[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.util.*;
2   import java.io.*;
3   import java.math.BigDecimal;
4   import java.math.BigInteger;
5   import java.math.MathContext;
6   class Main
7   {
8   	static class Pair implements Comparable
9   	{
10  		int first;
11  		double second;
12  		public Pair(int f, double s)
13  		{
14  			first = f;
15  			second = s;
16  		}
17  		public String toString()
18  		{
19  			return first + " " + second;
20  		}
21  		public int compareTo(Object o)
22  		{
23  			Pair p2 = (Pair)o;
24  			if(this.first < p2.first)
25  				return -1;
26  			if(this.first > p2.first)
27  				return 1;
28  			if(this.second < p2.second)
29  				return -1;
30  			if(this.second > p2.second)
31  				return 1;
32  			return 0;
33  		}
34  	}
35  	static int log2(double a)
36  	{
37  		return (int)(Math.log(a) / Math.log(2));
38  	}
39  	static long gcd(long a, long b)
40  	{
41  		long r;
42  		while((r = a % b) > 0)
43  		{
44  			a = b;
45  			b = r;
46  		}
47  		return b;
48  	}
49  	static int distance(int i1, int j1, int i2, int j2)
50  	{
51  		return Math.abs(i1 - i2) + Math.abs(j1 - j2);
52  	}
53  	public static void main(String[] args) throws IOException
54  	{
55  		int h = in.nextInt();
56  		int w = in.nextInt();
57  		int d = in.nextInt();
58  		int[][] ans = new int[h][w];
59  		for(int i = 0; i < h; i ++)
60  			Arrays.fill(ans[i], -1);
61  		for(int i = 0; i < h; i ++)
62  		{
63  			for(int j = 0; j < w; j ++)
64  			{
65  				boolean[] not = new boolean[4];
66  				int x = j + d;
67  				int y = i;
68  				for(; y <= i + d; x --, y ++)
69  				{
70  					if(x >= w)
71  						continue;
72  					if(y >= h)
73  						continue;
74  					if(ans[y][x] != -1)
75  					{
76  						not[ans[y][x]] = true;
77  					}
78  				}
79  				x = j + d;
80  				y = i;
81  				for(; y >= i - d; x --, y --)
82  				{
83  					if(x >= w)
84  						continue;
85  					if(y < 0)
86  						continue;
87  					if(ans[y][x] != -1)
88  					{
89  						not[ans[y][x]] = true;
90  					}
91  				}
92  				x = j - d;
93  				y = i;
94  				for(; y <= i + d; x ++, y ++)
95  				{
96  					if(x < 0)
97  						continue;
98  					if(y >= h)
99  						continue;
100  					if(ans[y][x] != -1)
101  					{
102  						not[ans[y][x]] = true;
103  					}
104  				}
105  				x = j - d;
106  				y = i;
107  				for(; y >= i - d; x ++, y --)
108  				{
109  					if(x < 0)
110  						continue;
111  					if(y < 0)
112  						continue;
113  					if(ans[y][x] != -1)
114  					{
115  						not[ans[y][x]] = true;
116  					}
117  				}
118  				for(int z = 0; z < 4; z ++)
119  				{
120  					if(!not[z])
121  					{
122  						ans[i][j] = z;
123  						break;
124  					}
125  				}
126  			}
127  		}
128  		char[] c = {'R', 'Y', 'G', 'B'};
129  		for(int i = 0; i < h; i ++)
130  		{
131  			for(int j = 0; j < w; j ++)
132  			{
133  				sop(c[ans[i][j] != -1 ? ans[i][j] : 0]);
134  			}
135  			sop("\n");
136  		}
137  	}
138  	static class FastReader
139      {
140          BufferedReader br;
141          StringTokenizer st;
142          public FastReader()
143          {
144              br = new BufferedReader(new InputStreamReader(System.in));
145          }
146          String next()
147          {
148              while (st == null || !st.hasMoreElements())
149              {
150                  try
151                  {
152                      st = new StringTokenizer(br.readLine());
153                  }
154                  catch (IOException  e)
155                  {
156                      e.printStackTrace();
157                  }
158              }
159              return st.nextToken();
160          }
161          int nextInt()
162          {
163              return Integer.parseInt(next());
164          }
165          long nextLong()
166          {
167              return Long.parseLong(next());
168          }
169          double nextDouble()
170          {
171              return Double.parseDouble(next());
172          }
173          String nextLine()
174          {
175              String str = "";
176              try
177              {
178                  str = br.readLine();
179              }
180              catch (IOException e)
181              {
182                  e.printStackTrace();
183              }
184              return str;
185          }
186      }
187      static FastReader in = new FastReader();
188      public static void sop(Object o)
189      {
190      	System.out.print(o);
191      }
192  }
```


**Question**: Does `(d,57)` have data dependence over `(x,105)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes