[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control 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  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

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


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

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


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

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

---

[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 line `61` have control dependence over line `105` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes