int[] v2 = new int[v.length];
37          System.arraycopy(v, 0, v2, 0, v.length);
38          Arrays.sort(v);
39          boolean r = Arrays.equals(v, v2);
40          if (!r) {
41              debug(v, v2);
42          }
43          return r;
44      }
45      private static String getRes(int n, List<P> paris) {
46          TreeSet<F> pfracs = new TreeSet<>();
47          TreeSet<F> nfracs = new TreeSet<>();
48          boolean hasDup = false;
49          for (int i = 0; i < n; i++) {
50              P p1 = paris.get(i);
51              for (int j = i + 1; j < n; j++) {
52                  P p2 = paris.get(j);
53                  int d1 = p1.a - p2.a;
54                  int d2 = p1.b - p2.b;
55                  if ((d1 == 0 && d2 == 0) || (d1 > 0 && d2 > 0)) {
56                      hasDup = true;
57                      break;
58                  }
59                  if (d1 != 0 && d2 != 0 && (d1 > 0 || d2 > 0)) {
60                      if (d1 < 0) {
61                          pfracs.add(new F(Math.abs(d1), Math.abs(d2)));
62                      }
63                      if (d2 < 0) {
64                          nfracs.add(new F(Math.abs(d1), Math.abs(d2)));
65                      }
66                  }
67              }
68              if (hasDup) break;
69          }
70          if (hasDup) {
71              return "IMPOSSIBLE";
72          }
73          F above = pfracs.isEmpty() ? null : pfracs.first();
74          F below = nfracs.isEmpty() ? null : nfracs.last();
75          if (above != null && below != null && above.compareTo(below) <= 0) {
76              return "IMPOSSIBLE";
77          }
78          if (above == null && below == null) {
79              return "1 1";
80          }
81          if (above == null) {
82              long z = below.num;
83              long r = below.denum;
84              long y = 0;
85              if (r % z == 0) {
86                  y = r / z;
87              } else {
88                  y = r / z + 1;
89              }
90              return "1 " + (y + 1);
91          }
92          if (below == null) {
93              long z = above.num;
94              long r = above.denum;
95              long x = 0;
96              if (r % z == 0) {
97                  x = r / z;
98              } else {
99                  x = r / z + 1;
100              }
101              return (x + 1) + " 1";
102          }
103          int gcd = gcd(above.denum, below.denum);
104          long b2 = above.denum / gcd;
105          long lcd = below.denum * b2;
106          long a = above.num * (lcd / above.denum);
107          long b = below.num * (lcd / below.denum);
108          long z = a - b;
109          long r = 2 * lcd;
110          long x = 0;
111          if (r % z == 0) {
112              x = r / z;
113          } else {
114              x = r / z + 1;
115          }
116          return x + " " + (x + 1);
117      }
118      private static int gcd(int a, int b) {
119          return (b != 0) ? gcd(b, a % b) : a;
120      }
121      private static final class P {
122          int a;
123          int b;
124          public P(int a, int b) {
125              this.a = a;
126              this.b = b;
127          }
128      }
129      private static final class F implements Comparable<F> {
130          int num;
131          int denum;
132          public F(int num, int denum) {
133              int gcd = gcd(num, denum);
134              this.num = num / gcd;
135              this.denum = denum / gcd;
136          }
137          @Override
138          public int compareTo(F o) {
139              int gcd = gcd(o.denum, denum);
140              long b2 = o.denum / gcd;
141              long lcd = denum * b2;
142              long a = num * (lcd / denum);
143              long b = o.num * (lcd / o.denum);
144              return Long.compare(a, b);
145          }
146      }
147      public static void debug(Object...args) {
148          System.out.println(Arrays.deepToString(args));
149      }
150  }
```


**Question**: Does `(d2,54)` have data dependence over `(nfracs,64)` in function `getRes`? If so, provide a trace.

**Output**:


Ground truth answer: yes